I. ll. Bi: 6.. iv. A I . $39.23“ 4W1. «ma... v.3. . A. , 1. V. #0 1...“! .IIL‘ I .- {4 “my? :14 avg“... -. 93% 1.: ad“... I he If. , 51. I: .5. .I- I. . fiIIuI. Fun). It: 3!». f , It!) 5.: 1.3;. I 1:. r. 1.55:1”... It... a an «an. I . , I I . I'III‘I QIEFJI “Shard?! .rfry.k5n.l.. . . . , THE? 8 STATE EIRLB IIIIIIIIIIIIIIIIIIIIIIII II III IIIIIIIIIIILIIIII 31293 01771 1130 LIBRARY Michigan State University This is to certify that the dissertation entitled SOFT (3%.qu EFFECT§ 0M ELECTRCWEAK Rosa/x} YJRQDMCTICM w‘ HADRCM CCLLISICJJS presented by CéA {DA fivA L A 13 has been accepted towards fulfillment of the requirements for 9913 . D degree in 70/7 Yfi/Kfi JX/MM Major professfr/ Date 5/19} “707 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1/” WW“ SOFT GLUON EFFECTS ON ELECTROWEAK BOSON PRODUCTION IN HADRON COLLISIONS By Csaba Balazs A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1999 ABSTRACT Soft Gluon Effects on Electroweak Boson Production in Hadron Collisions By Csaba Balazs Departures from the Standard Model (SM) are expected to emerge at colliders, and in order to disentangle signals of new physics we must predict the SM backgrounds precisely. In the forseeable future the two highest energy colliders operating will be the upgraded Fermilab Tevatron and the CERN Large Hadron Collider (LHC), both hadron-hadron machines. In hadronic collisions corrections from Quantum Chro- modynamics (QCD) tend to become large, and the sizable effects of the multiple soft—gluon emission must be included in the. theoretical description. This is achieved by the resummation of the large logarithmic contributions due to the gluon radiation. Extending the Collins-Soper—Sterman (CSS) resummation formalism in a renormal- ization group invariant manner, in this work, we present an uniform description of soft—gluon phenomena in a wide variety of hadronic initiated Electroweak (EW) pro- cesses, ranging from Drell—Yan type lepton—pair production through the single or pair production of colorless vector bosons (including the standard, or the vector bosons of the extended, unified gauge theories) to Higgs boson production. One of the outstanding open questions of the SM, which initiates new physics, is the underlying dynamics of the EW symmetry breaking (SB). It is commOn to assume the existence of (pseudo-) scalar bosons, either elementary or composite, associated with the EWSB, and the search for the(se) Higgs boson(s) has the highest priorities at the next generation of colliders. Even before its detection the Tevatron is able constrain the mass of the SM Higgs boson through the measurement of the top quark and Wi boson masses. The latter requires not only the detailed knowledge of the leptonic distributions from Wi decay, but also the same for the Z0 bosons. In this work, we give the resummed predictions and compare them to the next-to-leading- order predictions in detail. At the LHC the extraction of the Higgs signal will also be challenging, and the precise knowledge of the transverse momentum distributions of the Higgs decay products will be vital. To this end, we also present a resummed calculation for the Higgs boson background for the gold-plated pp(gg) ——> H 0X —> ZOZ0 or 77 production modes. Due to the top-quark-mass enhanced Yukawa coupling, the s-channel (pseudo- )scalar production via heavy quark—anti-quark annihilation can be an important new mechanism for discovering non-standard charged (and some neutral) scalar particles at hadron colliders. This work contains the calculation of the as QCD corrections to this s-channel production process, in particular, the production and decay rates of the charged top-pions of the topcolor models, and the charged Higgses of the generic two-Higgs doublet models. The direct extension to the s-channel production of the neutral (pseudo-)scalars (such as in the Minimal Supersymmetric Model with large tan fl and the topcolor models) from the bB-annihilation is also briefly discussed. Higgs boson production at hadron colliders associated with bottom quarks is also studied. It is found that strong, and model independent, constraints on the size of the ¢-b—5 coupling can be obtained for a wide range of Higgs boson masses. Based on these constraints the implications for supersymmetric models with large tan B, were studied. We concluded that the Tevatron and the LHC can put stringent bounds on these models, if the <15 —) b5 signal is not found. This work is dedicated to the Unknown Graduate Student who is doing research by its definition: being lost and learning the way as proceeding. iv ACKNOWLEDGMENTS I would like to express my gratitude to my thesis advisors, Wu-Ki Tung and C.-P. Yuan, who taught me almost everything that this work is based on. I am grateful to Wu-Ki for encouraging my graduate application, for his inspiration all through the years I spent at Michigan State University, for the continuous financial support I received from him, and for the freedom in research and thinking he provided me with. His deep knowledge of physics and mathematics motivated, stimulated, and sometimes provoked (in the good sense of the word) my work, teaching me to be always rigorous, thorough, complete, accurate, and careful when doing physics. I feel very fortunate to have C.-P. as my advisor as well. His encouragement and extensive help started me on the work in this thesis. His exceptional patience in discussing details of projects and his wide knowledge helped me through obstacles I could not overcome without him. His outstanding calculational and numerical abilities are woven through every line of this thesis. I learned from him to always do physics with enthusiasm and enjoyment. I am also indebted to him for his help in finding my first job. I also thank my thesis committee members, Chip Brock, Norman Birge, and Pavel Danielewicz, for their careful reading of the manuscript. While working on this thesis, I had extremely successful collaborations with Ed Berger, Lorenzo Diaz-Cruz, Hong—Jian He, Steve Mrenna, Wayne Repko, Tim Tait, and Jianwei Qiu. I am thankful to them since they literally contributed to this work. I am grateful to the high-energy theorists at Michigan State University: Dan Stump, Carl Schmidt, and Jon Pumplin for insightful discussions on various topics. I thank Chip Brock for encouraging me to participate in the tev_2000 activities and Joey Houston for his recognition and support of my work. High-energy experimentalists at Michigan State University have their impact on my work: Maris Abolins, Jim V Linemann, Harry Weerts; for this I am thankful to them. I wish to thank Glenn Ladinsky who left his legacy of resummed codes to me. I am grateful to the numerous users of the ResBos Monte Carlo event generator for providing me with useful feedback and helping me developing it, especially Michael Begel, Dylan Casey, Wei Chen, Mark Lancaster, Andre Maul, Jim McKinley, Pavel Nadolsky, Fan Qun, Willis Sakumoto, and John Wahl. I thank high-energy physicists outside of Michigan State University, who have influenced my work. Among the many: Uli Baur, who helped me to make my results on as public, Chris Hill, who invited me as summer visitor to Fermilab, Tao Han, Gregory Korchemsky, Steve Kuhlmann, Fred Olness, Alexander Pukhov, Xerxes Tata, and Marek Zielinski with whom I had fruitful discussions. Many thanks to Rolf Mertig, Hagen Eck, and Thomas Hahn, authors of the FeynC’alc and F eynArts M athemetz'ca. packages, of which I made extensive use while calculating the results of this work. My thanks also go to my colleagues in the Michigan State University high-energy group: Hung-Liang Lai for providing the EVIEX thesis format, Pankaj Agrawal, Jim Amundson, Doug Carlson, Dave Chao, Kate Frame, Chris Glosser, Jim Hughes, Fran- cisco Larios, Ehab Malkawi, Xiaoning Wang, and Mike Wiest for useful discussions. I thank Julius Kovacs for providing continuous support of my studies, Stephanie Hol- land and Debbie Simmons, coordinators of student affairs, Lorie Neuman, Jeanette Dubendorf, Lisa Ruess, Mary Curtis, the high-energy physics secretaries, for all their help. Finally, I thank my parents for their love and support, and my wife who helped casting this work into English and has sacrificed so much for my career. vi Contents LIST OF TABLES LIST OF FIGURES 1 Introduction 1.1 The Standard Model of Elementary Particles .............. 1.1.1 Electroweak Interactions ..................... 1.1.2 Strong Interactions ........................ 1.1.3 The Higgs Mechanism ...................... 1.2 The Quantum Nature of Gauge Fields ................. 1.2.1 Renormalization .......................... 1.2.2 Asymptotic Freedom and Confinement ............. 1.2.3 Factorization and Infrared Safety ................ 2 Soft Gluon Resummation 2.1 Lepton Pair Production at Fixed Order in a, ............. 2.1.1 The Collins-Soper Frame ..................... 2.1.2 General Properties of the Cross Section ............ 2.1.3 The Cross Section at 0(as) ................... 2.2 The Resummation Formalism ...................... 2.2.1 Renormalization Group Analysis ................ 2.2.2 Resummation Formula for Lepton Pair Production ...... 2.2.3 (9(a3) Expansion ......................... 2.2.4 A, B and C functions ...................... 2.2.5 Regular Contributions ...................... 3 Vector Boson Production and Decay in Hadron Collisions 3.1 Vector Boson Distributions ....................... 3.1.1 Vector Boson Transverse Momentum Distribution ....... vii xii cougar—Ira 12 12 14 19 24 24 25 27 30 34 34 38 44 49 51 54 54 59 3.1.2 Vector Boson Longitudinal Distributions ............ 72 3.1.3 The Total Cross Section ..................... 73 3.2 Lepton Distributions ........................... 76 3.2.1 Lepton Charge Asymmetry ................... 76 3.2.2 Transverse Mass Distribution .................. 81 3.2.3 Lepton Transverse Momentum .................. 84 3.2.4 Lepton Angular Correlations ................... 88 3.3 Conclusions ................................ 90 Photon Pair Production In Hadronic Interactions 94 4.1 Diphoton Production at Fixed Order .................. 95 4.2 Extension of the Resummation Formalism ............... 98 4.2.1 The Resummation Formula ................... 98 4.2.2 Resummation for the qq —) 'yy subprocess ........... 101 4.2.3 Contributions From qg Subprocesses .............. 104 4.2.4 Fragmentation Contributions .................. 105 4.2.5 Resummation for the gg —) ’77 subprocess ........... 111 4.3 Numerical Results ............................. 115 4.3.1 Tevatron Collider Energies .................... 115 4.3.2 Fixed—Target Energy ....................... 122 4.4 Conclusions ................................ 125 Gauge Boson Pair Production at the Upgraded Tevatron and at the LHC 130 5.1 Introduction ................................ 130 5.2 Analytical Results ............................ 133 5.2.1 The CSS Resummation Formalism for Z0 Pair Production . . 133 5.2.2 The M, qg and (79 —> ZOZOX subprocesses ........... 136 5.3 Numerical Results ............................ 139 5.3.1 Z0 pair production at the LHC ................. 141 5.3.2 ZO Pair Production at the upgraded Tevatron ......... 145 5.3.3 Diphoton production at the LHC ................ 148 5.3.4 Diphoton production at the upgraded Tevatron ........ 152 5.4 Conclusions ................................ 155 Charged Scalar Production at Hadron Colliders 158 6.1 Introduction ................................ 158 viii L1,. L}, Lu CC? US: 6.2 Charged (Pseudo-)Scalar Production via Charm-Bottom Fusion . . . 159 6.2.1 Fixed-Order Calculation ..................... 159 6.3 Soft-Gluon Resummation ......................... 168 6.4 Hadronic Decays of Charged (Pseudo-)Scalars to 0(03) ........ 174 6.5 Generalization to Neutral (Pseudo-)Scalar Production via ()5 Fusion . 184 6.6 Summary of the Analytic Results .................... 191 6.6.1 Partonic processes C?) -+ ¢+X .................. 191 6.6.2 Partonic processes gc, 95 —> ¢+X ................ 193 6.7 Conclusions ................................ 194 A Probing Higgs Bosons with Large Bottom Yukawa Coupling at Hadron Colliders 197 A.1 Introduction ................................ 197 A2 Constraints On MSSM Parameters and Implications for Models of Soft- breaking of SUSY ............................. 200 A.2.1 Bottom Yukawa Couplings and the MSSM Higgs Sector . . . . 201 A22 Constraints on MSSM from qbbb production at Tevatron and LHC 205 A.2.3 Interpretation of results for Models of soft breaking parameters 215 A3 Conclusions ................................ 218 B “Intrinsic M” and Soft Gluon Resummation 221 B.1 Introduction ................................ 221 B2 Smearing pT Distributions of Direct Photons .............. 224 B3 ‘ pT-Smearing and Soft gluon Resummation ............... 228 LIST OF REFERENCES 237 ix List of Tables 1.1 2.1 3.1 3.2 5.1 5.2 5.3 Electroweak quantum numbers of the first family of fermions ...... Vector boson parameters and couplings to fermions. The f f’ V vertex is defined as i7#[gL(1—75)+gg(1+75)] and 3w 2 sin 6w (cw = cos 0w) is the sine (cosine) of the weak mixing angle: sin2(6w(MZo))m = 0.2315. Q, is the fermion charge (62,, = 2/3,Qd = —1/3,Q,, = 0,Qe— = —1), and T3 is the eigenvalue of the third component of the S U (2) L generator (T; = 1/2, T51: —1/2, T; =1/2,T;“ = —1/2). ............ List of PDF’s used at the different models of calculations. The values of the strong coupling constants used with the CTEQ4L and CTEQ4M PDF’s are ag)(Mzo) = 0.132 and ag2)(Mzo) = 0.116 respectively. . . . Total cross sections of p}? —> (W+or Z °)X at the present and upgraded Tevatron, calculated in different prescriptions, in units of nb. The finite order total cross section results are based on the calculations in Ref. [85]. The 0mg) results were obtained from Ref. [65]. The “EB” signs refer to the matching prescription discussed in the text ...... Total cross sections of diphoton and Z0 boson pair production at the LHC and the upgraded Tevatron, in units of pb. The kinematic cuts are described in the text. The “69” sign refers to the matching prescription discussed in the text. .......................... Resummed cross sections of the subprocesses for Z0 boson pair pro- duction at the LHC and the upgraded Tevatron, in units of pb. The kinematic cuts are described in the text. ................ Cross sections of the subprocesses for diphoton production at the LHC and the upgraded Tevatron, in units of pb. The resummed qg —> 77X rate includes the fragmentation contribution. The 0(a3) 99 —> 77 rates were calculated using both the CTEQ4L and CTEQ4M PDF’s. The kinematic cuts are described in the text. ............. 41 59 76 140 142 6.1 6.2 A.1 8.1 Cross sections in fb for charged top-pion production in the TopC model at the upgraded Tevatron and the LHC are shown, by using four dif— ferent CTEQ4 PDFs. They are separately given for the LO and NLO processes, and for the qq‘ -—) ¢+X and qg —-> ¢+X sub-processes. At the upgraded Tevatron the top number is for 177.45 = 200 GeV, the middle is for rm), 2 300 GeV, and the lowest is for m¢ = 400 GeV. At the LHC the tOp number is for m¢ = 400 GeV, the middle is for rm), 2 700 GeV, and the lowest is for md, = 1 TeV. ................... Cross sections in fb for neutral Higgs boson production in the MSSM with tan [3 = 40, at the upgraded Tevatron and the LHC, are shown for four different CTEQ4 PDFs. They are separately given for the LO and NLO processes, and for the bb —+ AUX and by —+ AOX sub-processes. For the upgraded Tevatron the top number is for m A = 200 GeV, the middle is for mA = 300 GeV, and the lowest is for m A = 400 GeV. For the LHC the top number is for mA = 400 GeV, the middle is for mA = 700 GeV, and the lowest is for m A = 1 TeV. .......... Comparison of the neutral MSSM Higgs couplings to up—type (U = u, c, t) and down-type (D = d, s, b; e, p, 'r) fermions and to the gauge- boson (V = W, Z) pairs. The ratios to the corresponding SM couplings are shown, which are determined by angles ,8 and a at the tree-level. The amount of the smearing needed and the amount provided by the non-perturbative function to make theory agree with experiment for different direct photon experiments. Values of A are taken from Ref. [4]. .................................. xi 173 189 201 L 18 C. k" List of Figures 1.1 3.1 3.2 3.3 3.4 3.5 Running of the QCD coupling constant as, for different boundary val- ues at ,u = mg. The QCD prediction is compared to current experi- ments. (Reproduced from Ref. [37].) .................. Total W+ production cross section as a function of the parameter Q?” (solid curve). The long dashed curve is the part of the 0(03) cross section integrated from Q?” to the kinematical boundary, and the short dashed curve is the integral from QT = 0 to Q?” at 0(a5). The total cross section is constant within 10‘5 % through more than two order of magnitude of Q7553”. ....................... The low and intermediate QT regions of the W+ and Z 0 distributions at the Tevatron, calculated in fixed order C(03) (dotted) and 0(039) (dash-dotted), and resummed (1,1,1) (dashed) and (2,1,2) (solid) [cf. Table III]. The cross-over occurs at 54 GeV for the C(05), and at 69 GeV for the 0mg) Wi distributions. The matching between the resummed and the fixed order disrtibutions becomes much smoother 57 at 0mg) than at 0(as). The situation is very similar for the Z0 boson. 62 The ratio of the 0(a3 + 0%) Y piece (solid curve, R“) and Rm in- cluded), and the 0(as) Y piece (dashed curve, R“) included) to the resummed (2,1,1) distribution for W+ and Z0 bosons. ........ The K y-factor: ratio of the 0(ag+a§) Y piece (Rm and 12(2) included) to the (9(as) Y piece (Rm included). The curves are plotted for Q 2 MV and y = —2.0 (solid), —1.0 (long dash), 0.0 (short dash), 1.0 (dash-dot) and y =20 (dash-double-dot). ............... The fixed order perturbative 0mg) to 0(a5) K -factor as the function of QT. The curves are plotted for Q = My and y = —2.0 (solid), —1.0 (long dash), 0.0 (short dash), 1.0 (dash-dot) and y 22.0 (dash-double- dot). ................................... xii 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 The ratio R(QT, Q0), with yo = 0, for W+ and Z 0 bosons as a function of QT. For W+, solid lines are: Q0 2 78 GeV (upper) and 82 GeV (lower), dashed: Q0 = 76 GeV (upper) and 84 GeV (lower), dotted: Q0 = 70 GeV (upper) and 90 GeV (lower). For Z0 bosons, solid lines: Q0 2 88 GeV (upper) and 92 GeV (lower), dashed: Q0 2 86 GeV (upper) and 94 GeV (lower), dotted: Q0 2 80 GeV (upper) and 100 GeV (lower) ................................. The ratio R(QT,y0), with Q0 : My, for W+ and Z0 bosons as a function of QT. .............................. The ratio Ross as a function of Q?” for W+ and Z 0 bosons. The fixed order (0(as) short dashed, 0mg) dashed) curves are ill-defined in the low QT region. The resummed (solid) curves are calculated for g2 : 0.38 (low), 0.58 (middle) and 0.68 (high) GeV2 values. ........ Transverse momentum distributions of W+ and Z0 bosons calculated with low (long dash, 92 = 0.38 GeV2), nominal (solid, 92 = 0.58 GeVQ) and high (short dash, 92 = 0.68 GeV2) 92 non-perturbative parameter values. The low and high excursions in 92 are the present one standard deviations from the nominal value in the Ladinsky-Yuan parametriza- tion. .................................... Longitudinal :rp distributions of Z 0 bosons produced at the Tevatron. The NLO (dashed) curves overestimate the rate compared to the re- summed (solid) ones, because kinematic cuts enhance the low QT re- gion where the NLO and resummed distributions are qualitatively dif- ferent. Without cuts, the NLO and the resummed :rp distributions are the same. ................................. Rapidity distributions (resummed: solid, NLO: dashed) of Z0 bosons produced at the Tevatron with the kinematic cuts given in the text. . Lepton charge asymmetry distributions. (a) Without any kinematic cuts, the NLO (long dashed) and the resummed 0(03) (solid) curves overlap and the LO (short dashed) curve differs somewhat from them. (b) With cuts (QT < 30 GeV, p?” > 25 GeV), the effect of the different QT distributions renders the lepton rapidity asymmetry distributions different. The two resummed curves calculated with 92 = 0.58 and 0.78 GeV2 cannot be distinguished on this plot. .............. Distributions of positron rapidities from the decays of W+’s produced at the Tevatron, predicted by the resummed (solid) and the NLO (dashed) calculations with the same kinematic cuts as for the asym- metry plot. ................................ Transverse mass distribution for W+ production and decay at the 1.8 TeV Tevatron. .............................. Transverse momentum distributions of p9; from W+ and Z0 decays for the NLO (dashed) and the resummed C(05) (solid) calculations. Resumming the initial state multiple soft-gluon emission has the typical effect of smoothening and broadening the Jacobian peak (at p? = My/2). The CDF cuts are imposed on the W+ distributions, but there are no cuts on the Z0 distributions ................. xiii 66 68 71 73 74 78 86 3.16 3.17 3.18 3.19 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Balance in transverse momentum ApT = [pT | - [pT | and angular cor- relation 2 = —]7;~1 - 5;:2/[max(p‘T‘,pg? ]2 of the decay leptons from Z0 bosons produced at the Tevatron. .................... The correlation between the lepton azimuthal angles near the region A¢ ~ 7r for pp —+ (Z0 —) e+e‘)X. The resummed (solid) distribution gives the correct angular correlation of the lepton pair. The NLO (dashed lines) distribution near A¢ = 7r is ill-defined and depends on Q7389 calculation). The two N LO distributions were calculated with Q?” = 1.2 GeV (long dash) and Q?” = 2.0 GeV (short dash). ........ Distribution of the 6+ polar angle cos(0*) in the Collins-Soper frame from Z0 decays at the Tevatron with cuts indicated in the text. (the scale for separating soft and hard gluons in the NLO Transverse momentum distribution of virtual photons in mi —-> 7* —> 6+6“ events predicted by ResBos (solid curve) and ISAJ ET (histogram), calculated for the invariant mass range 30 GeV < Q < 60 GeV at the 1.8 TeV Tevatron. ............................ Feynman diagrams representing the leading order and next—to—leading order contributions to photon pair production in hadron collisions. The shaded circles signify the production of long—distance fragmentation photons, which are described by the fragmentation function 07H]. Comparison of the parton—level and Monte Carlo fragmentation con- tributions at the Tevatron. The upper and lower curves of the same type show the contribution before and after an isolation cut. The left figure shows the transverse momentum of the photon pair QT. The right figure shows the light—cone momentum fraction carried by the fragmentation photon. .......................... The predicted distribution for the invariant mass of the photon pair M77 from the resummed calculation compared to the CDF data, with the CDF cuts imposed in the calculation ................. The predicted distribution for the transverse momentum of the photon pair QT from the resummed calculation compared to the CDF data, with the CDF cuts imposed in the calculation .............. The predicted distribution for the difference between the azimuthal angles of the photons Arty, from the resummed calculation compared to the CDF data, with the CDF cuts imposed in the calculation. . . . The predicted distribution for the invariant mass of the photon pair M.77 from the resummed calculation compared to the DO data, with the DO cuts imposed in the calculation. ................ The predicted distribution for the transverse momentum of the photon pair QT from the resummed calculation compared to the DO data, with the D0 cuts imposed in the calculation. ............. xiv 88 89 90 96 107 115 117 118 119 120 4.8 4.9 4.10 4.11 4.12 4.13 4.14 5.1 5.2 5.3 5.4 5.5 The predicted distribution for the difference between the azimuthal angles of the photons A437, from the resummed calculation compared to the D0 data, with the DO cuts imposed in the calculation. The predicted distribution for the invariant mass of the photon pair M7, from the resummed calculation. The additional cut QT < MW has been applied to reduce the theoretical uncertainty. ........ The predicted distribution for the transverse momentum of the photon pair QT from the resummed calculation. The additional cut QT < MW has been applied to reduce the theoretical uncertainty. The lower solid curve shows the prediction of the pure NLO (fixed-order) calculation 121 122 for the qrj and qg subprocesses, but without fragmentation contributions. 123 The predicted distribution for the difference between the azimuthal angles of the photons Aqfiw from the resummed calculation. The ad- ditional cut QT < M.” has been applied to reduce the theoretical uncertainty. ................................ The predicted distribution for the invariant mass of the photon pair M.” from the resummed calculation appropriate for pN —) 77X at \/§=31.5 GeV. .............................. The predicted distribution for the transverse momentum of the photon pair QT from the resummed calculation appropriate for pN —> 77X at 73:31.5 GeV. .............................. The predicted distribution for the difference between the azimuthal angles of the photons Act.” from the resummed calculation appropriate for pN ——> 77X at x/S=31.5 GeV. .................... A representative set of Feynman diagrams included in the NLO calcu- lation of Z0 pair production. ...................... Transverse momentum distribution of Z0 pairs from qt? + qg partonic initial states at the LHC. The 0(as) (dotted) and the asymptotic (dash-dotted) pieces are coincide and diverge as QT —> 0. The re- summed (solid) curve matches the C(03) curve at about QT = 320 GeV. The resummed qrj contribution (excluding the qg contribution) is shown as dashed line. ......................... Invariant mass and rapidity distributions of Z 0 boson pairs, and trans- verse momentum distributions of the individual Z 0 bosons at the LHC. The resummed contribution of the q6 + qg —) ZOZOX subprocess is shown by the solid curve, and of the qq —-) ZOZOX subprocess by the dashed curve. The leading order distribution of mi —> ZOZ0 is shown by the dash-dotted curve. ........................ The integrated cross section for Z 0 boson pair production at the LHC. The resummed and the 0(03) distributions are shown in solid and dashed lines, respectively. ........................ Same as Fig. 5.3 except for the upgraded Tevatron. .......... XV 124 125 126 127 136 143 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 6.1 6.2 6.3 Same as Fig. 5.4 except for the upgraded Tevatron. .......... Same as Fig. 5.2 except for the upgraded Tevatron. .......... Invariant mass and rapidity distributions of photon pairs, and trans- verse momentum distributions of the individual photons at the LHC. The total resummed contribution (upper solid), and the resummed qt? + (19 —> 77X (dashed), <16 -+ 77X (dotted), 99 —> 779 (dash- dotted), as well as the fragmentation (lower solid) contributions are shown separately. The qrj —> 77 C(02) distribution is shown in the middle solid curve. ............................ The integrated cross section for photon pair production at the LHC. The resummed and the C(08) distributions are shown in solid and dashed lines, respectively. ........................ Transverse momentum distribution of photon pairs at the LHC. The total resummed contribution (upper solid), the resummed qij + qg —> 77X (dashed), qq —> 77X (dotted), 99 —+ 77X (dash-dotted), as well as the fragmentation (lower solid) contributions are shown separately. Same as Fig. 5.8 but for the upgraded Tevatron. ........... Same as Fig. 5.10 but for the upgraded Tevatron. ........... Same as Fig. 5.9 but for the upgraded Tevatron. ........... Representative diagrams for charged or neutral (pseudo-)scalar (dashed line) production from quark-antiquark and quark-gluon collisons at 0(ag) and 0(039): (a) leading order contribution; (b—d) self-energy and vertex corrections (with counter term); (e) real gluon radiation in qi-fusion; (f-g) s— and t-channel gluon-quark fusions. ......... Estimated current 3a-bounds in the TopC model and 2HDM-III: (a) the 30 upper bound on the top-pion Yukawa coupling Cfi’; (b) the 30 lower bound on the top-pion decay constant; (Here, in (a) and (b), the solid curves are derived from the combined LEP/SLD data of R?” = 0.21656 :l: 0.00074 while dashed curves are from the same 30 combined experimental error but with the central Rb-value equal to REM = 0.2158.) (c) the Rb-predictions of 2HDM-III with coupling {it = 1.0 and 1.5 (solid curves) and the 30 Rb-bounds (dashed lines). Top-pion production cross sections at the present Tevatron, upgraded Tevatron, and the LHC. For each collider we show the NLO cross section with the resummed running Yukawa coupling (solid), and with one-loop Yukawa coupling (dashed), as well as the LO cross section with resummed running Yukawa coupling (dotted) and with tree-level (dash-dotted) Yukawa coupling. The cross sections at \/-S_ = 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the x/S = 2 TeV curves. ......................... xvi 148 149 150 153 154 155 156 161 172 174 6.4 6.5 6.6 6.7 6.8 6.9 6.10 Cross sections for the charged top-pion production in the TopC model at the present Tevatron, upgraded Tevatron and the LHC. The N LO (solid), the qq’ (dashed) and qg (dash—dotted) sub-contributions, and the LO (dotted) contributions are shown. Since the qg cross sections are negative, they are multiplied by —1 in the plot. The cross sections at x/S = 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the x/S = 2 TeV curves. .............. The K -factors for the 45" production in the TopC model are shown for the NLO (K = UNLO/ULO, SOlld IIDGS), qtj’ (K = O'qu/O'Lo = (0L0 + Aoqu)/0Lo, dashed lines), and qg (K = —A0qg/0L0, dash-dotted lines) contributions, at the upgraded Tevatron (a) and the LHC (b). The ratios of NLO cross sections computed by four different sets of CTEQ4 PDFs relative to that by the CTEQ4M for charged top-pion production at the upgraded Tevatron (a) and the LHC (b). ..... Transverse momentum distributions of charged top-pions produced in hadronic collisions. The resummed (solid) and O(a,) (dashed) curves are calculated for m¢ = 200, 250, and 300 GeV at the upgraded Teva- tron (a), and for m¢ = 250, 400, and 550 GeV at the LHC (b). Total decay widths of ¢+ and BRs of (15+ —+ tb in the TopC model and 2HDM. (For the 2HDM, the BR_of the W+h0 channel is also shown, which is complementary to the tb channel.) In Fig. (a) and (b), the NLO (solid) and LO (dashed) curves differ only by a small amount. In Fig. (c), the K -factor, which is defined as the ratio_of the NLO to the LO partial decay widths, is shown for the q§+ —-) tb (solid) and —> cb (dashed) channels. The sample results for the 2HDM in this figure are derived for the parameter choice (63, {3) = (1.5, 1.5), a = 0, and (1m, mA) 2 (120, 1200) GeV. ...................... Invariant mass distribution of t-B and f-b pairs from ¢i (signal) and Wi“ (background) decays at the Tevatron Run-II for the TopC model (a), and 2HDM with Higgs mixing angles a = 0 (b), and a = 7r/ 2 (c). We show the signal for m¢ = 200, 250, 300 and 350 GeV. The solid curves show the results from the N LO calculation, and the dashed ones from the LO analysis. .......................... Invariant mass distribution of t-E and f—b pairs from (ti (signal) and W2“ (background) decays at the LHC for the TopC model (a), and for the 2HDM with the Higgs mixing angles a = 0 in (b), and a = 7r / 2 in (c). Here the charged pseudo-scalar or scalar mass are chosen as the typical values of m¢ = 250, 400 and 550 GeV. The solid curves show the results by the NLO calculation, while the dashed ones come from the LO analysis. ............................. xvii 175 176 177 178 178 179 6.11 6.12 6.13 6.14 6.15 6.16 Invariant mass distributions of W+-h° and W“-h0 pairs from (15* (3- channel resonance) and W3“ (s-channel non-resonance) decays at the Tevatron Run-II, and at the LHC, for the 2HDM with Higgs mixing angles a = 0. We show the signal for m¢ = 200, 250 and 300 GeV at the Tevatron (a), and for me; = 250, 400 and 550 GeV at the LHC (b). The solid curves show the results of the N LO calculation, and the dashed ones of the LO analysis. ..................... LO and NLO cross sections for the neutral Higgs A0 production in the MSSM with tanfi = 40, at the Tevatron and the LHC. (a) For each collider we show the NLO cross sections with the resummed running Yukawa coupling (solid) and with one-loop Yukawa coupling (dashed), as well as the LO cross sections with resummed running Yukawa coupling (dotted) and with tree-level Yukawa coupling (dash- dotted). (b) The NLO (solid), the bb (dashed) and by (dash-dotted) sub-contributions, and the LO (dotted) contributions are shown. Since the bg cross sections are negative, they are multiplied by —1 in the plot. The cross sections at \/§ 2: 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the \/S = 2 TeV curves. (c) The NLO cross sections with QCD running Yukawa coupling (solid curves) and those with additional SUSY correction to the running cou- pling are shown (upper dashed lines for the Higgs-mixing parameter ,u 2 +500 GeV and lower dashed lines for p = —500 GeV). ...... The K -factors for the A0 production in the MSSM with tan [3 = 40 are shown for the NLO (K = UNLQ/ULo, solid lines), bb (K = aha/0L0 = (0L0+A0b5)/0L0, dashed lines), and by (K = —Aabg/0Lo, dash-dotted lines) contributions, at the upgraded Tevatron (a) and the LHC (b). The ratios of NLO cross sections computed by four different sets of CTEQ4 PDFs relative to that by the CTEQ4M for neutral pseudo- scalar (A0) production in the MSSM with tan ,6 = 40, at the upgraded Tevatron (a) and the LHC (b). ..................... Transverse momentum distributions of pseudo-scalar A0 produced via hadronic collisions, calculated in the MSSM with tanfi = 40. The resummed (solid) and C(03) (dashed) curves are shown for m A = 200, 250, and 300 GeV at the upgraded Tevatron (a), and for mA 2 250, 400, and 550 GeV at the LHC (b). ................... Cross_ sections for the neutral b—pion 7r? or b—Higgs hg production via the bb—fusion in the TopC model at the Tevatron and the LHC. The NLO (solid), the qrj’ (dashed) and qg (dash-dotted) sub-contributions, and the LO (dotted) contributions with resummed running Yukawa coupling are shown. Since the qg cross sections are negative, they are multiplied by —1 in the plot. The cross sections at x/S = 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the VS = 2 TeV curves. ......................... xviii 181 188 1190 191 192 193 A.1 A2 A3 A.4 A5 A6 A7 B.1 Bottom Yukawa couplings to the MSSM Higgs bosons and the mass differences, mA -— rm. and my — mA, as a function of mA for tanfl values: 2.0, 5.0, 10.0, 20.0, 30.0, 40., 50.0. In (a), ybbh is in solid and 31be is in dashed, and tan B decreases from top to bottom curves. In (b), m A - mh is in solid, my — mA is in dashed, and tan [3 increases from the top to bottom curves. Here, all the SUSY soft-breaking mass parameters are chosen to be 500 GeV. ................. The same as the previous figure, but in (a)-(b), we change the right- handed stop mass to 200 GeV, and in (c)—(d), we use the “LEP II Scan A2” set of SUSY parameters. ...................... 95% CL. exclusion contours in the m A-tan 6 plane of the MSSM. The areas above the four boundaries are excluded for the Tevatron Run II with the indicated luminosities, and for the LHC with an integrated lu- minosity of 100 fb‘l. The soft SUSY breaking parameters were chosen uniformly to be 500 GeV in Fig. (a), while the inputs of the “LEP II Scan A2” are used for the Fig. (b) in which LEP II excludes the left area of the solid curve. ......................... Discovery and exclusion contours in the m A-tan 5 plane of the MSSM for the LHC with an integrated luminosity of 100 fb'l. The area above the lower boundary is excluded at 95% CL, while the upper boundary is the 50 discovery contour. The soft SUSY breaking parameters were chosen uniformly to be 500 GeV. .................... 95% CL. exclusion contours in the mh-tan 5 plane of the MSSM. The areas above the four boundaries are excluded for the Tevatron Run II with the indicated luminosities, and for the LHC with an integrated luminosity of 100 fb"1. LEP II can exclude the area on the left-hand side of the solid curve in the lower plot. ................ 95% CL. exclusion contours in the mA-mh plane of the MSSM. The shaded areas indicate the excluded regions for the Tevatron Run II with integrated luminosities, 2, 10, 30 fb‘l, and for the LHC with an integrated luminosity of 100 fb"1, as those in the previous figures. LEP II can exclude the area on the left-hand side of the solid curve in the lower plot. .............................. The running of the bottom quark mass 7711,0112) as a function of the renormalization scale #3. The solid curve shows the QCD evolution alone. The dashed curve further includes the supersymmetric correc- tions to the “effective” running mass, for tan 6 = 30. All soft SUSY breaking parameters have been fixed as 500 GeV. The dotted curve includes the SUSY corrections but with the sign of the Higgs-mixing parameter ,u flipped. ........................... Experiment numbers are listed at the top of each plot. Short dashed line: NLO transverse momentum distribution. Long dashed line: NLO theory convoluted with the Fourier transform of the non-perturbative function used in the Drell-Yan resummation formalism. ........ xix 203 204 206 208 209 211 212 228 B.2 a) The invariant mass versus the average Wi boson transverse mo- mentum in the resummed calculation. b) The center of mass energy versus the average W:t boson transverse momentum in the resummed calculation. 235 II II ' ' x ‘ tn . . 5*] .. .r - *7 . '3‘ v-.. I. I ’ .._ ‘r A—4 .‘ \‘, v‘ .1 A. '7' 7- s. I . f" -. I“; . V Q‘. ‘\ I '~. Chapter 1 Introduction 1.1 The Standard Model of Elementary Particles The Standard Model (SM) [2] of particle physics is the refined essence of our wisdom of the microscopic world. It embodies most of our knowledge about the smallest constituents of our universe. It unifies three of the four, known fundamental forces: the electromagnetic, weak and strong interactions, within a compact, economic frame- work. It describes a wide range of the observed physical phenomena, most everything1 other than gravity. Its validity is tested daily by numerous experiments with a preci- sion of many decimal points [3]. Since it is the underlying theoretical structure of our work, in this Section we highlight some of the most important features of the SM. From the mathematical standpoint the SM is a relativistic, local, non-Abelian quantum field theory (QFT) [4, 5]. The framework of the theory is based on the following physical assumptions [6]: — space-time symmetry: the theory respect Poincare invariance, — point-like elementary particles, — locality: local (point-like) interactions, i.e. no actions at a distance, 1The simplest component of the SM, QED alone describes ”all of chemistry, and most of physics” as P. A. M Dirac put it [1]. — causality: commutativity of space-like separated observables, — unitarity: quantum mechanical evolution conserving probability, and — renormalizability: predictions of the theory to be free of infinities. The above principles constrain the theory quite uniquely and grant the SM substan- tial predicting power. Theories which surpass the SM either discard one of these assumptions (string theory [7], non-local QFT’S [8], effective field theories [9], theo- ries of gravity [10]) and / or try to improve on them (supergravity [11], supersymmetry [12]). Field theories are customarily expressed within the Lagrangian formalism [13], in which a single central quantity, the action density or Lagrangian embodies all the dynamical information about the physical system. The Lagrangian is built from quantum fields which represent the fundamental particle types. In the SM the differ- ent types of particles fall into two categories. The fermions, which are the building blocks of matter, and the bosons which are the force mediators. The fermions of the SM are further split into two branches: leptons and quarks. The former participating only in the electroweak interaction, while the latter also engaged by the strong force. The essence of the SM is in its forces, that is, in the interactions between the elementary fields. This, in short, is called the dynamics. The dynamics of the SM is dictated by symmetries [14], transformations of the Lagrangian which leave it invari- ant. One of the central results of the field theories, Noether’s theorem [15], connects symmetries with conserved physical quantities. This result enables us to relate the mathematical description with the observed reality, to discover the symmetries of Nature using its invariants. Among the symmetries of the SM, the most important ones are the space—time dependent or local, so called, gauge symmetries. Symmetry transformations form groups [16, 17], and symmetries are usually re- ferred to by their group. The symmetry of the SM is given by its semi-simple group: 30(3, 1) x SU(3)C x SU(2)L x U(1)y. Here 50(3, 1) represents the Poincare invari- ance and the rest the gauge symmetries. The Poincare group is usually implicit, it is understood that the SM to be Lorentz invariant. The structure of the Lagrangian respecting the above symmetry is discussed in the following subsections. 1.1.1 Electroweak Interactions The electroweak (EW) sector of the SM is invariant under the transformations of the gauge group: SU(2)], x U (1)y [18, 19]. Its unbroken subgroup (cf. Section 1.1.3), the U (1) EM group, emerged historically in Quantum Electrodynamics (QED) [20], where it represented the conservation of the electric charge. Similarly, the U (1);» symmetry of the SM, had remained unbroken, would conserve the hypercharge Y. The S U (2) group also appeared empirically in the theory of the weak interactions [21], originally describing a fermionic flavor symmetry. The invariants of the S U (2) L group are the square, T2, and the third component, T3, of the weak isospin generator. These group invariants are connected by the Gell-Mann—Nishijima relation: Y Q=T3+—2—, where Q is the electric charge. The fermions of the SM are manifested by the fundamental representation of the SU(2)L x U (1),» gauge group. The left handed components of the fermions are assumed to transform as doublets, and the right handed ones as singlets under S U (2) L. This is signified in the following notation: (+1/2) $1.: ”01’ I ¢R=80ig)- —1/2 a. . > Fermion Q T3 Y V}: 0 % —1 8L —1 —§ —1 : e .1. <3: 4 % eg —1 0 —2 UR i 0 i (1,. —; 0 —§ Table 1.1: Electroweak quantum numbers of the first family of fermions. IIere 1’75 1+75' 2 ya: so and 99:2: represent the left and right handed components of the fermion spinors (p, with 75 = Meryl/7273 in four space-time dimensions, and the Dirac matrices, 7“, defined by their anti-commutation relation: {7’27"} = 29"”, with 9“” being the space-time metric tensor2. The weak isospin eigenvalues, 0 and :I:1 / 2, show the transformation property of the given fermion field under 5' U (2) L. In the first family sofa/2) = 112,1“, 9924/2) 2 eL,dL, and (pg?) 2 eR,uR,dR, where V8 and e" are the leptons, and u and d are the quarks. Table 1.1 shows the most important quantum numbers of the first family members. In the SM the neutrinos are massless, which means that (pg?) = 113 is decoupled from the theory. This pattern is repeated in the second and third families of the leptons and quarks, where e is replaced by u and T, and u, d by c, s and t, b respectively. 2Throughout this work we use the 9"” = diag(1, —1, —1, —1) metric. Since fermions are assigned into the fundamental representation of the gauge group, their infinitesimal gauge transformation properties are: (WAR 2 (2'9 T,0,(.1:) + igl§0(a:)) t/JLJZ. Here 9 and gt are the gauge couplings, Tj and Y are the generators of S U (2) L and U (1)y, respectively, 0,-(22) and 0(33) are space-time dependent transformation param- eters, and j = 1,2,3 an isospin index. Since SU(2)L and U(1)y are Lie groups [17], the generators are defined by the Lie algebra of the S U (2)L x U(1)y group, through their commutators mm = Wm [T,, Y] = 0 for any i. The fully anti-symmetric unit tensor fijk gives the structure constants of the S U (2) L group. The generators can be represented by two the dimensional Pauli matrices: Ti = 01/2. Invariance requirement under the gauge and Poincare groups, and renormalizabil- ity constrain the fermionic part of the Lagrangian to: £17 = EI’YHIDLM/Jf, + JQ’YpiDRu'I/sz- Implicit summation is implied over the double fermion family indices f, as well as the double Lorentz indices p. The gauge-covariant derivatives are given by . Y 1),, = au—igmv;—ig/§B,,, DR“ 2 au—igIQBM. The gauge bosons W1. and B appear as the consequence of the gauge invariance requirement, which also fixes their transformation properties: . . ..k . k (WV; : 6,,6’(x) — g6" 9’(:r)W#, (SBp = 8,,0(.1:). From the above we can infer that the gauge bosons belong to the adjoint represen- tation of the gauge group. The kinetic term of the gauge bosons is written in the form 1 u 1 i iw £1) 1‘: —ZBW,B“ — ZWWJW‘ , where the field strength tensors defined as W,j,, apwj — aw; + g 61'ng Wf, B,” = aflBV—ausp. The Yang-Mills [22] nature of the electroweak gauge fields is reflected in the fact that Ly contains the self-interactions of the gauge bosons, due to the third term of the tensor W;,,. This terms is required by invariance under the S U (2) L, a non-Abelian transformation group. The W; and B,u fields are the, so called, electroweak (or interaction) eigenstates of the gauge bosons. Their physical counterparts, the Wi, Z 0 particles and the photon (A0), given by the following transformations: u \/2 1 Z3 2 cos 0,, W3 — sin 6w B”, A2 = sianW3+cosélem (1.1) where 0w is the weak mixing angle, and gl x/g2 + 9'?" After expressing the Lagrangian in terms of the physical fields, we also find that the sin 0w 2 charged fermions couple to the photon field by Q [6, where e = gsinfiw, which is the strength of the electromagnetic interactions. It is quite remarkable that the gauge and Poincare symmetries, together with the requirement of the renormalizability fully constrain the interactions between the fermions and the gauge bosons and leave only two independent free parameters in the electroweak sector. They are the group couplings g and gl. 1.1.2 Strong Interactions The strong interacting sector of the SM is called Quantum Chromodynamics, QCD in short. QCD was developed along the lines of QED, except from the beginning it was clear that the gauge group must be more involved. The SU (3) symmetry was originally proposed to describe nuclear interactions as a flavor symmetry, and was only later identified as a gauge (color) symmetry [23]. It is straightforward to extend the gauge symmetry of the electroweak sector of the SM Lagrangian to include the S U (3)0 sector by requiring that the quarks transform as: Y 61b“; 2 (igstafla(a:) + igT,0,(:1:) + igl§6(2:)) i/JLfl. Above, 9, is the gauge coupling and ta are the generators of the SU(3)C group, 00(93) are the gauge transformation parameters, and a = 1, ..., 8 a color index. The SU(3)C generators satisfy the algebra: [tan tbl = ifabctca and commute with the electroweak generators. The structure constants of the S U (3)0 are denoted by fab“. The three dimensional representation of the generators are the Gell-Mann matrices: to = A,/2. Just like in the EW case, gauge and Poincare invariances and renormalizability constrain the covariant derivatives of the quarks: . Y DLu = (9,, — z'gstanjl — ig 71W: — igl-2—BM, DR, = a, —— igstaGZ — igIQB,,. (1.2) The introduction of the gluon field G: is the necessity of the gauge invariance. It is assigned to the adjoint representation of the S U (3)0 group, and under the group transformations it transforms like 66': = 8,,6“(:r) -— gsfab66b(:r)Gf,. The kinetic term of the Lagrangian is 1 __Ga Ga“), Lg: 4 p.11 with the field strength tensor 0:. = 6.0:: — 0.0:: + gmbcazag. The strong interactions also have self interacting gauge bosons due to the non-Abelian nature of the symmetry group SU(3)C. QCD introduces only one additional parameter into the SM: 9,, the gauge coupling strength between quarks and gluons. In principle it is possible to introduce another, so called 6 term, into the QCD Lagrangian which satisfy all the symmetry requirements: 2 gs A £9 = $06” “"Gf'cAGZV, where e""“" is the fully antisymmetric unit tensor. This term violates P and CP conservation, where P stands for the parity and C for the charge conjugation discrete transformation. From the measurement of the electric dipole moment of the neutron one concludes that the 0 parameter must be very small (9 < 10‘9). This term has no significance for this work, therefore we simply ignore it. 1.1.3 The Higgs Mechanism In the above discussion of the EW and QCD interactions all the fields, representing different types of fundamental particles, are massless. Since fermion mass terms de- scribe transitions between left- and right- handed chirality states, and the different handed fermions have different transformation properties under S U (2) L the simplest fermion mass terms are forbidden by the SM gauge group. Gauge boson masses are not allowed either, since even in the simplest gauge group U (1) a term like 771238po breaks the gauge symmetry. This seemed to conflict the observation that the weak interactions are very short ranged, which originally hinted the existence of massive mediators [24]. To resolve this problem, the spontaneous breaking of the gauge sym- metry was proposed [25]. The essence is that while the Lagrangian can be kept invariant under the S U (2) L x U (1)y transformations, one can break the symmetry of the vacuum, a result of which the gauge bosons can be rendered massive. This maneuver is called the Higgs mechanism. The spontaneous symmetry breaking (SSB) is implemented in the SM by the introduction of an elementary scalar which has a non-vanishing vacuum expectation value (vev) [18]. As an added bonus the same mechanism offers a possibility to generate the fermion masses [26]. The idea of the Higgs mechanism is based on the Goldstone theorem [27]. In order to achieve the symmetry breaking, customarily, a SU(2)L doublet, complex scalar field is introduced: The Nambu-Goldstone fields (pi and goo are non-physical since their degrees of freedom will be absorbed by the W:t and Z0 gauge bosons as longitudinal polarizations. The 10 physical Higgs field H has the following electroweak quantum numbers: «was» where v is a real parameter. This implies that the vev’s of the SU(2)L, and U (1),» generators, and the charge operator acting on the (I) field are: ’U (:1:>=”<>¢0. >=fi¢0. >=((T3+§)<1>)=o, 3 where a, are the Pauli matrices. Namely, SU(2)L x U (1),, is broken by the Higgs vacuum but the electromagnetic symmetry U (1) EM is intact. In order to show that simultaneously with the breaking of S U (2) L and U (1)y, the gauge bosons will acquire a mass, the Lagrangian of the scalar sector is written in an SU(2)L x U(1)y invariant form: ,2 2 £7, = (1),,)T (D“(I>) — /\ (ole — 92—) , where D” 2 DZ of Eq.(1.2). The quadratic terms for the gauge bosons emerge from the first term. After diagonalizing the bosonic mass matrices and using Eq. (1.1) we arrive at the mass relations: ,/ ‘2 I? ,—— mWZ'g—2’ga mzzi—fZ—LQ—Ba msz 2A9 and m A = 0, as expected. Fermion masses can be generated trough Yukawa interaction terms between the fermions and the field. Utilizing that the scalar field is an SU(2)L doublet, the 11 fermion Yukawa terms (shown here only for the first generation of fermions) can be written in a gauge invariant form: Ly = —)\,,(EL 3;) (I) uR — MW). 31,) (Dd); — AalveL EL) (I) 6,; + hermitian conjugate, where (I) = 2iT2* is the charge conjugate of (I), and we assumed that neutrinos are massless. Yukawa mass terms induce the possibility of physically observable mixing between fermions, and there is empirical evidence that the fermions of different families mix. This mixing can be represented as follows. The electroweak (gauge) eigenstates are expressed in terms of the mass eigenstates through a unitary rotation UL 11L d1. dL CL = U}: CL , SL = U: 81, , etc, tL t1, bL bL where we denote the spinors of the mass eigenstates by bold letters. After the diago- nalization of the fermion mass matrices the mixing among the leptons (assuming zero neutrino masses) or the right handed quarks can be absorbed into the definition of the fields. After redefining the left-handed up type quarks such that their mass matrix is diagonal, the remaining effect can be described by the Cabibbo—Kobayashi—Maskawa (CKM) weak mixing matrix [28], V = UZlUfi, relating the mass and electroweak eigenstates of the left handed down type quarks as (£21,335), 2 V”, (til/2)». The indices f, f' = 1, 2, 3 run over the three families. The free parameters associated with the spontaneous symmetry breaking are the Higgs boson mass m”, and the vev '0. Besides, the Yukawa couplings Aflam, are not restricted by the symmetries of the SM, so none of the fermion masses are predicted, which increases the number of the free parameters by 9. After the re-phasing of the quark fields 3 independent angles and a phase parametrizes the unitary CKM matrix. In total, there are 19 free parameters in the SM, including the QCD 0 parameter. f 04' 0,.. ~ A— :T'V'v' 4... ‘5' . .. _, g, . 01H] f'v fflyr ' " ‘a w 12 While the gauge sector of the SM is well established, the symmetry breaking mech- anism is awaiting confirmation from the experiments. The Higgs boson, up to date, is a hypothetical particle, and the Higgs potential has hardly any experimental con- straints [29]. There are also alternative formulations of the spontaneous symmetry breaking which avoid the introduction of an elementary scalar [30]. 1.2 The Quantum Nature of Gauge Fields The detailed description of the quantization process of gauge fields is laid out in many textbooks [4, 5]. Here we only highlight those results which are necessary for the understanding the rest of this work. Because we want to focus on the partial summation of the perturbative series, to introduce the basic definitions and to set up a simple example, in this Section first we examine the strong coupling constant, as the running coupling between quantum fields. Then we explain the basic ideas of factorization which connects the hadronic level cross sections to the parton cross sections which are calculable from the SM Lagrangian. 1.2.1 Renormalization In the pioneering days of QED it became evident that beyond the lowest order cal- culations individual terms appear to contain infinities. Ultraviolet (UV) singularities arise because of the local nature of the field theory, combined with the assumption that the theory is valid at all energy scales. When particle 100ps are shrunk to a single space-time point, that is the momentum of the particle in the loop is taken to be infinity, the corresponding integral in the Feynman rules is divergent. Similarly, when a massless particle, e.g. a photon or a gluon, forms a loop and its momentum vanishes, the Feynman integral over the loop momentum is singular. This latter is a type of the infrared (IR) singularities. lll A ....,..~ I“ y. “1.9.. ; 49.5.. p . u“. Ill: II”; V . . ‘ ‘ w: 1 ‘ 1 1: Millie 13 In a field theory we encounter two typical attributes when calculating physical ob- servables using the method of the perturbative expansion. The first is the systematic redefinition of the parameters of the theory, order by order in the perturbation series. This feature is the consequence of the perturbative method, and already present in quantum mechanics. The second property, specific to field theories, are the appear- ance of the infinities. These two features are connected in renormalization, when the redefinition of the parameters is tied to the removal of the infinities. Renormalization is only possible if the infinities are universal, which is the remarkable case in the SM. In order to execute the renormalization procedure in a quantum field theory, first we have to regularize its infinities. There were several regularization methods pro- posed in the literature [31]. Today the most commonly used regularization method is dimensional regularization [32] , which preserves the symmetries of the SM La- grangian, most importantly, the Lorentz and gauge invariances. The idea of dimen- sional regularization is based on the simple fact that integrals which are singular in a given number of dimension, can be finite in another. In the framework of the dimen- sional regularization we assume that the number of dimensions in the loop-integrals D, i.e. the number of the space-time dimensions, differs from four. We then calculate all integrals in unspecified dimensions D, where all the results are finite. After the removal of the singular terms from the results we can safely take the D —-> 4 limit, and obtain meaningful physical predictions. The procedure of relating the measurable parameters to the parameters of the initial Lagrangian, while systematically removing the UV divergences, by the intro- duction of suitable counter terms into the Lagrangian order by order in the pertur- bation theory, is called renormalization [33]. If the counter terms can be absorbed in the original terms of the Lagrangian by multiplicative redefinition of the couplings (g), masses (m), gauge fixing parameters (6) or the normalization of the fields them- .‘ ‘ a LA..- ir. :r. :1, 'L'. l t “I ..., . .‘a a' i; F .3, i IAN] I Ii" . . ”H. t 14 selves, then we say that the theory is renormalizable. If an infinite number of counter terms needed to define all measurables (in all perturbative orders), then the theory is not renormalizable. Historically, it was the proof of the renormalizability of the Yang-Mills theories with spontaneous symmetry breaking [34] which opened the door in front of the S U (2) L x U (1)y gauge theory toward today’s SM. After the renormalization procedure the parameters, 9, m, E, etc., and the fields of the original, unrenormalized Lagrangian are referred to as bare ones, and the redefined parameters gr = Zglga mr : ZTT-ll/2m) 51' : Zé—lé’ etc-a and the fields as renormalized ones. There is a finite arbitrariness in the definition of the singular terms which are removed by renormalization. This is fixed by the renormalization scheme. Throughout this work we use the MS scheme, unless it is stated otherwise. 1.2.2 Asymptotic Freedom and Confinement It is a generic feature of the parameters of a quantum field theory that they exhibit a dependence on the (energy or distance) scale at which the theory is applied. This dependence leads to crucial consequences in Yang-Mills theories, for it results in asymptotic freedom and hints confinement in QCD. In this Section we outline these features, which play important roles in our resummation calculations. When calculating a physical quantity beyond the lowest order, we encounter in- finities. After regularization these infinities can be isolated either in additive or multiplicative fashion. Taking the QCD coupling constant as an example, we can write: 9 = u"Zg(/t) gr(u)- (1-3) This In: ." , . Inl mg] p: l"’\ L I ii” vpy [7‘ A .A'«AA mitt ”he: ¢ .J A—-—{ 15 The factor if is introduced to preserve the mass dimension of the 4 dimensional bare coupling constant 9, and e = (4 — D) / 2. The above equation also reflects that in any process of regularization we inevitably introduce a scale into the theory. In the case of dimensional regularization this energy scale is the renormalization scale it. The renormalization constant Z9 embodies the infinities, while renormalized coupling g, is finite. Renormalizability of the SM exhibits itself in that Z9 is process independent. The evolution of the renormalized parameter as a function of the renormalization scale can be described by the transformation 9.00 = 2m, u’) 94M). where 2901, u’) = Z; 1(It) 3901’)- These transformations form an Abelian group which is called the renormalization group (RG). The trivial requirement that the bare parameters of the Lagrangian are indepen- dent of the renormalization scale is expressed, for the coupling, as: £9._ d” 0. This and the relation between the bare and renormalized coupling lead to the differ- ential equation dgr(u) ,u dp = B(.9r)- (14) This equation is the renormalization group equation (RGE) for the coupling [35]. From the RGE and the relation of the bare and renormalized coupling we deduce the relation between the 6 function and the renormalization constant Z9: __ e H ngU‘) ”(m— g.( +2900 du I ’ .' T,.,1 Vi 1.1 I. i‘ “ AL L. “12,; 16 The above equation is the RGE for the renormalization constant 29 and, in general, it is written as 7 ___ I” (£2901) g 2901) dl‘ , where 79 is called the anomalous dimension (of the coupling, in this specific case). The renormalization constant is calculable order by order in the perturbation theory by the requirement that the counter terms must cancel the singular terms arising in the given order. In the lowest non-trivial order in the coupling, for the S U (N ) gauge coupling one finds [5]: 2901) = 1 0(5)“)- T g?(p)11N — 4TRNf 1 + 47r2 24 :3 Here N is the number of dimensions of the fundamental representation, TR is the normalization of the trace of the SU (N) generators (Tr(t“tb) = (WTR), and N f is the number of the light fermionic flavors (i.e. m f < ,u). With the aid of the last two equations we can determine the lowest order coefficient in the perturbative expansion of the B function a a 2 a 3 My.) = —g..—5. —g. (—) a —g. (—) a. +0(g.a4). 7r 1r 7r where we introduced the running coupling oz(,u) = g3(u)/(47r), in the fashion of the QED fine structure constant. The lowest order coefficient is identified to be: 5 _ 11N — 4TRN, 1“ 12 ' The higher coefficients, B2 and B3, in the expansion of the B (9,) function were also calculated [36]. By solving the RGE, using the lowest order truncation of the B function, the running coupling can be expressed in terms of B1 a< ) a” it = 2, (1 1+ lfil III-[£5 7r #0 17 where do and B0 are boundary value parameters. It is customary to introduce a single parameter A = hoe—"((zmao), and to write 0'01)“ 71' 2- _ t 51 111‘]?- Equation (1.4) and its solution for the coupling has a striking physical meaning: the coupling depends on the energy scale of the interaction. From the solution we immediately see that as the scale It increases the coupling strength decreases: lim 0(a) = 0. p—ioo This is the phenomenon of asymptotic freedom, a necessity of a Yang-Mills type interaction. In QCD it means that the higher the energy we probe the quarks at, the least they interact. Asymptotically they are free at high energies, that is at short distances. On the other hand, as they separate, or at lower energies, their interaction becomes strong. This implies that when quarks separate the color field becomes so strong between them that it will be possible spontaneously create a new quark pair. This way quarks can never be separated by more that a few femto—meters. They are confined within hadronic bound states. In order to determine the coupling constant we have to calculate at least one physical quantity, e.g. a total scattering cross section, and compare the calculation with the measurement of the cross section. In practice, the calculation is necessarily truncated at a certain finite order. For the truncated series to be a good approxima- tion of the full sum, the expansion variable, the coupling constant, has to be small. We also want to keep the logarithms of the renormalization scale in the coefficients of the expansion small. These two conditions constrain the value of the renormalization 18 0.5 l: m 7 T r 1—71' 1' f T fiT r' Y? "I. SQ, ‘ “0(0) ”3. mm . m 2 ‘ I- It (DJ 0 .. C1,: '1. 8 i 1 .1! g m I I n T 9’: m m 7‘ ] 0.3 — " 'g " if"; I“ in +0 ‘ b d- '0 '33 .0 ...a 0 1 I l‘I.‘ ‘ o 8 '8 T .0 \g ; g 0.2 ” ..... la 8 3 d ....... Q 4 I -.'.‘.'.'.',‘," - U) in“. 5 I r .............................. .3. . 0.1 r I ’ ..IIIKIII: al.(l/|z)=0-118i0-003 O b l 1 I 1 A l l l 1 L A i l I A A l l l O 100 Q/[GeV] Figure 1.1: Running of the QCD coupling constant as, for diflerent boundary values at p = mg. The QCD prediction is compared to current experiments. (Reproduced from Ref. [37].) scale close to the typical physical scale of the process at band. Fig. 1.1 shows the running of the QCD coupling, a,(Q) = gf(Q)/(47r), as the function of the physical energy scale Q = u, in the experimentally accessible region for the boundary values a,(mz) = 0.118 :l: 0.003. The running, predicted by QCD, is in agreement with the experiments within the uncertainties. From our argument it is non-trivial that 01,01) is a smooth function of the energy, since the value of the coefficient B1 is chang- ing abruptly as the function of N f at every quark mass threshold. The continuous definition of 01,01) is achieved by requiring a non-continuous A parameter. The A parameter in QCD, AQCD(Nf, Nerd"), becomes the function of the number of active (mq < u) quark flavors N f, and the perturbative order Nordera up to which the B function was calculated, to ensure a smooth matching at the quark thresholds. As an my" '7' ..o.t.n. I‘ .5 5 c ‘ I f. ‘, n. 19 example we show a set of AQCD(Nf, Norder)/GeV values which is recently in use [38]: N,=3 N,=4 N,=5 N,:6 No.0,... = 1 0.272 0.236 0.181 0.094 NW, = 2 0.354 0.298 0.202 0.081 The running coupling is one of the simplest examples of a resummed quantity. It is calculated by the reorganization of the perturbative series, such that the largest all order contributions are included in the new leading terms. The partial summation of the perturbative series implemented in this fashion is called resummation, which is the main theme of this work. Since the running coupling is a resummed quantity, when used in a fixed order expression it automatically includes certain all order effects in the calculation, and as such, it improves the quality of the prediction. 1.2.3 Factorization and Infrared Safety The QCD Lagrangian describes the interaction of quarks and gluons. On the other hand, because of confinement, we observe only hadrons. According to the parton model [39], hadrons are composed from partons: quarks and gluons. Although, nu- merical implementations of QCD suggest that the theory describes confinement, we do not have a mechanism to derive low energy (long distance) properties from the QCD Lagrangian. In order to describe interactions involving hadrons at high ener- gies we use the tool of factorization. Factorization is the idea of the separation of the short distance physics from the long distance physics in high energy collisions. The short distance physics is described by perturbative QCD. The long distance effects are showed to be universal, parametrized, and extracted from measurements. In order to illustrate the plausibility of factorization in hadronic scattering cross sections, we first highlight the calculation of partonic cross sections. When calculating 1., l. \4 I g I 'r I 4. O r? L. . . ,. L‘ r .r ’I‘ ,1.‘ .1§ )- Va t.‘.‘r 20 a partonic cross section beyond the leading order, after UV renormalization, and cancellation of the IR singularities between real emission and virtual diagrams the cross section still contains collinear singularities. These collinear singularities, on the other hand, can be isolated as multiplicative factors, just like in the case of UV renormalization. Similarly to Eq. 1.3, the partonic scattering cross section can be written as d0.mag-—>X({3}) = [01 dam [01 (11:2 ¢b1/a.($1./1)d501b2—Ix({3}.$1.$2.#)¢b2/a2(£132,#). (1.5) where summation over the double partonic indices (2,- is implied, 61,15an is the hard scattering cross section of two partons b1 and b2, and ¢b/a(:r, p) can be interpreted as the probability of the collinear emission of parton b from parton a with a momentum fraction 3: at energy it, and {3} represents the collection of the kinematical variables relevant to d& (e.g. the center of mass energy of the collision, etc.), and X is some arbitrary final state. The hard cross section 5b102—Ix and the distributions qfib/a are calculable in perturbative QCD order by order.3 The above form is useful because, as it can be shown by direct calculation for a given process, all the non-cancelling infrared singularities associated with the emission of collinear partons are universal and can be defined into the distributions 45b)“ and the hard scattering cross section 50102» X is infrared safe (i.e. finite). When calculating 501024 X at higher orders of the perturbation theory, its infrared safety can be maintained by absorbing its collinear singularities into the distributions (two, [40]. In this procedure there is a freedom of shifting part of the hard cross section into the distributions. This introduces the factorization scale, [2, dependence into the definition of both 6%,wa and ¢b/a- Just like in the case of the UV renormalization, there are different scheme choices for the parton densities, depending on the amount of the finite contributions arbitrarily 31n the lowest order of the perturbation theory the distributions are just delta functions: ¢b/a($a#) = 6505(1 - x) + 0(a.(u),#)- 21 defined into (bb/a. This factorization of the infrared behavior is a factorization of the short and long—distance physics, because the fact that (3010st is infrared safe implies that it cannot depend on physics which is associated by long distances. The long distance physics is described by the universal distributions (lb/a, which can be shown to be independent of the short distance features captured by 51.1024 X. We can show by explicit calculation that Eq. (1.5) generalizes to hadronic cross sections: d0h1h2—§X({8}) = fol 611171 [01 (1132 fbl/h1($1,fl) d60102—)X({8}7$1a$2au) fbg/h2(£172,#). (1.6) where fa/h($, p) is a universal (process independent) function which gives the proba- bility of finding parton a in a hadron h with a momentum fraction of a: at the scale ,u. (While the introduction of the partonic distributions is somewhat heuristic in the parton model, they are defined formally in the operator product expansion in terms of expectation values of non-local operators.) The factorization theorem [41], generally formulated in Eqs. (1.5) and (1.6) is an assumption, until it is proven process by pro- cess, and order by order of the perturbation theory that the infrared singularities of the hard cross section can be factorized into the parton distributions, and an infrared safe hard cross section can be defined. The proofs are given in the literature for the most important processes as deep inelastic scattering or the Drell-Yan lepton pair production. After it is proven, the factorization theorem is used to predict cross sec- tions involving hadrons. Comparing these predictions to the experiments, the parton distributions then can be extracted. Although, presently we do not know how to calculate the parton distribution functions of hadrons, fa/h(:r, u), from the first principles, since their scale dependence is governed by perturbative QCD, based on the RG technique we can calculate their 22 evolution with the energy scale )1: “dfa/hcvvp) 1% du 0€ where the splitting functions, Pa._1,(:r,a,(u)), are calculable order by order of the P... (g am) 10.16,”). perturbation theory [42], since they describe the short distance physics of partonic splitting. (The analog of this equation can be written for partonic distributions db/a, where the splitting functions are calculated by perturbative methods.) For the case of gluon radiation their C(03) expressions are: 1+ 22 Pji2k(z) : CF< ) t + 1-2 1022c) = ; [22+(1— 2):]. (1.7) where the “+” prescription for arbitrary functions f (z) and 9(2) is defined as [31 d2 (f(Z))+ g(z)z/01dz f(Z) [@(Z — $)g(z) — 9(1)] , with 0, ifrr<0, e($)_{1’ifx->-O’ the unit step function. (It is assumed that f (z) is less singular than 1/ (1 — .2)2 at z = 1.) Similarly to the case of the running coupling, the parton distributions, being solutions of the above RGE, include resummed collinear contributions. The renormalization of the ultraviolet singularities and the factorization of the in- frared divergences can be viewed in parallel [43]. The ultraviolet singularities absorbed into the renormalization constants Z,-, just like the collinear divergences absorbed into the parton distributions fa/b, that is in the factorization the parton distributions play the role of the renormalization constants, with the added complication that they also depend on the momentum fraction 2‘, besides the energy scale it. The evolution equations, describing the scale dependence of the renormalization constants and the 23 parton distributions can also be casted into analogous forms, with the anomalous dimensions 7,- identified with the splitting functions Panb. Chapter 2 Soft Gluon Resummation 2.1 Lepton Pair Production at Fixed Order in as In this work we apply and extend the resummation formalism developed by Collins, Soper and Sterman (CSS) [44, 45, 46], which resums large logarithmic contribu- tions, arising as the consequence of the initial state multiple soft gluon emission. We calculate these resummed corrections to processes of the type hlhg -—> 121ng, where h,- are hadrons, 01 and 02 is a pair of leptons, photons or Z0 bosons, and X is anything else produced in the collision undetected. More specifically, we con- sider lepton pair production through electroweak vector boson production and decay: hlhg —+ V(—> €1f2)X (V is Wi, Z0 or a virtual photon), and vector boson pair pro- duction hlhg —> VVX with V being photon or Z0 boson. To resum the contributions from the soft gluon radiation, we use the results of the various fixed order calcula- tions of these processes, which are found in the literature [47, 48, 49, 50, 51]. In this Chapter we review the lepton pair production through W i, Z 0 or virtual photon (7“) production and decay in some detail, to introduce a pedagogical example identifying technical details which are most important for the resummation calculation. 24 25 2.1.1 The Collins-Soper Frame Since the Collins-Soper-Sterman resummation formalism gives the cross sections in a special frame, called the Collins-Soper (CS) frame [52], we give the detailed form of the Lorentz transformation between the CS and the laboratory (lab) frames. The lab frame is defined as the center-of-mass frame of the colliding hadrons 111 and hg. In the lab frame, the cartesian coordinates of the hadrons are: . \/§ pfl1,h2(lab) : T(1v0103i1)I where \/S is the center-of-mass energy of the collider. The CS frame is the special rest frame of the vector boson in which the z axis bisects the angle between the hl hadron momentum ph, (CS) and the negative hg hadron momentum —ph,(CS) [53]. To derive the Lorentz transformation A“,,(lab —> CS) that connects the lab and CS frames (in the active view point): p“(CS) 2 A“,,(lab —> CS) p"(lab), we follow the definition of the CS frame. Since the invariant amplitude is independent of the azimuthal angle of the vector boson (43y), without loosing generality we start from a lab frame in which (by is zero. First, we find the boost into a vector boson rest frame. Then, in the vector boson rest frame we find the rotation which brings the hadron momentum ph, (CS) and negative hadron momentum -Ph2(C S ) into the desired directions. A boost by B = —(i'(lab)/q0 brings four vectors from the lab frame (with (by = 0) into a vector boson rest frame (rest). The matrix of the Lorentz boost from the lab frame to the rest frame, expressed explicitly in terms of q“ is A“,,(lab ——> rest) = (IO -Q;I‘ 0 "(13 Q , 3 Q 0 0 Q 0 _. 3 9:11 1132: ‘1 qoiQ 0 Q + q°+Q where Q = \/(q0)2 — Q3~ — (q3)2 is the vector boson invariant mass, and the transverse mass is defined as MT = «Q? + Q%. After boosting the lab frame hadron momenta into this rest frame, we obtain pfihhz (rest) = A”,,(lab —> rest) pzl,h2(lab) = [5 (00¢? _Q_:r_~q°+Q4=q3 0 (iQ-q3)(q0+Q):t(q3I2) 2 Q ’ Q q°+Q ” Q(q°+Q) ’ and the polar angles of pfil (rest) and —p’,:2 (rest) are not equal unless QT = 0. (In the above expressions the upper signs refers to hl and the lower signs to hz.) In the general QT aé 0 case we have to apply an additional rotation in the rest frame so that the z-axis bisects the angle between the hadron momentum ph, (CS) and the negative hadron momentum —ph,(CS). It is easy to verify that to keep 1572.112 in the 122 plane, this rotation should be a rotation around the y axis by an angle a = [00° + MT>/(MT(q° + Q))]- Thus the Lorentz transformation from the lab frame to the CS frame is A“,,(lab —> CS) 2 A”,\(rest ——> CS) A",,(lab —) rest). Indeed, this transformation results in equal polar angles 6,“th = arctan(QT / Q). The inverse of this transformation takes vectors from the CS frame to the lab frame is: A“,,(CS —> lab)=(A",,(lab——>CS))"1= cIOMT QOQT 0 QBQ 1 QTil/IT M% 0 0 W5 0 0 QMT 0 q3MT q3QT 0 (1062 The kinematics of the leptons from the decay of the vector boson can be described 27 by the polar angle 0 and the azimuthal angle d), defined in the Collins-Soper frame. The above transformation formulae lead to the four-momentum of the decay product fermion (and anti-fermion) in the lab frame as p p“ : g ((15 +sin0cos¢ X“+sindsind Y“ +cos€ Z“), : qu_pu, P where q" = (MT cosh y. QT cos ¢v. QT sin 49v, MT sinh y), u _ u —u M72‘ 11 X — —QTMT (q+n +q_n —§ ), Z” = "1“ (‘1+”” “ Q—fi”) 9 MT Y“ = Euuaflgzan. (2.1) Here, qi = %(q0 :l: q3), y = %1n(q+/q_), n” = %(1,0,0,1), ft” = $—§(1,0,0, —1) and the totally anti-symmetric unit tensor is defined as 50123 = —1. 2.1.2 General Properties of the Cross Section Along the lines of the factorization theorem (cf. Eq. 1.6), the inclusive differential cross section of the lepton pair production in a hadron collision process ’1th —+ V(-—> €1g2)X IS written as d0h1h2 V(—> f 1(72)X, and fa”, are the parton distribution functions. The variable Q denotes the invariant mass, y the rapidity and QT the transverse momentum of the lepton pair, while 6 28 and d are the polar and azimuthal angles of the lepton 21 in the rest frame of V. The indices a and b denote different parton flavors (including gluon), and V may stand for Wi, Z0 or 7*. The partonic momentum fractions :17,- are related to these independent variables as a: _ MTey a: _ MT€_y l — ' a 2 —— ' 3 \/§ \/§ and 351,2 21,2 — '— (1,2 The partonic cross section Bab is calculated as . 1 d0ab= 1 ‘ I 29,. 6(8 + t + u — C22) dozdogdydcosadqs. The averaged square of the invariant amplitude I . [Mab(S, t, u, IL)]2 = — Z [Mab(8, t, u, Itf)]2, c c s 5 ”07171720717? colors where "3,0 and nib are the number of spin and color degrees of freedom of partons a and b. 29 Using the relation 3 = {1628, where \/S is the center of mass energy of the collider, the differential cross section of the lepton pair production in a fixed order of the strong coupling is written as d0(h1h2 "‘k V(—) €1g2)X) _ 1 Z [1 _d£1_ 1% dQ2dQ%dydcosdd¢ F d — 2107r4S at x. 5. x. 52 X fa/h1(€11 #)]Mab(3, t1 U1“)I26(3 + t + u — Q2) fb/h'z (£21 ’1'): In the above equation both the invariant amplitude Mab(8,t,u, 11.) and the parton distribution functions fa/h(:1:, ,u) depend on the renormalization scale H=C4Q and the corresponding value, g,(u) = 47ra8(/1), of the QCD coupling. Normally one sets the constant 0., to be of order 1, to avoid large logarithms ln(Q/u), that would otherwise spoil the usefulness of a low—order perturbative approximation of Mab. The squared invariant amplitude of the partonic level process is calculated con- tracting the hadronic and leptonic tensors H “V and LW: 2 V [M..(s.t, 21ml = H5. (k. z,Q)L,..(z..12.o) where, for the qt] —> V(—> 81(72)g process k, 1, ll, [2 are the quark, anti-quark, lepton and anti-lepton four-momenta, respectively. The detailed from of the hard cross sec- tion is found in the next Section. The hadronic tensor Hub is calculated by evaluating the 0(a5) real and virtual gluon emission diagrams and adding together these con- tributions. In order to keep track of the vector boson polarization one has to evaluate the symmetric and anti-symmetric parts of both the hadronic and leptonic tensors. Since the calculation of the leptonic tensor is trivial we focus on the hadronic tensor. The hadronic tensor can be split into a sum of a symmetric and an anti-symmetric tensor H::(k,l,Q) : Hcl‘ijslki va) + H:;A(kv la Q) 30 Here Hf: 3(k, I, Q) depends on symmetric combinations of the four-momenta and the metric tensor. The calculation of the symmetric part of the hadronic tensor H::S(k, I, Q) is straightforward and does not depend on the prescription of the 75. This is because one does not have to evaluate a trace that contains '75 to calculate H::S(k,l,Q). But the anti-symmetric part of the hadronic tensor involves traces containing single 75’s. These traces are, by definition, proportional to the fully anti- symmetric tensor e‘wafi, so that the calculation of H 5: A(k, l, Q) lead to the form H:;A(k, i, Q) = h1(3, t, u, 622%.!ka _+_ h2(8, t, u, Q2)EuulQ + 6 h3(8, t, u, Q2)E’wkl. The last term in this equation would violate the Ward identities in four dimension. Although proportional to e = (4 — D) / 2, after the phase-space integration it con- tributes to the four-dimensional cross section if ’23 does not vanish. 2.1.3 The Cross Section at (9(a5) To correctly extract the distributions of the leptons, we have to calculate the pro— duction and the decay of a polarized vector boson. The C(03) QCD corrections to the production and decay of a polarized vector boson can be found in the literature [48], in which both the symmetric and the anti-symmetric parts of the hadronic ten- sor were calculated. Such a calculation was, as usual, carried out in general number (D) of space-time dimensions, and dimensional regularization was used to regulate infrared (IR) divergences because it preserves the gauge and the Lorentz invariances. Since the anti-symmetric part of the hadronic tensor contains traces with an odd number of 75’s, one has to choose a definition (prescription) of 75 in D dimensions. It was shown in a series of papers [54, 55] that in D 76 4 dimension, the consistent 75 prescription to use is the t’Hooft-Veltman prescription [54]. Since in Ref. [48] a different prescription [56] was used, we give below the results of our calculation in the t’Hooft-Veltman 75 prescription. (fl 31 For calculating the virtual corrections, we follow the argument of Ref. [57] and im- pose the chiral invariance relation, which is necessary to eliminate ultraviolet anoma- lies of the one loop axial vector current when calculating the structure function. Applying this relation for the virtual corrections we obtain the same result as that in Refs. [48] and [58]. The final result of the virtual corrections gives MTBornMViT't + MLirt'MBOT‘n : as 47w“ 1 (__2__§ 2_) 2 CF?” ( Q2) P(1—€) £2 €+7r 8 [Mgorn] , (2.2) where e = (4 — D) / 2, p is the t’Hooft mass (renormalization) scale, and CF 2 4/ 3 in QCD. The four-dimensional Born level amplitude is 16624 (Q2 - M1242 + Q4fo/Mx2/ x [(9% + gm: + fiaco + (9% — gm — $.43] , (2.3) [MBornI2 : where we have used the LEP prescription for the vector boson resonance with mass My and width PV. The angular functions are £0 = 1 + cos2 6 and A3 = 2cos 0. The initial state spin average (1 /4), and color average (1 / 9) factors are not yet included in Eq. (2.3). When calculating the real emission diagrams, we use the same (t’Hooft-Veltman) 75 prescription. It is customary to organize the 0(a’g) corrections by separating the lepton degrees of freedom from the hadronic ones, so that d0(h1h2 _) V(—) €1Z2)X) real emission _ aS(Q)CF Q2 dQZdde2Tdcos 0d¢ _ (27r)3S (Q2 — IVE/)2 + Q‘Tfi/ME, 0(05) x 2 f1 g I 91—6—2 9" cheese?) 7:.(QT,Q,z1.z-.) A.(6,¢), 0,11,: 1 51 x2 £2 with zl = 331/51 and 22 2 1172/52. The dependence on the lepton kinematics is carried by the angular functions 1 1 . £0 = 1 + cos2 0, A0 = -2—(1 — 3cos2 6), A1: sin 20cos¢, A2 = isln‘zflcos 245, 32 A3 = 2 cos 0, A4 = sin 0cos qb. In the above differential cross section, 2' = —1,...,4 with A_1 2 £0; and 9‘ = (9% + 9%)(f2 +ffi) for 2' = —1,0, 1,2; gi = (gi — 9%,)(ffi — f},) for z' = 3,4. The parton level cross sections are summed for the parton indices (1, b in the following fashion 2 5,, 7;, = Z (5,, 7;, + [1,, 7,, + 5,0 7;, + 5,0 7,}, + £0, 73, + £0, 73“,) . a,b q=u,d,s,c,b The partonic luminosity functions Eab are defined as £q§(€li€27Q2) I fq/h1 (6.1an) fq/h2(€‘23Q2), etc, where fq/h1 is the parton probability density of parton q in hadron hl, etc. The squared matrix elements for the annihilation sub-process qrj —> VG in the CS frame, including the 6 dependent terms, are as follows: 7;? = i4114“,” ‘ (t + “)26)’ 7;, ___ m:$%(T+(u,t)—(Q2+Sl2€)’ 731:7 : giaggT—(u’tXl—d, 7:; z 52% (T+(u,t) _ (Q2 — 201931622 — t)u2€) , 7:; = %%(T_(u,t)+Q2(“_t)€)' For the Compton sub-process qG —> Vq, we obtain 7"1 = :1 (71(3) 1‘) — (3 + “)26)’ q su 7% z 7’2 _“1%((Q2_u)2+(Q2+3)2_(s+u)26), (:0 — 2 su MT 7.2 = iQTQ(2(Q2—u>2—2++( >< Q2 )6)’ —2 7% ==ZE%%CM@V-$+TM&W—UW‘IWQ' In the above equations, the Mandelstam variables: 3 = (k + [)2, t = (k — Q)2, and u = (l — Q)2 where k, l and q are the four momenta of the partons from hadrons h1, h2 and that of the vector boson, respectively, and Ti(t, u) = (Q2 —— t)2 :l: (Q2 — u)2. All other necessary parton level cross sections can be obtained from the above as summarized by the following crossing rules: i=—Lar2 i=a4 7272:7213 Tar-“Gt: éq= th(u+-)t) Gl'q=-q1G(UHt) new: 7:.2=- .2 Tqu=7290 éfi=_72;0’ with the only exceptions that TC}, 2 —7;IG (u <—-> t) and Ta“, = 73%; (u H t). These results are consistent with the regular pieces of the Y term given in Section 2.2.5 and with those in Ref. [49]. In the above matrix elements, only the coefficients of £0 and A3 are not suppressed by QT or Q?“ so they contribute to the singular pieces which are resummed in the CSS formalism. By definition we call a term singular if it diverges as Q;2 x [1 or ln(Q2/Q§~)] as QT —> 0. Using the t’Hooft-Veltman prescription of 75 we conclude that the singular pieces of the symmetric (£0) and anti-symmetric (A3) parts are the same, and o -—1 . _ 2 = . :3- .‘I ._ 2 : _ $130 7;, 6(s+t+u Q) (£120 7;, 6(s+t+u Q) sqq, a _1 — 2 : ‘ _ 2 : Qlirgo 7'0, 6(s+t+u Q ) (21220 703,, 6(s+t+u Q ) SGq, 34 where Sqé : Q1 [2 6(1— 21) 6(1— 2:2)(111 -Q—: _. :3) 2 T T 2 +6(1-— 21) (1+z22) +6(1— .22) (11+ 212) — ((1 — z1)6(1 — .22) + (1 — 22)6(1— 21))6] + O (621—), T — 1[(zf+ (1—zf))5(1—22)—6(1—z2)e]+0( 1 ) sG — — — q 62% QT As QT —> 0, only the £0 and A3 helicity cross sections survive as expected, since the 0mg) differential cross section contains only these angular functions [of Eq. (2.3)]. 2.2 The Resummation Formalism 2.2.1 Renormalization Group Analysis In hard scattering processes, the dynamics of the multiple soft—gluon radiation is predicted by resummation [59]—[60]. Since our work is based on the Collins-Soper- Sterman resummation formalism [44, 45, 46], in this Section we review the most important details of the proof of this formalism. After the pioneering work of Dok- shitser, D’Yakonov, and Troyan [61], it was proven by Collins and Soper in Ref. [44] that besides the leading logarithms [62] all the large logarithms, including the sub- logs in the order-by-order calculations, can be expressed in a closed form (resummed) for the energy correlation in e e collisions. This proof was generalized to the single vector boson V production process hlhz —> VX, which we consider as an example. Generalization of the results to other processes is discussed later. As a first step, using the factorization theorem, the cross section of the process is 35 written in a general form 010011};2 —> VX) _ dQ2inrdy _ 161 The hard part of the cross section is calculated order by order in the strong coupling 1d d / 61/152 52 Zfa/m( (51,14) Tab(QaQTazla zmwl fb/h2(§2’“)' 1' x2 a,b Tab(Q,QT,Zl,Z2,/Jx) : Z (:9)nT Til?)(Q QTazlaZZ ”A n=O At some fixed order in as the coefficients T8,") can be split into singular and regular parts T‘")(QQT,21,Z2,u)= T(i‘6(Q QT121’Z21“)6(QT)+ 211—1 2 Z Tig’m)(Q7QTaZ1722aH)-Q—1;lnm(g?)+RR(n)(Q7QTazlaz23#) m=0 where R512) contains terms which are less singular than Q7"? and 6(QT) as QT —> 0. The perturbative expansion of the cross section defined in this manner is not useful when a?Q;~21nm (Q2/an) is large, i.e. in the QT —-) 0 limit. This is expressed in saying that the large log’s are not under control when the perturbative series is calculated in terms of T ,5]: ). To gain control over these large log’s, that is to express the cross section in terms of coefficients which do not contain large log’s, the summation of the perturbative terms is reorganized. The cross section is rewritten in the form d0(h1h2 —> VX) _ szindy — where W(b,Q,$1,fE2) accommodates the singular logarithmic terms in the form of bBZQ‘T'EW(b7 Q1131) $2) + Y(QT3 Q, 1:19 (1:2) ln(b2Q2). The ”regular” terms are included in the Y piece Y(QT)Q)$11$2): 11d 00 s 11 L1 (16611 $2 f—ig fa/h1(€17u) g)('c:r—)1IRR(MJ,)(Q QT,Z},ZQ,[L) fb/h2(£2a p.) The 17111.: n. 0115'.) that i1 Where C 4 1115311. 11: Wimft‘ u in the 111 Th? dljs 36 The Fourier integral form was introduced to explicitly conserve QT [62]. Based on the form of the singular pieces in the QT —> 0 limit [46], it is assumed that the x1 and 3:2 dependence of W factorizes W(Q,b, 331,132) : chh1(Qabaxl) th2(Qab1$2) Q3) .7 where CJ-h is a convolution of the parton distribution with a calculable Wilson coeffi- cient, called C function d wow) = 2: /. 1 f— 0.43.1), 11,621 men), where a sums over incoming partons, and j denotes the quark flavors with charge Q, in the units of the positron charge. It is argued , based on the work in Ref. [45], that W obeys the evolution equation 3 ~ ~ 31,, Q2W(Q1 (’1 $1,132) = [K (1211,9300) +0 (62/11, 9300)] W(Q, 12,111.22»), where K (by, 9301.)) and G (Q/u, gs(p)) satisfy the renormalization group equations (RGE’S) piK (1211,9300) = —w (9361)), 11536 (1911,9300) = +7.. (9.0»). The anomalous dimension 7K can be determined from the singular part of the cross section [63]. These RGE’s allow the control of the log’s, since using them one can change the scale it in K (by, g,(p)) and G (Q/u, 9301)) independently, such that it is the order of 1 / b and Q respectively, so they do not contain large log’s. The solution of the RGE’s constructed in this fashion contain the functions A and B, which one uses to rewrite the evolution equation of W 6 WW(Q,5,,$1,$2)= 37 {/Cszfl—p2 _2 [A( 93(_ MCI)+B (93(CZQ)1CI’C2)]}W(Q’bv$1’x2)° 01/!) The A and B functions can be approximated well, calculating them order by order in the perturbation theory, because they are free from the large log’s. The only remaining task is to relate the A and B functions to the cross section. This is done by solving the evolution equation of W. The solution is of the form ~ C1 W(Qaba$1,$2)= _8”’(ch’ Cz)l’V(—C—1b' b 11,172), 2 where the Sudakov exponent is defined as 022Q2 d—z 2 8(Q,b,cl,cz)=/C,/b, —,‘.‘—,[A(g J1 )c,)1n( Off? )+B (9301 )101302) 1 From this we can write the cross section in terms of the A, B and C functions d0(h1h2 —') WiJY) _ dQ2 eridy — 1 .~ ~ (C C W [dzbelQT'b e—S(Q’b’cl’C2) ZCJ'( ,,b (111) th2(— HI) 1132) Q2- 1 "1 C b C b + Y(QT)Q)x13$2)' This important expression is the resummation formula which we make vital use in the rest of this work. The remaining slight modification of the resummation formula ensures that the impact parameter b does not extend into the b 2 1/AQCD region, where perturbation theory is invalid. To achieve this the b dependence of W is replaced by b _ 1/1+(b/bm..>2’ which is always smaller than bmaz. This arbitrary cutoff of the b integration is com- pensated by the parametrization of the non-perturbative region with the introduction 38 of a non-perturbative function for each quark and anti-quark flavors j and E 2 WjIzP(baQaQOa$17$2) : exp [—Fl(b)1n(g_2' 0 )‘F,/,,,(x1,b) — FF/h2($2’b) 1 where the functions F1, Fj flu and FIE/1:2 have to be determined using experimental data. The In (Qz/Qg) term is introduced to match the logarithmic term of the Su- dakov exponent and its coefficient is expected to be process independent. For the production of vector bosons in hadron collisions another formalism was presented in the literature to resum the large contributions due to multiple soft gluon radiation by Altarelli, Ellis, Greco, Martinelli (AEGM) [64]. The detailed differences between the CSS and AEGM formulations were discussed in Ref. [66]. It was shown that the two are equivalent up to the few highest power of ln(Q2/Q:2,~) at every order in as for terms proportional to Q712, provided as in the AEGM formalism is evalu- ated at b,2;,/b2 rather than at Q2. A more noticeable difference, exceptlthe additional contributions of order Q“2 included in the AEGM formula, is caused by different ways of parametrizing the non-perturbative contribution in the low QT regime. Since, the CSS formalism was proven to sum over not just the leading logs but also all the sub- logs, and the piece including the Sudakov factor was shown to be renormalization group invariant [45], we only discuss the results of CSS formalism in the rest of this work. 2.2.2 Resummation Formula for Lepton Pair Production Due to the increasing accuracy of the experimental data on the properties of vector bosons at hadron colliders, it is no longer sufficient to consider the effects of multiple soft gluon radiation for an on-shell vector boson and ignore the effects coming from the decay width and the polarization of the massive vector boson to the distributions of the decay leptons. Hence, it is desirable to have an equivalent resummation formal- 39 ism [67] for calculating the distributions of the decay leptons. This formalism should correctly include the off-shellness of the vector boson (i.e. the effect of the width ) and the polarization information of the produced vector boson which determines the angular distributions of the decay leptons. In this Section, we give our analytical re- sults for such a formalism that correctly takes into account the effects of the multiple soft gluon radiation on the distributions of the decay leptons from the vector boson. To derive the building blocks of the resummation formula, we use the dimensional regularization scheme to regulate the infrared divergencies, and adopt the canonical- 75 prescription to calculate the anti-symmetric part of the matrix element in D- dimensional space-time.1 The infrared-anomalous contribution arising from using the canonical-75 prescription was carefully handled by applying the procedures outlined in Ref. [69] for calculating both the virtual and the real diagrams.2 The resummation formula for the differential cross section of lepton pair produc— tion is given in Ref. [67]: d0(h1h2 —) V(—') €1g2)X) _ 1 Q2 dQ2 dy erzrdcosflqu m — 487rS (Q2 — M3,)2 + Q‘Tfi/Mg 1 _- ~ ~ ~ X {(2702 /d2bezQT'b Z];WJI—c(bunQa$19$2701¢1017021C3) WJ¥P(b7Qa$lax2) .7, + Y(QT7Q3xl)$2,01¢,C4)} - (2.4) In the above equation the parton momentum fractions are defined as 2:1 2 eyQ/JS and $2 = e‘yQ/x/S, where x/S is the center-of-mass (CM) energy of the hadrons h] and h2. For V = W:t or Z 0, we adopt the LEP line-shape prescription of the resonance behavior, with My and PV being the mass and the width of the vector boson. The renormalization group invariant quantity W,,—,(b), which sums to all orders in as all 1In this prescription, 75 anti-commutes with other 7’s in the first four dimensions and commutes in the others [54, 55, 68]. 2In Ref. [69], the authors calculated the anti-symmetric structure function F3 for deep-inelastic scattering. 40 the singular terms that behave as Q7“? x [1 or ln(Q2/Q%~)] for QT —> 0, is ij(bi Q) $111216, ¢9 C17 02: CB) : exp {—S(b) Q7011 02)} I f/jk '2 XC{[ j/hi ($1)Ck/h2($2) +Ck/h1($l) Cj/h2(x2)] 53+) (aem(C2Q) 03(02Q)7 6) + [Ci/hi($1) C117/hazel”) CE/h1($1) Cj/h2(552)] Jig—)(aem(C2Q)’a3(C2Q)36)} - (2.5) Here the Sudakov exponent 8 (b, Q, C1,C2) is defined as (3ng dflz CZQ2 8(b, Q, 01,02) 2’] T” [A (a3(/1),Cl)ln(2_2 2 cw .11 )+B(as(n 10.32)]. (2.6) The explicit forms of the A, B and C functions and the renormalization constants C,- (z'=1,2,3) are summarized in Section 2.2.3. The 16,, coefficients are given by V- _ Cabibbo — Kobayashi — Maskawa matrix elements for V = W i, (2 7) Jk “ 53* for V = Z0,7*. ' In Eq. (2.5) Ci/h denotes the convolution of the Cm Wilson coefficients with the parton distributions fa/h Cz/h($ )=Z/ld: Cza<-,b,# :0; 01.02) full: (5,” = $3) - (2-8) In this notation we suppressed the b and p dependences of Ci/h, which play the role of generalized parton distributions, including (through their b dependence) transverse effects of soft—gluon emission. In the above expressions j represents quark flavors and it stands for anti-quark flavors. The indices (1 and b are meant to sum over quarks and anti-quarks or gluons. Summation on these double indices is implied. In Eq. (2.5) f-(i) are kinematic factors that depend on the coupling constants and the polar angle of the lepton £1 ;K+)(a,m(02Q), a,(C2Q) 0): (9% + 9%)(ff, + f%)(1 + 13082 9) 17:8 )(aem(C2Q) (“(0262) 9): (92"9f1)(f12. _ fftXZ COS 6) 41 V Mv (GeV) I‘v (GeV) 9L 9R 7 0.00 0.00 gofsw/z ngsw/2 Wi 80.36 2.07 g/(2x/i) 0 Z0 91-19 2-49 9(T3—Qfsi.)/(20w) -9Qfs2w/(26w) Table 2.1: Vector boson parameters and couplings to fermions. The f f” V vertex is defined as i7,,[gL(1 — 75) + gR(1 + 75)] and 3w = sin 0w (cw = cos 0",) is the sine (cosine) of the weak mixing angle: sin2(6w(Mzo))m = 0.2315. Q f is the fermion charge (Qu = 2/3,Qd = -1/3,Q,, = 0,Qe— = —1), and T3 is the eigenvalue of the third component of the SU(2)L generator (T5‘ = 1/2, Tgf’ = —1/2, T; = 1/2, T3” = —1/2). The couplings fLJ; and gL,R are defined through the KIK—QV and the qrj’V vertices, which are written respectively, as ”u ”L“ — ’75) + fR(1+ ’75)] and 1'7“ [9L(1 — 75) + 93(1+ 75)]- For example, for V = W+,q = u, (j’ = (I, 61 = 118, and (72 2 6+, the couplings 9% = ff 2 GFMEV/fi and 9% = f}?2 = 0, where Gp is the Fermi constant. The detailed information on the values of the parameters used in Eqs. (2.4) and (2.5) is given in Table 2.1. In Eq. (2.4) the magnitude of the impact parameter b is integrated from 0 to 00. However, in the region where b > 1/AQCD, the Sudakov exponent S(b,Q,Cl,C'2) diverges as the result of the Landau pole of the QCD coupling ozS(;1) at y. = AQCD, and the perturbative calculation is no longer reliable. As discussed in the previous section, in this region of the impact parameter space (i.e. large b), a prescription for parametrizing the non-perturbative physics in the low QT region is necessary. Following the idea of Collins and Soper [70], the renormalization group invariant 42 quantity Wr k(b) is written as where :+(\/1+(b/bma,,) Here W- k(b. ) is the perturbative part of Wjfib) and can be reliably calculated by perturbative expansions, while Wfi-[P (b) is the non-perturbative part of Wjfib) that cannot be calculated by perturbative methods and has to be determined from exper- imental data. To test this assumption, one should verify that there exists a universal functional form for this non-perturbative function W111) (b). This expectation is based on the general feature that there exists a universal set of parton distribution functions (PDF’s) that can be used in any perturbative QCD calculation to compare it with experimental data. The non-perturbative function was parametrized by (cf. Ref. [46]) NP Q2 WE (baQ1Q01xla‘T2) =exp [F1() (Q2 where F1, Fj/h1 and Fig/hm, have to be first determined using some sets of data, and )— Fj/h1($1,b) — FE/h2($21b)l , (2.9) later can be used to predict the other sets of data to test the dynamics of multiple gluon radiation predicted by this model of the QCD theory calculation. As noted in Ref. [46], F1 does not depend on the momentum fraction variables $1 or 2:2, while Fj/hl and Fig/M in general depend on those kinematic variables.3 The ln(Q2/Qg) dependence, associated with the F1 function, was predicted by the renormalization group analysis described in Section 2.2.1. It is necessary to balance the ln(Q2) de- pendence of the Sudakov exponent. Furthermore, F1 was shown to be universal, and its leading behavior (~ b2) can be described by renormalon physics [71]. Various sets of fits to these non-perturbative functions can be found in Refs. [72] and [73]. 3Here, and throughout this work, the flavor dependence of the non-perturbative functions is ignored, as it is postulated in Ref. [46]. 43 In our numerical results we use the Ladinsky-Yuan parametrization of the non- perturbative function (cf. Ref. [73]): WJ€P(b1Q1Q01$11$2) 2 exp [91 b2 — 9252 111 (220 where 91: 0.11+88§ GeV2, 92 = 0.581%; GeV2, 93 = —1.5“:8;[ GeV'1 and Q0 2 ) —glg3bln(100x1x2) , (2.10) 1.6 GeV. (The value bmax = 0.5 GeV‘1 was used in determining the above g,’s and in our numerical results.) These values were fit for CTEQ2M PDF with the canonical choice of the renormalization constants, i.e. CI = 03 = 26-75 (7,3 is the Euler constant) and 02 = 1. In principle, for a calculation using a more update PDF, these non-perturbative parameters should be refit using a data set that should also include the recent high statistics Z0 data from the Tevatron. This is however beyond the scope of this work. In Eq. (2.4), W17, sums over the soft gluon contributions that grow as Q? x [1 or ln(Q2/Q§~)] to all orders in (13. Contributions less singular than those included in Wig should be calculated order-by-order in as and included in the Y term, introduced in Eq. (2.4). This would, in principle, extend the applicability of the CSS resummation formalism to all values of QT. However, as to be shown below, since the A, B, C, and Y functions are only calculated to some finite order in as, the CSS resummed formula as described above will cease to be adequate for describing data when the value of QT is in the vicinity of Q. Hence, in practice, one has to switch from the resummed prediction to the fixed order perturbative calculation as QT g Q. The Y term, which is defined as the difference between the fixed order perturbative contribution and those obtained by expanding the perturbative part of W]; to the same order, is given by Y(QT1Q $1,$2,9, ¢,C4)= [$151 dél 1 453:0: 2:1(QS(:4Q)) 1:2 62 "_1 44 331 1132 X fa/h1(€11 C4Q) R5115) (QT) Q, __1 —) 51 52 9, (1)) fb/h2(§2, C4Q): (2-11) where the functions R512) contain contributions which are less singular than Q522 x [1 or ln(Q2/Q%)] as QT ——> 0. Their explicit expressions and the choice of the scale 0.; are summarized in Section 2.2.5. Within the Collins—Soper-Sterman resummation formalism ijc(b) sums all the singular terms which grow as a’S‘Q;2lnm(Q2/Q§~) for all n and 0 S m 5 2n — 1 provided that all the A("), B(") and CW4) coefficients are included in the pertur- bative expansion of the A, B and C functions, respectively. This was illustrated in Eqs. (A.12) and (A.13) of Ref. [66]. In our numerical results we included A“), A”), B“), 3(2), 0(0) and C“), which means we resummed the following singular pieces [66]: do (L+ 1) 22 34 46 ——~ a +aL+aL+aL+... cm 0% {S S S S + 02; +agL2 +a§L4 + }, (2.12) where L denotes ln(Q2/Q%~) and the explicit coefficients multiplying the logs are suppressed. The lowest order singular terms that were not included are Q;2(L + 1)(a3§ + 0145L2 + ...) . Also, in the Y term we included R“) and [2(2) (cf. Eq.(2.11)), which are derived from the fixed order oz, and (if calculations [65, 66]. The results of the usual (9(05) calculation can be obtained by expanding the above CSS resummation formula to the as order, which includes both the singular piece and the Y term. Details are given in Sections 2.2.3 and 2.2.5, respectively. 2.2.3 0(as) Expansion In this section we expand the resummation formula, as given in Eq. (2.4), up to C(03), and calculate the QT singular piece as well as the integral of the 0(a3) corrections 45 from O to PT. These are the ingredients, together with the regular pieces to be given in Section 2.2.5, needed to construct our NLO calculation. First we calculate the singular part at the C(05). By definition, this consist of terms which are at least as singular as Q? x [1 or 1n(Q2/Q§~)]. We use the perturbative expansion of the A, B and C functions in the strong coupling constant as as: Maw-0,01) = §(“‘9(’-)) Arm), 11:1 7? B(GS(/1),Cucz) = Eff”) B""(CI,C2), (2.13) n=l 00 n n C Cja(z’b1#101102) = 2(9—M) C§0)(Z,b,/,l,-CTI). 11:0 71' 2 The explicit expressions of the Am), B(") and CW coefficients are given in Section 2.2.4. After integrating over the lepton variables and the angle between 5 and QT, and dropping the regular (Y) piece in Eq. (2.4), we obtain . d0 00 11111 —— = -— 1 2 _ 2 QT—w dQ2dde% s (Q MV) [meg C2 2 02 2 . _ _ x BM: (“31’ 37622:) fF/hz (1% in?!) + J H k} +0(QT1)1 where we have substituted the resonance behavior by a fixed mass for simplicity, and [0 d1] 77 J00?) €_S('7/QT,Q,C1,C.2) defined 00 as4 [45] 47roz2 ... 00 = W, for V = ’7 , 2 00 = La—ZlV-kl" forV=Wi 2 J 1 a 4For our numerical calculation (within the ResBos Monte Carlo package), we have consistently used the on-shell scheme for all the electroweak parameters in the improved Born level formula for including large electroweak radiative corrections. In the V = Z0 case, they are the same as those used in studying the ZO-pole physics at LEP [75]. 46 W20: 00 = __TTZKI — 4]Qj[8€y)2 +1][‘/jk[2a fOI~ V 2 20. 12.9ch jk Here a is the fine structure constant, SW (cw) is the sine (cosine) of the weak mixing angle 0w, Qj is the electric charge of the incoming quark in the units of the charge of the positron (e.g. Qup = 2/3, Qdown = —-1/3, etc.), and ij is defined by Eq. (2.7). To evaluate the integral over 1) = bQT, we use the following property of the Bessel functions: dF07) jooodnnJoM) 17(0) = -/OoodnnJ1(n) 71-77, which holds for any function F (77) satisfying [7] J1(77) F (17)]3o = 0. Using the expansion of the Sudakov exponent 8(b, Q, Cl, Cg) = 5(1)(b, Q, C1, Cg) + 0(0429) with 2 1 02 2 c2 2 8(1)(b,Q,Cl,C2) : 9129.). [5,z1(1)(01)1n2 (Cg/Ob?) + B(1)(CI,CQ) 1n (TIE/Q19» , and the evolution equation of the parton distribution functions d fi/h($v#2) 08W) (1) 2 2 W Z 2,”. (Pj4—a ® fa/h) (mall )+ 0((15), we can calculate the derivatives of the Sudakov form factor and the parton distribu- tions with respect to 77: i -S(’l/QTaQ,Cl,C2)___2aS(Q2) (1) 03592772 (1) 2 dne _ n W A (Cmn 012% +B (01,02) +0615). 03433:) __ jas(Q2)(P(1) d d—Tlfj/hW n2 — n 271’ J+—a®fa/h)($’Q2)+0(a§)' Note that as itself is expanded as as(u2)_as(Q2) 03(6)?) 2 #2 3 2 T. -——2. _2a(——27r )1n(@)+00 2 %5(Q2 - M3)2WIQ% 013:? ) {[ fj/h,(rv1,Q2) (Pa—b <59 fb/hz) ($2,532) + (Ijje—a ® fa/hl) ($1, Q2) fE/h2 ($2) Q2)] 1 c2 2 . + [A( )(Cl)ln(—C'12_/2lggg) + B(I)(Cva‘2)] fj/h1 (1:1an) fF/h2($2aQ2) , _ 1 + g H k} + 0mg, E)’ (2.15) for arbitrary Cl and 02 constants. If Cl is not equal to Cgbo then, when QT is of the order of Q, the arbitrary log terms 1n(C12/(C§bg)) can potentially be larger than ln(Q2/Q§~). Therefore, to properly describe the QT distribution of the vector boson in the matching region, i.e. for QT ~ Q, Eq. (2.15) has to be used to define the asymptotic piece at 0(a5). This asymptotic piece is different from the singular contribution derived from a fixed order perturbative calculation at C(05) which is given by __6!0__ _ dozdydot QM 48 2 %6(Q2 _ A45)27TIQ% 03:? ) {[ fj/h1(-T1,Q2) (PE—b ® fb/hg) ($2,Q2) + (PJV—a ® fa/hl) ($1, Q2) fZ/hz (1:2, Q2)] +[Am1n (g; )+ B ] fj/h.(x1,o'~’) f;/,.,(x2,Q2) + j+—> E} T + 0mg, ), (2.16) E where A“) = CF and B“) = —3CF/2. Compared to the general results for A(1)(C1) and B(I)(C'1,C2), as listed in Section 2.2.4, the above results correspond to the special case of 01 = Cgbo. The choice of Cl 2 Cgbo = C3 2 b0 2 26"”? is usually referred to as the canonical choice. Throughout this work, we use the canonical choice in our numerical calculations. To derive the integral of the C(05) corrections over QT, we start again from the resummation formula [Eq. (2.4)] and the expansion of the A, B and C functions [Eq. (2.13)]. This time the evolution of parton distributions is expressed as mow?) = fj/h(x,o2)+ 13924. 2) meg), with f(1)($, #2) = 05(Q2)1n(__2) (Pu) ®f )(x Q2) J/h 2,” Q2 Jé—a a/h a a where summation over the partonic index a is implied. Substituting these expansions in the resummation formula Eq. (2.4) and integrating over both sides with respect to Q? We use the integral formula, valid for an arbitrary function F (b): iii/P72 dQT / d2b e'QT 5F(b) 31;]: db PTJ1(bPT)F(b), together with Eq. (2.14) to derive d0 00 PI; 2 2 2 [0 dQT W — §5(Q — Mv) 2 2 X {(1 _ 05(7er ) [2A(1)1n2(g—T)+B(1)ln(g%)]) fj/h‘($1’Q2) fF/h2($2’Q2) 49 2 2 _fll 1n (g) [(Pfl—a ® fa/hl) ($1) Q2) fE/hz (1‘2) Q2) 27r 13% —fj/h1 (1171, Q2) (PL—b ‘3 fb/hzz) (552’ 622)] 2 +9—S(—flQ—) [(Cfi) <8 fa/h1)($13Q2) fE/h2("72’Q2) +fj/h1(331a Q2) (Cg) ‘8 fb/h2) (x2, Q2)” _ 13% +j e k + [0 dQ’é Y(QT,Q,:r1,x2), (2.17) where 11:1 2 eyQ/x/S and 2:2 2 e‘yQ/x/S. Equations (2.16) and (2.17) (together with the regular pieces, discussed in Section 2.2.5) are used to program the 0(a3) results. 2.2.4 A, B and C functions For completeness, we give here the coefficients A, B and C utilized in our numerical calculations. The coefficients in the Sudakov exponent are [46, 72]. A“)(Cl) = CF, .. 67 7T2 5 b (2) : _ _ _ _ _ _ _0_ A (Cl) C}? [(36 12) NC 181V; 251111 (01)] , 3 Cb (1) ___ ___ 2 0 B (01,02) CF[ 2 2111 ( Cl ):l, 2 B(2l(Cl,Cg) = CF (CF ("I — % — 3<(3)) + N0 (117.2 — — + -<(3)) + NF 1 2 17 67 772 5 Cgbo 7%“ +12)—[(18— 6)NC‘9Nf]‘“(c. )+ 2 b0 2 3 2);,[111 (5)411 (02)—-2-1n(C2)]}. where N f is the number of light quark flavors (m, < Qv, e.g. N f = 5 for Wi or Z0 production), 0;: = tr(tata) is the second order Casimir of the quark repre- sentation (with ta being the SU(NC) generators in the fundamental representation), [31 = (11NC — 2Nf)/12 and ((22) = 2:172.” is the Riemann zeta function, and 50 ((3) z 1H202 FOI' QCD, NC = 3 and CF = 4/3. The (75-2) coefficients up to n = 1 are: 01 C;2)(Zabapa—CT2) : 6jk6(1_z)a quantg) : o, (1) Cl _ 1 1 lib (1) C], (z,b,n, —) _. (5,ka {5(1 — z) — E11163) Pfltz) Cl _ W2 23 51— —12(—— 3/4) __— +( 2)]: n [30026 +4 16 , (1) CI _ 1 lib <1) Cja(2,b,fl,C2) — 22(1 z) 1n(b0)P.-._G(z), where 1),-(Bah) are the leading order DGLAP splitting kernels [42] given in Section 1.2.3, and j and k represent quark or anti-quark flavors. The constants Cl, 02 and C3 E ,ub were introduced when solving the renormaliza- tion group equation for ij. Cl enters the lower limit fl = 01 / b in the integral of the Sudakov exponent [cf. Eq. (2.6)], and determines the onset of the non-perturbative physics. The renormalization constant Cg, in the upper limit fl = CgQ of the Su- dakov integral, specifies the scale of the hard scattering process. The scale it = Cg / b is the scale at which the C functions are evaluated. The canonical choice of these renormalization constants is CI = C3 = 26-73 E be and Oz 2 C4 = 1 [45]. We adopt these choices of the renormalization constants in the numerical results of this work, because they eliminate large constant factors within the A, B and C functions. After fixing the renormalization constants to the canonical values, we obtain much simpler expressions of A11), B11), A12) and Bl”. The first order coefficients in the Sudakov exponent become A“)(Cl) = CF, and 30>(01 = (20,02 = 1) = —3Cp/2. 51 The second order coefficients in the Sudakov exponent simplify to 67 7r2 5 2 (2) = Z ___ 2 :_3_ (2.212% E ) B (01 b0,02 1) C:(4 16 3<(3 )) +CpNC 3671 48 + 2((3) 1 1_7 The Wilson coefficients C]? for the parity-conserving part of the resummed re- sult are also greatly simplified under the canonical definition of the renormalization constants. Their explicit forms are b C 2 1 . (1) _0 _1 _ _ . _ _ _ _ 2 _- C'J-k (z ,=b,,u b’ C: —b()) — 6 {3(1 2:) + 36(1 z)(7r 8)} and b C 1 (1) _0 1 _ = _ _ CJG(z,b,p= 1),—02— b0) 2z(1 z). The same Wilson coefficient functions CJ-a also apply to the parity violating part which is multiplied by the angular function A3 = 2 cos 6. 2.2.5 Regular Contributions The Y piece in Eq. (2.4), which is the difference of the fixed order perturbative result and their singular part, is given by the expression ”QT,62.131.1'2,61,420,)z];611—.£1 1_d_’52_ i (0461162)) 2:2 62 ”:1 7r X fa/h1(€laC4Q) Rab)(QT1Qa 21, 22, 91¢) fb/h2(€2, C4Q), (218) where z,- = 113/5,- (1 = 1,2). The regular functions R(b) only contain contributions which are less singular than 627:2 x [1 or ln(Q2/Q%~)] as QT —> 0. Their explicit expressions for hlhg —> V(—> 61(72)X are given below. The scale for evaluating the regular pieces is C4Q. To minimize the contribution of large logarithmic terms from higher order corrections, we choose C4 = 1 when calculating the Y piece. 52 We define the qq’ V and the (31ng vertices, respectively, as 17.1910 — 75) + 93(1 + 75)] and i7.[f1.(1 — 15) + fn(1+ 75)]. For example, for V = W+,q = u, q’ :2 d, 81 2 V8, and (72 2 8+, the couplings 9% = f}, = GFMEV/fi and 9f? 2 f}z = 0, where CF is the Fermi constant. Table 2.1 shows all the couplings for the general case. In Eq. (2.18), 2 [(92 + 92>(f2 + f2)R‘.‘” + (92 — 92.102: — 1.21111] , where the coefficient functions R2” are given as follows5: .- .- T (1,21) Q T.(u,t) Q2 jk __ J]: + ~. __ 2 R, —r [10+ 8 6(s+t+u Q)[A0+A2+—QTT+(t,u)A1]—M%, 12%".c = rjEA3 + T+(t’ 11) 6(3 + t + u — Q2) Q2 Q 2622 T.(t.u) x {a (1'4? ‘ 11"“ ' QTMTT+(t.u)A4}’ 2 2 Gj£0 _ 3572?" T+(:'7 3) (S R?J'=r (s+t+u—Q2) x {W[A0+A2]+ Q (Q2 - u)2 + T_(u. t) T+(u) 8) A1}, ——'I-‘ T+(u18) R57]:_TGJA3_QI_ M6(s+t+u-Q2) . . x{g—; [.27 (#22222? 4) +1113 ..[22213H1114l with 5Note that in Ref. [67] there were typos in R]; and R5” . 53 2 2 _60—21) 1+z;2 —6(1~z2) 1+2:l ’ 1’22 + 1_Zl + 7.01 _ Q2 {__sz M _Q—‘éus where Ti(t,u) = (Q2 — t)2 :i: (Q2 — u)2. The Mandelstam variables 3,13, 11 and the and 6(s+t+u— Q2) — [zf + (1 —— z1)2] 6(1— 22)}, angular functions £0, A,- are defined in Sections 2.1.2 and 2.1.3. The ij coefficients are defined by Eq. (2.7). For a 23' and b = G: |V3¢|2 = 2 |ij|2 where j and k are light quark flavors with opposite weak isospin quantum nfimbers. Up to this order, there is no contribution from gluon-gluon initial state, i.e. R82; = 0. The remaining coefficient functions with all possible combinations of the quark and gluon indices (for example Rkj, Raj or R10, etc.) are obtained by the same crossing rules summarized in Section 2.2.3. Having both the singular and the regular pieces expanded up to 0(a3), we can construct the NLO Monte Carlo calculation by first including the contribution from Eq. (2.17), with PT = Q5916”, for QT < Q7533”. Second, for QT > Q53", we include the 0(a3) perturbative results, which is equal to the sum of the singular [Eq. (2.16)] and the regular [Eq. (2.18)] pieces up to C(05). (Needless to say that the relevant angular functions for using Eqs.(2.16) and (2.17) are £0 = 1 + cos26 and A3 = 2cos 0, cf. Eq. (2.3).) Hence, the NLO total rate is given by the sum of the contributions from both the QT < Q?” and the QT > 62;” regions. Chapter 3 Vector Boson Production and Decay in Hadron Collisions 3.1 Vector Boson Distributions At the Fermilab Tevatron, about ninety percent of the production cross section of the Wi, Z 0 bosons and virtual photons is in the small transverse momentum region, where QT S 20 GeV (hence 62%. << Q2). In this region the higher order perturbative corrections, dominated by soft' and collinear gluon radiation and of the form Q? $11 35;] "um ag lnm(Q§./Q2), are substantial because of the logarithmic enhancement [45]. ("12". are the coefficient functions for a given n and m.) As we discussed, these corrections are divergent in the QT —+ 0 limit at any fixed order of the perturbation theory. After applying the renormalization group analysis, these singular contribu- tions in the low QT region can be resummed to derive a finite prediction for the QT distribution to compare with experimental data. In this Chapter we discuss the phenomenology predicted by the resummation for- malism. To illustrate the effects of multiple soft gluon radiation, we also give results predicted by a next-to—leading order (NLO, C(05)) calculation. As expected, the resummed and the NLO predictions of observables that are directly related to the transverse momentum of the vector boson will exhibit large differences. These ob- 54 55 servables, are for example, the transverse momentum of the leptons from vector boson decay, the back-to—back correlations of the leptons from Z0 decay, etc. The observ- ables that are not directly related to the transverse momentum of the vector boson can also show noticeable differences between the resummed and the NLO calculations if the kinematic cuts applied to select the signal events are strongly correlated to the transverse momentum of the vector boson. Due to the increasing precision of the experimental data at hadron colliders, it is necessary to improve the theoretical prediction of the QCD theory by including the effects of the multiple soft gluon emission to all orders in as. To justify the importance of such an improved QCD calculation, we compare various distributions predicted by the resummed and the NLO calculations. For this purpose we categorize measurables into two groups. We call an observable to be directly sensitive to the soft gluon resummation effect if it is sensitive to the transverse momentum of the vector boson. The best example of such observable is the transverse momentum distribution of the vector boson (do/dQT). Likewise, the transverse momentum distribution of the decay lepton (do/dpéw) is also directly sensitive to resummation eflects. The other examples are the azimuthal angle correlation of the two decay leptons (Aci‘lZZ), the balance in the transverse momentum of the two decay leptons (p53 — 135?), or the correlation parameter z = #5751 - fif”/[max(pTel, 197572)]? These distributions typically show large differences between the NLO and the resummed calculations. The differences are the most dramatic near the boundary of the kinematic phase space, such as the QT distribution in the low QT region and the A45“? distribution near 7r. Another group of observables is formed by those which are indirectly sensitive to the resummation of the multiple soft gluon radiation. The predicted distributions for these observables are usually the same in either the resummed or the NLO calculations, provided that the QT is fully integrated out in both cases. Examples of indirectly sensitive quantities 56 are the total cross section a, the invariant mass Q, the rapidity y, and 33p (2 2q3/ x/S) of the vector bosonl, and the rapidity y‘ of the decay lepton. However, in practice, to extract signal events from the experimental data some kinematic cuts have to be imposed to suppress the background events. It is important to note that imposing the necessary kinematic cuts usually truncate the range of the QT integration, and causes different predictions from the resummed and the NLO calculations. We demonstrate such an effect in the distributions of the lepton charge asymmetry A(y€) predicted by the resummed and the NLO calculations. We show that they are the same as long as there are no kinematic cuts imposed, and different when some kinematic cuts are included. They differ the most in the large rapidity region which is near the boundary of the phase space. To systematically analyze the differences between the results of the N L0 and the resummed calculations we implemented the 0(ag) (LO), the C(05) (N L0), and the resummed calculations in a unified Monte Carlo package: ResBos (the acronym stands for Resummed Vector Boson production). The code calculates distributions for the hadronic production and decay of a vector bosons via h1h2 —> V(—> €1Z2)X, where h] is a proton and hg can be a proton, anti-proton, neutron, an arbitrary nucleus or a pion. Presently, V can be a virtual photon 7* (for Drell-Yan production), Wi or Z0. The effects of the initial state soft gluon radiation are included using the QCD soft gluon resummation formula, given in Eq. (2.4). This code also correctly takes into account the effects of the polarization and the decay width of the massive vector boson. It is important to distinguish ResBos from the parton shower Monte Carlo pro- grams like HERWIG [78], ISAJET [76], PYTHIA [77], etc., which use the backward radiation technique [79] to simulate the physics of the initial state soft gluon radia- 1Here (13 is the longitudinal-component of the vector boson momentum q" [cf. Eq. (2.1)]. f 1111.11 [Sulld [rum {rum thrm. lion his): tion Of fl; lParf. lf‘tfi? _ Wa; . [11121 57 0T0! (pb) 7‘ 30000 ' 20000 ’ \ 10000 - \ -10000 ~ ,’ -20000} / 0.1 0:5 1 5 1b. Q15” (GeV) Figure 3.1: Total W+ production cross section as a function of the parameter Q?” (solid curve). The long dashed curve is the part of the C(03) cross section integrated from Q?” to the kinematical boundary, and the short dashed curve is the integral from QT = 0 to Q?” at 0(ag). The total cross section is constant within 10‘5 ‘70 through more than two order of magnitude of Q1551” . tion. They are frequently shown to describe reasonably well the shape of the vector boson distribution. On the other hand, these codes do not have the full resumma- tion formula implemented and include only the leading logs and some of the sub—logs of the Sudakov factor. The finite part of the higher order virtual corrections which leads to the Wilson coefficient (C) functions is missing from these event generators. ResBos contains not only the physics from the multiple soft gluon emission, but also the higher order matrix elements for the production and the decay of the vector boson with large QT, so that it can correctly predict both the event rates and the distributions of the decay leptons. In a NLO Monte Carlo calculation, it is ambiguous to treat the singularity of the vector boson transverse momentum distribution near QT = 0. There are different ways to deal with this singularity. Usually one separates the singular region of the phase space from the rest (which is calculated numerically) and handles it analytically. 58 We choose to divide the QT phase space with a separation scale 62%”. We treat the QT singular parts of the real emission and the virtual correction diagrams analytically, and integrate the sum of their contributions up to Q31”. If QT < 62;” we assign a weight to the event based on the above integrated result and assign it to the QT = 0 bin. If QT > Q5931", the event weight is given by the usual NLO calculation. The above procedure not only ensures a stable numerical result but also agrees well with the logic of the resummation calculation. In Fig. 3.1 we demonstrate that the total . . . . s . . cross section, as expected, 1s 1ndependent of the separation scale QT” in a w1de range. As explained above, in the QT < C2318” region we approximate the QT of the vector boson to be zero. For this reason, we choose Q?” as small as possible. We use Q?" = 0.1 GeV in our numerical calculations, unless otherwise indicated. This division of the transverse momentum phase space gives us practically the same results as the invariant mass phase space slicing technique. This was precisely checked by the lepton charge asymmetry results predicted by DYRAD [80], and the NLO [up to 0(ag)] calculation within the ResBos Monte Carlo package. To facilitate our comparison, we calculate the NLO and the resummed distribu- tions using the same parton luminosities and parton distribution functions, EW and QCD parameters, and renormalization and factorization scales so that any difference found in the distributions is clearly due to the different QCD physics included in the theoretical calculations. (Recall that they are different models of calculations based upon the same QCD theory, and the resummed calculation contains the dynamics of the multiple soft gluon radiation.) This way we compare the resummed and the NLO results on completely equal footing. The parton distributions used in the different order calculations are listed in Table 3.1. In Table 3.1 and the rest of this work, we denote by resummed (2,1,2) the result of the resummed calculation with A and B calculated to 0% order, C to as, and R to (1% order, that is with AW”, 8“”, 59 Fixed order Resummed 0%) was) 0(aé) (1,0,0) (1,0,1) (2,1,2) PDF CTEQ4L CTEQ4M CTEQ4M CTEQ4L CTEQ4M CTEQ4M Table 3.1: List of PDF’s used at the different models of calculations. The values of the strong coupling constants used with the CTEQ4L and CTEQ4M PDF’s are ag1)(Mzo) = 0.132 and ag2l(MZo) = 0.116 respectively. 0““) and, 1302) included [cf. Section 2.2.4]. Similarly, resummed (1,0,1) includes A“), B“), CM, and R“), and resummed (1,0,0) includes A“), 3(1), 0(0) without the Y piece. Unless specified otherwise, hereafter we use A(1,2), 30,2), C(O’I) and, R052) in our resummed calculation. In the following, we discuss the relevant experimental observables predicted by these models of calculations using the ResBos code. Our numerical results are given for the Tevatron, a pp collider with \/§ 2 1.8 TeV, and CTEQ4 PDF’s unless specified otherwise. 3.1.1 Vector Boson Transverse Momentum Distribution According to the parton model the primordial transverse momenta of partons en- tering into the hard scattering are zero. This implies that a 7*, Wi or Z0 boson produced in the Born level process has no transverse momentum, so that the LO QT distribution is a Dirac-delta function peaking at QT = 0. In order to have a vector boson produced with a non-zero QT, an additional parton has to be emitted from the initial state partons. This happens in the QCD process. However the singularity at QT = 0 prevails up to any fixed order in as of the perturbation theory, and the transverse momentum distribution d0/dQ% is proportional to (2712 x [1 or ln(Q§~/Q2)] at small enough transverse momenta. The most important feature of the transverse momentum resummation formalism is to correct this unphysical behavior and render 60 da/dQ§~ finite at zero QT by exponentiating the QT singular logs. The CSS formalism itself is constructed to do even more than that. By including the regular Y contribution, it interpolates between the low and the high QT regions smoothly, provided that the A, B, C functions and the Y contribution are evaluated to all orders in 015.2 The Y piece is defined as the difference of the fixed order perturbative result and its QT singular (asymptotic) part which grows as Qfx [1 or ln(Q%/Q2)] when QT —> 0. In the QT << Q region, the ln(Q/QT) terms are large and the perturbative distribution is dominated by these singular logs, that is the perturbative and the asymptotic parts are about the same. Consequently, for low QT, the exponentiated asymptotic pieces, i.e. the CSS piece, dominates over the Y piece. In the QT ~ Q region the ln(QT/Q) terms are small, and the perturbative part is dominated by its other terms. The CSS and the asymptotic terms cancel each other leaving the perturbative piece to dominate the high QT region. The cancellation between the perturbative and asymptotic pieces is always exact (by definition) in the low QT region order by order in as, and the formalism is well defined for low QT, no matter in which order the A, B, C functions and Y are known. On the other hand, the cancellation between the CSS and the asymptotic pieces in the high QT region becomes better if the asymptotic piece is calculated in higher order in as. This is because the CSS piece contains logs in all order (cf. Eq.(2.12)) while the asymptotic part only up to a fixed order in as. The above will be clearly illustrated later in Fig. 3.2. Consequently, the CSS formalism must break down for QT ,2, Q, since A, B, C and Y are known only up to a finite order. Although the matching is built into the formalism, in practice it is still necessary 2Strictly speaking, this is only true when the energy of the collider is much larger than Q, because the resummed and the perturbative pieces are evaluated at different a: values. The former depends on 2:; and :32 defined in Eq. (2.4), however the latter depends on {1 and (2 (cf. Eq. (2.11)) in which the energy carried away by the emitted gluon is also included. 61 to specify a matching prescription which provides a smooth transition between the resummed and the fixed order perturbative results. In Fig. 3.2 we show the resummed (1,1,1) (resummation with A(1),B(1), 0““), and R“) included) and the fixed order 0(as) QT distributions for W+ and Z0 bosons. As shown, the resummed (1,1,1) and the fixed order curves are close to each other for Q/ 2 < QT < Q, and they cross around QT ~ Q/ 2. Based on this observation we adopt the following procedure for calculating the fully differential cross section da/szdQTdy. For QT values below the crossing points Q?“‘Ch(Q, y) of the resummed and the fixed order QT distributions, as the function of Q and y, we use the resummed cross section, and above it we use the fixed order perturbative cross section. The resulting do/szdQTdy distribution is continuous, although not differentiable with respect to QT right at the matching points Q?““”‘(Q,y). The differential cross section da/dQT, on the other hand, is completely smooth since it has no specific matching point. Most importantly, the above prescription does not alter either the resummed or the fixed order perturbative distributions in the kinematic regions where they are proven to be valid. To improve the theory prediction for the QT distribution, we also include the effect of some known higher order (at O(oz§)) corrections to the Sudakov factor [72], the Y piece [66], and the fixed order perturbative cross section [65]. As we described, the Y piece plays an essential role in the matching between the resummed and the fixed order QT distributions which are dominated by the Y piece when QT is in the matching region. To emphasize this in Fig. 3.3 we show the ratio of the Y piece to the total resummed (2,1,1) cross section. This ratio can be larger than one because the CSS piece, which is the difference between the total resummed cross section and the Y piece, can be negative for large enough QT(Z Q / 2). As indicated, the Y contribution is small for QT < 30 GeV. At QT = 30 GeV it only contributes by about 25% to da/dQT. The total contribution of the Y-term to f030 GeV dQT (do/dQT) is less than 62 -119 - 1400» do 7 dQT dQT 50 (pk/Ge V) ’20” ’ (ph/GeV) 1000 1 40 80!) . 3” 600 » 20 400 200 . m 0 - n ‘ ) ‘ . . n 30 40 50 ()0 70 80 Q7'(G¢'V) _ d9 ' ' V ' ' . do. ,, dQT 800 - ‘ ‘ dQT " ’ (pb/GeV) (Pb/69V) 30 . . our) » 25 . 20 _ 400 . ‘ )5 > 200 » ’” ‘ 5 . 0 a . _ ‘ L . ,1 0 . 0 5 ’0 ’5 2” 25 3” 30 40 50 60 70 80 QflGeV) Q1'(GeV) Figure 3.2: The low and intermediate QT regions of the W+ and Z 0 distributions at the Tevatron, calculated in fixed order 0(015) (dotted) and 0mg) (dash-dotted), and resummed (1,1,1) (dashed) and (2,1,2) (solid) [cf. Table III]. The cross-over occurs at 54 GeV for the C(05), and at 69 GeV for the 0mg) Wi distributions. The matching between the resummed and the fixed order disrtibutions becomes much smoother at 0mg) than at 0((13). The situation is very similar for the Z0 boson. a percent. Therefore, in the region of QT < 30 GeV, the CSS piece dominates. We can also define the K y-factor as the ratio of the Y pieces calculated at the 0(05 +03%) to that at the 0(as). The Ky-factor is plotted in Fig. 3.4 as the function of QT and y for W:t and Z0 bosons (for Q = My). As shown, when QT is between 30 to 80 GeV, the Ky factor is about 10% unless the rapidity of the W and Z bosons become large (i.e. [1;] > 2). Similarly, in Fig. 3.5, we show the K p factor, which is defined as the ratio of the fixed order differential cross sections calculated at the 0(a5 + 01%) to that at the 0((15), as a function of QT, and y for Q = My. As expected, when [y] is large, i.e. near the boundary of the available phase space, the K p factor can be large. 63 (dc/dQT)Res 0.5 0.25 0 10 50 3}) 4‘0 50 Q7'(GeV) (do/dQ ) ’4‘ ‘ #__7;Y- 20 I (dc/dQT)Res 12) Ill / / 1 - l / 0.8 » ,’ / 0.6 » ,’ / / 0.4 > / J / 0.2 . / / o 10 2}) 3b Jo 50 QrfGeV) Figure 3.3: The ratio of the 0(as + 01%) Y piece (solid curve, Rm and 3(2) included), and the 0(a3) Y piece (dashed curve, B“) included) to the resummed (2,1,1) distri- bution for W+ and Z0 bosons. The variation as a function of Q; for QT > 50 GeV is small, of the order of 10%. In Fig. 3.2 we also show the resummed (2,1,2) (with A(1’2),B(1’2), C(O’I), and 302) included) and the fixed order O(a§) QT distributions. Joining these distributions at the triple differential cross section level defines the resummed 0(ag) distribution in the whole QT region. As shown, while in 0(as) the matching takes place at lower QT values leaving a noticeable kink in the cross section, in 0mg) the matching occurs at larger QT values and is much smoother. This happens because, as discussed above, the cancellation between the CSS and the asymptotic pieces becomes better. In summary, the CSS resummation formalism is constructed in such a manner 64 30 40 50 60 70 80 KYfQT) 1.14» 1.12 1.1 - 1.011 » 1.06 > 30 30 5}) so 711 80 QT ( Ge V) Figure 3.4: The Ky-factor: ratio of the (9(a3 + 0%) Y piece (Rm and 12(2) included) to the 0(013) Y piece (Rm included). The curves are plotted for Q = My and y = —2.0 (solid), —1.0 (long dash), 0.0 (short dash), 1.0 (dash-dot) and y 22.0 (dash-double—dot). that if the A, B, C functions and the R coefficients were calculated to all order then the matching would be completely natural in the sense that the resummed cross section would blend into the fixed order one smoothly and no additional matching prescription would be necessary. However, in a practical calculation, because A, B, C and Y are only known to some finite order in as, the matching prescription described above is necessary. Using this procedure, we discuss below a few other interesting results calculated from the resummation formalism. We find that in the resummed calculation, after taking out the resonance weight- “‘Ff 65 1.6 KMQT) 1.55 1 ~ 1.5 » 1.45 40 50 60 70 80 90 I 00 QflGeV} KMQT) [.3 > 40 50 60 70 80 90 l 00 QT(GeV) Figure 3.5: The fixed order perturbative wag) to 0(03) K -factor as the function of QT. The curves are plotted for Q = My and y = -2.0 (solid), —1.0 (long dash), 0.0 (short dash), 1.0 (dash-dot) and y =2.0 (dash-double-dot). ing factor Q2/((Q2 — Mfi)2 + Q‘Tfi/Mfi) in Eq. (2.4), the shape of the transverse momentum distribution of the vector boson V for various Q values in the vicinity of My is remarkably constant for QT between 0 and 20 GeV. Fixing the rapidity y of the vector boson V at some value yo and taking the ratio do szdedy d0 dQQdedy QzQOsyzyO 3 R(QT, Q0) 2 Q=Mv1y=yo we obtain almost constant curves (within 3 percent) for Q = My :t 10 GeV (cf. RerQo) VII-\‘x- . I B a 1.41 ............................. “(+- 1.3» 1.2» 1.1L ____________________ 1+ 0.9» ________________________ 0.81 0.7 . """""" - ----------- . ------------ . .......... . 0 5 10 15 20 QT(GeV) R(Q7\Q0) ’4 ‘ Y ~- ' 1.... -------------------- 2?-” 1.2» 1.1» —————————————————————— 1. 0.9» _________________________ 0.11» ______________________ 0 ------ 3 10 13 2‘0 QT(G€V) Figure 3.6: The ratio R(QT, Q0), with yo = 0, for W+ and Z0 bosons as a function of QT. For W+, solid lines are: Q0 2 78 GeV (upper) and 82 GeV (lower), dashed: Q0 = 76 GeV (upper) and 84 GeV (lower), dotted: Q0 2 70 GeV (upper) and 90 GeV (lower). For Z0 bosons, solid lines: Q0 = 88 GeV (upper) and 92 GeV (lower), dashed: Q0 2 86 GeV (upper) and 94 GeV (lower), dotted: Q0 = 80 GeV (upper) and 100 GeV (lower). Fig. 3.6) for V = W+ and 20. The fact that the shape of the transverse momentum distribution shows such a weak dependence on the invariant mass Q in the vicinity of the vector boson mass can be used to make the Monte Carlo implementation of the resummation calculation faster. This weak dependence was also used in the DG W mass analysis when assuming that the mass dependence of the fully differential W boson production cross section factorizes as a multiplicative term [81]. Similarly, we 67 define the ratio d0 szdedy do dQQdedy Q=Mv.y=yo R(QT, yo) = ’ QZAJV 1y=0 to study the QT shape variation as a function of the vector boson rapidity. Our results are shown in Fig. 3.7. Unlike the ratio R(QT, Q0) shown in Fig. 3.6, the distributions of R(QT,y0) for the W35 and Z0 bosons are clearly different for any value of the rapidity yo. To utilize the information on the transverse momentum of the W+ boson in Monte Carlo simulations to reconstruct the mass of the W+ boson, it was suggested in Ref. [82] to predict QT(W+) distribution from the measured QT(Z°) distribution and the calculated ratio of QT(W+) and QT(Z°) predicted by the resummation calcula- tions [45, 66], in which the vector boson is assumed to be on its mass-shell. Unfor- tunately, this idea will not work with a good precision because, as clearly shown in Fig. 3.7, the ratio of the W+ and Z0 transverse momentum distributions depends on the rapidities of the vector bosons. Since the rapidity of the W+ boson cannot be accurately reconstructed without knowing the longitudinal momentum (along the beam pipe direction) of the neutrino, which is in the form of missing energy carried away by the neutrino, this dependence cannot be incorporated in data analysis and the above ansatz cannot be realized in practice for a precision measurement of MW.3 Only the Monte Carlo implementation of the exact matrix element calculation (in- cluded in ResBos) can correctly predict the distributions of the decay leptons, such as the transverse mass of the Wi boson, and the transverse momentum of the charged 3If a high precision measurement were not required, then one could choose from the two—fold solutions for the neutrino longitudinal momentum to calculate the longitudinal momentum of the W3: boson. R(QT.)‘0) .............................. I» \0 = +1.0 . ‘ \ w+ 0.8 ’_ - ‘ ~ ‘ ‘\-- — : '_' I '__- _., _ _ yo = —1.0 l 0.6 ’ yo = +2.0 ‘ 0.4» l 0.2 » ‘ 0 5 f0 [5 2‘0 23 30 R(Q7LV0) v 20 I. 0.8- y0=-l.0,+l.0 ‘ 0.6 » l 0.4 .v y = —2.0.+2.0 0.2 » '0 ‘ 0 3 10 15 2‘0 2‘5 30 Figure 3.7: The ratio R(QT, yo), with Q0 = My, for W+ and Z0 bosons as a function 0f QT- lepton, so that they can be directly compared with experimental data to extract the value of MW. We comment on these results later in this section. Another way to compare the results of the resummed and the NLO calculations is given by the distributions of U(QT > Q$i“)/0Toml, as shown in Fig. 3.8. We defined the ratio as min Qmu U(QT > QT ) 1 f T dQT d0(h1h2 —> V), Ross E om dQT 0’Total 0'Total where Q?” is the largest QT allowed by the phase space. In the NLO calculation, U(QT > Q33”) grows without bound near Q32lin = 0, as the result of the singular behavior 1 / Q? in the matrix element. The NLO curve runs well under the resummed of l 211‘ ‘1 11.1 Ill f‘i den 69 one in the 2 GeV < Q?Jin < 30 GeV region, and the QT distributions from the NLO and the resummed calculations have different shapes even in the region where QT is of the order 15 GeV. With large number of fully reconstructed Z 0 events at the Tevatron, one should be able to use data to discriminate these two theory calculations. In view of this result it is not surprising that the DO analysis of the as measurement [83] based on the measurement of U(W + 1 jet) /0(W + 0 jet) does not support the NLO calculation in which the effects of the multiple gluon radiation are not included. We expect that if this measurement was performed by demanding the transverse momentum of the jet to be larger than about 50 GeV, at which scale the resummed and the NLO distributions in Fig. 3.2 cross, the NLO calculation would adequately describe the data. To show that for QT below 30 GeV, the QCD multiple soft gluon radiation is important to explain the DO data [83], we also include in Fig. 3.8 the prediction for the QT distribution at the order of 01%. As shown in the figure, the 0% curve is closer to the resummed curve which proves that for this range of QT the soft gluon effect included in the erg calculation is important for predicting the vector boson QT distribution. In other words, in this range of QT, it is more likely that soft gluons accompany the W:t boson than just a single hard jet associated with the vector boson production. For large QT, it becomes more likely to have hard jet(s) produced with the vector boson. Measuring Ross in the low QT region (for QT S Q / 2) provides a stringent test of the dynamics of the multiple soft gluon radiation predicted by the QCD theory. The same measurement of R035 can also provide information about some part of the non- perturbative physics associated with the initial state hadrons. As shown in Fig. 3.9 70 l v . - v r RCSS l 0.8 r 0.6 1 ‘X \ 4 0.41 \ \ 0.2' “s‘ \ ~ _____ 5 10 l 5 20 25 30 ‘ v v fl I Rcss \ 0.8 ' 0.6 ’ “ \ 04 * \\\\ \ \ \ . . \ 0.21 x . - ---- ---‘ 5 10 15 20 2:5 30 Figure 3.8: The ratio R035 as a function of Q?" for W+ and Z0 bosons. The fixed order (0(a3) short dashed, 0mg) dashed) curves are ill-defined in the low QT region. The resummed (solid) curves are calculated for 92 = 0.38 (low), 0.58 (middle) and 0.68 (high) GeV2 values. and in Ref. [73], the effect of the non-perturbative physics on the QT distributions of the W3c and Z0 bosons produced at the Tevatron is important for QT less than about 10 GeV. This is evident by observing that different parametrizations of the non-perturbative functions do not change the QT distribution for QT > 10 GeV, although they do dramatically change the shape of QT for QT < 10 GeV. Since for Wi and Z0 production, the ln(Q2/Qg) term is large, the non-perturbative function, as defined in Eq. (2.9), is dominated by the F1(b) term (or the 92 parameter) which is supposed to be universal for all Drell-Yan type processes and related to the physics of 71 _ (19.7 1200 dQT (pb/GeV) 1000 * 800 ’ 600 400 > 200 5 10 [5 2‘0 55 30 Q'rfGeV) d0 ,. ' ' . ' * dQT 700 ’ I“ b/GeV ’ 1’ (p ) 600 [ 1 .' \ 1 \ ‘ . 1 ' \ ‘. 500 f l \ \\ 1 ‘ ‘1 \ s 400» 1 \\ ' \ 1 \ 300 r 200 1 [00 0 A A A A A 5 l0 [5 20 25 30 QT (GeV) Figure 3.9: Transverse momentum distributions of W+ and Z0 bosons calculated with low (long dash, g2 = 0.38 GeV2), nominal (solid, 92 = 0.58 GeV?) and high (short dash, 92 = 0.68 GeVQ) 92 non-perturbative parameter values. The low and high excursions in 92 are the present one standard deviations from the nominal value in the Ladinsky-Yuan parametrization. the renormalon [71]. Hence, the measurement of R053 cannot only be used to test the dynamics of the QCD multiple soft gluon radiation, in the 10 GeV < QT < 40 GeV region, but may also be used to probe this part of non-perturbative physics for QT < 10 GeV. It is therefore important to measure R055 at the Tevatron. With a large sample of Z 0 data at Run 2, it is possible to determine the dominant non-perturbative function which can then be used to calculate the Wi boson QT distribution to improve the accuracy of the MW and the charged lepton rapidity asymmetry measurements. The transverse momentum distribution of W i bosons within a modified CSS 72 formalism was also calculated by in Ref. [84]. The conclusions and results, when they overlap, agree with ours. 3.1.2 Vector Boson Longitudinal Distributions The resummation of the logs involving the transverse momentum of the vector boson does not directly affect the shape of the longitudinal distributions of the vector bosons. A good example of this is the distribution of the longitudinal momentum of the Z0 boson which can be measured at the Tevatron with high precision, and can be used to extract information on the parton distributions. It is customary to plot the rescaled quantity 22p = 2q3/\/—S_, where q3 = sinh(y)(/Q2 + Q§~ is the longitudinal momentum of the Z0 boson measured in the laboratory frame. In Fig. 3.10, we plot the distributions predicted in the resummed and the NLO calculations. As shown, their total event rates are different in the presence of kinematic cuts. (Although they are the same if no kinematic cuts imposed.) This conclusion is similar to that of the y‘ distributions, as discussed in Sections 3.1.3 and 3.2.1. Without any kinematic cuts, the vector boson rapidity distributions are also the same in the resummed and the NLO calculations. This is so because when calculating the 3; distribution the transverse momentum QT is integrated out so that the integral has the same value in the NLO and the resummed calculations. On the other hand, experimental cuts on the final state leptons restrict the phase Space, so the difference between the NLO and the resummed QT distributions affects the vector boson rapidity distributions. This shape difference is very small at the vector boson level, as shown in Fig. 3.11. 73 21}; 400 - (pb) 300 - 200- 100 1 0‘75 -05 025 A 0 0.15 0.1 5‘ 0.15 xF Figure 3.10: Longitudinal 17p distributions of Z0 bosons produced at the Tevatron. The NLO (dashed) curves overestimate the rate compared to the resummed (solid) ones, because kinematic cuts enhance the low QT region where the NLO and re- summed distributions are qualitatively different. Without cuts, the NLO and the resummed mp distributions are the same. 3.1.3 The Total Cross Section Before we compare the distributions of the decay leptons, we examine the question whether or not the QT resummation formalism changes the prediction for the total cross section. In Ref. [85] it was shown that in the AEGM formalism, which differs from the CSS formalism, the C(05) total cross section is obtained after integrating their resummation formula over the whole range of the phase space. In the CSS formalism, without including the C and Y functions, the fully inte- grated resummed result recovers the 005%) cross section, provided that QT is inte- grated from zero to Q. This can be easily verified by expanding the resummation formula up to (9(015), dropping the Cm and the Y pieces (which are of order C(05)), and integrating over the lepton variables. It yields P72 2 d0 _ 00 2 [2 40 - (Pb) 1 30 ~ 20' 10- -3 -2 A .1 0‘ 1 A 2 3 y Figure 3.11: Rapidity distributions (resummed: solid, NLO: dashed) of Z0 bosons produced at the Tevatron with the kinematic cuts given in the text. x1<1~ee11w212—34181110311 X 1371110131an ) fig/1120321622) 2 1321) In (4%) [(Pj... e 1.11..) (:11, Q2) 1.1.0102) + fj/h1($1,Q2) (PB-b ® fb/hz) ($21Q2)] +j H E} a (3'1) where PT is the upper limit of the QT integral and we fixed the mass of the vector boson for simplicity. To derive the above result we have used the canonical set of the C,- (2' = 1, 2, 3) coefficients (cf. Section 2.2.4). When the upper limit PT is taken to be Q, all the logs in the above equation vanish and Eq. (3.1) reproduces the Born level (0(ag)) cross section. Similar conclusion holds for higher order terms from the expansion of the resummed piece when C and Y are not included. This is evident because the singular pieces from the expansion are given by do 1 °° 2" 1 The integral of these singular terms will be proportional to ln(Q2 / P721) raised to some 75 power. Again, for PT 2 Q all the logs vanish and the tree level result is obtained. Including 0(1) and the Y contribution changes the above conclusion and leads to a different total cross section, because C“) contains the hard part virtual corrections and Y contains the hard gluon radiation. Combining the resummed (1,1,1) and the fixed order C(05) distributions, the above described matching prescription provides us with an C(05) resummed total cross section with an error of (9(af), as shown in Ref. [66]. In practice this translates into less than a percent deviation between the resummed C(as) total cross section and the inclusive NLO (0(as)) calculation. This can be understood from the earlier discussion that if the matching were done at QT equal to Q, then the total cross section calculated from the CSS resummation formalism should be the same as that predicted by the NLO calculation, provided that C“) and Y“) are included. However, this matching prescription would not result in a smooth curve for the QT distribution at QT = Q. The matching procedure described above causes a small (about a percent) difference between the (9(015) resummed and the NLO total cross sections. This difference indicates the typical size of the higher order corrections not included in the NLO total cross section calculation. The total cross section predicted from the various theory calculations are listed in Table 3.2. As shown in Table 3.2, as far as the total rate is concerned, there is hardly any observable difference between the predicted results from the resummed (2,1,1) matched to the fixed order perturbative C(05) and the resummed (2,1,2) matched to the fixed order perturbative 0mg), although the latter gives a smoother QT curve, as shown in Fig. 3.2. Kinematic cuts affect the total cross section in a subtle manner. It is obvious from our matching prescription that the resummed 0(ag) and the fixed order 0(015) curves in Fig. 3.2 will never cross in the high QT region. On the other hand, the 76 Em Fixed Order Resummed Experiment (TeV) 0101’.) was) (1,1,1) (2,1,1) (2,1,2) (Ref. 1861) 63 C(03) EB C(05) G3 001%) W+ 1.8 8.81 11.1 11.3 11.3 11.4 11.5 :t 0.7 W+ 2.0 9.71 12.5 12.6 12.6 12.7 Z0 1.8 5.23 6.69 6.79 6.79 6.82 6.86 :l: 0.36 Z0 2.0 6.11 7.47 7.52 7.52 7.57 Table 3.2: Total cross sections of pp ——> (W+or Z°)X at the present and upgraded Tevatron, calculated in different prescriptions, in units of nb. The finite order total cross section results are based on the calculations in Ref. [85]. The (Mag) results were obtained from Ref. [65]. The “EB” signs refer to the matching prescription discussed in the text. resummed (9(0129) total cross section is about the same as the fixed order 0(a3) cross section when integrating QT from 0 to Q. These two facts imply that when kinematic cuts are made on the QT distribution with QT < Q, we will obtain a higher total cross section in the fixed order 0(013) than in the resummed C(afi) calculation. In this Chapter we follow the CDF cuts (for the W+ boson mass analysis) and demand QT < 30 GeV [87]. Consequently, in many of our figures, to be shown below, the fixed order 0(a3) curves give about 3% higher total cross section than the resummed ones. 3.2 Lepton Distributions 3.2.1 Lepton Charge Asymmetry The CDF lepton charge asymmetry measurement [88] played a crucial role in con- straining the slope of the u/d ratio in recent parton distribution functions. It was shown that one of the largest theoretical uncertainty in the Wi mass measurement 77 comes from the parton distributions [89], and the lepton charge asymmetry was shown to be correlated with the transverse mass distribution [90]. Among others, the lepton charge asymmetry is studied to decrease the errors in the measurement of MW com- ing from the parton distributions. Here we investigate the effect of the resummation on the lepton rapidity distribution, although it is not one of those observables which are most sensitive to the QT resummation, i.e. to the effect of multiple soft gluon radiation. The definition of the charge asymmetry is d_0 _ _d: d d _ Ao) = ——j; if, . _ + _— dy+ dy_ where 3).. (31-) is the rapidity of the positively (negatively) charged particle (either vector boson or decay lepton). Assuming CP invariance,4 the following relation holds: do do W 1: Key). Hence, the charge asymmetry is frequently written as do do 211—(y Ol-d—(9<0) A1): y y y 1% >0)+513( <0) dyy dyy For the charge asymmetry of the vector boson (wt) or the charged decay lepton (8*), the fixed order and the resummed 0(a3) [or C(01%)] results are the same, provided that there are no kinematic cuts imposed. This is because the shape difference in the vector boson transverse momentum has been integrated out and the total cross sections are the same up to higher order corrections in as. In Fig. 3.12(a) we show 4Here we ignore the small CP violating effect due to the CKM matrix elements in the SM. 78 A!” . (5 )0] -0.1 41.2 1 -0.3 1 -0.4 1 My”) 0.11 -0.l 1 -0.2 1 -0.3 1 -0.4 - Figure 3.12: Lepton charge asymmetry distributions. (a) Without any kinematic cuts, the NLO (long dashed) and the resummed 0(a3) (solid) curves overlap and the LO (short dashed) curve differs somewhat from them. (b) With cuts (QT < 30 GeV, p?” > 25 GeV), the effect of the different QT distributions renders the lepton rapidity asymmetry distributions different. The two resummed curves calculated with 92 = 0.58 and 0.78 GeV2 cannot be distinguished on this plot. the lepton charge asymmetry without cuts for CTEQ4M PDF. The NLO and the resummed curves overlap, although they differ from the 0(a%) prediction. On the other hand, when kinematic cuts are applied to the decay leptons, the rapidity distributions of the vector bosons or the leptons in the fixed order and the resummed calculations are different. Restriction of the phase space implies that only part of the vector boson transverse momentum distribution is sampled. The differ- ence in the resummed and the fixed order QT distributions will prevail as a difference 79 in the rapidity distributions of the charged lepton. We can view this phenomenon in a different (a Monte Carlo) way. In the rest frame of the W*, the decay kinematics is the same, whether it is calculated up to (9(a5) or within the resummation formalism. On the other hand, the Wi rest frame is different for each individual Monte Carlo event depending on the order of the calculation. This difference is caused by the fact that the QT distribution of the Wi is different in the 0(a3) and the resummed calculations, and the kinematic cuts select different events in these two cases. Hence, even though the QT distribution of the W:t is integrated out, when calculating the lepton rapidity distribution, we obtain slightly different predictions in the two calcu- lations. The difference is larger for larger lye], being closer to the edge of the phase space, because the soft gluon radiation gives high corrections there and this effect up to all order in as is contained in the resummed but only up to order of as in the NLO calculation. Because the rapidity of the lepton and that of the vector boson are highly correlated, large rapidity leptons mostly come from large rapidity vector bosons. Also, a vector boson with large rapidity tends to have low transverse mo- mentum, because the available phase space is limited to low QT for a Wi boson with large |y|. Hence, the difference in the low QT distributions of the NLO and the re- summed calculations yields the difference in the y‘ distribution for leptons with high rapidities. Asymmetry distributions of the charged lepton with cuts using the CTEQ4M PDF are shown in Fig. 3.12(b). The applied kinematic cuts are: QT < 30 GeV, p?” > 25. These are the cuts that CDF used when extracted the lepton rapidity distribution from their data [88]. We have checked that the ResBos fixed order 0(as) curve agrees well with the DYRAD [80] result. As anticipated, the O(a%), C(as) and resummed results deviate at higher rapidities (lyel > 1.5).5 The deviation between the N LO and 5As indicated before, here and henceforth, unless specified otherwise, by a resummed calculation 80 the resummed curves indicates that to extract information on the PDF in the large rapidity region, the resummed calculation, in principle, has to be used if the precision of the data is high enough to distinguish these predictions. Fig. 3.12(b) also shows the negligible dependence of the resummed curves on the non-perturbative parameter 92. We plot the result of the resummed calculations with the nominal g2 = 0.58 GeV2, and with 92 = 0.78 GeV2 which is two standard deviations higher. The deviation between these two curves (which is hardly observable on the figure) is much smaller than the deviation between the resummed and the NLO ones. There is yet another reason why the lepton charge asymmetry can be reliably pre- dicted only by the resummed calculation. When calculating the lepton distributions in a numerical 0(a3) code, one has to artificially divide the vector boson phase space into hard and soft regions, depending on — for example — the energy or the QT of the emitted gluon (e.g. qfi —-) W+ hard or soft gluon). The observables calculated with this phase space slicing technique acquire a dependence on the scale which separates the hard from the soft regions. For example, when the phase space is divided by the QT separation, the dependence of the asymmetry on the scale Q?” can be comparable to the difference in the 0(a5) and the resummed results. This means that there is no definite prediction from the NLO calculation for the lepton rapidity distribution. Only the resummed calculation can give an unambiguous prediction for the lepton charge asymmetry. Before closing this section, we also note that although in the lepton asymmetry distribution the NLO and resummed results are about the same for [316+] < 1, it does not imply that the rapidity distributions of the leptons predicted by those two theory models are the same. As shown in Fig. 3.13, this difference can in principle be observable with a large statistics data sample and a good knowledge of the luminosity we mean our resummed 0(a%) result. 81 150 1 1001 501 .1. ye Figure 3.13: Distributions of positron rapidities from the decays of W+’s produced at the Tevatron, predicted by the resummed (solid) and the NLO (dashed) calculations with the same kinematic cuts as for the asymmetry plot. of the colliding beams. 3.2.2 Transverse Mass Distribution Since the invariant mass of the W2t boson cannot be reconstructed without knowing the longitudinal momentum of the neutrino, one has to find a quantity that allows an indirect determination of the mass of the Wi boson. In the discovery stage of the W35 bosons at the CERN Sp'pS collider, the mass and width were measured using the transverse mass distribution of the charged lepton-neutrino pair from the W1 boson decay. Ever since the early eighties, the transverse mass distribution, mT = fipfipfil — cos Ade”), has been known as the best measurable for the extraction of both MW and fly, for it is insensitive to the transverse momentum of the Wi boson. The effect of the non—vanishing vector boson transverse momentum on the my distribution was analyzed [91, 92] well before the QT distribution of the W:t boson was correctly calculated by taking into account the multiple soft gluon radiation. 82 Giving an average transverse boost to the vector boson, the authors of Ref. [91] concluded that for the fictive case of FW 2 0, the end points of the transverse mass distribution are fixed at zero: do/dm?r(m% = 0) 2 do/dm%(m% = MEV) = 0. The sensitivity of the mT shape to a non-zero QT is in the order of ((QT/Mw)2> 21% without affecting the end points of the my distribution. Including the effect of the finite width of the W:t boson, the authors in Ref. [92] showed that the shape and the location of the Jacobian peak are not sensitive to the QT of the Wi boson either. The non-vanishing transverse momentum of the Wi boson only significantly modifies the mT distribution around mT = 0. Our results confirm that the shape of the Jacobian peak is quite insensitive to the order of the calculation. We show the NLO and the resummed transverse mass distributions in Fig. 3.14 for Wi bosons produced at the Tevatron with the kinematic cuts: QT < 30 GeV, p?” > 25 GeV and |ye+l < 3.0. Fig. 3.14(a) covers the full (experimentally interesting) mrp range while Fig. 3.14(b) focuses on the mT range which contains most of the information about the Wi mass. There is little visible difference between the shapes of the N LO and the resummed mT distributions. On the other hand, the right shoulder of the curve appears to be “shifted” by about 50 MeV, because, as noted in Section 3.1.3, the total cross sections are different after the above cuts imposed in the NLO and the resummed calculations. At Run 2 of the Tevatron, with large integrated luminosity (~ 2 fb'l), the goal is to extract the Wi boson mass with a precision of 30-50 MeV from the mT distribution [89]. Since MW is sensitive to the position of the J acobian peak [92], the high precision measurement of the Wi mass has to rely on the resummed calculations. The extraction of MW from the transverse mass distribution has some drawbacks. The reconstruction of the transverse momentum p51 of the neutrino involves the mea- surement of the underlying event transverse momentum: p’T" = _p'T‘ ._ 1517;131:011 _ 83 (IO me 50 , (a) (pb/CeV) 40 » 30 1 / 20 > z 10» , do dm T (pb/Ge V) 50 1 40- 30 1 201 75.5 110 110.5 111 111.5 Figure 3.14: Transverse mass distribution for W+ production and decay at the 1.8 TeV Tevatron. "underlymg event. This resolution degrades by the number of interactions per crossing (N 1c) [89]. With a high luminosity (~ 100 fb‘l) at the 2 TeV Tevatron (TEV33) [93], N [c can be as large as 10, so that the Jacobian peak is badly smeared. This will lead to a large uncertainty in the measurement of MW. For this reason the systematic pre- cision of the mT reconstruction will be less at the high luminosity Tevatron, and an MW measurement that relies on the lepton transverse momentum distribution alone could be more promising. We discuss this further in the next section. The theoretical limitation on the MW measurement using the mgr distribution comes from the dependence on the non-perturbative sector, i.e. from the PDF’s and 84 the non-perturbative parameters in the resummed formalism. Assuming the PDF’s and these non-perturbative parameters to be independent variables, the uncertainties introduced are estimated to be less than 50 MeV and 10 MeV, respectively, at the TEV33 [81, 94]. It is clear that the main theoretical uncertainty comes from the PDF’s. As to the uncertainty due to the non-perturbative parameters (e.g. 92) in the CSS resummation formalism, it can be greatly reduced by carefully study the QT distribution of the Z0 boson which is expected to be copiously produced at Run 2 and beyond. The MW measurement at the LHC may also be promising. Both ATLAS and CMS detectors are well optimized for measuring the leptons and the missing ET [94]. The cross section of the W+ boson production is about four times larger than that at the Tevatron, and in one year of running with 20 fb"1 luminosity yields a few times 107 W ——> 211 events after imposing similar cuts to those made at the Tevatron. Since the number of interactions per crossing may be significantly lower (in average NIC = 2) at the same or higher luminosity than that at the TEV33 [94], the Jacobian peak in the mT distribution will be less smeared at the LHC than at the TEV33. Furthermore, the non-perturbative effects are relatively smaller at the LHC because the perturbative Sudakov factor dominates. On the other hand, the probed region of the PDF’s at the LHC has a lower value of the average :1: (~ 10‘3) than that at the Tevatron (~ 10’2), hence the uncertainty from the PDF’s might be somewhat larger. A more detailed study of this subject is desirable. 3.2.3 Lepton Transverse Momentum Due to the limitations mentioned above, the transverse mass method may not be the only and the most promising way for the precision measurement of MW at some future hadron colliders. As discussed above, the observable mT was used because of 85 its insensitivity to the high order QCD corrections. In contrast, the lepton transverse momentum (pg) distribution receives a large, 0((QT/MW)) ~ 10%, perturbative QCD correction at the order as, as compared to the Born process. With the re- summed results in hand it becomes possible to calculate the p? distribution precisely within the perturbative framework, and to extract the Wi mass straightly from the transverse momentum distributions of the decay leptons. Just like in the 111T distribution, the mass of the Wi boson is mainly determined by the shape of the distribution near the J acobian peak. The location of the maximum of the peak is directly related to the Wi boson mass, while the theoretical width of the peak varies with its decay width I‘yy. Since the Jacobian peak is modified by effects of both QT and PW, it is important to take into account both of these effects correctly. In our calculation (and in ResBos) we have properly included both effects. The effect of resummation on the transverse momentum distribution of the charged lepton from W+ and Z0 decays is shown in Fig. 3.15. The NLO and the resummed distributions differ a great amount even without imposing any kinematic cuts. The clear and sharp Jacobian peak of the NLO distribution is strongly smeared by the finite transverse momentum of the vector boson introduced by multiple gluon radia- tion. This higher order effect cannot be correctly calculated in any finite order of the perturbation theory and the resummation formalism has to be used. One of the advantages of using the p55 distribution to determine MW is that there is no need to reconstruct the p; distribution which potentially limits the precision of the mT method. From the theoretical side, the limitation is in the knowledge of the non-perturbative sector. Studies at DO [95] show that the p51 distribution is most sensitive to the PDF’s and the value of the non-perturbative parameter 92. The p5w distribution is more sensitive to the PDF choice, than the mT distribution _dg__ dee + (pb/GeV) 50 - 40 1 30 1 20 1 I01 25 30 3‘5 40 45 50 55 d9- - 1 V V I " W (1P7!+ 10 ' I, | 20 (pb/GeV) 0 10 20 3‘0 40 5‘0 ... pTe ( Ge V) Figure 3.15: Transverse momentum distributions of p? from W+ and Z0 decays for the NLO (dashed) and the resummed 0(015) (solid) calculations. Resumming the initial state multiple soft-gluon emission has the typical effect of smoothening and broadening the Jacobian peak (at p51+ = My / 2). The CDF cuts are imposed on the W4r distributions, but there are no cuts on the Z0 distributions. is. The uncertainty in the PDF causes an uncertainty in MW of about 150 MeV, which is about three times as large as that using the mT method [95]. A 0.1 GeV2 uncertainty in 92 leads to about AMW = 30 MeV uncertainty from the p51 fit, which is about five times worse than that from the mT measurement [95]. Therefore, to improve the MW measurement, it is necessary to include the 20 data sample at the high luminosity Tevatron to refit the gi’s and obtain a tighter constrain on them from the QT distribution of the Z0 boson. The DC study showed that an accuracy of A92 2 0.01 GeV2 can be achieved with Run 2 and TeV33 data, which would 87 contribute an error of AMW < 5 MeV from the p; [95]. In this case the uncertainty coming from the PDF’s remains to be the major theoretical limitation. At the LHC, the p51 distribution can be predicted with an even smaller theoretical error coming from the non-perturbative part, because at higher energies the perturbative Sudakov factor dominates over the non-perturbative function. It was recently suggested to extract MW from the ratios of the transverse momenta of leptons produced in Wi and Z 0 decay [96]. The theoretical advantage is that the non-perturbative uncertainties are decreased in such a ratio. On the other hand, it is not enough that the ratio of cross sections is calculated with small theoretical er- rors. For a precision extraction of the W3: mass the theoretical calculation must be capable of reproducing the individually observed transverse momentum distributions themselves. The W35 mass measurement requires a detailed event modeling, under- standing of detector resolution, kinematical acceptance and efficiency effects, which are different for the W:t and Z 0 events, as illustrated above. Therefore, the ratio of cross sections can only provide a useful check for the Wi mass measurement. For Drell-Yan events or lepton pairs from Z 0 decays, additional measurable quan- tities can be constructed from the lepton transverse momenta. They are the distribu- tions in the balance of the transverse momenta (Am 2 [13%| — [575? I) and the angular correlation of the two lepton momenta (z = —p;,‘;1 - fifiz/[max(p;1,p§2)]2). It is ex- pected that these quantities are also sensitive to the effects of the multiple soft gluon radiation. These distributions are shown in Figure 3.16. As shown, the resummed distributions significantly differ from the N LO ones. In these, and the following fig- ures for Z0 decay distributions, it is understood that the following kinematic cuts are imposed: Q72:o < 30 GeV, peT+’e_ > 25 GeV and |y8+1e—| < 3.0, unless indicated otherwise. d9_ "(1‘ ' l (pb/GeV) 1' I 1 1| 20» 1 " ‘ l ' I [ 1 ' ' l H ] 15’ f If \ I, l. \\ 10- 1 I' \ ll [I / Il \ S / [l \ / ’ 1' ‘ \ __-—-"'”’ f \\\‘—..__1 0 -10 -5 0 5 l0 l5 AI71'(GeV) do, 2000» 20 'x z ’ 1 I (M) , ‘| I [500* I ‘l ‘ ’ 1 ’ 1 ’ 1 1000» 1’ 1 I / 500- I’ z / 0 _ fl — 06 0.7 0:8 0:9 1 Z . . ..g .... Figure 3.16: Balance in transverse momentum Am 2 lpT‘I — [pé’] and angular cor- 1' ——*‘1.“‘3 ‘1 52211111 1 f Z°b d d re atlon z — pT pT /[max(pT, pT)] o t e ecay eptons rom osons pro uce at the Tevatron. 3.2.4 Lepton Angular Correlations Another observable that can serve to test the QCD theory beyond the fixed-order perturbative calculation is the difference in the azimuthal angles of the leptons £1 and Z72 from the decay of a vector boson V. In practice, this can be measured for 7*orZO ——> €102. We show in Fig. 3.17 the difference in the azimuthal angles of 6+ and e‘ (Adah—L measured in the laboratory frame for Z 0 —> e+e‘, calculated in the NLO and the resummed approaches. As indicated, the NLO result is ill-defined in the vicinity of Ad) ~ 71, where the multiple soft-gluon radiation has to be resummed 89 7d0'_1200 * dA¢e+e- (pb/rad) ””0“ 8001 600L 400- 2001 44+ AALLL 028‘ 2.215 2:9 2.05 5 3.05 A 3.1 Ade”- (rad) Figure 3.17: The correlation between the lepton azimuthal angles near the region Ad ~ 7r for pp -—> (Z0 —+ e+e‘)X. The resummed (solid) distribution gives the correct angular correlation of the lepton pair. The NLO (dashed lines) distribution near Ad 2 7r is ill-defined and depends on Q?” (the scale for separating soft and hard gluons in the NLO calculation). The two NLO distributions were calculated with Q59.” = 1.2 GeV (long dash) and Q?” = 2.0 GeV (short dash). to obtain physical predictions. Another interesting angular variable is the lepton polar angle distribution cos 9"“ in the Collins-Soper frame. It can be calculated for the Z0 decay and used to ex- tract sin2 0w at the Tevatron [97]. The asymmetry in the polar angle distribution is essentially the same as the forward-backward asymmetry A FB measured at LEP. Since AFB depends on the invariant mass Q and around the energy of the Z0 peak AFB happens to be very small, the measurement is quite challenging. At the hadron collider, on the other hand, the invariant mass of the incoming partons is distributed over a range so the asymmetry is enhanced [98]. The potentials of the measurement deserve a separated study. In Fig. 3.18 we show the distributions of cos 0"f predicted from the NLO and the resummed results. 90 do [ v ‘r v 1 1 v V T f 4 dcosB‘j 120 ' (Pb) : 1 100 ~ ~ \ T / 801 *~-— 1 l 601 f 40- 20 075 0.5 50:25 0 L 0.25 0.5 “0‘75 c050” Figure 3.18: Distribution of the e+ polar angle cos(0*) in the Collins-Soper frame from Z 0 decays at the Tevatron with cuts indicated in the text. 3.3 Conclusions With a 100 pb—1 luminosity at the Tevatron, around 2 x106 Wi and 6 x 105 Z 0 bosons are produced, and the data sample will increase by a factor of 20 in the Run 2 era. In view of this large event rate, a careful study of the distributions of leptons from the decay of the vector bosons can provide a stringent test of the rich dynamics of the multiple soft gluon emission predicted by the QCD theory. Since an accurate deter- mination of the mass of the Wi boson and the test of parton distribution functions demand a highly precise knowledge of the kinematical acceptance and the detection efficiency of Wi or Z0 bosons, the effects of the multiple gluon radiation have to be taken into account. In this work, we have extended the formalism introduced by Collins, Soper and Sterman for calculating an on-shell vector boson to include the ef- fects of the polarization and the decay width of the vector boson on the distributions of the decay leptons. Our resummation formalism can be applied to any vector boson V where V = 7*, Wi, Z0, W’, Z’, etc., with either vector or axial-vector couplings 91 to fermions (leptons or quarks). To illustrate how the multiple gluon radiation can affect the distributions of the decay leptons, we studied in detail various distributions for the production and the decay of the vector bosons at the Tevatron. One of the methods to test the rich dynamics of the multiple soft gluon radiation predicted by the QCD theory is to measure the ratio Ross E 393%? for the Wi and Z0 bosons. We found that, for the vector boson transverse momentum less than about 30 GeV, the difference between the resummed and the fixed order predictions (either at the as or a% order) can be distinguished by experimental data. This suggests that in this kinematic region, the effects of the multiple soft gluon radiation are important, hence, the QT distribution of the vector boson provides an ideal opportunity to test this aspect of the QCD dynamics. For QT less than about 10 GeV, the distribution of QT is largely determined by the non-perturbative sector of QCD. At the Tevatron this non-perturbative physics, when parametrized by Eq. (2.10) for Wi and Z0 production, is dominated by the parameter 92 which was shown to be related to properties of the QCD vacuum [71]. Therefore, precisely measuring the QT distribution of the vector boson in the low QT region, e.g. from the ample Z 0 events, can advance our knowledge of the non-perturbative QCD physics. Although the rapidity distributions of the leptons are not directly related to the transverse momentum of the vector boson, they are predicted to be different in the resummed and the fixed order calculations. This is because to compare the theoreti- cal predictions with the experimental data, some kinematic cuts have to be imposed so that the signal events can be observed over the backgrounds. We showed that the difference is the largest when the rapidity of the lepton is near the boundary of the phase space (i.e. in the large rapidity region), and the difference diminishes when no kinematic cuts are imposed. When kinematic cuts are imposed another important difference between the results of the resummed and the NLO calculations is the pre- 92 diction of the event rate. These two calculations predict different normalizations of various distributions. For example, the rapidity distributions of charged leptons (y‘i) from the decays of Wi bosons are different. They even differ in the central rapidity region in which the lepton charge asymmetry distributions are about the same (cf. Figs. 3.12 and 3.13). As noted in Ref. [90], with kinematic cuts, the measurement of MW is correlated to that of the rapidity and its asymmetry through the transverse momentum of the decay lepton. Since the resummed and the NLO results are differ- ent and the former includes the multiple soft gluon emission dynamics, the resummed calculation should be used for a precision measurement of MW. In addition to the rapidity distribution, we have also shown various distributions of the leptons which are either directly or indirectly related to the transverse momentum of the vector boson. For those which are directly related to the transverse momentum of the vector boson, such as the transverse momentum of the lepton and the azimuthal correlation of the leptons, the resummation formalism predicts significant differences from the fixed order perturbation calculations in some kinematic regions. The details were discussed in Section 3.2. As noted in the Introduction, a full event generator, such as ISAJ ET, can predict a reasonable shape for various distributions because it contains the backward radia- tion algorithm [79], which effectively includes part of the Sudakov factor, i.e. effects of the multiple gluon radiation. However, the total event rate predicted by the full event generator is usually only accurate at the tree level, as the short distance part of the virtual corrections cannot yet be consistently implemented in this type of Monte Carlo program. To illustrate the effects of the high order corrections coming from the virtual corrections, which contribute to the Wilson coefficients C in our resumma- tion formalism, we showed in Fig. 3.19 the predicted distributions of the transverse momentum of the Drell-Yan pairs by ISAJET and by ResBos (our resummed cal- <12 10 dQT 1 (pb/GeV) 8 _ 6 . 4 . i 21 0 Figure 3.19: Transverse momentum distribution of virtual photons in pp —> 7* —> e+e‘ events predicted by ResBos (solid curve) and ISAJET (histogram), calculated for the invariant mass range 30 GeV < Q < 60 GeV at the 1.8 TeV Tevatron. culation). In this figure we have rescaled the ISAJET prediction to have the same total rate as the ResBos result, so that the shape of the distributions can be directly compared. We restrict the invariant mass of the virtual photons Q to be between 30 and 60 GeV without any kinematic cuts on the leptons. If additional kinematic cuts on the leptons are imposed, then the difference is expected to be enhanced, as discussed in Section 3.2.1. As clearly shown, with a large data sample in the future, it will be possible to experimentally distinguish between these two predictions, and, more interestingly, to start probing the non-perturbative sector of the QCD physics. Chapter 4 Photon Pair Production In Hadronic Interactions An increasing amount of prompt diphoton data is becoming available from the Teva— tron collider and the fixed—target experiments at Fermilab [99, 100, 101]. After the upgrade of the Fermilab Tevatron the amount of the di-gauge boson data increases by a factor of 20, and the LHC by three order of magnitude. A comparison of the data to the calculation of the diphoton production rate and kinematic distributions provides a test of many aspects of perturbative quantum chromodynamics (pQCD). Furthermore, understanding the diphoton data is important for new physics searches. For example, diphoton production is an irreducible background to the light Higgs boson decay mode 11 ——> 77. The next—to—leading order (NLO) cross section for the pp —> 77X process [102] was shown to describe well the invariant mass distribution of the diphoton pair after the leading order (LO) 99 —+ 77 contribution (from one—loop box diagrams) was included [103, 50]. However, to accurately describe the distribu- tion of the transverse momentum of the photon pair and the kinematical correlation of the two photons, a calculation has to be performed that includes the effects of initial—state multiple soft—gluon emission. In this Section, we extend the Collins—Soper—Sterman (CSS) soft gluon resumma- 94 95 tion formalism, to describe the production of photon pairs. This extension is similar to the formalism deve10ped for describing the distribution of the leptons from vector boson decays in Chapter 2, because the final state of the diphoton process is also a color singlet state at LO. Initial—state multiple soft—gluon emission in the scattering subprocesses qrj, qg and 99 —> 77X is resummed by treating the photon pair 77 sim- ilarly to the Drell—Yan photon 7*. In addition, there are contributions in which one of the final photons is produced through a long—distance fragmentation process. An example is qg ——> 1m followed by the fragmentation of the final state quark q —> 7X. An earlier study of soft—gluon resummation effects in photon pair production may be found in Ref. [104]. The results of the calculation are compared with CDF [99] and D0 [100] data taken at the collider energy \/— = 1.8 TeV. A prediction for the production rate and kinematic distributions of the diphoton pair in proton—nucleon interactions at the fixed—target energy x/S = 31.5 GeV, appropriate for the E706 experiment at Fermilab [101], is also presented. 4.1 Diphoton Production at Fixed Order The leading order (LO) subprocesses for diphoton production in hadron interactions are of order aim, where am, denotes the electromagnetic coupling strength. There are three classes of LO partonic contributions to the reaction ’1th —+ 77X , where hl and hz are hadrons, illustrated in Fig. (4.1a)—(4.1c). The first (4.1a) is the short—distance ch -—) '17 subprocess. The second (4.1b) is the convolution of the short—distance qg ——> 170 subprocess with the long—distance fragmentation of the final state quark q ——) *yX. This is a LO contribution since the hard scattering is of order aemas, while fragmentation is effectively of order aem/as. Here, as denotes the QCD coupling —-—’VVV‘- —AM/‘ (a) 11 (b) 11 ——"\/\/V" UHF—9M o --->a,r-®’\/L ] ‘VVV‘ (c) 2] (d) 5;] 11 456% o —’\/\N\ —W (e) 11 (1) 1( W m ———’VVV" —ar—% (9) 11 (h) 3]] WW ‘4‘?“ Figure 4.1: Feynman diagrams representing the leading order and next—to—leading order contributions to photon pair production in hadron collisions. The shaded circles signify the production of long—distance fragmentation photons, which are described by the fragmentation function D,,_q. strength. Class (4.1b) also includes the subprocess qr'j —-> 79 convoluted with the fragmentation g —) 7X. Finally, there are LO contributions (4.1c) involving subpro- cesses like qq —-> qq, where both final state quarks fragment q —) 7X. The transverse momenta of the photons are denoted {in and 15's,, and the transverse momentum of the pair is QT = 1‘57, + p‘T,. In the absence of transverse momentum carried by the incident partons, the LO process (4.1a) provides QT = 0. With the added assumption of collinear final—state fragmentation, (4.1b) provides QT = pp, + 5T2 = (1 — z)p'T,, where photon 2 carries a fraction 2 of the momentum of the final—state quark. Given a lower limit on the magnitude of the transverse momentum p} of each photon, the total cross section at LO is finite. The next—to—leading order (NLO) subprocesses for diphoton production are of order agmas. One class of one—100p Feynman diagrams (4.1d) contributes by inter- fering with the tree level diagram (4.1a). Real gluon emission (4.1e) is also present at NLO. The subprocess qg ——> y'yq contains a singular piece (4.1f) that renormalizes the 97 lower order fragmentation (4.1b) and a piece (4.1g) that is free of final—state collinear singularities. Finally, subprocesses like qq —> qqyy contain a regular piece involv- ing photon emission convoluted with a fragmentation function (4.1h) and pieces that renormalize the double fragmentation process (4.1c). The regular 3—body final state contributions from (4.1e), (4.1f), and (4.1g) provide QT = —p‘Tj, where j represents the final—state quark or gluon. The full set of NLO contributions just described is free of final—state singularities, and the total integrated cross section at NLO is finite for a finite lower limit on each p51. Higher order calculations in as improve the accuracy of predictions for total cross sections involving quarks or gluons when only one hard scale Q is relevant. For hlhz —) 77X, this scale can be chosen proportional to the invariant mass of the photon pair, Q = M7,, which is about equal to 2p} for two well separated photons in the central rapidity region. For kinematic distributions that depend on more than one scale, a NLO calculation may be less reliable. This is the case for the distribution of the transverse momentum of the photon pair, QT = |QT|. Similarly to the Drell—Yan pair, at fixed Q, the behavior of the fixed order differential cross section for small QT is gram—Q} 2:6,) gjlnm(§—.T)+0(Q— 5,112.5) 14.1) 11:0 m: o where "'11", are calculable perturbatively. The structure of Eq. (4.1) indicates that the fixed order QCD prediction is reliable when QT 2 Q, but becomes less reliable when QT << Q, where ln(Q2/Q%) becomes large. In the region QT << Q, the photon pair is accompanied by soft and/or collinear gluon radiation. To calculate distributions like— 761T reliably in the region QT << Q, effects of multiple soft gluon emission must be taken into account. The contributions (4.1e) and (4.1g) exhibit singular behavior that can be tamed by resummation of the effects of initial—state multiple soft—gluon 98 radiation to all orders in as. Other contributions that do not become singular as QT —> 0 do not need to be treated in this manner. Fragmentation contributions like (4.1b) are found to be small in magnitude after isolation restrictions are imposed on the energy of the hadronic remnant from the fragmentation. Therefore, contributions like (4.1c) and (4.1h) are ignored in this work. Gluon fragmentation to a photon can be ignored, since its magnitude is small. 2 2 The subprocess 99 ——> 77, involving a quark box diagram, is of order aemas. While formally of even higher order than the NLO contributions considered so far, this LO gg contribution is enhanced by the size of the gluon parton distribution 3 8 correction leads to resummation of the function. Consideration of the order agma gg subprocess in a manner analogous to the qci resummation. 4.2 Extension of the Resummation Formalism 4.2.1 The Resummation Formula To improve upon the prediction of Eq. (4.1) for the region QT << Q, perturba- tion theory can be applied using an expansion parameter a? ln"(Q2/Q%), with n == 0,. . . , 2m — 1, instead of 0;". The terms a3” ln"(Q2/Q%) represent the effects of soft gluon emission at order a2". Resummation of the singular part of the perturbative series to all orders in as by Sudakov exponentiation yields a regular differential cross section as QT —-> 0. The differential cross section in the CSS resummation formalism for the production of photon pairs in hadron collisions is given by Eq. (2.4), with the changes of My 2 0 and FV = 0: d0(h1h2 —-) 7172X) _ 1 i dQ2 dy dQQT dcosO dd — 487rS' Q2 99 1 .~ ~ ~ ~ X {w/d2081QTb waij(anvxla-T2561 (15501102103) W’s/F(baanlaak) M + Y(QT7Q5$17$2505¢1C4)} ' (42) The variables Q, 3;, and QT here denote the invariant mass, rapidity, and transverse momentum of the photon pair in the laboratory frame, while 0 and d are the polar and azimuthal angle of one of the photons in the Collins—Soper frame [52]. The initial— state parton momentum fractions are defined as :01 = eyQ/JS, and 2:2 2 e’yQ/x/B, and VB is the center—of—mass (CM) energy of the hadrons hl and h2. The renormalization group invariant quantity VII/0(0) sums the large logarithmic terms a? ln"(sz2) to all orders in as. For a hard scattering process initiated by the partons z' and j, Wij (b7 Q1 $111.25 6: ¢9 Cla C2103) : exp {—Sij (1)) Q1 015 02)} X [Ci/h1(x1)Cj/h2($2) + Cj/hi ($1)Ci/h2 (372)] $10 (aem (CQQ)1 (13(02Q), 01 ¢)(43) where the Sudakov exponent S,j(b,Q,C1,C2) is defined by Eq. (2.6), and 0,7,,(1') denotes the convolution of the perturbative Wilson coefficient functions C5,") with parton distribution functions fa/h as given by Eq. (2.8). The Aij and 8,5 coefficients of the Sudakov exponent and the functions Ci]- are calculated perturbatively in powers of as/7r as in Eq. (2.13). The kinematic factor f},- will be defined later for each particular partonic process. The dimensionless constants C1, C2 and C3 E 110 were introduced in the solution of the renormalization group equations for Wij- The constant C1 determines the onset of non-perturbative physics, C2 specifies the scale of the hard scattering process, and ,u = C3/b is the factorization scale at which the C5,") functions are evaluated. We use the conventional choice of the renormalization constants: C1 = C3 = 2e‘7E _=_ 00 and C2 = C4 = 1 [46], where 75 is the Euler constant. 100 As in the case of lepton pair production, in Eq. (4.2), the impact parameter b is to be integrated from 0 to 00. However, for b 2 bmm which corresponds to an energy scale less than l/bmx, the QCD coupling a,([1 ~ l/b) becomes so large that a per- turbative calculation is no longer reliable, and non-perturbative physics must set in. The non-perturbative physics in this region is described by the empirically fit function 0 ,/1 + (b/bmy’ where bmax is a phenomenological parameter used to separate long and short distance W3,” [105, 73], and Wij is evaluated at a revised value of b, b, = physics. With this change of variable, 0... never exceeds bmax; 0mx is a free parameter of the formalism [107] that can be constrained by other data (e.g. Drell—Yan). The function Y in Eq. (4.2) contains contributions in the full NLO perturbative calculation that are less singular than Q? or Q1}2 ln(Q2/Q§~) as QT ——> 0 (both the factorization and the renormalization scales are chosen to be C4Q). It is the difference between the exact perturbative result to a given order and the result from Wz‘j expanded to the same fixed order (called the asymptotic piece). The function Y restores the regular contribution in the fixed order perturbative calculation that is not included in the resummed piece Wij- It does not contain a contribution from final—state fragmentation, which is included separately as described in Section 4.2.4. The CSS formula Eq. (4.2) contains many higher—order logarithmic terms, such that when QT ~ Q, the resummed differential cross section can become negative in some regions of phase space. In this calculation, the fixed—order prediction for the differential cross section is used for QT 2 Q whenever it is larger than the prediction from Eq. (4.2). The detailed properties of this matching prescription can be found in Ref. [108]. 101 4.2.2 Resummation for the ch —+ 77 subprocess For the qq —-> 77 subprocess, the application of the CSS resummation formalism is similar to the Drell-Yan case qr‘j (I) —> V“ ——> €11.72, where (1 and £2 are leptons produced through a gauge boson V“ [108]. Since both processes are initiated by qr‘j ('l color singlet states, the A(1),A(2) and B“) functions in the Sudakov form factor are identical to those of the Drell-Yan case when each photon is in the central rapidity region with large transverse momentum and is well separated from the other photon. This universality can be understood as follows. The invariants .5, f and '11 are defined for the (100602) —1 70907094) subprocess as § = (101+ 192V, 5: (191 - 193V, 0 = (112 — pay- The transverse momentum of each photon can be written as p} = \/tu—/5. When p} is large, f and {1 must also be large, so the virtual—quark line connecting the two photons is far off the mass shell, and the leading logarithms due to soft gluon emission beyond the leading order can be generated only from the diagrams in which soft gluons are connected to the incoming (anti—)quark. To obtain the 3(2) function, it is necessary to calculate beyond NLO, so it is not included in this calculation. However, the Sudakov form factor becomes more accurate when more terms are included in Aij and Bij. Since the universal functions A]? depend only on the flavor of the incoming partons (quarks or gluons), A326) can be appropriated from Drell—Yan studies, and its contribution is included in this work. To describe the effects of multiple soft—gluon emission, Eq. (4.2) can be applied, where i and 3' represent quark and anti—quark flavors, respectively, and fij = 26,j(g% + 9%)2(1+ cos2 0)/(1 — cos2 0). 102 The couplings 9L,R are defined through the qrj7 vertex, written as 1711 ML“ — ’15)+ 912(14— 75)], with 9L = 9;; = eQ f / 2, and eQ; is the electric charge of the incoming quark (Q, = 2 / 3, Q; = —1/ 3). The explicit forms of the A and B coefficients used in the numerical calculations are: A0111)(C'1) = CF, e _ fl_fi _3 _ A Aqq (Cl) — CF [(36 12) NC 18Nf 201111 (Cl , B;;)(C1,C‘2) :: CF [—2 — 21H (Céf0)], where N; is the number of light quark flavors, NC = 3, CF 2 4/3, and 01 = (11NC — 2Nf)/12. To obtain the value of the total cross section to N LO, it is necessary to include the Wilson coefficients CE?) and CS). These can be derived from the full set of LO contributions and NLO corrections to 77 production [50]. After the leading order and the one-loop virtual corrections to qq‘ —> 77 and the tree level contribution from ch ——> 779 are included, the coefficients are: E 019121.11. (:2) = 5,,511 — z), C)°a)(z,b1u,%;) = 0, 01101.11}? = 6..Cp[-;—<1—z)—511;1n (€09) 11.9.12) +M1—a[4&(i%f4“)+§+éa}. MA) After factorization of the final—state collinear singularity, as described below, the real 103 emission subprocess qg ——> 77q yields: 11> fl -1 _ _ EL” 11> 0,012.12, 11, C2) — ,zu 2) 1n ( ,0) 5.612). In the above expressions, the splitting kernels [42] are given by Eq. 5.7. For photon pair production, the function V is 2 “A 71 at (FWD) — 2) , Fvirtw) = (2+ ” )ln2(v)+(2+1_v)ln2(1—v) 1— v v +( 11 +1—0) l2( )+ln2(1—v)—3+27r2 1 — 11 11 n 11 3 v 1 — 11 +2 (111(0) + ln(1 — v) + 1) + 3 (1 _ ln(1 — v) + 11101)) 14.5) where v = —f1/.§, and ii = —.§(1 + cos 0) / 2 in the qr] center-of~mass frame. Because of Bose symmetry, Fvi"(v) = F “”0 — v). A major difference from the Drell—Yan case (pr = —8 + 7r2) is that V77 depends on the kinematic correlation between the initial and final states through its dependence on {1. and f. The non-perturbative function used in this study is the empirical fit [73] qugp(b’ Q, Q0, 1131, 1132) = exp {—9102 '- 9202111 ('2%') " 91930111 (1002:1232) ’ 0 where 91 = 0.11i3;3§ GeV2, 92 = 0.5818} GeVz, 93 = —1.5‘_L83] GeV-l, and Q0 2 1.6 GeV. (The value 0mm, 2 0.5 GeV'1 was used in determining the above gi’s and for the numerical results presented in this work.) These values were fit for the CTEQ2M parton distribution function, with the conventional choice of the renormalization con- stants, i.e. C1 = C3 = 00 and C2 = 1. In principle, these coefficients should be refit for the CTEQ4M distributions [109] used in this study. The parameters of Eq. (2.9) were determined from Drell—Yan data. It is assumed that the same values should be applicable for the 77 final state. 104 4.2.3 Contributions From qg Subprocesses As described in Section 4.1, the complete NLO calculation of diphoton production in hadron collisions includes photons from long—distance fragmentation processes like (4.1b) and short—distance processes like (4.1f) and (4.1g). The latter processes yield a regular 3—body final state contribution, while the former describes a photon recoiling against a collinear quark and photon. The singular part of the squared amplitude of the q(p1)g(p2) —> 7(p3)7(p4)q(p5) subprocess can be factored into a product of the squared amplitude of q(p1)g(p2) —> 7(p3)q(p4+5) and the splitting kernel for q(p4+5) —+ 7(p4)q(p5). In the limit that the emitted photon 7(p4) is collinear with the final state quark q(p5): lim IM (q(p1)g(p2) —> 7(p3)7(p4)q(ps))l2 = P4 “1’5 82 p4 . P5 '74—q () IM ((131)9(192) —> 7(p3)q(p4+5))[2. (4.6) A similar result holds when 193 and 1);, become collinear and/or the quark is re- placed with an anti—quark. Conventionally, the splitting variable 2 is the light—cone momentum fraction of the emitted photon with respect to the fragmenting quark, z— — p4 +/(p4 + p5 +), where pf = (p (0)+ pm) /\/5. (p [0) is the energy and pm is the longitudinal momentum component along the moving direction of the fragmenting quark in the qg center-of-mass frame.) Alternatively, since the final state under con- sideration contains only a fragmenting quark and a spectator, a Lorentz invariant Splitting variable can be defined as: [110] Pi'Pk Pj 'Pk +P1'Pk +Pi'Pj 221— In this notation, 2' = 5 is the fragmentation quark, j is the fragmentation photon, and k is the prompt spectator photon. When the pair 2' j becomes collinear, 2 becomes the flu ('0. Plt‘ IF 0 L773 105 the same as the light—cone momentum fraction 2 carried by the photon. Aside from the color factor, P”) (z) in Eq. (4.6) is the usual DGLAP splitting kernel for q —> gq P(1)(z)=(1+(1_ 2)?)- (_— ‘Y (I Z The regular contribution qg ——> 77g (4.1f) is defined by removing the final—state, collinear singularity from the full amplitude of the partonic subprocess. The matrix element squared for (4.1f) can be written [110]: IM ((19 —> wolf... = 82 P4 'P5 IM ((19 —> 77061.1( 1379., (5) IM (q(p1)g(p2) -+ 7(103)q(204+5))l2 - (4-7) After the final—state collinear singularity is subtracted, the remainder expresses the regular 3—body final state contribution 77g. This remainder, as shown in (4.1g), contains terms that diverge when QT ——> 0 which should be regulated by renormalizing the parton distribution at the NLO. The contribution from this divergent part is included in the resummed qtj cross section in 01%), as shown in Eq. (4.2.2). The part that is finite as QT -—> 0 is included in the function Y. When QT 2, Q, Eq. (4.7) describes the NLO contribution from the qg —> 77g subprocess to the QT distribution of the photon pair. The subtracted final—state collinear singularity from the NLO qg ——> 'y'yq subprocess is absorbed into the fragmentation process (4.1b). 4.2.4 Fragmentation Contributions Final—state photon fragmentation functions D7/,-(z, 11%.) are introduced in an analo- gous manner to initial—state parton distribution functions f,- /h, (:12, It?) Here, z(a:) is the light—cone momentum fraction of the fragmenting quark (incident hadron) carried by the photon (initial—state parton), and [JF([£[) is the final state (initial state) frag- mentation (factorization) scale. The parton—level cross section for the fragmentation 106 contribution (4.1b) is evaluated from the general expression for a hard scattering to a parton m, which then fragments to a photon: . 1 . m_ i do : -2—§|M(p1p2 —> p3 . . .pm)]2d( 2)[PS]dzD7/m(z, 11%). Here, .M is the matrix element for the hard scattering subprocess, d(’"“2) [PS] is the m - 2—body phase space, and an integral is performed over the photon momentum fraction 2 weighted by the fragmentation function Dv/m(z, #37). Since fragmentation is computed here to L0 only, the infrared divergences discussed by Berger, Guo and Qiu are not an issue [111]. The fragmentation function D714, obeys an evolution equation, and the leading— logarithm, asymptotic solution D5}; is [50]: DLL(z,ui~) = aemln i215:— D“) (z), '7/(1 27r AEQCD "ii—q Q2(2.21 — 1.28z + 1.2922)20'049 _ zDgflAz) = q 1 _ 163ln(1 _ z) + 0.0020(1 — z)2'oz 1‘54, (4.8) where AQCD is the QCD scale for four light quark flavors. As shown in Fig. 4.2, the collinear approximation made in defining D7,” leads to kinematic distributions with an unrealistic sensitivity to kinematic cuts, such as cuts to define an isolated photon. The Monte Carlo showering method goes beyond the collinear approximation used in solving the evolution equation for the fragmentation function D7+"Q' In Monte Carlo calculations, the probability for photon emission is determined from the splitting function P7,_Q(z), which is a collinear approximation. However, the kinematics are treated by assigning a virtuality to the fragmenting quark whose value lies between the hard scale of the process and a phenomenological cutoff ~ 1 GeV. This cutoff replaces the parameter AQCD in Eq. (4.8). Most importantly, gluon emission can 107 Pp 5 GeV. Mk2 (upper) no isolation (lower) isolalim do/QT (ph/GcV) 8 \__ \ a '5. ... | ~. N.“ s I u. ....... ..... ..... .- - 1 1 --------- - 1035;123:121 o o as s 75 10 us as 115 20 as as 01(GcV) aoo - —— Pun-aloud 250 f ~ - vim-um: L ----- Who-r L Ppsocvm'la m L (upper) no isolation [ (lower) isolation § [ 13 “° f o ---------- ‘U Figure 4.2: Comparison of the parton—level and Monte Carlo fragmentation contri- butions at the Tevatron. The upper and lower curves of the same type show the contribution before and after an isolation cut. The left figure shows the transverse momentum of the photon pair QT. The right figure shows the light—cone momentum fraction carried by the fragmentation photon. 108 be incorporated into the description of final state fragmentation. Because there is no collinear approximation in the kinematics, kinematic distributions do not exhibit the unrealistic behavior of the parton—level calculation. The “correctness” of either approach can be judged only after a careful comparison of their respective predictions. The collinear approximation becomes an issue because of the experimental def- inition of isolated photons. Experimentally, an isolation cut is necessary to sepa— rate prompt photons from various hadronic backgrounds, including 7r° and 1) me- son decays. The separation between a particle j and the photon is expressed as R,- = (fl?) — 773-)2 + ((15 — 45,-)2, where the coordinates HUD) and d)(¢j) are the pseu- dorapidity and azimuthal angle of the photon (particle j ) At hadron colliders, the standard isolation criterion is that the sum of excess transverse energy ET contained inside a cone of size R0 centered on the photon candidate is below a cutoff E33", 2 E} < E371”. The sum is over each particle j. Since the resummed CSS piece §f E530. Above QT = Elf", the perturbative corrections contained in the function Y are affected by isolation. On the other hand, because of the collinear approximation, the parton— level fragmentation calculation based on Eq. (4.8) does not depend on the isolation cone R0; the hadronic remnant of the fragmentation (4.1b) always satisfies R < R0. Hence, for this case, QT = (1 -— z)13'rp,, and the isolation cut reduces to a step function requirement 6MB?" — QT). The parton—level calculation of the fragmentation contribution at the Tevatron based on the fragmentation function D,,_q(z, a?) has been compared with a Monte Carlo estimate based on PYTHIA [77]. For the parton—level calculation, the scale 109 up 2 M77 is used. For the PYTHIA calculation, the scale is up = x/g, and hadroniza- tion is not performed, so that no photons arise from 7r° or n meson decays, for example. For this comparison, the invariant mass \/§ of the hard—scattering subprocess is lim- ited to 20 < \/§ < 50 GeV in both approaches, and the photons are required to satisfy p} > 5 GeV and I777] < 2. These kinematic cuts are chosen to increase the statistics of the PYTHIA calculation, while reflecting the kinematic region of interest for a comparison with data. PYTHIA can simulate the QED and QCD showering of the final—state quark as well as the QCD showering of the initial—state quark and gluon. To isolate the effect of initial—state gluon radiation, PYTHIA calculations were performed with and without the QCD initial—state radiation (i.e. by preventing space—like showering). In neither case is initial—state QED radiation simulated. It is possible for the partons produced in initial—state showering to develop time—like showering. Any photons produced from this mechanism are discarded, since they are formally of higher—order than the contributions considered here. Such contributions, however, might be necessary to understand photon pairs with small invariant mass and small QT. Figure 4.2 is a comparison of kinematic quantities from the parton—level and Monte Carlo calculations. The left-side of Fig. 4.2 shows the QT—distribution for the parton—level (solid), PYTHIA with initial—state radiation of gluons (short—dashed), and PYTHIA without initial—state radiation (long—dashed) calculations. Each curve is plotted twice, with and without an isolation cut E?” = 4 GeV and R0 = 0.7. Before the isolation cut, the total parton-level fragmentation cross section is approximately 50% higher than the Monte Carlo cross section. After isolation, the total cross sections are in good agreement, even though the parton—level calculation is discontinuous at QT = E§§°. The effect of initial—state gluon radiation in the PYTHIA calculation is to increase QT without compromising the isolation of the photons. 110 The right—side of Fig. 4.2 shows the distribution of the light—cone momentum fraction z of the quark carried by the fragmentation photon (for this figure, 2 is defined in the laboratory frame). After isolation, the parton—level contribution is limited to z > 0.55 by kinematics, whereas the Monte Carlo contribution is more uniformly distributed between 0 and 1. For the PYTHIA result, 2 is calculated with respect to the final state quark before showering. In the showering process, some energy—momentum can be exchanged between the final state prompt photon and the fragmenting quark, since the quark is assigned a virtuality. As a result, the effective 2 value can extend beyond the naive limit 2 = 1. The conclusions of this comparison are: (1) after isolation, the total cross sec- tions from the parton—level and Monte Carlo fragmentation calculations are in good agreement, and (2) the Monte Carlo kinematic distributions (e.g. QT and z) are not very sensitive to the isolation cut. For these reasons, the Monte Carlo estimate with initial—state radiation is used to account for the (4.1b) contribution in the final results. Furthermore, with initial—state radiation, the PYTHIA calculation includes the leading effects of a full resummation calculation of the qg —) 'yq process. It is approximately equivalent to performing a resummation calculation in the CSS for- malism with quantities A“) and B“) calculated for a qg initial state and the LO Wilson function. One final comparison was made with the Monte Carlo calculation by treating the subtracted term in Eq. (4.7), with P“) replaced by U(1) defined in Eq. (4.8), as a 3—body matrix element. The collinear divergence was regulated by requiring a separation R0 between the photon and quark remnant for all QT. This calculation agrees with PYTHIA in the shape and normalization of various distributions, except when QT < Em, where there is a substantial difference. 111 4.2.5 Resummation for the 99 —> 77 subprocess A resummation calculation for the 99 -—> 77 subprocess is included in the theoretical prediction. The L0 contribution comes from one—loop box diagrams of order agma2 in S perturbative QCD. At present, a full NLO calculation, of 0(agmai), for this process is not available. Nevertheless, the resummation technique can be applied to resum part of the higher order contributions and improve the theoretical prediction. The exact NLO gg ——> 779 calculation must include gluon emission from the internal quark lines of the box diagram, thus generating pentagon diagrams. However, such diagrams do not generate large logarithms when the final state photons have large transverse momentum, are in the central rapidity region, and are well separated from each other. All the large logarithms originate from the diagrams with soft gluons coupling to the initial—state gluons. Similarly, the exact NLO qg ——> 77q calculation, of 0(agmag), must include contributions involving a box diagram with one incoming gluon off shell. Large logarithms only arise from soft gluon emission off the initial—state quark or gluon. The leading logarithms due to initial—state radiation are universal, and the A“) function calculated for the resummed 99 —> H process [112, 113] or the color singlet part of the gg —> QQ process [60] can be applied directly to the resummed gg ——> 77 calculation, since these subprocesses have the same QCD color structure. When the transverse momentum of the photon pair is much smaller than its invariant mass, i.e. QT << Q, and each photon has large transverse momentum, then the box diagram of the hard scattering subprocess gg —> 77 can be approximated as a point—like interaction (multiplied by a form factor which depends on §,f and 11). This approximation ignores pentagon diagrams in the 99 —-> 77g subprocess and the virtuality of intermediate quarks in the qg ——> 77q subprocess. It does not have the complete structure of the hard process, but it does contain the most important 112 logarithmic terms from initial state gluon radiation. Under such an approximation, the sub-leading logarithmic terms associated with B“), A”), and 0(1) of Eqs. (4) and (5) can be included in the resummation calculation. These functions were calculated for the 99 —> H process [112, 113]. Without a complete 0(agmai’) calculation, the exact Wilson coefficient function C“) is not known. Since part of the exact 0(1) function must include the piece for the 99 —> H process, it is included to estimate the possible NLO enhancement to the production rate of the gg subprocess. After these ingredients are incorporated into Eq. (4.2), the resummed kinematics of the photon pair from the gg —> 77 subprocess can be obtained. The distribution of the individual photons can be calculated approximately from the LO angular dependence of the box diagram. The above approximation certainly fails when QT is of the order of Q. In the absence of a complete 0(agmag) calculation of the gg ——+ 779 and qg —> 77q subpro- cesses, it is not possible to estimate the uncertainties introduced by the approxima- tion. In the limit of QT << Q, the approximation should be reliable, since the soft gluon approximation is applicable. In the same spirit, the approximate function Y for photon pair production is taken from the results of the perturbative piece for the 99 —) Hg and gq —> Hq processes [112, 113]. In summary, the resummed distributions of the photon pair from the gg subprocess in the region of QT << Q can be described by Eq. (4.2), with z" = j = g, and .799 = NC|M99_,77(s,t,u)|2/211. Here, |./’i/lgg_,.,.,(s,t,u)|2 is the absolute square of the invariant amplitude of the gg —) 77 subprocess [103] summed over spins, colors, and the fermion flavors in the box loop, but without the initial—state color (1 /82), spin (1/22) average, and the final—state identical particle (1/ 2) factors. The A and B functi ills l Only the” reg”: 0f [l ”‘81) Sub] Hiak MU] .4 113 functions used in the calculation for the 99 initial state are AQQKCH) = 0,423, C A5729)(Cl) = C_:A(%(Cl), Cl BE.?(C..Cg) = 2 [31n(C2b0) — a]. The L0 and NLO Wilson coefficients, extracted from the 99 ——> H subprocess, are: Cf," g),(zb,;ug—1) C2, 6(1— 2), C(0)(Z,b;,€1-fl) = 0 qg 02 C533,)(z ,b,,%; p) = —ln(%§)Pg,_g(z)+6(I—z){1711—+3—:: —31n2(C(:;0)+31n(Cf;0) +(2ol — 3) 1n (If—3)} (1,1 “(mg ,u) = —ln (£509) Pg._.,(z)+§z. Since the N LO pentagon and off—shell box diagram calculations are not included, the Wilson coefficients CS) are expected to predict accurately the total cross section only when QT << Q, the transverse momenta of the individual photons are large, and their rapidities are small. Under the approximation made above, the resummed 99 result increases the integrated rate by about a factor of 2, for kinematic cuts typical of the Tevatron, as compared to the lowest order (one—100p calculation) perturbative result. This comparison suggests that the full NLO contribution of the gg initiated subprocess is large. Because it is necessary to impose the condition QT < Q to make the above approximations valid, the 99 resummed result presented in this work probably underestimates the rate when QT is large or the separation of the azimuthal 114 angle (Aqb) between the two photons is small. This deficiency can be improved only by a complete 0(agmaf) calculation. At the Tevatron, the 99 contribution is important when the invariant mass (M77 : Q) of the two photon pair is small. Because of the approximation made in the 99 calculation beyond the LO, the prediction will be more reliable for the data with larger Q. A more detailed discussion is presented in the next section. The full calculation of the 99 contribution in the CSS formalism depends also upon the choice of non-perturbative functions. However, the best fits to the parametriza- tions are performed for qr] processes [105, 73]. Two assumptions were studied: (2') the non-perturbative functions are truly universal for mi and 99 processes, and (22') the non-perturbative functions obey the same renormalization group properties as the A functions for each type of process (which are universal for all qr] or 99 subprocesses), so the coefficient of the 1n (%3) term in the non-perturbative function Eq. (2.9) is scaled by CA/Cp relative to that of the qq process. Specifically, the different assumptions are: (Z)Wghg/P(biQ1Q0axlim2) : W'qup(b,Q,Q0,$1,$2) .. ~ ~ , C (22) ngzp(biQ7 Q07‘T17x2) : WqI;P(b)QaQO)$IaJ/‘2) WIth g2 __f C_:g2 The numerical values of 91,92, and 93 are listed following Eq. (2.9). These two as- sumptions do not exhaust all possibilities, but ought to be representative of reason- able choices. Choice (ii) is used for the results presented in this work. The effect of different choices is discussed in the next Section. 1 15 W Resummation at \/S=l .8 TeV IYrITYYUIrYYYIVYV'IYI'TIVYTYIVYYTTYYYH % 1 —- Total 1 g -------- qq+_qg 1 @ ...-....:.;.::.:. :2 .1 g 000000 frag B 7new | ll<9 ‘ 1: pp . n . ¢CDF ] _A O \- - 5"." ... u ”a: - '- ‘- '- - '- ----- a... v u, . o. . Q 10 - "--2. 501M160” 7o ‘ so MW(GeV) Figure 4.3: The predicted distribution for the invariant mass of the photon pair M77 from the resummed calculation compared to the CDF data, with the CDF cuts imposed in the calculation. 4.3 Numerical Results 4.3.1 Tevatron Collider Energies Two experimental collaborations at the Tevatron pp collider have collected diphoton data at \/S = 1.8 TeV: CDF [99], with 84 pb‘l, and DO [100], with 81 pb‘l. The kinematic cuts applied to the resummed prediction for comparison with the CDF data are p} > 12 GeV and I777] < 0.9. For DO , the kinematic cuts are p33 > 14 GeV and pr]? > 13 GeV, and |777| < 1. For CDF, an isolation cut for each photon of R0 = 0.7 and E5?" = 4 GeV is applied; for DO , the cut is R0 = 0.4 and E?” = 2 GeV. Other ingredients of the calculation are: (2') the CT EQ4M parton distribution functions, (ii) the NLO expression for (1,, (iii) the NLO expression for am, and (iv) the non-perturbative coefficients of Ladinsky-Yuan [73]. The predictions for the CDF cuts and a comparison to the data are shown in 116 Figs. 4.3—4.5. Figure 4.3 shows the distribution of the photon pair invariant mass, do/dMW vs. M77. The dashed—dot curve represents the resummation of the 99 subprocess, which is the largest contribution for MW 3, 30 GeV. The long—dashed curve represents the full qq resummation, while the short—dashed curve is a simi- lar calculation with the gluon parton distribution function artificially set to zero. Schematically, there are contributions to the resummed calculation that behave like q —> gql ® qlq -—) 77 and g -—> qq, ® qlq ——> 77. These contributions are contained in the terms proportional to PJi2k(z) in Eq. (4.4) and PJ-(26(z) in Eq. (4.2.2), re- spectively. The full qq resummation contains both the qq and qg contributions. The short—dashed curve is calculated by setting 0):; = 0 and retaining only the qq‘ con- tribution in the function Y. Since the short—dashed curve almost saturates the full qq + qg contribution, except at large QT or small A43, the qg initiated subprocess is not important at the Tevatron in most of phase space for the cuts used. The frag- mentation contribution is denoted by the dotted line. The sum of all contributions including fragmentation is denoted by the solid line. After isolation, the fragmen- tation contribution is much smaller than “direct” ones, but contributes 2 10% near the peak. The uncertainty in the contribution of the fragmentation process can be estimated by comparing the Monte Carlo result with a parton-level calculation, as shown in Fig. 4.2. Figure 4.4 shows the distribution of the transverse momentum of the photon pair, do/dQT vs. QT. Over the interval 5 S. QT g 25 GeV, the contribution from the 99 subprocess is comparable to the qq+qg subprocess. The change in slope near QT = 20 GeV arises from the 99 subprocess (dot-dashed line) for which QT S. M.” is required in our approximate calculation. The peak near QT 2 1.5 GeV is provided mostly by the qq + qg (long—dashed line) subprocess. In general, the height and breadth of the peak in the QT distribution depends on the details of the non-perturbative function 117 W Resummation at VS=1.8 TeV TTrT—Yfi'YTT—TIVYI‘YVYIYYIY‘YYYUTTjYVITTTr dG/dQT (pb/GcV) ppm GeV. |n7|<.9 . ¢CDF .2 o f 1 141 . . -.o s u . o. 550. ~~. \ "- ' ‘- 10 ~ - -- i P ‘. "u ° ... O I >- ‘. o ‘- o .‘ "O -1 . 's. lllLlileLJlll1141l4Jl_LLL.l—111_LL£L11.L_1111L11_1LLJE 0 51015 20 25 30 35 4o 45 50 QT(GeV) Figure 4.4: The predicted distribution for the transverse momentum of the photon pair QT from the resummed calculation compared to the CDF data, with the CDF cuts imposed in the calculation. in Eq. (4.2). The effect of different non-perturbative contributions may be estimated if the parameter 92 is varied by :i:2o. For QT > 10 GeV, the distribution is not sensitive to this variation. The height and the width (half—maximum) of the peak change by approximately 20% and 35%, respectively, but the integrated rate from 0 to 10 GeV is almost constant. The peak of the distribution (which is below 5 GeV), shifts approximately +0.5 GeV or —0.6 GeV for a +20 or -—20 variation. The mean QT for QT < 10 GeV shifts at most by 0.4 GeV. For 99 resummation, it is not clear which parametrization of the non-perturbative physics should be used. However, the final effect of the two different parametrizations outlined in Eq. (4.9) is minimal, shifting the mean QT for QT < 40 GeV by about 0.4 GeV. The parametrization (ii) is used in the final results, so that the coefficient 92 is scaled by CA/CF relative to the qq non-perturbative function. Figure 4.5 shows do/dA¢ vs. Aqb, where Arb is the azimuthal opening angle be- 1 1 8 w Resummation at ‘iS=l .8 TeV Y r1 T f' Y 1" I Y Y I Y Y I Y I fiY—I T T Y T T 1' .5 O N IIIY E Total ‘3 L -------- qfi+qg B ................ qq ¢CDF . 3.10 ~ -------- gg : .9 ...... 1 3 i \ . .4 g i o .- o u o .- ‘0 ,0 C O 00000 ....... A AAAALLAAALAAJJA A AAA AJA‘ALALALQLAJA ’— A .20 . O I 10 'xililiLiilriALJALLJliAiiliiii 1.5 2 2.5 3 1 A4) (radians) Figure 4.5: The predicted distribution for the difference between the azimuthal angles of the photons Ac),7 from the resummed calculation compared to the CDF data, with the CDF cuts imposed in the calculation. tween the two photons. The change in slope near Agb = 7r / 2 is another manifestation of the approximations made in the treatment of the 99 contribution (dot—dashed line). The height of the distribution near Agb 9: 7r is also sensitive to the details of the non-perturbative function. In the absence of resummation or NLO effects, the 99 box contribution supplies QT = 0 and A43 2 7r. In this calculation, as explained earlier, the NLO contribution for the 99 subprocess is handled in an approximate fashion. For the cuts listed above, the total cross section from the complete 99 resummed calculation, including the function Y, is 6.28 pb. If the resummed CSS piece is used alone, the contribution is reduced to 4.73 pb. This answer can be compared to the contribution at LO. For the same structure functions, the LO gg cross section for the CDF cuts is 3.18 pb for the scale choice Q = M7,. Therefore, the effect of including part of the NLO contribution to the 77 process is to approximately double the LO 99 box contribution to the cross 119 w Resummation at ‘JS=1.8 TeV Yfiij’YYIYTIVI—rIVVI[VIIIVIYTIYYTFITYfi’ do/dlvgfl (pb/GcV) p,,1>13.85, 14.90 GeV |n7| 14 GeV, p33 > 13 GeV and 777 < 1.0, an “equivalent” set of cuts is used in the theoretical calculation: p33 > 14.9 GeV, p]? > 13.85 GeV, and n7 < 1.0 [114]. The effect of this “equivalent” set is to reduce the theoretical rate in the small M.” region. While the agreement in both shapes and absolute rates is generally good, there 120 w Resummation at VS=1.8 TeV YU1Y'YYT Tr]!'1I'VYUVIYY"rYVVYIYj'TITITYT‘ do/dQT (pb/GeV) _t o O. o"-., . a"... lllLLlilltlJllLllllllllllllJllLLJLLLlllllJlllll. 0 5 10 15 20 25 3O QT(GcV) Figure 4.7: The predicted distribution for the transverse momentum of the photon pair QT from the resummed calculation compared to the DO data, with the DO cuts imposed in the calculation. are some discrepancies between the resummed prediction and the data as presented in these plots. At small QT (Fig. 4.4) and large Ad) (Fig. 4.5), where the CDF cross section is large, the theoretical results are beneath the data. Since this is the kinematic region in which the non-perturbative physics is important, better agreement can be obtained if the non-perturbative function is altered. In Fig. 4.6, the calculated M77 distribution is larger than the DO data at large MW, while the calculation appears to agree with the CDF data in Fig. 4.3. The small discrepancy in Fig. 4.6 at large values of M77 is not understood. (The systematic errors of the data, which are about 25% [114], are not included in this plot.) On the other hand, Figs. 4.7 and 4.8 show that the resummed calculation is beneath the data at large QT or small Ad). The discrepancies in Figs. 4.7 and 4.8 may result from the approximations made in the 99 process (notice the kinks in the dot—dashed curves). A complete N LO calculation for the gg subprocess is needed, and may improve the comparison with data for small figu of th the] .10. Sum. fort {Hm Snlaj Died flirt SUhd t[011. prod in ii I l 121 w Resummation at VS=1.8 TeV 102F'rTTIfv'IIIYTI'T"i.1v'i*"Tl': :5: — Total U """" 951mg 2 r ................ ,,, ..D 310 :— ........ gg ¢Do 'e' E ...... frag 3 '0 L Pn.2>l3.85.14.9oocv .] In’| 0 divergence in the fixed—order calculation is removed. However, there is also a marked difference 122 w Resummation at ‘18:] .8 TeV . 2. ,fi. IITVV'IYYTTTT Yfifi‘IIIIYVY" p.].>12 GeV. ln7|<.9 ‘ Q1. 3 GeV and |77"| < 0.75. No photon isolation is required. The same phenomenological inputs are used for this calculation as for the calculation at collider energies. The Be nucleon target is treated as having an admixture of 4 / 9.01 proton and 5.01 / 9.01 neutron parton distribution functions. The A dependence effect appears to be small in the prompt photon data (the effect is parametrized as A“ and figu Dair app] dirtl With The 1 123 W Resummation at 45:18 TeV fl 1 Y Y 1 T 1 I I I I I I I 1 V j r T Y Y Y T T do/dQT (pb/GeV) ppm GeV, |n*l<.9 ] QT 3GeV where the event rate is small. Since the LO qq rate does depend on as, and the L0 99 rate is proportional to (13(02MW), the gg rate increases for a smaller (3'2 value, but the qq rate remains about the same. In conclusion, the E706 data can be used to constrain the non-perturbative functions associated with the qq and 99 hard processes in hadron collisions. 125 W Resummation at 43:31.5 GeV dc/dMW (pb/GeV) ...L o l l 10 m TTrT Figure 4.12: The predicted distribution for the invariant mass of the photon pair MW from the resummed calculation appropriate for pN —-> 77X at JS=31.5 GeV. 4.4 Conclusions Prompt photon pair production at fixed target and collider energies is of interest in its own right as a means of probing the dynamics of strong interactions. The process is of substantial interest also in searches for new phenomena, notably the Higgs boson. In this Chapter, a calculation is presented of the production rate and kinematic distributions of photon pairs in hadronic collisions. This calculation incorporates the full content of the next—to—leading order (NLO) contributions from the qq‘ and qg initial—state subprocesses, supplemented by resummation of contributions to these subprocesses from initial state radiation of soft gluons to all orders in the strong coupling strength. The computation also includes important contributions from the gg box diagram. The gg contributions from initial—state multiple soft gluons are re- summed to all orders, but the NLO contribution, of 0(agmag), to the hard scattering subprocess is handled in an approximate fashion. The approximation should be reli- 126 W Resummation at 48:31.5 GeV 5 YTWYIYI TTYYTiITTi‘ffrVYfiI[TYTY—IIVfTFWY dCS/dQT (pb/GeV) .0 9.0.0.0 on a: Vamp-2 .°.° can .° N Figure 4.13: The predicted distribution for the transverse momentum of the photon pair QT from the resummed calculation appropriate for pN -+ 77X at x/S=31.5 GeV. able at relatively small values of the pair transverse momentum QT as compared to the invariant mass of the photon pair M77. At collider energies, the 99 contribution is comparable to that of the qq and qg contributions over a significant part of phase space where M.” is not large, and its inclusion is essential. The exact 0(a2ma3) corrections to the 99 box diagram should be calculated to test the validity of the approximations made in this calculation. Finally, the calculation also includes long— distance fragmentation contributions at leading order from the subprocess qg —> 7q, followed by fragmentation of the final quark, q —> 7X. After photon isolation, frag- mentation plays a relatively minor role. The fragmentation contribution is computed in two ways: first, in the standard parton model collinear approximation and second, with a Monte Carlo shower simulation. This overall calculation is the most complete treatment to date of photon pair production in hadronic collisions. Resummation plays a very important role particularly in the description of the behavior of the QT 127 w Resummation at VS=31.5 GeV 1' *r Tfi‘r I j Tfi I I I I I I I I I ...A O l do/dAtb (pb/radian) 10 2.8 2.4 2.6 Ad) (radians) Figure 4.14: The predicted distribution for the difference between the azimuthal angles of the photons Aqfiw from the resummed calculation appropriate for pN —) 77X at $82315 GeV. distribution at small to moderate values of this variable, where the cross section takes on its largest values. The resummed calculation is necessary for a reliable prediction of kinematic distri- butions that depend on correlations between the photons. It is a significant improve- ment over fixed—order NLO calculations that do not include the effects of initial—state multiple soft—gluon radiation. Furthermore, even though the hard scattering qq and qg subprocesses are computed to the same order in the resummed and fixed—order NLO calculations, the cross sections from the two calculations can differ after kine- matic cuts are imposed [108]. The results of the calculation are compared with data from the CDF and DO col- laborations, and the agreement is generally good in both absolute normalization and shapes of the distributions in the invariant mass MW of the diphoton system, the pair transverse momentum QT, and the difference in the azimuthal angles Act. Discrep- 128 ancies with CDF results at the smallest values of QT and Ad) near 7r might originate from the strong dependence on the non-perturbative functions in this kinematic re- gion. In comparison with the DO data, there is also evidence for disagreement at intermediate and small values of A45. The region of intermediate Ad), where the two photons are not in a back—to—back configuration, is one in which the full treatment of three body final—state contributions of the type 773' are important, with j = q or g. The distributions in Figs. 5 and 8 suggest that an exact calculation of the NLO contribution associated with the 99 initial channel would ameliorate the situation and will be necessary to describe data at future high energy hadron colliders. Predictions are also presented in the work for pN ——> 77X at the center—of— mass energy 31.5 GeV, appropriate for the E706 fixed—target experiment at Fermilab. The large QT and small Ad) behavior of the kinematic distributions is dominated by the resummation of the gg initial state. Non-perturbative physics controls the QT distribution, and neither the perturbative Sudakov nor the regular NLO contribution plays an important role, except in the very large QT region where the event rate is small. For the E706 kinematics, the requirement QT < Q is generally satisfied. Therefore, the approximate 99 calculation presented in this work should be reliable. In this calculation, the incident partons are assumed to be collinear with the in- cident hadrons. A recurring question in the literature is the extent to which finite “intrinsic” [CT may be required for a quantitative description of data [117, 101]. An important related issue is the proper theoretical specification of the intrinsic com- ponent [118]. In the CSS resummation formalism, this physics is included by prop- erly parametrizing the non-perturbative function W” P (b), which can be measured in Drell—Yan, W, and Z production. Because photons participate directly in the hard scattering, because their momenta can be measured with greater precision than that of hadronic jets or heavy quarks, and because the 77 final state is a color singlet, 129 the reaction m? —> 77X may serve as a particularly attractive laboratory for the un- derstanding of the role of intrinsic transverse momentum. The agreement with data on the QT distributions in Figs. 4.4 and 4.7 is suggestive that the CSS formalism is adequate. However, the separate roles of gluon resummation and the assumed non-perturbative function in the successful description of the QT distributions are not disentangled. In the non—perturbative function of Eq. (2.9), the dependence on b (and, thus, the behavior of do/dQT at small QT) is predicted to change with both Q and the values of the parton momentum fractions am. At fixed Q, dependence on the values Of the x.- translates into dependence on the overall center—of—mass energy of the reaction. As data with greater statistics become available, it should be pos- sible to verify these expectations. In combination with similar studies with data on massive lepton—pair production (the Drell—Yan process), it will be possible to deter- mine whether the same non—perturbative function is applicable in the two cases, as is assumed in this work. The diphoton data may allow a study of the non-perturbative as well as the perturbative physics associated with multiple gluon radiation from the gluon—initiated hard processes, which cannot be accessed from Drell—Yan, W*, and Z data. With this knowledge, it may be possible to improve calculations of single photon production and other reactions sensitive to gluon—initiated subprocesses. In the DO data analysis [100], an asymmetric cut is applied on the transverse momenta (p}) of the two photons in the diphoton event. This cut reduces the effect of multiple gluon radiation in the event. To make the best use of the data for probing the interesting multiple gluon dynamics predicted by the QCD theory, a symmetric p} cut should be applied. Chapter 5 Gauge Boson Pair Production at the Upgraded Tevatron and at the LHC 5. 1 Introduction The underlying dynamics of the electroweak symmetry breaking sector of the Stan- dard Model (SM) awaits understanding. The principal goal of the CERN Large Hadron Collider (LHC) is to shed light on this open question. The direct searches at the CERN Large Electron Positron (LEP) collider have constrained the mass of the SM Higgs boson to be higher than 90 GeV [119]. Furthermore, global analyses of electroweak data [29] and the values of the top quark and the W:t boson masses [120] suggest that the SM Higgs boson is light, less than a few hundred GeV. Argu- ments based on supersymmetry (SUSY) also indicate that the lightest Higgs boson is lighter than the top quark [121]. Hence, the existence of a light Higgs boson is highly possible. It has been shown in the literature that a SM like Higgs boson with a mass less than or about 180 GeV can be detected at the upgraded Fermilab Tevatron via p15 —> Wi(—> fiu) H(—> bb,7'+7") [89], or p13(gg) ——> H(—> W‘W“ —> Eujj and 130 131 EVEV) [122], and a SUSY Higgs boson can be detected in the Wih and hbb modes [123]. To observe a light (my < 120 GeV) neutral Higgs boson at the LHC, the most promising detection mode is the diphoton channel H —) 77 [124] via the production process pp(gg) ——> HX. In the intermediate mass (120 GeV < my < 2 mg) region, the Z 0* Z 0 channel is also useful in addition to the 77 channel [124]. If the Higgs boson is heavier than twice the mass of the Z 0 boson, the gold-plated decay mode into two Z0 bosons (which sequentially decay into leptons) [124] is the best way to detect it. At the LHC, as at any hadron—hadron collider, initial—state radiative corrections from Quantum Chromodynamics (QCD) interaction to electroweak processes can be large. Some fixed order QCD corrections have been calculated to the Higgs signal and to its most important backgrounds [125, 126, 127, 128]. The next-to-leading order (NLO) corrections to the total cross section of pp(gg) —> H X have been found to be large (50- 100 %) [125], and the largest contribution in the fixed order corrections results from soft gluon emission [129]. This signals the slow convergence of the perturbative series, and the importance of still higher order corrections. Furthermore, the fixed order corrections fail to predict the transverse momentum distributions of the Higgs boson and its decay products correctly. The knowledge of these distributions is necessary to precisely predict the signal and the background in the presence of various kinematic cuts, in order to deduce the accurate event rates to compare with theory predictions [130]. To predict the correct distribution of the transverse momentum of the photon or Z0 pair and the individual vector bosons, or the kinematical correlation of the two vector bosons produced at hadron colliders, it is necessary to include the effects from the initial—state multiple soft—gluon emission. In this Section, we present the results of our calculation for the most important continuum backgrounds to the Higgs boson signal detected at hadron colliders. The distributions of the Higgs boson signal for the hlhg —-> H —> 77X and ZOZOX processes, including the soft—gluon effects, will 132 be discussed in our future work [131]. In scattering processes involving hadrons, the dynamics of the multiple soft—gluon radiation can be described by the resummation technique. We extend the Collins— Soper-Sterman (CSS) resummation formalism [44, 45, 46] to describe the production of photon and Z 0 pairs. This extension is analogous to our recent resummed calcula- tion of the hadronic production of photon pairs [132]. In comparison, an earlier work [133] on the soft—gluon resummation for the qq —> Z 0Z 0X process did not include the complete N LO corrections. In the present Section, the effect of initial—state multiple soft—gluon emission in qij —> ZOZOX is resummed with the inclusion of the full NLO contributions, so that the inclusive rate of the Z0 boson pair production agrees with the NLO result presented in Ref. [126]. Furthermore, we also include part of the higher order contributions in our results by using the CSS resummation formalism. The collected diphoton data at the Tevatron, a pp collider with center-of—mass energy /S = 1.8 TeV with 84 and 81 pb‘l integrated luminosity (for CDF and DO ), are in the order of 103 events per experiment [134, 135]. After the upgrade of the Tevatron, with \/S = 2 TeV and a 2 fb"1 integrated luminosity, about 4 x 104 photon pairs can be detected, and more than 3 x 103 Z 0 boson pairs can be produced. At the LHC, a JS = 14 TeV pp collider with a 100 fb‘l integrated luminosity, we expect about 6 x 106 photon and 1.5 x 106 Z0 pairs to be produced, after imposing the kinematic cuts described later in the text. This large data sample will play an important role in the search for the Higgs boson(s) and new physics that modifies the production of the vector boson pairs (e.g., by altering the vector boson tri—linear couplings [136]). The rest of this Chapter is organized as follows. Section 5.2 briefly summarizes the extension of the CSS resummation formalism to the ZOZ0 pair production. In 133 Section 5.3 the numerical results of the resummed and fixed order calculations are compared for various distributions of the photon and Z 0 pairs produced at the LHC and the upgraded Tevatron. 5.2 Analytical Results 5.2.1 The CSS Resummation Formalism for Z0 Pair Produc- tion When QCD corrections to the Z0 boson pair production cross section are calculated order by order in the strong coupling constant as, the emission of potentially soft gluons spoils the convergence of the perturbative series for small transverse momenta (QT) of the Z0 boson pair. In the QT << Q region, the cross section can be written as [46] do °° 2"”1 nvm m 2 1 ——22-—- (awe). where Q is the invariant mass of the Z0 boson pair, and the coefficients ”on, are perturbatively calculable. At each order of the strong coupling the emitted gluon(s) can be soft and / or collinear, which yields a small QT. When the two scales Q and QT are very different, the logarithmic terms lnm(Q2/Q2T) are large, and for QT << Q the perturbative series is dominated by these terms. It was shown in Refs. [44, 45, 46] that these logarithmic contributions can be summed up to all order in as, resulting in a well behaved cross section in the full QT region. The resummed differential cross section of the Z 0 boson pair production in hadron collisions is written, similarly to the cross sections of the lepton pair production [137], or photon pair production [132], in the form: d0(h1h2 —> ZOZOX) _ 1 A dQ2 dy dQ§~dcosfld¢ _ 487rS Q2 134 1 .~ ~ ~ ~ X {W/dee‘QT'b ZWijU’nQi$11$2161¢aCl,2,3) Wigyp(bianlim2) iaj + Y(QT7Q7x1)x2363 (15: C4)} (5'1) In this case, the variables Q, y, and QT are the invariant mass, rapidity, and transverse momentum of the Z 0 boson pair in the laboratory frame, while 0 and (,b are the polar and azimuthal angle of one of the Z0 bosons in the Collins-Soper frame [52]. The factor 4mzz Q2 originates from the phase space for producing the massive Z 0 boson pair. The parton 5: 1— momentum fractions are defined as .731 = eyQ/x/S, and 21:2 2 e‘yQ/x/S, and \/S is the center—of-mass (CM) energy of the hadrons hl and hz. The renormalization group invariant function W,j(b) sums the large logarithmic terms a?ln"‘(b2Q2) to all orders in as. For a scattering process initiated by the partons 2' and j, {470(1), 62,171, $2, 9, d), 01,2,3) = exp {—Sij(ba Q, 012)} X [Ci/h1($1, b,01,2,3,t,U)Cj/h2(172, 5,0123, L U) + Cj/hl (131, 5, 01,2,3, U, t) 61/112052, 5, 01,2,3, U, 75)] >< Jig-(61(02Q), 03(0262), 9, ¢)- (5-2) Here the Sudakov exponent Sij(b,Q,Cl,2) is defined in Eq.(2.6). In Eq. (5.2), fl]- originates from the hard scattering process, and will be given later for specific initial state partons. Eq. (2.8) defines Ci/h(a:) which denotes the convolution of the perturba- tive Wilson coefficient functions Cm with parton distribution functions (PDF) fa/h(f) (describing the probability density of parton a inside hadron h with momentum frac- tion 5). The invariants s, t and u are defined for the q(p1)q(p2) —> Z°(p3)Z°(p4) suhprm: . I ll'llll .5 ~ lhr- as huh immdi; canonh where ' For lS (Next the [WI [‘16 ill; perturl lnCalt-u Dimes] WUfk ll and 2‘ data. 135 subprocess as s = (131 +192)”, t= (p1 —- 19:02, u = (192 — 1902, (5-3) with s+t+u= 2m2z. The functions AU, Bij and Cij are calculated perturbatively in powers of (rs/7r . as indicated by Eq. (2.13). The dimensionless constants Cl, 02 and 0;, E 11b were introduced in the solution of the renormalization group equations for Wij- Their canonical choice is CI = C3 = 2e‘7E E b0, C2 = 01 /b0 2 1, and C4 = CQ 2 1 [46], where 7g is the Euler constant. For large b, which is relevant for small QT, the perturbative evaluation of Eq. (5.2) is questionable. Thus in Eq. (5.1), W),- is evaluated at b. = b/\/1 + (b/bmax)2, so that the perturbative calculation is reliable. Here bmax is a free parameter of the formalism [46] that has to be constrained by other data (e.g. Drell—Yan), along with the non- perturbative function W3 P (b) which is introduced in Eq. (5.1) to parametrize the incalculable long distance effects. Since the qq —> 77 or ZOZO, and the qq' —-> V —-> 813’ processes have the same initial state as well as the same QCD color structure, in this work we assume that the non-perturbative function 1717,],y P (b), fitted to existing low energy Drell-Yan data [73], also describes the non-perturbative effects in the qq -) 77 and Z 0 Z 0 processes. Needless to say, this assumption has to be tested by experimental data. The function Y in Eq. (5.1) contains contributions from the NLO calculation that are less singular than 1/Q2T or ln(Q2/Q§~)/Q§~ as QT -—> 0. This function restores the regular contribution in the fixed order perturbative calculation that is not included in the resummed piece W”. In the Y function, both the factorization and the renor- malization scales are chosen to be C4Q. The detailed description of the matching (or “switching”) between the resummed and the fixed order cross sections for QT ~ Q 136 (a) (b) (0) KW Figure 5.1: A representative set of Feynman diagrams included in the NLO calculation of Z0 pair production. 1 l 5 if; 1.1.1 can be found in Ref. [137]. 5.2.2 The qq, qg and (jg ——> Z”Z0X subprocesses The largest background to the Higgs boson signal in the ZOZ0 channel is the contin- uum production of Z0 boson pairs via the qq —> Z°Z0X partonic subprocess [138]. The next—to~leading order calculations of this process are given in Refs. [126, 127]. A representative set of Feynman diagrams, included in the NLO calculations, is shown in Fig. 5.1. The application of the CSS resummation formalism for the qq ——> Z 0ZOX subprocess is the same as that for the case of qq —-—> 77X [132]. The A“), A”) and B“) coefficients in the Sudakov exponent are identical to those of the Drell—Yan case. This follows from the observation that to produce a heavy Z “ boson pair, the virtual—quark line connecting the two Z0 bosons in Fig. 5.1 is far off the mass shell, and the leading logarithms due to soft gluon emission beyond the leading order can only be generated from the diagrams in which soft gluons are connected to the incoming (anti~)quark. This situation was described in more detail for di-photon production [132]. 137 The resummed cross section is given by Eq. (5.1), with z' and j representing quark and anti—quark flavors, respectively, and 21+coszt9 fry-(9,95,61,05) — 2611(91. + W 1:33—25 The left- and right-handed couplings 9L3 are defined through the qer0 vertex, which is written as 17,, [gL(1 — 75) + 93(1 + 75)], with T — s2 32 9 JT& and 9R = -g wa- cw 26w 91, = (5.4) Here 9 is the weak coupling constant, T3 is the third component of the SU(2)L gener- ator (T3 = 1 / 2 for the up quark Qu, and -1/2 for the down quark Qd), 3w (cw) is the sine (cosine) of the weak mixing angle, and Q f is the electric charge of the incoming quark in the units of the positron charge (Q11 = 2/3 and Q) = —1 / 3). The values of these parameters will be given in the next section. The explicit forms of the A and B coefficients, used in the numerical calculations are: Aggml) = CF, ' 67 2 5 b Ag?(Cl) 2 CF (— — 1') NC “ —Nf _ 251m (1)] ’ 36 12 18 Cl [_ l where N f is the number of light quark flavors, NC = 3 is the number of colors in QCD, CF 2 4/3, and ,61:(11NC — 2Nf)/12. To obtain the value of the total cross section to NLO, it is necessary to include the Wilson coefficients CS” and Cu) ,1 , which can be derived similarly to those for diphoton production in Chapter 4. The results are: C 0):)(2, b, 11, 67;, t, u) = 6jk6(1 — z), 138 Cighzybm, Ciatm) = 0, 012312.111, g1...) = 6.1.01 {$(1 — z) - 51— In (fif—b) 1393M +6(1— z) [— ln2 (fie—3(4) + £31) + 19—6] }, 05.1mm, g1 11) = $41 — z) — ln (5;?) 3126(2). (5.6) In the above expressions, the splitting kernels are [42] 1 2 Pfflko) = Cp( +2) and + l—z 3120(2) = $[zz+(1-z)2]. (5.7) For Z0 boson pair production the function V in Eq. (5.5) is given by 7r2 + to 3 t2-l-u2 V(t, u) = Vzozo(t, u) = -4 + (F(t, u) + F(u, t) — 2). The definition of the function F (t, u) is somewhat lengthy and can be found in Ap- pendix C of Ref. [126] (cf. Eqs. (C1) and (C2)). The function V(t, u) depends on the kinematic correlation between the initial and final states through its dependence on t and a. In the mz ——> 0 limit, F(t, u) + F(u, t) reduces to F"""(t, u) of the diphoton case which is given in Ref. [132].1 The non-perturbative function used in this study is [73] W451)“, Q1 Q0, $1,172) = exp [“91132 — 92b2 111 (5%") " 9193b1n(100$1$2) a 0 with 91 = 0.11 Ger, 92 = 0.58 GeV2, 93 = —1.5 GeV", and Q0 = 1.6 GeV. These values were fit for the CTEQ2M parton distribution function, with the canonical 1This is connected to the fact that as m z —> 0 the virtual corrections of the Z 0 pair and diphoton productions are the same (up to the couplings), which is apparent when comparing Eq.(ll) of Ref. [139] and Eq.(l2) of Ref. [126], after including a missing factor of 1/ (167m) in the latter equation. 139 choice of the renormalization constants, i.e. CI = C3 = b0 and 02 = 1, and b,,,” = 0.5 GeV"1 was used. In principle, these coefficients should be refit for CTEQ4M distributions [38] used in this study. We have checked that using the updated fit in Ref. [140] does not change largely our conclusion because at the LHC and Tevatron energies the perturbative Sudakov contribution is more important compared to that in the low energy fixed target experiments. Before concluding this section we note that for the diphoton production, we use the formalism described in Ref. [132] to include the gg —+ 77X contribution, in which part of the higher order corrections has been included via resummation. Since a gauge invariant calculation of the 99 —> Z°Z°g cross section in the SM involves diagrams with the Higgs particle, we shall defer its discussion to a separate work [131]. 5.3 Numerical Results We implemented our analytic results in the ResBos Monte Carlo event generator [137]. As an input we use the following electroweak parameters [3]: G). = 1.16639 x 10-5 GeV—2, mg 2 91.187 GeV, (5.8) 1 mw — 80.41 GOV, a(mz) — m. In the on-shell renormalization scheme we define the effective weak mixing angle miv - 2 eff_ sm 6 —1— , w pm2z with _ 213% (1 7roz(mz) )4. p — mQZ — fiGmev In Eq.(5.4), the coupling of the Z0 boson to fermions, g, is defined using the improved Born approximation: 92 = 4\/§GF(Cfo)2m§p, 140 Di-boson Collison Ecm Fixed Order 0(a2) Resummed produced type (TeV) CTEQ4L CTEQ4M EB 0(05) ZOZ0 pp 14 9.14 10.3 14.8 ZOZ0 pp 2 0.91 1.01 1.64 77 pp 14 22.1 24.5 60.8 77 pp 2 8.48 9.62 22.8 77 pp 1.8 6.30 7.15 17.0 Table 5.1: Total cross sections of diphoton and Z 0 boson pair production at the LHC and the upgraded Tevatron, in units of pb. The kinematic cuts are described in the text. The “EB” sign refers to the matching prescription discussed in the text. with Ci,” 2 \/1 — sin2 03,”, the cosine of the effective weak mixing angle. (In Eq. (5.4) cm is identified with can.) We use the NLO expression for the running strong and electroweak couplings (13(11) and a(,u), as well as the NLO parton distribution function CTEQ4M (defined in the modified minimal subtraction, i.e. M_S, scheme), unless stated otherwise. Furthermore, in all cases we set the renormalization scale equal to the factorization scale: 11;; = up = Q. Table 5.1 summarizes the total rates for the leading order (LO), i.e. 0(a2), and resummed photon- and ZO-pair production cross sections for the LHC and the Tevatron. For the lowest order calculation we show results using LO (CTEQ4L) and NLO (CTEQ4M) parton distributions, because there is a noticeable difference due to the PDF choice. As it was discussed in Ref. [137], the resummed total rate is expected to reproduce the C(03) rate, provided that in the resummed calculation the A“), B“) and C“) coefficients and the (9(as) Y piece are included, and the QT distribution is described by the resummed result for QT g Q and by the 0(a3) result for QT > Q. In our present calculation we added the A”) coefficient to include the most important higher order corrections in the Sudakov exponent. Our matching prescription (cf. 141 Ref. [137]) is to switch from the resummed prediction to the fixed-order perturbative calculation as they cross around QT ~ Q. This switch is performed for any given Q and y of the photon or Z 0 boson pairs. In the end, the total cross section predicted by our resummed calculation is about the same as that predicted by the N LO calculation. The small difference of those two predictions can be interpreted as an estimate of the contribution beyond the NLO. 5.3.1 Z0 pair production at the LHC In the LHC experiments the H —> ZOZ0 channel can be identified through the decay products of the Z0 bosons. The detailed experimental kinematic cuts for this process are given in Ref. [124]. Since the aim of this work is not to analyze the decay kine- matics of the background, rather to present the effects of the initial—state soft—gluon radiation, following Ref. [126] for the LHC energies we restrict the rapidities of each Z0 bosons as: lyz | < 3.0. We do not apply any other kinematic cuts. The total rates are given in Table 5.1. Our 0(a2) rates are in agreement with that of Ref. [126] when calculated using the same PDF. We expect the resummed rate to be higher than the (9(1),) rate due to the inclusion of the Am term. Indeed, our K factor, defined as the ratio of the resummed to the LO rate using the same PDF in both calculations, is higher than the naive soft gluon K -factor (Km,ye = 1+87ra3(Q) / 9 ~ 1.3) of Ref. [141], which estimates the NLO corrections to the production rate of qq -—> ZOZOX in the DIS (deep-inelastic scattering) scheme. Our K -factor approaches the naive one with the increase of the center-of-mass energy, as expected. The rates for the different subprocesses of the Z 0 boson pair production are given in Table 5.2. At the LHC the qg —+ Z°Z°X subprocess contributes about 25% of the qq + qg —-> ZOZOX rate. The K—factor is defined as the ratio U(qq‘ + qg —> ZOZOX)/o(qq‘ —+ ZOZO), which is about 1.4 for using CTEQ4M PDF. 142 Em, Collison qq —> Z°Z°X qg —-> ZOZOX (TeV) type 14 pp 10.9 3.91 2 pp 1.62 0.02 Table 5.2: Resummed cross sections of the subprocesses for Z 0 boson pair production at the LHC and the upgraded Tevatron, in units of pb. The kinematic cuts are described in the text. Figs. 5.2-5.3 show our results for proton-proton collisions at the LHC energy, \/S = 14 GeV. Fig. 5.2 shows the transverse momentum distribution of Z0 pairs. The NLO (0(as)) prediction for the qq + qg -—> ZOZOX subprocesses, shown by the dotted curve, is singular as QT ——> 0. This singular behavior originates from the contribution of terms which grow at least as fast as 1/Q2T or ln(Q2/Q§~)/Q§.. This, so-called asymptotic part, is shown by the dash-dotted curve, which coincides with the 0(03) distribution as QT -—> 0. After exponentiating these terms, the distribution is well behaved in the low QT region, as shown by the solid curve. The resummed curve matches the 0(a3) curve at about QT = 320 GeV. Following our matching prescription described in the previous section, we find that this matching takes place around QT = 300 GeV, depending on the actual values of Q and y. Fig. 5.2 also shows that at the LHC there is a substantial contribution from qg scattering, which is evident from the difference between the solid and dashed curves, where the dashed curve is the resummed contribution from the qq —-> Z°Z°X subprocesses. In Fig. 5.4 we give the integrated distributions, defined as Qmax d (10’ _ f T 0 (5.9) dos" ‘ Q?“ T dQT’ where Q?“ is the largest QT allowed by the phase space. In the NLO calculation, this distribution grows without bound near Q?“ = 0, as a result of the singular behavior Tal.-"‘"-- 4 .- .o ..--— I I I dcr/dQ,zz (pb/GeV) Q Q I . C Q . C Q ‘ Q Q .. Q 1O 1111l111il ALJIJ .....1 . O 50 100 150 200 0121(GeV) 111.11.11.11. 250 300 350 Figure 5.2: Transverse momentum distribution of Z 0 pairs from qq+qg partonic initial states at the LHC. The C(08) (dotted) and the asymptotic (dash-dotted) pieces are coincide and diverge as QT —> 0. The resummed (solid) curve matches the 0(03) curve at about QT = 320 GeV. The resummed qq contribution (excluding the qg contribution) is shown as dashed line. of the scattering amplitude when QT —-> 0. It is clearly shown by Fig. 5.4 that the QT distribution of the resummed calculation is different from that of the NLO calculation. The different shapes of the two curves in Fig. 5.4 indicates that the predicted Z 0 pair production rates, with a minimal value of the transverse momentum QT, are different in the two calculations. This is important at the determination of the background for the detection of a Higgs boson even with moderately large transverse momentum. For Q?“ = 50 GeV, the resummed cross section is about 1.5 times of the NLO cross section. The invariant mass and the rapidity distributions of the Z0 boson pairs, and the transverse momentum distribution of the individual Z0 bosons are shown in Fig 5.3. When calculating the production rate as the function of the Z0 pair invariant mass, we integrate the QT distribution for any Q, and y. When plotting the transverse 144 O O u U! I 1 dO/dQu (pb/GeV) da/dyn (pb/GeV) . p . 8 0.02 _ 0.015 ‘ 0.01 : 0.005 ’ -" ...1...-1...1--1..1.1 200 300 400 500 600 700 O 012 (GeV) 1 .. ,.. ., ...., -, .. .--f,.-.fi.. . da/dp.z (pb/Gev) 10 111.121 .1.l.1_.l . . l . . 1 . l 1 1 1 1 l . . . 0 50 100 150 200 250 300 350 400 of (GeV) Figure 5.3: Invariant mass and rapidity distributions of Z 0 boson pairs, and transverse momentum distributions of the individual Z0 bosons at the LHC. The resummed contribution of the qq + qg —-) ZOZOX subprocess is shown by the solid curve, and of the qq —> ZOZOX subprocess by the dashed curve. The leading order distribution of qq —> ZOZ0 is shown by the dash-dotted curve. 145 Q N T J- do/dQ A 1 J 1 1 1 l L A J AL A A 300 A ‘ 550 ’ 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 O 50 100 150 200 250 min 01’ (GeV) Figure 5.4: The integrated cross section for Z0 boson pair production at the LHC. The resummed and the 0(03) distributions are shown in solid and dashed lines, respectively. momentum distributions of the individual Z0 bosons, we include both of the Z0 bosons per event. In the shape of the invariant mass and rapidity distributions we do not expect large deviations from the NLO results. Indeed, the shape of our invariant mass distribution agrees with that in Ref. [126]. However, the resummed transverse momentum distribution P7? of the individual Z0 bosons is slightly broader than the NLO distribution (not shown in Fig 5.3, cf. Ref. [126]). This is expected because, in contrast with the NLO distribution, the resummed transverse momentum distribution of the Z0 boson pair is finite as QT —> 0 so that the P7? distribution is less peaked. 5.3.2 Z0 Pair Production at the upgraded Tevatron After the upgrade of the Fermilab Tevatron, there are more than 3 x 103 Z 0 boson pairs to be produced. Since this data sample can be used to test the tri-linear gauge boson couplings [136], we also give our results for the upgraded Tevatron with proton—anti- 146 proton collisions at a center-of-mass energy of 2 TeV. Our kinematic cuts constrain the rapidity of both of the Z0 bosons such that [312 I < 3. Both the LO and resummed total rates are listed in Table 5.1. The ratio o(qu + qg —-) ZOZOX)/o(qq —> Z°Z°) is about 1.6, which is larger than the naive soft gluon K -factor of 1.3. Table 5.2 shows that qg —-> ZOZOX partonic subprocess contributes only a small amount (about 3%) at this energy, in contrast to 25% at the LHC. Figs. 5.5—5.6 show the resummed predictions for the upgraded Tevatron. The invariant mass and rapidity distributions of Z0 boson pairs, and the transverse mo- mentum distribution of the individual Z0 bosons are shown in Fig. 5.5. The solid curve shows the resummed contributions from the qq+qg —+ Z 0Z 0X subprocess. The resummed contribution from the qq —> ZOZOX subprocess is shown by the dashed curve. The leading order qq —> ZOZ0 cross section is also shown, by the dash-dotted curve. The invariant mass distribution of the qq‘ + qg —+ ZOZOX subprocess is in agreement with the NLO result of Ref. [126], when calculated for \/_ = 1.8 TeV. From this figure we also find that the contribution from the qg ——+ Z 0ZOX subprocess at the energy of the Tevatron is very small. In Fig. 5.7 we compare the NLO and resummed distributions of the transverse momentum of the Z0 pair. The figure is qualitatively similar to that at the LHC, as shown in Fig. 5.2. The resummed and the NLO curves merge at about 100 GeV. The resummed contribution from the qq —-> ZOZOX subprocess is shown by the dashed curve, which clearly dominates the total rate. In Fig. 5.6 we show the integrated distribution 010/an7?” for Z0 boson pair pro- duction at the upgraded Tevatron. The figure is qualitatively the same as that for the LHC (cf. Fig. 5.4). The NLO curve runs well under the resummed one in the Q?” < 80 GeV region, and the QT distributions from the NLO and the resummed oar/do“ (pb/GeV) 147 >vv—v*f[vvvvIvvvvIvvvvlvaw—Ivvvvrvvvav~v—y—yr‘ 0.3 ’— do/dy“ (pb) 0.7 :- 1.-.1 111. 0.6 l . 0.5 P ...., A 0.4 :- 0.3 n. ‘ - . 0 .1 a . 0.2 — ,. ., . t . u . .‘ ‘. 4 LA 1.. .1-...1. 1.. ..:'~< ' .... .-......- 200 300 400 500 600 700 -2 - 1.5 -l -0.5 . 6 Qzz (GeV) ya do/dp} (pb/Gev) 1 .-.1.1.1 1. ..-- .11. 1 O 50 100 150 200 250 300 350 400 IO. ..1..1 91’ (GeV) Figure 5.5: Same as Fig. 5.3 except for the upgraded Tevatron. 148 T da/dQ L l A L A l L PL! 1 A l l .. ..1...1...1. 0 20 40 60 80 100120 140 01"“ (GeV) Figure 5.6: Same as Fig. 5.4 except for the upgraded Tevatron. calculations have different shapes even in the region where QT is of the order 60 GeV. For Q?" = 30 GeV, the resummed rate is about 1.5 times of the NLO rate. 5.3.3 Diphoton production at the LHC Photon pairs from the decay process H —> 77 can be directly detected at the LHC. When calculating its most important background rates, we impose the kinematic cuts on the final state photons that reflect the optimal detection capabilities of the ATLAS detector [124]: p} > 25 GeV, for the transverse momentum of each photons, [y'll < 2.5, for the rapidity of each photons, and p§~/(p1T + p?) < 0.7, to suppress the fragmentation contribution, where p; is the transverse momentum of the photon with the higher pT value. We also apply a AR = 0.4 separation cut on the photons, but our results are not 149 da/dQ,zz (pb/GeV) I I I l l l 1 A 1.1 [A 1111L4L1 l 1 LALAIL o” 20 4o 60 so 100‘ 120 140 0122 (GeV) Figure 5.7: Same as Fig. 5.2 except for the upgraded Tevatron. sensitive to this cut. (This conclusion is similar to that in Ref. [132].) The total rates and cross sections from the different partonic subprocesses are presented in Tables 5.1 and 5.3. We have incorporated part of the higher order contributions to this process by including Am in the Sudakov factor and 05,? in the Wilson coefficient functions (cf. Ref. [132]). Within this ansatz, up to 0(a2), the gg —) 77X rate is about 24 pb, which increases the total K -factor by almost 1.0. The leading order gg —-) 77 rate, via the box diagram, is about 22 pb and 14 pb for using the LO PDF CT EQ4L and the NLO PDF CTEQ4M, respectively. The large difference mainly due to the differences in the strong coupling constants used in the two calculationszz CTEQ4L requires as(mz) = 0.132, while for CTEQ4M as(mz) = 0.116. The ratio 0016+ 09 -+ wxvmq —> ’77) is 1-5, and 0(99 —> 7790/0015 -+ ’W) is about 1- Hence, the ratio of the resummed and the C(02) rates, is quite substantial. Figs. 5.8-5.9 show our predictions for distributions of diphotons produced at the 2When using the CTEQ4L PDF, we consistently use the LO running coupling constant as. 150 rTf‘fi ‘7'" v‘r‘fivv vvva—rv da/dy" (pb) do/dQ" (pb/Gev) 10'- 10* p" (GeV) Figure 5.8: Invariant mass and rapidity distributions of photon pairs, and transverse momentum distributions of the individual photons at the LHC. The total resummed contribution (upper solid), and the resummed qq + qg —) 77X (dashed), qrj —> 77X (dotted), 99 —+ 779 (dash-dotted), as well as the fragmentation (lower solid) con- tributions are shown separately. The qc‘j —-> 77 0(02) distribution is shown in the middle solid curve. 151 Em Collison qc'j —> 77X qg —> 77X gg —) 77 99 —> 779 Fragmen- (TeV) type 4L 4M tation 14 pp 20.5 16.6 22.3 14.4 23.9 6.76 2 pp 9.68 4.81 6.02 4.34 8.26 2.15 Table 5.3: Cross sections of the subprocesses for diphoton production at the LHC and the upgraded Tevatron, in units of pb. The resummed qg —) 77X rate includes the fragmentation contribution. The (9(af) 99 —> 77 rates were calculated using both the CTEQ4L and CTEQ4M PDF’s. The kinematic cuts are described in the text. LHC. In Fig. 5.8 we plot the invariant mass and rapidity distribution of the photon pairs, and the transverse momentum distribution of the individual photons. When plotting the transverse momentum distributions of the individual photons we include both photons per event. The total (upper solid) and the resummed ch + qg —> 77X (dashed), qq -—> 77X (dotted), 99 —+ 77X (dash-dotted), and the fragmentation (lower solid), as well as the leading order qq —-> 77 (middle solid) contributions are shown separately. The ratio of the resummed and the LO distributions is about 2.5 which is consistent with the result in Table 5.1. The relative values of the contribu- tions from each subprocesses reflect the summary given in Table 5.3. Fig. 5.10 shows various contributions to the transverse momentum of the photon pair. At low QT values (QT << Q), the qq —+ 77X contribution is larger than the qg ——> 77X contribution, while at high QT values (QT > Q), the qg —> 77X subprocess becomes more important. The 99 contribution dominates the total rate in low QT region, and the kink in the 99 curve at about 50 GeV indicates the need for the inclusion of the complete 0(a2) gg —) 77g contribution. (Recall that our prediction for the 99 contribution at 0(a3) only holds for small QT, where the effect of the initial-state soft-gluon radiation is relatively more important for a fixed Q.) 152 ‘ M T do/dQ . . r . . . . 1 . . . . 1 h . . . l . . . . 0 5O 1 00 1 50 200 250 Qt” (03") Figure 5.9: The integrated cross section for photon pair production at the LHC. The resummed and the 0(as) distributions are shown in solid and dashed lines, respectively. In Fig. 5.9 we give the integrated cross section as the function of the transverse momentum of the photon pair produced at the LHC. Similarly to the Z0 pair pro- duction, there is a significant shape difference between the resummed and the NLO curves in the low to mid QT region. For Q?" = 50 GeV, the resummed rate is about 1.5 times of the NLO rate. 5.3.4 Diphoton production at the upgraded Tevatron In Ref. [132], we have presented the predictions of the CSS resummation formalism for the diphoton production at the Tevatron with \/§ 2 1.8 TeV, and compared with the data [134, 135]. In this Section, we show the results for the upgraded Tevatron with \/§ 2 2.0 TeV. We use the same kinematic cuts which were used in Ref. [132]: p} > 12 GeV, for the transverse momentum of each photons, Iy7| < 0.9, for the rapidity of each photons. 153 da/dQ," (pb/GeV) . . . I '- 41—er 1.41L1L1I~u111 . . . . . . .“ l 0 25 50 75 100 1 25 1 50 175 200 0," (GeV) Figure 5.10: Transverse momentum distribution of photon pairs at the LHC. The total resummed contribution (upper solid), the resummed qc'j + qg -—) 77X (dashed), qrj —+ 77X (dotted), 99 —-> 77X (dash-dotted), as well as the fragmentation (lower solid) contributions are shown separately. An isolation cut of AR = 0.7 is also applied. The total cross sections and the rates of the different subprocesses are given by Tables 5.1 and 5.3. The ratio of the qq' + (19 —> 77X and mi -—> 77 rates is about 1.5, similar to that at the LHC. The leading order rate for the 99 —> 77 subprocess is about 6.0 pb and 4.3 pb for using CTEQAL and CTEQ4M PDF, respectively. The NLO rate for gg —+ 779 is estimated to be 8.3 pb, using the approximation described in Ref. [132], which is about the same magnitude as the leading order qq —> 77 rate. From our estimate of the NLO 99 rate, we expect that the complete C(03) contribution will be important for photon pair production at the Tevatron. Figs. 5.11—5.13 show our results for photon pairs produced at the upgraded Teva- tron. The resummed predictions for the invariant mass and rapidity distributions of the photon pairs, and the transverse momentum distribution of the individual photons do/dQ"(pb/Gev) 154 I l Yfi_T l l 3 2°r— """ T"I I l F1'l"'l" I a . ‘ _ 4 V ; Suits; '0 u'. ‘. \ :, - b r ‘ U 15 F- . [1’ » ‘4 r ‘- 12.5 r;- -1 10 :- 10 - i 15% 5 5 C, -2 I b 10 t I 2.5 i: > E I L o L 4 -- O 20 -1 -O.8 —O.6 ~O.4 -0.2 O 0.2 0.4 0.6 0.8 Figure 5.11: y" O vrTfivl v'viw'v'vv V'V vvl da/dp((pb/GeV) 9" (GeV) Same as Fig. 5.8 but for the upgraded Tevatron. . . LALAJAAL do/dQ" (pb/GeV) .0 . a .O o a .0 . 0.... . u ., 1o lllll‘lllllllLLJlllllJ_L lllll I‘L43LJ_AAIT..MAL1 1 0 20 30 40 50 50 7O 80 90 07" (GeV) Figure 5.12: Same as Fig. 5.10 but for the upgraded Tevatron. are shown in Fig. 5.11. In Fig. 5.12 we also plot the contributions to the transverse momentum of the photon pair from the q6 + qg —¥ 77X (dashed), qc‘j -+ 77X (dot- ted), gg —> 779 (dash-dotted), and the fragmentation (lower solid) subprocesses, separately. The leading order ch —-> 77 cross section (middle solid) is also plotted. In the low QT region, the 99 and the qq rates are about the same, and the qg rate becomes more important in the large QT region. Furthermore, after imposing the above kinematic cuts, the fragmentation contribution is found to be unimportant. Fig. 5.13 shows the integrated QT distribution. The qualitative features of these distributions are the same as those predicted for the LHC. For Q?“ = 10 GeV, the resummed cross section is about twice of the NLO cross section. 5.4 Conclusions In this Section we studied the effects of the initial—state multiple soft-gluon emis- sion on the total rates and various distributions of the most important background 156 dO/dQT ‘0 Figure 5.13: Same as Fig. 5.9 but for the upgraded Tevatron. processes (pp, p}? —> 77X, ZOZOX) to the detection of the Higgs boson at the LHC. We applied the extended CSS formalism to resum the large logarithms induced by the soft—gluon radiation. We found that for the qrj and qg initiated processes, the total cross sections and the invariant mass distributions of the photon and Z 0 boson pairs are in agreement with the fixed order calculations. From our estimate of the N LO rate of the gg initiated process, we expect that the complete C(03) contribution will be important for photon pair production at the Tevatron. We showed that the resummed and the NLO transverse momentum distributions of the Z0 and photon pairs are substantially different for QT g Q / 2. In terms of the integrated cross sec- tion above a given QW", this difference can be as large as 50% in the low to mid-range of Q?“. Using the resummation calculation, we are able to give a reliable prediction of the QT and any other distribution in the full kinematical region at the LHC and the Tevatron, even in the presence of kinematic cuts. Since the bulk of the signal is in the low transverse momentum region, we conclude that the difference between 157 the NLO and resummed predictions of the background rates will be essential when extracting the signal of the Higgs boson at hadron colliders. Chapter 6 Charged Scalar Production at Hadron Colliders 6. 1 Introduction The top quark (t), among the three generations of fermions, is the only one with a large mass as high as the electroweak scale. This makes the top the most likely place to discover new physics beyond the Standard Model (SM). In a recent study [142], it was proposed that, due to the top-mass enhanced flavor mixing Yukawa coupling of the charm (c) and bottom (b) with a charged scalar or pseudo-scalar (96*), the s-channel partonic process chéb —> oi, can be an important mechanism for the production of ¢i at various colliders. From the leading order (LO) calculation [142], it was demonstrated that the Fermilab Tevatron Run-II has the potential to explore the mass range of the charged top-pious up to about 300—350 GeV in the topcolor (TopC) models [143, 144]. In this Chapter, we compute the complete next-to-leading order (NLO) QCD corrections to the process qq’ —> 65*, which includes the one-loop virtual corrections and the contributions from the additional 0(as) processes, qti’ —> ¢ig and qg —> q’afii- (6-1) 158 159 The decay width and branching ratio (BR) of such a (pseudo-)scalar are also included up to NLO to estimate the event rates. The QCD resummation of multiple soft-gluon radiation is also carried out, which provides a better prediction of the transverse momentum distribution of the (pseudo-)scalar particle. We shall choose the TopC model [143] as a benchmark of our analysis. The generalization to the generic type- III two-Higgs doublet model (2HDM) [145, 146] is straightforward since the QCD- corrections are universal.1 The direct extension to the production of neutral (pseudo- )scalars via b6 fusion is studied in the Minimal Supersymmetric SM (MSSM) [147, 12] with large tanfi and in the TopC models with U (1)-tilted large bottom Yukawa coupling [143, 144]. 6.2 Charged (Pseudo-)Scalar Production via Charm- Bottom Fusion 6.2.1 Fixed-Order Calculation We study charged (pseudo-)scalar production via the top-mass-enhanced flavor mix- ing vertex c-b-(,/)i [142]. The corresponding Yukawa coupling can be generally defined as CLE+CRR in which E = (1—75)/2 and R = (1+75)/2. The total cross sections for the (15+ production at hadron colliders (cf. Fig 1) can be generally expressed as o (6th -—> ¢+X) = Z [1 dml f1; (1:132 [fa/Incl”,Q2)f5/h2($21Q2) + (a H 18)] 603(afi __f ¢+X)a 0,5 To Ii (6.2) where To = mg/S, 231,2 =,/To a”, 772,» is the mass of 66*, \/§ is the center-of-mass energy of the ’1th collider, and fa/h(:c, Q2) is the parton distribution function (PDF) 1We note that the finite part of the counter term to the q-(j’-d>°'i Yukawa vertex is renormalization scheme- and model-dependent. 160 of a parton a with the factorization scale Q. The quantity 605 is the partonic cross section and has the following LO contribution for 66 —> 45+ (cf. Fig. la) [142]: 71' 635; = 60,65,360 — ?)80 , 60 a 12 (ICLI2 + ICRIQ) , (63) C41) where f = mi/S‘ with ’s‘ the center-of—mass energy of the sub-process, and the terms suppressed by the small mass ratio (may/m,”2 have been ignored. Since we are interested in the inclusive production of the scalar (15, it is natural to choose the factorization scale Q to be its mass m¢, which is of 0(102‘3) GeV and much larger than the mass of charm or bottom quark. Hence, in this work, we will treat 6 and b as massless partons inside proton or antiproton and perform a NLO QCD calculation with consistent sets of PDFs [148, 149, 150]. The NLO contributions are of 0(a3), which contain three parts: (i) the one- 1oop Yukawa vertex and quark self-energy corrections (of. Fig. 1b-d); (ii) the real gluon emission in the qi-annihilations (cf. Fig. 16); (iii) 3- and t-channel gluon- quark fusions (cf. Fig. lf-g). The Feynman diagrams coming from permutations are not shown in Fig. 1. Unlike the usual Drell-Yan type of processes (where the sum of the one-loop quark-wavefunction renormalization and vertex correction gives the ultraviolet finite result), we need to include the renormalization for the Yukawa coupling (3),) which usually relates to the relevant quark mass (mqj), i.e., we have to add the counter term at the NLO (cf. Fig.1d) besides the contribution from the usual wavefunction renormalization ZWW = é-(Zql + Zqz) (cf. Fig.lc). This applies to the Yukawa interactions of the SM and MSSM Higgs bosons as well as the top-pious in the TopC models. It is clear that, for flavor-mixing vertex c-b-qf):t in the TopC model [cf. Eq. (6.10) below], the counter-term of the Yukawa coupling is equal to the top quark mass counter-term 6mt/mt, which we determine from the top—quark mass renormalization in the on-shell scheme so that mt is the pole mass of 161 (a) (b) (c) (d) (6) (I ) (9) Figure 6.1: Representative diagrams for charged or neutral (pseudo-)scalar (dashed line) production from quark-antiquark and quark-gluon collisons at 0(ag) and 0(039): (a) leading order contribution; (b-d) self-energy and vertex corrections (with counter term); (e) real gluon radiation in qq"-fusion; (f—g) s- and t-channel gluon-quark fusions. the top quark. In other cases such as in the general 2HDM (type-III) [146] and the TopC models (with b—Higgs or b—pions) [151], some of their Yukawa couplings are not related to quark masses or not of the above simple one-to-one correspondence, and thus have their independent counter terms (63},- / yj). In addition to the virtual QCD- loop corrections, the contributions of the real gluon emission from the initial state quarks have to be included (cf. Fig. 1e). The soft and collinear singularities appeared in these diagrams are regularized by the dimensional regularization prescription at D = 4 - 26 dimensions. After summing up the contributions of virtual gluon-loop and real gluon-radiation (cf. Fig. lb-e), the ultraviolet and soft singularities separately cancel. The collinear singularities are still left over and should be absorbed into the renormalization of the PDF [58]. (The MS renormalization scheme is used in our calculation.) Finally, the gluon-quark fusion sub-processes (cf. Fig. 1f—g) should also be taken into account and computed at general dimension-D. All these results are separately summarized in Section 6.6. The hadron cross sections become regular after renormalizing the Yukawa coupling and the PDFs in (6.2), which are functions of the renormalization scale )1 and the 162 factorization scale pp(= VQQ). The partonic NLO cross section 6§€0 (015 -—> ¢+X) contains the contributions A3q¢(ch’ -> (12+, (15+ 9), A6q9(qg —) ¢+q’), and Aaqgfig _) ¢+<7’)= 1—2 1 7 Am = CF [4 (1 +?2) (#1) — 21+: 1n? + 2 2 + (31-2-9 6(1—?)+2(1-?)[ +2P§2q(?)lng%’, (64) _ A (1 r)? m2 1 A A Aocgygg — Pq(}_)g( ) [ln ln Q—g’ — 1(1 — T) (3 — 77'), A 1+?2 A 1 A A P§2q(7)=CF(1_?) , Pq((1—)g(7-)=§[T2+(1_T)2]a + where i" = mg/Ks‘ and CF = 4/3. The mass counter term for the Yukawa vertex renormalization is determined in the on-shell scheme, i.e., m=—CFQS [3(é-7E+1n47r)+§2[, (6.5) mt 47f in the TopC model. Here, the bare mass mm and the renormalized mass m, are related by mm = mt + 6m; and mt 2 175 GeV is taken to be the top-quark pole mass. The finite part of the mass counter term is Q = 3 In [112 / mf] + 4 in the TopC model, where Q 2 O for p 2 mte‘2/3 2 90 GeV. In the following, we shall choose the QCD factorization scale up (set as the invariant mass \/Q7) and the renormalization scale )1 to be the same as the scalar mass, i.e., \/Q7 = 11 = m¢, which means that in (6.4) the factor In (mg/Q2) vanishes and the quantity Q becomes 9 = 3ln [mi/mg] + 4 . (6.6) For the case of 772,» >> mt, the logarithmic term In (mg/m?) becomes larger for 771,» >> 7m, and its contributions to all orders in as In (mg/m?) may be resummed by 163 introducing the running Yukawa coupling yt(u), or correspondingly, the running mass mt(]u). In the above formula, mt is the pole mass (m?01 2 175 GeV) and is related to the one-loop running mass via the relation [152] —1 0 3C 2 0 O C O mt(p) 2: mt(mf 1) 1 — Judy.) 1n # mt(m[’ 1) = mf l [1 + £04m? 1) pol ’ 471’ mt (6.7) Using the renormalization group equation, one can resum the leading logarithms to all orders in as [153] and obtains map) = m.(m:’°‘) [%] M” , (6.8) with n f = 6 for p > mt . Thus, to include the running effect of the Yukawa coupling, we can replace the (m§’°l)2-factor (from the Yukawa coupling) inside the square of the S-matrix element [up to O(a,)] by the running factor mm.) {1 + 2% [1+ iln (#7)?” = mf(p) [1 + We], (6.9) where the logarithmic term in the bracket [ - ] is added to avoid double-counting with the resummed logarithms inside mf(u). It is clear that this [1 + (CF01, (p) /21r) Q] factor will cancel the Q-term inside the NLO hard cross section A66; in Eq. (6.4) at 0(03), so that the net efl'ect of the Yukawa vertex renormalization (after the resummation of leading logarithms) is to replace the relevant tree-level on-shell quark mass (related to the Yukawa coupling) by its MS running mass [cf. Eq. (6.8)] and remove the Q—term in Eq. (6.4). When the physical scale 11 (chosen as the scalar mass m¢) is not much larger than mt, the above running effect is small since the ln(p/mt) factor in the Yukawa counter-term Jmt/mt is small. However, the case for the neutral scalar production via the b5 annihilation can be different. When the loop correction to the (150-65 Yukawa coupling contains the logarithm ln(p/mb), which is much larger 164 than ln(p/mt), these large logarithms should be resummed into the running coupling, as we will do in Section 4. In the TopC model, there are three pseudo-scalars, called top-pions, which are predicted to be light, with a mass around of 0(100 ~ 300) GeV. The relevant Yukawa interactions for top-pious, including the large tR-cR flavor-mixing, can be written as2 [142] _mttanfl[ 67;; = iKgRKggEtRng’ + fiKfiéKfibLfiij+ (6.10) inchKggKECR’R'? + fiKggKgLEEbLW? '1' 11C] , where tan 6 = W ~O(4—1.3) with the top—pion decay constant vt~0(60— 150) GeV, and the full vacuum expectation value (VEV) v 2 246 GeV (determined by the Fermi constant). The analysis from top-quark decay in the Tevatron tf events sets a direct lower bound on the charged top-pion mass to be larger than about 150 GeV [154, 143]. The existing low energy LEP/SLD measurement of Rb, which slightly lies above the SM value by about 0.90 [155], also provides an indirect constraint on the top-pion Yukawa coupling 6‘” = (fimt/v) tan 6 due to the one—loop contribution of charged top-pions to Rb. However, given the crude approximation in estimating the top-pion loops (with all higher loop orders ignored) and the existence of many other sources of contributions associated with the strong dynamics, the indirect Rb constraint is not conclusive [143]. For instance, it was shown that the 30 Rb bound from the one-loop top-pion correction can be fully removed if the top-pion decay constant 11, is increased by a factor of 2 (which can be the typical uncertainty of the Pagels—Stokar estimate) [143, 156]; also, the non-perturbative contributions of coloron- 2As pointed out in Ref. [142], an important feature deduced from (6.10) is that the charged top- pion 1r? mainly couples to the right-handed top (t3) or charm (CR) but not the left-handed top (t L) or charm (CI), in contrast to the standard W-t-b coupling which involves only n. This makes the top-polarization measurement very useful for further discriminating the signal from the background events. 165 exchanges can shift the Rb above its SM value [143] and tend to cancel the negative top-pion corrections. Due to these reasons, it is clear that the inconlusive Rb-bound in TopC models should not be taken seriously. Nevertheless, to be on the safe side, we will uniformally impose the roughly estimated Rb-constraint in our current analysis of the TopC model, by including only the (negative) one-loop top-pion contribution as in Ref. [156].3 As shown in Fig. 6.2a, the current 30 Rb-bound requires a smaller tOp—pion Yukawa coupling, Cfi’ ~ 1.3 — 2 (or, tan 6 ~ 1.3 - 2), for the low mass region of mm: ~200—500 GeV. Since the top-pion decay constant vt is related to tan 6, this also results in a constraint of vt to be around 150~100 GeV when m”? ~ 200 — 500 GeV (cf. Fig. 6.2b). The usual Pagels-Stokar estimate of 12¢ (by keeping only the leading logarithm but not constant terms) gives v¢~64-—97 GeV for the topC breaking scale A ~ 1 — 10 TeV, where a typical factor of 2 ~3 uncertainty in the calculation of v? is expected [157, 143]. This estimate is slightly lower than the Rb-constrained vt-values in Fig. 6.2b, but is still in reasonable consistency (given the typical factor of x/2~ \/3 error in the leading logarithmic Pagels-Stokar estimate of 1),). In (6.10), K UL, R and K DL, R are defined from diagonalizing the up- and down-type quark mass matrices Mg and MD: KLLMUKUR = 3“, KLLMDKDR = M3“ , with M8“: diag(mu,mc,mt) and Mg”: diag(md,m,,mb). For the class-I TopC models [151], we have constructed [142] a realistic and attractive pattern of KUL and K DI, so that the well-constrained Cabibbo-Kobayashi-Maskawa (CKM) matrix V (= K f1 LK 0;) can be reproduced in the Wolfenstein-parametrization [158] and all potentially large contributions to the low energy data (such as the K J? , D-D and B—B mixings and the b —+ 37 rate) can be avoided [142]. We then found that the 3However, it is important to keep in mind that such a rough Rb-bound is likely to over-constrain the top-pion Yukawa coupling since only the negative one-100p tOp-pion correction (but nothing else) is included in this estimate. A weaker Rb-bound will less reduce the top-pion Yukawa coupling and thus allow larger production rates of charged top-pious at colliders which can be obtained from our current analysis by simple re-scaling. 166 right-handed rotation matrix KUR is constrained such that its 33 and 32 elements take the values as [142] K3,, 2 0.99—0.94, KER S 1/ 1 — KgR2 2 0.11—0.33, (6.11) which show that the tR-cR flavor mixing can be naturally around 10 — 30%. For the current numerical analysis we make the benchmark choice [142] based upon the above TopC model and consider C]? = C§§(Rb constrained), CE: = Cfi’KffR 2 C]: x 0.2, C]? = CZ” = 0. (6.12) It is trivial to scale the numerical results presented in this Chapter to any other values of Cu; when needed. Unless specified otherwise, we use CTEQ4M PDF [38] to calculate the rates. Note that CTEQ4M PDFs are consistent with the scheme used in the current study which treats the initial state quarks as massless partons in computing the Wilson coeflicient functions. The only effect of the heavy quark mass is to determine at which scale Q this heavy quark parton becomes active.4 In our case, the scale Q = 7m, > ma, mb. In Fig. 6.3, we present the total cross sections for the charged top-pion production as functions of its mass, at the Tevatron (a [915 collider at 1.8 and 2 TeV) and the LHC (a pp collider at 14 TeV). We compare the improvements by including the complete NLO results [cf. (6.4)] and by including the resummed running Yukawa coupling or running mass [cf. (6.8)]. For this purpose, we first plot the LO total cross sections with the tree-level Yukawa coupling [dash—dotted curves, cf. (6.3) and (6.12)] and with the resummed running Yukawa coupling or running mass [dotted curves, cf. (6.3) and (6.8)]; then we plot the NLO cross sections with the one-loop Yukawa coupling [dashed curves, cf. (6.4)] and with the resummed running Yukawa coupling 4This is the Collins-Wilczek-Zee (CWZ) scheme [159]. 167 or running mass [solid curves, of. (6.4), (6.8) and (6.9)] . We see that at the LHC there is a visible difference between the pure LO results with tree-level Yukawa coupling (dash-dotted curves) and other N LC and/ or running-coupling improved results. But at the Tevatron, the LO results with running Yukawa coupling (dotted curves) are visibly smaller than the results in all other cases for m¢ > 300 GeV. This shows that without the complete NLO calculation, including only the running Yukawa coupling in a LO result may not always warrant a better improvement. Finally, the comparison in Fig. 6.3 shows that the resummed running Yukawa coupling or top-mass [cf. Eq. (6.8)] does not generate any significant improvement from the one-loop running. This is because the top-mass is large and as ln (mg/m?) is small for m¢ up to lTeV. Thus, the improvement of the resummation in (6.8) has to come from higher order effects of as In (mg/m3). However, as to be shown in Section 6.5 the situation for summing over powers of as In (mg/mg) is different due to mb << mt, mas. Fig. 6.4 is to examine the individual NLO contributions to the charged top- pion production via the qrj’ and qg sub-processes, in comparison with the full NLO contributions.5 The L0 contributions are also shown as a reference.6 [Here q de- notes the heavy charm or bottom quark] In this figure, there are three sets of curves for the charged tOp-pion production cross sections: the highest set is for the LHC (VS 2 14 TeV), the middle set is for the upgraded Tevatron (\/§ 2 2TeV), and the lowest set is for the Tevatron Run I («S = 1.8 TeV). The L0 cross sections are plotted as dotted lines while the NLO cross sections as solid ones. The dashed lines show the contributions from the ch’-fusion sub-processes, and the dash-dotted lines describe the contributions from the qg-fusion sub-processes. The qg-fusion cross sec- tions are negative and are plotted by multiplying a factor of —1, for convenience. For 5Unless specified, qg includes both qg and q‘g contributions. 6With the exception of Figs. 6.3, 6.8, and 6.12, we only show our numerical results with the resummed running Yukawa coupling or running mass. 168 a quantitative comparison of the individual NLO contributions versus the full NLO results, we further plot, in Fig. 6.5, the ratios (called K -factors) of the different NLO contributions to the LO cross section by using the same set of CTEQ4M PDFs. The solid lines of Fig. 6.5 show that the overall NLO corrections to the pp,pp —+ ¢iX processes are positive for 1nd, above ~150 (200) GeV and lie below ~15 (10)% for the Tevatron (LHC) in the relevant mass region. This is in contrast with the NLO cor- rections to the W:t boson production at hadron colliders, which are always positive and as large as about 25% at the Tevatron [137]. The reason of this difference origi- nates from the differences in the Aaqq-I and Aoqgm for gift and Wi production. While in the case of Wi production the positive Aoqq: piece dominates, in the case of (pi production the size of negative Aaqmq piece becomes comparable with that of the positive Aoqq: such that a non-trivial cancellation occurs. While it is reasonable to take the renormalization and the factorization scales to be m4, for predicting the inclusive production rate of ¢+, it is desirable to estimate the uncertainty in the rates due to different choices of PDFs. For that purpose, we examine a few typical sets of PDFs from CTEQ4, which predict different shapes of charm, bottom and gluon distributions. As shown in Table 6.1 and Fig. 6.6, the uncertainties due to the choice of PDF set are generally within 220% for the relevant scalar mass ranges at both the Tevatron and the LHC. 6.3 Soft-Gluon Resummation The a, corrections to the (pseudo—)scalar production involve the contributions from the emission of virtual and real gluons, as shown in Figs. 1(b), (0) and (e). As the result of the real gluon radiation, the (pseudo-)scalar particle will acquire a non- vanishing transverse momentum (QT). When the emitted gluons are soft, they gener- 169 ate large logarithmic contributions of the form (in the lowest order): asln'" (Q2/Q§~) /Q§~, where Q is the invariant mass of the (pseudo-)scalar, and m = 0,1. These large logarithms spoil the convergence of the perturbative series, and falsify the 0(as) prediction of the transverse momentum when QT << Q. To predict the transverse momentum distribution of the produced (pseudo-)scalar, we utilize the Collins-Soper—Sterman (CSS) formalism [44, 45, 46], resumming the logarithms of the type a: ln'" (Q2/Q§~) /Q%, to all orders n in a, (m = 0, ..., 2n — 1). The resummation calculation is performed along the same line as for vector boson production in Chapter 3. Here we only give the differences from that given in Chapter 3. But for convenience, we also list the A“), A”), and B“) coefficients of the Sudakov exponent, which have been used in the current analysis: A<1>(Cl) = CF, BWC. =bo.C2 =1) = -30., 2 (6.13) 2 _ _ 67 7r 5 A()(Cl—b0)—Cp[(—36——12) C— [ 18"f ’ where CF 2 4/ 3 is the Casimir of the fundamental representation of SU(3), NC = 3 is the number of SU(3) colors, and n, is the number of light quark flavors with masses less than Q. In the above we used the canonical values of the renormalization constants Cl 2 b0, and 02 = 1. To recover the O (as) total cross section, we also include the Wilson coefficients C [1) among which CS) differs from the vector boson production (here i denotes quark 80’ or antiquark flavors, and a = q,- or gluon g). Explicitly, C§2)(z,b, 14, 01/02) = 6316(1 — z), 05342:, b,u,C1/Cz) = o, 1 1 ,ub C§;)(Z,b, [1,, 01/02) = (SjkCF {5(1 — Z) — a—Fln (3;) [DJ-(2,42) 170 +6(1—z) [—ln2( Cl 6_3/4) +B+—9—]}, boC'2 4 16 1 b (:39, b, M, 01/02) = 52(1 — z) — In (a?) 1930(2), (6.14) where 1),-(29 is the 0(as) gluon splitting kernels [42] given in Section 1.2.3. In the above expressions, V = pr = —8 + #2 for the vector boson production [137], and V 2 V4, = 7r2 for the (pseudo-)scalar production, when using the running mass given in Eq. (6.8) for the Yukawa coupling. Using the canonical values of the renormalization constants, ln(,ub/b0) vanishes, because [u = Cl / b = b0 / b. The only remaining difference between the resummed formulae of the vector boson and (pseudo-)scalar production is in the regular (Y) terms, which comes from the difference of the 0 (as) real emission amplitude squares (of, the definitions of 7:51 and 7;;1 in Section 2.1.3 and Eqs. (6.22) and (6.25) of this Chapter). The non- perturbative sector of the CSS resummation (the non-perturbative function and the related parameters) is assumed to be the same as that in Ref. [137]. As described in Ref. [137], the resummed total rate is the same as the 0(a3) rate, when we include 0,9,) and Y“), and switch from the resummed distribution to the fixed order one at QT = Q. When calculating the total rate, we have applied this matching prescription. In the case of the (pseudo-)scalar production, the matching takes place at high QT ~ Q values, and the above matching prescription is irrelevant when calculating the total rate because the cross sections there are negligible. Thus, as expected, the resummed total rate differs from the 0(a3) rate only by a few percent. Since the difference of the resummed and fixed order rate indicates the size of the higher order corrections, we conclude that for inclusive (pseudo—)scalar production the 0(a3) corrections are likely much smaller than the uncertainty from the parton distribution functions (cf. Fig.6.5). 171 In Fig. 6.7, we present the numerical results for the transverse momentum dis- tributions of the charged top-pions (in TopC model) and the charged Higgs bosons (in 2HDM) produced at the upgraded Tevatron and the LHC. The solid curves show the resummation prediction for the typical values of m¢. The dashed curves, from the 0(as) prediction, are irregular as QT —> 0. The large difference of the transverse momentum distributions between the results from the resummation and fixed-order analyses throughout a wide range of QT shows the importance of using the resumma- tion prediction when extracting the top-pion and Higgs boson signals. We also note that the average value of QT varies slowly as m¢ increases and it ranges from 35 to 51 GeV for m¢ between 250 and 550 GeV at the 14 TeV LHC, and from 23 to 45 GeV for md, between 200 and 300 GeV at the 2 TeV Tevatron. 172 ‘ V Y ‘ V Y Y Y ")- T Y Y I V Y r 1 2w ' a 3 b 3 3.5 F TopColor ( )~3( ) TopColor 1 3 ._ ’ ' 2* 1 i ." I ”allowed [50" 2.5 L Rf—30/ I 6 9: t x‘ 4» 8 U 2 E ,/ R.”—30€: \ 5' 15 F f 3:” ‘ ~. q ‘00 . 1. _ an“ . U allowed it R:- 3 ~. . ‘ . 1 “I ‘-1 :1 ~ : (c) 2HDM m 0.21 A l A A J l 4 4 L l J A A l 4 L A 200 400 600 800 woo Mass of Charged Higgs (GeV) Figure 6.2: Estimated current 3a-bounds in the TopC model and 2HDM-III: (a) the 30 upper bound on the top-pion Yukawa coupling Cfi’; (b) the 30 lower bound on the top-pion decay constant; (Here, in (a) and (b), the solid curves are derived from the combined LEP/SLD data of R?“ = 0.21656 :1: 0.00074 while dashed curves are from the same 30 combined experimental error but with the central Rb-value equal to REM = 0.2158.) (c) the Rb-predictions of 2HDM-III with coupling {a = 1.0 and 1.5 (solid curves) and the 30 Rb-bounds (dashed lines). 173 Collider Upgraded Tevatron (2 TeV) LHC (14 TeV) Process \ PDF 4A1 4M 4A5 4HJ 4A1 4M 4A5 4HJ 367 382 376 387 5380 5800 6060 5890 L0 42.6 43.7 41.5 46.6 863 901 896 906 6.88 7.05 6.56 8.38 235 240 323 241 370 402 412 407 5430 6080 6510 6170 NLO 45.6 48.6 47.9 51.6 912 976 997 981 7.70 8.21 7.89 9.56 255 266 264 268 551 584 585 590 7530 8290 8740 8400 qq —> ¢+X 64.5 67.4 65.5 71.7 1210 1280 1290 1290 10.6 11.1 10.5 13.0 331 341 335 343 -180 —181 —174 -183 —2100 —2200 —2240 -2230 qg——>¢+X —19.2 —18.9 —17.5 —19.9 —299 —302 —293 -303 —2.94 —2.86 —2.59 —3.34 —76.0 —74.7 —70.6 —75.0 Table 6.1: Cross sections in fb for charged top-pion production in the TopC model at the upgraded Tevatron and the LHC are shown, by using four different CTEQ4 PDFs. They are separately given for the L0 and NLO processes, and for the ch —+ ¢+X and qg —) ¢+X sub-processes. At the upgraded Tevatron the top number is for m¢ = 200 GeV, the middle is for m4, = 300 GeV, and the lowest is for m4, = 400 GeV. At the LHC the top number is for m¢ = 400 GeV, the middle is for m¢ = 700 GeV, and the lowest is for me, = 1 TeV. 174 Io ""'."'Yr""r YVYTYYV IYYIVIVVY‘VT—YT‘IYTYYfi 4 3 3 Top-pion Production 1': . . 102: (CTEQ4M) 4 i . 1o . 99"” X 4, . u 0 o ‘- vs-urev Tevatron ”x 4 \-. 1 AAA._A_ ALLAIA.‘+A_ 122. .221...’“. ....1.2 1..2.1 10100 200 300 400 500 600 700 800 900 1000 m, (GeV) Figure 6.3: Top-pion production cross sections at the present Tevatron, upgraded Tevatron, and the LHC. For each collider we show the NLO cross section with the resummed running Yukawa coupling (solid), and with one-loop Yukawa coupling (dashed), as well as the LO cross section with resummed running Yukawa coupling (dotted) and with tree-level (dash-dotted) Yukawa coupling. The cross sections at x/S = 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the \/§ 2 2 TeV curves. 6.4 Hadronic Decays of Charged (Pseudo-)Scalars to 0(as) In the TopC models, the current Tevatron data from the top quark decay into charged top-pion (77?) and b-quark already requires the mass of 7r,i to be above ~150 GeV [143, 154]. In the current analysis, we shall consider mm > mt + mb, so that its dominant decay channels are 775‘ —+ tb, cb. The decay width of 77?: (= (25*), including the 0(as) QCD corrections, is given by [160,161] 175 3 joJ'IY—v—rfTrfiIvarIvvrrlrvr I—v—rrvlvvvulvva'vv v 3 ._ . Top-pion Production o < 10 ‘ (CTE04M) _, " " LHC = ------- :t 10 ‘ ........ pp —> 0 X 1 .......... vs - 14 TeV 1 ......................... 10.1 . 'xTevotron L - i , g: 10 2 1;... pp—> 9 x 4 ‘ ' vs - 2 TeV ‘ 10" ......... 1:'~.;"“1.....'-1..'1.1....1....1....1.... 1002003004005006007008009001000 m,(GeV) Figure 6.4: Cross sections for the charged top-pion production in the TopC model at the present Tevatron, upgraded Tevatron and the LHC. The NLO (solid), the qrj’ (dashed) and qg (dash-dotted) sub-contributions, and the LO (dotted) contributions are shown. Since the qg cross sections are negative, they are multiplied by —1 in the plot. The cross sections at x/S = 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the \/S = 2 TeV curves. _ 0150p I391.0(9)) — I‘1.0(Q) [1+ 277 me) = ,;;—,9 (19.12 + 16512) (1 — 02, 9 72], 2 (6.15) + [3111 “—2 + 4 — Q] _ 2 2 7" 72— (1—7‘) +(1-—r)(3-—7r+2r)ln1 mt _2(1-r)2[.1%-—T_T) )—ln(1—T)ln1:1'], r _ 2M2 (1 — r in which Q = JQ’ is the invariant mass of 41*. The small bottom and charm masses are ignored so that r E (mt/m¢)2 for tb final state and r = O for cb final state. Thus, for 05* —~> oh, the quantity R reduces to ’R. = 17/2 - Q. In Fig. 6.8, we present the results for total decay widths of 06+ and branching ratios of W" —) t5 in the TopC model and the 2HDM. For the 2HDM, we also show the branching ratios of the W+h0 channel, which is complementary to the t5 channel. The NLO (solid) and 176 3 2 .'*"“""""..*"""""""". 2 ‘5 I Tevatron (o) I l LHC (b) 1 LE "3? pb—n'x if pp-N'x 1 ‘3 X I VS-ZTeV ll VS-HTeV 1.6 - 4— ~ 1.6 p .....-........:: 4 ‘P ............ .0- a 1 - d 1.4 — "/ “’ “L“ 1.4 : ]: 006/010 ] . d» 4 1.2 — 3- — 1.2 . 4r 1 s “100/010 17 0100/0110 ~ 1 0.8 l 3'— 4 0.0 0.6 l «”- 4 0.6 »- ......... -Ao../o.. - 0.4 ~ ................... 4 3“ ....... Two/0m - 0.4 4L 1 24 l l ..Lx. l _LLLLAJ 100 200 300 200 400 000 800 1000 m.(GeV) Figure 6.5: The K -factors for the 43+ production in the TopC model are shown for the NLO (K = UNLQ/O’Lo, solid lines), qq’ (K = qur/O’Lo = (0L0 + Aaq¢)/0Lo, dashed lines), and qg (K = -A0qg/0Lo, dash-dotted lines) contributions, at the upgraded Tevatron (a) and the LHC (b). LO (dashed) curves difl'er only by a small amount. In the same figure, the K ~factor, defined as the ratio of the NLO to the LO partial decay widths, is plotted for the 03+ -) t5 (solid) and —> cf) (dashed) channels. Here, the sample results for the 2HDM are derived for the parameter choice: a = 0 and (M1,, M A) = (100, 1200) GeV. With the decay width given above, we can study the invariant mass distribution of tb for the s-channel ¢+-production: , (9215/46) Br [¢+—> t6] , 7r [(QZ— mgr + (Q2P4/m¢)2] , do 2 615 [hlhg —+(¢+X)—> tBX] = a [h1h2-*¢+(Q)X] (6.16) where I}, and Br [¢+—+ t5] are the total decay width of (15+ and the branching ratio of (11+ —> t5, respectively, which are calculated up to the NLO. We note that the one-loop box diagrams with a virtual gluon connecting the initial state quark and final state quark (from the hadronic decay of 45) have vanishing contribution at 0(as) because 177 1,2 ......... ,4... ,;,..fifiqwqr- 1.2 g Tevatron 5 +1., LHC " .‘h {”5 " «sszrev ,frfl vs=14rev ““5 b . . . ‘1 . 1.1 - — ~ \ 1 1.1 . 07504140,: ] , \creoms , 1.05 — \-._ 1 r \x, 51.05 . 5..., ] .. 91504140 \ ] - ........... Ms cream ............ .2 I x.“ ‘5,“ 1 \4. crsow CTEQ4A5 “N4: ' ‘ 0.95 e _________ ~ 1 e -------- - 0.95 0.9 _ CTEQ4A1 1 F - 0'9 ' CTE04A1 0.85 — 1 — 4 0.85 (a) . (b) l .......11.... '.1-..1...132.1422 100 200 300 200 400 600 800 1000 m , (GeV) Figure 6.6: The ratios of NLO cross sections computed by four different sets of CTEQ4 PDFs relative to that by the CTEQ4M for charged top-pion production at the upgraded Tevatron (a) and the LHC (b). the scalar 06 is color-neutral. In Fig. 6.9a and Fig. 6.10a, we plot the invariant mass distribution for t-B and f-b pairs from (19* (top-pion signal) and Wi" (background) decays in the TopC model. In these plots, we have included the NLO contributions, as a function of Q, to the W3” background rate at the Tevatron and the LHC. The overall K -factor (after averaging over the invariant mass Q) including both the initial and final state radiations is about 1.4 (1.34) for the Tevatron (LHC) [162]. The total rate of Wi“ up to the NLO is about 0.70 [0.86] pb and 11.0 pb at the 1.8 [2] TeV Tevatron and the 14 TeV LHC, respectively. 178 0 10 20 30 4O 5O 50 7O 80 90 100 A -1 V‘I'V'I‘YVYYIYVYVIWY—TTIY"'I'Y‘IIVVV'IVfi—TTYVYY > 10 1 0 , , . 0 ., _ Top-pron productlon E 10. ''''' Tevatron VS - 2Tev 1 e - ............... , C510- .......................... 1:1 ......................... \ ............................. o - ........................... -o 10 ........................... ' ....1....1....1..L-1....1....1....1....1...-1. .. . , .. , ' , " 101 1 1 7 TT Tfjfil 1 '1"— Charqed H1990 production ] ‘ " LHcvs=14rev 1 Figure 6.7: Transverse momentum distributions of charged top-pious produced in hadronic collisions. The resummed (solid) and 0(as) (dashed) curves are calculated for md, = 200, 250, and 300 GeV at the upgraded Tevatron (a), and for 1114, = 250, 400, and 550AGeV at the LHC (b). 5 . 2 o : I 0 V 2 1.075 O 5‘ '° , : l; 9 I V 1.05 10 ‘ .3 . . 4 1.025 1 . 1 . 1 1 . J 1 l 1 - . 1“ ' Y 1 frv I v v v ‘V'Y’ 3 0.9 TopC o’—>tb 0.975 L a) 0'8 3 . . 0.7 .OéthC-i _ .1 0.95 0.6 ‘ ’ 1 0.5 2HDM , 0.925 0.4 2 I 0.3 "-“5 . ' 0.9 0'2 (b) : . . 1 1 1 1 1 A l . A 200 400 600 800 200 400 600 800 1000 Figure 6.8: Total decay widths of 03+ and BRs of 45+ —> t5 in the TopC model and 2HDM. (For the 2HDM, the BR of the W+h° channel is also shown, which is complementary to the t5 channel.) In Fig. (a) and (b), the NLO (solid) and LO (dashed) curves differ only by a small amount. In Fig. (c), the K -factor, which is defined as the ratio of the NLO to the LO partial decay widths, is shown for the 05+ -—> t5 (solid) and —-) cf) (dashed) channels. The sample results for the 2HDM in this figure are derived for the parameter choice (£5, £3) = (1.5, 1.5), a = 0, and (mh, mA) = (120, 1200) GeV. 179 ' YI’IYVVITY'I'irWrT—Y—TI—V—TYrfi‘l‘IVTI[TY O 5‘ ) TopColor model Tevatron VS == 2 TeV 1 _n O 'f llAAlln.AlAAAlAAAlAAAlAAAJAA+1AA YVTIYVVIYWVIVVVIIVYIVII“!IIVIVIIIY da/dQ (fb/GeV) Two Higgs doublet model « Tevatron VS - 2 TeV —: 10; . «=0 Two Higgs doublet model Tevatron VS 8 2 TeV a - 11/ 2 n L 4 AALAAL ’1.1...L...1...1...1..2L...1.22LL. . 180 200 220 240 250 280 300 320 340 350 Q(GeV) Figure 6.9: Invariant mass distribution of t—E and f—b pairs from (12* (signal) and W*‘ (background) decays at the Tevatron Run-II for the TopC model (a), and 2HDM with Higgs mixing angles a = 0 (b), and a = 77/2 (c). We show the signal for m4, = 200, 250, 300 and 350 GeV. The solid curves show the results from the NLO calculation, and the dashed ones from the LO analysis. 180 A103 .2..j ...,-2..,...4,.-.. 7’ (°) T CI 001 0 opoorm e , } 1° LHCVS=14TeV a .. 3 V o 10 ‘O E 11.1-1..11_l....l.1.1 U .,....,..fi,.4..,....,.... 102?“) Two Higgs doublet model : LHcvs-141ev 10 a=0 ‘1 lg L4 1 1 121% 1_ Two Higgs doublet model ‘° LHC vs - 14 TeV 1 a - 11/2 3 1o 1, A . l . A 1 . A 1 A 1 1 4 200 300 400 500 600 700 0 (GeV) Figure 6.10: Invariant mass distribution of t-B and {-0 pairs from 45* (signal) and W** (background) decays at the LHC for the TopC model (a), and for the 2HDM with the Higgs mixing angles a = 0 in (b), and a = 77/2 in (0). Here the charged pseudo-scalar or scalar mass are chosen as the typical values of m4, = 250, 400 and 550 GeV. The solid curves show the results by the NLO calculation, while the dashed ones come from the LO analysis. 181 175 200 225 250 275 300 325 350 375 400 9 T'r’T—Y‘Y‘V—rYTYrTf‘FT'IVVVVI Y I'V'rrVVrWITV—r 1 8 10 r W 11' production 1 E Tevatron VS = 2 TeV C ; 2HDM (a-0) O 'o 1 _ _ \ E E b : : 'U » . » 4 1o". . ...1.1.2L.L..1...11....1....1.... ' I I I ' ' I ' I 'f‘ 'fi T'firw ' I ' ' ' f 10 E W h' production 1 r 1 t LHcvs-urcv * 10 g 2HDM (6-0) .3 C l 1 .— (b) 10"1 1 . . 1 1 1 1 - 0 (GeV) Figure 6.11: Invariant mass distributions of W‘l-hO and W’-h0 pairs from (0* (.9- channel resonance) and Wi“ (s-channel non-resonance) decays at the Tevatron Run- II, and at the LHC, for the 2HDM with Higgs mixing angles a = 0. We show the signal for m4, = 200, 250 and 300 GeV at the Tevatron (a), and for m4, = 250, 400 and 550 GeV at the LHC (b). The solid curves show the results of the NLO calculation, and the dashed ones of the LO analysis. Before concluding this section, we discuss how to generalize the above results to the generic 2HDM (called type-III [146]), in which the two Higgs doublets (1)1 and (1)2 couple to both up- and down-type quarks and the ad hoc discrete symmetry [163] is not imposed. The flavor-mixing Yukawa couplings in this model can be conveniently ' formulated under a proper basis of Higgs doublets so that <1>= (O,v/\/2)T and <2>= (O, 0)T. Thus, the diagonalization of the fermion mass matrix also diagonalizes the Yukawa couplings of (1)1, and all the flavor-mixing effects are generated by Yukawa couplings (17,51 and 17,? ) of 2 which exhibit a natural hierarchy under the ansatz [145,146] 1712*” = €3’D\/m.-mj/<1> (6.17) with {3’0 ~ 0(1). This ansatz highly suppresses the flavor-mixings among light quarks and identifies the largest mixing coupling as the one from the t-c or c—t tran- 182 sition. A recent renormalization group analysis [164] shows that such a suppression persists at the high energy scales. The relevant Yukawa interactions involving the charged Higgs bosons H i are [142]: £60 = 11+ [2; (“gm bL — E (VT/D)“, b,; + 67,:(175V)d, bL — q (vi/0)., b,2 ] + he. 2 H+ [t-IE 17,? bi, + “CT; 17,?“ bL] + h.c. + (small terms), (6.18) where Ft? = {g x (fimt/v) 2 fig, and 17,? = ff: x (WM) 2 {g x 9% , in which 6% ~ 0(1) is allowed by the current low energy data [146, 165]. As a result, the Yukawa counter term in Fig. 1d involves both 6m, and 6mc. Consequently, we need to replace the NLO quantity 9 in the finite part of the Yukawa counter term [cf. the definition below (6.4)] by Q(2HDM) = 3111 [mg/(mtmcfl + 4 , (6.19) for the type—III 2HDM. In the relevant (bi-c-b coupling of this 2HDM, we note that, similar to the case of the TopC model, only the right-handed charm is involved [142], i.e., ct; = 62" = 0, cf: = {go/5mm), cg: 2 {g x 9%. (6.20) where the parameters (55, (g) are expected to be naturally around 0(1). We have ex- amined the possible constraint of {g from the current Rb data and found that the typ- ical values of {5 ~ 1.0—1.5 are allowed for mgi 2 200 GeV (cf. Fig. 6.2c).7 The pro- duction cross section of H i in this 2HDM can be obtained by rescaling the result of the TopC model according to the ratio of the coupling-square [Cf,§(2HDM)/C§§(TopC)]2 ~ [0.095,’fi/0.2€§$’(T0pC)]2 (which is about 1/7 for {g = 1.5 and the charged scalar mass around 400 GeV). 7Our calculation of Rb in the 2HDM-III is consistent with those in Ref. [166] and Ref. [165] (though a larger {g-value was chosen in the numerical analysis of Ref. [165]). 183 Finally, we note that there are three neutral Higgs bosons in the 2HDM, the CP- even scalars (ho, H 0) and the CP-odd pseudo-scalar A0. The mass diagonalization for h0 and H 0 induces the Higgs mixing angle a. The low energy constraints on this model require [146, 165] mh, my S mgi 3 mA or mA _<_ my: 3 mh, mg. For the case of my: > mho + MW, the H i —> Wih0 decay channel is also open. Taking, for example, a = 0 and (mh, mA) 2 (120,1200)GeV, we find from Fig. 6.8b that the tb and Wh0 decay modes are complementary at low and high mass regions of the charged Higgs boson H i. In Figs. 6ch and 6.10b—c, we plot the invariant mass distributions of t5 and {b pairs from H * (signal) and W3“ (background) decays in the 2HDM at the 2TeV Tevatron and the 14 TeV LHC, with the typical choice of the parameters: (55, 6g) 2 (1.5, 1.5) in Eq. (6.20), (mh, mA) 2 (120, 1200) GeV, and a = 0 or 7r/ 2. [A larger value of 53 will simultaneously increase (reduce) the BR of tb (Who) mode] We see that, due to a smaller c-b-Hi coupling [cf. (6.20)], it is hard to detect such a charged Higgs boson with mass my: > 250 GeV at the Tevatron Run-II. We then examine the potential of the LHC for the high mass range of H i. Similar plots are shown in Figs. 6ch for a = 0 and a = 7r/ 2, respectively. When cosa is large (e.g., a = 0), the branching ratio of the tb-channel decreases as my: increases (cf. Fig. 6.8b), so that the LHC does not significantly improve the probe of the large mHi, range via the single-top mode (cf. Fig. 6.10b). In this case, the WihO channel, however, becomes important for large mgi, as shown in Fig. 6.11 (cf. Fig. 6.8b, for its decay branching ratios) since the H i—WJF-ho coupling is proportional to cos a [146]. On the other hand, for the parameter space with small cosa (e.g., a = 7r/2), the Wih0 channel is suppressed so that the single-top mode is important even for large mass region of H #8 This is illustrated in Fig. 6.10c at the 8Note that the H i-W:F-H 0 coupling is proportional to sin a and is thus enhanced for small cos a. In this case, the WH0 mode may be important provided that H 0 is relatively light. We will not further elaborate this point here since it largely depends on the mass of H 0. 184 LHC for a = 7r/ 2. In order to probe the whole parameter space and larger mgi, it is important to study both tb and Wh0 (or WHO) channels. 6.5 Generalization to Neutral (Pseudo-)Scalar Pro- duction via bf) Fusion The QCD corrections are universal so that the generalization to the production of neutral scalar or pseudo-scalar 6)” via the b5 fusion is straightforward, i.e., we only need to replace (6.19) by Q(¢°b5) = 31n [mg/mg] + 4, (6.21) in which m), is the mass of 43°. The finite piece of the Yukawa renormalization [cf. the quantity 0 in (6.5)] is scheme-dependent. We can always define the ¢°-b—5 Yukawa coupling as flmb/v times an enhancement factor K so that the Yukawa counter term is generated by 6mb/mb.9 After resumming the leading logarithmic terms, [as ln(m§,/m§)]n, via the renormalization group technique, the net effect of the Yukawa renormalization is to change the Yukawa coupling or the related quark- mass into the corresponding M—S running coupling or mass, as discussed in the previous section. The b5 decay branching ratios of the neutral Higgs bosons in the MSSM with large tanfl are almost equal to one [168]. The same is true for the b—Higgs or b- pion in the TopC model [143]. It has been shown that at the Tevatron, the b5 di-jet final states can be properly identified [169]. The same technique developed for studying the resonance of the coloron or techni-p in the b5 decay mode [169] can also be applied to the search of the neutral Higgs bosons with large bottom Yukawa 9This specific definition works even if the Yukawa coupling is not related to any quark mass. For instance, the bottom Yukawa couplings of the b-Higgs and b-pion in the TopC model [143, 167] are independent of quark masses because the b-Higgs does not develop VEV. 185 coupling. When the neutral scalar or pseudo-scalar cf)” is relatively heavy, e.g., in the range of 0(250 - 1000) GeV, the QCD di-jet backgrounds can be effectively removed by requiring the two b-jets to be tagged with large transverse momenta (PT) because the PT of each b-jet from the (15° decay is typically at the order of m¢ / 2. Hence, this process can provide complementary information to that obtained from studying the (15%?) associate production [170, 167, 171]. We first consider the production of the neutral Higgs boson 4)”, which can be either A0, h", or H 0, in the MSSM with large tan B, where the corresponding Yukawa couplings to bf) and T+T‘ are enhanced relative to that of the SM since 310/ng is equal to tan 6, — sin a/ cos 6, or cos a/ cos B, respectively, at the tree-level. In the large tan 6 region, the MSSM neutral Higgs bosons dominantly decay into b5 and 7+7"— final states, which can be detected at the hadron colliders. In comparison with the recent studies on the 45%?) [167] and ¢°T+T‘ [172] associate production, we expect the inclusive ¢° production via the bB—fusion would be more useful for m4, being relatively heavy (e.g., m4, 2 200 — 300 GeV) because of the much larger phase space as well as a better suppression of the backgrounds in the high PT region. The total L0 and NLO cross sections for the inclusive production process pp, p}? —> AOX at the Tevatron and the LHC are shown in Figs. 6.12a and b, in parallel to Figs. 6.3 and 6.4 for the case of charged top-pion production. Here, we have chosen tanfl = 40 for illustration. The cross sections at other values of tan B can be obtained by multiplying the scaling factor (tan 6/40)2 . From Fig. 6.12a, we see a significant improvement from the pure LO results (dash-dotted curves) by resumming over the large logarithms of mi/mg into the running Yukawa coupling. The good agreement between the LO results with running Yukawa coupling and the NLO results is due to a non-trivial, and process- dependent, cancellation between the individual 0(as) contributions of the bf) and by sub-processes. In contrast to the production of the charged top-pion or Higgs boson 186 via the initial state CS or Eb partons, the neutral Higgs boson production involves .the b5 parton (densities. The K ~factors for the ratios of the NLO versus LO cross sections of pfi/ pp ——> AOX are presented in Fig. 6.13 for the MSSM with tan 6 = 40. The main difference is due to the fact that the individual contribution by the 0(as) bg-fusion becomes more negative as compared to the case of the charged top-pion production shown in Fig. 6.5. This makes the overall K -factor of the NLO versus LO cross sections range from about —(16~17)% to +5% at the Tevatron and the LHC. In parallel to Table 1 and Fig. 6.6, we have examined the uncertainties of the CTEQ4 PDFs for the AO-production at the Tevatron and the LHC, and the results are summarized in Table 6.2 and Fig. 6.14). The transverse momentum (QT) distributions of A“, produced at the upgraded Tevatron and at the LHC, are shown in Fig. 6.15 for various A0 masses (mA) with tan 5 = 40. The solid curves are the result of the multiple soft-gluon resummation, and the dashed ones are from the 0(as) calculation. The shape of these transverse momentum distributions is similar to that of the charged top-pion (cf. Fig. 6.7). The fixed order distributions are singular as QT —+ 0, while the resummed ones have a maximum at some finite QT and vanish at QT = 0. When QT becomes large, of the order of m A, the resummed curves merge into the fixed order ones. The average resummed QT varies between 25 and 30 (40 and 60) GeV in the mass range of m A from 200 to 300 (250 to 550) GeV at the Tevatron (LHC). We also note that for large tan B, the SUSY correction to the running 450-1343 Yukawa coupling is significant [173] and can be included in a way similar to our recent analysis of the ¢°b5 associate production [167]. To illustrate the SUSY correction to the b-Yukawa coupling, we choose all MSSM soft-breaking parameters as 500 GeV, and the Higgs mixing parameter 11 = :L-500 GeV. Depending on the sign of p, the SUSY correction to the 650-55 coupling can either take the same sign as the QCD 187 correction or have an opposite sign [167]. In Fig. 6.12c, the solid curves represent the NLO cross sections with QCD correction alone, while the results including the SUSY corrections to the running bottom Yukawa coupling are shown for p 2 +500 GeV (upper dashed curves) and p = —500 GeV (lower dashed curves). As shown, these partial SUSY corrections can change the cross sections by about a factor of 2. The above results are for the inclusive production of the CP-odd Higgs boson A0 in the MSSM. Similar results can be easily obtained for the other neutral Higgs bosons (h0 and H 0) by properly rescaling the coupling strength. We also note that in the large tan ,6 region, there is always a good mass-degeneracy between either h.0 and A0 (in the low mass region with m A f. 120 GeV) or H 0 and A0 (in the high mass region with mA 2. 120 GeV), as shown in Figs. 10 and 11 of Ref. [167]. 188 0 (pb) Tevatron 2TeV ."'~.-1....1.........11 .4 ....,-...,0. .‘ (b) ‘9 LHC 14 TeV '9 ... ........... ..... .... ... - ..... a ......... ..... u LHC14TcV i -.u‘ - Tevatron 1o Nun" 270V 0000000000 "000600000000“.‘000 m, (GeV) Figure 6.12: L0 and N LO cross sections for the neutral Higgs A0 production in the MSSM with tan 6 = 40, at the Tevatron and the LHC. (a) For each collider we show the NLO cross sections with the resummed running Yukawa coupling (solid) and with one-loop Yukawa coupling (dashed), as well as the LO cross sections with resummed running Yukawa coupling (dotted) and with tree-level Yukawa coupling (dash-dotted). (b) The NLO (solid), the b5 (dashed) and by (dash-dotted) sub-contributions, and the LO (dotted) contributions are shown. Since the by cross sections are negative, they are multiplied by —1 in the plot. The cross sections at \/S = 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the x/S = 2 TeV curves. (c) The NLO cross sections with QCD running Yukawa coupling (solid curves) and those with additional SUSY correction to the running coupling are shown (upper dashed lines for the Higgs-mixing parameter p = +500 GeV and lower dashed lines for u = —500 GeV). 189 Collider Upgraded Tevatron (2 TeV) LHC (14 TeV) Process \ PDF 4A1 4M 4A5 4HJ 4A1 4M 4A5 4HJ 2020 1900 1660 1920 18100 19800 16600 17900 L0 166 153 129 163 1520 1440 1280 1440 19.9 18.2 15.0 21.7 258 238 206 238 1810 1780 1620 1800 17100 17400 16700 17500 N L0 160 154 134 164 1520 1470 1350 1470 20.3 19.3 16.4 22.9 265 250 222 251 3040 2900 2590 2930 25400 25400 24100 25600 qtj ——> (boX 253 237 203 251 2140 2050 1850 2050 31.0 28.8 24.0 33.8 364 339 298 340 -1230 —1120 —970 —1130 —8320 —8010 —7370 —8050 qg —> ¢°X -92.9 -—83.1 —69.0 —87.5 -623 —575 —505 —574 —10.6 -—9.42 —7.59 —10.9 -98.8 -88.8 —75.8 -88.7 Table 6.2: Cross sections in fb for neutral Higgs boson production in the MSSM with tanfl = 40, at the upgraded Tevatron and the LHC, are shown for four different CTEQ4 PDFs. They are separately given for the L0 and NLO processes, and for the b5 —> AOX and by —> AOX sub-processes. For the upgraded Tevatron the top number is for m A = 200 GeV, the middle is for m A = 300 GeV, and the lowest is for m A = 400 GeV. For the LHC the top number is for m A = 400 GeV, the middle is for m A = 700 GeV, and the lowest is for m A = 1 TeV. 190 L 2 v v 11 l v rrv 1' Y Y errvyv‘FYTv WTVY 2 2 1 .. 1 8 L Tevatron (a) .. LHC (b) . u. "8. pb-)A°X i.” pp-)A°X 1 ‘3 x . VS=2T¢V ;: V5814TeV 1.6 ~ —- — 1.6 ’ ....... 05/0“, ‘F ...... 1 1.4 — _0 - 14 : .: 0.5/0w 1.2 — 1; . 12 . 1. 1 ~ — a 1 . dun/am :. Gnu/aw : 0.8 — -» ~ 0.8 1 0.6 P‘, .......... --->:: A 0.6 : —A0 0 ....... 0: ------ -A0 a I 0.4 ~— J “’ ; ------ .‘.'./..‘." ...... : 0.4 ’1.-+.m._1.r-. , ”11.1.1.1.1...‘ 100 200 300 200 400 600 800 1000 mA (GeV) Figure 6.13: The K -factors for the A0 production in the MSSM with tanfi = 40 are shown for the NLO (K = O’NLQ/O'LQ, solid lines), bb (K = aha/0L0 = (0L0 + Adm/01,0, dashed lines), and by (K = —A0bg/0Lo, dash-dotted lines) contributions, at the upgraded Tevatron (a) and the LHC (b). We then consider the large bottom Yukawa coupling of the neutral b—Higgs (hg) and b—pion (712) in the TopC model [143, 151, 167]. The new strong U (1) force in this model is attractive in the channel but repulsive in the <5b> channel. Thus, the top but not the bottom acquires dynamical mass from the vacuum. This makes the t-Yukawa coupling (yt) super-critical while the b—Yukawa coupling (yb) sub-critical, at the TopC breaking scale A, i.e., yb(A) 0S. ycmzm f. yt(A) , which requires yb being close to yt and thus naturally large. Our recent renormalization group analysis [167] shows that the relation yb(p)~yt(p) holds well at any scale ,u below A. For the current numerical analysis, we shall choose a typical value of yb(mt) 2 yt(mt) z 3, i.e., |cgb| = ICfi’I 2 3/\/2. In Fig. 6.16, we plot the production cross sections of hg or 7r}: at the Tevatron and the LHC. This is similar to the charged top-pion production in Fig. 6.4, except the non-trivial differences in the Yukawa couplings (due to the different tree-level values and the running behaviors) and the charm versus bottom parton luminosities. 191 1.2 ....-.fl. ...fi,.-.,.-.,.,. g Tevatron .4 » LHC l {“5 ” «s-zrev fir «s-14Tev * 1" h CTEQ4HJ 4i ‘ . creams . '1‘ 1.05 b '. -._....: l'\ 0‘8 .,-':‘ ' ....... x 4’ ”u ",u ' < t"‘-l.\-.. ¢+X The spin- and color-averaged amplitude-square for the Ch —-> (19+ 9 process is _ 27TCF 2 WI 3 The individual contributions (from the virtual loop and real gluon emission) to A 0.95 0.9 0.85 2 2 26 t a (13(ICLI +|CR| )[t [(1—6) (E+?+2)+2 A‘o "Y‘Y' VVVVVVVVV I VVVVVVVV l‘VY'I'VV'IY'VV‘V'YVIYTYVY % o :i. . .. A' production in the MSSM §1o'z- ........ Tevatron VS - 2 TeV v ' ............... tang-4o c5 -3 ......................... 1310 ................... \ ................. b . ..................... 13 ....................... -4. ................. 10 (°) -------------- ....1....1....1....1....1....1....1..'I‘.~ Y‘EYI I I lY'fiT"' A’ production in the MSSM -I 10 ..... 10' ................................ (b) ------------------------- ...1...1 1 .1._n 1-:.-"I'1-1-.,.L O 20 4O 50 50 100 120 140 01 (69") Figure 6.15: Transverse momentum distributions of pseudo-scalar A0 produced via hadronic collisions, calculated in the MSSM with tan fl = 40. The resummed (solid) and 0(as) (dashed) curves are shown for m A = 200, 250, and 300 GeV at the upgraded Tevatron (a), and for mA = 250, 400, and 550 GeV at the LHC (b). the NLO partonic cross section are: 2 6 _ 2 MW“: aoa‘CF (4”) F(1 6) [ 3Jr31T——2]c5(1—‘r‘), l°°p 27r Q2 F(l — 2c) _52 3 “virtual A 03017 e F(1 —e) [ 3 ] A = ___—__— ——— — 1 —- Account 00 2,” (4”) F0. _ 26) 6 Q 6( T), Argal _ A “‘0‘” 4W2 e F(1-€)[Z _A § _A _3111A —1 Aacb — 00 27r ( Q2 ) F(l _ 26) 626(1 7’) + 6(5(1 7') 6Pq,_q('r)CI;~ _ A A2 +4 (1+?) (M) 41“. lnr+2(1—?)], l — 'r + 1 — 7' 1 + $2 1 + e2 3 (1) A = = __ _ _ " PqHI(T) Cp(1_?)+ CF [(1-$)++26(1 7)], (6.23) where the standard plus prescription ( - ~)+ is given by 1 1 [0 da€(a) [x(a)]+ = [O dax [6(a) — {(1)}. (6.24) In (6.23), the infrared 5i? poles cancel between Mm!“ and A63“. The term A8132,“ from the virtual loop actually contains two types of l poles inside [ - -] : —3—— + 3- 6 6UV €112 193 3 ..fi....,fi-..,fi..,...-,...-,.-..fi..,,,. . 3; 10 ‘ b-Higgs or b-pion Production g b ., ._ (CTEQ4M) ; 1° LHC 1 2 """" ‘° ...... pp->¢ix 3 . .......... vs - 14 TeV 3 10 ' .................... .5 1 Tevatron ............. E -1 .... -, - i 1 1o ‘. ‘ 99‘"¢ X "i VS - 2 TeV 5 -2 1o .3 10— mug...» ....1;.:".....fit...‘xl....1....1....l 1 00 200 300 400 500 600 700 800 900 1 000 m. (GeV) Figure 6.16: Cross sections for the neutral b-pion 712 or b-Higgs hg production via the bB-fusion in the TopC model at the Tevatron and the LHC. The NLO (solid), the qrj’ (dashed) and qg (dash-dotted) sub-contributions, and the LO (dotted) contributions with resummed running Yukawa coupling are shown. Since the qg cross sections are negative, they are multiplied by -1 in the plot. The cross sections at x/S = 1.8 TeV (thin set of lowest curves) are multiplied by 0.1 to avoid overlap with the \/S = 2 TeV curves. with 60v = —e,;; E e = (4 — D) / 2 > 0. Also, the ——E- pole inside the Yukawa counter- real term contribution Afivm‘m is ultraviolet while the +— pole inside A665 is infrared 6 count (IR). We see that the contribution Afivmua“ from the counter term of the Yukawa count virtual loop (which is absent coupling is crucial for cancelling the UV divergence from A6 virtual 1 in the usual Drell-Yan type processes), while the soft — divergences between Afiloop e 1 and A6331 cancel. Finally, the Z collinear singularity inside A6331 will be absorbed into the re-definition of the PDF via the quark-quark transition function Pq(}_)q(?). All the finite terms are summarized in Eq. (6.4). 6.6.2 Partonic processes go, 95 —> ¢+X The spin- and color-averaged amplitude-square for the gc, 95 ——> ¢+X process is 7rd, 3 IMP = 3—(1—:€—) (ICLl2 + ICRl2) [12‘ [(1 — €) (£?+ g — 2) - 2AAA ] . (6.25) 194 The 0(as) partonic cross section for the quark-gluon fusions is given by: Area; _ A asCF 4W2 6 AM (Liz. (1) A AUG-”9 _ 00 27r (Q2) [( 151"(1—2€)+h1 7° P""9(T) 1 - A +Z("3 + 7r)(1 -— 7)] , (626) P“) (r) = a? + (1 — W], (Ii-'9 1 where it is clear that the collinear -€- singularity will be absorbed into the re-definition of the PDF via the gluon-splitting function Péggfi). The final result is finite and is given in Eq. (6.4). 6.7 Conclusions In summary, we have presented the complete 0(as) QCD corrections to the charged scalar or pseudo-scalar production via the partonic heavy quark fusion process at hadron colliders. We found that the overall NLO corrections to the pfi/pp ——> (15* processes are positive for m¢ above ~150 (200) GeV and lie below ~15 (10)% for the Tevatron (LHC) in the relevant range of 7nd, (cf. Fig. 6.5). The inclusion of the NLO contributions thus justifies and improves our recent LO analysis [142]. The uncertainties of the NLO rates due to the different PDFs are systematically examined and are found to be around 20% (cf. Table 1 and Fig. 6.6). The QCD resummation to include the effects of multiple soft-gluon radiation is also performed, which provides a better prediction of the transverse momentum (QT) distribution of the scalar qbo’i, and is important for extracting the experimental signals (cf. Fig. 6.7). We confirm that the Tevatron Run-II (with a 2 fb’1 integrated luminosity) is able to explore the natural mass range of the top-pions up to about 300—350 GeV in the TopC model [143, 151]. On the other hand, due to a possibly smaller (bi-b-c coupling in the 2HDM, we show that to probe the charged Higgs boson with mass above 200 GeV in 195 this model may require a high luminosity Tevatron (with a 10 — 30 fb’l integrated luminosity). The LHC can further probe the charged Higgs boson of the 2HDM up to about 0(1)TeV via the single-top and W‘th0 (or WiHO) production. The complementary roles of the tb and Wih0 channels in the different regions of the Higgs mass and the Higgs mixing angle a are demonstrated. We have also analyzed a direct extension of our NLO results to the neutral (pseudo-)scalar production via the bb-fusion for the neutral Higgs bosons (A0, ho, H 0) in the MSSM with large tan 6, and for the neutral b—pion (712) or b-Higgs (hg) in the TopC model with U (1)-tilted large bottom Yukawa coupling. In comparison with the ¢°b5 associate production [167], this inclusive ¢°~production mechanism provides a complementary probe for a neutral Higgs boson (with relatively large mass), whose decay products, e.g., in the b5 or T7" channel, typically have high transverse momenta (~ m¢/ 2) and can be effectively detected [169]. This is particularly helpful for the discovery reach of the Tevatron. Further detailed Monte Carlo analyses at the detector level should be carried out to finally conclude the sensitivity of the Tevatron Run-II and the LHC via this process. We mention that a recent paper [174] also studied the QCD corrections for the neutral Higgs production b5 —+ H 0 within the SM, and partially overlaps with our work as the pure NLO QCD correction is concerned. The overlapped part is in general agreement with ours except that we determine the counter term of the Yukawa coupling (expressed in terms of the relevant quark mass) by the on-shell scheme (cf. Refs. [160, 161]) while Ref. [174] used MS scheme. After resumming the leading logarithms into the running mass or Yukawa coupling, the two results coincide. Note that the apparent large 0(as) correction derived in Ref. [174] is due to the fact that it only includes the contribution from the bf) sub-process, which is part of our complete 0(as) contribution. The inclusion of the NLO contribution from the bg sub-process, which turns out to be negative and partially cancels the b5 contribution, 196 yields a typical size of 0(as) correction to the production rate of a neutral Higgs boson produced via heavy quark fusion. The by sub-process is identified as 0(1/ ln[mH/mb]) instead of 0(as) correction in Ref. [174]. Appendix A Probing Higgs Bosons with Large Bottom Yukawa Coupling at Hadron Colliders A. 1 Introduction A major task for all future high energy colliders is to determine the electroweak symmetry breaking (EWSB) mechanism for generating the (W1, Z 0) masses and the mechanism for the fermion mass generation [175]. Whether the two mechanisms are correlated or not is an interesting and yet to be determined issue. Given the large top quark mass (mt = 175.6 :t 5.5 GeV [176]), as high as the EWSB scale, it has been speculated that the top quark may play a special role for the EWSB. One of such ideas is that some new strong dynamics may involve a composite Higgs sector to generate the EWSB and to provide a dynamical origin for the top quark mass generation (e.g., the top-condensate/top—color models [144]). Another idea is realized in the supersymmetric theories in which the EWSB is driven radiatively by the large top quark Yukawa coupling [177]. In the minimal standard model (SM) there is only one Higgs doublet, which leaves a physical neutral scalar boson as the remnant of the spontaneous EWSB. The Yukawa 197 198 couplings of the SM Higgs are determined from the relevant SM fermion masses di- vided by the vacuum expectation value (vev), v 2 246 GeV. Thus, aside from the coupling to the heavy top quark, all the other SM Yukawa couplings are highly sup- pressed, independent of the Higgs boson mass. For the top-condensate/topcolor type of models [144], with a composite Higgs sector, the new strong dynamics associated with the top sector plays a crucial role for generating the large top mass and (possibly) the W, Z boson masses. As to be discussed below, in this scenario, the interactions of the top-bottom sector with the composite Higgs bosons are different from that in the SM. Due to the infrared quasi-fixed-point structure [178] and the particular boundary conditions at the compositeness scale, the bottom Yukawa coupling to the relevant Higgs scalar is naturally as large as that of the top in such models. In the minimal supersymmetric extension of the SM (MSSM) [179], there are two Higgs doublets, whose mass spectrum includes two neutral scalars (h and H), a pseudoscalar (A) and a pair of charged scalars (H i). The MSSM Higgs sector contains two free parameters which are traditionally chosen as the ratio of the two Higgs vev’s (tanfl = vu/vd) and the pseudoscalar mass (m A). A distinct feature of the MSSM is that in the large tan 6 region the Higgs Yukawa couplings to all the down-type fermions are enhanced by either tanB or 1/ cos 13. Among the down-type fermions, the bottom quark has the largest mass and so the largest Yukawa coupling. Thus, it represents a likely place where new physics could reveal itself experimentally. This common feature, the large bottom Yukawa coupling relative to that of the SM, present in both types of the (conceptually quite distinct) models discussed above, serves as the theoretical motivation for our analysis. Since the third family I) quark, as the weak isospin partner of the top, can have large Yukawa coupling with the Higgs scalar(s) in both composite and supersymmetric models, we recently proposed [180] to use the b quark as a probe of possible non- 199 standard dynamics in Higgs and top sectors. In fact, because of the light b mass (2: 5 GeV) relative to that of the top (2 175 GeV), the production of Higgs boson associated with b quarks (1915/ pp -> abbb —+ bbbb) may be experimentally accessible at the Fermilab Tevatron1 or the CERN Large Hadron Collider (LHC)2, even though the large top mass could render associated Higgs production with top quarks (pp/pp —> otfl infeasible. As we will show, this makes it possible for the Run II of the Tevatron and the LHC to confirm the various models in which the b-quark Yukawa coupling is naturally enhanced relative to the SM prediction. However, if the gbbb signal is not found, the Tevatron and the LHC can impose stringent constraints on the models with either a composite or a supersymmetric Higgs sector, in which the Yukawa coupling of the Higgs boson(s) and bottom quark is expected to be large. In [181], this reaction was explored at the LHC and the Tevatron3 to probe the supersymmetry (SUSY) parameters of the MSSM. The conclusion was that, with efficient b—tagging, useful information concerning the MSSM could be extracted from either the LHC or a high luminosity Tevatron. In this work, we expand upon earlier results [180] in which it was concluded that even the much lower integrated luminosity of the Tevatron Run II can provide useful information through this reaction, provided an optimized search strategy is employed. This analysis is based on the model-independent results in Ref. [182], where the relevant backgrounds were calculated and an effective search strategy to extract a signal from the backgrounds were presented. In what follows, after deriving the exclusion contours on the m A-tan 6 plane of the MSSM, we analyze its implication on the supergravity[183] and gauge-mediated[184] models of soft SUSY breaking that naturally predict a large tan b. The comparison of our Tevatron Run II results with the LEP II bounds (from the Z h and hA channels) 1A pp? collider with J3 = 2 TeV. 2A pp collider with J3 = 14 TeV. 3In [181], a 1.8 TeV p13 Tevatron was assumed with an integrated luminosity of 30 fb‘l, and the squark mixings of the MSSM were neglected. 200 is also presented, illustrating the complementarity of our analysis to other existing Higgs search strategies. A.2 Constraints On MSSM Parameters and Impli- cations for Models of Soft-breaking of SUSY Supersymmetry (SUSY) is one of the most natural extensions of the SM, mainly because of its ability to solve the hierarchy problem, as well as for its capacity to imitate the current experimental success of the SM, despite the plethora of the in- troduced new particles and free parameters [183]. The Minimal Supersymmetric SM (MSSM) [179] requires a two Higgs doublet extension of the SM [175] together with the corresponding supersymmetric partners. The model includes all renormalizable interactions that respect the standard gauge group SU(3)C <8 SU(2)L (81 U (1)y and supersymmetry. In order to prevent potentially dangerous baryon and lepton number violating interactions, invariance under a discrete R-parity4 is also required. To be compatible with data, supersymmetry has to be broken. The breaking of SUSY is parametrized by the general set of soft-breaking terms, which, in principle, should be deduced from a specific underlying model for SUSY breaking, such as the supergrav- ity [183] and gauge-mediated [184] models. In this section, we discuss the potential of the Tevatron and the LHC to test MSSM via measuring the production rate of gbbb mode. We shall also discuss the implication of this result on various supergravity and gauge-mediated models of soft SUSY-breaking. 201 Higgs A h H yy /y(S)M cot ,8 , cos a/ sin 6 sin a/ sin 6 yD/ySDM tan 5 — sin a/ cos 6 cos 01/ cos 6 9¢vv/y$i‘/Av 0 sin(fi — a) cos(fi — a) Table A.1: Comparison of the neutral MSSM Higgs couplings to up-type (U = u, c, t) and down-type (D = d, s, b; e, ,u, r) fermions and to the gauge-boson (V = W, Z) pairs. The ratios to the corresponding SM couplings are shown, which are determined by angles 6 and a at the tree-level. A.2.1 Bottom Yukawa Couplings and the MSSM Higgs Sec- tor In the MSSM, the Higgs couplings to the SM fermions and gauge bosons involve two new free parameters at the tree-level, which are the vacuum angle 6(5 arctan vu/vd) and the Higgs mixing angle a. These couplings are shown in Table A.1. We see that the MSSM Higgs couplings to the gauge boson pairs are always suppressed relative to that of the SM, while their couplings to the down(up)-type fermions are enhanced in the large (small) tan 6 region. These enhanced Yukawa couplings are of great phenomenological importance for the Higgs detection and especially for probing the associated new dynamics in the top-bottom sector. Alternatively, we can choose tan 6 and the pseudoscalar mass m A as two free parameters. Then, at the one-loop level, a can be calculated from —1 61 cos 2b ’ tan 2a 2 tan 26 (mg, + m2z) [m?4 — mg + (A.1) with a 6 (—7r / 2, 0). Here the parameter Q represents the dominant top and stop loop corrections which depend on the fourth power of the top mass rm and the logarithm 4The R-parity is defined in such a way that SM particles are even under R and their superpartners are odd. 202 of the stop mass ME: 4 M2 3GFm‘ log (—t, + 1) . (A.2) 6‘ = m m. Note that for large tan 5, the bottom and sbottom loop corrections can also be impor- tant. Hence, in our numerical analysis below we have included the complete radiative corrections with full mixing in the stop and sbottom sectors, and the renormalization group improvements are also adopted.5 As shown in Table A.1, the A-b—b coupling has no explicit a dependence. The bottom Yukawa couplings ybbh and ybe are a- and fl-dependent, their magnitudes relative to the SM prediction are displayed in Fig. A.1a as a function of mA for various tan ,6 values. It shows that for mA above ~ 110 GeV, the h-b-b coupling quickly decreases, approaching to the SM value for all tan fl, while the H-b—b coupling increases for large tan 6. Therefore, we expect that in the large tanB region, the production rate of Abb or hbb can be large for small m A, while the rate of Abb or H bb are enhanced for large m A. Whether the signals of the two Higgs scalars (A and h or A and H) can be experimentally resolved as two separate signals (e.g., two bumps in the bb invariant mass distribution) depends on their mass degeneracy. The MSSM Higgs boson mass spectrum can be determined by taking the second derivative on Higgs effective potential with respect to the Higgs fields. At tree- level, the resulting Higgs masses obey the relations: mh 3 mg cos 2fi, mg 3 my, mh 3 mA 3 my, and mw S mHi. However, these relations are substantially modified by radiative corrections [186]. Including the important contributions from top and stop loops, the masses of h and H can be written as: mi,” = g {(M2 +12) 4: 5We have used the HDECAY program [185] to compute the Higgs masses, couplings and decay branching ratios. Figur. differi- lllll. dEClP; dash? soft-l) be 203 50 60 70 80 90100 200 300 400 Figure A.1: Bottom Yukawa couplings to the MSSM Higgs bosons and the mass differences, mA - m], and my — mA, as a function of mA for tan 6 values: 2.0, 5.0, 10.0, 20.0, 30.0, 40., 50.0. In (a), W». is in solid and ybe is in dashed, and tan6 decreases from top to bottom curves. In (b), mA — mh is in solid, my — mA is in dashed, and tan6 increases from the top to bottom curves. Here, all the SUSY soft-breaking mass parameters are chosen to be 500 GeV. [(M2 + 602 — 46¢ (mzz cos2 6 + m3, sin2 6) — 4'")?sz cos2 26] %} , (A.3) where M 2 E mzz + m3, and the parameter at is defined in (A.2). To analyze the Higgs mass degeneracies, we plot the mass differences mA-mh and mH-mA in Fig. A.1b using the complete radiative corrections to the Higgs mass spectrum [185]. We see that for the large tan 6 values, the pseudoscalar A is about degenerate in mass with the lighter neutral scalar h below ~ 120 GeV and with the heavier neutral H above ~ 120 GeV. This degeneracy indicates that the ¢bb signal from the MSSM generally contains two mass-degenerate scalars with similar couplings, and thus results in a stronger bound on tan 6 by about a factor of \/2. Finally, we note that in Fig. A.1, all soft-breaking mass parameters were chosen to be 500 GeV. Various choices of SUSY soft-breaking parameters typically affect these 204 Figure A.2: The same as the previous figure, but in (a)-(b), we change the right- handed stop mass to 200 GeV, and in (c)-(d), we use the “LEP II Scan A2” set of SUSY parameters. quantities by about 10%-30%. To illustrate these effects, we plot in Fig. A.2 the same quantities, but changing the right-handed stop mass to M,- = 200 GeV in (a—b), and in (c-d) we use the “LEP2 II Scan A2” set of SUSY parameters6 for comparison. Because the MSSM predicts a large bottom quark Yukawa coupling for large tan 6, and the mass of the lightest neutral scalar has to be less than ~ 130 GeV, we expect that the Tevatron and the LHC can test this model via measuring the ¢bb production rate. In the following, we shall discuss the range of the mA-tan6 plane that can be explored at various colliders. Some models of SUSY soft-breaking predict a large tan 6 with light Higgs scalar(s), and thus predict a large ¢bb rate. Without observing such a signal, one can impose a stringent constrain on the model. This will also be discussed below. 6The parameters mo and M2 are fixed at 1 TeV, u is chosen to be —100 GeVand m; = 175 GeV. The scalar trilinear coupling A, is fixed at J6 TeV, corresponding to the maximal left- and right- handed top-squark mixing. Detailed prescription about this set of parameter scan can be found in Ref. [187, 188]. 205 A.2.2 Constraints on MSSM from (bbb production at Teva- tron and LHC To use the model-independent result of Km," obtained in Ref. [182] to constrain the m A-tan 6 plane in the MSSM, one needs to calculate the SUSY Higgs boson masses, decay branching ratios, and their couplings to the bottom quark for a given set of the soft breaking parameters. In the following numerical analysis, we use the HDECAY code to include the full mixings in the stop / sbottom sector with QCD and electroweak radiative corrections [185]. For simplicity, we assume that the superpartners are all heavy enough so that the decays of the Higgs bosons into them are forbidden. Under this assumption, we find that the decay branching ratio of h -> bb is close to one for the relevant region of the parameter space. As explained above, we combine signals from more than one scalar boson provided their masses differ by less than Ame”, which is the maximum of the experimental mass resolution (cf. footnote-7) and the natural decay width of the scalar boson. Since the results of Ref. [182] are given in terms of the minimal enhancement factor Kmin defined in Eq.(l) of that reference, we need to convert them into exclusion bounds in the m A-tan 6 plane of the MSSM, in case that a signal is not found. The bound on tan 6min (with the possible mass degeneracy included) can be derived from that on K min (fOr a single scalar) by requiring 4, 2 tan26 BR(A —> bb) :39: (Amexp — [AMA¢|) (0%)“) BR(d) —> bb) 2 K3,,“ , (A.4) where 3),?“ and yg’ denote the b quark Yukawa coupling in the SM and the MSSM (with (b = h or H), respectively. Inside the argument of the 6-function, AMM, is the mass difference between A and qb. Thus, the equality sign in the above relation determines the minimal value tan 6min for each given K min. 206 tonB tonfi Figure A3: 95% CL. exclusion contours in the mA-tan6 plane of the MSSM. The areas above the four boundaries are excluded for the Tevatron Run II with the indi- cated luminosities, and for the LHC with an integrated luminosity of 100 fb”1. The soft SUSY breaking parameters were chosen uniformly to be 500 GeV in Fig. (a), while the inputs of the “LEP II Scan A2” are used for the Fig. (b) in which LEP II excludes the left area of the solid curve. 207 To estimate the exclusion regions in the m A-tan6 plane, a set of soft breaking parameters has to be chosen, which should be compatible with the current data from the LEP II and the Tevatron experiments, while not much larger than 1 TeV. For simplicity, we choose all the soft SUSY breaking parameters (and the Higgs mixing parameter-u) to be 500 GeV as our “default” values, i.e., Msoft = 500 GeV. In Fig. A.3a, we show the 95% CL. exclusion contours in the m A-tan 6 plane derived from the measurement of [215/pp —> qibb —> bbbb, using this “default” set of SUSY parameters. The areas above the four boundaries are excluded for the Tevatron Run II with the indicated luminosities, and for the LHC with an integrated luminosity of 100 fb‘l. Needless to say, different choice of SUSY parameters, such as the mass and the mixing of the t0p squarks and the value (and sign) of the parameter u, would modify this result. To compare the potential of the Tevatron and the LHC in constraining the MSSM parameters via the ¢bb (<15 2 h, A, H) production to that of the LEP II experiments via Z (15 and hA production, we consider one of the “benchmark” parameter scans discussed in [187, 188], which is called the “LEP II Scan A2”12 set. For this set of SUSY parameters, the LEP II exclusion contour [187, 188] at the 95% CL. is displayed in Fig. A.3b, for a center-of—mass energy of 200 GeV and an integrated luminosity of 100 pb‘1 per LEP II experiment. As shown in Fig. A.3b, the Tevatron Run II result, in comparison with the LEP II result, can already cover a substantial region of the parameter space with only a 2 fb‘l luminosity. Thus, detecting the ¢bb signal at hadron colliders can effectively probe the MSSM Higgs sector, especially for models with large tan 6 values. Furthermore, for m A Z. 100 GeV, Tevatron Run II is complementary with LEP II, because the latter is not sensitive to that region of parameter space. The LHC can further probe the MSSM down to tan6 ~ 7 (15) for mA < 400 (1000) GeV. This is also shown in Fig. A.4 using the “default” set of SUSY parameters, in which the region above the upper curve is the 957°C.L. lllll_l_llllJ_4l_llJJ_1J_LllJlIlllllllJJ_l_lJ_l_I_Lll_lJll_Ll 200 400 600 800 1 000 MA (GeV) Figure A.4: Discovery and exclusion contours in the mA-tan6 plane of the MSSM for the LHC with an integrated luminosity of 100 fb‘l. The area above the lower boundary is excluded at 95% C.L., while the upper boundary is the 50 discovery contour. The soft SUSY breaking parameters were chosen uniformly to be 500 GeV. discovery contour at the 50 level for the LHC with an integrated luminosity of 100 fb‘l, and the area above the lower curve can be excluded at 95% C.L., if a signal is not found. For completeness, we also present the exclusion contours in the mh-tan6 and m A-tan 6 planes for both the “default” and the “LEP II Scan A2” sets of SUSY pa- rameters. They are shown in Figs. A5 and A6. Again, we see that the Tevatron Run II and the LHC bounds sensitively and complementarily cover the MSSM parameter space in contrast with the LEP 11 results. We have also studied the bounds with the “LEP II Scan A1” inputs [187, 188] and found that the exclusion contours from the ¢bb production are similar to those with “LEP II Scan A2” inputs. The most noticeable difference is that the theoretically allowed range for mh becomes smaller 209 (a) ‘D. E (b) 93. .6 1010" " Figure A5: 95% CL. exclusion contours in the mh-tan 6 plane of the MSSM. The ar- eas above the four boundaries are excluded for the Tevatron Run II with the indicated luminosities, and for the LHC with an integrated luminosity of 100 fb‘l. LEP II can exclude the area on the left-hand side of the solid curve in the lower plot. 210 by about 10 GeV in the “Scan A1” set, as compared to the “Scan A2” inputs. Even though our analyses, described above and in Ref. [182] are quite different from that of Ref. [181], the final bounds at the LHC happen to be in qualitative agreement. We also note that our bounds on the mA-tan 6 plane improve considerably the one obtained in Ref. [189], in which the 1915 ——> qibb —> T+T‘bb production rate at the Tevatron Run I data was compared to the MSSM prediction. Though, we do not choose to explicitly present projected results for the Tevatron Run I data, we encourage our experimental colleagues to pursue this analysis on the existing Run I data sample, as it seems likely that one could obtain useful information even with the lower luminosity and collider energy of Run I as well as a somewhat lower b—tagging efficiency. Before concluding this subsection, we remark upon the effects on our bounds from the possible radiative corrections to the ¢bb production process.7 As mentioned in Sec. II of Ref. [182], one of the dominant correction is from the next-to—leading order (NLO) QCD 100ps, which are not currently available for the ¢bb signal and background cross sections. However, aside from the QCD corrections to the ¢bb vertex (part of that can be simply included into the running of the ¢bb Yukawa coupling or the running b-mass), there are pentagon 100ps formed by the virtual gluons radiated from an initial state quark (gluon) and re—absorbed by the final state b quark with the ¢bb vertex included in the loop. Such QCD corrections are not factorizable so that a consistent improvement of our results is impossible before a full NLO QCD analysis is completeds. Putting aside the complexity of the full NLO QCD corrections, we briefly comment upon how the radiative corrections to the running ¢bb Yukawa coupling affect our final bounds. At the one-loop level, the relation between the pole 7This point has also been recently discussed in Ref. [190]. 8Such a full NLO QCD calculation is beyond the scope of our current study. A systematic calculation for this is in progress [191]. 211 400 p—IVT—YTIIITIIITTVfiTIIVYWYT” H'IIIUI‘IIIIIIIIIIIY (a) 300 200 100 lllllllllllllllllllilllllllllllllllllllllJJLLLLlL 80 90 100 110 120 130 Mb 400 ...‘lrrrrrrrrlrrrrrrrrrIrvrirrrrrIrrrTTTrrt]" ‘ (b) ' LEP II Scon A2 ‘ 300 200 100 Figure A6: 95% CL. exclusion contours in the mA-m,l plane of the MSSM. The shaded areas indicate the excluded regions for the Tevatron Run II with integrated luminosities, 2, 10, 30 fb‘l, and for the LHC with an integrated luminosity of 100 fb‘l, as those in the previous figures. LEP II can exclude the area on the left-hand side of the solid curve in the lower plot. 212 VIIIIIIIITIUYITTIIIYTTITTIYYIrTTITrTTII IIIIIIIII 8 ; QCD running alone (solid) _ QCD running + MSSM, p = 500 (dosh) ‘ QCD running + MSSM, ,u = -500 (dot) - a: l 1 ‘fi ‘. u ... .- ... I... u oooooooooooooo ...... .................... no») (GeV) 44 l Figure A7: The running of the bottom quark mass mb(pg) as a function of the renor- malization scale [13. The solid curve shows the QCD evolution alone. The dashed curve further includes the supersymmetric corrections to the “effective” running mass, for tan 6 = 30. All soft SUSY breaking parameters have been fixed as 500 GeV. The dotted curve includes the SUSY corrections but with the sign of the Higgs-mixing parameter p flipped. quark mass m?“ and the MS QCD running mass at the scale [1 = mg“ is: 4a8 mpol _l mq(m§°') = mg‘” [I + ——-;?q-—Z] . (A.5) When running upward to any scale 11, 0 c a, )1 7r m,(,.) = m.(m:; ') e [ (p31 1 , , (A6) c [03(mq )/7r] where (cf. Ref. [185]) c[:c] = (23:.:/6)12/23 [1 + 1.17595] for mg“ < u < meI, C(51)) = (7:1:/2)4/7 [1 + 1.398117] for 11 > mf‘”. Numerically, mt(mf°1) 2 166 GeV and mb(mf°l) 2 3GeV. This QCD correction, Eq. (A.6), alone will reduce the running mass mb(u) by about 40% from the scale 213 p = mime 2 5 GeV up to the weak scale of 0(200) GeV (cf. the solid curve of Fig. A?) This is however not the full story. The complexity comes from the finite SUSY threshold correction in the large tan 6 region which are potentially large [192, 193, 194]. As shown in Ref. [193], the dominant one-loop SUSY correction contributes to running b-mass a finite term so that mb(p) at the SUSY scale ,u E m; = Msoft is multiplied by a constant factor 1/[1 + Ab(SUSY)] , which appears as a common factor in the bottom Yukawa couplings of all three neutral Higgs bosons. For large tan 6, Ab(SUSY) contains the following tan 6-enhanced terms from sbottom-gluino and stop-chargino loopsg, Ab(SUSY) .—_ (Amb)5§+ (Amby-X mb mb —- tan6 8 = H377" {393771.117 (m5,m5,m§) + lytAtF (mf’mf’m _ggMZ’F (mf’m2’”)]l ’ (A.7) with the function F defined as: :rylnsr/y + yzln y/z + 21: In z/rr F(fi’fi’fi) :— (x-mo-zxz—x) ' In these equations, the MSSM Higgs parameter a should not be confused with the usual renormalization scale 113. In (A.7), we have assumed, for simplicity, mass degen- eracy for the top and bottom squarks, respectively. Eq. (A.7) shows that the SUSY correction to the running m, is proportional to tan 6 and )1. Thus, this correction is enhanced for large tan 6 and non-negligible in comparison with the QCD corrections. Also changing the sign of 11 will vary the sign of the whole correction Ab(SUSY) and implies that the SUSY correction can either increase or decrease the running b-mass at the energy scale around of 0(Msoft). Normally, when defining the running 9We thank K. Matchev for discussing his published results in Ref. [193], and to him and W.A. Bardeen for discussing the issue of the running b-mass. Our convention of the MSSM Higgs parameter p differs from that of Ref. [193] by a minus sign. 214 coupling using the Collins-Wilczek-Zee (CWZ) scheme [159], only the jun-dependent contributions are included while all the [LR-independent terms are absorbed into the corresponding Wilson coefficient functions. However, since the 113—independent con- tribution Ab(SUSY) is not small for large tan 6 and a full NLO SUSY calculation is not yet available, we include Ab(SUSY) to define an “effective” running coupling/ mass of b—quark even below the SUSY threshold scale Msoft. This could give a rough esti- mate on the large SUSY loop corrections from the cb-b-b vertex. Obviously, when [13 is much smaller than M30“, the CWZ scheme should be used. Hence, in Fig. A.7, we only show the “effective” running mass mb(p) down to about 100 GeV which is the relevant energy scale and the mass scale (~ Mmggs) considered in this work. Fig. A.7 illustrates that due to the SUSY correction the “effective” running b mass at the weak scale can be either larger or smaller than the SM QCD running value (~ 3 GeV), depending on the choice of the sign of u parameter (and also other soft-breaking parameters such as the trilinear coupling At and masses of the gluino, gaugino, stop and sbottom). For the “default” set of SUSY parameters used in our analysis, all the soft-breaking parameters are set to 500 GeV for simplicity. It happens to be the case that the SM QCD and SUSY corrections nearly cancel each other so that the “effective” running mass mb is very close to the pole mass value (~ 5 GeV) for the scale above ~ 100 GeV (cf. upper curve of Fig. A.7). For comparison, in our anal- ysis using the “LEP II Scan A2” set of SUSY parameters, the SM QCD and SUSY corrections do not cancel and tend to reduce the “effective” running b-mass or the Yukawa coupling yb(u) which results in a weaker bound for the Tevatron Run II and the LHC, as shown in Fig. A.3b.10 The difference in the exclusion contours shown in Figs. A.3a and A.3b, derived from the measurement of the (,bbb production rate at hadron colliders, is mainly due to the difference in the “effective” running coupling or 10A detailed analysis of these effects at the Tevatron Run II is currently underway [190]. 215 mass (including the QCD and SUSY corrections), as described above.11 We therefore conclude that a full NLO QCD calculation is important for a consistent improvement of our current analysis. A.2.3 Interpretation of results for Models of soft breaking parameters The MSSM allows for a very general set of soft SUSY-breaking terms and thus is specified by a large number of free parameters (2 124 [179]), though only a com- plicated subset of this parameter space is consistent with all current experimental results. It is therefore important to understand the mechanism of supersymmetry breaking (which presumably occurs at a high energy scale [195]) and to predict the soft parameters at the weak scale from an underlying model. Many alternative ideas about how supersymmetry might be broken, and how this will result in the low energy soft breaking parameters exist in the literature, including the supergravity inspired (SUGRA) models and gauge—mediated SUSY breaking (GMSB) models. In this sec- tion we examine the sensitivity of the ¢bb process to probe a few models of SUSY breaking, concentrating for the most part on the popular SUGRA and GMSB models. However, there are also other interesting ideas to which the ¢bb process may provide interesting information, because these models naturally prefer a large tan 6. A few examples include the SO(10) grand unification theories [196] (SO(10) GUTS); the infrared fixed-point scenario [197]; and also a scenario with compositeness, the “more minimal supersymmetric SM” [198]. llFfom Fig. A.3, it is also clear that, for the LHC bounds, the SUSY correction has much less impact since the relevant tan 6 values become much lower, around of 0(2 - 15). 216 Supergravity Models with large tan 6 The supergravity inspired (SUGRA) models [199] incorporate gravity in a natural manner, and solve the problem of SUSY breaking through the introduction of a hidden sector, which breaks SUSY at a very high scale [~ 0(10“) GeV] and interacts with the MSSM fields only gravitationally. This model offers an exciting glimpse into the possible connection between the heavy top quark and the EWSB by allowing radiative breaking of the electroweak symmetry, in which the large t0p quark Yukawa coupling can drive one of the Higgs masses negative at energies ~ mz. In the limit of large tan 6 the bottom and tau Yukawa couplings can also play an important role [200]. Under the assumption that the gravitational interactions are flavor-blind, this model determines the entire SUSY spectrum in terms of five free parameters (at the high energy scale of the SUSY breaking) including a common scalar mass (rho), a common gaugino mass (M), a common scalar tri-linear interaction term (A), tan 6, and the sign of the Higgs mixing parameter (sgn(,u)). The weak scale particle spec- trum can then be determined by using the renormalization group analysis to run the sparticle masses from the high scale to the weak scale. Though large tan 6 is not required by the minimal SUGRA model, it can naturally be accommodated, as demonstrated in [200, 201], where it was found that a large tan 6 also generally requires that the pseudoscalar Higgs mass be light (m A S 200 GeV for tan 6 Z 30), because the enhanced b and T Yukawa couplings act through the renormalization group equations to reduce the down-type Higgs mass term at the weak scale, thus resulting in a light Higgs spectrum. Since the importance of the b and 7 effects in the RG analysis increases with larger tan 6, as tan 6 increases the resulting m A decreases, making the large tan 6 scenario in the SUGRA model 217 particularly easy to probe through the ¢bb process. From the limits on the m A-tan 6 plane derived above, it thus seems likely that from the data of the Tevatron Run II with 2 fb’1 of integrated luminosity, a large portion of the minimal SUGRA model with tan 6 2 20 may be excluded. Gauge-mediated SUSY Breaking Models with Large tan 6 Models with GMSB break SUSY at a scale which is typically much lower than that present in the SUGRA models. The supersymmetry is generally broken in a hidden sector which directly couples to a set of messenger chiral superfields. This induces a difference in mass between the fermion and scalar components of the messenger fields, which in turn generates masses for the gauginos and sfermions of the MSSM fields via loops involving the ordinary gauge interactions [202, 203]. A generic feature of this scenario is that because of the relatively low scale of SUSY breaking, the gravitino acquires a much smaller mass than in the SUGRA scenarios, and is generally the lightest supersymmetric partner (LSP). While specific models of GMSB vary as to their assumptions and relevant parameters, generally what must be assumed is the field content of the messenger sector (including transformation properties under the gauge group and number of multiplets in the theory) and the scale at which SUSY is broken in the hidden sector. The minimal GMSB models can also result in a radiative breaking of the EWSB, through the renormalization group evolution of the Higgs masses (driven by the large top Yukawa coupling) from the effective SUSY breaking scale to the weak scale. As in the SUGRA model case, this evolution can drive the mass term of the up—type Higgs negative at the weak scale, thus breaking the electroweak symmetry. In fact, because the effective SUSY breaking scale in a GMSB model is typically much lower than in the SUGRA model (and thus closer to the weak scale), in order for the proper EWSB 218 [to occur, it was demonstrated in [204] that a large tan6 is required (about 30—40). However, because the effective SUSY breaking scale is typically much lower than in the SUGRA model, the large effects of the b and r on the Higgs mass running do not reduce the Higgs spectrum to the degree that occurs in the SUGRA model with large tan 6, and thus result in a heavier A with mass of about 400 GeV. So, this particular model would only be explored through the qbbb process at the LHC. However, more general analyses of the GMSB scenario [205, 206, 207, 208], introducing more degrees of freedom in the messenger sector than the minimal model, can allow for large tan 6 and relax m A to be as low as about 200 GeV. Thus, through the ¢bb production these more general models can be first probed at the Tevatron for the relevant mass range, and then largely explored at the LHC. A.3 Conclusions It remains a challenging task to determine the underlying dynamics of the electroweak symmetry breaking and the flavor symmetry breaking. Either fundamental or com- posite Higgs boson(s) may play a central role in the mass generation of the weak gauge bosons and the fermions. The heavy top quark, with a mass on the same order as the scale of the electroweak symmetry breaking, suggests that the top quark may play a special role in the mechanism of mass generation. In this work, we have shown that in the typical models of this type, the bottom quark, as the weak isospin partner of the top quark, can also participate in the dynamics of mass generation, and serves as an effective probe of the possible new physics associated with the Higgs and top sectors. We have presented a model-independent analysis on Higgs boson production in association with bottom quarks, via the reaction 1213/ pp —-> gbbb -—> bbbb, at the Tevatron 219 Run II and the LHC. Using the complete tree level calculation with an estimated QCD k-factor of 2, we derive the exclusion contour for the enhancement factor (in the coupling of ¢bb relative to that of the SM) versus the Higgs mass m4, at the 95% C.L., assuming a signal is not found. We apply these results to analyze the constraints on the parameter space of the MSSM (in the large tan 6 region) with naturally large bottom Yukawa couplings. To confirm the MSSM, it is necessary to detect all the predicted neutral Higgs bosons h, H, A and the charged scalars H i. From LEP II, depending on the choice of the MSSM soft-breaking parameters, the current 95% CL. bounds on the masses of the MSSM Higgs bosons are about 70 GeV for both the CP-even scalar h and the CP- odd scalar A [209]. It can be improved at LEP II with higher luminosity and maximal energy, but the bounds on the Higgs masses will not be much larger than ~ mg for an arbitrary tan 6 value. The Wh and WH associated production at the Tevatron Run II can further improve these bounds, if a signal is not observed. At the LHC, a large portion of parameter space can be tested via pp —> tf + h(—+ ’77) + X, and pp —+ h(—-> ZZ") + X, etc. [210, 211]. A future high energy e+e‘ collider will fully test the MSSM Higgs sector [212, 213] through the reactions 6+6“ —> Z + h(H), A + h(H), H +H ", etc. In this work, we demonstrate that studying the ¢bb channel at hadron colliders can further improve our knowledge on MSSM. The exclusion contours on the m A-tan 6 plane of the MSSM shows that the Tevatron and the LHC are sensitive to a large portion of the parameter space via this mode. It therefore provides a complementary probe of the MSSM Higgs sector in comparison with that from LEP II. The implications of these bounds in the parameter space on both the supergravity and the gauge-mediated SUSY breaking models are further discussed. We find that the ¢bb process can effectively test the models with either scheme of the SUSY soft-breaking in the large tan 6 scenario. 220 In conclusion, the stringent constraints obtained by studying the associated ¢bb production at hadron colliders for the supersymmetric models show the utility of this search mode. However, much work remains to be done. For example, we have found that a b-trigger is essential for this analysis. In fact, the CDF group at the Fermilab has demonstrated that it is possible to detect events with four or more jets and two or more b—tags [214], which can significantly be improved with the implementation of b-trigger at the Run II of the Tevatron [215]. However, the large QCD 4-jet background rate at the LHC can potentially make triggering on the bb¢(—> bb) events difficult, though it is expected that b-trigger would becomes more efficient for a heavier (pseudo-)scalar (15. (This is because the b—jets from the decay of a heavier (13 are more energetic, and its QCD background rate drops rapidly as a function of the transverse momentum of the triggered jet.) We hope that the interesting results afforded by studying this channel will stimulate interest in working on these problems in our experimental colleagues. We have found that the ¢bb process complements Higgs searches in other channels, and thus it is expected that experimental searches for this signature at the Tevatron Run II (and possibly beyond) and the CERN LHC will provide interesting and im- portant information about the mechanism of the electroweak symmetry breaking and the fermion mass generation. Appendix B “Intrinsic kT” and Soft Gluon Resummation B. 1 Introduction For many years, it has been a common practice to “smear” the transverse momentum (pT) distribution of the Drell-Yan pairs or the direct photons calculated in perturba- tive QCD to obtain a better agreement between the theoretical predictions and the experimental data. The argument of smearing the pT distribution is that any parton inside a hadron generally has a non-zero transverse momentum which is called the “intrinsic transverse momentum” (or “intrinsic M”) of the parton. The concept of intrinsic transverse momentum evolved in the literature from the covariant parton model of Landshoff, Polkinghorne and Short [216] to the [CT dependent parton dis- tributions of Collins and Soper [217]. Intrinsic kT emerges from the Fermi motion of partons within the hadron. Since the Fermi motion is uncorrelated with the direction of the acceleration, it leads to momentum fluctuations in the transverse directions. By definition, this ab initio transverse momentum has a non-perturbative nature and typically is of the order of AQCD, i.e. a few hundred MeV. It was however found that the amount of “smearing” needed to make the theoretical prediction of the pT distribution agree with data in some cases could be as large as a few GeV which 221 222 is much larger than AQCD. To understand this, we need the full armory of QCD including both the perturbative and the non—perturbative part. In the low pT re- gion, which is the region where a perturbative calculation will deviate the most from experimental data, one needs to take into account the effects of multiple (soft and collinear) gluon radiation from both the initial and the final states. For extremely low pT regions, the non-perturbative effects of the QCD theory, e.g. effects due to higher twist operators, should also be included when comparing with the measurements. In this work we make use of the soft gluon resummation formalism developed by Collins and Soper to estimate the amount of smearing needed to apply onto a perturbative QCD calculation so that its prediction would agrees with that from a resummation calculation. For the Drell-Yan process, in which only the initial state involves QCD colors, Collins, Soper and Sterman (CSS) gave a set of comprehensive formula to account for the effects on the pT distribution of the lepton pair due to the multiple soft (and collinear) gluon emissions in the initial state. In the CSS formalism, the perturbative part of the multiple gluon radiation effects is summarized by the perturbative Sudakov form factor and some Wilson coefficients (i.e. C—functions); the non—perturbative part of QCD physics which is relevant to the low pT distribution is parametrized by some non-perturbative functions which at the present time can only be fitted to some existing Drell-Yan data. The moral is that if the CSS formalism is the correct parametrization of the multiple gluon effects, then the same form of the non- perturbative functions (fitted to particular sets of data) should also be applied to the other Drell-Yan data at different energies of the colliders and / or invariant masses of the lepton pairs. The effects due to initial state gluon radiations should be universal to any pro- cess involving the same kinds of incoming partons with similar QCD color structures 223 in the events. As we saw earlier, the CSS formalism describes well the transverse momentum distribution of wt, Z-bosons and diphotons at the Tevatron. Although the direct photon process does not have the same kinds of incoming partons or color structures as those contributing to the Drell-Yan process, we expect that the dom- inant effects on the low pT distribution of the direct photon would come from the initial state radiation whose non-perturbative contributions are more or less the same as those measured in the Drell-Yan process. As far as we know, there is no proof for this statement. It is only our educated guess that the non-perturbative behavior of the partons inside a hadron would be similar. Of course, the calculable perturba- tive Sudakov form factors would distinguish an initial gluon parton from an initial quark (or anti-quark) parton, and should be properly included in the CSS formalism for calculating the pT distributions. Although we lack theoretical proof, there are indications from phenomenology which seem to confirm the need of the resummation of the initial state QCD radiation for the direct photon process for low pT. Monte Carlo simulations (using Bauer-Reno’s MC or PYTHIA) indicate that initial state radiation improves the finite order QCD results. There are also efforts to derive CSS type formulas with partial success. In this work, we do not intend to do a complete calculation on the pT distribu- tion of the direct photon using the CSS formalism, which as described in the above, involves both the perturbative Sudakov form factor and the non-perturbative func- tions. We only apply the non-perturbative part of the CSS formalism to explain the amount of pT-smearing needed to make a “smeared” next-to—leading order (NLO) QCD perturbative calculation agree with the experimental data of the direct photon in its transverse momentum distribution. We show that for the low energy fixed- target experiments, the non-perturbative part of the CSS resummation formalism alone can explain the required amount of low-pT-smearing. However, to explain the 224 high energy collider data, such as those at the Tevatron, the perturbative part of the Sudakov form factor should also be included in the calculation. We defer the complete calculation, which separately includes the effects from the perturbative Sudakov form factor associated with qq (qrj) or gq (gc‘j) initiated direct photon production processes, to our future work. B.2 Smearing PT Distributions of Direct Photons In this section we show that it is possible to resolve the discrepancy between the theoretical prediction and the experimental data of the pT distribution of the direct photon measured in fixed target experiments. This is achieved by applying the CSS resummation formalism and by including only the contribution from the non-perturbative functions of the formalism, which (in the direct photon context) has just one free parameter to be determined by experimental data. Our approach is to assume the approximative universality (i.e. insensitivity to the flavors of the active partons) of the non-perturbative function in the CSS resummation. Hence, the same non-perturbative function that is used to describe the Drell-Yan process can also be used for the direct photon production to improve the next-to-leading order predictions. What follows in this section is rather intuitive and is not backed up by full theoretical understanding or precise calculations. It is a fact that the transverse momentum distribution of the photons produced directly at hadron-hadron collisions deviates systematically from the experimental measurement. The NLO QCD calculations [218] give less steep curves than the ex- perimental data. A recent global analysis of the problem [219], examining both fixed target and collider experiments, concluded that neither new parton distributions nor new fragmentation functions can resolve the problem, since the deviation occurs at 225 different a: values in experiments with different energies. In the above study the au- thors also exclude the possibility that the deviation can be entirely accounted for the scale choice in the calculations. It was shown in Ref. [219] that convoluting the theoretical pT distribution with a Gaussian resolution function can raise the cross section at the low pT end so that it agrees with the experimental data better. This is because smearing a falling distribu- tion by a Gaussian can steepen its original shape. Writing the smeared pT distribution as d 1 00 d NLO , . a “ [mdpaidpr— “(”7“)” (13.1) Eff—AM defines A as the quantitative measure of the smearing. The amount of the smearing needed differs for different center of mass energies (see the first and second columns of Table B.1). There are several puzzling details here. First, there is no theoretical reason to introduce an ad hoc smearing of the NLO results. Secondly, the amount of the intrinsic kT that the partons acquire because of the Fermi motion within the hadron is in the order of AQCD, which is a few hundred MeV. The amount of smearing that brings theory and experiment to agreement is more than that. Finally, even if one invents a mechanism to introduce some type of a smearing, one has to find a smearing function that has a dependence on S which is the square of the center of mass energy of the hadron-hadron collision, and make sure that the smearing does not ruin the agreement at higher pT’s. To overcome all the above difliculties we propose the application of the non- perturbative function of the CSS resummation as a smearing device. This simple idea seems to be plausible because the non-perturbative function has a Gaussian form and depends on the center of mass energy of the process. Although we do not 226 have a proof, after noticing that the non-perturbative function acts like a smearing factor at the Drell-Yan process (for details see the next section), we may as well try it at the direct photon process. For processes with low energies the non-perturbative function is usually parametrized in a simple Gaussian form 1 WNPfQT, Q, Q0) = m exp {—Qi/‘lGflQ, Qol}, (32) with G2(Q,Qo) = 91 + ggln(Q/2Qo), 91 = 0.11GeV2, g2 = 0.58 GeV2 and Q0 2 1.6 GeV. We can easily check whether the amount of the smearing from the non- perturbative function comes close to what phenomenology demands. First we sub- stitute VS? for Q in 02(Q,Q0). This substitution is inspired by the relation Q = m 2 \/§ that holds in the case of the Drell-Yan process, with VS be- ing the center of mass energy of the colliding hadrons, and 22,4, 123 the longitudinal momentum fractions of the incoming partons. Since these latter quantities are not measured at the direct photon production we take :1: as a free parameter and let data determine its value. The best fit to the fixed target experimental data gives :2: = 0.3. Now we can calculate the numerical value of 02(Q, Q0) for the six different experiment. The results are shown in the last column of Table 8.1. We see at first sight that the non-perturbative function provides about the right amount of smearing needed to bring the theory prediction and the experimental data in a better agreement. Examining Table B.1 more closely, we observe that the smear- ing from the non-perturbative function falls short as the center of mass energy of the collider increases. Hence, for the fixed target experiments at a higher energy and for the collider experiments, the contribution from the non-perturbative function alone cannot account for all the amount of smearing needed. This can also be seen from Fig. B.1. Substituting the non-perturbative function 227 Experiment \/S (GeV) A (GeV) ‘/2G(Q,Qo) (GeV) WA70 23 < 1 1.0 UA6 24 g 1 1.1 E706 31 1-2 1.2 R806 63 2-3 1.5 UA2 630 3—4 2.2 CDF 1800 3-4 2.5 Table 8.1: The amount of the smearing needed and the amount provided by the non-perturbative function to make theory agree with experiment for different direct photon experiments. Values of A are taken from Ref. [4]. into the convolution equation (B.1), the theoretical distributions can be rendered close to the experimental ones as it is shown in Fig. B.1. (Among the fixed target experiments, UA6ppdata are not shown, but they have the same characteristics as UA6pp data.) The N LO QCD curve is shown in short dashes. The long dashed curve is the convolution of the theory with the non-perturbative function. As anticipated, the agreement between the data and the theoretical prediction improves in low the pT region after including the smearing effects for all the low energy fixed target experiments. However, the convolution (i.e. the pT-smearing) is less effective for the collider data (not shown). As discussed above, at collider energies the perturbative Sudakov form factor arising from resumming over the large logs which are due to the initial state radiation will be important besides the non-perturbative effects, and it is not included in the present study. In conclusion, the above results suggest that there is an important similarity be— tween the Drell-Yan and the direct photon processes. The roots of this correspondence are found in the initial state radiation, which is present in both cases. 228 e706 ua6_pp [ ' ' T 60 - 1 E‘ 50 0. 150 \\ . Q‘s. L '3 9,“, v3 40 g 100 ‘ \ .1; 30 \*\ U” ‘\ E] “:3 "‘ ‘~ \ H 20 ‘.i\ Ln. 50> “‘\i 3 \\\\ '5’ ‘ r. '0 10 A.“ \f‘“ , “Wu. 0’ - .--.-‘T"+j~---—A-r— 4 0 A __L - - A *«....___. 4.5 5 5.5 6.5 7 4.25 4.5 4.75 5 5.25 5.55.75 p p T T wa70_yeq r806 - -, - 40 ,\ 500. “ e. i} e. 01 D‘ ‘3. ‘31 4oo~.\ IO 30' \‘\ 1 'U f“ —\~ ‘] 1 300L RS 20' 0.“ g ‘ \\ E 0* 200» n] _H ., .,, ‘ U) 10 ‘\°~ it” ‘\ \ 'o “-t 'o 100» at“ .K‘Ou “It- 0 ‘ f~.——_-—‘ 0 - . ‘t'h‘T‘“Q-r—t—-r——t 4.5 5.5 6 5 6 7 8 9 10 p p T T Figure 8.1: Experiment numbers are listed at the top of each plot. Short dashed line: NLO transverse momentum distribution. Long dashed line: NLO theory convoluted with the Fourier transform of the non-perturbative function used in the Drell-Yan resummation formalism. B.3 pT-Smearing and Soft gluon Resummation In this section we demonstrate that “pT-smearing” naturally arises within the resummation formalism of Collins, Soper and Sterman [44, 45, 46]. It is the same type of smearing that is needed to make the theoretical prediction agree with the experimental data in pT distribution of the direct photon produced in fixed target experiments. The canonical example of the CSS formalism is the hadronic production of elec- troweak vector bosons. In the Drell-Yan type of process the kinematical variables of the vector boson V (real or virtual) are selected to be its invariant mass Q, rapidity 229 y, transverse momentum QT, and azimuthal angle 45v, measured in the laboratory frame. The kinematics of the leptons from the decay of the vector boson can be de- scribed by the polar angle 0 and the azimuthal angle 4’), defined in the Collins-Soper frame [52]. The fully differential cross section of the vector boson production and decay is given by the resummation formula: da(AB —> V(—> 11")X) _ 1 Q2 dQ2 dy dQ% day dcos0d¢ _ 967r2S (Q2 —- M3,)2 + Mara 1 .~ .- ~ X {W/dzbelQTb :ij(bt1QazA1$Bag1¢)WjIZP(b1Q1$A1$B) j,k + Y(QT1Q1$A1$3101¢)}' (8'3) flere ij(b1Q1 $A1z3161¢) = wjk($A1 $31 61 (15) exp {—S(b1Q)}1 (B4) with the Sudakov exponent 50. Q) = [2:5, 9:7 [1n (%—) AMA/1)) +B(a.(m) . (13.5) and 331,3 = flexp (:l:y). In the above expressions j represents quark flavors and it stands for anti-quark flavors. The definition of 17) together with further details can be found in Chapter 2. In Eq. (B.3), the impact parameter b is to be integrated from 0 to 00. However, for b larger than a certain bmax, corresponding to energy scales less than 1/bmax, the QCD coupling a, becomes so large that a perturbative calculation is no longer reliable. To avoid this, W is evaluated at b,,, with b = b * \/1+(b/bma.)2 (8.6) such that b,, never exceeds bmax. Thus bmam is the theoretical scale that separates perturbative and non-perturbative physics in the resummation formalism. Just like 230 other theoretical scales (e.g. renormalization or factorization scales), b,,m defines a somewhat arbitrary separation. Choosing bum = 1 / 2 GeV‘1 (as usual) instead of 1/AQCD shuffles parts of the low QT perturbative physics related to the initial state radiation into the non-perturbative function. Thus the non-perturbative function contains more than just the intrinsic partonic kT. WNP is needed in the formalism in order to suppress the contribution of 1773-], from the large b (small QT) region and to parametrize our ignorance of the non-perturbative phenomenon there. Further- more, since physics cannot depend on b,,m, WNP carries an implicit b,,,” dependence through the non-perturbative parameters 9,- defined below. Generally W’”3 has the structure WJ'IZP(b1 Q1 Q01 $141 1'3) : exp {—SNP(b1 Q1 Q01 $A1xB)}1 (B'7) with the non-perturbative exponent SNP(b, Q, Q0, ...,, 2:3) = 121(1)) ln (9;) + h,,..(.:.., b) + Iii/Bag, b). (38) 0 Here hl, hj/A and hie/B cannot be calculated using perturbation theory, they must be determined using experimental data. This particular structure of the non-perturbative exponent SNP is inspired by the form of the Sudakov form factor which in turn is derived by renormalization group arguments discussed in Chapter 2. Since for large values of the impact parameter the non-perturbative function has to cancel the con- tribution of W, the structure of the non-perturbative exponent is determined by the Sudakov exponent, that is, by perturbative resummation physics. We restrict our attention to the small QT region, where resummation effects may be important and are detectable. In the (2% << Q2 region the Y-term in Eq. (B.3) is negligible. Dropping the Y piece and rewriting ij in a form of a Fourier transform1 1We use the same symbol for the Fourier transformed function, and we distinguish the Fourier transformed quantity only by displaying the different argument. 231 we obtain da(AB —> V(—+ 11")X) _ 1 Q? do? dy dog. day dcosocza W, _ 967r2S (Q? — M3,)? + [VIE/Ff, 1 .~ ~ 1 ~~ __ d2b zQT-b E : (_ / d2 I —:Q -b W I ) X {(2%)2/ e ch 27f QTe T ]k(QT1Q1$A1$3101¢) X lefclp(b1Q1$/41$B)} (B9) The b integral can formally be carried out, resulting in the appearance of the inverse Fourier transform of the non-perturbative function. Introducing the convolution no- tation for the remaining integral, the first part of the resummation formula in QT space simplifies to d0(AB —+ V(—> lf’)X) _ 1 Q2 dQ2 dy dezr dqbv dcosfldqfi r684, _ 967r23 (Q2 — M3,)2 + Mfif‘f, 1 ~ X W (ij 8’ WJ'IZP) (QTinmAava 9,¢) (B-IO) where (X) denotes the convolution and is defined by (ij ® WJ'IZP) (QT1Q1 (It/11333161 ¢) = Z / szfl‘ ij(Q”T1 Q1 $141331 01 (b) WJ'IZP(QT — Q’Ta Q1$A1 $3). (B-ll) 3,1: Eq. (B.10) expresses that the resummed cross section at low QT region is given by the convolution of the exponentiated singular part of the hard scattering cross section with the non-perturbative function. It can be viewed as if the hard cross section were smeared by the non-perturbative function. The non-perturbative function is approximated with the flavor independent, Gaus- sian parametrization [73, 72] SNP(b1Q1Q01x/l1 $8) : 01(33/11 $3”) + G2(Q1Q0)b2 With 01 = 91931n (100:1:Ax3) and G2 = g1+ g21n(-2%-) (B.12) o 232 Recent analysis gives 91 = 0.11GeV2, g2 = 0.58 GeV2 , 93 = -—1.5GeV_1 and Q0 2 1.6GeV [73]. The Fourier transform of the above non-perturbative function (for simplicity taken without the linear term) is 1 W”P(QT.Q,QO) = men) {—Qi/mxaoa}. (B13) This is the form that we used to smear the direct photon NLO distribution. The resummation formula (B.10) indicates that the effect of resumming the initial state soft gluon radiations is twofold. First, it exponentiates contributions that are singular as QT —+ 0 and therefore contain large logs (1n(QT/Q)) of the finite order cross section, into the Sudakov form factor exp(—S). Second, it smears the above perturbative Sudakov form factor with the non-perturbative function WNP. The amount of the smearing provided by the non-perturbative function is larger than the amount of the intrinsic kT because the non-perturbative function contains parts of the perturbative multiple soft gluon emission effects in the low QT region. Note that the separation between the perturbative Sudakov form factor and the non-perturbative function depends on the arbitrary parameter bmax used in the CSS formalism. Hence, the non-perturbative function also contains some perturbative physics. We expect a similar underlying mechanism of the initial state radiation to be present in the direct photon process. That would alter the shape of the transverse momentum distribution of the direct photon in the low pT region. This is proven in the agreement between the experimental data and the predicted low-pT distribution after including the smearing effects due to the non-perturbative part of the CSS resummation formalism. For the direct photon process, however, the color flow of the initial state partons is different from that for the Drell-Yan process. This slightly complicates the above resummation results and affects the non-perturbative function given above. Even though, for example, the value of the 9,- parameters may change, 233 the form of the non-perturbative function should be the same. Our phenomenological parameter a: partly accounts for these changes. At low energies, due to lack of phase space, the multi gluon initial state radiation is suppressed. Mathematically this means that the contribution of the perturbative Sudakov form factor is less important. In the low energy region, expansion of W in term of as can be approximated by the first few terms, since ln(QT/Q) will not be large. (The two characteristic scales QT and Q are of the same order in magnitude.) The first few terms in the expansion of W are the singular parts of the fixed order cross section, which dominate the low QT behavior. In the CSS resummation formalism, this singular part is smeared by the non-perturbative function. This suggests that one can improve the theoretical prediction of a fixed order perturbative calculation at the low pT region by smearing it with the non-perturbative function described above. This is indeed what we see in Table 1 for low energy fixed target data. According to the resummation formula (B.3) the effect of the non-perturbative function on W is larger at the low QT (large b) end. As QT increases (b decreases) the non-perturbative function approaches unity. The Sudakov form factor, on the other hand, increases its contribution with increasing QT (decreasing b), which becomes evident after expanding it in as: S(b, Q) = % ln (2?) [A1 ln (bi?) + Bl] + 0(0‘3). (B.14) This means that at higher direct photon pT region the effect of the Sudakov factor would also be important. This effect might just cancel the smearing effect of the non- perturbative function leaving the higher pT data virtually unchanged, which would explain why kT smearing has to be performed only in the lowest pT regions. For data from experiments with higher center of mass energies, the initial state multiple gluon radiation effects are important. This also means that the perturbative 234 Sudakov form factor is important. In this case, the full resummation calculations should be carried out to describe the p7 distribution of the direct photon produced at high energy hadron colliders, such as the Tevatron. For a rough estimate of the effect of a Sudakov type of form factor at high energy direct photon experiments we turn to the resummation formula again. We observe that writing (B.3) into a form of a convolution can be repeated grouping terms in a slightly different way. Taking the Sudakov factor out from W and putting it into WNP, the argument that leads to the convolution form (B.10) can be repeated without a change. Now that the smearing function contains the Sudakov part we ask the question, whether this effective smearing function has a broader width, that is whether it provides an additional smearing what the collider experiments need. To answer this we have to evaluate the (two dimensional) Fourier transform of the non-perturbative function multiplied with the Sudakov factor evaluated at b... The simplest way of doing this is to perform the integrations numerically. The result agrees with our expectations. The effective smearing function has an approximate Gaussian shape and a width Aeff larger than of the non-perturbative function, with the increase depending on the center of mass energy. We can interpret this result as an indication that the missing pT smearing may well come from the Sudakov form factor. To further illustrate the power of the Fourier transformed form of the resumma- tion formula, we attempt some simple estimates on observables that are related to the transverse momentum of the W-boson in hadron collisions. Consider the average transverse momentum (QT) of the weak vector boson produced in hadron collisions. Inserting the Fourier transform (B.13) of the non-perturbative function into formula (B.10) the only QT dependence of the right hand side comes from the exponential term of the non-perturbative function exp(—(QT -— Q’T)2 / 4G). Since the largest con- tribution in the Q’T integral comes from the region where Q’T ~ QT we approximate 235 9 ' p pBar -> W 16 . Legacy preliminary Legacy preliminary y = 0.0 y = 0.0 V O r I _A d M b r 1 Average 01- (GeV) 8 Average 01 (GeV) O I O 1 at r 5 " 1 4 101 1o2 0 (GeV) Ecru (GeV) Figure B.2: a) The invariant mass versus the average Wi boson transverse momentum in the resummed calculation. b) The center of mass energy versus the average W:h boson transverse momentum in the resummed calculation. the Q’T2 term by Q%. To calculate the average transverse momentum we make use of the integral £1 Qi’ Under the above approximations we obtain that the average QT is proportional to [0 +°° dQ'T 0’. exp {Q’TQr/2G2} = (8.15) G2(Q,Q0). (A more elaborate numerical calculation supports this.) The result is given on Fig. B.2a. We find that the average QT grows as log of Q. Using the first order equality Q = m we can obtain a similar conclusion for the dependence of the average transverse momentum on the center of mass energy of the hadron colliders. Of course, the S dependence will not be purely logarithmic, since ij (and a more thorough parametrization of the non-perturbative function) makes it more complicated. Nevertheless, as Fig. B.2b shows, even the S dependence of (QT) is not very far from being logarithmic. This result differs from what one would infer from a finite order perturbative calculation which would predict that (QT) is proportional to Ecm = x/S. In conclusion, we have shown that the discrepancy between the theoretical pre- 236 diction and the experimental data of the pp distribution of the direct photon can be resolved for the fixed target experiments after smearing the NLO calculation by the non-perturbative function that is suggested by the CSS resummation formalism. To improve the phenomenological method described in this work for high energy collider experiments, one needs to include the effects on pT distribution of direct photons from the perturbative Sudakov form factor in the CSS resummation formalism. 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