RESBOS2: PRECISION RESUMMATION FOR THE LHC ERA
By
Joshua Paul Isaacson

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics—Doctor of Philosophy
2017

ABSTRACT
RESBOS2: PRECISION RESUMMATION FOR THE LHC ERA
By
Joshua Paul Isaacson
With the precision of data at the LHC, it is important to advance theoretical calculations
to match it. Previously, the ResBos code was insufficient to adequately describe the data
at the LHC. This requires an advancement in the ResBos code, and led to the development
of the ResBos2 package. This thesis discusses some of the major improvements that were
implemented into the code to advance it and prepare it for the precision of the LHC.
The resummation for color singlet particles is improved from approximate NNLL+NLO
accuracy to an accuracy of N3 LL+NNLO accuracy. The ResBos2 code is validated against
the calculation of the total cross-section for Drell-Yan processes against fixed order calculations, to ensure that the calculations are performed correctly. This allows for a prediction
of the transverse momentum and φ∗η distributions for the Z boson to be consistent with the
√
data from ATLAS at a collider energy of s = 8 TeV. Also, the effects of choice of resummation scheme are investigated for the Collins-Soper-Sterman and Catani-deFlorian-Grazzini
formalisms. It is shown that as long as the calculation of each of these is performed such that
the order of the B coefficient is exactly 1 order higher than that of the C and H coefficients,
then the two formalisms are consistent. Additionally, using the improved theoretical prediction will help to reduce the theoretical uncertainty on the mass of the W boson, by reducing
the uncertainty in extrapolating the dσW distribution from the data for the dσZ distribution
dpT

dpT

by taking the ratio of the theory predictions for the Z and W transverse momentum.
In addition to improving the accuracy of the color singlet final state resummation calculations, the ResBos2 code introduces the resummation of non-color singlet states in the

final state. Here the details for the Higgs plus jet calculation are illustrated as an example
of one such process. It is shown that it is possible to perform this resummation, but the
resummation formalism needs to be modified in order to do so. The major modification that
is made is the inclusion of the jet cone-size dependence in the Sudakov form factor. This
result resolves, analytically, the Sudakov shoulder singularity. The results of the ResBos2
prediction are compared to both the fixed order and parton shower calculations. The calculations are shown to be consistent for all of the distributions considered up to the theoretical
uncertainty. As the LHC continues to increase their data, and their precision on these observables, the ability to have analytic resummation calculations for non-color singlet final
states will provide a strong check of perturbative QCD.
Finally, the calculation of the terms needed to match to N3 LO are done in this work.
Once the results become sufficiently publicly available for the perturbative calculation, the
ResBos2 code can easily be extended to include these corrections, and be used as a means
to predict the total cross-section at N3 LO as well.

Dedicated to: My parents and Laurel DiPucchio.

iv

ACKNOWLEDGMENTS

Throughout the process of obtaining my Ph.D, there have been many influential people to
help me along the way. My thanks goes to all of you who have helped me along the way to
achieve my dream of becoming a physicist.
My parents have played an important role in helping me to learn all of the skills I would
need to be successful in school from an early age. Additionally, they have always been there
for me in times of struggle throughout my education. I would not have been able to make it
through graduate school without them.
Also, Laurel DiPucchio has been a great support in my times of struggle throughout this
process. I know that I have caused her so much stress with this work, and I couldn’t be more
thankful to have her with me throughout this journey. I am thankful for all the sacrifices
she has made for me and continues to make for me.
I would also like to thank my thesis advisor, C.-P. Yuan, for giving me the tools needed
to preform this research, and as a guiding hand throughout. The main discussions that we
have had over the years has helped to develop me as a physicist, and taught me how to
approach complex research questions in a way to make them less daunting. C.-P.’s guiding
idea that his Ph.D students should teach him something new by the end of their work, has
really helped me to understand the topics that are presented in this thesis to the depth that
I now do, along with many other topics that I have learned along the way. His dedication
to always learning new ideas being presented in the community, and to develop my skills
in such a way to provide a needed insight into the world of physics has become my guiding
principle as I move forward in my career.
In addition to C.-P., I would like to acknowledge the help from the rest of the professors
v

in the High Energy Group at MSU. Specifically, Wade Fisher, Joey Huston, Carl Schmidt,
Wayne Repko, Andreas von Manteuffel, and Chip Brock, for their discussions of physics
topics throughout my career at MSU. Wade always had an open door to come and discuss
topics with, and his knowledge of statistics has been invaluable to me. Joey Huston has
always made sure that I know what is going on in the community that I can contribute to,
and has helped me to understand the way experimentalists think about the same problems
I am trying to tackle. This has allowed me to approach the problems, and the solutions in
a way that benefits not only the theorists, but are also helpful to experimentalists. Carl
Schmidt has always been available to discuss the details of calculations with when I would
get stuck on some of the more technical details. Wayne Repko was always interested to
learn about the status of my work, and to provide thought provoking questions to myself
and officemates that really made us think about the underlying physics. While Andreas
von Manteuffel arrived late in my career, his helpful discussions since his arrival has helped
me to understand the complexities of higher loop calculations, and has given me many
interesting directions of research as I begin my post doctoral career. Finally, Chip Brock
has been paramount in helping me develop my communication skills with experimentalists
throughout our discussions over the intricate details of Parton Distribution Functions and
their need to be improved for the precision of the LHC.
My coworkers throughout my time at MSU have been a valuable resource in terms of
learning what is happening in the field, outside my immediate focus. I would especially like
to thank Kirtimaan Mohan, Jan Winter, and Chris Willis. Kirtimaan has always provided
advice when I would be struggling with my own research, and also helped me to learn about
effective field theories and developing Beyond the Standard Model theories to understand the
anomalies found at the LHC. Chris was the experimentalist who I had the most discussions
vi

with, and without him I would not have the same level of appreciation for the work the
experimentalists in this field do that I now have. Finally, Jan has been helpful in terms of
helping me to work through different difficult situations that have arisen while trying to get
my Ph.D. He always has ideas on things to look into, and is always willing to help look into
the setup of the numerical calculations to make sure everything comes out correctly. His
help with getting the Sherpa predictions throughout this work has been invaluable.
Finally, I would like to thank Kim Crosslan and Brenda Wenzlick for being the amazing
people that they are. Both of them have been helpful in navigating the bureaucracy of MSU
and the wider physics community,

vii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

Chapter 1 Standard Model . . . . . . . . . . . .
1.1 Electroweak . . . . . . . . . . . . . . . . . . .
1.1.1 Higgs Mechanism . . . . . . . . . . . .
1.1.2 Broken Electroweak Sector . . . . . . .
1.1.3 Electroweak Precision Observables . .
1.1.4 Electroweak Parameters . . . . . . . .
1.2 Renormalization and Regularization . . . . . .
1.3 Quantum Chromodynamics . . . . . . . . . .
1.3.1 Introduction . . . . . . . . . . . . . . .
1.3.2 Asymptotic Freedom and Confinement
1.3.3 Soft and Collinear QCD . . . . . . . .
1.3.4 Factorization . . . . . . . . . . . . . .

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1
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32
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Chapter 2 Experimental Measurements
2.1 Basic Concepts . . . . . . . . . . . . .
2.2 Detecting Particles . . . . . . . . . . .
2.2.1 Trackers . . . . . . . . . . . . .
2.2.2 Electromagnetic Calorimeters .
2.2.3 Hadronic Calorimeters . . . . .
2.2.4 Muon Spectrometers . . . . . .
2.3 Physics Objects . . . . . . . . . . . . .
2.3.1 Jets . . . . . . . . . . . . . . .
2.3.1.1 B-Tagging . . . . . . .
2.3.2 Leptons . . . . . . . . . . . . .
2.3.3 Missing Transverse Energy . . .

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44
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Chapter 3 Resummation . . . . . . . . . . . .
3.1 Collins-Soper Frame . . . . . . . . . . . . .
3.2 Fixed-Order Calculations . . . . . . . . . . .
3.2.1 Real Corrections . . . . . . . . . . .
3.2.2 Virtual Correction . . . . . . . . . .
3.3 NLO DY Total Cross-Section . . . . . . . .
3.3.1 Breakdown of Fixed-Order . . . . . .
3.4 Resummation Formalisms . . . . . . . . . .
3.4.1 Collins-Soper-Sterman Formalism . .
3.4.2 Catani-deFlorian-Grazzini Formalism

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58
59
63
66
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80

viii

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3.5

3.6

3.4.3 Comparison of CSS to CFG . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.4 Non-Perturbative Contribution . . . . . . . . . . . . . . . . . . . . . 83
The Asymptotic Piece and Obtaining Fixed-Order Cross-Sections from Resummed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.1 O (Îąs )
 Singular, Asymptotic Piece, and NLO Result . . . . . . . . . . 94
3.5.2 O Îąs2 Singular, Asymptotic, and the NNLO Total Cross-Section . . 97
Scale Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Chapter 4 Z Boson Resummed Predictions
4.1 Z pT Distribution . . . . . . . . . . . . . .
4.2 Z φ∗η Distribution . . . . . . . . . . . . . .
4.3 Angular Functions . . . . . . . . . . . . .

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105
105
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113

Chapter 5 W Mass . . . . . . .
5.1 SM EW Precision Fit . . . .
5.2 Experimental Measurement
5.3 ResBos2 Results . . . . . . .

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119
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125

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Chapter 6 Color Singlet Boson Plus Jet Resummation
6.1 Higgs Plus Jet Resummation . . . . . . . . . . . . . . .
6.1.1 Virtual Corrections . . . . . . . . . . . . . . . .
6.1.2 Jet Corrections . . . . . . . . . . . . . . . . . .
6.1.3 Collinear Corrections . . . . . . . . . . . . . . .
6.1.4 Soft Corrections . . . . . . . . . . . . . . . . . .
6.1.5 Total Real Correction . . . . . . . . . . . . . . .
6.1.6 Resummation Calculation . . . . . . . . . . . .
6.1.7 Results . . . . . . . . . . . . . . . . . . . . . . .
6.1.7.1 Comparison to Fixed Order . . . . . .
6.1.7.2 Comparison to Parton Showers . . . .
6.1.8 Future of Higgs Plus Jet Resummation . . . . .

at Hadron
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Colliders128
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Chapter 7 Conclusion . . . . . . .
7.1 Color Singlet Resummation . .
7.1.1 Resummation Schemes .
7.1.2 Z Boson Predictions . .
7.1.3 W Boson Predictions . .
7.2 Non-Color Singlet Resummation
7.3 Final Remarks . . . . . . . . . .

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APPENDICES . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX A The Dirac Equation and Îł Matrices . .
APPENDIX B Spin, Helicity, and Chirality . . . . . .
APPENDIX C Standard Model Higgs Boson and Higgs
APPENDIX D Structure of SU (3) . . . . . . . . . . .
APPENDIX E QCD Feynman Rules . . . . . . . . . .
ix

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158
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. . . . . . .
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Mechanism
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163
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APPENDIX F Calculation Details . . . . . . . . . . . . . . . . . . . . . . . . . . 191
APPENDIX G Higgs Plus Jet Resummation Calculation Details . . . . . . . . . . 218
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

x

LIST OF TABLES

Table 1.1: Standard Model of Particle Physics . . . . . . . . . . . . . . . . . . . . . .

3

Table 1.2: Electroweak quantum number for the first family of fermions . . . . . . .

17

Table 1.3: The current best fit results for the S, T, U parameters, along with their
uncertainties and correlation coefficients. Reproduced from [1] . . . . . .

18

Table 2.1: Representative cross section scales (cross sections are not exact). Reproduced from Ref. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Table 3.1: The organization of fixed-order and resummed calculations. Going across
a row is the fixed-order calculation included at a given order of Îąs , while
going down the columns is the resummed calculation up the the given power
of the logs included. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Table 3.2: The non-perturbative functions parameters fitting results. Here, Nf it is
the fitted normalization factor for each experiment. . . . . . . . . . . . .

86

Table 3.3: Total inclusive NLO Drell-Yan Cross-sections for 66 GeV < Mll < 116
GeV. MCFM calculation using version 8.0 [3]. FEWZ calculation using
version 3.1 rc [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

Table 3.4: Total inclusive NNLO Drell-Yan Cross-sections for 66 GeV < Mll < 116
GeV. MCFM calculation using version 8.0 [3]. FEWZ calculation using
version 3.1 rc [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Table 5.1: Electroweak fit. The fourth column gives the fit results without using
any experimental or phenomenological estimate for the parameter when
performing the fit. The fifth column is the same as the fourth, but ignores
all theory uncertainties. The table is reproduced from Ref. [5]. . . . . . . 121
Table 5.2: The values are reported in GeV, and taken from Ref. [5] . . . . . . . . . . 121
Table F.1: Table of the results of calculating the area of a d-Sphere . . . . . . . . . . 196

xi

LIST OF FIGURES

Figure 1.1: The two different scenarios for the value of Âľ2 . . . . . . . . . . . . . . .

8

Figure 1.2: The best fit in the T S plane with U free to float on the left and U fixed
to zero on the right. Additionally, mH = 126 GeV and mt = 173 GeV in
this fit. Reproduced from [1] . . . . . . . . . . . . . . . . . . . . . . . .

19

Figure 1.3: An example of a loop diagram, p1 , p2 , and p3 are the momentums of the
external particles, and k is the momentum of the loop. . . . . . . . . . .

21

Figure 1.4: The first order correction to the strong coupling constant can be calculated
from this vertex correction diagram. . . . . . . . . . . . . . . . . . . . .

29

Figure 1.5: The comparison between the theoretical calculation of the strong coupling
compared to experimental data. The plot is reproduced from Ref. [6] . .

32

Figure 1.6: Electron scattering off of a proton with an energy of 188 MeV. Results
show data is inconsistent with a point-like proton. Reproduced from [7].

33

Figure 1.7: Depiction of the F2 proton structure function, showing Bjorken scaling.
This was the motivation for the parton model proposed by Richard Feynman. Plot is reproduced from [8]. . . . . . . . . . . . . . . . . . . . . . .

34

Figure 1.8: Test of the Callan-Gross relation, with K0 = F2 /(2xF1 ) − 1. Results are
consistent with spin-1/2 predictions. Reproduced from [9]. . . . . . . . .

36

Figure 1.9: The splitting kernels that occur at Îąs . Note that to obtain the missing
kernels, take fermions to anti-fermions and vice-versa. . . . . . . . . . . .

37

Figure 1.10: Plots of the CT14 NNLO Parton Distributions at 2 GeV (left) and 100
GeV (right). The plots are reproduced from Ref. [10] . . . . . . . . . . .

40

Figure 1.11: (a) Anti-kt algorithm. (b) kt Algorithm. (c) Cambridge-Aachen algorithm. Reproduced from Ref. [11] . . . . . . . . . . . . . . . . . . . . .

43

Figure 2.1: Depiction of particles traveling through the CMS Detector. Reproduced
from Ref. [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Figure 2.2: Fractional energy loss per radiation length in lead as a function of electron
or positron energy. Reproduced from Ref. [6] . . . . . . . . . . . . . . . .

50

xii

Figure 2.3: Performance curves obtained for different b-tagging algorithms using simulated data. (Left) The mis-identification for light jets. (Right) The
mis-identification for c jets. Plots are reproduced from Ref. [13] . . . . .

55

Figure 3.1: Depiction of the Collins-Soper Frame. Reproduced from [14]. . . . . . . .

60

Figure 3.2: The diagrams that contribute to the Îąs correction to Drell-Yan. (a) is
the born diagram. (b) is the virtual coupling correction. (c) is the wave
function correction. (d) and (e) are the real corrections . . . . . . . . . .

64

Figure 3.3: Plot showing fixed-order versus resummed predictions. The dashed curves
correspond to fixed-order calculations, and the solid curves correspond to
resummed predictions. The plot is reproduced from Ref. [15] . . . . . . .

73

Figure 3.4: A diagrammatic representation of the factorized cross-section for DrellYan, broken into a soft, collinear, and hard factor. The soft factor is
labeled by the S, the collinear factors are labeled by the C’s, and the
hard factor is labeled by the H . . . . . . . . . . . . . . . . . . . . . . .

75

Figure 3.5: Fit to the differential cross section for Drell-Yan lepton pair production
in hadronic collisions from E288 Collaboration [16]. . . . . . . . . . . . .

87

Figure 3.6: Fit to the Drell-Yan data from the E605 Collaboration [17]. . . . . . . .

88

Figure 3.7: Fit to the Drell-Yan data from the R209 Collaboration [18]. . . . . . . .

88

Figure 3.8: Fit to the Tevatron Run I data from the CDF and D0 Collaborations [19,
20]. The fits include only the A(1,2) , B (1,2) , and C (1) contributions. . .

89

Figure 3.9: Fit to the Tevatron Run II data from the CDF and D0 Collaborations [21,
22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Figure 3.10: Comparison of the SIYY1 and SIYY2 fit to the ATLAS data. . . . . . .

89

Figure 3.11: ∆χ2 distribution scanning g2 parameter in SIYY1 fit (left), and ∆χ2
distribution scanning g2 parameter in SIYY2 fit (right). . . . . . . . . .

90

Figure 4.1: Comparison of pll
T data from ATLAS compared to ResBos predictions,
reproduced from Ref. [23] . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Figure 4.2: Comparison of the ResBos2 calculation with a scale choice of Âľ2 = Mll2 +p2T
to the ATLAS data given in Ref. [23], focusing on the high transverse
momentum region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

xiii

Figure 4.3: Comparison of the CSS and CFG formalisms, focusing on only the small
transverse momentum, and not including the Y -piece. . . . . . . . . . . . 108


Figure 4.4: Comparison of the fixed order piece up to O Îąs2 to the asymptotic piece


up to O Îąs2 . The cutoff on the transverse momentum of both calculations
is set to 2 GeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108


Figure 4.5: Comparison of the Asymptotic piece to O Îąs3 to the N3 LL resummed
piece. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Figure 4.6: Comparison of both the CSS(left) and CFG(right) prediction to data. The
PDF and scale uncertainty is given by the error bands. The lighter error
band is the PDF uncertainty, and the darker error band is the combination
of the scale and PDF uncertainty. . . . . . . . . . . . . . . . . . . . . . . 109
Figure 4.7: Comparison of the fixed order calculation of Z+jet to NNLO compared
to the ATLAS data. Reproduced from [24]. . . . . . . . . . . . . . . . . 111
Figure 4.8: The frame definition for φ∗η , reproduced from [25]. . . . . . . . . . . . . . 111
Figure 4.9: Comparison of φ∗η data from ATLAS compared to ResBos predictions,
reproduced from Ref. [23] . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Figure 4.10: Comparison of both the CSS(left) and CFG(right) prediction to data. The
PDF and scale uncertainty is given by the error bands. The lighter error
band is the PDF uncertainty, and the darker error band is the combination
of the scale and PDF uncertainty. . . . . . . . . . . . . . . . . . . . . . . 113
Figure 4.11: Comparison of the fixed order calculation of Z+jet to NNLO compared
to the ATLAS data. Reproduced from [24]. . . . . . . . . . . . . . . . . 114
Figure 4.12: The predictions for A2 − A0 for the DYNNLO code [26], at NLO and
NNLO. Reproduced from [27]. . . . . . . . . . . . . . . . . . . . . . . . . 116
Figure 4.13: Theoretical prediction from the ResBos2 program for the angular coefficients of A0 , A2 , A4 , and the breaking of the Lam-Tung Relationship for
the CSS Scheme. The lighter error band is the PDF uncertainty, and the
darker error band is the combination of the scale and PDF uncertainty. . 117
Figure 4.14: Theoretical prediction from the ResBos2 program for the angular coefficients of A0 , A2 , A4 , and the breaking of the Lam-Tung Relationship for
the CFG Scheme. The lighter error band is the PDF uncertainty, and the
darker error band is the combination of the scale and PDF uncertainty. . 118

xiv

Figure 5.1: Results of the EW Fit, reproduced from Ref. [5] . . . . . . . . . . . . . . 123
Figure 5.2: The ratio of the normalized transverse momentum distributions of the W
boson to the Z boson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Figure 5.3: An example of the χ2 distribution for an error set, others are similar to
this. The central input mass was MW = 80.358 GeV. The minimum here
occurs around 80.372, a shift of 14 MeV. . . . . . . . . . . . . . . . . . . 127
Figure 6.1: Feynman diagrams for the gluon-gluon scattering channel to a Higgs Boson.135
Figure 6.2: Rapidity dependence of the hard factor for Higgs+Jet for different scale
choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Figure 6.3: The differential cross sections of Higgs boson plus one jet production at
the LHC as functions of the total transverse momentum q⊥ (left) and
the Higgs boson transverse momentum PH⊥ (right). The resummation
predictions (resum) with resummation scale set to be PJ⊥ (black line) and
Q (green line) respectively are compared to the LO result from MCFM
(pink line) with non-zero q⊥ , and the NLO result from MCFM (red line)
which is the production rate of Higgs boson plus two separate jets up to
one-loop in QCD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Figure 6.4: The central predictions(left panel) and with theoretical uncertainties (right
panel) for the inclusive jet multiplicities as predicted by fixed-order calculations, resummed calculations, NNLO and NLO Monte Carlos. The
bottom panel is divided up into three subplots all showing the ratios with
respect to the POWHEG NNLOPS prediction. The upper of these plots contains the HEJ and SHERPA NNLOPS ratios, while the middle one includes all
NLO merged predictions (Madgraph5 aMC@NLO, Herwig 7.1 and SHERPA)
and the lower one shows all those listed in the bottom left legend of the
main panel. Reproduced from [28]. . . . . . . . . . . . . . . . . . . . . . 151
Figure 6.5: The Higgs boson transverse momentum in the presence of at least one
jet central predictions(left) with uncertainty bands(right). The ratio plot
panel is divided into six parts where the upper four exhibit the ratios to
the POWHEG NNLOPS result while the lower two show them to the NLO
calculation for h + 1 jet as provided by GOSAM+SHERPA. Reproduced
from [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

xv

Figure 6.6: The leading jet transverse momentum distribution for h + ≥ 1-jet production, to the right (left) shown with (without) the uncertainty bands provided by the various calculations. The part below the main plot contains
four ratio plots taken wrt. the NNLO result of the BFGLP group following
the same strategy for grouping the predictions as before (NNLOPS versus
NLO ME+PS versus fixed-order and resummation results). Reproduced
from [28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Figure 6.7: The rapidity distribution for the leading jet in h + ≥ 1-jet production,
shown without (left) and with (right) theoretical uncertainties. Ratio plots
are displayed in the lower part of the plot using the POWHEG NNLOPS result
for Higgs boson production as their reference. Predictions are grouped,
from top to bottom, according to the categories NNLOPS, ME+PS at NLO
and NLO fixed order as well as resummation. Reproduced from [28]. . . 154
Figure 6.8: The transverse momentum of the Higgs-boson-leading-jet system in the
presence of at least one jet. For better visibility, results are shown without
(left) and with (right) theoretical uncertainties. The plot layout exactly
corresponds to that of Figure 6.7, except for the extended ŷ-axis range in
the ratio plots. Reproduced from [28]. . . . . . . . . . . . . . . . . . . . 156
Figure 6.9: Noramlized distributions of the vector boson fusion and gluon-fusion contributions to the Higgs boson
√ plus two jets production in the typical kinematics at the LHC with S = 13T eV , where the jet transverse momenta
k1⊥ = k2⊥ = 30GeV , yj1 = −yj2 = 2 and yh = 0: as functions of the
total transverse momentum q⊥ (left); the total rate as function of the
upper limit of q⊥ (right). Reproduced from [29]. . . . . . . . . . . . . . 157
Figure F.1: The hadronic virtual correction Feynman Diagram to Drell-Yan at NLO.

193

Figure F.2: Multiplicity distribution as function of transverse momentum in semiinclusive hadron production in deep inelastic scattering compared to the
experimental data from HERMES Collaboration at Q2 = 3.14GeV2 . . . . 203
Figure F.3: Connection of a soft/collinear gluon between two colored particles.

xvi

. . . 214

Chapter 1
Standard Model
The Standard Model (SM) of Particle Physics is almost a complete description of all of the
phenomena observed in high energy physics, including interactions of the strong and electroweak forces. With the discovery of the Higgs Boson on July 4th, 2012 [30, 31, 32], the
standard model is now complete. Although experimental data appears to agree with the
SM quite precisely, there are certain phenomena that cannot be explained in the framework
of the SM (such as gravity [33], dark matter [34, 35, 36], neutrino masses [37], and the
matter-antimatter asymmetry [38]). Additionally, there are some problems that arise from
a theoretical standpoint, such as the hierarchy problem [39, 40] and the strong CP problem [41]. Therefore, the SM is anticipated to be an effective theory of some complete higher
energy theory. The precision of agreement between the experimental results and theoretical
predictions forces the discovery of any new physics to either be through searches at high
energies or through the study of small effects on specific observables. To study small effects
on observables requires precise experimental measurements. Observables that can be measured precisely and that are sensitive to the parameters of the SM are considered precision
observables. One such observable is the g − 2 of the electron, which is currently measured
to a precision of ≈ 10−13 and calculated theoretically to a similar precision. Despite the
success of the SM, there are many unanswered questions pertaining to particle physics, such
as the exact nature of the Higgs Boson, and what new physics can explain dark matter. In

1

order to search for new physics, the parameters of the SM must be measured as precisely as
possible and any deviations from the expected value will hint at the presence of new physics.
Calculational accuracy of the SM can be improved in either of two forces: the electroweak
force, which unifies both the weak and electromagnetic forces (details can be found in Section 1.1), and the strong force (details can be found in Section 1.3). These predictions are
calculated using Quantum Field Theory (QFT). QFT is a mathematical framework based
on the physically motivated assumptions that the theory incorporates Poincare invariance,
point-particles, local interactions (no actions at a distance), causality, unitarity (the quantum mechanical evolution conserves probability), and is free of divergences (also known as
a renormalizable theory) [42]. These restrictions define the underlying principles of the
standard model.
The standard model is traditionally expressed via the Lagrangian formalism, in which
all physical interactions are encoded into a Lagrangian. The form of the Lagrangian, is
determined by demanding the Lagrangian be invariant under a set of local symmetries.
These symmetry transformations correspond to locally conserved quantities as detailed by
Noether’s Theorem [43]: Noether’s theorem states that for each symmetry of the Lagrangian
there is a corresponding conserved quantity of the form,

∂L
γι ,
∂ q̇α

(1.1)

where L is the Lagrangian density, q̇α is the time derivative of a coordinate, and γα is
some function (note the use of Einstein summation notation). Through Noether’s theorem,
quantities that are experimentally conserved are used to introduce symmetries of the SM
Lagrangian and greatly constrain its form. The SM symmetries are the Poincare symmetry,

2

First Generation
u
d
e
νe

Fermions
Second Generation
c
s
Âľ
ν¾

Third Generation
t
b
τ
ντ

Bosons
Îł
g
W Âą, Z
H

Table 1.1: Standard Model of Particle Physics

SO(3, 1), and the gauge symmetries SU (3)C ×SU (2)L ×U (1)Y . Generally, a gauge symmetry
is a symmetry that is space-time dependent, or local. Gauge symmetries are responsible for
all of the SM forces and as such can be broken into two groups: the SU (2)L × U (1)Y gauge
groups generate the electroweak force, while the SU (3) gauge group generates the strong
force.
In addition to the gauge symmetries that define the forces of the standard model, the
fermionic content of the standard model needs to be introduced by hand, but must still obey
the symmetries of the standard model. Therefore, for a particle to interact with the gauge
field it must be in a non-singlet representation of the group, or in other words, has to have
a special transformation property under the group symmetry1 . The full particle content of
the standard model can be found in Table 1.1. The details for the electroweak sector can be
found in the following section (Sec. 1.1), and the details of the strong force can be found in
the section on QCD (Sec. 1.3).
The following sections discuss the Electroweak sector and the QCD sector of the SM. Additionally, in the Electroweak section the Higgs Boson and spontaneous symmetry breaking
is discussed. The symmetry groups that make up the SM will be discussed, along with the
experimental evidence that supports these observations through the discussion of conserved
1 In the SM outside of the gauge fields and the Higgs boson, all other particles are fermions. Details of
the equation of motion for fermions can be found in App. A.

3

quantities as described in Noether’s theorem.

1.1

Electroweak

The electroweak force is the unification of the electromagnetic force, and the weak nuclear
force. The quantum description of electromagnetism (QED) arose in the 1920s and 1930s,
through work by Dirac, Pauli, Heisenberg, and Fermi [44, 45, 46, 47]. One of the major accomplishments in QED was the calculation of the electron gryomagnetic ratio, known as g−2.
Currently, the most accurate measurement was done by Hanneke et. al. [48], obtaining a
value of g/2 = 1.00115965218073(28). This is consistent with the theoretical prediction given
as: g/2 = 1.00115965218182(6)(4)(2)(78), where the uncertainties in the theoretical prediction, arise from the eighth order QED correction, tenth order QED correction, Hadronic
Electroweak corrections, and the Atomic Physics determination of Îą, respectively [49].
In 1896, Henri Becquerel discovered β-decay [50], based off of the penetration of the
decays through material. In modern language, β decay is the process in which a proton
turns into a neutron and emits a positron and a neutrino. This phenomenon is unable to be
explained in QED, and thus a new theory was needed to explain it. At first, Fermi developed
the four point interaction to describe β-decays, and introduced a new particle known as the
neutrino, in order to conserve momentum [51]. In the 1970s, Weinberg, Glashow, and Salam
developed a theory that could explain both QED and β decay through a single theory,
the Electroweak theory [52, 53, 54]. In addition, it predicted that β decay is mediated
by a W boson, and further predicted the existence of a massive neutral gauge boson, Z.
The first evidence of the neutral current in this theory was discovered by the Gargamelle
collaboration in 1973 in neutrino scattering [55]. The direct detection of the W and Z

4

bosons was first done by the UA1 and UA2 collaborations in 1983 at the Super Proton
Synchrotron [56, 57, 58, 59]. Finally, ’t Hooft and Veltman showed that the Electroweak
theory is renormalizable in 1972 [60]. The discussion that follows explains the theory that
was proposed, along with the definition of renormalizable.
Any fermion can be described by the direction of its spin in relation to the direction of
its momentum. If its spin is parallel to its momentum, then the particle is said to be righthanded, and if its spin is anti-parallel to its momentum then the particle is left-handed2 .
The electroweak (EW) sector of the standard model is described by the gauge symmetries of
SU (2)L ×U (1)Y . Experimentally, it was discovered that the weak force (β-decay for example)
only interacts with left-handed particles. Secondly, the U (1)Y gauge group interacts with
a particle’s hypercharge. The SU (2)L group has three gauge mediating bosons, W i where
i = 1, 2, 3, and the U (1)Y group has one gauge mediating boson, B.
To understand how the gauge bosons interact with the fermions, consider how the infinitesimal gauge transformations affect the fermions. Namely,



Y
0
δψL,R = igTi θi (x) + ig θ(x) ψL,R ,
2

(1.2)

where g and g 0 are the gauge couplings, Ti and Y are the generators of for SU (2)L and
U (1)Y respectively, and θi (x) and θ(x) are space-time dependent transformation variables.
The commutators for the two groups are given by:




Ti , Tj = iijk Tk ,

(1.3)

[Ti , Y ] = 0 for all i,

(1.4)

2 For more details on spin and the difference between spin and helicity one can reference Appendix B

5

where ijk is the fully anti-symmetric tensor in three dimensions, and Ti = σi /2, where σi
are the two dimensional Pauli matrices. A Lagrangian composed of fermionic fields that
remains invariant under this transformation rule is:

/
Lf = iψ̄ Dψ,

(1.5)

/ = Îł Âľ DÂľ , and D is the covariant derivative which
where ψ is the fermionic field, ψ̄ = ψ † γ 0 , D
is defined in such a way to maintain the gauge invariance. The covariant derivative for the
electroweak sector is:
Dµ = ∂µ − igPL Ti Wµi − ig 0

Y
BÂľ .
2

(1.6)

In addition to dynamical fermionic terms, the SM Lagrangian also requires dynamical
bosonic terms. The gauge bosons transform under the adjoint representation of the gauge
groups, according to:

δWµi = ∂µ θi (x) − gijk θj (x)Wµk ,

(1.7)

δBµ = ∂µ θ(x).

(1.8)

The kinetic term for the gauge bosons (which is invariant under the above transformations) is given as:
1
1
LV = − B µν Bµν − W iµν Wiµν ,
4
4

6

(1.9)

where B ¾ν and W i¾ν are known as the field strength tensors given by:

B µν = ∂ µ B ν − ∂ ν B µ ,

(1.10)

W iµν = ∂ µ W iν − ∂ ν W iµ + gijk W jµ W kν .

(1.11)

Because SU (2)L is non-Abelian, the W -boson kinetic Lagrangian contains self-interaction
terms. Additionally, the interaction Lagrangian between fermions and these gauge fields is
given by:
g0
/ R − ψ̄L
Lint = − ψ̄R YR Bψ
2

 0

g YL
g~ ~
/ + T ¡W
/ ψL ,
B
2
2

(1.12)

where ψ̄L/R and ψL/R are the left and right handed fermion wavefunctions, YR/L is the
right- and left-handed hypercharge respectively.
However, these gauge bosons are all massless. Introducing a mass term by hand explicitly
breaks the gauge symmetry. This issue is resolved through the use of the Higgs Mechanism
and spontaneous symmetry breaking.

1.1.1

Higgs Mechanism

The Higgs Mechanism is a means of breaking spontaneously the Electroweak SU (2)L ×U (1)Y
gauge symmetry, and through which massless gauge fields and massless Nambu-Goldstone
bosons combine to produce massive bosons [61, 62, 63, 64]. To illustrate this mechanism,
consider a U (1) gauge example.
A symmetry can be spontaneously broken by simply adding a complex scalar field (ÎŚ),

7

V (φ)

V (φ)

Im(φ)

Im(φ)
Re(φ)

Re(φ)
(a) The unbroken vacuum, i.e. µ2 ≥ 0

(b) The broken vacuum, i.e. Âľ2 < 0

Figure 1.1: The two different scenarios for the value of Âľ2
with a Lagrangian given by:

L = (∂µ Φ)† ∂ µ Φ − µ2 |Φ|2 − λ|Φ|4 .

(1.13)

The physical interpretation of this Lagrangian depends on the sign of Âľ2 . When Âľ2 > 0, the
symmetry remains unbroken. The classical potential of this case is illustrated in Fig. 1.1a.
The minimum of the vacuum state occurs at the origin and furthermore, the Lagrangian
expanded around small oscillations of the minimum reproduces Eq. 1.13. However, when
Âľ2 < 0, the minimum of the vacuum is no longer unique as shown in Fig. 1.1b, with a
minimum given by:
hÎŚ2 i0

¾2 iθ ν 2 iθ
=− e = e ,
2Îť
2

(1.14)

where θ is any real number, for simplicity θ will be taken to be zero. The U (1) degeneracy
of this minima reflects the U (1) symmetry of the initial Lagrangian. Defining a shifted field:

8

Ρ + iΞ
Φ0 = Φ − hΦi0 = √ ,
2

(1.15)

the Lagrangian takes the form of:

1
1
L = (∂µ η)(∂ µ η) + (∂µ ξ)(∂ µ ξ) + µ2
2
2

 4

Ρ
Ρ3 Ρ2Ξ 2
ΡΞ 2
Ξ4
ν2
2
+
+
+Ρ +
+ 2−
(1.16)
ν
ν
4
4ν 2
2ν 2
4ν

and looking at small oscillations about the minimum of the potential (keeping terms that
are quadratic in the fields), the following Lagrangian is obtained:

1
1
LSO = (∂µ η∂ µ η + 2µ2 η 2 ) + ∂µ ξ∂ µ ξ,
2
2

(1.17)

plus a constant term. There are two particles in the Lagrangian, one of which is massless (ξ), and one which has a mass given by mη = −2µ2 > 0. The ξ particle is called a
Nambu-Goldstone Boson(NGB), and the general outcome is known as the Goldstone phenomenon [62, 65]. In general, there exists one NGB for each broken generator of the original
symmetry group. As such, when applied to the Standard Model, the breaking of symmetry
groups of SU (2)L × U (1)Y to U (1)Q will generate three NGBs.
The appearance of NGBs during the Higgs Mechanism significantly impacts the massless
gauge fields. For simplicity, consider the U (1) example again, but now extend the Lagrangian
with a gauged U (1) symmetry:

1
L = |Dµ Φ|2 − µ2 |Φ2 | − λ|Φ|4 − F µν Fµν ,
4

(1.18)

φ +iφ
where Φ = 1√ 2 is a complex scalar field, Dµ is the covariant derivative as defined pre2

9

viously, and F ¾ν is the field strength tensor for a U (1) symmetry as defined in Eqs. 1.9,
1.10. Consider when Âľ2 < 0. In this case, the absolute minimum corresponds to the vacuum
expectation value given as:

h|ÎŚ|2 i0 =

−µ2 iθ ν 2 iθ
e = e .
2Îť
2

(1.19)

To produce a viable quantum theory, the complex scalar field must be expanded around this
minimum. By choosing the vacuum expectation value to be real and positive (θ = 0), and
expanding around the minimum, ÎŚ is given by the following formula:

ÎŚ=

ν + φ1 + iφ2
√
.
2

(1.20)

After plugging this into Eq. 1.18, the Lagrangian becomes:
1
1
1
q2ν 2
L = (∂ µ φ1 ∂µ φ1 + 2µ2 φ21 ) + ∂ µ φ2 ∂µ φ2 − F µν Fµν + qνAµ ∂ µ ξ +
A¾ Aν .
2
2
4
2

(1.21)

In addition to the massive scalar resulting from the Goldstone phenomenon, the ”photon”
gains a mass term. Furthermore, the gauge transformation given by:

1
∂µ ξ,
qν
ν+Ρ
Φ → Φ0 = e−iξ/ν Φ = √ ,
2

Aµ → A0µ = Aµ +

(1.22)
(1.23)

greatly simplifies the Lagrangian to:
1
1
q 2 ν 2 0 0ν
L = (∂µ η∂ µ η + 2µ2 η 2 ) − F µν Fµν +
AÂľ A ,
2
4
2
10

(1.24)

plus terms that are not of interest to this discussion. Note there are only two fields appearing
in the above equation: a massive scalar field (mη = −2µ2 ) and a massive ”photon” field
(mA0 = qν). There is no Nambu-Goldstone Boson field. This is colloquially described as
the massless gauge boson “eating” the Nambu-Goldstone Boson in order to become massive.
Details of how this simple example may be extended to an SU (2) × U (1) symmetry can be
found in Appendix C.
The above mechanism resolves the issue of describing massive gauge bosons in the Standard Model, yet leaves the fermions massless. To resolve this issue, consider a fermionic
mass term:
Lmf = mf ψ̄R ψL + h.c.,

(1.25)

where mf is the mass of the fermion, and h.c. is the hermitian conjugate of the previous term.
The mass term mixes left-handed and right-handed fermions. This explicitly breaks the
SU (2)L symmetry because ψ̄R ψL does not form an SU (2)L singlet, and thus the Lagrangian
does not respect SU (2)L symmetry. This can be resolved by introducing a Higgs doublet
under transformations of SU (2),


Ό=

φ+
v+H+iφ0
√
2



,

(1.26)

which is already in the appropriate form to expand about a classical vacuum and subsequently
including the following SU (2)L invariant interaction in the SM Lagrangian:

Lint = y ψ̄R Φ† ψL + h.c.

11

(1.27)

When expanded, the above interaction generates terms that are related to the interaction
of the fermion to the Higgs boson, and the Goldstone bosons associated with the W and
Z bosons, along with a mass term. The interactions with the Goldstone bosons can be
absorbed into the W and Z interactions, and are discussed below. The interaction of the
Higgs boson with the fermion and the fermion’s mass term are given by”

Lint = yu ūR

(v + H + iφ0 )
√
uL + h.c.
2

(1.28)

This includes a fermionic mass term (compare the above to Eq. 1.25), where the mass may
be identified as mu = yu √v .
2

Therefore, the Higgs Mechanism introduces masses into the Standard Model without
entirely destroying the symmetry structures admitted by gauge symmetries.

1.1.2

Broken Electroweak Sector

After symmetry breaking through the Higgs Mechanism, the gauge bosons now obtain a
mass. Defining the electric charge as:

Q = T3 +

Y
,
2

(1.29)

where T3 is the third component of the weak isospin generator, Y is the hypercharge, and Q
is the electric charge of the operator. Applying the isospin, hypercharge, and electric charge
operators to the ground state of the SU (2)L ⊗ U (1)Y after spontaneous symmetry breaking

12





 0 
hφi0 = 
gives the following results:
√ 
ν/ 2



√
0 1  0  ν/ 2
=
 √  = 

1 0
ν/ 2
0


 

√
0 −i  0  −iν/ 2
=
 √  = 

i 0
ν/ 2
0


 

1 0   0   0 
=
 √  = 
√ 
0 −1
ν/ 2
−ν/ 2


 0 
= +1hφi0 = 
√ 
ν/ 2


  
0   0  0
Y + 1
=
 √  =  .
0
Y −1
ν/ 2
0


T1 hφi0

T2 hφi0

T3 hφi0

Y hφi0

Qhφi0





(1.30)

(1.31)

(1.32)

(1.33)

(1.34)

In the above equations, only the combination of the electric charge corresponds to a zero,
implying that this symmetry is unbroken, while the remaining symmetries are broken. In
other words, the original four bosons correspond to broken symmetries, but the linear combination given in Eq. 1.29 is unbroken. This combination can be recognized as the photon,
and remains massless, while the other three gain masses. Furthermore, the W 1 and W 2
bosons can be exchanged for the W + and W − bosons through the following relationship:

W¾¹

Wµ1 ∓ iWµ2
√
=
.
2

13

(1.35)

This combination is similar to the raising and lowering operators for angular momentum often
seen when discussing the hydrogen atom in introductory quantum mechanics3 . Plugging this
transformation back into the Lagrangian results in a mass term of the form:

g2ν 2  + 2
−
2
|WÂľ | + |WÂľ | ,
4

(1.36)

0
and the W mass can be defined as: MW = gν
2 . Additionally, the WÂľ and BÂľ bosons mix

into two experimentally observable bosons using the orthogonal combinations:
−g 0 B µ + gWµ3
ZÂľ = p
g 2 + g 02
gB Âľ + g 0 WÂľ3
AÂľ = p
,
g 2 + g 02

(1.37)
(1.38)

where g is the coupling of the SU (2)L gauge group, and g 0 is the coupling of the U (1)Y
gauge group. It is convenient to relate these couplings through the use of a mixing angle,
given by:
g 0 ≡ g tan θw .

(1.39)

Using the above equation the relationship can be rewritten as:

Zµ = cos θw Wµ3 − sin θw Bµ

(1.40)

A¾ = sin θw W¾3 + cos θw B¾ ,

(1.41)

3 The angular momentum of the hydrogen atom also has an SU (2) symmetry.

14

where θw is the experimentally measured weak mixing angle4 . Finally, the interaction Lagragian in Eq. 1.12 can be rewritten into the charge current interaction and the neutral current
interactions. The charged current in the lepton sector is given by:


g  /+
−
/
Lint,W = − √ ν̄ W PL e + ēW PL ν ,
2

(1.42)

where ν, e are the neutrino and electron wavefunctions respectively, and PL is the left-handed
projection operators. The left(right)-handed projection operators project the fermion onto
a left(right)-handed state. The projection operators are defined as:

PL =

1 − γ5
,
2

PR =

1 + Îł5
,
2

(1.43)

where Îł5 = iÎł0 Îł1 Îł2 Îł3 . The coupling of the W boson can be identified in terms of measurable
observables as

2
G F MW
√
2

, where GF is the Fermi constant, and will be discussed in detail later.

The neutral current interaction for the lepton sector is given by:

Lint,neutral = p

gg 0
g 2 + g 02

/
ēAe

p
g 2 + g 02
/ Lν
−
ν̄ ZP
2


1
g 2 − g 02
02
/ Re +
/ Le .
+p
−g ēZP
ēZP
2
g 2 + g 02

(1.44)

Identifying the coupling of the photon with the fundamental electric charge e, gives:
gg 0
e= p
.
g 2 + g 02
4 Details of measuring this angle can be found in Sec. 1.1.3.

15

(1.45)

It is important to note that this also gives a definitive prediction for the Z boson mass given
by:
MZ2 =

2
MW
,
cos θw

(1.46)

which implies that the Z boson mass must be greater than the W boson mass, which is
confirmed experimentally.
Similar to angular momentum in the hydrogen atom, the square of the weak isospin
generator (T 2 ) and the third component of the isospin generator can be used to distinguish
the compnents of the doublet after symmetry breaking. All of the fermions are in the
fundamental representation of the U (1)Y gauge group, and are each assigned a hypercharge.


T
Furthermore, the left-handed fermions combine into doublets of SU (2) (i.e. ψ = uL dL
for the left-handed up and down quarks), while the right-handed fermions are singlets. The
quantum numbers for the first generation of fermions are summarized in Table 1.2. It is
important to note that while νR is included in Table 1.2, it does not interact with anything,
and is therefore also acceptable to completely drop right-handed neutrinos from the theory.
However, if the neutrinos are to have a Dirac mass term, then right-handed neutrinos need
to be included in the theory. As of the writing of this work, neutrinos are known to have
mass, but the origin of mass is unknown5 . The quantum numbers for the first generation
are also the same for the respective members in the second and third generation of fermions.

1.1.3

Electroweak Precision Observables

In the electroweak sector, a set of three measurable quantities known as the S, T , and U
parameters are able to test the nature of the electroweak theory [67]. The parameters are
5 For additional details on the status of neutrinos, the reader is referred to [66].

16

Fermion

Q

T3

Y

e
νL
eL
uL
dL
e
νR
eR
uR
dR

0
-1

1
2
− 12
1
2
− 12

-1
-1

0
0
0
0

0
-2

2
3
− 13

0
-1
2
3
− 13

1
3
1
3

4
3
− 23

Table 1.2: Electroweak quantum number for the first family of fermions

extracted from the precise data from the LEP experiment and include the recent Higgs
mass measurement by ATLAS and CMS [1]. The parameters are defined using the oblique
(vacuum polarization) corrections to the gauge bosons. Current measurements are consistent
with the Standard Model prediction.
The vacuum polarization functions for the Electroweak bosons can be expanded in terms


of the four momentum transfer, q 2 . Keeping only up to O q 2 , the vacuum polarizations
can be expressed as:

 
 
Πγγ q 2 = q 2 Π0γγ (0) + O q 4 ,
 
 
ΠZγ q 2 = q 2 Π0Zγ (0) + O q 4 ,
 
 
2
2
0
ΠZZ q = ΠZZ (0) + q ΠZZ (0) + O q 4 ,
 
 
ΠW W q 2 = ΠW W (0) + q 2 Π0W W (0) + O q 4 ,

(1.47)
(1.48)
(1.49)
(1.50)

where Π0 represents a derivative of the vacuum polarization with respect to q 2 . Furthermore,
the constant terms for Πγγ and ΠZγ are zero due to the renormalization conditions of the
Electroweak sector, namely that the photon should have zero mass and that the mass matrix
for the mixing of the photon and Z is diagonal. In the above equations, there are six different
17

Variable
S
T
U

Value
0.03 Âą 0.10
0.05 Âą 0.12
0.03 Âą 0.10

S Correlation

T Correlation

0.89
-0.54

-0.83

Table 1.3: The current best fit results for the S, T, U parameters, along with their
uncertainties and correlation coefficients. Reproduced from [1]

quantities, Π0γγ (0), Π0Zγ (0), ΠZZ (0), Π0ZZ (0), ΠW W (0), and Π0W W (0). Three of these can
be fixed by input parameters, in this work they are the fine structure constant (Îą), the Fermi
coupling constant (GF ), and the Z boson mass. This leaves three parameters that can be
measured. These are the S, T, U parameters, and are defined as:

ÎąS =

4s2w c2w



Π0ZZ


c2w − s2w 0
0
(0) −
ΠZγ (0) − Πγγ (0) ,
s w cw

ΠW W (0) ΠZZ (0)
−
,
2
MW
MZ2


αU = 4s2w Π0W W (0) − c2w Π0ZZ (0) − 2sw cw Π0Zγ (0) − s2w Π0γγ (0) ,
ÎąT =

(1.51)
(1.52)
(1.53)

where sw and cw are the sine and cosine of the weak mixing angle respectively. In most
observables, the effect of the U parameter is small, and it also predicted to be small in most
new physics models. Therefore, typically a two parameter space is used with U being set to
zero. The fit to current data for both U non-zero and U zero can be seen in Figure 1.2a and
1.2b, respectively. The current best fit values for S, T, U are given in [1], and reproduced
in Table 1.3. The SM predictions for the S, T, U parameters are all zero, and the data
currently is consistent with the SM.
The W mass is an important observable in constraining these parameters, and the discussion of how to precisely measure this observable can be found in Chap. 5

18

0.4

T

T

0.5
68%, 95%, 99% CL fit contours, U free
(SM : MH=126 GeV, m t =173 GeV)

0.5
0.4

ref

ref

0.3

0.3

0.2

0.2

0.1

0.1

0
-0.1

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.4

0.4

0.5

-0.5
-0.5

G fitter
-0.4

-0.3

-0.2

-0.1

0

S

0.1

0.2

0.3

0.4

B
SM

Sep 12

-0.4

B
SM

SM Prediction
with MH ∈ [100,1000] GeV

-0.3
Sep 12

G fitter

MH

-0.2

SM Prediction
with MH ∈ [100,1000] GeV

-0.4

SM Prediction
MH = 125.7 Âą 0.4 GeV
mt = 173.18 Âą 0.94 GeV

-0.1

MH

-0.3

-0.5
-0.5

0

SM Prediction
MH = 125.7 Âą 0.4 GeV
mt = 173.18 Âą 0.94 GeV

-0.2

68%, 95%, 99% CL fit contours, U=0
(SM : MH=126 GeV, m t =173 GeV)

0.5

S

(a)

(b)

Figure 1.2: The best fit in the T S plane with U free to float on the left and U fixed to zero
on the right. Additionally, mH = 126 GeV and mt = 173 GeV in this fit. Reproduced from
[1]

1.1.4

Electroweak Parameters

While this work focuses on the QCD corrections for given processes, the processes themselves
contain Electroweak couplings. Therefore, it is important to discuss the parameters in the
electroweak sector, the different schemes to set the parameters, and the scheme and values
of input parameters used for the results in this work.
In the electroweak sector, typically, the values of MZ , the fine structure constant Îą, and
Fermi’s constant GF are used as input parameters. The reason that these three are chosen
is due to the fact that these are the most precisely measured parameters. It is possible to
trade out GF for MW , and other such exchanges. These would result in slightly different
predictions that should only differ by higher order corrections. In this work, the choice of
parameters follows that detailed in Section 3.1 of [68], and are detailed below for simplicity.
The constant width approach is used, and thus the W and Z masses and widths need to be
adjusted from that measured in the s-dependent width approach [69, 70]. This results in the

19

inputs for the W and Z boson to be:

MZ = 91.1535GeV,
MW = 80.358GeV,

ΓZ = 2.4943GeV
ΓW = 2.084GeV,

(1.54)
(1.55)

where ΓZ,W is the width of the Z and W boson respectively. Additionally, the value of α (0)

2 
√
MW
2
is replaced by the effective coupling αG = 2GF MW 1 − 2 /π. The input value for
F

MZ

GF = 1.1663787 × 10− 5 GeV−2 . In all numerical results, these input parameters are used
for all the calculations, to ensure that any difference is not due to different choices of input
parameters.
There are three schemes for the measurement of the EW parameters that are discussed
in the literature, the on-shell scheme [71], the M S scheme, and the sin θw scheme. In the
on-shell scheme, the weak mixing angle is fixed to be given by the following equation to all
orders:
sin2 θw = 1 −

2
MW

MZ2

.

(1.56)

In the MÂŻS scheme, all parameters develop a scale dependence and the parameters run as a
function of the hard scale. Finally, the effective angle scheme determines the mixing angle
by using Z pole observables to obtain the Z couplings to fermions, and thus fixing sin θw .
These schemes will have small deviations in the calculated parameters.

1.2

Renormalization and Regularization

In Quantum Field Theories, calculations are usually performed as a series expansion of
a small coupling constant. However, when calculating an observable beyond the leading
20

p2

p3
k
p1
Figure 1.3: An example of a loop diagram, p1 , p2 , and p3 are the momentums of the
external particles, and k is the momentum of the loop.6

term in the expansion, the theory generates infinities. Because the physical observables are
finite, physicists developed a procedure to ensure that predictions are similarly finite. This
procedure consists of regularization and renormalization.
Regularization is a means to isolate the divergences in the theoretical calculation. The
present work focuses on Dimensional Regularization, the most commonly used regularization
scheme [72]. In Dimensional Regularization, field theory equations are calculated in D =
4 − 2 spacetime dimensions, where  is eventually taken to 0. The divergences will show
up in the form of −n poles, where n is a positive integer. Consider the Feynman Diagram
in Fig. 1.3. From conservation of momentum, the values of p1 , p2 , and p3 are all related,
but k is left arbitrary. Therefore, all possible values for k need to be considered, resulting
in an integral over the four momentum of k. In general the set of one-loop diagrams can be
expressed in integrals of the form:

Z

dD k N (k 2 )
,
(2π)D (k 2 − ∆)n

(1.57)

where the numerator and denominator must each be constructed exclusively from even pow6 Drawn using TikZ-Feynman [73]

21

ers of k, since any odd powers of k will integrate to zero since it would be an integration
of an odd function over symmetric bounds. If the power of k in the denominator is greater
than that in the numerator by at least 4, then the subsequent integral is divergent in 4
dimensions as the loop momentum goes to infinity. This divergence is known as an ultraviolet(UV) divergence. In dimensional regularization, this divergence is regulated by moving to
a dimension slightly smaller than 4 and results in simple poles in . Similarly, it is possible
that a divergence appears as the loop momentum goes to zero. Such a divergence is called
an infrared(IR) divergence, and is discussed in more detail in Section 1.3.3. For now, note IR
divergences are regulated in dimensions larger than 4. Any poles in the resulting calculation
need to be removed in order to obtain a physical result, this is known as renormalization.
Additionally, Dimensional Regularization causes couplings to gain a mass dimension. However, this is undesirable, so the coupling is traditionally modified to be dimensionless, and
an additional dimensionful parameter Âľ is introduced to ensure the proper dimensions in the
Lagrangian (g 2 → g 2 µ4−D ). The above discussion can be generalized to higher order loops,
by modifying Eq. 1.57 to:
Z Y
n
dD ki N (k12 , ..., kn2 )
,
(2π)D D(k12 , ..., kn2 )

(1.58)

i=1

where the ki ’s are the loop momenta, and N and D are some function of the momenta
squared.
Renormalization modifies the Lagrangian parameters from so-called bare parameters into
renormalized parameters. When redefining these bare parameters into renormalized parameters, the UV divergences are absorbed. Because the renormalized parameters must be able to
absorb a divergent contribution yet obtain a finite result, the bare parameters must also be
divergent. There are many different renormalization schemes. This work utilizes the Modi-

22

fied Minimal Subtraction (M S) Scheme [74, 75]. To illustrate the renormalization procedure,
consider the bare QED Lagrangian:

1
L = − F µν Fµν + iψ¯0 ∂/ψ 0 − m0 ψ¯0 ψ 0 − e0 ψ¯0 A/0 ψ 0 ,
4

(1.59)

where m0 is the bare electron mass, e0 is the bare electron charge, ψ 0 is the bare electron
field, and A0 is the bare photon field. By defining renormalized quantities, the Lagrangian
can be rewritten in terms of renormalized quantities plus counterterms. To begin, rewrite
the bare electron field according to:

1
ψR = √ ψ0,
Z2

(1.60)

where ψ R is the renormalized field, and Z2 is some infinite number. Because the calculations
will be performed order by order, it is more convenient to express Z2 as 1 (its tree level value)
plus a counterterm. Thus, Z2 can be written as 1 + δ2 , where δ2 is the counterterm, and
is expressed as a Taylor series expansion in the coupling of the theory. Furthermore, the
process can be repeated with the mass, thereby yielding an additional counterterm δm and
a renormalized mass mR . These terms are used to absorb the divergences in the calculation
as well as some finite portion of the calculation. The form of the finite piece absorbed is
determined by the renormalization scheme used. As mentioned above, this work utilizes the
M S scheme, wherein counterterms are chosen to remove the aforementioned poles, along with


constants proportional to ln 4πe−γE , arising from any loop calculation performed using
dimensional regularization. The renormalization of the coupling constant will be discussed
in relation to QCD in detail in Subsection 1.3.2, and is discussed in terms of the running of

23

the coupling, in which the coupling gains an energy dependence.
In all of the above discussion, it was assumed all the divergences could be absorbed by a
finite number of counterterms. A QFT for which this is true is said to be a renormalizable
theory. However, it is not guaranteed that any theory can be renormalized using a finite
number of counterterms. To show that a theory is completely renormalizable is quite complicated. t’Hooft [76] showed this is true for spontaneously broken non-Abelian gauge theories,
thus paving the path for the Standard Model. A rough sketch of proving that a theory is
renormalizable is described below. Firstly, it is necessary to show that it is true not only for
one loop calculations, but for all loop orders. Bogoliubov, Parasiuk, Hepp, and Zimmerman
developed a theorem describing the conditions needed for a QFT to be renormalizable to all
orders [77, 78, 79]. The BPHZ Theorem states a theory is renormalizable if all divergences in
the theory can be removed by counterterms corresponding to superficially divergent one-loop
irreducible amplitudes. Therefore, it is sufficient to show that there are only a finite number
of superficially divergent one-loop processes and that the number of superficially divergent
amplitudes does not increase with increasing orders of perturbation theory. Showing there
are a finite number of counterterms at one-loop is a straight forward calculation. Ensuring
no new superficially divergent amplitudes appear at higher perturbative orders is more difficult, and involves a proof by induction. The details of this proof for the Standard Model
are in Ref. [80].

24

1.3
1.3.1

Quantum Chromodynamics
Introduction

The theory of Quantum Chromodynamics was developed in order to study the strong force
that holds hadrons together. As mentioned at the start of this chapter, there are 6 quarks
that interact through the strong force.
Gell-Mann and Zweig introduced the quark model to describe the many hadrons being
observed [81, 82]. However, the model possessed a major problem. Consider the ∆++
particle:

∆++ = |u↑ u↑ u↑ i.

(1.61)

Because the up quarks are identical and all spin up, multiple fermions occupy the same
state, and this spin 3/2 particle violates Fermi-Dirac statistics. This problem was solved
by introducing what is now called “color charge”. Using this new idea, the ∆++ can be
expressed with a completely anti-symmetric expression by including three different colors of
quarks, and requiring the new object to be colorless:

1 X
↑ ↑ ↑
∆++ = √
ijk |ui uj uk i.
6 ijk

(1.62)

Motivated by the color theory of optical light, colorless refers to the fact that the ∆++
contains one of each color (named red, green, and blue). Another way to create a colorless
object is to have one color and the same anti-color (e.g. red and anti-red). Quarks are

25

defined in a color triplet as follows:




 |qr i 




|qi =  |qg i  .




|qb i

(1.63)

It is important to note that experimentally, observables must depend on the complete triplet
e.g.,


O ∟ hq|O|qi ∟ ||qr i|2 + ||qg i|2 + ||qb i|2 .

(1.64)

The above equation is invariant under the transformation q → q 0 = U q, where U is an
unitarity 3 x 3 matrix (U † U = 1). This matrix includes Abelian transformations that do
not mix qr , qg , and qb . These transformations are not desired (this type of transformation
is included in QED). To single out the transformations between qr , qg , and qb the matrix is
required to be special (det(U ) = 1).
These transformations form the group of special unitary transformations of degree 3,
labeled SU(3). For a group SU(N), the number of generators are given by N 2 − 1. Therefore,
SU(3) has 8 generators resulting in 8 different force mediators (gluons). Let us define the
a
Gell-Mann matrices7 Îťa , and the T matrices, T a = Îť2 .

The properties that follow here are general to SU(N):
1. The T a are traceless and hermitian:

Tr(T a ) = 0

(T a )† = T a

7 A representation for the Gell-Mann matrices can be found in Appendix D.

26

(1.65)

2. Lie-Algebra of SU(N):

[T a , T b ] = ifabc T c

[Îťa , Îťb ] = 2ifabc Îťc

(1.66)

3. Structure Constants:
f abc

= −2i Tr



[T a , T b ]T c







(1.67)

4. Traces:
Tr



T aT b



1
= δ ab
2

Tr

Îťa Îťb

= 2δ ab

(1.68)

5. Fierz Identity:
a
Tija Tkl

1
=
2



1
δil δjk − δij δkl
N

(1.69)

The covariant derivative for quarks under the strong force is given by:

Dµ = ∂µ − igs Ta Gaµ ,

(1.70)

where gs is the strong coupling constant and Ga is the gluonic field. The introduction of
the gluonic field is needed to ensure that a local SU (3) transformation of the quark fields
remains invariant. Under the gauge transformation, the gluonic fields transform as:

Gaµ → Gaµ + ∂µ θa (x) − gs f abc θb (x) Gcµ .

(1.71)

The kinetic term in the Lagrangian for the gluon fields is given by:

1
¾ν
LG = − Gaµν Ga ,
4
27

(1.72)

where the field strength tensor is given by:

Gaµν = ∂µ Gaν − ∂ν Gbµ + gs f abc Gbµ Gcν .

(1.73)

Since SU (3) is a non-Abelian group, the gluons are allowed to self-interact. The Lagrangian for the cubic and quartic coupling are given by:

i
1 h
Lc = − gs fabc (∂ µ Aa,ν ) Abµ Acν − fabc (∂ ν Aa,µ ) Abµ Acν ,
2
0

0

Lq = gs2 fabc fab0 c0 A¾,b Aν,c Ab¾ Acν ,

(1.74)
(1.75)

respectively. The fact that the gluons self-interact leads to many interesting properties of
QCD. These properties are asymptotic freedom and confinement, which are discussed in the
following section.8

1.3.2

Asymptotic Freedom and Confinement

In Quantum Field Theories, the coupling in the theory is dependent upon the energy scale
at which the theory is being calculated. In this section, the low energy and high energy
properties of Quantum Chromodynamics are discussed.
Asymptotic freedom is the phenomenon that occurs in QCD at high energies, while
confinement is the phenomenon that occurs in QCD at very low energies. Both of these
phenomena are the result of the running of the strong coupling constant. As mentioned in
Subsection 1.2, the couplings in the gauge theory gain an energy dependence when performing
loop corrections to the coupling (an example diagram for corrections to a coupling is shown
8 Details on the derivation of the QCD Feynman Rules can be found in App. E.

28

Figure 1.4: The first order correction to the strong coupling constant can be calculated
from this vertex correction diagram.
in Figure 1.4). The bare coupling should be independent of the scale of the calculation,
which leads to the following equation:
dÎąbare
∂α(µ2 )

 = β (ι) ,
=0→
dÂľ
∂ ln µ2

(1.76)

g2
where α = 4π
, with g being the coupling of the theory, and β is given by:

 Îą 2


Îą
β (α) = −α β0 − α
β1 + O ιι2 .
π
π

(1.77)

For QED, the leading term in the β-function is given by:

QED

β0

2
=− .
3

(1.78)

With this running of coupling constant, at low energy scales the coupling is small, but as the
energy is taken to be larger and larger, the coupling grows. At a certain point the coupling
divergences, this is referred to as the Landau pole [83], and occurs around 10286 eV for QED.

29

For QCD, the leading term in the β-function is given by:

QCD

β0

=

11CA − 2nf
,
12

(1.79)

where CA is the number of colors in the theory, and nf is the number of active flavors.
The number of active flavors is determined by the scale at which the calculation is being
performed. For example, if Âľ2 < m2c then nf = 3, and if m2b < Âľ2 < m2t then nf = 5.
Furthermore, it is interesting to note that if nf is less than 6, than the value of β0 is greater
than zero. In QCD, this condition is satisfied, which leads to a running of the coupling
which is the opposite of QED. So while the QED coupling grows as a function of energy,
the QCD coupling decreases as a function of energy. This leads to a running coupling
that asymptotically approaches zero at high energies, and therefore QCD asymptotically
approaches a free theory [84, 85]. On the other hand, the coupling grows for small energy
scales and eventually becomes non-perturbative (αs > 4π), and it is no longer reasonable
to describe QCD using quarks and gluons, but rather using hadrons. Calculations in this
regime are done either using Chiral Perturbation Theory [86] or Lattice QCD [87].9 Since
the coupling becomes so strong the quarks and gluons become confined into the hadrons
that are used to describe the theory, this describes the phenomena of confinement found in
QCD.
It is possible to analytically solve the running of the coupling at the lowest order, the
9 Both ChPT and Lattice QCD are beyond the scope of this work. For additional information, the reader
is referred to [86, 88, 89] for ChPT and to [87, 90, 91, 92, 93] for Lattice QCD.

30

result given in terms of β0 is:
Îą (Âľ0 )

Îą (Âľ) =
1+

Îą(Âľ0 )
π β0 ln

Âľ2
Âľ20

,

(1.80)

where Âľ is the scale the coupling is being evaluated, and Âľ0 is the input scale at which the
coupling was measured at. While β1 and β2 10 have been calculated [94], it is not possible to
solve the running of the coupling analytically, and is traditionally done numerically.
Given the value of the coupling at a given scale, it is possible to predict the coupling at
any other scale. For QCD, the experimental measurements of the coupling are shown to be
consistent with the theoretical predictions as seen in Fig. 1.5. Typically, the starting scale
is chosen to be the mass of the Z boson, and is given as Îąs (MZ ) = 0.118 Âą 0.0011 [6]. As
mentioned previously, the value of nf is dependent upon the scale that one is evaluating the
coupling at, and thus leads to the need to create a matching procedure in order to ensure that




the coupling is a continuous function. This results in choosing αs mq −  = αs mq + 
at the scale of mq .
When using the renormalization group equation (Eq. 1.76) to obtain the running of the
coupling, the large logarithmic terms that appear at all orders are resummed in order to
improve the theoretical prediction of this quantity. Later in this work, there will be a large
focus on the methodology of resummation in the case of the transverse momentum of a given
system.
10 The expressions for β and β can be found in App. E.
1
2

31

Figure 1.5: The comparison between the theoretical calculation of the strong coupling
compared to experimental data. The plot is reproduced from Ref. [6]

1.3.3

Soft and Collinear QCD

As mentioned in the previous section, all physical results must be finite. However, during
the calculations there can be an IR divergence that is the result of two particles being
emitted collinear to each other, or in the case of a massless particle, the momentum can
be very small, which is referred to as a soft divergence. To see these divergences from a
mathematical standpoint consider the denominator of a massless and massive propagator
given as:
1
1
=
,
2
2p ¡ k
(p + k)

(1.81)

1
1
=
,
2
2
2
(p + k) − m
2p · k + p + k 2 − m2

(1.82)

and

respectively. In the massless case, in the limit that k µ → 0 or k µ → apµ , where a is some
scaling constant, the propagator diverges. However, when considering the second equation,
32

Figure 1.6: Electron scattering off of a proton with an energy of 188 MeV. Results show
data is inconsistent with a point-like proton. Reproduced from [7].

only the limit that k µ → apµ results in a divergence. This is a result of having a momentum
cutoff due to the mass of the particle.

1.3.4

Factorization

The QCD cross-sections can be broken into a long distance piece and a short distance piece.
While the short distance piece is the major discussion of the rest of this work, here the long
distance physics needed in order to compare a theoretical calculation with data is briefly
discussed. The two major long distance physics phenomena that will be discussed here are
the parton distribution functions (PDFs), and hadronization along with jet algorithms.
The parton model (Richard Feynman, 1969) [95] is the idea that a proton is composed
of many smaller pieces, that are essentially free point-like particles. This was supported by
data showing that indeed the proton was not a point particle and the validation of Bjorken
33

Figure 1.7: Depiction of the F2 proton structure function, showing Bjorken scaling. This
was the motivation for the parton model proposed by Richard Feynman. Plot is
reproduced from [8].

34

Scaling [96] (See Fig. 1.7). If the proton was a point particle, then proton electron collisions
should follow the predictions made by Mott Scattering [97]. However, as seen from the results
of [7] and reproduced in Fig. 1.6, it is clear that electron-proton scattering does not follow
the prediction for Mott Scattering. To resolve this issue, proton structure functions are
introduced. For further simplicity the calculations are performed in the ”infinite momentum
frame,” where pµ ≈ (E, 0, 0, E) with E  Mp , where Mp is the mass of the proton. It is
then possible to write the scattering as the following cross-section:



d2 σ
4πα2 
1−y 
2
2
2
2
=
1 + (1 − y) F1 (x, Q ) +
F2 (x, Q ) − 2xF1 (x, Q ) ,
x
dxdQ2
Q4

(1.83)

0
where Q2 is the energy transferred by the photon being exchanged, y = 1 − EE , with E 0

the outgoing energy of the electron, and E the incoming energy of the electron. Also,
x=

Q2
,
2Mp (E−E 0 )

and F1 and F2 are the two structure functions that appear. Furthermore,

there exists the Callan-Gross relation [98], that states for a spin- 12 particle, F2 = 2xF1 . This
can be tested experimentally, and the results can be seen in Fig. 1.8. It is clear that the
results are close to zero and thus the proton is made of spin-1/2 particles. However, this is
not the complete story, and at higher energies there is a deviation away from zero as a result
of higher order QCD corrections, and interactions with the gluon.
From these results, the picture of the proton becomes one of a set of valence quarks, sea
quarks, and gluons. When calculating a theoretical prediction, it is necessary to know the
probability of removing a given parton from the proton at a set energy. These distributions
are known as Parton Distribution Functions (PDFs) and a detailed discussion follows in the
subsequent section. However, because these distributions are related to the proton composition at low energies, it is non-perturbative. Therefore, to obtain a calculation, factorization

35

Figure 1.8: Test of the Callan-Gross relation, with K0 = F2 /(2xF1 ) − 1. Results are
consistent with spin-1/2 predictions. Reproduced from [9].

was introduced. The idea behind factorization is that one can introduce an arbitrary scale
to separate the non-perturbative contributions from the perturbative contributions. Because
the scale is arbitrary, the final result should be independent of scale, if the calculation can
be performed to all orders. However, this is not the case and therefore a scale dependence
is introduced, which is used for estimating theoretical uncertainties, referred to as the factorization scale (ÂľF ).
The factorization of the long distance physics from the short distance physics is given
schematically for deep inelastic scattering (e + p → e + X) as:

Z
σ=

f (x)σ̂dx,

(1.84)

where σ is the total measured cross-section, f (x) is the PDF, and σ̂ is the calculation of
a quark interacting with an electron. A similar technique is used to describe final state
quarks. In this case, since only hadrons are physically observed, there needs to be a function
that takes quarks into hadrons. This is known as the fragmentation function, and is again
defined using an arbitrary scale. Additional details for fragmentation functions can be found
36

xp

xp
(1-x)p

p

(a) Quark splitting to
quark and gluon

(b) Quark splitting to
gluon and quark

xp
p

(1-x)p

p

xp
(1-x)p

p

(c) Gluon splitting into
quark and anti-quark

(1-x)p

(d) Gluon splitting into
gluon and gluon

Figure 1.9: The splitting kernels that occur at Îąs . Note that to obtain the missing kernels,
take fermions to anti-fermions and vice-versa.
in [99, 100]. However, if the details of the hadrons in the final state are unimportant, a jet
algorithm can be used to create theoretically manipulable objects. Details on jets and jet
algorithms are provided below.

Parton Distribution Functions
In order to perform calculations at a hadron collider, it is necessary to use Parton Distribution
Functions. However, it is possible to extend the basic purely non-perturbative picture of the
PDFs to include the improved parton model. In the improved parton model, it is possible
to calculate the energy dependence of the PDFs. In order to calculate these effects, consider
diagrams of the forms found in Fig. 1.9. The results of these calculations can be represented

37

by the following equations at leading order:

1 + z2
3
Pqq (z) = CF
+ δ(1 − z) ,
(1 − z)+ 2


1 + (1 − z)2
Pqg (z) = CF
,
z

1 2
Pgq (z) =
z + (1 − z)2 ,
2 

z
1−z
Pgg (z) = 2CA
+
+ β0 δ(1 − z),
(1 − z)+
z


(1.85)
(1.86)
(1.87)
(1.88)

1
where Pij is the splitting kernel from j to i. Also, the (1−z)
is the plus-distribution and
+

is discussed in detail in Appendix F. Similar to the summation that was performed for the
running of the coupling, it is possible to define bare PDFs which are independent of scale.
This results in a set of 2Nf +1 coupled differential equations, known as the full form DGLAP
evolution equations [101, 102, 103], given by:





Z
x
2
∂ 
 ιs (¾2 ) 1 dz Pqq (z) Pqg (z) q( z , ¾ )

=


,
2π
∂ ln µ2
z z
x
2
2
g(x, Âľ )
Pgq (z) Pgg (z)
g( z , Âľ )
q(x, Âľ2 )



(1.89)

up to O (Îąs ). To solve these equations, it is convenient to define a new set of PDFs, defined
using the valence quark, singlet, non-singlet, and gluon contributions. There are Nf valence
quark contributions, Nf − 1 non-singlet contributions, and one of both the singlet and gluon
contributions. The valence quark distribution is

qiV = qi − q̄i ,

38

(1.90)

the non-singlet distribution is

qiF

=

i−1
X

(qn + q̄n − qi − q̄i ),

(1.91)

n=1

and the singlet distribution is
N

qS =

f
X

(qn + q̄n ).

(1.92)

n=1

This change of variables leads to a set of 2Nf − 1 decoupled and two coupled differential
equations given up to O (Îąs ) as:
∂qiV
∂ ln µ2
∂qiF
∂ ln µ2
∂q S
∂ ln µ2
∂g
∂ ln µ2

Îąs (Âľ2 )
=
Pqq ⊗ qiV
2π
Îąs (Âľ2 )
Pqq ⊗ qiF
2π

Îąs (Âľ2 ) 
S
=
Pqq ⊗ q + 2Nf Pqg ⊗ g
2π

Îąs (Âľ2 ) 
=
Pgq ⊗ q S + Pgg ⊗ g
2π
=

(1.93)
(1.94)
(1.95)
(1.96)

Additionally, there are constraints on the solutions implemented by sum rules. The momentum sum rule forces that the total momentum of the partons to be the momentum of the
hadron,
XZ 1

xf (x) = 1.

q,q̄,g 0

39

(1.97)

(a)

(b)

Figure 1.10: Plots of the CT14 NNLO Parton Distributions at 2 GeV (left) and 100 GeV
(right). The plots are reproduced from Ref. [10]

The other sum rule is the valence quark sum rule, which for a proton is given by:
Z 1
(fu − fū ) = 2,

(1.98)

(fd − fd¯) = 1,

(1.99)

0

Z 1
0

Z 1
(fq − fq̄ ) = 0,

for q 6= u, d.

(1.100)

0

The DGLAP equations can be extended order by order in QCD, and currently have been
calculated to NNLO [104, 105]. An example of a PDF can be found in Figure 1.10, where
the energy dependence can be seen between the two subfigures.

40

Jet Algorithms
A true observable must be an infrared safe observable which means the observable is unchanged by the addition of a soft or collinear contribution. If this is not the true, then the
result will suffer from IR divergences and is known as an IR unsafe observable. In the case of
final state partons, an infrared safe observable is constructed by collecting the partons into
jets. The method of combining the final state partons into jets is known as a Jet Algorithm,
and must satisfy the following two conditions:
1. If two particles become collinear then the observable must not change:

On (p1 , p2 , ..., pn ) → On−1 (p1 + p2 , ..., pn )

(1.101)

2. If one particle is soft then the observable must not change:

On (p1 , p2 , ..., pn ) → On−1 (p2 , ..., pn )

(1.102)

Currently, there is one major class of jet algorithms used to ensure the final state is infrared
safe. This class of algorithms is known as sequential recombination algorithms, which contains anti-kt , kt , and Cambridge-Aachen algorithms [106, 107, 108, 11]. For this class of
algorithms, the methodology of forming jets is the same, but the measure to determine when
to combine two subjets together varies from one to the other. The steps of the algorithm is
as follows:
1. Calculate the distance between all sets of two partons (dij ) and between all partons
and the beam (diB )

41

2. Find the smallest distance. If dij is the smallest, combine i and j into a subjet, else if
diB is smallest remove subjet i from the list and mark as a jet
3. Repeat until all particles are clustered into a set of jets
The definition of the distance varies from one algorithm to another, but can be represented
in general by:



2p 2p Rij
dij = min kti , ktj
R
2p

diB = kti ,

(1.103)

where, kti is the transverse momentum of the ith particle, p is the power factor that depends
on the scheme (-1 for anti-kt , 1 for kt , and 0 for Cambridge-Aachen), Rij is the distance
2 = (y − y )2 + (φ − φ )2 , and R is a parameter that
between the two particles given by Rij
i
j
i
j

determines the radius of a typical jet. The anti-kT , kT , and Cambridge-Aachen algorithms
can be compared by looking at how they cluster the same event as shown in Figure 1.11. It
is interesting to note that the anti-kT algorithm tends to have the most cone like jets. In
recent years, there has been work to begin delving into the sub-structure of jets to obtain
more information about them. While that discussion is beyond the scope of this work, the
reader is referred to [109, 110, 111, 112, 113, 114] for some of the recent advancements.

42

(a)

(b)

(c)

Figure 1.11: (a) Anti-kt algorithm. (b) kt Algorithm. (c) Cambridge-Aachen algorithm.
Reproduced from Ref. [11]

43

Chapter 2
Experimental Measurements
Since the true test of any theory is how it compares to experimental data, it is important to
understand how the data is collected and analyzed. The purpose of this chapter is to discuss
the most important pieces needed to understand the data used in future chapters, and how
it was obtained. The main focus will be only on applications to hadron colliders, however,
some of the topics are general to all colliders. In this chapter, the needed concepts that are
general to all hadron colliders will first be introduced, along with the details that are unique
to both the ATLAS and CMS detectors at the LHC.

2.1

Basic Concepts

There are many experimental concepts and methods that are general to all hadron colliders.
These include the types of data that are collected, the variables that are used to describe
the momenta of the particles, and methods of determining particle identities.
Firstly, data is collected in terms of events. An event is an interesting collision between
the two incoming hadrons that gets recorded for further analysis. To compare the events
that are collected by the experiments, it is necessary to convert either the experimentally
measured events into cross-sections, or take the theoretically calculated cross-sections and
convert that to the number of expected events. In either case, there is a conversion factor

44

Cross Section
1b
30 mb
50 mb
50 nb
20 pb
100 fb

Process
cross section of heavy nuclei
proton size (πr2 , where r = 1 fm)
proton-proton inelastic cross-section at the LHC
W and Z boson production at the LHC
inclusive Higgs boson production at the LHC: pp → H + anything
inclusive ttH production at the LHC: pp → tt̄H + anything

Table 2.1: Representative cross section scales (cross sections are not exact). Reproduced
from Ref. [2]
that is used, defined as the luminosity. The instantaneous luminosity is a quantity that
measures the intensity, density, and thickness of the beams. The integrated luminosity is
simply the integral of the luminosity over a given period of time. The conversion between
events and cross-section is given by:

σ = N × Lint ,

(2.1)

where σ is the cross-section, N is the number of events, and Lint is the integrated luminosity.
This however is not the complete picture. This assumes that for every single bunch crossing
all of the data is recorded. Since the detector does not cover the full 4π region of the collisions
an additional factor, called the efficiency (), needs to be included in Eq. 2.1. The efficiency of
the gaps in the detector is modeled through programs such as GEANT4 [115]. Additionally,
due to the large number of collisions that occur at these colliders, it is impossible to record
every collision. Therefore, there are minimum cuts placed on the momenta of the final state
particles, to ensure that the recorded events are of physical interest. This reduces the crosssection and needs to be included in the efficiency factor. In order to understand the need
for the momentum cuts, some examples of cross-sections are listed in Table 2.1.

45

Figure 2.1: Depiction of particles traveling through the CMS Detector. Reproduced from
Ref. [12]

2.2

Detecting Particles

When recording events, only certain particles can actually be recorded by the detector.
These particles are relatively stable, with lifetimes that allow the particle to make it from
the collision point to the detector. This reduces the number of particles that can be detected
to a list of only seven particles. These include: electrons, muons, photons, pions, kaons,
protons, and neutrons. The remaining particles that are produced in the hard interaction
either cannot be detected (neutrinos), or decay before they hit the detector (top quarks, b
hadrons). A depiction of a cross-section of the CMS detector at the LHC is shown in Fig. 2.1
to show which pieces of the detector each of the particles interact with. In the previously
mentioned figure, the curved tracks are due to charged particles traveling through a toroidal
magnetic field. The direction of the curvature is used to determine whether the particle
is positively or negatively charged. Additionally, the curvature of the tracks allows the

46

momentum of the particle to be calculated, recalling from classical Electromagnetism:

r=

p
,
qB

(2.2)

where r is the radius, p and q are the momentum and charge of the particle respectively, and
B is the strength of the magnetic field.
Another handle on the energy and momentum of the particles is through the use of
calorimeters. The calorimeters are classified into two classes, electromagnetic and hadronic.
These two classes focus on two different mechanisms to stop the particles. The electromagnetic calorimeter focuses on stopping lighter particles (electrons and photons) through the
use of electromagnetic interactions. The hadronic calorimeter is used to stop hadrons, as the
name implies, through the use of nuclear interactions. The typical property to define the
material used in a calorimeter is the radiation length (X0 ) and nuclear interaction length
(ÎťI ) for the electromagnetic and hadronic calorimeters respectively. The typical size of an
electromagnetic calorimeter is 15-30 X0 , and for an hadronic calorimeter 5-8 ÎťI [6]. These
detectors tend to be segmented in both azimuthal and rapidity segments in order to obtain
direction information and improve reconstruction. The details of how each calorimeter works
can be found in Sections 2.2.2, 2.2.3.

2.2.1

Trackers

The inner most layer of the detector is known as the tracking system. This layer is used to
obtain track paths of charged particles and aid in vertex reconstruction. Furthermore, the
tracker is helpful in determining the momentum of charged particles as previously mentioned.
The curvature of the path allows for momentum and charge information to be collected. How47

ever, the momentum and charge determined becomes inaccurate at very high momentum,
due to the radius becoming too large to detect. The ability to reconstruct the vertex allows
the ability to distinguish between particles that come from the hard process versus some
background event (underlying event) or from a secondary decay. The beams consist of large
bunches of protons, and many may collide in each crossing. An underlying event is another
interaction that occurred in the crossing, but is not related to the hard interaction being
studied. A secondary decay results in a displaced vertex, and usually arises from events
containing b or t quarks. Experimentalists have developed techniques to determine massive
particles such as b and t quarks from these displaced vertices called ”tagging”. Details on
tagging can be found in Section 2.3.1.1.
At ATLAS, the tracking system is formed from three sub-systems. The first layer is the
pixel detector [116], which is made up of 92 million silicon pixels covering a rapidity range
of |Ρ| < 2.5. As particles pass through the pixels, ionization occurs by creating electron/hole
pairs. The resolution of the tracks formed by this system are 10µm in the r − φ plane and
115Âľm along the z-axis. The next layer is the semiconductor tracker [117, 118], which covers
the same rapidity region. However, this system is formed by strips of silicon instead of
pixels. This system provides position resolution of 17µm in the r − φ plane and 580µm along
the z-axis. Finally, the last sub-system is the transition radiation tracker [119, 120, 121],
which covers a rapidity range of |Ρ| < 2. The transition radiation tracker contains a gaseous
medium used for producing ionizing radiation needed to obtain tracking information. The
resolution of this system is approximately 170µm in the r − φ plane, but does not provide
information about the particles position along the z-axis. Through the use of vertex finding
algorithms and the tracking data, the resolution of the location of the vertex is about 10Âľm
in the r − φ plane, and between 36-40µm in the z-axis [122].
48

At CMS, the tracking system also consists of multiple sub-systems [123, 124]. The first
layer is again a silicon pixel detector containing 66 million pixels with a resolution of the
tracks at the 10-15µm level in the r − φ plane, and a resolution of 150µm in the z-direction,
covering the rapidity range of |Ρ| < 2.5. The second layer is a silicon strip tracker, consisting
of 70 m2 silicon micro-strips. The strip tracker also covers the same rapidity range as the
pixel detector, and has a resolution on position of approximately 20µm in the r − φ plane,
and a resolution of about 500Âľm in the z-axis. The CMS collaboration decided to not use a
gas system for position measurements in order to have their tracking system all of the same
material. The combination of all of these systems, along with the algorithms implemented
to determine track information results in a vertex resolution of 10-12Âľm in all three spacial
directions [125].

2.2.2

Electromagnetic Calorimeters

The method of stopping particles in electromagnetic calorimeters is through electromagnetic
showers. The primary method of energy loss is through ionization, but Møller scattering,
Bhabha scattering, and annihilation also play a role, as seen in Fig. 2.2 for lead [6]. The
energy resolution of an electromagnetic calorimeter (σ/E) can be parameterized as:

σ
a
c
= √ ⊕b⊕ ,
E
E
E

(2.3)

where E is the energy of the particle in GeV, and ⊕ means that the terms are added in
√
quadrature. For the LHC, the electromagnetic calorimeter resolution is: 2.8%/ E ⊕ 0.3% ⊕
√
12%/E for CMS [126] and 10%/ E ⊕ 0.7% ⊕ 0.170/E for ATLAS [127]. The response
of electromagnetic calorimeters is well understood, and is simulated accurately in both the
49

Figure 2.2: Fractional energy loss per radiation length in lead as a function of electron or
positron energy. Reproduced from Ref. [6]

EGS4 [128] and GEANT [115] Monte Carlo detector simulators.

2.2.3

Hadronic Calorimeters

Unlike electromagnetic calorimeters, hadronic calorimeters are more difficult. The size of the
calorimeter needs to be approximately 30 times larger than the electromagnetic calorimeter
in order to absorb the same fraction of the particle’s energy. Additional complications in
measuring energy deposits arise from large fluctuations in neutron production, undetectable
energy loss to nuclear disassociation, and other effects [6]. At both the ATLAS and CMS
detector, the hadronic calorimeters are behind the electromagnetic calorimeters. The type
of calorimeters used at the LHC are known as sampling calorimeters. This means that the
calorimeter is made up of alternating layers of dense absorbers and plastic scintillators.
The absorber layers allow for the hadrons to interact, creating secondary particles. These

50

particles form a hadronic shower. When the particles pass through the scintillators, light is
created and then through the use of wavelength-shifting fibers are transmitted to electronic
readout equipment [6].
Furthermore, the resolution of hadronic calorimeters are lower than the resolution of
electromagnetic calorimeters, due to the fact that the hadronic showers are broader than
electromagnetic showers. The energy response of hadronic calorimeters tend not to be linear
since the enegy deposited arises from both hadronic and electromagnetic showers. This
further adds to the complexity of measuring final state jets. The energy resolution at CMS
is given by a stochastic term, a = 84.7%, and a constant term, b = 7.4% [129]. The energy
resolution at ATLAS is given by a stochastic term, a = 41.9%, a constant term, b = 1.8%,
and with c = 1.8 [130].

2.2.4

Muon Spectrometers

As mentioned in Sec. 1, muons are second generation leptons. Muons have a lifetime of
2.2Âľs, and are about 200 times heavier than an electron. Due to these properties, muons
escape from the colliders with minimal energy loss as it transitions through the trackers
and calorimeters. Therefore, it is necessary to have additional detectors outside of these
systems to gain additional information on the muons in order to accurately determine their
momentum.
The momentum and charge of a muon is determined by the sagitta of the tracks formed
in magnetic fields. A sagitta is the distance from the center of a circular arc to the center of
the base. The measurement of the sagitta in the inner tracker is combined with that from
the muon spectrometer in order to improve the precision of the measurement.
In the ATLAS detector, the muon spectrometer is made up of four main sub-systems.
51

These systems consist of the Monitored Drift Tube (MDT) chambers, the Cathode Strip
Chambers (CSC), the Resistive Plate Chambers (RPC), and the Thin Gap Chambers (TGC) [131].
These sub-systems can be divided into two sub-groups. The first group, consisting of the
MDT chambers and the CSC, is responsible for high precision tracking and momenta measurements. The second group, consisting of the RPC and the TGC, is used as inputs for
the trigger system. The goal of the precision tracking components are to be able to measure
the momentum of a 1 TeV muon to an accuracy of 10%, resulting in a requirement on the
accuracy of the sagitta to be measured to a resolution of approximately 50Âľm. The MDT
chambers cover the rapidity region |Ρ| < 2.7, and are used to obtain the z component of the
position, while the CSC is responsible for the R component in the range of 2.0 < |Ρ| < 2.7
for muon rapidity. The need for the CSC is to measure the high flux of muons in the forward direction, which is unable to be handled by the MDT chambers. Finally, the RPC
and TGC do not have very accurate position determination, but are optimized instead for
fast response times in order to interface with the muon trigger electronics. This allows for
accurate bunch-crossing identifications in order to properly reconstruct events. However,
this means that these systems need to provide coverage over |η| < 2.4 and over the entire φ
range.
In the CMS detector, the muon spectrometer is made up of three main sub-systems.
These systems consist of a RPC, a CSC, and a Drift Tube Chambers (DT) [132]. The
RPCs are used in both the barrel and the endcap solely for the purpose of being used
for the trigger system. The response time is approximately 1.3ns, guaranteeing a precise
identification of the bunch crossing. The endcaps use the CSC to provide precise time and
position measurements, and are able to handle the high flux and complex magnetic field.
This system is able to obtain a spatial resolution by approximately 100 − 240µm. Finally,
52

the DT chambers are used in the barrel region to obtain precise position measurements, and
from a single hit is able to obtain a precision of approximately 190Âľm. Combining these
systems into many stations, CMS is able to obtain a resolution on the momentum between
5 − 13% for momentums of 1 TeV if the inner tracker information is also used.

2.3

Physics Objects

Experimentally, all that is measured are energy deposits. Therefore, it is important to
separate these contributions into the fundamental particles that lead to the energy deposits,
in order to study the underlying physics. The major objects that these deposits or lack of
a deposit can be grouped into: jets, leptons, and missing energy. It is also possible to use
tracks to measure particles with a short but non-negligible lifetime through the use of offset
vertices from the major interaction point, this is known as tagging.

2.3.1

Jets

The definition of jets for experimentalists, is a collimated spray of hadrons that fall within a
given cone. Typically, the reconstruction of jets at the LHC is through the use of the anti-kt
algorithm, see Section 1.3.4 for details. The dominant uncertainties arise from the absolute
energy scale and the jet energy resolution, or in other words, the relationship between jets
formed in simulations and from those actually reconstructed in the detector [2]. To help
validate this, a comparison between dijet events, and photon plus jet events are compared.
In addition to understanding the jets as a complete object, it is important at the LHC to
begin to probe into the substructure of jets. Some of these techniques are described below.

53

2.3.1.1

B-Tagging

There are a few different algorithms that are implemented in tagging the decay of b hadrons.
These include techniques that range from looking at signed significance of the decay length
to ones that use both secondary vertex and impact parameters [133]. These are combined
using multivariate methods to obtain the best discrimination possible.
One class of algorithms are known as lifetime-based tagging, which is based on the fact
that a b hadron with a pT ≈ 50 GeV will travel approximately 3mm before decaying [133].
This will leave behind a displaced vertex inside the tracker from which the charge particles
produced from the decay of the b hadron lead back to. The algorithms in this class try
to identify this topology in the events, and are broken into two sub-categories: impact
parameter algorithms, and vertex algorithms.
Impact parameter algorithms compute the impact parameters with respect to the primary
vertex. The transverse impact parameter is assigned to separate the tracks from the decay
from the hard interaction tracks. The sign of the momentum is positive if the secondary
vertex is in front of the primary vertex, and negative otherwise. An implementation of this
algorithm is known as JetProp [134], which was used at LEP and the Tevatron.
The vertex algorithm uses a three dimensional reconstruction of the vertex formed by
the decay products of the b hadron. The invariant mass of the charged tracks are used to
help reject vertices due to decays of Ks and Λ particles, along with photon conversions [133].
An example implementation of an algorithm which uses the secondary vertex to tag b-jets is
known as JetFitter [135].
Another class of algorithms that are used to tag b-jets is muon-based algorithms. Muons
are produced from the decay of the b hadrons either directly, or indirectly through the decay

54

Figure 2.3: Performance curves obtained for different b-tagging algorithms using simulated
data. (Left) The mis-identification for light jets. (Right) The mis-identification for c jets.
Plots are reproduced from Ref. [13]

of the subsequent c hadrons. The efficiency of these types of algorithms tend to be lower
than that of lifetime-based algorithms, due to the fact that the branching ratio to muons is
only about 20%. However, the Soft Muon Tagger used by ATLAS does not use any lifetime
information, therefore making it a complementary method [133]. This method looks for
muons that are within ∆R(jet, µ) < 0.5 and other additional selection criteria.
These sets of algorithms are then combined using multivariate techniques as previously
mentioned. The results for the combined tagging efficiency to mis-identified jets and tagging
efficiency to mis-tagging c jets at CMS can be found in Fig. 2.3.

2.3.2

Leptons

When measuring leptons, there are a few sup-groups that are used in order to classify them
more throughly. This list consists of prompt, non-prompt, tight, loose, and fake leptons.
Prompt leptons are those that come from the decays of a W , Z, Îł, top, Higgs, or some
new physics and not associated with a jet. A lepton that is associated with a jet is the result
of the decay of a short lived hadron and is known as a non-prompt lepton.
55

The identification of tight and loose leptons refer to the set of selection criteria that are
applied to an experimental object during reconstruction. Tight criteria are used to help
ensure that the object is more likely to be a prompt lepton, and therefore prompt leptons
should pass tight selection criteria with a very high efficiency and non-prompt should not.
Loose selection criteria would be able to reconstruct both prompt and non-prompt leptons,
but also have the chance to identify an object that is not a lepton as a lepton. This leads to
the last identity, fake leptons. Since the tight selection criteria will not guarantee that the
object is a prompt lepton let alone an electron at all, a fake lepton is used to describe such
objects. Experimentally, there is not much that can be done to handle fake leptons. The
best method is to estimate the probability of fake leptons passing the tight lepton criteria
and to remove the contributions from the final result on average. However, removing fake
leptons is impossible on an event-by-event basis.

2.3.3

Missing Transverse Energy

As mentioned above, there are means to detect charged particles, photons, and neutral
hadrons. However, particles that are neutral and have lifetimes that are long enough to
escape from the detector before decaying are not detected at all. In the SM, the only such
particle is the neutrino, but there may exist additional particles that also have these properties. It is still possible to interpret the energy in the transverse direction that these particles
had, and to know its azimuthal angle, but not its rapidity. The transverse components
are simply obtained through the use of conservation of momentum, since the beams only
have momentum along the beam axis, the total transverse momentum must sum to zero.
Complications arise for this measurement due to gaps in the detector, and imperfect energy resolution. The accuracy on the missing transverse energy at ATLAS has a stochastic
56

behavior with a = 0.55 [136] and CMS has a stochastic behavior with a = 0.63 [137].
The particles that are inferred from the missing transverse energy measurement are neutrinos, dark matter candidates, and other exotic particles that may or may not exist. In
this work, the missing transverse energy will be solely used for neutrinos in determining the
W boson mass (Chap. 5). The W boson mass is an important measurement that strongly
depends on both resummation and an ability to accurately measure the missing transverse
energy.

57

Chapter 3
Resummation
When performing fixed-order calculations, the calculation is organized by the power of Îąs .
fixed-order calculations make sense when each term in the series is smaller than the previous
term. However, there are certain phase space points that result in each subsequent term
being larger than the previous one, causing the breakdown of the fixed-order calculation.
To resolve this, resummation is introduced. Resummation reorganizes that calculation by
noticing that there are certain terms that appear at every order in Îąs [138]. These terms
 
2
that appear in a specific form at each order are logarithms of two scales, e.g. log Q2 . The
qT

number of logarithmic terms included in the calculation is denoted by leading log for having
only the leading term, and adding next-to for each additional log term included. Table 3.1
shows the difference between the organization of a fixed-order calculation and a resummed
calculation.
NLL

NNLL

LO
NLO
NNLO
..
.

LL
1
Îąs log2
Îąs2 log4
..
.

Îąs log
Îąs2 log3
..
.

Îąs
2
Îąs log

N k LO

Îąsk log2k

αsk log2k−1

..
.
k
αs log2k−2

NNNLL

...

Îąs2
..
.

...

αsk log2k−3

...

N k LL

Îąsk

Table 3.1: The organization of fixed-order and resummed calculations. Going across a row
is the fixed-order calculation included at a given order of Îąs , while going down the columns
is the resummed calculation up the the given power of the logs included.

58

This chapter introduces the Collins-Soper frame, which resummation is typically performed in, followed by the calculation of Drell-Yan to next-to-leading order as an example
of a fixed-order calculation. Additionally, it will be shown how the fixed-order calculation
breaks down at each order in perturbation theory for the transverse momentum of the vector
boson. This will then lead into the derivation of the resummation formalism, a discussion of
the resummation scheme, and the non-perturbative function. Reproducing the fixed-order
calculation from the resummed calculation is an important cross-check of the resummation
formalism, and also the asymptotic piece needs to be obtained to allow the resummed piece
to properly match to the fixed-order result at high transverse momentum. Finally, the scale
dependence of the resummation formalism is calculated.

3.1

Collins-Soper Frame

The Collins-Soper (CS) Frame is a special reference frame in which qT resummation is
typically performed [139]. Here the details of the Lorentz transformations between lab and
CS frame are given for completeness. The lab frame is the the center of mass frame of the
colliding hadrons. In this frame, the momentum of the two colliding hadrons are given by:
√
Âľ
ph ,h
1 2

=

S
(1, 0, 0, Âą1) ,
2

where h1 , h2 are the two incoming hadrons and

(3.1)

√
S is the center of mass energy of the

collider. The Collins-Soper frame is a special rest frame of the vector boson. In this frame,
the z-axis is defined to bisect the angle between the hadron momentum ph1 , and the negative
of the hadron momentum ph2 in the CS frame, shown in Fig. 3.1.

59

Figure 3.1: Depiction of the Collins-Soper Frame. Reproduced from [14].

Âľ

Define the Lorentz transformation between the lab and CS frames as Λν :

Âľ

Âľ

pCS = Λν pνlab .

(3.2)

Since the amplitude is independent of the φ angle in the lab frame, without loss of
generality, the φ angle will be taken to be zero. The transformation can be performed in
two-steps. First, the lab frame is boosted to the rest frame of the vector boson. Secondly,
the frame is then rotated such that the z-axis bisects the angle formed by the momentum
Âľ

Âľ

ph and −ph .
1
2
The boost factor from the lab frame to the rest frame is defined by β~ = q~q , where ~q is
0
the 3 momentum, and q0 is the energy of the vector boson in the lab frame. Boosting by β~
brings any four vector from the lab frame to the rest frame of the vector boson. The explicit
Lorentz transformation matrix in terms of the momentum of the vector boson in the lab

60

frame, is given by:




0
−qT
0
−q 3 
 q


2
3


qT
q
q
T
−qT Q +

0
0 +Q
0 +Q 
1 
q
q
Âľ
,
Λν = 

Q
 0

0
Q
0


 2 

3


q
qT q 3
−q 3
0
Q
+
0
0
q +Q

where Q =

q

q0


2

− qT2 − q 3


2

(3.3)

q +Q

is the invariant mass of the vector boson, q3 is the mo-

mentum along the z-axis, and qt is the transverse momentum of the vector boson in the lab
frame.
After the boost, a rotation is applied to ensure that the z-axis bisects the angle formed
between ph1 and −ph2 . Once the boost given in Eq. 3.3 is performed, the momentum of the
incoming hadrons are given by:

Âľ
ph ,h
1 2

√
S
=
2






2 !
±Q − q 3 q 0 + Q ± q 3
q 0 ∓ q 3 qT q 0 + Q ∓ q 3


,−
, 0,
.
Q
Q q0 + Q
Q q0 + Q

(3.4)

In the case that qT is zero, there is no needed rotation. However, in general qT 6= 0, and
an additional rotation is needed. To keep the hadronic momentum in the x − z plane, the
rotation should be made around the y-axis. The angle for the rotation is given by:

h 
 

i
0
0
Îą = arccos Q q + MT / MT q + Q
,

where MT is the transverse mass given by MT =

(3.5)

q
Q2 + qT2 .

Combining the two transformations given in Eqs. 3.3 and 3.5 into a single Lorentz
transform gives the full transformation from the CS to the lab frame, which is given by the
61

Lorentz matrix:


0
 q MT

q0q



q3Q

0
T





2

0
0 

−1
1 qT MT MT

Âľ
Âľ
Λν (CS → lab) = Λν (lab → CS)
=

.

QMT 
 0
0
QMT
0 




3
3
0
q MT q qT
0
q Q

(3.6)

The kinematics of the leptons from the decay of the vector boson are completely determined by two angles (polar and azimuthal) defined in the Collins-Soper frame. Taking the
lepton momentum in the CS frame, and using the transformation given in Eq. 3.6, gives the
lepton momentum in the lab frame. The momentum for the fermion and the anti-fermion
in the lab frame are given by:

pÂľ

Q
=
2

 Âľ

q
Âľ
Âľ
Âľ
+ sin θ cos φX + sin θ sin φY + cos θZ
, p̄µ = q µ − pµ ,
Q

where pµ (p̄µ ) is the (anti-)fermion momentum, q µ is the vector boson momentum in the
lab frame, X, Y , and Z are the axes in the CS frame given by the lab frame momenutm.
The anti-fermion momentum is simply obtained by conservation of momentum. The vector
boson momentum and axes in the lab frame are defined as:

q µ = (Mt cosh y, qt cos φV , qt sin φV , Mt sinh y) ,
!
2
M
Q
Xµ = −
q+ nµ + q− n̄µ − 2t q µ ,
qt Mt
Q
ZÂľ =

1
(q+ nµ − q− n̄µ ) ,
2
Mt

Y ¾ = ¾νιβ

qν
Z ι Xβ ,
Q

(3.7)

62



where, q± = √1 q 0 ± q 3 , y = ln (q+ /q− ), nν = √1 (1, 0, 0, 1), n̄ν = √1 (1, 0, 0, −1),
2
2
2
and ¾νιβ is the completely anti-symmetric Levi-Civita tensor. With the derivation of the
Collins-Soper frame, the act of resummation can now be performed. Next the motivation
for resummation will be discussed, followed by the means to perform resummation.

3.2

Fixed-Order Calculations

In Quantum Field Theory, calculations are performed order by order in perturbation theory.
Calculating the first term in the series is referred to as leading order (LO). Adding an
additional term is referred to as next-to-leading order (NLO), etc. The Drell-Yan process
(pp → e+ e− ) is calculated to NLO below in order to illustrate the steps of a fixed-order
calculation.
The Feynman diagrams that represent this calculation are shown in Fig. 3.2. Feynman
diagrams are a means to pictorially represent the calculation of the scattering matrix element.
To simplify the calculation, the hadronic matrix element is separated from the leptonic matrix
element. The leptonic matrix element is the same for both the leading order calculation and
the next-to-leading order calculation, and the diagram is represented by the right half of
each diagram of Fig. 3.2. The result for the leptonic martix element is given by:

h 



2
L¾ν = 4 2 fL2 + fR
l1µ l2ν + l2µ l1ν − gµν (l1 · l2 )


i
2
2
ρ
σ
+ fL − fR iµνρσ q l12 ,

(3.8)

where qµ = l1µ + l2µ , l12µ = l1µ − l2µ , l1µ /l2µ is the momentum of the lepton and anti-lepton
respectively, and fL/R are the left and right handed couplings respectively.

63

(a)

(b)

(c)

(d)

(e)

Figure 3.2: The diagrams that contribute to the Îąs correction to Drell-Yan. (a) is the born
diagram. (b) is the virtual coupling correction. (c) is the wave function correction. (d) and
(e) are the real corrections

64

The leading order calculation for the hadronic matrix element is done in d-dimensions,
with d = 4 − 2, since this dependence will be needed to obtain the correct result after
calculating the virtual correction. The hadronic matrix element is given by:

H ¾ν

h 



Âľ
Âľ
2
2
= 4 2 gL + gR p1 pν2 + pν1 p2 − g µν p1 · p2


i
2 − g 2 iµναβ p p
+ gL
2ι 1β ,
R

(3.9)

where gL/R are the left and right handed coupling of the quarks to the vector boson, p1 is
the momentum of the incoming quark, and p2 is the momentum of the incoming anti-quark.
At next-to-leading order, both the virtual corrections and real corrections are required.
The virtual corrections contain a virtual gluon connecting the two initial state partons (vertex correction, Fig. 3.2b), as well as a virtual gluon connecting one initial parton to itself
(self-energy, Fig. 3.2c). However, in Dimensional Regularization the self-energy diagrams do
not contribute if the quark is massless, which is the case here. The real correction contains
a real gluon being radiated off of one of the external legs, and also a gluon splitting into
a quark anti-quark pair, as seen in Figs. 3.2d, and 3.2e respectively. Both of these contributions are required in order to ensure that the soft divergences cancel according to the
KLN theorem [140, 141]. The KLN theorem states that in the SM that all IR divergences
that arise from loop integrals (virtual corrections), must be canceled by phase space corrections. In addition to these two contributions, the renormalization of the Parton Distribution
Function needs to be included to ensure that the final result is completely IR finite.

65

3.2.1

Real Corrections

In performing the calculations of the born contribution, the virtual contribution, and the
real contribution, it is possible to separate the leptonic piece of the matrix element from
the hadronic piece of the matrix element. First, consider the vector boson gluon final state
hadronic matrix element. The matrix element is the sum of the left hand sides of the diagrams
in Figs. 3.2d, and 3.2e. Defining the Mandelstam variables as: t = (k − q)2 , u = (q − l)2 ,
and s = (k + l)2 , the hadronic matrix element is expressed by:

h


8  2
Âľ
Âľ 

2
gL + gR
p2 pν2 + p1 pν1 −4g 2 (1 − )
ut

2 
2 
Âľ
ν
¾ν
2
2
+4sq q  − g
q −u + q −t
(1 − )

H ¾ν =





 

 
Âľ
−g µν 2q 2 s − 2tu  + p1 q ν + q µ pν1 2 q 2 − t (1 − ) + 2 −q 2 + u 


 

i

 
Âľ
Âľ
Âľ 

+ p2 q ν + q µ pν2 2 q 2 − u (1 − ) + 2 −q 2 + t  + p1 pν2 + p2 pν1 4q 2 
h
i
8  2
2
¾νιβ
¾νιβ
+ i gL − gR 
p1ι qβ (2 (s + u)) + 
p2α qβ (−2 (s + t)) .
(3.10)
ut

The hadronic matrix element can be separated into a symmetric and anti-symmetric piece
under the change of p1 → −p1 , p2 → −p2 . The symmetric piece is thus:

H [S]¾ν

h


8  2
2
Âľ
ν
Âľ
ν
2
=
g + gR (l l + k k ) −4g (1 − )
ut L

2 
2 
Âľ
ν
¾ν
2
2
+4sq q  − g
q −u + q −t
(1 − )

 


 
2
2
2 q − t (1 − ) + 2 −q + u 
 


 

i
+ (lµ q ν + q µ lν ) 2 q 2 − u (1 − ) + 2 −q 2 + t  + (k µ lν + lµ k ν ) 4q 2  ,
−g µν





(3.11)

2q 2 s − 2tu

 + (k ¾ q ν

+ q¾kν )

66

and the anti-symmetric piece is:

H [A]¾ν =

h
i
8  2
2
i gL − gR
µναβ kα qβ (2 (s + u)) + µναβ lα qβ (−2 (s + t))
ut

(3.12)

With both the leptonic and hadronic matrix element, the amplitude squared is:
64 Q2
|M |2qq̄ = H µν Lµν =
ut 2


  
2 
2 


2
2
2
2
2
2
2
2
gL + gR fL + fR
a
t−Q
+ u−Q
+ 4 (k ¡ l12 ) + (l ¡ l12 )


 −a (t + u)2 − 4 (k · l12 + l · l12 )2


 



i
2 − g2
2 − f2
2 (k · l ) − 4 u − Q2 (l · l ) ,
+ gL
f
4
t
−
Q
12
12
R
L
R

(3.13)

where a = 1 in 4-dimensions, and a = 1 −  in d-dimensions. Finally, the color average factor
( 13 × 13 ) and the spin average factor ( 12 × 12 ), in addition to a sum over the initial state colors
(3) needs to be included in the final result.
To obtain the calculation for quark-gluon initial states, crossing symmetry can be used,
which is defined by the following transformation for this process:

p1 → p 2 ,

q → q,

p 1 → q − p1 − p2 ,

u → s,

s → t,

t → u.

− (p1 + p2 − q) → p1 ,

(3.14)
(3.15)

Additionally, the color and spin factors need to be modified since there is now a gluon in
1
the initial state. In d-dimensions, the spin factor goes from 12 × 12 → 2(1−)
× 12 , and the

color factor is 13 × 13 → 18 × 13 . Finally, since there is an exchange of an outgoing particle
with an incoming anti-particle, the final result needs to be multiplied by (−1). This gives

67

the amplitude squared as:

|M |2gq



  
2 
2 
1 Q2  2
2
2
2
2
2
= 64
gL + gR fL + fR
a − u−Q
− s−Q
su 2




2
2
2
2
−4 (k · l12 + l · l12 ) + (l · l12 ) +  a (u + s) + 4 (k · l12 )


 




2 − g2
2 − f2
2 (l · l ) − 4 s − Q2 (k · l + l · l )
+ gL
f
−4
u
−
Q
.
12
12
12
R
L
R
(3.16)

Similarly, for the anti-quark-gluon initial states, again crossing symmetry can be taken
advantage of to obtain the same result as Eq. 3.16, but with a change of sign for the antisymmetric piece. With the real corrections calculated, the virtual corrections are needed to
complete the NLO correction to the Drell-Yan process.

3.2.2

Virtual Correction

Finally, to obtain the complete NLO correction to the total cross-section to the Drell-Yan
process, the virtual corrections need to be calculated. The diagram that contributes a
non-zero result up to one-loop in the strong coupling are given in Fig. 3.2b. When using
dimensional regularization to calculate the loop contributions, the loops on massless external
legs lead to a contribution of zero, since the loop integral obtained is a scaleless integral, and
thus must be zero. From a physics standpoint, this can be seen by the fact that the UVdivergences exactly cancel the IR-divergences in dimensional regularization (U V = −IR ).
Therefore, only one diagram remains to be calculated. Additionally, when performing the
calculation, only the terms proportional to Îąs need to be considered. From this, the virtual
matrix element squared does not contribute, but the interference of the one-loop diagram

68

with the tree-level diagram does. The hadronic matrix element is given by:

 

¾ν
2 + g 2 (k µ lν + k ν lµ − g µν (k · l))
Hvirtual = 4 (1 + 2f ()) 2 gL
R



2 − g 2 µναβ l k
−2i gL
ι β ,
R

(3.17)

where
Îąs
f () =
C
4π F



4πµ2
Q2



1
Γ (1 − )



2
3
2
− 2 − −8+π .



(3.18)

Details of obtaining f () can be found in App. F. The one-loop matrix element squared is
thus:

|M |2vitrual = 16 (1 + 2f ()) Q4



 



2 + g2
2 + f2
2 (θ) + g 2 − g 2
2 − f 2 2 cos (θ) .
gL
f
2a
−
1
+
cos
f
R
L
R
L
R
L
R
(3.19)

Eventually, the limit of  to zero will be taken and the divergences in f () will cancel with
those in the real correction once the additional radiation is integrated out and also with the
corrections to the PDF.

3.3

NLO DY Total Cross-Section

It is convenient to express the matrix elements in terms of angular functions in the CollinsSoper Frame. At this order, the matrix element can be expressed using the following angular

69

functions:

L = 1 + cos2 θ,

1
2
A0 =
1 − cos θ ,
2

(3.20)

A1 = sin 2θ cos φ,

(3.22)

A2 =

1 2
sin θ cos 2φ,
2

(3.21)

(3.23)

A3 = cos θ,

(3.24)

A4 = sin θ cos φ.

(3.25)

Rewriting the real matrix elements in terms of these angular functions gives the following
for the q q̄ channel:




2 − t 2 + Q2 − u 2 2
Q
Q
|M |2qq̄ = 64
ut
2



 

 
2
2
2
2
2
2
2
2
gL + gR fL + fR H1 − H̃1  + gL − gR fL − fR H2 ,

(3.26)

where the H functions are given by:

H1
H̃1
H2


2
− Q2 − t
qT Q
=L+ 2
(A0 + A2 ) +
A

2

2 2
2
2 1
Q + qT
Q2 − u + Q2 − t Q + qT

2
qT2
Q2 + s
(t + u)2
=L
+
(A0 + A2 )

2

2

2

2
2
Q2 − u + Q2 − t
Q2 − u + Q2 − t Q2 + qT

2

2
Q2 − u − Q2 − t
2Q
2qT
=q
A3 +
A4 .

2

2 q
Q2 − u + Q2 − t
Q2 + qT2
Q2 + qT2
qT2

Q2 − u


2

(3.27)

(3.28)

(3.29)

There is a similar set of relations for the quark-gluon and anti-quark-gluon initial state
processes, see App. F. It is important to note that the coefficient of A0 and A2 is identical at
70

NLO. This is known as the Lam-Tung relation [142, 143, 144]. A more detailed discussion of
the angular functions for Drell-Yan, the breaking of the Lam-Tung relation, and a comparison
of the ResBos2 code to LHC data can be found in Sec. 4.3.
In the real diagram, the divergences come from when the gluon becomes soft or collinear
to an initial parton. In other words when the transverse momentum of the vector boson goes
to zero. In the limit qT → 0, the terms proportional to 12 for the q q̄ channel are:
q
T

|M |2singular







Q2  2
2
2
2
2
2
2
2
= 64
gL + gR fL + fR K1 L + gL − gR fL − fR 2K2 A3 , (3.30)
2

with
1
Q2
K1 = 2 δ (1 − zA ) δ (1 − zB ) ln
qT
qT2
!
!
2
1 + zA
1
+ 2
δ (1 − zB )
1 − zA
qT

!

3
−
2

!

+


− 2 ((1 − zA ) δ (1 − zB )) + zA ↔ zB ,
qT
!
!
2
Q2
3
K2 = 2 δ (1 − zA ) δ (1 − zB ) ln
−
2
qT
qT2
!
!
2
1 + zA
1
+ 2
δ (1 − zB ) + zA ↔ zB ,
1 − zA
qT

(3.31)

(3.32)

+

where zA/B is the partonic momentum fraction of the parton from hadron A/B respectively,
1
and (1−z)
is the plus-function and its properties can be found in App. F. There are similar
+

expressions for the quark-gluon and anti-quark-gluon initial states (App. F). The most important difference between the quark-anti-quark and (anti-)quark-gluon expressions is that
there are no terms proportional to δ(1 − zA )δ(1 − zB ) for those with gluons in the initial

71

state. Integrating out the transverse momentum of the vector boson results in the poles that
are needed to cancel the double poles in the virtual correction. In order to obtain a full cancellation of the poles, the parton distribution functions need to be renormalized. Once the
poles for the PDF renormalization are included, then all of the poles will be canceled, and a
finite result will remain. To obtain the singularities from the real correction, an integral over
the phase space is needed. Details on d-dimensional phase space integrals can be found in
App. F. The results of the phase space integration are shown below for the singular terms.


αs 4πµ2
1
2
2 (1 + z)2
=
σ
C
δ
(1
−
z)
−
0 F
π −Q2
Γ (1 − ) 2
 (1 − z)+
dQ2




  ln (1 − z) 
1 + z2
2
+4 1 + z
−2
ln z .
1−z
1−z
+

dσqRq̄

(3.33)

Similarly, performing the phase space integration over the virtual result gives the following
results:
dσqVq̄

Îąs
= σ 0 CF
2
π
dQ



4πµ2
−Q2



1
Γ (1 − )




2
3
2
− 2 − − 8 + π δ (1 − z) .



(3.34)

Comparing Eqs. 3.33 and 3.34 the terms proportional to 1/2 cancel, as required by the
KLN theorem. However, there remains a divergence which is canceled by the definition of
the parton distribution functions. The singularities of the PDFs are:

2
Îąs
CF

π

!
(1 + z)2
3
+ δ (1 − z) ,
(1 − z)+ 2

(3.35)

which exactly cancels the remaining poles. Combining the real and virtual correction results
in the total cross-section for Drell-Yan at NLO. At this order, the result is completely UV

72

Figure 3.3: Plot showing fixed-order versus resummed predictions. The dashed curves
correspond to fixed-order calculations, and the solid curves correspond to resummed
predictions. The plot is reproduced from Ref. [15]
finite. At higher corrections in Îąs this is no longer true, and requires the strong coupling to
be renormalized as discussed in Section 1.3.2. Details on the calculations to NNLO can be
found in [145], and parts of the calculations to N3 LO can be found in [146, 147, 148].

3.3.1

Breakdown of Fixed-Order

When using fixed-order calculations, certain distributions contain integrable singularities,
resulting in nonsense theoretical predictions in certain kinematical regions. For example, the
transverse momentum distribution of the vector boson in Drell-Yan displays such properties.
This is seen in the qT distribution at NLO and NNLO for Drell-Yan in Fig. 3.3.
As the calculation is performed at higher and higher orders the small qT region will
73

oscillate, as seen in Fig. 3.3 comparing the red dashed (NLO) and blue dashed (NNLO)
predictions. As the orders increase, the result continues to oscillate, but will never reach a
stable result until all orders of the calculation are included. The solution to this problem is
the introduction of the resummation formalism, which sums up the large logarithmns that
appear at each order in the fixed-order calculation.

3.4

Resummation Formalisms

The dynamics of multiple soft-gluon radiation in scattering processes is treated through the
use of the resummation formalism [149, 150, 151, 152, 153]. There are many applications
of resummation at modern colliders. In this work, the focus will be on the treatment of
transverse momentum resummation. The formalism was originally shown to be possible for
all the large logarithms (leading and subleading) to all orders by Collins, Soper, and Sterman [138]. The formalism developed in their work will be referred to as the CSS Formalism.
A more recent formalism was developed by Catani, de Florian, and Grazzini, which is known
as the CFG Formalism [154]. The details of the two formalisms are explained in Sec. 3.4.1
and Sec. 3.4.2 respectively. The differences between the two formalisms are highlighted in
Sec. 3.4.3. The remainder of this section will focus on the general outline of qT resummation.
Firstly, resummation is a means to relate the different scales of a multi-scale process to
a single scale, which also removes the large logarithms that result from the large difference
between the scales. Therefore, the first step is to factorize the cross-section calculation into
the different scale regions that are involved in the calculation. The regions that are important
to this work are known as the hard factor, the soft factor, and the collinear or jet factors. A
diagrammatic representation of each piece for the Drell-Yan process can be seen in Fig. 3.4,

74

C

H

S

C

Figure 3.4: A diagrammatic representation of the factorized cross-section for Drell-Yan,
broken into a soft, collinear, and hard factor. The soft factor is labeled by the S, the
collinear factors are labeled by the C’s, and the hard factor is labeled by the H
and can be expressed as:

dσ
dQ2 dydqT2

∝ H(µ)S(µ, µR )C1 (µ, µF )C2 (µ, µF )J(µ, Rµ0 ),

(3.36)

where H is the hard factor, S is the soft factor, C1 and C2 are the collinear factors for each
incoming hadron, and J is the jet factor. The remainder of the section is to discuss the
calculation of the soft factor, and derive the well known Sudakov factor. Starting from the
fixed-order calculation up to the nth order in Îąs , the result can be split into a singular piece,
 
Q2
1
m
and a regular piece. The singular piece are terms that are proportional to 2 log
2
qT

qT

(m = 0, 1, ..., 2n − 1) and δ(qT ), and the regular terms are less singular than those previously
 
Q2
1
m
n
mentioned. As detailed in Sec. 3.3.1, this calculation breaks down when Îąs 2 log
2
qT

qT

becomes large.
To resolve this issue, the logarithms need to be summed to all orders to obtain a finite
result in the limit qT → 0, and remove all large logarithms from the final result. In order to
perform the resummation correctly, the cross-section needs to be Fourier transformed into

75

impact parameter (b) space. In impact parameter space, the total transverse momentum is
explicitly conserved [155]. After the Fourier transform, the cross section can be expressed
as:
dσ
1
=
2
dQ2 dqT dy
(2π)2

Z

~

d2 bei~qT ¡b W̃ (b, Q, x1 , x2 ) + Y (qT , Q, x1 , x2 ),

(3.37)

where W̃ contains the resummation of the singular pieces of the cross section, and Y contains
the regular pieces of the cross section defined by taking the fixed-order calculation and
subtracting the asymptotic piece. The asymptotic piece contains the terms that are at least
as singular as 12 in the fixed-order calculation in the limit qT → 0. The calculation of the
q
T

asymptotic piece up to Îąs3 can be found in Secs. 3.5.1- 3.5.2.
By studying the form of the singular piece, the x1 and x2 dependence in W̃ can be
factorized into:

W̃ (b, Q, x1 , x2 ) =

X

Cj (b, Q, x1 ) Cj (b, Q, x2 ) W̃ (b, Q) ,

(3.38)

j

where Cj is a convolution of the PDFs with a collinear Wilson coefficient, with the convolution defined as:
x

X Z 1 dz
Cj =
Cja
, b, Âľ, Q fa (z, Âľ) ,
z
z
a x

(3.39)

where Cja is the Wilson coefficient, fa is the PDF, the sum a runs over all incoming partons, and j represents the parton that enters into the hard cross section calculation. These
functions are the collinear factors as previously mentioned. The remaining term contains
the hard factor, and the soft factors.

76

W̃ is determined by solving the evolution equation [156]:

∂
W̃ (Q, b) = [K (b¾, gs (¾)) + G (Q/¾, gs (¾))] W̃ (Q, b) ,
∂ log Q2

(3.40)

where K (bÂľ, gs (Âľ)) and G (Q/Âľ, gs (Âľ)) satisfy the renormalization group equations (RGEs),

d
K (bµ, gs (µ)) = −γK (gs (µ)) ,
d log Âľ
d
G (b/Âľ, gs (Âľ)) = ÎłK (gs (Âľ)) ,
d log Âľ

(3.41)
(3.42)

where ÎłK is the anomalous dimension, calculated from the singular terms of the cross section []. Through the RGE equations, K (bÂľ, gs (Âľ)) and G (b/Âľ, gs (Âľ)) can be evolved independently to scales of order 1/b and Q respectively, removing all large logarithms from the
calculation. After solving these equations, the A and B functions can be defined such that
Eq. 3.40 can be rewritten as:

∂
W̃ (Q, b) = −
∂ log Q2

Z C 2 Q2 2
dÂľ
2
2
C 2 /b2 Âľ
1

C 2 Q2
A (gs (Âľ) , C1 ) log 2 2 + B (Âľ) , C1 , C2
Âľ

!!
W̃ (Q, b) ,
(3.43)

where C1 and C2 are arbitrary constants of integration arising from solving the RGEs. It
is possible to calculate the values for the A and B functions order by order in perturbation
theory.
Finally, to obtain a result that can be used to make predicitons of the cross section, the
evolution equation of W̃ needs to be solved. The solution can be written as

W̃ (Q, b) =

e−S(Q,b) W̃

77




C1
,b ,
C2 b

(3.44)

where S is known as the Sudakov factor, and is given by
Z C 2 Q2 2
dÂľ
2
S (Q, b) =
2
C 2 /b2 Âľ
1

C 2 Q2
A (gs (Âľ) , C1 ) log 2 2 + B (Âľ) , C1 , C2
Âľ

!
.

(3.45)

Putting all of the results above together, the resummed cross section can by written as:

dσ
H
=
2
dQ2 dqT dy
(2π)2

Z

X
~
d2 bei~qT ·b e−S(Q,b)
C


j

j

 

C1
C1
, Q, x1 Cj
, Q, x2 +Y (qT , Q, x1 , x2 ).
C2 b
C2 b
(3.46)

This is the general form for transverse momentum resummation. However, this form is not
the final form used in calculations, due the fact that when the impact parameter becomes
large, the scale of resummation goes below ΛQCD . Therefore the calculation becomes nonperturbative. To prevent using a scale below ΛQCD , the b∗ prescription is introduced, where:

b

b∗ = r

1+

b2
2
bmax

,

(3.47)

where bmax is chosen such that 1/bmax is of order ΛQCD . The lower bound of the Sudakov
integral is then modified from C12 /b2 to C12 /b2∗ . This functional form prevents b∗ from ever
being large than bmax , preventing scales below ΛQCD . However, this causes the prediction
to be inaccurate at low qT , since a piece is removed by the b∗ prescription. To resolve this,
a non-perturbative function needs to be introduced.
There are many different proposals for the form of the non-perturbative function [157,
158, 159, 160]. In this section, the general concepts of the non-perturbative function will be
covered. The method of obtaining this function is through fits to data. It is believed that
the non-perturbative function should be universal, and only depend on the color structure

78

of the initial states. Details on a specific proposed non-perturbative function can be found
in Section 3.4.4. This then gives the final form of the resummation formalism in a scheme
independent way as:

X H Z
dσ
~
=
d2 beiq~T ·b e−Spert e−SN P C ⊗ fi C ⊗ fj ,
2
2
2
(2π)
dQ dqT dy
i,j

(3.48)

where Spert is the Sudakov factor, while SN P is the non-perturbative Sudakov factor. Finally,
up to this point the integration coefficients (C1 , C2 , and C3 ) were left to be arbitrary. The
canonical choice for these scales are given by C1 = b0 , C2 = 1, and C3 = b0 , where b0 =
2e−γE . In Section 3.6, the relationship between the canonical scale choice and any arbitrary
choice is calculated. The theory uncertainty due to the missing higher order corrections can
be estimated by modifying the values of C1 , C2 , and C3 as discussed later in this chapter.

3.4.1

Collins-Soper-Sterman Formalism

So far, the resummation formalism has been developed in a resummation scheme independent
way. Here, the Collins-Soper-Sterman Formalism is introduced [138]. In this formalism, the
hard matrix element, H, is taken to be 1, with no corrections as a function of Îąs , and the B
and C coefficients become process dependent. The A, B, and C coefficients can be expanded
as a series in Îąs as:

A=

X  Îąs n

A(n) ,

(3.49)

B (n) ,

(3.50)

X  Îąs n (n)
Cij .
π
n

(3.51)

n

B=

X  Îąs n
n

Cij = δij +

π

79

π

For Drell-Yan, the coefficients for A up to Îąs3 , B up to Îąs2 and C up to Îąs are given with
the canonical scale choice as [161, 162, 163, 164, 165, 166]:

A(1) = CF ,
1
A(2) = CF
2

(3.52)
2





67 π
5
−
CA − Nf ,
(3.53)
18
6
9

 N2


CF Nf
55
11ζ3 11π 4 67π 2 245
f
(3)
2
A = CF
ζ3 −
−
+ CA
+
−
+
2
48
108
24
720
216
96


−7ζ3 5π 2 209
+CA Nf
+
−
,
(3.54)
12
108 432
3
B (1) = − CF ,
2



π2
3
11 2 193 3
(2)
2
B = CF
−
− 3ζ3 + CF CA
π −
+ Îś3
4
16
36
48
2


2
17 π
+ CF Nf
−
,
24 18


1
1
(1)
Cqq (z) = CF (1 − z) + δ(1 − z) CF π 2 − 8 ,
2
4
1
(1)
Cqg (z) = z(1 − z),
2
(1)

(1)

(1)

Cqq̄ (z) = C 0 (z) = C 0 (z) = 0,
qq
q q̄

(3.55)

(3.56)
(3.57)
(3.58)
(3.59)

where CF = 4/3, CA = 3, and Nf is the number of active quarks. The results for B (3) can
be found in Ref. [167], and for C (2) can be found in Ref. [168].1

3.4.2

Catani-deFlorian-Grazzini Formalism

Catani, deFlorian, and Grazzini realized that the behavior of soft gluons is independent of
the hard process, and developed a resummation formalism in which the hard factor which is
process dependent can be pulled out of the Fourier Transform [169]. This then leads to the
1 The results for B (3) and C (2) can be found in App. F.

80

calculation in impact parameter space only depending on the initial state partons, and not
the hard factor. Like in CSS, the A, B, and C functions can be expanded as a series in Îąs .
However, in addition to these three, the hard factor H is not fixed to one, but can also be
expanded as a series in Îąs . In the CFG formalism, the A, B, and C coefficients are given
by:

A(1) = CF ,

(3.60)



67 π 2
5
−
CA − Nf ,
(3.61)
18
6
9

 N2


CF Nf
55
11ζ3 11π 4 67π 2 245
f
(3)
2
A = CF
ζ3 −
−
+ CA
+
−
+
2
48
108
24
720
216
96


−7ζ3 5π 2 209
+CA Nf
+
−
,
(3.62)
12
108 432
A(2)

1
= CF
2



3
B (1) = − CF ,
2


−17 88
(2)
2
B = (−3 + 24ζ2 − 48ζ3 ) CF +
− ζ2 + 24ζ3 CF CA
3
3



2 16
+
+ Μ2 CF Nf /16 + CF β0 Μ2 ,
3
3
1
(1)
Cqq (z) = CF (1 − z),
2
1
(1)
Cgq (z) = CF z,
2
1
(1)
Cqg (z) = z(1 − z),
2
(1)

(1)

(3.63)

(3.64)
(3.65)
(3.66)
(3.67)

(1)

Cqq̄ (z) = C 0 (z) = C 0 (z) = 0.
qq
q q̄

(3.68)

The relationship for obtaining the A, B, and C coefficients in the CFG formalism from
the coefficients in the CSS formalism can be found in Sec. 3.4.3. The hard factor is process

81

dependent, and for Drell-Yan are given as:

H DY (1) = CF

π

up to O Îąs2

[154].


−4 ,
(3.69)
2



59ζ3 1535 215π 2
π4
1 2
511 67π 2 17π 4
DY
(2)
H
= C F CA
−
+
−
+ CF −15ζ3 +
−
+
18
192
216
240
4
16
12
45


1
+
C N 192ζ3 + 1143 − 152π 2 ,
(3.70)
864 F f

3.4.3




Comparison of CSS to CFG

The conversion between the CSS and CFG Formalisms can be given using the all orders
relations [169]:

F (z)
Cab

BcF

=

h

i1
F
Ha 2

Cab (z),

(3.71)

d ln HcF
= Bc − β
,
d ln Îąs

(3.72)

where the F superscript is used to indicate which pieces are process dependent, and β is
the function that describes the running of Îąs 2 . This can be expanded order by order to
give a conversion between explicit CSS and CFG coefficients. It is important to note that
the A coefficients are always universal, and B (1) is also universal (only depends on the color
2 See Section 1.3.2 for a detailed discussion of the β function.

82

structure of the initial state). The conversions up to N3 LL resummation are listed below:

1 (1)F
(1)
(z) = Cab (z) + δab δ(1 − z) Ha ,
2


1 (1)F (1)
1
1  (1)F 2
(2)F
(2)
(2)F
Cab (z) = Cab (z) + Ha Cab (z) + δab δ(1 − z)
Ha
−
Ha
,
2
2
4
(1)F

Cab

(2)F

Bc

(3)F
Bc

with β0 =

(2)

= Bc
=

(3)
Bc

(1)F

(3.73)
(3.74)

+ β0 Hc

,

(3.75)

(1)F
+ β1 Hc



1  (1)F 2
(2)F
+ 2β0 Ha
−
Ha
,
2

(3.76)

11CA −2Nf
12

and β1 =

2 −5C N −3C N
17CA
A f
F f
.
24

Comparisons between the numerical results, and how each compare to data can be found
in Section 4.1 and 4.2.

3.4.4

Non-Perturbative Contribution

As mentioned above, the resummation formalism needs to be cut-off at large values of impact
parameter. Originally, the functional form was taken to be solely for the calculation of DrellYan. However, recent work in the transverse-momentum dependent PDF community (TMD)
propose that the functional form should be universal.
The most successful form for the non-perturbative function is the Brock-Landry-NadolskyYuan (BLNY) fit to the transverse momentum dependent Drell-Yan lepton pair production
in hadronic collisions [157, 170]. The BLNY fit parameterizes the non-perturbative form
factors as (g1 + g2 ln(Q/2Q0 ) + g1 g3 ln(100x1 x2 )))b2 in the impact parameter space with x1
and x2 being the longitudinal momentum fractions of the incoming nucleons carried by the
initial state quark and antiquark. These parameters are constrained from the combined
fit to the low transverse momentum distributions of Drell-Yan lepton pair production with

83

4GeV < Q < 12GeV in fixed target experiments and W/Z production (Q ∟ 90GeV) at
the Tevatron. However, this parameterization does not apply to the semi-inclusive deep inelastic scattering (SIDIS) processes measured by HERMES and COMPASS collaborations:
extrapolating the above parameterization down to the typical HERMES kinematics where
Q2 is around 3GeV2 , the transverse momentum distribution of hadron production in the
experiments cannot be described [171, 172].
A parametrization form that can consistently describe the Drell-Yan data and SIDIS
data in the CSS resummation formalism with a universal non-perturbative TMD function is
proposed below. In order to describe the SIDIS data, it is necessary to modify the original
BLNY parameterization. In the original BLNY parameterization, there is a strong correlation between the x-dependence and the Q2 -dependence [157, 170], since x1 x2 = Q2 /S
where S is the center of mass energy squared of the incoming hadrons. Therefore, the xdependence will be separated out, and assumed to follow a power law behavior: (x0 /x)Îť .
These two parameterizations (logarithmic and power law) differ strongly in the intermediate
x range. Secondly, the ln Q term in the non-perturbative form factor is modified by following
the observation of Ref. [171, 172], which has shown that a direct integral of the evolution
kernel can describe the SIDIS and Drell-Yan data of Q2 range from a few to hundreds of
GeV2 . Direct integration of the evolution kernel leads to a functional form proportional to
ln(b/b∗ ) ln(Q), instead of b2 ln(Q2 ). Therefore, the experimental data will be fit with the
proposed non-perturbative function defined as:



g1 b2 + g2 ln(b/b∗ ) ln(Q/Q0 ) + g3 b2 (x0 /x1 )λ + (x0 /x2 )λ ,

(3.77)

where Q20 = 2.4GeV 2 , x0 = 0.01 and Îť = 0.2, and are fixed in the fit. This new non-

84

perturbative function will be referred to as the Sun-Isaacson-Yuan-Yuan (SIYY) non-perturbative
function. The x-dependence is motivated by a saturation picture for parton distributions at
small-x. This functional form also has mild dependence on x in the intermediate x-range as
compared to the original BLNY parameterization.
After obtaining the TMD non-perturbative function from the fit to the Drell-Yan data,
the fit is applied to the transverse momentum distributions in SIDIS processes from HERMES
and COMPASS.
In the BLNY parameterization, the g2 term is responsible for the Q2 dependence, which is
modified in order to describe the Drell-Yan and SIDIS processes simultaneously. At small-b,
the above function reduces to b2 behavior, which is consistent to the power counting analysis
in Ref. [160]. However, at large b, the logarithmic behavior will lead to different predictions
from BLNY.
To constrain the values of g1 , g2 , and g3 a global fit is performed on the Drell-Yan data.
The data that is included in the fit is listed below:
• Drell-Yan lepton pair production from fixed target hadronic collisions, including R209,
E288 and E605 [16, 18, 17].
• Z boson production in hadronic collisions from Tevatron Run I and Run II [19, 20, 21,
22].
In total, 7 Drell-Yan data sets from 3 fixed target experiments and 4 Tevatron experiments
are included. While the LHC data was not included in the fit, the fit is compared to data
from the LHC afterwards, and is shown to be in good agreement. In total, there are 140
experimental data points, and g1 , g2 , and g3 as free parameters in the global fit, with
bmax = 1.5GeV −1 . In the numerical calculations, the CT10-NLO parton distributions at
85

Parameter
g1
g2
g3
E288
(28 points) (Norm Err = 0.25)
E605
(35 points) (Norm Err = 0.15)
R209
(10 points) (Norm Err = 0.1)
CDF Run I
(20 points) (Norm Err = 0.04)
D0 Run I
(10 points) (Norm Err = 0.04)
CDF Run II
(29 points) (Norm Err = 0.04)
D0 Run II
(8 points) (Norm Err = 0.04)
χ2
χ2 /DOF

SIYY1 fit
0.200
0.810
0.0204
Nf it = 0.82
χ2 = 52.6
Nf it = 0.86
χ2 = 63.5
Nf it = 1.02
χ2 = 3
Nf it = 1.06
χ2 = 10
Nf it = 0.93
χ2 = 7
Nf it = 0.990
χ2 = 30
Nf it = 0.94
χ2 = 3.7
169
1.21

SIYY2 fit
0.18084
0.16741
0.00323
Nf it = 0.757
χ2 = 38
Nf it = 0.824
χ2 = 61
Nf it = 0.956
χ2 = 5
Nf it = 1.048
χ2 = 9.3
Nf it = 0.94
χ2 = 6.3
Nf it = 0.992
χ2 = 26.2
Nf it = 0.939
χ2 = 3.6
150
1.07

Table 3.2: The non-perturbative functions parameters fitting results. Here, Nf it is the
fitted normalization factor for each experiment.
the scale µ = b0 /b∗ are used [173]. An additional fitting parameter (Nf it ) is assigned for
each experiment to account for the luminosity uncertainties.
In Figs. 3.5, 3.6, 3.7, 3.8, and 3.9, the best fits to the Drell-Yan data from E288, E605,
and R209 Collaborations, and Z boson production from the CDF and D0 Collaborations at
the Tevatron Run I and II are shown. The fitting results and χ2 distributions are listed in
Table 3.2. The plots and the table show that the SIYY non-perturbative function provides
a reasonable fit to all 7 experimental data with 3 non-perturbative parameters g1,2,3 and 7
independent normalization factors.
In the above fit, SIYY2 is an update fit to the BLNY fit, with the choice of bmax = 1.5
instead of the traditional value of 0.5. This fit is updated as well to estimate the uncertainty

86

E288: 6<Q<7 GeV
dσ
dp2dy

Legend
Data
SIYY1
SIYY2

3

2.5

1.1

Legend
Data
SIYY1
SIYY2

T

T

dσ
dp2dy

E288: 5<Q<6 GeV

1
0.9
0.8

2
0.7
0.6
1.5
0.5
0.4

1

0

0.2

0.4

0.6

0.8

1

1.2

0.3
0

1.4
p (GeV)

0.2

0.4

0.6

0.8

1

1.2

T

T

dσ
dp2dy

E288: 8<Q<9 GeV

0.35

Legend
Data
SIYY1
SIYY2

T

Legend
Data
SIYY1
SIYY2

0.4

T

dσ
dp2dy

E288: 7<Q<8 GeV
0.16

0.14

0.3

0.12

0.25

0.1

0.2

0.08

0.15

0.06

0

1.4
p (GeV)

0.2

0.4

0.6

0.8

1

1.2

0.04
0

1.4
p (GeV)

0.2

0.4

T

0.6

0.8

1

1.2

1.4
p (GeV)
T

Figure 3.5: Fit to the differential cross section for Drell-Yan lepton pair production in
hadronic collisions from E288 Collaboration [16].
in the non-perturbative fit due to the parameterization of the non-perturbative function.
Among these parameters, the most important one, relevant to the LHC W and Z boson
physics, is g2 which controls the Q2 dependence in the non-perturbative form factors. From
the χ2 distribution in scanning the g2 parameter as shown in the left panel of Fig. 3.11, its
uncertainty is given as:
g2 = 0.81 Âą 0.06 (at 90% C.L.) .

(3.78)

In order to demonstrate the sensitivity of g2 on different experiments, in the right panel of
Fig. 3.11, the ∆χ2 distributions are plotted as functions of g2 from separate data sets: one
from the Drell-Yan experiment E288, the combined contribution from all other Drell-Yan
experiments, and one from Tevatron Z-boson experiments. From this figure, it is clear that
the strongest constraints come from the precise Drell-Yan data at fixed target experiments,

87

dσ
dp2dy

E605: 8<Q<9 GeV

1
0.9

Legend
Data
SIYY1
SIYY2

T

Legend
Data
SIYY1
SIYY2

T

dσ
dp2dy

E605: 7<Q<8 GeV
1.1

0.45

0.4

0.8

0.35

0.7
0.3
0.6
0.25
0.5
0.2

0.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4
p (GeV)

0

0.2

0.4

0.6

0.8

1

1.2

T

1.4
p (GeV)
T

E605: 11.5<Q<13
dσ
dp2dy

dσ
dp2dy

E605: 10.5<Q<11.5 GeV

Legend
Data
SIYY1
SIYY2

T

T

Legend
Data
SIYY1
SIYY2

0.08

0.07

0.07

0.06

0.06

0.05

0.05

0.04

0.04
0.03
0.03
0

0.2

0.4

0.6

0.8

1

1.2

1.4
p (GeV)

0

0.2

0.4

0.6

0.8

1

1.2

T

1.4
p (GeV)
T

dσ
dp2dy

E605: 13.5<Q<18 GeV
Legend
Data
SIYY1
SIYY2

T

0.03

0.025

0.02

0.015

0.01

0

0.2

0.4

0.6

0.8

1

1.2

1.4
p (GeV)
T

Figure 3.6: Fit to the Drell-Yan data from the E605 Collaboration [17].

R209: 5<Q<8 GeV
dσ
dp2dy

dσ
dp2dy

R209: 8<Q<11 GeV

T

100

Legend
Data
SIYY1
SIYY2

T

Legend
Data
SIYY1
SIYY2

120

14

12

10

80

8
60
6
40
4
20
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2
0

1.8
2
p (GeV)
T

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8
2
p (GeV)
T

Figure 3.7: Fit to the Drell-Yan data from the R209 Collaboration [18].

88

Legend
Data
SIYY1
SIYY2

25

20

dσ
dp

dσ
dp

T

D0 Run 1

T

CDF Run 1
22

Legend
Data
SIYY1
SIYY2

20
18
16

15

14
12

10

10
8

5
6
0

1

2

3

4

5

6

7

8

9
10
p (GeV)

0

1

2

3

4

5

6

7

8

9
10
p (GeV)

T

T

Figure 3.8: Fit to the Tevatron Run I data from the CDF and D0 Collaborations [19, 20].
The fits include only the A(1,2) , B (1,2) , and C (1) contributions.

D0 Run 2
T

Legend
Data
SIYY1
SIYY2

22
20

dσ
dp

dσ
dp

T

CDF Run 2
24

Legend
Data
SIYY1
SIYY2

20
18
16

18
16

14

14

12

12

10

10
8
8
6

6

4

4
0

2

4

6

8

10

12

14
p (GeV)

0

2

4

6

T

8

10

12

14

16

18

20
p (GeV)
T

Figure 3.9: Fit to the Tevatron Run II data from the CDF and D0 Collaborations [21, 22].

-1
1 dσ
σdp (GeV )

ATLAS
Legend
Data
SIYY1
SIYY2

T

0.06

0.05

0.04

0.03

0.02

0

2

4

6

8

10

12

14

16

18
20
p (GeV)
T

Figure 3.10: Comparison of the SIYY1 and SIYY2 fit to the ATLAS data.

89

∆ χ2

80

Data
Total

70

E288

∆χ2

Drell-Yan w/o E288

60

Data

60

Tevatron-Z

Total
E288

50

Drell-Yan w/o E288

50

Tevatron-Z

40

40

30

30

20

20

10

10

0

0
0.65

0.7

0.75

0.8

0.85

0.9

0.95

g

0.12

1

0.14

0.16

0.18

0.2

0.22

g

2

2

Figure 3.11: ∆χ2 distribution scanning g2 parameter in SIYY1 fit (left), and ∆χ2
distribution scanning g2 parameter in SIYY2 fit (right).
i.e., the E288 experiment. It is also interesting to note that, although the Tevatron data for
Z-boson production is the most precise data, it does not pose a strong constraint on the g2
parameter. This can be understood as a result of the dominance of the perturbative contribution to the transverse momentum distribution of W/Z boson productions at these collider
energies. A similar observation was made in Ref. [174, 159] with a different prescription to
introduce the non-perturbative form factors. However, in the original BLNY fit, the Gaussian form of the g2 term (g2 b2 ln Q) leads to a stronger effect on the W/Z boson production
as shown in Ref. [158]. The comparison of this non-perturbative fit to the SIDIS data can
be found in App. F. As seen in the figures, the updated Gaussian fit is also consistent with
the data, and the two fits give a good estimate of the uncertainty due to the form of the
non-perturbative function.

3.5

The Asymptotic Piece and Obtaining Fixed-Order
Cross-Sections from Resummed Results

The asymptotic piece contains the terms that are at least as singular as 12 , and can be
q
T

obtained in two manners. One way to obtain the asymptotic piece is to take the limit of
90

qT goes to zero of the real correction. However, this explicitly involves integrating out all
the kinematic variables of the additional radiations, with the requirement that the boson
P
qT is the desired value, i.e. adding in a δ (2) (q~T − i p~T i ), where i is over all additional
radiations. The other method of obtaining the asymptotic piece is to perform an expansion
of the resummation formalism to a fixed-order in Îąs . The expansion of the resummation
formalism reproduces all of the divergences that exist in the real correction, and thus the
two should cancel as qT goes to zero. The second method is detailed below up to Îąs3 .
An important cross check of the calculations, is to ensure that when the appropriate
expansions are made, the fixed-order total cross section is reproduced. In order to calculate
the fixed-order total cross section three pieces are needed, the singular piece below pcut
T , the
cut
real correction for qT > pcut
T , and the regular piece below pT . The regular piece is the

difference between the real correction and the asymptotic piece. The contribution to the
total cross-section from the regular piece is small if pcut
T is taken to be small. Throughout
this work, pcut
T will be chosen to be small such that the regular piece provides a negligible
contribution and will therefore be ignored. For the leading order calculation, obtaining
the fixed-order calculation from the resummed result is trivial, and therefore the first order
discussed will be NLO. Throughout the following calculation, without loss of generality, the
choices for the resummation constants (C1 , C2 , and C3 ) will be chosen to be the canonical
values for simplicity (C1 = C3 = b0 , C2 = 1). As will be discussed in the following section
(Sec. 3.6), all C1 , C2 , and C3 dependence should cancel in the expansion of the resummation
formalism to some fixed-order in Îąs . Therefore, the asymptotic, singular, and real piece do
not depend on the choice of resummation constants.
The asymptotic piece consists of terms that are at least as singular as qT−2 . The singular
piece consists of terms that are at least as singular as qT−2 plus it also includes the δ(qT ) terms.
91

The expansion of the A, B, C, and H coefficients to O (Îąsn ) can be explicitly found up to


O Îąs3 in Section 3.4.1, Section 3.4.2, and Appendix F for the CSS and CFG formalisms. The
expansion of both the CSS and CFG formalism result in the same singular and asymptotic
piece, so it is sufficient to only consider the CSS formalism. The lepton variables and angle
between ~b and ~qT are integrated out to simplify the discussion, but do not modify the results.
After these simplifications, the resummation formalism becomes:
Z ∞


dσ
1
−S η/qT ,Q
lim
∝
dΡΡJ
(Ρ)
e
0
qT →0 dQ2 dydq 2
2πqT2 0
T




2
2
2
2
× C ⊗ fj x1 , qT /η C ⊗ fk̄ x2 , qT /η + j ↔ k̄,

(3.79)

where terms that are not of importance in the derivation have been dropped, and terms that
are less singular than 12 or δ(qT ) have also been dropped. The asymptotic piece is obtained
qT

by integrating over Ρ = bqT . This can be performed by using the following integration by
parts identity:
Z ∞
0

Z ∞
dηηJ0 (η) F (η) = −

0

dΡΡJ1 (Ρ)


dF (Ρ)
,
dΡ

which is true given that the boundary term vanishes, ΡJ1 (Ρ) F

(η)∞
Ρ=0

(3.80)


= 0 . Additionally,

the following integral results will be important in obtaining both the asymptotic and singular

92



piece up to O Îąs3 :

Z ∞
0

dΡJ1 (Ρ) lnm






1, if m = 0






2



ln Q2 , if m = 1


qT





2



ln2 Q2 , if m = 2


qT

! 


2
Ρ 2 Q2
=
ln3 Q2 − 4ζ(3), if m = 3
2
2

qT
b0 q T





2
2



ln4 Q2 − 16ζ(3) ln Q2 , if m = 4


qT
qT





2
2


ln5 Q2 − 40ζ(3) ln2 Q2 − 48ζ(5), if m = 5,



qT
qT






Q2
Q2
Q2

ln6 2 − 80ζ(3) ln3 2 − 288ζ(5) ln 2 + 160ζ(3)2 , if m = 6,
qT

qT

qT

(3.81)

where b0 = e−γE , and γE is the Euler constant. Up through m = 2 is needed for the O (αs )


calculations, through m = 4 is needed for the O Îąs2 calculations, and all of the above will


be needed for the O Îąs3 calculations.
Secondly, the singular piece is obtained by taking the integral of qT for the O (Îąsn )
corrections from 0 to pcut
T , and is calculated in a manner similar to the asymptotic piece,
with one key modification. Instead of using the integration by parts identity, the order of
integration between qT and b is interchanged, giving us the following relationship:


Z pcut 2
T

1
(2π)2 0

dqT2

Z

~
d2 bei~qT ¡b F

Z ∞


1
cut F (b) .
(b) =
dbpcut
J
bp
1
T
T
2π 0

(3.82)

Finally, the calculation of the real corrections is needed. The additional jets then need
to be integrated out, and the singularities need to be canceled in order to obtain a finite
prediction. For NLO this is straightforward, and there is an analytic form. However, at
93

NNLO this is not possible, and an external code with the parameters tuned to match that
of the ResBos2 code will be used to obtain this contribution. Additionally, to ensure the
external code is tuned correctly, it will be validated against the total cross-section at NLO.
Once the functional form for C (3) is calculated, the ResBos2 code along with the Z plus jet
to NNLO calculation will be able to predict the N3 LO total cross-section.

3.5.1

O (Îąs ) Singular, Asymptotic Piece, and NLO Result

In this subsection, the computational details of obtaining the singular and asymptotic piece
needed for the NLO calculation are shown. Firstly, to obtain the asymptotic piece the


Sudakov factor is expanded to O (Îąs ), S(b, Q) = S (1) (b, Q) + O Îąs2 , with S (1) given as:

S (1) (b, Q)

Îąs Q2
=
π


"

#
1 (1) 2 Q2 b2
Q 2 b2
(1)
A ln
+ B ln 2
.
2
b20
b0

(3.83)

Also, the convolution of the PDF with the C function needs to be expanded to O (Îąs ).
At O (Îąs ), the C function does not appear in the asymptotic piece at this order, but will
appear at higher orders. For the asymptotic piece, at this order it is sufficient to use the
PDF evolution equation given as:


 
dfj (x, Âľ2 )
Îąs (Âľ2 )  (1)
2 + O Îą2 .
=
P
⊗
f
x,
Âľ
a
s
j←a
2π
d ln Âľ2

(3.84)

Using the integration by parts identity in Eq. 3.80, requires the derivatives with respect to
Ρ of the Sudakov factor and the PDF, given as:
"
#
 
2Ρ2
d −S(η/q ,Q) −2 αs (Q2 )
Q
T
e
=
A(1) ln 2 2 + B (1) + O Îąs2 ,
dΡ
Ρ
π
b0 q T

94

(3.85)

and


 
d  2 2 2  −2 αs (Q2 )  (1)
fj x, b0 qT /Ρ =
Pj←a ⊗ fa x, µ2 + O αs2 .
dΡ
Ρ
2π

(3.86)

Combining these results with those from Eq. 3.81, the final results for the asymptotic piece
is given as:




dσ
σ0 1 ιs (Q2 ) nh 
2
2
lim
=
f j x1 , Q
Pk̄←b ⊗ fb x2 , Q
S 2πqT2
π
qT →0 dQ2 dydq 2
T


i


+ fk̄ x2 , Q2 Pj←a ⊗ fa x1 , Q2
(3.87)
"
#
)

 

 
Q2
+ 2 A(1) ln 2 + B (1) fj x1 , Q2 fk̄ x2 , Q2 + j ↔ k̄ + O αs2 .
qT
For simplicity, it is useful to introduce the following definition,
n
∞ 2n−1 
dσ
σ0 1 X X X ιs (¾2 )
(i,j) m
=
n Cm ln
2
2
2
S
π
dQ dydqT
2πqT i,j n=1 m=0

Q2
qT2

!
,

(3.88)

which becomes very useful for organization beyond O (Îąs ). The definition above differs from
s
that found in Ref. [175] by expanding in factors of απs instead of α
2π , and the overall factor
1 instead of 1 . Using these definitions the above results are given as:
for the 12 term is 2π
π
q
T

(i,j)
1 C1

= 2A(1) fi fj ,

(i,j)
1 C0




= 2B (1) fi fj + fj (Pi←b ⊗ fb ) fi Pj←a ⊗ fa .

Secondly, calculating the singular piece begins by expanding the Sudakov factor in the
same manner. However, instead of using the evolution of the PDF as a shortcut to obtain

95

the derivative with respect to Ρ, the PDF is directly expanded to O (ιs ) as:

  

 
2


Îą
Q
Âľ2
s
2 + O Îą2 ,
fj x, Âľ2 = fj x, Q2 +
ln
P
x,
Q
j→a⊗fa
s
2π
Q2








(3.89)

where Âľ2 is the factorization scale3 . It will be convenient to again introduce the following
definition:


Z pcut 2
T
0

n
∞ 2n 
dσ
σ0 X X X ιs (¾2 )
(i,j) m
2
dqT
=
ln
n Vm
2
2
S
π
dQ dydqT
i,j n=0 m=0

Q2
qT2

!
(3.90)

Combining this with Eq. 3.82 we obtain the following results using the above definition of
(i,j)
n Vm :

(i,j)
0 V0

= fi fj ,

(i,j)
1 V2

1
= − A(1) fi fj ,
2



1
= −B (1) fi fj − fj (Pi←b ⊗ fb ) + fi Pj←a ⊗ fa ,
2




(i,j)
(1)
(1)
= fj Ci←b ⊗ fb + fi Cj←a ⊗ fa .
1 V0
(i,j)
1 V1

Finally, the regular terms and real corrections can be found in Ref. [176] in Section 2.2.5.
By combining these pieces, the NLO prediction can be calculated, and is implemented into
the ResBos2 code. As mentioned above, there is a pcut
T that is artificially introduced to
separate the singular region from the regular region. The numerical result at NLO is for the
total cross section for Drell-Yan in the invariant mass region of 66 GeV < Mll < 116 GeV
√
at a collider energy of S = 8 TeV. Furthermore, comparisons to other publicly available
codes tuned to have the same Electroweak parameters as defined in Ref. [68], and mentioned
3 The higher order expansion for the PDF is detailed in App. F

96

NLO Calculation
ResBos2 (qTcut = 0.1GeV)
ResBos2 (qTcut = 3GeV)
MCFM
FEWZ

Cross-Section
1113.3 pb
1115.1 pb
1112.6 pb
1113.2 pb

Table 3.3: Total inclusive NLO Drell-Yan Cross-sections for 66 GeV < Mll < 116 GeV.
MCFM calculation using version 8.0 [3]. FEWZ calculation using version 3.1 rc [4]

in Sec. 1.1.4, and using the CT14nnlo PDF [10], can be seen in Table 3.3.

3.5.2



O Îąs2 Singular, Asymptotic, and the NNLO Total CrossSection

At NNLO, the procedure above is extended to the next order in Îąs . The most important
difference from NLO is that the expansion of Îąs (Âľ2 ) needs to be considered. This expansion


is given by the following to O Îąs3 :

Îąs

(Âľ2 )
π

Îąs (Âľ2R )
Îąs (Âľ2R )
=
−
π
π

!2

Îąs (Âľ2R )
β0 ln 2 +
π
ÂľR
Âľ2

!3
β02 log2

Âľ2
Âľ2R

Âľ2

!

− β1 log 2 +O
ÂľR



Îąs4



(3.91)
where β0 and β1 are defined in App. E, and ¾R is the renormalization scale4 . The Sudakov


factor then can be expressed as S = S (1) + S (2) + O Îąs3 with:

S (2) (b, Q)

Îąs2 Q2
=
π2


"

#
1 (2) 2 Q2 b2
Q2 b2
(2)
A ln
+ B ln 2
,
2
b20
b0

4 The complete derivation of the result can be found in Appendix F

97

(3.92)

,

and after expanding Îąs , the Sudakov factor is
!
2 b2
1 (1) 2 Q2 b2
Q
A ln
+ B (1) ln 2
2
b20
b0
 Îą 2 1
Q2 b2 Âľ2
1
Q 2 b2 1
Q2 b2
s
+
A(1) β0 ln2 2 ln R2 + A(1) β0 ln3 2 + A(2) ln2 2
π
4
6
2
Q
b0
b0
b0
!
 
1
Q2 b2 Âľ2
1
Q2 b2
Q2 b2
+ B (1) β0 ln 2 ln R2 + B (1) β0 ln2 2 + B (2) ln 2
+ O Îąs3 .
2
4
Q
b0
b0
b0
Îąs
S=
π

(3.93)

A similar expansion as above is carried out for the convolution of the C function with
the PDF. The results can then be expressed as:
!!
Âľ2F
Q2 b2
ln 2 + ln 2
Q
b0
!
Âľ2
1  Îąs 2
Q 2 b2
+
4β0 C (1) ⊗ f ln 2 + ln F2
8 π
Q
b0
!
!
!
2
2
2 2
2 b2
2 b2
2 b2
Âľ
Âľ
Âľ
Q
Q
Q
− 2β0 P (1) ⊗ f ln 2 + ln F2
ln 2 + ln R2 + β0 P (1) ⊗ f ln 2 + ln F2
Q
Q
Q
b0
b
b0
! 0
Âľ2
Q2 b2
+ 4C (1) ⊗ P (1) ⊗ f ln 2 + ln F2 + 8C (2) ⊗ f
Q
b0
!2
!
2
2
 
2
2
2
2
Âľ
Âľ
Q b
Q b
+ P (1) ⊗ P (1) ⊗ f ln 2 + ln F2
+ 2P (2) ⊗ f ln 2 + ln F2  + O αs3 ,
Q
Q
b0
b0

Îąs
C ⊗f =f +
π

1
C (1) ⊗ f − P (1) ⊗ f
2

(3.94)
where ÂľF is the factorization scale.

98

Following the procedure used at NLO, the n Cm ’s for the asymptotic piece, are given as:

2
= −2 A(1) fi fj ,





(i,j)
(1)
(1)
(1)
= −6A B + 2A β0 fi fj − 3A(1) fj (Pi←b ⊗ fb ) fi Pj←a ⊗ fa ,
2 C2
!
2

2


Âľ
(i,j)
= A(1) β0 ln R2 + 2A(2) − 2 B (1) + B (1) β0 fi fj + 4A(1) C (1) ⊗ fi fj
2 C1
Q




Âľ2F
(1)
(1)
(1)
(1)
− 2A
P ⊗ fi fj ln 2 − 4B
P ⊗ fi fj
Q







+ β0 P (1) ⊗ fi fj − P (1) ⊗ P (1) ⊗ fi fj − P (1) ⊗ fi P (1) ⊗ fj + i ↔ j,
!
2

2
Âľ
(i,j)
= 4 A(1) Μ(3) + B (1) β0 ln R2 + 2B (2) fi fj
2 C0
Q
!
2




Âľ
+ B (1) 4 C (1) ⊗ fi fj − 2 P (1) ⊗ fi fj ln F2
Q
!
2




Âľ
− β0 4 C (1) ⊗ fi fj − 2 P (1) ⊗ fi fj ln R2
Q




 

Âľ2
+ 2 C (1) ⊗ P (1) ⊗ fi fj + 2 C (1) ⊗ fi P (1 ) ⊗ fj − P (1) ⊗ P (1) ⊗ fi fj ln F2
Q


 Âľ2


+ P (1) ⊗ fi P (1) ⊗ fj ln F2 + P (2) ⊗ fi fj + i ↔ j,
Q
(i,j)

2 C3

where the i ↔ j corresponds to the same terms with i and j interchanged. These results
above are consistent with the results of [177]. Also, using the expansions above, the singular

99

piece is given by

1  (1) 2
A
fi fj ,
8




 
1 (1) (1) 1 (1)
1 (1)  (1)
(i,j)
(1)
=
A B − A β0 f i f j + A
P ⊗ fi fj + P ⊗ fj fi ,
2 V3
2
6
4


!  (1) 2
2
B
ÂľR
β0 B (1) A(2) 
(i,j)
 1
= − β0 A(1) ln
+
−
−
 fi fj
2 V2
8
4
8
4
Q2
(i,j)

2 V4

=

!
!




Âľ2F
1 (1)
B (1) β0
(1) ⊗ f + 1 P (1) ⊗ P (1) ⊗ f f
+
A ln
+
−
f
P
j
i
i
j
4
2
8
8
Q2




1
1  (1)
− A(1) C (1) ⊗ fi fj +
P ⊗ fi P (1) ⊗ fj + i ↔ j,
2
8
!
!
2

2 1


(2)
Âľ
B
(i,j)
(1)
(1) ln
(1) C (1) ⊗ f f
R −
V
=
−ζ(3)
A
−
β
B
f
f
−
B
2 1
i j
i
j
4 0
2
Q2
!



Âľ2F  (1)
1 (1)
1  (1)
1  (1)
(1) ⊗ f f
+ B ln
P
⊗
f
f
+
β
C
⊗
f
f
−
C
⊗
P
i
j
i
j
i
j
2
2 0
2
Q2
!
!
2

 1


Âľ2F  (1)
Âľ
1
(1) ⊗ f
(1) ⊗ P (1) ⊗ f f
F
+ ln
P
⊗
f
P
+
ln
P
i
j
i
j
4
4
Q2
Q2
!


Âľ2R  (1)
1
1  (2)
− β0 ln
P
⊗
f
f
−
P
⊗
f
i
j
i fj + i ↔ j,
4
4
Q2




1
(i,j)
(1)
(1)
(1)
(1)
(1)
=
β ζ(3)A − ζ(3)A B
fi fj − ζ(3)A
P ⊗ fi fj
2 V0
3 0
!
!

 1

Âľ2F  (1)
Âľ2F  (1)
1
(1)
(1)
− ln
C ⊗ fi P ⊗ fj − ln
C ⊗ P ⊗ fi jj
2
2
Q2
Q2
!



Âľ2F  (1)
1
1  (1)
(1)
+ β0 ln
C ⊗ fi fj +
C ⊗ fi C ⊗ fj
2
2
Q2
!

Âľ2F  (1)
1
2
+ β0 ln
P ⊗ fi fj
8
Q2
!
!

 1

Âľ2F  (1)
1 2 Âľ2F  (1)
(1)
2
(1)
+ ln
P ⊗ fi P ⊗ fj + ln
P ⊗ P ⊗ fi fj
8
8
Q2
Q2
!
!
2
2




Âľ
Âľ
1
(1)
(2)
F
R
− β0 ln
ln
P ⊗ fi fj + C ⊗ fi fj
4
Q2
Q2
!
2


Âľ
1
(2)
F
− ln
P ⊗ fi fj + i ↔ j.
4
Q2
100

NNLO Calculation
ResBos2 (qTcut = 1.5GeV)
ResBos2 (qTcut = 4.5GeV)
MCFM
FEWZ

Cross-Section
1111.0 pb
1115.6 pb
1116.6 pb
1111.0 pb

Table 3.4: Total inclusive NNLO Drell-Yan Cross-sections for 66 GeV < Mll < 116 GeV.
MCFM calculation using version 8.0 [3]. FEWZ calculation using version 3.1 rc [4]

Similar to the NLO calculation, the results can be compared to that from other publicly
available tools. Everything in the comparison is the same as it was in the NLO calculation,
with the exception that the real correction is obtained from SHERPA [178]. The results are
shown in Tab. 3.4.


The results for the O Îąs3 calculation can be found in the App. F.

3.6

Scale Dependence

In the resummation formalism, there exists three constants that are a result of solving the
renormalization group equations. These constants are arbitrary, and should therefore not
appear in the expansion of the resummation formalism to a fixed-order in Îąs . This implies
that the resummation coefficients, A, B, C, and H should depend on these parameters. Here
the calculation is done in the CSS scheme, but the CFG scale dependence can be obtained
using Eq. 3.73.
To obtain the coefficients, the resummation formalism is expanded for arbitrary scales
and is compared to the canonical choice, (C1 = C3 = b0 , and C2 = 1). In other words:

W (b, Q, C1 , C2 , C3 ) |O(Îąn ) = W (b, Q, C1 = b0 , C2 = 1, C3 = b0 ) |O(Îąn ) ,
s

s

101

(3.95)

where the definitions of C1 , C2 , and C3 can be found in the scale dependent resummation
formalism given as:
!
!
Z C 2 Q2 2
C22 Q2
dÂľ
2
W = exp −
A(Âľ; C1 ) log
+ B(Âľ; C1 , C2 )
2
Âľ2
C12 /b2 Âľ




C1 C3
C1 C 3
C ⊗ f a x1 ,
,
C ⊗ f b x2 ,
,
.
C2 b
C2 b

(3.96)



performing the series expansion of the previous equation, and using Eq. 3.95, to O Îąs3 , the
scale dependence is given by:

A(1) = A(1,c)

(3.97)

!
2
b
0
A(2) = A(2,c) − β0 A(1,c) log
C12
 
 

b0
b0
(3)
(3,c)
2
(1,c)
2
A =A
+ 4β0 A
log
− 2 log
β1 A(1,c) + 2β0 A(2,c)
C1
C1
!
b2 C 2
B (1) = B (1,c) − A(1,c) log 0 22
C1
!
2C 2
b
B (2) = B (2,c) − A(2,c) log 0 22
C1

 

b0
(1,c)
2
(1,c)
2
(1,c)
+ β0 2A
log
− 2A
log (C2 ) + 2B
log (C2 )
C1
!
2C 2
b
B (3) = B (3,c) − A(3,c) log 0 22
C1

 


b0
(1,c)
2
(1,c)
(1,c)
+ 2β1 A
log
+ log (C2 ) B
−A
log (C2 )
C1

 


4 2
b0
(1,c)
3
2
(1,c)
(1,c)
− β0 2A
log
+ log (C2 ) 2A
log (C2 ) − 3B
3
C1

 


b0
(2,c)
2
(2,c)
(2,c)
+ 4β0 A
log
+ log (C2 ) B
−A
log (C2 )
C1

102

(3.98)
(3.99)
(3.100)

(3.101)

(3.102)

C (1)

=

(1,c)
Cja (ξ) + δja δ(1 − ξ)

1
− A(1,c) log2
4

b20 C22
C12

!

1
+ B (1,c) log
2

b20 C22
C12

C2
1 (1)
− Pja log 23
2
b0

!!

(3.103)

!
!
2 Âľ2
2C 2
b
b
1
0 2 log
F
C (2) =
− β0 A(1,c) log2
4
C12
b20
!
!
!
2 Âľ2

2
2C 2
2C 2
b
b
b
1
1
F
0 2
+ β0 B (1,c) log 0 22 log
+
A(1,c) log4
2
32
C1
b20
C12
!
!
2C 2
2C 2
b
b
1
1
0 2 − A(1,c) B (1,c) log3
0 2 + 1 B (2,c) log
− β0 A(1,c) log3
2
12
8
2
C1
C12
!
!
1 (2,c) 2 b20 C22
1  (1,c) 2 2 b20 C22
1
− A
log
+
B
log
+ β0 B (1,c) log2
2
2
4
8
4
C1
C1
!
!
b20 C22
1 (1,c) (1,c)
1 (1,c) (1,c) 2 b20 C22
+ B
Cja log
− A
Cja log
2
4
C12
C12


C2
1 (1,c)
1 (2)
(1,c)
(1)
+ β0 Cja − Cjb ⊗ Pba − Pja log 23
2
4
b0
!
2
2C 2
C
b
C2
1 (1,c) (1)
1
(1)
(1)
+ A
Pjb ⊗ Pba log2 23 − B (1,c) Pja log 0 22 log 23
8
4
b
C1
b0
!0
b20 C22
C32 β0 (1) 2 C32
1
(1)
+ A(1,c) Pja log2
log
− Pja log 2 ,
8
4
C12
b20
b0
(2,c)
Cja (ξ) + δja δ(1 − ξ)

!
b20 C22
C12
!!
b20 C22
C12

(3.104)

Comparing these results to that from Ref. [179], it is important to note the differences in
the definition of β0 and β1 . In Ref. [179], the β functions are β0 = (11CA − 2nf )/6 and
2 − 5C n − 3C n )/6, while here β = (11C − 2n )/12 and β = (17C 2 −
β1 = (17CA
0
1
A f
F f
A
f
A

5CA nf − 3CF nf )/24. Note that this result is consistent with Ref. [179], except for the scale
dependence in C (2) . Additionally, the calculation is extended to include A(3) and B (3) . The
maximum uncertainty for the Sudakov factor arises for the choice C1 = b0 /2 and C2 = 2
and C1 = 2b0 and C2 = 1/2, which can be understood from the fact that this has the largest
impact on the value of the Sudakov integral. The dependence of C3 for the uncertainty is
more complicated, because it deals with the complex energy and x-dependence of the PDFs.
103

With the these scale dependence calculations, and the calculations of the previous sections
for the asymptotic piece, the comparison of the ResBos2 code to the LHC data for W and Z
physics is possible to a higher precision then previously. The comparison to the LHC data
can be found in Chap. 4 and 5, for the Z and W results respectively.

104

Chapter 4
Z Boson Resummed Predictions
At the LHC, one of the most important precision Standard Model processes to study is DrellYan, specifically, with a focus on the Z boson mass peak. The precision of the data at the
LHC is at the sub-percent precision for the normalized distributions, and a few percent for
unnormalized distributions due to the luminosity uncertainty at the Z-peak. The transverse
momentum distribution is important for the study of soft gluon resummation. However, a
new observable was recently proposed at the Tevatron which has been shown to be more
accurate, known as φ∗η [180, 25] and is discussed in Sec. 4.2.

4.1

Z pT Distribution

As mentioned in Chapter 3, the transverse momentum of the Z boson requires resummation
to describe the small pT region. With the precision of the LHC data, the current ResBos
code is insufficient, and the order of resummation needs to be increased, from NNLL to
N3 LL. A comparison of the calculation using NNLL resummation to the data at ATLAS at
√

s = 8 TeV can be seen in Fig. 4.1. Clearly, there is a disagreement between the theory

and the data. There are two regions of disagreement between the theory prediction and
the data. These two regions can be categorized as: the intermediate transverse momentum
region, between about 10 GeV and 100 GeV, and the high transverse momentum region,

105

T

1/σ dσ/dp ll RESBOS / Data

1.4
1.3
1.2

ATLAS

s = 8 TeV, 20.3 fb-1

Data - statistical uncertainty
Data - total uncertainty
RESBOS uncertainty
66 GeV ≤ mll < 116 GeV, |y | < 2.4
ll

1.1
1
0.9
1

10

102 pll [GeV]
T

Figure 4.1: Comparison of pll
T data from ATLAS compared to ResBos predictions,
reproduced from Ref. [23]

above 100 GeV. The resolution to each of these disagreements arises for different physics
reasons.
Firstly, in the high transverse momentum region, the ResBos code used the invariant
mass of the lepton pair for the factorization and renormalization scale. However, when the
transverse momentum is above the mass of the Z boson, the major contribution comes from
the process of a Z boson with one hard jet. In this process, the hard jet will have an energy
close to that of the Z boson, and the scale that appears in the calculation is the transverse
mass of the Z boson, defined as:
Âľ2 = Mll2 + p2T ,

(4.1)

where Mll is the invariant mass of the lepton pair, and pT is the transverse momentum of the
lepton pair. Changing the scale to the transverse mass results in better agreement between
the theory prediction and the data as seen in Fig. 4.2.
However, in the intermediate transverse momentum region simply changing the scale

106

1dσ
σdq (pb/GeV)
t

10−5

10−6

10−7

Theory
CSS321

10−8
1.2
1.15

200

300

400

500

600

700

800

200

300

400

500

600

700

800

900
qt (GeV)

1.1
1.05
1
0.95
0.9
0.85
0.8

900

qt (GeV)

Figure 4.2: Comparison of the ResBos2 calculation with a scale choice of Âľ2 = Mll2 + p2T to
the ATLAS data given in Ref. [23], focusing on the high transverse momentum region
to the transverse mass does not resolve the problem. Therefore, the precision of the theory
calculation needs to be improved. The order of the resummation calculation needs to be done


to N3 LL, which means that the A coefficient is done to O Îąs3 , the B coefficient is done to




O Îąs3 , and the C coefficient is done to O Îąs2 . The order of B needs to be one order higher
than that of the C coefficient in order to have good agreement between the CSS and CFG
prediction as seen in Fig. 4.3. In the previously mentioned figure, only the W -piece is included
to focus solely on the difference between the two schemes (the Y -piece is the same in both
schemes). The fixed order calculation for Z+jet at NLO is obtained from the Sherpa [178]
program to correctly predict the angular distributions which is discussed in Section 4.3.
Since the fixed order calculation contains a jet, the transverse momentum of the Z boson
is required to be greater than 2 GeV to remove the singularity at pT = 0. Additionally, it
has been shown that at this order the fixed order and asymptotic calculations cancel to a
sub-percent accuracy for pT ≈ 2 GeV, as shown in Fig. 4.4. Finally, the matching of the
resummed prediction to the fixed order prediction needs to be performed at high transverse
momentum. In the ResBos2 code, matching occurs when the asymptotic expansion cancels
with the resummed calculation, i.e. W − A = 0. As seen in Fig. 4.5, there are multiple

107

T

dσ
dp

100

CSS Prediction
CFG Prediction

80

60

40

Ratio to CSS

20

1.008

0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50
p (GeV)

1.006
1.004
1.002
1
0.998
0.996
0.994
0.992
0.99

T

dσ (pb/GeV)
dp

Figure 4.3: Comparison of the CSS and CFG formalisms, focusing on only the small
transverse momentum, and not including the Y -piece.

T

102

Fixed Order

Asymptotic Piece

10

1

10

20

30

40

50

60

70
p (GeV)
T



Figure 4.4: Comparison of the fixed order piece up to O Îąs2 to the asymptotic piece up to


O Îąs2 . The cutoff on the transverse momentum of both calculations is set to 2 GeV.
crossing points, and the first crossing past the peak is used as the matching condition.
Above the matching point, the prediction is set to be only the fixed order prediction. In
this matching scheme, the crossing occurs approximately around 20 GeV for the invariant
mass of the lepton pair near the Z peak, and in the central rapidity region, for a resummed


order of N3 LL and an asymptotic order of O Îąs3 . With the resummation calculation to


N3 LL, asymptotic piece to O Îąs2 , the fixed order to NNLO, and the matching procedure
as described above, the ResBos2 code can be compared to the data.
The comparison of the ResBos2 code for both the CSS and CFG schemes to the ATLAS

108

T

dσ (pb/GeV)
dp

Resummed

10

Asymptotic Piece

1

0

10

20

30

40

50

60

70
p (GeV)
T

T

10-1

1 dσ
σ dp

1 dσ
σ dp

T



Figure 4.5: Comparison of the Asymptotic piece to O Îąs3 to the N3 LL resummed piece.

10-2
-3

-3

10

10

10-4

10-4

-5

10

-6

-5

10

-7

10-7

-8

10
1.3

10

-6

10

10

-8

10
1.3

Ratio to Theory

Ratio to Theory

10-1

10-2

1.2
1.1
1
0.9
0.8
1

10

102

1.2
1.1
1
0.9
0.8

*

φη

1

10

102

*

φη

Figure 4.6: Comparison of both the CSS(left) and CFG(right) prediction to data. The
PDF and scale uncertainty is given by the error bands. The lighter error band is the PDF
uncertainty, and the darker error band is the combination of the scale and PDF uncertainty.

109

data [23] can be found in Fig. 4.6. In the figure, both the scale and PDF uncertainties
are shown. As mentioned previously, the high transverse momentum region is resolved
by switching the scale of the calculation, and this is supported by the results shown in
the previously mentioned figure. Additionally, by increasing the order to N3 LL accuracy,
the intermediate transverse momentum is improved. There exists a calculation for Z+jet
up to NNLO [24], however, their results are not public enough yet to perform a detailed


comparison to the asymptotic piece up to O Îąs3 . However, in their calculation, the kfactor is approximately flat above the matching region, in which the fixed order is the only
calculation. To approximate these corrections at high transverse momentum, this k-factor is
included. One step for the future of the ResBos2 calculation, is to perform a matching to the
NNLO Z+jet calculation, when the results are public enough to check that the asymptotic
piece cancels exactly in the limit of the transverse momentum going to zero. The results from
the NNLO Z+jet calculation for the same data set as compared to above can be found in
Fig. 4.7, and shows improvement in the intermediate transverse momentum region over the
NLO result. Therefore, the matching to this prediction would further improve the prediction
of the ResBos2 code.

4.2

Z φ∗η Distribution

The φ∗η observable is a new observable proposed at the Tevatron [180, 25] that only depends on the angular distribution of the final state leptons, but directly correlates with the
transverse momentum of the Z boson. The definition of this observable is given as:

φ∗η


= tan

π − ∆φ
2

110





sin θη∗ ,

(4.2)

�������

���

����

�������������������

����

��������������

��������������������������������

���������

���
���
���
���
���
���
���
���

���������������������

���
���
���
���
���
���
���
���

����������������������

���
���
���
���
���
���
���
���

�����������������������

ďż˝

ďż˝

��

��

��
���������

���

���

���

Figure 4.7: Comparison of the fixed order calculation of Z+jet to NNLO compared to the
ATLAS data. Reproduced from [24].

Figure 4.8: The frame definition for φ∗η , reproduced from [25].
where ∆φ is the azimuthal separation of the two leptons and θη∗ is the measurement of the
scattering angle with respect to the proton beam direction in the rest frame of the Z boson.




−
+
Ρ
−η
θη∗ is given in terms of lab observables as cos θη∗ = tanh
, where η − and η + are
2
the pseudorapidities of the negatively and positively charged lepton respectively, see Fig. 4.8
for a depiction of the frame. Taking the limit of the transverse momentum to zero, the φ∗η
prediction can be approximated by:

φ∗η ≈

qT
sin φCS ,
Mll
111

(4.3)

1/σ dσ/dφ*

Ρ

RESBOS / Data

1.4
1.3
1.2

ATLAS

s = 8 TeV, 20.3 fb-1

Data - statistical uncertainty
Data - total uncertainty
RESBOS uncertainty
66 GeV ≤ mll < 116 GeV, |y | < 2.4
ll

1.1
1
0.9
10-2

10-1

1

φ*

Ρ

Figure 4.9: Comparison of φ∗η data from ATLAS compared to ResBos predictions,
reproduced from Ref. [23]

where φCS is the φ angle in the Collins-Soper Frame, qT is the transverse momentum of the
Z boson, and Mll is the invariant mass of the lepton pair. Using the equation above, the
correspondence between the φ∗η distribution and the transverse momentum distribution may
be approximated, e.g. for Mll ≈ MZ , the range of 10− 3 ≤ φ∗η ≤ 0.1 radians corresponds to
a transverse momentum from 0.1 GeV to 10 GeV. From Eq. 4.3, the features can be mapped
to Fig. 4.9, and therefore can be explained in the same manner as before. Looking first at
the high φ∗η region, the disagreement is again resolved by a change of scale, from Mll to
MT . For the intermediate φ∗η region (0.1 ≤ φ∗η ≤ 1), the calculation needs to be improved
as detailed above. Again Sherpa is used to obtain the perturbative prediction for Z+jet at
NLO for the transverse momentum above 2 GeV. The matching is done in the same way
as mentioned above, again matching in the transverse momentum distribution. Since the
results are fully differentiable, the φ∗η distribution can be calculated from the lepton and
anti-lepton momentums.
Similar to the discussion above for the transverse momentum distribution of the Z boson,
112

1 dσ
σ *
dφη

1 dσ
σ *
dφη

10

1

1

10-1

10-1

10-2

10-2

-3

-3

10

Ratio to Theory

10

Ratio to Theory

10

1.3
1.2
1.1
1
0.9
0.8
10-4

-3

10

10-2

10-1

1.2
1.1
1
0.9
0.8
10-4

*

1

1.3

φ10
Ρ

-3

10

10-2

10-1

*

1

φ10
Ρ

Figure 4.10: Comparison of both the CSS(left) and CFG(right) prediction to data. The
PDF and scale uncertainty is given by the error bands. The lighter error band is the PDF
uncertainty, and the darker error band is the combination of the scale and PDF uncertainty.
the improvement from the ResBos2 code is again noticeable. The PDF and scale uncertainties
for the ResBos2 prediciton of φ∗η can be found in Fig. 4.10. The improvement in the large
φ∗η region is again due to using the correct scale for the calculation. Also, the improvement
in the intermediate region is due to the increased accuracy of the ResBos2 code over the
ResBos code. Finally, the Z+jet to NNLO fixed order calculation [24] compared to the data
can be found in Fig. 4.11. The matching cannot yet be completed to this order, until the
results of the group become more public. However, once the matching can be performed,
the ResBos2 prediction should be even further improved.

4.3

Angular Functions

In Drell-Yan, the angular distributions of the charged lepton pairs allow for an additional
handle on precision QCD studies. The fully differential cross-section describing the kinematics of the leptons can be decomposed into nine harmonic polynomials in the Collins-Soper
Frame [181, 182, 183, 184]. It is convenient to factor out the unpolarized cross-section, and
write the differential cross-section as a function of the harmonic polynomials and dimension-

113

�������

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����

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���������

����������������������

���
���

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���
���
���
���
���
���
���

����������������

���
���
���
���
���
���
���

����������������

��������������

������������

���
���
���
���
���
���
���

����������������

���
���
���
���
���
���
���

����������������

���
���
���
���
���
���
���

����������������

���
���
���
���
���
���

���

���

ďż˝

ďż˝

ďż˝

��

��

Figure 4.11: Comparison of the fixed order calculation of Z+jet to NNLO compared to the
ATLAS data. Reproduced from [24].

114

less angular coefficients, A0−7 as:

dσ
dσ
=
2
dpT dydQ d cos θdφ dpT dydQ2

 1 

2
2
1 + cos θ + A0 1 − 3 cos θ + A1 sin 2θ cos φ
2

(4.4)

1
+ A2 sin2 θ cos 2φ + A3 sin θ cos φ + A4 cos θ
2

2
A5 sin θ sin 2φ + A6 sin 2θ sin φ + A7 sin θ sin φ .

It is important to note that only A4 is non-zero in the limit of pT goes to zero. There
is a well established relationship that states that A0 − A2 = 0, known as the Lam-Tung
relation [142, 143, 144], and is expected to hold up to O (Îąs ), but is known to break down at


O Îąs2 as will be shown later. The coefficients A5,6,7 are zero at NLO, and small at NNLO,
and thus will not be included in the comparisons to data. The coefficients A3 and A4 depend
on the relationship between the vector and axial couplings and are therefore sensitive to the
weak mixing angle (sin2 θW ).
To obtain the theory predictions for each of the angular coefficients, the moments of each
coefficient are calculated by:
R
hPi (cos θ, φ)i =

Pi (cos θ, φ) dσd cos θdφ
R
,
dσd cos θdφ

(4.5)

where the Pi ’s are the angular functions associated with each Ai respectively. There is a

115

Figure 4.12: The predictions for A2 − A0 for the DYNNLO code [26], at NLO and NNLO.
Reproduced from [27].
direct relationship between each moment and the angular coefficient given by:


1
3
2
hP0 (cos θ, φ)i = h 1 − 3 cos θ i =
2
20
1
hP1 (cos θ, φ)i = hsin 2θ cos φi = A1 ,
5
1
hP2 (cos θ, φ)i = hsin2 θ cos 2φi = A2 ,
10
1
hP3 (cos θ, φ)i = hsin θ cos φi = A3 ,
4
1
hP4 (cos θ, φ)i = hcos θi = A4 ,
4
1
hP5 (cos θ, φ)i = hsin2 θ sin 2φi = A5 ,
5
1
hP6 (cos θ, φ)i = hsin 2θ sin φi = A6 ,
5
1
hP7 (cos θ, φ)i = hsin θ sin φi = A7 .
4



2
A0 −
,
3

(4.6)

The precision of the LHC shows the breaking of the Lam-Tung relation at large transverse
momentum [27], as predicted by the NNLO calculation by DYNNLO [26] of the angular
116

A2

A0

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
1

10

102

1

p (GeV)

10

102

p (GeV)
T

0.4

A0-A2

A4

T

0.35

0.4

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

-0.05

-0.05

-0.1
1

10

102

-0.1
1

p (GeV)
T

10

102

p (GeV)
T

Figure 4.13: Theoretical prediction from the ResBos2 program for the angular coefficients
of A0 , A2 , A4 , and the breaking of the Lam-Tung Relationship for the CSS Scheme. The
lighter error band is the PDF uncertainty, and the darker error band is the combination of
the scale and PDF uncertainty.
functions in Drell-Yan, see Fig. 4.12. The original ResBos code is unable to appropriately
predict the angular coefficients to the precision required by the LHC, due to the fact that the
k-factor obtained using the calculation by Arnold and Kauffman [177] can only be applied
to the symmetric and anti-symmetric leading order angular functions (L0 and A4 ), but not
to the other angular distributions. To include these corrections in the ResBos2 code, the
fixed order prediction is obtained by using the Z+jet prediction at NLO from the Sherpa
code [178], which includes the full angular dependence.
From the prediction in Figs. 4.13 and 4.14, the ResBos2 prediction shows the LamTung relationship breaking down, as expected at this order. Additionally, A4 is non-zero as
expected. There is some disagrement with the data, but it should improve with the inclusion
of higher order corrections.

117

A2

A0

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
1

10

102

1

p (GeV)

10

102

p (GeV)
T

0.4

A0-A2

A4

T

0.35

0.4

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

-0.05

-0.05

-0.1
1

10

102

-0.1
1

p (GeV)
T

10

102

p (GeV)
T

Figure 4.14: Theoretical prediction from the ResBos2 program for the angular coefficients
of A0 , A2 , A4 , and the breaking of the Lam-Tung Relationship for the CFG Scheme. The
lighter error band is the PDF uncertainty, and the darker error band is the combination of
the scale and PDF uncertainty.

118

Chapter 5
W Mass
One of the most interesting places to look for new physics signals is in the W mass. The precision of the Standard Model prediction of the mass is at the 0.01% level, and the experimental
measurement is of similar precision. In order to improve the experimental measurement of
the W boson, it is important to understand the transverse momentum of the W boson,
and its relationship with the Z boson transverse momentum. Currently, the dominate uncertainty in the direct measurement of the W mass at the LHC arises from the theoretical
predictions of the transverse mass of the W boson, or the transverse momentum of the lepton. In this chapter, the Standard Model prediction from Electroweak precision tests will be
discussed, followed by a discussion of the experimental measurements at the Tevatron and
the LHC. Finally, the improvements to the ResBos2 calculation for the W mass measurement
are introduced.

5.1

SM EW Precision Fit

Using the Electroweak input scheme defined by on MZ , GÂľ , and Îą, the W boson mass is a
predicted parameter. The Standard Model calculation is detailed in Ref. [185], and is given
by:
2
MW

1−

2
MW

MZ2

!

πα
=√
(1 + ∆r) ,
2GÂľ

119

(5.1)

with the loop corrections contained in ∆r. At the one-loop order, the calculation of ∆r is
given as:
c2
∆r = ∆α − W
∆ρ + ∆rrem (MH ) ,
s2W

(5.2)

M2
where c2W = W
, s2W = 1 − c2W , and sW is the Weinberg angle defined in Chap. 1 as the
M2
Z

rotation angle needed to diagonalize the neutral boson sector of the Standard Model. The
correction to α, given by ∆α is due to light fermions and is proportional to the logarithm of
their masses. Additionally, there is a modification to the ρ parameter as defined in Sec. 1.1.3.
Finally, the last term contains all the dependence of the Higgs boson mass.
The calculation of ∆r has been done to two-loops in the Electroweak coupling, contains
the fermionic contributions [186, 187, 188] and the bosonic contributions [189, 190, 191,


192]. As for the QCD corrections to ∆r, they are known to O ααs2 [193, 194, 195, 196].


Furthermore, ∆ρ has been calculated to O ααs3 [197, 198, 199]. The fit to all of the
Electroweak observables was done at NNLO in Ref. [5], and a summary of the results can be
found reproduced in Table 5.1. The precision obtained by the global fit on the mass of the
W boson sets the goal for the LHC. It is possible to break down the uncertainty of the fit to
the W mass measurement into the contributions from the top mass, the theory uncertainty
of the top mass, the Z mass, ∆αhad , αs , the Higgs mass, and the theory uncertainty of the
W mass. These contributions are broken down in Table 5.2. The largest uncertainties on
the indirect determination of the W mass come from both the experimental and theoretical
determination of the top quark mass, followed by the theory uncertainty on the W mass, and
then the experimental measurement of the mass of the Z boson. The LHC should provide an
improvement on the experimental measurement of the top quark mass, and should improve
significantly at a future electron-positron collider. There is not going to be any improvement

120

Parameter

Input Value

MH [GeV] a 125.14 Âą 0.24

Fit Result

w/o exp. input

w/o exp. input
or theory unc.

125.14 Âą 0.24

93+25
−21

93+24
−20

MW [GeV]
ΓW [GeV]
MZ [GeV]
ΓZ [GeV]

80.385 Âą 0.015
2.085 Âą 0.042
91.1875 Âą 0.0021
2.4952 Âą 0.0023

80.364 Âą 0.007
2.091 Âą 0.001
91.1880 Âą 0.0021
2.4950 Âą 0.0014

80.358 Âą 0.008
2.091 Âą 0.001
91.200 Âą 0.011
2.4946 Âą 0.0016

80.358 Âą 0.006
2.091 Âą 0.001
91.200 Âą 0.010
2.4945 Âą 0.0016

mt [GeV]

173.34 Âą 0.76

173.81 Âą 0.85

b
170.0+2.3
−2.4

177.0 Âą 2.3 b

a

Average of the ATLAS [200] and CMS [201] measurements, ignoring any correlation
between systematic uncertainties
b The theoretical top-mass uncertainty is excluded.
Table 5.1: Electroweak fit. The fourth column gives the fit results without using any
experimental or phenomenological estimate for the parameter when performing the fit. The
fifth column is the same as the fourth, but ignores all theory uncertainties. The table is
reproduced from Ref. [5].
Source
mt
δtheory mt
MZ
∆αhad
Îąs
MH
δtheory MW

Value
0.0046
0.0030
0.0026
0.0018
0.0020
0.0001
0.0040

Table 5.2: The values are reported in GeV, and taken from Ref. [5]

on the experimental measurement of the Z boson at the LHC or at a future electron-positron
collider, due to the fact that the LEP measurements are very precise [202]. Finally, the
experimental measurement of the W mass ideally should be close to the current uncertainty
of the indirect measurement by the end of the LHC running [203, 204]. The details of the
experimental goals and the limits that are expected are discussed in the following section.

121

5.2

Experimental Measurement

The experimental measurement of the W mass is one of the most important measurements
at a hadron collider. The most precise channel to study the W mass at the LHC comes from
the process, pp → eν + X. However, the fact that there is a neutrino in the final state leads
to the inability to directly reconstruct the invariant mass of the lepton-neutrino system.
Therefore, two different observables are proposed to study the mass of the W boson. The
first observable is the transverse mass of the W boson defined as:

/ (1 − cos θ) ,
MT2 = 2pT E
l T

(5.3)

/ T is the missing transverse
where pT is the transverse momentum of the final state lepton, E
l

momentum (attributed to the neutrino), and θ is the angle between the lepton and missing
transverse momentum. The other observable is the transverse momentum of the lepton.
Both of these observables are not accurately predicted by a fixed order calculation due to
the dependence on the transverse momentum of the W boson, which requires resummation.
In measuring the W mass, an appropriate prediction of the Jacobian peak region for
the lepton is important to obtain the correct mass. One of the most important features in
determining the W mass is the tail of the distribution, and the Jacobian peak. Having an
accurate theoretical prediction is required to match the shape of the data.
The traditional method of obtaining the W mass is through the use of a template fit
technique. To estimate the uncertainty that arises from the PDF, the following 4 step
method is used [205]1 :
1 For real data, a similar procedure is used to estimate the central value, but replacing the pseudodata
with the actual data.

122

Figure 5.1: Results of the EW Fit, reproduced from Ref. [5]

123

1. Generate the observable to be used for the fitting for fixed W mass for each PDF error
set of interest. Treat the generated observable as pseudodata.
2. Generate the mass templates for the distributions used to fit the pseudodata as a
function of the W boson mass using the central prediction for the PDF.
3. Given the PDF set i, and the mass template j, calculate the χ2 for each combination
of i and j, given by:

χ2i,j =

1
Nbins

Nbins

X
k=1

2
j
Ok − Oki

2  j 2 ,
i
σk + σk


(5.4)

where Ok and σk are the value of the observable and the standard deviation of it in a
given bin (k), respectively.
4. The minimum of the χ2i,j distribution for each PDF set i gives the mass j that the
PDF best fits. The difference from the central value is the shift in the mass induced
by PDF i. Combining all of the PDF sets, using the method to obtain the uncertainty
as defined for the given PDF set, the allowed mass range is obtained.
This procedure will be demonstrated below for the CT14nnlo PDF set [10] for the lepton
transverse momentum. Similar results are obtained if the transverse mass of the W boson
is used. The CTEQ PDFs are based off of the Hessian method. Therefore, each eigenvector
does not give the full uncertainty, and the uncertainty needs to be calculated by using the

124

master equation as given in [206] and below:
v
u n
uX


2
+
∆X = t
max Xi+ − X0 , Xi− − X0 , 0 ,

(5.5)

i

v
u n
uX


2
−
∆X = t
max X0 − Xi+ , X0 − Xi− , 0 ,

(5.6)

i

where i goes over each eigenvector, and the Âą refers to whether the shift is along the positive
or negative direction respectively.

5.3

ResBos2 Results

One important prediction to make is the ratio of the transverse momentum of the Z boson
to the transverse momentum of the W boson. With an accurate prediction of this ratio, the
experimentalists can use data driven methods to obtain the transverse momentum predictions
of the W boson through:
 dσ

)
 dp(W

dσ
T th  dσ

=  dσ
,
 (Z)
(W )
dpT
dp
(Z)
T data
dpT

where

dσ
(W )
dpT

(5.7)

th

is the data driven prediction for the W boson transverse momentum,

the theory prediction for the W boson transverse momentum, dσ
(Z)
for the Z boson, and

dσ
(Z)
dpT
data

dpT

dσ
(W )
dpT
th

is

is the theory prediction
th

is the Z boson transverse momentum data. An accurate

prediction of this ratio allows to attempt to apply a transverse momentum cut on the W
boson. This cut reduces the theoretical uncertainty arising from the PDFs greatly [205].
The normalized ratio (RW/Z ) is shown in Fig. 5.2 for the ResBos2 prediction.

125

T

RW/Z(p )

1.06
1.04
1.02
1
0.98
0.96
0.94
0.92
0

5

10

15

20

25

30
p (GeV)
T

Figure 5.2: The ratio of the normalized transverse momentum distributions of the W boson
to the Z boson.
The ResBos code can be used to calculated the theoretical uncertainty of the PDFs, as
described in the section above. An example of the χ2 distribution for one error set can be
found in Fig. 5.3. After calculating the shift for all of the eigenvectors, and using the Hessian
master equation as given in Eq. 5.5, the CT14nnlo PDF uncertainty can be calculated. For
√
a collider center of mass energy of 13 TeV, the W mass uncertainty is given as δ + = 37
MeV and δ − = 34 MeV.

126

χ2

500
450
400
350
300
250
200
150
80260

80280

80300

80320

80340

80360

80380

80400

80420

80440
W Mass (MeV)

Figure 5.3: An example of the χ2 distribution for an error set, others are similar to this.
The central input mass was MW = 80.358 GeV. The minimum here occurs around 80.372,
a shift of 14 MeV.

127

Chapter 6
Color Singlet Boson Plus Jet
Resummation at Hadron Colliders
Soft gluon resummation for the class of color singlet boson plus jet processes has its own
interest in perturbative QCD. To deal with the divergence in low transverse momentum hard
processes, the transverse momentum resummation formalism is employed [138]. However, the
qT resummation formalism has been mainly applied to color-neutral particle production, such
as inclusive vector boson W/Z and Higgs boson productions. Extensions to jet production in
the final state has been limited, not only because of the technical issues associated with the
jets in the final state, but also because jets carry color. Therefore, the soft gluon interactions
are more complicated than those for color neutral particle production. Nevertheless, there
has been progress made in the last few years on the qT resummation for dijet production in
hadronic collisions [207, 208, 209]. Here the details of expanding the calculation to single
jets plus a color singlet boson are examined.
The resummation formula for V + j resummation can be summarized as
X
d5 σ
=
σ0
dyV dyj dPT2 d2 q⊥
ab

"Z

d2~b⊥ −i~q ·~b
e ⊥ ⊥ Wab→cd (x1 , x2 , b⊥ ) + Yab→cd
(2π)2

#
,

(6.1)

where yV and yj are rapidities for the boson and the jet, respectively, PT is the jet transverse

128

momentum, and ~q⊥ = P~V ⊥ + P~J is the total transverse momentum of the boson and the jet.
The first term W contains the all order resummation, the second term Y comes from the
difference between the fixed order corrections and the asymptotic piece, and σ0 represents
the normalization of the differential cross section. Higgs plus jet qT resummation will be
used as an example of the procedure for calculating the resummation results for this class of
processes.

6.1

Higgs Plus Jet Resummation

The effective Lagrangian in the heavy top quark mass limit is used to describe the coupling
between the Higgs boson and gluon given by:

Lef f = −

ιs a a¾ν
F F
H,
12πv µν

(6.2)

where v is the vacuum expectation value, H the Higgs field, F ¾ν the gluon field strength
tensor, and a the color index.
To begin, the W term can be written as:

2
Wgg→Hg (x1 , x2 , b) = Hgg→Hg (Q)x1 fg (x1 , µ)x2 fg (x2 , µ)e−S(Q ,b⊥ ) ,

(6.3)

2
Wgq→Hq (x1 , x2 , b) = Hgq→Hq (Q)x1 fg (x1 , µ)x2 fq (x2 , µ)e−S(Q ,b⊥ ) + x1 ↔ x2 ,

(6.4)

2
Wqq̄→Hg (x1 , x2 , b) = Hqq̄→Hg (Q)x1 fq (x1 , µ)x2 fq (x2 , µ)e−S(Q ,b⊥ ) + x1 ↔ x2 ,

(6.5)

at the next-to-leading logarithmic (NLL) accuracy. Here Q2 = s = x1 x2 S and represents
the hard momentum scale, b0 = 2e−γE , with γE being the Euler constant, fa,b (x, µ) are the
parton distributions for the incoming partons a and b, and x1,2 are momentum fractions of
129

the incoming hadrons carried by the partons. Beyond the NLL, a C function associated with
the parton distribution functions will also need to be included.
The Sudakov form factor can be written as

SSud

(Q2 , b


Z C 2 Q2 2   2 
dÂľ
Q
1
2
ln
A + B + D ln 2 ,
⊥) =
2
Âľ2
R
C 2 /b2 Âľ
1

(6.6)

⊥

where R represents the cone size of the jet. Here the parameters A, B, D can be expanded perturbatively in αs , for example in the gg → Hg channel, A = CA απs , B =
s
−2CA β0 απs , and D = CA α
2π .

The hard coefficient H can be calculated order by or-

der, from the leading Born diagrams. For example, the gluon-gluon channel is given by:


H (0) = s4 + t4 + u4 + m8h /(stu) [210, 211], where s = Q2 , t and u are the Mandelstam
variables for the partonic 2 → 2 process.

6.1.1

Virtual Corrections

Firstly, W (b) needs to be calculated at the one-loop order and shown that it can be factorized
into the parton distribution, jet, soft, and hard factors. The virtual corrections have been
calculated in the literature, see for example Refs. [210, 212]. For convenience, the results are
summarized below to demonstrate the resummation of the Higgs plus jet system. Since the
virtual diagrams are proportional to δ(qT ), the b-space result can be expressed as

(1)v

Wab

(bT ) = x1 f (x1 , µ) x2 f (x2 , µ) Γvab ,

(6.7)

where f1 and f2 are the PDFs, x1 and x2 are the momentum fractions of the partons from
each hadron, and the expressions for Γvab can be extracted from the calculations in the

130

references mentioned above. For the gluon-gluon channel,

Γvgg =

(0) Îąs CA
Hgg

"

3
1
− 2+



P2
Q2
2 ln 2 + ln J2
Âľ
Âľ

!

2π
!2
PJ2
1
Q2 PJ2
Q2 Q2 11 2
+
ln 2
− 2 ln 2 ln 2 − 2 ln
ln
+ π
2
−t −u
6
Âľ
Âľ
Âľ




 u 
M2
t
+ 2Li2 1 − 2 + 2Li2
+
2Li
2
Q
M2
M2

2 
2 


 #
t̃
t̃
ũ 2
ũ 2
+ ln 2
− ln
+ ln 2
− ln
−t
−u
M
M
(1)

+ δHgg ,

(6.8)

(1)

(0)

where ũ, t̃ = M 2 − u, t respectively, and δHgg is the finite term not proportional to Hgg
and is given by:

(1)
δHgg

i
1
M2 h
2
2
= (N − nf ) N (N − 1)
stu + M (st + su + tu) .
3
stu

(6.9)

For the quark-gluon channel,

Γvqg

=







2
−u 10
−2 2
−u
−t
s
4CA β0
+ ln 2 −
+ CA 2 −
ln 2 − ln 2 − ln 2
2π

3

Âľ

Âľ
Âľ
Âľ

(0) Îąs
Hqg

s
u
−u
−t
s
ũ
ũ
+2 ln 2 ln + ln2 2 − ln2 2 − ln2 2 − 2 ln2
+ 2 ln2 2
t
−u
Âľ
Âľ
Âľ
Âľ
M

 u 
2
π
14
+4Li2
+ 18ζ (2) −
+
2
3
3
M



−4 2
−u
−u
s
−u
+CF
−
−2 ln 2 + 3 − 2 ln2 2 + 2 ln2 + 6 ln 2
2

t

Âľ
Âľ
Âľ





2
M
t
t̃
t̃
2π 2
2
2
+4Li2 1 −
+ 4Li2
− 2 ln
+ 2 ln
+ 4ζ (2) −
− 16
s
−t
3
M2
M2
(1)

+ δHqg ,

(6.10)

131

(1)

(1)

again with δHqg analogous to δHgg above, given by:

(1)

δHqg = (CA − CF ) CA CF (s + t) .

(6.11)

Finally, for the quark-anti-quark channel,

Γvqq̄





2
s
10
−2 2 sµ2
=
4CA β0
− 2 ln 2 +
+ CA 2 − ln

3

ut
Âľ



2
ut
s
−t
−u
M
ln2 2 + 2 ln2 2 − 2 ln2 2 − 2 ln2 2 + 4Li2 1 −
s
sÂľ
Âľ
Âľ
Âľ




π2
−4 2
s
−14ζ (2) −
− 8 +CF
−
−2 ln 2 + 3
2
3


Âľ


 u 
−t −u
s
ut
s
ut
t
2
−18 ln 2 ln 2 + 6 ln 2 + 2 ln 2 + 4 ln 2 ln 2 + 4Li2
+ 4Li2
Âľ
Âľ
Âľ
Âľ s
Âľ
sÂľ
M2
M2

t̃
ũ
t̃
ũ
2
−2 ln2
− 2 ln2
+ 2 ln2 2 + 2 ln2 2 + 36ζ (2) − π 2 − 16
−t
−u
3
M
M
(0) Îąs
Hqq̄
2π

(1)

+ δHqq̄ ,

(6.12)

(1)

with δHqq̄ given as:
(1)

δHqq̄ = (CA − CF ) CA CF (−t − u) .

(6.13)

It is important to note the 12 and 1 divergences that appear in the above equations. These

will be canceled by the divergences in the remaining terms.
At NLO, the strong coupling needs to be renormalized, yielding the following contribution:
Îąs CA
2π

 2 −  
Q
3
−
2β0 .
2

Âľ

Here, the renormalization scale is set to Q2 to simplify the final expression.

132

(6.14)

6.1.2

Jet Corrections

Next, since this process contains a final state jet, the jet function needs to be applied in
order to cancel out the collinear divergences from the final state. In order to perform this
calculation, the Narrow Jet Approximation (NJA) is applied as derived in the calculation of
inclusive jet production [213]. This approximation allows an analytic result for the jet production cross section to be obtained. For completeness, the quark and gluon jet contributions
are listed below:

Jg
Jq

"
Îąs CA 1
1
=
+
2
2π


"
Îąs CF 1
1
=
+
2
2π



!
#
PJ2 R2
2β0 − ln
+ Ig ,
Âľ2
!
#
PJ2 R2
3
− ln
+ Iq ,
2
Âľ2

(6.15)

where I g,q depends on the jet algorithm used. For the kt -family of jet algorithms the values
of I q and I g are,

1
Ig =
2

P 2 R2
ln J 2
Âľ

!2

1
=
2

PJ2 R2
ln
Âľ2

!2

Iq

P 2 R2 67 3 2 23
− 2β0 ln J 2 +
− π − Nf ,
9
4
54
Âľ

(6.16)

3 PJ2 R2 13 3 2
− ln
+
− π .
2
2
4
Âľ2

(6.17)

Their contributions to W (b⊥ ) can be included in the following manner,

(1)J

Wgg (b⊥ ) = x1 fg (x1 , µ)x2 fg (x2 , µ)Jg ,
(1)J

Wqg (b⊥ ) = x1 fq (x1 , µ)x2 fg (x2 , µ)Jq ,
(1)J

Wqq̄

(b⊥ ) = x1 fq (x1 , µ)x2 fq̄ (x2 , µ)Jg .

133

(6.18)
(6.19)
(6.20)

Again, there are divergences proportional to 1 as expected for a collinear jet function, which
will be canceled with similar contributions from the soft corrections, see Sec. 6.1.4.

6.1.3

Collinear Corrections

Thirdly, the collinear radiation associated with the incoming partons needs to be calculated.
These contributions can be written in terms of parton splitting functions:
2
dx0 0
0 , ¾) Ξ(1 + Ξ ) ,
x
f
(x
q
x0
(1 − ξ)+
Z


Îąs 1
dx0 0
0
2
Pgq = 2 2 CF
x fq (x , µ) 1 + (1 − ξ)
,
x0
2π q⊥


Z
Îąs 1
dx0 0
2Ξ
2(1 − ξ)
0
Pgg = 2 2 CA
x fg (x , ¾)Ξ
+
+ 2ξ(1 − ξ) ,
x0
(1 − ξ)+
Ξ
2π q⊥
Z


Îąs 1 1
dx0 0
0 , µ) ξ 2 + (1 − ξ)2 .
Pqg = 2 2
x
f
(x
g
x0
2π q⊥ 2

Îąs 1
Pqq = 2 2 CF
2π q⊥

Z

(6.21)
(6.22)
(6.23)
(6.24)

where Ξ = x/x0 . Here the Ξ 6= 1 contributions are taken into consideration, whereas the
Ξ = 1 part will be evaluated together with the soft gluon radiation contribution. These
terms introduce divergences of the form 1 Pab into Wab (b).

6.1.4

Soft Corrections

For soft gluon radiations, the leading power expansion can be applied to derive the dominant
contribution by the Eikonal approximation, as discussed in Sec. F. This analysis has been
applied in Ref. [214] to obtain the leading double logarithmic contributions to dijet production. For outgoing quark, antiquark, and gluon lines, the Eikonal approximation gives the

134

Figure 6.1: Feynman diagrams for the gluon-gluon scattering channel to a Higgs Boson.
following Feynman rules:
Âľ

2ki
g,
2ki ¡ kg + i

Âľ

−

Âľ

2ki
2ki
g,
g,
2ki ¡ kg + i
2ki ¡ kg + i

(6.25)

respectively, where ki represents the momentum of the outgoing particles. Similarly, for
incoming quark, antiquark, and gluon lines, the rules are:
Âľ

2p1
g,
2p1 ¡ kg + i

Âľ

−

Âľ

2p1
2p1
g,
g
2p1 ¡ kg + i
2p1 · kg − i

(6.26)

respectively, where p1 represents the momentum for the incoming particle, g is the strong
coupling constant, and kg is the momentum of the soft gluon.
In the gluon-gluon scattering channel, the relevant Feynman diagrams can be found in

135

Fig. 6.1, from which the following contributions for soft gluon radiation can be calculated,

g2

Z


d3 kg
C 
δ (2) (q⊥ − kg⊥ ) A Sg (p1 , p2 ) + Sg (k1 , p1 ) + Sg (k1 , p2 )
3
2
(2π) 2Ekg

(6.27)

where Sg (p, q) is a short-handed notation for

Sg (p, q) =

2p ¡ q
.
p ¡ kg q ¡ kg

(6.28)

In order to evaluate the contributions from soft gluon radiation, the phase space of the
gluon with restriction that gluon transverse momentum leads to the imbalance between the
Higgs-jet system needs to be integrated out, i.e.,

g2

Z

d3 kg
δ (2) (q⊥ − kg⊥ )Sg (p1 , p2 ) .
(2π)3 2Ekg

(6.29)

The derivation of the above term is straightforward, by noticing that the lower limit in the
longitudinal momentum fraction integral, is given by:
Z 1

dx 1
,
2
xmin x kg⊥

(6.30)

where x is the momentum fraction of p1 carried by the soft gluon. Because of momentum
k2

conservation, the lower limit for the x-integral is given by: xmin = g⊥
. Therefore, the
Q2
above integral leads to the following leading contribution,
1
Q2
ln
.
2
2
q⊥
q⊥

136

(6.31)

Substituting the above equation into Eq. (6.29) results in,

g2

Z

d3 kg
Îąs 1
δ (2) (q⊥ − kg⊥ )Sg (p1 , p2 ) = 2 2
3
(2π) 2Ekg
2π q⊥

Q2
2 ln 2
q⊥

!
.

(6.32)

The other terms are not as straightforward to calculate, because the simple phase space
integral will contain jet contributions which have already been taken into account by the
jet functions above (Sec. 6.1.2). Therefore, the jet contributions need to be subtracted in
order to avoid double counting. That means the phase space integral has to exclude the jet
region, noting that this exclusion does not depend on the jet algorithm at this order. This
is because, this contribution arises from the soft gluon radiation, whereas the jet algorithm
focuses on the collinear gluons associated with the jet.
As a general discussion, take the example of one term, given by:

Z

d3 kg (2)
δ (q⊥ − kg⊥ )Sg (k1 , p1 ) =
2Ekg

Z

d2 k

g⊥

δ (2) (q

Z
⊥ − kg⊥ )

dΞ1
2
, (6.33)
ξ1 (kg⊥ − ξ1 k1⊥ )2

where k1⊥ represents the transverse momentum for the final state jet, and ξ1 = kg ·p1 /k1 ·p1 .
Clearly, there is a collinear divergence associated with the jet, if the gluon radiation is within
the jet cone a collinear divergence will be generated. In order to regulate this collinear jet
divergence, the phase space integral can be limited to require that the gluon radiation be
outside of the jet cone. Under this restriction, there will be no divergence associated with
the jet, instead, the jet size R will be introduced to regulate the collinear divergence from
the jet.
There are different ways to regulate the above integral. The central point is to identify
the jet cone requirement. In the above example, a divergence exists when kg is parallel to

137

k1 , which means the invariant mass of k1 + kg is small. The out of cone radiation requires
that the invariant mass has a minimum,

(k1 + kg )2 > Λ2 .

(6.34)

Clearly, Λ depends on the jet size, if (k1 + kg )2 is smaller than Λ2 , its contribution needs
to be excluded, because it belongs to the jet contribution calculated in the previous section.
Following a similar analysis as for the jet contribution, the size of Λ can be worked out. For
example, if the kinematics of k1 and kg are substituted into the above equation, the results
are:



2 ,
(k1 + kg )2 ≈ k1⊥ kg⊥ ey1 −yg + eyg −y1 − 2k1⊥ kg⊥ cos(φ1 − φg ) ≈ k1⊥ kg⊥ R1g

(6.35)

where y1 and yg are rapidities for k1 and kg , φ1 and φg are the azimuthal angles, for
the jet and additional gluon respectively, and R1g represents the seperation of k1 and kg
q
(R1g = (y1 − yg )2 + (φ1 − φg )2 ). In other words, if R1g is smaller than R, the gluon
radiation will be considered inside the jet cone and vice versa. Therefore, in the phase
space integral of Eq. (6.33), the following kinematic restriction is imposed: Θ(2k1 · kg − Λ2 )
with Λ2 = k1⊥ kg⊥ R2 . Equivalently, a slight off-shell-ness for the jet momentum k1 can
2 R2 . By doing so there is no need
be adopted to regulate the divergence: k12 = m21 = k1⊥

to impose any kinematic constraints, and the phase space integral can be carried out in a
straightforward manner. The choice of m21 is made to ensure (k1 + kg )2 is always larger than

138

Λ2 , shown by,

(k1 + kg

)2

q
y1 −yg + eyg −y1 
 − 2k k
2 + m2 k
=
k1⊥
(6.36)
1⊥ g⊥ cos(φ1 − φg )
1 g⊥ e


q
q
2
2
2
2
2 + m2 − k
≈
k1⊥ + m1 kg⊥ (∆y) + k1⊥ kg⊥ (∆φ) + 2kg⊥
k1⊥
.
1⊥
1

2 R2 , (k + k )2 is guaranteed to be larger than Λ2 for any values of
By choosing m21 = k1⊥
g
1

∆y and ∆φ. In addition,the narrow jet approximation, i.e., R → 0 limit, is taken. In this
limit, the phase space cut-off technique results in the same leading contributions in terms of
ln(1/R), as the introduction of a mass term.
After adding an off-shell-ness to the jet momentum, the above integral can be written as

Z

dΞ1
2
.
2 (1 + R2 ) + k 2 − 2ξ k
ξ1 ξ12 k1⊥
¡
k
1
1⊥
g⊥
g⊥

(6.37)

To further integrate, the azimuthal angle of the jet is averaged over but the azimuthal angle
is fixed for kg⊥ , giving:
Z
dξ1 π dφ
2
,
2
2
2
2
ξ1 0 π ξ1 k1⊥ (1 + R ) + kg⊥ − 2ξ1 k1⊥ kg⊥ cos(φ)
Z
dΞ1
1
r
=
.
2
Ξ1
2
2
2
2
2
2
2
ξ1 k1⊥ (1 + R ) + kg⊥ − 4ξ1 k1⊥ kg⊥
Z

(6.38)

Again, the integration over ξ1 has a lower limit, and taking the limit of q⊥  Q and R → 0,
the leading power contribution is:
"
#
 
1 1
Q2
t
1
ln 2 + ln
+ ln 2 .
2 2
u
R
q⊥
q⊥

139

(6.39)

Therefore, the final result for the Sg (k1 , p1 ) term can be written as

g2

Z

"
 #
2
d3 kg
Îą
1
Q
1
t
s
δ (2) (q⊥ − kg⊥ )Sg (k1 , p1 ) = 2 2 ln 2 + ln 2 + ln
. (6.40)
3
u
(2π) 2Ekg
2π q⊥
q⊥
R1

Evaluation of the other terms follow the same procedure, and the results are summarized
below:

Îąs 1
Sg (p1 , p2 ) ⇒
2
2π 2 q⊥

Q2
2 ln 2
q⊥

!
,

"
!#
 
Îąs 1
Q2
1
t
1 2 1
π2
Sg (k1 , p1 ) ⇒
ln 2 + ln 2 + ln
+
ln 2 +
,
2
u
2
6
2π 2 q⊥
q⊥
R1
R1
"
!#
u
Îąs 1
Q2
1
1 2 1
π2
Sg (k1 , p2 ) ⇒
ln 2 + ln 2 + ln
+
ln 2 +
,
2
t
2
6
2π 2 q⊥
q⊥
R1
R1

(6.41)
(6.42)
(6.43)

where the  terms are kept since they contribute to the finite piece after Fourier transforming
into b⊥ -space1 .

6.1.5

Total Real Correction

Combining the soft and collinear gluon radiation together, the asymptotic behavior at small
qT can be obtained. The gluon channel is given by:
(0)

Îąs Hgg
2π 2 qT2

Z



dz1 dz2
z1 fg (z1 ) z2 fg (z2 ) δ (ξ2 − 1) ξ1 P (1) (ξ1 ) + ξ1 ↔ ξ2
z1 z2

!!
Q2
1
1 2 1
π2
+ δ (ξ1 − 1) δ (ξ2 − 1) CA 2 ln 2 − 4β0 + ln 2 + 
ln 2 +
,
2
6
R
R
qT

1 Details on Fourier transforming into b-space can be found in App. F

140

(6.44)

(0)

where Ξi = xi /zi , xi is the momentum fraction of the gluon from the proton, and Hgg is the
tree level contribution. Similar equations exist for the gluon-quark and the quark-anti-quark
channel. Combining the three channels gives:

Îąs 1
2π 2 qT2

Z

n

dz1 dz2
z1 fa (z1 ) z2 fb (z2 ) δ (ξ2 − 1) ξ1 P (1) (ξ1 ) + ξ1 ↔ ξ2
z1 z2
"
!
Q2
1
(0)
+ δ (ξ1 − 1) δ (ξ2 − 1) Hgg CA 2 ln 2 − 4β0 + ln 2
R
qT
Q2
3
u
1
(CA + CF ) ln 2 − 2β0 − CF + (CA − CF ) ln + CF ln 2
2
t
R
qT

!

Q2
3
u
1
(CA + CF ) ln 2 − 2β0 − CF + (CA − CF ) ln + CF ln 2
2
t
R
qT
!#)
Q2
1
(0)
+Hqq̄ 2CF ln 2 − 3CF + CA ln 2
.
R
qT

!

(0)
+Hqg
(0)
+Hgq

(6.45)

This will be compared to the fixed order calculation in the limit qT goes to zero in Sec. 6.1.7.
Now that the complete asymptotic piece is calculated, the Fourier Transform into b-space
is performed. After the Fourier Transform, the poles from all of the different pieces cancel.
The gluon-gluon channel is shown below, but similar results exist for the other channels, and
can be found in App. G. The poles for the virtual correction are given by:

Îąs CA
H (0)
2π

3
1
− 2+



P2
Q2
2 ln 2 + ln J2
Âľ
Âľ

!!
,

(6.46)

the coupling renormalization is given in Eq. 6.14, and needs to be multiplied by the tree
level matrix element (H (0) ). The poles for the jet, collinear, PDF renormalization, and soft

141

corrections are given by:

Îąs CA
H (0)
2π

1
1
+
2



P 2 R2
2β0 − ln J 2
Âľ

!!
,



Îąs CA 1
δ (ξ2 − 1) ξ1 Pgg (ξ1 ) + ξ1 ↔ ξ2 ,
2π 




Îą
C
s A1
H (0)
δ (ξ2 − 1) ξ1 Pgg (ξ1 ) + 2β0 δ (1 − ξ1 ) + ξ1 ↔ ξ2 ,
2π 

Îąs CA 2
2 Âľ2
1
1
(0)
H
+ ln
− ln 2 ,
2π
 R
2  Q2
− H (0)

(6.47)
(6.48)
(6.49)
(6.50)

respectively. All the terms above are proportional to δ(ξ1 − 1)δ(ξ2 − 1) unless otherwise
noted. Combining Eqs. 6.46, 6.14, and 6.50, it is clear that all of the poles cancel, and the
remaining result is finite as required.

6.1.6

Resummation Calculation

After Fourier Transforming into b-space, the finite contribution at the one-loop order for the
gluon-gluon channel is given as:

(1)
Wgg→Hg (b)

(

i
b2 h
(1)
ln 2 0 2 δ(ξ2 − 1)ξ1 Pgg (ξ1 ) + (ξ1 ↔ ξ2 ) + δ(ξ1 − 1)δ(ξ2 − 1)
2π
b µ̄


!2 

2
2
2
2
Q b⊥
Q b⊥ 
1
(1)

× − ln 2
+ 4β0 − ln 2 ln 2  + Hgg δ(ξ1 − 1)δ(ξ2 − 1),

R
b0
b0
=

(0) Îąs CA
Hgg

(6.51)
where the integral over the parton distributions is implicit, and similar results exist for the
gluon-quark and quark-anti-quark channels. The hard coefficient, H (1) for the gluon-gluon

142

channel is:

(1)
Hgg

!
2
Q
Q2
1
Q2
−t −u
=
ln2
+
2β
ln
+
ln
ln
− 2 ln
ln
0
2
2
2
2
2
2π
s
s
R
PT
PT R
PT
!
!
!
 


2
m
t̃
t̃
ũ
ũ
+ ln2
− ln2
+ ln2
− ln2
+ 2Li2 1 − 2h
−t
−u
Q
m2h
m2h
!
!
#
t
u
67 π 2 23
(1)
+2Li2
+
2Li
+
+
− Nf + δHgg ,
(6.52)
2
2
2
9
2
54
mh
mh
(0) Îąs CA
Hgg

"

(1)

where t̃ = m2H − t, ũ = m2H − u, and δHgg can be found in Sec. 6.1.1. performing the
calculation for the gluon-quark channel results in:

(1)
Hgq

=

(0) Îąs
Hgq

2π

(

"

1 2
CA
ln
2

µ̂2
2
PJ⊥

!
+ ln

2
PJ⊥
µ̂2

!
ln

u
t

+ ln

2
PJ⊥
µ̂2

!


ln

s
µ̂2



!
#
2
−t −u
−u
µ̂2
u
ũ
ũ
7
4π
−2 ln 2 ln 2 − 4β0 ln 2 − 6β0 ln 2 + 2Li2
− ln2
+ ln2 2 + +
2
−u
3
µ̂
µ̂
µ̂
¾̃
mh
mh 3
"
!
2
2
PJ⊥
PJ⊥
1 2
µ̂2
3
µ̂2
1
µ̂2
u
s
+ CF
ln
+
ln
+
ln
ln
−
ln
ln
−
ln
ln
2
2 R2
2
2
2 PJ⊥
t
R2 PJ⊥
µ̂2
µ̂2
µ̂2
PJ⊥
!
!
!
 
m2h
−u
t
t̃
t̃
3
2
2
+3 ln 2 + 2Li2 1 −
+ 2Li2
−
ln
+
ln
−
s
−t
2
µ̂
m2h
m2h


5π 2
(1)
−
+ 20β0 + δHgq ,
(6.53)
6

143

(1)

where δHqg can again be found in Sec. 6.1.1. Finally, the quark-anti-quark channel is given
by:

(1)
(0) Îąs
Hqq̄ = Hqq̄
2π

(

"

1 2
CA
ln
2

µ̂2
2
PJ⊥

!
+ ln

2
PJ⊥
µ̂2

!
ln

u
t

+ ln

2
PJ⊥
µ̂2

!


ln

s
µ̂2



!
−t −u
−u
µ̂2
u
−2 ln 2 ln 2 − 4β0 ln 2 − 6β0 ln 2 + 2Li2
µ̂
µ̂
µ̂
¾̃
m2h
#
ũ
ũ
7 4π 2
− ln2
+ ln2 2 + +
−u
3
mh 3
"
!
2
2
PJ⊥
PJ⊥
1 2
µ̂2
3
µ̂2
1
µ̂2
u
s
+CF
ln
+
ln
+
ln
ln
−
ln
ln
−
ln
ln
2
2 R2
2
2
2 PJ⊥
t
R2 PJ⊥
µ̂2
µ̂2
µ̂2
PJ⊥
!
!
!
#
 
2
m2h
−u
t
t̃
t̃
3
5π
+3 ln 2 + 2Li2 1 −
+ 2Li2
− ln2
+ ln2
− −
2
s
−t
2
6
µ̂
mh
m2h
(1)

+20β0 } + δHgq ,

(6.54)

(1)

where δHqq̄ is given in Sec. 6.1.1. With the above results, it is possible to calculate the
resummation result in the CFG formalism by solving the evolution equations as discussed
in Sec. 3.4. This results in a similar Sudakov factor to that for color singlet final states.
however, the cone size needs to be introduced. This gives the following form:
Z µˆ2

dÂľ2
S(b) =
2
b2 /b2 Âľ
0



s
1
A ln 2 + B + D ln 2
Âľ
R


,

(6.55)

where R is the cone size for the jet. The coefficients for A, B, and D depend on the process,
and can be expanded order by order in perturbation theory. For the gluon-gluon channel,
A(1) = CA , B (1) = −2β0 , and D = CA . For the gluon-quark channel, A(1) = 12 (CA + CF ),
B (1) = −β0 − 34 CF − 12 (CA − CF ) ln ut , and D = CF . Finally, for the quark-anti-quark
channel, A(1) = CF , B (1) = − 32 CF , and D = CA .

144

The dependence of ln 12 in the Sudakov factor can be understood by looking at the
R
soft radiation, which is what is contained in the Sudakov factor. As mentioned during the
calculation of the soft factors (Sec. 6.1.4), the phase space needed to be separated to ensure
that the soft gluons inside the jet were not double counted. This can also be understood by
considering the contribution of the soft gluons inside the jet cone to the transverse momentum
of the Higgs plus jet system. If the gluon falls within the jet cone, then the momentum of
the gluon contributes to the jet, and does not contribute to the imbalance of transverse
momentum between the jet and the Higgs boson. Instead, if it falls outside the cone it does
contribute to the imbalance in transverse momentum.
It is interesting to note that many of the logarithms in H (1) can be eliminated if the
resummation scale µ̂ is chosen to be PJ⊥ . To illustrate this point, the ratio of H (1) /H (0)
is plotted as functions of the Higg’s rapidity (yH ) as shown in Fig. 6.2 with the jet rapidity
fixed at yj = 0. This result shows that H (1) is much larger than H (0) in the large yH
region if µ̂2 = s. In contrast, the ratio of H (1) /H (0) becomes less sensitive to yH with
2 . This is because when the difference of y and y becomes large, the invariant
µ̂2 = PJ⊥
H
J

mass Q2 of the Higgs boson and the leading jet can become much larger than the transverse
momentum of the jet. Hence, the scale of the results will be taken to be µ̂ = PJ⊥ in order
to resum the large logarithms in the perturbative contributions. In the following, this scale
choice is adopted in the theory predictions for comparison to fixed order and parton showers.
However, results with µ̂2 = s will be shown for the sake of comparison, and to support the
2 .
choice of µ̂2 = PJ⊥

145

(0)

H /H

3

g g -> H g at Âľ =Q
g g -> H g at Âľ =PT
g q -> H q at Âľ =Q
g q -> H q at Âľ =PT

(1)

S=8 TeV
RJet=0.5

2.5

y

Jet
T

=0

P > 30 (GeV)
2

p + p -> H + Jet

1.5

1

0.5
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

yH

Figure 6.2: Rapidity dependence of the hard factor for Higgs+Jet for different scale choices

6.1.7

Results

The above resummation formula to compute the differential and total cross sections of the
Higgs boson plus jet will be used to compare to both fixed order and parton shower results.
While data exists for this process, it is currently not at an accuracy that will be able to
discriminate against different predictions. In the numerical calculations shown, only the
gluon-gluon and the gluon-quark channel are used. This is a valid approximation because
the contribution for the quark-anti-quark channel is less than 1%, while the gluon-gluon and
gluon-quark channels make up 71% and 29% respectively. The anti-kT algorithm is used in
defining the jet with a cone size of R = 0.5.
Firstly, before the results are given, a cross check of the total cross-section is performed.
This cross check is done in a manner similar to that described in Sec. 3.5. In this cross check,

146

it is evident that using the narrow jet approximation results in a 2% discrepancy between the
fixed order result from MCFM, and the result from the calculation described here. This discrepancy varies as a function of R supporting the fact that it is due to the narrow jet approximation. Therefore, the correction is modeled by an additional R-dependent function inside of
H (1) . The correction is obtained by fitting the difference in cross section to a quadratic in R,




s C R − 1.1R + 23.3R2 and H (0) αs C − 0.8 ∗ R + 22.3R2
giving a correction of H (0) Îą
A
F
2π
2π
for final state gluon and quark jets respectively. Additionally, for the resummation to be
valid, the qT needs to be smaller than PJ⊥ , since the qT is defined as the vector sum of the
transverse momentum of the Higgs boson and the leading jet. If qT is greater than PJ⊥ it
is not clear what the leading jet is, and the fixed order calculation needs to be used in this
region. Finally, since A(2) is global and only depends on the color factor in the initial state,
this is also included into the calculation.

6.1.7.1

Comparison to Fixed Order

In Fig. 6.3, the comparison of the Higgs plus jet system at the LHC is shown for the total
transverse momentum and the transverse momentum of the Higgs. The fixed order calculation is given by MCFM, while the resummation calculation is shown for the two scales
discussed above.
For the total transverse momentum distribution, the leading order (LO) MCFM prediction is from the first non-zero prediction of the transverse momentum of the Higgs plus
jet system, and the NLO result is the one-loop correction to that. In other words, the LO
prediction is given by Higgs plus two jets, while the NLO prediction is given by Higgs plus
two jets at NLO. The MCFM NLO prediction is cut-off at 20 GeV due to the numerical difficulties of obtaining an accurate prediction below this value. Comparing the total transverse
147

dσ/dPH

dσ/dq

103

Full LO from MCFM
Full NLO from MCFM
resum at Âľ=PJ
resum at Âľ=Q

Full NLO from MCFM
resum at Âľ=PJ
resum at Âľ=Q

120

100

102

p + p -> H + Jet
PJ > 30 (GeV)
-4.4 < y < 4.4
Jet
RJet=0.5
S=8 TeV

10

p + p -> H + Jet

80

PJ > 30 (GeV)
-4.4 < y

60

Jet

< 4.4

RJet=0.5
40

S=8 TeV

1
20

10-1
0

20

40

60

80

100

120

140

160

180

0

200

20

40

60

80

100

120

140

160

180

200

PH (GeV)

q (GeV)

Figure 6.3: The differential cross sections of Higgs boson plus one jet production at the
LHC as functions of the total transverse momentum q⊥ (left) and the Higgs boson
transverse momentum PH⊥ (right). The resummation predictions (resum) with
resummation scale set to be PJ⊥ (black line) and Q (green line) respectively are compared
to the LO result from MCFM (pink line) with non-zero q⊥ , and the NLO result from
MCFM (red line) which is the production rate of Higgs boson plus two separate jets up to
one-loop in QCD.

148

momentum distribution of the resummed and fixed order calculation, it is seen that the fixed
order fails to describe the small qT region. Additionally, it is clear that the scale choice of
µ̂ = Q results in a calculation that is too hard (too large of a cross-section for large qT ), and
can never be matched to fixed order. On the other hand, the scale choice of µ̂ = PJ⊥ gives
a much more reasonable prediction.
For the transverse momentum of the Higgs boson, only the NLO prediction from MCFM
is shown. This prediction is given by the Îąs corrections to the Higgs plus jet result. Again,
the resummation calculation is given for the two different scale choices. The choice for
µ̂ = Q is too hard in the high transverse momentum region as also seen in the total transverse
momentum distribution. Here, there is a discontinuity at the value of pH⊥ = pJcut , known as
the Sudakov shoulder singularity. This is due to a integrable singularity that arises when the
leading jet is close to back to back to the Higgs boson, or in other words, when qT approaches
zero. This is resolved in the resummation calculation, by appropriately resumming the
logarithms to all orders that result in the divergence to give a finite result.

6.1.7.2

Comparison to Parton Showers

Similar to the fixed order calculation, the comparison of the resummation result can be
compared to that from parton shower predictions. In this study, the Higgs boson was set to
125 GeV and left undecayed, the MMHT2014nlo68clas118 PDF set was used [215]. Finally,
the theoretical uncertainties were estimated by varying the scales by factors of 12 and 2
around the central scale, but left the renormalization scale (µ̂) fixed to PJ⊥ as discussed
above. The setup and details of the other predictions used in this comparison can be found
in [28]. The jets for this calculation are taken to have a cone size of R = 0.4, with pJ⊥ > 30
GeV and |yJ | < 4.4. The codes that are used in this comparison are: SHERPA [178] plus
149

GOSAM [216, 217], MINLO [218], BFGLP [219], HEJ [220, 221, 222, 223, 224], Herwig 7.1 [225],
Madgraph5 aMC@NLO [226], POWHEG-BOX [227], STWZ [228], and SHERPA NNLOPS [229, 230].
Exact details of the setups for each of the individual codes can be found in [28].
Firstly, the total rates for the jet multiplicities are compared across all of the codes,
shown in Fig. 6.4. The ResBos2 code is able to predict both the total inclusive rate for Higgs
along with the inclusive rate for Higgs plus 1 jet. Comparing across all of the codes, it is
clear that the ResBos2 prediction is consistent with the others as expected from the fixed
order comparison above.
The differential distributions that are compared are the transverse momentum of the
Higgs, the leading jet, the Higgs plus leading jet system, and the rapidity of the leading jet.
Here the transverse momentum of the Higgs requires the presence of at least one hard jet.
The comparison of the transverse momentum of the Higgs boson in the presence of an
additional jet can be seen in Fig. 6.5. The fixed order calculations again show the Sudakov
shoulder singularity, but the all orders calculations do not have this singularity. In the ratio
plot, at the high transverse momentum, there is a deviation of the resummation calculation
from that of POWHEG due to a different choice of scales. In the RESBOS2 calculation, the
central scale is Âľ0 = 12 mh as mentioned above, but the scale choice for POWHEG is Âľ0 =
q
1 m2 + p2 . This scale choice softens the high transverse momentum tail as discussed in
2
T
h
detail in Sec. 4.1.
Next, the comparison of the leading jet transverse momentum and rapidity distributions
can be found in Fig. 6.6 and Fig. 6.7, respectively. Both of these distributions do not contain
any large Sudakov effects, and therefore all the calculations should be close to that of the fixed
order calculation. This is clearly seen in the aforementioned figures, with all the calculations
falling within each others uncertainty bands. Additionally, the size of the uncertainty bands
150

1

STWZ
ResBos 2
Sherpa h Nnlo
BFGLP hj Nnlo
GoSam+Sherpa
NLO
hj
NLO
hjj
hjjj
NLO

1
MiNLO
MiNLO
MiNLO

Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1
Hej
STWZ
ResBos 2
Sherpa h Nnlo
BFGLP hj Nnlo
GoSam+Sherpa
NLO
hj
NLO
hjj
hjjj
NLO

nNLO
nNLO

MiNLO
MiNLO
MiNLO

nNLO
nNLO

Ratio to Powheg

10−1
1.4

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to Powheg

0.6

1.2
1
0.8

1.2
1
0.8

0.6

0.6

1.4

1.4
Ratio to Powheg

Ratio to Powheg
Ratio to Powheg

Inclusive jet multiplicity

10 1

10−1
1.4

Ratio to Powheg

10 2

LH15 pp → h + jets comparison

Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1
Hej

10 1

dσ/dn j [pb]

Inclusive jet multiplicity
LH15 pp → h + jets comparison

dσ/dn j [pb]

10 2

1.2
1
0.8
0.6

1.2
1
0.8
0.6

≥0

≥1

≥2

≥3

≥0
nj

≥1

≥2

≥3
nj

Figure 6.4: The central predictions(left panel) and with theoretical uncertainties (right
panel) for the inclusive jet multiplicities as predicted by fixed-order calculations, resummed
calculations, NNLO and NLO Monte Carlos. The bottom panel is divided up into three
subplots all showing the ratios with respect to the POWHEG NNLOPS prediction. The upper
of these plots contains the HEJ and SHERPA NNLOPS ratios, while the middle one includes
all NLO merged predictions (Madgraph5 aMC@NLO, Herwig 7.1 and SHERPA) and the lower
one shows all those listed in the bottom left legend of the main panel. Reproduced from
[28].
are consistent with each other and are of the order of 20%. Again the difference in the
high transverse momentum tail between the POWHEG and RESBOS2 calculations arise from
the difference in scale choices. For the rapidity, the calculation is consistent throughout
the rapidity range, and the offset between the POWHEG and RESBOS2 calculations is due to
151

10−1

ResBos 2
BFGLP hj Nnlo

10−2

1

Higgs boson transverse momentum (n j ≥ 1)
Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

10−1

ResBos 2
BFGLP hj Nnlo

10−2

GoSam+Sherpa hj
NLO
MiNLO
nNLO

GoSam+Sherpa hj
NLO
MiNLO
nNLO

Ratio to Powheg

10−3
1.4

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to Powheg

0.6

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to Powheg

0.6

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to Powheg

0.6

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to hj Nlo

0.6

1.2
1
0.8

1.2
1
0.8

0.6

0.6

1.4

1.4
Ratio to hj Nlo

Ratio to hj Nlo

Ratio to Powheg

Ratio to Powheg

Ratio to Powheg

Ratio to Powheg

10−3
1.4

Ratio to hj Nlo

LH15 pp → h + jets comparison

Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

dσ/dp ⊥ [pb/GeV]

Higgs boson transverse momentum (n j ≥ 1)
LH15 pp → h + jets comparison

dσ/dp ⊥ [pb/GeV]

1

1.2
1
0.8
0.6

1.2
1
0.8
0.6

0

50

100

150

200
p⊥ (h) [GeV]

0

50

100

150

200
p⊥ (h) [GeV]

Figure 6.5: The Higgs boson transverse momentum in the presence of at least one jet
central predictions(left) with uncertainty bands(right). The ratio plot panel is divided into
six parts where the upper four exhibit the ratios to the POWHEG NNLOPS result while the
lower two show them to the NLO calculation for h + 1 jet as provided by GOSAM+SHERPA.
Reproduced from [28].
152

10−1

STWZ
ResBos 2
BFGLP hj Nnlo

10−2

1

Leading jet transverse momentum (n j ≥ 1)
Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

10−1

STWZ
ResBos 2
BFGLP hj Nnlo

10−2

GoSam+Sherpa hj
NLO
MiNLO
nNLO

GoSam+Sherpa hj
NLO
MiNLO
nNLO

Ratio to hj Nnlo

10−3
1.4

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to hj Nnlo

0.6

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to hj Nnlo

0.6

1.2
1
0.8

1.2
1
0.8

0.6

0.6

1.4

1.4
Ratio to hj Nnlo

Ratio to hj Nnlo

Ratio to hj Nnlo

Ratio to hj Nnlo

10−3
1.4

Ratio to hj Nnlo

LH15 pp → h + jets comparison

Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

dσ/dp ⊥ [pb/GeV]

Leading jet transverse momentum (n j ≥ 1)
LH15 pp → h + jets comparison

dσ/dp ⊥ [pb/GeV]

1

1.2
1
0.8
0.6

1.2
1
0.8
0.6

40

60

80

100

120

140

160

180
200
p⊥ ( j1 ) [GeV]

40

60

80

100

120

140

160

180
200
p⊥ ( j1 ) [GeV]

Figure 6.6: The leading jet transverse momentum distribution for h + ≥ 1-jet production,
to the right (left) shown with (without) the uncertainty bands provided by the various
calculations. The part below the main plot contains four ratio plots taken wrt. the NNLO
result of the BFGLP group following the same strategy for grouping the predictions as
before (NNLOPS versus NLO ME+PS versus fixed-order and resummation results).
Reproduced from [28].
the fact that the total cross-sections differ by about 5%, which is within the theoretical
uncertainty at this order, as seen when the uncertainty bands are overlaid.

153

3
2.5
2
1.5

4
Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

3.5
3
2.5
2
1.5

ResBos 2
1

ResBos 2
1

GoSam+Sherpa hj
NLO
MiNLO
nNLO

0.5

Ratio to Powheg

0
1.4

1.2
1
0.8

1.2
1
0.8
0.6

1.4

1.4
Ratio to Powheg

0.6

1.2
1
0.8

1.2
1
0.8

0.6

0.6

1.4

1.4
Ratio to Powheg

Ratio to Powheg
Ratio to Powheg

GoSam+Sherpa hj
NLO
MiNLO
nNLO

0.5

0
1.4

Ratio to Powheg

LH15 pp → h + jets comparison

Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

3.5

dσ/dy [pb]

Leading jet rapidity
LH15 pp → h + jets comparison

dσ/dy [pb]

Leading jet rapidity
4

1.2
1
0.8
0.6

1.2
1
0.8
0.6

0

0.5

1

1.5

2

2.5

3

3.5

4

0
y( j1 )

0.5

1

1.5

2

2.5

3

3.5

4
y( j1 )

Figure 6.7: The rapidity distribution for the leading jet in h + ≥ 1-jet production, shown
without (left) and with (right) theoretical uncertainties. Ratio plots are displayed in the
lower part of the plot using the POWHEG NNLOPS result for Higgs boson production as their
reference. Predictions are grouped, from top to bottom, according to the categories
NNLOPS, ME+PS at NLO and NLO fixed order as well as resummation. Reproduced from
[28].
Finally, the transverse momentum of the Higgs plus jet system is examined. The comparison between the different results can be found in Fig. 6.8. Here comparing the RESBOS2
calculation to POWHEG has some interesting features that need to be discussed. In the high
transverse momentum region, there is a discrepancy between the two calculations which arise
from the fact that in the tail of the distribution, the RESBOS2 calculation only is matched to
154

the LO prediction for this observable as described in Sec. 6.1.7, and the agreement should be
improved if the matching is done to a higher order prediction. Additionally, in the RESBOS2
calculation the logarithms dealing with the jet cone size are resummed as mentioned in
Sec. 6.1.6. This results in a broader Sudakov peak than the other parton shower predictions, and an upward shift in the Sudakov peak value. When the LHC data becomes precise
enough, this difference will be a strong test of the resummation formalism versus the parton
shower model.
Overall, the parton shower predictions and the resummed predictions are consistent with
each other. This was expected, since the formal accuracy of the two predictions for all
observables with the exception of the transverse momentum of the Higgs plus jet system
are of the same order. The major difference arises in the prediction of the Higgs plus jet
system transverse momentum, due to the dependency of the jet cone size in the RESBOS2
prediction. Once the LHC data becomes precise enough, these codes and their differences
can be tested against data.

6.1.8

Future of Higgs Plus Jet Resummation

In this section, some brief future steps are discussed. Firstly, there has been some work to
begin to extend this calculation from Higgs plus one jet to Higgs plus two jets [29]. Additionally, at higher order calculations for the Higgs plus one jet system non-global logarithms
are introduced into the calculation.
The calculation of the Higgs plus two jets system is an important calculation to separate
the vector boson fusion (VBF) production of the Higgs boson from the gluon fusion production of the Higgs boson. As shown in [29], the peak of the transverse momentum of the
system peaks at drastically different values. This can be used to apply an additional cut
155

10−1

10−2

ResBos 2
GoSam+Sherpa hj
NLO
MiNLO
nNLO

1
0.5

1
0.5

1
0.5

0

20

40

60

80

100

120
140
p⊥ (hj1 ) [GeV]

Ratio to Powheg

10−3
1.5

Ratio to Powheg

Ratio to Powheg

1.5

Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

10−1

1.5

Ratio to Powheg

Ratio to Powheg
Ratio to Powheg

1.5

Transverse momentum of Higgs and leading jet system (n j ≥ 1)

10−2

ResBos 2
GoSam+Sherpa hj
NLO
MiNLO
nNLO

10−3
1.5

1

LH15 pp → h + jets comparison

Powheg NnloPs
Sherpa NnloPs
MG5_aMC FxFx
Sherpa MePs@Nlo
Herwig 7.1

dσ/dp ⊥ [pb/GeV]

Transverse momentum of Higgs and leading jet system (n j ≥ 1)
LH15 pp → h + jets comparison

dσ/dp ⊥ [pb/GeV]

1

1.5

1
0.5

1
0.5

1
0.5

0

20

40

60

80

100

120
140
p⊥ (hj1 ) [GeV]

Figure 6.8: The transverse momentum of the Higgs-boson-leading-jet system in the
presence of at least one jet. For better visibility, results are shown without (left) and with
(right) theoretical uncertainties. The plot layout exactly corresponds to that of Figure 6.7,
except for the extended ŷ-axis range in the ratio plots. Reproduced from [28].
on the Higgs production events to increase the purity of the VBF signal. The normalized
distributions for this observable can be seen in Fig. 6.9.
Finally, the goal is to extend the Higgs plus one jet calculation to NNLL accuracy.
However, going beyond NLL accuracy introduces what are known as non-global logarithms.
Non-global logarithms are logarithms that may not appear at each order in the calculation,
and do not have a specific form to predict the higher order coefficients of them. Therefore,

156

Figure 6.9: Noramlized distributions of the vector boson fusion and gluon-fusion
contributions to√the Higgs boson plus two jets production in the typical kinematics at the
LHC with S = 13T eV , where the jet transverse momenta k1⊥ = k2⊥ = 30GeV ,
yj1 = −yj2 = 2 and yh = 0: as functions of the total transverse momentum q⊥ (left); the
total rate as function of the upper limit of q⊥ (right). Reproduced from [29].
it is not possible to resum these logarithms to all orders [231, 232, 11, 233]. This results in
additional complications that need to be considered.

157

Chapter 7
Conclusion
The improvement of the ResBos code from the precision used for the Tevatron, to that of
the new ResBos2 code, will allow for precision resummation calculations at the LHC. These
improvements come from many different improved predictions, as mentioned above. Below
a review of the improvements to the predictions, along with the experimental comparisons
are summarized for the reader.
First, the improvements to color singlet resummation will be reviewed. In this section,
the difference between schemes will again be pointed out. Also, the predictions for both the
Z and W boson at the LHC will be reviewed. Finally, the future steps for this project will
be outlined.
Afterwards, the addition of non-color singlet resummation calculation will be reviewed.
In this section, the calculation for the Higgs plus jet system will be reviewed, along with
some mentions of other works using these ideas. Finally, the future prospects for this new
type of qT resummation will be emphasized.

158

7.1
7.1.1

Color Singlet Resummation
Resummation Schemes

Two main schemes in qT resummation are again the Collins-Soper-Sterman and the Catanide-Florian-Grazzini schemes. The differences between these schemes have both an order by
order, and an all orders relationship, to map from one scheme to the other. It was shown that
as long as the calculation is preformed such that the orders of the B, C, and H coefficient
are such that the order of B is one higher than C and H, then the two calculations are in
very good agreement. This is supported by the conversion between the two schemes as seen
in Eq. 3.73, and the results of Fig. 4.3.

7.1.2

Z Boson Predictions

Previously at the LHC, the prediction of the ResBos code was unable to reproduce the
data. The resolution to this issue took many different steps as outlined above. These steps,
included the improvement of the precision of the calculation, and the change of the scale of
the fixed order calculation.
The improvement in the precision of the calculation, was from NNLL to N3 LL, and the
fixed order was improved from approximate NNLO to NNLO, with a k-factor included in
the high transverse momentum region to match the NNLO Z plus jet calculation. One
improvement that will be made to the ResBos2 code is the matching fully with the NNLO Z
plus jet calculation. This will be included once the results of the NNLO Z plus jet calculation
become sufficiently available.
The other improvement was the change of the fixed order scale from the invariant mass
of the lepton pair to the transverse mass of the lepton pair, defined in the text above. This
159

choice of scale makes more sense for the high transverse momentum region due to the fact
that at high transverse momentum, the energy of the jet is comparable to the invariant mass
of the lepton pair.
Upon using these improvements to the ResBos prediction, the agreement between the
theory prediction and the data is greatly improved. Further improvements will arise with
the matching to the higher order fixed order calculation. However, this improvement will
only help in the intermediate region.
One more improvement that can be made to the ResBos code, is to further improve the
non-perturbative piece. Currently, the non-perturbative piece is fixed as a function of the
resummation scales. This has recently been believed to give too conservative an estimate of
the uncertainty of the prediction at small transverse momentum. Therefore, another future
improvement is to include the resummation scale variations into the non-perturbative fit,
allowing the non-perturbative function to depend on these scales, as makes physical sense.

7.1.3

W Boson Predictions

The ResBos2 predictions for the W -transverse momentum, and the ability to use the code
to help predict the W mass at the LHC will be helpful in understanding the EW sector of
the SM. As mentioned in the text above, the indirect measurement is much smaller than
the direct measurement of the W mass. It has been proposed to use resummation tools to
improve the direct measurement, by predicting the ratio of the W transverse momentum to
the Z transverse momentum. With these, the experimentalists will be able to reduce the
uncertainties in the W mass measurement. These reductions will allow the measurement to
be competitive with the indirect measurement.
Additionally, the ResBos2 prediction allows for a template fit of the experimental mea160

surements of the transverse momentum of the leptons to the theoretical prediction. This
will allow for a precise determination of the W mass.
Some future improvements that could be included into the ResBos2 prediction are again
similar to that of the Z boson, from the matching to the higher order fixed order calculation,
and improving the non-perturbative piece. While these may not have a large effect on the
determination of the W boson mass, there are not any improvements that can be made in
the theoretical prediction in the near future. The major improvements that would help,
included calculations to much higher orders, which are not feasible at the writing of this
thesis.

7.2

Non-Color Singlet Resummation

Recently, there has been work done to allow for non-color singlet qT resummation in QCD.
The first of these being the dijet calculation, and another related to the work of this thesis
being the Higgs plus jet resummation as mentioned previously. With these calculations
completed, a new set of processes are now able to be calculated that previously were thought
not possible.
As mentioned above, the Higgs plus jet calculation was an important step in understanding QCD, and an important contribution towards obtaining the Higgs plus two jet
resummation calculation. The Higgs plus two jet calculation is important in the study of
the Higgs boson, in allowing one to separate the gluon-gluon fusion production mechanism
of the Higgs boson from the EW production of the Higgs boson. Additionally, it is shown
that the current prediction for the qT resummed prediction for the Higgs plus jet system is
consistent with the parton shower results. However, there is the one major difference in that

161

the analytic resummation knows about the cone size of the jet, resulting in a shape that
differs from the parton shower codes. When the data becomes more precise, a great test of
this method would be to see if the data can distinguish whether the paron shower method,
or the analytic resummed method is correct.

7.3

Final Remarks

In conclusion, the improvement of the qT resummation calculations in QCD are included
in the ResBos2 code. With these improvements, the calculations are now at a level that is
sufficient for the precision required at the LHC. In the future, the ResBos2 will continue to
improve building off of the work of this thesis, and the ideas that have formed as a result of
this work.

162

APPENDICES

163

APPENDIX A
The Dirac Equation and Îł Matrices
The Dirac equation is the equation of motion to describe fermions, and is given by:



ψ̄ i∂ µ γµ + m ψ = 0,

(A.1)

where ψ is the wavefunction of a fermion, m is the mass of that fermion, and γ¾ 1 are the
Dirac matrices defined by their anti-commutation relation:

{γ ¾ , γ ν } = 2g ¾ν ,

(A.2)

where g µν is the metric tensor defined using the mostly negative choice, g µν = diag(1, −1, −1, −1),
such that squares of time-like four momenta result in positive results.
The Îł or Dirac matrices are used for infinitesimal transformations of spinors under spatial
rotations and Lorentz boosts. There are many representations for these matrices. The Weyl
representation is given by:




 0 1
γ0 = 
,
1 0


 0
γi = 
−σ i

σi
0



,

(A.3)

1 In Eq. A.1, it is common to rewrite ∂ µ γ as ∂.
/ This slashed notation will be used through the rest of
Âľ
this work. Also, additional details on the Îł matrices, along with one representation of them can be found in
App. A

164

where σ i are the Pauli matrices, and 1 is the 2 × 2 Identity matrix. Additionally, a fifth γ
matrix can be defined as:




−1 0
γ 5 = iγ 0 γ 1 γ 2 γ 3 = 
.
0 1

(A.4)

Important relationships of Îł matrices are:
1. γ ¾ γ¾ = 414 ,
2. γ µ γ ν γµ = −2γ ν ,
3. γ ¾ γ ν γ ρ γ¾ = 4g νρ 14 ,
4. γ µ γ ν γ ρ γ σ γµ = −2γ σ γ ρ γ ν ,
5. trace of any odd product of Îł matrices excluding Îł 5 is zero,
6. trace of any odd product of Îł matrices times Îł 5 is also zero,
7. Tr (γ ¾ γ ν ) = 4g ¾ν ,
8. Tr (γ µ γ ν γ ρ γ σ ) = 4 (g µν g ρσ − g µρ g νσ + g µσ g νρ ),




9. Tr γ 5 = Tr γ ¾ γ ν γ 5 = 0,
10. Tr (γ µ γ ν γ ρ γ σ ) = −4iµνρσ ,
11. Tr (γ ¾1 ¡ ¡ ¡ γ ¾n ) = Tr (γ ¾n ¡ ¡ ¡ γ ¾1 ),
where 14 is the 4 × 4 unit matrix and µνρσ is the completely anti-symmetric Levi-Civita
Tensor. When moving beyond 4 dimensions, the contraction identities are modified as follows:
165

1. γ ¾ γ¾ = d1d ,
2. γ µ γ ν γµ = −(d − 2)γ ν ,
3. γ µ γ ν γ ρ γµ = 4g νρ − (4 − d)γ ν γ ρ ,
4. γ µ γ ν γ ρ γ σ γµ = −2γ σ γ ρ γ ν + (4 − d)γ ν γ ρ γ σ .
The trace identities not involving Îł 5 are not modified since they are independent of dimensionality. There are many complications of dealing with Îł 5 in dimensions not equal to 4.
There have been many discussions on the appropriate way to handle this situation. The
reader is referred to [234, 235] for detailed discussions on this topic.

166

APPENDIX B
Spin, Helicity, and Chirality

Spin
~
~ For a fermion, S
~ = sigma
Spin is the eigenvalue of S.
2 . For a single particle, the spin

and the angular momentum operators are the same. The scalar spin s is the eigenvalue in
~ 2 = s(s + 1). When saying a particle is spin-1/2 refers to the value of s.
S

Helicity
Helicity is the projection of the spin on the direction of momentum. Helicity eigenstates are
~

p
given by the operator H = S¡~
, and exist for any spin particle.
|~
p|

Chirality
Chirality is a concept that only exists for fermionic particles. A particle is chiral if it is
not symmetric under a mirror symmetry. This leads to the definition of left-/right-handed
particles. The projection of the fermionic wavefunction to the left-/right-handed components
is given by:
PL =

1 + Îł5
,
2

PR =

167

1 − γ5
.
2

(B.1)





 ψL 
In the Weyl representation, the spinors can be represented by  . In the massless limit
ψR
chirality and helicity are the same.

168

APPENDIX C
Standard Model Higgs Boson and
Higgs Mechanism
As discussed in Chapter 1, the Higgs Mechanism is used to introduce masses for vector
bosons, in a manner that preserves the gauge theory. There, the calculation was demonstrated for a Higgs boson added to QED. The full details for adding a Higgs boson into
SU (2)L × U (1)Y is detailed below.
First, the addition of a scalar doublet is required to be added to the Standard Model,
given by:


1 φ1 + iφ2 
φ= √ 
,
2
φ3 + iφ4

(C.1)

which is a multiplet of SU (2)L timesU (1)Y . The hypercharge of the above doublet is chosen
to be Y = 1.
The potential for the scalar doublet that is responsible for the spontaneous breaking of
the symmetry is the generalized form from that in Chapter 1, and is given as:




2
V (φ) = µ2 φ† φ + λ φ† φ ,

169

(C.2)

where Âľ2 < 0. With a Lagrangian given as:



L = (Dµ φ)† Dµ φ − V (φ) ,

(C.3)

where DÂľ is the covariant derivative given as:

1
~ Âľ + ig 0 1 Y BÂľ .
Dµ = ∂µ + ig ~τ · W
2
2

(C.4)

The vacuum state (φ0 ) is given by:
 
1 0
φ0 = √   ,
2
v

(C.5)

where v is known as the vacuum expectation value. With this choice of the vacuum, SU (2)L ×
U (1)Y is broken, but since the vacuum has charge Q = I3 + 12 Y = 0, there remains a U (1)EM
symmetry as required.
The Lagrangian above can be rewritten into terms of the physical gauge bosons (W + , W − , Z, γ).
The W Âą boson can be related to W 1 and W 2 , by the following relationship:


1 
W ± = √ W 1 ∓ iW2 .
2

(C.6)

Additionally, the Z and Îł can be rewritten in terms of the W 3 and B bosons as given by:



1
A= p
g 0 W 3 + gB ,
g 2 + g 02


1
Z=p
gW 3 − g 0 B .
g 2 + g 02

170

(C.7)
(C.8)

Combining these relationships, the Lagrangian for the scalar can be rewritten as:




 1 

2

2 
(Dµ φ)† Dµ φ = v 2 g 2 W + + g 2 W − + g 2 + g 02 Z 2 + 0 · A2 ,
8

(C.9)

plus terms that involve the Higgs boson. The rest of the results follow as discussed in
Chapter 1.

171

APPENDIX D
Structure of SU (3)
A given representation of the Gell-Mann matrices is given by the following 8 matrices:

Îť1

Îť4

Îť7

0


= 1


0

0


= 0


1

0


= 0


0



1
0
0
0
0
0
0
0
i






0
0 −i 0
1 0 0 










,
Ν2 =  i 0 0  ,
λ3 = 0 −1 0 ,
0










0
0 0 0
0 0 0





1
0 0 −i
0 0 0 










,
Îť
=
,
Îť
=




,
5
6
0
0 0 0 
0 0 1 





0
i 0 0
0 1 0



0
1 0 0 


1 



,
λ8 = √ 0 1 0  .
−i


3



0
0 0 −2

(D.1)

The structure functions are completely anti-symmetric in the choices of a, b, and c, and
are a generalization of the Levi-Cevita Tensor from SU (2) to SU (3). They are defined as:
√
1
3
f123 = 1, f147 = f165 = f246 = f257 = f345 = f376 = , f458 = f678 =
,
2
2

(D.2)

and in general they are zero, unless they contain an odd number from the set of {2, 5, 7}.

172

APPENDIX E
QCD Feynman Rules

Fermion Lagrangian
To find the interaction terms for the fermions, we start from the free Lagrangian. For all of
this, we will assume that the fermions are massless. It is easy for one to extend the following
to include fermion masses.

Lf = ψ̄i∂/ψ

(E.1)

We will now require that the Lagrangian remain invariant under a SU(3) transformation.
To begin, a local SU(3) transformation can be written as:

a a
X = e−iT ρ (x)

(E.2)

Where ρa (x) are all real. We can then act this transformation on the above Lagrangian:

ψ → Xψ



∂µ ψ → X∂µ ψ + ∂µ X ψ

(E.3)

To bring back the desired SU(3) invariance, we introduce the covariant derivative which
is defined such that:
Dµ ψ → XDµ ψ
173

(E.4)

Working through the equations above to find the definition of the covariant derivative,
one finds that the form the covariant derivative must take is given by:

Dµ = ∂µ + igs T a Aaµ

(E.5)

Where in the above equation, gs is the strong coupling constant, and AaÂľ are the vector
boson fields for the strong force, known as gluons.
Therefore, the SU(3) invariant Lagrangian for fermions is defined as:

a


/
Lf = ψ̄ i∂/ − gs T a A

ψ

(E.6)

Looking at this equation, one can see where the interaction between the quarks and the
gluons arises. Details on the interaction and the Feynman rules for QCD will be calculated
in later sections.

Vector Boson Lagrangian
The vector bosons, or gluons, are in the adjoint representation of SU(3). Therefore, under a
SU(3) transformation the fields transform as:

a
a
b
c
A0a
µ = Aµ − ∂µ ρ (x) + gs fabc ρ (x)Aµ

(E.7)

We can also define the field strength tensor for QCD in an anologous way to that of QED,

174

but we have to include additional terms since unlike QED, QCD is non-Abelian.

a = ∂ Aa − ∂ Aa + g f abc A A
F¾ν
¾ ν
ν ¾
s
¾,b ν,c

(E.8)

Also, similar to QED, the kinetic part of the Lagrangian is formed from a Lorentz invariant using two field strength tensors.

1
a
Lkin = − F µν,a Fµν
4

(E.9)

To simplify the expressions, we can define: Aµ ≡ −iT a Aaµ and
a . Using these definitions, we can rewrite the kinetic part of the Lagrangian
Fµν ≡ −iT a Fµν

by using the following relations:

Fµν = ∂µ Aν − ∂ν Aµ + gs [Aµ , Aν ] =

1
[D¾ , Dν ]
gs

(E.10)

Where Dµ is the covariant derivative from Eq. E.5. Using the trace property of the T a ’s,
one can rewrite the Lagrangian as:

1
Lkin = − Tr(Fµν Fµν )
2

(E.11)

Putting the fermion Lagrangian E.6 together with the kinetic term for the boson Lagrangian E.9, one obtains the complete Lagrangian for QCD.



1
a + ψ̄ iD
/ −m ψ
LQCD = − F µν,a Fµν
4

175

(E.12)

QCD Feynman Rules
In order to find the Feynman Rules for QCD, we first need to expand the kinetic term of
the QCD Lagrangian in order to find the quartic, cubic, and quadratic terms for the vector
bosons in the Lagrangian.




a = ∂ µ Aν,a − ∂ ν Aµ,a + g f abc Aµ Aν
a − ∂ Aa + g f ab0 c0 A
F ¾ν,a F¾ν
∂
A
A
s
¾ ν
ν ¾
s
¾,b0 ν,c0
b c
0 0
= ∂ µ Aν,a ∂µ Aaν − ∂ µ Aν,a ∂ν Aaµ + gs f ab c ∂ µ Aν,a Aµ,b0 Aν,c0

− ∂ ν Aµ,a ∂µ Aaν + ∂ ν Aµ,a ∂ν Aaµ
0 0
Âľ
Âľ
− gs f ab c ∂ ν Aµ,a Aµ,b0 Aν,c0 + gs f abc Ab Aνc ∂µ Aaν − gs f abc Ab Aνb ∂ν Aaµ
0 0 Âľ

+ gs2 f abc f ab c Ab Aνc A¾,b0 Aν,c0

(E.13)

The equation above can be further simplified by noting:
∂ µ Aν,a ∂µ Aaν = ∂ ν Aµ,a ∂ν Aaµ and ∂ µ Aν,a ∂ν Aaµ = ∂ ν Aµ,a ∂µ Aaν
We can also interchange the A-fields since we represented them in terms of group components instead of matrices. Reorganizing the above equation and simplifying, one obtains:

a = 2 ∂ µ Aν,a ∂ Aa − ∂ µ Aν,a ∂ Aa 
F ¾ν,a F¾ν
¾ ν
ν ¾
h
i
+ 2gs fabc (∂ µ Aν,a ) Abµ Acν − fabc (∂ ν Aµ,a ) Abµ Acν
0

0

+ gs2 fabc fab0 c0 A¾,b Aν,c Ab¾ Acν

176

(E.14)

Putting this back into Eq. E.12, the Lagrangian can be written as:

1  ¾ ν,a
∂ A ∂µ Aaν − ∂ µ Aν,a ∂ν Aaµ
2
i
1 h
− gs fabc (∂ µ Aν,a ) Abµ Acν − fabc (∂ ν Aµ,a ) Abµ Acν
2
0 0
1
− gs2 fabc fab0 c0 Aµ,b Aν,c Abµ Acν
4


+ ψ̄ i∂/ − m ψ − gs T a Aaµ ψ̄γ µ ψ

LQCD = −

(E.15)

Now that we have the Lagrangian in this form, we can begin calculating the Feynman
Rules for QCD. We will calculate the Feynamn Rules for:
1. Gluon propagator
2. Triple gluon coupling
3. Quartic gluon coupling
4. Fermion propagator
5. Fermion-gluon vertex

Gluon Propagator
To find the gluon propagator Feynman Rule, one must first find all the quadratic terms in
AaÂľ . These terms are as follows:

1  ¾ ν,a
∂ A ∂µ Aaν − ∂ µ Aν,a ∂ν Aaµ
2
i
1 Âľ,a h 2
=− A
∂ gµν δab + ∂µ ∂ν δab Aν,b
2

−

(E.16)

However, to find the propagator one needs to find the inverse of this operator, but due
to gauge invariance this operator does not have an inverse. Therefore, in order to obtain the
177

gluon propagator we must first break the gauge invariance of the Lagrangian. To do this,
we will insert a new term by hand, known as the gague fixing term. Here we will use the
following gauge breaking term:

Lgf = −


2
1
1
∂µ Aµ,a = − Aaµ (−∂ µ ∂ ν δab ) Abν
2Îą
2Îą

(E.17)

Combining these two equations, we are able to find an operator that is invertible. The
new term that we need to invert is given by:



1 Âľ,a 2
1
− A
∂ gµν δab + (1 − )∂µ ∂ν δab Aν,b
2
Îą

(E.18)

The propagator is found by taking two derivatives with respect to the Fourier transform
of the Lagarangian.

P−1



∂ 2 F(L)
1
2
= −i
= −i −k gαβ δcd + (1 − )kα kβ δcd
Îą
∂Acα ∂Adβ

(E.19)

By observation, it is easy to see that the form that the inverse of this equation must take
the form of:


k¾ kν
P = i Ag¾ν + B 2
δab
k
One can solve the equations for A and B to obtain the result for the propagator.

Âľ

PP−1 = 1 = gσ δab

178

(E.20)


k¾kν
=
+B 2
δab
k




1
1
Âľ
Âľ
2
Âľ
Âľ
⇒ − Ak gσ − Bk kσ + A 1 −
k kσ + B 1 −
k ¾ kσ = gσ
Îą
Îą




1
1
2
⇒ − Ak = 1
−B+A 1−
+B 1−
=0
Îą
Îą


k 2 gνσ

1
− (1 − )kν kσ
Îą

1
⇒A = − 2
k



Ag ¾ν

(E.21)

1
B = (1 − α) 2
k

Plugging the solutions to A and B back into Eq. E.20, we obtain the Feynman Rule for
the gluon propagator.



(1 − α) kµ kν
δab
= i 2 −gµν +
k
k2

(E.22)

Triple Gluon Coupling
To find the triple gluon coupling, first we need to find all the terms that are cubic in the
gauge fields. We will also be using the all incoming momentum convention when calculating
the Feynman Rules. These terms are as follows:

1
L3 = − gs fabc [∂ µ Aν,a − ∂ ν Aµ,a ] Abµ Acν
2

(E.23)

We then follow a similar procedure to that used for the propagator. We need to find:

Γ(3) =

−i∂ 3 F(L3 )
n
∂Alα ∂Am
β ∂Aγ

(E.24)

The array of (l, m, n) has 3! = 6 permutations, but due to fabc being antisymmetric,
we can reduce the number of terms that need to be consider 3!
2 = 3 cases. In deriving the
Feynman rule, we will go through each of the three cases to determine it. Also, the gluons

179

will have the momenta p, k, and q, as seen in the diagram of the vertex below.
1. Consider (l, m, n) → (a, b, c):
The Fourier transform brings down the momentum associated with the gluon that has
the derivative acting on it. So in this case we get factors of (ip).




 −i






−i
gs fabc δmb g βµ δnc g γν δal gαν ipµ − gµα δal (ipν ) =
gs flmn ipβ gαγ − ipγ gαβ
2
2

2. Consider (l, m, n) → (b, c, a):
Similar to above, but here the momentum factor is (ik).




 −i




−i
gs fabc δlb g αµ δmc g βν δan gγν ikµ − gµγ δan (ikν ) =
gs fnlm (ikα ) gβγ − ikβ gαγ
2
2

Note: fnlm = −flnm = flmn
3. Consider (l, m, n) → (c, a, b):
Here the momentum factor is (iq)




 −i




−i
gs fabc δnb g γµ δlc g αν δan gβν iqµ − gµβ δan (iqν ) =
gs fmnl iqγ gαβ − (ipα ) gγβ
2
2

Putting the three terms together along with manipulating the structure constants to
match, one obtains the Feynman rule for the 3 point function as:

h
i
= gs flmn (q − k)α δβγ + (k − p)β δγα + (p − q)γ δαβ

180

(E.25)

Quartic Gluon Coupling
The last piece that includes on the gauge field are terms that contain quartic terms in the
fields. There is one such term shown below:

1
L4 = − gs2 fabc fade Aµ,b Aν,c Adµ Aeν
4

(E.26)

Therefore, following the same approach in the sections above, we will need to calculate
the following term in order to get the Feynman rule:

Γ(4) =

i∂ 4 F(L4 )
n
∂Akα ∂Alβ ∂Am
γ ∂Aσ

(E.27)

Here we can see that there will be 4! = 24 different permutations that can occur for the
indices (k, l, m, n). Again, due to the properties of the structure function we are able to
relate (k, l, m, n) to (k, l, n, m), (k, m, l, n), and (k, n, m, l).
However, to illustrate the relationship, we will still show the work for all 24 permutations.
1.

• Consider (k, l, m, n) → (b, c, d, e)

i
i
− gs2 fabc fade δbk g µα δcl g νβ δdm gµγ δen gνσ = − gs2 fakl famn gαγ gβσ
4
4

• Consider (k, l, m, n) → (b, c, d, e)

i
i
− gs2 fabc fade δbk g µα δcl g νβ δdn gµσ δem gνγ = − gs2 fakl fanm gασ gβγ
4
4

181

• Consider (k, m, l, n) → (b, c, d, e)

i
i
− gs2 fabc fade δbk g µα δcm g νγ δdl gµβ δen gνσ = − gs2 fakm faln gαβ gγσ
4
4

• Consider (k, m, n, l) → (b, c, d, e)

i
i
− gs2 fabc fade δbk g µα δcm g νγ δdn gµσ δel gνβ = − gs2 fakm faln gασ gβγ
4
4

• Consider (k, n, m, l) → (b, c, d, e)

i
i
− gs2 fabc fade δbk g µα δcn g νσ δdm gµγ δel gνβ = − gs2 fakn faml gαγ gβσ
4
4

• Consider (k, n, l, m) → (b, c, d, e)

i
i
− gs2 fabc fade δbk g µα δcn g νσ δdl gµβ δem gνγ = − gs2 fakn falm gαβ gγσ
4
4

2. From here out, we will just write the result for each term. This can be done by
recognizing the pattern of the terms above.
• Consider (l, m, n, k) → (b, c, d, e)

i
− gs2 falm fank gβσ gγα
4

• Consider (l, m, k, n) → (b, c, d, e)

i
− gs2 falm fakn gαβ gγσ
4
182

• Consider (l, n, m, k) → (b, c, d, e)

i
− gs2 faln famk gβγ gασ
4

• Consider (l, k, n, m) → (b, c, d, e)

i
− gs2 falk fanm gβσ gαγ
4

• Consider (l, n, k, m) → (b, c, d, e)

i
− gs2 faln fakm gαβ gγσ
4

• Consider (l, k, m, n) → (b, c, d, e)

i
− gs2 falk famn gβγ gασ
4

3.

• Consider (m, n, k, l) → (b, c, d, e)

i
− gs2 famn fakl gαγ gβσ
4

• Consider (m, n, l, k) → (b, c, d, e)

i
− gs2 famn falk gβγ gασ
4

183

• Consider (m, k, n, l) → (b, c, d, e)

i
− gs2 famk fanl gγσ gαβ
4

• Consider (m, l, k, n) → (b, c, d, e)

i
− gs2 faml fakn gαγ gβσ
4

• Consider (m, k, l, n) → (b, c, d, e)

i
− gs2 famk faln gβγ gασ
4

• Consider (m, l, n, k) → (b, c, d, e)

i
− gs2 faml fank gγσ gαβ
4

4.

• Consider (n, k, l, m) → (b, c, d, e)

i
− gs2 fank falm gβσ gαγ
4

• Consider (n, k, m, l) → (b, c, d, e)

i
− gs2 fank faml gγσ gαβ
4

184

• Consider (n, l, k, m) → (b, c, d, e)

i
− gs2 fanl fakm gασ gβγ
4

• Consider (n, m, l, k) → (b, c, d, e)

i
− gs2 fanm falk gαγ gβσ
4

• Consider (n, l, m, k) → (b, c, d, e)

i
− gs2 fanl famk gγσ gαβ
4

• Consider (n, m, k, l) → (b, c, d, e)

i
− gs2 fanm fakl gασ gβγ
4

Combining the above 24 terms together, and simplifing the expressions, we get the following for the Feynman Rule:

=

−igs2 [fakl famn gαγ gβσ − gασ gβγ




+fakm faln gαβ gσγ − gασ gβγ






+fakn falm gαβ gσγ − gασ gβγ ]

185

(E.28)

Fermion Propagator
Now we will consider terms in the Lagrangian containing fermions. The first term that we
will calculate is the fermion propagator. Unlike the gluon propagator, the fermion propagator
does not have the issue of being zero due to gague invariance. It is therefore, a straightforward calculation. The term needed in the Lagrangian is:



Lf = ψ̄ i∂/ − m ψ

(E.29)

The propagator is found by the same method as above. In other words, we need to
calculate:

P=

∂ 2 F Lf
−i
∂ ψ̄∂ψ


 !−1
(E.30)

Preforming this calculation, one obtains:


−1
P = i p/ − m
=

i(p/ + m)
i
= 2
p/ − m
p − m2

(E.31)

Where to get to the last equation, we multiplied the top and the bottom by p/ + m to
bring it to the typical form. Thus, the Feynman rule for the Fermion propagator is given by:

i(p/ + m)
= 2
p − m2

186

(E.32)

Fermion-Gluon Vertex
The remaining term that has not been accounted for in the Lagrangian thus far is the
interaction term between gluons and fermions. This term is given by:

Lint = gs T a Aaµ ψ̄γ µ ψ = gs Aaµ ψ̄i γ µ (T a )ij ψj

(E.33)

Where in the second equation we wrote in the indices corresponding to the color of the
fermions. The index a goes from 1 to 8, where the indicies i, j can be 1,2, or 3. Again, we
preform the same procedure as before:

Γint =

i∂ 3 F(Lint )
∂ ψ̄∂ψ∂Aaµ

(E.34)

This gives us the Feynman rule, which is shown below:

= igs γ¾ (T a )ij

(E.35)

Note: The order of (ij) is important because T a is not a symmetric matrix.

Fadeev-Popov Ghost Fields
Arising from an artifact of insiting on Lorentz invariance and unitarity for massless spin-1
particles are unphysical particles known as Fadeev-Popov ghosts. These also arise in QED,
but since the theory is Abelian, they do not couple to the photon. Here in the non-Abelian
theory they do couple to the gluon, but only appear at higher orders, since they are non-

187

physical. There are ways to derive the Feynman rules to not have any ghost fields knwon as
Axial gauges and a discussion on these can be found in [236].
The gauge fixing term can be expressed as:

1 a 2
(c )
2Îą

(E.36)

LF P = χ̄a Mab χb

(E.37)

Lgf = −

The ghost Lagrangian is defined as:

a
where Mab = δc b . We can see that there will be one ghost and one anti-ghost for each of
δΛ

the gluons. Therefore, there will be 8 ghost fields and 8 anti-ghost fields. Let us examine
how the ca fields change under a local gauge transformation.

ca = ∂ µ Aaµ → ∂ µ (Aaµ + δAaµ ) = ca + δca

(E.38)

To find the equation for Mab we will need to calculate the δC a term in the above equation.
The derivation is shown in the equations below.

δC a =∂ µ (δAaµ ) = ∂ µ (−∂µ Λa + gs fabc Λb Acµ )
= − ∂ 2 Λa + gs fabc ∂ µ (Λb Acµ )
h

i
= −∂ 2 δ ab + gs fabc Acµ ∂ µ + ∂ µ Acµ Λb
⇒Mab =

δca
2δ + g f
c ∂ µ + ∂ µ Ac 

=
−∂
A
s
ab
abc
Âľ
Âľ
δΛb

188

(E.39)

(E.40)

Plugging the above result into Eq. E.37, we get the resulting ghost Lagrangain.

h

i
LF P =χ̄a −∂ 2 δab + gs fabc Acµ ∂ µ + ∂ µ Acµ χb


= ∂µ χ̄a (∂ µ χa ) − gs fabc (∂ µ χ̄a ) Acµ χb

(E.41)

Now we will begin to calculate the Feynman rules for the ghost fields. We begin with the
calculation of the propagator.

P=

−i∂ 2 F(−χ̄a ∂ 2 δab χb )
∂ χ̄a ∂χb

!−1

1
= −i 2 δab
k

(E.42)
(E.43)

The only remaining Feynman rule for QCD to calculate is the ghost-gluon interaction
term. This term is given by the same procedure as above, giving:

= gs fabc pÂľ

(E.44)

It is important to note that the anti-ghost fields are not the anti-particles of the ghost
fields.

189

QCD β Functions
The β functions in QCD arise from the calculation of the renormalization of the strong
coupling constant. They have been calculated up to four-loops [237, 238]. The results up to
that order are listed below:

11
4
CA − TF nF ,
(E.45)
3
3
34 2
20
= CA
− CA TF nF − 4CF TF nF ,
(E.46)
3
3
2857 3
1415 2
158
44
=
CA −
CA TF nF +
CA TF2 n2F + CF TF2 n2F
54
27
27
9
205
−
C C T n + 2CF2 TF nF ,
(E.47)
9 A FF F



17152 448
4204 352
424
2
2
2
=CA CF TF nF
+
ζ3 + CA CF TF nF −
+
Îś3 +
C T 3 n3
243
9
27
9
243 A F F




7073 656
7930 224
1232
2
2
2
2
+ CA CF TF nF
−
Îś3 + CA TF nF
+
Îś3 +
CF TF3 n3F
243
9
81
9
243






39143 136
150653 44
1352 704
3
4
2
2
2
CA TF nF −
+
Îś + CA
− ζ3 + CF TF nF
−
Îś
81
3 3
486
9
27
9 3




dabcd
dabcd
dabcd
dabcd
512 1664
704 512
3
2
F
A
F
F
+ 46CF TF nF + nF
−
Îś + nF
−
+
Îś
NA
9
3 3
NA
9
3 3


dabcd
dabcd
80 704
A
A
+
− +
Îś ,
(E.48)
NA
9
3 3

β0 =
β1
β2

β3

where the coefficients in the β equations can be expressed in the specific values for an SU (N )
group are:

1
TF = ,
2

CF =

N2 − 1
,
2N

abcdA
dabcd
N (N 2 + 6)
F d
=
,
NA
48

CA = N,

abcdF
dabcd
N 4 − 6N 2 + 18
F d
=
,
NA
96N 2

abcdA
dabcd
N 2 (N 2 + 36)
A d
=
,
NA
24

190

NA = N 2 − 1.

(E.49)

APPENDIX F
Calculation Details

Plus Functions
The Plus function is a distribution to handle singularities of functions that occur for x = 1.
Given the function F (x) which is singular for x = 1, the plus function is defined as:
!

Z 1−β
F (x)+ ≡ lim

β→0

F (x)Θ(1 − x − β) − δ(1 − x − β)

dyF (y) .

(F.1)

0

When a plus function is convoluted with a test function, the final result is well defined. For
example, given a test function G(x), the convolution is given by:
Z 1

Z 1
dxF (x) (G(x) − G(1)) .

dxF (x)+ G(x) =
0

(F.2)

0

Additionally, a property of the plus function is:
Z 1
dxF (x)+ = 0,
0

191

(F.3)

which can be easily derived with a choice of G(x) = 1. Finally, it is possible to obtain the
result for a lower bound that is non-zero by:
Z 1

Z 1
dxF (x) (G(x) − G(1)) + G(1)

dxF (x)+ G(x) =
a

Z a

a

F (x).

(F.4)

0

NLO Real Corrections to Drell-Yan
Similar to the quark-anti-quark channel, the singular terms for the (anti-)quark-gluon channel are given by:



|M |2gq



 2  






Q
2 + g2
2 + f 2 K L − g2 − g2
2 − f 2 K (2A ) ,
= 64
gL
f
f
3 0
4
4
R
L
R
L
R
L
R
2
sing
(F.5)

with


1  2
K3 = 2 zA
+ (1 − zA )2 δ (1 − zB )
qT
1
−  2 δ (1 − zB ) zA (1 − ZA ) + zA ↔ zB ,
qT

1  2
K4 = 2 zA
+ (1 − zA )2 δ (1 − zB ) + zA ↔ zB .
qT

(F.6)
(F.7)

Loop Integrals
Here a brief discuss of loop integrals is given, along with the calculation of the loop integral
involved in calculating the NLO correction to Drell-Yan. For additional discussions of loop
integrals, and the more modern approach that has been developed to tackle the calculations
of higher number of loops can be found in [239, 240, 241].

192

p1
q

p1 + k
q
Îł

k
p2 − k

q
p2

Figure F.1: The hadronic virtual correction Feynman Diagram to Drell-Yan at NLO.
For Drell-Yan at NLO accuracy, there is only one virtual diagram to calculate if dimensional regularization is used. The hadronic component of the calculation is given the
following Feynman diagram in Fig. F.1. Since this diagram is O (Îąs ), the matrix element


squared will be of O Îąs2 . Since this is a higher order correction, only the cross term of the
virtual diagram and the tree level diagram are used up to NLO. The result is obtained from


†
2Re Mborn Mvirt . Using the Feynman rules, the virtual correction is given as:

2Re



†
Mborn Mvirt



dd k

Z
=
CF

(2π)d
igιβ
k2



Tr p/2

γι

i



p/2 − k/

i
2

(p2 − k) (p1 + k)2



γ¾



p/1 + k/



γβ p

gµν (2π) gs2 Q2 e2 ,

/1

γν



(F.8)

where CF is the color factor, gs is the strong coupling constant, Q is the charge of the quark,
and e is the electric charge. Consider the integral:

Z



 


Tr p/2 γ α p/2 − k/ γ µ p/1 + k/ γ β p/1 γ ν
.
k 2 (p2 − k)2 (p1 + k)2
(2π)d
dd k

193

(F.9)

Using the Feynman parameter trick1 , gives:

Z
I=



 


Tr p/2 γ α p/2 − k/ γ µ p/1 + k/ γ β p/1 γ ν
dxdy
.
(2π)d k 2 (1 − x − y) + (p2 − k)2 x + (p1 + k)2 y
dd k

(F.10)

The denominator of the above equation can be simplified, using the substitution: lÂľ =
Âľ
Âľ
k µ − p2 x + p1 y, and the definition q 2 = 2p1 · p2 , giving:

D = l2 + q 2 xy.

(F.11)

Now consider the numerator of Eq. F.10, the numerator can be simplified, using the substitution for l given above, to:

h
i
N = −4q 4 (1 − y − x) (1 − ) + xy − 22 xy
+8

1 − 2 2 2
q l − 8 (1 − ) q 2 l2 .
2 (2 − )

(F.12)

Combining the numerator and the denominator and putting the integrals back, gives three
unique integrals to calculate. The first integral is:
Z 1

dd l 1 − y − x


(2π)d l2 + q 2 xy 3
0
0




−i
4π  1
1
1
2
=
+ + 4 + O () ,
2 Γ (1 − ) 2 
16π 2 q 2 −q 2

I1 =

Z 1−x

dx

Z

dy

1 Details on the Feynman parameter trick can be found in Appendix A.4 of [42]

194

(F.13)

the second is:
Z 1

dd l
l2


(2π)d l2 + q 2 xy 3
0
0



 
−i 2 −  4π 
1
1
3 7
=
+ +
+ O 2
,
2
2
2
Γ (1 − ) 2 2
2
16π
−q

I2 =

Z 1−x

dx

Z

dy

(F.14)

and finally, the third is:
Z 1

dd l
xy


d
(2π) l2 + q 2 xy 3
0
0

 
 
−i 1 4π  1 3
=
+
+ O 2
.
2
2
2
2
2
16π q 2 −q

I3 =

Z 1−x

dx

Z

dy

(F.15)

Plugging all of the results back into Eq. F.10, gives:
iq 2
I=
16π 2






4π 
1
−2 3
2
− − 8 + π + O () .
Γ (1 − ) 2

q2

(F.16)

Finally, substituting the result of the integral back into Eq. F.8, gives the desired result
found in Eq. 3.18.

Phase Space In D-Dimensions
When working outside of d = 4−2 dimensions, the integrals over the phase space need to be
modified. The major calculation that is needed, is the area of a unit sphere in d dimensions.

195

This can be calculated by using the following trick:
√ 
d
π =

Z

dxe−x

2



d

Z

dd x exp −

=

d
X


x2i 

i=1

Z ∞

Z
=

Solving for

R

dÎŁd

2
dxxd−1 e−x

Z
=

0


dÎŁd

1
Γ
2

 
d
.
2

dÎŁd gives:
Z
dÎŁd =

2π d/2
.
Γ (d/2)

(F.17)

Using Eq. F.17 reproduces the expected results for the familiar integer dimensions as seen
in Table F.1.
d
1
2
3
4

Γ (d/2)
√
π
√1
π/2
1

R

dÎŁd
2
2π
4π
2π 2

Table F.1: Table of the results of calculating the area of a d-Sphere

Resummation Coefficients to N3LL
The coefficients up to NNLL can be found in Section 3.4.1 and 3.4.2 for the CSS and CFG
formalisms respectively. For the additional terms that appear at N3 LL, they can be found
in [167, 168], and are reproduced here for ease.


The B anomalous dimension in CSS at O Îąs3 is given as:

B3DY

=

Îł2DY

− γ2r

+ β1 cDY
1



1  DY 2
DY
+ 2β0 c2 −
c
,
2 1

196

(F.18)

substituting the numbers into the equation above, the numerical result is given as:

B3DY = 114.98 − 11.27nf + 0.32n2f ,

(F.19)

where nf is the number of active flavors. Note the above equation differs from that in [167],
Îąs
s
since the expansion in [167] is for Îą
4π , while this work uses π .

The hard-collinear coefficient at the NNLO for vector boson production is given by five
different initial states: q q̄, q q̄ 0 , qq, qq 0 , and qg. These coefficients are given as:

(2)
2Cqq̄ (z) + δ (1 − z)



DY (2)
Hq



3  DY (1) 2 CF  2
DY (1)
−
Hq
+
π − 8 Hq
4
4

1
DY (1)
+ CF Hq
(1 − z)
2
"
#

2 

CF2
π2 − 8
DY (2)
= Hqq̄←qq̄ (z) −
δ (1 − z)
+ π 2 − 10 (1 − z) − (1 + z) ln z ,
4
4
(F.20)
1 DY (1)
(2)
DY (2)
Cqg (z) + Hq
z (1 − z) = Hqq̄←qg (z)
4 


  π2
CF
1
2
−
z ln z +
1−z +
− 4 z (1 − z) ,
4
2
4
(2)

DY (2)

(2)

DY (2)

Cqq (z) = Hqq̄←qq (z) ,

(F.21)
(F.22)

C 0 (z) = H
(z) ,
qq
q q̄←qq 0

(F.23)

(2)

C 0 (z) = Hqq̄←qq̄0 ,
q q̄

(F.24)

197

where the H coefficients are given by:

DY (2)
Hqq̄←qq̄ (z)





7Îś3 101
1
59ζ3 1535 215π 2
π4
= CA CF
−
+
−
+
−
δ(1 − z)
2
27
1−z +
18
192
216
240

1 + z2
Li (1 − z)
Li (z) log(z) 1
+
− 3
+ Li3 (z) − 2
− Li2 (z) log(1 − z)
1−z
2
2
2

1
1
1 2
π2
3
2
−
log (z) − log (1 − z) log(z) + π log(1 − z) −
24
2
12
8





1
1
1
+
−
11 − 3z 2 ζ3 −
−z 2 + 12z + 11 log2 (z)
1−z
4
48


1  2
π2z
−
83z − 36z + 29 log(z) +
36
4



Li2 (z) 1
z + 100 1
+ (1 − z)
+ log(1 − z) log(z) +
+ z log(1 − z)
2
2
27
4
 



14
1
1
+ C F nF
+
192ζ3 + 1143 − 152π 2 δ(1 − z)
27 1 − z + 864



1 + z2
1
+
log(z)(3 log(z) + 10) +
(−19z − 37)
72(1 − z)
108
 

1
511 67π 2 17π 4
2
+ CF
−15ζ3 +
−
+
δ(1 − z)
4
16
12
45

1 + z 2 Li3 (1 − z) 5Li3 (z) 1
3Li2 (z) log(z)
+
−
+ Li2 (z) log(1 − z) +
1−z
2
2
2
2

3
1
1 2
5Îś3
2
2
+ log(z) log (1 − z) + log (z) log(1 − z) − π log(1 − z) +
4
4
12
2


3
2π 2 29
1
+ (1 − z) −Li2 (z) − log(1 − z) log(z) +
−
+
(1 + z) log3 (z)
2
3
4
24
 



1
1
1 2
2
2
+
−2z + 2z + 3 log (z) +
17z − 13z + 4 log(z)
1−z 8
4

z
− log(1 − z)
4



  Li (z) 1
1
π2
2
2
+ CF
(1 − z) 2z − z + 2
+ log(1 − z) log(z) −
z
6
6
36




1
1
+
(1 − z) 136z 2 − 143z + 172 −
8z 2 + 3z + 3 log2 (z)
216z
48


1  2
1
3
+
32z − 30z + 21 log(z) + (1 + z) log (z) ,
(F.25)
36
24


198

DY (2)
H
(z)
q q̄←q q̄ 0

DY (2)
Hqq̄←qq (z)




1
π2
2
= CF
(1 − z) 2z − z + 2
Li2 (z) + log(1 − z) log(z) −
12z
6


1
1
+
(1 − z) 136z 2 − 143z + 172 + (1 + z) log3 (z)
432z
48





1
1
2
2
2
−
8z + 3z + 3 log (z) +
32z − 30z + 21 log(z) ,
96
72



= CF

1
CF − CA
2

(F.26)




1 + z 2 3Li3 (−z)
1
Li (−z) log(z)
+ Li3 (z) + Li3
− 2
1+z
2
1+z
2



Li2 (z) log(z)
1
1
1
−
log3 (z) − log3 (1 + z) + log(1 + z) log2 (z)
2
24 
6 
4

2
π
3Îś3
Li2 (z) 1
15
+
log(1 + z) −
+ (1 − z)
+ log(1 − z) log(z) +
12
4
2
2
8


 π2
1
1
− (1 + z) Li2 (−z) + log(z) log(1 + z) + (z − 3) + (11z + 3) log(z)
2
24
8





1
π2
2
+ CF
(1 − z) 2z − z + 2
Li2 (z) + log(1 − z) log(z) −
12z
6



1
1  2
+
(1 − z) 136z 2 − 143z + 172 −
8z + 3z + 3 log2 (z)
432z
96



1
1
2
3
+
32z − 30z + 21 log(z) + (1 + z) log (z) ,
(F.27)
72
48
−

DY (2)
(z)
q q̄←qq 0

H

DY (2)
(z)
q q̄←q q̄ 0

=H

199

,

(F.28)

DY (2)
Hqq̄←qg (z)





1
(1 − z) 11z 2 − z + 2 Li2 (1 − z)
12z


  Li (1 − z) 1
1
3
2
3
+ 2z − 2z + 1
− Li2 (1 − z) log(1 − z) +
log (1 − z)
8
8
48


1

  3Li (−z) Li3 1+z
Li (−z) log(z)
1
3
+ 2z 2 + 2z + 1
+
− 2
−
log3 (1 + z)
8
4
8
24

1
1 2
1
2
+
log (z) log(1 + z) + π log(1 + z) + z(1 + z)Li2 (−z) + zLi3 (z)
16
48
4


1
3
149z 2
2
− zLi2 (1 − z) log(z) − zLi2 (z) log(z) −
2z + 1 ζ3 −
2
8
216



1  2
1
−
44z − 12z + 3 log2 (z) +
68z 2 + 6π 2 z − 30z + 21 log(z)
96
72
2
π z 43z
43
1
1
+
+
+
+ (2z + 1) log3 (z) − z log(1 − z) log2 (z)
24
48
108z 48
2
1
1
− (1 − z)z log2 (1 − z) + z(1 + z) log(1 + z) log(z)
8
4 
1
35
+ (3 − 4z)z log(1 − z) −
16
48


Li (1 − z) Li3 (z) 1
+ CF
2z 2 − 2z + 1
ζ3 − 3
−
+ Li2 (1 − z) log(1 − z)
8
8
8

= CA

+
+
−
+
−

−

Li2 (z) log(z)
1
1
−
log3 (1 − z) +
log(z) log2 (1 − z)
8
48 
16
1
log2 (z) log(1 − z)
16


3z 2
1  2
1 
3
2
−
4z − 2z + 1 log (z) +
−8z + 12z + 1 log2 (z)
8
96
64

1 
5
11z 1
−8z 2 + 23z + 8 log(z) + π 2 (1 − z)z +
+ (1 − z)z log2 (1 − z)
32
24
32
8 
1
1
9
(1 − z)z log(1 − z) log(z) − (3 − 4z)z log(1 − z) −
,
(F.29)
4
16
32

DY (2)
Hqq̄←gg (z)

z
=−
2



1
1 − z + (1 + z) log(z)
2

200


,

(F.30)

where Lik (z) (k = 2, 3) are the polylogarithm functions,
Z z
dt
Li2 (z) = −
ln(1 − t) ,
0 t

Z 1
dt
Li3 (z) =
ln(t) ln(1 − zt) ,
0 t

(F.31)

and the H factors are the scheme dependent resummation factors. For CSS, H is 1 to all
orders, while for CFG, H has Îąs dependence.

Fit to SIDIS Data
The universality of the parton distribution is a powerful prediction from QCD factorization. According to the TMD factorization, the universality of the TMD parton distributions
should exist between SIDIS and Drell-Yan processes as well. Therefore, the non-perturbative
functions determined for the TMD parton distributions from the Drell-Yan type of processes
shall apply to that in the SIDIS. Of course, the transverse momentum distribution of hadron
production in DIS processes also depends on the final state TMD fragmentation functions,
which need to be determined by fitting to existing experimental data. Following the universality arguments, the following parameterizations for the non-perturbative form factors for
SIDIS process can be assumed, in contrast to Eq. 3.77 for Drell-Yan process,

(DIS)

SN P

=

g1 2
b + g2 ln (b/b∗ ) ln(Q/Q0 ) + g3 b2 (x0 /xB )λ +
2

gh 2
b .
zh2

(F.32)

In the above parameterization, g1 and g2 have been determined from the experimental data of
Drell-Yan lepton pair production. The factor of 1/2 in front of the g1 term is due the fact that
there is only one incoming hadron in the SIDIS process, while there are two incoming hadrons
in the Drell-Yan process. Although there has been evidence from recent studies [242, 243] that
201

gh could be different for the so-called favored and dis-favored fragmentation functions, them
will be taken to be the same in this study for simplicity. When more precise data become
available, a global analysis with two separate gh parameters may need to be preformed.
In principle, g1 , g2 , and gh may be fitted to both Drell-Yan and SIDIS data simultaneously. However, the SIDIS data from HERMES and COMPASS mainly focus in the relative
low Q2 range. Because of that, the theoretical uncertainty of the CSS prediction is not well
under control, particularly, from the Y -term contribution. There have been several successful
phenomenological studies to describe the experimental data from HERMES and COMPASS
experiments, using the leading order TMD formalism [242, 244]. The goal of this paper
is to check if the non-perturbative form factors determined in the Drell-Yan process can be
applied to the SIDIS processes. As shown in Ref. [171], fitting both is not possible with
the original BLNY or KN fit, where it was found that the extrapolation of these fits to
the kinematic region of HERMES and COMPASS is in conflict with the experimental data.
However, it will be shown that the SIYY form will be able to extend to SIDIS experiments
from HERMES and COMPASS Collaborations.
Therefore, in the following, the parameters (g1,2 ) are set to the fitted values from the
Drell-Yan data and compared to the SIDIS data for consistency. In Fig. F.2, the comparisons
between the theory predictions with gh = 0.042 and the SIDIS data from HERMES is
given, with a total χ2 around 180. This parameter is consistent with previous analysis
when the leading order TMD formalism is considered [242, 244]. It is also consistent with
the TMD formalism with truncated evolution effects in Ref. [171]. The differential cross
section for the SIDIS process depends on the hadron fragmentation functions, for which
the parameterization from the new DSS fit [245, 246] is used here. A normalization factor
about 2.0 in the calculation of the multiplicity distributions shown in Fig. F.2 is used, which
202

Multiplicity

Multiplicity

4.5

9
8

4

7

3.5

6

3

5
4

0.2 < zh < 0.3

2.5

Q2=3.14GeV2

2

3
2

1.5

HERMES (e+p -> π++X)

HERMES (e+p -> π++X)
t

Multiplicity

t

Multiplicity

Q2=3.14GeV2

1
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
p (GeV)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
p (GeV)

1.6

0.3 < zh < 0.4

7

6

1.4
5
1.2
4
1

0.4 < zh < 0.6

0.8

Q2=3.14GeV2

0.6

HERMES (e+p -> π++X)

3

Q2=3.14GeV2
2

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
p (GeV)

HERMES (e+p -> π-+X)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
p (GeV)
t

Multiplicity

t

Multiplicity

0.2 < zh < 0.3

1.2

3

1
2.5
0.8
2

1.5

0.6

0.3 < zh < 0.4

0.4 < zh < 0.6

Q2=3.14GeV2
1

0.4

HERMES (e+p -> π-+X)

Q2=3.14GeV2
HERMES (e+p -> π-+X)

0.2
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
p (GeV)

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
p (GeV)
t

t

Figure F.2: Multiplicity distribution as function of transverse momentum in semi-inclusive
hadron production in deep inelastic scattering compared to the experimental data from
HERMES Collaboration at Q2 = 3.14GeV2 .

203

accounts for theoretical uncertainties from higher order corrections for both differential and
inclusive cross sections 2 . Here, the Y -term contribution is not included, for the reasons
mentioned above.
The figures in Sec. 3.4.4 and Fig.F.2 clearly illustrate that the SIYY non-perturbative
function is a universal non-perturbative TMD function which can be used to describe both
Drell-Yan lepton pair production and semi-inclusive hadron production in DIS processes in
the CSS resummation framework. Also, note that that the new functional form for the
non-perturbative function is crucial to achieve this conclusion as given in Eqs. (3.77) and
(F.32).

Îąs Expansion
The Taylor series expansion of Îąs around some fixed scale Âľ0 is given by:
Îą (Âľ) 
Îą (Âľ) 
∂ sπ 
Îąs (Âľ)
Îąs (Âľ0 )
µ2 1 ∂ 2 sπ
=
+
log 2 +
log2

2 

2
π
π
∂ log µ µ=µ
µ0 2 ∂ log µ2 µ=µ0
0

 
Âľ2
+ O Îąs3 . (F.33)
2
Âľ0

To obtain the final result, we need to calculate the derivatives. However, the first derivative
is exactly related to the β function for the running of the coupling,
Îą (Âľ)

∂ sπ
= β (ιs (¾)) ,
∂ log µ2

(F.34)

2 Compared to the leading order TMD fit of Ref. [244] where there is no normalization factor, the C (1)
coefficient is large and negative in the CSS resummation application to the SIDIS. Phenomenologically, that
is the reason to include a factor of 2 in the comparison to the SIDIS data. This could be improved if the
differential cross section (instead of multiplicity distributions) are measured in the future.

204

with
β (αs (µ)) = −


∞ 
X
Îąs (Âľ) k+1
k=1

π

βk−1 .

(F.35)

For the second derivative,


Îą (Âľ)
∞
X
∂ 2 sπ
∂β (αs (µ))
Îąs (Âľ) k
β (αs (µ)) = −β (αs (µ))
(k + 1)
βk−1 .

2 =
Îą (Âľ)
π
∂ log µ2
∂ sπ
k=1

(F.36)



Inputing these results into Eq. F.33, and keeping terms only up to O Îąs3 , the expansion of
Îąs is:

Îąs (Âľ)
Îąs (Âľ0 )
=
π
π

Îąs (Âľ0 )
Âľ2
1−
β0 log 2 +
π
Âľ0




Îąs (Âľ0 ) 2
π

Âľ2
Âľ2
β02 log2 2 − β1 log 2
Âľ0
Âľ0

!



+ O Îąs3



!

(F.37)

PDF Expansion
The Taylor series expansion of the parton distribution function around some fixed scale ÂľF
is given by:

∂fi (x, µ) 
Âľ2
fi (x, Âľ) = fi (x, ÂľF ) +
log
∂ log µ2 µ=µ
Âľ20
F


 
2
1 ∂ 2 fi (x, µ) 
Âľ2
1 ∂ 3 fi (x, µ) 
2
3 Âľ + O Îą4 .
+
log
+
log




s
2 ∂ log µ2 2 µ=µ
6 ∂ log µ2 3 µ=µ
Âľ2F
Âľ2F
F
F

205

(F.38)

.

To obtain the final result, the derivatives need to be calculated. However, the first derivative
is exactly related to the DGLAP evolution equations,

∂fi (x, µ)
Îąs (Âľ) (1)
=
P ⊗ fi (x, µ)
2
2π
∂ log µ




 
Îąs (Âľ) 2 (2)
Îąs (Âľ) 3 (3)
+
P ⊗ fi (x, µ) +
P ⊗ fi (x, µ) + O αs4 ,
2π
2π

(F.39)

When calculating the second derivative, it is also necessary to expand Îąs (Âľ) around ÂľR


as shown in the previous section. Only up to O Îąs3 will be keep to keep the equations
short. The second derivative is given as,

 Îą 2  1
∂ 2 fi (x, µ)
1
s
(1)
(1)
(1)
P ⊗ P ⊗ f (µ) − β0 P ⊗ f (µ)

2 =
π
4
2
∂ log µ2
 Îą 3 1
1 (2)
Âľ2 (1)
s
(1)
(2)
(1)
2
+
P ⊗ P ⊗ f (µ) + P ⊗ P ⊗ f (µ) + β0 ln 2 P ⊗ f (µ)
π
8
8
ÂľR
!
 
1
Âľ2 (1)
1
1
− β0 ln 2 P ⊗ P (1) ⊗ f (µ) − β1 P (1) ⊗ f (µ) − β0 P (2) ⊗ f (µ) + O αs4 ,
2
2
2
ÂľR
(F.40)
and the third derivative is given as,
 Îą 3 
∂ 3 fi (x, µ)
3
s
β02 P (1) ⊗ f (µ) − β0 P (1) ⊗ P (1) ⊗ f (µ)

3 =
π
4
∂ log µ2

 
1 (1)
(1)
(1)
+ P ⊗ P ⊗ P ⊗ f (µ) + O αs4 .
8

(F.41)



Inputing these results into Eq. F.38, and keeping terms only up to O Îąs3 , the expansion of

206

the PDF is:
1 Îąs (ÂľR ) Âľ2 (1)
ln 2 P ⊗ f (µF )
2
π
ÂľF

2
Âľ2F (1)
Îąs (ÂľR )
1
Âľ2
1
Âľ2
+
− β0 ln 2 ln 2 P ⊗ f (µF ) − β0 ln2 2 P (1) ⊗ f (µF )
π
2
4
ÂľF
ÂľR
Âľ
! F
1
Âľ2
1 Âľ2
+ ln2 2 P (1) ⊗ P (1) ⊗ f (µF ) + ln 2 P (2) ⊗ f (µF )
8
4 ÂľF
ÂľF

3
Îąs (ÂľR )
1 2 Âľ2 (1)
1 2 Âľ2 (2)
+
ln 2 P ⊗ P (2) ⊗ f (µF ) +
ln 2 P ⊗ P (1) ⊗ f (µF )
π
16
16
ÂľF
ÂľF

f (Âľ) = f (ÂľF ) +


 1 2 Âľ2 2 Âľ2F (1)
Âľ2F (1)
Âľ2
ln
P
⊗
f
Âľ
P ⊗ f (µF )
f + β0 ln 2 ln
2
Âľ2F
Âľ2R
ÂľF
Âľ2R
Âľ2F (1)
Âľ2F (1)
1
Âľ2
1
Âľ2
2
(1)
− β0 ln 2 ln 2 P ⊗ P ⊗ f (µF ) − β1 ln 2 ln 2 P ⊗ f (µF )
4
2
ÂľF
ÂľR
ÂľF
ÂľR
1
+ β02 ln2
2

1
+ β02 ln3
6

Âľ2 (1)
1
P ⊗ f (µF ) − β0 ln3
2
8
ÂľF

Âľ2 (1)
1
P ⊗ P (1) ⊗ f (µF ) − β1 ln2
2
4
ÂľF

Âľ2 (1)
P ⊗ f (µF )
Âľ2F

2
2
Âľ2 (1)
(1) ⊗ P (1) ⊗ f (µ ) − 1 β ln µ ln µF P (2) ⊗ f (µ )
P
⊗
P
F
F
2 0 Âľ2F
Âľ2F
Âľ2R
!
 
2
2
1
Âľ
1
Âľ
− β0 ln2 2 P (2) ⊗ f (µF ) + ln 2 P (3) ⊗ f (µF ) + O αs4 .
4
8 ÂľF
ÂľF

+

1 3
ln
48

207

(F.42)



O Îąs3 Asymptotic Piece


Continuing on to O Îąs3 , the calculations become extremely complex. The asymptotic piece
can be given by:

1  (1) 3
=
A
fi fj ,
4



2 5
5
5
5 2(1)
(i,j)
(1)
(1)
(1)
(1)
= A
B f i f j − β0 f i f j + f i P ⊗ f j + A
fj P ⊗ fi ,
3 C4
4
3
8
8


2 7
7
(i,j)
(1)
(1)
(1)
2
=A
B
− β0 B − A(2) + β0 fi fj + 2B (1) fj P (1) ⊗ fi − β0 fj P (1) ⊗ fi
3 C3
3
3

1
1
+ P (1) ⊗ fi P (1) ⊗ fj + fj P (1) ⊗ P (1) ⊗ fi
2
2
!
!!

2
1
Q2
Q2
(1)
(1)
(1)
− A
fj C ⊗ fi − fj P ⊗ fi log
− β0 fi fj log
+ i ↔ j,
2
Âľ2F
Âľ2R
!


3 (2)
3 (2)
Q2
(i,j)
(2)
(1)
2
(1)
2
= 2β0 A + B
β0 − A
+A
− B + β1 − 2β0 log
3 C2
2
2
Âľ2R
!!
Q2
(1)
+3β0 B log
Âľ2R


3
1  (1) 3 3  (1) 2
3
(1)
−5 A
Îś3 +
B
− β0 B
fi fj − A(2) fj P (1) ⊗ fi
2
2
2
!!
2
3
Q
+ A(1) 5β0 fj C (1) ⊗ fi + B(1) −3fj C (1) ⊗ fi − fj P (1) ⊗ fi log
2
Âľ2F
!
2
3 (1)
3
Q
− C ⊗ fi P (1) ⊗ fj − fj C (1) ⊗ P (1) ⊗ fi + β0 fj P (1) ⊗ fi log
2
2
Âľ2F
!
!
2
2
3 (1)
Q
3
Q
− P ⊗ fi P (1) ⊗ fj log
− P (1) ⊗ P (1) ⊗ fi fj log
4
4
Âľ2F
Âľ2F
!
!
2
Q
3
+3β0 fj P (1) ⊗ fi log
− fj P (2) ⊗ fi
4
Âľ2R
(i,j)
3 C5

3
3
+ β02 fj P (1) ⊗ fi − β0 P (1) ⊗ fi P (1) ⊗ fj − β0 fj P (1) ⊗ P (1) ⊗ fi
4
4
3  (1) 2
+
B
fj P (1) ⊗ fi
2 

3 (1)
3
(1)
(1)
(1)
(1)
(1)
+B
−3β0 fj P ⊗ fi + P ⊗ fi P ⊗ fj + fj P ⊗ P ⊗ fi
4
4
208

3
1
+ P (1) ⊗ P (1) ⊗ fi P (1) ⊗ fj + fj P (1) ⊗ P (1) ⊗ P (1) ⊗ fi
8
8
+ i ↔ j,
(i,j)
3 C1

−2β0 A(2) log

=

+B (1)


+ A(1)

Q2
Âľ2R

β1 − 2β02 log
2  40

!



+ A(3) − 2B (1) B (2) + 2B (2) β0 + 2 B (1)
Q2
Âľ2R

!!
+ A(1) β02 log


Q2
Âľ2R

!
− β1 log

2

Q2
Âľ2R

Q2
Âľ2R

β0 log

!

!!



β − 10B (1) ζ3 fi fj + A(2) 2fj C (1) ⊗ fi + fj P (1) ⊗ fi log
3 0
!
2
Q
1
− 4β02 fj C (1) ⊗ fi − 2β02 fj P (1) ⊗ fi log
− P (2) ⊗ fi P (1) ⊗ fj
2
2
ÂľR

Q2
Âľ2F

!!

1
− C (1) ⊗ P (1) ⊗ fi P (1) ⊗ fj − C (1) ⊗ fi P (1) ⊗ P (1) ⊗ fj
2
!
2
3 (1)
Q
1
− P ⊗ P (1) ⊗ fi P (1) ⊗ fj log
− 2B (2) fj P (1) ⊗ fi − fj P (1) ⊗ P (2) ⊗ fi
2
4
4
ÂľF
1
1
− fj P (2) ⊗ P (1) ⊗ fi − fj C (1) ⊗ P (1) ⊗ P (1) ⊗ fi
4
2
!
1
Q2
(1)
(1)
(1)
− fj P ⊗ P ⊗ P ⊗ fi log
4
Âľ2F
!!

2
Q2
(1)
(1)
(1)
+ B
−2fj C ⊗ fi − fj P ⊗ fi log
Âľ2F
+ 3β0

C (1)

⊗ fi

P (1)

1
⊗ fj + β0 P (1) ⊗ fi P (1) ⊗ fj
2

log

Q2
Âľ2F

!
+ 2 log

1
+ β0 fj P (2) ⊗ fi + 3fj C (1) ⊗ P (1) ⊗ fi + β0 fj P (1) ⊗ P (1) ⊗ fi log
2
!
Q2
(1)
(1)
+ β0 fj P ⊗ P ⊗ fi log
Âľ2R

209

Q2
Âľ2R
!

Q2
Âľ2F

!!

+ B (1)
+β0 fj

P (1)

+4β0 fj
−fj

−2C (1)

⊗ fi log

P (1)

P (1)

⊗ fi

P (1)
Q2
Âľ2F

⊗ P (1)

⊗ fi log

⊗ fi

P (1)

⊗ fj log

Q2
Âľ2F

!

!

Q2
Âľ2R

⊗ fi log

⊗ fj

− P (1)

!
− fj P (2) ⊗ fi − 2fj C (1) ⊗ P (1) ⊗ fi
Q2
Âľ2F

!

+6β0 fj C (1) ⊗ fi + β0 fj P (1) ⊗ fi

log
Q2

Q2
Âľ2
! F

!
+ 4 log

Q2

!!!

Âľ2R
Q2

1 (1)
1
P ⊗ fi P (1) ⊗ fj log2
+ fj P (1) ⊗ P (1) ⊗ fi log2
2
4
4
ÂľF
Âľ2F
!
!
1
Q2
Q2
(1)
2
(1)
(1)
+ β0 fj P ⊗ fi log
+ C ⊗ fi P ⊗ fj log
2
Âľ2F
Âľ2F
!
1
Q2
+ fj P (2) ⊗ fi log
2
Âľ2F
!
!
!
2
2
2
Q
Q
Q
+fj C (1) ⊗ P (1) ⊗ fi log
− 2β0 fj P (1) ⊗ fi log
log
2
ÂľF
Âľ2F
Âľ2
!! R
Q2
+C (1) ⊗ fi C (1) ⊗ fj + 2fj C (2) ⊗ fi − 4β0 fj C (1) ⊗ fi log
Âľ2R

2
+ β1 fj P (1) ⊗ fi − 10 A(1) ζ3 fj P (1) ⊗ fi
+ A(1)

+ i ↔ j,
(i,j)
3 C0

= 2C (1) ⊗ fj fi B (2) + P (1) ⊗ fj log

!
fi B (2) + 2C (1) ⊗ fi fj B (2)

!
2
3
Q
+ P (1) ⊗ fi log
fj B (2) + P (1) ⊗ fj P (1) ⊗ P (1) ⊗ fi log2
8
Âľ2F
!
!
2
2
3 (1)
Q
Q
+ P ⊗ fi P (1) ⊗ P (1) ⊗ fj log2
+ P (1) ⊗ fj log2
fi β02
8
Âľ2F
Âľ2R
!
!
Q2
Q2
(1)
2
(1)
2
+ 4C ⊗ fj log
fi β0 + P ⊗ fi log
fj β02
2
2
ÂľR
ÂľR
Q2
Âľ2F

!

Q2
Âľ2F

210

!

+ 4C (1)

⊗ fi log

Q2
Âľ2R

!
fj β02 + C (2) ⊗ fj P (1) ⊗ fi + C (2) ⊗ fi P (1) ⊗ fj

1
1
+ C (1) ⊗ fj P (2) ⊗ fi + C (1) ⊗ fi P (2) ⊗ fj
2
2
+ C (1) ⊗ fj C (1) ⊗ P (1) ⊗ fi + C (1) ⊗ fi C (1) ⊗ P (1) ⊗ fj
!
!
2
2
1 (1)
Q
1
Q
+ P ⊗ fj P (2) ⊗ fi log
+ P (1) ⊗ fi P (2) ⊗ fj log
2
2
Âľ2F
Âľ2F
!
!
2
2
Q
Q
+ P (1) ⊗ fj C (1) ⊗ P (1) ⊗ fi log
+ P (1) ⊗ fi C (1) ⊗ P (1) ⊗ fj log
Âľ2F
Âľ2F
!
!
2
2
1 (1)
Q
1
Q
+ C ⊗ fj P (1) ⊗ P (1) ⊗ fi log
+ C (1) ⊗ fi P (1) ⊗ P (1) ⊗ fj log
2
2
Âľ2F
Âľ2F
!
2
1 (1)
Q
+ P ⊗ P (1) ⊗ P (1) ⊗ fj log2
fi − 2β1 C (1) ⊗ fj fi
8
Âľ2F
1
1
+ P (3) ⊗ fj fi + C (1) ⊗ P (2) ⊗ fj fi + C (2) ⊗ P (1) ⊗ fj fi
4
2
!
2
1 (1)
Q
1
+ P ⊗ P (2) ⊗ fj log
fi + P (2) ⊗ P (1) ⊗ fj log
2
4
4
ÂľF
!
2
1 (1)
Q
+ C ⊗ P (1) ⊗ P (1) ⊗ fj log
fi − β1 P (1) ⊗ fj log
2
Âľ2F
!
2
1 (1)
Q
+ P ⊗ P (1) ⊗ P (1) ⊗ fi log2
fj − 2β1 C (1) ⊗ fi fj
8
Âľ2F

Q2
Âľ2F

!

Q2
Âľ2R

fi
!
fi

1
1
+ P (3) ⊗ fi fj + C (1) ⊗ P (2) ⊗ fi fj
4
2
!
1 (1)
Q2
1
(2)
(1)
(2)
+ C ⊗ P ⊗ fi fj + P ⊗ P ⊗ fi log
fj + P (2) ⊗ P (1) ⊗ fi log
2
4
4
ÂľF
!
!
2
2
1
Q
Q
+ C (1) ⊗ P (1) ⊗ P (1) ⊗ fi log
fj − β1 P (1) ⊗ fi log
fj
2
2
ÂľF
Âľ2R
!
2
1
Q
+ P (1) ⊗ fi P (1) ⊗ fj log2
β0 − 4C (1) ⊗ fi C (1) ⊗ fj β0
2
2
ÂľF
!
!
2
2
Q
Q
− C (1) ⊗ fj P (1) ⊗ fi log
β0 − C (1) ⊗ fi P (1) ⊗ fj log
β0
2
ÂľF
Âľ2F
!
!
2
2
Q
Q
− 2C (1) ⊗ fj P (1) ⊗ fi log
β0 − 2C (1) ⊗ fi P (1) ⊗ fj log
β0
2
ÂľR
Âľ2R

211

Q2
Âľ2F

!
fj

!
!
Q2
Q2
⊗ fi
⊗ fj log
log
β0
Âľ2F
Âľ2R
!
!
1 (1)
Q2
Q2
(1)
2
(2)
(1)
(1)
+ P ⊗ P ⊗ fj log
fi β0 − 4C ⊗ fj fi β0 − C ⊗ P ⊗ fj log
fi β0
4
Âľ2F
Âľ2F
!
!
Q2
Q2
(2)
(1)
(1)
− P ⊗ fj log
fi β0 − 2C ⊗ P ⊗ fj log
fi β0
Âľ2R
Âľ2R
!
!
!
Q2
Q2
1 (1)
Q2
(1)
(1)
(1)
2
− P ⊗ P ⊗ fj log
log
fi β0 + P ⊗ P ⊗ fi log
f j β0
4
Âľ2F
Âľ2R
Âľ2F
!
!
2
2
Q
Q
− 4C (2) ⊗ fi fj β0 − C (1) ⊗ P (1) ⊗ fi log
fj β0 − P (2) ⊗ fi log
f j β0
2
ÂľF
Âľ2R
!
!
!
2
2
2
Q
Q
Q
− 2C (1) ⊗ P (1) ⊗ fi log
fj β0 − P (1) ⊗ P (1) ⊗ fi log
log
f j β0
2
2
ÂľR
ÂľF
Âľ2R
!
!

 1
2
2
Q
1
Q
+ B (1)
P (1) ⊗ fi P (1) ⊗ fj log2
+ P (1) ⊗ P (1) ⊗ fj fi log2
2
2
4
ÂľF
Âľ2F
!
!
2
1
Q2
1 (1)
2 Q
+ P (1) ⊗ P (1) ⊗ fi fj log2
+
P
⊗
f
f
β
log
j i 0
4
2
Âľ2F
Âľ2F
!
!
1
Q2
1 (2)
Q2
+ P (1) ⊗ fi fj β0 log2
+
P
⊗
f
f
log
j i
2
2
Âľ2F
Âľ2F
!
!
2
2
Q
Q
+C (1) ⊗ fj P (1) ⊗ fi log
+ C (1) ⊗ fi P (1) ⊗ fj log
2
ÂľF
Âľ2F
!
!
2
2
Q
Q
+C (1) ⊗ P (1) ⊗ fj fi log
+ C (1) ⊗ P (1) ⊗ fi fj log
2
Âľ
Âľ2F
! F
1
Q2
+ P (2) ⊗ fi fj log
+ 2C (1) ⊗ fi C (1) ⊗ fj
2
2
ÂľF
!
!
!
!
2
2
2
2
Q
Q
Q
Q
−2P (1) ⊗ fj log
fi β0 log
− 2P (1) ⊗ fi log
fj β0 log
2
2
ÂľR
ÂľF
Âľ2R
Âľ2F
!
!
!
2
2
Q
Q
+2C (2) ⊗ fj fi + 2C (2) ⊗ fi fj − 4C (1) ⊗ fj log
fi β0 − 4C (1) ⊗ fi log
fj β0
2
ÂľR
Âľ2R
!

2
2
Q
+ A(1)
4C (1) ⊗ fj ζ(3)fi + 2P (1) ⊗ fj log
ζ(3)fi + 4C (1) ⊗ fi fj ζ(3)
2
ÂľF
!
!
Q2
+2P (1) ⊗ fi log
fj Îś(3)
Âľ2F
− 2P (1)

P (1)

212



28
f β ζ(3)P (1) ⊗ fj
3 i 0



+ B (1) −8P (1) ⊗ fj ζ(3)fi − 8P (1) ⊗ fi fj ζ(3) − 2P (1) ⊗ P (1) ⊗ fj fi ζ(3)

28 (1)
(1)
(1)
−2P ⊗ P ⊗ fi fj ζ(3) + P ⊗ fi fj β0 ζ(3)
3
!
Q2
+ fi fj −4 log
β0 B (2) + 2B (3)
2
ÂľR
!
!!





2
2
2
Q
Q
(1)
2
2
(1)
(1)
+ B
2 log
β0 − 2β1 log
+ A
−8ζ(3) B
Âľ2R
Âľ2R




3
56
(1)
2
+ β0 Μ(3) B
− 8β0 ζ(3) − 12 A(1) ζ(5) + 8A(1)+(2) ζ(3)
3
!
!

2
2
Q
−8 A(1) log
β0 Μ(3) ,
Âľ2R
+



A(1)

−4P (1) ⊗ fi ζ(3)P (1) ⊗ fj +



Similarly, the singular piece to O Îąs3 can be calculated in a similar method. However,
C (3) has not yet been calculated, and involves a complex three-loop calculation. Therefore,
it is not possible to obtain the Îąs3 cross-section at this point, and there is no need for the
singular piece to be calculated.

Eikonal Approximation
Eikonalization in the Collinear Limit
Given a collinear gluon of momentum l connecting two fermions of momentum p1 and p2 ,
as shown in Fig. F.3, the matrix element can be given as:



p/1 − /l


p/2 − /l
M = v̄ (p1 ) γα
¡¡¡
Îł Îą u (p2 ) ,
(p1 − l)2
(p2 − l)2
213



(F.43)

p1

(p1 − l)

(p2 − l)

p2

l

Figure F.3: Connection of a soft/collinear gluon between two colored particles.
where the dots represent any allowed processes between the two fermions. If l is collinear to
p2 , then


p/2 − /l γ α u (p2 ) = (p2 − l)+ γ − γ α u (p2 ) ,

(F.44)

where the coordinate system being used is the light-cone coordinates, and the +/− refers to
the plus or minus component of the momentum defined as: pÂą = E Âą pz . Consider the Îł
matrices in the above equation. If α = −, then γ − u (p2 ) = 0 from the on-shell condition. If
α = T , then γ − γ T = −γ T γ − from the anti-commutation relations, and therefore also gives
zero. Hence, the only non-vanishing term is for Îą = +, giving:


γ − p/1 − /l
(p2 − l)+ γ − γ +
M = v̄ (p1 )
¡
¡
¡
u (p2 ) .
(p1 − l)2
(p2 − l)2

(F.45)

Using the on-shell condition and commutation relation, the part before the dots can be
simplified to:
v̄ (p1 )

2p−
1
(p1 − l)2

,

(F.46)



+ Îą
−
T T
and furthermore, if p2 = p+ , 0, 0 , and pÎą
1 = p1 n̄ + p1 nα + p1 n , where n̄µ = (1, 0, 0),
nÂľ = (0, 1, 0), and nT = (0, 0, 1), then the equation can be shown to factorize, giving:

v̄ (p1 )

nÎą
.
(−n · l) + i

214

(F.47)

Eikonalization in the Soft Limit
Similar to the collinear limit, the process can be factorized if the gluon connecting the
particles is soft. Again, starting from Eq. F.43, taking the limit l ← 0, gives:




p/2 − /l α
pÎą
2
γ u (p2 ) →
.
(−p2 · l) + i
(p2 − l)2

(F.48)

Which can be calculated by using the on-shell condition, and properties of the Îł matrices. It
is important to note, that in calculations the sign of the  term is important to keep straight,
and must always be positive when using this approach. Again, the final result factorizes the
left-hand side of Fig. F.3 from the right-hand side.

Fourier Transform Details
The Fourier Transform in d − 2 dimensions from transverse momentum space to impact
parameter space is given by:

Z

~

eikT ·b 2−2
d
kT =
(kT2 )Îą

 2 +α−1
b
Γ (1 −  − α)
πα
.
4π
Γ (α)

(F.49)

To obtain the Fourier Transform for functions of the form 12 logn kT2 , derivatives with respect
kT

to Îą are taken of the previous equation, and then Îą is taken to one. The derivation of the
above equation can be found below, and requires the use of the following identity:

(kT2 )−α

Z ∞
2
1
−xkT
=
xα−1 e
dx,
Γ (α) 0

215

(F.50)

using this identity, the first equation is derived as follows:

Z

~

~

eikT ·b 2−2
1
d
kT =
2
Îą
Γ (α)
(kT )

Z

~
eikT ·b d2−2 k

Z ∞

Z ∞
T

2
−xkT
dx

xα−1 e

0

Z ∞
2
1
ik b cos(θ)−xkT
α−1
=
x
dx dΩ
kT2−2−1 e T
dkT
Γ (α) 0
0
Z ∞
Z ∞
Z
2
1
ik b cos(θ)−xkT
2−2−1
α−1
=
x
dx
kT
dkT dΩ2−2 sin−2 θe T
Γ (α) 0
0


Z ∞
Z ∞
2
1
2π 1−
1 2 2
−xkT
2−2−1
α−1
=
x
dx
kT
dkT
0 F1 1 − ; − b kT e
Γ (α) 0
Γ (1 − )
4
0
 2 +α−1
b
Γ (1 −  − α)
=
πα
,
4π
Γ (α)
Z



where 0 F1 1 − ; − 14 b2 kT2 is the confluent hypergeometric function. Using this relationship,
we can easily derive the Fourier Transform of all the terms needed to check the asymptotic


contribution up through O Îąs3 . Here are a list of the results for completeness:
 2 
b
=π
Γ (−)
(F.51)
4π
 2 


b
b2
(1)
I = −π
Γ (−) γE + log − ψ0 (−)
(F.52)
4π
4
!
 2 

2
2
2
b
b
π
I (2) = π
Γ (−)
γE + log − ψ0 (−) −
+ ψ1 (−)
(F.53)
4π
4
6
 2 

3


b
b2
π2
b2
(3)
I = −π
Γ (−)
γE + log − ψ0 (−) −
γE + log − ψ0 (−)
4π
4
2
4



b2
+3ψ1 (−) γE + log − ψ0 (−) − 2 (ψ2 (1) + ψ2 (−))
(F.54)
4
I (0)

216

 2 

4

2
b
b2
b2
2
=π
Γ (−)
γE + log − ψ0 (−) − π γE + log − ψ0 (−)
4π
4
4

2
b2
π4
2
−π ψ1 (−) + 6ψ1 (−) γE + log − ψ0 (−) +
+ ψ3 (−)
4
60


b2
−4 (ψ2 (1) + ψ2 (−)) γE + log − ψ0 (−)
(F.55)
4
 2 

5

3
b
b2
5π 2
b2
(5)
I =π
Γ (−)
γE + log − ψ0 (−) −
γE + log − ψ0 (−)
4π
4
3
4

3


b2
b2
2
+10ψ1 (−) γE + log − ψ0 (−) − 10π γE + log − ψ0 (−) ψ1 (−)
4
4
!


2
b2
π2
−10 (ψ2 (1) + ψ2 (−))
γE + log − ψ0 (−) + ψ1 (−) −
4
6
 2
2 

π
b2
+15
+ ψ1 (−)
γE + log − ψ0 (−)
6
4




b2
b2
−30ζ (4) γE + log − ψ0 (−) + 5 γE + log − ψ0 (−) ψ3 (−)
4
4



− ψ4 (1) + ψ4 (−)
(F.56)
I (4)



R d2−2 qt
~
where I (n) =
logn qt2 e−iq~t ·b , and ψn is the nth polygamma function defined as:
2
q
t

ψn (z) =

dn+1
log (Γ (z))
dz n+1

217

(F.57)

APPENDIX G
Higgs Plus Jet Resummation
Calculation Details
The poles for the quark-gluon channel are given for the virtual, renormalization, jet, collinear,
PDF renormalization, and soft corrections as:




24
−2 2
−u
−t
s
β + CA
−
ln 2 − ln 2 − ln 2
2π  0

2
Âľ
Âľ
Âľ



4
2
−u
+CF − 2 −
−2 ln 2 + 3
,


Âľ
Îąs 18
− H (0)
2β ,
2π  0
!!
2 R2
P
Îą
C
2
2
3
s F
H (0)
+
− ln J 2
,
2π
2  2
Âľ
Îąs
H (0)





αs −2
CF δ (ξ2 − 1) ξ1 Pqq (ξ1 ) + CA δ (ξ1 − 1) ξ2 Pgg (ξ2 ) ,
2π 





Îąs 2CF
3
2CA
(0)
H
Pqq (ξ1 ) + δ (xi1 − 1) +
Pgg (ξ2 ) + 2β0 δ (xi2 − 1) ,
2π

2






Îąs
1
1 Âľ2
2
t
1
(0)
H
2 (CA + CF ) 2 + ln
−
(CA − CF ) ln + CF ln 2
,
2π

s

u

R
H (0)

218

(G.1)
(G.2)
(G.3)
(G.4)
(G.5)
(G.6)

‘respectively. Similarly, for the quark-anti-quark channel, the poles are in the same order as
above, and are given as:






1
−2 2 sµ2
−4 2
s
24β0 + CA
− ln
+ CF
−
−2 ln 2 + 3
,
2π


ut

2
2
Âľ


αs −6CA
(0)
H
2β0 ,
2π

!!
2
P
Îąs
1
1
1
H (0) 2CA 2 +
2β0 − ln T2 + ln 2
,
2π


Âľ
R
Îąs
H (0)



αs −2
CF δ (ξ2 − 1) ξ1 Pqq (x) + ξ1 ↔ ξ2 ,
2π  

Îąs 2CF
3
(0)
H
Pqq (ξ1 ) ξ1 + δ (ξ1 − 1) + ξ1 ↔ ξ2 ,
2π 
2




2
Îąs
2
2 Âľ
1
1
(0)
H
2 CF
+ ln
− CA ln 2 .
2π
s

2 
R
H (0)

(G.7)
(G.8)
(G.9)
(G.10)
(G.11)
(G.12)

In both of the above set of equations, all the terms are proportional to δ(ξ1 − 1)δ(ξ2 − 1)
unless otherwise noted. Looking at the above equations, all of the poles cancel, leaving a
finite result, and validating the results of the calculations.

219

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