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Mult iple Parton Radiation in Hadroproduct ion
at Lepton-Hadzon Colliders

presented by

PAVEL M. NAH)LSKY

has been accepted towards fulfillment
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MULTIPLE PARTON RADIATION IN

HADROPRODUCTION
AT LEPTON—HADRON COLLIDERS

By
Pavel M. N adols‘ky

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Physics & Astronomy

2001

ABSTRACT
MULTIPLE PARTON RADIATION IN

HADROPRODUCTION
AT LEPTON-HADRON COLLIDERS

Pavel M. Nadolsky

Factorization of long— and short-distance hadronic dynamics in perturbative Quan—
tum Chromodynamics (QCD) is often obstructed by the coherent partonic radiation,
which leads to the appearance of large logarithmic terms in coefficients of the pertur—
bative QCD series. In particular, iarge remainders from the cancellation of infrared
singularities distort theoretical predictions for angular distributions of observed prod-
ucts of hadronic reactions. In several important cases, the predictive power of QCD
can be restored through summation of large logarithmic terms to all orders of the per-
turbative expansion. Here I discuss the impact of the the coherent parton radiation
on semi-inclusive production of hadrons in deep inelastic scattering at lepton-proton
colliders. Such radiation can be consistently described in the b-space resummation
formalism, which was originally developed to improve theoretical description of pro-
duction of hadrons at e+e’ colliders and electroweak vector bosons at hadron-hadron
colliders. I present the detailed derivation of the resummed cross section and the
energy flow at the next-to-leading order of perturbative QCD. The theoretical results
are compared to the experimental data measured at the ep collider HERA. A good

agreement is found between the theory and experiment in the region of validity of the

resummation formalism. I argue that semi-inclusive deep inelastic scattering (SIDIS)
at lepton-hadron colliders offers exceptional opportunities to study coherent parton
radiation, which are not available yet at colliders of other types. Specifically, SIDIS
can be used to test the factorization of hard scattering and collinear contributions at
small values of :1: and to search for potential crossing symmetry relationships between
the properties of the coherent radiation in SIDIS, 6+6" hadroproduction and Drell-

Yan processes.

To Sunny, my true love and inspiration

iv

ACKNOWLEDGMENTS

My appreciation goes to many people who helped me grow as a physicist. Fore—
most, I am deeply indebted to my advisors C.-P. Yuan and Wu-Ki Tung, who made
my years at MSU a truly enriching and enjoyable experience. I feel very fortunate
that C.-P. and Wu-Ki agreed to work with me when I first asked them to. Tradi-
tionally C.-P. warns each new graduate student about the challenges that accompany
the career in high energy physics. In my own experience, I found that benefits and
satisfaction from the work in the team with C.-P. and Wu-Ki outweigh all possible
drawbacks.

Both of my advisors have spent a significant effort and time to teach me new
valuable knowledge and skills. They also strongly supported my interest in the sub-
jects of my research, both morally and through material means. I am wholeheartedly
grateful for their guidance and support. The remarkable scope of vision, vigor, and
patience of C.-P., together with profound knowledge, careful judgment and precision
of Wu-Ki, inspire me as model personal qualities required for a scientist.

I owe my deep appreciation to my coauthor Dan Stump, who spent many focused
hours working with me on the topics in this thesis. Dan’s proofreading was the main
driving force behind the improvements in my English. A significant fraction of my
accomplishments is due to the possibility of open and direct communication between
the graduate students, MSU professors, and members of the CT EQ Collaboration.
My understanding of resummation formalism was significantly enhanced through dis-
cussions with John Collins, Jianwei Qiu, Davison Soper, and George Sterman. At
numerous occasions, I had useful exchanges of ideas with Edmond Berger, Raymond
Brock, Joey Huston, Jim Linnemann, Fred Olness, Jon Pumplin, Wayne Repko, and
Carl Schmidt. Many tasks were made easy by the interest and help from my fellow

graduate students and research associates, most of all from Jim Amundson, Csaba

Balazs, Qinghong Cao, Dylan Casey, Chris Glosser, Hong—Jian He, Shinya Kanemura,
Stefan Kretzer, Frank Kuehnel, Liang Lai, Simona Murgia, Tom Rockwell, Tim Tait,
and Alex Zhukov.

Throughout this work I was using the CT EQ Fortran libraries, which were mainly
developed by W.-K. Tung, H.-L. Lai and J. Collins. The numerical package for
resummation in SIDIS was developed on the basis of the programs Legacy and ResBos
written by C. Balazs, G. Ladinsky and C.-P. Yuan. S. Kretzer has provided me
with the Fortran parametrizations of the fragmentation functions. Some preliminary
calculations included in this thesis were done by Kelly McGlynn.

I am grateful to Michael Klasen and Michael Kramer for physics discussions and
invitation to give a talk at DISZOOO Workshop. I learned important information about
the BFKL resummation formalism from Carl Schmidt, J. Bartels and NP. Zotov. I
thank Gunter Grindhammer and Heidi Schellman for explaining the details of SIDIS
experiments at HERA and TEVATRON colliders. I also thank D. Graudenz for the
correspondence about the inclusive rate of SIDIS hadroproduction, and M. Kuhlen for
the communication about the HZTOOL data package. I enjoyed conversations about
semi-inclusive hadroproduction with Brian Harris, Daniel Boer, Sourendu Gupta,
Anna Kulesza, Tilman Plehn, Zack Sullivan, Werner Vogelsang, and other members
of the HEP groups at Argonne and Brookhaven National Laboratories, University of
Wisconsin and Southern Methodist University. I am grateful to Brage Golding, Joey
Huston, Vladimir Zelevinsky for useful advices and careful reading of my manuscript.

I am sincerely grateful to Harry Weerts, who encouraged me to apply to the MSU
graduate program and later spent a significant effort to get me in. During my years
at Michigan State University, I was surrounded by the friendly and productive at-
mosphere created for HEP graduate students by a persistent effort of many people,

notably Jeanette Dubendorf, Phil Duxbury, Stephanie Holland, Julius Kovacs, Lorie

vi

Neuman, George Perkins, Debbie Simmons, Brenda Wenzlick, Laura Winterbottom
and Margaret Wilke. My teaching experience was more pleasant due to the interac-
tions with Darlene Salman and Mark Olson.

Perhaps none of this work would be completed without the loving care and enthu-
siastic encouragement from my wife Sunny, who brings the meaning and joy to each
day of my life. The completion of this thesis is our joint achievement, of which Sunny’s
help in the preparation of the manuscript is only the smallest part. My deep gratitude
also goes to my parents, who always support and love me despite my being overseas.
The memory of my grandmother who passed away during the past year will always

keep my heart warm.

vii

Contents

1 Introduction

2 Overview of the QCD factorization
2.1 QCD Lagrangian and renormalization ..................
2.2 Asymptotic freedom ...........................
2.3 Infrared safety ...............................
2.4 Two-scale problems ............................

2.4.1 Resummation of soft and collinear logarithms .........

2.4.2 QCD at small :1: ..........................

3 Resummation in semi-inclusive DIS: theoretical formalism
3.1 Kinematical Variables ..........................
3.1.1 Lorentz scalars ..........................
3.1.2 Hadron frame ...........................
3.1.3 Photon-hadron center-of-mass frame ..............
3.1.4 Laboratory frame .........................
3.1.5 Parton kinematics .........................
3.2 The structure of the SIDIS cross-section ................
3.3 Leading-order cross section ........................

3.4 The higher-order radiative corrections ..................

viii

12
14
22
22
35

39
41
42
44
49
53
59
60
62

3.4.1 Factorization of collinear singularities at 0(a5) ........ 66
3.4.2 All-order resummation of large logarithmic terms ....... 70
3.5 Hadronic multiplicities and energy flows ................ 78

3.6 Relationship between the perturbative and resummed cross-sections.

Uncertainties of the calculation ..................... 82

3.6.1 Matching ............................. 82

3.6.2 Kinematical corrections at (1T x Q ................ 84

4 Resummation in semi-inclusive DIS: numerical results 89
4.1 Energy flows ................................ 93
4.1.1 General remarks ......................... 93

4.1.2 Comparison with the data .................... 98

4.1.3 How trustworthy is the resummed z-fiow at large qT? ..... 104

4.2 Normalized distributions of charged particle production ....... 107
4.3 Discussion and conclusions ........................ 117

5 Azimuthal asymmetries of SIDIS observables 120
5.1 Large logarithmic corrections and resummation ............ 122
5.2 Asymmetry of energy flow ........................ 127

A The perturbative cross-section, finite piece and z-flow distribution 13]

B 0(a5) part of the resummed cross section 135

Bibliography 140

ix

List of Figures

1.1

1.2

2.1

2.2
2.3

2.4

3.1

(a) Production of hadronic jets at e+e‘colliders; (1)) Production of
lepton pairs at hadron-hadron colliders .................

Semi-inclusive deep inelastic scattering .................

Factorization of collinear singularities in completely inclusive electron-
hadron DIS ................................
The space-time picture of hadroproduction at 6+6“ colliders .....
The structure of infrared singularities in a cut diagram D for the
energy-energy correlation in the axial gauge ..............
Examples of the finite soft subdiagrams: (a) the subdiagrams that are
connected to J A, J B by one or several quark lines; (b) the subdiagrams
that are connected to H .........................
The space-time picture of Drell-Yan process ..............

The ladder structure of the DIS cut diagrams .............

Geometry of the particle momenta in the hadron frame ........

17
21

25

45

3.2

3.3
3.4
3.5

3.6

3.7

4.1

4.2

(a) In the current fragmentation region, the hadron-level cross section
can be factorized into hard partonic cross sections 3m, parton distribu—
tion functions Fa/A(£a, ,up), and fragmentation functions DB/b(§b, up).
(b) In the target fragmentation region, the hadrons are produced through
the mechanism of diffractive scattering that depends on “diffractive
parton distributions” Ma,B/A(€a, (3, ,up). ................
Particle momenta in the hadronic center-of-mass (hCM) frame . . . .
Particle momenta in the laboratory frame ...............
The variables qT and go as functions of the angles 63, 903. Solid lines
are contours of constant qT for qT / Q ranging from 0.1 (the innermost
contour) to 3.0. Dashed lines are contours of constant (,0 for (p ranging
from 7r/10 to 37r/4. The contour (p = 1r coincides with the BB-axis.
The plots correspond to EA = 820 GeV, E = 27 GeV, Q = 6 GeV,
:5 = 0.01 (upper plot) and :1: = 0.001 (lower plot). ...........
Feynman diagrams for semi-inclusive DIS: (a) LO; (b—d) N LO virtual
diagrams; (e-f) N LO real emission diagrams ..............
The contours of the integration over {0, {b for (a,b,c) the perturbative

cross-section; (d) the asymptotic and resummed cross-sections

The average q% as a function of :1: and z in the charged particle pro-
duction at Q2 = 28 — 38 GeVz. The data points are extracted from
published distributions (przr) vs. mp [60,63] using the method described

in Section 4.2 ................................

 

\/(q%Ez) / (X32) reconstructed from distributions d (ET) /d77cm in bins
of a: and Q2 [65]. .............................

xi

47
49
53

58

62

88

91

92

4.3

4.4

4.5

4.6

4.7

4.8

Comparison of the NLO perturbative and resummed expressions for
the z-flow distribution with the existing experimental data from HERA
[64]. The data is for (1:) = 0.0049, (Q2) = 32.6 GeVz. The resummed
curve is calculated using the parametrization 1 of S [V P . CTEQ4M PDFs
[91] were used ................................
Comparison of the resummed z-fiow (solid curve) in the current region
of the hCM frame with the data in the low-Q2 bins from Refs. [65] (filled
circles) and [64] (empty circles). For the bin with (Q2) 2 33.2 GeV2
and (as) = 0.0047, the fixed-order 0((15) contribution for the factoriza-
tion scale a = Q is shown as the dashed curve. ............
Comparison of the resummed z-flow in the current region of the hCM
frame with the data in the high-Q2 bins from Ref. [65]. For the bin
with (Q2) 2 617 GeV2 and (2:) = 0.026, the C(05) contribution for
up = Q is shown as a dashed curve. ..................
The hCM pseudorapidity distributions of the transverse energy flow in
the current fragmentation region. The data are from [64]. CTEQ4M
PDFs and the parametrizationl of Si”) were used. ..........
The dependence of the C(05) prediction for the total charged particle
multiplicity on the value of the separation scale q}. The calculation is
done for (W) = 120 GeV, (Q2) 2 28 GeV2. ..............
The distributions (a) (19%) vs. xp and (b) ((1%) vs. 23,: for the charged
particle multiplicity at (W) = 120 GeV, (Q2) 2 28 GeV2. The ex-
perimental points for the distribution (pzT) vs. x1: are from Fig. 3c of
Ref. [60]. The “experimental” points for the distribution ((1%) vs. 113;:
are derived using Eq. (4.19). The solid and dashed curves correspond

to the resummed and NLO (u = Q) multiplicity, respectively ......

xii

99

100

101

106

110

111

4.9

4.10

5.1

5.2

The dependence of the charged particle multiplicity on the transverse
momentum 197 of the observed particles in the hCM frame. The data
points are from [60]. The solid and dashed curves correspond to the
resummed and NLO multiplicities, respectively. ............
The dependence of the charged particle multiplicity on the Feynman
variable 12;: in the hCM frame. The solid curve corresponds to the
resummed multiplicity. The dashed, lower dotted and upper dotted
curves correspond to the NLO multiplicity calculated for n = Q, 0.5Q

and 2Q, respectively. ...........................

Comparison of the (9((1 5) prediction for the ratio (cos (,0)/ (cos 2m) with
the ratio of experimentally measured values of (cos (,0) and (cos 290) from
[61]. The error bars are calculated by adding the statistical errors of
(cos (p) and (cos 290) in quadrature. Systematic errors are not included.
The theoretical curve is calculated for (1:) = 0.022, (Q2) 2 750 GeVQ,
using the CTEQ5M1 parton distribution functions [90] and fragmen-
tation functions by S. Kretzer from [88]. ................
Energy flow asymmetries (ET cos cp)(qT) and (ET cos 2<p)(qT) for (a)
a: = 0.0047, Q2 = 33.2 Gev2 and (b) a: = 0.026, Q2 = 617 GeV2. The
Figure shows predictions from the resummed (solid) and the 0(a3)

(dashed) calculations. ..........................

xiii

113

116

125

Chapter 1

Introduction

Since its foundation in 1970’s, perturbative Quantum Chromodynamics (PQCD) has
evolved into a precise theory of energetic hadronic interactions. The success of the
QCD theory in the quantitative description of hadronic experimental data originates

from the following fundamental ideas:

1. Hadrons are not elementary particles. As it was first shown by the quark model
of Gell-Mann and Zweig [1], basic properties of the observed low-energy hadronic
states are explained if hadrons are composed of a few “constituent quarks” with
spin 1 / 2, fractional electric charges and new quantum numbers of flavor and
color [2]. If the hadron constituents (partons) are bound weakly at some energy,
they can possibly be detected in scattering experiments. The parton model of
Feynman and Bjorken suggested that the pointlike hadronic constituents may
reveal themselves in the wide-angle scattering of leptons off hadronic targets [3].
The first direct experimental proof of the hadronic substructure came from
the observation of the Bjorken scaling [4] in the electron-proton deep—inelastic
scattering [5]; subsequently the quantum numbers of partons were tested in a

variety of experiments [6].

2. The elementary partons of QCD are “current” quarks, which interact with one
another through mediation of non-Abelian gauge fields (gluons) [7]. These gauge
fields are introduced to preserve the local SU (3) symmetry of the quark color
charges, in accordance with the pioneering work on non-Abelian gauge sym-
metries by C. N. Yang and R. L. Mills [8]. Remarkably, the QCD interactions
weaken at small distances because of the anti-screening of color charges by
self-interacting gluons [9]. Due to this feature (asymptotic freedom) of QCD,
probabilities for parton interactions at distance scales smaller than 1 GeV"1 can
be calculated as a series in the small QCD running coupling (13. In the opposite
limit of large distances, as grows rapidly, so that the QCD interactions be-
come nonperturbative at the scale of about 0.2 GeV"1. Such scale dependence
of the QCD coupling explains why the partons behave as quasi-free particles
when probed in the energetic collision, but eventually are confined in colorless

hadrons at the later stages of the scattering.

3. Because of the parton confinement, quantitative calculations within QCD re-
quire systematic separation of dynamics associated with short and long distance
scales. The possibility for such separation is proven by factorization theorems
[10—15]. With time, the factorization was proven for observables of increas-
ing complexity. In 1977, G. Sterman and S. Weinberg introduced a notion of
infrared-safe observables, which are not sensitive to the details of long-distance
dynamics [16]. A typical example of an infrared-safe observable is the cross-
section for the production of well-separated hadronic jets at an e+e‘ collider.
It was shown that infrared-safe observables can be systematically described by
means of PQCD. As a next step, factorization was proven for inclusive observ-
ables depending on one large momentum scale Q2. In the limit Q2 —+ 00,

such observables can be factorized into a perturbatively calculable hard part,

2

describing energetic short-range interactions of hadronic constituents, and sev-
eral process-independent nonperturbative functions, relevant to the complicated

strong dynamics at large distances.

The proof of factorization is more involved for hadronic observables that depend
on several momentum scales (e.g., differential cross sections). The complications
stem not so much from the complex dependence of the cross sections on kinematical
variables, but from the presence of logarithms 1n r, where r is some dimensionless
function of the kinematical parameters of the system. For instance, r may be a ratio
of two momentum scales P1 and P2 of the system, r = P1 / P2. Near the boundaries
of the phase space, the ratio r can be very large or very small, in which case the
convergence of the series in the QCD coupling as can be spoiled by the largeness of
terms proportional to powers of In r. Hence the factorization cannot be proven as
straightforwardly as in the case of the inclusive observables, because its most obvious
requirement — sufficiently rapid convergence of the perturbative series — is violated.

To restore the convergence of PQCD, one may have to sum the large logarithmic
terms through all orders of as. This procedure is commonly called resummation.
Logarithmic terms of one rather general class appear due to the QCD radiation along
the directions of the observed initial- or final-state hadrons (collinear radiation) or
the emission of low-energy gluons (soft radiation). Such logarithms commonly affect
observables sensitive to the angular distribution of the hadrons. In several important
processes, the soft and collinear logarithms can be consistently resummed through
the use of the formalism developed by J. Collins, D. Soper and G. Sterman (C88).

The original resummation technique was proposed in Ref. [17] to describe angular
distributions of back-to-back jets produced at e+e‘ colliders (Fig. 1.1a). Subsequent
developments of this technique and its comparison to the data on the e+e‘ hadropro-

duction were presented in Refs. [18—20]. In Ref. [21] the resummation formalism was

3

Jet A

6+ 7* 20 §
’ §
6 §
Jet B
(a)
Hadron A \
\
\\ 1
Y 1W“: ’20 1
§
T
2

Hadron B

////////

(b)

Figure 1.1: (a) Production of hadronic jets at e+e’colliders; (b) Production of lepton

pairs at hadron-hadron colliders

extended to describe transverse momentum distributions of lepton pairs produced
at hadron-hadron colliders (Fig. 1.1b). In the subsequent publications [22—27], this
technique was developed to a high degree of numerical accuracy. Currently the re-
summation analysis of this type is employed in the measurements of the mass [28] and
the Width [29] of the W—bosons produced at the p13 collider TEVAT RON . With some
modifications, this resummation formalism is also used to improve PQCD predictions
for the production of Higgs bosons [30] and photon pairs [31] at the Large Hadron
Collider (LHC).

The hadroproduction at e+e‘colliders and the lepton pair production at hadron-
hadron colliders (Drell-Yan process) are the simplest reactions that require resumma-
tion of the soft and collinear logarithms. In both reactions, the interaction between
the leptons and two initial- or final-state hadronic systems is mediated by an elec-
troweak boson V with a timelike momentum. The CSS resummation formalism can
also be formulated for reactions with the exchange of a spacelike electroweak vector
boson [32,33]. In this work, I discuss resummation in the semi-inclusive production
of hadrons in electron-hadron deep-inelastic scattering, which is the natural analog of
e+e" hadroproduction and Drell-Yan process in the spacelike channel. The reaction
of semi-inclusive deep-inelastic scattering (SIDIS) e + A —+ e + B + X, where A and
B are the initial- and final-state hadrons, respectively, is shown in Figure 1.2.

As in the other two reactions, in SIDIS the multiple parton radiation affects an-
gular distributions of the observed hadrons. The study of the resummation in SIDIS
has several advantages in comparison to the reactions in the timelike channels. First,
SIDIS is characterized by an obvious asymmetry between the initial and final hadronic
states, so that the dependence of the multiple parton radiation on the properties of
the initial state can be distinguished clearly from the dependence on the properties of

the final state. In contrast, in 8+6_ hadroproduction or the Drell—Yan process some

 

Figure 1.2: Semi-inclusive deep inelastic scattering

details of the dynamics may be hidden due to the symmetry between two external
hadronic systems. Notably, I will discuss the dependence of the resummed observables
on the longitudinal variables a: and 2, which can be tested in SIDIS more directly than
in 6+6_ hadroproduction or Drell-Yan process.

Second, SIDIS can be studied in the kinematical region covered by the measure-
ments of the hadronic structure functions F,(:r, Q2) in completely inclusive DIS. The
ongoing DIS experiments at the ep collider HERA probe F,(a:, Q2) at :1: down to 10“”,
which are much smaller than lower values of it reached at the existing hadron-hadron
colliders. The region of low as, which is currently studied at HERA, will also be
probed routinely in the production of wit, Z 0 and Higgs bosons at the LHC. At such
low values of 2;, other dynamical mechanisms may compete with the contributions

from the soft and collinear radiation described by the CSS formalism. The study of

the existing SIDIS data provides a unique opportunity to learn about the applicabil-
ity of the CSS formalism in the low-a: region and estimate robustness of theoretical
predictions for the electroweak boson production at the LHC.

Last, but not the least, is the issue of potential symmetry relations between the
resummed observables in SIDIS, e+e’ hadroproduction and Drell-Yan process. In
SIDIS, the dynamics associated with the initial-state radiation may be similar to the
initial-state dynamics in the Drell-Yan process, while the final-state dynamics may
be similar to the final-state dynamics in e+e‘hadroproduction. It is interesting to
find out if the data support the existence of such crossing symmetry.

The results presented here were published or accepted for publication in Ref. [35—
37]. The remainder of the thesis is organized as follows. In Chapter 2, I discuss
the basics of factorization of mass singularities in hadronic cross sections. Then I
review the general properties of the Collins-SOper-Sterman resummation formalism
and illustrate some of its features with the example of hadroproduction at e+e‘
colliders.

In Chapter 3, I apply the resummation formalism to semi-inclusive deep inelastic
scattering. Guided by the similarities between SIDIS, e+e‘ hadroproduction and
Drell-Yan process, I introduce a set of kinematical variables that are particularly
convenient for the identification and subsequent summation of the soft and collinear
logarithms. I also identify observables that are directly sensitive to the multiple
parton radiation. In particular, I argue that such radiation affects the dependence of
SIDIS cross sections and hadronic energy flow on the polar angle in the photon-proton
center-of-mass frame. Next I derive the 0(a3) cross section and obtain the 0(ag)
coefficients for the resummed cross sections and the hadronic energy flow.

In Chapter 4, I compare the results of the CSS resummation formalism and 0(a3)

fixed-order calculation with the data from the ep collider HERA. I show that the CSS

resummation improves theoretical description of various aspects of these data. I also
discuss the dependence of the resummed observables on the longitudinal variables
a: and z. I show that the HERA data are consistent with the rapid increase of
nonperturbative contributions to the resummed cross section at a: S, 0.01. I discuss
the potential dynamical origin of such low-a: behavior of the CSS formula.

Finally, in Chapter5 I discuss the impact of the multiple parton radiation on az-
imuthal asymmetries of the SIDIS cross sections. I show that the CSS resummation
formalism can be used to distinguish reliably between perturbative and nonperturba-
tive contributions to the azimuthal asymmetries. I also suggest to measure azimuthal

asymmetries of the transverse energy flow, which provide a clean test of PQCD.

Chapter 2

Overview of the QCD factorization

Perturbative calculations in Quantum Chromodynamics rely on a systematic proce-
dure for separation of short- and long-distance dynamics in hadronic observables. The
proof of feasibility of such procedure naturally leads to the methods for improvement
of the convergence of the perturbative series when this convergence is degraded by
infrared singularities of contributing subprocesses. Here I present the basics of the
factorization procedure. The omitted details can be found in standard textbooks on

the theory of strong interactions, e.g., Refs. [38-41].

2.1 QCD Lagrangian and renormalization

Low-energy hadronic states have internal substructure. They are composed of more
fundamental fermions (quarks) that are bound together by non-Abelian gauge forces.
The quanta of the QCD gauge fields are called gluons. Quantum ChromoDynamics

(QCD) is the theory that describes strong interactions between the quarks. In the

classical field theory, the QCD Lagrangian density in the coordinate space is

agape) = Ewe—9M. Hume
f

1
4

A
FffiFaafi — Ema/4:? + 5a (6mm ' 3 — 90¢;de ' Ab) Cd, (2-1)

where 1,!) f1(:r), Af,‘(:r) and ca(:c) are the quark, gluon and ghost fields, respectively;
F5” (:13) E 00.45 — om: - 900,0.43/15 (2.2)

is the gauge field tensor; —A(na - A3)2/2 is the term that fixes the gauge 17 - A = 0.
The vector 77" is equal to the gradient vector 6" in covariant gauges (60A: = 0) or an
arbitrary vector n“ in axial gauges (naAg‘ = 0). The color indices l, m vary between
1 to NC (where NC = 3 is the number of colors), while the color indices a, b, c, d vary
between 1 and NC2 — 1. The index f denotes the flavor (i.e., the type) of the quarks,
which is conserved in the strong interactions. The remaining parameters in LQCD(:c)
are the QCD charge 9 and the masses of the quarks m f.

The QCD Lagrangian is invariant under the gauge transformations of the S U (No)

color group:

we) 4 (Joanna); (23)
neon) —> U<9<x>>T.A:(x>U-‘<o<x))+3<aaU(e(w>>)U-1<6<x)), (2.4)

9
where the 113—dependent unitary operator U (6(zr)) is
U(6(a:)) E e‘iTagah‘). (2.5)

(T 0),", and COM are generator matrices and structure constants of the color group.

10

The commutators of the matrices (Ta);m are

[Tau Tb] = iCabcTc- (2.6)

The quark fields «of; and gauge fields A: are vectors in the fundamental and adjoint
representations of S U (N), respectively.

In the quantum theory, 212,, A“, ca are interpreted as “bare” (unrenormalized) op-
erators of the corresponding fields; 9 and m, are interpreted as the “bare” charge
and masses. The perturbative calculation introduces infinite ultraviolet corrections
to these quantities. In order to obtain finite theoretical predictions, EQCD has to be
expressed in terms of the renormalized parameters, which are related to the “bare”
parameters through infinite multiplicative renormalizations.

If the ultraviolet singularities are regularized by the continuation to n = 4 — 26,
e > 0 dimensions [42], the renormalized parameters (marked by the subscript “R”)

are related to the bare parameters as

$11201) = 2,3100%, (2-7)
3120‘) = 2,1100%, (2-8)
c0301) = 251006... (2-9)
91201) = Zg“(#)#“g, (2-10)
mmm) = Z;‘(#)m;, (2-11)

where Z), Z A, Z6, Z9, and Zn are perturbatively calculable renormalization constants.
In the dimensional regularization, the renormalized parameters depend on an auxil-
iary momentum scale an, which is introduced to keep the charge 9 dimensionless in

n 75 4 dimensions. In Eqs. (2.7-2.11) the renormalized parameters and the constants

11

Z, are expressed in terms of another scale a, which is related to an as
a2 = 47re'751ri. (2.12)

Here 73 = 0.577215... is the Euler constant.

2.2 Asymptotic freedom

The further improvement of the theory predictions for physical observables is achieved
by enforcing their invariance under variations of the scale ,u, i.e., by solving renormal-
ization group (RG) equations. Consider an observable S that depends on N external

momenta pf, i = 1,. . . , N. If the renormalized expression for S is

S (91201), {mm(u)}a {201-}, u)

(where “{. . . }” denotes a set of parameters), then the RG-improved expression for S

S (601), {771101)}, {pi}, M), (2-13)

where §(,u) and m;(,a) are the running QCD charge and quark masses. By solving

the equation for the independence of S from ,a,

#525 (gm), {mini}, {1).}, u) = o, (2.14)

we find the following differential equations for 9(a) and fizfm):

12

Bow)

#7,,— = new». (2.15)
LE“) = —7m;<ga)>m;a>. (216)

The approximate expressions for the functions 5(9) and ym(g) on the r.h.s. of
Eqs. (2.15) and (2.16) are found from the a—dependence of the fired-order renor-

malized charges and masses:

 

{301300) Maggi”), (2.17)

_ 1 #87n%R(a)
2mla(#) 6a

 

7mf(ga(u)) (2'18)

The renormalization group analysis of the QCD Lagrangian suggests that the
interactions between the quarks weaken at high energies, i. e., that Quantum Chro-
modynamics is asymptotically free in this limit. Indeed, the perturbative series for

the function 6(9) is

{3(9) = mi (217%)]: fit. (2.19)

where as E 92/47r is the QCD coupling. In the modified minimal subtraction (W)

regularization scheme [43], the lowest-order coefficient 61 in Eq. (2.19) is given by

11 4
fir = E-CA-ETRNf, (2.20)

where N; is the number of active quark flavors, CA = NC = 3, and TR 2 1/2. By

13

solving Eq. (2.15), we find that

 

513 (u) _ 515040)

'_ _ 2 o (2.21)
1+ %£,81 ln 9-7
1r #0

This equation proves the asymptotic freedom of QCD interactions: for six known
quark generations, 61 > 0 and

lim C750!) 2 0.

11,—)00

Higher-order corrections to the beta-function do not change this asymptotic behavior.
Eq. (2.21) also shows that @sUt) has a pole at some small value of a. The position of

this pole can be easily found from the alternative form of Eq. (2.21),

_ _ 47r
as(,a) — 51 111(l‘2/A2QCD) [1 + . . .]. (2.22)

 

In Eq. (2.22), AQCD is a phenomenological parameter, which is found from the analysis
of the experimental data. The most recent world average value of AQCD for N f = 5
and 0(a‘fg) expression for the fi-function is 208f§§ MeV [44]. According to Eq. (2.22),
as(a) becomes infinite when a = AQCD. This feature of the QCD running coupling

obstructs theoretical calculations for hadronic interactions at low energies.

2.3 Infrared safety

Due to the asymptotic freedom, the calculation of QCD observables at large a can
be organized as a series in powers of the small parameter §(a). To find out when the

perturbative calculation may converge rapidly, consider the formal expansion of the

14

RG-improved expression (2.13) for the observable S in the series of 9(a):

s = «Papa, mania“ ([11,73] , [MD aka). (2.23)

2
k=0 'u

In this expression, the function <I>({p,—}, {inf}, ,a) includes all coefficients that do not
depend on the order of the perturbative calculation (for instance, the phase space
factors). The mass dimension of <I>({p,}, {772,}, a) is equal to the mass dimension of
S. The sum over k on the right-hand side is dimensionless. The coefficients of the
perturbative expansion S (k) depend on dimensionless Lorentz—invariant combinations
of the external momenta pf, the mass parameter a, and the running quark masses
772,01). There are indications that the perturbative series in Eq. (2.23) are asymptotic
[45], so that it diverges at sufficiently large k. However, the lowest few terms of this
series may approximate S sufficiently well if they do not grow rapidly when k increases.

The factors that control the convergence of Eq. (2.23) can be understood in a
simpler case, when all Lorentz scalars p,- - p,- in Eq. (2.23) are of the same order Q2.

Then Eq. (2.23) simplifies to

 

S = <1>({pi}, Wife) :0: 5“” ('3; {7711002 ]) 9%)- (224)

2
k=0 'u

When Q2 >> A500, we can choose a N Q to make g7(,a) small. This choice also elim-
inates potentially large terms like ln(Q2/a2) from the coefficients Sm. In addition,
let’s assume that Q is much larger than any quark mass mf(a) on which S depends.
For instance, S may be dominated by contributions from the a, d, s quarks, whose

running masses are lighter than 200 MeV at a = 2 GeV [44]. At ,u' > 2 GeV, the

15

quarks become even lighter due to the running of mi:

TIM/1') = 771101) 6XP {- [[1 ggvmflfll} < 771101), (2-25)

since in QCD

info) = 333,930}? + O(o‘z§) > 0. (2.26)

Here OF E (N3—1)/(2NC) = 4/3.

When the quark masses vanish, many observables, which are finite if 77m 75 0,
acquire infrared singularities. These singularities are generated from the terms in
the perturbative coefficients that are proportional to the logarithms ln(m}/a2). The
expansion in the perturbative series (2.24) makes sense only for those observables S
that remain finite when mf(a)2/p2 —> 0.

There are two categories of observables for which the perturbative expansion (2.24)
is useful. In the first case, the coefficients S (k) are finite and analytically calculable

when a —-> oo :

<{—}><><{<——>}>

Such observables are called infrared-safe [16]. For instance, the total and jet pro-
duction cross sections in e+e" hadroproduction are infrared-safe. In this exam-
ple, hadrons appear only in the completely inclusive final state. According to the
Kinoshita-Lee-Nauenberg (KLN) theorem [46], such inclusive states are free of in-
frared singularities, so that the finite expressions for the total and jet cross sections

can be found from the massless perturbative calculation.

16

 

Figure 2.1: Factorization of collinear singularities in completely inclusive electron-

hadron DIS

In the second case, S (k) are not infrared-safe, but all mass singularities of S (k) can
be absorbed (factorized) into one or several process-independent functions. These
functions can be measured in one set of experiments and then used to make predictions
for other experiments.

To understand which singularities should be factorized, notice that there are two
classes of the infrared singularities in a massless gauge theory: soft singularities and
collinear singularities. The soft singularities occur in individual Feynman diagrams
when the momentum k“ carried by some gluon line vanishes (k" ~ Aft“, where /\ —+ 0
and K.“ are fixed). The soft singularities cancel at each order of @501) once all Feynman
diagrams of this order are summed over.

In contrast, the collinear singularities occur when the momenta 12’; and p5 of two
massless particles are collinear to one another , i.e., when p1 - p2 —> 0. Since one or

both collinear particles can be simultaneously soft, the class of the collinear singu-

17

larities partially overlaps with the class of the soft singularities. The soft collinear
singularities cancel in the complete fixed-order result just as all soft singularities
do. On the contrary, the singularities due to the collinearity of the particles with
non—vanishing momenta do not cancel and should be absorbed in the long-distance
phenomenological functions.

As an illustration of the factorization of the purely collinear singularities, consider
the factorized form for the cross section of inclusive deep inelastic scattering e+A -7—>

e + X (where A is a hadron) in the limit Q2 —> oo:

_ dahard (L' Q
dZdQ2 ‘- Z/x (16“de sz (a 5(Q) {a #7) Fa/A(€aa/1'F) +0 (Q17). (2.28)

This representation and notations for the particle momenta are illustrated in Fig. 2.1.

In Eq. (2.28), Q2 E —q2 is the large invariant mass of the virtual photon 7*,
a: E Q2 / (2(1)); - q)). These variables are discussed in more detail in Subsection 3.1.1.
dagard/ (do: dQQ) is the infrared-safe (“hard”) part of the cross-section for the scattering
e + a -—> e + X of the electron on a parton a. Fa/A(€a, up) is the parton distribution
function (PDF), which absorbs the collinear singularities subtracted from the full
parton- -level cross section to obtain dohard/ (d2: dQ2). In the inclusive DIS, all collinear
singularities appear due to the radiation of massless partons along the direction of
the initial-state hadron A. The final state is completely inclusive; hence, by the KLN
theorem, it is finite.

The collinear radiation in the initial state depends only on the types of a and A
and does not depend on the type of the particle reaction. Therefore, Fa/A(§a, up)
is process-independent. It can be interpreted as a probability of finding a massless
parton a with the momentum {019’}, in the initial hadron with the momentum pi. To

obtain the complete hadron-level cross section, we sum over all possible types of a

18

(a = g, u, a, (1, cl, . . . ) and integrate over the allowed range of the momentum fractions
5. (Massey)-

In Eq. (2.28), both the “hard” cross sections dSZOTd/(dcc dQQ) and the parton dis-
tribution functions Ira/AK“, up) depend on an arbitrary factorization scale up, which
appears due to some freedom in the separation of the collinear contributions included
in Fa/A(£a, up) from the “hard” contributions included in do’mrd/(da: dQ2). Of course,
the complete hadron-level cross section on the lbs. of Eq. (2.28) should not depend
on up. Hence the ftp-dependence of Fa/A(€a,;1p) should cancel the ftp-dependence of
the hard cross section. This requirement is used to find Dokshitser-Gribov-Lipatov-

Altarelli-Parisi (DGLAP) differential equations [47], which describe the dependence

0f Fa/A(5a,l1F) on HF:

 

dFa (5a,
[AF/1:! (IF) :(Z’P ab ® Fb/A) (Em/1F). (2.29)

b

Here be(§,u) are “spacelike” splitting functions that are currently known up to
0(ag) [48]. They describe evolution of partons with spacelike momenta. The convo-

lution in Eq. (2.29) is defined as

(f ®9)(:v.u) a / f($/€.u)g(€.u)g§ (2.30)

f .

A similar approach can be used to derive factorized cross sections for reactions with
observed outgoing hadrons. Such cross sections depend on fragmentation functions
(FFs) Ds/b(€, up), which absorb the singularities due to the collinear radiation in
the final state. The fragmentation function can be interpreted as the probability of
finding the hadron B among the products of fragmentation of the parton b. The

variable E is the fraction of the momentum of b that is carried by B. In the presence

19

of F F s, the hadron-level cross section becomes dependent on yet another factorization
scale up. Similarly to the PDFs, the dependence of the FPS on [1.0 is described by

the DGLAP evolution equations:

dDB/b(€b, #0)
MD at
#D

 

= 2(03/1. 8’ pill.) (gm/1D): (2-31)

0

where ’PZMC, ,a) are the “timelike” splitting functions.

As in the case of the renormalization scale a, it is natural to choose the factor-
ization scales up and up of order Q to avoid the appearance of the potentially large
logarithms 1n (Q/up) and In (Q/uD) in the “hard” cross section. I should emphasize
that the factorized cross sections are derived under the assumption that all Lorentz
scalars p,- - 19,- are of order Q2, so that a: in Eqs. (2.28) is sufficiently close to unity.
When some scalar product p,- - 11,- is much larger or smaller than Q2, the convergence
of the perturbative series for the hard cross section is worsened due to the large loga-
rithms of the ratio p,- - pj/Q2. This is a general observation that applies to any PQCD
calculation. In some cases, the predictive power of the theory can be restored by the
summation of the large logarithms through all orders of the perturbative expansion.
In particular, the resummation of the large logarithms is required for the accurate
description of the angular distributions of the final-state particles, including angular
distributions of the final-state hadrons in SIDIS. In the next Section, I discuss general
features of such resummation on the example of angular distributions of the jets in

e+e" hadroproduction.

20

Production of hadrons in 2° decays

a) No QCD radiation
q

/
/

C]
b) QCD radiation

 

Figure 2.2: The space-time picture of hadroproduction at e+e‘ colliders

21

2.4 Two-scale problems

2.4.1 Resummation of soft and collinear logarithms

To understand the nature of the problem, consider the process e+e‘ —+ Z0 —> jets
(Fig. 1.1a). The space-time picture of this process is shown in Figure 2.2. Let us
assume that the Zo-bosons are produced at the resonance (Ee+ + E8.- = M 2) at rest
in the laboratory frame. In e+e‘ hadroproduction, the hadronic decays are initiated
predominantly by the direct decay of the ZO-boson into a quark-antiquark pair. The
QCD radiation off the quarks produces hadronic jets, which are registered in the
detector.

If no additional hard QCD radiation is present (Fig. 2.2a), the decay of the Z0
boson produces two narrow jets escaping in the opposite directions in the lab frame.
The typical angular width of each jet is of the order AQCD / E A, 3 << 1, where EAB a:
M z/ 2 are the energies of the jets. The quarks may also emit energetic gluons, in
which case the angle between the jets is not equal to 7r (Fig.2.2b). If the angle 6
in Fig. 2.2b is large, the additional QCD radiation is described well by the rapidly
converging series in the small perturbative parameter“ as(MZ)/7r. But when 6 —> 0,
the higher-order radiation is no longer suppressed, because the smallness of a s(M z) / 7r
is compensated by large terms ln"(02/4)/62, p 2 0 in the hard part of the hadronic
cross section. Therefore, the calculation at any fixed order does not describe reliably
the shape of the hadronic cross section when 6 —> 0.

To illustrate this point, consider the hadronic energy-energy correlation [49], de-

fined as

 

 

d): 1 “2” W2 do
E — d . 2. 2
dcoso Mg/0 EA], “EB EAEBdEAdEBdcosd ( 3 )

 

‘From here on, I drop the “bar” in the notation of the running as.

22

In the limit 6 —-> 0, but 6 5A 0 ,dE/dcos6 behaves as

 

d2 1 °° as(Mz) * 2"“ 62
z — —— .m1 m — , .
dcos6 040 62 g ( 7r ) "12:0 c), n 4 (2 33)

 

where ckm are calculable dimensionless coefficients. Additionally there are virtual
corrections to the lowest order cross section, which contribute at 6 = 0. Suppose we
truncate the perturbative series in Eq. (2.33) at k = N. If N increases by 1 (that
is, if we go to one higher order in the series of as), the highest possible power of
the logarithms lnm(62/4) on the r.h.s. of Eq. (2.33) increases by 2. Therefore, the
theoretical prediction does not become more accurate if the order of the perturbative
calculation increases. Equivalently, the energy-energy correlation receives sizeable
contributions from arbitrarily high orders of as.

To expose the two-scale nature of this problem, let us introduce a spacelike four-

vector qf‘ and a momentum scale qT as

uQ'PB pQ'PA

 

 

u — u
t APA 'PB BPA 'PB ( )
(1% E win. > 0. (2-35)

where q“, pi, p73 are the momenta of the ZO-boson and two jets, respectively. The
vector qf‘ is interpreted as the component of the four-momentum q“ of the Z 0-boson

that is transverse to the four-momenta of the jets; that is,

(It ‘ PA 2 91 'PB = 0- (2.36)

The orthogonality of qf‘ to both pf, and p]; follows immediately from its defini-
tion (2.34).

23

In the laboratory frame,

q“ = (ll/12.5); (2.37)
Pi; = EA“: 77A); (2.38)
P; : EB(11 _fiB)1 (2.39)

where EA, 733,; and EB, —r'is are the energies of the jets and the unity vectors in the
directions of the jets, respectively. The large invariant mass q2 = M; of the Z 0-boson
can be associated with the QCD renormalization scale Q2. Let the z-axis be directed
along r‘iA. Then qT coincides with the length of the transverse component (YT of qf‘:

6 6
(1ft : (—A[Z tan 5, (1T, 0, —ll[Z tan é) .

At the same time

q§~ _1—cos6

————— 2.40
Q2 1+cos6’ ( )
and
2 2 2
- 21-9; 9_
lflQ2—4(1+6+OH)' (2.41)

We see that the problems at 6 -—> 0 arise due to the large logarithmic terms

lnm(Q%/Q2)/ (1% when (If/Q2 << 1:

 

 

 

«12 .. 9: r9.
dcos6 9—»0 2 dq% (”40
_ 1 00 05(Q) k 2k—l I m (122‘

24

 

Figure 2.3: The structure of infrared singularities in a cut diagram D for the energy-
energy correlation in the axial gauge

where

2

The origin of these logarithms can be traced back to the presence of infrared
singularities in the QCD theory. Before considering these singularities, notice that
the energy-energy correlation is sufficiently inclusive to be infrared-safe. Therefore,
the complete expression for the energy-energy correlation is finite at each order of
as(u). On the other hand, the infrared singularities do appear in individual Feynman
diagrams. According to the discussion in Section 2.3, these singularities are due to
the emission of soft gluons.’r Although the soft singularities cancel in the sum of all
Feynman diagrams at the given order of as, this cancellation leaves large remainders

lnm(q%/Q2)/q% if qT is small.

lThe purely collinear singularities do not appear because of the overall infrared safety of the
energy-energy correlation.

 

25

Fortunately, not all coefficients cjm, in Eq. (2.42) are independent. Refs. [50,51]
suggested that the leading logarithmic subseries in Eq. (2.42) and in analogous ex-
pressions in SIDIS and Drell-Yan process can be summed through all orders of as.
The possibility to sum all logarithmic subseries in Eq. (2.42) and restore the conver-
gence of the series in as was proven by J. Collins and D. Soper [17]. Schematically,

Eq. (2.42) can be written as [23]

 

d2 _1_{
dqgw qT—)O (1%
(15(L-f'1)
+ a§(L3 +L2+L+1)

+ a; (L5 +L4+L3+L2+L+1)

+ ...], (2.44)

where L E ln(q%/Q2), and the coefficients 2czm/(7er2) are not shown. This series

can be reorganized as

 

1% (”.40 ~ —1%-{asZ1 + aSZg +. ..,}

where
asZI ~ as(L+1)+a§(L3+L2)+a§(L5+L4)+... |A1,Bi.Co;
.32. ~ a§(L+ 1) +a§<L3 +L2) +~- IAN-32:61;
ang ~ a§(L+1)+... |A3,Bg,C2;

(2.45)

In Eq. (2.45), the right-hand side shows the new coefficients Ak,Bk,Ck_1 that are

required to calculate each new subseries ang. The complete subseries ang can be

26

reconstructed as soon as the coefficients Ak, Bk,Ck_1 are known from the calculation
of the term a§(L + 1). Each successive subseries ang in Eq. (2.45) is smaller by
as than its predecessor, so that as regains its role of the small parameter of the
perturbative expansion.

The rule that makes the resummation of the subseries a’ng possible follows from
(a) the analysis of the structure of the infrared singularities in the contributing Feyn-
man diagrams at any order of as(u) and (b) the requirement that the full energy-
energy correlation is infrared-safe and gauge- and renormalization-group invariant.

The structure of the infrared singularities can be identified from the analysis of
analytic properties of the Feynman diagrams with the help of the Landau equations
[52—54] and the infrared power counting [14,38, 55]. This structure for some con-
tributing cut diagram D is illustrated by Figure 2.3. Throughout this discussion the
axial gauge C - A = 0 is used.1 In D we can identify two jet parts JA, J3, the hard
vertex H, and possibly the soft subdiagram 89. By their definition, the jet parts J A
or J3 are the connected subdiagrams of D that describe the propagation of nearly
on-shell massless particles inside the observed jets. Each of the particles in the jet

part J A has a four-momentum pf that is proportional to the momentum of the jet A:

pfi‘ = 6.195.. (2.46)
where
0 g s,- g 1, and Za- = 1. (2.47)

Similar relations hold for the momenta of the particles in the jet part J B.

 

*The discussion of the infrared singularities in covariant gauges can be found, for instance, in
Ref. [38].

27

 

a) bi

Figure 2.4: Examples of the finite soft subdiagrams: (a) the subdiagrams that are
connected to JA, J3 by one or several quark lines; (b) the subdiagrams that are

connected to H

Both jets originate from the hard vertex H that contains contributions from the
highly off-shell particles. In the axial gauge the jet parts are connected to H only
through the single quark lines. Since the hard scattering happens practically at one
point, H depends only on Q2 and not on q?“

After the jets are created, they propagate in different directions with the speed
of light. Due to the Heisenberg uncertainty principle, these jets, which are separated
by large distances, do not interact with one another except by the exchange of low
momentum (soft) particles. The infrared singularities, which are associated with the
long-distance dynamics, can occur only in the jet parts or the soft subdiagram. This
observation can be refined by the dimensional analysis of the Feynman integrals in
the infrared limit [14, 38, 55], which shows that the infrared singularities are at most

logarithmic. Also, those soft subdiagrams that are attached to the jet parts J A and

28

J3 with one or more quark propagators (Fig. 2.4a) or are connected to H (Fig. 2.4b)
are finite.

To summarize, the infrared singularities of any individual Feynman diagram reside
in the soft parts of the jets J .4, J B and in subdiagrams Sg that are connected to J A, J B
by soft gluon lines (cf. Fig. 2.3). Both types of singularities contribute at qT = 0 (i.e.,
q" = pt; + 11%), in agreement with the expectation that the small-qT logarithms are

remainders from the cancellation of such singularities. Therefore, at small qT the

distribution dE/dq% naturally factorizes as

22

qu (IT—>0

H(Q2) cout COBut 8((13‘, Q2), (248)

where H (Q2) is the contribution from the pointlike hard part, 8(q%,Q2) is the all-
order sum of the large logarithms, and C0”; collect finite contributions from the jet
parts. Clearly, Cf,” = cg“ due to the symmetry between the jets.

The factorized formula is proven by considering the Fourier-Bessel transform of

d2/dq% to the space of the impact parameter b, which is conjugate to if. Explicitly,

 

 

 

where
=Ze§c0“‘( ()(Jl,c2 )c;, (C C2)e -Sb00102>. (2.50)
In Eqs. (249,2.50),
Se..- = Q2 (2.51)

29

is the square of the center-of-mass energy of the initial-state electron and positron;
2]. denotes the summation over the active quark flavors (i.e., j = u,1’1,d,d, . ..)

7

E} are the couplings of the quarks to the ZO-bosons§; (Go/Se+e-—) 21.6? is the Born
approximation for the hard part H (Q2). The Fourier-Bessel transform of the shape
factor S(q%, Q2) is given by e'sw'Q’Cl’CZ), where S (b, Q, 01,02) is called the Sudakov
function. At b2 << A5200 (i.e., in the region of applicability of perturbative QCD),

the Sudakov function is given by the integral between two momentum scales of the

order Q and 1 / b, respectively:

 

03¢)2 d—z C2 2
limsao,c.,o.) = / _—‘:(A(as<m,cl)1n i? weaning»),
b—>0 Gil/b2 H [1:

(2.52)

where .A and B can be calculated in PQCD. Cl and 6'2 are arbitrary constants of the
order 1 that determine the range of the integration in S (b, Q). The undetermined
values of these constants reflect certain freedom in separation of the collinear-soft
contributions included in S P (b, Q) from the purely collinear contributions included in
C-functions. At each order of as, changes in S1D (b, Q) due to the variation of C1, C2
are compensated by the opposite changes in the C—functions. Hence the perturba-
tive expansion of 1717;; does not depend on these constants. However, the complete
form-factor H72 in Eq. (2.50) does have residual dependence on (71,02 because of the
exponentiation of the terms depending on Cl and 02 in exp (—S (b, Q, Cl, 02)). The

variation of 01,02 allows us to test the scale invariance of the separation of soft and

 

§For the up quarks,

@- e l-4sin26
"sinzow 2 3 W ’

where e is the charge of the positron and 6w is the weak mixing angle. For the down quarks,

E'*—e—- --1-+Zsin26
J-sin26w 2 3 W'

30

 

collinear contributions in the W‘s-term.

At b2 > A5271), the behavior of S is determined by complicated nonperturbative
dynamics, which remains intractable at the current level of the development of the
theory. At large b the Sudakov function S is parametrized by a phenomenological
function S N P (b, Q), which has to be found from the comparison with the experimental
data. When Q —> 00, the sensitivity of the resummation formula to the nonpertur-
bative part of S (b, Q) is expected to decrease.

Now suppose that the experiment identifies a hadron H A in the jet J A and a hadron
B in the jet J B. Let 2A1; be the fractions of the energies of the jets J A and J3 carried
by H A and H B, respectively. The cross section of the process e+e‘ L0) H AH BX is
no longer infrared-safe because of the collinear singularities due to the fragmentation
into the hadrons H A and H 3. Nonetheless, in the limit qT —> 0 the cross section

do/(dzAdzBdfi) factorizes similarly to Eqs. (249,250):

daHAHB
dzAdzsdqg.

 

a d25 .- ~~
: 5 +0— /-(-§7T_)§equ.bw/HAHB (biz/4128): (2'53)

 

qT—+O

where at b —> 0

”1111130) 211.23) 2:82- x

.7

(Z DHA/a @633”) (ZAJJMID) (Z DHB/b @ngyt) (28,1), vole-S;
“ b

()
a,b=g,(u), d,

. (u-H)
J: ,d,.... (2.54)

The only major difference between the form-factor WI“ H3 for the hadron pair pro-

duction cross section do/(dzAdzqu%) and the form-factor W; for the energy-energy

31

 

correlation dZ/dfi. is the presence of the fragmentation functions DH/a(§, ,a), which
absorb the collinear singularities due to the final-state fragmentation into the observed
hadrons H A, H B. The FFs are convolved with the coefficient functions
C:,',“(§, Cl, Cg, up, b), which absorb finite contributions due to the perturbative collinear
radiation.

The same resummation technique can also be applied to the production of vector
bosons (e. g., virtual photons 7*, which decay into lepton-antilepton pairs) at hadron-
hadron colliders (Fig. 1.1b). In this process, the four-vector q‘,“ is introduced using
the same definition (2.34), where now pA and p3 denote the momenta of the initial
hadrons A and B. The scale qT is just the magnitude of the transverse momentum pT

of 7“ in the center-of-mass frame of the hadron beams (Fig. 2.5), since in this frame

qé‘ = (0.pT.0.0)-

Therefore, the b—space resummation formalism [21] applies to the production of vector
bosons with small transverse momenta. The cross section for the production of the
virtual photon ’y“ at qT —> 0 can be factorized as

_ 06 (Pl;
— SAB (271')2

do7

—— "175117 b .w , 2. 5
dQ2dydq% e 7( iTAaTB), ( 5 )

 

(yr—>0

where Q2 and y are the virtuality and rapidity of 7* in the lab frame, 334,3 E Egg-e“,

and

W7(baan$B)[b_)0 = :6? (C321 ® Fa/A) (13A. 1)./1) (Cl: ‘8 Fb/B) ($3.1). tile—S-

a.b.J'

(2.56)

In F177, e,- are fractional electric charges of the quarks (ej = 2/3 for up quarks and

32

—1/3 for the down quarks). (6;: ® Fa/A) (22,4, b, u) and ((331281 Fb/B) (.133, b, a) are the
jet parts corresponding to the incoming hadrons A and B. They are constructed from
the perturbatively calculable coefficient functions C23 (5, b, u) convolved with the PDFs
for the relevant partons. The perturbative part of the Sudakov function in Eq. (2.56)
has the same functional dependence as in Eq. (2.52) for e+e’-hadroproduction. As in
the case of W2 and WHAHBi the large-b behavior of W, should be parametrized by a
phenomenological function.

To conclude, the b—space resummation formalism was originally derived to de-
scribe the production of hadrons at e+e" colliders [17] and production of electroweak
vector bosons at hadron-hadron colliders [21]. The possibility to apply the same
formalism to SIDIS relies on close similarities between the three processes. First,
hadronic interactions in all three processes are described by the same set of Feynman
diagrams in different crossing channels. Second, multiple parton radiation dominates
each of the three processes when the final-state particle escapes closely to the direc-
tion predicted by the leading-order kinematics. The formalism for the resummation

of such radiation can be formulated in Lorentz-invariant notations, so that it can be

continued from one process to another.

33

Vector boson production at
hadron-hadron colliders

—p

13 —P

a) No QCD radiation

q 51'
Assn,

b) QCD radiation

 

Figure 2.5: The space-time picture of Drell-Yan process

34

 

 

 

 

 

 

 

 

Figure 2.6: The ladder structure of the DIS cut diagrams

2.4.2 QCD at small 3:

According to the discussion in Section 2.3, the convergence of the series in as(u)
depends on the absence of very large or small dimensionless quantities in the pertur-
bative coefficients. In particular, the dimensionless variable :1: in the inclusive DIS
cross section should not be too close to zero: otherwise the hard part of the DIS cross
section contains large logarithms lnm(1/:r), which compensate for the smallness of
as(Q). These logarithms are different from the logarithms lnm #2 resummed by the
DGLAP evolution equations. As a result, the factorization of the DIS cross section in
the hard cross section and PDFs (cf. Eq. (2.28)) may experience difliculties at small 1:.

The large logarithms ln’"(1/:z:) are resummed in the formalism of Balitsky, Fadin,
Kuraev and Lipatov (BFKL) [56]. The BFKL and DGLAP pictures for the history of
the parton probed in the hard scattering are quite different. Both types of formalisms

resum contributions from the cut ladder diagrams shown in Figure 2.6. In this Fig-

35

ure, each “rung” R), is a two-particle irreducible subdiagram that corresponds to the
radiation off the probed parton line (see Refs. [38,57] for more details). The vertical
propagators correspond to the quarks or the gluons that are parents to the probed
quark. The momenta pigmn flow from the parent hadron to the probed quark. The
momenta kl‘,2,...,n flow through the rungs and are sums of the momenta of the radiated

particles. The conservation of the momentum in each rung implies that

prPf-f-l—kzfla 2:17H'1n9 (2.57)

where pf: H E pfi. In the reference frame where the hadron A moves at the speed of

light along the z—axis, :1: coincides with the ratio of the plus components of p3 and p53:

+
.1: = p—fl, (2.58)
PA
where
hi 2 k0 4: 14“. (2.59)

The DGLAP equation arises from the resummation of the ladder diagrams corre-
sponding to the collinear radiation along the direction of the hadron A. The radiating
parton remains highly boosted at each rung of the ladder. At the same time, the trans-
verse momentum carried away by the radiation grows rapidly from the bottom to the
top of the ladder. The DGLAP equation corresponds to the strong ordering of the

transverse momenta flowing through the rungs R; that is,

Q2 >> 1:3.1 > k?” >> --- >> k3." >> 43,00, (2.60)

36

while

pf~p§~m~pi§~p2>>0 (2.61)
and

pf~p§~---~p;~p.1~0- (2.62)

On the other hand, the BFKL formalism describes the situation in which the QCD

radiation carries away practically all energy of the probed parton. In this case,

I)? << 19'2“ << ---<< 1): << 19:, (2.63)
and

191’ >> 192" >> ~->> I); >> 192. (2.64)

In addition, the BFKL picture imposes no ordering on the transverse components

of kf‘:
15;, ~ 15.3., ~ ... ~ k3,, >> A300. (2.65)

As a result, the probed quark is likely to have a significant transverse momentum
throughout the whole process of evolution, which is impossible in the DGLAP picture.
Due to its large kT, the radiating parton is off its mass shell at any moment of its
evolution history, so that the BFKL radiation cannot be factorized from the hard
scattering. As another consequence of the kT-unordered radiation, the BFKL picture

implies broad angular distributions of the final-state hadrons, while in the DGLAP

37

picture the hadrons are more likely to belong to the initial- and final-state jets.

Since the BFKL approach applies to the limit :2: —> O and ka >> A2201), it corre-
sponds to asymptotically high energies of hadronic collisions. So far, the experiments
have produced no data that would definitely require the BFKL formalism to explain
them. In particular, the behavior of the inclusive DIS structure functions in the
low a: region at HERA agrees well with the 0mg) predictions of the traditional fac-
torized formalism and disagrees with the steep power-law growth predicted by the
leading-order solution of the BFKL equation [56].

The situation is not so clear for some less inclusive observables, which deviate
from the low-order predictions of PQCD. Specifically, SIDIS in the small-a: region
is characterized by large higher-order corrections. Some of these corrections can be
potentially attributed to the enhanced [CT-unordered radiation at 1: —> 0. If this is
indeed the case, the effects of the [CT-unordered radiation may be identified by observ-
ing the changes in the angular distributions of the final-state hadrons or “intrinsic kT”
of the partons. In order to pinpoint these effects, good understanding of the angular
dynamics in the traditional DGLAP picture is needed. Such understanding can be
achieved in the framework of the small-qT resummation formalism, which systemati-
cally describes angular distributions of the hard, soft and collinear radiation. Hence
it can be naturally used to organize our knowledge about the angular patterns of the

DGLAP radiation and search for the effects from new low-:2: QCD dynamics.

38

Chapter 3

Resummation in semi-inclusive DIS:

theoretical formalism

Deep-inelastic lepton-hadron scattering (DIS) is one of the cornerstone processes to
test PQCD. Traditionally, the experimental study of the fully inclusive DIS process
6 + A —+ e + X, where A is usually a nucleon, and X is any final state, is used to
measure the parton distribution functions (PDFs) for A. These functions describe the
long-range dynamics of hadron interactions and are required by many PQCD calcula-
tions. During the 1990’s, significant attention has been also paid to various aspects of
semi-inclusive deep inelastic scattering (SIDIS), for instance, the semi-inclusive pro-
duction of hadrons and jets, e+A —% e+B+X and e+A -—> e+jets+X. In particular,
the H1 and ZEUS collaborations at HERA, European Muon Collaboration at CERN,
and the E665 experiment at Fermi National Accelerator Laboratory performed ex-
tensive experimental studies of the charged particle multiplicity [58—63] and hadronic
transverse energy flows [64,65] at large momentum transfer Q. It was found that
some aspects of the data, e.g., the Feynman a: distributions, can be successfully ex-

plained in the framework of PQCD analysis [66,67]. On the other hand, applicability

39

of PQCD to the description of other features of the process is limited. For example,
the perturbative calculation in lowest orders fails to describe the pseudorapidity or
transverse momentum distributions of the final hadrons. Under certain kinematical
conditions the whole perturbative expansion as a series in the QCD coupling may fail
due to the large logarithms discussed in Section 2.4.

To be more specific, consider semi-inclusive DIS production of hadrons of a type B.
At large energies, one can neglect the masses of the participating particles. In semi-
inclusive DIS at given energies of the beams, any event can be characterized by two
energy scales: the virtuality of the exchanged vector boson Q and the scale qT intro-
duced analogously to e+e‘ hadroproduction and Drell-Yan process (cf. Section 2.4).
The scale qT is also related to the transverse momentum of B. The expansion in
the series of as is justified if at least one of these scales is much larger than AQCD.
However, the above necessary condition does not guarantee fast convergence of per-
turbative series in the presence of large logarithmic terms. If A2200 << Q2, q% << Q2,
the cross sections are dominated by the soft and collinear logarithms logm (q%/Q2) ,
which can be resummed in the framework of the small-qT resummation formalism
(Subsection 2.4.1). In the limit A2200 < q%, Q2 << q§a (photoproduction region)
PQCD may fail due to the large terms logm (Cf/(1%), which should be resummed into
the parton distribution function of the virtual photon [68]. Finally, even in the region
A2200 < q% ~ Q2 one may encounter another type of large logarithms corresponding
to events with large rapidity separation between the partons and/or the hadrons.
This type of large logarithms can be resummed with the help of the Balitsky—Fadin-
Kuraev-Lipatov (BFKL) formalism (Subsection 2.4.2).

In this Chapter I discuss resummation of soft and collinear logarithms in SIDIS
hadroproduction e + A —) e + B + X in the limit AZQCD << Q2, (1% << Q2. The

calculations are based on the works by Meng, Olness, and Soper [33,34], who analyzed

40

the resummation technique for a particular energy distribution function of the SIDIS
process.‘ This energy distribution function receives contributions from all possible
final—state hadrons and does not depend on the specifics of fragmentation.

Here the resummation is discussed in a more general context compared to [33,34]:
namely, I also consider the final-state fragmentation of the partons. Using this for-
malism, I discuss the impact of soft and collinear PQCD radiation on a wide class of
physical observables including particle multiplicities. The calculations will be done
in the next-to-leading order of PQCD. In the next Chapter, I compare the resumma-
tion formalism with the H1 data on the pseudorapidity distributions of the transverse
energy flow [64,65] and ZEUS data on multiplicity of charged particles [60] in the
7*]? center-of-mass frame. Another goal of this study is to find in which regions of
kinematical parameters the CSS resummation formalism is sufficient to describe the
existing data, and in which regions significant contributions from other hadroproduc-
tion mechanisms, such as the BFKL radiation [56], higher-order corrections including
multijet production with [68] or without [69,70] resolved photon contributions, or

photOproduction showering [71], cannot be ignored.

3.1 Kinematical Variables

I follow notations which are similar to the ones used in [33,34]. In this Section I
summarize them.

I consider the process

e+Aae+B+X an

 

‘The general features of the resummation formalism in semi-inclusive DIS were first discussed by
J. Collins [32].

41

where e is an electron or positron, A is a proton (or other hadron in the initial state),
B is a hadron observed in the final state, and X represents any other particles in the
final state in the sense of inclusive scattering (Fig. 1.2). I denote the momenta of A
and B by pf, and p73, and the momenta of the electron in the initial and final states
by l“ and 1'”. Also, q” is the momentum transfer to the hadron system, q" = l” — 1’”.
Throughout all discussion, I neglect particle masses.

I assume that the initial electron and hadron interact only through a single photon
exchange. Contributions due to the exchange of Z-bosons or higher-order electroweak
radiative corrections will be neglected. Therefore, q“ also has the meaning of the 4-
momentum of the exchanged virtual photon 7*; q" is completely determined by the
momenta of the initial- and final-state electrons. In many respects, DIS behaves as
scattering of virtual photons on hadrons, so that the theoretical discussion of hadronic

interactions can often be simplified by considering only the photon-proton system.

3. 1 . 1 Lorentz scalars

For further discussion, I define five Lorentz scalars relevant to the process (3.1). The

first is the center-of—mass energy of the initial hadron and electron VSCA where

S... :— (pA +1)2 = 2m - z. (3.2)

I also use the conventional DIS variables :1: and Q2 which are defined from the mo-

mentum transfer q“ by

Q2 E —q2 = 22 - 6’, (3-3)

42

Q2
21M ° (1'

 

a: (3.4)

In principle, :1: and Q2 can be completely determined in an experimental event by
measuring the momentum of the outgoing electron.

Next I define a scalar 2 related to the momentum of the final hadron state B by

Z___PB'PA : 2$PB°PA

q ' PA Q2 (35)

The variable .2 plays an important role in the description of fragmentation in the final
state. In particular, in the quark-parton model (or in the leading order perturbative
calculation) it is equal to the fraction of the fragmenting parton’s momentum carried
away by the observed hadron.

The next relativistic invariant q% is the square of the component of the virtual

photon’s 4-momentum q” that is transverse to the 4-momenta of the initial and final

hadrons:

(1% = «156... (3.6)
where

(12‘ = q" - 19" LE;- - "LB/1- (3-7)

APA'PB BPA'pB.

As discussed in Subsection2.4.1, the momentum qf‘ plays the crucial role in the re-
summation of the soft and collinear logarithms. In particular, a fixed-order PQCD
cross-section is divergent when qr —> 0, so that all-order resummation is needed to
make the theory predictions finite in this limit. According to Eqs. (3.5,3.7) qf = 0 if

and only if p]; = z (:ztp’:1 + q“) . Hence the resummation is required when the final-state

43

hadron B apprbximately follows the direction of 235,4 + (f.

In the analysis of kinematics, I will use three reference frames. The most obvious
frame is the laboratory frame, or the rest frame of the experimental detector. The
observables in this frame are measured directly, but the theoretical analysis is com-
plicated due to the varying momentum of the photon-proton system. Hence I will
mostly use two other reference frames, the center-of-mass frame of the initial hadron
and the virtual photon (hadronic c.m., or hCM frame), and a special type of Breit
frame which I will call, depending on whether the initial state is a hadron or a parton,
the hadron or parton frame. As was shown in Ref. [33], the resummed cross section
can be derived naturally in the hadron frame. On the other hand, many experimental
results are presented for observables in the hCM frame. These observables are not
measured directly; rather they are reconstructed from directly measured observables
in the laboratory frame. I will use subscripts h, cm and lab to denote kinemati-
cal variables in the hadron, hCM or laboratory frame. Below I discuss kinematical

variables in all three frames.

3. 1.2 Hadron frame

Following Meng et al. [33,34] the hadron frame is defined by two conditions: (a)
the energy component of the 4-momentum of the virtual photon is zero, and (b) the
momentum of the outgoing hadron B lies in the .732 plane. The directions of particle
momenta in this frame are shown in Fig. 3.1.

In this frame the proton A moves in the +2 direction, while the momentum transfer

6' is in the —z direction, and q0 is 0:

(1;: = (0) 0: 01_Q)3 (38)
p71,}; = 5% (1) 0103 1) - (3.9)

44

 

 

Figure 3.1: Geometry of the particle momenta in the hadron frame

The momentum of the final-state hadron B is

2 2

p _ ZQ qT 2qT QT
pB,h — 7(1+52-,—62—,0,52- — 1). (3.10)
The incoming and outgoing electron momenta in the hadron frame are defined in

terms of variables 1/2 and go as follows [83]:

[Z = % (cosh 1p, sinh (1) cos (0, sinh 1,!) sin (,0, —1) ,
3]," = % (cosh 1p, sinh 2/1 cos (,0, sinh 1/1 sin (0, +1) .

(3.11)

Note that cp is the azimuthal angle of [7,, or 8-], around the Oz-axis. 112 is a parameter

45

of a boost whiCh relates the hadron frame to an electron Breit frame in which I?“ =

(Q/2,0,0, —Q/2). By (3.2) and (3.11)

2& 2
:yA_1=§‘1’ (an)

 

cosh 2,!) =

where the conventional DIS variable y is defined as

Q2

1758A.

 

y (3.13)

The allowed range of the variable y in deep-inelastic scattering is 0 S y g 1 (see
Subsection 3.1.4); therefore 2,0 3 O.
The transverse part of the virtual photon momentum qf‘ has a simple form in the

hadron frame; it can be shown that

u__£%_ 0_& 3M
qt,h_( Q) (1T) 7 Q). (‘ )

In other words, qT is the magnitude of the transverse component of (Rh- The trans-

verse momentum pT of the final-state hadron B in this frame is simply related to qT,

by
PT = ZQT- (3.15)

Also, the pseudorapidity of B in the hadron frame is

0
77,, E — log (tan -%’3) = log a. (3.16)

The resummed cross-section will be derived using the hadron frame. To transform

the result to other frames, it is useful to express the basis vectors of the hadron frame

46

(T“, X ”, Y”, Z“) in terms of the particle momenta [34]. For an arbitrary coordinate

frame,

q‘”+2:1:p"j4
Q 3
1 p“ (12
Xl1=__§_#_[1l]”,
QT( Z q + Q2 IPA
Y“ = e“"""Z,,Tan,

p.
Z“=——q. -7
Q (31)

If these relations are evaluated in the hadron frame, the basis vectors T“, X “, Y”, Z "

are (1,0,0,0), (0,1,0,0),(0,0,1,0),(0,0,0,1), respectively.

 

 

a) Fragmentation of the quark current b) Fragmentation of the target hadron

Figure 3.2: (a) In the current fragmentation region, the hadron—level cross section
can be factorized into hard partonic cross sections 3M, parton distribution functions
Fa/A ((0, up), and fragmentation functions Dig/“Eb, up). (b) In the target fragmenta-
tion region, the hadrons are produced through the mechanism of diffractive scattering

that depends on “diffractive parton distributions” Man/AK... (3, up).

47

The limit of small qT, which is the most relevant for our resummation calculation,
corresponds to the region of large negative pseudorapidities in the hadron frame.
Hence the resummation affects the rate of the production of the hadrons that follow
closely the direction of the virtual photon. The region of negative 77,, is often called the
current fragmentation region, since the final-state hadrons are produced due to the
interaction of the virtual photon with the quark current. In the current fragmentation
region, hadroproduction proceeds through independent scattering and subsequent
fragmentation of partons. Therefore, in this region the hadron-level cross section
03A can be factorized in the cross sections 8,", for the electron-parton scattering
e+a —> e+ b+X, the PDFs Fa/A(€a, up), and the FFs D3/b(€b, up) (cf. Figure 3.2a).
The formal proof of the factorization in the current region of SIDIS can be found in
[32, 7 2].

In the Opposite direction 77;. >> 0 (qT —) +00) contributions from the current frag-
mentation vanish. Rather the produced hadron is likely to be a product of fragmenta-
tion of the target proton, which moves in the +z-direction (cf. Eq. (3.9)). According
to Eq. (3.5), such hadrons have 2 z 0. The target fragmentation hadroproduction is
described by a different approach, which relies on factorization of the hadron-level
cross section into cross sections of parton subprocesses and diffractive parton distri-
butions Ma,B/A(EQ,CB, up) (cf. Figure 3.2b). These distributions can be interpreted
as probabilities for the initial hadron A to fragment into the parton a, the hadron B,
and anything else. {a and CB denote fractions of the momentum of A that are carried
by the parton a and the hadron B, respectively. The distributions Ma,B/A({a, (3, up)
(also called fracture functions) were introduced in Refs. [73,74] and used in [67, 75—77]
to describe various aspects of SIDIS with unpolarized and polarized beams. The fac—
torization of cross sections in the target fragmentation region was formally proven in

the scalar field theory [78] and in full QCD [79, 80]. The recent experimental studies

48

of the diffractive scattering at HERA are reviewed in [81]. The detailed discussion
of diffractive scattering and interesting models [82] that are applied for its analysis is

beyond the sc0pe of this work.

 

 

Figure 3.3: Particle momenta in the hadronic center-of-mass (hCM) frame

3.1.3 Photon-hadron center-of-mass frame

The center-of-mass frame of the proton A and virtual photon 7* is defined by the
condition 5,4,6," + (fem = 0. The relationship between particle momenta in this frame

is illustrated in Fig. 3.3. As in the hadron frame, the momenta (fan and 5,1,6", in the

49

hCM frame are directed along the 02 axis. The coordinate transformation from the
hadron frame into the hCM frame consists of (a) a boost in the direction of the virtual
photon and (b) inversion of the direction of the Oz axis, which is needed to make the
definition of the hCM frame consistent with the one adopted in HERA experimental
publications. In the hCM frame the momentum of 7“ is

w2 _ Q2 w2 + Q2
u = _— ——
qcm ( 2W )0? 0) 2W 3 (3.18)

where 14’ is the hCM energy of the 7‘1) collisions,

W2 E (19.4 + q)2 = Q2 (i — 1) Z 0- (3-19)

Since all energy of the 7*p system is transformed into the energy of the final-state
hadrons, W coincides with the invariant mass of the B + X system.

The momenta of the initial and final hadrons A and B are given by

W2 + Q2 W2 + Q2
I‘ _ __ __
A,cm _ ( Zl/V 7 01 01 21V a (3.20)
pilgfim : (E8, E8 Sin 68.6"!) 0) EB COS 68,cm>a (3.21)
where
W2 + (12
E = —-—T— 3.22
B Z 2W ’ ( )
W2 - (1%
COS 63‘6", = m. (3.23)

The hadron and hCM frames are related by a boost along the z-direction, so that

50

the expression for the transverse momentum of the final hadron B in the hCM frame

is the same as the one in the hadron frame,

PT = ZQT- (3.24)

Also, similar to the case of the hadron frame, the relationship between qT and the

pseudorapidity of B in the hCM frame is simple,

qT = We‘"“". (3.25)

Since the directions of the z-axis are opposite in the hadron frame and the hCM
frame, large negative pseudorapidities in the hadron frame (QT —> 0) correspond to

large positive pseudorapidities in the hCM frame. Hence multiple parton radiation

effects should be looked for in SIDIS data at qT/ Q S 1, or

 

(I:

1 _
72cm 2, 1n ( x) > 2. (3.26)

The boost from the hadron to the hCM frame also preserves the angle (,0 between
the planes of the hadronic and leptonic momenta, so that the momenta l”, l’” of the

electrons in the hCM frame are

[gm = {Ill/17((W2 + Q2) cosht/2 + W2 - Q2), E:f—sinhibcos (,0,

—% sinh z/Jsin {’0’111/_V-((W2 + Q2) + (W2 — Q2) cosh 1b) }; (3.27)
I _ __1_ 2 2 _ 2 2 _Q_ '

leg—{4W((W +Q)cosh1/) W +Q),2srnhi,bcoscp,

—9 sinh Ibsin —1—(—W2 — Q2 + (W2 - Q2) COSh 1P) (3 28)

51

Finally I wOuld like to mention two more variables, which are commonly used in
the experimental analysis. The first variable is the flow of the transverse hadronic

energy

ET E Etot Sin 66m) (3.29)

where Em is the total energy of the final-state hadrons registered in the direction of
the polar angle 06",. The measurement of ET does not require identification of indi-
vidual final-state hadrons; hence ET is less sensitive to the final-state fragmentation.

The second variable is Feynman 1:, defined as

 

2 z 2
Clip E [aim = z < — q—T) . (3.30)

In (3.30) 1923‘", is the longitudinal component of the momentum of the final-state
hadron in some frame. For small values of qT, i.e., in the region with the highest

rate,

(Up x Z. (3.31)

52

 

 

Figure 3.4: Particle momenta in the laboratory frame

3.1.4 Laboratory frame

In the laboratory frame, the electron and proton beams are collinear to the 02 axis.
The definition of the HERA lab frame is that the proton (A) moves in the +2 direction
with energy E A, and the incoming electron moves in the —z direction with energy E.

The momenta of the incident particles are

piJab : (El/17030) EA) 2 (3.32)

53

15:1,, = (ED, 0, —E). (3.33)

We can use (32,332) and (3.33) to express the Mandelstam variable SM in terms of

the energies E A, E in the lab frame:
SBA = 4EAE. (3.34)

The outgoing electron has energy E’ and scattering angle 6 relative to the —z
direction. I define the Gas-axis of the HERA frame in such a way that the outgoing

electron is in the Oxz-plane; that is,
1ng = (E', —E' sin 6, 0, —E' cos 6). (3.35)

The four-momentum q“ = l“ — 1’“ of the virtual photon that probes the structure of

the hadron is correspondingly
qfiw = (E —- E', E' sin 0, 0, —E + E' cos 6) . (3.36)

The scalars a: and Q2 are completely determined by measuring the energy and the

scattering angle of the outgoing electron:

Q2 = 2EE’(1 — c056), (3.37)

_ EE’(1 -— cos 0)
— EA [2E — E’(1 + cos 6)]'

 

x (3.38)

Rather than working directly with E’ and 6 (or Q2 and 3:), it is convenient to introduce

54

another pair of variables y and fl:

Q2 _ 2E — E’(1+cos€)

 

 

 

 

 

 

y E 51:56,; 2E , (3.39)
and
2 E 2EE’ 1— 6
,8; $A=‘/ ( C05). (3.40)
Q 2E—E’(1+c086)
The variable 3; satisfies the constraints
w2
SySL BAD
SeA

where W' is defined in the previous subsection. The relationship (3.41) can be derived

easily by rewriting y as

 

2(p4 ' 1’) TeA
= 1 _ _— = 1 3.42
y 88/1 + SeA, ( )
where
7L42(n4—FV~ (34%

Eq. (3.41) follows from the geometrical constraints on TBA for the fixed invariant mass

W2 of the final-state hadrons:

W2 — S... g 211.,1 g 0. (3.44)

The observed hadron (B) has energy EB and scattering angle 63 with respect to

55

the +2 direction, and azimuthal angle 903; thus its momentum is
pa“), = (E3, E3 sin 63 cos (03, EB sin 63 sin (03, EB cos 63). (3.45)

The scalars z and q% depend on the momentum of the outgoing hadron:

z _ 6EB(1— cos 6B)

 

 

 

 

) 3.46
Q ( )
2
q; = ESEO 1 — cos 7] . (3-47)
In Eq. (3.47) 7 is the angle between 15'}; and 11:15}; + (7 (cf. Fig. 3.4);
1 1 — 2
E0 E Q( +( Wfi ) (3.48)

2K3

is the energy component of mpg, + q“. Define 6.. to be the polar angle of mp5; + q“ :
33pm” + qfilb E E0 (1, sin 6..., 0, cos 6.), (3.49)
where

c0t% = 6\/1— y. (3.50)

The angle 7 in Eq. (3.47) can be easily expressed in terms of the angles 6., 6B, and

(p8, as

cosy = cos 6.. cos 63 + sin 6... sin 63 cos 903. (3.51)

56

Finally, the azimuthal angle (,0 of the lepton plane in the hadron frame (cf.

Eqs. (3.11)) is related to the lab frame variables as

2
(IT 1 298
_ ____ t _ . .
1 y+ ;2 2coh 2 (352)

 

_ Q
cosg0——

1
ZQTvl-Z/

Figure 3.5 shows contours of constant qT and cp in the plane of the angles 63 and $3.
The point qT = 0 corresponds to 63 = 6..., 993 = 0, in agreement with Eqs. (347,351).
According to these equations, qT depends on 903 through cos (03, which is a sign-even
function of (pa. Thus each pair of qT, (,0 determines (pg up to the sign, so that the

contours in Figure 3.5 are symmetric with respect to the replacement 903 ——> —ch.

57

—100i
—150:

—50§
-1oo}
—150§

  

 

,A+A*‘***““4 ‘98:deg
~~. 100 125 150 175

 

 

 

 

 

x=0.001
6. = 156°

Figure 3.5: The variables qT and cp as functions of the angles 63, «pg. Solid lines
are contours of constant qT for qT/Q ranging from 0.1 (the innermost contour) to
3.0. Dashed lines are contours of constant (p for cp ranging from 7r/ 10 to 37r/4. The
contour (,0 = 7r coincides with the 6B-axis. The plots correspond to EA 2 820 GeV,

E = 27 GeV, Q = 6 GeV, :1: = 0.01 (upper plot) and a; = 0.001 (lower plot).

58

3.1.5 Parton kinematics

The kinematical variables and momenta discussed so far are all hadron-level variables.
Next, I relate these to parton variables.
Let 0. denote the parton in A that participates in the hard scattering, with mo-

mentum

P2? = 6.11351. (3.53)

Let (2 denote the parton of which B is a fragment, with momentum

292‘ = p’é/éb- (3.54)

The momentum fractions 5,, and 5;, range from 0 to 1. At the parton level, I introduce

the Lorentz scalars 53, E, 21} analogous to the ones at the hadron level

 

 

A Q2 a:

a; = = _, 3.55
2pa - q 6a ( )

A pb ' pa Z

2 = = —, 3.56
q ' pa {b ( )

6% = -61‘7u- (3.57)

Here {131 is the component of q“ which is orthogonal to the parton 4-momenta pg and

p
pb)

@‘Pa=§t'Pb=‘-0-

59

Therefore,

 

(1°Pb [Q'Pa
31w 14 _; .

=q“- p
apa'pb bpa'pb

(3.58)

In the case of massless initial and final hadrons the hadronic and partonic vectors qf‘

coincide,

55‘ = (1% (3-59)

3.2 The structure of the SIDIS cross-section

The knowledge of five Lorentz scalars SeA, Q, qT, 3:, z and the lepton azimuthal
angle cp in the hadron frame is sufficient to specify unambiguously the kinematics of
the semi-inclusive scattering event 6 + A ——> e + B + X. In the following, I will discuss

the hadron cross-section (103,1, which is related to the parton cross-section dfiba by

 

 

(10134 [16b dfb dial, d35a(/1F,HD)
—D a)“ A A '
dIdZdQ2dqr§~dQO 2: B/b (Eb) #0) I Ea Fa/xl (6H F)dIdZdQ2dq%d(p

(3.60)

Here Fa/A(.fa, [1]?) denotes the distribution function (PDF) of the parton of a type a in
the hadron A, and Bra/5(6), up) is the fragmentation function (FF) for a parton type
b and the final hadron B. The sum over the labels a, b includes contributions from all
parton types, i.e., g,u,fl,d,d, . .. . In the following, a sum over the indices 2', j will
include contributions from active flavors of quarks and antiquarks only, i.e., it will
not include a gluonic contribution. The parameters ,up and ,uD are the factorization
scales for the PDFS and FFs. To simplify the following discussion and calculations, I

assume that the factorization scales up, up and the renormalization scale )1 are the

60

same:

up = pl) 2 p. (3.61)

The analysis of semi-inclusive DIS can be conveniently organized by separating
the dependence of the parton and hadron cross-sections on the leptonic angle (,0 and
the boost parameter 6) from the other kinematical variables x, 2, Q and qT [83].
This separation does not depend on the details of the hadronic dynamics. Following
[34], I express the hadron (or parton) cross-section as a sum over products of func-
tions of these lepton angles in the hadron frame AAA/MP), and structure functions

I’VE/1(1):) 2) Q21 (1%) (01‘ pf/ba (E) 37 Q2: (1%: “)1 respectively):

4

 

 

d0!“ 2 :p 2 2
= "f , H A / . 2
dEdZszdq%d(p p21 BA($aQO 1(11) phi/Np): (36 )
d3 (H) 4
b“ = ”'7’ “: A 2 ? .A / . .
(IEdEszdq%dcp p22; ba(CL,ZaQaq1‘a/l) p(7#a99) (3 63)

The coefficients ”VBA (or ”IT/5a) of the angular functions Ap(z,/), (,0) are independent of
one another.

At the energy of HERA, hadroproduction via parity-violating Z-boson exchanges
can be neglected, and only four out of the nine angular functions listed in [34] con-

tribute to the cross-sections (3.62-3.63). They are

A1 2 1+ cosh2 (Z), A3 = — cosrpsinh 21/),

A2 = —2, A4 = cos 290 sinh2 1/1. (3.64)

61

A >—€ I

d) e) t)

Figure 3.6: Feynman diagrams for semi-inclusive DIS: (a) LO; (b—d) NLO virtual

diagrams; (e-f) NLO real emission diagrams

Out of the four structure functions, Ilia for the angular function A1 = 1 +cosh2 1,!)
has a special status, since only llj'ba receives contributions from the lowest order of
PQCD (Figure 3.6a). At C(03), only the contribution to the 117b,, structure function

diverges in the limit qT —) 0.

3.3 Leading-order cross section

Consider first the (9mg) process of the quark—photon scattering (F ig.3.6a). This
process contributes to the total rate of SIDIS at the leading order (LO). There is no

LO contribution from gluons. Due to the conservation of the 4-momentum in the

62

parton-level diagram, at this order

p]; = p: + (111. (3.65)
This condition and Eqs. (358,3.59) imply that
(1% = --m - q: = 0. (3.66)
Also the longitudinal variables arel
fa = 17, 56 = Z,
5? : ’2? = 1, (3.67)
so that the momentum of the final-state hadron B is
19% = Z (1562+ <1”). (3-68)

Since both quarks and electrons are spin-1/2 particles, the LO cross section is

proportional to 1 + cosh2 z!) E A1(1/),<p) (Callan-Gross relation [84]). Hence the LO
cross section is

(13:) _ UOFI6(" )A1(?/),99)
dzddequ%dc,0 QT

— ———6,-6261—E(51—3,
L0 58A 2 J]( )( )

 

(3.69)

tTo obtain Eqs. (3.67), consider, for instance, Eq. (3.65) in the Breit frame for u = O and 3. By
using explicit expressions (3.8-3.10,3.53,3.54) for the parton and hadron momenta at qT = 0, we find

Q(1 A
‘2‘ 5“?)
Q 1 A
3(5“2+Z)

Eqs. (3.67) are solutions for this system of equations.

0,

0.

63

where (YT E (0,qT, 0,0) in the hadron frame. In Eq. (3.69) the parameter 00 collects

various constant factors coming from the hadronic side of the matrix element,

2

or Q2 (g) (3.70)

2 47rseAx2 2
The factor E that comes from the leptonic side is defined by

621

(3.71)

6, denotes the fractional electric charge of a participating quark or antiquark of the
flavor j; e,- = 2/3 for up quarks and -1/3 for down quarks.

The L0 cross section (3.69) does not explicitly depend on Q2, but rather on a:
and z. This phenomenon is completely analogous to the Bjorken scaling in completely
inclusive DIS [4], i.e., independence of DIS structure functions from the photon’s vir-
tuality Q2. Just as in the case of inclusive DIS, the scaling of the LO SIDIS cross
section is approximate due to the dependence of PDFs and FFs on the factorization
scale p. This scale is naturally chosen of order Q, the only momentum scale in the LO
kinematics. When u it: Q varies, the PDFs and FFs change according to Dokshitser-
Gribov-Lipatov-Altarelli-Parisi (DGLAP) differential equations (2.29,2.31). By solv-
ing the DGLAP equations, one sums dominant contributions from the collinear ra-
diation along the directions of the hadrons A and B through all orders of PQCD.
Formally, the scale dependence of the PDFs and FFs is an 0(a5) effect, so that it is on
the same footing in the LO calculation as other neglected higher-order QCD correc-
tions. By observing the dependence of the LO cross section on p, we can qualitatively
test the importance of such neglected corrections. One finds that this dependence is
substantial, so that a calculation of 0(a5) corrections is needed to reduce theoretical

uncertainties. Let us now turn to this more elaborate calculation.

64

3.4 The higher-order radiative corrections

The complete set of (9(075) corrections to the SIDIS cross section is shown in
Figs. 3.6b-f. These corrections contribute to the total rate at the next-to-leading
order (NLO).

At this order, one has to account for the virtual corrections to the LO subprocess
(6) 7* —+(€_1) (Figs.3.6b-d), as well as for the diagrams describing the real emission
subprocesses (I 7“ —% (7 g and g7" —> qr], with the subsequent fragmentation of the
final-state quark, antiquark or gluon (Figs. 3.6e-f). The explicit expression for the
0(a5) cross section is given in Appendix A.

Due to the momentum conservation, the momentum of the unobserved final-state

partons (6.9. the gluon in Fig. 3.6e) can be expressed in terms of q”, pg, pg:
19: = <1“ + 103‘— pl:- (372)

When there is no QCD radiation (pg; 2 0), the momentum of b satisfies the leading-
order relationship 1)]: 2 pg +q“, so that (1T 2 0. If qT / Q << 1, the perturbative parton-
level cross section is dominated by the term with p = 1. In the limit q%/Q2 —> 0, but

QT ¢ 0, lfim behaves as 1 /q} times a series in powers of as and logarithms ln(q%/Q2),

 

k2k—1
17,, z ”OF’ i. 00 9—5 23“”)(3 2) 16'" 12T— (3.73)
27TSCA (1% k=l 7T mzo b0 Q2 ,

figmfifz‘) are generalized functions of the variables 5? and 3.

where the coefficients
Obviously, the coefficient of the order a’g in Eq. (3.73) coincides with the most diver-
gent part of the 0(a’g) correction to the SIDIS cross section from the real emission
subprocesses. This coefficient will be called the asymptotic part of the real emission

correction to 1173,, at 0(a’g).

65

Convergence of the series in (3.73) deteriorates rapidly as qT/ Q —> 0 because of
the growth of the terms (q?) lnm(q%/Q2). Ultimately the structure function 117),), has
a non-integrable singularity at (11 = 0. Its asymptotic behavior is very different from
that of the structure functions 2’3’4I7ba, which are less singular and, in fact, integrable
at qT = O. This singular behavior of 117b,, is generated by infrared singularities of
the perturbative cross section that are located at (IT = 0. Indeed, according to the
discussion in Section 2.3, the diagrams with the emission of massless particles generate
singularities when the momentum pf of one of the particles is soft (pf —> 0) or
collinear to the momentum pg of another participating particle (p1 gpg = 0). The soft
singularities in the real emission corrections cancel with the soft singularities in the
virtual corrections. For instance, at C(05) the soft singularities of the diagrams shown
in Fig. 3.6e-f cancel with the soft singularities of the diagrams shown in Fig. 3.6b—d.
The remaining collinear singularities are included in the PDFs and FFs, so that they
should be subtracted from 117“.

There exist two qualitatively different approaches for handling such singularities.
The first approach deals with the singularities order by order in perturbation theory;
the second approach identifies and sums the most singular terms in all orders of
the perturbative expansion. In the next two Subsections, I discuss regularization of

infrared singularities in each of these two approaches.

3.4.1 Factorization of collinear singularities at 0(a5)

Let us begin by considering the first approach, in which singularities are regularized
independently at each order of the series in as. The singularity in the 0(a3) part of
the asymptotic expansion (3.7 3) can be regularized by introducing a “separation scale”
q? and considering the fixed-order cross section separately in the regions 0 g qT S qr}?

and qT > qS. The value of q5 should be small enough for the approximation (3.73)
T T

66

to be valid over the whole range qT S q)?

The quantity q75~ plays the role of a phase space slicing parameter. In the region
0 g QT g (175., we can apply the modified minimal subtraction (M—S) factorization
scheme [43] to take care of the singularities at qr = 0. In the W scheme, the
regularization is done through continuation of the parton-level cross section to n =
4 — 26, e > 0 dimensions [42]. The n-dimensional expression for the 0(a3) part of

the asymptotic expansion (3.73) of 1173,,(57‘, 3, Q2, (1%) is

 

 

           

If, _ 27w. ‘1‘" 00F: a_s 1 _:5 x
[M 2 0(05) — 3 27rSeA 7r 2—q% 1’] 6310637
[6(1— z){1(P )+ P1911(E}+{P,]11(z)+ P11( (2)} 6(1 — 33)
A A Q2 2
+ 26(1—z)6(1—a:) (:Flog—q———2 —Cp +0(‘:r—S ,q (1.7.) (3.74)

Here the color factor CF 2 (N2 — 1)/(2Nc) = 4/3, Nc = 3 is the number of quark
colors 1n QCD. The functions 19(1) (5) entering the convolution integrals in (3.74) are

the unpolarized (9(075) splitting kernels [47]:

 

 

P5346) = 1:2] , (3.75)
qulfi) = $(11—2g+2§), (3.76)
199(5)“) = CF1 +11; g). (3.77)

The “+”-prescription in P551“) regularizes P53“) at f = 1; it is defined as

[0 d6 meme) 2 / due) (9(a) — 9(1)).

The scale parameter an in (3.74) is introduced to restore the correct dimensionality

67

of the parton-level cross section dEbG/(ddedQqu;_2d¢) for n 75 4. The soft and
collinear singularities appear as terms proportional to 1/62 and 1/6 when n —> 4. The
soft singularity in the real emission corrections cancels with the soft singularity in the

virtual corrections. At 0(05), the virtual corrections (Fig. 3.6b-d) evaluate to

 

 

 

dam, _ _35: 47mg, 6 1 2 + 3 +8 x
dccdde2dq%d<,0 UiTt,O(CYs) _ 27r F Q2 F(1 — 6) 62 6
d3)",
x drrddequ%d<p L0, (378)

 

where the LO cross section is given in Eq. (3.69).

The remaining collinear singularities are absorbed into the partonic PDFs and F Fs.
When the partonic PDFs and FFs are subtracted from the partonic cross section
013, the remainder is finite and independent of the types of the external hadrons.
We denote this finite remainder as (d3),,a,d. The convolution of (£6)th with the
hadronic PDFs and FFs yields a cross section for the external hadronic states A and
B. The “hard” part depends on an arbitrary factorization scale ,1 through terms like
P756) ln()u/K), where P5303) are splitting functions, and K is some momentum scale

in the process. The scales )1 and (1,, are related as

#2 =2 fire—”pi.

The dependence on the factorization scale in the hard part is compensated, up to
higher-order terms in as, by scale dependence of the long-distance hadronic functions.
After the cancellation of soft singularities and factorization of collinear singulari-
ties, one can calculate analytically the integral of (117%),de over the region

0 g qT S q?. At 0(a5) this integral is given by

68

S)2

/(0T
0

The L0 and NLO structure functions are

2 1" _ 00F! 2 1"Lo QSIANLO
qu( x/ba)hard-_ m26j{ ffbag' +7: /ba,j } (379)

j

 

 

117,23 = 6(1—2)6(1—?E)6b,-6,-a, (3.80)
117”“) — J C in? Q2 —30 ln—i 6(1—3)6(1—6)66
2 1’ (6)2 F ((1%)? 1’1 ,.
l—i— 6 -’~76-P(1)A- P11)"6 —"6.
+ n 432 (1~)b3 j.(z)+ bj(z)(1:r)3a
(qu)
+ 6(1 — 2)6,,c;’;<11(6) + cg;‘<11(2)8(1 — 6)6,,. (3.81)
The coefficient functions CZZ’OM(1)(€) that appear in 19,560 are given by
in on 1
c5? (€)=c..‘1”(€) = 6.ch §(1—€)-26(1—€)], (3.82)
in 1
c)? (6) = 560-6), (3.83)
on C
at? ‘(o = {a (3.84)

Now consider the kinematical region qT > (195:, where the approximation (3.74) no
longer holds. In this region, ( 1lza)hard should be obtained from the exact NLO result.

With this prescription, the integral over qi‘} can be calculated as

 

/max (1%“ d 2 daba _
0 91‘ dEdEdQqugdqa ‘

(a?)2 A "13"qu A
A1(16a90){/0 dq§~ (1%“)hard + j; dq3~ (1%0)hard}

<75?)2

4 max 2
+ 2 AM, ,0) /0 QT at); (917M) , (3.85)
p22

where max q} is the maximal value of q% allowed by kinematics. The first integral

69

on the right-hand side is calculated analytically, using the approximation (3.79); the
second and third integrals are calculated numerically, using the complete perturbative
result of the order 0(a5). The numerical calculation is done with the help of a Monte
Carlo integration package written in the style of the programs Legacy and ResBos
used earlier for resummation in vector boson production at hadron-hadron colliders

[25).

3.4.2 All-order resummation of large logarithmic terms

A significant failure of the computational procedure in (3.85) is that it cannot be
applied to the description of the qT-dependent differential cross sections. Indeed,
the cancellation of the infrared singularities is achieved by integration of the cross
section over the region 0 S qT s qfi. However the shape of the qT distribution is
arbitrary and depends on the choice of the parameter q? that specifies the lowest qT
bin 0 _<_ (1T 3 (175.. The fundamental problem is that the terms in (3.73) with small
powers of as do not reliably approximate the complete sum in the region qT << Q.
This problem justifies the second approach to the regularization of the singularities
at qT = 0, in which large logarithms in (3.73) and virtual corrections at qT = O are
summed to all orders. A better approximation for 1%,, at qT / Q << 1 is provided by the
Fourier transform of a fi-space function $341), Q3, 3, p), which sums the dominant

terms in (3.73) and virtual corrections through all orders of as:

 

A A A a F d2!) 1.- .72 A A
1160(2), 2, Q2, q%, ”)[W = 5%] (271')28 QT bl’Vba(baQa$a 21/1) (386)

Here 6 is a vector conjugate to (YT, and b denotes the magnitude of 5. Hence 117b,, at

70

all values of qT can be approximated by
114.4533, 622413134) = 11443536243314) W + 131.432.6236), (3.87)

where 117),), is the difference between the 0(a5) expression for 1F)”, (cf. Appendix A)
and C(05) asymptotic part (3.74), taken at n = 4. This difference is finite in the
limit qT ——> O.

The complete hadron-level resummed cross section can be obtained by including
the finite parton structure functions for p = 2, 3, 4 and convolving the parton-level

structure functions with PDFs and FFs (cf. Eqs. (360-363)):

 

 

dUBA _ 0051A1(¢,Sfll/ (125

2'“ -5~
dmdde2dq§.dcp mum — 33A 2 (27,)26‘” I’VBA(b,Qa11»',Z)+YBA.

(3.88)

In this equation, the hadron-level b—dependent form-factor FEB/((1), Q, CC, 2) is the sum

of convolutions of parton-level form-factors WMUJ, Q, 133,?) with the PDFs and FFs:

WBA = EBB/“81W“ ®Fa/A- (3.89)
a,b

YBA denotes the complete finite piece,

4
YBA E 1YEA + Z ”VB/1147201580), (3-90)
p=2
where
1YEA E Z DB/b ® 13766 ® Fa/A- (3-91)
a,b

71

The explicit expression for YBA is presented in Appendix A.
At small b and large Q (i.e., in the region where perturbative dynamics is ex-
pected to dominate) the general structure of W3A(b, Q, 11:, z) can be found from first

principles [17, 21]:

W3,“ b, Q, 3:, z) = 253(198/(2 @ng )(z,b)(C;~:,l ®Fa/A)($,b)e_SBA(b'Q). (3.92)
1'

According to the discussion in Section 2.4, the form—factor WBA is the all-order sum
of the large logarithms, which remain after the cancellation of soft singularities and
factorization of collinear singularities. The soft contributions are included in the
Sudakov function SBA(b, Q). At small b, SBA(b, Q) does not depend on the types of

the external hadrons and looks like

 

 

 

0.3622 72 2 2
33AM): [W %— (A(as(m,cl>1n C :9 +B<asm who») 25PM),
(3.93)
with
00 _ k
Ammo.) = ZAk<Cl)(“if“)), (3.94)
k=1
00 _ k
8(as(n),cl,cg) = g8k<ctcz)(“if“)) . (3953)

Contributions from the collinear partons are included in the functions Ci” (1’13, b, p)

and C°"‘(3, b, it). These functions can also be expanded in series of as / 7r, as

k

 

c;"( (:12, b, p) =Zc:"(k’(§s ,201,C Motif”) (3.96)

) ,

72

00 k
Cf”t( (z, b, p.) : Zijut(k)( z ,C1,C2, #b)(g—S7—:-fl) . (3.97)

k=0

According to Eq. (3.93), the integration in SP (b, Q) is performed between two
scales Cl/b and C'2Q, where Cl and 02 are constants of order 1. These constants
also appear in terms proportional to 6(1 — 32?) or 6(1 — E) in the C—functions. The
complete factor W(b, Q) is approximately independent from Cl and C2. In addition,
the C-functions depend on the factorization scale )1 that separates singular collinear
contributions included in the PDFS and FFs from the finite collinear contributions
included in the C-functions. To suppress certain logarithms in 0(05) parts of the Ci"

and Com functions, it is convenient to choose

C1 = 26—78 E ()0, (3.98)
C2 = 1, (3.99)

where 75 = 0.577215... is the Euler constant.
Using our NLO results, I find the following expressions for the coefficients Ak(C 1),
812(01, Cg) and the C-functions:

A1 = CF, (3.101)

8—3/401)
B = 2C. lo . 3.102
1 F 8( [9002 ( )

 

To the same order, the expressions for the C-functions are

73

0 L0:

6532‘“) (3, Mb)
63’2“”(290

in(0) _
ng

o NLO:

in 1 A
Cjk( )(‘Ta lab)

6'1"“)(3, 1w)

1!)

outl A
Cjk ( )(Zalub)

outl A
ng ( )(Zal'tb)

 

: 6.7k6(1 — 53‘),
out(0) _ .
Cg]. — O,
C. A A 13b
2 —2—’:(1 — ) PJJ)(1)log(-5;)
A 23 2 53/402
_ Cp6(1—:r)(fé+log ( b002 )),
1A A A ab
= §I(1_ ’1?)— Pégl)(:r) log(-l—);),
C. A A [1b
= -—2£(1 — z) — qul(z) log(%)
A 23 8 3/401
_ 026(1 —2)(1—6+lo (W )),
_ CF: (1) c: I‘b
._ —2—4. qu (~) l0 (Eb—0)

(3.103)
(3.104)

(3.105)

(3.106)

(3.107)

(3.108)

(3.109)

In these formulas, the indices j and k correspond to quarks and antiquarks, and

g to gluons. In Appendix B I show that the expansion of the integral over b in

Eq. (3.88) up to the order C(05), with perturbative coefficients given in Eqs. (3.101-

3109), reproduces the small-QT limit of the fixed-order 0(a5) cross section discussed

in Subsection 3.4.1.

Due to the crossing relations between parton-level SIDIS, vector boson production,

and e+e' hadroproduction, the (fin-functions are essentially the same in SIDIS and

the Drell-Yan process; and the Com-functions are essentially the same in SIDIS and

74

e+e" hadroproduction. At NLO the only difference stems from the fact that the
momentum transfer q2 is spacelike in DIS and timelike in the other two processes.
Hence the virtual diagrams Figs. 3de differ by 772 for spacelike and timelike q2.
Correspondingly, 63:20) for SIDIS does not contain the term (7r2 / 3)6 (1 — ’15), which is
present in the 6;:(1)-function for the Drell-Yan process. Similarly, Cfi‘m) for SIDIS
does not contain the term (7r2 / 3)6 (1 — 2), which is present in the Cigtutfunction for
e+e‘ hadroproduction.

Up to now, I was discussing the behavior of the resummed cross-section at short
distances. The representation (3.92) should be modified at large values of the variable

b to account for nonperturbative long-distance dynamics. The authors of Ref. [21]

suggested the following ansatz for W321 which is valid at all values of b:

1473110). Q. 17, Z) = Z €§(DB/b ‘3 C§}‘tl(z, MW}: 8’ Fa/A)($a bade—SBA- (3-110)

.7

Here the variable

1,, E b (3.111)

2
1+ (2.3...)

serves to reproduce the perturbative solution (3.92) at b << bmaz, with bmam z

 

 

0.5 GeV"1, and turn off the perturbative dynamics for b 2 bmax. Furthermore, the
Sudakov factor is modified, being written as the sum of the perturbatively calculable
part S P (b.., Q) given by Eq. (3.93), and a nonperturbative part, which is only partially

constrained by the theory:

SBA(b,Q,:c, z) = SP(b.,Q,:1:, z) + ng(b, Q,:r, 2). (3.112)

75

An explicit solution for the function 52’ f,(b Q, 21:, 2) has not been found yet.
Nonetheless, the renormalization properties of the theory require that the Q depen-
dence in the nonperturbative Sudakov term be separated from the dependence on the

other kinematical variables, i.e.,

S”.f<b,9,x,z)— — gt‘hb 2: z) +9310 2: z)log— Q

Q0, (3.113)

with Q0~ 1 GeV. The theory does not predict the functional forms of 93 (b, 110,2)
and 913 23,,(b x ,,2) so these must be determined by fitting experimental data. In ad-
dition, if SB ,f,(b Q, :17, z) indeed describes long—distance dynamics, it should vanish
or be much smaller than SP(b,Q,:1:,z) in the perturbative region b < bmaz. In the
analysis of the experimental results, we may find that the fit to the data prefers a
parametrization of S N P (b) that is not small in comparison to the perturbative part
of W322 at b < bmax. Such observation will be an evidence in favor of important
dynamics that is not included in the b-space resummation formula with coefficients
calculated at the given order of PQCD. Therefore, this work uses an interpretation
of S N P that is broader than its original definition in [21]. SNP will parametrize not
only large-b physics, but additional contributions to W321 at all values of b that are
not included in the perturbative part of W322. In the following parts of the thesis I
will test whether the data are consistent with the assumption that these additional
contributions are small in comparison to the perturbative part of W322 when b < bmax.

Before ending this section, I would like to comment on a subtle difference be-
tween Ci" and Com. While the initial-state coefficient functions C;:(1)(EE,C'1, C2, b, )1)
given in Eqs. (3.106) and (3.107) depend on the factorization scale 0 through a fac-
tor ln[/1b/b0], the final— state functions C‘m (1)(z, Cl, C2, b, )1) given in Eqs. (3.108) and

(3.109) depend instead on ln[ub/(b03)]. The additional term oc In? in the func—

76

tions C§:t(1)(§,Cl,C2,b, 11) becomes large and negative when E —> 0, so that it can

significantly influence the C(03) contribution at small values of 3. As a result, the re-
summed total rate tends to be lower than its fixed-order counterpart for z 5, 0.1. This
issue is discussed in more detail in Section 4.2. Similarly, the 0(a5) part of the NLC)

1‘7NLO -

structure function bad- 1n (3.81) depends on )1 through a logarithm 1n [)12 / (2.1%)?)

The appearance of the additional terms oc In E in the functions Cg," 1(1) and 117,550
reflects the specifics of separation of the (9(05) “hard” cross section (damn) from the
collinear contributions to the FF 3 in the VS factorization scheme. The easiest way to
see the specific origin of the 1n 2 terms is to notice that the dependence on the param-
eter 11,, in the n-dimensional expression (3.74) for 117245323, Q2, q%) comes through a
factor (27rpn/E)4‘", rather than through a more conventional (27rpn)4‘". In its turn,
2 appears in Oahu/EV”, because the WI—S-scheme prescribes to continue to n — 2 di-
mensions the transverse momentum ET of the outgoing parton, rather than the vector
(j'T = 52/? relevant to the resummation calculation. It is this factor that generates
the p-dependent logarithmic terms In [ab/(b03)] in the functions C$t(1)(’z‘, Cl, C2, b, ,u)
and 1%? £0. The C;Z(1)-functions do not include In 3 because they are evaluated along
the direction 3 = 1 in the phase space. In contrast, nothing forbids such a term in
the functions C519“), in which ”i can be anything between 2 and 1. Moreover, the

ln 3 terms are needed to reproduce MS expressions for 0(05) coefficient functions in

completely inclusive DIS [85) by integration of d’o‘ba/(szddedqg) over qT and 3.

77

3.5 Hadronic multiplicities and energy flows

Knowing the hadronic cross-section, it is possible to calculate the multiplicity of the
process, which is defined as the ratio of this cross-section and the total inclusive DIS

cross-section for the given leptonic cuts:

1 do
dam /da:dQ2 dzdde2dq%dcp'

 

Multiplicity 2 (3.114)

Both the cross-section and the multiplicity depend on the properties of the final-state
fragmentation. The analysis can be simplified by considering energy flows which do
not have such dependence. A traditional variable used in the experimental literature

is a transverse energy flow (ET) in one of the coordinate frames, defined as

 

 

1 da(e+A—>e+B+X)
= dd) E . .
<ET)<DB 0101 23:be B T (“DB (3 115)

This definition involves an integration over the available phase space (DB and a sum-
mation over all possible species of the final hadrons B. Since the integration over (1)3
includes integration over the longitudinal component of the momentum of B, the de-
pendence of (ET) on the fragmentation functions drops out due to the normalization

condition

2/ zDB/b(z)dz =1. (3.116)
B

Instead of (ET), I will analyze the flow of the variable .3. This flow is defined

as [86]

 

 

(122 1 d0(e+A—>e+B+X)
= d . 3.117
d2: sz dq%d¢ 23:19.... Z drr dz dQ2 dq%d<p Z ( )

78

I prefer to use 22 rather than (ET) because (ET) is not Lorentz invariant, which
complicates its usage in the theoretical analysisf. Since qT is related to the pseudora-
pidity in the hCM frame via Eq. (3.25), and the transverse energy of a nearly massless

particle in this frame is given by
ET % PT = ZQT: (3'118)

the experimental information on dZZ / (d1: dQ2 dq%) can be derived from the hCM
frame pseudorapidity (7km) distributions of (ET) in bins of :1: and Q2. If mass effects

are neglected, we have

d(ET) _ 3 dEZ
2 _ 2Q?" 2 2 '
(£de dncmdgo dl‘dQ qudgo

 

 

(3.119)

By the factorization theorems of QCD, the hadron-level z-flow 22 can be written

as the convolution of a parton-level z-flow 232 with the PDFS,

 

 

d__EaF d2 2)“ )
ddedequdw 22/1 £0 Fa/A (607,” LF)(1.7L\ Aszdqdep (3.120)

Similarly to the SIDIS cross section, the z-flow can be expanded in a sum over the

leptonic angular functions Ap(1l), cp):

am 4 A
dmdQ2dq%dcp :23” [z/ “($11622 (17‘1“ HA) pWJ 90,) (3.121)

 

where the structure functions ”1720(53, Q2, q%, 11) for the z-flow are related to the struc-

ture functions ”17,2265, 3, Q2, q%, )u) for the SIDIS cross section by

 

IThe z-fiow 22 is related to the energy distribution function 2 calculated in [33] as 532 =
(21E A/ Q2)E. Here E A is the energy of the initial hadron in the HERA lab frame.

79

1
”123(E,Q2.q%,u) = E: / 24311433203439). (3.122)
b 0

The resummed z-flow is calculated as

 

dzz UOFI A1(1/1180)/___ (125 225"“
= e2 W b Y2, . 2
ddequ§.d<p SM 2 (27r)2 ( Q $)+ (31 3)
where
W (I), Q, .i) =Ze§cgute -5 (“91 (c;3®F,/2)(x,b.,,i). (3.124)

As in the case of the resummation of hadronic cross sections, only the structure
function 1V“; for the angular function A1 = 1 + cosh2 1,!) has to be resummed.

The functions C}: in (3.124) are the same as in (3.92). The coefficient C2“ is

 

7 2 —3/4
7‘ 126 CI). (3.125)

01$
coutz1 —C (——————
z + 71' F 16 3 n C200
The parameter b.., given by (3.111) with b22202 = 0.5 GeV—l, is introduced in (3.124)
to smoothly turn off the perturbative dynamics when b exceeds bmax. The term Y2

11 (3.123) is the difference between the complete fixed-order expression at C(05) for

d2 2/(dde2dqff dip) and its most singular part in the limit (1T —> 0; that IS,

 

 

d2; dEZ
z = — . .12
Y dde2dq%d<p (ddei’dfidgo) (3 6)
asym

The asymptotic part calculated to (9(015) is

80

 

 

 

(12,: = GORE 1 A1(¢,§0)
ddequ%d<p 36,4 71' 2q3. 27r

2:2
X 8]-

.7

 

{(19.5.1) 9 41/119, 11) + (19.5.” 9 Fg/A>(x.10}

2 3 2
T

Similar to (3.112), the z-flow Sudakov factor 82 is a sum of perturbative and

nonperturbative parts,
5.01 e. x) = S”(b., Q, m) + $211.24). (3.128)

The NLO perturbative Sudakov factor SP is given by the universal x-independent
expression (3.93). As in the case of SIDIS multiplicities, the renormalization group
invariance requires that the dependence of S 2’ P on In Q be separated from the depen-

dence on other variables:

S§P(b, Q,:1:) = g(ll(b,:1:) + 9(2)(b, 11:) log Q2. (3.129)

0
In principle, the z-fiow Sudakov factor S2(b, Q,:1:) is related to the Sudakov factors
SBA(b, Q,:c, z) of the contributing hadroproduction processes e + A —> e + B + X

through the relationship

I
e—S‘w’x) = —T— 2:]zdze‘SBAUbQ’x’z)(DB/1J ®C§';‘)(z,b.,/1). (3.130)
Cgu (b*’ H) B ’

In practice, the efficient usage of this relationship to constrain the Sudakov factors
is only possible if the fragmentation functions and the hadronic contents of the final

state are accurately known. I do not use the relationship (3.130) in my calculations.

81

3.6 Relationship between the perturbative and re-
summed cross-sections. Uncertainties of the cal-
culation

In the numerical calculations, some care is needed to treat the uncertainties in the
definitions of the asymptotic and resummed cross sections, although formally these

uncertainties are of order (9((05/7r)2,q;1).

3.6. 1 Matching

The generic structure of the resummed cross-section (3.88), calculated up to the order

0((aS/W)N)1is
0132,", = 0,2; + W"). (3.131)

In (3.131), the W-piece receives all-order contributions from large logarithmic terms

1 00 as k 2k-1 .m m
k=1 m=0

The Y—piece is the difference of the fixed-order perturbative and asymptotic cross-

sections:

Y(N) = 0W) — 0W) (3.133)

pert asym ‘

In the small-qT region, we expect cancellation up to terms of order 0(a’SV1'1/7rN“)
between the perturbative and asymptotic pieces in (3.133), so that the W-piece dom-

inates the resummed cross-section (3.131). On the other hand, the expression for

82

the asymptotic piece coincides with the expansion of the W—piece up to the order
0(ag/7rN), so that at large qT the resummed cross-section (3.131) is formally equal
to the perturbative cross-section up to corrections of order 0(0/5v +1 / 7r” +1).

In principle, due to the cancellation between the perturbative and asymptotic
pieces at small qT, and between the resummed and asymptotic piece at large qT, the
resummed formula 022322,, is at least as good an approximation of the physical cross-
section as the perturbative cross-section apart of the same order. However, in the NLO
calculation at qr >> Q it is safer to use the fixed order cross-section instead of the
resummed expression. At the NLO order of 015, the difference between the W-piece
and the asymptotic piece at large qT may still be non-negligible in comparison to the
perturbative piece. In particular, due to the fast rise of the PDFs at small :13, the
resummed and asymptotic pieces receive large contributions from the small-2: region,
while the perturbative piece does not (see the next Subsection for details). Therefore,
the resummed cross-section 0mm may differ significantly from the NLO cross-section
0pm. This difference does not mean that the resummed cross-section agrees with the
data better than the fixed-order one. At (1T 2 Q, the NLO cross-section is no longer
dominated by the logarithms that are resummed in Eq. (3.131). In other words, the
resummed cross-section (3.131) does not include some terms in the NLO cross-section
that become important at qT 2 Q. For this reason, at qT > Q the resummed cross-
section may show unphysical behavior; for example, it can be significantly higher the
NLO cross section or even oscillate if the W—term changes rapidly near the boundary
between the perturbative and nonperturbative regions.

As the order of the perturbative calculation increases, the agreement between the
resummed and the fixed-order perturbative cross-sections is expected to improve. In-
deed, such improvement was shown in the case of vector boson production [25], where

one observes a smoother transition from the resummed to the fixed-order perturbative

83

cross—section if the calculation is done at the next-to-next-to—leading order. Also, at
the NNLO the switching occurs at larger values of the transverse momentum of the
vector boson than in the case of the NLO.

Since the fixed-order result is more reliable at qT 2, Q, the switching from the
resummed to the fixed-order perturbative cross-section should occur at qT x Q. How-
ever, there is no unique point at which this switching happens. Similarly, it is not
possible to say beforehand which of the two cross sections agrees better with the data
in the region qT 2 Q. In SIDIS at small :13, the NLO z—fiow underestimates the data at
qT 2, Q, while the resummed z-flow is in better agreement. Therefore, it makes more
sense to use the resummed z-flow in this region, without switching to the fixed-order
piece. On the other hand, in the charged particle production one has to switch to the

NLO cross section at (1T x Q in order to reproduce the measured pT-distributions.

3.6.2 Kinematical corrections at qT x Q

In this Subsection I discuss the differences between the kinematics implemented in
the definitions of the asymptotic and resummed cross-sections, and the kinematics of
the perturbative piece at non-zero values of qT.

Let us first discuss the NLO approximation to the hadronic cross section (3.60).

The integrand of the NLO cross section contains the delta-function

6 [53‘ — G. — 1) (:A- — 1)] = $ch [(5. — 2:)(51. — z) — mg, , (3.134)

which comes from the parton-level cross-section (A.2). Depending on the values of
:1:,z,Q2,q§., the contour of the integration over {a and 5;, determined by (3.134) can
have one of three shapes shown in Fig. 3.7a,b,c. For qT << Q the integration proceeds

along the contour in Fig. 3.7a, and the integral in (3.60) can be written in either of

84

two alternative forms

 

 

dUBA _/1 alga ...,. 2 2
ddedQ2dqgwd90 — ( MBA<€aa€b1$azaQ ,qT,<p)

 

£a)min {a .- x
1 dgb A A 2 2

= (6) éb _ 2MBA(€a1£b;$1 Z1 Q 3qT, ‘10)) (3135)
b min

where

MBA(€G)€b; $131 Q21 q%) (p) Z

00171 OISM

——:1:z Z DB/b(€b)Fa/A(€a)
a,b

47TS€A 7T

4

 

”fba(35. 3. Q2, (1%)14120/1. 90)- (3-135)
1

p:

The lower bounds of the integrals are

 

 

w
(éalmin = 1_ Z + 3:, (3.137)
2
'U)
(5b)min 1 _ £13 + Z: (3.138)
with
w 2 95m. . (3.139)

Alternativel the cross-section can be written in a form svmmetric with res ect
, y

to a: and z,

 

 

 

6103/1 [1 déa /1 déb
: .140
323.33.123.12 .... 4. — 4M“ + .... a — zMBA’ (3 )

where the integrals are calculated along the branches RP and RQ in Fig. 3.7a, re-

85

spectively. AS (IT —-> 0,

(€0)min '9 III, (€b)min —> Z)

and the contour PRQ approaches the contour of integration of the asymptotic cross-
section shown in Fig. 3.7d. The horizontal (or vertical) branch contributes to the
convolutions with splitting functions in Eq. (3.74) arising from the initial (or final)
state collinear singularities, while the soft singularities of Eq. (3.74) are located at the
point {a = :12, {b = 2.

On the other hand, as qT increases up to values around Q, the difference between
the contours of integration of the perturbative and asymptotic cross-sections may
become significant. First, as can be seen from (3.140), in the perturbative piece 5,,
and {b are always higher than :1: + w or z + w , while in the asymptotic piece they
vary between :1: or z and unity. At small :1: (or small z) the difference between the
phase spaces of the perturbative and asymptotic pieces may become important due
to the steep rise of the PDFs (or F F3) in this region. Indeed, for illustration consider
a semi-inclusive DIS experiment at small 3:. Let qT/ Q = 0.5, z = 0.5, and x = 10‘4;
then a: + w = 3.2 - 10’3 >> :10 2 10'4. In combination with the fast rise of the PDFs
at small 11:, this will enhance the difference between the perturbative and resummed
cross-sections.

Second, for :1: or 2 near unity, it could happen that :1: + w 2 1 or z + w 2 1, which
would lead to the disappearance of one or two branches of the integration of the
perturbative piece (Fig. 3.7b,c). In this situation the phase space for nearly collinear
radiation along the direction of the initial or final parton is suppressed. Again, this
may degrade the consistency between the perturbative and asymptotic piece, since

the latter includes contributions from both branches of the collinear radiation. For-

86

tunately, the :1: — z asymmetry of the phase space in SIDIS is not important in the
analysis of the existing data from HERA, since it covers the small-:1: region and is less
sensitive to the contributions from the large 2 region, where the rate of the hadrOpro-
duction is small.

The numerical analysis below includes a correction that imitates the phase space
contraction in the low-a: region. This correction is incorporated by replacing it in

Eqs. (3.74, 3.88) by the rescaled variable

2 2
.].
Q (1T1

Q, (3.141)

55:

This substitution simulates the phase space contraction of the perturbative piece. At
small qT, the rescaling reproduces the exact asymptotic and resummed pieces, but at

larger qT it excludes the unphysical integration region of {a z :1:.

87

 

 

 

 

 

 

 

 

 

a
F5
_ .
_ _
_ _
111111 IIIIIIIIII _ IIIIIIIIII .l
_ _
. _ n
_ _ .m
_ _ )
I lllllllllll L IIIIIIIIII pram.“
II. _ (\
7R _
I
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. II _
_ I _
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I
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. I.
11117 11111111111 y 1111111111 x
_ _
_ _
_ _
.1 Z
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kc)
_ _
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. _ Q.
IIIIIfl IIIIIIIIIIIIIII fil. IIIII 1
I. _
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_ I . _
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_ II . _ F n
_ R _ _ ...m.
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P. llllllllll ArerLl 111E;
_ .I. (x
1111... 1111111111111 Lr1.1.11 111 x
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b)

a)

 

 

 

 

 

 

Em e
.w
. _ t
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.
_ “Q m
liblq. 1111111111111111111111 .I t
I’_ ml“
7, o.
— I
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u x ) (a. m
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llllfllill llnllJ lllllll . lllll l .1
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I'll—IIIIIIIIIIP IIIIII Ir. lllll x t
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88

Chapter 4

Resummation in semi-inclusive DIS:

numerical results

Despite the abundance of experimental publications on SIDIS, none of them presents
dependence of SIDIS observables on the variable qT. Hence qT-distributions, which
are sensitive to the multiple parton radiation, have to be derived from the published
data on less direct distributions. The qT-distributions for some of the HERA data
were reconstructed for the first time in [35,36]. In this Chapter, I concentrate on the

analysis of the qT-distributions for the z—flow (3.117)

_1_ d2:
amt ddequgr’

which can be derived from published pseudorapidity distributions for the transverse

energy flow in the hCM frame [64, 65]. I will also discuss several observables, including

the average value of q%, that were measured in the production of light charged hadrons
[60, 63].

Reconstruction of the qT-dependence reveals an interesting trend in the data:

89

namely, the average qT (or average q%) increases rapidly when either a: or 2 decreases.
This trend is illustrated in Figs. 4.1 and 4.2. Figure 4.1 shows the average q} in the
charged particle production for several bins of :1: and 2 at 28 g Q2 S 38 GeVZ. The
procedure of reconstruction of (q%) is described in detail in Section 4.2. As can be
seen from Figure 4.1, (q%) in the ZEUS data sample ( (:13) = 1.94-10‘3) is several times
higher than ((1%) at the same value of (z) and larger values of (11:) in the E665 data
sample ((512) = 0.07 — 0.29). (q?) increases even faster when (2) decreases and (2:) is
fixed. For instance, at (3;) = 1.94 - 10‘3 ((1%) increases from 3 GeV2 at (z) = 0.775 to

82 GeV2 at (z) = 0.075.

 

A similar trend is apparently present in the behavior of the quantity \/(q%.)32) / (22),
which was derived from the data for the distributions d (ET) /d77cm published in [65].

This quantity is shown in Figure 4.2 as a function of Q2 and 112* At each value

 

 

of Q2, \/(q%22) / (22) becomes larger when :1: decreases. Also, \/(q%22) / (2,.) is

roughly constant along the lines of constant y = Q2/5135“ (i.e., the lines parallel to

 

the kinematical boundary y = 1). Larger values of \/(q§.Ez) / (22) at smaller as are
the evidence of “broader” distributions dEz/qu. In the subsequent Sections, I discuss
this phenomenon in the context of the qT-resummation formalism.

In this Chapter I assume that the angle (,0 is not monitored in the experiment,
so that it will be integrated out in the following discussion. Correspondingly, the
numerical results for c103,, / (dxddequ%) will not depend on terms in Eqs. (3.62-
3.63) pr0portional to the angular functions A3 and A4, which integrate to zero. The

dependence on the azimuthal angle (,0 is discussed in more detail in Chapter 5.

 

 

‘The distributions \/(q%.22) / (232) were derived by converting distributions d(ET) /dncm in
dEZ/dfi. with the help of Eq. (3.119) and then averaging (@322) and (2;) over the experimen-
tal bins of (11. In each bin of qr, central values of 2; and q§~ were used. This procedure provides a
reasonable estimate for \/(q§.21) / (2;) if the experimental qT-bins cover all available range of QT-
Figure 4.2 shows \/(q§.2z) / (2;) for the “low-Q” data set of from [65], which satisfies this criterion.

 

 

90

Q’ = 28—38 GeV2

 

2
">10 . . 2:0.075
Q)
9 L z=o.05.11,+
f) , 2:0.125 1
6.
v , 2:0.185
1 2:0.27
<1 +
10 _— l z=0.385 z=o.15 ¢ ¢+¢
I I 2:0.55
z=0.3+++
2:0.775 4i

ji
N
ll
0
01
+
+
_._ +
+9—

 

 

 

z = 0.7 |
1 - 1
ZEUS, DESY—95—221 E665 doto
02:28 csev2 02:31—38 CeV2
~ 1 1 1 14 1 1 1l 1 1 1 1 1 1 1 11
-3 —2 —1
1O 1O 10 X

Figure 4.1: The average (1% as a function of :1: and z in the charged particle production
at Q2 = 28 - 38 GeVz. The data points are extracted from published distributions

(1)?) vs. sap [60,63] using the method described in Section 4.2.

91

 

 

 

 

 

N>
(D
o 35
N .. 60
o
39 26
24
45 38 27 (< qu :, >/< z,>)‘/2
DESY—99—091, low-Q bins
1 L11]! 1 J+LILJII 1 1 iillllL 1
10‘4 10’3 10‘2

X

 

Figure 4.2: \/(q§.22) / (Zz) reconstructed from distributions d (ET) /dncm in bins of
:1: and Q2 [65].

92

4. 1 Energy flows

4.1.1 General remarks

As was discussed at length in Subsection 3.4.2, knowledge of the resummed SIDIS
cross section can be used to predict the pseudorapidity spectrum of the transverse
energy flow in the hadron Breit frame or the hCM frame. It is advantageous to study
the energy flows, because they are less dependent on the specifics of the final-state
fragmentation of the scattered partons into the observed hadrons. I therefore start the
presentation of the numerical results with the comparison of the resummation formal-
ism to the experimentally measured pseudorapidity distributions for the transverse
energy flow in the hCM frame.

I consider the data on (1 (ET) /dncm which has been published in [64,65]. I consider
seven bins ofzr and Q from [64] (10 3 (Q2) 3 50 GeVz, 3.7-10‘4 S (:13) S 4.9-10’3)
and two sets of bins of :r and Q from ]65] (“low-622” set covering 13.1 < (Q2) <
70.2 GeVz, 8 x 10‘5 < (:13) < 7 x 10’3 and “high-Q2” set covering 175 < (Q2) <
2200 Gev2 and 0.0043 < (1:) < 0.11).

The experimental distributions d (ET) /d7)cm at a fixed value of W 2 = Q2(1 — :13) / :1:

can be converted into the distributions dEz/qu using Eqs. (3.253119):
(IT : l/Ve—nc'" (4.1)

and

d2. d(ET)

1
11de2qu ‘ Edzdwdnm' (4'2)

The “derived” data for dEz /(d:1:dQ2dq%) can be compared with the resummed z-flow

93

(3.123), which is calculated as

(123 7r d2b .. ~~
—_—Z__: F 1 h2 —— IQT’b [z z o
11de2qu S... —00 l ( + COS 11’) / (27,)28 W (11.62.11) + Y , (4 3)
where
2(b Q,CII) =32 :e2coute —S (b ,Q, m) (cm ® Fa/A)($, b“ '11,) (44)

The Sudakov factor 52 in Eq. (4.4) is

Sz(b,Q,$) = SP(511Q1$) + Sfp(b,Q.:v).

where the perturbative part SP is given by Eq. (3.93), and a realistic parametriza-
tion of the nonperturbative part S§VP(b,Q,:r) can be obtained by comparison with
experimental data at low and intermediate values of Q, especially with the measured
pseudorapidity distributions at Q m 3 -— 20 GeV. At high Q, we expect the data to
be dominated by the perturbatively calculable parton radiation and be less sensitive
to the nonperturbative effects incorporated in Sfp(b, Q, :12). According to the renor-

malization group invariance argument, Sf”)

1n Q:

includes a part that is proportional to

swam) = 9011.22) +g<2>(b,x>log%, (4.5)

where the parameter Q0 1% 1 GeV"1 prevents 1n Q/Qo from being negative in the
region of validity of PQCD. In the following analysis, I use two parametrizations of
SQ’PU), Q, 51:), which I will call parametrizations 1 and 2.

o Parametrization 1 was proposed in our paper [35] with D. Stump and

94

 

C.-P. Yuan based on the analysis of the data in Ref. [64]:

5513(1), Q. as) = 1,2 [910(1) + %(g<2>(b, Q)|DY + 9(2)|.+..- (1), Q)) ], (4.6)

where 9(2)(b,Q)] DY and 9(2)(b,Q)]e+e_ are Q-dependent terms in the nonper-
turbative Sudakov factors in Drell-Yan process and e+e" hadroproduction. The
parametrization of the function 9(2)(b, Q) in Eq. (4.6) is suggested by the cross-
ing symmetry between SIDIS, the Drell-Yan process and e+e‘ hadroproduction.

Due to this symmetry, the functions g(z)(b, Q) in these processes may be related

as [33]

(900,000,, + 11%, 0116..-). (4.7)

1
2 _
9( )(b1 Q)[SIDIS " '2'

If the relationship (4.7) is true, then the function g(2)(b, Q) in SIDIS is com-
pletely known once parametrizations for the functions 9(2) (b, Q) in the Drell-Yan
and 9(2) (b, Q) in e+e‘ hadroproduction processes are available. In practice, the
only known parametrization of the nonperturbative Sudakov factor in the e+e‘
hadroproduction was obtained in Ref. [19] by fitting the resummation formula
to the data at Q = 27 GeV. Most of the (ET) data from HERA correspond to
significantly smaller values of Q, where the usage of the parametrization [19] is
questionable. In addition, the known parametrizations of the nonperturbative
Sudakov factors for the Drell-Yan [22,24—27] and e+e" hadroproduction [19]

processes correspond to slightly different scale choices:

01 = ()0, Cg = 1 (4.8)

95

 

H“ 7]

 

and
C1 = 00, CQ 2 €_3/4, (4.9)

respectively. Therefore, the known functions 9(2)]DY (b) and g(2)]e+e_ (b) are
not 100% compatible and in principle should not be combined to obtain g<2l(b)
for SIDIS. In the numerical calculation, I have used the functions 9(2)(b)]DY
from [22] and 9(2)(b)]e+e_ from [17], despite the fact that g(2)(b)] DY was fitted
to Drell-Yan data using a different C2 value than in g<2l(b) [e+e_. Explicitly, the
Q-depcndent part 9(2)(b, Q) in Eq. (4.6) is

b C
9(2)“)1 Q) = $112 (0.4810g(§%—) + 5.32Cplog(—) log( 2Q )). (4.10)

 

0 b... C[1 Q0

In Eq. (4.10), the constants are CI = 26’”, C2 = 6’3/4,Q0 = 1 GeV. The
variable b... is given by Eq. (3.111), with bmaz : 0.5 GeV—l.

The functional form of g(l)(b, 2:) in terms of b and :1: was parametrized as

0.58
,/53

 

g(1>(b, 1:) = (—4.58 + )112, (4.11)

where the numerical coefficients were determined by fitting the experimental
data. These data cover a limited region of :1: and Q2 (10 S Q2 g 50 GeV2,
3.7 - 10"1 g :1: g 4.9 - 10‘3 ), so that the parametrization 1 should not be
used away from this region. Also, the dependence of S§P(b,Q,:1:) on Q can-
not be determined reliably using exclusively the data from Ref. [64], since all
pseudorapidity distributions in this publication are presented in a small range
of Q m 2 — 6 GeV. This circumstance motivated us to model the Q-dependent

terms in the parametrization 1 by using the crossing relationship (4.7) instead

96

of trying to find these terms from the comparison with the data.

Parametrization 2 overcomes several shortcomings of the parametrization 1.
The parametrization 2 was proposed in [36], where the analysis of Ref. [35] was
repeated using the latest and more comprehensive data on the transverse energy
flow [65]. From our analysis, we found that the data from Refs. [64,65] are con-
sistent with the following representative parametrization of the nonperturbative

Sudakov factor:

1-—:1:)3

537(1), Q, 1:) :12 (0.013( + 0.19111 9— + C) , (4.12)

11? Q0

where the parameter Q0 is fixed to be 2 GeV to prevent 1n Q/Qo from being
negative in the region of validity of PQCD, and where we set C = 0 for reasons

explained later.

The H1 Collaboration presented pseudorapidity distributions of the transverse
energy flow for Q2 up to 2200 GeVQ. However, the data at such high Q2 is
rather insensitive to the nonperturbative dynamics because of the poor resolu-
tion of the H1 detector in the region of large Q2 and mm. Thus the H1 data at
very high Q2 is not informative about the QQ-dependence of 5513(1), Q, 3:) either.
Fortunately, the H1 Collaboration presented distributions in two bins at inter-
mediate values of (Q2), namely (Q2) = 59.4 GeV2 and (Q2) = 70.2 GeV2. To-
gether with the data from Refs. [64,65] at lower values of Q, these distributions
provide the first direct tests of the Q2-dependence of Sf," P (b, Q,a:). Therefore
the parametrization 2 of Sf P (b, Q,:1:) includes a numerical value for the coef-
ficient of In Q/Qo, which yields reasonable agreement with all of the analyzed
data. The resulting value for this coefficient differs noticeably from its model

expression in the parametrization 1. However, we should not draw too strong a

97

 

conclusion from this difference, because it might be caused by ambiguities in the
separation of Q2 dependence and a: dependence in the existing data. To draw a
strong conclusion about the crossing symmetry model, experimental pseudora-
pidity distributions in a larger range of :1: at intermediate values of Q2, as well
as improvements in the knowledge of the nonperturbative Sudakov factor in the

e+e‘-hadroproduction will be needed.

4.1.2 Comparison with the data

The numerical results below were obtained using the parameters of the HERA electron-
proton collider. The energies of the proton and electron beams are taken to be
equal to 820 and 27.5 GeV, respectively. All calculations were performed using
CTEQ5M1 parton distribution functions [90] and the parametrization2 of the non-
perturbative Sudakov function Sf”) (Eq. (4.12)), unless stated otherwise. The theo-
retical results in Figs.4.3-4.5 were obtained using the kinematical correction to the
asymptotic and resummed cross-sections at non-zero qT, which was discussed in Sub-
section 3.6.2. The factorization and renormalization scales of the perturbative and
asymptotic pieces are all set equal to 11 = Q. The resummed piece was calculated
using 01 2 b0, CQ 2 1, )1 = bo/b, where he E 26—75.

In Fig. 4.3, I present the comparison of the existing data from [64] in one of the
bins of :c and Q2 ((23) = 0.0049, (Q2) = 32.6 GeVQ) with the NLO perturbative and
resummed z-fiows. Figure 4.3 demonstrates two important aspects of the NLO qT
distribution (dashed curve): namely, the NLO z-flow exceeds the data at small qT and
is below the data at gr 2 Q. In fact, the deficit of the NLO prediction in comparison
with the data at medium and large qT (qT 2 5 GeV) is present in the entire region
of :1: and Q2 that was studied.

As I discussed in Section 3.6, one can trust the resummed calculation only for

98

reasonably small values of qT / Q. For large values of qT, the fixed-order perturbative
result is more reliable. This means that the NLO resummation formalism will not
give an accurate description of the data for qT >> Q due to the small magnitude of

the NLO perturbative z-flow in this region.

 

  
  

 

 

 

T 0.3 .
i _ i NLO
O . : CTEQ4M
5‘0 25 t E x = 0.0049
'0 - " 1 Q = 5.7 GeV
\. ‘-.
N '.
D I-
;5 .
b 0.2 -
\ .
0.15
resummed
0.1
0.05
1
I ...... \
OPillllLlllllllllllllllllL111111111ll111111;.l.:.l-:.l-:-l.]

 

 

 

0 1 2 3 4 5 6 7 8 9 10
qu GeV

Figure 4.3: Comparison of the N LO perturbative and resummed expressions for the
z-flow distribution with the existing experimental data from HERA [64]. The data
is for (:13) = 0.0049, (Q2) = 32.6 GeVz. The resummed curve is calculated using the
parametrization 1 of SS”. CTEQ4M PDFs [91] were used.

99

(1/0101) de/dQ‘h GeV-1

(1/0101) de/dqh Gev‘l

(1/0101) dzz/dQTr GeV-]

0.3

0.2

0.1

0.3

0.2

0.1

0.3

 

 

 

 

 

    

 

 

 

- 0’ = 13.1 sev'

: x = 0.00036

[- l l l 1 l I 1 l l

0 5 1O
0’ = 14.9 Gev'
x = 0.0023

0 10 20
0’ = 33.2 (313V2

x = 0.0047

 

 

 

 

qr, GeV

0.3

     

0.1

 

 

0’ =14.1oev’

x = 0.00063

 

 

 

 

 

0’ = 28.8 GeV’
11 = 0.00093

 

 

 

 

0 10 20
E 0’ = 59.4 GeVz
3+ 1: = 0.002
+4
1 J 1 11h

 

 

10 20

q., GeV

0

0.3

0.2

0.1

0.1

0.1

 

[I‘ll

rT

I

 

l I
.0—

o’ =14.1cev'
x = 0.0011

 

1 l l l J I l l

 

5 10

 

     

Q2 = 31.2 GeV’
1: = 0.0021

 

 

10 20

 

    

0’ = 70.2 GeV’
x = 0.007

 

 

10

20
q.. GeV

Figure 4.4: Comparison of the resummed z-flow (solid curve) in the current region of

the hCM frame with the data in the low-Q2 bins from Refs. [65] (filled circles) and [64]

(empty circles). For the bin with (Q2) = 33.2 GeV2 and (:13) = 0.0047, the fixed-order

C(05) contribution for the factorization scale 11 = Q is shown as the dashed curve.

100

 

 

 

 

 

 
   
 

 
      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

'1'
-1

i “ 0' =17scev' 10-1 0’ =253 sev’ 10 o’ =283Gev1
0_ X: 0.01 x= 0.026
5' -2
Q 1‘2 10 E
w" 0 3
“o -3-
159 —3 —-—- E
\\ _2 l 10 l l _4_
:10 1111 11 111 1141110 11111111111

0 10 0 20 40 0 20 40 60
7 -1 -1 -1
a 10 0’=511cev' 10 o’=5170ev' 10 o’=682c;ev'
O x = 0.012 x = 0.026 x = 0.076
*" —2
810
\.
‘8’ —3
,310 “*—
15‘"
>1O-41111111111LJL 10—4 1111111111
V 0 25 50 0 20 40
_ 1 qT,GeV
'> 10 0’ =2200 oev'
Q)
(9 x= 0J1
621-2
U 0
\.
w -3
6‘" -4
\10 llllllLlJllll
V 0 25 50

qT,GeV

Figure 4.5: Comparison of the resummed z-flow in the current region of the hCM
frame with the data in the high-Q2 bins from Ref. [65]. For the bin with (Q2) =
617 GeV2 and (:13) = 0.026, the C(03) contribution for up = Q is shown as a dashed

curve.

101

The excess of the data over the NLO calculation at large qT (cf. Figs. 4.3-4.5) can
be interpreted as a signature of other intensive hadroproduction mechanisms at hCM
pseudorapidities nm 5 2. A discussion of the cross-sections in this pseudorapidity
region is beyond the scope of this thesis. There exist several possible explanations
of the data in this region, for instance, the enhancement of the cross-section due to
BF KL showering [56] or resolved photon contributions [68,71]. It is clear, however,
that better agreement between the data and the theory, in a wider range of 17m, will be
achieved when next-to-next-to-leading order contributions, like the ones contributing
to (2+1) jet production [69], are taken into account.

On the other hand, Figs.4.3-4.5 illustrate that the resummed z-flow is in better
agreement with the data, over a wide range of qT/Q, but also lies below the data
if qT/Q significantly exceeds unity. The better consistency between the resummed
z-fiow and the data suggests that the resummed z-flow should be used up to values of
qT / Q ~ 1 — 4, 1'. 12., without switching to the fixed-order expression. This procedure
was followed in the derivation of our numerical results.

Let us discuss the features of the data presented in Figs. 4.4 and 4.5. First, the data
in the low-Q2 bins is significantly influenced by nonperturbative effects and therefore
is sensitive to the details of the parametrization of S§P(b, Q, 2:). This feature can be
seen from the abundance of data points around the maximum of the qT-distribution,
where the shape is mainly determined by SivP(b, Q,$). Also, the low-Q2 data from
HERA is characterized by small values of 2:, between 10’4 and 10”. For the theory
to be consistent with the data from Ref. [64] in this range of 2:, the nonperturbative
Sudakov factor must increase rapidly as 2: —> 0, at least as l/fi. Such m-dependence
is implemented in the parametrization 1 of Si”). In our newer analysis, we found that

growth of Sf? (b, Q, 2:) as 1/2: at small 2: is in better agreement with the more recent

data from [65].

102

 

Second, the data in the high-Q2 bins of Fig. 4.5 shows a behavior that is qualita-
tively different from Fig. 4.4. In the region covered by the experimental data points,
the qT distribution is a monotonically decreasing function of qT, which shows good
agreement with the resummed z-flow over a significant range’1 of qT. In the region
qT < 10 GeV, i.e., where the maximum of the qT distribution is located and where
nonperturbative effects are important, the experimental qT-bins are too large to pro-
vide any information about the shape of dEZ/qu. Thus, as mentioned earlier, the
published high-Q2 z-flow data from Ref. [65] is not sensitive to the dynamics described
by the nonperturbative Sudakov factor 89’ P (b, Q, 2:).

A third comment is that most of the high-Q2 data points in Fig. 4.5 correspond to
(2:) > 104. If the resolution of the H1 measurements at high Q2 were better in the
small-QT region, then the high-Q2 data would also reveal the behavior of 83’ P (b, Q, :r)
at large 2:. But, as mentioned before, the published data in the high-Q2 bins are not
very sensitive to the shape of the z-flow at small qT. Therefore it is not possible to
impose any constraints on Sf P (b, Q, 2:) at large values of 2, except that it should be
positive, S? P (b, Q,2:) > 0. For this reason we have chosen the x-dependent part of
S? P (b, Q, 2:) in the parametrization 2 such that Sf”) (b, Q, 2:) grows approximately as
1/2: as 2: —> 0 and is positive for all 21. For the same reason, we chose C = 0 in the
parametrization 2. Although the most general parametrization of S? P (b, Q, 2:) can
have C aé 0, the current data cannot distinguish between the parametrization 2 with
C = 0 and C 31E 0, as long as the value of C is not very large.

Finally, Fig. 4.6 shows the results of our calculation presented as the hCM pseudo-
rapidity distributions of the transverse energy flow (ET). This quantity is obtained by

the transformation (3.119). The small-qT region, where the resummation formalism

 

[I point out once again that both the C(05) and resummed z-fiow lie below the data at very
large (17, in all bins of 2: and Q2 in Figures 4.4 and 4.5.

103

 

 

is valid, corresponds to large pseudorapidities. In this region, the agreement between
our calculation and the data is good. At smaller pseudorapidities (larger qT), one
sees the above-mentioned excess of the data over the perturbative N LO calculation.
In the (ET) vs. 176", plot, this excess is magnified because of the factor q% in the

transformation (4.2).

4.1.3 How trustworthy is the resummed z—flow at large qT?

As noted earlier, the 0(015) fixed-order z-flow is much larger than the data in the
region qT/ Q << 1 and smaller than the data in the region qT/ Q 2, 1. In the small-qT
region, the resummed z—flow is, by its construction, more reliable than the fixed-order
result. In the large-qT region, the resummed z-flow, with the kinematical correc-
tion (3.141) included, is also in better agreement with the data than the fixed-order
calculation. But theoretically, the resummed z-flow at large qT/ Q is not absolutely
trustworthy, because it does not include those parts of the fixed-order z-flow that are
subleading in the limit qT —> 0, but which might be important at large qT. If the
NLO result were in a good agreement with the data at large qT, it would be justified
to consider it a more reliable prediction in this region. But since the (9(015) contribu-
tion is systematically smaller than the data, higher-order corrections are presumably
necessary in order to describe the region qT 2, Q reliably.

A systematic approach for improving the theoretical description of the large-qT
region would require inclusion of the complete 001;) terms in both the fixed-order
and resummed z-flows. But because such a calculation is not available, it might be
beneficial to use the resummed z-flow as a better theoretical prediction both in the
region qT / Q < 1, where application of the resummation formalism is fully justified,
and for qT up to several units of Q, where the resummed z—flow agrees with the data

better than the fixed—order one. Then the use of the resummed qT-distributions of

104

the z-flow will provide more reliable predictions for other observables relevant to the
SIDIS process.

As an example, resummation can improve the reliability of the theoretical predic-
tion for the azimuthal asymmetry of the z-flow. The b—space resummation formalism
affects only the coefficient IVM of the angular function A1(1/1,<p). This coefficient
is the one that dominates the cp-integrated z—flow in the small-qT region, where the
energy flow is the most intense. 0n the other hand, the main goal of the mea-
surement of angular asymmetries is to study structure functions other than 1V”,
e.g., those corresponding to the angular functions A3(1,b,1p) = —cos<,osinh 21b and
A4(1/1, cp) = cos 21p sinh2 (b. By using a better approximation for the coefficient leA,
it is possible to measure the coefficients “V” more reliably. Conversely, by knowing
that the all-order resummation effects are important in the region of small qT and by
concentrating on the region where qT is of the order Q or larger, one may find angular
asymmetries that are well approximated in the lowest orders of PQCD. The impact

of resummation on the angular asymmetries is discussed in more detail in Chapter 5.

105

 

 

 

     

   

  

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

% - x=0.0049 ' x-0.0021 ' x=0.00093
o o’=32.6 Gev' E]— o'a-sos Gev' :_[_ o’-28.a sev'
E " I.
9 2 '
"O I-
\~ :
L'J 1
“O
s
b
\ O 1 1 1 l 1
> 3 _ _ _
<1) x=0.0023 _ x-0.001 1 _ x=0.00063
O t O’=14.5 GcV' - O'-14 CaV' _ Q'-14.2 GeV'
. - 1
'0 - -
\. : '
L1J 1 I.—
‘0 _
69 _
\ O l h 1 1 1 14
'— 2 4 6 2 4 6
> 3 _
8 . x=0.00037 nu" 776'"
- o'-13.1 oev'
. 1++
F? 2 "
'0 1—
\. :
L1J 1 _
'O -
5 I
> 0
2 4 6
77:11.

Figure 4.6: The hCM pseudorapidity distributions of the transverse energy flow in
the current fragmentation region. The data are from [64]. CTEQ4M PDFs and the

parametrizationl of SS”) were used.

106

4.2 Normalized distributions of charged particle pro-
duction

Let us now turn to the discussion of particle multiplicities. Although the resummation
formalism, as outlined in Chapter 3, can describe the cross section for any massless
final-state particle (provided that the fragmentation functions for this particle are
known), in this Section I will concentrate on distributions of the charged particle

multiplicity, defined as

1 da(A+e—>hi+e+X)
Utot d9 .

 

(4.13)

Here 0 is some kinematical variable, such as the variable q% in Eq. (3.6), the transverse
momentum pT of the final-state charged particle in the hCM frame, or the Feynman

variable 2: p,

 

2 z 2
IFE p‘l/ifm =Z( —-g-T—) . (4.14)

Our calculation assumes that the charged particles registered in the detector are
mostly charged pions, kaons and protons. Therefore the cross section for charged
particle production can be calculated using (3.60) with the replacement of the frag-

mentation functions Dig/11(0), 11) by

Dh.)b(§b, 11) 2 2 DB /1.(£1., 11). (4.15)

Bz‘n’i 11(3: apifi

The fragmentation functions DB/b(€b, 11) are known reasonably well only for £1; Z

0.05 — 0.1 [87—89]. Thus, the formalism presented here is applicable to the production

of charged particles with sufficiently large energies, i.e., for 2 Z 0.05.

107

Certain experimental distributions are readily available from the literature [59, 60,
62,63], such as da/de, da/dzp, as well as distributions for the average transverse
momentum (12%). However, the “experimental” qT distributions must currently be
derived from pseudorapidity distributions by using Eq. (3.25). Although the distri-
butions da/de and (12%.) are quite sensitive to resummation effects, they cannot be
interpreted as easily as the distributions da/qu, primarily because the distributions
da/de and (12%.) mix resummation effects at small values of qT with perturbative
contributions from the region qT / Q 2, 1. The most straightforward way to study the
effects of multiple parton radiation would be to consider the qT (or pseudorapidity)
distributions that satisfy the additional requirement 2 > 0.05 — 0.1 and that are or—
ganized in small bins of Q2 and 2:. Unfortunately, such distributions have not been
published yet. Although Ref. [59] presents distributions d0(p+ e -—> h:t + e +X)/dncm
for some values of 2c and Q2, these distributions are integrated over the full range of
2:. Therefore, they are sensitive to the uncertainties in fragmentation functions, mass
effects,1 and contributions from diffractive scattering.

Because the experimental qT distributions are unavailable, we have decided to
undertake a simpler analysis than the one presented for the energy flow. Our goal

here is to understand how the multiple parton radiation could affect various aspects

 

3 Our calculation assumes that all participating particles, including the final-state hadrons, are
massless. Because of this assumption, the production of final-state hadrons with z = 0 is allowed.
However, in realistic SIDIS experiments there is a non-zero minimal value of 2 determined by the
finite mass of the observed hadron. It follows from the definition (3.5) of z and Eqs. (3.20, 3.21) for
the initial and final hadron momenta in the y‘p c.m. frame, that

P; m8
'0'" > —— .
W — W ’ (4 16)

 

Z :
where

+ _ z
pB,cm _ E3167” + pB.cm'

According to Eq. (4.16), the mass of the final-state hadron should be included if z ~ m3 / W ~
AQC D/ W. Hence, our massless calculation is not suited for the analysis of the distributions da(A +
e —+ hi + e + X )/ dncm from [59], which are sensitive to such small values of z.

108

 

2p 5‘ “-—

of charged particle production. For this purpose we focused our attention on data
from the ZEUS Collaboration [60], which presents the charged particle multiplicity
in a phase-space region characterized by the mean values (W) = 120 GeV, (Q2) 2
28 GeV2, and the additional constraint 2 > 0.05. These values of (W) and (Q2)
translate into an average value of 2: = 1.94 x 10‘3. A simple model for nonperturbative
effects at small qT will demonstrate that resummation describes qualitative features
of this set of experimental data better than the fixed—order calculation.

In all of the cases presented, the strategy is to compare the resummed multiplicity
to that from the next-to-leading order calculation. In the numerical analysis, the
multiplicity was calculated using the CTEQ5M1 PDFs [90] and the FFs from [88]. For
the resummed multiplicity, the “canonical” combination 01 2 b0, C2 = 1, 11 = bo/b
was used. The NLO cross section was calculated according to Eq. (3.85), for the
factorization scale 11 = Q. As explained in detail in Section 3.4.1, the integration
of the NLO term over qT is done separately over the regions 0 g qT _<_ q; and
(17 > (159., where q? is a particular type of a phase space slicing parameter. The
final results should not depend on the exact value of q75~ provided that it is chosen in
the region where the 0(a5) part of the next-to—leading-logarithmic expansion (3.73)
approximates well the exact NLO cross section. In practical calculations, q; cannot
be chosen to be too small, because the numerical calculation becomes unstable due
to large cancellations between the integrals over the regions 0 S gr 3 q? and qT 2
qfa. The NLO prediction for the integrated charged particle multiplicity achgd/otot
at (W) = 120 GeV, (Q2) 2 28 GeV2 is practically independent of gig. in the region
1 f, q? S 2.5 GeV (cf. Figure 4.7). The NLO distributions shown in the subsequent
Figures were calculated for gig. = 1.2 GeV, which lies within the range of stability of
17¢th /0101. As in the case of the z-flow, the resummed charged particle multiplicity

may suffer from matching ambiguities at qT / Q ~ 1.

109

 

Chosen
value

 

------J

10

1

0.5 1 1.5 2 235 3
(1353p, GeV

 

 

Figure 4.7: The dependence of the 0(a5) prediction for the total charged particle

multiplicity on the value of the separation scale q?. The calculation is done for

(W) = 120 GeV, (Q?) = 28 GeVz.

In Section 4.1, we found that the resummed z-flow is in better agreement with the
experimental distributions than the NLO z-flow, for the whole range qT / Q S, 2 — 4.
That result suggests that it might be preferable to use the resummed z—flow in the
whole range qT / Q S, 2-4 as a better theoretical prediction, until the O (01%) prediction
for the z-flow in the region qT/ Q 2, 1 becomes available. In the case of the charged
particle multiplicity, the resummed cross section, which is calculated according to the

formula

 

dO'BA _UOE/ (120

dxddequg. _ 7rS A (2w)2ei§T.bWBA (b15131 Z1 Q) + YBA: (4.17)

110

 

(.0!me

<p.’>(hCM). Gev’

N

 

 

 

 

 

 

 

 

 

 

0.5
0.8
0.7
0.6
0.5
0.4
0.3
0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2.9

F F

a) b)

Figure 4.8: The distributions (a) (12%.) vs. xp and (b) ((1%) vs. 2:): for the charged
particle multiplicity at (W) = 120 GeV, (Q2) = 28 GeVZ. The experimental points
for the distribution (12%.) vs. 23;: are from Fig. 3c of Ref. [60]. The “experimental” points
for the distribution ((1%) vs. mp are derived using Eq. (4.19). The solid and dashed
curves correspond to the resummed and NLO (11 = Q) multiplicity, respectively.

overestimates the experimentally measured rate for the production of charged parti-
cles with 127 > 2 GeV. This discrepancy indicates that the resummed cross section in
the region qT/Q ,2 1 is too high, so that switching to the perturbative cross section
in this region is in fact required. Therefore, we have chosen to use the resummed
cross section for (1T 3 5 GeV and switch to the next-to-leading order cross section for
qT Z 5 GeV.

As in the case of the z-flow, the shape of the qT distribution for the charged particle
multiplicity at small values of qT depends strongly on the unknown nonperturbative
Sudakov factor S N P (b, Q, 2:, 2). For the purposes of this study, we introduced a pre-
liminary representative parametrization of the nonperturbative Sudakov factor for

the fixed values of 2: = 1.94 X 10‘3 and Q2 = 28 GeVz, i.e., the values that coincide

111

 

 

with the average values of 2: and Q2 in [60]. This z-dependent parametrization is

51"” (0, Q2 = 28 GeV2,x =1.94 x 10-3, 2) = 112 (0.18 + 0897—1728) . (4.18)
Since the ZEUS Collaboration did not publish pseudorapidity distributions for the
charged particle multiplicity (1 /awt)da/dncm in bins of varying 2, we had to deduce
information about the z-dependence of SNP from the less direct distribution of (11%)
vs. 2:}: presented in Fig. 3c of [60]. This distribution, known as a “seagull” for its char-
acteristic shape (Fig. 4.8a), can be converted into the more illustrative distribution of
(q%) vs. 2:): (Fig. 4.8b). Since the major portion of the registered events comes from
the region q%/W2 << 1, or 2:p z z, a first estimate of the experimental data points

for the distribution of (q?) vs. 27;: can be obtained by assuming that

A

103) (11%)

(GT) z W % <$F>21

 

(4.19)

where (231?) denotes central values of 2:1: in each bin in F ig.4.8a.§ We refer to the
resulting values as “derived data”.

Note that the shapes of (12%) vs. 23,: and ((1%) vs. 231: are quite different. The
transformation from Fig. 4.8a to Fig. 4.8b shows immediately that the wing-like shape
of the distribution of (11%) vs. :01: should be attributed to a purely kinematical effect,
namely an extra factor 1/2:2 which is absent in the distribution of (q?) vs. mp. Once
this extra factor is removed, we see from Fig. 4.8b that (q%) increases monotonically
and rapidly as 2 approaches zero. In other words, the qT distribution broadens rapidly
when 2 decreases. This behavior is approximately realized by the simple z-dependent

nonperturbative Sudakov factor 3” P (b, Q, 2:, 2) given in Eq. (4.18).

 

§In principle, a more accurate experimental distribution (qr?) vs. 2:): can be determined by its
direct measurement.

112

 

10

 

(1/ 01.1) dU/dpr. 1/GeV

  

10

IITIITT

 
 

I

I

10

-l]'-

 

 

UIIII
I

\
llllllllllllllllllllllllllllllllllllij“ lllllllll

0 0511.5 2 2.5 3 3.5 4 4.5 5
p.,GeV

 

Figure 4.9: The dependence of the charged particle multiplicity on the transverse
momentum pT of the observed particles in the hCM frame. The data points are from
[60]. The solid and dashed curves correspond to the resummed and NLO multiplici-

ties, respectively.

The parametrization of S N P (b, Q, 2:, 2) was chosen to maximize the agreement be-
tween the resummed distribution of (q?) vs. mp and the “derived data” (cf. Fig. 4.8b).
Figure 4.8b shows that the resummed calculation is in better agreement with the data

points than the NLO expression. We have found it difficult to reproduce the rapid

113

growth of (q%) as 23p ——> 0 in either approach. In the future, it will be interesting to
see how a more precise theoretical study will be able to explain adequately this rapid
growth of (q%) in the region 2:p —-) 0, assuming that the actual experimental data for
the (q%) vs. mp distribution resemble the “derived data” discussed above.

The resummation also significantly affects the pT dependence of the charged par-
ticle multiplicity. In Fig.4.9 we present the distribution (1/atot)da/dpq~. We see
that resummation effects must be included to describe the shape of this distribution
at pT S 1 GeV. Furthermore, resummation also improves the agreement between
the theory and the experiment in the whole range of m. Through Eq. (3.24), the
improved description of the qT distribution in the small-qT region translates into a
better agreement with the pT distribution in the whole range of pT. Just as in the
case of the z-flow, the fixed-order calculation gives a rate that is too small compared
to the data, which implies that higher-order corrections are important. Until the
complete 0(039) corrections are available, the resummation formalism, which already
accounts for the most important contributions in the region of the phase space with
the highest rate (i.e., at small qT), serves as a better theoretical prediction in the
whole range of pr.

Finally, Fig. 4.10 shows the sup-distribution for the charged particle multiplicity
(1 /0,ot)d0'/d2:p. We see that both the resummed and fixed—order distributions are
in reasonable agreement with the data and with earlier published theoretical results
for the (9(015) xp-distributions [66]. For the fixed—order multiplicity, we present two
additional curves corresponding to different choices of the factorization scale 11 in
(3.60); the lower and upper dotted curves correspond to p. :: 0.5Q and 2Q, respec-
tively. Note that the scale dependence of the N LO multiplicity increases when 2 —+ 0.
Also note that the resummed multiplicity is significantly lower than the data in the

two lowest bins of xp ((2):) = 0.075 and 0.125), but consistent with the NLO mul-

114

 

tiplicity within the uncertainty due to the scale dependence. Such behavior of the
resummed multiplicity results from the dependence of the C(05) coefficient functions
C§:t(1)(3, C1, C2, b.., 11) on the additional term In? which was given in Eqs. (3.108) and
(3.109) and discussed at the end of Subsection 3.4.2. This negative logarithm domi-
nates the Comm-functions at very small values of 3. Similarly, the integral (3.79) of

the NLO cross section over the lowest bin 0 S q% S ((1%)2 depends on In? through

the terms

2
_ __ _ 6 .p. ~ . _ _, 6,, ,
2W1n(gq591)2<6(1 Z) 02 20(1) + Pb2(z)6(1 ) J )

as given in (3.81). Numerically, this dependence is less pronounced than in the re-
summed cross section. For 2 S, 0.1, the growing scale dependence of the multiplicity
in the C(05) calculation indicates that unaccounted higher-order effects become im-
portant and are needed to improve the theory predictions. For example, including
the (9mg) coefficient C]? in the resummed calculation will be necessary to improve

the description of the charged particle multiplicity in the small-z region.

115

O
N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

u D
X ..
'0 I
\ _
b .
.0
§ ‘1"
Q —
‘— I
V10 7 *.— .......
3 +
" 'uuvuuuuh
~ 4
........ 1.......
L
_fi fi -
l _— '
I- l ................
-1
10 .-
bl l I l l l l I l l l l l l l l l l l l l l l l 1 l l I 1 l l I l l l l l 14 LL L

 

 

 

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 )((3.9
F

Figure 4.10: The dependence of the charged particle multiplicity on the Feynman
variable 2:p in the hCM frame. The solid curve corresponds to the resummed mul-
tiplicity. The dashed, lower dotted and upper dotted curves correspond to the NLO

multiplicity calculated for ,u = Q, 0.5Q and 2Q, respectively.

116

4.3 Discussion and conclusions

The results in this Chapter demonstrate that multiple radiation of soft and collinear
partons influence a large class of observables and can be described with the help of
the CSS resummation formalism [32,33]. Multiple parton radiation affects hadropro-
duction in the current region of deep-inelastic scattering, i.e., for large pseudorapidity
of the final-state hadrons in the photon-proton c.m. frame.

Although the resummation formalism needs further development, in particular in
the procedure for matching the resummed curve to the perturbative result in the
transition region, it already improves the agreement between the theory and the data
and provides interesting insights about qualitative features of SIDIS. The formalism
describes well the behavior of the transverse energy flows measured at HERA [64,65]
in the region of large hCM pseudorapidity 17m, 2 3.0. At smaller pseudorapidities
the NLO rate falls below the existing data. Evidently, this is a signature of the
importance of the NNLO corrections, which were not studied in this paper. The
resummation formalism describes the pseudorapidity distributions of the transverse
energy flow more accurately than the NLO calculation; this formalism also has good
potential to improve the description of various distributions of particle multiplicity.

The presented analysis shows that the experimentally measured qT distributions
for the energy flow broaden rapidly as 2: —-> 0. This rapid broadening of the qT dis-
tributions can be realized if the nonperturbative Sudakov factor in the resummed
energy flow increases as 1 /:1: Similarly, the qT distribution for the charged particle
multiplicity broadens rapidly when 2 —1 0, which is consistent with the nonperturba-
tive Sudakov factor increasing as 2‘”. The SIDIS nonperturbative Sudakov factors
at small values of x and z are therefore qualitatively different from the known non-

perturbative Sudakov factors for vector boson production and e+e‘ hadroproduction,

117

which do not depend on the longitudinal variables at all. The rapid growth of the
nonperturbative Sudakov factor in SIDIS might indicate that the ep collider HERA
tests the resummation formalism in a new dynamical regime, which was not yet stud-
ied at colliders of other types. In particular, the CSS formula adopted here assumes
the usual DGLAP physics for the evolution of initial and final state partons [47], in
which the radiation of unobserved collinear partons is kT-ordered. The broadening
of the qT distributions may be a result of the increasing importance of kT-unordered
radiation in the limit 2: —> 0. The growth of the nonperturbative Sudakov factor S N P
as 2: decreases may be caused by the increase of the intrinsic transverse momentum
of the probed partons due to such radiation.

There are several theoretical aspects of the resummation formalism that can be
clarified when more experimental data are published. Perhaps the largest uncertainty
in the predictions of the resummation formalism comes from the unknown nonpertur-
bative contributions, which in the b—space formalism are included in the nonpertur-
bative Sudakov factor S N P (b) I have presented simple parametrizations of S N P (b)
for the transverse energy flow (cf. Eqs. (46,412)) and charged particle multiplicity
(cf. Eq. (4.18)). These parametrizations were found by fitting the resummed energy
flow and charged particle multiplicity to the data from Refs. [64,65] and Ref. [60],
respectively. Experimental measurements outside the range of those data will make
it possible to further improve these parametrizations and, hence, the accuracy of the
resummation formalism.

The most straightforward way to study S N P (b) is by measuring the variation of
the (IT spectra of physical quantities due to variations of one kinematical variable,
with other variables fixed or varying only in a small range. For the energy flow, it
would be beneficial to obtain more data at 2: > 10”, where the predictions of the

resummation formalism can be tested more reliably, without potential uncertainties

118

due to the small-2: physics. Another interesting question is the dependence of the
nonperturbative Sudakov factor on the virtuality Q of the vector boson. This de-
pendence can be tested by studying the qT spectra in a range of Q with sufficient
experimental resolution in the current fragmentation region. Finally, to study effects
of multiple parton radiation on semi-inclusive production of individual hadrons, it
will be interesting to see the qT spectra for particle production multiplicities with
the additional constraint 2 > 0.05 ~ 0.1, i.e., in the kinematical region where the

parametrizations of the fragmentation functions are known reasonably well.

119

Chapter 5

Azimuthal asymmetries of SIDIS

observables

In a recent publication [61] the ZEUS Collaboration at DESY-HERA has presented
data on asymmetries of charged particle (11*) production in the process 6 + p 1+
e + hi + X, with respect to the angle (0 defined as the angle between the lepton
scattering plane and the hadron production plane (of hi and the exchanged virtual
photon). This angle is shown in Figure 3.3. The azimuthal asymmetries, (cos (p) and
(cos 21,0), as functions of the minimal transverse momentum p6 of the observed charged

hadron h:h in the hadron-photon center-of-mass (hCM) frame, are defined as

271' d
(cosn >( )_ f dd) f0 d‘pCOS n‘pdndzdogdprdn
W “C — fd<I>f2"d —————da
0 (pdxddezdedgo

 

, (5.1)

with n = 1,2. In terms of the momenta of the initial proton pi, the final-state

hadron 10%, and the exchanged photon q“, the variables in (5.1) are Q2 = —q,,q",

2: = Q2/2(pA ~q), and z = (pA -pB)/(pA-q). fd<I> denotes the integral over 2:, z,Q2,pT
within the region defined by 0.01 < :1; < 0.1, 180 Gev2 < Q2 < 7220 GeV2, 0.2 <

120

z < 1, and pr > 126. Nonzero (cos 21p) comes from interference of the helicity +1 and
—1 amplitudes of the transverse photon polarization; and nonzero (cos (,0) comes from
interference of transverse and longitudinal photon polarization.

More than 20 years ago it was proposed to test QCD by comparing measured
azimuthal asymmetries to the perturbative predictions [92]. However, it was also
realized that nonperturbative contributions and higher-twist effects may affect the
comparison [72,93—95]. For example, intrinsic kT might be used to parametrize the
nonperturbative effects [93], and indeed ZEUS did apply this idea to their analysis
of the data. The relative importance of the nonperturbative effects is expected to
decrease as p1 increases. Thus, the azimuthal asymmetries in semi-inclusive deep-
inelastic scattering (SIDIS) events with large [)7 should be dominated by perturbative
dynamics.

By comparing their data to the PQCD calculation at the leading order in as [96—
98], the ZEUS Collaboration concluded that the magnitude of the measured asymme-
try (cos (p) exceeds the theoretical prediction for pc < 1 GeV, and (cos 290) is system-
atically above the theoretical prediction for pc > 1.25 GeV. ZEUS also estimated the
possible nonperturbative contribution, by introducing a transverse momentum [CT of
the initial-state parton in the proton, and similarly of the final-state hadron due to
nonperturbative fragmentation. It was found that this nonperturbative contribution
is negligible for (cos 21,0). For (cos 1p), the nonperturbative contribution can be sizable
(up to 20%), but it is not large enough to account for the difference between the data
and the 0(015) calculation at low pc. Hence, it was suggested that the discrepancy in
(cos 1,0) may be caused by large higher-order corrections.

From the comparison to the PQCD calculation at the leading order in as [96,97],
the ZEUS Collaboration concluded that the data on the azimuthal asymmetries at

large values of pa, although not well described by the QCD predictions, do provide

121

clear evidence for a PQCD contribution to the azimuthal asymmetries. In this Chap—
ter, the ZEUS data is discussed in a framework of QCD resummation formalism
[17 , 21, 32, 33, 35—37] that takes into account the effects of multiple soft parton emis-
sion. The discussion targets two objectives. First, it is shown that the analysis of
(cos (,0) and (cos 290) based on fixed-order QCD is unsatisfactory because it ignores
large logarithmic corrections due to soft parton emission. In addition, perturbative
and nonperturbative contributions are mixed in the transverse momentum distri-
butions, so that the presented data does not clarify the dynamical mechanism that
generates the observed asymmetries. Second, I make two suggestions for improvement
of the analysis of the ZEUS data. I show that perturbative and nonperturbative con-
tributions can be separated more clearly in asymmetries depending on a variable qT
related to the pseudorapidity of the final hadron in the hCM frame. I also suggest to
measure the asymmetries of the transverse energy flow that are simpler and may be
calculated reliably. I present numerical predictions for the asymmetries of transverse

energy flow. These predictions are the most important result in this Chapter.

5.1 Large logarithmic corrections and resummation

The resummation formalism applied here was discussed in Chapter 3. It describes
production of nearly massless hadrons in the current fragmentation region, where the
production rate is the highest. In this region, transverse momentum distributions are
affected by large logarithmic QCD corrections due to radiation of soft and collinear
partons. The leading logarithmic contributions can be summed through all orders of
PQCD [32, 33,35, 36] by applying a method originally proposed for jet production in

.]..

e e“ annihilation [17] and vector boson production at hadron-hadron colliders [21].

According to Eq. (3.62), the spin-averaged cross section for SIDIS in a parity-

122

conserving channel, e. g., '7‘ exchange, can be decomposed into a sum of independent

contributions from four basis functions A,,(1,b, 1,0) of the leptonic angular parameters

1/1. <2 [34l=

do
dzddequgdcp

 

4
= Z ”We. 2,0219%)pr190]

10:1

Here 1,!) is the angle of a hyperbolic rotation (a boost) in Minkowski space; it is
related to the conventional DIS variable y, by y = Qz/xSeA = 2/ (1 + cosh 1/2). The
angular basis functions are A1 = 1 + cosh2 1/2, A2 = —2, A3 = — coscpsinh 210, A4 =
cos 21p sinh2 10. Of the four structure functions 9V, only 1V and 2V contribute to the
denominator of (5.1), i.e., the cp-integrated cross section. Of these two terms, 1V is
more singular, and it dominates the rate. According to the discussion in Chapter 3,
the singular contributions in 1V can be conveniently explored by introducing a scale
qT related to the polar angle (63”,) of the direction of the final hadron (B) in the
hCM frame:

QT : Q V l/il? — 1 exp (—770m)1 (52)

where mm is the pseudorapidity of the charged hadron in the hCM frame. In the
limit qT —-> 0, the structure function 1V is dominated by large logarithmic terms; it
has the form of 22:1(05/71)’: ZZZ] 11W") lnm(q%/Q2), where 110"") are some gener-
alized functions. To obtain a stable theoretical prediction, these large terms must be
resummed through all orders of PQCD. The other structure functions 231“V are finite
at this order; they will be approximated by fixed-order (9(013) expressions.

In Eq. (5.1), the numerator of (cos 1p) or (cos 21,0) depends only on the structure

function 3V or 4V, respectively. The measurement of (cos (,0) or (cos 21,0) must be

123

combined with good knowledge of the (p-integrated cross section, 2'. e., the denominator
of (5.1), to provide experimental information on the structure function 3V or 4V.
Thus it is crucial to check whether the theory can reproduce the cp-integrated cross
section as a function of pT before comparing the prediction for (5.1) to the data. But
Figure 4.9 shows that the 0((15) fixed-order cross section is significantly lower than the
data from [60] in the range of pT relevant to the ZEUS measurements. This difference
signals the importance of higher-order corrections and undermines the validity of the
C(05) result as a reliable approximation for the numerator of Eq. (5.1).

On the other hand, the resummation calculation with a proper choice of the non-
perturbative function yields a much better agreement with the experimental data for
the pT-distribution from [60]. One might try to improve the theoretical description
of the ZEUS data using resummation for the denominator of Eq. (5.1). However, the
resummation calculation for do/(dzddequ%d<,0) in the phase space region relevant
to the ZEUS data is currently not possible, largely because of the uncertainty in the
parametrization of the nonperturbative contributions in this region. The impact pa-
rameter (b-space) resummation formalism [32,33] includes a nonperturbative Sudakov
factor , which contains the effects of the intrinsic transverse momentum of the initial-
state parton and the nonperturbative fragmentation contributions to the transverse
momentum of the final-state hadron (cf. Eq. (3.112)). Without first determining this
nonperturbative factor, e.g., from other measurements, it is not possible to make a
trustworthy theoretical prediction for the denominator of Eq. (5.1) and, hence, these

azimuthal asymmetries.

124

 

 

 

 

 

 

 

/\ 0
S C _[__
N —+—
8 -1- —1—
U
V
k .
3'2? / ]
(I)
o ) —.._
Q .
V -3:' /
-4}
_5; —..—
_6.11111111111111I11111111111111111111111111111
0 0.25 0.5 0.75 1.0 0.75 1.5 1.75 2.0 2.25

pc(hCM), GeV

Figure 5.1: Comparison of the C(03) prediction for the ratio (cos (,0) / (cos 290) with
the ratio of experimentally measured values of (cos (p) and (cos 210) from [61]. The
error bars are calculated by adding the statistical errors of (cos (,0) and (cos 21,0) in
quadrature. Systematic errors are not included. The theoretical curve is calculated
for (2:) = 0.022, (Q2) = 750 GeVZ, using the CTEQ5M1 parton distribution functions

[90] and fragmentation functions by S. Kretzer from [88].

The azimuthal asymmetries measured by ZEUS may also be sensitive to uncertain-
ties in the fragmentation to hi in the final state. Indeed, the cross section in Eq. (5.1)
includes convolutions of hard scattering cross sections with fragmentation functions
(FFs), integrated over the range 0.2 < z < 1. Although the knowledge of FFs is
steadily improving [87—89], there is still some uncertainty about their z-dependence
and flavor structure for the range of Q relevant to the ZEUS measurement. Therefore
the most reliable tests of the theory would use observables that are not sensitive to

the final-state fragmentation. The asymmetries (cos mp) would be insensitive to FFs

125

if the dependence on the partonic variable 3 were similar in the hard parts of the
numerator and denominator of Eq. (5.1), so that the dependence on the FPS would
approximately cancel. It is shown in Appendix A that the partonic structure function
117, which dominates the denominator of (5.1), contains terms proportional to 1/22
that increase rapidly as 2 decreases. However, the most singular terms in the partonic
structure functions 3'41} are proportional to 1/?. Therefore, the dependence on the
FPS does not cancel in the azimuthal asymmetries.

A curious fact appears to support the suggestion that the theoretical predictions
for (cos n90) depend significantly on the fragmentation functions. While each of the
measured asymmetries, (cos cp) and (cos 210), deviates from the 0(ag) prediction, the
data actually agree well with the 0(a5) prediction for the ratio (cos (,0) / (cos 290), as
shown in Fig. 5.1. The error bars are the statistical errors on (cos 1,0) and (cos 20)
combined in quadrature; this may overestimate the experimental uncertainty if the
two errors are correlated. Since this ratio depends only on the numerators in Eq. (5.1),
which are less singular with respect to ’2 than the denominator, the dependence on
the fragmentation functions may be nearly canceled in the ratio. The good agreement
between the C(03) prediction and the experimental data for this ratio supports our
conjecture that the fragmentation dynamics has a significant impact on the individual
asymmetries defined in Eq. (5.1).

The final remark about the azimuthal asymmetries in Eq. (5.1) is that the pr (or
pc) distributions are not the best observables to separate the perturbative and non-
perturbative effects. The region where multiple parton radiation effects are important
is specified by the condition q§~/Q2 << 1. But the W distributions are smeared with
respect to the qT distributions by an additional factor of 2, because pT = z qT. Thus
the whole observable range of pT is sensitive to the resummation effects in the region

of qT of the order of several GeV. A better way to compare the data to the PQCD

126

prediction is to express the azimuthal asymmetries as a function of qT, not pT. Then
the comparison should be made in the region where the multiple parton radiation is

unimportant, i.e., for qT/Q ,3 1.

5.2 Asymmetry of energy flow

Next, I describe an alternative test of PQCD, which is less sensitive to the above
theoretical uncertainties: measurement of the azimuthal asymmetries of the trans-

verse energy fiow. In the hCM frame, the transverse energy flow can be written as

[33—36,86]

4

dET f , 2 2 r
ddequ%d¢ : Z pl’ET(-E1 Q 7(1T)‘4p(11/j1 (10) (O3)

0:1

 

Unlike the charged particle multiplicity, the energy flow does not depend on the
final-state fragmentation. According to the results in Chapter 4, a resummation cal-
culation can provide a good description for the experimental data on the cp-integrated

ET-flow. A new class of azimuthal asymmetries may be defined as

277 dB
1 (1‘1) fo COSWWW
211 (113 °
f (1(1) [0 dde2qu2 dgo C190

 

(ET cos 019M911 = (5-4)

The structure functions ”VET for the ET-flow can be derived from the structure func-
tions ”V for the SIDIS cross section using Eq. (3.122). Similar to the case of the par-
ticle multiplicities, the asymmetries (ET cos (0) and (ET cos 210) receive contributions
from 3VET and 4VET, respectively. But, unlike the previous case, the denominator
in Eq. (5.4) is approximated well by the resummed ET-flow. Thus these asymmetries
can be calculated with greater confidence.

Figure 5.2 shows the prediction for the azimuthal asymmetries (ET cos (,0) and

127

 

 

(ET cos 290) as functions of qT for (a) a: = 0.0047, Q2 = 33.2 G8V2 in the left plots
and (b) :2: = 0.026, Q2 = 617 GeV2 in the right plots. The asymmetries are shown
in qT-bins that are obtained from the experimental pseudorapidity bins for the (,0-
integrated ET-fiow data from Ref. [65]. The upper x-axis shows values of the hCM
pseudorapidity 17cm that correspond to the values of qT on the lower x-axis. For each
of the distributions in Fig. 5.2, the structure functions 3VET and 4V;_.;,. were calculated
at leading order in QCD, i.e., 0(05). The solid and dashed curves, which correspond
to the resummed and 0(a5) results respectively, differ because the structure function
1VET in the denominator of (5.4) differs for the two calculations. The resummed
go-integrated ET-flow is closer to the data than the fixed-order result, so that the
predictions made by PQCD for the subleading structure functions 3VET and 4V3,.
will be confirmed if the experimental azimuthal asymmetries agree with the resummed
distributions.

The discussion in Section 4.1 shows that in the region qT ~ Q the resummed
(p-integrated ET-flow is larger than the 0(a5) prediction. This explains why the
asymmetries for qT ~ Q are smaller for the resummed denominator than for the
0(a5) denominator. In the region qT/ Q < 1, the asymmetries are determined by
the asymptotic behavior of the fixed-order and resummed partonic structure functions
”VET. As qT —> O, the 0(a5) structure functions ( l171.3T)0(0,3), 3V5,” and 4173,. behave
as 1/q%, 1/qT and 1, respectively. Thus, asymptotically, the ratios 3"‘VETM IVETbmS)
go to zero, although the qT distribution for the asymmetry (ET cos (p) is quite large
and negative for small, but non-vanishing qT (cf. Fig. 5.2). Resummation of IVET
changes the qT-dependence of the denominator, which becomes nonsingular in the
limit qT —-> 0. Consequently, the asymmetry (ET cos cp) with the resummed denomi-
nator asymptotically grows as 1/qT (i.e., in accordance with the asymptotic behavior

of 3173,). Hence neither the fixed-order nor the resummed calculation for (ET cos cp)

128

is reliable in the low—qT region, so that higher-order or additional nonperturbative
contributions must be important at qT —) 0. The asymptotic limit for the resummed
(ET cos 2gp) remains finite, with the magnitude shown in Fig. 5.2. Since the magnitude
of (ET cos 290) is predicted not to exceed a few percent, an experimental observation
of a large asymmetry (ET cos 290) at small qT would signal the presence of some new
hadronic dynamics, e. 9., contributions from T—odd structure functions discussed in
[95].

Figure 5.2 shows that the predicted asymmetry (ET cos cp)(qT) at qT/Q = 1 is
about 1—2% for the resummed denominator, while it is about 2—4% for the (9(a5)
denominator. The asymmetry (ET cos 290)(q7~) at qT/Q = 1 is about 1.5-2% or 3.5-
5%, respectively. Both asymmetries are positive for qT ~ Q. According to Fig. 5.2a,
the size of the experimental qT bins (converted from the n bins in [65]) for low or in-
termediate values of Q2 is small enough to reveal the low-qT behavior of 3"‘VET with
acceptable accuracy. However, for the high-Q2 events in Fig. 5.2b, the experimental
resolution in qT may be insufficient for detailed studies in the low-qT region. Nonethe-
less, it will still be interesting to compare the experimental data to the predictions of
PQCD in the region qT/Q z 1, and to learn about the angular asymmetries at large
values of Q2 and 2:.

To conclude, the azimuthal asymmetry of the energy flow should be measured
as a function of the scale qT. These measurements would test the predictions of the
PQCD theory more reliably than the measurements of the asymmetries of the charged

particle multiplicity.

129

 

 

 

65 4 3 2 n(hCM)
I l

ooooooooooooooooooooo

 

 

 

 

    

 

 

 

O
0
-.
V‘I}_1TTVY‘YVVI
'1

 

O AAhAlJAkAl AllA‘AAlAALl

O
bl
O?
CD
:23
6‘.

 

 

   

 

Flt/0:1

.
-
AllAAALinAAl+LAAlAAAAlALLA

 

 

.........................

 

 

 

 

 

 

A ‘50. L 460
qt. GeV

 

Figure 5.2: Energy flow asymmetries (ET cos cp)(qT) and (ET cos 299)(qT) for (a) :r =
0.0047, (,22 = 33.2 GeV2 and (b) a: = 0.026, Q2 = 617 GeV2. The Figure shows

predictions from the resummed (solid) and the 0(03) (dashed) calculations.

130

Appendix A

The perturbative cross-section, finite

piece and z-flow distribution

In this Appendix, I collect the formulas for the NLO parton level cross~sections
d’o‘ba/(dEdEdQ2dq%dcp), which were originally obtained in [98].
According to Eqs. (3.60,3.63), the hadron level cross-section doBA / (dzdde2dq%d<p)

is related to the parton-level structure functions ”I71”, as

 

dUBA 4
= A
dxddequ%d¢ Z; pWJ, 99) x
1 d6 1 dga A A A
X Z] Ema/MSW] E—Fa/Mémxt)”Wa(rv,z,Q2,q§~.u)- (Al)
a,b Z .‘L‘ a

At non-zero qT, the parton cross—section receives the contribution from the real

emission diagrams (Fig. 3.6e-f), so that ”171,0 can be expressed as

pA
fba

 

OOF‘ 9‘25 [9%

47rSeA 7r Q2 — (i _ 1) (i — 1)] pfba(5,3,Q2,(1:2r), (A2)

(E Z

with the same notations as in Chapter 3. In this Equation, the qu structure func-

131

 

 

tions are

A A 2 _ M —
pfjk($a Z) Q2, qT) : 20F$Zۤ6jk pfjki

 

where
_ 4
lfjk : 622qu (£72- + (Q2 _ (1%)?) + 6;
2fjk Z 2(4 fjk) = 4;
f = i— (Q2 + «1%).

QQT

The C—1 9 structure functions are

pfjg($a “Q (1T) 5 {5(1— fie? pf—jgi

where
2

1‘. _ 9_ _L__2_ _3_Z.
f,g — (1%(5232 EE+2)+10 a? 3,
2fig : 2(4-19)=81

_ 2 9:).
3 2

. : __ +

f]. QqT(2(Q (1%)—

(-)
Finally, the g 4 structure functions are

”fgj(§3, 3, Q2, (1%) E chfll - 36? ”fgj,

132

(A6)

(A.7)

 

where

 

lfgj : 1~ ( (Q4 A 2 + (Q2 — 6302) +6;

 

Q2q% ff? 1 — z)
2ng = 2(4fgj) = 4;
:3ng : Q_~(IT (Q2 + 6T) (A 8)
In (A.8),
«1T —- 1237",. (A9)

The indices j and k correspond to a quark (antiquark) of a type j or k, the index g
corresponds to a gluon.

The finite part YEA of the (9(a5) cross section (A.1) is

“WE/.16? Tag/.5. >f f‘Ta/Aemmx

x ”Rba(E,E,Q2,qT, ,u). (A.10)

 

 

For p = 1, the functions ”Rho are

A A 2 1 1 A A
‘Rjk(x,z,Q2,q:2r) = ‘5 [53% _ (a? _ 1) (3" 1)] lfjk($,z,Q2,Q%)

1
— gdjke§{6(1—E)Pg(fi)+P(1)z(A)6(1—a;)
+ QCF6(1 — 2)5(1 — a) (log, 9—2— — §) }; (All)
QT 2
1 . A A 2 2 _ (1% 1 1 1 2 2
R..<x.z,Q .qT) — 6(52—2-(532—1) (71)] was, so .qT)
— éegau—apgm; (A12)

133

 

 

1 AA 2 2 (1% 1 1 1 AA 2 2
jo(.’L',Z,Q,qT) = 6[@‘2‘—(—_1) (§_1):| fgj($1zaQ1qT)

1 e——2P<1>(zA)5(1 — :5) (A.13)

21 94
QT

For p = 2,3,4,

q?
212,422 ,Q2, (12.): a (QT; —(% — 1) (é— 1)] 213,422, 422,112.). (A.14)

From (3.60), it is possible to derive the perturbative z-flow distribution,

 

 

de = 2/1 zdz dUBA
ddequ%d¢ — B 2m." dxddequ%d¢

 

F d___a_F 4

(A.15)

It depends on the same functions pfba(fr?, 3, Q2, q?) with the parton variable 33 deter-
mined by the 6-function in (A2),
A 1 — 35

Z = (q%/Q2 — 1) is + 1' (A16)

 

134

Appendix B

(9(a3) part of the resummed cross

section

In this Appendix, I demonstrate that the C(05) part of the W—term in the resummed
cross section (3.88) coincides with the small—qT approximation of the factorized 0(as)
fixed-order cross section. Correspondingly the complete 0(a5) part does not depend
on the scales 01 / b and CzQ separating collinear-soft and collinear contributions to
the resummed cross section.

At qT << Q, the resummed cross section (3.88) is dominated by the W—term:

 

 

 

 

(103,4 ~ 00EA1(1/),(P)/ d2b iq'T-Eiv _
dCIIddequgdcp resum ~ SeA 2 (27028 IVE/1(1)) _
0on A1(1/),<P)/ €125 1M~ 5
Q _
5.1 2622 (27028 Wme)’ (8'1)
where
B _=. Q5. (B.2)

135

According to Eq. (3.92), at b -—> 0

W310), Q, :8, Z) = Z 63(133/1 ® ijytflz, I1) (Ci: ‘3 Fa/A)($, b)e"SPQ’Q)- (B-3)

3'

The perturbative Sudakov factor SP and the C—functions in fig/1(1), Q,x, 2) can be

expanded up to 0(ag) using Eqs. (3.93,3.101-3.109)

 

0ng 2 2 2
5201.62) a / 3‘._Q—.(A(as(n>,cl>1nQ;Q +B(as(fi),Ci,C2)) z

 

 

 

 

02/112 H
A c2 2b2 _ 2b?
~‘jf (7112 2—3, +Bl(as(u>,ci,cg)ln Q2Q ———-—) + was
~ 35 2 C§Q2b2 _ 3 C22Q2b2 _ Cfbo C22Q2b2 .
CF (211102 2111 012 2111 Ci 1n C12 , (B.4)

 

 

' C 3
(Ci: ‘3 Fa/A)($,b,#) = Fa/A($,u) (1+ 5;ng (—ln2 1 + — 1n Cl ))

 

 

Czbo 2 Cgbo
013 (1 m ”b (1) -

+7 ((6.. ®Fa/A)(ft u)- (111— b0 __jaP ®Fa/A)($1#)) 1 (8'5)

C 3 C

out _ __ _ 2 1 — 1
(DB/bQQCbJ' )(z,b,u) _ [DB/“3’!” (10+ 7r SFC ( 11102120 + 21nC2b0))

011 b

+aS((DB/b ® Cm t)(31#) — (DB/b @111 —2/\20Pb(j1))(za #2)) a (B-G)

where the functions cgam ”((2)0111 are defined in Eqs. (3.82-3.84). In these equations the

running of as, which is an effect of 0%), is neglected. Inserting the C(05) represen-

tations (B.4-B.6) in Eq. (B.3), we obtain the 0(a5) expression for W311“), Q,a:, z) :

_ 2
—E ejx

o(0:5) j

Wang

Q,Q’xiz)

 

1 2 3 2
{DB/1(Zyflle/A($,u) [1_ QECF (21112 £2 “ '2‘1’1 11%)]
+¥ [((DB/b ® C(1-)0ut)(21#) - (DB/b @111 [ziQ—E‘EJ P('1))(Z1#)) Fj/AW l‘)

136

+DB/J(Z#) (((®Fa/A)($#)(1H[Qf]Pg®Fa/&~’E)(afll)]}-

(3.7)

This expression does not depend on the constants C1, C2, so that the only factorization
scale in Eq. (B7) is p. The Fourier-Bessel transform of W3A(b, Q, 2:, z) to the qT-space

can be realized by using relationships

 

 

_d2___b2 ~ ~
e-iQT'b : -' .
de ... - b2 7r[1L1
——e_““"'b ln — = —— ' B.9
/ (2w)? b3 q%~ ( )
de .1 ~ I)2 2 lug
6"‘1T'b 1n2 — = — [ T] , B.10
/(2W)2 b8 7r (1% + ( )

where the “+”—prescription is defined as

f quT mama) = [2" dcp /+ququf(qT) (9(a) — 9(6)). (8.11)

Hence the small-qT approximation for the 0(as)cross-section is

03‘7“; :6; x

X (6(QT)F6($1Z:QJ,#)+F+($azaQaqT1#))a(B'12)

dUBA
dxddequ%d<p

 

 

 

0(03 )9qT—’0

where

F5(CU, Z, Q3 ’1’) : DB/j(za.u’)E]/A($7H) +

i—S- { ((DB/b 69 6mm ”(2,11)" (DB/b @111 [2 Q] lejl))(z M) Fj/A($’”)

+ DB/j(Z,H) ((Ca (1)m ®Fa/A)($ M)" (In [

g] Pjg) ® Fa/A)($i #1)) }; (BB)

137

1 C23
F+($,Z,Q,QT,H) : ZWQZ?

{2CFDB/j(zau)Fj/A(xal‘) (fig-In 312—] ’2 [22?] ) + [Q—% ]x
+ + +

X ((DB/b ® leJ-l))(Z,M)Fj/A($,H) + DB/j(z'/”)(Pj(¢l) ‘59 Fa/A)( 17 H) )-(} (-B 14)

 

The function F5(:c,z,u) contributes at qT = 0; it receives contributions from the
leading order scattering, evolution of the PDFs between the scales p and Q, evolution
of the FPS between the scales u and 2Q, and C(05) coefficient functions cm“, C(1)out.
The function F+(:r,z,Q,qT,p) is just a regularized asymptotic part (3.74) of the
C(05) fixed-order structure function 1VBA (cf. Eq. (3.74)).

The 0(a3) cross section (8.12) can be integrated over the lowest bin of qT,

O S q% _<_ (q?)2 << Q2. The resulting integral is

 

 

 

(QT) dUBA_00F1 A1( (221) w)
d
/0 (IT dzddeQdfi dcp 21:62 X

meow—>0 56A
x (F5(z,z,o,u>+F;<x, 2,515,110 (8.15)
where
F;(:z:z £2311):
_ 35.1).( )F-( )01232__31;Q2_
27r B/J zvfl J/A 113M F n (qu n((1%)2

2
+ 1n (-——Qs)2 [(DB/b ‘59 PS) )(2 Hle/A($ u)

'l‘ DB/j (z) H)(Pj(¢l) ® Fa/A)($a #4)] } (B'lfi)

This expression agrees with Eqs. (3.79-3.81). Technically, this integration can be

138

 

easily realized by going back to the b—space and using relationships

3

qT gig" s
/ Jo(qu)quqT = flow), (317)
0

+00 b2
/ J1(ab)lnb—2bdb = —ln (a2), (B.18)
0 0

+00 ()2

J1(ab)ln2 b—2bdb = 1n2 (a2). (B.19)

o o

139

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