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<11) LIBRARY
Michigan State
University

 

 

 

TM: is to certify that ‘he

dissertation entitled

HEAVY QUARK PmDUCTION IN
PERIURBATIVE mD AT HERA

presented by
Xiaoning Wang

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Phys iCS

C5315; $712

. \I
Major professor

Date August 6 , 1998

 

MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

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HEAVY QUARK PRODUCTION IN PERTURBATIVE QCD AT HERA
By

Xiaoning Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1998

ABSTRACT

HEAVY QUARK PRODUCTION IN PERTURBATIVE QCD AT HERA

By

Xiaoning Wang

Heavy quark (charm quark, bottom quark) production in deep inelastic scattering
(DIS) has becomes an increasingly important area of research as new data from high

energy collider experiments are available.

The conventional perturbative Quantum Chromodynamics (PQCD) calculation
methods are not general enough to cover heavy quark production at all energies,
because this is a “two-large-scale problem”. Existing results tend to have a large

QCD scale dependence and the predictions do not agree with data well in some cases.

A unified PQCD formalism that is valid for all energy range was defined. This
unified scheme is often refered as the ACOT scheme, which is actually a composite
of two simple renormalization schemes. For charm quark DIS production, the two
simple schemes are the 3-fiavor scheme and the 4-flavor scheme. The 3-flavor scheme
is exactly the same as the conventional PQCD approach and applies in the quark
mass threshold region. The 4-flavor scheme treats the heavy quark as an additional
parton flavor inside the proton and includes new partonic scattering processes which

2
essentially resums the large logarithmic terms In — This scheme applies at higher

m2.
energy scales. In both schemes, charm mass is kept in the calculation. At the inter-
mediate energy scale region, a set of matching conditions are defined to provide the

transition between the two different renormalization schemes.

We implement the ACOT scheme calculation for charm productions in DIS. In this
implementation, we calculate both the 3-flavor contributions and the 4-flavor contri-
butions up to 0(a,). Our implementation uses the Monte Carlo method to perform
the phase space integrations. Results for inclusive structure functions and differen-
tial distribution functions of charm DIS production at HERA are presented. Our
results show that the ACOT scheme calculation is well behaved in the perturbative

expansion and the predictions agree with experiment very well.

For My Family.

iv

ACKNOWLEDGMENTS

First and foremost I would like to express my deepest gratitude to my thesis
advisor, Wu-Ki "Dung, for his constant supports and tireless guidance throughout my
research. Without his inspiration and insights, I would never have finished anything

nor learned so much.

I owe great thanks to Jim Amundson and Carl Schmidt, both of whom participated
in this project. Without their helps in theory and programming, this work would not

have been possible.

I am also very grateful to the members of my Thesis Committee: Jon Pumplin,
for his time and effort in proofreading my draft and correcting my English; Harry
Weerts, Tim Beers and Scott Pratt for their patient and careful evaluation of the

manuscript.

I thank C.—P. Yuan and Hung-Liang Lai for many fruitful discussions in both

physics and other common interests.

I also thank my colleagues in the Physics Department: Chris Glosser for correcting
my English in my first draft; Tom Rockwell, Csaba Balasz, Mike Wiest, Hongjian He,
Ming Lei and Jian Huang for many stimulating conversations that make these years

enjoyable.

I must thank my parents for their encouragement and help over the years. Without
their support, I would not be where I am today. Last, but not at all least, I express
my heartfelt thanks to Ming, the love of my life. For without her support, not only

would this dissertation be still incomplete, I would be incomplete as well.

Contents

LIST OF TABLES viii
LIST OF FIGURES ix
1 Introduction to the Standard Model 1
1.1 Strong Interaction ............................. 2
1.2 Electroweak Interaction .......................... 4
1.3 Higgs Sector and Spontaneous Symmetry Breaking .......... 7
1.4 Yukawa Interaction and the CKM Matrix ................ 10
1.5 Gauge Interactions ............................ 12
1.6 Problems with the Standard Model ................... 13
2 Deep Inelastic Scattering, QCD Parton Model and Asymptotic Fme-
dom 15
2.1 Deep Inelastic Scattering ......................... 16
2.2 The Naive Parton Model ......................... 21
2.3 ‘QCD and Hadron Physics ........................ 26
2.4 Factorization and the QCD Improved Parton Model .......... 30
2.5 Other Hadronic Interactions ....................... 34
2.6 The Global Analysis for Parton Distribution Emotions ........ 37
3 Heavy Quark Production Mechanisms 39
3.1 Heavy Quark Production in Collider Physics .............. 40
3.2 Conventional Methods for Heavy Quark Production Calculation . . . 46
3.3 The Three-flavor Scheme ......................... 51
3.4 The Four-flavor Scheme ......................... 52
3.5 The Composite Scheme — The Unified Approach to Charm Production 56

vi

Implementation of the ACOT Scheme Calculation 59

4.1 The Calculation Formalism ........................ 60
4.2 Heavy Quark Mass Effects on the Kinematics ............. 68
4.3 The Next-to—Leading Order Production Mechanism .......... 71
4.4 Leading Order Calculations ....................... 77
4.5 Next-to—Leading Order Calculations ................... 78
Results of Inclusive and Differential Distributions of Charm Quark

Production at HERA 83
5.1 Implementation of the Calculation .................... 84
5.2 Inclusive Charm Structure Function ................... 91
5.3 Differential Distributions ......................... 95
Summary and Outlook 108

Helicity Amplitudes of Heavy Quark Deep Inelastic Scattering Pro-

ductions Via Neutral Current Interactions 110
A.1 7"/Z + C -+ C .............................. 110
A.2 7"/Z+g—> 0+6, HC1 ......................... 111
A.3 7"/Z + C —+ C +g, HE2 ......................... 113
From Helicity Amplitudes to Cross Sections 117
3.1 7"/Z + C —> C, HEl ........................... 120
B2 7"/Z + C -+ C, HE2 ........................... 121
3.3 7‘/Z + g -+ C + U, HC1 ......................... 122
BA 7"/Z + C -) C +g, HE2 ......................... 124
LIST OF REFERENCES 127

vii

THES;3

of...

< 1121/? )

lllllllllllllllllllllllllillllllllllllllllllllllllllllllllllll

3 1293 01766 702 7

LIBRARY

Michigan State
University

 

 

 

This is to certify that ‘he

dissertation entitled

HEAVY QUARK PRODUCTION IN
PERTURBATIVE QCD AT HERA

presented by
Xiaoning Wang

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in PhVS iCS

CL)\—r («-44 K7
Wu-Ki Tung 4W2?“

Major professor V

Date August 6 , 1998

MSU is an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

 

 

 

 

Jul unu

- all 5". u. -1 sun. ..

PLACE lN RETURN Box to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

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1m macs-914

HEAVY QUARK PRODUCTION IN PERTURBATIVE QCD AT HERA
By

Xiaoning Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1998

ABSTRACT

HEAVY QUARK PRODUCTION IN PERTURBATIVE QCD AT HERA

By

Xiaoning Wang

Heavy quark (charm quark, bottom quark) production in deep inelastic scattering
(DIS) has becomes an increasingly important area of research as new data from high

energy collider experiments are available.

The conventional perturbative Quantum Chromodynamics (PQCD) calculation
methods are not general enough to cover heavy quark production at all energies,
because this is a “two-large-scale problem”. Existing results tend to have a large

QCD scale dependence and the predictions do not agree with data well in some cases.

A unified PQCD formalism that is valid for all energy range was defined. This
unified scheme is often refered as the ACOT scheme, which is actually a composite
of two simple renormalization schemes. For charm quark DIS production, the two
simple schemes are the 3-fiavor scheme and the 4-fiavor scheme. The 3-fiavor scheme
is exactly the same as the conventional PQCD approach and applies in the quark
mass threshold region. The 4-fiavor scheme treats the heavy quark as an additional
parton flavor inside the proton and includes new partonic scattering processes which

2
essentially resums the large logarithmic terms In — This scheme applies at higher

1712 °
energy scales. In both schemes, charm mass is kept in the calculation. At the inter-
mediate energy scale region, a set of matching conditions are defined to provide the

transition between the two different renormalization schemes.

We implement the ACOT scheme calculation for charm productions in DIS. In this
implementation, we calculate both the 3—flavor contributions and the 4—flavor contri-
butions up to O(a,). Our implementation uses the Monte Carlo method to perform
the phase space integrations. Results for inclusive structure functions and differen-
tial distribution functions of charm DIS production at HERA are presented. Our
results show that the ACOT scheme calculation is well behaved in the perturbative

expansion and the predictions agree with experiment very well.

For My Family.

iv

ACKNOWLEDGMENTS

First and foremost I would like to express my deepest gratitude to my thesis
advisor, Wu-Ki Tung, for his constant supports and tireless guidance throughout my
research. Without his inspiration and insights, I would never have finished anything

nor learned so much.

I owe great thanks to Jim Amundson and Carl Schmidt, both of whom participated
in this project. Without their helps in theory and programming, this work would not

have been possible.

I am also very grateful to the members of my Thesis Committee: Jon Pumplin,
for his time and effort in proofreading my draft and correcting my English; Harry
Weerts, Tim Beers and Scott Pratt for their patient and careful evaluation of the

manuscript.

I thank C.—P. Yuan and Hung—Liang Lai for many fruitful discussions in both

physics and other common interests.

I also thank my colleagues in the Physics Department: Chris Glosser for correcting
my English in my first draft; Tom Rcckwell, Csaba Balasz, Mike Wiest, Hongjian He,
Ming Lei and Jian Huang for many stimulating conversations that make these years

enjoyable.

I must thank my parents for their encouragement and help over the years. Without
their support, I would not be where I am today. Last, but not at all least, I express
my heartfelt thanks to Ming, the love of my life. For without her support, not only

would this dissertation be still incomplete, I would be incomplete as well.

Contents

LIST OF TABLES viii
LIST OF FIGURES ix
1 Introduction to the Standard Model 1
1.1 Strong Interaction ............................. 2
1.2 Electroweak Interaction .......................... 4
1.3 Higgs Sector and Spontaneous Symmetry Breaking .......... 7
1.4 Yukawa Interaction and the CKM Matrix ................ 10
1.5 Gauge Interactions ............................ 12
1.6 Problems with the Standard Model ................... l3
2 Deep Inelastic Scattering, QCD Parton Model and Asymptotic Free-
dom 15
2.1 Deep Inelastic Scattering ......................... 16
2.2 The Naive Parton Model ......................... 21
2.3 QCD and Hadron Physics ........................ 26
2.4 Factorization and the QCD Improved Parton Model .......... 30
2.5 Other Hadronic Interactions ....................... 34
2.6 The Global Analysis for Parton Distribution Functions ........ 37
3 Heavy Quark Production Mechanisms 39
3.1 Heavy Quark Production in Collider Physics .............. 40
3.2 Conventional Methods for Heavy Quark Production Calculation . . . 46
3.3 The Three-flavor Scheme ......................... 51
3.4 The Four-flavor Scheme ......................... 52
3.5 The Composite Scheme —— The Unified Approach to Charm Production 56

vi

Implementation of the ACOT Scheme Calculation 59

4.1 The Calculation Formalism ........................ 60
4.2 Heavy Quark Mass Effects on the Kinematics ............. 68
4.3 The Next-to—Leading Order Production Mechanism .......... 71
4.4 Leading Order Calculations ....................... 77
4.5 Next-to—Leading Order Calculations ................... 78
Results of Inclusive and Differential Distributions of Charm Quark

Production at HERA 83
5.1 Implementation of the Calculation .................... 84
5.2 Inclusive Charm Structure Function ................... 91
5.3 Differential Distributions ......................... 95
Summary and Outlook 108

Helicity Amplitudes of Heavy Quark Deep Inelastic Scattering Pro-

ductions Via Neutral Current Interactions 110
A.1 7"/Z + C —+ C .............................. 110
A.2 7‘/Z+g—> 0+6, HC1 ......................... 111
A.3 7‘/Z + C —) C +g, HE2 ......................... 113
From Helicity Amplitudes to Cross Sections 117
8.1 7"/Z + C —i C, HEI ........................... 120
8.2 7‘/Z + C -+ C, HE2 ........................... 121
3.3 7"/Z +9 -+ C + C, HC1 ......................... 122
BA 7"/Z + C -> C +g, HE2 ......................... 124
LIST OF REFERENCES 127

vii

List of Tables

1.1 Boson Masses ............................... 2
1.2 Lepton and Quark Masses ........................ 3
1.3 SU (2) representation of the fermions .................. 5
1.4 Quantum numbers of the fermion spectrum ............... 5
2.1 MS renormalization parameters ..................... 28

viii

List of Figures

1.1
1.2

2.1
2.2
2.3
2.4
2.5
2.6
2.7

3.1

3.2

3.3

3.4

3.5

3.6

4.1
4.2
4.3

Electroweak charged current and neutral current interactions ..... 13
Gauge boson self-interactions ...................... 14
Deep inelastic scattering I + h -) l’ + X ................. 17
The parton model interpretation of e + P —+ e + X process ...... 23
Born level e + q -> e + q scattering ................... 24
QCD running coupling a, with n, = 5. ................. 30
Vector boson production A -i- B -) V + X ............... 34
Partonic process (1 + 7 -> Wi/Z for vector boson production . . . . 35
Jet production ............................... 36
Scale dependence of the integrated b-quark p1 distribution at 630 GeV

(dashed lines) and at 1800 GeV (solid lines), for difl'erent values of p?"'. 42

CDF data on the integrated b—quark 19. distribution, compared to the
results of N LO QCD ............................ 43
D0 data on the integrated b—quark pr distribution, compared to the
results of NLO QCD ............................ 44
Linear comparison between experimental data and theory for the inte-
grated b-quark 1), distribution ....................... 45
Three-flavor scheme production mechanism. (a) is 7" + g —) c + E

process(heavy quark creation: HC), (b) is one loop 7" + g -) c + 2!
process, (c) is 7" +9 —) g+ c+7§ process, ((1) is 7" +0 -) a+c+E
process. Only one diagram for each process is plotted .......... 53
Four flavor production processes. (a) is 7" + 0/5 -+ c/‘c’ process(heavy
quark excitation: HE), (b) is one loop 7" + 6/6 -) c/‘c' process, (c) is
7" -i- c/E —» g + c/E process. (d) is 7" + g —-) c + ‘0' process (heavy

quark creation: HC), Only one diagram for each process is shown. . . 54
The factorization for the deep inelastic scattering ........... 61
The factorization of the structure functions ............... 62
Deep inelastic scattering: vector boson and nucleon scattering . . . . 69

ix

4.4

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

Processes contributing to the leading ACOT scheme calculation. (a),
(b), (c) only present in the 4-flavor region, ((1), (e), (f), (g), (h) presents
in both the 3-flavor scheme region and the 4-flavor scheme region.
HC1:(d), HE1:(a), HE2:(b),(c), GF2:(e), HC2:(f),(g),(h). Only one
diagram for each process is plotted. ...................

F; for Q2 = 100 and me = 1.5 GeV from 1(')"‘c —-) c) + (7‘0 —) gc)
processes. Solid line is this work’s result. Dashed line is the calculation
of Reference [69] ..............................

F; for a: = 0.1 and me = 1.5 GeV from 1(7% -) c) + (7's —) gc)
processes. Solid line is this work’s result. Dashed line is the calculation
of Reference [69] ..............................

F; for Q2 = 10 and me = 1.5 GeV from l(7"‘c -> c) + (7"c —) gc)
processes. Solid line is this work’s result. Dashed line is the calculation
of Reference [69] ..............................

F; for Q2 = 10 and me = 0.3 GeV from l('y"c -) c) + (7‘0 -) gc)
processes. Solid line is this work’s result. Dashed line is the calculation
of Reference [69] ..............................

p-dependence of F2°(:r = 0.01,Q = 10 GeV) in various calculational
schemes. no is defined by Equation. 5.9. ................

Measured D‘ cross section at Zeus [63] with experimental cuts (de-
scribed in the text) compared with our calculation as a function of the
pole mass me. The shaded band represents the 1 — 0‘ experimental
errors. ...................................

Total F; for various values of Q2. Solid lines: ACOT. Dashed lines:
three-flavor NLO calculation of Reference [38]. Dotted lines: three-
flavor LO calculation. Solid points: Zeus 95 preliminary [65]. Open
points: H1 [64]. ..............................

F; at a: = 0.1, solid line: This work, dashed line: LO ACOT calcula-
tion, dotted line: LO three-flavor calculation ..............

Contributions of the various subprocesses to F§(.r, Q) at fixed a: = 0.01.
The sign convention is such that the total cross section is HEl + HC1
- HCIsub + HE2 - HEzsub + GF2 ....................

Contributions of the various subprocesses to F§(:r:, Q) at fixed (,22 = 7
GeV. The sign convention is such that the total cross section is HEl
+ HC1 - HClwb -i- HE2 - HE2,ub + GF2 ...............

5.11 1), distribution compared with data from Zeus 95. The experimental

5.12

5.13

5.14

cuts are described in the text. .....................

Q2 distribution compared with data from Zeus 95. The experimental
cuts are described in the text. ......................

W distribution compared with data from Zeus 95. The experimental
cuts are described in the text. .....................

17 distribution compared with data from Zeus 95. The experimental
cuts are described in the text. .....................

74

89

90

91

92

99

101

102

102

103

104

105

106

A.1 7"/Z + g —t 0+ '5 in the CE center of mass frame ............. 113
A2 7"/Z + c —+ g + c in the gc center of mass frame ............. 115

xi

Chapter 1

Introduction to the Standard
Model

The Standard Model [1, 2] is the fundamental theory behind today’s high energy
physics phenomenological and experimental studies. It consists of electroweak theory
[3] and quantum Chromodynamics (QCD) theory [4], and has been very successful in

describing and predicting experimental results.

The Standard Model is a local gauge theory [5] with SU(3)C x SU(2)L x U(1)y
symmetry. Color SU(3)C; symmetry is associated QCD, which describes the elemen-
tary strong color interaction. SU(2)L X U(l)y is associated with electroweak theory,

which describes the electromagnetic and weak interactions.

The strong and electroweak interactions are mediated by gauge bosons. Eight
gluons with different color quantum numbers mediate the strong interaction, and
four vector bosons , W*, Z0 and 7 mediate the electroweak interaction. The gauge

bosons and their properties are listed in Table 1.1.

Leptons and quarks are the fundamental fermonic constituents of matter. Both
leptons and quarks engage in electroweak interactions through exchanges of Wi', Z
and 7 gauge bosons, but only quarks engage in the strong interactions as a conse-

quence of their color-charges. There are three generations of quarks and leptons in the

l

Table 1.1: Boson Masses

Particle Symbol Mass (GeV) Charge Spin Force

Photon 7 0 0 1 Electromagnetic Force
W Boson W:t 80.33 :1: 1 Charged Weak Force
Z Boson 20 91.187 0 1 Neutral Weak Force
Gluon G 0 0 1 Strong Force

 

Stande Model, and their masses are generated by spontaneous symmetry breaking

through Yukawa interaction. The masses are listed in Table 1.2.

All the standard model particles have been discovered except the Higgs boson.

The detailed properties of each particle can be found in the Particle Data Book [6].

1.1 Strong Interaction

Under SU(3) color symmetry, each quark flavor is represented by a color triplet,

grad
\I’q = qgreen a
qbluc

and gluons are associated with 3 x 3 — 1 = 8 SU(3) group generators. Quarks and

gluons interact with each other by exchanging gluons.

The strong interaction is described by the QCD Lagrangian:
L — 1F" F‘”" - '"D°q" 11
QCD _ -2 pu + z]:QI017 pfl f1 ( ' )

where

17;, = 0,0; — 6.0;”, — g,f.-,,.G{,0,’: (1.2)

Table 1.2: Lepton and Quark Masses

 

Flavor Symbol Mass (GeV)

Electron neutrino u.3 0

Electron 6 0.00051 First
Up quark 11 0.002 to 0.008 Generation
Down quark d 0.005 to 0.015

Muon neutrino up 0

Muon p 0.106 Second
Charm quark c 1.0 to 1.6 Generation
Strange quark s 0.1 to 0.3

Tau neutrino u, 0

Tau 7' 1.78 Third
Top quark t 180 Generation
Bottom quark b 4.1 to 4.5

 

is the field strength tensor for gluon fields CL, 1' = 1, 2, - - - , 8, and q? is the f-th (f =

1, . - -, 6) quark flavor (u,d,c,s,t,b) with color index a, a = 1,2,3.
0 ' A2113 1' .
Dpfi = (Dp)a[3 = 0,,605 + 1g,-2—Cr‘p (1.3)

is the quark covariant derivative and g, is QCD SU(3) running gauge coupling con-
stant. fijk (i, j, k = 1, - - . ,8) are SU(3) group structure constants, and they satisfy
the Lie algebra

[A‘, ,v'] = 21' f.,-,.,\*,

where 3, i = 1, - - - ,8, are the eight SU(3) group generators for the fundamental

representation. The A matrices are listed in Equation 1.4.

H

CFO
QC
COO
V
V
N
I
A A

It is easy to see from the QCD Lagrangian that the color interactions are diagonal
in the flavor indices, but in general change the quark colors. There are no mass terms

in Equation 1.1, as they will be generated later by spontaneous symmetry breaking.

1.2 Electroweak Interaction

SU(2)L and U(2)y represent weak isospin and weak hypercharge symmetries respec-

tively. The weak hypercharge Y is specified according to the formula

1
Q=T3+§Y

in electroweak theory to incorporate the electric charge Q and unify the electromag-

netic force and weak force.

The SU(2) symmetry is chiral, which means the left-handed fermion field ($1, =

1 3751/1) transforms differently from the right-handed fermion field ($3 = 1 .2751”:

Under SU(2) symmetry, the left-handed fermions behave as SU(2) doublets while the

 

 

right-handed fermions behave as singlets. There is no right-handed neutrino in the

Standard Model. The SU(2) representation of fermions are listed in Table 1.3.

Table 1.3: SU (2) representation of the fermions

 

' t
quarks; (3)149 (:)L, (b)L; "R1 d3, CR, 8R1 til: I”!

. ”c ”n V? .
leptons. ( e )L, (L‘ )L, ( r )L, 812, #1:, TR

 

 

 

 

 

 

Table 1.4: Quantum numbers of the fermion spectrum

Chirality Q T T3 Y 0

m 0 1/2 1/2 -1 o
«2,, -1 1/2 -1/2 -1 0
11;, 2/3 1/2 1/2 1/3 r,g,b
dL -1/3 1/2 -1/2 1/3 r,g,b
e3 -1 0 0 -2 0
11,; 2/3 0 0 4/3 r,g,b
d]; -1/3 0 O -2/3 r,g,b

 

The quantum numbers of the first generation of fermions are listed in Table 1.4,
where T is the weak isospin, and T3 is the third component of T. The other two

generations of fermions have exactly the same quantum numbers as those listed in

Table 1.4.

The electroweak interaction is described by Lagrangian:

Lelectrowealc = Lfermion + Lgauge + LHiggs + LYukawa - (15)

The fermion part is
3
Lyemgm = 2 (fimLi7"D,,qu+ImLi7“D,.lmL+fim3i7"D,,qu+Im3i7"D,,lmR), (1.6)
m=l

where m is the family index. Since the right-handed fermions do not couple to weak

isospin, their covariant derivative is
1),, = 6,, + 1321-143,“

where 91 is the U (1) gauge coupling constant, and Bu is the U (1) gauge field. The

corresponding covariant derivative for the left-handed fermions is

T:

2

g;

Dp=6u+12

YB“ + £92 Wg,

where 92 is the .S' U (2) L gauge coupling constant, and W3, j=1, .. -, 3 are the SU(2)],

gauge fields. 1'" are the Pauli matrices and shown in Equation 1.7.
1' _ 0 1 r _ 0 -1' 1' _ 1 0
1" 10 ’ 2‘ i o ’ 3" 0 —1 (1.7)

The gauge Lagrangian is

Lmc = -11?" F‘”" — game" (1.8)

4"”

where
BM = BpBV — 0V8”
F;, = apwj — aw; — gze‘ikwgwf
The gauge bosons will gain their masses through the spontaneous symmetry breaking

mechanism. The discovery of their mass eigenstates, the W35 and Z bosons, at CERN

by the UAl [7] and UA2 [8] groups in 1983 confirmed the electroweak theory.

7

1.3 Higgs Sector and Spontaneous Symmetry Break-
ing

The Lagrangian we discussed so far only contains massless gauge bosons and fermions.
Explicit mass terms in the Lagrangian are not allowed because if there were such bare
mass terms, the electroweak gauge invariance would be violated. On the other hand,
massless gauge bosons are not acceptable for the weak interactions since these are
known to be short-ranged. Hence, to have a sensible theory of massive elementary
particles and to be able to explain the short-ranged weak interactions, the gauge
invariance must be broken somehow. In the Standard Model, the Higgs mechanism
was introduced to account for the spontaneous symmetry breaking. The idea is that
instead of the Lagrangian, it is the vacuum that does not respect gauge invariance,

which induces effective masses for the propagating particles.

The Higgs Lagrangian is
ngg. = (D"¢)‘D,.¢ - V(¢), (1-9)

+
where qb = (:0 ) is the complex Higgs scalar. The Higgs scalar field is a doublet
under SU(2) and has U(1) charge Y,» = l. The gauge covariant derivative is

7‘

2

1),, = 3,, + £19,, + igg

2 w;

V(¢) is the Higgs potential and takes the form of

V(¢) = u2¢’¢ + A(¢*¢)2, A > 0, #2 < 0-
When A > 0, p2 < 0, it is easy to see that the ground state of the Higgs potential
2 _ 2
can be produced when vacuum expectation value, < ¢l¢ >0 = 1%, with v = —’\flm

When the Higgs field approaches its classical ground state, i.e., classical vacuum

0
state, 410 = (7'3) , the generator L1, L2 and L3 — Y are spontaneously broken,

ale. L1 4’0 at 0, L2 430 76 0, (L3 - Y/2) 450 7i 0. On the other hand, the electric
charge Q = L3 + :2,- is still conserved, that is, Q¢o = (L3 + Y/2) $0 = 0. Thus
after introducing the Higgs mechanism, the electroweak SU(2)L x U(l)y symmetry is

spontaneously broken down to U(1)Q.

Using the Kibble transformation, Higgs scalar can be rewritten as

s=(t‘3)=-J-§s‘E-’-s“‘(.fa) as»

where H is a neutral Hermitian field which will eventually turn out to be the physical

Higgs field. The three 1p fields are Goldstone bosons and will disappear from the

physical spectrum in the unitary gauge where

¢=—\;—§(v-i(-)H)’ (1'11)

but the Goldstone bosons essentially reappear as the longitudinal degree of freedom

of the gauge boson after the symmetry breaking.

In the unitary gauge, the covariant kinetic energy for the scalar field becomes:

[Du¢)t(Dp¢) = %(0,v + H) [%TiW; + 223- p]? (1) 3H)

)+g_21B”]2(v-EH)

(1.12)

gm;

2

1 W3 flw-
= §(O,U + H)

fiw+ W3

Working out the kinetic energy term for the gauge bosons, we have

92 ‘9192 3

m _ 1 W"

(D,.¢)1(Du¢) 4(7)2wgw u+§v2(W3,B,.)( 2 ) ( 3,, )+H terms.
‘9192 .91

(1.13)

This essentially generates the mass terms for the gauge bosons. The H terms in
Equation 1.13 includes the kinetic energy terms for the Higgs boson which we will

not discuss here.

In Equation 1.13, the neutral gauge boson masses are not diagonal in the basis of
W3 and the weak hypercharge B field, so a diagonalizing transformation is performed.
As the results of the transformation, two new fields, Z boson and photon 7 field, are

defined by the mixing of W3 and B.

Z = —sin9wB+cosi9wW3, (1.14)

A = cosflwB-i-sinflst, (1.15)

where 0w is called the weak angle and defined by tan 0w = £1. W‘t is defined as
2

Wi = —‘}_-2-(W‘ :1: 1W2). (1.16)

After the redefinition of the gauge boson fields, the kinetic energy term becomes

{[42

(Dp¢)l(D“¢) —) Meywfl‘W; + 2 Z"Z,, + H terms, (1.17)

from which the masses of the W and Z bosons can be easily obtained.

The masses of the four gauge bosons are

 

MW = 93—” (1.18)
_ 2 2 2= MW

Mz — 2 91+92 COSQW (1.19)

M, = o (1.20)

The W and Z bosons were discovered at CERN by UA1 [7] and UA2 [8] in 1983,
and their masses and properties are in very good agreement with the standard model

predications.

10

The Higgs mass is not predicted by the Standard Model because the A parameter
in Higgs potential term is unknown. Although various theoretical and experimental
limits seem to suggest that 60 GeV < M” < 0(600) GeV, Higgs remains the last

particle in the Stande Model to be found.

1.4 Yukawa Interaction and the CKM Matrix

The fermions gain their masses through Yukawa interaction by coupling with the

Higgs doublet. In the unitary gauge, the Yukawa Lagrangian is

 

3
+H v+H
"L name = —0 P“ _v 0 "I'-0 Pd do
Y I: m§=1 Qm,L m,n( \/§ )un,R qm,L m,n( fl ) 11.3
.0 U+H
+1", r5“, — 9, +H.C. 1.21
.L .( \f2- )6 ,R ( )

after the spontaneous symmetry breaking. In Equation 1.21, m,n = 1,2,3 are the

generation indices, and

.. = ((32)1(::)’(::))1

1. = ((”"‘)’(""")1(”"))s
8!. #1. TL
HR = "R, CRatR
d3 = (in, SR, be
en = BR, I13, 712- (1-22)

In Equation 1.21 and Equation 1.22, we have used fermion weak eigenstates in the
Lagrangian, and since I‘m," is a 3 x 3 matrix and generally not diagonal, it means there
are possible mixings among different families in Yukawa interaction. Alternatively,

we can perform unitary transformations on the left-handed and right-handed fermion

11

fields separately and rewrite the Yukawa Lagrangian as

3
_ . _ __ -325. .
LYukawa — §?¢L( mi 2M“, H)” iR+qliL(—m 8_2MWH)d:R
- gmi I
+I';L(—m;— mH)€,~R+ HUG
3 '__ 197"“
= 21.1 —m.- 211w ——H)¢.-, 11.23)
i=4

where 1,1).- can be identified as mass eigenstates of the fermions and m.- = 5.5 f,- as
the corresponding masses. However, since f,- is not predicated by the Standard Model,

the measured fermion masses are used as parameters in the Lagrangian instead.

From Equation 1.21 and 1.23, we can see that generally the electroweak eigenstates
of fermions are different from the mass eigenstates. However, this does not afl'ect the
lepton sector since the neutrinos are massless and their fields can be redefined without
affecting the Lagrangian. In the quark sector, the charged quark current, taking at,

as the an e basis and u’ as the mass basis, is
8 8 L

J“

charge

= 51,7de = E’L7”SZngd’L

and there is a mixing between different generations of quarks. By convention, the

three charge 2/ 3 quarks u, c and t are unmixed:

11:11-

All the mixing is therefore expressed in terms of a (3 x 3) unitary matrix V: SutSd

which operates on the charge (-1/ 3) quarks d, s and b:

d ' Vud Vus Vub d
s E V“) Vc, Vcb 8 (1-24)
b Weak If“! Via Kb b Mass

12

The quark mixing matrix V is called Cabibbo-Kobayashi-Maskawa (CKM) [9, 10,

11] matrix and can be parameterized by four parameters, 012, 013, 923 and 613.
012013 . s12013 . 813645“
V = —812€23 - 01232381387'6” 012023 - 312323313645.u 823013 (1°25)
812823 - 6126238138-“’" —012823 - 812C2:3«‘31:1*3-"’l3 023013

Here 0,-1- = cos 0,-1- and 3,-1- = sin 0,3, with i, j = 1, 2, 3 being the family label. The com-
plex phase introduced by 613 in the CKM matrix signals the existence of CP violation
in the Standard Model. In the limit of 623 = 013 = 0, the third generation decouples,

and the CKM matrix reduces to the usual Cabbibo matrix in GIM mechanism [10].

1 .5 Gauge Interactions

The major tests of the electroweak theory involve gauge interactions mediated by 7,
Wi and Z bosons. The charged current weak interaction mediated by W boson is in-
corporated into the Standard Model from the original four-fermi interaction, and the
7 mediated electromagnetic interaction is incorporated from quantum electrodynam-
ics. From the Standard Model SU(2) x U(1) local gauge theory, the neutral current
weak interaction mediated by Z boson was successfully predicted and confirmed in

the experiments.

The charged current interaction Lagrangian is given by

_ EL - t +
L——2‘/§(J{;,W,, +J,';,W,,), (1.26)

where Ja, is the weak current. The charged current weak interaction has been success-
fully tested in a large variety of weak decays, and it has been used to measure CKM
matrix elements. The neutrino-hadron scattering processes such as VpN -+ p'X as
shown in Figure 1.1 have been used as a probe the structure of the hadrons and QCD

as well.

13

”It [1 Ve ”6 e e

W Z 7*. Z

d u,c,t d

Figure 1.1: Electroweak charged current and neutral current interactions

The neutral current interaction Lagrangian is given by

2 2
+
L = ——Vg’292Jg (— sin awn) + cosowwwg) , (1.27)

where J; is the weak neutral current. Since there was no evidences for flavor-changing
neutral currents in early experiments, GIM mechanism was introduced along with the
prediction of the charm quark. The weak neutral current was discovered at CERN in
1973 and at Fermilab shortly after, and in 1974 .I/1t1 was discovered at Brookhaven
and SLAC as the lowest energy bound states of the charm quark. Since then, the
neutral weak current has been extensively studied in many difl'erent interactions such
as VeN —+ ueX as shown in Figure 1.1, and these have been the primary quantitative

test of the unification part of the Standard Model.

The self-interactions of the gauge bosons have not been extensively tested. Some
typical interactions predicted by the Standard Model are shown in Figure 1.2. These

tests will be very important for the Standard Model study and the Higgs search.

1.6 Problems with the Standard Model

The Standard Model is a very successful theory and its predictions have been consis-

tent with all experimental results so far. However, because it has too many arbitrary

14

9 g 9 9
9 9 9 9 9
9 g g 9
9 9 9

Figure 1.2: Gauge boson self-interactions

parameters, few believe it is the final fundamental theory. Not counting the assump-
tion that neutrinos are massless, there are 18 free parameters in the minimal Standard
Model Lagrangian, nine of which are fermion masses (mu, md, me, 111,, mg, m5, me,

mp, m,), four are CKM matrix parameters (012, 013, 923, 613), four are electroweak pa-
rameters (6, MW, 0w, M”), and the last one is the QCD strong coupling constant a,.
In addition, there is no explanation in the theory why there are three generations of
fermions, or what is the origin of the CKM flavor mixing. Also, the Standard Model is
a complicated direct product of three sub—groups SU(3) x SU(2) x U(1) with separate
gauge couplings, and there is no explanation why only the electroweak interaction is

chiral.

Various new theories have been proposed to extend the Standard Model, for ex-
ample, SUSY, GUT, superstring theory, etc. But so far, there is no concrete exper-
imental evidence to support any of these new models, and despite its shortcomings,

the Standard Model seems to agree with experiments amazingly well.

14

g 9
9 g 9 9
9 9 9 9 9
9 g g 9
9 9 9

Figure 1.2: Gauge boson self-interactions

parameters, few believe it is the final fundamental theory. Not counting the assump-
tion that neutrinos are massless, there are 18 free parameters in the minimal Standard
Model Lagrangian, nine of which are fermion masses (mu, m4, mc, m,, mt, m5, me,

mp, m,), four are CKM matrix parameters (012, 013, 023, 613), four are electroweak pa-
rameters (e, MW, 0w, M H), and the last one is the QCD strong coupling constant 01,.
In addition, there is no explanation in the theory why there are three generations of
fermions, or what is the origin of the CKM flavor mixing. Also, the Standard Model is
a complicated direct product of three sub-groups SU(3) X SU(2) x U(1) with separate
gauge couplings, and there is no explanation why only the electroweak interaction is

chiral.

Various new theories have been proposed to extend the Standard Model, for ex-
ample, SUSY, GUT, superstring theory, etc. But so far, there is no concrete exper-
imental evidence to support any of these new models, and despite its shortcomings,

the Standard Model seems to agree with experiments amazingly well.

Chapter 2

Deep Inelastic Scattering, QCD
Parton Model and Asymptotic
Freedom

That QCD is the theory of hadrons and the strong interaction has been universally
accepted today, and in virtually every experimental analysis and theoretical calcu-
lation, QCD plays an important role. However, in the early sixties, because of the
bewildering spectrum of baryons, mesons and their resonances, it was by no means
obvious that a theory of strong interaction would succeed at all. Douglas B. Lenat,
one of the foremost computer scientists today, in describing the reason of artificial
intelligence as his career choice in the Sixties, once said [12], “ I got far enough along
in mathematics to realize I would not be one of the world’s great mathematicians . . .
I got far enough along in physics to realize that in some sense it was all built on sand
People would walk around with ever-growing chest pocket cards of elementary
particles which really means resonances that were found but not understood. Things

were just happening that divorced themselves from physical reality. ”

While a systematic basic theory for the strong interaction seemed out of reach at
that time, much progress was still made in hadron physics, for example, Gell-Mann’s

constitute quark model successfully explained hadron and meson resonances. How-

15

16

ever, it was not until the SLAC—MIT [13] eXperiment in the late sixties that strong
interaction theory and hadron physics really began their exciting development. The
experiments clearly showed that the proton possessed charge substructure of a spatial
size much smaller that proton itself, and suggested an incoherent scattering process
between the lepton and the constituent substructure. The “Bjorken scaling” [14] ob-
served in the experiment was successfully explained by the naive parton model. Since
a “color” quantum number for the quark was required in the constituent-quark model,
it was quickly recognized the theory could be a SU(3)C color Yang-Mills gauge theory.
The theory was later found to possess a number of important prOperties, one of the
most crucial one being asymptotic freedom, which means the coupling decreases with
an increase in the measured energy scale. With asymptotic freedom, the separation
of long distance physics and short distance physics, that is, factorization, is verified,

and the QCD parton model is established in the context of quantum field theory.

2.1 Deep Inelastic Scattering

Deep inelastic scattering (DIS) plays a crucial role in our understanding of the hadron
structures. In sixties, the SLAC-MIT experiment of electron-nucleon scattering gave
us the first evidence that strong interactions become weak at short distances, and
today, the structure function results from DIS give us the most precise tests of the
theory and most accurate data used to determine the momentum distributions of

partons inside hadrons.

Consider the process

((11) + h(p) -+ 1’11’) + X , (2.1)

as illustrated in Figure 2.1, where we label the initial-state lepton of momentum k"

by l(k), the initial-state hadron of momentum p" by h(p), the final-state lepton of

17

momentum k'” by l’ (k’), and the inclusive hadronic final state by X. The lepton l
and the hadron h interact through the exchange of vector boson V which can be a 7,

W", or Z boson. The momentum of the exchanged vector boson is labeled as

q" = k'” — k" . (2.2)

1(k) 1’(k’)

V

W X

h(p)

 

 

Figure 2.1: Deep inelastic scattering l + h —-> I’ + X

Because of the point-like electroweak interaction between the vector bosOn and the
lepton, the cross section for this process can be written in term of hadron structure
functions,

dak' 0‘], u» k Vh
' 23w) 4712(42 _ m2V)2L1v( .q)W,... (9,9) . (2.3)

 

where cv is the coupling constant, Lfi',’ is the leptonic tensor and W31," is the hadronic
tensor. Notice that the leptonic part and the hadronic part are separated in Equa-
tion 2.3 and the only connection is vector boson of momentum transfer q.” So lepton-
hadron scattering can also be viewed as a vector boson scattering on a hadron with
center of mass energy W = (q + p)2. Note that W equals the square of the final

hadronic state invariance mass.

Lf‘; can be easily calculated from electroweak theory:

Lit/1&9) = nTrUfl‘tzw- CPI/1] , (2-4)

18

where I‘m is the electroweak vertex connecting lepton i to vector V and the outgoing
lepton 1’, but with the factor 0%, removed. To average over lepton spin, we set 11 equal
to 1/2 for unpolarized ei or 14*, and 1 for 11(0). For example, for photon exchange

at e+p->e+X,

L""(k,f1) = 212191.11 - W - We" + +1. - q 9*") (2.5)

The hadronic tensor is defined in term of the electroweak current operators,

W‘Vh’usq) )=—):(h1p)lj"* (0)|X)(X|J'.Y(0)Ih(p))X(27r)‘6‘(p+q—px), 12.6)

where sum is done on the final inclusive hadronic state. Unlike leptonic sector,
(lefi’ (0)|h(p)> is not calculable and W” is usually defined in term of several struc-

ture functions or form factors.

The scattering process is deeply inelastic provided the magnitude of momentum

2
transfer Q2 = -q2 and p . q are both large while their ratio a: = 21? q is fixed.

On the other hand, when the invariant mass of the inclusive final hadronic state

 

W = —Q2 + 2p - q + M}, —2 Mg, taking electron-proton scattering as an example,
the proton is mostly probed by long—wavelength photons with modest values of Q2,
and as a result, the proton may be excited to various resonances and quickly decay
into different baryons and mesons afterwards. In elastic scattering, proton will not
be'broken into other hadrons, and the process could be described by e(E1) + P —-)
e(E2) + P with W = Mg. In this case, WW can be expressed in term of two form
factors, F1 and F2, which are related to the proton charge and magnetic moment
distributions. Contracting the tensor indices in L’“’ and WW, we can derive the

Rosenbluth Formula:

d0 a2 E2 20 Q2 20
55"“ ‘ (4Efsin4g) E1 {(F‘ ' 4MgF22)m82 OS'2' ' 2mm1 +"F2) 8’“ 2}
(2.7)

 

19

Rosebluth Formula describes the elastic scattering between the electron and the pro-

ton.’ F1 and F2 in Equation 2.7 only depend on the scattering angle 0, and E2 is
. El
1 + (2E1/Mp) sin2 (9/2)‘

However, when Q2 becomes large enough, the proton is bombarded by short-

 

fixed by the elastic scattering kinematics as

wavelength, energetic photons (or W, Z bosons when Q2 reaches their mass thresh-
old) and will break up. To describe this more complicated deep inelastic scattering
interaction, a set of six independent basis tensors is needed. So, in deep inelastic

scattering, Wz,” can be written as

Vh _ Pupv scafluupaqfi
Wit” — —g’wW1 + wwg - [WW3

qpqu prv + 911171; prv - gupu
—W ———W ————W , 2.8

where the scalar coefficient functions W,- are the invariant hadron structure functions.

In photon (7") and proton deep inelastic scattering, due to parity conservation

and electromagnetic current conservation,
9"W,fL" =9"W,fL" = 0

it. can be shown that

 

W3- = o (2.9)
. 2

W4 = (22—q)2W2+?'W1 (2.10)

W — 2”'"W 211

5 _ - Q2 2 ( ' )

W6 = o (2.12)

and hadron tensor W3,” can be simplified as

W59 = —19"" — 32-35% figs" — qus‘xp" - %ZQ")W2 12.13)

20

in this special case.

In deep inelastic scattering, several standard kinetic variables are usually used,

 

 

p2 = M:
u = 5%:531—532
2 2
x = ZE-qzth(g—E2)
s = (k+p)2=ME+g—:

l-x
a:

 

W = (q + p)2 = M3 + Q2 (2-14)

where E1 and E2 are energies of the incoming and outgoing lepton respectively in the

rest frame of the proton.

In Equation 2.8, {W4, W5, W6} terms are usually ignored because their contribu-
tion to the cross section are proportional to 0(mf/Q2) after contracting with the
leptonic tensor. Also, the proton structure functions {W.-} are usually replaced by

three dimensionless structure functions {E}, where

F1($,Q2) = W1(st2)a (2'15)
F2(a:,Q2) = fiW2(z,Q2), (2.16)
F3($,Q2) = fiww’). (217)

Contracting the leptonic tensor and hadronic tensor in Equation 2.3, the differen-
tial cross section for DIS can be expressed in terms of the dimensionless variables a:

and y and the structure functions {17,-},

do
dxdy

 

 

2
= MVP-W1 + <1 — y - Aggyw. + my - 95M] , (2.18)

21

where 6" is :l:1 for Wi exchange and zero for the photon exchange, and

 

M E
IV.7 = 87ra2—é1r, (2.19)
wt 2 MhE
= . .2
N 7'“ 29.111Z ow“;2 + MEL)? (2 0)
Or alternatively, the cross section in the hadron target rest frame is
do .
m = N” [2W1(W')(x, q2) sm2(0/2) + W2(W.')(:c, q2) cos2(0/2)
+6VW§VM (x, q2)-E—l-A-;—--Ei sin2(0/2)] (2.21)
)1

Comparing with Rosebluth Formula in Equation 2.7, W1,W2 and W3 are functions
of 1:,q2, or equivalently, 0, q2. In Equation 2.21, E2 is not kinematically fixed by El

and the scattering angle 0 as in Equation 2.7.

The striking feature of early deep inelastic scattering experiments results was
that for Q2 2 2 GeVz, the structure function F,(x, Q2) become functions of a: only,
nearly independent of Q2. This property is called “Bjorken scaling” [14]. It was
originally postulated by Bjorken and later dramatically confirmed by the SLAC-MIT

experiment. The “Bjorken scaling” is nicely explained by the Naive Parton Model.-

2.2 The Naive Parton Model

The naive parton model [17, 18] assumes that a hadron is made of point-like on-
shell particles which are called partons. Each of these partons carries some fraction
5 of the proton momenta. In deep inelastic scattering, the striking vector boson
interacts incoherently with the partons one at a time, which means during the short
distance interaction, the other partons are present merely as spectators and there is

no interference with the scattering mechanism.

22

The parton model does not make predictions about the distributions of the partons
inside hadrons, nor does it explain how the partons eventually hadronize into baryons

and mesons. That information must be extracted from experimental data.

Mathematically, the fundamental relation of the parton for deep inelastic scatter-

ing can be written as

dammq) = $1.1“ dogma.) me). (222)
where dam (p, q) is the inclusive cross section for lepton-hadron scattering, and daggm
is the Born level, elastic lepton-parton scattering cross section. daggm (5p, q) is calcu-
lable in perturbative QCD theory. The functions <15, m (6) are the parton distribution
functions (PDF), which describe the probability of finding a parton of flavor f with
momentum fraction 6 in the hadron. These parton distribution functions are not cal-
culable at the present time because they involve non-perturbative physics and must
be extracted from experimental data. Obviously the parameter £ in (bf/N“) is be-
tween zero and one, and because of the total momentum conservation, the patron

distribution functions must satisfy
1
2 f0 45 5 ¢1/~(€) = 1. (2.23)
1‘

Early deep inelastic scattering experiments also confirmed the Callan-Gross rela—
tion which states that the scaling functions are related by 2xF1(:c) = F2(a:). This
relation reflects the fact that the charged partons inside the proton carry spin 1/2,
which are exactly the quark constitutes of the proton in the Cell-Mann quark model
[19]. Hence by summing over all contributing partons in the proton, the quantum

numbers of the proton should be recovered. It follows that

[01 ( u(a:) — 11(1) )dx = 2, (2.24)

23

and
1 _
[o ( d(a:) — (1(3) )dx = 1 (2.25)

number sum rules must be satisfied by the parton distribution functions.

 

' Figure 2.2: The parton model interpretation of e + P —> e + X process

The parton model interpretation of the deep inelastic scattering can be pictured as
Figure 2.2. Notice in Equation 2.22, there is no interference between different flavors
and different momentum fractions 5, and the total cross section is just an incoherent
sum of the cross sections of all the available partons and momentum fractions. This
“incoherence” feature implies that parton distribution functions are universal and
independent of short distance scattering processes. There is no explanation for this
in the naive parton model, and it is invoked as an assumption. In QCD improved

parton mode, it is proved order by order in the context of perturbative QCD.

To calculate am using Equation 2.22, we need to calculate the lepton-parton
scattering a” first. The lepton-parton scattering process is shown in Figure 2.3.
(loan can be calculated eaSily for this 2 —> 2 scattering process since the electroweak
couplings between the vector boson and the quark are point-like couplings. Similar to

Equation 2.3 where daw') was written in term of hadronic tensor WW, we can factor

24

out the lepton sector and write dam) as

dsk’ qzc“:

(’f)_

 

where wW is the partonic tensor. Upon factoring out the lepton tensor in Equation 2.3
and 2.26, we can get the parton model relation between hadronic tensor and partonic

tensor,

WSW = z [13:- wt!) «mm (2.27)

q q

Figure 2.3: Born level e + q —+ e + q scattering
For photon mediated electron-proton deep inelastic scattering process, similar to
Equation 2.13,
cm u qpqu P ' q
wpu = _(g,._ q” —)w w1+ —(P" - —9")(P - -q-2-q”)w2- (2-28)

A calculation of the Born level photon parton elastic scattering process Figure 2.3

gives the results for ngl,

1 d3p’
(em) = _ __ 2 I 4 4 I __ _
w”, 8“ (2W)323¢ eftrm my 1))(27r) <5 (p p q)

 

1 . q qu 2

. . 1
+0». - (1.2—24x12. — qu%g)e§;5(l - x) (2.29)

25

where's, is the fractional charge of the parton. We then have, from Equation 2.28,
(n 1 2 mm 2 mi
")1 = 58,6(1_ 1‘) , ")2 = 8,6(1"' 107. (2.30)
Or using dimensionless structure function,

F”): -e,6(l— x) , F2”) = e}6(1— :r) (2.31)

Substitute win or F5”) into Equation 2.27, it is easy to find
2xF‘”’(x )= F‘"’( (x)= 2: Q, 2 «we: r) (232)

Despite of its success in explaining the “Bjorken scaling” of the proton structure
functions, the naive parton model can not be accepted as a complete theory of hadron
physics. First, more accurate experimental data show that DIS structure functions
only scale approximately instead of the naive parton model’s perfect scaling predic-
tion, and the naive parton model can’t explain this scaling violation. Furthermore, its
assumption of free partons inside hadron can not be confirmed either experimentally
or theoretically, and the naive parton model itself does not give any insight about
why the assumption is true either. A calculation [20] based on the measured structure

function data reveals that
/ d2: a:( )+ do: )+ -(x) + 3(2) ) g 0.54 (2.33)

instead of 1. The other 46% momentum of the proton is carried by neutral particles,
that is, the gauge boson of the SU(3) theory, gluons. With the discovery of asymp-
totic freedom in SU(3) theory, QCD became the candidate for the strong interaction.
Later, the factorization theorem was proved in the context of the perturbative theory,
and QCD not only provides a theoretical basis for the naive parton model, but also

improves on it in many aspects. Today, the QCD improved parton model has become

26

the cornerstone of most applications of perturbative QCD to observable phenomena.
In the next two sections, we will first discuss QCD asymptotic freedom and then the

factorization theorem.

2.3 QCD and Hadron Physics

The QCD Lagrangian, including the mass terms, is

~ 1
LQCD = -ZF(:)F(O)’W + ;w 1/J(1‘)’(D m) — mq6gj)¢}
Fifi) = 611021—600 -g.f.b.G‘Gf,

p
a
i

X-
(Dp)ij = 5.53,. +19. JG" (2-34)

2
as described in Chapter One. However, in quantum field theory [15], we need an
extra gauge fixing term to quantize the Lagrangian since the part of the Lagrangian
quadratic in the gauge field has no inverse. This, depending on the gauge fixing term
chosen, may necessitate a ghost term to satisfy the unitary requirement. Conventional

gauge fixing terms choices can be one of the following:

Lcovariant = -§:(60Ga )2 (2.35)
1
Latin! — -2—A(n' '2G) (2.36)

where A is the gauge parameter. When A = 1, the covariant gauge becomes the
familiar Feynman gauge. When A = 0, n2 = 0, the axial gauge is called the light-
cone gauge. Covariant gauge fixing term requires a ghost Lagrangian which is given
by

L91... = 3w“'(D£‘w°). (2-37)
where 17" is a complex scalar ghost field that obeys Fermi statistics. From the complete
QCD Lagrangian,

LQCD = iQCD + Lgauge—fizing + Lghost , (2-38)

27

the Feynman rules for QCD can be derived.

The theory, of course, must be renormalized [16]. Using dimensional regulariza-
tion, the integration of the two, three and four point functions of the quark, gluon
and ghost fields can be carried out in d dimensions in which the integrals become
finite and the singularities are exhibited as poles in e = (4 — d)/2. Denoting Zi’s
as the renormalization factors, the renormalized fields G, q, 17 and the renormalized

parameters g, m can be written as

yo)" = u—‘Zr‘zgflgm’m (2.39)
m = z;1m<°>, (2.40)
G‘ = 2;”20‘0), (2.41)
q. = 2;”2 5,0), (2.42)
17., = 23-1/2179), (2.43)
A = z;1,\(°) (2.44)

where p is a mass scale parameter introduced in dimensional regularization to keep

the Lagrangian dimension correct in d dimension.

Renormalization requires the ultraviolet divergence terms (the pole terms in e) to
be absorbed into theoretical bare quantities, and different ways to do this result in
different renormalization schemes. In QCD calculations, 191—8- renormalization scheme
is usually chosen. In this scheme, one chooses various Z, in such a way that the %
terms are subtracted along with a fixed finite constant term. Some of the Z.- to one

loop order are listed in Table 2.1.

One of the most crucial features of QCD is that it is an asymptotically free theory,
which means the strength of the strong coupling decreases as the momentum scale

at which it is defined increases. Asymptotic freedom can be derived from renormal-

-28

Table 2.1: Mg renormalization parameters

 

 

 

 

 

 

 

 

 

 

 

92 1 17 3A 4
z, 1+ 161:2 (e 7" +111(47”) NC(12 4 ) 3””
22 1-i(l-7E+ln( 411%) [CPA]

g? 16 2 A
23"16—2 (- - 7.; +1n(4vr)) IVA-62 — 5) — 3.41.]
~ g2 ( 3 A ’
Z3 1+T6— -l-—7E+ln(41r)) Nc(Z-Z)

1
2... 1+ 1%: 2 - — 7,; +1n(41r))[3cp]
Z, l + i—g-z— (- - 73 + Eln(41r)) [Nc(— — -2-n 1T3]

 

 

 

 

ization group equations (RGE), which originate from the fact that the S-matrix is
independent of the scale [1 and can be computed in terms of either bare quantities
or renormalized quantities. In fig scheme, the relation between the bare and the

renormalized coupling constant is

you) = #“Z;’go, (2.45)

where Z, = Zl' 1Z3” .The beta functionfl(g) is defined as

 

6
My) = 6mm»... (2-46)
From Equation 2.45 and
2—Z ( - )6 3Z (2 47)
”a” 9 9(903p» _ (9)69 99 '

6(9) can easily be solved. Using Table 2.1, to the one loop order,

I 2
My...) = —;§3—,;(13Nc - 5M- (2.48)

For QCD, NC = 3, n, = 6, therefore, 6(9) is negative. A negative 3 function means

the renormalized coupling will decrease with the increase of the renormalization scale,

29

and thus QCD is an asymptotically free theory. In the case of QED, on the contrary,
the coupling will increase as the renormalization scale increases since the fl function

is positive as shown in Equation 2.49.

 

 

 

 

e3
3(e(u)) = 1—2;r_2- (2-49)
To two loop order, a more complicated calculation yields
_ g" 11NC—2n,_ 153—61911,+7
fl Inln(p )
0.(#2) = 0300‘2) [1 - 050012) 2 4 firm/1A2 QCD + 0(a:o(#2))] (2-51)
where
47r
a 2 = , 2.52
50(l" ) fl11n(p2/Agcp) ( )
and
51 = (11Nc — 2nf)/3 , ,52 = 102 — 38nf/3; (2.53)

In Figure 2.4, we plot the two loop order strong coupling a, as a function of
the energy scale it with different choices of AQCD. The plots clearly show that as
energy scale [1 increases, the strong coupling 0, decreases: a, -+ 0 as p —) 00.
This property is called asymptotic freedom, and it is with asymptotic freedom that
everything begins. Without it, there will be no natural explanation in the quantum

field theory for the parton model.

AQCD in Equation 2.51 and 2.52 is introduced to cut off the integration when
solving the differential equations. For Q2 >> A500, (1, is small and perturbative QCD
can be applied. For Q2 ~ A500, (1, is large, which means the gluons and quarks are
strongly coupled together and order by order perturbative QCD expansions will not

work in this situation.

30

 

0.4 - ----.-,
.

 

0.35 P ‘I‘Vx‘ A .1

01.01.63»)

 

 

 

 

Figure 2.4: QCD running coupling a, with n, = 5.

2.4 Factorization and the QCD Improved Parton
Model

Although the naive parton model can not be accepted as a complete theory of hadron
physics, much of its structure remains in perturbation theory. This has to be at-

tributed to the property of factorization [21].

Factorization permits cross sections of high energy scattering processes to be writ-
ten as a convolution of a hard scattering cross section and a remainder which contains
the low energy physics. The former contains only the high energy and momentum
components, and because of asymptotic freedom, it can be calculated order by order
in perturbation theory. The latter piece describes non-perturbative physics, and is
described by the process independent, universal parton distribution function. For
deep inelastic scattering, factorization theorem can be written as

do’”(p,q) z] d£d0“°’(£p.q.u MA) )¢./~(e,u ) (2.54)

0:9 9

31

where do" is the hard scattering piece, aha/N is the parton distribution function, and a
includes quarks and gluons. Comparing with Equation 2.22 of the naive parton model,
QCD scale ,1 dependency is introduced after the renormalization and factorization are
carried out, a, dependence originates from the order by order perturbative expansions.
As before, 6 is the momentum fraction of the hadron that a parton carries. Parton
distribution functions ¢a/N(£, p2) depend on both 5 and energy scale p. The energy
scale dependency in (pa/N (£, 112) originates from the factorization scale introduced
when long distance physics is separated from the short distance physics. The proofs
of factorization theorem require a detailed examination of all the dangerous regions

of phase space in Feynman diagrams and is beyond the scope of this thesis.

A remarkable result of factorization is that measuring parton distribution func-
tions at one value of 112 allows us to predict their values at all other values of 112, as
long as the p2 are large enough so that the perturbation theory is applicable. This
ability is related to the freedom in choosing the renormalization and factorization
scales in the proofs of the factorization theorem. In order to perform the factor-
ization, we have to introduce the artificial scale p2 which separate the high energy
physics and low energy physics. However, no physical quantity can depend on the
particular value chosen for this scale. In Equation 2.54, we notice the left side of
the equation has no QCD scale p2 dependence, which is exactly the way a physical
observable should be, so

d IN_ d (la) _
dpa .. (1"(0 49¢) -0. g (2.55)

Thus, the p dependence in hard scattering cross section 0““) and parton distribution
functions (bu/N must compensate each other. Although ¢a/N can not be calculated
in perturbation theory, however, the perturbative partonic distribution functions,

(Pa/“$112), which represents the probability of finding a parton a in a parton b with

32

a momentum fraction 6 of the longitudinal momentum of the parent parton, can
be calculated order by order. The 450/), functions are not physical quantities like
(pa/N, but they essentially represent the evolutions of the parton distribution functions
due to the parton splittings in the parton-parton collinear configuration. In QCD,
which graphs contain these collinear singularities depend on the gauge choice. In the
light cone gauge, the graphs responsible are the “ladder” diagrams. Systematically
calculating all the “ladder” diagrams in the light cone gauge, the variations of the

parton distributions with the changes of scale p can be obtained,

d a 2 l 1'
$043142) = -—32(:—)/x ? 9i(£a#2)qu(E) + G(£,p)qu(%)] (2.56)

for quark (or antiquark) distributions q,-, and

d 2 1
Watt: [12) = Egg—)1 25-]; (145911513096?) + G(£,fl)PGG(%)] (2.57)

for gluon distribution G. These are known as Gribov-Lipatov-Altarelli-Parisi (GLAP)
evolution equations [22, 23, 24], and the evolution kernel PqG(x), PGq(:r), and PGG(:r)
are called splitting functions. The splitting functions can be calculated order by order

in perturbation theory, for example,

as
P,,(z, (1,) = P;3)(z) + gpmz) + - -~ (2.58)

Based on quark number conservation and momentum conservation in the splittings

of quarks and gluons, the splitting functions must satisfy,

qu=Pfi3 PqG=P§G, (2.59)
l
[0 dz P,,(z) = o, (2.60)
I
[0 dz 2 [qu(z) + qu(z)] == 0, (2.61)

[01 dz 2 [2nIPqG(z) + PGG(;1:)] = 0 (2.62)

33

The lowest order approximations to the evolution kernel are,

 

 

 

 

 

P<°)(z) = Cp (11::)++-:-6(1-x)], (2.63)
P(0)(z ) = TR [:1:2 +(1-x) 2] (2.64)
P(°)(z) = C [1+ (1’3”?) (2.65)
P(0)(z) = 2Nc [(15104- + 1;”: +x(l —x)]
+5(1 - at) (“NC 24mm, (2.66)
where
N3—1
CF: 2N. ’
T3 = 31—).

2

The methods of QCD improved parton model can also be applied to the frag-
mentation process, which describes the decay of a parton into hadrons. In this case,
fragmentation function DH/,~(£, 1.12) is defined as the probability of a parton i decaying
into hadron H which carries a fraction 6 of the parton momentum. The evolution

function for fragmentation functions can be derived similar to the GLAP equation,

r

-d-D"(z,#2) = Mfg- Df(£.u2)qu(£

a:
27f é . 6) + Df(€1p)qu(E)] (2.67)

 

_DH(.,,.2) = Mfg); Dfi’(£,u2)qu(£

H 2

Currently, the fragmentation functions are mostly extracted from e+e‘ annihilation

data.

34

2.5 Other Hadronic Interactions

Besides the deep inelastic scattering, QCD parton model has been successfully applied
to other processes such as vector boson production, jet production, and direct photon
productions etc. For each of these processes, theoretical calculations are performed
using the factorization theorem, and the results so far are in good agreement with

experimental data.

 

Figure 2.5: Vector boson production A + B -) V + X

Vector Boson production (VBP) in hadron A hadron B collision
A(p) + B(p’) —> V(q) + X -> 1(k) + l’(k’) + X, (2.69)

as shown in Figure 2.5, is the simplest process for large transverse momentum re-
actions with two colliding hadrons in the initial state. The vector boson generated
during the hard scattering, '7, W*, or Z, is usually detected through its leptonic

decay products.

The factorization for inclusive vector boson production can be written as,

 

do", I 1 l I I dflv 6 1£’p,1 9
#(papaq) =§¢/0‘ (16/0 d£¢a/A(£1I‘2)¢b/B(£ 1’12) 0““ pdqz q ,1) . (2.70)

35

where a, b are the two partons from the two colliding hadrons and 5, 5’ are their
momentum fractions respectively. From Equation 2.70, we see that the hard scattering
can be calculated in d&.+b.>y+x, and the universal parton distribution functions, just
as in deep inelastic scattering, describe the long distance physics and are factored out
from the hard scattering. The lowest order contribution to the hard scattering in
vector boson production is the process q + 7 -) W, Z or 7", as shown in Figure 2.6.
Vector Boson Production was first analyzed by Drell and Yan, so it is also called

Drell-Yan(DY) [25] process sometimes.

9

W35, Z

ql

Figure 2.6: Partonic process q + E" -) W‘t /Z for vector boson production

Vector Boson Production is a complimentary process to deep inelastic scattering
and it provides a lot of useful information needed in the parton distribution function
analysis. Recently, it has been used as a precision test for electroweak theory in W

and Z boson productions [26].

Another important application of the QCD improved parton is the jet production
in hadron collisions. Jet production is the dominant hard scattering activity in hadron
collisions because of the strong coupling between quarks and gluons. Jets are formed
when. the colored final state quarks and gluons in the hard scattering hadronize to

the observable color neutral particles. The validity of the QCD improved parton

36

model for the description of large transverse momentum hadron-hadron interaction
got dramatic qualitative confirmation when the clear jet events were found by the

UA1 and UA2 experiments [27, 28].
For single jet inclusive process, the jet production cross section can be written as

do , ' (16
—(AB 4 gets) =2 [dxadxb¢a/A(xaap2)¢b/B($b1p2)—§(ab "" Cd), (271)
JP? abcd d1),

where p, is the transverse momentum relative to the beam axis of the scattered

partons, and it is given by

)

2

P: = (2-72)

Q>| Q)

§ = (17., + P02 is the squared center of mass energy of the parton subprocess, f =
(12,, - 19..)2 is the t-channel energy exchange 11 = (pa — p.,¢)2 is the u-channel energy
exchange. A list of the lowest order partonic scattering processes between quarks and

gluons in jet production is shown in Figure 2.7.

iii
3:
:64

119%

Figure 2.7: Jet production

QCD improved parton model has also been applied into direct photon production,
6+8” jet production etc. Furthermore, not only are inclusive quantities calculated

through the factorization theorem, differential distributions have also been calculated

37

for various scattering processes. The results usually are in good agreement with the

experimental data.

2.6 The Global Analysis for Parton Distribution
Functions

The factorization theorem, based on the QCD parton model described before pro-
vides the foundation for analyzing high energy hard scattering processes. There are
two basic ingredients of calculations used for comparing theoretical predictions with
experiments: (1) the perturbatively calculated scattering cross sections involving the
fundamental partons, leptons, and gauge bosons; and (2) the parton distributions
inside the incoming hadrons. The universal, i.e. process independent, parton distri-
butions functions (PDF’s) are derived from the analysis of data in a variety of hard

scattering processes, but governed by the renormalization group equations.

With the wealth of data and corresponding theoretical calculations from various
processes, global QCD analyses have become possible. In such an analysis there are
two main goals. The first is to determine the parton distribution functions as precisely
as possible, and the second is to explore whether or not the parton level theoretical
calculations in perturbative QCD constitute a consistent theoretical framework to
account for all the available experimental data. Here we briefly lay out the essential

elements of performing a global QCD analysis used by the CTEQ collaboration.

0 A well-defined physical measurable can be written in terms of the convolution

of parton distributions and the hard cross sections by the factorization theorem;

0'th = f®6 . (2.73)

38

o The hard cross sections can be calculated order by order in 04,:
6 :2 0120”,, . (2.74)

o Patton distributions evolve in ,u according to the renormalization group equa-
tions;

d a,
(£711.45 — 5P ® 4’ , (2°75)

where the splitting function P is calculable order by order in 0,.

0 Since the initial parton distributions are in a non-perturbative physics regime
and not calculable, their initial conditions are parameterized at the scale )4 = Q0

with certain functional forms: ¢(p = (20,33) = (150(17)-

o AQCD is needed for the calculation of a,

With experiments on the one hand and parameter space (e.g. initial parton distribu-
tion parameters and AQCD) on the other, based on QCD theory, CTEQ performs a
least x2 fit by adjusting parameters to obtain parton distributions and the correspond-
ing a, in consistency with data. Most of the modern global analyses [29, 30, 31, 32, 33]

use both the hard cross section 6 and the splitting function P in NLO.

Chapter 3

Heavy Quark Production
Mechanisms

The QCD parton model we discussed in Chapter 2 allows one to relate the non-
calculable hadronic structure functions to the calculable partonic structure functions
involving only elementary particles. This is achieved through the factorization theo-
rem which separates the long distance physics from the short distance physics. The
conventional QCD factorization theorem works well in one large scale problems such
as inclusive deep inelastic scattering where Q2 of the probing vector boson is the only
large scale. However, in the case of heavy quark production, there exists an additional
scale -— quark mass. For multiple-scale [34] problems like heavy quark production,
the conventional approaches are often plagued by large logarithmic terms in the cal-
culations. These large logarithmic terms are the results of ratios among the multiple
energy scales associated with the collision process. Since these scales can vary sig-
nificantly, the logarithmic terms can be very large in some kinematic region. The
existence of the large logarithmic terms in calculations often signals a breakdown in

the perturbative expansion.

In this chapter, we will discuss various approaches in the heavy quark production

calculation and introduce the ACOT scheme [41]. The ACOT scheme is a composite

39

40

renormalization scheme applicable over all energy range, and it is an example of
resummation methods which have been successfully used to handle the multiple scale

problems.

3.1 Heavy Quark Production in Collider Physics

The study of heavy quark production has become an increasingly important area
of theoretical and experimental research. It not only provides us with critical new
tests of perturbative QCD, but also gives us a tool to probe new physics beyond
the Standard Model. By heavy quark, here we mean the quark whose mass m” is
significantly larger than AQCD so that perturbative QCD is applicable at its mass
scale. In the Standard Model, this includes the charm (me 2 1.5 GeV), bottom
(m), 9: 5 GeV) and top quarks (m, 2: 175 GeV). The existence of heavy quarks with
different masses allows us to probe perturbative QCD in regions of different energy
scales, where the relative impact of radiative corrections and non-perturbative effects

are very different.

The top quark was discovered by GDP and D0 at the Fermilab Tevatron collider
in 1995 [35, 36]. It is the heaviest of the known heavy quarks. The top quark
production cross section has roughly theipredicted magnitude at Tevatron, and the
existing study of the kinematic distributions show qualitative agreements with QCD
predictions. More studies are being pursued in top quark physics to further test the
underlying strong interaction dynamics and possible new physics. However, in this
work, we will not discuss the top quark because its mass is too high for our formalism

to make any practical differences from the conventional method.

While there are only a few top quark events found until now, there are plenty of

bottom and charm quarks produced at Tevatron and LEP. The electron-proton col-

41

lider HERA has also begun to produce data on charm photo-production and electro-
production. These experiments provide us an invaluable tool for quantitative QCD
study. They can be very useful either as a probe of the nucleon structure, or as a test
of perturbative QCD itself at different energy scales. Furthermore, these experiments
will help us to search for signals of new physics, or study backgrounds to new physics.

For example:

0 Charm productions in charged-current interactions in lepton-hadron scattering
have been used to probe the strange quark content of the proton and measure

the CKM matrix parameter V”.

0 Bottom production cross sections at Tevatron are being extensively studied to
improve the reliability of estimations of the 5 rates at the LHC, where b’s can be
used to measure CP violation and probe possible b-meson rare decays predicted

by several theories beyond the Stande Model.

0 Inclusive b production in high energy hadronic collisions are critical to the

searches for Higgs at hadronic colliders because the QCD process gg —> bb

are the main background events to the H -) ()5 process.

Although perturbative QCD theory has been successfully applied to many hard
scattering processes at collider energies and most calculations agree well with experi-
mental data, the theoretical results for heavy quark production are not very satisfying
[37]. First, the next-to—leading order corrections to the leading order results are very
large, in the case of bottom hadroproductions, often up to 50% to 100%. Second, the
theoretical reSults show a strong dependence on the renormalization and factorization
scales. The scale dependence for the next-to—leading order (NLO) calculation of the

bottom hadroproduction is plotted in Figure 3.1. The dependences are far from being

 

3.0 V V V l I' V V V I T Y 1' V I V T Vi V V V V Y I T r Y T I j Y T V I Y Y V v
4 <
1

2.5 p?“ = 10 GeV -.r p?” = 20 GeV -I

Solid: 1800 GeV

Dashes: 630 GeV

'V'T'TTY‘V‘VV‘r' 77"

VTT'

 

0M“. lyl<1: u)/0(p¥“". lyl<1: I‘o)

 

 

 

 

 

u/uo fl/l‘o
Figure 3.1: Scale dependence of the integrated b—quark [)7 distribution at 630 GeV
(dashed lines) and at 1800 GeV (solid lines), for different values of p?‘".
flat. Also, the NLO is not flatter than the LO. Third, the experimental data tend to
lie on the upper side of the theoretical predications. This can be seen in Figure 3.2
and Figure 3.3 where bottom productions at Tevatron are plotted. Notice the y—axis

is in logarithmic scale.

For an easier comparison of the results, the N LO theoretical results and experi-
t

mental data can be present on a linear scale plot [37] in the form of 1):?- as in

Theory
Figure 3.4. In Figure 3.4, we also include the UA1 data. The central line in the
figure is the central theoretical prediction result where QCD scale [1 is chosen as
#0 = «m2 + p?. The upper and lower lines are the upper theory with p = po/Z
and the lower theory with p = 2110 respectively. The dot-dashed straight lines are

constant fits to the ratios, weighed by the inverse of the experimental uncertainties.

43

 

 

 

 

5 .
10 E; I I I T I I I I I I I I I I I I I I I I I I I I ‘5‘
Upper theory: 5
+ mb=4.5 GeV, p=po/2. MR8125 -
A 104 v ‘ 5 J V x Central theory: '3
g .8” mb=4.75 GeV, u=#o. MRSA'
b u B-d/vK Lower theory: .
g 103 ,— [J e+x mb=5 GeV, ”=2“, MRSA' _:
:53 E (””3" #o=‘/(m§+P$) E
A _ 0 §# x -‘
a " e‘X ‘
a.

r; 102 5— u‘x ‘5
E .. e+X E
101 —- Data: CDF -—.
E p}? -* b+X, Vs=1.8 TeV, |y|<l :

100 I l L l I l l l l l l l l l l L l L L 1 4L l 141
0 10 20 30 4O 5O

Ptmm (63V)

Figure 3.2: CDF data on the integrated b—quark p, distribution, compared to the
results of NLO QCD.

It is easy to see that independently of the beam energy, the data are higher by a

factor of about 2 than the default prediction based on ,u = no.

The large next-to—leading order corrections and the significant scale dependence of
the NLO results for bottom ‘hadroproductions are symptoms of uncertainties due to
neglected large contributions from even higher order processes. The possible existence
of large corrections from higher order contributions results in bad convergence of the
perturbative expansion and casts doubts on the NLO calculation formalism. In the
case of the charm electroproduction where the same conventional perturbative QCD
formalism has been used to carry out the N LO calculations, the same kind of problems
also exist [37, 38, 39, 40], although the results at the current experimental range are

somewhat better behaved than those of bottom hadroproduction shown above.

44

 

 

   
   
  
   
 

 

 

105E....l...f]....l....l...g
: Upper theory: 5
- m,=4.5 GeV, p=po/2. MR3125 -
104 E Central theory: "5
A M; mb=4.75 GeV, ,u=,u0, MRSA'
'2 ~ % Lower theory: .1
: 103 E— ‘ rnb=5 GeV, “=2“, MRSA' 1
3.. E u0=\/(m,2,+p¥) i
53' I I
§ 102 :— 1
b : :
E 3
: .1 Incl p. :
101 E— Data: D0 (preliminary) —E
E p5 -» b+X, \/s=1.8 TeV. |y|<1 3

100 L4 1 1 l 1 1 1 1 l 1 1 1 1J 1 1 1 1 l 1 1 1
O 10 20 30 4O 50

p?“ (GeV)

Figure 3.3: D0 data on the integrated b—quark p1 distribution, compared to the results

of NLO QCD.

45

 

C0
IIIIIIIII

 

 

 

 

 

p—
p-

 

IIIIIIIII VIII rTrr
I

 

 

 

 

 

IIIT ITIII

 

Data/Theory

 

 

 

 

 

 

 

 

O 1 l 1 1 I1 1 1 1 L111 1 I 1 1 1 1 I 1 1 1 1 I 1 1 J :

E CDF <o>=2s4 E
a}: _ ..................... 1‘
2:— —‘
1: ‘
0:1 1 1 111 1 11 I111 1 I 1 1 1 1 I I 1 l 1I11 1 1“

o 10 20 so 40 50 so

I)?“ (GeV)
Figure 3.4: Linear comparison between experimental data and theory for the inte-
grated b-quark p, distribution.

Recent measurements of charm production in the deep inelastic scattering at
HERA [43, 44] has shown that charm final states account for up to 25% of the total
cross section in the small a: region. To study the details of the charm production
mechanisms in deep inelastic scattering and extract useful information on the charm
and gluon contents of the proton, we must have a more reliable theoretical formalism
and calculation than the currently existing conventional QCD results so that we can
effectively explore HERA’s wide kinematical range. Clearly, the formalism and calcu-
lation methods of the conventional perturbative QCD for the heavy quark production

must be‘carefully reexamined [41, 42].

46

3.2 Conventional Methods for Heavy Quark Pro—
duction Calculation

For heavy quark production, the existence of the quark mass m H makes the problem
considerably more subtle than that of light parton(jet) production. Conventional per—
turbative QCD (PQCD) theory is formulated in terms of zero-mass quark-partons.
For processes depending on one hard scale such as Q or P“ the well-known factor-
ization theorem then provides a straightforward procedure for order-by-order pertur-
bative calculations, as well as an associated intuitive parton picture interpretation of
the perturbation series. Heavy quark production represents a challenge in PQCD
because the heavy quark mass, m), (H = c, b), provides an additional hard scale

which complicates the perturbative series.

The two conventional methods for PQCD calculation of heavy quark processes
effectively treat these multiple-scale problems as if they are one-scale problems: (i)
in the parton model approach, the zero-mass parton approximation is applied to a
heavy quark calculation as soon as the typical energy scale of the physical process
(Q) is above the mass threshold m H, leaving Q as the only hard scale in the problem;
and (ii) in the heavy quark approach, the quark H is always treated as a “heavy”
particle, the mass parameter m H is explicitly kept along with Q (as if they are of the
same order) and H is never considered as a parton. Clearly, these two treatments
represents two diametrically opposite ways of reducing the two-scale problem to a

one-scale problem.

In the parton model approach, the cross section for heavy quark production in

deep inelastic scattering can be'written as

01N-+HX = Z ¢iv($,fl) ® 61a—1Hx(§1Q1#)I-,NE=0 - (3-1)

a=active partons

In Equation 3.1, p is the factorization and renormalization scale, and Q is the hard

47

scattering scale set by the probing vector boson. 6).,on is the perturbatively cal-
culable hard cross section for the I + a —) H + X hard scattering process where I
is the incoming lepton, a is the initial parton, H is the final state heavy quark and
X represents anything else in the final state. The parton label a is summed over
all possible active parton species. In the parton model approach, whether the heavy
quark H is included in the sum or not depends on the energy scale )1. For example,
in charm quark production, the active partons are u, d, s and 9 when p is below
the charm mass threshold, however, when p is above this threshold, charm quark
becomes an active parton and the active parton species then include u, d, s, g and
c. The masses of the partons in this approach are all set equal to zero, including the
heavy quark mass m H. The advantage of the parton model approach is that it is quite
easy to implement. The hard cross section 61,.on is calculated in the limit of zero
mass for all the partons, and it is made infra-red safe by dimensional regularization
in the MS scheme. The parton distribution functions ¢‘1',, (1:, p) are extracted from
global analyses. Their p—dependence is determined by QCD renormalization group

equations.

The parton model approach is routinely used in most high energy calculations
such as global analysis of parton distributions in EHLQ [46], MRS [45] and CTEQ
[31], as well as in all analytic or Monte Carlo programs for generating Standard Model
and new physics cross sections. For the light partons a ={g,u,d, 3}, ma —-> 0 is a
valid approximation for all hard scale Q (since, by definition, Q >> ma). However, for
a heavy quark H, it is a reasonable approximation only in the high energy regime )1 ~
Q > m H; and it clearly becomes unreliable in the intermediate region Q ~ 0(m”).
So this approach can only be reliably applied when )1 >> m”. In the threshold
region, the effects of heavy quark mass cannot be ignored and the zero quark mass

approximation becomes questionable. It is because of this reason that for the study of

48

heavy quark production where the quark mass effects are very important, this method

is seldomly used.

The heavy quark approach, on the other hand, treats the heavy quark as a real
“heavy” particle that only appears in the final state — much in the same way as
top quark is treated. In this approach, only light partons are included in the initial
state and the number of parton flavors 12., is kept at a fixed value regardless of the
energy scales involved. An important feature of the heavy quark approach is that the
heavy quark mass mg is kept exactly in the hard cross section ézanyx. Comparing
with Equation 3.1, the cross section for inclusive heavy quark production using this
approach can be written as

GIN-mx = Z ¢iv(1‘,#) ® 610-)HX(§9Q$ "111111), (3-2)

a=light partons only
where the sum over parton a only includes light massless partons. For charm pro-
duction in deep inelastic scattering, no matter what the energy scale is, the partons

a only include u, d, s and g and the number of flavor na is fixed at 3.

The heavy quark approach is conceptually simple and well defined. The hard
cross section éahgxfi, Q, my, p) can be calculated order by order with appropriate
prescriptions for subtracting various divergences. Broadly speaking, divergences due
to the light parton are removed using the MS counter terms, whereas those due to
the charm quark are removed using the BPHZ zero-momentum subtraction counter
terms. The NLO calculations using the heavy quark approach requires considerable
amount of work, but they have been carried out for both electroproduction [38, 39, 40]
and hadroproduction [47, 48].

Since the heavy quark approach has played a dominant role in the NLO calcula-
tions of the production of heavy quarks, it has been routinely used in most recent

heavy quark production phenomenological studies. As expected, this approach works

49

well when Q ~ m H because in the mass threshold region, we effectively have a one
scale problem. Typically, the perturbative hard cross section 61.,on calculated using
the heavy quark approach will contains logarithm factors of the form (1201) log'" (Re).
When p ~ m 3, these terms are under control and the perturbative expansion is well
behaved. However, when p > m H, these logarithmic terms become quite large and
the perturbative expansion is no longer consistent because the truncated perturbative
series in the heavy quark approach has left out important physics effects. Therefore,
its predictions should only be reliable over some range of Q2. Unfortunately, we do
not know a priori how large that range is. Recent estimates [49, 50] comparing the
differences between calculations carried out in different schemes, suggest that Q ~ 20
GeV marks the limit of this range for electro-production of charm quarks. However,
the criterion used is not definitive; the boundary depends necessarily on the process
(e.g., charged/ neutral current leptoproduction, hadroproduction, etc.) as well as on
the variable a: . In this situation, the validity of the perturbative expansion using
the heavy quark approach becomes questionable. In fact, this has been known since
the next-to-leading order (N LO) calculations in the heavy quark approach were com-
pleted. As we have seen in Section 3.1 where the N LO results for bottom hadropro-
ductions calculated using the heavy quark approach were quoted, the next-to—leading
order‘corrections are often of the same numerical magnitude as the leading order
result, and the uncertainty of the theoretical calculation, as measured by the depen-
dence of the calculated cross section on the unphysical scale parameter p, is as large
in NLO as in L0 — contrary to what is expected from a good perturbation expan-
sion. Experimentally, comparisons also show that the measured charm and bottom

production cross sections do not agree with the NLO theoretical predictions very well.

A more careful study reveal that the results may not be all that surprising after

all. For charm quark and bottom quark production, the condition y ~ mg is not

50

well satisfied in most collider experiments. In fact, the current experimental range for
the lepto- and hadro-production of those heavy quarks mostly lie in a region between
those appropriate for the parton model approach ([1 >> my) and the heavy quark
approach (p ~ my). To make reliable predictions and study the QCD mechanisms
about heavy quark productions in detail, a well defined theory which can be applied

over the full energy scale is needed.

The clue for solving this problem can be obtained from examining the conven-
tional massless QCD theory. In the heavy quark approach, when Q >> my , the
logarithmic terms in 6 become large and are infra-red unsafe. The “mass singular”
term as mi” -§ 00 for heavy quark is equivalent to mg -+ 0 in the massless QCD
theory. In massless QCD theory, these infra-red unsafe terms are resumed into parton
distribution functions [22, 23, 24]. The same method of resummation can also be ap-
plied in heavy quark production: The large logarithms of the form a: lnm(-rf;) can
be resumed to all orders in a, into the parton distribution function ¢fi(r, p) for the
heavy quark H. After the resummation, the H parton should be included in the sum
over parton flavors — it participates in the hard scattering on the same footing as
the other partons. Also, the infra-red unsafe large logarithmic terms are subtracted
from the hard scatter cross section 6 and the remaining hard cross section becomes
infra-red safe as —Q— -+ 00. This observation leads to a natural solution of heavy

mH
quark production problem over the full energy range — the ACOT scheme [41].

In the following sections, we will focus the discussion on charm neutral current
production in deep inelastic scattering, although the discussion and method apply
to other heavy quark productions as well. We will use 7" to refer 7" and Z boson
generically. First, we will define two simple renormalization schemes for charm elec-
troproduction, the three-flavor scheme and the four-flavor scheme. Then, we will

define the ACOT scheme, which is actually a composite scheme composed of the

51

three-flavor scheme and the four-flavor scheme.

3.3 The Three-flavor Scheme

The 3-flavor scheme is an example of the application of the heavy quark approach
to the problems of charm quark production. This scheme is the one used in Refer-
ence [38, 39, 40, 51] to calculate charm production to NLO, i.e. C(03). It is precisely
defined by choosing to work with only 3 active quark flavors, consisting of the light
quarks, and using the subtraction procedure of Reference [52]. The prescription for
subtracting ultra-violet divergences encountered in the calculation of the partonic
structure functions and distribution functions depends on the particle that produces
the divergence. Divergences involving the light partons a are removed using the MS
counter terms, whereas those involving the charm quark c are removed by the BPHZ
zero-momentum subtraction counter terms. This ultra-violet subtraction scheme has
the nice feature that the charm quark explicitly decouples as its mass becomes large.
In particular, the operators which make up the charm quark distribution function
are suppressed by powers of order Az/mf. Since these terms are power-suppressed in
the “heavy quark” mass, they are usually excluded from the 3-flavor scheme parton

picture, which usually represents leading-twist dynamics.

In practice then the partonic calculations in this scheme are done by considering
only diagrams with the massive charm quark in the final state and no charm quark
distribution functions in the initial state. The light parton distributions always evolve
according to the 3-flavor GLAP equation, irrespective of the renormalization scale p.
The parton distribution functions defined in this scheme will be restricted to the light
parton a, (a, a’ = 9, q, (7), sector, and they will be denoted by 3(1)}. In the perturbative

calculation, the perturbative partonic distribution functions 3432' contain 6'1 pole

52

terms which are due to collinear singularities. The lowest order (LO, C(03)) process
contributing to the calculation of the partonic structure functions in this scheme, to
be denoted by of, is the 7‘9 —> c5 “heavy-flavor creation” (HC) process (also known
as boson-gluon fusion), corresponding to the diagrams of Fig.(3.5a). It is finite. The
next-to-leading order (NLO) contribution consists of the l-loop virtual corrections
to 7'9 —) 06 (cf. Fig.(3.5b)), plus the real partonic HC processes 7‘9 —) 059 (cf.
Fig.(3.5c)) and Ya —) 060 (cf. Fig.(3.5d)). The collinear divergences which appear
in the calculation of the 0(03) partonic structure functions 30359 and 30:5“ arise
from splitting of massless light partons in the collinear configuration, and take the
form of e“ pole terms, precisely corresponding to those appearing in 343 :' mentioned
above. That is, the partonic structure functions have the factorized structure shown
in Equation 3.2, and the hard cross section functions 6,, will be free from 6‘1 collinear
singularities.

As mentioned in the last section, hard cross sections calculated in this scheme
contain p0wers of ln(Q2/mf). The perturbative expansion should be accurate at en-
ergy scales not too far above threshold, or Q2 ~ m3, where ln(Q2/m§) is of order
1. However, at high Q2 >> mc the perturbative expansion parameter is effectively
a, ln(Q2/m§), and the large logarithm factor spoils the convergence of the perturba-

tive series. In other words, the “hard cross sections” 6,, defined in this scheme are
me

Q

finite, but not infra-red safe in the limit —) 0 —- they contain “mass singularities”

in this sense.

3.4 The Four-flavor Scheme

In order to better deal with the large logarithms at high energies associated with

mass singularities, it is more useful to use the 4-flavor scheme. In this scheme the

53

9
a

( )
I g a
(c) (01)

Figure 3.5: Three-flavor scheme production mechanism. (a) is 7“ + g —} c + E
process(heavy quark creation: HC), (b) is one loop 7" + g -) c + 5 process, (c) is
7" + g —+ g + c + 2 process, ((1) is 7" + a —) a + c + 6 process. Only one diagram for
each process is plotted.

9 (b)

 

renormalization of an and the perturbative partonic function 43:, (a,b = g,q,ij,c)
is carried out using dimensional regularization and the MS counter terms for all

Feynman diagrams, while keeping the full quark mass dependence in the Lagrangian.

Charm distribution functions calculated in this scheme, 4&2: are not suppressed as
in the 3-flavor scheme, but contain powers of ln(mc/p), along with possible 6’1 poles.
Because of the different subtraction procedures used in the two schemes, even the light
parton distributions “(if ,I, l’ = q, 6, 9 will differ from 343], by a finite renormalization
in general. Because renormalization constants in the HS subtraction procedure are
independent of mass, the evolution kernels for the “$2 parton distributions will be

the same as the corresponding ones in the familiar zero-mass 4-flavor case. This is

54

a significant convenience. The perturbative parton distribution functions 465: have

been calculated to NLO in Reference [50].

c/c i: c/c
c/c c/c
(b)
' C
(d)

 

c/c 9

(c)

Figure 3.6: Four flavor production processes. (a) is 7‘ + c/E —) c/E process(heavy
quark excitation: HE), (b) is one loop 7'+c/Z —) c/Z process, (c) is 7‘+c/E -) g+c/E
process. ((1) is 7" + g -) c+ E process (heavy quark creation: HC), Only one diagram
for each process is shown.

Since charm also has a parton interpretation in this scheme, the set of partonic
processes are expanded to include those involving charm initial states. The L0
partonic process in the 4-flavor scheme is the 7‘c —§ c “heavy-quark excitation” (HE)
process (Fig.(3.6a)). N LO contributions to charm production in the 4-flavor scheme
come from the l-loop virtual corrections to HE y‘c —) c (Fig.(3.6b)), and from the
real HE 7"c -+ gc and HC 7'9 -) cc processes (Fig.(3.6c,d)). Partonic cross sections

a. calculated beyond L0 in this subtraction scheme will contain, as in the 3—flavor

scheme, both 6‘1 poles due to collinear singularities associated with light degrees

55

of freedom and powers of ln(Q/mc) due to collinear configurations associated with
the heavy degree of freedom. The important difference compared to the 3-flavor
case, is: these potentially large logarithm terms also appear in the 4-flavor parton
distributions “(132. Hence, they will be systematically subtracted out from a,I when we
evaluate the hard cross sections 6, As a result, 6.. will be free from both types of
collinear “singularities” (in quotes since the logarithms become singular only in the
zero-mass limit). In effect, all logarithmic factors ln(Q/mc) in a. will be replaced

by ln(Q/p) in 6., (with accompanying finite subtractions), and the latter is infra-red
E
Q

limit, and is expected to give a much more reliable description of the physics of charm

safe in the —-) 0 limit. Thus, the 4-flavor scheme has a well-defined high energy

production at large Q than the 3-flavor scheme.

As formulated above, the hard cross sections still contain finite charm-mass de-
pendence, i.e. 6,,I = 6412,33 %,p). Being infra-red safe, the limit 6,(a:,Q,mc, p)
46r°=°(x,Q,p) as mc/Q —) 0 is well defined. In this limit, the 4-flavor scheme
with non-zero charm mass reduces to the conventional parton model scheme. As em-
phasized in Reference [41], however, the factorization of potentially dangerous ln(mc)
terms does not require taking the me -+ 0 limit in the infra-red safe coefficient func—
tions. The conventional practice of always setting m6 = 0 in the hard cross section
6,,(x,Q, p) is a convenience, not a necessity; it results from the use of dimensional
regularization of the zero-mass theory as a simple way to classify and to remove the
collinear singularities. For a “heavy quark” with non-zero mass me, this convenient
method of achieving infra-red safety is not a natural one (as it is for light flavors),
since mc itself already provides a natural cutoff. In other words, the theory has no
real collinear “singularities” associated with the charm quark, and the universal (i.e.

process-independent) and potentially large mass-logarithms can be factorized system-

atically as outlined above. In fact, by keeping the charm quark mass dependence,

56

this scheme can be extended down to lower values of Q with much more reliable re-
sults than in the zero-mass case—it has the built-in characteristics to approximate

the 3-flavor calculation in the region above threshold [41, 53].

Since the charm quark distributions are explicitly included in the 4-flavor scheme,
and since me is not much larger than a typical non-perturbative scale such as the nu-
cleon mass, one can allow for the existence of a possible nonperturbative (“intrinsic”)
charm component inside a hadron at a low energy scale, say Qo— as the boundary
condition for evolution to higher scales, just like the other light flavors. This is a

possibility not permitted in the 3-flavor scheme by assumption.

3.5 The Composite Scheme — The Unified Ap-
proach to Charm Production

Both the 3-flavor and the 4-flavor schemes described above are valid schemes for
defining the perturbative series of charm production in principle. They are equivalent
if both are carried out to all orders in the perturbation series. At a given finite
order, they differ by a finite renormalization of the parton distribution functions, as
well as] the strong coupling 0,. From the physics point of view, the 3-flavor scheme
provides a more natural and accurate description of the production mechanism near
the threshold (Q2 ~ m3), whereas the 4-flavor scheme does the same in the high
energy regime (Q2 >> m2).

It becomes obvious then that a unified program to calculate charm production
must involve a composite scheme consisting of: (i) the 3-flavor scheme, applied in the
threshold region; (ii) the 4—flavor scheme, applied at higher energy scales; and (iii)
a set of matching conditions to effect the transition between the two schemes at an

intermediate scale (say, #1:) where they are comparable to each other. The existence

57

of an appropriate transition region, where the difference between the two schemes is
small (i.e. it is of a higher order in a, with no large logarithms), is important. As
demonstrated in Reference [41] and mentioned above, the 4-flavor scheme has the
required feature of approximating the 3-flavor scheme results as Q -) me from above;
whereas the conventional parton model approach does not. The transition from the
3-flavor to the 4oflavor scheme involves performing the requisite finite renormalization
(“matching”) on a, and 457v at some scale p = Me, and using the appropriate 01,, (257,,

and 6. in the cross section calculation in the two respective regions.

This composite scheme described above constitutes the ACOT scheme [41]. It was
implemented at the order a} level in Reference [41], and now rigorously established in
Reference [53]. It is a more precise formulation of the commonly accepted zero-mass
parton picture with effective quark flavor number increasing with the energy scale—
hence the often used term “variable-flavor—number” scheme. As emphasized above, it
is more precisely a composite scheme, consisting of two simple schemes with different

numbers of active quark flavors, and a set of appropriate matching conditions.

Formally, the ACOT procedure is based on the CWZ renormalization scheme [52]
which provides a natural transition from the mass threshold region [1 ~ 0(mfl) to
the high energy region [1 > m H. To switch from one region to another across the
threshold, finite renormalization matching conditions are needed to make the schemes
equivalent in the domain of overlap )1 ~ my region where they are equally valid for
practical low order calculations. The transition between two schemes can, in principle,
be carried out at any scale [1 ~ m H. The explicit formulas of the finite renormalization

coefficients [54, 55, 50] are:

30 m2
40801) = 30,01) [1 - #111 7:29- + C(03) (3.3)

58

and

‘¢7v(x,u) = 3¢lv(z1#) + 0 + 0(03)

4 _ 3 3000‘) 133 2

¢iv(=r1u) —- #206,”) + —-61r 111,12 ¢iv(x1u) + 0(01.)
3 a 2 (1

445516,”) = o + $11153, §<22+(1—z)2)3¢:v(§.m + was)

(3.4)
The ACOT scheme is defined to keep all infra-red safe m H-dependent effects in the
hard cross sections so that there is no loss of accuracy when [1 ~ m”. This is
accomplished by defining 61“.,HX(§,Q,mH,p) as the full 6;a_,Hx(§,Q,mH,,u) with

mass m H singularities subtracted.

The ACOT scheme is designed to be applicable over all energy range. It coincides
with those of the three-flavor scheme in its region of applicability, p ~ my, and
reduces to those of the four-flavor scheme model in the asymptotic energy regime
p > my. In the middle region where p is larger but not far larger than my, the
quark mass effects are kept in a consistent manner and the ACOT scheme provides a
good approximation to the physical cross section. Furthermore, when the finite terms
in the logarithmic resummation are properly calculated, the ACOT scheme will agree
with the zero-mass parton approach in the limit of mg -+ 0. We will present the
detailed implementation of the ACOT scheme and its applications in the next two

chapters.

Chapter 4

Implementation of the ACOT
Scheme Calculation

In this chapter, we will apply the ACOT scheme to the problem of charm quark
PI‘Oduction in deep inelastic scattering. This scheme has been applied, at leading
Order, to the total inclusive structure functions by M. Aivazis et. a1. [56, 41]. To ef-
fectively study heavy quark production mechanisms and extract useful informations
about the charm and gluon contents of the proton from experimental data, both
higher order calculations and some differential distributions are needed. Higher order
Calculations are necessary because these contributions can not only give us better
theoretical predictions to compare with experiments, but also give us an indication
how well the perturbative expansion converges in the theory, which will be an im—
Portant self-consistency test. Difl'erential cross section distributions are important
because these results will help us to further study and differentiate different heavy
quark production mechanisms and probe possible intrinsic charm contents inside the

proton.

In this work, we will implement the next logical extension of the ACOT scheme
Calculation — we will extend the calculation to include all contributions at order

O(a,). We will also extend the calculation to include differential distributions. This

59

60

is done by using the phase-space splicing method to separate different kinematic
regions, and the Monte Carlo numerical integration method to handle the phase
space integration. Also to compare with experimental data, fragmentation functions
for the final state charm quark will be introduced to evolve the quarks into mesons.
While the formalism is applicable to both charged current interactions and neutral
current interactions, we will focus on charm deep inelastic scattering production in

neutral current interactions, especially at HERA.

4.1 The Calculation Formalism

For the calculations of heavy quark production in the deep inelastic scattering
11(k)+ N (P) —> 12(k') + H (p’) + X (Px) , (4.1)

the factorization theorem [57, 53] states that the dominant contributions to the

hadronic cross section have the factorized form of Figure 4.1 with

dallN—“zflx( ¢¢Ilv ® dafga-HQHX

q,P,mH,'°°)

l
= [o d€¢7v(£1#) d6"“""”x(k1q1#210a(#)1mu:'“l

 

_ 1 ' 45 .1
— 2A(3,M,2.,0) /?¢~(€1u2)lMal2dF1 (4.2)

In Equation 4.2, Ma is the hard scattering matrix element and 2A(s, M N, 0) is the flux

 

factor with A(a, b, c) = £02 + b2 + c2 — 2ab — 2bc -— 2ac). The label ‘a’ is summed
implicitly over all active parton species. In the ACOT scheme, whether the heavy
quark H is included or not depends on the energy scale 11 — In the three-flavor
scheme region where p < p,,,,.e,hdd, a includes u,d,s and 9, whereas in the four-
flavor scheme region where u > ”threshold, a includes u, d, 3,6 and g. The threshold

parameter ”may,“ should be in the region where both the three-flavor scheme and

61

H (10')

 

Figure 4.1: The factorization for the deep inelastic scattering

the four—flavor scheme are valid for practical calculations. The heavy quark mass

dependence is kept in d6 after the factorization of the large logarithmic terms.

In Equation 4.2, the phase space factor dl" is

d3k’ d3p’ II (13p:c
(21r)32k(,(21r)32E’ (21r)32E; '

 

dI‘ = (2«)‘6‘(p+k - k’ —p’— 212;) (4.3)

Using the kinematic variables defined in Equation 2.14, we can simplify the lepton

 

k’
h —. '
p ase space factor (27032“) Then the cross section becomes
Where
(131), d3pl
I __ 4 4 _ I_ I a:

As described in Chapter Two, the leptonic factor can be factored out from the
Cross section (16"N‘"2 ”X and (mm-”3”" . The factorization theorem for heavy quark
production in the deep inelastic scattering can then be written in terms of hadronic

tensor structure function

WLNHHX(q7P,mH1°H) = Z¢IIV®LDX:-,HX
a

62

 

 

 

 

 

Figure 4.2: The factorization of the structure functions

1
= 2/0 ééqbfvasl‘)0X:-)Hx(klaQa/12aaa(fl)am”’°H)

(4.6)
as depicted in Figure 4.2.

Writing the hadronic tensor W’” and the partonic tensor w” in terms of the
Structure functions as in Equation 2.8 and substituting them into Equation 4.6, we
Can obtain the relations between the hadronic invariant functions W,- and the partonic
invariant functions w,- based on the relation between the hadronic momentum P“ and
the partonic momentum p“. In the case of massless parton and massless hadron

target, p“ = 5P”, and the relations between W,- and w,- are simply

WV"””X(41P1°--)= 22/01 52$¢7v(£.#)wl’“”x(p1 (MW, 0101), - - -)1 i=11213
«1 (4.7)
However, because of the existence of the quark mass in heavy quark production, when
the initial state parton a is a heavy quark, its four-momentum p“ is not proportional
to the proton momentum P“. Thus, equation 4.7 is not applicable. In this case, since

the vectors P, p and q are collinear, p can be parameterized as

P" = €pP" + £116)“ 1 (4.3)

63

where f p and {q are rather complicated functions of the masses and the convolution
variable 5. As a result of this heavy quark mass effect, the relation between the W.-

and the w, is rather complicated and it has a general form of

l .
W1VN"HX(q,P,--.)=Z/o gafidém) xc}x w}’°"”x(p,q,m,p,---) i=1,2,3
(4.9)

where c;- are complicated coefficient functions of relevant kinematic variables [56].

A much better way to express the hadronic and partonic tensors in the presence

of non-zero mass quarks is to use the helicity structure functions, which are defined

as

F1 = 6’1"(P1<1)Wuy(P1q1°-°)EK(P.Q) (4-10)
and

f1 = 6T0), a)ww(p1 r11 - ' 061(1), q), (4-11)
for the hadron and parton respectively. 6", A = +, 0, -, is the polarization vector of

the probing vector boson.

To relate F; with f,\, the relations between the eflP, q) and 6’;(p,q) must be
obtained first. This is where the simplification of the helicity approach comes from
—— the two sets of polarization vectors 6‘;(P, q) and c‘flp, q) are identical even in the
presence of nucleon and parton masses. The reason for this equivalence is that the
polarization vectors for a vector boson with momentum q only depends on the plane
defined by q and a reference momentum, which in this case can be either p or P.
Since (q, PPM”) and (q, pmrm) define the same plane, 6’; (P, q) = 6: (p, q). Thus,

the factorization theorem for helicity structure functions can be written as:

arm q) - WLNW - 61"” = z a s (cm q) ma”: - cap, a) (4.12)

64

FLIPq’m )=Z‘/old£ _¢N( Ev” )"fA(paQa ) (4°13)

As shown in Equation 4.13, the helicity structure functions have a much simpler

factorization form than the invariant structure functions.

Using the helicity method will not only enable us to use the simple factorization
formula, but also simplify the calculation of the scattering matrix elements. In the
hadron sector, the helicity method is able to take full advantage of the basic chiral
coupling between the vector boson and the quark and utilize the symmetries among
different helicity scattering amplitudes. In the lepton sector, the helicity approach
results in a very simple form of the lepton current because of the simple lepton
vertex function and the helicity conservation of massless leptons. For neutral current
interactions, considering the two possible helicities of the incoming lepton (L, R for
electron, L for neutrino, and R for anti-neutrino) separately enables us to separate
the lepton sector and parton sector at the amplitude level and add the contributions
of the photon and the Z boson coherently. For charged current interactions, the

separation is natural because the W boson only couples to the left chiral currents.

Upon the separation of the left and right handed incoming lepton currents and
extraction of a factor of Q—24 from the matrix element, the chiral coupling of the vector
bosons (7", Z) to the quark in neutral current interactions can be written as —ie7“gfl
where a = qL, qR labels the left and right chiral couplings of the quark to the neutral
bosons and b = 8L, eR labels the left and right handed incoming lepton currents. g2

can be obtained as the following:

 

 

 

 

geL = Q — ‘i + Sin? 9W ___Q2 T39 " Q? Sinz “W
9" q sin 9w cos 0w Q2 + Mg Sill 9w 003 9W
QeL = Q _ -2 + sin“ 0W Q2 'Qq sin“ 0W
“R q sin 9w cos 9w Q2 + Mg sin 9w COS 6w

65
cf! __ Q _ Sill? 9w Q2 T3,, - Qq sin2 9w
qu — q Sill 9w COS 9w Q2 + M g sin 9w cos 9w

- 2 2 - 2
cl! sm 9w Q —Qq s1n 9w
— ——-—— . 4. 4
qu Q“ (sinflw cosflw) (Q2+M§) (sinflw cosfiw ( 1 )

In Equation 4.14, Q, is the fraction charge of the quark, T3,, is the quark’s third

 

 

 

 

component of the weak isospin, and 0w is the weak angle. The coupling g: effectively
adds the photon and Z boson contributions to the hadron current at the amplitude

level after the lepton currents are factored out.

For a polarized incoming lepton beam with left and right polarization pl, and

p3 (pl, + pa = 1) respectively, the cross section 4.4 can be reorganized into

_ 622/ 0143 d6 1
do — —32n,Q,dxdy2—1r/?¢~(£)

(m * Iii‘JtI” +pn * W12?) an", (4.15)
where the lepton currents are

ji‘ = <k’-|7"|k->

jg = <k’+|7"|k+> (4.16)

and L, R denotes the lepton’s left and right handed helicity.

The lepton current can be expanded directly in terms of the vector boson po-
larization vectors if they are defined in the same plane. However, in this work, we
will define the polarization vectors 6? of the vector boson in the boson and proton
collinear frame with either (q, P) or (q, p) as the reference momenta. Generally, we
assume the hadron sector is in x — 2 plane and the angle between the hadron current
plane and the lepton current plane is (I). Then the polarization vectors defined in the

hadron plane can be rotated to the lepton sector plane and becomes

(-q2)P“ + (P - (1)0”

65(1), (1) \/(-Q"’) [0" ° (1)2 - <12le

6mg) = fl
8"“
61(1’10) = 754014-11 210)
e+i$
e’i(P,q) = —(0,+1,+i,0). (4.17)

J2"

The lepton current can then be expanded in terms of these polarization vectors

. _ f— sinhz/J _ costh—l _ cosh1/)+1
15L " 2Q2 «'2- 6:; I \/§ )6?!- (—\/§ )d]

= ‘/2QzD-L€’-‘, i: +, —,o
.y _ sinh fizfl _ (COSh1/J + 1 _ coshrb -1 ]
JeR "' V2Q2 5:; ' —\/§ )6“; (——\/§ )6”:

= «262213.357, i=+, —,0 (4.18)

The hyperbolic functions originate from a rotation inside the lepton plane when the
reference momentum is changed from (q, P) to (q, 11). The rotation is actually

a Lorentz boost since it is carried out in the a: — t plane. It is easy to get that
2P - (k + k')

coshd: -_- A[—Q2,P2,Pf

from the kinematics. In the laboratory frame, coshrb

 

El, + E13

simplifies to W.

Defining helicity partonic structure functions to” as
‘1' - " J" — l J‘Jj‘dl" ' ' — o (419)
w _ Epwpueu _' '24; a “9] - +1—a a -

where J i = GLJ" and substituting Equation 4.18 into Equation 4.15, the cross section

can now be written as

(16: 4: 2Q——y2d:r dyd—¢ Eff —¢N( £,p2 (pl, * D-LDLw'ga +123 * DFDRwéO) (4.20)

67

Implicitly included in w” of Equation 4.19 and 4.20 are the sums of all possible
partonic helicity scattering contributions. The diagonal helicity structures 10“ are
the usual helicity helicity functions f,-, i = +1, —1,0, as defined in Equation 4.11.

The factorization formula for the hadron helicity structure function F). is

F1040”): [01%¢(€1#2)(pr.*ff(£1u’,02,a.)+pn*ff(£,#21Q21a.)) . (4.21)

Remember that this factorization formula is much simpler in helicity basis than in
invariant basis.
For heavy quark production at HERA where unpolarized electron beam scattering

on proton, p1, = p); = —, so the cross section and the structure functions simplify to

 

2
da- dyd¢ _/01_ “ii-(11;, 2)(0fong + Dflofwga) (4.22)
and
l
Fi(x,Q"’) = é [0 $445,142) (ff(£1#21Q2.a.) + ff(€1u2,Q21as)) (423)
respectively.

For the experimental results, the invariant structure functions F1,“ are usually
published instead of the helicity structure functions F+,_,o. However, it is easy to
convert between helicity basis and invariant basis once the vector boson polarision

vectors are defined. Applying Equation 4.17 to equation 4.10, we obtain

+Qz
F+ = W1+m17W3

__ Q2
F. — W1 2M 1+-— -—W3

2
F0 = —W1 + (1+ %)W3 (4.24)

68

or equivalently,

£3 = §<F++PL>
a = JgffliWO't +F—+——;’F')
F3 = (Q—2”:—,,)<F+—F.), (4.25)

P-q
JP-P°

In this work, we will focus on charm quark production in deep inelastic scattering,

 

where u =

especially in the H1 and ZEUS experiments at HERA. Equation 4.22 and 4.23 are
the basic formulas we will use to calculate the cross sections and structure functions.
The ACOT scheme will be used to maintain an accurate description of heavy quark
production mechanisms from the mass threshold to the asymptotic energy region. For
each subprocess, all possible helicity amplitudes J ‘ will be calculated to obtain the

helicity structure functions, w” and the various differential cross section distributions.

4.2 Heavy Quark Mass Effects on the Kinematics

In this section, we will briefly discuss some kinematic efl'ects resulting from the heavy

quark mass and the hadron target mass.

For the heavy quark production in lepton-hadron deep inelastic scattering process
11(k) + N(P) —-) 12(k’) + H(p') + X(Px), (4.26)

the actual underlying scattering process is
V(q) + N(P) -> H(p') + X(Px), (4.27)

where a space-like vector boson V strikes a nucleon N, as shown in Figure 4.3. Since

 

 

N(P)

Figure 4.3: Deep inelastic scattering: vector boson and nucleon scattering

the scattering really occurs between the vector boson V(q) and the nucleon N (P), it
is more natural to use the collinear coordinate frame where q and P are collinear in
the z-axis and the t — 2 plane is defined by 4 vectors (q, P) instead of (k, P). Following

the modern formulation of the factorization theorem, we specify the particles’ four

 

momenta by their light-cone coordinate components (33*, 2:1, 2:2, 2."), where x”: =
o 3
a: :1: 1: , . . .
fl instead of the usual (2:0, 2:1, 2:2, :03). Thus in this q, P collinear frame, we

have

P" = (P+,0, —

II
A
l

d
"U
+
P

4“ (4.28)

where P+ is arbitrary and 17 is specified by the equation:

2
2QP=-n—-17M2.

70

2 .
Since 3 = 575—6, the relation between 17 and :1: can be solved as:

1 1 M2

- = _ _ __ 4.29

x 17 Q2 ( )
or equivalently,

2
.1. = 1 1 + 91. (4.30)

1, a; + 4713 Q2
It is easy to see 17 is the generalization of the usual Bjorken .7: with the presence of

target mass M.

In the class of collinear reference frames where t — 2 plane is defined by (q, P), a
specific frame is specified by a given choice of P+. For instance, setting P+ = M / J2,
we obtain the laboratory frame with the z axis along qt Setting P+ -) 00, we get the

infinite momentum frame which is often used to derive QCD factorization theorem.

In the QCD parton model, the initial parton a carries a fraction 5 of the nucleon

momentum.

 

p“ = (61”,5, ml ) . (4.31)
2513+

+
where 5 = €5,2— and m1 is the initial state parton mass. Assume the final state
P .

roton

threshold is s". due to the final state heavy quarks, then

33 = (p + (1)2 = (Q2 + gmfflg- - 1) Z 4» (4-32)

It is easy to find the threshold value for 5 from the above equation,

(Q2 - "I? + §u1)+ A(-Q21mf1§¢h)

61h = If 2Q2 9 (4'33)

 

where

 

A(a, b, c) = (/(a2 + b2 + (:2 -— 2ab — 2bc - 2ac). (4.34)

71

So due to the heavy quark mass effect, the initial parton momentum fraction {’8 range
is 1 2 £ 2 (1), instead of 1 Z 6 Z 0 . For the leading order charm production partonic
process q + k1(m1) -) k2(m2), 5.), = mg and

n (02 - m? + m3) + A {-02, m1. m3]
2Q2 °

4m 2
(1+ (1+ Q2 —)) , when m1 = mg. (4.35)

4. 3 The Next- to-Leading Order Production Mech-
anism

 

{M =

_’

Nlfi

Experiments observe baryons and mesons instead of quarks and gluons in the detec-
tors. Sometimes the data are converted into theoretical quantities which have less
dependence on the non-perturbative physics. For example, the deep inelastic scat-
tering inclusive D’i meson cross sections a” data are almost always converted into
a charm production cross section 6‘. On the other hand, results for the final state
hadrons are also often published. For example, the differential cross section distri-

1 d6 D . . . .
bution— .To compare wnth expenmental data, the theoretical calculation of

—Da dpD

1 da‘

0‘ dpf
the fragmentation function, the cross section formula Equation 4.22 becomes

 

needs to be convolved with the c —) D“ fragmentation functions. Including

 

2
do” = 8:52da:dy§;5 d¢ 1 664505 45% p2) (D-LD-LwLa + DFDnga) Df(z,p2)dz.
(4.36)

D.

Notice that for total inclusive quantities such as a , we can use

OD. = a“ - PHD. , (4.37)

where PHD. is the charm fragmentation probability into D“ meson, to convert inclu-

sive charm results to inclusive meson results. So in total inclusive cases, the difference

72

between the meson cross section and quark cross section is only a trivial constant
factor. As a result, the fragmentation function is often omitted in the theoretical
presentation. However, to make the following discussions clear, we will have the frag-
mentation functions explicit in the formulas in this section, although we will often
refer to both parton distribution functions and fragmentation functions generically
as parton distribution functions. Also we will use H to denote the final state hadron

and c to denote the charm quark.

For charmed meson H production in deep inelastic scattering,
UVN-+H( (x Wq? =az¢a® ®6Va-+b @615, (438)

The exact nature of the factorization of the physical cross section into the three
pieces on the right-hand side of Equation 4.38 depends on the scheme used to define
the parton distributions. The physical cross section is independent of any calcula-
tion scheme; therefore, the subtraction scheme which is used to define the parton
distributions (157,, also uniquely defines the hard cross sections 6. Since the ACOT
scheme is a composite scheme based on the CWZ renormalization, different subtrac-
tion procedures are used in the different energy regions. Within a given scheme, the
hard cross sections 6V‘H" are obtained as follows: (i) Start with cross section ova-’5
similar to the left-hand side of Equation 4.38 but with parton targets and calculate
them in perturbative theory in the given renormalization scheme (i.e. with specific
counter-terms); (ii) Independently, calculate the set of process-independent pertur-
bative partonic distribution functions 43: in the same renormalization scheme, using
either the (moment space) operator-product expansion or, equivalently, the (:r-space)

bi-local operator definition of the distribution functions; (iii) Verify that all diver-

gences and potentially large logarithms appearing in UV‘H" can be factorized into the

73

universal (132 functions, in the manner of Equation 4.38,

0V“"‘(Q2, as, m.) =2 43': ® 6”“ a J: ; (4.39)

a,d

(iv) Systematically invert Equation 4.39 to solve for the finite hard cross section
6V'H“, which is then used in Equation 4.38 for calculating the physical cross section.
There are two points to note: (i) The inversion of Equation 4.39 order-by-order in
the perturbation series is equivalent to subtracting the singularities contained in $2
from aV‘H"; (ii) There is no need to set the quark mass(es) to zero anywhere in the

above procedure.

For the ACOT scheme, the leading contributions to heavy quark production in

leptoproduction are depicted diagrammatically in Figure 4.4.

The relevant hard scattering processes (with the associated fragmentation) are

listedbelow.
V+G-+c+6 ; c—iH : HC1
V+c—)c ; c->H : HEl
V+c—)c+G ; c-)H : HE2 (440)
V+q-+q+G ; G-+H : GF2 '
V+G—>c+E+G ; c—)H : HC2
V+q—>c+E+q ; c-+H : HC2

where

o HC1 corresponds to the (2 -> 2) heavy-flavor creation process;
0 BE] corresponds to the (2 —) 1) heavy-flavor excitation process;

0 HE2 corresponds to the (2 —) 2) HE process;

GF2 corresponds to the (2 —) 2) light parton scattering process, followed by

gluon fragmentation into H;

HC2 corresponds to the (2 —-) 3) HC process;

74

j—— 6/? %7 C/C

c/c
(

c/c
(.) b)
6/?) c
9 C
(C) 9 (d)
’ g (n
1 $2 a
(h)

Figure 4.4: Processes contributing to the leading ACOT scheme calculation. (a), (b),
(c) only present in the 4-flavor region, ((1), (e), (f), (g), (h) presents in both the 3-
flavor scheme region and the 4-flavor scheme region. HC1:(d), HE1:(a), HE2:(b),(c),
GF2:(e), HC2:(f),(g),(h). Only one diagram for each process is plotted.

in”

(a)

75

The order of magnitude of the various process contributions depends on how the
parton distribution functions, especially the heavy quark parton density, are treated.
In conventional applications of perturbative QCD with light partons, the parton dis-
tributions 457v at some relatively low scale p = Q0 are regarded as non-perturbative
input; they are assumed to be of order unity. In practice, it is found that the gluon
distribution dominates at small a: and the. valence u and d distributions dominate
at large 1:. When a charm quark participates in the interaction, (1)31, (:12, p) is usu—
ally assumed to be generated through PQCD evolution. This is called “radiatively
generated” charm. The parton distribution function ¢‘,’v(:r, u) should therefore be of
order a, in the region immediately above charm threshold. As mentioned in the last
chapter, it is possible, however, that a non—perturbative component of charm also
exists inside the hadron at the scale p = me, both on general grounds (since m6 is
not so much larger than the QCD scale) and from the point of view of specific model
calculations [59]. This is another advantage of the ACOT scheme over the current
routinely used three-flavor scheme. Since the three-flavor scheme assumes from the
very beginning that there is no heavy quark distribution inside the proton at any
energy scale, there is no way for it to accommodate a non-perturbative heavy quark
distribution. On the contrary, the ACOT scheme can naturally accommodate a non-
perturbative charm parton distribution ¢fv(:r,p = m.) 95 0 and allow the size and

shape of this component of hadron structure to be determined phenomenologically.

As a rough guide to the order of magnitude of the various factors which enter
into the master equation, Equation. 4.38, we shall assume (1);, (m,n) to be of order
a, compared to the dominant. parton distributions, whether the non-perturbative

component is present or not:

(15%; ~ 0(a.). (4.41)

Although this assumption could fail in the case of a large non-perturbative component

76

of charm, it appears to be an extremely reasonable and safe starting point to take.
This is because even if a truly quantitative comparison between theory and experi-
ment has yet to be carried out, the fact that current three-flavor calculations are in
qualitative agreement with the first measurements of the charm structure function F;
makes it unlikely that the non-perturbative component could be anomalously large.
Reasonable dynamic models also suggest that the non-perturbative charm compOnent

is not more than a few percent. Similar considerations suggest for the fragmentation

functions:
(If ~ 0(1)
d3 ~ 0(a,) (4.42)
d? ~ 0(af)

The numerical suffix in Equation 4.40 to the label of each term corresponds to the
estimated order of magnitude of the term in powers of effective 0,, counting all the

factors in the convolution 45 ® 3 (8 (I.

As previous claimed, the ACOT scheme formalism contains the conventional par-
ton model approach as a special case. This can be seen as follows: From the traditional
zero-mass parton point of view, HEl process represents the LO 0(02) contribution.
HC1, HE2 and GF2 represent the NLO C(04) contribution. The only difference is
that the mass singularities associated with charm are subtracted by mass-subtraction
term in this scheme instead of the KITS subtraction in the parton model approach. In
the ACOT scheme, the mass dependence is kept without taking m.c —) 0. Appropri-
ate choices of the finite terms in the mass-subtraction enable the hard cross section

6(mc, - - ) in this scheme to agree with the standard zero-mass results when m -+ O.

In the following sections, we will discuss the neutral current charm production in

deep inelastic scattering based on Equation 4.23 and 4.22:

e2y

do 87rQ2

 

61¢ 1 a L L i' n R z“
dxdyé; [o -€—¢N(£.u2) (p.- D. mg... + D.- D.- wt.)

77

mm”) = éfgflélu”)(ff(£.u2.02.a.)+ff(£,p’.Q2,a.)) (4.43)

Helicity current J ‘ results for the each contributing process will be presented].

4.4 Leading Order Calculations

The ACOT scheme leading order processes include the leading order flavor excitation
process (HEl), leading order flavor creation process (HC1) and the double counting
term (HC 1.“). The leading order heavy quark structure functions in this scheme were
calculated by M. Aivazis el. al.[56, 41, 58]. However, that calculation was limited to
the inclusive structure functions only and did not provide any information about the

final state particle differential distributions.

The leading order 7"/Z + c -> c is very simple and its contribution to the cross
section is infra-red safe. There are only four non-vanishing helicity amplitudes:
71/2- + Cl, —) on, 71/24. + Ca -) cL, 75/20 + cL —-) cL and 75/20 + cu —§ 63 in
this process. The latter two are proportional to the quark mass and will disappear
when quark mass becomes zero because of the helicity conservation in the massless
limit. The explicit expressions of the helicity amplitudes are listed in Appendix A.
Since this process is a 2 -) 2 lepton hadron scattering process, the azimuthal angle 53
between the lepton sector and hadron sector dependence in Equation 4.17 is trivial,

and the parton momentum 6 integration in Equation 4.22 and 4.23 for this process

4 2
shrinks into a 6 function with parton momentum fraction {0 = g (1 + (l (1 + g; l.

The helicity amplitudes for the partonic process 7" / Z + g -) 0+5 can be calculated

 

in the CE center-of-mass frame. For helicity structure functions, this gives

.. 1 . . dcosédd;
.1 = __ t 1* __ .
327r2/JJ fl 2 27r’ (4 44)

1The author thanks Carl Schmidt for making the results available.

 

8

 

78

2 - ..
where fl = 1 — 4—215 with s = 02 (g- — I) , 0 is the scattering angle and 45 is

the angle'between the lepton plane and the hadron plane. The lower limit Eu: on

4m2

Q;

amplitudes from this process are also listed in Appendix A.

 

the convolution variable 5 is 17(1 + ) due to the mass effects. The helicity

Although the partonic helicity structure function It)” for 7" / Z + g —) 0+ 6 process
in Equation 4.44 is finite after integrating over the phase space, it contains terms
proportional to ln(;12 / mg) As we pointed out before, these terms are already factored
into the charm distribution function 433,, of the 4-flavor scheme and their contributions
have been resummed into the flavor excitation processes HEl. To correct the double
counting between process HEl and HC1, we must subtract off the gluon-to—charm
splitting term in the evolution of the charm parton density, and this leads to the

subtraction term HClsub:

.. .. 2
wag... = my x 5:; In {5; / %¢t(e,u2)a.(%°). (4.45)

where w? is the leading order HEl helicity structure function and splitting function
P¢9(z) = %(22 + (1 — z)2). Essentially, the charm distribution function resums the
collinear logarithmic terms to all orders and the subtraction term is used to correct

the double counting of the first leg in the resummation.

4.5 Next-to—Leading Order Calculations

The next-to—leading order contributions to the heavy quark deep inelastic scattering
include the one loop level flavor excitation process (HE2V), the next-to-leading order
flavor excitation process (HE2), the double counting term (HE2sub), the gluon frag-
mentation process (GF2) and the next-to-leading order flavor excitation processes

(HC2). ,

79

The virtual corrections to '7‘ / Z + c —> c can be summarized in the vertex function.

If the lowest order vertex is —ie7"(fv + L475), then the one loop order corrections

 

. ' uv
would have a general form of -ie (7"(F1v + 171,475) + 102mg” ng) . Calculating both

C

wave function renormalizations and vertex loop corrections in the CWZ scheme, one

 

 

 

 

obtains
m = $10 .MIIH» (4.46)
F14 = 22:; quUl-Iz) (4-47)
EN = ‘3; ,fv(212) (4.48)
where
_ 47”; c 1 +162 fl-l
11 — (mg) r<1+e){;[—1—(2fl )In(fl—;—,)]-2
+(1+5? 1 52—1
)[_-m(§._1.;1)_-1n(%_;1)1n(4_fl,_)
5+1 fl— 1
+Li2(—— 24 ) Li2(-—— 25 )]} (4.49)
2 _.
I. = "$5111 14%;) (4.50)

The separation of 11 and [2 terms is due to the reason that I1 term is proportional
to the leading order vertex —‘ie7"(fv + fA'ys) while [2 term is not. In Equation 4.46,
0,, is 4/3 for QCD and fy and fA are the vector and axial vector couplings of the

2 .
42;" , and Li2(:c) is the

 

vector boson with the quark. In Equation 4.49, fl = l -

usual Spence function defined by
Li2(z = _ [oi—1““ ’2) z. (4.51)

l .
The ; term in Equation 4.49 is the infrared unsafe piece, and it will cancel agamst

the'soft gluon contributions from HE2 process.

80

With the renormalized vertex coupling coefficients Fly, F1 ,4 and F2V available, the

helicity amplitudes can be easily calculated as in HEl process.

The helicity amplitudes for partonic process 7"/Z + c —> c + 9 can be calculated
similar to the 7"/Z + g —> c + 6 process. Actually, due to the crossing symmetry
between the outgoing and incoming momenta, the calculation can be simplified by
using results from 7"/Z + g -+ c + 5 process. The final results are also listed in

Appendix A.

After integrating over the final state phase space, the '7‘/Z + c —) c + g partonic
helicity structure function w” in Equation 4.44 has a E term. This term comes
from the part of the phase space where a very soft gluon g is emitted. This infrared
divergence is proportional to helicity structure function for the leading order process
HE] w}? and cancels against the $- term in the HE2., loop diagram. Because the
integration diverges when-the gluon is soft, the Monte Carlo method cannot be applied
to the whole phase space region. In our implementation we use the phase space
splicing method [60] which allows us to isolate the soft singular poles. In the soft
gluon phase space region, eikonal approximations for the amplitudes and analytical
integration using dimensional regularization are used to perform the integration. For
all the other phase space region, normal Monte Carlo method is used. The cancellation
of soft singularities takes place when the contributions from the renormalized virtual

diagrams are added. This structure function for the soft integration region is

 

 

 

 

 

 

,C 4 2 2 1 1 2 l
"W = "’0'“ 202N°F<1+6X 2:)‘(5%)2£{2[1’ if 111499]
1 2 1 2 . -1
+[1+ g; (1n(—{T)—%+Lz((g+l)2)
32-1 3+1 2 5+1 .
+21n( 4fl )ln(E-:-f)+ln (IE-Tl )]}, (4.52)

where wo is the leading order HEl process helicity structure function and 6 is a

81

parameter used to separate the soft region from the phase space. To obtain a correct
and stable result, 6 must be small enough so that the eikonal approximations are
valid for the analytical integration, and it also must be large enOugh so that the
Monte Carlo integration will not be trapped in the divergence region, Although both
soft and non-soft region integrations could strongly depend on 6, the sum of the two

should not depend on 6. This can be used as a self consistence test.

Similar to the HC1 case, there is also a double counting between the HEl and
HE2. As expected, to correct the double counting between process HE2 and HEl,
we must subtract off the charm-to-gluon splitting term in the evolution of the charm

parton density, and this leads to a subtraction term HClwbz

 

a- 5' 0. d6 ~ 60
40.532... = wo’ x 2—7; / mewag). (4.53)
where (60¢ is the splitting function defined by
~ 4 l + 22 p2
¢cc(z) — § [(1_ 2) (In a? -1— 21n(1— x))]+ (4.54)

The 7" + q —> q + 9 vector boson and light parton scattering process is similar
to HE2 process except the q quark mass is zero. So its helicity amplitudes can be
obtained by simply setting the quark mass to zero in the helicity amplitudes for HE2
process. The gluon fragmentation term d;(z, #2) is defined by

21P¢g(z) ln(1-::), (4.55)

2
27r 6

(1302.442) =

and it should be evolved through the GLAP equation in order to resum the logarithmic
terms to all orders. Then the evolved fragmentation function is convoluted with the
7" + q -) q + g two-to-two parton process to obtain its contributions to the charm

quark production cross section.

In this work, we have not included the HC2 processes 7“ + g —> c + 'c' + g and

7" + q —-> c + E + q. The logarithmic contributions for the inclusive cross section from

82

these processes have been resummed in the flavor excitation processes. When the
energy scale is much larger than the charm mass threshold, total HC2 contributions
are well represented in our HEl , HE2 terms. However, when the energy scale is around
the threshold, the logarithmic terms are small and we have left out some important
contributions from the non-logarithmic terms. Also, for differential distributions, the
HC2 processes have additional kinematic configurations that are not available to the
lower order processes, and thus have a more accurate description to the exclusive

states of the final state hadrons.

Our eventual goal is to achieve next-to—leading order accuracy throughout the
whole energy scale range by including the HC2. processes in our ACOT scheme cal-
culation. However, our results in this work show that both the inclusive results and
differential distributions agree well with experiments. This implies that although
HC2 contributions can be important in some cases, our present calculation already
captures most of the important physics needed to interpret current data. In the
next chapter, we will present our results and compare with other calculations and

experiment data.

Chapter 5

Results of Inclusive and
Differential Distributions of Charm
Quark Production at HERA

In this chapter, we will use the ACOT scheme formalism we described before to

calculate inclusive F; and differential distributions for charm productions at HERA.

HERA(Hadron-Elektron-Ring-Anlage) is the world’s first electron(positron)-proton
collider. It can operate with either electron or positron beams. The current beam
energies are E, = 27.6 GeV for electrons and Ep = 820 GeV for protons. The center
of mass energy J? = m = 301 GeV. The H1 [61, 62] and ZEUS [63] detec-
tors in the ep interaction regions detect the the scattered electron and the emerging
hadrons. Recently, charm quark production data from neutral current deep inelastic
scattering became available from H1 [64] and ZEUS [65]. In particular, substantial
samples of D"*(2010) and D°(1864) mesons have been obtained. In this chapter, we
will present the ACOT scheme results for charm quark and D“ meson production in

the :1: and Q2 region covered by the HERA collider.

First, we will discuss some issues related the actual programming implementation.
The choice of parton distributions is determined by the renormalization scheme. For

the ACOT calculations, we use CTEQ4M parton distributions. Our actual program-

83

84

ming implementation uses the phase space splicing method to separate infrared diver-
gences from other finite pieces and use the Monte Carlo method [66] tonumerically
carry out the phase space integration. Following the introduction of the implementa-
tion, we will present the inclusive F2c results. Our results agree well with experiments
and are much more efficient than the conventional three-flavor scheme calculation.
Since we use a Monte Carlo method tointegrate the final state phase space, we can
easily incorporate experimental cuts. We will show results of differential distributions

'da do at d . .
a i for the D‘ meson with approprlate expenmental cuts. These

dpr’ (10” M" (11)
results also agree well with experimental data.

 

5.1 Implementation of the Calculation

To calculate inclusive structure functions, we need to sum all the subprocess contri-

butions we discussed in the last chapter. Thus, we get

F§(Q2$xa") = w®155 ~ “'
+(¢._ w®‘¢;—¢C®1¢:)®°éx (51)
+¢¢®1f§j .

+¢9®1f$®dg

where the In (L) terms in the f“ factors are kept intact, and the needed subtrac-

C

tion terms are explicitly grouped with the leading 2-)1 term with the same kinematics.

In Equation 5.1,

 

1~c _ a, 4 1 + :1:2 p2 _ _ _
1436 = (a /2«)2P (x) In if: (5 3)
g 9 9""1 mg '

a, T 2
1d; = —£%r)—£(x2 +(1— z)2) ln £7. (5.4)

85

Note that in this chapter, we will use c to denote the heavy quark, i.e., charm quark,

and use H to denote charm hadrons, such as D‘.

Parton distribution functions are scheme-dependent quantities. Appropriate choices
of parton distribution functions are crucial to the implementation of ACOT scheme
calculation and meaningful comparisons between physical predictions from different
schemes. The collinear divergences appearing in a particular factorization scheme
must be canceled by properly defined parton distribution functions (and, in general,
fragmentation functions). The parton distribution functions ¢7v(x, p) needed for this
work can be found from the CTEQ [29, 31] distributions. For all the numerical
results from ACOT scheme calculations presented below, we use the CTEQ4M distri-
butions. In order to compare results from LO and NLO three-flavor scheme, we use
the CTEQ4F3 distributions for the latter calculations. The CTEQ4M and CTEQ4F3
distributions are obtained from global analysis of the same data sets, using the same
procedure, but in the two distinct renormalization and factorization schemes respec-
tively [31]: CTEQ4M is defined in the CWZ scheme and CTEQ4F3 is defined in the

three-flavor scheme.

The fragmentation functions df(:r, 11) describe the process in which a final state
parton (1 fragments into a charm hadron H. These functions are needed for the cal-
culation of the differential distributions because it is the cross sections of D meson
that are measured in the experiments. The fragmentation functions are similar to
the parton distribution functions, and they need to be evolved from suitable initial
functions at some scale 110 according to the GLAP equations We will use the frag-
mentation functions obtained Cacciari and Greco el/ al. [67] in our calculation. For

a given hadronic charm final state H, we have

df(xa#'0) = (mimic) 8’ Df($,#o) (5-5)

86

where the partonic charm fragmentation functions {df,; a = u, d, s, g, c) are considered
perturbatively calculable, and Df(:1:, #0) is a nonperturbative function that could
be extracted from experiments similar to the global analysis for parton distribution

functions. In particular, Reference [67] gives, to order 0,:

d:(:r,po) = 6(1 - x) + a’(u0)CF [14.12 (In “(2) — 2ln(1— x) — 1)] (5.6)

 

21r 1 — .2: 111—3 +
_ as(#0)TF 2 2 ”(2)
d;(x,flo) - T“ + (1 —$) “Pm—3 (5-7)
dflflhflo) = d§($,flo) = d§($,#o) = 0 (5-8)

where Tp = 1/2 and Cp = 4/3. Note that, although d§(:r,po) contains a delta
function, Equation 5.5 makes df,’ (x, no) a well-defined function for any H. The total
inclusive cross section represented by F; is obtained, in principle, by summing over
all H : 2),, df(a:, p0) = d§(:r, #0)- For our numerical calculation of F2“, we only need
to convolute GLAP evolved d;(z,p2) with 7"/Z + q -) q + 9 process because F;
is totally inclusive in the final charm state hadrons. Since Dg’ does not affect the
inclusive structure functions, we will discuss it later in the next section when we

discuss differential distributions.

In the implementation of the parton distribution functions, we have to satisfy the
matching conditions 3.3 and 3.4 when the number of flavors switches between 3 and
4. It was pointed out in Reference [68] that, at order 01,, the finite renormalization of
both a,(/.1) and (1)“ (x, p) of Equation 3.3 and 3.4 vanish if the matching scale is chosen
to be 11,, = 1m. Thus, it is tempting to introduce a single coupling function a,(p),
coinciding with 30:,(11) for p < me and with 40,01) for u > me, which would then be
continuous across the transition point. The same applies to the parton distribution
functions ¢“(.r,p). In spite of this convenience, however, there are good reasons to

perhaps consider choosing a transition scale other than 11.. = me. First, the continuity

87

of a,(p) and 43“ (1:, 11) across the point pc = mc does not hold beyond leading order, as
shown by recent explicit NLO calculation [50, 55]. Secondly, intuitive considerations
indicate that the physical threshold for charm production should be at a higher scale,
say 2m, Furthermore, unless there is really a non-perturbative component‘of charm
inside the nucleon, the partonic interpretation of charm, built into the 4-flavor scheme,
becomes a physically natural picture only at a scale higher than me. However, no
matter what the choice of no is, to have a smooth transition across the threshold and
effective applicability at all energy level for the ACOT scheme calculation, #0 should
always be the same order of magnitude as the charm mass. Following the ACOT
leading order calculation, we also choose no to be 171C in this work. In keeping with
the choice of the matching point in our overall calculation, we also choose p0 = 171.6

for the fragmentation functions.

We now consider the calculation of F; due to the individual subprocesses in Equa-
tion 5.1. Only 7" is explicitly used in the process descriptions although the descrip-

tions apply to both 7" and Z.

0(7"‘c —) c) + (7‘9 -> 05) — Subtraction : These terms comprise the original ACOT
calculation [56, 41]. With non-zero me, they are all finite. The helicity am-
plitudes are listed in the Appendix A. The implementation of the new Monte
Carlo calculation is straightforward. We have verified that the new Monte Carlo

program reproduces the original ACOT results in detail.

1(7% -> c) + (7"c -) gc) — Subtraction : The relevant helicity amplitudes for these
processes are listed in Appendix A. Individual terms in these 0 (a,) virtual and
real corrections to the LO 7'0 —) 0 process contain soft divergences after the
renormalization. In the Monte Carlo implementation, we use the phase-space

splicing [60] method to achieve the proper cancellation of the soft divergences

88

between the real and virtual parts. First, the d—dimensional, two-body phase
space of the 7*c -+ gc process is divided into two domains according to the
softness of the emitted gluon. A theoretical parameter 6 is introduced in the
programming for this purpose. Second, the 2 —) 2 matrix matrix element in
the soft gluon domain is approximated by a simpler form through the eikonal
approximation, and then the approximated matrix element is integrated over the
soft gluon phase space analytically. Finally, the partially integrated result from
the soft gluon phase space region is added to the renormalized virtual 7‘c —§ c
contributions to explicitly cancel the soft :- pole. Both the reminder left after
the cancellation of the :15- pole and the matrix element from the non-soft gluon
region of the 7*c —-) gc phase space are finite. The two separate contributions
can be integrated numerically through the normal Monte Carlo method. While
individually, each of the two contributions depends on the arbitrary theoretical
parameter 6, the sum of them should be independent of 6. This has been used
as a self consistency test for our programming. An appropriate choice of 6 is
important. If 6 is too small, then the numerical Monte Carlo integration would
be trapped in the singularity region of the phase space. On the other hand, if 6
is too large, then the eikonal approximation used to simplify the computation
would be invalid. Our extensive tests show our results are flat in an appropriate

range of 6 we haven chosen.

For double-checking, we also implemented an analytic calculation based on the
formulas by Hoffmann and Moore [69]. The total F; of 7"c —) gc and one loop

level To -> c processes are compared.

As shown in Figure 5.1 and 5.2, the two calculations agree quite well with each
other over the full :1: and Q range, with the exception of small values of Q/mc

and Bjorken 3:. This is more evident in Figure 5.3. The difference can be

89

 

-0.04 -

11‘.‘ -o.os~

-0.08 -

 

-0.1 L 'Our result' —— -
'Hoffmann & Moore' ---------

 

 

 

0.12 . . . x4144 . . . 1....1 A . . .....1 I . . ....
0.0001 0.001 0.01 0.1 1

X

Figure 5.1: F; for Q2 = 100 and m6 = 1.5 GeV from 1('7"c —) c) + (7‘c -+ gc) pro-
cesses. Solid line is this work’s result. Dashed line is the calculation of Reference [69]
understood as due a different treatment of the charm quark kinematics adopted
by Reference [69] in deriving their formulas. When m6 is small compared to
Q, this effect is expected to go away. This is exactly what we see in Figure 5.4

where me is set to smaller value — 0.3 GeV.

(7"q —) gq) -(g —> 05) : The treatment of the gluon-fragmentation term requires some
care due to kinematics of zero-mass partons. The integrated cross section of
7" + q —) q + 9 process would have an infrared singularity arising from the
integration in the region f = (p, — 122“")2 -> 0. But this potential singularity
is outside the physical region when the proper kinematic limit for the entire
process, t > tmgn = 4m?, is applied. The resulting finite expression, however,
will have a logarithm factor Ing- due to the me cutoff of the phase space

me

integration. This appears, on the surface, to contradict our claim of infra-red

90

 

o v V f v V v V T

 

 

 

 

'bur rasult' —-;—

'Hoflmann 81 Moore' ---------
-0.002 - ~
-0.004 - q
11‘.“ -0.008 ~ 1
-0.008 ~ «
-0.01 + a

-0.012 ‘ ‘ 1 . ‘ - ‘ ‘ l *
1 10

0

Figure 5.2: F; for :1: = 0.1 and me = 1.5 GeV from l('y"c —-) c) + (7"c -) gc) processes.

Solid line is this work’s result. Dashed line is the calculation of Reference [69]
safety of the ACOT scheme calculation. The resolution of this dilemma lies in
the observation that the c-quark dynamically mixes with all the others as an
active quark flavor in the ACOT scheme. The particular ln( 7110:) factors seen
here cancel with corresponding ones appearing in charm-loop contributions to
light quark final state contributions to the total F2. The total structure function
F2 is well defined to all orders. However, because the charm-loop diagram
contribution to light quark final states does not contribute to Ff, the charm
contribution to F2, i.e., F2", is not well defined at high orders in the ACOT
scheme. In practice, for this order of a, calculation, the entire contribution
due to the gluon fragmentation subprocess is so small in the current HERA
kinematic range, we do not need to worry about this problem. However, it will

become a relevant issue when the calculation is extended to order 03, since this

term is intimately related to the resummation of final-state collinear logarithms

91

 

-0.002

I

-o.004 -

11‘.“ -o.oos-

41.oos r

 

-0.01 '4” 'Our result' —— -
'Hoffmann 81 Moore' ---------

 

 

A A A A A JmA A l

 

-o.o12 . - W
0.001 0.01 0.1

Figure 5.3: F; for Q2 = 10 and me = 1.5 GeV from 1(7% -> c)+('y‘c -§ gc) processes.
Solid line is this work’s result. Dashed line is the calculation of Reference [69]

in the NLO calculation.

5.2 Inclusive Charm Structure Function

We start by considering inclusive measurements, i.e., the inclusive structure function,

Ff, and the total cross section for the production of charm quarks, e. g., 0".

In principle, the calculation contains three scale choices: the renormalization scale,
the parton distribution factorization scale and the fragmentation scale. As is custom-
ary, we set the three to be equal. The scale should be dominated by the heavy quark
mass for small values of Q but insensitive the heavy quark mass at large Q. We make

the scale ansatz
1(Q2 + T2) for Q > m
2 Q2 c (5.9)

m2 for Q _<_ me

C

ON

 

-0.005

-0.01

0.015

it.” -0.02
0.025

-0.03

I
I
I
I
I
o 035
C y

 

 

 

 

-0.04 . . Mu ......U 4 “-1-..
0.0001 0.001 0.01 0.1 1

X

Figure 5.4: F; for Q2 = 10 and me = 0.3 GeV from 1(7"6 —+ c)+(7‘c —) gc) processes.
Solid line is this work’s result. Dashed line is the calculation of Reference [69]

In Figure 5.5 we display the p—dependence of F; at :1: == 0.01,Q = 10 GeV,
using no as the reference value. Adding the higher-order terms to the leading ACOT
results increases the cross section without substantially changing the p—dependence.
In either the LO‘ACOT or the NLO ACOT result case the p dependence is weaker
than the LO three-flavor calculation; it is approximately the same as the NLO three-
flavor calculation. We expect that the full 0(03) calculation will improve the a-
dependence after the contributions from the higher order flavor creation processes

with experimental cuts are added.

Once we have made a scale choice, the only parameters in the calculation are
AQCD and the charm quark pole mass, me. For the former, we use 11% = 0.202 GeV.
However, the charm quark mass is not as well determined. In Figure 5.6 we show the

dependence of the D“ cross section on the charm quark mass mc . The shaded band

93

 

 

 

 

 

 

 

 

 

0.2 I I
This
0.15 >< ACOT -
FFN NLO
1.9 0.1 Wm -
FFN LO
0.05 ~ 4
o J l
0.5 1 1.5 2

11410

Figure 5.5: p—dependence of F§(:r = 0.01,Q = 10 GeV) in various calculational
schemes. no is defined by Equation. 5.9.

represents measured cross section from ZEUS [63] within 1-0 experimental errors.
The cross section is measured in the region 1 GeV’2 3 Q2 S 600 GeVz, 0.04 S y S 0.7,
1.5 GeV S p,” S 10 GeV, and InDI _<_ 1.5. These cuts are easily applied to our Monte
Carlo implementation. The total integrated D" cross section is related to the charm
cross section by a fragmentation probability constant, which we take to be 0.26 [70].
Once the cuts are applied, however, there will be some residual dependence on the

D“ fragmentation function in the result.

The measured cross section is consistent with the typical range of estimates of
the charm quark pole mass within the 1-0 range, although lower values of me are
disfavored. Based on this analysis we use me = 1.5 GeV for our other calculations.
For consistent implementation of this comparison, we have used parton distribution

functions fit for each individual value of me.

94

 

 

 

 

 

g a ' d
O
4 )- -
2 I- q
o l l l
1.2 1.3 1.4 1.5 1.6
"'0

Figure 5.6: Measured D" cross section at Zeus [63] with experimental cuts (described
in the text) compared with our calculation as a function of the pole mass me. The
shaded band represents the l —- 0 experimental errors.

Figure 5.7 shows our results for F; for various values of Q2. Our calculation is in
good agreement with the data, as is the three-flavor NLO calculation. The differences
between the two calculations are well within the experimental uncertainties. Since our
calculation only involves order 0, terms, this means the same result is obtained with
significant efficiency and economy compared to the conventional order a? three-flavor
NLO calculation. The CPU time required by our calculation is nearly an order of
magnitude smaller than required by the three—flavor NLO calculation. The efficiency
is due to the fact that the major contributions of the complicated NLO three-flavor
formula are from those large logarithmic terms which have been neatly resumed into

the flavor excitation processes in our formalism.

Notice that while the simple three-flavor L0 is substantially smaller than either

the three-flavor NLO calculation or ours, our results are very similar to the three-

 

 

 

 

 

 

95

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

o’neov’ ohms“? ahzseov’
0.0 "m, m... m... 0.0 m... m... m...
0.4 1- -1
1: [ 1:
01 :. .
0‘ ~ 4 1 o 1 ‘ 0‘ 4 1
10“ 10" 10 10' 10° 10“ 10” 10“ 10‘ 10° 10" 10" 10 10‘ 10°
R I X
o’usow’ o'.1rooov'
0.. ""1 "'"w "'"1 "
0.4 -1
if." 'h?‘
0.2 -
o ‘4 4 4 1 o ‘ 4 ‘
10 1o 10 10' 10° 10" 10“ 1o 10“ 10°
H X H

Figure 5.7: Total F; for various values of Q2. Solid lines: ACOT. Dashed lines: three-
flavor N LO calculation of Reference [38]. Dotted lines: three-flavor LO calculation.
Solid points: Zeus 95 preliminary [65]. Open points: H1 [64].

flavor N LO results throughout most of the available range of Q2. This also implies
that the flavor excitation processes where charm c is treated as an active parton
effectively resums the most important contributions from the higher order processes.
The small differences between our calculation and the three-flavor N LO calculation are
illustrative and easily understood. The largest difference between the two calculations
occurs at the smallest values of Q2. The logarithms resumed by our calculation are
not so large in this region and the three-flavor NLO calculation is superior because
it includes the contributions from the 0(03) HC2 processes 7" + g —> c + E + g and
7" + q —) c + E + q. We can see, however, for the majority of the parameter space
explored at HERA energies the two calculations are practically indistinguishable.
At the highest measured energies the two calculations begin to diverge. Here the

logarithms begin to dominate the calculation. In this case our calculation is the

96

most reliable, although more experimental data and smaller error bars are needed to

practically distinguish the two calculations.

Figure 5.8 shows the comparison between the LO ACOT F; and F2” from this work
at a: = 0.1. The small correction that the NLO ACOT processes has in the whole
energy range indicates the ACOT scheme perturbative expansion is well behaved.
Notice the relatively large correction from the LO three-flavor result which indicates
that the ACOT scheme calculation resums all the important contributions from the

higher order, more complicated three-flavor heavy quark creation HC2 processes.

 

0.“ r r 1 f r T f

0.03

0&5 1-

0.02

t.

 

0.015 )-

0.01

0.“!1- .

 

 

 

 

1 4 1 1 L m 1 1 1
0 1° 20 a co m 00 70 I) W 100
a

Figure 5. 8. F; at :r- — 0.1, solid line: This work, dashed line: LO ACOT calculation,
dotted line: LO three-flavor calculation

Figure 5.9 shows the results of our calculation for F§(:r, Q) for fixed :1: = 0.01.
Near threshold the HCl (photon-gluon fusion) piece dominates. In this region the
leading-order subtraction piece tends to cancel the contribution of the HE] (heavy—
flavor excitation) piece. As Q increases, so do the log (%8) terms. The effect is that
the HEl piece quickly starts to dominate the cross section. The logarithmic term is
present in the HC1 piece also, so the HC1 piece also grows rapidly. However, the

subtraction term, which removes the would-be doubly-counted logarithm cancels the

97

majority of the HC1 term. The net efl'ect is that the cross section is completely

dominated by the HE] term at large Q.

 

0.3 v v v v v v v I v v *r v r v 1 v

'I’f‘

 

 

 

 

0‘ A A 4 A A A A J A A A A L A A A
0

Figure 5.9: Contributions of the various subprocesses to Ff(:r, Q) at fixed a: = 0.01.
The sign convention is such that the total cross section is HEI + HC1 - HCIM, +
HE2 - HE2.ub + Cl“ 2

The new features of this calculation (compared with Reference [56]) are the GF2
and HE2 contributions, along with the corresponding subtraction. The new contribu-
tions are considerably smaller than the lower-order HEI and HC1 pieces, indicating
that the perturbation series is converging as it should. At this (relatively large) value
of x the fragmentation (GF2) contribution is the most important factor for large Q.
The net effect of the HE2 contribution is very small after the subtraction is included.
Be aware that the relative sizes of the contributions are strongly x-dependent. How-
ever, it is a general feature that the HE2 and GF2 contributions are small corrections

to the HEl and HC1 contributions for reasonable values of a: and Q.

98

(if-100001!”

 

o.“ v v v v r 1 Ya I

in?

 

 

 

 

 

 

10" 10" 10"

Figure 5.10: Contributions of the various subprocesses to F§(:r, Q) at fixed Q2 = 7
GeV. The sign convention is such that the total cross section is HEl + H0] - HCl..,b
+ HE2 - HE2wb + GF2

In Figure 5.10, we see the same interplays between various subprocesses. For fixed
Q, the large a: limit correspond to the partonic hard scattering energy approaches
to the final state threshold and every contribution becomes small. For :1: -+ 0, the

structure function rises significantly.

5.3 Differential Distributions

Because we have a Monte Carlo implementation of our calculation we are able to
generate predictions for differential distributions involving final-state charm mesons.
We are also able to directly incorporate experimental cuts into our calculation. This
is an important advantage. Our calculation produces somewhat different results from

three-flavor calculations in the small 11, region. If the experiments have to extrapolate

99

their data over the typically unobserved small p, region, the results may depend
more on differences between our calculations and another calculation than on the
observed data. It is much better to directly compare the unextrapolated data with

the appropriate theory.

In order to make predictions for mesons, as opposed to quarks, we need to incor-
porate the fragmentation of charmed quarks into charmed mesons. We have used the

Peterson [71] form for the fragmentation of the charm quark into charmed mesons,

 

(.) A
d5” (2) = z[1—1/z — e/(l — 3)]2’ (5'10)

with e = 0.02 [72, 73, 74] and A such that the branching fraction B(c —) D") = 0.26
[70]. The “11p” in Equation 5.10 reflects non-perturbative nature of the fragmentation
function. The Peterson form has the advantage of being widely used. Unfortunately,
it violates the scaling behavior of QCD as found by heavy quark effective theory.
However, inasmuch as it is merely a functional form fit to experimental data, it is

perfectly satisfactory for our purposes.

The fit to Equation 5.10 in Reference [73] includes a convolution with the c —> c

fragmentation function in Equation 5.6, i.e.,

d?“’(z,u) = 432.11) s 40%) (5.11)

c,np

The fit includes data from J? = 10.6 GeV and ([3- = 91.2 GeV, so the perturbative
evolution is substantial. The perturbative evolution only has a small effect on our

calculation for HERA energies. Nonetheless, we include it for consistency.

There is an ambiguity in defining the momentum fraction of a heavy quark meson,
z, for heavy quark fragmentation. We use the light-cone coordinate scaling variable,
[)3 = (pf, to scale the momenta for fragmentation kinematics in the lab frame. Harris

and Smith [51] use a different prescription for fragmentation in their 3-flavor scheme

100

NLO calculation. In their prescription the 3-momenta scale such that f)"; = (fie. They
then adjust the energy such that p2,) = mfg. We have verified that our results are
insensitive to the differences between the two prescriptions. It is may be possible,
however, to come up with an observable which depends more strongly on the exact

prescription, so one must be careful when comparing results of different calculations.

Although the Monte Carlo approach allows us to plot arbitrary distributions, some
care is required in choosing and interpreting them. First, the factorization scheme
upon which we rely has been proven only for total cross sections. Differential distri-
butions may still require other resummation in some kinematic regions. Second, our
results for differential distributions at the quark level are singular in some regions.
The simplest example is the p,-distribution due to the leading order flavor excitation
process HEI 7" + c -) c. The quark p. is described by a Dirac delta function at
zero. The same problem also happens in jet calculations. In hadron-hadron collisions
this is rarely noticed because the singular region is experimentally inaccessible. In ep
collisions, however, the singular region is visible in the lab frame. Fortunately, exper-
iments measure mesons, not quarks. Once our quark-level calculation is convoluted
with meson fragmentation, all our distributions are physically smooth. Actually, the
presence of such effects probably indicates a need for a small-p, resummation at some

level.

To compare to our calculations of differential distributions, we have used prelim-
inary data from Zeus 95 [75] for experimental data because it is substantially more
detailed than similar previously published distributions from Zeus and H1. We ex-
pect similarly detailed data from H1 in the near future. In Figure 5.11 we compare
our predicted 1), distribution with the Zeus 95 preliminary data. Notice that this and
the following differential distributions in this chapter are subject to experimental cuts

GeV2 5 Q2 S 600 GeV), 0.04 S y S 0.7, 1.5 GeV 5 pf" S 10 GeV, and [nD'l _<_ 1.5.

101

The distribution in Q2 is free from the potential problems in the p, distribution. We

 

 

 

 

 

101 f . :
1
0
(53' 10
a 3
.E.
5
2 -1
.3 10 E ‘3
10'2 . 1 . {1
10

91(0) [GeV]

Figure 5.11: 11, distribution compared with data from Zeus 95. The experimental cuts
are described in the text.
compare our results with the Zeus results in Figure 5.12. We also compare our predic-
tions with experimental results for the distributions in 170° and W in Figure 5.14 and
5.13, respectively. The center-of-mass energy of the virtual photon—proton system,
W, is given by

W2 = mi, + mi — 1). (5.12)

These distributions do not involve any further subtleties in the calculation. Unfor-
tunately, the shape of the distributions is more a function of cuts than physics, so
they serve primarily as a demonstration that our calculation is consistent with the

experimental results under the appropriate cuts.

More differential distributions are possible, but they may involve further sub-

tleties. For example, the evolution included in the GLAP equations involves an

102

 

 

 

 

10‘. . . fl '''' '
10° r i '
E 1 r I
(D 10' r 7
B r i ‘
.E.
E 10'2 r I ‘
b .
u p
10'3 r 7
-4 ’ 1 , A A m .1 f4
10 1 1 1 A A AAA A
10° (1191 2 102
[GeV]

Figure 5.12: Q2 distribution compared with data from Zeus 95. The experimental
cuts are described in the text.

integration over the remaining final state particles. For this reason we are unable
to extract charm-charm correlations, such as the azimuthal angle distribution, in the
final state. Measuring the longitudinal momentum distribution of the charmed par-
ticles in the final state presents another problem. While the the distributions are
not singular, they depend strongly on the assumptions that go into our definition of
the momentum fraction, the light-cone scaling prescriptions, and the frame in which
the momentum scaling is implemented. These problems leave us without a definitive

prediction for the longitudinal momentum distribution.

103

 

0.08 T r r

0.06

1
l

 

 

0.04

 

do/dW [nb/GeV]

0.02

 

 

 

 

o 1 1 1
50 100 1 50 200 250
W [GeV]

Figure 5.13: W distribution compared with data from Zeus 95. The experimental
cuts are described in the text.

 

 

 

 

 

 

 

4.5 1 I I I U
4 . v ’ 1
3.5 _ -
.—-. 3 P 0 1 q
.0
‘5 2.5 7 1. . 0 "
g 2 . ' .
.0 .
1.5 . /—\‘\ .
1 ~ _ .
0.5 - .
0 1 1 1 1 1
-2 -1 ,0 1 2
n(D)[GeV]

Figure 5.14: 1) distribution compared with data from Zeus 95. The experimental cuts
are described in the text.

Chapter 6

Summary and Outlook

We discussed the physics of heavy quark production at deep inelastic scattering. First,
we introduced the ACOT scheme and pointed out that the three-flavor scheme calcula-
tion, which has been widely used for recent heavy quark production phenomenological
study, can not be reliably applied when the relevant energy scale is far larger than the
heavy quark mass. Then, we demonstrated our formalism by calculating the inclu-
sive F; and various charm meson distribution functions at HERA using the helicity

method and the Monte Carlo integration. Our results agree well with experiments.

This work extends the original ACOT leading order calculation by adding the
terms which are required to give a full N LO calculation at high energies and imple-
menting our calculation in a Monte Carlo analysis which allows to calculate differential

distributions and incorporate experimental cuts.

Extending our work to the full NLO at high energies brings our calculation to the
level of accuracy of the other theoretical inputs. to the CTEQ global QCD analysis,
and unlike the three—flavor N LO calculations, ours is valid to arbitrarily high energies.
At HERA energies our calculation gives similar results to the NLO three-flavor cal-
culation for inclusive quantities, but much more efficiently. The calculation is much

simpler and the resulting program runs faster. This will useful for future global QCD

104

105

analysis where efficient computation is very important. At the theoretical level, we
point out that we have identified the numerically most important contributions to
heavy quark production. We also point out that the smallness of the corrections in-
cluded in this work indicates that our perturbation series is in fact well-behaved. One
might worry that the fairly large NLO/LO ratio seen in the three-flavor scheme in-
dicates significant corrections from even higher order processes. However, the ACOT

scheme does not have this problem.

By generating results for differential distributions we have shown that our calcu-
lation also does a reasonable job of predicting the details of heavy quark production.
In this case we do expect the three-flavor NLO to have something of an edge. That
calculation includes 2 -) 3 kinematics; ours does not. Nonetheless, our predictions
are in reasonable agreement with data from HERA. By incorporating experimental
cuts in the Monte Carlo we are able to ensure that we are comparing our calculation
directly with the data and not with the details of a different calculation needed to

extrapolate the experimental data to all of phase space.

Having established that our calculation does a reasonable job in describing the
existing HERA data, we are now in a position to explore further in several directions.
We can extend the ACOT results to include all 0(03) terms. Such a calculation
would include all the advantages of both the current ACOT calculation and the three—
flavor NLO calculation, especially, for the differential distributions. With more data
available in the future, we will be able to use ACOT calculations to extract parton
informations in the global analysis of the parton distribution functions. Another es-
pecially interesting question is whether the proton contains a non-perturbative charm
component or not. The ACOT scheme is the only existing scheme which can address

this problem in a self-consistent way.

Appendix A

Helicity Amplitudes of Heavy
Quark Deep Inelastic Scattering
Productions Via Neutral Current
Interactions

In this appendix, we list the helicity amplitudes of the contributing processesl. The

notation is as following:

C denotes a heavy quark with mass 111,, i.e. charm quark, and Q = \/—q2 where

q is the 7"/Z momentum.

A.1 7*/Z+C—> C

For the 7'/Z + C —-) C process with arbitrary vector boson and quark couplings

. 5‘” ° pu '
—ie [(Frv + FrA'Ysl’Y” + 102mg” F2v + 12mg” F211], the helicity amplitude J '(q; 19;),

where i, j,k are the helicities of the vector boson, the initial state quark, the final

 

 

state quark respectively, are:

J-(qLa 43) = -ie\/2Q(F1v + 1721/ ‘- 351.4)

J+(Qn, 9,1,) = -ie\/§Q(-F1v - F2V — 334)

1The author thanks Carl Schmidt for making the results available.

 

106

 

 

 

J0(QL1q’L) = -i€Q( Q FIV - 231‘31‘21/ - 2ZcflF24)

 

 

 

 

. 2mc
JO(QR1q’R) = —38Q( Q Fly - 231CF2V + chflFzA) , (A.1)
2
In the equations, 3 = 1 + 43:," .

A.2 wz+g —1 0+6, HCl

The helicity amplitudes J’(gj, qk, Q7) = quJ’(L,gJ-,q),,q1) + gan’(R,g,-,qk,q1) listed
below are calculated in the CE center of mass frame, as plotted in Figure A.1. The
L, R indicate the chirality of the current and 9,11,, gqn are defined in 4.14 with electron
helicity implicitly included. i, j, 10,! are the helicities of the vector boson, the gluon,

the final state quark, and the final state antiquark, respectively.

J+(L19+,QL,§L) = A—+(—BSC+C+32)
J+(L19+19L,§R) = A++("BC2+C+SC)
J+(L,g+,qR.<7L) = A--(-BS’-C+SC)
J+(L19+,QR,§R) = A—+(-BSC+C+02)
”(19.941.41.00 = A-+(-Bsc+C-c2)
J+(Rag+,qL,§R) = A--(—Bcz—C—SC)
J+(R,9+1QR,<7L) = A+4(—BS2+C—30)
J+(R19+,QR,§R) = A-.(-Bsc-c_é2)
J_(L19+1<1L,(IL) = A—+("’BSC)

J-(L1g‘l-1qLaqR) = A++(+BS2)

108

J-(L,9+,<IR,¢7L) = A---(+B<?2)

J—(L1g+1QR1(IR) = A-+(—BSC) (At-2)
J-(R19+1QL,§L) = A—+(-BSC)

J 1319+, (11., in) = A—'—(+382)

J-(Rag'HQIiaq-L) = A++(+302)
J-(R,9+, (In, (711) = A—+(“BSC)

J°(L, 9+, qL, 971,) = 2‘1/2A_+[B(+s2/D + 062) — DC+sc]
J0(L19+1 (11., (712) = 2Il/2I4++lB(‘I‘SC/D " D80) + DC+82I
J°(L,g+, qR, a) = 2-1/2A_-[B(-sc/D + Dsc) + DC+c2]
J0(L19+1QR,§R) = 2.1/2/‘l-«LIBFCZ/D "' 032) " DC+SCI
J°(R,g+, qL, m) = 2'1/2A_+[B(—cz/D - D32) + DC_sc]
J°(R,g+, qL, q“) = 2'1/2A--[B(+sc/D — Dsc) - DC_32]
J°(R,g+, qR, 171,) = 2‘1/2A++[B(-sc/D + Dsc) — DC.c2]

J°(R,g+,qg,q'g) = 2_1/2A_+[B(+82/D + DCz) '1' DC_SC] ,

 

 

where
A — ‘2i69aT“[(1 :1: 3X13; 5)]1/2
fl - (1 - 52 cos? 0') ’
B = (Echarm/Egluon)flSln9’ = (1- fl/€)flsin0’ ,
ct = lzl: fleosO’ ,
A = 2 {_1
3 Q (,7 ) ,

and s = sin(0’ / 2), c = cos(0’ / 2). 0’ is the scattering angle as plotted in Figure A.1.

109

C

Figure A.1: 7‘/Z + g —) c + E in the CE center of mass frame.

The remaining processes can be obtained by

J’([L.R],gj,qt,r71.) = J”([R1L]ag-j,qm(7&)

J’(IL,RI,9j,qL,7R) = ‘J-i(IR1L]19—MR,€7L) - (AA)
A.3 772+ C .4 0+9, HE2

The helicity amplitude J‘(gj,qk,q]) = gLJ‘(L,g,-,qk,q]) + gRJ‘(R,gj,q1.,q]) listed
below are calculated in the gc center of mass frame, as plotted in Figure A.2. The
L,R indicate the chirality of the current and 9“,qu are defined in 4.14 with the
electron helicity implicitly’included. i, j, [0,! are the helicities of the vector boson, the

gluon, the initial state quark, and the final state quark, respectively.

J+(L19+1QL1q’L) = -A++(83)
rusted.) = -A.-(s2c)(1+B)
J+(L1g+1QL1q,R) = A-+(CS’)
”(1119441412140 = A--(082)(1+B)

J+(R1 9+1 gin qIL) = 0

where

J+(R,g+.qR.Q'L)
J+(Rag+1QL1q’R)
J+(R1 9+1 (In, 4’11)
J-(L,9+,QL,€1’L)
J—(L1g-l-1QRanL)
J'(L,g+,qL,q'R)
J-(L19+aQR,QIR)
J'(R,g+,qL,a/L)
J-(R1g-i-1QR1q’L)
J-(R,9+,QL,QIR)
J‘(R,g+,qn,q’3)
J0(L,9+19L14’L)
J0(L1g+1QR1q’L)
J0(L1g+1QL1q,R)

J0 (1119+th4’12)
10(R19+1QL1(IIL)
J0(R19+1QR1 q’L)
J0(R19+1‘1L19’3)

J0(R1g+1 qR1q’R)

A14 =

110

A_+(2cs2)B

0

-A++(8)(Bc2 + D)
A++(802)(1 - B)
A+—(C’)
A_+(s2c)(1 — B)
A-_(sc2) (M)
A-_(sc2)B

0

—A+_(c)(-B.92 + D)

o

_2—1/2A++cs2(—2qo + B(qo + P))/ Q
2'1/2A+_s’c(200 + B(Qo - p))/Q _
2‘1/2A_+s(qo(s2 - 0’) - p + B(qo + p)c")/ Q
2‘1/2A__c(qo(s2 - c2) — p + 3(90 - p)s’)/ Q
2‘1/2A_-82€B(90 - 19)/Q

—2"/2A-+S2CB(QO + p)/ Q

—2“/2A+-s(qo — zo)(Bc2 + D)

2_1/2A++S(q0 + P)(-’BS2 + D)1

—21'eg,T“[(E,’,/E,,)(1 :1: ma 1 (3)1”

 

(1— flcosfi’)

B = P/Elv

111

 

_ p”-lu
D _ pm'lu,
. f 2 5 2
s = —-—l +1—-m,
(17 )Q ( n)
. 2
s—m
E — —,
I 2%;
l §+m2
E = ,
P 2“;
E _ §+Q2+m2
P 2% 1
s—Qz—m2
= —, A.7
qO 2“; ( )

and s = sin(0’/2), c = cos(0’/2). 0’ is the 7'/Z(q) + c/E(p) —) g(l) + c/E(p’) scatter-
ing angle, p“ is the 4-momentum of the initial state quark, p’” is the 4-momentum
of the final state quark and 1" is the 4-momentum of the gluon, The meanings of

qo, p, p’, E1» E], can be read from Figure A.2.

g (E), p’ sin 0’, 11' cos 0’, p’)

  

’ I - ’ I ’ I
C (Ep, —p smB , —p c030 , —p)
Figure A.2: 7"/Z + c —> g + c in the gc center of mass frame.

The remaining processes can be obtained by

J‘([L1 R11 gj1 qL1¢L) = J—i([R1 Ll’g‘j’ q”? q’R)

112

Ji([L1R]1gj1QL1q’R) = _J-i([R1leg-j’QR1q'L)° (A.8)

Appendix B

From Helicity Amplitudes to Cross
Sections

In this appendix, we derive the cross section formula we used for this work. Most
of the equations are already presented in the main text, but here we collect these

scattered formulas together for easier reference.

From the factorization theorem, the cross section for deep inelastic scattering

11(k)+N(P) 412(k’)+H(P’)+X(pz) is

 

l;N—+lgHX_ dé. 2 2
do _2A(3M§,,O)./ —¢~(4. )IMaldI‘ (8.1)

with phase space factor DI‘ as

431.4 13,1 (131);

 

_ 4 4 _ I _ I _ I
dr ’ (2") 5 00+ k k ” zp’)(27r)32k6 (271)3213' (270321;; ' (3'2)
Simplifying Equation 8.1 by using the standard variables used in DIS,
d310, _ MEI—J—yda: dy g2
(27r)32k(’, — 81r2 y21r
A(s,M§,,0) = 2MNE,,,
we obtain
da— — 3—2—dxy .ryd di’ 27?] d6 7,)(5 )|M (2111", (3.3)

113

114

with

3

 

for the hadronic final state.

Using the helicity method, we can write the cross section as

_ y d¢ d6.
do — Weds/57;]?- N(o

(p.11. 0:1th ”+11.“ 11:31:: 2) II", (13.5)

where eL, eR represent the left-handed lepton and the right-handed lepton respec-
tively. The sum of pet, and Pen, which represent a polarized lepton beam, is obvious
1. For neutral current interaction, we can extract a factor (72" from the matrix ele-
ment squared. The vector boson and quark coupling can then be written as -—ie7"gf,
where a = qL, qR labels the left and right chiral couplings of the quark to the neutral
bosons and b = eL, eR labels the left and right handed incoming lepton currents. 92

can be obtained as the following:

 

 

 

 

 

 

 

 

9.1. = Q _ +411sz _9; T,_Q,...2ow

9" q sin 9w cos 0w Q2 + Mg sin 0w COS 0W
get. = Q - "i + Sinz 9W Q__2_ —Q, Sinz 9’”

9” 9 sin 0w cos 0w Q2 + Mz sin 0w COS 0w

.11 _ Q _ sin2 9w Q__’__ Ta. — Q. 311.2 0w
9"" _ q sin 0w cos 9w Q2 + M 2 sin 9w COS 0w

. 2 2 - 2

.11 _ _ 8'" 9W _2__ ‘Qq 3‘“ ”W B 6

90R — Q9 (sinélw cosflw) (Q2 + Mg) (sinflw cosflw) ' ( . )

In Equation 8.6, Q,, is the fraction charge of the quark, T3,, is the quark’s third
component of the weak isospin, and 0w is the weak angle. The coupling 9: effectively
adds the photon and Z boson contributions to the hadron current at the amplitude

level after the lepton currents are factored out.

115

Defining the vector boson polarization vectors 62‘ as

 

_ (- 42)P"+(P (1)0”
611(P1Q) - Fq2)[ (P q) 2—q2P2]
esmq) = 7%-
e44;
6:(P,q) = 7-2-(01'1'11—510)
cum) = i—Zm, +1, +1, 0). (13.7)

The lepton currents can be expanded as

.” = 2 smh1/2 _ coshzb-l _ cosh/2+1 J
M. V25? _1/5 £5 (_fl )4 ( fl ) -
= «2020,94, i=+,—,0

.,, _ sinhzl)
1.3 — J27 —ea‘-( (TM-(7)61]

= @054, .=+,-,o- (13-8)

Then, the cross section can be written as

2 dyd¢ dE‘iv
do = 16y+Q2M W] {#2)

(Pol. * DfLDiiJiL,.J3£:( + Pen :1 DfRDfflijJfga) dI" (13.9)

where Ji = GLJ". For heavy quark production at HERA where unpolarized electron
. . i 1 .

beam scattermg With proton, p1, = 1);; = 5, so the cross sect1on and the structure

functions simplify to

2
do- ye

" 3215422“

 

dd . . . .
“245/0 (MW/(5’ #2) (Df‘DfLJ:L..J.z,. +D:”D§”J;R,.J:E..) d1"
(13.10)

116

and
l 1 d6 e e e at
FACE Q2) = 8_7|'./0 ?¢(£,H2) (DALDXL JcL, 0J3. L,a + DARDARJeRpJe Alta) dI" ' (B'll)

Now, we will present the more detailed cross section formulas for different pro-
cesses. We use L and R to represent the left and right handed chiral couplings
respectively. The hadronic currents will be expanded with the hadronic amplitudes
listed in Appendix A. The phase space factor (11" will also be presented in terms of

the kinematic variables introduced in Appendix A.

3.1 721/2 + C _> c, HEl

2

_ ye 61¢ ldgC 2 cL eLi eR eRi
d0 — dedyz—W-lo' ?¢N(£,p)(D' Dj JeL Cg'] [MC-+1) Dj JeR, CJeEC)dl-‘

 

_ ye2 (if—‘2 _ mos-£0) 1
87rQ2d2—1r/olc?¢g(£’”u) 2‘/1+4m2/Q2*2*

((08% (92” (L qua) +92 J°(R (11,911)»2

(03" * (92" (L,qn,q:z) +9sz J0(1‘?,qn.rm)))2
(02" * (92" J'(L,qL,qn) +5172" J’(R.q:.,<m)))2
(01" * (92L J+(L,q:z,qz.) +97% J+(R aqu)»

(193R * (92” J°(L 41.41:.) +97: ”,J°(R 9L,QL)))2
(DoR * (92” J°(L 912,113) +93” J"(R,qn,qzaz)))2
(131“ (92“ J (L qmm) +972 "'J (R QL14R)))2

(012*(g22 J+(L qn,q,.)+gn2 J+<R 93,4Llll2) (8.12)

+ +

+++++

The helicity amplitude Js are defined as in Appendix A with

Ji(‘1j, 91:) = gLJ‘(LanaQ;c) + QRJi(Ra‘Ij1 Gig), (3'13)

117

where L,R represent the left and right chiral couplings respectively. Changing the

V-A coupling to chiral couplings in Equation A.1 gives the appropriate helicity am-
4mg

Q2

 

plitudes used in Equation 3.12. In Equation refapple, £0 = g (1 + (1 +
with
1 l 1 M},

5:255" 47526?

3.2 wz + C _. C, HE2

This one is similar to the above one, except J 3, defined in Appendix A.1 , are calculated

using renormalized couplings:

 

 

_ie [(Fw + 53,737» + 22:4" 112,, + ’2qu Fm] (3.14)
with
F1v = :7: qu(Il+I2) (13-15)
F... = g; JAM—12) (13.16)
m = :7; mm» (3.17)
where

 

7r 2 2 _
11 = (ii—)2 I‘(1+6){%[-1-(1+fl)ln(-fi——1-)]—2

 

 

 

 

3 2fl fl+1
+(1l $2) [-§1n(g—;—i) - gmg; i)ln(fl:fl'.1>
+Li2(321;—1) — L12(5—2‘31)] }, (8.18)
12 = :2”; hug—1%). (3.19)

 

118

L12 (1:) is the usual Spence function defined by

Li2(x) = —/0L l—M—lz-—L)dz , (3.20)

4m2

andfl=1+—Q—2-.

B.3 7*/Z+g —-> 0+5, HCl

In Equation B.21, fl = l— 41:2, as defined 1n Appendix A. 2.

2
ye d¢ ldé 2 eL cl. i 3 2R :8 1' 3
do = 32”, dexdyfi- f0 -£—¢3,(§,p )(D, D, JCL,9J¢’L,9+D,- D,- JemJjfl‘g)dI"

 

 

_ dyLLLL d£ fl Tr[T"T"] 1 dcosO’di;
8-L7r—Q:d filo]: —("¢L"€’ 2*) 32 “4“” 8 ”‘2’“ 2 21K“

[(DoL * (92L J0(L,L 9+aQL (IL) +9izL JO(R19+1QL1‘7L))
+DL+L * (92L J+(L 9+,(1L,QL) +972 LJ+(R 9+,QL,QL))* L
+ DiL * (92L JLUthQLfiL) +97% J-(R19+19Laq-L)) * :LLLY
+ (DoL * (gLL (L,g+,QL,§R) +gizL Jo(Rag+aQL,€IR))
+ Di” * (92” J+(L 9+,qL,qR) +93 J+(R 9+,qL.qn)) *6 "4’
+DL-L* *.9i.(L J (L, 9+aQL,QR) +QRL J (R, 9+,QL,QR))*€+LL)L
+ (DBL * (92" (L,g+,q:z,€1'1.) +972” (R.g+,qn,(h))
+DiL * (92L J+(L 9+,QRJIL) +gnL J+(R 9+,QR,<1L)) *6 LL"
+ DL—L * (92L J—(L,9+,QR.§L) +9izL J-(Rag+,QR,§L)) * BLLLY
+ (DoL * (QLL (Lag+,QR,§R) +97% JL(R.9+aQR,§R))

-54,

+ Di" * (92L J+(L,g+,qn,r7n) +5172" J+(R,9+1‘1R"JR)) *6

119

+ D5." * (92” J'(L1y+,qu, (in) +1171” JL(R19+1QR1(7R))* 6+“)L

(052 1 (922 (11.11-41.171) + 922 J°(R,g-.q,., 3.))

+ DLL * (92L J+(L1g-1QL1qL)+9;zLJ+(R1g—1QL1QL)) *eLLL

+ DL—L * (92L JL(L19—1QL1§L) +gizL JL(R19—1QL,§L)) * eLLL)L

(DoL *(gLL JLIL19-1QL1QR)+972L JL(R 9—1QL1QR»

+ DLL * (92L JL(L19—1QL1§R)+971LJLIR19-1QL1QR))*€LLL

+ DL—L * (92L JL(L19-1QL1§R) +97% JL(R19—1QL1§R)) * €4,141)?

(DoL *(gLL JL(L19-1QR1QL)+93L JL(R19-1QR19L))

+ DLL * (gLL JL(L1 9—1QR1QL) +QRL JL(R1 9—1QR1qLLll *3 LL

+ D‘." * (92" J'(L,g—.qn,fiz.) +97% J‘(R,g—,qR.r7L)) *e‘L'L’)L

(DoL *(QLL JL(L19-1QR1QR)+9LL JL(R1g-1QR1QR))
+D‘L*(gi" J+(L 9—103143) +91: JL’(R g-1QR1QR» *6 ’4’"

+ D‘.” * (92" JL(L19-1QR,(7R)+971L J‘(R1g—,qn,tin)) *e+‘¢)L

(DSLH (92L (L19+1QL1§L) +92? JL(R19+1QL,§L))

+ DiLL*9(LLLJL(L19+1QL 9L) +9L LLJL(R1 9+1QL1QL))* LLL

+ Di“ (97.” J’(L1y+,qL,§L) +93” J (R 9+1QL19L))*:+LL)2

(193“ * (92” J°(L 94.143411) +971" (R.g+,qL,<7n))

+ DLLL* (92L JL(L19+1QL1QR) +9LLLJL(R19+1QL1§R)) *CLLL

+ 02.2132” J (L 9.41.1111) +91” (12.111.41.13) 192111)”

(DOLL * (QLLL (Lg-1143161.) +QRLL J0(R19+1QR19L))

+ DLLLL'“ *(QL LL J+(L1g+1QR1QL) +9LLLJL(R19+1QR1QL)) *eLLL

+ DL—LL * (92L JL(L19+1QR19L) +9LLLLJ (R19+1QR,§L)) * eLLL)L

(D3L* (92L (L19+1(IR1(IR) +9LLL1JL(R 9+1QR1QR»

120

+ D1”*(92”J+(L19+191219n)+92“J (R1 91119111912» *6
+Df." * (92 J (L1 91119111911) +92" J (R1 91119111910) * e+'¢)L
+ (DOLL * (9LL (L19-19L19L)+9§L JL(R19-19L19L))
+0?" (9LL JL(L19—19L19L) +9LLJL1(R 9-19L19L)) *3 LLL
+ DL—L* *9LLL( JL1(L 9-19L19L) +9L LLLJ (R1g-1QL19L» * BLLL)2
+ (DER * (92” J°(L1 9-191.193) +92" J°(R1 9-191.193»
+ Di” * (92 J+(L1 9—19L19n) +92 RJ+(R1 9-191.193» *6 "4’"
+ Di“ *9(L ‘3 J (L19—19L19n) +92R’J (R19-19L19n)) *e+'¢)2
+ (DER * (92" (L1 9-1911191.) +92 J°(R1 9-1912191.»
+ DLL" *(9L LL JL(L1 9—19319L) +9L LJL(R1 9-1911191.» *3 LLL
+ Di” * (92" J (L1 9—19121911) +92 ”‘J (R19-19n1 91.)) * e+"")L
+ (DEL * (92L (1119-1931912) +97: (1319—1931910)
+ DLLL“ (92L JL(L19-19J219R) +9LLLJL(R19-19R19R)) *BLLL

+ DL—L * (92L JL(L19—19R19R) +9RL JL(R19-19319R)) * eLLL) 2] - (13-21)
13.4 7*/Z+C —1 0+9, HE2

The definitions of 1/9 and E, in Equation B22 are in Appendix A.3.

 

2
ye d¢ ldé C cL eL i c cl! 3' L‘
d0 = WdardyE/é ?¢N(€’”2) (D. Dj JCL gJeLg+DitLDj JeR‘ngfl’g) d1"
_ ye? dydqb 115% 2 213, 1 Tr[T“T“] 1 dcose'y‘:
LL87rdQ2L Elolfd’ W)7'E3—211*L24“‘L 3 ”'2” 2211*

[(DeL :0: (gLL J0(L1L9+19L19L)+gRL J0(R,g+,qL,qL))

+ D"’* (92 J+(L19+19L19L)+92" J+(R1 9+19L19L)) *6 "'4’

121

+ D:L .. (92L J’<L,g+.q2,cm + 92L J'(R.g+.q2.q,.» =~ eL‘L)”
(03L *(QLL J0”! 9+a9L,9R)+93L J0(R 9+19La9R»
+ DLL *(921’ J+(L 9+,9L,9R) +9nL J+(Ra9+aQL,9R)) * e—LL
+ 011’ * (92L J-(L,9+,9L,9R) +9sz J-(R,9+,9L,9R)) * 3+LL)2
(Do L2L*(9 J°(L 9+,9n.92) +922L J°(R 9+.9u,92))

+DLL * (92L JL(L 9+,9u,9z.) +92 LJL(R 9+.9n,9'z.)) *6 “LL
+ Di,” * (92L J-(ng-h‘Iqu) +9sz J-(Rvg-HQqu-L» * eL‘L)2
(DoL *(92L (L,9+,9n.9n)+92" J°(R.9+.9n,9n))
+ DLL * (92L JL(L 9+,9n.9n) +92 LJL(R 9+,9n.9‘n)) *8 "LL

+D:L *gmL J (L 9+,qu,qn) +97% 18,94,912, 93)) * 6+9)?
(DEL * (9LL Jo“, 9-,9L,9L)+9RL J0(R, 9-,9L,9L))

+DLL * (92L JL(L 9—,92,9L) +9L LJL(R 9—,9L.91.))* ’L‘L
+ BL." * (92L J‘(L,g_,qz.,92) +95% rung—92,92» * :L‘L)L
(03L ,. (92L (L 9-91.93) +9 LJ°<R 9-91.93»
+ DLL * (92L JL(L 9—.9L,9n) +9L JLL(R 9—,9L,9n)) *6 "LL
+ DLL * (92L J—(ng-aQL,9R) +92L J‘(R.9—,92.9n)) * eLLL)2
(03L * (92L (L,g-,qR,qL) +97: J0(R,9—,9R,9L))
+ DLL * (92L J+(L 9-,QR,9L) +93 LJ+(R 9-,QR,9L))* ELL
+ Dc—L * (92L J-(L,9—,QR,9L) +9? J-(R,9-,QR,9L)) * L’LLLL)2
(Do LLL*(9 J0(L 9—.912,9R)+9RL 10(3 9—,93,9R))

+DLL * (92L JL(L 9-41293) +9L LJL(R 9-,9n,9'n)) * 6 LL
+ DL_L * (92L J"(L,9—,9n,9n) +92L J‘(R.9-.9n,9n)) * eLL‘L)2

(D33 * (9L“ J0(L,9+,9L,9L) +972” JO(R,9+,9L,9L))

122

+ DLLL *(92” JL(L2 9+292292) +92” JL(R2 92292292» *6 L
+ DL.” * (92L JL(L29+292292) + 92” JL(R29+292292)) * 6LLL)L
(DoLL * (92L JL(L2 9+29229n) +92“ JL(R2922922922))

_.‘¢

+D2LL * (92L JL(L2 922292292) + 92" JL(R29+292292)) * 6

+ DL—LL * (928 J (L 9+29L29R) +93” J-(R29+29L29R)) *eLLLL)2

(Don * (9LLL (L2 9422932 9L) +912 LL2JLL(R 9+29R29L))

+ D2“ * (92L JL(L2 9229222 92) +92 ”JL(R2 922922292» *6 LLL

+ DL_LL * (92” J (L 9222922292) +92 ’LJL (R2 922292292» * 6LLL)L

(D LL2LL*(9 JL(L2 9+29229n)+92“ JL(R2 9229222922»

+ DL’L * (92“ JL(L2 92229222922) +92 "JL(R2 9222922292» *6 LLL

+DLL * (92” J (L 922922922) +92" J (R2 92292292» * eLLL)L

(DELL * (92" (L2 9—29L29L) +93 LLJLL(R2 9-29L29L))

+ 013* *9(L "JL(L2 9-29L29L) +922 LL2JL(R 9-29L29L)) *3 _LLL

+ DL.” * (92” JL(L29—292292) + 92” J (R 9—292292)) * 6LLL)L

(DELLLL (92R (L29-29L29R) +911“ JL(R2 9—29L29R))

+ DLLL* (92” JL(L2 9—2922922) +92 ”JL(R2 9—29229n)) *6 LL

+ DL-LL* *9(L LLJ (L2 9-29L29R) +972 LL-J (R2 9-29L29R)) *BLLLL)L

(Don * (9LLL (L29-29R2 9L) +9723 (R29—29R29L))

+ D2“ *92( LL2JL(L 9—2922292) +92 LL2JL(R 9-2922292» *6 LL
+DLL 2 (22" J (L 2,222,222) +92 ”J (R 2,222,222» eL+‘L)

(DBLL’L (97.3 (L29—29329R) +9RLL JL(R2 9-2911293»

+ D2“ *(92 LL2JL(L 9—292292) +92 LL2JL(R 9—2932922» *6 LL

220:“ (.22 “r (L g- 22,22)+22L J (92-,22.q2))2e+‘L)L]. (8.22)

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