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LIBRARY

it. itchigan State
University

«um-r? 'f“ mv'

 

 

This is to certify that the

dissertation entitled
Higgs Boson Product ion in Hadron Hadron Colliders
presented by
CHRISTOPHER JAMES GLOSSER
has been accepted towards fulfillment

of the requirements for

P1"! . D . degree in Physics

4%on

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Date 08/15/01

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6/01 c-JCIRCIDatoDuopss-nts

 

HIGGS BOSON PRODUCTION IN
HADRON-HADRON COLLIDERS

By

Christopher James Glosser

AN ABSTRACT OF 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

2001

Professor Carl R. Schmidt

ABSTRACT

HIGGS BOSON PRODUCTION IN
HADRON-HADRON COLLIDERS

By

Christopher James Glosser

The Higgs boson is the last remaining component of the Standard Model of Particle
Physics to be discovered. Within the context of the Standard Model, the Higgs boson
is responsible for the breaking of the SUL(2) x U(1) gauge symmetry and provides a
mechanism for the generation of the masses of the corresponding gauges bosons, the
Wi and the Z. In addition, this same mechanism provides masses for the leptons
and quarks via Yukawa couplings. Therefore, the discovery and subsequent study of

the Higgs boson and its properties is of the highest priority in particle physics today.

The dominant production mechanism for Higgs bosons in hadron-hadron col-
liders is via gluon-gluon fusion, in which gluons fuse via a virtual top quark to produce
a Higgs. At the Large hadron Collider, which has a center of mass of approximately
14 TeV, one expects 4000-40000 of these particles to be produced via this process
in one year of running. At leading order in Quantum Chromodynamics (QCD), the
process gg —+ H produces a Higgs with no momentum component transverse to the
beam axis. Higgs bosons of non-trivial transverse momentum may be produced if the

the Higgs recoils against one or more emitted partons.

In this work we seek to compute precisely the rate at which these Higgs par-
ticles are produced as a function of the transverse momentum and the rapidity at
the Large Hadron Collider. This calculation is carried out to next leading order in

the QCD strong coupling 0,. In particular, this calculation includes the dominant

gluon-initiated mechanism, which is responsible for the majority of the cross section.
We work in the infinite top mass approximation, and argue that this is very precise

in the domain of applicability.

We find an increase in the cross section by a factor of 1.9 - 2.0 in the interme-
diate transverse momentum range, where this theory applies. We also find that the
renormalization scale dependence is much reduced in our calculation as compared to

the leading order prediction.

I would like to dedicate this work my parents, Natalie Miller and Donn Miller, and

to my friend Kelly R. Carmichael, for their unending support and encouragement.

I would like to dedicate this work to the memory of James W. Glosser, who instilled

a love and passion for science in his son, and hence made this work possible.

iv

ACKNOWLEDGMENTS

I would like to acknowledge the following people. Carl R. Schmidt, for his useful
advice and guidance, Tim Tait for his interesting topical discussions, Tom Rockwell
and Jim Amundson for their endless patience with me in computer matters, Wu-
Ki Tung, Chien-Peng Yuan, Csaba Balazs, Pavel Nadolsky, Hung-Jian He for their

physics input, and finally, Jim Linnemann for his mentorship.

Contents

List of Tables x
List of Figures xii
1 Introduction: The Standard Model of Particle Physics 1
1.1 Yang-Mills Gauge Theory ........................ 2
1.2 The Standard Model ........................... 7
1.2.1 Spin 1 Gauge Bosons ....................... 7

1.2.2 Spin 1/2 Fermions ........................ 8

1.3 Electroweak Symmetry Breaking ..................... 10
1.3.1 The Higgs Boson ......................... 12

1.3.2 The Masses of the W and Z Bosons ............... 13

1.3.3 Fermion Masses and the CKM Matrix ............. 14

2 The Higgs Boson: Phenomenological Overview 18
2.1 Vacuum Stability and Triviality ..................... 19
2.2 Electroweak Precision Data and the Rho Parameter .......... 20

21

2.3 Phenomenology at Colliders .......................

vi

2.3.1 Width and Branching ratios ................... 21

2.3.2 Cross Sections ........................... 23

3 Perturbation Theory 26

3.1 Divergences and Regularization Schemes ................ 27

3.1.1 Divergences ............................ 27

3.1.2 Historical Regularization Schemes ................ 28

3.1.3 Dimensional Regularization ................... 28

3.2 Renormalization .............................. 31

3.2.1 Modified Minimal Subtraction .................. 32

3.3 Quantum Chromodynamics ....................... 33

3.3.1 The Naive Parton Model ..................... 33
3.3.2 Perturbative Quantum Chromodynamics: Improved Parton Dy-

namics ............................... 35

3.4 Helicity Amplitudes and Color Ordering ................ 39

4 The Higgs Boson Cross Section and Transverse Momentum Spec-
trum: Leading Order Calculation 41
4.1 The Gluon Fusion Cross Section ..................... 42
4.2 Low Energy Effective Lagrangian .................... 46
4-3 The Leading Order p i spectrum in the large mm, limit ........ 48

49

4.3.1 Kinematics and Notation .....................

4.3.2 Analytic Procedure ........................ 50

vii

4.3.3 Numerical Procedure and Results ................

5 The Higgs Boson Transverse Momentum Spectrum: Gluonic Radia-

tive Corrections

5.1 Notational Conventions ..........................

5.2 Cross Section at NLO ..........................

5.3 Angular Integration of Real Emission Piece ...............
5.3.1 99 —> H 99 .............................

5.4 Extraction of Soft Divergences and Collinear Renormalization

5.5 Virtual Contribution ...........................

5.6 Cancellation of Soft Singularities and

Renormalization of Ultraviolet Singularities ..............
5.7 Numerical Procedure and Results ....................
5.7.1 1); Spectrum and the K Factor .................
5.7.2 Rapidity Spectrum ........................

5.7.3 Small py Behavior ........................

6 Conclusions

A Matrix Elements
A.1 Helicity Amplitudes ............................
A.2 Real Contributions ............................
A.2.1 H —> 999 .............................

A22 H ——> gggg .............................

viii

52

56

58

60

61

62

68

70

71

72

73

77

78

81

83

84

A.3 Virtual ................................... 88

 

 

A.3.1 H -—> ggg ............................. 88

B Formulas for Angular Integration 90
B.1 General Expression for terms of the form W18.” ............ 90

i‘ 13” '23
B.2 Expressions for terms of the form 9,, 15.," ................ 92
‘ 13" 1213

C Results of Angular Integration 95
01 Expression for Regular Pieces of H ——) gggg Matrix element ...... 95
C.1.1 m(1‘,2+,3+,4+) ......................... 97

C.1.2 m(3",4+,1+,2+) ......................... 99

C13 m(1‘,3+,2+,4+) ......................... 101

C14 m(3‘, 2+,4+, 1+) ......................... 102

D Formulas for Cross Sections 104
D1 Expression for a 2 -—> 2 Cross Section .................. 104
D2 Evaluating the Convolutions ....................... 105
D.2.1 Evaluating Terms With “+” Functions ............. 107
References 107

ix

List of Tables

1.1 Gauge. Boson Properties ......................... 7

1.2 Fermion Properties ............................ 8

1.3 Fermion Quantum Numbers ....................... 10
x

List of Figures

2.1

2.2

2.3

4.1

4.2

4.3

4.4

4.5

5.1

5.2

5.3

5.4

Higgs Width vs Mass ........................... 21
Higgs Branching Ratios .......................... 23
Higgs production rates at the LHC ................... 25
Feynman Diagram for Higgs Production through Gluon Fusion . . . . 43
Comparison of the infinite quark mass approximation with the mmp 2174

GeV calculation. ............................. 44

Feynman diagrams for Higgs production in the large m L limit ..... 50

Comparison of the exact leading-order p 1 spectrum with the effective

theory calculation. ............................ 53
The Higgs pi spectrum at leading order ................ 54
Variation of the pi spectrum with scale ................. 74
Scale dependence at p i = 50GeV .................... 75
K factor at the LHC (x/S = 14 TeV) .................. 76
NLO Higgs p i spectrum at the LHC (\/S = 14 TeV) for various values

77

of the Higgs rapidity, y” .........................

xi

5.5

5.6

5.7

Rapidity Spectrum at the LHC (VS 2 14 TeV). The dotted lines are

the leading order calc, and the solid lines are the NLO calc. .....
Rapidity dependence of the K factor (VS 2 14 Tel") .........

Small p i dependence of the cross section, y” = 0. ...........

xii

79

80

Chapter 1

Introduction: The Standard Model

of Particle Physics

The Standard Model (SM) of Particle Physics is the most stringently tested theoretical
description of the world in which we live. It has succeeded in describing all subatomic
phenomenology seen to date, to an incredible degree of accuracy [1]. However, even
with all of the successes of the SM, a number of theoretical questions remain. Of these
the most crucial is the mechanism in which the electroweak symmetry is broken. Over
the past twenty years or so, a number of different methods have been proposed to
generate this symmetry breaking. All models using a perturbative approach to this
symmetry breaking require at least one additional scalar electroweak isospin doublet.
The phenomenological manifestation of this scalar is generically dubbed a “Higgs

boson.”

In order to understand why such a field is necessary, a pedagogical review of
the Standard Model and its gauge theory is beneficial. From this discussion, we will
come to understand the role of the Higgs field in maintaining gauge invariance, as
well as understanding some of the theoretical problems that a lone Higgs doublet

generates.

It is crucial to understand the experimental signatures of the Higgs with the
next generation of high energy experiments. As the current phenomenological under-
standing of the Higgs is plagued with large theoretical uncertainties, new calculations
are required to improve the present predictions. With the discovery of the Higgs and
a detailed theoretical and experimental description of its prOperties, we will gain our

first insight into the question of electroweak symmetry breaking.

1.1 Yang-Mills Gauge Theory

The entire dynamics of the SM is the result of a particular manifestation of Yang
Mills gauge theory [2], which was inspired by General Relativity. In Yang-Mills gauge
theory, we note that measurable quantities are prOportional to the absolute square of
a complex field, and thus the general theory should be invariant under a general phase
transformation. That is, we may demand that the fermion field \IJ(1:,,) be invariant
under the transformation

\Il(;r,,) —> exp(ia)\ll(x,,). (1.1)

Since from quantum mechanics, we know that all observables are computed from
expectation values of some differential operator, it is quite trivial to see that these
observables will remain unchanged under this phase rotation. This invariance is

known as a global gauge symmetry.

Now, we may require that the theory be invariant under a local gauge symme-
try, in which the phase parameter is a function of the space time co—ordinates at”, by
making the replacement

a —> a(:r,,). (1,2)

A typical fermion Lagrangian is of the form

LF 2: ‘Il(x,,)(i@—m)\11(x,,). (1.3)

It is quite trivial to see that the Lagrangian is not invariant under such a trans-
formation, since the derivative in the kinetic energy may act directly on the a(.r,,).

However, if we. replace the derivative with
D“ = 0,, + igA,,(:r) (1.4)
and demand that the field A" transform as
1
.4#(.IT) ——> A,,(17)+-§0,,(a(;r#)), (1.5)

we see that the latter term cancels the anomalous term resulting from the differen-
tiation of the phase. The phase parameter a(:r) is, in this case, a scalar function of

the space-time variables. This is known as an obelion gauge symmetry.

Now, we wish to know what will happen if we give the fermion field some
gauge index, i, and demand that it. transform under a local gauge symmetry

\II,(.L') —> [exp(iaa(r,,)T“)]..

1.7

where we imagine that \II,(r) transforms under some gauge group G. The matrices
T“ are the gauge group generators of some representation of the gauge group G.
This is an example what is commonly called a non-abelz’an gauge transfonnatz’on.
Again, it is trivial to see that kinetic terms in the Lagrangian will be spoiled by this

transformation. One then notes that the kinetic terms may be rescued, provided that

 

the conventional kinetic term 26,, is replaced with the so-called covariant derivative,
[Dflllj :- (2,5,1 + ZQTSA:(.L‘). (1.7)

This expression is completely analogous to equation (1.4). The gauge field 4f,(:r) is

required to transform under the gauge transformation as

a a iabr b a a 1 iafr, C
tweak <~>Ti..{ .,A,.<Jr>—E0.a.}ie WT it- (1.8)

..
I

in order to cancel the term coming from the differentiation of the phase in the kinetic:
term. After making the replacement 8 —) D, we find that the covariant fermion

Lagrangian for a non-abelian gauge theory,
Lap 2: \II(.r,,)(i$ - 7n.)\II(:r,,), (1.9)

is completely invariant under the transformation (1.6).

In order to have nontrivial dynamics, kinetic terms for the gauge fields must

be introduced. These kinetic terms must satisfy the following properties:

0 They must be gauge invariant.
O The resultant quantum theory must be renormalizable. 1

0 There must be one term which is quadratic in 0“.

All of these constraints are satisfied by the tensor F a where

pu’

a a 1
Tr“, = floppy] (1.10)

 

 

1It is possible to construct models with “higher dimension operators” which are not renormaliz-
able in the conventional sense. The divergent 100p terms are then cut off at some arbitrary scale at
Which new physics comes into play. These theories are beyond the scope of the discussion, however,
as they do not play a role in conventional Standard Model dynamics.

= T°{0,,A,‘j — 6W4: — gfabCAgAg}, (1.11)
which allows us to write down a kinetic term for the gauge fields,
1 11
LG}? 2 —ZTr[F,,,,F" ]. (1.12)

Note that a general gauge field is self-interacting because of the self-coupling term

proportional to fabc in the tensor F fly. 2

So far, everything is completely consistent, although we have been considering
only the classical limit of the theory. If we wish to consider a Quantum Field Theory,
in which the fields themselves consist of discrete numbers of excitations, we must be
more careful in order to ensure that the unitarity of the S-matrix is not violated. There
are two different procedures for quantizing a field theory, the canonical approach and
the path integral 3 approach. After naive quantization of the theory, one finds that
quantum corrections apparently spoil the unitarity of the S-matrix, for particular
gauge choices. In order to compensate, one then adds ghost fields (scalar fields that

obey fermion statistics) to cancel the non-unitary terms in the Lagrangian.

While this may seem ad-hoc, the ghost terms may be derived using an explicit
gauge fixing constraint in the path integral in what is known as F adeev-Popov gauge
fixing, in which the ghost terms are generated spontaneously when the gauge in the

path integral is fixed [4] Consider the path integral for a non-abelian field AZ,

/DA exp [iS[A]] 2 /DA exp [—E/d‘fi: ((qu)2)[ . (1.13)

This path integral infinitely overcounts the field configurations, since it sums over

 

 

2It should also be noted that a CP violating term —%Tr[€pupananV] is also consistent With the

listed conditions. Again, this is beyond the scope of the discussion .
3For an introduction to the path integral, a field theory text such as reference [3] IS useful.

 

the gauge “copies” that represent the same physics. To remove this overcounting, we
must establish a gauge choice by constraining the field A through some functional
relationship F [A] = 0. This is known as fixing the gauge. As an example, the familiar

Lorentz gauge is fixed through the constraint FM]: 8,,A“”— — 0.

To fix our choice of gauge, we begin by inserting an expression for 1 into the

path integral,

 

(1.14)

1— _ fDac5(F[4 ])Det [81130]] ,

where

1
AZ" 2 Ac" + —D,,oz"‘. (1.15)
.9

We may freely change variables in the path integral from A —> A0,. Since the inte-
gration measure DA and the action S [A] are gauge invariant, everything in the path
integral now is a function of Au and therefore we may, without loss of generality,
remove the label a. The resulting expression is

art-4)

a

 

fD-lexp is ]=(/Da) /D4exp 25(4110 (F [4])Det[ ], (1.16)

where the term f Da contributes an infinite overall constant to the normalization.

The factor Det [8F [A] / 80] can be evaluated by introducing the ghost field. It
is instructive to work this out explicitly for the Lorentz gauge. Since the gauge field

. . . . 0 ~ I ,a ' r r . . '
vanes mfimtemmally as (Aa)f, — (A )2 ~ 5D,,a , the determlnant may be revr ritten,

196495.421] = net [1340,] (1.17)
9

8a
: [Dc/DE exp [i/d4175(—8”Du)c], (1.18)

where the c is an anti-commuting scalar field4, and therefore a ghost. We have thus

 

 

“Anticommuting fields become Grassman variables in the path integral representation.

 

 

 

Boson Symbol Mass(GeV) Charge Color Isospin

gluon g 0 0 Ad joint No
photon 7 0 0 N o N o
W ”"4, 80.41 :l:e N 0 Yes
Z Z 91.187 0 No Yes

Table 1.1: Gauge Bosons

generated the Fadeev-Popov ghost term as a result of rigorously fixing the gauge.
The c fields are indeed ghosts, and have no asymptotic states. They do, however,
contribute as closed loops in diagrams. Moreover, the infinite gauge repetitions have

been systematically eliminated in the path integral, and the gauge has been fixed.

1.2 The Standard Model

The Standard Model is, in essence, a Yang-Mills gauge theory, consisting of three
“copies” of fermions, called flavors, interacting with a set of gauge bosons dictated by
the direct product of groups G :2 SUC(3) x SU1(2) x Uy(1). As was seen before, for
each of these symmetries we may associate a dynamical gauge field. Therefore, there
is a gluon field G; that interacts with the SU(3) “color” quantum number, A Weak
Boson field W7, that interacts with the isospin quantum number (I), and an abelian

field B“ that interacts with the Hypercharge (Y).

1.2.1 Spin 1 Gauge Bosons

Since the Electroweak symmetry is broken, we will observe admixtures of its fields,
W; and B“, in nature. The 4 fields combine into two electric charge eigenstates, Wi,
a massive chargeless boson, the Z, and a massless chargeless boson, the photon 7. A

list of the general properties of these particles may be found in Table 1.1

 

Fermion Symbol Mass(GeV) Charge Color

electron e .000511 -—1e Singlet
muon a .106 —- 1e Singlet
tau 7 1.78 ——1e Singlet
electron neutrino 12,, < 15 x 10‘9 0 Singlet
muon neutrino 14, < .00017 0 Singlet
tau neutrino VT < .0182 0 Singlet
up u .001 to .005 Se Triplet
charm c 1.15 to 1.35 ge Triplet
top t 175 3e Triplet
down (1 .003 to .009 —%e Triplet
strange s .075 to .170 —§e Triplet
bottom b 4.0 tp 4.4 —§e Triplet

Table 1.2: Standard Model Fermions

1.2.2 Spin 1 / 2 Fermions

There are three different generations of fermions. The first generation consists of
the electron, the electron neutrino, the up quark and the down quark. The second
consists of the muon, the muon neutrino, the charm quark, and the strange quark.
The third generation consists of the tau, the tan neutrino, the top quark and the

bottom quark.

The particles of different generations may be grouped into subsets of particles
which are indistinguishable, save their masses and Yukawa couplings (with the Higgs).
These are known as families. For instance, an electron, muon and tau have exactly
the same isospin and charge characteristics, and hence belong to the same family.

Table 1.2 organizes the various known fermions by their family5.

Since the left and right handed components couple differently to the elec-

troweak sector of the theory, it is beneficial to list the separate components and their

 

5The masses in Table 2 were obtained from reference [1]. The quark masses are the so-called
“current-quark masses.”

transformation properties in a separate table. The lepton doublets contain a left
handed, -1 charge piece, which is “isospin down” (this refers to the eigenvalues of
the 03 matrix), and a charge zero component that has “isospin up”, to be identified
with the e,;1,r (former) and their respective neutrinos (latter). We then arrange the

components into isospin doublets,
LL : , (1.19)

with identical expressions for the remaining two families. The Hypercharge (Y) may
be computed directly from the charges and the isospin eigenvalue, using the for-
mula Q = I + )2: (the charge here is in units of e). We find that the left-handed
e,p,r,z/e,1/,,,and VT have a hypercharge of -1, while the right-handed e,/1,and T have a

hypercharge of -2.

We also arrange the left-handed quarks into an isospin doublet

/ 11L

dz.
The left-handed quarks transform just as the left-handed leptons do under SU1(2),
and have a hypercharge Y = g. This, of course, implies that both components have
nontrivial interactions with the electromagnetic field. The right-hand components

of the up and down type components have hypercharges of g and —§ respectively.

Again, these may be computed directly from the charge formula, Q = I + )5.

All quark species transform as triplets in the fundamental representation of the
SUC(3) color force, while all leptons transform as singlets, and thus do not interact

strongly. That is, each of the quark components may be written as a three component

Fermion Q I Y Color

L L (-1,0) Doublet —1 Singlet
Q1, ( +2 / 3,-1 / 3) Doublet 1 / 3 Triplet
113 +2 / 3 Singlet 4 / 3 Triplet
d3 -1 / 3 Singlet —2 / 3 Triplet
l R -1 Singlet -2 Singlet

Table 1.3: Fermion Quantum Numbers

spinor,

Qr
Q: Qg , (1.21)

Qb

where Q may be any of the quark fields, Q), a, or d,.. The leptons do not couple to
the color sector and hence are all singlets under SUC(3). The quantum numbers of

the standard model particles are conveniently summarized in Table 1.3.

One may notice that, in dividing terms up into left and right-handed states,
and by demanding that these states transform differently under the various gauge
groups, that terms which mix the left and right handed terms are expressedly forbid-
den. Since mass terms do exactly this, they are forbidden in (Unbroken) Electroweak
gauge theory. Thus we must break the Electroweak gauge symmetry somehow in

order to have a realistic model of the Standard Model Particles.

1.3 Electroweak Symmetry Breaking

The most glaring problem with the current particle physics phenomenology is the
existence of massive particles. Electroweak gauge invariance forbids hard mass terms

of the form #282 which is obviously not invariant under a gauge transformation

3

B —> B + 8A. Also, the left and right-handed fermions, which transform differently

10

under S U](2) and Uy(1), seem to preclude the existence of any mass terms for the
fermions, since mass terms mix the right and left-handed components, once again

spoiling gauge invariance.

We wish to introduce these terms in a gauge invariant way. The first clue
is that we must somehow include three additional degrees of freedom, one for each
additional spin state of the three massive bosons. This seems to indicate that another

field is necessary to compensate for the additional degrees of freedom.

The general way to break the electroweak symmetry is through spontaneous
symmetry breaking. To spontaneously break a symmetry, one introduces a field with
a vacuum state which is not invariant under the symmetry in question. One then ex—
pands the field around this vacuum. Since the kinetic energy terms in the Lagrangian
of this field will, in general, contain covariant derivatives (which in turn contain cou-
plings to the fields W,Z), expanding the field around its vacuum expectation value

(VEV), will introduce mass terms for the gauge bosons.

If we also allow the field to interact with fermions via a fermion-antifermion-
scalar interaction known as a Yukawa coupling, we find that we have also sponta—
neously generated masses for the fermions. The mass of this field which we have
introduced can be arbitrary (at least at tree level) in the SM. We therefore have a

theoretical answer to all inconsistencies in the Standard Model.

A scalar field which has these properties is generically dubbed a Higgs field.
In the SM, the Higgs is a complex isospin doublet, so that it transforms under both
the hypercharge and isospin rotations. Since this introduces four degrees of freedom,
there is one left over degree of freedom after the Electroweak symmetry is broken.
The phenomenological consequence of this is the existence of a massive, chargeless

scalar called the Higgs Boson.

One final caveat concerns the quark masses. The quark masses are not as well

11

defined as those of the other fermions, and so they must be treated with some care.
Masses in a quantum field theory are basically due to the self coupling of the field.
These, like other couplings in the theory are affected by the renormalization process.
Since most masses in the SM are measurable parameters (that is, not predefined by
the theory), particle masses are generally empirically defined as being the masses in
the absence of other forces. The problem is of course that the quarks are confined
inside hadrons and mesons, and thus are never “free particles” in the classic sense.
For a quark, this limit never exists, so the idea of a “rest mass” is poorly defined.
Quarks that have large Yukawa couplings to the Higgs may still have a rest mass

defined, as long as it is larger than the strong coupling scale, AQCD.

1.3.1 The Higgs Boson

To begin our discussion of the Higgs field, we write down the conventional Lagrangian

for a standard model Higgs boson. It is

L” = (D,,<I>)l(D"<I>) —- A(]<I>|2 -— 32—)”, (1.22)

where (I) is complex scalar isospin doublet, and thus has hypercharge +1. The covari-
ant derivative in this case is
282

i 7i 7g (1
[Dulv = 3,5,,- + 701.].i/i,#(;1:)+ 31608,,(1‘), (1.23)

where we include both the isospin field ll"; as well as the hypercharge field By. Now,

we parameterize the isoscalar doublet as

1 ¢ +2452
(1) = 7 1 ; (1.24)
2 U+¢3+i¢>4

12

 

or, writing this as a gauge rotation,

(I 0
(I): . —'—'6,.: . 12’
exp< 2., (2)) v”, < .)
V5

The dynamical field H (.13) is the Higgs boson field. We see that for a particular
gauge choice (commonly called the unitary gauge), we may eliminate three degrees
of freedom. These degrees of freedom will show up as the necessary third degree of
freedom for each of the gauge fields W;, B,“ leading to the rather quaint analogy that
the gauge fields have “eaten” three of the components of the Higgs doublet, leaving

the Higgs boson field as an undiscovered dynamical field.

1.3.2 The Masses of the W and Z Bosons

As a consequence of choosing the field (I) to have a nontrivial vacuum, numerous
phenomenological characteristics of the SM emerge from the various couplings of the
weak gauge fields with the Higgs field. Moreover, gauge invariant couplings with
the quark and lepton fields generate mass terms for these particles as well, without
breaking the overall gauge invariance of the theory. We begin by writing the most
general parameterization for the model. The mass terms for the gauge bosons arise
from the following term in the Lagrangian,

1 . - i ri a
1‘1“ (920:11’I'ftxlE) + 915ijBZ(JI7)) (.QQUjkW,1(-rl+ gléjkBp($)) (I). (126)

If we redefine the fields as

71 ' ,,/'2
ll” i 211,,

W1. = ————. 1.2
14 fl ( 7)
90211 = 92W: “' 918117 (1.28)

13

90A” = 9114",? + 928... (1-29)

where ya 2 Mg? + 9%, we find that the following mass term arises:
M 2 WWV” 1M.2 Z 2 1 30
W’ ( p. p ) + i Zi Iii ' ( ' )

whereas the field A,, massless. This expression now contains explicit formulas for the

gauge boson masses

(”13V '2 92(71’), (1-31)
2 2 v? 2 2 v2
JMZ = .QO(—4‘):(91+92)(‘4—)1 (132)

Hence, we identify the fields [if and Z ,, as massive phenomenological gauge particles;
whereas the field A” maintains a local U (1) symmetry and is identified with the
photon field. It is straightforward to derive the expression for the electromagnetic
charge e = 97:33 by rewriting the covariant derivative in terms of the field A,,. It is

also customary to define the Weinberg angle, sin 0W E gl/go.

1.3.3 Fermion Masses and the CKM Matrix

One may also postulate interactions between the Higgs field and the fermions of the

standard model. We may write this Yukawa interaction with the fermions:

LYukawa : (ygméfquiLT+(CHC))
+ (ysmfl’teiQi‘HC-CJ)

+ (ynggong‘ + (0.0)), (1.33)

14

where n and m run over the fermion families. The field 4)" is related to the complex
conjugate of the Higgs field by
(i : —za,,<1>*. (1.34)

This expression is the most general term that we may write. Note that there is no
requirement that the Yukawa matrix yum be diagonal or real. It will, in general, mix
electroweak eigenstates of the fermions. This has the implication that CP symmetry

is violated in a general theory with a Higgs.

First, let’s consider the leptons. Since the neutrinos are considered to be
massless, and have only a left hand component, we may freely redefine the fields
such that the mass matrix is diagonal. The three diagonal coefficients of the mass
matrices are then interpreted as the masses of the leptons. The diagonal Yukawa
matrix elements are then the masses of the three leptons with the factor v/V2 scaled
out; that is, m, 2: ygv / V2 Note that this also implies that the Higgs boson couples
to leptons with strength y, :2 V2m1/v; that is, the coupling is directly proportional
to the lepton mass. This is true of the Higgs coupling to all particles, and is of great

importance in phenomenology, as we shall see in the next section.

The up and down mass matrices, due to their additional complexity, may not
be simultaneously diagonalized. It is conventional to choose a basis in which the up
type quarks are diagonal, and the down type quarks are off—diagonal. One then finds
that since the ”7* terms mix quark mass eigenstates, the W Boson violates “quark

family number.”

Let’s consider this situation in a little more detail. Since in the Standard
Model the only coupling which actually mixes isospin-up and isospin-down fermions
is the coupling to wit, we find family mixing terms in the couplings between the

fermions and the W boson terms. That is, the W couples to “weak eigenstates”

15

which differ from mass eigenstates by the rotation matrix, V:

d vial ‘rus "rub d
S : I’fcd I'cs [Cb 8 ' (135)

, Ytd Vts Ytb b
W eak M ass

This is the so called Cabbibo—Kobayashi-Maskawa (CKM) matrix [5] This matrix

has the standard parameterization

612613 312013 813613
(1.36)

“312023 — 012023813813 612023 —' 812823513813 323013 1

312823 — 612023643613 ”012323 - 812023813813 023013

where Ci]- : cos (Qij),s,j = sin (6,,), and Cij = exp(—i6,;j). In the Standard Model,
these angles are not constrained by any symmetry, and must be determined experi-
mentally. It may not be possible to make these same arguments in extended models,
and one may consider searching for deviations in the CKM matrix as an indication

of physics beyond the SM. See [6] for an excellent discussion of these indicators.

If the neutrino sector of the theory contains a Dirac mass term, we must forego
the previous analysis, as one cannot simultaneously diagonalize the electron and neu-
trino mass matrices unless at least one of the two is completely massless. Recent
measurements of neutrino oscillations [7] do seem to suggest that the neutrino species
have a mass difference (and therefore, masses), which of course must be accounted for
theoretically. One may postulate the existence of an additional right-handed compo—
nent, which will in general have its own coupling to the Higgs. After diagonalizing
the charged leptons, the neutrino components are left with off-diagonal terms. These
terms, while extremely important to astrophysics, do not contribute much to con-

ventional high energy physics collider experiments, where the approximation that the

16

 

 

 

1

neutrino is massless is certainly viable.

17

Chapter 2

The Higgs Boson:

Phenomenological Overview

The Higgs boson is currently the only missing component of the Standard Model.
Precise knowledge of Higgs physics will most likely give the first glimpses of the un-
derlying structure responsible for electroweak symmetry breaking. Since the couplings
of the Higgs to the various particles in the Standard Model are already determined,
any deviation from the rates computed using these couplings could be an indication

of new physics.

However, in order to examine these rates, we must first find a Higgs and mea-
sure its mass. The Higgs mass is not fixed by any theoretical argument in the Stan-
dard Model. In fact, a tree-level analysis quickly reveals that there are no constraints

whatsoever on the Higgs mass.

At the l-loop and higher orders, however, constraints do arise on the Higgs
mass. A renormalization group analysis reveals that the Higgs mass cannot be too
large or too small relative to the Weak scale. The former results in triviality, while

the latter in vacuum instability. The Higgs mass also appears in 1-loop corrections

18

to eé ——> Z —-> f f , and so by precisely measuring the cross section to this process,
we may constrain the Higgs mass indirectly, as was done for the top quark before its

discovery in 1995 [8, 9].

2.1 Vacuum Stability and Triviality

We begin our analysis of Higgs phenomenology by considering some theoretical issues
concerning the mass of the Higgs. As mentioned before, the classical theory of the
Higgs and its couplings to the other Standard Model particles does not in any way
restrict the Higgs mass. However, radiative corrections cause the VEV and the quartic
coupling to run, and can cause problems if the Higgs mass is too far removed from

the other electroweak parameters.

One-loop corrections to the Higgs effective potential lead to vacuum stability
constraints. These result from the fact that the asymmetric electroweak-violating
vacuum could decay into a symmetric one, therefore destroying the universe as we
know it. Typically, these analyses put lower bounds on the Higgs mass which are far

less then the current mass bound from LEP.

Triviality arises from the fact that the quartic coupling A in a4 theory exhibits
a Landau pole [10], and therefore the theory becomes strongly coupled at some high

energy scale. To see this, consider the expression for A running from some energy

M11) 2 A0 (1+ 3&4? log (EE))— . (2.1)

Since A0 is related to the Higgs mass through the VEV (i.e. mi, 2 Av2), the re-

scale 110 to u,

quirement that the coupling remain finite below some relevant scale puts an upper
constraint on the Higgs mass. If the theory is to remain perturbative up to the Planck

Scale, then the Higgs mass cannot be more than about 1 TeV.

19

Since this behavior is essentially nonperturbative, such calculations are best
done on the lattice. 1 A recent lattice study [11] has placed an upper bound on the

Higgs mass of about 460 GeV, if no new physics arises before then 2.

2.2 Electroweak Precision Data and the Rho Pa-

rameter

The masses, decay widths and cross sections for W and Z also receive corrections

from virtual Higgs 100ps. The result is that the p parameter,

2

mW
: a 2.2
p m2z cos2(6lw) ( )

 

which is 1 at tree level, picks up the following corrections from virtual t0p and Higgs

100ps:
3 62 m? 3 g2

647r'2 sin2 6w 772%,, 327r'3

ta112(6w) log (77-353) + . . .. (2.3)

 

 

p=1+

All of the fermion loop corrections to p vary quadratically with their masses , so

it is easy to see that the top gives the dominant contribution. Therefore, a precise

knowledge of 9w mw, mg, and mt can be used to constrain the Higgs mass.

The major problem with indirectly measuring the Higgs mass is that the cor-
rections to the p parameter vary only logarithmically with M H, rather than quadrat-
ically, as do the top corrections. It is hence very difficult to have enough statistics

and eliminate enough systematic uncertainty to put a precise constraint on the mass.

As of this writing, the upper limit from the electroweak precision data is 224.8

 

 

lLattice Gauge Theory is an attempt to formulate quantum field theories numerically by placing

them on a discrete space-time lattice, and evaluating the path integral numerically.
2Reference [10] is an excellent introduction to these theoretical issues, including the theoretical

issues surrounding the Higgs phenomenon in extended models.

20

 

100

10

1
S‘
0
S2
L—

0.1

0.01

0.001

 

 

 

 

 

’ I— fi fi

L
1
l

L 1

T '1
J

r 1
1

l

r 1

L

_L _L 41
1 00 200 300 400 500
m, (GeV)

Figure 2.1: Higgs Width vs Mass

GeV. The lower bound on the Higgs mass is most strongly constrained by the direct

search at LEP[12]. They find a lower bound of 114.1 GeV at 95% confidence level.

2.3 Phenomenology at Colliders

In order to search for the Higgs in a collider experiment, we must have a precise

understanding of how it is produced, and how it decays. This is the main topic of

this section.

2.3.1

Width and Branching ratios

A graph of the Higgs width as a function of mass is provided in Figure 2.1 3. As

can be plainly seen, the width increases dramatically around 160 GeV. This dramatic

 

 

3This graph and the branching ratio graph (figure 2.2) were generated using the program HDE-

CAY [131

21

increase in the width is due to the fact that the Higgs is now massive enough to decay
into weak bosons, with which it has a comparatively strong coupling. Higgs particles
that can decay into weak bosons have a collider phenomenology dramatically different
than those which cannot decay to weak bosons. The former are referred to in this

work as heavy Higgs, and the latter intermediate mass Higgs.

Looking at the branching ratios for a Higgs in the intermediate mass range
(Figure 2.2) , we see why its detection is so problematic. As noted in the last section,
the Higgs couples to particles with a coupling strength that is proportional to the
particle mass. Therefore, it prefers to decay to the heaviest particles for which it is
kinematically allowed. For the intermediate mass Higgs, it decays most of the time
into bottom quarks, which are swamped by QCD background events. If one wishes
to look for the Higgs through the bb channel, one must be able to efficiently tag
bottom quarks, via vertex detection. It is not clear that bottom tagging will have the

resolution necessary to see Higgs events.

About 18% of the time, the Higgs will decay into taus, charms or gluons. The
decay to tans is also a very promising mode, since taus usually decay into electroweak
particles. Observations of Higgs decays to charm quarks is basically out of the ques-
tion, since the ability of detectors to tag charm quarks is extremely unreliable. Gluons

are out of the question as well, since they may not be tagged at all.

The mode H —> 77, which arises through loop effects, is quite rare, with a
branching ratio of 10*. This, in particular, is quite unfortunate, since this mode
offers the best chance for detection in a hadron-hadron collider [14, 15]. This process
has been called the “golden mode”, since the energy resolution of these events in a
particle detector is relatively fine and its background is fairly small. One can hope to
measure the rate to 7’5 and look for the characteristic enhancement in the transverse

momentum spectrum, provided that one has a precise prediction of the spectrum of

22

 

 

 

 

11. WW -1
[ 227
1 .31
.' _ it
0.1 F7771: gg El
. . C -' e ’ i
Branching Ratlo l- , ' . -
0.01 ;- 4
c . i
g 7.7.271 3° 1

100 200 300 400 500

m), (GeV)

Figure 2.2: Higgs Branching Ratios

background events.

The situation is better for a heavy Higgs, which decays predominately to W’s,
Z ’s or t0p quarks. While the production rate for a Higgs in this mass range is just
too small to measure at the Tevatron, these rates can presumably be measured at the

LHC, provided that the Higgs mass is within this range.

2.3.2 Cross Sections

Overall, the best way to search for a Higgs is with a lepton collider, such as LEP, since
one has a precise knowledge of the initial state energies of the interacting particles.
At these machines, one begins to expect Higgs particles around an energy of ECOM =
Mz + M H, with the Higgs radiating from the final state Z boson. The decay into
jets or leptons is relatively clean, allowing the invariant mass of such events to be
reconstructed. If the Higgs mass were within the accessible range of LEP, then it

most surely would have been discovered. This allows a lower bound to be placed

23

 

 

on the Higgs mass, which is currently 114.1 GeV. Recently, a recent analysis by the
ALEPH collaboration revealed an excess in events at LEP at around 206 GeV [12].
These events, while still preliminary, are very suggestive of a Higgs with M H z 115

GeV.

Producing a Higgs boson at a hadron—hadron collider is generally not difficult.
In fact, even at the Tevatron (FNAL), there are most likely dozens of Higgs produced
each year that the laboratory takes data (depending on the Higgs mass). The real
problem is detection; if the mass is less than 135 GeV, the Higgs decays predominately
into bb pairs. Since these particles generally show up in the detector as jets, this
process is swamped by QCD background events. T herfore, for Higgs bosons below
135 GeV, the most likely channel for discovery at the Tevatron is the associated
production channels pp —+ HZ and pp —+ H W. For these channels, tagging of b-
quarks is crucial. For my > 135 GeV, the W W" decay mode (where W" represents
an off-shell W boson) becomes available. The actual reach at the Tevatron at Run II
depends on the total integrated luminosity, of course. A recent study [16] found that
for 20 fb—l, the Tevatron can exclude a Higgs at 95% confidence level up to 180 GeV,

and could discover it at the 30 level if the mass is below 125 GeV.

If the Higgs boson is not discovered at the Tevatron, then the Large Hadron
Collider (LHC) at CERN, which is due to start runs around 2006, will find the Higgs
particle. The main advantage of the LHC is that there will simply be more events,
due to the extremely large luminosity of this machine, as well as the enhanced cross
section. This allows for a much better chance for detection, especially in the H —+ 77
mode. We will therefore focus our discussion on this device, where the discovery of

the Higgs will most likely happen.

There are a number of ways that a Higgs boson can be produced at a hadron

collider. One notable mechanism is the production rate through the gluon fusion

24

 

 

  
    

 

 

 

P Yj T 'j Th WI T fiat—I'M V—‘ T I j ftW I j r I r [M 'j "rT fl fl—r fj— ‘—] '—V T

, L. o(pp—>H+X) [pb] -

1° 1 \/s = 14 TeV 1

t , M, = 175 GeV 1

\
10 r 1,1.“ CTEQ4M
E .:'-.,\\\‘~"‘- _
[- '~ '\\\\ ...........

I E‘ \ C \ .............. :

E ~ 3 \ ________________ .

-1 E .\ \s \ _ ...... z

10 E- .,‘;,-.,::::$ \ \\ \qq —>HW 1.

E \. \ ‘\ \ x g

-2 [: ‘\:‘ ‘ ‘ -4

10 r ' " ---4--‘..:..,~ - i

: ...... \ MW."~-_"'§ a

-3 _ ......... ‘ ‘~ ‘ ‘wg~;'..;.; .......... gge 61‘9““ 1

10 F ............... ‘ ~ :~ ‘-:..;..: WT:

- _ a .......... ‘~~ ~a

E gg»qq—>Hb6 .............. (H142 ‘ ~ -.

-4 1' "Q ........ i

10 1 L 1 11;; LA I n .4 l L: l 1 1.114 A A I 1 A IJLILL L411 ALL LLLLL”J.AL LA 1

0 200 400 600 800 1000

MH [GeV]
Figure 2.3: Higgs production rates at the LHC

process, gg —+ H, which dominates the overall production rate. Other ways to pro-
duce a Higgs generally fall into the category of radiation from electroweak bosons,
production through electroweak boson fusion, and radiation from heavy quarks. The

expected values of these cross sections at the LHC are given in Figure 2.3 4.

Clearly, the process gg —> H dominates the total production rate at the LHC.
Therefore, we will concentrate our discussion on this particular production mecha-
nism. This mechanism is not only the most dominant production mechanism, but
is the least understood in a quantitative sense. In particular, we will show that the
QCD corrections to the overall cross section as well as the differential cross section

are sizeable.

 

4We thank Michael Spira for the generous use of this plot, as well as the program HDECAY
[14, 13].

25

Chapter 3

Perturbation Theory

If one wishes to search for a Higgs particle in a high energy collision, one needs to
have an accurate prediction of its behavior in such an event. For instance, we would
like to know in what kinematic regions we should should search for decay products.
In order to do so, we are going to have to calculate observables using the theoretical

apparatus of Quantum Field Theory.

Integrations over momentum in such theories are plagued with divergences of
several types. We will therefore have to come up with a consistent way to render

these integrals finite, so that the divergences may be dealt with in a rigorous manner.

Once the divergences are regularized, we need a systematic procedure for elim-
inating them from our predictions. Fortunately, in a well defined theory, these diver-
gences will either cancel amongst themselves or we will be able to factorize them into
universal parameters of the theory in a process known as renormalization. These
procedures, if applied rigorously and correctly, will allow us to compute precise,

divergence-free expressions for observables in particle accelerators.

26

3.1 Divergences and Regularization Schemes

3.1. 1 Divergences

When one calculates an amplitude in QCD or any other field theory, one typically gets
expressions which are poorly behaved in the large energy region of phase space. These
are known as ultraviolet (UV) divergences, since they correspond to a divergence in
the number of high energy (frequency) fluctuations. An example of an integral which
contains these types of divergences is

A2 2 2 _ 2
lim 1(1)— : lim log (Lil). (3.1)

Awaoo 0 p2 — 1712 A—+oo

Clearly this expression diverges in the limit that the regularization parameter A goes

to infinity.

In addition, if the mass of the particles is small, two additional types of diver-
gences appear, which are collectively called infrared (IR) singularities. First, there is
a proliferation in the number of particles generated at low energy. This is known as
a soft singularity, called such since it causes little recoil when emitted from a more
energetic particle. In a renormalizable quantum theory such as QCD, these infrared
singularities will cancel in the calculation of any properly defined “infrared safe” ob—
servable quantity. We will return to the question of what it means to be an “infrared

safe” observable later.

The other type of infrared singularity that one may have is called a collinear or
mass singularity, which occurs, for instance, in QCD. A collinear singularity occurs
when the momenta of two massless particles line up. To be more precise, as the
invariant mass of the two distinct particles becomes zero, the resultant jets that are

produced when the particles hadronize become indistinguishable from one another.

27

Therefore, one does not really know whether a jet of hadrons came from one particle
or two or more that were collinear. We will discuss how to handle such divergences

in the section on perturbative QCD.

3.1.2 Historical Regularization Schemes

In this subsection, we shall discuss two regularization schemes that were used histori-
cally to quantify the ultraviolet and infrared divergences . We will see that they both

lead to consistency problems.

In the fictitious mass scheme, one introduces a fictitious mass term into the
integration to regularize the soft and collinear divergences. The problem with this is
that since mass terms violate gauge invariance, gauge anomalies will in general arise
in calculations, and their removal is quite subtle. For this reason, this fictitious mass

scheme is rarely used in modern calculations.

The UV divergences were historically handled using the Pauli— Villars regular-
ization scheme (PV). PV regularizes the UV divergences through the introduction of
a massive scalar ghost field, which converts the problem terms into logs of the PV
mass term Apv. The ghost fields must be introduced in a gauge-invariant manner,

which can become quite complicated for all but the simplest theories.

3.1.3 Dimensional Regularization

Almost all current calculations use dimensional regularization (DR) to regularize the
divergences. Using dimensional regularization has a number of advantages. Since no
extraneous fields are introduced, unitarity is manifest, and the gauge invariance of

the theory is preserved.

Let us explore the basic ideas of dimensional regularization. In a divergent

28

field theory, we expect to have terms arise in calculating observables that are poorly
behaved in the large (UV) or small (IR) energy limit. Equation 3.1 contains examples
of both of these types of divergences First, it diverges as A ——> 00. This is the
aforementioned ultraviolet divergence. Second, the integral also contains an infrared

divergence, as m ——> 0.

The main idea of dimensional regularization is that divergent integrals of this
type may be rendered finite by analytically continuing the number of space-time
dimensions to 4 — 26 dimensions. The d(p2) terms which come from the radial piece
of the phase space in 4 dimensions, are replaced with the analogous piece in 4 — 25
dimensions:

2 . (1042)

) ' <-— 6 .— -c ’ 6
d4p .=_ Zz—d(p2)dF —> d1 2 p 2: (pi)1 —2—dF . (3.2)

Where dl‘6 is the 4 — 2e dimensional angular measure. The net result of our efforts
is that the divergent p2 integral, which arises from evaluating the phase space in 4

dimensions, picks up an additional factor 114‘:

00 1 .2 2 “
f f (1’ l B. , (3.3)
o p2 — nr2 112

which is well defined for e > 0. The [1 term, for now, maintains the dimensionality of

the original expression.

We may also regularize the IR divergence in a similar manner. Eliminating
the mass term in equation (3.3), and noting that we must carefully separate the IR

divergent and the UV divergent regions of phase Space, we obtain

1: 4:2)“ . 1:: entire)“

1 m2 -61R 1 m2 ‘fuv
—61R (F) _ —€UV (E) , (34)

 

 

 

 

ll

 

29

where we have defined Cuv > 0 and em < 0 to represent the ultraviolet and infrared
poles, respectively. In QCD, we also find that poles arise from the angular integration
piece, where the internal partons become collinear. These poles in 6 also contribute

infrared poles to the matrix element.

In dimensional regularization, then, we find that we have traded a poorly
behaved log for a pole of the form, —%. These poles must either cancel amongst
themselves (which is the case for the infrared poles), or they must cancel against
renormalization counterterms (as is the case for ultraviolet or collinear poles), which
we will discuss shortly. If everything works as it should, we should be able to analyt-
ically continue the complete expression, including counterterms, from c = $00 ——> 0
without hitting any poles. That is, all poles of the form (~e -—- n)”, where n is a

integer must cancel 1. We may therefore set 6 ——> 0 with good conscience.

Now that we have examined the basic ideas of dimensional regularization,
let us examine the finer details. Immediately, we see an ambiguity in this scheme.
It is not immediately obvious how to handle quantities that are only well-defined
in integer dimensions, like the number of spins, nor is it obvious what to do with
antisymmetric tensors like 75 (which depends crucially on the integer nature of Nd,,,,).
The most common prescription is to continue the number of spins of bosons by setting

Nspin = 2(1 — e), and by demanding that:
0 75 anticommutes with 7°, 71, 72, 73, and
O commutes with all other 7’s.

This convention was developed by t’Hooft and Veltman [17], and is the standard way

of handling this problem. Gauge invariance is manifest in this scheme, and there

 

 

ln/2 is the degree of divergence of the theory, that is, a quadratically divergent theory will
typically have terms like (6 -— I)”. All of the interactions of the Standard Model save that of the
Higgs are logarithmically divergent, so all of these superficially higher order divergences must cancel
amongst themselves.

30

 

 

are precisely two chirality states (defined in terms of projection Operators) for the
fermions. Thus, this scheme avoids any anomalous degrees of freedom which could
arise with other “generalizations” of 75. By setting the number of gluon helicities
equal to Nspin = 2(1 — c), we avoid having to separate out the gW tensors that arise
from the internal spin sums. This way of handling the helicities, where the gauge
boson spins are manifestly continued to 2 — 2e and the number of fermion helicities

is fixed to be 2, is known as Conventional Dimensional Regularization.

3.2 Renormalization

Having regularized the divergent pieces, we now seek to eliminate them from our
predictions in a manner which is consistent with our theory. We will take advantage
of the freedom that our theory provides to absorb these divergent terms into the
definitions of the wave function normalization, the definition of the charge, and the
definition of the mass. That is, we have the freedom to fix our normalizations to

whatever we wish. We may, for example, make the replacements

(1‘) —> 2W,
e —> Zee, (3.5)

711,1, —> meg.

These reparameterizations will allow us to absorb the ultraviolet divergences into
these definitions. Provided that we do this consistently, we arrive at a well-defined,
finite prediction for our observable. The gauge invariance of the theory will in general
impose relations between these normalization constants, but in general, we still have

Significant room to absorb the ultraviolet divergences.

31

 

 

f ’ -_~__.- 4

3.2.1 Modified Minimal Subtraction

Even with the gauge restrictions, we find that we have significant freedom to repa-
rameterize the theory. We must therefore set down conventions for the inclusion of
the UV counterterms. These terms, which arise because of a redefinition of the nor—
malization constants in the field theory, are somewhat ambiguous. This ambiguity
arises since renormalization group analysis only defines how one modifies the charge
to O[6“]. We wish to remove the poles in 6, but we also have the freedom to add any

finite piece to the renormalized charge that we wish. Consider

1
(0) (I): <0) 1_933 (_ 0))
g 49 g ( 47,0 E+016] , (3.6)

where the coefficient [30 is the first renormalization group coefficient. The details of
this coefficient depends on the symmetry of the group, and the number of fermions
interacting with the gauge bosons. O[e°] represents terms that are unspecified by any

gauge or other restrictions.

In modified minimal subtraction (MS), we keep finite terms which tend to arise
from the finite phase space integration in the dimensional regularization procedure
outlined above. Instead of subtracting just the e pole, we subtract off the piece

1 (4n)‘ 0: ,
(0) (1) _ (0) _ 5 3
g —> g — g (1 6 _—r(1 6,717,111) (3.7)

in MS. Again, this is a just a convention which is used to remove the extraneous
universal pieces that tend to complicate calculations in dimensional regularization.
There are a number of other ways of expressing the ITS counterterm, which are
technically different from this one at 0[e]. However, since these O[e] terms will

eventually be set to zero in a next-to-leading order calculation, the same result is

obtained.

32

 

In principle, the subtraction scheme should not matter when one is calculating
an observable. In practice, however, different subtraction schemes can cause signifi-
cant variations in the final result at a fixed order in perturbation theory. One must
also consistently use the same subtraction scheme to calculate the parton distribution
functions from the raw deep inelastic scattering (DIS) data [18], as well as to calculate
the UV counterterm from the charge renormalization. If one wishes to do a calcu—
lation in a different scheme, one must either know how to convert the result in that
scheme back into MS, or one must be able to show that the scheme produces the same
results at that order of perturbation theory. Since it is fairly involved to do either
of these things, most calculations are done using the modified minimal subtraction

scheme.

3.3 Quantum Chromodynamics

Quantum Chromodynamics is the theory of the strong interaction at high energies.
This model introduced dynamics to the quarks of the Cell-Mann SU(3)flavor model,
and preposed a carrier of the strong nuclear force, the gluon. We shall begin by
discussing the Naive Parton Model, which explains the behavior of the hadrons in a
qualitative sense, and then move on to perturbative QCD, which allows one to compute

much more precise expressions for collider observable quantities.

3.3.1 The Naive Parton Model

We shall begin the discussion of Quantum Chromodynamics (QCD) by introducing
the parton model, due to Bjorken and Feynman. The parton model was introduced
in an effort to explain the deep inelastic scattering data, which seemed to indicate

that the proton had a definite substructure. As originally proposed, the parton model

33

suggested that hadrons consisted of massless point particles which could then interact
electroweakly with the scattering electrons to produce jets (semi-collinear collections
of hadrons). The model originally had no other interactions built in to explain any
of the strong coupling effects seen. This model is known as the naive parton model,
and ignores any perturbative QCD effects. Indeed, this model did seem to describe
the data in a qualitative fashion, and it was later deduced from the DIS data that the
components of the proton that were interacting with the electron were spin 1/2 ob-
jects. This is precisely what one would expect it these “partons” were to be identified

with the quarks of the Cell-Mann SU(3)fiavor model.

The naive parton model assumes the high energy limit, in which both quarks
and hadrons are massless. To calculate a cross section in DIS, we convolve a partonic
cross section with a form factor fH_,q(§). In other words, the hadronic cross section
is

1
dUeH—mx' :/0 défH—rq(£)d(}q(€)7 (3-8)

where we have introduced fH_)Q(.T), the process-independent parton distribution func-
tions (PDF). These empirically measured functions represent the probability of the
interacting parton having a fraction 23 of the parent hadron’s momentum. In the
limit that both particles are massless, the hadron momentum is given in terms of the

proton momentum as

19,, = 61)”. (3.9)

The hadron-hadron cross section is defined in a similar fashion, where the
structure functions are deemed to be universal, that is, process independent. We
may then measure the parton distribution functions through DIS and apply them to

any process we wish. The hadron-hadron cross section is then given by

1 1 .
dO'HH’aX :/0 déafH—mfialfo débfflab(€b)d0‘ab(€aa€b)° (3.10)

34

 

 

In equation (3.8), q may be a quark or an anti-quark, and in equation (3.10), a and
b may be any species of parton. These expressions must be summed over the parton

species, q, a, and 0, since partons may not be directly observed.

3.3.2 Perturbative Quantum Chromodynamics: Improved Par-

ton Dynamics

Ideally, one would like to extend the qualitative successes of the naive parton model
with a dynamic theory. Such a theory would have the same qualitative behavior of the
naive parton model, but with enough computational power to make precise analytic

predictions of observable quantities.

We find that, if we allow the quarks of the Cell-Mann model to interact dy-
namically via an unbroken SU(3) Yang—Mills theory, we succeed in extending the
parton model. This theory contains generalizations of the hadron structure functions
mentioned in the previous subsection, and allows improvements of the predictions of

that model by the dynamic interaction with the SU(3) gluon field.

This model based on the SU(3) gauge theory is called Quantum Chromody-
namics (QCD). QCD is an (unbroken) SU(3) gauge theory in which the quarks of
Cell-Mann’s flavor model interact with spin-1 bosons known as gluons. There are
eight of these, interacting with the quarks via a three—dimensional quantum number
known as color. One may show that such a theory satisfies asymptotic freedom, in

which the coupling becomes very small at high energy scales.

This theory is also interesting, since it gives an explanation as to why the
quarks are never seen in nature. The coupling a, of the SU(3) gauge theory actually
diverges for a small energy scale, indicating that any perturbation theory is poorly

defined in this limit. In quantum mechanics, small energies are equivalent to large

35

distances. This implies that any attempt to pull quarks away from one another results
in a steady increase in energy, until the available energy is enough to create a quark
anti-quark pair out of the vacuum. In order to examine such behavior, one must be

able to calculate exact expressions, without resorting to perturbation theory.

If the interaction is over a very small distance (large energy scale), one may use
perturbation theory, since a, is small over these distances. Such an expansion, and
the requisite renormalization of the PDF’S and the coupling constant, are known as
perturbative QCD. For very “hard” scattering processes, where a significant amount
of energy is exchanged among the hadrons, we may calculate corrections to the naive
parton model. This consists of calculating a parton-level cross section in standard

perturbation theory, then inserting the results into the convolution equations above.

QCD, like all gauge theories, is poorly behaved in the ultraviolet region. Wave
function renormalization removes the ultraviolet singularities that occur. In addition,
two other types of singularities occur, collinear (or mass) singularities, and soft (or
infrared) singularities. The soft singularities must cancel when all of the pieces of the
cross section are added together, but the collinear singularities do not cancel against
any other pieces of the calculation. However, the collinear poles are universal (process

independent), and may be absorbed into the (universal) structure functions. That is,

fh—)p(Z) ——> /01dLr/xfhnpz(a:/z,up)Kpr_,p(z), (3.11)

where the convolution kernel K p_,p1(z) is

 

a, 47TH}; 6 PA '(3) .
[(1,141,021) 7: 6(1— Z)(Sppi — (57E) ( #F ) fffi?) +finlte. (3.12)

The convolution kernel may be thought of as being analogous to the normalization

constants Z in the ultraviolet renormalization procedure. The splitting kernels P(z)

36

in equation (3.12) are

 

Fri—“1(3) =3 §(E§—) + 26(1— 2) (3.13)
Page) = $1351) (3.14)

1'" Z Z 11 72f 5
Pg—ig(z) :- 6( Z +W+z(l—Z))+(3*—3')d(l—Z) (3.1)
P9442) = %(z2+(1-—z)2). (3.16)

In the preceding relations, we have introduced the “+” prescription, which is defined

through the relation

1 1 1 9(z)-9(1)
[dz—Ig(z)=/O dz——(—1———— (3.17)

for any arbitrary function 9(3).

In equation (3.12) we have introduced another dimensional byproduct of the
renormalization process, the factorization scale up. We have also made explicit the
separate scales introduced in ultraviolet and infrared regularization processes; the
ultraviolet renormalization scale will be referred to as #3. In practice, one generally
takes these scales to be the same, though there is no theoretical argument that they
should be taken to be the same. When we have set these scales equal to each other,

we will refer to the resulting parameter as n, to emphasize it’s generic nature.

The physical interpretation of the factorization scale is somewhat different
than that of the renormalization scale. Indeed, the factorization scale represents the
cutoff scale for so—called “hard processes.” In other words, it quantifies exactly how
much of the soft QCD radiation is to be included in the partonic matrix element, and
how much is to be absorbed into the parton distributions. The renormalization scale

represents the cutoff scale at both the large energy end, where we expect new physics

37

to come into play, and the low end, where we absorb the remaining IR singularity

into the quantum mechanical excitations.

Now, in order to cancel the soft poles that occur in the matrix element, we
must carefully define our observables. In general, an observable 0 may be a function
of the number of particles, as well as the kinematics of the system. In order for an

observable to be infrared safe, we guarantee that (for N and N + 1 parton systems):

0N+1(E1 —> 0) 2 ON. (3.18)

That is, 0(N + 1) is infrared safe if, as the energy of one of the partons vanishes,
0(N + 1) becomes identical to the analogous function with one less parton, 0(N).
Similarly, an observable is collinear safe if for two particles with momenta p1, p2, we
have that

()N+1(Slg -—> 0) =3 ON, (3.19)

where (512 = ([11 + p2)2). If 0 satisfies these two conditions, then the cross-section is

guaranteed to be finite to N L0 in perturbative QCD.

This process may be generalized. We may calculate cross sections and other
observables to any order in perturbation theory that we wish. These calculations tend
to be quite complicated, as the additional complexity of the SU(3) gauge group tends
to generate large numbers of contributing graphs. This is especially true in processes
that involve large numbers of gluons, since gluons can couple not only to fermions,

but to other gluons as well.

38

3.4 Helicity Amplitudes and Color Ordering

In recent years, many techniques have been developed to simplify the computation of
multi-parton amplitudes in QCD [19, 20]. By using techniques such as helicity am-
plitudes and gauge-invariant color orderings, one may simplify a calculation greatly.
This technique significantly reduces the number of diagrams that one has to evaluate

in computing, for instance, gluon amplitudes.

In order to avoid calculating things redundantly, one may make use of some
of the symmetry properties of QCD. In the approximation that all of the partons are
massless, the spinor states of the partons become true helicity eigenstates. Since some
of these states are related to other states by charge conjugation, Bose symmetry, or
parity, one avoids calculating redundant helicity states. Also, each non-cyclic color
projection is gauge invariant from other non—cyclic projections, so we may reduce the

number of terms by calculating a single cyclically related color projection.

Another simplification lies in the fact that these projections of color and he-
licity are gauge invariant, so that one may choose a different gauge to evaluate the
partonic matrix element for each different helicity state. It is generally convenient to
express polarization vectors in terms of massless Weyl spinors [21, 22]. The polariza-

tion vectors for a gluon of momentum p may be written

 

_ (irll'r"lqi)
9““)— fi<¢qlpi>’ (3'20)
where
link) = 1:7 U(1)) (3.21)

is a massless spinor, and q is some arbitrary massless reference momentum such that

q - p 75 0. Since a change in q is equivalent to adding a piece proportional to p, we

39

have basically absolute freedom to choose q. The terms proportional to p will then

cancel in gauge invariant amplitudes via the Ward identities.

Using these expressions, the number of terms one needs to calculate a QCD
observable is reduced considerably. However, ambiguities arise at NLO, where we
would like to analytically continue the number of dimensions. The number of spins
states in a conventional IV—‘IS calculation is generally taken to be 2(1 -— e) which is
quite clearly an abstraction. If we consider actual helicity states, the number of
external particles must be two exactly. Internal (loop) particles may have either,
but are generally taken to have 2(1 —— e) helicity states. Therefore, when calculating
expressions using helicity states, one must know how to alter the resultant expressions
in order to obtain the conventional IVI—S expression. We will encounter this issue later,

during our calculation of the Higgs differential cross section at NLO in QCD.

40

Chapter 4

The Higgs Boson Cross Section and
Transverse Momentum Spectrum:

Leading Order Calculation

One. of the most important quantities that will be measured at the LHC is the pro-
duction rate of the Higgs [14, 23, ‘24]. Therefore, a precise theoretical understanding
of the rate of Higgs production is critical to any attempts to search for the particle.
Since the cross section is dominated by the gluon-fusion process [14], this process
must be calculated as precisely as possible, in order to precisely determine how many

of these particles will be produced in a year of running.

Aside from having a good idea of the overall production rate of the Higgs, the
transverse momentum spectrum and the rapidity spectrum of the Higgs boson is of
great interest to the LHC project. Since the detectors themselves must ideally be
designed around the kinematic regions where one expects to see the Higgs, having a

reliable prediction for these differential cross sections is of particular importance.

Unfortunately, QCD observables are notoriously unreliable at leading order,

41

and these must generally be calculated to NLO (at least) in order to have a reli-
able prediction for their behavior. Moreover, leading order QCD calculations tend
to exhibit a strong dependence on the factorization/ renormalization scale, a, which

reflects this theoretical uncertainty.

The focus of this work is to calculate a reliable numerical estimate of the trans-
verse momentum / rapidity spectrum. In order to improve the existing calculation, we
will add to it radiative corrections which will hopefully have less scale dependence,
and therefore less theoretical uncertainty. Since the procedure for calculating this
differential cross section at NLO includes calculating the LO spectrum, it is instruc-
tive to begin with the much simpler LO calculation. We will then show in the next
chapter how to modify this expression in order to obtain the differential cross section

at NLO.

We will assume in this calculation that the dominant contributions to the
transverse momentum spectrum come from the aforementioned gluon-fusion process.
We will calculate the LO production rate both for a finite top quark mass, and in
the limit that the t0p mass goes to infinity (the large mt limit) . We will then argue
that in the Higgs mass range under consideration, the finite mass calculation and
the infinite-mass approximation give virtually the same result, whereas the QCD
corrections give very large contributions to both the transverse momentum spectrum

as well as the overall production rate.

4.1 The Gluon Fusion Cross Section

The gluon fusion process proceeds through a virtual t0p quark loop as represented

by the Feynman diagram in figure 4.1. As this process is well known in the literature

42

 

9 W
Figure 4.1: Feynman Diagram for Higgs Production through Gluon Fusion

[25], we will just present the result

,2 2
0. |.4|26<1— ’1’!) . (4.1)
8

0:——
256m,2

The form factor A is

A(T)=Tll+(1-T)f(7‘ll (42-)

where
f(r) : arcsin2 (fl) fOT T < 1, (4.3)
W) = —i (10s(l_ iii) ‘1'”) f0” >1, (4'4)

and r : 4'77130p/7‘n3l. In the limit of a large quark mass, this form factor converges to
a constant

A(T —+ 00) = —— (4.5)

which results in the cross section in the limit of a large quark mass;

 

. a: mi
0(T 7 0°) : 576%12‘26 (1 _ T) ’ (4'6)

A graph of the exact hadronic cross section is given in Figure 4.2 for a t0p
mass of 174 GeV. In addition, the corresponding infinite mass limit is plotted on the

same graph as a reference. Clearly, both graphs are qualitatively the same for an

43

 

100 . r* ‘n ’r“ ”w r" T’ ”i
[ approx. —+— 1
. exact ------- l

0 (pb)

 

 

 

 

0.1 l l l l l l L
1 00 1 50 200 250 300 350 400 450 500
mH (GeV)

Figure 4.2: Comparison of the infinite quark mass approximation with the mmp =174
GeV calculation.

44

intermediate mass Higgs (my z 100—200 GeV), which is the region that we will be
considering. If we assume that the LHC will run at a luminosity of approximately
z 300 fb‘1 yr’1 [1], then we may compute the expected number of Higgs produced
in one year of continuous running by multiplying this figure by the cross section.
Looking at the data on this graph, we see that the LHC will produce approximately
0.4 — 4 x 106 Higgs bosons in a year of running, depending on the Higgs mass. We

can conclude that Higgs production will be quite prolific at this machine.

The close correspondence of the finite calculation with the infinite mass limit
calculation seems to suggest that in the infinite quark mass limit, the dynamics of
the loop freezes out, allowing us to replace the loop with an effective coupling. As
long as the relevant energy scales in the problem do not become too large, we can be
confident that the results of this calculation give a fairly good approximation to the

true cross section.

Even in the region where the energy scale is approximately the top quark mass,
the infinite mass approximation is quite good. The O[m,‘0%,] give additive corrections

[26] which behave as

 

2
7 m 1
0(mwp) z 0(mt0p = 00) (I + 1-5 (2771:0p) ) (4.7)

so that for a Higgs mass with a mass as large as 200 GeV, the exact leading-order
expression differs from the large mmp approximation by z 15%. Since every exper-
imental indicator tells us that the Higgs is very likely below this limit, we can with

good faith apply this effective theory to the inclusive cross section.

We may also note that the cross section calculated in the large mtop limit
underestimates the Higgs mass over the entire range of plotted masses. As we argued

earlier, it is highly improbable that the Higgs mass is outside of this range. We

45

therefore expect that the infinite mass effective theory will be quite good over the
entire range of applicable masses. Moreover, recent work [27, 28] seems to suggest
that the ratio of the next—to—leading order calculation to the leading order calculation
is more or less independent of the top quark mass. Therefore, one may compute the
next-to—leading order corrections using the infinite top mass limit, divide that value
by the leading order, infinite mass limit result, and apply the resulting enhancement

to the leading order expression with confidence.

4.2 Low Energy Effective Lagrangian

Since the large mtop approximation is quite valid in our leading order calculation of
the cross section, a systematic way to calculate the various Higgs observables using
this approximation is desirable. We may therefore propose a low energy effective

theory to do gluon fusion calculations 1.

This idea of freezing the top quark inside the loop, and replacing it with an

effective coupling leads to the following effective Lagrangian,

 

1 '9 u
g = "1 (3:!) (1+ A) HG“ G,,.,, (4.8)

where the top-loop dynamics have been replaced by an effective coupling 2, £35. In
order to be valid, the relevant energy scales of the process must be less than the top
mass, and even outside that neighborhood this approximation generally underesti-

mates the cross section, so that one errs on the conservative side. We may derive the

—¥

1Aside from the original leading order calculation [291’ The radiative corrections using this infinite
Tntop limit have also been computed [26, 30]. Moreover. progress has recently been made on the
NNLO calculation [31, 32l-

2The effective coupling 013/31”) receives a finite renormalization at each order in perturbation
theory. We represent this with a constant A, which is zero at leading order. there is a contribution

from A at NLO which one must take care to include.

46

following Feynman rules from this effective low energy Lagrangian:

The Higgs—gluon-gluon coupling is

up

Hl‘l’ (171.122) = “i445ab(P1P-2 9"” — pip-z). (4-9)

the Higgs-three gluon coupling is

Hgllfflhvpzapsl = gAfabc ((1)1 — pg)" 9"" + (p2 —— pg)“ 9"" + (1)3 — p1)"g’“’), (4.10)

and the Higgs-four gluon coupling is

Hl‘bfipwnpz. pm) = 1'9“] fabefcde (Wow — y""9"")
facefbdc (91“)ng _ gpogup)

fadefbce (.9ng _ gppguafl, (411)

where A 2 (33:5) (1+ A) is the effective coupling and fabc are the SU(3) group

coefficients in the adjoint representation. These are represented graphically as:

    

uvap
0, )u Habcd (1.0

 

In the preceding expressions and diagrams, the gluon a has color a, momentum pl,

47

and helicity index ,u, with analogous relations for the remaining particles. Note that
these rules become those for QCD if one simply takes the limit pH —+ 0 and replaces

the coupling A with 1.

4.3 The Leading Order p j spectrum in the large

mm, limit

By far, the most significant contribution to Higgs production is through the gluonic
coupling via a virtual top quark 100p. Not only is the rate enhanced by the large
top Yukawa coupling, it is also enhanced considerably by the large number of gluons
(in comparison to quarks) inside the proton. Therefore, in computing the transverse
momentum spectrum, we shall restrict our calculation to radiative corrections of the
99 —> H cross section. In particular, those that produce a nontrivial transverse
momentum will be considered. When we refer to the Leading Order or Tree Level
contribution, we mean the lowest order in perturbation theory which produces this

nontrivial contribution.

At a proton-proton collider such as the LHC, by far the most dominant con-
tribution is the process gg —> Hg. This is simply because the gluon distributions
are much larger than the quark distributions at these energies. The quark anti-quark

mechanism is also small, since the antiquark distributions are also very small.

We would like to use our effective theory to calculate this process, but we
have provided no arguments that the effective theory will be any good whatsoever
in this markedly different calculation. Therefore, we will compare it to the finite top
mass calculation done in reference [33]. Indeed, we will present the argument that
the effective theory does quite nicely as long as the mass scales in the problem (in

this case, m H and p j) are fairly low in comparison to the t0p quark mass.

48

4.3.1 Kinematics and Notation

We begin by defining our kinematics, and establishing our notation for the calculation.

As is the convention of parton model, we relate the hadron momenta, Pa, Pb to the

parton momenta pb, p), via momentum fractions, Eu, 5),;

pazfa Pa» széb Pb.

(4.12)

We will sometimes use the convenient shorthand a 2 pa, b 2 pb for the parton level

momenta. It is important to note that the final state parton momentum will be called

Q in this work. This differs from many similar calculations in which the momentum

of the final-state massive particle is referred to as Q. We therefore have the following

momentum conservation relation;

Pa +Pb = P11 + Q-

We shall define the parton IV’Iandelstam invariants in the usual fashion:

3 : (pa + pb)2:

t = (Pa - pn)2. u = (pi — on)?-

If we define the hadronic lVlandelstam variables analogously

s = (P, + By, T : (P. — py)2. U = (Pb — P502,

these are related to the partonic variables via

3 : {061257

t=m§,(1—€a)+€bT, u:m%(1_€“)+€bU‘

49

(4.13)

(4.14)

(4.15)

(4.16)

 

 

Figure 4.3: Feynman diagrams for Higgs production in the large m ( limit.

If we choose to evaluate the Higgs momentum in the hadron-hadron center
of momentum (COM) frame, as is usually the convention, we may re—express the

hadronic variables as;

_ 2
S — Econ!)
._ 2 . _ 2 ’ '
T — mH — "MECOM exp (-yu). U — m), — ijCOM exp (3)”).

(4.17)

Here, the rapidity of the Higgs in the COM frame is denoted y”, and my denotes the
transverse mass (2 «mg, + pi). Therefore the partonic momenta can be expressed

in terms of the traditional external variables and the momentum fractions;

, _ 2
b — 5415111300111)
__ 2 _ 2
t — mH — {aijCOM exp (—yH), U — my — {bm1ECOM exp (UH)-

(4.18)

4.3.2 Analytic Procedure

The Higgs p 1 spectrum has three contributing partonic processes at leading order,

99 —> Hg, qq —> Hg, and gq ——> H q. These may be derived from the graphs in Figure

50

4.3 using the Feynman rules outlined above. We find the following expressions for

the (helicity-averaged) matrix elements in d = 4 — 26 dimensions

 

 

 

 

—_ 24 403 s4 + t4 + u4 + m8 .
111 2 = S _ H __ 2
l 9949”] 256(1— (5)2 97m2 {(1 6) sin 4€ m”}’ (419)
_ 16 a3 t2 + 242 tn
1‘! -._.) 2 = 3 — — _ / .
l qq 9”] 9(36) 7!, v2 { 8 (1 6) 26 S } 3 (‘71 20)
and
_ 16 a3 52 + U2 su
1‘ _* 2 : s. — - . .
l 19" 4”] 9(96)(1— c) 7r 42] —t (1 ‘l 26 a} (4 21)

Because of CP invariance, the process gq ——> qH is identical to the expression for
yr} —> q‘H, and the matrix element for qq‘ —> gH may be derived from the latter
expression using crossing symmetry. The appropriate spin and color averaging factors

must, of course, be substituted. These have been included in the preceding expression.

The general expression for a 2 to 2 partonic differential cross section is given

by
d6 6 (Q2)

= .1712. (4.22)
dpi dy 167r s

 

 

 

The hadronic cross section is related to the partonic cross section via the double

convolution, as outlined in Chapter 3;

(1043 I 22/1 dfaf.4—.i(€a,HFl fol d5be—>j(§b,Mldéijlgmébl (423)
i j 0

Using the expression for cross section found in the appendix D and the expression for

the partonic cross section, Eq. (4.22), we may derive the following expression for the

differential cross section:

, 1 1 1 f 2 ,.
W = 22/ dlfafA_.4(€a,/lp)/ dEbe—41(€b,/1F)5(Q2) 167:3 Mn] - (4-24)
dpi dyH z. j o 0

51

Noting that the double convolution of the delta function may be written (also Ap-

pendix D)

 

 

df l 1
Aldfa/O débé ()Q2 l") 0 (5b , V
1+ {05 + T Mfr + r— {b5 + U — Mfr, (4.20)
where
:1: : _~ ex i' ,
i \/§ p( yH): (.126)

we arrive at our final expression for the differential cross section,

 

 

dappaHX da; 1 (11.55
2—2( ....S.T ..S.U_M))

|._)1,,|2. (4.27)

Xf.44.:(€a. #FlfB—U' (5": “Fl 16143

Note that there is only one degree of freedom in the integration, as the momentum

fractions are fixed by the on-shell condition in each term.

4.3.3 Numerical Procedure and Results

The convolution (4.27) must be evaluated numerically since the parton distribution
functions are empirically determined. We shall use the CTEQ5L parton distributions

[34], and we shall choose our scale to be the transverse energy, m 1- The number of

(light) quarks n; is 5 throughout the calculation.

The finite-mass calculation proceeds identically to the one outlined above. One
simply replaces the effective theory expression for the matrix element with the exact
result [33]. Because the mass of the top quark is so large, we expect the large mtop
limit to be quite good out to very large transverse momentum.

Figure 4.4 displays a graph of the transverse momentum spectrum at zero

rapidity for a Higgs of mass m), = 115 GeV calculated using the full mtop dependence

52

 

 

 

 

 

 

1 I I l l I , 1 1 1
infinite quark mass ‘
mtop = 174 GeV ....... .
0.1 -
if
3;
I
>
u
d‘
‘0
3 0.001 _-
'0
0.0001 _-
‘7 “‘4
19'05 l ' l 1 l l l l 1
0 50 100 1 50 200 250 300 350 400 450 500

pt

Figure 4.4: Comparison of the exact leading-order p 1 spectrum with the effective
theory calculation.

(dotted line). Superimposed over this graph is the identical calculation performed
with the simplification that the fermion mass is taken to be infinite. It is plain to see
that the agreement is quite good out to the very large p _L region, which is roughly
about 300 GeV. There is really no reason to suspect that the behavior of any of the
higher-order corrections will behave any differently than this.

It is advantageous to break the infinite—mass limit calculation up into the
various contributions from its partonic constituents, so that we may see which partonic
graphs contribute the most to the cross section. A plot of the various contributions

to the p 1 spectrum at leading order for a Higgs with a mass of 115 GeV is given in

Figure 4.5. All of the terms exhibit the expected divergent behavior as p _L goes to
zero, with the exception of the s-channel piece, qq ——> Hg.

As can be plainly seen in the graph, the dominant contribution to the total

53

 

 

 

 

 

0.1 _-
é‘ 0.01 :-
I
a
I
>
“O
a"
3 0.001 -
O .
‘0
0.0001 : q
19'05 J l l ............................... INN-.4—
0 50 100 150 200

Figure 4.5: The Higgs p) spectrum at leading order

cross section (solid black line) comes from the gluon initiated process, gg —> Hg
(long dashes). This contribution is roughly an order of magnitude larger than the
next largest contribution, qg —> Hg (dashed line), in the small transverse momentum
region. Even in the large p _L region, the gluon initiated process is twice as large as the

gluon-quark initiated one. The process qr'j ——> Hg is completely negligible compared

to these other two processes.

To summarize, the gluon fusion process dominates at the LHC, and working
with an effective field theory in which the heavy top loop is replaced with an effective
vertex is a quite good approximation. Even for the inclusive cross section, the N LO
pieces are found to produce enhancements which are much larger than the difference
between the finite mtop calculation and this infinite limit, at least for Higgs masses
in the expected mass range, 100-200 GeV. Since the differential cross sections at

leading-order tend to be even more unreliable than the leading—order cross sections,

54

it is imperative that we include some of these next-to-leading order terms in order to

improve our estimate for the differential cross section.

55

Chapter 5

The Higgs Boson Transverse
Momentum Spectrum: Gluonic

Radiative Corrections

In the previous chapter, we found that the dominant production mechanism for pro-
ducing a Higgs boson at the LHC was that of gluon fusion, and that this process
is well represented by an effective theory in which the massive quark in the 100p is

replaced by an effective coupling between the Higgs boson and two or more gluons.

Moreover, the dominant partonic process is the gluon initiated gg -—> Hg
process, which is responsible for about 90% of the total cross section. This is primarily

due to the large number of gluons in the protons compared to the quarks.

As we have mentioned previously, leading-order calculations in QCD are at
best an order of magnitude approximation to the actual process. This is reflected in
a large scale dependence in the leading order cross section. If one wishes to have a
more precise prediction for the cross section, one needs to carry out the calculation

of the differential cross section to next-to—leading order in perturbative QCD. It is

56

hoped that the resulting calculation will have less scale dependence, and therefore,

less theoretical uncertainty.

There are several different methods for doing this. One notable method is
using a Monte-Carlo event generator to do the entire integration [35]. Since the
matrix element contains soft and collinear divergences, one must take care to extract
the divergent pieces and integrate them in 4 — 26 dimensions in order to cancel the

divergences in the virtual pieces. A calculation of this sort has been done [36].

Nonetheless, it is advantageous to do the entire calculation analytically for
several reasons. First, one would like to calculate the process using conventional MS,
and show that it agrees with the Monte Carlo calculation. Second, one needs an
analytic formula in order to expand the matrix element around small transverse mo-
mentum. This is necessary to calculate the Bl?) coefficient in Collins-Soper-Sterman
resummation scheme, and compare it to the result calculated in a different manner
in reference [37]. To have a complete next to leading order calculation, good for

all transverse momentum, the intermediate p 1 calculation included herein should be

combined with a resummed curve good in the small p) limit [38, 39, 40, 41].

In this chapter, we will be computing the dominant piece of the NLO correc-
tions, the partonic process represented by gluonic radiation from the leading order
process, gg —-> Hg. This, we will argue, dominates the NLO cross section in the

intermediate p 1 region with which we are concerned.

To do this analytically, we will use the procedure outlined in Chapter 3. Since,
in these corrections, we will be required to renormalize ultraviolet divergences and
factorize collinear divergences, we will have to carefully compute the matrix elements
in d = 4 — 26 dimensions. The parton distributions that we will be using are calcu-
lated from deep inelastic scattering data using the MS factorization/ renormalization

scheme, so we must use the same scheme in order to obtain a meaningful result. Once

57

we have a finite expression, the convolutions with the structure functions may be eval-
uated numerically to obtain the cross section as a function of the various kinematic

parameters.

5.1 Notational Conventions

The notations that we will be using are in complete analogy to those of the previous
section. The 4-momenta of the hadrons shall be taken to be Pa and Pb, and the
4—momenta of the partons shall be denoted pa, 1),, (2 60,3, PM) . The Higgs boson
momentum shall be denoted H and the mass m”, the transverse momentum p L, and
the rapidity y 3. Therefore, the hadronic (S, T, U) and partonic (s, t, u) Mandelstam

variables have precisely the same form as the previous chapter.

It is also convenient to define crossed momenta, p112 = —pa,b, and to label the
final state partons 193, p4. This allows us to express the invariants of the 2 —> 3 process

in terms of generalized Mandelstam variables,

Sn = (192‘ +Pj)2, Sijk = (Pi +15 +I’kl2- (5.1)

The invariant mass Q2 of the final state gluons and the Higgs invariants t = (a — H )2,

u = (b — H )2 are related to these via

Q2 = 534 = (P3 + 1902, t: 5234 = (P2 +113 +1902, (5.2)

and

u = 5134 2 (191+ P3 +p4)2° (53)

58

One may then show that the Higgs mass M is related to the remaining variables by
M2 = s + t + u — Q2, (5.4)

and that the transverse momentum p i of the Higgs may be expressed as

Pi : (t_Q)(U"Q)__Q2, (5.5)

S
tu — Q217l'12
8 .

 

We will find it convenient to define the parameters Q L, V as follows:

QT = \/Q2 +191,

V = QQ/Qi- (5-7)

The physical interpretation of Q L is that it is the transverse mass of the gluon pair.

Finally, we will also use 5, and S”, which are defined as

Su : 513+Sl4 : U—Q2, (5.8)

St = 523 + 5‘24 = t _ Q2: (5.9)
and we will occasionally use the parameter S H,

SH = S123 + S124 2 (”2 + Q2 ‘ 3- (5.10)

59

5.2 Cross Section at NLO

To next-to-leading order, the two-to-two (2 —> 2) differential cross section is:
d ‘0’ 5 11
0NL0:d02_+2+d0-2_)2+d02_23. (. )

Here, (10222 is the tree-level contribution, (10222 is the expression for the 1-100p
virtual corrections, and da2o_,3 is radiative corrections to the leading order process.
If we include a collinear counterterm (10252 and a UV counterterm d02_,2, then the

cross section may be written as

do

an? / f “a“ “222221222

. (10.0 (101(1) dUUV. dam) d0
2 2—)>'2 2—>2 ‘2——>‘2 ‘2—>3 2*)3 «-
(6(Q)( dt + dt + dt )+ dtdu + dtd:}' (0'12)

 

 

 

Since these are to be computed using the conventional dimensional regular-
ization, we must be careful to include all e dependence in the latter two terms. Note

that the expression for the 2 —> 2 differential cross section becomes

d 2 4 ,..2 ‘ M2
(’2 .2 = 7‘2” . [2 I . (5.13)
dt P1 / 167rs F(1—e)

The expressions from the other two 2 —> 2 terms (the virtual corrections and UV

counterterm) are identical to this one with the exception that one must substitute

the requisite matrix elements.

The contribution coming from the radiative corrections (those involving 5 par—

ticles) may be written

d02913 _ 47r/12 6 47m2 C 1 1
dt du — pi Q2 P(1 —- 2c) (21r)(47r)2

60

 

 

 

—2c [MP

x 2/Ond6(sin(6))1‘2‘/0nd¢(sin(qb)) 167w" (5.14)

The contribution from the Altarelli-Parisi term is

 

 

($2153 _ 47TH 6 as (47r)‘ 1 x
dt d'u — pi 27r eF(1-—~ r.)2167rs'2
‘61 a . __
/ : P(Za)5(za(s + t —— 1112) + ’u)[1l/I(€aza,fb)|2 + (t <—-> u). (5.15)
0 a

In the preceding expressions, we must note that we have suppressed the sum over

parton species for clarity.

The angles 6’, d) in the expression for the 2 —> 3 cross section will be the polar
angles of the final state partons in the Q2 rest frame. The 2 axis in this frame will
be defined by either initial state parton or the Higgs boson, whichever is convenient.
See Appendix B for a further discussion on the explicit evaluation of these angular

integrals. Once these integrations are calculated, we need only concern ourselves with

the end result.

5.3 Angular Integration of Real Emission Piece

To calculate the real contribution, we must integrate the two angular degrees of
freedom occuring in the matrix element. While there are a large number of terms to

integrate, the calculation lends itself to an algorithm very nicely. The basic procedure

for the angular integration is the following:

1. Partial fraction the amplitudes using the relationship amongst the various Man-
delstam invariants. This will reduce all of the integrals to the form (const) x
531.53.

is some collection of kinematic invariants that does not depend on the angular

n or (const) x 535;", for which standard formulae exist. Here, const

61

variables 6 and gb.

2. Extract the terms proportional to 1 / (1 +5) and verify that they cancel amongst
themselves. This verifies that there are no quadratically divergent pieces in the

amplitude (which is true in general for QCD).

3. Extract the universal —1/€ piece and verify that it cancels algebraically against

the collinear counterterm.

4. Regularize the soft divergences (Q2 —> 0), and extract the universal soft pole.

This will cancel against the soft pole in the virtual piece.

Once this is done, the real matrix element is integrable over the convolution space

outlined in Appendix D. The final integral must, of course, be done numerically.

As discussed in chapter 3, the helicity amplitudes obtained in the literature are
computed using gluons with two polarizations. To use VS, we must have expressions
for the matrix elements with 2 — 26 gluon polarizations. Therefore, in order to use
m, we will need to calculate the 0[e] corrections to these expressions due to higher
dimensional gluon spin effects. This calculation is much simpler than the original

matrix element calculation, since we need only to calculate the matrix element in the

collinear limit. We will explain this shortly.

5.3.1 gg —-> Hgg

We now restrict ourselves to the process 99 "’ Hgg. To begin, we first break the
cross section into two sections, a piece which represents the sum over the (+ + ++)

and (— — ++) helicity configurations, and a piece which represents a sum over the

62

odd-helicity components, (— + ++). Let

 

~(0) - ..
daggfiHgg : daggfiflgg + (10994399 (5.16)
dtdu dtdu 1 dtdu 2’
with
((592—1ng PI ‘6 dd)
——‘- = F 1— .. ———, x
( dtdu 1 ( 6) 47m;2 167r23‘2
2 ()M(1+2+3+4+))2 +1.11(1-2+3+4-)|2+
|M(1+2‘3+4‘)|2 + |11v~1(1+2+3‘4")|2), (5.17)
and

 

déggaflgg _ P21 #6 (M)
( dtdu 2 — N1 6) 47ml? 167r232X

2 ([Zl»1(1—2+3+4+)[2 + 3 cyclic spin perms). (5.18)

In these expressions, the remaining parts of the 2 —> 3 phase space d<I> may be

succinctly written
1 F(1—e) 4m? 6
= ______. —.— 1.9“), 1119
(M) 872211“ _ 2E) ( Q2 ( (o l

where (19“) is the 4 — 2e angular integration measure,

we _ _1__ [0"d0(sin(9))1-26 /".1¢(sin(¢))*22 (5.20)

—27r 0

which is defined in Appendix B.

First, we define

a2 N2 a, 2 47w ‘5 47w 6 1
0. = S . C 2 ‘7 “—5— ———- (5.21)
2567r s2 N: —1 37r 116 Q [)2 F(1 — 2e)

 

63

Then, the angular integral of the first contribution in (5.16) yields
W Z 40. x _ (1) A118+§4A+Sj+Sf +sz><
dtdu 1 c 32 Q2 pi

[(1+ V)(1— 1/)_6 + (1 — u)I/"(1+ figs] +

 

(401/3 — 421/2 + 30,, -— 11)

 

53+Sf( 676+1—I/
$26221)?L 18 6
(1/4 +(1—1/)4 — 1))

6

 

21/ 6 17
—— — _ '— . I22
+4(1__V+ln(1 u))+3u+3} (5 )

Noteworthy are the terms that behave like BEE/51 and 171;, which diverge as p 1 —> 0,
.L

as well as the universal % pole.

For the (—,+, +, +) terms, we write the results as a sum of the universal
divergent pieces and an analytic piece which is finite as Q and p 1 go to zero. We
find from the symmetries of the matrix element that there are in actuality only 4

independent terms, and that the sum over the odd-helicity components may written

Z

colors,(—,+,+,+)

 

1
1M]2 —> 4 {(m(1—, 2+, 3+, 4+) + -2-m(1—, 4+, 2+, 3+) + (u H t)>

+2 m(3—,4+,1+,2+) + m(3—,2+,4+,1+)}. (5.23)

Using this relationship, we may write

d‘
(W) 2 202 {[2 (2(1, 2, 3, 4) + 2Q(3,4, 1, 2) + Q(1,4,2,3) + Q(3,2,4,1)]
2

+(u <——> t)}, (5.24)

where the Q’s are defined as the angular integrations of the helicity subamplitudes,

e.g.
d9 _ _
0(1,2,3,4)=/27;Im(1+,213 ,4 W. (5.25)

64

with similar relationships for the other subamplitudes.

The angular integration over the (1-, 2+, 3+ 4+) helicity configuration may
be written as a sum over two terms, a term that is finite as Q, 6 —+ 0 which we shall

call 520(1, 2, 3, 4), and a term which contains the pole structure, which we shall call

Qdiv.(112v 37 4); i‘e'?
(2(1, 2, 3, 4) = (20(1, 2,3,4) + 9,2,0, 2,3, 4). (5.26)

The second term in this equation is

1 1 M2 2 4 s 2 4 M4133 2
9.....(1.2,3.4) = [(-—Q) +(fl +t4—2u( Q

 

—;Q232pi u M2 — u)(1\12 — t)

1 211/1123 (1
c s Qiump — t)(1'11'2 — u)

 

— V)"1"‘, (5.27)

and the term 90(1, 2, 3, 4) may be found in Appendix C. The matrix element with the
same color ordering, but with the final-state gluons having the opposing helicities,

integrates to

4 .- 4 2 2 4
+ 1 1112(Q2 — t) 3Q p

2Q28Al4(1112 — u)3 _ _1_ 2M4(M2 — u)‘3( _ 11)-1“
_ (M2 — t)tu e (M? — 031qu
11 676 M8 + s4 f M4 (5.28)
(7 ‘ ) Q232Qi — 362262“:

 

5 T3—
The term with the complicated pole structure (1324) yields

4 2 4 , 2
2 1 M 2Q2 5p] 4 _ spL
Qdi’”‘(1’3’2’4) : {762232111 [( u + u H (1 tu )

M41336)? 2 2(1 — u)-1-'~'M4t3
11(2112 -— u)(M? - t) e s QiuUVI'2 — t)(M2 — u)

 

 

—2

65

 

2 2t3 1 7r2
__ — —e 2—
e usQiV (1+6 6 )}

for the term where the initial state particles have the opposing helicities, and

1 1 .9 4 1112 2—t 4 22 4
Qdiv.(312a411) : {—2QT2P2L [(u—UQZ) +< (Qt )> +((tS?S-2L)t)

2Q23M4(A12—u)3 3P1 ,8 3 4
‘ (112-1)... ‘EW +9 +52]

 

 

 

1211!“ [V12— 3 1M8 8 4 2
__ 2( “(1)2 (1_ V)—l—c _ _ +? + 8 U—l—c(1 + 62:)
c (Ii/I — t)stuQ] 6 sQitu 6
+(u H t)} (5.30)

for the terms that have the final-state gluons with opposing helicities. Again, the

expressions for the 520’s may be found in Appendix C.

In using the helicity amplitudes to calculate the amplitude squared, we have
neglected terms of O[c] which appear naturally from the sum over spins. If these O[c]
spin pieces are proportional to a collinear singularity, they will contribute to the finite
pieces left over from the collinear cancellation, and hence, the cross section. Therefore,

it is necessary to calculate these terms in order to complete the next-to-leading order
calculation.

While it may seem formidable to calculate these terms, one realizes that one
only needs the terms proportional to collinear poles in the matrix element, since the
remaining terms vanish when the limit 5 ——> 0 is taken. The correction term may be
calculated in this manner. For the (1234) color ordering, we find the correction term

to be

2
N2 514523 — 513524)2 ( A12 ) 1 "
2 _ _ , 2 2 __ + — . 0.31
AICOT.| — 26 (‘4) 9 (1V2 _ 1) ( S34 51343234 5122 ( )

2 /S§4 is evaluated in the limit

 

 

 

 

In this expression, everything save the (514523 - 513524)

66

$34 ——> 0. The remaining divergent pieces (as $23 —> 0 and SH —> 0) may be obtained
from this one by cyclically permuting the momenta in the above expression for the
amplitude, and taking the subsequent limit. Upon integrating this term, we find the

following correction to the (1234) piece,

. V2
Z / (g— "fr—2%) "MI—cor.2 1234 — _(—25 (‘4)2 92M) (T1 + T2 + T3), (5.32)

poles

 

where
1 Pi M1
T — -—- 1 + -,— , 5.33
1 3 SQ2 < pi ( )
__ 2 x14 4
T2 = 3 (92—) (1+ 1 j“) , (5.34)
and
T3 : T201. H t). (5.35)

The integral over the (1342) ordering yields precisely the same result. The (1423)

piece simply gives

‘2 2 V2 .
Z [(d9)51.0.|1123 (44.4) g (Ti—T17) (213nm. (5.36)

poles

Note that, when added to the previous expression, all of the extraneous pole behavior

of the form 6/Q2 is eliminated.

67

5.4 Extraction of Soft Divergences and Collinear

Renormalization

The expression for the real piece contains collinear singularities, which must be ab-

sorbed into the PDF’S. The expression for the splitting function is

d 1 1(12 1 (191714 Z l _C 1 '
U /0 I b b( ) IAIdf)b)|2(€aaZé-b’)(47r) 0(z(t—Q2)_t)

STITAP : 16713 .2— 22% F[1— 2e]
1 [1 11310.3 Pa’—>a(3)
0

+16“ 2 dflmfif]Izlfif’gl‘zkéauéb)(4W)“6(z(u—Q2)—u)- (5.37)

 

Using the 6 function to evaluate the z integrals gives the following:

 

 

1. (10 _ (477)—6 £03 Pb’——>b(zt) IAI‘EOb) |2(£av 2Ttéb’)

5 (1t AP _ 167m 6 27r F[l -— 26] —t
(47f)_€ 105 Pa’—+a(:u) Ifildf)l))l2(€uza’sfb) (5 38)
167m 6 27? F[1 — 26] —u ’

where the parton momentum fractions are defined via the delta functions to be

t. 'u,

 

[’u t, “u, — Q2
For the process in question, the matrix elements above become
~ 5])2I 4
. 0 ' _ ‘ C ‘ c ___ . . ‘ , _
IA’ISEQLQHIZKM thb’) ‘ 4(Nc2 _ 1)? 2582])1
while the splitting kernel Pg_,g(zt) is
4 4
(1 + (1 —Zt) +21) M30504”. (5.41)

 

Pg—>g(zt) :

Z; (1 '— Zg)+
The second term in (5.38) is related to this one by u H t.

68

Fr

 

The entire pole behavior of the matrix element may be expressed as

 

 

 

 

3 d0 = 0c _ 1 Pg‘_,9(zt) A18 + (3,3)4 + t4 + (.':t.5'[)'_2L/t)4
dt du Nee —t zfszpi
. . 4
_1 12“ 1+ 71262 _1 M8 + (.zts)‘1 «(t4 + (.ztspi/f)
6 6 2,2321)in
1 11 67 118+ z.e4+t4 3.21.4
_T(_+—6)l (”)2' 2+‘(19P1/l +(t—Hl)
2 6 18 3,321)in
+Reg., (542)

where “Reg.” represents all of the remaining nonsingular (as Q2 ——> 0) pieces of the

cross section. The functions P939 are defined to be

1+z4+(1—z)4)
2(1—z)

 

( (3) =1”Vc(

9‘29

(5.43)

This term, when multiplied by the Q‘Z‘ implicit in 0,, contains both soft and collinear
poles. We must take care to extract the poles from this term, and cancel them
accordingly. In addition, the term in the second line is also poorly behaved as Q2 —> 0,

although it is free of any collinear divergences.

In order to extract the infrared pole from the expression (5.42), so that we may
cancel the remaining (collinear) pole against the Altarelli-Parisi expression (5.38), we

make use of the identity

(—)—<< + (i—z)““"‘{<1i.>.—m(—->}

This allows us to write the pole term as

2 1 2 ‘ 130 111
sd(I = a 1+}r1+— E_1_ + ”+26— X
dtdu 6 12 £2 2 —t 6* c 5

_t “ M8 + s4 + t4 +214
(——) 6(622)

.2. 2
H2 '5 P1

 

 

 

 

69

1 Pg_,g(zt) M8 + (2,5)4 + t4 + (aspi/t)4
Nee —t Z1282pi

+1 log(1—2t) _10 _@E ( 1 ) x
-t l—zt + g t 1‘3t +

1 ragga) .118 + (2,3)4 + t4 + (2,3112, /t)4
Noe —t 323321’1

+1 log(1 — 2t) _ 10g _th‘_‘L ( 1 ) x
—t 1— Zt + #2t 1— 3t +

M8 + (my + t4 + (ztspi/t)4
2382121

 

 

 

 

 

 

 

 

 

111 1 M8 2 4 t4 z." 2 t“
____( ) +( t") {+2 +( [5111/ l + (t _> u)
2 6 1— z) + ztszpi(—t)
+Reg., (5.44)
where Pg_,g(z) is the gluon splitting kernel, and
1 + 24 + (1 —- z)4 r
Pgflge) = N. Z . (0.45)

The second line now cancels identically against the gluonic contribution to the Altarelli-
Parisi counterterm; equation (5.38). The remainder is finite with the exception of the
infrared pole proportional to the delta function. It should be noted that while this
expression is now well behaved for any invariant mass Q2, it is poorly behaved as

[XL—“>0.

5.5 Virtual Contribution

The second term that contributes to the NLO cross section is the one-loop virtual
correction [42]. This expression contains both ultraviolet and soft divergences. The ul-
traviolet divergence cancels against the charge renormalization counterterm, whereas

the soft divergence cancels against the soft divergence in the real contribution.

70

 

The complete 1—loop correction, including the ultraviolet renormalization piece,

may be written as follows:

 

 

 

do 4 1 + 7r2 L' MB + s“ + t" + u.“ +
3 Z 0c — IE
dtdu 3 sin
4 n 32')2 1112 + AI“: .112 —. + 2
_( _ 4) of 11 (5(2 9 pi). (5.46)
3 NC 82pl

In this expression, U6 is the universal singular piece, which has been renormalized as

described in Chapter 3 and Appendix B. Explicitly, it is

1 p2 6 11.2 f 112 E
U. = ——. —— + -— + —
62 [(—t) (—u s
2 2 2 '2 2
+ log (Zn—IL) + log <——:—7L) +10g(__27_r_1,,_>
s m” — t m” — u.
l s 1 —t l 3 lo —u 10 —u l —t
— o ——.— o — — 0 —. —.— — —-—.— 0 ~—
g mi, g mg, g mi, g mi, g mi, g mi,
2 2 2
, . m
+2Lig(1—Tfl-)+2Li2( 72"” )+2Li2( .2 ” ) 4
s m H — t m ,, — a

In this expression, we have rewritten the dilogarithm functions Li-z using several

7r2 11 3
+——+— E

 

 

 

transformation formulae. The above expression is completely real over the physical

range of the parameters.

5.6 Cancellation of Soft Singularities and
Renormalization of Ultraviolet Singularities
When the contribution from the virtual matrix element (5.47) is combined with the

term proportional to 6(Q2) from (5.42), we find that all of the singular terms cancel,

Up to terms of order 71,, which we are neglecting. Therefore the finite remainder of

71

the cancellation of the soft singularities Ureg is

3 18 NC 2 t i
2 2 2 2 m2
mH mu H
—— l
+log( s) + Og(m§,—t) +log(m2 Z—u)2
s —t u ——t
~—l — l —— —l l l ——
00(2) 00(2) 00(-mi- .)00(m .) 0% 700(2)
2 2
+2Li2(1——m—”)+2L12( m )+2L..(;'0 )
s mH—t mH—u
+

C(71)], (5.48)

2 67 11 4
Ureg : +1 +_+——+-IO g(g)2 +1—‘10g(-%—st)

 

 

 

 

where we have set 71, equal to zero. To complete the cancellation of the soft poles, the
soft singularities resulting from the process gg -—> H qr} must be also included. This

piece contributes the remaining nf/6 piece required to cancel the (complete) 60 pole.

For the present calculation, we will assume that these fermionic pieces do
cancel, and that the finite remainder contributes very little to the overall cross section.
With this simplification, we take 130 ——> NC11/6, and the pole cancellation is complete.
While this will probably give us a very good approximation to the cross section, the
gluon initiated process 99 —> H gr] is probably of the same order of magnitude as the

pure gluonic process [43], and must eventually be included in a complete analysis.

5.7 Numerical Procedure and Results

In order to generate a transverse-momentum spectrum that may be observed in na-
ture, we must numerically evaluate the convolutions of the preceding quantities with
the parton distribution functions. We choose to use the CTEQ5 structure functions
[34]. The convolutions of the matrix elements with these structure functions must

be evaluated numerically. We use a simple Gaussian integration routine, as found in

72

Numerical Recipes for C [44].

With this code, we may generate the differential cross section for any transverse
momentum or rapidity (assuming of course that these are kinematically allowed). The
process is to be plotted as observed at the LHC, a proton—proton collider running at
a COM energy of \/§ 2 14 TeV. Unless otherwise stated, we shall use a Higgs mass

of 115 GeV in our evaluations.

The leading order cross section, when shown, was computed using the CTEQSL
distributions, together with the leading order definition of as. The next-to—leading
order cross section will always be computed using the NLO definitions of the CTEQ
PDF’S, namely the CTEQSMl distributions. In particular, we have taken care to
define the K factor as it is traditionally defined, as the ratio of the N LO cross section
(with NLO PDF’s and as) to the LO cross section (with LO PDF’S and as). Also,
all contributions involving quarks have been ignored in this calculation 1. We do

however, keep 72,, z 5 in the parton distributions and the strong coupling 0,.

We shall begin with a discussion of the transverse momentum spectrum, fo-
cusing on how it differs from the leading order behavior. Then, the scale dependence
is addressed. We close by looking at the dependence on rapidity and the behavior of

the cross section at small transverse momentum.

5.7.1 pi Spectrum and the K Factor

Theoretically, the dramatic scale dependence of the cross section at LO should be
reduced significantly at N LO. Figure 5.1 shows the cross section at N LO superimposed
over the cross section at leading order over a wide range of rapidity. We have chosen
to plot bands in this case in which the scale has been varied from .SmL —> 2mi.

The leading order graph (light grey) clearly varies more over the chosen scale range

 

1For the sake of consistency, we use only the gluon-initiated graphs in the LO piece.

73

 

do

 

 

 

 

 

dPrdYB’ L
2“.
GeV (-
.01 k .01
C
l
##gmx._mr+ 4 mm L. #1141 gig iiiii .001
25 50 75 100 125 150 175 200

Transverse Momentum, GeV

Figure 5.1: Variation of the p 1 spectrum with scale

than the next-to-leading order graph. Again, this improvement is exactly what one
expects for an NLO calculation, as the scale dependence represents the theoretical
uncertainty in the calculation. As we improve our calculation order-by-order in QCD,

our dependence on this artificial parameter should decrease.

The graph of Figure 5.2 displays the dependence of the next-to-leading-order
cross section on scale (dashed line) compared with the scale dependence of the leading-
order calculation (solid line). We display the ratio of the cross section at a scale
it = Ami, normalized to the cross section for /\ = 1. The quantity plotted, then, is

basically the fractional uncertainty in each distribution which is purely due to theory.

As can be plainly seen in the graph, the leading order cross section increases
monotonically as /\ —> 0, whereas the NLO cross section appears to be approaching

a peak around dry/dag = 1.4. Eventually, the NLO graph will turn over and begin

74

 

fiLO.E=.—502;=E :'_'
NLO, pt=50' yH:0 .......

*7 fi '7 7 fi
3.5 a a - r ~

0/00

 

 

 

o + % J; I A 4* A #1

0.1 1 1o
HF/mt

 

Figure 5.2: Scale dependence at p2 = 50GeV

to drOp to zero as the scale approaches AQCD. However, it is very clear the NLO
calculation represents a dramatic improvement in the theoretical uncertainty due to
scale dependence. For instance, for a typical range .5 < A < 2,0, the leading order
cross section varies from 75% to 140% of the /\ = 1 value, whereas the NLO cross

section only varies from 85% to 115%.

In looking at Figure 5.1, it is clear that the enhancement over the leading order
approximation is very significant. The graph of the NLO cross section divided by the
LO cross section (called the K Factor) is, perhaps more interesting, since it entails
explicitly the magnitude of these corrections relative to the LO approximation. This
is displayed in Figure 5.3.

The K Factor as a function of p 1 is fairly flat down to about 25 GeV, where
it begins a dramatic decline. This is to be expected, since for small p 1, the NLO p 1

spectrum behaves roughly like log3 (p 1 /mH) / pi, which diverges to negative infinity,

while the leading order expression goes to positive infinity for small p 1-

75

 

 

 

 

 

I I l
2.4 _ K= NLO/LO — .
2.2 - _
2 _ ’11
S? 1.8 - _
I
>
x‘
1.6 - ..
1.4 - q
1.2 - -
l l l 1
O 50 100 150 200

pt

Figure 5.3: K factor at the LHC (\/§ 2 14 TeV)

We should note that our expression disagrees substantially with the previous
Monte Carlo calculation [36]. Whereas their K factor is about 1.5, ours is roughly
1.9. The most likely reason for this discrepancy is that we have not included the
Quark graphs in our calculation, whereas the Monte Carlo calculation has these pieces
included. It should be noted that our K factor is computed at y” = 0, whereas the
Monte Carlo calculation integrates over y”. As we shall soon see, the K factor appears
to exhibit a minimum at y” = 0, and therefore the integral over rapidity is unlikely
to be the source of the disagreement. Were we to compute the integrated K factor,
it would most likely disagree by an even wider margin. However, before claiming a
discrepancy with the calculation in reference [36], we must include the quark pieces

in our calculation, or at least establish that their contribution is non-negative.

76

 

 

 

 

 

1 I
I l
Rapidity: .
Y=0 —* I
0.1 :-
I
>
'0
d:
'C
D
U
0.01 '-
0.001 1 1 I -
0 50 100 150 200

pt

Figure 5.4: NLO Higgs py spectrum at the LHC (\/§ 2 14 TeV) for various values
of the Higgs rapidity, y”

5.7 .2 Rapidity Spectrum

Kinematically, it is simply less probable to produce a massive particle at large rapidity
unless the transverse momentum is small. Therefore, the cross section should fall off
rapidly for large rapidity. Figure 5.4 is a graph of the complete pL spectrum at next-
to—leading order for various rapidities, y” = 0, 1, 2, which are the solid, dashed, and
dotted lines respectively. The graphs indeed decrease rapidly as the absolute value of

the rapidity is increased.

Figure 5.5 is a plot of the complete next-to—leading order rapidity spectrum
at the LHC for p 1 values of 50 GeV and 100 GeV. This graph displays the classic
peak at 3111 = 0, and rapidly drops to zero as the edges of accessible phase space is

approached. That is, the maximum rapidity of a Higgs Boson is governed by

(P1 + m1) EXP III/HI < \/§ (5.49)

77

 

 

 

 

 

l l I [ l I ‘
NLO ——
0.1 - L0 -------
I 0.01 _
> .
D
d-
'O
O
'0
0.001 2
0.0001 “ l 1 1 l 1 1 I .
'4 '3 -2 '1 0 l 2 3 4
YH

Figure 5.5: Rapidity Spectrum at the LHC (\/S = 14 TeV). The dotted lines are the
leading order calc, and the solid lines are the NLO calc.

beyond which it is kinematically forbidden to produce a Higgs.

The rapidity dependence of the K factor is plotted in Figure 5.6. The first
thing to note is that the graph is very flat as a function of rapidity, indicating that the
phase space limitations more or less govern the shape of this graph. Also, note the
minimum that we mentioned earlier at y” = 0, indicating that an integral over the
rapidity will further enhance our expression for the K factor. Were we to integrate

OV'er yH, this would most likely increase our K factor slightly. Again, this is interesting

due to our disagreement with [36].

5.7.3 Small p 1 Behavior

The transverse momentum spectrum calculation at any fixed order in perturbation

theory is poorly behaved for very small p 1, as it is dominated by logarithms of the

78

 

 

 

 

l I I I I l i
2.4 - ‘
2.2 - q
2 V
1.8 - d
X

1.6 - ._
1.4 - q
1.2 - ..

1 1 1 1 1 1 1 1
4 3 2 -1 0 1 2 3 4

YH

Figure 5.6: Rapidity dependence of the K factor (VS 2 14 TeV)

transverse momentum divided by the Higgs mass in this limit. At each order in
perturbation theory, the coefficient of the leading divergence changes sign, therefore

rendering the predictive power of traditional perturbation theory useless in this limit.

For our calculation, we expect to see exactly this. The cross section should
display a peak in the small p 1 region, after which the cross section starts to diverge to

negative infinity. We expect that our calculation will be a fairly good approximation
beyond this peak.
Our plot, Figure 5.7, of the small p 1 region displays just this behavior (dotted

line). Again we have superimposed the leading order calculation (solid line) as a

reference. Our calculation displays a prominant peak around 2.1 GeV for our standard
choice of the Higgs mass, 115 GeV.

If one wishes to have a good prediction of the cross section in the small p 1

FEgion, one may resum the large logarithmic behavior via the techniques developed

79

 

2.5 .

=0)

1.5 - 2

doldp2dyH(yH

 

 

1
8 10

0)]—

b

 

 

0
2
D!

Figure 5.7: Small p 1 dependence of the cross section, y” = (l.

by Collins, and Soper [45], and Sterrnan [46]. The first order of business for a cal-
culation of this nature would be to expand our analytic result, and extract the 8(2)

resummation coefficient. This could then be used to verify the result of reference

[37]. In order to have a complete small transverse momentum calculation, one needs
to analytically calculate the contributions from the process 99 —> chH in the small
pi limit.

This work is a major part of a larger calculation. Whereas we have the dom-

inant part of the cross section, the gluonic piece, completed, we need the quark

contributions to have a complete and reliable prediction. One also wishes to have
a reliable calculation at small transverse momentum, which involves calculating the

resummation coefficient mentioned above. Including the quarks IS work in progress,

as is the computation of the resummation coefficient from the asymptotic expansion

of our analytic results.

80

Chapter 6

Conclusions

In this work we have seen how the Standard Model of particle physics, while close to
being complete from a phenomenological standpoint, still lacks a crucial piece, the
Higgs Boson. The advent of the LHC, which in effect will be a “Higgs factory”, will
allow us to study this component of the SM in considerable detail. If we are to under-
stand the mechanism behind electroweak symmetry breaking, we must understand

the behavior of a pure Standard Model Higgs as much as possible.

One of the most important properties to understand is the transverse momen-
tum spectrum of the Higgs at the LHC. Primarily, one needs to know what to expect
in a detector as far as the decay product energies, among other things. Since higher

order corrections to p 1 spectra from QCD radiation tend to be quite significant, one

may not consider the LO process to be reliable.

As we have seen, the results herein have verified that perturbative corrections
to the Higgs boson transverse momentum spectrum are very significant. The terms
that we have calculated enhance the cross section by a factor of approximately 2
in the large transverse momentum region. In addition, we find that the theoretical

uncertainty due to varying the renormalization scale is far less than the leading order

81

result, indicating, that our calculation is more reliable. This result also agrees with
the expected asymptotic behavior in the small p 1 region, and agrees qualitatively
with the Monte Carlo calculation by DeFlorian et.al. [36]. In order to precisely
compare their the results with this calculation, the NLO quark terms, which have

been neglected, must be included.

In the next ten to twenty years, much will change as to our understanding of the
fundamental building blocks of nature. In particular, many clues to the fundamental
phenomenon behind electroweak symmetry breaking lie in wait to be uncovered. The
discovery of the phenomenon behind the electroweak symmetry breaking mechanism
will probably be the first significant increase in our understanding of particle physics
since the development of the Standard Model in the 1970’s. Discovering the Higgs,

and understanding its properties, is the first crucial step.

82

Appendix A

Matrix Elements

In this appendix, we include the expressions for the (unintegrated) matrix elements
for the 2 —-> 2 and 2 ——> 3 gluonic parton processes, computed in the large quark
mass limit. Moreover, we include the expressions for the 2 —> 2 process H —% gqq, so
that the complete leading-order transverse momentum spectrum may be computed
in the large mmp limit. H —> ggg and H —> gqfi amplitudes have been known in the
literature for a while [33, 26], as have the expressions entailing the process H ——> gggg
[47]. Moreover, the amplitudes for the virtual corrections to the 2 ——> 2 partonic
processes H —> 999 and H —> gqfi may be found in reference [42]. For simplicity, we
give the matrix elements for all particles, except the Higgs, in the final state. The
matrix elements relevant to our calculation can be obtained from these by crossing

symmetry.

Since we will be using color-ordered helicity amplitudes in our calculation
we will begin by reviewing the standard notation and identities associated with the
decomposition of QCD amplitudes into color-ordered, gauge-invariant subamplitudes.

We will then present the matrix elements relevant to this calculation in their entirety.

83

A.1 Helicity Amplitudes

We begin by presenting the standard notation used in calculations of amplitudes, and
list a series of identities useful in simplifying these same amplitudes 1. We introduce

the following notation for (massless) Dirac spinors,

UMP) : [Pil- (All

In addition, we shall use the following notation for spinor products,

fl—(1))‘U+(q) = (p—|q+>=(pc17). (A?)

admit—((1) = <p+|q->=[PQI- (A3)

The momenta p, q are assumed to be massless, and therefore these may also be iden-

tified with relativistically invariant chirality states.

The polarization vectors of the gluons are parameterized in a dual basis [21, 22]

,. _ is 41101111» ,
4.1.12 “ «02 <12: 1m ’ (A4)

 

where p is the momentum of the gluon, and k is some arbitrary massless reference

vector.

Products of these spinors are found to satisfy the following identities (for

massless p, q):

(pq>lqpl = 210-11 (11.5)

09(1)" = 8311(1) - 9)lqpl (A.6)

 

1Those who are new to helicity spinor techniques will find references [19] useful.

84

 

————(1 127% = lpi><1vit (11.7)

From this last relation, we see that we may derive the following identity,

1
(Hal — p2><-p2| + p3><+p3| - p4) - - - <—p..| + 191) = §Tr {(1 + 1's)1511$2153164 . . .15..) .
(A.8)
This last relation is particularly useful in computing absolute squares of matrix ele-

ments, and converting them into traces over products of gamma matrices.

A.2 Real Contributions

In order to calculate the p 1 spectrum to next-to-leading order, we need to write down
all the separate partonic contributions to this process. They can be broken up into
three separate terms; the leading order or tree-level process which consists of Higgs to
3-partons , the virtual or 1-loop correction to this process, and the 4-parton process,
which we refer to as the real corrections. The (unintegrated) expressions for these

have been computed in works outlined below.

One may compute the spin- and color-averaged matrix element squared by
simply summing over colors, adding up the 2” spin terms, then dividing by the
number of spins and colors of the particles in the initial state. That is, if N02,, are the

number of spins and Cay, are the number of colors of the initial-state partons a and

b,then

I
— CaCbiValVb

 

lM(a, b; 1, 2, . . . , 71.)]? Z [M(a’\“,b’\b; 1'" , 2A2, . . . , n"")|2, (11.9)

where the A,- represent the helicity of particle 2', and the sum is implicitly over the

spins and colors, unless noted otherwise. Generally, the amplitudes for the process

85

 

involving initial state partons may be computed using crossing symmetry, and being

careful to note that certain terms can pick up a sign via equation (A.6).

A.2.1 H —> ggg

We shall begin by writing down our expressions for the Higgs production matrix
elements. The matrix element with the full c-dependence was first computed by S.
Dawson [26]. It is fairly straightforward to compute from the conventional Feynman

rules (together with the non-conventional ones from our low energy Higgs model).

There are 2 independent subprocesses that contribute to the leading-order

expression. The H —> ggg piece is given by

 

AfabcA'I4
M 1+,2+,3+ g ” , A.10
( ) fi(12)(23)(31) ( )
A 2 2 2 3
11,1(1-,2+,3+) .__. _g_fb_[_3]_ (All)

V5l12ll13l '

The remaining contributions may be obtained through parity and charge conjugation

operations. If we sum this over spins and colors, we find

92.42NC(NE — 1)

M H——> ( 2=
XI ( 991M 512513523

 

(5?2 + sf, + 5:3, + 1113). (11.12)

The remaining subprocess that we will consider for the leading-order transverse

momentum spectrum is H —+ qq‘g. The matrix element for this subprocess is given

by

ingA [13]2

111(1+,2-,3+)=— \/2 [12]. (4.13)

This may be also be easily summed over spins and colors

922112 (Ng —- 1) U2 +1.2
4 s '

 

2 WW -> quH2 = (A.14)

86

 

It should be noted that each of these amplitudes were computed using the conventional

normalization for the SU(3) generators, Tr[TaTb] = £6225.

A.2.2 H —> gggg

In order to facilitate the calculation, it is beneficial to break up the subamplitude

into color-ordered helicity sabamplz'tudes

111(1*0,20‘0,3*0,4*0) = 2.492 2 Tr[T“TbTCTd]m(1’\1,2*2,3A3,4"4). (A.15)

colors

These amplitudes are cyclically symmetric in the colors, they are invariant under
reversal of the color indices, and they satisfy the dual Ward Identities [19]. There are

three independent helicity subamplitudes for this process:

 

 

 

 

 

 

 

+ + + ,+ __ M:
m“ ’2 03 ’4 )— (12)(23>(34)(41>’ (M6)
_ . _ came—>224]:mam—>123]?
7”“ ’2 ’3 ’4 l‘ “ 8.2.3.251. 5.2.8.2323
_<1—115112—>2134122 [241 {S £10110:
5.3.5.153. [121(23><34>[411 2“ (41>
__ H4..
+534“————‘“—<1 (“162; >“l24lS234}1 (A17)
(A.18)
and
m(1‘, 2—,3+,4+) = — <12>4 [3414 (A.19)

(12)(23)(34)(41) — [12]]23][34][41]'
The remaining terms can be related to these by using the charge conjugation and

time reversal transformations, and the dual Ward identities.

87

A.3 Virtual

The expression for the virtual H —> 999 amplitude was calculated in reference [42].
We define the amplitudes in precisely the same way that we defined the leading-order
pieces, in terms of helicity amplitudes. The resulting expressions are then related to
the leading-order amplitudes via a coefficient. These are outlined in the following

subsections. To construct the contribution to the amplitude squared, use the formula

[11(0) + MW2 2 Z |M<°>|2 + 2 2 114.11”) (11(1))‘1 + . .. (A.20)

spins spins

In this expression, 111(0) and M (I) represent generic L0 and 1-loop amplitudes. For

clarity, the spin dependence has been suppressed in this equation.

A.3.1 H —> ggg

The expressions for the virtual H ——> 999 subamplitudes are

 

 

 

 

(1) + + + (0) + + + 08. 471/12 ( NU
111 (1,2 ,3) = AI (1 ,2 ,3 )4—7;71(:_1fi;) “ c .
1V , 3:31523+531512+512523]
+§(‘ C 7 '1’) M},
2 c
, _ . _ + 3: 47W) yr 1121
M“)(1 ,2+,3+) = 111<O>(1 ,2+,3 )47rrr(_M%l 1.0 ( )
1 531512]
- NC — Tl ,
+3( f) 5023
where the prefactor is
2
_ r<1+e>r2<1—e> z _I‘_(1_::_>_[.+.21), (1.22)
T“ — F(1—2€) F(1—26) 3

88

and

U = 12[_(—M§,)<_(—M',22)6_(—M§2)0 +1?
6 ‘512 ‘523 -531 2
‘512 ‘523 —312 —331 —S-~ —S
-1( )1(—-)-'( .) (—-')- ( 2") (—'“)
n —M,2, n —Mf, n —M,2, 1" —M,?, I“ 411222 I“ —M},

. 512 . S23 . 5'11
-2L(.__)_2 .(__.)_ (-——‘;—).
12 A01}, L12 1 Ali, 2ng 1 111,2, (A 23)

 

 

 

 

 

 

It should be noted that all of the singular behavior (including as pj -—> 0) is contained

within terms proportional to U.

Since we are working with an effective theory, we obtain a finite renormaliza-

tion at each order of perturbation theory,
111‘” —> M“) + (A +362) 111(0) , (.424)

where A is the finite renormalization of the effective H 99 operator, at this order

in perturbation theory, and (59 is the gauge—coupling counterterm. Using the MS

subtraction scheme, the counterterm is

 

103 6 111VC n;
52 = —;7r(1+€)(47r) [ 6 "5], (A25)

and the finite renormalization coefficient is

A = 11 (30%;). (A-26)

89

Appendix B

Formulas for Angular Integration

In this appendix, we give detailed formulas for the angular integration of all the terms

appearing in the partial-fractioned matrix element. There are two different cases:
0 when the two invariants involve massless particles, and
0 when one of the invariants contains a mass term.

Momentum conservation may be used to reduce all of the matrix elements into sums

of terms of this form, multiplied by a coefficient which has no angular dependence.

2 1
Bl General Expressmn for terms of the form #512. 32.3

' ' ° ' ‘ - 00 ' 0 ' . . 0 two
We begin by evaluating integrals 1nvolv1ng two massless invarlants. There are

forms that occur, and these are related by a u ——> (1 — V) symmetry:

(13.1)

 

1
mu m n (C)
m ’)(u)=S2,S,/dfl 32225224,
(13.2)

 

1
m,n _ m n ('5) .
Q( )(l—V)’Su St [d9 88323

90

We recall the explicit defintion of V,

1
u = ~2—(1—cos(00))

‘2
z 20— (B...

Those that involve the other combinations of the invariants are related to these via

the freedom to relabel the final state parton indices, that is exchange 3 <—-> 4.

First, we evaluate the phase space in the Q2 rest frame. Then without los-
ing any generality, we perform this integration assuming that the invariant $13 is

independent of the azimuthal angle to The following definitions are useful:

11213 = 513/(313 + S”), (B4)
2 513/5211 (B5)
to” = 51.1/(513 + S”), (B.6)
: 511/511, (B.7)
can 2 523/(523 + 524), (B8)
: 523/3“ (89)
1112.; = SIM/(523 + 524), (B.10)
= 5.22/32. (8.11)

Taking the gluon a to lie along the z axis, we find

(4)13 =3 -;-(1— COS(913)), (8.12)

and

(1123 = $0 — cos(6lo) cos(0) — sin(60) sin(0) cos(¢)). (B.13)

91

The integrals (B.2) become

900%) = / 291—612222—m2222—n (13.14)

Q(m”‘)(1—V) = fdQ(“)w13‘mw23‘". (13.15)

These may be evaluated in general in terms of hypergeometric functions [48]. The

explicit expression (for arbitrary e) is

F2(1—6)F(1—n—6)F(1—m—e)
Q (V) F(1— 26) F(2 — n — m — 26) 1’2“”, n, 6 V) ( )

 

One may easily prove that the relations for 3 H 4 yield precisely the same result.

. 1
B.2 Expressmns for terms of the form 5" Sm
13 123

For these expressions, a considerable simplification is attained if one rotates the pre-
ceding frame so that the Higgs momentum lies in the direction of the z axis. In the

Q2 rest frame, this is equivalent to a frame where a + b = :3 la + b|. The rotation

angle is then given by
—B (Bin

COS(X) 2 W1

and therefore the invariant 5123(4) is given by

5123(4) Z A ¥ V B2 + C? COS 9H1 (B.18)
where we have defined
1 (13.19)
A = -2—(23 + S, + 5,),
B = $(Su + Stcos 60), (B20)

92

1
C : 5(Su + St COS 60). (8.21)

The kinematic invariants 513, etc. take on the requisite form

51.
$13 = 7 (1 — cos xcos 6H — sin xsin 0” cos ab), (8.22)
S
523 2 5t (1 — cos xcos 0H — sin xsin 9” cos a). (8.23)
(B.24)

where the other terms are related by B —) —B.

We therefore seek to evaluate expressions of the form
1222'" = / (11232720052213. (B25)

All of the other angular form factors may be obtained from these by either exchanging

B <—-> —B or u H t. Their explicit expressions are (to O[€0])

 

 

 

 

 

1,1 _ 1 0.1 (i) 1 l (A + B’COS XV B 26
Q” _ A+B’cost‘ (V)+ 2a A+B’cosx 0g Az—B’2 ( ' )

2,1 _ 1 0,1 (i) 1 X B27)
0” _ (A + B’ cos x)2Q‘ (V) + 2a (A + B’ COS Xlz (

 

A2 — 8’2 + A2 — B’2
I 12 0,1
12102 1 120026) + (AB COS,X+ 82) 9‘ M] (8-28)
A+B’cosx ‘ (A+Bcosx) a
1 [AB’cosX + B’2 (A + B’cosx)2 _ 2(B'Si" ’02 ](B.29)

[I (A + 3’ cos 2102 28’2 + 2AB’ cos X]
0g

 

 

 

 

 

 

 

 

 

2? (A + B’ cos >03 0g .4? — B’2 (A + 8’ cos x)3
”2,2 = 9202(11) + (3(B’ cos x)2 + 28’ cos x(A + B’ cos )0) x
H (A + 3’ cos x)2 (A + B’ COS X)4
9930/) log W) 4(B’ sin x)2
( 2a + a2 — 02(A + 8' COS X)4
r2
+ B (8.30)

a‘2(A2 — B'i’)(A + B’ cosxl"

93

1 A+B’

121,)“ = ——1
213' 0g A — ,
2,0 1 B
9,, = ——-
A2 _ 30
92—1 2 i l
2 B’ [(B + Acos ”QED — cos x]
S23“ 2 E. ' , ‘
1 B’ [(B + A cos Xmfio — cos xflh’o]
Q 9_2 = i I
H B’ [(B + Acos >092” — cos xflgfl]
(aSin X)2 12
H 61—232 [(3 — Ami)" +14]
(2,} = — [(B'+Aco S2’—1
B’ sx) 2H — cos lefll
(aSin X)2 I2 ‘
—_2B'2 [(3 —AZ)12§)0 —1+2A§21’°
92—3 : a BI H
B’ [( + A cos ”OZ—2 — cos xfl2f2]
3 (a sin x)2
§—-—B,2 [(30 — Am“ 4 a
92:3 a H +1 (2,, +B'(—)cosx]
H : —- [(B’ + A C ‘ ill—2 3
B’ ,os x)QH — cos xflE—2]
3 (a sin x)2
5—812 [(80 — AM?“ + 49"" ’ a
H ‘ H +B(§)cosxfl}}°],

where B’ = qa/B"2 + C2

 

94

(B31)
(B32)
(B33)
(B34)
(B35)
(B36)
(B37)
(B38)
(B39)
(B40)
(B41)

(B42)

Appendix C

Results of Angular Integration

C.1 Expression for Regular Pieces of H —> 9999

Matrix element

In this section, we give the detailed results for the non-singular parts of complicated

(+ — ——) helicity configurations. Recall that the sum over spins of the H ——> 9999

amplitude may be written

2 2 2
_ . as N (N ”1) /\1 /\2 3A3 4A4 2+
|M|2 = (43013)2 (——) ————.———2 lm(1 ,2 . , )l
31m 16(N2 — l)2 hel.{

(7n(1*1,3’\2,4"3,2*4)|2+|m(1*1,4*2,2*3,3*4)|2}. (C.1)

We write the expression as

 

as 2 N2 (N2 — 1)
) 8(N2—1)‘2

{7n++++ + Tn++__ + m+___} , ((3.2)

+ + + 2
m++++ =|m(1+12+a3+14+)l2 + l7n(1+13+14+12+)|2 + lm(1+’4 72 ’3 )l ’ (0'3)

95

where

m++-—=|m(1+»2+,314‘)!2 + |m(1+, 3“,41‘,2')|2 + lm(1+, 4+, 2‘,3‘)|2 +
|7r2(1+,2—,3+,4—)|2 + |m.(1+,3-,4+, 2—)|2 + |m(1+,4-, 2+, 3—)|2 +

mm, 2", 3‘, 4+)|2 + |m(1+, 3‘, 4‘, 2+)|2 + |m(1+, 4‘, 2‘, 3+)|2. (C4)

The angular integrals of the preceding expressions are presented in their entirety in
chapter 4. The remaining pieces, which we shall refer to as the odd helicity pieces,

are

771+++_=|m(1+, 2+, 3+, 4”)|'2 + Im-(1+, 3+, 4+, 2-)"; +l7n(1+14+72+13_)|2 +
|m(1+, 2+, 3‘, 4+)I2 + |m(l+, 3+, 4‘, 2+)I2 + |m(1+,4+, 2‘,3+)l'2 +
|m(1+, 2‘, 3+, 4+)I“2 + lm(l+, 314+, 2+)I2 + |m(1+,4', 2+, 3+)]2 +

|m(1+, 2'23", 4‘)I'2 + |m(1+,3_, 4‘,2‘)l2 + |1'r1.(1+,4_,2_,3’)|2. ((3.5)

We define the normalized angular integrals of these expressions by
Q+++_ : 2/ (19“) m+++_. (C.6)

dQ‘ is the angular measure in 4 — 26 dimensions from appendix B,

/ d9“) = .1— /" d6 (5111(0))(1—2‘> / dq’b(sin(¢))(‘2‘). ((3.7)
27r o 0

Using the various symmetries, as well as the fact that we may freely exchange

the partons 3 and 4 (since this does not affect the outcome of the angular integration),

we may rewrite this as

Q+++_ ——— 212(1, 2,3,4) + 29(3,4, 1,2) + (20,3, 2,4) + 0(3,2,4, 1) + (t H U), ((3.8)

96

where

90.1.31): / 49‘ ,Im(r,j+,k+,l+)12. (<19)

Finally, we break these into a fairly simple singular term Q.(i, j, k, 1) (given in
chapter 4), and a nonsingular piece (20(2) j, k, l), which contains no 6 or Q2 singular—
ities. We have chosen to include a number of (universal) nonsingular factors in the

expressions for {2.03,}, k, l).

C.1.1 m(1-,2+,3+,4+)

First, let us define

 

 

 

 

(C.10)

Then we may write the integral over the angular phase space as

120(1,2,3,4) _—. F(0)(.s,t,11.) + F‘”(s,t,u), (011)
where

8M4 2 2 8M4s 411125,, M2 (~4M28+5H2)

F(0)(Stu):— QPJ.._ .u+________+ M2 +
l a u4 3112 su .su,
Q2.+ c.22—M‘2)S..)2 3 4
log(( 9 ($1255.22 ) (M + (lg-PS“) ) +

 

su (3 +51.) SH

. 2
t3 10g( (Q23+)(:22:_:122)St) ) t310g(ff:§—{:T7) (Q21! +85” +113”)
“ 41

 

 

 

 

 

 

 

s (s+s.) SH + Q23u(s+Su)SH
3142 M8Q8+t4u4+s4pfi_ 2M4t3
lOg(M'ZSu2) (22.<321441)J_'2 sup)? (5+Su) (3+St)
M8+(s+S.J:__f t3
—Su(S+Su) SH 8(S-i-St) S};
t3 (Q2u+SSH+USH)), ((3.12)
— Q2su(s+Su) SH

97

and

 

 

 

 

 

 

 

 

 

 

 

 

 

2 ‘2 _ Q . 1
F(1)(stu) = M (7'S+M ( 1+Sn)) +10g(1—‘_LZ)C,,+2M4I‘(_2‘2)(u,t)
, 5 s2 77 82
2M4 (3+Sul (I‘(_1,1)(u,t)—I‘(_1,1)(t,u))
s25”
2M4 1"—21 (U1tl+P—2 (tau) 6M4I‘ - t
+ ( ( ,) 2 ( ,1) )+ (—1,2)(u, ). (0.13)
8 SH 3
The angular form factors in F(1)(s, t, u) are
1 1 .7 .
F «(1115) : log(f‘i—él x 43 (5.3+spt2—3Q2) (2M2Q2—u(t+u))
(—2’2) ’ ZSHU 2_ 2 2
(5;; 43M
431196221)? (53+.pg_.Q2)2_432Q2p,2
2 . 2
(SH2-4SM2) (SH2—48M2)
2
s (2 M2622 —u (t2+ 1.)) (e14)
(SH2-4SM2) M2
1 .
(2M2Q2—u(t+u)) 55+3P42—3Q2 log(1—2,1) C
P -12 (u t) = + 2 2 ( -15)
( l ’ (SH2_4SM2)M2 SH ~4SM SH"
r ( log(l—fil) s2 (2114262?-—u(t+u))2 233P?Q2M2
_ 1,t = —- >< - 2
( 2’” ‘ ) Sm; (sf—43M?)2 (—4M23+SH2)
. 2
+ sgsH—zsusu +SH232szi-(Su3H—isul , (C.16)
(—4M23+SH2) 2 (3H2 —-43M2)
and
s (2M2Q2 —u (t+u)) log(i—iil) -23"+SUSH (C.17)

 

 

I‘(_1,1)(u,t) = —4M23+SH2.

S” (—4M23 + 5H2) 7?

98

Finally, the coefficient C” is in this case

—21:3(1+u)2 2
C = _ _ 2 3 2
,, (228115”, (223115”? [ 2M t (2Q +t+u)

+M6 (——4Q4 +6Q2u) +2M4Q2 (412—216—62+Q2 (t+u))].(C.18)

 

Note the existence of a “naive” Q2 pole QO( 1, 2, 3, 4). It is superficial, since the limit

as Q2 —> 0 yields a finite result.

C.1.2 m(3-,4+,1+,2+)

Again, we write the integral over the angular phase Space as a sum of logs,

5'12 10g[1_+_11

90(3, 4,1,2) : C0 + C1 + C1} 771—1)

10% 1712—52
(M2 — Q2 + s)3 O (Q2 s + (—M2 + Q2) 5,.)2
3(1)]2 — t)u g M2 3 Sn2

 

 

((3.19)

The coefficients C,- in this expression are

2 . 4.2 16 4s 27 4—12Q2s+13s2
(:0 : 63Q3+1709+8Qt49 + g + Q 312 +
s l
3Q6—18Q48+5233 s2 432 ll (Q6—3Q48+3Q282—.S'3)+
3Q28t +3_u—2+3Q2u 3Q2tu
1162432 38622.92 2Q2s3 —9Q2s+7482 —9Q2s2+553+
_ _ 2
3514 3&3 3145;3 35)2 232148) 3
22Q6+66Q48+129Q282—5283+—18Q48+27Q s —lls +
3Q2sSt 3Q2U51
3
235. 235. 235, 5.2 25}.2+11(24s2 34Q2.32_2Q2s3+
s + 3112 + (.2211 +3—1;§+3Q2u 3.9,,4 381,3 3tSu
2
2Q2SSt 8(3Q2+268) 3Q232—7S3+3S§¢+ St2+
S3 + 352 3tSu2 S. 25..
u u 2 2__ .3
22Q6+66Q4s+93Q232—433+—6Q4s+15Q s 113 +
3QQSSu 3Q2tSu 2 2)S
4 4 —28
11(Q8+4Q63+6Q432+4Q233+23)+(11Q +2632ng 1+
6Q283¢Su u

 

 

 

 

 

 

 

 

 

 

 

99

2112—-._~52 255, 24 4~ , ‘2 4 2
( Q s) ,+ + Q55 _10Q35,+3135,+24Q 5,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 Q2 s 5, s t4 T S? t‘1
17 8,2 8 Q4 5.13 _ 8 Q2 5:13 5113 + 8 (4 Q4 511 -' Q2 8 Sn)
2 5:2 s t4 s t3 s t2 t3 +
11 Q4 5, + 34 Q2 s 5, — 26 .52 5, _ 2 (6Q4 5, —— c2235,-13325,) +
Q2 s S, Q2 st
27 Q4 5, — 66 Q2 3 51+ 26 s2 51 + 2 (11 Q2 5112 — 13 5 5'12)
3 s t2 3 Q2 3 St
2 (27 Q2 5,2 - 8 s 5,2) go2 5,2 + 26 5 5,2 16 (Q4 51:2 - Q2 3 5qu
— 33t2 + 3Q2.st + 5t3 ’
(C20)
and
N
C1 1 (C21)

 

: Cyst“ 11(1112 —- t),

where the numerator N1 is

N1 = ((1112 — Q2)1 + s“) 5,3 (s + 5,) — Q10 (6 s3 —- 5, (—2 .52 + 2 s 5, + 5,2)

+(4 s + 5,)2 S, + 3 (4s + 5,) 8,2 + 35,")

+622 5,2 (6 35 + 2 s4 (7 5, + 3 5,)
+33 (34 5,2 + 34 5, 5, — 6 5,?)
+3 (5, + 5,)2 (115,2 + 22 5, 5, — 5 5,?)
+1032 (5, + 5,) (3 5,“2 + 4 5, 5, — 512)
+(5, + 5,)3 (5,2 + 5 5, 5, — 53))

+626 (2 35 + 6 5,5 + 5,4 (31.5 +13 5,)
+5,3 (40 32 + 20 s 5, —- 2 5,?)
—5, (23+Su) (4153 +1435, +1135u2+35u3)
-2 5, (3 + 5,) (7 s3 + 27's2 5, + 2335,2 + 65,3)

+512 (1633 — 2511 (1132 + 2585“ + 953»)

100

+6245, (4 5,5 + 5,4 (293 + 175,) + 25,”2 (2s + 5,) (1232 + 9.95, — 5,2)
+2 5,3 (29 52 +3235, +95?)
+2 5, (3+ 5,) (333 — 55, (.s+5,) (23+5,)) +
(.9 + 5,) (6 .54 — S, (1253 + ‘28 .52 S, + 163 5.12 + 35113)»
_2Q8 (.254 _ 5,3 (73+5,) +5,2 (—3s2+838,+55,2)+
5, (53+35,) (32+25, (25+5,))

+(s+S,) (633+S, (1632+3S, (4s+S,)))).

 

 

 

 

(C22)
Finally, The coefficient C,, in equation (019) is
C 10 (Q2 - 2 8) s (W2 - Q2)4 + 8“) (1‘ %)
7) : , + ‘h +
tu Q2St’u
4-9(Q“-3Q23+332) 25h(—4(Q2—38)s—6SS,+S,2)
. + . (C23)
tu Sh s t '11,

(3.1.3 m(1-,3+,2+,4+)

Recalling S H = A12 + 3 — Q2, we write the result of the integration over the angular

phase space as

 

 

 

 

 

, 2 ,2 , 4 2 2 [4 log 1___'l
5200,33,,” = 2M; _16M 4Q p, +412 (1,)
SU u SH
8112 5t2
1 ————-— 1 ————- C
+ “3 1, (3622— (212—6225,?)
(Q2 —t) (M2—u) 5, g 5312123

 

2 (M8 + (M2 — a“) 1 (8622 - (1112 — Q2) so?)
+(M2—t)(Q2—u)uSH og( SEM23 '
((3.24)

101

The coefficients are

Cleys (11121—11) 1143,, x [—2 (1M2 —.5) t3u4+21l12t3u3 (2Q2+t+2u)

+4 A110 (2 Q6 — 2Q4 u + 2622 122 — 11.3)

 

—4M8 (Q4 (t—5u) u+Q6 ((+311)

—u3 (15+311)+622-21.2 (t+6u))
—4 M4 112 (Q (12— u 2)+

Q2 (t+2u) (2t2+u2 ) —u2 (2t2+tu+u2))
+4212” (Q6 (t + u) + Q4 (12 +111 — 4112)

+2Q2n (2t2+tu+3u2) —122 (2t2+2tu+3112))] (C25)

and

2t3 (1M.2 (2Q2—2t—u)+t (—-Q2+t+u))

. . (C26)
3 (ill2 — if) (Q2 — t) (‘2 M2 — t — u) u

 

C2

(3.1.4 m(3-,2+,4+,1+)

The last independent helicity amplitude is m(3",2+,4+,1+). We write the finite

remainder of the the azimuthal integration as

 

 

 

 

906,214,” = $C0+C1 logfigg
+C2 log( (Q2 3 -1821: E11522) 511)?) + (u <—-> t), (C27)
where
C, : _( (M2—1)3 ) _ (2t+u) (12+tu+u)21116 (Q4t-{t—(47t—u)u2)
t(—1V12+u) stu 3 u

102

+M2 (—4t3+3tu2+3u3) + M4 (26221 (t+u)+u (14,2+4tu_3u2))

 

 

 

 

Stu.2 stu3
_(—2M2 +t+u)3
suSu ’ (C28)
5:13
(3. = _ ..
2 ((112 — t) u 5,)’ (C 29)

and finally, the coefficient Co, which is symmetric under the interchange u <—> t:

8M4 — 4M2t 2M2 Q2 (4M4 — M21— 3(2) 8M4Q2t 71114622

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CO 2 5 + t2 _ st3 511,3 ml2
M2 (414(1— 11)2 + t2 (t + u) (t + 321) + M21 (—12 + ta + 4112))
+
st2u2
1112 Q4 (—8 11121114 +8113 (—4t + u) + 8 M4 (i4 + 214)) 6 M“ t2
+ + 4
5 t‘1 u“ S,
2 (3 A!“ Q2 — 2312021) A12 qu 21l12 qu
_ stu 125,4 115,“
-—10 1112 Q“ 11 — 3 A12 t2 u + Q4112 + 4 62215112 + {2112
+ ,
31
6 NH (2 M2 +1) 2t (Q2 + 2t) u 10 M4 + 411.12t+t2
_ 3 + 3 + 2
St St St
211—141 (M2 — '11,) 4M2 (M2 +1) 21114t+1112 (—8M2 + t) 8,
SU St? _ tSt 3U. 53 t3
GQ“ (M2 —1)'2 602 (M2 — t) (Q2 +1) 4 M2 — 8 M2 Q2 + Q4
5,4 ’ 5,3 + Sf
22144622 2 (2M2 — 5'1622)1‘,+t2
+"‘_‘_—t 51‘2" — Su2 Su‘Z
+2(11121 (t— 311.) +1 (Q2 + 2t) u +1114 (—2t+u))- (030)
tuSu

103

Appendix D

Formulas for Cross Sections

Here we provide standard formulas for the evaluation of 2 —> 2 and 2 —> 3 processes
for 2 partons in the initial state, 1 massive particle in the final state, and 1 or 2
massless particles in the final state. We also list convenient variable transformations

for the convolutions, which will allow us to directly evaluate the “+” functions.

D.1 Expression for a 2 —> 2 Cross Section

The 2 —> 2 contribution to the 19, may be written

 

daLO 1 1 [1 do 2
_______ = _ , a d __5 , D.1)
dp1 dyH S/o d€,f(€) 0 €bf(5b)3dt (Q) (
where

Q2=s+t+u-M2 (D2)
and

$27. = 41 2|M|2, (D3)
1r5

104

where we have suppressed the parton indices for clarity.

The expression for the N LO cross section may be defined likewise:

 

 

dUNLo 1 f1 1 dog .
——_ :- __ d a/ d 6 2 .
d])_L dyH S 0 g 0 6” Sdt (1U (Q )7 (D 4)
where
day} 1 (19 —
Z _ -- A” 2. .
(it du 42.52 27rl 1' (D 5)

D2 Evaluating the Convolutions

First, we must evaluate the expression

fol din [)1 dfb 0(Q2) : A1 (160,41 dgb 6(faéb — 176a "" IUEb + 7), (D.6)

2 -’I' 7712 —U 1712 . . .
where, as before, 2T 2 3%, 17v = 415—, and T = 3.21. Evaluating these in the

hadron-hadron COM frame, we find that these expressions become

(‘2
"IL
-H
IT: 56y,

2 6+2”. (D.7)

mi

S

:1? U

Substituting expressions (D.7) for Q2, we find

2
m mi P1
fag!) — $T€a - :L‘Ufb + T : (file—y” — 7—?)(gb€+yfl — 7g) ‘ '37, (D8)
which has the implication that
2 ~41! _ L“ Pi (D.9)

(teem - .3?ng fl) > S-

105

 

We may break this up into four different phase space regions, each defined by (5, beiy” ——
mi/x/S) greater than or less than p 1 / \f3— . One case of the four is automatically for-
bidden because it implies p 1 < 0. After evaluating the H-function for each term, we

may recombine two of the regions into one region, leaving us with

/01 (15a [01 déb 0(Q2) : A: (1g, A1150) (15, + £1— df, Alb) (1,5,, ([110)

The lower limits of the internal integrals in this expression are solutions to the equa-

tion Q2 = 0 for each of the variables. Explicitly, these are

{(450) = (413T — T)€a — T(1 ‘51:),

 

 

517U — {a
:1: r —— T —— T 1 —-
60(6)) : ( 1, )5b ( {(2), (D11)
513T — (Eb
whereas the outer limits .Ti are
/__+;_\ _ (11.12)

x4 = exp(iyn)( 3

It is quite trivial to convert these terms back into the Q.2 variable. If this is done, the

expression for the phase space in reference [49, 50] is recovered identically:

1 1 2 1 d6, “‘1 Q: l__d§b__ 421(93). D.13
/0‘d€a/(; (jg-(26(Q)—> /0 d(S)+ 2:_€b"-75T/(l S ( )

r+ {a — £130

 

In this formula, the limits of integration A,- are

A1 :- €a(1 — SET) ‘- IL'U + Ta (D14)

A2 = 17+ (5b - 513T) ‘ 231/5!) + 7' (DIS)

. . ' h'
Note that it is quite trivial to evaluate the 6(Q2) of the 2 ——> 2 phase space usmg t is

106

 

form of the convolution.

D.2.1 Evaluating Terms With “+” Functions

In the regularization of the soft singularities, terms of the form f (z,,,)+ arise, and it

is beneficial to change variables from d(Q2) to dz“, in order to easily evaluate these

“+” functions. Recall the definition of 2“,:

—t, u

z = ———.—.
t’" —t,u + Q2

(D.16)

Now, consider 2:, (a completely analogous analysis will follow for 2,). It is beneficial to

first convert to the phase space (D.13) as an intermediate step. We wish to evaluate

the Jacobian for the change of variables. This is summarily expressed as

4“ . 1 d3 (“02 1 dzu (—u)2
d 2 —)/ .7 —)/ . 7
[1 (Q ) 7 (.4) sf —t0 MA) 2'5 —U~0

~f

 

 

where to = t|Q2:0 and 210 = 14,2220. Thus

1 l
/ dfa/ dae(e2)—+
0 0
1 1 d6, (1:4,(—t)'2 1 1 (15b [1 dZt(—t)

2
Aria!) zt

 

 

 

 

1
g 2+ {a — fL‘U /z,(.41) Z? —to S I- {b — 1137‘ “to

(13.17)

(D.18)

The evaluation of “+” functions in this basis is quite straightforward. Converting the

Phase space into 2, proceeds along similar lines.

107

 

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