Parton Densities in Quantum Chromodynamics: Gauge invariance, path-dependence and Wilson lines 9783110439397, 9783110430608, 9783110430684, 9783110430615

Citation preview

Igor O. Cherednikov, Frederik F. Van der Veken Parton Densities in Quantum Chromodynamics

De Gruyter Studies in Mathematical Physics

Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, Sa˜ o Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia

Volume 37

Igor O. Cherednikov, Frederik F. Van der Veken

Parton Densities in Quantum Chromodynamics

Gauge Invariance, Path-Dependence, and Wilson Lines

Mathematics Subject Classification 2010 35-02, 65-02, 65C30, 65C05, 65N35, 65N75, 65N80 Authors Dr. Igor O. Cherednikov Universiteit Antwerpen Departement Fysica Groenenborgerlaan 171 2020 Antwerpen Belgium [email protected] Dr. Frederik F. Van der Veken CERN CH 1211 GENEVA 23 Switzerland [email protected]

ISBN 978-3-11-043939-7 e-ISBN (PDF) 978-3-11-043060-8 e-ISBN (EPUB) 978-3-11-043068-4 Set-ISBN 978-3-11-043061-5 ISSN 2194-3532 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

8

© 2017 Walter de Gruyter GmbH, Berlin/Munich/Boston Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck.  Printed on acid-free paper Printed in Germany www.degruyter.com

Preface The application of quantum chromodynamics (QCD) in practice implies computation of various hadronic and vacuum gauge-invariant path-dependent matrix elements that often contain complicated systems of Wilson lines and loops. The latter may include light-like segments, (semi-)infinite parts, simple obstructions such as cusps, that is the points where the path itself is continuous, while the derivative is not, and self-intersections, which make a path nonplanar. The purpose of this book is to give a systematic pedagogical introduction into the quantum field theory approach to quantitative analysis of Wilson path-ordered exponentials in QCD and its applications of this formalism to the study of gauge-invariant quark and gluon correlation functions, which can be associated with the three-dimensional transverse momentum-dependent parton density functions, commonly known nowadays as TMD pdfs or simply TMDs. The strong interest in TMD pdfs is due, in the first place, to the rapid theoretical and experimental development and impressive recent results achieved in the study of the three–dimensional structure of the nucleon, which suggests that not only the longitudinal fraction of the struck parton momenta, normally associated with the Bjorken-x variable, but also the two transversal components k⊥ = (kx , ky ) are taken into account. A new era in the investigation of the quark and gluon contents of nucleons has been launched in the research programmes dealing with high-energy semi-inclusive reactions with polarized and unpolarized hadrons, where the transverse motion and the spin–orbit correlations of the partons are directly accessible. Understanding the partonic structure of nucleons beyond the collinear approximation calls for an appropriate development of the theory. In classical inclusive processes, such as deep-inelastic ep-scattering (DIS) or electron–positron annihilation to hadrons, where no more than one hadron is identified in the initial state, the so-called collinear QCD factorization approach is applicable. The latter suggests that the longitudinal (parallel to a large light-like momentum in a suitable system) momenta of the patrons are intrinsic (non-perturbative), while their transverse momenta can be created by perturbative radiation effects (parton showers). In less inclusive processes, such as the Drell–Yan lepton pair production, semi-inclusive DIS, hadron–hadron annihilation to jets, Higgs and heavy-flavor production, where two or more hadrons in the initial or final state are detected, one is tempted to go beyond the collinear approximation. The reason is that now the momenta of the particles participating and detected in the process entail a nonplanar kinematical setting, which makes it natural to keep not only the collinear but also the transverse components of the parton momenta unintegrated. The so-called transverse momentum dependent factorization framework is believed to be a promising tool in these situations. It is expected to

VI

Preface

provide a unifying QCD-based framework with both mechanisms of the transverse momentum creation taken into account, that is intrinsic (essentially non-perturbative) as well as the perturbative radiation in parton showers. From the phenomenological prospective, observation of large single-spin asymmetries in experiments with polarized hadrons certainly demands the development of relevant theoretical tools that must include the non-perturbative intrinsic transverse momenta of the partons. The use of TMD parton densities as non-perturbative input provides such a framework because they contain explicit correlations between partonic transverse momenta and the orbital momenta and spins of the nucleons. Moreover, the TMD factorization approach is considered to be a promising tool for the QCD study of some unpolarized high-energy processes in the specific regimes. Namely, the Drell–Yan vector boson production for low-qT and the high-energy hadronic collisions with fixed momentum transfer at small Bjorken-x, where the gluon longitudinal momentum fractions become small, while the transverse momentum components dominate, give us examples of the TMD-related regimes accessible at the Large Hadron Collider (LHC). The TMD approach is also applicable to unpolarized processes with sensitivity to polarized gluon distributions as well as the Higgs, jet and heavy flavor production processes at the LHC. Among other currently operating and planned facilities with the most promising TMD-related experimental programmes we name the following ones: – Relativistic Heavy Ion Collider (RHIC in Brookhaven National Laboratory) hosts various reactions with polarized protons and nuclei. – One-third of the already approved experiments for the 12 GeV Upgrade of the Thomas Jefferson National Accelerator Facility (JLab) are devoted to the investigation of three-dimensional structure of the nucleon with strong TMD-related programme. – Planned Electron-Ion Collider (EIC) is designed as a high luminosity machine with particularly interesting TMD experiments with polarized hadrons. Technically, our text is meant to be a continuation of our previous monograph: – I.O. Cherednikov, T. Mertens and F.F. Van der Veken: Wilson lines in quantum field theory”, De Gruyter, Berlin (2014) where we are mostly concerned about the mathematical foundations of Wilson loops and geometrical properties of the generalized loop space. In the present book, we start with identifying and explaining the most important concepts and ideas that we shall use in the main body of the text and then develop ab initio calculation techniques applicable to generic piecewise-linear Wilson lines. We present also the practical tools for its implementation. Emphasis is put on the issues of gauge-invariance of nonlocal path-dependent QCD correlation functions with different Wilson lines keeping in mind their connection with the geometrical properties of generalized loop space. The present volume can be used as a primer and an introductory text to the advanced

Preface

VII

expositions presented in the following fundamental books, which deal partially with similar topics: – R. Gambini and J. Pullin: “Loops, knots, gauge theories and quantum gravity”, Cambridge (1996) – Y. Makeenko: “Methods of contemporary gauge theory”, Cambridge (2002) – J. Collins: “Foundations of perturbative QCD”, Cambridge (2011) The following key topics will be in the center of our exposition: – Integrated and unintegrated (transverse momentum-dependent) parton density functions – Normal-, time- and path-ordering and -ordered exponentials in quantum mechanics and quantum field theory – Abelian and non-Abelian Wilson lines and loops – QCD factorization in inclusive and semi-inclusive hadronic processes – Path dependence and its consequences in the practical calculations of gaugeinvariant correlation functions with Wilson lines Antwerp, December 2016 I.O. Cherednikov F.F. Van der Veken

Contents 1 1.1 1.2

Introduction 1 Main Properties of QCD 1 Principal Tools to Work with QCD in the High-Energy Regime

2

2.2.6 2.3

Particle Number Operators in Quantum Mechanics and in Quantum Field Theory 5 Quantum Mechanics 5 Time Evolution of Classical Systems 5 Hilbert Space and Operators 9 From Classical to Quantum Mechanics: The Heisenberg Picture 10 The Schrödinger Picture 11 Time Evolution in the Dirac Picture 13 Scattering Matrix in the Dirac Picture, Time and Path Ordering 14 Path Ordering 17 Connection Between the Heisenberg and Dirac Pictures 19 Correlation Functions in Quantum Field Theory 20 Correlation Functions in the Heisenberg Picture 20 Correlation Functions in the Dirac Pictures 23 Positive and Negative Frequency Decomposition 24 Quantum Harmonic Oscillator: Particle Number Representation 27 Creation and Annihilation Operators and Normal Ordering 29 Wick’s Theorems: Normal and Time Ordering 30 Summary 32

3 3.1 3.1.1 3.1.2 3.1.3 3.2 3.3

33 Geometry of Quantum Field Theories Parallel Transport and Wilson Lines 33 The Parallel Transporter 33 Non-Abelian Paths 36 The Covariant Derivative 40 The Gauge Field Tensor and Wilson Loops Summary 47

4 4.1 4.1.1 4.1.2

Basics of Wilson Lines in QCD A Wilson Line Along a Path Properties of Wilson Lines Path Ordering 50

2.1 2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.1.8 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5

49 49 50

41

2

X

4.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.4 4.5 4.5.1 4.5.2 4.6 5 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.2 5.2.1 5.2.2 5.2.3 5.2.4 5.3 5.3.1

Contents

Piecewise Wilson Lines 53 Wilson Lines on a Linear Path 58 Bounded from Below 58 Bounded from Above 60 Path Reversal 61 Finite Wilson Line 65 Infinite Wilson Line 69 External Momenta 73 Relating Different Path Topologies 74 Piecewise Linear Wilson Lines 77 Path Functions 81 Diagrams with Final-State Cuts 83 Eikonal Approximation 87 91 Gauge-Invariant Parton Densities Revision of Deep Inelastic Scattering 91 Kinematics 91 Invitation: The Free Parton Model 93 The Parton Model 95 Parton Distribution Functions 100 Operator Definition for PDFs 104 Gauge-Invariant Operator Definition 107 Semi-inclusive Deep Inelastic Scattering 111 Conventions and Kinematics 111 Structure Functions 113 Transverse Momentum-Dependent PDFs 115 Gauge-Invariant Definition for TMDs 118 Evolution of TMDs 123 About the Rapidity Cut-offs 126

129 6 Simplifying Wilson Line Calculations 6.1 Advanced Colour Algebra 129 6.1.1 Calculating Products of Fundamental Generators 130 6.1.2 Calculating Traces in the Adjoint Representation 135 6.2 Self-Interaction Blobs 138 6.2.1 2-Gluon Blob 138 6.2.2 3-Gluon Blob 141 6.3 Wick Rotations 144 6.3.1 Regular Wick Rotation 144 6.3.2 Wick Rotation with Wilson Lines 147 6.3.3 Light-Cone Coordinates: Double Wick Rotation 150 6.4 Wilson Integrals 151 6.4.1 2-Gluon Blob Connecting Two Adjoining Segments 154

Contents

A

Brief Literature Guide

163

164 B Conventions and Reference Formulae B.1 Notational Conventions 164 B.2 Vectors and Tensors 165 B.3 Spinors and Gamma Matrices 166 B.4 Light-Cone Coordinates 169 B.5 Fourier Transforms and Distributions 171 B.6 Lie Algebra 174 B.6.1 Representations 174 B.6.2 Properties 175 B.6.3 Useful Formulae 178 B.7 Summary of the Noether Theorems 180 B.8 Feynman Rules for QCD 181 184 C Integrations C.1 Reference Integrals 184 C.1.1 Algebraic Integrals 184 C.1.2 Logarithmic Integrals 185 C.1.3 Cyclometric Integrals 186 C.1.4 Gaussian Integrals 186 C.1.5 Discrete Integrals 187 C.2 Special Functions and Integral Transforms C.2.1 Gamma Function 188 C.2.2 Beta Function 190 C.2.3 Polylogarithms 190 C.2.4 Elliptic K-Function 192 C.2.5 Integral Transforms 192 C.3 Dimensional Regularization 195 C.3.1 Euclidian Integrals 195 C.3.2 Wick Rotation and Minkowskian Integrals C.4 Path Integrals 198 C.4.1 Properties 198 Bibliography Index

211

201

188

197

XI

1 Introduction The purpose of the first two chapters of our book is to identify and explain the most important concepts, which we shall need to study gauge-invariant path-dependent quantum correlation functions with Wilson lines. They provide, therefore, an extended technical introduction to the forthcoming material.

1.1 Main Properties of QCD Quantum chromodynamics (QCD) is the quantum field theory of strong interaction. The QCD Lagrangian LQCD = Lquark + Lgluon + L quark–gluon

(1.1)

contains the terms describing kinematics of quark and gluon fields, separately, and their interaction and self-interaction (for gluons). The most efficient approach to deal with a quantum field theory is to consider interactions as “weak” (in certain sense) perturbations and to use the Feynman diagram techniques to evaluate matrix elements, which can be associated with measurable quantities. For the theory of electrons and photons, quantum electrodynamics (QED), this approach works more or less straightforwardly. In contrast to QED, QCD possesses some remarkable properties, which make the direct application of this methodology impossible. – Confinement: The QCD Lagrangian is formulated in terms of the quark (fermion) and gluon (boson) fields, which are considered then as the fundamental degrees of freedom of the strong interaction. However, these objects do not exist as physical particles. At the same time, there are no QCD fields for the physical strongly interacting particles – hadrons. Therefore, the QCD matrix elements should be calculated for the physical (hadronic) states, while the operators correspond to the quark and gluon fields. –

Running coupling: After renormalization, the QCD coupling !s starts running, that is it changes with characteristic energy scale. The same happens to the electromagnetic charge in QED, but the behaviour is different: in QED, the coupling decreases together with the energy scale, while the strong coupling grows up if the energy goes down. This behaviour implies the property of asymptotic freedom: at high energy or, equivalently, at small distance, the quarks and gluons can be considered as free particles and one can expect that the perturbative approach is applicable.

Hence, we see that one can work with QCD as with a “normal” quantum field theory by making use of the perturbative expansion only in the asymptotic freedom regime. However, any hadronic process (even at very high energy) contains not only large

2

1 Introduction

energy scale, but also a remarkably small one, the hadronization scale, at which quarks and gluons recombine to make up real hadrons and any perturbative methods are not applicable.

1.2 Principal Tools to Work with QCD in the High-Energy Regime Any scattering process in which strongly interacting particles – hadrons – participate, even those which occur at high centre-of-mass energy s = (E1 + E2 )2 , include also dependence on a low-energy scale, which can be associated with typical hadron mass Mh . The large s, however, does not say us anything about the momentum scale which characterizes the partonic subprocess and which sets up the scale for the running coupling. The energy of a probe which allows us to penetrate a hadron to reveal its partonic substructure is called hard. If in a given process, there is no other energy scale which can be treated as “hard” in the above sense, then the process is considered as soft. An important example of such process is given by, e.g. the elastic hadron–hadron scattering. The methodology of analysis of such reactions goes beyond pure perturbative techniques and will not be considered here. We shall focus on the so-called hard hadronic processes in the high-energy regime. For an inclusive scattering reaction in the t-channel, P1 + P2 → P1′ + anything

(1.2)

it means that besides the large centre-of-mass energy s = (P1 + P2 )2 ≫ Mh2 ,

(1.3)

where Mh is a hadronic mass scale, such as the proton mass, there is also the large momentum transfer t = BP12 = (P1′ – P1 )2 ≫ Mh2 .

(1.4)

Given the Heisenberg uncertainty relation, the large momentum transfer suggests that we can “measure” the intrinsic structure of the hadron with the spatial resolution Br2 ∼ BP1–2 = t–1 .

(1.5)

If t is large enough, we can access to the partonic subprocesses, where the strong coupling is small and the perturbation theory can be reasonably justified. Still, we have to specify how the low-energy part of the total reaction (1.2) should be taken into account. The QCD factorization approach provides an

1.2 Principal Tools to Work with QCD in the High-Energy Regime

3

appropriate method to merge the high- and low-energy regimes and perform efficient calculations. – In a few words, the idea of QCD factorization consists in the consistent separation of large-distance (essentially nonperturbative, hadronization level) S and small-distance H (perturbatively calculable matrix elements, partonic level) contributions to a given process. The latter is being said to be factorizable if the differential cross section can be presented as the convolution of the hard and soft parts d3 = Hsmall

distance

⊗ Slarge distance .

(1.6)

It is important to note that S is expected to be universal, while H depends on a particular process. –

Contribution of the soft part of the factorization formula (1.6) can be presented in terms of parton density functions1 (PDFs), which accumulate information about the intrinsic structure of hadrons. More precise, the PDFs determine the probability distributions of quarks and gluons confined in the hadron. This is essentially the basic assumption of the parton model. The parton model is, however, not equivalent to QCD. The consistent construction of QCD-improved parton picture and the appropriate factorization scheme call for a suitable field-theoretical operator definitions for the PDFs.

Parton density functions in the momentum space can formally be obtained from the correlation functions of the appropriate quantum field operators of the following generic form2 :  I(k) =

d4 z –ikz γ e P|8H (z)Uz,0 8H (0)|PH , (20)4

(1.7)

where the field operators 8, 8 and the hadronic vectors of state |P are taken in γ the Heisenberg representation and U(z;0) is a Wilson line or a system of lines which connects the points z and 0 with an arbitrary trajectory γ and make the correlation function (1.7) gauge-invariant ⎛ γ

U(z;0) = Pexp ⎝ig

z

⎞ d& , ta Aa, (& )⎠ .

(1.8)

0

The matrix elements (1.7) are associated with the hadronic expectation values of the parton number operators 1 Also parton distribution or fragmentation functions. 2 Here and in what follows we discuss mostly quark correlation functions.

4

1 Introduction

I(Q) ∼ a†Q aQ  = NQ ,

(1.9)

where NQ stands for the operator which returns the number of particles possessing quantum numbers {Q} (momenta, spins, colours, etc.) in a given state. The latter property connects the field-theoretical definition (1.7) with the intuitive ideas of the original parton model. Nevertheless, equation (1.7) is too symbolic to make real calculations and useful predictions possible. The rest of the book is devoted to the identification of the most important issues and development of the methods to deal with them.

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory 2.1 Quantum Mechanics 2.1.1 Time Evolution of Classical Systems In classical mechanics, the idea of a particle and a number of particles is intuitively trivial. One assumes that a system of N point-like particles is described by dynamical variables {A } which are the functions of 2N independent variables: the generalized coordinates {qi (t)} and momenta {pi (t)}: A = A (t, {qi }, {pi }).

(2.1)

The set of variables {A (t0 )} provides then an instantaneous snapshot of the system at some time point t0 . The problem of analytical mechanics is to determine the state of the system at another instant t given that {A (t0 )} is known. Formally, the time evolution of the dynamical variables obeys the first-order differential equation in partial derivatives  dA ∂A  ∂A ∂qi ∂A ∂pi = + + . (2.2) dt ∂t ∂qi ∂t ∂pi ∂t i

We accept as an axiom that among the various possible dynamical variables, there exists one which plays a crucial role. Let this function be the Hamiltonian of the system H(q, p, t), so that the action is given by the integral

t2 S[t1 , t2 ] =

dt t1



 pi (t)˙qi (t) – H(q, p, t) .

(2.3)

i

To complete the picture, we postulate that between the instants t1 and t2 , the system evolves in time following the trajectory in the phase space (q, p) which delivers a local extremum to the action (2.3): $S[q, p] = S[q + $q, p + $p] – S[q, p] = 0,

(2.4)

$qi (t1 ) = $qi (t2 ) = 0,

(2.5)

$pi (t1 ) = $pi (t2 ) = 0.

(2.6)

given that for each i

Note that the action S is a functional of the trajectories in the phase space (q, p). From the definition of the action (2.3) and variational principle (2.4), assuming that

6

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

the variations $qi and $pi are arbitrary and independent, one obtains the following equations of motion ∂H(q, p) , ∂pi ∂H(q, p) p˙ i = – . ∂qi q˙ i =

(2.7)

Therefore, the time evolution (2.2) of a function A , provided that the equations of motion (2.7) are fulfilled and the initial condition is given A (t0 ) = A(0)  ,

(2.8)

dA (q, p, t) ∂A (q, p, t) = + {A , H} , dt ∂t

(2.9)

is determined by the Hamiltonian:

where the Poisson bracket is defined as1   ∂A1 ∂A2 ∂A1 ∂A2 {A1 , A2 } = – . ∂qi ∂pi ∂pi ∂qi

(2.10)

i

The Poisson bracket has the following useful properties: it is antisymmetrical {A1 , A2 } = –{A2 , A1 }

(2.11)

{A1 , {A2 , A3 }} + {A2 , {A3 , A1 }} + {A3 , {A1 , A2 }} = 0.

(2.12)

and satisfies the Jacobi identity

In particular, the Poisson brackets of the coordinate and the momentum variables are {qi , pj } = $ij , {qi , qj } = 0,

(2.13)

{pi , pj } = 0. Anticipating quantization of the classical system, one says that a variable A1 commutes with another variable A2 if their Poisson bracket is equal to zero. Therefore,

1 Note that the symbol for the Poisson bracket is the same as for the anticommutator (see e.g. equation (B.17)). This might be a bit confusing as the Poisson bracket has the same signs as a regular commutators, but the difference should be clear from context.

2.1 Quantum Mechanics

7

the following statement holds: An explicitly time-independent function A (q, p) is conserved in time A (q, p) = const if (1) it Poisson-commutes with the Hamiltonian of the system: {A , H} = 0,

(2.14)

and (2) the equations of motion (2.7) are satisfied. For example, if the Hamiltonian (as a dynamical variable) is time-independent, the system possesses constant energy. This statement is a particular case of the Noether theorem, which relates the symmetry properties of a system with the existence of a number of conserved quantities. The conservation of energy corresponds in this context to the invariance of the system with respect to the temporal translations, while the conservation of momentum follows from the spatial translation invariance. An alternative but equivalent picture is given within the Lagrangian formalism. In this case, one starts with a function of the generalized coordinates and their derivatives in time L = L(q, q˙ , t),

(2.15)

and the action reads t dt L(q, q˙ , t),

S[q] =

(2.16)

t0

and the equations of motion stem from the stationarity condition ∂L d ∂L – = 0, ∂qi dt ∂ q˙ i

(2.17)

given that the variations $qi are arbitrary and independent: $S[q] = S[q + $q] – S[q] = 0,

(2.18)

given that $qi (t0 ) = $qi (t) = 0. Let us emphasize that for quantum systems the Hamiltonian and the Lagrangian formalisms are not directly equivalent. Having in mind that classical systems should undergo a quantization procedure, we work in the Hamiltonian approach. Let us consider an object which is particularly convenient for quantization, namely a one-dimensional harmonic oscillator with the

8

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

unity mass. Its Hamiltonian reads 1 1 H(q, p) = 92 q2 + p2 , 2 2

(2.19)

where 9 is a parameter. The phase space of this system is two dimensional, and their time evolution of the coordinate and the momentum obey the equations of motions: dq(t) = {q, H} = p, dt dp(t) = {p, H} = –92 q. dt

(2.20) (2.21)

These two first-order differential equations can be combined to give one of the secondorder equations

d2 2 + 9 q(t) = 0, dt2

(2.22)

or,

d2 2 + 9 p(t) = 0. dt2

(2.23)

The solutions to these equations can be presented as the linear combinations of the time-dependent variables 1 q(t) = √ (a+ (t) + a– (t)), 29  9 + p(t) = i (a (t) – a– (t)). 2

(2.24) (2.25)

These variables obey the following time evolution: a(±) (t) = a(±) 0 exp(±i 9t) , a(±) 0

(±)

=a

(2.26)

(0) , 9 > 0.

It is straightforward to check that the Poisson bracket of the a± -functions is given by {a– , a+ } = –i ,

(2.27)

and the Hamiltonian (2.19) can be presented as H(a+ , a– ) =

 1 + – a a + a– a+ . 2

(2.28)

2.1 Quantum Mechanics

9

The latter representation has a direct analogy in quantum mechanics, where it allows us to express the Hamiltonian of a system in terms of the particle number operator. In its turn, equation (2.25) resembles the decomposition of the field operators in creation and annihilation operators within the second quantization framework in quantum field theory, which we shall introduce and discuss in the next sections. 2.1.2 Hilbert Space and Operators Quantum mechanics suggests that all information about the state of a given system is accumulated in a ray in a Hilbert space. That is to say, if at the initial moment of time the state of the system can be identified with a vectors of state h ∈ H, the same state is represented by a set of vectors ! ⋅ h in the same Hilbert space. Let us briefly list the most important definitions and properties of these objects. A space H consisting of the elements {hi } ∈ H, which are called vectors, is linear if an operation “+” is defined, such that for all elements hi and for all complex numbers !i h1 + h2 = h2 + h1 ; (h1 + h2 ) + h3 = h1 + (h2 + h3 ); !1 (h1 + h2 ) = !1 h1 + !2 h2 ; !1 (!2 h1 ) = (!1 !2 )h1 .

(2.29)

There is also a null vector 0 ∈ H, such that for all hi ∈ H hi + 0 = hi .

(2.30)

A linear space H is called pre-Hilbert if the scalar product h1 , h2  is defined, which is a complex number and has the following properties: h1 , h2  = h2 , h1 † ; !1 h1 , h2  = !1 h1 , h2 ; h1 + h2 , h3  = h1 , h3  + h2 , h3 ; h1 , h1  ≥ 0; hi , hi  = 0 if and only if hi = 0.

(2.31)

Given the scalar product is defined, one can introduce a norm as a real number:  (2.32) ||hi || = hi , hi . Consider a sequence of the elements {hk } ∈ H. The sequence is called convergent if there exists an element h ∈ H, such that lim ||hk – h|| = 0.

k→∞

(2.33)

10

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

A sequence {hl } ∈ H is called fundamental if lim ||hli – hlj || = 0.

li ,lj →∞

(2.34)

Finally, a pre-Hilbert space H is called Hilbert if each fundamental sequence in it is convergent. Then the time evolution and other measurable properties of the system is determined by quantum operators acting in the Hilbert space. Namely, for each observable quantity, there is a Hermitian operator A = A† ,

(2.35)

which generates a complete set of orthonormal eigenvectors {hAk } and real eigenvalues ak , such that AhAk = ak hAk , A h†A k , hl  = $kl  hAi , h†A i  = 1.

(2.36)

i

The probability to the system in a given state J to have an observable value ak for a measurable quantity associated with the operator A is given by 2 |h†A k , h| ,

(2.37)

so h†A k , h can be interpreted as a probability amplitude. 2.1.3 From Classical to Quantum Mechanics: The Heisenberg Picture The original Heisenberg’s idea of the correspondence between the classical and quantum descriptions consists in the replacement of Poisson brackets for the classical dynamical variables with commutators of quantum operators. In other words, instead of equation (2.14), one introduces the canonical commutation relations between operators of coordinate and momentum2 [qi , pj ] = qi pj – pj qi = i $ij ,

(2.38)

[qi , qj ] = [pi , pj ] = 0 .

(2.39)

Hence, time evolution of an arbitrary operator AH (q, p, t) is given by the commutator with the Hamiltonian, in analogy to equation (2.9): i

∂ AH (q, p, t) = [AH (q, p, t), H]. ∂t

2 Here and in what follows we put h¯ equal to 1 for convenience.

(2.40)

11

2.1 Quantum Mechanics

Subindex H stands here to mark operators and vectors of state in the Heisenberg representation anticipating the discussion below. Note that in contrast to the classical coordinates and momenta (q, p), neither the quantum operators A nor the vectors of state J can be directly observable. In the previous section, we postulated that physical quantities in quantum mechanics are associated with average values of the operators A(t) = J† (t)A(t)J(t),

(2.41)

where J are the elements of a Hilbert space. The functions (2.46) must be representation-invariant whatever picture is used to construct the operators and the vectors of state. The Heisenberg picture is defined by the time evolution law (2.40) for the operators AH and the requirement of the time independence of the vectors of state ∂ JH = 0. ∂t

(2.42)

Therefore, the time evolution of the amplitude (2.46) in the Heisenberg representation is given by i

d d AH (t) = J†H i AH (t) JH dt dt = J†H [AH (t), HH ]JH .

(2.43)

2.1.4 The Schrödinger Picture An alternative approach to the evaluation of the time evolution of quantum mechanical observables can be formulated in the Schrödinger picture, which suggests that operators are time independent while vectors of state change in time. Within this picture, time evolution of a quantum system is determined by the Schrödinger equation i

∂ JS (t) = HS JS (t), ∂t

(2.44)

where HS is the Hamiltonian in the Schrödinger representation. The time evolution of operators in this picture is trivial: ∂ AS (t) = 0 → AS = const. ∂t Still the amplitudes, i.e. the average values of the operators, do depend on time:

(2.45)

12

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

i

d d d AS (t) = i J†S (AS JS ) + (J†S AS )i JS dt dt dt = J†S (t)[AS , HS ]JS (t).

(2.46)

The two representations can be related to each other via a unitary transformation. Consider the time evolution operator in the Schrödinger picture U(t) = ei HS t

(2.47)

assuming that the Hamiltonian is time-independent ∂t H = 0. Let us demonstrate that vectors of state in the Heisenberg can be obtained from the Schrödinger ones: JH = U(t)JS (t).

(2.48)

The average value of an operator A then reads AS (t) = J†S (t)AS JS (t) = J†H ei HS t AS e–i HS t JH .

(2.49)

Therefore, the operators in the two pictures can be related as AH (t) = ei HS t AS e–i HS t .

(2.50)

It is easy now to check that the operators (2.50) depend on time according to equation (2.40): i

 ∂ ∂  i HS t AS e–i HS t AH (t) = i e ∂t ∂t 

= –HS e

i HS t

AS e

–i HS t

+e

i HS t

∂ i AS e–i HS t ∂t

– ei HS t AS HS e–i HS t

(2.51)

= [AH , HH ], given that the Hamiltonian remains intact by this transformation so that HH = HS . In its turn, the vector of state becomes time-independent after the transformation (2.48):  ∂  i HS t ∂ JS (t) JH = i e ∂t ∂t i HS t JS (t) + HS ei HS t JS (t) = 0. = –HS e

i

(2.52)

To summarize, in the simplest case of time-independent Hamiltonians, one can equivalently describe the time evolution of quantum mechanical observables A(t) by means of the Heisenberg or the Schrödinger picture. The former suggests that the vectors of state are fixed at the initial instant of time and the entire time evolution of the averages A(t) is due to the time-dependent operators, while the latter let the

2.1 Quantum Mechanics

13

operators be constant in time and the vectors of state being time-dependent. However, reality is more complicated: The Hamiltonians are not usually time-independent, so that neither the pure Schrödinger representation nor the Heisenberg one provide the reliable basis to calculations. An “intermediate” framework comes to the rescue which can also be obtained from the Heisenberg and Schrödinger representation by unitary transformation. 2.1.5 Time Evolution in the Dirac Picture Consider the time evolution of a system in the Schrödinger representation i

∂ JS (t) = HS JS (t) ∂t

(2.53)

with the time-dependent Hamiltonian H(t), which can be written as HS (t) = HS0 + HSint (t).

(2.54)

Here H0 stands for the time-independent free Hamiltonian and Hint (t) is considered as a dynamical time-dependent “interaction” part. Note that the solution to equation (2.53) cannot be presented in the form e–i HS t because the Hamiltonian may not commute with itself at different moments of time: [HS (t′ ), HS (t′′ )] ≠ 0,

(2.55)

so that the exponential e–i Ht is ill-defined. For that reason, let us define a timedependent vector of state via the exponential of the free Hamiltonian 0

ei HS t JS (t) = J(t).

(2.56)

This representation is known as the Dirac or interaction picture. In what follows, vectors of state and operators without indices will stand for the quantities defined in this picture. Let us find the differential equations which determine the time evolution of operators and vectors of state in the Dirac picture. For the vector of state (2.56), we have  ∂ ∂  i H0 t J(t) = i e S JS (t) ∂t ∂t   0 ∂ i H0S t 0 = –HS e JS (t) + ei HS t i JS (t) ∂t

i

14

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

    0 0 = –HS0 ei HS t JS (t) + ei HS t HS0 + HSint (t) JS (t)   0 0 0 = ei HS t HSint (t) e–i HS t ei HS t JS (t) = Hint (t)J(t),

(2.57)

where the interaction Hamiltonian in the Dirac representation Hint is defined by the unitary transformation of the Schrödinger operator 0

0

Hint (t) = eiHS t HSint e–iHS t .

(2.58)

Obviously the free Hamiltonians coincide in all three representations 0 HS0 = HH = H0 .

(2.59)

Given the relation between the Dirac and Schrödinger Hamiltonians (2.58), we set up the transformation law for an arbitrary operator in the Dirac picture 0

0

A(t) = ei HS t AS e–i HS t .

(2.60)

It is now straightforward to find the time evolution equation for the Dirac operators, taking into account that the Schrödinger operators AS are time independent: i

0 0 ∂ A(t) = –H0 A(t) + ei HS t AS H0 e–i HS t ∂t = [A(t), H0 ].

(2.61)

An important conclusion is that the time evolution of the operators in the Dirac picture is determined by the free Hamiltonian, which significantly simplifies calculations and makes the Dirac picture favourable in the situations where the separation of the free and the dynamical interaction terms in the total Hamiltonian is possible and the latter can be presented in the form (2.54). Note that the interaction Hamiltonian in the Dirac picture also obeys the time evolution law (2.61) i

∂ int H (t) = [Hint (t), H0 ]. ∂t

(2.62)

Now we are in a position to obtain the explicit solution of the time evolution equation for the vectors of state in the Dirac picture (2.57).

2.1.6 Scattering Matrix in the Dirac Picture, Time and Path Ordering We have to find the time evolution operator U(t, t0 ) which implements the translation of the initial state J(t0 ) at the given instant t0 to another moment of time t:

2.1 Quantum Mechanics

J(t) = U(t, t0 )J(t0 ).

15

(2.63)

It is natural to demand and easy to demonstrate that this operator has the following properties: U(t′′ , t′ )U(t′ , t0 ) = U(t′′ , t0 ), U(t, t0 ) = U –1 (t0 , t) = U † (t0 , t).

(2.64)

The normalization condition obviously reads U(t0 , t0 ) = 1.

(2.65)

By formal integration of equation (2.57), we obtain t J(t) – J(t0 ) = –i

dt′ Hint J(t′ ).

(2.66)

t0

Let us assume that the interaction induced by the term Hint is, in certain sense, “weak”. This assumption is natural for an important set of the scattering problems, where the initially free particles perturbatively interact in a finite period of time and become again free at the asymptotically large time scale. In other words, the interaction is considered as a perturbation. Let us present the vector of state in the form J(t) = J(0) (t) + J(1) (t) + J(2) (t) + ⋯ + J(n) (t) + ⋯

(2.67)

and set the initial condition J(0) (t) = J(t0 ).

(2.68)

Substituting the expansion (2.67) into the integral equation (2.66), we obtain J(t) = J(0) (t) + J(1) (t) + ⋯ t = J(t0 ) – i

  dt′ Hint (t′ ) J(0) (t′ ) + J(1) (t′ ) + J(2) (t′ ) + ⋯ .

(2.69)

t0

We can solve this equation in each order of the expansion (2.67). Therefore, we obtain, for the first two orders, a trivial result: J(0) (t) = J(t0 ), t J(1) (t) = –i t0

⎡

⎢ dt′ Hint (t′ )J(0) (t′ ) = –i ⎣

t

t0

⎤ ⎥ dt′ Hint (t′ )⎦ J(t0 ).

(2.70)

16

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

The next order results are not that straightforward: t

dt′ Hint (t′ )J(1) (t′ )

J(2) (t) = –i t0

t = (–i)2

⎡ ⎢ dt′ Hint (t′ ) ⎣

t0

t′

⎤ ⎥ dt′′ Hint (t′′ )⎦ J(t0 ).

(2.71)

t0

Having in mind that t′ and t′′ are both just integration variables and can be, therefore, safely reshuffled, we write ⎡

t

⎢ dt′ Hint (t′ ) ⎣

t0

⎤

t′

⎥ dt′′ Hint (t′′ )⎦ =

t0

t

⎡ ⎢ dt′′ Hint (t′′ ) ⎣

t0

t′′

⎤ ⎥ dt′ Hint (t′ )⎦ .

(2.72)

t0

Hence, t t0

⎡ ⎢ dt′ Hint (t′ ) ⎣

⎡

1⎢ ⎣ 2

t′

⎤ ⎥ dt′′ Hint (t′′ )⎦ =

t0 t′

t

dt′

t0



dt′′ Hint (t′ )Hint (t′′ ) +

t0

t

dt′′

t0

t′′

⎤ ⎥ dt′ Hint (t′′ )Hint (t′ )⎦ .

(2.73)

t0

It is convenient to introduce at this point the time-ordered product of two noncommuting operators T [A1 (t1 )A2 (t2 )] = ((t1 – t2 )A1 (t1 )A2 (t2 ) + ((t2 – t1 )A2 (t2 )A1 (t1 ).

(2.74)

Then we have t

′

int ′

t′

dt H (t ) t0

1 dt H (t ) = 2 ′′

int ′′

t0

t t

dt′ dt′′ T [Hint (t′ )Hint (t′′ )].

(2.75)

t0 t0

It is easy now to check the nth order term in the series (2.67), which is determined by the integral of the time-ordered product of n Hamiltonian operators in the Dirac representation: 1 n!

t

t dt1 . . . dtn T [Hint (t1 ) . . . Hint (tn )].

... t0

t0

(2.76)

2.1 Quantum Mechanics

17

Finally, the time evolution operator U(t, t0 ), which solves equation (2.63), is given by the time-ordered exponential, the latter being defined by means of the infinite series ⎞ ⎛ t ⎟ ⎜ U(t, t0 ) = T exp ⎝–i Hint (t′ )dt′ ⎠ t0

t =1–i

(–i)2 dt H (t ) + 2 ′

int ′

t0

⋯+

(–i)n n

t t

dt′ dt′′ T [Hint (t′ )Hint (t′′ )]

t0 t0

t

t dt1 ⋯dtn T [Hint (t1 )⋯Hint (tn )] + ⋯

⋯ t0

(2.77)

t0

The time-evolution operator (2.77) which corresponds to the transition from the initially free state at t0 = –∞ to another free state at t = +∞ is called scattering matrix or S-matrix: ⎛ ⎞ ∞ (2.78) S = U(∞, –∞) = T exp ⎝–i Hint (t′ )dt′ ⎠ . –∞

This object plays a crucial role in the construction of perturbatively calculable quantum correlation functions in the Dirac representation.

2.1.7 Path Ordering The time evolution operator (2.77) implements translations from one point on the time ray to another one lying on the same ray, the Hamiltonian being the generator of the time translations. In other words, the Hamiltonian enables the infinitesimal transport of a function from one point to another on the temporal ray:   ∂ + i H(t) J(t) = 0. (2.79) ∂t One can formally define an analogue of the covariant derivative, which one introduces in gauge theories Dt =

∂ + i H(t), ∂t

(2.80)

where the Hamiltonian provides a connection of two functions J taken at different instants of time. The transport equation then reads Dt J(t) = 0,

(2.81)

18

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

which solution is given by ⎛ ⎜ J(t) = T exp ⎝–i

t

⎞ ⎟ Hint (t′ )dt′ ⎠ J(t0 ).

(2.82)

t0

This view can be easily generalized to the more complicated trajectories than a straight line. Consider a function J(z) defined in the Minkowskian space–time. The parallel transport of the function J from a point x to another point y along a given path γ , which is parameterized as z, ∈ γ : z, (t) = v, t z, (0) = x, z, (1) = y,

(2.83)

is defined by the parallel transport equation D, J(z)|γ = 0,

(2.84)

where the connection is implemented by an operator A D, =

∂ + i A, (z), z ∈ γ ∂z,

(2.85)

Solution to equation (2.84) is given by ⎡ J(y)|γ = P exp ⎣–i

y

⎤ A, (z)dz, ⎦ J(x)

x

⎡ = P exp ⎣–i

1

γ

⎤

A, (z)v, dt⎦ J(x),

0

(2.86)

γ

where instead of the time-ordering T , we have the path ordering P[A1 (z1 )A2 (z2 )]|γ = ((z1 – z2 )A1 (z1 )A2 (z2 ) + ((z2 – z1 )A2 (z2 )A1 (z1 ),

(2.87)

given that the points at the trajectory γ are ordered in the sense of the parameterization (2.83). The transporter ⎡ U(y;x) |γ = P exp ⎣–i

y x

⎤ A (z)dz, ⎦ ,

(2.88) γ

provides a simple example of Wilson line, that is path-ordered path-dependent exponential of a connection field A. Another useful representation of the transporter (2.88) is given by

2.1 Quantum Mechanics

⎡ P exp ⎣–i

y

⎤ A, (z)dz, ⎦ =

x



 1 + i A, (z)dz, ,

19

(2.89)

γ

which is equivalent to equation (2.88).

2.1.8 Connection Between the Heisenberg and Dirac Pictures Equation (2.60) states the relation between the operators in the Schrödinger and Dirac pictures. Taking into account that the Hamiltonian with interaction term is time-dependent, it is natural to think that the connection between the Dirac and Heisenberg pictures should be less straightforward. Suppose that a Dirac vector of state can be obtained from a Heisenberg one by means of a unitary operator V(t) J(t) = V(t)JH .

(2.90)

J(t) = U(t, t0 )J(t0 )

(2.91)

By virtue that

with U given by equation (2.77), and that JH is time-independent, we have V(t) = U(t, t0 )V(t0 ),

(2.92)

where V(t0 ) = V0 is a constant to be determined. It is now convenient to re-write the above formulas in the Schrödinger representation: 0

J(t) = ei HS t JS (t) , JH = ei HS t JS (t).

(2.93)

Hence, we obtain 0

ei HS t = U(t, t0 )V0 ei HS t .

(2.94)

We know that U(t0 , t0 ) = 1, therefore, the constant V0 can be set to 0

V0 = ei HS t0 e–i HS t0 .

(2.95)

To do the next step, an assumption is in order. It is natural to demand that in a scattering problem, the interaction switches on “adiabatically”, that is in the infinite past t = –∞ the interaction term in the Hamiltonian was equal to zero Hint , which gives V0 = 1 and V(t) = U(t, –∞).

(2.96)

20

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

We establish thus the following relation between the Dirac and Heisenberg vectors of state: J(t) = U(t, –∞)JH ,

(2.97)

which implies the connection between the corresponding operators A(t) = U(t, –∞)AH (t)U † (t, –∞).

(2.98)

2.2 Correlation Functions in Quantum Field Theory Quantum field theory can be formally introduced as quantum mechanics with infinite number of degrees of freedom. In other words, we let the coordinates {q } and momenta {p },  = (1, N) to be defined on a space–time lattice {x } and then take the continuum limit N → ∞. For the real scalar field the new ‘coordinates’ 6(x) and ‘momenta’ 0(x) must obey the equal-time commutation relations3 [6(x, t), 0(x′ , t)] = i$(9–1) (x – x′ ), [6(x, t), 6(x′ , t)] = [0(x, t), 0(x′ , t)] = 0.

(2.99)

2.2.1 Correlation Functions in the Heisenberg Picture To describe the dynamics of a classical field, one follows the same line as in the case of classical mechanics. The Lagrangian of the real scalar field reads 1 L[6(x), ∂6(x)] = ∂, 6 ∂ , 6 – V[6], 2

(2.100)

and the action is given by  S=

d4 x L[6, ∂6].

(2.101)

The equations of motion follow from the minimal action principle: one assumes that the system undergoes the evolution which delivers local minimum to the action functional $S = 0.

(2.102)

Therefore, ∂, ∂ , 6 + V ′ [6] = 0 , V ′ [6] =

d V[6] . d6

(2.103)

3 We work in 9-dimensional space–time to anticipate the dimensional regularization for Feynman loop integrals, which we shall use in the text.

2.2 Correlation Functions in Quantum Field Theory

21

A free theory contains only quadratic self-interactions 1 V0 [6] = m2 62 2

(2.104)

  2 ∂ + m2 6(x) = 0 , ∂ 2 ≡ ∂, ∂ , .

(2.105)

and the equation of motion is

Within the quantization procedure, the fields 6(x, t) are set to obey the canonical commutation relations (2.99). The fields are now time-dependent operators so that the Heisenberg picture is assumed. To define their dynamics, we first introduce the free Hamiltonian   ˙ – L = 1 0 2 + ∇6 + m2 62 . H[6, 0] = 0 6 2

(2.106)

The Heisenberg equations of motion read ∂ 6(x) = [6(x), H], ∂t ∂ i 0(x) = [0(x), H], ∂t

i

(2.107)

which can be shown to be equivalent to the equations of motion in the Lagrangian approach (2.105). Let us define the classical Green’s function of the Klein–Gordon differential   operator ∂ 2 + m2 in 9-dimensional space–time as 

 ∂ 2 + m2 G(x – x′ ) = –i $(9) (x – x′ ).

(2.108)

In the momentum space, we have 

d9 q i qz! e G(q), (20)9  d9 q ∂ 2 G(z) = G(q), (–q2 )ei qz! (20)9 G(z) =

(2.109)

which gives ! G(q) =

q2

i . – m2

(2.110)

Note that equation (2.110) does not define yet the Green’s function ! G(q) since the rhs is 2 2 singular at q = m and a prescription how to deal with this pole is needed. Formally, several prescriptions are allowed, each of which delivers a solution to the equation G(q) = –i. (–q2 + m2 )!

(2.111)

22

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

For example, the advanced and retarded prescriptions can be defined as ! G(q)adv/ret =

i , (q0 ± i 0)2 – q2 – m2

(2.112)

which yield the correlation functions in the coordinate space G(x – x′ )adv = ((x′ – x)[6(x), 6(x′ )]0, G(x – x′ )ret = –((x – x′ )[6(x), 6(x′ )]0,

(2.113)

respectively. We are more interested in the Feynman prescription ! G(q)F =

q2

i , – m2 + i0

(2.114)

which is associated with the vacuum expectation value of the time-ordered product GF (x – x′ ) = T[6(x)6(x′ )]0

(2.115)

It is instructive to demonstrate explicitly that GF (x – x′ ) indeed solves equation (2.108), given that the fields 6 and their time derivatives obey the commutation relations ˙ ′ , t)] = i $(9–1) (x – x′ ), [6(x, t), 6(x ˙ t), 6(x ˙ ′ , t)] = 0. [6(x, t), 6(x′ , t)] = [6(x,

(2.116)

Let us act with the Klein–Gordon operator to the expectation value    ∂ 2 + m2 T[6(x)6(x′ )]0 = ∂ 2 + m2 J†0 T[6(x)6(x′ )]J0



(2.117)

given that the equation of motion is valid for the vacuum state  ∂ 2 + m2 6(x)J0 = 0.



(2.118)

Before we proceed, note that the spatial part of the Klein–Gordon operator yields 

   –∇2 + m2 T[6(x)6(x′ )]0 = T[ –∇2 + m2 6(x)6(x′ )]0 ,

(2.119)

and only the temporal differential operator ∂02 produces a non-trivial result. One has then  " #  ∂02 T 6(x)6(x′ ) = ∂0 ∂0 [((x0 – x0′ )6(x)6(x′ ) + ((x0′ – x0 )6(x′ )6(x)] .

(2.120)

2.2 Correlation Functions in Quantum Field Theory

23

Because ∂0 ((x0 – x0′ ) = –∂0 ((x0′ – x0 ) = $(x0 – x0′ ),

(2.121)

and, by virtue of the commutation relations (2.116) $(x0 – x0′ )[6(x), 6(x′ )] = 0,

(2.122)

we obtain " ∂0 $(x0 – x0′ )6(x)6(x′ ) + ((x0 – x0′ )∂0 6(x)6(x′ ) # –$(x0 – x0′ )6(x′ )6(x) + ((x0′ – x0 )6(x′ )∂0 6(x) ˙ 6(x′ )]. = ((x0 – x0′ )∂02 6(x)6(x′ ) + ((x0′ – x0 )6∂02 6(x) + $(x0 – x0′ )[6(x),

(2.123)

The ∂02 6-terms sum up with the spatial contribution (2.119) to yield the Klein–Gordon equation   T[ ∂02 – ∇2 + m2 6(x)6(x′ )]0 = 0,

(2.124)

while the temporal terms give 

 ∂ 2 + m2 T[6(x)6(x′ )]0

= –i $(x0 – x0′ )$(9–1) (x – x′ ) = –i $(9) (x – x′ ).

(2.125)

The relations (2.113) can be proven similarly.

2.2.2 Correlation Functions in the Dirac Pictures The above results are obtained within the Heisenberg representation of the quantum field operators 6(x). Now we are in a position to derive the calculation-friendly representation of the functions, which are initially defined in the Heisenberg picture. The goal of any quantum field theory is to provide the tools to evaluate various correlation functions, which are commonly written in terms of the ordered expectation values of products of quantum operators defined at different space–time points. Consider a time-ordered vacuum correlation of n Heisenberg operators {Ak } in a given state J 1 H 2 H n G(t1 , . . . , tn ) = T [AH 1 (t )A2 (t ) . . . An (t )] 1 H 2 H n = J† T [AH 1 (t )A2 (t ) . . . An (t )]J.

(2.126)

Equation (2.98) allows us to re-write it in the Dirac representation: G(t1 , . . . , tn ) = J† T [U † (t1 , –∞)A1 (t1 )U(t1 , –∞)⋅ ⋅ U † (t2 , –∞)A2 (t2 )U(t2 , –∞) . . . U(tn , –∞)An (tn )U(tn , –∞)]J.

(2.127)

24

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

Using the properties of the time evolution operator U † (t, –∞) = U † (–∞, ∞)U(∞, t) = S† U(∞, t),

(2.128)

U(∞, t) = U(t, –∞)U(–∞, ∞) = U(t, –∞)S,

(2.129)

where S is defined by equation (2.78), we obtain G(t1 , . . . tn ) = J† S† T [U(∞, t1 )A1 (t1 )U(t1 , t2 )A2 (t2 ) . . . An (tn )U(tn , –∞)]J.

(2.130)

Taking into account that under the fixed ordering (time ordering in this case), the operators commute and, therefore, can be re-shuffled, we collect and re-sum all U(ti , tj ), which yield G(t1 , . . . tn ) = J† S† T [A1 (t1 )A2 (t2 ) . . . An (tn )U(∞, –∞)]J = J† S† T [A1 (t1 )A2 (t2 ) . . . An (tn )S]J.

(2.131)

Equation (2.131) provides a basis for the construction of a reliable perturbation theory which allows one to evaluate correlation function of the generic form (2.126) considering the interaction Hint as a weak perturbation. The point is that the time evolution of the operators in the Dirac representation is determined by the free Hamiltonian, so that one can use the free propagators and vertices to construct the Feynman rules and to introduce the diagram techniques, while the effects of the interaction are accumulated in the S-matrix S, the latter being constructed from the free fields as well. It is worth noting that for the vacuum expectation value, that is for J = J0 , equation (2.131) can be written as 1 H n J†0 T [AH 1 (t ) . . . An (t )]J0 =

J†0 T [A1 (t1 ) . . . An (tn )] SJ0 J†0 SJ0

(2.132)

by virtue that the vacuum, i.e. the ground state of the system, is non-degenerate and J†0 SJ0 = ei!

(2.133)

produces only a phase factor, which can be subtracted.

2.2.3 Positive and Negative Frequency Decomposition Equation (2.131) would be sufficient for construction of perturbative approach to evaluate quantum correlation functions if the interaction Hamiltonian could be presented by a single local quantum operator Hint (x). However, this is not the case, interactions are induced by products of several operators defined in the same space-time point, for example,

2.2 Correlation Functions in Quantum Field Theory

int

H (x) =

 l



25



+l

6i (x) i

.

(2.134)

l

$ The products of local fields 6i (x) are per se ill-defined as we shall see now. Consider, as in the previous discussion, the free Klein–Gordon equation for a real scalar field (∂ 2 + m2 )6(x) = 0,

(2.135)

and try to extract from it as much as possible information about the field itself. The Fourier transform 

d9 q i qx ! e 6(q), (20)9 6† (x) = 6(x) → ! 6(q) = ! 6(–q), 6(x) =

(2.136) (2.137)

allows us to find the solution to equation (2.135) in terms of the singular delta-function and an arbitrary smooth function 6(q) ! 6(q) = $(q2 – m2 )6(q).

(2.138)

Hence, we have in the Minkowskian (3 + 1)D space–time 

d4 q i qx 2 e $(q – m2 )6(q). (20)4

6(x) =

(2.139)

Taking into account that 



 F[fi ] F[f ] $(f ) = , f (yi ) = 0, ′ f (y) f ′ (yi )

(2.140)

 $(q0 + q2 + m2 ) q2 + m2 ) ((q0 ) + ((–q0 ), 2q0 |2q0 |

(2.141)

dyF(y)$[f (y)] =

df

i

and 2

2

$(q – m ) =

$(q0 –



we find the following representation of the solution of equation (2.135) 6(x) = 6+ (x) + 6– (x),

(2.142)

26

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

where 

d4 q ±i qx 2 e $(q – m2 )((q0 )6(±q) (20)4  d3 q e±i qx ±  6 (q), = (20)4 29q  6± (9q , q) 6± (q) =  , 9q = |q0 | = + q2 + m2 . 29q 6± (x) =

(2.143) (2.144) (2.145)

We can re-write equation (2.142) in the following form: 

  d3 q  e–i qx aq + ei qx a†q , (20)3 29q

6(x) =

(2.146)

which gives for the conjugated momentum  0(x) =



d3 q (20)3 29q

  (i 9q ) –e–i qx aq + ei qx a†q ,

(2.147)

making use of the operators aq , a†q normalized such that the commutation relations are satisfied [6– (q), 6+ (q′ )] = $(3) (q – q′ ), ±

±

′

[6 (q), 6 (q )] = 0

(2.148) (2.149)

and [aq , a†q′ ] = $(3) (q – q′ ), [aq , aq′ ] =

[a†q , a†q′ ]

= 0.

(2.150) (2.151)

Note that the equations of motion for the time-dependent operators aq (t) = e–i 9q t aq , a†q (t) = ei 9q t a†q

(2.152)

∂t2 aq (t) + 92q aq (t) = 0,

(2.153)

∂t2 a†q (t) + 92q a†q (t) = 0  9q = q2 + m2 ,

(2.154)

read

which is equivalent to the harmonic oscillator equation for each q. We observe, therefore, a similarity between the decomposition of the coordinate and momentum in classical and quantum mechanics in the positive and negative frequency components

2.2 Correlation Functions in Quantum Field Theory

27

for the harmonic oscillator, equation (2.25), and the solutions of the free Klein–Gordon equation in the field theory, equations (2.146) and (2.147). One can draw a useful conclusion that a quantum field can be consistently considered as an infinite ensemble of harmonic oscillators defined in every point of space–time. By making use of the representation (2.142), we can re-write some important operators in terms of the elementary operators aq , a†q . For example, the free Hamiltonian reads H[6]

 # " 1 d3 x (∂t 6)2 + (∇6)2 + m2 62 2    1 = d3 q 9q 6+ (q)6– (q) + 6– (q)6+ (q) , 2 =

(2.155) (2.156)

where 6± (q) are defined in equation (2.145). The interpretation of a quantized field as a continuous infinite ensemble of harmonic oscillators suggest that the vacuum state can be defined as 6– (q)J0 = 0.

(2.157)

On the other hand, it is natural to demand that the vacuum expectation value of the energy E0 = H0 = J† HJ0

(2.158)

is equal to zero. However, it is easy to observe that by virtue of the commutation relations (2.149), the Hamiltonian (2.156) returns an infinite value. The same is true also for the vacuum expectation value of the momentum and for other operators which are supposed to have zero average in the ground state. To fix this problem, we introduce another ordering prescription which makes the operator product in one space–time point well defined.

2.2.4 Quantum Harmonic Oscillator: Particle Number Representation As we have noticed in Section 2.2.3, expectation values of the products of quantum operators taken in the same space–time point are ill-defined in general case. The simplest example of how this problem can be treated is provided by the quantized one-dimensional harmonic oscillator. Making use of the coordinate q and momentum p decomposition in the positive and negative frequency functions a± 1 q(t) = √ (a+ (t) + a– (t)), 29  9 + p(t) = i (a (t) – a– (t)), 2

(2.159) (2.160)

28

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

the Hamiltonian can be written as 1 1 1 H[q, p] = 92 q2 + p2 = 9(a– a+ + a+ a– ). 2 2 2

(2.161)

Equation (2.161) is still classical. Upon quantization, the functions a± (t) get transformed to the operators a† , a, which obey the commutation relation [a, a† ] = 1,

(2.162)

 1 1 , H[a, a† ] = 9(aa† + a† a) = 9 N + 2 2

(2.163)

and the Hamiltonian reads

where N = a† a. This operator obviously commutes with the Hamiltonian, which means that its eigenstates are in the same time the Hamiltonian eigenstates. The latter are determined by the stationary Schrödinger equation HJn = En Jn ,

(2.164)

where n are natural numbers and the energy level are given by  1 . En = 9 n + 2

(2.165)

In its turn, the eigenvalues of the operator N for the same eigenvectors Jn are NJn = nJn .

(2.166)

The operators a, a† act to these eigenstates to give √ aJn = nJn–1 , √ a† Jn = n + 1Jn+1 .

(2.167) (2.168)

Equations (2.168) entail the interpretation of the a and a† operators as the annihilation and creation operators, respectively. Indeed, they affect the vectors of state increasing or decreasing the number of elementary quanta, each of which bears the energy share 9. It is natural to define the vacuum state by the action of the annihilation operator aJ0 = 0,

(2.169)

2.2 Correlation Functions in Quantum Field Theory

29

and to construct any higher state iteratively J1 = a† J0 , a† a† a† J2 = √ J1 = √ J0 2 2 ... a†n Jn = √ J0 . n!

(2.170)

Finally, the expectation value of the operator N = a† a in the state Jn equals the number of elementary quanta J†n N Jn = n.

(2.171)

In general case, a state with quanta of different sorts marked from 1 to N is given by †n

Jn1 ,n2 ,...nN = 1≤≤N

a  √ J0 . n !

(2.172)

Therefore, N can be naturally interpreted as the particle number operator.

2.2.5 Creation and Annihilation Operators and Normal Ordering Consider the vacuum expectation value of the Hamiltonian (2.163): 1 H[a, a† ]0 = 9. 2

(2.173)

This implies that the energy of the ground state differs from zero. A natural way to treat this issue is to subtract the “vacuum energy”, which cannot be observed anyway Evac =

 1 9i , 2

(2.174)

i

where indices i label oscillators from a given system. In term of the creation and annihilation operators, this corresponds to re-writing the Hamiltonian of the system of quantum oscillators in the form   H[a, a† ] = 9i a†i ai = 9i Ni . (2.175) i

i

In other words, the normal ordering of the operators is suggested: all creation operators must stand to the left of all annihilation operators. Denote this operation by N : N [H] =

 i

9i a†i ai .

(2.176)

30

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

It is evident that vacuum expectation values of a normal-ordered operators equal to zero N [A]0 = 0

(2.177)

for any A.

2.2.6 Wick’s Theorems: Normal and Time Ordering In general case, a product of two operators A1 and A2 is called normal ordered if all positive frequency components of their decomposition Ai = A+i + A–i stand to the left of all the negative components: N [A1 A2 ] = N [(A+1 + A–1 )(A+2 + A–2 )] = A+1 A+2 + A+1 A–2 + A+2 A–1 + A–1 A–2 .

(2.178)

By construction, the vacuum expectation value of a normal-ordered product equals to zero. The normal ordering is suggested in what follows for all products of operators defined in one point. In particular, the S-matrix is defined by the normal-ordered interaction Hamiltonian   ′ int ′ S = T exp i dt N [H (t )] . (2.179) Perturbative expansion of correlation functions in the Dirac picture (2.131) may include, therefore, several overlapping ordering prescriptions. Let us list some useful theorems which allow us to deal with those objects. First a trivial observation: the difference between the simple product of two operators and the normal ordered one is a number and is given by the vacuum expectation value of this product A1 (x)A2 (x′ ) – N [A1 (x)A2 (x′ )] = A1 (x)A2 (x′ ),

(2.180)

A1 (x)A2 (x′ ) = A1 (x)A2 (x′ )0 .

(2.181)

The number A1 (x)A2 (x′ ) is called a contraction of the two operators. Define also a time contraction of two operators: A1 (x)A2 (x′ ) = T [A1 (x)A2 (x′ )] – N [A1 (x)A2 (x′ )],

(2.182)

A1 (x)A2 (x′ ) = T [A1 (x)A2 (x′ )]0 .

(2.183)

31

2.2 Correlation Functions in Quantum Field Theory

Let us define a normal product with a single contraction: N [A1 . . . Ai . . . Aj . . . An ] =Ai Aj N [A1 . . . Ai–1 Ai+1 . . . Aj–1 Aj+1 . . . An ].

(2.184)

This definition can be easily generalized to a normal product with multiple contractions. The following statements hold for boson fields (their extension to fermion fields is straightforward and can be found in the textbooks cited in the bibliography): 1. The first Wick theorem states that a product of n operators is equal to their normal product plus all possible normal products with contractions: A1 . . . An = N [A1 . . . Ai . . . Aj . . . An ] + +





N [A1 . . . Ai . . . Aj . . . An ]

(2.185)

i≠j

N [A1 . . . Ai . . . Aj . . . Ak . . . Al . . . An ] + further permutations

(2.186)

i≠j≠k≠l

2.

The second Wick theorem concerns relation between time-ordered and the normal-ordered products and states that a time-ordered product of n equals the normal-ordered product plus the normal-ordered product with all possible permutations of the normal products with contractions T [A1 . . . An ] = N [A1 . . . Ai . . . Aj . . . An ] +



N [A1 . . . Ai . . . Aj . . . An ]

(2.187)

N [A1 . . . Ai . . . Aj . . . Ak . . . Al . . . An ] + further permutations

(2.188)

i≠j

+

 i≠j≠k≠l

Note that for a time-ordered product of several normal-ordered products T [N [A1 . . . An ] N [B1 . . . Bn ] . . . N [F1 . . . Fn ]]

3.

(2.189)

the second Wick theorem is valid given that the time contractions of the operators belonging to the same normal product are not taken into account. It is now easy to see that for the vacuum expectation values of time-ordered products the third Wick theorem holds: T [N [BA1 . . . An ]]0 =

 T [BA1 . . . Ai . . . An ]0 . i

(2.190)

32

2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory

2.3 Summary Let us briefly summarize the most important results of this chapter. 1. It is convenient to formally describe a quantum system and introduce relevant operators in the Heisenberg representation. This picture suggests, however, that the time evolution of the operators is highly non-trivial and is determined by the full Hamiltonian. Therefore, the correlation functions in the Heisenberg picture are difficult to evaluate in practice. 2. The Dirac picture, which is connected with the Heisenberg one by unitary transformation, allows us to present correlation functions and the S-matrix matrix elements in terms of the free fields. The interactions can be taken into account perturbatively and a consistent calculation framework is possible. 3. Given that interactions are normally defined by products of several quantum operators taken in one space–time point, the normal ordering is used to make sense to such products. Hence, the complete computation scheme for the correlation functions in the Dirac picture includes the perturbative expansion of the S-matrix and application of the Wick theorems to disentangle the products of normal- , timeor path-ordered products of the operators in terms of the vacuum expectation values of the Green functions. The latter are associated with free propagators of the quantum fields. 4. This approach allows us to properly define and, at least, in principle, to evaluate various matrix elements of the form  I(k) =

d4 z –ikz e 6H (z)6H (0)H , (20)4

(2.191)

which can be related to probability distributions or particle density functions.

3 Geometry of Quantum Field Theories In this section, we will show how one can construct the Yang–Mills Lagrangian from geometric arguments and past knowledge from Quantum Mechanics. From experiment, it is known that matter particles, like leptons and quarks, have spin 1/2, obeying Fermi–Dirac statistics and the Pauli exclusion principle. And from Quantum Mechanics, we know that such particles obey the Dirac equation. We start by only considering a Dirac field, as in our simple-minded Ansatz we have no idea yet about the true nature of gauge fields, only knowing that they act on particles that are charged under the su(n) symmetry. Our basic building brick is the free Dirac Lagrangian, on which we will impose invariance under local phase rotations.

3.1 Parallel Transport and Wilson Lines We know the free Dirac Lagrangian is not gauge invariant. Consider a local su(n) phase rotation a (x)ta

8(x) → e±ig !

8(x),

(3.1)

where g is just a constant (which will be identified later as the coupling constant), and we leave the sign unspecified. The antiparticle field 8 transforms with an opposite sign in the exponent: a (x)ta

8(x) → 8(x)e∓ig !

.

The mass term is behaving nicely and remains unchanged under the transformation: a (x)ta ±ig !a (x)ta

–m 88 → –m 8 e∓ig !

e

8 = –m 88 .

However, the derivative term is giving problems, as it pulls out a factor ∂, ! from the exponential: / → i 8∂8 / – g 8 ∂! / a ta 8 . i 8∂8

3.1.1 The Parallel Transporter The standard way to proceed is to introduce a so-called gauge field Aa, that transforms in such a way that it cancels the problematic terms. But instead of introducing a field ad hoc, we investigate the problem at hand a bit deeper, and see if we can pinpoint the erratic behaviour (and solve it) purely by geometric arguments.

34

3 Geometry of Quantum Field Theories

Actually, it is not surprising that the derivative spoils local transformations, as it is not a local but a bi-local operator, viz. it is defined in two space–time points instead of one. The definition of the derivative of 8(x) along the direction of a vector n, is namely  1 8(x + :n) – 8(x) . :→0 :

n, ∂, 8 = lim

(3.2)

This definition is not well defined, as 8(x + :n) and 8(x) obey different transformation laws. In other words, there doesn’t exist a sensible transformation for the quantity ∂, 8. If we would have an object that is able to transport the transformation properties of a field at a point x to those of a field at a point y, we could use it to adapt the derivative to have a single transformation. Let us assume that we have found such a quantity U(x ; y) that is scalar, only depending on x and y, and transforms under the symmetry in equation (3.1) as a (y)ta

U(y ; x) → e±ig !

a (x)ta

U(y ; x) e∓ig !

,

(3.3)

U(y ; x) 8(x) .

(3.4)

so we can use it to transport a field at x to a field at y: a (y)ta

U(y ; x) 8(x) → e±ig !

For this reason it is often called a parallel transporter, or a comparator. Other common names are a gauge link or a Wilson line. We will mostly stick to the latter naming convention. In principle, the requirement in equation (3.3) is the only constraint on U so far (besides that it should be scalar), leaving a whole list of functions as possible candidates. In an attempt to narrow down this list, we will add some additional constraints that seem logical to enforce. First of all, transporting a field from x to y, and then from y to z, should yield the same result as transporting it directly from x to z. Hence the parallel transporter should be transitive: U(z ; y) U(y ; x) = U(z ; x) .

(3.5)

This is not yet a rigorous definition, but it is accurate enough to help us understand the structure of U a bit more. Next, if we place the transporter in between a bi-local product 8(y)8(x) to make it invariant, we can use equation (3.5) to split it at some point z: 8(y)U(y ; x) 8(x) = 8(y)U(y ; z) U(z ; x) 8(x) . But we could as well redefine 8 by absorbing the transporter, i.e.

(3.6)

3.1 Parallel Transport and Wilson Lines

35

N

J(z) = U(z ; x) 8(x), allowing us to write the bi-local product as J(z)J(z). By writing out the barred field: #† " J(z) = U(z ; y) 8(y) = 8(y) U(z ; y) , and identifying with equation (3.6), we see that the dagger operation switches the endpoints: " #† U(z ; y) ≡ U(y ; z) .

(3.7)

Furthermore, moving a field from x to y, and then back to x, should have no final effect. This immediately implies that U should be unitary: def

U(x ; y) U(y ; x) = 1

⇒

#† U(y ; x) U(y ; x) ≡ 1

"

(3.8)

Every unitary object can be represented as a pure phase, i.e. U(y ; x) = e±i gf (y,x) , N

(3.9)

where f is a real function, and the sign in front of ig is for convenience the same as the one chosen in the transformation equation (3.1), and we extracted the same factor g from f (y, x) as we did before from !a . Because of the transitivity of U , this function has to be transitive with respect to addition, in particular f (y, x) ≡ f (y, z) + f (z, x). Also, because of the Hermiticity of U , the function f has to be antisymmetric in its arguments, viz. f (y, x) = –f (x, y). This suggests something of the form ?

f (y, x) = f (y) – f (x) . All these prerequisites are typical for a path-dependent function. This is a function that takes a coordinate as an argument, but is evaluated at the endpoints of the path. For a path C, we can write this as C : z , = x , . . . y,

⇒

%y def f (y, x) = f C(z)%x .

(3.10)

We thus succeeded in limiting the list of possible candidates for U(y ; x) to all pure phases that are functions of a path connecting x and y. This is illustrated in Figure 3.1,

36

3 Geometry of Quantum Field Theories

y

x

Figure 3.1: As a parallel transporter transforms in function of its path endpoints only, all paths shown will give rise to equivalent U(y ; x) ’s, shifting a field at x to a field at y.

where every path leads to a different parallel transporter that is a valid candidate for U(y ; x) . This is why we added the label C to f ; each path can have its own function. There is no restriction on the possible paths, except that they should all be continuous and have the same endpoints. From the physical point of view, this makes a lot of sense. Intuitively, it feels logical that, when transporting a field from one point to another, this is done in a continuous way without abrupt jumps, i.e., along a continuous path in space–time. As the only property we are interested in is its transformation–depending only on the endpoints, we end up with an infinite set of possible parallel transporters, each one defined along a different path, and all equally valid. Whenever it is needed to make a distinction between parallel transporters on different paths, we can write U(yC ; x) to identify the chosen path. 3.1.2 Non-Abelian Paths Now we have to stop for a moment, because there is a real caveat that we overlooked. From the transitivity requirement in equation (3.5), we deduced that f (y, x) = f (y, z) + f (z, x). However, because the transformation exponentials in equation (3.3) are possibly Lie algebra valued functions, it is to be expected that f is Lie algebra valued as well. But in that case, we cannot simply split the exponential, i.e. ef (y,z)+f (z,x) ≠ ef (y,z) ef (z,x) , but we have to use the Baker–Campbell–Hausdorff formula instead, involving chained commutators of f at different space–time points. We decide to turn a different road, and order the f along the path. More specifically, in the expansion of the exponential, all f are ordered in such a way that the f that is first on the path (having the largest path parameterization parameter) is written leftmost. We will use the symbol P in front of the exponential to show that it is a pathordered exponential. Path-ordering was briefly introduced in (2.1.7) and is treated in much more detail in Section 4.1 on page 50 and onwards. Then the transitivity property is valid:

3.1 Parallel Transport and Wilson Lines

y

y

y C2ʹ

C2 C

z

z C1

x

C1ʹ

x

U(Cy ; x) = U C2

( y ; z)

37

x ʹ ≠ U C2

U(zC1; x)

(y ; z)

C1ʹ U(z ; x)

Figure 3.2: When a parallel transporter is split in a point z that lies on its path, it can be written as the product of the two new parallel transporters, i.e. it is transitive. This is not correct when z lies outside the original path.

P ef (y,x) = P ef (y,z) P ef (z,x) .

(3.11)

There is, however, another caveat: this property is only valid if z already lies on the path connecting x and y because we cannot blindly split the path in a point outside of it, see Figure 3.2. The correct definition of the transitivity property is then ∀ C 1 + C2 = C :

C

C

U(yC ; x) = U(y2; z) U(z1; x) .

(3.12)

Note that it is not impossible to use the transitivity rule with a point outside the path. C If we multiply both sides of the last equation to the left with U(z 2; y) , we get C

C

U(z2; y) U(yC ; x) = U(z1; x) ,

(3.13)

because †  C C C C U(z2; y) U(y2; z) = U(y2; z) U(y2; z) = 1. We can interpret this as saying that the path C1 can be “split” at a point outside the path if, and only if, the path returns on top of itself to cancel the superfluous part, i.e. if C1 = C – C2 .

(3.14)

This is illustrated in Figure 3.3. Let us now try to parameterize f C for a given path C. We approximate the path connecting x and y by dividing it into n infinitesimal linear segments, as illustrated in Figure 3.4: f C(y, x) ≈

n 

f C(xi+1 , xi )

(3.15)

38

3 Geometry of Quantum Field Theories

y

C2 z

z C

C1 x

Figure 3.3: The transitivity rule can be also

x

applied to points outside the path C on the condition that the path returns on itself, cancelling the overlapping segments.

C2 U C(z1; x) = U (y UC ; z) (y ; x)

y

xn x n–1 x n–2

C

C x2 x1 x

Figure 3.4: Any path C can be discretized by dividing it into n linear segments. In the limit n → ∞, the original path is recovered.

x0

From symmetry considerations, we expect f C to depend in the infinitesimal limit on the centre of the segment:1 f C(xi+1 , xi ) = ! fC ?

x

+ xi  . 2

i+1

Yet this cannot be correct, as now f C is symmetric instead of antisymmetric. This was to be anticipated as we discarded information, going from a function of two variables to one of one variable. If we choose to describe a linear line segment [x, y] in function of its centre (x + y)/2 , we have to add the separation vector y – x as a second variable. But instead of making ! f C dependent on the separation vector, we interpret it as though its structural form itself changes, in function of the direction it is evaluated over. In other words, we promote ! f C to a vector function which we will call A, and contract it with the separation vector. Then f C is defined as f C(y, x) ≈

n 

(xi+1 – xi ), A,

x

+ xi  . 2

i+1

1 Using Ito-calculus, one can show that taking the centre of the segment is even a necessity in order to get a correct definition.

3.1 Parallel Transport and Wilson Lines

39

Note that we dropped the label C from A, , because the path dependence is now fully moved to the coordinates xi and xi+1 , while the structural form of A, is path independent. In other words, the same A, is used for every parallel transporter, for any path, in contrast with f C , which is different for every path. The path dependence manifests itself through the factor (xi+1 – xi ), in front, in the index , of A, , and in the argument (xi+1 + xi )/2 of A .2 As we discussed before, f C (y, x) is expected to be Lie algebra valued , because the transformation of U is Lie algebra valued as well. It is then logical to put def the Lie dependence inside A, , i.e. A, = Aa, ta . Taking the limit n → ∞, the discrete formula becomes a line integral (see equation (C.5c)), i.e.:  C f (y, x) = dz,Aa, (z) ta . (3.16) C

This is in perfect accordance with our physical intuition of a path with fixed endpoints. Inserting this result in equation (3.9) gives us the final definition for the parallel transporter: def

U(y ; x) = P e

±ig

&y x

dz,Aa, (z) ta

.

(3.17)

This newly constructed definition for the parallel transporter allows us to interpret the transformation rules in equation (3.3) as a transformation of A, because the latter is the only dynamic component of U . To see this, we take the parameters from the exponentials in equation (3.3) and insert them in the integral using the gradient theorem (see equation (C.14)): y ± ig!(y) ∓ ig!(x) = ±ig

dx,∂, !.

(3.18)

x

This implies that we have to define the transformation of the vector function Aa, as follows: Aa, (x) → Aa, (x) + ∂, !a .

(3.19)

Note that the sign in the transformation of Aa, is always a plus, independent of the sign chosen in equation (3.1) because ∂, ! has the same sign as A, (compare equation (3.18) with equation (3.17) to see this). This result is only approximately true because we assumed that we can simply combine the exponentials. This is however not true in

2 It is to realize that although we intentionally named A, (x) as to simplify its identification with the gauge field later on, at the moment, it is nothing but a generic vector function of x, without specifying its further structure.

40

3 Geometry of Quantum Field Theories

the non-Abelian case, as the presence of Lie generators makes the field behave like operators (see the end of Appendix B.1 for a short discussion on the usage of operators in this book). It is generally known that for any two operators X, Y with Y invertible, the following relation holds: eYXY

–1

= YeX Y –1 ,

which, applied on the transformation of a Wilson line, gives a (y)ta

e±ig !

=Pe

a (x)ta

U(y ; x) e∓ig !



y a a & a a ±igexp±ig ! (y)t dz,Aa, (z) ta e∓ig ! (x)t x

.

(3.20)

We can simplify this result by partial integration and the gradient theorem:

Pe

±ig

&y x

  a a a a dz, e±ig ! (z)t Aa, (z) ta ± gi ∂, e∓ig ! (z)t

,

from which we deduce the correct transformation rule for a non-Abelian field:  i a a a a ±ig !a (x)ta a a A, (x)t → e A, (x) t ± ∂, e∓ig ! (x)t , g

(3.21)

(3.22)

which indeed reduces to equation (3.19) for Abelian fields.

3.1.3 The Covariant Derivative Let us return to the original goal for which the comparator was constructed, viz. formulating a sensible definition for the derivative in equation (3.2). This is now straightforward; we simply transport the field from x to x + :n. The result is called a covariant derivative: n, D, 8 = lim def

:→0

 1 8(x + :n) – U(x+:n ; x) 8(x) . :

(3.23)

Because the comparator connects two points that are separated by an infinitesimal distance :n, we can expand its definition in equation (3.17) up to first order in :. Applying the discrete definition of the line integral (equation (C.14)), this leads to , A (x+ 1 :n) , 2

U(x+:n ; x) ≈ P e±i:gn

≈ 1 ± i:gn, A, (x).

(3.24)

The covariant derivative is then given by D, 8 = ∂, 8 ∓ ig Aa, ta 8,

(3.25)

3.2 The Gauge Field Tensor and Wilson Loops

41

and it indeed transports the transformation of the field:     D, 8 → ∂, e±ig! 8 ∓ ig A, + ∂, ! e±ig! 8   = ±ig ∂, ! e±ig! 8 + e±ig! ∂, 8   ∓ igA, e±ig! 8 ∓ ig ∂, ! e±ig! 8 = e±ig! D, 8. To derive the transformation rule of the covariant derivative itself, we simply insert 1 = e±ig!(x) e∓ig!(x) between D, and 8. We see that it transforms similarly to a parallel transporter U(x ; x) on a closed path:3 D, → e±ig!

a (x)ta

D, e∓ig!

a (x)ta

.

(3.26)

From this, we can also express the transformation rule for A, in a different way: i a a a a Aa, ta → ± e±ig! t D, e∓ig! t . g

(3.27)

With help from the covariant derivative, we can now define a Dirac Lagrangian that is invariant under local transformations like those in equation (3.1):   LDirac = 8 iD/ – m 8.

(3.28)

The vector field part in the covariant derivative gives rise to an interaction term between the Dirac fields and the vector field: / LIDirac = ±g 8A8.

(3.29)

This is the main result of our approach: By making the derivative a well-defined mathematical object, we let a vector field emerge naturally in the form of interaction terms with the Dirac field. Of course, this vector field will be identified as the su(n) gauge field, but let’s not be too rash in our conclusion. There are still some missing parts in our approach.

3.2 The Gauge Field Tensor and Wilson Loops In a Quantum Field Theory, the next step to proceed after defining a classical Lagrangian is to quantize the theory so that every function of space–time coordinates 3 We are not insinuating that the covariant derivative is a special type of U(x ; x) . But we do observe that they have the same transformation behaviour, which is to be expected as D, is constructed from U .

42

3 Geometry of Quantum Field Theories

inside the Lagrangian will be interpreted as a particle field. However, our Lagrangian isn’t “complete” yet. To understand this, note that the quantization procedure is split into two parts per field: – Quadratic terms are treated as the dynamics for the field and quantized, –

Remaining terms are treated as interactions using perturbation theory.

As we are now constructing the Lagrangian at a global scale, we conclude that we are missing kinetic terms for the field A, . Of course, the standard approach is to continue in a heuristic manner. We prefer however to let the kinetic terms emerge in the Lagrangian in a natural and elegant fashion, in the same way the interaction term emerged in the previous section. We want to base our approach on geometrical arguments only, starting from the parallel transporter. Because the kinetic terms can only contain A, fields, we have to start with a gauge-invariant version of the Wilson line (as we have no other fields to balance the transformation rules). If we evaluate a line on a closed path and trace it, i.e. Uloop = U(x ; x) = Tr P e±ig def

'

, C dz A, (z) ,

(3.30)

it is automatically invariant:   Tr U(x ; x) → Tr e±ig!(x) U(x ; x) e∓ig!(x) = Tr U(x ; x) .

(3.31)

Such an object is called a Wilson loop, and it contains – as we will show – all dynamics of the vector field. We use Stokes’ theorem to transform the line integral over a vector into a surface integral over the gradient: (  dz ⋅ A(z) = d3,- ∂[, A-] , (3.32) C

G

where G is the surface that is spanned by the closed path C. Note that because the path is oriented, the surface is oriented as well. The orientating of the normal of the surface follows the corkscrew-rule: making a fist, if your fingers follow the path, your thumb points in the normal direction of the surface. This is illustrated in Figure 3.5. dσ C Ʃ

dx . A C

dσ

= Ʃ

Figure 3.5: Stokes’ theorem relates a line

integral over a closed path C with a surface integral over the enclosed surface G.

3.2 The Gauge Field Tensor and Wilson Loops

43

Just as we can parameterize a curve in function of a parameter + as C : z, (+)

⇒

dz, = d+

∂z, , ∂+

(3.33)

we can parameterize a surface in function of two parameters + and *: G : z, (+, *)

d3,- = dz,∧ dz- = 2d+d*

⇒

∂z[, ∂z-] . ∂+ ∂*

So we write the surface integral as   ∂z, ∂z-  ∂, A- – ∂- A, . d+d* ∂+ ∂+

(3.34)

(3.35)

G

Our next move is a bit peculiar. We have to find a loop that makes sense from a physical point of view, i.e. it should be as general as possible. The most natural case is to choose a “zero”-loop, infinitesimally small, starting from and ending in a point x. To achieve this, we discretize space–time and define our theory on a lattice with grid spacing :. Discretizing space–time is only allowed in an Euclidian space, so we make our space Euclidian by doing a Wick rotation (see Section 6.3): def

z0 = izE0 , ∂

def

(3.36a)

def

(3.36b)

def

(3.36c)

z = zE ,

0 def

= i∂E0 ,

∂ = ∂ E,

def

A = AE ,

A0 = iA0E , which changes a vector product by ,

v, w, = –vE , wE , but leaves a matrix product invariant, i.e., ,-

9,- 1,- = 9E ,- 1E . Then we can rewrite the Wilson loop as ⎫ ⎧  , ⎨  ⎬ ∂zE ∂zE-  ∂, AaE - – ∂- AaE , ta . Uloop = Tr P exp ±ig d+d* ⎭ ⎩ ∂+ ∂*

(3.37)

±

A short remark: For the Wick rotation to be valid, we assume that A, is well-behaving on the contour C, especially that it doesn’t introduce poles that would hit the Wick rotation (invalidating the result). Similarly, we assumed space–time and the loop G to be continuously enough for Stokes’ theorem to hold. In fact, Stokes’ theorem is only well-defined for smooth paths and manifolds (which is now not the case), but we use an extension to Stokes’ theorem that is well-defined for piecewise smooth paths. There

44

3 Geometry of Quantum Field Theories

x+

x

+

Figure 3.6: On a lattice, the smallest loop possible is a

x+

planar square with sides equal to the lattice spacing.

is a strong mathematical background for this extension, but this would go too far beyond the scope of an introductory approach to Quantum Field Theory, so we don’t pay much attention to it and assume all necessary conditions to be satisfied. A ‘zero’-loop is of course the smallest loop possible; on a lattice, this is a rectangular planar loop spanning the lattice spacing, as is illustrated in Figure 3.6. We will naturally choose our coordinate system along the grid, such that the sides of the square loop lie along the basis directions. We can parameterize such a loop as: ,

G : zE (+, *) = x, + n, + + ! n, *,

+, * = 0 . . . :,

(3.38)

n, are perpendicular basis vectors (i.e. n⋅! n ≡ 0). It is necessary to expand where n, and! equation (3.37) up to second order because the first order vanishes due to the tracelessness of the Lie generators tr ta = 0. Even in the Abelian case—where the only generator is the identity with non-vanishing trace—the first-order terms vanish, by cancellation, which is easy to prove. Ignoring the constant first term, the expansion gives

1 Uloop ≈ –g 2 n,! n -n1! n3 ⋅ ⋅ 2

+1 ,*1

:

   d+2 d*2 ∂, AaE - – ∂- AaE , ∂1 AaE 3 – ∂3 AaE 1 ,

d+1 d*1 0

0

where the factor 1/2 comes from the trace of the generators tr ta tb = 1/2$ab . We satisfied the path ordering requirement by chaining the integrals (see Section 4.1.2 for more information on how this works). Note that we can simplify the factor n,! n -n1! n 3 by collecting similar vectors. The tensor product n, n- can be represented by a matrix that is zero everywhere except on the diagonal entry of the directional vector, where it is one, i.e.

3.2 The Gauge Field Tensor and Wilson Loops

⎛

1 ⎜ , - ⎜0 n1 n1 =˙ ⎜ ⎝0 0 ⎛ 0 ⎜ , - ⎜0 n3 n3 =˙ ⎜ ⎝0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1 0

⎞ 0 0⎟ ⎟ ⎟, 0⎠ 0 ⎞ 0 0⎟ ⎟ ⎟, 0⎠ 0

⎛

0 ⎜ , - ⎜0 n2 n2 =˙ ⎜ ⎝0 0 ⎛ 0 ⎜ , - ⎜0 n4 n4 =˙ ⎜ ⎝0 0

0 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

45

⎞ 0 0⎟ ⎟ ⎟, 0⎠ 0 ⎞ 0 0⎟ ⎟ ⎟. 0⎠ 1

If we take, e.g. the infinitesimal square loop spanned by the n1 and n2 vectors, the integrand automatically becomes 

  ∂1 AaE 2 – ∂2 AaE 1 ∂1 AaE 2 – ∂2 AaE 1 ,

in other words, both factors get the same indices. As this starts to look as a sum over indices, we try to exploit this further. In a four-dimensional Euclidian space, there are , , , , , 12 independent planes, viz. the planes spanned by n1 n-2 , n1 n-3 , n1 n-4 , n2 n-3 , n2 n-4 and , n3 n4 , and the planes oppositely oriented to them. There is no reason why one plane would be preferred over the other, so we define our Wilson loop as the sum of all 12 square loops, one for each independent infinitesimal plane. Note that summing these planes gives a straightforward result: 

, 1

ni ni ! nj-! nj3 = $,1 $-3 – g ,13- ,

(3.39)

planes

where g,-13 = 1, only when , = - = 1 = 3. Because both factors of the integrand are already antisymmetric in ,- resp. 13, only the first term of the rhs will contribute, and we can just make the contractions. Then the second order term in the expansion becomes +1 ,*1

: Uloop ≈ –g

2

  , a d+2 d*2 ∂, AaE - – ∂- AaE , ∂ , A-E a – ∂ - AE .

d+1 d*1 0

(3.40)

0

Because : is an infinitesimal parameter, going to zero in the continuum limit, we can approximate the integrals with help from equation (C.5c). The innermost integral then equals 

%  , - a - , a  , - a - , a % d+2 d*2 ∂ AE – ∂ AE ≈ +1 *1 ∂ AE – ∂ AE %%

+1 ,*1 0

,

n x+ 21 +1 n+ 21 *1!

46

3 Geometry of Quantum Field Theories

so that the outermost integral becomes : 0

  %% d+1 d*1 +1 *1 ∂, AaE - – ∂- AaE , %%



%

, a  %%

∂ , A-E a – ∂ - AE

%

n x+ 21 +1 n+ 21 *1!

n x++1 n+*1!

%  %%  , - a - , a % :4  a a % ∂ AE – ∂ AE %% ≈ . ∂, AE - – ∂- AE , % 4 n n x+ : n+ : ! x+ : n+ : ! 2

2

4

4

Because the arguments of the fields in both factors are the same in the limit : → 0, we already drop the linear parts in : from the arguments. We then have Uloop ≈ –g 2

2 %   :4  % ∂, AaE - – ∂- AaE , % + O :4 . x 4

(3.41)

Unfortunately, these are not the only terms of order :4 . If we expand the exponential further to third and up to fourth order, additional terms of order :4 emerge. We won’t show the calculation here, as it is trivial to do but really long, but just give the result instead. The extra terms are ∓g 3 f abc AaE , AbE - ∂ , A-E a –

1 4 abx xcd a b , c - d g f f AE , AE - AE AE . 4

So we can conclude that, up to an irrelevant constant term in front,   1 ,- a Uloop ≈ –g 2 :4 FEa ,- FE + O :5 . 4

(3.42)

Now comes the tricky part. We cannot simply take the continuum limit : → 0, as the action, the fields, and the coupling constant are subject to rescalings and renormalization to be able to reproduce the correct continuum theory. When summing over all lattice points (i.e. when integrating over x), we have to divide by the lattice spacing to the fourth, i.e. :4 , before taking the limit : → 0. See, e.g. Ref. [62] for a profound treatment on the continuum limit. So finally, after rescaling the coupling constant, moving back to the continuum theory and un-Wick rotating, we find 1 a ,- a Uloop ≡ – F,F , 4

(3.43)

up to an irrelevant constant term (which will be subtracted from the action anyway). The gauge field tensor is given by a F,= ∂, Aa- – ∂- Aa, ± g f abc Ab, Ac- .

(3.44)

This is exactly what we hoped for. We have shown how the kinetic terms for the gauge field naturally emerge from the Wilson loop. Because the latter is gauge invariant by definition, the gauge field tensor is automatically gauge invariant as well.

3.3 Summary

47

3.3 Summary The results of this chapter show us that every gauge theory has a deep geometric structure, in fact, that every gauge theory is a geometric effect in se. Starting only from the free Dirac Lagrangian and the demand of local phase invariance, we constructed a full gauge theory. More specifically, the requirement for the derivative to be well-behaving, viz. having a sensible local definition instead of bi-local, leads to – the construction of a parallel transporter, or Wilson line, and with it the introduction of the gauge field. –

the transformation rules for the gauge field.

–

the definition of the covariant derivative, and hence with it the description of interactions between matter fields and the gauge fields.

–

the construction of a gauge-invariant Wilson loop, and the demonstration that it is intimately related to the gauge field dynamics and kinetic terms.

Note that the last statement is a direct indication that the gauge sector of any gauge theory can be fully recast in function of Wilson loops. This methodology, firmly based on the mathematics of loop space, replaces coordinate and momentum dependence with path dependence. The full Yang–Mills Lagrangian for a su(n) symmetry is thus given by   1 LYM = 8 iD/ – m 8 – Tr F,- F ,- . 2

(3.45)

For the rest of this book, we will stick to the convention of a positive sign in the gauge transformation exponential. The opposite convention is easily recovered by making the substitution g → –g. This sign convention propagates through the definition of the covariant derivative and the gauge field tensor: D, = ∂, – i g Aa, ta ,

(3.46a)

a = ∂, Aa- – ∂- Aa, + g f abc Ab, Ac- , F,-

(3.46b)

and the local gauge transformations 8 → ei g ! 8 → 8e Aa, ta →

a (x)ta

8,

–i g !a (x)ta

(3.47a) ,

(3.47b)

i i g !a (x)ta a a D, e–i g ! (x)t , e g

(3.47c)

D, → ei g !

a (x)ta

i g !a (x)ta

F,- → e

D, e–i g ! F,- e

a (x)ta

,

(3.47d)

–i g !a (x)ta

(3.47e)

,

48

3 Geometry of Quantum Field Theories

which leave the Lagrangian invariant. Four final remarks: – Note that these gauge transformation rules immediately invalidate mass terms for the gauge field; indeed, terms of the form m2 Aa, A, a are not gauge invariant. –

This is not the most general Lagrangian possible that conserves a su(n) symmetry. We could add terms of the form :,-13 Tr F ,- F 13 . However, these terms are parity violating, and hence beyond the scope of this book.

–

Note that the covariant derivative and the gauge field tensor can be related by " # D, , D- = –igF,- .

–

The minus sign between the derivative terms inside the gauge field tensor (equation (3.44)) is a direct result of the wedge product in the infinitesimal surface element in Stokes’ theorem (equation (3.34)); hence, it is a pure geometrical effect.

4 Basics of Wilson Lines in QCD We saw in the previous chapter that a Wilson line is a path-ordered exponential constructed from the gauge fields (see equation (3.17)) that transforms bi-locally (see equation (3.3)). It is an object that emerges naturally in gauge theories from geometrical arguments to cure the definition of the derivative, but this is not its only application. Because of its bi-local transformation properties, it is often used as a parallel transporter to render non-local terms gauge invariant, especially in quantum chromodynamics (QCD) calculations concerning validation of factorization schemes, and in calculations for constructing or modelling parton density function (PDF) (see Chapter 5). For these reasons, Wilson lines deserve some special attention – which is why we investigate them with great detail in this chapter and the next. We focus on piecewise linear Wilson lines – which are vastly the most commonly used and the only ones used in this book – and will derive their properties and Feynman rules, and construct a framework meant to simplify perturbative calculations. Finally, in the last section, we briefly motivate the importance of Wilson lines by explaining the eikonal approximation – one of the main applications of Wilson lines.

4.1 A Wilson Line Along a Path A Wilson line is a path-ordered exponential of a line integral of the gauge field along a given path C: U

C

=Pe

ig

& C

d z , A , (z)

.

(4.1)

The sign convention, which was discussed in Chapter 3, also manifests itself in the definition of the Wilson line. Choosing a positive sign in the gauge transformation of the particle field – as we do in this book – results in a positive sign in the path-ordered exponential. This follows from its behaviour under gauge transformations, as it has to have the correct sign to cancel the transformation of the particle field, in order to parallel transport it to a different spacetime point (see equations (3.3), (3.4) and (3.18)). The Wilson line we constructed in Chapter 3 is valid for any group theory. We focus on QCD, and as such every Wilson line we will use from now on will be an SU(3) group element, expressed in the fundamental or the adjoint representation. If no statement about the representation is made, we assume it to be in the fundamental. The physical interpretation of a Wilson line becomes clear by expanding the exponential:  ∞  1 , , n C U = (4.2) (ig ) P dzn n⋯ dz1 1 A,n(zn ) ⋯ A,1(z1 ), n! n=0 C

That is, the nth order in the expansion represents a radiation of n gluon fields as in Figure 4.1. To save writing space, we often use the shorthand notation N

a

Ai = A,ii (zi ).

(4.3)

50

4 Basics of Wilson Lines in QCD

y

C ...

A4

A5

An An−1

A3

x

A2 A1

Figure 4.1: Illustration of the nth order term in the C expansion of the Wilson line U(y . The n-radiated ; x) fields Ai (using the short-hand notation defined in equation (4.3)) can be radiated at any point (due to the fact that they are integrated over along the path), but are ordered such that +n ≥ +n–1 ≥ ⋅ ⋅ ⋅ ≥ +2 ≥ +1 .

The point zi at which the field Ai is radiated is integrated over the full path to get all possible configurations. However, path-ordering adds the additional constraint that all fields should remain in the same order, i.e. the field Ai has to be radiated between Ai–1 and Ai+1 . The full exponential is thus a resummation of all possible radiations from the path. We can interpret this as a full gauge effect along the path; this can be e.g. the nett effect of a particle moving in an external medium. Resumming all gluons, a Wilson line can also represent, e.g. a fast-moving quark (when in the fundamental representation) or a fast-moving gluon (when in the adjoint representation). In this case, we assume the quark resp. gluon not to deviate after radiating a gluon. This is called the eikonal approximation and is treated in more detail in Section 4.6.

4.1.1 Properties of Wilson Lines In Chapter 3, we constructed a Wilson line by the requirement to satisfy a set of properties. As we discovered during the derivation, these properties led to the natural interpretation of a Wilson line being a functional of a path. We just list them here again for easy reference: " #† A Wilson line is unitary: U C U C = 1.

(4.4a) "

Path reversion equals Hermitian conjugation: U C = U C

It is path-transitive, i.e. if C = C1 + C2 , then U = U

C1

U

# –C † C2

.

.

(4.4b) (4.4c)

It transforms in function of its endpoints only: U(y ; x) → eig !

a (y)ta

U(y ; x) e–ig !

a (x)ta

.

A Wilson loop is gauge invariant: Uloop → Uloop .

(4.4d) (4.4e)

4.1.2 Path Ordering The symbol P in equation (4.1) denotes path ordering, ensuring that the gauge fields are ordered in such a way that the first fields on the path are written leftmost. When as-

4.1 A Wilson Line Along a Path

51

sociating a diagram with this formula, we will use the same convention as with Dirac lines: we read them from right to left. Getting a bit ahead, we have already written the gauge fields in a reversed order, from n to 1, such that when drawing a Wilson line on a path from left to right, we can notate the gauge fields as 1 to n (from left to right, see Figure 4.1). This ensures that An is the first field on the path (having the highest value for its parameter +, see below). The Wilson line still has two open indices, in the fundamental or the adjoint depending on the representation chosen for the Lie generators (remember that we defined the gauge fields with the generators absorbed, A, = Aa, ta ). In case of a Wilson loop, these indices are traced. It is convenient to parameterize the path C in function of a one-dimensional parameter +: C : z, (+),

+ = a . . . b,

(4.5)

where z(a) and z(b) are the start, resp. endpoints of the path. Then we can formally write out the path-ordering requirement, e.g. for two fields we have P A,1(z1 )A,2(z2 ) = ( (+1 –+2 ) A,1 (+1 ) A,2 (+2 ) + ( (+2 –+1 ) A,2 (+2 ) A,1 (+1 ) for z,1 (+1 ) and z,2 (+2 ). This can easily be generalized to more than two fields, by chaining an appropriate number of (-functions: 

P A,1(z1 ) . . . A,n(zn ) =

n–1

3(+1 ,...,+n )

/ where

3(+1 ,...,+n )

( (+i+1 –+i ) A,1 (+1 ) . . . A,n (+n ) ,

(4.6)

i=1

represents a sum over all possible permutations of +i . Note that in the

case of classical Abelian fields, all fields commute, and we can sum all (-functions. Then the path-ordering symbol can just be ignored: P A,1(z1 ) ⋯ A,n(zn ) = A,1(z1 ) ⋯ A,n(zn ). In that case, every term in the expansion of the exponential is just a power of the same integral:

Uabelian = P exp

ig

& C

dz, A, (z)

⎞n ⎛  ∞  1 = (ig )n ⎝ dz, A, (z)⎠ . n! n=0 C

52

4 Basics of Wilson Lines in QCD

But of course, we are mainly interested in non-Abelian fields, as in this book we will be using Wilson lines in QCD. Calculating a line integral is easiest by parameterizing the path as in equation (4.5). Then we make a change of variables: dz, → d+

dz, , d+

in order to rewrite the path-ordered exponential as

C U(b;a)

⎧ b ⎫ ⎨  ⎬  , ′ = P exp ig d+ z A, (z) . ⎩ ⎭

(4.7)

a

We can effectuate the path ordering in the expansion using equation (4.6). This will manifest itself as a chaining of the parameters +i in the upper integration borders:  P

d+n ⋯ d+1

dz,n (+n ) dz,1 (+1 ) ⋯ d+n d+1 b  +n +n–1 +2 dz,n (+n ) dz,1 (+1 ) = n! d+n d+n–1 d+n–2 ⋯ d+1 ⋯ . d+n d+1 a

a

a

(4.8)

a

This literally tells us what we anticipated: that the ith gauge field (with parameter +i ) has to be radiated between the (i – 1)th and the (i + 1)th gauge field because with these integration borders the parameters satisfy +i+1 ≥ +i ≥ +i–1 . Note that equation (4.8), is only valid for integrands of the form     A,n z(+n ) . . . A,1 z(+1 ) ,

(4.9)

that is, products of the same vector field function, depending on different variables. We cannot use it, e.g.       A,n z(+n ) . . . ∂- A,i z(+i ) . . . A,1 z(+1 ) , because the interchange symmetry is broken by the derivative. It is possible to move the chaining to the lower integration borders, but in this case, we need to flip the order of the parameters (to ensure that we can keep the order of the radiated gluons as it is, i.e. from n to 1):

4.2 Piecewise Wilson Lines

 P

d+n ⋯ d+1

dz,n (+n ) dz,1 (+1 ) ⋯ d+n d+1 b b b b dz,n (+n ) dz,1 (+1 ) = n! d+1 d+2 d+3 ⋯ d+n ⋯ . d+n d+1 a

+1

+2

53

(4.10)

+n–1

It is straightforward to check these formulae in an intuitive way, by verifying that in both cases +n ≥ +n–1 ≥ ⋅ ⋅ ⋅ ≥ +2 ≥ +1 . Using the integrand in equation (4.9), the full expression is then automatically path ordered. Depending on the specific path calculation, equation (4.8) or (4.10) might be easier to use. In order to investigate how different path structures influence a Wilson line, it is preferable to separate the path content from the gauge field content. Luckily, this can be easily done by making a Fourier transform. The path content is then fully described by the following integrals:  n  , ′  , ′  1 In = (4.11) (ig )n P d+1 . . . d+n z1 1 . . . zn n ei ki ⋅ zi . n! Note that although we use the common convention for Fourier transforms (with a negative sign in the exponent for the inverse transform, see equations (B.57)),1 we preferred to make the integration substitution k → –k, to make the link with common literature concerning Wilson lines (cf., e.g. in Ref. [85]). The nth-order term of the Wilson line expansion is then given by  9 d kn d9 k1 Un = ⋯ A, (–kn ) ⋯ A,1 (–k1 ) In . (4.12) 9 (20) (20)9 n The negative signs in the arguments are a result from the integration substitution explained above. They remind us that the results we will derive are defined for momenta pointing outwards in a Feynman diagram. Also, remember that the fields are ordered from n to 1 to allow them to be read from left to right.

4.2 Piecewise Wilson Lines In general, most interesting and dynamically rich paths will not be smooth, but contain cusps. These are points in the path where the path is continuous but its derivative is not, i.e. the path looks cracked. This is illustrated in Figure 4.2. The reason that cusps are more compelling is that they don’t occur naturally, but are the result of external driving forces. For example, if the Wilson line represents a resummed quark, a cusp can be the effect of an interaction with a hard photon. Cusps hence contain all information on the dynamics of a system. 1 This implies that positive momenta are pointing inwards in a Feynman diagram.

54

4 Basics of Wilson Lines in QCD

C2

C3

Figure 4.2: A path with cusps. Although the full path is not

smooth, the three cusps divide the path into four smooth segments. The full path C can be approached as a piecewise path with regions C1–4 .

C4 C1

At first sight, paths with cusps might seem a bit problematic from a mathematical point of view, as a general path is supposed to be smooth, i.e. continuously differentiable. What saves the day is the transitivity property of a Wilson line, because it allows us to split the path at the cusp and continue with a product of two Wilson lines. For example, in Figure 4.2, we have three cusps that divide the path into four segments. In other words, the path on which a Wilson line is evaluated can be piecewise, as long as each segment is smooth (in particular, each segment should be defined over an interval that is not a single point). Note that we don’t even need the restriction that the segments should be joined because we can always define a piecewise function (with possibly disconnected segments) in function of adjoining intervals in the parameterization parameter: ⎧ ⎪ f 1 (+) + = a1 . . . a2 , ⎪ ⎪ ⎪ ⎪ ⎨f 2 (+) + = a2 . . . a3 , f (+) = . (4.13) ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎩ M f (+) + = aM . . . aM+1 . We will use capitalized Latin letters for the index referring to segments. Let us consider a piecewise smooth path, consisting of M continuously differentiable segments. We would like to be able to express the integrals In that contain all path information (see equation (4.11)) in function of the same integrals but expressed over each segment separately. The first-order integral over the Jth segment only involves the Jth part of f : S1J

aJ+1 aJ+1 = d+ f (+) = d+ f J (+). aJ

(4.14)

aJ

Then of course the first-order integral I1 can be trivially expressed in function of the first-order segment integrals, as it is just their sum: ⎞ ⎛ aM+1 aM+1 a2 a3 ⎟ ⎜ d+ = ⎝ + + ⋯ + ⎠ d+, a1

a1

I1 =

M  J=1

a2

S1J .

aM

(4.15)

4.2 Piecewise Wilson Lines

55

The calculation of the second-order integral is a bit more tricky, as there are two points of particular interest. First, because in equation (4.11), every incarnation of e–iki ⋅ z gets another index i, we need to introduce this dependence in f , which we will do with a lower index: +1

aM+1

I2 =

d+1 fi (+1 ) a1

d+2 fi+1 (+2 ), a1

where of course in this case i = 1, but we left it open for the sake of generality. The second point is that in the definition of I2 , the inner integral has a variable upper border, and thus is a piecewise function itself: ⎧+ &1 ⎪ ⎪ +1 = a1 . . . a2 , ⎪ d+2 fi+1 (+2 ), ⎪ ⎪ a1 ⎪ +1 ⎪ & ⎪ + ⎪ 1 ⎨S1 (i + 1) + d+2 fi+1 (+2 ), +1 = a2 . . . a3 , a2 d+2 fi+1 (+2 ) = . (4.16) ⎪ .. ⎪ ⎪ a1 ⎪ ⎪ ⎪ M–1 ⎪ &+1 / J ⎪ ⎪ S (i + 1) + d+2 fi+1 (+2 ), +1 = aM . . . aM+1 . ⎩ aM

J=1

The outermost integral will be split as well, combining the appropriate regions: +1

aM+1

d+1 fi (+1 ) a1

a2 =

d+2 fi+1 (+2 ) a1

+1

d+1 fi (+1 ) a1

aM+1

a3 d+2 fi+1 (+2 ) + ⎛

a1

⎜ d+1 fi (+1 ) ⎝

+⋯+

⎛ ⎜ d+1 fi (+1 ) ⎝S1 (i + 1) +

a2 M–1 

SJ (i + 1) +

J=1

aM

+1

⎞

+1

⎞ ⎟ d+2 fi+1 (+2 )⎠

a2

⎟ d+2 fi+1 (+2 )⎠ .

aM

This can be simplified by using the notation for the first-order segment integral and by introducing the notation for the second-order segment integral:

S2J (i)

aJ+1 +1 J = d+1 d+2 fiJ (+1 )fi+1 (+2 ), aJ

I2 =

M  J=1

(4.17)

aJ

S2J (1) +

J–1 M  

S1J (1)S1K (2).

(4.18)

J=2 K=1

Note that S2 only depends on the Jth segment; no mixing occurs. This will be true to all orders and is exactly what we hoped for: We can express the full path-ordered integral as path-ordered integrals over the separate segments. Also note that the

56

4 Basics of Wilson Lines in QCD

argument of successive segment integrals (which is the incarnation index of f ) is simply incrementing; this will also be true to all orders, i.e. only terms of the form J

J

J

Sm1 1 (i)Sm2 2 (i + 1) . . . Smkk (i + k) will appear. In what follows, we will drop this argument of S, as it is trivial to deduce as long as we keep the ordering of the Ss fixed. Although we used the expression for the path ordering given by equation (4.8), the whole derivation is equally valid when using the chaining of the integration borders as given in equation (4.10). The extension to higher orders is trivial but paper-consuming, so we just give the results:

I3 =

M 

S3J +

J=1

I4 =

M 

  J–1  K–1 M  S1J S1K S1L , S1J S2K + S2J S1K +

J=2 K=1

S4J

+

J=1

+

J–1  M  

(4.19a)

J=3 K=2 L=1

J–1  M  

 S1J S3K

+

S2J S2K

+

S3J S1K

J=2 K=1

J–1  K–1  L–1 M  

S1J S1K S1L S1O

(4.19b)

J=4 K=3 L=2 O=1

+

J–1  K–1  M  

 S1J S1K S2L

+

S1J S2K S1L

+

S2J S1K S1L

,

(4.19c)

J=3 K=2 L=1

⎡

⎛

In =

n ⎢  ⎢⎝ ⎢ ⎣ i=1

i j=1

⎞⎤ i J $ j All terms of the form S ⎜ lj ⎟⎥  ⎟⎥ ⎜ j=1 ⎠ ⎟⎥ . ⎜ i / ⎠⎦ ⎝ Jj =i–j+1 such that l = n j J0 –1=M Jj–1 –1

⎛

⎞

(4.19d)

j=1

It is straightforward to write out the nth order integral for any n. All we have to do is to make all possible combinations of Si s that give n internal f s and adding the correct number of sum symbols while keeping the ordering. It is also possible to give a recursive definition:

In (M) =

M  J=1

SnJ +

M  n–1  J=2 i=1

SiJ In–i ( J – 1).

(4.20)

4.2 Piecewise Wilson Lines

57

The last two equations literally translate to a Wilson line; just replace every S with a U , for instance:2 U3 =

M 

J–1 J–1  M  M  K–1   U3J + 2 U(1J U2)K + U1J U1K U1L ,

J=1

where

 UnJ =

ig 160 4

J=2 K=1

(4.21)

J=3 K=2 L=1

n  d4 k1 . . . d4 kn A,1 (k1 ) . . . A,n (kn ) SnJ .

(4.22)

Note that the ordering of the U J remains important, as the momentum integration runs over the Si ’s and the fields, which are non-commutative due to the colour generators. The physical interpretation of the nth-order formula is a collection of all possible diagrams for n-gluon radiation from an M-segment Wilson line, as is illustrated in figure (Figure 4.3) for three gluons radiated from a line with four linear segments. Note the manifest path ordering: the U J are path ordered by definition, and the sums are such that the gluon from segment J is radiated before K which is radiated before L (here, we literally see that a Wilson line is read from right to left, as the order of J, K and L is flipped). Consider now the product of e.g. three Wilson lines, labelled U A , U B and U C . Expanding the exponentials and collecting terms of the same order in g we get   U A U B U C = 1 + U1A + U1B + U1C   + U1A U1B + U1A U1C + U1B U1C + U2A + U2B + U2C  + U1A U1B U1C + U1A U2B + U1A U2C + U1B U2C + U2A U1B  + U2A U1C + U2B U1C + U3A + U3B + U3C + ⋯ , which equals, up to third order, the sum of equations (4.14), (4.18) and (4.19a). In other words, we can equate a product of Wilson lines to one line with several segments. The proof easily generalizes to all orders. Note that the order of the segments is reversed w.r.t. the order of the product (because we read the lines from right to left), e.g. the product U A U B U C is a line with first segment C, second segment B and last segment A, i.e. U A U B U C = U ABC , where we read the order of the segments in the rhs from right to left. 2 We use the brace notation for tensor symmetrization, i.e. U(1J U2)K Equation (B.12).

=

1 2



 U1J U2K + U2J U1K . See

58

4 Basics of Wilson Lines in QCD

M UJ = 3

J

J

+

J

+

J

+

J=1

M

K

J−1

J

+

U1J U2K = J=2 K=1

K

+

M

K

J−1

J

J

+

U2J U1K = J=2 K=1

K

+

M

J−1

K−1

L

J

K J

U1J U1K U1L = J=3 K=2

K

J

+ K

+

K

+ K

+

L

J

K

J

K J

+

J

J

+

K

K J

+

J

K

J

+

L

K

J

+

LK

J

L=1

Figure 4.3: Correspondence between equation (4.21) and all possible diagrams for 3-gluon radiation on a 4-segment Wilson line. Path ordering is manifestly conserved.

4.3 Wilson Lines on a Linear Path The results from the former section are general results, i.e. valid for any path. Let us now turn our focus towards paths built from linear segments — as these are the most commonly used in literature — and derive Feynman rules for the different linear topologies. For every segment, there exist four possible path structures: it can be a finite segment connecting two points, it can be a segment connecting ±∞ and a point r, , or it can be a fully infinite line connecting –∞ with +∞. We will investigate them case by case. 4.3.1 Bounded from Below We start with a path going from a point a, to +∞ along a direction n, . Such a path can be parameterized as z, = a, + n, +,

+ = 0 . . . ∞.

(4.23)

59

4.3 Wilson Lines on a Linear Path

Using the path ordering as defined in equation (4.10), we can write equation (4.11) as3 Inl.b.

,1

n

= (ig) n ⋯ n

,n

ia ⋅

/ j

e

kj

/ ∞∞ ∞ i (n ⋅ kj +i')+j ⋯ d+1 ⋯ d+n e j , 0 +1

(4.24)

+n–1

where the terms +i' in the exponential (with ' > 0 infinitesimal) are needed to make the integral convergent. Solving this integral is straightforward. First, we calculate the innermost integral, which is just the Fourier transform of a Heaviside (-function: ∞

i ei(n ⋅ kn +i')+n–1 , n ⋅ kn + i'

d+n ei(n ⋅ kn +i')+n =

+n–1

(4.25)

Note that if we would have used equation (4.8) instead of equation (4.10), this result would have contained two terms — of course valid as well but much more difficult. We can summarize the effect of the innermost integral as a factor 1 n ⋅ kn

(4.26)

and an extra term n ⋅ kn in front of +n–1 . The next integral will then give a factor 1 n ⋅ kn + n ⋅ kn–1

(4.27)

and so on. In other words, we simply get Inl.b.

n

,1

= (ig) n . . . n

,n

ia ⋅

e

/ j

kj n j=1

i n⋅

n /

,

(4.28)

kl + i'

l=j

There is a small subtlety in this result, as we absorbed some factors in front of the i' into ' (which is something we are allowed to do as the limit ' → 0 is implicitly assumed). We can reconstruct the result in equation (4.28) with the following Feynman rules: k

Propagator: External point:

= aμ

k

+∞

Line to infinity: j

(4.29a)

= eia ⋅ k ,

(4.29b)

= 1,

(4.29c)

  = ig n, ta ij .

(4.29d)

i k

Wilson vertex:

i , n ⋅ k + i'

μ, a

3 The symbol n is used both as an index (in the nth-order expansion) and as a directional vector. The difference should be clear from context.

60

4 Basics of Wilson Lines in QCD

n

n

aμ

j=1

kj

j=2

kj

kn–1 + kn

...

k1

...

k2

kn

kn–2

kn

kn–1

Figure 4.4: n-Gluon radiation for a Wilson line going from a, to +∞.

These Feynman rules are for momenta that start in the external point and point outwards from the Wilson line (see the discussion above equation (4.12)). If one or more momenta are inwards, the correct Feynman rule can be trivially retrieved by making the substitution ki → –ki . As an illustration, the resulting nth-order diagram is drawn in Figure 4.4. 4.3.2 Bounded from Above The logical next step is to investigate a path that starts at –∞ and now goes up to a point b, , which we parameterize as z, = b, + n, +,

+ = –∞ . . . 0.

(4.30)

In this case, it is easier to reverse the integration variables as in equation (4.8), which then gives the integral for a Wilson line with upper bound: Inu.b.

,1

n

= (ig) n . . . n

,n

ib ⋅

/

e

j

+2

0 +n kj  

⋯

d+n . . . d+1 e

in ⋅

/ j

kj +j

.

(4.31)

–∞–∞ –∞

The calculation goes as before, giving Inu.b.

n

,1

= (ig) n . . . n

,n

ib ⋅

/

e

j

kj

n j=1

–i n⋅

j /

,

(4.32)

kl – i'

l=1

which differs from equation (4.28) only in the accumulation of momenta in the denominators (bottom-top instead of top-bottom) and the sign of the convergence terms. The Feynman rules derived before remain valid if we make the substitution k → –k in the Wilson line propagators, but not in the external point. Then it is straightforward n–1

k1

−∞ k1

k1 + k2 k2

... k3

...

j=1

kn–1

n

kj

j=1

kj bμ

kn

Figure 4.5: n-Gluon radiation for a Wilson line going from –∞ to b, . Path flow is opposite to momentum flow.

4.3 Wilson Lines on a Linear Path

61

to draw the nth-order diagram for a Wilson line going from –∞ to b, , as demonstrated in Figure 4.5. The path still flows from left to right, but now the momenta are opposite to the path flow. However, the main idea remains the same: momenta start from the external point and are spread over the outgoing gluons.

4.3.3 Path Reversal Let us now investigate what changes when we reverse the path of a Wilson line. First of all, the integration borders are of course interchanged because the path flows from the final point to the initial point. This is the same as keeping the integration borders as they are, and flipping the sign in the exponent. But the most important is that the order of the fields is reversed because the field that normally lies first on the path will be encountered last when following the reversed path flow. This is the idea of anti-path ordering P, defined such that the field with the highest value for + is written rightmost instead of leftmost. The reversed Wilson line is thus given by ⎛ U(a ; b) = Pexp ⎝–ig

b

⎞ dz, A, ⎠ .

(4.33)

a

It comes as no surprise that this is exactly the same as the Hermitian conjugate, as this was one of the properties imposed during the derivation of the Wilson line in Chapter 3. Note that the reversal of the field ordering is not only a logical step when reversing the path, but also a direct result of the Hermitian conjugate because (An . . . A1 )† = A†1 . . . A†n .

(4.34)

A(k)† = A(–k)

(4.35)

By using the fact that

is a Hermitian function4 , and making the substitution k → –k, the relation to the reversed path becomes apparent. We thus have indeed † U(a ; b) = U(b ; a) .

(4.36)

But of course it would be desirable to express the Hermitian conjugate line in function of normal path-ordered fields, such that we can use the same Feynman rules as before.

4 Because A(x) is real.

62

4 Basics of Wilson Lines in QCD

Let’s see how, e.g. a Wilson line from –∞ to b, behaves when Hermitian conjugated (remember that in equation (4.12) we defined the regular Wilson line with opposite momenta, i.e. with factors Ai (–ki )): ⎡ ⎤†  9 ∞ ⎢ n / d kn ⎢ n ib ⋅ kj † U(b = n ⋅ A(–k ) . . . n ⋅ A(–k ) e ig ⎢ ( ) n 1 ; –∞) ⎣ (20)9 n=0

=

=

 9 n / d kn n ⋅ A(k1 ) . . . n ⋅ A(kn ) e–ib ⋅ kj (–ig )n 9 (20) n=0

n j=1

 n

(–ig )

n=0

n⋅

j /

kl – i'

⎥ ⎥ ⎥ ⎦

l=1

∞ 

∞ 

–i

n

i

j=1

n⋅

j /

kl + i'

l=1 n / d9 kn ib ⋅ kj n ⋅ A(–k ) . . . n ⋅ A(–k ) e 1 n (20)9

n j=1

–i n⋅

j /

,

kl – i'

l=1

where in the last step, we made the integration substitution ki → –ki . In order to make the identification with equation (4.12), we have to relabel the fields by doing 1 → n, 2 → n – 1, . . . , n → 1, which gives † U(b ; –∞)

∞   d9 k1 d9 kn = ⋯ A, (–kn ) ⋯ A,1 (–k1 ) Inl.b. † , (20)9 (20)9 n n=0 n

n /     Inl.b. † = (ig )n –n,1 ⋯ –n,n eib ⋅ kj

j=1

–n ⋅

i n /

. kl + i'

l=j

We see now that this is the expansion of a Wilson line from b, to +∞, but with opposite n, .5 The same will be true for a Wilson line from a point to +∞, so we can write: % % † U(+∞ (4.37a) ; a) = U(a ; –∞) %n → –n , % % † U(b . (4.37b) ; –∞) = U(+∞ ; b) % n → –n

We will indicate the direction of n in a Feynman diagram with a blue arrow on the Wilson line, where going from left to right implies a positive n. This also indicates the direction of the path flow: an arrow from right to left implies a negative n, 5 The important fact to realize here is that what defines whether a Wilson line is going from –∞ to a, /j or from a, to +∞ is how the momenta are summed in the denominator. For the former, it is l=1 , and /n for the latter, l=j .

4.3 Wilson Lines on a Linear Path

n

bμ

j=1

63

n

kj

j=2

kj

kn–1 + kn

...

kn

+∞

k1

...

k2

kn–2

kn–1

kn

† Figure 4.6: Reversing the path of a Wilson line is the same as taking the Hermitian conjugate U(b . ; –∞) If we want to express this in standard path ordering, we have to make the substitution n → –n (shown by the arrow on the double line) and change the path into a line going from b, to +∞.

implying a Hermitian conjugate, implying a reversed path (from right to left).6 With this convention, we can draw the reversed version of U(b ; –∞) as in Figure 4.6. Let us have a look at how the Feynman rules change when making the substitution n → –n. First for the Wilson line propagator, we see that it gets complex conjugated when the momentum flow is opposed to the path direction: k

k

=

i , n ⋅ k + i'

=

–i , n ⋅ k – i'

k

=

–i , n ⋅ k – i'

(4.38a)

=

i . n ⋅ k + i'

(4.38b)

k

The vertex coefficient only depends on the path direction (not on the momentum direction): j k μ, a

i

j

= ig n, (ta )ij ,

k μ, a

i

= –ig n, (ta )ij . (4.39)

On the other hand, the sign in the exponent for an external point doesn’t depend on the direction of the path flow, but only on the momentum direction as compared to the point itself: k

k rμ = rμ

k

k

k rμ = rμ

k rμ = rμ

= k =

= e i r ·k ,

(4.40a)

= e− i r ·k .

(4.40b)

k rμ = rμ

6 Note that the substitution n → –n is not the same as a path reversal. To appreciate the difference, remember that for a linear path z, = r, + n, +, so the substitution n → –n changes a path from –∞ to 0 into one from +∞ to 0 (ignoring the difference between path ordering and anti-path ordering). But the reversed path goes in the direction –n; this is why, we can use the blue arrow to denote both. The difference is maybe subtle, but cannot be neglected.

64

4 Basics of Wilson Lines in QCD

Most of the time, we will drop the arrow indicating the path flow on the Wilson line, as it obscures readability, and assumes – unless specified otherwise – the path flowing from left to right. We now introduce a shorthand notation to denote the path structure for a Wilson line segment. We represent the two structures that we calculated first by U(+∞ ; a)

=

N

,

(4.41a)

U(b ; –∞)

N

.

(4.41b)

=

Note that there is a subtlety in our drawing conventions. Until now, we’ve only drawn small pieces of a segment in order to illustrate the Feynman rules. But here we give a schematic representation of a full segment (including gluons). Confusion might especially appear between the depiction of the Feynman rule for an external point and this representation, however, the correct interpretation should be clear from the context. Furthermore, from now on, we will mostly use the latter notation. For the reversed path, there is some ambiguity in the interpretation. Combining equations (4.36) and (4.37), we can write % % U(a ; +∞) = U(a ; –∞) %

n → –n

.

(4.42)

In both sides of the equation, the blue arrow is pointing from right to left. However, in the lhs, the line touches +∞ and in the rhs, it touches –∞. The lhs gives the correct physical picture, while the rhs gives the correct calculational picture (assuming we keep all fields in standard path-ordering). We choose the latter, so keep in mind that this is not a correct physical representation: % % U(a ; +∞) = U(a ; –∞) % %n → –n % U(–∞ ; b) = U(+∞ ; b) % n → –n

=

N

,

(4.43a)

N

.

(4.43b)

=

This helps avoiding calculational mistakes, as the former representation would seem to suggest that U(+∞ ; a) and U(a ; +∞) are related by a simple sign change in n. This is not enough, one also has to change the accumulation of momenta in the denomin/ /j ator of the propagator, from nj kl (equation (4.28)) to 1 kl (equation (4.32)). A trick to remind this correctly is to remember that path reversing equals Hermitian conjugation, and the latter is easily demonstrated in our schematic notation using a “mirror relation”:

(

†

)

=

,

(

which is literally the same as equations (4.37).

†

)

=

,

(4.44)

4.3 Wilson Lines on a Linear Path

65

4.3.4 Finite Wilson Line Next, we investigate a Wilson line on a finite path, going from a point a, to a point b, , where now the direction is defined by n, =

b, – a, . b – a

We parameterize this as z, = a, + n, +,

+ = 0 . . . b – a .

Of course, we can simply use equation (4.4c) to split the line at ±∞, i.e. U(b ; a) = U(b ; +∞) U(+∞ ; a) = U(b ; –∞) U(–∞ ; a) , but in what follows we will do a brute-force calculation giving the same result, and this for three reasons: – It is an extra, practical check of the transitivity formula. –

Following our calculations, we will see that this is the natural and only way to calculate a finite line in momentum space, i.e. so far no easier solutions exist.

–

Halfway the calculation, we will need to solve a recursive relation, which we can re-use when calculating the fully infinite line.

Because there are no infinities at the borders, it doesn’t matter whether we choose equation (4.10) or (4.8), as both are equally difficult but will give the same results. Choosing equation (4.10), we write the segment integral as

Infin. =

n

,1

(ig) n ⋯ n

,n

ia ⋅

e

/ j

kj

b–a  +n

+2 n / ⋯ d+n ⋯ d+1 ein ⋅ kj +j .

0

0

(4.45)

0

Dropping the factors in front of the integral, we find a recursion relation: b–a 

 –i  i(b–a) ⋅ k1 e –1 , n ⋅ k1 0  –i fin. fin. Infin (k1 ,...,kn ) = In–1 (k1 +k2 ,...,kn ) – In–1 (k2 ,...,kn ) . n ⋅ k1 I1fin (k1 )

=

d+ ein ⋅ k1 + =

(4.46a)

(4.46b)

To ameliorate notational clarity, we will drop the factors n, (from the fractions) and i(b – a), (from the exponent) in the next calculation. The first few orders are easily calculated:

66

4 Basics of Wilson Lines in QCD

 i(–i)    (–i)2 ek2 – 1 , ek1 +k2 – 1 + k1 (k1 + k2 ) k1 k2     (–i)3 i(–i)2 = ek1 +k2 +k3 – 1 + ek2 +k3 – 1 k1 (k1 + k2 )(k1 + k2 + k3 ) k1 k2 (k2 + k3 )   i2 (–i) ek3 – 1 , + (k1 + k2 )k2 k3   (–i)4 = ek1 +k2 +k3 +k4 – 1 k1 (k1 +k2 )(k1 +k2 +k3 )(k1 +k2 +k3 +k4 )   i(–i)3 + ek2 +k3 +k4 – 1 k1 k2 (k2 +k3 )(k2 +k3 +k4 )     i3 (–i) i2 (–i)2 ek4 – 1 ek3 +k4 – 1 + + (k1 +k2 )k2 k3 (k3 +k4 ) (k1 +k2 +k3 )(k2 +k3 )k3 k4

I2fin = I3fin

I4fin

and so on. We see that every term has a fraction where a part of the momenta is accumulated from below, and the other part is accumulated from above. We can thus express the nth order term exactly as (reintroducing the factors n, and (b – a), ):

Infin. =

n–1 

⎛ i(b–a) ⋅

⎝e

n / m+1

⎞

kj

m=0

⎛

⎜ ⎜ – 1⎠ ⎜ ⎝

⎞⎛ m j=1

n⋅

i m /

kl

⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

⎞ n

–i

j=m+1

l=j

n⋅

j /

kl

⎟ ⎟ ⎟. ⎠

(4.47)

l=m+1

This can be simplified further. First of all, we have exactly 2n terms, of which n have no exponential and can thus be summed to simplify into one term (these are all terms corresponding to the –1 term in parentheses). Note that if we sum these n terms and add the m = n term, we get zero: ⎛ n ⎜  ⎜ ⎜ ⎝

m=0

⎞⎛ m j=1

n⋅

i m / l=j

kl

⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

⎞ n

j=m+1

–i n⋅

j /

kl

⎟ ⎟ ⎟ = 0, ⎠

(4.48)

l=m+1

which is easy to prove by induction. This is known as the eikonal identity, and is especially useful in the case of Abelian fields, because then it tells us that – for a given diagram where two Wilson lines are connected to each other with n/2 photons (or gluons when ignoring colour) – if we sum the possible emission partitions between the two lines, the result is automatically zero.7 Using the eikonal identity, we can replace the sum of the n terms by the opposite of the m = n term:

7 We don’t even have to sum over all possible crossings. Any given diagram connecting the photons is represented as a product of $-functions and propagators and is factorized out of this calculation.

67

4.3 Wilson Lines on a Linear Path

⎛ –

⎞⎛

n–1 ⎜  ⎜ ⎜ ⎝

m j=1

m=0

n⋅

i m /

⎞

⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

kl

n

–i

j=m+1

j /

n⋅

l=j

kl

⎟ ⎟ ⎟= ⎠

n j=1

n⋅

i m /

, kl

l=j

l=m+1

The important observation is now that the last term is also the m = n term for the full sum including the exponential (equation (4.47)), as in this case the exponential vanishes:8 e

i(b–a) ⋅

n / n+1

kj

= e0 = 1.

This then gives ⎛ Infin. =

n–1 

e

i(b–a) ⋅

n / m+1

kj

m=0

⎞⎛

⎜ ⎜ ⎜ ⎝

m j=1

n⋅

i m / l=j

⎛ =

n 

ei(b–a) ⋅

/n

m+1 kj

m=0

⎜ ⎜ ⎜ ⎝

kl

⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

m j=1

n⋅

⎞ n

–i

j=m+1

j /

n⋅

l=m+1

⎞⎛

i m /

kl

⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

kl

⎟ ⎟ ⎟+ ⎠

j=m+1

n⋅

l=j

j=1

n⋅

j /

kl

i n /

, kl

l=j

⎞

–i

n

n

⎟ ⎟ ⎟. ⎠

l=m+1

Reintroducing the factors in front, we see that the exponential simplifies into

e

ia ⋅

/n

1 kj

n 

e

i(b–a) ⋅

n /

m+1

m=0

kj

=

n 

eia ⋅

/m

1 kj eib ⋅

/n

m+1 kj .

m=0

We can thus finally write the path content integral for a finite line as ⎛ Infin. =

n ⎜  m=0

⎜ ⎜ ⎝

⎞⎛ m j=1

ig n,j n⋅

i m /

kl

⎟⎜ ⎟⎜ eia ⋅ kj ⎟ ⎜ ⎠⎝

⎞ n j=m+1

ig n,j

l=j

–i j / n⋅ kl

⎟ ⎟ eib ⋅ kj ⎟ . ⎠

(4.49)

l=m+1

Using the fact that this kind of chained sum can in general be written as a product of two infinite sums:

∞

∞ n ∞     Am Bn–m = An Bn , n=0 m=0

n=0

n=0

/ 8 Remember that by definition bj=a f (j) = 0 if a > b, this is an “empty sum”. The same is true for $b multiplication: j=a f ( j) = 1 if a > b.

68

4 Basics of Wilson Lines in QCD

we can transform equation (4.49) into a product of two half-infinite Wilson lines:9 ⎞⎛ ⎞ ⎛ ∞ 

∞ ⎜ ⎜ Infin. = ⎜ ⎝

n

–i

ig n,j n⋅

n=0 j=1

j /

kl

∞ ⎟ ⎜ ⎟⎜ eib ⋅ kj ⎟ ⎜ ⎠⎝

n

ig n,j n⋅

n=0 j=1

i n /

kl

⎟ ⎟ eia ⋅ kj ⎟ . ⎠

l=j

l=1

To make the identification with two half-infinite Wilson lines, we will manually add the convergence terms in the fraction (we can do this without problem because in the infinitesimal limit, they are zero anyway), but to be consistent, they have to have the same sign in both products:10 ⎞⎛ ⎞ ⎛ ∞ 

∞ ⎜ ⎜ Infin. = ⎜ ⎝

n

–i

ig n,j n⋅

n=0 j=1

j /

kl + i'

∞ ⎟ ⎜ ⎟⎜ eib ⋅ kj ⎟ ⎜ ⎠⎝

n

i

ig n,j n⋅

n=0 j=1

n /

kl + i'

⎟ ⎟ eia ⋅ kj ⎟ , ⎠

l=j

l=1

which is literally the same as two lower bound Wilson lines: % † U(b ; a) = U(b ; –∞) %n→–n U(+∞ ; a) = U(+∞ ; b) U(+∞ ; a) = U(b ; +∞) U(+∞ ; a) .

(4.50)

Of course, we can just as well insert convergence terms with a negative sign, that is, ⎞⎛

⎛ ∞ 

∞ ⎜ ⎜ Infin. = ⎜ ⎝

n

–i

ig n,j n⋅

n=0 j=1

j /

kl – i'

⎞

∞ ⎟ ⎜ ⎟⎜ eib ⋅ kj ⎟ ⎜ ⎠⎝

n

n=0 j=1

i

ig n,j n⋅

n /

kl – i'

⎟ ⎟ eia ⋅ kj ⎟ , ⎠

l=j

l=1

which gives us two upper-bound Wilson lines: % U(b ; a) = U(b ; –∞) U(+∞ ; a) %n→–n = U(b ; –∞) U(a ; –∞) [†] = U(b ; –∞) U(–∞ ; a) ,

(4.51)

proving the arbitrariness of the transitivity property. When putting this relations in a schematic form, it is easiest to represent the one where the finite line is cut at +∞ because then the line is literally torn into two: μ

a

b

μ

=

a

μ

μ

b .

9 There is a small subtlety here: In equation (4.49), the propagators are ordered from 1 to n. But of course the fields are ordered from n to 1 as explained in equation (4.12). So basically, when including the momentum integrals over the fields, the two products switch places. 10 The reasoning behind it is that the correct place to introduce these convergence terms is not here – at the end of the calculation – but at the start of the calculation in the exponent of equation (4.45), /n i.e. ei(n ⋅ kj ±i')+j . Then after doing the full calculation, both products in equation (4.49) will have convergence terms with the same sign.

4.3 Wilson Lines on a Linear Path

69

The reversed path is simply the Hermitian conjugate (note that the Hermitian conjugation also flips the order of the two lines in the rhs): μ

a

b

μ

b

=

μ

μ

a .

So the things that change after reversing the path are the external points. This is of course logical, as we could interpret it as a normal finite line from b, to a, , for which the former schematic relation holds. 4.3.5 Infinite Wilson Line Finally, the last possible path structure for a linear segment is a fully infinite line, going from –∞ to +∞ along a direction n, and passing through a point r, . Such a path can be parameterized as z , = r , + n, +

+ = –∞ ⋅ ⋅ ⋅ +∞.

(4.52)

Using equation (4.8), we can write the segment integral as Ininf.

n

,1

= (ig ) n . . . n

,n

ir ⋅

e

/ j

+∞ kj 

+2

...

–∞

d+n . . . d+1 ein ⋅

n /

kj +j

.

–∞

Naively, one could think that Ininf. consists of n –1 integrals that evaluate to the Fourier transform of a Heaviside (-function (see equations (B.62)), n ⋅ –i k–i' , and one integral, the outermost that evaluates to a Dirac $-function. This would give the following result (again dropping the factors in front of the integral for convenience): ⎛ ⎞

n ⎜ n–1 ⎟  –i ⎟ n ,1 inf. ,n ⎜ In = (ig ) n . . . n ⎜ (4.53) kj . ⎟ 20 $ n ⋅ j ⎝ ⎠ / j=1 n ⋅ kl – i' l=1

However, there is one caveat. When we explicitly write the convergence terms used in the n – 1 innermost integrals, we see that the outermost integral doesn’t equal a $-function at all, but is badly divergent:  n / +∞ i n ⋅ kj –i' +n d+n e .

(4.54)

–∞

This is the Fourier transform of e'+n , which is divergent.11 In other words, either we drop the convergence terms (' = 0), making the $ integral representation convergent 11 The only square-integrable linear exponential functions are e–'|+| and e–'+ ((+).

70

4 Basics of Wilson Lines in QCD

but making all n – 1 innermost integrals divergent, or we add the convergence terms in order to make the innermost integrals convergent, but then we lose the $ function representation and are stuck with a divergent Fourier transform. Simply using the convergence terms for the n–1 innermost integrals, and then setting them to zero for the last integral won’t do, as there is no reason to believe that we are allowed to take the limit ' → 0 halfway. Furthermore, we will need the convergence terms in the Wilson line propagators when doing momentum integrations12 . We will show that it is not difficult to make a mathematically correct all-order proof, based on solving the same recursion relation as we encountered in the calculation of the finite line. The regularized path runs from r, – .2 n, to r, + .2 n, (with . > 0), and is parameterized as follows:  2 . , , z. = r + tanh + = –∞ . . . + ∞ (4.55) + n, , . 2 If we take the limit . → 0, we recover the same parametrization as in equation (4.52). The innermost integral equals: 

+2 I1inf.

d+1 sech2

=

  . . i 2 (n ⋅ k1 –i') tanh 2 +1 +1 e . 2

–∞

   . –i i 2 (n ⋅ k1 –i') tanh 2 +2 –i 2 (n ⋅ k1 –i') –e . e. n ⋅ k1 – i'  2 –i +2 →+∞ → 2 i sin . (n ⋅ k1 – i') . n ⋅ k1 – i'

=

The factor sech2 is the integration measure that comes from the reparameterization of the path  ′ dz, → z, d+. Note that we added the convergence terms i', despite the fact that at first sight they don’t seem necessary. However, intuitively one can expect that they are in fact indispensable, as the regularization of the path acts on the outermost “$”-integral and not on the innermost “(”-integrals, leaving the latter unregularized. We will indeed confirm their necessity in the next step. To proceed, we observe that for higher orders, the integrals obey a recursion relation (again dropping the factors in front):  –i –i .2 (n ⋅ k1 –i') inf. inf. inf. In (k1 ,...,kn ) = In–1 (k2 ,k3 ,...,kn ) . I (k1 +k2 ,k3 ,...,kn ) – e (4.56) n ⋅ k1 –i' n–1 This relation looks a lot like equations (4.46). In fact, the result is very similar and can be simplified into 12 A consistent approach is to regularize the path, as is calculated up to second order in Ref. [151]. However, their proof is based on a not so rigorous use of the Riemann–Lebesgue lemma.

71

4.3 Wilson Lines on a Linear Path

Ininf.

= 2i

n–1  m=0

e

–i .2 n ⋅

m / 1

kj –i'



 n  2 ⋅ sin kj –i' n⋅ . m+1

m 1

i m / n ⋅ kl –i' j

n m+1

–i n⋅

j /

.

kl –i'

m+1

(4.57) Note how the convergence terms –i' ensure that the exponent converges nicely in the > limit . → 0; they are indeed indispensable. All terms thus vanish in this limit, except the m = 0 term, where the exponent equals 1. We now move from the regularized path back to the original path by taking this limit: ⎛ ⎞

n ⎜ n–1 ⎟  –i ⎟ n ,1 inf. ,n ⎜ In = (ig ) n . . . n ⎜ kj – i' . (4.58) ⎟ 20 $ n ⋅ j ⎝ ⎠ / 1 n⋅ 1 kl – i' 1

As this is the same result as in equation (4.53), we have shown that there is no need to regularize the path and that the naive calculation leads to correct results, although seemingly divergent at first sight. A few words on the emergence of the $-function however. We use here the concept of a nascent $-function, which is any function $. with infinitesimal parameter . > 0, that has the weak limit lim $. (x) ≅ $(x). >

(4.59)

. →0

This weak limit relates $. and $ not by equality, but by the sifting property: lim >

+∞ dx $. (x)f (x) = f (0).

(4.60)

. →0 –∞

In other words, for all practical purposes, we can treat the weak limit of a nascent delta function as a normal delta function. One can construct such a nascent $-function from any function g that is absolutely integrable and has total integral equal to 1 by defining  1 x $. (x) = g . (4.61) . . As the sinc function has total integral equal to  sin x = 0, dx x we can easily construct a nascent $-function from it: x

1 sin . . $. (x) = 0 x

(4.62)

72

4 Basics of Wilson Lines in QCD

We still have one encumbrance to overcome, namely that in our result the argument of the sine has an infinitesimal (but non-zero) complex shift –i', while the $-function and nascent $-functions are only defined for real arguments. Luckily, the former steps can be proven to be valid for complex shifts as well. First note that  sin (x – i') dx =0 (4.63) x – i' from which it is straightforward to show that   +∞ 1 sin x – i'/. dx f (x – i') = f (0) > 0 x – i' . →0 lim

(4.64)

–∞

In other words, the sifting property still holds after making a small complex shift (at least for this type of nascent delta functions). We thus can make the identification:   n /

sin .2 n ⋅ kj – i' n  1 lim ≅ 0$ n⋅ kj – i' (4.65) n > / . →0 1 n ⋅ kj – i' 1

leading to the final result in equation (4.58). Still one word of caution: as mentioned before, this weak limit doesn’t ensure that it equals a $-function, but merely shows that the sifting property holds. This implies that when using equation (4.58), we are not allowed to use the integral representation of the $-function (the latter wouldn’t make any sense, as it is a divergent integral). The correct way to make use of a $-function with a complex argument, is to only use it in conjunction with the sifting property. Returning to the infinite Wilson line, we can get an equivalent definition by starting from equation (4.10):

n–1 n  i n ,1 inf. ,n In = (ig ) n . . . n 20 $ n ⋅ kj + i' . (4.66) n / 1 n⋅ 1 kl + i' j

We conclude that the correct way to draw an infinite Wilson line is to put all radiated gluons on one side from the point r, , where the line piece connecting the point to a gluon is a cut propagator having the following Feynman rule: k

Cut propagator:

= –$(n ⋅ k + i'),

(4.67)

where –$(x) is defined in equation (B.55). There are hence two ways two draw a Feynman diagram for an infinite Wilson line, i.e. having all gluons radiated before or after the point r, . This is illustrated in Figure 4.7.

73

4.3 Wilson Lines on a Linear Path

n–1

k1

k1 + k2

···

−∞ k1

k2 n–1

n–1

kj

j=1

−∞

···

k3 kj

j=1

j=1

kn–1

kn kn–1 + kn

···

j=2

rμ

n

kj

kj +∞ rμ

kn +∞

k1

k2

···

kn–2

kn–1

kn

Figure 4.7: Two possible diagrams for n-gluon radiation from a Wilson line going from –∞ to +∞. The upper diagram corresponds to equation (4.58) and the lower one to equation (4.66).

4.3.6 External Momenta Sometimes it is useful to write the Feynman rule for external points in momentum representation. To achieve this, we Fourier transform the full Wilson line in all of its external points. Consider, e.g., the simple line

U=

which consists of four segments (remember that the finite segment is split into two). It has two external points, or vertices, that each contribute a factor ei kr on each side of the vertex: r1 q 1 k1

k2

ei k1·r 1eiq1·r1

q2 r2

ei k2·r 2eiq2·r–2

The Wilson line in momentum space is defined as the Fourier transform in every vertex:  def

U momentum =

d9 r1 d9 r2 e–iP1 ⋅ r1 e–iP2 ⋅ r2 U coordinate ,

(4.68)

74

4 Basics of Wilson Lines in QCD

These integrations will give rise to $-functions: P1 q1 q2

k1

k2 P2 δ(ω)(P2−k2−q2)

δ(ω)(P1−k1−q1)

So we can simply replace the Feynman rule for the external point with the demand of momentum conservation at every vertex, with an additional external momentum per vertex.

4.4 Relating Different Path Topologies In the former section, we have seen that there are eight possible linear path structures; two connecting a point to ±∞, next a finite line and finally a fully infinite line, all with normal or reversed path flow. We won’t treat the fully infinite line anymore in the remainder of this book because it has a particular way of dealing with it, as infinite lines won’t often appear as segments of a piecewise Wilson lines and are thus less relevant for what follows in the next section.13 The remaining six path structures are not independent. We have seen in the previous subsections that if we choose the two path topologies from equations (4.41),14 n

,1

= (ig ) n ⋯ n

,n

ir ⋅

/ j

e

kj

n j=1

= (ig )n n,1 ⋯ n,n e

ir ⋅

/ j

i n⋅

n /

,

(4.69a)

,

(4.69b)

kl + i'

l=j kj

n j=1

–i n⋅

j /

kl + i'

l=1

we can express the remaining four in function of them: % % = , %n→–n % = ,

(4.70a) (4.70b)

n→–n

= =



⊗ ⊗

, % %

a↔b

(4.70c) .

(4.70d)

13 Technically, they are relevant when considering several infinite Wilson lines and treating these as one line with multiple segments, but we avoid these scenarios as they complicate the formalism. 14 Remember that n is defined along the path flow (from starting point to ending point). A reversed path arrow thus always denotes –n.

4.4 Relating Different Path Topologies

75

Note that in equations (4.69), we deliberately chose two structures that have positive convergence terms +i' so that all calculations have the same type of poles. But these two structures aren’t fully independent either, as they are related by a sign difference and an interchange of momentum indices: = (–)n

% %

(k1 ,...,kn )→(kn ,...,k1 )

.

(4.71)

We can exploit this relation when making a full calculation, i.e. connecting the Wilson line to a blob. This blob can be constructed from any combination of Feynman diagrams, but cannot contain other Wilson lines. If one is interested in interactions between different Wilson lines, it is sufficient to treat the different lines as different segments of one line (as is explained in the end of Section 4.2). We will name the blob depending on the number of gluon lines that connect it to the Wilson segment. Valid blobs are, e.g. a gluon propagator connected to the Wilson segment (a 2-gluon blob), a gluon connecting a quark to the Wilson segment (a 1-gluon blob). Note that the naming of the blob isn’t always faithful to the number of gluons participating in the process. For example, in the case of a gluon being connected to a segment (a self-energy diagram), this is clearly one gluon, but we will refer to it as a 2-gluon blob, as two gluon lines enter the blob:

It is a matter of convention, and we chose this one as it helps categorizing the blobs. In the next section, we will research how to calculate diagrams with piecewise Wilson lines, but first we investigate how to connect a blob to one segment. For the structure given in equation (4.69a), this is (again abusing the path integration measure notation): 

... F

= t

an

...t

a1

d9 k1 l.b. a1 ... an d9 kn . . . I F (k1 , . . . , kn ), (20)9 (20)9 n ,1 ... ,n a ...a

where we absorbed the gluon propagators into the blob F,11...,nn . Furthermore, we always define the blob as the sum of all possible crossings; it is thus symmetric under the simultaneous interchange of Lorentz, colour and momentum indices. Because every Lorentz index of F is contracted with the same vector n, , it is automatically symmetric in these. The combination of these symmetries implies that an interchange of momentum variables is equivalent to an interchange of the corresponding colour indices. In particular, it is now straightforward to relate equation (4.69b) to

76

4 Basics of Wilson Lines in QCD

equation (4.69a) when connected to a symmetrized blob:  F

d9 k1 l.b. an ...a1 d9 kn . . . I F (k1 , . . . , kn ). (20)9 (20)9 n ,1 ...,n

= (–)n tan . . . ta1

Note that the only difference is an interchange of the colour indices. Often the blob has a factorable colour structure, i.e. ...an F,a11..., (k1 , . . . , kn ) = ca1 ...an F,1 ...,n (k1 , . . . , kn ). n

(4.72)

If we then define the following notations: c = tan ⋅ ⋅ ⋅ ta1 ca1 ...an , c=t

an

a1 an ...a1

⋅⋅⋅t c

(4.73a)

,

(4.73b)

we can simply write ...

c

=

F

... F

...

,

F

n

= (−) c

... F

(4.74a) .

(4.74b)

The red, “photon-like” wavy lines are just a reminder that there is no colour structure left in the blob. In other words, when changing the structure of a Wilson line, we don’t have to redo the calculation of the integral! The difference is merely some colour algebra, changing c into c and a sign difference. Remember that due to the fact that we read a Wilson line as a Dirac line, i.e. from right to left, we have to order the generators in equation (4.73a) from n to 1. On the other hand, the blob is written from left to right – which is a matter of choice, as it is fully symmetrized15 – leading to the difference in ordering between tan . . . ta1 and ca1 ...an . For a factorable blob example, take, e.g. the 3-gluon vertex:   F = g f a1 a2 a3 (k1 – k2 )1 D,1 - (k1 )D-,2 (k2 )D1,3 (k3 ) + cross⋅ ,

15 The important fact to realize is that indeed the order doesn’t matter for a fully symmetrized blob, but it should of course have the same ordering in its momenta, i.e. we identify a1 with k1 . Because most references in literature write simple blobs from left to right, we keep this convention for the blob for the sake of simplicity.

4.5 Piecewise Linear Wilson Lines

77

with colour structure ca1 a2 a3 = f a1 a2 a3

c = ta3 ta2 ta1 f a3 a2 a1 = –ta3 ta2 ta1 f a1 a2 a3 = –c.

⇒

This clearly implies that = ( − )3 c

= c

=

.

(4.75)

So, the 3-gluon vertex is path topology-invariant. Of course, a lot of blob structures won’t be colour factorable, but we can always write these as a sum of factorable terms: ...an F,a11..., (k1 , . . . , kn ) = n



a ...an

ci 1

Fi ,1 ...,n (k1 , . . . , kn ),

i

such that we can repeat the same procedure as before ...

ci i

F ...

,

Fi

n

ci

(–)

i

F

...

... Fi

(4.76a) .

(4.76b)

In conclusion: whatever the structure of the Wilson line to calculate a given diagram, we can retrieve it from the calculation of the same diagram with a Wilson line bounded from below, using straightforward colour algebra. For a trivial structure containing only one segment, the gain is not that big, but for a line consisting of several segments – as we will see in the next section and the next chapter – this trick can save us quite some calculation time.

4.5 Piecewise Linear Wilson Lines Now we turn our attention to piecewise Wilson lines, using the results from Section 4.2. When connecting a n-gluon blob to a piecewise Wilson line, the n gluons aren’t necessarily all connected to the same segment; other diagrams are possible as well, where the n-gluons are divided among several segments. This is the physical interpretation of formula equation (4.19d). As mentioned before, the UiJ aren’t commutative in se due to the non-Abelian nature of the fields. However, when connected to the same symmetrized blob that is summed over all crossings, they can be treated as if they were. This implies that multiple-segment terms can be related by straightforward substitution, e.g. % % U1K U2J = U2K U1J % , (4.77) (rK ↔rJ ,nK ↔nJ )

78

4 Basics of Wilson Lines in QCD

etc. To get all diagrams connecting a given n-gluon blob to a piecewise line, we have to calculate exactly p(n) diagrams, the partition function from combinatorics,16 independent of the number of segments M. When connecting, e.g. a 4-gluon blob, we need to calculate exactly five diagrams. These are constructed from the following segments (cf. equation (4.19b)): U4J , U3J U1K , U2J U2K , U2J U1K U1L and U1J U1K U1L U1O . They are the easiest represented schematically: J

K

J

F

F L

K

K

J

J

F O

L

F

K

J

F

(4.78)

In addition, there are three more diagrams that can be related using equation (4.77), built from the segments U1J U3K , U1J U2K U1L and U1J U1K U2L . Now what about external momenta? We saw in Section 4.3.6 that if we want to express the Wilson line in momentum space, we have to add an additional external momentum to every vertex and apply momentum conservation. However, most of the time a vertex connects two or more segments, and to be able to use our framework, we need Feynman rules that are defined per segment (not per vertex). Luckily, this can be easily achieved by using the fact that a Fourier transformation transforms a product in a convolution: Fk [ f (r)g(r)] = Fk [ f (r)] ⊗ Fk [ g(r)].

(4.79)

This means that we can associate per segment the Fourier transform of an external point, i.e. we replace the Feynman rule for an external point with External point:

P

k

=– $(9) (P – k).

(4.80)

An “empty” segment – with no gluon radiated from it – then naturally gains a –$(P). After connecting the blob, we make a convolution over the segments that are connected to the same external point. 16 The partition function, p(n), is the number of integer partitions of n; for example, 4 = 4, 3 + 1, 2 + 2, 2 + 1 + 1, or 1 + 1 + 1 + 1. Thus p(4) = 5. Other examples are p(3) = 3, p(5) = 7 and p(6) = 11.

4.5 Piecewise Linear Wilson Lines

79

In the TMD framework (see Chapter 5), it is common to express Wilson lines only partially in momentum space. More specific, the plus and minus components are expressed in coordinate space, while the transversal components are expressed in momentum space. The Feynman rule is in this case replaced with External point:

P

k r

= –$(9–2) (P⊥ – k⊥)eir

+ k– +ir– k+

,

(4.81)

where the convolution is now only over P⊥ . Let us briefly sketch the steps needed to do a full calculation in this framework. We illustrate each step with an easy example, viz. the calculation of all self-interactions of the Wilson line with the following path structure: r1 U = n1

n2

r2 n4 ,

n3 r3

where the path flow is assumed from left to right. The steps to undertake are: – List the segments that form the Wilson line and their corresponding path con, stants: the segment direction nK , the external point rK± and the external momentum P⊥K . For the given example, these are (from left to right): – n1 , P1 , r1 , n2 , P1 , r1 , n2 , P2 , r2 , n3 , P2 , r2 , n3 , P3 , r3 , n4 , P3 , r3 . The first segment has a minus sign in its direction because we used relation equation (4.70a). –

Define the process under consideration and identify all possible blob structures for the process, ordered by the number of gluons interchanged between the blob and the Wilson line. For the NNLO17 self-interaction example, there are three blobs: the 2-gluon blob, which is the gluon propagator at NLO, the 3-gluon blob, which is the three gluon vertex, and the 4-gluon blob, which consists of two gluon propagators at LO.

17 Note that there is a difference between referring to the order of the process and the order of the blob. As every gluon radiated from the blob already contributes a factor g, the total order of a diagram conn necting an n-gluon blob to a Wilson line will be !s/2 plus the order of the blob. For example, connecting 3 a NLO 4-gluon blob gives a diagram at N LO.

80

4 Basics of Wilson Lines in QCD

–

For every blob, list all possible diagrams. For the 4-gluon blob, these are listed in equation (4.78a). The 2-gluon blob has two diagrams, and the 3-gluon blob has three. Note that this step is independent on the content of the blob (only dependent on the number of interchanged gluons), and thus independent on the process.

–

For every diagram, separate out the dependence on the path, like we did in the previous section by factorizing out the colour structure. We will develop a more formal approach in the next section (see, e.g. equation (4.83)). Next, apply the Feynman rules and calculate the momentum integrals in the diagram.

–

Apply the specific path structure to the relevant diagrams, and sum all diagrams   according to equation (4.19d). If external momenta are used, assign a –$(9–2) PK⊥ to all external points that do not participate in the diagram, and make a convolution over duplicate external momenta. Let’s illustrate the latter with an example. The diagram connecting the 2-gluon blob to two different segments will be a func, , tion of P⊥J , P⊥K , rJ± , rK± , nJ and nK (it will also depend on the type of path structure, but let us ignore this for now): ,

,

W2 (P⊥J , P⊥K , rJ± , rK± , nJ , nK ),

N

=

Consider now the following contribution:

.

The third external point isn’t participating, so it gets a – $-function. The gluon isn’t connecting the same external point, so no convolution is needed. The result for this term is hence simply     – $(9–2) P⊥3 W2 P⊥1 , P⊥2 . But if we consider the contribution

,

we need to do a convolution because now P⊥J = P⊥K . This can be easily done, by making the substitutions P⊥J → P⊥J – q⊥ and P⊥K → q⊥ , and integrating over q ⊥ . The result for this term is then –$

(9–2)



P⊥2



–$

(9–2)



P⊥3





  d9–2 q⊥ W2 P⊥1 – q⊥ , q⊥ . (20)9–2

4.5 Piecewise Linear Wilson Lines

81

Note that because this diagram essentially forms a tadpole, that is, P1

,

momentum conservation demands the incoming momentum to vanish as well,   i.e. it will give –$(9–2) P⊥1 . So we could have ignored the convolution from the beginning. But this is only true for 2-gluon blobs. Consider, e.g. the 4-gluon blob diagram ,

this is no tadpole, so we need to make a double convolution. The result is then   – $(9–2) P⊥3



  d9–2 q⊥1 d9–2 q⊥2 W4 P⊥1 – q⊥1 , q⊥1 , P⊥2 – q⊥2 , q⊥2 . 9–2 9–2 (20) (20)

The good thing about this framework is that the results from the second, third, and fourth steps are independent of the structure of the Wilson line. Furthermore, we can already calculate the convolution integrals when calculating the momentum integrals such that we have the result ready. In other words, once we calculated the three possible diagrams connecting a 3-gluon vertex to a piecewise Wilson line, we can retrieve this result for any Wilson line structure and never need to recalculate it again; we only need to change the colour factors and the way the different diagrams combine.

4.5.1 Path Functions Now what about the different path structures, as defined in equations (4.69)? We can use the same trick as in the end of the former section, viz. a sign change and an interchange of the corresponding colour indices. For instance:

= F

.

(−)2 F |a

3

a4

The easiest way to implement this on a general basis, is to define a function I per diagram for a given blob, that gives the colour structure in function of the path type, hence it depends on the segment index J. For the leading order 2-gluon blob, this is straightforward. When the gluon is connected to the same segment, flipping

82

4 Basics of Wilson Lines in QCD

the segment makes no difference because we have a factor (–)2 , and the colour interchange has no effect ($ba = $ab ). When the gluon connects two different segments, flipping one of the segments gives a sign difference because of the factor (–)1 . So the leading order 2-gluon blob has the following path function: : I( J) = CF ,

(4.82a)

: I( J, K) = (–)6J +6K CF ,

(4.82b)

where 6J is a number representing the type of the path: 1 6J =

0

J=

1

J=

.

(4.83)

Keep in mind that in our original definition of the Wilson line in equation (4.1), colour indices are not yet traced, meaning that Equations (4.82) should still be multiplied with a unit matrix 1Nc × Nc . Similarly, we find for the leading order 3-gluon blob: : I( J) = –i

Nc CF , 2

Nc CF , 2 Nc : I( J, K, L) = –i(–)6J +6K +6L CF . 2

: I( J, K) = –i(–)6J +6K

(4.84a) (4.84b) (4.84c)

For non-factorable blobs, we use the same trick as in Equations (4.76a,b), by giving I an extra index to identify the sub diagram it belongs to. Let us introduce a new notation, to indicate a full diagram but without the colour content, in which a blob is connected to m Wilson line segments, with ni gluons connected to the ith segment: J ...J

 N

Wnmm ...n11 =

d9 kn d9 k1 l.b. Jm l.b. J . . . I . . . In1 1 F,1 ...,n1 +⋯+nm , (20)9 (20)9 nm

(4.85)

where we will write the indices from right to left to be consistent with the Wilson line being read from right to left. It is important to be consistent in the choice of the “base” structure, from which all other linear topologies can be derived. We have chosen the lower bound Wilson line as the base, which can be seen in the integrals I l.b. . If we would have chosen e.g. the upper bound line, also the definition in equation (4.83) would change.

83

4.5 Piecewise Linear Wilson Lines

Returning to the 4-gluon blob, we can now write the full result for a factorable blob using equation (4.19b): U4 =

M 

I4 W4J

J

+

J–1  M  

J–1  M  K–1   I3 1 W3K1J + I2 2 W2K2J + I2 1 1 W2L1K1 J

J=2 K=1

+

J=3 K=2 L=1

J–1  M  K–1  L–1 

I1 1 1 1 W1O1L1 K1 J + symm,

J=4 K=3 L=2 O=1

where the symmetrized diagrams I1 3 W1KJ3 , I1 2 1 W1L2K1 J and I1 1 2 W1L1K2 J are calculated using equation (4.77), interchanging also the 6J . In other words: I1 3 (K, J) W1K3J = I3 1 ( J, K) W3J 1K , I1 2 1 (L, K, J) W1L2K1 J I1 1 2 (L, K, J) W1L1K2 J

= =

I2 1 1 (K, L, J) W2K1L1 J , I2 1 1 ( J, K, L) W2J 1K1L .

(4.86a) (4.86b) (4.86c)

For a non-factorable blob, every term is just replaced by a sum over subdiagrams, for example, I4 W4J →



Ii4 W4i J .

(4.87)

i

It is important to realize that both the I and W can be calculated independent of the path structure, giving a result depending on nJ , rJ and 6J . We will call the latter the path constants, which fully determine a piecewise linear path. If we have made a full calculation for a given path, we can easily port the result to another path, simply by inputting the new path constants.

4.5.2 Diagrams with Final-State Cuts So far we have only calculated amplitudes. To get probabilities from these, we can do this in the standard way, viz. squaring diagrams and combining them order by order (squared terms and interference terms), or we could treat the squared diagram as one Wilson line – with double the number of segments – where the segments to the right of the cut are the Hermitian conjugate of those to the left. The choice is a matter of personal taste. We choose to continue with the latter case, where we now have three

84

4 Basics of Wilson Lines in QCD

distinct sectors of diagrams: a sector Uleft where the blob is only connecting segments left of the cut, a sector Uright where the blob is only connecting segments right of the cut (this is just the Hermitian conjugate of the former, but possibly with different path parameters rJ , nJ and 6J ), and a sector Ucut where the blob is connecting segments both left and right of the cut. In other words: U = Uleft + Ucut + Uright .

(4.88)

For the first two nothing changes, the calculations go as before. For the example of the 4 4-gluon blob, the first sector Uleft is almost exactly equal to equation (4.86a), but the sums run up only to Mc , the number of segments before the cut, instead of M. The last 4 sector Uright is simply the Hermitian conjugate of this, starting at Mc +1 :18 4 Uright = M  Mc +1

+

I†4 W4† J +

J–1  M  

J–1  M  K–1   I†3 1 W3† 1K J + I†2 2 W2† 2K J + I†2 1 1 W2† 1L1K J

Mc +2 Mc +1

J–1  M  K–1  L–1 

Mc +3 Mc +2 Mc +1

I†1 1 1 1 W1†1O1L1 K J + symm.

Mc +4 Mc +3 Mc +2 Mc +1

For the remaining sector Ucut , we need to define a cut blob. Given a blob, several possible cut blobs might exist, depending on the number of gluons to the left and right of the cut. For example, the leading order 4-gluon cut blobs are given by =

+ cross.

(4.89a) =

+

+ cross.

(4.89b) where the crossings are to be made on the sides of the cut separately. Also note that when the cut blob is more complex, it should be summed over all possible cut locations. Consider the fermionic part of the NLO 2-gluon cut blob: =

+

+

.

(4.90) Note that even if a blob has no lines crossing the cut, it will be considered a cut blob as long as it has gluons on both the left and the right side, as e.g. the first term in the

18 At first sight, one might expect that Hermitian conjugation also flips the order of the segments, but as explained in the paragraph above equation (4.77), they can be treated as commutative.

4.5 Piecewise Linear Wilson Lines

85

rhs of equation (4.89b). As the blob itself connects the left and the right sector (even if internally it doesn’t literally), this blob is associated to the sector Ucut . Now concerning the latter sector, we investigate how many diagrams are added in comparison to equation (4.78) due to the cut. First note that a Wilson line segment itself is never cut.19 A semi-infinite line (lower bound or upper bound) cannot be cut due to the symmetric nature of a squared amplitude, and although a finite line can be cut, we can always write it as a convolution of two semi-infinite lines, placing the cut in between.20 Another remark is that we cannot simply use relation equation (4.77) as before, because it could change the cut topology. Cut diagrams are sorted depending on how its gluons are distributed on the left resp. right side of the cut, and connected to the appropriate cut blob. For instance, the second diagram of equation (4.78), namely W341 , can be cut in one way only, connecting the Wilson line to the cut blob in equation (4.89a). But the fourth diagram, W241 1 , can be cut in two ways: cutting with one gluon on the left (written as W241|1 and connected to the blob in 4 equation (4.89a)), or cutting with two gluons on the left (written as W2|1 1 and connected to equation (4.89b)). Other cut topologies can be related by Hermitian conjugation when switching left and right sides, for example, KJ †JKL W1L1|2 = W2|1 1 .

(4.91)

In the case of the 4-gluon blob, the following diagrams have to be added to equation (4.78):

(4.92)

19 A cut line does appear in the context of infinite Wilson lines as we saw in equation (4.67), but this is a different type of cut (not a final-state cut), and anyway we are not (yet) including infinite Wilson lines in this framework. 20 In the TMD framework, it is common to associate a cut finite line with a true cut propagator, but this is merely a matter of naming conventions. For example, in Section 5.1.6, we make the definition of a collinear PDF (a cut diagram itself) gauge invariant by adding a finite Wilson line that is cut. In literature (see, e.g. Ref. [85]), it is then common to also integrate over the exponential coming from equation (4.29b) leading to a delta function, which in turn can be interpreted as the Feynman rule for a cut propagator (see equation (4.67)). We prefer to avoid this approach, as it is more general to leave a, unintegrated. We will however adapt the same pictorial representation of a cut finite Wilson line, see, e.g. Figure 5.8, but we remind ourselves that it is just a cut between two lower-bound segments.

86

4 Basics of Wilson Lines in QCD

Now we have the necessary ingredients to write the cut sector for the 4-gluon blob:

4 Ucut =

Mc  M  

  KJ KJ W3|1 + h.c. + W2|2

Mc +1 1

+

Mc  J–1  M  

Mc  K–1 M     KJ KJ W2L1|1 + + symm. + W1L1|2 h.c.

Mc +2 Mc +1 1

+

Mc +1 2

Mc J–1  M  K–1  

W1O1L1|1K J +

Mc +3 Mc +2 Mc +1 1

+

Mc  J–1  L–1 M   Mc +2 Mc +1 2

Mc  K–1  L–1 M   Mc +1 3

2

1

h.c.

1

LKJ W1O1|1 1

(4.93)

1

Although this might look quite complex, note that it only is the way how to combine the diagrams that is a bit involving. And even then, it is a matter of good bookkeeping, making it look a lot worse than it is. One final remark: in equation (4.88), we assumed that the sectors Uleft and Uright only use regular blobs, while the sector Ucut only uses cut blobs. While the latter is always true – as we cannot connect a segment before the cut with a segment after the cut without including the cut line – the former is only partially correct. It is possible that the left (or right) sector connects to a blob that is cut before or after all gluons. Let us illustrate this. Consider a trivial example, namely the interaction between a Wilson line and a quark line. Ignoring the self-interactions of the quark, we have the following possible probability diagrams at NLO:

This means, we have four different blobs, namely two 1-gluon blobs,

=

=

,

(4.94a)

,

(4.94b)

4.6 Eikonal Approximation

87

and two 2-gluon blobs:

=

,

(4.95a)

=

.

(4.95b)

The two 1-gluon blobs and the first 2-gluon blob will be used in the Uleft and Uright sectors, while the second 2-gluon blob is the only blob used in the Ucut sector. Note that it is not for every blob possible to draw a cut fully to the right or left. For example, the LO 2-gluon self-interaction blob doesn’t have this possibility because if we would try to draw such a blob with the cut fully to the left, that is,

=

,

(4.96)

we immediately see that this is not a valid Feynman diagram, as the amplitude on the left side would represent two real gluons popping out from nothing. We thus conclude that self-interaction blobs cannot have cuts fully to the left or right.

4.6 Eikonal Approximation Before we delve into the calculational techniques to work with Wilson lines – as we will do in the next chapter – we will motivate the usefulness of Wilson lines by one of their most important applications, namely as a resummation of soft and collinear gluons. In the eikonal approximation, we assume a quark with momentum large enough to neglect the change in momentum due to the emission or absorption of a soft gluon. Even after multiple soft interactions, it won’t deviate much from its path, which we then take to be unaltered. Such a quark is called eikonal.

p

p–q 1

p–q 1 –q 2 F

q1 μ, a

q2 v, b

Figure 4.8: An incoming quark radiating two

soft gluons.

88

4 Basics of Wilson Lines in QCD

Let us investigate this a bit further. We take an incoming (hence real) quark with momentum p that radiates two soft gluons with momentum q1 and q2 . This is illustrated in Figure 4.8 (where the blob represents all possible diagrams connected to the quark propagator). This diagram is equal to F

i(p/ – q/1 – q/2 ) i(p/ – q/1 ) i g tb γ i g ta γ , u(p). (p – q1 – q2 )2 + i0 (p – q1 )2 + i0

(4.97)

Making the soft approximation is the same as neglecting q/i with respect to p/ , and q2i with respect to p ⋅ qj , giving F

i p1 γ 1 γ i p3 γ 3 γ , i g tb i g ta u(p), –2 p ⋅ q1 – 2 p ⋅ q2 + i0 –2 p ⋅ q1 + i0

where we used p2 = 0 because this is the momentum of a real quark. Because of the Dirac equation (B.19) in momentum representation, i.e. p/ u(p) = 0, we can add a term ip3 γ , γ 3 to the numerator of the rightmost fraction: F

i p1 γ 1 γ i p3 {γ 3 , γ , } i g tb i g ta u(p). –2 p ⋅ q1 – 2 p ⋅ q2 + i0 –2 p ⋅ q1 + i0

(4.98)

Next we use the anticommutation rule equation (4.17) and write the momentum as p3 = |p| n3 , with n3 a normalized directional vector, in order to get F

i p1 γ 1 γ –in, i g tb i g ta u(p). –2 p ⋅ q1 – 2 p ⋅ q2 + i0 n ⋅ q1 – i0

(4.99)

Because the rightmost fraction doesn’t contain any Dirac structure anymore, we can repeat the same steps on the leftmost fraction. This gives F

–i –i i gn, ta u(p). i gn- tb n ⋅ (q1 + q2 ) – i0 n ⋅ q1 – i0

(4.100)

What we see is that the Dirac propagators have been replaced by Wilson line propagators, and the Dirac-gluon couplings by Wilson vertices. By using the eikonal approximation, we literally factorized out the gluon contribution from the Dirac part. Of course this remains valid when radiating more gluons. In the latter case, the resulting formula is straightforward: (i g )n F tan . . . ta1 u(p)

n ,2 –in,n n,1 ... . n / n ⋅ (q1 + q2 ) – i0 n ⋅ q1 – i0 n ⋅ qi – i0

This is exactly the result for an incoming bare quark connected to the blob, multiplied with a Wilson line going from –∞ to 0 (we know that the external point has to be zero because there is no exponential): F U(0 ; –∞) u(p).

(4.101)

4.6 Eikonal Approximation

89

However, when using the momentum representation for the external point, the latter gives a factor –$(9) (q), so that we can write this relation as an exact convolution:  F(p)

d9 q U (q) u(p – q) = F(p) (U ⊗ u) (p), (20)9

(4.102)

where q can be interpreted as the sum of the gluon momenta. This is illustrated in the diagram in Figure 4.9. Note that equations (4.101) and (4.102) don’t give a bare multiplication either, because the tai are placed between the u(p) of the external quark and the blob. Writing out the Dirac and Lie indices makes this clear:21  j "!  a   i F "$ t n . . . ta1 ji u(p) !

i g n,1 i g n,n ... . n / n ⋅ q1 + i' n ⋅ qi + i'

From this result, we introduce the concept of an eikonal quark. This is a quark that is only interacting softly with the gauge field and thus doesn’t deviate from its straight path. It can be understood as a bare quark convoluted with a Wilson line to all orders: % 2 % 2 % % i ij %8eik. = U(0 ; –∞) ⊗ %8 j .

(4.103)

In other words, the net effect of multiple soft gluon interactions on an eikonal quark is just a colour rotation (nothing but a phase). It is common to denote an eikonal quark with a double line, but this gives rise to ambiguities: the double line was already used to denote a Wilson line propagator. These are, although related, not the same. The eikonal line represents a quark (carrying spinor indices) resummed with soft gluon radiation to all orders, while the Wilson line propagator represents gluon radiation at a specified order (not necessarily soft), still to be multiplied with the quark (carrying no spinor indices itself). In short, Wilson line propagators are used in the calculation of an eikonal line. To appreciate the difference, have a look at equation (4.103): the % 3 ij eikonal quark is the combination U(0 ; –∞) ⊗ %8 j , while the Wilson line propagators are components of U(0 ; –∞) .

≈ ...

...

Figure 4.9: A quark radiating n soft gluons can be represented as a bare quark multiplied with a Wilson line going from –∞ to 0.

21 Note that we replace the prescription term + i0 with the term +i'. The reasoning behind this is that while in the case of the quark propagator this term is literally just a pole prescription, in the case of a Wilson line propagator it also acts as an infrared regulator.

90

4 Basics of Wilson Lines in QCD

Soft limit ...

Figure 4.10: In the soft limit, a bare quark can be represented as an eikonal quark.

To avoid confusion, we will draw an eikonal line in red, and always explicitly draw an arrowhead (representing the quark’s momentum flow): p

% 2 % An eikonal line, i.e. %8ieik. (p) , !

q

A Wilson line propagator, i.e.

i . n ⋅ q + i'

(4.104a) (4.104b)

But keep in mind that these notations are commonly interchanged in literature. Using our notation for the eikonal line, we can write down the eikonal approximation diagrammatically as in Figure 4.10. A last remark: In the derivation of the eikonal approximation, more specifically equation (4.98), we used the fact that the quark in question is external, by adding a term γ , p/ u(p) = 0. This is a crucial step, without which we wouldn’t have been able to resum all gluons into a Wilson line, i.e. Wilson lines as a resummation of gluon radiation can only appear next to quarks that are on-shell. It is possible to resum gluon radiation into a Wilson line even if it is not soft. For example, in the collinear approximation, we allow for large radiated momenta q which are collinear to p, i.e. if p, = |p| n, , then q, = |q| n, in the same direction. The Dirac equation tells us that p/ u(p) = 0 and thus n/ u(p) = 0, which implies we can add a term γ , q/ u(p) to equation (4.97). If we keep the quasi on-shell constraint, q2 ≈ 0 as compared to p ⋅ q, this again leads to a Wilson line, but this time with possibly big q momentum components (as long as they are collinear to p).

5 Gauge-Invariant Parton Densities In this chapter, we will give a brief review of the basics of the gauge-invariant path-dependent approach to unintegrated parton densities, i.e. we investigate situations where the common collinear factorization is no longer adequate and has to be replaced by a new factorization approach that introduces transverse momentum dependence in the parton density functions (or parton distribution functions, or PDFs for short). The main goal is to have a relevant description for processes that are not fully inclusive, but where e.g. more than one hadron is detected in the initial or the final state. In such a process we cannot integrate over k⊥ because the final hadron will have a manifest k⊥ -dependence. We start this chapter with a revision of deep inelastic scattering (DIS), where we will construct a gauge-invariant operator definition for the PDF using Wilson lines. Then, we move to a less inclusive experimental setup, viz. semi-inclusive deep inelastic scattering (SIDIS), and introduce PDFs that are k⊥ -dependent, the so-called transverse-momentum dependent PDFs (or TMDs), and construct operator definitions for these as well. In the last section of this chapter, we investigate the evolution of TMDs, and construct equations that will be used similarly to the Dokshitzer–Gribov– Lipatov–Altarelli–Parisi (or DGLAP) equations in the collinear case.1

5.1 Revision of Deep Inelastic Scattering Before we start with the investigation of the TMD framework, we will review the theory behind DIS in a formal way. This will allow us to construct gauge-invariant operator definitions for the PDFs, which will show to be indicative for the construction of TMDs later on.

5.1.1 Kinematics DIS is the most straightforward process to probe the insides of a hadron. An electron is collided head-on with a proton (or whatever hadron), destroying it maximally. The kinematic diagram is shown in Figure 5.1. We will always neglect electron masses. The centre-of-mass (CoM) energy squared s is then given by: s = (P + l)2 = m2p + 2P ⋅ l ,

(5.1)

1 The formalism of TMDs has gone through a lot of evolution in the last decades. Originally introduced in, e.g. Refs [94, 96–99], later adapted into a gauge-invariant approach [21, 35, 39, 46, 92, 126, 143, 144, 146], and the most recent definitions are found in, e.g. Refs [15–17, 19, 20, 43, 85, 86, 177]. For an introduction to the issues of the renormalization of TMDs, see, e.g. Refs [78–82, 198].

92

5 Gauge-Invariant Parton Densities

l′ electron

l q k

proton

X

P

Figure 5.1: Kinematics of deep inelastic electron–proton scattering.

and q is the momentum transferred by the photon: ′

q, = l, – l , . N

(5.2)

Because q2 = 2Ee Ee′ (cos (ee′ – 1) ≤ 0, we define Q2 = – q2 ≥ 0. The invariant mass of the final state X is then given by N

m2X = (P + q)2 = m2p + 2P ⋅ q – Q2 .

(5.3)

In order for the photon to probe the contents of the proton, it should have a wavelength + > m2p ) and inelastic (m2X >> m2p ) scattering. The two Lorentz invariants of interest in the process are Q2 and P ⋅ q, but it is convenient to use the variables Q and xB instead, where N

xB =

Q2 2P⋅q

(5.4)

is called the Bjorken-x. Unless necessary to avoid confusion, we will always drop the index “B”, just remember that x always denotes the Bjorken-x (and thus not a general fraction, see further). Its kinematics restrain x to lie between Q2 /s + Q2 (neglecting   terms of O M 2 /Q2 ) and 1 (the elastic limit). Another useful variable is P⋅q , P⋅l Q2 . =  x s – m2p N

y=

(5.5a) (5.5b)

In the rest of the frame this equals y = (E – E′ )/E, the fractional energy loss of the lepton. It is not an independent variable because   Q2 = x y s – m2p . Let us finish this subsection on kinematics with two trivial relations:

(5.6)

93

5.1 Revision of Deep Inelastic Scattering

2x P⋅l =

Q2 , y ′

l⋅q = –l ⋅q = –

(5.7a) Q2 . 2

(5.7b)

′

The latter can be demonstrated by calculating (l – q)2 = l 2 = 0.

5.1.2 Invitation: The Free Parton Model A parton is a terminology used to denote any point-like constituent of the proton, being quarks, antiquarks or gluons. The parton model (PM) describes the proton as a black box containing an undetermined amount of such partons. The mutual interactions of these partons have large timescales compared to the interaction with the photon, allowing us to separate the latter from the former. For instance, inside the proton, a gluon could fluctuate into a quark–antiquark pair. The photon would enter the proton and kick out one of the quarks, much faster than the pair can recombine. The pair looks “frozen” to the photon: because of the much larger timescale of the parton interactions, all dynamics are hidden for the photon. From the latter’s viewpoint, all partons are hence “free”. As we will see in equation (5.9), in DIS, the momentum of a parton is defined as a fraction . of the original parent hadron. The size of this fraction will define the dominant type of parton that arises in this regime. We have three types: Valence quarks:

10–2 ≤ . ≤ 1. These are the quarks that normally form the parent hadron. For example, for a proton there are three valence quarks, viz. two up quarks and a down quark.

Sea quarks:

10–4 ≤ . ≤ 10–2 . These quarks always come in a quark– antiquark pair, and are created from a virtual gluon.

Gluons:

10–8 ≤ . ≤ 10–2 . At large x, gluons can be neglected as the valence quarks are by far the dominant partons. However, at small x gluons quickly dominate all partons and one can neglect all quarks.

Note that the regions in . are just vague approximations, as there is no way to sharply separate the different parton types based on their momentum fraction. Furthermore, at higher energy scales, the gluons become dominant much faster. Also note that the existence of both valence and sea quarks makes that the quark number operator is not well-defined, as it is not conserved. It is convenient to let the short-distance process – the interaction between the photon and one of the partons – be named the hard part, which we will often denote

94

5 Gauge-Invariant Parton Densities

hard

parton model

soft

Figure 5.2: DIS in the FPM. The virtual photon strikes one of the quarks, while the other two quarks are left unharmed and don’t influence the process anyhow.

with a hat, e.g. sˆ is the hard CoM energy squared. In contrast to this stands the soft part, which – as we will see in later sections – contains all interactions at large distances. For now, we can make an intuitive distinction: everything inside the proton is soft, everything outside the proton plus the interaction point – the photon and the struck parton – is hard. This is illustrated in the left picture of Figure 5.2. Later on, we will give a more rigorous formulation for this distinction. The PM thus describes DIS without the strong interaction participating, as all effects of the strong force are absorbed in the proton, without giving any clue for the structure of the latter except one, viz. that we can extract a parton from it. Before we really delve into the PM, we try to get a general idea by investigating an extreme case: the Free Parton Model (FPM). In this toy model, the proton has no dynamic structure, but merely consists of exactly three quarks, totally unaware of each other’s existence. From the point of view of the photon, it doesn’t matter how the proton structure looks, be it in the FPM or the regular PM, it just hits a parton like it would hit any electromagnetically charged particle, ignoring all other structure in the proton. The leading order hard part of DIS is therefore genuine electron-quark scattering, which we can describe similarly to electron-muon scattering.2 This is illustrated schematically in Figure 5.2. Of course, at timescales much larger than the process, the remains of the proton and the struck parton will hadronize into jets, as free quarks can only exist for a short amount of time due to the asymptotic freedom of QCD. The differential cross section for (unpolarized) e– ,+ scattering can be calculated by basic QED techniques and equals   40!2 s d3  – + y2 e , → e– ,+ = , 1 – y + dy Q4 2

(5.8)

1 is the electromagnetic fine-structure constant (see equation (B.2)). where ! ≈ 137 The only difference between the cross section for e– ,+ scattering and that for e– q± scattering is the charge of the quark:

2 Note that we deliberately choose e– ,+ scattering over e– e+ scattering, because the latter also contains a diagram where the two electrons annihilate into a virtual photon, which has no correspondence with e– q scattering.

5.1 Revision of Deep Inelastic Scattering

95

  40!2 sˆ d3ˆ  – ± y2 e q → e– q± = e2q , 1 – y + dy Q4 2 but now sˆ = (l + k)2 , the centre-of-mass energy squared of the electron and the quark. In order to relate the hard cross section to the full cross section, we define the quark momentum as a fraction of the proton momentum: k = .P

0 < . < 1,

(5.9)

such that sˆ = . s,

yˆ = y .

For the outgoing quark to be on-shell, we have the requirement (k + q)2 ≈ 2. P ⋅ q – Q2 ≡ 0 , ⇒ . ≡ x. In this case, the on-shellness constraint fixes the momentum fraction to equal the Bjorken variable, but this is certainly not a general result. The Bjorken-x is a kinematical constraint defining the process, while . is nothing more than a momentum fraction (totally independent of the process). Keeping both x and . as independent variables (which will simplify comparisons with later results), the electron-quark cross section is given by  d3 3ˆ q 40!2 s y2 2 (5.10) = eq . $(x – . ). 1–y+ dx dy d. Q4 2 Going to the electron–proton cross section is obvious in the FPM. We simply integrate over all possible quark fractions . and make a weighted sum over the three quarks:  d3 3ˆ q d2 3FPM 1  d. = , dx dy 3 q dx dy d.  40!2 s 1 2 y2 e . (5.11) = x 1–y+ Q4 2 3 q q 5.1.3 The Parton Model Let us redo our intuitive derivation from the previous section in a more formal way. We will treat the proton as a “black box” (contrary to the FPM representation where it is an exact packet of three partons), which we deeply probe with a highly virtual photon. This is depicted in Figure 5.3. We know that in the PM, it is assumed that the photon interacts with one constituent of the proton only (a quark, an antiquark, or at higher orders possibly a gluon), on a timescale sufficiently small to allow the struck

96

5 Gauge-Invariant Parton Densities

X

Figure 5.3: DIS to all orders: a photon hitting a proton and

breaking it.

parton to be considered temporarily “free”. To motivate this quantitatively, we write the components of the proton momentum P and the parton momentum k in light-cone (LC) coordinates (see Appendix B.4): P, = P+ ,

m2p 2P+

  k, = k+ , k– , k ⊥ .

, 0⊥ ,

In the rest frame of the proton, the distribution of its constituents is isotropic, i.e. all components of p, are of the order ≲ mp . In the limit P+ → ∞, the so-called infinite-momentum frame (or IMF for short), the only remaining component of the proton momentum is its plus-component. The parton naturally follows the proton in the boost. Then the 4-momenta become:   , PIMF = P+ , 0– , 0⊥ ,

  , kIMF ≈ k+ , 0– , 0⊥ .

The parton’s transverse component p⊥ ∼ mp can be trivially neglected when compared to p+ → ∞. The ratio of the plus momenta is boost invariant, so that we can write .=

k+ , P+

⇒

,

,

kIMF = . PIMF .

As long as we can boost to a frame where P+ is the only remaining large component of the proton momentum, the parton is fully collinear to the parent proton and can thus considered to be “free”. From now on, we will always parameterize the proton momentum and the struck quark momentum based on the dominantly large P+ : ,

+

P = P ,

m2p 2P+



k2 + k2⊥ k = .P , , k⊥ , 2. P+

,

, 0⊥ ,

+

(5.12)

where we can safely assume k2 , k2⊥ m2p

→ 1 .

(5.24)

Hence, we can expand the hadron tensor as: W ,- = A g ,- + B qˆ , qˆ - + C qˆ , tˆ- + D tˆ, qˆ - + E tˆ, tˆ- + iF%,-13 tˆ1 qˆ 3 , where the scalar functions A, . . . , F only depend on m2p , Q2 and x (because there are no other invariants in the proton system). In the case of polarized hadrons, the spin vector S, and its combinations should be added to the basis. Next, we impose current conservation, which requires ∂, J , = 0. Applying this to equation (5.19), we find qˆ , W ,- = W ,- qˆ - = 0. This condition gives: A ≡ B,

C ≡ D ≡ 0.

W ,- should also be Hermitian and time-reversal invariant, and for the electromagnetic and the strong force, it should be parity invariant as well. By using the transformation matrix

99

5.1 Revision of Deep Inelastic Scattering

⎛

-

D,

1 ⎜0 ⎜ =⎜ ⎝0 0

0 –1 0 0

0 0 –1 0

⎞ 0 0⎟ ⎟ ⎟. 0⎠ –1

(5.25)

we can write out these conditions (adding spin-dependence for general reference): Hermiticity: parity reversal: time reversal:

∗ W,(q, P, S) ≡ W-, (q, P, S), 1 D, D-3 W,- (q, P, S) 1 ∗ D, D-3 W,(q, P, S)

≡ W,- (! q, ! P, –! S), ≡ W,- (! q, ! P, ! S).

(5.26a) (5.26b) (5.26c)

where ! q, = $,0 q0 – $,i qi . The effect of these conditions is that A, . . . , F should be real functions, and the parity-reversal requirement sets F = 0. But parity is not conserved in weak interactions; in that case, F is allowed to have a non-zero value. We can rewrite W ,- as (taking S = 0 again):  1  ,,(5.27) W ,- = – g⊥ FT (x, Q2 ) – tˆ, tˆ- FL (x, Q2 ) – i%⊥ FA (x, Q2 ) , 2x where FT = –2x A,

FL = 2x (A + E),

FA = 2x F.

These are called the transversal resp. longitudinal resp. axial structure functions of the proton. They are non-perturbative (and thus non-calculable) objects, which have to be extracted from experiment. In parallel to these, a different notation is also used in literature: 1 FT , 2x 1 F2 = 2 (FL + FT ) , * 1 FA , F3 = x *2 F1 =

FT = 2x F1 ,

(5.28a)

FL = *2 F2 – 2x F1 ,

(5.28b)

FA = x *2 F3 .

(5.28c)

We can express the hadron tensor in function of these other structure functions as well:  2 * *2 ,W ,- = –g⊥ F1 + tˆ, tˆ(5.29) F2 – F1 + i %⊥,- FA . 2x 2 The difference between FT , FL , FA and F1 , F2 , F3 is just a matter of historic convention. However, there exist different conventions for the normalization of the structure functions, if so often differing by a factor of 2 or 2x. We follow the same convention as e.g. in [172], as we believe it to be the most commonly accepted one. The structure functions can be extracted from the hadronic tensor by projecting with appropriate tensors:

100

5 Gauge-Invariant Parton Densities

1 ,F1 = – g⊥ W,- , 2  x  ,F2 = 2 2tˆ, tˆ- – g⊥ W,- , * 2i ,F3 = – 2 %⊥ W,- , * ,-

,-

FT = –x g⊥ W,- ,

(5.30a)

FL = 2x tˆ, tˆ- W,- ,

(5.30b)

,-

FA = –2x i %⊥ W,- ,

(5.30c)

,-

where g⊥ and %⊥ are just the transversal metric and transversal antisymmetric tensor: ,- def

g⊥ = g ,- + qˆ , qˆ - – tˆ, tˆ- ,

(5.31)

,- def %⊥ = %,-13 tˆ1 qˆ 3

(5.32)

.

For the rest of this book, we will ignore weak interactions, dropping FA from the hadronic tensor. Combining the result from the leptonic and the hadronic tensor, we get       2Q2 y2 2 2 L,- W ,- = x, Q + (1 – y) F x, Q F 1 – y + . T L x y2 2 Plugging this result in equation (5.15) gives us the final expression for the unpolarized cross section for electron–proton Deep inelastic scattering (neglecting terms of order

m2p ): Q2

d2 3 40!2 s = dxdy Q4

 1–y+

y2 2



     FT x, Q2 + (1 – y) FL x, Q2

(5.33)

If we compare this with the result in equation (5.11), we find the following structure functions for the FPM:   1  2 e , (5.34a) FTFPM x, Q2 = x 3 q q   (5.34b) FLFPM x, Q2 = 0 . 5.1.4 Parton Distribution Functions In Section 5.1.2, we succeeded in deriving a lowest order result for the cross section, starting from a static proton. On the other hand, in Section 5.1.3, we followed a more formal approach, without any assumptions about the proton structure but one: that we can separate the hard interaction from the proton contents. This is the concept of factorization: in any process containing hadrons we try to separate the perturbative hard part (the scattering Feynman diagram) from the non-perturbative part (the hadron contents). The latter is not-calculable from the first principles in QCD, and consequently it has to be described by a PDF that gives the probability to find a parton with momentum fraction . in the parent hadron. However, one has to proceed

5.1 Revision of Deep Inelastic Scattering

101

with caution because factorization has not been proven but for a small number of processes, including electron–positron annihilation, DIS, SIDIS and Drell–Yan. The PDF is literally the object that describes the proton as a black box. You give it a fraction . and it returns the probability to hit a parton carrying this longitudinal momentum fraction when you bombard the proton with a photon. It is commonly written as fq (. ), where q is the type of parton for which the PDF is defined. There are thus seven PDFs, one for each quark and antiquark, and one for the gluon. Parton distribution functions are not calculable; they have to be extracted from experiment. We can however calculate their evolution equations, such that we can evolve an extracted PDF from a given kinematic region to a new kinematic region. It is a probability density, but it is also a distribution in momentum space; by plotting the PDF in function of x, one gets a clear view of the distribution of momenta of the partons in the proton. Furthermore, we assume that the PDF only depends on . , and not e.g. on the parton’s transverse momentum. This doesn’t mean that we automatically neglect the struck parton’s transverse momentum component! But because we don’t identify any hadron in the final state, and because we have to sum over all final states and integrate out their momenta (the final-state cut), any transverse momentum dependence in the PDF or the hard part is integrated out. Factorization in DIS – also called collinear factorization because of the collinearity of the struck quark to the proton – is a factorization over x (plus an energy scale). We can write this formally as     d3 ∼ fq x, ,2F ⊗ Hˆ x, ,2F , dx which is just a schematic. We will treat the technical details later. Whenever information on the transverse momentum is needed, e.g. when identifying a final hadron like in SIDIS, collinear factorization won’t do, and k⊥ -factorization is needed instead, where a transverse momentum-dependent PDF, or TMD for short, is convoluted with the hard part:     d3 ∼ fq x, k⊥ , ,2F ⊗ Hˆ x, k⊥ , ,2F . dx Formally, a PDF and a TMD should be related by integrating out the transverse momentum dependence:  fq (. ) = d2 k⊥ fq (. , k⊥ ) , however, QCD corrections make this equality invalid. In the parton model, the concept of (collinear) factorization can be painlessly implemented:

102

5 Gauge-Invariant Parton Densities

PM

d3 ≡

 q

 x d. fq (. ) d3ˆ q , .

= fq ⊗ d3ˆ q . N

(5.35a) (5.35b)

& Note that this is not the common convolution definition, i.e. d4 f (4)g(t – 4). This is because the latter is a convolution as defined in Fourier space. In QCD, a lot of theoretical progress has been made by the use of Mellin moments. These form an advanced mathematical tool, which would take use too long to delve into. The thing to keep in mind is that the type of convolution as in equations (5.35) is a convolution in Mellin space (see equations (B.65) for the definition of the Mellin transform). We can express the structure functions of the proton in terms of the structure functions of the quark, where the latter are defined at leading order as def q Fˆ T (x) = x e2q $– (1 – x) .

(5.36)

To do this, we use equations (5.35) to relate the electron-quark cross section in equation (5.10) with the electron–proton cross section in equation (5.33). Then it is easy to show that FTPM



x, Q

2



=

 q

=



1

d. fq (. ) Fˆ T

q

x

e2q xfq (x) ,

 x , .

(5.37a) (5.37b)

q

  FLPM x, Q2 = 0 .

(5.37c)

Note that FTPM does not depend on Q2 ! This is called the “Bjorken scaling” prediction: the structure functions scale with x, independently of Q2 . Because this prediction is a direct result from the parton model, it should be clearly visible in leading order (up to first-order QCD corrections, where the Bjorken scaling is broken). This is indeed confirmed by experiment3 (see Figure 5.5). Also note that by comparing equations (5.37) to equations (5.34), we can easily find the quark PDFs in the FPM: fqFPM (x) =

1 , 3

(5.38)

which is exactly what the initial assumption for the FPM is: the proton equals exactly three quarks, thus the probability of finding one of those is always one-third per quark, regardless the value of x. A small remark on the difference between structure functions and PDFs. A structure function emerges in the parametrization of the hadronic tensor, the latter 3 Data from Eur. Phys. J. C75 (2015) 580

5.1 Revision of Deep Inelastic Scattering

103

H1 and ZEUS Combined Data

σ 20

105

XBJ = 0.00008 ( x 2 ) XBJ = 0.00013 ( x 219) 18 XBJ = 0.0002 ( x 2 ) 17 XBJ = 0.00032 ( x 2 ) 16 XBJ = 0.0005 ( x 2 ) XBJ = 0.0008 ( x 215) 14 XBJ = 0.0013 ( x 2 )

XBJ = 0.002 ( x 213) XBJ = 0.0032 ( x 212) 11

XBJ = 0.005 ( x 2 ) 10 XBJ = 0.008 ( x 2 )

1000

XBJ = 0.013 ( x 29) 8 XBJ = 0.02 ( x 2 ) 7 XBJ = 0.032 ( x 2 )

XBJ = 0.05 ( x 26) XBJ = 0.08 ( x 25)

10

4 XBJ = 0.13 ( x 2 ) 3 XBJ = 0.18 ( x 2 ) 2 XBJ = 0.25 ( x 2 )

XBJ = 0.4 ( x 21)

0.1

XBJ = 0.65

10

100

1000

104

105

Q2 (GeV2)

Figure 5.5: Modern DIS data from DESY, loosely fitted. Note how the data points almost don’t change in function of Q2 . Hence Bjorken scaling is preserved.

being process dependent. If we have a look at its definition for DIS in Equation 5.19, we see that the hadronic tensor contains information both on the proton content and the photon hitting it. This is illustrated in Figure 5.4, where the blob represents the hadronic tensor, describing the process of a photon hitting a (black box) proton. As a structure function is just a parametrization of the hadronic tensor, the same applies to it. If we change the process to, say, deep inelastic neutrino scattering, our structure functions change as well, because now they describe the process of a W ± or Z 0 boson hitting a proton. But the main idea behind factorization is that, inside the structure functions, we can somehow factorize out the proton content (which is process independent) from the process-dependent part. This is shown in Figure 5.6, where the smaller blob now represents a quark PDF. The factorization of structure functions in the parton model is demonstrated in equations (5.37). The initial factorization ansatz, equations (5.35), is required to be valid for any cross section, given a unique set of PDFs, i.e. the PDFs are universal. We can extract these PDFs in one type of experiment, like electron DIS, and reuse them in another experiment like neutrino DIS. In contrast with the structure

104

5 Gauge-Invariant Parton Densities

W μν fq

Figure 5.6: Difference between structure functions and

PDFs.

Figure 5.7: Factorization in DIS at leading order.

functions, PDFs emerge in the parametrization of the quark correlator – as we will see in Section 5.1.5 – which is universal by definition.

5.1.5 Operator Definition for PDFs We can assume that the photon scatters off a quark with mass m inside the proton, if Q2 is sufficiently large. The final state can therefore be split in a quark with momentum p and the full remaining state with momentum pX . Constructing the (unpolarized) hadronic tensor for this setup is straightforward. First, we remark that pulling a quark out of the proton at a space–time point (0+ , 0– , 0⊥ ) is simply 8! (0)|P. Then, we construct the diagram for the hadronic tensor, the so-called handbag diagram, step by step: k X

ν

=

X | ψα(0) |P ,

=

% 3  "! 5 % X %8! (0)% P , u+" (p) γ -

∼

% 25 % % 3 " ,  #"! 4 %% % P %8" (0)% X X %8! (0)% P , γ p/ + m γ -

p

k X

ν k

p

μ k

5.1 Revision of Deep Inelastic Scattering

105

where we omitted the prefactor, sums and integrations over X and p and the $function. Including these, the full hadronic tensor is given by W ,- =

where

 

   1  2  d3 p eq d4 zei(P+q–pX –p)⋅z 40 q (20)3 2p0 X % 25 % % 3 " ,  #"! 4 %% % P %8" (0)% X X %8! (0)% P , × γ p/ + m γ -

(5.39)

is defined as a sum and integration over a full set of states:  

|XX| = 1 ,

 

N

=

 X

(5.40)

d3 pX = (20)3 2EX

 X

 d4 pX +  2 $– pX – m2X , (20)4

(5.41)

where   N     $–+ p2X – m2X = 20 $ p2X – m2X ( p0 .

(5.42)

Next we replace the integral over p in the hadronic tensor with an on-shell condition: 

d3 p (20)3 2p0

 →

 d4 p +  2 $– p – m2 . 4 (20)

We introduce the momentum k = p – q, giving W ,- =

    1  2  d4 k +  2 2 (k + q) eq $ – m d4 z ei(P–k–pX )⋅z 40 q (20)3 X % 25 % % 3 " ,  #"! 4 %% % P %8" (0)% X X %8! (0)% P . × γ k/ + q/ + m γ -

Now we use the translation operator and the completeness relation to rewrite W ,- as        1 2 (5.43a) W ,- = eq d4 k $+ (k + q)2 Tr Iq γ , k/ + q/ γ - , 2 q q

 N

I!" =

% 2 d4 z –ik⋅z 4 %% % P e (z)8 (0) %8 %P . ! " (20)4

(5.43b)

I is the quark correlator, which will be used as a basic building brick to construct PDF. Note that its Dirac indices are defined in a reversed way, this is done deliberately to set the trace right. This result is quite a general result, valid for a range of processes.   Neglecting terms of O 1/Q , we can approximate the $-function in equation (5.43a) as   $ (k + q)2 ≈ P+ $ (. – x) ,

106

5 Gauge-Invariant Parton Densities

which sets . ≡ x. This then gives W ,- ≈

  1 2 P+  eq Tr Iq (x) γ , k/ + q/ γ - , 4 q P⋅q

where the integrated quark correlator is defined as  I(x) = dk– d2 k⊥ I (x, k– , k⊥ ) ,  % 2 4 % 1 + – % % dz– e–ixP z P %8" (0+ , z– , 0⊥ )8! (0)% P . = 20

(5.44)

(5.45)

A last simplification that we can make is to assume that the outgoing quark is moving largely in the minus direction; k, +q, ≈ k– +q– . This is easily understood in the infinite momentum frame, where the quark ricochets back after being struck head-on by the photon. However, it is a valid simplification in any frame, which can be shown by making a Q1 expansion of W ,- . With this assumption, we get

+  k2 + k2⊥ P+  + P – +q k/ + q/ ≈ γ P⋅q P⋅q 2. P+ ≈ 1, giving the finalresult  for the unpolarized hadron tensor in DIS at leading twist (this means up to O Q1 ): W ,- ≈

 1 2  q eq Tr I (x) γ , γ + γ - . 4 q

(5.46)

Now, let us investigate the unintegrated quark correlator equation (5.43b) a bit deeper. Since it is a Dirac matrix, we can expand it in function of Lorentz vectors, pseudovectors and Dirac matrices. The variables on which it depends are p, , P, and S, (the latter is a pseudovector in the case of fermionic hadrons). Our basis is then (see equation (B.27)) spanned by 7 , , ,8 7 8 p , P , S ⊗ 1, γ 5 , γ , , γ , γ 5 , γ ,- , where γ ,- = γ [, γ -] . To continue, we impose two necessary conditions on the hadron tensor, namely that it should be Hermitian and that parity should be conserved: Hermiticity: Parity:

I(p, P, S) ≡ γ 0 I† (p, P, S)γ 0 ,   I(p, P, S) ≡ γ 0 I ! p, ! P, –! S γ0 .

(5.47a) (5.47b)

These conditions will reduce the expansion significantly. For instance, the integrated quark correlator can be expanded up to leading twist as

5.1 Revision of Deep Inelastic Scattering

I(x, P, S) =

 # " 1 1 f1 (x)γ – + g1L (x)SL γ 5 γ – + h1 (x) S/ T , γ – γ 5 , 2 2

107

(5.48)

where the three integrated PDF f1 , g1L and h1 are the unpolarized resp. helicity resp. transversity distributions. They can be recovered from the quark correlator by projecting on the correct gamma matrix:  1  f1 = Tr I γ + , 2  1  g1L = Tr I γ + γ 5 , 2  1  h1 = Tr I γ +i γ 5 . 2

(5.49a) (5.49b) (5.49c)

5.1.6 Gauge-Invariant Operator Definition A general Dirac field transforms under a non-Abelian gauge transformation as (see equation (3.47a)): 8(x) → ei g!

a (x)ta

8(x)

–i g!a (x)ta

8(x) → 8(x) e

(5.50a) .

(5.50b)

As a result, the quark correlator is not gauge-invariant:  I→

% 2 d4 z –ik⋅z 4 %% % –i g!a (z)ta i g!a (0)ta P e (z) e e 8 (0) %8 %P . ! " (20)4

But we know from equation (4.4d) that a Wilson line U(x ; y) transforms as U(x ; y) → ei g!

a (x)ta

U(x ; y) e–i g!

a (y)ta

.

Hence, the following definition for the quark correlator is gauge invariant:  def

I=

% 2 d4 z –ik⋅z 4 %% % P e (z) U 8 (0) %8 %P . ! (z ; 0) " (20)4

(5.51)

Note that the gauge transformation of U only depends on its endpoints. Although the latter are fully fixed by the quark correlator, there is still freedom in the choice of the path, influencing the result. The gauge-invariant correlator is thus path dependent, because of the path dependence of the underlying Wilson line. This will play a big role when working with the k⊥ -dependent correlator, which we will investigate further in Section 5.2. Although the requirement of gauge invariance for the correlator leaves the path unspecified, it is the precise development of factorization proofs that uniquely dictates

108

5 Gauge-Invariant Parton Densities

(a)

(b)

Figure 5.8: (a) The gauge invariant quark correlator function, with a cut Wilson line. (b) The Wilson lines inside the definition of the correlator account for the resummation of soft gluons.

which path should be used in the definition of PDF. For the integrated quark correlator the path is separated along the z– direction, as in equation (5.45), which leads to a straightforward Wilson line structure:4 

1 I(x) = 20

dz– e–ixP –i g

U(z– ; 0) = Pe

&z 0

+ z–

% 2 4 %   % % P %8" 0+ , z– , 0⊥ U(z– ; 0) 8! (0)% P ,

–

d+ A+ (0+ , +, 0⊥ )

(because n–⋅A = A+ ).

In the LC gauge, we have A+ = 0 and thus U – = 1, reducing the quark correlator to the definition in equation (5.45). As long as one stays in the A+ = 0 gauge, it is perfectly valid to neglect the Wilson line inside the PDF. The line is a finite line, so using the transitivity property, we can split it at +∞ (see, e.g. equation (4.50)) and write it as †  – – U(z– ; 0) = U(+∞ ; z) U(+∞ ; 0) .

(5.52)

It is common to draw the Wilson line as a finite line being cut. Following the discussion on page 85, we know that the cut passes in between two semi-infinite lines, but we keep the representation of a cut finite line for convenience. This is illustrated in Figure 5.8. Remember from equation (4.103) that a quark dressed with a Wilson line can be considered an eikonal quark, essentially being a quark with soft and collinear gluon resummation. The physical interpretation for the quark correlator is nothing different: it represents all soft and collinear interactions between the struck quark and the proton. We inserted the Wilson line somewhat ad-hoc: we were looking for an object having the correct transformation properties to make the quark correlator gauge invariant, and the Wilson line happens to be such an object. It is however not so difficult to prove this in a more formal way, using the eikonal approximation. Consider the diagram in Figure 5.9, where one soft gluon before the cut connects the struck quark with the blob. The hadronic tensor is then (see also equation (5.46)):

4 In the context of PDF, Wilson lines are commonly called gauge links. We won’t use this terminology in this book.

5.1 Revision of Deep Inelastic Scattering

p−

109

p

k−

Figure 5.9: A first order correction to the PDF.

W

,-

∼



1 e2q Tr 2

q

 IA1 (k, k – ) γ , γ + γ 1

p/ – / + m γ(p – )2 – m2 + i:

,

where the quark–quark–gluon correlator is given by IA (k, k – ) =

1 2



% 2 d4 z d4 u –ik⋅z –i⋅(u–z) 4 %% % P %8" (z) gA1 (u)8! (0)% P . e e 4 4 (20) (20)

Remember that we have an on-shell quark so that we can use the eikonal approximation. The γ + is what’s left of the real quark; after making the sum over polarization states, 

us (p)us (p) = p/ + m

→

p– γ + ,

(5.53)

so we can use γ + as though it were an u(p) on which to perform the eikonal approximation (as in equation (4.98)). Then we can make the approximation γ +γ 1

p/ – / + m –n1 ≈ γ+ . 2 2 n⋅ – i: (p – ) – m + i:

(5.54)

† This is indeed a Wilson line propagator.An important remark: the definition of U(+∞ ; z) also incorporates an exponential coming from the Feynman rule for the external point. + – This exponential has been extracted from U (it is e–ixP z ), but this remains valid by momentum conservation. The choice to extract the exponential from the Wilson line is by historic convention. It is straightforward to generalize this to any number of gluons, where gluons on the left of the cut will be associated with a line from 0 to +∞, and gluons on the right of the cut with a line from +∞ to z. In other words:

W ,- ∼

 q

 1  e2q Tr I1 (x) γ , γ + γ - , 2

where now the quark–quark–gluon correlator is resummed to all orders: 1 I= 20

 dz– e–ixP

+ z–

% 2 4 %   –† % % – P %8" 0+ , z– , 0⊥ U(+∞ ; z) U(+∞;0) 8! (0)% P .

(5.55)

110

5 Gauge-Invariant Parton Densities

This is indeed the anticipated result. Using equation (5.49), we can give a gaugeinvariant formulation of the unpolarized integrated quark parton density function: fq/p (x) =

1 40

 dz– e–ixP

+ z–

% 2 4 % % % –† + – P %8(z– ) U(+∞ γ U 8(0) %P , ; z) (+∞ ; 0)

(5.56)

where the subscript in fq/p is a common convention to denote “the integrated quark PDF for a quark with flavour q inside a proton”. But what about the gluon PDF? Until now, we totally ignored the possibility of the photon hitting a gluon inside the proton, because it is a higher order interaction. But while we are moving towards a more realistic approach of QCD, we cannot ignore gluon densities any further. A photon can hit a gluon by interchanging a quark. This is the boson–gluon fusion process mentioned before, and is illustrated in Figure 5.10. To construct the integrated gluon PDF, we start in the LC gauge A+ = 0 such that we can ignore Wilson lines for now. There is a constraint equation on A– relating it to the transverse gauge field, implying that the latter are the only independent fields. Then following the same derivation as in Section 5.1.5, we find (see Ref. [145] for the original derivation): 1 fg/p (. ) = 20

 dz– . P+ e–i. P

+ z–

% 2 4 % % % P %Aia (z– )Aia (0)% P .

The factor . P+ is typical for fields with even-valued spin. To make this gauge invariant, we cannot simply insert a Wilson line as before, because the gauge fields transform with an extra derivative term. However, the gauge field density F ,- transforms without such a derivative. We can easily relate the two: a F,= ∂, Aa- – ∂- Aa, + gf abc Ab, Aca F+i = ∂+ Aai

⇒ Aai =

(A+ = 0) 1 F+i , ∂+

(5.57)

which we can use to redefine the gluon PDF. Inserting a Wilson line (in the adjoint representation, as it has to couple to gluons) gives our final result for the integrated gluon PDF:

X Figure 5.10: Boson–gluon fusion in DIS.

5.2 Semi-inclusive Deep Inelastic Scattering

fg/p (. ) =

1 20



% 2 dz– –i. P+ z– 4 %% +i b – A ba % +i a P e (z ) U F (0) %F %P . – (z ; 0) . P+

111

(5.58)

There are a few subtleties when dealing with gluon PDF (like the difference between the Weizsäcker–Williams and dipole gluon distributions, see Ref. [107]), but discussing these issues would lead us too far away from our main topics of interest.

5.2 Semi-inclusive Deep Inelastic Scattering Collinear factorization is a well explored and experimentally verified framework, but it only works when integrating out all final states. Keeping these final states, i.e. fully exclusive DIS, would maximally break collinear factorization. In this section, we investigate an intermediate solution, where we identify exactly one hadron in the final state, and integrate out all other states. This is called SIDIS. Because there is no restriction on the momentum of the final hadron, it can acquire a transversal part. To put it more formally: in DIS, we were able to describe our process on a plane, because it only has two independent directions, viz. the direction of the incoming proton (which is parallel to the incoming electron) and the direction of the outgoing electron. We have chosen a frame where the plus and minus components of the momenta span this plane, such that the transversal components are zero. In SIDIS, a third direction emerges from the momentum of the identified hadron, which doesn’t necessarily lie in the plane spanned by the incoming and outgoing electron. In this frame, the final hadron will have a non-zero transverse momentum component. As we will discover in this section, the breaking of collinear factorization is not an insurmountable task to overcome; we can adapt our factorization framework to allow for k⊥ -dependence, such that the convolution between the hard part and the PDF – now also dependent on k⊥ , and thus from now on called a TMD – is a convolution over k⊥ . In this book, we will not delve into the technicalities for k⊥ -factorization, as they are quite intricate and would lead us too far.

5.2.1 Conventions and Kinematics Different conventions exist in literature concerning the naming of the different TMD and azimuthal angles. We will follow the Trento conventions, as defined in Ref. [21]. Furthermore, concerning the labelling of momenta, we will follow the same convention as used in Ref. [43]. In a SIDIS process, we have an electron with momentum l that scatters of a proton with momentum P. The mediated photon has momentum q, and hits a parton with momentum k, that has momentum p after scattering (i.e. p = k + q). The struck parton then fragments into a hadron with momentum Ph . This is shown in Figure 5.11. Note that we now have two density functions; one that represents the probability to find a parton in the proton (the TMD), and one that

112

Electron

5 Gauge-Invariant Parton Densities

l′

l q

Ph

p

k Proton

X

P

Figure 5.11: Kinematics of

semi-inclusive deep inelastic electron–proton scattering.

represents the probability for a parton to fragment in a specific hadron (the fragmentation function [FF]). For simplicity, we assume the final hadron to be a spin 0 hadron, like a pion. We define a new Lorentz invariant z: z=

P⋅Ph . P⋅q

(5.59)

The value for z can be measured in experiment; it will approximate the fractional momentum of the detected hadron relative to its parent parton, in the same way x approximates the fractional momentum of the struck quark relative to the parent proton. Intuitively, we can add an FF Dq (z) to equation (5.37), giving a PM collinear estimate for F2 in SIDIS:  e2q x f q (x) Dq (z) , (5.60) F2PM = q

which gives us using equation (5.33) a first estimate for the SIDIS cross section:  d3 3 40!2 s y2  2 q eq x f (x) Dq (z) . (5.61) ≈ 1 – y + dx dy dz Q4 2 q Another important variable is the azimuthal angle 6h , which is defined as cos 6h = –

ˆl⋅Ph , |Ph⊥ |

where |Ph⊥ | is the length of the transversal component of the momentum of the outgoing hadron: 9 , |Ph⊥ | = –g⊥ ,- Ph Ph- . The geometrical construction of the azimuthal angle is shown in Figure 5.12. We can now construct the cross section: !2 d6 3 = L,- W ,- , dx dy dz d6h dP2h⊥ 2z x s Q2 where we approximated d3 Ph ≈ dz d2 Ph⊥

Eh . z

(5.62)

5.2 Semi-inclusive Deep Inelastic Scattering

Ph

113

l l′

Ph

Lepton

plane

Tran sve rsal plan e

ϕh

Lepton

plane

e

n n pla

Hadro

Figure 5.12: In the rest frame of the proton, Ph⊥ is the projection of Ph onto the plane perpendicular to the photon momentum. The azimuthal angle 6h is the angle between Ph⊥ and the lepton plane.

5.2.2 Structure Functions The hadronic tensor is defined as W ,- = 40 3 N

=

1 40

  

% % % 3 5 % 35 $(4) (P + q – pX – Ph ) P %J †, (0)% X, Ph X, Ph %J - (0)% P ,

X

% 35 % % 3 5 % d4 r eiq⋅r P %J †, (r)% Ph Ph %J - (0)% P .

(5.63)

As we will see in Section 5.2.3, this is a bit simplistic as we cannot integrate out the X states without affecting Ph , but the general idea is correct. Note that because we do not integrate over Ph (we measure it in the final state), we cannot drop the state |Ph Ph |. This leads to an important difference as compared to the hadronic tensor in DIS, viz. that we cannot naively impose the same constraints as before, because time-reversal invariance isn’t automatically satisfied. We can restore this invariance by changing it slightly, namely we require invariance under the simultaneous reversal of time and of initial and final states. For the parametrization of the hadronic tensor, we use the same orthonormal basis as before, but now we have an additional physical vector at our disposal, which we can use to construct the fourth basis vector: ,-

Ph N g hˆ , = ⊥ . |Ph⊥ |

(5.64)

114

5 Gauge-Invariant Parton Densities

hˆ , is a space-like unit vector: hˆ , hˆ , = –1 .

(5.65)

Watch out, as although we normalized this vector, it is not fully orthogonal! We have as expected ˆ tˆ = 0 , h⋅

ˆ qˆ = 0 , h⋅

but it is not orthogonal to ˆl, : ˆ ˆl = cos 6h . h⋅

(5.66)

This is a deliberate choice, because now we have the azimuthal dependence hardcoded inside our new basis. Note that ,

– ˆl⊥ %⊥ ,- hˆ - = sin 6h ,

(5.67)

which implies that 6h is fully defined in the region 0 . . . 20. We can parameterize W ,in the same way as we did before, now with hˆ added. This gives (for the unpolarized case): W ,- =

z  ,cos 6 –g⊥ FUU,T + tˆ, tˆ- FUU,L + 2tˆ(, hˆ -) FUU h x    cos 26 sin 6 ,+ 2hˆ , hˆ - + g⊥ FUU h – 2itˆ[, hˆ -] FLU h

(5.68)

The subscript UU denotes a structure function for an unpolarized beam on an unpolarized target, while the labelling in function of 6h will be motivated by contracting with the lepton tensor from equation (5.17): L,- W ,- =

4zs y

 1–y+

y2 2

FUU,T +



cos 6h

1 – y(2 – y) cos 6h FUU cos 26h

+ (1 – y)FUU,L + (1 – y) cos 26h FUU

  sin 6 + + y 1 – y sin 6h FLU h .

cos 6

sin 6

As anticipated, FUU h has a factor cos 6h in front, and so on. Note that FLU h is the structure function for a longitudinally polarized lepton beam (on an unpolarized proton target), which is confirmed by the factor + in front (originating from the last term in the lepton tensor equation (5.17)). The cross section is then given by equation (5.62): d6 3 2!2 = 2 dx dy dz d6h dPh⊥ x y Q2

 1–y+

y2 FUU,T + (1 – y)FUU,L 2

5.2 Semi-inclusive Deep Inelastic Scattering

115

 sin 6 cos 26 + + y 1 – y sin 6h FLU h + (1 – y) cos 26h FUU h   cos 6 + 1 – y(2 – y) cos 6h FUU h ,   d3 3 40!2 y2 ! ! F + (1–y) F = 1–y + UU,T UU,L , dx dy dz x y Q2 2

(5.69a) (5.69b)

where we integrated over Ph⊥ in the last step, which got rid of the 6h -dependence. The tilde structure functions are the integrated versions:  ! FUU,T (x, z, Q2 ) = d2 Ph⊥ FUU,T (x, z, Q2 , Ph⊥ ) , (5.70) and similarly for ! FUU,L . From the logical demand,  h

dz z

d3 3SIDIS d2 3DIS ≡ , dx dy dz dx dy

we can relate the SIDIS structure functions to the DIS structure functions:      dz z ! FUU,T x, z, Q2 ≡ FT x, Q2 , h



    dz z ! FUU,L x, z, Q2 ≡ FL x, Q2 .

(5.71a) (5.71b)

h

5.2.3 Transverse Momentum-Dependent PDFs We can construct the diagram for the hadronic tensor following the same step-by-step procedure we used in DIS (see Section 5.1.5), this time adding an FF, as is illustrated in Figure 5.13. Remember that the amplitude for extracting a quark from a proton with momentum P is 8! (0)|P . Then, the amplitude for a quark fragmenting in a hadron with momentum Ph is of course Ph |8! (0) . Ph ν k

p

Ph p

μ k Figure 5.13: Leading order diagram for the hadronic tensor

in SIDIS.

116

5 Gauge-Invariant Parton Densities

So we simply have ν k X

=



γ-

% 3 "! 5 % X %8! (0)% P ,

Ph ν

p Y

k X

=

% 2  5 % % 4 % 3 "! % % X %8! (0)% P . Y, Ph %8" (0)% 0 γ -

The QED-vertex adds a $-function, and making the final-state cut adds two final-state sums (using the notation defined in equation (5.40)) and two $-functions: W ,- =

     1 2 eq d4 kd4 p $(4) (P–k–pX ) $(4) (Ph +pY –p) $(4) (k+q–p) 2 q X Y % % 2 5 % % 3 4 % % 2 5 % % 34 % % % % × P %8% X γ , 0 %8% Y, Ph Y, Ph %8% 0 γ - X %8% P .

Next, we will separate the proton content from the fragmentating hadron content, applying on each the same steps as before (expressing the $-function as an exponential, using the translation operator and the completeness relation). Then we get the general leading order result:    1 2 W = e d4 kd4 p $(4) (k+q–p) tr I(k, P)γ , B(p, Ph )γ - , 2 q q  % 2 d4 r –ik⋅r 4 %% % I!" (k, P) = P e (r)8 (0) %8 %P , ! " 160 4  % 2 % 3 4 %% d4 r –ip⋅r 5 %% % Ph Ph %8 (r)%% 0 . 0 8 B!" (p, Ph ) = e (0) ! " 160 4 ,-

(5.72a)

(5.72b) (5.72c)

Next, we choose a frame where the parton in the TMD carries a fraction . of the proton’s plus momentum, and where the final hadron carries a fraction & of the fragmentating parton’s minus momentum, i.e.

k2 + k2⊥ k = .P , , k⊥ 2 . P+ ,

+

,

 2 p + p2⊥ Ph– p = & , , p⊥ , 2Ph– & ,

(5.73)

5.2 Semi-inclusive Deep Inelastic Scattering

such that we can write (neglecting terms that are

1 Q

117

suppressed):

  $(4) (k + q – p) ≈ $ k+ + q+ $ (q– – p– ) $(2) (k⊥ + q⊥ – p⊥ ) ,  1 1 1 (2) ≈ + – $ (. – x) $ – $ ( k ⊥ + q ⊥ – p⊥ ) . P Ph & z We transform the integral measures as d4 k = P+ d. dk– d2 k⊥ ,

d4 p = dp+ d&

Ph– 2 d p⊥ . &2

Then we can rewrite the hadronic tensor as     ,e2q d2 k⊥ ztr I (x, k⊥ , P) γ , B (z, k⊥+q⊥ , Ph ) γ - , W =

(5.74)

q

where we defined the k⊥ -dependent correlators as 

% 2 d3 r –ixP+ r– +ik⊥ ⋅r⊥ 4 %% % + – P e , r , r )8(0) %8(0 %P , ⊥ 80 3  3 – % 2 % 3 4 %% + – 1 d r –i Ph r+ +ik⊥ ⋅r⊥ 5 %% N % Ph Ph %8(r , 0 , r ⊥ )%% 0 . z 0 8(0) B (z, p⊥ , Ph ) = e 2z 80 3 N

I (. , k ⊥ , P ) =

(5.75a) (5.75b)

We can parameterize the quark correlator and fragmentator functions in terms of TMD and FF, precisely as we did with the quark correlator in the case of DIS. Keeping only the contributions at leading-twist, we obtain the following unpolarized TMD and FF [20]: i 1 k/ ⊥ – I (. , k⊥ ) = f1 (. , k⊥ ) γ – + h⊥1 (. , k⊥ ) γ , 2 2 mp i 1 k/ ⊥ + B (& , k⊥ ) = D1 (& , k⊥ ) γ + + H1⊥ (& , k⊥ ) γ . 2 2 mh

(5.76a) (5.76b)

The correlator is built from the unpolarized TMD f1 (. , k⊥ ) and the so-called Boer– Mulders TMD h⊥1 (. , k⊥ ). The fragmentator is built from the unpolarized TMD FF D1 (& , k⊥ ) and the so-called Collins function H1⊥ (& , k⊥ ). If we plug this result in equations (5.62) and (5.69b), and use the approximation q⊥ ≈ –

Ph⊥ , z

(5.77)

we get the factorization formula for the unpolarized transversal structure function in SIDIS: FUU,T =

 q

q

q

e2q x f1 ⊗ D1 ,

(5.78)

118

5 Gauge-Invariant Parton Densities

where we defined the convolution over transverse momentum as   1 q q q q 2 2 (2) f1 ⊗ D 1 = d k ⊥ d p ⊥ $ k⊥ –p⊥ – Ph⊥ f1 (x, k⊥ ) D1 (z, p⊥ ) , z

(5.79)

which is a regular convolution with “open” variable z1 Ph⊥ . Other structure functions arise when convoluting polarized TMD (which arise when the target hadron is polarized), which we don’t treat in this book. This factorization formula is however not yet fully rigorous, and we will improve it in Section 5.3.

5.2.4 Gauge-Invariant Definition for TMDs Just as was the case in the previous section for DIS, our TMD and FF defined so far (equations (5.75)) are not gauge invariant, and are only valid in the LC gauge A+ = 0. Gauge invariance can be restored by inserting a Wilson line:  I (. , k⊥ , P) =

% 2 d3 r –i. P+ r– +ik⊥ ⋅r⊥ 4 %% % P e U 8(0) %8(r) %P , (r ; 0) 80 3

(5.80)

where now the space–time point separation no longer lies on the LC, i.e. the Wilson line has to connect the point (0+ , 0– , 0⊥ ) with the point (0– , r+ , r ⊥ ). But a Wilson line is path dependent, implying that different path choices give different results. How do we choose a path, or at least motivate our choice? Just as was the case for the collinear PDF, the gauge invariance requirement doesn’t put any constraints on the path. The correct path can however be retrieved by explicit calculations of a full process. Different processes might require different paths, that can be quite complex (see, e.g. Refs [18, 60, 61]). This is an active topic of interesting research these days, as it is intimately bound to the validity of TMD factorization. In the collinear case, we were able to interpret the Wilson line as a colour rotation on the quark, making it an eikonal quark. We then split the Wilson line into two parts at infinity. This splitting had two advantages, first that we could associate a line with the quarks on each side of the cut diagram separately, and second that we could use easy Feynman rules. In the TMD definition for unpolarized SIDIS, we can do something analogous. We add a light-like line to each quark:   + – – U(+∞ – , 0 ; 0– , 0 ) 8 0 , 0 , 0 ⊥ , ⊥ ⊥  –†  8 0+ , r– , r ⊥ U(+∞ – , r ; r– , r ) . ⊥ ⊥

But now we have because of the transverse separation –† – U(r ; 0) ≠ U(+∞ – , r ; r– , r ) U(+∞– , 0 ; 0– , 0 ) . ⊥ ⊥ ⊥ ⊥

(5.81a) (5.81b)

5.2 Semi-inclusive Deep Inelastic Scattering

119

So we need a Wilson line to connect the transverse “gap”, i.e. –† ⊥ – U(r ; 0) = U(+∞ – ; r– ) U(r ; 0 ) U(+∞– ; 0– ) . ⊥ ⊥

We will split this line at +∞⊥ for the same reasons as before. Adding this to equations (5.81) gives   + – ⊥ – U(+∞ – , +∞ ; +∞– , 0 ) U(+∞– , 0 ; 0– , 0 ) 8 0 , 0 , 0⊥ , ⊥ ⊥ ⊥ ⊥  –†  ⊥† 8 0+ , r– , r ⊥ U(+∞ – , r ; r– , r ) U(+∞– , +∞ ; +∞– , r ) , ⊥ ⊥ ⊥ ⊥

(5.82a) (5.82b)

leading to the final definition for the gauge-invariant TMD quark correlator:  I=

% 2 d3 r –ixP+ r– +ik⊥ ⋅r⊥ 4 %% % !† ! P e U U 8(0) %8(r) %P , (+∞ ; r) (+∞ ; 0) 80 3

(5.83a)

⊥ – U!(+∞ ; 0) = U(+∞ – , +∞ ; +∞– , 0 ) U(+∞– , 0⊥ ; 0– , 0⊥ ) , ⊥ ⊥

(5.83b)

† –† ⊥† U!(+∞ ; r) = U(+∞– , r ⊥ ; r– , r ⊥ ) U(+∞– , +∞⊥ ; +∞– , r ⊥ ) .

(5.83c)

What about the physical interpretation? Consider again the 1-gluon exchange as depicted in Figure 5.9. We saw in equation (5.54) that the net contribution for a soft or collinear gluon is a factor  g

p/ – / d4  A/ ≈ –g 160 4 (p – )2 + i%



d4  A– , 160 4 – – i%

(5.84)

,

p the direction of the where , is the momentum of the exchanged photon and n, = |p| outgoing quark. We were able to make this simplification because in the correlator this correction stands to the right of a factor u(p), such that we can make use of the fact u(p)p/ = 0:

  u(p)A/ p/ = u(p) A/ p/ + p/ A/ = 2u(p)p⋅A .   because when , is collinear to p, , also u p/ – / = 0. As we saw before, the contribution in equation (5.84) calculated to all orders leads to the light-like Wilson line. In the collinear case, this was the end of the story. But now that we are in the TMD case, we cannot simply take the exchanged gluon to be collinear, instead we get an additional contribution:    / ⊥ / ⊥ p/ – / d4  d4  A– d4  A / g A (5.85) ≈ –g + g 160 4 (p – )2 + i% 160 4 – – i% 160 4 2p ⋅  + 2⊥ – i% The first term will again induce a resummation of collinear gluons, i.e. a light-like Wilson line, but the second term is something new. To understand what this term means, we will try to simplify it further (following the same steps as in Ref. [35]). First of all, we exponentiate it using the integral representation from equations (C.10):

120

5 Gauge-Invariant Parton Densities

 g

/⊥ /⊥ d4  A =g 160 4 2p⋅ + 2⊥ – i%



d4  160 4

∞

2

/ ⊥ / ⊥ e–i!(2p⋅+⊥ –i%) . d! A

(5.86)

0

Now using the fact that p ⋅  = p– + (the quark momentum lies fully along the minus direction), the + and – integrations are just a Fourier transform of A, , putting the field at A, (z+ = 0+, z– = 2!p–, ⊥ ). We see that in the IMF limit p– → ∞, the field values become   A, z+ = 0+, z– = ∞–, ⊥ ,

(5.87)

and the dependence on ! disappears. The !-integration is then trivial, giving  g

d2 ⊥ / / ⊥ (0+, ∞–, ⊥ ) 2 ⊥ . A 40 2 ⊥ – i%

(5.88)

Now it is important to realize that because the field strength vanishes, A, has to be a pure gauge,5 such that we can write it as a gradient potential:   Ai 0+, ∞–, z⊥ = ∇i 6 (z⊥ ) ,

(5.89)

which has a mixed Fourier transform:   Ai 0+, ∞–, ⊥ = ii 6 (⊥ ) .

(5.90)

This allows us to further simplify the integral, using /⊥ /⊥ = –2⊥ :  – ig

d2 ⊥ 2 6 (⊥ ) 2 ⊥ = –ig 6 (z⊥ = 0⊥ ) . 2 40 ⊥ – i%

(5.91)

Using equation (5.89), we can write equation (5.91) as a line integral: ∞ – ig 6 (0⊥ ) = ig

  d2 z⊥ ⋅A⊥ 0+, ∞–, z⊥ .

(5.92)

0

This is exactly the first-order term of a Wilson line on a random path that lies in the transverse plane at z+ = 0+ , z– = ∞– . To see how the higher orders resum into a Wilson line, we investigate the n-gluon case. We can use the same steps up to equation (5.88), such that the transversal part of a general n-gluon contribution is given by

5 A pure gauge is a field, that can be written as a pure gauge transform: A, = U † ∂, U . Hence, for a pure gauge field, we can always choose a gauge in which the field vanishes.

5.2 Semi-inclusive Deep Inelastic Scattering

 gn

d2 1⊥ 40 2



d2 2⊥ ... 40 2



d2 n⊥ / / ⊥ (1⊥ ) 2 1⊥ A 40 2 1⊥ – i% n /

A/ ⊥ (2⊥ )

121

/1⊥ + / 2⊥ (1⊥ + 2⊥ )2 – i%

. . . A/⊥ (n⊥ )  n /

i=1

i⊥

/i⊥ 2

.

(5.93)

– i%

i=1

But now we cannot use the same trick as in equation (5.91) because of the summations of momenta. To solve this, we move to coordinate space, using i = –20i 2⊥



d2 z⊥ i⊥ ⋅z⊥ zi e = –20i 40 2 z2⊥



d2 z⊥ i⊥ ⋅z⊥ e ∇i ln |z⊥ | , 40 2

(5.94)

where an implicit mass parameter is assumed to make the logarithm dimensionless. Putting this formulation in equation 5.93 and doing the momentum integrals (which just result in a chaining of $-functions), we get: 

n

(–ig )n i=1

d2 zi⊥  n A/⊥ (z1⊥ )∇ / 1 ln |z1⊥ – z2⊥ | ⋯ 40 2 A/⊥ (z(n–1)⊥ )∇ / n ln |zn⊥ | . / n–1 ln |z(n–1)⊥ – zn⊥ |A/⊥ (zn⊥ )∇

(5.95)

Next, we replace all A⊥ by 6, integrate by parts to move the two derivatives to the logarithm, and use ∇ / 12 ln |z1⊥ – z2⊥ | = –∇12 ln |z1⊥ – z2⊥ | = –20 $(2) (z1⊥ – z2⊥ ) .

(5.96)

This gives finally (–ig )n

1 n 6 (0⊥ ) . n!

(5.97)

We can write this as a product of line integral with change borders, cancelling the factor n!: 1 (–ig )n 6n (0⊥ ) = (ig )n n!

∞ 0

z(n–1)⊥ z1⊥  2 2 d z1⊥ ⋅A⊥ (z1⊥ ) d z2⊥ ⋅A⊥ (z2⊥ ) ⋯ d2 zn⊥ ⋅A⊥ (zn⊥ ) , 0

0

(5.98) which is indeed the all-order resummation of a Wilson line on a random path that lies in the transverse plane at z+ = 0+ , z– = ∞– . So in the end, inside the TMD, we have both a resummation of collinear gluons – coming from the line parts U – and U –† – and a resummation of soft transversal gluons, coming from the U! and U!† parts. Note, however, that by choosing an appropriate

122

5 Gauge-Invariant Parton Densities

(0+,+∞−,+∞ )

n n− (0+,r –,r )

(0+,0–,0 )

(0+,r –,r )

(0+,+ ∞−,0 )

(0+,0–,0 )

Figure 5.14: Structure of the Wilson lines in the TMD definition.

(a)

(b)

Figure 5.15: (a) In SIDIS, the longitudinal Wilson line inside the fragmentation function represents a resummation of soft and collinear gluons connected to the incoming quark. (b) In Drell–Yan, the longitudinal Wilson line inside one TMD represents a resummation of soft and collinear gluons connected to the parton extracted from the other TMD.

gauge, it is possible to cancel the contribution of one type of these lines, e.g. in the LC gauge only the transversal parts remain (with advanced or retarded prescriptions the LC gauge can cancel the transverse segment as well). Of course, the same reasoning can be repeated for the FF, but then the light-like Wilson lines will lie in the plus direction. This is illustrated in Figure 5.15a. To end this section, we give an example for the use of Wilson lines in the Drell– Yan process. In this setup, two protons (or a proton and an antiproton) are collided and create a photon or weak boson by quark–antiquark annihilation. We thus need two TMDs, which are both in the initial state. The longitudinal part of the Wilson line used to make the TMD gauge invariant represents a resummation of gluons connected to the parton struck from the other TMD. This is illustrated in Figure 5.15b. Because of the fact that the Wilson line now represents initial state radiation, the line structure will be different. More specifically, the path will flow towards –∞ before returning, as shown in Figure 5.16. This has an important consequence: two out of the eight (unpolarized and polarized) TMD are T-odd and will have a sign change with this line structure as compared to SIDIS (the Boer–Mulders TMD from equation (5.76) is an example). This would imply that TMDs are process dependent, and not universal as they ought to be. So far it is not experimentally verified whether these

5.3 Evolution of TMDs

123

T-odd TMD have a non-zero value, although there is a growing amount of evidence that they exist (in particular the so-called Sivers function). They are not universal, but they are manageable because the non-universality can usually be calculated (like the sign change in DY w.r.t. SIDIS). The latter is however not yet confirmed by experiment.

5.3 Evolution of TMDs One thing that remains to be defined before we can really use TMD in experiment, is their evolution. In the collinear case, the evolution of PDF is fully governed by the DGLAP equations. To do something similar for TMD, we first have to take a new look at our crude factorization formula in equation (5.78), because we will have to adapt it due to the singularity structure of the TMD involved. As the Wilson line structure is now such that the light-like lines do not overlap (see Figure 5.14), we are left with overlapping LC divergences (originating from the double pole in equation (6.72)) and additional IR divergences. The latter can be managed by extracting all soft contributions into a so-called soft factor S. It is typically calculated as a vev of the complex square of two Wilson lines with one cusp. This is illustrated in Figure 5.17.

(0+,+∞−,+∞ )

(0+,r –,r )

(0+,+ –∞−,0 )

(0+,0–,0

Figure 5.16: Structure of Wilson lines in the DY

TMD definition. Because of the initial-state interactions in DY, the direction of the Wilson line is reversed, going now towards –∞.

)

ñ

ñ

n

n

soft limit

Figure 5.17: In the soft limit, all possible soft contributions between the quark legs are resummed into on-LC Wilson line segments. This gives rise to the soft function as depicted on the right.

124

5 Gauge-Invariant Parton Densities

On the other hand, the LC divergences can be managed in different ways. The most common way is to regulate the “on-LC-ness” of the LC Wilson line segment by slightly putting it off-LC and introducing a regulator & :6 def

&=

k⋅n , |n|

(5.99)

that measures how much off-LC the segment is. As always, k is the momentum of the struck parton before the interaction, and n is the direction of the Wilson line segment (the same direction as the struck quark after the interaction). The on-LC segment is retrieved in the limit & → ∞ (which is equivalent to n2 → 0). A similar regulator is introduced for the LC-divergences in the FF: def

&h =

p⋅! n , |! n|

(5.100)

These two regulators will act as rapidity cut-offs, softening the overlapping divergences. The TMD and FF then gain an extra dependence on & resp. &h :   f x, k⊥ , & , ,2 ,

  D z, p⊥ , &h , ,2 .

(5.101)

Now we can give a rigorous definition for the factorization of the SIDIS cross section: %2 ∂ 2 3γ ∗ p %% ∼ H(,2F )% 2 ∂z∂q⊥

 d2 k⊥ d2 p⊥ d2 l⊥ $(2) (k⊥ –p⊥ +l⊥ +q⊥ ) ×



      e2q f˜ q x, k⊥ , & , ,2F D˜ q z, p⊥ , &h , ,2F S l⊥ , ,2F .

(5.102)

q

This is illustrated in Figure 5.18. The hard part is perturbatively calculable and normalized, % % %H(,2 )%2 = 1 + O (!s ) , F

(5.103)

and does not contain real radiation.This factorization formula is only valid at small transverse momenta (of the order of the proton mass or smaller), and has been proven at one loop and leading twist.7 The factorization formula can be simplified by Fourier transforming into impact parameter space, as the $-function makes that all densities depend on the same impact parameter b⊥ . Furthermore, it is convenient to redefine the TMD and FF in order to absorb the soft function: 6 Not to be confused with the & momentum fraction in Ph– = & p– , that is integrated over in the FF. 7 In the regime with large transversal momenta, regular collinear factorization can be applied.

5.3 Evolution of TMDs

125

D H

H

S

Figure 5.18: Factorization in SIDIS: the bull diagram. All

f

IR divergences are absorbed in the soft factor S, that hence only interacts with the TMD and FF. Note that there is no real radiation coming from the hard part.

  def q     f q x, b⊥ , & , ,2F = Rf b⊥ , ,2F f˜ q x, b⊥ , & , ,2F ,   def q     Dq z, b⊥ , &h , ,2F = RD b⊥ , ,2F D˜ q z, b⊥ , &h , ,2F . q

(5.104a) (5.104b)

q

where Rf and RD are two unspecified functions that combine into the soft function S: q

q

Rf ⋅ RD = S. There are some subtleties in doing this – as one also has to divide out all self-energy contributions – but we don’t delve into these technicalities. The factorization formula is then:       ∂ 2 3γ ∗ p %%  2 %%2 ∼ H , e2q f q x, b⊥ , & , ,2F Dq z, b⊥ , &h , ,2F . (5.105) d2 b⊥ e–ib⊥ ⋅q⊥ F 2 ∂z∂q⊥ q We can now derive evolution equations in a similar way as in the DGLAP case, viz. by demanding that the cross section is independent on the factorization scale ,F and the rapidity cut-offs & and &h . It can then be shown that ∂ ln f (x, b, & , ,) = K(b, ,) , ∂ln & ∂ ln f (x, b, & , ,) = γF (& , ,) , ∂ln ,

(5.106a) (5.106b)

which are known as the CS evolution equations. The same equations apply for D as well (with & replaced by &h ), as for any TMD or FF. K(b, ,) is the CS kernel, and γF is the anomalous dimension of the TMD. The kernel is perturbatively calculable but only for small q⊥ (but this is something we don’t mind, as at large q⊥ TMD-factorization is broken anyway). The kernel and the anomalous dimension satisfy the following RGE: ∂K(b, ,) ∂γF (& , ,) = = γK (!s ) , ∂ln , ∂ln &

(5.107)

where γK is the anomalous dimension of the kernel. It is easily calculated at one-loop to equal:

126

5 Gauge-Invariant Parton Densities

γK = –2

  !s CF + O !2s . 0

(5.108)

We will show in the later chapters that there is an easy relation between the anomalous dimension of the kernel and the cusp anomalous dimension: γK ≡ –2Acusp ,

(5.109)

where the factor 2 arises from the fact that there are 2 cusps in the squared amplitude (between the incoming quark and the outgoing Wilson line). We can solve the CS evolution equations for f : f (& , b, ,) = f (&0 , b, ,0 ) e

,   & & & K(b,,0 ) ln & + d ln ,′ γF (& ,,′ )–γK (,′ ) ln ′ , 0 , 0

,

(5.110)

where the exponential factor evolves the TMD from (&0 , ,0 ) to (& , ,). 5.3.1 About the Rapidity Cut-offs Before we end this chapter, we make a few remarks on the rapidity cut-offs. First, we note that they have a significant physical meaning, namely they disentangle different gluon contributions and sort them in the right density, the TMD or the FF. Let us illustrate this a bit more elaborately. Suppose the incoming quark is in the LC plus direction, then the outgoing quark is in the LC minus direction. Naively, we would assume that all radiation from the incoming quark goes into the TMD, and all radiation from the outgoing quark goes into the FF:

But this is of course not true, as the Wilson line in the TMD is in the LC minus direction, and resums gluons that are collinear to the incoming quark:

127

5.3 Evolution of TMDs

But a gluon that is radiated from the incoming quark ánd collinear to the incoming quark, is absorbed by the TMD’s evolution and hence also enters the TMD:

So we conclude that it doesn’t matter from which line the gluon is radiated, but that it is its direction that matters, i.e. collinear to the incoming quark. In other words, all gluons radiated in the LC plus direction go into the TMD. A similar reasoning can be applied to the FF, such that all gluons radiated in the LC minus direction go into the FF. To know what to do with gluons that are not collinearly radiated, we look at their rapidity Y. The LC plus direction is associated with Y = +∞ and the LC minus direction with Y = –∞, so it is tempting to absorb all gluons with positive Y into the TMD and all gluons with negative Y into the FF. But then we are absorbing too much, because gluons with small rapidity (positive or negative) are hard, and have to remain in the hard part. This is exactly what & and &h do, as they separate these different regions:

Y < &h → FF ,

(5.111a)

&h < Y < & → H ,

(5.111b)

& < Y → TMD ,

(5.111c)

from which also naturally follows & &h ≡ Q2 ,

(5.112)

because hard gluons are parameterized by Y ≤ Q. We can show this pictorially: FF

H

TMD Y

ζh

ζ

Another remark about the rapidity cut-offs is that – as we mentioned when introducing them – there are different ways to treat the LC divergences. The main reason why one would like to use a different method is that calculations with off-LC segments are much more involving than with on-LC segments (as we experienced ourselves when calculating the results in equations (6.72) and (6.82)).

128

5 Gauge-Invariant Parton Densities

One method is to adapt the renormalization procedure to subtract the double pole as well, as is commonly done with Wilson loops. However, as this has to be done consistently on all contributions of the TMD, it quickly becomes quite cumbersome. Another method – which is quite recent – is to extract besides the soft factor a collinear factor from the squared amplitude as well. It will have a similar singularity structure, and all rapidity divergences cancel out when combining diagrams.

6 Simplifying Wilson Line Calculations In Chapter 4, we deeply investigated piecewise linear Wilson lines. We derived Feynman rules for the eight different possible linear topologies (four types plus their path reversals) and discovered how to relate them to each other. Finally, by collecting a set of diagrams per blob, we wish to connect to a given Wilson line; we were able to develop a framework to significantly simplify calculations if the Wilson line has a lot of segments. Moreover, and this was the most important result, from the moment, we have made a calculation with a given blob, we can easily port the result to any piecewise linear Wilson line. However, there is a fly in the ointment. Calculating these general integrals is quite complicated, because we have to keep the path constants (nJ, , r,J , and 6J ) as general as possible, while normally in a calculation, these are fixed in such a way that the integral simplifies a lot (e.g. n2 = 0, n⊥ = 0). This chapter is devoted to a few tricks that can help in simplifying the calculations encountered with piecewise linear Wilson lines. In Section 6.1, we develop an advanced technique to simplify any product or trace of fundamental Lie generators. As we separated out the colour structure in the previous chapter, this section can be a big help in calculating these factors. In Section 6.2, we show how to properly define the contents of the blobs introduced in the previous chapter. We do this by using the example of self-interaction blobs – which contain no external lines and hence represent interactions between the Wilson line and itself – for a 2-gluon blob and a 3-gluon blob. In Section 6.3, we explore the validity of a Wick rotation in the context of Wilson lines and adapt the formulation where necessary. And finally, in Section 6.4, we make the first steps to calculate a common general Wilson line integral and narrow it down to an easy example, a 1-gluon cusp correction, which we fully calculate for general segment directions, be it purely light-like, transversal, or mixed.

6.1 Advanced Colour Algebra The first step of any calculation in the piecewise framework is the calculation of path functions (see, e.g. equations (4.82)). These functions contain a colour part and a part related to the line structure. The colour part is given by (see equation (4.73a)): c = tan . . . ta1 ca1 ... an . The n-generator product is simply the colour structure of the nth-order expansion term of the Wilson line, while the factor ca1 ... an is the colour structure of the blob. Let’s see if we can somehow simplify this term.

130

6 Simplifying Wilson Line Calculations

6.1.1 Calculating Products of Fundamental Generators Similarly to what we did with the Fierz identity in equation (B.77), any product of colour generators can be written as a linear combination of the identity operator and the generators, because the latter spans the full product space. In other words: ta1 ta2 . . . tan = Aa1 a2 ... an 1 + Ba1 a2 ... an b tb . As Aa1 a2 ... an and Ba1 a2 ... an b are just coefficients, tracing the rhs simply gives Aa1 a2 ... an tr (1), because the single generator after Ba1 a2 ... an b is traceless. In a similar way, we can recover Ba1 a2 ... an b by multiplying the product with a generator before making the trace, that is   tr ta1 ta2 . . . tan = Nc Aa1 a2 ... an ,   1 tr ta1 ta2 . . . tan tb = Ba1 a2 ... an b . 2 But the lhs of the latter can also be calculated as one order higher, i.e. ta1 ta2 . . . tan tb = Aa1 a2 ... an b 1 + Ba1 a2 ... an bc tc , giving   tr ta1 ta2 . . . tan tb = Nc Aa1 a2 ... an b , ⇒ Aa1 a2 ... an ≡

1 a1 a2 ... an B . 2Nc

Only one of these is linearly independent. We will adopt the notation Ca1 a2 ... an = Nc Aa1 a2 ... an , N

(6.1)

with C standing for “colour factor”. Note the difference with equation (4.72), i.e. a capital C is the colour factor that represents an n-generator product, while a lower case c represents the colour structure of a given blob. We can now rewrite the first equation as follows:

1

ta1 . . . tan = Ca1 ... an + 2 Ca1 ... an b tb , Nc   Ca1 ... an = tr ta1 . . . tan .

(6.2a) (6.2b)

As the colour factor identically equals the trace, it naturally has the same properties, namely cyclicity and Hermiticity: Ca1 a2 ... an = Ca2 a3 ... an a1 = . . . , Ca1 a2 ... an = C

an ... a2 a1

.

(6.3a) (6.3b)

6.1 Advanced Colour Algebra

131

The first colour factors are straightforward to calculate: C0 = Nc , a

C = 0, 1 Cab = $ab , 2 1 Cabc = habc . 4

(6.4a) (6.4b) (6.4c) (6.4d)

To calculate higher orders, we use equation (B.74) to deduce a recursion formula for traces in the fundamental representation – and hence for colour factors – by applying it on the last two generators (the last two indices of a colour factor): Ca1 ... an =

$an–1 an a1 ... an–2 han–1 an b a1 ... an–2 b C + . C 2Nc 2

(6.5)

This gives for instance 1 ab cd 1 ab x x cd $ $ + h h , 4Nc 8  1 1  abc de h $ + $ab hcde + hab x hx c y hy de , = 8Nc 16 1 ab cd ef 1 ab x x c y y d z z ef = $ $ $ + h h h h 32 8Nc 2  1  ab x x cd ef h h $ + habc hdef + $ab hcd x hx ef . + 16Nc

Cabcd = Cabcde Cabcdef

One extremely useful observation is that inner summations only appear between consecutive hs, and never with a $. This allows us to define the following shorthand notations: N

$ = $ai ai+1 , N

h = hai ai+1 ai+2 , N

hh = hai ai+1 x hx ai+2 ai+3 , N

hhh = hai ai+1 x hx ai+2 y hy ai+3 ai+4 , ⋯ so that we can rewrite the former result as (note that the order of the $s and hs is significant because of the indices): 1 C2 = $ , 2 1 C3 = h , 4

(6.6a) (6.6b)

132

6 Simplifying Wilson Line Calculations

1 1 $$ + hh , 4Nc 8 1 1 C5 = ( h$ + $h ) + hhh , 8Nc 16  1 1 1  $$$ + hh $ + h h + $ hh + hhhh . C6 = 2 16Nc 32 8Nc

C4 =

(6.6c) (6.6d) (6.6e)

If we generalize this to an nth order trace, we get from equation (6.5): n

for n even : C

a1 ... an

=

2 –1 

1

i=0

2 2 +i Nc

n –i–1 2

n

all allowed $, h combinations , built from 2i hs

n–3

for n odd : C

a1 ... an

=

2 

i=0

1 2

n–1 n+1 +i 2 Nc 2 –i–1

(6.7a)

all allowed $, h combinations . (6.7b) built from 2i + 1 hs

where the $, h combinations need to have n open indices, using $

two open indices ,

h

three open indices ,

hh

four open indices ,

hhh

five open indices ,

etc. and where it is forbidden to put any $ or h in between two contracted hs. So for instance, h$h and hhh are not allowed. With this in mind, we can tackle any trace without the need for recursive calculations. For example, Ca1 ... a10 =

1 32Nc

$$$$$ + 4

1 64Nc 3



hh $$$ + $hh $$ + $$hh $ + $$$hh

+ h h $$ + h $ h $ + h $$ h + $ h h $ + $ h $ h + $$ h h ) +



1 128Nc

2

hhhh $ $

+ $ hhhh $ + $ $ hhhh + hhh h $ + hhh $h + $ hhh h + h hhh $+ + h $ hhh + $ h hhh + hh hh $ + hh $ hh + $ hh hh + hh h h + h hh h  1  + h h hh + hhhhhh $ + $hhhhhh + hhhhhh + h hhhhh 256Nc  1 hhhhhhhh . + hhhh hh + hh hhhh + hhh hhh + 512

6.1 Advanced Colour Algebra

133

This looks quite complex, but is in its essence not that difficult, as we only have six “types” of terms, depending on the number of hs in a certain term. The result will simplify drastically if there exist symmetries between the indices of Ca1 ... a10 . As a result from equations (6.5), we can use a trick to double check our result, namely that the total number of terms should equal the (n – 1)th Fibonacci number (counting 0 as the zeroth Fibonacci number). Indeed, for the tenth-order trace, we have 34 terms. Making contractions over the indices of one colour factor is straightforward using the formulae in the end of Appendix B.6. For example, Cabab =

1 ab ab 1 ab x x ab $ $ + h h . 4Nc 8

The last term can be calculated by using equation (B.86e): hab x hx ab = hab x hx ba = –

4 bb $ = –8CF , Nc

giving Cabab =

1  2  1 Nc – 1 – CF = – CF . 4Nc 2

(6.8)

Similarly, if we flip the last two indices of the colour factor, we have Cabba =

1 ab ba 1 ab x x ba $ $ + h h . 4Nc 8

This time the last term can be calculated by using equation (B.86d):   hab x hx ba = 4 Nc 2 – 2 CF , giving Cabba = Nc CF2 .

(6.9)

Because of the cyclicity, these two are the only independent fully contracted fourthorder colour factors. We will conclude this subsection with a list of properties for the colour factors. First, a listing of some common contractions of the first-to-fifth-order colour factors: Caa = Nc CF , 1 Caxxb = CF $ab , 2

Caba = 0 , Caxbx = –

1 ab $ , 4Nc

(6.10a) (6.10b)

134

6 Simplifying Wilson Line Calculations

1 Cabab = – CF , 2

Cabba = Nc CF2 ,

Cabcxx = CF Cabc ,

Cabxcx = –

Caxxyy = 0 ,

Caxyxy = 0 ,

C

axyyx

= 0,

Cabcxdx = –

C 1 abcd C , 2Nc

abcdxx

Cabxcdx

(6.10c)

1 abc C , 2Nc

(6.10d) (6.10e)

abcd

= CF C , 1 1 abcd = $ab $cd – C , 8 2Nc

(6.10f) (6.10g)

and the sixth-order colour factors: 1 Cabxxyy = CF2 $ab , 2 1 Cabxyyx = CF2 $ab , 2 1 ab $ , Caxbyxy = 8Nc 2 1 Caxxbyy = CF2 $ab , 2 1 ab Caxybyx = $ , 8Nc 2 CF Cabacbc = , 4Nc

CF ab $ , 4Nc CF ab =– $ , 4Nc CF ab =– $ , 4Nc

Cabxyxy = –

(6.11a)

Caxbxyy

(6.11b)

Caxbyyx

Caxybxy =

(6.11c)

1 Nc 2 +1 ab $ , 8 Nc 2

(6.11d)

Caabbcc = Nc CF3 ,

(6.11e)

1 Caabcbc = – CF2 , 2 Nc 2 + 1 Cabcabc = CF . 4Nc

Cabccba = Nc CF3 ,

(6.11f) (6.11g)

We also list some contractions between two colour factors: Cax Cxb = Caxy Cyxb

1 ab $ , 4 Nc 2 –2 ab = $ , 8Nc

Cab Cab = Cabc Cabc

Nc CF , 2 CF =– , 2

1 ab $ , 4Nc Nc 2 –2 = CF , 4

Caxy Cxyb = –

(6.12a)

Cabc Ccba

(6.12b)

and between a colour factor and a standard colour constant: Nc 1 , CF 2 Nc Nc 2 =i CF , 2

Cab ta tb = Cabc f abc

Cabcd f abc = 0 ,

Nc ab $ , 4 Nc Cabxy f xyc = i habc , 8 Nc 3 Cabcd f abx f xcd = – CF . 4 Caxy f xyb = i

(6.13a) (6.13b) (6.13c)

For contractions with longer colour factors, we can simply use the Fierz identities (see equation (B.76)). Expressed in function of colour factors, these are

6.1 Advanced Colour Algebra

1 1 a1 ... an C , Ca1 ... am xam+1 ... ap xap+1 ... an = Ca1 ... am ap+1 ... an Cam+1 ... ap – 2 2Nc Ca1 ... am xxam+1 ... an = CF Ca1 ... an , 1 1 a1 ... am am+1 ... an C C . Ca1 ... am x Cxam+1 ... an = Ca1 ... an – 2 2Nc

135

(6.14a) (6.14b) (6.14c)

6.1.2 Calculating Traces in the Adjoint Representation In the adjoint representation, equation (6.2a) is not true, because the set {1, T a } isn’t sufficient to reproduce all possible products of adjoint generators. Still, it would be useful to find a method to calculate adjoint traces. Unfortunately, this is not so trivial, as we have no useful expression for the anticommutation relations – which we need to get a recursion relation as in equation (6.5). Instead of a brute-force calculation, we will relate traces in the adjoint representation to trace in the fundamental using a nifty trick. First, note that in general, the product space of the fundamental and the anti-fundamental is isomorphic to the sum space of the adjoint and the identity: F ⊗ F ≃ A ⊕ 1,

(6.15)

from which we can derive (UA denotes “the group element U expressed in the adjoint representation”):   tr (UA ) = tr (UF ) tr UF – 1 .

(6.16)

For the trivial group element U = 1, we get indeed dA = d2F – 1. To calculate the nth order trace, it is sufficient to take U =

n $

a a

etR !i , expand it, and

i

compare terms of the same order in !i . Furthermore, we can use UF = UF† = UF–1 , which implies et

a1 a1 !1

. . . et

an an !n

= e–t

an !an n

. . . e–t

a1 !a1 1

.

For example, the fourth-order trace in the adjoint can be calculated as follows:  a a b b c c d d  d d c c b b a a  a a b b c c d d tr eT !1 eT !2 eT !3 eT !4 = tr et !1 et !2 et !3 et !4 tr e–t !4 e–t !3 e–t !2 e–t !1 – 1 ,

136

6 Simplifying Wilson Line Calculations

       !a1 !b2 !c3 !d4 tr T a T b T c T d = !a1 !b2 !c3 !d4 tr ta tb tc td Nc + Nc tr td tc tb ta             + 2 tr ta tb tr tc td + 2 tr ta tc tr tb td + 2 tr ta td tr tb tc . Using this trick, we can calculate any trace in the adjoint representation in function of traces in the fundamental representation. Also note that we can derive equations similar to equation (6.16) using different representation combinations. For example in SU(3), we have 3 ⊗ 3 ≃ 6 ⊕ 3, implying   tr (U2F ) = tr (UF ) tr (UF ) – tr UF . Now back to the adjoint generators. We can generalize their trace as        tr T a1 . . . T an = Nc tr ta1 . . . tan + (–)n tr ta1 . . . tan +

n–2  m=2

(–)n–m

    n! tr t(a1 . . . tam | tr tam+1 . . . tan )o . m! (n – m)!

(6.17)

(6.18)

We introduced two new notations: first, we have the “conjugated” trace, which is simply the trace in reversed order:     tr ta1 . . . tan = tr tan . . . ta1 . The only thing that changes when reversing a trace of fundamental generators is that every h gets replaced by its complex conjugate h (hence the notation tr). The result can then be simplified further using relations as h–h = 2i f ,   hh + hh = 2 dd – f f , etc. The second notation we introduced, ( | )o , is an “ordered” symmetrization which for a general tensor M is defined as:

137

6.1 Advanced Colour Algebra

m! (n – m)! M (a1 ... am | am+1 ... an )o = n! ⎛ ⎞ all permutations for which the first m ⎜ ⎟ × ⎝M a1 ... an + indices and the last n–m indices are ⎠ . ordered with respect to (a1 . . . an )

(6.19)

For instance, M (ab | N cd)o =

 1  ab cd M N + M ac N bd + M ad N bc + M bc N ad + M bd N ac + M cd N ab . 6

One handy property is that when A and B are commutative, we have: A(a1 ... am | Bam+1 ... an )o = B(a1 ... an–m | Aan–m+1 ... an )o , e.g. ( $| h)o = ( h| $)o . To conclude, let us list some traces:1 a a2

CA1

a a2 a3

CA1

a a2 a3 a4

CA1

a a2 a3 a4 a5

CA1

a a2 a3 a4 a5 a6

CA1

= Nc $ ,  Nc  = h–h , 4  N   1 c = hh + hh , $$ + 3( $$) + 2 8     )o  1  ( | = h – h $ + $ h – h + 10 $ h – h 8  Nc  + hhh – hhh , 16  1   1  ( ) $$$ + 15 $$$ + = hh + hh $ 4Nc 16      ) ( | o + hh + hh + $ hh + hh + 15 $ hh + hh   Nc  ( | )o – 20 h h + hhhh + hhhh , 32

(6.20a) (6.20b) (6.20c)

(6.20d)

(6.20e)

where we introduced the notation a ... an

CA1

  N = tr T a1 . . . T an ,

in analogy with the fundamental representation.

1 Note that ( $| $)o = ( $$) , for any number of $s.

(6.21)

138

6 Simplifying Wilson Line Calculations

6.2 Self-Interaction Blobs Now that we have developed some useful tools for working with the colour algebra, we make a list of path functions for some common blobs. We will list the Wilson selfinteraction blobs – i.e. all blobs that connect the Wilson line to itself, not to, e.g. a fermion – up to next-to-next-to-leading-order (NNLO). These are the blobs that are used in the calculation of soft factors and Wilson loops. Note that we always take the convention that gluon momenta are pointing outwards from the Wilson line, into the blob.

6.2.1 2-Gluon Blob The 2-gluon blob up to NNLO is simply the gluon propagator plus one-loop and twoloop corrections:

=

+

+

+

.

So the formula for the blob is simply given by F,ab1 ,2 (k1 , k2 ) = $ab D,1 - (k1 ) –$(9) (k1 + k2 ) B -,2 (k1 ) ,

(6.22)

where B resums all corrections: " # 1 NLO 31 32 NNLO B-, (k) = g-, + iFNLO (k)FNLO (k) + ⋅ ⋅ ⋅ D , (k) , -1 (k) – F-31 (k)D 32 1 (k)+ iF-1

(6.23)

and where F is the 1PI diagram. We anticipated the fact that we can factor out the colour structure to be $ab . This is only logical, as there are no other elements in the algebra that have exactly two adjoint indices open (and that cannot be reduced to something proportional to $ab ). This also means that the path function defined in equations 4.82 is indeed valid, because the colour structure is (see equation (4.73a)): c = ta tb $ab = CF 1 , and what is more, it is valid to all orders for the 2-gluon self-interaction blob, following our reasoning above. Connecting this blob to a Wilson line is trivial using the methods developed in Section 4.5. This gives: U2 =

M  J

I2 ( J)W2J +

J–1 M   J=2 K=1

I1 1 ( J, K)W1JK1 ,

(6.24)

6.2 Self-Interaction Blobs

139

where I and W are given by equations (4.82) and (4.85): J

I2 (J) = CF ,

k2 :

k1

(6.25a)



d9 k1 d9 k2 l.b. J I (k1 , k2 ) F,1 ,2 (k1 , k2 ) , (20)9 (20)9 2  1 d9 k , , 1 D, - (k) B -,2(k) , = –ig2 nJ 1 nJ 2 ,2: ' (20)9 nJ ⋅k + i' 1

W2J =

K

J

k1

k2

W1JK1 =



:

I1 1 (J, K) = (–)6J +6K CF ,

(6.25b)

d9 k1 d9 k2 l.b. K l.b. J I (k1 ) I1 (k2 ) F,1 ,2 (k1 , k2 ) , (20)9 (20)9 1 ,

,



= –g 2 nK1 nJ 2 ,2:

  i rK –rJ ⋅k

d9 k e 1 D, - (k) B -,2(k) , 9 K J (20) n ⋅k + i' n ⋅k – i' 1

where 6J is defined in equation (4.83), and Inl.b. J is given in equation (4.28). In Sections 6.3 and 6.4, we will investigate some calculational tools to do these momentum integrations. Equation (6.25) are of course still gauge-invariant statements, as all gauge-dependent contents are contained in D,- and B,- . Now let us investigate the contents of the 2-gluon blob a bit more precise. The one-loop diagrams are the fermion loop, ghost loop (in non-axial gauges), and the two possible gluon loops:

N

=

+

+

+

,

which are: % % iFNLO -1 -2 %

ferm.

% % iFNLO -1 -2 %

ghost

% % iFNLO -1 -2 %

3-gluon

  g 2 2: d9 q  -2 tr γ BF (q)γ -1 BF (q + k) , , 9 2 (20)  d9 q = g 2 Nc ,2: (q + k)-1 q-2 GF (q + k)GF (q) , (20)9  g2 d9 q -1 11 31 = – Nc A D11 12 (q)D31 32 (q+k)A-2 12 32 , 2 (20)9 =

(6.26a) (6.26b) (6.26c)

140

6 Simplifying Wilson Line Calculations

A-13 = g -1 (q – k)3 – g 13 (2q + k)- + g 3- (q + 2k)1 ,  %  -1 -3  2: d9 q NLO % 2 -1 -2 13 1 2 iF-1 -2 % = –ig Nc g g – g g , D13 (q) . 4-gluon (20)9

(6.26d)

In Feynman gauge (and using dimensional regularization), it is not so difficult to calculate these next-to-leading-order (NLO) contributions. The sum of all contributions equals2     !s 5Nc – 2Nf 1  Nc 2 2 iFNLO = i g k – k k + ln 40 + ln , – γ – I , + ,, E ,0 12 : 12 ⎡ ⎤ Nf  1  1 2 ln B(mq )⎦ , + x(1 – x) ln B(0) – x(1 – x) I = dx⎣Nc 4 q 0

where N

B(m2 ) = m2 – x(1 – x)k2 . The k2 -integration that emerges when connecting the blob will be quite complicated due to the logarithm of square roots. For this reason, we leave the x-integration undefined until after the momentum integration. The NNLO 1PI diagram is a lot more involving. We will not calculate it – as we don’t need it – but just list its 15 subdiagrams: N

=

+

+

+

+

+

+

+

+

+

+

+

+

+

+

.

We already deduced from first principles that at any order, the colour structure has to be proportional to $ab . Using equations (B.83)–(B.88), we can easily double-check this: 2 There is a very important caveat: when calculating blob diagrams that will be inserted in other diagrams, it is always preferable not to expand the ultraviolet poles, because a priori the parent diagram and its poles are not known, i.e. we don’t know yet to which order we have to keep the finite terms. Leaving : unexpanded avoids this problem.

6.2 Self-Interaction Blobs

141

  1 ab tr ta tx tb tx = – $ , 4Nc   Nc tr tb tx ty f ayx = –i $ab , 4   Nc tr ty tz f axy f bzx = – $ab , 2 f xay f ycz f zbw f wcx =

f avw f xby f ywz f zvx =

Nc 2 ab $ , 2 Nc 2 ab $ , 2

f awv f bzw f xzy f yvx = Nc 2 $ab ,

f xay f ycz f zbw f wcx =

Nc 2 ab $ , 2

f vaw f wbz f xzy f yvx = Nc 2 $ab , and similarly for the seven remaining diagrams.

6.2.2 3-Gluon Blob The 3-gluon blob up to NNLO is simply the 3-gluon vertex plus one-loop corrections: =

+

+

.

So, the formula for the blob is given by (k , k2 , k3 ) F,abc 1 ,2 ,3 1 = f abc D,1 -1 (k1 )D,2 -2 (k2 )D,3 -3 (k3 ) $–9 (k1 + k2 + k3 ) A-1 -2 -3 (k1 , k2 , k3 ) , where – resums all corrections: LO+B NLO A,-1 (k, p, q) = ALO ,-1 (k, p, q) + A,-1 (k, p, q) + A,-1 (k, p, q) + . . . # " 1 , . ALO ,-1 (k, p, q) = g g,- (k–p) + g-1 (p–q) + g1, (q–k)

(6.27a) (6.27b)

142

6 Simplifying Wilson Line Calculations

Again, we anticipated the fact that we can factor out the colour structure to be f abc . The same arguments as for the 2-gluon blob are however no longer valid, as now there exists a second structure – independent of f abc – with three adjoint indices, namely dabc . It just happens that up to first loop, the colour structure only depends on f abc , of which we took advantage to factor it out. Just remember that this might no longer be possible at second loop (if this is indeed the case, we will have to split the blob as a sum of two colour-factorable sub-blobs, as explained in Section 4.4). Again, the path function defined in equations (4.84) is indeed valid, because the colour structure is (see equation (4.73a)): c = tc tb ta f abc = Ccba f abc

1 Nc

+ 2Ccbax f abc tx = –i

Nc CF 1 , 2

where we used equation (6.2a) to write the generator product as a sum of colour factors and used the relations in equations (6.13) to simplify the result. Remember that we need to read a Wilson line as a Dirac line, i.e. from right to left, to get the correct result. Here, this can be seen in the order of the generators; reversing the order would give a minus sign difference. Investigating the colour structure, we have three different possibilities: all gluons connected to one segment, two gluons to one segment and one to another, or all to a different segment. Reversing the direction of a segment line is the same as flipping the order of the colour indices that are connected to it and multiplying with –1 for each gluon. For the situation, where all gluons are connected to one segment, flipping this segment leaves the result invariant (see equation (4.75)). For the situation 1+2, flipping the segment with one gluon only gives a sign difference (because reversing one index doesn’t change anything): (–)1 c = i

Nc CF 1 , 2

and flipping the segment with two gluons reverses these indices, also giving a sign difference: (–)2 tc tb ta f acb = Ccba f acb

1 Nc

+ 2Ccbax f acb tx = i

Nc CF 1 . 2

And for the situation 1 + 1 + 1, flipping any segment trivially gives a sign difference. So, these effects can be combined into the path function as given in equations (4.84). The full Wilson line result can be calculated using the methods developed in Section 4.5: U3 =

M  J

I3 W3J +

J–1  M  

I2 1 W2JK1 + J ↔ K

J=2 K=1

J–1  K–1 M   J=3 K=2 L=1

where I and W are now given by equations (4.84) and (4.85):

I1 1 1 W1JKL 11 ,

(6.28)

143

6.2 Self-Interaction Blobs

J

:

k3

k1

W3J =



I3 (J) = –i

Nc CF , 2

(6.29a)

d9 k1 d9 k2 d9 k3 l.b. J I (k1 , k2 , k3 ) F,1 ,2 ,3 (k1 , k2 , k3 ) , (20)9 (20)9 (20)9 3

 9 1 d k1 d9 k2 1 , , , 1 = ig 3 nJ 1 nJ 2 nJ 3 ,3: ' (20)9 (20)9 nJ ⋅k1 –i' nJ ⋅ (k1 +k2 ) –i' D,1 -1 (k1 )D,2 -2 (k2 )D,3 -3 (k1 + k2 ) A-1 -2 -3(k1 , k2 ,–k1 –k2 ) , K

J

k1

I2 1 ( J, K) = –i(–)6J +6K

: k3

Nc CF , 2

(6.29b)



d9 k1 d9 k2 d9 k3 l.b. K l.b. J I (k1 ) I2 (k2 , k3 ) F,1 ,2 ,3 (k1 , k2 , k3 ) , (20)9 (20)9 (20)9 1    9 i rK –rJ ⋅k1 9 k d k e d 1 1 1 2 , , , = –g 3 nK1 nJ 2 nJ 3 ,3: 9 9 K J J (20) (20) n ⋅k1 +i' n ⋅k1 –i' n ⋅ (k1 +k2 ) –i'

W2JK1 =

D,1 -1 (k1 )D,2 -2 (k2 )D,3 -3 (k1 + k2 ) A-1 -2 -3(k1 , k2 ,–k1 –k2 ) , L k1

K

J

I1 1 1 (J, K, L) = –i(–)6J +6K +6L

: k3

Nc CF , 2

(6.29c)

 9 d k1 d9 k2 d9 k3 l.b. L l.b. J W1JKL = I (k1 ) I1l.b. K (k2 ) I1 (k3 ) F,1 ,2 ,3 (k1 , k2 , k3 ) , 11 (20)9 (20)9 (20)9 1 

=

, , , –g 3 nL1 nK2 nJ 3 ,3:

  i rL –rJ ⋅k

  i rK –rJ ⋅k

1 2 d9 k1 d9 k2 e e 1 (20)9 (20)9 nL ⋅k1 +i' nK ⋅k2 +i' nJ ⋅ (k1 +k2 ) –i'

D,1 -1 (k1 )D,2 -2 (k2 )D,3 -3 (k1 + k2 ) A-1 -2 -3(k1 , k2 ,–k1 –k2 ) .

where 6J is defined in equation (4.83), and Inl.b. J is given in equation (4.28). Again, these results are still gauge-invariant. Now let us investigate the contents of the 3-gluon blob a bit more precise. There are two one-loop contributions, viz. a “pure” contribution where the vertex point is substituted by a loop, and a contribution where the propagator in one of the legs is evaluated at NLO. The latter is really straightforward to calculate, because the result was already calculated in equation (6.23):

144

6 Simplifying Wilson Line Calculations

NLO LO NLO LO ALO+B ,-1 (k, p, q) = B,3 (k)A3-1(k, p, q) + B-3 (p)A,31(k, p, q) NLO (k)ALO + B13 ,-3(k, p, q) .

(6.30)

The pure contribution is however a bit more complicated. The loops are – just as was the case for the propagator – a fermion loop, ghost loop, and the two possible gluon loops: N

=

+

+

+

,

This result is quite challenging to calculate and spans several pages (see, e.g. Refs [28, 102]). It is instructive to take out the pole part, as this is the most interesting part. In covariant gauges, it is given by: UV ANLO (k, p, q) = ,-1

   1 !s 2 2 3 Nf – Nc + . ALO ,-1(k, p, q) , : 40 3 3 4

(6.31)

where . = 0 in Feynman gauge. Note that when choosing the gauge defined by  8 Nf .= –1 , (6.32) 9 Nc which is . =

8 9

for Nf = 6, there are no ultraviolet divergences.

6.3 Wick Rotations In this section, we will investigate the possibility to use a Wick rotation when doing integrations with Wilson lines. We cannot blindly make the substitution k0 = ikE0 as in Appendix C.3, because the rotation might hit the poles. To see what we mean with this statement, let us first investigate how a regular Wick rotation works.

6.3.1 Regular Wick Rotation Naively, one could think that in a Minkowskian integral, 

d9 k f (k2 )

(6.33)

the substitution k0 = ikE0 would suffice to change it into an Euclidian integral  i

  d9 kE f –kE2 ,

(6.34)

6.3 Wick Rotations

145

but this is of course not true, as a complex substitution changes the contour of the integration (in the same way a real substitution can change the integration borders). The transformation is hence only valid if we can prove that the integration over the real axis equals the one over the complex axis. For most calculations in quantum field theory, this is trivial, as they will primarily contain integrals where the integrand is a combination of Feynman propagators. These can always be brought into the form 

d9 k 1   , (20)9 k2 – B + i0 n

(6.35)

by completing the square and using Feynman parameterization. This expression has two manifest poles of order n:3   k0 = ± k2 + B – i0 ≈ ± k2 + B ∓ i0 .

(6.36)

These poles lie in the second and fourth quadrant (the numbering of the quadrants follows the angular magnitude, i.e. anticlockwise, starting in the upper right quadrant). Note that even when B < –k2 , the poles will lie in the second and fourth quadrants, because then   k0 = ± k2 + B – i0 ≈ ±i –k2 – B ∓ 0 . If we now choose the contour as in Figure 6.1, the contour integral vanishes because it doesn’t enclose any poles. We then have: 

( = C

II

 +

CR

 +

C1

 ≡0

+ CI

 ⇒

C2

 =–

CR

.

(6.37)

CI

I CI

C1

−Ek + i0 CR

Figure 6.1: The contour chosen for a Wick rotation. Because it

Ek – i0 C2 III

IV

doesn’t enclose any poles, the integral vanishes. If  contour  the integrand behaves as O 1/k2 , the real integral CR equals the imaginary integral CI . Note that although this is a contour with a self-intersection, we can split it at (0, 0) into two valid contours, giving the same result (see Figure 6.3).

3 Where the second step is made using the expansion

√

x – i0 ≈

√

x–

i0 √ 2 x

and absorbing

√ x in i0.

146

6 Simplifying Wilson Line Calculations

  The integrations over the arc segments vanish because the integrand is of O 1/(k0 )2n . The minus sign in front of the integral over CI flips its borders, so that we indeed have 

 9 d kE 1 1 d9 k n = (–) i     . (20)9 k2 – B + i0 n (20)9 kE2 + B n

(6.38)

We dropped the pole prescription in the rhs as it is no longer needed.   Note that with this contour, it is indeed required that the integrand is of O 1/(k0 )2 , because exponential damping won’t be sufficient. Suppose that we have an integration of the form  d9 k k0 ik⋅x   e , 9 (20) k2 – B + i0 2   which is of O 1/k0 . To calculate the integrations over C1 and C2 , we use analytic continuation to parameterize k0 in polar representation as k0 = Rei( . The k0 integration over C1 then becomes 0

2 lim i R

R→∞

d( ei (

0

Rei ( R2 e2i (

∼ lim e–R x

2

0

– k – B + i0

0 sin (

R→∞

ei R(cos (+i sin ()x e–i k⋅x

.

The sign of sin ( is fixed, because ( = 0...

0 2

⇒

sin ( > 0 .

Hence, this integral only vanishes for x0 > 0. On the other hand, the integration over C2 has ( = –0/2 . . . – 0 and thus sin ( < 0, so this integral only vanishes for x0 < 0. We cannot have both x0 > 0 and x0 < 0, at the same time, so exponential damping cannot make the integrations over the arcs vanish.4 There are two requirements to beallowed  to make a Wick rotation: – The integrand should scale as O (k01 )2 , –

The integrand should only have poles in the second and fourth quadrants.

It is however possible to relax the second requirement by adapting the Wick rotation formula a bit, as we will see in what follows. 4 Unless we forge some unphysical exponential of the form ei|k

0 |x0 –i x⋅k

.

6.3 Wick Rotations

147

6.3.2 Wick Rotation with Wilson Lines Propagators from Wilson lines will introduce a linear dependence on k0 in the denominator. Consider the NLO self-energy of a Wilson line segment (see equation (6.25a)):  2 1 = –ig 2 nJ k2 '

J k1



1 d9 k 1 , (20)9 nJ ⋅k + i' k2 + i0

(6.39)

where the gluon propagator is expressed in Feynman gauge. If (nJ )0 = 0, the Wilson line segment doesn’t introduce additional poles, and we can safely make a Wick rotation. So from now on, we suppose that (nJ )0 ≠ 0. Then the integrand has three poles:  k0 = ± k2 – B ∓ i0 ,

k0 =

1 (n⋅k – i') . n0

(6.40)

The problematic pole is the last one, as it can lie in all quadrants, depending on the sign of n0 and n ⋅ k, as illustrated in Figure 6.2. The troublesome quadrants are the first and the third; the integral has its poles in these quadrants when n⋅k ≤ 0. We can separate these values by splitting the integral in three parts using 1 = ((n⋅k)+((–n⋅k)+ $(n⋅k):5 

 d9 k ((n⋅k) d9 k ((–n⋅k) 1 1 + 9 2 9 2 J J (20) n ⋅k + i' k + i0 (20) n ⋅k + i' k + i0  d9 k $(n⋅k) 1 + . (20)9 nJ ⋅k + i' k2 + i0

(6.41)

The first integral has no poles in the first and third quadrants and can hence be Wick rotated without problem. The second integral can be calculated by using the Residue theorem: 

( = C

 ≡ 20i Res

+ CR



CI

⇒

 ≡ 20i Res –

CR

.

(6.42)

CI

However, here, we are skipping an important step: the contour in Figure 6.1 is only valid because we can split it into two contours without self-intersections. The lower left contour is defined clockwise, so its residue gains a minus sign (this doesn’t matter in case of a regular Wick rotation, as then the residues are zero anyway). So, we split the contour into a positive contour C + and a negative contour C – , as in Figure 6.3:

5 If n ⋅ k = 0, the pole lies on the contour, which is treated differently. For this reason, we chose to define ((0) = 0.

148

6 Simplifying Wilson Line Calculations



( = C+

CR+



(

= 



= CR

 ≡ 20i Res+ –

⇒

CI+

CR+



CI–

CR–

,

(6.43b)

CI–



= 20i Res+ – 20i Res– –

(6.43a)



≡ –20i Res– –

⇒ CR–



, CI+



≡ –20i Res–

+ CR+

 ≡ 20i Res+

+ CR–

C–

 +

.

(6.43c)

CI

We will never have Res+ and Res– at the same time, as these are two versions of the same residue, depending on the sign of n0 (see Figure 6.2), which we assumed to be non-zero (because if it is zero, we don’t need to do this calculation anyway as we can just Wick-rotate the integral without problem). So, we can write the residue as:      20i Res+ – 20i Res– = ( –n0 – ( n0 20i Res ,

(6.44)

where the full residue is given by  20i Res = i

d9–1 k n0 ((–n⋅k) 2 , 9–1 2  (20) (n⋅k – i') – n0 k

(6.45)

Hence, we can write  20i Res – 20i Res = –i +

–

% 0% %n % ((–n⋅k) d9–1 k 2 . 9–1 2  (20) (n⋅k – i') – n0 k

(6.46)

We dropped the i% pole-prescription, as it is no longer needed (the integration over k0 has been done). We cannot drop the i', as it will act as a soft regulator. We can repeat the same calculation for the third integral in equation (6.41), but because the pole then lies on the contour, the residue only contributes a factor 0 instead of 20:

n0 0 n·k 0

n0 0 n·k 0

n0 0 n·k 0

n0 0 n·k 0

Figure 6.2: The pole of the Wilson propagator can lie in any quadrant, depending on the signs of n0 and n⋅k. The problematic poles are those in the first and third quadrants. These are the poles with n⋅k < 0. The troublesome poles are marked in red.

149

6.3 Wick Rotations

C1

C +I C R+ C R–

Figure 6.3: When there are extra poles in the first or third

C2

quadrant, we need to split the contour in its subcontours. The lower left contour, C – , is evaluated clockwise, hence its Residue gains a minus sign.

C –I

 0i Res+ – 0i Res– = –i

% 0% %n % $(n⋅k) d9–1 k 2 . 9–1 2  (20) (n⋅k – i') – n0 k

(6.47)

Combining the three terms in equation (6.41), we can write a Wick rotation with a Wilson line propagator as a regular Wick rotation plus a correction term WP : 

 9 1 1 d9 k d kE 1 1 – iWP , = i (20)9 nJ ⋅k + i' k2 + i% (20)9 nJ ⋅kE – i' kE2 E % 0%   %n % d9–1 k 1 WP = $(n⋅k) . ((–n⋅k)+   (20)9–1 (n⋅k–i')2 – n0 k 2 2

(6.48a) (6.48b)

  It is interesting to see that the factor ((–n⋅k) + 1/2$(n⋅k) in the definition of WP can be replaced by a single (-function, if we adopt the convention ((0) = 1/2. Equation (6.48) easily generalize to the integral of a Wilson line propagator and any function f that doesn’t have poles in the first or third quadrants: 

 9   d9 k d kE 1 1 f (k) = –i f ikE0 , kE – iWP , (20)9 nJ ⋅k + i' (20)9 nJ ⋅kE – i' E

(6.49)

where now WP is given by (assuming f is symmetric in k0 ):  WP =

%  % d9–1 k 1 1 % % ((–n⋅k) + $(n⋅k) f (k)%% % (20)9–1 %n0 % 2

.

(6.50)

k0 → 10 (n⋅k–i') n

However, in general, WP is quite difficult to calculate – especially in dimensional regularization – because of the angular part in n ⋅ k. In Section 6.4, we will investigate a more brute-force approach. We can rewrite equation (6.50) as (adopting the convention $(0) = 1/2):

150

6 Simplifying Wilson Line Calculations



d9 k f (k) $–+ (n⋅k + i') , (20)9

WP =

(6.51)

which we can interpret as the “real emission” of a Wilson line segment. We have argued before (on page 85) that there is no such thing as a cut Wilson segment, so this interpretation cannot be rigorous. Indeed, the $+ implies a (-function with a complex   shift, i.e. ( n0 k0 + i' , which is not well defined (and we have no trick to deal with it as we did have in the case of a complex $-function). So, we leave equation (6.51) as a vague physical interpretation without mathematical rigour, at most a curious coincidence.

6.3.3 Light-Cone Coordinates: Double Wick Rotation When using light-cone (LC) coordinates, we have to make two consecutive Wick rotations, as now the first two components of the momentum have a positive sign. Let us investigate again a general Feynman propagator integral as in equation (6.35): 

d9 k 1 . 9 2 (20) (k – B + i0)n

Let us first rotate the k+ component. It has a pole in k+ =

 1  2 k + B – i0 , 2k– ⊥

which lies in the second or fourth quadrants, as long as B ≥ –k2⊥ . In the latter case, we can safely Wick rotate it by identifying k+ = ikE+ : 

1 d9 k =i (20)9 (k2 – B + i0)n



d9–1 k dkE+ 1  n . (20)9 + 2ikE k– – k2⊥ – B + i0

But now, the pole of k– is given by k– =

   1  2 1   2 – i k k + B – i0 = + B –0 , ⊥ ⊥ + + 2i kE 2kE

(6.52)

which now lies in the first or third quadrant for B ≥ –k2⊥ . Luckily, this doesn’t pose a problem, as this is a second independent integration, and we can just choose a different contour, like in Figure 6.4. Then, we have (

 =

C

 +

CR

 +

C1



CI

 ≡0

+ C2

⇒

 =–

CR

, CI

(6.53)

6.4 Wilson Integrals

II

151

I CI C2 CR Figure 6.4: The contour chosen for the second Wick rotation

C1 III

when working in LC coordinates. Because of the shift k+ = ikE+ , the poles have switched quadrants, so the contour needs to be flipped.

IV

as before. But now, we don’t have to switch the borders of the integral over CI , so we retain the minus sign. So after making the second Wick rotation by identifying k– = ikE– , we have: 

d9 k 1 = (–)n (20)9 (k2 – B + i0)n



1 d9 kE . (20)9 (kE2 + B)n

The only difference with a Wick rotation in Cartesian coordinates is the lack of the i in front. This result was only valid for B ≥ –k2⊥ . But in case B < –k2⊥ , we just start with the second contour and end with the original contour, to get the same result. So for any   function, f that scales as O 1/k2 , we have: 

d9 k f (k2 ) = (20)9



d9 kE f (–kE2 ) . (20)9

(6.54)

Adding a Wilson line propagator will however complicate the calculation, as we will get two independent correction terms, one per rotation.

6.4 Wilson Integrals One common integral when dealing with Wilson lines is the following one:  I=

d9 k (20)9

n i=1

1 ni ⋅ (k + Ki ) + Ai + i3i '

m

1 ei r⋅k , 2 k + Pj + Bj + i0

 j=1

(6.55)

constructed from n “linear propagators” (Wilson line propagators) and m “squared propagators” (regular Feynman propagators). The +i0 are merely pole prescriptions, but the ±i' act as soft regulators and can have positive or negative signs. We encapsulated their sign into the 3i = ±1 in front, such that we can assert that ' > 0. We also naturally assume n ≥ 1, m ≥ 1, and Ki , Pi , Ai , Bi ∈ R and will use LC coordinates. Furthermore, Ai and Bj cannot depend on k.

152

6 Simplifying Wilson Line Calculations

As this framework to work with Wilson lines has been developed only recent, general results have not yet been reached. In this section, we briefly sketch the first steps to solve this general integral and continue with a simpler case, namely that of the LO 2-gluon blob at r = 0 (i.e. the two segments are connected). The first step in calculating the integral in equation (6.55) is to use Schwinger parameterizations (see equations (C.10)), and write it as I=

n

 3i (–i)n+m

d9 k (20)9

∞ d!1 . . . d!n d"1 . . . d"m 0 i

e

n / i

3i !i [ni ⋅(k+Ki )+Ai ]+i

 m  n / / 2 "j (k+Pj ) +Bj +ir⋅k– !i ' j

i

.

Note that we already took the limit i0 → 0. We can do this without problem, as it is merely a prescription, telling us which contour to use when doing the integration. Although we didn’t use contour integration, the prescription served its purpose: the sign of the Schwinger parameterization follows the sign of the pole prescription. By completing the square, we rewrite the exponential as

i

2 / / / / 2 2    "j Pj "j Pj 3 i !i n i r ( 3i !i ni ) / "j k + / + / + / –i / –i "j "j 2 "j 2 "j 4 "j / / / 2 3i !i "j ni ⋅Pj "j Pj ⋅r r 3i !i ni ⋅r / / –i / –i –i –i / "j "j 4 "j 2 "j      !i ' . +i 3i !i (ni ⋅Ki + Ai ) + i "j Pj2 + Bj –

After making the shift / / "j Pj 3 i !i n i r + / + / , k→k+ / "j 2 "j 2 "j the k-integration is just a Gaussian, which we can solve by making a Wick rotation: 

 9 d kE –i(/ "j )k2 d9 k i(/ "j )k2 E , e = i e 9 (20) (20)9  :–2 i "j (40i): . =– 2 (40)

To be mathematically rigorous, we have to mention that a purely imaginary Gaussian integral is divergent. Luckily, this is easily solved by regulating the integral with an infinitesimal negative shift:

6.4 Wilson Integrals



" # d9 kE –i (/ "j )–i$ k2 E e (20)9

lim

%→0

153

(6.56)

The validity of regulating an integral (and taking the limit $ → 0) in the middle of a calculation can be questioned. But in dimensional regularization, it can be proven that all complex Gaussian integrals are well defined, even when purely imaginary [89]. The result so far is I=

n

n+m+1

3i (–i)

(40i): (40)2

∞ d!1 . . . d!n d"1 . . . d"m

 :–2 "j ei E ,

(6.57)

0

where the exponent is given by

E=

n  i

2 / "j Pj !i [3i (ni ⋅Ki + Ai ) + i'] + + Bj – / "j j / / / / 2 3i !i "j ni ⋅Pj "j Pj ⋅r r2 3i !i ni ⋅r ( 3 i !i n i ) / / / – – / – – – / . "j "j 4 "j 4 "j 2 "j m 





"j Pj2

(6.58)

/  The appearance of a lot of factors of the form "j is a hint for the next step: we will / make the substitution "j = γj L, such that "j = L (see equations (C.11)). The integral then becomes ∞    I = N d!1 . . . d!n dγ1 . . . dγm dL $ 1 – γj Lm+:–3 eiE ,

(6.59)

0

where N and the exponent are

n

3i (–i)n+m+1

N = E=

n  i

(40i): , (40)2

!i [3i (ni ⋅Ki +Ai ) +i'] + L

(6.60) m 

   2 γj Pj γj Pj2 +Bj – L

j

2   1   – 3i !i γj ni ⋅Pj – γj Pj ⋅r . r+ 3i !i ni – 4L

(6.61)

From this point on, things are getting a bit more difficult, as the exponent contains terms linear in L, but terms linear in 1/L as well. We continue with a much easier situation, viz. that of an LO 2-gluon blob connecting two adjoining segments.

154

6 Simplifying Wilson Line Calculations

6.4.1 2-Gluon Blob Connecting Two Adjoining Segments The integral we need to calculate is given by (see equation (6.25b)):

,

,

  i rK –rJ ⋅k



d9 k e 1 D, - (k) . 9 K J (20) n ⋅k + i' n ⋅k – i' 1

W1JK1 = –g 2 nK1 nJ 2 ,2:

(6.62)

This is a specific form of our “master” integral, with n = 2, Bj = 0 ,

Ki = 0 ,

Ai = 0 ,

3K = +1 ,

3J = –1 .

m = 1, r = rK – rJ ,

Pj = 0 ,

(6.63a) (6.63b)

Our result so far in this specific example is then

W1JK1 = –ig 2 nK ⋅nJ ,2:

(40i): (40)2

∞

d!1 d!2 d" ":–2 eiE ,

(6.64)

0

where the exponent is given by E=–

2 1  !1 nK – !2 nJ + r + i!1 ' + i!2 ' . 4"

(6.65)

We now make the substitution 1 d! " = – 2 d" . "

1 ! "= , " This gives (dropping the factors in front) ∞ I=

i!

2

d!1 d!2 d! "! "–: e– 4 "(!1 nK –!2 nJ +r)

–!1 '–!2 '

.

0

This is a complex A representation (see equations (C.6)), which is convergent for 0 < : < 1. The result is now ∞ I = A(1 – :) (–4i)

1–:

 d!1 d!2 !21 n2K + !22 n2J + r2

0

–2!1 !2 nK ⋅nJ + 2!1 nK ⋅r – 2!2 nJ ⋅r Next, we again use x-L parameterization:

:–1

e–!1 '–!2 ' .

6.4 Wilson Integrals

155

!1 = xL , !2 = (1 – x)L , L = 0...∞, x = 0...1, d!1 d!2 = dxdL L , which gives 1 1–:

I = A(1 – :) (–4i)

∞ dx

0

 #2(:–1) –L' "  dL L L xnK – xnJ + r e ,

(6.66)

0

where x is just a shorthand notation: N

x = 1–x.

(6.67)

Now, we apply the additional demand that the Wilson segments are adjoining, viz. rK = rJ and hence r = 0. The L integration is then again a Gamma function integral representation and gives a factor '–2: A(2:): 

1 I = A(1 – :)A(2:) '2

:

1 1–:

(–4i)

2(:–1)  dx xnK – xnJ .

(6.68)

0

In the case of on-LC segments, i.e. n2K = n2J = 0, the x-integral is just a Beta function (see equation (C.7)): 1



dx xnK – xnJ 0

2(:–1)

:–1  = –2nK ⋅nJ

1

dx x:–1 x:–1 ,

0

:–1  = –2nK ⋅nJ B(:, :) , :–1 A(:)A(:)  . = –2nK ⋅nJ A(2:)

(6.69)

We just have to collect all missing factors from the intermediate steps to get the on-LC result. From equation (6.64), we have a factor –ig 2 nK ⋅nJ ,2:

(40i): . (40)2

Next, we have from equation (6.68) a factor  1 : –4iA(1 – :)A(2:) – 2 , 4i' and last we have to subtract some finite terms, for which we follow the minimal sub¯ (which is a division by S: , see equation (C.17)) that gives an extra traction scheme (MS) A(1 – :)/(40): . These three factors together give:

156

6 Simplifying Wilson Line Calculations

 : !s 1 ,2 – nK ⋅nJ A(1 – :)A(1 – :)A(2:) – 2 . 0 4'

(6.70)

The full on-LC result is then %

W1JK1 %LC

 : !s nK ⋅nJ ,2 = . A(1 – :)A(1 – :)A(:)A(:) 20 2 '2

(6.71)

Expanding in function of the regulator gives our final result for the 2-gluon blob at LO connecting two adjoining on-LC segments: % W1JK1 %LC

   2 !s 1 1 02 nK ⋅nJ ,2 nK ⋅nJ ,2 1 = + ln + ln 2 + ln + ln 2 + . 20 :2 : 2 ' 2 2 ' 3

(6.72)

Note that the convention in equation (C.17) subtracts the most finite terms possible when having a double pole. This is due to the fact that A(1 – :)A(1 – :)A(:)A(:) =

  1 02 + + O :2 . 2 : 3

(6.73)

If we would use the regular convention as in equation (C.18), the subtraction would be less strong because : (e–γE ) A(1 – :)A(:)A(:) =

1 1 02 – 2γE + 2γE2 + + O (:) , 2 : : 4

which leaves an extra pole term and an extra term with γE . Also note that although the result seems to be divergent in the limit nK ⋅ nJ → 0, this is perfectly finite. The seemingly divergent behaviour is an artefact from the regulation. If nK ⋅ nJ = 0, the original contribution in equation (6.25b) is zero before we need to start a regulation procedure. Now, we will repeat this calculation, starting from equation (6.68), but with offLC segments. If n2K , n2J ≠ 0, we can parameterize the scalar product in function of a Minkowskian angle 7 between the two segments: def

cosh 7 =

n K ⋅ nJ % %, |nK | %nJ %

(6.74)

so that we can rewrite  N

xnK – xnJ

2

≡ x2 n2K – 2xxnK nJ cosh 7 + x2 n2J ,

(6.75)

where now nK = |nK |. We will rewrite the x-integral in function of a new angle 8 that we introduce by the relation,

6.4 Wilson Integrals

xnK sinh 8 = xnJ sinh(7 + 8) .

157

(6.76)

Or in other words, nJ sinh(7 + 8) , nK sinh 8 + nJ sinh(7 + 8) nK sinh 8 x= , nK sinh 8 + nJ sinh(7 + 8)   xnK – coth 7 , 8 = arcoth csch 7 xnJ 1 xnK – xnJ e–7 = ln , 2 xnK – xnJ e7

(6.77a)

x=

(6.77b) (6.77c) (6.77d)

where we used the hyperbolic sum rule sinh(7 + 8) = sinh 7 cosh 8 + sinh 8 cosh 7 . We can now simplify 

xnK – xnJ

2

= x2 n2K – 2xxnK nJ cosh 7 + x2 n2J , n2K n2J sinh2 7

= 

2 .

nK sinh 8 + nJ sinh(7 + 8)

To make the integral substitution, we first note that sinh 7

d8 = –nK nJ 

xnK – xnJ

8(x = 0) = –7

2 dx ,

8(x = 1) = 0 ,

giving eventually: 1

2(:–1)  dx xnK –xnJ

0

0  2:–1  –2: = – nK nJ sinh 7 d8 nK sinh 8 + nJ sinh(7 + 8) . –7

To calculate this integral, we expand it in :: 0

 –2: d8 nK sinh 8 + nJ sinh(7 + 8)

–7

–7 = 7 + 2: 0

" # d8 ln nK sinh 8 + nJ sinh(7 + 8) .

(6.78)

158

6 Simplifying Wilson Line Calculations

Using the exponential representation of the hyperbolic sine, i.e. sinh 8 =

 1 8 e – e–8 , 2

(6.79)

we rewrite the argument of the logarithm as  "  #  1  ln nK sinh 8 + nJ sinh(7+8) = ln e8 nK +nJ e7 – nK +nJ e–7 e–28 , 2     = – ln 2 + 8 + ln nK +nJ e7 – nK +nJ e–7 e–28 . The integral over the first terms is trivial. Using the shorthand notations a = nK +nJ e7 , b = nK +nJ e–7 , the integral over the logarithm can be done by making the substitution t = –b e–28 . We have (see equation (C.2e)):

–

–b

1 2

dt

ln(a + t) t

–b e27

  27   1 be b ln be27 ln a – Li2 – ln b ln a + Li2 , 2 a a      1  nK +nJ e–7 nJ +nK e7 = 7 ln nK +nJ e7 + Li2 – Li . 2 2 nK +nJ e7 nJ +nK e–7

=

We can require nK and nJ to have the same length, i.e. |nK | ≡ |nJ |, because it is just a path parameterization where only their direction matters (and their length is not zero). Then, the result simplifies into  1   7 ln nK + 7 ln 1 + e7 – Li2 e7 – Li2 e–7 . 2 So, the final result for the x-integration is then 1

 

dx xnK – xnJ 0

2(:–1)

=–

n2K n2J sinh2 7

:

nK nJ sinh 7

7 (1 + :H) ,

   1 Li2 e7 – Li2 e–7 . H = 2 ln 2 + ln n2K + 2 ln 1 + e7 + 7 – 7

(6.80a) (6.80b)

Now, we add the missing factors from the intermediate steps (see equation (6.70)). We will write nK ⋅nJ in function of the Minkowskian angle, i.e. nK ⋅nJ = nK nJ cosh 7, which will combine with the 1/sinh 7 into a coth 7. This gives

159

6.4 Wilson Integrals

: 2 2 2 n K nJ !s 2 , 7 coth7 A(1 – :)A(1 – :)A(2:) – sinh 7 2 (1 + :H) . 0 4 '

(6.81)

To allow for an easy comparison with the on-LC case, we rewrite the sinh inside the : exponential as  2   –n2K n2J sinh2 7 = n2K n2J 1 – cosh2 7 = n2K n2J – nK ⋅nJ . Expanding in function of :, the result becomes: %

W1JK1 %LC



2  n2K n2J – nK ⋅nJ !s 1 ,2 = 7 coth7 + ln + ln 2 + H . 20 : 4 '

(6.82)

When comparing to the on-LC result in equation (6.72), we see that the most important difference is that in the on-LC case, there is a double pole in : that is not present in the off-LC case. This is true in general: light-like Wilson line segments will introduce additional divergences. Indeed, if one of the segments – or both – goes on-LC, the angle becomes infinite: 7 = arcosh

nK ⋅nJ ∼ – log(|nK | |nJ |) |nK | |nJ |

on-LC

→

∞.

(6.83)

This is the key manifestation of LC-divergences. There is no way of retrieving the on-LC-result from the off-LC result, as we are missing a double pole and mixed terms. Another possibility is a situation where one of the segments is on-LC and the other is not. This is a bit complicated, as the Minkowskian angle is still not well defined, but neither can we use the Beta function integral representation as we were able to in the on-LC case (see equation (6.69)). Luckily, using a bit of trickery, we can do something similar using the incomplete Beta function (see equation (C.7c)). We start again from equation (6.68) but assume now that n2J = 0. It doesn’t matter which segment we take on-LC, the results are the same, but when choosing n2J = 0, the term with x drops from the calculation, which is easier. The x-integration is now: 1 0

2(:–1)  dx xnK – xnJ =

1

:–1  dx x2 n2K – 2xx nK ⋅nJ ,

0



= 2nK ⋅nJ

1 :–1 0

dx x

:–1

:–1

2nK ⋅nJ +n2K x–1 2nK ⋅nJ

.

160

6 Simplifying Wilson Line Calculations

Next, we make the substitution t=

2nK ⋅nJ +n2K N x =! nx, 2nK ⋅nJ

⇒

dx =

1 dt , ! n

which gives 1



dx xnK – xnJ

2(:–1)

0

:–1 1 = 2nK ⋅nJ ! n: 

!n

dt t:–1 (t – 1):–1 ,

0

:–1 1 = 2nK ⋅nJ B(! n; :, :) . ! n: 

(6.84)

B(! n; :, :) is the incomplete Beta function. It has a series expansion given by equation (C.7f):

∞  1 : B(: + 1, m + 1) m+1 : ! B(! n; :, :) = ! . n) 1 + n (1 – ! n : B(2:, m + 1) m=0

(6.85)

We can expand the fraction of the two Beta functions, which we have to do up to second order in : (because we have a double pole, one in front of the Beta function expansion and on from A(2:) in equation (6.68)): B(: + 1, m + 1) A(: + 1) A(2: + m + 1) = , B(2:, m + 1) A(2:) A(: + m + 2)

 m 1 1 1 1 2 ≈ 2: . + 2: – m+1 m+1 k (m + 1)2

(6.86)

k

The infinite sums running over m are just straightforward convergent series. The first and the last are just the first and the second polylogarithms (see equations (C.8a) and (C.8)), while the second sum is a bit less trivial as it is a chained sum: ∞ ∞  ! nm+1  ! nm n) = – ln(1 – ! n) , = = Li1 (! m + 1 m=1 m m=0 ∞ ∞   ! ! nm+1 nm = = Li2 (! n) , 2 (m + 1) m2 m=0 m=1 ∞ m  ! nm+1  1 1 2 n) . = ln (1 – ! m+1 k 2 m=0 k

So, the expansion of the incomplete Beta function is now: " # 1 : n) + :2 ln2 (1 – ! n): 1 – 2: ln(1 – ! n) – 2 :2 Li2 (! n) . B(! n; :, :) ≈ ! n (1 – ! :

(6.87a) (6.87b)

(6.87c)

6.4 Wilson Integrals

161

Putting everything together, we have (adding the factor in front from equation (6.70)): 1 !s A(1 – :)A(1 – :)A(2:) – 20 :



n2K ,2 4 '2

:



2nK ⋅nJ 1 + 2 : ln – 2 nK

 2!

+:B ,

(6.88)

with ! B = ln

2

2nK ⋅nJ – 2 nK

– 2Li2

2nK ⋅nJ + n2K 2nK ⋅nJ

.

Giving the final result, %

W1JK1 % 1 LC 2

!s =– 40



  2 nK ⋅nJ 1 ,2 1 + + ln 2 + fin. , ln :2 : ' n2K

(6.89)

where the finite terms are given by

2 2 2  n ⋅n  2 n n 1 , K J ln2 – – ln2 K + ln K – ln 2 2 4 2 4 '

2nK ⋅nJ + n2K 02 (nK ⋅nJ )2 ,2 + . + ln ln 2 – 2Li2 4 ' 2nK ⋅nJ 2

(6.90)

To conclude, general results are not that straightforward to calculate as they might seem at first sight, because the integrals at hand are quickly getting involving. On the other hand, once calculated in a given gauge, they never have to be calculated again as the results can be easily applied to any Wilson line structure using the framework developed in the previous chapter.

A Brief Literature Guide

In the present book we use the standard operator approach to introduce the concept of quantum parton density function. Basic texts on the operator formulation of quantum field theory, which we have used in our exposition, are: N. N. Bogolyubov and D. V. Shirkov (1959). “Introduction to the theory of quantized fields.” Intersci. Monogr. Phys. Astron. 3, 1–720. S. S. Schweber (1961). “An Introduction to relativistic quantum field theory.” Row, Peterson and Co. 1-905 A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski (1975). “Methods of quantum field theory in statistical physics.” Dover Publ., Inc., New York 1–352 References in main Bibliography are given in alphabetical order. For the reader’s convenience, we give below a short classification of the references divided in four major topics. I. Books and textbooks on general questions of quantum mechanics, quantum field theory, and mathematical methods relevant for our exposition: [3, 50, 51, 73, 77, 85, 113–115, 122, 148, 161, 175, 189, 195, 207, 209] II. Theoretical problems of TMD PDFs; structure of Wilson lines in the operator definitions; factorization breaking; renormalization and evolution; pathdependence: [14–16, 20-22, 35, 39, 41, 42, 44, 45–49, 52, 57–61, 78–82, 86–99, 104, 106, 112, 117– 120, 123–126, 137–140, 142–147, 168, 169, 173, 177, 183, 186, 188, 193, 194, 198, 200, 201, 208] III. Experimental results and phenomenology of TMD-related processes; scientific programs of planned facilities; numerical tools in high-energy physics; higherorders and resummation in perturbative QCD calculations: [1, 2, 4–10, 13, 17–19, 28-34, 40, 43, 53, 56, 84, 100–102, 105, 109–111, 127–135, 155–157, 159, 160, 167, 180, 187, 192, 196, 199, 202, 211, 212] IV. Wilson lines and loops in quantum field theory; mathematical properties of loop space; Wilson lines in small-x physics: [74–76, 83, 103, 107, 108, 116, 121, 141, 149–154, 158, 162–166, 170–172, 174, 176, 178, 179, 181, 182, 190, 191, 197, 203–206, 210, 213]

B Conventions and Reference Formulae B.1 Notational Conventions We will use a few different equal sign, to clarify some of our statements. In particular: = “. . . is equal to . . . ” , ≡

“. . . is required to be equal to . . . ” ,

def

=

“. . . is defined as . . . or . . . is defined to equal . . . ” ,

sup

“assume . . . to be equal to . . . ”

= N

= ?

=

“. . . is written as . . . ” , “. . . is maybe equal to . . . ” (statement has still to be verified) .

In general, we use the same conventions as in Ref. [189]. We will never use the comma notation to denote derivatives (as it is too easily confused with misplaced commas). We will work in natural units: h¯ = c = %0 = kB = 1 .

(B.1)

This means in particular that the electromagnetic fine structure constant is given by !=

1 g2 ≈ . 40 137.04

(B.2)

Although ! is originally only associated with the electromagnetic force, it is common to define a similar constant for the strong force: N

!s =

g2 . 40

(B.3)

Concerning indices and general variable namings, we try to be as consistent as possible (which is sometimes difficult due to the limited amount of characters available in the alphabet). In particular, we will use: r, s, t, u, v, w, x, y, z

for coordinates,

k, l, p, q

for momenta,

,, -, 1, 3

for Minkowskian indices,

i, j, k, l

for Euclidian indices, and for the spatial part of 4-vectors, and for fundamental indices (Lie algebra), and for enumerations in sums and products,

B Conventions and Reference Formulae

a, b, c, d, e, f , g

for adjoint indices (Lie algebra),

x, y, z, w

for summations of adjoint indices (Lie algebra),

!, ", γ , $

for Dirac indices,

n

for any integer greater than zero,

165

and for any directional 4-vector, J, K, L, O

for Wilson line segments.

B.2 Vectors and Tensors For the Minkowski metric, we take the common convention ⎛

g ,-

1 ⎜0 ⎜ =⎜ ⎝0 0

0 0 –1 0 0 –1 0 0

⎞ 0 0⎟ ⎟ ⎟, 0⎠ –1

(B.4)

where Greek indices run over 0, 1, 2, 3 (for t, x, y, z). To denote only the spatial components, we use Roman indices, like i, j, etc. We use the Einstein notation convention throughout the whole book, meaning that repeated indices are to be summed over. A 4-vector is denoted in italic, a 3-vector in bold and a 2-vector (the transversal components) in bold and with a subscript ⊥:       p, = p0 , p1 , p2 , p3 = p0 , p = p0 , p⊥ , p3 ,

(B.5)

while a length is mostly denoted in italic, be it a length of a 4-, 3- or 2-vector. The difference should be clear from context, but when needed for clarity, we use |p| and % % %p⊥ %. The scalar product is fully defined by the metric: x⋅p = x0 p0 – x⋅p .

(B.6)

This implies that we can define a vector with a lower index as       p, = g,- p- = p0 , –p1 , –p2 , –p3 = p0 , –p = p0 , –p⊥ , –p3 ,

(B.7)

x⋅p = x, p, .

(B.8)

such that

Note that the index switches places when moving the coordinate to the denominator, as is the case for the derivative: ∂ N = ∂, . ∂x,

(B.9)

166

B Conventions and Reference Formulae

The position 4-vector combines time and 3-position, while the 4-momentum combines energy and 3-momentum: x, = (t, x )

p, = (E, p) .

(B.10)

A particle that sits on its mass-shell (on-shell for short) has p2 = E2 – |p|2 = m2 .

(B.11)

All real particles (having timescales and distances larger than the quantum level) are on-shell. Last we define the symmetrisation (. . .) and antisymmetrization [. . .] of a tensor as  1  ,A + A-, , 2  1  ,A – A-, . = 2

A(,-) =

(B.12a)

A[,-]

(B.12b)

A rank-2 tensor has the peculiar property that it can be split exactly in its symmetric and antisymmetric parts A,- = A(,-) + A[,-] .

(B.13)

This is in general not true for tensors of higher rank. Symmetrizing an antisymmetric tensor returns zero, this implies: A(,-) B[,-] = 0 .

(B.14)

It is straightforward to generalize the definition of symmetrization to tensors of higher rank:  1  ,1 ⋅⋅⋅ ,n A + all permutations , n!  1  ,1 ⋅⋅⋅ ,n A = – all odd perm. + all even perm. . n!

A(,1 ⋅⋅⋅ ,n ) =

(B.15a)

A[,1 ⋅⋅⋅ ,n ]

(B.15b)

B.3 Spinors and Gamma Matrices Any field with half-integer spin, i.e., a Dirac field, anticommutes: 8(x)8(y) = –8(y)8(x),

x ≠ y.

(B.16)

We define gamma matrices by their anticommutation relations 7

8 γ , , γ - ≡ 2 g ,- 1 ,

(B.17)

B Conventions and Reference Formulae

167

with the following additional property: 

γ,

†

= γ 0 γ , γ 0.

(B.18)

Then, we can define the Dirac equation for a particle field 8: 

 i∂/ – m 8 = 0.

(B.19)

where the slash is a shortcut notation for ∂/ = γ , ∂, .

(B.20)

We can identify an antiparticle field with 8 if we define 8 = 8† γ 0,

(B.21)

which satisfies a slightly adapted Dirac equation: i∂, 8 γ , + m8 = 0.

(B.22)

We can expand Dirac fields in function of a set of plane waves: 8(x) = us (p) e–ip⋅x s

8(x) = v (p) e

+ip⋅x

(p2 = m2 , p0 > 0), 2

2

0

(p = m , p < 0),

(B.23a) (B.23b)

where s is a spin-index. If we define u = u† γ 0 ,

v = γ 0 v† ,

(B.24)

we can find the completeness relations by summing over spin: 

us (p)us (p) = p/ + m,

(B.25a)

vs (p)vs (p) = p/ – m.

(B.25b)

s

 s

We will identify u with an incoming fermion, u with an outgoing fermion, v with an incoming antifermion, v with an outgoing antifermion.

168

B Conventions and Reference Formulae

If we define N

i ,- 13 γ, γ- γ1 γ3 , % 4!  1 , γ γ – γ -γ , , = 2

γ 5 = iγ 0 γ 1 γ 2 γ 3 = – γ ,- = γ [, γ -] N

(B.26a) (B.26b)

we can construct a complete Dirac basis:

1, γ , , γ ,- , γ , γ 5 , γ 5 .

(B.27)

We will identify 1 with a scalar, γ , with a vector, γ ,- with a tensor, γ , γ 5 with a pseudo-vector, γ 5 with a pseudo-scalar. Furthermore, γ 5 has the following properties: 

γ5

†



= γ5 ,

γ5

2

8 γ 5, γ , = 0 .

7

= 1,

(B.28)

Let’s list some contraction identities for gamma matrices in 9 dimensions: γ , γ, = 9 ,

(B.29a)

, -

-

γ γ γ, = (2 – 9)γ , , - 1

γ γ γ γ, = 4 g , - 1 3

-1

(B.29b) - 1

+ (9 – 4)γ γ ,

3 1 -

(B.29c) - 1 3

γ γ γ γ γ, = –2γ γ γ + (4 – 9)γ γ γ .

(B.29d)

And some trace identities: tr (1Dirac ) = 4 , tr (odd number of γ s) = 0 ,   tr γ , γ - = 4 g ,- ,     tr γ , γ - γ 1 γ 3 = 4 g ,- g 13 – g ,1 g -3 + g ,3 g -1 ,     tr γ ,1 γ ,2 ⋯ γ ,n–1 γ ,n = tr γ ,n γ ,n–1 ⋯ γ ,2 γ ,1 . Let us finish this section by listing some useful relations:

(B.30a) (B.30b) (B.30c) (B.30d) (B.30e)

B Conventions and Reference Formulae

k/ k/ = k2 ,

k/ p/ + p/ k/ = 2p⋅k , 2

γ , k/ = 2k, – k/ γ , , 2

p/ k/ p/ = 2p⋅k p/ – p k/ ,

p/ k/ q/ p/ = 2p⋅q p/ k/ – 2p⋅k p/ q/ + p k/ q/ ,

k/ p/ q/ + q/ p/ k/ = 2p⋅q k/ – 2q⋅k p/ + 2k⋅p q/ .

169

(B.31a) (B.31b) (B.31c)

B.4 Light-Cone Coordinates Light-cone (LC) coordinates form a useful basis to represent 4-vectors. For a random vector k, , they are defined by  1  k+ = √ k0 + k3 , 2  1  0 – k = √ k – k3 , 2  1 2 k⊥ = k , k .

(B.32a) (B.32b) (B.32c)

We will represent the plus-component first, i.e.   k, = k+ , k– , k ⊥ .

(B.33)

One often encounters in literature the notation (k– , k+ , k⊥ ), but this is merely a matter of convention. The factor √12 normalizes the transformation to unit Jacobian, such that d9 k = dk+ dk– d9–2 k⊥ .

(B.34)

It is straightforward to show that the scalar product has the form k⋅p = k+ p– + k– p+ – k⊥ ⋅p⊥ , 2

+ –

k = 2k k –

k2⊥ .

(B.35a) (B.35b)

This implies that the metric becomes off-diagonal: ⎛

,-

gLC

0 ⎜1 ⎜ =⎜ ⎝0 0

⎞ 1 0 0 0 0 0⎟ ⎟ ⎟. 0 –1 0 ⎠ 0 0 –1

(B.36)

We will drop the index LC when clear from context. Note that this basis is not orthonormal. Note also that ,

g,- g-1 = $1 ,

g ,- g,- = 4 ,

(B.37)

170

B Conventions and Reference Formulae

just like the Cartesian metric. We can also define two light-like basis vectors:   , n + = 1+ , 0 – , 0 ⊥ ,   n,– = 0+ , 1– , 0⊥ .

(B.38a) (B.38b)

These are light-like vectors, and maximally non-orthogonal: n2+ = 0,

n2– = 0,

n+ ⋅n– = 1 .

(B.39)

Watch out, as lowering the index switches the light-like components because of the structure of the metric:   n+ , = 0+ , 1– , 0⊥ ,   n– , = 1+ , 0 – , 0 ⊥ ,

(B.40a) (B.40b)

such that they project out the other light-like component of a vector: k⋅n+ = k– ,

k⋅n– = k+ .

(B.41)

In other words, we can write k = (k⋅n– ) n+ + (k⋅n+ ) n– – k2⊥ .

(B.42)

For Dirac matrices in LC coordinates, we have 7

8 γ + , k/ = 2 g +, k, = 2k+

⇒ γ + k/ = 2k+ – k/ γ + .

(B.43)

Note that 

γ+

2

1 7 + –8 γ ,γ = 1, 2

= (γ – )2 = 0 ,

such that equation (B.17) remains valid in LC coordinates. We can use the light-like basis vectors to construct a metric for nothing but the transversal part: ,-

g⊥ = g ,- – 2 n+ n-) – ⎛ 0 0 0 ⎜0 0 0 ⎜ =⎜ ⎝ 0 0 –1 0 0 0 (,

⎞

0 0⎟ ⎟ ⎟. 0⎠ –1

(B.44a)

(B.44b)

B Conventions and Reference Formulae

171

Note that ,-

,

,

,-

g⊥ g⊥ -1 = $1 – n+ n– 1 – n,– n+ 1 ,

g⊥ g⊥ ,- = 2 .

(B.45)

Last, we can define an antisymmetric metric: ,-

%⊥ = %+–,⎛ 0 ⎜0 ⎜ =⎜ ⎝0 0

⎞

0 0 0 0 0 0⎟ ⎟ ⎟, 0 0 1⎠ 0 –1 0

(B.46a)

(B.46b)

where we adopt the convention %0123 = %+–12 = +1.

B.5 Fourier Transforms and Distributions The Heaviside step function is defined as 1 ((x) =

0, x < 0

1,

x>0

,

(B.47)

and is undefined for x = 0 (sometimes it is hard-coded to equal 0, 1 or 1/2, but we leave it undefined). It can be used to limit integration borders: +∞ +∞ dx ((x – a) f (x) = dx f (x) , –∞ +∞

a

dx ((a – x) f (x) = –∞

(B.48a)

a

dx f (x) .

(B.48b)

–∞

The Dirac $-function is defined as the derivative of the (-function:  d $(x) = ((x) , ⇒ dx $(x) = 1 , dx

(B.49)

and is zero everywhere, except at x = 0. A generalization to n dimensions is straightforward:  dn x $n (x) = 1 . (B.50) The most important use of the Dirac $-function is the sifting property, which follows straight from equation (B.50):  dn x f (x) $n (x – t) = f (t) . (B.51)

172

B Conventions and Reference Formulae

However, for non-infinite borders, the sifting property gains additional (-functions: b dx $(x – c) f (x) = ((b – c) ((c – a) f (c) .

(B.52a)

a

Similar properties can be derived for the (-function: b

b

b

dx ((x–c) f (x) = ((a–c)

dx f (x) + ((b–c) ((c–a)

a

a

b

b

(B.52b)

dx f (x) .

(B.52c)

c

c

dx ((c–x) f (x) = ((c–b) a

dx f (x) ,

dx f (x) + ((b–c) ((c–a) a

a

When dealing with on-shell conditions, we often encounter the combination of a Heaviside ( and a Dirac $ function. To save space, we define the shorthand notation   N $+ p2 – m2 = $(p2 – m2 ) ((p0 ) .

(B.53)

When working in LC-coordinates, the integration over p0 is replaced by an integration over p+ ; hence, in this case, we define the shortcut as   N $+ p2 – m2 = $(p2 – m2 ) ((p+ ) .

(B.54)

Another shorthand notation we will often use to save space is N

$–(n) (x) = (20)n $(n) (x) ,

(B.55)

because a $-function is often accompanied with powers of 20. The combination of the two is trivial: N

$–+ (x) = 20 $(p2 – m2 ) ((p+ ) .

(B.56)

When dealing with Fourier transforms, we will use the following conventions: ! f (k) =

 

f (x) =

d4 x f (x) eik⋅x ,

(B.57a)

d4 k ! f (k) e–ik⋅x . (20)4

(B.57b)

The tilde will always be omitted, as the function argument specifies clearly enough whether we are dealing with the coordinate or momentum representation. Note that due to the Minkowski metric, Fourier transforms over spatial components have the signs in their exponents flipped:

B Conventions and Reference Formulae

 f (x) = ! f (k) =



173

d3 k ! f (k) eik⋅x , (20)3

(B.58a)

d3 x f (x) e–ik⋅x ,

(B.58b)

and the same for two-dimensional Fourier transforms. When necessary to emphasize the Fourier transform itself, we will use the notation  N Fk [f (x)] = d4 x f (x) eik⋅x , (B.59)  Fx–1 [f

N

(k)] =

d4 k f (k) e–ik⋅x . (20)4

An “empty” Fourier transform gives a $-function:  Fk [1] = dn x eik⋅x ≡ (20)n $(n) (k) ,  dn k –ik⋅x Fx–1 [1] = e ≡ $(n) (x) , (20)n

(B.60)

(B.61a) (B.61b)

and the Fourier transform of 1/k leads to a Heaviside (-function:   +∞ 1 1 1 dk Fx ≡ ((x) = – lim e–i kx , %→0 20i k k + i% = – lim

%→0

1 20i

–∞ +∞

dk

1 ei kx , k – i%

(B.62a)

(B.62b)

–∞

where the integration should be made by choosing the appropriate complex contour. A Dirac $-function having a complex argument is in general not well defined, as its exponential representation is divergent:1    dn k –ik(x+iy) ? –1 ek y .  (B.63) $n (x + iy) = e = F x (20)n But we will allow this notation anyway, because sometimes a function acts as a nascent $: -function, which implies that–in combination with the sifting property–it behaves exactly the same as a regular $-function (mostly under certain conditions). It is possible that such nascent $: -functions allow for complex arguments and still retain their sifting property (see, e.g. the discussion of the infinite Wilson line on page 72). It is for these situations that we allow the notation of a complex $-function, but we keep in mind that it can only be used together with the sifting property and has no exponential representation. 1 The only non-divergent Fourier transform of a linear real exponential is that of e–a|x| with a > 0.

174

B Conventions and Reference Formulae

To conclude this section, we list two other common transformations. First, we have the Laplace transform: f (x) =

1 20i ∞

c+i ∞

dn s f (s) es⋅x , c–i ∞

dn x f (x) e–s⋅x ,

f (s) =

(B.64a)

(B.64b)

0

and second the Mellin transform: 1 f (x) = 20i

c+i∞ 

dn s f (s) x–s ,

(B.65a)

c–i∞

∞ dn x f (x) xs – 1 ,

f (s) =

(B.65b)

0

which is quite common in quantum chromodynamics (QCD) (e.g. it is the driving transform behind the convolution in the so-called DGLAP equations). Both for the inverse Laplace integral and the inverse Mellin integral, c is chosen such that it is bigger than all singularities in f (s).

B.6 Lie Algebra B.6.1 Representations Let’s revise some basic color algebra. As is well known, the group which governs QCD is SU(3), but for the sake of generality, we list some basic rules and derive some properties for SU(N) (more specifically, for su(n), the corresponding Lie algebra of SU(N)). The latter is fully defined by dA = N 2 – 1 linearly independent Hermitian generators ta and their commutation relations [ta , tb ] = i f abc tc ,

(B.66)

where the f abc are real and fully antisymmetric constants (the so-called structure constants). The structure constants themselves satisfy the Jacobi identity: f ab x f x cd – f ac x f x bd + f ad x f x bc = 0 .

(B.67)

From a mathematician’s point of view, any set of generators (not necessarily Hermitian) that satisfy the commutation relations and the Jacobi identity define a Lie algebra. In practice, we will work with representations of the algebra, where the generators are represented by dR × dR Hermitian matrices, with dR as the dimension of the

B Conventions and Reference Formulae

175

representation. Two representations of particular interest are first the fundamental representation which has dimension dF = N.2 It has the additional unique property that its matrices, if complemented with the identity matrix, form a set 

 1, ta .

that acts as a basis for the generator products under matrix multiplication. The second important representation is the adjoint representation, which is constructed from the structure constants: 

Ta

 bc

= –i f abc ,

and has dimension dA = N 2 – 1. We will make the distinction in notation by writing the fundamental with lowercase t and the adjoint with uppercase T. Note that in literature, several different notations exist (e.g. tF and tA ). B.6.2 Properties All matrices are traceless in every representation:   tr ta = 0 . The trace of two matrices is zero if they are different:   tr ta tb = DR $ab .

(B.68)

(B.69)

DR is a constant depending on the representation. In the fundamental representation, this is by convention almost always DF = 21 . Summing all squared matrices gives an operator that commutes with all other generators (and combinations of generators), the so-called Casimir operator t a t a = CR 1 ,

(B.70)

Again, CR is a constant depending on the representation. Both constants can be easily related   ta ta = CR 1 ⇒ tr ta ta = CR tr (1) = CR dR ,     tr ta tb = DR $ab ⇒ tr ta ta = DR $aa = DR dA , ⇒

(B.71a) (B.71b)

CR D R = . dA dR

2 A small remark: when working in QCD, It is common to denote the dimension of the fundamental representation by Nc , as it represents the number of colors used in the theory. In this section of the appendix, we will use the notation N to keep it general, but in the body of this book, we will use the notation Nc to enhance interpretation.

176

B Conventions and Reference Formulae

These properties are valid for any representation, not only the adjoint. Let us now list the constants for the fundamental and the adjoint representation: DF =

1 , 2

CF = DF

DA = 2DF dF = N , dA N 2 – 1 = , dF 2N

CA = DA = N , dA = d2F – 1 = N 2 – 1 .

dF = N ,

Because in the fundamental representation, we have a basis that spans its full product space, we can derive additional properties that are not valid in other representations. First of all, the anticommutator has to be an element of the algebra, and thus a linear combination of the identity and the generators: {ta , tb } =

1 ab $ 1 + dabc tc . N

(B.72)

The constant in front of the identity was calculated by taking the trace and comparing to equation (B.69), while dabc can be retrieved, as well as f abc , from i  a b c tr [tR , tR ]tR , DR   = 2tr {ta , tb }tc .

f abc = –

(B.73a)

dabc

(B.73b)

Equation (B.73a) is valid in any representation, while equation (B.73b) only makes sense in the fundamental representation. Because almost every calculation ends with a full colour trace, having the identity matrix written implicitly is dangerous, as one might forget to add the factor N coming from its trace. For this reason, we will often write 1 N together, as it is a factor that is trace-normalized to 1. It is easy to check that the dabc are fully symmetric and that they vanish when contracting any two indices: daab = dbaa = daba = 0 . It is interesting to note that in SU(2), all dabc vanish. By combining the commutation rules with the anticommutation rules, we can find another useful property: 7 a b8 " a b# t ,t + t ,t t t = 2 1 ab 1 1 abc c = $ + h t , 2 N 2 a b

(B.74)

B Conventions and Reference Formulae

177

where we defined habc = dabc + i f abc .

(B.75)

habc is Hermitian and cyclic in its indices: habc = h abc

h

aab

h

bac

bca

=h

baa

=h

cba

=h

cab

=h

aba

=h

acb

=h

,

, = 0.

A last useful property is the Fierz identity  a  a 1 1 t ij t kl = $il $jk – $ij $kl . 2 2N

(B.76)

It is straightforward to prove this identity; first, we write a general element of the fundamental representation as X = c0 1 + ica ta ,

(B.77)

which is true only in the fundamental representation.3 The c0 and ca are easily calculated: 1 tr(X ) , N   a c = –2i tr Xta .

c0 =

We then get the requested by calculating

∂(X)ij ∂(X)kl

= $ik $jl . The Fierz identity is especially

handy to rearrange traces containing contractions:  1  1 tr(ABC) , tr A ta B ta C = tr(AC) tr(B) – 2 2N  a a tr t B t = CF tr(B) ,     1 1 tr A ta B tr C ta D = tr(ADCB) – tr(AB) tr(CD) , 2 2N

(B.78a) (B.78b) (B.78c)

where A, B, C, D are expressions built from ta s. But it can be used for standard products as well, e.g. ta A ta =

N 1 1 tr(A) – A. 2 N 2N

3 This is because in other representations the set {1, tFa } doesn’t span the full product space.

(B.79)

178

B Conventions and Reference Formulae

B.6.3 Useful Formulae To conclude, we list–without proof–some useful properties of the constants f abc , dabc and habc . Most of these are easy to prove by straightforward calculation, but see [105] for a nice approach using tensor products for the more difficult ones. The Jacobi identity can be extended to include dabc , now with all positive signs: f ab x dx cd + f ac x dx bd + f ad x dx bc = 0 .

(B.80)

If we want to find a Jacobi identity with habc , we have to add the Hermitian conjugate due to the sign change: f ab x hx cd + f ac x hx bd + f ad x hx bc = 0 .

(B.81)

A Jacobi identity with only dabc also exists, but only in SU(3): dab x dx cd + dac x dx bd + dad x dx bc =

 1  ab cd $ $ + $ac $bd + $ad $bc . 3

(B.82)

Contracting any of the structure constants with a generator gives   f ab x tx = –i ta tb – tb ta , dab x tx = ta tb + tb ta – $ab hab x tx = 2ta tb – $ab hab x tx = 2tb ta – $ab

1 N

1

N

1 N

(B.83a) ,

(B.83b)

,

(B.83c)

.

(B.83d)

Contracting any of the structure constants with two generators gives N f a xy tx ty = i ta , 2 2–4 N da xy tx ty = ta , 2N 2 ha xy tx ty = – ta , N N2 – 2 a ha xy tx ty = t . N

(B.84a) (B.84b) (B.84c) (B.84d)

Contracting any of the structure constants with three generators gives N2 1 CF , 2 N N2 – 4 1 dx yz tx ty tz = CF , 2 N f x yz tx ty tz = i

(B.85a) (B.85b)

B Conventions and Reference Formulae

1 , N   1 hx yz tx ty tz = N 2 – 2 CF , N

hx yz tx ty tz = –2CF

179

(B.85c) (B.85d)

Tracing any two structure constants gives f x a y f y b x = –N$ab , d

xay ybx

f

= 0, N 2 – 4 ab $ , N N 2 – 2 ab =2 $ , N 4 = – $ab , N N 2 – 2 ab =2 $ . N

(B.86a) (B.86b)

d x a y dy b x =

(B.86c)

hx a y hy b x

(B.86d)

hx a y hy b x hx a y hy b x

(B.86e) (B.86f)

And tracing any three structure constants gives N f x a y f y b z f z c x = – f abc , 2 N abc xay ybz zcx f =– d , d f 2 N 2 – 4 abc xay ybz zcx d d f = f , 2N N 2 – 12 abc dx a y d y b z d z c x = d , 2N 2 N – 3 abc hx a y hy b z hz c x = 2 h , N   2 hx a y hy b z hz c x = – 2habc + habc , N  2  abc hx a y hy b z hz c x = – h + 2habc , N 2–3 N hx a y hy b z hz c x = 2 habc . N

(B.87a) (B.87b) (B.87c) (B.87d) (B.87e) (B.87f) (B.87g) (B.87h)

Tracing four structure constants quickly becomes messy, so we only list a few f and d combinations:  1  ab cd f x a y f y b z f z c w f w d x = $ad $bc + $ $ + $ac $bd 2  N  ad x x bc (B.88a) + dad x dx bc , + f f 4   N ad x x bc (B.88b) f d – dad x f x bc , f x a y f y b z f z c w dw d x = 4

180

B Conventions and Reference Formulae

 N2 – 8 1  ac bd $ $ – $ab $cd – f ad x f x bc 2 4N N – dad x dx bc , 4 1  ab cd ac bd  N  ad x x bc ad x x bc  , = $ $ –$ $ – f f +d d 2 4

f x a y f y b z dz c w dw d x =

f x a y dy b z f z c w dw d x

(B.88c) (B.88d) (B.88e)

B.7 Summary of the Noether Theorems We make a short summary of the Noether theorems. Given the following transformation: x, → x, + :a X , a , a

6i → 6i + :

Iai



+ ∂, :

a



(B.89a) ,a Ki ,

(B.89b)

the Noether theorems state that if the Lagrangian remains invariant up to a divergence, i.e.   $L ≡ ∂, :a K , ,

(B.90)

we can construct the following quantities (the Noether tensor resp. Noether current resp. Noether charge): ∂L , a K , ∂∂- 6i i  $L , a ∂L , , a def ∂L a J = I + K – ∂- 6i – L $- X - a –K , a , ∂∂, 6i i $6i i ∂∂, 6i  def Qa = d3 x J 0 a ,

F ,- a =

def

(B.91a) (B.91b) (B.91c)

that are conserved ∂, ∂- F ,- a ≡ 0 , ,a

∂, J

(B.92a)

≡ 0,

(B.92b)

˙a

Q ≡ 0,

(B.92c)

∂- F -, a ≡ J , a .

(B.93)

and satisfy the additional relation

Furthermore, the equations of motion are given by $L a I ≡ ∂, $6i i



$L , a K $6i i

,

(B.94)

B Conventions and Reference Formulae

181

where the variational derivative is given by ∂L $L def ∂L = – ∂, . $6i ∂6i ∂∂, 6i

(B.95)

In case of a local internal symmetry, the defined charges don’t have a physical inter,a pretation and can be ignored. In case of a global symmetry, we have Ki = 0, implying that there is no Noether tensor (and the relation in equation (B.93) is invalidated). The equations of motion then simplify into $L ≡ 0. $6i

B.8 Feynman Rules for QCD The full Lagrangian for QCD is given by 2   1 / L = 8 i∂/ – m 8 – ∂, Aa- – ∂- Aa, + g 8A8 4   1 – g f abc ∂, Aa- A,b A-c – g 2 f abx f xcd Aa, Ab- A,c A-d , 2

(B.96)

where A/ = Aa, γ , ta . The sum over gluon polarization states depends on the gauge, and equals (where in both equations we have made the additional gauge choice . = 0): N



%, (k)%- (k) = –g ,-

(Lorentz)

(B.97a)

(LC)

(B.97b)

pol



%, (k)%- (k) = –g ,- +

pol

2k(, n-) n⋅k

where the LC gauge is defined by the vector n– as n– ⋅A = A+ = 0. The Lagrangian gives rise to the following (extensive) list of Feynman rules (see next page): p i, s p i, s p i, s p i, s k

μ, a k

μ, a

=

usi (p) (initial)

(B.98a)

=

usi (p) (final)

(B.98b)

=

vsi (p)

(B.98c)

=

vis (p) (final)

(B.98d)

=

%a, (k)

(B.98e)

=

%a, (k) (final)

(initial)

(initial)

(B.98f)

182

B Conventions and Reference Formulae

p i p i

a, μ

k

a

=

   $ij $–+ p2 – m2 p/ + m

=

LC

k

a, μ

j

b, ν

k

a, μ

i $ij

b, ν

k

a, μ

=

Lorentz

k

p/ + m p2 – m2 + i:

j

=

(B.98i)

(B.98j)

(B.98k)

LC

=

   k(, n-) –$ab $–+ k2 g ,- – 2 k⋅n

(B.98l)

=

i $ab k2 + i%

=

i n⋅k + i'

(B.98n)

=

$(n⋅k – + i')

(B.98o)

=

i gEM ,:EM γ , $ij

(B.98p)

=

  ig,: γ , ta ji

(B.98q)

=

g,: f abc k,

=

  i g,: n, ta ji

(B.98s)

=

eir⋅k

(B.98t)

Lorentz

b, ν

k, n

(B.98h)

  –$ab $–+ k2 g ,-

b, ν

b

  –i $ab k, k,– 1 – . g ( ) k2 + i% k2   k(, n-) k2 k, k–i $ab ,– 2 g + . k2 + i% k⋅n (k ⋅ n)2

(B.98g)

=

(only Lorentz gauges)

(B.98m)

k, n

j

i μ

j

i μ, a k

a

c

(only Lorentz gauges)

(B.98r)

μ, b i

k, n

j k

μ, a k r

+∞ =

1

(no momentum flow)

(B.98u)

183

B Conventions and Reference Formulae

ρ, c

ν, b p

q k μ, a

ν, b

=

(B.98v)

" –ig 2 ,2: f abx f xcd (g ,1 g -3 –g ,3 g -1 ) – f acx f xbd (g ,3 g -1 –g ,- g 13 ) # + f adx f xbc (g ,- g 13 –g ,1 g -3 )

(B.98w)

ρ, c

= μ, a

" g,: f abc g ,- (k–p)1 + g -1 (p–q), # + g 1, (q–k)-

σ, d

Furthermore: – Momentum conservation is imposed at every vertex. –

Loop momenta have to be integrated with an additional factor 1/(20)4 .

–

Fermion loops (hence ghost loops as well) add an additional factor –1.

–

All Feynman rules are complex conjugated when on the right side of a final-state cut. Additionally, the 3-gluon vertex and the ghost vertex change sign if on the right side of a final-state cut.

–

The final result has to be divided by the symmetry factor of the diagram. For a cut diagram: multiply with a symmetry factor for each side.

–

For each set of k indistinguishable particles in the final state, divide by k! .

–

Impose momentum conservation between initial and final states. 9 Divide by the flux factor. It is 4 (p1 ⋅ p2 )2 – m21 m22 when there are exactly two incoming particles.

–

C Integrations In this appendix, we list some common techniques to solve integrals and give some reference formulae as well.

C.1 Reference Integrals We start with regular integrals, sorted by type. Note that we omitted the constant term +c that appears in indefinite integrals because of space constraints.

C.1.1 Algebraic Integrals The easiest type of algebraic integrals is binomial integrals: m   m  m 1 nk+1 m–k dx xn +! = a , x k nk+1 0  m+1   m  1 nk+1 m+1–k m+1 k a , x dx xn +! xn = m+1 nk+1 k 0  m  r    n  m  p r  m r 1 x +b = dx x +! xnk+pl+1 am–k br–l . k l nk+pl+1



(C.1a)

(C.1b)

(C.1c)

k=0 l=0

Next, we have rational integrals. First, we list some properties of complex logarithms: i 1 – ix ln 2 1 + ix 1+ x 1 artanh x = ln 2 1–x ln (–1) = i 0 + 2i k0, arctan x =

ln (i) =

i0 + 2i k0, 2

(C.1d) (C.1e) ln (1) = 2i k0 ln (–i) = 3

i0 + 2i k0 2

(C.1f) (C.1g)

For the correct derivation of complex logarithms, use polar representation: i = 0 ei( 2 +2k 0) ⇒ ln (i) = i02 + 2ik0. We evaluate logarithms in the region [0, 20]. Rational integrals will almost always result in a combination of polynomials and/or logarithms: 

1 i 1 – ix = ln , 1 + x2 2 1 + ix  1 1+ x 1 = ln , dx 1 – x2 2 1 – x dx

(C.1h) (C.1i)

C Integrations



√ ⎧ √1 ln √B–(2ax+b) , ⎪ ⎪ B+(2ax+b) ⎨ B



B > 0,

1 –2 = 2ax+b dx 2 , √ ax +bx+c ⎪ ⎪ ⎩ √i ln √B–i(2ax+b) , B

B = 0,

b√ 2a B

x 1 dx 2 = a1 ln (2ax+b) + ba 2ax+b , ax +bx+c ⎪   ⎪ ⎩ 1 ln ax2 +bx+c – √ ib 2a

dx 

xn x–a

= an ln(x – a) +

k=0

ak

xn–k n–k

√

ln √B+(2ax+b) ,

2a –B

n–1 

(B = b2 – 4ac)

(C.1j)

B < 0,

B+i(2ax+b)

⎧   1 ⎪ ln ax2 +bx+c + ⎪ ⎨ 2a



185

B–(2ax+b)

ln

√ √–B–i(2ax+b) , –B+i(2ax+b)

B > 0, B = 0,

(C.1k)

B < 0,

,

(C.1l)

 k an xn n–1 = n a ln x – a + + ak–1 xn–k , ( ) (x – a)2 a–x n–k n–1

dx  dx

xn m $ (x–ai )

=

m  i=1

an $ i ln(x–ai ) + (ai –aj ) j≠i

k=1 n–1  m  k=1 i=1

ak xn–k $ i . (ai –aj ) n–k

(C.1m) (C.1n)

j≠i

Algebraic integrals with real exponents lead to a Beta function (see equation (C.7a)) or an incomplete Beta function (see equation (C.7c)): ∞ dx

x! = a1+!–" B(" – ! – 1, ! + 1) , (x + a)"

(C.1o)

dx

 (x + a)! b–a 1+!–" = (b – a) B ; " – ! – 1, ! + 1 . b+c (x + b)"

(C.1p)

0

∞ c

C.1.2 Logarithmic Integrals The most common logarithmic integrals contain a logarithm and a polynomial:

 n+1 k    n+1 1 x (–!)n+1–k n+1 n ln |x+a| – x – (–!) , (C.2a) dx x ln |x+!| = n+1 k 1 ⎛ ⎞  n/2 2k+1 n–2k % x+a % % x+! %    1 x ! n odd % % % ⎝ xn+1 –!n+1 ln %% ⎠, (C.2b) dx xn ln % % = %+2 x–! n+1 x–a 2k+1 0 ⎛ ⎞  n/2 2k n+1–2k % x+! % % % % 2 2%  x ! % % n even 1 ⎝ n+1 % x+a % n n+1 ⎠. (C.2c) x ln % dx x ln % % = % + ! ln %x –a % + x–! n+1 x–a k 1 We also list a few integrals of a logarithm divided by x:  ln(x – a) 1 dx = ln2 (x – a) , x–a 2   x–b ln(x – a) a ≠ b = ln(x – b) ln(b – a) – Li2 , dx x–b a–b

(C.2d) (C.2e)

186

C Integrations



 1  ln(x – a) ln(x – a) + 1 , = – 2 (x – a) x–a   ln(x – a) a ≠ b 1  x–a dx = ln(x – b) – ln(x – a) , (x – b)2 b–a  x–b  1 ln(x – a) a ≠ b 1 = ln(x – b) ln(b – a) – ln2 (x – a) dx (x – a)(x – b) b–a 2   x–b Li2 , a–b   ln(x + a) a ≠ b ≠ c 1 = ln(x – b) ln(b – a) – ln(x – c) ln(c – a) dx (x – b)(x – c) b–c   x – c  x–b –Li2 + Li2 , a–b a–c dx

(C.2f) (C.2g)

(C.2h)

(C.2i)

where Li2 is the dilogarithm, a specific form of the polylogarithm Lis (see Appendix C.2.3). Note that the distinction between a = b and a ≠ b is mathematically not needed, but we prefer to explicitly state the different integral as the limit b → a is not so obvious. C.1.3 Cyclometric Integrals We list two important cyclometric integrals:   1  dx arctan x = x arctan x – ln 1 + x2 , 2   1  dx artanh x = x artanh x + ln 1 – x2 . 2

(C.3a) (C.3b)

C.1.4 Gaussian Integrals Gaussian integrals are by far the most common integrals in physics in general. We list the most common one-dimensional integrals:  +∞ 0 –ax2 dx e = , a

(C.4a)

–∞

 +∞ 0 b2 +c –ax2 +bx+c dx e = e 4a , a –∞

∞

2

dx x2n e–ax = 0

∞

2

dx x2n+1 e–ax = 0

∞

n –ax2 +bx+c

dx x e 0

1 (2n – 1)!! 2 2n an

(C.4b) 

  0 1 A n + 21 , = a 2 an+ 21

1 n! 1 A (n + 1) = , 2 an+1 2 an+1

  k+1 n   1 b2 +c  n b n–k A 2 4a = e , k+1 2 2a k a 2 0

(C.4c)

(C.4d)

(C.4e)

187

C Integrations

and the most common multidimensional integrals: 

 n

d xe

–xi Aij xj

= 

 

ij x

dn x x,1 ⋅ ⋅ ⋅ x,n e–xi A

j

(C.4f)

1 bi A–1 bj +c 0n , e 4 ( )ij det A  0n n even = A–1 A–1 ⋅ ⋅ ⋅ A–1 ,n–1 ,n ) , det A (,1 ,2 ,3 ,4

ij x +bi x +c j i

dn x e–xi A

0n , det A

=

(C.4g) (C.4h)

where always only the symmetric part of A contributes to the determinant. One special integral is one that is encountered often when variables are chained:  dx1 ⋅ ⋅ ⋅ dxn e

"

i+ (x1 –a)2 +(x2 –x1 )2 +⋅ ⋅ ⋅+(b–xn )2

6

# =

1 n+1



i0 +

n

+

2

ei n+1 (b–a) .

(C.4i)

Gaussian integrals can also be expressed as integrations over complex variables. Then the square root is gone and an in is added: 

ij z j

dn z dn z e–zi A 

ij z +wi z +zi u j i i

dn z dn z e–zi A

(20 i)n , det A (20 i)n wi (A–1 )ij uj = . e det A =

(C.4j) (C.4k)

C.1.5 Discrete Integrals For completeness and because we need it in Chapter 3, we give the discrete approximation to the integral over an infinitesimal line segment [a, a + :]: a+:  : dx f (x) ≈ :f a + . 2

(C.5a)

a

A macroscopic integral can then be approximated as a sum of such infinitesimal segments: b dx f (x) ≈ a

n x + x   i i–1 (xi –xi–1 )f , 2

(C.5b)

i

where x0 = a and xn = b. We can use the same method to discretize line integrals: a+:  : dx, f, (x) ≈ :, f, a + , 2

(C.5c)

a

b a

dx, f, (x) ≈

n x + x   i i–1 (xi –xi–1 ), f, . 2 i

(C.5d)

188

C Integrations

C.2 Special Functions and Integral Transforms In this section, we list some common integral relations and transforms, and special functions.

C.2.1 Gamma Function The Gamma function is probably the most well-known special function. It is defined as ∞ def

dt tz–1 e–t

A(z) =

Re(z) > 0 .

(C.6a)

0

For n ∈ N0 , it is related to the factorial A(n) = (n – 1)!

(C.6b)

and has a similar “factorial” property, but for all z: A(z + 1) = z A(z) .

(C.6c)

We can also relate it to the double factorial (now for n ∈ N, i.e. n can be 0):  √ (2n – 1)!! 1 , = 0 A n+ 2 2n

(C.6d)

where the double factorial multiplies every second number: def

a!! = a(a – 2)(a – 4) ⋯,

(C.6e)

a! ≡ a!! (a – 1)!!.

(C.6f)

which gives the logical relation

An important value of the Gamma function is A

 √ 1 = 0, 2

(C.6g)

which is just a re-expression of a Gaussian integral. There are four inter-related reflection formulae:

C Integrations

0 , sin(0z) 0 A(z)A(–z) = – , z sin(0z)   1 0 1 A –z = , A z+ 2 2 cos(0z) A(z)A(1 – z) =

189

(C.6h) (C.6i) (C.6j)

 1 1 2z–1 A(z)A z + A(2 z) = √ 2 . 2 0

(C.6k)

The Gamma function has poles in 0 and all negative integers. It can however be expanded around these poles:  (–)n 1 n ∈ N0 , (C.6l) A(: – n) = + 8(0) (n+1) + O(:) , n! : where 8(0) is the digamma function, defined as the logarithmic derivative of the Gamma function: def

8(0) (z) =

A′ (z) . A(z)

(C.6m)

For integer values n > 0, it equals 8(0) (n) = –γE +

n–1  1 j=1

j

.

(C.6n)

The Gamma function is especially useful when solving integrals, as we can often express the integral at hand in the form of equation (C.6a). For easy reference, we list a representation with a quadratic exponential: ∞

1 ! 2 dt t!–1 e–t = A , 2 2

Re(!) > 0 .

(C.6o)

0

We can extend equation (C.6a) to allow for a complex contour. This gives the following complex Gamma representations, that are bound to strict convergence criteria: ∞

dt t!–1 ei(A+i B)t = A(!) i! (A + i B)–!

∀A, B > 0, Re(!) > 0 ,

(C.6p)

∀A, B > 0, Re(!) > 0 ,

(C.6q)

∀A, 0 < Re(!) < 1 .

(C.6r)

0

∞ 0

dt t!–1 e–i(A–i B)t = A(!)(–i)! (A + i B)–! ∞ 0

dt t!–1 e±i At = A(!)(±i)! A–!

190

C Integrations

Note that integrals with an exponent ei(A–i B) or e–i(A+i B) are divergent for B > 0. For easy reference, we also list some complex quadratic representations: ∞ 0 ∞

0

! 1 ! ! 2 dt t!–1 ei(A+i B)t = A i 2 (A + i B)– 2 2 2

! ! 1 ! 2 dt t!–1 e–i(A–i B)t = A (–i) 2 (A – i B)– 2 2 2

∞

! ! 1 ! 2 dt t!–1 e±i At = A (±i) 2 A– 2 2 2

∀A, B > 0, Re(!) > 0 ,

(C.6s)

∀A, B > 0, Re(!) > 0 ,

(C.6t)

∀A, 0 < Re(!) < 2 .

(C.6u)

0

C.2.2 Beta Function The Beta function is defined as 1 def

B(!, ") =

dt t!–1 (1 – t)"–1 ,

Re(!), Re(") > 0.

(C.7a)

0

It can be expressed in terms of A-functions: B(!, ") ≡

A(!)A(!) . A(! + ")

(C.7b)

When solving complex integrals, it is sometimes convenient to use the incomplete Beta function, which is defined as z def

B(z; !, ") =

dt t!–1 (1 – t)"–1 ,

(C.7c)

0

such that B(1; !, ") ≡ B(!, ") .

(C.7d)

A useful property is its mirror symmetry: B(x; !, ") = B(1 – x; ", !) ,

(C.7e)

which follows directly from the definition. Especially helpful is its series expansion (see [3]):

∞  B(! + 1, n + 1) n+1 1 ! " . (C.7f) x B(x; !, ") = x (1 – x) 1 + ! B(! + ", n + 1) n=0 C.2.3 Polylogarithms When integrating logarithms, the result will often depend on the so-called polylogarithm, which is defined by its series expansion:

C Integrations

def

Lis (z) =

∞ k  z k=1

191

.

(C.8a)

Lis (t) . t

(C.8b)

ks

It can also be defined as a recursive integral: z Lis+1 (z) =

dt 0

For s ≤ 1 and s being an integer, the polylogarithm can be expressed as a regular function: Li1 (z) = – ln(1 – z), z , Li0 (z) = 1–z  ∂ n z . Li–n (z) = z ∂z 1 – z

(C.8c) (C.8d) (C.8e)

The polylogarithm of 0 is always 0 itself, and the polylogarithm of 1 equals the Riemann & -function: Lis (0) = 0,

(C.8f)

Lis (1) = & (s).

(C.8g)

Of particular interest is the following asymptotic behaviour: Lis (e: ) = A(1 – s)(–:)s–1 ,

(C.8h)

which is valid for |:| → 0 and Re(s) < 1. The polylogarithm emerges naturally in the solution to Bose–Einstein and Fermi–Dirac integrals: ∞ dk

ks = A(s + 1) Lis+1 (e, ), –1

(C.8i)

ek–,

0

∞ dk

ks ek–, + 1

= –A(s + 1) Lis+1 (–e, ),

(C.8j)

0

where A is just the Gamma function. The most common polylogarithm is the dilogarithm Li2 . Three particular values are: Li2 (1) = & (2) =

02 , 6

Li2 (0) = 0,

Li2 (–1) = –

02 . 12

(C.8k)

It satisfies some additional useful properties: 1 Li2 (z) + Li2 (–z) = Li2 (z2 ), 2 02 Li2 (z) + Li2 (1 – z) = – ln z ln(1 – z), 6

(C.8l) (C.8m)

192

C Integrations

 02 1 1 Li2 (z) + Li2 = – – ln2 (–z), z 6 2  1 2 1 Li2 (1 – z) + Li2 1 – = – ln z, z 2 1 02 2 Li2 (–z) – Li2 (1 – z) + Li2 (1 – z ) = – – ln z ln(1 + z). 2 12

(C.8n) (C.8o) (C.8p)

However, only the first two relations are valid for the full complex plane. C.2.4 Elliptic K-Function The elliptic K-function is defined as 20 def

d> 9

K(k) =

0

1

.

(C.9a)

1 – k sin2 >

It is divergent for k = 1 and becomes complex for k > 1. A related integral is 20 d>  0

1

= √

a + b cos >

4 a+b

 K

2b , a+b

(C.9b)

which is divergent for a = ±b. C.2.5 Integral Transforms An integral transform that we will use a lot is the so-called Schwinger parameterization, which is a complex exponential integral representation for a denominator (where k ∈ R and % > 0): ∞

1 = –i k + i%

d! ei(k+i %)! ,

(C.10a)

d! e–i(k–i %)! .

(C.10b)

0

1 =i k – i%

∞ 0

We can summarize this if we define 3 = ±1 to be the sign in front of i:: 1 = –i 3 k + i 3%

∞

d! ei 3(k+i 3%)! .

(C.10c)

0

Note that we can use lower half integrals as well: 1 = –i k + i% 1 =i k – i%

0

d! e–i(k+i %)! ,

(C.10d)

–∞ 0

–∞

d! ei(k–i %)! .

(C.10e)

C Integrations

193

It is possible to define a similar parameterization with a real exponential, but then the sign of k matters: 1 = k

∞ d! e–k! ,

k > 0,

(C.10f)

k < 0.

(C.10g)

0

1 =– k

∞ d! ek! , 0

Having to split up an expression in two terms in function of the sign of k is a bit cumbersome, which is why we won’t use the latter parameterization. Another parameterization we will often use is the so-called xL-parameterization that is used to simplify integrals of the form ∞∞ d! d" f (!, ") .

(C.11a)

0 0

The trick is to use the parameterization ! = xL,

" = (1 – x)L ,

d! d" = L dL dx .

(C.11b)

The integral then simplifies into 1

L dx

0

  dL L f xL, (1 – x)L ,

(C.11c)

0

which is often easier to solve when starting with the L integration. It can be easily generalized to any number of integrations: 1

L dx1 . . . dxn

(n – 1)! 0

dL $ 1 –

n 

xi

  Ln–1 f x1 L, . . . , xn L .

(C.11d)

0

The normalization comes from the fact that 1

dx1 . . . dxn $ 1 –

n 

xi

=

1 . (n – 1)!

(C.11e)

0

One word of caution, however, is that the $-function will chain its influence in all the xi integrations. This is due to the fact that for non-infinite borders, the sifting property gains additional (-functions (see equations (B.52)). This implies in general:

194

C Integrations

1



1 dx1 . . .

0

dxn $ 1 –

n 

f (x1 , . . . , xn )

xi

0

1 = 0

n–2 /

1– xi 1–x1 1–x  1 –x2 n–1  dx1 dx2 dx3 . . . dxn–1 f x1 , . . . , xn–1 , 1– xi , 0

0

(C.11f)

0

1 dt1 dt2 . . . d tn–1 t1n–2 t2n–3 . . . tn–2

= 0

× f (1–t1 , t1 (1–t2 ), t1 t2 (1 – t3 ), . . . , t1 t2 . . . tn–1 ) .

(C.11g)

To get to the last step, we used the transformation ⎛ xi = ⎝

i–1

⎞ tj ⎠ (1 – ti ) .

j

Each of these three expressions can be more easy to use or not, depending on the structure of f . If we combine the Schwinger and the xL-parameterization, we get the well-known Feynman parameterization: 1 1 = AB

1 dx

1 . (xA + (1 – x)B)2

(C.12a)

0

We can easily generalize this for n fractions, each with a power mi :  A

m

A1 1

n /

mi

1 = n A(m1 ) . . . A(mn ) . . . Am n



1

dx1 . . . dxn $ 1 –

n 

0

xi

m –1

x1 1 . . . xnmn –1 n / mi . (C.12b) / xi Ai

Whether we use the Schwinger parameterization, the xL-parameterization or the Feynman parameterization is situation dependent, but remains above all a matter of personal taste. Closely related to these parameterizations is the fact that we can use the Gamma function to our advantage: 1 1 = ! A A(n)

∞

dt t!–1 e–A t ,

0

which can be a huge simplification when A is not too complicated.

(C.13)

C Integrations

195

Totally unrelated to the integral transformations, but still worth mentioning, an important property of line integrals is the gradient theorem: b

dx, ∂, f (x) ≡ f (b) – f (a) .

(C.14)

a

C.3 Dimensional Regularization Here, we will list some common loop integrals as a quick reference. Most common loop integrands can be transformed into the form (k2 – B)–n , for which we will give a set of solutions. Furthermore, integrands with momenta in the numerator can be simplified as well. First of all, terms with an odd power of k vanish by symmetric integration. Also by symmetry arguments, we can replace k2 ,g , 9  ,- 13  k4 g g + g ,1 g -3 + g ,3 g -1 . k, k- k1 k3 → 9(9 + 2) k, k- →

(C.15a) (C.15b)

The most straightforward way to use dimensional regularization is when the integrand only depends on k2 . Because then we can make a Wick rotation, move to spherical coordinates, and calculate the angular part separately: 

2 d9–1 K = 9   . (20)9 (40) 2 A 92

(C.16)

Dimensional regularization is often accompanied by a subtraction scheme. We will only use the MS scheme and mostly with Collins’ convention of dividing the result by a factor S: =

(40): . A(1 – :)

(C.17)

The more common regular subtraction is done by dividing S: = (40e–γE ): ,

(C.18)

but we prefer Collins’ convention as it works better with the double poles that originate from light-like segments. C.3.1 Euclidian Integrals Let us now list some common Euclidian integrals in dimensional regularization. On the second line of each integral, we give the condition for it to be divergent and the expansion in the poles for the latter case.

196 

C Integrations

  1 A n – 92 9 –n 1 d9 kE B2 ,   = (20)9 kE2 + B n (40) 92 A(n)





d ≥ 2n d even



(C.19a)

⎛

⎞

d

d 2 –n  1 (–) 2 –n ⎜1 ⎟ = + – γ + ln 40 – ln B⎠,   ⎝ E d d : j (n–1)! –n ! 2 (40) 2 d

B 2 –n

  kE2 1 9 A n – 92 – 1 9 +1–n d9 kE = , B2   (20)9 kE2 + B n (40) 92 2 A(n)



(C.19b)

⎛



d +1–n 2

⎞

 1 9 (–) ⎜1 ⎟ + ln 40 – ln B⎠, d  ⎝ – γE + 2 : j (n–1)! +1–n ! (40) 2    9 kE4 1 9(9+2) A n – 92 – 2 9 +2–n d kE , (C.19c) B2   = (20)9 kE2 + B n (40) 92 4 A(n) ⎛ ⎞ d +2–n 2  d +2–n d –n  1 ⎜1 ⎟ B2 9(9+2) (–) 2 d ≥ 2n–4 = –γE + + ln 40 – ln B⎟ d  ⎜ d ⎝ ⎠. d even 4 j (n–1)! 2 +2–n ! : (40) 2 d ≥ 2n–2 d even

=

B

d +1–n 2

d +1–n 2

d 2

We can generalize an Euclidian dimensionally regulated integral to real positive values of the exponents as 

     2 ! 9 9 kE 9 +!–" A ! + 2 A " – ! – 2 d9 kE 1   . = B2 (20)9 k2 + B" (40) 92 A 92 A (") E

(C.20)

From this we can deduce that 

 ! d9 kE kE2 = 0 ,

(C.21)

"→0

because in the denominator we have A(") → 1/", hence the fraction goes to 0 in this limit. This is valid ∀! ≥ 0. For any function that only depends on the square of the momenta, we can write 

d9 kE ,1 , k . . . kEn f (kE2 ) = (20)9 E $(,1 ,2 . . . $,n–1 ,n )

A

1 9

(40) 2

n

+ 21



2 √ 0 A 2 + n2  92

∞

df kE9–1+n f (kE2 ) ,

(C.22)

0

for n even (for odd n the integral is 0). Note that in the case of a Minkowskian integral, the $-functions are replaced with g ,- s.

197

C Integrations

C.3.2 Wick Rotation and Minkowskian Integrals Calculating Minkowskian loop integrals can be straightforwardly done by Wick rotating the momenta to Euclidian space, by making the substitution def

k0 = i kE0 ,

k2 = –kE2 .

(C.23)

There are some intricalities with Wick rotations, as one has to make sure not to cross the poles. See Section 6.3 for a digression on this topic. Furthermore, one has to be consistent in the whole formulation. For example, $-functions change as well under a Wick rotation: $(n) (k) = i$(n) E (kE ) .

(C.24)

To see this, we move to the exponential representation of the $-function:  $(n) (k) =

 n dn x i k⋅x d xE –i kE ⋅xE e = i e = i $(n) E (kE ) . (20)n (20)n

The Minkowskian loop integrals are then the same as the Euclidian ones, up to a possible sign difference: 

  d9 k (–)n A n – 92 9 –n 1 B2 ,   =i 9 (20)9 k2 – B n A(n) (40) 2





⎛

⎞ d –n 2  1 (–) B ⎜1 ⎟ =i + ln 40 – ln B⎠,  d  ⎝ – γE + d : j (n–1)! –n ! (40) 2 2 d –n 2

d 2

  (–)n+1 9 A n – 92 – 1 9 +1–n k2 d9 k , B2   =i 9 (20)9 k2 – B n A(n) (40) 2 2





d ≥ 2n d even



(C.25a)

d ≥ 2n–2 d even



(C.25b)

⎛

⎞  1 9 B (–) ⎜1 ⎟ =i + ln 40 – ln B⎠, d  ⎝ – γE + d 2 : j (n–1)! +1–n ! 2 (40) 2 d +1–n 2

d +1–n 2

d 2

  (–)n 9(9+2) A n – 92 – 2 9 +2–n k4 d9 k , B2   =i 9 (20)9 k2 – B n 4 A(n) (40) 2



d ≥ 2n–4 d even

⎛

=i

B

d +2–n 2

(40)

d 2

(C.25c) d +2–n 2

⎞

1 ⎜1 ⎟ 9(9+2) (–) –γE + + ln 40 – ln B⎟ d  ⎜ ⎝ ⎠. 4 j (n–1)! 2 +2–n ! : d 2

198

C Integrations

We list some other common Minkowskian integrals: 

 9 9 i d9 k ln(k2 – a) = – a2 , 9 A – 9 (20) 2 (40) 2  b2 i d9 k ak2 –i b⋅k – 92 4a e = e , 9 a 9 (20) (40) 2    9 d k i A 92 – ! 1 1 –ib⋅k e = . 9 9 ! 2 (20)9 (–k2 )! A(!) 2 4 0 (–b ) 2 –!

(C.26a) (C.26b) (C.26c)

Two transversal integrations: 

1 1 d9–2 k⊥ –i !k⊥2 e = , 9 9 (20)9–2 (40) 2 –1 (i !) 2 –1    9–2 A 92 – ! – 1 d k⊥ 1 i k⊥ ⋅b⊥ 1 1 e = . 9 –1 9 2 ! (20)9–2 (k⊥2 )! A(!) 4 02 (b⊥ ) 2 –!–1

(C.27a) (C.27b)

One integral that frequently appears after making a (dimensionally regulated) transverse momentum integration is 

dk+ dk– i(a k+ k– + p+ k– + p– k+ + i %) 1 1 – i p+ p– e . = e a (20)2 20 a

(C.28)

C.4 Path Integrals C.4.1 Properties Every path integral is required to be linear and translation invariant. A rotation of the fields gives an extra determinant in front. These properties can be written together as 

    D6 aF[6] + bG[6] = a D6 F[6] + b D6G[6] ,   D6 F[L6 + 7] = det L D6 F[6] ,

(C.29a) (C.29b)

where we used the short-hand notation  N

L6 =

d4 x L(y, x)6(x) ,

(C.30)

inside the functional F. We will keep using this short-hand, just remember that the fields are integrated over their coordinates. The real scalar Gaussian path integral is given by

C Integrations



1 D6 e6K6 = NG √ , det K  N 6K6 = – d4 xd4 y 6(x)K(x, y)6(y) , √ NG = lim 0 n . n→∞

199

(C.31a) (C.31b) (C.31c)

If there is an extra linear term in the Gaussian exponent, we can complete the square: 

1

D6 e–6K6+J6 = e 4 JK

–1 J

 D6 e–6K6 .

(C.32)

We can use this property to calculate Gaussian integrals with extra field factors in front of the exponential:  D6 61 ⋅ ⋅ ⋅ 6n e

–6K6

 $ $ 1 JK –1 J %% 4 = ⋅⋅⋅ e D6 e–6K6 , % J=0 $J1 $Jn  (n – 1)!! –1 K(i1 i2 ⋅ ⋅ ⋅ Ki–1 D6 e–6K6 . = n n–1 in ) 22

(C.33) (C.34)

Complex scalar Gaussian path integrals are calculated in a similar way:  D6D6 e–6K6 =

NG , det A

NG = lim (20 i)n , n→∞   –1 –6K6+J6+J6 D6D6 e = eJK J D6D6 e–6K6 ,  D6D6 6i1 6j1 ⋅ ⋅ ⋅ 6in 6jn e–6K6 =

$ $ $ $ JK –1 J %% ⋅⋅⋅ e % J,J=0 $Ji1 $J j2 $Jin $J jn  × D6D6 e–6K6 .

(C.35a) (C.35b) (C.35c)

(C.35d)

In analogy with the discrete integration, we can give a path integral definition for the functional $-function:  & 4 $ (6) = D9 ei d x 6(x)9(x) . (C.36)

Bibliography [1] [2] [3] [4] [5] [6]

[7]

[8]

[9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19] [20]

[21]

G. A AD et al. (2012). “Study of jets produced in association with a W boson in pp collisions at √ s = 7 TeV with the ATLAS detector.” Phys. Rev. D85, 092002. arXiv:1201.1276 [hep-ex] G. A AD et al. (2015). “Measurements of the W production cross sections in association with jets with the ATLAS detector.” Eur. Phys. J. C752, 82. arXiv:1409.8639 [hep-ex] M. ABRAMOWITZ and I. STEGUN (1965). Handbook of Mathematical Functions. Dover Publications, New York. A. ACCARDI et al. (2012). “Electron Ion Collider: The Next QCD Frontier - Understanding the glue that binds us all.” . arXiv:1212.1701 [nucl-ex] A. AIRAPETIAN et al. (2009). “Observation of the naive-T-odd Sivers effect in deep-inelastic scattering.” Phys. Rev. Lett. 103, 152002. arXiv:0906.3918 [hep-ex] R. ANGELES-MARTINEZ et al. (2015). “Transverse Momentum Dependent (TMD) parton distribution functions: status and prospects.” Acta Phys. Polon. B4612, 2501–2534. arXiv:1507.05267 [hep-ph] M. ANSELMINO, M. BOGLIONE, U. D’ALESIO, E. LEADER, et al. (2006). “The general partonic structure for hadronic spin asymmetries.” Phys. Rev. D73, 014020. arXiv:hep-ph/0509035 [hep-ph] M. ANSELMINO, M. BOGLIONE, J. HANSSON, and F. MURGIA (1996). “Polarized inclusive leptoproduction, lN → hX and the hadron helicity density matrix, 1(h): Possible measurements and predictions.” Phys. Rev. D54, 828–837. arXiv:hep-ph/9512379 [hep-ph] M. ANSELMINO, M. BOGLIONE, and F. MURGIA (1995). “Single spin asymmetry for p↑ p → 0X in perturbative QCD.” Phys. Lett. B362, 164–172. arXiv:hep-ph/9503290 [hep-ph] M. ANSELMINO, E. LEADER, and F. MURGIA (1997). “Single spin asymmetries in DIS.” Phys. Rev. D56, 6021–6024. arXiv:hep-ph/9610407 [hep-ph] I. Y. AREFEVA (1980). “Quantum contour field equations.” Phys. Lett. B93, 347–353. I. AREFEVA (1980). “Non-Abelian Stokes formula.” Theor. Math. Phys. 43, 353. [Teor. Mat. Fiz. 43, 111(1980)]. E.-C. ASCHENAUER et al. (2015). “The RHIC spin program: Achievements and future opportunities.” . arXiv:1501.01220 [nucl-ex] E. AVSAR (2011). “On the understanding and use of ’unintegrated’ parton distributions in small-x QCD.” Int. J. Mod. Phys. Conf. Ser. 04, 74–84. arXiv:1108.1181 [hep-ph] E. AVSAR (2012). “TMD factorization and the gluon distribution in high energy QCD.” arXiv:1203.1916 [hep-ph] S. M. AYBAT, J. C. COLLINS, J.-W. QIU, and T. C. ROGERS (2012). “The QCD evolution of the Sivers function.” Phys. Rev. D85, 034043. arXiv:1110.6428 [hep-ph] S. M. AYBAT, A. PROKUDIN, and T. C. ROGERS (2012). “Calculation of TMD evolution for transverse single spin asymmetry measurements.” Phys. Rev. Lett. 108, 242003. arXiv:1112.4423 [hep-ph] A. BACCHETTA, C. BOMHOF, P. MULDERS, and F. PIJLMAN (2005). “Single spin asymmetries in hadron-hadron collisions.” Phys. Rev. D72, 034030. arXiv:hep-ph/0505268 [hep-ph] A. BACCHETTA (2011). “Transverse-momentum-dependent parton distributions (TMDs).” AIP Conf. Proc. 1374, 29–34. arXiv:1012.2315 [hep-ph] A. BACCHETTA (2014). “Transverse momentum distributions.” URL http://www2.pv.infn.it/ ~bacchett/teaching/Bacchetta_Trento2012.pdf. Lecture notes at the ECT* doctoral training programme. A. BACCHETTA, U. D’ALESIO, M. DIEHL, and C. A. MILLER (2004). “Single-spin asymmetries: The Trento conventions.” Phys. Rev. D70, 117504. arXiv:hep-ph/0410050 [hep-ph]

202

[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]

[43]

[44] [45]

Bibliography

A. BACCHETTA, M. DIEHL, K. GOEKE, A. METZ, et al. (2007). “Semi-inclusive deep inelastic scattering at small transverse momentum.” JHEP 02, 093. arXiv:hep-ph/0611265 [hep-ph] I. BALITSKY (1996). “Operator expansion for high-energy scattering.” Nucl. Phys. B463, 99–160. arXiv:hep-ph/9509348 [hep-ph] I. BALITSKY and A. TARASOV (2015). “Evolution of gluon TMD at low and moderate x.” Int. J. Mod. Phys. Conf. Ser. 37, 1560058. arXiv:1411.0714 [hep-ph] I. BALITSKY and A. TARASOV (2015). “Rapidity evolution of gluon TMD from low to moderate x.” JHEP 10, 017. arXiv:1505.02151 [hep-ph] I. BALITSKY and V. M. BRAUN (1989). “Evolution equations for QCD string operators.” Nucl. Phys. B311, 541–584. I. BALITSKY and L. LIPATOV (1978). “The Pomeranchuk singularity in quantum chromodynamics.” Sov. J. Nucl. Phys. 28, 822–829. J. S. BALL and T.-W. CHIU (1980). “Analytic properties of the vertex function in gauge theories. 2.” Phys. Rev. D22, 2550. R. D. BALL et al. (2015). “Parton distributions for the LHC Run II.” JHEP 04, 040. arXiv:1410. 8849 [hep-ph] V. BARONE, A. DRAGO, and P. G. RATCLIFFE (2002). “Transverse polarisation of quarks in hadrons.” Phys. Rept. 359, 1–168. arXiv:hep-ph/0104283 [hep-ph] V. BARONE and E. PREDAZZI (2002). High-energy particle diffraction. Theoretical and Mathematical Physics. Springer, Berlin. T. BECHER and M. NEUBERT (2011). “Drell-Yan production at small qT , transverse parton distributions and the collinear anomaly.” Eur. Phys. J. C71, 1665. arXiv:1007.4005 [hep-ph] T. BECHER, M. NEUBERT, and D. WILHELM (2012). “Electroweak Gauge-Boson production at small qT : Infrared safety from the collinear anomaly.” JHEP 02, 124. arXiv:1109.6027 [hep-ph] T. BECHER, M. NEUBERT, and D. WILHELM (2013). “Higgs-Boson production at small transverse momentum.” JHEP 05, 110. arXiv:1212.2621 [hep-ph] A. V. BELITSKY, X. JI, and F. YUAN (2003). “Final state interactions and gauge invariant parton distributions.” Nucl. Phys. B656, 165–198. arXiv:hep-ph/0208038 [hep-ph] A. BELITSKY (2012). “A note on two-loop superloop.” Phys. Lett. B718, 205–213. arXiv:1207. 1924 [hep-th] A. BELITSKY (2012). “OPE for null Wilson loops and open spin chains.” Phys. Lett. B709, 280–284. arXiv:1110.1063 [hep-th] A. BELITSKY, G. KORCHEMSKY, and E. SOKATCHEV (2012). “Are scattering amplitudes dual to super Wilson loops?” Nucl. Phys. B855, 333–360. arXiv:1103.3008 [hep-th] A. BELITSKY and A. RADYUSHKIN (2005). “Unraveling hadron structure with generalized parton distributions.” Phys. Rept. 418, 1–387. arXiv:hep-ph/0504030 [hep-ph] Z. BERN, L. J. DIXON, F. FEBRES CORDERO, S. HOECHE, et al. (2013). “Next-to-leading order W + 5-Jet Production at the LHC.” Phys. Rev. D881, 014025. arXiv:1304.1253 [hep-ph] D. BOER, M. G. A. BUFFING, and P. J. MULDERS (2015). “Operator analysis of pT -widths of TMDs.” JHEP 08, 053. arXiv:1503.03760 [hep-ph] D. BOER, W. J. DEN DUNNEN, C. PISANO, M. SCHLEGEL, et al. (2012). “Linearly polarized Gluons and the Higgs transverse momentum distribution.” Phys. Rev. Lett. 108, 032002. arXiv:1109.1444 [hep-ph] D. BOER, M. DIEHL, R. MILNER, R. VENUGOPALAN, et al. (2011). “Gluons and the quark sea at high energies: Distributions, polarization, mertens, tomography.” A report on the joint BNL/INT/JLab program. arXiv:1108.1713 [nucl-th] D. BOER, R. JAKOB, and P. J. MULDERS (2000). “Angular dependences in electroweak semiinclusive leptoproduction.” Nucl. Phys. B564, 471–485. arXiv:hep-ph/9907504 [hep-ph] D. BOER and P. J. MULDERS (1998). “Time-reversal odd distribution functions in leptoproduction.” Phys. Rev. D57, 5780–5786. arXiv:hep-ph/9711485 [hep-ph]

Bibliography

[46] [47] [48]

[49] [50]

[51] [52] [53] [54] [55] [56]

[57]

[58]

[59]

[60]

[61] [62] [63] [64] [65] [66] [67]

203

D. BOER, P. MULDERS, and F. PIJLMAN (2003). “Universality of T-odd effects in single spin and azimuthal asymmetries.” Nucl. Phys. B667, 201–241. arXiv:hep-ph/0303034 [hep-ph] D. BOER and W. J. DEN DUNNEN (2014). “TMD evolution and the Higgs transverse momentum distribution.” Nucl. Phys. B886, 421–435. arXiv:1404.6753 [hep-ph] D. BOER, W. J. DEN DUNNEN, C. PISANO, and M. SCHLEGEL (2013). “Determining the Higgs spin and parity in the diphoton decay channel.” Phys. Rev. Lett. 1113, 032002. arXiv:1304.2654 [hep-ph] D. BOER and C. PISANO (2015). “Impact of gluon polarization on Higgs boson plus jet production at the LHC.” Phys. Rev. D917, 074024. arXiv:1412.5556 [hep-ph] N. N. BOGOLYUBOV, A. A. LOGUNOV, A. I. OKSAK, and I. T. TODOROV (1990). General principles of quantum field theory. Dordrecht, Netherlands: Kluwer (1990) 694 p. (Mathematical physics and applied mathematics, 10). N. N. BOGOLYUBOV and D. V. SHIRKOV (1959). “Introduction to the theory of quantized fields.” Intersci. Monogr. Phys. Astron. 3, 1–720. C. J. BOMHOF and P. J. MULDERS (2008). “Non-universality of transverse momentum dependent parton distribution functions.” Nucl. Phys. B795, 409–427. arXiv:0709.1390 [hep-ph] F. BRADAMANTE (2012). “New COMPASS results on Collins and Sivers asymmetries.” Nuovo Cim. C035N2, 107–114. arXiv:1111.0869 [hep-ex] R. A. BRANDT, A. GOCKSCH, M. A. SATO, and F. NERI (1982). “Loop space.” Phys. Rev. D26, 3611. R. A. BRANDT, F. NERI, and M.-A . SATO (1981). “Renormalization of loop functions for all loops.” Phys. Rev. D24, 879. S. J. BRODSKY, F. FLEURET, C. HADJIDAKIS, and J. P. LANSBERG (2013). “Physics opportunities of a fixed-target experiment using the LHC beams.” Phys. Rept. 522, 239–255. arXiv:1202.6585 [hep-ph] S. J. BRODSKY, D. S. HWANG, and I. SCHMIDT (2002). “Final state interactions and single spin asymmetries in semiinclusive deep inelastic scattering.” Phys. Lett. B530, 99–107. arXiv: hep-ph/0201296 [hep-ph] M. G. A. BUFFING, A. MUKHERJEE, and P. J. MULDERS (2012). “Generalized universality of higher transverse moments of quark TMD correlators.” Phys. Rev. D86, 074030. arXiv:1207.3221 [hep-ph] M. G. A. BUFFING, A. MUKHERJEE, and P. J. MULDERS (2013). “Generalized universality of definite rank gluon transverse momentum dependent correlators.” Phys. Rev. D88, 054027. arXiv: 1306.5897 [hep-ph] M. G. A. BUFFING and P. J. MULDERS (2015). “Color effects for transverse momentum dependent parton distribution functions in hadronic processes.” Few Body Syst. 566-9, 337–342. arXiv:1410.6345 [hep-ph] M. BUFFING and P. MULDERS (2014). “Color entanglement for azimuthal asymmetries in the Drell-Yan process.” Phys. Rev. Lett. 1129, 092002. arXiv:1309.4681 [hep-ph] M. C ASELLE (2000). “Lattice gauge theories and the AdS/CFT correspondence.” Int. J. Mod. Phys. A15, 3901–3966. arXiv:hep-th/0003119 [hep-th] S. C ATANI, M. CIAFALONI, and F. HAUTMANN (1990). “Gluon contributions to small-x heavy flavor production.” Phys. Lett. B242, 97–102. S. C ATANI, M. CIAFALONI, and F. HAUTMANN (1991). “High-energy factorization and small-x heavy flavor production.” Nucl. Phys. B366, 135–188. S. C ATANI, M. CIAFALONI, and F. HAUTMANN (1993). “High-energy factorization in QCD and minimal subtraction scheme.” Phys. Lett. B307, 147–153. S. C ATANI, F. FIORANI, and G. MARCHESINI (1990). “Small-x behavior of initial state radiation in perturbative QCD.” Nucl. Phys. B336, 18–85. S. C ATANI and F. HAUTMANN (1993). “Quark anomalous dimensions at small x.” Phys. Lett. B315, 157–163.

204

[68]

[69] [70] [71] [72]

[73] [74] [75] [76] [77] [78]

[79]

[80]

[81]

[82]

[83] [84] [85] [86] [87]

[88]

Bibliography

S. C ATANI and F. HAUTMANN (1994). “High-energy factorization and small-x deep inelastic scattering beyond leading order.” Nucl. Phys. B427, 475–524. arXiv:hep-ph/9405388 [hep-ph] S. C ATANI and M. GRAZZINI (2011). “QCD transverse-momentum resummation in gluon fusion processes.” Nucl. Phys. B845, 297–323. arXiv:1011.3918 [hep-ph] S. C ATANI, M. GRAZZINI, and A. TORRE (2014). “Transverse-momentum resummation for heavy-quark hadroproduction.” Nucl. Phys. B890, 518–538. arXiv:1408.4564 [hep-ph] H.-M. CHAN and S. T. TSOU (1986). “Gauge theories in loop space.” Acta Phys. Polon. B17, 259. S. CHATRCHYAN et al. (2012). “Measurement of the rapidity and transverse momentum √ distributions of Z bosons in pp collisions at s = 7 TeV.” Phys. Rev. D85, 032002. arXiv:1110. 4973 [hep-ex] T. P. CHENG and L. F. LI (2000). Gauge theory of elementary particle physics: Problems and solutions. Oxford, UK: Clarendon (2000) 306 p. I. CHEREDNIKOV and T. MERTENS (2015). “Fréchet derivative for light-like Wilson loops.” Phys. Lett. B741, 71–76. arXiv:1401.2721 [hep-th] I. O. CHEREDNIKOV, J. LAUWERS, and P. TAELS (2014). “On a Wilson line approach to the study of jet quenching.” Eur. Phys. J. C74, 2721. arXiv:1307.5518 [hep-ph] I. O. CHEREDNIKOV and T. MERTENS (2014). “On geometric scaling of light-like Wilson polygons: Higher orders in !s .” Phys. Lett. B734, 198–202. arXiv:1404.6713 [hep-th] I. O. CHEREDNIKOV, T. MERTENS, and F. F. VAN DER VEKEN (2014). Wilson lines in quantum field theory. De Gruyter Studies in Mathematical Physics. De Gruyter, Berlin. I. CHEREDNIKOV, A. KARANIKAS, and N. STEFANIS (2010). “Wilson lines in transverse-momentum dependent parton distribution functions with spin degrees of freedom.” Nucl. Phys. B840, 379–404. arXiv:1004.3697 [hep-ph] I. CHEREDNIKOV and N. STEFANIS (2008). “Renormalization, Wilson lines, and transversemomentum dependent parton distribution functions.” Phys. Rev. D77, 094001. arXiv:0710. 1955 [hep-ph] I. CHEREDNIKOV and N. STEFANIS (2008). “Wilson lines and transverse-momentum dependent parton distribution functions: A Renormalization-group analysis.” Nucl. Phys. B802, 146–179. arXiv:0802.2821 [hep-ph] I. CHEREDNIKOV and N. STEFANIS (2009). “Renormalization-group properties of transversemomentum dependent parton distribution functions in the light-cone gauge with the Mandelstam-Leibbrandt prescription.” Phys. Rev. D80, 054008. arXiv:0904.2727 [hep-ph] I. CHEREDNIKOV and N. STEFANIS (2011). “Transverse-momentum-dependent parton distributions at the edge of the lightcone.” Int. J. Mod. Phys. Conf. Ser. 4, 135–145. arXiv:1108.0811 [hep-ph] M. CIAFALONI (1988). “Coherence Effects in Initial Jets at Small q2 /s.” Nucl. Phys. B296, 49–74. P. CIPRIANO, S. DOOLING, A. GREBENYUK, P. GUNNELLINI, et al. (2013). “Higgs boson as a gluon trigger.” Phys. Rev. D889, 097501. arXiv:1308.1655 [hep-ph] J. COLLINS (2011). Foundations of perturbative QCD, Cambridge monographs on particle physics, nuclear physics and cosmology, vol. 32. Cambridge University Press, Cambridge. J. COLLINS (2014). “TMD theory, factorization and evolution.” Int. J. Mod. Phys. Conf. Ser. 25, 1460001. arXiv:1307.2920 [hep-ph] J. COLLINS and J.-W. QIU (2007). “kT factorization is violated in production of high-transversemomentum particles in hadron-hadron collisions.” Phys. Rev. D75, 114014. arXiv:0705.2141 [hep-ph] J. COLLINS and T. ROGERS (2015). “Understanding the large-distance behavior of transversemomentum-dependent parton densities and the Collins-Soper evolution kernel.” Phys. Rev. D917, 074020. arXiv:1412.3820 [hep-ph]

Bibliography

[89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103]

[104] [105] [106] [107] [108]

[109]

[110] [111]

205

J. C. COLLINS (1984). Renormalization. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge. J. C. COLLINS (1989). “Sudakov form-factors.” Adv. Ser. Direct. High Energy Phys. 5, 573–614. arXiv:hep-ph/0312336 [hep-ph] J. C. COLLINS (2002). “Leading twist single transverse-spin asymmetries: Drell-Yan and deep inelastic scattering.” Phys. Lett. B536, 43–48. arXiv:hep-ph/0204004 [hep-ph] J. C. COLLINS and F. HAUTMANN (2000). “Infrared divergences and nonlightlike eikonal lines in Sudakov processes.” Phys. Lett. B472, 129–134. arXiv:hep-ph/9908467 [hep-ph] J. C. COLLINS and F. HAUTMANN (2001). “Soft gluons and gauge invariant subtractions in NLO parton shower Monte Carlo event generators.” JHEP 03, 016. arXiv:hep-ph/0009286 [hep-ph] J. C. COLLINS and D. E. SOPER (1981). “Back-to-back jets in QCD.” Nucl. Phys. B193, 381. J. C. COLLINS and D. E. SOPER (1982). “Back-to-back jets: Fourier transform from B to K-transverse.” Nucl. Phys. B197, 446–476. J. C. COLLINS and D. E. SOPER (1982). “Parton distribution and decay functions.” Nucl. Phys. B194, 445. J. C. COLLINS, D. E. SOPER, and G. F. STERMAN (1982). “Does the Drell-Yan cross-section factorize?” Phys. Lett. B109, 388. J. C. COLLINS, D. E. SOPER, and G. F. STERMAN (1983). “Factorization for One Loop Corrections in the Drell-Yan Process.” Nucl. Phys. B223, 381. J. C. COLLINS, D. E. SOPER, and G. F. STERMAN (1985). “Transverse momentum distribution in Drell-Yan Pair and W and Z Boson production.” Nucl. Phys. B250, 199. M. CZAKON (2015). “Precision top-quark physics with applications.” Nucl. Part. Phys. Proc. 261-262, 115–129. M. CZAKON, P. FIEDLER, and A. MITOV (2013). “Total top-quark pair-production cross section at Hadron Colliders through !4s .” Phys. Rev. Lett. 110, 252004. arXiv:1303.6254 [hep-ph] A. I. DAVYDYCHEV, P. OSLAND, and O. TARASOV (1996). “Three gluon vertex in arbitrary gauge and dimension.” Phys. Rev. D54, 4087–4113. arXiv:hep-ph/9605348 [hep-ph] C. DI BARTOLO, R. GAMBINI, and J. GRIEGO (1993). “The Extended loop group: An Infinite dimensional manifold associated with the loop space.” Commun. Math. Phys. 158, 217–240. arXiv:gr-qc/9303010 [gr-qc] M. DIEHL, T. KASEMETS, and S. KEANE (2014). “Correlations in double parton distributions: effects of evolution.” JHEP 05, 118. arXiv:1401.1233 [hep-ph] P. DITTNER (1971). “Invariant tensors in SU(3).” Commun. Math. Phys. 22, 238–252. Y. L. DOKSHITZER, D. DIAKONOV, and S. I. T ROIAN (1978). “On the transverse momentum distribution of massive lepton pairs.” Phys. Lett. B79, 269–272. F. DOMINGUEZ, C. MARQUET, B.-W. XIAO, and F. YUAN (2011). “Universality of unintegrated gluon distributions at small x.” Phys. Rev. D83, 105005. arXiv:1101.0715 [hep-ph] F. DOMINGUEZ, A. H. MUELLER, S. MUNIER, and B.-W. XIAO (2011). “On the small-x evolution of the color quadrupole and the Weizsäcker-Williams gluon distribution.” Phys. Lett. B705, 106–111. arXiv:1108.1752 [hep-ph] S. DOOLING, P. GUNNELLINI, F. HAUTMANN, and H. JUNG (2013). “Longitudinal momentum shifts, showering, and nonperturbative corrections in matched next-to-leading-order shower event generators.” Phys. Rev. D879, 094009. arXiv:1212.6164 [hep-ph] S. DOOLING, P. GUNNELLINI, F. HAUTMANN, and H. JUNG (2013). “Nonperturbative corrections and showering in NLO-matched event generators.” arXiv:1304.7180 [hep-ph] S. DOOLING, F. HAUTMANN, and H. JUNG (2014). “Hadroproduction of electroweak gauge boson plus jets and TMD parton density functions.” Phys. Lett. B736, 293–298. arXiv:1406.2994 [hep-ph]

206

Bibliography

[112] M. G. ECHEVARRIA, T. KASEMETS, P. J. MULDERS, and C. PISANO (2015). “QCD evolution of (un)polarized gluon TMDPDFs and the Higgs qT -distribution.” JHEP 07, 158. arXiv:1502.05354 [hep-ph] [113] L. D. FADDEEV and A. A. SLAVNOV (1980). “Gauge fields. Introduction to quantum theory.” Front. Phys. 50, 1–232. [Front. Phys.,1(1991)]. [114] R. P. FEYNMAN (1951). “An operator calculus having applications in quantum electrodynamics.” Phys. Rev. 84, 108–128. [115] R. D. FIELD (1989). Applications of perturbative QCD, frontiers in physics, vol. 77. AddisonWesley, Reading, MA. [116] R. GAMBINI and J. PULLIN (2000). Loops, knots, gauge theories and quantum gravity. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK. [117] M. GARCÍA ECHEVARRÍA, A. S. IDILBI, and I. SCIMEMI (2012). “Factorization theorem for Drell-Yan at low qT and transverse momentum distributions on-the-light-cone.” JHEP 1207, 002. arXiv: 1111.4996 [hep-ph] [118] M. GARCÍA ECHEVARRÍA, A. S. IDILBI, and I. SCIMEMI (2013). “Soft and collinear factorization and transverse momentum dependent parton distribution functions.” Phys. Lett. B726, 795–801. arXiv:1211.1947 [hep-ph] [119] M. GARCÍA ECHEVARRÍA, A. S. IDILBI, and I. SCIMEMI (2014). “A unified treatment of the QCD evolution of all (Un-)polarized TMD functions: Collins function as a study case.” Phys. Rev. D90, 014003. arXiv:1402.0869 [hep-ph] [120] M. GARCÍA ECHEVARRÍA, A. S. IDILBI, and I. SCIMEMI (2014). “On rapidity divergences in the soft and collinear limits of QCD.” Int. J. Mod. Phys. Conf. Ser. 25, 1460005. arXiv:1310.8541 [hep-ph] [121] R. GILES (1981). “The reconstruction of gauge potentials from Wilson loops.” Phys. Rev. D24, 2160. [122] L. GRIBOV, E. LEVIN, and M. RYSKIN (1983). “Semihard processes in QCD.” Phys. Rept. 100, 1–150. [123] F. HAUTMANN (2001). “Cancellation of Sudakov logarithms in radiative decays of quarkonia.” Nucl. Phys. B604, 391–404. arXiv:hep-ph/0102336 [hep-ph] [124] F. HAUTMANN (2002). “Heavy top limit and double logarithmic contributions to Higgs production at m2H /s much less than 1.” Phys. Lett. B535, 159–162. arXiv:hep-ph/0203140 [hep-ph] [125] F. HAUTMANN (2006). “Power-like corrections and the determination of the gluon distribution.” Phys. Lett. B643, 171–174. arXiv:hep-ph/0610078 [hep-ph] [126] F. HAUTMANN (2007). “Endpoint singularities in unintegrated parton distributions.” Phys. Lett. B655, 26–31. arXiv:hep-ph/0702196 [HEP-PH] [127] F. HAUTMANN (2009). “Unintegrated parton distributions and applications to jet physics.” Acta Phys. Polon. B40, 2139–2163. [128] F. HAUTMANN (2012). “Energy flow observables in Hadronic collisions.” arXiv:1205.5411 [hep-ph] [129] F. HAUTMANN and H. JUNG (2008). “Angular correlations in multi-jet final states from k⊥ -dependent parton showers.” JHEP 10, 113. arXiv:0805.1049 [hep-ph] [130] F. HAUTMANN and H. JUNG (2008). “Recent results on unintegrated parton distributions.” Nucl. Phys. Proc. Suppl. 184, 64–72. arXiv:0712.0568 [hep-ph] [131] F. HAUTMANN and H. JUNG (2012). “Collinearity approximations and kinematic shifts in partonic shower algorithms.” Eur. Phys. J. C72, 2254. arXiv:1209.6549 [hep-ph] [132] F. HAUTMANN and H. JUNG (2014). “Precision DIS data and TMD parton distributions.” arXiv:1411.7240 [hep-ph] [133] F. HAUTMANN and H. JUNG (2014). “Transverse momentum dependent gluon density from DIS precision data.” Nucl. Phys. B883, 1–19. arXiv:1312.7875 [hep-ph]

Bibliography

207

[134] F. HAUTMANN and H. JUNG (2015). “Hadron structure beyond collinear level and precision DIS measurements.” EPJ Web Conf. 90, 07004. [135] F. HAUTMANN, H. JUNG, and S. T. MONFARED (2014). “The CCFM uPDF evolution uPDFevolv Version 1.0.00.” Eur. Phys. J. C74, 3082. arXiv:1407.5935 [hep-ph] [136] F. HAUTMANN and D. E. SOPER (2001). “Color transparency in deeply inelastic diffraction.” Phys. Rev. D63, 011501. arXiv:hep-ph/0008224 [hep-ph] [137] F. HAUTMANN and D. E. SOPER (2007). “Parton distribution function for quarks in an s-channel approach.” Phys. Rev. D75, 074020. arXiv:hep-ph/0702077 [hep-ph] [138] A. IDILBI, X.-D. JI, J.-P. MA, and F. YUAN (2004). “Collins-Soper equation for the energy evolution of transverse-momentum and spin dependent parton distributions.” Phys. Rev. D70, 074021. arXiv:hep-ph/0406302 [hep-ph] [139] A. IDILBI and I. SCIMEMI (2011). “Singular and regular gauges in soft collinear effective theory: The introduction of the New Wilson line T.” Phys. Lett. B695, 463–468. arXiv:1009.2776 [hep-ph] [140] A. IDILBI and I. SCIMEMI (2011). “The T-Wilson line.” AIP Conf. Proc. 1343, 320–321. arXiv:1012.4419 [hep-ph] [141] S. V. IVANOV, G. P. KORCHEMSKY, and A. V. RADYUSHKIN (1986). “Infrared asymptotics of perturbative QCD: Contour gauges.” Yad. Fiz. 44, 230–240. [Sov. J. Nucl. Phys.44,145(1986)]. [142] A. JAIN, M. PROCURA, and W. J. WAALEWIJN (2012). “Fully-unintegrated parton distribution and fragmentation functions at perturbative kT .” JHEP 04, 132. arXiv:1110.0839 [hep-ph] [143] X.-D. JI, J.-P. MA, and F. YUAN (2004). “QCD factorization for spin-dependent cross sections in DIS and Drell-Yan processes at low transverse momentum.” Phys. Lett. B597, 299–308. arXiv: hep-ph/0405085 [hep-ph] [144] X.-D. JI, J.-P. MA, and F. YUAN (2005). “QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum.” Phys. Rev. D71, 034005. arXiv:hep-ph/0404183 [hep-ph] [145] X.-D. JI, J.-P. MA, and F. YUAN (2005). “Transverse-momentum-dependent gluon distributions and semi-inclusive processes at hadron colliders.” JHEP 0507, 020. arXiv:hep-ph/0503015 [hep-ph] [146] X.-D. JI and F. YUAN (2002). “Parton distributions in light cone gauge: Where are the final state interactions?” Phys. Lett. B543, 66–72. arXiv:hep-ph/0206057 [hep-ph] [147] X. JI, P. SUN, X. XIONG, and F. YUAN (2015). “Soft factor subtraction and transverse momentum dependent parton distributions on the lattice.” Phys. Rev. D91, 074009. arXiv:1405.7640 [hep-ph] [148] N. P. KONOPLEVA and V. N. POPOV (1981). Gauge Fields. Chur, Switzerland: Harwood (1981) 264p. [149] G. P. KORCHEMSKY and A. V. RADYUSHKIN (1986). “Loop space formalism and renormalization group for the infrared asymptotics of QCD.” Phys. Lett. B171, 459–467. [150] G. KORCHEMSKY and A. RADYUSHKIN (1987). “Renormalization of the Wilson loops beyond the leading order.” Nucl. Phys. B283, 342–364. [151] P. KOTKO (2014). “Wilson lines and gauge invariant off-shell amplitudes.” JHEP 1407, 128. arXiv:1403.4824 [hep-ph] [152] Y. V. KOVCHEGOV (1999). “Small-x F(2) structure function of a nucleus including multiple pomeron exchanges.” Phys. Rev. D60, 034008. arXiv:hep-ph/9901281 [hep-ph] [153] E. A. KURAEV, L. N. LIPATOV, and V. S. FADIN (1976). “Multi - reggeon processes in the Yang-Mills theory.” Sov. Phys. JETP 44, 443–450. [154] E. KURAEV, L. LIPATOV, and V. S. FADIN (1977). “The Pomeranchuk singularity in nonabelian gauge theories.” Sov. Phys. JETP 45, 199–204. [155] J. P. LANSBERG et al. (2015). “Spin physics and TMD studies at A Fixed-Target Experiment at the LHC (AFTER@LHC).” EPJ Web Conf. 85, 02038. arXiv:1410.1962 [hep-ex]

208

Bibliography

[156] H. T. LI, C. S. LI, D. Y. SHAO, L. L. YANG, et al. (2013). “Top quark pair production at small transverse momentum in hadronic collisions.” Phys. Rev. D88, 074004. 1307.2464 [157] Y. LI, S. MANTRY, and F. PETRIELLO (2011). “An exclusive soft function for Drell-Yan at next-to-next-to-leading order.” Phys. Rev. D84, 094014. arXiv:1105.5171 [hep-ph] [158] R. LOLL (1992). “Loop approaches to gauge field theory.” Theor. Math. Phys. 93, 1415–1432. [Teor. Mat. Fiz.93,481(1992)]. [159] G. LUISONI and S. MARZANI (2015). “QCD resummation for hadronic final states.” J. Phys. G4210, 103101. arXiv:1505.04084 [hep-ph] [160] Y.-Q. MA and J.-W. QIU (2015). “QCD Factorization and PDFs from Lattice QCD Calculation.” Int. J. Mod. Phys. Conf. Ser. 37, 1560041. arXiv:1412.2688 [hep-ph] [161] YU. MAKEENKO (2005). Methods of contemporary gauge theory. Cambridge University Press. [162] Y. MAKEENKO and A. A. MIGDAL (1981). “Quantum chromodynamics as dynamics of loops.” Nucl. Phys. B188, 269. [163] YU. M. MAKEENKO and A. A. MIGDAL (1980). “Selfconsistent areas law in QCD.” Phys. Lett. B97, 253–256. [164] Y. MAKEENKO and A. A. MIGDAL (1979). “Exact equation for the loop average in multicolor QCD.” Phys. Lett. B88, 135. [165] S. MANDELSTAM (1962). “Quantum electrodynamics without potentials.” Annals Phys. 19, 1–24. [166] S. MANDELSTAM (1968). “Feynman rules for electromagnetic and Yang-Mills fields from the gauge independent field theoretic formalism.” Phys. Rev. 175, 1580–1623. [167] M. L. MANGANO, M. MORETTI, F. PICCININI, R. PITTAU, et al. (2003). “ALPGEN, a generator for hard multiparton processes in hadronic collisions.” JHEP 07, 001. arXiv:hep-ph/0206293 [hep-ph] [168] S. MANTRY and F. PETRIELLO (2010). “Factorization and resummation of Higgs Boson differential distributions in soft-collinear effective theory.” Phys. Rev. D81, 093007. arXiv:0911.4135 [hep-ph] [169] S. MANTRY and F. PETRIELLO (2011). “Transverse momentum distributions from effective field theory with numerical results.” Phys. Rev. D83, 053007. arXiv:1007.3773 [hep-ph] [170] G. MARCHESINI and B. R. WEBBER (1992). “Final states in heavy quark leptoproduction at small x.” Nucl. Phys. B386, 215–235. [171] G. MARCHESINI (1995). “QCD coherence in the structure function and associated distributions at small x.” Nucl. Phys. B445, 49–80. arXiv:hep-ph/9412327 [hep-ph] [172] A. D. MARTIN (2008). “Proton structure, partons, QCD, DGLAP and beyond.” Acta Phys.Polon. B39, 2025–2062. arXiv:0802.0161 [hep-ph] [173] S. MEISSNER, A. METZ, and K. GOEKE (2007). “Relations between generalized and transverse momentum dependent parton distributions.” Phys. Rev. D76, 034002. arXiv:hep-ph/0703176 [HEP-PH] [174] M. B. MENSKY (1981). “The group of paths in gravitation and gauge theory.” In *Moscow 1981, Proceedings, Quantum Gravity*, 527-546. [175] M. B. MENSKY (1983). “Group of paths: Measurements, fields, particles.” Nauka, Moscow [in Russian] . [176] M. B. MENSKY (1984). “Group theoretical derivation of path integrals.” Theor. Math. Phys. 57, 1095–1105. [Teor. Mat. Fiz.57,217(1983)]. [177] S. MERT AYBAT and T. C. ROGERS (2011). “TMD factorization and evolution for TMD correlation functions.” Int. J. Mod. Phys. Conf. Ser. 04, 97–105. [178] T. MERTENS (2013). “Generalized loop space and TMDs.” PoS Hadron2013, 135. [179] T. MERTENS and P. TAELS (2013). “Evolution of light-like Wilson loops with a self-intersection in loop space.” Phys. Lett. B727, 563–567. arXiv:1308.5296 [hep-ph] [180] S. MOCH, J. A. M. VERMASEREN, and A. VOGT (2005). “The longitudinal structure function at the third order.” Phys. Lett. B606, 123–129. arXiv:hep-ph/0411112 [hep-ph]

Bibliography

209

[181] A. H. MUELLER, B.-W. XIAO, and F. YUAN (2013). “Sudakov double logarithms resummation in hard processes in the small-x saturation formalism.” Phys. Rev. D8811, 114010. arXiv:1308.2993 [hep-ph] [182] A. H. MUELLER (1994). “Soft gluons in the infinite momentum wave function and the BFKL pomeron.” Nucl. Phys. B415, 373–385. [183] P. J. MULDERS and J. RODRIGUES (2001). “Transverse momentum dependence in gluon distribution and fragmentation functions.” Phys. Rev. D63, 094021. arXiv:hep-ph/0009343 [hep-ph] [184] P. J. MULDERS and R. D. TANGERMAN (1996). “The complete tree level result up to order 1/Q for polarized deep inelastic leptoproduction.” Nucl. Phys. B461, 197–237. [Erratum: Nucl. Phys.B484,538(1997)]., arXiv:hep-ph/9510301 [hep-ph] [185] B. U. MUSCH, P. HAGLER, M. ENGELHARDT, J. W. NEGELE, et al. (2012). “Sivers and Boer-Mulders observables from lattice QCD.” Phys. Rev. D85, 094510. arXiv:1111.4249 [hep-lat] [186] B. U. MUSCH, P. HAGLER, J. W. NEGELE, and A. SCHAFER (2011). “Exploring quark transverse momentum distributions with lattice QCD.” Phys. Rev. D83, 094507. arXiv:1011.1213 [hep-lat] [187] P. M. NADOLSKY, C. BALAZS, E. L. BERGER, and C. P. YUAN (2007). “Gluon-gluon contributions to the production of continuum diphoton pairs at hadron colliders.” Phys. Rev. D76, 013008. arXiv:hep-ph/0702003 [HEP-PH] [188] G. PARISI and R. PETRONZIO (1979). “Small transverse momentum distributions in hard processes.” Nucl. Phys. B154, 427–440. [189] M. E. PESKIN and D. V. SCHROEDER (1995). An introduction to quantum field theory. Advanced Book Program. Westview Press, Boulder, CO. [190] A. M. POLYAKOV (1979). “String representations and hidden symmetries for gauge fields.” Phys. Lett. B82, 247–250. [191] A. M. POLYAKOV (1980). “Gauge fields as rings of glue.” Nucl. Phys. B164, 171–188. [192] J. P. RALSTON and D. E. SOPER (1979). “Production of Dimuons from high-energy polarized proton proton collisions.” Nucl. Phys. B152, 109. [193] T. C. ROGERS (2013). “Extra spin asymmetries from the breakdown of transverse-momentumdependent factorization in hadron-hadron collisions.” Phys. Rev. D881, 014002. arXiv:1304. 4251 [hep-ph] [194] T. C. ROGERS and P. J. MULDERS (2010). “No generalized TMD-factorization in hadro-production of high transverse momentum hadrons.” Phys. Rev. D81, 094006. arXiv:1001.2977 [hep-ph] [195] A. SCHWARZ (1975). “Mathematical foundations of quantum field theory.” Atomizdat, Moscow [in Russian] . [196] D. W. SIVERS (1990). “Single spin production asymmetries from the hard scattering of point-like constituents.” Phys. Rev. D41, 83. [197] N. STEFANIS (1984). “Gauge invariant quark two point green’s function through connector insertion to O(!s ).” Nuovo Cim. A83, 205. [198] N. STEFANIS and I. CHEREDNIKOV (2009). “Renormalization-group anatomy of transverse-momentum dependent parton distribution functions in QCD.” Mod. Phys. Lett. A24, 2913–2923. arXiv:0910.3108 [hep-ph] [199] I. W. STEWART, F. J. TACKMANN, and W. J. WAALEWIJN (2010). “The quark beam function at NNLL.” JHEP 09, 005. arXiv:1002.2213 [hep-ph] [200] P. SUN, B.-W. XIAO, and F. YUAN (2011). “Gluon distribution functions and Higgs boson production at moderate transverse momentum.” Phys. Rev. D84, 094005. arXiv:1109.1354 [hep-ph] [201] F. F. VAN DER VEKEN and N. G. STEFANIS (2016). “Identifying the universal part of TMDs.” Few-Body Systems, 1

210

Bibliography

[202] H. VAN HAEVERMAET, P. CIPRIANO, S. DOOLING, A. GREBENYUK, et al. (2014). “Higgs boson as a gluon trigger: the study of QCD in high pile-up environments.” PoS DIS2014, 163. arXiv:1407. 2815 [hep-ph] [203] F. F. VAN DER VEKEN (2013). “Evolution and dynamics of cusped light-like Wilson loops.” PoS Hadron2013, 134. arXiv:1405.4017 [hep-ph] [204] F. F. VAN DER VEKEN (2013). “Evolution and dynamics of cusped light-like Wilson loops.” Nuovo Cim. C36, 89–94. arXiv:1302.6765 [hep-th] [205] F. F. VAN DER VEKEN (2015). “A new approach to piecewise linear Wilson lines.” Int. J. Mod. Phys. Conf. Ser. 37, 0029. arXiv:1411.3372 [hep-ph] [206] F. F. VAN DER VEKEN (2015). “Piecewise linear Wilson lines.” EPJ Web Conf. 85, 02004. arXiv: 1409.3190 [hep-ph] [207] J. A. M. VERMASEREN, A. VOGT, and S. MOCH (2005). “The third-order QCD corrections to deep-inelastic scattering by photon exchange.” Nucl. Phys. B724, 3–182. arXiv:hep-ph/0504242 [hep-ph] [208] W. VOGELSANG and F. YUAN (2007). “Hadronic dijet imbalance and transverse-momentum dependent parton distributions.” Phys. Rev. D76, 094013. arXiv:0708.4398 [hep-ph] [209] A. VOGT, S. MOCH, and J. A. M. VERMASEREN (2004). “The three-loop splitting functions in QCD: The Singlet case.” Nucl. Phys. B691, 129–181. arXiv:hep-ph/0404111 [hep-ph] [210] K. G. WILSON (1974). “Confinement of quarks.” Phys. Rev. D10, 2445–2459. [211] H. X. ZHU, C. S. LI, H. T. LI, D. Y. SHAO, et al. (2013). “Transverse-momentum resummation for top-quark pairs at hadron colliders.” Phys. Rev. Lett. 1108, 082001. arXiv:1208.5774 [hep-ph] [212] R. ZHU, P. SUN, and F. YUAN (2013). “Low transverse momentum heavy quark pair production to probe gluon tomography.” Phys. Lett. B727, 474–479. arXiv:1309.0780 [hep-ph] [213] I. P. ZOIS (1998). “On Polyakov’s basic variational formula for loop spaces.” Rept. Math. Phys. 42, 373–384. arXiv:hep-th/9703069 [hep-th]

Index Bjorken scaling 102 Bjorken-x 92 canonical commutation relations 10 Collins-Soper equation 126 covariant derivative 40 cusp anomalous dimension 126 deep inelastic scattering (DIS) 91 – semi-inclusive (SIDIS) 111 – azimutal angle 112 delta function: see nascent delta function eikonal 87 – quark 89 equations of motion 7 Green’s function – Klein-Gordon operator 21 – advanced 22 – retarded 22 – Feynman 22 Hilbert space 9 infinite momentum frame 96 lattice 43 nascent delta-function 71 operator ordering – time 16 – path 18 – normal 19 parallel transport 34 particle number operator 29 parton – gluon 93 – sea quark 93 – valence quark 93 – free parton model 94

parton density function (PDF) 3, 100, 110 – evolution 101, 123 – gauge-invariance 107 – gluon 110 – operator definition 104 – quark 106 – transverse momentum dependent (TMD, TMD PDF) 104 Poisson bracket 6 probability amplitude 10 quantum chromodynamics (QCD) – confinement of colour 1 – factorization 2, 100 – high-energy regime 2 – Lagrangian 1 – running coupling 1 quantum mechanical representation – Heisenberg 10 – Schroedinger 11 – Dirac 13 rapidity cutoff 126 scattering matrix (S-matrix) 17 semi-inclusive deep inelastic scattering (SIDIS): see deep inelastic scattering soft factor 124, 125 Stokes’ theorem 42 structure function 100 tensor – leptonic 97 – hadronic 97, 113 Trento conventions 111 vacuum state 27 Wick’s theorem – first 31 – second 31 – third 31

212

Index

Wilson line – comparator 34 – expansion 49 – Feynman rules 59 – finite 65 – in Quantum Mechanics 18

– parallel transporter 34 – path-reversal 61 – piecewise linear 54 – properties 50 – transverse 121 Wilson loop 42–46, 51, 138

De Gruyter Studies in Mathematical Physics Volume 36 Alexander B. Borisov Nonlinear Dynamics: Non-Integrable Systems and Chaotic Dynamics, 2016 ISBN 978-3-11-043938-0, e-ISBN (PDF) 978-3-11-043058-5, e-ISBN (EPUB) 978-3-11-043067-7, Set-ISBN 978-3-11-043059-2 Volume 35 Vladimir K. Dobrev Invariant Differential Operators: Volume 1 Noncompact Semisimple Lie Algebras and Groups, 2016 ISBN 978-3-11-043542-9, e-ISBN (PDF) 978-3-11-042764-6, e-ISBN (EPUB) 978-3-11-042780-6, Set-ISBN 978-3-11-042765-3 Volume 34 Abram I. Fet Group Theory of Chemical Elements, 2016 ISBN 978-3-11-047518-0, e-ISBN (PDF) 978-3-11-047623-1, e-ISBN (EPUB) 978-3-11-047520-3, Set-ISBN 978-3-11-047624-8 Volume 33 Alexander M. Formalskii Stabilisation and Motion Control of Unstable Objects, 2015 ISBN 978-3-11-037582-4, e-ISBN (PDF) 978-3-11-037589-3, e-ISBN (EPUB) 978-3-11-039282-1, Set-ISBN 978-3-11-037590-9 Volume 32 Victor V. Przyjalkowski, Thomas Coates Laurent Polynomials in Mirror Symmetry, 2017 ISBN 978-3-11-031109-9, e-ISBN (PDF) 978-3-11-031110-5, e-ISBN (EPUB) 978-3-11-038299-0, Set-ISBN 978-3-11-031111-2 Volume 31 Christian Bizouard Geophysical Modelling of the Polar Motion, 2016 ISBN 978-3-11-029804-8, e-ISBN (PDF) 978-3-11-029809-3, e-ISBN (EPUB) 978-3-11-038913-5, Set-ISBN 978-3-11-029810-9

www.degruyter.com