Cancellation of infrared divergences in QCD

Daniel Wennberg

Physics specalization project Submission date: January 2014 Supervisor: Michael Kachelrieß

Norwegian University of Science and Technology Department of Physics Abstract

The scattering process hadrons is considered at next-to- + − leading order, . It is demonstrated푒 푒 → that the cross sections for both the real gluon emission풪(훼푠) and virtual gluon exchange have infrared singu- larities. Using dimensional regularization it is shown that the divergent terms cancel exactly when adding the contributions, such that the total cross section for hadrons is finite at this order. This result, and + − the general implications푒 푒 → of infrared divergences and their cancellation, is discussed in light of the Kinoshita-Lee-Nauenberg theorem and the concept of infrared safety.

2 Contents

List of figures 5

List of tables 6

List of symbols and notation 7

Preface 9

1 Introduction 10 1.1 Perturbative quantum chromodynamics ...... 10 1.2 Infrared divergences ...... 13

2 Cancellation of infrared divergences in hadrons 14 + − 2.1 Leading order amplitude for 푒 푒 →...... 15 + − 2.2 Cross sections for hadrons푒 푒 → . 푞 .푞 ...... 19 + − 2.3 Real gluon emission:푒 푒 → at leading order ...... 21 + − 2.4 Virtual gluon exchange:푒 푒 → 푞푞푔 at next-to-leading order . . 26 + − 2.5 Cancellation: hadrons푒 푒 → at 푞 next-to-leading푞 order . . . . 34 + − 2.6 The ultraviolet푒 divergence푒 → ...... 35 3 Discussion 38 3.1 Infrared divergences in general quantum field theories . . . . 38 3.2 The optical theorem and cutting rules ...... 41 3.3 Infrared safety in QCD ...... 44

4 Summary 46

A Feynman rules for QED and QCD 47 A.1 General remarks ...... 47 A.2 QED ...... 49 A.3 QCD ...... 51 A.4 Polarization ...... 52 A.5 Some results from ...... 53 SU(푁푐) 3 Contents

B Properties of gamma matrices and Dirac spinors 56 B.1 Gamma matrices ...... 56 B.2 Dirac spinors ...... 60

C Dimensional regularization 66 C.1 Important identities ...... 68 C.2 Phase space in dimensions ...... 69 푑 Bibliography 72

4 List of figures

2.1 Leading order diagram for ...... 16 − + 2.2 Leading order diagrams for푒 푒 → 푞푞 ...... 22 − + 2.3 Diagrams for at one푒 푒 loop→ 푞.푞푔 ...... 27 − + 푒 푒 → 푞푞 3.1 Forward scattering ...... 42 + − + − 3.2 Cut diagrams for 푒 푒 hadrons→ 푒 푒 ...... 43 ∗ 훾 → A.1 Example of a Feynman diagram ...... 47

5 List of tables

1.1 Masses of quarks and leptons ...... 12

A.1 Building blocks of QED Feynman diagrams ...... 50 A.2 Lines in QCD Feynman diagrams ...... 51 A.3 Vertices in QCD Feynman diagrams ...... 52

6 List of symbols and notation

The coupling constant of quantum chromodynamics; . 2 푔푠, 훼푠 훼푠 ≡ 푔푠 /4휋 em The positive elementary electric charge and coupling constant of quan- 푒, 훼 tum electrodynamics; em . The electron charge is . 2 훼 ≡ 푒 /4휋 −푒 The electric quantum number of a quark ; the quark charge is . 푒푞 푞 푒푒푞 The complex conjugate of the number : . Applies ∗ ∗ 푧 elementwise to vectors, matrices etc. 푧 = 푥 + 푖푦 푧 = 푥 − 푖푦 The real part of the complex number : . ∗ ℜ{푧} 푧 ℜ{푧} = (푧 + 푧 )/2 The imaginary part of the complex number : . ∗ ℑ{푧} 푧 ℑ{푧} = (푧 − 푧 )/2푖 The identity matrix. 푰 The transpose of the matrix : . 푇 푇 퐴 퐴 (퐴 )푖푗 = (퐴)푗푖 The Hermitian adjoint of the matrix : . † † ∗ 푇 퐴 퐴 퐴 = (퐴 ) The metric tensor in Minkowski space, with signature . 휇휈 휂 (+, −, −, −) The spatial part of a 4-vector : . 0 0 1 2 3 1 2 3 풂 푎 = (푎 , 풂) = (푎 , 푎 , 푎 , 푎 ) 풂 = (푎 , 푎 , 푎 ) The scalar product of two spatial vectors: . 푖 푖 풂 ⋅ 풃 풂 ⋅ 풃 = 푎 푏 The scalar product of a spatial vector with itself: . 2 2 풂 풂 = 풂 ⋅ 풂 The length of a spatial vector: . Also used for Euclidean lengths 2 ̃푎 of vectors in arbitrary dimensions.̃푎= √풂 The inner product of two 4-vectors: . 휇 0 0 푎 ⋅ 푏 푎 ⋅ 푏 = 푎 푏휇 = 푎 푏 − 풂 ⋅ 풃 The inner product of a 4-vector with itself: . 2 2 휇 0 2 2 푎 푎 = 푎⋅푎 = 푎 푎휇 = (푎 ) −풂 The energy fraction of particle in the COM frame: with 2 푥푖 . 푖 푥푖 = 2푝푖 ⋅ 푄/푄 푄 = ∑푖 푝푖 7 List of symbols and notation

The Dirac gamma matrices. 휇 훾 The Dirac adjoint of a Dirac spinor . † 0 휓 휓 = 휓 훾 휓 The Feynman slash, denoting the contraction of a 4-vector and the 푎 gamma matrices: . / 휇 /푎 = 푎휇훾 Dirac spinor for a fermion with spin state and momentum . 푠 푢 (푝) 푠 푝 Dirac spinor for an antifermion with spin state and momentum . 푠 푣 (푝) 푠 푝 A component of a 4-vector , specified by a greek index 휇 0 1 2 3 푎 , known as a푎 Lorentz = (푎 , 푎 index., 푎 , 푎 ) The index may be lowered 휇by ∈ contraction {0, 1, 2, 3} with the metric tensor: . 휈 푎휇 = 휂휇휈푎 A spatial component of a 4-vector , specified by a latin 푖 0 1 2 3 푎 index . 푎 = (푎 , 푎 , 푎 , 푎 ) 푖 ∈ {1, 2, 3} The number of color degrees of freedom for quarks. For real-world 푁푐 quarks, . 푁푐 = 3 The Gell-Mann matrices, listed in equation (A.15). 푎 휆 The generators of the fundamental representation of . For 푎 푇 defined as SU(푁푐) 푁푐 = 푎 푎 3 푇 = 휆 /2 The structure constants of the generator algebra: . 푎푏푐 푎 푏 푎푏푐 푐 푓 [푇 , 푇 ] = 푖 푓 푇 The Dynkin index of the generator algebra: . For 푎 푏† 푎푏 푇퐹 and , . Tr [푇 푇 ] = 푇퐹훿 푎 푎 푁푐 = 3 푇 = 휆 /2 푇퐹 = 1/2 The color factor of the generator algebra: . For 푎 푎† 퐶퐹 and , . (푇 푇 )푖푘 = 퐶퐹훿푖푘 푁푐 = 3 푎 푎 푇 = 휆 /2 퐶퐹 = 4/3 The Feynman propagator for a gauge boson with momentum . 휇휈 퐷퐹 (푝) 푝 The Feynman propagator for a fermion with momentum . 푆퐹(푝) 푝 A general scattering amplitude. ℳ The average over initial spin and color states and sum over final spin 2 ∣ℳ∣ and color states of . 2 ∣ℳ∣ The Einstein summation convention applies to all kinds of indices unless otherwise stated, e.g. . For Lorentz indices, there must always be one upper and one lower푎푖푏푖 = summation ∑푖 푎푖푏푖 index: . 휇 휇 휇 휈 Units are chosen such that . 푎 푎휇 = ∑휇 푎 푎휇 = ∑휇휈 휂휇휈푎 푎 ℏ = 푐 = 1

8 Preface

This report is the outcome of my work in the physics specialization project, TFY4510, carried out during the fall semester of 2014 under supervision of Prof. Michael Kachelrieß at the Department of Physics at the Norwegian University of Science and Technology (NTNU). The objective of the project has been to learn about the phenomenon of infrared divergences and how they cancel when defining observables satisfying particular requirements, mainly in the context of quantum chromodynamics and with electron-positron annihilation to quarks as the process of study. Infrared divergences have not been treated in detail during the course of my studies, and the topic was therefore largely unknown to me before embarking on this project. The main motivation for studying this is to prepare for a possible Master’s thesis studying annihilations in a wino-like Dark Matter candidate model. As the project work has consisted solely of gaining acquaintance with the subject matter, there are no new results to present. The report therefore amounts to a thorough presentation of established theoretical results, reflecting my level of knowledge as of finishing this project. The aim has been to give a comprehensive and largely self-contained presentation of the main ideas and derivations, including a significant amount of background material that I had to learn or review in order to get through the calculations. Much of this has been relegated to appendices in an attempt to obtain a readable and reasonably well-structured text, but I still consider it an integral part of my project work.

9 1 Introduction

Quantum chromodynamics (QCD) is the theory describing the strong interac- tions between the quarks and gluons, the fundamental particles which make up baryons such as the proton and neutron, and mesons such as the pion. The quarks and gluons carry a quantum number which is specific to QCD, known as color charge. QCD is a quantum field theory described by the Lagrangian [1, pp. 21-22]

(1.1) 푎 푎 1 푎휇휈 푎 ℒQCD = 휓푞,푖 (푖휕/ − 푚푞) 훿푖푗휓푞,푗 − 푔푠휓푞,푖푇푖푗퐴/ 휓푞,푗 − 퐹 퐹휇휈 , Here, is the Dirac spinor field for a quark flavor labeled4 , with mass , while 휓푞 is the field strength tensor for the gauge field 푞 describing푚 the푞 푎 푎 gluons.퐹휇휈 The indices label the color charge basis퐴휇 states of the quarks, while the index푖, 푗 ∈ {1, 2, 3} labels color charge basis states for the gluons. The terms are further푎 ∈ {1, discussed … , 8} in appendix A. Quarks also carry electric charge and are subject to interactions in quantum electrodynamics. These interactions are not present in equation (1.1) but are discussed in appendix A.2. Finally, quarks interact weakly, but weak interactions will not be discussed in this text.

1.1 Perturbative quantum chromodynamics

When applying the usual tools of perturbative quantum field theory to the QCD Lagrangian, the Feynman rules in appendix A.3 appear. Compared to for example quantum electrodynamics (QED), there is however a significant complication when predicting experimental outcomes using this formalism: perturbative quantum field theory describes scattering processes where the initial and final states consist of free particles in the fundamental fields in the Lagrangian, but free quarks or gluons are never observed in experiments. On the contrary, color charged particles appear only in bound states such as baryons and mesons, which are color singlets. This property is known as color confinement. One way to get an understanding of this is to look at the effective coupling constant of QCD, obtained by renormalizing the theory. The procedure of

10 1.1. Perturbative quantum chromodynamics renormalization is not discussed here, but is treated in any text on quantum field theory, and discussed in detail for the case of QCD in [1, pp. 80-105]. One main result is that the parameters appearing in the bare Lagrangian in equation (1.1), such as , do not equal to the values measured in experiments. In fact, the measured parameters푔푠 will depend on the energy scale at which the process occurs. For the QCD coupling constant, one usually considers rather 2 than itself, following the example of the fine structure constant훼푠 = used푔푠 /4휋 in QED, em 푔푠 . To leading order in perturbation theory, one finds the following 2 훼expression= 푒 /4휋 for the running of the coupling [1, p. 31]:

(1.2) 2 1 푠 훼 (푄 ) = 2 2 . 훽0 ln (푄 /훬푄퐶퐷) Here, is the momentum transfer in the reaction of interest, is a con- stant, and푄 is an experimentally determined훽0 > parameter 0 called the QCD훬푄퐶퐷 scale.∼ 풪(200) MeV Performing a computation in perturbative quantum field theory essentially amounts to making a series expansion in the coupling constant , and only computing the first few terms. This only makes sense if 훼푠(and since equation (1.2) is only a leading order approximation derived훼푠 ≪ from 1 the same perturbation theory, this expression is only valid in the same regime). But from equation (1.2), it is clear that requires . This shows that 2 2 perturbative QCD may be applicable훼푠 ≪ 1 to high-energy,푄 ≫ short-distance 훬푄퐶퐷 processes, but cannot be used at lower energies and larger distances where the coupling is strong. It is precisely these low-energy processes that account for the binding of quarks and gluons into hadrons which can be detected, and it is therefore not at all obvious how to connect calculations in perturbative QCD to experiment. There are in general two possible solutions to this problem. One is to factor- ize a QCD process into a perturbative part which describes the specific short- distance interactions relevant for the process, and a universal non-perturbative part describing the transition from perturbatively free quarks and gluons to hadrons, starting from the perturbative final state. The non-perturbative part cannot be calculated analytically from the Lagrangian, but may be measured experimentally. The justification of this approach relies on the fact that the perturbative and non-perturbative phenomena occur on very different energy, time and distance scales, in many cases isolating them from affecting each other to a significant degree. For a number of quantities there exist formal proofs that the factorization is well-defined [2], but such issues are beyond the scope of this text. The other approach, which is intimately related to the main topic of this text, is to define the measured observables such that they are insensitive to the effects of long-distance interactions. Such observables are called infrared safe. This concept will be defined and discussed in chapter 3.

11 1.1. Perturbative quantum chromodynamics

Table 1.1: Masses of quarks and leptons. The two uncertainties on the top mass give statistical and systematical uncertainties, respectively. The data is retrieved from [4].

Particle Mass 푚/MeV +0.7 푢 2.3−0.5 +0.5 푑 4.8−0.3 푠 95 ± 5 3 푐 (1.275 ± 0.025) ⋅ 10 3 푏 (4.18 ± 0.03) ⋅ 10 3 푡 (173.21 ± 0.51 ± 0.71) ⋅ 10 푒 0.510998928 ± 0.000000011 휇 105.6583715 ± 0.0000035 휏 1776.82 ± 0.16 An example of an infrared safe quantity which will be discussed in great detail is the total cross section for the production of hadrons from the annihila- tion of a positron and an electron. To leading order in electroweak interactions, the electron and positron annihilate to a virtual photon or -boson, which then forms a quark-antiquark pair: . So푍 far this process + − ∗ does not involve QCD, and is described푒 푒 perturbatively→ 훾 /푍 → 푞푞 in electroweak theory. The -state then undergoes hadronization via non-perturbative processes. To leading푞푞 order in both electroweak theory and QCD, this is the only per- turbative -scattering process that undergoes hadronization, so the total + − cross section푒 푒 for must be identical to the total cross sec- + − ∗ tion for hadrons푒 푒 →at 훾 /푍 this → order. 푞푞 Although this cross section gives no + − details about푒 푒 which→ particles or momenta to expect in the hadronic final state, it is interesting by virtue of being a testable prediction derived using only perturbative techniques. An unrelated, but important feature of perturbative QCD is seen by con- sidering the masses of fundamental particles, shown in table 1.1. With on the order of , it is clear that also implies that 훬푄퐶퐷 2 2 2 200. Assuming MeV that no 푄quarks≫ 훬푄퐶퐷 and leptons are produced푄 ≫ 2 2 2 2 2 (this푚푢, 푚 is푑, guaranteed 푚푠 , 푚푒 , 푚휇 if ), it is푐, 푏, therefore 푡 a very휏 good approximation 2 2 in perturbative QCD푄 to< assume (2푚푐) that all particles are massless. This greatly simplifies many calculations. This introduction to perturbative QCD has been mostly based on heuristic reasoning and appeal to intuition. For a more rigorous discussion, see e.g. [3].

12 1.2. Infrared divergences

1.2 Infrared divergences

One may obtain higher-order approximations of the cross section for + − hadrons by including QCD interactions between the particles in the푒 pertur-푒 → bative final state. The lowest order correction involves a free gluon emitted from one of the quarks, such that the process considered now is . + − Another possibility is to let the -pair emit and absorb a virtual gluon푒 푒 → before 푞푞푔 leaving the perturbative domain,푞푞 thus going to higher order in the ampli- tude for . Although these processes have different final states + − on the perturbative푒 푒 → 푞푞 level, they both contribute to the total cross section for hadrons. + − 푒 푒 When→ computing these amplitudes, however, a serious complication arises: the gluon is massless and can be emitted with arbitrarily small momentum, and as will be shown, the amplitude for diverges as the gluon + − energy tends to . The divergence survives푒 the푒 → phase 푞푞푔 space integral, leaving the cross section0 for the process infinite. This phenomenon is an example of an soft divergence, that is, a divergence due to contributions from a low- momentum limit. In the approximation of massless quarks, the amplitude also diverges when the gluon is emitted parallel to the quark; this is called a collinear divergence. Together, they are known as infrared divergences. These divergences are fundamentally different from the ultraviolet divergences often encountered in loop integrals in quantum field theory, which stem from high- energy contributions. The procedure of renormalization used to handle the ultraviolet divergences does not eliminate infrared divergences. However, an infrared divergence also appears in at next-to- + − leading order, when the energy of the virtual gluon tends푒 푒 to→ 푞.푞 This diver- gence has the opposite sign of the divergences in 0 , and when + − computing the total cross section of hadrons,푒 these푒 → divergences 푞푞푔 cancel + − exactly, leaving cross section finite.푒 The푒 → main objective of this project is to show this cancellation by explicitly calculating the necessary amplitudes and cross sections and regularizing the infinities. This calculation is carried out in chapter 2. The specific process of cancellation discussed here is just an example of a more general phenomenon. Infrared divergences always show up in theories with massless particles, they appear at all orders, and in many cases it is possible to show rigorously that they also cancel at each order when computing quantities involving sums over properly defined sets of initial and final states. From a physical perspective, it is imperative that singularities can be avoided in one way or the other when computing physically measurable quantities, and this is where the concept of infrared safety becomes relevant. The general theorems and implications of infrared divergences and infrared safety, as well as their interpretation in the context of QCD, will be discussed in chapter 3.

13 2 Cancellation of infrared divergences in hadrons + − 푒 푒 → This chapter contains the mathematical derivations demonstrating the cancel- lation of QCD infrared divergences in annihilation to hadrons. First, the + − amplitude for the finite lowest order contribution,푒 푒 , is computed, + − and then a formalism is developed for efficient computation푒 푒 → 푞 of푞 the necessary cross sections. This is used along with dimensional regularization to show how infrared divergences in hadrons cancel to first order in . The + − presentation closely follows [1].푒 푒 → 훼푠 To get a preliminary intuition for the cancellation mechanism, consider first the scattering amplitude for the process , which can be expanded + − in the QCD coupling as 푒 푒 → 푞푞 훼푠 (2.1) (0) (1) ℳ푞푞 = ℳ푞푞 + 훼푠ℳ푞푞 + ⋯ . As will be shown, is finite, while contains infrared singularities. (0) (1) Similarly, the processℳ푞푞 can beℳ expanded푞푞 as + − 푒 푒 → 푞푞푔 (2.2) (0) ℳ푞푞푔 = √훼푠ℳ푞푞푔 + ⋯ . Compared to equation (2.1), this amplitude contains an extra factor of , or equivalently , due to the free gluon emission. As will be shown, 푠also (0)푔 contains infrared√훼푠 singularities. ℳ푞푞푔 Note in passing that, as a general rule in this text, a subscript like or on a quantity denotes the final state of the process the quantity pertains푞푞 푞푞푔 to. This unambiguously identifies the process in question, since the only initial state considered is . A superscript like or means the that the + − quantity is computed푒 at푒 a specific order in the(0) QCD perturbation(1) expansion, with leading order starting at , and all powers of factored out to facilitate straightforward comparison of(0) the order of terms from훼푠 different expansions. These amplitudes correspond to processes with different final states and cannot be added directly, but both contribute to the total cross section for

14 2.1. Leading order amplitude for + − 푒 푒 → 푞푞 hadrons. Using equation (A.1), one can write + − 푒 푒 → hadrons (2.3) 휎 = 휎푞푞 + 휎푞푞푔 + ⋯ , (2.4) 1 2 휎푞푞 = ∫ 푑훷2 ∣ℳ푞푞∣ , 4퐼 (2.5) 1 2 휎푞푞푔 = ∫ 푑훷3 ∣ℳ푞푞푔∣ . Expanding the squared amplitudes4퐼 to first order in , one finds 훼푠 2 (2.6) 2 (0) (0) (1)∗ ∣ℳ푞푞∣ = ∣ℳ푞푞 ∣ + 2훼푠ℜ {ℳ푞푞 ℳ푞푞 } + ⋯ , 2 (2.7) 2 (0) ∣ℳ푞푞푔∣ = 훼푠 ∣ℳ푞푞푔∣ + ⋯ . Although absolute squares of amplitudes must be positive, individual interfer- ence terms such as the one in equation (2.6) may be negative. For the infrared divergences to cancel to order , one must find that 훼푠 2 finite (2.8) (0) (1)∗ (0) ∫ 푑훷2 2ℜ {ℳ푞푞 ℳ푞푞 } + ∫ 푑훷3 ∣ℳ푞푞푔∣ = . Of course, this will only show cancellation to first order in . The next term in equation (2.6) is , and it will contain positive infrared훼푠 singularities, 2 (1) 2 as will the leading훼푠 order|ℳ푞푞 | term for . These may then cancel + − against interference terms of order 푒 푒in equation→ 푞푞푔푔 (2.7), and so on. Only the 2 cancellation to first order will be rigorously훼푠 derived here, but the general issue will be discussed in chapter 3.

2.1 Leading order amplitude for + − The leading order contribution to electron-positron푒 푒 annihilation→ 푞푞 to hadrons, , is given by the Feynman diagram in figure 2.1. Strictly + − ∗ 푒speaking,푒 → 훾 the→ similar 푞푞 process with the virtual photon replaced by a ∗ boson also participates, but this contribution is strongly훾 attenuated except푍 around the pole of the boson propagator at momentum transfer 2 2 [4, 5, p. 337].푍 Here, this contribution will be neglected, thus푄 implicitly= 푀푍 ≈ 2 (91assuming GeV) a momentum transfer far from this resonance. The diagram in figure 2.1 consists of QED vertices only, and using the rules in appendix A the amplitude is straightforward to write down:

(0) 푠′ + 푠 − 휇휈 푟 푟′ (2.9) 푖ℳ푞푞 = 푣 (ℓ ) (푖푒훾휇) 푢 (ℓ ) 푖퐷퐹 (푄) 푢 (푞) (−푖푒푒푞훾휈) 훿푖푗푣 (푞) 푠′ + 푠 − 휇휈 푟 푟′ = [푖푒푣 (ℓ ) 훾휇푢 (ℓ )] 푖퐷퐹 (푄) [−푖훿푖푗푒푒푞푢 (푞) 훾휈푣 (푞)] , where are the 4-momenta of the electron, positron, quark and anti- − + quark,ℓ respectively,, ℓ , 푞, 푞 while is the four-momentum transfer, − + 푄 = ℓ + ℓ = 푞 + 푞 15 2.1. Leading order amplitude for + − 푒 푒 → 푞푞

− 푒 , 푠 푞, 푟, 푖

∗ 훾

+ 푒 , 푠′ 푞, 푟′, 푗 Figure 2.1: Leading order diagram for . The labels denote − + spin polarizations, while are color indices.푒 푒 → 푞푞 푠, 푠′, 푟, 푟′ 푖, 푗 and is the (gauge-dependent) photon propagator from equation (A.7), 휇휈 repeated퐷퐹 (푄) here for convenience:

휇휈 휇 휈 2 (2.10) 휇휈 −휂 + (1 − 휉) 푄 푄 /푄 푖퐷퐹 (푄) = 푖 2 . 푄 + 푖휀 The are Dirac spinors, explained in appendix B.2. The enforces color푢, conservation 푣, 푢, 푣 on the photon-quark-antiquark vertex. 훿푖푗 For later convenience, the lepton and hadron currents and can be defined as the incoming and outgoing currents coupling to the퐽퐿 photon퐽푋 propa- gator, respectively. For this diagram, one finds

(2.11) 푠′ + 푠 − (퐽퐿)휇 = 푖푒푣 (푝 ) 훾휇푢 (푝 ), (2.12) (0) 푟 푟′ 푞푞 푖푗 푞 휈 (퐽 )휈 = −푖훿 푒푒 푢 (푞) 훾 푣 (푞) . As several different hadron currents will be considered throughout the report, they are labeled with subscripts and superscripts identifying the process and expansion order they pertain to. The lepton current will be the same in all computations, so it is simply referred to . Contracting the momentum transfer퐽퐿 with the lepton and hadron cur- 휇 rents gives 푄

휇 푠′ + + − 푠 − (2.13) 푄 (퐽퐿)휇 ∝ 푣 (ℓ )(/ℓ + /ℓ ) 푢 (ℓ ) 푠′ + 푠 − = 푣 (ℓ ) (−푚푒 + 푚푒) 푢 (ℓ ) = 0 , 휈 (0) 푟 푟′ (2.14) 푞푞 푄 (퐽 )휈 ∝ 푢 (푞) (/푞 + /푞) 푣 (푞) 푟 푟′ = 푢 (푞) (푚푞 − 푚푞) 푣 (푞) = 0 .

16 2.1. Leading order amplitude for + − 푒 푒 → 푞푞 Here the Dirac equations in momentum space,

(2.15) (푝/ − 푚) 푢 (푝) = (푝/ + 푚) 푣 (푝) = 0 , (2.16) 푢 (푝) (푝/ − 푚) = 푣 (푝) (푝/ + 푚) = 0 , were used. They are discussed in detail in appendix B.2 as equations (B.35), (B.36), (B.60) and (B.61). Thus, the gauge dependent term in the photon propagator will not con- tribute to the amplitude, as predicted by gauge invariance, and one can there- fore elect to work in the Feynman gauge , where 휉 = 1 휇휈 (2.17) 휇휈 −휂 푖퐷퐹 (푄) = 푖 2 . The amplitude may then be written 푄

휇 (2.18) (0) 푖 (0) 푖ℳ푞푞 = − 2 (퐽퐿)휇 (퐽푞푞 ) . In most experimental setups, the푄 beam of particles in the initial state is un- polarized and the spin and color states of the final particles are never measured. The squared amplitude describing this process is thus obtained by averaging over initial state spins, and summing over final state spins and colors, the squared amplitudes for the spin- and color-dependent amplitude, as follows:

2 2 (0) 1 (0) ∣ℳ푞푞 ∣ = ∑ ∣ℳ푞푞 ∣ (2.19) 4 푠,푠′,푟,푟′,푖,푗 1 (0)휇휈 = 4 퐿휇휈퐻푞푞 , where the overbar denotes matrix elements푄 that are spin- and color-averaged as described. The lepton and hadron tensors and are defined as 휇휈 퐿휇휈 퐻푞 ̅푞 (2.20) 1 ∗ 퐿휇휈 = ∑ (퐽퐿)휇 (퐽퐿)휈 , 4 푠,푠′ 휇 휈 (2.21) 휇휈 ∗ 퐻푞 ̅푞 = ∑ (퐽푞) ̅푞 (퐽푞) ̅푞 . 푟,푟′,푖,푗 To compute these tensors, it is convenient to make use of the completeness relations for Dirac basis spinors, given in equations (B.71) and (B.72) and repeated here for convenience:

(2.22) 푠 푠 ∑ 푢 (푝) 푢 (푝) = 푝/ + 푚 , 푠 (2.23) 푟 푟 ∑ 푣 (푝) 푣 (푝) = 푝 − 푚 . 푟 / 17 2.1. Leading order amplitude for + − 푒 푒 → 푞푞 The lepton tensor evaluates to

† 2 푠′ + 푠 − 푠′ + 푠 − 4퐿휇휈 = 푒 ∑ [푣 (ℓ ) 훾휇푢 (ℓ )] [푣 (ℓ ) 훾휈푢 (ℓ )] 푠,푠′ 2 푠′ + 푠 − 푠 − 푠′ + = 푒 ∑ 푣 (ℓ ) 훾휇푢 (ℓ ) 푢 (ℓ ) 훾휈푣 (ℓ ) 푠,푠′ (2.24) 2 푠′ + 푠 − 푠 − 푠′ + = 푒 ∑ Tr [푣 (ℓ ) 훾휇푢 (ℓ ) 푢 (ℓ ) 훾휈푣 (ℓ )] 푠,푠′ 2 푠′ + 푠′ + 푠 − 푠 − = 푒 ∑ Tr [푣 (ℓ ) 푣 (ℓ ) 훾휇푢 (ℓ ) 푢 (ℓ ) 훾휈] 푠,푠′ 2 + − Here, equation (B.59)= 푒 isTr used [(/ℓ in− the 푚푒) first 훾휇 ( step./ℓ + The 푚푒) resulting 훾휈]. expression consists of two sandwiches of spinors and spinor-space matrices, which both evaluate to spinor-space scalars, and it is therefore trivially equivalent to the spinor-space trace of the same expression, as shown in step two. The cyclicity of the trace (equation (B.7)) is then exploited in the third step to reorganize the expression such that the completeness relations can be used to give the final expression. Using the same manipulations, the hadron tensor is found to be (2.25) (0)휇휈 2 휇 휈 The extra factor퐻 of푞푞 =comes 푁푐(푒푒 from푞) Tr [(the/푞 sum+ 푚푞) over 훾 ( color/푞 − 푚 states푞) 훾 ]. in the final state. Following appendix푁 A,푐 the number of colors will be kept general throughout the computations, such that (2.26) ∑ 훿푖푗훿푖푗 = 푁푐 . Using results from appendix푖,푗 B.1, the traces may be computed:

2 푒 +휌 −휎 2 2 퐿휇휈 = ℓ ℓ Tr [훾휌훾휇훾휎 훾휈] − 푒 푚푒 Tr [훾휇훾휈] 4 2 +휌 −휎 2 (2.27) = 푒 [ℓ ℓ (휂휌휇휂휎휈 − 휂휌휎 휂휇휈 + 휂휌휈휂휇휎 ) − 푚푒 휂휇휈] 2 + − − + + − 2 = 푒 [ℓ휇 ℓ휈 + ℓ휇 ℓ휈 − (ℓ ⋅ ℓ + 푚푒 ) 휂휇휈] 2 + − − + 2 휇 휈 휇 휈 휇휈 = 푒 [ℓ ℓ + ℓ ℓ − (푄 /2) 휂 ], (2.28) 2 (0)휇휈 휇 휈 휇 휈 2 휇휈 Here, the퐻푞 following푞 = 4푁 identites푐 (푒푒푞) [푞 have푞 + been푞 푞 used,− (푄 which/2) 휂 follow]. from the relativistic energy-momentum relation (equation (B.32)):

2 + − 2 푄 = (ℓ + ℓ ) (2.29) + 2 − 2 + − = (ℓ ) + (ℓ ) + 2ℓ ⋅ ℓ + − 2 = 2 (ℓ ⋅ ℓ + 푚푒 ), (2.30) 2 2 푄 = 2 (푞 ⋅ 푞 + 푚푞). 18 2.2. Cross sections for hadrons + − 푒 푒 → Using the equations (2.29) and (2.30), the final expression for the matrix element then becomes

2 2 2 (0) 4푁푐 (푒 푒푞) + − − + (2.31) ∣ℳ푞푞 ∣ = 4 [2 (ℓ ⋅ 푞) (ℓ ⋅ 푞) + 2 (ℓ ⋅ 푞) (ℓ ⋅ 푞) 푄 2 2 2 + 푄 (푚푒 + 푚푞)]. 2.2 Cross sections for hadrons + − It should be clear from the discussion푒 푒 above→ that the factorization of the matrix element into lepton and hadron tensors and is not limited to . 휇휈 2 On the contrary, such factorizations are퐿 possible휇휈 퐻 in any process where|ℳ푞 the푞| interaction between two fermion currents is mediated by a single photon in all included diagrams. In the case of hadrons, it is thus a general feature + − of the matrix elements to any order푒 in푒 QCD,→ as long as only the leading order in QED is considered. Moreover, the lepton tensor will be the same in all these calculations. Given the magnitude of the QED퐿휇휈 coupling at em , this −2 should be a good approximation. Therefore, only leading-order훼 contributions∼ 10 in QED will be considered here. Thus, the cross section for , where is a state of quarks and + − gluons, can in general be written푒 푒 → 푋 푋

(2.32) 1 1 휇휈 휎푋 = 2 ⋅ 4 퐿휇휈 ∫ 푑훷푋 퐻푋 , where only the hadron tensor2푄 and푄 phase space measure depends on the fi- nal state. The flux factor in equation (A.2) has been used together with the approximation of massless particles, discussed in section 1.1. From this point, all calculations will assume massless particles, . Moreover, all calculations will take place in spacetime dimensions푚푒 = 푚푞 = 0to facilitate dimensional regularization, using the푑 = results 4−2휀 in appendix C. Hence, the metric tensor contracts to , and the coupling constants will be 휇 accompanied by a mass scale 휂in휇 = accordance 푑 with equation (C.3). 휀 Due to gauge invariance in휇 QED, one must have and 휇휈 휇휈 , such that the -term푄 in휇 the퐻 photon= 푄휈퐻 propagator= 0 휇휈 휇휈 휇 휈 drops푄휇퐿 out.= 푄휈 In퐿 section= 0 2.1, this was(1 explicitly − 휉)푄 푄 shown on the level of currents in equations (2.13) and (2.14). When going to higher order, the cancellation is not guaranteed to happen for individual diagrams, but when summing all diagrams of a given order the gauge-dependent term must disappear.1 1In this particular case the gauge-dependent term would drop out even if it was only canceled by one of the tensors, and it was confirmed in section 2.1 that does the job. However, gauge invariance must hold for all QED processes, including for example퐿휇휈 the process where the matrix element factorizes as , so any current coupling to a photon must 휇휈 cancel the gauge-dependent term when∝ summing 퐻휇휈퐻 all contributions at a given order.

19 2.2. Cross sections for hadrons + − 푒 푒 → In equation (2.32) the momenta of the final state particles are integrated out, so the only free variable is , and the only quantities available to carry 2 the Lorentz indices of are 푄 and . Hence 휇휈 휇휈 휇 퐻 휂 푄 (2.33) 휇휈 2 휇휈 2 휇 휈 ∫ 푑훷 퐻 = 퐻1 (푄 ) 휂 + 퐻2 (푄 ) 푄 푄 . Contracting this with and imposing gauge invariance shows that 휇 2 . Thus, defining푄 , one has 퐻1(푄 ) = 2 2 2 2 −푄 퐻2(푄 ) 퐻(푄 ) = −퐻1(푄 )/(푑 − 1) 휇 휈 (2.34) 휇휈 1 휇휈 푄 푄 2 ∫ 푑훷 퐻 = (−휂 + 2 ) 퐻 (푄 ) . 푑 − 1 푄 This can be solved for by contracting with , since 2 퐻(푄 ) 휂휇휈 휇 휈 (2.35) 휇휈 푄 푄 − 휂휇휈 (−휂 + 2 ) = 푑 − 1 , 푄 such that (2.36) 2 휇휈 퐻 (푄 ) = −휂휇휈 ∫ 푑훷 퐻 . Substituting equation (2.34) into equation (2.32) and using the gauge invariance condition for , one finds 퐿휇휈 (2.37) 1 휇휈 2 휎푋 = 6 (−휂 퐿휇휈) 퐻푋 (푄 ). 2(푑 − 1)푄 Using equation (2.27) and equation (2.29) gives

(2.38) 휇휈 2휀 2 2 휂 퐿휇휈 = 휇 푒 푄 (1 − 휀) . The cross section can thus be expressed

2휀 2 휇 푒 1 − 휀 2 휎푋 = 4 퐻푋 (푄 ) (2.39) 2푄 em3 − 2휀 2휀 2휋훼 휇 1 − 휀 2 = 4 퐻푋 (푄 ). 푄 3 − 2휀 Here, em . This way, the computation of any cross section contributing 2 to the total훼 = cross 푒 /4휋 section for hadrons at leading order in QED is reduced + − to finding the corresponding푒 푒 hadron→ tensor, contracting it with the metric tensor and integrating over the relevant phase space. Using this formalism, the cross section for at leading order is + − straightforward to compute. The first step is to푒 find푒 → 푞푞

2 (2.40) (0)휇휈 2휀 2 휂휇휈퐻푞푞 = 4푁푐휇 (푒푒푞) 푄 (1 − 휀) . 20 2.3. Real gluon emission: at leading order + − 푒 푒 → 푞푞푔 This does not depend explicitly on any of the outgoing momenta, so the phase space integral is trivial by specializing to the center-of-momentum (COM) frame and using equation (C.35):

(0) 2 (푑) (0)휇휈 퐻푞푞 (푄 ) = − ∫ 푑훷2 휂휇휈퐻푞푞 (2.41) 2 2휀 2 휀 4푁푐휇 (푒푒푞) 푄 휋 ̃푞 (1 − 휀) 훤 (1 − 휀) = ( 2 ) . 4휋 2 훤 (2 − 2휀) ̃푞 √푄 Here, . In the COM frame, . 2 2 0 2 2 0 2 0 2 Hence, ̃푞 = 풒 = (푞 ) − 푚푞 = (푞 ) ̃푞= 푞 = √푄 /2 휀 em 2 (2.42) (0) 2 2 2 4휋휇 (1 − 휀) 훤 (1 − 휀) 퐻푞푞 (푄 ) = 2훼 푁푐푒푞푄 ( 2 ) . Putting it all together, the cross section for푄 훤 (2at leading− 2휀) order becomes + − 푒 푒 → 푞푞 em 2 2휀 2 2 휀 2 (2.43) (0) 4휋훼 휇 푁푐푒푞 4휋휇 (1 − 휀) 훤 (1 − 휀) 휎푞푞 = 2 ( 2 ) . 푄 푄 3 − 2휀 훤 (2 − 2휀) As a side remark, note that the cross section has mass dimension , 2(휀−1) or equivalently, length dimensions, and is thus an area푀 in the usual sense only2 for− 2휀 = 푑 −. 2 This is a consequence of appending the mass scale to all couplings휀 = in 0 the computation, and if is literally interpreted as 휀 the spacetime휇 dimension, it is consistent with the definition푑 of the scattering cross section as the proportionality constant relating the number of scattering events in a differential spacetime region to the number of targets present in this region times the number of scatterers passing through the region per cross sectional area of the incident beam, as explained in e.g. [6, p. 159].2 On the other hand, in [1] the mass dimension is only appended to the couplings in the hadron current and not those in the휇 lepton current, and thus the cross section is always 2-dimensional. This is unproblematic in this particular case, but the principle seems difficult to generalize and this approach will not be followed here. The final results are of course identical in the limit. 휀 → 0 2.3 Real gluon emission: at leading order + − The leading order Feynman diagrams푒 for푒 the→ process 푞푞푔 is shown in + − figure 2.2. Following the discussion in the last section,푒 only푒 → the 푞푞푔 hadron tensor needs to be computed in order to evaluate the cross section. Applying the Feynman rules from appendix A, the sum of the hadron currents in these two

2In spatial dimensions, the cross section of a beam traveling along a single dimension is a 푑-dimensional − 1 quantity, and so is the scattering cross section. 푑 − 2

21 2.3. Real gluon emission: at leading order + − 푒 푒 → 푞푞푔

− − 푒 , 푠 푞, 푟, 푖 푒 , 푠 푞, 푟, 푖

(푞 + 푔) ∗ 훾 ∗ 푔, 푡, 푎 푔, 푡, 푎 훾 −(푞 + 푔)

+ + 푒 , 푠′ 푞, 푟′, 푗 푒 , 푠′ 푞, 푟′, 푗 Figure 2.2: Leading order diagrams for . The labels − + denote spin polarizations, while are color푒 푒 indices.→ 푞푞푔 The momentum푠, 푠′, of 푟, 푟′,the 푡 internal quark flows in the same푖, direction 푗, 푎 as the fermion current, as indicated by the arrowhead.

diagrams is found to be

(0) 2휀 푎 ∗휎 푠 푞푞푔 푞 푠 푖푗 푡 √훼 (퐽 )휇 = −푖휇 푒푒 푔 푇 휀 (푔) (2.44)

푟 ⎡ /푞 + 푔/ − (/푞 + 푔)/ ⎤ 푟′ × 푢 (푞) ⎢훾휎 2 훾휇 + 훾휇 2 훾휎 ⎥ 푣 (푞) . (푞 + 푔) (푞 + 푔) Here, is the 4-momentum⎣ of the gluon. As explained on page⎦ 14, all powers of are푔 factored out of quantities defined at a specific order in the perturbation expansion.훼푠 This time, the hadron tensor includes a sum over the polarization and color states of the gluon in addition to everything else:

휇 휈 (0)휇휈 (0) (0)∗ 훼푠퐻푞푞푔 = 훼푠 ∑ (퐽푞푞푔) (퐽푞푞푔 ) (2.45) 푟,푟′,푖,푗,푡,푎 휇휎 휈휌 ∗ (0) (0)∗ = 훼푠 ∑ 휀푡,휎 (푔) 휀푡,휌 (푔) ∑ (퐽푞푞푔) (퐽푞푞푔 ) . 푡 푟,푟′,푖,푗,푎 Here, the gluon polarization vector was factored out to isolate the polarization sum, which is evaluated in equation (A.14). However, using equations (B.25), (B.26), (B.36) and (B.60) and the vanishing mass of all particles, one finds

휇휎 (0) 푟 ⎡ /푞 + 푔/ 휇 휇 /푞 + 푔/ ⎤ 푟′ (퐽푞푞푔) 푔휎 ∝ 푢 (푞) ⎢푔/ 2 훾 − 훾 2 푔/⎥ 푣 (푞) ⎣ (푞 + 푔) (푞 + 푔) ⎦ (2.46) 푟 푔//푞 휇 휇 /푞푔/ 푟′ = 푢 (푞) [ 훾 − 훾 ] 푣 (푞) 2푞 ⋅ 푔 2푞 ⋅ 푔 푟 /푞푔/ 휇 휇 푔//푞 푟′ = 푢 (푞) [(푰 − ) 훾 − 훾 (푰 − )] 푣 (푞) = 0 , 2푞 ⋅ 푔 2푞 ⋅ 푔 22 2.3. Real gluon emission: at leading order + − 푒 푒 → 푞푞푔 so the second term in equation (A.14) falls out, and the polarization sum may be replaced by (2.47) ∗ ∑ 휀푡,휎 (푔) 휀푡,휌 (푔) → −휂휎휌 . This could be anticipated from푡 the discussion in appendix A.4: the final state contains only a single gluon, so there is no external gluon pair coupling to a triple-gluon vertex in any diagram. The result is that

휇휎 휈 (2.48) (0)휇휈 (0) (0)∗ 푠 푞푞푔 푠 푞푞푔 푞푞푔 훼 퐻 = −훼 ∑ (퐽 ) (퐽 )휎 . 푟,푟′,푖,푗,푎 This expression also includes a sum over products of the , which equals the 푎 trace from equation (A.20): 푇푖푗

∗ (2.49) 푎 푎 푎 푎† ∑ 푇푖푗 (푇푖푗) = Tr [푇 푇 ] = 퐶퐹푁푐 . 푖,푗,푎 The only remaining sum is that over the spinor indices. As in section 2.1, this can be rewritten as a trace over gamma matrices.

(0)휇휈 푟 휎 /푞 + 푔/ 휇 휇 /푞 + 푔/ 휎 푟′ 퐻푞푞푔 ∝ ∑ 푢 (푞) [훾 훾 − 훾 훾 ] 푣 (푞) 푟,푟′ 2푞 ⋅ 푔 2푞 ⋅ 푔 푟′ 휈 /푞 + 푔/ /푞 + 푔/ 휈 푟 × 푣 (푞) [훾 훾휎 − 훾휎 훾 ] 푢 (푞) (2.50) 2푞 ⋅ 푔 2푞 ⋅ 푔 휎 /푞 + 푔/ 휇 휇 /푞 + 푔/ 휎 = Tr[/푞 (훾 훾 − 훾 훾 ) 2푞 ⋅ 푔 2푞 ⋅ 푔 휈 /푞 + 푔/ /푞 + 푔/ 휈 × /푞 (훾 훾휎 − 훾휎 훾 )] 2푞 ⋅ 푔 2푞 ⋅ 푔 The hadron tensor will be contracted with when finding the cross section, and doing it right away will simplify the traces.휂휇휈 Defining (2.51) 휎 휇 푡푞푞 = Tr [/푞훾 (/푞 + 푔)/ 훾 /푞훾휇 (/푞 + 푔)/ 훾휎 ], (2.52) 휎 휇 푡푞푞 = Tr [/푞훾 (/푞 + 푔)/ 훾 /푞훾휎 (/푞 + 푔)/ 훾휇], (2.53) 휇 휎 푡푞푞 = Tr [/푞훾 (/푞 + 푔)/ 훾 /푞훾휇 (/푞 + 푔)/ 훾휎 ], (2.54) 휇 휎 푡푞푞 = Tr [푞훾 (푞 + 푔) 훾 푞훾휎 (푞 + 푔) 훾휇], the contracted hadron tensor/ can/ be/ written/ / /

2 (0)휇휈 4휀 −훼푠휂휇휈퐻푞푞푔 = 퐶퐹푁푐휇 (푒푒푞푔푠) (2.55) 푡푞푞 푡푞푞 푡푞푞 + 푡푞푞 ⎡ ⎤ × ⎢ 2 + 2 − ⎥ , (2푞 ⋅ 푔) (2푞 ⋅ 푔) where . ⎣(2푞 ⋅ 푔) (2푞 ⋅ 푔) ⎦ 푄 = 푞 + 푞 + 푔 23 2.3. Real gluon emission: at leading order + − 푒 푒 → 푞푞푔 Here, the infrared divergences appear for the first time. The contracted hadron tensor, and hence the scattering cross section , diverge as (0) or . By inspecting the calculations performed휎푞푞푔 so far, one may푞 ⋅ 푔 also → 0 realize푞 ⋅ 푔 that → 0 the denominators in equation (2.55) would consist of these factors even without assuming massless quarks. Writing

√ 2 (2.56) √ 푚푞 푞 ⋅ 푔 = 퐸푞퐸푔 (1 − 훽푞 cos 휃푞푔) , 훽푞 = √1 − 2 , 퐸푞 it is clear that the divergences occur as . Additionally,⎷ if such that is allowed and , the expression퐸푔 → 0 will diverge as푚푞 = 0 or 퐸푞 =. The 0 singularities at훽푞 zero= 1 energy are examples of soft singularities,휃푞푔 → 0 while퐸푞 → 0the singularity at zero angle is a collinear singularity. The traces can be simplified and computed using equation (B.22):

2 푡푞푞 = 4 (1 − 휀) Tr [/푞 (/푞 + 푔)/ /푞 (/푞 + 푔)]/ 2 휇휈 휌휎 휇휌 휈휎 휇휎 휈휌 (2.57) = 16 (1 − 휀) 푞휇 (푞 + 푔)휈 푞휌 (푞 + 푔)휎 [휂 휂 − 휂 휂 + 휂 휂 ] 2 2 = 16 (1 − 휀) [2 (푞 ⋅ (푞 + 푔)) (푞 ⋅ (푞 + 푔)) − 푞 ⋅ 푞 (푞 + 푔) ] 2 The trace= 32 (1can − 휀) be[(푞 found ⋅ 푔) by(푞 ⋅ replacing 푔)] . with in , so . The cross terms are푡 more푞푞 complicated, but using equation푞 푞 (B.20),푡푞푞 one푡푞푞 finds= 푡푞푞

푡푞푞 + 푡푞푞 = 16 (푞 + 푔)휌 (푞 + 푔)훿 (푞휏푞휆 + 푞휏푞휆) 휌훿 휏휆 휌휏 훿휆 휌휆 훿휏 × [ (−2 + 휀 (1 + 휀)) 휂 휂 + 휀 (1 − 휀) (휂 휂 + 휂 휂 )] (2.58) = 16[ (−2 + 휀 (1 + 휀)) 2 (푞 ⋅ 푞) ((푞 + 푔) ⋅ (푞 + 푔)) + 휀 (1 − 휀) 2 (푞 ⋅ (푞 + 푔)) (푞 ⋅ (푞 + 푔)) + 휀 (1 − 휀) 2 (푞 ⋅ (푞 + 푔)) (푞 ⋅ (푞 + 푔)) ] 2 = −32 (1 − 휀) [ (푞 ⋅ 푞) 푄 − 2휀 (푞 ⋅ 푔) (푞 ⋅ 푔) ] . Here, the relations and were used 2 2 2 2 to simplify the expression.푞 = 푞 = Inserting 푔 = 0 the푄 traces= 2푞 in ⋅ 푞 equation + 2푞 ⋅ 푔+ (2.55) 2푞 ⋅푔 gives

2 (0)휇휈 4휀 −훼푠휂휇휈퐻푞푞푔 = 8퐶퐹푁푐휇 (푒푒푞푔푠) (1 − 휀) (2.59) 2 푞 ⋅ 푔 푞 ⋅ 푔 (푞 ⋅ 푞) 푄 × [(1 − 휀) ( + ) + − 2휀] . To simplify the phase space integral,푞 ⋅ 푔 the푞 ⋅ COM 푔 frame(푞 ⋅ 푔) (energy푞 ⋅ 푔) fractions are used. They are defined in equation (C.36), and repeated here for convenience:푥푖

(2.60) 2푝푖 ⋅ 푄 푥푖 = 2 . 푄 24 2.3. Real gluon emission: at leading order + − 푒 푒 → 푞푞푔 For the massless quarks and gluons, this implies that

(2.61) 2푞 ⋅ 푔 1 − 푥푞 = 2 , 푄 (2.62) 2푞 ⋅ 푔 1 − 푥푞 = 2 , 푄 (2.63) 2푞 ⋅ 푞 1 − 푥푔 = 2 , 푄 (2.64) 푥푞 + 푥푞 + 푥푔 = 2 . Inserting this and using equation (C.37) to evaluate the phase space integral in equation (2.36) gives

em 2 2 2 2휀 (0) 2 훼 훼푠퐶퐹푁푐푒푞푄 4휋휇 (1 − 휀) 훼푠퐻푞푞푔 (푄 ) = ( 2 ) 휋 푄 훤 (2 − 2휀) 1 1 1 푞 푞 휀 × ∫0 푑푥 ∫1−푥푞 푑푥 [(1 − 푥푞) (1 − 푥푞) (푥푞 + 푥푞 − 1)]

푞 푞 푔 ⎡ 1 − 푥 1 − 푥 2 (1 − 푥 ) ⎤ × ⎢(1 − 휀) ⎜⎛ + ⎟⎞ + − 2휀⎥ 1 − 푥푞 1 − 푥푞 푞 푞 ⎣ ⎝ ⎠ (1 − 푥 ) (1 − 푥 ) (2.65)⎦ The collinear and soft quark divergences now appear in the limit where either or , while the soft gluon divergence manifest as the double pole when푥푞 → 1 both푥푞 limits→ 1 are taken simultaneously. To calculate the integral, substitute . The Jacobian is then , and the integral becomes 푥푞 = 푥, 푥푞 = 1 − 푣푥 푥

1 1 푥 ∫ 푑푥 ∫ 푑푣 2 휀 0 0 [푥 (1 − 푥) 푣 (1 − 푣)] 푥푣 1 − 푥 2 (1 − 푣) × [(1 − 휀) ( + ) + − 2휀] 1 − 푥 푥푣 (1 − 푥) 푣 훤 (2 − 휀) 훤 (1 − 휀) 훤 (−휀) = 2 (1 − 휀) (2.66) 훤 (3 − 3휀) 2 3 훤 (2 − 휀) 훤 (−휀) 훤 (1 − 휀) + 2 − 2휀 훤 (2 − 3휀) 훤 (3 − 3휀) 3 훤 (1 − 휀) 2 1 1 1 1 = ( 2 − − ( − 2 − 2휀)) 훤 (1 − 3휀) 1 − 3휀 휀 휀 2 − 3휀 휀 3 훤 (1 − 휀) 2 3 19 107 2 = ( 2 + + − 휀 + 풪 (휀 )) . 훤 (1 − 3휀) 휀 휀 2 4 Each term in the integral factorizes into a product of two Euler -function integrals, equation (C.16). All of the integrals are convergent for 훽 , which 휀 < 0 25 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞 is typical of infrared divergences: they are regularized when going to higher spacetime dimensions . The expressions have been manipulated further using equation (C.12). 푑 > 4 This gives

em 2 2 2 2휀 (0) 2 훼 훼푠퐶퐹푁푐푒푞푄 4휋휇 (1 − 휀) 훤 (1 − 휀) 훼푠퐻푞푞푔 (푄 ) = ( 2 ) 휋 푄 훤 (2 − 2휀) 2 훤 (1 − 휀) 2 3 19 107 2 × ( 2 + + − 휀 + 풪 (휀 )) (2.67) 훤 (1 − 3휀) 휀 휀 2 4 2 휀 (0) 2 훼푠퐶퐹 4휋휇 = 퐻푞푞 (푄 ) ( 2 ) 2휋 푄 2 훤 (1 − 휀) 2 3 19 × ( 2 + + + 풪 (휀)) . Thus, the scattering cross훤 (1 section − 3휀) for휀 휀 2 at leading order is + − 푒 푒 → 푞푞푔 em 2휀 2 2 2휀 2 (0) 2훼 훼푠휇 퐶퐹푁푐푒푞 4휋휇 (1 − 휀) 훤 (1 − 휀) 훼푠휎푞푞푔 = 2 ( 2 ) 푄 푄 3 − 2휀 훤 (2 − 2휀) 2 (2.68) 훤 (1 − 휀) 2 3 19 × ( 2 + + + 풪 (휀)) 훤 (1 − 3휀) 휀 휀 2 2 휀 2 (0) 훼푠퐶퐹 4휋휇 훤 (1 − 휀) 2 3 19 = 휎푞푞 ( 2 ) ( 2 + + + 풪 (휀)) These expressions2휋 show explicitly푄 훤 the(1 − how 3휀) the휀 infrared휀 2 divergences manifest as poles as . The total hadrons cross section can only be finite at + − if these휀 → poles 0 are canceled푒 푒 → exactly by another contribution at the same 풪(훼order.푠)

2.4 Virtual gluon exchange: at + − next-to-leading order 푒 푒 → 푞푞 The next-to-leading order diagrams for are shown in figure 2.3, + − with only the hadronic part included. As푒 푒 explained→ 푞푞 by equation (2.6), the -contribution to the cross section is given by the interference term be- 풪(훼tween푠) these diagrams and the leading order diagram calculated in section 2.1. In the formalism used here, the first step is to write the total hadron current as an expansion in . To order , 훼푠 풪(훼푠) (2.69) (0) (1) 푞푞 푞푞 푠 푞푞 (퐽 )휇 = (퐽 + 훼 퐽 + ⋯)휇 . Here, is the leading-order current from equation (2.12), and is the (0) (1) sum of퐽푞 the푞 currents from the diagrams in figure 2.3. The function 훼푠퐽푞푞 from 2 퐻(푄 ) 26 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞

푞, 푟, 푖 푞, 푟, 푖 푞, 푟, 푖

푔 + 푞

∗ 푔 ∗ 푔 ∗ 푔 훾 훾 훾 푔 − 푞

Figure 2.3: Diagrams for푞, 푟′,the 푗 hadron current in푞, 푟′, 푗 at one loop.푞, 푟′, The 푗 la- − + bels denote spin polarizations, while are color푒 푒 → indices. 푞푞 The momentum of the푟, 푟′ internal quarks flow in the same direction푖, 푗 as the fermion current, as indicated by the arrowhead. equation (2.36) can also be expanded:

(2.70) 2 (0) 2 (1) 2 퐻푞푞 (푄 ) = 퐻푞푞 (푄 ) + 훼푠퐻푞푞 (푄 ) + ⋯ , where is the leading-order term given by equation (2.42). By combining (0) the definition퐻푞푞 of the hadron tensor from equation (2.21), and of 휇휈 2 from equation (2.36), one finds that퐻 the푞푞 next term is an interference퐻(푄 term) between the leading and next-to-leading order currents:

휇 휈 (1) 2 (푑) (0) (1)∗ 훼푠퐻푞푞 (푄 ) = −훼푠휂휇휈 ∫ 푑훷2 ∑ [ (퐽푞푞 ) (퐽푞푞 ) 푟,푟′,푖,푗 휇 휈 (2.71) (1) (0)∗ + (퐽푞푞 ) (퐽푞푞 ) ]

휇 (푑) (0)∗ (1) 푠 2 푞푞 푞푞 = −훼 ∫ 푑훷 ∑ 2ℜ {(퐽 ) (퐽 )휇}. 푟,푟′,푖,푗 In the first two diagrams in figure 2.3, the virtual gluon loops back onto the quark it was emitted from. The corresponding factor in the amplitude is known as the quark self-energy , and for massless quarks it is given in the Feynman gauge as −푖훴(푞)

푑 휇휈 2휀 2 푎 푎 푑 푔 휂 훾휇 (푔/ + /푞) 훾휈 −푖훴 (푞) = −휇 푔푠 푇푖푘푇푘푗 ∫ 푑 2 2 (2.72) (2휋) 푔 (푔 + 푞) 푑 2휀 2 푑 푔 푔/ + /푞 = 2퐶퐹훿푖푗휇 푔푠 (1 − 휀) ∫ 푑 2 2 , 푔 (푔 + 2푞 ⋅ 푔) where equations (A.19) and (B.22) and the(2휋) fact that is an external, on-shell momentum were used. The integral in equation (2.72)푞 can be rewritten using

27 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞 the results in appendix C:

푑 푑 푔 푔/ + /푞 ∫ 푑 2 2 (2휋) 푔 (푔 + 2푞 ⋅ 푔) 푑 1 푑 푔 푔/ + /푞 = ∫ 푑 ∫0 푑훼 2 2 2 (2휋) (훼 (푔 + 2푞 ⋅ 푔) + (1 − 훼) 푔 ) 푑 푑 푔 1 푔 + 푞 / / (2.73) = ∫ 푑 ∫0 푑훼 4 (2휋) (푔 + 훼푞) 푑 1 푑 푘 1 = ∫0 푑훼 (1 − 훼) /푞 ∫ 푑 4 (2휋) 푘 푑 푖/푞 푑 푘퐸 1 = ∫ 푑 4 2 (2휋) 푘퐸 ∞ 푖/푞 훺푑 푑−5 = 푑 ∫ 푑푘퐸̃ 푘퐸̃ . 2 0 Here, the denominator(2휋) was first rewritten using the Feynman parameter inte- gral from equation (C.18), and then by completing the square and using . 2 The integration variable was changed to and the term proportional푞 = 0 to discarded, since it is antisymmetric푘 under = 푔 + reflection 훼푞 through the origin and/푘 thus must vanish upon integration. Finally, the -integral was rotated to Euclidean space as per equation (C.7) and rewritten푘 as a single-variable integral by exploiting the isotropy of the integrand and using equation (C.9). The final expression in equation (2.73) reveals that this integral is unde- fined for any value of the dimension : it has an infrared divergence due to contributions as for ,푑 and an ultraviolet divergence due to contributions as 푘퐸̃ →for 0 푑 ≤. This 4 is related to the fact that a massless, on-shell quark does푘퐸̃ → not ∞ provide푑 ≥ an 4 intrinsic scale for the integral, since . 2 Following the conventions in dimensional regularization, such an integral푞 = 0is defined to be zero [7, p. 172]. Somewhat heuristically, this can be justified by splitting the integral at an arbitrary scale and assuming different values of in the two regimes such that both integrals훬 are convergent: 푑

∞ 훬 ∞ −2휀 −2휀′ (2.74) 푑−5 푑−5 푑′−5 훬 훬 퐸̃ 퐸̃ 퐸̃ 퐸̃ 퐸̃ 퐸̃ ∫0 푑푘 푘 = ∫0 푑푘 푘 + ∫훬 푑푘 푘 = − . Here, to regulate the infrared divergence, while−2휀 −2휀′to regulate the ultraviolet휀 < 0 divergence. Of course, the right hand side >휀′ would > 0 only equal the left hand side if , and by analytic continuation one might thus say that the integral vanishes.휀′ = 휀 This means that the first two diagrams in figure 2.3 do not contribute, and is given by the third diagram only. This diagram is known as the vertex (1) 퐽correction.푞푞

28 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞 This result was derived in the Feynman gauge, but holds in any covariant gauge. The gauge-dependent term in the gluon propagator gives a term proportional to the integral

푑 푑 푔 푔/ (푔/ + /푞) 푔/ ∫ 푑 4 2 (2휋) 푔 (푔 + 2푞 ⋅ 푔) 푑 1 (2.75) 푑 푔 2 (1 − 훼) 푔/ (푔/ + /푞) 푔/ = ∫ 푑 ∫0 푑훼 6 (2휋) (푔 + 훼푞) 1 푑 1 − 휀 2 푑 푘 1 = − ∫ 푑훼 2 (2훼 (1 − 훼) + (1 − 훼) ) /푞 ∫ 푑 4 . 0 2 − 휀 Here, the denominator was rewritten using equation (C.19),(2휋) integration푘 vari- able changed to , and terms proportional to an odd power of discarded. The gauge-dependent푘 = 푔 + 훼푞 term thus contains the same -integral as푘 equation (2.73), and is therefore set to zero by the same argument.푘 Strictly speaking, this could have been inferred from the absence of a scale in the denominator of equation (2.75), without further manipulations. An important result used when simplifying the expression above is the following: consider the integral

푑 푑 (2.76) 푑 푘 2 휇 휈 휇휈 푑 푘 2 2 ∫ 푑 푓 (푘 ) 푘 푘 = 휂 ∫ 푑 푓 (푘 ) 푔 (푘 ). where the function(2휋)is isotropic in -space as(2휋) indicated by the -dependence. 2 The Lorentz indices푓 of the integrated푘 quantity can only be carried푘 by , and 휇휈 it must therefore be possible to rewrite the integral as shown on the휂 right, with an unknown function . Contracting both sides with shows that one can set . Hence,푔 in integrals over all of -space,휂휇휈 one may always 2 2 substitute푔(푘 ) = 푘 /푑 푘 (2.77) 휇 휈 1 2 휇휈 푘 푘 → 푘 휂 , provided the other factors in the integrand푑 are isotropic. In particular, one may perform the following replacements:

(2.78) 1 − 휀 2 /푘/푞/푘 → − 푘 /푞 , 2 − 휀 (2.79) 1 2 (푘 ⋅ 푞) (푘 ⋅ 푞) → 푘 (푞 ⋅ 푞) , where the former was used in the last2 (2 step − 휀) in equation (2.75), and the latter will become important below. The hadron current of the third diagram in figure 2.3 is given by (1) 훼푠퐽푞푞

푑 푟 푟′ (2.80) (1) 3휀 2 푎 푎 푑 푔 푢 (푞) 훤휇푣 (푞) 푠 푞푞 푞 푠 훼 (퐽 )휇 = −휇 푒푒 푔 푇푖푘푇푘푗 ∫ 푑 2 2 2 , (2휋) 푔 (푔 + 푞) (푔 − 푞) 29 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞 where3

휌 휎 휌휎 푔 푔 훤휇 = 훾휌 (푔/ + /푞) 훾휇 (푔/ − /푞) 훾휎 (휂 − (1 − 휉) 2 ) (2.81) 푔 휎 1 − 휉 = 훾 (푔/ + /푞) 훾휇 (푔/ − /푞) 훾휎 − 2 푔/ (푔/ + /푞) 훾휇 (푔/ − /푞) 푔/ . 푔 Hence, the interference in equation (2.71) can be written

휇 2 (0)∗ (1) 4휀 푞푞 푠 푞푞 퐹 푐 푞 푠 ∑ −2 (퐽 ) (훼 퐽 )휇 = 2푖퐶 푁 휇 (푒푒 푔 ) 푟,푟′,푖,푗 (2.82) 푑 휇 푑 푔 Tr [훾 /푞훤휇/푞] × ∫ 푑 2 2 2 . (2휋) 푔 (푔 + 푞) (푔 − 푞) The trace can be split into two terms corresponding to the two terms in . The first trace can be evaluated using equation (B.20): 훤휇

휇 휎 Tr [훾 /푞훾 (푔/ + /푞) 훾휇 (푔/ − /푞) 훾휎 /푞] = 16푞휌 (푔 + 푞)휏 (푔 − 푞)훿 푞휆 휌훿 휏휆 휌휏 훿휆 휌휆 훿휏 × [ (−2 + 휀 (1 + 휀)) 휂 휂 + 휀 (1 − 휀) (휂 휂 + 휂 휂 )] (2.83) = 16[ (−2 + 휀 (1 − 휀)) (푞 ⋅ (푔 − 푞)) (푞 ⋅ (푔 + 푞)) + 휀 (1 − 휀) (푞 ⋅ (푔 + 푞)) (푞 ⋅ (푔 − 푞)) + 휀 (1 − 휀) (푞 ⋅ 푞) ((푔 + 푞) ⋅ (푔 − 푞)) ] 4 2 2 2 = 8 (1 − 휀) [푄 − 2푔 ⋅ (푞 − 푞) 푄 − 4 (푔 ⋅ 푞) (푔 ⋅ 푞) + 휀푔 푄 ] Here, the relations and were used to simplify the 2 2 2 expression. The second푞 = trace,푞 = corresponding 0 푄 = 2푞 to ⋅ the푞 gauge-dependent term in , is found by applying equation (B.24): 훤휇

휇 Tr [훾 /푞푔/ (푔/ + /푞) 훾휇 (푔/ − /푞) 푔//푞] (2.84) = −2Tr [(푔/ + /푞) 푔//푞 (푔/ − /푞) 푔//푞] + 2휀Tr [/푞푔/ (푔/ + /푞) (푔/ − /푞) 푔//푞] To proceed, note that since , 2 2 푞 = 푞 = 0 (2.85) 2 3Note that the gauge(푔/ + parameter/푞) 푔//푞 = (푔/here + /푞) specifies (푔/ + /푞) the/푞 gauge = (푔 used + 푞) for/푞 QCD, . which is not the same as the gauge used for QED and휉 discussed in sections 2.1 and 2.2. Although the final result should be invariant with respect to the QCD gauge also, the gauge-dependent term is kept here for generality and as an extra check on the computations.

30 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞 Similarly identities hold for and . Hence, /푞푔(/ 푔/ + /푞) (푔/ − /푞)푔//푞 휇 Tr [훾 /푞푔/ (푔/ + /푞) 훾휇 (푔/ − /푞) 푔//푞] (2.86) 2 2 = −2 (1 − 휀) (푔 + 푞) (푔 − 푞) Tr [/푞/푞] 2 2 2 = −4 (1 − 휀) (푔 + 푞) (푔 − 푞) 푄 . The factors and cancel against the same factors in the de- 2 2 nominator in(푔 the + 푞) -integral,(푔 − so푞) the gauge-dependent term is proportional to 푔 푑 (2.87) 푑 푔 1 ∫ 푑 4 , which is set to zero for the same reason(2휋) as푔 the self-energy integrals. Thus, the gauge-dependency drops out. The denominator in equation (2.82) can be rewritten using equation (C.19):

1 2 2 2 푔 (푔 + 푞) (푔 − 푞) 1 1−훼 2 (2.88) = ∫ 푑훼 ∫ 푑훽 3 0 0 2 2 2 [훼 (푔 + 푞) + 훽 (푔 − 푞) + (1 − 훼 − 훽) 푔 ] 1 1−훼 2 = ∫ 푑훼 ∫ 푑훽 3 . 0 0 2 2 [(푔 + 훼푞 − 훽푞) + 훼훽푄 ] The -integral is then solved by changing integration variable to . Substituting푔 this variable change into equation (2.83) gives 푘 = 푔+훼푞−훽푞

휇 휎 Tr [훾 /푞훾 (푔/ + /푞) 훾휇 (푔/ − /푞) 훾휎 /푞] 4 = 8 (1 − 휀) [ (1 − 훼 − 훽 + (1 − 휀) 훼훽) 푄 2 (2.89) − 2푘 ⋅ [푞 − 푞 − (1 − 휀) (훼푞 − 훽푞)] 푄 2 2 − 4 (푘 ⋅ 푞) (푘 ⋅ 푞) + 휀푘 푄 ] 2 2 2 (1 − 휀) 2 → 8 (1 − 휀) 푄 ⎢⎡(1 − 훼 − 훽 + (1 − 휀) 훼훽) 푄 − 푘 ⎥⎤ , 2 − 휀 where the third line was obtained⎣ by using the replacement from equation⎦ (2.79) and discarding the term proportional to , both justified by the fact that the expression will be integrated over all of푘 -space. The relevant -integrals 푘 푘

31 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞 evaluate to

푑 ∞ 푑−1 푑 푘 1 푑−1 훺푑 푘퐸̃ ∫ 푑 3 = (−푖) 푑 ∫ 푑푘퐸̃ 3 2 2 0 2 2 (2휋) [푘 + 훼훽푄 ] (2휋) [푘퐸̃ + 훼훽푄 ] (푑/2)−3 푑−1 2 1 (−푖) (훼훽푄 ) 2−(푑/2) (푑/2)−1 = 푑/2 ∫0 푑푥 푥 (1 − 푥) (4휋) 훤 (푑/2) 휀 푖 4휋 1 1 = 2 (− 2 ) 훤 (1 + 휀) 휀 2 2 (4휋) 푄 (훼훽) 훼훽푄 (2.90)

푑 2 ∞ 푑+1 푑 푘 푘 푑−1 훺푑 푘퐸̃ ∫ 푑 3 = (−푖) 푑 ∫ 푑푘퐸̃ 3 2 2 0 2 2 (2휋) [푘 + 훼훽푄 ] (2휋) [푘퐸̃ + 훼훽푄 ] (푑/2)−2 푑−1 2 1 (−푖) (훼훽푄 ) 1−(푑/2) (푑/2) = 푑/2 ∫0 푑푥 푥 (1 − 푥) (4휋) 훤 (푑/2) 휀 푖 4휋 1 2 − 휀 = 2 (− 2 ) 훤 (1 + 휀) 휀 2 (4휋) 푄 (훼훽) 휀 (2.91) Here, equations (C.8) and (C.9) were used to get single-variable integrals, and the integration variable was changed to to put them on 2 2 −1 Euler -function form, equation (C.16). 푥 = (1 + 푘퐸̃ /훼훽푄 ) Combining훽 the results in equations (2.88) to (2.91), the interference term becomes:

휇 (0)∗ (1) 푠 푞푞 푠 푞푞 훼 ∑ −2 (퐽 ) (훼 퐽 )휇 푟,푟′,푖,푗 휀 em 2 (2.92) 2휀 2 2 4휋휇 = −16훼 훼푠휇 퐶퐹푁푐푒푞푄 (− 2 ) (1 − 휀) 훤 (1 + 휀) 푄 2 1 1−훼 1 1 − 훼 − 훽 + (1 − 휀) 훼훽 (1 − 휀) × ∫ 푑훼 ∫ 푑훽 휀 ⎢⎡ − ⎥⎤ 0 0 (훼훽) 훼훽 휀 The - and -integrals decouple into⎣ Euler -function integrals upon⎦ substi- 훼 훽 훽

32 2.4. Virtual gluon exchange: at next-to-leading order + − 푒 푒 → 푞푞 tuting : 훽 = (1 − 훼)푣 2 1 1−훼 1 1 − 훼 − 훽 + (1 − 휀) 훼훽 (1 − 휀) 휀 ⎡ ⎤ ∫0 푑훼 ∫0 푑훽 ⎢ − ⎥ (훼훽) ⎣ 훼훽 휀 ⎦ 1 1 −휀 1−휀 −휀 1 − 푣 1 = ∫ 푑훼 ∫ 푑푣 훼 (1 − 훼) 푣 [ + (1 − 휀) (2 − )] 0 0 훼푣 휀 2 2 (2.93) 훤 (−휀) 훤 (1 − 휀) 1 = ⎜⎛ + (1 − 휀) (2 − )⎟⎞ ⎝훤 (2 − 2휀) 훤 (3 − 2휀) 휀 ⎠ 2 1 훤 (1 − 휀) 2 1 = ( 2 − ) 2 훤 (1 − 2휀) 휀 (1 − 2휀) 휀 2 1 훤 (1 − 휀) 2 3 2 = ( 2 + + 8 + 16휀 + 풪 (휀 )) . 2 훤 (1 − 2휀) 휀 휀 There is no explicit - or -dependency left, so the phase space integral is trivial in the COM frame using푞 푞 equation (C.35), and one finds

em 2 2 2 2휀 (1) 2 −훼 훼푠퐶퐹푁푐푒푞푄 4휋휇 휀 훼푠퐻푞푞 (푄 ) = ℜ{ ( 2 ) (−1) 휋 푄 2 (1 − 휀) 훤 (1 − 휀) 훤 (1 + 휀) 훤 (1 − 휀) × 훤 (2 − 2휀) 훤 (1 − 2휀) (2.94) 2 3 2 × ( 2 + + 8 + 16휀 + 풪 (휀 )) } 휀 휀 2 휀 (0) 2 −훼푠퐶퐹 4휋휇 휀 = 퐻푞푞 (푄 ) ( 2 ) ℜ {(−1) } 휋 푄 2 훤 (1 + 휀) 훤 (1 − 휀) 2 3 × ( 2 + + 8 + 풪 (휀)) . 훤 (1 − 2휀) 휀 휀 The next-to-leading order contribution to the cross section for can + − thus be written 푒 푒 → 푞푞

2 휀 (1) (0) −훼푠퐶퐹 4휋휇 휀 훼푠휎푞푞 = 휎푞푞 ( 2 ) ℜ {(−1) } (2.95) 2휋 푄 2 훤 (1 + 휀) 훤 (1 − 휀) 2 3 × ( 2 + + 8 + 풪 (휀)) . 훤 (1 − 2휀) 휀 휀 This clearly diverges as . Before proceeding to show that divergences cancel in the total 휀 →hadrons 0 cross section, it is a good idea to pause + − for a moment and consider푒 푒 → what kinds of divergences the poles in this equa- tion (2.95) actually represent. By comparing with equation (C.16), it is clear that equation (2.90) is convergent in an interval including . The result then enters the first term in the integrand in equation (2.93),휀 and= 0 the integral

33 2.5. Cancellation: hadrons at next-to-leading order + − 푒 푒 → of this term is convergent only for . Thus, the first term on the fourth line in equation (2.93) must represent휀 infrared < 0 divergences. On the other hand, equation (2.91) is obviously logarithmically divergent for , and only converges for . The result is used in the second term푑 in= the4 integrand in equation (2.93),휀 > 0 and this integral places no further restrictions on for convergence. Hence, the second term in the fourth line in equation (2.93)휀 (the -term) represents an ultraviolet divergence. However, the procedure−1/휀 of cancellation about to be demonstrated actually removes all of these divergences, although it was never expected to work for ultraviolet divergences, which are normally removed by renormalization. However, having QCD vertex corrections renormalizing the QED coupling is physically unacceptable, since these corrections would apply to quarks but not leptons, and thus enable processes violating conservation of electric charge. It may seem like the ultraviolet divergence is really an infrared divergence in disguise, and it will certainly be treated as such when showing the cancellation. Section 2.6 contains a short discussion of why this may be considered plausible.

2.5 Cancellation: hadrons at next-to-leading + − order 푒 푒 → The total cross section for hadrons at is given by the sum up to + − of the cross sections푒 of푒 all→ processes 풪(훼푠) that can be defined at + − 풪(훼this푠 order.) Here, is any state of quarks and푒 gluons.푒 → 푋 There are no other such processes than the푋 two considered so far, and hence the cross section may be written

hadrons (0) (0) (1) 2 휎 = 휎푞푞 + 훼푠 (휎푞푞푔 + 휎푞푞 ) + 풪 (훼푠 ) 2 휀 2 (0) 훼푠퐶퐹 4휋휇 훤 (1 − 휀) 2 3 19 = 휎푞푞 {1 + ( 2 ) [( 2 + + + 풪 (휀)) 2휋 푄 훤 (1 − 3휀) 휀 휀 2 휀 훤 (1 + 휀) 훤 (1 − 3휀) 2 3 − ℜ {(−1) } ( 2 + + 8 + 풪 (휀)) ] 훤 (1 − 2휀) 휀 휀 2 푠 + 풪 (훼 )}. (2.96) The factor can be expanded as follows: 휀 (−1) 휀 푖휋휀 (−1) = 푒 (2.97) 2 3 휋 2 푖휋 3 4 = 1 + 푖휋휀 − 휀 − 휀 + 풪 (휀 ), such that 2 6

2 (2.98) 휀 휋 2 4 ℜ {(−1) } = 1 − 휀 + 풪 (휀 ). 2 34 2.6. The ultraviolet divergence

The three -functions in the second term in equation (2.96) may be expanded using equation훤 (C.15). For the denominator, the geometric series expansion must also be used: −1 2 (1 − 푥) = 1 + 푥 + 푥 + ⋯

2 −1 1 휋 2 2 3 = [1 + 2훾퐸휀 + ( + 2훾퐸) 휀 + 풪 (휀 )] 훤 (1 − 2휀) 3 2 휋 2 2 = 1 − (2훾퐸휀 + ( + 2훾퐸) 휀 ) (2.99) 3 2 2 휋 2 2 3 + (2훾퐸휀 + ( + 2훾퐸) 휀 ) + 풪 (휀 ) 3 2 휋 2 2 3 = 1 − 2훾퐸휀 − ( − 2훾퐸) 휀 + 풪 (휀 ). 3 Multiplying this with the straightforward expansions of the factors in the numerator gives

2 (2.100) 훤 (1 + 휀) 훤 (1 − 3휀) 휋 2 3 = 1 + 휀 + 풪 (휀 ), and thus, 훤 (1 − 2휀) 2 (2.101) 휀 훤 (1 + 휀) 훤 (1 − 3휀) 3 ℜ {(−1) } = 1 + 풪 (휀 ). This is exactly the result needed훤 for(1 − the 2휀) divergent terms in hadrons to vanish, and hence the limit can be taken, giving the finite result휎 휀 → 0

hadrons (0) 3훼푠퐶퐹 2 휎 = 휎푞푞 (1 + + 풪 (훼푠 )) (2.102) em 4휋 2 2 4휋훼 푁푐푒푞 3훼푠퐶퐹 2 = 2 (1 + + 풪 (훼푠 )) . 3푄 4휋 Thus, it has been shown that the total cross section for hadrons in + − the massless quark approximation is without soft and collinear푒 푒 → divergences at , despite the fact that the individual contributions from real gluon emission풪(훼푠) and virtual gluon exchange both suffer from this.

2.6 The ultraviolet divergence

As mentioned in section 2.4, the virtual gluon exchange cross section (1) contains what appears to be an ultraviolet divergence, which nevertheless훼푠휎푞푞 cancels by the same mechanism as the infrared divergences. Some insight on this conundrum can be gained by once again considering the self-energy diagrams, postponing for the moment the definition stating that the scaleless

35 2.6. The ultraviolet divergence integral vanishes. Using equation (2.74), the self-energy may be written in the Feynman gauge as

2 휀′ 2 휀 훼푠퐶퐹훿푖푗 ⎡ 4휋휇 1 4휋휇 1 ⎤ −푖훴 (푞) = 푖/푞 ⎢( 2 ) − ( 2 ) ⎥ (2.103) 4휋 훬 휀′훤 (1 − 휀′) 훬 휀훤 (1 − 휀) ⎣ ⎦ 훼푠퐶퐹훿푖푗 1 1 = 푖/푞 [ − + 풪 (휀′ − 휀)] . 4휋 휀′ 휀 By convention, half of the self-energy is canceled by wave function renormal- ization [1, p. 111, 8, p. 36]. The contributing current from the two first diagrams in figure 2.3 is thus

휀 푟 ⎡(−푖훴 (푞)) /푞 −/푞 (−푖훴 (−푞))⎤ 푟′ 훼푠 (퐽훴)휇 = 휇 푒푒푞푢 (푞) ⎢ 2 훾휇 + 훾휇 2 ⎥ 푣 (푞) (2.104) ⎣ 2 푞 푞 2 ⎦ 휀 푖훼푠퐶퐹훿푖푗휇 푒푒푞 1 1 푟 푟′ = [ − + 풪 (휀′ − 휀)] 푢 (푞) 훾휇푣 (푞) . 4휋 휀′ 휀 As in equation (2.74), the unprimed is used in the term that was regular- ized with and contains the infrared휀 divergence, while is used in the ultraviolet divergent term. The interference term entering is then 휀 < 0 휀′(1) 훼푠휎푞푞

(0)∗ 휇 훼푠 ∑ −2 (퐽 ) (퐽훴)휇 푟,푟′,푖,푗 (2.105) em 2휀 2 1 1 = − 8훼 훼푠퐶퐹푁푐휇 푄 [ − + 풪 (휀′ − 휀)] . 휀′ 휀 The term in the vertex correction interference term that was regularized in with , and hence contains the ultraviolet divergence, is found by including 휀only > 0 the second term under the integral in equation (2.92). Expanding such that only the divergence is explicitly written, it reads

em (2.106) (0)∗ 휇 (1) 2휀 2 1 ⎡ 푠 푞푞 ⎤ 푠 퐹 푐 ⎢훼 ∑ −2 (퐽 ) (퐽 )휇⎥ = 8훼 훼 퐶 푁 휇 푄 [ + 풪 (1)] . 푟,푟′,푖,푗 휀′ ⎣ ⎦푈푉 A primed is used since the divergence is ultraviolet. Summing휀′ these two terms leads to a remarkable result: the ultraviolet divergence is canceled exactly, but replaced by an infrared divergence with the exact same expression. It was this infrared divergence, then, that was canceled by adding the real gluon emission and the virtual gluon exchange cross sections together – the ultraviolet divergence had already been canceled by summing the different diagrams for virtual gluon exchange. The same result is derived in an arbitrary gauge in [8, pp. 39-41], using Pauli-Villars regularization. Dimensional regularization makes the mechanism

36 2.6. The ultraviolet divergence at work here somewhat opaque in comparison, but an interpretation is that the definition in effect admits analytic continuation of integrals 푑 2 −푛 regularized∫ for 푑 푘 (푘 ) to = 0 , such that a pole appearing to be ultraviolet may in fact be infrared.휀 > This0 휀 is < also 0 discussed in e.g. [9, p. 498].

37 3 Discussion

The appearance and cancellation of infrared divergences are ubiquitous phe- nomena in theories with massless particles. There is a lot to learn about physically measurable quantities, both in quantum field theories in general and in QCD in particular, from understanding the nature of these singularities and under which conditions they cancel. These issues will be discussed in this chapter.

3.1 Infrared divergences in general quantum field theories

A scattering process emitting massless particles in the final state, such as depicted in figure 2.2, will in general have a factors like the following in the denominator of the amplitude:

2 2 (푘 + 푝) − 푚 = 2푘 ⋅ 푝 (3.1) 2 푚 = 2퐸푘퐸푝(1 − 훽푝 cos 휃푘푝), 훽푝 = √1 + 2 . 퐸푝 Here, and denote the momenta of the massless particle and the particle it was emitted푘 푝 from, respectively, and is the mass of the latter. Since these are the momenta of particles in the final푚 state, they are on-shell with and 2 , giving the simplified form. The on-shell condition for the푘 massless= 0 2 푝particle= 푚 allows , or equivalently, , and unless some factors cancel against factors in푘 = the 0 numerator, the amplitude퐸푘 = 0 will clearly have a singularity here. Moreover, integrating over phase space will not eliminate the divergence when working in spacetime dimensions, hence the cross section inherits the singularity. When푑 = 4 such that , singularities also appear as or . These푚 → are 0 the soft and훽푝 collinear→ 1 singularities, investigated 퐸in푝 detail→ 0 for휃푘푝 a→ specific 0 example in the previous chapter. However, this simple argument shows that the phenomenon is universal: in massless theories, scattering cross sections blow up when a massless particle is emitted with low momentum or parallel to the emitter.

38 3.1. Infrared divergences in general quantum field theories

This is actually not too surprising. In the -limit, the virtual particle with momentum that existed before the푘 → emission 0 is put on-shell – it becomes real. This푘 may + 푝 be interpreted as the particle being free to propagate for an unlimited time and distance before the emission actually happens, and this violates the fundamental assumptions used when setting up the perturbative expansion in quantum field theories. Specifically, when deriving the LSZ reduction formula, which relates scattering amplitudes to the Greens functions that can be derived directly from the Hamiltonian, it is assumed that the fields are free of interactions in the far past and future, , due to the increasing distance between the particles [6, pp. 111-116].푡 → In ±∞ the limit of low energy, however, the wavelength of a massless particle becomes infinite, so this freedom can never be obtained. In this sense, final states with a fixed number of massless particles are not well-defined in perturbative quantum field theory. This can also be seen by considering the similar process where the massless particle is exchanged as a virtual particle rather than emitted, as in figure 2.3. The -limit puts all three particles in the loop on the mass-shell, and hence the loop푘 → 0 interaction may take an arbitrarily long time and cover arbitrarily long distances. Thus, the distinction between exchange and emission of the massless particle vanishes. This indicates that when computing a quantity incorporating the -limit of one of these processes, the other should also be taken into account.푘 = 0 One can also consider this from an experimentalists perspective. A detector has a finite energy resolution, so there is no way to actually detect a particle with arbitrarily low momentum in an experiment. From this point of view, the singularities in the cross section are not a problem at all, the mistake was rather to integrate over the phase space regions where the singularities appear. The contribution from momenta smaller than some cutoff should instead be regarded as a part of the cross section for the similar process without emission of the massless particle, since this is what it would look like to the detector. Particles below the detection limit are called soft particles. These arguments are readily extended to cover collinear singularities when both particles are massless. In this case, the distinction between a particle with momentum and two collinear particles with with momenta and vanishes. 푝 휆푝 (1 − 휆)푝 These arguments explain how, for massless theories, the concept of external states with a definite number of particles must be abandoned in order to avoid infrared singularities. Although each amplitude in the perturbation expansion is constructed from such states, a physically measurable quantity can only be computed by including the contribution from a particular external state on equal footing with all states obtained by adding any number of soft and collinear particles to it. The external states of processes without infrared singularities therefore do not consist of isolated particles, but rather of jets made up of any number of nearly parallel particles with total momentum

39 3.1. Infrared divergences in general quantum field theories equal to the particles they replace, along with a cloud of soft particles that escape detection. A formalism making the mathematical statement of these ideas straight- forward was introduced by [10]. A general measurement from a scattering experiment with massless particles may be expressed as

(3.2) 1 푑휎푛 ℐ = ∑ ∫ 푑훷푛 풮푛 (푝1, … , 푝푛). 푛 푛! 푑훷푛 Here, labels the number of particles in a perturbative final state, and is the differential푛 cross section for scattering into this state. The measurement푑휎푛/푑훷푛 is assumed to not distinguish different kinds of particles, hence there is only a single differential cross section per , and each term is divided by to avoid overcounting. The measured quantity푛 is defined by the weights 푛! . For the total cross section, the weights are simply 풮푛(푝1, …. , The푝푛) requirement for infrared safe measurements is that the풮 weights푛(푝1, … ,are 푝푛) equal = 1 in all soft and collinear limits, that is,

(3.3) 풮푛+1 (푝1, … , (1 − 휆) 푝푛, 휆푝푛) = 풮푛 (푝1, … , 푝푛), and similar for all other pairs of momenta. For , this ensures that a single particle is considered equal to two collinear0 < 휆

The Kinoshita-Lee-Nauenberg theorem The discussion above is presented in a heuristic fashion, but the claims are sup- ported by rigorous proofs. The Kinoshita-Lee-Nauenberg theorem (KLN) [12, 13] states that in any quantum field theory, the total transition rate between states of equal energy is free of infrared divergences in the limit of mass- less particles. Formally, the theorem can be stated as follows (similar to the presentation in [7, pp. 364-370, 14, pp. 440-447]):

1This particular observable is not often used in current experiments, but is used as an example here because it easy to understand.

40 3.2. The optical theorem and cutting rules

Let denote the set of states with energy in the interval 퐷(퐸,0 and, 휀) be the -matrix of the theory. Then,퐸 the total transition퐸0 − rate 휀 < 퐸 < 퐸0 + 휀 푆 푆 (3.4) 2 푃 (퐸0, 휀) = ∑ |⟨푎|푆|푏⟩| 푎,푏∈퐷(퐸0,휀) is free of infrared singularities in the limit of massless particles. Proofs of the theorem may be found in all of the listed references. The significance of the theorem is that it shows that for observable quanti- ties satisfying some criterion, all quantum field theories are free of infrared divergences in the limit of massless particles. Moreover, this holds at each order in perturbation theory. However, in general one is not only required to sum over final states, as done in the calculation in chapter 2, but also over initial states. This was avoided in the hadrons cross section because + − the initial state did not participate in QCD푒 푒 interactions.→ The KLN theorem involves summation over all states with the same to- tal energy, which is a somewhat unfamiliar configuration, and much more inclusive than the observables described by equation (3.3). However, in [15, pp. 548-553], the proof is modified to show that the same result holds for con- figurations such as Sterman-Weinberg jets and other measurements satisfying equation (3.3) as well. The requirement to sum over initial states then enters via the differential cross sections, which must be defined by averaging over the squared amplitudes and flux factors for all possible initial states related to the nominal initial state by additional soft or collinear particles. It is worth noting that in QED, if the electron mass is non-zero, infrared safe quantities can be obtained by only summing over final states. This is known as the Bloch-Nordsieck theorem [16]. In QCD, however, this does not hold, as shown by e.g. [17], so summation over initial states will in general be required.

3.2 The optical theorem and cutting rules

A convenient way of visualizing and interpreting the relation between infrared divergent quantities whose sum is finite is obtained using the optical theo- rem. For two-particle scattering, the optical theorem can be stated as follows (adapted from [18, p. 455]):

(3.5) ℑ {ℳ (푘1푘2 → 푘1푘2)} = 2퐼 ∑ 휎 (푘1푘2 → 푋) , 푋 where are the momenta of the two incoming particles and is the flux factor from푘1, 푘2 equation (A.2). The theorem states that the imaginary퐼 part of the forward scattering amplitude for a two-particle state is proportional to the sum over the total cross section for this initial state. A forward scattering amplitude is a scattering amplitude where the final state is identical to the initial state.

41 3.2. The optical theorem and cutting rules

− − 푒 푒

+ + 푒 푒 Figure 3.1: Forward scattering at em . The dashed line + − + − 2 shows an example cut. 푒 푒 → 푒 푒 풪(훼 훼푠)

An intuitive understanding of this theorem may be obtained by consid- ering forward scattering Feynman diagrams which include loops where the kinematic conditions allow the particles to be put on-shell. Figure 3.1 shows an example of a diagram for at em . When performing + − + − 2 the loop integrals in this amplitude,푒 푒 → some 푒 푒 of the풪(훼 virtual훼푠) particles will be put on-shell in certain regions of the space of loop momenta. However, when a propagator goes on-shell, an imaginary delta function appears:

(3.6) 1 ℑ{ } = −휋훿 (푥) . The result is that when solving푥 + the 푖휀 loop momentum integrals, the regions where particles in the loop go on-shell contributes an imaginary part to the scattering amplitude. Moreover, the delta function makes this contribution integral appear very similar to a phase space integral, and in fact it will be pro- portional to the cross section of the scattering process at em , + − 2 where is the final state consisting of the on-shell particles푒 푒 → in 푋 the풪(훼 loop. The훼푠) state 푋must contain all particles along some cut through the diagram, and it must푋 be possible to put all these particles on-shell simultaneoulsy. An example of a cut is shown by the dashed line in figure 3.1. Thus, to calculate the imaginary part of the amplitude for a single Feynman diagram such as the one in figure 3.1, one may proceed as follows: 1. Identify all possible cuts through the intermediate states in the diagram such that the particles along the cut may be put on-shell simultaneously. 2. For each cut, calculate the cross section of the interference between the half-diagram to the left of the cut and the mirror image of the half- diagram to the right of the cut.

42 3.2. The optical theorem and cutting rules

Figure 3.2: Cut diagrams for hadrons at . The similar diagrams ∗ with the quark and antiquark훾 swapped→ are not풪(훼 drawn.푠)

3. Add all these cross sections together, and multiply with half of the flux factor for the initial state.

This is known as the cutting rules, and may be thought of as the optical theorem in action, broken down to specific orders and diagrams. The claims will not be proved here, but they are not difficult to demonstrate for simple processes such as the one in figure 3.1, see e.g. [18, pp. 453-466]. By virtue of the optical theorem and cutting rules, the diagram in figure 3.1 is very closely related to the cross sections computed in chapter 2. To minimize clutter, the external states will be amputated in the following, such that + − cuts of forward scattering diagrams푒 푒 for at em are considered instead. ∗ Figure 3.2 shows the two different ways훾 to cut풪(훼 each훼푠 of) the two possible loop configurations at this order. The top line contains the same loop as figure 3.1 cut in two different ways: the first cut corresponds to the interference terms between the two diagrams in the real gluon emission diagrams in figure figure 2.2, and the second cut corresponds to the interference between the leading order diagram in figure 2.1 and the vertex correction in figure 2.3. The bottom line contains a similar loop where the gluon is emitted and absorbed by the same quark. Here, the first cut corresponds to the squared amplitude of one of the real emission diagrams in figure 2.2, while the second corresponds to the interference between the leading order diagram in figure 2.1 and a self-energy insertion in figure 2.3.

43 3.3. Infrared safety in QCD

Since QED and QCD are renormalizable theories, the amplitude for a loop diagram such as the one in figure 3.1 is expected to be finite.2 The cutting rules relate the imaginary part of this amplitude to the sum of cross sections computed in chapter 2, containing infrared divergences, and this provides an alternative argument for why the divergences should cancel when the cross sections are summed. Moreover, it admits an alternative interpretation of the origin infrared divergences: they appear in quantities corresponding to incomplete loop momentum integrals. This picture is of course not more fun- damental than other interpretations presented in this chapter, but it provides a neat way to organize the diagrams and identify groups of terms whose sum must be finite, since they are different cuts of the same loop. This picture also strengthens the viewpoint that in the infrared limit, the distinction between real emission and virtual exchange vanishes: these two kinds of cuts corre- spond to adjacent regions in the space of loop momenta, and the infrared limit corresponds to the boundary between these regions. Cut diagrams are used in the original proof of the KLN theorem in [12]. The cut diagrams explained here are not sufficient for this purpose, however, since they cannot handle infrared divergences in the initial state.

3.3 Infrared safety in QCD

The discussion in the previous sections applies to quantum field theories in general. When considering quantum chromodynamics in particular, there is an additional concern related to the increasing strength of the QCD coupling at long distances, and in particular confinement. In particular, the arguments relying on the properties of detectors do not translate directly to QCD, since the final states in perturbative QCD are never observed in experiment, as explained in section 1.1. On the other hand, the concept of jets containing any number of nearly parallel particles with definite total momentum is most welcome: this is de- scription is quite appropriate to the hadronic final states observed in QCD scattering experiments, although the internal dynamics of the QCD jets, lead- ing to hadronization, seems to be more complex than in theories such as QED. The essential observation described in section 3.1, explaining infrared divergences and leading to the definition of infrared safe observables and jets, is that sensible measurements must be insensitive to interactions over long time and distance scales, since otherwise it is not possible to separate the interacting fields from the free external states. In QCD the idea of free external states made up of fundamental particles is invalid, but if an observable is truly insensitive to long-distance interactions, the composition of the external states should not

2There is no need to worry about infrared divergences in figure 3.1 diagram, since the process is not considered at an order where photon emission or exchange in the external states cannot be accomodated.

44 3.3. Infrared safety in QCD matter. Thus, infrared safe quantities calculated in perturbative QCD may be compared directly to experiment without considering any non-perturbative dynamics. This is not to say that infrared safe observables are the only interesting observables in QCD. In particular, any observable distinguishing different kinds of hadrons obviously depends on long-distance interactions, and does not fit into the scheme described by equation (3.3). The solution in this case is to factorize the process into a perturbative and a non-perturbative part as described in section 1.1, where the perturbative part depends on the particular high-energy, short-distance interactions involved, while the non-perturbative part is universal and completely independent of short-distance physics. In this approach the perturbative part must still be infrared safe in order to be calculable in perturbation theory, and all the long-distance sensitivity is placed in the non-perturbative part [3].

45 4 Summary

In the total cross section for the scattering process hadrons , the + − contributions at have been shown to contain infrared푒 푒 singularities,→ both from the real gluon풪(훼 emission푠) and virtual gluon exchange diagrams. However, by using dimensional regularization to safely manipulate the infinities, it has been demonstrated that the divergent terms cancel, such that the total cross section is finite. By considering the causes of these singularities, one is led to the conclusion that well-defined measurements computed in perturbative quantum field the- ory cannot in general have a definite number of massless particles in the initial and final states. In order to obtain finite observables one must instead take care to define them such that they are insensitive to low-energy, long-distance interactions. Observables satisfying this property are called infrared safe. This naturally leads to the concept of jets, collections of an arbitrary number of nearly parallel particles with definite total momentum, which can be related to experimental measurements. The Kinoshita-Lee-Nauenberg theorem demon- strates that for suitably defined observables, infrared divergences will always cancel, to all orders. In the particular case of quantum chromodynamics, infrared safety defines the set of observables for which theoretical values may be calculated from perturbation theory alone. However, due to the non-perturbative strength of the QCD coupling at low energies and long distances, and in particular the phenomenon of confinement, many relevant observables will necessarily depend on long-distance interactions. By factorizing the process into a pertur- bative and a non-perturbative part, however, the short-distance physics may still be described by infrared safe calculations in perturbation theory.

46 A Feynman rules for QED and QCD

This appendix lists the Lagrangian and Feynman rules for quantum electrody- namics (QED) and quantum chromodynamics (QCD), mainly following the conventions in [1]. Using the parametrization from [19], the sign conventions are . 휂푠 = 휂푒 = 휂퐺 = +1 A.1 General remarks

A Feynman diagram is a pictorial representation of a scattering process, which can be translated directly to a term in the power series expansion of the associ- ated scattering amplitude , where the expansion parameter is the coupling constant of the theory, e.g.ℳ em for QED and for QCD. 2 2 In the diagram, particles are훼 represented= 푒 /4휋 by lines joined훼푠 together= 푔푠 /4휋 at vertices, and the number of vertices in the diagram is related to the order of the term it represents. The initial and final states in the scattering process are represented by open-ended lines at each end of the diagram, called external lines, while lines terminating at a vertex in both ends are called internal lines and are said to represent virtual particles. A typical example, describing a process similar to the one discussed in section 2.1, is shown in figure A.1.

Figure A.1: An example of a Feynman diagram

47 A.1. General remarks

The convention here is that time runs from left to right, so the initial states come in from the left and the final states exit to the right. Four different kinds of lines are used to represent the different particles in QED and QCD:

A fermion of any kind. The arrowhead shows the direction of the asso- ciated fermion current, hence lines with arrows pointing back in time represent the corresponding antifermion. Since the current is conserved, any fermion line pointing towards a vertex must be balanced by a line pointing away from it.

A photon, the gauge boson of QED.

A gluon, the gauge boson of QCD.

A ghost, an unphysical particle appearing in QCD when working in certain gauges. The ghosts only appear as internal lines. Ghosts obey fermionic statistics, and are therefore drawn with an arrowhead.

The translation from a diagram to a term in the scattering amplitude is done according to the following general rules

• Each external line is replaced by a solution to the equation of motion for the corresponding free field, e.g. a Dirac spinor for a fermion line.

• Each internal line is replaced by the propagator for the corresponding free field.

• Each vertex is replaced by a vertex factor derived from the interaction Lagrangian of the theory.

• 4-momentum is conserved at each vertex. Any momenta left undeter- mined by this are integrated over using the measure , or in 4 4 spacetime dimensions, . 푑 푝/(2휋) 푑 푑 푑 푑 푝/(2휋) • For each closed loop of fermion lines, a factor of is appended. −1 • If identical fermions exist in the external states, a diagram obtained by crossing two of the external fermion lines while leaving the initial and final states unchanged acquires a factor of compared to the uncrossed diagram. −1 • For fermions, the order of the terms must be correct in order for the spinors and matrices to contract correctly. For each connected string of fermion lines, the external state spinors, vertex factors and propagators

48 A.2. QED

must be written down in the order they appear when following the string from end to end against the direction of the arrowheads. For a closed fermion loop, the starting and ending point is arbitrary, but a trace must be taken over the final product. (Alternatively, all spinor-related quantities may be written with explicit spinor indices that match at each vertex.)

• In some cases, a symmetry factor must be appended if the number of different ways to connect the vertices and obtain an identical diagram is not as large as the prefactor from the Taylor expansion of the action exponential and any numerical coefficients from the interaction term in the Lagrangian. This is a rather technical point that is easiest to understand through practice. See [15, pp. 265-267] for a comprehensive review.

When the scattering amplitude has been computed to the desired order, the differential cross section forℳ the process is found taking the absolute square of the amplitude, dividing푑휎 by the flux factor , and appending the phase space measure for a final state with particles:4퐼 푑훷푛 푛 (A.1) 1 2 푑휎 = ∣ℳ∣ 푑훷푛 . In this text only two-particle collisions4퐼 are considered. The flux factor is then [1, p. 427, 6, pp. 158-160]

2 2 (A.2) 4퐼 = 4√(푝when1 ⋅ 푝2) − (푚1푚2) 2 2 2 ≈ 2푠, 푚1, 푚2 ≪ 푠 = (푝1 + 푝2) . Phase space measures are discussed in appendix C.2.

A.2 QED

The Lagrangian for the quantum electrodynamics of a fermion with with mass and electric charge is 푚푓 푒푒푓

1 휇휈 ℒQED = 휓푓 (푖퐷/ − 푚푓 ) 휓푓 − 퐹 퐹휇휈 (A.3) 4 1 휇휈 = 휓푓 (푖휕/ − 푚푓 ) 휓푓 − 푒푒푓 휓푓 퐴휓/ 푓 − 퐹 퐹휇휈 , where is the spinor field of the fermion, is4 the gauge field for the photon,휓 the푓 (푥) photon field strength tensor is defined퐴휇 as 퐹휇휈 (A.4) 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 , 49 A.2. QED

Table A.1: Building blocks of QED Feynman diagrams

Incoming lines Outgoing lines Internal lines Vertex

푝 ⟶ 푝 ⟶ 푝 ⟶ = 푢 (푝) = 푢 (푝) = 푖푆퐹 (푝) = 푣 (푝) = 푣 (푝) 휇 = −푖푒푒푓 훾휇 휇 ∗휇 휇휈 휇 = 휀 (푝) 휇 = 휀 (푝) 휇 휈 = 푖퐷퐹 (푝)

and the gauge covariant derivative is 퐷휇 (A.5) 퐷휇 = 휕휇 + 푖푒푒푓 퐴휇 . The constant is the (positive) elementary electric charge, and is the quantum number associated푒 with electric charge for the fermion under푒푓 consideration. For electrons and other charged leptons, , while for quarks with for the quarks and 푒푓 = −1 for the quarks푒푓 = 푒푞 This푒푞 = leads 2/3 to the푞 Feynman = 푢, 푐, 푡 diagram building푒푞 = −1/3 blocks shown푞 = 푑, in 푠, table 푏 A.1. The Dirac spinors for the external fermions are further explained in ap- pendix B.2. The fermion푢, 푣 propagator is

(A.6) 푝/ + 푚 푖푆퐹 (푝) = 푖 2 2 . 푝 − 푚 + 푖휀 The 4-vector specifies the polarization for an external photon with 휇 4-momentum휀 ,(푝) and is discussed further in appendix A.4. The gauge푝 field carries redundant degrees of freedom, and a gauge is a 휇 set of constraints selected퐴 to eliminate this redundancy (see appendix A.5 for a brief discussion of this concept). Expressions relating to are in general 휇 different in different gauges. Covariant gauges are defined퐴 by the constraint , and the photon propagator can then be expressed as 휇 휕휇퐴 = 0

휇휈 휇 휈 2 (A.7) 휇휈 −휂 + (1 − 휉) 푝 푝 /푝 푖퐷퐹 (푝) = 푖 2 , 푝 + 푖휀 where is an arbitrary parameter. Popular choices include the Feynman gauge, ,휉 and the Landau gauge, . 휉 =The 1 -term in the denominator휉 = 0 of the propagators is included as a pre- scription푖휀 for how to treat the poles at or when integrating. The 2 2 2 limit should be taken at the end푝 of= a 푚 computation.푝 = 0 The -term is often dropped휀 → to 0 save writing when doing calculations, but is implicitly푖휀 assumed whenever relevant.

50 A.3. QCD

Table A.2: Lines in QCD Feynman diagrams

Incoming lines Outgoing lines Internal lines

푝 ⟶ 푝 ⟶ 푝 ⟶ = 푢 (푝) = 푢 (푝) 푖 푗 = 푖푆퐹 (푝) 훿푖푗 2 = 푣 (푝) = 푣 (푝) 푎 푏 = 푖훿푎푏/(푝 + 푖휀) 휇 ∗휇 휇휈 휇 = 휀 (푝) 휇 = 휀 (푝) 휇, 푎 휈, 푏 = 푖퐷퐹 (푝) 훿푎푏 A.3 QCD

The Lagrangian for the quantum chromodynamics of a single quark flavor labeled , with mass and color degrees of freedom, is 푞 푚푞 푁푐

1 푎휇휈 푎 ℒQCD = 휓푞,푖 (푖퐷/푖푗 − 푚푞훿푖푗) 휓푞,푗 − 퐹 퐹휇휈 (A.8) 4 푎 푎 1 푎휇휈 푎 = 휓푞,푖 (푖휕/ − 푚푞) 훿푖푗휓푞,푗 − 푔푠휓푞,푖푇푖푗퐴/ 휓푞,푗 − 퐹 퐹휇휈 , where is the spinor field of the quark , with the4 index representing휓푞,푖(푥) the quark color charge basis states,푞 is the gauge푖 ∈field {1, for … , the푁푐} 푎 gluon, with the index representing퐴휇 the gluon color basis 2 states, the gluon field strength푎 ∈ {1, … tensor , 푁푐 − 1} is 푎 퐹휇휈 (A.9) 푎 푎 푎 푎푏푐 푏 푐 퐹휇휈 = 휕휇퐴휈 − 휕휈퐴휇 − 푔푠 푓 퐴휇퐴휈 , and the gauge covariant derivative is defined as 퐷휇 (A.10) 푎 푎 (퐷휇)푖푗 = 훿푖푗휕휇 + 푖푔푠푇푖푗퐴휇 , For the quarks constituting the hadrons seen in experiment, the number of color degrees of freedom is found to be . However, is often kept general throughout theoretical calculations.푁푐 = 3 푁푐 The factors and matrices are discussed in appendix A.5.The constant 푎푏푐 푎 is the coupling푓 constant of quantum푇 chromodynamics. 푔푠 This leads to the Feynman diagram building blocks shown in tables A.2 and A.3. In covariant gauges, defined by the condition , a set 푎휇 of unphysical ghost particles appear to compensate for unphysical휕휇퐴 = 0degrees of freedom that are not eliminated in these gauges. The main text does not consider any processes to the orders where ghosts become relevant, but they are included here for completeness. The symbols have the same meaning here as in the QED case, but as shown, the푢, 푣, propagators 휀, 푆퐹, 퐷퐹 acquire an extra color-conserving -factor in the QCD case. A color-conserving must also be appended to the훿 QED vertex factor when the fermions are quarks.훿푖푗

51 A.4. Polarization

Table A.3: Vertices in QCD Feynman diagrams

푖

푎 휇, 푎 = −푖푔푠푇푖푗훾휇

푗 푏

휇, 푎 푎푏푐 휇 푝 ↙ = −푔푠 푓 푝

푐 휈, 푏 ↘ 푝2 푎푏푐 휇휈 휌 = −푔푠 푓 [ 휂 (푝1 − 푝2) 휈휌 휇 휇, 푎 2 3 푝3 ↗ + 휂 (푝 − 푝 ) ← 푝1 휌휇 휈 + 휂 (푝3 − 푝1) ] 휌, 푐 휌, 푐 휈, 푏 2 푒푎푏 푒푐푑 휇휌 휈휎 휇휎 휈휌 = −푖푔푠 [ 푓 푓 (휂 휂 − 휂 휂 ) 푒푎푐 푒푑푏 휇휎 휈휌 휇휌 휈휎 + 푓 푓 (휂 휂 − 휂 휂 ) 푒푎푑 푒푏푐 휇휈 휌휎 휇휌 휈휎 + 푓 푓 (휂 휂 − 휂 휂 )] 휎, 푑 휇, 푎 A.4 Polarization

As mentioned above, an external photon or gluon is described by a polarization 4-vector . This is the coefficient in a plane wave solution to the classical equation휀(푝) of motion for the gauge field [6, p. 96]: 휇 퐴 (A.11) 휇 휇 −푖푝⋅푥 ∗휇 푖푝⋅푥 퐴 (푥) ∝ 휀 (푝) 푒 + 휀 (푝) 푒 . The polarization therefore satisfies the momentum space equivalent of any equation satisfied by the field . In covariant gauges one has , 휇 휇 translating to the constraint 퐴 . For a general 4-momentum 휕(implying휇퐴 = 0 non-zero mass), this gives three푝⋅휀(푝) polarization = 0 degrees of freedom,푝 which can be represented by an orthonormal basis:

(A.12) 휇 ∗휇 휀푟 (푝) ⋅ 휀푟′ (푝) = −훿푟푟′ , 푟 ∈ {1, 2, 3} . 52 A.5. Some results from SU(푁푐) Let the decomposition be such that denotes the polarization along the direction of travel: , and 휀3(푝) . However, if the mass 0 0 is zero, such as for휺 gluons3(푝) ∝ and 풑 photons,휀3(푝) = 풑 ⋅ 휺3(푝)/푝and , and it is clear 2 that equation (A.12) can no longer be satisfied푝 = 0 when휀3 including(푝) ∝ 푝 . Thus, for massless particles longitudinal polarization is unphysical, and푟 = only 3 two polarization degrees of freedom remain, . Following [1, pp. 76-79], this can be enforced explicitly by adding푟 a∈ second {1, 2} constraint , where is any 4-vector satisfying . This clearly also implies휀(푝) ⋅ 푛 = 0 , so this푛 constraint defines a new class푛 ⋅ 푝 of ≠ gauges 0 called physical gauges.푛 ⋅ 퐴 = 0 When calculating matrix elements, one frequently needs to find the polar- ization sum

휇 ∗휈 휇휈 휇 휈 휇 휈 ∑ 휀푟 (푝) 휀푟 (푝) = 퐴 (푝, 푛) 휂 + 퐵 (푝, 푛) 푝 푝 + 퐶 (푝, 푛) 푛 푛 (A.13) 푟 휇 휈 휇 휈 + 퐷1 (푝, 푛) 푝 푛 + 퐷2 (푝, 푛) 푛 푝 . This form of the right hand side could be written down by considering the only available independent variables and quantities carrying a Lorentz index. Applying the conditions , and equation (A.12), the sum is found to be 푝 ⋅ 휀푟(푝) = 0 푛 ⋅ 휀푟(푝) = 0

휇 휈 휇 휈 (A.14) 휇 ∗휈 휇휈 푝 푛 + 푛 푝 2 푝휇푝휈 ∑ 휀푟 (푝) 휀푟 (푝) = −휂 + − 푛 2 . 푟 푛 ⋅ 푝 (푛 ⋅ 푝) In physical gauges, the numerator of the gluon and photon propagator would look similar to this, and since these gauges eliminate all unphysical degrees of freedom, this would eliminate the need for ghost particles in QCD (at the expense of imposing a gauge condition that is not Lorentz covariant). On the other hand, not explicitly eliminating the unphysical degrees of freedom for external gluons (that is, neglecting -dependent terms in equation (A.14) when applied to initial and final states)푛 would introduce the need to consider additional diagrams with external ghost pairs in processes involving dia- grams where external gluon pairs couple to triple gluon vertices. The middle ground when working in covariant gauges is to use the covariant propagator from equation (A.7) for internal gluons, and including diagrams with internal ghosts as necessary, while for external gluons using the polarization sum in equation (A.14) and thus avoiding the need for external ghosts.

A.5 Some results from 푐 The fields appearing in the LagrangiansSU(푁 ) in equations (A.3) and (A.8) carry some redundancy in their description. This is to say that the fields may be transformed in certain ways without changing the value of the Lagrangian, and thus without changing the physics. This is called gauge symmetry, and is the origin of the name “gauge field” for the photon and gluon fiels , and of 퐴휇 53 A.5. Some results from SU(푁푐) the concept of gauges introduced while discussing the propagator and 휇휈 the polarization vectors. 퐷퐹 Gauge symmetries are defined by symmetry groups. The gauge group of QED is , while the gauge group of QCD is for quarks with color degreesU(1) of freedom. No properties of SU(푁, 푐) or groups in 푁general푐 will be derived here, but some importantU(1) resultsSU(푁 will푐) be stated. For a physicist-friendly treatment of Lie groups in general, see e.g. [20]. For a discussion in the context of QCD, see [1]. That the gauge symmetry of QCD is means that the QCD La- grangian, equation (A.8), is invariant underSU(푁 a transformation푐) of the quark field in the fundamental representation of , provided 휓that푞 = the (휓푞,1 gauge, … , 휓 field푞,푁푐 ) transforms in the adjoint representationSU(푁푐) of the 푎 푎 same group (with an퐴 additional휇 = 푇 퐴휇 inhomogeneous term if the transformation is local, that is, varies continuously between spacetime points). These transforma- tion properties are not explicitly applied in the main text, but their significance is that they explain how a gauge field with degrees of freedom must 2 appear when requiring that a Lagrangian is푁 invariant푐 − 1 under a local transformation of a matter field with degrees of freedom. This explainsSU(푁푐) the ranges of the color indices . For푁푐 real-world quarks, such that . 푖, 푗, 푎 푁푐 = 3 2 푁푐 −A 1 general = 8 element of may be expressed as , 푎 푎 where the are realSU(푁 parameters푐) and the matrices푈(휃)are = known exp(푖휃 as푇 the) 푎 2 푎 generators of휃 푁푐 −, and 1 are the same as the appearing푇 in equations (A.8) 푎 and (A.10). InSU(푁 the푐) fundamental representation,푇 the elements of are unitary -matrices , with determinant SU(푁푐) . † † Hence,푁 the푐 × generators 푁푐 are푈, Hermitian 푈푈 = 푈 푈 and = tracecless푰 matricesdet(푈) = +1 푎 . The most common representation of푁 the푐 × generators 푁푐 of 푇 = 푎† 푎 푇is , Tr [푇 ] =, where 0 are the Gell-Mann matrices listed in equation (A.15).SU(3) 푎 푎 푎 푇 = 휆 /2 휆 0 1 0 0 −푖 0 1 0 0 1 ⎜⎛ ⎟⎞ 2 ⎜⎛ ⎟⎞ 3 ⎜⎛ ⎟⎞ 휆 = ⎜1 0 0⎟ 휆 = ⎜푖 0 0⎟ 휆 = ⎜0 −1 0⎟ ⎝0 0 0⎠ ⎝0 0 0⎠ ⎝0 0 0⎠ 0 0 1 0 0 −푖 (A.15) 4 ⎜⎛ ⎟⎞ 5 ⎜⎛ ⎟⎞ 휆 = ⎜0 0 0⎟ 휆 = ⎜0 0 0 ⎟ ⎝1 0 0⎠ ⎝푖 0 0 ⎠ 0 0 0 0 0 0 1 0 0 6 ⎜⎛ ⎟⎞ 7 ⎜⎛ ⎟⎞ 8 1 ⎜⎛ ⎟⎞ 휆 = ⎜0 0 1⎟ 휆 = ⎜0 0 −푖⎟ 휆 = ⎜0 1 0 ⎟ , 0 1 0 0 푖 0 √3 0 0 −2 The algebra⎝ of the⎠ generators⎝ is defined⎠ by the structure⎝ constants⎠ : 푎푏푐 푓(A.16) 푎 푏 푎푏푐 푐 For with generators represented[푇 , 푇 ] = 푖as 푓 above,푇 . the structure constants are antisymmetricSU(3) in all indices, and up to permutation of indices the non-zero

54 A.5. Some results from SU(푁푐) values are:

123 푓 = 1 , (A.17) 458 678 √3 푓 = 푓 = , 2 147 165 246 345 257 1 푓 = 푓 = 푓 = 푓 = 푓 = . For the fundamental representation of , the2 generators may be chosen such that the structure factors are antisymmetricSU(푁푐) in all indices. The normalization of the generators can then be written

(A.18) 푎 푏† 푎 푏† 푎푏 푇푖푗푇푗푖 = Tr [푇 푇 ] = 푇퐹훿 , (A.19) 푎 푎† 푎 푎† 푇푖푗푇푗푘 = (푇 푇 )푖푘 = 퐶퐹훿푖푘 . Clearly, (A.20) 푎 푎† 푎 푎† 2 Here, is called푇푖푗 the푇푗푖 = Dynkin Tr [푇 푇 index] = and푇퐹 (푁 may푐 − 1) be = chosen 퐶퐹푁푐 . freely, and is called the푇퐹 color factor. Since the generators are Hermitian, the daggers in퐶퐹 the above identities may be removed without altering the results. For with generators represented as above, SU(3)

(A.21) 1 푇퐹 = , 2 (A.22) 4 퐶퐹 = . 3

55 B Properties of gamma matrices and Dirac spinors

This appendix reviews properties of the Dirac gamma matrices and Dirac spinors. All derivations are performed in a representation-independent man- ner, inspired by [21], but the presentation focuses on the properties needed in the main text.

B.1 Gamma matrices

The gamma matrices are any set of -matrices , where is a Lorentz 휇 index, satisfying the anticommutation4 × relations 4 훾 휇 (B.1) 휇 휈 휇휈 {훾 , 훾 } = 2휂 푰 . The objects acted upon by these matrices are spinors, presented in appendix B.2, and the entries are referred to as where are called spinor indices. 휇 More explicitly, Equation (B.1)훾훼훽 can be written훼, 훽

0 2 (훾 ) = 푰 , (B.2) 푖 2 (훾 ) = −푰 , 푖 ∈ {1, 2, 3} , 휇 휈 휈 휇 훾 훾 = −훾 훾 , 휇 ≠ 휈 . From this, it is clear that can only have eigenvalues , while the can 0 푖 only have eigenvalues 훾. ±1 훾 A similarity transformation±푖 leaves equation (B.1) 휇 휇 휇 −1 invariant and thus relates different훾 representations→ 훾′ = 푆훾 푆 of the gamma matrices. The most common representations are those where is Hermitian and the 0 푖 anti-Hermitian, and . A convenient훾 way of expressing훾 0 † 0 푖 † 푖 this is (훾 ) = 훾 (훾 ) = −훾 (B.3) 휇 † 0 휇 0 Such representations are related(훾 by) =unitary 훾 훾 훾similarity. transformations 휇 . 훾 → 휇 휇 † † −1 훾′ = 푈훾 푈 , 푈 = 푈 56 B.1. Gamma matrices

The restriction in equation (B.3) is related to the origin of the gamma matrices as coefficients in the Dirac Hamiltonian

(B.4) 0 퐻 = 훾 (휸 ⋅ 풑 + 푚) . Equation (B.1) can be derived by requiring this Hamiltonian to satisfy the relativistic energy-momentum relation . The Hermiticity condi- 2 2 2 tions, equation (B.3), are needed for to퐻 be= Hermitian. 풑 + 푚 Only representations satisfying equation (B.3) are considered퐻 here. It is convenient to define the matrix

(B.5) 5 0 1 2 3 훾 = 푖훾 훾 훾 훾 . It satisfies the following properties:

5 † 5 (훾 ) = 훾 , (B.6) 5 2 (훾 ) = 푰 , 5 휇 {훾 , 훾 } = 0 . This is straightforward to show by applying equations (B.2) and (B.3). The 5 matrix is closely related to the concept of parity [5, pp. 235-238], but here it훾 is only used as a convenient tool to prove trace identities.

Traces It is often necessary to compute traces of products of gamma matrices. The trace of a matrix is the sum of its diagonal entries: . The trace operation is obviously linear, and it is also invariantTr under [퐴] =cyclic ∑푖 퐴 permutations푖푖 of the factors in its argument:

(B.7) Tr [퐴1퐴2 ⋯ 퐴푁−1퐴푁] = Tr [퐴푁퐴1퐴2 ⋯ 퐴푁−1]. This is known as the cyclicity of the trace. The gamma matrices are traceless, as seen by

휇 5 2 휇 Tr [훾 ] = Tr [(훾 ) 훾 ] 5 휇 5 (B.8) = −Tr [훾 훾 훾 ] 5 2 휇 = −Tr [(훾 ) 훾 ] 휇 = −Tr [훾 ]. Here, the properties from equation (B.6) are used in the first two steps by inserting an identity in the form of and then anticommuting. A cyclic 5 2 permutation is performed in the third(훾 step.)

57 B.1. Gamma matrices

By a similar argument, the trace of a product of any odd number of gamma matrices must vanish:

휇1 휇푁 5 2 휇1 휇푁 Tr [훾 ⋯ 훾 ] = Tr [(훾 ) 훾 ⋯ 훾 ] 푁 5 휇1 휇푁 5 (B.9) = (−1) Tr [훾 훾 ⋯ 훾 훾 ] 푁 5 2 휇1 휇푁 = (−1) Tr [(훾 ) 훾 ⋯ 훾 ] 푁 휇1 휇푁 The case of two matrices is straightforward:= (−1) Tr [훾 ⋯ 훾 ].

휇 휈 휇휈 (B.10) Tr [훾 훾 ] = 휂 Tr [푰] 휇휈 since by cyclicity, = 4휂 , . 휇 휈 휈 휇 휇 휈 In the generalTr case, [훾 훾 equation] = Tr [훾 (B.1)훾 ] can = Tr be [{훾 applied, 훾 }/2] repeatedly to reduce a trace of gamma matrices to a sum of traces of gamma matrices by anticommuting푁 one matrix through the rest of the푁 product. − 2 By recursive application of this algorithm, the trace of any (even) number of gamma matrices may be found. For , the algorithm works as follows: 푁 = 4 휇 휈 휌 휎 휇휈 휌 휎 휈 휇 휌 휎 Tr [훾 훾 훾 훾 ] = 2휂 Tr [훾 훾 ] − Tr [훾 훾 훾 훾 ] 휇휈 휌 휎 휇휌 휈 휎 휈 휌 휇 휎 (B.11) = 2 (휂 Tr [훾 훾 ] − 휂 Tr [훾 훾 ]) + Tr [훾 훾 훾 훾 ] 휇휈 휌 휎 휇휌 휈 휎 휇휎 휈 휌 = 2 (휂 Tr [훾 훾 ] − 휂 Tr [훾 훾 ] + 휂 Tr [훾 훾 ]) 휈 휌 휎 휇 The last term on the− right Tr [훾 hand훾 훾 side훾 ]. is equal to the expression on the left hand side by cyclicity. Substituting equation (B.10) gives the result: (B.12) 휇 휈 휌 휎 휇휈 휌휎 휇휌 휈휎 휇휎 휈휌 Similarly, Tr [훾 훾 훾 훾 ] = 4 (휂 휂 − 휂 휂 + 휂 휂 ).

휇 휈 휌 휎 훿 휆 휇휈 휌휎 훿휆 휇휈 휌훿 휎휆 휇휈 휌휆 휎훿 Tr [훾 훾 훾 훾 훾 훾 ] = 4 ( 휂 휂 휂 − 휂 휂 휂 + 휂 휂 휂 휇휌 휈휎 훿휆 휇휌 휈훿 휎휆 휇휌 휈휆 휎훿 − 휂 휂 휂 + 휂 휂 휂 − 휂 휂 휂 (B.13) 휇휎 휈휌 훿휆 휇휎 휈훿 휌휆 휇휎 휈휆 휌훿 + 휂 휂 휂 − 휂 휂 휂 + 휂 휂 휂 휇훿 휈휌 휎휆 휇훿 휈휎 휌휆 휇훿 휈휆 휌휎 − 휂 휂 휂 + 휂 휂 휂 − 휂 휂 휂 휇휆 휈휌 휎훿 휇휆 휈휎 휌훿 휇휆 휈훿 휌휎 Generalizing to spacetime+ 휂 dimensions,휂 휂 − 휂 it is휂 still휂 possible+ 휂 휂 define휂 ). an algebra satisfying equation푑 (B.1). When doing dimensional regularization as discussed in appendix C, the intent is always to take the limit in the end, and one can therefore assume [1, p. 433]. Thus, equations푑 → 4 (B.10), (B.12) and (B.13) may be used forTr dimensional [푰] = 4 regularization. The only important difference is that the contraction of the metric tensor with itself is changed, , and this must be kept in mind when contracting the traces 휇 with휂휇 = other푑 = 4 quantities.− 2휀

58 B.1. Gamma matrices

Contractions The Lorentz index on the gamma matrix may be raised and lowered by the metric tensor in the same way as for vectors:

(B.14) 휈 휂휇휈훾 = 훾휇 . Thus, contracted gamma matrix products are straightforward to simplify using the anticommutation relations. Some examples are shown here, calculated in dimensions: 푑 = 4 − 2휀 휇 1 휇 휇 (B.15) 훾 훾휇 = {훾 , 훾휇} = 휂휇푰 2 = 푑푰 = (4 − 2휀) 푰 , 휇 휌 휇휌 휌 휇 (B.16) 훾 훾 훾휇 = (2휂 − 훾 훾 ) 훾휇 휌 휌 = (2 − 푑) 훾 = −2 (1 − 휀) 훾 , 휇 휌 휎 휇휌 휌 휇 휎 훾 훾 훾 훾휇 = (2휂 − 훾 훾 ) 훾 훾휇 (B.17) 휎 휌 휌 휎 = 2훾 훾 + 2 (1 − 휀) 훾 훾 휎휌 휌 휎 = 4휂 − 2휀훾 훾 . 휇 휌 휎 휆 휇휌 휌 휇 휎 휆 훾 훾 훾 훾 훾휇 = (2휂 − 훾 훾 ) 훾 훾 훾휇 휎 휆 휌 휌 휆휎 휎 휆 (B.18) = 2훾 훾 훾 − 훾 (4휂 − 2휀훾 훾 ) 휎 휆 휆휎 휌 휌 휎 휆 = 2 (훾 훾 − 2휂 ) 훾 + 2휀훾 훾 훾 휆 휎 휌 휌 휎 휆 = −2훾 훾 훾 + 2휀훾 훾 훾 . Combining these relations, one may even resolve “intertwined” contractions like

휇 휌 휎 휏 훿 휆 휌훿 휏 휆 훾 훾 훾 훾 훾휇훾 훾휎 훾 = − 8휂 훾 훾 (B.19) 휌휏 훿 휆 휌 훿 휏 휆 + 4휀 (2휂 훾 훾 + 훾 훾 훾 훾 ) 2 휌 휏 훿 휆 − 4휀 훾 훾 훾 훾 . The trace of this expression is

휇 휌 휎 휏 훿 휆 휌훿 휏휆 (B.20) Tr [훾 훾 훾 훾 훾휇훾 훾휎 훾 ] = 16[ (−2 + 휀 (1 + 휀)) 휂 휂 휌휏 훿휆 휌휆 훿휏 + 휀 (1 − 휀) (휂 휂 + 휂 휂 )]. The contraction of a the gamma matrices with a 4-vector is expressed by the so-called Feynman slash:

휇 (B.21) /푎 = 푎휇훾 0 0 = 푎 훾 − 풂 ⋅ 휸 . Note that slashed vectors are matrices, and do not in general commute with other slashed vectors or gamma matrices.

59 B.2. Dirac spinors

Slashed vectors may appear in contracted products of gamma matrices like those shown above:

휇 휇 휌 (B.22) 훾 /푎훾휇 = 푎휌훾 훾 훾휇 = −2 (1 − 휀) /푎 , (B.23) 휇 훾 /푎/푏훾휇 = 4 (푎 ⋅ 푏) 푰 − 2휀/푎/푏 . (B.24) 휇 훾 /푎/푏/푐훾휇 = −2/푐/푏/푎 + 2휀/푎/푏/푐 . Products of slashed vectors may be rewritten using the anticommutation relations:

휇 휈 (B.25) /푎/푏 + /푏/푎 = 푎휇푏휈 {훾 , 훾 } = 2 (푎 ⋅ 푏) 푰 , (B.26) 2 /푎/푎 = 푎 푰 . For real-valued vectors, equation (B.3) may be applied directly to the corre- sponding slashed vector:

† 0 0 (B.27) /푎 = 훾 /푎훾 0 0 = 푎 훾 + 풂 ⋅ 휸 . Combined with equation (B.21), this yields

(B.28) † 0 0 /푎 + /푎 = 2푎 훾 . B.2 Dirac spinors

The modern form of the Dirac equation is found by substituting the Dirac Hamiltonian (equation (B.4)) into the Schrödinger equation , using the position space momentum operator , and multiplying푖휕푡휓 = 퐻휓 with to 0 set the temporal and spatial derivatives풑 on = equal −푖훁 footing. The result is 훾 (B.29) (푖휕/ − 푚) 휓 = 0 . where , and is a spinor, which may be viewed as a complex vector 휇 with components휕/ = 훾 휕휇 where휓 is a spinor index. A Dirac spinor휓훼 describes훼 a free fermion with mass and -momentum , and is푢(푝) related to a plane-wave solution of the푚 Dirac4 equation: 0 푝 = (푝 , 풑) (B.30) −푖푝⋅푥 휓 (푥) = 푢 (푝) 푒 . Substituting this into equation (B.29) and dividing by gives the Dirac equation in momentum space: exp{−푖푝 ⋅ 푥} (B.31) (푝/ − 푚) 푢 (푝) = 0 . 60 B.2. Dirac spinors

Multiplying with shows that in order to find a non-trivial solution for , the -momentum푝/ + 푚 must satisfy the relativistic energy-momentum relation:푢(푝) 4 (B.32) 2 2 which is satisfied(푝/ if + 푚) (푝/ − 푚) 푢 (푝) = (푝 − 푚 ) 푢 (푝) = 0 , (B.33) 0 2 2 2 2 A -momentum satisfying this(푝 ) constraint= 풑 + 푚 is≡ written 퐸 . . 4There is an ambiguity in the definition of 푝 = (퐸, 풑) . Hence, for a 2 2 given momentum there are two sets of solutions퐸 = ± to√풑 equation+ 푚 (B.31) – one with and one풑 with . This is resolved by interpreting a particle with -momentum퐸 > 0 and energy퐸 < 0 as an antiparticle with -momentum . To4 accomplish푝 this transparently,퐸 < 0 another set of Dirac spinors4 is introduced:−푝 , describing a free antifermion with -momentum . The full 푣(푝)set of ∝ Dirac 푢(−푝) spinors thus describes the plane waves4 푝

−푖푝⋅푥 (B.34) {⎧푢 (푝) 푒 , 휓 (푥) = 푖푝⋅푥 {⎨푣 (푝) 푒 , now only allowing solutions with⎩ . Substituting into equation (B.29) gives the two momentum-space Dirac퐸 > equations 0 for particle and antiparticle spinors:

(B.35) (푝/ − 푚) 푢 (푝) = 0 , (B.36) (푝 + 푚) 푣 (푝) = 0 . In order to discover the properties/ of these spinors, the first step is to find expressions for them in the case that , such that . Equations (B.35) and (B.36) then reduce to 풑 = 0 퐸 = 푚 (B.37) 0 (훾 − 1) 푢 (푚, 0) = 0 , (B.38) 0 (훾 + 1) 푣 (푚, 0) = 0 . Hence, these spinors are eigenvectors of , with eigenvalue for 0 and for . It has been shown in훾 appendix B.1 that +1are the푢(푚, only 0) eigenvalues−1 푣(푚, of 0), and that it is traceless, hence there must±1 be two linearly 0 independent eigenvectors훾 for each eigenvalue. Let and be two sets of 푠 푠 orthonormal eigenvectors, one for each eigenvalue, indexed휉 휒 by : 푠 = ±1 (B.39) 0 푠 푠 푠 † 푠′ 푠푠′ 훾 휉 = 휉 , (휉 ) 휉 = 훿 , (B.40) 0 푠 푠 푠 † 푠′ 푠푠′ 훾 휒 = −휒 , (휒 ) 휒 = 훿 . Since is Hermitian, the eigenspaces are orthogonal: 0 훾 (B.41) 푠 † 푠′ (휉 ) 휒 = 0 . 61 B.2. Dirac spinors

The orthogonal decomposition within each eigenspace can be made explicit, but this is not necessary here. The zero-momentum Dirac spinors must be proportional to these eigen- vectors:

(B.42) 푠 푠 푢 (푚, 0) ∝ 휉 , (B.43) 푠 −푠 푣 (푚, 0) ∝ 휒 . The minus sign in equation (B.43) is conventional. The index on and is then interpreted as the spin state of the fermions. 푠 푢 푣 For general momenta, it is straightforward to show that the spinors satisfy equations (B.35), (B.36), (B.42) and (B.43) if they are defined as

(B.44) 푠 푠 푢 (푝) ∝ (푝/ + 푚) 휉 , (B.45) 푠 −푠 푣 (푝) ∝ (−푝/ + 푚) 휒 ,

Normalization To determine a normalization for the spinors, consider the expression . 푠 † 푠′ Since , this is equal to . Thus,(휉 only) 푝휉 the 0 푠 푠 푠 † 0 0 푠′ 푠 † † 푠′ / Hermitian훾 휉 = part 휉 of , the -term,(휉 can) 훾 contribute.푝훾 휉 = (휉 The) 푝 휉 same is true for . 0 / / 푠 Hence, 푝/ 퐸훾 휒

푠 † 푠′ 푠 † 0 푠′ (B.46) (휉 ) 푝휉/ = 퐸 (휉 ) 훾 휉 푠푠′ = 퐸훿 , 푠 † 푠′ 푠 † 0 푠′ (B.47) (휒 ) 푝휒/ = 퐸 (휒 ) 훾 휒 푠푠′ = −퐸훿 . Equation (B.28) gives , such that † 0 푝/ + 푝/ = 2퐸훾 † 0 (B.48) (±푝/ + 푚) (±푝/ + 푚) = (2퐸훾 − (푝/ ∓ 푚)) (푝/ ± 푚) 0 = 2퐸훾 (푝/ ± 푚) . Combining these results gives

푠 † 푠′ 푠 † 0 푠′ (B.49) 푢 (푝) 푢 (푝) ∝ 2퐸 (휉 ) 훾 (푝/ + 푚) 휉 푠푠′ = 2퐸 (퐸 + 푚) 훿 , 푠 † 푠′ −푠 † 0 −푠′ (B.50) 푣 (푝) 푣 (푝) ∝ 2퐸 (휒 ) 훾 (푝/ − 푚) 휒 푠푠′ = 2퐸 (퐸 + 푚) 훿 .

62 B.2. Dirac spinors

The spinors are then defined as

(B.51) 푠 (푝/ + 푚) 푠 푢 (푝) = 휉 , √퐸 + 푚 (B.52) 푠 (−푝/ + 푚) −푠 푣 (푝) = 휒 , √ such that the normalization reads 퐸 + 푚

(B.53) 푠 † 푠′ 푠푠′ 푢 (푝) 푢 (푝) = 2퐸훿 , (B.54) 푠 † 푠′ 푠푠′ 푣 (푝) 푣 (푝) = 2퐸훿 . This also shows that different spin states are orthogonal for any momentum, when the spinor is treated as a complex vector.

The Dirac adjoint Frequently when working with Dirac spinors, the Dirac adjoint is used rather than the Hermitian adjoint. The Dirac adjoint of a spinor is defined as 휓 (B.55) † 0 휓 = 휓 훾 . The rationale for introducing the Dirac adjoint is that it can be used to construct quantities that are well-behaved under Lorentz transformations. Examples are the inner product , which is a Lorentz scalar, and the current , 휇 which is a Lorentz 4-vector.휑휓 Corresponding quantities defined using휑훾 do휓 † not behave as nicely. This is obviously a consequence of the transformation휑 properties of spinors under Lorentz transformations, but these properties will not be discussed here. Many useful quantities can be defined by sandwiching a matrix expression between a spinor and a Dirac adjoint, in general expressed as for spinors and and a matrix . When squaring scattering amplitudes휑 훤휓 it is often 휓necessary휑 to find the complex훤 conjugate of such expressions. This can be expressed as

∗ † 0 † [휑 훤휓] = [휑 훾 훤휓] † † 0 (B.56) = 휓 훤 훾 휑 † 0 0 † 0 = 휓 훾 훾 훤 훾 휑 = 휓 훤휑 , where is defined for any matrix as 훤 (B.57) 0 † 0 훤 = 훾 훤 훾 .

63 B.2. Dirac spinors

If can be expressed as a product of gamma matrices, , one finds 휇 휈 훤 훤 = 훾 ⋯ 훾

0 휈 † 휇 † 0 훤 = 훾 (훾 ) ⋯ (훾 ) 훾 (B.58) 0 휈 † 0 0 0 0 휇 † 0 = 훾 (훾 ) 훾 훾 ⋯ 훾 훾 (훾 ) 훾 휈 휇 = 훾 ⋯ 훾 . This leads to the following convenient identity:

(B.59) 휇 휈 ∗ 휈 휇 [휑훾 ⋯ 훾 휓] = 휓훾 ⋯ 훾 휑 . It is also useful to find the adjoint of the Dirac equations in momen- tum space, equations (B.35) and (B.36), which is straightforward using equa- tion (B.27):

(B.60) 푢 (푝) (푝/ − 푚) = 0 , (B.61) 푣 (푝) (푝/ + 푚) = 0 . Inner products As mentioned in the previous section, one can define a Lorentz-invariant inner product of two spinors and as . This is similar to the usual ∗ inner product on a complex휓 vector휑 space,휑휓 = but (휓휑) uses the Dirac adjoint in place of the Hermitian adjoint. To find the various inner products of and , consider the expres- 푠 푠 sion 푢 (푝) 푣 (푝)

† 0 0 (±푝/ + 푚) 훾 (±푝/ + 푚) = 훾 (푝/ ± 푚) (푝/ ± 푚) (B.62) 0 2 2 = 훾 (푝 ± 2푚푝/ + 푚 ) 0 = 2푚훾 (±푝/ + 푚) , Comparing with equation (B.48), it is clear that replacing and with and in equations (B.53) and (B.54) amounts푠 to† replacing푠 a† 푠 푠 푢 (푝) 푣 (푝) factor푢 of(푝) with푣 a(푝) factor of and inserting a minus sign for the spinors: 퐸 푚 푣 (B.63) 푠 푠′ 푠푠′ 푢 (푝) 푢 (푝) = 2푚훿 , (B.64) 푠 푠′ 푠푠′ 푣 (푝) 푣 (푝) = −2푚훿 . Thus, different spin states are orthogonal also with respect to the spinor inner product. Particle and antiparticle states are also orthogonal, since

(B.65) † 0 0 (±푝/ + 푚) 훾 (∓푝/ + 푚) = 훾 (푝/ ± 푚) (푝/ ∓ 푚) = 0 . such that (B.66) 푠 푠′ 푠 푠′ 푣 (푝) 푢 (푝) = 푢 (푝) 푣 (푝) = 0 . 64 B.2. Dirac spinors

Completeness relations Consider the matrices

(B.67) 푠 푠 푃푢 (푝) = ∑ 푢 (푝) 푢 (푝) , 푠 (B.68) 푠 푠 푃푣 (푝) = ∑ 푣 (푝) 푣 (푝) . 푠 The four - and -spinors for a given momentum form a complete basis for the spinor푢 space.푣 A general spinor can therefore be푝 written

(B.69) 푟 푡 휓 = ∑ 푎푟푢 (푝) + ∑ 푏푡푣 (푝) . 푟 푡 Applying to and using equations (B.63) and (B.64) gives 푃푢(푝) 휓

푟 푡 푃푢 (푝) (∑ 푎푟푢 (푝) + ∑ 푏푡푣 (푝)) 푟 푡 (B.70) 푠 푠푟 = 2푚 ∑ 푢 (푝) (∑ 푎푟훿 + ∑ 푏푡 ⋅ 0) 푠 푟 푡

푟 = 2푚 (∑ 푎푟푢 (푝)) . 푟 Hence, must be the projection operator into the subspace spanned by 푃.푢(푝)/2푚 Similarly, is the projection operator into the subspace 푠 spanned푢 (푝) by . Comparing−푃푣(푝)/2푚 with equations (B.35) and (B.36), it is therefore 푠 clear that 푣 (푝) (B.71) 푠 푠 ∑ 푢 (푝) 푢 (푝) = 푝/ + 푚 , 푠 (B.72) 푠 푠 ∑ 푣 (푝) 푣 (푝) = 푝 − 푚 . 푠 / These are the completeness relations for Dirac spinors.

65 C Dimensional regularization

To regularize divergent integrals in quantum field theory, an approach that is often used is to compute the integrals in an arbitrary -dimensional spacetime with timelike dimension and spacelike dimensions.푑 Then, setting 1 , the divergences of푑 the − 1 integrals in 4-dimensional spacetime 푑manifest = 4 − as 2휀 poles when . Certain quantities need휀 → to 0 be reinterpreted in order to make sense when generalizing the number of dimensions. Most obviously, the contraction of the metric tensor in Minkowski space with itself becomes

(C.1) 휇휈 휇 휂 휂휇휈 = 휂휇 = 푑 = 4 − 2휀 , instead of the usual . 휇 In addition, measures휂휇 = must 4 be taken to ensure consistency in the dimension of the Lagrangian and the action. The action is the integral of the Lagrangian over all spacetime: (C.2) 푑 푆 = ∫ 푑 푥ℒ , and it should have the same dimensions as Planck’s constant, which is set to in this text. Since the dimension of is obviously , where 푑 푑 푑 ℏ =denotes 1 a length dimension, the dimension푑 푥 of must be[푑 푥] = 퐿 . −푑 푑 퐿Here, is a mass dimension, and the relation ℒ is a[ℒ] consequence = 퐿 = 푀 of −1 setting푀 . 푀 = 퐿 Fromℏ = this 푐 = one 1 can deduce the dimensions of the fields and coupling con- stants in equations (A.3) and (A.8). In partuclar, one finds that . However, being physical parameters, the coupling[푔푠 constants] = [푒] = 2−(푑/2) 휀 푀should be= defined 푀 independently of the spacetime dimension. An arbitrary mass scale is therefore introduced, and when doing dimensional regulariza- tion one performs휇 the replacement

휀 (C.3) 푒 → 휇 푒 , 휀 푔푠 → 휇 푔푠 , leaving the values of and unchanged. 푒 푔푠 66 In dimensions, -dimensional integration measures such as 푑 푑 must be푑 used in place푑 of the usual 4-dimensional ones. For integrals푑 푝/(2휋) of the form

푑 푑−1 (C.4) 푑 푝 푓 (푝) +∞ 푑퐸 푑 풑 푓 (퐸, 풑) 푛 푛 ∫ 푑 2 = ∫−∞ ∫ 푑−1 2 2 , (2휋) (푝 − 퐴 + 푖휀) 2휋 (2휋) (퐸 − 풑 − 퐴 + 푖휀) it is often beneficial to avoid the poles by rotating from Minkowski to Euclidean space, defining the Euclidean vector , where and is the time-component of the 4-vector 푝퐸 = (푝0,퐸, 풑). Thus, 푝0,퐸 = −푖퐸 퐸 . 2 2 2 2 The term in equation (C.4) entails an푝 = analytic (퐸, 풑) continuation푝퐸 ≡푝 of0,퐸 the+ 풑-integral= −푝 into the complex푖휀 plane to avoid the poles at , and퐸 a closed 2 integration contour with no poles inside is obtained퐸 = ±√풑 by+ going 퐴 up the real axis, down the imaginary axis, and closing the contour in the first and third quadrants (assuming that has no poles in the interior of this curve). Thus, 푓 (퐸, 풑)

(C.5) +∞ 푑퐸 푓 (퐸, 풑) +푖∞ 푑퐸 푓 (퐸, 풑) 푛 푛 ∫−∞ 2 2 − ∫−푖∞ 2 2 = 0 . 2휋 (퐸 − 풑 − 퐴 + 푖휀) 2휋 (퐸 − 풑 − 퐴 + 푖휀) Making the substitution gives 푝0,퐸 = −푖퐸 (C.6) +∞ 푑퐸 푓 (퐸, 풑) +∞ 푑푝0,퐸 푓 (푖푝0,퐸, 풑) 푛 푛 ∫−∞ 2 2 = 푖 ∫−∞ 2 2 . 2휋 (퐸 − 풑 − 퐴 + 푖휀) 2휋 (−푝0,퐸 − 풑 − 퐴 + 푖휀) Hence,

푑 푑 (C.7) 푑 푝 푓 (퐸, 풑) 푑 푝퐸 푓 (푖푝0,퐸, 풑) ∫ 푑 2 푛 = 푖 ∫ 푑 2 푛 . (2휋) (푝 − 퐴 + 푖휀) (2휋) (−푝퐸 − 퐴 + 푖휀) Note that the -term is no longer significant if . On the other hand, if the rotation푖휀 is not very helpful, since poles퐴 remain > 0 in the integrand. The 퐴solution < 0 in this case is to rotate the spatial axes in their respective complex planes instead, defining the euclidean vector 풑 with , such that . The poles lie on opposite sides푝퐸 = of (퐸, the 풑 axes퐸) in the풑퐸 =-plane −푖풑 2 2 compared푝퐸 to= the 푝 -plane, so the integration contour must now go푝푖up the imaginary axis and퐸 close in the second and fourth quadrants. The equivalent of equation (C.6) therefore acquires a negative sign, so the rotation introduces a factor of per axis. The final result is −푖 푑 푑 (C.8) 푑 푝 푓 (퐸, 풑) 푑−1 푑 푝퐸 푓 (퐸, 푖풑퐸) ∫ 푑 2 푛 = (−푖) ∫ 푑 2 푛 . (2휋) (푝 − 퐴 + 푖휀) (2휋) (푝퐸 − 퐴 + 푖휀) Here the -term is insignificant as long as . 푖휀 퐴 < 0 67 C.1. Important identities

The main benefit from rotating to Euclidean space is that the integrands are often isotropic there, enabling further simplification:

푑 ∞ (C.9) 푑 푝퐸 2 훺푑 푑−1 2 퐸 퐸 퐸 퐸 ∫ 푑 푓 (푝 ) = 푑 ∫0 푑 ̃푝 ̃푝 푓 ( ̃푝 ) (2휋) (2휋) where is the Euclidean length of , and is the volume of the 2 1 unit sphere퐸̃푝 = √푝퐸 in dimensions. The value푝퐸 of 훺is푑 stated here without 푑−1 derivation: 푆 푑 훺푑 푑/2 (C.10) 2휋 훺푑 = . A proof can be found in e.g. [22]. 훤(푑/2)

C.1 Important identities

A number of mathematical identities come in handy when working in di- mensions. Some of the most important are given here without proof. 푑

The Euler - and -functions The Euler -function훤 훽 is defined as 훤 ∞ (C.11) 푧−1 −푡 훤 (푧) = ∫ 푑푡 푡 푒 , ℜ{푧} > 0 , and extended to the whole complex0 plane by analytic continuation. It satisfies the following identities: (C.12) 훤 (푧 + 1) = 푧훤 (푧) , 2푧−1 (C.13) 2 1 훤 (2푧) = 훤 (푧) 훤 (푧 + ). For integer , √휋 2 푛 (C.14) The first few terms in series expansion훤 (푛) = (푛 of − 1) !around . are 훤(푧) 푧 = 1 2 (C.15) 휋 1 2 2 훤 (1 + 휀) = 1 − 훾퐸휀 + ( + 훾퐸) 휀 + 풪 (휀 ), where is the Euler-Mascheroni12 2 constant. The훾 Euler퐸 = 0.577216-function … is defined as 훽 1 푥−1 푦−1 (C.16) 훽 (푥, 푦) = ∫0 푡 (1 − 푡) 푑푡 ℜ{푥}, ℜ{푦} > 0 . 훤 (푥) 훤 (푦) = , 1Note that the unit sphere훤 ( is푥 the + 푦) -dimensional surface of the unit ball. Also, note that volume is used in a general sense(푑 here. − 1) For example, is what would normally be called the surface area of the unit sphere . 3 2 훺 = 4휋 푆 68 C.2. Phase space in dimensions 푑 Feynman parameter integrals The general Feynman parameter integral can be written [1, p. 429]

푛1−1 푛푘−1 1 1 훤 (푛1 + ⋯ + 푛푘) 훼1 ⋯ 훼푘 훿 (1 − ∑푖 훼푖) 푛1 푛푘 1 푘 푛1+⋯+푛푘 = ∫0 푑훼 ⋯ 푑훼 . (C.17)퐴1 ⋯ 퐴푘 훤 (푛1) ⋯ 훤 (푛푘) (훼1퐴1 + ⋯ + 훼푘퐴푘) The special cases used in the main text are those with two or three factors in the denominator:

(C.18) 1 1 푑훼 = ∫ 2 , 퐴퐵 0 (훼퐴 + (1 − 훼) 퐵) (C.19) 1 1 1−훼 2 = ∫0 푑훼 ∫0 푑훽 3 . 퐴퐵퐶 (훼퐴 + 훽퐵 + (1 − 훼 − 훽) 퐶) C.2 Phase space in dimensions

The differential phase space푑 of outgoing particles in the usual spacetime dimensions, with -momenta 푛 , is 4 4 푝푖, 푖 = 1, ..., 푁 푁 푁 4 (C.20) 4 (4) 푑 푝푖 2 2 0 푑훷푛 = (2휋) 훿 ⎜⎛푄 − ∑ 푝푖⎟⎞ ∏ 4 2휋훿 (푝푖 − 푚푖 ) 훩 (푝푖 ). 푖=1 푖=1 (2휋) Here, is the total -momentum⎝ ⎠ and is the Heaviside step function. This푄 expression4 is readily generalized훩 to spacetime dimensions: 푑 푁 푁 푑 (C.21) (푑) 푑 (푑) 푑 푝푖 2 2 0 푑훷푛 = (2휋) 훿 ⎜⎛푄 − ∑ 푝푖⎟⎞ ∏ 푑 2휋훿 (푝푖 − 푚푖 ) 훩 (푝푖 ). 푖=1 푖=1 (2휋) Using the relation ⎝ ⎠ (C.22) 훿(푥 − 푥푖) 훿(푔(푥)) = ∑ , where the are the roots of , gives푖 ∣푔′(푥푖)∣ 푥푖 푔(푥) 0 2 (C.23) 2 2 0 훿 (푝푖 − 퐸푖 (풑푖 )) 훿 (푝푖 − 푚푖 ) 훩 (푝푖 ) = 2 , 2퐸푖 (풑푖 ) with (C.24) 2 2 2 푖 √ 푖 Thus, upon substituting 퐸 (풑 ) = in풑 the+ 푚 integrand,. the following is an 0 2 equivalent phase space measure:푝푖 = 퐸푖(풑푖 )

푁 푁 푑−1 (C.25) (푑) 푑 (푑) 푑 풑푖 푑훷푛 = (2휋) 훿 ⎜⎛푄 − ∑ 푝푖⎟⎞ ∏ 푑−1 2 . ⎝ 푖=1 ⎠ 푖=1 (2휋) 2퐸푖 (풑푖 ) 69 C.2. Phase space in dimensions 푑 2 particles In the case of two outgoing particles, the phase space integral of a function can be written 푓 (풑1, 풑2) 푑−1 푑−1 푑 풑1푑 풑2 퐼 = ∫ 푑−2 2 2 4(2휋) 퐸1 (풑1) 퐸2 (풑2) 0 2 2 (푑−1) × 훿 (푄 − 퐸1 (풑1) − 퐸2 (풑2)) 훿 (푸 − 풑1 − 풑2) 푓 (풑1, 풑2) (C.26) 푑−1 푑 풑1 = ∫ 푑−2 2 2 4(2휋) 퐸1 (풑1) 퐸2 ((푸 − 풑1) ) 0 2 2 × 훿 (푄 − 퐸1 (풑1) − 퐸2 ((푸 − 풑1) )) 푓 (풑1, 푸 − 풑1). In the COM frame, and . To proceed, assumptions 0 2 must be made on the form푸 = of 0 푄 = √.푄 In general,= √푠 even a scalar integrand such as a matrix element may have푓 (풑1, a −풑 directional1) dependence on through its dependence on other vectors, e.g. the momenta of incoming particles.풑1 If it is isotropic, however, one may define and write, using equation (C.9), 2 ̃푝= √풑1

∞ 푑−2 (C.27) 훺푑−1 ̃푝 푑 ̃푝 2 2 퐼 = ∫ 푑−2 2 2 훿 (√푠 − 퐸1 ( ̃푝 ) − 퐸2 ( ̃푝 )) 푓 ( ̃푝). 0 4(2휋) 퐸1 ( ̃푝 ) 퐸2 ( ̃푝 ) Let be the argument of the -function, such that 푔( ̃푝) 훿 (C.28) 2 2 푔( ̃푝)= √푠 − 퐸1 ( ̃푝 ) − 퐸2 ( ̃푝 ), ̃푝 ̃푝 푔′( ̃푝)= − 2 − 2 (C.29) 퐸1 ( ̃푝 ) 퐸2 ( ̃푝 ) 2 2 퐸1 ( ̃푝 ) + 퐸2 ( ̃푝 ) = − ̃푝 ⎜⎛ 2 2 ⎟⎞ . 퐸1 ( ̃푝 ) 퐸2 ( ̃푝 ) The solutions to are ⎝ ⎠ 푔( ̃푝)= 0 (C.30) 1 2 2 2 ±̃푝 = ± √푠 + 2푠(푚1 + 푚2) + (푚1 − 푚2) . Only the positive solution2√푠 contributes in the integral, which becomes

푑−3 (C.31) 훺푑−1 +̃푝 퐼 = 푑−2 푓 (+ ̃푝 ). 4(2휋) √푠 Substituting , the spherical volume can be written 푑 = 4 − 2휀 (3/2)−휀 (C.32) 2휋 훺3−2휀 = . 훤 ((3/2) − 휀) 70 C.2. Phase space in dimensions 푑 Using equations (C.12) and (C.13), one can rewrite

3 2(휀−1) 훤 (3 − 2휀) 훤 ( − 휀) = 2 √휋 (C.33) 2 훤 (2 − 휀) 2휀−1 훤 (2 − 2휀) = 2 √휋 . Thus, 훤 (1 − 휀) 휀 (C.34) 1 휋 ̃푝 훤 (1 − 휀) 퐼 = ( 2 ) 푓 ( ̃푝), where the subscript is4휋 dropped̃푝 √ and푠 훤 (2is − understood 2휀) to be subject to the constraints from -momentum+ conservation.̃푝 From this expression4 the fully integrated two-particle phase space factor in dimensions can be isolated. It is valid in the COM frame for integrands푑 = 4 − that2휀 are isotropic in the momentum, and reads:

휀 (C.35) (푑) 1 휋 ̃푝 훤 (1 − 휀) 훷2 = ( 2 ) . 4휋 ̃푝 √푠 훤 (2 − 2휀) 3 particles In the case of three outgoing particles, it is most convenient to give the phase space integral in terms of the energy fractions in the COM frame, defined as

(C.36) 2푝푖 ⋅ 푄 푥푖 = 2 . 푄 The phase space integral is not derived here, but cited from [1, p. 434]. The result is

2 2휀 (푑) 푄 4휋 1 ∫ 푑훷3 푓 (풑1, 풑2, 풑3) = 3 ( 2 ) 2 (4휋) 푄 훤 (2 − 2휀) 1 1 푓 (푥1, 푥2, 푥3) 1 2 × ∫ 푑푥 ∫ 1 푑푥 휀 0 1−푥 [(1 − 푥1) (1 − 푥2) (1 − 푥3)](C.37)

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