Applications of Effective Field Theory Techniques to Jet Physics by Simon

Applications of Effective Field Theory Techniques to Jet Physics

Simon M. Freedman

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Applications of Eﬀective Field Theory Techniques to Jet Physics

Simon M. Freedman Doctor of Philosophy Graduate Department of Physics University of Toronto 2015

In this thesis we study jet production at large energies from leptonic collisions. We use the framework of eﬀective theories of Quantum Chromodynamics (QCD) to examine the properties of jets and systematically improve calculations.

We first develop a new formulation of soft-collinear effective theory (SCET), the appropriate effective theory for jets. In this formulation, soft and collinear degrees of freedom are described using QCD fields that interact with each other through light-like Wilson lines in external currents. This formulation gives a more intuitive picture of jet processes than the traditional formulation of SCET. In particular, we show how the decoupling of soft and collinear degrees of freedom that occurs at leading order in power counting is explicit to next-to-leading order and likely beyond.

We then use this formulation to write the thrust rate in a factorized form at next-to-leading order in the thrust parameter. The rate involves an incomplete sum over ﬁnal states due to phase space cuts that is enforced by a measurement operator. Subleading corrections require matching onto not only the next-to-next-to leading order SCET operators, but also matching onto subleading measurement operators. We derive the appropriate hard, jet, and soft functions and show they reproduce the expected subleading thrust rate.

Next, we renormalize the next-to-leading order dijet operators used for the subleading thrust rate. Constraints on matching coeﬃcients from current conservation and reparametrization invariance are shown. We also discuss the subtleties involved in regulating the infrared divergences of the individual loop diagrams in order to extract the ultraviolet divergences. The results can be used to increase the theoretical precision of the thrust rate.

Finally, we study the (exclusive) k⊥ and C/A jet algorithms in SCET. Regularizing the virtualites and rapidities of the individual graphs, we are able to write the O(αs) dijet cross

ii section as the product of separate hard, jet, and soft contributions. We show how to reproduce the Sudakov form factor to next-to-leading logarithmic accuracy previously calculated by the coherent branching formalism. Our result only depends on the running of the hard function, and we comment that regularizing rapidities is not necessary in this case.

iii Dedication

To my patient parents and wife

iv Acknowledgements

I would like to start by giving my sincere thanks to my supervisor Michael Luke, for advising and encouraging me throughout my degree. The work in this thesis would not have been possible without my collaborators: William Man-Yin Cheung and Ray Georke. I would also like to thank Saba Zuberi and Andrew Blechman for teaching me a lot about EFTs during my first few years. As well, I would like to thank my committee members, Bob Holdom and Pierre Savard for keeping me on track, and non-committee member Erich Poppitz for teaching me about the Standard Model. I must also thank my fellow and former graduate students/zombies Catalina Gomez and Santiago Amigo, for many useful discussions and terrible jokes. I would also like to thank my families, both new and old. I owe my new family, the Chiu’s, for all their support and great meals throughout the years. My parents, I would like to thank for bearing with me through the long years of wondering when I would finally graduate and my seemingly insane babbling about my research. My sister, I would like to thank for being my best- “man” and for fighting the good fight while I do less practical things. And my grandparents, despite the weekly questions about when I would graduate, I would like to thank them for their strength and passion that I hope I inherited. Finally, I reserve an unimaginable amount of gratitude to my wife Melissa, for her unwa- vering support and encouragment, and an endless energy to keep my spirits high. This work was supported by NSERC and the University of Toronto (and viewers like you).

v Contents

1 Introduction 1 1.1 Hadronic Jet Physics ...... 3 1.2 Factorization From Eﬀective Field Theory ...... 5 1.3 Organization of the Thesis ...... 7

2 Effective Theories of QCD 8 2.1 Review of Quantum Chromodynamics ...... 11 2.2 Heavy Quark Effective Theory ...... 12 2.3 Soft-Collinear Effective Theory ...... 14 2.3.1 Examples of SCET Currents ...... 19 2.4 Conclusion ...... 21

3 SCET, QCD, and Wilson Lines 22 3.1 Introduction ...... 22 3.2 Label SCET Formulation ...... 23 3.3 SCET as QCD Fields Coupled to Wilson Lines ...... 26 3.3.1 Dijet Production at Leading Order ...... 27 3.3.2 Subleading Corrections to Dijet Production ...... 31 3.4 Heavy-to-Light Current ...... 36 3.5 Conclusions ...... 39

4 Subleading Corrections To Thrust Using Eﬀective Field Theory 41 4.1 Introduction ...... 41 4.2 Review of SCET ...... 43 4.3 Leading Order Calculation ...... 45 4.4 Next-to-Leading Order Calculation ...... 50 4.4.1 Measurement Operator ...... 50 4.4.2 Factorization ...... 53 4.5 Conclusion ...... 58 4.6 Appendix: Dijet Operators ...... 59 4.7 Appendix: Jet and Soft Operators ...... 61

vi 5 Renormalization of Subleading Dijet Operators in Soft-Collinear Eﬀective Theory 70 5.1 Introduction ...... 70 5.2 SCET and NLO Operators ...... 73 5.2.1 Constraining the NLO Operators ...... 77 5.3 Infrared Regulator ...... 80 5.3.1 The Delta Regulator ...... 80 5.3.2 Gluon mass ...... 82 5.4 Anomalous Dimensions ...... 83 5.5 Conclusion ...... 88

6 The Exclusive kT and C/A Dijet Rates in SCET with a Rapidity Regulator 89 6.1 Introduction ...... 89 6.2 Review of Previous Work ...... 92 6.3 Next-to-Leading-Order calculation ...... 95 6.4 Next-to-leading logarithm summation ...... 97 6.5 Discussion ...... 102 6.6 Conclusion ...... 103 6.7 Appendix: General Rapidity Anomalous Dimension ...... 103

7 Conclusion 105 7.1 Future Directions ...... 106

vii Chapter 1

Introduction

The Standard Model describes the strong and electroweak interactions at low energies. Elec- troweak symmetry is spontaneously broken at low energies by the Higgs mechanism resulting in four particles, three of which are Nambu-Goldstone bosons that become the longitudinal modes of the massive weak bosons. The fourth particle is a fundamental scalar called the Higgs boson that couples to massive Standard Model particles with a strength proportional to their mass. Only recently was a scalar particle with the Higgs boson’s quantum numbers observed at the Large Hadron Collider (LHC) with a mass of 125.36 ± 0.37(stat.) ± 0.18(syst) GeV [1] and +0.26 +0.26 125.02 −0.27 (stat.) −0.15 (syst.) GeV [2] as measured by the ATLAS and CMS collaborations respectively. The discovery of the Higgs boson gives a portal to explore the electroweak symmetry breaking mechanism. In order to explore this mechanism further, the LHC is increasing the luminosity, and increasing the collision energy from the current 8 TeV to 13 TeV energy. While the increased luminosity and collision energy allows for larger statistics and smaller distance scales to be probed, the large hadronic background in the form of jets will still make precision measurements at the LHC difficult. Improvements in the understanding of jets from the theory side will become more important in order to search for new physics and test the Standard Model. In this thesis, we will study jets in simpler leptonic colliders using effective theory techniques with the future goal of applying this understanding to the LHC environment. The majority of hadron production and interactions occur due to Quantum Chromody- namics (QCD). The Lagrangian of QCD is gauged under SU(3) and describes the interactions between coloured quarks and gluons. However, the final states observed at detectors are hadrons, which are colour singlet bound states of the quarks and gluons held together by long distance non-perturbative effects. The discrepancy between the degrees of freedom of the QCD Lagrangian and the observed final states is due to the property of asymptotic freedom. The

QCD coupling as a function of energy αs(Q), which describes the strength of the interactions between quarks and gluons, increases at low energy to the point where it is no longer a suitable expansion parameter. The energy scale where the coupling is large enough that the theory becomes non-perturbative is above ΛQCD ≈ 250 MeV in the MS regularization scheme. These non-perturbative eﬀects make theoretical predictions involving QCD interactions challenging.

1 Chapter 1. Introduction 2

Studying large energy quark and gluon production requires understanding the large energy sprays of hadrons called jets. In order to simplify the analysis we will concentrate on high energy lepton collisions, which share many key similarities with hadron collisions for jet production at small distance scales. Leptonic collisions provide a cleaner environment for calculations because effects from parton distribution functions, initial state radiation, and beam remnants are either non-existent or suppressed by the electroweak coupling and can be mostly ignored. The only processes that need be considered at leading order in the electroweak coupling are the large energy collision and the hadronization of the final states. The strategy for calculating jet production is to factorize the different scales in the process. Factorization helps to restore predictive power by separating the long distance non-perturbative physics of the hadronization process, from the various short distance dynamics of the large energy particle collision. The short distance physics can then be calculated in perturbation theory where the prediction is limited by the expansion in αs(Q) 1. The hadronization process describes the long distance evolution of quarks and gluons into hadrons such as π’s and K’s. This process is model dependent due to the non-perturbative QCD effects. The processes we will be considering have the form ee¯ → V → X for some intermediate vector boson V and final state X. A factorized rate will take the form

2 2 dσ(ee¯ → V (Q ) → X) = dˆσ(ee¯ → V (Q ) → Xˆ) × Shad(Xˆ → X), (1.1) where Q ΛQCD is the large centre-of-mass energy, dˆσ is the perturbative rate, and the hadronization process is described by Shad. The perturbative rate describes the production of the partonic structure of the final state Xˆ from the initial hard interaction. The hadronization process models how the actual final state X is produced from the partonic structure calculated ˆ 2 by the perturbative rate. The partonic final state X = qq¯ + qqg¯ + O(αs) includes all possible partons that can make the final state X after hadronization. The separation on the right side of (1.1) is due to the different time scales associated with hadronization, which occurs at

O(1/ΛQCD), the hard interaction, which occurs at O(1/Q), and even longer time scales of the subsequent gluon radiation, being well separated. Figure 1.1 shows an illustration of the process being described and what types of dynamics each of the functions above describe. Factorization allows us to consider the perturbative rate separately and ignore the effects of hadronization until we are ready to compare to experiment. Although the perturbative rate dˆσ is simpler to understand than the hadronization process, calculating the perturbative rate is complicated by the existence of further scales in the final state. We will concentrate on calculating jet-like final states that have two or three scales, typically associated with the mass of the jet. While all these scales are well above ΛQCD, their existence leads to logarithms of the ratio of these scales. These logarithmic enhancements ruin the perturbative expansion in αs(Q) 1 when the ratio of scales is large, limiting the precision of theoretical predictions. Perturbative QCD methods exist to sum these logarithmic enhancements (see for example [3–7]); however, they rely on a detailed understanding of the Chapter 1. Introduction 3

e¯ dˆσ

Shad

Figure 1.1: Schematic drawing of an ee¯ → V → X collision. The parts described by the perturbative rate and the hadronization process are shown by the large boxes. The shaded blobs represent the hadrons produced in this process. graphs used in the calculation and can be diﬃcult to extend to higher orders in the ratio of scales. Eﬀective Field Theory (EFT) techniques provide another framework for summing logarithms that is often simpler. In this thesis we use EFT techniques to calculate jet rates. We will introduce a new formulation of the EFT commonly used for calculating jets that makes factorization explicit. We then show an example of how this formulation can be applied to factorization at higher orders and begin the process of an improved measurement of αs(MZ ). We also demonstrate a calculation for a jet process where factorization in the EFT calculation appears to require the introduction of a new regulator.

1.1 Hadronic Jet Physics

The hadronization and showering processes will smear the high energy partons produced in the hard scattering into a spread of low mass, collimated hadrons in the detector. This can make it difficult to determine the seed of a particular cluster of hadrons. However, studying these collimated bunches of hadrons, called jets, can give us information about their parent particles and the processes that produced them. This can provide a useful test of QCD, but are also important to understand as a background to many interesting processes since any interaction with the strong force will be accompanied by jets in the detector. In order to make concrete calculations, a jet must have a specific definition. Different definitions will obviously divide the final state into different looking jets. We will distinguish between jet algorithms and jet shapes in this thesis. A jet algorithm is a procedure to combine multiple partons or hadrons in the final state into a specific number of jets. Examples of jet algorithms are the kT and anti-kT algorithms [8], with the latter being the default jet algorithm at the LHC and the former explored in Chapter 6. A jet shape instead returns a Chapter 1. Introduction 4

(a) Virtual (b) Real

Figure 1.2: The real and virtual diagrams for the αs contribution of V → hadrons. continuous parameter that describes the configuration of final state hadrons or partons. We will be concerned with jet shapes that have a kinematic region where the hadrons in the detector are collimated. Thrust [9] is an example of a jet shape that will be explored in Chapter 4 and used as motivation in Chapter 5. The requirements for a proper jet definition are outlined in [10]. Of particular importance for theoretical calculations is the need for the jet definition to be free of infrared (IR) and collinear divergences at all orders in perturbation theory. This is an obvious condition that makes the rate calculable and thereby predictable. We can understand its significance by analyzing the one-loop contribution to a jet rate from a vector boson decay. The one-loop diagrams are shown in Figure 1.2. The virtual diagram in Figure 1.2a can be written as Z αs dθ dE − , (1.2) 4π sin θ E where the energy of the internal gluon is E and θ is the emitted angle from the quark. The above has an IR or soft divergence when E → 0 and a collinear divergence when the angle θ → 0, π. Both divergences are due to the massless fermions going on-shell. These divergences must be cancelled by corresponding divergences from the contribution of the real emissions. The real emission diagrams for qqg¯ production are shown in Figure 1.2b. Squaring and integrating over phase space gives a contribution in the limit of a low energy gluon of Z αs dθ dE , (1.3) 4π J sin θ E where J is the restriction from the specific jet definition being used. Comparing (1.2) and (1.3) we see the IR and collinear divergences in the virtual diagram will be cancelled by the real diagram so long as the jet definition is insensitive to arbitrarily soft and collinear emission. The virtual diagrams at each subsequent order in αs have the same IR and collinear divergences as (1.2). These divergences must also be cancelled by a corresponding divergence in the real gluon emission graphs at the appropriate order in αs. Therefore, in order to make theoretical calculations, jet definitions can only restrict the upper limits of E and θ. Such jet definitions are called IR and collinear safe. The phase space restrictions also gives rise to a logarithmic enhancement in the perturbative Chapter 1. Introduction 5 rate. One of the simplest phase space restrictions that can be made is to limit the invariant mass of the quark-gluon pair E sin θ to be less than some mass scale M ∗. This restriction on 2 the external gluon momentum leads to an αs ln (M/Q) term in the O(αs) rate, where Q is the total energy of the system. The form of this logarithm is a general feature of all jet definitions.

The cancellation of the IR and collinear divergences in the M → 0 limit to all orders in αs means the perturbative rate will have the form X X n m M M dˆσ = dˆσnmα ln + O , (1.4) s Q Q n m≤2n where dˆσnm are O(1) constants. This is called the Sudakov or double logarithmic enhancement. In general, the kinematic region that leads to boosted jets is when M Q. The logarithmic enhancement in (1.4) will ruin the perturbative expansion in αs(Q) 1 in the particular limit where M is small enough that αs(Q) ln(M/Q) ∼ O(1). The series is instead an expansion in the large logarithms, which naively does not converge and leads to large theoretical errors. Understanding how to sum these large logarithms is important for making accurate theoretical predictions. In the next section we will summarize how EFT techniques approach this summation.

1.2 Factorization From Eﬀective Field Theory

The EFT approach to summing the logarithmic enhancements in (1.4) is to factorize the perturbative rate into pieces that each depend on a single scale. EFTs are constructed to do this automatically by expanding in the ratio of the scales of interest, similar to a multipole expansion. This disentangles the physics at each of the scales and allows the rate to be written in terms of operators that only depend on a single scale. The usual Renormalization Group Equations (RGE) can then sum the logarithmic enhancements as desired. The calculation can be made arbitrarily accurate by going to higher loop orders and/or higher orders in the ratio of scales. Both of these eﬀects can be included systematically in the EFT technique. To understand how EFTs can factorize a rate, we consider a process involving two scales Q and M such that there is a hierarchy Q M. The factorized rate will be written in terms of two pieces, one of which depends on Q and the other on M. The rate requires calculating the square matrix element

X Z dˆσ(I(Q2) → F ) ∼ d4x e−iQxhI|J †(x)|F ihF |J(0)|Ii, (1.5) F where I and F are the initial and ﬁnal states respectively, Q is the initial energy, and J is a current in the full theory. For totally inclusive processes such as B → Xsγ, there are no restrictions on the sum over ﬁnal states as the current enforces the b → s transition. The

∗ This deﬁnition serves only as an example and violates the criteria of [10] beyond αs. Chapter 1. Introduction 6 matrix element squared can be related through unitarity arguments to the matrix element of the imaginary part of the time-ordered product Z 4 −iQx † Tˆincl = −i d x e T {J (x),J(0)}. (1.6)

This is the well-known optical theorem and is due to forward scattering amplitudes developing a branch cut when intermediates states go on-shell. Alternatively, jets are semi-inclusive processes that sum over all possible particles in the final state but have restrictions on their allowed momentum that cannot be enforced by the current. This means the sum over final states in (1.5) is only a partial sum and the optical theorem is no longer valid. For these processes, projectors are introduced to remove the restrictions on the sum by only allowing final states that have the correct momentum configuration. The rate is then related to the imaginary part of the time-ordered product Z 4 −iQx † Tˆsemi-incl = −i d x e TJ (x)McJ J(0), (1.7)

where McJ is the projector for a specific jet definition J . Here the on-shell states that develop branch cuts in the forward scattering amplitudes will have phase-space restrictions from the projectors. The degrees of freedom in the effective theory will not be able to probe all the components of the small vertex displacement xµ ∼ 1/Q in (1.6) and (1.7). The operator Tˆ, which is non-local in all components, is replaced with operators that are local in the appropriate components of xµ for the appropriate degrees of freedom. Therefore, the full theory calculation is matched onto (semi-)local effective theory operators Oi by

1 1 Tˆ → C0(Q)O0(M) + C1(Q)O1(M) + C2(Q)O2(M) + ..., (1.8) Q Q2 where Ci(Q) are the matching coefficients. The subscript denotes the suppression in Q since the EFT includes higher dimensional operators due to insertions of derivatives and extra fields. This is known as the operator product expansion for inclusive processes. For semi-inclusive processes, the projector McJ must also be matched onto the EFT degrees of freedom. These latter will be more thoroughly examined in Chapter 4. The right-hand side of (1.8) is the desired factorized form as implied by the arguments of the matching coefficients and operators. The effective operators are constructed to reproduce all the IR physics of the full theory around the scale M so only depend on this scale. The ultraviolet (UV) physics above the scale Q is reproduced by the matching coefficients, which are the coupling constants of the effective theory. Factorizing the rate into pieces that each depend on a single scale splits the large ln(M/Q) in (1.4) into ln(M/µ)’s and ln(Q/µ)’s, where µ is a renormalization scale. The effective operators will give the M dependent logarithms and the matching coefficients will give the Q dependent logarithms. Each of these logarithms can Chapter 1. Introduction 7 be minimized for the appropriate choice of µ and the RGE is then used to run between the different scales. Running between the scales will sum the logarithmic enhancements in (1.4). The advantage of using EFTs comes from automating the process of splitting the logarithms and summing them. Subleading logarithms can be systematically included by calculating higher loop effects in the RGE. Systematic improvements in M/Q can be included by using the higher dimensional operators in (1.8) and any logarithmic enhancements to these subleading effects can also be summed using the RGE. When there are more than two scales in the process, a sequence of matching at each scale onto a new EFT and running to the next scale is required. Two examples of multiple scales are B → Xsγ when the photon’s energy is close to B meson mass, and jet rates. In these cases there are three correlated scales that we must run between. The correlation of the scales makes factorization more difficult than the usual two-scale factorization described above; however, the effective theory approach of matching and running is the same.

1.3 Organization of the Thesis

The rest of the thesis will be organized in the following way. In Chapter 2, we give a more detailed review of EFTs. We also introduce heavy quark effective theory (HQET), the EFT relevant for inclusive B decays, and the traditional formulation of soft-collinear effective theory (SCET), the EFT relevant for jet observables. In Chapter 3, we derive a new way of describing SCET as QCD fields coupled to Wilson lines. In Chapter 4, we use this formulation to derive a factorization theorem at subleading orders for the thrust jet shape. In Chapter 5, we renormalize the next-to-leading order operators necessary for the subleading thrust factorization theorem with the purpose of summing the large logarithms. In Chapter 6, we examine the k⊥ and C/A jet algorithms and discuss how a rapidity regulator restores the separation of soft and collinear graphs in SCET at one-loop. We conclude the thesis in Chapter 7 and motivate future work to be examined. We also note the following regarding notation and repeated information within this thesis. The names and notation used for some operators in Chapters 3 & 4 were changed in Chapter 5; although, within each chapter the notation is self-consistent. As well, a summary of Chapter 3 is given in the second section of Chapters 4 & 5, making each chapter in this thesis self-contained. Chapter 2

Eﬀective Theories of QCD

Effective field theories (EFTs) are used for calculating processes with multiple scales by only describing the dynamics relevant below each scale. Large momentum transfers from massive or highly off-shell particles above each scale cannot be resolved by low energy particles and only occur through loops. The EFT is constructed by removing these higher energy degrees of freedom from the description of the dynamics of the process. Only the relevant low energy degrees of freedom remain in the theory. Using this principle, EFTs aim to provide a framework to capture all the relevant low energy dynamics, while also providing a technique for increasing the precision of predictions. A process that involves a hard scale Q and a soft scale M has an expansion in M/Q when these scales are widely separated. Above the scale Q, the dynamical degrees of freedom include “heavy” fields with characteristic momentum p2 > Q2. The heavy fields may be particles with masses greater than Q, such as Four-Fermi Theory, or may be off-shell particles with large virtuality p2 ∼ Q2, such as the EFTs introduced in Sections 2.2 and 2.3. Below the scale Q, there are only “light” degrees of freedom that have a characteristic momentum p2 Q2. The heavy fields can only be produced in loops and are removed from the low energy theory. The

Lagrangian for the full theory LH , which describes the interactions of all the fields, is matched onto the EFT described by X On(φL) LH −→ L(φL) = Cn . (2.1) Qn−4 n>0 where φL are the light fields. The EFT operators On depend only on the light fields and are characterized by their mass dimension n. The factor of Q is introduced to make the matching coefficients Cn dimensionless. The strategy for writing the EFT Lagrangian is to write all possible operators involving the light fields that have the correct quantum numbers and symmetries. These effective operators will reproduce all the IR physics of the full theory, including the M dependence. The matching coefficients Cn captures our ignorance of the physics above the scale Q, which acts as an UV

8 Chapter 2. Effective Theories of QCD 9

n − 4 Name: > 0 irrelevant, non-renormalizable = 0 marginal, renormalizable < 0 relevant, super-renormalizable

Table 2.1: Three cases of operators. cut-off on the effective theory. If the full theory is known, the matching coefficients can be found by subtracting matrix elements in the full and effective theory. By construction, the subtraction will be the difference between the UV of the theories since both theories have the same IR. The sum in (2.1) makes it appear there are an infinite number of operators in the effective theory to be accounted for. This would make it impractical for calculational purposes. However, we can ignore most operators when working to a particular order in the M/Q expansion. There are three cases of effective operators as described in Table 2.1. The irrelevant operators are so named because their contribution falls as M/Q or faster based on dimensional analysis. The effective operators depend only on the low energy degrees of freedom so all amputated Green’s functions will have dimension M n−4. The 1/Qn−4 in front of the operator in (2.1) ensures the contribution from an irrelevant operator is (M/Q)n−4 1 with n > 4. These operators can typically be ignored unless greater accuracy is required or they are the leading operator in the expansion. The marginal operators are conformally invariant in the classical theory as they have no scale dependence in d = 4 dimensions. This conformal invariance is broken by quantum corrections which introduces a new scale to the theory and gives the coupling constant a dependence on energy. The coefficient will have a Landau pole and the coupling becomes non-perturbative. This occurs either in the UV or IR, depending on the details of renormalization. However, the energy dependence is logarithmic, meaning the theory is perturbative so long as the energy scale is reasonably far away from the non-perturbative scale. The final case of operators are the relevant operators. Relevant operators can cause problems in an effective theory because they have a positive dependence in the large scale Q in the EFT 2 2 Lagrangian. An example is a scalar mass term that would enter as Q C2φL with C2 ∼ O(1). Such a field has a mass above the cut-off of our theory and should be integrated out unless C2 is finely tuned to be small [11]. Even if M 2 was used in the Lagrangian, quantum corrections 2 2 would bring M C2 → Q C2 because Q is the cut-off scale of the EFT. EFTs with relevant operators are said to suffer from a naturalness problem unless extra symmetries in the C2 → 0 limit protect the matching coefficient from becoming too large. A classic example of an effective theory is Four-Fermi Theory, which is the low energy theory of weak interactions. Suppose we are interested in non-leptonic b → cud¯ decay. The lowest order diagram in the Standard Model is mediated by a W boson and is shown in Figure 2.1a.

The W boson has a large mass MW compared to the b quark mass mb, so the scales can be Chapter 2. Effective Theories of QCD 10

d u W b u b

c c (a) Full Theory (b) Eﬀective Theory

Figure 2.1: Diagrams for non-leptonic b → c transition in the Standard Model and Four-Fermi Theory. The square represents an eﬀective operator insertion.

ordered as Q ∼ MW mb ∼ M & mc. The amplitude for this process is

ig 2 −ig √2 ∗ µ µν ¯ ν VcbVud (¯cLγ bL) 2 2 dLγ uL (2.2) 2 q − MW

1 where g2 is the weak coupling constant, Vij are CKM matrix elements, and ψL = 2 (1 − γ5)ψ. The momentum qµ is the momentum transfer between the b and c quarks. The full theory decay has a non-local interaction due to the W boson propagator. The non-locality of the propagator 2 is a distance of order 1/MW , which is beyond the 1/mb resolution of the low mass ﬁnal states. 2 2 2 2 By expanding the propagator in (2.2) in powers of q /MW ∼ O(mb /MW ), these interactions are replaced by local operators in the eﬀective theory as seen in Figure 2.1b. The lowest order operator is the six-dimensional operator

∗ µ ¯ O6I = VcbVud (¯cLγ bL) dLγµuL , (2.3) where we have left the CKM matrix elements in for convenience. The tree-level matching coefficient can be found by subtracting the amplitude in (2.2) from (2.3) and gives C = √ √ 6I 2 2 2 2 −i2 2GF MW + O(αs). Fermi’s constant is defined as GF = g2/(4 2MW ) and the MW is needed due to the convention in (2.1). This an irrelevant operator, but is the lowest dimensional operator that can describe non-leptonic b → c decay. Calculating one loop QCD corrections in both the Standard Model and the effective theory will give the O(αs) corrections to the matching coefficient. Keeping with the philosophy that we must write all possible operators that respect the symmetry of the theory, there is another possible six-dimensional operator [12]

∗ a µ ¯ a O6II = VcbVud (¯cLT γ bL) dLT γµuL , (2.4)

a where T is an SU(3) generator. This operator mixes with O6I under renormalization. Again, subtracting the amplitude (2.2) from the contribution of this operator gives the matching co- Chapter 2. Effective Theories of QCD 11

eﬃcient C6II = 0 + O(αs). We can also ﬁnd a higher dimensional operator by expanding the 2 2 amplitude in (2.2) to O(q /MW ), which gives

∗ a µ 2 ¯ a O8 = VcbVud (¯cLT γ bL)(iD) dLT γµuL (2.5) √ 2 2 2 with matching coefficient C8 = i2 2GF MW +O(αs). This operator is suppressed by O(mb /MW ) compared to the leading order operator in (2.3) due to the derivative D insertions. There will be many other operators at this order that can be written down; however, these operators are 4 4 only required when accuracy of O(mb /MW ) is necessary. In the next sections we will begin with a brief review of QCD and then review two effective theories of QCD relevant for this thesis: heavy-quark effective theory (HQET), and soft-collinear effective theory (SCET).

2.1 Review of Quantum Chromodynamics

The strong sector of the Standard Model is described by Quantum Chromodynamics (QCD). QCD is a gauge theory with an SU(3) symmetry that couples to quarks in the fundamental representation. The Lagrangian for QCD is

X 1 µν LQCD = ψ¯(i /D − m)ψ − Tr (G Gµν) (2.6) 2 flavour where ψ is a Dirac spinor and m is the mass of the field. The covariant derivative is Dµ = µ µ 2 µ aµ a ∂ − igA and αs = g /(4π) is the coupling constant of QCD. The gluon field is A = A T where the repeated colour indices a = 1,..., 8 are summed and T a are the generators of SU(3). The field strength is denoted by Gµν = (i/g)[Dµ,Dν] and the trace in (2.6) is over colours. µνκλ We have neglected the CP violating term GµνGκλ and gauge fixing terms as they are unimportant to this thesis. As described in Chapter 1, QCD exhibits asymptotic freedom, whereby the coupling constant grows rapidly in the IR limit. This can be seen from the one loop running of the coupling

αs(Q) αs(µ) = (2.7) β0αs(Q) 1 − 2π ln(Q/µ) where β0 = 11CA/3 − 2nf /3 and nf is the number of active ﬂavours of quarks. The coupling −2π/(αs(Q)β ) constant has a Landau pole at µ = ΛQCD = Qe 0 ≈ 250 MeV in the MS renormalization scheme, where (2.7) diverges. QCD becomes strongly coupled before this scale. The energy dependence of the coupling at high energies away from the Landau pole has been con- ﬁrmed experimentally as shown in the plot of Figure 2.2. In this thesis, we will be concerned with energy scales well above ΛQCD, where the coupling constant is small and a perturbative expansion is valid. Chapter 2. Effective Theories of QCD 12

0.5 April 2012

α 3 s(Q) τ decays (N LO) Lattice QCD (NNLO) 0.4 DIS jets (NLO) Heavy Quarkonia (NLO) – e+e jets & shapes (res. NNLO) Z pole fit (N3LO) pp –> jets (NLO) 0.3

0.2

0.1

QCD α s (Μ Z ) = 0.1184 ± 0.0007 1 10 100 Q [GeV]

Figure 2.2: Values of αs at diﬀerent energy scales Q. The shaded line represents the prediction from perturbative QCD. Plot taken from [13].

2.2 Heavy Quark Eﬀective Theory

HQET is an effective theory that describes the interaction of a single heavy quark such as a bottom or charm quark with light gluons and quarks. Unlike in Four-Fermi theory, where the W boson was completely removed from the effective theory, HQET does not remove the heavy quark entirely. Instead, the trajectory of the heavy quark is fixed by removing large momentum transfers. The theory has been used extensively in describing B meson decays and interactions [14]. We give a brief review of its derivation here.

In the limit where the mass of the heavy quark mQ → ∞ the heavy quark behaves as a static time-like colour source. The heavy quark’s momentum can be parameterized as

µ µ µ pQ = mQv + k , (2.8)

µ µ where v is the fixed trajectory. The residual momentum k mQ describes small perturba- tions away from this trajectory, and vµ will act as a label distinguishing different heavy quark fields. The label is conserved in all interactions since no light field can give a large momentum change to the heavy quark.

The large component of momentum is removed from the heavy quark ﬁeld by writing the spinor as X −imQv·x X −imQv·x Q(x) = e Qv(x) = e (hv(x) + Hv(x)) (2.9) v v Chapter 2. Effective Theories of QCD 13

where the two ﬁelds are deﬁned by the projectors P± = (1 ± /v)/2

hv(x) = P+Qv(x) Hv(x) = P−Qv(x). (2.10)

The decomposition means derivatives acting on the hv and Hv fields bring down momentum of O(k) only. Substituting the quark field in (2.9) into the QCD Lagrangian in (2.6) removes the mass term for the hv field, while the Hv field has a mass of 2mQ. Therefore, the Hv field is above the cut-off of our effective theory mQ and can be removed using the equations of motion

1 Hv(x) = i /Dhv(x). (2.11) iv · D + 2mQ

This gives the leading order Lagrangian for the heavy quark

LHQET = h¯viv · Dhv + O(1/mQ), (2.12) where the higher order terms can be found by expanding (2.11) further. As expected, the labels are conserved and cannot be changed by any interaction above. The leading order Lagrangian also has a spin-flavour symmetry due to the absence of spin matrices and references to the specific flavour of quark being described [12, 14]. This was expected due to the heavy quark being a static colour source.

The leading order Feynman rules can also be obtained by expanding the QCD Feynman rules for a heavy quark propagator, and gluon emission from a heavy quark. The QCD quark propagator can be expanded in mQ to give

/pQ + mQ i i 2 2 = P+ + O(1/mQ), (2.13) pQ − mQ v · k which is the propagator for a heavy quark in (2.12). Similarly, we can consider the Feynman rule for the emission of a soft gluon from a heavy quark ﬁeld. Each heavy quark ﬁeld is accompanied µ with a P+ projector due to the above propagator, so the usual γ vertex becomes

µ µ µ µ P+γ P+ = P+(P−γ + v ) → v , (2.14) where we have absorbed the final P+ into the definition of the heavy quark field. This is of course also the same interaction term we found in (2.12).

Subleading interactions can be found by either expanding the Feynman rules further or substituting the 1/mQ corrections to the equation of motion in (2.11) into the QCD Lagrangian. Using the latter approach, the subleading terms are

2 αβ (1) ¯ D⊥ ¯ gσαβG LHQET = −hv hv − hv hv, (2.15) 2mQ 4mQ Chapter 2. Effective Theories of QCD 14

where σαβ = i[γα, γβ]/2. The superscript represents the suppression by 1/mQ. The second term explicitly breaks heavy quark spin symmetry. Although only the tree-level coupling constants can be derived using the equation of motion, the coupling constant for the first term is correct to all orders in αs due to reparametrization invariance [15]. Reparametrization invariance will be discussed in Chapter 5 in the context of jet production. The coefficient for the second term differs beyond tree-level, which can be accounted for by including a matching coefficient a(µ) = 1 + O(αs) and calculating loop corrections in both QCD and HQET.

2.3 Soft-Collinear Eﬀective Theory

Soft-collinear eﬀective theory describes the interactions between highly boosted low invariant mass collimated jets of particles. In this section we will introduce the traditional formulation of SCET [16–20]∗. We will re-derive SCET in an alternative way in Chapter 3. We also give two examples of currents that are required to describe processes where SCET is useful. The approach to deriving the SCET Lagrangian is similar to derivation of the HQET La- grangian. Because all the particles being described are massless, no particles are integrated out of the theory, just as no particles were removed in HQET. Instead, the trajectory of the total sum of the collimated jets of particles is ﬁxed, although the trajectory of an individual particle within a jet can be changed by another particle in the jet. It is convenient to introduce light-cone coordinates in order to describe the momentum of these particles. We introduce two light-like vectors nµ = (1, nˆ)n ¯µ = (1, −nˆ) (2.16) wheren ˆ is a unit three-vector. The two vectors have the properties that n2 = 0 =n ¯2 and n · n¯ = 2. Any four-vector can be decomposed into these coordinates as

n¯µ nµ n¯µ nµ pµ = p · n + p · n¯ + pµ ≡ p+ + p− + pµ ≡ (p+, p−, ~p ). (2.17) 2 2 ⊥ 2 2 ⊥ ⊥

Energetic particles with small invariant masses are boosted in the centre-of-mass frame and will have their momentum dominated by one component. In terms of the notation above, the momentum of such a particle scales as

µ 2 pn ∼ Q(λ , 1, λ). (2.18)

As usual, Q denotes the large energy of the total system, and the subscript n refers to the particle travelling in the nµ direction. The small parameter λ 1 is the expansion parameter of the eﬀective theory. A particle that has momentum scaling as in (2.18) is called an n- collinear particle and has a virtuality of O(λ2Q2). The collinear particles alone are not enough to reproduce the IR of QCD [16]. Soft degrees of freedom that communicate between diﬀerent

∗We call this formulation “label SCET” in Chapter 3 but refer to it as the traditional formulation in subsequent chapters. Chapter 2. Effective Theories of QCD 15 collinear sectors are also present and have momentum scaling

µ 2 2 2 ps ∼ Q(λ , λ , λ ). (2.19)

These particles have a virtuality of O(λ4Q2), which is much smaller than that of the collinear particles. Often these particles are called ultra-soft to distinguish them from particles scaling as Q(λ, λ, λ), which can arise in certain processes. In this thesis, unless it is otherwise stated, we will mean soft to refer to ultra-soft particles.

A collinear particle is treated similar to a heavy quark in the previous section. The momentum is similarly parameterized by µ µ µ pn =p ˜ + k (2.20) where the residual momentum kµ ∼ λ2Q and the label momentum

nµ p˜µ =p ˜− +p ˜µ ∼ Q(1, 0, λ) (2.21) 2 ⊥ contains the large components of the momentum. Labels will be conserved separately in SCET just as the velocity label in HQET was conserved. To derive the collinear quark Lagrangian

Lξξ, the large label is removed from the QCD spinor and the ﬁeld is decomposed into

X −ip˜·x X −ip˜·x ψ(x) = e ψn,p˜(x) + qs(x) = e (ξn,p˜(x) + ζn,p˜(x)) + qs(x). (2.22) p˜ p˜

The two n-collinear ﬁelds ξn,p˜ and ζn,p˜ are deﬁned using the projectors Pn = (/n/n¯)/4 and

Pn¯ = (n/n/¯ )/4 as

ξn,p˜(x) = Pnψn,p˜(x) ζn,p˜(x) = Pn¯ψn,p˜(x) (2.23) such that /nξn,p˜ = 0 = nζ/¯ n,p˜. The soft quark field is included in the decomposition but is subleading [18,19]. Unlike in HQET, where the two-component hv spinor includes only creation operators, both ξn and ζn spinors have creation and annihilation operators. There are also two types of gluons: one for collinear and one for soft. We also remove the large label momentum from the collinear field and write a QCD gluon field as

µ X −iq˜·x µ µ A (x) = e An,q˜ + As . (2.24) q˜

This is unlike HQET where there was only one type of gluon ﬁeld. However, similar to HQET, 2 derivatives acting on the ξn, ζn, An, and soft ﬁelds now pull down momentum of O(λ ).

The collinear quark Lagrangian is found by substituting the ﬁelds in (2.22) and (2.24) into the massless quark Lagrangian of QCD

X −i(˜p0−p˜)·x ψi¯ /Dψ = e ξ¯n,p˜0 (in · D)ξn,p˜ + ζ¯n,p˜0 (¯n · p˜ + in¯ · D))ζn,p˜ + mixed terms. (2.25) {p˜} Chapter 2. Effective Theories of QCD 16

µ µ P iq˜·x µ µ The covariant derivative iD = i∂ + g q˜ e An,q˜ + gAs also includes a label that we have suppressed and the sum is over all label momentum. For a fixed label momentum, the ξn,p˜ fields are massless, whereas the ζn,p˜ fields have massn ¯ · p˜, which is of order the cut-off Q. This is similar to the massive Hv field in HQET. Therefore, we use the equation of motion to remove the ζn,p˜ field from our theory. The equation of motion for these heavy fields are

1 n/¯ ζn,p˜(x) = (P/ + /D ) ξn,p˜(x), (2.26) n¯ · (P + iD) ⊥ ⊥ 2 where we have introduced the “label operator” Pµ. The label operator acts on any collinear

ﬁeld φn,p˜ to pull down the label momentum

µ µ P φn,p˜ =p ˜ φn,p˜. (2.27)

Label momentum conservation is also implicitly understood in (2.26).

The equation of motion is inhomogeneous in λ scaling because the covariant derivative includes partial derivatives, which pull down residual momentum, and contains soft gluons, which are both suppressed compared to the label operator and collinear gluon. We can deﬁne µ a homogeneous collinear derivative Dn that has components

in¯ · Dn =n ¯ ·P + gn¯ · An,q˜ µ µ µ iDn⊥ = P⊥ + gAn,q˜⊥ (2.28)

in · Dn = in · ∂ + gn · An,q˜.

The partial derivative in the third line is because n ·P = 0 from the deﬁnition of the label momentum in (2.21). The equation of motion in (2.26) can be expanded in λ using the collinear derivative and the O(λ0) collinear quark Lagrangian becomes (0) X −iP·x ¯ 1 n/¯ Lξξ = e ξn,p˜ in · D + i /Dn⊥ i /Dn⊥ ξn,p˜. (2.29) in¯ · Dn 2 {p˜}

The Feynman rules of the theory are given in Figure 2.3. All the momentum transfers of O(Q2) have been removed from the eﬀective theory. Only the total label momentum is conserved, not the label of individual collinear particles, due to the sum over labels in (2.29). This is seen by the third diagram in Figure 2.3. The full momentum is conserved at each vertex in this theory due to the seperate conservation of label and residual momentum.

The 1/(in¯ · Dn) in the SCET Lagrangian couples a collinear quark to an arbitrary number of collinear gluons when the Lagrangian is expanded in g, as seen by the fourth diagram of Figure 2.3. We can re-write this term using Wilson lines to make this explicit. The deﬁnition Chapter 2. Effective Theories of QCD 17

p n/¯ n¯ · p˜ = i 2 2 (n · p)(¯n · p˜) − p˜⊥

µ, a k n/¯ = igT anµ p p0 2

µ, a q ⊥ 0 ⊥ 0 ! a γµ p/˜⊥ p/˜⊥γµ p/˜⊥p/˜⊥ n/¯ = igT nµ + + − n¯µ p p0 n¯ · p˜ n¯ · p˜0 n¯ · p˜ n¯ · p˜0 2

ig2T aT b γµ p/˜ p/˜0 γν p/˜0 p/˜0 n/¯ , a,b µ, aν, µ ν ⊥ ⊥ ν ⊥ ⊥ µ ⊥ ⊥ µ ν = γ⊥γ⊥ − n¯ − 0 n¯ + 0 n¯ n¯ q n¯ · (˜p − q˜) n¯ · p˜ n¯ · p˜ (¯n · p˜)(¯n · p˜ ) 2 2 b a ν 0 µ 0 0 0 ig T T γ p/˜⊥ p/˜ γ p/˜ p/˜ n/¯ p p + γν γµ − ⊥ n¯µ − ⊥ ⊥ n¯ν + ⊥ ⊥ n¯νn¯µ n¯ · (˜q +p ˜0) ⊥ ⊥ n¯ · p˜ n¯ · p˜0 (¯n · p˜)(¯n · p˜0) 2

(0) 2 Figure 2.3: Feynman rules for Lξξ in (2.29) up to O(g ). Collinear quarks are denoted by dotted lines and collinear gluons are denoted by springs with lines.

of a Wilson line is " # X 1 W = exp −g n¯ · A n ·P n perm n¯ ∞ m am a1 X X (−g) n¯ · An,q˜ ··· n¯ · An,q˜ = m 1 (2.30) m! (¯n · q˜ ) ··· (¯n · (˜q + ··· +q ˜ ) m=0 perm 1 1 m where we are summing over permutations and the label operator only acts within the square brackets. In position space the Wilson line is deﬁned as†

Z x Wn(x) = P exp ig dsn¯ · An(¯ns) (2.31) −∞ where P defines the path-ordering. A Wilson line represents a colour source travelling along a µ semi-infinite trajectory in then ¯ direction. The equation of motion in¯ · DnWn = 0 leads to the simplification † f(in¯ · Dn) = Wnf(¯n ·P)Wn. (2.32)

†The definition here is slightly different than the definition given in Chapter 3 and beyond; however, it is the definition used most often in the traditional formulation of SCET, so we use it here. Chapter 2. Effective Theories of QCD 18

We can use this identity to simplify the Lagrangian to (0) X −iP·x 1 † n/¯ L = e ξ¯n,p˜ in · D + i /D Wn W i /D ξn,p˜. (2.33) ξξ n⊥ n¯ ·P n n⊥ 2 {p˜}

The Lagrangian is non-local due to the Wilson lines and 1/(¯n ·P), which is a total shift. The Lagrangian still couples to an arbitrariy number of gluons, but this is now made explicit by the existence of the Wilson lines.

The Wilson line in the Lagrangian arise from integrating out off-shell propagators in the n¯µ direction. This may appear odd because we have not explicitly introduced any fields in the n¯ direction. However, it is due to decomposing the QCD quark field into ξn and ζn fields. A boost in the nµ direction transforms a Dirac spinor to

η − η ψn(x) → e 2 ξn + e 2 ζn, (2.34) where η is the pseudorapidity, which is large and enhances the ξn degrees of freedom. The µ ζn components of the field become the colour source in then ¯ direction for the ξn fields and are described by a Wilson line. Absorbing the labels back into the fields in (2.33) and taking µ µ Dn → D gives back the full QCD Lagrangian in light-cone quantization [21]. Obviously, the expansion in (2.34) and thus (2.22) is frame-dependent since in the boosted frame, the degrees of freedom of the n-collinear quarks are homogeneous. This is the motivation of Chapter 3.

There is still another simpliﬁcation we can do to the Lagrangian in (2.33). The in · D term in the Lagrangian contains a soft gluon, which couples as shown in the second diagram of Figure 2.3. This mixing of soft and collinear degrees of freedom can be removed explicitly from the Lagrangian by introducing a soft Wilson line

Z x Yn(x) = P exp ig dsn · As(ns) , (2.35) −∞ which represents a colour source travelling along nµ. The collinear ﬁelds can then be redeﬁned as µ µ † ξn,p˜ → Ynξn,p˜ An,q˜ → YnAn,p˜Yn . (2.36)

Using this redeﬁnition, the soft gluon can be removed from the Lagrangian using the equation of motion (in · ∂ + gn · As)Yn = 0. The leading order SCET collinear quark Lagrangian then becomes (0) 1 † n/¯ L = ξ¯n in · Dn + i /D Wn W i /D ξn. (2.37) ξξ n⊥ n¯ ·P n n⊥ 2 We have removed the sum over labels and instead impose label conservation implicitly at each vertex. The form of the Lagrangian above is explicitly factorized because it only depends on collinear degrees of freedom. Chapter 2. Effective Theories of QCD 19

The full SCET Lagrangian in this formulation is

X (0) (0) (0) LSCET = Ls + Lξξ + Lcg + Lgf + O(λ) (2.38) collinear where the sum is over all collinear sectors needed to describe the process being considered. The Lagrangian for the collinear gluon can be derived in a similar manner as the collinear quark

(0) 1 µν L = − Tr G Gnµν (2.39) cg 2 n µν µ µ where Gn = (i/g)[Dn,Dn] only depends on collinear ﬁelds. There is no expansion in the soft sector, so soft ﬁelds couple as massless QCD

1 µν Ls =q ¯si /Dqs − Tr G Gsµν, (2.40) 2 s where the derivative only involves soft ﬁelds. We do not write the gauge-ﬁxing Lagrangian

Lgf involving ghost terms for the sake of brevity. The soft Lagrangian is exact to all orders in λ, whereas both the collinear quark and gluon Lagrangian have subleading terms that can be obtained similar to HQET. However, the subleading terms in the collinear Lagrangian include terms that explicitly couple collinear and soft degrees of freedom, even after the field redefinition of (2.36). The explicit coupling between sectors will be further discussed in Chapter 3. The decoupling of the soft and collinear sectors means soft and collinear graphs can be calculated separately. However, the sum over labels in the collinear Lagrangian leads to a complication in calculating loop diagrams. For example, the integral for an internal particle of momentumq ˜ + k will have the form Z X 4 d k In,q˜(k), (2.41) q˜6=0 where we have explicitly specified that the label momentum can never be small. The sum is often impractical for calculations and can be replaced by a “binning” procedure that subtracts a piece when the label momentum is small. This procedure has the form Z Z Z Z Z X 4 4 4 4 ˜ 4 d k In,q˜(k) → d q In(q =q ˜ + k) − d k In(k) = d q In − d k Ino/, (2.42) q˜6=0 where the Iñ is called the “naive” integral and the second integral is called the “zero-bin” [22]. The combination of the naive and zero-bin integrals properly reproduces the IR of QCD.

2.3.1 Examples of SCET Currents

Two examples of processes where SCET is useful are B → Xsγ in the shape function region and dijet production, which are both shown at tree-level in Figure 2.4. The shape function region Chapter 2. Effective Theories of QCD 20

µ Eγn¯ γ γ s Qµ b

(a) B → Xsγ (b) Dijet Production

Figure 2.4: Examples of processes where SCET is the appropriate theory. The W boson has already been removed for B decays as symbolized by the circle-cross vertex.

is the kinematic limit where the energy carried away by the photon is Eγ ∼ O(mb) as seen in Figure 2.4a. Dijet production in this thesis will refer to the production of exclusively two jets as seen in Figure 2.4b. In both these examples, there are one or more boosted coloured fermions produced making SCET the relevant effective theory to describe the process. A heavy-to-light current describes b → sγ decays in the shape function region. The strange quark is boosted along a direction opposite to the photon in the B meson’s rest frame due to momentum conservation. An SCET collinear quark field will describe the strange quark and the b quark will be described by an HQET heavy quark field. A Wn Wilson line is required by gauge invariance and will describe the interactions of the heavy quark with collinear gluons. From gauge invariance, the leading order operator is

−i(P+Eγ n¯−mbv)·x Jhl = e ξ¯nWnΓhv (2.43) where Γ is a general Dirac matrix. After the ﬁeld redeﬁnition in (2.36) the current becomes [16, 18] −i(P+Eγ n¯−mbv)·x Jhl = e ξ¯nWnΓYnhv. (2.44)

Both the leading order current and the leading order Lagrangian in (2.38) explicitly decouple the soft and collinear degrees of freedom. The interactions between sectors are reproduced by the Wilson lines, which describes the total colour of each sector. However, currents suppressed by λ, as well as the subleading Lagrangian, will explicitly couple the soft and collinear degrees of freedom as seen in [23–25]. A dijet operator describes the production of two-jet final states. Off-shell photons of virtuality Q2 can produce quark–anti-quark pairs that are boosted in opposite directions. This leads to a dijet process where the energy of each of these fermions are of order Q. SCET will describe this process using two collinear sectors, one for each of the directions the fermions are boosted. These sectors can be labelled as nµ andn ¯µ for the directions they travel. The leading order dijet operator after the field redefinition is

−i(P−Qn/2−Qn/¯ 2)·x ¯ † † † O2 = e ξnWnYn ΓYn¯ Wn¯ ξn¯ (2.45) Chapter 2. Effective Theories of QCD 21

where Wn¯ and Yn¯ are deﬁned similarly to Wn in (2.30) and Yn in (2.35) respectively. As in the heavy-to-light currents in the previous paragraph, the Wilson lines are necessary for gauge invariance and describe the total colour of the other sectors. The current also explicitly decouples the soft and collinear sectors as does the leading order Lagrangian. Subleading corrections to this current have not been derived for this formulation of SCET; however, it is expected that the soft and collinear sectors will explicitly couple in subleading currents. In Chapter 3 we will show a new formulation of SCET where subleading Lagrangians and currents maintain the explicit decoupling of sectors. This theory will have an explicit SU(3) symmetry for each sector.

2.4 Conclusion

We have reviewed the characteristics of EFTs and given an example of how a heavy particle is removed from the low energy theory. After reviewing QCD, we introduced two eﬀective theories where large momentum transfers were removed. HQET was introduced to describe the dynamics of a heavy quark ﬁeld. We also introduced the traditional formalism of SCET using two-component spinors and label momentum. Two process where SCET is used, B → Xsγ in the shape function region and dijet production, were described and the necessary leading order external currents were shown. In this formulation of SCET, both the Lagrangian and current are expanded λ and both break the leading order decoupling of soft and collinear sectors. Chapter 3

SCET, QCD, and Wilson Lines

In this chapter we point out that the collinear expansion in the SCET Lagrangian in Chapter 2 is unnecessary, and that the SCET Lagrangian may instead be written as multiple decoupled copies of QCD. The interactions between the sectors in full QCD are reproduced in the eﬀective theory by an external current consisting of QCD ﬁelds coupled to Wilson lines. We illustrate this picture with two examples: dijet production and B → Xsγ. The text in this chapter is reproduced in [26].

3.1 Introduction

SCET [16–21] describes the interaction of low invariant mass jets of particles which are highly boosted with respect to one another. SCET is an expansion in inverse powers of the highly boosted energy. At leading order in the SCET expansion, a field redefinition may be used to manifestly decouple the soft and collinear degrees of freedom from one another at the operator level [19]. Interactions between different soft and collinear sectors are reproduced in the currents of the effective theory by lightlike Wilson lines. This simplification is the basis of factorization theorems in SCET, allowing differential cross sections to be written as convolutions of independent soft and collinear pieces. While factorization theorems have been well-studied using traditional QCD approaches, the manifest decoupling of soft and collinear pieces at the level of the Lagrangian in SCET both dramatically simplifies the study of factorization theorems, and allows power corrections in inverse energy to be studied in a systematic way. In standard formulations of SCET [16–21,24], there is an inherent asymmetry in the treatment of soft and collinear degrees of freedom. While, for example, soft quark fields are identical to four-component QCD quark fields, collinear quark fields are described by two-component spinors with complicated nonlocal interactions. On general grounds, this asymmetry must be spurious: QCD is Lorentz invariant, and dimensional regularization is a Lorentz invariant regulator. One may therefore always boost to a reference frame in which the energy of the collinear fields is small, and the collinear quark fields are described by four-component QCD fields. Thus, the SCET description of collinear fields must be equivalent to that of full QCD.

22 Chapter 3. SCET, QCD, and Wilson Lines 23

This is not a new observation. It was observed in [21] that the Feynman rules of collinear SCET fields are equivalent to those of QCD in light-cone quantization [27], and this equivalence has been used to simplify calculations in the collinear sector of the theory [16,17,28]. In [29], it was formally proven at leading order in power corrections that SCET is equivalent to multiple copies of QCD coupled to Wilson lines when the field redefinition of [19] is used to decouple soft from collinear fields. However, beyond leading order the approach was less clear.

In this chapter, we argue that this picture may be extended to all orders in the SCET expansion. We show that the soft and collinear sectors of SCET may individually be described by a separate copy of the full QCD Lagrangian, and that these sectors are decoupled from one another to all orders in the SCET expansion. The interactions between the sectors in full QCD are reproduced by the interactions between the individual sectors and the external current, which consists of QCD ﬁelds coupled to Wilson lines. In particular, soft-collinear mixing terms in the Lagrangian do not arise in the theory; their eﬀects are accounted for by subleading corrections to the external current, whose form is similar to that of subleading twist shape functions [30, 31].

In order to motivate this picture, we derive the subleading operators for two speciﬁc phenom- + − ena; e e → dijet production and B → Xsγ. In Section 3.2 we review the standard derivation of SCET. In Section 3.3.1 we present our approach for dijet production at leading order, while in Section 3.3.2 we derive the new subleading operators for dijet production. In Section 3.4 we present a similar analysis for B → Xsγ, and in Section 3.5 we present our conclusions.

3.2 Label SCET Formulation

In the approach to SCET introduced in [16–20], collinear ﬁelds are described by eﬀective two- component spinors, ξn,p˜, where n denotes the (lightlike) direction of motion,p ˜ is a label which denotes the large components of the collinear momentum,

nµ p˜µ ≡ n¯ · p˜ +p ˜µ (3.1) 2 ⊥ and the collinear quark momentum is pµ =p ˜µ + kµ. We will refer to this approach as “label SCET” to denote the removal of the large label momentum, and to distinguish it from the approach of [21,24], in which label momentum was not removed, but the collinear quarks were still treated as two-component spinors. The SCET Lagrangian for the collinear quark field is obtained by integrating out the two small components of the field and expanding in powers of λ2 ∼ kµ/n¯ · p. This procedure results in the effective Lagrangian for the n-collinear quark

(0) (1) (1) Lξ = Lξξ + Lξξ + Lξq + ..., (3.2) Chapter 3. SCET, QCD, and Wilson Lines 24 where the superscript refers to the suppression in λ [21, 23, 24, 32–34]. The leading order term (0) Lξξ is (0) X −iP·x ⊥ 1 † ⊥ n/¯ L = e ξ¯ 0 in · D + i /D Wn W i /D ξn,p˜ (3.3) ξξ n,p˜ n n¯ ·P n n 2 {p˜}

a a µ µ µ where the covariant derivative Dµ = ∂µ − igT Aµ, A = As + An contains both soft and µ collinear gluons, Dn only contains n-collinear gluons, Wn is a Wilson line built out of collinear µ An ﬁelds in then ¯ direction, and the “label operator” P pulls down the large label momentum (1) of the collinear ﬁelds. The subleading operator Lξξ describes higher order corrections to the (0) interactions in Lξξ , while the subleading operator

(1) ¯ 1 Lξq = ξn ig [in¯ · Dn, i /Dn⊥] Wnqus + h.c (3.4) in¯ · Dn is the leading operator which couples collinear and soft quarks. Performing the field redefinitions on collinear quark and gluon fields [19]

(3) (0) ξn,p˜(x) = Yn (x, ∞)ξn,p˜(x) a µ (8)ab (0) b µ An,p˜(x) = Yn (x, ∞)An,p˜ (x), (3.5)

(R) where Yn are Wilson lines built out of soft As fields defined below, it may be shown that all dependence on soft gluons disappears from the leading order Lagrangian (3.3), so soft and collinear fields manifestly decouple at leading order in SCET. The collinear and soft lightlike Wilson lines in position space are defined as

Z n ·(y−x) ! (R) 2 a a Wn (x, y) = P exp −ig ds n¯ · An(x +ns ¯ )TR 0 Z n¯ ·(y−x) ! (R) 2 a a Yn (x, y) = P exp −ig ds n · As (x + ns)TR (3.6) 0 where R labels the SU(3) representation.∗ Under a gauge transformation, the Wilson lines transform as

(R) (R) (R) (R)† Wn (x, y) → Uc (x)Wn (x, y)Uc (y) (R) (R) (R) (R)† Yn (x, y) → Us (x)Yn (x, y)Us (y) (3.7)

(R) where Uc,s is either a collinear or soft gauge transformation for representation R. Note that (R) (R)† (R) (R) (R) Wn (x, y) = Wn (y, x), and similarly for Yn , and that Wn (x, y) and Yn (x, y) corre- (R) (R) spond to colour charge R propagating from y to x. Also note that Wn and Yn couple then ¯ and n components of the corresponding gluons, respectively; this notation is used to be

∗In the SCET literature [19], the fundamental Wilson lines (R = 3) are typically denoted by W and Y , and the adjoint Wilson lines (R = 8) are denoted by Wab and Yab, where a, b = 1,..., 8. Chapter 3. SCET, QCD, and Wilson Lines 25 consistent with the SCET literature. At leading order, performing the ﬁeld redeﬁnitions (3.5), the current for dijet production in the full theory QCD −iQ·x ¯ J2 = e ψ(x)Γψ(x) (3.8)

(where Γ is an arbitrary Dirac structure and Q is the external momentum) may be written in the factorized form in the eﬀective theory

J˜(0) −iQ·x ¯(0) (3) ∞ (3)† ∞ (3) ∞ (3)† ∞ (0) 2 = e ξn,p˜1 (x)Wn (x, )Yn (x, )ΓYn¯ (x, )Wn¯ (x, )ξn,¯ p˜2 (x) (3.9) where the Wn¯ and Yn¯’s are lightlike Wilson lines defined analogously to (3.6). Label momentum (3) conservation is enforced at each vertex. The collinear Wilson line Wn arises from integrating (3) out the interactions of n-collinear fields withn ¯-collinear fields and similarly for Wn¯ . Each sector is therefore decoupled at leading order and described by QCD fields coupled to Wilson lines. In this form it is manifest that all interactions between the different sectors occur via Wilson lines, as was formally shown in [29]. The redefined quark fields ξ(0) do not transform under soft gauge transformations, so the soft fields only couple to the Wilson line Y . Physically, this corresponds to the fact that soft fields cannot deflect the worldline of a highly energetic quark, and so they only see the direction and gauge charge of the collinear degrees of freedom (much the same way that in Heavy Quark Effective Theory [14], soft degrees of freedom only see the velocity and gauge charge of heavy quarks). Similarly, in a frame in which the n-collinear quark fields are soft, the soft andn ¯-collinear fields are recoiling in the opposite direction; thus the n-collinear quark fields can only resolve the total gauge charge of the combined soft and n¯-collinear fields via the Wilson line Wn (and similarly for then ¯-collinear fields). At higher orders in the expansion, however, label SCET looks more complicated, and the (1) operator decoupling is no longer manifest. In particular, interactions such as Lξq , in which soft and collinear sectors couple directly instead of via Wilson lines [33], make the extension of the arguments in [29] to higher orders unclear. In the next section we show how the leading order picture can be easily extended, by reformulating the theory using QCD fields. Another formulation of SCET [21,24] replaces the removal of label momentum with a multipole expansion in soft position. Our formulation of SCET more closely resembles this formulation than label SCET. However, we will diverge from the [21,24] treatment of collinear quarks, which are two-component spinors giving mixed collinear-soft Lagrangian terms at subleading orders similar to label SCET. Also, the non-Abelian nature of SCET requires the introduction of a Wilson line R(x) [24]. Without the R Wilson line, soft transformations of collinear fields gives higher order in λ pieces due to the soft and collinear fields being at different positions. The R Wilson line redefines the collinear fields so they transform homogeneously in λ under soft transformations. However, after the field redefinition (3.5), collinear fields no longer transform under the soft gauge group, and the R Wilson line is not needed. In our formulation, soft Chapter 3. SCET, QCD, and Wilson Lines 26 and collinear fields are decoupled and each sector does not transform under the other so the R Wilson line will be unneeded.

3.3 SCET as QCD Fields Coupled to Wilson Lines

Despite the complexity of the leading order n-collinear Lagrangian (3.3) and the corresponding Feynman rules, it is equivalent to the QCD Lagrangian [21, 29]. This is not unexpected: as long as one is just describing soft fields or collinear fields in one direction, there is no Lorentz- invariant expansion parameter, and one could just as easily work in a frame where the energy is small, in which case it is obvious that there is no effective field theory description and QCD is the appropriate theory. The large boost of a collinear quark only has physical meaning when it is coupled to fields with large relative momentum via an external current, such as in e+e− → qqX¯ or B → Xsγ. The purpose of SCET is to describe the interactions in such situations between fields whose relative momentum is greater than the cutoff of the theory. We therefore begin with the starting point that in the absence of an external current, each sector (collinear in each relevant direction and soft) can be described by LQCD, since QCD is Lorentz invariant. Therefore, the all-orders SCET Lagrangian is

X i LSCET = LQCD, (3.10) i=s,nj where j runs over all relevant collinear directions. LSCET then consists of a separate copy of the QCD Lagrangian for each sector, each with a separate gauge symmetry. All interactions between the diﬀerent sectors will be described by the external current, which for dijet production takes the form " # SCET −i Q x·(n+¯n) (0) (0) 1 X (1i) (1i) 2 J = e 2 C O + C O + O(λ ) (3.11) 2 2 2 Q 2 2 i

(0) (1i) where O2 is O(1) and the O2 ’s are O(λ), and we have pulled out the phase corresponding to the momentum of the external current. This is the only place the λ expansion enters in this formulation of SCET. As discussed in the previous section, fields in one sector only resolve the direction and colour charge of fields in other sectors; hence, the sectors can only interact with each other via SCET Wilson lines. The current J2 therefore decouples into separately SU(3)-invariant pieces representing each sector, each of which describes QCD fields coupled to Wilson lines. At (0) leading order the current O2 is equivalent to the usual leading order SCET current (3.9). The (1i) subleading operators O2 are constructed from Wilson lines with derivative insertions, in a similar manner as higher twist corrections to light-cone distribution functions [30, 31]. We will show that we can do this to subleading order for dijet and heavy-to-light currents with nonlocal operators. It will prove unnecessary to introduce large label momenta, since Chapter 3. SCET, QCD, and Wilson Lines 27

p1 p1

q q

p2 p2

(a) (b)

Figure 3.1: QCD vertex for dijet production. In this and all other figures, the quark, antiquark and gluon momenta are denoted p1, p2 and q. these are frame-dependent. Instead, we follow [16] and [21, 24] and implement the appropriate multipole expansion through the coordinate dependence of the currents. We first work out the leading order operators to illustrate our picture in the next section, and then describe the subleading corrections. We demonstrate that all such corrections may be accounted for by subleading corrections to the current, rather than direct interactions between the different sectors (such as the collinear-soft quark interaction term Lξq). This is the principal result of this chapter.

3.3.1 Dijet Production at Leading Order

Consider the process e+e− → qqX¯ , which contributes to dijet production. The external current carries momentum nµ n¯µ Qµ = Q + Q , (3.12) e+e− 2 2 where Q is large compared to the invariant mass of the jets so SCET is the appropriate theory.

The O(αs) graphs contributing to this process in QCD are shown in Figure 3.1. The SCET expansion of a given graph depends on the relative scaling of the momenta: n- 2 2 collinear momenta scale like pn ∼ Q(λ , 1, λ),n ¯-collinear momenta like pn¯ ∼ Q(1, λ , λ) and soft 2 2 2 † momenta like ps ∼ Q(λ , λ , λ ) . To match amplitudes onto SCET, we expand the relevant graphs with the appropriate scalings in powers of λ, including the various energy-momentum conserving delta functions. In particular, the full theory energy-momentum conserving delta function is

(4) (4) µ µ (2) δQCD(Q; p) ≡ δ (Qe+e− − p ) = 2 δ (Q − p · n) δ (Q − p · n¯) δ (~p⊥) (3.13) where pµ is the four-momentum of the ﬁnal state. Splitting pµ into n-collinear,n ¯-collinear and soft momenta, µ µ µ µ p = pn + pn¯ + ps (3.14)

† µ We use light-cone coordinates, where p = (p · n, p · n,¯ ~p⊥) and n · n¯ = 2. Chapter 3. SCET, QCD, and Wilson Lines 28 and expanding in powers of λ gives at leading order the SCET energy-momentum conserving δ function (4) (4) µ ∂ (4) 2 δQCD(Q; p) = δSCET(Q; pn, pn¯) + ps⊥ µ δSCET(Q; pn, pn¯) + O(λ ) (3.15) ∂pn⊥ where (4) (2) δSCET(Q; pn, pn¯) = 2 δ (Q − pn · n¯) δ (Q − pn¯ · n) δ (~pn⊥ + ~pn¯⊥) (3.16) and the first term in (3.15) is O(1) and the second is O(λ). Note that soft momenta are unconstrained by overall energy-momentum conservation in the effective theory. This expansion differs from the label SCET derivation, which replaces (3.16) with label conservation δp,˜ p˜0 , and which conserves momentum exactly in the effective theory. Higher order terms in the expansion of (3.13) are accounted for by higher order corrections in SCET. The expansion (3.15) can be understood in calculations as expanding QCD phase-space in SCET momentum, where subleading phase-space effects are incorporated into the subleading current through the higher multipole moments. Such was the case when considering phase-space of jets at O(λ0) [35, 36]. We can write the external production current (3.11) in terms of four-component QCD spinors

ψn and ψn¯. The leading order operator is h i h i h i (0) ¯ (3) ∞ (3) ∞s (3) ∞s (3) ∞ O2 (x) = ψn(xn¯)Pn¯ΓWn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ ) Wn¯ (xn , xn)Pn¯ψn¯(xn) (3.17) (0) with C2 = 1 + O(αs). This has a similar form as (3.9), with the difference that the collinear fields are four-component spinors, and the positions of the fields

∞ xn = (0, x · n,¯ ~x⊥) xn = (0, ∞, ~x⊥) ∞ xn¯ = (x · n, 0, ~x⊥) xn¯ = (∞, 0, ~x⊥)

∞s ∞s xn = (0, ∞, 0) xn¯ = (∞, 0, 0), (3.18) are chosen to obtain the correct momentum conservation (3.16). Note the coordinate xn¯ conserves p · n¯ momentum, and similarly for xn. We have also deﬁned the usual projectors

/n/n¯ n/n/¯ Pn = Pn¯ = (3.19) 4 4 so at leading order in λ the external current (3.17) only couples to the large components of the external quark spinors. However, the collinear quark fields evolve via QCD, which couples all (0) four components of the field. The one-gluon Feynman rules for O2 are shown in Figure 3.2. The terms in each square bracket of (3.17) each transform under a separate SU(3) symmetry, corresponding to the various sectors of the theory‡ and represents a different decoupled sector

‡We ignore possible gauge transformations at ∞ since we use covariant gauge, which is “regular”. For complications that arise in “singular” gauges, see [37, 38]. In our formulation, the necessary extra Wilson lines should occur naturally in the matching. Chapter 3. SCET, QCD, and Wilson Lines 29

p1 p1 p1 p1

p2 p2 p2 p2

ig −

(0) Figure 3.2: One-gluon Feynman rules for O2 . The SCET energy-momentum conserving delta function δSCET has been omitted but is implied. Springs are soft gluons, springs with lines are collinear gluons, and solid lines are fermions. Lines angled up are n-collinear, angled down are n¯-collinear, and horizontal are soft. The dashed lines represent the emission from a Wilson line.

(a) n-collinear (b) soft (c)n ¯-collinear

(0) Figure 3.3: Physical picture of O2 as seen in each of the three sectors. The dashed lines represent Wilson lines and the solid lines represent fermions.

giving the physical picture of Figure 3.3, which we explain below. (0) It is straightforward to show that the one-gluon matrix element of O2 reproduces the QCD amplitude at leading order in λ. The one-gluon amplitudes in Figure 3.1 in QCD are

α α a 2p1 + γ /q ∗ (4) iMa = −igT u¯(p1) Γv(p2)α(q)δQCD(Q; p1 + p2 + q) (3.20) 2p1 · q and α α a 2p2 + /qγ ∗ (4) iMb = igT u¯(p1)Γ v(p2)α(q)δQCD(Q; p1 + p2 + q) (3.21) 2p2 · q where Γ is the Dirac structure of the external current. The corresponding leading order contributions in SCET comes from an n-collinear quark,n ¯-collinear antiquark, and a gluon which is either soft or collinear, each of which gives a diﬀerent result in SCET.

We ﬁrst consider the case in which the n-collinear quark emits an n-collinear gluon. Using the Dirac Equation /pnu(pn) = 0 to write /pn,⊥ n/¯ u(pn) = 1 + Pnu(pn) (3.22) n¯ · pn 2 Chapter 3. SCET, QCD, and Wilson Lines 30

and similarly for v(pn¯), it is straightforward to show that

α α α α u¯(p1) (2p1 + γ /q)Γv(p2) =u ¯(p1) (2p1 + γ /q) Pn¯ΓPn¯v(p2) + O(λ) (3.23) so we can expand (3.20) as

α α a 2p1 + γ /q ∗ (4) iMan = −igT u¯(p1) Pn¯ΓPn¯v(p2)α(q)δSCET(Q; p1 + q, p2) + O(λ). (3.24) 2p1 · q

With the projectors Pn¯ now surrounding the Dirac structure Γ, this is precisely the amplitude (0) in the eﬀective theory for a q-¯q pair to be produced by O2 , followed by the emission of an n-collinear gluon from the n-collinear quark through the usual QCD vertex. It is useful to compare this with the expression for the same graph in label SCET:

α α 0 a ¯ α γ⊥/p1⊥ (/p1⊥ + /q⊥)γ⊥ (/p1⊥ + /q⊥)/p1⊥ α n/¯ iMan = −igT ξn,p˜1 (k1) n + + + n¯ n¯ · p1 n¯ · (p1 + q) n¯ · p1 n¯ · (p1 + q) 2 /n n¯ · (p1 + q) ∗ × 2 Γξn,¯ p˜2 (k2)n,q˜ α(k3) 2 n · (k1 + k3)n ¯ · (p1 + q) + (p1⊥ + q⊥) (4) × δn¯·(˜p1+˜q),Qδn·p˜2,Qδ0,p˜1⊥+˜p2⊥+˜q⊥ δ (k1 + k2 + k3) (3.25) where the ﬁrst factor in parentheses is the collinear quark - collinear quark - collinear gluon vertex, the second is the collinear quark propagator in label SCET, and the ξ ﬁelds are two- component spinors. Some straightforward Dirac algebra shows that this is indeed equivalent to the expression (3.24); however, the more complicated Feynman rules of label SCET, arising from the fact that the collinear spinors are 2-component objects rather than 4-component spinors obeying (3.22), makes the intermediate expression considerably more complicated. Expanding the amplitude in which an n-collinear gluon is emitted from ann ¯-collinear antiquark, (3.21), in powers of λ gives

α a n¯ ∗ (4) iM = igT u¯(p1)Pn¯ΓPn¯v(p2) (q)δ (Q; p1 + q, p2) + O(λ) (3.26) bn n¯ · q α SCET where we have used the expansions

2 2p2 · q = (p2 · n)(q · n¯) + O(λ ) ∗ ∗ 2 2p2 · (q) = (p2 · n)( (q) · n¯) + O(λ ). (3.27)

In SCET, the n-collinear quark does not couple to then ¯-collinear antiquark directly, but rather (3) ∞ (0) to the Wilson line Wn (xn¯, xn¯ ) in O2 , and this amplitude is reproduced in the eﬀective theory by the graph in which the Wilson line emits the n-collinear quark. The interactions (3.24) and (3.26) of the n-collinear gluon is represented in Figure 3.3a by a QCD quark ﬁeld in the n direction and a Wilson line in then ¯ direction. Similarly, the amplitudes in Figure 3.1 are reproduced forn ¯-collinear gluons in SCET by a Chapter 3. SCET, QCD, and Wilson Lines 31

(3) ∞ gluon emitted from a semi-inﬁnite n-collinear Wilson line Wn¯ (xn, xn ), and the usual QCD Feynman rules for gluon emission, respectively. Then ¯-collinear gluon interaction is represented in Figure 3.3c. Finally, the amplitude for soft gluon emission from the quark and antiquark lines is obtained by expanding the sum of the two previous graphs for soft gluon momentum,

α α a n n¯ ∗ (4) iMs = −igT − u¯(p1)PnΓPn¯v(p2) (q)δ (Q; p1, p2) + O(λ), (3.28) n · q n¯ · q α SCET which is the amplitude for gluon emission from a fundamental and anti-fundamental Wilson (3) ∞s (3) ∞s line, Yn (xn , 0) and Yn¯ (0, xn¯ ) respectively, represented in Figure 3.3b. (0) Thus, we have shown that O2 as deﬁned in (3.17) reproduces the leading order QCD qqg¯ production amplitudes. In the next section, we will show how O(λ) operators arise as (0) generalizations of O2 and the physical picture of Figure 3.3.

3.3.2 Subleading Corrections to Dijet Production

At leading order, the external current is written as a product of QCD fields coupled to Wilson lines. Higher order corrections to the current are therefore expected to have the same structure, but with insertions of derivatives and additional fields, in the same way that subleading twist shape functions and parton distributions are related to the leading order operators [30, 31]. (1i) Defining the external current to subleading order in (3.11) where the O2 ’s are O(λ), it is (1i) (1i) straightforward to determine the required operators O2 and coefficient functions C2 by carrying out the expansion of the previous section to higher orders in λ. Starting with the emission of an n-collinear gluon, we can expand the QCD amplitudes (3.20) and (3.21) to O(λ):

a α ∗ (4) 2 iM(a,b)n = −igT u¯(p1)Γ(a,b)nv(p2)α(q)δSCET(Q; p1 + q, p2) + O(λ ) (3.29) where

α α αµ α α α (2p1 + γ /q) ∆ (¯n, p1) n/¯ (2p1 + γ /q) n/¯ (/p1⊥ + /q) Γan = Pn¯ΓPn¯ − γµ⊥ΓPn¯ + ΓPn¯ p1 · q n¯ · (p1 + q) 2 2p1 · q 2 n¯ · (p1 + q) and α α α n¯ n¯ n/¯ /p1⊥ 1 αµ /n Γbn = − Pn¯ΓPn¯ − ΓPn¯ + ∆ (¯n, q)Pn¯Γγµ⊥ (3.30) n¯ · q n¯ · q 2 n¯ · p1 Q 2 where we have deﬁned n¯αpµ ∆αµ(¯n, p) = gαµ − (3.31) n¯ · p and we have used the expansion

∗ /n αµ ∗ 2 /q/ (q) = (¯n · q) γµ ∆ (¯n, q) (q) + O(λ ) (3.32) 2 ⊥ α Chapter 3. SCET, QCD, and Wilson Lines 32 as well as the spinor expansion (3.22). The sum of the graphs is

α α α α α (2p1 + γ /q) n¯ 1 αµ /n Γan + Γbn = Pn¯ΓPn¯ − Pn¯ΓPn¯ + ∆ (¯n, q)Pn¯Γγµ⊥ 2p1 · q n¯ · q Q 2 α α α n/¯ α n¯ (2p1 + γ /q) n/¯ + γ⊥ + /p1⊥ ΓPn¯ + (/p1⊥ + /q)ΓPn¯ . (3.33) 2 n¯ · q 2p1 · q 2

The first two terms of (3.33) are O(1), while the remaining terms are O(λ), and are reproduced in the effective theory by the operators h i h i (1an) ¯ (3) ∞ (3) ∞s (3) ∞s O2 = ψn(xn¯)Pn¯Γi /D⊥(xn¯)Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ ) (3) ∞ /n × W (x , xn) ψn¯(xn) n¯ n 2 ←− h i (1bn) n/¯ (3) ∞ (3) ∞s (3) ∞s O = ψ¯n(xn¯) i /D (xn¯)ΓW (xn¯, x ) Y (x , 0)Y (0, x ) 2 2 ⊥ n n¯ n n n¯ n¯ h (3) ∞ i × Wn¯ (xn , xn)Pn¯ψn¯(xn) (3.34) where the covariant derivatives are defined in the usual way

µ µ a µa D ψn,n,s¯ = ∂ ψn,n,s¯ − igT An,n,s¯ ψn,n,s¯ (3.35) to only couple the corresponding gluon fields to n-collinear,n ¯-collinear, and soft quarks, respectively. The one-gluon Feynman rules for these operators are given in Figure 3.6. The last term in (3.33) corresponds in the effective theory to a gluon emitted from the n-collinear (1bn) quark leg via LQCD after the insertion of the subleading operator O2 . From (3.33), we find (1an) (1bn) C2 = −1 + O(αs) and C2 = 1 + O(αs).

We can perform a similar expansion for soft gluon emission. Expanding the amplitude (3.20) for soft momentum qµ, including the multipole expansion (3.15), gives

α a n µ ∂ 2p1⊥µ αµ iMas = −igT u¯(p1)Pn¯ΓPn¯v(p2) 1 + q⊥ µ + ∆ (n, q) (3.36) n · q ∂p1⊥ Q n · q α n ∗ (4) 2 + u¯(p1)(n/p/¯ ΓPn¯ + Pn¯Γ/p /n) v(p2) (q)δ (Q; p1, p2) + O(λ ). 2Q n · q 1⊥ 2⊥ α SCET

(1bn) The second line in (3.36) is reproduced in the eﬀective theory by O2 , followed by emission of a soft gluon oﬀ the soft Wilson line Yn, and so has already been accounted for. The term proportional to ∆αµ(n, q) requires the introduction of the operator

Z ∞ h ←− i (1cns) ¯ µ (3) ∞ O2 (x) = −i dt ψn(xn¯)Pn¯ΓiD ⊥Wn (xn¯, xn¯ ) (3.37) 0 h ←− i h i (3) ∞s (3) (3) ∞s (3) ∞ × Yn (xn , tn)iD ⊥µ(tn)Yn (tn, 0)Yn¯ (0, xn¯ ) Wn¯ (xn , xn)Pn¯ψn¯(xn) Chapter 3. SCET, QCD, and Wilson Lines 33 while the higher order term in the multipole expansion of the momentum requires the operator h i h ←− i (1δs) ¯ (3) ∞ (3) ∞s µ µ (3) ∞s O2 (x) = Q ψn(xn¯)Pn¯ΓWn (xn¯, xn¯ ) Yn (xn , 0)(x⊥µD⊥ + D ⊥x⊥µ)Yn¯ (0, xn¯ ) h (3) ∞ i × Wn¯ (xn , xn)Pn¯ψn¯(xn) . (3.38)

(1cns) (1δs) From (3.36) we ﬁnd C2 = −2 + O(αs) and C2 = 1 + O(αs).

In addition to subleading corrections to the leading order amplitudes, at subleading order additional processes – soft quark and n-collinear antiquark emission – occur. In the standard SCET approach, these arise from subleading terms in the eﬀective Lagrangian which directly couple the various sectors. In our formulation, the only coupling between the diﬀerent sec- SCET tors occurs via the external current J2 , so these processes are also described in SCET by (1i) subleading operators O2 .

Consider Figure 3.1a where the gluon is n-collinear and the quark is soft. Expanding the amplitude (3.20) for these kinematics gives

αµ a ∆ (¯n, q) /n ∗ (4) 2 iMasq = −igT u¯s(p1) γµ⊥ΓPn¯v(p2)α(q)δSCET(Q; q, p2) + O(λ ). (3.39) n · p1 2

The amplitude is reproduced in SCET by the subleading operator

Z ∞ (1dns) h aµν (8)ab ∞ i O2 (x) = −i dt n¯µigGn (xn¯)Wn (xn¯, xn¯ ) 0 (8)bc ∞s (3) c /n (3) ∞s × Y (x , tn)ψ¯s(tn)Y (tn, 0)T γν ΓY (0, x ) n n n 2 ⊥ n¯ n¯ h (3) ∞ i × Wn¯ (xn , xn)Pn¯ψn¯(xn) . (3.40)

The structure of (3.40) can be understood by generalizing the arguments that led to (3.17). The physical picture of (3.40) is shown in Figure 3.4. The n-collinear gluon recoiling against the soft quark andn ¯-collinear antiquark in an SU(3) adjoint looks like a gluon coupled to an adjoint Wilson line, giving the first factor in (3.40) and the picture Figure 3.4a. Then ¯- collinear sector sees no difference between an antiquark recoiling against an n-collinear quark and recoiling against an n-collinear gluon and a soft quark in a relative fundamental state, so the third factor is unchanged from (3.17) and gives the picture Figure 3.4c. Finally, the soft sector has fundamental and anti-fundamental Wilson lines emitted by the current as usual, but then the fundamental emits a soft quark and becomes an adjoint Wilson line (the n-collinear (1dns) gluon) as pictured in Figure 3.4b. From (3.39) we find C2 = 1 + O(αs).

The situation is similar for emission of ann ¯-collinear gluon recoiling against an n-collinear quark-antiquark pair. Expanding the amplitudes (3.20) and (3.21), we ﬁnd the leading order Chapter 3. SCET, QCD, and Wilson Lines 34

(a) n-collinear (b) soft (c)n ¯-collinear

(1dns) Figure 3.4: O2 as seen in each of the three sectors. The single and double dashed lines represent fundamental and adjoint Wilson lines respectively.

terms cancel between the two diagrams, giving the amplitude

n/¯ n/¯ ! a ∗ 2 γµ⊥ΓPn Pn¯Γγµ⊥ 2 µα (4) iMnnn¯ = igT α(q)¯u(p1) − v(p2)∆ (n, q)δSCET(Q; p1 + p2, q) n¯ · p1 n¯ · p2 + O(λ2) (3.41)

These terms are reproduced in SCET by the operators

Z ∞ (1en) ¯ d n/¯ (3) O2 (x) = −i dt ψn(xn¯ + tn¯)T γν⊥ΓWn (xn¯ + tn,¯ xn¯) 0 2 i h i (8)dc ∞ (8)cˆb ∞s (8)ˆbb ∞s ×Pnψn(xn¯)Wn (xn¯ + tn,¯ xn¯ ) Yn (xn , 0)Yn¯ (0, xn )

h (8)ba ∞ aµν i × nµWn¯ (xn , xn)igGn¯ (xn) (3.42) and Z ∞ (1en¯ ) ¯ n/¯ (3) O2 (x) = −i dt ψn(xn¯)Pn¯Γγν⊥ Wn (xn¯, xn¯ + tn¯) 0 2 i h i d (8)dc ∞ (8)cˆb ∞s (8)ˆbb ∞s × T ψn(xn¯ + tn¯)Wn (xn¯ + tn,¯ xn¯ ) Yn (xn , 0)Yn¯ (0, xn )

h (8)ba ∞ aµν i × nµWn¯ (xn , xn)igGn¯ (xn) . (3.43)

(1en¯ ) (1en) (1en¯ ) O2 is illustrated in the three frames in Figure 3.5. From (3.41) we ﬁnd C2 = −C2 = −1 + O(αs).

There are an additional four operators, deﬁned analogously to the above operators, that Chapter 3. SCET, QCD, and Wilson Lines 35

(a) n-collinear (b) soft (c)n ¯-collinear

(1en¯ ) Figure 3.5: O2 as seen in each of the three sectors.

arise due to corresponding corrections to then ¯ sector:

h i (1an¯ ) n/¯ (3) ∞ (3) ∞s (3) ∞s O (x) = ψ¯n(xn¯) W (xn¯, x ) Y (x , 0)Y (0, x ) 2 2 n n¯ n n n¯ n¯ h (3) ∞ ←− i × Wn¯ (xn , xn)i /D ⊥(xn)ΓPn¯ψn¯(xn) h i h i (1bn¯ ) ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ ) (3) ∞ /n × W (x , xn)Γi /D (xn) ψn¯(xn) n¯ n ⊥ 2 Z ∞ h i (1cns¯ ) ¯ (3) ∞ O2 (x) = −i dt ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) (3.44) 0 h i (3) ∞s (3) µ (3) ∞s × Yn (xn , 0)Yn¯ (0, tn¯)iD⊥(tn¯)Yn¯ (tn,¯ xn¯ ) h (3) ∞ i × Wn¯ (xn , xn)iDµ⊥(xn)ΓPn¯ψn¯(xn) Z ∞ h i (1dns¯ ) ¯ (3) ∞ O2 (x) = −i dt ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) 0 (3) ∞s n/¯ c (3) (8)cb ∞s × Y (x , 0)Γγν T Y (0, tn¯)ψs(tn¯)Y (tn,¯ x ) n n ⊥ 2 n¯ n¯ n¯ h (8)ba ∞ aνµ i × Wn (xn , xn)igGn (xn)nµ .

These operators have the same matching coeﬃcients as their n sector counterparts. The one- (1i) gluon Feynman rules for the operators O2 are shown in Figure 3.6. Loop calculations still require a zero-bin subtraction [22], which serves to ﬁx the double counting in the same way as the standard SCET.

+ − Thus, we have shown the primary result of this chapter for e e → dijet at O(αs): O(λ) SCET eﬀects can be written as QCD ﬁelds coupled to Wilson lines, where each sector is decoupled into a separate SU(3) gauge theory. The only expansion is in the current, and the subleading operators and physical pictures are generalizations of the O(λ0) operator (3.17) and physical picture Figure 3.3. Chapter 3. SCET, QCD, and Wilson Lines 36

p1 p1 p1 p1

(1 an) (1 an¯ )

p2 p2 p2 p2

p1 p1 p1 p1

p2 p2 p2 p2

p1 p2 p1

p1 p1

p2 p2

(1i) Figure 3.6: One-gluon Feynman rules for NLO dijet operators O2 . The notation is the same (1bn) as Figure 3.2. The rule for O2 has been split into two diagrams, depending on whether the gluon is emitted from the vertex or the Wilson line.

3.4 Heavy-to-Light Current

A similar analysis may be carried out for B → Xsγ decay in the shape function region,

2 1 − y ∼ ΛQCD/mb ∼ λ (3.45) where y = 2Eγ/mb is the scaled energy of the photon. In this region the light ﬁnal-state hadrons are constrained to form a jet, and SCET is the appropriate EFT. The SCET analysis of this process has been carried out to O(λ2) [23,25,32,39]. In this section we present the operators up to O(λ) in order to show how the picture introduced in this chapter matches the standard SCET results. The arguments are analogous to the dijet analysis. However, now there is only one collinear sector and one soft sector, and a copy of the Heavy Quark Eﬀective Theory (HQET) Lagrangian is necessary. Again, the collinear and soft Lagrangian is not expanded in λ and Chapter 3. SCET, QCD, and Wilson Lines 37

p p

q q pb pb (a) (b)

Figure 3.7: One-gluon contributions to the matching of the QCD vertex for B → Xsγ. The quark, gluon, and heavy quark momentum are p, q, and pb respectively. only the EFT current " # m SCET −i b (n+(1−y)¯n)·x (0) (0) 1 X (1i) (1i) 2 J = e 2 C O + C O + O(λ ) (3.46) h h h m h h b i and the HQET Lagrangian are expanded in λ. The phase in (3.46) corresponds to the removal µ µ µ of the large b-quark momentum, mb(n +n ¯ )/2, and the outgoing photon momentum, Eγn¯ .

The relevant O(αs) graphs are in Figure 3.7 and have amplitudes

α α B a 2pb − /qγ ∗ (4) iMa = igT u¯(p)Γ u(pb)α(q)δQCD(mb, y; p + q + pb) (3.47) 2pb · q and α α 2p + γ /q (4) iMB = −igT au¯(p) Γu(p )∗ (q)δ (m , y; p + q + p ). (3.48) b 2p · q b α QCD b b in QCD. The amplitude expansions are similar to the dijet case. The collinear and heavy quark spinors are expanded using the Dirac Equation. The collinear quark is done in (3.22) and the 2 heavy quark expansion is u(pb) = 1 + O(λ ) hv(k)where hv is an HQET heavy quark ﬁeld satisfying (/n + n/¯)hv = 2hv. As before, the QCD momentum conserving δ function,

(4) (2) δQCD(mb, y; p = 2 δ (mb − p · n¯)) δ (mb(1 − y) − p · n) δ (~p⊥) , (3.49) is expanded onto the SCET momentum conserving δ function,

(4) (4) δQCD(mb, y; p) = δSCET(mb, y; pn, ps) + ..., (3.50) where

(4) (2) δSCET(mb, y; pn, ps) = 2 δ(mb − pn · n¯)δ(mb(1 − y) − (pn + ps) · n)δ (~pn,⊥) (3.51) and the higher moments are reproduced by higher orders of the SCET current (3.46). The µ 2 residual b quark momentum k ∼ λ mb is included in ps with the appropriate sign. Unlike in the dijet case, all components of the collinear momentum and the ps · n component of the soft momentum are constrained. The leading order expansion of (3.47) and (3.48) are reproduced Chapter 3. SCET, QCD, and Wilson Lines 38 by the operator h i h i (0) ¯ (3) ∞ (3) ∞s Oh (x) = ψn(x)Pn¯Γ Wn (x, xc ) Yn (xn¯ , xs)hv(xs) (3.52)

(0) with Ch = 1 + O(αs), which is similar to the leading order label SCET operator [16]

(0) mb (0) ˜ −i 2 (n+(1−y)¯n)·x ¯ (3) (3) Jh = e ξn,p˜(x)Wn (x, ∞)ΓYn (∞, x)hv(x). (3.53)

The difference between (3.52) and (3.53) are the use of four-component spinors for the collinear fields and the fields are at the positions

⊥ x = (x · n, x · n,¯ x ) xs = (0, x · n,¯ 0)

∞ ⊥ ∞s xc = (−∞, x · n,¯ x ) xn¯ = (0, ∞, 0). (3.54)

The positions are chosen to reproduce (3.51).

The O(λ) expansion of the amplitudes is done in the same way as in the previous section, so we omit the details. We ﬁnd the following subleading operators: (1ac) /n (3) ∞ h (3) ∞s i O (x) = ψ¯n(x)Pn¯Γi /D (x) W (x, x ) Y (x , xs)hv(xs) h ⊥ 2 n c n n¯ ←− (1bc) n/¯ (3) ∞ h (3) ∞s i O (x) = ψ¯n(x) i /D (x)ΓW (x, x ) Y (x , xs)hv(xs) h 2 ⊥ n c n n¯ Z 0 h ←− i (1cs) ¯ (3) ∞ Oh (x) = −i dt ψn(x)Pn¯iD ⊥µ(x)ΓWn (x, xc ) (3.55) −∞

h (3) ∞s ←−µ (3) i × Yn (xn¯ , xs + tn)iD ⊥(xs + tn)Yn (xs + tn, xs)hv(xs) Z 0 (1ds) h aµν (8)ab ∞ i Oh (x) = −i dt n¯µigGn (x)Wn (x, xc ) −∞ (8)bc ∞s c /n (3) × Y (x , xs + tn)ψ¯s(xs + tn)T γν ΓY (xs + tn, xs)hv(xs) n n¯ 2 ⊥ n h i h ←− −→ i (1δs) ¯ (3) ∞ (3) ∞s µ Oh (x) = mb ψn(x)Pn¯ΓWn (x, xc ) Yn (xn¯ , xs) D ⊥x⊥µ + x⊥µ D ⊥ (xs)hv(xs) .

These operators are the analogous dijet operators with only one collinear sector. The opera- (1cs,1s) tors Oh have integration limits −∞ and 0 since the b-quark is coming in from −∞ (as opposed to the dijet case where the partons are outgoing to +∞). The matching coeﬃcients (1ac,1bc,1δs) (1ds) (1cs) are Ch = 1 + O(αs), Ch = −1 + O(αs) and Ch = 2 + O(αs).

The formulation of SCET introduced in this chapter must be equivalent to standard SCET (1ds) formulations at subleading orders. For example, the soft quark operator Oh is reproduced (1) ˜(0) by the time-ordered product of Lξq in (3.4) and the leading order current Jh of (3.9), whose Feynman rules are shown in [32]. A particularly simple example of a subleading standard SCET Chapter 3. SCET, QCD, and Wilson Lines 39 operator that can be re-written as one of our operators is the O(λ2) label SCET current [21,24]

(2A) 1 ∞ J = ξ¯(x) [in · D(x)Wc(x, x )] hv(xs) + ..., (3.56) in¯ · D c which is identical to the operator we ﬁnd at O(λ2)

Z 0 h i (2ac) ¯ (3) (3) ∞ Oh = −i mb dt ψn(x)Pn¯Wn (x, x + tn¯)in · D(x + tn¯)ΓWn (x + tn,¯ xc ) −∞

h (3) ∞s i × Yn (xn¯ , xs)hv(xs) (3.57)

(2ac) 2 with Ch = 1 + O(αs). The “...” in (3.56) refer to other O(λ ) terms. The equivalence between (3.56) and (3.57) can be shown using the relations [18, 21]

1 1 W (3)(x, x∞) = W (3)(x, x∞) (3.58) in¯ · D n c n c in¯ · ∂ and 1 Z 0 φ(x) = −i dt φ(x + tn¯), (3.59) in¯ · ∂ −∞ and the ﬁeld redeﬁnition (3.5). The relations (3.58) and (3.59) can be used to re-write (3.56) as

(2A) ∞ 1 ∞ ∞ J = ξ¯(x)Wc(x, x ) Wc(x , x)[in · D(x)Wc(x, x )] hv(xs) c in¯ · ∂ c c Z 0 h i ¯(0) (0) (0) (0) ∞ = −i dt ξ (x)Wc (x, x + tn¯) in · D (x + tn¯)Wc (x + tn,¯ xc ) −∞

∞s × Yn(xn¯ , xs)hv(xs) (3.60)

(2ac) = Oh where the ﬁeld redeﬁnition (3.5) and ξ(x) = Pnψ(x) were used in the second line. The other operators in [21, 24] can be shown to be equivalent using the same trick.

3.5 Conclusions

We have demonstrated how SCET can be written as a theory of separate, decoupled sectors of QCD by explicitly performing the matching of the external current at tree level in αs to subleading order in λ for both dijet production and B → Xsγ. Interactions between different sectors are reproduced in SCET by Wilson lines. We believe this makes the SCET picture more transparent: instead of a complicated collinear Lagrangian that couples two-component collinear quarks to soft fields, the Lagrangian is just multiple QCD copies. The only expansion in λ occurs in the currents, which are QCD fields coupled to Wilson lines that represent the colour flow of the other sectors. The subleading currents are generalizations of the leading order Chapter 3. SCET, QCD, and Wilson Lines 40 currents akin to higher twist corrections to light-cone distribution functions. Corrections to leading-order factorization theorems should be simpler since the manifest decoupling of sectors that occurred at leading order now exists to all orders. Chapter 4

Subleading Corrections To Thrust Using Eﬀective Field Theory

In this chapter we calculate the subleading corrections to the thrust rate. We use SCET to factorize the rate and match onto jet and soft operators that describe the degrees of freedom of the relevant scales. We work in the perturbative regime where all the scales are well above ΛQCD. The thrust rate involves an incomplete sum over ﬁnal states that is enforced by a measurement operator. Subleading corrections require matching onto not only the higher dimensional dijet operators, but also matching onto subleading measurement operators in the eﬀective theory.

We explicitly show how to factorize the O(αsτ) thrust rate into a hard function multiplied by the convolution of the vacuum expectation value of jet and soft operators. Our approach can be generalized to other jet shapes and rates. The text in this chapter is reproduced in [40].

4.1 Introduction

Jet shapes are examples of observables with multiple scales, which can give rise to logarithmic enhancements in the fixed order rate that ruin the perturbative expansion in the strong coupling constant. Effective Field Theories (EFTs) separate the scales by expanding in their small ratio and matching onto operators that describe the degrees of freedom at each scale. The logarithmic enhancements are then summed by using the renormalization group to run between the scales restoring perturbative control of the rate. The effects of the subleading corrections from the small ratio of scales are systematically calculated in the EFT by matching onto higher dimensional operators. The appropriate theory for describing jet shapes with narrow jets is SCET [16–21, 26]. Deriving factorization theorems that separate the scales is straightforward at leading order (LO) in SCET due to the explicit decoupling of the collinear and soft degrees of freedom. Subleading corrections to SCET have previously been used to study subleading corrections to B decays [23]; however, jet shapes are complicated by kinematic cuts placed on the phase space.

41 Chapter 4. Subleading Corrections To Thrust Using EFT 42

The goal of this chapter is to demonstrate how to factorize the subleading corrections to jet shapes using SCET by considering the example of the thrust observable. Thrust [9] is deﬁned by

1 X τ = min Ei + ~t · ~pi,Ei − ~t · ~pi , (4.1) Q i∈X where Q is the initial energy, X is the final state, and the thrust axis ~t is the unit vector that P ~ maximizes i∈X |t · ~pi|. When τ 1 the final state is a pair of back-to-back jets with small invariant mass. Thrust is a convenient observable to illustrate how to calculate subleading jet shapes in SCET due to its simple phase space. We will concern ourselves only with e+e− → γ∗ → X in order to reduce the contribution from initial state radiation and restrict ourselves to vector currents. It is possible to generalize to weak process but for simplicity, we will not consider axial currents in this chapter. The LO in τ thrust rate was calculated using SCET in [41,42]. The rate was written in the factorized form H × h0|J|0i ⊗ h0|J¯|0i ⊗ h0|S|0i + O(τ), (4.2) where the convolution is defined in Section 4.3. The LO rate was factorized by matching onto SCET and expanding the final state phase space imposed by (4.1) in the SCET power √ counting. The jet operators, J and J¯, describe the physics at the intermediate scale τQ, the soft operator, S, describes the physics at the soft scale τQ, and the hard function, H, describes the physics above the hard scale Q. Factorization allows the large ln τ’s to be summed by separately renormalizing the jet and soft operators. This was done in [42]. We are interested in extending the results of [41, 42] to include the O(τ) corrections to the rate. We restrict our calculation to the perturbative regime when the soft scale is well above

ΛQCD. We will ignore the effects of hadron masses, which have been discussed in [43]. We will show how the O(τ) rate can be factorized as in (4.2) using SCET with the appropriate subleading jet and soft operators. These subleading operators are generalizations of the LO operators, and properly separate the scales while having consistent power counting. We leave renormalizing these operators to a future work. Understanding how to incorporate subleading phase space effects is important for writing a factorization theorem beyond LO. We will show how subleading phase space is accounted for in the effective theory by 1) consistent expansion of the cuts using the SCET power counting, and 2) insertions of subleading operators that account for the QCD momentum conservation expansion. Both these effects must be accounted for in order to calculate the O(τ) corrections to the rate and reproduce the perturbative QCD result at O(αsτ) in [44]. The rest of the chapter is organized as follows: in Section 4.2 we review the description of SCET in [26] and explain our reasoning for using this formulation over the formulation in [16–20]. In Section 4.3 we review the LO calculation and introduce the notation used in the rest of the chapter. We calculate the O(αsτ) rate in Section 4.4 and demonstrate how to write Chapter 4. Subleading Corrections To Thrust Using EFT 43 it in a factorized form. We conclude in Section 4.5. The full list of operators is reserved for the appendices. 2 2 2 √ 2 2 2 For simplicity we use the notation LH ≡ ln(µ /Q ), LJ ≡ ln(µ /(Q τ) ), LS ≡ ln(µ /(Qτ) ), 2 andα ¯s ≡ αsCF /(2π) where CF = (NC − 1)/(2NC ) for NC colours.

4.2 Review of SCET

Soft-Collinear Eﬀective Theory describes the interactions between highly boosted and low energy degrees of freedom. Three types of ﬁelds are required for calculating thrust: n-collinear, n¯-collinear, and soft, which have characteristic momentum scaling∗ in terms of the SCET expansion parameter λ 1

µ 2 µ 2 µ 2 pn ∼ Q(λ , 1, λ), pn¯ ∼ Q(1, λ , λ), ks ∼ Qλ (4.3) respectively. The two collinear sectors describe fields that are highly energetic and moving in opposite directions, while the soft sector describes low energy fields. The SCET Lagrangian and operators are derived by expanding the interactions between the different fields in λ. Momenta denoted by k will be O(λ2)Q unless otherwise specified. We will use the SCET formulation of [26]. In this formulation, soft and collinear fields are described by QCD in the absence of an external current. Therefore, the SCET Lagrangian is

n n¯ s LSCET = LQCD + LQCD + LQCD, (4.4)

m which has no subleading contributions and each LQCD will describe ﬁelds from the m sector only. The interactions between the sectors are contained in the external currents. The QCD vector current µ µ J2 (x) = ψ¯(x)γ ψ(x) (4.5) is matched onto the SCET dijet operators

µ (0) (0)µ 1 X (i) (i)µ J2 → C O + C O + ... (4.6) 2 2 Q 2 2 i≥1 where the superscripts denote the suppression in λ and the ellipses represent higher dimensional (i) operators. The operators and their tree-level matching coeﬃcients C2 are found by expanding the diagrams in Figure 4.1 in λ for n-collinear,n ¯-collinear, and soft ﬁelds. At LO, the (anti-)quark must be (¯n-)n-collinear and the gluon can be either collinear or

∗ µ + − + − The momentum p = (p , p , p⊥) is deﬁned in light-cone coordinates by p ≡ p · n = E − ~p · ~n, p ≡ p · n¯ = µ µ + n¯µ − nµ E + ~p · ~n, and p⊥ = p − p 2 − p 2 where n · n¯ = 2. Chapter 4. Subleading Corrections To Thrust Using EFT 44

(a) (b)

Figure 4.1: QCD vertex diagrams required for dijet production at O(αs). soft. The LO dijet operator is [26] h i h i h i (0)µ ¯ (3) ∞ (3) ∞s µ (3) ∞s (3) ∞ O2 (x) = ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) Yn (xn , 0)γ Yn¯ (0, xn¯ ) Wn¯ (xn , xn)Pn¯ψn¯(xn) , (4.7) with one-loop matching coeﬃcient [45]

2 2 2 (0) 1 2 µ 3 µ π 2 C (µ) = 1 − α¯s ln + ln + 4 − + O(α ) (4.8) 2 2 −Q2 2 −Q2 12 s and MS counterterm

2 (0) 1 3 1 µ 2 Z (µ) = 1 +α ¯s − − − ln + O(α ). (4.9) 2 2 2 −Q2 s

The interactions between the diﬀerent sectors are reproduced by Wilson lines deﬁned in a representation R

Z n ·(y−x) ! (R) 2 a a Wn (x, y) = P exp −ig ds n¯ · An(x +ns ¯ )TR 0 Z n¯ ·(y−x) ! (R) 2 a a Yn (x, y) = P exp −ig ds n · As (x + ns)TR (4.10) 0

(R) (R) with similar deﬁnitions for Wn¯ and Yn¯ . The projectors Pn = (/n/n¯)/4 and Pn¯ = (n/n/¯ )/4 are required from the expansion of the Dirac equation. The positions are deﬁned as

∞ ∞s xn = (0, x · n,¯ ~x⊥) xn = (0, ∞, ~x⊥) xn = (0, ∞, 0)

∞ ∞s xn¯ = (x · n, 0, ~x⊥) xn¯ = (∞, 0, ~x⊥) xn¯ = (∞, 0, 0) (4.11) and are necessary to conserve the appropriate components of momentum that respect (4.3). Each square bracket in (4.7) is a separately gauge invariant piece, which means the sectors explicitly decouple from one another. The physical interpretation of the operator is given in [26]. Chapter 4. Subleading Corrections To Thrust Using EFT 45

The next-to-leading order (NLO) dijet operators and matching coefficients were found in [26] and are reproduced in Section 4.6. The operators are generalizations of the LO operator (4.7) with appropriate derivative insertions. For O(τ) corrections to thrust, we will also need the N2LO operators, which are found by following the approach of [26]. These operators and their matching coefficients are also shown in Section 4.6. Both the NLO and N2LO dijet operators explicitly decouple the sectors in the same way as the LO operator. A more widely used formulation of SCET [16–20] separates the collinear momentum into p =p ˜+k, where k ∼ λ2Q andp ˜ ∼ Q, λQ are the residual and label momentum respectively. The large label momentum is removed from all interactions leading to the Lagrangian being a non- local expansion of two-component spinors with soft and collinear fields explicitly interacting at NLO. Although, the formalisms of [16–20] and [26] are equivalent, they approach momentum conservation differently. This is important when cuts are placed on phase space such as for the thrust rate. In the approach of [26], momentum is not conserved and the subleading (δ) O2 operators in Section 4.6 account for the expansion of QCD momentum conservation using SCET momentum power counting. In the approach of [16–20], label and residual momentum are separately conserved meaning momentum is exactly conserved. The equivalent action of (1δ) the O2 operator comes from the subleading kinetic interaction [46]

(1) X ⊥ 1 ⊥ /n L = ξ¯n,p˜(x)i/∂ P/ ξn,p˜0 (x) + h.c (4.12) ξξ gs=0 P¯ 2 p,˜ p˜0

µ where ξn,p˜ is a two-component n-collinear spinor with label momentump ˜, and P is an operator that pulls down label momentum. Insertions of this term into time-ordered products is equivalent to expanding the on-shell condition (˜p + k)2 = 0 in λ. These terms are not necessary for inclusive phase space calculations such as subleading B decays [23]. However, they will be important when cuts are placed on phase space such as in the subleading thrust rate and (1δ) reproduce the action of the O2 operator in time-ordered products. In this chapter we choose to use the formalism of [26]. The Lagrangian of [26] has simpler (1δ) (1) Feynman rules and only one insertion of O2 is required instead of an insertion of Lξξ for each collinear ﬁeld. The explicit decoupling of sectors at the operator level also makes it easier to derive a subleading factorization theorem.

4.3 Leading Order Calculation

The LO thrust distribution was calculated using the approach of [16–20] in [42] and was written in the factorized form (4.2). In this section we review the calculation using the formalism of [26], which gives an equivalent form of the answer. In the next section we generalize this description to calculate the O(τ) rate. Chapter 4. Subleading Corrections To Thrust Using EFT 46

The thrust rate is the cumulate of the distribution Z Z 1 0 dσ 0 4 −iQ·x µ† ˆ R(τ) = dτ 0 θ(τ − τ ) = d xe h0|J2 (x)MQCD(τ)J2µ(0)|0i (4.13) σ0 dτ

µ µ µ where Q = (Q/2)(n +n ¯ ) is the momentum of the incoming photon and σ0 is the Born cross-section. The measurement operator, Mˆ QCD(τ) [41, 47], acts on states |Xi

ˆ MQCD(τ)|Xi ≡ MQCD(τ, {pX })|Xi (4.14) to project only those ﬁnal states that give a thrust value τ. When taking the cuts of diagrams, the function MQCD(τ, {pX }) generates the appropriate phase space by restricting the momentum of the particles {pX }.

To factorize the rate, we ﬁrst match the QCD currents and measurement operators onto SCET dijet and measurement operators. The matching of the QCD currents onto SCET dijet operators was discussed in Section 4.2. The QCD measurement operators are matched onto SCET measurement operators in a similar manner

ˆ (0) (1) (2) 3 MQCD(τ) → Mc (τ) + Mc (τ) + Mc (τ) + O(λ ), where the superscripts refer to the suppression in λ. The SCET measurement operators are found by expanding the thrust constraints implemented by MQCD using the SCET momentum scaling (4.3).

Thrust is measured with respect to the thrust axis, ~t, defined below (4.1). The definition of the thrust axis in SCET has an expansion in λ and is written as ~t = ~t (0) + O(λ2). The SCET momentum power counting defines the LO thrust axis ~t (0) = −~n [41], where we have chosen the

−~n axis to be exactly along the totaln ¯-collinear momentum (i.e. ~pn¯⊥ ≡ 0). The overall sign of ~t is unimportant as seen in (4.1). The sectors decouple in the LO measurement operator [41,42]

(0) (0) (0) (0) Mc (τ) = Mcn (τn) ⊗ Mcn¯ (τn¯) ⊗ Mcs (τs) (4.15) because the thrust axis is independent of any individual particle. The convolution above is deﬁned as Z f1(τ1) ⊗ f2(τ2) ⊗ f3(τ3) ≡ dτ1dτ2dτ3θ(τ − τ1 − τ2 − τ3)f1(τ1)f2(τ2)f3(τ3). (4.16) Chapter 4. Subleading Corrections To Thrust Using EFT 47

Using the LO deﬁnition of the thrust axis, the action of the measurement operators are

P − d−2 2 ! p 1 X |~pi⊥| M(0)(τ, {p}) = i i δ τ − n Q Q − i pi P + d−2 2 ! (0) p 1 X |~pi⊥| M (τ, {p}) = i i δ τ − n¯ Q Q + i pi 1 (0) (0) M(0)(τ, {k}) = δ τ − n¯ · k + n · k (4.17) s Q +t −t where the sums are only over the momentum in each sector. We have deﬁned

µ X µ ~ ~ (0)µ (2)µ 4 k±t = ki θ(±ki · t ) = k±t + k±t + O(λ ) (4.18) i as the total soft momentum in the ±~t hemisphere. The LO deﬁnition is

(0)µ X µ ~ ~(0) X µ ~ k±t = ki θ(±ki · t ) = ki θ(∓ki · ~n), (4.19) i i where in both (4.18) and (4.19) the sum is over all soft particles. The d = 4 − 2 dependent prefactors in the collinear sectors come from choosing the collinear fields to be in the nµ andn ¯µ directions. At LO these prefactor have no affect, but are important for the O(τ) corrections. By matching the QCD operators in (4.13) onto the SCET operators, the LO thrust rate is written as Z (0) 2 d −iQ·x (0)µ† (0) (0) R(τ) = |C2 | d xe h0|O2 (x)Mc (τ)O2 µ(0)|0i + O(τ) (4.20) where the O(τ) corrections will be calculated by the subleading in λ operators. The rate is factorized by matching above the operator product onto jet and soft operators Z d −iQ·x (0)µ† (0) (0) (0) (0) ¯(0) (0) d x e O2 (x)Mc (τ)O2µ (0) = C J (τn) ⊗ J (τn¯) ⊗ S (τs) (4.21) with matching coefficient C(0). As usual, the superscripts on the jet and soft operators refer to their suppression in λ. The rate can then be written in the desired factorized form

(0) (0) 2 (0) H (µ) = |C2 (µ)| C (µ) (4.23) is the product of the matching coeﬃcients. The explicit decoupling of n-collinear,n ¯-collinear, and soft degrees of freedom in the dijet Chapter 4. Subleading Corrections To Thrust Using EFT 48

0 xn

(a) h0|J (0)|0i (b) h0|J (0)|0i (c) h0|J (0)|0i (d) h0|J (0)|0i

n n

n¯ n¯

(e) h0|S(0)|0i (f) h0|S(0)|0i (g) h0|S(0)|0i (h) h0|S(0)|0i

Figure 4.2: One-loop diagrams of (4.25). Solid lines represent fermions, dashed lines represent Wilson lines with the colour ﬂowing in the direction of the arrows, and the dots represent Lagrangian insertions. The type of soft Wilson lines are labelled in Figure 4.2e. The cut is distinguished by the bold vertical dashed line. The contributions to J¯(0) look identical to the contributions of J (0) after a rotation of 180◦.

and measurement operators makes ﬁnding the appropriate jet and soft operators straightforward. Using the Fierz identity to separate the spin and colour indices, we ﬁnd the operators are

1 Z + n/¯ (0) + d−2 −iQx ¯ (3) ∞s (0) (3) ∞ J (µ, τ) = Tr dx d x⊥e ψn(0)Wn (0, xn¯ ) Mcn (τ)Wn (xn¯ , xn¯)ψn(xn¯) NC 2 1 Z − /n ¯(0) − d−2 −iQx ¯ (3) ∞ (0) (3) ∞s J (µ, τ) = Tr dx d x⊥e ψn¯(xn)Wn¯ (xn, xn ) Mcn¯ (τ)Wn¯ (xn , 0)ψn¯(0) NC 2

(0) 1 (3) ∞s (3) ∞s (0) (3) ∞s (3) ∞s S (µ, τ) = Tr Yn¯ (xn¯ , 0)Yn (0, xn )Mcs (τ)Yn (xn , 0)Yn¯ (0, xn¯ ), (4.24) NC where the trace is over spins and colour. These operators give the same Feynman rules as those found in [42]. The matching coeﬃcient in (4.21) is most easily found by comparing the vacuum expectation value of both sides. The real emission contributions to the one-loop vacuum expectation value of J (0) and S(0) are pictured in Figure 4.2. They are calculated by cutting the diagrams along the vertical dashed lines and applying the measurement operator to the ﬁelds passing through this cut [42]. The virtual diagrams are scaleless and thus zero in Chapter 4. Subleading Corrections To Thrust Using EFT 49

MS. The vacuum expectation value of the jet and soft operators are [42]

2 (0) 2 2 3 2 3 7 π h0|J (µ, τ)|0i = δ(τ) 1 +α ¯s + LH + + L + LH + − 2 2 H 2 2 2 2 3 θ(τ) +α ¯s − − − 2LJ 2 τ + h0|J¯(0)(µ, τ)|0i = h0|J (0)(µ, τ)|0i (4.25) 2 (0) 2 2 2 π 4 θ(τ) h0|S (µ, τ)|0i = δ(τ) 1 +α ¯s − 2 − LH − LH + +α ¯s + 4LS , 6 τ + where we have included the zero-bin procedure [22], which accounts for the double counting between the collinear and soft operators. The matching coeﬃcient C(0)(µ) = 1 [42] meaning the hard function is

2 (0) 2 7π 2 H (µ) = 1 +α ¯s L − 3LH − 8 + + O(α ). (4.26) H 6 s √ As expected, the jet and soft operators separate the τQ and τQ scales. The hard function describes the physics above the cut-oﬀ Q, of the eﬀective theory.

Matrix elements of the jet and soft operators also do not need any further expansion in τ. This is required to ensure that operators of diﬀerent orders do not mix. We note that this is diﬀerent than the results in [35], which considered the exclusive JADE two-jet rate at LO.

This rate has the same O(αs) phase space as the thrust rate in QCD. In [35], the phase space was not consistently expanded in λ and an expansion in τ was required after the phase space integration. A subleading zero-bin procedure is necessary to ensure that the LO operators do not contribute to subleading corrections. In this chapter and in [42], the measurement operator is consistently expanded in λ so matrix elements of the jet and soft operators automatically have consistent power counting.

0 The thrust rate at O(αsτ ) is calculated by substituting the hard function and the vacuum expectation values of the jet and soft operators into (4.22). The rate is found to be

2 2 π 2 R(τ) = 1 +α ¯s −2 ln τ − 3 ln τ + − 1 + O(α , τ), 3 s which reproduces the rate found in perturbative QCD at this order [42]. By separately renormalizing the jet and soft operators the large ln τ’s were summed in [42]. In the next section we follow the same procedure to ﬁnd the O(αsτ) correction to the rate and write it in a factorized form analogous to (4.22). Chapter 4. Subleading Corrections To Thrust Using EFT 50

4.4 Next-to-Leading Order Calculation

The results are extended to include the O(τ) corrections by systematically matching the QCD current and measurement operator in (4.13) onto subleading SCET dijet and measurement operators. The O(τ) thrust rate in SCET is written as Z X (i)∗ (j) 4 −iQ·x (i)†µ (k) (j) 2 R(τ) = C2 C2 d xe h0|O2 (x)Mc (τ)O2µ (0)|0i + O(τ ). (4.27) i+j+k≤2 √ From the LO calculation λ ∼ τ, so we need the O(λ2) SCET operators as illustrated by the constraints on the sum. One can explicitly check that the O(λ) corrections, which would give √ O( τ) corrections, vanish. As in the previous section, we want to write (4.27) in a factorized form by matching onto subleading jet and soft operators. These operators will be generalizations of the LO operators and their matrix elements must have a consistent power counting in τ as in the LO case. We will only do the tree-level matching, which is all that is necessary to calculate the O(αsτ) rate. The subleading dijet operators are written in Section 4.6 and explicitly decouple the sectors. We must also ﬁnd the subleading measurement operators, Mc(1,2)(τ) in (4.15). We will show in the next section that the action of the subleading measurement operators also decouples the sectors, analogously to (4.15). The explicit decoupling of the sectors in the dijet and measurement operators makes writing the rate in the desired factorized form in Section 4.4.2 straightforward.

4.4.1 Measurement Operator

The action of the subleading measurement operators are found by first expanding the definition of the thrust axis and then finding this expansion’s effect on the measurement of thrust. The thrust axis is defined as the unit vector that maximizes the sum below (4.1). The sum is maximized when the thrust axis is in the direction of the hemisphere with the largest three- momentum, ~p+t [41]. Therefore, the thrust axis is written in SCET as the expansion

~p+t ~t = = ~t (0) + ~t (2) + ~t (4) + O(λ6), (4.28) |~p+t| where the superscripts refer to the suppression in λ.

In order to ﬁnd ~p+t, we ﬁrst note that the n-collinear andn ¯-collinear particles are always in opposite hemispheres. SCET momentum power counting enforces this at LO and the zero-bin procedure will enforce this at all orders in λ. Therefore, the total momentum of the hemisphere will be

~p+t = ~pn¯ + ~k+t, (4.29)

µ µ where pn¯ is the totaln ¯-collinear momentum and k+t is the total soft momentum in the +~t hemisphere as deﬁned in (4.18). The expansion of (4.18) and (4.28) allows us to iteratively Chapter 4. Subleading Corrections To Thrust Using EFT 51

(i) (i)µ solve for both ~t and k±t . The subleading corrections to the thrust axes are

(0) 2~k ~(2) +t⊥ t = + (4.30) pn¯ (0) (0) (2) 2(~k )2 (2~n · ~k + k−) 2~k ~(4) +t⊥ +t n¯ ~ (0) +t⊥ t = + 2 ~n + + 2 k+t⊥ + + (pn¯ ) (pn¯ ) pn¯

− µ where kn¯ is the totaln ¯-collinear momentum in the n direction. The subleading correction to the total soft momentum in each hemisphere

(2)µ (2)µ 1 X µ ~ (0) ~ ~ k+t = −k−t = + ki 2k+t⊥ · ki⊥ δ(−~n · ki) (4.31) pn¯ i is found by inserting the ~t (2) into (4.18). The ﬁrst equality above is because the sum of the soft momentum in the two hemispheres must be O(λ2).

We note we could have instead used −~p+t = ~pn + ~k−t in (4.28), where ~pn is the total n- collinear momentum and ~k−t is the total soft momentum in the −~t hemisphere. However, the deﬁnition in (4.29) is simpler due to our choice of ~n in Section 4.3 such that ~pn¯⊥ = ~0. The apparent asymmetry in the labelling of nµ andn ¯µ in the resulting phase space will be accounted (1δ) (2δ⊥) ~ for by the O2 and O2 operators in Section 4.6. The choice of ~pn¯⊥ ≡ 0 means the Feynman rules of these operators involve ∂/∂~pn⊥’s only. The subleading measurement operators are found by substituting the corrections to the thrust axis into (4.1). We ﬁrst consider the contribution from an n-collinear particle with momentum pi. As discussed in the second paragraph of this section, Ei + ~pi · ~t is always the minimum for each n-collinear particle. Therefore, the n-collinear sector contributes

1 X (0) 1 (2) (4) X 5 (Ei + ~pi · ~t ) + ~t + ~t · ~pi + O(λ ) (4.32) Q Q i i to the thrust, where the sum is only over n-collinear particles. The ﬁrst term reproduces the action of the LO measurement operator in (4.17). The t(2) term gives the NLO correction and the ﬁrst term of t(4) in (4.30) gives the N2LO correction. This power counting is due to p⊥i ∼ O(λ)Q for n-collinear particles. The contribution to thrust from then ¯-collinear particles is found in a similar way. Here, the Ei − ~pi · ~t is always the minimum so then ¯-collinear sector contributes

1 X (0) 1 (4) X 5 (Ei − ~pi · ~t ) − ~t · ~pi + O(λ ), (4.33) Q Q i i where the sum is only overn ¯-collinear particles. The ﬁrst term is the LO contribution and the 2 (2) second term is the N LO contribution. There is no ~t term because we have set ~pn¯⊥ = ~0 by our choice of ~n. Chapter 4. Subleading Corrections To Thrust Using EFT 52

Unlike the collinear particles, soft particles can be in either hemisphere. The minimum of

Ei ± ~t · ~ki is determined by which hemisphere the soft particle is in. Therefore, the soft sector contributes

1 (0) (0) 1 (2) (2) (0) (0) n · k +n ¯ · k + n · k +n ¯ · k + ~t (2) · ~k − ~k + O(λ6). (4.34) Q +t −t Q +t −t +t −t to the total thrust. The ﬁrst line is the LO contribution in (4.17) and the remaining terms are all N2LO.

The action of the subleading measurement operators is found by Taylor expanding the contribution to thrust in λ. We incorporate the NLO and N2LO corrections from the collinear sectors in (4.32) and (4.33) by writing the action of the subleading measurement operators as

α (1n) pn⊥ ∂ (0) Q (0) M (τ, {pn, pn¯, ks}) = Mn (τn, {pn}) ⊗ + Mn¯ (τn¯, {pn¯}) Q ∂τn pn¯ (0)α ! −2k+t⊥ (0) ⊗ M (τs, {ks}) Q s α β 2 ! 2 (2na) pn⊥pn⊥ ∂ (0) Q (0) M (τ, {pn, pn¯, ks}) = 2 2 Mn (τn, {pn}) ⊗ +2 Mn¯ (τn¯, {pn¯}) Q ∂τn pn¯ (0)α (0)β ! 2k+t⊥k+t⊥ (0) ⊗ M (τs, {ks}) Q2 s − 2 p ∂ Q (0) M(2nb) { } n M(0) { } ⊗ M { } (τ, pn, pn¯, ks ) = n (τn, pn ) + 2 n¯ (τn¯, pn¯ ) Q ∂τn (pn¯ ) ~ (0) 2 ! −(k+t⊥) (0) ⊗ M (τs, {ks}) (4.35) Q2 s + (2¯nb) (0) pn¯ ∂ (0) M (τ, {pn, pn¯, ks}) = Mn (τn, {pn}) ⊗ Mn¯ (τn¯, {pn¯}) Q ∂τn¯ ~ (0) 2 ! (k+t⊥) (0) ⊗ M (τs, {ks}) . Q2 s

M(2na) comes from the second term in the expansion of the NLO correction in (4.32). The N2LO corrections from the soft sector in (4.34) are incorporated by the subleading measurement operators (2si) (0) Q (0) (2si) M (τ, {pn, pn¯, ks}) = Mn (τn, {pn}) ⊗ + Mn¯ (τn¯, {pn¯}) ⊗ Ms (τs, {ks}) (4.36) pn¯ Chapter 4. Subleading Corrections To Thrust Using EFT 53 where

(0) (0) 2~k · ~k (2s1) +t⊥ +t⊥ ∂ (0) Ms (τ, {ks}) = Ms (τs, {ks}) Q ∂τs (0) (0) −2~k · ~k (2s2) +t⊥ −t⊥ ∂ (0) Ms (τ, {ks}) = Ms (τs, {ks}) Q ∂τs + · (2) (2s−t) pn¯ n k+t ∂ (0) Ms (τ, {ks}) = 2 Ms (τs, {ks}) (4.37) Q ∂τs + · (2) (2s+t) pn¯ n¯ k−t ∂ (0) Ms (τ, {ks}) = 2 Ms (τs, {ks}). Q ∂τs

+ (2s±t) The pn¯ in the action of the soft measurement operators M are introduced to cancel the + pn¯ in (4.31). The actions of the measurement operators (4.35) and (4.36) define the NLO and N2LO measurement operators Mc(1,2)(τ). As the brackets suggest, the sectors explicitly decouple in the action of the subleading measurement operators. This is due to the corrections to thrust depending only on the total momentum of each collinear sector and not any individual particle. While it is possible to formally write the subleading measurement operators using the energy- flow operator [41,42] and not just their actions in momentum space, we see no reason to do so: only their actions in momentum space are necessary in calculations. We note that the measurement operators in this section are found using the formalism of [26] and would be different if we used the SCET formalism of [16–20]. It was suggested in [41] the subleading measurement operators could be found using the subleading terms in the SCET Lagrangian of [16–20]. While we do not explicitly check this, we note that the breaking of the explicit decoupling of soft and collinear fields in the subleading Lagrangian would complicate factorization.

4.4.2 Factorization

The explicit decoupling of the sectors in the subleading dijet and measurement operators makes it straightforward to factorize the subleading corrections to the rate. In order to factorize the rate, each operator product in (4.27) is matched onto the appropriate jet and soft operators Z d (i)† (k) (j) X (i,j,k) (l) ¯(m) (n) d x O2 (x)Mc (τ)O2 (0) = Cα J (τn) ⊗ J (τn¯) ⊗ S (τs) (4.38) α(l,m,n) (i,j,k) (0) ¯(0) (0) + τ C0 J (τn) ⊗ J (τn¯) ⊗ S (τs),

(i,j,k) 2 with matching coeﬃcients Cα,0 . The last line is made N LO by the explicit τ in front. The integer α ≥ 1 labels the combination of jet and soft operators and we require i + j + k = 2 = Chapter 4. Subleading Corrections To Thrust Using EFT 54 l + m + n. The O(τ) rate can then be written in the factorized form

(i,j,k) (i)∗ (j) (i,j,k) Hα,0 (µ) = C2 (µ)C2 (µ)Cα,0 (µ). (4.40)

This generalizes the LO factorization (4.22) to incorporate the N2LO corrections and is the main result of our chapter. While it is possible to calculate the O(αsτ) rate directly from (4.27), the factorized form will allow the jet and soft operators to be renormalized separately.

Below we will use a few examples to demonstrate how the jet and soft operators in (4.38) are found. The full list of operators and their matching coeﬃcients are found in Section 4.7. For the sake of brevity, we only write those operators that contribute to the O(αsτ) rate. We omit the phase space integrals when calculating matrix elements of the operators to avoid potentially confusing the reader.

A) i = 1an, j = 1bn, k = 0 : The operators are found in (3.34). The left-hand side of (4.38) is

Z h i h i d −iQ·x 1 (3) ∞ (3) ∞s (3) ∞s − d xe ψ¯n¯(xn)W (xn, x ) Y (x , 0)Y (0, x ) Q n¯ n n¯ n¯ n n

h (3) ∞ ←−β1 β1µ i (0) (0) (0) × Wn (xn¯ , xn¯)iD ⊥ (xn¯)Γ1 ψn(xn¯) Mcn (τn) ⊗ Mcn¯ (τn¯) ⊗ Mcs (τs) (4.41) h ←− i h i h i 1 β2 β2µ (3) ∞s (3) ∞s (3) ∞s (3) ∞s × ψ¯n(0)iD (0)Γ W (0, x ) Y (x , 0)Y (0, x ) W (x , 0)ψn¯(0) Q ⊥ 2 n n¯ n n n¯ n¯ n¯ n

αµ /n α µ αµ n/¯ α µ µ where Γ1 = 2 γ⊥γ Pn and Γ2 = 2 γ⊥γ Pn¯ and γ⊥ is deﬁned in Section 4.7. The 1/Q’s come from the deﬁnition of the operator expansion (4.6) and the negative sign comes from taking the hermitian conjugate. The square brackets denote the separately gauge invariant pieces of each sector. As for the LO factorization, we use the Fierz identity to separate the spin and colour indices and match onto the appropriate jet and soft operators

(1an,1bn,0) (2abn) ¯(0) (0) (1an,1bn,0) (0) ¯(0) (0) C1 J (τn) ⊗ J (τn¯) ⊗ S (τs) + τ C0 J (τn) ⊗ J (τn¯) ⊗ S (τs). (4.42)

The LO operators J (0), J¯(0) and S(0) are deﬁned in (4.24) and the subleading jet operator is

1 Z + (2abn) + d−2 −iQx ¯ α (3) ∞s J (µ, τ) = 2 Tr dx d x⊥e ψn(0)iD⊥(0)Wn (0, xn ) (4.43) Q NC n/¯ (0) (3) ∞ × Mc (τ)W (x , xn¯)iD⊥α(xn¯)ψn(xn¯). (4.44) 2 n n n¯ Chapter 4. Subleading Corrections To Thrust Using EFT 55

(a) (b) (c)

(2abn) Figure 4.3: Matrix element h0|J |0i that give non-zero τ values at O(αs).

The operator is suppressed by λ2 due to the derivative insertions. The vacuum expectation value of this operator is shown in Figure 4.3. In order to find the matching coefficients, we calculate the vacuum expectation value of (4.41) and (4.42). The Feynman rules for the dijet operators were written in [26], and we find the vacuum expectation value of (4.41) is

1 2 − α¯sτ + O(α ). (4.45) 2 s

The vacuum expectation values of J (0), J¯(0), and S(0) were found in (4.24) and the diagrams in Figure 4.3 lead to

(2abn) 1 2 h0|J (µ, τ)|0i = α¯s + O(α ). (4.46) 2 s Here and in the calculations below we have included the zero-bin procedure. As expected, the matrix element of the J (2abn) operator is suppressed by τ ∼ λ2 compared to the LO jet operator. (1an,1bn,0) (1an,1bn,0) 2 The matching coeﬃcients are found to be C1 = −1+O(αs) and C0 = 0+O(αs) meaning the hard functions are

(1an,1bn,0) H1 (µ) = 1 + O(αs) (1an,1bn,0) 2 H0 (µ) = 0 + O(αs). (4.47)

Therefore, the contribution of (4.41) to the rate can be written in the factorized form (4.39), with the appropriate subleading jet operator. (2δn) B) i = 2δn, j = k = 0 : The operator O2 is shown in (4.59) and accounts for the matching of QCD momentum conservation onto SCET momentum conservation at O(λ2). The (2δn) contribution where j = 2δn in (4.38) means taking O2 (0), which vanishes due to the explicit x dependence in the dijet operator. As in the previous example, we use the Fierz identity to factorize the contribution from this operator and match onto Z d (2δn)† (0) (0) (2δn,0,0) (2δn) ¯(0δn) (0) d x O2 (x)Mc (τ)O2 (0) = C1 J (τn) ⊗ J (τn¯) ⊗ S (τs)

(2δn,0,0) (0) ¯(0) (0) + τ C0 J (τn) ⊗ J (τn¯) ⊗ S (τs), (4.48) Chapter 4. Subleading Corrections To Thrust Using EFT 56 where the new jet operators are

1 Z + (2δn) + d−2 −iQx ¯ (3) ∞s J (τ) = Tr dx d x⊥e ψn(0)Wn (0, xn ) QNC n/¯ (0) (3) ∞ ←− × Mc (τ)W (x , xn¯) in · D + in · D (xn¯)ψn(xn¯) (4.49) 2 n n n¯ iQ Z − ¯(0δn) − d−2 −iQx − ¯ (3) ∞ J (τ) = Tr dx d x⊥e x ψn¯(xn)Wn¯ (xn, xn ) NC

/n (0) (3) ∞s × Mc (τ)W (x , 0)ψn¯(0). 2 n¯ n¯ n¯

The derivatives in J (2δn) pull down the O(λ2) component of the n-collinear momentum leading 2 − (2δn) to a λ suppression compared to the LO jet operator. The x dependence of O2 is put into J¯(0δn) because only then ¯-collinear ﬁelds depend on this coordinate. We have introduced the Q’s so the operators will have the correct mass dimensions. The diagrams of the vacuum expectation value of J (2δn) and J¯(0δn) are the same as the diagrams of the vacuum expectation value of J (0) and J¯(0) in Figure 4.2. As before, the matching coeﬃcient is most easily determined by comparing the vacuum (2δ ) − n ¯(0δn) expectation value of both sides of (4.48). The explicit x in both O2 and J , which is + (0) a derivative ∂/∂pn¯ in momentum space, acts on the d-dependent prefactor in Mn¯ of (4.17). Therefore, the vacuum expectation value of the left-hand side of (4.48) is 4 2 α¯sτ − − 4LJ − 3 + O(α ). (4.50) s and the vacuum expectation value of the jet operators are (2δn) 2 3 2 h0|J (µ, τ)|0i =α ¯s − − 2LJ − + O(α ) 2 s

(0δn) h0|J¯ (µ, τ)|0i = 2δ(τ) + O(αs). (4.51)

(2δn,0,0) Comparing both sides of (4.48) we ﬁnd the matching coeﬃcients are C1 (µ) = 1 + O(αs) (2δn,0,0) 2 and C0 (µ) =α ¯s + O(αs). We see in this example why the last line of (4.38) is necessary: (0) the d−2 from the derivative acting on Mn gives an extra term in (4.50) when it is analytically continued to d = 4 compared to h0|J¯(0δn)|0i, which has no poles at d = 4. The contribution of this operator product is thus factorized in the form of (4.39) with hard functions

(2δn,0,0) H1 (µ) = −1 + O(αs) (2δn,0,0) 2 H0 (µ) = −α¯s + O(αs). (4.52)

Again, the matrix elements of the jet operators have the appropriate power counting and the logarithm in (4.51) is minimized at the jet scale, as expected.

C) i = 1δs, j = 0, k = 1n : As a ﬁnal example, we show an insertion of the subleading Chapter 4. Subleading Corrections To Thrust Using EFT 57 measurement operator. The contribution of the NLO measurement function, k = 1n, vanishes at O(αs) unless i = 1δs. This contribution accounts for the phase space we neglected at LO due to not conserving soft momentum. We factorize this operator product by matching Z (1δ )† (0) (1δ ,0,1n) d s (1n) s (−2δsMn) ¯(0M1) (4δsM) d x O2 (x)Mc (τ)O2 (0) = C1 J (τn) ⊗ J (τn¯) ⊗ S (τs)

(1δs,0,1n) (0) ¯(0) (0) + τ C0 J (τn) ⊗ J (τn¯) ⊗ S (τs). (4.53)

The decoupling of the sectors in Mc(1n) means the NLO measurement operator can be treated identically as the LO measurement operator in (4.21) and gets pulled through when we use the Fierz identity. Therefore, the appropriate jet and soft operators are

1 Z + (−2δsMn) + d−2 −iQx ¯ (3) ∞s J (µ, τ) = Tr dx d x⊥e x⊥αψn(0)Wn (0, xn ) QNC n/¯ ←−α ∂ (0) (3) ∞ × ∂ Mc (τ)W (x , xn¯)ψn(xn¯) 2 ⊥ ∂τ n n n¯ Z ∞ Z iQ − (3) ¯(0M1) − d−2 −iQx ¯ ∞ J (µ, τ) = Tr dt dx d x⊥e ψn¯(xn)Wn¯ (xn, xn ) (4.54) NC 0

/n (0) (3) ∞s × Mc (τ)W (x , nt)ψn¯(nt) 2 n¯ n¯ n¯ ←− (4δsMn) 1 (3) ∞s (3) ∞s (2n)α S (µ, τ) = Tr Yn (xn , 0) iD⊥α + iD ⊥α (0)Yn¯ (0, xn¯ )Mcs (τ) QNC

(3) ∞s (3) ∞s × Yn¯ (xn¯ , 0)Yn (0, xn ) where (0)α 2k M(2n)α(τ, {k}) = +t⊥ M(0)(τ, {k}). (4.55) s Q s

(1δs) (−2δsMn) The explicit x⊥ dependence in O2 is put into J because we chose the ~n axis such ~ µ (1n) that ~pn¯⊥ ≡ 0. The pn⊥ in Mn acts as a total derivative at the cut so becomes a derivative † −1 R ∞ at inﬁnity . We have also used the identity [18, 21] (in · ∂) φ(x) = 0 dt φ(x + nt) to write + ¯(0M1) the 1/pn¯ in (4.35) as a displacement in the J operator. The power counting suggests J (−2δsMn) ∼ λ−2 will be enhanced compared to the LO jet operator. However, it is always convoluted with S(4δsMn) ∼ λ4, meaning the contribution to the rate will be O(λ2) as expected. The diagrams for the vacuum expectation value of the jet and soft operators have the same picture as Figure 4.2 and give

(−2δsMn) 0 h0|J (µ, τ)|0i = 2δ (τ) + O(αs)

(0M1) h0|J¯ (µ, τ)|0i = δ(τ) + O(αs) (4.56)

(4δsMn) 2 h0|S (µ, τ)|0i = −4¯αsτ + O(αs), which have the expected power counting in τ. The matching coeﬃcients are found to be

†We use a covariant gauge so the gauge ﬁeld vanishes at inﬁnity. Chapter 4. Subleading Corrections To Thrust Using EFT 58

(1δs,0,1Mn) (1δs,0,1Mn) 2 C1 (µ) = 1 + O(αs) and C0 (µ) = 0 + O(αs) and the contribution from this operator can be written in the factorized form of (4.39) with hard functions

(1δs,0,1Mn) H1 (µ) = 1 + O(αs) (1δs,0,1Mn) 2 H0 (µ) = 0 + O(αs). (4.57)

This example shows how the decoupling of the measurement operators makes it straightforward to factorize their contribution. The factorization of the rest of the subleading dijet and measurement operators follows in the same way as the above examples. The explicit decoupling of the sectors makes finding the jet and soft operators a matter of using the Fierz identity to separate spinor and colour indices. The required jet and soft operators are written in Section 4.7. These operators are generalizations of the LO operators found in Section 4.3. The matching coefficients are found by comparing the vacuum expectation values of (4.38). These matrix elements are pictured in Figure 4.4 of Section 4.7 and their values are shown in Table 4.1 of the Appendix. The appropriate matching coefficients are shown in Table 4.2. Combining the results of this Table, we find the O(αsτ) rate is

2 2 π 2 2 R(τ) = 1 +α ¯s −2 ln τ − 3 ln τ + − 1 + τ(2 ln τ − 4) O(α , τ ). (4.58) 3 s

We can compare these results with those found using perturbative QCD in [44]. Although the results in [44] are for the exclusive two-jet rate using the JADE algorithm, the O(αs) QCD phase space is the same as the phase space for thrust, as mentioned above. Therefore, the full O(αs) result calculated in [44] can be compared to (4.58). Summing the τ ln τ’s requires renormalizing the jet and soft operators, which we do not do here.

4.5 Conclusion

We have shown how to systematically calculate the O(τ) corrections to the thrust rate in the perturbative regime using SCET. The rate was factorized and written as the convolution of the vacuum expectation value of jet and soft operators. Each operator has consistent power counting and depends on a diﬀerent scale associated with the rate. The appropriate jet and soft operators are found in this work, as well as the matching coeﬃcients. The O(αsτ) rate was calculated and reproduced the rate found using perturbative QCD. The rate was factorized by matching the QCD currents and measurement operators onto SCET dijet and measurement operators, which in the formulation of [26], explicitly decouple the n-collinear,n ¯-collinear, and soft degrees of freedom. The non-local product of the dijet and measurement operators were then matched onto jet and soft operators that separately describe each of these degrees of freedom. The approach illustrated here can be applied to Chapter 4. Subleading Corrections To Thrust Using EFT 59 other jet observables to calculate subleading corrections. We are currently exploring subleading corrections to the continuous angularity observables of which thrust is an example.

4.6 Appendix: Dijet Operators

The LO dijet operator was given in (4.7). In this section we write the NLO and N2LO dijet operators necessary for calculating the O(αsτ) thrust rate. They are found by following the SCET formulation of [26]. The NLO corrections are described in [26] and in chapter 3 along with their matching coeﬃcients and Feynman rules. We also require the operators describing N2LO corrections and do so in the same way as NLO operators.

The operators describing the N2LO corrections to the n-collinear sector are h i h i (2an) ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = ψn(xn¯)Pn¯Γin · D(xn¯)Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ )

h (3) ∞ i × Wn¯ (xn , xn)Pn¯ψn¯(xn) Z ∞ ←− (2bn) ¯ n/¯ /n (3) O2 (x) = −i dt ψn(xn¯ + tn¯)i /D ⊥(xn¯ + tn¯) Γ Wn (xn¯ + tn,¯ xn¯) 0 2 2 i h i h i (3) ∞ (3) ∞s (3) ∞s (3) ∞ ×i /D⊥(xn¯)Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ ) Wn¯ (xn , xn)ψn¯(xn) Z ∞ h (2An) ¯ (3) αβ O2 (x) = −i dt ψn(xn¯)Pn¯Wn (xn¯, xn¯ + tn¯)Γgσ⊥αβGn (xn¯ + tn¯) (4.59) 0 i h i h i (3) ∞ (3) ∞s (3) ∞s (3) ∞ × Wn (xn¯ + tn,¯ xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ ) Wn¯ (xn , xn)Pn¯ψn¯(xn) h ←− i h i (2δn) ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = Q ψn(xn¯)Pn¯Γ n · D + n · D (xn¯)Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ )

h − (3) ∞ i × x Wn¯ (xn , xn)Pn¯ψn¯(xn) with matching coeﬃcients

(2an) 1 (2bn) C = − + O(αs) C = −1 + O(αs) 2 2 2

(2An) 1 (2δn) C = + O(αs) C = 1 + O(αs). (4.60) 2 2 2

(2δn) We use the same notation as in [26] and chapter 3. The Q in O2 is required because of the Chapter 4. Subleading Corrections To Thrust Using EFT 60 deﬁnition in (4.6). The operators describing corrections to then ¯-collinear sector are h i h i (2an¯ ) ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ )

h (3) ∞ i × Wn¯ (xn , xn)Γin¯ · D(xn)Pn¯ψn¯(xn) Z ∞ h i h i (2bn¯ ) ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = −i dt ψn(xn¯)Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ ) 0 (3) ∞ ←− (3) n/¯ /n × W (x , xn)i /D (xn)W (xn, xn + tn) Γ i /D(xn + tn)ψn¯(xn + tn) n¯ n ⊥ n¯ 2 2 Z ∞ h i h i (2An¯ ) ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = −i dt ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ ) 0 h (3) ∞ αβ (3) i × Wn¯ (xn , xn + tn)gGn¯ (xn + tn)σ⊥αβΓPn¯Wn (xn + tn, xn)ψn¯(xn) h i h i (2δn¯ ) + ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = Q x ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) Yn (xn , 0)Yn¯ (0, xn¯ )

h (3) ∞ ←− i × Wn¯ (xn , xn)Γ n¯ · D +n ¯ · D (xn)Pn¯ψn¯(xn) (4.61) with matching coeﬃcients the same as their n-collinear counterparts. The intuitive picture of the operators in (4.59) and (4.61) is similar to the pictures presented in [26]. The N2LO corrections for soft gluon emission are

Z ∞ h i (2Ans) ¯ (3) ∞ O2 (x) = −i dt ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) 0 h i h i (3) ∞s µν (3) (3) ∞s (3) ∞ × Yn (xn , tn)gσµνGs (tn)Yn (tn, 0)ΓYn¯ (0, xn¯ ) Wn¯ (xn , xn)Pn¯ψn¯(xn) Z ∞ h i (2Ans¯ ) ¯ (3) ∞ O2 (x) = −i dt ψn(xn¯)Pn¯Wn (xn¯, xn¯ ) 0 h i h i (3) ∞s (3) µν (3) ∞s (3) ∞ × Yn (xn , 0)ΓYn¯ (0, tn¯)gσµνGs (tn¯)Yn¯ (tn,¯ xn¯ ) Wn¯ (xn , xn)Pn¯ψn¯(xn) (4.62) with matching coeﬃcients (2Ans) (2Ans¯ ) 1 C = C = + O(αs). (4.63) 2 2 2 The corrections from the expansion of the momentum conserving delta function are described Chapter 4. Subleading Corrections To Thrust Using EFT 61 by h i h ←− i (2δs+) + ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = Q x ψn(xn¯)Pn¯ΓWn (xn¯, xn¯ ) Yn (xn , 0) n¯ · D +n ¯ · D Yn¯ (0, xn¯ )

h (3) ∞ i × Wn¯ (xn , xn)Pn¯ψn¯(xn) h i h ←− i (2δs−) ¯ (3) ∞ (3) ∞s (3) ∞s O2 (x) = Q ψn(xn¯)Pn¯ΓWn (xn¯, xn¯ ) Yn (xn , 0) n · D + n · D Yn¯ (0, xn¯ )

h − (3) ∞ i × x Wn¯ (xn , xn)Pn¯ψn¯(xn)

(2δs⊥) Q h (3) ∞ i O (x) = x x ψ¯n(xn¯)Pn¯ΓW (xn¯, x ) 2 2 ⊥α ⊥β n n¯ h ←− ←− i h i (3) ∞s α β β α (3) ∞s (3) ∞ × Yn (xn , 0) D⊥D⊥ + D ⊥ D ⊥ Yn¯ (0, xn¯ ) Wn¯ (xn , xn)Pn¯ψn¯(xn) Z ∞ h ←− i (2δcsn) ¯ ν (3) ∞ O2 (x) = −i dt x⊥µψn(xn¯)Pn¯ΓiD ⊥(xn¯)Wn (xn¯, xn¯ ) (4.64) 0 h ←− ←− i (3) ∞s (3) µ µ (3) ∞s × Yn (xn , tn)iD ⊥ν(tn)Yn (tn, 0)(D⊥ + D ⊥)(0)Yn¯ (0, xn¯ ) h (3) ∞ i × Wn¯ (xn , xn)Pn¯ψn¯(xn) Z ∞ h i (2δcsn¯ ) ¯ (3) ∞ O2 (x) = −i dt x⊥µψn(xn¯)Pn¯Wn (xn¯, xn¯ ) 0 h ←− i (3) ∞s µ µ (3) (3) ∞s × Yn (xn , 0)(D⊥ + D ⊥)(0)Yn¯ (0, tn¯)iD⊥ν(tn¯)Yn¯ (tn,¯ xn¯ ) h (3) ∞ i × Wn¯ (xn , xn)iDν⊥(xn)ΓPn¯ψn¯(xn) with matching coeﬃcients

(2δs+) (2δs−) (2δs⊥) C2 = C2 = 1 + O(αs) C2 = 1 + O(αs)

(2δcsn) (2δcsn¯ ) C2 = C2 = −2 + O(αs). (4.65)

We do not require the N2LO corrections from soft quark emission for an N2LO calculation. As expected, the subleading operators can be written as separate n-collinear,n ¯-collinear, and soft pieces. The Feynman rules for the LO and NLO operators at O(αs) were shown in [26]. For the sake of brevity, we do not show the N2LO Feynman rules here. We are only concerned with vector currents in the above calculations, so Γ = γµ.

4.7 Appendix: Jet and Soft Operators

The jet and soft operators are found by doing the matching in (4.38). It is convenient to use the basis α αβ /n n/¯ α /n n/¯ /n α n/¯ α /n n/¯ 1, γ , γ5, σ , , , γ γ5, , , γ , γ , γ5, γ5 (4.66) ⊥ ⊥ 2 2 ⊥ 2 2 2 ⊥ 2 ⊥ 2 2 Chapter 4. Subleading Corrections To Thrust Using EFT 62

sn tn

n¯ + tn¯ snx ¯

(a) h0|J (08)|0i (b) h0|J (2e)|0i (c) h0|S(2dns)|0i (d) h0|S(08)|0i

Figure 4.4: Diagrams for the operators describing a soft quark and the quark and anti-quark in the same sector. The double dashed lines represent Wilson lines in the adjoint representation. Reverse the fermion arrows for h0|J (2f)|0i. The ﬁlled boxes highlight a Wilson line changing representation.

µ µν µν i µ ν µν µν for the Dirac matrices. We have deﬁned γ⊥ = g⊥ γν and σ⊥ = 2 [γ⊥, γ⊥] where g⊥ = g − µ ν ν µ αβ αβµν (n n¯ + n n¯ )/2. For use later, we aslo deﬁne ⊥ = nµn¯ν . The jet and soft operators will be parity even scalars due to only considering vector currents.

The leading order jet operators required at O(αs) are

1 Z + n/¯ (0) + d−2 −iQx ¯ (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e ψn(0)Wn (0, xn¯ ) Mcn (τ) NC 2 (3) ∞ × Wn (xn¯ , xn¯)ψn(xn¯) αβ Z + (08) g⊥ n¯µn¯ν + d−2 −iQx µαa (8)ab ∞s (0) J (µ, τ) = dx d x⊥e Gn (0)Wn (0, xn )Mcn (τ) (4.67) QNC (8)ba¯ ∞ νβa¯ × Wn (xn¯ , xn¯)Gn (xn¯)

Q Z + n/¯ (0δn¯ ) + d−2 −iQx + ¯ (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e x ψn(0)Wn (0, xn¯ ) Mcn (τ) NC 2 (3) ∞ × Wn (xn¯ , xn¯)ψn(xn¯) (4.68)

1 Z + ←− n/¯ (0cnsδs) + d−2 −iQx ¯ α (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e x⊥αψn(0)D ⊥(0)Wn (0, xn ) Mc (τ) NC 2 (3) ∞ × Wn (xn¯ , xn¯)ψn(xn¯).

(0cnsδs) Although J has a D⊥ suppression, there is an enhancement by an explicit x⊥ so the operator is LO. The vacuum expectation value of the operator J (08) is shown in Figure 4.4a. The other operators have the same diagrams as Figure 4.2 with the appropriate derivative insertions. The Q’s are introduced by dimensional analysis. There are also the enhanced jet Chapter 4. Subleading Corrections To Thrust Using EFT 63 operators

2 Q Z + n/¯ (−2δs) + d−2 −iQx 2 ¯ (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e (x⊥) ψn(0)Wn (0, xn¯ ) Mcn (τ) NC 2 (3) ∞ × Wn (xn¯ , xn¯)ψn(xn¯) (4.69) ←−α 1 Z + n/¯ ∂ ∂ (−2δsMn) + d−2 −iQx ¯ (3) ∞s ⊥ (0) J (µ, τ) = Tr dx d x⊥e x⊥αψn(0)Wn (0, xn ) Mcn (τ) NC 2 Q ∂τ (3) ∞ × Wn (xn¯ , xn¯)ψn(xn¯) ←− 1 Z + n/¯ n¯ · ∂ ∂ (−2M2n ) + d−2 −iQx ¯ (3) ∞s (0) J b (µ, τ) = Tr dx d x⊥e ψn(0)Wn (0, xn ) Mcn (τ) NC 2 Q ∂τ (3) ∞ × Wn (xn¯ , xn¯)ψn(xn¯) where the derivatives act at the cut. These operators are always associated with O(λ4) soft operators so do not give an enhancement to the thrust rate. We have neglected the contribution (−2na) 2 from M because it vanishes at O(αs). The required N LO jet operators are

1 Z + n/¯ (2abn) + d−2 −iQx ¯ α (3) ∞s (0) J (µ, τ) = 2 Tr dx d x⊥e ψn(0)D⊥(0)Wn (0, xn ) Mcn (τ) Q NC 2 (3) ∞ × Wn (xn¯ , xn¯)D⊥α(xn¯)ψn(xn¯) 2 Z ∞ Z (2e) g + d−2 −iQx+ (8)ab ∞ J (µ, τ) = Tr dsdt dx d x⊥e Wn (xn¯ , xn¯ + tn¯) QNC 0 (3) µ n/¯ b (0) × ψ¯n(xn¯)W (xn¯, xn¯ + tn¯)γ γ⊥α T ψn(xn¯ + tn¯)Mc (τ) n ⊥ 2 n

¯b n/¯ (3) (8)¯ba ∞s × ψ¯n(sn¯)T γ γ W (sn,¯ 0)ψn(0)W (sn,¯ x ) 2 ⊥α ⊥µ n n n 2 Z ∞ Z (2f) g + d−2 −iQx+ (8)ab ∞ ¯ b J (µ, τ) = Tr dsdt dx d x⊥e Wn (xn¯ , xn¯ + tn¯)ψn(xn¯ + tn¯)T QNC 0 (3) n/¯ α µ (0) × W (xn¯ + tn,¯ xn¯) γ γ ψn(xn¯)Mc (τ) n 2 ⊥ ⊥ n

n/¯ (3) ¯b (8)¯ba ∞s × ψ¯n(0)γ γ W (0, sn¯)T ψn(sn¯)W (tn,¯ x ) ⊥µ ⊥α 2 n n n 1 Z + n/¯ (2an) + d−2 −iQx ¯ (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e ψn(0)in · D(0)Wn (0, xn ) Mcn (τ) QNC 2 (3) ∞ × Wn (xn¯ , xn¯)ψn(xn¯) ∞ 1 Z Z + (2An) + d−2 −iQx ¯ (3) αβ αβ J (µ, τ) = Tr dt dx d x⊥e ψn(0)Wn (0, tn¯)g⊥ Gn (tn¯) QNC 0

(3) ∞s n/¯ (0) (3) ∞ × W (tn,¯ x ) γ5Mc (τ)W (x , xn¯)ψn(xn¯) n n 2 n n n¯ 1 Z + n/¯ (2δn) + d−2 −iQx ¯ (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e ψn(0)Wn (0, xn¯ ) Mcn (τ) NC 2 (3) ∞ ←− × Wn (xn¯ , xn¯) in · D + in · D (xn¯)ψn(xn¯) (4.70) Chapter 4. Subleading Corrections To Thrust Using EFT 64 and their hermitian conjugates. The vacuum expectation value of J (2abn) was pictured in Figure 4.3. Although J (2e) and J (2f) look complicated, they have simple diagrams shown in Figure (2An) αβ 4.4b. The γ5 in J is necessary because ⊥ is parity odd. The vacuum expectation value of these operators are shown in Table 4.1a.

The J¯ operators are similar to the J operators. The leading order operators are

1 Z − /n ¯(0) − d−2 −iQx ¯ (3) ∞ (0) (3) ∞s J (µ, τ) = Tr dx d x⊥e ψn¯(xn)Wn¯ (xn, xn ) Mcn¯ (τ)Wn¯ (xn , 0)ψn¯(0) NC 2 αβ Z − (08) g⊥ nµnν − d−2 −iQx µαa (8)ab ∞ (0) J¯ (µ, τ) = dx d x⊥e G (xn)W (xn, x )Mc (τ) Q n¯ n¯ n n¯

(8)ba¯ ∞s νβa¯ × Wn¯ (xn¯ , 0)Gn¯ (0) Z ∞ Z iQ − (3) /n (0) ¯(0M1) − d−2 −iQx ¯ ∞ J (µ, τ) = Tr dt dx d x⊥e ψn¯(xn)Wn¯ (xn, xn ) Mcn¯ (τ) NC 0 2 (3) ∞s × Wn¯ (xn , tn)ψn¯(tn) (4.71) 2 Z ∞ Z ∞ Z iQ − (3) /n (0) ¯(0M2) − d−2 −iQx ¯ ∞ J (µ, τ) = Tr dt ds dx d x⊥e ψn¯(xn)Wn¯ (xn, xn ) Mcn¯ (τ) NC 0 0 2 (3) ∞s × Wn¯ (xn , (t + s)n)ψn¯((t + s)n)

iQ Z − /n ¯(0δn) − d−2 −iQx − ¯ (3) ∞ (0) J (µ, τ) = Tr dx d x⊥e x ψn¯(xn)Wn¯ (xn, xn ) Mcn¯ (τ) NC 2

(3) ∞s × Wn¯ (xn , 0)ψn¯(0).

There is only one enhanced operator that needs to be considered

Z ←− 1 − (3) /n in · ∂ ∂ (−2¯n ) ¯(−2Mn¯b ) − d−2 −iQx ¯ ∞ b J (µ, τ) = Tr dx d x⊥e ψn¯(xn)Wn¯ (xn, xn ) Mcn¯ (τ) NC 2 Q ∂τ

(3) ∞s × Wn¯ (xn , 0)ψn¯(0). (4.72) Chapter 4. Subleading Corrections To Thrust Using EFT 65

l h0|J (l)(µ, τ)|0i 0 δ(τ) m h0|J¯(m)(µ, τ)|0i 08 −δ(τ) 0 δ(τ) 0δn −2δ(τ) 08 −δ(τ) 0cnsδs −2δ(τ) 0 0M1 δ(τ) −2δs 2δ (τ) 0 0M2 δ(τ) −2δsMn δ (τ) 0δn −2δ(τ) − 0 2Mnb δ (τ) 0 −2Mn¯b δ (τ) 2abn α¯s/2 2abn¯ α¯s/2 2an α¯s(−2/ − 2LJ − 9/2) 2an¯ α¯s(−2/ − 2LJ − 9/2) 2An α¯s 2An¯ α¯s 2δn α¯s(−2/ − 2LJ − 3/2) 2δn¯ α¯s(−2/ − 2LJ − 3/2) 2e α¯s(−2/ − 2LJ + 2) (b) 2f α¯s(−2/ − 2LJ + 2) (a)

n h0|S(n)(µ, τ)|0i 0 δ(τ)

08 δ(τ)

2cnsδs α¯s(1/ + LS − 1)

2δcns α¯s(1/ + LS − 1)

2dns α¯s(−1/ − LS + 1)

2dns¯ α¯s(−1/ − LS + 1)

2Ans α¯s(2/ + 2LS − 2)

2δsn α¯s(2/ + 2LS − 2)

2Ms1 −4¯αs

2Ms+t −4¯αs

4δs⊥ −4¯αsτ

4δsMn 4¯αsτ

4Mnb 2¯αsτ

4Mn¯b −2¯αsτ (c)

Table 4.1: Relevant vacuum expectation values of the jet and soft operators for the O(αsτ) thrust rate. The operators distinguished by primes give the same values. Chapter 4. Subleading Corrections To Thrust Using EFT 66

The N2LO operators are

1 Z − /n ¯(2abn¯ ) − d−2 −iQx ¯ α (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e ψn¯(xn)iD⊥Wn¯ (xn, xn ) Mcn¯ (τ) QNC 2 (3) ∞ × Wn¯ (xn , xn)iD⊥α(xn)ψn¯(0)

1 Z − /n ¯(2an¯ ) − d−2 −iQx ¯ (3) ∞s (0) J (µ, τ) = Tr dx d x⊥e ψn¯(xn)Wn¯ (xn, xn ) Mcn¯ (τ) QNC 2 (3) ∞ ←− × Wn¯ (xn , 0)in¯ · D(0)ψn¯(xn) (4.73) ∞ 1 Z Z + /n ¯(2An¯ ) + d−2 −iQx ¯ (3) ∞ (0) J (µ, τ) = Tr dt dx d x⊥e ψn(xn)Wn¯ (xn, xn )γ5 Mcn¯ (τ) QNC 0 2 (3) ∞ αβ αβ (3) × Wn¯ (xn¯ , tn)g⊥ Gn¯ (tn)Wn (tn, 0)ψn¯(0) 1 Z − ←− /n ¯(2δn¯ ) − d−2 −iQx ¯ (3) ∞s J (µ, τ) = Tr dx d x⊥e ψn¯(xn) in¯ · D + in¯ · D (xn)Wn¯ (xn, xn ) NC 2 (0) (3) ∞ × Mcn¯ (τ)Wn¯ (xn , 0)ψn¯(0)

The diagrams are similar to those found for the J operators. For example, J (08) is the horizontal reﬂection of Figure 4.4a. The vacuum expectation values of these operators are shown in Table 4.1b. The diﬀerence in the required J¯ operators compared to the J operators is because we have chosen the ~n axis to be anti-parallel to then ¯-collinear sector.

The leading order soft operators are

(0) 1 (3) ∞s (3) ∞s (0) (3) ∞s (3) ∞s S (µ, τ) = Tr Yn¯ (xn¯ , 0)Yn (0, xn )Mcs (τ)Yn (xn , 0)Yn¯ (0, xn¯ ) (4.74) NC

(08) 1 (8)ab ∞s (8)bc ∞s (0) (8)cd ∞s (8)da ∞s S (µ, τ) = 2 Tr Yn¯ (xn¯ , 0)Yn (0, xn )Mcs (τ)Yn (xn , 0)Yn¯ (0, xn¯ ) NC which only diﬀer in the representation of the Wilson lines. The vacuum expectation value of Chapter 4. Subleading Corrections To Thrust Using EFT 67

S(08) is pictured in Figure 4.4d. The N2LO operators are

Z ∞ ←− (2cnsδs) i (3) ∞s α α (3) ∞s (0) S (µ, τ) = Tr dtYn¯ (xn¯ , 0) iD ⊥ + iD⊥ (0)Yn (0, xn )Mcs (τ) QNC 0 ←− (3) ∞s α (3) (3) ∞s × Yn (xn , tn)iD ⊥(tn)Yn (tn, 0)Yn¯ (0, xn¯ ) Z ∞ ←− (2δcns) (3) ∞s α α (3) S (µ, τ) = Tr dtYn¯ (xn¯ , 0) iD ⊥ + iD⊥ (0)Yn (0, tn)D⊥α(tn) QNC 0 (3) ∞s (0) (3) ∞s (3) ∞s × Yn (tn, xn )Mcs (τ)Yn (xn , 0)Yn¯ (0, xn¯ ) g2 Z ∞ (2dns) (8)ab ∞s ¯ b (3) S (µ, τ) = 2 Tr dsdt Yn (xn , sn)ψs(sn)T Yn (sn, 0) QNC 0

(3) ∞s /n (0) (3) ∞s (3) c (8)ca ∞s × Y (0, x ) Mc (τ)Y (x , 0)Y (0, tn)T ψs(tn)Y (tn, x ) n¯ n¯ 2 s n¯ n¯ n n n g2 Z ∞ (2dns¯ ) (8)ab ∞s ¯ b (3) S (µ, τ) = 2 Tr dsdt Yn¯ (xn¯ , sn¯)ψs(sn¯)T Yn¯ (sn,¯ 0) QNC 0

(3) ∞s n/¯ (0) (3) ∞s (3) c (8)ca ∞s × Y (0, x ) Mc (τ)Y (x , 0)Y (0, tn¯)T ψs(tn¯)Y (tn,¯ x ) n n 2 s n n n¯ n¯ n¯ Z ∞ (2Ans) gnαn¯β (3) ∞s (3) αβ (3) ∞s (0) S (µ, τ) = Tr dt Yn¯ (xn¯ , 0)Yn (0, tn)Gs (tn)Yn (tn, xn )Mcs (τ) QNC 0 (3) ∞s (3) ∞s × Yn (xn , 0)Yn¯ (0, xn¯ ) Z ∞ (2Ans¯ ) gnαn¯β (3) ∞s αβ (3) (3) ∞s (0) S (µ, τ) = Tr dt Yn¯ (xn¯ , tn¯)Gs (tn¯)Yn¯ (tn,¯ 0)Yn (0, xn )Mcs (τ) QNC 0 (3) ∞s (3) ∞s × Yn (xn , 0)Yn¯ (0, xn¯ ) ←− (2δsn¯ ) 1 (3) ∞s (3) ∞s (0) S (µ, τ) = Tr Yn¯ (xn¯ , 0) in¯ · D + in¯ · D (0)Yn (0, xn )Mcs (τ) QNC

(3) ∞s (3) ∞s × Yn (xn , 0)Yn¯ (0, xn¯ ) ←− (2δsn) 1 (3) ∞s (3) ∞s (0) S (µ, τ) = Tr Yn¯ (xn¯ , 0) in · D + in · D (0)Yn (0, xn )Mcs (τ) QNC

(3) ∞s (3) ∞s × Yn (xn , 0)Yn¯ (0, xn¯ ) (4.75)

There are also N2LO operators involving the measurement function expansion that we write as

(2Mi) 1 (3) ∞s (3) ∞s (2i) (3) ∞s (3) ∞s S (µ, τ) = Tr Yn¯ (xn¯ , 0)Yn (0, xn )Mcs (τ)Yn (xn , 0)Yn¯ (0, xn¯ ) (4.76) NC where

(0) 2(~k )2 ∂ M(s1)(τ, {k}) = +t⊥ M(0)(τ, {k}) s Q2 ∂τ s (2) p+n¯ · k ∂ M(s+t)(τ, {k}) = n¯ +t M(0)(τ, {k}) (4.77) s Q2 ∂τ s

+ (2) where the pn¯ cancels in the deﬁnition of k+t in (4.31). The contribution from the other measurement operators in (4.36) vanish at O(αs). Chapter 4. Subleading Corrections To Thrust Using EFT 68

There are also N4LO operators that contribute to the N2LO rate due to convoluting with enhanced jet operators. These operators are

←− ←− (4δs⊥) 1 (3) ∞s α α (3) ∞s (0) S (µ, τ) = Tr Yn¯ (xn¯ , 0) D ⊥ D ⊥α + D⊥D⊥α Yn (0, xn )Mcs (τ) NC (3) ∞s (3) ∞s × Yn (xn , 0)Yn¯ (0, xn¯ ) ←− (4δsMn) 1 (3) ∞s α α (3) ∞s (2n)α S (µ, τ) = Tr Yn (xn , 0)(iD⊥ + iD ⊥)Yn¯ (0, xn¯ )Mcs (τ) QNC

(3) ∞s (3) ∞s × Yn¯ (xn¯ , 0)Yn (0, xn ) (4.78)

(4Mi) 1 (3) ∞s (3) ∞s (4i) (3) ∞s (3) ∞s S (µ, τ) = Tr Yn¯ (xn¯ , 0)Yn (0, xn )Mcs (τ)Yn (xn , 0)Yn¯ (0, xn¯ ) NC where

(0) −(~k )2 M(4nb)(τ, {k}) = +t⊥ M(0)(τ, {k}) s Q2 s (0) (~k )2 M(4¯nb)(τ, {k}) = +t⊥ M(0)(τ, {k}). (4.79) s Q2 s

The vacuum expectation value of the soft operators are shown in Table 4.1c. The operators in Section 4.6 are matched onto combinations of the operators in this section. The matching coeﬃcients are found by taking the vacuum expectation value of both sides of

(4.38) and are shown in Table 4.2. Using this Table and Table 4.1 the O(αsτ) rate in (4.58) can be calculated. Chapter 4. Subleading Corrections To Thrust Using EFT 69

(i,j,k) (i,j,k) (i, j, k) (l, m, n) C1 H1 R(τ)

(1an, 1bn, 0) (2abn, 0, 0) −1 +1 α¯sτ/2 † (1bn, 1an, 0) (2abn, 0, 0) −1 +1 α¯sτ/2

(2an, 0, 0) (2an, 0, 0) 1 −1/2 α¯sτ(−1/ − LJ − 9/4) † (0, 2an, 0) (2an, 0, 0) 1 −1/2 α¯sτ(−1/ − LJ − 9/4)

(2An, 0, 0) (2An, 0, 0) −1/2 −1/4 −α¯sτ/4 † (0, 2An, 0) (2An, 0, 0) −1/2 −1/4 −α¯sτ/4

(2δn, 0, 0) (2δn, 0δn, 0) 1 1 α¯s(4/ + 4LJ + 7)

(1en, 1en, 0) (2e, 08, 08) −1/2 −1/2 α¯sτ(−1/ − LJ − 2)

(1fn, 1fn, 0) (2f, 08, 08 −1/2 −1/2 α¯sτ(−1/ − LJ − 2)

(1an¯ , 1bn¯ , 0) (0, 2abn, 0) −1 +1 α¯sτ/2 † (1bn¯ , 1an¯ , 0) (0, 2abn¯ , 0) −1 +1 α¯sτ/2

(2an¯ , 0, 0) (0, 2an¯ , 0) 1 −1/2 α¯sτ(−1/ − LJ − 9/4) † (0, 2an¯ , 0) (0, 2an¯ , 0) 1 −1/2 α¯sτ(−1/ − LJ − 9/4)

(2An¯ , 0, 0) (0, 2An¯ , 0) −1/2 −1/4 −α¯sτ/4 † (0, 2An¯ , 0) (0, 2An¯ , 0) −1/2 −1/4 −α¯sτ/4

(2δn, 0, 0) (2δn, 0δn, 0) 1 1 α¯s(4/ + 4LJ + 7)

(1δs, 1cns, 0) (0cnsδs, 0, 2cnsδs) 1/2 1 α¯sτ(−2/ − 2LS − 2)

(2δcsn, 0, 0) (0cnsδs, 0, 2δcns) 1/2 1 α¯sτ(−2/ − 2LS − 2)

(1dns, 1dns, 0) 08, 0, 2dns) 1 1 α¯sτ(1/ + LS + 1)

(1dns¯ , 1dns¯ , 0) (08, 0, 2dns¯ ) 1 1 α¯sτ(1/ + LS + 1)

(2Ans, 0, 0) (0, 0, 2Ans) 1 1/2 α¯sτ(1/ + LS + 1) † (0, 2Ans, 0) (0, 0, 2Ans) 1 1/2 α¯sτ(1/ + LS + 1)

(2Ans¯ , 0, 0) (0, 0, 2Ans¯ ) 1 1/2 α¯sτ(1/ + LS + 1) † (0, 2Ans¯ , 0) (0, 0, 2Ans¯ ) 1 1/2 α¯sτ(1/ + LS + 1)

(2δs−, 0, 0) (0, 0δn, 2δs−) 1 1 α¯sτ(−4/ − 4LS − 4)

(2δs⊥, 0, 0) (−2δs, 0, 4δs⊥) 1/2 1/2 4¯αsτ

(1δs, 0, 1n) (−2δsMn, 0M1, 4δsMn) 1 1 4¯αsτ

(0, 0, 2nb) (−2M2nb , 0M2, 4Mnb ) 1 1 −2¯αsτ

(0, 0, 2¯nb) (0, −2Mn¯b , 4Mnb ) 1 1 2¯αsτ

(0, 0, 2s1) (0, 0M , 2Ms1) 1 1 −4¯αsτ

(0, 0, 2s+t) (0, 0M , 2M+t) 1 1 −4¯αsτ

(i,j,k) (i,j,k) (i, j, k) C0 H0 R(τ)

(2δn, 0, 0) −4¯αs −4¯αs −4¯αsτ

(1en, 1en, 0) α¯s α¯s α¯sτ

(1fn, 1fn, 0) α¯s α¯s α¯sτ

(2δn¯ , 0, 0) −4¯αs −4¯αs −4¯αsτ

(1dns, 1dns, 0) −α¯s −α¯s −α¯sτ

(1dns¯ , 1dns¯ , 0) −α¯s −α¯s −α¯sτ

(2δs−, 0, 0) 4¯αs 4¯αs 4¯αsτ

Table 4.2: The operators and matching coefficients for the O(αs) factorization. The dagger means the operator is the hermitian conjugate. At O(αsτ), there is at most one set of subleading jet and soft operators to be matched onto, so α = 1. The table at the bottom gives the matching coefficients for the last line of (4.38). Values of the hard functions defined in (4.40), and the contribution to R(τ) are also given. Chapter 5

Renormalization of Subleading Dijet Operators in Soft-Collinear Eﬀective Theory

In this chapter we calculate the anomalous dimensions of the next-to-leading order dijet operators in the SCET formulation of Chapter 3. We introduce a small gluon mass to regulate the infrared divergences of the individual loop diagrams in order to properly extract the ultraviolet divergences. We discuss this choice of infrared regulator and contrast it with the δ-regulator. Our results can be used to increase the theoretical precision of the thrust distribution. The text in this chapter is reproduced in [48].

5.1 Introduction

Perturbative calculations of jet observables involve multiple scales. In the kinematic region where all the scales are much greater than ΛQCD but the ratio of these scales is small, often called the “tail” region, the rate is perturbative in both the strong coupling constant αs and the ratio of the scales involved. However, the rate includes large logarithms of the ratio of these scales at each order in perturbation theory. These large logarithms limit the precision of theoretical predictions. EFT techniques provide a framework to sum the terms enhanced by the logarithms using renormalization group equations (RGE). This framework also contains a systematic procedure for including higher order eﬀects in the small ratio of scales using subleading operators, allowing for logarithms suppressed by this small ratio to be summed in addition to those at leading order in the ratio. These techniques can be used to improve the precision of the theoretical predictions. In this chapter we renormalize the next-to-leading order dijet operators in SCET with the purpose of using the RGE to sum the logarithms suppressed by the ratio of scales. We will use the SCET operators introduced in the formulation of [26,40], in which the QCD dynamics of jets are described by multiple decoupled copies of QCD, and the

70 Chapter 5. Renormalization of Subleading Dijet Operators in SCET 71

EFT expansion only enters in the external currents. Our results are useful for any observable requiring dijet operators; however, we will use the concrete example of the thrust rate to illustrate their usefulness. Thrust [49] is a useful jet shape observable for precision studies of high energy collisions, in ∗ particular for measuring αs(MZ ) from LEP data . Thrust is deﬁned as

1 X τ = min Ei + ~t · ~pi,Ei − ~t · ~pi , (5.1) Q i∈X where X is the final state, Q is the total energy, and ~t is chosen to maximize the sum. The integrated rate of the differential thrust distribution is defined by

Z τ 1 0 dσ R(τ) = dτ 0 , (5.2) σ0 0 dτ where the Born rate is σ0. We will call this the thrust rate in the following. A perturbative calculation of the thrust rate in the tail region where (Λ /Q) τ 1 involves three relevant QCD √ scales: the hard scattering scale Q, the intermediate scale τQ, and the soft scale τQ. The rate can be written as an expansion R(τ) = R(0)(τ) + τR(1)(τ) + O(τ 2) in this region, where the superscripts refer to the suppression in τ, with R(0) and R(1) referring to O(τ 0) and O(τ) (i) rate respectively. Each of the R (τ) terms in the thrust rate has an expansion in αs of the form

(0) X X (0) n m R (τ) = Rnmαs ln (τ), n m≤2n (1) X X (1) n k R (τ) = Rnk αs ln (τ), (5.3) n k≤2n−1

(i) where the Rnm are O(1) constants and the large logarithms ln τ 1 are due to the separation of scales. The highest logarithmic power for the O(τ) rate is suppressed by an additional power 0 0 of αs relative to the O(τ ) rate. When αs ln τ ∼ O(1) the O(τ ) rate becomes a divergent sum in increasing powers of ln τ, spoiling the expansion in αs(Q) 1. Although the O(τ) rate has an overall suppression by τ compared to the O(τ 0) rate, the rate is similarly a sum in increasing powers of the logarithm. Therefore, in order to restore a perturbative expansion in αs for both the O(τ 0) and O(τ) rates, the logarithms must be summed. The O(τ 0) thrust rate has already been calculated to N3LL accuracy and included the ﬁxed order O(τ) rate at O(αs) [51]. In order to increase the theoretical precision in the tail region, the leading logarithms in the O(τ) rate can become more important than further increasing the 0 logarithmic accuracy in the O(τ ) rate. Therefore, if the precision of the αs(MZ ) measurement is to be improved, these former contributions to the thrust rate will need to be calculated. The appropriate EFT for describing thrust is SCET [16–21,24,26]. SCET includes collinear

∗See [50] and previous works by this collaboration. Chapter 5. Renormalization of Subleading Dijet Operators in SCET 72 and ultrasoft (usoft) fields that reproduce both the highly boosted, and low energy degrees of freedom that are relevant in the tail region. The expansion parameter of SCET is usually √ denoted by λ. For thrust λ ∼ τ, meaning the O(τ) corrections require next-to-next-to-leading order in λ (N2LO) corrections to the effective theory†. We use a formulation of SCET in which QCD fields are coupled to Wilson lines [26]. Each of the sectors (usoft and collinear) interact amongst themselves via QCD, while the interactions between sectors are described by Wilson lines in appropriate representations. This picture has been shown explicitly to N2LO by doing a tree-level matching from QCD [40]. We contrast this formulation with the traditional approach to SCET, which has a Lagrangian expansion and mixes the various sectors [16–21, 24]. SCET can sum the large logarithms in (5.3) by factorizing the rate and using the RGE to run from the hard scale to the soft scale. The QCD operators are first matched onto the appropriate SCET dijet operators at the hard scale Q. For the O(τ 0) rate we use the LO dijet operators. The O(τ) rate requires the NLO and N2LO dijet operators, which are then run to √ the intermediate scale τQ using the RGE. At the intermediate scale, the dijet operators are matched onto soft operators with the help of a factorization theorem. The Wilson coefficients of the soft operators, often called the jet function, are run to the soft scale τQ. The sequence of matching and running sums the large logarithms in the rate. Recently, a factorization theorem has been shown for the O(τ) rate [40] that makes this possible. The appropriate dijet operators and the tree-level matching coefficients were derived, as well as the appropriate soft operators. By solving the RGE for the operators in [40] the large logarithms in the O(τ) rate can be summed. In this chapter we begin this process by calculating the anomalous dimensions of the NLO dijet operators in SCET. Summing all the logarithms in the O(τ) rate of (5.3) also require the N2LO dijet operators, which we leave for future work. To compute the anomalous dimensions of the subleading effective operators we first compute their counterterms. We regulate using the MS scheme and include a separate infrared (IR) regulator to ensure the 1/ poles are ultraviolet (UV) divergences. The decoupling of the collinear and usoft sectors, manifest in the formulation of [26], means the IR cannot be regulated using a fermion off-shellness because the usoft sector will not be changed by this regulator. We identify two possible IR regulators that will regulate the formulation of [26]: the δ-regulator and a gluon mass. The δ-regulator [52] is similar to off-shellness but also modifies the Feynman rules of the usoft Wilson lines. Unfortunately, the regulator introduces additional terms that make the calculation unnecessarily complicated. We will demonstrate this in Section 5.3.1. A gluon mass does not introduce any additional terms, meaning fewer calculations are needed. However, this is done at the expense of introducing unregulated divergences in individual diagrams that only cancel if all the diagrams are added together before integrating. Either choice of regulator is equivalent since the counterterms do not depend on the IR regulator. We chose to use a gluon mass.

†Unless otherwise stated, LO, NLO, and N2LO refer to the expansion in λ. Chapter 5. Renormalization of Subleading Dijet Operators in SCET 73

The rest of the chapter is organized as follows: In Section 5.2 we brieﬂy summarize the SCET formulation of [26] and write the operators used in this calculation. We note that it was necessary to generalize the operators of [26] in order to account for the mixing that occurs under renormalization. In Section 5.3 we discuss our choice of using a gluon mass as an IR regulator over the δ-regulator. We present the anomalous dimensions for the NLO operators in Section 5.4 and conclude in Section 5.5.

5.2 SCET and NLO Operators

In the kinematic region where thrust is dominated by collimated jets of light, energetic particles, SCET is the appropriate description. It is convenient to introduce light-cone coordinates for describing the momentum of the highly boosted particles. In light-cone coordinates the momentum is decomposed into two light-like components described by the vectors nµ andn ¯µ as n¯µ nµ pµ = p · n + p · n¯ + pµ . (5.4) 2 2 ⊥ The vectors nµ andn ¯µ satisfy n2 = 0 =n ¯2 and n · n¯ = 2. A boosted particle with p · n¯ ∼ Q will be described by n-collinear fields in the effective theory. Similarly, a boosted particle with p · n ∼ Q will be described by ann ¯-collinear field. The perpendicular momentum of a collinear µ particle p⊥ ∼ λQ is suppressed compared to the hard scale. We must also include usoft fields that have no large components of momentum and whose momentum scales like pµ ∼ λ2Q. We follow the approach of [26] in deriving the NLO SCET dijet operators. Since particles in the same sector have no large momentum transfers, the interactions within each sector are governed by QCD. Consequently, the Lagrangian has no expansion in λ and can be written as

n n¯ us LSCET = LQCD + LQCD + LQCD, (5.5)

i th ‡ where LQCD is the QCD Lagrangian involving only i -sector ﬁelds. The interactions of particles in diﬀerent sectors are described by external currents. Since these interactions involve large momentum transfers, the external currents can be organized into an expansion in λ. When computing the thrust rate in the limit τ 1, the relevant external currents are dijet operators, which can be determined by matching the full QCD current

" Z # (0) (0) 1 X (1i) (1i) ψ¯(x)Γψ(x) = e−iQ(n+¯n)·x/2 C O (x) + {dtˆ} C ({tˆ})O (x, {tˆ}) + O(λ2) 2 2 Q 2 2 i (5.6) for a general Dirac structure Γ. The phase corresponding to the external momentum has been pulled out. The superscripts in the dijet operators refer to the suppression in λ and the 1/Q is included because the subleading operators are higher dimensional. We have introduced a set of

‡The approach of including decoupled copies of QCD for each sector has also been used to study factorization in QCD [53, 54]. Chapter 5. Renormalization of Subleading Dijet Operators in SCET 74 dimensionless shift variables {tˆ} = {Qt} that were not included in [26]; it will become apparent below that this shift corresponds to a displacement along a light-like direction describing the position of a derivative insertion. This generalization is needed in order to properly describe the mixing of operators under renormalization.

The leading order operator in (5.6) is [26] h i h i h i (0) ¯ (3) ∞ (3) ∞n (3) ∞n¯ (3) ∞ O2 (x) = ψn(xn¯)Wn (xn¯, xn¯ ) Yn (xs , 0)Γ(0)Yn¯ (0, xs ) Wn¯ (xn , xn)ψn¯(xn) , (5.7) and its matching coeﬃcient is [45, 55]

2 2 2 (0) αsC µ µ π C (µ) = 1 + F − ln2 − 3 ln − 8 + (5.8) 2 4π −Q2 −Q2 6 where µ is the renormalization scale. The Dirac structure is

Γ(0) = Pn¯ΓPn¯ (5.9) with projectors Pn = (/n/n¯)/4 and Pn¯ = (n/n/¯ )/4. The subscripts on the ﬁelds denote the sector of the ﬁeld. Each of the square brackets in (5.7) are independently gauge invariant and corresponds to a separate sector. The Wilson lines in the R representation

Z n·(y−x)/2 ! (R) a a −s Wn (x, y) = P exp −ig dsn¯ · An(x +ns ¯ )TRe 0 Z n¯·(y−x)/2 ! (R) a a −s Yn (x, y) = P exp −ig dsn · As (x + ns)TRe , (5.10) 0 represent a light-like colour source corresponding to the total colour of the other sectors (the symbol P indicates path-ordering). The in the deﬁnition above gives the proper i pole (R) (R) prescription. The Wn¯ and Yn¯ Wilson lines are deﬁned similarly. The positions in (5.7)

∞ xn = (0, x · n,¯ x⊥) xn = (0, ∞, x⊥) ∞ xn¯ = (x · n, 0, x⊥) xn¯ = (∞, 0, x⊥) (5.11)

∞n ∞n¯ xs = (∞, 0, 0) xs = (0, ∞, 0), come from multipole expanding the total momentum conservation constraint in λ and is needed to ensure consistent power-counting at each order in λ.

The NLO operators are found by including O(λ) corrections in the interactions between the Chapter 5. Renormalization of Subleading Dijet Operators in SCET 75 sectors [26]. The operators that describe the modiﬁcation to the n-collinear sector are h i (1an) ¯ (3) µ (3) ∞ O2 (x, t) = ψn(xn¯)Wn (xn¯, xn¯ +nt ¯ )iD⊥(xn¯ +nt ¯ )Wn (xn¯ +nt, ¯ xn¯ ) h (3) i h (3) i × Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) W (x∞, x )ψ (x ) n s (1an) n¯ s n¯ n n n¯ n h i (1bn) ¯ (3) µ (3) ∞ O2 (x, t) = ψ(xn¯)Wn (xn¯, xn¯ + tn¯)iD⊥(xn¯ + tn¯)Wn (xn¯ + tn,¯ xn¯ ) h (3) i h (3) i × Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) W (x∞, x )ψ (x ) n s (1bn) n¯ s n¯ n n n¯ n h ←− i (1Bn) ¯ (3) ∞ µ O2 (x) = ψ(xn¯)Wn (xn¯, xn¯ )i ∂ ⊥ (5.12) h (3) i h (3) i × Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) W (x∞, x )ψ (x ) n s (1bn) n¯ s n¯ n n n¯ n h ←− i (1cn) ¯ (3) µ (3) ∞ O2 (x, t1, t2) = ψn(xn¯)Wn (xn¯, xn¯ + t2n¯)iD ⊥(xn¯ + t2n¯)Wn (xn¯ + t2n,¯ xn¯ ) h ←− i (3) ∞n µ (3) (3) ∞n¯ × Yn (xs , t1n)iD ⊥(t1n)Yn (t1n, 0)Γ(1cn)Yn¯ (0, xs ) h (3) ∞ i × Wn¯ (xn , xn)ψn¯(xn)

(1dn) h aµν (8)ab ∞ i O2 (x, t) = ign¯µGn⊥ (xn¯)Wn (xn¯, xn ) h i × (8)bc ∞n ¯ c (3) ν (3) ∞n¯ Yn (xs , tn)ψs(tn)T Yn (tn, 0)Γ(1dn)Yn¯ (0, xs ) h (3) ∞ i × Wn¯ (xn , xn)ψn¯(xn) h i h i (1en) aµν (8)ab ∞ (3)ddˆ ∞n (8)dcˆ ∞n¯ O2 (x, t) = ign¯µGn⊥ (xn¯)Wn (xn¯, xn ) Yn (xs , 0)Yn¯ (0, xs ) h i (8)cb ∞ ¯ b (3) × Wn¯ (xn , xn + tn)ψn¯(xn + tn)T Γ(1en)Wn (xn + tn, xn)ψn¯(xn) h i (1δ) ¯ µ (3) ∞ O2 (x) =Q ψn(xn¯)x⊥Wn (xn¯, xn¯ ) h ←− i h i (3) ∞n µ µ (3) ∞n¯ (3) ∞ × Yn (xs , 0)Γ(1δ)(D⊥ + D ⊥)(0)Yn¯ (0, xs ) Wn¯ (xn , xn)ψn¯(xn) , where the Dirac structures are

µ µ /n µ n/¯ µ Γ =Pn¯Γγ Γ = γ ΓPn¯ Γ(1c ) =Pn¯ΓPn¯ (1an) 2 (1bn) 2 n µ /n µ µ /n µ Γ = γ ΓPn¯ Γ = γ ΓPn¯ Γ(1δ) =Pn¯ΓPn¯. (5.13) (1dn) 2 ⊥ (1en) 2 ⊥

The covariant derivative is defined as Dµ(x) = ∂µ − igT aAaµ(x) and only couples the gluon to the corresponding sector on which it acts. The field strength tensor is defined as igGaµν = abc bµ cν abc f [A ,A ] where f are the SU(3) structure constants. The derivative in the (1Bn) operator is strictly a partial derivative and not a covariant derivative because we are working in a covariant gauge where the gauge transformations at infinity vanish. The Q in front of the (1δ) operator is required dimensionally. The matching coefficients for the operators listed above Chapter 5. Renormalization of Subleading Dijet Operators in SCET 76 are [26]

(1an) (1bn) C2 (tˆ) = −δ(tˆ) + O(αs) C2 (tˆ) = δ(tˆ) + O(αs)

(1Bn) (1cn) C2 = 1 + O(αs) C2 (tˆ2, tˆ1) = 2iθ(tˆ1)δ(tˆ2) + O(αs) (5.14)

(1dn) (1en) C2 (tˆ) = −2iθ(tˆ) + O(αs) C2 (tˆ) = iθ(tˆ) + O(αs) (1δ) C2 (tˆ) = 1 + O(αs), which are all dimensionless. The factors of i ensure the convolution in (5.6) is real. The NLO operators explicitly decouple the sectors, just as in the LO operator. These operators differ from the LO operators by a D⊥ insertion at an arbitrary point along a Wilson line (for example the (1an) operator) or by a change in the field content and Wilson line representation (for example the (1en) operator). The operators in (5.12) are generalizations of the NLO operators in [26, 40]. We find the form in (5.12) is necessary to properly renormalize the operators, since different values of the parameters can mix under renormalization. We have also slightly changed the definition of the (1bn) operator and included the (1Bn) operator, which makes the operator basis in (5.12) diagonal under renormalization. As was done in [26], we can compare the operators in (5.12) with the subleading operators in other formulations of SCET, such as in [56]. In [56] the subleading heavy-to-light currents were renormalized. While the dijet and heavy-to-light operators obviously differ in the usoft andn ¯-collinear sectors, the modifications to the n-collinear sector from the vector currents and subleading Lagrangian insertions in [56] only differ from the corresponding operators in (5.12) by the appropriate Dirac structure basis. This will serve as a way for us to compare the anomalous dimensions we calculate in Section 5.4 with the results of [56]. ˜(i) We find it more convenient to work with the Fourier transformed operators O2 defined as Z Z (1i) dtˆ ˆ (1i) dt (1i) O˜ (x, u) = e−iutO (x, tˆ) = Q e−iQutO (x, t) 2 (2π) 2 (2π) 2 Z ˆ ˆ (1i) dt2 dt1 −i(tˆ2u2+tˆ1u1) (1i) O˜ (x, u2, u1) = e O (x, tˆ2, tˆ1). (5.15) 2 (2π) (2π) 2

The matching in (5.6) is written in terms of these operators as Z Z (1i) (1i) ˜(1i) ˜(1i) d{tˆ}C2 ({tˆ})O2 ({tˆ}) = d{u}C2 ({u})O2 ({u}) (5.16) where Z ˜(1i) iutˆ (1i) C2 (u) = dtˆ e C2 (tˆ)

Z ˆ ˆ ˜(1i) i(u2t2+u1t1) (1i) C2 (u2, u1) = dtˆ2dtˆ1 e C2 (tˆ2, tˆ1). (5.17)

The u’s are momentum fractions at the vertex of the external current. For collinear momentum Chapter 5. Renormalization of Subleading Dijet Operators in SCET 77

0 ≤ u ≤ 1 due to momentum conservation, while for usoft momentum 0 ≤ u < ∞ because usoft momentum is not conserved at the vertex. The Fourier transformation of the NLO operators are h i ˜(1an) ¯ ˆ µ (3) ∞ O2 (x, u) = ψn(xn¯)δ(u − in · D)iD⊥(xn¯)Wn (xn¯, xn¯ ) h (3) i h (3) i × Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) W (x∞, x )ψ (x ) n s (1an) n¯ s n¯ n n n¯ n h i ˜(1bn) ¯ ˆ µ (3) ∞ O2 (x, u) = ψ(xn¯)δ(u − in · D)iD⊥(xn¯)Wn (xn¯, xn¯ ) h (3) i h (3) i × Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) W (x∞, x )ψ (x ) n s (1bn) n¯ s n¯ n n n¯ n h ←− i ˜(1cn) ¯ ˆ µ (3) ∞ O2 (x, u2, u1) = ψn(xn¯)δ(u2 − in · D)iD ⊥(xn¯)Wn (xn¯, xn¯ ) ←− ←− (3) ∞n µ ˆ (3) ∞n¯ × Yn (xs , 0)iD ⊥(0)δ(u1 − in · D)Γ(1cn)Yn¯ (0, xs )

h (3) ∞ i × Wn¯ (xn , xn)ψn¯(xn) h i ˜(1dn) aµν (8)ab ∞ O2 (x, u) = ign¯µGn⊥ (xn¯)Wn (xn¯, xn ) (5.18) ←− × (8)bc ∞n ¯ c − · ˆ ν (3) ∞n¯ Yn (xs , 0)ψs(0)T δ(u in D)Γ(1dn)Yn¯ (0, xs )

h (3) ∞ i × Wn¯ (xn , xn)ψn¯(xn) h i h ˆ ˆ i ˜(1en) aµν (8)ab ∞ (3)dd ∞n (8)dc ∞n¯ O2 (x, u) = ign¯µGn⊥ (xn¯)Wn (xn¯, xn ) Yn (xs , 0)Yn¯ (0, xs ) ←− (8)cb ∞ ¯ b ˆ × Wn¯ (xn , xn)ψn¯(xn)T Γ(1en)δ(u − in · D)ψn¯(xn) where Dˆ µ = Dµ/Q is a dimensionless covariant derivative. The tree-level matching coeﬃcients up to O(αs) corrections are

˜(1an) ˜(1bn) C2 (u) = −1 C2 (u) = 1 ˜(1cn) 2 ˜(1dn) 2 ˜(1en) 1 C2 (u2, u1) = − C2 (u) = C2 (u) = − (5.19) u1 u u

The (1Bn) and (1δ) are independent of tˆ so are not transformed.

5.2.1 Constraining the NLO Operators

We restrict ourselves to the electromagnetic current Γ = γλ in this chapter. This current is both CP invariant and conserved. We will show how we can exploit these two properties to constrain the NLO SCET operators. We will also show how we can use the ambiguity in deﬁning the nµ andn ¯µ directions to make further constraints. First we use CP invariance to expand the list of operators to include corrections to the n¯-collinear sector. The action of CP is equivalent to switching n andn ¯ and then taking the complex conjugate. Therefore, the NLO corrections to then ¯-collinear sector can be obtained Chapter 5. Renormalization of Subleading Dijet Operators in SCET 78 for free from the operators in (5.12). The operators are

(1a ) h i h (3) i O n¯ (x, t) = ψ¯ (x )W (3)(x , x∞) Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) 2 n n¯ n n¯ n¯ n s (1an¯ ) n¯ s h (3) ∞ ←−µ (3) i × Wn¯ (xn , xn + tn)iD ⊥(xn + nt)Wn¯ (xn + nt, xn)ψn¯(xn) (1b ) h i h (3) i O n¯ (x, t) = ψ¯ (x )W (3)(x , x∞) Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) 2 n n¯ n n¯ n¯ n s (1bn¯ ) n¯ s h (3) ∞ ←−µ (3) i × Wn¯ (xn , xn + tn)iD ⊥(xn + nt)Wn¯ (xn + nt, xn)ψn¯(xn) (1B ) h i h (3) i O n¯ (x) = ψ¯(x )W (3)(x , x∞) Y (3)(x∞n , 0)Γµ Y (0, x∞n¯ ) 2 n¯ n n¯ n¯ n s (1bn¯ ) n¯ s h (3) ∞ i × i∂⊥µWn¯ (xn , xn)ψn¯(xn) h i (1cn¯ ) ¯ (3) ∞ O2 (x, t2, t1) = ψn(xn¯)Wn (xn¯, xn¯ ) h i (3) ∞n (3) µ (3) ∞n¯ × Yn (xs , 0)Γ(1cn¯ )Yn¯ (0, t1n¯)D⊥(t1n¯)Yn¯ (t1n,¯ xs ) h (3) ∞ µ (3) i × Wn¯ (xn , xn + t2n)iD⊥(xn + t2n)Wn¯ (xn + t2n, xn)ψn¯(xn) h i (1dn¯ ) ¯ (3) ∞ O2 (x, t) = ψn(xn¯)Wn (xn¯, xn¯ ) h i × (3) ∞n ν (3) c (8)bc ∞n¯ Yn (xs , 0)Γ(1dn¯ )Yn¯ (0, tn¯)T ψs(tn¯)Yn¯ (tn, xs ) (5.20) h aµν (8)ab ∞ i × ignµGn¯⊥ (xn)Wn¯ (xn, xn¯ ) h i (1en¯ ) ¯ (3) b (8)bc ∞ O2 (x, t) = ψn(xn¯)Wn (xn¯, xn¯ + tn¯)Γ(1en¯ )T ψn(xn¯ + tn¯)Wn (xn¯ + tn,¯ xn¯ ) h i h i (3)cdˆ ∞n (8)ddˆ ∞n¯ aµν (8)ad ∞ × Yn (xs , 0)Yn¯ (0, xs ) ignµGn¯⊥ (xn)Wn¯ (xn, xn¯ ) with Dirac structures

Γµ = Γµ Γµ = Γµ Γ = Γ (1an¯ ) (1bn) (1bn¯ ) (1an) (1cn¯ ) (1cn) µ µ n/¯ µ µ n/¯ Γ = Pn¯γ Γ Γ = Pn¯γ Γ . (5.21) (1dn¯ ) ⊥ 2 (1en¯ ) ⊥ 2 µ The (1δ) is already CP invariant since the x⊥ can be moved into either collinear sector. CP invariance guarantees the matching coeﬃcients of the (1in) and (1in¯) are equal

(1in) (1in¯ ) C2 = C2 (5.22) for i = {a, b, c, d, e, B}. The Fourier transform of the operators in (5.20) are similar to those in (5.18), and we avoid writing them down for the sake of brevity. In the following we will use

CP invariance to avoid talking about the (1in¯) operators unless it is necessary. λ Next, we can exploit the conservation of the electromagnetic current ∂λψ¯(x)γ ψ(x) = 0. As was discussed in [57], the EFT dijet operators must also be conserved at each order in λ §. The only operators in (5.12) that are not conserved by themselves are the (1an), (1bn), and (1B(n,n¯))

§We would like to thank Ilya Feige and Ian Moult for this observation Chapter 5. Renormalization of Subleading Dijet Operators in SCET 79 operators. All the other NLO operators are conserved up to O(λ2). Therefore, conservation of the current requires (1an) (1bn) (1Bn) (1Bn¯ ) C2 = −C2 C2 = C2 (5.23) to all orders in αs.

Finally, we can exploit Reparametrization Invariance (RPI) [15,58] to constrain the matching coeﬃcients. RPI has been discussed extensively for heavy-to-light currents in the traditional SCET formulations [32, 56] but has not previously been discussed in the SCET formulation we use. However, insight can be drawn from the traditional SCET formulations due to the equivalence of the two approaches.

The n-collinear ﬁelds represent particles boosted in the nµ direction, where nµ is a vector we specify when matching from QCD onto SCET. Then ¯-collinear particles are described similarly. However, an n-collinear particle does not travel exactly along the nµ direction, and will generically have a momentum perpendicular to nµ of order λ. In fact, we could have chosen a slightly diﬀerent nµ, for example 0µ µ µ n = n + ⊥, (5.24)

0µ where ⊥ ∼ O(λ). In this case an n-collinear particle also appears to be boosted along the n direction and has relative perpendicular momentum of order λ. Therefore, it should not matter to the physical result whether we include an n-collinear sector or an n0µ-collinear sector. We µ µ µ can make use of this equivalence by applying the variation n → n +⊥ to the operators in the n-collinear sector and enforcing that they cancel order-by-order in λ. This provides constraints on the matching coeﬃcients that must hold to all orders in αs.

(R) Using the equation of motion for a Wilson line n · DWn = 0 and a fermion /Dψ = 0, the variation of the LO operator is

µ µ µ (0) n →n +⊥ (0) O2 (x) −−−−−−−→ O2 (x) h i h i h i ¯ (3) ∞ (3) ∞n (3) ∞n¯ (3) ∞ + ψn(xn¯)Wn (xn¯, xn¯ ) Yn (xs , 0)δ(Γ(0))Yn¯ (0, xs ) Wn¯ (xn , xn)ψn¯(xn) · h ←− i (¯n x)⊥µ (3) ∞ (3) ∞n µ µ (3) ∞n¯ + ψ¯n(xn¯) W (xn¯, x ) Y (x , 0) D + D Γ Y (0, x ) (5.25) 2 n n¯ n s (0) n¯ s h (3) ∞ i 2 × Wn¯ (xn , xn)ψn¯(xn) + O(λ ), where n/¯ /⊥ δ(Γ ) = ΓPn¯. (5.26) (0) 2 2

Only the left projector is transformed because the Dirac structure is Γ(0) = Pn¯1 ΓPn2 where µ µ n1 and n2 are the light-like directions of the two sectors. However, matching enforces that n ≡ n1 =n ¯2, so the transformed projector reduces to (5.26).

It is straightforward to show that the (1δ) and (1Bn) operators cancel the variations in Chapter 5. Renormalization of Subleading Dijet Operators in SCET 80

(5.25) if their matching coeﬃcients are constrained to be

(0) (1δ) (1Bn) C2 = C2 = C2 (5.27) to all orders in αs. We note this is similar to what was found in [56] for heavy-to-light currents. We will check the constraints in (5.23) and (5.27) when we renormalize the NLO operators. The anomalous dimensions, being the kernels of a linear integro-diﬀerential equation, are expected to be equal if two operators are constrained to be the same up to a multiplicative constant. We will see this in Section 5.4.

5.3 Infrared Regulator

In order to extract counterterms from loop diagrams we must be able to differentiate between UV and IR divergences. Introducing a small mass scale to serve as an IR cut-off allows us to regulate the IR separately from dimensional regularization and ensures that all the 1/ poles in the loop integrals are UV divergences. A common scheme is to introduce a small fermion off-shellness, as was done in [16,45]. However, in the SCET approach of [26] where the sectors explicitly decouple, a fermion off-shellness leaves the Wilson lines unchanged and will not properly regulate the usoft sector of the LO operator¶. A regulator that produces similar results to a fermion off-shellness is the δ-regulator [52]. The δ-regulator modifies the Feynman rules of both the usoft and collinear sectors thereby regulating the IR of the SCET approach we use in this chapter. However, when there is more then one external leg in a single sector, the δ-regulator introduces extra terms that complicate the calculation. Using a gluon mass to regulate the IR avoids these additional terms, although the individual diagrams will contain unregulated divergences, which cancel in the total sum of diagrams. We have chosen to use a gluon mass as our IR regulator, and in this section we will contrast some of the details of the two approaches. In this and following sections we will use a condensed notation for representing the Feynman diagrams considered in our calculations. As an example to illustrate the notation, Figure 5.1 shows the diagrams for n-collinear quark andn ¯-collinear anti-quark production using the LO dijet operator. This notation becomes especially useful when considering subleading operators with a gluon in the final state, as the number of diagrams grows considerably.

5.3.1 The Delta Regulator

The δ-regulator was introduced when considering SCET with massive gauge particles to help make the loop integrals of individual diagrams converge [52]. The construction is similar to using a fermion oﬀ-shellness and can be used to regulate the IR of the SCET formulation of [26].

¶In the traditional approach to SCET [16–20] the LO operator does not explicitly decouple until after a field redefinition, which does not affect the counterterms. Chapter 5. Renormalization of Subleading Dijet Operators in SCET 81

p1  p1                                        =       ≡                           p2 p2

(a) In                                         =       ≡                          

(b) Ius                                         =       ≡                          

(0) Figure 5.1: Relevant graphs for the renormalization of the O2 operator. Solid lines and dashed lines are fermions and Wilson lines respectively. We decompose the diagram on the left into the contribution from each sector in the middle three diagrams. We can also compactify the notation by only showing the sector that has the one-loop contribution, as shown on the right.

This makes it an obvious choice for regulating the NLO operators in (5.12). However, the δ- regulator requires extra terms when there is more than one external leg that do not appear when using a gluon mass to regulate the IR. As an example to show where these extra terms arise, we renormalize the LO dijet operator with ann ¯-collinear anti-quark and an n-collinear quark and gluon in the final state. The diagrams are shown in Figure 5.2. The δ-regulator regulates the IR by inserting a small mass term into the Lagrangian for each field. The particles are brought off-shell by maintaining the massless equations of motion p2 = 0. The Feynman rules for the Wilson lines are also modified to incorporate this off-shellness. The Feynman rules for the propagators and Wilson lines are [52] α a 1 n¯i TRj 2 and (5.28) (pi + k) − ∆i k · n¯i − δj,n respectively. The momentum of the internal particle is k and ∆i is the mass term inserted Chapter 5. Renormalization of Subleading Dijet Operators in SCET 82

                                                     

(a) In (b) In¯ (c) Ius

(0) Figure 5.2: One-loop diagrams for O2 with an external n-collinear gluon, using the compact notation of Figure 5.1.

into the Lagrangian. The Feynman rule for the Wilson line is for a particle in the jth-sector th k with colour TRj emitting a particle in the i -sector . The shift in the Wilson line is δj,n =

(2∆j)/((ni · nj)(¯nj · pi)). The regulator naively breaks the explicit decoupling into usoft and collinear ﬁelds. However, once all the diagrams and their zero-bins have been accounted for, the result does factorize [52]. The modiﬁcation to the Feynman rule of the Wilson line in (5.28) leads to extra terms in the calculation of the diagrams. For example, the usoft diagram in Figure 5.2 leads to the integral Z 2 d 1 CF + CA CA 2ig κ d k 2 − , (5.29) (k − ∆g)(¯n · k + δq,¯ n¯) n · k − δq,n n · k − δg,n

2 γ d where κ = (µ e E ) /(2π) . The extra CA terms account for the internal usoft gluon being emitted by or before the external n-collinear gluon. These extra terms are necessary to cancel all the mixed UV/IR divergences from the n-collinear diagrams. Then ¯-collinear diagram will also require extra diagrams. However, as expected, the final result reproduces the expected LO anomalous dimension and is very similar to using a fermion off-shellness in a theory that does not decouple usoft and collinear fields.

5.3.2 Gluon mass

Another scheme that can be used to regulate the IR is to introduce a small gluon mass. Un- fortunately, massive bosons introduce an obstacle in SCET: the individual diagrams are often unregulated in dimensional regularization [52]. However, the sum of all the diagrams from a particular operator must still be well-regulated by a gluon mass [52]. As an example, we show how a gluon mass can be used to calculate the anomalous dimension of the LO operator. The

kWe note the colour structure was not in the original δ-regulator definition but is necessary when looking at O(g3) processes. Chapter 5. Renormalization of Subleading Dijet Operators in SCET 83 necessary diagrams are shown in Figure 5.1. The n-collinear diagram gives the integral Z 2 d n¯ · (p1 − k) In = 2ig CF κ d k 2 2 2 (k − M )(p1 − k) (¯n · k) − 1− Z p1 − − 2 d/2 −2 dk k = −2g CF κπ Γ()M − 1 − − (5.30) 0 k p1 where M is the gluon mass. The second line above is found by doing the k+ integral by contours − and then the k⊥ integral. The final integral diverges as k → 0 for all dimensions. The usoft diagram gives the integral Z 2 d 1 Ius = 2ig CF κ d k (k2 − M 2)(n · k)(−n¯ · k) Z ∞ − 2 d/2 −2 dk = −2g CF κπ Γ()M − (5.31) 0 k after doing the same integrals as the n-collinear diagram. This integral diverges as k− → 0 and ∞. Then ¯-collinear diagram gives the integral Z 2 d n · (p2 − k) In¯ = 2ig CF κ d k 2 2 2 (k − M )(p2 − k) (¯n · k) Z ∞ p+ M −2 2 d/2 − 2 − −2 − − + − = 2g CF κπ Γ() dk 2 − + ( M + ( k p2 ) ) + (5.32) 0 M + k p2 1 − again doing the same integrals as the n-collinear diagram. The first term above diverges as − k → ∞. As usual, we must also subtract a zero-bin Ino/ = Ius = Ino¯/ for each of the collinear sectors [59]. Therefore, the sum of the diagrams is

In + In¯ − Ius. (5.33)

Each of the divergences in the above integrals cancel in the sum and we can ﬁnd the anomalous dimension 2 αsC −Q 3 γ = F ln − . (5.34) 2(0) π µ2 2 This is the well-known result for the anomalous dimension of the LO dijet operator [45]. Although the δ-regulator would avoid unregulated divergences in intermediate steps, it requires keeping track of additional terms. We chose to calculate the counterterms using a gluon mass and expect a δ-regulator to give the same results.

5.4 Anomalous Dimensions

In order to run the NLO Wilson coeﬃcients in (5.14) from the high scale Q to any other scale below Q, we must solve the RGE. To do so we must renormalize the NLO operators and Chapter 5. Renormalization of Subleading Dijet Operators in SCET 84 calculate their anomalous dimensions. The renormalized operators (R) and bare operators (B) are related by Z ˜(1i)(B) X ˜(1i)(R) O2 (µ; x, {u}) = {dv}Z2(1ij)(µ; {u, v})O2 (x, {v}) (5.35) j where Z2(1ij) is the counterterm matrix extracted from the UV divergences of the Green’s functions of the operator. In general, the continuous set of operators can mix within each label u and with other operators j. The independence of µ of the renormalized operators leads to an integro-diﬀerential equation for the bare operators Z d (1i) X (1j) O˜ (B)(µ; x, {u}) = − {dv}γ (µ; {u, v})O˜ (B)(µ; x, {v}). (5.36) d ln µ 2 2(1ij) 2 j

The anomalous dimension is calculated from the counterterms

X Z d γ (µ; {u, v}) = − {dw}Z−1 (µ; {u, w}) Z (µ; {w, v}). (5.37) 2(1ij) 2(1ik) d ln µ 2(1kj) k

The corresponding equation for the Wilson coeﬃcients Z d (1i) X (1j) C˜ (µ; {u}) = {dv}C˜ (µ; {v})γ (µ; {v, u}) (5.38) d ln µ 2 2 2(1ij) j is the RGE that must be solved. 2 The operators in (5.18) are written in a diagonal basis in i, j up to O(αs) corrections meaning  Z2(1i) if i = j Z2(1ij) = (5.39) 0 if i =6 j .

The counterterms can be written perturbatively as

αs (1) Z (µ; {u, v}) = δ({u − v}) + Z (µ; {u, v}) + O(α2). (5.40) 2(1i) 4π 2(1i) s

The anomalous dimension will also be diagonal in i, j and the lowest order contribution will be ∂ ∂ (1) γ2(1i)(µ; {u, v}) = 2 αs − 2 Z2(1i)(µ; {u, v}). (5.41) ∂αs ∂ ln µ

The ﬁrst term comes from the renormalization of the coupling constant g(R) = g(B)µ−2. We will suppress the explicit dependence on µ in the anomalous dimension for the sake of more concise notation. The diagrams for the calculation of the anomalous dimensions of the NLO operators are shown in Figure 5.3. We must consider a gluon in the ﬁnal state for most of the operators Chapter 5. Renormalization of Subleading Dijet Operators in SCET 85

                                                           

(1an) (1bn) (a) O2 and O2                                                            

(b) O(1δ)    2                                                         

(c) O(1cn)    2                                                         

(d) O(1dn)      2                                                       

(1en) (e) O2

Figure 5.3: Diagrams for the NLO operators. Each bracket represents the one-loop graph from a separate sector. Going from left to right, the diagrams are the n-collinear, usoft, and n¯-collinear sectors. The box vertex represents the derivative insertion. Chapter 5. Renormalization of Subleading Dijet Operators in SCET 86

as these operators have a gluon in the final state at tree-level. The (1Bn) operator can be renormalized in a frame where the total perpendicular collinear momentum is non-zero and it has the same diagrams as the LO operator in Figure 5.1. We use the background field method [60] to maintain gauge invariance under renormalization. The background field method −1/2 ensures Zg = ZA , which properly renormalizes the derivative insertions and the Wilson lines. Extracting the UV divergences from the diagrams lead to the following anomalous dimensions

2 αsδ(u − v)θ(¯v) −Q 3 CA γ (u, v) = CF ln − + lnv ¯ + (1an) π µ2 2 2 αs CA uv + CF − u¯ θ(1 − u − v) π 2 u¯v¯ uv + u + v − 1 + θ(¯u)θ(¯v)θ(u + v − 1) uv αsC v¯ − uv u¯ − uv − A u¯ θ(¯u)θ(u − v) + θ(¯v)θ(v − u) 2π uv¯ vu¯ 1 uθ¯ (¯u)θ(u − v) vθ¯ (¯v)θ(v − u) + + u¯v¯ u − v v − u +

γ(1bn)(u, v) = γ(1an)(u, v) 2 αsC 3 −Q γ = F − + ln = γ (5.42) (1Bn) π 2 µ2 (0) 2 αsδ(u1 − v1)δ(u2) 3 −Q CA γ (u2; u1, v1) = CF − + ln + ln v1 (1cn) π 2 µ2 2 αsC δ(u2) θ(v1 − u1)θ(u1) θ(u1 − v1)θ(v1) − A + π v1 − u1 u1 − v1 + θ(u − v ) θ(v − u ) − 1 1 − 1 1 u1 v1 2 αsδ(u − v) CF −Q 1 γ (u, v) = − + CA ln + ln(v) − (1dn) π 2 µ2 2 αs CA 1 vθ(u − v)θ(v) uθ(v − u)θ(u) − CF − + π 2 v u − v v − u + 2 αsδ(u − v)θ(¯v) CF −Q γ (u, v) = + CA ln + ln(v) − 1 (1en) π 2 µ2 αs CA 1 − CF − θ(¯v)θ(v − u)uv¯ + θ(¯u)θ(u − v)vu¯ π 2 vv¯ uvθ¯ (¯u)θ(u − v) uvθ¯ (¯v)θ(v − u) + + , u − v v − u +

whereu ¯ = 1 − u andv ¯ = 1 − v. We have used a generalized symmetric plus-distribution ﬁrst introduced in [56], which was denoted by square brackets as in [ ]+. The formal deﬁnition of Chapter 5. Renormalization of Subleading Dijet Operators in SCET 87 this distribution is

[θ(u − v)q(u, v) + θ(v − u)q(v, u)]+ d Z 1+v Z 0 ≡ − lim θ(u − v − β) dw q(w, v) + θ(v − u − β) dw q(v, w) , (5.43) β→0 du u u which is the same as the distribution deﬁned in [56] when u, v ≤ 1. The above deﬁnition is also valid when u, v > 1, which was not required in [56]. Equation (5.42) is our main result.

We can compare the results for γ2(1an) with [56]. The (1an) operator in (5.12) is similar to the NLO vector heavy-to-light current in [56]. As expected, the anomalous dimension for these two operators are the same for the non-diagonal terms. They only disagree in the diagonal terms by the diﬀerence of the LO dijet and heavy-to-light operator, which is expected. Also, the anomalous dimensions of the (1an) and (1bn) operators are the same, as expected from current conservation in (5.23). We can also check that γ2(1in¯ ) = γ2(1in) as expected from CP invariance. Finally, the (1Bn), (1δ), and (0) operators all have the same anomalous dimension as expected from RPI.

A ﬁnal check is to compare the anomalous dimension of the (1en) and (1dn) operators. From

(5.12) we see the (1dn) operator is the limit of the (1en) operator when the quark becomes usoft. 2 Therefore, we expect in the limit where u ∼ λ ∼ v in the (1en) anomalous dimension to recover ∗∗ the (1dn) anomalous dimension. This is indeed the case as seen in (5.42) . The NLO operators have a cusp in the usoft light-like Wilson lines at xµ = 0. Therefore, the anomalous dimension depends on at most a single logarithm and can be written in the form

2 C −Q NC γ (µ; u, v) = δ(u − v)Γ (αs) ln + γ (αs; u, v). (5.44) 2(1i) (1i) µ2 (1i)

The coeﬃcient of the logarithm is proportional to the universal cusp anomalous dimension

Γcusp(αs) [61], which means it is possible to perform NLL summation without going to higher 2 loops. This universal form of (5.44) is conﬁrmed up to O(αs) corrections in (5.42).

Obviously, solving the RGE analytically is straightforward for the (1Bn) and (1δ) operators because the anomalous dimension is the same as the LO dijet operator. However, solving the

RGE analytically for the other operators is more difficult. The non-diagonal terms in the (1an) and (1bn) RGE were solved in [56] by exploiting that the non-diagonal terms in the anomalous dimensions can be written as f(u, v)S(u, v) where S(u, v) is a symmetric function. For example, f(u, v) =u ¯ for the (1an) operator and 1/(vv¯) for the (1en) operator. The authors of [56] were able to expand in an infinite set of Jacobi polynomials with the appropriate weight functions in order to diagonalize the anomalous dimensions and solve the RGE. We expect that a similar solution will work for the (1an), (1bn) and (1en) operators. However, the (1cn) and (1dn) operators are qualitatively different due to the limits on the labels, and a different strategy may

∗∗This limit must be taken carefully, since the u → O(λ2) limit does not commute with the limit in the deﬁnition of the plus distribution. Chapter 5. Renormalization of Subleading Dijet Operators in SCET 88 be required. In any case, we believe it may be more practical to solve the RGE numerically, and we leave this for future work.

5.5 Conclusion

In order to increase the accuracy of the αs(MZ ) measurement the O(τ) corrections are becoming important. Just like for the O(τ 0) rate, the O(τ) rate includes large logarithms that must be summed. We describe how this can be done using SCET and the factorization theorem in [40]. The required operators in the O(τ) factorization theorem must be renormalized so they can be run from the hard scale to the usoft scale. The running can be done in two stages. First the NLO and N2LO dijet operators in SCET must be renormalized. These operators are then run from the hard scale to the intermediate scale. In the next step, the soft operators introduced in [40] will be renormalized and run from intermediate scale to the usoft scale. This sequence of running and matching will sum all the large logarithms in the O(τ) rate. In this chapter, we have started the first step by renormalizing the NLO dijet operators. Although we have used thrust as a concrete example of an application, our results is applicable to any observable requiring dijet operators. Because we use the SCET formulation of [26] we cannot use fermion off-shellness to regulate the IR. Instead we have used a gluon mass, which leads to individual diagrams being unregulated. However, the sum of all the diagrams from a given operator is well-defined, as expected. The UV divergences are extracted by looking at the 1/ poles allowing us to calculate the anomalous dimensions of the NLO operators. We have checked our results with similar operators for the heavy-to-light currents in [56] and find good agreement. We leave renormalizing the N2LO dijets operators and the soft operators to future work. Although we have calculated the anomalous dimensions of the NLO operators, and investigated the possibility of solving the RGE analytically, we believe that it may be more practical to solve it numerically, which we leave for future work. Chapter 6

The Exclusive kT and C/A Dijet Rates in SCET with a Rapidity Regulator

We study the (exclusive) kT and C/A jet algorithms using effective field theory techniques. Regularizing the virtualities and rapidities of graphs in SCET, we are able to write the next- to-leading-order dijet cross section as the product of separate hard, jet, and soft contributions. For the C/A algorithm, we show how to reproduce the Sudakov form factor to next-to-leading logarithmic accuracy previously calculated by the coherent branching formalism. Our resummed expression only depends on the renormalization group evolution of the hard function, rather than on that of the hard and jet functions as is usual in SCET. We comment that regularizing rapidities in this case is necessary for assessing effects of scale variations, but not for obtaining the resummed expression. The text in this chapter is reproduced in [62].

6.1 Introduction

Jets are important for understanding the background to new physics being investigated at the Large Hadron Collider. Jet production is a multiscale process that involves the large energy of the jet, Q, and its small invariant mass, mjet, given by the details of the jet deﬁnition. A 2 2 hierarchy of scales Q mjet gives rise to large logarithms of the form L ≡ ln(Q /mjet) 1 in perturbative calculations. These logarithms manifest in the jet production rate in the form

∞ 2n X X n m R = Rnmαs L , (6.1) n=0 m=0 where αs is the strong coupling constant. Even when αs 1, the large logarithms will ruin 2 perturbation theory when αsL ∼ 1. Well-known perturbative QCD (pQCD) techniques based on factorization theorems [5] and

89 Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 90 the coherent branching formalism [3] can sum these logarithms by writing the series (6.1) as

R = C(αs)Σ(αs,L), (6.2) where

∞ X n C(αs) = Cnαs , (6.3) n=0

ln Σ(αs,L) = Lf0(αsL) + f1(αsL) + αsf2(αsL) + ....

The coeﬃcient function C(αs) contains no large logarithms L, while Σ(αs,L) sums the logarithms. The f0 term sums the leading logarithms (LL), the f1 term sums the next-to-leading logarithms (NLL), and the fn≥2 terms sum the subleading logarithms. In this chapter we will always refer to the logarithmic order in the exponent (6.3) as opposed to the logarithmic order in the perturbative rate (6.1).

An example of a jet definition is the (exclusive) kT jet algorithm [6,63], proposed to resolve the exponentiation issue of the earlier JADE algorithm [6,64,65]. The kT and JADE algorithms combine final-state partons into jets using a distance measure yij for all pairs of final-state partons {i, j}. If the smallest yij is smaller than some pre-determined resolution parameter yc, then that pair of partons are combined and all the yij’s are re-calculated. The procedure is repeated until all yij > yc, and these pseudo-partons are then called jets. The kT algorithm measure for e+e− → jets is

2 2 min(Ei ,Ej ) yij = 2(1 − cos θij) (6.4) Q2 where Q is the centre-of-mass energy, θij the angle between the ﬁnal-state pair, and Ei,j their respective energy. We are interested in a two-jet ﬁnal state where the cut parameter is small. √ Jets in the yc 1 region have small mass mjet ≈ ycQ Q, which gives rise to large logarithms

L ≡ ln(1/yc). The kT dijet production rate has been calculated using the coherent branching formalism to full LL accuracy in [6, 63] and partial NLL accuracy in [66]. Clustering eﬀects 2 2 among multiple gluon emissions generate unsummed logarithms that start at O(αsL ) in the exponent [49] and ruin the NLL summation of [66].

Another jet definition, the C/A algorithm [4] is defined similarly to the kT algorithm but avoids clustering effects. Two measures are used in this case: vij = 2(1 − cos θij) and the kT algorithm measure yij in (6.4). Pairs of partons are ordered based on vij, but only the pair with the smallest vij is combined when their yij < yc. At O(αs), the kT and C/A algorithms give the same dijet rate at leading order in yc. However, clustering effects do not show up in the C/A algorithm to ruin the NLL summation [49]. Therefore, the result in [66] gives full NLL accuracy for the C/A dijet production. EFT techniques offer another approach to summing the large logarithms. Using EFTs has Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 91 the advantage of using the renormalization group (RG) to sum the large logarithms, as well as providing a systematic approach to power corrections. In [35] the kT dijet rate was calculated using SCET to next-to-leading order (NLO). SCET [16–20, 26] describes QCD using highly boosted “collinear” fields and low energy “soft” fields. SCET has previously been successful in calculating jet shapes [42,67], where it automatically separated the hard scattering interaction from the highly boosted interactions in the jets and from the soft radiation between them. Such a separation allows the rate to be written as the convolution of hard, jet (one for each of the dijets), and soft functions, R = H × J × J¯ × S, (6.5) each of which depends on a different scale. These functions are then run individually to a common scale for logarithm summation. The authors of [35], however, were unable to use dimensional regularization to regulate the individual NLO collinear and soft graphs of the kT dijet rate, making it unclear how to write the rate as separate jet and soft functions as in (6.5). The C/A dijet rate was not discussed in [35], but its NLO calculation faces identical problem as the kT dijet rate. Recently [68,69] a new regulator capable of regulating these divergences has been proposed. This new “rapidity regulator” effectively places a cut on the rapidities of the fields [69], enabling the rate to be written as separate scheme dependent jet and soft functions. The rapidity regulator was used to sum logarithms in the jet broadening event shape [42,68,69], which has a similar issue at NLO to the kT and C/A dijet rates. The introduction of the rapidity regulator opens up the possibility of the RG running in another scale ν, in analogy to the usual RG running scale µ of dimensional regularization. We propose to extend the work of [35] using the new rapidity regulator and investigate how to write the kT dijet rate as the product of hard, jet, and soft functions as in (6.5). Our work provides an application of the rapidity regulator to jet algorithms. Our calculation also applies to the C/A algorithm as the two dijet rates are identical at NLO. As in [6, 63, 66] we assume a factorization theorem for both the kT and C/A dijet rates, which allows us to interpret the SCET collinear and soft graphs as the jet and soft functions that are run using the RG. We can then use the RG to attempt to sum the large logarithms. We find that we reproduce the coherent branching formalism result for the kT and C/A dijet rates [66], but that neither approach sums the logarithms generated by the clustering effects for the kT dijet rate [49]. A similar result was recently found for the inclusive kT algorithm [70]. We expect our result to sum all the logarithms in the C/A dijet rate as in the coherent branching formalism, since clustering effects are absent in this case [49].

Reproducing the coherent branching formalism result for the kT and C/A dijet rates using SCET only requires the running of the hard function to NLL accuracy. The jet and soft functions act as a single soft function S = J × J¯ × S that reproduces the infrared physics of QCD and depends only on a single soft scale. For NLL accuracy, defining separate scheme dependent jet and soft functions using the rapidity regulator is only required to examine the Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 92 effects of scale variations, but not for obtaining the resummed expression. The rest of the chapter proceeds as follows: in Section 6.2 we review the NLO results and issues of [35], and in Section 6.3 we show how the rapidity regulator solves these issues. In Section 6.4 we show our final result with NLL summation excluding clustering effects and compare with the coherent branching formalism result. We discuss the interpretation of our results and the utility of the rapidity regulator in Section 6.5. Finally we conclude in Section 6.6.

6.2 Review of Previous Work

The kT algorithm was previously studied using SCET in [35]. SCET is the appropriate EFT to describe QCD with highly boosted massless fields. Collinear fields describe the boosted particles, and soft fields describe the low-energy particle exchanges. The interactions within each sector (soft, collinear in each direction) decouple from one another and are described by a copy of QCD [26]. The interactions between sectors in the full theory are reproduced in the currents via Wilson lines [17–20, 26]. The appropriate SCET operator for dijet production where n andn ¯ are respectively the light-like directions of the jets is [20, 71] h i h i ¯ † † O2 = ξnWn Yn ΓYn¯ Wn¯ ξn¯ (6.6) where ξn,n¯ is a two-component n- orn ¯-collinear spinor. The Wilson lines are defined in momentum space as

X −g W = exp n¯ · A n ·P n perm n¯ X −g Y = exp n · A , (6.7) n ·P s perm n

µ with Wn¯ and Yn¯ deﬁned analogously. Here P is the momentum operator that acts on the gluon

fields. The fields As,An, and An¯ represent soft, n-, andn ¯-collinear gluon fields respectively. The matching between QCD and SCET is well known [45] and gives the matching coefficient

2 2 2 αsCF 1 2 µ 3 µ π C2(µ) = 1 + − ln − ln − 4 + + ... (6.8) 2π 2 −Q2 2 −Q2 12 and MS counterterm

2 αsCF 1 3 1 µ Z2(µ) = 1 + + + ln + .... (6.9) 2π 2 2 −Q2

The ellipses denote higher orders in αs. The matching coeﬃcient reproduces the ultraviolet Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 93

(a) (b)

µ Figure 6.1: QCD diagrams for real emission. In SCET the gluon can either be soft (p3 = + − ~ µ + − µ + − (k3 , k3 , k3⊥)), n-collinear (p3 = (k3 , p3 , ~p3⊥)) orn ¯-collinear (p3 = (p3 , k3 , ~p3⊥)).

(UV) physics of QCD. The e+e− → γ∗ → dijet rate is calculated in SCET by summing the collinear and soft diagrams and integrating over the appropriate phase space. Generally the rate is written in the form 1 dσdijet = H × (J × J¯ × S) (6.10) σ0 dyc 2 2 P 2 ∗ where σ0 = (4πα /Q ) f ef is the Born cross section . The soft contribution S describes the interaction of the soft fields, while the hard function H captures the physics of the hard initial 2 interaction. The hard function is defined to be H = |C2| . The jet contributions J and J¯ describe the interactions of the n- andn ¯-collinear fields respectively. For perturbative calculations, the contributions in (6.10) are individually written as

F (µ) = 1 + F (1)(µ) + F (2)(µ) + ... (6.11)

¯ (n) n where F = H, J, J,S and F is the O(αs ) term. The two QCD diagrams that contribute to real emission at NLO are shown in Figure 6.1. In SCET, the gluon can either be soft, n-, or n¯-collinear, resulting in six graphs that must be summed. We write all momenta in lightcone µ + − coordinates q = (n · q, n¯ · q, ~q⊥) ≡ (q , q , ~q⊥). We adopt the convention of [35] and use the symbol k Q for soft momentum, and p ∼ Q for large momentum. Contributions from the NLO collinear and soft graphs in dimensional regularization are given by integrating the corresponding diﬀerential cross sections over the relevant phase space PSF [35] Z (1) αsCF + − 2 S (µ) = f dk3 dk3 + − 1+ 2π PSS (k3 k3 ) Z + − − − − ˜(1) αsCF + − (k3 p3 ) p3 Q − p3 J (µ) = f dk3 dp3 + (1 − ) + 2 − 2π PSn Qk3 Q p3 Z + − (1) αsCF dk3 dk3 J0 (µ) = 2 f + − 1+ (6.12) 2π PS0 (k3 k3 )

∗For dijet rates via a Z0, only the Born cross section is modiﬁed. This is irrelevant for our calculation. Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 94

(a) (b) (c)

Figure 6.2: NLO kT dijet phase space for (a) n-collinear gluon (b) soft gluon and (c) zero-bin. These plots are taken from [35], and the bold arrows indicate that the plots extend to inﬁnity. The C/A dijet phase space is identical.

n-collinear zero-bin soft + + − k3 k3 p3 + − 2 + + − 2 min − , − 2 < yc k3 k3 < ycQ k3 (k3 + k3 ) < ycQ p3 (Q−p3 ) − √ − √ − + − 2 p3 < ycQ k3 < ycQ k3 (k3 + k3 ) < ycQ − √ p3 > Q(1 − yc)

Table 6.1: Phase space constraints for NLO real emission for the kT and C/A algorithms. The constraints are plotted in Figure 6.2.

where J˜ is referred to as the naive collinear graph and J0 the zero-bin. The “true” collinear contribution requires a zero-bin subtraction [22] and is deﬁned as J(µ) = J˜(µ) − J0(µ). We 2 γ have introduced f ≡ µ e E /Γ(1 − ) for later convenience. Then ¯-collinear graph is the same as the n-collinear graph at NLO, J¯(1)(µ) = J (1)(µ).

The relevant NLO phase space constraints in SCET are found by applying the kT measure (6.4) to the qqg¯ ﬁnal state and expanding in k p, Q. The phase space constraints for the C/A algorithm are identical to the kT constraints. At leading order in power counting the fermions must be collinear, and we deﬁne nµ to be in the direction of the quark. The constraints for a soft and n-collinear gluon are shown in Table 6.1 and plotted in Figure 6.2. The constraints for an n¯-collinear gluon are the same as those for an n-collinear gluon with “+” and “−” interchanged. In [35] it was found that the NLO soft graph can be written as

2 γE 2 Z 1 x − (1) αsCF e µ (1 − 2 ) S (µ) = −2 2 dx + ..., (6.13) π Γ(1 − ) ycQ 0 x where the ellipses denote terms that are properly regulated in dimensional regularization. The integral in (6.13) is not regularized as x → 0, and this means that interpreting the soft function as the sum of the soft graphs as in (6.10) is not well deﬁned. However, it was noted in [35] that Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 95 the NLO zero-bin can be written as

γE 2 Z 1 (1) αsCF e µ dx J0 (µ) = − 2 + ..., π Γ(1 − ) ycQ 0 x where again the ellipses denote terms that are properly regularized. The x → 0 divergence in (1) ¯ this integral is the same as the soft graph. Because J0 (µ) enters into both J(µ) and J(µ) dijet with a relative minus sign compared to the soft graph, the total rate (1/σ0)(dσ /dyc) is kT (C/A) properly regularized at NLO as expected. Decomposing the rate as separately regularized jet and soft functions as in (6.10), where J and S are respectively the collinear and soft graphs, is therefore not possible using pure dimensional regularization. The issue of separately well-defined functions comes from how phase space is being divided between the collinear and soft graphs in this scheme. The soft ± + − 2 graph is being integrated over the region k3 → 0 while keeping k3 k3 ≤ ycQ . This is a highly boosted region, and is more naturally associated with the jet function than the soft function. The jet broadening rate has a similar issue in SCET as shown in [42]. As pointed out in [35], the soft graph can be regulated using a different scheme such as a cut-off regulator. The cut-off regulator removes the contribution of the aforementioned region from the soft graph and regulates the integral in (6.13). The jet broadening rate can also be regularized using a cut-off. The cut-off regulator, however, is not very attractive as it is not gauge invariant, making it hard to run using the RG. It is also unclear how to define it in the naive collinear calculation. Another scheme also studied in [35] is to use off-shellness as an infrared regulator, while using dimensional regularization to regulate the UV. Here, the small quark and anti-quark off-shellness regulates the integrals in (6.13) and (6.14). However, the resulting collinear and soft contributions – including the virtual diagrams – are not individually infrared finite, even though these infrared divergences cancel in the total NLO rate as expected. Therefore it is again unclear how to interpret these as the jet and soft functions of (6.10). In the next section we use the recently introduced rapidity regulator [68, 69] to separate the low energy theory into jet and soft functions associated with the collinear and soft fields respectively.

6.3 Next-to-Leading-Order calculation

In this section we show how all the divergences in the phase space of the soft graphs are tamed with the introduction of the rapidity regulator [68,69]. The rapidity regulator was used to solve the similar issue and sum the logarithms in jet broadening [68, 69]. The regulator acts as an energy cut-off in a similar way that dimensional regularization acts as a cut-off on the mass scale of loop momenta [72]. The form is similar to dimensional regularization and also maintains gauge invariance [69], unlike a cut-off regulator. We will show in this section that using the Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 96 rapidity regulator splits the NLO collinear and soft graphs into separately finite pieces. This allows us to interpret the jet and soft functions as the collinear and soft interactions respectively.

The rapidity regulator modiﬁes the momentum-space deﬁnition of the Wilson lines (6.7) to [68, 69]

X −g |n¯ · P|−η W = exp w2 n¯ · A n ·P −η n perm n¯ ν " #! X −g |2P3|−η/2 Y = exp w n · A . (6.14) n ·P −η/2 s perm n ν

Here P3 pulls down the component of momentum in the spatial direction of the jet. The new parameter η acts similarly to in dimensional regularization. The parameter w is a bookkeeping parameter that is set to one at the end [68, 69]. Implementing the rapidity regulator modiﬁes the NLO collinear and soft graphs to

Z + − + − −η (1) αsCF 2 dk3 dk3 k3 − k3 S (µ, ν) = 2 w f + − 1+ (6.15) 2π PSs (k3 k3 ) ν Z + − − " − − − −ν# ˜(1) αsCF + − (k3 p3 ) p3 2 Q − p3 p3 J (µ, ν) = f dk3 dp3 + (1 − ) + 2w − 2π PSn Qk3 Q p3 ν Z + − − −η (1) αsCF 2 dk3 dk3 k3 J0 (µ, ν) = 2 w f + − 1+ . 2π PS0 (k3 k3 ) ν

Note that the phase space constraints PSF are not aﬀected. The pure dimensional regularized functions are recovered in the η → 0 limit.

Calculating the collinear and soft graphs is now straightforward. As has been previously demonstrated [69], we must expand in η before . As we are considering the yc 1 region, all terms subleading in yc are also suppressed.

The naive NLO collinear graph is

2 2 (1) αsCF 2 π 1 1 Q yc 2 1 J˜ (µ, ν) = 4w 1 − − ln 2 − + ln 2 + − ln w (2 + ln yc) − . 2π 12 2 µ2 2 (6.16)

We leave in w for now and will set it to one at the end. The logarithms cannot be minimized at any one scale because we have not yet included the zero-bin subtraction. The NLO zero-bin contribution is

2 2 2 2 (1) αsC 2 1 ycQ 2 ycQ ycQ ycQ J (µ, ν) = F w2 − + ln + ln − ln ln . (6.17) 0 2π η ν2 η µ2 µ2 ν2

Subtracting the zero-bin from the naive collinear graph gives the true (bare) collinear contri- Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 97 bution

2 αsC π 1 J B(1)(µ, ν) = F 4w2 1 − − ln 2 − + ln 2 2π 12 2 2 2 1 Q yc 1 1 Q 1 + − ln 2w2 + 1 − ln − . (6.18) µ2 η 2 ν2 2 √ The collinear logarithms can be minimized at µJ = ycQ and νJ = Q. Then ¯-collinear contribution J¯(µ, ν) is exactly the same as J(µ, ν) at this order in αs. The NLO soft graph can be calculated similarly. The extra η-dependent piece in (6.15) regulates the divergence of (6.13). The NLO (bare) soft graph is

2 2 2 2 αsC ycQ π 1 2 ycQ 1 ycQ SB(1)(µ, ν) = F w2 ln2 − + 2 − + ln − ln , (6.19) 2π µ2 3 η ν2 µ2 √ where the logarithms are minimized at the scales µS = ycQ = νS. Note that the dimensional regularization scale of the soft graph is equal to that of the collinear graph, µJ = µS. Putting the collinear and soft graphs together, as well as the matching coeﬃcient (6.8) and the counterterm (6.9), the kT and C/A relative dijet rate is

dijet 1 dσk (C/A) T = H(µ)J(µ, ν)J¯(µ, ν)S(µ, ν) (6.20) σ0 dyc 2 αsCF 2 π 2 = 1 + − ln yc − 3 ln yc + − 1 − 6 ln 2 + O(α ), 2π 6 s which exactly reproduces the pQCD [6, 65, 66]. All the graphs must be evaluated at the same (µ, ν). Notice that the ν dependence must cancel between the collinear and soft graphs because H is ν-independent. This is a general result and means that, when added together, the η dependence of the J, J¯ and S counterterms must vanish [68]. We ﬁnd that unlike in [35], we can deﬁne the jet and soft functions in (6.10) as the collinear and soft interactions respectively. In the next section we show how to sum the logarithms using the RG by running each function individually. We then compare the summed expression to the coherent branching formalism result.

6.4 Next-to-leading logarithm summation

We wish to calculate both f0 and f1 of (6.3) to sum the logarithms and compare to [66]. Because we have two UV regulators, the jet and soft functions now run through a two-dimensional (µ, ν) space. The renormalized function F is deﬁned in terms of the bare function F B and counterterm B ZF as F = ZF F . Therefore, the anomalous dimensions in the two directions of the (µ, ν) Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 98 space are found using µ ∂ ∂ γF (µ, ν) = − + β(αs) ln ZF ∂ ln µ ∂αs ν ∂ ∂ γ (µ, ν) = − + β(w) ln ZF (6.21) F ∂ ln ν ∂w

2 where F = H, J, S. The running of the coupling constant β(αs) = −2αs+O(αs) is well-known and β(w) = −ηw/2 exactly [69]. The counterterms of the jet and soft functions are found from (6.18) and (6.19) to be

2 2 αsCF 2 4 2 µ 4 µ0 ZS =1 + − + ln + ln + ... 2π 2 η ν2 η µ2 2 2 αsCF 2 3 1 Q 2 µ0 ZJ =1 + + − ln − ln + ... (6.22) 2π η 2 ν2 η µ2

2 2 where we have set w = 1, µ0 = ycQ , and the ellipses here denote higher orders in αs. The −1 2 2 hard function counterterm ZH ≡ |Z2| = ZSZJ as expected. The NLO anomalous dimensions are

2 2 2αsC µ αsC 3 ν γµ = F ln γµ = F + ln S π ν2 J π 2 Q2 2 2 2αsC µ0 αsC µ0 γν = F ln γν = − F ln S π µ2 J π µ2 2 µ αsCF µ γ ≡ γH = − 3 + 2 ln . (6.23) H π Q2

The hard anomalous dimension in the ν direction vanishes identically because Z2 is ν-independent. For consistency in the running, we must have

− ~γH = 2~γJ + ~γS, (6.24) µ ν ~ ~ d d where ~γF = (γF , γF ) = −∇ ln ZF with ∇ ≡ µ dµ , ν dν . From (6.23), we see that these conditions are satisﬁed at NLO.

The anomalous dimensions allow the functions to be run to any scale. However, unlike in the usual case of only using dimensional regularization to regulate the UV, the hard, jet, and soft functions are now scalar functions deﬁned over a two-dimensional (µ, ν) space. Path independence of running is equivalent to the curl of ~γF vanishing. This vanishing curl gives the condition d d µ γν (µ, ν) = ν γµ (µ, ν), (6.25) dµ F dν F which, along with (6.24), must be satisﬁed to all orders in αs. We show in the Appendix that Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 99 the soft ν anomalous dimension can be written as Z µ 0 ν ν dµ d µ 0 γS(µ) = γS(µ0) + 0 ν γS (µ , ν) , (6.26) µ0 µ dν where the general form of the soft µ anomalous dimension is taken to be

2 µ µ γ (µ, ν) = ΓS[αs(µ)] ln + γS[αs(µ)]. (6.27) S ν2

ν Here ΓS is called the cusp anomalous dimension. The γS(µ0) contains no logarithms and all ν the logarithmic dependency of γS(µ) is determined by the µ anomalous dimension. A similar expression to (6.26) appears in [69]. The hard anomalous dimension has a similar form as (6.27) with ν = Q [42]. The jet anomalous dimension is completely constrained by the hard and soft anomalous dimensions from (6.24). We can use the above to solve the RG equations and sum the logarithms. Each function

F (µ, ν) must be evolved from the scale that minimizes its logarithms (µF , νF ) to a common scale. The solution to the RG equations gives the running of each function

R µ2 dµ µ R ν2 dν ν µ γF (µ,ν2) ν γF (µ1,ν) F (µ2, ν2) = F (µ1, ν1)e 1 µ e 1 ν (6.28) where we have chosen to run in ν ﬁrst but path independence is guaranteed with the use of (6.26). The summed rate for the path in Figure 6.3 is therefore

dσdijet 1 kT (C/A) = H(µH )J(µJ , νJ )J¯(µJ , νJ )S(µS, νS) σ0 dyc ωH (µH ,µJ ) ωS (µJ ,µ0) µJ νS × eKH (µH ,µJ ) (6.29) Q νJ where we have run to a general (µ, ν), and used the consistency equations (6.24) and path independence (6.26) to write everything in terms of the hard and soft running. Terms subleading to NLL accuracy have been suppressed. Because of path independence, we can choose any other path and get the same NLL terms. The summed rate is both µ- and ν-independent, as expected.

The running kernels in (6.29) are deﬁned as

0 ΓF β1 αs(µ2) ωF (µ1, µ2) = − ln r + K − (r − 1) β0 β0 4π 0 0 γF 2πΓF r − 1 − r ln r KF (µ1, µ2) = − ln r − 2 (6.30) 2β0 β0 αs(µ1) β1 1 − r + ln r β1 + K − + ln2 r , β0 4π 8πβ0

n n where we denote r = αs(µ1)/αs(µ2). The coeﬃcients ΓF and γF are given from the general Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 100

Q H µH γH µ γµ(ν) γµ(ν) µ S J 0 ν ν µ γS(µ0) γJ (µ0) γS (νs) µ γJ (νJ ) µS S µJ J ν νS ν νJ

Figure 6.3: Running each function from (µF , νF ) to (µ, ν).

form of the anomalous dimension (6.27) as

2 αs 0 αs 1 ΓF [αs(µ)] = Γ + Γ + ... 4π F 4π F 2 αs 0 αs 1 γF [αs(µ)] = γ + γ + ... (6.31) 4π F 4π F where from (6.23) we can read oﬀ

0 0 ΓH = −8CF γH = −12CF 0 0 ΓS = 8CF γS = 0. (6.32)

The β-function of the coupling constant αs also has an expansion αs αs 2 β[αs(µ)] = −2αs β0 + β1 + ... (6.33) 4π 4π where

11CA 2nf β0 = − 3 3 2 34CA 10CAnf β1 = − − 2CF n . (6.34) 3 3 f

The two-loop running in the coupling constant αs(µ) gives

2 2 αs(Q) αs(Q)β0 µ αs(Q)β1 αs(Q)β0 µ = 1 + ln 2 + ln 1 + ln 2 . (6.35) αs(µ) 4π Q 4πβ0 4π Q

The factor 67 π2 10 K ≡ − CA − n (6.36) 9 3 9 f Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 101

1 NLO NLO/LL

0.8 NLO/NLL

0.6

0.4

0.2

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Figure 6.4: Plots of (6.38) for LL and NLL accuracy. NLO is the order in αs the coeﬃcient function C(αs) of (6.3) is taken to. The multipole expansion breaks down before L → 0 or yc → 1, and the inclusion of NLO terms in C(αs) improves accuracy of the curve over this region. The procedure for calculating the error bands is described in the text.

1 0 is the well-known ratio of the one- and two-loop cusp anomalous dimensions, K = ΓF /ΓF [42, 66], and is required for the NLL summation. Choosing the scales that minimize the logarithms in the hard, jet, and soft functions

µH = Q µJ = µS = µ0

νJ = Q νS = µ0 (6.37) simpliﬁes (6.29) to

dijet dσ ωH (Q,µ0) 1 kT (C/A) KH (Q,µ0) µ0 = H(Q)J(µ0,Q)J¯(µ0,Q)S(µ0, µ0)e , (6.38) σ0 dyc Q which sums the large logarithms to partial NLL accuracy for the kT dijet rate and full NLL accuracy for the C/A dijet rate. From the above equation we can see that only the RG of the hard function is required for the summation to NLL accuracy. The action of running in rapidity cancels between the jet and soft functions. We will elaborate more on this issue in the following section.

We can now ﬁnd the functions f0 and f1 of (6.3) from (6.38). The LL summation comes Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 102

0 from setting γH = K = β1 = 0. For the C/A dijet rate, the NLL summation comes from the 0 terms proportional to a single power of γH , K, and β1. Therefore,

0 ΓH ln(1 − x) f0 = − 1 + (6.39) 2β0 x 0 0 0 γH ΓH K x ΓH β1 x + ln(1 − x) 1 2 f1 = ln(1 − x) + 2 + ln(1 − x) − 3 + ln (1 − x) 2β0 2β0 1 − x 2β0 1 − x 2 where x ≡ αsCF β0L/(4π). Using (6.32) we see that the functions agree exactly with the coherent branching formalism result [66]. We plot the summed rate in Figure 6.4 as a function of ln(1/yc). The maximum jet production is at around yc ' 0.2.

The error bars in Figure 6.4 are found by varying the scales µH,J,S and νJ,S in (6.29) by 2 and 1/2 of their values in (6.37). We vary the jet and soft scales together to maintain the

µJ ≈ µS scaling. Varying the νF scales without varying µJ produces no error due to the exponent ωS(µJ = µ0, µ0) = 0. We take a naive approach to estimate the correlated errors by varying µJ and νF together, and taking the geometric mean of the resulting percent errors.

6.5 Discussion

That only the RG of the hard function is necessary to reproduce the coherent branching formalism result of [66] to NLL accuracy suggests the kT and C/A dijet rate should be written as dijet 1 dσk (C/A) T = H(µ)S(µ). (6.40) σ0 dyc Here the new soft function S(µ) = J(µ)J¯(µ)S(µ) is the combined collinear and soft graphs and is well defined at NLO in pure dimensional regularization as seen in [35] and Section 6.2. This new soft function is also infrared finite, as shown by using off-shellness to regulate the infrared divergences of the collinear and soft graphs [35]. By running the functions between µH = Q √ and µS = ycQ = µS,J the is reproduced to NLL accuracy for the C/A algorithm. By choosing to run along the particular path in Figure 6.5, it is clear that only the combined collinear and soft graphs are required for reproducing the resummed expression (6.38). Along this path, the general form of the ν anomalous dimension (6.26) becomes

ν X (m) m γS(µ0) = αs(µ0) γ˜S αs (µ0), (6.41) m≥0 which contains no large logarithms. For NkLL accuracy only the m ≤ k terms are required. (0) However, in general theγ ˜S term, which is required for NLL accuracy, vanishes as seen in the 2 kT and C/A dijet rate above and all the cases in [69]. For N LL accuracy, therefore, only the (1) hard running and theγ ˜S term are required. However, understanding the dependence on the scale µ0 requires the rapidity regulator as discussed in [69]. Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 103

Figure 6.5: Running the functions along a particular path. Note that, up to NLL accuracy, the summed result is independent of the path chosen and the ﬁnal (µ, ν) point.

6.6 Conclusion

We have studied the (exclusive) kT and C/A dijet rates using effective theory methods, and shown how to reproduce the coherent branching formalism result to NLL accuracy. We must use the rapidity regulator if we wish to separate the NLO rate into regularized jet and soft functions. We have demonstrated how to sum logarithms to LL and NLL accuracy using the rapidity regulator in a path independent way, which can be generalized to any process that has a factorization theorem. We comment that the rapidity regulator is necessary for evaluating effects of scale variations, but not for summing the large logarithms to NLL accuracy in the example of the C/A dijet rate. The same resummed result can be achieved if we consider the combined jet and soft function and run to the common jet and soft scale. We also find that using SCET with a rapidity regulator does not account for clustering effects and cannot improve the coherent branching formalism result for the kT dijet rate. A more complicated SCET-like theory may be able to properly account for these clustering effects, however, we do not explore such a theory in this thesis.

6.7 Appendix: General Rapidity Anomalous Dimension

Here we show, using the soft function as an example, how to obtain (6.26), which allows us to sum to NLL accuracy. Our argument relies on factorization of scales, the consistency condition

(6.24), the vanishing curl (6.25), the general form of γH , and that the anomalous dimensions are deﬁned perturbatively in αs. Factorization means that the anomalous dimensions of each function are sensitive only to scales relevant to it. Therefore, the ν dependence of ~γJ and ~γS will only be of the form ln(ν/Q) and ln(ν/µ0) respectively. The consistency condition (6.24) requires that all ν dependence of µ ν ~γJ and ~γS must cancel to all orders in perturbation theory. As γH is cusp-like and γH vanishes, Chapter 6. The Exclusive kT and C/A Dijet Rates in SCET . . . 104

Figure 6.6: Running the soft function in a rectangular path back to itself. Naively this results in a LL phase when ν2 ν1.

µ ν γF can have at most a linear dependence on ln(ν/νF ) and γF must have no ν dependence. The appearance of ln(µ/µ0) in ~γF , on the other hand, is not constrained. These logarithms can show up in arbitrary powers, as long as they cancel one another in the sum ~γS + 2~γJ to reproduce ~γH . Fortunately these logarithms have negligible eﬀect on NLL summation. This fact is made clear by the particular path shown in Figure 6.5, where these logarithms vanish in ν γS(µ0). Because of this and path independence, we suppress these terms in (6.27). ν µ The fact that γS is independent of ν to all orders in αs is also ﬁxed by the form of γS in (6.27) and the vanishing curl (6.25). Taking this general form and applying ν(d/dν) to both sides of (6.25) yields d d µ ν γν (µ, ν) = 0. (6.42) dµ dν S ν This means that ν(d/dν)γS(µ, ν) is independent of µ and in particular αs(µ). Such terms do ν not exist in perturbation theory, unless γS(µ, ν) is independent of ν. ν µ The full µ dependence of γS can therefore be obtained from γS via integrating (6.25):

Z µ 00 ν ν 0 dµ d µ 00 γS(µ) = γS(µ ) + 00 ν γS (µ , ν) . (6.43) µ0 µ dν

0 ν 0 In (6.26) we choose µ = µ0 such that all logarithms in γS(µ ) vanish and only the non- logarithmic terms remain. If (6.43) is not used, then running the soft function in the closed path shown in Figure 6.6 would result in a large phase that spoils the LL accuracy of the results when ν2 ν1. Chapter 7

Conclusion

Testing large energy QCD requires understanding how jets are produced. We have discussed the difficulties in calculating jet rates for large energy collisions. Jets are important to be able to analyze properly because they provide good tests of QCD as well as a large background to events at colliders. On the theoretical side, jet rates are interesting to calculate because they involve multiple scales that are usually correlated. Separating the physics at each of these scales is challenging. We have explored how to use Effective Field Theory (EFT) techniques to separate these scales and improve theoretical results by calculating the subleading order effects. While we are only interested in the perturbative rate where the QCD coupling constant is small, the hierarchy of scales in jet rates leads to large logarithmic enhancements in the perturbative rate. We have discussed how EFTs provide a framework for separating the physics at each of these scales and systematically improving calculations. The appropriate EFT for jet physics is soft-collinear effective theory (SCET). We introduced two formulations of SCET: the first where the large energy of the jet is removed from the theory leaving behind two-component spinors that couple to Wilson lines. The second formulation did not remove the large energy and instead described the theory as QCD fields coupled to Wilson lines. While both formulations are equivalent, the second formulation, discussed in Chapter 3, explicitly separates the various sectors at each order in the small jet mass over large energy expansion.

Thrust is a jet shape that can be used to make accurate measurements of αs(MZ ). The rate had previously been factorized at leading order in the thrust parameter and the logarithmic enhancements had been summed to a high degree. We showed in Chapter 4 how using the explicit separation of collinear and soft ﬁelds in the formulation of SCET in Chapter 3, a factorization theorem could be derived at subleading orders. This factorization was the ﬁrst ever factorization of a jet rate at subleading orders. The subleading rate also has enhancements from large logarithms that ruins the perturbative expansion in the QCD coupling constant αs(Q) 1. These logarithms can be summed by solving the Renormalization Group Equations (RGE) for the operators set out in Chapter 4. In Chapter 5 we started to renormalize the next-to-leading order operators in Chapter 3. We found that this was most easily done with a gluon mass regulator that leads to divergent integrals

105 Chapter 7. Conclusion 106 from each sector, but are convergent in the sum of all diagrams.

Finally, we discussed the kT and C/A jet algorithms in the context of SCET. Unlike the thrust observable, the kT and C/A two-jet rates depend only on two invariant scales. This leads to the soft and collinear graphs being individually divergent despite expecting them to separate at O(αs). A rapidity regulator is introduced that properly regulates the divergences appearing in these graphs. This allows the individual graphs to be calculated separately. We show how the general form of the anomalous dimensions of the soft and jet operators at all orders in αs. The general form can be written in such a way that leading-logarithmic accuracy does not require the introduction of any new regulators. The leading-logarithmic result can instead be obtained by running only the hard function, which is the square of the SCET Wilson coeﬃcient. The rapidity regulator is only required for determining how the rate depends on the renormalization scales.

7.1 Future Directions

We showed using the SCET formulation in Chapter 3 that the soft and collinear fields decoupled to all orders in power counting when matching from QCD at the ultraviolet (UV) scale Q. However, we are uncertain why it was necessary to introduce the soft fields at the UV scale in the first place in order to properly reproduce the infrared (IR) of QCD. As was discussed in Chapter 2, knowledge of the existence of multiple IR scales is not required for matching at the UV scale. These scales are only necessary after running the EFT theory down to the next IR scale. In SCET, the soft fields are typically at a scale well below the collinear fields as seen in (2.18) and (2.19). In fact, soft fields can be described as a subset of collinear fields in this momentum scaling. However, the inclusion of soft fields at Q is currently necessary for reproducing the IR of QCD in all formulations of SCET, which is counterintuitive. A future direction would be to expand the work of Chapter 3 so soft fields are only introduced at a lower IR scale. The necessity for the introduction of a new regulator in the perturbative calculation for jet rates involving only two scales, such as in Chapter 6, is also unexpected. The new regulator appears to be required to maintain the scale independence of the rate. The resulting rapidity logarithm appears to be an artifact of matching onto SCET when there are only two scales instead of the usual three scales in rates such as thrust. When there are only two scales, the UV scale and an IR soft scale, the soft and collinear fields both become heavy at the same soft scale and must be removed from our description. SCET is then matched onto a new effective theory. When there are three scales, the soft and collinear fields become heavy and are removed at two widely separated scales. The rapidity logarithm may be due to a matching condition at the lowest soft scale that is not typically necessary for calculations involving three scales, such as thrust. Finally, the work in Chapters 4 and 5 make it possible to increase the precision of the mea- Chapter 7. Conclusion 107

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