On Soft Theorems and Asymptotic Symmetries in Four Dimensions

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a dissertation presented by Temple Mu He to The Department of Physics

in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics

On Soft Theorems and Asymptotic Symmetries in Four Dimensions

Abstract

It was recently discovered that soft theorems, Ward identities associated to asymptotic symmetries, and the memory effect in gravity and gauge theories are in fact mathematically equivalent. This is a rather remarkable correspondence, as each of these subjects in theoretical physics has been ex- tensively but separately studied for many decades. Since then, this correspondence has been widely exploited to explore the infrared structure of various theories, and has been particularly fruitful both in giving new insights towards a potential holographic description of scattering amplitudes in asym- pototically flat spacetimes as well as possibly resolving the black hole information paradox.

In this thesis, I will chronicle the early progress made in establishing the equivalence between the soft theorems and asymptotic symmetries of gravity and gauge theories in four spacetime dimensions. Specifically, I will demonstrate that the leading soft theorems in 4D quantum electrodynamics (QED), nonabelian gauge theory, N = 1 supersymmetric QED, and gravity in asymptotically flat spacetimes all correspond to Ward identities of asymptotic symmetries in those theories. I will lastly use the subleading soft graviton theorem to make progress towards constructing a one-loop exact operator, whose insertion into scattering amplitudes generates the Virasoro-Ward identities associated to a energy-momentum tensor in a 2D conformal field theory.

iii Contents

0 Introduction 1

1 Quantum Electrodynamics 6 1.1 Asymptotic Expansion at I + ...... 8 1.2 Large Gauge Transformations ...... 11 1.3 Canonical Formulation ...... 13 − 1.4 Asymptotic Structure at I ...... 18 I + I − 1.5 Matching − to + ...... 21 1.6 Quantum Ward Identity ...... 22 1.7 Soft Photon Theorem ...... 24

2 Nonabelian Gauge Theory 32 2.1 Conventions and Notation ...... 35 2.2 Asymptotic Fields and Symmetries ...... 38 2.3 Soft Gluon Theorem ...... 41 2.4 Holomorphic Soft Gluon Current ...... 44 2.5 Antiholomorphic Current ...... 53

3 N = 1 Supersymmetric Quantum Electrodynamics 56 3.1 Aspects of N = 1 Gauge Theories ...... 59 3.2 Soft Photino Theorem ...... 63 3.3 Asymptotic Symmetries ...... 69 3.4 Soft Photino Theorem from Asymptotic Fermionic Symmetries ...... 88

4 Gravity in Asymptotically Flat Spacetimes 95 4.1 Supertranslation Generators ...... 97 4.2 Supertranslation Invariance of the S-Matrix ...... 107 4.3 The Soft Graviton Theorem ...... 110 4.4 From Momentum to Asymptotic Position Space ...... 113 4.5 Soft Graviton Theorem as a Ward identity ...... 118

5 Virasoro Symmetry of 4D Quantum Gravity 121 5.1 Tree-Level Energy-Momentum Tensor ...... 124 5.2 One-Loop Correction to the Subleading Soft Graviton Theorem ...... 130 5.3 One-Loop Correction to the Energy-Momentum Tensor ...... 133

iv 6 Conclusion 140

References 150

v This thesis is dedicated to my family.

vi Acknowledgments

The journey I embarked upon to obtain my PhD is certainly one that is not traveled alone. I would like to begin by thanking my advisor Professor Andy Strominger, who has been a constant beacon of inspiration and a bottomless fountain of knowledge for me these past few years. To this day, I remain awestruck at the seemingly infinite amount of time and energy you have to always provide careful guidance and life advice not just to me, but to all your graduate students, and I thank you for being my guide in my journey through asymptotic symmetries and soft theorems.

I am also indebted to Professor Craig Heinke, who spent countless hours mentoring me in astro- physics research during my junior year in high school and introduced me to the world of theoretical physics research. In college, I had the privilege to work under Professor Shamit Kachru and his former postdoc Alexander Westphal, both of whom put an incredible amount of faith in me and opened my eyes to a world of possibilities in high energy theory research. I was later able to broaden my research horizons by working under Dr. Salman Habib, who not only served as an excellent mentor, but also a caring friend who educated me in the arts of wine tasting, cooking, and classical

Indian music. To all of you, thank you for the faith and trust you put in me, for which without I would never have made it this far.

Furthermore, I would like to thank Professors Stefan Vandoren and Matt Headrick, along with their postdocs Javier Magan and Bogdan Stoica, as well as Ning Bao, who took me under their wings

vii during my graduate school years and gave me the opportunity to pursue research in the rich topic of entanglement entropy. I also would like to particularly thank Prahar Mitra, who would always pa- tiently explain subtle aspects of physics to me as we worked on a multitude of projects together. Of course, I also extend a gratitude of thanks to my numerous other collaborators Shawn Cui, Thomas

Dumitrescu, Patrick Hayden, Dan Kapec, Slava Lysov, Achilleas Porfyriadis, Ana-Maria Raclariu, and Michael Walter.

Next, I would like to thank the various members of the strings group at Harvard, including Min- jae Cho, Scott Collier, Mboyo Esole, Daniel Harlow, Daniel Jafferis, Patrick Jefferson, David Kolch- meyer, Ying-Hsuan Lin, Alex Lupsasca, Monica Pate, Abhishek Pathak, Tom Rudelius, Shu-Heng

Shao, Lily Shi, Burk Schwab, Cumrun Vafa, David Vegh, Yifan Wang, Xi Yin, and Sasha Zhiboe- dov, from whom I learned so much about both life and physics through discussions over coffee and meals. I also would like to thank Stephen Chan, Shubhayu Chatterjee, Shiang Fang, Wenbo Fu, Alex

Ji, Rebecca Krall, John Liu, Andy Lucas, Aditya Parikh, Janos Perczel, Jing Shi, Tony Tong, and

Matthijs Vakar for all the entertainment they have brought into my life over the years. Of particular importance is Monica Kang, who is a limitless source of mirth and magic in my life.

Lastly, I want to thank my family, to whom I am eternally grateful for. I want to thank my brother

Rolland He for being my oldest friend and companion; I want to thank my grandma Bangyuan Xie, who was my sole guardian during the first four years of my life and nurtured my curiosity; most of all, I want to thank my parents Harry He and Wendy Wu, who sacrificed everything to begin anew in America and give me a fresh start filled with educational opportunities. Thank you for everything, and thank you for teaching me the importance of being a better person above all else.

viii ix 0 Introduction

In theoretical physics, symmetries have throughout history played an important role in the understanding of the world around us. Classically, a symmetry of a field theory is, in the most naive and straightforward sense, a field transformation that leaves the equations of motions invariant. For instance, spacetime translation is often one such symmetry, which is simply the statement that the laws of physics are the same regardless of both time and location. Quantum mechanically, a symme-

1 try of a quantum field theory (QFT) can be thought of as a Ward identity. In other words, if Q is the charge that generates the symmetry, we expect

⟨out|(QS − SQ)|in⟩ , (1)

i.e. Q commutes with the S-matrix.

One particular set of symmetries that are of great interest are asymptotic symmetries, which are those whose charges act nontrivially on the physical states. This class of symmetries differs from the usual gauge symmetries, whose actions on fields vanish at infinity and therefore act trivially on physical states. For this reason, asymptotic symmetries are also sometimes called “large gauge symmetries.”1 Historically, asymptotic symmetries were first studied in the context of gravity in the

1960s by Bondi, Metzner, van der Burg, and Sachs [1, 2]. By analyzing gravitational radiation at null infinity in asymptotically flat spacetimes, they discovered the BMS group, which is an infinite- dimensional enhancement of the symmetries of special relativity, otherwise known as the Poincare group. This was a rich and unexpected discovery, and general relativists studied its far-reaching consequences for over half a century.

At roughly the same time, Weinberg was extending upon the work done by Bloch, Nordsieck, and Low in [3–5] by studying a class of theorems in quantum electrodynamics (QED) and gravity known as soft theorems [6]. These theorems demonstrated the universal relation between scattering processes involving a massless particle with vanishing energy to those without the massless particle,

1This definition of large gauge symmetries allows for topologically trivial symmetries as well.

2 and are crucial for maintaining the consistency of QFT. As a result, a generation of particle physicists delved into these soft theorems, and their research has been instrumental in both the theoretical understanding of QFTs as well as the experimental analysis of scattering cross-section areas at the

Large Hadron Collider.

As there were ostensibly no connections between soft theorems and asymptotic symmetries, the two subjects flourished independently. However, in the early 2010s, Strominger began exploring a possible relationship between the two seemingly disparate subjects [7,8], and surprisingly discovered that these two subjects were in fact mathematically identical, namely that the Ward identity associated to the asymptotic symmetries of a theory was precisely the corresponding soft theorem. Perhaps just as surprising, Strominger and Zhiboedov worked out a year later that there was in fact also a mathematical equivalence between the BMS group and gravitational memory as well [9]. Gravita- tional memory was first studied in [10] in the 1970s, and it describes the phenomenon in which the spatial seperation between two points is permanently altered due to the passage of a gravitational wave. Although the memory effect is incredibly small and has escaped detection from LIGO thus far, there is hope that it will be detected in the near future by accumulating the data from numerous gravitational wave measurements from LIGO [11].

Since the discovery of the equivalence between asymptotic symmetries, soft theorems, and gravitational memory, there has been an explosion of activity in exploiting this equivalence and studying the consequences. This triplet became known as the “infrared triangle” [12], as they are related to the infrared dynamics of their corresponding theories. In the past, physicists have primarily focused on leading soft theorems, which are soft theorems that only involved the leading terms in the univer-

3 sal relation between scattering processes with and without the low-energy massless particle. How- ever, with the advent of the infrared triangle, every soft theorem is a consequence of a symmetry, and hence focus has also turned to studying the subleading, and in some cases, the sub-subleading terms in the universal relation. These research endeavors have given rise to a wealth of knowledge and understanding for gravity and gauge theories that we previously did not possess. By using either the asymptotic symmetry or a soft theorem to derive the corresponding memory effect, which may in principle be observable, it is hoped that all these theoretical efforts will eventually be vindicated by experiments.

Perhaps one of the most exciting prospects of studying this infrared triangle, and indeed one of the original motivations that began this enterprise, is its potential to produce a holographic correspondence between 4D asymptotically flat spacetimes and a 2D conformal field theory (CFT). In particular, it was discovered in [13, 14] that the extended BMS group, which includes an additional

Virasoro symmetry, gives rise to the subleading soft graviton theorem. The existence of this Virasoro symmetry suggests that the 4D scattering amplitudes can be thought of as 2D CFT correlators living on the asymptotic sphere at null infinity. If successful, this correspondence will give us novel insight into quantum gravity in our own universe, in the same manner the AdS/CFT correspondence gave us invaluable tools towards understanding quantum gravity in anti-de Sitter (AdS) spacetimes.

The goal of this modest thesis is certainly not to paint a comprehensive portrait of all the over- whelming progress that has been made in all the topics listed above; rather, I will provide aspects of my own research that helped contribute towards the understanding of the infrared triangle in 4D gravity and gauge theories. As my research has exclusively been focused on the relationship between

4 asymptotic symmetries and soft theorems, we will not delve further into the memory effect. In the first chapter, I will illustrate the connection between asymptotic symmetries and soft theorems in the simplest conceptual framework, that of four-dimensional QED. The analysis will then be gener- alized to 4D nonabelian gauge theories in chapter 2. In chapter 3, we illustrate the supersymmetric version of the soft photon theorem by deriving the N = 1 soft photino theorem and its corresponding asymptotic symmetries. Next, we turn to gravity in chapter 4, and show how the famous soft graviton theorem derived by Weinberg in the 1960s is precisely the Ward identity associated to the

BMS group that was discovered concurrently. Lastly, we reveal in chapter 5 the progress towards writing down a one-loop exact stress tensor that generates the Virasoro symmetry in the extended

BMS group, with hopes that this will ultimately lead us to a holographic correspondence between scattering amplitudes in asymptotically flat spacetimes and correlators in a CFT.

5 1 Quantum Electrodynamics

Although the general equivalence between soft theorems and asymptotic symmetries initially emerged in the context of gravity [7, 8, 13–29], it is conceptually much simpler to first demonstrate this equivalence in the context of QED. Soft theorems, as already mentioned, are relations between n- and

(n + 1)-particle scattering amplitudes, where the extra particle is soft, i.e. has vanishing energy. This relation between scattering amplitudes is linear, and thus can be recast as an infinitesimal symme-

6 try of the S-matrix. It is gratifying that in some cases the resulting symmetries have turned out to be known space-time or gauge symmetries. For example Weinberg’s soft graviton theorem [6, 30] is equivalent to a symmetry of the S-matrix generated by a certain diagonal subgroup of the product

− of BMS supertranslations acting on past and future null infinity, I + and I [8].

This equivalence relation is of interest for several reasons. It reveals that soft theorems exist and are universal because they arise from a symmetry principle. Moreover, it imparts observational meaning to Minkowskian asymptotic symmetries, which have at times eluded physical interpretation. The framework has proven useful for establishing new symmetries [14] and new soft theorems [13, 16, 17]. In the quantum gravity case, the symmetries provide the starting point for any attempt at a holographic formulation [31], which we hope to elaborate more on in chapter 5. In the gauge theory case, they are potentially useful for improving the accuracy of collider predictions [32].

The purpose of the present chapter is to argue that the soft photon theorem in massless QED can be understood as a new asymptotic symmetry.1 The symmetry is generated by “large” U(1) gauge transformations which approach an arbitrary function ε(z, z) on the conformal sphere at I but are

− constant along the null generators, even as they antipodally cross from I to I + through spatial infinity. Except for the transformation where ε is a constant, these symmetries are spontaneously broken in the conventional vacuum used in perturbative QFT, and the soft photons appear as Gold- stone modes living on the sphere at the boundary of I . Indeed, these large U(1) gauge symmetries are the precise analogs of BMS supertranslations in gravity, discovered half a century earlier!2

1The equivalence between the soft photon theorem and the corresponding asymptotic symmetry also holds for massive QED [33, 34], but the analysis more involved and beyond the scope of this thesis. 2It would be interesting to systematically derive this large U(1) symmetry group using the type of asymp-

7 This chapter is organized as follows. In section 1.1 we describe the classical final data formulation at I +. Section 1.2 gives the asymptotic symmetries, which are the QED analogs of BMS symmetries, and constructs the associated charges. In section 1.3, the commutators at I + are given and the charges of section 1.2 are shown to generate the symmetries. This turns out to require a careful treatment of the Goldstone modes and boundary conditions at the boundaries of I +. Section 1.4 gives

− − the corresponding formulae for I . In section 1.5, we give conditions which tie the data of I to that of I +, thereby defining the scattering problem. The conditions are shown to break the sepa-

− rate asymptotic symmetries on I + and I into a diagonal subgroup that preserves the S-matrix elements. In section 1.6, the quantum Ward identity of this symmetry is shown to relate scattering amplitudes with and without a soft photon insertion. Finally, in section 1.7, we show that this Ward identity is the soft photon theorem.

1.1 Asymptotic Expansion at I +

In this subsection we consider the canonical final data formulation of U(1) electrodynamics coupled to massless charged matter at future null infinity (I +). It is convenient to adopt retarded coordinates

2 − 2 − 2 ds = du 2dudr + 2r γzzdzdz , (1.1) totic analysis employed in the original derivation of the BMS group [1, 2]. Instead, we shall argue that the non- triviality of the symmetries is established by their equivalence to the soft photon theorem.

8 = − = 2 3 where u t r is the retarded time and γzz (1+zz)2 is the round metric on the conformal sphere.

Thus I + is the null S2 ×R boundary at r = ∞ with coordinates (u, z, z), while its own boundaries

+ are at u = ±∞ denoted as I± .

The bulk equations of motion for U(1) gauge theory are

∇νF 2 M (1.2) νμ = e jμ ,

F A − A M ∇μ M where μν = ∂μ ν ∂ν μ and jμ is the conserved matter current obeying jμ = 0. The equations (1.2) have a gauge symmetry under which

δεbAμ = ∂μεb , (1.3)

where εbhas periodicity

εb ∼ εb+ 2π . (1.4)

To study QED, we will use the retarded radial gauge, which is

Ar = 0 , (1.5)

A | u I + = 0 . (1.6)

3 = iφ θ We have set z e tan 2 .

9 While this choice is suited for studying QED, we will in the next chapter adopt a different gauge when studying nonabelian gauge theories.

We now wish to expand the fields around I +, the regime that is of most interest to us. The radi- ∫ I + F zF ation flux through is proportional to I + u uz. To ensure that the radiation flux is nonzero

+ + and finite, we require Az ∼ O(1) near I . Following (1.6), we also require Au ∼ O(1/r) near I .

Thus, we have the expansions

∑∞ (n) A (u, z, z) A ( , , , ) = ( , , ) + z , z r u z z Az u z z n = r n 1 (1.7) ∑∞ (n) 1 A (u, z, z) A (r, u, z, z) = A (u, z, z) + u . u r u rn+1 n=1

+ −2 The leading terms in the field strengths near I are then Fzz = O(1), Fur = O(r ), Fuz = O(1),

−2 and Frz = O(r ), with coefficients

Fzz = ∂zAz − ∂zAz ,

Fuz = ∂uAz , (1.8) − (1) Frz = Az ,

Fur = Au .

Note that the fields F and A live on I + and have no r dependence. In terms of these fields, the leading constraint equation becomes

2 γzz∂uAu = ∂u (∂zAz + ∂zAz) + e γzzju , (1.9)

10 where we defined

[ ] 2 M ju(u, z, z) ≡ lim r j (u, r, z, z) . (1.10) r→∞ u

As we are only interested in configurations which revert back to the vacuum in the far future, we can set

Fur|I + = Fuz|I + = 0 . (1.11) + +

For any given ju, we can determine from (1.9) and (1.11) Au in terms of Az and Az. Thus, we will take

+ 1 Az and Az to be coordinates on the asymptotic phase space Γ . Subleading terms in the r expansions of all the other equations of motion then determine the expansion of Aμ in terms of the final data Az and matter current.

1.2 Large Gauge Transformations

The gauge conditions (1.5) and (1.6) leave unfixed residual gauge transformations generated by an arbitrary function approaching εb = ε(z, z) on the conformal sphere at r = ∞. We will refer to these as “large gauge transformations”, which act on Γ+ via

δεAz(u, z, z) = ∂zε(z, z) . (1.12)

11 These comprise the asymptotic symmetries of QED considered in this chapter, and the associated charge is [31]

∫ ∫ [ ] + 1 2 1 2 2 Qε = d zγzzεFru = dud zε ∂u (∂zAz + ∂zAz) + e γzzju , (1.13) 2 + 2 + e I− e I where we obtained the second equality by integrating by parts, assuming the final charge relaxes to

I + + zero at + , and using the constraint (1.9). For the special case ε = 1, Q1 is just the total initial electric charge, which obeys

∫ + 2 (1.14) Q1 = dud zγzzju . I +

On the other hand, for the choice ε(z, z) = δ2(z − w), we have the fixed-angle charge

∫ ∞ + 1 2 Qww = du [∂u (∂wAw + ∂wAw) + e γwwju] . (1.15) e2 −∞

This is the total outgoing electric charge radiated into the fixed angle (w, w) on the asymptotic S2.

The first term is a linear “soft” photon contribution to the fixed-angle charge,4 which doesn’t con-

+ tribute to the total charge Q1 as it is a total derivative. The second term is the accumulated matter charge flux at the angle (w, w) and is responsible for generating large gauge transformations on mat-

+ ter fields. Indeed, for an arbitrary ε(z, z), it can be shown that Qε generates the large gauge transfor-

4Throughout this thesis, by “soft” we mean that the momentum is strictly zero, as opposed to just small.

12 mation on matter fields via

[∫ ] [ ] ′ + 2 ′ − Qε , Φ(u, z, z) = du d wεγwwju , Φ(u, z, z) = qε(z, z)Φ(u, z, z) , (1.16) I + where Φ is any massless charged matter field operator on I + with charge q.

1.3 Canonical Formulation

The commutators, or equivalently a non-degenerate symplectic form, on the radiative phase space

+ ≡ { } ΓR Fuz, Fuz are constructed, for example, in [31,35]. The non-vanishing ones are

[ ] 2 ( ) ′ ie ′ F (u, z, z), F ′ (u , w, w) = ∂ δ u − u δ2(z − w) . (1.17) uz u w 2 u

Integrating and fixing the integration constants by antisymmetry yields

[ ] 2 ( ) ′ ie ′ A (u, z, z), A (u , w, w) = − Θ u − u δ2(z − w) , (1.18) z w 4 where Θ(x) = sign(x). Given (1.16), one might expect that symmetry transformations on the gauge

+ fields are then generated by commutators with Qε using (1.18). However, an explicit computation instead yields

[ ] + i Q , A (u, z, z) = ∂ ε(z, z) ≠ iδεA (u, z, z) , (1.19) ε z 2 z z

13 1 1 which is off by a factor of 2 . As we will later see in Chapter 4, a similar factor of 2 was encountered in the construction of the BMS supertranslation operator in [15].

In order to resolve this discrepancy, we must give a more precise description of the phase space

+ + + Γ . In particular, we must both specify boundary conditions on Az at the boundaries I± of I and include the soft photon zero modes. The boundary values of the fields are denoted by

± ≡ ±∞ (1.20) Az (z, z) Az(u = , z, z) .

We consider here the sector of the phase space with no long-range magnetic fields, namely

| + = . (1.21) Fzz I± 0

± I + In other words, the connections Az are flat on ± . We will implement (1.21) as constraints. These constraints are not preserved by the commutators (1.18), which hence must be modified according to

Dirac’s procedure. A unique modification is determined by the continuity condition

[ ] [ ] ± ′ ′ A (z, z), Aw(u , w, w) = lim Az(u, z, z), Aw(u , w, w) , z u→±∞ (1.22) [ ] [ ] +( , ) − −( , ), ±( , ) = +( , ) − −( , ), ( ′, , ) , Az z z Az z z Aw w w ′ lim Az z z Az z z Aw u w w u →±∞ and the vanishing of equal-u commutators. The solution to (1.21) is given by

± 2 (1.23) Az (z, z) = e ∂zφ±(z, z) .

14 ± Of course, the constant modes of φ± cannot be determined from Az , but it natural and useful to

2 include them by simply treating φ±(z, z) as unconstrained fields on S . The commutators satisfying

(1.22) are then

[ ] ′ i 1 φ±(z, z), A (u , w, w) = ∓ , w 8π z − w (1.24) [ ] i 2 φ (z, z), φ−(w, w) = log |z − w| . + 4πe2

+ Using (1.23), the charge Qε can now be written as

∫ ∫ ( ) + 2 − 2 (1.25) Qε = 2 d zε∂z∂z φ+ φ− + dud zγzzεju . S2 I +

It then immediately follows from (1.24) and (1.25) that

+ [Qε , Az(u, z, z)] = i∂zε(z, z) , (1.26) + i [Q , φ±(z, z)] = ε(z, z) , ε e2 and that the charges satisfy the Abelian algebra

[ ] + + (1.27) Qε , Qε′ = 0 .

This demonstrates that on the constrained phase space defined by (1.21), the modified commutators properly generate the large gauge transformations.

15 1 Periodicity of ε implies that φ− lives on a circle of radius e2 :

2π φ− ∼ φ− + . (1.28) e2

Furthermore, we note exponentials of φ− obey

[ ] 2 2 + ine φ− − ine φ− Qε , e = nεe , (1.29)

and have (in our conventions) integer charges n. Such operators do not in themselves create physical states. Rather states with charge n are created by products of these operators with neutral matter- sector operators. This is virtually the same operator product decomposition familiar in 2D CFT when factoring a U(1) current algebra boson, or in 4D soft collinear effective field theory (SCET) involving the so-called jet field [36, 37].

A vacuum wave function for the Goldstone mode, which we take to be φ−, can be defined by the condition

φ−(z, z)|0⟩ = 0 . (1.30)

However, (1.26) implies that the large gauge symmetries are broken in this vacuum, and that the symmetries transform (1.30) into more general φ− eigenstates obeying

φ−(z, z)|α⟩ = α(z, z)|α⟩ . (1.31)

16 Up to an undetermined normalization, the inner products are

∏ ( ) ′ ′ ⟨α|α ⟩ = δ α(z, z) − α (z, z) . (1.32) z,z

Other zero-energy states are given by the expression

∫ ∫ 2 |β⟩ = Dα e2i d z∂zα∂zβ|α⟩ , (1.33)

which obey ∫ +∞ duFuz|β⟩ = ∂zβ|β⟩ (1.34) −∞ and ∫ 2 Q+|β⟩ = d2zε∂ ∂ β|β⟩ . (1.35) ε e2 z z

Thus, any state with β = constant has unbroken large gauge symmetry. These vacua are annihilated by the zero mode and are not the ones usually employed in QED analyses: it might be of interest to consider scattering in such states.5

Let us restate and summarize this section. The classic expression (1.17) is a non-degenerate symplectic form on the phase space of radiative modes with non-zero frequency. The soft photon, ∫ 2 − i.e. zero frequency mode duFuz = e ∂z(φ+ φ−), is orthogonal to this form and has no symplectic partner among these radiative modes. We remedy this by adding the boundary degree

− of freedom Az and constructing a symplectic form which pairs it with this zero mode. This is

5Such states are related to the vacua considered in [38, 39] and studied more in [40].

17 − done consistently with the constraint (1.21) on both Az and the conjugate zero mode represent- ing the absence of long range magnetic fields. The resulting non-degenerate phase space, denoted by { } + Γ = Fuz(u, z, z), φ+(z, z), φ−(z, z) , then consists of the usual (non-zero frequency) radiative

− modes, together with the zero-momentum soft photon φ+(z, z) φ−(z, z) and the canonically conjugate periodic real boson φ−(z, z).

1.4 Asymptotic Structure at I −

− A similar structure exists near I and is needed to discuss scattering. In this subsection we recap

− the requisite formulae. I is at r = ∞ with v fixed in advanced coordinates, given by

2 − 2 2 (1.36) ds = dv + 2dvdr + 2r γzzdzdz .

These are related to (1.1) by the coordinate transfomation u → v − 2r, r → r, and z → −1/z. The last relation means that points on S2 with the same value of z in retarded and advanced coordinates are antipodal. This coordinate choice is natural in the conformal compactification of Minkowski

− space, where there are null generators of I which run from I to I + through spatial infinity. We have chosen coordinates so that points anywhere on I with the same value of (z, z) lie on the same null generator. In advanced radial gauge Ar = 0 = Av|I − and the fields have the large r expansion

− 1 − A (r, v, z, z) = B (v, z, z) + O(r 1) , A (r, v, z, z) = B (v, z, z) + O(r 2) . (1.37) z z v r v

18 Analogous to the case on I +, the leading order constraint equation is

− − 2 (1.38) γzz∂vBv = ∂v (∂zBz + ∂zBz) e γzzjv ,

where now

[ ] 2 M jv(v, z, z) = lim r j (v, r, z, z) . (1.39) r→∞ v

− Unfixed large gauge transformations are parameterized by ε (z, z), under which

− δε− Bz = ∂zε . (1.40)

Similar to before, the associated charge is

∫ ∫ ∞ [ ] − − 1 2 − 1 2 − 2 Qε− = d zγzzε Bv = dvd zε ∂v (∂zBz + ∂zBz) + e γzzjv , (1.41) 2 I − 2 −∞ e + e and the boundary values of the fields are defined to be

± ≡ ±∞ (1.42) Bz (z, z) Bz (v = , z, z)

19 with constraints

± − ± (1.43) ∂zBz ∂zBz = 0 .

This is solved by

± 2 (1.44) Bz = e ∂zψ±

for some periodic function ψ±. Employing the same methods and assumptions as our analysis near

I +, the commutators consistent with (1.43) are

[ ] ′ i 1 ψ±(z, z), B (u , w, w) = ∓ , w 8π z − w (1.45) [ ] i 2 ψ (z, z), ψ−(w, w) = log |z − w| . + 4πe2

These in turn imply

− − [Qε− , Bz(v, z, z)] = i∂zε (z, z) , (1.46) − i − [Q − , ψ±(z, z)] = ε (z, z) . ε e2 { } − The incoming phase space is then Γ = Gvz(v, z, z), ψ+(z, z), ψ−(z, z) , where Gvz = ∂vBz.

20 I + I − 1.5 Matching − to +

− The classical scattering problem is to find the map from Γ to Γ+, i.e. to determine the final data ( ) ( ) I + I − Fuz, φ− on which arises from a given set of initial data Gvz, ψ+ on . Given a field strength everywhere on Minkowski space, this data is so far determined only up to the large gauge

− − transformations which are generated by both ε and ε and act separately on Γ+ and Γ . Clearly

− there can be no sensible scattering problem without imposing a relation between ε and ε . Any relation between them should preserve Lorentz invariance. Under an SL(2, C) Lorentz transformation parameterized by ζz ∼ 1, z, z2, one finds

z z δζψ+ = (ζ ∂z + ζ ∂z)ψ+ , (1.47) z z δζφ− = (ζ ∂z + ζ ∂z)φ− .

This symmetry is preserved by the natural requirement

(1.48) ψ+(z, z) = φ−(z, z) ,

which in turn requires

− ε(z, z) = ε (z, z) , (1.49)

21 as well as the generalization to finite gauge transformations. Note that, because of the antipodal

− identification of the null generators of I + and I , this means the gauge parameter is not the limit of a function which depends only on the angle in Minkowskian (r, t) coordinates. Rather, it goes to the same value at the beginning and end of light rays crossing through the origin of Minkowski space.6 Thus, ε is a function on the space of null generators of I .

Both the gauge field strength and the charge current are invariant under these symmetries. The phases they generate on matter fields are classically unobservable. Hence, they have little import for the usual discussion of classical scattering. It simply (antipodally) equates the final data for φ− with that of the initial data for ψ+. However, in the quantum theory, where phases matter, they have significant consequences to which we now turn.

1.6 Quantum Ward Identity

In this section, we consider the consequences of the large gauge symmetry on the semi-classical S-

in out matrix. Let us denote an in (out) state comprised of n (m) particles with charges qk (qk ), incoming ⟩ in out 2 | ⟩ ≡ in ··· in ⟨ | ≡ at points zk (outgoing at points zk ) on the conformal sphere S by in z1 , , zn ( out

⟨ out ··· out| S ⟨ | S | ⟩ z1 , , zm ). The -matrix elements are then denoted as out in . The quantum version of the classical invariance of scattering under large gauge transformations is

( ) ⟨ | +S − S − | ⟩ out Qε Qε in = 0 . (1.50)

6Such gauge transformations were considered in [41].

22 The semi-classical charge obeys the quantum relations (from (1.13) and (1.41))

∑m ⟨ out ··· out| + ⟨ out ··· out| + out out out ⟨ out ··· out| z1 , , zm Qε = z1 , , zm F [ε] + qk ε(zk , zk ) z1 , , zm , = k 1 (1.51) ⟩ ⟩ ⟩ ∑n − in ··· in − in ··· in in ··· in in in in Qε z1 , , zn = F [ε] z1 , , zn + z1 , , zn qk ε(zk , zk ) , k=1 where

∫ ( ) + ≡ − 2 − F [ε] 2 d w∂wε∂w φ+ φ− , ∫ (1.52) ( ) − ≡ − 2 − F [ε] 2 d w∂wε∂w ψ+ ψ− .

Defining

− F[ε] ≡ F+[ε] − F [ε] (1.53)

and the time-ordered product

− : F[ε]S : = F+[ε]S − SF [ε] , (1.54)

(1.50) becomes

[ ] ∑n ∑m ⟨ | S | ⟩ in in in − out out out ⟨ | S | ⟩ out : F[ε] : in = qk ε(zk , zk ) qk ε(zk , zk ) out in , (1.55) k=1 k=1

23 This Ward identity relates the insertion of a soft photon with polarization and normalization given in (1.53) into any S-matrix element to the same S-matrix element without a soft photon insertion.

For an incoming state which happens to be the vacuum, (1.51) reduces to

− | ⟩ − | ⟩ Qε 0in = F [ε] 0in . (1.56)

− Hence Qε does not annihilate the vacuum unless ε = constant, implying that all but the constant mode of the large gauge symmetries are spontaneously broken. Moreover (1.46) identifies ψ+ as the corresponding Goldstone boson.

This result may seem surprising for the following reason. Soft photons are labelled by a spatial direction and a polarization. This suggests two modes for every point on the sphere, which is twice the number predicted by Goldstone’s theorem. However, as we will later demonstrate, the positive- and negative-helicity modes are not independent, thereby reducing the two modes for every point on the sphere to the single one predicted in Goldstone’s theorem.

1.7 Soft Photon Theorem

Finally, we will in this subsection show that the Ward identity (1.55) is the soft photon theorem in disguise. In order to do so, we must now rewrite everything in momentum space. The first step is to

± expand the soft photon operators F [ε], expressed above as weighted integrals over the conformal sphere at I , in terms of the standard plane wave in and out creation and annihilation operators.

24 Momentum eigenmodes in Minkowski space are usually described in flat coordinates

ds2 = −dt2 + d⃗x · d⃗x , (1.57)

which are related to the retarded coordinates in (1.1) by

2rz r (1 − zz) t = u + r , x1 + ix2 = , x3 = , (1.58) 1 + zz 1 + zz where⃗x = (x1, x2, x3) satisfies⃗x · ⃗x = r2. At late times and large r, the wave packet for a massless particle with spatial momentum centered around⃗p becomes localized on the conformal sphere near the point (z, z) with

ω ⃗p = ωbx = (z + z, −i (z − z) , 1 − zz) , (1.59) 1 + zz

b ⃗x where x = r . The momentum of massless particles may equivalently be characterized either by

μ (ω, z, z) or by p . At late times t → ∞, the gauge field Aμ becomes free and can be approximated by the mode expansion

∫ ∑ 3 [ ] d q 1 ∗ · † − · A (x) = e εα (⃗q)aout(⃗q)eiq x + εα(⃗q)aout(⃗q) e iq x , (1.60) μ (2π)3 2ω μ α μ α α=± q

25 0 where q = ωq = |⃗q| and α = ± are the two helicities. The creation and annihilation operators on

I + out† out , aα and aα , obey

[ ] ( ) out out ′ † 3 3 − ′ (1.61) aα (⃗q), aβ (⃗q ) = δαβ(2π) (2ωq)δ ⃗q ⃗q

for ωq > 0; as we will soon see, for ωq = 0 the positive and negative helicities are linearly depen-

in in† − dent. Similarly, a± and a± annihilate and create incoming photons on I . In terms of (w, w) the polarization tensors have components

1 ε+μ(⃗q) = √ (w, 1, −i, −w) , 2 (1.62) − 1 ε μ(⃗q) = √ (w, 1, i, −w) , 2

±μ which satisfy qμε (⃗q) = 0.

To expand the gauge field near I + recall that

Az(u, z, z) = lim Az(u, r, z, z) . (1.63) r→∞

μ Using Az = ∂zx Aμ, the mode expansion in (1.60), and the stationary phase approximation, we find

√ ∫ ∞ [ ] i 2e − † ( , , ) = − out( b) iωqu − out( b) iωqu . (1.64) Az u z z 2 dωq a+ ωqx e a− ωqx e 8π 1 + zz 0

26 Defining the energy eigenmodes

∫ ∞ ω ≡ iωu Nz (z, z) due Fuz , (1.65) −∞ we obtain

√ ∫ ∞ [ ( ) ( )] ω − 1 2e out b − out b † Nz (z, z) = dωqωq a+ (ωqx)δ ω ωq + a− (ωqx) δ ω + ωq . (1.66) 4π 1 + zz 0

When ω > 0 (ω < 0), only the first (second) term contributes. We define the zero mode by the

Hermitian expression

( ) 0 1 ω −ω (1.67) Nz (z, z) = lim Nz + Nz . ω→0+ 2

It follows that

√ [ ] 0 − 1 2e out b out b † (1.68) Nz (z, z) = lim ωa+ (ωx) + ωa− (ωx) . 8π 1 + zz ω→0+

− Similarly on I we define

∫ ∞ √ [ ] 0 ≡ iωv 1 2e in b in b † Mz (z, z) dve Gvz = lim ωa+(ωx) + ωa−(ωx) , (1.69) −∞ 8π 1 + zz ω→0+

in in† − where a± and a± annihilate and create incoming photons on I .

27 It follows from (1.65), (1.68), and (1.69) that

∫ ∫ [ ] ∞ ∞ 2 0 − 0 − e 1 Nz Mz = duFuz dvGvz = F , (1.70) −∞ −∞ 4π z − w

1 where F[ε] is defined in (1.53). Setting ε(z, z) = z−w , the Ward identity (1.55) becomes

[ ] ∑m ∑n e2 qout qin ⟨out| :(N0 − M0) S : |in⟩ = k − k ⟨out| S |in⟩ . (1.71) z z 4π z − zout z − zin k=1 k k=1 k

Using (1.68) and (1.69), the above equations become

[ ] [ ] ∑m out ∑n in ⟨ | out b S | ⟩ √e qk − qk ⟨ | S | ⟩ lim ω out a+ (ωx) in = (1 + zz) out in out in , (1.72) ω→0+ 2 z − z z − z k=1 k k=1 k where we have used the fact

[ ] [ ] ⟨ | S in b † | ⟩ ⟨ | out b S | ⟩ lim ω out a−(ωx) in = lim ω out a+ (ωx) in . (1.73) ω→0+ ω→0+

We now wish to compare (1.72) with the soft photon theorem in its conventional form [42], which is given by

[ ( )] M { in} { out} lim ω + pγ; pk , pk ω→0+ [ ] (1.74) ∑m out out · ε+( ) ∑n in in · ε+( ) ( ) ωqk pk pγ − ωqk pk pγ M { in} { out} = e lim out in pk , pk . ω→0+ p · pγ p · pγ k=1 k k=1 k

28 ( ) M { in} { out} Here, pk , pk is the momentum space scattering amplitude of n (m) incoming (outgo- ( ) in out in out M { in} { out} ing) particles with charges qk (qk ) and momenta pk (pk ), and + pγ; pk , pk is the same amplitude with one additional outgoing positive-helicity soft photon with momentum pγ. In our conventions,

( ) ( ) M { in} { out} ⟨ | S | ⟩ M { in} { out} ⟨ | out ⃗ S | ⟩ (1.75) pk , pk = out in , + pγ; pk , pk = out a+ (pγ) in .

Using the parametrization of the momenta discussed earlier, i.e.

( ( ) ) zin + zin −i zin − zin 1 − zinzin (pin)μ = Ein 1, k k , k k , k k , k k 1 + zinzin 1 + zinzin 1 + zinzin ( k k (k k ) k k ) out out − out − out − out out out μ out zk + zk i zk zk 1 zk zk (pk ) = Ek 1, out out , out out , out out , 1 + zk zk 1 + zk zk 1 + zk zk ( ) (1.76) μ z + z −i (z − z) 1 − zz pγ = ω 1, , , , 1 + zz 1 + zz 1 + zz 1 ε+(p ) = √ (z, 1, −i, −z) , μ γ 2 we find

( ) M { in} { out} lim ω + pγ; pk , pk ω→0+ [ ] ∑m ∑n (1.77) e qout qin ( ) = √ (1 + zz) k − k M {pin}, {pout} . 2 z − zout z − zin k k k=1 k k=1 k

This is precisely (1.72), thereby demonstrating that the Ward identity (1.55) implies the soft photon theorem. To show that the soft photon theorem (1.74) implies the Ward identity (1.55), it suffices to

29 1 run the argument backwards with ε = z−w . However, since any function ε(z, z) can be written as

∫ 1 1 ε(w, w) = d2zε(z, z)∂ , (1.78) 2π z z − w and F[ε] is linear in ε, the soft photon theorem implies (1.55) for any ε(z, z).

We conclude this chapter by demonstrating that the positive- and negative-helicity soft photons are not linearly independent. As this is most easily seen in the (z, z) coordinates, we will start with the soft photon theorem in the parameterization given in (1.72). Consider now the amplitude involving the following linear combination of the positive-helicity soft photons:

∫ [ ] { } O 1 2 1 1 out b (z, z) = (1 + zz) d w ∂w lim ωa+ (ωy) , (1.79) 2π z − w 1 + ww ω→0+ where by points towards (w, w). Insertions of this operator is given by (1.72) as

[ ] ∑m ∑n e qout qin ⟨out| O(z, z)S |in⟩ = √ (1 + zz) k − k ⟨out| S |in⟩ . (1.80) 2 z − zout z − zin k=1 k k=1 k

This is precisely the soft photon theorem for a negative-helicity soft photon with momentum point- ing towards (z, z). We therefore conclude that the linear combination

∫ [ ] out b out 1 2 1 a+ (ωy) a− (ωbx) − (1 + zz) d w ∂ (1.81) 2π z − w w 1 + ww has no poles and decouples from the S-matrix at leading order. In the more familiar momentum

30 space variables, this is

∫ out 1 1 + cos θq out a− (ωbp ) + ( ) dΩ ( ) a (ωbq). (1.82) γ q + b · b 2 + 2π 1 + cos θpγ ε (pγ) q where the integral is over the angular distribution of bq. This completes our discussion of soft theorems and asymptotic symmetries in four-dimensional QED.

31 2 Nonabelian Gauge Theory

In this chapter, we will describe the n-particle scattering amplitudes An of any four-dimensional

2 quantum field theory (QFT4) as a collection of n-point correlation functions on the two-sphere (S ) with coordinates (z, z), explicitly written as

An = ⟨O1(E1, z1, z1) ··· On(En, zn, zn)⟩ , (2.1)

32 | | where Ok creates (if Ek < 0) or annihilates (if Ek > 0) an asymptotic particle with energy Ek at the

2 I ± point (zk, zk) on the asymptotic S at null infinity ( ). The alternate description (2.1) is obtained from the usual momentum space description by simply trading the three independent components

μ 2 − 2 of the on-shell four momentum pk (subject to pk = mk) with the three quantities (Ek, zk, zk).

The Lorentz group SL(2, C) acts as the global conformal group on the asymptotic S2 according to az + b z → , (2.2) cz + d

with ad − bc = 1. Hence, in this respect, Minkowskian QFT4 amplitudes resemble Euclidean two- dimensional conformal field theory (CFT2) correlators. It is natural to ask what other properties

QFT4 scattering amplitudes, expressed in the form (2.1), have in common with conventional CFT2 correlators, and more generally whether a holographic relation of the form Minkowskian QFT4 =

1 Euclidean CFT2 might plausibly exist when gravity is included. In this chapter, we will consider tree-level scattering of massless particles in 4D nonabelian gauge theories with gauge group G.A salient feature of all such amplitudes is that soft gluon scattering is controlled by the soft gluon theorem [48]. A prescription is given for completing the hard S-matrix (in which all external states

̸ S have Ek = 0) to an -correlator which includes positive-helicity soft gluons at strictly zero energy.

It is shown that the content of the soft gluon theorem at tree-level is that the positive-helicity soft gluon insertions are holomorphic 2D currents which generate a 2D G-Kac-Moody algebra in the S- correlator! Turning the argument around, the soft gluon theorem can be derived as a tree-level Ward

1The results of [13, 14, 43–47] all suggest that for quantum gravity scattering amplitudes the SL(2, C) Lorentz symmetry (2.2) is enhanced to the infinite-dimensional local 2D conformal symmetry.

33 identity of the Kac-Moody symmetry.

Moreover, we will show that the Kac-Moody symmetries are equivalent to the asymptotic symmetries of 4D gauge theories described in [7]. They are CPT-invariant gauge transformations, which are independent of advanced or retarded time and take angle-dependent values on I . CPT invariance requires that the gauge transformation at any point on I + equals that at the PT antipode

− on I . Such transformations act nontrivially on the asymptotic physical states and comprise the asymptotic symmetry group. These are the gauge theory analogs of BMS transformations in asymptotically flat gravity [1, 2,8,14, 15, 31, 45–47,49, 50].

As we showed in the last chapter, the asymptotic symmetries of QED are spontaneously broken in the perturbative vacuum and the soft photons are the resulting Goldstone bosons. Analogously, the standard rules of Yang-Mills perturbation theory presume a trivial flat color frame on I . We will soon see that this trivial frame is not invariant under the non-constant Kac-Moody transformations and the large gauge symmetry is spontaneously broken, with the soft gluons being, as expected, the corresponding Goldstone bosons.

However, the nonabelian interactions of Yang-Mills theory lead to some surprising new features that are not present in the analysis of QED. As was pointed out by S. Caron-Huot, the double-soft limit of the S-matrix involving one positive- and one negative-helicity gluon is ambiguous. The result depends on the order in which the gluons are taken to be soft. Hence, a prescription must be given for defining the double-soft boundary of the S-matrix. We adopt the prescription that positive helicities are always taken soft first. With this prescription, there is one holomorphic G-Kac-

Moody symmetry from positive-helicity soft gluons, but not a second one from negative-helicity soft

34 gluons.

The soft gluon theorem has well-understood universal corrections due to IR divergences which appear only at one loop, see e.g. [37]. These will certainly affect any extension of the present discussion beyond tree-level. Since an infinite number of relations among S-matrix elements remain, an asymptotic symmetry may survive these corrections. However, it is not clear if it can still be understood as a Kac-Moody symmetry, and we will not address this issue further.

This chapter is organized as follows. Section 2.1 establishes our notation and conventions. In section 2.2, we introduce the various asymptotic fields used in this chapter and discuss the asymptotic symmetries of nonabelian gauge theories. The standard proof of the tree-level soft gluon theorem is reviewed in section 2.3. In section 2.4, we show that the soft gluon theorem is the Ward identity of a holomorphic Kac-Moody symmetry which can also be understood as an asymptotic gauge symmetry. Finally, in section 2.5, we show that the double-soft ambiguity of the S-matrix obstructs the appearance of a second antiholomorphic Kac-Moody symmetry.

2.1 Conventions and Notation

We consider a nonabelian gauge theory with group G and associated Lie algebra g. Elements of G in representation Rk are denoted by gk, where k labels the representation. The corresponding Hermi- tian generators of g obey

a b abc c (2.3) [Tk, Tk] = if Tk ,

35 where a = 1, ··· , [[g]] and the sum over repeated Lie algebra indices is implied. The adjoint ele-

G a a − abc ments of and generators of g are denoted by g and T , respectively, with (T )bc = if . The real antisymmetric structure constants fabc are normalized so that

facdfbcd = δab = tr[TaTb] . (2.4)

A Aa a The four-dimensional matrix-valued gauge field is μ = μT , where a μ index here and hereafter refers to flat Minkowski coordinates in which the metric is

ds2 = −dt2 + d⃗x · d⃗x , (2.5)

with⃗x = (x1, x2, x3) satisfying⃗x · ⃗x = r2. As in the QED case, we describe I + using retarded coordinates

2 − 2 − 2 ds = du 2dudr + 2r γzzdzdz , (2.6)

= − = 2 I − where u t r and γzz (1+zz)2 is the round metric on the sphere, and describe using advanced coordinates

2 − 2 2 ds = dv + 2dvdr + 2r γzzdzdz , (2.7) where v = t + r. Recall that the point with coordinates (r, v, z, z) in advanced coordinates is antipodally related to the point with coordinates (r, u, z, z) in retarded coordinates.

As we are working with a nonabelian theory, the field strength corresponding to Aμ is now given

36 to be

F A − A − A A F a a (2.8) μν = ∂μ ν ∂ν μ i[ μ, ν] = μνT ,

and the theory is invariant under gauge transformations

−1 −1 Aμ → gAμg + ig∂μg ,

→ (2.9) φk gkφk ,

M → M −1 jμ gjμ g ,

M where φk are matter fields in representation Rk and jμ is the matter current that couples to the

b gauge field. The infinitesimal gauge transformations with respect to εb = εbaTa (where g = eiε) are

δεbAμ = ∂μεb− i[Aμ, εb] ,

ba a (2.10) δεbφk = iε Tkφk ,

M − M b δεbjμ = i[jμ , ε] .

Finally, the bulk equations that govern the dynamics of the gauge field are

[ ] ∇νF − Aν F 2 M νμ i , νμ = gYMjμ , (2.11)

where ∇μ is the covariant derivative with respect to the spacetime metric.

37 2.2 Asymptotic Fields and Symmetries

In this section, we establish our conventions for the asymptotic expansion around I , specify the gauge conditions and boundary conditions, and describe the residual large gauge symmetry. We begin by choosing to work in the temporal gauge, as opposed to the retarded radial gauge adopated in the previous chapter:

Au = 0 . (2.12)

In this gauge, we can expand the gauge fields near I + as

Az(r, u, z, z) = Az(u, z, z) + O (1/r) , (2.13) 1 A (r, u, z, z) = A (u, z, z) + O (1/r3) , r r2 r where the leading behavior of the gauge field is chosen so that the charge and energy flux through

I + is finite. The full four-dimensional gauge field is determined by the equations of motion in terms of Az(u, z, z), which forms the boundary data of the theory.

2 The leading behavior of the field strength is Fur = O(1/r ) and Fuz, Fzz = O(1), with leading

38 coefficients

Fur = ∂uAr ,

Fuz = ∂uAz , (2.14)

Fzz = ∂zAz − ∂zAz − i[Az, Az] .

We will be interested in configurations that revert to the vacuum in the far future, i.e.

Fur|I + = Fuz|I + = Fzz|I + = 0 . (2.15) + + +

This immediately implies we can write

−1 Uz ≡ Az|I + = iU∂zU , (2.16) + where U(z, z) ∈ G. As in the QED case, a residual gauge freedom near I + is generated by an arbitrary function ε(z, z) on the asymptotic S2. These create zero-momentum gluons and will be referred to as large gauge transformations. Under finite large gauge transformations, U → gU. We also define the soft gluon operator

∫ ∞ Nz ≡ duFuz = Uz − Az|I + . (2.17) −∞ −

39 − We now perform the analogous analysis near I . The temporal gauge condition implies

Av = 0 . (2.18)

A O −1 A 1 O −3 We expand the gauge fields as z = Bz + (r ) and r = r2 Br + (r ), and the field strength

2 has leading behaviour Fvr ∼ O(1/r ) and Fvz, Fzz = O(1), with leading coefficients

Gvr = ∂vBr ,

Gvz = ∂vBz , (2.19)

Gzz = ∂zBz − ∂zBz − i[Bz, Bz] .

Configurations that begin from the vacuum in the far past satisfy

| − = | − = | − = , (2.20) Gur I− Gvz I− Gzz I− 0

and the four-dimensional gauge field is uniquely determined by the boundary data Bz(v). Residual

− − gauge freedom near I is generated by an arbitrary function ε (z, z) on the asymptotic S2, and

(2.20) implies

−1 V ≡ | − = V∂ V . (2.21) z Bz I− i z

40 − We also can define the soft gluon operator on I to be

∫ ∞ Mz ≡ dvGvz = Bz|I − − Vz . (2.22) −∞ +

The classical scattering problem, i.e. determining the final data Az(u) given a set of initial data

− Bz(v), is defined only up to the large gauge transformations generated by both ε and ε that act separately on the initial and final data. Clearly, there can be no sensible scattering problem without

− imposing some relation between ε and ε . To do this, we match the gauge field at i0. Lorentz invariant matching conditions are

Az|I + = Bz|I − , (2.23) − + which is preserved by

− ε(z, z) = ε (z, z) . (2.24)

2.3 Soft Gluon Theorem

In this section, we review the standard proof of the tree-level soft gluon theorem. For simplicity, we consider a theory with only scalar matter in Rξ-gauge:

[ ] ∑ ( ) ( ) [( ) ] 1 † 1 2 L = − tr F F μν − D φ Dμφ − tr ∂μA + L . (2.25) 4g2 μν μ k k 2ξg2 μ gh YM k YM

41 L The ghost action gh will be irrelevant at tree-level. From this action, we can determine the propaga- tors as

p [ ] p −ig2 δab μ ν − ij μ; a ; = YM μν − ( − ) p p , = iδ . ν b p2−iε η 1 ξ p2 p2−iε (2.26)

The vertex Feynman rules are

ρ; c q μ; a 1 abc μν − ρ νρ − μ ρμ − ν p = g2 f [η (k p) + η (p q) + η (q k) ] , k YM ν; b

[ μ; a ν; b i abe cde μρ νσ μσ νρ = − 2 f f (η η − η η ) gYM

+ facefbde (ημνηρσ − ημσηνρ) ] ρ; c ade bce μν ρσ μρ νσ σ; d + f f (η η − η η ) , (2.27)

μ; a

p μ μ a = i (k + q )(Tk)ij , j i k q

i ν; b ( ) = − μν a b iη TkTk ij .

j μ; a

Furthermore, every external gluon is accompanied with a Feynman rule factor of gYM.

We now consider the amplitude involving only external scalars, given by the Feynman diagram

42 ′ pn pm

··· ···

(2.28)

p ′ 1 p1 M where the kth scalar particle is in representation Rk. We shall denote this amplitude as . Now,

μ consider the same amplitude with an additional outgoing soft gluon of momentum pγ , color index a, and polarization εμ(pγ) satisfying the gauge condition pγ · ε(pγ) = 0. We denote this by

Ma,ε 0 → (pγ). The dominant diagrams in the soft pγ 0 limit are

pm+1 a; pγ pm+1 pm pm+1 a; pγ

pm ∑m a; pγ m∑+n pm . = . + p . . k (2.29) . k=1 p k=m+1 . . k .

pm+n p1 pm+n p1 pm+n p1

0 → + In the limit of pγ 0 , we then get

[ ] [ ] ∑m · m∑+n · 0 a,λ pk ελ(pγ) a pk ελ(pγ) a ∗ lim p M (pγ) = gYM T − (T ) M , (2.30) 0→ + γ · b k · b k pγ 0 pk pγ pk pγ k=1 k=m+1

bμ where λ is the helicity of the gluon, pλ is defined as

pμ bμ ≡ γ , (2.31) pγ 0 pγ

43 a M and Tk acts on the kth index on . Although we derived (2.30) in the context of scalar matter, it is in fact generally true for any type of matter.

2.4 Holomorphic Soft Gluon Current

In this section, we show that the soft theorem for outgoing positive-helicity gluons (or equivalently incoming negative-helicity gluons) is the Ward identity of the holomorphic large gauge transformations and takes the form of a holomorphic G-Kac-Moody symmetry acting on the S2 on I .

Let Ok(Ek, zk, zk) denote an operator which creates or annihilates a colored hard particle with

̸ 2 I 2 energy Ek = 0 crossing the S on at the point zk. We denote the standard n-particle hard amplitudes by

An(z1,..., zn) = ⟨O1 ··· On⟩U=1 . (2.32)

There are no traces here, so An has n suppressed color indices. Since the gauge field vanishes at infin-

2 −1 3 ity, the asymptotic S has a flat connection Uz = iU∂zU , where U ∈ G. In order to compare the color of particles emerging at different points on the S2, this connection must be specified. The

U = 1 subscript here indicates the fact that the standard perturbation theory presumes the trivial

4 connection Uz = 0.

2For instance, for scalar particles ∫ ∞ 4π [ ] ( , , ) = − iEku∂ ( , , , ) . Ok Ek zk zk due u →∞lim rφk u r zk zk Ek −∞ r

3 U Uz should not be confused with z (defined in (2.16)). 4 For vanishing Uz an outgoing configuration with a red quark at the north pole and a red bar quark at the south pole is a color singlet state which can be created by a colorless incoming state. For more general choices of Uz this will not be the case.

44 The hard S-matrix has soft boundaries where gluon momenta vanish. We wish to give a prescription to extend, or “compactify” the S-matrix to a larger object that includes these boundaries. Since zero-energy gluons are not obviously either incoming or outgoing, the S-matrix so compactified is not obviously a matrix mapping in states to out states. Hence we will refer to the compactified

S-matrix as the S-correlator.5

2.4.1 Soft Gluon Current Insertion

In this subsection, we will show that insertions of the soft gluon current Jz, defined by

(∫ ∫ ) 4π 4π ≡ − − − (2.33) Jz 2 (Nz Mz) = 2 dvGvz duFuz , gYM gYM into the hard tree-level S-matrix are determined by the soft gluon theorem. We begin by noting for

2 λ = + and pk = 0, the soft theorem (2.30) reads

[ ( ) ] m∑+n 1 − a out ηk a ∗ εb lim ω ⟨out| : a pγ S : |in⟩ = ⟨out| :(T ) S : |in⟩ , (2.34) z + + − k gYM ω→0 z zk k=1

√ b− 1 μ − 2 − where εz = r ∂zx εμ = 1+zz and ηk = 1 for outgoing particles and 1 for incoming particles.

a Next, to determine what is the effect of inserting the soft current Jz , we want to compute it in terms of creation and annihilation operators. This is done by studying the asymptotic expansion of the

5In the abelian examples of gravity and QED [8, 14, 15, 51], it is possible to view the S-correlator as a conventional S-matrix. However, the noncommutativity (see (2.61)) of the multi-gluon soft limits persists even if ′ one gluon is outgoing (q0 > 0) and the other incoming (q 0 < 0). This means that the soft limit on an out state does not commute with the soft limit on an in state, creating difficulties for the reinterpretation of the S-correlator as an S-matrix.

45 gauge field.

At late times t → ∞ (or equivalently u → ∞ in the retarded coordinates), the gauge field Aμ becomes free and can be approximated by the mode expansion

∫ ∑ 3 [ ] d q 1 ∗ · † − · Aa (x) = g εα(q) aa (⃗q)outeiq x + εα(q)aa (⃗q)oute iq x , (2.35) μ YM (2π)3 2ω μ α μ α α=± q

0 where q = ωq = |⃗q| and α = ± are the two helicities. The creation and annihilation operators obey

[ ] ( ) a out b† ⃗′ out ab 3 3 − ′ (2.36) aα(⃗q) , aβ (q ) = δαβδ (2π) (2ωq)δ ⃗q ⃗q .

a To determine Az , recall

Aa(u, z, z) = lim Aa(u, r, z, z) . (2.37) z r→∞ z

Aa μAa Using z = ∂zx μ, the mode expansion in (2.35), and the stationary phase approximation, we find

∫ ∞ [ ] ig − − † a( , , ) = − YM εb a ( b)out iωqu − a ( b)out iωqu . (2.38) Az u z z 2 z dωq a+ ωqx e a− ωqx e 8π 0

46 Using this, we immediately determine

∫ ∞ [ ( ) ( )] − ω,a −gYM b− a b out − a† b out iωAz = εz dωqωq a+(ωqx) δ ω ωq + a− (ωqx) δ ω + ωq . (2.39) 4π 0

When ω > 0 (ω < 0) only the first (second) term contributes. Using this with (2.17) then yields

[ ] a −gYM b− a b out a† b out (2.40) Nz = εz lim ωa+(ωx) + ωa− (ωx) . 8π ω→0+

− Similarly on I , we find

[ ] a −gYM b− a b in a† b in (2.41) Mz = εz lim ωa+(ωx) + ωa− (ωx) , 8π ω→0+

in in† − where a± and a± annihilate and create incoming gluons on I . Crossing symmetry of S-matrix amplitudes implies that an outgoing positive-helicity gluon has the same soft factor as an incoming negative-helicity gluon up to a sign. This implies

Nz + Mz = 0 (2.42)

when inserted into scattering amplitudes. Furthermore, by using (2.33) with (2.40) and (2.41), inser-

a tions of Jz produces the result

[ ] ⟨ | aS | ⟩ 1 b− ⟨ | a b out S | ⟩ out : Jz : in = εz lim ω out : a+ (ωx(z, z)) : in . (2.43) gYM ω→0+

47 Using (2.34), we arrive at

m∑+n η ∗ ⟨ | : aS : | ⟩ = k ⟨ | :( a) S : | ⟩ . out Jz in − out Tk in (2.44) z zk k=1

We would like to express our results in the language of correlators. For the conventional soft gluon theorem (2.30), this is given by

∑n p · ε ⟨ ( ) ··· ( ); a( , ε)⟩ = k ⟨ ( ) ··· a ( ) ··· ( )⟩ + O( 0) , O1 p1 On pn O q U=1 gYM · O1 p1 TkOk pk On pn U=1 q pk q k=1 (2.45) where Oa(q, ε) = tr [TaO(q, ε)] creates or annihilates, depending on the sign of q, a soft gluon with

μ a momentum q and polarization ε , and Tk is a generator in the representation carried by Ok. Note that gauge invariance of the theory requires that the right hand side vanishes when ε = q, which means

∑n ⟨ ··· a ··· ⟩ O1(p1) TkOk(pk) On(pn) U=1 = 0 , (2.46) k=1 which is global color conservation. Lastly, (2.44) in the language of correlators is

∑n 1 ⟨ a ··· ⟩ = ⟨ ··· a ··· ⟩ , Jz O1 On U=1 − O1 TkOk On U=1 (2.47) z zk k=1

a ≡ a · → where Jz tr [T Jz]. The collinear q pk 0 singularities of (2.45) become the poles at z = zk in

a (2.47). The soft pole in (2.45) is absent in (2.47) simply because the definition of Jz involves the zero

48 mode of the field strength rather than the gauge field and hence an extra factor of the soft energy.

2.4.2 Kac-Moody symmetry

Since ∂zJz = 0 away from operator insertions, Jz is a holomorphic current. Consider a contour

a a C and an infinitesimal gauge transformation ε (z) which is holomorphic (∂zε = 0) inside C. It follows from (2.47) that

∑ ⟨ ··· ⟩ ⟨ ··· ··· ⟩ JC(ε)O1 On U=1 = O1 εk(zk)Ok On U=1 , (2.48) k∈C

a a where εk(zk) = ε (zk)T , k I dz JC(ε) ≡ tr [εJz] , (2.49) C 2πi and the sum k ∈ C includes all insertions inside the contour C. Moreover, from the soft theorem with multiple Jz insertions one finds

∑ ⟨ ··· ⟩ ⟨ ··· ··· ⟩ ⟨ ··· ⟩ JC(ε)JwO1 On U=1 = JwO1 εk(zk)Ok On U=1 + ε(w)JwO1 On U=1 , (2.50) k∈C where the last term is added only when w is also inside C.

We observe that (2.50) is a very familiar formula in two-dimensional conformal field theory. It is the Ward identity of a holomorphic Kac-Moody symmetry for the group G. The absence of a term with no Jw on the right hand side of (2.50) indicates that the Kac-Moody level is zero (at tree-level).

Hence, the S-correlators for any massless theory with nonabelian gauge group G transform under a

49 holomorphic level-zero G-Kac-Moody action.

2.4.3 Asymptotic Symmetries

In this subsection, the Kac-Moody symmetry is identified with holomorphic large gauge symmetry of the gauge theory. According to (2.9) under the action of the asymptotic symmetry transformation

→ Ok(zk, zk) Uk(zk, zk)Ok(zk, zk) , (2.51)

S where Uk acts in the representation of Ok. It follows -correlators for general U are simply related to those for U = 1

⟨ a i1 · · ·⟩ ab i1j1 ···⟨ b j1 · · ·⟩ Jz O1 U = U(z, z) U1(z1, z1) Jz O1 U=1 . (2.52)

To compare the asymptotic symmetry action (2.52) with the Kac-Moody action (2.48), consider infinitesimal complexified transformations of the form

U(z, z) = 1 + iε(z) + ··· , (2.53)

which are holomorphic inside the contour C and vanish outside. In that case (2.52) linearizes to

∑ ⟨ ··· ⟩ ⟨ ··· ··· ⟩ δε O1 On U=1 = i O1 εk(zk)Ok On U=1 , (2.54) k∈C

50 where the operator insertions could also include a postive helicity soft gluon. Comparing with (2.48) we see that

− iδε⟨O1 ··· On⟩U=1 = ⟨JC(ε)O1 ··· On⟩U=1 . (2.55)

Hence, JC(ε) generates holomorphic asymptotic symmetry transformations

∫ δ C(ε) = 2 εa , J d zγzz a (2.56) DC δU

where DC is the region inside C and Ua is the Lie algebra element corresponding to U, that is, U =

a a eiU T .

Let C0 be any contour that divides the incoming and outgoing particles. For ε holomorphic on

C the incoming side of 0, the corresponding JC0 (ε) is then the charge that generates the asymptotic symmetries on the incoming state. If ε is holomorphic and non-constant on the incoming side of C0, it extends to a meromorphic section that must have poles on the outgoing side whose locations we denote w1,..., wp. We may also evaluate the contour integral by pulling it over the outgoing state.

Equating this with (2.48), we find, for any meromorphic section ε

∑p − ⟨ ··· ⟩ = − ⟨ [ε ] ··· ⟩ , iδε O1 On U=1 tr Jw wi O1 On U=1 (2.57) i=1 where

[ε ] = [ε ] . (2.58) tr Jw wi Res tr Jw w→wi

51 This is another form of the soft gluon theorem. It states that S-correlators are invariant under the asymptotic symmetries up to insertions of the soft gluon current. The appearance of the inhomoge- nous term on the right hand side implies that the U = 1 vacuum spontaneously breaks the symmetry. The soft gluons are the associated Goldstone bosons. Indeed, when p = 0, i.e. when ε is a globally holomorphic function on the sphere (and therefore a constant), we have

δε⟨O1 ··· On⟩U=1 = 0 , (2.59)

which is precisely (2.46). This indicates that the subgroup of constant global asymptotic color rota- tions is not spontaneously broken, as expected.

One might think that the Kac-Moody symmetry does not capture all of the asymptotic symmetry group, since the transformations are restricted to be holomorphic within some contour C. However,

S this is an irrelevant restriction. The -correlator identities depend only on the n values εk = ε(zk)

C of ε at the n operator insertions. For any choice of εk there exists a holomorphic ε(z) inside some such that ε(zk) = εk at the positions of operator insertions. Hence, the holomorphicity does not preclude consideration of any gauge transformation on Fock space states, and all nontrivial relations among S-correlation functions can be derived from the Kac-Moody symmetry. In particular, the soft gluon theorem (2.47) is itself a Ward identity of the the Kac-Moody symmetry.

52 2.5 Antiholomorphic Current

We have seen that positive-helicity soft gluon currents Jz generate a holomorphic Kac-Moody sym-

a metry. Naively one might expect that negative-helicity soft gluon currents Jz generate a second Kac-

Moody symmetry that is antiholomorphic. This turns out not to be the case for a very interesting reason.

Consider a boundary of the S-matrix near which two gluons become soft. One finds

′ ′ An+2(p1,..., pn; q, ε, a; q , ε , b) ∑n ∑n ′ ε · p ε · p = 2 k j ⟨ ··· a ··· b ··· ⟩ gYM · ′ · O1 TkOk Tj Oj On U=1 q pk q pj (2.60) k=1 j=1 ∑n ′ · · ′ 2 abc ε pj ε q c 0 ′0 − ig f ⟨O ··· T O ··· O ⟩ = + O(q , q ) , YM q′ · p q · q′ 1 j j n U 1 j=1 j where the above limit has been computed by taking q → 0 first. Surprisingly, the right hand side actually depends on the order of limits and

[ ] ′ ′ lim, lim An+2(p1,..., pn; q, ε, a; q , ε , b) q→0 q′→0 ( )( ) ∑n ′ ′ ′ ε · p ε · q ε · p ε · q = 2 abc k − k − ⟨ ··· a ··· ⟩ igYMf · · ′ ′ · · ′ O1 TkOk On U=1 (2.61) pk q q q q pk q q k=1 ( ) ′ + O q0, q 0 .

In the special case that the helicities are the same, the right hand side of the above expression vanishes and the limits commute. In this case, the S-matrix can be extended to its soft boundaries un-

53 ambiguously. When the helicities are not the same, the value of the S-matrix at the soft boundary is ambiguous. In terms of currents, taking the positive-helicity gluon to zero first gives

ifabc JaJb ∼ − Jc , (2.62) z w z − w w while in the other order we have ifabc JaJb ∼ − Jc . (2.63) z w z − w z

Thus, the extension (or “compactification”) of the S-matrix to all soft boundaries requires a prescription. For our purposes, we adopt the prescription that positive-helicity gluon momenta are always taken to zero before negative-helicity gluon momenta. With this prescription, it follows from

a a (2.62) that the current Jz generates a Kac-Moody symmetry, under which Jz transforms in the ad-

a a a joint, while Jz itself does not generate a symmetry. A prescription which treats Jz and Jz symmetri- cally yields no symmetry, while taking negative-helicity momenta to zero first gives one antiholomor-

a phic Kac-Moody symmetry generated by Jz .

The situation is reminiscent of three-dimensional Chern-Simon gauge theory on a manifold with a boundary parameterized by (z, z). A priori, one might have expected Az and Az to generate both holomorphic and antiholomorphic G-Kac-Moody symmetries. However a more careful analysis reveals that boundary conditions must be chosen to eliminate one or the other. Indeed, this may be

a 2 more than an analogy. The current Jz has no time dependence and lives on the S at the boundary of the 3-manifold I , and the addition of a θF ∧ F term to the 4D gauge theory action induces a Chern-

54 Simons term on I . It would be interesting to understand how such a term affects the present analysis.

55 3 N = 1 Supersymmetric Quantum

Electrodynamics

In the previous chapters, we demonstrated that the universal soft behavior of gauge boson amplitudes is traced back to the existence of infinitely many symmetries that act on asymptotic scattering states at Minkowskian null infinity, i.e. asymptotic symmetries, whose Ward identities are equivalent

56 to the soft theorems. Typically, these asymptotic symmetries can be viewed as large gauge transformations, which do not vanish at infinity and therefore act non-trivially on physical states.1

It is natural to ask whether soft theorems for massless particles that are not gauge bosons have similar interpretations in terms of asymptotic symmetries. Here we will explore this question in the context of rigid supersymmetric gauge theories, where the gauge fields are accompanied by massless

1 N spin- 2 superpartners. For simplicity, we confine our attention to U(1) gauge theories with =

1 supersymmetry and massless charged matter in four dimensions. The U(1) photon Aμ has an electrically neutral, fermionic superpartner – the photino Λα – whose couplings to charged matter are related to those of the photon by supersymmetry. The soft photino theorem takes the general form

s Mn+1 → S Mn as p → 0 , (3.1) where S is a non-trivial fermionic soft operator, which acts on the external states of the hard amplitude Mn in a universal fashion.

In this chapter, we establish the existence of infinitely many fermionic asymptotic symmetries,

2 parametrized by a chiral spinor-valued function χα(z, z) on S , whose Ward identities give rise to the soft photino theorem. The corresponding anticommuting charges F [χ] act on the asymptotic fields at null infinity. However, unlike the infinity of bosonic charges E [ε], they are not a subgroup of any obvious symmetry of the Lagrangian.2 The usual Lagrangian only displays a finite number

1We are using the terminology where large gauge transformations are assumed to act non-trivially on physical states, because they do not vanish sufficiently rapidly at the boundary of spacetime. However, they may be topologically trivial, i.e. deformable to the identity gauge transformation. 2 The asymptotic symmetries related to the magnetic generalization of the leading soft photon theo-

57 of manifest fermionic symmetries – the global supersymmetries generated by Qα and Qα˙ . It is perhaps surprising that even rigid supersymmetric gauge theories can support an infinite number of fermionic asymptotic symmetries.3 By contrast, this is expected in supergravity, where local supersymmetry is a gauge symmetry [55,56].

Under the action of F [χ] we find that the photino Λα shifts inhomogenously. Hence these symmetries are spontaneously broken, and the soft photini are interpreted as the corresponding Gold- stone fermions. Interestingly, supersymmetry relates the fermionic charges F [χ] to the bosonic charges E [ε]. We find (see (3.76) below),

{ } { } α F E α F ζ Qα, [χ] = i [ζ χα] , Qα˙ , [χ] = 0 . (3.2)

Here the supersymmetry transformation is parametrized by a commuting, constant spinor ζα, E and χα(z, z) is also taken to be commuting. The charges [ε] commute with Qα and Qα˙ .

The soft photon theorem implies that the insertion of a zero-momentum, positive-helicity photon into a scattering amplitude can be interpreted as the Ward identity for a U(1) Kac-Moody current, which transforms in a (1, 0) representation of the SL(2, C) conformal symmetry acting on the S2 at null infinity [7, 51]. Similarly, we will see that the insertion of a positive-helicity photino rem [52] or the subleading soft photon theorem [53] are also not manifest at the level of the Lagrangian. 3A similar phenomenon occurs in three-dimensional, supersymmetric Chern-Simons theory in the presence of a suitably supersymmetric boundary, which supports a supersymmetric Kac-Moody current algebra. (As we will see below, the asymptotic symmetries E [ε] and F [χ] also give rise to just such a current algebra.) The bosonic Kac-Moody symmetries are conventional gauge transformations that do not vanish at the boundary. The Kac-Moody fermions can be understood as a remnant of the full super gauge symmetry that is present before fixing Wess-Zumino (WZ) gauge (see for instance [54]). It is plausible that our asymptotic symmetries F [χ] have a similar interpretation, but we will not show it here. Instead, we will exhibit the charges F [χ] directly in WZ gauge and explore their properties.

58 1 2 behaves like a ( 2 , 0) current on S . The two currents are related by supersymmetry, as was the case for the charges in (3.2).

We organize this chapter as follows. In section 3.1, we begin by reviewing basic aspects of abelian gauge theories with N = 1 supersymmetry, focusing on the structure of the supermultiplet that con-

J KF tains the electric current μ, which couples to the photon, and its fermionic superpartner α, which couples to the photino. In section 3.2, we derive the tree-level soft photino theorem. In section 3.3 we analyze the classical dynamics of the supersymmetric gauge theory near null infinity. This is fa- cilitated by a convenient choice of coordinates and spinor basis, in which the asymptotic behavior of massless fields near null infinity is simply related to their quantum numbers with respect to the conformal group that governs the deep IR behavior of the theory. After reviewing the results of [51] on the asymptotic dynamics of the photon and the associated bosonic charges E [ε], we repeat the analysis for the photino and construct the fermionic asymptotic charges F [χ]. Lastly, in section 3.4 we show that the Ward identity for the fermionic symmetries F [χ] reproduces the soft photino theorem derived in section 3.2.

3.1 Aspects of N = 1 Gauge Theories

Unless stated otherwise, we will use the conventions given by Wess and Bagger in [57] to study U(1) gauge theories with N = 1 supersymmetry. After fixing Wess-Zumino (WZ) gauge, the vector multiplet V is given by ( ) V = Aμ , Λα , Λα˙ , D . (3.3)

59 Here Aμ is the U(1) gauge field (i.e. the photon) with field strength Fμν = ∂μAν − ∂νAμ. It is subject to conventional U(1) gauge transformations, which remain unfixed in WZ gauge. The spin-

1 2 superpartner of the photon is the photino, which is described by a left-handed Weyl fermion Λα and its right-handed Hermitian conjugate Λα˙ . The vector multiplet also contains a real scalar D, which is a non-propagating auxiliary field.

In WZ gauge, the non-vanishing (anti-) commutators of the component fields in the vector multi-

V plet with the supercharges Qα, Qα˙ are given by

[ ] [ ] A − α˙ A α Qα, μ = σμαα˙ Λ , Qα˙ , μ = Λ σμαα˙ , (3.4a) { } { } D − μν F D − μν F Qα, Λβ = εαβ i (σ )αβ μν , Qα˙ , Λβ˙ = εα˙ β˙ i (σ )α˙ β˙ μν , (3.4b) [ ] [ ] D − μ α˙ D − α μ Qα, = iσαα˙ ∂μΛ , Qα˙ , = i∂μΛ σαα˙ . (3.4c)

The dynamics of the gauge multiplet is described by a Lagrangian Lgauge, which is invariant (up to a total derivative) under the supersymmetry transformations in (3.4),

1 i 1 L = − F F μν − Λ σμ∂ Λ + D2 + (higher-derivative terms) . (3.5) gauge 4e2 μν e2 μ 2e2

In addition to the standard two-derivative kinetic terms for the gauge multiplet, we are allowing for the possibility of higher-derivative terms, e.g. terms such as F4 + (fermions), which arise in supersymmetric Born-Infeld actions. The soft theorems discussed below remain valid in the presence of such terms.

60 The interaction of the gauge field Aμ with matter proceeds through a conserved current Jμ, which resides in a real linear multiplet J ,4

( ) J KB KF KF J μJ = , α , α˙ , μ , ∂ μ = 0 . (3.6)

KB KF KF Here is a real scalar, while α and its Hermitian conjugate α˙ are left- and right-handed Weyl

J KB KF KF spinors. Unlike μ, neither nor α, α˙ obey a differential constraint, i.e. they are not conserved currents. All fields in the current supermultiplet J are gauge invariant. Their non-vanishing supersymmetry transformations take the following model-independent form,

[ ] [ ] KB KF KB KF Qα, = i α , Qα˙ , = i α˙ , (3.7a) { } ( ) { } ( ) KF − μ J KB KF μ J − KB Qα˙ , α = iσαα˙ μ + i∂μ , Qα, α˙ = iσαα˙ μ i∂μ , (3.7b)

[ ] ( ) [ ] ( ) ˙ , J = − β∂νKF , , J = ∂νKFβ . Qα μ 2 σμν α β Qα˙ μ 2 σμν α˙ β˙ (3.7c)

At first order, the interaction of the fields in the vector multiplet V with matter proceeds via the following universal couplings to the operators in the current multiplet J ,5

μ F F B Lint = −A Jμ − iΛK + iΛ K − DK + (higher order) . (3.8)

4In superspace, a real linear multiplet is described by a real superfield J that satisfies the constraints 2J 2J D = D = 0, where Dα, Dα˙ are the usual super-covariant derivative operators defined in [57]. δS ∫ 5 For instance, this means that J (x) = − int , where S = d4x L . μ δAμ(x) int int

61 The higher-order terms are required by gauge invariance and supersymmetry.

In general, the current multiplet J encodes all couplings of the gauge theory to charged matter, as well as possible self-interactions due to higher-derivative terms, such as those indicated in (3.5).

For simplicity, we will take all matter fields to reside in massless chiral multiplets. Most of the results below only rely on general properties of the current multiplet J , e.g. its supersymmetry transformations (3.7), but do not depend on the detailed form of the interaction terms. Nevertheless, it is helpful to keep in mind the simplest theory in this class, which consists of a single massless, mini- mally coupled chiral multiplet of charge q, with canonical kinetic terms and no superpotential or higher-derivative interactions.6 In this theory, the operators in the current multiplet J are given by

KB = qΦΦ , (3.9a) √ √ KF KF α = q 2 ΦΨα , α˙ = q 2 ΦΨα˙ , (3.9b) ( ←→ ) Jμ = q iΦ DμΦ + Ψ σμΨ . (3.9c)

Here Φ, Ψα are the propagating component fields in the chiral multiplet (their Hermitian conjugates Φ, Ψα˙ reside in an anti-chiral multiplet) and Dμ = ∂μ − iqAμ is the gauge-covariant derivative.

In our conventions, the electric charge E is given by

∫ 3 E = d x J0 , (3.10)

6 This theory is quantum mechanically anomalous. The anomaly can be cancelled by including additional chiral multiplets with suitable U(1) charge assignments.

62 and the statement that Φ, Ψα both have charge q means that

[E , Φ(x)] = −qΦ(x) , [E , Ψα(x)] = −qΨα(x) . (3.11)

This implies that a state |Φ⟩ ∼ Φ(x)|0⟩ created by Φ(x) has charge −q, since E |Φ⟩ = −q|Φ⟩.

3.2 Soft Photino Theorem

We are interested in tree-level scattering amplitudes involving particles in the gauge and matter multiplets. An n-point amplitude Mn is specified by n asymptotic one-particle states |f, p, s⟩, where p is the four-momentum of the particle and s is a spin or helicity label. The label f denotes the particle type, i.e. the field f(x) that creates the state, |f⟩ ∼ f(x)|0⟩.7 For instance, we write |F, p, ±⟩ for a photon of momentum p and helicity ±1. Similarly, |Λ, p, −⟩ and |Λ, p, +⟩ are photini of momen-

− 1 1 tum p and helicity 2 and + 2 , respectively. We normalize one-particle states as follows,

′ ′ ′ ′ ⟨ | ⟩ 3 0 ′ ′ (3) − f, p, s f , p , s = (2π) (2p ) δf,f δs,s δ (⃗p ⃗p ) . (3.12)

{ } M | , , ⟩ m We can then write the n-point amplitude n that describes m incoming particles fi pi si i=1 { } − | , , ⟩ n scattering into n m outgoing particles fi pi si i=m+1 as

Mn = ⟨m + 1 ; ... ; n S 1 ; ... ; m⟩ , (3.13)

7 In this notation, ⟨f| ∼ ⟨0|f(x), while |f⟩ = f(x)|0⟩ and ⟨f| ∼ ⟨0|f(x), where f(x) is the Hermitian conjugate of f(x).

63 where S is the scattering matrix and we have introduced the shorthand

|i⟩ = |fi, pi, si⟩ (i = 1,..., n) . (3.14)

Mout,+ We now want to study scattering amplitudes n+1 involving a soft outgoing photino Λ of

8 momentum pn+1 and positive helicity, as well as n other hard particles, i.e. those given by

⟨ ⟩ Mout,+ S n+1 = m + 1 ; ... ; n ; Λ, pn+1, + 1 ; ... ; m . (3.16)

In order for the amplitude to be non-zero, the total number of fermions involved in the scattering process (including the photino) must be even. We are interested in the leading behavior of this amplitude when the photino momentum is taken to zero, pn+1 → 0. This arises from single insertions

F F of the interaction terms −iΛK + iΛ K ⊂ Lint in (3.8) that attach only to external lines. For a ⟨ F positive-helicity photino, insertions of −iΛK do not contribute, since Λ, ps, + Λα(0) |0⟩ = 0.

8 We add the photino to the out state by acting with its annihilation operator as follows, ⟨ ⟨

m + 1 ; ... ; n ; Λ, pn+1, + = m + 1 ; ... ; n aΛ,+(pn+1) . (3.15)

64 Therefore, the amplitude obeys the following soft theorem,

⟨ Mout,+ −→ − | ⟩ × n+1 Λ, pn+1, + Λα˙ (0) 0 ( ∑ ∑n − ˙ ⟨ ⟩ i σi Fα (−1) ⟨i| K (0) |f, pi, s⟩ m + 1 ; ... ; f, pi, s ; ... ; n S 1 ; ... ; m 2pn+1 · pi f,s i=m+1 ) ∑m − ⟨ ⟩ ˙ i σi Fα − (−1) m + 1 ; ... ; n S 1 ; ... ; f, pi, s ; ... ; m ⟨f, pi, s| K (0) |i⟩ . 2p + · p i=1 n 1 i

(3.17)

Fα˙ Here (−1)σi is a fermion sign factor that comes from anticommuting K across multi-particle states.9

The photino wavefunction is given by

⟨ | ⟩ Λ, pn+1, + Λα˙ (0) 0 = eηα˙ (pn+1) . (3.18)

10 Here ηα˙ (p) is a standard spinor-helicity variable corresponding to a null momentum p,

μ pμσαα˙ = ηα(p)ηα˙ (p) . (3.19)

Fα˙ We must now evaluate the matrix elements of the fermionic operator K between single-particle states, in the forward limit. In general, the matrix elements of such an operator may be model-

9 ⟨ | For an out state m + 1 ; ... ; i ; ... ; n we define σi to be the number of fermionic states in positions i + 1 through n. For an in state |1 ; ... ; i ; ... ; m⟩ we define it to be the number of fermionic states in positions 1 through i − 1. 10 ⟩ α˙ α˙ In terms of the usual angle and square brackets, we have ηα(pi) = i α and η (pi) = i] .

65 Fα˙ dependent. However, K resides in the same supermultiplet (3.6) as the conserved electric current Jμ, whose forward matrix elements are universal. Explicitly, we can evaluate the following commutation relation from (3.7),

{ } ( ) KF μ J − KB Qα, α˙ = iσαα˙ μ i∂μ , (3.20)

B between single-particle states in the forward limit, where we can drop the total derivative ∂μK .

Lorentz invariance and current conservation completely determine the matrix elements of the electric current Jμ(x) between states of equal momentum (i.e. in the forward limit) in terms of their electric charge to be (see for instance [42], chapter 10)

⟩ ′ ′ ⟨ | J − ′ ′ f, p, s μ(0) f , p, s = 2qf pμδff δss . (3.21)

It follows { } ⟩ F ′ ′ μ ⟨ | K − ′ ′ f, p, s Qα, α˙ f , p, s = 2iqf pμσαα˙ δff δss . (3.22)

The appearance of δff′ on the right-hand side shows that only single-particle states that reside in the

KF same supermultiplet can can lead to non-vanishing matrix elements for α˙ . When the supercharges act on the left or the right, they lead to other states in this supermultiplet, in a way that is completely

KF determined by representation theory. This can be used to derive all matrix elements of α˙ between massless or massive single-particle states of arbitrary spin.

Here we explicitly work this out for a massless chiral multiplet Φ, Ψα of charge q, and its conju-

66 gate anti-chiral multiplet Φ, Ψα˙ of charge −q. The relevant single-particle states are

|Φ, p⟩ , |Ψ, p, −⟩ and |Φ, p⟩ , |Ψ, p, +⟩ . (3.23)

On these states, the supersymmetry algebra is represented as follows,11

√ | ⟩ | ⟩ | −⟩ Qα˙ Φ, p = 0 , Qα Φ, p = 2i ηα(p) Ψ, p, , √ | −⟩ − | ⟩ | −⟩ Qα˙ Ψ, p, = 2i ηα˙ (p) Φ, p , Qα Ψ, p, = 0 . (3.24)

The action of the supercharges on the conjugate anti-chiral states is obtained by exchanging Qα ↔

| ⟩ ↔ | ⟩ ↔ | −⟩ ↔ | ⟩ Qα˙ , Φ, p Φ, p , ηα(p) ηα˙ (p), and Ψ, p, Ψ, p, + .

We can now implement the procedure described after (3.22) to obtain all non-vanishing matrix

KF elements of α˙ ,

√ ⟨ |KF | −⟩ ⟨ |KF | ⟩ Φ, p α˙ (0) Ψ, p, = Ψ, p, + α˙ (0) Φ, p = 2qηα˙ (p) . (3.25)

Substituting into (3.17), we obtain the final form of the soft photino theorem,

√ ∑n √ ∑m Mout,+ −→ qi F M − qi F M n+1 2ie ( i n) 2ie ( i n) . (3.26) η(p + )η(p ) η(p + )η(p ) i=m+1 n 1 i i=1 n 1 i

11 This follows√ from the non-vanishing commutation√ relations for a free chiral multiplet, { } μ [Qα, Φ] = i 2Ψα and Qα˙ , Ψα = 2σαα˙ ∂μΦ, as well as the mode expansions of the fields in terms of creation and annihilation operators.

67 Here the qi are the electric charges of the asymptotic states. The n-particle amplitude Mn is ob-

Mout,+ tained from n+1 by deleting the photino, but since it has an odd number of fermion external states, it vanishes. The non-vanishing n-point amplitude FiMn is obtained from Mn by acting on the ith single-particle state with a fermionic operator F , which satisfies

⟨Φ, p|F = −⟨Ψ, p, −| , ⟨Ψ, p, +|F = ⟨Φ, p| , (3.27) F |Φ, p⟩ = |Ψ, p, +⟩ , F |Ψ, p, −⟩ = −|Φ, p⟩ .

The action of F on all other single-particle states vanishes, and we take F to act from the right on out states and from the left on in states. Since F is a fermionic operator, it picks up a sign whenever it moves past another fermionic operator or state. This accounts for the factors (−1)σi in (3.17).

So far we have only discussed an outgoing soft photino of positive helicity. The negative-helicity case can similarly be shown to satisfy

√ ∑n ( ) √ ∑m ( ) Mout,− −→ − qi F †M qi F †M n+1 2ie i n + 2ie i n , η(p + )η(p ) η(p + )η(p ) i=m+1 n 1 i i=1 n 1 i (3.28)

† where the fermionic operator F is the Hermitian conjugate of the operator F defined in (3.27).

Finally, the soft theorems for ingoing photini can be obtained from (3.26) and (3.28) by crossing symmetry.

68 3.3 Asymptotic Symmetries

3.3.1 Kinematics

2 a b We are studying scattering from past to future null infinity in flat Minkowski space, ds = ηabdy dy

− with ηab = +++. In order to discuss physics at null infinity, it is convenient to choose a different set of coordinates xμ = (u, r, z, z), in which the Minkowski metric takes the form

ds2 = −dudr + r2dzdz . (3.29)

Note in particular that this is a different choice of coordinate than the standard retarded coordinates used in the previous chapters. These coordinates degenerate at r = 0, but this will not affect our discussion of the asymptotic regions at large r. The transformation to conventional flat coordinates ya is given by

1 y0 = (u + r (1 + |z|2)) , 2 r y1 = (z + z) , 2 (3.30) ir y2 = − (z − z) , 2 1 y3 = − (u − r (1 − |z|2)) . 2

The null coordinates −∞ < u, r < ∞ are real, while z ∈ C is complex. Future and past null

− infinity I + and I are located at r → +∞ and r → −∞, respectively. Both have topology R ×

69 S2, where R is a null direction parametrized by u, while the S2 is spacelike. The complex variable z is an angular coordinate, which covers all of S2, except one point that will not be important for us.

Note that a light ray traversing Minkowski space hits the same angular coordinate z at both I +

− and I , i.e. points with the same angular coordinate are identified by the antipodal map.12

+ + The future and past boundaries of I are at u → ±∞ and will be denoted by I± . Given a

+ + field O(u, z, z) on I , its boundary values at I± are given by the following limits, assuming they exist,

O|I + = lim O(u, z, z) . (3.31) ± u→±∞

− − Similarly, the boundaries of I at u → ±∞ are I± , and the corresponding boundary values of

− − − − O ( , , ) I O | − I a field u z z on are denoted by I± . We will generally label fields on using the same symbols as the corresponding fields on I +, but with an extra superscript − . Spatial infinity is

I + I − pinched between − and + .

In order to discuss spinors, it is convenient to choose the vielbein

∂ya ea dxμ = dya = dxμ , (3.32) μ ∂xμ with ya(xμ) as in (3.30), since it leads to a vanishing spin connection. Therefore, covariant deriva-

12 The (u, r, z, z) coordinates are simply related to other useful coordinate systems. For instance, they can ′ ′ ′ ′ be obtained from standard retarded coordinates (u , r , z , z ), in which

′2 ′ ′ ′ 4r ′ ′ ds2 = −du 2 − 2du dr + dz dz , (1 + |z′|2)2

′ = ′ = 1 ′ = → + → − 1 by setting u cu, r 2c r, z cz and taking the constant c 0 . By further redefining r r , we obtain the coordinates that were used in [58] to study energy and charge correlators at null infinity.

70 tives of spinors coincide with ordinary partial derivatives. This makes it straightforward to adapt standard flat-space formulas (e.g. supersymmetry transformations) to (u, r, z, z) coordinates. An- other simplification comes from using a helicity basis for spinors. In the frame (3.32), we define the following linearly independent, commuting, left-handed basis spinors,

     1   0  (+)   (−)   ξα (z) =   , ξα =   , (3.33) z 1

(+) (−) as well as their right-handed complex conjugates ξα˙ (z) and ξα˙ . (We will often suppress the de-

(+) (+) pendence of ξα and ξα˙ on z and z.) They satisfy

− (+)α (−) − (+) ( )α ˙ ξ ξα = 1 , ξα˙ ξ = 1 . (3.34)

(−) (+) (−) (+) Note that ξα is (covariantly) constant, while ∂zξα = ξα and ∂zξα = 0. All σ-matrices can be

(±) (±) expressed as bilinears in ξα and ξα˙ .

(+) (+) The spinors ξα , ξα˙ are position space analogues of the momentum space spinor-helicity vari-

μ ables ηα(p), ηα˙ (p) introduced in (3.19). In order to see this, we parametrize the null momentum p as follows,

( ) μ 2 2 p ∂μ = 2ω∂r = ω (1 + |z| ) ∂y0 + (z + z) ∂y1 − i (z − z) ∂y2 + (1 − |z| ) ∂y3 . (3.35)

Here ω > 0 has units of energy, but it only coincides with the energy p0 when z = 0. It follows

71 from (3.35) that

μ (+) (+) p σμαα˙ = 2ωξα (z)ξα˙ (z) , (3.36) so that the spinor-helicity variables defined in (3.19) are given by13

√ √ (+) (+) ηα(p) = 2ωξα (z) , ηα˙ (p) = 2ω ξα˙ (z) . (3.37)

The Lorentz group SL(2, C) is isomorphic to the global conformal group in two Euclidean dimensions, which acts on the asymptotic S2 parametrized by z. Infinitesimal Lorentz transformations are therefore parametrized by two-dimensional holomorphic vector fields Yz = a + bz + z2 with a, b, c ∈ C. The corresponding Lorentz transformation LY is given by

( ) ( ) 1 ( ) u u Lμ∂ = ∂ Yz + ∂ Yz (u∂ − r∂ ) + Yz − ∂2Yz ∂ + Yz − ∂2Yz ∂ . (3.38) Y μ 2 z z u r 2r z z 2r z z

The SO(2) rotation that stabilizes the null vector field pμ in (3.35) is generated by the following infinitesimal Lorentz transformation,

J = i (z∂z − z∂z) . (3.39)

13 The relation in (3.19) only defines ηα(p) up to a little group phase, which is fixed by (3.37).

72 Under such a rotation, the spinors in (3.33) transform as follows,14

(±) i (±) (±) i (±) L ξ = ± ξ , L ξ = ∓ ξ . (3.40) J α 2 α J α˙ 2 α˙

(∓) (±) ± 1 This shows that the spinors ξα and ξα˙ have definite helicity 2 , and hence they are chiral spinors on the asymptotic S2 parametrized by z.

Given a field F ··· ˙ ···˙ that transforms in the (j, j) representation of the Lorentz group (α1 α2j)(β1 β2j) C × ∈ 1 Z SL(2, ) = SU(2) SU(2), with j, j 2 , it is natural to expand it in terms of the helicity

(±) (±) eigenspinors ξα , ξβ˙ ,

j ∑m ∑ − − (+) ··· (+) (−) ··· (−) (+) ··· (+) ( ) ··· ( ) F ··· ˙ ···˙ = ξ ξ ξ ξ ξ ˙ ξ˙ ξ˙ ξ˙ F(m,m) . (α1 α2j)(β β ) (α1 αj+m αj+m+1 α2j) (β β + β + + β ) 1 2j − 1 j m j m 1 2j m= j m=−j (3.41)

As we will discuss in examples below, the coefficient fields F(m,m) obey simple falloff conditions near null infinity. In order to state these conditions, we need to introduce a conformal scaling dimension Δ for the field F ··· ˙ ···˙ , even though the theory under consideration need not (α1 α2j)(β1 β2j) be conformally invariant. Nevertheless, we expect its long-distance behavior near null infinity to be governed by a conformally invariant IR fixed point, and we take Δ to be the scaling dimension

14 μ The Lie derivatives of left- and right-handed spinor fields Ψα and Ψα˙ along a Killing vector K are given by 1 L Ψ = Kμ∇ Ψ − ∇ K (σμν) βΨ , K α μ α 2 μ ν α β ˙ α˙ μ α˙ 1 μν α˙ β L Ψ = K ∇ Ψ − ∇ K (σ ) ˙ Ψ . K μ 2 μ ν β

73 of F ··· ˙ ···˙ at that fixed point. We are interested in abelian gauge theories, which are IR free, (α1 α2j)(β1 β2j) and Δ coincides with the engineering dimensions of the fieldF ··· ˙ ···˙ . The behavior of the (α1 α2j)(β1 β2j) I ± → ±∞ coefficient field F(m,m) near (i.e. for r ) is governed by Δ and its Lorentz quantum numbers m, m, ( ) 1 F (r, u, z, z) = O . (3.42) (m,m) |r|Δ−m−m

The quantity Δ − m − m is known as the collinear twist: it is the eigenvalue of the conformal generator D+M, which stabilizes the null vector field pμ.15 As a simple example, consider an IR free, massless scalar field Φ of scaling dimension Δ = 1. According to (3.42), its asymptotic expansion near I + is

1 − Φ(u, r, z, z) = φ(u, z, z) + O(r 2) , (3.43) r

− − and similarly near I , with φ → φ .

3.3.2 Summary of Photon Asymptotics

In this subsection, we summarize some of the photon asymptotics discussed in chapter 1 that will be useful in deriving the fermionic asymptotic symmetries. Because we are working in a slightly different coordinate system than that in chapter 1, all the relevant equations will be modified accordingly.

To satisfy finite energy flux requirements, the falloff conditions for the abelian field strength is

15 μ − μ − Here D = u∂u +r∂r is a dilatation, which satisfies [D, p ∂μ] = p ∂μ, and M = u∂u r∂r +z∂z +z∂z μ μ μ is a boost along p , which satisfies [M, p ∂μ] = p ∂μ.

74 given by

Fuz = O (1) , ( ) 1 F = O , ur 2 r (3.44)

Fzz = O (1) , ( ) 1 F = O . rz r2

+ This is consistent with the following asymptotic expansion for the gauge field Aμ near I :

1 − A (u, r, z, z) = A (u, z, z) + O(r 2) , u r u 1 − A (u, r, z, z) = A (u, z, z) + O(r 3) , (3.45) r r2 r

−1 Az(u, r, z, z) = Az(u, z, z) + O(r ) , which satisfies Maxwell’s equations

μ 2 ∇ Fμν = e Jν . (3.46)

Substituting (3.44) or (3.45) into this equation leads to the following asymptotic expansion for the

75 + electric current Jμ near I ,

1 − J (u, r, z, z) = j (u, z, z) + O(r 3) , u r2 u 1 − J (u, r, z, z) = j (u, z, z) + O(r 5) , (3.47) r r4 r 1 − J (u, r, z, z) = j (u, z, z) + O(r 3) . z r2 z

Substituting the asymptotic expansions (3.45) and (3.47) into the equations of motion (3.46)

1 and expanding in powers of r lead to a system of equations that determines (together with suitable

+ boundary conditions at I± ) the bulk gauge field Aμ in terms of the boundary data Az(u, z, z) and

+ the current Jμ, up to gauge transformations that vanish at I and hence do not act on physical states. In particular, we obtain the leading constraint equation on I +,

e2 ∂ F = ∂ (∂ A + ∂ A ) + j , (3.48) u ur u z z z z 2 u

where Fur is the leading term in the asymptotic expansion of Fur,

1 − F = F + O(r 3) , F = A + ∂ A . (3.49) ur r2 ur ur u u r

The boundary data on I + is acted on by an infinite number of asymptotic symmetries, parametrized by a function ε(z, z). The corresponding charges, which generate the symmetries, are denoted

76 by E [ε],16

[ ] [ ] E [ε], Az(u, z, z) = i∂zε(z, z) , E [ε], fq(u, z, z) = −qε(z, z)fq(u, z, z) . (3.50)

Here fq(u, z, z) denotes an arbitrary boundary field of charge q. For real ε, these symmetries can be viewed as large gauge transformations that do not vanish on I +.

In the absence of magnetic charges, it was shown in [51] that the E [ε] can be written as integrals

+ over I− , ∫

E − 1 2 [ε] = d z ε(z, z) Fur I + . (3.51) e2 −

+ By rewriting this as an integral of ∂uFur over all of I and using the constraint equation (3.48), the charges can be expressed as a sum of hard and soft contributions,

E [ε] = E h[ε] + E s[ε] , (3.52)

where the hard charge is given by

∫ ∫

E h 1 2 − 1 2 [ε] = dud z ε(z, z)ju d z ε(z, z)Fur I + (3.53) 2 e2 +

16 + We previously called this charge Qε in chapter 1. We switch notation here to avoid confusion with the supersymmetry charge Q.

with Fur I + = 0 in theories without massive particles, and the soft charge in (3.52) is given by +

∫ ∫ 1 4π E s[ε] = − d2z ε(z, z)(∂ js + ∂ js) , js(z, z) = − du ∂ A . (3.54) 4π z z z z z e2 u z

s Here we have defined the soft photon current jz, with normalization given in [59].

− − It is straightforward to repeat the preceding discussion near I . In particular, the charges E [ε]

− that generate the asymptotic symmetries at I are given by

∫

E − − 1 2 − [ε] = d z ε(z, z)Fur I − . (3.55) e2 +

− 2 − + Here F = lim r Fur is the boundary value of the electric field on I . The fields on I ur r→−∞ − and I satisfy Lorentz-invariant matching conditions at spatial infinity. Since we are considering gauge theories without magnetic charges, we follow [51] and demand that

− Az I + = A I − , (3.56a) − z +

− Fur I + = F I − . (3.56b) − ur +

Comparing with (3.51) and (3.55) leads to the conservation law

E [ε] = E −[ε] . (3.57)

For ε = 1 this reduces to electric charge conservation. Semiclassically, the conservation law (3.57)

78 amounts to a Ward identity for the tree-level S-matrix,

− E [ε]S − SE [ε] = 0 . (3.58)

In the next subsection, we will apply supersymmetry to these results to obtain the corresponding fermionic asymptotic symmetries.

3.3.3 Photino Asymptotics

The dynamics of the photino field Λα is governed by the following equations of motion, which can be derived from (3.5) and (3.8),

μαα˙ 2 Fα˙ σ ∂μΛα = e K . (3.59)

KF As in (3.41), we project the photino Λα and the fermionic source α, both of which transform

1 (±) as ( 2 , 0) under the Lorentz group, onto the basis spinors ξα defined in (3.33),

(+) (−) KF KF (+) KF (−) (3.60) Λα = ( 1 ,0) ξα + Λ(− 1 ,0) ξα , α = ( 1 , ) ξα + (− 1 , ) ξα . 2 2 2 0 2 0

Following the discussion around (3.42), the leading large-r behavior of the coefficient fields Λ(± 1 , ) 2 0

KF and (± 1 , ) is determined by their conformal scaling dimensions: Λα is a free fermion of scaling 2 0 3 KF 5 dimension ΔΛ = 2 , while α has dimension ΔKF = 2 . Applying (3.42) then leads to the following

79 asymptotic expansions near I +,

1 −2 Λ( 1 , )(u, r, z, z) = λ(+)(u, z, z) + O(r ) , 2 0 r (3.61) 1 O −3 Λ(− 1 ,0)(u, r, z, z) = λ(−)(u, z, z) + (r ) , 2 r2 and

F 1 −3 K 1 ( , , , ) = ( , , ) + O( ) , ( ,0) u r z z 2 k(+) u z z r 2 r (3.62) KF 1 O −4 (− 1 ,0)(u, r, z, z) = k(−)(u, z, z) + (r ) . 2 r3

As for the photon, the equations of motion (3.59) determine the bulk field Λα from the boundary

KF I + data λ(+) and the source α, as well as suitable boundary conditions at ± . In particular, the leading constraint equation on I + is given by,

e2 ∂ λ + ∂ λ − = k , (3.63) z (+) u ( ) 2 (+)

which determines the u-dependence of λ(−) in terms of λ(+) and k(+). As in the bosonic case, we

+ assume that the source k(+) vanishes at I± .

We would like to know how supersymmetry relates the fermionic boundary fields in (3.61) and (3.62) to the bosonic boundary fields in (3.45) and (3.47). Even though all four supercharges remain unbroken at null infinity, we will focus on supersymmetry transformations with constant spinor parame-

80 (−) ter ξα and their complex conjugates, which are generated by the following supercharges

− { } Q (−)α Q ( )α ˙ Q Q − = ξ Qα , = ξ Qα˙ , , = 4i∂u . (3.64)

They are the position space analogues of the supercharges that act non-trivially on massless particle representations, as in (3.24). The only non-vanishing commutators of Q with the boundary photon field Aμ are given by

[ ] Q, Az = λ(+) , (3.65a) [ ] Q, Ar = −λ(−) , (3.65b)

while the only non-vanishing anticommutators of Q with the boundary photino λ(±) are

{ } Q, λ(+) = 4i∂uAz , (3.66a) { } Q, λ(−) = −D − 2iFur + 2iFzz . (3.66b)

The action of Q on these fields can be obtained by taking the Hermitian conjugates of these formulas. Here Fzz = ∂zAz − ∂zAz is the large-r limit of Fzz, whose falloff is O(1), as discussed in (3.44). The auxiliary field D in the vector multiplet (3.3) is a Lorentz scalar with IR scaling di-

− D D D O 3 mension Δ = 2, which according to (3.42) falls off like = r2 + (r ). The fact that there are no residual powers of r in these formulas shows that the assumed large-r falloffs are consistent with

81 supersymmetry.

− It is straightforward to repeat the preceding discussion near I . The photino fields on I +

− and I must then be matched at spatial infinity. The appropriate matching conditions can be determined from the matching conditions (3.56) for the photon using supersymmetry. Combining the supersymmetry variation in (3.65a) with the matching condition for Az in (3.56a) leads to

− λ(+) I + = λ I − . (3.67) − (+) +

Similarly, the supersymmetry variation in (3.66b) and the matching condition for Fur in (3.56b) imply that the u-independent part of λ(−) should be matched across spatial infinity. However, the

constraint equation (3.63) implies that λ(−) + does not exist, since I−

( )

λ(−) → u −∂ λ(+) + + ℓ(z, z) as u → −∞ . (3.68) z I−

− Instead, we should match the u-independent term across spatial infinity, ℓ(z, z) = ℓ (z, z), which can be expressed in terms of λ(−) as follows,

− − − (1 u∂u) λ(−) I + = (1 u∂u) λ − I − . (3.69) − ( ) +

82 3.3.4 Fermionic Asymptotic Symmetries

− Consider the following fermionic charges on I + and I , for any complex-valued χ(z, z),

∫

F 1 2 − [χ] = d z χ(z, z)(1 u∂u) λ(−) I + , 2e2 − ∫ (3.70)

F − 1 2 − − [χ] = d z χ(z, z)(1 u∂u) λ(−) I − . 2e2 +

We can express them in a more covariant form by introducing a commuting, chiral spinor-valued function on S2,

(+) χα(z, z) = χ(z, z)ξα (z) . (3.71)

Using the expansion (3.60) and the falloffs (3.61), we can write

∫ [ ( )] 1 F [ ] = − 2 α( , ) ( − ∂ ) 2 ( , , , ) , χ 2 d z χ z z 1 u u →∞lim r Λα r u z z + (3.72) 2e r I−

− and similarly for F [χ]. Comparing the matching condition (3.69) to (3.70) implies the conservation law

− F [χ] = F [χ] , (3.73)

and hence a Ward identity for the tree-level S-matrix,

− F [χ]S − SF [χ] = 0 . (3.74)

83 In section 3.4 we will show that this identity gives rise to the positive-helicity soft photino theo-

† rem (3.26); the Hermitian conjugate charges F [χ] lead to the negative-helicity case (3.28). In the remainder of this section we establish several basic properties of F [χ].

Supersymmetry relates the fermionic symmetries F [χ] defined in (3.70) to the bosonic asymptotic symmetries E [ε] in (3.51). For instance, we can use (3.66b) to determine the anticommutators of the supercharges Q, Q that were singled out in (3.64) with F [χ],17

∫ { } { } Q F − i 2 E Q F , [χ] = d z χ(z, z) Fur I + = i [χ] , , [χ] = 0 . (3.75) e2 −

Note that the fermionic symmetry F [χ], with complex parameter χ(z, z), transforms into the bosonic symmetry E [ε] with the same parameter, ε(z, z) = χ(z, z). This shows that it is natural to allow complex ε(z, z), as was discussed in [7, 52, 59] and reviewed after (3.50). More generally, we can use (3.4) and (3.72) to express the commutator of an arbitrary supercharge with F [χ] in the covariant form quoted in (3.2),

{ } { } α F E α F ζ Qα, [χ] = i [ζ χα] , Qα˙ , [χ] = 0 . (3.76)

Here ζα is a commuting, constant spinor and χα(z, z) was defined in (3.71). It can similarly be shown that the bosonic charges E [ε] in (3.53) are annihilated by all supercharges. This is expected

17 Here we use the fact that D| + = F | + = 0. The first equation is obtained by solving for the auxil- I zz I B + iary field D in (3.3) in terms of the bosonic source K in (3.6), which is assumed to vanish at I± . The second equation follows from the fact that there are no magnetic charges.

84 from their interpretation as conventional gauge transformations that do not vanish at I +, since the latter commute with supersymmetry.

Following the discussion of the bosonic case around (3.51), we can express F [χ] as an integral over I + and use the constraint equation (3.63) to write it as a sum of hard and soft contributions,

F [χ] = F h[χ] + F s[χ] . (3.77)

The hard charge is given by

∫ ∫

F h 1 2 1 2 − [χ] = dud z χ(z, z)u∂uk(+) + d z χ(z, z)(1 u∂u) λ(−) I + . (3.78) 4 2e2 +

As in (3.53), the first term represents the contribution of massless charged particles that couple to the photino, while the second term is nontrivial only if there are massive charged particles passing through future timelike infinity. Since we are considering theories without massive particles, we

− 18 set (1 u∂u) λ(−) I + = 0. The supersymmetry transformation (3.66b) turns this condition +

into Fur I + = 0, which was imposed after (3.53). We can compute the following anticommutators + with the supercharges singled out in (3.64),

∫ { } i { } Q, F h[χ] = dud2z χ(z, z)j = iE h[χ] , Q, F h[χ] = 0 , (3.79) 2 u

18Following [33, 34], it should be possible to incorporate massive particles by appropriately taking into account their semiclassical photino field as they pass through timelike infinity.

85 + up to boundary terms at I± that involve the sources and hence vanish by assumption. In section 3.4 we will use these relations to determine the action of the hard charges F h[χ] on asymptotic scattering states.

The soft charges in (3.77) are given by

∫ ∫ 1 π F s[χ] = d2z ∂ χ(z, z) ωs , ωs = du u∂ λ . (3.80) 2π z e2 u (+)

Here we have defined a soft photino current ωs.19 Under a Lorentz transformation (3.38), it changes as follows, ( ) 1 δ ωs = ∂ Yz + Yz∂ + Yz∂ ωs , (3.81) Y 2 z z z

up to boundary terms that vanish as long as λ(+) asymptotes to a u-independent function of z, z suf-

+ 20 s ficiently rapidly at I± . The Lorentz transformation (3.81) shows that the soft photino current ω

C 1 is a two-dimensional field with SL(2, ) conformal weights h = 2 and h = 0, i.e. it is a left-moving

1 s spin- 2 current. Under the supercharges in (3.64), the soft photino current ω transforms into the

s soft photon current jz defined in (3.54) as follows,

{ } { } Q s s Q s , ω = ijz , , ω = 0 . (3.82)

In order to understand the action of the soft charges (3.80) on the photino, it is convenient to ∫ 19 s Note that the operator du ∂uλ(+), which is similar to soft photon current jz defined in (3.54), can be shown to vanish inside S-matrix elements by expressing it in terms of creation and annihilation operators and comparing to the soft photino theorem (3.26). ( ) 20 −(1+δ) It is sufficient to assume that λ(+) = λ(+) + + O |u| , with δ > 0, as u → ±∞. I

86 rewrite them as follows,21

∫ s 1 2 F [χ] = − lim dud z ∂zχ(z, z) cos(ωu)λ(+) . (3.84) 2e2 ω→0

In terms of creation and annihilation operators,

∫ √ ( ) F s √i 2 − † [χ] = lim ω d z ∂zχ(z, z) a (ω, z, z) a ,−(ω, z, z) . (3.85) 4 2eπ ω→0 Λ,+ Λ

This shows that F s[χ] acts on zero-momentum photini. Using this expression, as well as the mode expansion for λ(+) and the anticommutation relations for creation and annihilation operators, it can be checked that

{ } { } s s F [χ], λ(+)(u, z, z) = 0 , F [χ], λ(+)(u, z, z) = −∂zχ(z, z) . (3.86)

Thus, λ(+) shifts inhomogeneously whenever ∂zχ(z, z) ≠ 0. Just as in the bosonic case (3.50), we interpret this as spontaneous breaking of the corresponding charges F s[χ]. The u-independent part

† of λ(+) furnishes the corresponding Goldstone fermions. Similar comments apply to F [χ], which shifts λ(+) by −∂zχ(z, z).

21 Given a function f(u) such that lim f(u) exists, but is nonzero, we have the following identity, u→±∞ ∫ ∞ ∫ ∞ ′ du uf (u) = − lim du cos(ωu)f(u) , (3.83) −∞ ω→0 −∞ which amounts to integrating by parts but dropping the divergent boundary terms.

87 3.4 Soft Photino Theorem from Asymptotic Fermionic Symmetries

3.4.1 Fermionic Ward Identity for Scattering Amplitudes

In the previous section we argued for the existence of a fermionic asymptotic symmetry F [χ], which is classically conserved (see (3.73)) and hence leads to a Ward identity (3.74) for the tree-level S- matrix,

− F [χ]S − SF [χ] = 0 . (3.87)

We will now show that this Ward identity is nothing but the soft photino theorem for the case of an outgoing positive-helicity photino (equivalently, by crossing symmetry, an ingoing negative-helicity photino), which we repeat for convenience,

√ ∑n √ ∑m Mout,+ −→ qi F M − qi F M n+1 2ie ( i n) 2ie ( i n) . (3.88) η(p + )η(p ) η(p + )η(p ) i=m+1 n 1 i i=1 n 1 i

† Here pn+1 → 0 is the momentum of the soft photino. Analogously, the Ward identity for F [χ] leads to the soft photino theorem (3.28) for an outgoing negative-helicity photino.

We begin by translating (3.88) from momentum to position space. As explained in section 3.3.1, we can express the null momenta pi of the n + 1 external particles in terms of variables ωi, zi, zi, using the parametrization in (3.35). In particular, the spinor-helicity variables corresponding to the pi are given by (3.37), so that

√ α − η (pn+1)ηα(pi) = 2 ωn+1ωi (zn+1 zi) , (i = 1,..., n) . (3.89)

88 In this parametrization, the soft photino theorem can be written as follows,

√ ∑n ∑m Mout,+ −→ √qi 1 F M − √qi 1 F M 2ωn+1 n+1 ie ( i n) ie ( i n) . (3.90) ω z + − z ω z + − z i=m+1 i n 1 i i=1 i n 1 i

In order to reproduce this result, we take the matrix element of the Ward identity (3.87) between an m-particle in-state |1 ; ... ; m⟩ and an (n − m)-particle out-state ⟨m + 1 ; ... ; n|. All in- and outgoing particles (some of which could be photini) are hard, i.e. they have non-vanishing momenta.

Writing F [χ] = F h[χ] + F s[χ] as a sum of hard and soft contributions, as in (3.77), and similarly

− for F [χ], we obtain

− ⟨m + 1 ; ... ; n| F s[χ]S − SF s [χ] |1 ; ... ; m⟩ = (3.91) − − ⟨m + 1 ; ... ; n| F h[χ]S − SF h [χ] |1 ; ... ; m⟩ .

To proceed, we need to know the action of the soft and hard charges on asymptotic scattering states.

The soft charge was expressed in terms of photino creation and annihilation operators in (3.85). It creates an outgoing positive-helicity photino and an ingoing negative-helicity photino of zero momentum. Crossing symmetry implies that these two contributions lead to identical S-matrix elements, so that we can write the left-hand side of (3.91) as the ωn+1 → 0 limit of

√ ∫ ⟨ ⟩ i ωn+1 2 √ d w ∂ χ(w, w) m + 1 ; ... ; n ; Λ, p(ω + , w, w) S 1 ; ... ; m . (3.92) 2 2eπ w n 1

The action of the hard charges on asymptotic states will be derived section 3.4.2 below, where it is

89 shown that

⟩ ⟩ qf F h[χ] f, p(ω, z, z), s = − √ χ(z, z)F f, p(ω, z, z), s , 2 ω (3.93) ⟩ ⟩ † qf † F h [χ] f, p(ω, z, z), s = − √ χ(z, z)F f, p(ω, z, z), s . 2 ω

∈ { } F Here qf is the electric charge of the state labeled by f Φ, Φ, Ψ, Ψ . The operator and its Her-

† mitian conjugate F appear in the soft theorem (3.88). Its action on chiral and anti-chiral matter states was defined in (3.27).

1 If we choose χ(w, w) = , where z + is the z-value parametrizing the momentum p + of zn+1−w n 1 n 1 the soft photino, the Ward identity collapses to the soft theorem (3.88). As in the bosonic case [51], the argument can be reversed to deduce the Ward identity – and hence the underlying symmetries – from the soft theorem, which establishes their equivalence.

3.4.2 Action of the Fermionic Charges on Matter Fields

Here we show that the action of the hard fermionic charges F h[χ] on asymptotic states is given by (3.93), thereby completing the argument of section 3.4.1. We will do this by using the supersymmetry relations (3.79),

{ } { } Q, F h[χ] = iE h[χ] , Q, F h[χ] = 0 . (3.94)

90 h Here E [χ] are the hard bosonic charges, whose action (3.50) on boundary fields fq(u, z, z) of electric charge q is given by [ ] E [ε], fq(u, z, z) = −qε(z, z)fq(u, z, z) . (3.95)

Given the action of the supercharges Q, Q on charged boundary fields, we can extract the action of F h[χ] on such fields from (3.94) and (3.95).

For our present purposes, all charged fields reside in massless chiral or anti-chiral multiplets. A chiral multiplet consists of a complex scalar Φ, whose asymptotic expansion near I + was described in (3.43),

1 − Φ(u, r, z, z) = φ(u, z, z) + O(r 2) , (3.96) r

and a left-handed fermion Ψα, whose large-r behavior is identical to that of the photino Λα, which was discussed around (3.60) and (3.61),

1 (+) − Ψ (u, r, z, z) = ψ(u, z, z)ξ + O(r 2) . (3.97) α r α

If the chiral multiplet has charge q, then so do the boundary fields φ and ψ, i.e.

[ ] [ ] E h[ε], φ(u, z, z) = −qε(z, z)φ(u, z, z) , E h[ε], ψ(u, z, z) = −qε(z, z)ψ(u, z, z) . (3.98)

Given the asymptotic expansions (3.96) and (3.97), we obtain the following transformation rules for

91 the boundary fields φ, ψ under the supercharges Q, Q singled out in (3.64),22

√ [ ] [Q, φ] = 2iψ , Q, φ(u, z, z) = 0 , (3.99a) { } √ {Q, ψ(u, z, z)} = 0 , Q, ψ(u, z, z) = −2 2∂uφ(u, z, z) . (3.99b)

Given the transformation properties (3.98) and (3.99) of the chiral multiplet fields under the bosonic symmetry E h[ε] and the supersymmetries Q, Q, the commutators in (3.94) are only consistent if the fermionic symmetry F h[χ] acts as follows,

[ ] { } q F h[χ], φ(u, z, z) = 0 , F h[χ], ψ(u, z, z) = −√ χ(z, z)φ(u, z, z) . (3.100) 2

The first commutator can be understood as a consequence of the U(1)R symmetry that is expected to emerge at the superconformal IR fixed point that governs the dynamics near null infinity. Since

F [χ] is linear in the photino (see (3.70)), it has R-charge +1. (We take the R-charge of Qα to be −1.)

The electric and U(1)R charges of the first commutator in (3.100) are not consistent with any fermionic field in the chiral multiplet, and hence it must vanish.

For the anti-chiral multiplet of charge −q, which is described by taking the Hermitian conjugates

− 22 In principle the anticommutator {Q, ψ}, which falls off as O(r 1) at large r, could receive a contribution from the dimension 2 scalar auxiliary field F in the chiral multiplet, but according to (3.42) such a field − falls off as O(r 2) and hence it does not contribute.

92 of (3.96), (3.97), (3.98), and (3.99), the consistency of (3.94) requires

{ } F h[χ], ψ(u, z, z) = 0 , (3.101a) [ ] iq F h[χ], ∂ φ(u, z, z) = − √ χ(z, z)ψ(u, z, z) , (3.101b) u 2 2 { [ ]} Q, F h[χ], φ(u, z, z) = iqχ(z, z)φ(u, z, z) . (3.101c)

As above, the first equation (3.101a) is due to the electric and U(1)R charges of the fields. While (3.101b) shows that ∂uφ has a local transformation rule, it follows from (3.101c) that this does not lead to a local transformation rule for φ itself. If it did, then φ would be Q-exact, which is not the case because φ is the bottom component of the supermultiplet in (3.99).

The (anti-) commutators in (3.100), (3.101a), and (3.101b) are sufficient to establish the action

† of F h[χ] and F h [χ] on asymptotic states. Using the mode expansions and the fact that F h[χ] annihilates the vacuum,23 we find that

⟩ ⟩

F h[χ] Φ, p(ω, z, z) = F h[χ] Ψ, p(ω, z, z), + = 0 , ⟩ ⟩ q F h[χ] Ψ, p(ω, z, z), − = − √ χ(z, z) Φ, p(ω, z, z) , (3.102) 2 ω ⟩ ⟩ q F h[χ] Φ, p(ω, z, z) = − √ χ(z, z) Ψ, p(ω, z, z), + . 2 ω

Here the null momenta of the asymptotic states are parametrized in terms of ω, z, z as in (3.35).

We can express (3.102) in terms of the operator F , whose action on asymptotic states was defined

23 Recall that the soft charges F s[χ] are spontaneously broken, since they shift the photino as in (3.86) and hence do not annihilate the vacuum. However, this is not the case for the hard charges F h[χ].

93 in (3.27),

F |Φ, p⟩ = F |Ψ, p, +⟩ = 0 , F |Φ, p⟩ = |Ψ, p, +⟩ , F |Ψ, p, −⟩ = −|Φ, p⟩ . (3.103)

Since |Φ, p⟩ has charge q and |Ψ, p, −⟩ has charge −q, we can express (3.102) as follows,

qf F h[χ] |f, p, s⟩ = − √ χ(z, z)F |f, p, s⟩ , (3.104) 2 ω

where qf is the electric charge of the state. It is straightforward to repeat the preceding discussion for the Hermitian conjugate charges. They obey

† qf † F h [χ] |f, p, s⟩ = − √ χ(z, z)F |f, p, s⟩ , (3.105) 2 ω

† where the action of F on one-particle asymptotic states was defined in (3.27). Together with (3.104), this establishes the relations stated in (3.93).

94 4 Gravity in Asymptotically Flat Spacetimes

After exploring spin-1 gauge theories and their supersymmetric versions, we finally turn to gravity, which ironically was the first theory in which the connection between asymptotic symmetries and soft theories was established. Indeed, gravity in 4D asymptotically flat spacetimes was perhaps the first theory in which both its asymptotic symmetry group as well as its corresponding soft theorem were both known for over half a century. The soft graviton theorem was established by Weinberg

95 in 1965, and relates any S-matrix element in any quantum theory including gravity to a second S- matrix element which differs only by the addition of a graviton whose four-momentum is taken to zero [6]. Meanwhile, the asymptotic symmetry group of 4D asymptotically flat spacetimes was discovered by Bondi, van der Burg, Metzner, and Sachs in 1962 and was dubbed the BMS group

[1,2].

In this chapter, we will prove the conjecture put forth by Strominger in [8] that the quantum gravity S-matrix has an exact symmetry given by a certain infinite-dimensional “diagonal” subgroup

− of the direct product of the BMS group on I + with that on I . In the course of our demonstra-

± tion it is necessary to carefully define the physical phase spaces Γ of gravitational modes at past

− ± and future null infinity (I and I +). Γ must include, in addition to the Bondi news, all soft graviton degrees of freedom that do not decouple from the S-matrix, which are constrained by

± boundary conditions at the boundaries of I . The soft modes can be viewed as living on these boundaries, and the boundary conditions reduce their number by a crucial factor of 2. The reduced space of modes may then be transparently identified (from their transformation law) as nothing but the Goldstone modes of spontaneously broken supertranslation invariance. The relevant physical

± phase spaces Γ become simply the usual radiative modes plus the Goldstone modes.1 The boundary constraint entails a modification of the naive Dirac bracket. After this modification, canonical

± ± expressions for T are given which generate supertranslations on all of Γ . While there has been

1This is the minimal phase space required for a good action of supertranslations. We have not ruled out the possibility of further soft modes and a larger phase space associated to local conformal symmetries [43, 45–47, 60] which for example lie in the subleading components of the metric and will be considered in the next chapter.

96 ± much discussion of T over the decades, the construction of generators which act properly on the infrared as well as radiative modes is new.

In this chapter, we mainly consider only the case of pure gravity but expect the inclusion of mass-

± less matter or gauge fields to be straightforward. We present in section 4.1 the full I phase spaces

± Γ (including the boundary condition), construct the Dirac brackets and supertranslation gener-

± ators T , and identify the soft gravitons as Goldstone modes. Section 4.2 reviews the proposed

− matching condition between I + and I near spatial infinity and derives with the Ward identity assosciated with the diagonal supertranslations that preserve this relation. We review the standard derivation of the soft graviton theorem in momentum space in section 4.3. In section 4.4, we describe the transformation between the asymptotic description of section 4.2 and the momentum space description of section 4.3. Finally, in section 4.5, we show that Weinberg’s soft graviton theorem is the Ward identity following from diagonal supertranslation invariance.

4.1 Supertranslation Generators

In this section we construct the physical phase space, the symplectic form (or equivalently the Dirac

± bracket), and the canonical generator of supertranslations at I .

4.1.1 Asymptotic Vector Fields

Begin by considering asymptotically flat geometries in the finite neighborhood of Minkowski space defined in [61] and referred to as CK spaces. These have a large-r weak-field expansion near future

97 null infinity (I +) in retarded Bondi coordinates (see [45–47, 60] for details)

2 − 2 − 2 ds = du 2dudr + 2r γzzdzdz (4.1) 2m + B du2 + rC dz2 + rC dz2 − 2U dudz − 2U dudz + ··· , r zz zz z z where

1 U = − DzC . (4.2) z 2 zz

The retarded time u parameterizes the null generators of I + and (z, z) parameterize the conformal

2 ( , , ) = 2 S . The Bondi mass aspect mB and Czz depend on u z z , γzz (1+zz)2 is the round metric on

− the unit sphere, and Dz is the γ-covariant derivative. Near past null infinity I , CK spaces have a similar expansion in advanced Bondi coordinates given by

2 − 2 2 ds = dv + 2dvdr + 2r γzzdzdz − (4.3) 2m + B dv2 + rD dz2 + rD dz2 − 2V dvdz − 2V dvdz + ··· , r zz zz z z where

1 V = DzD . (4.4) z 2 zz

I + I + I + I − I − I − We denote the future (past) of by + ( − ), and the future (past) of by + ( − ).

± These comprise the boundary of I . Furthermore, we define the outgoing and incoming Bondi

98 news by

Nzz ≡ ∂uCzz , Mzz ≡ ∂vDzz . (4.5)

BMS+ transformations [1, 2] are defined as the subgroup of diffeomorphisms that acts nontrivially on the radiative data at I +. These include the familiar Lorentz transformations as well as the supertranslations. The latter are generated by the infinite family of vector fields2

1 f∂ − (Dzf∂ + Dzf∂ ) + DzD f∂ , (4.6) u r z z z r

2 + for any function f(z, z) on the S . BMS acts on Czz according to

L − 2 fCzz = f∂uCzz 2Dzf . (4.7)

− − Similarly BMS transformations act on I and contain the supertranslations parameterized by

− f (z, z):

− 1 − − − f ∂ + (Dzf ∂ + Dzf ∂ ) − DzD f ∂ , (4.8) v r z z z r under which

L − 2 − f− Dzz = f ∂vDzz + 2Dzf . (4.9) 2 1 The subleading in r terms depend on the coordinate condition: see [45–47,60].

99 4.1.2 Dirac Brackets on I

The Dirac bracket on the radiative modes (the non-zero modes of the Bondi news) at I + was found in [31,49, 50,62] to be

{ ′ } − − ′ 2 − Nzz(u, z, z), Nww(u , w, w) = 16πG∂uδ(u u )δ (z w)γzz , (4.10)

where G is Newton’s constant. The generator of BMS+ supertranslations on these modes is [31, 45–

47, 49,50,60,62]

∫ + 1 2 T (f) = d zγ f(z, z)mB + zz 4πG I− ∫ 1 [ ] = dud2zf γ N Nzz + 2∂ (∂ U + ∂ U ) , (4.11) 16πG zz zz u z z z z

+ {T (f), Nzz} = f∂uNzz , where in the second line we have used the constraints and assumed no matter fields.

Of course BMS+ transformations acting on the radiative modes alone do not comprise an asymptotic symmetry. One must act on a larger phase space Γ+ including some non-radiative modes. The

′ obvious guess is to identify this larger space with a space of Czz, and define a bracket for all (u, u ) by integrating (4.10) to

{ ′ } − ′ 2 − Czz(u, z, z), Cww(u , w, w) = 8πGΘ(u u )δ (z w)γzz , (4.12)

100 where Θ(x) = sign(x). However, if we use this, we find

{ + } − 2 ̸ L T (f), Czz = f∂uCzz Dzf = fCzz . (4.13)

The inhomogeneous term is off by a factor of 2, so either the bracket (4.12) or the generator (4.11) is incorrect. This problem did not seem to have been addressed previously.

We resolve this problem by motivating and imposing boundary conditions on Czz at the boundaries of I +, and then incorporating this boundary constraint into a modified Dirac bracket. Since the constraints apply only to the boundary degree of freedom, (4.12) will be unaltered unless either

′ u or u is on the boundary. However, this will turn out to give us exactly the missing factor of 2 in

(4.13). The supertranslation invariant boundary conditions are

[∂ − ∂ ] + = , zUz zUz I± 0 (4.14)

| + = . Nzz I± 0 (4.15)

Equivalently the first condition may be written

2 2 [ − ] + = . DzCzz DzCzz I± 0 (4.16)

It has a coordinate invariant expression in terms of the component of the Weyl tensor that is some-

101 times referred to as the magnetic mass aspect:

(0) | + = . Im Ψ2 I± 0 (4.17)

There are two related motivations for this constraint besides the fact that it (as we will see mo- mentarily) leads to a proper action of T+. Firstly, the boundary condition (4.14) is obeyed by CK

I + I + + + spaces [61]. Furthermore, operator insertions of [∂zUz] + and [∂zUz] + correspond to soft gravi- I− I− tons and have non-vanishing S-matrix elements (due to Weinberg poles) even though they are pure gauge. Therefore, they must be retained as part of the physical phase space. However, because these

I + I + + − + S poles cancel in the difference [∂zUz] + [∂zUz] + , this combination decouples from all -matrix I− I− elements and should not be part of the physical phase space. Our constraint (4.14) projects out these fully decoupled modes.

The general solution of the constraints (4.16) can be expressed as

2 | + = , Czz I− DzC (4.18) ∫ ∞ 2 duNzz = DzN , (4.19) −∞ where the boundary fields C, N are real. We may then take as our coordinates on phase space the

102 boundary and bulk fields3

+ Γ := {C(z, z), N(z, z), Czz(u, z, z), Czz(u, z, z)} . (4.20)

The arguments u of the bulk fields terms are restricted to non-boundary (i.e. finite) values only. The bulk-bulk Dirac brackets remain (4.12). A priori it is not obvious how one extends the bulk-bulk bracket (or equivalently the symplectic form) over all of Γ+. We do so by first imposing (4.19) as a relation between bulk-bulk and bulk-boundary brackets in the form

∫ ∞ 2{ } ′{ ′ } Dz N(z, z), Cww(u, w, w) = du Nzz(u , z, z), Cww(u, w, w) , (4.21) −∞ and then constraining the boundary-boundary bracket by continuity in the form

2 D {N(z, z), C(w, w)} = lim {N(z, z), Cww(u, w, w)} . (4.22) w u→−∞

The non-zero Dirac brackets following from the boundary constraints (4.15) and (4.16) are then

3 2 C and N each have four zero modes of angular momentum 0 and 1 which are projected out by Dz and hence do not appear in the metric. They might be omitted from the definition of Γ+ and do not play an important role in the present discussion. However, we retain them for future reference, since as will become apparent below, the C zero modes have an interesting interpretation as the spatial and temporal position of the geometry.

103 uniquely determined as4

{ ′ } − ′ 2 − Czz(u, z, z), Cww(u , w, w) = 8πGΘ(u u )δ (z w)γzz ,

{ ′ } − 2 | − |2 C(z, z), Cww(u , w, w) = 8GDw(S ln z w ) , (4.23) { ′ } 2 | − |2 N(z, z), Cww(u , w, w) = 16GDw(S ln z w ) ,

{N(z, z), C(w, w)} = 16GS ln |z − w|2 ,

′ where u, u are not on the boundary and

(z − w)(z − w) S ≡ . (4.24) (1 + zz)(1 + ww)

S is the sine-squared of the angle between z and w on the sphere and obeys

S D2 (S ln |z − w|2) = , w (z − w)2 (4.25) 2 2 | − |2 2 − DzDw(S ln z w ) = πγzzδ (z w) .

− I [∂ − ∂ ] − = Similarly, on , the constraints zVz zVz I± 0 can be solved by

∫ ∞ | 2 2 Dzz I − = DzD , dvMzz = DzM . (4.26) + −∞

4We note but do not pursue herein the interesting appearance of logarithms related to the four C and N 2 zero modes. These are projected out by acting with Dz and hence irrelevant to the supertranslation generators below.

104 − The coordinates on the phase space at I can then be taken as

− Γ := {D(z, z), M(z, z), Dzz(v, z, z), Dzz(v, z, z)} , (4.27)

where v is not on the boundary. The non-zero Dirac brackets are

{ ′ } − ′ 2 − Dzz(v, z, z), Dww(v , w, w) = 8πGΘ(v v )δ (z w)γzz ,

{ ′ } 2 | − |2 D(z, z), Dww(v , w, w) = 8GDw(S ln z w ) , (4.28) { ′ } 2 | − |2 M(z, z), Dww(v , w, w) = 16GDw(S ln z w ) ,

{M(z, z), D(w, w)} = 16GS ln |z − w|2 ,

′ where v, v are not on the boundary.

The demand of continuity (4.22) is not as innocuous as it looks because we see from (4.23) and

(4.28) that other brackets (in particular {Nzz, Cww}) are not continuous as u is taken to the boundary. We have not ruled out the possibility that there are inequivalent extensions of the symplectic

± form on the radiative phase space to all of Γ corresponding to inequivalent quantizations of the boundary sector. In an action formalism, this could arise from different choices of boundary terms.

However, an a posteriori justification of our choice is, as we will soon show, the fact that it leads to a

± realization of supertranslations as a canonical transformation on Γ .

105 4.1.3 Canonical Generators

The supertranslation generator may now be written in terms of bulk and boundary fields as

∫ ∫ ∫ + 1 2 1 2 zz 1 2 zz 2 2 T (f) = d zγ fmB = dud zfγ NzzN − d zγ fD D N , (4.29) + zz zz z z 4πG I− 16πG 8πG where the integral over infinite u in the first term is the Cauchy principal value. Using the brackets

(4.23), one finds

+ {T (f), Nzz} = f∂uNzz ,

{ + } − 2 T (f), Czz = f∂uCzz 2Dzf , (4.30) {T+(f), N} = 0 ,

{T+(f), C} = −2f , as desired.

− Similarly on I , we have

∫ ∫ − − 1 − 1 − T (f ) = dvd2zf γ M Mzz + d2zγzzf D2D2M , (4.31) 16πG zz zz 8πG z z

106 and

− − − {T (f ), Mzz} = f ∂vMzz ,

{ − − } − 2 − T (f ), Dzz = f ∂vDzz + 2Dzf , (4.32) − − {T (f ), M} = 0 ,

− − − {T (f ), D} = 2f , as desired.

At the quantum level supertranslations do not leave the usual in or out vacua invariant. Acting with T+, the last term in (4.29) is linear in the graviton field operator and creates a new state with a soft graviton. The energy of the new state is degenerate with the out vacuum but has different angular momentum. Hence, supertranslation symmetry is spontaneously broken in the usual vacuum.

− 1 The last line of (4.30) clearly identifies 2 C as the Goldstone mode associated with this symmetry breaking, and it is conjugate to the soft graviton zero mode N. This demonstrates that the construc-

± tion of a generator of supertranslations on I is possible, but subtle, and requires a careful analysis

± of the zero mode structure and boundary conditions on the boundaries of I .

4.2 Supertranslation Invariance of the S-Matrix

In this section we summarize the supertranslation invariance of the S-matrix conjectured in [8] as

− well as the associated Ward identity. The first step is to understand how I + and I may be linked near spatial infinity. In the conformal compactification of asymptotically flat spaces, the sphere at

107 spatial infinity is the boundary of a point i0. Null generators of I in the conformal compactifica-

− tion of asymptotically flat spaces run from I to I + through i0. We label all points lying on the same such generator with the same value of (z, z). This gives an “antipodal” identification of points

− on the conformal spheres at I with those on I +, exactly in the same manner as demonstrated in

I + I − the previous chapters. For CK spaces, we may identify geometric data on − with that at + via the continuity condition [8]

Czz|I + = −Dzz|I − , (4.33) − + or equivalently

− C (z, z) = −D+(z, z) . (4.34)

− In [8], it was conjectured that the “diagonal” subgroup of BMS+×BMS which preserves the continuity condition (4.33) is an exact symmetry of both classical gravitational scattering and the quantum gravity S-matrix. The diagonal supertranslation generators are those which are constant on the null generators of I , i.e.

− f (z, z) = f(z, z) . (4.35)

The conjecture states that S-matrix obeys

− T+(f)S − ST (f) = 0 . (4.36)

A Ward identity is then derived by taking the matrix elements of (4.36) between states with n incom-

108 in out I in out ing (m outgoing) particles at zk (zk ) on the conformal sphere at . These carry energies Ek (Ek ), where

∑m ∑n out in Ek = Ek (4.37) k=1 k=1

⟩ ⟨ out · · ·| in ··· by total energy conservation. We denote the out and in states by z1 , and z1 , . Choosing

1 f(w, w) = z−w , it was shown that the matrix element of (4.36) between such states implies

[ ] ⟩ ⟩ ∑m out ∑n in E E ⟨zout, · · ·| : P S : zin, ··· = ⟨zout, · · ·| S zin, ··· k − k , (4.38) 1 z 1 1 1 z − zout z − zin k=1 k k=1 k where the :: denotes time-ordering and the “soft graviton current” is defined by

(∫ ∞ ∫ ∞ ) 1 Pz ≡ dv∂vVz − du∂uUz . (4.39) 2G −∞ −∞

± Since Pz involves zero-frequency integrals over I it creates and annihilates soft gravitons with a certain z-dependent wave function. The supertranslation Ward identity (4.38) relates S-matrix elements with and without insertions of the soft graviton current. It can also easily be seen that

(4.38) implies the general Ward identities following from (4.36) for an arbitrary function f(z, z) [8].

109 4.3 The Soft Graviton Theorem

In this section, we specify our conventions and briefly review Weinberg’s derivation of the soft graviton theorem for the simplest case of a free massless scalar. Recall that Einstein gravity coupled to a free massless scalar is described by the action

∫ [ ] √ 2 1 S = − d4x −g R + gμν∂ φ∂ φ , (4.40) κ2 2 μ ν

2 where κ = 32πG. In the weak field perturbation expansion gμν = ημν + κhμν, and the relevant leading terms are

2 1 1 L = − R = − ∂ h ∂σhμν + ∂ h∂μh + ∂μh ∂ hνρ − ∂ hμν∂ h + ··· , grav κ2 2 σ μν 2 μ μν ρ μ ν [ ] (4.41) 1 √ 1 1 1 L = − −ggμν∂ φ∂ φ = − ∂μφ∂ φ + κhμν ∂ φ∂ φ − η ∂σφ∂ φ + ··· . s 2 μ ν 2 μ 2 μ ν 2 μν σ

μ 1 In harmonic gauge ∂ hμν = 2 ∂νh the Feynman rules take the form (see [63])

p ( ) μν = 1 + − −i , αβ 2 ημαηνβ ημβηνα ημνηαβ p2−iε

p = −i , p2−iε

(4.42)

μν ( ) p = iκ + − · . 2 p1μp2ν p1νp2μ ημνp1 p2 p1 p2

110 Now, consider an on-shell amplitude involving n incoming (with momenta p1, ··· , pn) and m out-

′ ··· ′ going (with momenta p1, , pm) massless scalars represented by the diagram

′ pn pm

··· ··· (4.43)

p ′ 1 p1

Consider the same amplitude with an additional outgoing soft graviton of momentum q and polar-

μ 1 μ ization εμν(q) satisfying the gauge condition q εμν = 2 qνε μ. The dominant diagrams in the soft q → 0 limit are ′ pn q pn pm pn q

′ ′ pm ∑m q ∑n pn . = . + pk . . ′ . k=1 k=1 . . pk .

′ ′ ′ p p1 p1 1 p1 p1 p1 (4.44)

111 Additional diagrams with the external graviton attached to internal lines cannot develop soft poles

[6]. The contribution of these diagrams to the near-soft amplitude is

∑m ′ ′ ′ ′ ′ −i M (q, p , ··· , p , p , ··· , p ) = M(p , ··· , p + q, ··· , p , p , ··· , p )( ) μν 1 m 1 n 1 k m 1 n ′ + 2 − = pk q iε k 1 [ ] ( ( )) iκ ′ ′ ′ ′ ′ ′ × p (p + q) + p (p + q) − η p · p + q 2 kμ k ν kν k μ μν k k ∑n ′ ′ −i + M( , ··· , , , ··· , + , ··· , ) p1 pm p1 pk q pn − 2 − (pk q) iε k=1 [ ( )] iκ × p (p − q) + p (p − q) − η p · (p − q) . 2 kμ k ν kν k μ μν k k

(4.45)

The soft graviton theorem is the leading term in q-expansion:

[ ] ∑m ′ ′ ∑n ′ ′ κ pkμpkν pkμpkν ′ ′ M ( , , ··· , , , ··· , ) = − M( , ··· , , , ··· , ) , μν q p1 pm p1 pn ′ · · p1 pm p1 pn 2 p q pk q k=1 k k=1

(4.46) where q → 0. While we reviewed the derivation here for a massless scalar, note that the prefactor in square brackets is a universal soft factor and does not depend on the spin of the matter particles.

Moreover, the expression is actually gauge invariant, since under a gauge transformation we have

δεμν = qμΛν + qνΛμ one finds

[ ] ∑m ∑n μνM μ ′ − M δε μν = κΛ pkμ pkμ = 0 (4.47) k=1 k=1

112 by momentum conservation. Hence, (4.46) is valid in any gauge.

4.4 From Momentum to Asymptotic Position Space

The supertranslation Ward identity (4.38) is expressed in terms of field operator Pz integrated along fixed-angle null generators of I . Weinberg’s soft graviton theorem (4.45), on the other hand, is expressed in terms of momentum eigenmodes of the field operators. In this section, in order to compare the two, we transform the field operator between these two bases.

Momentum eigenmodes in Minkowski space are usually described in the flat coordinates

ds2 = −dt2 + d⃗x · d⃗x . (4.48)

These flat coordinates are related to the retarded coordinates (4.1) by the transformation

t = u + r ,

2rz x1 + ix2 = , (4.49) 1 + zz r(1 − zz) x3 = , 1 + zz with⃗x = (x1, x2, x3) obeying⃗x · ⃗x = r2. At late times and large r, the wave packet for a massless particle with spatial momentum centered around⃗p becomes localized on the conformal sphere near the point ⃗x ω ⃗p = ωbx ≡ ω = (z + z, −iz + iz, 1 − zz) , (4.50) r 1 + zz

113 where⃗p · ⃗p = ω2. Hence, the momentum of massless particles may be equivalently characterized by

(ω, z, z) or pμ.

As t → ∞, the gravitational field becomes free and can be approximated by the mode expansion

∫ ∑ 3 [ ] d q 1 ∗ · † − · hout(x) = εα (⃗q) aout(⃗q)eiq x + εα (⃗q)aout(⃗q) e iq x , (4.51) μν (2π)3 2ω μν α μν α α=± q

0 where q = ωq = |⃗q|, α = ± are the two helicities, and

( ) out out ⃗′ † 3 3 − ⃗′ [aα (⃗q), aβ (q ) ] = δαβ(2ωq)(2π) δ ⃗q q . (4.52)

The outgoing gravitons with momentum q and polarization α, as in the amplitude (4.45), corre-

out spond to final-state insertions of aα (⃗q).

Let us now take (ωq, w, w) to parameterize the graviton four-momentum

ωq qμ = (1 + ww, w + w, −i (w − w) , 1 − ww) . (4.53) 1 + ww

± ± ± The polarization tensors may be written ε μν = ε με ν with

1 ε+μ(⃗q) = √ (w, 1, −i, −w) , 2 (4.54) − 1 ε μ(⃗q) = √ (w, 1, i, −w) . 2

114 ±μν ±μ These obey ε qν = ε μ = 0 and

√ √ 2rz (w − z) − − 2r (1 + wz) ε+ (⃗q) = ∂ xμε+ (⃗q) = , ε (⃗q) = ∂ xμε (⃗q) = . (4.55) z z μ (1 + zz)2 z z μ (1 + zz)2

In retarded Bondi coordinates, it follows from (4.1) that on I +,

1 out Czz(u, z, z) = κ lim h (r, u, z, z) . (4.56) r→∞ r zz

μ ν Using hzz = ∂zx ∂zx hμν and the mode expansion (4.51), we find

∫ ∑ 3 [ ] 1 μ ν d q 1 α ∗ out −iωqu−iωqr(1−cos θ) Czz = κ lim ∂zx ∂zx ε (⃗q) a (⃗q)e + h.c. . (4.57) r→∞ r (2π)3 2ω μν α α=± q where θ is the angle between⃗x and⃗q. The integrand oscillates rapidly at large r with stationary points at θ = 0, π. The stationary phase approximation to the integral over the momentum-space sphere then gives

∫ ∞ [ ] iκ − † = − out( b) iωqu − out( b) iωqu , (4.58) Czz 2 2 dωq a+ ωqx e a− ωqx e 4π (1 + zz) 0 where the contribution from the θ = π stationary point vanishes in the large r limit. Defining

∫ ∞ ω ≡ iωu Nzz(z, z) due ∂uCzz , (4.59) −∞

115 and using (4.58), we find

∫ ∞ [ ] ω − κ out b − out b † Nzz(z, z) = 2 dωqωq a+ (ωqx)δ(ωq ω) + a− (ωqx) δ(ωq + ω) . (4.60) 2π (1 + zz) 0

When ω > 0 (ω < 0), only the first (second) term contributes, and we obtain

κωaout(ωbx) Nω (z, z) = − + , zz 2π (1 + zz)2 (4.61) out † − κωa− (ωbx) N ω(z, z) = − , zz 2π (1 + zz)2 where we have taken ω > 0. In the case of the zero mode, we will define it in the following hermitian way:

(0) ≡ 1 ω −ω Nzz lim (Nzz + Nzz ) . (4.62) ω→0+ 2

It follows that

[ ] (0) − κ out b out b † Nzz (z, z) = 2 lim ωa+ (ωx) + ωa− (ωx) . (4.63) 4π (1 + zz) ω→0+

− A parallel construction is possible on I . Defining

∫ ∞ ω ≡ iωv Mzz(z, z) dve ∂vDzz , (4.64) −∞

116 we find for ω > 0

κωain(ωbx) Mω (z, z) = − + , zz 2π (1 + zz)2 (4.65) in † − κωa−(ωbx) M ω(z, z) = − , zz 2π (1 + zz)2

in in† − where a± and a± annihilate and create incoming gravitons on I . At ω = 0,

[ ] (0) ≡ − κ in b in b † Mzz (z, z) 2 lim ωa+(ωx) + ωa−(ωx) . (4.66) 4π (1 + zz) ω→0+

From (4.59) and (4.64) we have also

(0) 2 Nzz (z, z) = DzN , (4.67) (0) 2 Mzz (z, z) = DzM .

Defining

O ≡ (0) (0) 2 2 (4.68) zz Nzz (z, z) + Mzz (z, z) = DzN + DzM ,

the soft graviton current (4.39) can be written as

( ) I − I + 1 | + − | + 1 zz O Pz = Vz I − Uz I + = γ ∂z zz . (4.69) 2G − − 4G

117 4.5 Soft Graviton Theorem as a Ward identity

Equations (4.66)–(4.69) express the soft graviton current Pz in terms of standard momentum space creation and annihilation operators. Amplitudes involving the latter are given by Weinberg’s soft graviton theorem. In this section we simply plug this in and reproduce the supertranslation Ward identities.

We denote an S-matrix element of m incoming and n outgoing particles by

⟩ ⟨ out · · ·| S in ··· (4.70) z1 , z1 , ,

where the in (out) momenta are parametrized by zin (zout) as in (4.50). We now consider the S- ⟩ ⟨ out · · ·| O S in ··· matrix element z1 , : zz : z1 , with a time-ordered insertion. Using (4.66) and (4.68), this can be written as

⟩ [ ⟩ ⟨ out · · ·| O S in ··· − κ ⟨ out · · ·| out b S in ··· z1 , : zz : z1 , = 2 lim ω z1 , a+ (ωx) z1 , 4π (1 + zz) ω→0+ (4.71) ⟩] ⟨ out · · ·| S in b † in ··· + ω z1 , a−(ωx) z1 , .

out b † in b → 5 Here, we have used the fact that a− (ωx) (a+(ωx)) annihilates the out (in) state for ω 0. The first term is the S-matrix element with a single outgoing positive-helicity soft graviton with spatial momentum ωbx, while the second term is the S-matrix element with a single incoming negative-

5This holds even if for example the initial state contains soft gravitons because of the factor of ω in (4.52).

118 helicity soft graviton also with spatial momentum ωbx. The two amplitudes are equal, and we get

⟩ [ ⟩] ⟨ out · · ·| O S in ··· − κ ⟨ out · · ·| out b S in ··· z1 , : zz : z1 , = 2 lim ω z1 , a+ (ωx) z1 , . (4.72) 2π (1 + zz) ω→0+

The soft graviton theorem (4.46) with a positive-helicity outgoing graviton reads

[ ⟩] ⟨ out, · · ·| out(⃗)S in, ··· lim→ ω z1 a+ q z1 ω 0 [ [ ] ] ∑m ′ · + 2 ∑n · + 2 ⟩ κ ω pk ε (q) ω [pk ε (q)] out in = lim ′ − ⟨z , · · ·| S z , ··· . + · · 1 1 2 ω→0 p q pk q k=1 k k=1

(4.73)

Using the parametrization of the momenta discussed earlier

( ( ) ) zin + zin −i zin − zin 1 − zinzin pμ = Ein 1, k k , k k , k k , k k 1 + zinzin 1 + zinzin 1 + zinzin ( k k (k k ) k k ) out out − out − out − out out ′μ out zk + zk i zk zk 1 zk zk pk = Ek 1, out out , out out , out out , 1 + zk zk 1 + zk zk 1 + zk zk ( ) (4.74) z + z −i (z − z) 1 − zz qμ = ω 1, , , , 1 + zz 1 + zz 1 + zz 1 ε+μ(q) = √ (z, 1, −i, −z) , 2

119 and (4.72), we find

⟩ ⟩ 8G ⟨zout, · · ·| : O S : zin, ··· = ⟨zout, · · ·| S zin, ··· 1 zz 1 (1 + zz) 1 1 [ ( ) ( ) ] ∑m ∑n Eout z − zout Ein z − zin × ( k )( k ) − ( k )( k ) . z − zout 1 + zoutzout z − zin 1 + zinzin k=1 k k k k=1 k k k

(4.75)

Now, using (4.69), we can relate the insertion of Pz to that of Ozz:

⟩ ⟩ ⟨ out · · ·| S in ··· 1 zz ⟨ out · · ·| O S in ··· z1 , : Pz : z1 , = γ ∂z z1 , : zz : z1 , 4G [ ] ⟩ ∑m out ∑n in E E = ⟨zout, · · ·| S zin, ··· k − k 1 1 z − zout z − zin k=[1 k k=1 k ] ⟩ ∑m out out ∑n in in E z E z + ⟨zout, · · ·| S zin, ··· k k − k k . 1 1 1 + zoutzout 1 + zinzin k=1 k k k=1 k k

(4.76)

The very last square bracket vanishes due to total momentum conservation. We then have

[ ] ⟩ ⟩ ∑m out ∑n in E E ⟨zout, · · ·| : P S : zin, ··· = ⟨zout, · · ·| S zin, ··· k − k , (4.77) 1 z 1 1 1 z − zout z − zin k=1 k k=1 k which reproduces exactly the supertranslation Ward identity (4.38) derived in [8]. We can also run the above argument backwards to show that this supertranslation Ward identity implies Weinberg’s soft graviton theorem.

120 5 Virasoro Symmetry of 4D Quantum

Gravity

In addition to demonstrating the equivalence between the Ward identity associated to the supertranslations of the BMS group and the leading soft graviton theorem, there was recently much progress in exploring what is known as the subleading soft graviton theorem and what symmetries

121 it may correspond to. Whereas the leading soft graviton theorem is the claim that the Feynman diagram with a soft graviton equals the Feynman diagram without the soft graviton multiplied by a factor, whose leading term is the universal Weinberg pole, i.e. the O(1/ω) term, the subleading soft graviton theorem states that even the subleading O(1) term is universal. This theorem was first conjectured and proved at tree-level in [13], and it was recently demonstrated in [14] that this gives rise to, at tree-level, a Virasoro or “superrotational” symmetry that acts on the celestial sphere at null infinity in any theory of gravity in four asymptotically flat dimensions. This extra Virasoro symmetry was in fact already conjectured in [43–46, 64], and augments the BMS group into the

“extended” BMS group. Except for an anomaly arising from one-loop exact infrared (IR) divergences [18, 19, 26, 29, 65], this subleading soft theorem extends to the full quantum theory. How- ever, the implications of this anomaly for the Virasoro symmetry of the full quantum theory are not understood, and the exploration of such implications comprises the subject of the present chapter.

There are several open possibilities. One is that the Virasoro action is defined in the classical but not in the quantum theory. If so, anomalous symmetries still have important quantum constraints that would be interesting to understand. A second possibility is that the Virasoro action acts on the full quantum theory, but that the generators and symmetry action are renormalized at one-loop.

This is suggested by the discussion in [66], where it is pointed out that the very definition of a scattering problem in asymptotically flat gravity requires an infinite number of exactly conserved charges and associated symmetries, as well as by [21], which found that the anomaly vanishes with an alternate order of soft limits. A third possibility is that the implications can only be properly for- mulated in a Faddeev-Kulish [38, 67–71] basis of states (constructed for gravity in [39]), in which

122 case all IR divergences are absent. After all, IR divergences preclude a Fock-basis S-matrix for quantum gravity and, although we have become accustomed to ignoring this point, it is hard to discuss symmetries of an object which exists only formally!

In this chapter we give evidence which is consistent with, but does not prove, the second hypoth- esis, which states that the Virasoro action persists to the full quantum theory but requires the generators to have a one-loop correction. We use the recent construction of a 2D energy-momentum tensor Tzz found in [72, 73], where z is a coordinate on the celestial sphere, in terms of soft graviton modes. The tree-level subleading soft theorem [13] implies that insertions of Tzz in the tree-level

S-matrix infinitesimally generate a Virasoro action on the celestial sphere. At one-loop order, the subleading soft theorem has an IR divergent term with a known universal form. This spoils the Vira- soro action generated by Tzz insertions. However, we show explicitly that the effects of the IR divergent term can be removed by a certain shift in Tzz that is quadratic in the soft graviton modes. The possibility that this could be achieved by a simple shift is far from obvious and requires a number of nontrivial cancellations.

This does not demonstrate that there is a Virasoro action on the full quantum theory generated by a renormalized energy-momentum tensor, as there may also be an IR finite one-loop correction to the subleading soft theorem. At present, little is known about such finite corrections. In all cases which have been analyzed [18, 19], the finite part of the correction vanishes. Yet, there is no known argument that this should always be the case, and this remains an open issue for us.

The outline of this chapter is as follows. In section 5.1 we fix conventions and recall the construction of the tree-level soft graviton energy-momentum tensor. We then reproduce the derivation of

123 the one-loop exact IR divergent corrections to the subleading soft graviton theorem in section 5.2.

Finally, this divergence is rewritten in section 5.3 in terms of the formal matrix element of another quadratic soft graviton operator, effectively renormalizing the tree-level energy-momentum tensor.

5.1 Tree-Level Energy-Momentum Tensor

In this section, we review the derivation of the 2D tree-level energy-momentum tensor living on the celestial sphere at null infinity [72, 73]. Asymptotic one-particle states are denoted by |p, s⟩, where p is the 4-momentum and s is the helicity, and such states are normalized so that

( ) ′ ′ ′ ⟨ | ⟩ 3 0 ′ (3) − p , s p, s = (2π) 2p δss δ (⃗p ⃗p ) . (5.1)

An n-particle S-matrix element is denoted by

Mn ≡ ⟨out|S|in⟩ , (5.2)

where |in⟩ ≡ |p1, s1; ··· ; pm, sm⟩ and ⟨out| ≡ ⟨pm+1, sm+1; ··· ; pn, sn|. Consider the amplitude

M± ≡ ⟨ ± |S| ⟩ n+1(q) out; q, 2 in , (5.3)

consisting of n external hard particles along with an additional external graviton that has momen-

≡ 0 ≡ ± tum pn+1 q, energy pn+1 ω, and polarization εμν. Denoting the same amplitude without the

124 extra external graviton as Mn, the tree-level soft graviton theorem states that

[ ] M± (0)± (1)± O M lim + (q) = S + S + (q) n , (5.4) ω→0 n 1 n n where the leading and subleading soft factors are given by

∑n μ ν ± ∑n ± μ ± κ p p εμν(q) ± iκ εμν(q)p qλ (0) = k k , (1) = − k J λν , Sn · Sn · k (5.5) 2 pk q 2 pk q k=1 k=1

1 2 ≡ J λν respectively. As before, κ 32πG is the gravitational coupling constant, and k is the total angular momentum operator for the kth particle.

As we already explained in chapter 4, but will for convenience review here, asymptotically flat metrics in Bondi gauge take the form

2 − 2 − 2 ds = du 2dudr + 2r γzzdzdz (5.6) 2m + B du2 + rC dz2 + rC dz2 + DzC dudz + DzC dudz + ··· , r zz zz zz zz

≡ 2 2 where γzz (1+zz)2 is the round metric on the S and Dz is the associated covariant derivative. The coordinates (u, r, z, z) are asymptotically related to the standard Cartesian coordinates according to

1 x0 = u + r , xi = rbxi(z, z) , bxi(z, z) = (z + z, −i(z − z), 1 − zz) . (5.7) 1 + zz ± 1 (0) In chapter 4 we only focused on the leading term Sn .

125 A massless particle with momentum pk crosses the celestial sphere at a point (zk, zk). In the helicity basis, the particle momentum and polarization can be parameterized by an energy ωk and this crossing point. It follows from (5.7) that

( ) − − − μ zk + zk i(zk zk) 1 zkzk ··· pk = ωk 1, , , , k = 1, , n , 1 + zkzk 1 + zkzk 1 + zkzk

1 − 1 ε+(p ) = √ (−z , 1, −i, −z ) , ε (p ) = √ (−z , 1, i, −z ) , μ k 2 k k μ k 2 k k ( ) (5.8) z + z −i(z − z) 1 − zz qμ(z) = ω 1, , , ≡ ωbqμ(z) , 1 + zz 1 + zz 1 + zz

1 − 1 ε+(q) = √ (−z, 1, −i, −z) , ε (q) = √ (−z, 1, i, −z) , μ 2 μ 2

± ± ± where the soft graviton polarization tensor is taken to be εμν = εμ εν .

Now, the perturbative fluctuations of the gravitational field have a mode expansion given by

∫ ( ) ∑ 3 [ ] d q 1 ∗ · † − · hout x0,⃗x = εα (q) aout(⃗q)eiq x + εα (q)aout(⃗q) e iq x , (5.9) μν (2π)3 2ω μν α μν α α=± q

out † out where aα (⃗q) and aα (⃗q) are the standard creation and annihilation operators for gravitons obeying the commutation relations

[ ] out ⃗ out † 3 (3) ⃗ − (5.10) aα (p), aβ (⃗q) = (2π) (2ω)δαβδ (p ⃗q ) .

126 The transverse components of the metric fluctuations near I +are given by

( ) 1 μ ν out Czz(u, z, z) ≡ κ lim ∂zx ∂zx h u + r, rbx(z, z) . (5.11) r→∞ r μν

The large-r saddle-point approximation yields

∫ ∞ [ ( ) ) ] iκ − † ( , , ) = − εb out b iωqu − out( b iωqu , (5.12) Czz u z z 2 zz dωq a− ωqx e a+ ωqx e 8π 0 where

1 2 εb = ∂ xμ∂ xνε+ (q) = . (5.13) zz r2 z z μν (1 + zz)2

Note that (5.12) is an intuitively plausible result since it states that the graviton field operator at a point (z, z) on the celestial sphere has an expansion in plane wave modes whose momenta are aimed towards that point.

The Bondi news tensor Nzz = ∂uCzz has Fourier components

∫ ∫ ω ≡ iωu ω ≡ iωu Nzz due Nzz , Nzz due Nzz . (5.14)

The zero mode of the news tensor is defined by

( ) [ ( ) ( ) ] (0) ≡ 1 ω −ω − κ b out b out b † N lim N + N = εzz lim ωa− ωx + ωa+ ωx . (5.15) zz 2 ω→0 zz zz 8π ω→0

127 Similarly, the first zero energy moment of the news is defined by

( ) ( ) [ ( ) ( ) ] (1) ≡ − i ω − −ω iκ b out b − out b † N lim ∂ω N N = εzz lim 1 + ω∂ω a− ωx a+ ωx . (5.16) zz 2 ω→0 zz zz 8π ω→0

S → (0) All of these quantities have nonvanishing -matrix insertions even as ω 0. The operator Nzz projects onto the leading Weinberg pole in the soft graviton theorem [6], so its matrix elements are tree-level exact and are given by

(0) κ (0)− ⟨out| N S |in⟩ = − εbzz lim ωS ⟨out| S |in⟩ , (5.17) zz 8π ω→0 n where

∑n − κ ω (z − z ) (0) = − ( + ) k k . Sn 1 zz − (5.18) 2ω (z zk)(1 + zkzk) k=1

This was precisely the result we obtained in chapter 4.

(1) O In a similar fashion, Nzz projects onto the subleading (1) term in the soft graviton theorem. At tree-level, its matrix elements are given by

( ) iκ − ⟨out| N 1 S |in⟩ = εb S(1) ⟨out| S |in⟩ , (5.19) zz 8π zz n

128 where

[ ] ∑n 2 − κ (z − z ) 2h S(1) = k k − Γzk h − ∂ + |s |Ω . n − − zkzk k zk k zk (5.20) 2 z zk z zk k=1

z 2 In this expression, Γzz is the connection on the asymptotic S , hk and hk are the conformal weights given by

1 1 h ≡ (s − ω ∂ ) , h ≡ (−s − ω ∂ ) , (5.21) k 2 k k ωk k 2 k k ωk

2 and Ωz is the corresponding spin connection. As was demonstrated in [72], (5.19) implies that insertions of the operator

∫ ww 4i γ ( ) T ≡ d2w D3 N 1 (5.22) zz κ2 z − w w ww into the tree-level S-matrix reproduce the Ward identity for a 2D conformal field theory:

[ ] ∑n h h 1 ⟨out|T S|in⟩ = k + k Γzk + (∂ − |s |Ω ) ⟨out|S|in⟩ . zz − 2 − zkzk − zk k zk (5.23) (z zk) z zk z zk k=1

Our goal in this chapter is to derive the one-loop exact IR divergent correction to (5.22). ( ) ( ) ( ) + − √ ± z z 2 , = , ± = ± 1 − The zweibein is e e 2γzz dz dz and Ω 2 Γzzdz Γ zzdz .

129 5.2 One-Loop Correction to the Subleading Soft Graviton Theorem

The matrix element (5.17) is exact because the leading Weinberg pole in the soft graviton expansion is uncorrected. On the other hand, the subleading theorem which governs the O(1) terms in the soft graviton expansion does have quantum corrections that modify the matrix element (5.19)[18].

These corrections are known to be one-loop exact, and arise from IR divergences in soft exchanges between external lines. Indeed, they must be present in order to cancel (within suitable inclusive cross-sections) IR divergences that arise from the Weinberg pole. The divergent part of this one- loop correction was derived in [18], which we will now review.

The loop expansion of the n-particle scattering amplitude is

∑∞ M M(ℓ) 2ℓ n = n κ , (5.24) ℓ=0 where we factored out the κ2 term that comes along with each additional loop.3 In dimensional regu- larization with d = 4 − ε, the divergent part of the one-loop n-point graviton scattering amplitude is universally related to the tree amplitude according to [74, 75]

M(1) σn M(0) n = n , (5.25) div ε

3 n−2 (ℓ) In addition, there is a factor of κ in each Mn due to the n external lines.

130 with

∑n 1 μ2 σ ≡ − (p · p ) log . (5.26) n 4(4π)2 i j −2p · p i,j=1 i j

( ) − The O ε 1 singularity is due exclusively to IR divergences because pure gravity is on-shell one- loop finite in the UV and has no collinear divergences. Using (5.25) and applying the tree-level soft theorem involving a negative-helicity soft graviton, we obtain

( ) → M(1)− −−→q 0 σn+1 (0)− (1)− M(0) (5.27) n+1 (q) Sn + Sn n . div ε

We would like to expand the above equation in powers of the soft energy ω. To proceed, we separate

σn+1 into two terms, one with the soft graviton momentum q and one without:

∑n 2 ′ ′ 1 μ σ + = σ + σ , σ ≡ − (p · q) log . (5.28) n 1 n n+1 n+1 2(4π)2 i −2p · q i=1 i

( ) ′ O O 0 Note that σn+1 = (ω) as the log ω term vanishes by momentum conservation, while σn = ω . ( ) We then find, up to O ω0 ,

( ) ′ ( ) → M(1)− −−→q 0 (0)− (1)− M(1) σn+1 (0)−M(0) − 1 (1)− M(0) n+1 Sn + Sn n + Sn n Sn σn n , (5.29) div div ε ε

(1)− where Sn in the last term acts only on the scalar σn. The anomalous term consists of the last two ( ) terms on the right-hand side of the above equation and is O ω0 . It is a universal correction to the

131 subleading soft theorem from IR divergences.

Thus far, we have been focusing on the IR divergent part of the one-loop amplitude. However, the one-loop amplitude also has a finite piece:

M(1) M(1) M(1) n = n + n . (5.30) div fin

It is expected from [18] that

( ) → M(1)− −−→q 0 (0)− (1)− M(1) (1)−M(0) n+1 Sn + Sn n + ΔfinSn n , (5.31) fin fin

(1)− (1)− where ΔfinSn is the one-loop finite correction to the negative-helicity subleading soft factor Sn .

Given that the subleading soft graviton theorem is one-loop exact [18], it follows that the all-loop soft graviton theorem is

[ ( ′ ( ) )] − q→0 − − σ − 1 − − M −−→ S(0) + S(1) + κ2 n+1 S(0) − S(1) σ + Δ S(1) M , (5.32) n+1 n n ε n ε n n fin n n where the terms in square brackets proportional to κ2 are the IR divergent and finite parts of the anomaly.

(1)− Little appears to be currently known about ΔfinSn . In all explicitly checked cases, including all identical helicity amplitudes and certain low-point single negative-helicity amplitudes, it was demonstrated that there are no IR finite corrections to the subleading soft graviton theorem [18, 19],

(1)− implying ΔfinSn = 0 for these cases. Nevertheless, we are unaware of any argument indicating

132 that this term always vanishes, or on the contrary that its form is universal. In the absence of such information, we will restrict our consideration to the universal divergent correction given in (5.29).

5.3 One-Loop Correction to the Energy-Momentum Tensor

5.3.1 Derivation of Main Result

The one-loop corrections (5.32) to the subleading soft factor are expected to result in corrections to the tree-level Virasoro-Ward identity (5.23). In this section, we show that this is indeed the case.

Moreover, we find that the effects of the universal divergent correction in (5.29) can be eliminated by a corresponding one-loop correction to the energy-momentum tensor. That is, whenever we have

(1)− ΔfinSn = 0, the shifted energy-momentum tensor obeys the unshifted Virasoro-Ward identity

(5.23).

(1) The tree-level matrix elements of the operator Nzz are given by (5.19). At one-loop level, the matrix elements acquire a divergent correction of the form

( ( )) 3 ( ) ′ ⟨ | (1)S| ⟩ iκ b σn+1 (0)− − 1 (1)− ⟨ |S| ⟩ out Nzz in = εzz lim 1 + ω∂ω Sn Sn σn out in . (5.33) div 8π ω→0 ε ε

It immediately follows from (5.22) that the IR divergent one-loop correction to the Tzz Ward iden-

133 tity is given by

⟨out|ΔTzzS|in⟩ ∫ (5.34) ww [ ( )] − κ 2 γ 3 b ′ (0)− − (1)− ⟨ |S| ⟩ = d w D εww lim (1 + ω∂ω) σ + S (S σn) out in , 2πε z − w w ω→0 n 1 n n where

⟨ | S| ⟩| ≡ ⟨ | S| ⟩ out Tzz in div out ΔTzz in . (5.35)

It is far from obvious, but nevertheless possible, to rewrite this in terms of the zero modes of the

Bondi news. This computation is done explicitly in the next subsection, and we find that ΔTzz can be expressed as

∫ ww ( ( )) 2 γ ( ) ( ) ΔT = − d2w 2N(0)D N 0 + D N(0)N 0 . (5.36) zz πκ2ε z − w ww w ww w ww ww

Hence, the shifted energy-momentum tensor, given by

e Tzz = Tzz − ΔTzz , (5.37)

obeys the unshifted Ward identity

[ ] ∑n e hk hk 1 ⟨out|T S|in⟩ = + Γzk + (∂ − |s |Ω ) ⟨out|S|in⟩ zz − 2 − zkzk − zk k zk (5.38) (z zk) z zk z zk k=1

(1)− to all orders, whenever ΔfinSn = 0.

134 This result seems interesting for a number of reasons. First of all, note that while the renormalized soft factor contains logarithms and explicit dependence on the renormalization scale, such terms do not appear in the anomalous contribution to the energy-momentum tensor. Furthermore, the fact that the divergence takes the form of a matrix element involving only the local operators

(0) (0) Nww (w) and Nww (w) allows us to perform an “IR renormalization” of the operator Tzz by sub- tracting away the divergent operator. The form of the divergence, when rewritten in terms of the soft graviton operators, is reminiscent of the forward limit of a scattering amplitude. However, it remains to be seen whether or not there are finite corrections to the Ward identity (5.38) and, if so, whether or not they can be eliminated by a further finite shift of the energy-momentum tensor.

5.3.2 Explicit computation

In this subsection, we fill in the missing steps in going from (5.34) to (5.36) and explicitly compute the matrix elements of ΔTzz given in (5.34) by

⟨out|ΔTzzS|in⟩ ∫ (5.39) ww [ ( )] − κ 2 γ 3 b ′ (0)− − (1)− ⟨ |S| ⟩ = d w D εww lim (1 + ω∂ω) σ + S (S σn) out in . 2πε z − w w ω→0 n 1 n n

For convenience, we recall the expressions for the leading and subleading tree-level soft factors:

∑n μ ν ± ∑n ± μ ± κ p p εμν(q) ± iκ εμν(q)p qλ (0) = k k , (1) = − k J λν . Sn · Sn · k (5.40) 2 pk q 2 pk q k=1 k=1

135 (1)− J Since Sn acts on a scalar in (5.39), the action of kμν is given by

[ ] ∂ ∂ J σ = −i p − p σ . (5.41) kμν n kμ ∂ ν kν μ n pk ∂pk

Using (5.40), (5.41), and momentum conservation, it follows that

[ ( )] (1)− ≡ 1 ′ (0)− − (1)− ΔdivSn σn+1Sn Sn σn ε [ ] ∑n − 2 2 κ (p · ε ) pj · q − − μ = i (p · q) log − (p · ε )(p · ε ) log . 4(4π)2ε p · q j p · p i j −2p · p i,j=1 i i j i j

(5.42)

Momentum conservation implies that (5.42) is independent of both the soft energy ω and the renormalization scale μ. It follows that (5.39) becomes

∫ ww ( ) κ γ − ⟨out|ΔT S|in⟩ = − d2w D3 εb Δ S(1) ⟨out|S|in⟩ . (5.43) zz 2π z − w w ww div n

Before proceeding, it is useful to define the quantity

√ 2 εb ≡ ∂ bxi(w)ε+(q(w)) = , (5.44) w w i 1 + ww

136 so that εbww = εbwεbw. It is then straightforward to show that

( ) 2 b − Dw εwε = 0 ,

2 Dwq = 0 , (5.45) ( ) · − 2 w (pi ε ) (2) D Dw εbww = −2πωiδ (w − zi) , pi · bq where qμ = ωbqμ. Using the first of the above identities, we have

( ) 2 3 b · − · − μ Dw εww(pi ε )(pj ε ) log = 0 , (5.46) −2pi · pj which implies

( ) ( ) ∑n − 2 − κ (p · ε ) pj · q D3 εb Δ S(1) = D3 εb i (p · q) log . (5.47) w ww div n 4(4π)2ε w ww p · q j p · p i,j i i j

To evaluate this, we distribute the covariant derivatives via the product rule and first compute the term

( ) ∑n − 2 (p · ε ) pj · q D3 εb i (p · q) log w ww p · q j p · p i,j=1 i i j (5.48) ∑n [ ] pj · bq = −2πγ ω (p · bq)D δ(2)(w − z ) + 3(p · ∂ bq)δ(2)(w − z ) log , ww i j w i j w i p · bp i,j=1 j i

137 where we have used the last two identities of (5.45) along with momentum conservation. Similarly, using (5.45) and momentum conservation, we find

[ ( ) ( )] ∑n − 2 (p · ε ) pj · q 3 D D εb i (p · q) D log w w ww p · q j w p · p i,j=1 i i j [ ] ∑n − 2 2 (p · ε ) (pj · ∂wq) = 3 D εb i , (5.49) w ww p · q p · q i,j=1 i j and

( ) ∑n − 2 ∑n − 2 (p · ε ) pj · q (p · ε ) 2 εb i (p · q)D3 log = εb i (p · ∂ q)3 . (5.50) ww p · q j w p · p ww p · q (p · q)2 j w i,j=1 i i j i,j=1 i j

Finally, using momentum conservation and the relationship between the soft momenta and polarization vectors [73] ( ) + 1 b εμ = ∂w √ qμ , (5.51) γww we find ( ) ∑n 2 ∑n + 2 pj · ∂wbq (pj · ε ) = εb . (5.52) p · bq ww p · bq j=1 j j=1 j

Substituting (5.48), (5.49), and (5.50) into (5.47), and then using (5.17) along with (5.52), we find

∫ ww ⟨ [ ( )] ⟩ 2 γ ( ) ( ) ⟨out|ΔT S|in⟩ = − d2w out −2N 0 D N(0) + 3D N(0)N 0 S in zz πκ2ε z − w ww w ww w ww ww ∫ ww ⟨ [ ( )] ⟩ 2 γ ( ) ( ) = − d2w out 2N(0)D N 0 + D N(0)N 0 S in , πκ2ε z − w ww w ww w ww ww

(5.53)

138 which is precisely (5.36).

139 6 Conclusion

Since the discovery of the infrared triangle, there has been a flurry of research exploring the deep infrared structure of both gravity and gauge theories. In this thesis, we have concentrated on studying the equivalence between leading soft theorems and asymptotic symmetries in several types of theories in four dimensions. We began with QED, the simplest case conceptually, and showed that the Ward identity associated to the asymptotic symmetry group of QED is simply the soft photon

140 theorem. We then explored the nonabelian generalization of this and showed that the tree-level soft gluon theorem is nothing but the Ward identity for the asymptotic symmetry group of nonabelian gauge theory. Furthermore, we supersymmetrized QED and showed that the soft photino theorem of N = 1 supersymmetric QED implies the existence of an infinite-dimensional fermionic symmetry in the theory. This is particularly surprising, since whereas the asymptotic symmetry group in

QED and nonabelian gauge theory is a subgroup of the gauge symmetries of the theory, the number of supersymmetry generators in rigid N = 1 QED is finite. Therefore, this newly discovered infinite- dimensional fermionic symmetry cannot possibly be a symmetry of the Lagrangian. The nature of this symmetry is still a subject of active research.

After exploring the various gauge theories described above, we moved on to demonstrate the equivalence between the leading soft theorem and the Ward identity associated to the BMS supertranslations in chapter 4. It was not until the last chapter that we moved beyond leading soft theorems and considered the consequences of the subleading soft graviton theorem. This theorem was only recently discovered, and its equivalence to the superrotations of the extended BMS group was discovered almost immediately afterwards. Superrotations generate a Virasoro symmetry on the

4D scattering amplitudes in asymptotically flat spacetimes, thereby suggesting that there may be a

2D CFT that describes such scattering amplitudes. Such a candidate CFT would include an energy- momentum tensor, whose insertion generates the Virasoro symmetry, and we were able to provide evidence for its existence by determining the one-loop exact IR divergent portion of such an operator.

Despite our lack of emphasis on the subleading (and sub-subleading) soft theorems, they are

141 currently the focus of an extensive research enterprise, partially because such theorems were not discovered until fairly recently. The miracle of the infrared triangle is that given any theory in any dimension, we are able to garner information about any corner once we have discovered one corner.

Because for many theories, the subleading soft theorem is the most straightforward to establish, the infrared triangle has allowed physicists to discover new symmetries of old theories, e.g. see [53].

While the subleading soft graviton theorem corresponds to the previously conjectured Virasoro symmetry of 4D scattering amplitudes in asymptotically flat spactimes, many of these new theories do not have such straightforward interpretations, and it is a subject of active research to discover the meaning of these asymptotic symmetries, e.g. see [76, 77]. In fact, as we mentioned in chapter

3, even the asymptotic symmetries associated to the leading soft photino theorem in 4D N = 1 rigid supersymmetric QED do not currently have a well-understood interpretation in terms of a symmetry of the Lagrangian.

Furthermore, the infrared triangle has also been useful in studying theories with spacetime dimensions greater than four. For instance, it was a longstanding belief by many that the asymptotic symmetry group of asymptotically flat spacetimes with dimension d > 4 is not the BMS group, but rather the Poincare group [78, 79]. The reason is because the asymptotic symmetry group consists of the transformations that preserve the boundary conditions of the fields while changing the boundary data, and for the boundary conditions chosen for these spacetimes with d > 4 spacetime dimensions, the supertranslations were not able to preserve the boundary condition. This is particularly strange from the viewpoint of the infrared triangle, since the soft graviton theorem exists in arbitrary dimensions, so the corresponding asymptotic symmetries should as well. This discrepancy

142 was resolved in [80], where it was shown that if the boundary conditions are chosen appropriately, then the supertranslations survive in the higher-dimensional gravitational theories. This has led us to better understand the asymptotic structure of these higher-dimensional gravitational theories.

As was already mentioned in the introduction, the potential for a correspondence between 4D scattering amplitudes in asymptotically flat spacetimes and correlators of a 2D CFT is particularly exciting since it may provide the tools necessary to probe quantum gravity in our physical universe. Perhaps just as interesting is the possibility for the correspondence between soft theorems and asymptotic symmetries to resolve the black hole information paradox. The black hole information paradox essentially arises from the “no hair theorem”, which states that up to diffeomorphisms, the only conserved charges of an electrically neutral black hole are the Poincare charges – its energy, linear momentum, angular momentum, and boost charges. Therefore, regardless of how many conserved charges an object has before being thrown into a black hole, all information about it except the Poincare charges is lost once it falls into the black hole. However, as was emphasized throughout this thesis, progress in understanding the infrared structure of gravity in recent years has demonstrated that certain diffeomorphisms, such as the supertranslations and superrotatations, are in fact physical and hence should not be neglected. Such charges have been dubbed “soft hairs” due to their connection with the soft gravitons arising from the supertranslation and superrotation charges, and they are at the center of attention as physicists scrutinize the possibility of if, and how, the soft modes may account for all the information stored in a black hole [66, 81]. It is true that the soft modes may ultimately still fail to account for all the information stored in the black hole, but the fact that they are offering fresh insight into one of the oldest problems in modern theoretical physics

143 suggests that there is still much to learn. This thesis has mapped only the the initial few steps in this journey towards understanding the infrared triangle in a variety of gravity and gauge theories, and

I hope that this is sufficient to demonstrate the vastness of unexplored territory in infrared physics still waiting for us to explore and conquer.

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150 his thesis was typeset using LATEX, originally developed by Leslie Lamport Tand based on Donald Knuth’s TEX. The body text is set in 11 point Egenolff-Berner Garamond, a revival of Claude Garamont’s hu- manist typeface. The above illustration, Science Experiment 02, was created by Ben Schlitter and released under cc by-nc-nd 3.0. A template that can be used to format a PhD dissertation with this look & feel has been released under the permissive agpl license, and can be found online at github.com/suchow/Dissertate or from its lead author, Jordan Suchow, at [email protected].

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