Semiclassical Virasoro Symmetry of the Quantum Gravity S-Matrix

It is shown that the tree-level S-matrix for quantum gravity in four-dimensional Minkowski space has a Virasoro symmetry which acts on the conformal sphere at null infinity.


Introduction
BMS + transformations [1] comprise a subset of diffeomorphisms which act nontrivially on future null infinity of asymptotically Minkowskian space times, or I + . BMS − transformations act isomorphically on past null infinity, or I − . A particular 'diagonal' subgroup of the product group BMS + ×BMS − has recently been shown [2] to be a symmetry of gravitational scattering. Ward identities of this diagonal symmetry relate S-matrix elements with and without soft gravitons. These S-matrix relations are not new [3]: they comprise Weinberg's soft graviton theorem [4]. More generally, the connection to soft theorems provides a new perspective on asymptotic symmetries in Minkowski space [5].
Over the decades a number of extensions/modifications to the BMS group have been proposed: e.g. the Newman-Unti group [6], the Spi group [7] and the extended BMS group [8,9]. A criterion is needed to decide whether or not such extensions are 'physical'. Here we adopt the pragmatic approach that a Minkowskian asymptotic symmetry is physical if and only if it provides nontrivial relations among S-matrix elements. We will view these S-matrix relations as a definition of the symmetry.
In this paper we will show that, at tree-level, quantum gravity in asymptotically Minkowskian spaces in this sense has a physical Virasoro symmetry. The symmetry is implied by a recently proven soft theorem [10] and acts (diagonally) on the conformal S 2 at I.
Our story begins with a conjecture of Barnich, Troessaert and Banks (BTB) [8,9]. BMS + has an SL(2, C) Lorentz subgroup generated by the six global conformal Killing vectors (CKVs) on the S 2 at I + . Locally, BTB showed that all of the infinitely many CKVs preserve the same asymptotic structure at I + and are hence also candidate asymptotic symmetry generators. This larger set of vector fields was a priori excluded in the original work of BMS, who demanded that they be nonsingular everywhere on S 2 . This restriction cuts the Virasoro group down to a mere SL (2, C). BTB conjectured that the true asymptotic symmetry group of I + is the 'extended BMS + group' generated by all CKVs. However it has not been clear if or in what sense the singular CKVs truly generate physical asymptotic symmetries.
Herein we consider, in the spirit of [2], a certain diagonal subgroup of (extended BMS + )×(extended BMS − ), denoted X . Ward identities are derived for a Virasoro subgroup of X . They are found to involve a soft graviton insertion with the Weinberg pole projected out, leaving the finite subleading term in the soft expansion. These Ward identities are in turn shown to be implied by a conjectured [11] soft relation schematically of the form lim ω→0 M n+1 = S (1) M n . (1.1) Here M n+1 is an n + 1-particle amplitude with a certain (pole-projected) energy ω soft graviton insertion, and S (1) involves the soft graviton momentum as well as the energies and angular momenta of the incoming and outgoing particles. Details are given below. The proof [10] of (1.1) for tree-level gravity amplitudes then implies a semiclassical Virasoro symmetry for the case of pure gravity. This demonstrates that the singularities in the generic CKVs do not, at least in this context, prevent them from generating physical symmetries. 1 One might also hope to run the argument backwards and see to what extent the Virasoro symmetry of the S-matrix implies the soft relation (1.1). In the case of supertranslations, the argument can be run in both directions [3]. However here we encounter several obstacles, including the need for a prescription for handling the CKV singularities and some zero mode issues. We leave this to future investigations. 2 Hence at this point the existence of a Virasoro symmetry is potentially a weaker condition than the validity of the soft relation (1.1). 1 It may alternatively be possible to reach this conclusion without appealing to direct computations such as in [10] by carefully regulating the singularities and analyzing their effects. We do not attempt such an analysis herein. 2 The Virasoro charges constructed in [8] may be useful for this purpose.
The analysis of [2,3] related two structures which have been well-established and thoroughly studied over the last half-century: BMS symmetry and Weinberg's soft graviton theorem. Here the situation is rather different. We are relating two unestablished and understudied structures: asymptotic Virasoro symmetries and subleading soft graviton theorems.
We hope the relation will illuminate both. In any case it is a rather different enterprise! An important issue which we will not address is the quantum fate of the semiclassical Virasoro symmetry. Here the situation is currently up in the air. In [12,13] it was shown that, in a standard regulator scheme, (1.1) receives IR divergent quantum corrections (at one loop only), which also make the S-matrix ill-defined in this scheme. However in [14], the factor S (1) in (1.1) relating the 5 and 4 point amplitude was found to remain uncorrected at one loop in a scheme with the soft limit taken prior to removing the IR cutoff. 3 In the recent work [17,18] (see also [19]) it was shown that a properly defined S-matrix utilizing the gravity version of the Kulish-Faddeev construction [20] is free of all IR divergences.
This may be the proper context for the discussion, as it is hard to have a symmetry of an S-matrix without an S-matrix! Should it ultimately be found that (1.1) does receive scheme-independent corrections, one must then determine whether it implies a quantum anomaly in the asymptotic Virasoro symmetry (which is potentially weaker than (1.1)), or a quantum deformation in its action on the amplitudes. Clearly highly relevant, but not yet fully incorporated into this discussion, is the low-energy theorem of Gross and Jackiw [21] who use dispersion theory to show that there is no correction to the first three terms 4 of the Born approximation to soft graviton-scalar scattering. This generalized the classic lowenergy theorem for QED by Low [22]. Progress on the gravity version was recently made by White [19]. Clearly, there is much to understand!  [24,25]. Since the symmetry 3 [14] claims a result only for this one special case by direct computation. However, it has been suggested [15] that, using [16], a proof can be constructed in the scheme of [12,13] that all loop corrections to S 1 in (1.1) are linked to discontinuities arising from infrared singularities and hence in the scheme of [14] (with the soft limit taken first) all loop corrections would disappear along with the discontinuities. 4 S (0) , S (1) and S (2) in the notation of [14]. 5 They may not be the familiar ones from the study of unitary 2D CFT on the sphere. acts at the boundary, it is likely relevant to any holographic duality as long ago envisioned in [26].
This paper is organized as follows. Section 2 establishes notation and reviews a few salient formulae for asymptotically flat geometries. Section 3 describes the conjectured extended BMS ± symmetry following [8]. In section 4 we define the diagonal subgroup X of (extended BMS + )×(extended BMS − ) transformations, review Christodoulou-Klainerman (CK) spaces and define extended CK spaces by acting with X . A prescription is given to define classical gravitational scattering from I − to I + and shown to be symmetric under X . In section 5 the discussion of the quantum theory begins with the the action of extended BMS ± generators on in and out states. A Ward identity is then derived which is equivalent to infinitesimal Xinvariance of S. It relates amplitudes with and without a particular soft graviton insertion.
Finally in section 6 we give the detailed form of the soft relation (1.1) and show that it implies the X Ward identity.

Metrics
A general asymptotically Minkowskian metric can be expanded in 1 r around I + . In retarded Bondi coordinates it takes the form 6 where the first line is the flat Minkowski metric, γ zz (D z ) is the round metric (covariant derivative) on the unit S 2 and The Bondi mass aspect m B , the angular momentum aspect N z and C zz depend only on (u, z,z) and not r. The outgoing news tensor is defined by I + is the null surface (r = ∞, u, z,z). We use the symbol I + + (I + − ) to denote the future (past) boundary of I + at (r = ∞, u = ∞, z,z) ((r = ∞, u = −∞, z,z)). This is depicted in There is an analogous construction on I − with the metric given by The I − coordinate z in (2.4) is antipodally related to the I + coordinate z in (2.1) in the sense that, for flat Minkowski space, a null geodesic begins and ends at the same value of z.
Put another way, in the conformal compactifcation of asymptotically flat spaces, all of I is generated by null geodesics which run through spatial infinity i 0 . These generators have the same constant z value on both I + and I − . The incoming news tensor is defined by When expanding about flat Minkowski space we sometimes employ flat coordinates in which the flat metric takes the form These are related to Bondi coordinates in flat space by

Constraints
The data in (2.1) are related by the constraint equations G µν = T M µν , where T M µν is the matter stress tensor and we adopt units in which 8πG = 1. The leading term in the expansion of is the rescaled matter stress tensor which we have assumed falls of like 1 r 2 near I + . The G uz constraint gives Given the Bondi news, m B , N z and C zz are all determined up to u-independent integration constants which are discussed below. The I − constraints are

Extended BMS ± transformations
The extended BMS + group has been proposed [8,9] as the asymptotic symmetry group at I + of gravity on asymptotically flat spacetimes. It is generated by vector fields ξ + that locally preserve the asymptotic form (2.1) of the metric at I + All such vector fields near I + are of the form where f + is an arbitrary function on S 2 and here and elsewhere we suppress (in some cases metric-dependent) terms which are further subleading in 1 r and irrelevant to our analysis: see [8] for a recent treatment specifying these terms. Y + must be a conformal Killing vector on S 2 which obeys the equation Globally there are six real vectors fields in an antisymmetric matrix Y z µν obeying (3.3): These generate the six Lorentz boosts and rotations on I + . Locally there are infinitely many solutions of the form Y z ∼ z n with poles somewhere on the sphere. In their original work [1], BMS excluded these singular vector fields. However in this paper we shall explore the conjecture of [8,9] that all of these 'superrotations' should be included as part of the asymptotic symmetry group.
The extended BMS + group is a semi-direct product of superrotations with supertranslations. The supertranslations were recently analyzed in [2,3]. For notational brevity we henceforth consider only the superrotation subgroup which has f + = 0 in (3.2) and reduces to This maps I + to itself via Similarly on I − we have BMS − vector fields parametrized by Y − Infinitesimal BMS + transformations act on the Bondi-gauge metric components as  In this paper we are interested in extended BMS + ×BMS − transformations. We denote by X the subgroup of these transformations generated by vector fields asymptotic to (ξ + , ξ − ) on (I + , I − ) subject to (4.1), where now Y z is any of the infinitely many conformal Killing vectors on the sphere. Elements of X transform a solution (D zz , C zz ) of the gravitational scattering problem to a new one (D zz , C zz ) with different final and initial data. We will argue below that the new data is a new solution of the scattering problem.

Christodoulou-Klainerman spaces
We are interested in asymptotically flat solutions of the Einstein equation which revert to the vacuum in the far past and future. In particular we want to remain below the threshold for black hole formation. We will adopt the rigorous definition of such spaces given by Christodoulou and Klainerman (CK) [27] who also proved their global existence and analyzed their asymptotic behavior.
CK studied asymptotically flat initial data in the center-of-mass frame on a maximal spacelike slice for which the Bach tensor ε ijk D kl of the induced three-metric decays like r −7/2 (or faster) at spatial infinity and the extrinsic curvature like r −5/2 . This implies that in normal coordinates about infinity the leading part of the three-metric has the (conformally flat) Schwarzschild form, with corrections which decay like r −3/2 . CK showed that all such initial data which moreover satisfy a global smallness condition give rise to a global, i.e. geodesically complete, solution. We will refer to these solutions as CK spaces.
The smallness condition is satisfied in a finite neighborhood of Minkowski space, so this result established the stability of Minkowski space. Moreover many asymptotic properties of CK spaces at null infinity were derived in detail, see [28] for a summary. Here we note that the Bondi news N zz vanishes on the boundaries of I + as where G is Newton's constant and M is the ADM mass. Similar results pertain to I − .
In this paper we consider generalizations of pure gravity which include coupling massless matter which dissipates at late (early) times on I + (I − ) so that the system begins and ends in the vacuum. The CK analysis has not been fully generalized to this case, although there is no obvious reason analogs of (4.2)-(4.6) might not still pertain to a suitably defined neighborhood of the gravity+matter vacuum. In the absence of such a derivation (4.2)-(4.6) will simply be imposed, in the matter-coupled case, as restrictions on the solutions under consideration.

Classical gravitational scattering
The classical problem of gravitational scattering is to find the outgoing data at I + resulting from the evolution of given data on I − . We take the incoming data to be D zz (v, z,z) and the outgoing data to be C zz (u, z,z). The remaining metric components on I are then determined by constraints. We consider the geometries in the neighborhood of flat space defined by CK, which have m B = 0 (m − B = 0) at I + + (I − − ). In particular we remain below the threshold for black hole formation.
A CK geometry, as described in (t, r, θ, φ) coordinates, does not quite provide a solution to this scattering problem. To find the in (out) data, one must perform a coordinate transformation to ingoing (outgoing) Bondi coordinates and determine D zz (C zz ). This procedure is not unique: the coordinate transformations are ambiguous up to extended BMS ± transformations on I + or I − . D zz and C zz are not invariant under these transformations.
Hence a solution of the scattering problem requires a prescription for fixing this ambiguity.
A prescription to fix this ambiguity is to demand that It was shown in [2] that the falloffs (4.2)-(4.6) imply this is always possible. One may then integrate the constraint equations to determine D zz and C zz , which will not in general vanish at I + + and I − − . This prescription does not give all near-flat solutions of the scattering problem. Indeed, all such solutions are in the center-of mass frame and have vanishing ADM three-momentum.
However, given any solution of the scattering problem obeying (4.7), a new one with nonzero three-momentum may be obtained simply by acting with the boost element of X . More generally, our prescription to define gravitational scattering is to take all solutions obtained by doing arbitrary X transformations on the solutions obeying (4.7). We shall refer to such scattering geometries, complete with I ± data, as extended CK spaces. Acting with an arbitrary finite conformal transformation w(z) followed by an arbitrary finite supertranslation f on (4.7) leads to the asymptotic behaviors for large negative u and positive v 7 We also have the relations at all the boundaries of I ± ∂zN zz | I + ± = 0, = −2(∂ w z) 1/2 ∂ 2 w (∂ w z) −1/2 , which is the transformation law for a 2D CFT stress tensor. 8 Explicit expressions for the proper BMS ± charges as integrals of fields on I were worked out in detail in [3] and shown to generate the proper BMS ± symmetries. Expressions for the Virasoro charges Q ± are given in [8], but were not shown to generate the symmetries. In this paper such explicit expressions will not be needed: transformation laws for the states suffice.
Then the hard action is simply to act with ξ −µ ∂ kµ on each scalar particle

4)
Here −(1 + E k ∂ E k ) arises from the Fourier transform of v∂ v , and the coefficient of D z Y −z is shifted by one half as in [5] due to the r∂ r term in (3.7). For spinning particles we must To determine Q − S , note that the inhomogeneous transformation of the incoming Bondi news M zz is The action of Q − S on a state must implement this shift. It follows that Using the commutator [26] [ one concludes that, up to a total derivative commuting with M zz , This reproduces the linear term in the full expression for the charge given in [8]. 10 Q − S is a zero-frequency operator (because of the v integral) linear in the metric fluctuation.
Acting on the in -vacuum, it creates a soft graviton with polarization tensor proportional to D 3 z Y −z . The explicit form of the momentum space creation operator will be constructed below in subsection 5.3. Altogether then Q − maps the n-particle states into themselves plus an n-hard+1-soft state: More explicitly if we have a particle of helicity h, and Rindler energy −iv∂ v = E R , the parentheses in (5.4) are of the form Y z ∂ z + Yz∂z + h R D z Y z + h L DzYz where for helicity h, the 'conformal weights' (see e.g. [5] . 10 The formula in [8] differs by a total derivative which improves the large |v| behavior and may be essential in a more general context. The slightly simpler expression here is sufficient for the present purpose.
Similarly Virasoro transformations on I + are decomposed as and we denote out-states comprised of m particles with energies E k outgoing at points z k for k = n + 1, ...n + m by z n+1 , z n+2 , ...|. (5.11) One finds where

X -invariance of S
In this section we derive a quantum Ward identity from the assumption that X -invariance survives quantization. The quantum version of infinitesimal X invariance of classical gravitational scattering is, using (4.
where the k sum now runs over both in and out particles and again for spinning particles the Lie derivative replaces the ordinary one on the right hand side. This relates the derivatives of any S-matrix element to the same S-matrix element with a particular soft graviton insertion.
where θ is the angle between x and q. This integral is dominated for large r by the contribution near θ=0: Here,x is parameterized by (z,z) x ≡ (5.26) Define: Then from the large r saddle point expansion of (5.23), we have: with ω > 0 in both cases. We define N (1) zz as: (5.29) A mode expansion analogous to (5.24) can be defined for Dzz on I − which is out|S|in with soft graviton insertions. 12 Such insertions generically have Weinberg poles behaving as 1 ω . However the prefactor 1 + ω∂ ω projects out this pole, leaving the subleading O(ω 0 ) soft factor. Equation (5.33) and its hermitian conjugate are related to the Q S matrix element by out| : Q S S : |in Here J kλν ≡ L kλν + S kλν is the total ingoing orbital+spin angular momentum of the k th particle which obeys the global conservation law J kλν = 0. We note the (1+ω∂ ω ) prefactor on the left hand side projects out the would-be Weinberg pole accompanying a soft insertion. 12 Here we assume that |in and out| states contain no soft gravitons. 13 A single soft graviton insertion has the ω expansion For notational brevity we consider the contribution for negative polarization: the general formula has an S (1) with a general polarization tensor replacing (6.2). We will show that (6.1) implies the Ward identity (5.16), which in turn is equivalent to infinitesimal X -invariance of the S-matrix. Although the relation (6.1) potentially has wider validity, the only case in which it is known to be a theorem is tree-level gravitons [10]. Hence only for this case do we claim the results of this section imply a Virasoro symmetry.
Gauge invariance provides an important check on this formula. Amplitudes must vanish for pure gauge gravitons with polarizations ε µν Λ = q µ Λ ν + q ν Λ µ (6.3) for any Λ. Inserting this into (6.2) we find The first terms vanishes by global angular momentum conservation, while the second vanishes by antisymmetry of J kµν . This is very similar to the gauge invariance of the Weinberg pole, which vanishes due to global energy-momentum conservation or equivalently translational symmetry. The Weinberg soft theorem implies that this global translational symmetry is promoted to a local supertranslational symmetry on the sphere [3], because there is one symmetry for every angle q. In this section we will see a parallel story for rotational invariance: the soft relation (6.1) implies that rotations are promoted to a local superrotationalequivalently Virasoro -symmetry on the sphere.

(6.5)
One then finds for the orbital terms