BMS Symmetry of Celestial OPE

In this paper we study the BMS symmetry of the celestial OPE of two positive helicity gravitons in Einstein theory in four dimensions. Celestial OPE is obtained by Mellin transforming the scattering amplitude in the (holomorphic) collinear limit. Collinear limit gives the leading term of the celestial OPE. We compute the first subleading correction to the OPE by analysing the four graviton scattering amplitude directly in Mellin space. The subleading term can be written as a linear combination of BMS descendants with the OPE coefficients determined by BMS algebra and the coefficient of the leading term in the OPE. This can be done by defining a suitable BMS primary state. We find that among the descendants, which appear in the first subleading order, there is one which is created by holomorphic supertranslation with simple \emph{pole} on the celestial sphere.


Introduction
Operator product expansion (OPE) plays a central role in the non-perturbative formulation of conformal field theory. OPE is the statement that when two primary operators φ i and φ j come close to each other (inside a correlation function) we can replace the product φ i φ j by a sum over conformal families each of which contains a primary operator, say φ k and its descendants. In general, the OPE coefficient C k ij which multiplies the primary operator φ k , cannot be determined by conformal symmetry alone. But, once C k ij is specified, the coefficients of all the descendants of φ k are completely fixed by conformal invarince of the OPE. C k ij is known as the structure constant of the operator algebra. The goal of the present paper is to study this aspect of the OPE in case of Celestial Conformal Field Theory.
Celestial CFT is conjectured to be the holographic dual of quantum gravity in asymptotically flat space-time [4][5][6][7][8]. The observables of the celestial CFT are Mellin transformations of flat space scattering amplitudes [19][20][21][22][23][24][25][26]. Under Lorentz transformations, which act on the celestial sphere as global conformal group, Mellin amplitudes transform like correlation functions of a CFT. Now the correspondence between soft theorems [27,28] and Ward identities for asymptotic symmetries [29][30][31][32][33][34][35][36][37][38][39][40] show that the celestial CFT has, in fact, a much larger symmetry known as BMS [48,49]. BMS is an extension of the usual Poincaré group and consists of superrotations [32][33][34][35], which are local conformal transformations of the celestial sphere, and supertranslations, which are local angle-dependent space-time translations at null-infinity. Due to the presence of the supertranslations, the properties of the celestial CFT are somewhat different from usual CFT. For example, BMS algebra is not a direct product of holomorphic and antiholomorphic transformations because supertranslation generators have both holomorphic and antiholomorphic weights. As a result, at least naively, we do not expect holomorphic factorisation at the level of BMS representations. This is a major difference from usual CFTs.
A useful way to study various aspects of celestial CFT and representation theory of BMS algebra is through the construction of celestial OPE. OPE of two primary operators can be obtained by Mellin transformation of the collinear limit of flat space scattering amplitudes [1][2][3]. In the collinear limit, an (n+1) point function factorizes into an n point function times a universal splitting function [42][43][44]. By Mellin transformation of the splitting function one obtains the leading term in the celestial OPE and the structure constant of the celestial operator algebra. It is conceivable that the subleading terms in the OPE can be generated by Mellin transforming the subleading terms in the collinear expansion [45]. Now for celestial OPE what is remarkable is that one can obtain the structure constant by imposing the constraint coming from the subleading soft theorem in case of gluons and subsubleading soft theorem in case of gravitons [1]. This suggests that owing to an unusually large amount of global symmetry, algebraic techniques [1,[9][10][11][12] may play a crucial role in determining the structure of celestial correlation functions (or flat-space S-matrix elements). This, in particular, will require an understanding of BMS representation theory in the context of S-matrix theory or celestial amplitudes.
Motivated by this, in this paper we compute the first subleading correction to the (holomorphic) collinear limit directly in the Mellin space. The subleading terms in the collinear limit give the subleading terms in the celestial OPE. We focus on the tree level four graviton scattering amplitude in Einstein theory and compute subleading OPE of two positive helicity outgoing graviton primaries. Unlike in the case of 2-D CFT, the first correction to the leading order result contains the supertranslation descendant created by singular supertranslation of the form u → u + ǫ/z. We also show that the subleading OPE coefficients can be derived from the BMS algebra once we define a suitable notion of BMS primary state. This suggests the possibility that just like in the case of ordinary CFT, celestial OPE also organizes itself into representations of BMS algebra with the OPE coefficients of BMS descendants determined by BMS algebra. It will be very interesting to prove or disprove this in complete generality.

BMS Transformation
Let us now describe BMS transformations acting on a three dimensional space with coordinates (u, z,z) where u can be thought of as the retarded or Bondi time and (z,z) are the stereographic coordinates of the celestial sphere. At the end, when we derive the subleading OPE coefficients from the BMS algebra, we will restrict to the celestial sphere at u = 0.

Transformation of Fields
Under an infinitesimal conformal transformation (2.1), the primary field φ h,h (u, z,z) of weight (h,h) transforms as [8], Similarly for antiholomorphic transformation [8], For infinitesimal supertranslation given by (2.4) the transformation of the primary field is given by [8], At this point we would like to mention one useful point. From the transformation laws (2.7), (2.8) and (2.9) it is easy to check that if φ h,h (u, z,z) is a primary then so is (∂/∂u) n φ h,h (u, z,z), with weight (h + n/2,h + n/2).

Superrotation and Supertranslation Ward Identities
In celestial CFT correlation functions of the two dimensional primary operator φ h,h (z,z) are defined as Mellin transformation of flat space S-matrix elements [4,5], where ǫ i = ±1 for outgoing and incoming particles, respectively. The null momentum p(ω, z,z) is parametrised as, and σ denotes the helicity of the particle. Under (Lorentz) global conformal transformation, the L.H.S of (3.1) transforms as the correlation function of primary operators of weight (h i ,h i ), given by The action of global space-time translation on (3.1) was studied in [26]. The two dimensional field φ h,h (z,z) is the restriction of the three dimensional field φ h,h (u, z,z) to the u = 0 celestial sphere. Correlation function of the three dimensional fields φ h,h (u, z,z) is defined as [8], Under (Lorentz) global conformal transformation, the L.H.S of (3.4) transforms as, It is well known that Cachazo-Strominger subleading soft graviton theorem [28] is equivalent to the (superrotation) conformal Ward identity [35], where the stress tensor T (z) can be constructed as the shadow of the subleading soft graviton. In [35] the Ward-identity was derived for the two dimensional fields φ h,h (z,z), but the same derivation can be easily repeated for the fields φ h,h (u, z,z) and one obtains (3.6). For details please see the appendix. It is important to note that the stress tensor T (z) does not depend on the time coordinate u because it is constructed from a soft graviton and in the soft limit the time coordinate decouples.
The singular terms in the OPE between the stress tensor T (z) and the primary φ h,h (u, w,w) is given by [11], This is consistent with the transformation law (2.7) which one can check by using the standard 2-D CFT method. The OPE (3.7) gives the commutation relation, where the Virasoro generator L n is defined in the usual manner as, where c 0 is a contour around z = 0. Similarly, Weinberg's soft graviton theorem [27] is equivalent to the supertranslation Ward-identity given by [29,30,39], Here P (z) is the supertranslation current and can be written as, P (z) = − lim iλ→0 iλ∂G + ∆=1+iλ , where G + ∆ is the positive helicity graviton primary of weight ∆(= h +h). Again, P (z) has no u dependence.
The singular term in the OPE between P (z) and a matter primary φ h,h (u, z,z) is given by, This OPE is equivalent to the commutation relation, where the supertranslation generator P a,−1 , defined as [29], generates the holomorphic supertranslation u → u + ǫz a . The commutation relation (3.12) shows that, From the BMS commutation relation (2.6) one can check that the supertranslation generator P a,−1 has weight (−a − 1 2 , 1 2 ). So for a > −1 the holomorphic weight of P a,−1 is negative and it annihilates the primary operator φ h,h (0).

Supertranslation
Let us first consider the (holomorphic) supertranslation descendants.
Let us assume that the standard CFT form for the OPE between the supertranslation current P (z) and a matter primary φ h,h (u, z,z) holds, i.e, where the leading term is given by, From the conformal transformation laws (2.7) and (2.8) of primary fields, we know that i ∂ ∂u φ h,h (u, z,z) also transforms like a primary field of weight (h + 1/2,h + 1/2). We denote this field by The nonsingular terms of the OPE (4.1) define the (holomorphic) supertranslation which are new local fields created by singular The descendants can be defined by the usual contour integral formula, which follows from (4.1). Here c z is a contour around w = z. Later in the paper we will explicitly verify the existence of these descendants by taking the leading conformal soft limit of tree-level four graviton scattering amplitude in Einstein theory.
We now need to find out correlation functions with the insertion of the descendants . This can be computed in the standard way by using the Ward identity (3.10), and taking the limit w → z. In this limit we can use the OPE (4.1) and obtain, where the differential operator P −a,−1 (z) acting on correlation functions of primary operators is defined as, The rest of the (holomorphic) supertranslation descendants are of the form , where a, b > 1, will not appear in the OPE to the first subleading order and so we leave their discussion to future work. Mellin transform of graviton scattering amplitudes in string theory [25] is well defined without the time coordinate u. In this case the correlation function with the insertion of a holomorphic supertranslation descendant is given by a simple change of (4.7), where ǫ = ±1 for an outgoing and incoming particle, respectively and

Superrotation or Virasoro
The discussion of superrotation or Virasoro descendants is identical to that in 2-D CFT. The correlation function with the insertion of (L −n φ h,h )(u, z,z) is given by, This can be obtained by assuming the following OPE between the stress tensor T (z) and the primary field φ h,h (u, z,z),

OPE From Four Graviton Scattering Amplitude in Einstein Theory
In this section and the following we denote a graviton primary operator of scaling dimension For simplicity we focus on the four graviton tree-level scattering amplitude in Einstein theory, given by 1 where (1, 2) are incoming and (3, 4) are outgoing. We have also used the relations 2 where ǫ i = ±1 for an outgoing and an incoming particle, respectively. As in [1], we work in split signature so that we can treat z andz as independent real variables. It is also important that in split signature there is a non-zero three point function which is crucial for our purpose. In order to facilitate the (holomorphic) OPE expansion as z 3 → z 4 , withz 34 held fixed, we write the momentum-conserving delta function as, An n-graviton amplitude is multiplied by a factor of ( κ 2 ) n−2 where κ = √ 32πGN . To simplify the formulas we work in units where κ = 2. 2 We work in split signature and parametrize a null momentum p as p = ωq(z,z) = ω(1 + zz, z +z, z − z, 1 − zz). where Now we make a change of variable In terms of these new variables the Mellin amplitude can be computed easily and is given by, (5.7) In writing down this integral we have assumed the OPE limit, i.e, |z 34 | ≪ |z 13 |, |z 23 |, |z 14 |, |z 24 |. The details of the derivation are given in the Appendix.

Leading Term of the OPE
We can see from (5.7) that the expression inside the bracket multiplying the termz 34 z 34 is finite in the limit z 34 ,z 34 , u 34 → 0 and so we can Taylor expand around z 34 =z 34 = u 34 = 0. The leading term in the expansion is given by, (5.8) Now the three graviton amplitude in Mellin space, when there are two negative helicity incoming gravitons at 1 and 2 with weights ∆ 1 (= 1 + iλ 1 ) and ∆ 2 (= 1 + iλ 2 ) and one positive helicity outgoing graviton at 4 with weight ∆ 3 + ∆ 4 − 1(= 1 + iλ 3 + iλ 4 ), is given by, (5.9) Using (5.9) we can write the leading term (5.8) in the four point function as, where B(p, q) is the Euler beta function. This leading term corresponds to the leading term in the OPE given by, where ∆ i = 1+iλ i . In writing the last line of (5.11) we have used the fact that i ∂ ∂u 4 G + ∆ 3 +∆ 4 −1 (4) is a primary with weight (∆ 3 + ∆ 4 ). The leading answer (5.11) matches with [1].

Leading Conformal Soft Limit and Holomorphic Supertranslation Descendants
In the leading conformal soft limit iλ 3 → 0 [13-18, 24] the amplitude (5.7) simplifies and is given by, This can be rewritten as, where the three point function M 3 (1 − 2 − 4 + ) is now given by, Now we expand the R.H.S of (6.2) in powers of z 34 Using them we can write, This can be written in a more suggestive form as, where we have used (4.7) and the definition of the supertranslation current 3 We would like to emphasize that this equation is true only in the OPE limit z 3 → z 4 . (6.9) is essentially the OPE (4.1) between the supertranslation current P (z) and the graviton primary G + ∆ 4 (4), is defined in (5.9). In the above expression we have obtained the O(z 34 ) term by expanding the Dirac delta function appearing in the integrand in (5.7). We will now identify each term in the expansion as contribution coming from some descendant of the operator G + ∆ 3 +∆ 4 −1 (4) : 1) The first term can be written as, The second term can be written as, 3) Now we would like to point out one important point. The expression for the three point function M 3 (1 − 2 − 4 + ) contains the kinematic theta function Θ(z 42 /z 12 )Θ(z 14 /z 12 ) and so, when ∂/∂z 4 acts on it, it produces terms proportional to the delta functions δ(z 42 /z 12 ) and δ(z 42 /z 12 ). Now, as long as z 12 is finite, the delta function term is nonzero only if z 4 coincides with either z 1 or z 2 . But this is ruled out in the OPE limit and so we should set the (contact) terms, obtained by differentiating the theta functions, to zero. We have taken this fact into account in coming from the first line to the second line in (7.3).
3) The third term can be written as, The fourth term can be written as, Therefore at the level of 4-point function we get the following OPE, (7.6) We have boxed the descendant P −2,−1 G + ∆ 3 +∆ 4 −1 (4) just to emphasize the fact that it is not a Poincaré descendant. In the above equation we have not written an equality sign because some extra terms may be there which are not visible at the level of 4-point function. In the following section we will see that this is not the case.

OPE Coefficients From BMS Algebra
Let us start with the commutation relation between the supertranslation and (superrotation) Virasoro generators, In particular, for n = 0, this reduces to, We can see that the supertranslation generator P a,b has negative holomorphic or antiholomorphic scaling dimension for a > −1 or b > −1 or both. So, if φ h,h is a primary operator then the dimension of the operator [P a,b , φ h,h (0)] can be made arbitrarily negative by choosing a or b to be a large positive integer. This motivates us to set for a BMS primary operator φ h,h . Now, for generators P a,b with a > −1, b ≥ −1 or a ≥ −1, b > −1, condition (8.3) follows simply from the transformation law (2.4). But, there are generators P −a,b with a > 1, b > −1 or P a,−b with a > −1, b > 1, for which the condition (8.3) does not follow from the transformation law (2.4). They generate singular supertranslation on the holomorphic side and non-singular supertranslation on the antiholomorphic side and vice-versa. At this stage, except for the heuristic argument given below, the only justification for (8.3) for generators of this mixed type is that it gives the correct OPE coefficients, as we will see. The definition (8.3) of the BMS primary can be motivated by the following heuristic argument. Suppose the primary operator φ h,h (0) and (a subset of) its descendants 4 [P a,b , φ h,h (0)], appear in the OPE of two primaries φ h 1 ,h 1 (u, z,z) and φ h 2 ,h 2 (0). The operator [P a,b , φ h,h (0)] appears in the OPE with a prefactor proportional to Now, symmetry requires all descendants to be present in the OPE and this implies that the order of the pole in z orz cannot be bounded from above because a, b can be arbitrarily large positive integers. But, this is practically impossible because in the present situation the leading pole can be obtained by Mellin transforming the splitting function in the collinear limit [1][2][3]. So it seems reasonable to assume that the order of the pole in z orz will be bounded from above and therefore, we should impose the condition (8.3) on the primary φ h,h .
For the Virasoro generators we have the standard conditions that, and Therefore we can now consider the descendants generated by the raising operators only.

A Hilbert space picture
In the following discussion it will be convenient to assume a Hilbert space picture and define a vacuum |0 by the condition, We also define the state h,h as, using the above definitions the BMS-primary state condition can be written as, For the restricted class of supertranslation generators P a,b with both (a, b) > −1, condition (8.7) was also proposed in [41]. With this definition of the primary state the BMS descendants are given by, where n 1 ≥ n 2 ≥ .......n p > 0 and a i > 0, b i > 0. We do not put any order on the {a i } and {b i } because the supertranslation generators commute among themselves.

Primary descendants
Suppose h,h is a BMS-primary state. Then it is easy to check using the commutation relations that the state (P −1,−1 ) n h,h , with n ≥ 1, is also a BMS-primary with weight (h + n/2,h + n/2). We denote this state by, (P −1,−1 ) n h,h = h + n/2,h + n/2 (8.14)

Graviton-Graviton OPE
From now on our discussion will be confined to the OPE of two positive helicity gravitons denoted by G + ∆ (u, z,z). In our case, ∆ = 1 + iλ and so, At this point let us note that although the Mellin amplitudes in Einstein gravity are UV divergent, the OPE (7.6) has no singularity as u 34 → 0 and we can safely restrict the OPE to the celestial sphere at constant u, if we wish. From this point of view, the time coordinate u acts as a covariant regulator in the celestial CFT, which can be set to zero at the end of the calculation. For simplicity, we will do so. Once we set u = 0, modulo the singular prefactor, the OPE expansion becomes a Taylor series in z andz around the origin (z = 0,z = 0) and we can write the following (7.6), where G + ∆ 1 (z,z) = G + ∆ 1 (u = 0, z,z). Let us now enumerate the possible subleading terms in the OPE at O(z) and O(z). For this we remind ourselves that P −1,−1 is a (1/2, 1/2) operator. Taking this into account we get, 1) O(z) operators : 2) O(z) operators : If we keep powers of u then there will be an additional operator at O(u) given by (P −1,−1 ) 2 , which follows from the fact that u has scaling dimension (−1/2, −1/2). Now generically, at every order, there can be descendants and also new primaries. But here we focus on the contribution arising from a specific primary G + ∆ 1 +∆ 2 −1 (0). So let us write the first subleading order in the OPE as, Herec i is not the complex conjugate of c i . Also, for simplicity of notation, we have kept the dependence of the OPE coefficients (c i ,c i ) on the scaling dimensions of primary operators implicit.
We will now compute the coefficients (c i ,c i ) from the fact that both sides of the OPE must transform in the same way under BMS transformation. In order to do this it is more convenient to write the OPE as, where, The BMS generators act on the operators as, The first two relations are obtained by setting u = 0 in commutation relations (3.8). The last commutator is obtained from, by setting u = 0 and recongnizing that i∂ u φ h,h (u, z,z) is the BMS -primary (descendant) (P −1,−1 φ h,h )(u, z,z) with dimension (h + 1/2,h + 1/2). With this information one can readily compute the OPE coefficients, in the standard way, by applying the lowering operators to both sides of (8.20). Let us now state the equations for the OPE coefficients obtained in this way, 1) Applying P 0,−1 we get, 2) Applying P −1,0 we get, 3) Applying L 1 we get, This matches exactly with the OPE coefficients (7.6), obtained from graviton scattering amplitude, with the replacement λ 3 → λ 1 and λ 4 → λ 2 . We would like to point out that since the descendants P −2,−1 h 3 ,h 3 and P −1,−2 h 3 ,h 3 appear in the OPE (8.20), the above equations for the OPE coefficients probe the BMS algebra beyond the Poincaré subalgebra.
Before we conclude, we would like to point out that to arrive at the equations for the OPE coefficients we need commutators of the form, [L 1 , P −2,−1 ] h 3 ,h 3 = P −2,0 h 3 ,h 3 , which by the definition (8.10) of a primary state is equal to zero. The generator P −2,0 generates supertranslation of the mixed type, u → u + ǫz z . So we can see that the condition that generators of the mixed type should also annihilate a primary state is necessary to obtain the correct OPE coefficients starting from the BMS algebra, at least to the order we are working.

Virasoro Representations
Since Virasoro algebra is a subalgebra of the BMS algebra, one should be able to decompose a BMS representation into irreducible Virasoro representations. This is useful because in this way one can use the known results from the (highest-weight) Virasoro representation theory to compute some of the OPE coefficients. At the first subleading order this can be done in the following way.
Let us write the OPE at the first subleading order as, where we have setc 2 = 0 following (8.29). Among these states, the stateL −1 P −1,−1 h 3 ,h 3 is the antiholomorphic Virasoro descendant of the BMS primary P −1,−1 h 3 ,h 3 . We have two more states, given by L −1 P −1,−1 h 3 ,h 3 and P −2,−1 h 3 ,h 3 , one of which is a Virasoro descendant and the other one is a supertranslation descendant. So let us consider the state, The corresponding field can be written as, where the correlation function with the insertion of (P −2,−1 G h 3 ,h 3 )(z,z) is given by (4.9). Now, one can check using the BMS algebra and the definition of the BMS primary state h 3 ,h 3 that |ψ is a Virasoro primary but, not a BMS primary. For example, one can check that and so on. Now using the Virasoro primary |ψ we can rewrite the OPE (8.30) as, The OPE (8.34) now has the familiar structure of OPE in 2-D CFT. The first line consists of the Virasoro primary h 3 + 1/2,h 3 + 1/2 and its level 1 descendants. In the second line we have a new Virasoro primary |ψ with the (Virasoro) structure constant B(iλ 1 , iλ 2 )c 2 . The Virasoro structure constant B(iλ 2 , iλ 2 )c 2 cannot be determined by Virasoro algebra alone but, the coefficients c ′ 1 andc 1 are determined by Virasoro algebra. They are given by the known formulas, Using the values (8.21) for the scaling dimensions we get, from (8.36) and (8.37), that We can see that the value ofc 1 obtained in this way using Virasoro algebra, matches with the value (8.27) obtained using translational invariance of the OPE. Similarly, the value of c ′ 1 also matches with (8.35) once we substitute the values (8.26) and (8.28) of c 1 and c 2 , obtained using BMS algebra. This matching is a check of the overall consistency of the procedure.

Future Directions
The problem of extracting the celestial OPE from flat space scattering amplitudes consists of two parts. The first part is the (holomorphic or antiholomorphic) collinear expansion including subleading terms and the second part is the determination of the celestial correlation functions with the insertion of BMS descendants. The correlation functions involving superrotation or Virasoro descendants are well known from the work of Belavin-Polyakov-Zamolodchikov on two dimensional CFTs. The new objects are the supertranslation descendants. Among these, the simplest ones are the descendants {P −a,−1 φ , a > 1} created by singular (anti) holomorphic supertranslations. These are captured in a straightforward manner by supertranslation Ward-identity (3.10) following from Weinberg's soft graviton theorem. The correlation functions with the insertion of (anti) holomorphic supertranslation descendants follow from this Ward-identity and is given by (4.7) or (4.9). But, there are also descendants of the form {P −a,−b φ, a > 1, b > 1} created by supertranslations, which are neither purely holomorphic nor antiholomorphic. For example, if we want to go to higher order in the OPE expansion (8.19), then at O(zz) we encounter the descendant P −2,−2 G + ∆ 1 +∆ 2 −1 (0). Unless we know the Mellin amplitude with the insertion of this descendant, it will not be possible to extract the OPE at higher order. Although the Wardidentity (3.10) captures all the supertranslation descendants [29] -not just holomorphicit may require a more involved procedure to determine the correlation function of a general supertranslation descendant.
Another important point is that the BMS algebra may have (field-dependent) central extension [46,47]. In this paper the central extension does not play any role because in the first subleading order the Virasoro descendants {L −n φ, n > 1} do not appear, although they appear in the higher order of the OPE. The values of the OPE coefficients should depend on the central charge and perhaps one can determine the central charge by demanding that the OPE coefficients determined from the BMS algebra match with those obtained from the subleading terms in the collinear expansion of the Mellin amplitude. We hope to return to these problems in future.

Acknowledgements
SB would like to thank the participants of "Saha Theory Workshop 2020: Amplitudes and Correlators" where a part of this work was presented. SB would also like to thank

Mellin Transform of Four Graviton Amplitude and Holomorphic Collinear Expansion
In this appendix we discuss the Mellin transform of the tree-level four graviton scattering amplitude in Einstein theory in detail. The four graviton amplitude in momentum space is given by, where we have taken (1, 2) to be incoming and (3, 4) to be outgoing. We work in split signature so that we can treat z andz as independent real variables. The OPE limit, we are interested in, is z 3 → z 4 withz 34 held fixed. The corresponding Mellin amplitude is given by, (11.2) where ǫ i = ±1 for outgoing and incoming particles, respectively. Here we take (1, 2) to be incoming and (3,4) to be outgoing and so, ǫ 1 = ǫ 2 = −1 and ǫ 3 = ǫ 4 = 1.
Now, using the momentum conserving delta function we solve for ω 1 , ω 2 ,z 14 andz 24 and write, where N is a normalization constant which we will take to be 1 √ 8π 2 . If we do a formal Laurent expansion of a(p(ω, z,z), σ) in ω we will have a(ω, z,z, σ) = n S n (z,z, σ) ω n (11.15) One can easily check using the Lorentz transformation property of a(ω, z,z, σ) that the coefficients are "soft operators" which transform as primary of weight n and spin σ under the conformal group SL(2, C). Let us now substitute the expansion (11.15) into (11.14). Doing the integral we get where u = u − iδ and δ → 0 + . Let us notice that this integral is ill-defined for any n = 1 if the exponential factor e −iωu is not there. For the time being let us assume that n ≤ 1. Now as λ → 0 the singularity comes only from the n = 1 term and it is a pole of the Γ function. Around the pole we can write A(u, z,z, λ, σ) = −N 1 iλ S 1 (z,z, σ) + finite terms (11.17) Therefore the residue at the pole is the leading soft operator S 1 (z,z, σ), up to a sign. So we can write When inserted in an S-matrix element this leads to the leading conformal soft theorem. We would like to stress that this construction is valid only when all the operators are inserted in an S-matrix element and then we can meaningfully talk about analytic continuation in λ. In the same fashion we can talk about subleading (n = 0) conformal soft limit which is obtained in the limit λ → i: we have The insertion of such an operator will lead to the subleading conformal soft theorem without the contamination from the leading soft theorem. It is worth noticing that the appearance of a single power of u is determined by dimensional analysis because u transforms like a primary of dimension −1 and spin 0. Similarly we can write for p ≥ 0 We can see that in the absence of n > 1 terms in the soft expansion (11.15) the poles in the upper-half λ plane correspond to the IR behaviour of the scattering amplitude.
We're ready now to show how the Virasoro Ward identity arises from the subleading soft theorem. Let A n+1 ({p i } i∈{1,...,n} , q) be the n + 1 scattering amplitude involving n massless particles and one graviton of momentum q µ and polarization ǫ (±) µν (q). The soft limit for the graviton of momentum q → 0 gives [27,28] A n+1 ({p i } i∈{1,...,n} , q) → S where J λν,(±) j is the sum of the spin and angular momentum of the j-th particle. Let's focus, without loss of generality, on the minus helicity case. In the coordinates (ω, z,z) the subleading soft factor becomes − ∂z j (11.24) whereĥ j = 1 2 (σ j − ω j ∂ ω j ). As we have already explained the insertion of the operator S 1 defined in (11.18) in the scattering amplitudes will extract the subleading soft behaviour. Now supposing that we have n 1 incoming and n 2 outgoing hard particles (n 1 + n 2 = n) we can define the subleading soft graviton contribution to a (n + 1)-point scattering amplitude in the modified Mellin basis as follows 0| S 1 (z,z, −) A out (u j 2 , z j 2 ,z j 2 , λ j 2 , σ j 2 )     n 1 a out (p j 2 (u j 2 , z j 2 ,z j 2 ), σ j 2 ) (11.26) 6 We are using here the convention κ = √ 32πG = 2 as done in the main text of the paper.