Celestial Operator Products of Gluons and Gravitons
Monica Pate, Ana-Maria Raclariu, Andrew Strominger, Ellis Ye Yuan
aa r X i v : . [ h e p - t h ] O c t Celestial Operator Products of Gluonsand Gravitons
Monica Pate ∗‡ , Ana-Maria Raclariu ∗ , Andrew Strominger ∗ and Ellis Ye Yuan ∗† Abstract
The operator product expansion (OPE) on the celestial sphere of conformal primarygluons and gravitons is studied. Asymptotic symmetries imply recursion relations be-tween products of operators whose conformal weights differ by half-integers. It is shown,for tree-level Einstein-Yang-Mills theory, that these recursion relations are so constrain-ing that they completely fix the leading celestial OPE coefficients in terms of the Eulerbeta function. The poles in the beta functions are associated with conformally softcurrents. ∗ Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, USA † Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, Zhejiang, China ‡ Society of Fellows, Harvard University, Cambridge, MA, USA ontents
A Celestial OPEs from bulk three-point vertices 25B Higher order OPEs 26C Subleading soft gluon symmetry 27D Subsubleading soft graviton symmetry 29E Solving the recursion relations 30
The subleading soft graviton theorem implies that any quantum theory of gravity in anasymptotically flat four-dimensional (4D) spacetime has an infinite-dimensional 2D confor-mal symmetry [1,2]. This symmetry acts on the celestial sphere at null infinity, with Lorentztransformations generating the global SL (2 , C ) subgroup [3]. 4D scattering amplitudes in1 conformal basis transform like a collection of correlators in a 2D ‘celestial conformal fieldtheory’. Properties of the so-defined celestial CFTs have been extensively studied and dif-fer in ways which are not yet fully understood from those of conventional CFTs. Celestialoperator spectra were studied in [4–11] and celestial scattering amplitudes in [12–25].In a general celestial CFT, the operator spectrum is continuous, with one continuumfor every stable species of particles. Unstable particles decay before reaching infinity andare not part of the data on the celestial sphere. For a stable particle of spin s , a completebasis is given by celestial conformal primaries with conformal weights ( h, ¯ h ) = ( ∆+ s , ∆ − s ) and Re(∆) = 1 [5].In this paper we study the operator product expansion (OPE) of these celestial pri-maries. Poles in the celestial OPE for massless particles turn out to be Mellin transforms ofcollinear singularities in momentum space which can be computed with Feynman diagrams.The OPEs follow from the three-point vertices coupling the stable particles. We derive asimple and universal formula (12) relating the conformal weights in the operator productexpansion to the bulk scaling dimension of the three-point vertex.Celestial CFTs are subject to multiple infinities of asymptotic symmetry constraintsbeyond the familiar ones following from 2D conformal symmetry. These constraints have noanalogs in conventional CFTs. They follow from the leading and subsubleading soft gravitontheorems and, if there are gauge bosons, the subleading soft photon/gluon theorem. On theface of it, it would seem impossible for a collection of celestial amplitudes to satisfy additionalinfinities of constraints, but of course we know this seemingly overconstrained problem musthave a solution as many celestial amplitudes have been explicitly constructed. So far therehas been little study of the implications of these constraints.In this paper we show that the additional symmetry constraints have remarkable impli-cations for the operator product expansion. They imply recursion relations between prod-ucts of celestial operators whose conformal weights differ by a half-integer. We analyze indetail tree-level Einstein-Yang-Mills (EYM) theory and find that the recursion relations, to-gether with some analyticity assumptions, are so powerful that they completely determine(at least) all the conformal primary OPE coefficients of the leading poles in the operatorproduct expansion. They are given by Euler beta functions (ratios of Gamma functions)with arguments given by the conformal weights. We check that the direct but lengthier2eynman-diagrammatic computation yields the same beta functions.Inclusion of quantum, stringy or other corrections would introduce higher dimensionterms into the effective action. These may alter both the three-point vertices and the(sub)subleading soft theorems, and hence the subleading terms in the OPEs in accord withthe general formula (13) below. It will be interesting to study the symmetry constraintson OPEs in this more general context, as well as to extend the analysis beyond the leadingpoles.In a conventional (unitary, discrete) CFT, the operator spectrum and the conformalprimary OPEs fully determine the theory. Should an analogous result hold in celestialCFT, it would suggest that complete quantum theories of gravity are determined by thesesymmetry-constrained OPE coefficients. These are far fewer in number than the numberof possible terms in the effective Lagrangian. This resonates with similar findings in theamplitudes program [26–30]. It would be interesting to study further constraints amongthese OPEs from crossing symmetry.This paper is organized as follows. Section 2 contains conventions and useful formulae.Section 3 begins with a general derivation of the relation between the bulk dimension of thethree-point couplings and the conformal weights of the OPE. Subsection 3.1 considers thegluon OPE poles in tree-level Yang-Mills (YM) theory. The subleading soft gluon theorem isshown to imply recursion relations among the OPE coefficients, with the overall normaliza-tion fixed by the leading soft gluon theorem. For the collinear pole terms, these are uniquelysolved – subject to certain falloffs at large operator dimension – by Euler beta functions.Subsection 3.2 derives similar results, invoking the subsubleading soft graviton theorem, forthe graviton OPEs in Einstein gravity, while 3.3 derives the EYM gluon-graviton OPEs.In section 4, building on previous analyses of collinear limits of gravitons and gluons, wedirectly compute the collinear singularities in momentum space and then the OPE polesvia a Mellin transformation. This direct analysis fully agrees with the symmetry-derivedresults. We generalize our results for operators associated to incoming and outgoing parti-cles in section 5. The EYM OPEs are all summarized in section 5.3. Appendix A detailsthe relation between the bulk scaling dimension of a three-point vertex and the conformalweights entering the OPE. Appendix B presents the list of all OPE coefficients which can begenerated by higher-dimension operators. In appendices C and D we review the unbrokenglobal symmetries which are related to the subleading soft gluon theorem and subsubleading3oft graviton theorem and used to derive the recursion relations. In appendix E we solve therecursion relations for the beta function and spell out the regularity conditions which makethe solution unique.
In this section we give our conventions for celestial scattering amplitudes and collect someuseful formulae.Celestial amplitudes e A of massless particles are obtained from momentum-space ampli-tudes A (including the momentum-conserving delta function) by performing Mellin trans-formations with respect to the particle energies [5, 12] e A s ··· s n (∆ , z , ¯ z , · · · , ∆ n , z n , ¯ z n ) = n Y k =1 Z ∞ dω k ω ∆ k − k ! A s ··· s n ( ǫ ω , z , ¯ z , · · · , ǫ n ω n , z n , ¯ z n ) , (1)where the helicity s k = ± for gluons and s k = ± for gravitons. In order to writemomentum-space amplitudes as functions of ( ǫ k ω k , z k , ¯ z k ) , we parametrize the Cartesiancoordinate massless 4-momenta components as p µk = ǫ k ω k √ z k ¯ z k , z k + ¯ z k , − i ( z k − ¯ z k ) , − z k ¯ z k ) , (2)with µ = 0 , , , , ǫ k = ± for outgoing and incoming momenta respectively and helicitiesare defined with respect to outgoing momenta. In the following two sections we computeOPEs of outgoing states with ǫ k = 1 . We finally explain how to generalize the analysis tomixed incoming and outgoing OPEs in section 5. Color indices and in/out labels on celestialamplitudes are suppressed. We later use A to denote color-ordered partial amplitudes. Wenote that p · p = − ǫ ǫ ω ω z ¯ z , (3)where z = z − z , ¯ z = ¯ z − ¯ z . (4)For coordinates x µ = u∂ z ∂ ¯ z q µ ( z, ¯ z ) + rq µ ( z, ¯ z ) ,q µ ( z, ¯ z ) = 1 √ z ¯ z, z + ¯ z, − i ( z − ¯ z ) , − z ¯ z ) , (5)4he flat metric is ds = dx µ dx µ = − dudr + 2 r dzd ¯ z, (6)the celestial sphere is conformally mapped to the celestial plane and z k is the spatial locationat which a particle of momentum p k crosses I + . e A transforms as a correlator of n weight ( h k , ¯ h k ) = ( ∆ k + s k , ∆ k − s k ) primaries under conformal transformations of the celestial plane.In the next two sections we consider only OPEs between outgoing particles, and use O ∆ ,s to denote a generic such primary, O ± a ∆ for a primary gluon where s = ± (with a an adjointgroup index) and G ± ∆ for a primary graviton with s = ± . In section 5 we reintroduce theadditional label ǫ to distinguish between incoming and outgoing operators O ǫ ∆ ,s . (Wheneverthe label is absent, the operator is taken to be outgoing.) Group structure constants f abc obey the Jacobi identity f abd f dce + f bcd f dae + f cad f dbe = 0 , (7)and generators are normalized such that Tr( T a T b ) = g Y M δ ab , (8)where T a are in the fundamental representation. We work with the following polarizationvectors for massless spin-1 particles ε + k µ = 1 ǫ k ω k ∂ z k p µk , ε − k µ = 1 ǫ k ω k ∂ ¯ z k p µk , (9)and polarization tensors ε ± k µν = ε ± k µ ε ± k ν for massless spin-2 particles. These obey p · ε − = ǫ ω z , p · ε +2 = ǫ ω ¯ z . (10)Generically, the Mellin transform ω k -integrals converge only for restricted values of ∆ k .For example in gauge theory they converge on the unitary principle series with Re(∆) = 1 .However we will be interested in the celestial amplitudes for other complex values of ∆ k ,where we define them by analytic continuation. In this section we study OPEs of conformal primary gluon and graviton operators on thecelestial plane labeled by ( z, ¯ z ) . z and ¯ z will be varied independently. (These variables are5ndependent in (2,2) signature, for which the celestial plane becomes Lorentzian.) Moreoverwe consider only the ‘holomorphic limit’ z → with ¯ z , ¯ z fixed. Symmetry implies similarOPEs for ¯ z → with z , z fixed. However, order-of-limits subtleties arise when both z → and ¯ z → [31–33]. These are likely important for the structure of celestialamplitudes but are beyond the scope of this paper.Singularities in the celestial OPEs are the Mellin transforms of collinear divergences inthe momentum-space scattering amplitudes. This allows us to deduce some simple propertiesof the OPEs without any detailed computations. Collinear singularities arise when p || p formassless particles which couple via a three-point vertex to form a nearly on-shell internalparticle. The resulting propagator is proportional to p · p which, according to (3), divergesas z for z → . Hence two-operator OPE singularities are at most simple poles in z . Schematically the OPE of conformal primaries O ∆ ,s with conformal weights ( h, ¯ h ) =( ∆+ s , ∆ − s ) takes the form O ∆ ,s ( z , ¯ z ) O ∆ ,s ( z , ¯ z ) ∝ z O ∆ ,s ( z , ¯ z ) + order (cid:0) z (cid:1) . (11)Contributions to the OPE (11) arise from the three-point interaction vertices in theexpansion of terms in the bulk effective Lagrangian around flat space. Since gravitons andgluons have bulk scaling dimension one, these are characterized by bulk dimension d V =3 + m , where m is the number of spacetime derivatives. For example the most relevantgluon-gluon-graviton vertex h∂A∂A has d V = 5 , while the gluon-gluon-gluon vertex A∂AA has d V = 4 . The conformal weight ∆ of the operator on the right hand side of (11) can beinferred from d V . Each derivative leads to one extra factor of ω inside the Mellin transform(1), and therefore shifts ∆ up by one. Accounting for all the factors of ω (including twoin the internal propagator), one finds that the OPE of two operators of conformal weight ∆ and ∆ which couple via a three-point vertex of bulk dimension d V can only produce anoperator with conformal weight ∆ = ∆ + ∆ + d V − . (12)Details are in appendix A. Further, insisting on conformal invariance, one finds that the6ontribution to the OPE from a vertex of fixed d V must take the form O ∆ ,s ( z , ¯ z ) O ∆ ,s ( z , ¯ z ) ∼ d V − X n =0 c n,d V (∆ , s ; ∆ , s ) z n − ¯ z d V − − n O ∆ +∆ + d V − ,s + s +3+2 n − d V ( z , ¯ z ) . (13)Although in the most general case (13) is an infinite series when summing over d V , manyterms are eliminated when the spins range over limited values. For example in a theory withonly s = ± gluons the O + O + OPE in (13) reduces to the two terms O + a ∆ ( z , ¯ z ) O + b ∆ ( z , ¯ z ) ∼ if abc (cid:16) c dV − ,d V z d V / − ¯ z d V / − O + c ∆ +∆ + d V − ( z , ¯ z )+ c dV − ,d V z d V / − ¯ z d V / − O − c ∆ +∆ + d V − ( z , ¯ z ) (cid:17) . (14)In this paper we consider in detail only symmetry constraints on the leading ( n = 0 ) poleterms in EYM theory, for which there are only seven nonzero coefficients c ,d V with d V = 4 , .These are all completely fixed by asymptotic symmetries and summarized in section 5.3.Equally powerful symmetry constraints apply to all terms in the expansion (13), but themore intricate higher-order analysis is left to future investigation. In this section we consider pure renormalizable glue theory with d V = 4 . In this case, O + a ∆ ( z , ¯ z ) O + b ∆ ( z , ¯ z ) ∼ − if abc z C (∆ , ∆ ) O + c ∆ +∆ − ( z , ¯ z ) , (15) O + a ∆ ( z , ¯ z ) O − b ∆ ( z , ¯ z ) ∼ − if abc z D (∆ , ∆ ) O − c ∆ +∆ − ( z , ¯ z ) , (16)for some to-be-determined coefficients C (∆ , ∆ ) = C (∆ , ∆ ) and D (∆ , ∆ ) . O − O − isnonsingular in z . For gluons, the conformal primaries with Re(∆) = 1 are a complete basis Since there are no gauge and coordinate invariant d V < relevant operators in a theory with onlygluons or gravitons (except of course the cosmological constant, which we assume vanishes!), there are no z ¯ z singularities. Due to the lower limit in (13), the second term is absent for d V = 4 . Additional terms on the right hand side in the presence of gravitons are determined in subsection 3.3.An F term with d V = 6 would lead to an O − term on the right hand side of (15).
7f square-integrable wave packets [5]. We see that in the renormalizable theory the OPEs(15), (16) close on such operators.The OPE coefficients are subject to a number of symmetry constraints. The simplest istranslations P towards the ‘north pole’ of the celestial sphere, which involves a factor of ω inmomentum space. In a conformal basis, this symmetry shifts the operator dimension [7, 19]: δ P O ± a ∆ ( z, ¯ z ) = O ± a ∆+1 ( z, ¯ z ) . (17)Acting on both sides of (15) and (16) with δ P gives the recursion relations C (∆ , ∆ ) = C (∆ + 1 , ∆ ) + C (∆ , ∆ + 1) , (18) D (∆ , ∆ ) = D (∆ + 1 , ∆ ) + D (∆ , ∆ + 1) . (19)Such relations were also found in [11].Next, the leading conformally soft theorem is [20–22] lim ∆ → O + a ∆ ( z , ¯ z ) O ± b ∆ ( z , ¯ z ) ∼ − if abc (∆ − z O ± c ∆ ( z , ¯ z ) . (20)This implies poles in C and D with residues lim ∆ → (∆ − C (∆ , ∆ ) = lim ∆ → (∆ − D (∆ , ∆ ) = 1 . (21)Further, less familiar, constraints come from the subleading soft symmetry parametrizedby ( Y za , Y ¯ za ). Under these symmetries, the gauge field on I + shifts by [34] δ Y A az = u∂ z Y za , ¯ δ Y A a ¯ z = u∂ z Y ¯ za . (22)If the right hand side is nonzero, the symmetry is spontaneously broken. The unbrokensymmetries are the most useful for present purposes. These correspond to Y za = zǫ a , ǫ a and Y ¯ za = ¯ zǫ a , ǫ a for constant ǫ a . As shown in appendix C (see also [34]), for the globalsymmetry Y za = zǫ a conformal primary gluons transform as δ b O ± a ∆ ( z, ¯ z ) = − (∆ − ± z∂ z ) if abc O ± c ∆ − ( z, ¯ z ) . (23)Similarly for Y ¯ za = ¯ zǫ a we have ¯ δ b O ± a ∆ ( z, ¯ z ) = − (∆ − ∓ z∂ ¯ z ) if abc O ± c ∆ − ( z, ¯ z ) . (24)8ince they are unbroken, the Ward identities for these symmetries involve no soft insertions n X k =1 h O · · · δO k · · · O n i = 0 , n X k =1 h O · · · ¯ δO k · · · O n i = 0 . (25)We now extract the consequences of this global symmetry for the OPE (15). This iscomplicated by the appearance of derivatives in the transformation laws (23) and (24) whichmix up primaries and descendants, and therefore do not map the leading OPE relations (15)and (16) to themselves. These bothersome terms can be eliminated in ¯ δ by considering thespecial case ¯ z = ¯ z = 0 , where (15) still holds. (The z -analog of this trick cannot be usedto analyze the implications of δ symmetry because (15) blows up for z = z = 0 .) Actingwith ¯ δ d on both sides of (15) we get (∆ − if adc O + c ∆ − ( z , O + b ∆ ( z ,
0) + (∆ − O + a ∆ ( z , if bdc O + c ∆ − ( z , ∼ ∆ + ∆ − z C (∆ , ∆ ) f abc f cde O + e ∆ +∆ − ( z , . (26)Using the OPE again on the left hand side we obtain the consistency condition (∆ − C (∆ − , ∆ ) f adc f cbe + (∆ − C (∆ , ∆ − f bdc f ace = (∆ + ∆ − C (∆ , ∆ ) f abc f cde . (27)Applying the Jacobi identity (7) this implies (∆ − C (∆ − , ∆ ) = (∆ + ∆ − C (∆ , ∆ ) . (28)Under suitable assumptions spelled out in appendix E about boundedness and analyticityin ∆ , ∆ (basically that there are no poles other than those implied by the soft theorems),(28) together with the normalization condition (21) have the unique solution C (∆ , ∆ ) = B (∆ − , ∆ − , (29)where B is the Euler beta function B ( x, y ) = Γ( x )Γ( y )Γ( x + y ) . (30) Symmetry of C (∆ , ∆ ) under ∆ ↔ ∆ together with the subleading soft symmetry constraint (28)in fact imply the translation invariance relation (18), which therefore does not further constrain C . ¯ δ d on both sides of (16) gives a slightly different result because of the ± in (24).Instead of (28) we find two different recursion relations (∆ − D (∆ − , ∆ ) = (∆ + ∆ − D (∆ , ∆ ) , ∆ D (∆ , ∆ −
1) = (∆ + ∆ − D (∆ , ∆ ) . (31)Again, (31) together with the normalization condition (21), have the unique solution D (∆ , ∆ ) = B (∆ − , ∆ + 1) . (32)(29) and (32) agree with the expressions previously obtained in [20] by direct Mellin transformof the collinear singularities in momentum space. Here we see the OPE is entirely fixed bysymmetries.In fact there are further consistency conditions, which we did not need to use to fix C and D , but it can be checked that they are satisfied. One of these is that the OPEs haveproperly normalized poles at ∆ → corresponding to the subleading soft theorem. This isindeed manifest in (29) and (32). We have used here only a few global symmetries. There areinfinitely many more constraints from the infinity of soft symmetries. However these may allbe obtained by commuting the global symmetries with the local conformal symmetry, whichis manifestly built in to our construction and so their satisfaction is guaranteed. For gravitons in Einstein gravity the three-point vertex has d V = 5 . According to (13) thisleads to an OPE of the form G +∆ ( z , ¯ z ) G ± ∆ ( z , ¯ z ) ∼ ¯ z z E ± (∆ , ∆ ) G ± ∆ +∆ ( z , ¯ z ) , (33)for some to-be-determined coefficients E + (∆ , ∆ ) = E + (∆ , ∆ ) and E − (∆ , ∆ ) , while G − G − is nonsingular in the z → limit. As for the case of gluons, translation invarianceimplies the recursion relation E ± (∆ , ∆ ) = E ± (∆ + 1 , ∆ ) + E ± (∆ , ∆ + 1) . (34) A contribution of the form ¯ z z E ′ + (∆ , ∆ ) G − ∆ +∆ +4 to the G +∆ G +∆ OPE might for example be gener-ated by an R correction to the Einstein-Hilbert action. ∆ → is fixed by the the leading soft graviton theorem [24] lim ∆ → E ± (∆ , ∆ ) ∼ − κ − , κ = √ πG. (35)The subleading soft symmetry corresponds to 2D conformal transformations, which aregenerated by the shadow of G +0 [7, 10, 35, 36]. However, by working in a conformal basis, wehave already ensured that the OPE is conformally invariant, and no further constraints on E ± are obtained from the subleading soft symmetry.The role of the subleading soft gluon theorem in constraining gauge theory OPEs is hereplayed by the subsubleading soft graviton theorem, which implies further global symmetries.We show in appendix D that the relevant gravitational analog of the gauge theory relation(24) is ¯ δG ± ∆ ( z, ¯ z ) = − κ (cid:2) (∆ ∓ ∓ −
1) + 4(∆ ∓ z∂ ¯ z + 3¯ z ∂ z (cid:3) G ± ∆ − ( z, ¯ z ) . (36)However, to study the consequences of this symmetry on the OPE, we cannot directly set ¯ z = ¯ z = 0 in (33) because that will set the right hand side to zero and no useful relationwould be obtained. To avoid this we first differentiate with respect to ¯ z , and then set ¯ z = ¯ z = 0 . The positive helicity graviton OPE in (33) is then ∂ ¯ z G +∆ ( z , G +∆ ( z , ∼ E + (∆ , ∆ ) z G +∆ +∆ ( z , . (37)Equation (36) becomes ¯ δG +∆ ( z,
0) = − κ − − G +∆ − ( z, , (38)and in addition implies ¯ δ∂ ¯ z G +∆ ( z,
0) = − κ − ∂ ¯ z G +∆ − ( z, . (39)Invariance of the OPE (33) then holds if and only if (∆ + 1)(∆ − E ± (∆ − , ∆ ) + (∆ ∓ − ∓ E ± (∆ , ∆ − + ∆ ∓ + ∆ ∓ − E ± (∆ , ∆ ) . (40) Supertranslations are generated by the current P z = − κ lim ∆ → (∆ − ∂ ¯ z G +∆ . E ± (∆ , ∆ ) = − κ B (∆ − , ∆ ∓ . (41)In section 4.1 (see equations (57), (58)) we directly compute the Mellin transform of thenear-collinear graviton amplitudes and find complete agreement with (41).Additionally, the OPE coefficients E ± must have properly normalized poles at ∆ → and ∆ → − associated to the subleading and subsubleading soft graviton symmetries,respectively. As in the gauge theory case, we did not impose such conditions in our derivation,but find that our results are consistent with these conditions. In this section we consider OPEs involving both gravitons and gluons. The Einstein-Yang-Mills interaction (schematically hF ) has d V = 5 . The relevant term in (13) is G +∆ ( z , ¯ z ) O ± a ∆ ( z , ¯ z ) ∼ ¯ z z F ± (∆ , ∆ ) O ± a ∆ +∆ ( z , ¯ z ) . (42)Translation invariance again implies the recursion relation (18) for F ± . A second set ofrelations is determined from the global symmetry associated to subsubleading soft gravitontheorem, whose action on gluons is shown in appendix D to be ¯ δO ± a ∆ ( z, ¯ z ) = − κ (cid:2) (∆ ∓ − ∓
1) + 4(∆ ∓ z∂ ¯ z + 3¯ z ∂ z (cid:3) O ± a ∆ − ( z, ¯ z ) . (43)Consistency of the OPE with this symmetry requires (∆ + 1)(∆ − F ± (∆ − , ∆ ) + (∆ ∓ − ∓ F ± (∆ , ∆ − + ∆ ∓ − + ∆ ∓ F ± (∆ , ∆ ) , (44)where these relations are derived by studying the OPE of ∂ ¯ z G +∆ ( z , O ± a ∆ ( z , as in theprevious section. Fixing the normalization with the leading soft graviton theorem one finds F ± (∆ , ∆ ) = − κ B (∆ − , ∆ ∓ . (45)12n the presence of gravitons, the right hand side of the gluon OPE (16) can also receivea correction of the form O + a ∆ ( z , ¯ z ) O − b ∆ ( z , ¯ z ) ∼ − if abc z B (∆ − , ∆ + 1) O − c ∆ +∆ − ( z , ¯ z ) + δ ab ¯ z z H (∆ , ∆ ) G − ∆ +∆ ( z , ¯ z ) , (46)corresponding to the fact that two gluons can make a graviton. This new term might seemto violate the subleading soft gluon theorem. Indeed, we will find shortly that symmetryconstrains H to have a pole associated with the subleading soft gluon symmetry at ∆ = 0 .However, as shown in [37, 38], this theorem is corrected at tree-level in Einstein-Yang-Millstheory by the hF coupling! The known form of the correction in fact can be used to fix theconstant normalization of H .Translation invariance implies H obeys a recursion relation of the form (18), while thesubsubleading soft graviton theorem implies H obeys the recursion relation (∆ + 2)(∆ − H (∆ − , ∆ ) + ∆ (∆ + 1) H (∆ , ∆ − + ∆ + 2)(∆ + ∆ + 1) H (∆ , ∆ ) . (47)The properly normalized solution is H (∆ , ∆ ) = κ B (∆ , ∆ + 2) . (48)The symmetry-derived results (45) and (48) agree with the Mellin transforms of direct Feyn-man diagram computations found in the next section.The appearance of a graviton in the OPE of two gluons is presumably the boundarymanifestation of the still-enigmatic double-copy relation [39–41], in which gravity is thesquare of gauge theory. A remarkable discovery due to Stieberger and Taylor [32, 42] is thata pair of collinear gluons in a scattering amplitude can be replaced by a single graviton. Ifwe take ∆ = ∆ = 0 in (46), the right hand side contains G − which is the shadow of theboundary stress tensor. This is a Sugawara-like construction of the stress tensor from a pairof subleading soft currents. We leave these fascinating connections to future exploration. In this section we directly compute the celestial OPEs among gravitons and gluons in EYMby Mellin transforms of Feynman diagrams. We begin by reviewing the collinear limits of13auge and gravity amplitudes. The various OPEs are derived by Mellin transforming thecorresponding amplitudes in the collinear limit and found in all cases to agree with thesymmetry-inferred results summarized later in section 5.3. The OPEs among gluons werealready derived in this manner in [20]. Their computation confirms (29) and (32) and willnot be repeated here.
The collinear limits of gravity amplitudes were first derived in [43] and further developmentsare in [44, 45]. The leading divergence is generically protected against loop corrections [43].Here we specialize to a holomorphic collinear limit.Consider a tree-level n -graviton scattering amplitude. In the limit when z ij → forfixed ¯ z i , ¯ z j , the amplitude contains a universal piece which factorizes as lim z ij → A s ··· s n ( p , · · · , p n ) −→ X s = ± Split ss i s j ( p i , p j ) A s ··· s ··· s n ( p , · · · , P, · · · , p n ) , (49)where in the collinear limit P µ = p µi + p µj , ω P = ω i + ω j . (50)The collinear factor Split ss i s j ( p i , p j ) then takes the form [43] Split ( p i , p j ) = − κ z ij z ij ω P ω i ω j , Split − − ( p i , p j ) = − κ z ij z ij ω j ω i ω P , (51)with all other combinations of helicities vanishing. In the collinear limit, the celestial gravityamplitude e A becomes e A s ··· s n (∆ , z , ¯ z , · · · , ∆ n , z n , ¯ z n ) i || j −→ n Y k =1 Z ∞ dω k ω ∆ k − k X s = ± Split ss i s j ( p i , p j ) A s ··· s ··· s n ( p , · · · , P, · · · , p n ) + · · · . (52) At subleading order in z ij , (50) receives corrections, but these do not affect the leading singularitiesconsidered here. For a discussion of subleading terms see [32]. We work with the Einstein-Hilbert action normalized as S = κ R d x √− gR, g µν = η µν + κh µν . Thisyields the following leading soft factor S ± (0) = κ P k ( p k · ε ± ) p k · q .
14o simplify, we make the following change of variables, ω i = tω P , ω j = (1 − t ) ω P , (53)so that for example Z ∞ dω i ω ∆ i − i Z ∞ dω j ω ∆ j − j Split ( p i , p j ) = − κ z ij z ij Z dt t ∆ i − (1 − t ) ∆ j − Z ∞ dω P ω ∆ i +∆ j − P . (54)The t integral is immediately recognizable as the integral representation of the Euler betafunction, B ( x, y ) = Z dt t x − (1 − t ) y − , (55)whose origin is hence a splitting factor for the conformal weight between the two collinearexternal particles. Since the only t dependence on the right hand side of (52) comes from Split ss i s j ( p i , p j ) , one finds lim z ij → e A s ··· ··· ··· (∆ , z , ¯ z , · · · , ∆ i , z i , ¯ z i , · · · , ∆ j , z j , ¯ z j , · · · ) −→− κ z ij z ij B (∆ i − , ∆ j − e A s ··· ··· (∆ , z , ¯ z , · · · , ∆ i + ∆ j , z j , ¯ z j , · · · ) + order( z ij ) . (56)Since this holds in any celestial amplitude, it implies the leading OPE between two positivehelicity gravitons is G +∆ ( z , ¯ z ) G +∆ ( z , ¯ z ) ∼ − κ z z B (∆ − , ∆ − G +∆ +∆ ( z , ¯ z ) , (57)in agreement with (41). By similar arguments, one also finds the following leading OPEbetween opposite helicity gravitons G +∆ ( z , ¯ z ) G − ∆ ( z , ¯ z ) ∼ − κ z z B (∆ − , ∆ + 3) G − ∆ +∆ ( z , ¯ z ) , (58)again in agreement with (41). In order to derive graviton-gluon OPEs from collinear limits of EYM amplitudes, we herederive the collinear limits of conventional momentum-space amplitudes.15e start with the general Stieberger-Taylor formula which relates a momentum-spaceamplitude of n gluons and one graviton to a sum over color-ordered partial amplitudes of n + 1 gluons [46] A s ··· s n ; ± ( p , · · · , p n ; p ) = − κ n − X ℓ =1 ( ε ± ( p ) · χ ℓ ) A s ··· s ℓ ± s ℓ +1 ··· s n ( p , · · · , p ℓ , p, p ℓ +1 , · · · , p n ) , (59)where p i , i = 1 , ..., n are the momenta of the gluons, p is the momentum of the graviton, ε ( p ) is the polarization of a gluon of momentum p and χ ℓ = ℓ X k =1 p k . (60)This formula allows us to determine collinear graviton-gluon limits from collinear gluonlimits. The known leading collinear behavior of gluon amplitudes arises from adjacent gluonsin color-ordered partial amplitudes [47] lim z ij → A s ··· s n ( p , · · · , p i , p j , · · · , p n ) −→ X s = ± Split ss i s j ( p i , p j ) A s ··· s ··· s n ( p , · · · , P, · · · , p n ) , (61)where P was defined in (50) and the non-vanishing Split ss i s j ( p i , p j ) for collinear gluons aregiven by Split ( p i , p j ) = 1 z ij ω P ω i ω j , Split − − ( p i , p j ) = 1 z ij ω j ω i ω P . (62)Consider the collinear limit between a positive helicity gluon of momentum p i and a positivehelicity graviton. In the collinear limit, the leading order contributions from the right handside of (59) are just the two terms where the gluon of momentum p which replaces thegraviton is adjacent to the i th gluon: lim z i − z → A s ··· ··· s n ;2 ( p , · · · , p i , · · · , p n ; p ) −→ − κ (cid:2)(cid:0) ε + ( p ) · χ i − (cid:1) A s ··· s i − s i ··· s n ( p , · · · , p i − , p, p i , · · · , p n )+ (cid:0) ε + ( p ) · χ i (cid:1) A s ··· s i s i +1 ··· s n ( p , · · · , p i , p, p i +1 , · · · , p n ) (cid:3) −→ − κ ε + ( p ) · ( χ i − χ i − ) 1 z i − z ω P ω i ω A s ··· s i − s i +1 ··· s n ( p , · · · , p i − , P, p i +1 , · · · , p n ) . (63) Note that ε ± ( p ) · χ n = − ε ± ( p ) · p = 0 by momentum conservation, hence the sum in (59) can be takenfrom to n .
16e use (60) to further simplify χ i − χ i − = p i (64)and using (10), ε + ( p ) · ( − χ i − + χ i ) = ε + ( p ) · p i = ω i (¯ z i − ¯ z ) . (65)Putting it all together, we obtain the following collinear limit for a positive helicitygluon and graviton lim z i − z → A s ··· ··· s n ;2 ( p , · · · , p i , · · · , p n ; p ) −→− κ z i − ¯ zz i − z ω P ω A s ··· s i − s i +1 ··· s n ( p , · · · p i − , P, p i +1 · · · , p n ) . (66)By similar arguments, keeping only singular terms in z i − z , we obtain the followingcollinear graviton-gluon limit for the mixed helicity case lim z i − z → A s ···− ··· s n ;2 ( p , · · · , p i , · · · , p n ; p ) −→− κ z i − ¯ zz i − z ω i ωω P A s ··· s i − − s i +1 ··· s n ( p , · · · p i − , P, p i +1 · · · , p n ) . (67)Taking Mellin transforms, we find the leading OPEs G +∆ ( z , ¯ z ) O + a ∆ ( z , ¯ z ) ∼ − κ z z B (∆ − , ∆ ) O + a ∆ +∆ ( z , ¯ z ) ,G +∆ ( z , ¯ z ) O − a ∆ ( z , ¯ z ) ∼ − κ z z B (∆ − , ∆ + 2) O − a ∆ +∆ ( z , ¯ z ) , (68)which agree with equation (45).Now we compute the graviton contribution to the mixed helicity gluon OPE. Since weare interested in the contribution from G − to the O + O − OPE, consider the on-shell vertex V ( p , p , p ) = − iκδ a a (cid:2) ( ε +1 · ε − )( ε +3 · p )( ε +3 · p ) − ( ε +1 · p )( ε +3 · p )( ε − · ε +3 ) (cid:3) . (69)Here ε , ε are the polarizations of the positive and negative helicity gluons of momenta p , p and colors a , a respectively. ε ± µν = ε ± µ ε ± ν is the graviton polarization. Evaluating inour parametrization (2) and (9), the on-shell vertex becomes V ( p , p , p ) = − iκδ a a ω ω ¯ z . (70)17n (2 , signature, the result is non-vanishing and upon taking z = z = z , momentumconservation reduces to ω + ω + ω = 0 ,ω ¯ z + ω ¯ z + ω ¯ z = 0 . (71)Solving for ¯ z , we find ¯ z = ω ω + ω ¯ z + ω ω + ω ¯ z ⇒ ¯ z = ω ω + ω ¯ z . (72)Then, accounting for the graviton propagator, we find that the collinear singularity foropposite helicity gluons due to the EYM vertex (69) is Split − − ( p , p ) = κ z z ω ω P . (73)Taking a Mellin transform, we deduce that the O + a ∆ ( z ) O − b ∆ ( z ) OPE contains a term of theform δ ab κ z z B (∆ , ∆ + 2) G − ∆ +∆ ( z , ¯ z ) , (74)in agreement with the symmetry-derived result (48). In this section we generalize our results to account for the presence of both incoming andoutgoing particles. We introduce celestial operators O ǫ k ∆ k ,s k ( z k , ¯ z k ) = Z ∞ dω k ω ∆ k − k O s k ( ǫ k ω k , z k , ¯ z k ) (75)carrying an additional label ǫ k = ± which distinguishes between outgoing and incomingstates respectively. O s k ( ǫ k ω k , z k , ¯ z k ) are operators associated to the standard ‘out’ and ‘in’momentum eigenstates through the parametrization (2). Since the action of the translationoperator on ‘in’ and ‘out’ momentum eigenstates differs by a sign, the action of P on thecelestial operators generalizes to δ P O ǫ ∆ ,s ( z, ¯ z ) = ǫ O ǫ ∆+1 ,s ( z, ¯ z ) . (76)18ote, since the ‘in’ and ‘out’ labels of asymptotic states are directly related to charges ofthe corresponding operators under a global symmetry of the celestial CFT, these labels arenaturally a part of the celestial CFT data.Likewise, since the inverse of P appears in the relevant subleading gluon and subsub-leading graviton symmetry actions (see appendices C and D), the actions of these symmetries(125) and (133) generalize to ¯ δ a O ǫ k ∆ k ,s k ( z k , ¯ z k ) = h − ǫ k (∆ k − s k − z k ∂ ¯ z k ) T ak P − k − κ z k F + ak + κ z k G + ak i O ǫ k ∆ k ,s k ( z k , ¯ z k ) , ¯ δ O ǫ k ∆ k ,s k ( z k , ¯ z k ) = − κ ǫ k (cid:2) (∆ k − s k )(∆ k − s k −
1) + 4(∆ k − s k )¯ z k ∂ ¯ z k + 3¯ z k ∂ z k (cid:3) O ǫ k ∆ k − ,s k ( z k , ¯ z k ) , (77)where F and G are defined in appendix C. We now determine the OPE coefficients among outgoing and incoming gluons from (77).The case when both operators are incoming is mostly identical to the previously studiedcase with both operators outgoing since the symmetry constraints remain unchanged. Thatis, up to normalization, these OPE coefficients are solved by the Euler beta functions (29)and (32) for gluons of identical and opposite helicity respectively. We therefore consider theOPEs of outgoing and incoming gluons where as we will see, the constraints from symmetrydiffer.Generalizing (15) and (16), we begin with the ansatz O + a,ǫ ∆ ( z , ¯ z ) O + b, − ǫ ∆ ( z , ¯ z ) ∼ − ǫ if abc z (cid:2) C ′ (∆ , ∆ ) O + c,ǫ ∆ +∆ − ( z , ¯ z )+ C ′′ (∆ , ∆ ) O + c, − ǫ ∆ +∆ − ( z , ¯ z ) (cid:3) , (78) O + a,ǫ ∆ ( z , ¯ z ) O − b, − ǫ ∆ ( z , ¯ z ) ∼ − ǫ if abc z (cid:2) D ′ (∆ , ∆ ) O − c,ǫ ∆ +∆ − ( z , ¯ z )+ D ′′ (∆ , ∆ ) O − c, − ǫ ∆ +∆ − ( z , ¯ z ) (cid:3) . (79)Using the generalized action of the translation operator (76), we find the OPE coefficientsmust obey C ′ (∆ + 1 , ∆ ) − C ′ (∆ , ∆ + 1) = C ′ (∆ , ∆ ) ,C ′′ (∆ + 1 , ∆ ) − C ′′ (∆ , ∆ + 1) = − C ′′ (∆ , ∆ ) , (80)19nd D ′ (∆ + 1 , ∆ ) − D ′ (∆ , ∆ + 1) = D ′ (∆ , ∆ ) ,D ′′ (∆ + 1 , ∆ ) − D ′′ (∆ , ∆ + 1) = − D ′′ (∆ , ∆ ) . (81)As before, these recursion relations do not fully constrain the answer, so we turn to thesubleading soft gluon symmetry. Constraining (78) with the symmetry in (77) and followingthe logic in section 3.1, we obtain the following relations (∆ − C ′ (∆ − , ∆ ) f adc f cbe − (∆ − C ′ (∆ , ∆ − f bdc f ace = (∆ + ∆ − C ′ (∆ , ∆ ) f abc f cde , (∆ − C ′′ (∆ − , ∆ ) f adc f cbe − (∆ − C ′′ (∆ , ∆ − f bdc f ace = − (∆ + ∆ − C ′′ (∆ , ∆ ) f abc f cde , (82)which using the Jacobi identity reduce to (∆ − C ′ (∆ − , ∆ ) = (∆ + ∆ − C ′ (∆ , ∆ ) , − (∆ − C ′ (∆ , ∆ −
1) = (∆ + ∆ − C ′ (∆ , ∆ ) , (83)and (∆ − C ′′ (∆ − , ∆ ) = − (∆ + ∆ − C ′′ (∆ , ∆ ) , (∆ − C ′′ (∆ , ∆ −
1) = (∆ + ∆ − C ′′ (∆ , ∆ ) . (84)By shifting the arguments and taking a linear combination of the two constraints for eachOPE coefficient, one can verify that these new recursion relations imply the modified recur-sion relation (80) from translation symmetry. (83) and (84) are solved by C ′ (∆ , ∆ ) = − B (∆ − , − ∆ − ∆ ) ,C ′′ (∆ , ∆ ) = B (∆ − , − ∆ − ∆ ) , (85)where we have used the celestial soft gluon theorem, generalized for incoming and outgoingoperators, lim ∆ → O + a,ǫ ∆ ( z , ¯ z ) O + b, − ǫ ∆ ( z , ¯ z ) = − ǫ if abc z − O + c, − ǫ ∆ ( z , ¯ z ) (86)to fix the normalization. Note that both C ′ and C ′′ are fixed by (86) due to the symmetryof (78) under exchange of labels which implies that they have soft poles at ∆ , ∆ = 1 (∆ − D ′ (∆ − , ∆ ) = (∆ + ∆ − D ′ (∆ , ∆ ) , − ∆ D ′ (∆ , ∆ −
1) = (∆ + ∆ − D ′ (∆ , ∆ ) , (87) (∆ − D ′′ (∆ − , ∆ ) = − (∆ + ∆ − D ′′ (∆ , ∆ ) , ∆ D ′′ (∆ , ∆ −
1) = (∆ + ∆ − D ′′ (∆ , ∆ ) . (88)The leading soft gluon theorem implies lim ∆ → O + a,ǫ ∆ ( z , ¯ z ) O − b, − ǫ ∆ ( z , ¯ z ) = − ǫ if abc z − O − c, − ǫ ∆ ( z , ¯ z ) , (89)which together with the recursion relation (88) uniquely fixes D ′′ (∆ , ∆ ) = B (∆ − , − ∆ − ∆ ) . (90)On the other hand, (87) is solved by D ′ (∆ , ∆ ) = αB (∆ + 1 , − ∆ − ∆ ) (91)for some yet-to-be determined constant α . To fix α , consider the mixed-helicity gluon OPE,evaluated at ∆ = ∆ ≡ ∆ O + a,ǫ ∆ ( z , ¯ z ) O − b, − ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫ (cid:2) αB (cid:0) ∆ + 1 , − (cid:1) O − c,ǫ − ( z , ¯ z ) + B (cid:0) ∆ − , − (cid:1) O − c, − ǫ − ( z , ¯ z ) (cid:3) . (92)Taking ∆ → , which corresponds to a double soft limit of a scattering amplitude, we obtainan OPE among celestially soft operators lim ∆ → (∆ − O + a,ǫ ∆ ( z , ¯ z ) O − b, − ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫ
12 lim ∆ → (cid:2) α (∆ − O − c,ǫ − ( z , ¯ z ) + (∆ − O − c, − ǫ − ( z , ¯ z ) (cid:3) . (93)The above OPE is related to another OPE for celestially soft operators lim ∆ → (∆ − O + a,ǫ ∆ ( z , ¯ z ) O − b,ǫ ∆ ( z , ¯ z ) ∼ − if abc ǫz lim ∆ → (∆ − O − c,ǫ − ( z , ¯ z ) (94)21y the crossing relation for soft modes [31], which on the celestial sphere takes the form lim ∆ → (∆ − O ± a,ǫ ∆ ( z, ¯ z ) = − lim ∆ → (∆ − O ± a, − ǫ ∆ ( z, ¯ z ) . (95)Comparing the two, we find α = − . (96) We now confirm the symmetry-derived results from a momentum-space amplitude calcula-tion. As before, the OPE coefficients can be derived by Mellin transforming the collinearsplitting functions. For incoming and outgoing gluons these take the general form
Split s s ( p , p ) = 1 z ( ǫ ω ) α ( ǫ ω ) β ( ǫ ω + ǫ ω ) γ . (97)To evaluate Z ∞ dω Z ∞ dω ω ∆ − ω ∆ − Split s s ( p , p ) , (98)it is convenient to make the following change of variables ω = (1 − ǫ ǫ t ) ω P , ω = tω P , (99)where ω + ǫ ǫ ω = ω P . (100)For ǫ ǫ = − , (98) splits into two integrals such that the celestial OPE takes the form O + a,ǫ ∆ ( z , ¯ z ) O ± b,ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫ α + γ ǫ β (cid:20)Z ∞ dω P Z ∞ dt (1 + t ) ∆ − α t ∆ − β ω ∆ +∆ + α + β + γ − P O ± c ( ǫ ω P , z , ¯ z ) − Z −∞ dω P Z − −∞ dt (1 + t ) ∆ − α t ∆ − β ω ∆ +∆ + α + β + γ − P O ± c ( ǫ ω P , z , ¯ z ) (cid:21) = − if abc z ǫ α + γ ǫ β (cid:20)Z ∞ dt (1 + t ) ∆ − α t ∆ − β O ± c,ǫ ∆ +∆ + α + β + γ ( z , ¯ z )+( − γ Z ∞ dt t ∆ − α (1 + t ) ∆ − β O ± c, − ǫ ∆ +∆ + α + β + γ ( z , ¯ z ) (cid:21) , (101)22here to obtain the second line, we performed the change of variables ω P → − ω P and t → − (1 + t ) on the second term. Upon making a further change of variables t = u − u , wefind the remaining t -integrals once again take the form (55) so that the OPE coefficients aregiven by Euler beta functions O + a,ǫ ∆ ( z , ¯ z ) O ± b,ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫ α + γ ǫ β (cid:2) B (∆ + β, − ∆ − ∆ − α − β ) O ± c,ǫ ∆ +∆ + α + β + γ ( z , ¯ z )+( − γ B (∆ + α, − ∆ − ∆ − α − β ) O ± c, − ǫ ∆ +∆ + α + β + γ ( z , ¯ z ) (cid:3) . (102)For equal helicity gluons α = β = − γ = − and so the in/out OPE is O + a,ǫ ∆ ( z , ¯ z ) O + b, − ǫ ∆ ( z , ¯ z ) ∼ if abc ǫz (cid:2) B (3 − ∆ − ∆ , ∆ − O + c,ǫ ∆ +∆ − ( z , ¯ z ) − B (∆ − , − ∆ − ∆ ) O + c, − ǫ ∆ +∆ − ( z , ¯ z ) (cid:3) . (103)For opposite helicity gluons α = − β = γ = − and we find O + a,ǫ ∆ ( z , ¯ z ) O − b, − ǫ ∆ ( z , ¯ z ) ∼ if abc ǫz (cid:2) B ( − ∆ − ∆ + 1 , ∆ + 1) O − c,ǫ ∆ +∆ − ( z , ¯ z ) − B (∆ − , − ∆ − ∆ ) O − c, − ǫ ∆ +∆ − ( z , ¯ z ) (cid:3) , (104)which agree with the symmetry-derived OPEs. Analogous computations yield the gravitonand gluon-graviton in/out OPEs. We summarize the results in the following section. In summary, all the nonzero leading z poles for all possible configurations of incomingand outgoing gluon and graviton OPEs are determined by the asymptotic symmetries intree-level EYM. The equal helicity gluon OPEs are O + a,ǫ ∆ ( z , ¯ z ) O + b,ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫB (∆ − , ∆ − O + c,ǫ ∆ +∆ − ( z , ¯ z ) ,O + a,ǫ ∆ ( z , ¯ z ) O + b, − ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫ (cid:2) − B (∆ − , − ∆ − ∆ ) O + c,ǫ ∆ +∆ − ( z , ¯ z )+ B (∆ − , − ∆ − ∆ ) O + c, − ǫ ∆ +∆ − ( z , ¯ z ) (cid:3) . (105)23he mixed helicity gluon OPEs are O + a,ǫ ∆ ( z , ¯ z ) O − b,ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫB (∆ − , ∆ + 1) O − c,ǫ ∆ +∆ − ( z , ¯ z )+ κ z z δ ab B (∆ , ∆ + 2) G − ,ǫ ∆ +∆ ( z , ¯ z ) ,O + a,ǫ ∆ ( z , ¯ z ) O − b, − ǫ ∆ ( z , ¯ z ) ∼ − if abc z ǫ (cid:2) − B (∆ + 1 , − ∆ − ∆ ) O − c,ǫ ∆ +∆ − ( z , ¯ z )+ B (∆ − , − ∆ − ∆ ) O − c, − ǫ ∆ +∆ − ( z , ¯ z ) (cid:3) + κ z z δ ab (cid:2) B (∆ + 2 , − − ∆ − ∆ ) G − ,ǫ ∆ +∆ ( z , ¯ z )+ B (∆ , − − ∆ − ∆ ) G − , − ǫ ∆ +∆ ( z , ¯ z ) (cid:3) . (106)The graviton OPEs are G + ,ǫ ∆ ( z , ¯ z ) G ± ,ǫ ∆ ( z , ¯ z ) ∼ − κ z z B (∆ − , ∆ + 1 ∓ G ± ,ǫ ∆ +∆ ( z , ¯ z ) ,G + ,ǫ ∆ ( z , ¯ z ) G ± , − ǫ ∆ ( z , ¯ z ) ∼ κ z z (cid:2) B (∆ + 1 ∓ , ± − ∆ − ∆ ) G ± ,ǫ ∆ +∆ ( z , ¯ z )+ B (∆ − , ± − ∆ − ∆ ) G ± , − ǫ ∆ +∆ ( z , ¯ z ) (cid:3) . (107)The gluon-graviton OPEs are G + ,ǫ ∆ ( z , ¯ z ) O ± a,ǫ ∆ ( z , ¯ z ) ∼ − κ z z B (∆ − , ∆ + 1 ∓ O ± a,ǫ ∆ +∆ ( z , ¯ z ) ,G + ,ǫ ∆ ( z , ¯ z ) O ± a, − ǫ ∆ ( z , ¯ z ) ∼ − κ z z (cid:2) B (∆ + 1 ∓ , ± − ∆ − ∆ ) O ± a,ǫ ∆ +∆ ( z , ¯ z ) − B (∆ − , ± − ∆ − ∆ ) O ± a, − ǫ ∆ +∆ ( z , ¯ z ) (cid:3) . (108)From (8), we recall a factor of g Y M is absorbed in f abc . The ¯ z → celestial OPEs areobtained in a similar way by imposing the δ symmetry instead.The presence of higher-dimension operators due to quantum, stringy or other correctionsis expected to augment this list with the finite number of additions allowed by the generalformula (13). A list of all possible corrections in theories with only gluons and gravitons isgiven in appendix B. Acknowledgements
We are grateful to Nima Arkani-Hamed, Prahar Mitra, Sabrina Pasterski, Andrea Puhm,Shu-Heng Shao, Mark Spradlin and Tom Taylor for useful conversations. This work was24upported by NSF grant 1205550 and the John Templeton Foundation. M.P. acknowledgesthe support of a Junior Fellowship at the Harvard Society of Fellows.
A Celestial OPEs from bulk three-point vertices
In this appendix we relate the conformal weights of the operators which are allowed to appearin the OPE of two conformal primaries to the bulk dimensions of the corresponding three-point vertices. We consider a bulk three-point vertex among gluons and gravitons whichschematically takes the form V = ∂ m Φ ( x )Φ ( x )Φ ( x ) , (109)where Φ , Φ , Φ ∈ { A µ , h µν } , and we omitted Lorentz indices which should be contractedaccordingly. m is the total number of derivatives in the interaction, which are appropriatelydistributed among Φ , Φ , Φ . Since both gluons and gravitons have dimension , the netdimension of the vertex is d V = 3 + m. (110)Suppose Φ , Φ are taken to be outgoing external legs (on-shell states). In momentumspace, each derivative is associated with a factor of momentum. Upon parametrizing mo-menta as in (2), Mellin transforming with respect to ω and ω and taking the collinear limit z → , the celestial amplitude takes the general form e A = X α,β Z ∞ dω Z ∞ dω ω ∆ − ω ∆ − ω m + α ω β ω α + βP ω ω F α,β ( z , ¯ z , z , ¯ z ; · · · ) , (111)where we used momentum conservation and accounted for the Φ propagator. Since we’reworking in a collinear expansion, F α,β depends only on ω P , but not ω or ω independently.In general, the amplitude involves a sum over terms with different α, β . The details dependon the precise form of the interaction but turn out to be irrelevant in determining the scalingdimension of the allowed operators. Setting ω = ω P t, ω = ω P (1 − t ) , (112)the celestial amplitude becomes e A = X α,β B (∆ + m + α − , ∆ + β − Z ∞ dω P ω ∆ +∆ − mP F α,β ( z , ¯ z , z , ¯ z ; ω P , · · · ) . (113)25his allows one to read off the scaling dimension of the associated operator in the OPEexpansion ∆ − + ∆ − m = ⇒ ∆ = ∆ + ∆ + d V − , (114)where in the last equation we used (110). We therefore conclude that the primaries in the Φ , Φ OPE can be classified according to the dimension of the possible corresponding bulkthree-point vertices as in (13).
B Higher order OPEs
There is a finite number of primaries which contribute to the OPE (13) to any finite order inthe z expansion. To get a flavor of this, in this appendix we collect all possible single-poleor finite terms. For the gluon-gluon OPEs these are O + a ∆ ( z , ¯ z ) O + b ∆ ( z , ¯ z ) : 1 z O + c ∆ +∆ − ( z , ¯ z ) , ¯ z z O − c ∆ +∆ +1 ( z , ¯ z ) , ¯ z O + c ∆ +∆ +1 ( z , ¯ z ) , ¯ z O − c ∆ +∆ +3 ( z , ¯ z ) ,G +∆ +∆ ( z , ¯ z ) , ¯ z z G − ∆ +∆ +2 ( z , ¯ z ) , ¯ z G − ∆ +∆ +4 ( z , ¯ z ) ,O + a ∆ ( z , ¯ z ) O − b ∆ ( z , ¯ z ) : 1 z O − c ∆ +∆ − ( z , ¯ z ) , ¯ z O − c ∆ +∆ +1 ( z , ¯ z ) , ¯ z z G − ∆ +∆ ( z , ¯ z ) , ¯ z G − ∆ +∆ +2 ( z , ¯ z ) . (115)Operators on the right hand side of dimension ∆ +∆ − arise from the pure YM three-pointvertices while those of dimension ∆ + ∆ from three-point vertices in EYM (excluding thethree-gluon vertex). All other operators of dimensions ∆ + ∆ + n, n = 1 , ..., correspondto the following higher derivative vertices in order: F , RF , ∂ F , ∂ RF . Similarily, the finite or single pole terms in the graviton-graviton OPE are G +∆ ( z , ¯ z ) G +∆ ( z , ¯ z ) : ¯ z z G +∆ +∆ ( z , ¯ z ) , ¯ z G +∆ +∆ +2 ( z , ¯ z ) , ¯ z z G − ∆ +∆ +4 ( z , ¯ z ) , ¯ z G − ∆ +∆ +6 ( z , ¯ z ) ,G +∆ ( z , ¯ z ) G − ∆ ( z , ¯ z ) : ¯ z z G − ∆ +∆ ( z , ¯ z ) , ¯ z G − ∆ +∆ +2 ( z , ¯ z ) . (116)26perators of dimensions ∆ + ∆ + n, n = 2 , , arise from the following higher derivativevertices in order: R , R , ∂ R . The coefficient of the R term can be eliminated by fieldredefinition [48].The finite or single pole terms in the gluon-graviton OPEs are G +∆ ( z , ¯ z ) O + a ∆ ( z , ¯ z ) : ¯ z z O + a ∆ +∆ ( z , ¯ z ) , ¯ z z O − a ∆ +∆ +2 ( z , ¯ z ) , ¯ z O + a ∆ +∆ +2 ( z , ¯ z ) , ¯ z O − a ∆ +∆ +4 ( z , ¯ z ) ,G +∆ ( z , ¯ z ) O − a ∆ ( z , ¯ z ) : ¯ z z O − a ∆ +∆ ( z , ¯ z ) , O + a ∆ +∆ ( z , ¯ z ) , ¯ z O − a ∆ +∆ +2 ( z , ¯ z ) . (117)Operators of dimensions ∆ + ∆ + n, n = 2 , correspond to the higher derivative vertices RF and ∂ RF respectively. C Subleading soft gluon symmetry
Tree-level gauge theory amplitudes obey the soft relation (see [34] and references therein) A an +1 ( p , ..., p n ; q ) = (cid:0) J a (0) + J a (1) (cid:1) A n ( p , ..., p n ) + O ( q ) , (118)where J a (0) , J a (1) are the leading and subleading gluon soft factors and we suppressed all colorindices except for a , the one associated with the soft gluon. In this section we derive theaction of the subleading soft gluon symmetry on outgoing gluons in a conformal basis. Thesubleading soft gluon operators are J ± a (1) = n X k =1 i ε ± µ q ν J µνk q · p k T ak , (119)where T ak are the generators of the non-abelian gauge group in representation k . In theparametrization (2) and (9), (119) takes the form J − a (1) = n X k =1 z − ¯ z k (cid:18) − s k ω k + ∂ ω k + z − z k ω k ∂ z k (cid:19) T ak ,J + a (1) = n X k =1 z − z k (cid:18) s k ω k + ∂ ω k + ¯ z − ¯ z k ω k ∂ ¯ z k (cid:19) T ak . (120)27pon performing a Mellin transform we find J − a (1) = n X k =1 z − ¯ z k ( − h k + 1 + ( z − z k ) ∂ z k ) T ak P − k ,J + a (1) = n X k =1 z − z k (cid:0) − h k + 1 + (¯ z − ¯ z k ) ∂ ¯ z k (cid:1) T ak P − k , (121)where h k = 12 (∆ k + s k ) , ¯ h k = 12 (∆ k − s k ) , (122)and P − k implements the inverse shift on the k th operator to the one defined in (17). Treating z, ¯ z as independent complex variables, we can define the operators δ a ≡ I d ¯ z πi J − a (1) (0 , ¯ z ) = lim ∆ → ∆ I d ¯ z πi O − a ∆ (0 , ¯ z ) , ¯ δ a ≡ I dz πi J + a (1) ( z,
0) = lim ∆ → ∆ I dz πi O + a ∆ ( z, , (123)which have the following action on gluons δ a O ± b ∆ k ( z k , ¯ z k ) = − if bac (2 h k − z k ∂ z k ) O ± c ∆ k − ( z k , ¯ z k ) , ¯ δ a O ± b ∆ k ( z k , ¯ z k ) = − if bac (cid:0) h k − z k ∂ ¯ z k (cid:1) O ± c ∆ k − ( z k , ¯ z k ) . (124)Equations (124) define a global symmetry associated with the subleading soft gluon theoremand constrain the gluon OPE coefficients as in (27). J (1) receives corrections in the presence of gravitons. These can be deduced from thevertex (69) in which case (120) becomes J − a (1) = n X k =1 z − ¯ z k (cid:18) − s k ω k + ∂ ω k + z − z k ω k ∂ z k (cid:19) T ak + κ z − z k ¯ z − ¯ z k F − ak − κ z − z k ¯ z − ¯ z k G − ak ,J + a (1) = n X k =1 z − z k (cid:18) s k ω k + ∂ ω k + ¯ z − ¯ z k ω k ∂ ¯ z k (cid:19) T ak + κ z − ¯ z k z − z k F + ak − κ z − ¯ z k z − z k G + ak , (125)where F ± ak | p k , s k = ∓ , a k i = δ aa k | p k , s k = ∓ i , F ± ak | p k , s k = ± , a k i = 0 , G ± ak | p k , s k = ± i = δ aa k | p k , s k = ± , a k i , G ± ak | p k , s k = ∓ i = 0 . (126)This implies that (46) obeys lim ∆ → ∆ O + a ∆ ( z , ¯ z ) O − b ∆ ( z , ¯ z ) = if abc z ∆ O − c ∆ − ( z , ¯ z ) + κ z z δ ab G − ∆ ( z , ¯ z ) , (127)which fixes the normalization of the graviton OPE coefficient.28 Subsubleading soft graviton symmetry
In this section we derive the symmetry actions (36) and (43) (for outgoing particles) fromthe subsubleading soft graviton theorem.Tree-level gravity amplitudes were shown in [1] to obey the following soft relation A n +1 ( p , ..., p n ; q ) = (cid:0) S (0) + S (1) + S (2) (cid:1) A n ( p , ..., p n ) + O ( q ) , (128)where S (0) , S (1) and S (2) are the leading, subleading and subsubleading soft factors respec-tively. In this appendix we focus on the subsubleading soft factor, S (2) = − κ n X k =1 ε µν ( q ρ J ρµk )( q σ J σνk ) q · p k , (129)where ε µν and q are the polarization and momentum of the soft graviton and J i , p i are thetotal angular momenta and momenta of the hard particles. Using the parametrizations (2)of momenta and the angular momentum operators in [35], (129) can be shown to reduceto [25, 49] S − (2) = − κ n X k =1 ωω k z − z k )(¯ z − ¯ z k ) (cid:2) ( z − z k )( s k − ω k ∂ ω k ) − ( z − z k ) ∂ z k (cid:3) ,S +(2) = − κ n X k =1 ωω k z − z k )(¯ z − ¯ z k ) (cid:2) (¯ z − ¯ z k )( − s k − ω k ∂ ω k ) − (¯ z − ¯ z k ) ∂ ¯ z k (cid:3) (130)for negative and positive helicity soft gravitons respectively. In a conformal basis, (130)become e S − (2) = − κ n X k =1 z − z k ¯ z − ¯ z k (cid:2) h k (2 h k − − z − z k )2 h k ∂ z k + ( z − z k ) ∂ z k (cid:3) P − k , e S +(2) = − κ n X k =1 ¯ z − ¯ z k z − z k (cid:2) h k (2¯ h k − − z − ¯ z k )2¯ h k ∂ ¯ z k + (¯ z − ¯ z k ) ∂ z k (cid:3) P − k , (131)with h k , ¯ h k and P − defined in appendix C. Treating z, ¯ z as independent complex variables,we define the soft operators δ ≡ I d ¯ z πi ∂ z e S − (2) (0 , ¯ z ) = lim ∆ →− (∆ + 1) I d ¯ z πi ∂ z G − ∆ (0 , ¯ z ) , ¯ δ ≡ I dz πi ∂ ¯ z e S +(2) ( z,
0) = lim ∆ →− (∆ + 1) I dz πi ∂ ¯ z G +∆ ( z, , (132)29hich act on celestial operators as follows δ O ∆ k ,s k ( z k , ¯ z k ) = − κ (cid:2) h k (2 h k −
1) + 8 h k z k ∂ z k + 3 z k ∂ z k (cid:3) O ∆ k − ,s k ( z k , ¯ z k ) , ¯ δ O ∆ k ,s k ( z k , ¯ z k ) = − κ (cid:2) h k (2¯ h k −
1) + 8¯ h k ¯ z k ∂ ¯ z k + 3¯ z k ∂ z k (cid:3) O ∆ k − ,s k ( z k , ¯ z k ) . (133)Equations (133) define the action of the global symmetries associated with the subsubleadingsoft graviton theorem (36) and (43). They constrain the form of the graviton and graviton-gluon OPEs (33) and (42) as discussed in sections 3.2 and 3.3. E Solving the recursion relations
Consider a symmetric function of complex variables C (∆ , ∆ ) = C (∆ , ∆ ) which obeysthe recursion relation ∆ C (∆ , ∆ ) = (∆ + ∆ ) C (∆ + 1 , ∆ ) . (134)Provided C (∆ , ∆ )Γ(∆ + ∆ ) is holomorphic for Re(∆ ) > and bounded for Re(∆ ) ∈ [1 , , (134) has the unique solution C (∆ , ∆ ) = C (1 , B (∆ , ∆ ) . (135)This can be proven in the following way. Define a function f ( x ) ≡ C ( x, y )Γ( x + y ) . Then(134) becomes xf ( x ) = f ( x + 1) . (136)By Wieland’s theorem [50], (136) has the unique solution f ( x ) = f (1)Γ( x ) . (137)Eliminating f ( x ) , f (1) in terms of C ( x, y ) , C (1 , y ) we find C ( x, y ) = C (1 , y )Γ(1 + y )Γ( x )Γ( x + y ) . (138)For y = 1 , (138) implies C ( x, x ) = Γ( x ) C (1 , . (139)30ow replacing x with y and using symmetry in the arguments of C ( x, y ) we deduce that C ( x, y ) = C (1 ,
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