PLB34580S0370-2693(19)30260-610.1016/j.physletb.2019.03.063The Author(s)TheorySymmetries of celestial amplitudesStephanStiebergera⁎stephan.stieberger@mpp.mpg.deTomasz R.TaylorbaMax–Planck–Institut für Physik, Werner–Heisenberg–Institut, 80805 München, GermanyMax–Planck–Institut für PhysikWerner–Heisenberg–InstitutMünchen80805GermanybDepartment of Physics, Northeastern University, Boston, MA 02115, USADepartment of PhysicsNortheastern UniversityBostonMA02115USA⁎Corresponding author.Editor: N. LambertAbstractCelestial amplitudes provide holographic imprints of four-dimensional scattering processes in terms of conformal correlation functions on a two-dimensional sphere describing Minkowski space at null infinity. We construct the generators of Poincaré and conformal groups in the celestial representation and discuss how these symmetries are manifest in the amplitudes.Celestial amplitudes provide holographic imprints of four-dimensional scattering processes in terms of conformal correlation functions on a two-dimensional celestial sphere describing Minkowski space at null infinity [1]. In these amplitudes, Lorentz symmetry is realized as the SL(2,C) conformal symmetry of the celestial sphere. They are particularly interesting in the soft limit, when one or more particles carry zero energy. In this limit, the well-known soft theorems can be interpreted as Ward identities of 2D CFT currents associated to asymptotic symmetries of four-dimensional spacetime [2–4]. Beyond the soft limit, several examples of celestial amplitudes have been recently discussed in [5–7]. The underlying 2D CFT may be very complicated but it is worth studying because it could lead to a holographic description of gauge theories and of (at least) some aspects of perturbative quantum gravity in asymptotically flat spacetimes. In the present work, we address the question how four-dimensional Poincaré and conformal symmetries are realized at the level of 2D celestial amplitudes. We construct the symmetry generators in the celestial representation. One of their interesting features is the presence of operators that shift conformal dimensions. In particular, the momentum (spacetime translation) operators involve such shifts.We first review the steps leading from “old-fashioned” to celestial amplitudes. We will be considering the scattering processes involving massless gauge bosons and gravitons. Their asymptotic momenta, long before or after gauge/gravitational interactions take place, can be parametrized as(1)pμ=ωqμ,withqμ=12(1+|z|2,z+z¯,−i(z−z¯),1−|z|2), where ω are the (light-cone) energies and (z,z¯) are complex kinematic variables that determine momentum directions. On the other hand, the celestial sphere describing Minkowski spacetime at null infinity, from where these particles emerge and to where they head after they interact, is a Riemann sphere parameterized by complex coordinates. The starting point for constructing celestial amplitudes is the identification of kinematic variables (z,z¯) of Eq. (1) as the coordinates of points on celestial sphere. It follows that four-dimensional Lorentz group is realized as SL(2,C) conformal symmetry of celestial sphere,(2)z→az+bcz+d(ad−bc=1). In this framework, it is natural to replace the asymptotic in and out plane wave functions by the so-called conformal wave packets characterized by (z,z¯) and two-dimensional conformal weights (h,h¯), with the conformal spin J=h−h¯ identified as the helicity of the particle [8]. Their dimensions Δ=h+h¯ are restricted by the requirement of normalizability to the so-called principal series with Δ=1+iλ, λ∈R.At the level of scattering amplitudes, the change of asymptotic in and out basis from plane waves to conformal packets is accomplished by Mellin transformations with respect to the energies:(3)A˜{hn,h¯n}(zn,z¯n)=(∏n=1N∫0∞ωnΔn−1dωn)δ(4)(ω1q1+ω2q2−∑k=3Nωkqk)M(ωn,zn,z¯n), with the dimensions Δn=hn+h¯n dual to ωn. Here, M are the N-particle invariant matrix elements describing particles 1 and 2 scattering into N−2 final particles.11Our discussion applies though to all scattering channels. We omit the factor i(2π)4. For notation, conventions and a general introduction into the subject, see [9]. They depend on all quantum numbers, including internal gauge charges, and may contain some group-dependent (a.k.a. color) factors. In this case, we will be extracting purely kinematic “partial” (or “stripped”) amplitudes associated to individual Chan-Paton trace factors. Thus A˜(i,j,k,…) denotes the celestial amplitude associated to Tr(TaiTajTak⋯) [9].The celestial amplitudes defined in Eq. (3) transform under conformal SL(2,C) transformations like the correlation functions of N conformal primary fields with weights (hn,h¯n). It is clear that Lorentz invariance of underlying amplitudes must be reflected in conformal Ward identities [10]. This helps in identifying the Lorentz generators(4)L1≡M23+iM10=(1−z2)∂z−2zh,−M23+iM10=L¯1L2≡M20+iM13=(1+z2)∂z+2zh,−M20+iM13=L¯2L3≡M21+iM03=2(z∂z+h),−M21+iM03=L¯3, which obey the usual su(1,1) commutation relations(5)[L1,L2]=2L3[L2,L3]=2L1[L3,L1]=−2L2. Indeed, as a consequence of conformal Ward identities, any N-point celestial amplitude satisfies the requirements of Lorentz invariance(6)LIA˜N=L¯IA˜N=0,LI=∑n=1NLI,n, where LI,n, I=1,2,3, are the Lorentz transformations (4) acting on the coordinates of nth particle.In standard amplitudes, the momentum operator acts as multiplication by pμ written in Eq. (1). Multiplication by an ω energy factor yields a shift of conformal weights: (h,h¯)→(h+1/2,h¯+1/2), cf. Eq. (3). Hence the momentum generators are realized as(7)P0=(1+|z|2)e(∂h+∂h¯)/2P1=(z+z¯)e(∂h+∂h¯)/2P2=−i(z−z¯)e(∂h+∂h¯)/2P3=(1−|z|2)e(∂h+∂h¯)/2 and the momentum conservation reads(8)PμA˜N=0,Pμ=Pμ,1+Pμ,2−∑n=3NPμ,n, where Pμ,n, μ=0,1,2,3, act on the coordinates of nth particle. The operators (4) and (7) generate the Poincaré group. It is easy to check that all celestial amplitudes written explicitly in Refs. [5–7] are Poincaré invariant; they satisfy Eqs. (6) and (8). More details are given in the Appendix.While shifting conformal weights may seem as a trivial operation, it affects the ultra-violet behavior of Mellin transforms. Yang-Mills amplitudes are “mariginally” convergent, with the overall energy scale integral [5–7](9)∫0∞ω(∑n=1NΔn−N−1)dω=2πδ(∑n=1Nλn) (recall that Δn=1+iλn). A shift of conformal dimension Δn→Δn+1 induced by the momentum operator Pμ,n results in a linearly divergent integral. As shown in Ref. [7], such divergences can be avoided by treating the amplitudes as α′→0 limits of superstring amplitudes. This works because superstring theory is “supersoft” in the ultra-violet: all scattering amplitudes are exponentially suppressed at high energies. In Ref. [7], some four-point gravitational amplitudes have been discussed by using such a superstring embedding. Each power of the gravitational coupling constant (with mass dimension −1) brings an energy factor hence at the level of celestial amplitudes, it has the same effect as the momentum operator. Seen in this way, celestial gravitational amplitudes appear as Yang-Mills amplitudes translated in space-time. As an example, the well-known relation [11] between the Einstein-Yang-Mills amplitude with a single graviton G and pure gauge amplitudes can be written as(10)gA˜(1,2,…,N,G±±)=κ∑l=1N−1(ϵG±μXμ,l)A˜(1,2,…,l,G±,l+1,…,N), where(11)Xμ,l=Pμ,1+Pμ,2−∑n=3lPμ,n, while g and κ are the gauge and gravitational couplings, respectively. The polarization vectors are given by(12)ϵGμ+=12(z−w)(1+z¯w,w+z¯,−i(w−z¯),1−z¯w),ϵGμ−=(ϵGμ+)⁎, where w is a reference point on celestial sphere. The amplitude (10) does not depend on this point as a consequence of gauge invariance [11] which, in this case, follows from the Bern-Carrasco-Johansson relations [12]. It is remarkable that the relations (10) hold in full-fledged heterotic superstring theory, to all orders in the α′ expansion [13].At the tree-level, Yang-Mills theory is scale-invariant. Accordingly, the tree-level helicity amplitudes are invariant under four-dimensional conformal transformations [14]. The celestial representation of special conformal generators can be deduced in a similar way as in Ref. [14]. We find(13)K0=[∂z∂z¯+(z∂z+2h−1)(z¯∂z¯+2h¯−1)]e−(∂h+∂h¯)/2K1=[(z∂z+2h−1)∂z¯+(z¯∂z¯+2h¯−1)∂z]e−(∂h+∂h¯)/2K2=−i[(z∂z+2h−1)∂z¯−(z¯∂z¯+2h¯−1)∂z]e−(∂h+∂h¯)/2K3=[∂z∂z¯−(z∂z+2h−1)(z¯∂z¯+2h¯−1)]e−(∂h+∂h¯)/2. After computing all commutators one finds that indeed, the operators LI,L¯I,Pμ,Kμ of Eqs. (4), (7) and (13), supplemented by the dilatation generator(14)D=−i(h+h¯−1), generate full conformal group [10]. The simplest way of verifying Eq. (14) is by computing [K1,P1]=2iD.Due to the complicated structure of the special conformal generators, cf. Eq. (13), it is a tedious, although straightforward, exercise to show that the tree-level Yang-Mills amplitudes possess the symmetry(15)KA˜YM=0,K=Kμ,1+Kμ,2−∑n=3NKμ,l. On the other hand, it is trivial to see that they are dilatation invariant. Since Δ=1+iλ,(16)DA˜YM=0,D=∑n=1Nλn, due to the universal delta function (9) present in all tree-level Yang-Mills amplitudes [5–7].In this work, we explained how Poincaré and conformal symmetries are realized in celestial amplitudes. In this formalism, gravitational amplitudes appear from space-time translations of pure gauge amplitudes, indicating that celestial CFT will be helpful in studying connections between gauge theories and gravity.AcknowledgementsWe are grateful to Wei Fan, Angelos Fotopoulos, Sabrina Pasterski and Andy Strominger for useful conversations and communications. This material is based in part upon work supported by the National Science Foundation under Grant Number PHY–1620575. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.Appendix AWe wish to expand the argument why celestial amplitudes are invariant under translations generated by the momentum operators (7). To that end, we consider one particular generator,(17)P+=12(P0+P3)=e(∂h+∂h¯)/2,P+=P+,1+P+,2−∑n=3NP+,n. The net effect of P+ is to shift the conformal dimension Δ→Δ+1 or equivalently iλ→iλ+1. Formally, P+ acting on the amplitude (3), introduces the factor (ω1+ω2−∑k=3Nωk) under the Mellin integral. This factor is annihilated by the energy-conserving delta function, therefore P+A˜=0. More caution should be exercised however because of possible convergence problems of Mellin transforms, therefore it is a good idea to have a closer look at some specific examples.22We are grateful to Andy Strominger for suggesting this Appendix. One such example is the tree-level, four-gluon Yang-Mills MHV amplitude that was Mellin-transformed into a celestial form in Refs. [5–7]. In the notation of Ref. [7], it reads(18)A˜(−,−,+,+)=A(z,z¯,λ)J0(γ),γ=i∑n=14λn, where(19)A(z,z¯,λ)=δ(r−r¯)(z24z¯13)iλ1(z¯24z13)iλ3(z¯34z12)i(λ1+λ2)×(z14z¯32)i(λ2+λ3)4r3z¯122z342, with zij≡zi−zj and the real cross-ratio(20)r=r¯=z12z34z23z41 constrained to the kinematic domain of r>1. In Eq. (18),(21)J0(γ)=∫0∞ωγ−1dω, see also Eq. (9). Acting on the amplitude (18), each momentum generator P+ shifts γ→γ+1 and yields a factor rational in the z-coordinates. As a result,(22)P+A˜(−,−,+,+)=[z24z¯34z12z¯13+z14z¯34z12z¯32−z14z¯24z13z¯32−1]×A(z,z¯,λ)J0(γ+1). After simple algebraic manipulations using the reality constraint r=r¯, one finds that the expression inside the square bracket is zero. A similar argument can be repeated for the remaining momentum components, thus demonstrating translational invariance of the celestial MHV amplitude. 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