Soft Photon theorem in the small negative cosmological constant limit
SSoft Photon theorem in the small negativecosmological constant limit
Nabamita Banerjee, a Karan Fernandes, b and Arpita Mitra. a a Indian Institute of Science Education & Research Bhopal,Bhopal Bypass Road, Bhauri, Bhopal 420 066,Madhya Pradesh, India. b Harish-Chandra Research Institute,Chhatnag Road, Jhusi, Prayagraj 211019,Uttar Pradesh, India
E-mail: [email protected], [email protected],[email protected]
Abstract:
We study the effect of electromagnetic interactions on the classical soft the-orems on an asymptotically AdS background in 4 spacetime dimensions, in the limit of asmall cosmological constant or equivalently a large AdS radius l . This identifies 1 /l per-turbative corrections to the known asymptotically flat spacetime leading and subleadingsoft factors. Our analysis is only valid to leading order in 1 /l . The leading soft factorcan be expected to be universal and holds beyond tree level. This allows us to derive a1 /l corrected Ward identity, following the known equivalence between large gauge Wardidentities and soft theorems in asymptotically flat spacetimes. a r X i v : . [ h e p - t h ] M a r ontents /l correction of the large gauge Ward identity 28 Soft theorems are statements about quantum scattering amplitudes when one or more ofthe external particles go soft, i.e. their momenta k µ → . These theorems state that a( m + n ) point scattering amplitude A m + n , where m number of external particles go soft,is proportional to A n , the n point scattering amplitude involving the other hard particles.The proportionality factor is universal at leading order, irrespective of the details of theinteractions [1, 2]. The soft factor is also divergent at leading order in the soft momentaexpansion and depends on properties of the hard particles. The theorems are valid for anygauge invariant quantum field theory in any spacetime dimensions. In particular, the softphoton theorem follows from the U(1) gauge invariance [3, 4] and the soft graviton theoremfollows from the diffeomorphism invariance of quantum field theories [5–7]. Recent works[8–19] have extended soft theorems beyond leading orders, with results for subleading andsub-subleading soft theorems in gravitational and U(1) gauge theories. In four spacetimedimensions there exists an additional subtlety, in that the subleading soft factor divergesas the logarithm of the frequency due to the existence of asymptotically non-vanishing longrange interactions. To be precise, below we write the exact statement of the single softphoton theorem to subleading order in four spacetime dimensions [20]:– 1 – flatem = S flatem;leading + S flatem;subleading , with S flatem;leading = n (cid:88) a =1 q ( a ) (cid:15) µ p µ ( a ) p ( a ) .k , (1.1) S flatem;subleading = i n (cid:88) a =1 q ( a ) (cid:15) ν k ρ j ρν ( a ) p ( a ) .k = i ln ω − n (cid:88) a =1 q ( a ) (cid:15) ν k ρ (cid:18) c ρ ( a ) p µ ( a ) − c µ ( a ) p ρ ( a ) (cid:19) p ( a ) .k + · · · (1.2)where k µ and (cid:15) µ are respectively the momentum and polarization of the soft photon, while q ( a ) , p µ ( a ) and j µν ( a ) are the charges, asymptotic momenta and angular momenta of the n hardparticles. In going from the first to second equality of Eq. (1.2), we have an expansion interms of the classical trajectories of the hard particles. The individual trajectories involvelogarithmic contributions in four dimensions due to the presence of long range interactionsof the electromagnetic fields. We can expand the classical trajectories of these particles tofind the following leading order contribution in proper time r µ ( a ) ( t ) = η ( a ) p µ ( a ) m ( a ) + c µ ( a ) ln | t | + · · · , where t is the proper time along the particle trajectories, η ( a ) is +1 for incoming particlesand − m ( a ) are the particle masses and c µ ( a ) are coefficients whichdepend on the long range electromagnetic force. Using this expression for r µ ( a ) ( t ) in j µν ( a ) = r µ ( a ) ( t ) p ν ( a ) − r ν ( t ) p µ ( a ) then provides the term in the second line of Eq. (1.2) on replacing t with ω − . The additional terms not described in the second line of Eq. (1.2) are quantumcorrections. These terms can be ignored as long as they are much smaller than the classicalscattering contribution. This will be the case when the wavelength of the soft particlesare much larger than the impact parameter involved in the scattering and when the totalradiated energy is less than the energy of the scatterer. In such cases, we can derivethe universal contributions entirely from the low frequency limit of the (gauge invariant)classical radiative fields. This relation is provided by the classical soft photon theorem[20–22], lim ω → (cid:15) µ ˜ a µ ( ω , (cid:126)x ) = e iωR (cid:16) ω πiR (cid:17) D − ω S flatem = − i πR e iωR S flatem for D = 4 , (1.3)where ˜ a µ is the radiative component of the electromagnetic field in frequency space, D isthe spacetime dimension and R denotes the distance of the soft photon from the scatterer.A similar analysis holds for the soft graviton theorem [20, 23–25].– 2 –n interesting question to ask is: how does the above story change when we study a quan-tum field theory in an asymptotically non-flat background? Since the gauge invarianceremains intact even for asymptotically non-flat theories, we expect a version of soft the-orems to be valid in this case as well. Non-asymptotically flat backgrounds, particularlyof the kind of asymptotically Anti-de Sitter (AdS) or de Sitter (dS) types are of greatimportance in physics. AdS arises as an interesting gravity background for some exactcomputations in the context of String Theory and AdS/CFT conjectures. On the otherhand dS spacetime has its importance in cosmology. Thus, understanding aspects of softtheorems in these spacetimes are important. In this paper we shall look for classical softphoton theorems in asymptotically AdS spacetime, where the radius of the AdS space isconsidered to be large (we shall make this condition more precise in later sections). Ourresults, with slight modifications, are also valid for asymptotically dS spacetimes in thelarge radius limit.AdS (dS) is a solution of Einstein’s gravity with a negative (positive) cosmological constant.AdS spacetimes have an effective potential under which particles behave like being con-fined in a box. The null rays bounce back from the timelike boundary an infinite numberof times. This creates the main obstacle in defining the usual “in” and “out” states for aquantum field theory in AdS backgrounds. Thus, unlike in asymptotically flat theories, thedefinition of the usual scattering amplitudes [26–29] and hence a soft theorem is not knownfor quantum field theories defined in an asymptotically AdS spacetime . Instead we lookfor a possible soft factorization on taking the classical limit of scattering amplitudes in AdSbackgrounds. This gives us the analogue of classical soft theorems known from asymptoti-cally flat spacetime classical scattering processes. While computing the classical radiationprofile, for technical simplification, we consider the value of cosmological constant to besmall , or equivalently the radius l of AdS large and treat it as a perturbation parameterin our computations. Our results are exact up to order 1 /l of the AdS radius. Physically,we think of studying a scattering process in an asymptotically flat theory modified by asmall potential (inversely proportional to the square of the AdS radius). Thus our resultsprovide us perturbative corrections to order 1 /l of known results for classical photon andgraviton radiation profiles in the asymptotically flat Reissner-Nordstr¨om case [31]. Thedetails of the scattering process we consider will be discussed in later sections.Finally to study the “soft limit” of the classical radiation in asymptotically AdS spacetime,we consider a double scaling limit [32]: where the frequency of radiation and cosmologicalconstant simultaneously tend to zero, keeping their ratio finite. This is due to the fact thata radiation mode in a theory that asymptotes to AdS spacetime has a minimum frequencyinversely proportional to the size of the AdS and hence the frequency of the radiationcannot limit to a zero value. In other words, there is a mass gap in AdS that restricts theusual soft limit. Physically the double scaling limit implies that we consider the radiationlimits to a strictly soft one as the space is limiting to an asymptotically flat spacetime.By taking this limit, we find the classical soft photon theorem to leading and sub-leading a related work can be found in [30] – 3 –rder in an asymptotically AdS theory, in the large AdS radius limit.On asymptotically flat spacetimes, Weinberg’s soft theorems for scattering amplitudes areknown to be equivalent to Ward identities for large gauge transformations [4], [8],[33–36].These identities represent the soft charge conservation across null infinity I . For asymp-totically flat spacetimes, it is well known by now that the classical soft factor provide thesame leading (quantum) soft factor in the classical limit up to the usual gauge ambigu-ity. Hence we can as well recast the classical soft theorem in terms of the large gaugeWard identity. In our present study, as the classical soft photon theorem receives a 1 /l correction (to its flat space form) in taking the l → ∞ limit of AdS spacetimes, we ex-pect the equivalence to imply a 1 /l correction of the usual large gauge Ward identity.This physically amounts to deriving 1 /l perturbative modifications in the Ward identitythat correctly reproduces the modified classical soft theorem. A formal derivation of thelarge gauge Ward identity in AdS spacetime is complicated by the fact that there doesnot appear to be a unique large r saddle point corresponding to the low frequency result,unlike in asymptotically flat spacetimes. As mentioned, the 1 /l corrections to the softfactor results from a double scaling limit on the cosmological constant and frequency, thusreceiving contributions across different length scales. We can nevertheless infer this Wardidentity from the soft photon theorem, following the procedure used in [4] to demonstratethe equivalence on asymptotically flat spacetimes. We find that specific corrections of thesoft photon mode and gauge parameter that provide a Ward identity equivalent to theclassical soft photon theorem up to 1 /l corrections.The current paper generalizes our previous work [32] on the effect of the small AdS potentialon the classical soft graviton theorem by including an electromagnetic interaction. We alsofind the effect of the small AdS potential on the classical soft photon theorem. The paperis organised as follows: In section 2, we review basic properties of AdS Reissner-Nordstr¨omspacetime and then study its perturbations by introducing a charged point probe particle.In section 3, we have obtained the solution to the gauge and gravity radiations. Next insection 4, we study the soft limit and extract the classical soft photon factor from classicalradiation profile. In taking a similar limit, in section 5 we state the results for the classicalsoft graviton factor. It turns out that the charge of the central black hole, considered as thescatterer in the classical probe scattering process, has no explicit effect on the soft gravitonfactor. Finally in section 6, we find the Ward identity of large gauge transformations,perturbatively modified to 1 /l order using our classical soft photon factors that we derivedin section 4. We end the paper with a conclusion and some interesting open questions insection 7. Appendix A contains the computation details for the classical soft gravitonfactor. In this paper, we are interested in studying the classical soft photon theorem in asymptot-ically AdS backgrounds. To achieve this, we study the classical scattering of a charged andmassive probe particle by a Reissner-Nordstr¨om black hole placed in asymptotically AdS– 4 –pacetime in 4 spacetime dimensions. The equations of motions result from the action,which consists of the Einstein-Hilbert term and the Maxwell term, S = 116 πG (cid:90) d x √− g ( R − − π (cid:90) d x √− g F µν F µν . (2.1)In Eq. (2.1) R is the Ricci scalar for metric g µν , Λ is the cosmological constant, G isNewton’s constant and F µν = A ν,µ − A µ,ν is the field strength tensor of the electromagneticfield A µ . We use the standard convention of denoting partial derivatives by subscriptedcommas and covariant derivatives with semi-colons. Varying the action in Eq. (2.1) withrespect to the metric tensor one gets the Einstein equations, R µν − Rg µν + Λ g µν = 8 πGT EMµν , (2.2)where T EMµν = 14 π (cid:18) F µα F νβ g αβ − g µν F αβ F γδ g αγ g βδ (cid:19) . Similarly for the gauge field A µ we get the source-free Maxwell equations √− g π F µν ; ν = 0 . (2.3)The solutions of equations Eq. (2.2) and Eq. (2.3) for a static spherically symmetric space-time with mass M, charge Q and a negative cosmological constant Λ = − /l , provide themetric ds = − f ( r ) dt + dr f ( r ) + r ( dθ + sin θdφ ) , (2.4)with a gauge potential A = Qr . (2.5)The lapse function f ( r ) in global coordinates takes the form f ( r ) = 1 − GMr + GQ r − Λ r − GMr + GQ r + r l . (2.6)Since we are interested in studying the radiation emitted by the scattering of a probeparticle moving in an unbounded trajectory on the spacetime (from the point of view ofan asymptotic observer) we introduce isotropic coordinates. We refer the reader to [32] forfurther justification on choosing this particular coordinate system. In these coordinates,the resulting radiation will be isotropic in all spatial directions. We assume that the probeparticle with mass m ( (cid:28) M ) and charge q ( (cid:28) Q ) has a large impact parameter from theblack hole which implies M/r (cid:28)
Q/r (cid:28)
1. In addition, we also truncate our metricup to 1 /l terms, as we consider radiative solutions in the large cosmological constant limit.Therefore our result will be valid in the regime Q ≤ M (cid:28) r (cid:28) l . The Maxwell action is written using Heaviside units and further details can be found in the AppendixE of [37]. – 5 –he metric Eq. (2.4) in isotropic coordinates, expanded up to quadratic order in ρ , takesthe form ds = − g dt + g ij dx i dx j , (2.7)where g = − (cid:18) − GMρ + GQ ρ + ρ l (cid:19) , g i = 0 , g ij = δ ij (cid:18) GMρ − GQ ρ + ρ l (cid:19) . (2.8)Here ( i, j ) = 1 , ,
3, run over spatial directions and ρ = | (cid:126)x | . The isotropic coordinate ρ isrelated to the Schwarzschild coordinate ‘ r ’ by ρ = r (cid:18) − GMr + GQ r − r l (cid:19) (2.9)We now impose the assumptions discussed above to express the metric of Eq. (2.8) ina form relevant for our calculations. We set 8 πG = 1 in the following. Therefore wewill replace G by 1 / π in the remainder of the paper. The condition of a large impactparameter amounts to considering the leading order contribution of the gravitational andelectromagnetic potential, which we will denote by φ ( (cid:126)x ). The potential goes like r − andcan be defined either with respect to the mass or the charge. The analysis in this paper isindependent of either choice. We define φ ( (cid:126)x ) = − M πρ . (2.10)The gauge potential can then be expressed as A ( (cid:126)x ) = Qρ = − πQM φ ( (cid:126)x ) , (2.11)Retaining terms up to leading order in φ and 1 /l , we then find that the metric componentsin Eq. (2.8) take the form g = − (cid:18) φ + ρ l (cid:19) , g i = 0 , g ij = δ ij (cid:18) − φ + ρ l (cid:19) . (2.12)Note that the spacetime metric in Eq. (2.12) provides the leading AdS correction aboutan asymptotically flat spacetime and in isotropic coordinates it just behaves like the AdS-Schwarzschild metric with a gauge potential. The metric is however equivalent to themetric in Eq. (2.4) up to leading order in φ and 1 /l . In particular, the timelike boundaryof the full AdS spacetime is not part of the spacetime we have considered. In the remainingsections, we shall investigate the scattering of a charged probe particle in this backgroundgiven by the metric in Eq. (2.12) and gauge field in Eq. (2.11).– 6 – .1 Perturbations of Einstein-Maxwell equations We now linearly perturb the spacetime by introducing a point probe particle with mass‘ m ’ and charge ‘ q ’ moving along a worldline trajectory r ( σ ), [38–42] whose action is S P = − m (cid:90) dσ (cid:114) − g µν dr µ dσ dr ν dσ + q π (cid:90) dσA µ dr µ dσ . (2.13)where dr µ dσ = u µ is the tangent to the worldline of the probe and the metric is evaluated at r . Variation of Eq. (2.13) gives the following stress tensor T µν ( P ) 3 and current J µ ( P ) T µν ( P ) = 2 √− g δS P δg µν = m (cid:90) δ ( x, r ( σ )) dr µ dσ dr ν dσ dσ ,J µ ( P ) = 1 √− g δS P δA µ = q π (cid:90) δ ( x, r ( σ )) dr µ dσ dσ (2.14)where δ ( x, r ( σ )) is the covariant delta function. It is related to the flat spacetime deltafunction δ ( x − r ( σ )) via δ ( x, r ( σ )) √− g = δ ( x − r ( σ )) = δ ( t − r ( s )) δ (3) ( (cid:126)x − (cid:126)r ( σ )) . (2.15)and normalized as (cid:90) √− g δ ( x, r ( σ )) dσ = 1 . (2.16)The stress-energy tensor and current of the point particle induces a perturbation of thebackground metric and gauge potential, g µν → g µν + δg µν = g µν + 2 h µν ,A µ → A µ + δA µ = A µ + a µ . (2.17)The variations of Eq. (2.2) and Eq. (2.3) yield, δ ˜ G µν − δT hµν − δT aµν = T ( P ) µν (2.18) δ ( F µν ; ν ) = 4 πJ µ ( P ) , (2.19)where ˜ G µν = R µν − Rg µν + Λ g µν (2.20)In Eq. (2.18) we have split the total perturbation of the stress-energy tensor into twocomponents, one part δT hµν is due to the perturbation of the metric and another part δT aµν is due to the perturbation of the gauge potential. We have chosen σ to be proper time as measured in this spacetime and therefore u µ satisfies the relation g µν u µ u ν = − – 7 –n simplifying δ ˜ G µν , δT hµν , δT aµν and δ ( F µν ; ν ), we find δ ˜ G µν = − e µν ; αα + e µα ; αν + e ανα ; µ + (cid:16) R ν δ e δµ + R µδ e δν (cid:17) + 2 R ανµδ e δα − g µν e αβ ; αβ + g µν R αβ e αβ − Re µν + 2Λ e µν − Λ g µν eδT hµν = eT EMµν − π e µν F αβ F αβ − π g α(cid:15) g βδ e (cid:15)δ (cid:18) F αµ F βν − g µν F αγ F βγ (cid:19) δT aµν = 14 π g αβ (cid:18) f αµ F βν + f αν F βµ − g µν g γδ f αγ F βδ (cid:19) δ ( F µν ; ν ) = − g αρ g µν (cid:104) g βσ ( e ρσ F νβ ; α + e σν ; α F βρ ) − e ,ρ F να − f νρ ; α − F αν e ρβ ; β (cid:105) , (2.21)where we have denoted the perturbed electromagnetic field strength tensor by f µν = a ν,µ − a µ,ν , (2.22)and have introduced the trace-reversed metric perturbations e µν defined by e µν = h µν − hg µν ; h = g µν h µν = − e = − g µν e µν . (2.23)Substituting the first three expressions of Eq. (2.21) in Eq. (2.18), we find the followingexpression for the perturbed Einstein equation − T ( P ) µν = e µν ; αα − e µα ; αν − e ανα ; µ − (cid:16) R ν δ e δµ + R µδ e δν (cid:17) − R ανµδ e δα + Re µν − e µν + Λ g µν e − g µν R αβ e αβ + g µν e αβ ; αβ + 14 π g αβ (cid:18) f αµ F βν + f αν F βµ − g µν g γδ f αγ F βδ (cid:19) + eT EMµν − π e µν F αβ F αβ − π g α(cid:15) g βδ e (cid:15)δ (cid:18) F αµ F βν − g µν F αγ F βγ (cid:19) . (2.24)Similarly plugging the expression of δ ( F µν ; ν ) from Eq. (2.21) in Eq. (2.19) gives the per-turbed Maxwell equation − πg µν J ( P ) ν = g αρ g µν (cid:104) g βσ ( e ρσ F νβ ; α + e σν ; α F βρ ) − e ,ρ F να − f νρ ; α − F αν e ρβ ; β (cid:105) . (2.25)We will now express Eq. (2.24) and Eq. (2.25) about the background with the metricEq. (2.12) and gauge potential Eq. (2.11). We will also rewrite parts of the equations interms of the following quantities, k µ = e µν ; ν , b = − a , + a i,i . (2.26)The radiative components of gravitational and electromagnetic perturbations are spatial– 8 –n nature in isotropic coordinates. The spatial components of Eq. (2.24) are as follows, − T ( P ) ij = (cid:3) (cid:18)(cid:18) φ − ρ l (cid:19) e ij (cid:19) − k i,j − k j,i − (cid:32) φ ,k − ρ ,k l (cid:33) e ki ) ,j − (cid:32) φ ,k − ρ ,k l (cid:33) e kj ) ,i − ( k , − k l,l ) δ ij + 4 (cid:20) φe ij, + φ ,i e j, + φ ,j e i , + 12 (cid:18) φ ,ij − φ kk δ ij (cid:19) ( e + e ll ) (cid:21) + 2 δ ij ( φ ,kl e kl − φ ,k k k + 2 φk , ) + + δ ij l (cid:0) ρ ,kl e kl + ρ ,k k k + 2 ρ k , (cid:1) + 12 l (cid:20) ρ e ij, + ρ i e j , + ρ j e i , + ρ ij (2 e − e ll ) − δ ij ρ kk e ll + 32 ( ρ ki e kj + ρ kj e ki ) − ρ kk e ij + 32 ρ k e ij,k (cid:21) − QM ( f i φ ,j + f j φ ,i − δ ij f l φ ,l ) , (2.27)where (cid:3) = − ∂ + ∂ i . The spatial component of perturbed Maxwell equation Eq. (2.25) inthe AdS-Reissner-Nordstr¨om background gives, − πJ i ( P ) = − π (cid:18) φ − ρ l (cid:19) J ( P ) i = (cid:3) a i − b ,i + 3 ρ l ( a i, − a , i ) + (cid:18) φ − ρ l (cid:19) ( a i,kk − a k,ki ) − (cid:18) φ k + ρ k l (cid:19) f ik + 16 πQM (cid:18) e j φ ij + ( e ij, − e i ,j ) φ j + 12 ( e , − e kk, ) φ i + φ ,i k (cid:19) . (2.28)We now need to implement gauge choices. To this end, we adopt the following choice for k µ and b to simplify our equations Eq. (2.27) and Eq. (2.28), k µ = − (cid:18) φ k − ρ k l (cid:19) e kµ + 2 QM a φ ,µ (2.29) b = − (cid:18) φ k + ρ k l (cid:19) a k − (cid:18) φ + ρ l (cid:19) a k,k . (2.30)Using Eq. (2.29) one can simplify Eq. (2.27) as, (cid:3) (cid:18)(cid:18) φ − ρ l (cid:19) e ij (cid:19) + 4 (cid:20) φe ij, + φ ,i e j, + φ ,j e i , + 12 (cid:18) φ ,ij − φ kk δ ij (cid:19) ( e + e ll ) (cid:21) + 12 l (cid:20) ρ e ij, + ρ i e j , + ρ j e i , + ρ ij (2 e − e ll ) − δ ij ρ kk e ll + 32 ( ρ ki e kj + ρ kj e ki ) + 32 δ ij ρ kl e kl − ρ kk e ij + 32 ρ k e ij,k (cid:21) − QM (cid:20) a i, φ ,j + a j, φ ,i − δ ij a l, φ ,l + 2 a (cid:18) φ ,ij − δ ij φ kk (cid:19)(cid:21) = − πGT ( P ) ij . (2.31)Similarly for the perturbed Maxwell equation Eq. (2.28) we use both Eq. (2.29) and– 9 –q. (2.30) to get, − π (cid:18) − φ + 5 ρ l (cid:19) J ( P ) i = (cid:3) (cid:18)(cid:18) φ + ρ l (cid:19) a i (cid:19) + 4 (cid:20)(cid:18) φ + ρ l (cid:19) a i, + (cid:18) φ i + ρ i l (cid:19) a , (cid:21) + 2 (cid:18) φ ki + ρ ki l (cid:19) a k − (cid:18) φ kk + ρ kk l (cid:19) a i + 16 πQM (cid:20) e j φ ij + ( e ij, − e i ,j ) φ j + 12 ( e , − e kk, ) φ i (cid:21) (2.32)Note that for both the equations Eq. (2.31) and Eq. (2.32) we keep the terms up to leadingorder in φ and 1 /l . In the next section we will solve Eq. (2.31) and Eq. (2.32) for e ij and a i in frequency space using the worldline formalism. The result will involve a Green’sfunction for the 1 /l correction which was previously derived in [32]. To solve Eq. (2.31) and Eq. (2.32) we first briefly review the solution for perturbed scalarfield equation. For an arbitrary source f ( s ), the solution of the following scalar box equationin a curved spacetime ψ ; αα ( x ) = − (cid:90) δ ( x, r ( σ )) f ( σ ) dσ . (3.1)can be written as, ψ (1) ( x ) = ψ (0) ( x ) + δψ (0) ( x )= 14 π σ (cid:90) −∞ δ ( − Ω ( x, r ( σ ))) f ( σ ) dσ + 116 π (cid:90) (cid:112) − g ( y ) δ ( − Ω ( x, y )) d y σ (cid:90) −∞ δ (cid:48) ( − Ω ( y, r ( σ ))) F ( y, r ( σ )) f ( σ ) dσ . (3.2)where Ω ( x, r ( σ )) is the Synge world function and F ( x, r ( σ )) is the Ricci tensor dependentterm which arises from the derivative of the world function Ω( x, r ) = 12 ( u − u ) (cid:90) u u g αβ U α U β du, (3.3) F ( x, r ) = 1 u − u u (cid:90) u ( u − u ) R µν U µ U ν du . (3.4)In Eq. (3.3) and Eq. (3.4) we assume that the observer ( x ) and probe particle source ( r )are joined by a unique geodesic ξ α with affine parameter ‘u’ and U α = dξ α du is the tangentvector to the geodesic. Gravitational and electromagnetic radiation follow this path fromthe source to the observer. R µν in Eq. (3.4) gets the contribution from the black hole, For a more detailed discussion on the world function in the context of our derivation we refer the readerto [32]. – 10 –hich in our scattering approximation can be treated as a point particle with mass M andcharge Q . The integration limit is chosen up to σ instead of infinity ( ∞ ). This ensures that r µ ( σ ) lies outside the light cone centred at x µ and the contribution to scalar pertubation ψ only comes from the retarded part of the Green’s function.Expanding Eq. (3.1) in terms of the d’Alembertian operator we get, (cid:3) ψ (1) + 4 φ∂ t ψ (0) − ρ l ∂ t ψ (0) + 34 l ρ k ∂ k ψ (0) + 3 ρ l ∂ k ψ (0) = − (cid:90) δ ( x − z ( s )) f ( s ) ds + O (cid:0) R (cid:1) . (3.5)We can solve Eq. (3.5) by substituting Eq. (3.2) and performing a Fourier transformation˜ ψ (1) ( ω, (cid:126)x ) = (cid:90) dte iωt ψ (1) ( t, (cid:126)x ) . (3.6)The transformed field ˜ ψ can be perturbatively solved about flat spacetime. The solutionof ˜ ψ that are leading order in φ and 1 /l provide tail contributions to the flat spacetimeGreen’s function, which arises due to the black hole potential and AdS potential. Denotingthese tail terms as G M and G l , they have the solutions [31, 32] G M ( ω, (cid:126)x, (cid:126)r ) = − iM πω e iω | (cid:126)R | Γ ( (cid:126)x, (cid:126)r ) | (cid:126)R | − ∞ (cid:90) dv e iω ( v + | (cid:126)z | + ρ ( v )) ( v + | (cid:126)z | ) ρ ( v ) (3.7) G l ( ω, (cid:126)x, (cid:126)r ) = − iω e iω | (cid:126)R | ( (cid:126)x.(cid:126)r ) (3.8)where G M and G l correspond to the contribution due to the black hole mass and AdSradius respectively.The equations for the perturbed fields e ij and a i have additional contributions from thebackground apart from those in Eq. (3.5) due to their respective tensor and vector nature.These additional contributions provide terms in the frequency space solutions in terms ofthe derivatives of Eq. (3.7) and Eq. (3.8) − ∇ i ˜ (cid:3) G M ( ω, (cid:126)x, (cid:126)r ) = φ i e iω | (cid:126)R | | (cid:126)R | , −∇ i ∇ k ˜ (cid:3) G M ( ω, (cid:126)x, (cid:126)r ) = φ ik e iω | (cid:126)R | | (cid:126)R | , (3.9)and − ∇ i ˜ (cid:3) G l ( ω, (cid:126)x, (cid:126)r ) = ρ i e iω | (cid:126)R | | (cid:126)R | , −∇ i ∇ k ˜ (cid:3) G l ( ω, (cid:126)x, (cid:126)r ) = ρ ik e iω | (cid:126)R | | (cid:126)R | . (3.10)where ˜ (cid:3) = ( ω + ∂ i ).We can now derive the frequency space solutions ˜ e ij and ˜ a i . This requires substituting e µν and a µ in terms of their Fourier transformed fields ˜ e µν ( ω , (cid:126)x ) and ˜ a µ ( ω , (cid:126)x ) in Eq. (2.31)– 11 –nd Eq. (2.32). In the case of Eq. (2.31), we find − T ( P ) ij = (cid:90) dω e − iωt (cid:101) (cid:3) (cid:18)(cid:18) φ − ρ l (cid:19) ˜ e ij (cid:19) − (cid:90) dω e − iωt (cid:20) ω φ ˜ e ij + iω ( φ ,i ˜ e j + φ ,j ˜ e i ) − (cid:18) φ ,ij − φ ,kk δ ij (cid:19) (˜ e + ˜ e ll ) (cid:21) − l (cid:20) ω ρ ˜ e ij + iωρ i ˜ e j + iωρ j ˜ e i − ρ ij (2˜ e − ˜ e ll ) − δ ij ρ kl ˜ e kl + 12 δ ij ρ kk ˜ e ll −
32 ( ρ ki ˜ e kj + ρ kj ˜ e ki ) − ρ kk ˜ e ij − ρ k ˜ e ij,k (cid:21) + 2 QM (cid:90) dω e − iωt (cid:20) iω (˜ a i φ ,j + ˜ a j φ ,i − δ ij ˜ a l φ ,l ) − a (cid:18) φ ,ij − δ ij φ ,kk (cid:19)(cid:21) (3.11)where T ( P ) ij = m (cid:90) δ ki δ lj δ ( x − r ( σ ))1 + 2 φ ( (cid:126)r ) + r l dr k dσ dr l dσ dσ (3.12)Similar steps for the perturbed Maxwell equation Eq. (2.32) gives, − (cid:18) − φ + 5 ρ l (cid:19) πJ ( P ) i = − q (cid:90) δ ki δ ( x − r ( σ ))1 + φ ( (cid:126)r ) + r l dr k dσ dσ = (cid:90) dω e − iωt (cid:101) (cid:3) (cid:18)(cid:18) φ + ρ l (cid:19) ˜ a i (cid:19) − (cid:90) dω e − iωt (cid:2) (cid:0) ω ˜ a i φ + iω ˜ a φ ,i (cid:1) + ˜ a i φ ,kk − a k φ ,ik (cid:3) + (cid:90) dω e − iωt πQM (cid:18) ˜ e j φ ,ij − ( iω ˜ e ij + ˜ e i ,j ) φ j −
12 ( iω ˜ e − iω ˜ e kk ) φ i (cid:19) − (cid:90) dω e − iωt (cid:34) (cid:32) ω ˜ a i ρ l + iω ˜ a ρ ,i l (cid:33) + ˜ a i ρ ,kk l − ˜ a k ρ ,ik l (cid:35) (3.13)We get the solution for ˜ e ij from the scalar perturbation solution comparing ˜ ψ (0)0 ( ω , (cid:126)x ) with (cid:16) φ ( (cid:126)x ) − x l (cid:17) ˜ e (0) ij ( ω , (cid:126)x ) and replacing f ( σ ) with 2 mδ ki δ lj (cid:16) − φ ( (cid:126)r ) − r l (cid:17) dr k dσ dr l dσ .The zeroth order solution of ˜ e ij ( ω , (cid:126)x ) in frequency space is,˜ e (0) ij ( ω , (cid:126)x ) = m (cid:90) e iω ( r + R )4 πR v i v j dr dσ dr , (3.14)where we have denoted dr k dr as v k .Likewise, comparing ˜ ψ (0)0 ( ω , (cid:126)x ) to (cid:16) φ ( (cid:126)x ) + x l (cid:17) ˜ a i ( ω , (cid:126)x ) and replacing f ( σ ) with qδ ki (cid:16) − φ ( (cid:126)r ) − r l (cid:17) dr k dσ , we find the following zeroth order solution ˜ a (0) i ( ω , (cid:126)x ) in Fourierspace, ˜ a (0) i ( ω , (cid:126)x ) = q (cid:90) e iω ( r + R )4 πR v i dr , (3.15)– 12 –e can further compute the other components of gravitational and electromagnetic pertur-bations from ˜ e ij and ˜ a i , as they are related among themselves by the gauge fixing condition.It follows from our choice in Eq. (2.29) and Eq. (2.30) that Eq. (2.26) on flat spacetimesimplifies to e ij,j − e i , = 0 , e i,i − e , = 0 , (3.16) a , − a i,i = 0 . (3.17)These are simply the de Donder and Lorenz gauges in flat spacetime. By Fourier trans-forming Eq. (3.16) and using Eq. (3.14), we can now derive the following zeroth ordersolutions of ˜ e i and ˜ e ,˜ e (0) i ( ω , (cid:126)x ) = − m (cid:90) e iω ( r + R )4 πR v i dr dσ dr + O ( φ ) , ˜ e (0)00 ( ω , (cid:126)x ) = m (cid:90) e iω ( r + R )4 πR dr dσ dr + O ( φ ) . (3.18)Using the Fourier transform of Eq. (3.17), we can similarly use Eq. (3.15) to determine thesolution for electromagnetic perturbation ˜ a (0)0 ˜ a (0)0 ( ω , (cid:126)x ) = − q (cid:90) e iω ( r + R )4 πR dr + O ( φ ) . (3.19)Hence the gauge conditions determine all the lowest order expressions. To find the completesolution we first substitute all zeroth order solutions Eq. (3.14), Eq. (3.18), Eq. (3.15) andEq. (3.19) in all terms that are coefficients of φ and 1 /l in Eq. (3.11). We then makeuse of the expressions in Eq. (3.10) and Eq. (3.9) to determine the following solution for˜ e ij ( ω, (cid:126)x ),˜ e ij ( ω, (cid:126)x ) = m φ ( (cid:126)x ) − x l (cid:90) dr dr dσ e iω ( r + R )4 πR v i v j φ ( (cid:126)r ) + r l − (cid:90) dr e iωr (cid:90) d (cid:126)r (cid:48) δ (3) (cid:0) (cid:126)r (cid:48) − (cid:126)r (cid:0) r (cid:1)(cid:1) (cid:40) dr dσ mπ (cid:20) ω v i v j − iω (cid:0) v i ∇ j + v j ∇ i (cid:1) − (cid:0) (cid:126)v (cid:1) (cid:18) ∇ i ∇ j − δ ij ∇ (cid:19)(cid:35) + qQ πM (cid:20) iω (cid:16) v i ∇ j + v j ∇ j − δ ij v k ∇ k (cid:17) +2 (cid:18) ∇ i ∇ j − δ ij ∇ (cid:19)(cid:21) (cid:41) G M (cid:0) ω, (cid:126)x, (cid:126)r (cid:48) (cid:1) − (cid:90) dr e iωr (cid:90) d (cid:126)r (cid:48) δ (3) (cid:0) (cid:126)r (cid:48) − (cid:126)r (cid:0) r (cid:1)(cid:1) (cid:40) dr dσ m πl (cid:2) ω v i v j − iω ( v i ∇ j + v j ∇ i ) − (cid:0) − (cid:126)v (cid:1) ∇ i ∇ j − δ ij v k v m ∇ k ∇ m − δ ij v ∇ −
32 ( v k v j ∇ k ∇ i + v k v i ∇ k ∇ j ) + v i v j ∇ + 38 iω ( v i ∇ j + v j ∇ i ) (cid:21) (cid:41) G l (cid:0) ω, (cid:126)x, (cid:126)r (cid:48) (cid:1) . (3.20)– 13 –arrying out similar steps for ˜ a i ( ω, (cid:126)x ) we get,˜ a i ( ω, (cid:126)x ) = q φ ( (cid:126)x ) + x l (cid:90) dr e iω ( r + R )4 πR v i φ ( (cid:126)r ) + r l − (cid:90) dr e iωr (cid:90) d (cid:126)r (cid:48) δ (3) (cid:0) (cid:126)r (cid:48) − (cid:126)r (cid:0) r (cid:1)(cid:1) (cid:40) qπ (cid:20) ω v i − iω ∇ i + 14 v i ∇ − v k ∇ k ∇ i (cid:21) + dr dσ QmM (cid:20) v j ∇ i ∇ j + iω (cid:18) v i v j ∇ j + 16 ∇ i − (cid:126)v ∇ i (cid:19)(cid:21) (cid:41) G M (cid:0) ω, (cid:126)x, (cid:126)r (cid:48) (cid:1) , − q π l (cid:90) dr e iωr (cid:90) d (cid:126)r (cid:48) δ (3) (cid:0) (cid:126)r (cid:48) − (cid:126)r (cid:0) r (cid:1)(cid:1) (cid:40) ω v i − iω ∇ i + 18 v i ∇ − v k ∇ k ∇ i (cid:41) G l (cid:0) ω, (cid:126)x, (cid:126)r (cid:48) (cid:1) , (3.21)In the next section we explicitly carry out the soft expansion of ˜ a i following the prescriptiondescribed in [31]. We will see that the contribution to the soft photon factor due to theAdS radius will only come from the first line of Eq. (3.21). The terms in the last line ofEq. (3.21) give finite contributions in considering the double scaling limit of a vanishingfrequency and infinite AdS radius. Computing the quantum soft factor of a quantum scattering amplitude in asymptoticallyAdS spacetime is tricky. Asymptotic states cannot be defined in AdS as it has timelikeboundary and particle geodesics are periodic. Therefore we choose to calculate the softfactor from a classical prescription. One can compute the soft factors for photons or gravi-tons in asymptotically flat spacetimes by considering the classical limit of single/multiplesoft theorems arising from a quantum scattering process [22]. The same factors can also bederived from the low frequency classical radiation produced in a classical scattering pro-cess. The classical scattering is subject to the condition that the total energy carried bythe soft radiation is small compared to the energy carried by the scatterer. Finally in theclassical limit, the soft factor is extracted from the power spectrum of the low frequencyclassical radiation.Considering that the observer is far away from the probe, i.e. (cid:126)x (cid:29) (cid:126)r ( σ ) and taking thefrequency ω → (cid:15) α ˜ a α ( ω, (cid:126)x ) = N (cid:48) S em ( (cid:15), k ) , (4.1)where (cid:15) α is an arbitrary polarization vector of the photon, S em is the soft photon factorand ‘ k ’ denotes the momentum of soft photon. Similarly for the soft graviton factor onecan write, (cid:15) αβ ˜ e αβ ( ω, (cid:126)x ) = N (cid:48) S gr ( (cid:15), k ) . (4.2)– 14 –n Eq. (4.2), (cid:15) αβ is an arbitrary rank two polarization tensor of the graviton, S gr is the softgraviton factor and ‘ k ’ denotes the momentum of the soft graviton [22]. For both Eq. (4.1)and Eq. (4.2) R ≡ | (cid:126)x | , N (cid:48) ≡ − i π e iωR R , k ≡ − ω (1 , ˆ n ) , ˆ n = (cid:126)x | (cid:126)x | . (4.3)The soft factors S em and S gr have a term proportional to ω − at leading order in frequencyand another term proportional to ln ω − at subleading order . Equations Eq. (4.1) andEq. (4.2) can be considered as an alternate definition for the soft factor which can be easilycomputed from considering the soft limit of classical electromagnetic and gravitationalradiation profiles.In the case of asymptotically flat backgrounds, to calculate the soft factor one needs toconsider the large | t | limit and a suitable parametrization of (cid:126)r ( t ), where (cid:126)r ( t ) is the positionof the scattered probe particle at time t . As shown in [32], for computing the soft factor upto 1 /l in an asymptotically AdS spacetime, we can still consider the particle to follow anapproximately straight line geodesic for large values of t . In particular we do not requirethe correction to the particle trajectory due to the cosmological constant for computing thesoft factor up to this order in 1 /l . Thus we parameterize (cid:126)r ( t ) for asymptotic trajectoriesat large | t | in four spacetime dimensions as [22], (cid:126)r ( t ) = (cid:126)β ± t − C ± (cid:126)β ± ln | t | + finite terms , (cid:126)v = (cid:126)β ± (cid:18) − C ± t (cid:19) . (4.4)where C ± are constants and t denotes the proper time. The ln | t | terms are the contributionsfrom long range interaction forces which only exist in 4 spacetime dimensions. Eq. (4.4)can be written in the following covariant form, r µ ( a ) ( t ) = η ( a ) m ( a ) p µ ( a ) + c µ ( a ) ln | t | (4.5)where η ( a ) is positive (negative) for incoming (outgoing) particles and m ( a ) is the massof the a-th particle. We will consider the proper time as negative for incoming particlesand positive for outgoing particles. Using the parametrization Eq. (4.4), we will retainterms up to 1 /t as these are relevant in the soft expansion of ˜ e ij and ˜ a i . Using suitableintegrals given in [22] one can easily find the soft factors following the relation Eq. (4.1)and Eq. (4.2).For asymptotic AdS spacetimes, soft limits cannot merely imply a vanishing frequencylimit. Since there is no notion of null infinity in asymptotic AdS spacetimes and anymassless radiation gets bounced off an infinite number of times from spatial infinity, thefrequency of a radiation can never strictly go to zero. To define a “soft limit” in this caseone needs to consider a double scaling limit. In [32] the soft limit for AdS was defined assimultaneously taking the frequency of the radiation in AdS space to zero and the radius A detailed description of how to derive S gr is given in [21, 22, 24, 25]. – 15 –f AdS space to infinity i.e. ω → l → ∞ , keeping ωl fixed. Another interestingfeature of the metric Eq. (2.12) is that the effect of long range interactions are the same asin asymptotically flat spacetimes [32]. This happens because the perturbed fields alreadycontain terms up to 1 /l and we are interested in results up to this order only. For large‘ l ’, the effective potential near the boundary of our spacetime becomes polynomial andradiation can leak out from the boundary unlike the case on a globally AdS spacetime.To calculate the soft photon factor, we set r = t and simplify the tail terms in the Green’sfunction for (cid:126)x >> (cid:126)r ( σ )˜ G M ( ω, (cid:126)x, (cid:126)r ) = lim (cid:126)x (cid:29) (cid:126)r G M ( ω, (cid:126)x, (cid:126)r ) = iM πω (cid:34) ln (cid:18) | (cid:126)r | + ˆ n.(cid:126)rR (cid:19) + (cid:90) ∞| (cid:126)r | +ˆ n.(cid:126)r duu e iωu (cid:35) e iω ( R − ˆ n.(cid:126)r ) R , (4.6)˜ G l ( ω, (cid:126)x, (cid:126)r ) = lim (cid:126)x (cid:29) (cid:126)r G l ( ω, (cid:126)x, (cid:126)r ) = − iω e iω ( R − ˆ n.(cid:126)r ) ˆ n.(cid:126)r (4.7)We can then rewrite ˜ a i ( ω, (cid:126)x ) in the large | t | limit as˜ a i ( ω, (cid:126)x ) = ˜ a (1) i ( ω, (cid:126)x ) + ˜ a (2) i ( ω, (cid:126)x ) + ˜ a (3) i ( ω, (cid:126)x ) + ˜ a (4) i ( ω, (cid:126)x ) + ˜ a (5) i ( ω, (cid:126)x ) + ˜ a (6) i ( ω, (cid:126)x )+ ˜ a (7) i ( ω, (cid:126)x ) + ˜ a (8) i ( ω, (cid:126)x ) + ˜ a (9) i ( ω, (cid:126)x ) + ˜ a (10) i ( ω, (cid:126)x ) , (4.8)where the individual terms present in the above equation are defined as below:˜ a (1) i ( ω, (cid:126)x ) = q x l e iωR πR (cid:90) dt e iω ( t − ˆ n.(cid:126)r ) v i φ ( (cid:126)r ) + r l (4.9)˜ a (2) i ( ω , (cid:126)x ) = q π (cid:90) dt e iωt v k ∇ k ∇ i ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (4.10)˜ a (3) i ( ω , (cid:126)x ) = iqπ ω (cid:90) dt e iωt ∇ i ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (4.11)˜ a (4) i ( ω , (cid:126)x ) = − iM q π e iωR R ω (cid:90) dtv i (cid:26) ln | (cid:126)r (cid:48) | + ˆ n.(cid:126)r (cid:48) R e iω ( t − ˆ n.(cid:126)r (cid:48) ) + (cid:90) ∞| (cid:126)r (cid:48) | +ˆ n.(cid:126)r (cid:48) duu e iω ( t − ˆ n.(cid:126)r (cid:48) + u ) (cid:27) (4.12)˜ a (5) i ( ω , (cid:126)x ) = − q π (cid:90) dt e iωt v i ∇ k ∇ k ˜ G M ( ω, (cid:126)x, (cid:126)r ) (4.13)˜ a (6) i ( ω , (cid:126)x ) = 4 QmM (cid:90) dt dtdσ e iωt v k ∇ k ∇ i ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (4.14)˜ a (7) i ( ω , (cid:126)x ) = − i QmM ω (cid:90) dt dtdσ e iωt v i v k ∇ k ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (4.15)– 16 – a (8) i ( ω , (cid:126)x ) = − i Qm M ω (cid:90) dt dtdσ e iωt ∇ i ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (4.16)˜ a (9) i ( ω , (cid:126)x ) = i QmM ω (cid:90) dt dtdσ e iωt (cid:126)v ∇ i ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (4.17)˜ a (10) i ( ω , (cid:126)x ) = − q πl ω (cid:90) dt e iωt v i ˜ G l ( ω, (cid:126)x, (cid:126)r ) (4.18)˜ a (11) i ( ω , (cid:126)x ) = iq πl ω (cid:90) dt e iωt ∇ i ˜ G l ( ω, (cid:126)x, (cid:126)r ) (4.19)˜ a (12) i ( ω , (cid:126)x ) = − q πl (cid:90) dt e iωt v i ∇ k ∇ k ˜ G l ( ω, (cid:126)x, (cid:126)r ) (4.20)˜ a (13) i ( ω , (cid:126)x ) = q πl (cid:90) dt e iωt v k ∇ k ∇ i ˜ G l ( ω, (cid:126)x, (cid:126)r ) . (4.21)Here we have denoted ∇ i = ∂∂r i + ∂∂x i .We can think of ˜ a (2) i to ˜ a (5) i as contributions due to the scatterer black hole’s mass, ˜ a (6) i to ˜ a (9) i arising due to the black hole’s charge and ˜ a (10) i to ˜ a (13) i due to the AdS potential.The soft limit evaluation of ˜ a (2) i to ˜ a (9) i has been previously derived in [31]. Here we willevaluate ˜ a (1) i in the soft limit. This term will give us the contribution to the soft photonfactor of asymptotically flat spacetime due to the AdS spacetime.To evaluate the soft limit of ˜ a (1) i we will be using the following relation, e iω ( t − ˆ n.(cid:126)r ( t )) = 1 iω ∂ t ( t − ˆ n.(cid:126)r ( t )) ddt e iω ( t − ˆ n.(cid:126)r ( t )) = 1 iω − ˆ n.(cid:126)v ( t )) ddt e iω ( t − ˆ n.(cid:126)r ( t )) (4.22)Using Eq. (4.22) and carrying out an integration by parts, we have˜ a (1) i ( ω, (cid:126)x ) = − q x l e iωR πR iω (cid:90) dt e iω ( t − ˆ n.(cid:126)r ) ddt (cid:34) v i (1 − ˆ n.(cid:126)v ( t )) 11 + φ ( (cid:126)r ) + r l (cid:35) (4.23)Since we assume r l (cid:28) φ ( (cid:126)r ) + r l = 11 + φ ( (cid:126)r ) (1 − r l )Hence Eq. (4.23) simplifies to˜ a (1) i ( ω, (cid:126)x ) = − q x l e iωR πR iω (cid:90) dt e iω ( t − ˆ n.(cid:126)r ) ddt (cid:20) v i (1 − ˆ n.(cid:126)v ( t )) 11 + φ ( (cid:126)r ) (cid:21) + q l e iωR πR iω (cid:90) dt e iω ( t − ˆ n.(cid:126)r ) ddt (cid:20) v i (1 − ˆ n.(cid:126)v ( t )) r (cid:21) = A + A + A (4.24)– 17 –here A = − q e iωR πR iω (cid:90) dt e iω ( t − ˆ n.(cid:126)r ) ddt (cid:20) v i (1 − ˆ n.(cid:126)v ( t )) 11 + φ ( (cid:126)r ) (cid:21) (4.25) A = q x l e iωR πR iω (cid:90) dt e iω ( t − ˆ n.(cid:126)r ) ddt (cid:20) v i (1 − ˆ n.(cid:126)v ( t )) (cid:21) (4.26) A = q l e iωR πR iω (cid:90) dt e iω ( t − ˆ n.(cid:126)r ) ddt (cid:20) v i (1 − ˆ n.(cid:126)v ( t )) r (cid:21) (4.27)Note that we are keeping terms up to leading order in φ and 1 /l . In the asymptoticexpansion φ will take the form of, φ ( (cid:126)r ( t )) = − M π | (cid:126)r ( t ) | = ∓ M π | (cid:126)β ± | t + O (cid:0) t − (cid:1) , (4.28)where we have used the parametrization of r from Eq. (4.4). A is the contribution onasymptotically flat spacetimes and has been evaluated in [31]. Therefore we will concentrateon the other two integrals. To evaluate A , we will substitute v from Eq. (4.4) and performthe following expansion, v i (1 − ˆ n.(cid:126)v ) − = (cid:16) − ˆ n.(cid:126)β ± (cid:17) − β ± i (cid:34) − C ± t − ˆ n.(cid:126)β ± (cid:35) + O (cid:0) t − (cid:1) (4.29)Therefore Eq. (4.26) reduces to, A = q x l e iωR πR iω (cid:90) dt e − iω ˜ g ( t ) ddt β ± i (cid:16) − ˆ n.(cid:126)β ± (cid:17) (cid:32) − C ± t − ˆ n.(cid:126)β ± (cid:33) (4.30)with ˜ g ( t ) = − (1 − ˆ n.(cid:126)β ± ) t − C ± ˆ n.(cid:126)β ln t (4.31)In the ω → I = 1 ω + ∞ (cid:90) −∞ dte − iωg ( t ) ddt f ( t ) = 1 ω ( f + − f − ) + i ( a + k + − a − k − ) ln ω − + finite (4.32)where f ( t ) → f ± + k ± t , g ( t ) → a ± t + b ± ln | t | (4.33)Using Eq. (4.32), we find that the integral A in Eq. (4.30) evaluates to A = − iω qx l e iωR πR (cid:34) β + i − ˆ n.(cid:126)β + − β − i − ˆ n.(cid:126)β − (cid:35) + qx l e iωR πR ln ω − (cid:34) β + i − ˆ n.(cid:126)β + C + − β − i − ˆ n.(cid:126)β − C − (cid:35) (4.34)– 18 –o consider the soft limit, we will need to take the simultaneous limits ω → l → ∞ ,while keeping ωl fixed. Therefore, by defining ωl = γ , we find that Eq. (4.34) can bewritten as A = − i qx γ e iωR πR ω (cid:34) β + i − ˆ n.(cid:126)β + − β − i − ˆ n.(cid:126)β − (cid:35) + qx γ e iωR πR ω ln ω − (cid:34) β + i − ˆ n.(cid:126)β + C + − β − i − ˆ n.(cid:126)β − C − (cid:35) (4.35)We thus note that in the ω → A term is a finite contribution and does notprovide the divergent terms in the soft factor. It follows that divergent contributions dueto l corrections in the soft factor will arise from those integrals which at least fall off like ω − .The integrand in A involves the term v i (1 − ˆ n.(cid:126)v ( t )) r , which has the following expansion in tv i (1 − ˆ n.(cid:126)v ( t )) r = β ± i − ˆ n.(cid:126)β ± (cid:126)β ± (cid:34) t − C ± t ln | t | − C ± − ˆ n.(cid:126)β ± t + 2 C ± − ˆ n.(cid:126)β ± ln | t | (cid:35) + β ± i (cid:16) − ˆ n.(cid:126)β ± (cid:17) C ± (cid:126)β ± ˆ n.(cid:126)β ± (cid:34) − C ± ˆ n.(cid:126)β ± − ˆ n.(cid:126)β ± t (cid:35) + · · · , (4.36)where · · · in Eq. (4.36) refer to subleading terms that are O (cid:0) t (cid:1) and O (cid:16) ln | t | t (cid:17) , whichwill always lead to finite terms in the soft factor and are hence ignored. The terms in thesecond line of Eq. (4.36) can be directly substituted into Eq. (4.27) to provide the followingcontribution, which we denote as A (0)3 A (0)3 = − iq γ e iωR πR ω (cid:90) dt e − iω ˜ g ( t ) ddt C ± (cid:126)β ± ˆ n.(cid:126)β ± β ± i (cid:16) − ˆ n.(cid:126)β ± (cid:17) (cid:34) − C ± ˆ n.(cid:126)β ± − ˆ n.(cid:126)β ± t (cid:35) (4.37)In taking the simultaneous limits ω → l → ∞ of the A (0)3 integral in Eq. (4.37), theoverall coefficient ensures that we will have a non-divergent contribution in ω to the softfactor.To determine the other possible contributions of A , we need to differentiate Eq. (4.36)with respect to t and retain terms up to 1 /t . This gives us ddt (cid:20) v i (1 − ˆ n.(cid:126)v ( t )) r (cid:21) = β ± i − ˆ n.(cid:126)β ± (cid:126)β ± (cid:34) t − C ± ln | t | − C ± − n.(cid:126)β ± − ˆ n.(cid:126)β ± + 2 C ± − ˆ n.(cid:126)β ± t (cid:35) (4.38)– 19 –o apply the integral Eq. (4.32) we have to perform another integration by parts usingEq. (4.22) until we get the form of Eq. (4.33). By using the expression in Eq. (4.38) andsubsequently performing the integration by parts, we find A = − q l e iωR πR (cid:18) iω (cid:19) (cid:90) dt e − iω ˜ g ( t ) ddt (cid:34) − ˆ n.(cid:126)v ( t )) β ± i − ˆ n.(cid:126)β ± (cid:126)β ± (cid:32) t − C ± ln | t | − C ± − n.(cid:126)β ± − ˆ n.(cid:126)β ± + 2 C ± − ˆ n.(cid:126)β ± t (cid:33)(cid:35) = − q l e iωR πR (cid:18) iω (cid:19) (cid:90) dt e − iω ˜ g ( t ) ddt (cid:34) β ± i (1 − ˆ n.(cid:126)β ± ) (cid:126)β ± (2 t − C ± ln | t | + 1 t (cid:32) C ± − ˆ n.(cid:126)β ± (cid:33) (cid:16) n.(cid:126)β ± (cid:17) − C ± − ˆ n.(cid:126)β ± (4.39)We will now separate the A integral above into terms which can be evaluated usingEq. (4.32) and those which require an additional integration by parts. The terms in theparenthesis of Eq. (4.39) which are constant and proportional to t provide the followingcontribution which we denote as A (1)3 A (1)3 = − q γ e iωR πR (cid:90) dt e − iω ˜ g ( t ) β ± i (1 − ˆ n.(cid:126)β ± ) ddt C ± − ˆ n.(cid:126)β ± − t (cid:32) C ± − ˆ n.(cid:126)β ± (cid:33) (cid:16) n.(cid:126)β ± (cid:17) (4.40)Evaluating this integral in the soft limit using Eq. (4.32), we find a finite contribution thatis irrelevant to the soft photon factor. The remaining terms in the integrand of Eq. (4.39)require another differentiation followed by an integration by parts. This contribution,which we denote by A (2)3 , takes the form A (2)3 = − q l e iωR πR (cid:18) iω (cid:19) (cid:90) dt e − iω ˜ g ( t ) β ± i (1 − ˆ n.(cid:126)β ± ) (cid:18) (cid:126)β ± − (cid:126)β ± C ± t (cid:19) = iq γ e iωR πR ω (cid:90) dt e − iω ˜ g ( t ) β ± i (1 − ˆ n.(cid:126)β ± ) ddt (cid:32) (cid:126)β ± − (cid:126)β ± C ± t − ˆ n.(cid:126)β ± (cid:33) (4.41)This integral in the soft limit evaluates to A (2)3 = 1 γ (cid:34) iq ω e iωR πR (cid:40) β + i (1 − ˆ n.(cid:126)β + ) (cid:126)β − β − i (1 − ˆ n.(cid:126)β − ) (cid:126)β − (cid:41) − q ω − e iωR πR (cid:40) β + i (1 − ˆ n.(cid:126)β + ) C + (cid:126)β − β − i (1 − ˆ n.(cid:126)β − ) C − (cid:126)β − (cid:41)(cid:35) (4.42)– 20 –he divergent terms in this result contribute to the soft photon factor. From Eq. (4.42)we can write the soft factor following Eq. (4.1).The other radiative field terms, those from ˜ a (10) i to ˜ a (13) i , are either finite in the soft limitor are proportional to k i . Gauge invariance can be used to show that terms which go like k i do not contribute to the soft factor. The invariance of S em in Eq. (4.1) under (cid:15) µ → (cid:15) µ + k µ imposes the constraint k µ ˜ a µ = 0. This implies that ˜ a can be determined from ˜ a i andallows us to set (cid:15) = 0. In addition, the expressions for ˜ a µ are determined only up toa choice in gauge. Denoting the arbitrary gauge parameters by λ , we have the followinggauge transformations δ ˜ a µ = λk µ . (4.43)Using Eq. (4.43) in Eq. (4.1), we thus have k µ (cid:15) µ = 0. This provides the condition on thepolarization vector as k i (cid:15) i = 0.Therefore the 1 /l contribution to the soft photon factor arises only from Eq. (4.42). Onsubstituting in Eq. (4.1) we find S l em = q γ (cid:15) i (cid:34) − ω (cid:40) β + i (1 − ˆ n.(cid:126)β + ) (cid:126)β − β − i (1 − ˆ n.(cid:126)β − ) (cid:126)β − (cid:41) − i ln ω − (cid:40) C + β + i (1 − ˆ n.(cid:126)β + ) (cid:126)β − C − β − i (1 − ˆ n.(cid:126)β − ) (cid:126)β − (cid:41)(cid:35) (4.44)We can describe the above result in terms of the momentum and angular momentum ofthe probe particle, and the momentum of the soft photon. From Eq. (4.4) we find theasymptotic momenta (as t → ∞ ) to have the expressions p (1) = m (cid:113) − (cid:126)β − (cid:16) , (cid:126)β − (cid:17) , p (2) = − m (cid:113) − (cid:126)β (cid:16) , (cid:126)β + (cid:17) (4.45)The overall negative sign in p (2) reflects the convention for outgoing momenta. UsingEq. (4.4) we also have for the position four vector r (1) = (cid:16) t, (cid:126)β − t − C − (cid:126)β − ln | t | (cid:17) , r (2) = (cid:16) t, (cid:126)β + t − C + (cid:126)β + ln | t | (cid:17) , (4.46)with the coefficients of the logarithmic terms in the position vector, c µ ( a ) , taking the form c (1) = (cid:16) , − C − (cid:126)β − (cid:17) , c (2) = (cid:16) , − C + (cid:126)β + (cid:17) (4.47)Here for simplicity we are ignoring an overall constant term in the position vector, whichwill not contribute to the soft factor result (as it is a finite contribution in the t → ∞ limit).The angular momentum of the probe particle can be written as j µν ( a ) = r µ ( a ) p ν ( a ) − r ν ( a ) p µ ( a ) + spin = ( c µa p νa − c νa p µa ) ln | t | + · · · (4.48)– 21 –nd thus has the following non-vanishing components j i (1) = m (cid:113) − (cid:126)β − C − (cid:126)β − ln | t | , j i (2) = − m (cid:113) − (cid:126)β C + (cid:126)β + ln | t | (4.49)Lastly, the outgoing soft photon has the momentum k = − ω (1 , ˆ n ). Using this along withthe expressions in Eq. (4.45) and Eq. (4.49), and replacing ln | t | by ln ω − , we determinethat Eq. (4.44) can be written as the following sum S l em = S l (0)em + S l (1)em , (4.50)where S l (0)em and S l (1)em are the universal leading and subleading contributions respectivelyon AdS spacetimes, with the expressions S l (0)em = q l (cid:88) a =1 ( − a − (cid:15) µ p µ ( a ) p ( a ) .k (cid:126)p a ) (cid:0) p ( a ) .k (cid:1) , (4.51) S (1)em = i q l (cid:88) a =1 ( − a − (cid:15) ν k ρ j ρν ( a ) p ( a ) .k (cid:126)p a ) (cid:0) p ( a ) .k (cid:1) = i q l ln ω − (cid:88) a =1 ( − a − (cid:15) ν k ρ ( c ρa p µa − c µa p ρa ) p ( a ) .k (cid:126)p a ) (cid:0) p ( a ) .k (cid:1) (4.52)Apart from the factor (cid:126)p a ) ( p ( a ) .k ) appearing in each of the above sums, the expressions arethose for the universal leading and subleading soft factors on asymptotically flat spacetimes,in the case of the scattering of a single probe particle. The summation index values a = 1and a = 2 correspond to the outgoing and incoming configuration of the probe, respectively.It can be noted that the momentum soft factor expression is not covariant, due to theinvolvement of (cid:126)p a ) . This is a consequence of working in the ‘static gauge’ choice for thepolarization vector of the electromagnetic field. We would always be required to adopt somegauge in the probe scattering calculations on a fixed curved spacetime, which is needed toderive the AdS result. Furthermore, we see from the dependence of additional momentafactors that the above result on AdS spacetimes cannot be represented as an overall phaseof the asymptotically flat spacetime result.For completeness, we note that soft photon factor can be derived on asymptotically flatbackgrounds arising from contributions in A , ˜ a (3) i , ˜ a (4) i and ˜ a (5) i is [31], S flatem = − qω (cid:32) (cid:126)(cid:15).(cid:126)β + − ˆ n.(cid:126)β + − (cid:126)(cid:15).(cid:126)β − − ˆ n.(cid:126)β − (cid:33) − iq ln ω − (cid:32) C + − ˆ n.(cid:126)β + (cid:126)(cid:15).(cid:126)β + − C − − ˆ n.(cid:126)β − (cid:126)(cid:15).(cid:126)β − (cid:33) − iq M π ln ( ωR ) (cid:32) (cid:126)(cid:15).(cid:126)β + − ˆ n.(cid:126)β + − (cid:126)(cid:15).(cid:126)β − − ˆ n.(cid:126)β − (cid:33) . (4.53)In this case, the soft factor can be expressed in covariant form as given in [22].– 22 – Classical soft graviton factor
In this section, we will simply present the results on the contribution of the charge andmass of the scatterer black hole to the soft graviton factor. For a detailed calculation werefer the reader to Appendix A. The soft graviton factor can be split in two parts as, S gr ( (cid:15), k ) = S flatgr ( (cid:15), k ) + S l gr ( (cid:15), k ) , (5.1)where S flatgr ( (cid:15), k ) is the contribution due to the black hole mass and charge, which wasderived in [31]. The result in its covariant form can be expressed as [22] S flatgr ( (cid:15), k ) = (cid:88) a =1 (cid:15) µν p µ ( a ) p ν ( a ) p ( a ) .k + i (cid:88) a =1 (cid:15) µν p µ ( a ) k ρ p ( a ) .k . (5.2)In Eq. (5.2), p (1) and p (2) are the momenta of the probe particle before and after thescattering, while j (1) and j (2) are the angular momenta of the probe before and after thescattering. As in the previous section, we choose the convention that all momenta andangular momenta are positive for ingoing and negative for outgoing particles. The indices µ, ν, · · · run over all space-time coordinates in Eq. (5.2). The covariant expression has animplicit dependence on the masses and charges in the scattering process. The explicit formof the above equation on replacing the momenta and angular momenta has been providedin A.Further in [32], we derived the leading order contribution of cosmological constant to thesoft graviton factor on the AdS Schwarzschild black hole spacetime. Since to our orderof approximation, the metric in the isotropic coordinates does not receive any correctionfrom the charge of the black hole, the results for the soft factor remains the same. Herewe present this result in covariant form, in terms of the incoming and outgoing momentaof the probe particle S l gr = S l (0)gr + S l (1)gr , with S l (0)gr = 12 l (cid:88) a =1 (cid:15) µν p µ ( a ) p ν ( a ) p ( a ) .k (cid:126)p a ) (cid:0) p ( a ) .k (cid:1) (cid:32) (cid:126)p a ) p a ) (cid:33) ,S l (1)gr = i l (cid:88) a =1 (cid:15) µν p µ ( a ) k ρ j ρν ( a ) p ( a ) .k (cid:126)p a ) (cid:0) p ( a ) .k (cid:1) (cid:32) (cid:126)p a ) p a ) (cid:33) , (5.3)where S l (0)gr and S l (1)gr are respectively the universal leading and subleading contributionsto the soft graviton factor on AdS spacetimes. Note that for the four momentum we have p a ) = − m . The leading and subleading soft factor sums are, as in the soft photon resultsin Eq. (4.51) and Eq. (4.52), expressed in terms of a product of the flat spacetime resultand other momentum dependent terms. In the analysis of the previous sections we derived universal classical soft photon andgraviton theorems in an asymptotically AdS theory, to leading order 1 /l in the large– 23 –dS radius limit. We have derived the theorems in four spacetime dimensions, up tofirst subleading order in the soft momentum expansion. Considering the large AdS radiuslimit is important for computational simplifications as well as for defining a proper softlimit in AdS space. Our analysis can also be interpreted as 1 /l corrections of classicalsoft theorems in asymptotically flat spacetime. Thus we expect these AdS corrected softfactors derived in Eq. (4.51) and Eq. (5.3) to satisfy similar relations as the usual classicalsoft factors.By now, it is a known fact that classical soft factors partially reproduce the quantum softfactors. These results hold exactly at the level of tree and all loop scattering amplitudesfor the leading soft factor . On the other hand, it is also known that soft theoremswith the leading soft factor contribution manifest as certain Ward identities arising fromlarge gauge transformations on asymptotically flat spacetimes. We can thus anticipatethat soft theorems involving leading soft factor terms, with AdS corrections, should havea corresponding realization in a Ward identity. In this section, we concern ourselves withthe soft photon case. We first review the correspondence between the large gauge Wardidentity and leading soft factor for a theory on asymptotically flat spacetimes and thenderive the same for our case.For a residual gauge parameter at null infinity which satisfies the antipodal boundarycondition (cid:15) ( z , ¯ z ) | I + − = (cid:15) ( z , ¯ z ) | I − + (6.1)we can define the total charge for the Maxwell field on I + and I − in terms of boundarycontributions on asymptotically flat spacetimes Q + (cid:15) = (cid:90) I + − (cid:15) ∗ F , Q − (cid:15) = (cid:90) I − + (cid:15) ∗ F . (6.2)here ( z, ¯ z ) are coordinates on the Celestial 2-sphere. The above charges satisfy the followingconservation equation Q + (cid:15) − Q − (cid:15) = 0 . (6.3)We can now consider Maxwell’s equations in the presence of a source to re-express thecharges in Eq. (6.2) as the following integrals over I + and I − Q + (cid:15) = − (cid:90) I + dud z (cid:16) ∂ z (cid:15)F (0) u ¯ z + ∂ ¯ z (cid:15)F (0) uz (cid:17) + (cid:90) I + dud z(cid:15)γ z ¯ z j (2) u , (6.4) Q − (cid:15) = − (cid:90) I − dvd z (cid:16) ∂ z (cid:15)F (0) v ¯ z + ∂ ¯ z (cid:15)F (0) vz (cid:17) + (cid:90) I − dvd z(cid:15)γ z ¯ z j (2) v . (6.5) for subleading and sub-subleading parts, the quantum soft factor reduces to the classical one in theclassical limit [19] – 24 –n the above expressions γ z ¯ z is the metric on the 2-sphere and the superscripts on the fieldstrength tensors and currents represent the order of the coefficient in their corresponding r − expansions in asymptotically flat spacetimes. The first integrals in Q + (cid:15) and Q − (cid:15) areknown as the soft charges. The field strength tensors involved in the soft charges can beexpressed in terms of their soft modes. Further in the absence of asymptotic magneticfields and magnetic monopoles, we have the following condition F z ¯ z | I ±± = 0 . (6.6)In this case, it can be shown that we can define the soft modes of the field strength tensorsin terms of the following real scalars N ∂ z N + ( z , ¯ z ) = (cid:90) I + duF (0) uz , ∂ ¯ z N + ( z , ¯ z ) = (cid:90) I + duF (0) u ¯ z ∂ z N − ( z , ¯ z ) = (cid:90) I − dvF (0) vz , ∂ ¯ z N − ( z , ¯ z ) = (cid:90) I − dvF (0) v ¯ z . (6.7)Using Eq. (6.7) in Eq. (6.4) and Eq. (6.5) then gives us the total charge expressions Q + (cid:15) = 2 (cid:90) d z N + ∂ z ∂ ¯ z (cid:15) + (cid:90) I + dud z(cid:15)γ z ¯ z j (2) u , (6.8) Q − (cid:15) = 2 (cid:90) d z N − ∂ z ∂ ¯ z (cid:15) + (cid:90) I − dvd z(cid:15)γ z ¯ z j (2) v . (6.9)With these classical results, we can now describe the Ward identity. Given a scatteringprocess going from an incoming state on I − to a state on I + governed by an S -matrix S , the charge conservation in Eq. (6.3) now takes the form (cid:104) out | ˆ Q + (cid:15) S − S ˆ Q − (cid:15) | in (cid:105) = 0 , (6.10)where ˆ Q ± (cid:15) are operator versions of the expressions in Eq. (6.8) and Eq. (6.9). The integratedhard charges can be expressed as the following sum over hard charges in the incoming andoutgoing states (cid:90) I + dud z(cid:15)γ z ¯ z j (2) u = (cid:88) k =out Q k ( z k , ¯ z k ) (cid:15) out (6.11) (cid:90) I − dvd z(cid:15)γ z ¯ z j (2) v = (cid:88) k =in Q k ( z k , ¯ z k ) (cid:15) in (6.12)– 25 –he coordinates { z k , ¯ z k } denote the asymptotic positions of the hard particles on theCelestial 2-sphere. Using the above expressions in Eq. (6.8) and Eq. (6.9), we find that theoperators ˆ Q ± (cid:15) take the form (cid:104) out | ˆ Q + (cid:15) = 2 (cid:90) d z∂ z ∂ ¯ z (cid:15) (cid:104) out |N + + (cid:88) k =out Q k ( z k , ¯ z k ) (cid:15) out (cid:104) out | , (6.13)ˆ Q − (cid:15) | in (cid:105) = 2 (cid:90) d z∂ z ∂ ¯ z (cid:15) N − | in (cid:105) + (cid:88) k =in Q k ( z k , ¯ z k ) (cid:15) in | in (cid:105) . (6.14)Substituting Eq. (6.13) and Eq. (6.14) in Eq. (6.10) then gives us the following Wardidentity [4] 2 (cid:90) d z∂ z ∂ ¯ z (cid:15) ( z , ¯ z ) (cid:104) out |N + ( z , ¯ z ) S − SN − ( z , ¯ z ) | in (cid:105) = (cid:34) (cid:88) k =in Q k ( z k , ¯ z k ) (cid:15) in ( z , ¯ z ) − (cid:88) k =out Q k ( z k , ¯ z k ) (cid:15) out (cid:35) (cid:104) out |S| in (cid:105) (6.15)We can express the left hand side of Eq. (6.15) entirely in terms of either the soft photonmode N + or N − . This follows from the CPT invariance of matrix elements involving thein and out soft photons. This implies in particular that (cid:104) out |N + ( z , ¯ z ) S| in (cid:105) = −(cid:104) out |SN − ( z , ¯ z ) | in (cid:105) Hence Eq. (6.15) can be expressed as4 (cid:90) d z∂ z ∂ ¯ z (cid:15) ( z , ¯ z ) (cid:104) out |N + ( z , ¯ z ) S| in (cid:105) = (cid:34) (cid:88) k =in Q k ( z k , ¯ z k ) (cid:15) in ( z , ¯ z ) − (cid:88) k =out Q k ( z k , ¯ z k ) (cid:15) out (cid:35) (cid:104) out |S| in (cid:105) (6.16)An expression for ∂ z N + can be formally derived from a mode expansion of the gauge fieldsand a saddle point approximation [43] ∂ z N + = − π √
21 + z ¯ z lim ω → (cid:104) ωa out+ ( ω ˆ x ) + ωa out+ ( ω ˆ x ) † (cid:105) (6.17)We next consider the gauge parameter. A particularly convenient choice as considered in[43] for (cid:15) ( z , ¯ z ) is the following (cid:15) ( z , ¯ z ) = 1 z − z k (6.18)In particular, the derivative of Eq. (6.18) satisfies– 26 – ¯ w (cid:15) ( z , ¯ z ) = 2 πδ (2) ( z − w ) (6.19)By substituting Eq. (6.19) and Eq. (6.17) in Eq. (6.16), we then findlim ω → (cid:104) out | ωa + ( ω ˆ x ) S| in (cid:105) = (1 + z ¯ z ) √ (cid:34) (cid:88) k =out Q k z − z k − (cid:88) k =in Q k z − z k (cid:35) (cid:104) out |S| in (cid:105) (6.20)The large gauge Ward identity is formally derived as a property satisfied by quantizedfields of a given theory. However, we may also derive the Ward identity using the softphoton theorem. We recall that the soft photon theorem relates a scattering process witha soft photon insertion to the scattering process without the soft photon by a soft factor.Assuming for simplicity the insertion of the soft photon in the ‘out’ state, the theoremprovides the relation (cid:104) out | a + ( ω ˆ x ) S| in (cid:105) = S flatem (cid:104) out |S| in (cid:105) (6.21)where a + denotes the creation operator in the outgoing state with helicity ‘+’, S flatem is takento be the leading contribution to the soft photon factor S flat (0)em = (cid:34) (cid:88) k =out p ( k ) .(cid:15) + p ( k ) .q Q k − (cid:88) k =in p ( k ) .(cid:15) + p ( k ) .q Q k (cid:35) (6.22)and S is a generic scattering process in flat spacetime. Our analysis in previous sectionsconsidered a single probe particle with charge q . Thus in this case, we set Q k = q inEq. (6.22).We can now readily derive the Ward identity in Eq. (6.16) from Eq. (6.21). For thisderivation, we only need to use the expressions for the momenta of all the particles (hardand soft) and the polarization vector for the soft particle.For the soft particle and its polarization, we have the following conditions q µ q µ = 0 , q µ (cid:15) ± µ ( (cid:126)q ) = 0 , (cid:15) µα (cid:15) ∗ βµ = δ αβ , (6.23)where α , β represent the polarization directions ± . A natural choice for the null vector q µ is the following q µ = ω (cid:18) , (cid:126)xr (cid:19) = ω z ¯ z (1 + z ¯ z , z + ¯ z , − i ( z − ¯ z ) , − z ¯ z ) , (6.24)with ω the frequency of the soft particle. From this choice of q µ one can determine thepolarization vectors – 27 – + µ = 1 √ z , , − i , − ¯ z ) , (cid:15) − µ = 1 √ z , , i , − z ) (6.25)Since the hard particles are also massless, but of finite energy, using Eq. (6.24) we canchoose p µk = E (cid:126)p z ¯ z (1 + z ¯ z , z + ¯ z , − i ( z − ¯ z ) , − z ¯ z ) (6.26)If we now substitute Eq. (6.24), Eq. (6.25) and Eq. (6.26) in Eq. (6.22), we find S flat (0)em = (cid:34) (cid:88) k =out p ( k ) .(cid:15) + p ( k ) .q Q k − (cid:88) k =in p ( k ) .(cid:15) + p ( k ) .q Q k (cid:35) = 1 + z ¯ z √ ω (cid:34) (cid:88) k =out z − z k Q k − (cid:88) k =in z − z k Q k (cid:35) , (6.27)Thus on substituting Eq. (6.27) in Eq. (6.21), we getlim ω → √
21 + z ¯ z (cid:104) out | ωa + ( ω ˆ x ) S| in (cid:105) = (cid:34) (cid:88) k =out z − z k Q k − (cid:88) k =in z − z k Q k (cid:35) (cid:104) out |S| in (cid:105) (6.28)It can be noted that this result agrees with the Ward identity in Eq. (6.20).In the presence of AdS corrections of asymptotically flat spacetimes, we can determine thetree level soft theorem which involves the soft factor derived from our classical scatteringanalysis. However, a first principles derivation of the Ward identity using quantized fieldsis not particularly clear. As previously described, the reason for this is that the γ − correction to the asymptotically flat spacetime soft factor arises from a double scalinglimit. This implies that unlike asymptotically flat spacetimes, the γ − correction cannotresult from a unique saddle point approximation of the gauge fields. Rather, there canexist several contributions to γ = lω as we simultaneously take l → ∞ and ω →
0. Thus,while the saddle point approximation determines the soft mode N in terms of creation andannihilation operators on asymptotically flat spacetimes, this does not extend to possible1 /l corrections arising from the nearly flat limit of asymptotically AdS spacetimes. Thesecorrections could be determined by assuming that the equivalence of tree level soft theoremsand large gauge Ward identities on asymptotically flat spacetimes continues to hold undersmall corrections. As we will show in the following, this is highly constraining on theexpressions for the 1 /l corrections of the soft photon mode and residual gauge parameter. /l correction of the large gauge Ward identity We derived the leading soft photon factor correction S l (0)em from a classical scattering anal-ysis on an AdS spacetime in Eq. (4.51). As we are carrying out our analysis on flatspacetime (on assuming that we take l → ∞ in the AdS spacetime), the parametrizationsfor the momenta of the hard (and soft) massless particles and the polarization are the– 28 –ame as those given in Eq. (6.24), Eq. (6.25) and Eq. (6.26). Using these expressions inEq. (4.51), we have the the following result for the soft factor correction in terms of theasymptotic coordinates of the particles ( z , ¯ z ), ( z k , ¯ z k ) and the soft photon frequency S l (0)em = 14 l (cid:34) (cid:88) k =out p ( k ) .(cid:15) + p ( k ) .q (cid:126)p k ) (cid:0) p ( k ) .q (cid:1) Q k − (cid:88) k =in p ( k ) .(cid:15) + p ( k ) .q (cid:126)p k ) (cid:0) p ( k ) .q (cid:1) Q k (cid:35) = (1 + z ¯ z ) √ γ ω (cid:34) (cid:88) k =out (1 + z k ¯ z k ) (¯ z − ¯ z k ) z − z k ) Q k − (cid:88) k =in (1 + z k ¯ z k ) (¯ z − ¯ z k ) z − z k ) Q k (cid:35) , (6.29)where we have denoted l ω as γ .A consistent deformation of the soft photon theorem given in Eq. (6.21) on asymptoticallyflat spacetimes requires 1 /l corrections on either side of the equation. This leads to thefollowing identity for scattering processes on flat spacetimes (cid:104) out | a + ( ω ˆ x ) S| in (cid:105) + (cid:104) out | ˜ a + ( ω ˆ x ) S| in (cid:105) = (cid:16) S flat (0)em + S l (0)em (cid:17) (cid:104) out |S| in (cid:105) (6.30)with a separate theorem satisfied by the corrected components (cid:104) out | ˜ a + ( ω ˆ x ) S| in (cid:105) = S l (0)em (cid:104) out |S| in (cid:105) (6.31)In Eq. (6.30), S flat (0)em and S l (0)em are the universal leading contributions without and with 1 /l corrections respectively, and ˜ a + denotes a creation operator in the out state correspondingto the 1 /l correction. Note that the ˜ a + operator must be distinct from the flat space softphoton operator and in particular has a different dimension. We will make this point clearby defining ˜ a + = l a AdS+ later on.The way to interpret Eq. (6.30) is that there exists a scattering process in flat spacetime,and we insert a soft photon mode which involves a 1 /l contribution. This leads to acorresponding 1 /l correction of the soft factor. This result informs us that there mustbe 1 /l corrections of both ∂ w N and the gauge parameter (cid:15) appearing in the usual flatspacetime Ward identity given in Eq. (6.16). The reason for 1 /l corrections in ∂ w N alsofollows from 1 /l corrections of the soft photon operator.On the other hand, the soft factor informs us on the 1 /l correction of the gauge parameter (cid:15) ( z k , ¯ z k ). This must involve the terms appearing in the parenthesis of Eq. (6.29). However,we desire that the gauge parameter have some of the properties of the asymtotically flatspacetime gauge parameter given in Eq. (6.18). We accordingly define the total gaugeparameter to be – 29 – ( w , ¯ w ) = (cid:15) flat ( w , ¯ w ) + 1 l (cid:15) AdS ( w , ¯ w ) ; (cid:15) flat ( w , ¯ w ) = 1 ω − z ,(cid:15) AdS ( w , ¯ w ) = (1 + z ¯ z ) ( ¯ w − ¯ z ) (1 + w ¯ w ) ( w − z ) = (cid:15) flat ( w , ¯ w ) (1 + z ¯ z ) ( ¯ w − ¯ z ) (1 + w ¯ w ) ( w − z ) (6.32)We thus have defined (cid:15) AdS = Ω (cid:15) flat , where Ω = (1+ z ¯ z ) ( ¯ w − ¯ z ) (1+ w ¯ w ) ( w − z ) is a factor invariant underthe interchange of w ↔ z and ¯ w ↔ ¯ z . We note that this property ensures that (cid:15) AdS doeschange sign under the interchange of ω with z , as in the case of (cid:15) flat . This requirementled us to the include the factor (1 + w ¯ w ) in the definition of (cid:15) AdS apart from the termappearing in the parenthesis of Eq. (6.29).As noted above, we also have 1 /l corrections of the soft photon creation and annihilationoperators. We accordingly have a correction of ∂ z N ∂ w N = ∂ w N flat + 1 l ∂ w N AdS ; (6.33) ∂ w N flat = − √ π
11 + w ¯ w lim ω → (cid:104) ωa flat+ ( ω ˆ x ) − ωa flat − ( ω ˆ x ) † (cid:105) . (6.34)The goal is to determine the form of ∂ z N AdS such that the Ward identity for large gaugetransformations in Eq. (6.16) gives the soft photon theorem in Eq. (6.30), with the factorgiven in Eq. (6.29). By using the expressions from Eq. (6.32) and Eq. (6.33) in Eq. (6.16),and collecting the 1 /l coefficient, we find4 (cid:90) d w (cid:104) ∂ ¯ w (cid:15) flat ( w , ¯ w ) (cid:104) out | ∂ w N AdS ( w , ¯ w ) S| in (cid:105) + ∂ ¯ w (cid:15) AdS ( w , ¯ w ) (cid:104) out | ∂ w N flat ( w , ¯ w ) S| in (cid:105) (cid:105) = (cid:34) (cid:88) k =in (1 + z k ¯ z k ) (¯ z − ¯ z k ) (1 + z ¯ z ) ( z − z k ) − (cid:88) k =out (1 + z k ¯ z k ) (¯ z − ¯ z k ) (1 + z ¯ z ) ( z − z k ) (cid:35) (cid:104) out |S| in (cid:105) (6.35)Eq. (6.35) will agree with the correction to the soft photon theorem given in Eq. (6.31) if4 l (cid:90) d w (cid:104) ∂ ¯ w (cid:15) flat ( w , ¯ w ) (cid:104) out | ∂ w N AdS ( w , ¯ w ) S| in (cid:105) + ∂ ¯ w (cid:15) AdS ( w , ¯ w ) (cid:104) out | ∂ w N flat ( w , ¯ w ) S| in (cid:105) (cid:105) = − √ z ¯ z ) lim ω → (cid:104) out | ω ˜ a + ( ω ˆ x ) S| in (cid:105) (6.36)– 30 –his can only be true if ∂ w N AdS involves a sum of two parts – a term ∂ z N AdS1 whichprovides the ˜ a + contribution and another term ∂ z N AdS2 which only involves a flat+ . Wehence assume ∂ z N AdS = ∂ z N AdS1 + ∂ z N AdS2 (6.37)with ∂ w N AdS1 = h ( w , ¯ w ) lim ω → (cid:104) ω a AdS+ ( ω ˆ x ) − ω a AdS − ( ω ˆ x ) † (cid:105) (6.38) ∂ w N AdS2 = g ( w , ¯ w ) lim ω → (cid:104) ωa flat+ ( ω ˆ x ) − ωa flat − ( ω ˆ x ) † (cid:105) , (6.39)where ˜ a ± ( ω ˆ x ) = 1 l a AdS ± ( ω ˆ x ) , ˜ a ± ( ω ˆ x ) † = 1 l a AdS ± ( ω ˆ x ) † (6.40)We can now determine the expressions for h ( w , ¯ w ) and g ( w , ¯ w ) from the LHS of Eq. (6.35),which can be expressed as4 (cid:90) d w (cid:104) ∂ ¯ w (cid:15) flat (cid:104) out | ∂ w N AdS1 S| in (cid:105) + ∂ ¯ w (cid:15) flat (cid:104) out | ∂ w N AdS2 S| in (cid:105) + ∂ ¯ w (cid:15) AdS (cid:104) out | ∂ w N flat S| in (cid:105) (cid:105) = 4 (cid:90) d w∂ ¯ w (cid:15) flat (cid:104) out | ∂ w N AdS1 S| in (cid:105)− (cid:90) d w (cid:104) (cid:15) flat (cid:104) out | ∂ ¯ w ∂ w N AdS2 S| in (cid:105) + (cid:15) AdS (cid:104) out | ∂ ¯ w ∂ w N flat S| in (cid:105) (cid:105) (6.41)Hence the first term in the last line of Eq. (6.41) gives us4 (cid:90) d w∂ ¯ w (cid:15) flat (cid:104) out | ∂ w N AdS1 S| in (cid:105) = 8 π (cid:104) out | ∂ z N AdS1 ( z , ¯ z ) S| in (cid:105) (6.42)Using the expression from Eq. (6.38), we then find h ( z , ¯ z ) = − √ π (1 + z ¯ z ) , ⇒ ∂ z N AdS1 ( z , ¯ z ) = − √ π (1 + z ¯ z ) lim ω → (cid:104) ω a AdS+ ( ω ˆ x ) − ω a AdS − ( ω ˆ x ) † (cid:105) (6.43)We now also require that the last two terms in the last line of Eq. (6.41) cancel, whichdetermines the ∂ z N AdS2 contribution. From taking the ¯ w derivative of ∂ w N flat in Eq. (6.33)we have ∂ ¯ w ∂ w N flat = √ π w (1 + w ¯ w ) lim ω → (cid:104) ωa flat+ ( ω ˆ x ) − ωa flat − ( ω ˆ x ) † (cid:105) (6.44)– 31 –sing Eq. (6.44) and the expressions for (cid:15) AdS ( w , ¯ w ) and ∂ w N AdS2 from Eq. (6.32) andEq. (6.39) respectively, we then find that the terms in the last line of Eq. (6.41) to be (cid:90) d w (cid:104) (cid:15) flat (cid:104) out | ∂ ¯ w ∂ w N AdS2 S| in (cid:105) + (cid:15) AdS (cid:104) out | ∂ ¯ w ∂ w N flat S| in (cid:105) (cid:105) = (cid:90) d w(cid:15) flat (cid:34) ∂ ¯ w g ( w , ¯ w ) + √ w π (1 + z ¯ z ) ( ¯ w − ¯ z ) w − z ) (cid:35) (cid:104) out | ωa flat+ ( ω ˆ x ) S| in (cid:105) (6.45)Thus Eq. (6.45) vanishes if g ( w , ¯ w ) = − (cid:90) d ¯ w √ w π (1 + z ¯ z ) ( ¯ w − ¯ z ) w − z ) = √ w π (1 + z ¯ z ) ( ¯ w − ¯ z ) w ( w − z ) (6.46)Hence from Eq. (6.39) the result for ∂ w N AdS2 takes the form ∂ w N AdS2 = √ w π (1 + z ¯ z ) ( ¯ w − ¯ z ) w ( w − z ) lim ω → (cid:104) ωa flat+ ( ω ˆ x ) − ωa flat − ( ω ˆ x ) † (cid:105) (6.47)Thus the equation for the 1 /l corrected Ward identity on asymptotically flat spacetimesis2 (cid:90) d w (cid:104) ∂ ¯ w (cid:15) flat ( w , ¯ w ) (cid:104) out | ∂ w N AdS ( w , ¯ w ) S| in (cid:105) + ∂ ¯ w (cid:15) AdS ( w , ¯ w ) (cid:104) out | ∂ w N flat ( w , ¯ w ) S| in (cid:105) (cid:105) = (cid:34) (cid:88) k =in Q in k (cid:15) AdS ( z in k , ¯ z in k ) − (cid:88) k =out Q out k (cid:15) AdS ( z out k , ¯ z out k ) (cid:35) (cid:104) out |S| in (cid:105) , (6.48)which agrees with the corrected soft photon theorem in Eq. (6.31) by choosing (cid:15) ( w , ¯ w ) asin Eq. (6.32), and ∂ w N ( w , ¯ w ) as in Eq. (6.37).To summarize our main result in this section, we argued that 1 /l corrections in the softphoton factor arise due to a perturbation of the soft photon theorem on asymptoticallyflat spacetimes. The perturbed soft theorem was given in Eq. (6.31). Using the knownequivalence between soft theorems and large gauge Ward identities on asymptotically flatspacetimes, we could then derive a perturbed Ward identity which is given in Eq. (6.48).In the process of demonstrating this equivalence up to 1 /l , we determined the correctionsof the gauge parameter in Eq. (6.32) and the soft photon mode in Eq. (6.43) and Eq. (6.47)– 32 – Conclusion and open questions
Defining a quantum soft theorem in asymptotically AdS spaces is not only a technicallyinvolved problem but also is an unclear issue as the notion of asymptotic in and out statesare not well defined in AdS spacetime. Thus an alternative way to look for a possible softfactorization is required. The analysis of [21, 22] for asymptotically flat theories showedthat the soft factorization is also evident in classical radiation profiles. For asymptoti-cally flat spacetimes, the radiative parts of the electromagnetic and gravitational fieldsproduced in a classical scattering process provide the same leading quantum soft factor ob-tained from S matrix, up to the usual gauge ambiguity. Therefore, we looked for a similarbehaviour in asymptotically AdS systems with a small cosmological constant and found asimilar factorization by assuming the cosmological constant as a perturbation parameterover asymptotically flat gravity. Throughout our work, we assumed that massless radiationdie sufficiently fast at large spatial distances in our chosen coordinate system and do notbounce back at the spatial boundary.The soft factor arising from scattering processes on asymptotically AdS spacetimes can beexpected to have an interpretation in the boundary CFT dual to the spacetime. On asymp-totically flat spacetimes, soft theorems are equivalent to Ward identities arising from largegauge transformations. In Section 6, we used this relation to determine a Ward identitywhich recovers the soft photon factor including the leading 1 /l correction. In the presenceof 1 /l perturbative corrections to flat spacetime, the corrected soft photon theorem willgenerically take the form given in Eq. (6.30). We used this equation to determine thecorrected gauge parameter in Eq. (6.32), up to an overall factor. In order for Eq. (6.30) toresult from a perturbed flat spacetime Ward identity for large gauge transformations, weargued that the corrected soft photon mode N AdS must have a particular form. The softphoton mode must involve a component proportional to the flat spacetime creation and an-nihilation operators and another component with the corrected operators. The expressionsof these two components were derived in Eq. (6.43) and Eq. (6.47).More recently in [44], [30], the flat spacetime soft photon factor was derived from conformalWard identities on the boundary of global AdS spacetime. The derivation made use of therealization of flat spacetime as a patch near the center of the AdS spacetime in the large l limit. A description of bulk fields in terms of boundary operators, determined by using theHKLL formalism [45], can identify photon operators in the flat spacetime patch in termsof a U (1) current. A crucial step in this analysis requires taking the frequency spectrumof massless fields in AdS as ω flat l , where ω flat is the frequency in flat spacetime. Withthis assumption, the low frequency limit near the center of AdS spacetimes corresponds tothe soft limit in flat spacetime, on taking the simultaneous limit of l → ∞ and ω flat → l → ∞ limit, the flat spacetime soft photon factor relates acorrelation function with a soft conserved current insertion to a correlation function withoutthe inserted operator. It remains an open problem to derive our expressions for the soft– 33 –hoton and graviton factors in Section 5 from the boundary CFT on asymptotically AdSspacetimes. This can in principle be derived using the formalism in [44], [30], while nowretaining order 1 /l terms in the large AdS radius limit. We look forward to providing aconformal Ward identity derivation of the AdS corrected soft photon and graviton factorsin future work. Acknowledgements
We would like to thank Sayali Bhatkar and Arindam Bhattacharjee for useful discussions.Our work is partially supported by a SERB ECR grant, GOVT of India. Finally, we thankthe people of India for their generous support to the basic sciences.
A Classical soft graviton theorem
In this section we will consider the contribution of the scatterer black hole charge to thegravitational soft factor. For a detailed calculation of the gravitation soft factor as acontribution of potential due to the black hole mass and AdS radius we refer the reader to[31] and [32] respectively. S gr in D = 4 dimensions can be computed using, S gr = i πRe iωR (cid:15) ij ˜ e ij ( ω, (cid:126)x ) , (A.1)Similar to the the electromagnetic case, we let r = t , assume x >> r ( t ) and we will writeEq. (3.20) as,˜ e ij ( ω, (cid:126)x ) = ˜ e (1) ij ( ω, (cid:126)x ) + ˜ e (2) ij ( ω, (cid:126)x ) + ˜ e (3) ij ( ω, (cid:126)x ) + ˜ e (4) ij ( ω, (cid:126)x ) + ˜ e (5) ij ( ω, (cid:126)x ) + ˜ e (6) ij ( ω, (cid:126)x ) + ˜ e (7) ij ( ω, (cid:126)x )+ ˜ e (8) ij ( ω, (cid:126)x ) + ˜ e (9) ij ( ω, (cid:126)x ) + ˜ e (10) ij ( ω, (cid:126)x ) , (A.2)– 34 –here˜ e (1) ij ( ω, (cid:126)x ) = m e iωR π R − x l (cid:90) dt (cid:16) (cid:126)r ( t )) + r l (cid:17) dtdσ v i v j e iω ( t − ˆ n.(cid:126)r ( t )) + boundary terms , (A.3)˜ e (2) ij ( ω, (cid:126)x ) = m π (cid:90) dt dtdσ e iωt (1 + (cid:126)v ) (cid:18) ∇ i ∇ j − δ ij ∇ k ∇ k (cid:19) ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (A.4)˜ e (3) ij ( ω, (cid:126)x ) = − i M m π ω e iωR R (cid:90) dt dtdσ v i v j (cid:26) ln | (cid:126)r (cid:48) | + ˆ n.(cid:126)r (cid:48) R e iω ( t − ˆ n.(cid:126)r (cid:48) ) + (cid:90) ∞| (cid:126)r (cid:48) | +ˆ n.(cid:126)r (cid:48) duu e iω ( t − ˆ n.(cid:126)r (cid:48) + u ) (cid:27) , (A.5)˜ e (4) ij ( ω, (cid:126)x ) = iωmπ (cid:90) dt dtdσ e iωt ( v i ∇ j + v j ∇ i ) ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (A.6)˜ e (5) ij ( ω, (cid:126)x ) = − iω qQ πM (cid:90) dte iωt δ ij v k ∇ k ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (A.7)˜ e (6) ij ( ω, (cid:126)x ) = i qQπM (cid:90) dte iωt (cid:18) ∇ i ∇ j − δ ij ∇ k ∇ k (cid:19) ˜ G M ( ω, (cid:126)x, (cid:126)r ) , (A.8)˜ e (7) ij ( ω, (cid:126)x ) = iω qQ πM (cid:90) dte iωt ( v i ∇ j + v j ∇ i ) ˜ G M ( ω, (cid:126)x, (cid:126)r ) (A.9)˜ e (8) ij ( ω, (cid:126)x ) = − m πl (cid:90) dt dtdσ (cid:20) − (cid:0) − (cid:126)v (cid:1) ∇ i ∇ j − δ ij v k v m ∇ k ∇ m − δ ij v ∇ k ∇ k −
32 ( v k v j ∇ k ∇ i + v k v i ∇ k ∇ j ) + v i v j ∇ k ∇ k (cid:21) ˜ G l ( ω, (cid:126)x, (cid:126)r ) , (A.10)˜ e (9) ij ( ω, (cid:126)x ) = − m πl ω (cid:90) dt dtdσ e iωt v i v j ˜ G l ( ω, (cid:126)x, (cid:126)r ) , (A.11)˜ e (10) ij ( ω, (cid:126)x ) = m πl (cid:90) dt dtdσ e iωt ( v i ∇ j + v j ∇ i ) ˜ G l ( ω, (cid:126)x, (cid:126)r ) , (A.12)˜ e (1) ij ( ω, (cid:126)x ) to ˜ e (4) ij ( ω, (cid:126)x ) are contributions of the black hole mass. ˜ e (5) ij ( ω, (cid:126)x ) to ˜ e (7) ij ( ω, (cid:126)x )arises due to the charge of the black hole. ˜ e (8) ij ( ω, (cid:126)x ) to ˜ e (10) ij ( ω, (cid:126)x ) can be treated as thecontribution from the AdS radius.The contribution of black hole mass and charge to the soft factor for gravitation was– 35 –xplicitly derived in [31], S flatgr = i πRe iωR (cid:15) ij ˜ e ij ( ω, (cid:126)x )= − mω (cid:15) ij − ˆ n.(cid:126)β + (cid:113) − (cid:126)β β + i β + j − − ˆ n.(cid:126)β − (cid:113) − (cid:126)β − β − i β − j − im ln ω − (cid:15) ij (cid:113) − (cid:126)β (cid:40) C + (cid:32) − ˆ n.(cid:126)β + + 11 − (cid:126)β (cid:33) − M π | (cid:126)β + | (cid:126)β − − (cid:126)β (cid:41) β + i β + j − (cid:113) − (cid:126)β − (cid:40) C − (cid:32) − ˆ n.(cid:126)β − + 11 − (cid:126)β − (cid:33) + M π | (cid:126)β − | (cid:126)β − − − (cid:126)β − (cid:41) β − i β − j − im M π ln ( Rω ) (cid:15) ij − ˆ n.(cid:126)β + (cid:113) − (cid:126)β β + i β + j − − ˆ n.(cid:126)β − (cid:113) − (cid:126)β − β − i β − j − i qQ π ln ω − (cid:15) ij (cid:34) β + i β + j | (cid:126)β + | + β − i β − j | (cid:126)β − | (cid:35) + finite (A.13)To complete the analysis we need the expression for C ± , which can be determined fromconsidering the energy conservation equation. The energy of the probe particle can bewritten from the point particle action in Eq. (2.13) as, E = m | g | dtdσ − q π A (A.14)Expanding this expression in powers of t , we then find from the t coefficient the followingrelation of C ± with M and Q [31] C ± = ∓ M π | (cid:126)β ± | (1 − (cid:126)β ± ) ∓ qQ πm | (cid:126)β ± | (1 − (cid:126)β ± ) / . (A.15)Substituting for M and Q in Eq. (A.13) using Eq. (A.15) gives S flatgr = − mω (cid:15) ij − ˆ n.(cid:126)β + (cid:113) − (cid:126)β β + i β + j − − ˆ n.(cid:126)β − (cid:113) − (cid:126)β − β − i β − j − im ln ω − (cid:15) ij C + − ˆ n.(cid:126)β + (cid:113) − (cid:126)β β + i β + j − C − − ˆ n.(cid:126)β − (cid:113) − (cid:126)β − β − i β − j − im M π ln ( Rω ) (cid:15) ij − ˆ n.(cid:126)β + (cid:113) − (cid:126)β β + i β + j − − ˆ n.(cid:126)β − (cid:113) − (cid:126)β − β − i β − j (A.16)This result agrees with the classical limit of the soft graviton factor, up to the sublead-ing logarithmic contribution, in asymptotically flat spacetimes. In [32], the leading order– 36 –ontribution of cosmological constant to the soft factor was derived with the followingexpression S l gr = − m γ ω − (cid:15) ij − ˆ n.(cid:126)β ) β + i β + j (cid:126)β (3 − (cid:126)β )(1 − (cid:126)β ) − − ˆ n.(cid:126)β ) β − i β − j (cid:126)β − (3 − (cid:126)β − )(1 − (cid:126)β − ) − i m γ ln ω − (cid:15) ij β + i β + j (1 − ˆ n.(cid:126)β + ) C + (cid:126)β (3 − (cid:126)β ) (cid:16) − (cid:126)β (cid:17) − β − i β − j (1 − ˆ n.(cid:126)β − ) C − (cid:126)β − (3 − (cid:126)β − ) (cid:16) − (cid:126)β − (cid:17) (A.17) References [1] S. Weinberg, Phys. Rev. , B1049-B1056 (1964) doi:10.1103/PhysRev.135.B1049[2] S. Weinberg, Phys. Rev. , B516-B524 (1965) doi:10.1103/PhysRev.140.B516[3] D. Kapec, V. Lysov and A. Strominger, Adv. Theor. Math. Phys. , 1747-1767 (2017)doi:10.4310/ATMP.2017.v21.n7.a6 [arXiv:1412.2763 [hep-th]].[4] T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, JHEP , 112 (2014)doi:10.1007/JHEP10(2014)112 [arXiv:1407.3789 [hep-th]].[5] A. Strominger, JHEP , 152 (2014) doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229[hep-th]].[6] F. Cachazo and A. Strominger, [arXiv:1404.4091 [hep-th]].[7] M. Campiglia and A. Laddha, JHEP , 076 (2015) doi:10.1007/JHEP04(2015)076[arXiv:1502.02318 [hep-th]].[8] V. Lysov, S. Pasterski and A. Strominger, Phys. Rev. Lett. , no.11, 111601 (2014)doi:10.1103/PhysRevLett.113.111601 [arXiv:1407.3814 [hep-th]].[9] B. U. W. Schwab and A. Volovich, Phys. Rev. Lett. , no.10, 101601 (2014)doi:10.1103/PhysRevLett.113.101601 [arXiv:1404.7749 [hep-th]].[10] M. Campiglia and A. Laddha, Phys. Rev. D , no.12, 124028 (2014)doi:10.1103/PhysRevD.90.124028 [arXiv:1408.2228 [hep-th]].[11] E. Casali, JHEP , 077 (2014) doi:10.1007/JHEP08(2014)077 [arXiv:1404.5551 [hep-th]].[12] J. Broedel, M. de Leeuw, J. Plefka and M. Rosso, Phys. Rev. D , no.6, 065024 (2014)doi:10.1103/PhysRevD.90.065024 [arXiv:1406.6574 [hep-th]].[13] E. Conde and P. Mao, Phys. Rev. D , no.2, 021701 (2017)doi:10.1103/PhysRevD.95.021701 [arXiv:1605.09731 [hep-th]].[14] S. Chakrabarti, S. P. Kashyap, B. Sahoo, A. Sen and M. Verma, JHEP , 090 (2018)doi:10.1007/JHEP01(2018)090 [arXiv:1709.07883 [hep-th]].[15] S. Chakrabarti, S. P. Kashyap, B. Sahoo, A. Sen and M. Verma, JHEP , 150 (2017)doi:10.1007/JHEP12(2017)150 [arXiv:1707.06803 [hep-th]].[16] A. Laddha and P. Mitra, JHEP , 132 (2018) doi:10.1007/JHEP05(2018)132[arXiv:1709.03850 [hep-th]]. – 37 –
17] S. Atul Bhatkar and B. Sahoo, JHEP , 153 (2019) doi:10.1007/JHEP01(2019)153[arXiv:1809.01675 [hep-th]].[18] A. Addazi, M. Bianchi and G. Veneziano, JHEP , 050 (2019)doi:10.1007/JHEP05(2019)050 [arXiv:1901.10986 [hep-th]].[19] B. Sahoo, JHEP , 070 (2020) doi:10.1007/JHEP12(2020)070 [arXiv:2008.04376 [hep-th]].[20] B. Sahoo and A. Sen, JHEP , 086 (2019) doi:10.1007/JHEP02(2019)086[arXiv:1808.03288 [hep-th]].[21] A. Laddha and A. Sen, JHEP , 105 (2018) doi:10.1007/JHEP09(2018)105[arXiv:1801.07719 [hep-th]].[22] A. Laddha and A. Sen, JHEP , 056 (2018) doi:10.1007/JHEP10(2018)056[arXiv:1804.09193 [hep-th]].[23] A. Laddha and A. Sen, Phys. Rev. D , no.2, 024009 (2019)doi:10.1103/PhysRevD.100.024009 [arXiv:1806.01872 [hep-th]].[24] A. Laddha and A. Sen, Phys. Rev. D , no.8, 084011 (2020)doi:10.1103/PhysRevD.101.084011 [arXiv:1906.08288 [gr-qc]].[25] A. P. Saha, B. Sahoo and A. Sen, JHEP , 153 (2020) doi:10.1007/JHEP06(2020)153[arXiv:1912.06413 [hep-th]].[26] M. Gary and S. B. Giddings, Phys. Rev. D , 046008 (2009)doi:10.1103/PhysRevD.80.046008 [arXiv:0904.3544 [hep-th]].[27] J. Penedones, JHEP , 025 (2011) doi:10.1007/JHEP03(2011)025 [arXiv:1011.1485[hep-th]].[28] A. L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B. C. van Rees, JHEP , 095(2011) doi:10.1007/JHEP11(2011)095 [arXiv:1107.1499 [hep-th]].[29] L. Rastelli and X. Zhou, Phys. Rev. Lett. , no.9, 091602 (2017)doi:10.1103/PhysRevLett.118.091602 [arXiv:1608.06624 [hep-th]].[30] E. Hijano and D. Neuenfeld, JHEP , 009 (2020) doi:10.1007/JHEP11(2020)009[arXiv:2005.03667 [hep-th]].[31] K. Fernandes and A. Mitra, Phys. Rev. D , no.10, 105015 (2020)doi:10.1103/PhysRevD.102.105015 [arXiv:2005.03613 [hep-th]].[32] N. Banerjee, A. Bhattacharjee and A. Mitra, JHEP , 038 (2021)doi:10.1007/JHEP01(2021)038 [arXiv:2008.02828 [hep-th]].[33] T. He, V. Lysov, P. Mitra and A. Strominger, JHEP , 151 (2015)doi:10.1007/JHEP05(2015)151 [arXiv:1401.7026 [hep-th]].[34] Y. Hamada and G. Shiu, Phys. Rev. Lett. , no.20, 201601 (2018)doi:10.1103/PhysRevLett.120.201601 [arXiv:1801.05528 [hep-th]].[35] M. Campiglia and A. Laddha, JHEP , 287 (2019) doi:10.1007/JHEP10(2019)287[arXiv:1903.09133 [hep-th]].[36] S. Atul Bhatkar, JHEP , 110 (2020) doi:10.1007/JHEP10(2020)110 [arXiv:1912.10229[hep-th]].[37] R. M. Wald, doi:10.7208/chicago/9780226870373.001.0001 – 38 –
38] B. S. DeWitt and R. W. Brehme, Annals Phys. , 220-259 (1960)doi:10.1016/0003-4916(60)90030-0[39] P. C. Peters, “Perturbations in the Schwarzschild Metric,” Phys. Rev. , 938 (1966).doi:10.1103/PhysRev.146.938[40] P. C. Peters, Phys. Rev. D , 1559-1571 (1970) doi:10.1103/PhysRevD.1.1559[41] S. J. Kovacs and K. S. Thorne, Astrophys. J. , 252-280 (1977) doi:10.1086/155576[42] E. Poisson, A. Pound and I. Vega, Living Rev. Rel. , 7 (2011) doi:10.12942/lrr-2011-7[arXiv:1102.0529 [gr-qc]].[43] A. Strominger, [arXiv:1703.05448 [hep-th]].[44] E. Hijano, JHEP , 132 (2019) doi:10.1007/JHEP07(2019)132 [arXiv:1905.02729 [hep-th]].[45] A. Hamilton, D. N. Kabat, G. Lifschytz and D. A. Lowe, Phys. Rev. D , 066009 (2006)doi:10.1103/PhysRevD.74.066009 [arXiv:hep-th/0606141 [hep-th]]., 066009 (2006)doi:10.1103/PhysRevD.74.066009 [arXiv:hep-th/0606141 [hep-th]].