Subleading Soft Theorem for arbitrary number of external soft photons and gravitons

We obtain the subleading soft theorem for a generic theory of quantum gravity, for arbitrary number of soft photons and gravitons and for arbitrary number of finite energy particles with arbitrary mass and spin when all the soft particles are soft in the same rate. This result is valid at tree level for spacetime dimensions equal to four and five and to all loops in spacetime dimensions greater than five. We verify that in classical limit low energy photon and graviton radiation decouple from each other.

For specific theories subleading soft graviton theorem with two or arbitrary number of external soft gravitons has been derived [72][73][74][75][76][77][78][79][80]. Very recently, soft graviton theorem has been derived for a generic theory of quantum gravity with arbitrary number of external particles with arbitrary mass and spin, starting from 1PI effective action and covariantizing with respect to soft graviton background [70,71,81,82]. For a generic theory of quantum gravity multiple soft graviton theorem was derived in [82] when all the gravitons are soft in the same rate(simultaneous limit). In [83,84] the double soft graviton theorem was analysed from asymptotic symmetry point of view and symmetry interpretation was found for consecutive limit i.e. when one of the soft graviton is softer than the other one. A new direction was taken up in seminal papers [85][86][87] where classical limit of multiple soft graviton theorem was studied to obtain the long wavelength gravitational radiation emitted in classical scattering processes. Recently in four dimensions soft theorem was explored beyond tree level in [88] using extension of the infrared treatment developed in [89,90] and it was shown that new non-analytic terms appear in soft momentum expansion in the form of logarithmic functions of soft energy.
Our goal in this paper is to derive subleading soft theorem for a generic theory of quantum gravity, for arbitrary number of soft photons and gravitons and for arbitrary number of finite energy particles with arbitrary mass and spin when all the soft particles are soft in the same rate. This result can be applied in three possible directions which need to be pursued. One is to find the Ward identity interpretation of this soft theorem and understand the asymptotic symmetry. Another is to consider classical limit analogues to [85][86][87] and derive the power spectrum for long wavelength photon and graviton radiation in classical scattering in an example where both electromagnetic and gravitational interactions are important. The other is to test the CHY formula [91][92][93][94][95] for Einstein-Maxwell theory taking soft limit analogously as it was done for perturbative Einstein gravity in [80].
As it is well known there are IR divergences in D=4 and notion of S-matrix is itself ill-defined. As discussed in [82], this is seen in 1PI effective vertices containing soft particles, which have soft momentum factors in denominator. Thus, our result will hold only at tree level in D = 4, 5. In D > 5, there are no additional divergences even if loop momentum goes soft. So, our result holds to all loops. We hope that we can extend our analysis in D = 4 to extract logarithmic terms even for multiple external soft photon-graviton case generalising the strategy developed in [88].
We will combine soft graviton and photon into a single soft field and write down the soft theorem for this composite soft particle. We denote hard particle polarisations, momenta and charges by {ǫ i , p i , Q i } for i = 1, 2, · · · , N. Soft particle polarisations and momenta are denoted by {ξ r , k r } with ξ r (k r ) ≡ {e r (k r ), ε r (k r )} where e r , ε r respectively being the polarisation of soft photon and soft graviton for r = 1, 2, · · · , M. Our final result for soft theorem with N number of finite energy particles (hard particles) and M number of soft composite particles up to subleading order in soft momenta expansion is as follows: {p i · (k r + k u )} −1 M pp p i ; e r , k r ; e u , k u + M gg p i ; ε r , k r ; ε u , k u + M pg p i ; e r , k r ; ε u , k u + M pg p i ; e u , k u ; ε r , k r Γ α 1 ···α N , Q i e r,µ p µ i p i · k r , (1.2)
Since photon can couple to hard particles via its field strength in a non-universal way, the subleading soft part of the theorem depends on the theory and the theory dependent part is the N µν (i) (−p i ) factor which contains the effect of the non-minimal coupling term of the soft photon with two finite energy fields via field strength. Another point we note is that our multiple soft photongraviton theorem is dependent on spacetime dimension D, which appears in the expression of M pp .
As mentioned earlier, (1.1) is written in a combined way considering soft graviton and photon as two components of a single soft field. One can extract multiple soft photon theorem from (1.1) by setting all the graviton polarisations {ε r } to be zero. Similarly to get multiple soft graviton theorem from (1.1) one have to set photon polarisations {e r } to zero and the result agrees with [82].
The paper is organised as follows : we derive vertices and propagators needed to prove soft theorem in section 2 . In section 3 we write down the general strategy that will be used to manipulate the amplitudes to write them as soft theorems. Then, we derive the single, double and multiple soft theorems in successive sections 4 , 5 , 6 . Finally, we extract the soft theorem for arbitrary number of soft photons and gravitons and analyse it's classical limit to get long wavelength power spectrum of photon and graviton radiation in section 7 .

Evaluation of vertices and propagators
We will consider a generic theory of quantum gravity which is UV complete and background independent and is given in the form of 1PI effective action. For this generic theory to derive the relevant vertices and propagators we will follow the strategy developed in [70] [71] [81] [82]. First we will expand the 1PI effective action in terms of all the fields of the theory (including photon and graviton) around their vacuum values. Then we will gauge fix the action using Lorentz covariant gauge fixing condition to get well defined propagators having simple poles. Now the vertices having coupling with soft gravitons and/or photons are derived by covariantizing the gauge fixed action with respect soft photon and graviton background. We assume that all the fields carry tangent space indices and when covariantizing, we have to replace all the ordinary derivatives by covariant derivatives multiplied with inverse vielbeins in the soft photon-graviton background. We parametrize the metric in the following way : The vielbeins are similarly expanded as Here we use a, b, c, · · · as tangent space indices, µ, ν, ρ, · · · as curved space indices and all the indices are raised and lowered by the flat metric η. If {Φ α } represents all sets of fields present in the theory which belongs to some reducible representation of the local Lorentz group SO(1,D-1), then in the covariantization procedure a set of ordinary derivatives operating on Φ α transforms as where Here A µ represents the background U(1) gauge field and Q α is the U(1) charge of field Φ α . J ab is the spin angular momenta normalised in a way that when it operates on a covariant vector field φ c , it takes the form In the analysis of subleading soft theorem we need the spin connection only upto first order in S µν , When we covariantize two derivative terms, we get three new kind of couplings with soft fields: In (2.7) the first term containing Christoffel connection appears from the definition of left covariant derivative operation on a vector. The second term comes from the operation of left derivative on the gauge field contained in the right covariant derivative and the third term contains two gauge fields come from two covariant derivatives. The rest of the terms " · · · " represent the usual terms containing two derivatives, spin connections etc. which one can trivially get from the successive single derivative covariantization. In our analysis for multiple soft theorem upto subleading order in soft momenta, while covariantizing the terms having single soft field we need to keep upto first derivative of the soft field and for the terms having two soft fields we need to keep only the terms which do not have any derivatives on the soft fields. Also we can drop the terms containing Christoffel connection following the argument given in the appendix of [71]. The convention we follow is that all the particles are ingoing and in any diagram the thick lines represent hard particles and thin lines represent combined soft particles (photons and gravitons). The mode expansion of external soft photon and graviton fields with momentum k µ are A µ (x) = e µ (k) e ik.x , k µ e µ = 0 , (2.9) S µν (x) = ε µν (k) e ik.x , ε µν η µν = 0 , k µ ε µν = 0 = k ν ε µν . (2.10) Above we choose graviton polarisation to be symmetric traceless for simplicity, so that −det(g) is always one in the covariantized action. In diagram computations we treat soft photon and graviton fields are two components of a single soft field having polarisation ξ(k) = e(k), ε(k) .
To determine the vertices containing two finite energy fields and one or two soft particles we start by covariantising the quadratic part of 1PI effective action with the kinetic term satisfying : If Φ α is a grassmann odd field then the r.h.s of (2.12) contains an extra -ve sign but final result does not depend on this. The full renormalised propagator of the internal finite energy particle with momentum q µ is given by, where M is the renormalised mass of the particle.
Now the vertex associated with two finite energy fields and one soft field with polarisation and momentum (ξ, k) upto linear order in soft momentum, will be obtained from the following covariantized part of (2.11) In the above expression the first term inside the square bracket comes from the presence of gauge field within covariant derivative. Second term comes from the symmetric part of derivative operating on the gauge field when two covariant derivatives operate on the field Φ β . Source of third term is multiplication by inverse vielbein to transform flat space index to curved space index.
The fourth term comes from the spin connection part in covariant derivative. There is another source of three point vertex associated with soft photon coming from the non-minimally coupled field strength with two finite energy fields having most general form : where, Figure 1: 1PI vertex involving two finite energy particles and one soft particle.
The expression of 1PI vertex corresponding to Fig.(1), derived from (2.14) and (2.15) is : Now, global U(1) charge conservation in Γ (3) vertex implies Q α = −Q β ≡ Q. Using (2.12),(2.9),(2.10), (2.16) and making Taylor series expansion in k µ we write down the matrix form of Γ (3) up to linear order in k µ : (2.18) Next we want to derive the four point vertex involving two finite energy particles and two soft particles. The diagrams which contain this kind of vertex are already in subleading order relative to the diagrams having only Γ (3) vertices. So we need to evaluate this vertex only up to leading order in soft momenta. Hence we will drop terms having spin connection, Christoffel connection, derivative over gauge fields as well as the non-minimal coupling piece in the covariantized action. We need to pick terms quadratic in soft fields, i.e. photon-photon, photon-graviton and gravitongraviton fields. The part of the covariantized action from where we will get the required four point vertex involving two soft fields with polarisations and momenta (ξ(k 1 ), k 1 ) and (ξ(k 2 ), k 2 ) is:  Figure 3: An amputated Greens function with N-number of external finite energy particles and one soft particle. This diagram represents sum over the diagrams where the soft particle is not attached to any external leg.
The amputated Greens function of Fig.(3) can be evaluated by covariantizing the N particle amputated Greens function Γ(q 1 , q 2 , · · · , q N ) with respected to soft background field (photon and graviton). Since the diagrams that containΓ are already in subleading order, we only need its leading contribution.
The contribution of V (3) have three sources depending on the choice of two external soft particles (ξ 1 , ξ 2 ). When both the external soft particles are photons then the internal soft line has to be graviton and the contribution to V (3) with internal soft graviton carrying indices (ρ, σ) comes from the second part of (2.22), it is [96] : (2.23) When one of the external particle is photon and the other one is graviton then the internal soft particle has to be photon and the contribution to V (3) with internal soft photon carrying index µ from the second part of (2.22) turns out to be [96] : When both the external soft particles are gravitons then the internal soft particle has to be graviton and the contribution to V (3) with internal soft graviton carrying indices (µ, ν) from the first part of (2.22) in de Donder gauge turns out to be [82,96] : Hence the total contribution of the vertex in Fig.(4) contracted with some arbitrary off-shell polarisation tensorξ α ≡ (ê µ ,ε µν ), takes the form: Let us now focus on deriving propagators. The hard particle propagator is already given in (2.13). The soft graviton propagator with momentum k in de Donder gauge with endpoints (µ, ν) and (ρ, σ) takes the form: The soft photon propagator with momentum k in Feynman gauge with the endpoints µ and ν takes the form: (2.28)

Strategy for evaluation of diagrams
Before describing the strategy we first write down the identities which we have to use while computing diagrams. If the polarisation of i'th hard particle be ǫ i (p i ) then it satisfies the on-shell condition We can write down the hard particle propagator definition given in (2.13) for i'th particle, in matrix notation (removing the polarisation indices) as : Now operating one momentum derivative on above relations we get Operating another momentum derivative on (3.4) we get (3.6) Lorentz covariance of K and Ξ implies: To evaluate the diagrams we follow the steps : 1. First we write down the contribution of the diagram in terms of K and Ξ using the vertices and propagators derived in sec.
(2) with the hard particle polarisation tensors multiplied from the left.
2. Move all the J ab and (J ab ) T factors to the extreme right using (3.7), (3.8) and their derivatives so that the polarisation indices of these directly contract with Γ α 1 ,α 2 ,··· ,α N .
3. Expand all the K, Ξ and Γ α 1 ,α 2 ,··· ,α N in Taylor series in power of soft momenta up to the required order.
4. Now transfer the derivatives on K to Ξ to the maximal possible extent using (3.4), (3.6). For some special cases we need to move derivative from Ξ to K, we have to use (3.5).
5. At the end of the manipulation we need to use the on-shell condition (3.1).
4 Subleading soft theorem for one external soft particle In this section, we will write down soft theorem till subleading order in soft momentum for one external composite particle ( graviton and photon) going soft. The aim is to obtain the theorydependent terms in single soft theorem, as the universal pieces are well known in the literature. Let us start by defining the N-particle amplitude without soft insertion after stripping out the i'th particle polarisation tensor Using the expression for Γ (3) given in (2.18) and following the strategy described in sec.(3) up to subleading order we get where, Substituting (4.3) in (4.2) and using (3.1) after expanding Γ (i) (p i + k) up to linear order in k we get Figure 6: Diagram start contributing at subleading order being the soft particle is not attached to any external line. Fig.(6) one can easily read out from (2.21) to be

Contribution of
Now summing over contributions for soft insertion in different legs for Fig.(5) and adding the contribution (4.8) we get the subleading soft theorem for one external soft particle From here, we can recover the single soft graviton and photon theorems respectively by setting photon and graviton polarisation to be zero. We note that the theory dependent term N µν (i) (−p i ) come from the soft photon coupling via its field strength. We will eventually see that the same non-universal piece appears in the multiple soft theorem.
The above result is invariant under the following gauge transformations: for arbitrary scalar function λ(k) and arbitrary vector function ζ µ (k) satisfying ζ µ (k)k µ = 0.

Subleading soft theorem for two external soft particles
We consider two external soft particles with polarisations and and momenta (ξ 1 , k 1 ) ≡ (e 1 , ε 1 , k 1 ) and (ξ 2 , k 2 ) ≡ (e 2 , ε 2 , k 2 ). The leading contribution comes from the kind of diagrams having both the soft particles attached to external leg via Γ (3) vertices as shown in Fig.(7) and Fig.(8). The other diagrams shown in Fig.(9), (10) and (11) start contributing from subleading order in soft momenta. Figure 7: Diagram with two soft particles attached to same leg start contributing at leading order in soft momenta.
We begin with the evaluation of the contribution of Fig.(7), which in terms of vertices and propagators has the form : Using eq.(4.3), we get Now first substituting explicit form of N µν (i) (−p i ) from (4.4) in the above expression and then following the strategy described in Sec.(3) we get: To this, we need to add the contribution of the diagram analogous to Fig.(7) with (ξ 1 , k 1 ) ↔ (ξ 2 , k 2 ) exchange, . The diagram where two of the soft particles are attached to two different legs as shown in Fig.(8), one can easily compute following the procedure used for computation of A 1 . So using the result of (4.7) for i'th and j'th leg, the contribution of Fig.(8) becomes Figure 9: Diagram with one soft particle attached to external leg and another one attached to internal leg, starts contributing at subleading order.
To evaluate the contribution of Fig.(9) for i'th leg we can directly follow the derivation of A 1 , but we only need to keep the leading contribution here since the diagram starts contributing at subleading order. For soft insertions with polarisation ξ 2 , we can use the result of (2.21). Then the contribution of Fig.(9) turns out to be To this we have to add another diagram contribution with (ξ 1 , k 1 ) ↔ (ξ 2 , k 2 ) exchange of Fig.(9 Using the vertex expression from (2.20) and following the strategy given in sec.
(3) we get Figure 11: Diagram containing vartex having three soft particle interection starts contributing at subleading order. Fig.(11) also starts contributing from subleading order. Using the three soft particle vertex defined in (2.26) after stripping off the arbitrary off-shell polarisation tensor and contracting with proper internal soft propagator, the contribution turns out to be : Now we have to perform sum over all external legs i = 1, 2, · · · , N for the contributions and B 5 and sum over pair i, j = 1, 2, · · · , N for i = j for the contribution B 2 . Then adding all the contributions and organising in the standard form we get the expression of double soft theorem when both the particles become soft at the same rate, {p i · (k 1 + k 2 )} −1 M pp p i ; e 1 , k 1 ; e 2 , k 2 + M pg p i ; e 1 , k 1 ; ε 2 , k 2 + M pg p i ; e 2 , k 2 ; ε 1 , k 1 + M gg p i ; ε 1 , k 1 ; ε 2 , k 2 Γ . where M pp p i ; e 1 , k 1 ; e 2 , k 2 (5.14) One can obtain the double soft photon theorem by setting both the graviton polarisations to zero. The contact terms that survive have been clubbed within M pp . Similarly, M pg contains the contact terms that are present when one external graviton and one external photon are taken to be soft. This soft theorem is obtained by setting for example e 1,µ = 0 = ε 2,µν . Similarly, the double soft graviton theorem can be obtained by setting all the photon polarisations to be zero, which agrees with [82].
The statements of momentum, angular momentum and charge conservation which needs to be used in the intermediate stages to prove gauge invariance are: Under the gauge transformation for photon polarisation it is trivial to check that (5.11) is gauge invariant using charge conservation (5.19). Checking the gauge invariance under the gauge transformation of graviton is a little non-trivial. There in order to make use of (5.18), one has to pass p µ i through ∂/∂p jν , picking extra contribution δ ij η µν . The terms which do not vanish by themselves have the following variation under gauge transformation These terms add up to zero and other terms are invariant individually after using (5.18), (5.19) and (5.19) at different stages. Similarly we have following two variations adding up to zero, Analogous analysis goes through for gauge transformation of ε 2,µν . Also it is clear from above calculation that double soft photon, double soft graviton and soft gravtion-photon theorems are gauge invariant individually.

Subleading soft theorem for multiple soft particles
We will begin by writing down the multiple soft photon-graviton result for M number of soft particles which we will eventually prove: Let us start proving the leading part of (6.1), which can be written as: This proves (6.7). Now to prove the subleading soft theorem part let us start with the subleading part of Fig.(12).
Following the structures of (4.3) we can write the expression of Fig.(12) as: Now we will analyse the subleading terms one by one. First consider the contribution from k r ·k u terms in the denominator of propagators in the first line. When we expand the denominator of propagators in powers ofk r ·k u we need to pick up orderk r ·k u term from one of the propagator.
In the rest of the propagators we can putk r ·k u = 0 and for the rest of the coefficients we can only take the leading contribution. This term turns out to be  Q iẽs,µ p µ i +ε s,µν p µ i p ν i n r,u=1 r<uk r ·k u p i · (k r +k u ) −1 ǫ T i Γ (i) (p i ) . (6.12) Next consider the terms involvingẽ u andε u contracted withk r for r < u, appearing in the first and fifth terms within the square bracket of (6.10). Being subleading if we choose this kind of terms from u'th square bracket, from the rest of the square brackets of (6.10) we only need to pick up the leading terms. This turns out to be (6.14) To analyse the rest of the subleading terms in (6.10) we need to first substitutek r ·k u = 0 in the first line and remove the contractions ofẽ u,µ andε u,µν withk r,µ from the first and fifth terms of each square bracket. Then we need to expand all the K i and Ξ i up to subleading order in soft momenta. After Taylor expansion it is clear that from the second and sixth terms, the terms involving expansion of Ξ i vanish by successive application of ǫ T i K i (−p i ) = 0. The terms having contraction ofε u,µν withk ν r in the sixth terms of each square brackets vanish by the same logic. Similarly Q P (i) and Q G (i) being linear in soft momenta and K i (−p i ) sitting in front of it in the fourth and eighth terms of square bracket, these terms vanish by successive use of ǫ T i K i (−p i ) = 0. So, from (6.10) we are left with where, Now when we multiply all the terms and expand keeping terms only upto subleading order, ǫ T i operating on L s (γ) and L s (g) vanishes by (3.1) unless there is some other matrix between them. The matrices that can appear in between ǫ T i and L s (γ) and/or L s (g) are proportional to If we choose these kind of matrices from the r'th square bracket, then in the expression (6.15) the L s (γ) and L s (g) vanishes for s < r using (3.1). For s > r, L s (γ) and L s (g) terms are kept untouched. We need to also Taylor expand Γ (i) up to linear order in soft momenta and for the subleading order we have to keep only E s (γ) and E s (g) terms from the square brackets. This leads to Let us first focus on the non-universal term N µν (i) (−p i ) defined in (4.4). This contains a piece proportional to B µν (i) (−p i )Ξ i (−p i ). This part can be moved through L s (γ) and L s (g) in (6.18) can also be moved to the right by first using (3.5) and then (3.4). For the other terms having ∂K i (−p i ) we will not do anything. On the other hand we will move (J ab ) T part through the L s (γ) and L s (g) terms to the extreme right using (3.7) after expanding n s=r+1 E s (γ) + L s (γ) + E s (g) + L s (g) in the following way: E r+1 (γ) + L r+1 (γ) + E r+1 (g) + L r+1 (g) · · · E n (γ) + L n (γ) + E n (g) + L n (g) Now we combine the first term in the expansion (6.19) with (J ab ) T coefficient and with N µν i (−p i ) coefficient with the last two lines of (6.18). For the rest of the expansion we use (3.7) to move (J ab ) T to the right and use (3.5) and (3.4) the right. After combining similar terms we get: In expression (6.20) we have combined terms in three parts. In the first part we combined the terms where we could move the non-trivial matrices 1 to the extreme right. In the second part we have combined the terms where the non-trivial matrices are not moved at all. In the third part we have combined the terms where we have moved non-trivial matrices to one step right. Now let us write down the third part of (6.20) after possible Lorentz index contractions: Now it is quite obvious that second and fourth terms within the square bracket of above expression (6.21) completely cancels the second part of (6.20). So after cancellation we are left with the following terms in eq.(6.20) Let us first consider the first part of the above expression (6.22) and sum over all permutations of {ξ 1 ,k 1 } , {ξ 2 ,k 2 } · · · {ξ n ,k n }. Since the summation over r index part is already invariant under this permutation, sum over permutations effectively only acts on the propagator denominator factors and then using (A.1) the first part of (6.22) becomes Since this expression is already subleading, from the other hard particle legs we only have to keep leading contribution having form r∈A j (p j .k r ) −1 Q j e r,µ p µ j + ε r,µν p µ j p ν j . Now we have to sum over all possible distribution of soft particles among the hard legs. Then total contribution of this part With this contribution we will add the contribution coming from sum of the diagrams having one soft particle attached to n-hard particle amplitude via Γ and other (M − 1) soft particles attached to external hard particle legs via three point couplings Γ (3) , analogous to Fig.(9). Following the result of (5.6), the sum of the contributions from these kind of diagrams become Now let us analyse the second part of (6.22), where the (p i ·k r ) factor within the square bracket can be written as: p i ·k r = p i · (k 1 +k 2 + · · · +k r ) − p i · (k 1 +k 2 + · · · +k r−1 ) . (6.26) Substituting this expression in the second part of (6.22) it is easy to see that first term above cancels the propagator right to (ξ r ,k r ) insertion and second term above cancels the propagator left to (ξ r ,k r ) insertion. Now we have to sum over all permutations of {ξ r ,k r } but we will achieve this in the following steps. First we will fix all the point of insertions of soft particles in fig.(12) except the one carrying momentumk r and then sum over all possible insertion of (ξ r ,k r ) left to (ξ u ,k u ). Due to pairwise cancellation between terms (similar cancellation happens while proving Ward Identity for gauge theories) we will be left with only the term having insertion of (ξ r ,k r ) just left to (ξ u ,k u ). To write the second part of (6.22) in a convenient form we will relabel the polarisation and momenta of soft particles except the one having momentak r , from left to right as, (ξ 1 ,k 1 ), (ξ 2 ,k 2 ), · · · , (ξ u−2 ,k u−2 ), (ξ u ,k u ), (ξ u+1 ,k u+1 ), · · · , (ξ n ,k n ) . (6.27) Hence after this relabelling and summing over all (ξ r ,k r ) upto the position left of (ξ u ,k u ) for fixed r and u the second part of (6.22) reduces to where E s (γ), L s (γ), E s (g), L s (g) are the same as given in (6.16) and (6.17) with the tilde replaced by hat for the polarisations and momenta. Now with the above expression we also need to add the term with r and u exchange, which we can get just by interchanging µ and ρ indices in the term in (6.28). Then we have to sum over all permutations considering r-th and u-th soft particles as single unit. Before doing this let us analyse the remaining two kind of diagrams (13) and (14) which also contributes in the subleading order of soft momenta.
Insertion of two soft particles via Γ (4) vertex in Fig.(13) implies it has one less propagator than Fig.(12) and makes the contribution subleading. So from the rest of diagram we only need to pick leading contribution from the vertices and propagators. Rest of the diagram contribution contains (n − 2) number of Γ (3) Ξ i factors which one can write down using (4.3) and each of them will contribute E s (γ) + L s (γ) + E s (g) + L s (g) for s = 1, ..., n except for s = r, u. Then using (3.1) one can drop the L s (γ) + L s (g) factors appearing left to the vertex, where the soft particle with momentumk r +k u attached to the hard leg. In this way the hard particle polarisation ǫ T i moved just left to Γ (4) vertex and that can be evaluated similarly like (5.9). Hence the contribution of diagram(13) turns out to be k rk u · · · · · · Figure 14: diagram contributes in subleading order with two of the soft particles attached via V (3) vertex.
(6.31) Now we will sum over all permutations separately fork 1 ,k 2 , · · ·k u−2 andk u+1 ,k u+2 , · · ·k n and the corresponding polarisations keeping (r, u) fixed as a unit. Under this sum over permutations the only change happens in the first three lines of (6.31). Using (A.1) and combining E s (γ) + E s (g) , the first three lines of (6.31) turn out to be p i · (k r +k u ) Now we have to sum over all possible choice of r, u from {1, 2, · · · n} and add the contributions of (6.12) and (6.14). Consequently, we get M pp p i ;ẽ r ,k r ;ẽ u ,k u + M gg p i ;ε r ,k r ;ε u ,k u + M pg p i ;ẽ r ,k r ;ε u ,k u + M pg p i ;ẽ u ,k u ;ε r ,k r ǫ T i Γ (i) (p i ) = r,u∈A i r<u s∈A i s =r,u p i · k s −1 Q i e s,µ p µ i + ε s,µν p µ i p ν i p i · (k r + k u ) −1 M pp p i ; e r , k r ; e u , k u + M gg p i ; ε r , k r ; ε u , k u + M pg p i ; e r , k r ; ε u , k u + M pg p i ; e u , k u ; ε r , k r ǫ T i Γ (i) (p i ) . Q i e s,µ p µ i + ε s,µν p µ i p ν i p i · (k r + k u ) −1 M pp p i ; e r , k r ; e u , k u + M gg p i ; ε r , k r ; ε u , k u + M pg p i ; e r , k r ; ε u , k u + M pg p i ; e u , k u ; ε r , k r Γ(ǫ 1 , k 1 ; · · · ; ǫ N , k N ) . (6.34) Now summing over (6.7), (6.24), (6.25) and (6.34) we get the contribution given is (6.1). This proves multiple soft photon-graviton theorem up to subleading order in soft momenta. We can analogously prove the gauge invariance of the multiple soft theorem result eq.(6.1), following section 5.1 .
Eq.(7.7) is the classical version of soft theorem. The soft factors for graviton and photon have decoupled and as expected gravitational and electromagnetic radiations are independent in classical limit.
Acknowledgments: We are deeply thankful to Prof. Ashoke Sen for suggesting this problem, for insightful discussions and crucial ideas throughout the course of this work and for going through and correcting the draft with immense patience. We also want to thank Nabamita Banerjee and Dileep Jatkar for encouragement in pursuing this project. SB is grateful to HRI, Allahabad where this work has been completed. We express our gratitude to people of India for their continuous support to theoretical sciences.

A Summation Identities
To prove the multiple soft theorem in many intermediate stages we have used the following identities, which are proven in [82].