New Asymptotic Conservation laws for Electromagnetism

We study the subleading tail to the memory term in the late time electromagnetic radiative field generated due to a generic scattering of charged bodies. The universal nature of this term hints that it is tied to a soft theorem. We show that there exists an $\mathcal{O}(e^5)$ asymptotic conservation law related to the subleading tail term. This shows that the subleading tail arises from classical limit of a 2-loop soft photon theorem. Building on the $m=1$ \cite{1903.09133, 1912.10229} and $m=2$ cases, we propose existence of an $\mathcal{O}(e^{2m+1})$ conservation law for every $m$ which would imply a hierarchy of an infinite number of $m$-loop soft theorems. We also predict the structure of the $m^{th}$ order tail to the memory term that is tied to the corresponding classical soft theorem. We verify above proposals for $m=3$.


Introduction
Soft theorems are universal statements about quantum amplitudes in the limit when energy of one of the scattering particles is taken to be small [1][2][3][4][5][6]. Soft theorems are related to asymptotic symmetries [7][8][9][10][11][12][13][14][15][16]. In [9,10], the authors discussed the asymptotic conservation law corresponding to a subgroup of the U(1) gauge group called large gauge transformations. Classically the conservation law takes following form : The future charge Q + is defined at I + − i.e. the u → −∞ sphere of I + . Similarly, the past charge Q − is defined at I − + which is the v → ∞ sphere of I − . [9,10] showed that the Ward identity for S-matrix : Q + S − SQ − = 0 is equivalent to the leading soft theorem. λ + (x) that parameterises the large gauge transformation is an arbitrary function on the celestial sphere S 2 . The parameter at I − + is related to it via antipodal map λ + (x) = λ − (−x). Thus, we have a conservation law for every possible choice of λ. Different choices of λ corresponds to the different polarisations of the soft photon.
The asymptotic conservation law associated to large U(1) gauge transformations was proved generically in [17] by by evolving data from I − + to I + − using classical equations of motion at spatial infinity. It has been know for a long time that tree level amplitudes in QED admit a subleading soft theorem as well [1,2]. At subleading order, there are possible corrections due to non-minimal couplings [18]. The symmetry underlying the universal part of the subleading soft photon theorem was discussed in [19][20][21]. The corresponding classical asymptotic conservation law was discussed in [22]. In fact in [22], the authors proved a hierarchy of infinite number of conservation laws for classical electromagnetism. On the other hand, a hierarchy of infinite number of tree level soft theorems had been proved in [23,24]. In [22], the authors also provide evidence that suggests that the hierarchy of infinite number of conservation laws is equivalent to the hierarchy of infinite number of tree level soft theorems proved in [23,24]. Thus, tree level soft theorems in QED can be related to asymptotic conservation laws. The advantage of this approach is that it allows us to construct charges by studying classical configurations in the asymptotic regime of spacetime and identifying the modes that are conserved.
The leading soft photon theorem does not receive any loop corrections. [25][26][27] showed that there are loop corrections beyond the leading order term in four spacetime dimensions. In [28], the authors derived the subleading soft theorem for loop amplitudes. The subleading term is logarithmic in soft energy and is 1-loop exact. The logarithmic term is completely absent in tree level analysis. The asymptotic conservation law for this 1-loop soft theorem was proposed in [29] and the 1-loop soft theorem was recast as the Ward identity : Q + S−SQ − = 0 for the corresponding charge [29,30].
An interesting aspect is that soft theorems control the late time radiation emitted in any classical scattering process. The leading order term at late times has been studied extensivelyit is the so called memory effect [31][32][33][34][35]. The relation between the memory term and leading soft theorem was already shown in [36,37]. This relation between soft theorems and late time radiation was futher extended in [38]. In [38], the authors proposed a novel way of taking classical limit and used quantum multiple soft theorems [39] to show that soft radiation is controlled by the soft theorems. This is the frequency space analogue of the statement that soft theorems control the late time radiation. It is well known that in presence of massless particles, classical quantities get non-zero corrections from loops as well [41]. So, it is natural to ask if the 1-loop soft theorem affects the late time radiation. In [42], the authors showed that the 1-loop level soft theorem induces a term that falls as inverse power of time (u = t − r) in the late time radiation which is completely absent in tree level analysis i.e.
The u 0 -term is the memory term. The 1/u-term which is a tail to the memory term arises from 1-loop level soft photon theorem. In [43], the authors have given a prediction for the subleading tail ( log u u 2 ) to the gravitational memory term. In this paper, we compute the explicit form of the subleading correction to the memory term i.e. the log u u 2 -term in the late time electromagnetic radiation emitted in a generic classical scattering of charged bodies. The structure of this term turns out to be universal and is uncorrected even when we go to higher orders in e. So, we expect that this term should be tield to a soft theorem. As the log u u 2 -term is O(e 5 ), in particular we expect that it is related to a 2-loop soft theorem. To further this point, we demonstrate existence of a new asymptotic conservation law at O(e 5 ). We expect that this conservation law would lead to a new soft photon theorem at 2-loop order.
Based on the m = 1, 2 cases, we go ahead and propose an O(e 2m+1 ) asymptotic conservation law for every m in (82). As a non-trivial check, we verify the m = 3 conservation law. We leave it to further investigations to prove these O(e 2m+1 ) conservation laws following the analysis of [22] in presence of long range forces. Existence of these infinite number of conserved asymptotic charges Q m (at O(e 2m+1 ) ) hints towards existence of a hierarchy of an infinite number of soft theorems for QED. Ward identity of the charge Q m would correspond to a new soft theorem at every m-loop order that is m-loop exact.
Let us briefly discuss the implications on the late time radiation. The charge at O(e 2m+1 ) i.e. Q m is related to (log u) m−1 u m -tail in the late time radiation. Hence, the Q m -conservation laws would imply that following class of terms in the late time radiation are universal : In any process, these terms depend only on asymptotic properties of the participating particles and are independent of other details of scattering. '...' in above expression represent other terms of the form (log u) m u n such that m < n−1 and n > 1. These terms typically get corrected at all orders in coupling. Using the technique developed in [43], it can be shown that the fourier transform of (3) has following behaviour near ω → 0 : This is the classical version of soft theorems. The leading soft factor is given by . Similarly the classical subleading soft factors are given by : We predict that the form of S m is given by : n is total number of scattering bodies. Q i , p i are respectively the asymptotic charges and momenta of the scattering bodies. Q i , p i are defined including η i factors such that η i = 1(−1) for outgoing (incoming) particles. ǫ is the polarisation vector of the radiation. c i 's given in (36) represent the leading order effect of the long range force as seen in eq. (35). The form of F is fixed by long range force and is given in (90). We have explicitly checked the form of S m for m = 2, 3, 4. m = 1 was already studied in [42,43]. As discussed earlier, (4) can be obtained as classical limit of quantum soft thorems [38]. Hence, we expect that loop amplitudes would admit soft expansion similar to (4) such that the m-loop quantum soft factors reduce to S m 's in the classical limit. It would be interesting to explicitly obtain the m-loop soft photon theorems in the quantum theory and relate them to the Q m -conservation laws for m > 1. The 1-loop soft theorem [28] and its relation to m = 1 conservation law is already established [29,30]. We expect such conserved charges to exist in any general covariant theory of gravity as well.
Let us discuss the outline of this paper. In section 2, we review some known results to set up the background : we discuss the memory term, its relation to the leading soft factor and the corresponding conservation law. We prove the m = 1 conservation law and discuss its relation to the leading tail term in section 3. In section 4, we obtain the subleading tail to the memory term and discuss interesting aspects of this term. We show existence of conservation law at O(e 5 ) in section 5 and also discuss its relation to the subleading tail term. We give our proposal for O(e 2m+1 ) conservation law and summarise our results in section 6. The calculations for m = 3 case have been relegated to Appendix D.

Preliminaries
Our aim is to study late time radiation emitted in a general classical scattering process and obtain new asymptotic conservation laws obeyed in such a process. So, we need to consider a generic scenario : some n ′ number of charged bodies carrying velocities V µ i , charge e i and masses m i (for i = 1 · · · n ′ ) come in to interact for a finite time and within a finite region say a sphere of radius L around r = 0. This interaction could be of any sort. (n − n ′ ) number of final charged bodies with velcoties V µ i , charge e i and masses m i (for i = n ′ + 1 · · · (n − n ′ )) are produced as a result of the interaction .For r > L, the particles are apart enough so that only possible interactions between them would be the long range forces.
We need to calculate the asymptotic radiative field generated in such a generic process. We will carry out the calulations perturbatively in coupling e as well as in asymptotic parameters 1/r (or 1/t). It turns out that the internal structure of the scattering bodies is not relevant for calculating the late time radiation emitted in the scattering. The current corresponding to an extended object can be written as the contribution from pointlike object plus corrective terms that depend on internal structure of the object like its charge distribution. These corrective terms depend on derivatives of pointlike current and are subleading at large r. They do not contribute to late time radiation at all. Thus, the late time radiation is sensitive only to the total charge of the object. Hence, we can study scattering of point particles without any loss of generality [43]. We do not assume anything about the interaction, it could be of any kind or of any strength.

The memory term
In this section we will obtain the electromagnetic memory term [34,35,37] in radiative field. Since we need to find the asymptotic electromagnetic field, only the asymptotic part of trajectories of scattering particles is required. Let us first restrict ourselves to the leading order in coupling e, then we can ignore the effect of long range electromagetic interactions on the asymptotic trajectories. Hence an incoming particle has the trajectory (Greek indices will be used to denote 4d cartesian components) : . τ is an affine parameter, here T denotes the value of τ where particle exits the finite region L. Similarly, an outgoing particle has the trajectory : Hence, the asymptotic part of the current is given by : Here, we have labelled the incoming particles by i running from 1 to n ′ and outgoing particles by i running from n ′ + 1 to n . The radiation emitted during this process is given by A µ = −j µ . Using the retarded propagator, we get : We have added a superscript to note that we have ignored the bulk sources of radiation. Henceforth, we will drop this superscript but it should be always remembered that we are calculating only the asymptotic part of the field. To avoid clutter, let us first consider the electromagnetic field generated by the i th asymptotically free outgoing particle. At the end, we just need to sum over all particles. We denote it with a superscript (i) : The retarded root of the delta function δ(|x − x ′ | 2 ) is given by : Hence, Thus, the asymptotic field generated by the scattering process is given by : We consider a test charge placed at a distance which is sufficiently far from the 'scattering region' which is confined to a region of size 'L'. This test source will act as our detector. Let us calculate the radiative field at I + i.e. by taking the limits r → ∞ with t − r finite. We use retarded co-ordinate system to describe I + . The flat metric takes following form in this co-ordinate system (u = t − r) : We usex or (z,z) interchangeably to describe points on S 2 . We will often use following parametrisation of a 4 dimensional spacetime point : Here, q µ is a null vector that can be parameterised in terms of (z,z) as To find the radiative field around I + let us take the limit r → ∞ with u finite in eq.(9). Using (9) : where, 1 u ∞ denotes fall offs faster than any power law. Let us rewrite (11) a bit succinitly : Given above fall off for A µ , the behaviour of the Fourier transformed functionÃ µ at small ω turns out to be :Ã Thus, we see that the radiative field behaves as 1 ω as the frequency ω of the radiation is taken to 0. The coefficient of this term is proportional to the leading soft factor : We have used the common convention used for asymptotic quantities : such that η i = 1(−1) for outgoing (incoming) particles. e iωr 4πir is the overall normalization factor [38]. Let us discuss the effect of the 1 ω -mode on the test particle. Our test particle is placed at large r and large u. Let us denote its trajectory by x µ (σ). The equation of motion is given by (V µ = ∂x µ ∂σ ) : We will use capital latin alphabets to denote indices on S 2 . These indices take values (z,z). Using a co-ordinate transformation we have : V A = (∂ A x µ )V µ . We use (11) to find the leading term in F µν V ν . We get (noting that F In above equation, A 0 A is used to denote following mode in Thus, the velocity shift of the test particle is given by : ∆V A = e m ∆A 0 A . We note that this shift is in a plane transverse tor. Using (11), we can calculate the shift in gauge field : here, we have labelled the incoming bodies by i running from 1 to n ′ and outgoing bodies by i running from n ′ + 1 to n . This is the so called memory effect : the velocity of the test particle receives a kick due to the passage of electromagnetic radiation. As visible from the form of the expression, the amount of kick received is insensitive to the details of scattering. Let us use the oft-used basis for polarisation vectors [9]: Then we can write here we have defined ǫ µ A = (ǫ µ + , ǫ µ − ) for brevity. Thus ∆A A is proportional to the leading soft factor. The leading soft factor does not get any corrections from higher order terms in e. In (12), we have : We will see in the forthcoming sections that a ± µ is uncorrected even when we go to next order in e. The subleading behaviour is changed drastically as we go to next order in e.

Conservation law for the leading charge Q 0
Studying the expansion of any (n + 1)-point quantum amplitudes in the limit when energy (ω) of one of the scattering photons is taken to zero, the leading order term is inverse of ω. The coefficient of 1/ω is the n-point amplitude times a universal 'soft factor'. This is called the leading soft photon theorem : q is the leading soft factor. This is is the standard convention of writing soft factors where Q i = η i e i and p i = η i m i V i such that η i = 1(−1) for outgoing (incoming) particles.. ǫ is the polarisation vector of the soft photon and q = (1,q) is the direction of the soft momenta. Above equation is reminiscent of (13). 1 The leading soft theorem can be understood as a Ward identity for S-matrix : 10], where Q 0 is conserved : We will verify above conservation law in this subsection. Calculating the field strength from (11) : The leading order term in above expression is 1 r 2 . Using (10), the coefficient of this term can be written as : Performing a co-ordinate transformation : Next we need to derive the field configuration at past null infinity and compare the two expressions. We need to take the limit r → ∞ with v = t + r finite. In this co-ordinate system, 4 dimensional spacetime point can be parametrised as : Again,q µ is a null vector. Thus around I − , we have from (8) : , outgoing particles do not contribute to the field at I − . From (9), we get : Calculating the field strength, we get Using (26) along with above equation, we have rederived the 'conservation' law [9,17] : Here, F Hence it a function of the sphere coordinatesx. Similarly F

m = 1 conservation law
In this section we will obtain the asymptotic radiative field keeping the first order effect of long range forces and derive the O(e 3 ) conservation law proposed in [?, ?]. In the previous subsection, we rederived the conservation law that is equivalent to the leading soft photon theorem. We will follow similar strategy here. Loop amplitudes admit a subleading soft photon theorem : S 1-loop has been derived in [28]. Let us discuss the conservation law corresponding to the 1-loop log ω soft theorem. The log ω term is directly related to long range forces. In four space-time dimensions these 1 r 2 -forces are subtle as they induce logarithmic correction to the straight line trajectory at late times. Hence, the name long range forces. Due to the long range electromagnetic force, a charged particle continues to accelerate at late times and gives rise to new modes in the asymptotic field at O(e 3 ). In particular F rA gets a log because of the long range interaction between particles : Similarly around the past null infinity we have : [?] had proposed following conservation law : Let us find the late time radiation emitted by particles moving under the influence of long range forces and prove above statement. The acceleration experienced by asymptotic particles was already calculated in [28] and has been rederived in Appendix B. Due to the long range forces, the equation of trajectory of an outgoing particle i gets corrected to : From (111), we have : It should be noted that c i 's for an outgoing particle includes contribution only from outgoing particles. Similarly an incoming particle can interact only with incoming particles via the long range forces. Let us first find the asymptotic field produced by an outgoing particle i with the corrected trajectory given in (35). The current corresponding to the particle is modified to j . Hence we get : This equation includes O(e 3 ) corrections to the to eq.(9). Solving the δ-function condition is highly difficult because of the logarithmic correction. We solve it perturbatively. The details of the calculation are relegated to Appendix A. We quote the solution to the delta function constraint from (97) : Here, τ 0 is the zeroth order solution given in (8). Next, we need to use δ( The result of the integral is : Since above expression is vaild only to O(e 3 ), we can expand the denominator to O(e 3 ) as well.
Summing over all the incoming and outgoing particles, we get : It is worth recalling that above expression is valid in only asymptotic regions, we have ignored the contribution of the bulk sources. Let us first find the expansion of A σ around I + . Using (8), we have log τ 0 | I + ∼ log u + O(u 0 ). Then using (10) and the fact that V i , d i are O(r 0 ) parameters, we can find the leading order term in (38) : Using (10) in (41), we get X = −rq.V i + O(r 0 ). Substituting the limiting value of X in (40), we can read off the coefficient of the O( log u r 2 ) term in A σ : From here on, we just need to transform co-ordinates to get to F rA . We have : Next we need to derive the field configuration at past null infinity and compare above expression Using (27) the leading order term in (38) is : Using (8), we get log τ 0 | I − ∼ log r + O(r 0 ). Substituting in (40), we write down the coefficient of the O( log r r 2 ) term in A σ : Performing co-ordinate transformation : Thus, from (44) and (47) we can indeed check that the modes are equal under antipodal idenfication. The apparently extra minus sign in (47) is compensated by the factors of q µ . Finally we have shown that a generic scattering processe obeys following conservation law : We know that above modes are O(e 3 ). The charges are defined as These charges are parametrised by an S 2 vector field W A . It has been shown that the corresponding Ward identity for S-matrix i.e. Q + 1 S − SQ − 1 = 0 is equivalent to the 1-loop level log ω soft theorem [29,30].

Relation between the charge Q 1 and S 1
Next we find the precise relationship between the classical subleading soft factor S 1 and the conserved charge given by (48). In [40,43], it was shown that a log ω term in soft limit is mapped to a 1/u term in the late time field via Fourier transform. This 1/u term can be explicitly obtained using the asymptotic field in (40). Focussing on the 1/r-term of A σ , we get : p i , Q i have been defined in (15). Let us dicuss the qualitative picture first. Rewriting (49) a bit succinitly : We can compare above fall offs to the leading order radiative fall offs in (22). It is interesting to note that including even the first order correction in e has altered the late time profile appreciably. This is the so called tail memory effect [42,43]. The transverse velocity of a test particle V A at late times will exibit this 1/u behaviour before settling down to the final value determined by u 0 memory term. Let us note an important point about (50). This equation is not exact at O(e 3 ).
In (35), we have ignored O( 1 τ ) terms that are same order in e as the log τ but are suppressed at large τ . These terms have been represented as '...' in (50). If we keep all terms at O(e 3 ), then (50) takes following form : In the next section, we will study O(e 5 ) corrections to above equation and show that thre is a log u u 2 term which will dominate the O( 1 u 2 ) terms. Hence, the subleading tail to the memory term comes from O(e 5 ) corrections.
We can read off the soft factor from (50), using the results from [43]. Given (50), the Fourier transformed function has following behaviour at small ω : 4πir is the overall normalization factor [38]. Thus the soft factor is : We note that the coefficient of log ω in the classical field i.e. S 1 is only a part of S 1-loop that appears in quantum soft theorem (31). A part of S 1-loop vanishes in the classical theory. This point was already noted in [28].
Next we can relate the charge F [log u/r 2 ] rA to above soft factor using Maxwell's equations. The details of the calculations have been discussed in Appendix C. We quote the result here (121) : Above equation relates the charge on the left side while A on the right side is propotional to the (classical part of) 1-loop soft factor. Using (49), we have : To derive above expression, we also made use of the polarisation basis in (20). We see that A rz . Hence the charge is insensitive to a part of the soft factor. This implies that half of the soft factor is fixed by the asymptotic symmetry. This is similar to the tree level subleading soft theorem as well where D 2 z kills half of the tree level subleading operator. To summarise we observe that asymptotic symmetry fixes the −i i Q i ǫ.p i p i .q q.c i part of the soft factor. It should be noted that the rest of the soft factor i i Q i ǫ.c i can be fixed by demanding that the entire soft factor be gauge invariant. Thus the 1-loop soft factor is completely fixed by the asymptotic charge given in (48) and gauge invariance.

Subleading tail to the memory term
In this section we will derive the subleading tail to the memory term i.e. to (50). Let us first outline our steps. We have seen that a particle i undergoes acceleration under long range forces. By virtue of its acceleration, it radiates and the radiation is O(e 3 ). This radiation backreacts on the particles and produces O(e 4 ) deviation in the trajectories. Particles will in turn radiate at O(e 5 ) because of the second order deviation in trajectories and we will see that this produces the subleading tail to the memory term.
In Appendix B, we show that under the backreaction of the emitted radiation the asymptotic trajectories of the particles are corrected to : where (118) Next we find the resultant correction to radiative field. First let us find the field generated by i th outgoing particle using the corrected trajectory (55) in the Green function expression : where the current has been now modified to : We have to solve (x − x ′ ) 2 = 0 to second order in coupling e. The solution is given in (104) in Appendix A. Next we need to use : To study the subleading correction to the memory term, we focus on the leading 1 r term in A σ . To this order, the solution to the asymptotic field turns out to be : Now it only remains to subtitute the value of τ 2 . We get it from (105) : Substituting for τ 2 in (59), the leading 1 r term turns out to be : From above expression, we see that the subleading correction to the O(u 0 ) memory term is O( log u u 2 ). Summing over all the particles we get full contribution to the asymptotic field (Q i , p i have been defined in (15)) : This is the late time radiative field upto subsubleading correction. We have already discussed the leading term : the O(u 0 ) which produces the so called memory effect [34,35] and the 1 u -tail term [42,43]. As we have calculated in above equation, the subleading tail is of the form log u u 2 . This term is interesting because its coefficient is universal. Like the memory and tail term, it depends only on asymptotic properties of the interacting particles like momenta and charges. It is insensitive to the details of the scattering. And it is not corrected by higher order corrections in e. So like the memory term and its tail, we expect that the subleading tail is also tied to a soft theorem. Since it occurs at O(e 5 ), we expect it to be the classical limit of a 2-loop soft factor. Now let us discuss the Fourier transform of the log u u 2 -term. We rewrite (61) as : The large-u fall offs given in (62) fix the small energy behaviour. The Fourier transformed function has following behaviour at small ω [43] : Thus, from (61) we can get the coefficient of ω(log ω) 2 : This analysis hints at following soft expansion for 2-loop amplitudes : Here, we expect that S 2 is the classical limit of S 2-loop . We again note that subsubleading term in soft expansion of 2-loop amplitudes has not been calculated yet.
It is interesting to compare the form of S 2 with the tree level subsubleading term. Tree level amplitudes in minimally coupled QED admit following soft expansion : The terms clubbed in the square bracket are the subsubleading terms. The first term in the square bracket is universal and is fixed by asymptotic symmetry [22]. A µν is an arbitrary antisymmetric tensor depending on hard momenta. The form of A µν cannot be fixed by symmetry. This term is called remainder term. Factorisation into lower point amplitude is not guaranteed for this term. We note that we can obtain the the first term in S 2 replacing ∂ ν k → ic ν k in the tree level subsubleading soft factor. Interestingly this replacement rule holds for m = 3, 4 cases as well. The second half of S 2 has the same form as ǫ µ q ν A µν in (66). Thus, it seems natural to classifyÃ µν as a remainder term. We will make an interesting observation about this term when we study the m = 2 conservation law.

m = 2 conservation law
We have calculated the subleading tail to the memory term. The structure of this term turns out to be universal. In this section, we will find the asymptotic conservation law associated to the subleading tail term. Since, soft theorems are equivalent to asymptotic conservation laws this implies that the subleading tails is controlled by a soft theorem.
Including the O(e 5 ) terms leads to following expansion for F rA around future null infinity : We show in (123) in Appendix C that (log u) 2 r 3 mode is related to the subleading tail in the late time radiative field. Expansion around the past null infinity is given by : We will show that a generic classical scattering process obeys following conservation law : Next we will calculate the asymptotic field configuration to prove above conservation law. Let us rewrite (59) keeping the subleading corrections in 1 r since we need to go to O( 1 r 3 ).
We recall that τ 2 is given in (104). At I + , the charge is expected to be defined in terms of (log u) 2 r 3 -mode of A σ . Using (105), we get following expansion around I + : Hence, the (x−d i ).c i τ 2 + ... terms in the denominator of (70), have 1/u factors because of 1/τ 2 . The second and third term in the numerator also have similar behaviour. We will ignore these terms as they cannot contribute to (log u) 2 r 3 . So, we are left with following terms in (70) : We have used '∼' instead of '=' as we are ignoring certain terms in A σ that do not contribute to the 2-loop charge given in (69). Next we need to substitute the value of τ 2 . The full expression is given in (104). Let us retain only the logarithmic terms in τ 2 that are of relevance to us. We get : comes out to be : Next we need to sum over the contributions from all particles. Let us write down the coefficient at I + − . The contribution around I + − is from the incoming particles, thus we get : We just need to transform co-ordinates to go to F rA . Thus : Let us carry out the corresponding calculation at past null infinity. At I − , the term of our interest is the (log r) 2 r 3 -mode of A σ . Following earlier logic and using (106) analogous to (72), we get following expression at past null infinity we get : 1/2 , the limiting value at past null infinity turns out to be X| I − = rq.V i + O(r 0 ). Expanding above expression, obtain the (log r) 2 r 3 term in A σ : Next we just need to use appropiate co-ordinate transformations to arrive at F rA . We get : Using (75) and (77), we can write down the conservation law for these modes : It is important to note the minus sign. The charges are defined as (−x)| I − + and involve O(e 5 ) modes. We expect that the corresponding Ward identity for the S-matrix i.e. [Q 2 , S] = 0 would be related to the 2-loop ω(log ω) 2 soft theorem (65).

Relation between the charge Q 2 and S 2
We obtained the subsubleading classical soft factor S 2 in eq.(64) : We have shown that above soft factor is related to the conservation law : Let us go back to the coefficient of F in (75) : In the last line, we have used the basis given in (20) for polariation vectors. Comparing eq. (79) to eq. (81) we see that the second half of the soft factor i.e.
does not contribute to the m = 2 charge. This implies that this part of the soft factor is not controlled by asymptotic symmetry. Hence, we will classify 1 as a remainder term. But intriguingly the remainder term is also universal. We believe that the remainder term need to be understood better.

m th -order conservation law
Based on the m = 1, 2 conservation laws, we propose that there exists a conservation law for every m given by : We verify above statement for m = 3 in Appendix D. Thus, we expect that classical electromagnetism admits a hierarchy of infinite number of conservation laws. The m th level future charge is defined as These charges are O(e 2m+1 ). Hence in the quantum theory, these charges would give rise to an m-loop soft theorem for every m.
Let us discuss the corresponding universal terms in the late time radiation. When we include the effect of long range forces on the trajectory, the full correction to the trajectory is of the form : where c (m,n) iσ 's typically admit a series expansion in the coupling e. The logarithmic terms are produced only from the asymptotic regions and do not get any contribution from the bulk. Purely power law terms may depend on details of scattering. So, all the c (m,n) iσ 's are not universal. Substituting the corrected trajectories of (83) in the solution : we get following profile for the late time radiative field : As we discussed earlier typically [g m,n µ ] ± 's are not universal. Now we use the fact that (log u) n r n+1 -term in F rA in fixed by (log u) m−1 ru m -term in A σ by Maxwells equations. Hence we expect that (log u) m−1 ru m tail terms in A σ must be universal as they are related to asymptotic symmetry. To summarise we consider following class of terms in the late time radiation : ] ± is equal to b ± µ of (85) for m = 1. We expect that above terms are controlled by soft theorems. Here b (0) µ is the first order tail to the memory [42]. b (1) µ is the subleading tail we calculated in section 4. We have calculated the subsubleading tail i.e. b (2) µ in Appendix D. It should be noted that the universal terms in the late time field arise from following class of corrections to the equation of trajectory given in (83) : here we have defined (η i ) m f (m) iσ = c m,m iσ . η i = 1(−1) for outgoing (incoming) particles. So that f (1) iσ is the f iσ that appeared in the subleading tail calculated in section 4.
Using the technique developed in [43], it can be shown that the fourier transform of (86) has following behaviour near ω → 0 : Thus, the classical soft factors are given by : Based on the m = 1, 2 cases, we predict that the form of the soft factor for any m is given by : where . (90) iµ 's have been defined in (87). The indices have been enclosed within square brackets to denote that µ has to be antisymmetrised with ν m−1 2 . We have verified (89) for m = 3, 4 also. Let us first recall the m = 2 case i.e. eq.(64).
In section 5.1, we showed that the m = 2 asymptotic conservation law given in (69) controls the first term in (91). The F iµν -term given by iν ] is not by fixed by the asymptotic charge given in (69). Hence we classified this term as a remainder term. In Appendix D, we calculate the 3-loop F iµν 1 ν 2 -term to be : It can be concluded from the analysis of Appendix D that this term is not by fixed by the asymptotic charge Q 3 given in (129). We expect this to be true for all m's. Hence, we will call the F -term in (89) as a remainder term since it is not controlled by the Q m -conservation law given in (82). But this term is expected to be universal for all m's. The first term in (89) is fixed by the asymptotic charges in (82) and gauge invariance. It is interesting to compare (89) with the tree level coefficient. For tree level amplitudes the soft expansion is given by A n+1 ∼ S 0 ω + ∞ m=1 ω m S m tree . From [23,24] : The first term is universal and is fixed by asymptotic symmetry [22]. A µν 1 ···ν m−1 is an arbitrary tensor antisymmetric in µ and every ν i . This term is called remainder term. It depends on hard momenta and its form cannot be fixed by symmetry. Factorisation into lower point amplitude is not guaranteed for this term. The structure of the first term of (92) is very similar to that of the first term of (89). We note that we can obtain the the first term in (89) replacing ∂ ν k → ic ν k in the first term of (92). The structure of the remainder terms in the two is also similar.

Summary
In this paper we have proposed that classical scattering processes satisfy an O(e 2m+1 ) asymptotic conservation law for every m. We proved these laws for m = 1, 2, 3. We expect that this proposal for Q m -conservation law can be proved for generic m by incorporating the effects of long range force in the analysis of [22]. In the classical theory, the asymptotic charges imply existence of universal terms in low energy radiation emitted in the scattering process. These terms are seen as m th order tails in the late time radiative field. These tails have universal structure that is independent of the details of the scattering. The coefficient of these tails is expected to be given by (89) which we have verified for m = 2, 3, 4. For m ≥ 2 there are some remainder terms in these late time tails that are not captured by the asymptotic charges. An interesting question that arises here is to understand the universal nature of the remainder terms.
A natural question to ask is what is the implication of these charges for the quantum theory. We expect that these Q m charges imply existence of m-loop soft theorems for every m. But these soft theorems have not been studied in the literature before. So one needs to extend the calculations of [28] to higher loops and derive these m-loop soft theorems. A related question is to check that the classical limit of m-loop soft factors matches with our prediction given in (89) for generic m.
Several questions are in order about the conserved charges Q m .
• Most importantly, it needs to be checked if indeed all the {Q m , m ≥ 1} charges are independent or if they are related. And what is the algebra formed by these charges?
• What is the underlying symmetry? Are the charges related to large gauge transformations?
• For m ≥ 2 there exist remainder terms F µν 1 ···ν m−1 i that are not fixed by these Q m charges. These term too are expected to be universal for all m's and depend only on the electric charges and the asymptotic momenta of scattering particles. Can we extend Q m such that they reproduce the entire soft factor including the remainder terms?

Acknowledgements
I am deeply grateful to Alok Laddha for many insightful discussions.

A Perturbative solution
The Green function for d-Alembertian operator has δ([x − x ′ ] 2 ). We will find the solution of this delta function perturbatively in coupling e. Here, x ′µ (τ ) is the equation of trajectory that gets corrected as we go to higher orders in e. We will write down the perturbative solution for τ and see that it involves even powers of e.
At zeroth order, we have free particles : Hence, the root of delta function δ([x − x ′ ] 2 ) is given by : The sign of the square root has been chosen to ensure retarded boundary condition i.e. Θ(t−t ′ (τ )). Now, let us study above expression in the limit r → ∞ with u finite. Thus, around I + , using (10) we get : Now we take r → ∞ limit of (93) keeping v finite, using (27), we get : Next we include the leading order effect of long range electromagnetic force. We know that the first order correction to the trajectory is given by (35) : Using the corrected trajectory, the solution of delta function δ(|x − x ′ | 2 ) is given by : We have used V i .c i = 0. Noting that c µ is O(e 2 ), the RHS of above equation can be treated as a perturbation. Hence we substitute the zeroth order solution (93) in RHS of (95) that leads to following equation for τ : We ignored the c 2 i term as it is O(e 4 ). Now, above equation is just a quadratic equation in τ and the solution is given by : We have used a subscript 1 to denote that it includes the first order perturbative effects. We can expand the squareroot to O(e 2 ) : Here, we have defined 1/2 and τ 0 is given in (93). Thus, the first order solution is the zeroth order solution plus a perturbation : Expanding around I + , we get : Thus, in u → ±∞ limit, the correction to τ 0 is suppressed by log u u in addition to the suppression due to e 2 factor. Expanding (99) around I − , we get : A.1 Second order in perturbation Let us repeat above steps with second order efects of long range forces. The trajectory is corrected to (117) where f i ∼ O(e 4 ) : Hence at O(e 4 ), δ(|x − x ′ | 2 ) implies following equation for τ : We have used the fact that V i .c i = V i .f i = 0. Here, we have substituted the corrected solution (97) for the terms in the RHS. The second order solution is : We can expand the squareroot : We have used (99) for τ 1 to derive above expression. And as before . Now, let us study above expression in the limit r → ∞ with u finite. We have : The O( 1 r ) term starts at O(u 2 ). This produces O( u 2 r 3 )-term in A µ (see (70)). We see from (104) that there is a O( (log u) 2 r ) term, this contributes to the O( (log u) 2 r 3 )-term in A µ . We can expand (104) in large r limit keeping v finite to get :

B Effect of long range forces on asymptotic trajectories
Let us find the first order correction to equation of trajectory of particles in asymptotic regions. This calculation has been done in [28], we reproduce it here. The equation of trajectory of j th outgoing particle is given by : We need to find the field experienced by j due to all other particles. We have calculated the field strength in (24). We evaluate the field strength (24) at the position of the particle i.e. x = x j (τ ) : Here, we have not included any incoming particles as they cannot affect the outgoing particles. So, the trajectory of the j th particle is given by following equation : Here, we drop the O( 1 τ 3 ) correction. It is important to note that this another approximation we are making which is justified as we are working in large τ regime. But it should be noted above expression has corrections even at O(e 2 ). Thus, asymptotic trajectories of the particles are corrected to : where as given in [28], for outgoing particles : Above expression carries an extra minus sign compared to [28] because of difference in convention of η i . For i th incoming particle, j runs over the incoming particles :

B.1 Subleading correction to the equation of trjaectory
Next we find the subleading correction to the equation of trajectory (109). The corrections to (109) are of the form : Thus, log τ τ 3 represents the subleading correction (at large τ ) to the trajectory. This log τ τ 3 term arises from the log u r 3 mode in F µν . Using (40), we get : In above expression, we have retained only the relevant terms of F µσ (x). These modes in F µσ (x) are part of the radiation emitted by particles as the result of the particles undergoing acceleration under long range forces. Now as second order effect, a particle j will accelerate as a result of the radiation emitted. To find the effect of field generated by other particles at the poistion of j th particle we substitute : Substituting in (114), we get : There is one term depending on c j i.e. this term is produced due to acceleration of j th particle itself. Finally we have : Thus, asymptotic trajectories of the particles are corrected to : where For incoming particles, f µ i is negative of above expression. So, we have used η i f µ i .

C Maxwell'equations at future null infinity
The Maxwell's equations ∇ ν F σν = j σ can be expanded order by order around I + . We have already calculated the leading-r terms in the field strength in (24) : We substitute above fall-offs for the field strength components in Maxwell's equations, using Bianchi identities we get following equations : Here, we have used the fact that massive currents decay very fast at I + and the equations become homogenous. From (119), we have, Here, A 0 A denotes following behaviour in A A (x) : A A (x) ∼ A 0 A (u,x) + O( 1 r ). Above equation can be used to relate 1/u-term in A 0 A to the log u r 2 term in F rA . In particular, the z component of above equation gives us : Above equation is the one of our interest as it relates the charge on the left side while the right side involves A [1/u] z that is proportional to the 1-loop soft factor. Next we repeat above analysis for next order in 1 r . Maxwell's equations lead to : We substitute for ∂ u F D B is a function of sphere derivatives that contains upto fourth-order derivatives. Its exact form is not important for us. The takeaway point from this equation is that (log u) 2 term in F From above equation, it is clear that we need τ 3 only upto O(e 4 ). Since τ 3 = τ 2 + O(e 6 ), we will work with τ 2 . Using (105) : Hence, we have : Using this in (126), we see that the subsubleading correction to the u 0 term is (log u) 2 u 3 . So we have The coefficient of the (log u) 2 u 3 r -term is given by : iσ q.c i q.V i + 2f (2) iσ (q.V i ) 2 .
So, we can read off the classcial part of the 3-loop soft factor : This matches with (89) for n=3.

Conservation law
Next we will verify the conservation law : The solution of δ(|x − x ′ | 2 ) using (124) is given by : We know from (125) : At I + , we are interested in (log u) 3 r 4 term of A σ . To avoid clutter, let us write down only the relevant terms from (130) : We need to expand the squareroot in (132) to third order. At I + , τ 0 | I + ∼ u, hence we get the (log u) 3 r 4 -term : Thus : Let us carry out the corresponding calculation at past null infinity. Now we need to use in (132) the fact that τ 0 | I − ∼ r, hence we get the (log r) 3 r 4 -term. Hence, we get : Using appropiate co-ordinate transformations, we get Thus from (134) and (136), we get :