Classical sub-subleading soft photon and soft graviton theorems in four spacetime dimensions

Abstract

Classical soft photon and soft graviton theorems determine long wavelength electromagnetic and gravitational waveforms for a general classical scattering process in terms of the electric charges and asymptotic momenta of the ingoing and outgoing macroscopic objects. Performing Fourier transformation of the electromagnetic and gravitational waveforms in the frequency variable one finds electromagnetic and gravitational waveforms at late and early retarded time. Here extending the formalism developed in [1], we derive sub-subleading electromagnetic and gravitational waveforms which behave like u−2(ln u) at early and late retarded time u in four spacetime dimensions. We also have derived the sub-subleading soft photon theorem analyzing two loop amplitudes in scalar QED. Finally, we conjectured the structure of leading non-analytic contribution to (sub)n-leading classical soft photon and graviton theorems which behave like u−n(ln u)n−1 for early and late retarded time u.

A worldsheet for Kerr

“…Although amplitudes are rooted in quantum field theory, in recent years there has been an explosion of interest in applications of amplitudes to classical physics [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. This is motivated by the applicability of the results for elastic and inelastic [45][46][47][48][49][50][51][52][53][54][55][56][57] gravitational scattering of two massive particles to gravitational-wave physics [58][59][60][61][62][63][64][65][66][67][68][69][70][71][72], as JHEP03(2021)201 well as by the search for the underlying principles behind the double-copy relationship between gauge theory and gravity [4,…”

Soft theorems from boundary terms in the classical point particle currents

“…Enforcing current conservation gives us yet another justification for dropping the boundary term, and from this point forward we will always neglect such terms. Similar conclusions about this kind of integral have been drawn in [87], and a careful discussion of the i prescription turns out to be especially important when discussing logarithmic corrections to the soft theorems [88,89]. For us, the important thing to remember is that we should drop the oscillatory boundary term, leaving us with the following expression for the particle current:…”

“…For example in [85], the classical limit of the soft graviton theorem was taken up to sub-sub-leading order, in order to predict the classical radiation fields at early and late retarded times produced by scattering events. This prediction was then verified directly in the classical theory in five or more dimensions [86] and in four dimensions [87][88][89]. Interestingly, these latter works also identify logarithmic corrections to the soft theorems in four dimensions which arise beyond leading order because of long range forces between the particles in the in and out states asymptotically.…”

IR finite S-matrix by gauge invariant dressed states

“…It is discussed in [37] that this log behavior is related to the subleading soft factor proportional to log ω where ω is the energy of a soft photon. It would be interesting to incorporate the log deviation into the classical current (2.2), and see how the obtained dressed states are related to the subleading soft factor and also sub-subleading factor [39].…”