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.
Highlights
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 conjectured the structure of leading non-analytic contribution ton-leading classical soft photon and graviton theorems which behave like u−n(ln u)n−1 for early and late retarded time u
In [17] the authors made an observation that in the Feynman diagrammatics of loop amplitude if one replaces the Feynman propagator for virtual photon/graviton by it’s corresponding retarded propagator one gets only the classical soft factor at subleading order which is proportional to the classical electromagnetic/gravitational waveform
Summary
In a theory of quantum gravity, soft graviton theorem gives an amplitude with a set of finite energy external particles (hard particles) and one or more low energy external gravitons (soft gravitons), in terms of the amplitude without the low energy gravitons [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Classical limit of multiple soft graviton theorem determines the low frequency radiative mode of the gravitational waveform in terms of the momenta and spin of the macroscopic objects (scattering data) participating in the scattering process, without the detail knowledge of the interactions responsible for the classical scattering process [20,21,22]. We conjectured the structure of the leading non-analytic contribution of (sub)n-leading classical soft photon and soft graviton theorems in terms of some undetermined functions which reproduce u−n(ln u)n−1 tail memory. At sub-subleading order the leading non-analytic contribution of gravitational waveform has been conjectured in [1] which goes like u−2(ln u) as |u| → ∞ at late and early time. Relation between the soft factors and Ward identities associated with conserved quantities, the reader can look at the reference [56]
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