Soft theorems from boundary terms in the classical point particle currents

Abstract

Soft factorization has been shown to hold to sub-leading order in QED and to sub-sub-leading order in perturbative quantum gravity, with various loop and non-universal corrections that can be found. Here we show that all terms factorizing at tree level can be uniquely identified as boundary terms that exist already in the classical expressions for the electric current and stress tensor of a point particle. Further, we show that one cannot uniquely identify such boundary terms beyond the sub-leading or sub-sub-leading orders respectively, providing evidence that the factorizability of the tree level soft factor only holds to these orders. Finally, we show that these boundary terms factor out of all tree level amplitudes as expected, in a theory where gravitons couple to a scalar field.

Propagators in the Correlated Worldline Theory of Quantum Gravity

“…As a check on this result we can refreeze the metric field g µν (x) in (71) to some fixed configuration g µν (x); it is clear that we will then recover the conventional QFT result, ie., we will find that K(2, 1) → K φ (2, 1|g).…”

“…The structure of the CWL propagator K(Φ 2 , Φ 1 ) for a scalar field φ(x) between field configurations Φ 1 and Φ 2 , given in (71), is of course rather peculiar. To understand it better, it is illuminating to analyze it by doing a perturbative expansion in G N , in the same way that we did already for W; we now outline this.…”

“…From eqtn. (71) we see that a graphical construction of the perturbation expansion for the propagator K(2, 1) can be accomplished by three steps, which we summarize here for convenience: (i) for the contribution K n to the propagator, draw a set of "untethered" lines between start and end points φ (ii) Now insert all possible gravitational interactions between these n lines; This is done in accordance with the usual Feynman rules for conventional quantum gravity, since we are working inside a specific "tower", the n-th tower (ie., working with all diagrams involving n paths for the φ-field). Examples are shown in Fig.…”

“…2. Eikonal Expansion for K(2, 1) (ii) Eikonal Expansion for K(2, 1): Returning to our key result in (71) for K(2, 1), we wish again to do an expansion about the stationary phase saddle point.…”

From Noether's Theorem to Bremsstrahlung: a pedagogical introduction to large gauge transformations and classical soft theorems

“…In addition to these reviews, this work was particularly inspired by the discussion of the Noether theorems by Avery and Schwab in [15], the discussion of electromagnetic memory by Pasterski in [16], and the discussion of classical soft theorems in Fourier space by Laddha and Sen in [23,24]. See also [25][26][27][28].…”