The convolutional double copy: a case study with a point

Abstract

The double copy relates scattering amplitudes in gauge and gravity theories. It has also been extended to classical solutions, and a number of approaches have been developed for doing so. One of these involves expressing fields in a variety of (super-)gravity theories in terms of convolutions of gauge fields, including also BRST ghost degrees of freedom that map neatly to their corresponding counterparts in gravity. In this paper, we spell out how to use the convolutional double copy to map gauge and gravity solutions in the manifest Lorenz and de Donder gauges respectively. We then apply this to a particular example, namely the point charge in pure gauge theory. As well as clarifying how to use the convolutional approach, our results provide an alternative point of view on a recent discussion concerning whether point charges map to the Schwarzschild solution, or the more general two-parameter JNW solution, which includes a dilaton field. We confirm the latter.

A double copy for asymptotic symmetries in the self-dual sector

“…We chose not to display them in the diagram since they do not preserve the r → ∞ field fall-offs. 16…”

“…The idea of gravity as a double copy (DC) 1 has found applications in the study of numerous aspects of the gravitational theory, most notably scattering amplitudes, where much of the recent success has been driven by the identification of a duality between the color and kinematic factors, the so-called Bern-Carrasco-Johannson (BCJ) duality [4][5][6][7][8][9][10][11]. It has progressed to the study of solutions, both perturbative [12][13][14][15][16][17][18] and exact [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. Another important extension has been to find manifestations of it beyond gravity theory [36][37][38][39][40][41][42].…”

“…For example, it was shown in [40,44,58], working in the BRST formalism, that one can exploit it to obtain a gauge-mapping algorithm, which gives a gravity gauge fixing functional output from a YM gauge-fixing functional input. This has been extended to the study of classical solutions in [16]. However, this has so far only been understood at linear level -going beyond this, either perturbatively (or nonperturbatively in certain special cases), could prove very powerful in the DC construction of classical solutions and/or Lagrangians.…”

The Newman-Penrose map and the classical double copy

“…Progress in understanding this discrepancy has been made in [67], which pointed out how abelian and non-abelian gauge theory objects may be associated with the same object on the gravity side. Furthermore, by studying gravity as the convolution of left and right Yang-Mills theories [68], one may obtain a "convolutional double copy" which maps gauge and gravity solutions in the manifest Lorenz and de-Donder gauges [69]. The scenarios studied in the present work considered gravity solutions that, under the Newman-Penrose map, correspond to electromagnetic field configurations.…”

“…With a few exceptions [34,69], all classical double copy prescriptions, including the one presented here, rely on certain coordinate and gauge choices. Yet the relationship between diffeomorphism invariance and gauge redundancy under the double copy remains somewhat obscure.…”