One-loop amplitudes in Einstein-Yang-Mills from forward limits

Abstract

We present a method to compute the integrands of one-loop Einstein-Yang-Mills amplitudes for any number of external gauge and gravity multiplets. Our construction relies on the double-copy structure of Einstein-Yang-Mills as (super-)Yang-Mills with the so-called YM+φ 3 theory -pure Yang-Mills coupled to bi-adjoint scalars -which we implement via one-loop Cachazo-He-Yuan formulae. The YM+φ 3 building blocks are obtained from forward limits of tree-level input in external gluons and scalars, and we give the composition rules for any number of traces and orders in the couplings g and κ. One the one hand, we spell out supersymmetry-and dimension-agnostic relations that reduce loop integrands of Einstein-Yang-Mills to those of pure gauge theories. On the other hand, we present four-point results for maximal and half-maximal supersymmetry where all supersymmetry cancellations are exposed. In the half-maximal case, we determine six-dimensional anomalies due to chiral hypermultiplets in the loop.

One-loop diagrams with quadratic propagators from the worldsheet

“…We remark that such expansion has immediate applications to e.g. the study of cancellation of the anomaly in heterotic strings (or even type I/II superstrings) and their applications to field-theory amplitudes (see [35,38] for related discussions), since the only non-vanishing gauge variation of the parity odd part of their 6-pt correlators contains a term proportional to µ ν η µν . Let us end the section with an example of an n = 4 CHY integral with µ ν η µν which indeed gives quadratic propagators,…”

“…which contains an overall factor µ ν η µν − 2 but integrates to a rational expression upon loop momentum in D = 6 dimensions (see more in [38]).…”

“…One-loop CHY integrands in various theories have been organized as linear combinations of tree-level PT factors (2.13), for example for supersymmetric YM, GR, and EYM with various numbers of supercharges and their pure versions [35][36][37][38]80], which usually hold for arbitrary multiplicities and can be used to derive one-loop KLT-like relations and BCJ numerators with linear propagators. Based on the general construction of forward limit in higher dimensions [31,32], more results for other theories can be obtained similarly from their tree-level CHY formulae (see cf.…”

“…[21,22,32,[35][36][37]). Even though lower-point quadratic propagators could be obtained by brute-force manipulation of linear propagators, such as what have done in [38] for one-loop four-point amplitudes JHEP08(2022)240 in (half-)maximally supersymmetric Einstein-Yang-Mills (EYM) theory, a general method for all multiplicities is still missing despite lots of efforts [39][40][41][42][43][44][45][46][47]. In a recent paper [48], it was shown that the field-theory limits of homology invariants [49][50][51], which can be viewed as certain kinds of genus-one string correlators, can act as one-loop CHY integrands that produce quadratic propagators.…”

A note on multi-trace EYM amplitudes in four dimensions

“…These expansion formulas provided a new angle for understanding several related problems, including: constructing local Bern-Carrasco-Johansson (BCJ) [8] numerators [4,9,10], inducing new amplitude relations by gauge symmetries [4,[9][10][11][12], providing an off-shell approach to BCJ duality (and amplitude relations) [13,14] and evaluating EYM amplitudes in four dimensions [15]. Several discussions at loop amplitudes can also be found (see [16][17][18] for example).…”