Double Soft Theorem for Generalised Biadjoint Scalar Amplitudes

Abstract

We study double soft theorem for the generalised biadjoint scalar field theory whose amplitudes are computed in terms of punctures on CP k−1 . We find that whenever the double soft limit does not decouple into a product of single soft factors, the leading contributions to the double soft theorems come from the degenerate solutions, otherwise the non-degenerate solutions dominate. Our analysis uses the regular solutions to the scattering equations. Most of the results are presented for k = 3 but we show how they generalise to arbitrary k. We have explicit analytic results, for any k, in the case when soft external states are adjacent.

One-loop integrand from generalised scattering equations

“…Explicit evaluation of the amplitude for (k = 3, n = 6) helps us validate the double soft factor obtained in [49]. The adjacent double soft theorem for arbitrary k had an intriguing structure with the appearance of single soft factors for k−1 and k, with modified propagators.…”

“…Therefore, the residue on the factorisation channel t 6123 is equivalent to the CHY integral representation over the moduli space of 6-punctured Riemann sphere, M 0,6 . This becomes manifest if we choose a gauge where we fix, B Adjacent double soft theorem for arbitrary k Double soft limits of CEGM amplitudes have been studied in [49]. In this section, we present a summary of the double soft theorem in arbitrary (k, n) amplitudes when the two adjacent external states are taken to be soft simultaneously.…”

“…Contrary to the single soft case, here the single soft factor with shifted propagators realise the A 2 subalgebra as a facet of the cluster polytope. This A 2 subalgebra can be understood from the mutation of the six-point tree amplitude obtained in the forward limit corresponding to the variables X 1 , X3 , which are the variables that scale as τ 2 and correspond to the single soft factor in k = 2 with a composite label m = {56} going soft as discussed in [49].…”

“…Double soft limits of Gr (k, n) amplitudes have been explored in [49]. In Gr (k, n) amplitudes, the double soft factor contains 4 propagators and itself is the full amplitude.…”

“…The primed product denotes that k + 1 delta functions have been removed. Various properties of these amplitudes have been investigated in [43][44][45][46][47][48][49][50]. CEGM amplitudes have realisations in terms of the Grassmannian spaces Gr (k, n), and the k = 2 case reduces to the CHY construction.…”