The Coulomb-branch S-matrix from massless amplitudes

Abstract

We present a systematic method to extract the entire tree-level S-matrix on the Coulomb branch of N = 4 SYM from soft-scalar limits of on-shell amplitudes at the origin of moduli space. Massive amplitudes in the spontaneously-broken theory can thus be computed from on-shell amplitudes in the massless, unbroken theory. To check this correspondence, we first prove that soft and collinear divergences in the required massless amplitudes cancel for a judicious choice of soft-scalar momenta. We then explicitly verify our proposal in examples with arbitrarily many external legs and to all orders in the mass. As a byproduct, the construction leads to a massive CSW-like expansion that reproduces several known all-n results for Coulomb-branch amplitudes in an effortless way. We briefly discuss the extension of our method to loop integrands.

Integrands for QCD rational terms and $ \mathcal{N} = {4} $ SYM from massive CSW rules

“…We then recursively construct the missing boundary terms from a holomorphic all-line shift |i → |i + w bi |q . In particular, our derivation provides a (somewhat indirect) proof of the soft-limit construction of Coulomb-branch amplitudes proposed in [17,18].…”

“…Scattering amplitudes for gluons interacting with a charged massive scalar can be computed conveniently from the massive CSW rules given in [15]. These rules can also be understood as a special case of the massive CSW expansion on the Coulomb branch of N = 4 SYM [18], which we will derive in section 3. We emphasize that all momenta appearing in the CSW rules are strictly 4dimensional.…”

“…(See [16] for applications at the 4-and 5-point level.) Using this massive vertex expansion and the techniques developed in [18], we derive an extremely compact all-n expression for the integrand I ++•••+ of the (purely rational) all-plus amplitude in QCD. We also present a similarly compact all-n BCFW-like representation of the same integrand, which is manifestly free of spurious poles.…”

“…Coulomb-branch amplitudes have recently been studied (i) as an infrared regulator [30][31][32][33][34] for massless planar integrands in N = 4 [35][36][37][38], (ii) because they arise in the dimensional reduction of the massless maximally supersymmetric 6-dimensional (1, 1) theory [39][40][41][42][43][44], and (iii), in their own right, as the "simplest" massive field theory in 4 dimensions [45-47, 17, 18]. A direct construction of massive Coulomb-branch tree amplitudes and loop integrands from massless on-shell amplitudes was proposed in [17,18]. It was shown in [18] that this construction implies a certain massive vertex expansion for Coulomb-branch amplitudes, which we review in section 3.1.…”

“…A direct construction of massive Coulomb-branch tree amplitudes and loop integrands from massless on-shell amplitudes was proposed in [17,18]. It was shown in [18] that this construction implies a certain massive vertex expansion for Coulomb-branch amplitudes, which we review in section 3.1. In this paper, we derive this expansion for tree amplitudes and loop integrands from recursion relations.…”