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

Abstract

We use massive CSW rules to derive explicit compact expressions for integrands of rational terms in QCD with any number of external legs. Specifically, we present all-n integrands for the one-loop all-plus and one-minus gluon amplitudes in QCD. We extract the finite part of spurious external-bubble contributions systematically; this is crucial for the application of integrand-level CSW rules in theories without supersymmetry. Our approach yields integrands that are independent of the choice of CSW reference spinor even before integration.Furthermore, we present a recursive derivation of the recently proposed massive CSW-style vertex expansion for massive tree amplitudes and loop integrands on the Coulomb-branch of N = 4 SYM. The derivation requires a careful study of boundary terms in all-line shift recursion relations, and provides a rigorous (albeit indirect) proof of the recently proposed construction of massive amplitudes from soft-limits of massless on-shell amplitudes. We show that the massive vertex expansion manifestly preserves all holomorphic and half of the anti-holomorphic supercharges, diagram-by-diagram, even off-shell.

Constructing $$ \mathcal{N} $$ = 4 Coulomb branch superamplitudes

“…While the underlying origin is likely a vestige of dual (super)conformal invariance remaining on the Coulomb branch, we here leave an exploration of this to future work and instead prove the shift validity by using soft limits to extend the known behavior at the origin of moduli space. The idea that Coulomb branch component amplitudes may be found from soft limits of massless amplitudes with scalar insertions was proposed in [18], expanded upon in [19] and proven in [20]. The precise map is explained clearly around (4.3) of [19], but the details will not be necessary for us.…”

“…These nevertheless provide a further testing ground of the special symmetries and properties listed above and the extent to which they are deformed but not destroyed by Higgsing. Previous studies of massive amplitudes on the Coulomb branch have been made in [17][18][19][20], where a gamut of methods including soft limits, supersymmetric on-shell recursion and solutions to the supersymmetric Ward identities (SWIs) were proposed and used to compute some simple examples. Subsequently, some 4d tree-level amplitudes and loop integrands have been obtained by dimensional reduction from superamplitudes of the 6d N = (1, 1) SYM theory, for which dual conformal invariance has been established, despite the absence of conformality [21][22][23][24][25].…”

Cyclic Mario worlds — color-decomposition for one-loop QCD

“…[66]), summing up all the planar graphs with the specified routing of quark lines. Modern methods like generalized unitary [17,18,[73][74][75][76], (loop-level) BCFW [28,29,[77][78][79][80][81] or Q-cuts [82,83] simplify this computation for QCD as well as for supersymmetric extensions thereof.…”

Primary Feynman rules to calculate the ∊-dimensional integrand of any 1-loop amplitude

“…The case of QCD is particularly simple, and computations based on a standard Passarino-Veltman [56] decomposition are relatively easy [57,58]. In addition, for gluonic amplitudes, a super-symmetric decomposition relates the contribution of the rational terms to a scalar massive gluon running in the loop [59], which can also be computed by using massive Cachazo-Svrček-Witten Feynman rules [60,61,62].…”