Integrands for QCD rational terms and 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.

30 pages, many figures

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• Gravitational Effective Field Theory Islands, Low-Spin Dominance, and the Four-Graviton Amplitude
• Duals of Feynman Integrals, 2: Generalized Unitarity
• Constructing $\mathcal{N}=4$ Coulomb Branch Superamplitudes
• The Q-cut Representation of One-loop Integrands and Unitarity Cut Method
• Feynman Rules for the Rational Part of One-loop QCD Corrections in the MSSM
• Cyclic Mario Worlds -- Color-Decomposition for One-Loop QCD
• The dimension-shift conjecture for one-loop amplitudes
• A new Wilson line-based action for gluodynamics
• One-Loop Effective Action Approach to Quantum MHV Theory
• Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory