Abstract
Abstract We analyze the validity of BCFW recursion relations for currents of n − 2 gluons and two massive quarks, where one of the quarks is off shell and the remaining particles are on shell. These currents are gauge-dependent and can be used as ingredients in the unitarity-based approach to computing one-loop amplitudes. The validity of BCFW recursion relations is well known to depend on the so-called boundary behavior of the currents as the momentum shift parameter goes to infinity. With off-shell currents, a new potential problem arises, namely unphysical poles that depend on the choice of gauge. We identify conditions under which boundary terms are absent and unphysical poles are avoided, so that there is a natural recursion relation. In particular, we are able to choose a gauge in which we construct a valid shift for currents with at least n − 3 gluons of the same helicity. We derive an analytic formula in the case where all gluons have the same helicity. As by-products, we prove the vanishing boundary behavior of general off-shell objects in Feynman gauge, and we find a compact generalization of Berends-Giele gluon currents with a generic reference spinor.
Highlights
The phrase “on-shell technique” refers to methods of computing scattering amplitudes in which certain propagators are taken to their on-shell limits, and amplitudes are constructed from knowledge of the corresponding factorization properties
We analyze the validity of BCFW recursion relations for currents of n − 2 gluons and two massive quarks, where one of the quarks is off shell and the remaining particles are on shell
We identify conditions under which the boundary terms and unphysical poles vanish for massive fermion currents, so that the BCFW construction produces a recursion relation
Summary
Momenta of gluons are directed outward, while momenta of fermions are directed inward. We will be considering color-ordered amplitudes and off-shell currents with one massive fermion line, for example, iJ 1∗Q, 2Q, 3g, 4g, . We do not include the propagator for the off-shell leg in our definition. For this current, the quark line has its arrow pointing from leg 2 to leg 1. Where the round bracket |2) can be equal to either |2 or |2], depending on its spin This notation emphasizes the fact that the current is a spinorial object. Any current we construct with a specific gauge choice is expected to fit into a larger calculation, such as the one-loop computations of [4], in which all external legs are on-shell, so that after being combined with other ingredients computed in the same gauge, no trace of the gauge choice remains. We use the Lorentz vector Pi,j to denote the sum of color-adjacent momenta in increasing cyclic order, between and including legs i and j
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