Recursion Relations for Gauge Theory Amplitudes with Massive Particles

TL;DR

The paper develops tree-level recursion relations that seamlessly incorporate massive particles in gauge theories, extending the on-shell BCFW framework to massive scalars. It derives explicit 3-point primitive vertices and constructs compact 4-,5-,6-point amplitudes for gluons coupled to massive scalars, validating them against known results and diagrammatic calculations. A key result is the demonstration that the shifted amplitude A(z) vanishes at infinity for relevant helicity configurations, enabling boundary-term-free recursion. These massive-amplitude building blocks provide a practical foundation for computing one-loop, six-gluon amplitudes in non-supersymmetric theories using unitarity methods and generalize the on-shell recursion toolkit beyond massless cases.

Abstract

We derive general tree-level recursion relations for amplitudes which include massive propagating particles. As an illustration, we apply these recursion relations to scattering amplitudes of gluons coupled to massive scalars. We provide new results for all amplitudes with a pair of scalars and n < 5 gluons. These amplitudes can be used as building blocks in the computation of one-loop 6-gluon amplitudes using unitarity based methods.

Figures (5)

• Figure 1: Diagrammatic representation of the recursion relation. Arrows label the momentum flow.
• Figure 2: Representation of the 4-point amplitude using the recursion relation with $1$ and $2$ as the shifted momenta.
• Figure 3: The decomposition of the 5-point amplitude using the recursion relation with $1$ and $2$ as the shifted momenta.
• Figure 4: The decomposition of the 6 point amplitude using the recursion relation with $1$ and $2$ as the shifted momenta.
• Figure 5: In a Feynman diagram with marked particles $i,k,l$, the $z$ dependence flows along a unique three-legged path of propagators and vertices.