Recursion Relations for Gauge Theory Amplitudes with Massive Vector Bosons and Fermions

TL;DR

This work extends the BCFW on-shell recursion framework to tree-level amplitudes involving massive vector bosons and spinful fermions by formulating and computing single and double vector boson currents. It delivers compact, explicit results for single-vector currents up to six partons and for double-vector currents up to four partons across all helicities, plus general n-point currents for select helicity configurations. A key advancement is the generalization to massive particles with spin on internal lines, replacing helicity sums with standard spin sums, and validated through a concrete A_4(1_t,2,3,4_{ar{t}}) example that matches Feynman-diagram results. The approach offers an efficient, scalable route to tree-level SM amplitudes with massive gauge bosons and heavy quarks, with broad potential for beyond-Standard-Model extensions.

Abstract

We apply the on-shell tree-level recursion relations of Britto, Cachazo, Feng and Witten to a variety of processes involving internal and external massive particles with spin. We show how to construct multi-vector boson currents where one or more off-shell vector bosons couples to a quark pair and number of gluons. We give compact results for single vector boson currents with up to six partons and double vector boson currents with up to four partons for all helicity combinations. We also provide expressions for single vector boson currents with a quark pair and an arbitrary number of gluons for some specific helicity configurations. Finally, we show how to generalise the recursion relations to handle massive particles with spin on internal lines using $gg \to t\bar t$ as an example.

Figures (9)

• Figure 1: Decomposition of the five-parton amplitude after applying the recursion relation.
• Figure 2: Decomposition of the $n$-point vector boson current with $n-2$ positive helicity gluons.
• Figure 3: Decomposition of the $n$-point vector boson current with $n-3$ positive helicity gluons and two adjacent negative helicities.
• Figure 4: Decomposition of the $n$-point vector boson current with $n-3$ positive helicity gluons and two negative helicities, separated by a single gluon with positive helicity. The second contribution involves the NMHV current with adjacent negative helicities given in (4.21).
• Figure 5: Decomposition of the single vector boson current with three consecutive negative helicites. The recursion re-uses the current with two consecutive nehative helicites.