BCFW recursion for TMD parton scattering

TL;DR

This work generalizes the BCFW recursion to amplitudes with a single off-shell leg in Yang–Mills theories with fermions, addressing the necessary applicability conditions and the modifications to large-z behavior due to off-shell kinematics. It develops closed-form MHV expressions with off-shell insertions and provides a complete treatment of 5-point non-MHV amplitudes, including multiple valid shift schemes and cross-checks against numerical evaluation. The results supply practical, gauge-invariant analytic tools for off-shell amplitudes within HEF/TMD factorization frameworks and extend the utility of BCFW recursion beyond the fully on-shell regime. The accompanying Mathematica notebook reinforces reproducibility and enables efficient numerical use in high-energy factorization calculations.

Abstract

We investigate the application of the BCFW recursion relation to scattering amplitudes with one off-shell particle in a Yang-Mills theory with fermions. We provide a set of conditions of applicability of the BCFW recursion, stressing some important differences with respect to the pure on-shell case. We show how the formulas for Maximally-Helicity-Violating (MHV) configurations with any number of partons, which are well known in the fully on-shell case, are generalized to this kinematic regime. We also derive analytic expressions for all the helicity configurations of the 5-point color-stripped tree-level amplitudes for any of the partons being off the mass shell.

Figures (10)

• Figure 1: These are the only three possible modifications (modulo specular ones) leading from the $6$-gluon diagram to a diagram with 4-gluons and a fermion pair. We depict in red the propagators and vertices which are not asymptotically constant as functions of z. We observe that the large $z$ behaviour stays the same only if no fermion propagator is introduced by the switch; otherwise, it always improves by one power.
• Figure 2: The recursion for subleading amplitudes with an off-shell antifermion.
• Figure 3: The recursion for MHV amplitudes with an off-shell antifermion, shifting two gluons.
• Figure 4: The recursion for MHV amplitudes with an off-shell gluon, shifting the off-shell gluon itself and another gluon with positive helicity
• Figure 5: The recursion for $\mathcal{A}(g_1^+,g_2^+,\bar{q}^*,q^-,g_3^-)$ with $e^\mu = \frac{1}{2}\langle3| ^\mu|1]$.