Momentum shift and on-shell constructible massive amplitudes

We construct tree-level amplitude for massive particles using on-shell recursion relations based on two classes of momentum shifts: an all-line transverse shift that deforms momentum by its transverse polarization vector, and a massive BCFW-type shift. We illustrate that these shifts allow us to correctly calculate four-point and five-point amplitudes in massive QED, without an ambiguity associated with the contact terms that may arise from a simple ''gluing'' of lower-point on-shell amplitudes. We discuss various aspects and applicability of the two shifts, including the large-z behavior and complexity scaling. We show that there exists a ''good'' all-line transverse shift for all possible little group configurations of the external particles, which can be extended to a broader class of theories with massive particles such as massive QCD and theories with massive spin-1 particles. The massive BCFW-type shift enjoys more simplicity, but a ''good'' shift does not exist for all the spin states due to the specific choice of spin axis.