BCFW Recursion Relation with Nonzero Boundary Contribution

TL;DR

This work extends the BCFW on-shell recursion program to theories with nonzero boundary contributions by analyzing λφ^4 and scalar QCD with Yukawa couplings. By decomposing amplitudes into a boundary term A_b and a pole term A_pole, the authors show how boundary effects can be computed from lower-point on-shell data and incorporated into a generalized recursion A = A_b + A_pole. They provide detailed helicity-specific calculations and demonstrate consistency with standard Feynman-diagram results, including an explicit correspondence between boundary contributions and an auxiliary-field formulation. The findings broaden the applicability of on-shell methods, offering a practical framework for computing high-point amplitudes and informing potential extensions to loop-level rational pieces and S-matrix structure.

Abstract

The appearance of BCFW on-shell recursion relation has deepen our understanding of quantum field theory, especially the one with gauge boson and graviton. To be able to write the BCFW recursion relation, the knowledge of boundary contributions is needed. So far, most applications have been constrained to the cases where the boundary contribution is zero. In this paper, we show that for some theories, although there is no proper deformation to annihilate the boundary contribution, its effects can be analyzed in simple way, thus we do able to write down the BCFW recursion relation with boundary contributions. The examples we will present in this paper include the lambda-phi-four theory and Yukawa coupling between fermions and scalars.

Figures (11)

• Figure 1: $(a)$ The contribution from boundary. $(b)$ The contribution from pole part.
• Figure 2: pure scalar case: $(a)$ boundary term figure $(b)$ boundary term figure in view of auxiliary field
• Figure 3: General Feynman diagrams.
• Figure 4: Amplitude in total
• Figure 5: A 6-point Feynman diagram