A note on the boundary contribution with bad deformation in gauge theory

TL;DR

This work tackles nonzero boundary contributions arising from bad BCFW deformations in gauge theory by leveraging the ${\cal N}=4$ SYM framework. It develops an on-shell recursion that expresses boundary terms, associated with specific helicity configurations, in terms of pole contributions from related amplitudes via an ${\cal N}=4$ superfield expansion. Two concrete examples—a gluon MHV amplitude and a fermion-containing six-point amplitude—demonstrate that these boundary contributions are computable and consistent with known results, effectively showing a generalized cut-constructibility. The approach also sheds light on the use of nearby fermions for BCFW shifts and suggests extensions to gravity.

Abstract

Motivated by recently progresses in the study of BCFW recursion relation with nonzero boundary contributions for theories with scalars and fermions\cite{Bofeng}, in this short note we continue the study of boundary contributions of gauge theory with the bad deformation. Unlike cases with scalars or fermions, it is hard to use Feynman diagrams directly to obtain boundary contributions, thus we propose another method based on the ${\cal N}=4$ SYM theory. Using this method, we are able to write down a useful on-shell recursion relation to calculate boundary contributions from related theories. Our result shows the cut-constructibility of gauge theory even with the bad deformation in some generalized sense.

Figures (7)

• Figure 1: The four diagrams are these boundary contributions for $A_{0,4}$ with $t=s-1$ and $n=s+1$ using the recursion relation.
• Figure 2: The four diagrams are these boundary contributions for $A_{0,4}$ with $t=s-1$ and $n \neq s+1$ using the recursion relation.
• Figure 3: These four diagrams are the A part of boundary contributions for $A_{0,4}$ with $t\neq s-1$, $t\neq s+1$ and $n=s+1$ using the recursion relation.
• Figure 4: The diagram is the B part of boundary contributions for $A_{0,4}$ with $t\neq s-1$, $t\neq s+1$ and $n=s+1$ using the recursion relation.
• Figure 5: These diagrams are the A part of the boundary contributions for $A_{0,4}$ with $t\neq s-1$, $t\neq s+1$, $n\neq s+1$ and $n\neq s-1$ using the recursion relation.