N=1 Supersymmetric One-loop Amplitudes and the Holomorphic Anomaly of Unitarity Cuts

TL;DR

This work extends the holomorphic anomaly method for unitarity cuts from N=4 to N=1 supersymmetric one-loop amplitudes, focusing on a six-gluon non-MHV case with a chiral multiplet in the loop. It demonstrates that the holomorphic anomaly acting on cuts reproduces the collinearity operator's effect on the amplitude's imaginary part (consistent with the optical theorem) and shows that, for N=1, the resulting constraints on integral-coefficient functions are differential rather than algebraic. The authors derive differential equations for triangle and bubble coefficients, illustrate how physical inputs like collinear limits fix homogeneous solutions, and extend the analysis to higher-point amplitudes, highlighting the general structure and limitations of this approach. Overall, the paper provides a differential-equation framework for reconstructing N=1 one-loop amplitudes from cut information, with boundary conditions guiding unique determinations.

Abstract

Recently, it has been shown that the holomorphic anomaly of unitarity cuts can be used as a tool in determining the one-loop amplitudes in N=4 super Yang-Mills theory. It is interesting to examine whether this method can be applied to more general cases. We present results for a non-MHV N=1 supersymmetric one-loop amplitude. We show that the holomorphic anomaly of each unitarity cut correctly reproduces the action on the amplitude's imaginary part of the differential operators corresponding to collinearity in twistor space. We find that the use of the holomorphic anomaly to evaluate the amplitude requires the solution of differential rather than algebraic equations.