The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions
Roman N. Lee, Vladimir A. Smirnov
TL;DR
The paper extends the Dimensional Recurrence and Analyticity (DRA) method to multicomponent master integrals by showing that maximally cut unitarity integrals satisfy the homogeneous part of the dimensional-recurrence relations. This provides a practical route to construct the general homogeneous solution of coupled difference equations that arise for MMIs. Through a three-loop example for the static quark potential, the authors derive two independent homogeneous solutions from Mellin-Barnes representations, assemble the full solution with the inhomogeneous part, and obtain high-precision ε-expansions for the master integrals, validated by PSLQ against known constants. The work demonstrates that maximal cuts and MB-residue structures yield multiple independent solutions (one per pole-series), enabling analytic progress where traditional hypergeometric approaches fail. The approach promises broader applicability to higher-loop multicomponent sectors and deeper analytic control over master integrals.
Abstract
We consider the application of the DRA method to the case of several master integrals in a given sector. We establish a connection between the homogeneous part of dimensional recurrence and maximal unitarity cuts of the corresponding integrals: a maximally cut master integral appears to be a solution of the homogeneous part of the dimensional recurrence relation. This observation allows us to make a necessary step of the DRA method, the construction of the general solution of the homogeneous equation, which, in this case, is a coupled system of difference equations.