Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

TL;DR

This paper develops a comprehensive set of logarithmic vector fields for deriving integration-by-parts identities in Feynman integrals, avoiding dimension shifts and providing a complete mathematical framework applicable to any loop order.

Contribution

It introduces an explicit, complete set of solutions for IBP identities in Baikov representation using syzygies related to Gram determinants, with a rigorous proof of completeness.

Findings

Explicit generating set of solutions for IBP identities

Complete mathematical proof of the solution set's completeness

Applicable to any number of loops and external momenta

Abstract

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.