Integration-by-parts identities in the presence of measurement delta functions

Abstract

We derive a convenient form of integration-by-parts (IBP) identities for dimensionally regulated loop integrals that include a measurement constraint imposed by a delta distribution $\delta(\phi(\ell))$, where $\phi$ is a regular and generally non-linear function of the loop momentum. Non-trivial IBP relations can be generated directly by vectors tangent to the hypersurface $\phi=0$, leaving the measurement delta as an overall factor and avoiding derivatives of distributions. This construction provides a systematic route to differential distributions within reverse unitarity for non-linear measurement functions, in particular, for transverse-momentum distributions. We implement the construction and validate it for transverse-momentum distributions in $t\bar tH$ production at leading order and $t\bar t$ production at next-to-leading order in QCD.