Exact two-loop gravitational tail integrals and their ultraviolet-infrared structure

Abstract

Gravitational tail-of-tail corrections arise when radiation scatters twice from the mass-monopole field of its source. In the multipolar effective field theory, one of the two-loop topologies contributing to this process involves genuinely coupled loop momenta, preventing its evaluation through nested one-loop integrations. After tensor reduction, this nonfactorizing structure leads to a particularly challenging family of scalar integrals with unbounded numerator powers as the multipole rank increases. We derive exact closed-form expressions for the entire relevant integer numerator domain, valid in generic dimension and accommodating Feynman, retarded, and advanced prescriptions through consistent analytic continuation. Near $d=3+ε$, we also obtain explicit Laurent expansions through $O(ε)$, retaining the subleading information needed in higher-order radiative calculations. While these expansions determine the dimensional poles, identifying their ultraviolet (UV) or infrared (IR) origin requires a separate analysis of momentum regions. We carry out this analysis throughout the integral domain and apply it directly to the coupled tail-of-tail emission diagram for arbitrary electric and magnetic multipoles. In this way, we independently determine the UV and IR residues, and explicitly verify that the complete infrared pole structure, including both double and simple poles, agrees with the exponentiation of the one-loop gravitational tail. The UV residues, calculated directly from hard-momentum regions, reproduce the established multipole anomalous dimensions when combined with the known contributions of the remaining diagrams.