Multivariate elliptic kites and tetrahedral tadpoles

TL;DR

This paper analyzes complex two- and three-loop Feynman integrals with arbitrary masses, revealing their elliptic structures and providing compact formulas applicable across various gauge theories and mass configurations.

Contribution

It introduces a comprehensive analysis of elliptic Feynman integrals for kite and tetrahedral tadpole diagrams with arbitrary masses, including new compact formulae and number-theoretic insights.

Findings

Elliptic substructures are manageable and beneficial, not obstructive.

Provides explicit compact formulas for all mass cases.

Investigates number theory aspects of tadpole integrals.

Abstract

This work deals with two types of Feynman integrals in perturbative quantum field theory: the 2-loop 2-point kite, with 5 arbitrary internal masses, and its completion by a sixth propagator, to give a 3-loop tetrahedral tadpole, with 6 arbitrary masses. These general-mass cases cover broken and unbroken gauge theories, based on the Lie algebras U(1), SU(2) and SU(3), for the electromagnetic, weak and strong interactions. The elliptic substructure of these integrals should not be regarded as an obstruction. Rather, it is a bonus, thanks to the arithmetic-geometric mean of Gauss. Compact formulae are given, to handle all cases. Zero-mass limits are carefully considered. Anomalous thresholds of triangles in the kite pose no problem. The number theory of tadpoles is investigated, with intriguing results.