# Differential equations, recurrence relations, and quadratic constraints for $L$-loop two-point massive tadpoles and propagators

## TL;DR

This paper develops a comprehensive differential equation and recurrence relation framework for L-loop two-point massive tadpole integrals, expressing solutions via Lauricella functions and deriving quadratic constraints, advancing analytical methods in multiloop Feynman integrals.

## Contribution

The paper introduces a Pfaffian differential system for L-loop tadpole integrals and their relation to Lauricella functions, providing new tools for multiloop integral analysis.

## Key findings

- Derived differential equations with Pfaffian form for L-loop integrals.
- Expressed solutions in terms of Lauricella functions $F_C^{(L)}$.
- Obtained quadratic constraints for solutions near integer dimensions.

## Abstract

We consider $L$-loop two-point tadpole (watermelon) integral with arbitrary masses, regularized both dimensionally and analytically. We derive differential equation system and recurrence relations (shifts of dimension and denominator powers). Since the $L$-loop sunrise integral corresponds to the $(L+1)$-loop watermelon integral with one cut line, our results are equally applicable to the former. The obtained differential system has a Pfaffian form and is linear in dimension and analytic regularization parameters. In general case, the solutions of this system can be expressed in terms of the Lauricella functions $F_C^{(L)}$ with generic parameters. Therefore, as a by-product, we obtain, to our knowledge for the first time, the Pfaffian system for $F_C^{(L)}$ for arbitrary $L$. The obtained system has no apparent singularities. Near odd dimension and integer denominator powers the system can be easily transformed into canonical form. Using the symmetry properties of the matrix in the right-hand side of the differential system, we obtain quadratic constraints for the expansion of solutions near integer dimension and denominator powers. In particular, we obtain quadratic constraints for Bessel moments similar to those discovered by Broadhurst and Roberts.