CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators

TL;DR

CALICO introduces parametric annihilators as a general, representation-agnostic framework to derive linear relations and differential equations among parametric integrals. It links annihilator techniques with syzygy approaches, extends to multiple loop representations (Baikov, loop-by-loop Baikov, Lee–Pomeransky, Schwinger) and to duals, and uses efficient finite-field solvers to compute and reconstruct analytic identities. The paper provides a concrete public Mathematica package and demonstrates substantial gains in reducing loop-integral systems and generating master-integral differential equations, with cross-checks against traditional reduction methods. Overall, the work offers a scalable, versatile toolkit for high-precision perturbative calculations in quantum field theory and related areas, with broad potential across representations and dual formulations.

Abstract

We elaborate on the method of parametric annihilators for deriving integral relations. Parametric annihilators are differential operators that annihilate multivalued integration kernels appearing in suitable integral representations of special functions. We illustrate this approach in a way that applies to a broad variety of integral representations. We describe a method for computing parametric annihilators based on efficient linear solvers and use them to derive relations between a wide class of special functions related to important problems in high-energy physics. We also formulate a similar method for deriving differential equations satisfied by the independent integrals within an integral family. We show applications to several classes of special functions, including hypergeometric functions, loop integrals in various representations (including Baikov, loop-by-loop Baikov, Lee-Pomeransky and Schwinger representations) and duals of loop integrals. We finally present the public Mathematica package CALICO for computing parametric annihilators and its usage in several examples of high relevance in theoretical particle physics.

Figures (6)

• Figure 1: Double-box.
• Figure 2: Massive one-loop bubble.
• Figure 3: Integral families with regulated denominators considered in ref. Fontana:2023amt, for which connection matrices of inner integration layers have been checked against the method proposed in this paper.
• Figure 4: Pentabox contributing to top-pair plus Higgs production.
• Figure 5: Two-loop sunrise with different internal masses.