Symmetric $\epsilon$- and $(\epsilon+1/2)$-forms and quadratic constraints in "elliptic" sectors

TL;DR

This paper explores the existence of symmetric $(\,\epsilon+1/2)$-forms in differential systems related to multiloop calculations, revealing quadratic constraints on solutions that deepen understanding of their structure.

Contribution

It introduces the concept of symmetric $(\epsilon+1/2)$-forms for differential systems, providing a constructive method to identify these forms and derive quadratic constraints.

Findings

Symmetric $(\epsilon+1/2)$-forms are common in irreducible systems.

Quadratic constraints can be derived from these forms.

The approach applies to systems reducible and irreducible to $\epsilon$-form.

Abstract

Within the differential equation method for multiloop calculations, we examine the systems irreducible to $\epsilon$-form. We argue that for many cases of such systems it is possible to obtain nontrivial quadratic constraints on the coefficients of $\epsilon$-expansion of their homogeneous solutions. These constraints are the direct consequence of the existence of symmetric $(\epsilon+1/2)$-form of the homogeneous differential system, i.e., the form where the matrix in the right-hand side is symmetric and its $\epsilon$-dependence is localized in the overall factor $(\epsilon+1/2)$. The existence of such a form can be constructively checked by available methods and seems to be common to many irreducible systems, which we demonstrate on several examples. The obtained constraints provide a nontrivial insight on the structure of general solution in the case of the systems irreducible to $\epsilon$-form. For the systems reducible to $\epsilon$-form we also observe the existence of symmetric form and derive the corresponding quadratic constraints.