Symmetric ϵ- and (ϵ + 1/2)-forms and quadratic constraints in “elliptic” sectors.

Abstract

Abstract Within the differential equation method for multiloop calculations, we examine the systems irreducible to ϵ-form. We argue that for many cases of such systems it is possible to obtain nontrivial quadratic constraints on the coefficients of ϵ-expansion of their homogeneous solutions. These constraints are the direct consequence of the existence of symmetric (ϵ+1/2)-form of the homogeneous differential system, i.e., the form where the matrix in the right-hand side is symmetric and its ϵ-dependence is localized in the overall factor (ϵ + 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 ϵ-form. For the systems reducible to ϵ-form we also observe the existence of symmetric form and derive the corresponding quadratic constraints.