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

TL;DR

The work addresses the challenge of differential equations for multiloop master integrals that are not reducible to the standard $\epsilon$-form, focusing on elliptic sectors. It shows that many such systems admit a symmetric $(\epsilon+\tfrac{1}{2})$-form, and uses dimensional recurrence relations to derive quadratic constraints on the $\epsilon$-expansion coefficients of homogeneous solutions. The authors validate the approach on several elliptic cases (sunrise and nonplanar vertex); the resulting identities involve hypergeometric functions and elliptic integrals, revealing nontrivial structure in the general solution. The findings point to a potential geometric interpretation of these invariances and suggest broader applicability to irreducible differential systems beyond the cases studied.

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.

Figures (4)

• Figure 1: The sunrise topology.
• Figure 2: SummerTime code for calculation of $H_{\alpha,1,1,1}(y)$ at $y=2/3$.
• Figure 3: The two-loop nonplanar vertex topology, $k_1^2=k_2^2=0$, $q=k_1+k_2$.
• Figure 4: The three-loop sunrise topology.