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

Abstract

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

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

“…These relations are closely related to the ones described in Ref. [13] in the case when matrix A is symmetric and proportional to or + 1/2. In order to obtain these relations, we first note that the matrix A(D, µ) has the following symmetry…”

“…Using the symmetry properties of the matrix in the right-hand side of the differential system and following the path similar to that in Ref. [13], we obtain the bilinear relations between the solutions of the system for opposite signs of D and µ.…”

“…(6.2) corresponds to ( + 1/2)-form discussed in Ref. [13]. In particular, this form allows one to write down the -expansion of the solution near D = 1 in terms of Goncharov polylogarithms (see Appendix B for details).…”

Decomposition of Feynman integrals by multivariate intersection numbers

“…The former set of relations yields results that are equivalent to the known integration-by-parts identities (IBPs) [24], while the latter allow for a systematic classification of relations which, for certain type of integrals were originally detected within the application of number-theoretic methods to Feynman integrals, giving rise to interesting conjectures [25][26][27][28][29], proven to be true quite recently [30,31]. A special set of quadratic relations have been presented in [32], and it would be interesting to investigate if they can be classified as Twisted Riemann Period Relations [4].…”

Two-loop helicity amplitudes for gg → ZZ with full top-quark mass effects

“…Due to the massive top-quark loop in the corrections considered here, we expect the presence of functions beyond multiple polylogarithms, which makes the evaluations of master integrals considerably more challenging. While there has been significant progress concerning the analytic evaluation of Feynman integrals beyond polylogarithms [86][87][88][89][90][91], integrals of the type considered here remain a challenge. An alternative is the use of expansions to solve the differential equations numerically [41,[92][93][94][95].…”