Two-fold Mellin–Barnes transforms of Usyukina–Davydychev functions

Abstract

In our previous paper [Nucl. Phys. B 870 (2013) 243], we showed that multi-fold Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions may be reduced to twofold MB transforms. The MB transforms were written there as polynomials of logarithms of ratios of squares of the external momenta with certain coefficients. We also showed that these coefficients have a combinatoric origin. In this paper, we present an explicit formula for these coefficients. The procedure of recovering the coefficients is based on taking the double-uniform limit in certain series of smooth functions of two variables which is constructed according to a pre-determined iterative way. The result is obtained by using basic methods of mathematical analysis. We observe that the finiteness of the limit of this iterative chain of smooth functions should reflect itself in other mathematical constructions, too, since it is not related in any way to the explicit form of the MB transforms. This finite double-uniform limit is represented in terms of a differential operator with respect to an auxiliary parameter which acts on the integrand of a certain two-fold MB integral. To demonstrate that our result is compatible with original representations of UD functions, we reproduce the integrands of these original integral representations by applying this differential operator to the integrand of the simple integral representation of the scalar triangle four-dimensional integral J(1, 1, 1 − ε).

Integral representations combining ladders and crossed-ladders

“…[40]. Mathematically, the completely massless case is special (in D = 4), since here the "proper" (non-crossed) ladder graphs become conformally invariant, and possess closed-form expressions in terms of polylogarithms for any N [35,[41][42][43][44][45][46].…”

Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

“…As we have mention in Introduction, the integral relation (1) has a structure similar to decomposition of tensor product in terms of irreducible components, and the integral relation (10) has a structure similar to orthogonality condition. This observation suggests that behind MB transforms D (u,v) [ν 1 , ν 2 , ν 3 ] an integrable structure may exist [1,8,9]. It is remarkable that both the relations (1) and (10) are written for the Green functions, that is, the integrable structure should exist for the Green functions, too.…”

“…As it has been mentioned in Ref. [14], a hint for such a kind of relation (8) has appeared from the explicit calculation of a Green function in Ref. [23].…”

“…The explicit form of M (n) (z 2 , z 3 ) can be found in Refs. [1,8]. We do not need any use of the explicit form of those MB transforms.…”

“…Mellin-Barnes (MB) transforms of the momentum integrals corresponding to ladder diagrams at any loop order have been studied in Refs. [7,1,8,9] in four-dimensional case. In Ref.…”