Abstract
In this paper, we proceed to study properties of Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions. In our previous papers (Allendes et al., 2013 [13], Kniehl et al., 2013 [14]), we showed that multi-fold Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions may be reduced to two-fold MB transforms and that the higher-order UD functions may be obtained in terms of a differential operator by applying it to a slightly modified first UD function. The result is valid in d=4 dimensions, and its analog in d=4−2ε dimensions exits, too (Gonzalez and Kondrashuk, 2013 [6]). In Allendes et al. (2013) [13], the chain of recurrence relations for analytically regularized UD functions was obtained implicitly by comparing the left-hand side and the right-hand side of the diagrammatic relations between the diagrams with different loop orders. In turn, these diagrammatic relations were obtained using the method of loop reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and Usyukina. Here, we reproduce these recurrence relations by calculating explicitly, via Barnes lemmas, the contour integrals produced by the left-hand sides of the diagrammatic relations. In this a way, we explicitly calculate a family of multi-fold contour integrals of certain ratios of Euler gamma functions. We make a conjecture that similar results for the contour integrals are valid for a wider family of smooth functions, which includes the MB transforms of UD functions.
Highlights
Off-shell triangle-ladder and box-ladder diagrams are the only family of the Feynman diagrams which were calculated at any loop order, for example in d = 4 space-time dimensions [1, 2, 3, 4] with all indices equal to 1 in the momentum space representation (m.s.r.) and in d = 4 − 2ε spacetime dimensions with indices equal to 1 − ε on the rungs of ladders in the m.s.r. too [5, 6]
In the present paper we show that in the particular case when in the integrand of the contour integrals on the left hand sides of the diagrammatic relations the MB transforms of the UD functions stand, this chain of recurrent relations for the MB transforms of UD functions is produced by the contour integration
We showed in Ref.[14] [Nuclear Physics B 876 (2013) 322] that structure of the chain of recurrent relations for the Mellin-Barnes transforms of the analytically regularized UD functions guarantees the finiteness of the double-uniform limit when removing the analytical regularization
Summary
In the present paper we show that in the particular case when in the integrand of the contour integrals on the left hand sides of the diagrammatic relations the MB transforms of the UD functions stand, this chain of recurrent relations for the MB transforms of UD functions is produced by the contour integration. These contour integrals are calculated explicitly via the first and the second Barnes lemmas. In the papers we describe this family of functions and describe what kind of changes should be made for the contours of the integrals over complex variables for the case of other smooth functions different from certain ratios of Euler gamma functions.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
