Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin-Barnes Representation

Using a method mixing Mellin-Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary "$N$-point" functions for the simple case of zero-dimensional $\phi^4$ field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven non-perturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a non-perturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymptotic level. The Mellin-Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A numerical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes.Comment: v2: one reference added, one paragraph added in the conclusions, small changes in the text, corrected typos; v3: published versio

Asymptotic expansions of Feynman diagrams and the Mellin-Barnes representation

In this talk, we describe part of our recent work \cite{FGdeR05} (see also \cite{F05,G05}) that gives new results in the context of asymptotic expansions of Feynman diagrams using the Mellin-Barnes representation.Comment: Talk given at the $12^{th}$ High-Energy Physics International Conference on Quantum Chromodynamics, 4-8 July (2005), Montpellier (France

On Ruby's solid angle formula and some of its generalizations

Using the Mellin-Barnes representation, we show that Ruby's solid angle formula and some of its generalizations may be expressed in a compact way in terms of the Appell F4 and Lauricella Fc functions.Comment: 10 pages, accepted in Nucl. Instr. and Meth. in Phys. Res.

The L-R Correlator and its Chiral Condensates in the MHA and MHA+V' Approximations to Large-Nc QCD

In this talk, we describe part of a recent work on the correlation function of a V-A current with a V+A current in the framework of QCD in the limit of a large number of colours Nc. The discussion takes place within two successive approximations of this theory, called MHA and MHA+V'. Results concerning the evaluation of chiral condensates of dimension six and eight, as well as matrix elements of the Q7 and Q8 electroweak penguin operators, are given

Geometrical methods for the analytic evaluation of multiple Mellin-Barnes integrals

Two recently developed techniques of analytic evaluation of multifold Mellin-Barnes (MB) integrals are presented. Both approaches rest on the definition of geometrical objets conveniently associated with the MB integrands, which can then be used along with multivariate residues analysis to derive series representations of the MB integrals. The first method is based on introducing conic hulls and considering specific intersections of the latter, while the second one rests on point configurations and their regular triangulations. After a brief description of both methods, which have been automatized in the MBConicHulls.wl Mathematica package, we review some of their applications. In particular, we show how the conic hulls method was used to obtain the first analytic calculation of complicated Feynman integrals, such as the massless off-shell conformal hexagon and double-box. We then show that the triangulation method is even more efficient, as it allows one to compute these nontrivial objects and harder ones in a much faster way.Comment: 12 pages, 3 figures, presented by S. Friot at the XLV International Conference of Theoretical Physics "Matter to the Deepest", Warsaw, Poland, 17-22 September 2023 (extended version of the contribution to the proceedings published in Acta Physica Polonica B

Multiple Mellin-Barnes integrals and triangulations of point configurations

We present a novel technique for the analytic evaluation of multifold Mellin-Barnes (MB) integrals, which commonly appear in physics, as for instance in the calculations of multi-loop multi-scale Feynman integrals. Our approach is based on triangulating a set of points which can be assigned to a given MB integral, and yields the final analytic results in terms of linear combinations of multiple series, each triangulation allowing the derivation of one of these combinations. When this technique is applied to the computation of Feynman integrals, the involved series are of the (multivariable) hypergeometric type. We implement our method in the Mathematica package MBConicHulls.wl, an already existing software dedicated to the analytic evaluation of multiple MB integrals, based on a recently developed computational approach using intersections of conic hulls. The triangulation method is remarkably faster than the conic hulls approach and can thus be used for the calculation of higher-fold MB integrals as we show here by computing triangulations for highly complicated objects such as the off-shell massless scalar one-loop 15-point Feynman integral whose MB representation has 104 folds. As other applications we show how this technique can provide new results for the off-shell massless conformal hexagon and double box Feynman integrals, as well as for the hard diagram of the two loop hexagon Wilson loop.Comment: 16 pages, 4 figures, code repository https://github.com/SumitBanikGit/MBConicHull

Constant terms in threshold resummation and the quark form factor

We verify to order alpha_s^4 two previously conjectured relations, valid in four dimensions, between constant terms in threshold resummation (for Deep Inelastic Scattering and the Drell-Yan process) and the second logarithmic derivative of the massless quark form factor. The same relations are checked to all orders in the large beta_0 limit; as a byproduct a dispersive representation of the form factor is obtained. These relations allow to compute in a symmetrical way the three-loop resummation coefficients B_3 and D_3 in terms of the three-loop contributions to the virtual diagonal splitting function and to the quark form factor, confirming results obtained in the literature.Comment: 39 pages, no figure; version 2: same content, but improved presentation, with a new section devoted to the variety of resummation procedures; version 3: journal version, where a remark about the all orders validity of the conjecture in the DIS case is reporte

Asymptotics of Feynman Diagrams and The Mellin-Barnes Representation

It is shown that the integral representation of Feynman diagrams in terms of the traditional Feynman parameters, when combined with properties of the Mellin--Barnes representation and the so called {\it converse mapping theorem}, provide a very simple and efficient way to obtain the analytic asymptotic behaviours in both the large and small ratios of mass scales.Comment: References added. This is the version published in Physics Letters