Mellin-Barnes Integrals Research Articles

Multiple Mellin-Barnes integrals and triangulations of point configurations

Mellin-Barnes (MB) integrals are a well-known type of integrals appearing in diverse areas of mathematics and physics, such as in the theory of hypergeometric functions, asymptotics, quantum field theory, solid-state physics, etc. Although MB integrals have been studied for more than a century, it is only recently that, due to a remarkable connection found with conic hulls, N-fold MB integrals can be computed analytically for N>2 in a systematic way. In this article, we present an alternative novel technique by unveiling a new connection between triangulations of point configurations and MB integrals, to compute the latter. To make it ready to use, we have implemented our new method in the package oniculls.wl, an already existing software dedicated to the analytic evaluation of MB integrals using conic hulls. The triangulation method is remarkably faster than the conic hull approach and can thus be used for the calculation of higher-fold MB integrals, as we show here by testing our code on the case of the off-shell massless scalar one-loop N-point Feynman integral up to N=15, for which the MB representation has 104 folds. Among other examples of applications, we present new simpler solutions for the off-shell one-loop massless conformal hexagon and two-loop double-box Feynman integrals, as well as for some complicated 8-fold MB integrals contributing to the hard diagram of the two-loop hexagon Wilson loop in general kinematics. Published by the American Physical Society 2024

Hadronic Light-by-Light Corrections to the Muon Anomalous Magnetic Moment

We review the hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment. Upcoming measurements will reduce the experimental uncertainty of this observable by a factor of four; therefore, the theoretical precision must improve accordingly to fully harness such an experimental breakthrough. With regards to the HLbL contribution, this implies a study of the high-energy intermediate states that are neglected in dispersive estimates. We focus on the maximally symmetric high-energy regime and in-quark loop approximation of perturbation theory, following the method of the OPE with background fields proposed by Bijnens et al. in 2019 and 2020. We confirm their results regarding the contributions to the muon g−2. For this, we use an alternative computational method based on a reduction in the full quark loop amplitude, instead of projecting on a supposedly complete system of tensor structures motivated by first principles. Concerning scalar coefficients, mass corrections have been obtained by hypergeometric representations of Mellin–Barnes integrals. By our technique, the completeness of such kinematic singularity/zero-free tensor decomposition of the HLbL amplitude is explicitly checked.

Geometrical Methods for the Analytic Evaluation of Multiple Mellin–Barnes Integrals

Geometrical Methods for the Analytic Evaluation of Multiple Mellin–Barnes Integrals

Mellin–Barnes Integrals Related to the Lie Algebra u(N)

Mellin–Barnes Integrals Related to the Lie Algebra u(N)

To the Properties of One Fox Function

В работе рассматривается частный случай специальной функции Фокса с четырьмя параметрами, которая возникает в теории краевых задач для параболических уравнений c оператором Бесселя и дробной производной по времени. Целью исследования является получение некоторых рекуррентных соотношений, формул дифференцирования и интегрального преобразования рассматриваемой функции. При получении результатов работы в основном используется представление рассматриваемой функции через интеграл Меллина–Барнса. Также используются её асимптотические разложения при большом и малом значениях аргумента. С помощью указанного интегрального представления и некоторых известных формул для гамма-функции Эйлера, получены рекуррентные соотношения, связывающие функции с разными параметрами, а также функцию с её производной первого порядка. Получена формула дифференцирования n-го порядка. Исследуется несобственный интеграл первого рода, который содержит рассматриваемую функцию с двумя зависимыми параметрами из четырёх. Показывается, что этот несобственный интеграл может быть записан в терминах известной специальной функции Макдональда. При частных значениях параметров рассматриваемой в работе функции получаются некоторые известные элементарные и специальные функции. Результаты работы носят теоретический характер и будут полезны при исследовании краевых задач для вырождающихся параболических уравнений с производными дробного порядка по времени. The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin–Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the nth order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.

Multiple Mellin-Barnes integrals with straight contours

We show how the conic hull method, recently developed for the analytic and non-iterative evaluation of multifold Mellin-Barnes (MB) integrals, can be extended to the case where these integrals have straight contours of integration parallel to the imaginary axes in the complex planes of the integration variables. MB integrals of this class appear, for instance, when one computes the $\epsilon$-expansion of dimensionally regularized Feynman integrals, as a result of the application of one of the two main strategies (called A and B in the literature) used to resolve the singularities in $\epsilon$ of MB representations. We upgrade the Mathematica package MBConicHulls.wl which can now be used to obtain multivariable series representations of multifold MB integrals with arbitrary straight contours, providing an efficient tool for the automatic computation of such integrals. This new feature of the package is presented, along with an example of application by calculating the $\epsilon$-expansion of the dimensionally regularized massless one-loop pentagon integral in general kinematics and $D=4-2\epsilon$.

Wave Function for GL(n,ℝ) Hyperbolic Sutherland Model II. Dual Hamiltonians

Abstract Recently, we found Mellin–Barnes integrals, representing the wave function for $GL(n,{\mathbb {R}})$ hyperbolic Sutherland model. In present paper, we establish bispectral properties of this wave function with respect to dual Ruijesenaars–Macdonald operators.

Wave Function for GL(n, ℝ) Hyperbolic Sutherland Model

Abstract We obtain certain Mellin–Barnes integrals that present wave functions for $GL(n,{\mathbb {R}})$ hyperbolic Sutherland model with arbitrary positive coupling constant.

Sinc Numeric Methods for Fox-H, Aleph (ℵ), and Saxena-I Functions

The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on the functions in question. The approach’s pillars are the concept of a Mellin-Barnes integral and the Mellin representation of the given function. A Sinc quadrature is used in conjunction with a Sinc approximation of the function to achieve the numerical approximation of the Mellin-Barnes integral. The method converges exponentially and can handle endpoint singularities. We give numerical representations of the Aleph ℵ and Saxena I functions for the first time.

On the mixed-twist construction and monodromy of associated Picard–Fuchs systems

We use the mixed-twist construction of Doran and Malmendier to obtain a multi-parameter family of K3 surfaces of Picard rank $\rho \ge 16$. Upon identifying a particular Jacobian elliptic fibration on its general member, we determine the lattice polarization and the Picard-Fuchs system for the family. We construct a sequence of restrictions that lead to extensions of the polarization by two-elementary lattices. We show that the Picard-Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. Second, for the one-parameter mirror families of deformed Fermat hypersurfaces we show that the mixed-twist construction produces a non-resonant GKZ system for which a basis of solutions in the form of absolutely convergent Mellin-Barnes integrals exists whose monodromy we compute explicitly.

On the Laplace transform of the lognormal distribution: Analytic continuation and series approximations

We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal distribution are provided, one of which takes the form of a Mellin–Barnes integral. As a corollary, we obtain an integral expression for the characteristic function. We present two approximations for the Laplace transform of the lognormal distribution, both valid in ℂ∖(−∞,0]. In the last section, we discuss how one may use our results to compute the density of a sum of independent lognormal random variables.

Multiple Series Representations of N-fold Mellin-Barnes Integrals.

Mellin-Barnes (MB) integrals are well-known objects appearing in many branches of mathematics and physics, ranging from hypergeometric functions theory to quantum field theory, solid-state physics, asymptotic theory, etc. Although MB integrals have been studied for more than one century, until now there has been no systematic computational technique of the multiple series representations of N-fold MB integrals for N>2. Relying on a simple geometrical analysis based on conic hulls, we show here a solution to this important problem. Our method can be applied to resonant (i.e., logarithmic) and nonresonant cases and, depending on the form of the MB integrand, it gives rise to convergent series representations or diverging asymptotic ones. When convergent series are obtained, the method also allows, in general, the determination of a single "master series" for each series representation, which considerably simplifies convergence studies and/or numerical checks. We provide, along with this Letter, a Mathematica implementation of our technique with examples of applications. Among them, we present the first evaluation of the hexagon and double box conformal Feynman integrals with unit propagator powers.

Massive one-loop conformal Feynman integrals and quadratic transformations of multiple hypergeometric series

The computational technique of $N$-fold Mellin-Barnes (MB) integrals, presented in a companion paper by the same authors, is used to derive sets of series representations of the massive one-loop conformal 3-point Feynman integral in various configurations. This shows the great simplicity and efficiency of the method in nonresonant cases (generic propagator powers) as well as some of its subtleties in the resonant ones (for unit propagator powers). We confirm certain results in the physics and mathematics literature and provide many new results, some of them dealing with the more general massive one-loop conformal $n$-point case. In particular, we prove two recent conjectures that give the massive one-loop conformal $n$-point integral (for generic propagator powers) in terms of multiple hypergeometric series. We show how these conjectures, that were deduced from a Yangian bootstrap analysis, are related by a tower of new quadratic transformations in Hypergeometric Functions Theory. Finally, we also use our MB method to identify spurious contributions that can arise in the Yangian approach.

Fundamental solution of a multi-dimensional distributed order fractional diffusion equation

In this paper, we consider the multi-dimensional distributed order fractional diffusion equation in the whole space, the half space and the corridor space. We use the Titchmarsh theorem to show that the fundamental solutions are expressed in terms of the Laplace-type integrals and can be applied for the Gauss–Laguerre quadrature. The mathematical identities and their relationships for the fundamental solutions in different dimensions are also discussed, and the alternative representation is given in terms of the Mellin–Barnes integrals for the fundamental solution in the whole space.

Mellin-Barnes presentations for Whittaker wave functions

We obtain certain Mellin-Barnes integrals which present Whittaker wave functions related to classical real split forms of simple complex Lie groups.

Matrix model generating function for quantum weighted Hurwitz numbers

The KP -function of hypergeometric type serving as generating function for quantum weighted Hurwitz numbers is used to compute the Baker function and the corresponding adapted basis elements, expressed as absolutely convergent Laurent series in the spectral parameter. These are equivalently expressed as Mellin–Barnes integrals, analogously to Meijer G-functions, but with an infinite product of -functions as integral kernel. A matrix model representation is derived for the -function evaluated at trace invariants of an externally coupled matrix.

Hypergeometric Series and the Mellin-Barnes Integrals for Zeros of a System of Laurent Polynomials

In the article we present a criterion for convergence of the Mellin-Barnes integral for zeros of a system of Laurent polynomials. Also we give a hypergeometric series for these zeros

Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation

This paper is devoted to an in deep investigation of the first fundamental solution to the linear multi-dimensional space-time-fractional diffusion-wave equation. This equation is obtained from the diffusion equation by replacing the first order time-deri\-va\-ti\-ve by the Caputo fractional derivative of order $\beta, 0 <\beta \leq 2$ and the Laplace operator by the fractional Laplacian $(-\Delta)^{\frac\alpha 2}$ with $0<\alpha \leq 2$. First, a representation of the fundamental solution in form of a Mellin-Barnes integral is deduced by employing the technique of the Mellin integral transform. This representation is then used for establishing of several subordination formulas that connect the fundamental solutions for different values of the fractional derivatives $\alpha$ and $\beta$. We also discuss some new cases of completely monotone functions and probability density functions that are expressed in terms of the Mittag-Leffler function, the Wright function, and the generalized Wright function.

High-energy expansion of two-loop massive four-point diagrams

We apply the method of regions to the massive two-loop integrals appearing in the Higgs pair production cross section at the next-to-leading order, in the high energy limit. For the non-planar integrals, a subtle problem arises because of the indefinite sign of the second Symanzik polynomial. We solve this problem by performing an analytic continuation of the Mandelstam variables such that the second Symanzik polynomial has a definite sign. Furthermore, we formulate the procedure of applying the method of regions systematically. As a by-product of the analytic continuation of the Mandelstam variables, we obtain crossing relations between integrals in a simple and systematic way. In our formulation, a concept of “template integral” is introduced, which represents and controls the contribution of each region. All of the template integrals needed in the computation of the Higgs pair production at the next-to-leading order are given explicitly. We also develop techniques to solve Mellin-Barnes integrals, and show them in detail. Although most of the calculation is shown for the concrete example of the Higgs pair production process, the application to other similar processes is straightforward, and we anticipate that our method can be useful also for other cases.

High-precision numerical estimates of the Mellin-Barnes integrals for the structure functions based on the stationary phase contour

We present a recipe for constructing the effcient contour which allows one to calculate with high accuracy the Mellin-Barnes integrals, in particular, for the F3 structure function written in terms of its Mellin moments. We have demonstrated that the contour of the stationary phase arising for the F3 structure function tends to the finite limit as Re(z) → –∞. We show that the Q2 evolution of the structure function can be represented as an integral over the contour of the stationary phase within the framework of the Picard-Lefschetz theory. The universality of the asymptotic contour of the stationary phase defined at some fixed value of the momentum transfer square $Q_{0}^{2}$ for calculations with any Q2 is shown.