Academic literature on the topic 'Mellin-Barnes technique'

AbstractThe present paper contains three main results. The first is asymptotic expansions of Barnes double zeta-functions, and as a corollary, asymptotic expansions of holomorphic Eisenstein series follow. The second is asymptotic expansions of Shintani double zeta-functions, and the third is the analytic continuation of n-variable multiple zeta-functions (or generalized Euler-Zagier sums). The basic technique of proving those results is the method of using the Mellin-Barnes type of integrals.

Abstract We present a two-loop calculation of the supersymmetric circular Wilson loop in the $$ \mathcal{N} $$ N = 2* super Yang-Mills theory on the four-sphere. We develop an efficient framework for computing contributing Feynman graphs that relies on using the embedding coordinates combined with the Mellin-Barnes techniques for propagator-like integrals on the sphere. Our results exactly match predictions of supersymmetric localization providing a nontrivial consistency check for the latter in non-conformal settings.

Abstract We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.

The present era is one of precision in particle physics. To account for the lacunae in the otherwise successful Standard Model, observables are calculated to high precision in various theoretical models, which are then tested against experimental data to determine whether a given model is realised in nature. In perturbative quantum eld theoretical models, higher order calculations require the evaluation of multi-loop diagrams with multiple mass scales. Although an advanced technology has been developed to evaluate these loop integrals, the majority of techniques are still numerical in nature. In this thesis, we advance one technology that allows for the analytic evaluation of multi-loop diagrams with several mass scales, the Mellin-Barnes (MB) technique, by studying and applying it primarily in the context of three- avoured chiral perturbation theory (SU(3) ChPT). At two loop order, the expressions for the pion, kaon and eta masses and decay constants depend on 'sunset' diagrams, which appear with up to three independent masses, and the analytic evaluation of which provides us the backdrop on which we develop our techniques. The rst part of this work concerns itself with the development of the MB technology and its application to the mathematics of sunset diagrams. We begin by developing an approach that allows one to derive a minimal MB representation of a multi-loop multi-scale integral while retaining straight line contours throughout the derivation process. After reducing the variety of vector and tensor sunsets to a set of four scalar master integrals, this is then applied to evaluate all two mass scale con gurations of the sunset, including (for completeness) those not arising in the ChPT context. The same approach is used thereafter, with appropriate modi cations, to derive various MB representations of the three mass scale integrals appearing in SU(3) ChPT. Each of these integrals is evaluated for all accessible regions of convergence retaining their full dependence arising from dimensional regularization, and in the ! 0 limit for the expressions that converge with physical meson mass values. Formulae are also derived that allow one to expand these integrals to arbitrary order in . The second part of this work focusses on physical applications of the aforementioned results in ChPT. The sunset results are applied to obtain fully analytic expressions for m2 , m2 K, m2 , F , FK and F , which are subsequently truncated appropriately to obtain simpli ed representations that are particularly suitable for tting with lattice QCD data. Such a preliminary lattice t is performed for the expression FK=F to extract values of the low energy constants (LEC) Lr 5, Cr 14 + Cr 15 and Cr 15 + 2Cr 17. We also perform a numerical study of the meson masses and decay constants to examine the relative contributions of their various components, and to investigate their dependence on the values of the LEC. As another application of these analytic expressions, we nd an expansion of F and m2 in the strange quark mass in the isospin limit, and perform the matching of the chiral SU(2) and SU(3) low energy constants. A numerical study on this demonstrates the strong dependence of F on the LEC in the chiral limit. In the nal part of the thesis, we develop and demonstrate two methods of analytic continuation that may be used to obtain results when values of the mass parameters do not allow for convergence of Feynman integrals calculated using MB techniques. We apply the rst technique to the three mass scale sunsets, and therefore obtain the full set of results for these integrals, i.e. we get solutions for the sunsets for all possible values of the mass parameters. The same technique is then applied to analytically continue the results of the most general four mass scale sunset integral to obtain results which converge for physical values of the meson masses. We apply the second method of analytic continuation in a non-ChPT context to demonstrate the general applicability of the methods developed in this work. We rst calculate the complete result of a class of three-loop QED vacuum polarisation contributions arising from non-diagonal avour charged leptons to the g 􀀀 2 of each charged lepton, and then show how one may obtain the expression for the case with an external muon or tau leg from the results of the case of external electron leg by means of analytic continuation.

Noises are usually assumed to be Gaussian so that many existing signal processing techniques can be applied with no worry. However, in many real world natural or man-made systems, noises are usually heavy-tailed. It is increasingly desirable to address the problem of finding an opportune filter function for a given input noise in order to generate a desired output noise. By filtering theory, the probability density function of the output noise can be expressed by the integral of the product of the density of the input noise and the filter function. Adopting Mellin transformation rules, the Mellin transform of the unknown filter is determined by the Mellin transforms of the known density of the input noise and the desired density for the output noise. Finally, after the inversion, the Mellin-Barnes integral representation of the filter function is derived. The method is applied to compute the filter function to convert a Levy noise into a Gaussian noise.