We evaluate, analytically, a specific class of eighth--order and tenth--order QED contributions to the anomalous magnetic moment of the muon. They are generated by Feynman diagrams involving lowest order vacuum polarization insertions of leptons $l=e,μ$, and $τ$. The results are given in the form of analytic expansions in terms of the mass ratios $m_e/m_μ$ and $m_μ/m_τ$. We compute as many terms as required by the error induced by the present experimental uncertainty on the lepton masses. We show how the Mellin--Barnes integral representation of Feynman parametric integrals allows for an easy analytic evaluation of as many terms as wanted in these expansions and how its underlying algebraic structure generalizes the standard renormalization group properties. We also discuss the generalization of this technique to the case where two independent mass ratios appear. Comparison with previous numerical and analytic evaluations made in the literature, whenever pertinent, are also made.
In this paper, we present a new approach to the construction of Mellin-Barnes representations for Feynman integrals inspired by the Method of Brackets. The novel technique is helpful to lower the dimensionality of Mellin-Barnes representations in complicated cases, some examples are given.
Feynman integrals may be represented by the Mathematica packages AMBRE and MB as multiple Mellin-Barnes integrals. With the Mathematica package MBsums these Mellin-Barnes integrals are transformed into multiple sums.
We show how to evaluate one-dimensional Minkowski-region Mellin-Barnes representations arising from massive loop integrals, by modifying the contours of integration. We implement an exact solution to the differential equation determining the contours of stationary phase. We also present several simple approximations to these contours. Our approach points the way to more efficient computations of massless and massive Mellin-Barnes integrals in both Euclidean and Minkowski regions.
We calculate massive 5-propagator 2-loop integrals for operator matrix elements in the light-cone expansion, using Mellin-Barnes techniques and representations through generalized hypergeometric functions.
One of the two existing strategies of resolving singularities of multifold Mellin-Barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modified form. The corresponding algorithm is implemented as a Mathematica code MBresolve.m
The status of numerical evaluations of Mellin-Barnes integrals is discussed, in particular, the application of the quasi-Monte Carlo integration package QMC to the efficient calculation of multi-dimensional integrals.
We introduce techniques to treat numerically Mellin-Barnes integrals in physical regions, which arise in the need of the computation of Feynman integrals for the electroweak two-loop corrections to the pseudo observables at the Z-boson resonance.
The status of analytical evaluation of double and triple box diagrams is characterized. The method of Mellin-Barnes representation as a tool to evaluate master integrals in these problems is advocated. New MB representations for massive on-shell double boxes with general powers of propagators are presented.
Feynman diagrams may be evaluated by Mellin-Barnes representations of their Feynman parameter integrals in d=4-2\eps dimensions. Recently, the Mathematica toolkit AMBRE has been developed for the automatic derivation of such representations with a loop-by-loop approach. We describe the package and exemplify its use with the \eps-expansion of the massive one-loop QED vertex function.
The construction of Mellin-Barnes (MB) representations for non-planar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluation of MB representations for multiloop scalar and tensor Feynman integrals through infinite sums, preferably with analytic solutions. We shortly describe a strategy of further algebraic summation.
We show that at one-loop order, negative-dimensional, Mellin-Barnes' (MB) and Feynman parametrization (FP) approaches to Feynman loop integrals calculations are equivalent. Starting with a generating functional, for two and then for $N$-point scalar integrals we show how to reobtain MB results, using negative-dimensional and FP techniques. The $N-$point result is valid for different masses, arbitrary exponents of propagators and dimension.
Mellin-Barnes (MB) techniques applied to integrals emerging in particle physics perturbative calculations are summarized. New versions of AMBRE packages which construct planar and nonplanar MB representations are shortly discussed. The numerical package MBnumerics.m is presented for the first time which is able to calculate with a high precision multidimensional MB integrals in Minkowskian regions. Examples are given for massive vertex integrals which include threshold effects and several scale parameters.
The Mellin-Barnes technique to evaluate master integrals and the algorithm called FIRE to solve IBP relations with the help of Groebner bases are briefly reviewed. In FIRE, an extension of the classical Buchberger algorithm to construct Groebner bases is combined with the well-known Laporta algorithm. It is explained how both techniques are used when evaluating the three-loop correction to the static QCD quark potential. First results are presented: the coefficients of n_l^3 and n_l^2, where n_l is the number of light quarks.
An improved method is presented for the numerical evaluation of multi-loop integrals in dimensional regularization. The technique is based on Mellin-Barnes representations, which have been used earlier to develop algorithms for the extraction of ultraviolet and infrared divergencies. The coefficients of these singularities and the non-singular part can be integrated numerically. However, the numerical integration often does not converge for diagrams with massive propagators and physical branch cuts. In this work, several steps are proposed which substantially improve the behavior of the numeri…
A new approximation is proposed for the contour of the stationary phase of the Mellin--Barnes integrals in the case of its finite asymptotic behavior as ${\rm Re} z\to -\infty$. The efficiency of application of the proposed contour and the quadratic approximation to the contour of the stationary phase is compared by the example of the inverse Mellin transform for the structure function $F_3$. It is shown that, although for a small number of terms $N$ in quadrature formulas used to calculate integrals along these contours, the quadratic contour is more efficient, but for $N>20$ the asymptotic s…
The Mathematica toolkit AMBRE derives Mellin-Barnes (MB) representations for Feynman integrals in d=4-2eps dimensions. It may be applied for tadpoles as well as for multi-leg multi-loop scalar and tensor integrals. AMBRE uses a loop-by-loop approach and aims at lowest dimensions of the final MB representations. The present version of AMBRE works fine for planar Feynman diagrams. The output may be further processed by the package MB for the determination of its singularity structure in eps. The AMBRE package contains various sample applications for Feynman integrals with up to six external part…
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.
Multi-loop integrals can be evaluated numerically using Mellin-Barnes representations. Here this technique is applied to the calculation of electroweak two-loop correction with closed fermion loops for two observables: the effective weak mixing angle for bottom quarks, sin^2θ(eff,bb), and the branching ratio of the Z boson into bottom quarks, R(b). Good agreement with a previous result for sin^2θ(eff,bb) is found. The result for R(b) is new, and a simple parametrization formula is provided which approximates the full result within integration errors.
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.
We discuss the Mellin-Barnes representation of complex multidimensional integrals. Experiments frontiered by the High-Luminosity Large Hadron Collider at CERN and future collider projects demand the development of computational methods to achieve the theoretical precision required by experimental setups. In this regard, performing higher-order calculations in perturbative quantum field theory is of paramount importance. The Mellin-Barnes integrals technique has been successfully applied to the analytic and numerical analysis of integrals connected with virtual and real higher-order perturbativ…
During the last several years remarkable progress has been made in numerical calculations of dimensionally regulated multi-loop Feynman diagrams using Mellin-Barnes (MB) representations. The bottlenecks were non-planar diagrams and Minkowskian kinematics. The method has been proved to work in highly non-trivial physical application (two-loop electroweak bosonic corrections to the $Z \to b \bar{b}$ decay), and cross-checked with the sector decomposition (SD) approach. In fact, both approaches have their pros and cons. In calculation of multidimensional integrals, depending on masses and scales …
We present analytic expressions of all integrals required to complete the explicit evaluation of the real-virtual integrated counterterms needed to define a recently proposed subtraction scheme for jet cross sections at next-to-next-to-leading order in QCD. We use the Mellin-Barnes representation of these integrals in $4-2ε$ dimensions to obtain the coefficients of their Laurent expansions around $ε=0$. These coefficients are given by linear combinations of multidimensional Mellin-Barnes integrals. We compute the coefficients of such expansions in $ε$ both numerically and analytically by compl…
A number of irreducible master integrals for L-loop sunrise-type and bubble Feynman diagrams with generic values of masses and external momenta are explicitly evaluated via Mellin-Barnes representation.
I describe a package written in MATHEMATICA that automatizes typical operations performed during evaluation of Feynman graphs with Mellin-Barnes (MB) techniques. The main procedure allows to analytically continue a MB integral in a given parameter without any intervention from the user and thus to resolve the singularity structure in this parameter. The package can also perform numerical integrations at specified kinematic points, as long as the integrands have satisfactory convergence properties. I demonstrate that, at least in the case of massive graphs in the physical region, the convergenc…
The dual parametrization and the Mellin-Barnes integral approach represent two frameworks for handling the double partial wave expansion of generalized parton distributions (GPDs) in the conformal partial waves and in the $t$-channel ${\rm SO}(3)$ partial waves. Within the dual parametrization framework, GPDs are represented as integral convolutions of forward-like functions whose Mellin moments generate the conformal moments of GPDs. The Mellin-Barnes integral approach is based on the analytic continuation of the GPD conformal moments to the complex values of the conformal spin. GPDs are then…
The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin-Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples.
It is shown that with a precise determination of a few derivatives of the hadronic vacuum polarization (HVP) self-energy function $Π(Q^2)$ at $Q^2=0$, from lattice QCD (LQCD) or from a dedicated low-energy experiment, one can obtain an evaluation of the lowest order HVP contribution to the anomalous magnetic moment of the muon $a_μ^{\rm HVP}$ with an accuracy comparable to the one reached using the $e^+ e^-$ annihilation cross-section into hadrons. The technique of Mellin-Barnes approximants (MBa) that we propose is illustrated in detail with the example of the two loop vacuum polarization fun…
Multiple Mellin-Barnes integrals are often used for perturbative calculations in particle physics. In this context, the evaluation of such objects may be performed through residues calculations which lead to their expression as multiple series in powers and logarithms of the parameters involved in the problem under consideration. However, in most of the cases, several series representations exist for a given integral. They converge in different regions of values of the parameters, and it is not obvious to obtain them. For twofold integrals we present a method which allows to derive straightfor…
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 $ϕ^4$ 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 impro…
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 $ε$-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 $ε$ of MB repres…
The contributions to the anomalous magnetic moment of the lepton $L$ ($L=e\ , μ$ or $τ$) generated by a specific class of QED diagrams are evaluated analytically up to the eighth order of the electromagnetic coupling constant. The considered class of the Feynman diagrams involves the vacuum polarization insertions into the electromagnetic vertex of the lepton $L$ up to three closed lepton loops. The corresponding analytical expressions are obtained as functions of the mass ratios $r=m_l/m_L$ in the whole region $0 < r < \infty$. Our consideration is based on a combined use of the dispersion re…
Fermionic Bubble Loop in Cosmological Collider Revisited: Exact signals from spectral and Mellin-Barnes methods
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（RIKEN跨学科理论与数学科学中心（iTHEMS））；Department of Physics, Tsinghua University（清华大学物理系）；Cosmology, Gravity and Astroparticle Physics Group, Center for Theoretical Physics of the Universe, Institute for Basic Science（基础科学研究院宇宙理论物理中心宇宙学、引力与高能天体物理研究组）；Department of Physics, Institute of Science Tokyo（东京科学大学物理系）；Department of Physics and IPAP, Yonsei University（延世大学物理系及物理学与应用物理研究所）
The sixth order QED radiative corrections to lepton anomalies due to the fourth order vacuum polarization insertions within the Mellin-Barnes representation
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 is 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 nonre…
We address the problem of evaluation of multiloop Feynman integrals by means of their Mellin-Barnes representation. After a brief overview of available capabilities though open source toolkits and their application in various circumstances, we introduce a new code MBcreate which allows one to automatically deduce a concise Mellin-Barnes representation for a given parametric integral. A thorough discussion of its implementation and use is provided.
The method of Mellin-Barnes representation is used to calculate dimensionally regularized massive on-shell double box Feynman diagrams contributing to Bhabha scattering at two loops.
We calculate the O($\eps$)--term of the two--loop massive operator matrix elements for twist 2--operators, which contribute to the heavy flavour Wilson coefficients in unpolarized deep--inelastic scattering in the asymptotic limit $Q^2 \gg m^2.$ Our calculation was performed in Mellin space using Mellin--Barnes integrals and generalized hypergeometric functions. The O($\eps$)--term contributes in the renormalization at 3--loop order.
Some recent results on evaluating Feynman integrals are reviewed. The status of the method based on Mellin-Barnes representation as a powerful tool to evaluate individual Feynman integrals is characterized. A new method based on Groebner bases to solve integration by parts relations in an automatic way is described.
For the investigation of higher order Feynman integrals, potentially with tensor structure, it is highly desirable to have numerical methods and automated tools for dedicated, but sufficiently 'simple' numerical approaches. We elaborate two algorithms for this purpose which may be applied in the Euclidean kinematical region and in d=4-2*epsilon dimensions. One method uses Mellin-Barnes representations for the Feynman parameter representation of multi-loop Feynman integrals with arbitrary tensor rank. Our Mathematica package AMBRE has been extended for that purpose, and together with the packag…
Mellin-Barnes and sector decomposition methods are used to evaluate tensorial Feynman diagrams in the Euclidean kinematical region. Few software packages are shortly described and few examples demonstrate their use.
Mellin-Barnes (MB) representations have become a widely used tool for the evaluation of Feynman loop integrals appearing in perturbative calculations of quantum field theory. Some of the MB integrals may be solved analytically in closed form with the help of the two Barnes lemmas which have been known in mathematics already for one century. The original proofs of these lemmas solve the integrals by taking infinite series of residues and summing these up via hypergeometric functions. This paper presents new, elegant proofs for the Barnes lemmas which only rely on the well-known basic identity o…
Using the new non-analytic reconstruction method obtained from Mellin-Barnes properties, one can extract the value $m_c(\bar{\text{MS}}) = 1.12 \pm 0.08 \;\; \text{GeV}$ from experimental data of the radiation-corrected measured hadronic cross section to the calculated lowest-order cross section for muon pair production in the heavy-quark approximation.
I present an algorithm based on sector decomposition and Mellin-Barnes techniques to power expand Feynman integrals. The coefficients of this expansion are given in terms of finite integrals that can be calculated numerically. I show in an example the benefit of this method for getting the full analytic power expansion from differential equations by providing the correct ansatz for the solution. For method of regions the presented algorithm provides a numerical check, which is independent from any power counting argument.
