We present the Olsson.wl Mathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F-1 double hypergeometric series using the linear transformations of the Gauss F-2(1) hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D = 6 - 2 epsilon. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summary Program Title: Olsson.wl CPC Library link to program files: https://doi.org/10.17632/gc63xzwzz5.1 Licensing provisions: GNU General Public License v3.0. Programming language: Wolfram Mathematica version 11.3 and beyond. Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus. Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions. Companion package: ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

We present the Olsson.wl Mathematica package which aims to find linear transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson in J. Math. Phys. 5, 420 (1964) in order to derive the analytic continuations of the Appell F_1 double hypergeometric series from the linear transformations of the Gauss _2F_1 hypergeometric function. We provide a brief description of Olsson's method and demonstrate the commands of the package, along with examples. We also provide a companion package, called ROC2.wl and dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl.

Group Theory has become an invaluable tool in the physics community. Despite numerous introductory books, the subject remains challenging for beginners. Mathematica has emerged as a popular tool for research and education, offering various packages and built-in tools for Group Theory. However, these resources are often too scattered for effective educational use. This work aims to provide a comprehensive source to help beginning students grasp Group Theory concepts and their applications from a physicist's perspective, while also building familiarity with symbolic language. We present several example notebooks that succinctly cover well-known theories and demonstrate specific concepts, which can be easily adapted for educational purposes. We provide basic examples on finite, compact and non-compact groups, and motivate the use of these concepts in solving physics problems such as addition of angular momenta, modelling a system of qubits and the description of spacetime transformations.

We present new closed-form expressions for certain improper integrals of Mathematical Physics such as certain Ising, Box, and Associated integrals. The techniques we employ here include (a) the Method of Brackets and its modifications and suitable extensions to obtain the Mellin–Barnes representation. (b) The evaluation of the resulting Mellin–Barnes representations via the recently discovered Conic Hull method via the automated package MBConichulls.wl. Finally, the analytic continuations of these series solutions are then produced using the automated package Olsson.wl, based on the method of Olsson. Thus, combining all these recent advances allows for closed-form evaluation of the hitherto unknown B_3(s) , B_4(s) and related integrals in terms of multi-variable hypergeometric functions. Along the way, we also discuss certain complications while using the Original Method of Brackets for these evaluations and how to rectify them. The interesting case of C_5,k is also studied. It is not yet fully resolved for the reasons we discuss in this paper.

Quantum field theory was established about a hundred years ago and is the result of combining the principles of quantum mechanics and the special theory of relativity. Historically, some of the founding fathers were at high unease as intermediate calculations led to infinities. A different approach, sometimes called the analytic S-matrix or the bootstrap, relying on basic principles such as causality and unitarity, was put forward. A resurgence of these fields has taken place now since their power has not been tapped to the fullest, and are the most active fields of research in theoretical physics and spurring developments in computer algebra. In this article, after recalling the history of this field, we highlight some of the past and recent contributions from India.

Feynman integrals at any order of perturbation, in the Lee-Pomeransky representation, could be realised as a subset of Euler-Mellin integrals. Such integrals satisfy the Gelfand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. In an ongoing collaboration, we automate the derivation of the associated GKZ system for a given Feynman diagram from either its Lee-Pomeransky representation or its Mellin-Barnes representation. We also present the automation of two mathematically equivalent techniques, namely the Gröbner deformation method and the method of triangulations of point configurations to solve this system. We have implemented these in the Mathematica package FeynGKZ [1], which is the first public proof-of-concept software that evaluates Feynman integrals in the GKZ framework.

The transformation theory of the Appell $F_2(a,b_1,b_2;c_1,c_2;x,y)$ double hypergeometric function is used to obtain a set of series representations of $F_2$ which provide an efficient way to evaluate $F_2$ for real values of its arguments $x$ and $y$ and generic complex values of its parameters $a,b_1, b_2, c_1$ and $c_2$ (i.e. in the nonlogarithmic case). This study rests on a classical approach where the usual double series representation of $F_2$ and other double hypergeometric series that appear in the intermediate steps of the calculations are written as infinite sums of one variable hypergeometric series, such as the Gauss $_2F_1$ or the $_3F_2$, various linear transformations of the latter being then applied to derive known and new formulas. Using the three well-known Euler transformations of $F_2$ on these results allows us to obtain a total of 44 series which form the basis of the Mathematica package AppellF2, dedicated to the evaluation of $F_2$. A brief description of the package and of the numerical analysis that we have performed to test it are also presented.

The pseudoscalar particles pions, kaons and the $η$-particle are considerably lighter than the other hadrons such as protons or neutrons. Their lightness was understood as a consequence of approximate chiral symmetry breaking. This led to current algebra, a way to express the relations imposed by the symmetry breaking. It was realized by Weinberg that because of their low mass, it is possible to formulate a purely pionic (effective) field theory at experimental energies, which carries all information on the (non-perturbative) dynamics, symmetries, and their spontaneous breaking of quantum chromodynamics (QCD) and allows for systematic calculations of observables. In this review, we trace these developments and present recent activities in this field. We make the connection to other effective theories, more generally introduced by Wilson, as approximate field theories at low energies. Indeed, principles and paradigms introduced first for pions have become ubiquitous in particle physics and the standard model. Lastly, we turn to the latest development where the present (fundamental) standard model itself is considered as an effective field theory of a - yet to be formulated - even more fundamental theory. We also discuss important techniques that were developed in order to turn chiral perturbation theory into a predictive framework and briefly review some connections between lattice QCD and chiral perturbation theory (ChPT).

The method of brackets (MoB) is a technique used to compute definite integrals, that has its origin in the negative dimensional integration method. It was originally proposed for the evaluation of Feynman integrals for which, when applicable, it gives the results in terms of combinations of (multiple) series. We focus here on some of the limitations of MoB and address them by studying the Mellin-Barnes (MB) representation technique. There has been significant progress recently in the study of the latter due to the development of a new computational approach based on conic hulls [see B. Ananthanarayan et al., Multiple Series Representations of N-fold Mellin-Barnes Integrals, Phys. Rev. Lett. 127, 151601 (2021)]. The comparison between the two methods helps to understand the limitations of the MoB, in particular when termwise divergent series appear. As a consequence, the MB technique is found to be superior over MoB for two major reasons: 1. the selection of the sets of series that form a series representation for a given integral follows, in the MB approach, from specific intersections of conic hulls, which, in contrast to MoB, does not need any convergence analysis of the involved series, and 2. MB can be used to evaluate resonant (i.e. logarithmic) cases where MoB fails due to the appearance of termwise divergent series. Furthermore, we show that the recently added rule 5 of MoB naturally emerges as a consequence of the residue theorem in the context of MB.

Motivated by the foundational work of Tarasov, who pointed out that the algebraic relations of the type considered here can lead to functional reduction of Feynman integrals, we suitably modify the original method to be able to implement and automatize it and present a MATHEMATICA package AlgRel.wl. The purpose of this package is to help derive the algebraic relations with arbitrary kinematic quantities, for the product of propagators. Under specific choices of the arbitrary parameters that appear in these relations, we can write the original integral with all massive propagators in general, as a sum of integrals which have fewer massive propagators. The resulting integrals are of reduced complexity for computational purposes. For the one-loop cases, with all different and non-zero masses, this would result in integrals with one massive propagator. We also devise a strategy so that the method can also be applied to higher-loop integrals. We demonstrate the procedure and the results obtained using the package for various one-loop and higher-loop examples. Due to the fact that the Feynman integrals are intimately related to the hypergeometric functions, a useful consequence of these algebraic relations is in deriving the sets of non-trivial reduction formulae. We present various such reduction formulae and further discuss how, more such formulae can be obtained apart from the ones described here. The AlgRel.wl package and an example notebook Examples.nb can be found at GitHub. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). Funded by SCOAP3.

In the Lee-Pomeransky representation, Feynman integrals can be identified as a subset of Euler-Mellin integrals, which are known to satisfy Gel'fand-Kapranov-Zelevinsky (GKZ) system of partial differential equations. Here we present an automated package to derive the associated GKZ system for a given Feynman diagram and solve it in terms of hypergeometric functions using two equivalent algorithms, namely the triangulation method and the Gröbner deformation method. We present our code in the form of a Mathematica package FeynGKZ.wl which requires the softwares polymake, Macaulay2 and TOPCOM, and the packages AMBRE and Olsson.wl as dependencies. As applications of the package, we find series solutions to the GKZ systems of several one-loop and two-loop Feynman integrals. These are included in the file Examples.nb that can be downloaded along with the package from https://github.com/anant-group/FeynGKZ.

The analytic evaluation of multi-scale Feynman integrals is difficult due to the presence of various scales of the problem. When exact calculation is very difficult or impossible, systematic approximations may help. The strategy of expansion by regions is a useful method for obtaining the asymptotic analysis of multi-scale Feynman integrals. In this talk, we present a novel method for the identification of regions associated with multi-scale Feynman integrals.

In this special issue being brought in the centenary year of the birth of Yoichiro Nambu, we exemplify on his discovery of spontaneous symmetry breaking in elementary particle physics, and review precision pion physics in the present era. The notion of spontaneous symmetry breaking in elementary particle physics was introduced by Nambu, and found a realization in the strong interaction sector. It allows one to view the pions as the approximate Nambu–Goldstone bosons of spontaneously broken axial-vector symmetries associated with the (near) masslessness of quarks. Inspired by the phenomenon of superconductivity of condensed matter physics, Nambu found this application in a remarkable tour de force. Nambu’s work in collaboration with G. Jona-Lasinio gave a dynamical model where such pions may arise. Pions today play the role of being sensitive probes of the ground state of quantum chromodynamics, the Lagrangian field theory of the strong interactions with (confined) quark and gluon degrees of freedom, and whose ground state spontaneously breaks the approximate chiral symmetry. The presence of non-zero quark masses renders the symmetries approximate, and yet the properties of the low-energy sector can both be described and measured at high precision both in experiment and on the lattice. Notable physical quantities include the neutral pion lifetime and pion scattering lengths. An important role of pions is their contribution to the hadronic radiative corrections to the anomalous magnetic moment of the muon, which is being measured at high precision at Fermilab. We review some of the important aspects of the state of the art. We also say some words about the outstanding contributions of the recently departed Murray Gell-Mann who was a pioneer in the field initiated by Nambu.

We discuss the prospects for improving the precision on the hadronic corrections to the anomalous magnetic moment of the muon, and the plans of the Muon g-2 Theory Initiative to update the Standard Model prediction.

We determine the strange quark mass (m(s)) and quark mixing element broken vertical bar V-us broken vertical bar, and their joint determination from the Cabibbo suppressed hadronic tau decays in various perturbative schemes. We improve this analysis compared to the previous analysis based on the optimal renormalization or the renormalization group summed perturbation theory (RGSPT) scheme by replacing the theoretical longitudinal contributions with phenomenological parametrization; the RGSPT coefficients are used for the dimension-4 Adler functions. The improved analysis results in the extraction of m(s) (2 GeV) = 98 +/- 19MeV and vertical bar V-us broken vertical bar = 0.2191 +/- 0.0043 from the RGSPT scheme.

We determine the strange quark mass (m_s) and quark mixing element | V_us|, and their joint determination from the Cabibbo suppressed hadronic τ decays in various perturbative schemes. Compared to the previous analysis based on the optimal renormalization or the renormalization group summed perturbation theory (RGSPT) scheme, we have improved this analysis by replacing the theoretical longitudinal contributions with phenomenological parametrization, and the RGSPT coefficients are used for the dimension-4 Adler functions. The improved analysis results in the extraction of m_s(2 GeV)=98±19 MeV and | V_us|=0.2191±0.0043 from the RGSPT scheme.

The Method of Brackets (MoB) is a technique used to compute definite integrals, that has its origin in the negative dimensional integration method. It was originally proposed for the evaluation of Feynman integrals for which, when applicable, it gives the results in terms of combinations of (multiple) series. We focus here on some of the limitations of MoB and address them by studying the Mellin-Barnes (MB) representation technique. There has been significant process recently in the study of the latter due to the development of a new computational approach based on conic hulls (see Phys. Rev. Lett. 127, 151601 (2021)). The comparison between the two methods helps to understand the limitations of the MoB, in particular when termwise divergent series appear. As a consequence, the MB technique is found to be superior over MoB for two major reasons: 1. the selection of the sets of series that form a series representation for a given integral follows, in the MB approach, from specific intersections of conic hulls, which, in contrast to MoB, does not need any convergence analysis of the involved series, and 2. MB can be used to evaluate resonant (i.e. logarithmic) cases where MoB fails due to the appearance of termwise divergent series. Furthermore, we show that the recently added Rule 5 of MoB naturally emerges as a consequence of the residue theorem in the context of MB.

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 three-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.