arXiv subjects

A. I. Onishchenko

Publications and source records attributed to A. I. Onishchenko.

At least 19 recordsLinked to original sources

arXiv2026

Numerical analytical continuation of multivariate hypergeometric functions

We present a general framework for the high-precision numerical evaluation of multivariate hypergeometric functions defined as solutions of holonomic systems of partial differential equations. Our approach adapts and extends methods originally developed for multi-loop Feynman integrals to the setting of hypergeometric functions of many variables. In particular, we construct Pfaffian systems for arbitrary multivariate hypergeometric functions by applying the Laporta reduction algorithm to suitable systems of differential relations. Next, we construct a numerical scheme based on the Frobenius method, which allows us to compute local power-series solutions with controlled precision and to transport them along prescribed paths in the space of variables. A central part of the paper is devoted to a systematic analysis of multivaluedness and branch structure: we show how the Frobenius method can be used to access different Riemann sheets in a controlled way and to track changes of the solution under analytic continuation around singular loci.

math-ph↗

arXiv2025

Five W-boson amplitude = near-null decagon

We study a five-leg scattering amplitude on the special Coulomb branch of planar N=4 super Yang-Mills theory. We reach this point of the moduli space of scalar vacuum expectation values by considering six-dimensional N=(1,1) super Yang-Mills theory and reducing it down to four space-time dimensions with extra-dimensional momenta being nonvanishing. This branch is characterized by massive external W-bosons and massless internal gluons propagating in loops. We analyze the five W-boson amplitude in the kinematics when their masses are much smaller than all Mandelstam-like invariants. This is what we dub the near mass-shell limit. We perform calculations to two-loop order in 't Hooft coupling, making use of recent advances in analytic calculations of required Feynman integrals. Our findings confirm exponentiation of infrared logarithms and enable us to conjecture a concise all-order expression for the amplitude in question. We further analyze its duality to the `square root' of a five-point correlation function of infinitely-heavy half-BPS operators, known as the decagon. By considering the near-null limit for inter-operators distances, we verify that the two objects coincide. This observation corroborates the novel Coulomb amplitudes/heavy correlator duality previously observed for four W-boson amplitudes and Sudakov form factors.

hep-th↗

arXiv2025

$\texttt{PrecisionLauricella}$: package for numerical computation of Lauricella functions depending on a parameter

We introduce the $\texttt{PrecisionLauricella}$ package, a computational tool developed in Wolfram Mathematica for high-precision numerical evaluations of Lauricella functions with indices linearly dependent on a parameter, $\varepsilon$. The package leverages a method based on analytical continuation via Frobenius generalized power series, providing an efficient and accurate alternative to conventional approaches relying on multi-dimensional series expansions or Mellin--Barnes representations. This one-dimensional approach is particularly advantageous for high-precision calculations and facilitates further optimization through $\varepsilon$-dependent reconstruction from evaluations at specific numerical values, enabling efficient parallelization. The underlying mathematical framework for this method has been detailed in our previous work, while the current paper focuses on the design, implementation, and practical applications of the $\texttt{PrecisionLauricella}$ package.

cs.MS↗

arXiv2025

High-precision numerical evaluation of Lauricella functions

We present a method for high-precision numerical evaluations of Lauricella functions, whose indices are linearly dependent on some parameter $\varepsilon$, in terms of their Laurent series expansions at zero. This method is based on finding analytic continuations of these functions in terms of Frobenius generalized power series. Being one-dimensional, these series are much more suited for high-precision numerical evaluations than multi-dimensional sums arising in approaches to analytic continuations based on re-expansions of hypergeometric series or Mellin--Barnes integral representations. To accelerate the calculation procedure further, the $\varepsilon$ dependence of the result is reconstructed from the evaluations of given Lauricella functions at specific numerical values of $\varepsilon$, which, in addition, allows for efficient parallel implementation. The method has been implemented in the $\texttt{PrecisionLauricella}$ package, written in Wolfram Mathematica language.

hep-th↗

arXiv2024

Expansion of hypergeometric functions in terms of polylogarithms with nontrivial variable change

Hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. Often we have hypergeometric functions with indices linear dependent on a small parameter with respect to which one needs to perform Laurent expansions. Moreover such expansions are desirable to be expressed in terms of well known functions which can be evaluated with arbitrary precision. To solve this problem we use the differential equation method and the reduction of corresponding differential systems to canonical basis. Specifically we will be interested in the generalized hypergeometric functions of one variable together with Appell and Lauricella functions and their expansions in terms of Goncharov polylogarithms. Particular attention will be given to the case of rational indices of considered hypergeometric functions when the reduction to canonical basis involves nontrivial variable change. The article comes with a Mathematica package Diogenes, which provides algorithmic implementation of the required steps.

hep-th↗

arXiv2022

Three-loop photon spectral density in QED

We calculate three-loop photon spectral density in QED with $N$ different species of electrons. The obtained results were expressed in terms of iterated integrals, which are either reduce to Goncharov's polylogarithms or can be written in terms of one-fold integrals of harmonic polylogarithms and complete elliptic integrals. In addition we provide threshold and high-energy asymptotics of the calculated spectral density. It is shown, that the use of the obtained spectral density correctly reproduces separately calculated moments of corresponding photon polarization operator.

hep-ph↗

arXiv2022

Non-planar elliptic vertex

We consider the problem of obtaining higher order in regularization parameter $ε$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct integration of parametric representations in terms of iterated integrals. Taking non-planar elliptic vertex as an example we show that in addition to two mentioned methods one can use analytical solution of differential equations in terms of power series. Moreover, in the last case it is possible to obtain the exact in $ε$ results expressible either in terms of generalized hypergeometric or Kampé de Fériet functions

hep-ph↗

arXiv2022

On series and integral representations of some NRQCD master integrals

We consider new ways of obtaining series and integral representations for master integrals arising in the process of matching of QCD to NRQCD. The latter results are exact in space-time dimension $d$. In addition, we discuss series expansion of the obtained results at fixed values of $d$.

hep-ph↗

arXiv2021

Massive Two-Loop Heavy Particle Diagrams

We determine the master integrals for vertex and propagator diagrams that appear in effective field theories containing heavy fields. The integrals involve at least one heavy line, and the standard lines include an arbitrary mass scale. The evaluation is done analytically with modern techniques. We employ the methods of differential equations and dimensional recurrence relations to evaluate said integrals up to two-loop order.

hep-ph↗

arXiv2021

Anomalous dimensions of twist 2 operators and $\mathcal{N}=4$ SYM quantum spectral curve

We present algorithmic perturbative solution of $\mathcal{N}=4$ SYM quantum spectral curve in the case of twist 2 operators, valid to in principle arbitrary order in coupling constant. The latter treats operator spins as arbitrary integer values and is written in terms of special class of functions -- products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions. It is shown that this class of functions is closed under elementary operations, such as shifts, partial fractions, multiplication by spectral parameter and differentiation. Also, it is fully sufficient to solve arising non-homogeneous multiloop Baxter and first order difference equations. As an application of the proposed method we present the computation of anomalous dimensions of twist 2 operators up to four loop order.

hep-th↗

arXiv2020

Master integrals for bipartite cuts of three-loop photon self energy

We calculate master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the $4m$ cut, reduce to Goncharov's polylogarithms. The master integrals for $4m$ cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for $4m$ cut.

hep-ph↗

arXiv2020

Massive kite diagrams with elliptics

We present the results for two-loop massive kite master integrals with elliptics in terms of iterated integrals with algebraic kernels. The key ingredients are new integral representations for sunset subgraphs in $d=4-2ε$ and $d=2-2ε$ dimensions together with differential equations for considered kite master integrals in $A+Bε$ form. The obtained results can be easily generalized to all orders in $ε$-expansion and show that the class of functions defined as iterated integrals with algebraic kernels may be large enough for writing down results for a large class of massive Feynman diagrams.

hep-ph↗

arXiv2020

Pentagon OPE resummation in N=4 SYM: hexagons with one effective particle contribution

We present the technique for resummation of flux tube excitations series arising in pentagon operator expansion program for polygonal Wilson loops in N=4 SYM. Here we restrict ourselves with contributions of one-particle effective states and consider as a particular example NMHV 6 particle amplitude at one-loop. The presented technique is also applicable at higher loops for one effective particle contributions and has the potential for generalization for contributions with more effective particles.

hep-th↗

arXiv2019

$ε$-regular basis for non-polylogarithmic multiloop integrals and total cross section of the process $e^+e^-\to 2(Q\bar Q)$

We argue that in many physical calculations where the "eliptic" sectors are involved, one can express the results via iterated integrals with almost all weights being rational. Our method is based on the existence of $ε$-regular basis, which is akin to the $ε$-finite basis defined in Ref. [hep-ph/0601165]. As a demonstration of our technique, we calculate the photon contribution to the total cross section of the production of two $Q\bar Q$ pairs in the electron-positron collisions.

hep-ph↗

arXiv2019

DGLAP and BFKL equations in $\mathcal{N}=4$ SYM: from weak to strong coupling

DGLAP and BFKL equations are among the cornestones of the contemporary QCD. Moreover, they also played an important role in the recent studies of integrability structure of $\mathcal{N}=4$ SYM. Here, we review the results obtained along this way together with a brief account of approaches and methods used.

hep-th↗

arXiv2019

ABJM quantum spectral curve at twist 1: algorithmic perturbative solution

We present an algorithmic perturbative solution of ABJM quantum spectral curve at twist 1 in sl(2) sector for arbitrary spin values, which can be applied to, in principle, arbitrary order of perturbation theory. We determined the class of functions -- products of rational functions in spectral parameter with sums of Baxter polynomials and Hurwitz functions -- closed under elementary operations, such as shifts and partial fractions, as well as differentiation. It turns out, that this class of functions is also sufficient for finding solutions of inhomogeneous Baxter equations involved. For the latter purpose we present recursive construction of the dictionary for the solutions of Baxter equations for given inhomogeneous parts. As an application of the proposed method we present the computation of anomalous dimensions of twist 1 operators at six loop order. There is still a room for improvements of the proposed algorithm related to the simplifications of arising sums. The advanced techniques for their reduction to the basis of generalized harmonic sums will be the subject of subsequent paper. We expect this method to be generalizable to higher twists as well as to other theories, such as N=4 SYM.

hep-th↗

arXiv2019

NNLO corrections to false vacuum decay rate in thin-wall approximation

Recently it was discovered that the Standard Model vacuum may suffer an essential metastability near the Planck scale. In this regard it makes sense to study in more detail the decay processes of the metastable vacuum and to develop methods for their more accurate analysis. In this article we develop a technique to calculate two loop radiative corrections to false vacuum decay rate in thin-wall approximation and apply it to false vacuum decay in scalar quantum field theory with cubic and quartic interactions. The results obtained use dimensional regularization and given in two different renormalization schemes: Coleman-Weinberg and $\overline{MS}$.

hep-ph↗

arXiv2018

Two-loop corrections to false vacuum decay in scalar field theory

We consider radiative corrections to false vacuum decay in a four-dimensional scalar field theory with cubic and quartic potential. Using planar thin wall approximation we were able to get analytical expression for the decay rate up to two loop order. The results obtained employ dimensional regularization and $\overline{MS}$ renormalization scheme.

hep-ph↗