Prior art and originality#

This page lists the mathematical methods and prior software used by ramanujan_qft, based on the October 2026 literature review.

Summary#

Part of ramanujan_qft

Status

Closest prior art

Ramanujan’s Master Theorem (rmt)

classical (1910s; proof by Hardy)

Amdeberhan et al. 2012

Mellin residue engine RACE (mellin.product_integral)

known method

Marichev; Adamchik–Marichev 1990; Slater’s theorem; Fikioris 2007; SymPy meijerint; mpmath meijerg

“Taylor coefficients = residues of the Mellin transform”

known

Ramanujan / Hardy; Flajolet–Gourdon–Dumas 1995

Choosing which side to close the contour; perturbing at coinciding poles

known

mpmath meijerg (series=1/2), hypercomb; MBConicHulls

RMT / Method of Brackets for Feynman integrals

known

Gonzalez–Moll–Schmidt 2015; Prausa 2017; Gonzalez–Kondrashuk–Moll–Recabarren 2022

B0 as Γ(ε)(m²)^(−ε) ₂F₁(ε,1;3/2;w) and its ε-expansion

known

standard one-loop literature (Boos–Davydychev and later)

C0(0,0,s; m,m,m) = −(2/s) arcsin²(√(s/4m²))

known

Higgs → γγ triangle function

‘t Hooft–Veltman C0

known

‘t Hooft & Veltman 1979; Denner 1993; LoopTools

Sunset Im part as a phase-space integral; twice-subtracted dispersion relation

known

dispersive two-loop literature; TSIL computes the sunset for arbitrary masses

MoB series for the sunset with three masses

known

Ananthanarayan–Friot–Ghosh 2020

The package implements established methods. Its contributions are the Python interfaces, numerical evaluation and validation:

  • an arbitrary-precision Mellin residue evaluator in Python;

  • an a-priori check of which expansion closes the contour legitimately, plus cancellation control and explicit errors for unmet accuracy targets (SeriesConvergenceError, ValueError);

  • LoopTools-convention one-loop functions with adjustable precision;

  • a test suite validated against LoopTools and independent integrations.

The implementation does not establish a new integration identity or a general performance advantage over existing tools.

The Mellin-convolution method (what RACE implements)#

  • O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables, Ellis Horwood, 1983 (Russian original 1978). This is the algorithmic method: write the integral as a Mellin convolution of Meijer-G functions and expand the resulting G-function into residue series (Slater’s theorem). Its history is recounted in Oleg Marichev: On the occasion of the eightieth anniversary, arXiv:2509.25246.

  • V. S. Adamchik, O. I. Marichev, The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system, Proc. ISSAC ‘90 (Tokyo), ACM 1990, pp. 212–224. This is the computer-algebra realisation, which also underlies Mathematica’s definite integration. (REDUCE bibliography)

  • A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 3: More Special Functions, Gordon & Breach, 1990. Tables and validity conditions for ∫ G·G dx.

  • G. Fikioris, Mellin-Transform Method for Integral Evaluation: Introduction and Applications to Electromagnetics, Morgan & Claypool, 2007, DOI 10.2200/S00076ED1V01Y200612CEM013.

  • N. Bleistein, R. A. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart & Winston, 1975 (Dover 1986). Mellin-convolution asymptotics. See also DLMF §2.5.

  • Ph. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: Harmonic sums, Theor. Comput. Sci. 144 (1995) 3–58. This is the correspondence between expansions and poles (direct and converse mapping).

  • G. Lovat, S. Celozzi, Rapidly convergent series and closed-form expressions for a class of integrals involving products of spherical Bessel functions, J. Comput. Appl. Math. 445 (2024) 115802. It reports residue-type series “at least 1000 times faster than specific quadrature schemes”, for the inputs studied in that paper.

  • Software.

    • SymPy sympy.integrals.meijerint: symbolic integration by Mellin convolution of Meijer-G functions (Luke §5.6.2; Prudnikov–Brychkov–Marichev; K. Roach).

    • mpmath meijerg: numerical G-function by Slater series, selecting series=1 or 2 (closing the contour on either side); hypercomb perturbs parameters at raised precision when poles coincide.

    • In tests, SymPy returns exact closed forms in 0.03–0.4 s for the atlas pairs, e.g. ∫e^(−x²)cos x dx = √π e^(−1/4)/2 and ∫J₀(x)cos(x/2) dx = 2√3/3.

Ramanujan’s Master Theorem and the Method of Brackets in HEP#

  • T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Harrison, V. H. Moll, A. Straub, Ramanujan’s Master Theorem, Ramanujan J. 29 (2012) 103.

  • I. Gonzalez, V. H. Moll, I. Schmidt, Ramanujan’s Master Theorem applied to the evaluation of Feynman diagrams, Adv. Appl. Math. 63 (2015) 214.

  • I. Gonzalez, L. Jiu, V. H. Moll, An extension of the method of brackets. Part 2, Open Math. 18 (2020) 983–995, DOI 10.1515/math-2020-0062. Combines the method of brackets with the Mellin transform of a second factor, which is RACE’s “expand one factor, use the Mellin transform of the other”.

  • M. Prausa, Mellin–Barnes meets Method of Brackets: a novel approach to Mellin–Barnes representations of Feynman integrals, Eur. Phys. J. C 77 (2017) 594, arXiv:1706.09852.

  • I. Gonzalez, I. Kondrashuk, V. H. Moll, L. M. Recabarren, Mellin–Barnes integrals and the method of brackets, Eur. Phys. J. C 82 (2022), DOI 10.1140/epjc/s10052-021-09977-x, arXiv:2108.09421.

  • Z. P. Bradshaw, O. Atale, A generalized Ramanujan Master Theorem and integral representation of meromorphic functions, arXiv:2408.08725.

  • B. Ananthanarayan, S. Banik, S. Friot, S. Ghosh, Multiple series representations of N-fold Mellin–Barnes integrals, Phys. Rev. Lett. 127 (2021) 151601, arXiv:2012.15108 (MBConicHulls; resonant/logarithmic cases included).

  • B. Ananthanarayan, S. Friot, S. Ghosh, New series representations for the two-loop massive sunset diagram, Eur. Phys. J. C 80 (2020), arXiv:1911.10096.

  • S. Friot, D. Greynat, E. de Rafael, Asymptotics of Feynman diagrams and the Mellin–Barnes representation, hep-ph/0505038.

Mellin–Barnes and other HEP integral tools#

  • M. Czakon, Automatized analytic continuation of Mellin–Barnes integrals (MB.m), Comput. Phys. Commun. 175 (2006) 559, hep-ph/0511200.

  • M. Ochman, T. Riemann, MBsums – a Mathematica package for the representation of Mellin–Barnes integrals by multiple sums, Acta Phys. Polon. B 46 (2015) 2117, arXiv:1511.01323. Contour closing plus residue sums inside a HEP tool.

  • S. P. Martin, D. G. Robertson, TSIL: a program for the calculation of two-loop self-energy integrals, Comput. Phys. Commun. (2006), DOI 10.1016/j.cpc.2005.08.005, hep-ph/0501132.

Research directions#

Extensions need explicit convergence arguments and independent benchmarks against tools suited to the target integrals, such as AMFlow, pySecDec, FIESTA, TSIL and Mellin–Barnes packages. Possible directions include:

  1. An open-source Python counterpart of MBConicHulls / MBsums. Automatically derive and sum the convergent multiple series of N-fold Mellin–Barnes representations of Feynman integrals, with certified error control. MBConicHulls and MBsums are Mathematica packages.

  2. Crossing convergence boundaries in physical kinematics. Stitch series from different regions, or hand exact series values to differential-equation solvers as high-precision boundary data. This is a recognised bottleneck.

  3. Products of three or more factors that are not a product of two Meijer-G functions (bivariate Fox-H). Possible targets are configuration-space propagators, Bessel moments, and Hankel transforms of power spectra.

  4. Rigorous foundations of the Method of Brackets. Prove when its heuristic rules agree with the Mellin–Barnes / residue answer.

Attribution and performance claims#

Cite the software version and the mathematical methods used in a calculation. Performance comparisons should specify inputs, precision, accuracy checks and work budgets; measurements on selected products do not rank whole libraries.