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REVIEW 3 major objections 6 minor 107 references

Numerical computation of Fox functions

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes a practical Sinc-quadrature route to multivariate Mellin–Barnes integrals—Fox functions—and shows that contiguity relations shifting the decay parameter λ cure the slow convergence that afflicts Feynman integrals in…

desk verdict Genuinely useful contiguity and decomposition machinery for multivariate Fox functions, but the multivariate Sinc error control is an admitted heuristic and the physical-region examples drift, so the paper deserves a serious referee rather than immediate acceptance. read the letter →

arxiv 2506.09597 v1 pith:7HC27TFT submitted 2025-06-11 hep-ph math-phmath.MP

Giampiero Passarino (Turin U. , INFN , Turin) This is my paper · ORCID
classification hep-phmath-phmath.MP MSC 81T99 PACS 12.60.-i11.10.-z14.80.Bn02.30.Gp
keywords FoxH-functionMellin-BarnesintegralsSincnumericalmethodsFeynmancontiguityrelationsKorobovlatticerulesLauricellafunctionsphysicalregion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that multiple Mellin–Barnes integrals, viewed as multivariate Fox H-functions, can be evaluated numerically to high accuracy with Sinc quadrature, including in the physical region where these integrals are notoriously slow. The obstacle is that when the phase of a variable reaches απ/2, exponential decay turns into power-law decay controlled by the parameter λ, and the integral may converge only barely. The proposed cure is to apply contiguity relations that rewrite the integral as a combination of Fox functions with λ lowered by one unit, repeating until the decay is fast enough. The framework is developed for univariate, bivariate, and multivariate Fox functions, extended to modified functions containing digamma factors for infrared-divergent Feynman integrals, and tested against exact closed forms and randomized Korobov lattice results. A sympathetic reader would care because this is a residue-free numerical path to multi-loop Feynman integrals directly in the physical region.

What carries the argument

The central object is the multivariate Fox H-function, an r-fold Mellin–Barnes contour integral whose integrand is a quotient of Euler Gamma functions, optionally multiplied by digamma or polygamma factors in the modified version used for infrared-divergent integrals. The load-bearing parameters are α and β, which control exponential falloff; ρ, which fixes the convergence radius; and λ, which controls the power-law decay and must satisfy βσ+λ<−1 in the critical phase direction. Two numerical engines carry the argument: the Sinc approximation, which maps each integration line to the real axis and converts the integral to a weighted lattice sum, and randomized Korobov lattice rules, used as an independent check. The methodological key is the contiguity trick: because the integrand is a Gamma-function quotient, identities such as zΓ(z)=Γ(z+1) can be combined to express a slowly decaying integral as a sum of faster-decaying ones with λ lowered by one, with residues collected from the shifted contour.

What would settle it

Evaluate a multivariate Mellin–Barnes integral with a known closed form in the critical physical-phase region, for example I2 of Eqs. (169)–(170) at z1=−1/4 and z2=1/3+0.01i, using the paper's Sinc lattice and stopping rule. If the difference between consecutive approximants drops below tolerance while the error relative to the exact value E2 remains above it, the central accuracy claim fails at that point. A sharper test is a case with α=2, β=0 and λ only slightly below −1 with no contiguity shift applied: if the stabilized Sinc value disagrees with an independent pole-residue sum beyond the claimed accuracy, the method's practical claim is falsified.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the standard failure mode of numerical Mellin–Barnes integration—loss of exponential convergence at the critical phase, leaving only a |t|^{βσ+λ} power tail—can be handled inside the Fox-function language. The paper identifies λ as the control parameter that contiguity relations can move, and derives shifts λ→λ−1 for univariate, bivariate, and multivariate cases by recombining Gamma-function ratios with linear combinations of contiguous parameter choices. At each shift the contour moves and residue sums are subtracted, so the analytic value is preserved while the tail decays faster. The paper then shows through many worked examples that Sinc lattices, with the step size and truncation chosen from the strip of analyticity, deliver digits that stabilize as N increases, and it cross-checks those digits against randomized Korobov lattice integration and exact closed forms for Lauricella-type examples.

Load-bearing premise

The numerical scheme rests on the assumption that each integrand is analytic in a sufficiently wide strip around the integration contour, that comparing consecutive Sinc approximants is a reliable error estimate in many dimensions, and that useful contiguity relations exist for the parameter sets that arise in practice.

Editorial extensions

If this is right

  • Feynman integrals in the physical region become accessible to direct Sinc-quadrature integration rather than contour closing and multivariate residue summation.
  • Slowly convergent Mellin–Barnes integrals can be accelerated systematically: repeated contiguity shifts reduce λ by one until the power-law tail is fast enough for stable Sinc results.
  • In the tested multivariate examples, Sinc lattices reach target accuracy with fewer integrand evaluations than Korobov lattice rules, with the gap widening as the number of variables grows.
  • Modified Fox functions with digamma factors provide a single framework for ε-expansions of infrared-divergent Feynman integrals, including the coefficients of the expansion.
  • The convergence criteria imported from the H-function literature give practical tests for when a multivariate Mellin–Barnes integral converges, and when it diverges away from positive real variables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the λ→λ−1 shift is as general as it appears, it could be combined with differential-equation or sector-decomposition pipelines, since it accelerates MB tails without altering the underlying analytic structure of the integral.
  • The same machinery likely extends to MB integrals with irrational or incommensurate Gamma-function coefficients, where reduction to Meijer G-functions is impossible; the paper's ratio-of-Gamma asymptotic expansion gives a concrete starting point.
  • A direct test of the error heuristic would be to compare the Sinc-plus-contiguity result for a known physical-region integral, such as the exact logarithms in Section 8, with the difference between consecutive approximants, measuring whether the heuristic underestimates the true error.
  • The factorization/decomposition technique for Lauricella functions could be iterated to split a high-dimensional Fox function into products of lower-dimensional ones, potentially converting exponentially costly Sinc lattices into few-dimensional integrations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript develops numerical methods for multivariate Mellin-Barnes integrals written as Fox H functions, with applications to Feynman integrals. It introduces Sinc quadrature rules adapted to the exponential decay of Gamma-function integrands, compares them with randomized Korobov lattice rules, and proposes contiguity relations that lower the power-decay parameter lambda to cure slowly convergent physical-region integrals. It also discusses modified Fox functions with digamma insertions for epsilon expansions, analytic continuation above thresholds, factorization/decomposition of high-dimensional Fox functions, and efficient evaluation of Gamma and polygamma functions. The paper presents a large collection of one-, two-, three-, four-, and six-fold numerical examples, several of which are checked against exact results or an independent Korobov integrator.

Significance. If the proposed scheme is as accurate as claimed, it would offer a practical numerical route to Feynman integrals in the physical region, where Mellin-Barnes integrands have only power-law decay and naive quadrature is slow. The paper has real strengths: the contiguity relations are derived from Gamma identities rather than fitted; several one- and two-dimensional examples converge to exact values or to an independent Korobov routine (e.g., Eq. (160), Eq. (165)); the treatment of digamma-inserted Fox functions supports epsilon expansions; and the factorization/decomposition section gives a plausible strategy for reducing the effective integration dimension. However, the central practical claim of "high accuracy" for multivariate Sinc methods is not supported by a rigorous error bound, and the numerical evidence in the physical-region regime is thin and partly worrisome. The manuscript itself explicitly disclaims rigorous multivariate error control in Section 8, and the negative-z examples in Section 12 show drift rather than clean convergence. For these reasons the paper requires substantial revision before the central claim can be accepted.

major comments (3)
  1. [§8, "Accuracy of Sinc approximation"] The paper's central conclusion states that Mellin-Barnes integrals "can be approximated, with high accuracy, by using Sinc numerical methods," but Section 8 explicitly says that for the two-dimensional integral in Eq. (160) "the convergence is not at stake but the error cannot be determined rigorously" and that the optimal choice of Sinc lattice parameters "remains an open problem." For r >= 2 there is no multivariate analogue of the one-dimensional error bound in Eq. (155), and the proposed error control is only comparison of consecutive approximants. Since the multivariate case is the main subject of the paper, this gap is load-bearing: the high-accuracy claim is not established for the general case the paper advertises.
  2. [§12, Example 6 (table following Eq. (341))] The most directly relevant physical-region example, with z1 = -2.11 and four integration variables, shows the Sinc real part drifting as 62.9799396 -> 62.9370547 -> 62.9329160 as the number of integrand calls grows from 11M to 331M, while the independent OKROBV result is 62.98(9). The later Sinc values move away from the comparison value, and a consecutive-difference error surrogate would not detect this behavior. Example 4 (Eq. (332)) similarly reports only N = 30 for a negative-z case and differs from OKROBV at the 1e-4 level. These examples do not support the conclusion that Sinc methods give controlled high accuracy in the physical region; the manuscript needs either a rigorous multivariate error estimate or a systematic numerical validation protocol (varying N and the lattice parameters h, d, and comparing against independent high-precision references) for exactly this regime.
  3. [§12, Example 7 (Eq. (343))] The six-fold example is presented as evidence that the method extends to higher dimensions, but the independent reference is too imprecise to validate the claimed accuracy: OKROBV returns (0.459 +/- 0.073) x 10^-3 for the real part and (-0.186 +/- 0.017) x 10^-4 for the imaginary part, with relative errors of roughly 16% and 9%. The Sinc values are listed without error estimates or lattice parameters, and the text notes that the results follow "only after a fine tuning of the corresponding parameters." Without reporting h, d, M_j, and N for each run, or providing the code, the reader cannot assess whether the quoted agreement is robust or the result of parameter selection. This is a load-bearing reproducibility issue for the claimed high-dimensional capability.
minor comments (6)
  1. [§8, Eq. (143)] The notation M = h alpha+/alpha- N i is undefined: the bracket should be specified as the integer part, and the formula should distinguish the step-size h from the integer-part brackets to avoid confusion.
  2. [§12, Example 7] The text says the consecutive differences for 10^3 x ReH are "0.0052, 0.0019 and 0.008," but the table gives 0.0008 for the last difference; this typo should be corrected.
  3. [§12, Examples 4 and 6] The Sinc runs in these tables are identified only by N or by the number of calls; the corresponding lattice parameters h, d, and M_j are not reported, which makes the results hard to reproduce or to compare with the parameter-selection discussion in Section 8.
  4. [Throughout] There are numerous typographical slips, e.g., "intoduce" in Section 8, "condider" in Section 12, and "akas" in the Conclusions; a careful proofreading pass is needed.
  5. [§9, Eq. (181)-(182)] The derivation of the asymptotic phase of Gamma(x + i y) would benefit from a reference or a brief explanation of the expansion of the sine integral representation, since the displayed formula is used to justify the oscillation count but is not otherwise derived.
  6. [§10, Eq. (223)] The notation d_{k,j} is used both as coefficients in Eq. (223) and as integration differentials elsewhere; please introduce a distinct symbol or state the convention explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Sinc benchmarks are external and contiguity relations are derived; the multivariate error-control caveat is a correctness risk, not circularity.

full rationale

The derivation chain is self-contained relative to the paper's numerical claim. The Sinc approximation is imported from F. Stenger's external theory [16,17], and the convergence parameters alpha, beta, lambda are defined directly from the Gamma-function argument structure, not fitted to the integrals being evaluated. The numerical examples are benchmarked against OKROBV, an independent randomized Korobov routine, and against closed analytic forms (e.g. Eqs. (165)-(166), (169)-(170), and the Appell/Lauricella decompositions in Section 8). Contiguity relations that lower lambda by one unit are derived algebraically rather than tuned to target results; for example, the recurrence construction around Eqs. (23)-(31) and the Appell relation in Eq. (172) are explicit identities, not fits. Author self-citations, such as Ref. [7] for the Feynman-to-Fox dictionary and Ref. [45] for a fuller description of the physical-region procedure, supply background and organization, but the numerical benchmarks do not depend on those cited results for their validity; the analytic-continuation content needed here is derived in Section 6 via the Sokhotski-Plemelj theorem and locally verified. The paper's own limitation in Section 8, stating that for multidimensional integrals 'the error cannot be determined rigorously' and that optimal Sinc lattice parameters 'remains an open problem,' is an important correctness/robustness caveat, weakening the strength of the 'high accuracy' claim, but it is not circularity. No fitted parameter is renamed a prediction, no definition reduces to the target result, and no load-bearing conclusion is forced by a self-citation chain.

Assumptions & free parameters 1 free parameters · 6 assumptions · 1 invented entities

No physical constants are fitted; the free parameters are quadrature tuning choices. Background mathematical theorems are imported from standard references. The only newly introduced object is the H_psi notation, which has an explicit definition and numerical checks.

free parameters (1)
  • Sinc lattice parameters (N, M_j, h, d) = h = sqrt(d/(alpha_+ N)), N up to 1000, d around 0.2
    Hand-chosen quadrature parameters that control the Sinc grid; they have no physical meaning and are tuned by comparing successive approximants.
assumptions (6)
  • standard math Multivariate Fox convergence theorem of Hai and Srivastava, Eqs. (53)-(55) in Section 4
    Unproved background theorem, cited from Ref. [3], that governs convergence of the multivariate Mellin-Barnes integrals.
  • standard math Exponential decay of the Gamma function on vertical contours, |Gamma(x+iy)| ~ exp(-pi|y|/2)|y|^(x-1/2)
    Standard asymptotic used to set the Sinc decay rates alpha_+ and alpha_- in Section 8.
  • domain assumption Sinc approximation error bounds in the univariate case extend in practice to the multivariate case
    The paper explicitly states in Section 8 that the multivariate error cannot be determined rigorously and that the choice of Sinc points is not optimal; the practical use of Sinc with variable precision assumes similar exponential convergence.
  • domain assumption Analyticity of the integrand in a strip D_d of width d around the integration contour
    Required for the Sinc approximation; in physical-region kinematics the nearby poles narrow the strip, and the paper tunes d to manage this.
  • domain assumption Sokhotski-Plemelj-Fox and Hadamard finite-part treatment provides the correct analytic continuation above the normal threshold
    Used in Section 6 to continue Mellin-Barnes representations into the physical region; relies on distribution theory and is not re-derived.
  • standard math The series in Eq. (219) convergences sufficiently fast, after acceleration, for the factorization decomposition
    The series is a 2F1 at unit argument; the paper gives convergence conditions and uses Ames or Shanks acceleration to handle logarithmic convergence.
invented entities (1)
  • Modified Fox function H_psi with digamma insertions independent evidence
    purpose: Compact representation of epsilon expansions of infrared-divergent Feynman integrals
    Defined explicitly by the contour integral in Eqs. (68)-(69) and tested numerically against known integrals; it is a mathematical convention rather than a new physical degree of freedom.

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Pith. "Pith review of Numerical computation of Fox functions." pith.science (2026). https://pith.science/paper/7HC27TFT

@misc{pith2026250609597,
  author       = {Pith},
  title        = {Pith review of: Numerical computation of Fox functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7HC27TFT}},
  note         = {Machine review of arXiv:2506.09597}
}
read the original abstract

In this work we discuss techniques for the numerical computation of Fox functions that represent Feynman integrals. Illustrative examples based on Sinc numerical methods and Quasi-Monte Carlo methods are given

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