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arxiv: 2606.26691 · v1 · pith:I3RMCTJQnew · submitted 2026-06-25 · ✦ hep-ph · hep-th· physics.comp-ph

CHESS: CHEbyshev pSeudo-Spectral transport for Feynman integral differential equations

Yuanche Liu , Yang Zhang This is my paper

Pith reviewed 2026-06-26 04:35 UTC · model grok-4.3

classification ✦ hep-ph hep-thphysics.comp-ph
keywords Feynman integralsdifferential equationsChebyshev spectral methodsmaster integralsepsilon expansionnumerical transportpseudo-spectral collocationmulti-scale integrals
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The pith

CHESS package solves epsilon-factorized Feynman DEs via Chebyshev-Lobatto collocation on pulled paths with rapid node convergence.

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

The paper presents a Wolfram Language package that pulls epsilon-factorized differential equations for Feynman master integrals onto a chosen one-dimensional path and solves the resulting transport problem numerically. It assembles the pulled matrix from constant matrices and scalar one-form pullbacks, then applies Chebyshev-Lobatto spectral collocation, sparse assembly, sequential epsilon expansion, and residue regularization at endpoints. Benchmarks on large multi-scale families demonstrate quick convergence with node count, agreement with independent references, shorter wall times than local-series alternatives, and lower memory use on the biggest systems. A reader would care because high-precision multi-loop integrals remain a computational bottleneck, and a stable one-dimensional transport method could reduce the cost of evaluating them.

Core claim

The central claim is that Chebyshev pseudo-spectral collocation applied to the pulled transport matrix of epsilon-factorized differential equations yields an efficient high-precision solver for one-dimensional transport of Feynman master integrals, as evidenced by rapid convergence with increasing nodes and agreement with reference data in benchmarks of large multi-scale integral families.

What carries the argument

Chebyshev-Lobatto spectral collocation on the pulled differential matrix, with sequential epsilon-expansion propagation and residue-based endpoint regularization.

If this is right

  • Large multi-scale integral families become computable with fewer discretization nodes while preserving high precision.
  • Wall-clock times become shorter than those of fixed local-series methods in direct comparisons of the same systems.
  • Process-tree memory usage drops for the largest benchmark families relative to the local-series baseline.
  • Spurious regular singular endpoints are handled automatically through residue regularization without manual intervention.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pulled-matrix plus spectral-collocation structure could be applied to other classes of linear differential systems that admit an epsilon factorization.
  • Sparse assembly of the collocation matrix might combine with existing reduction algorithms to further lower the cost of multi-scale calculations.
  • Sequential propagation in the epsilon expansion suggests a natural way to obtain results at several orders simultaneously from a single transport run.

Load-bearing premise

The differential equations must be epsilon-factorized and the transport matrix must be assemblable from constant matrices and precomputed scalar pullbacks of the one-forms.

What would settle it

For a known large multi-scale integral family, increase the number of Chebyshev nodes and check whether the numerical results fail to converge to the independent reference values within the claimed precision or require unexpectedly many nodes.

Figures

Figures reproduced from arXiv: 2606.26691 by Yang Zhang, Yuanche Liu.

Figure 1
Figure 1. Figure 1: Schematic double-pentagon topology used for the physical-region DP transport [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Genuine three-loop five-point integral families: Pentagon–Box–Box (PBB), Box– [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Grouped-bar comparison of wall time, process-tree peak RSS, and accuracy [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two-loop non-planar triangle family (b) of Ref. [42]. Thick internal lines denote [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
read the original abstract

We present CHESS (CHEbyshev pSeudo Spectrum), a Wolfram Language package for high-precision one-dimensional transport of {\epsilon}-factorized differential equations for Feynman master integrals. The solver works with the matrix obtained by pulling a differential one-form to a chosen path. This matrix may be supplied directly, or assembled from constant matrices and precomputed scalar pullbacks of the one-forms. The program combines Chebyshev-Lobatto spectral collocation, sparse matrix assembly, sequential propagation in the {\epsilon}-expansion, and residue-based regularization of spurious regular singular endpoints. Benchmarks for large multi-scale integral families show rapid node convergence and agreement with independent reference data where such data are available. In the fixed local-series comparison used here, the Chebyshev transports also give shorter wall times; the reported process-tree memory usage is comparable for the smaller parallel runs and lower for the largest benchmark system in that comparison.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents CHESS, a Wolfram Language package implementing Chebyshev-Lobatto pseudo-spectral collocation for one-dimensional transport of ε-factorized differential equations for Feynman master integrals. The solver operates on the pulled transport matrix (supplied directly or assembled from constant matrices plus precomputed scalar one-form pullbacks), using sparse assembly, sequential ε-expansion propagation, and residue-based regularization at spurious regular singular points. Benchmarks on large multi-scale families report rapid convergence with node count, agreement with independent reference data, shorter wall times than fixed local-series methods, and lower process-tree memory usage on the largest system.

Significance. If the numerical performance claims hold under the stated operating conditions, the work supplies a practical high-precision tool for multi-scale Feynman integral families where the differential equations have already been prepared in ε-factorized form. The explicit demonstration of node convergence, wall-time and memory advantages relative to local-series transport, and the provision of a reusable Wolfram package constitute concrete strengths for the numerical methods literature in this area.

major comments (2)
  1. [Abstract] Abstract: the reported rapid node convergence, agreement with reference data, and performance gains (shorter wall times and lower memory on the largest benchmark system) are demonstrated exclusively under the precondition that the input differential equations are already ε-factorized and that the transport matrix can be supplied or assembled exactly as described (constant matrices plus scalar one-form pullbacks). No separate evidence, procedure, or discussion is given on how generally or automatically this precondition can be met for arbitrary large multi-scale families; this assumption is load-bearing for the transferability of the benchmark advantages.
  2. [Abstract] Abstract and benchmarks description: no error bars or quantitative uncertainty measures are reported on the agreement with independent reference data, and no explicit derivation or validation of the residue-based regularization for spurious regular singular endpoints is provided; both omissions directly affect the strength of the central numerical claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive comments on our manuscript. We address each major comment below, clarifying the scope of the work and indicating revisions where appropriate to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the reported rapid node convergence, agreement with reference data, and performance gains (shorter wall times and lower memory on the largest benchmark system) are demonstrated exclusively under the precondition that the input differential equations are already ε-factorized and that the transport matrix can be supplied or assembled exactly as described (constant matrices plus scalar one-form pullbacks). No separate evidence, procedure, or discussion is given on how generally or automatically this precondition can be met for arbitrary large multi-scale families; this assumption is load-bearing for the transferability of the benchmark advantages.

    Authors: The manuscript and package focus exclusively on the numerical transport step for differential equations that have already been prepared in ε-factorized form, as stated in the title, abstract, and introduction. This precondition is standard in the current literature on Feynman integrals, where ε-factorization is handled by separate algorithms and packages (e.g., those based on integration-by-parts reduction and symbol methods). The benchmarks and performance claims are presented under these operating conditions, which define the intended use case for CHESS. We will revise the abstract and add a short paragraph in the introduction to explicitly state the scope and reference representative works on the factorization step, thereby clarifying transferability without claiming to address the factorization problem itself. revision: yes

  2. Referee: [Abstract] Abstract and benchmarks description: no error bars or quantitative uncertainty measures are reported on the agreement with independent reference data, and no explicit derivation or validation of the residue-based regularization for spurious regular singular endpoints is provided; both omissions directly affect the strength of the central numerical claims.

    Authors: We acknowledge that quantitative uncertainty measures (such as maximum absolute differences or relative errors with respect to reference values) are not tabulated in the current benchmarks section, even though visual agreement and node-convergence plots are shown. Similarly, while the residue-based regularization procedure is described operationally in the methods, an explicit derivation and additional validation tests are not provided. Both points are valid and we will revise the manuscript to include (i) tabulated quantitative discrepancy measures for the benchmark families where independent data exist and (ii) an expanded subsection (or short appendix) deriving the regularization and presenting targeted numerical tests on spurious singularities. These additions will be made without altering the core claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical solver benchmarked against independent references

full rationale

The paper presents CHESS as a numerical solver for one-dimensional transport of already epsilon-factorized differential equations, using Chebyshev-Lobatto collocation on a pulled transport matrix that is either supplied or assembled from constants and precomputed scalar pullbacks. The load-bearing claims are benchmark results (node convergence, agreement with independent reference data, wall-time and memory comparisons) obtained by running the solver on large multi-scale families and comparing outputs to external references. No derivation chain reduces a result to its own inputs by construction, no fitted parameters are relabeled as predictions, and no load-bearing premise rests on self-citation chains or uniqueness theorems imported from the authors' prior work. The epsilon-factorized precondition is stated explicitly as the solver's operating mode rather than derived internally.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard properties of Chebyshev spectral methods and the assumption that the input DEs are already epsilon-factorized; no new physical entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption The differential equations for the master integrals are epsilon-factorized and can be represented by a matrix obtained via pullback along a chosen path.
    Stated directly in the abstract as the input format the solver works with.
  • domain assumption Chebyshev-Lobatto collocation combined with sequential epsilon-expansion and residue regularization produces convergent high-precision solutions.
    Core numerical assumption underlying the reported rapid node convergence.

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