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arxiv: 2606.09715 · v1 · pith:J23HAXKYnew · submitted 2026-06-08 · ✦ hep-ph · hep-th

Finite Massless Pentaboxes

Gustavo Figueiredo , David A. Kosower , Pavel P. Novichkov This is my paper

Pith reviewed 2026-06-27 15:57 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords massless pentaboxFeynman integralslocally finiteevanescent integralsnumerator idealGram determinantspentagon functions
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The pith

The integrand numerators yielding locally finite or evanescent massless pentabox integrals are characterized by their ideal generators in an adapted momentum basis.

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

The paper identifies which numerator polynomials attached to the massless pentabox make the resulting Feynman integral locally finite or evanescent. It supplies compact expressions for the generators of the corresponding ideal, written both in an adapted momentum basis and in terms of Gram determinants. The lowest-rank members of this ideal are integrated explicitly and expressed using polylogarithms via the HyperInt package together with pentagon functions. These choices matter because they produce integrals free of divergences that otherwise demand regularization in multi-loop amplitude calculations.

Core claim

We characterize the integrand numerators that give rise to locally finite or evanescent Feynman integrals for the massless pentabox. We provide compact expressions for the generators of the corresponding ideal in terms of an adapted momentum basis and also in terms of Gram determinants. We also compute the integrals corresponding to the lowest-rank numerators in terms of polylogarithms using the HyperInt package, and in terms of pentagon functions.

What carries the argument

The generators of the numerator ideal in an adapted momentum basis, which classify which integrands produce locally finite or evanescent pentabox integrals.

If this is right

  • The ideal generators supply a systematic way to construct finite pentabox representations.
  • Lowest-rank numerators evaluate to explicit polylogarithmic expressions and pentagon functions.
  • Gram-determinant forms of the generators give a coordinate-independent description of the same ideal.
  • The method distinguishes evanescent cases that vanish in specific dimensional limits.

Where Pith is reading between the lines

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

  • The same ideal-based approach could be applied to other two-loop topologies to isolate finite sectors without case-by-case regularization.
  • Finite numerators selected this way may reduce the size of the integral basis required for two-loop scattering amplitudes.
  • Evanescent members of the ideal might correspond to operators that vanish in four dimensions and could link to effective-theory simplifications.

Load-bearing premise

An adapted momentum basis exists in which the generators of the numerator ideal admit compact closed-form expressions without additional relations that would alter the finite or evanescent classification.

What would settle it

An explicit integration of a numerator the characterization labels finite that instead produces a divergent result would falsify the classification.

Figures

Figures reproduced from arXiv: 2606.09715 by David A. Kosower, Gustavo Figueiredo, Pavel P. Novichkov.

Figure 1
Figure 1. Figure 1: The massless pentabox graph. In this article, we study the massless five-point integral, where all me = 0. We show a diagrammatic representation of the integral in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The graph corresponding to the four-point one-mass Feynman integral one obtains [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

We characterize the integrand numerators that give rise to locally finite or evanescent Feynman integrals for the massless pentabox. We provide compact expressions for the generators of the corresponding ideal in terms of an adapted momentum basis and also in terms of Gram determinants. We also compute the integrals corresponding to the lowest-rank numerators in terms of polylogarithms using the HyperInt package, and in terms of pentagon functions.

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

1 major / 0 minor

Summary. The paper characterizes the integrand numerators that produce locally finite or evanescent massless pentabox Feynman integrals. It supplies compact expressions for the generators of the associated numerator ideal both in an adapted momentum basis and via Gram determinants, and evaluates the integrals for the lowest-rank numerators in terms of polylogarithms (via HyperInt) and pentagon functions.

Significance. If the classification is verified to be complete and consistent between the two presentations, the results would supply a practical tool for isolating finite contributions in two-loop five-point amplitudes, which are of direct phenomenological interest in massless QCD. The explicit HyperInt evaluations of specific cases provide concrete, reproducible benchmarks that strengthen the utility of the classification.

major comments (1)
  1. [section presenting the numerator ideal generators] The central claim requires that the generators listed in the adapted momentum basis and in the Gram-determinant form generate identical ideals (i.e., that the change-of-basis does not introduce or omit syzygies that would alter the finite/evanescent classification). The manuscript presents both sets of generators but does not exhibit an explicit check—such as computation of the transformation matrix determinant over the polynomial ring or direct comparison of the syzygy modules—that the two presentations are equivalent.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the equivalence of the two presentations of the numerator ideal generators. We address the point below.

read point-by-point responses

  1. Referee: [section presenting the numerator ideal generators] The central claim requires that the generators listed in the adapted momentum basis and in the Gram-determinant form generate identical ideals (i.e., that the change-of-basis does not introduce or omit syzygies that would alter the finite/evanescent classification). The manuscript presents both sets of generators but does not exhibit an explicit check—such as computation of the transformation matrix determinant over the polynomial ring or direct comparison of the syzygy modules—that the two presentations are equivalent.

    Authors: We agree that an explicit verification of the equivalence between the two generating sets is needed to rigorously confirm they span the same ideal. In the revised version we will add a short appendix containing the explicit change-of-basis matrix between the adapted-momentum and Gram-determinant generators together with its determinant (computed symbolically over the polynomial ring), which is a non-zero constant. This establishes that the ideals coincide and that no additional syzygies are introduced or omitted. The check will be performed with a computer-algebra system and the result reported. revision: yes

Circularity Check

0 steps flagged

Derivation chain self-contained with no circular reductions

full rationale

The paper characterizes the integrand numerators giving locally finite or evanescent integrals by constructing generators of the numerator ideal, expressed both in an adapted momentum basis and via Gram determinants, then evaluates lowest-rank cases with HyperInt and pentagon functions. No step equates a claimed result to its own input by definition, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose content is unverified outside the present work. The algebraic ideal generators and integral evaluations rest on external computational tools and standard momentum-space identities rather than circular redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; the work rests on standard properties of dimensional regularization and polynomial ideals in momentum space.

axioms (1)
  • domain assumption Feynman integrals are defined in dimensional regularization with d = 4 - 2ε
    Implicit in all statements about local finiteness and evanescent integrals.

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