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arxiv: 2301.04852 · v4 · submitted 2023-01-12 · ✦ hep-ph

Note on the evaluation of one type scalar three-point integral extracted from the Higgs boson decay

Jin Zhang This is my paper

Pith reviewed 2026-05-24 10:33 UTC · model grok-4.3

classification ✦ hep-ph
keywords scalar three-point integralHiggs to gg decayone-loop amplitudeanalytic evaluationtwo internal massesFeynman integralslogarithmic integral
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The pith

Complete analytic expressions are given for the 1/x log(quadratic) integral from the one-loop Higgs to gg amplitude with two internal masses.

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

The paper evaluates the amplitude of a scalar one-loop three-point diagram with two different internal masses that contributes to the leading-order decay of the Higgs boson into two gluons. The central technical step is to compute an integral whose integrand is the reciprocal of the integration variable multiplied by the logarithm of a quadratic polynomial in that variable. Fully closed-form results are supplied for the two cases distinguished by whether that quadratic has no real roots or two distinct real roots. These expressions make the amplitude analytic rather than numerical in the relevant kinematic regions. Applications to radiative decays of heavy quarks in the Standard Model and in a singlet vector-like model are noted.

Core claim

The integral of the form (1/x) log(quadratic) admits complete analytic results when the quadratic has either no real roots or exactly two different real roots, obtained directly from the one-loop three-point diagram with unequal masses.

What carries the argument

The integral whose integrand is the reciprocal of the integration variable times the logarithm of a quadratic function of that variable, evaluated by cases on the roots of the quadratic.

If this is right

  • The one-loop amplitude for Higgs decay to two gluons becomes fully analytic when the two internal masses differ.
  • The same closed forms apply without modification to the radiative decays of heavy quarks in the Standard Model.
  • The expressions can be inserted into calculations of the corresponding processes in a singlet vector-like model.

Where Pith is reading between the lines

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

  • The case distinction on real roots may simplify numerical checks of related loop integrals that contain similar logarithms.
  • If the method extends beyond this specific diagram, it could reduce the need for numerical libraries in other Higgs and quark decay channels.
  • The two-root case may connect to known dilogarithm reductions that appear in other one-loop calculations.

Load-bearing premise

The integrand must take exactly the form of the reciprocal of the integration variable multiplied by the logarithm of a quadratic function, as it arises in this specific scalar one-loop diagram.

What would settle it

Perform a direct numerical quadrature of the integral for concrete parameter choices in each root case and compare the result to the claimed closed-form expression.

Figures

Figures reproduced from arXiv: 2301.04852 by Jin Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: Massive triangle with two massless external lines. T [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

Motivated by the Higgs boson decaying to $gg$ at leading order approximation, the amplitude of scalar one loop three-point diagram with two different internal masses are evaluated and fully analytic results are obtained. The main ingredient of the evaluation is a integral in which the integrand is product of the reciprocal of the integral variable and a logarithm whose argument is a quadratic function of the general form. Complete analytic results for the two cases that there is no real root and there are two different real roots for the logarithm are presented. Applications of the results in this paper to radiative decays of the heavy quarks in the Standard Model and in the singlet vector-like model are discussed.

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 manuscript evaluates a scalar one-loop three-point integral with two different internal masses arising in the leading-order Higgs boson decay to gluons. The central object is the integral whose integrand is (1/x) times the logarithm of a quadratic polynomial in the integration variable; fully analytic closed-form results are presented for the two cases in which this quadratic has no real roots or two distinct real roots.

Significance. If the derivations are correct, the results supply exact analytic expressions for a class of integrals that appear in Higgs phenomenology and in radiative decays of heavy quarks, both in the Standard Model and in singlet vector-like extensions. Such closed forms can reduce reliance on numerical integration in precision calculations.

major comments (1)
  1. [Abstract] Abstract: the paper explicitly limits its 'complete analytic results' to the no-real-root and two-distinct-real-roots cases, yet omits the repeated-root case (discriminant D=0). Because the applications to Higgs and heavy-quark decays can reach kinematic points where D=0, the omission leaves the utility of the results for the stated applications incomplete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment regarding the scope of the analytic results. We address the point raised below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the paper explicitly limits its 'complete analytic results' to the no-real-root and two-distinct-real-roots cases, yet omits the repeated-root case (discriminant D=0). Because the applications to Higgs and heavy-quark decays can reach kinematic points where D=0, the omission leaves the utility of the results for the stated applications incomplete.

    Authors: We agree that the repeated-root case (D=0) is relevant for a complete treatment, since certain kinematic points in the Higgs boson decay to gluons and in radiative heavy-quark decays can reach this boundary. In the revised manuscript we will add the corresponding closed-form expression, which follows either by taking the appropriate limit of the two-distinct-roots result or by direct integration when the quadratic polynomial possesses a double root. This addition will ensure the results cover all three possible configurations of the logarithm argument. revision: yes

Circularity Check

0 steps flagged

Direct analytic evaluation of defined integral; no circularity

full rationale

The paper derives closed-form expressions for a specific integral (reciprocal times log of quadratic) by case analysis on the discriminant of the quadratic argument. This is a standard, self-contained mathematical evaluation using elementary methods; no parameters are fitted to data and then relabeled as predictions, no self-definitional loops appear, and no load-bearing results are imported solely via self-citation. The two cases treated (no real roots, two distinct real roots) are explicitly delimited by the problem statement itself. The omitted D=0 case is a completeness gap but does not create circularity in the presented derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical techniques for evaluating one-loop integrals in quantum field theory; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard techniques exist for analytic evaluation of Feynman integrals of the form involving 1/x times log of quadratic
    The evaluation proceeds from the given integrand form using established methods in the field.

pith-pipeline@v0.9.0 · 5628 in / 1301 out tokens · 31575 ms · 2026-05-24T10:33:34.489126+00:00 · methodology

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Reference graph

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