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arxiv: 2606.03055 · v2 · pith:ECA2HVJUnew · submitted 2026-06-02 · ✦ hep-ph

The Pion Gravitational Form Factors and the Trace Anomaly in QCD Factorization

Claudio Corian\`o , Hsiang-nan Li , Dario Melle This is my paper

Pith reviewed 2026-06-28 09:46 UTC · model grok-4.3

classification ✦ hep-ph
keywords pion gravitational form factorstrace anomalyQCD factorizationTJJ vertexlattice QCDSudakov resummationdilaton sum rule
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The pith

The trace anomaly from the non-Abelian TJJ vertex contributes to the pion gravitational form factors, canceling in A_π(Q²) but dominating the trace form factor.

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

The paper calculates the pion gravitational form factors in QCD factorization by incorporating the trace-anomaly component generated by the non-Abelian TJJ vertex. It combines a Sudakov-resummation-improved hard kernel with an anomaly form factor taken from momentum-space conformal field theory and the perturbative dilaton sum rule. Comparison to lattice QCD data reveals a hierarchy in which the isolated anomaly cancels in A_π(Q²) while the complete TJJ insertion lowers the leading-order result at small Q², the anomaly remains important in D_π(Q²), and it supplies the dominant TJJ piece to the trace form factor. A reader would care because these form factors encode how energy and momentum are distributed inside the pion and because the anomaly term tests a fundamental feature of QCD.

Core claim

The calculation shows that the trace anomaly component generated by the non-Abelian TJJ vertex cancels when isolated in the form factor A_π(Q²), while the full TJJ insertion lowers the leading-order curve at small momentum transfer squared Q². The anomaly is important in D_π(Q²) and supplies the dominant TJJ contribution to the trace form factor. This refined projection hierarchy is obtained after matching the Sudakov-improved factorization result to lattice QCD data.

What carries the argument

The non-Abelian TJJ vertex that generates the trace-anomaly component of the gravitational form factor.

Load-bearing premise

The anomaly form factor is correctly given by the momentum-space conformal field theory suggestion and the perturbative dilaton sum rule.

What would settle it

A lattice QCD calculation at small Q² that finds no lowering of the leading-order curve once the full TJJ insertion is included, or that shows the anomaly does not dominate the trace form factor, would falsify the reported hierarchy.

Figures

Figures reproduced from arXiv: 2606.03055 by Claudio Corian\`o, Dario Melle, Hsiang-nan Li.

Figure 1
Figure 1. Figure 1: Factorized pion GFF amplitude with the non-Abelian [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sudakov-resummation-improved pion GFFs obtained for [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We study the pion gravitational form factor in QCD factorization, focusing on the trace-anomaly component generated by the non-Abelian \(TJJ\) vertex. The calculation combines a Sudakov-resummation-improved pion hard kernel with the anomaly form factor suggested by momentum-space conformal field theory and by the perturbative dilaton sum rule. Comparison with lattice QCD data shows a refined projection hierarchy: the isolated anomaly cancels in the form factor \(A_\pi(Q^2)\), while the full \(TJJ\) insertion lowers the leading-order curve at small momentum transfer squared \(Q^2\); the anomaly is important in \(D_\pi(Q^2)\), and it gives the dominant \(TJJ\) contribution to the trace form factor.

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 examines the pion gravitational form factors in QCD factorization, with emphasis on the trace-anomaly contribution arising from the non-Abelian TJJ vertex. It combines a Sudakov-resummation-improved hard kernel with an anomaly form factor taken from momentum-space conformal field theory and the perturbative dilaton sum rule. Lattice comparisons are reported to establish a refined hierarchy: the isolated anomaly cancels in A_π(Q²), the full TJJ insertion lowers the leading-order result at small Q², the anomaly is important for D_π(Q²), and it supplies the dominant TJJ piece of the trace form factor.

Significance. If the external anomaly form factor accurately encodes the relevant TJJ matrix element, the work would clarify how the QCD trace anomaly projects onto hadronic gravitational form factors and would supply a concrete factorization-based interpretation of existing lattice data.

major comments (1)
  1. [Abstract] Abstract: the central hierarchy (cancellation of the isolated anomaly in A_π(Q²), lowering by the full TJJ insertion at small Q², importance in D_π(Q²), and dominance in the trace form factor) is obtained only after inserting an anomaly form factor taken from prior CFT and dilaton-sum-rule suggestions. No derivation or error control of this input is performed inside the QCD factorization framework itself; if the imported form factor deviates from the true non-Abelian TJJ contribution in the pion channel, every reported projection fails. This assumption is therefore load-bearing and requires explicit justification or independent lattice validation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on our manuscript. We respond to the major point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central hierarchy (cancellation of the isolated anomaly in A_π(Q²), lowering by the full TJJ insertion at small Q², importance in D_π(Q²), and dominance in the trace form factor) is obtained only after inserting an anomaly form factor taken from prior CFT and dilaton-sum-rule suggestions. No derivation or error control of this input is performed inside the QCD factorization framework itself; if the imported form factor deviates from the true non-Abelian TJJ contribution in the pion channel, every reported projection fails. This assumption is therefore load-bearing and requires explicit justification or independent lattice validation.

    Authors: We agree that the anomaly form factor is an external input adopted from momentum-space CFT and the perturbative dilaton sum rule, with no derivation or error control performed inside the QCD factorization framework. The factorization method is applied to the Sudakov-improved hard kernel for the pion, while the trace-anomaly piece is matched from these external calculations because a self-contained derivation of the non-Abelian TJJ matrix element within the same framework is not available. The lattice comparisons test the combined predictions rather than isolating the input. We will revise the manuscript (partial revision) to state the external origin more explicitly in the abstract and main text and to discuss the associated assumptions and uncertainties. revision: partial

Circularity Check

0 steps flagged

No significant circularity: anomaly form factor imported from external CFT/dilaton suggestions; results benchmarked against lattice QCD

full rationale

The derivation imports the anomaly form factor as an external input suggested by momentum-space conformal field theory and the perturbative dilaton sum rule, then inserts it into a Sudakov-resummation-improved factorization kernel to obtain A_π(Q²), D_π(Q²) and the trace form factor. These are compared directly to lattice QCD data, which serves as an independent external benchmark. No equation or step reduces the output to the input by construction, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The central hierarchy claim therefore remains falsifiable against the lattice results rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the external modeling of the anomaly form factor and on the validity of the QCD factorization framework for gravitational operators; no free parameters or new entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption The anomaly form factor is given by momentum-space conformal field theory and the perturbative dilaton sum rule.
    Used to supply the trace-anomaly component of the TJJ vertex.

pith-pipeline@v0.9.1-grok · 5657 in / 1256 out tokens · 27561 ms · 2026-06-28T09:46:03.988496+00:00 · methodology

discussion (0)

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