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arxiv: 2606.10993 · v1 · pith:OVO5HQWAnew · submitted 2026-06-09 · ✦ hep-ph · hep-th

The Gravitational Form Factor of the Pion in Perturbative QCD with a Dilaton Interaction

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

Pith reviewed 2026-06-27 12:12 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords gravitational form factorspiontrace anomalyQCD factorizationdilatonTJJ correlatorD-termSudakov resummation
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0 comments X

The pith

The QCD trace anomaly induces a dilaton-like scalar contribution that significantly modifies the pion gravitational form factors at large momentum transfer.

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

This paper examines the pion's gravitational form factors using QCD factorization applied to the non-Abelian TJJ correlator. The correlator captures the perturbative effects of the trace anomaly, which adds a scalar dilaton-like term to the hard-scattering kernel. By separating quark and gluon sectors while respecting gauge-fixing and Slavnov-Taylor identities, the analysis incorporates Sudakov resummation and a Gaussian transverse-momentum-dependent wave function for the pion. These steps regulate endpoint divergences and yield predictions showing that the anomaly corrections change the large-momentum behavior of the form factors, especially in the trace sector and D-term. Readers would care because this reveals how quantum anomalies influence the gravitational structure of light hadrons in a calculable regime.

Core claim

We demonstrate that the non-Abelian TJJ correlator, which couples the local QCD energy-momentum tensor to two external gluon fields, explicitly encodes the perturbative effects of the trace anomaly and induces a scalar, dilaton-like contribution to the hard-scattering kernel. After performing a careful separation of the quark and gluon sectors that accounts for modifications from gauge-fixing terms and Slavnov-Taylor identities, the anomaly-induced corrections are shown to significantly modify the behavior of the pion GFFs at large momentum transfer, leaving a unique imprint on the trace sector and the D-term.

What carries the argument

The non-Abelian TJJ correlator encoding the trace anomaly effects as a scalar dilaton-like contribution to the hard kernel in the QCD factorization of pion GFFs.

If this is right

  • The D-term receives significant anomaly-induced modifications at large momentum transfer.
  • The trace sector of the GFFs exhibits a distinctive imprint from the dilaton-like interaction.
  • Phenomenological predictions for pion GFFs at intermediate and large momentum transfers are altered by these corrections.
  • Consistency with Slavnov-Taylor identities is maintained through the separation of quark and gluon sectors.

Where Pith is reading between the lines

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

  • If the modifications hold, lattice calculations of pion GFFs at high momentum may need to account for these anomaly effects to match perturbative predictions.
  • This approach could be extended to other mesons or baryons to study anomaly impacts on their gravitational properties.
  • Future high-energy experiments probing pion structure might detect the unique trace sector signature.

Load-bearing premise

The Sudakov resummation framework combined with a Gaussian model for the pion's transverse-momentum-dependent wave function regulates soft-gluon endpoint divergences and yields realistic predictions at intermediate and large momentum transfer.

What would settle it

A high-momentum-transfer measurement or lattice QCD result for the pion D-term or trace sector that shows no deviation from standard perturbative QCD calculations without the dilaton-like anomaly term would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.10993 by Claudio Corian\`o, Dario Melle, Hsiang-nan Li, Leonardo Torcellini.

Figure 1
Figure 1. Figure 1: Diagrams of the hard scattering contribution to the pion GFF corresponding to the order [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Soft-overlap contribution diagrams to the pion form factor. The left panel shows the lowest-order contribution, [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: TJJ contribution to the GFF of the pion. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Q2 = −t. In the momentum form factor Aπ, the isolated anomaly cancels and does not generate a separate contribution. The full T JJ insertion nevertheless lowers the leading Sudakov curve at small Q2 , improving the local overlap with the lattice data in the first kinematic window. The curves are obtained for a2(µ0) = 0.2, µ0 = 1 GeV, mq = 0.33 GeV and βπ = 0.24 GeV−1 . ● Hackett et al. LO pQCD (Sudakov) LO… view at source ↗
Figure 5
Figure 5. Figure 5: In the Dπ form factor the anomaly gives an important contribution. The displayed full T JJ correction is more modest and should not be read as the complete order-α 2 s hard kernel; additional non-anomalous terms may improve the comparison with the data. The curves are obtained for a2(µ0) = 0.2, µ0 = 1 GeV, mq = 0.33 GeV and βπ = 0.24 GeV−1 . 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The trace form factor Θπ is the channel in which the anomaly is most significant: it dominates the T JJ contribution at both low and high momentum transfer. Treating the anomaly as a constrained contribution in the hard part improves the trace prediction and helps the interpretation of the Dπ form factor. The curves are obtained for a2(µ0) = 0.2, µ0 = 1 GeV, mq = 0.33 GeV and βπ = 0.24 GeV−1 . 27 [PITH_FU… view at source ↗
Figure 7
Figure 7. Figure 7: Effective pole representation of the anomalous [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
read the original abstract

We investigate the pion gravitational form factors (GFFs) at intermediate and large momentum transfer within the framework of QCD factorization. Our analysis centers on the non-Abelian $TJJ$ correlator, which couples the local QCD energy-momentum tensor to two external gluon fields and explicitly encodes the perturbative effects of the trace anomaly. We demonstrate how this quantum anomaly induces a scalar, dilaton-like contribution to the hard-scattering kernel. To ensure field-theoretic consistency, a careful separation of the quark and gluon sectors is performed, accounting for the modifications introduced by gauge-fixing terms and Slavnov-Taylor identities on the off-shell gluonic structure. To obtain realistic phenomenological predictions and regulate soft-gluon endpoint divergences in the hard kernel, we implement the Sudakov resummation framework coupled with a Gaussian model for the pion's transverse-momentum-dependent wave function. We show that the resulting anomaly-induced corrections significantly modify the behavior of the pion GFFs at large momentum transfer, leaving a unique imprint on the trace sector and the $D$-term.

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 / 2 minor

Summary. The manuscript investigates the pion gravitational form factors (GFFs) in perturbative QCD factorization, focusing on the non-Abelian TJJ correlator that encodes the trace anomaly. It introduces a scalar, dilaton-like contribution to the hard-scattering kernel, performs a separation of quark and gluon sectors consistent with gauge-fixing and Slavnov-Taylor identities, and employs Sudakov resummation together with a Gaussian model for the pion TMD wave function to regulate soft-gluon endpoint divergences. The central claim is that the resulting anomaly-induced corrections significantly modify the GFFs at large momentum transfer, producing a distinctive imprint on the trace sector and the D-term.

Significance. If the derivation and numerical implementation hold, the work would supply a concrete mechanism by which the QCD trace anomaly leaves an observable signature in hadronic GFFs at accessible momentum transfers. The explicit treatment of the TJJ correlator and the separation of sectors constitute a technical strength; the combination of factorization with Sudakov resummation is a standard and controlled approach when the wave-function model is independently validated.

major comments (2)
  1. [phenomenological implementation and numerical results] The central claim that anomaly-induced corrections produce a 'significant' and 'unique' modification to the trace-sector GFFs and D-term at large Q² rests on the Gaussian TMD ansatz both regulating endpoint divergences and preserving the correct power counting after Sudakov resummation. No robustness test against alternative transverse-momentum profiles (e.g., power-law tails) is presented; a different endpoint behavior would alter the relative weight of the dilaton-like scalar piece in the hard kernel.
  2. [TMD model and parameter choice] The Gaussian TMD wave-function parameters are introduced both to tame soft-gluon divergences and to generate phenomenological predictions. It is not shown whether these parameters are determined from independent observables or whether they are effectively tuned to the same GFF data against which the anomaly imprint is later compared, raising a potential circularity issue for the claimed modification.
minor comments (2)
  1. [Abstract] The abstract states that corrections 'significantly modify' the GFFs but supplies neither explicit expressions for the modified hard kernel nor numerical values or plots; the reader must reach the body of the paper to locate the quantitative evidence.
  2. [formalism] Notation for the separation of quark and gluon sectors and the precise definition of the dilaton-like scalar contribution should be collected in a single equation or table for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [phenomenological implementation and numerical results] The central claim that anomaly-induced corrections produce a 'significant' and 'unique' modification to the trace-sector GFFs and D-term at large Q² rests on the Gaussian TMD ansatz both regulating endpoint divergences and preserving the correct power counting after Sudakov resummation. No robustness test against alternative transverse-momentum profiles (e.g., power-law tails) is presented; a different endpoint behavior would alter the relative weight of the dilaton-like scalar piece in the hard kernel.

    Authors: The Gaussian TMD ansatz is a standard choice in the Sudakov-resummed factorization framework for the pion, as it ensures the correct exponential suppression of the soft endpoint region while preserving the leading power counting. The dominant regulation of endpoint divergences arises from the Sudakov factor itself rather than the specific transverse profile. Nevertheless, we acknowledge that an explicit robustness check would strengthen the presentation. In the revised manuscript we have added a paragraph discussing the stability of the anomaly-induced correction under moderate variations of the TMD width parameter (within the range constrained by independent observables) and note that the qualitative imprint on the trace sector persists for profiles that maintain the same leading-power behavior after resummation. revision: partial

  2. Referee: [TMD model and parameter choice] The Gaussian TMD wave-function parameters are introduced both to tame soft-gluon divergences and to generate phenomenological predictions. It is not shown whether these parameters are determined from independent observables or whether they are effectively tuned to the same GFF data against which the anomaly imprint is later compared, raising a potential circularity issue for the claimed modification.

    Authors: The Gaussian parameters are taken from independent determinations in the literature, specifically from global fits to the pion electromagnetic form factor and the leading-twist pion distribution amplitude performed in earlier works using the same Sudakov framework. They are not adjusted to reproduce the gravitational form factors. We have inserted a new paragraph in Section 3 of the revised manuscript that explicitly states the provenance of these parameters, quotes the numerical values and their sources, and verifies consistency with other low-energy observables, thereby eliminating any appearance of circularity. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper derives the scalar dilaton-like contribution to the hard kernel from the non-Abelian TJJ correlator and trace anomaly in perturbative QCD, with separation of quark/gluon sectors via gauge-fixing and Slavnov-Taylor identities. Sudakov resummation plus Gaussian TMD wave function is introduced explicitly as a regulator for soft-gluon endpoints to enable phenomenology; this is a standard ansatz choice, not a parameter fitted to the GFF outputs or the anomaly modification itself. No equations reduce the claimed large-Q^2 imprint on trace-sector GFFs and D-term to the model inputs by construction, and no load-bearing self-citations or uniqueness theorems are invoked in the provided text. The framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard QCD factorization, the identification of the trace anomaly with a scalar kernel term, and the choice of a Gaussian TMD model whose parameters are not independently constrained in the abstract.

free parameters (1)
  • Gaussian TMD wave-function parameters
    Introduced to model the pion transverse-momentum dependence and regulate endpoint divergences
axioms (2)
  • domain assumption QCD factorization applies to the off-shell TJJ correlator at intermediate and large momentum transfer
    Invoked as the framework for the entire analysis
  • domain assumption Sudakov resummation plus Gaussian model sufficiently regulates soft-gluon divergences
    Required to obtain finite phenomenological predictions
invented entities (1)
  • dilaton-like scalar contribution no independent evidence
    purpose: To encode the trace-anomaly effect in the hard-scattering kernel
    Induced by the non-Abelian TJJ correlator; no independent experimental handle supplied in the abstract

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discussion (0)

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Reference graph

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