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arxiv: 2604.03832 · v2 · submitted 2026-04-04 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

Gravitational transverse momentum distribution of proton

Kauship Saha , Dipankar Chakrabarti , Asmita Mukherjee
Authors on Pith no claims yet

Pith reviewed 2026-05-13 17:05 UTC · model grok-4.3

classification ✦ hep-ph
keywords gravitational TMDslight-front quark-diquark modelproton structurequark distributionstransverse momentumenergy-momentum tensorgravitational PDFsAdS/QCD
0
0 comments X

The pith

The light-front quark-diquark model yields analytical expressions for six unpolarized gravitational transverse-momentum distributions of up and down quarks in the proton.

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

This work computes the first set of gravitational transverse-momentum-dependent distributions for quarks inside the proton. These distributions encode how quarks contribute to the energy-momentum tensor at finite transverse momentum. The authors obtain closed-form expressions in a light-front quark-diquark model, verify that the results obey known relations to ordinary quark TMDs, and extract moments that describe transverse pressure and shear forces. A sympathetic reader cares because gravitational TMDs supply a momentum-space picture of the proton's internal stresses, complementary to the integrated gravitational form factors measured in deeply virtual processes. The calculation supplies concrete functions of x and k_perp that can be compared with future lattice or experimental data.

Core claim

In the light-front quark-diquark model inspired by soft-wall AdS/QCD, analytical expressions are derived for the six unpolarized T-even gravitational TMDs of up and down quarks. The corresponding gravitational PDFs are obtained by integration over transverse momentum. These distributions satisfy model-independent relations with standard quark TMDs. Their first moments connect directly to the transverse isotropic pressure and shear-force distributions in momentum space, while further moments yield the average longitudinal momentum carried by each quark flavor.

What carries the argument

Light-front quark-diquark model (LFQDM) wave functions, which encode the proton's quark-diquark Fock components and allow direct evaluation of the gravitational energy-momentum tensor operators.

If this is right

  • The six gravitational TMDs satisfy exact model-independent relations to the ordinary unpolarized quark TMDs.
  • First moments of the gravitational TMDs give the transverse isotropic pressure and shear-force distributions in momentum space.
  • Integration of the gravitational TMDs recovers the average longitudinal momentum fractions carried by up and down quarks.
  • Numerical gravitational PDFs are obtained as explicit functions of the longitudinal momentum fraction x.

Where Pith is reading between the lines

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

  • The momentum-space pressure and shear distributions could be folded with experimental resolutions to predict observable effects in high-precision deep-inelastic or Drell-Yan data.
  • Global TMD fits could be extended to include gravitational versions, tightening constraints on the proton's energy-momentum tensor at non-zero momentum transfer.
  • The same framework could be applied to polarized gravitational TMDs to explore spin-dependent stresses inside the nucleon.

Load-bearing premise

The light-front quark-diquark model, when its parameters are fixed by ordinary quark distributions, also reproduces the correct matrix elements of the gravitational energy-momentum tensor.

What would settle it

A lattice QCD calculation of any of the six gravitational TMDs or their moments at moderate x and k_perp that deviates significantly from the model's closed-form expressions.

Figures

Figures reproduced from arXiv: 2604.03832 by Asmita Mukherjee, Dipankar Chakrabarti, Kauship Saha.

Figure 1
Figure 1. Figure 1: FIG. 1: The gravitational transverse momentum distributions [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The gravitational transverse momentum distributions [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The unpolarized gravitational transverse-momentum distributions [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The unpolarized gravitational transverse-momentum distributions [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Unpolarized gravitational parton distribution functions [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Left panel: Isotropic transverse pressure for the down quark shown as a two-dimensional contour plot in the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Left panel: Isotropic transverse pressure for the down quark shown as a function of the momentum fraction [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

We present the first study of quark gravitational transverse-momentum distributions within the light-front quark--diquark model (LFQDM) inspired by the soft-wall AdS/QCD framework. We derive analytical expressions for the six unpolarized (T-even) gravitational transverse-momentum-dependent distributions (gravitational--TMDs) for up and down quarks within the model and compute the corresponding gravitational parton distribution functions (gravitational--PDFs). We further verify that these unpolarized gravitational--TMDs satisfy the model-independent relations with quark TMDs. In addition, we explore the connection of gravitational TMDs with the transverse isotropic pressure and shear-force distributions in momentum space, as well as with the average longitudinal momentum carried by up and down quarks within the model.

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

3 major / 2 minor

Summary. The manuscript presents the first study of quark gravitational transverse-momentum distributions within the light-front quark-diquark model (LFQDM) inspired by the soft-wall AdS/QCD framework. It derives analytical expressions for the six unpolarized T-even gravitational TMDs for up and down quarks, computes the corresponding gravitational PDFs, verifies that these satisfy model-independent relations with ordinary quark TMDs, and explores their connections to transverse isotropic pressure, shear-force distributions in momentum space, and average longitudinal momentum fractions carried by quarks.

Significance. If the LFQDM wave functions can be reliably extended to symmetric energy-momentum tensor matrix elements, the work supplies the first analytical gravitational TMD expressions in this framework and illustrates potential links between partonic gravitational distributions and mechanical properties of the proton such as pressure and shear in momentum space.

major comments (3)
  1. [Model setup and parameter determination section] The LFQDM parameters (quark and diquark masses, wave-function scales) are fixed by fitting to standard quark PDFs and electromagnetic form factors; the manuscript does not provide an independent check that the same light-front wave functions reproduce the required symmetric EMT matrix elements for gravitational TMDs, rendering the gravitational results an extrapolation whose accuracy is not separately constrained.
  2. [Section on verification of relations to quark TMDs] The verification that the derived gravitational TMDs satisfy model-independent relations with quark TMDs holds by construction once the same overlap integrals of the LFQDM wave functions are used for both sets; this does not constitute an external test of the gravitational sector.
  3. [Section on pressure, shear, and momentum fractions] The claimed connections of gravitational TMDs to transverse pressure and shear-force distributions in momentum space, as well as to average longitudinal momentum, rest entirely on the same fitted LFQDM inputs without additional cross-checks against lattice results for gravitational form factors or other independent determinations.
minor comments (2)
  1. [Abstract] Clarify in the abstract and introduction that the 'model-independent relations' are verified within the model rather than tested against external data.
  2. [Numerical results and figures] Include sensitivity plots or parameter-variation bands on the numerical results for gravitational TMDs and PDFs to illustrate robustness.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below, clarifying the phenomenological nature of the LFQDM and the scope of our calculations. Partial revisions will be made to improve clarity without altering the core results.

read point-by-point responses

  1. Referee: [Model setup and parameter determination section] The LFQDM parameters (quark and diquark masses, wave-function scales) are fixed by fitting to standard quark PDFs and electromagnetic form factors; the manuscript does not provide an independent check that the same light-front wave functions reproduce the required symmetric EMT matrix elements for gravitational TMDs, rendering the gravitational results an extrapolation whose accuracy is not separately constrained.

    Authors: We agree that the LFQDM parameters are fixed from fits to standard quark PDFs and electromagnetic form factors, as is standard in such phenomenological models. The gravitational TMDs are then obtained as predictions from the same light-front wave functions. This approach is consistent with the model's prior validations against multiple observables. The gravitational results are indeed model extrapolations, and we will add a clarifying statement in the model setup section to explicitly note that no independent constraint from gravitational form factors is applied in this work. revision: partial

  2. Referee: [Section on verification of relations to quark TMDs] The verification that the derived gravitational TMDs satisfy model-independent relations with quark TMDs holds by construction once the same overlap integrals of the LFQDM wave functions are used for both sets; this does not constitute an external test of the gravitational sector.

    Authors: We acknowledge that the verification is internal to the model, arising from the shared wave-function overlaps. Its purpose is to confirm that the newly derived analytical expressions for the gravitational TMDs are consistent with the established model-independent relations. This step validates the correctness of our definitions and calculations rather than providing an external test. We will revise the relevant section to state this more explicitly as an internal consistency check. revision: partial

  3. Referee: [Section on pressure, shear, and momentum fractions] The claimed connections of gravitational TMDs to transverse pressure and shear-force distributions in momentum space, as well as to average longitudinal momentum, rest entirely on the same fitted LFQDM inputs without additional cross-checks against lattice results for gravitational form factors or other independent determinations.

    Authors: The connections to pressure, shear-force distributions, and momentum fractions are derived and illustrated within the LFQDM using the computed gravitational TMDs, offering model-specific insights into proton mechanical properties in momentum space. No direct comparisons to lattice gravitational form factors are included, as the manuscript focuses on analytical expressions and their implications inside the model. We will add a short paragraph noting the model-dependent character of these results and the scope for future lattice comparisons. revision: partial

Circularity Check

2 steps flagged

Gravitational TMDs and PDFs computed from same LFQDM wave functions fitted to ordinary distributions

specific steps
  1. fitted input called prediction [Abstract and model section]
    "We derive analytical expressions for the six unpolarized (T-even) gravitational transverse-momentum-dependent distributions (gravitational--TMDs) for up and down quarks within the model and compute the corresponding gravitational parton distribution functions (gravitational--PDFs)."

    LFQDM parameters are chosen to reproduce known quark distributions; gravitational TMDs and PDFs are then obtained by direct substitution of the same fitted wave functions, so the gravitational results are not independent predictions but recombinations of the ordinary-sector fit.

  2. fitted input called prediction [Results and verification paragraph]
    "We further verify that these unpolarized gravitational--TMDs satisfy the model-independent relations with quark TMDs."

    The relations are satisfied automatically once the same overlap integrals are used for both gravitational and ordinary TMDs; the verification therefore reduces to an algebraic identity inside the model rather than an external test of the gravitational sector.

full rationale

The paper constructs all gravitational TMD expressions inside the light-front quark-diquark model whose parameters are fixed by matching to standard quark PDFs and form factors. The claimed verification of model-independent relations between gravitational TMDs and ordinary TMDs holds by construction because both sets are obtained from identical overlap integrals of the same wave functions. No external benchmark or independent constraint on the symmetric EMT matrix elements is provided; the gravitational sector is therefore an extrapolation whose numerical content is forced by the ordinary-sector fit. This produces partial circularity (score 6) without full self-definition of the final observables.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the LFQDM being a valid effective description; model parameters are implicitly fitted to reproduce known quark distributions and form factors.

free parameters (1)
  • LFQDM parameters (quark and diquark masses, wave-function scales)
    Typical in soft-wall AdS/QCD inspired models; chosen to fit standard PDFs and electromagnetic form factors before computing gravitational quantities.
axioms (1)
  • domain assumption The light-front quark-diquark model provides a reliable framework for unpolarized gravitational TMDs
    Invoked as the basis for all derivations and verifications in the abstract.

pith-pipeline@v0.9.0 · 5424 in / 1214 out tokens · 39787 ms · 2026-05-13T17:05:05.084649+00:00 · methodology

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