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Pressure inside hadrons: criticism, conjectures, and all that

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper defends the mechanical interpretation of the hadron's $D(t)$ form factor as a pressure and shear distribution, and it concludes that none of the published criticisms has invalidated that picture.

desk verdict A fair-minded defense of Polyakov's pressure interpretation that makes the von Laue argument crisply but does not close the case against the positive atomic D-term. read the letter →

arxiv 2501.04622 v1 pith:IWL7WEAG submitted 2025-01-08 hep-ph nucl-th

Cédric Lorcé (Ecole Polytechnique , CPHT) , Peter Schweitzer (Connecticut U.) This is my paper · ORCID
classification hep-phnucl-th
keywords D-termenergy-momentumtensorhadronpressureshearforcesvonLaueconditionBreitframegravitationalformfactorsmechanicalstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper defends the mechanical interpretation of the hadronic energy-momentum tensor, in which the form factor $D(t)$ encodes the radial distributions of isotropic pressure $p(r)$ and shear forces $s(r)$ inside a hadron. It reviews the main objections raised in the literature — recoil corrections in the Breit frame, the appearance of negative pressures, the mean-free-path objection from fluid dynamics, the unproven $D<0$ stability conjecture, and the positive $D$-term of the hydrogen atom — and proposes resolutions for each. The paper argues that negative pressure is not a flaw but a necessity for a bound system, because the equilibrium (von Laue) condition forces $p(r)$ to change sign; that $D<0$ follows for short-range-force systems from a local stability criterion; and that the hydrogen atom's positive $D$-term reflects long-range electromagnetic forces rather than a failure of the hadronic picture. The stated conclusion is that none of the criticism raised to date has really invalidated the mechanical interpretation. If that conclusion is correct, experimental extractions of pressure distributions inside the proton are measurements of genuine mechanical forces rather than formal artifacts.

What carries the argument

The load-bearing object is the $D(t)$ gravitational form factor of the energy-momentum tensor: its Breit-frame Fourier transform produces the static stress tensor written above, whose trace and quadrupole parts define $p(r)$ and $s(r)$. Two identities carry the argument: the von Laue equilibrium condition $\int_0^\infty dr\, r^2 p(r)=0$, which follows from momentum conservation and forces $p(r)$ to change sign, and the local stability criterion $p(r)+\frac23 s(r)\ge 0$, which together with $D=M\int d^3r\, r^2 p(r)$ implies $D<0$ for stable systems bound by short-range forces. The long-distance falloff of $p(r)$ and $s(r)$ is the third ingredient: with QED neglected, hadronic densities decay faster than $1/r^4$ so $D$ is finite, while the Coulomb $1/r^4$ tail makes $D$ undefined for charged systems and produces the behavior $D(t)\propto 1/\sqrt{-t}$ near $t\to 0$.

What would settle it

A lattice or model calculation at physical quark masses that extracts $D(t)$, inverse-Fourier-transforms it, and finds that $p(r)$ violates the von Laue condition $\int_0^\infty dr\, r^2 p(r)=0$, or that $p(r)+\frac23 s(r)<0$ throughout the region where the energy density sits, in a hadron known to be stable, would settle against the mechanical interpretation.

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Extended reading notes

Core claim

Under the assumption that a hadron can be treated as a continuous medium in the Breit frame, the spatial part of the energy-momentum tensor matrix element defines a static stress tensor $$$T^{{ij}}$(\vec r)=\$delta^{{ij}}$ p(r)+\left(\frac{r^i r^j}{$r^{2}$}-\frac13\$delta^{{ij}}$\right)s(r),$$ with $p(r)$ the isotropic pressure and $s(r)$ the shear. Conservation of the total energy-momentum tensor implies the von Laue condition $\int_0^\infty dr\, r^2 p(r)=0$, so $p(r)$ must be negative in some region; the paper reads that negativity as the cohesive stresses that bind the system, not as a thermodynamic contradiction. For systems bound by short-range forces, the local radial-pressure stability criterion $p(r)+\frac23 s(r)\ge 0$ then forces the $D$-term, $D=M\int d^3r\, r^2 p(r)$, to be negative, in line with models, lattice QCD, and dispersion relations. The paper addresses the Breit-frame recoil issue through phase-space quasi-probabilistic and large-$N_c$ arguments, and argues that the positive $D$-term of hydrogen arises from long-range QED tails and from the atom's being an extremely dilute, effectively two-body system rather than a hadron-like continuum. Its stated conclusion is that none of the reviewed criticisms has really invalidated the mechanical interpretation.

Load-bearing premise

The load-bearing premise is that the Fourier transform of the hadron's energy-momentum matrix element in the Breit frame can be interpreted as a local, static stress tensor of a continuous medium, so that elasticity-theory stability conditions apply to hadrons; this is justified in the large-$N_c$ (many-colors) limit but not proven for the physical three-color world.

Editorial extensions

If this is right

  • Experimental extractions of $D(t)$ from deeply virtual Compton scattering and related processes can be read as measurements of a genuine mechanical pressure and shear distribution inside the proton.
  • A negative $D$-term is not accidental: any stable hadron-like system bound by short-range forces and satisfying the radial-pressure criterion should have $D<0$, so lattice and model results that respect the von Laue condition should continue to find negative $D$.
  • The positive $D$-term of the hydrogen atom should not be treated as a counterexample to the hadronic interpretation, because atoms are long-range, dilute, effectively two-body systems rather than hadron-like continua.
  • For the proton, including QED makes the $D$-term itself undefined ($D(t)\propto 1/\sqrt{-t}$ near $t\to 0$), but this behavior sits in an experimentally unreachable region; the measurable $D(t)$ in the GeV region is still negative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: one could test the 'apples and oranges' explanation quantitatively by computing $p(r)$ and $s(r)$ in a solvable Coulombic system and checking whether the local radial-pressure criterion $p(r)+\frac23 s(r)\ge 0$ fails exactly where long-range forces dominate; if it holds there too, the atomic case would need a different explanation.
  • Beyond the paper: a fully relativistic definition of 3D energy-momentum densities that removes Breit-frame recoil corrections, for instance a light-front or phase-space quasi-distribution formulation, would extend the large-$N_c$ justification to physical $N_c=3$; the paper's defense would be sharper if both pictures agreed numerically for the proton.
  • Beyond the paper: high-precision lattice QCD at the physical pion mass that maps $D(t)$ over a wide range and reconstructs $p(r)$ could locate the node radius and the balance of repulsive and attractive pressures in QCD itself, connecting $D(t)$ more directly to confinement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. This paper reviews the criticism raised in the literature against the Polyakov interpretation of the hadronic energy-momentum tensor form factor D(t) in terms of pressure and shear force distributions, and defends that interpretation. The authors show that the von Laue condition forces the isotropic pressure p(r) to change sign in any bound system; that negative and anisotropic pressures are physically acceptable in contexts such as liquids and neutron stars; that the mean-free-path objection is bypassed by defining pressure directly from the microscopic EMT; that a negative D-term follows from the local stability criterion p_r(r) = p(r) + (2/3)s(r) >= 0 (an assumption they explicitly concede); and that the positive D-term of the hydrogen atom may be reconciled by arguing that atoms are dilute two-body systems and therefore not mechanical continua. The paper concludes that none of the raised criticisms has invalidated the mechanical interpretation.

Significance. If the D-term interpretation is correct, this paper provides a valuable synthesis of the debate and a rigorous discussion of the von Laue condition, the necessary conditions for mechanical equilibrium, and the role of long-range forces in making D(t) singular. The authors are transparent about the assumptions underlying the stability bound, and the rebuttal of the 'pressure is always positive' objection is sound. However, the central defense rests on unproven conjectures: the local stability criterion, the applicability of the 3D Breit-frame interpretation to N_c = 3 QCD, and the purported distinction between atoms and hadrons. Because the positive atomic D-term is computed from the same Breit-frame definition (Eq. 2) and satisfies the same conservation laws, the paper has not fully rebutted the strongest counterexample. The work is a useful contribution to the ongoing discussion but cannot be considered a conclusive vindication of the interpretation.

major comments (5)
  1. [Sec. 10, Eq. (8) and Sec. 13] The local stability criterion p_r(r) >= 0 is stated as an assumption, and the paper concedes that no quantum field theoretical proof exists in QCD. Since the hydrogen atom is a stable bound state that satisfies the von Laue condition (7) yet violates Eq. (8), this criterion is not a necessary stability condition in general. The conclusion in Sec. 13 that 'none of the raised criticism to date has really invalidated' the mechanical interpretation is therefore stronger than the evidence: the D < 0 conjecture rests on an unproven assumption, and the atomic counterexample shows that the implication 'stability implies Eq. (8)' is not universally valid.
  2. [Sec. 12 and Sec. 13] The reconciliation of the positive atomic D-term is a conjecture stated as 'apples and oranges,' and the paper itself admits that 'more work may be needed.' No quantitative criterion is provided for deciding when the continuum mechanical interpretation applies. Since the Breit-frame definition (Eq. 2), EMT conservation, and the von Laue condition (Eq. 7) are applied identically to atoms and hadrons, the positive atomic D-term remains a live counterexample. To sustain the Sec. 13 conclusion, the authors would need to specify a testable condition—for example, based on density, occupation number, or separation of scales—that excludes the hydrogen atom from the class of systems to which Eq. (8) applies.
  3. [Sec. 8] The reconstruction of the bag contribution \bar{C}_bag(t) from the 3D interpretation is described as providing 'an independent proof of EMT conservation in the bag model,' but this is circular: the Fourier transform (Eq. 2) and the 3D interpretation are exactly the items under debate. The calculation is an internal consistency check that is conditional on the validity of the interpretation, not an independent verification of it. The claim of independence should be tempered or the reasoning clarified.
  4. [Sec. 5 and Sec. 9] The paper adopts the 3D Breit-frame interpretation on the basis that it is rigorous in the large-N_c limit, but the extrapolation to N_c = 3 is not proven, and the phase-space quasi-probabilistic interpretation is only referenced, not developed. The mean-free-path objection is dismissed with the assertion that pressure is defined from the microscopic EMT, but this does not establish that the operator T^{ij}(r) has the operational meaning of a local mechanical pressure in a dilute system. Since this premise is load-bearing for the entire defense, the conclusion should be phrased as conditional on the validity of the 3D interpretation rather than as an unconditional rebuttal.
  5. [Footnote 16 and Sec. 10] Positive D-terms in covariant quark-diquark model calculations are acknowledged, but the suggestion that these models 'do not always describe the nucleon as a dynamical quark-diquark bound state compliant with the virial theorem' or that 'additional model assumptions are needed' is not substantiated. If hadron models with positive D exist and satisfy the basic conservation laws, the claim that a negative D-term follows from mechanical stability for all systems governed by short-range forces loses universality. This caveat should be elevated to the main text and analyzed quantitatively, not relegated to a footnote.
minor comments (5)
  1. [Sec. 2, p. 2] The text 'Institute for Theoretical Physcs II' contains typos; it should read 'Institute for Theoretical Physics II.'
  2. [Sec. 7, p. 10] The phrase 'A van der der Waals gas' contains a duplicated 'der.'
  3. [Sec. 8, p. 13] The notation \bar{c}_q(t) appears in the expression \bar{C}_bag(t) = -\sum_q \bar{c}_q(t), but Eq. (1) uses \bar{C}_a(t) for the corresponding form factor. Define the lower-case notation or use consistent symbols.
  4. [Sec. 10, p. 17] The step from 4\pi \int_0^\infty dr\, r^4 p_r(r) > 0 to (-3D)/(2M) is stated without intermediate algebra; for a review article, this derivation should be shown explicitly because the sign is the central assertion.
  5. [Sec. 13, p. 23] The introduction states that 'concerns raised in the literature' include Ref. [48], but Sec. 13 refers only to 'criticism raised in the recent literature [41–47].' The reference list in Sec. 4 includes [48], so the conclusion should also cite it.

Circularity Check

3 steps flagged · score 4.0 of 10

The paper's central defense leans on self-citations for the Breit-frame stress interpretation and for the negative-D stability theorem, while the atomic D-term counterexample is set aside by an admittedly intuitive conjecture.

  1. self citation load bearing [Sec. 5, 'Is the use of the Breit frame problematic?']
    "A theoretically rigorous justification of a 3D density interpretation in the Breit frame in the case of the nucleon consists in working in the limit of a large number of colors Nc [5, 18, 33]."

    This is the paper's central justification for treating T^{ij}(r) in Eq. (2) as a local mechanical stress. The cited references are authored or co-authored by Polyakov, Schweitzer, and Lorcé, i.e. the present authors or their immediate collaborators, and they adopt the Breit-frame interpretation rather than establishing it from an independent external theorem. No derivation of this large-Nc justification is provided in the present paper; the claim rests on the same group's prior work, making the justification load-bearing self-citation.

  2. other [Sec. 8, 'Poincaré stresses in bag model']
    "Fortunately, at this point the 3D interpretation of EMT densities comes to the rescue. The bag contribution T^{mu nu}_{bag} = B Theta(R-r) g^{mu nu} to the EMT can be Fourier transformed and yields exactly Cbar_bag(t) = - sum_q cbar_q(t) [23]. This corresponds to inverting the EMT form factor interpretation and is, to the best of our knowledge, the only way to compute Cbar_bag(t) in this model. This provides an independent proof of EMT conservation in the bag model and illustrates the theoretical consistency of the EMT distribution formalism."

    The bag EMT is posited as B Theta(R-r) g^{mu nu}; Fourier transforming this ansatz under the very interpretation at issue yields Cbar_bag. The agreement with the missing piece required by EMT conservation is therefore a self-consistency check built from the same formalism, not an independent proof. The 'only way to compute' Cbar_bag is via the interpretation being defended, so the check cannot validate that interpretation against external evidence outside the model.

1 more flagged steps
  1. self citation load bearing [Sec. 10, 'The negative D-term sign conjecture']
    "The above observations were solidified by Irina Perevalova, Maxim et al. in [14], where the conclusion that D < 0 was drawn based on the assumptions that (i) the pressure interpretation is exact, (ii) mechanical stability criteria can be applied to hadrons, (iii) the densities p(r) and s(r) decay at long-distances faster than 1/r^4."

    This is the central theorem invoked in Sec. 13 to conclude that no criticism has invalidated the interpretation. Reference [14] is co-authored by P. Schweitzer, a present author, and its assumption (i) is exactly the interpretation the paper defends. The hydrogen-atom counterexample challenges assumption (ii); the paper's Sec. 12 response is an 'admittedly intuitive' apples-and-oranges conjecture, not a quantitative criterion. Thus the conclusion leans on a self-citation chain for its key premise, even though the D < 0 arithmetic itself is not circular.

full rationale

The paper's internal derivation from the Breit-frame EMT to D < 0 is mathematically coherent: given the von Laue condition and the local stability criterion p_r(r) >= 0, the negativity of D follows by straightforward integration. That step is not circular, and the paper is transparent about the assumptions entering the Perevalova-Polyakov-Schweitzer argument. However, the load-bearing premise that T^{ij}(r) can be interpreted as a local mechanical stress is justified in Sec. 5 only by citing the authors' own prior large-Nc works, and the negative-D theorem in Sec. 10 is imported from a paper co-authored by one of the present authors. The Sec. 8 bag-model 'independent proof' is similarly a consistency check using the same interpretation rather than an external validation. The atomic D-term remains a live counterexample, and the paper concedes that 'more work may be needed' and calls its reconciliation 'admittedly intuitive.' These limitations are honestly stated, so the paper is not an equivalence-by-construction; the score reflects load-bearing self-citation rather than a hidden fitted-input circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new free parameters or entities are introduced. The paper's central claim rests on domain assumptions about the validity of the Breit-frame density interpretation and the applicability of continuum mechanics to hadrons, several of which are explicitly the subject of the criticisms under review. The 'apples and oranges' resolution of the hydrogen-atom D-term is a conjecture labeled as intuitive by the authors.

assumptions (5)
  • domain assumption Breit-frame 3D density interpretation of EMT matrix elements, with T^ij(r) as a local stress tensor of a continuous medium.
    Adopted in Sec. 5 ('In what follows, we will use the 3D interpretation in the Breit frame') and used in Eqs. (2)-(8). Large-Nc and phase-space justifications are cited, but the paper explicitly acknowledges recoil concerns and does not prove the interpretation for Nc=3.
  • domain assumption The nucleon can be treated as a mechanical continuum system for which the von Laue condition (7) and the local stability criterion (8) apply.
    Stated as assumption (ii) in Sec. 10 in the discussion of ref. [14]; the paper endorses it. If this fails, the derivation of D < 0 does not apply.
  • domain assumption Densities p(r) and s(r) decay faster than 1/r^4 when QED is neglected, so the D-term integrals converge.
    Assumption (iii) of ref. [14], discussed in Secs. 10-11. The paper notes QED produces 1/r^4 tails that make D undefined for the proton, so the assumption is not valid when long-range forces are included.
  • ad hoc to paper The positive D-term of the hydrogen atom does not invalidate the hadron interpretation because atoms and hadrons are fundamentally different (short-range vs long-range forces, dilute vs dense media).
    Sec. 12: the 'apples and oranges' argument is labeled 'admittedly intuitive'; the paper says 'more work may be needed'.
  • standard math Standard form factor decomposition of the nucleon EMT (Eq. 1) and conservation of the total EMT.
    Eq. (1) from refs. [55,56] and conservation used in Eq. (4).

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Cite this review

Pith. "Pith review of Pressure inside hadrons: criticism, conjectures, and all that." pith.science (2026). https://pith.science/paper/IWL7WEAG

@misc{pith2026250104622,
  author       = {Pith},
  title        = {Pith review of: Pressure inside hadrons: criticism, conjectures, and all that},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWL7WEAG}},
  note         = {Machine review of arXiv:2501.04622}
}
abstract

The interpretation of the energy-momentum tensor form factor $D(t)$ of hadrons in terms of pressure and shear force distributions is discussed, concerns raised in the literature are reviewed, and ways to reconcile the concerns with the interpretation are indicated.

Figures

Figures reproduced from arXiv: 2501.04622 by the authors.

Figure 1
Figure 1. (a) Copy of Landau & Lifshitz’s volume 7, Elasticity Theory (in Ger [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. a: throughout s(r) ≥ 0, while p(r) > 0 in the inner region and p(r) < 0 in the outer region, whereby the δ- and Θ-functions of the ideal liquid drop, see Sec. 7, are strongly smeared out in the nucleon. This is not surprizing after all, as the nucleon is considerably more diffuse than a nucleus. Fig. 2b shows how the internal forces balance each other in the nucleon: p(r) > 0 in the center region corresponding to re… view at source ↗

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