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arxiv: 2605.04163 · v1 · submitted 2026-05-05 · ✦ hep-ph · hep-lat· hep-th

Pressure-Energy Equations of State of the Nucleon

Keh-Fei Liu This is my paper

Pith reviewed 2026-05-08 17:53 UTC · model grok-4.3

classification ✦ hep-ph hep-lathep-th
keywords nucleon structurepressure-energy equation of stategravitational form factorsenergy-momentum tensorQCD confinementtrace anomalyLambda-CDM cosmologysuperconducting vortices
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The pith

Pressure-energy equations of state in the nucleon match those in cosmology and type-II superconductors.

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

The paper derives pressure-energy equations of state inside the nucleon from gravitational form factors that encode matrix elements of the energy-momentum tensor together with its conservation. Two components appear: a static pressure equal to the negative of the trace part of the energy density, interpreted as arising from depletion of gluon and quark condensates and central to QCD confinement, plus a dynamic pressure equal to one over the spatial dimension of the traceless energy density. These two balance to give the total pressure distribution. The identical relations hold for vortices in type-II superconductors, where they arise from Cooper-pair condensate depletion, and in the Lambda-CDM cosmology, where they arise from the cosmological constant. A reader would care because the work identifies a shared pressure-energy structure across particle physics, condensed matter, and the expanding universe.

Core claim

The pressure-energy equations of state in the nucleon are derived from the gravitational form factors, which parameterize matrix elements of the energy-momentum tensor (EMT), together with EMT conservation. There are two distinct components in the pressure and energy densities. The static pressure distribution arising from the Lorentz trace part of the EMT, as manifested in the spatial stress 1/3 T^{ii}, is equal to minus the corresponding trace part of the energy density. This trace-anomaly and sigma-term induced pressure plays a fundamental role in the confinement dynamics of QCD. In contrast, the dynamic pressure distribution from the traceless part of the spatial stress tensor equals 1/d

What carries the argument

gravitational form factors parameterizing matrix elements of the energy-momentum tensor together with EMT conservation, separating trace and traceless contributions

If this is right

  • The static component of pressure inside the nucleon is induced by the trace anomaly and sigma term and is essential to QCD confinement.
  • Total pressure in the nucleon is the sum of a negative static term and a positive dynamic term scaled by spatial dimension.
  • The same pressure-energy relations govern vortices in type-II superconductors via Cooper-pair condensate depletion.
  • The identical equations of state appear in the Lambda-CDM model via the cosmological constant.

Where Pith is reading between the lines

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

  • Condensate depletion may supply a general mechanism for negative-pressure contributions across different physical systems.
  • The relations could be checked in other QCD bound states or in analog condensed-matter systems.
  • Lattice simulations of the gravitational form factors could directly test the predicted spatial profiles of the two pressure components.

Load-bearing premise

The static pressure arises purely from depletion of gluon and quark condensates through the stress-volume relation and fully accounts for confinement dynamics.

What would settle it

A lattice QCD calculation or experimental extraction of the pressure and energy density profiles inside the nucleon in which the static pressure is not equal to the negative of the trace part of the energy density.

Figures

Figures reproduced from arXiv: 2605.04163 by Keh-Fei Liu.

Figure 1
Figure 1. Figure 1: Left panel: 4πr2 p(r)/M and 4πr2 ptr(r)/M plotted separately as functions of r. Right panel: the total 4πr2 p(r)/M as a function of r. The latter agrees with the result obtained from the lattice calculation of De(r) [44]. The combined pressure p(r) = p(r) + ptr(r), shown in the right panel of view at source ↗
read the original abstract

The pressure-energy equations of state in the nucleon are derived from the gravitational form factors, which parameterize matrix elements of the energy-momentum tensor (EMT), together with EMT conservation. There are two distinct components in the pressure and energy densities. The static pressure distribution arising from the Lorentz trace part of the EMT, as manifested in the spatial stress 1/3 $T^{ii}$, is equal to minus the corresponding trace part of the energy density. This relation may be interpreted as resulting from the depletion of the gluon and quark condensates through the stress-volume relation. This trace-anomaly and sigma-term induced pressure plays a fundamental role in the confinement dynamics of QCD. In contrast, the dynamic pressure distribution from the traceless part of the spatial stress tensor equals 1/d of the corresponding traceless part of the energy density, where d is the spatial dimension. The total pressure is balanced by these two components of the pressure. We point out that the same pressure-energy relations also hold for vortices in type-II superconductors, where the static pressure-energy relation arises from the depletion of the Cooper-pair condensate. Furthermore, these equations of state are identical to those in the $\Lambda$CDM model of cosmology, where the static pressure-energy relation arises from the cosmological constant.

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 derives the pressure-energy equations of state for the nucleon from the gravitational form factors parameterizing matrix elements of the energy-momentum tensor (EMT) together with the conservation of the EMT. It identifies two distinct components: a static pressure distribution from the Lorentz-trace part of the EMT (manifested in 1/3 T^{ii}) satisfying p_static = -ε_static, interpreted as arising from depletion of gluon and quark condensates via the stress-volume relation and playing a fundamental role in QCD confinement; and a dynamic pressure from the traceless part satisfying p_dynamic = ε_dynamic / d (d spatial dimension). The total pressure is the sum of these balanced components. Formal analogies are drawn to the pressure-energy relations in type-II superconductor vortices (from Cooper-pair condensate depletion) and the ΛCDM cosmological model (from the cosmological constant).

Significance. If the derivations are correct, the work provides a transparent decomposition of the nucleon's internal pressure and energy densities directly from EMT properties, underscoring the contribution of the trace anomaly and sigma term. The noted formal equivalences to superconductivity and cosmology may encourage cross-field analogies, though the primary value is in clarifying the static vs. dynamic split rather than introducing new computational results or falsifiable predictions.

major comments (2)
  1. [Abstract and static-pressure discussion] The central interpretive claim (abstract and discussion of static component) that the trace-anomaly/sigma-term pressure 'plays a fundamental role in the confinement dynamics of QCD' and arises specifically from condensate depletion is load-bearing for the paper's physical significance but is not supported by any explicit derivation, lattice comparison, or stability integral within the manuscript; the EMT relations alone do not establish dominance over other non-perturbative contributions such as dynamical gluon exchanges.
  2. [Derivation of the two-component EOS] The assertion that the derived relations 'fully capture' the pressure balance without additional corrections (as implied by the two-component decomposition) relies on the assumption that the EMT matrix elements and conservation exhaust the relevant physics; this is not tested against known nucleon EMT sum rules or explicit model calculations (e.g., bag model or lattice GFFs) that might reveal higher-order terms.
minor comments (2)
  1. [Dynamic pressure paragraph] The spatial dimension d is introduced without explicit statement that d=3 for the nucleon; adding this clarification in the dynamic-pressure paragraph would improve readability.
  2. [Discussion section] The manuscript would benefit from a brief comparison table or reference to existing nucleon pressure distributions from lattice QCD or other GFF parametrizations to contextualize the new relations.

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, clarifying the scope of our derivations and indicating textual revisions that will be incorporated to better distinguish between exact relations and interpretive statements.

read point-by-point responses

  1. Referee: [Abstract and static-pressure discussion] The central interpretive claim (abstract and discussion of static component) that the trace-anomaly/sigma-term pressure 'plays a fundamental role in the confinement dynamics of QCD' and arises specifically from condensate depletion is load-bearing for the paper's physical significance but is not supported by any explicit derivation, lattice comparison, or stability integral within the manuscript; the EMT relations alone do not establish dominance over other non-perturbative contributions such as dynamical gluon exchanges.

    Authors: We agree that the link between the static pressure component and QCD confinement is presented as an interpretive perspective rather than a result derived solely from the EMT relations. The equality p_static = -ε_static follows directly from the trace part of the EMT and conservation; the condensate-depletion interpretation is motivated by the established connection of the trace anomaly to gluon and quark condensates, but the manuscript does not demonstrate dominance over other mechanisms such as dynamical gluon exchanges. We will revise the abstract and discussion sections to explicitly label this as an interpretive suggestion based on the sigma term and trace anomaly, without implying exclusivity or providing a stability integral. This change will be implemented in the revised manuscript. revision: yes

  2. Referee: [Derivation of the two-component EOS] The assertion that the derived relations 'fully capture' the pressure balance without additional corrections (as implied by the two-component decomposition) relies on the assumption that the EMT matrix elements and conservation exhaust the relevant physics; this is not tested against known nucleon EMT sum rules or explicit model calculations (e.g., bag model or lattice GFFs) that might reveal higher-order terms.

    Authors: The two-component decomposition and the pressure-energy relations are obtained exactly by separating the EMT into its Lorentz-trace and traceless parts and applying conservation; within the gravitational form factor parameterization, these relations therefore exhaust the contributions from each sector by construction. The total pressure balance follows directly and does not rely on additional assumptions beyond EMT properties. We do not claim that the EMT framework excludes all possible higher-order effects in a model-specific sense; any such effects would be absorbed into the form factors. While explicit comparisons to the bag model, lattice GFFs, or sum-rule checks would be valuable for numerical illustration, they lie beyond the analytic scope of the present work. We will add a brief clarifying statement in the manuscript noting that the relations are general consequences of the EMT and that model-dependent validations remain for future study. revision: partial

Circularity Check

0 steps flagged

No circularity: relations follow from standard EMT conservation and decomposition

full rationale

The paper derives the static and dynamic pressure-energy relations (p_static = -epsilon_static for the trace component and p_dynamic = epsilon_dynamic / d for the traceless component) explicitly from the decomposition of the energy-momentum tensor into Lorentz-trace and traceless parts together with the conservation law ∇_μ T^{μν} = 0. These are independent, externally established principles of relativistic field theory and do not rely on redefinition of terms, fitted parameters renamed as predictions, or any ansatz introduced via self-citation. Gravitational form factors serve only to parameterize the relevant matrix elements; the algebraic relations between pressure and energy densities follow directly from the conservation identity without circular reduction. The condensate-depletion interpretation and analogies to type-II vortices or ΛCDM cosmology are presented as optional physical pictures and formal similarities, not as load-bearing steps in the derivation. No uniqueness theorem, self-citation chain, or renaming of known results is invoked to force the central equations. The derivation chain is therefore self-contained against standard EMT properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest primarily on established properties of the energy-momentum tensor and its conservation in quantum field theory, with no new free parameters or postulated entities introduced.

axioms (2)
  • standard math Conservation of the energy-momentum tensor
    Invoked to relate pressure and energy density components from gravitational form factors.
  • domain assumption Gravitational form factors parameterize the matrix elements of the EMT
    Standard parameterization in QCD for nucleon structure.

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