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Pressure-energy equations of state of the nucleon

Keh-Fei Liu

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Phys. Rev. D 114, 034042 – Published 21 August, 2026

DOI: https://doi.org/10.1103/1hsy-xvqx

Abstract

The pressure-energy equations of state in the nucleon are derived from the gravitational form factors, which parametrize matrix elements of the energy-momentum tensor (EMT), together with EMT conservation. The pressure and energy densities naturally separate into two distinct components. The static pressure distribution, arising from the Lorentz trace part of the EMT, as manifested in the spatial stress 13Tii, is equal to minus the corresponding trace part of the energy density. This relation may be interpreted as a consequence of the full or partial depletion of the gluon and quark condensates through the pressure-volume relation. The resulting trace anomaly–and sigma term–induced pressure is shown, through its volume scaling, to play a pivotal role in QCD confinement. In contrast, the dynamic pressure distribution, associated with the traceless part of the stress tensor, equals 1/d of the corresponding traceless part of the energy density, where d is the spatial dimension. These static and dynamic components together satisfy the pressure-balance condition in the nucleon. We further show that the same pressure-energy equations of state hold for vortices in type-II superconductors, where the static pressure-energy relation originates from the depletion of the Cooper-pair condensate. Moreover, these equations of state are identical to those of the ΛCDM cosmological model, where the static pressure-energy relation is generated by the cosmological constant.

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Phys. Rev. D 114, 034042– Published 21 August, 2026
  • Received 6 May 2026
  • Accepted 3 August 2026
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