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Fermion Thermal Field Theory for a Rotating Plasma (with Applications to Neutron Stars)

Alberto Salvio

TL;DR

The paper develops a comprehensive framework for fermionic thermal field theory in rotating plasmas with general equilibrium density matrices that include angular momentum. It constructs a robust (quasi-)free fermion basis, derives rotated ensemble averages and thermal propagators, and then formulates a path-integral approach that accommodates general fermion–scalar theories under rotation and chemical potentials. Key results include explicit expressions for rotation-deformed Fermi surfaces, general production-rate formulas for weakly-coupled fermions, and a striking finding that neutrino production via direct URCA processes can diverge as the rotation speed approaches the light-speed bound in finite-volume systems, while beta equilibrium remains rotation-insensitive. The work provides a versatile toolkit for studying particle production and transport in rotating compact objects such as neutron stars, with potential extensions to accretion disks and laboratory rotating plasmas.

Abstract

This paper provides a systematic and complete study of thermal field theory with fermion fields of any kind for generic equilibrium density matrices, which feature arbitrary values not only of temperature and chemical potentials, but also average angular momentum. This extends a previous study that focused on scalar fields, to all fermion-scalar theories. Both Dirac and Majorana fermions and both Dirac and Majorana masses are covered. A general technique to compute ensemble averages is provided. Path-integral methods are developed to study thermal Green's functions (with an arbitrary number of points) in generic interacting fermion-scalar theories, which cover both the real-time and imaginary-time formalism. These general results are applied to physical situations typical of neutron stars, which are often quickly rotating: the Fermi surface and Fermi momentum, the average energy, number density and angular momentum for degenerate fermions and particle production (such as neutrino production from rotating neutron stars, e.g. pulsars). In particular, it is shown that the neutrino production rate due to the direct URCA (DU) processes grows indefinitely as the angular velocity approaches the inverse linear size of the plasma and, therefore, rotation can significantly increase this rate.

Fermion Thermal Field Theory for a Rotating Plasma (with Applications to Neutron Stars)

TL;DR

The paper develops a comprehensive framework for fermionic thermal field theory in rotating plasmas with general equilibrium density matrices that include angular momentum. It constructs a robust (quasi-)free fermion basis, derives rotated ensemble averages and thermal propagators, and then formulates a path-integral approach that accommodates general fermion–scalar theories under rotation and chemical potentials. Key results include explicit expressions for rotation-deformed Fermi surfaces, general production-rate formulas for weakly-coupled fermions, and a striking finding that neutrino production via direct URCA processes can diverge as the rotation speed approaches the light-speed bound in finite-volume systems, while beta equilibrium remains rotation-insensitive. The work provides a versatile toolkit for studying particle production and transport in rotating compact objects such as neutron stars, with potential extensions to accretion disks and laboratory rotating plasmas.

Abstract

This paper provides a systematic and complete study of thermal field theory with fermion fields of any kind for generic equilibrium density matrices, which feature arbitrary values not only of temperature and chemical potentials, but also average angular momentum. This extends a previous study that focused on scalar fields, to all fermion-scalar theories. Both Dirac and Majorana fermions and both Dirac and Majorana masses are covered. A general technique to compute ensemble averages is provided. Path-integral methods are developed to study thermal Green's functions (with an arbitrary number of points) in generic interacting fermion-scalar theories, which cover both the real-time and imaginary-time formalism. These general results are applied to physical situations typical of neutron stars, which are often quickly rotating: the Fermi surface and Fermi momentum, the average energy, number density and angular momentum for degenerate fermions and particle production (such as neutrino production from rotating neutron stars, e.g. pulsars). In particular, it is shown that the neutrino production rate due to the direct URCA (DU) processes grows indefinitely as the angular velocity approaches the inverse linear size of the plasma and, therefore, rotation can significantly increase this rate.

Paper Structure

This paper contains 11 sections, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. (Quasi-)free fields
  3. Computing ensemble averages
  4. Thermal propagator
  5. Fermion path integral
  6. Some applications
  7. Fermi momentum and Fermi surface
  8. Weakly-coupled fermion production
  9. Direct URCA processes in rotating neutron stars
  10. Summary and conclusions
  11. Integral of products of Bessel functions

Figures (2)

  • Figure 1: Average energy density (upper left plot), average angular momentum density per unit of distance from the rotation axis (upper right plot) and average number density (lower left plot) as a function of the (rotational) velocity parameter $v$ in the case of a single Dirac fermion with mass $\mu$, a single chemical potential $\mu_B$ and $T\ll \mu$ (a relevant case for neutron stars). In the lower right plot it is shown how the average number density depends on $\mu_B$.
  • Figure 2: Diagrams representing the "non time-ordered" 2-point functions, Eq. (\ref{['SigmaForward']}) on the left and Eq. (\ref{['SigmaBackward']}) on the right, which are relevant for (anti)neutrino production via the DU processes in (\ref{['DirectUrca']}) in terms of the "non time-ordered" 2-point functions of $n$, $p$ and $l$. The Kobes-Semenoff circling notation KSKS2 is used.