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Fermionic Dispersion Relations in the Standard Model at Finite Temperature

C. Quimbay, S. Vargas-Castrillon

TL;DR

This work computes the one-loop finite-temperature dispersion relations for quarks, charged leptons, and neutrinos in the Minimal Standard Model across both electroweak phases (broken and unbroken). Using the real-time thermal-field theory formalism, it derives the self-energy structure and solves the dispersion equations, revealing four quasi-particle branches for quarks and charged leptons and two branches for neutrinos, with explicit flavor non-degeneracy from Yukawa couplings, CKM mixing, and masses. A key finding is that thermal effective masses are smaller in the broken phase, and an energy gap of order the fermion mass appears between the hyperbolic branches, with the top-quark sector showing distinctive temperature-dependent behavior. Gauge invariance is established at leading order in temperature, while subleading contributions introduce a small, controllable gauge dependence, consistent with Ward identities and prior results in QCD-like theories. These results illuminate how the thermal medium near the electroweak transition shapes fermion propagation and could impact electroweak baryogenesis scenarios.

Abstract

We compute the one-loop dispersion relations at finite temperature for quarks, charged leptons and neutrinos in the Minimal Standard Model. The dispersion relations are calculated in two different plasma situations: for a vacuum expectation value $\upsilon$ of the Higgs field $\upsilon \neq 0$ (broken electroweak symmetry) and for $\upsilon=0$ (unbroken electroweak symmetry). The flavour and chiral non-degeneracy of the quasi-particle spectrum is studied. Numerical results show that the thermal effective masses for fermions in the broken phase have a smaller value than those in the unbroken phase. The temperature dependence of the top quark and electron neutrino thermal effective masses is also presented. Gauge invariance of one-loop dispersion relations is studied.

Fermionic Dispersion Relations in the Standard Model at Finite Temperature

TL;DR

This work computes the one-loop finite-temperature dispersion relations for quarks, charged leptons, and neutrinos in the Minimal Standard Model across both electroweak phases (broken and unbroken). Using the real-time thermal-field theory formalism, it derives the self-energy structure and solves the dispersion equations, revealing four quasi-particle branches for quarks and charged leptons and two branches for neutrinos, with explicit flavor non-degeneracy from Yukawa couplings, CKM mixing, and masses. A key finding is that thermal effective masses are smaller in the broken phase, and an energy gap of order the fermion mass appears between the hyperbolic branches, with the top-quark sector showing distinctive temperature-dependent behavior. Gauge invariance is established at leading order in temperature, while subleading contributions introduce a small, controllable gauge dependence, consistent with Ward identities and prior results in QCD-like theories. These results illuminate how the thermal medium near the electroweak transition shapes fermion propagation and could impact electroweak baryogenesis scenarios.

Abstract

We compute the one-loop dispersion relations at finite temperature for quarks, charged leptons and neutrinos in the Minimal Standard Model. The dispersion relations are calculated in two different plasma situations: for a vacuum expectation value of the Higgs field (broken electroweak symmetry) and for (unbroken electroweak symmetry). The flavour and chiral non-degeneracy of the quasi-particle spectrum is studied. Numerical results show that the thermal effective masses for fermions in the broken phase have a smaller value than those in the unbroken phase. The temperature dependence of the top quark and electron neutrino thermal effective masses is also presented. Gauge invariance of one-loop dispersion relations is studied.

Paper Structure

This paper contains 19 sections, 167 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Some useful definitions
  3. Real part of the fermion self-energy
  4. Fermionic dispersion relations
  5. Quark dispersion relations
  6. Broken phase
  7. Unbroken phase
  8. Charged-lepton dispersion relations
  9. Broken phase
  10. Unbroken phase
  11. Neutrino dispersion relations
  12. Broken phase
  13. Unbroken phase
  14. Remarks about gauge invariance
  15. Numerical results
  16. ...and 4 more sections

Figures (10)

  • Figure 1: Generic $Z$-tadpole diagram
  • Figure 2: Generic gauge and scalar boson diagrams
  • Figure 3: Dispersion relations at $T=100$ GeV for the $s$ quark in the broken (solid curves) and the unbroken (dotted curves) phases, neglecting quark mixing. The upper (lower) dotted curves correspond to left (right) chirality branches. An energy gap ($\simeq 0.1$ GeV) appears between the minimum and maximum of the hyperbolic branches.
  • Figure 4: Dispersion relations at $T=100$ GeV for the $c$ quark in the broken (solid curves) and the unbroken (dashed curves) phases, neglecting quark mixing. The energy gap between the hyperbolic branches is $\simeq 1.0$ GeV. The thermal effective masses for the branches in the broken phase are smaller that those in the unbroken phase.
  • Figure 5: Variation with temperature of the thermal effective masses for the top quark in the broken phase.
  • ...and 5 more figures