Hydrodynamic Transport Coefficients in Relativistic Scalar Field Theory

TL;DR

The paper demonstrates that hydrodynamic transport coefficients in a weakly coupled relativistic scalar field theory can be computed from first principles by summing an infinite class of ladder diagrams, yielding a linear integral equation for an effective vertex that matches a linearized Boltzmann equation with temperature-dependent mass and scattering amplitudes. This equivalence holds across temperatures, including regimes where the mean free path is short, and is extended to include inelastic processes essential for the bulk viscosity. Numerically, the shear viscosity scales as $\eta \sim T^3/\lambda^2$ at high temperature, while the bulk viscosity exhibits more intricate temperature dependence, with high-$T$ behavior governed by scale anomalies and mass corrections; when cubic interactions are present, the bulk viscosity scales as $\zeta \sim T^3/\sqrt{\lambda}$ in certain regimes. Overall, the work provides a rigorous bridge between quantum-field theoretic and kinetic descriptions of transport in hot, weakly coupled scalar systems, with concrete calculations for both shear and bulk viscosities.

Abstract

Hydrodynamic transport coefficients may be evaluated from first principles in a weakly coupled scalar field theory at arbitrary temperature. In a theory with cubic and quartic interactions, the infinite class of diagrams which contribute to the leading weak coupling behavior are identified and summed. The resulting expression may be reduced to a single linear integral equation, which is shown to be identical to the corresponding result obtained from a linearized Boltzmann equation describing effective thermal excitations with temperature dependent masses and scattering amplitudes. The effective Boltzmann equation is valid even at very high temperature where the thermal lifetime and mean free path are short compared to the Compton wavelength of the fundamental particles. Numerical results for the shear and the bulk viscosities are presented.

Figures (38)

• Figure 1: A typical cut diagram in a scalar $\lambda\phi^4$ theory.
• Figure 2: One-loop self-energy diagrams in a scalar $g\phi^3{+}\lambda\phi^4$ theory. At high temperatures, the diagrams (a) and (b) produce a thermal mass squared of order $\lambda T^2$. The contribution of the diagram (c) to the thermal mass is ${\cal O}(g^2 T/m_{\rm th})={\cal O}(\lambda^{3/2} T^2)$.
• Figure 3: Cut two-loop self-energy diagrams in a scalar $g\phi^3{+}\lambda\phi^4$ theory which produce a thermal width of order $\lambda^2 T$.
• Figure 4: A typical one-loop cut diagram for the calculation of a Wightman function. The black dot at each end represents the external bilinear operator.
• Figure 5: Nearly pinching poles in the product $G(k)^2$. The heavy line along the real axis represents the integration contour.