Shear viscosity of a relativistic scalar field from functional renormalization

TL;DR

The paper develops a non-perturbative FRG framework to compute the shear viscosity of a relativistic real scalar field by flowing the effective action with dissipative branch-cut structures. Using a minimal truncation that includes Landau damping and a momentum-dependent vertex, the authors derive flow equations for the mass, damping, vertex damping, and viscosity, solving them numerically to obtain η(T) across parameter ranges. They demonstrate that Landau damping agrees with perturbative results in the weak-coupling limit and recover relativistic η ~ T^3 scaling in the high-temperature regime, while also exposing nonperturbative corrections at larger λ and nonrelativistic behavior at low T. The approach provides a controlled, nonperturbative pathway to transport coefficients and offers a platform for extensions to more realistic theories such as QCD matter in heavy-ion collisions.

Abstract

Renormalization group flow equations of the fluid dynamical shear viscosity transport coefficient of a relativistic real scalar field are derived. The flowing effective action contains branch cut contributions to the self energy and interaction vertex in the symmetric phase. We demonstrate how the flow equation method can systematically extend the perturbative resummation schemes. We show that our truncation is in that sense a minimal scheme in which a reliable viscosity coefficient is obtained.

Figures (12)

• Figure 1: The $\Gamma^{(m,n)\mu_1\nu_1\ldots\mu_n\nu_n}$ vertex's diagrammatic representation. Each solid line represents an external scalar field, while a wavy line corresponds to an external metric field.
• Figure 2: Analytic structure of the propagator given by \ref{['eq:inverse_propagator_truncation']} on the left and the retarded propagator on the right at finite $\Omega_p^2=m_k^2+k^2+\mathbf{p}^2$ and $\gamma_k$ for the complex frequency $z$.
• Figure 3: Shape of the spectral density for different ratios of $\gamma_k/\Omega_p$.
• Figure 4: Analytic structure of the tadpole diagram at $\gamma_k>0$ in the complex frequency plane. The red lines indicate branch cuts, while isolated poles from the Bose distribution are represented by red circles. The blue paths show the integration contour $\mathcal{C}$ in the left panel, corresponding to the Matsubara sum \ref{['eq:tadpole_matsubara_sum_contour_matsubara_poles']}. The modes at zero, as well as $\pm \mathrm{i} \omega_m$ are excluded due to the presence of a branch cut. An equivalent contour corresponding to \ref{['eq:sum_tadpole_intermediate']} is shown in the right panel, enclosing only the branch cuts.
• Figure 5: Dependence between the flow equations. A red arrow indicates a necessary existence of the prerequisite.