Dynamical viscosity of nucleating bubbles

We study the viscosity corrections to the growth rate of nucleating bubbles in a slightly supercooled first order phase transition in 1 + 1 and 3 + 1 dimensional scalar field theory. We propose a microscopic approach that leads to the non-equilibrium equation of motion of the coordinate that describes small departures from the critical bubble and allows to extract the growth rate consistently in a weak coupling expansion and in the thin wall limit. Viscosity effects arise from the interaction of this coordinate with the stable quantum and thermal fluctuations around a critical bubble. In the case of 1 + 1 dimensions we provide an estimate for the growth rate that depends on the details of the free energy functional. In 3 + 1 dimensions we recognize robust features that are a direct consequence of the thin wall approximation that trascend a particular model. These are long-wavelength hydrodynamic fluctuations that describe surface waves. We identify these low energy modes with quasi-Goldstone modes which are related to surface waves on interfaces in phase ordered Ising-like systems. In the thin wall limit the coupling of this coordinate to these hydrodynamic modes results in the largest contribution to the viscosity corrections to the growth rate. For a φ4 scalar field theory at temperature T < Tc the growth rate to lowest order in the quartic self-coupling λ is Ω = √ 2 Rc [ 1− 0.003λTξ ( Rc ξ )2] with Rc ; ξ the critical radius and the width of the bubble wall respectively. We obtain the effective non-Markovian Langevin equation for the radial coordinate and derive the generalized fluctuation dissipation relation, the noise is correlated on time scales Ω−1 as a result of the coupling to the slow hydrodynamic modes. We discuss the applicability of our results to describe the growth rate of hadron bubbles in a quark-hadron first order transition.