Relaxing Near the Critical Point

Abstract

Critical slowing down of the relaxation of the order parameter is relevant both in early the universe and in ultrarelativistic heavy ion collisions. We study the relaxation rate of the order parameter in an O(N) scalar theory near the critical point to model the non-equilibrium dynamics of critical fluctuations near the chiral phase transition.A lowest order perturbative calculation (two loops in the coupling lambda) reveals the breakdown of perturbation theory for long-wavelength fluctuations in the critical region and the emergence of a hierarchy of scales with hard q>T, semisoft T >> q >> lambda T and soft lambda T>>q loop momenta which are widely separated for weak coupling. The non-perturbative resummation in the large N limit reveals the renormalization of the interaction and the crossover to an effective 3D-theory for soft momenta.The effective 3D coupling goes to the Wilson-Fisher 3D fixed point in the soft limit.The relaxation rate of the order parameter for wave vectors lambda T >>k>> k_{us} or near the critical temperature lambda T>>m_T>>k_{us} with the ultra soft scale k_{us} = [(lambda T)/(4pi)] exp[-(4pi/lambda)] is dominated by classical semisoft loop momentum leading to Gamma(k,T) = lambda T/(2 pi N). For wavectors k<< k_{us} the damping rate is dominated by hard loop momenta and given by Gamma(k,T)=4 pi T/[3N ln(T/k)]. Analogously, for homogeneous fluctuations in the ultracritical region m_T<<k_{us} the damping rate is given by Gamma_0(m_T,T)=4 pi T/[3N ln(T/m_T)]. Thus critical slowing down emerges for ultrasoft fluctuations where the rate is lambda-independent. The strong coupling regime and the shortcomings of the quasiparticle interpretation are discussed.