Relaxing near the critical point

Critical slowing down of the relaxation of the order parameter has phenomenological consequences in early universe cosmology and in ultrarelativistic heavy ion collisions. We study the relaxation rate of long-wavelength fluctuations 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 λ) 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>>λT and soft λT>>q loop momenta which are widely separated in the weak coupling limit. A non-perturbative resummation is implemented to leading order in the large N limit which reveals the infrared renormalization of the static scattering amplitude and the crossover to an effective three dimensional theory for the soft loop momenta near the critical point. The effective three dimensional coupling is driven to the Wilson-Fisher three dimensional fixed point in the soft limit. This resummation provides an infrared screening and for critical fluctuations of the order parameter with wave vectors λT>>k>>k_us or near the critical temperature λT>>m_T>>k_us with the ultrasoft scale k_us=(λT/4π)exp[-4π/λ] the relaxation rate is dominated by classical semisoft loop momentum leading to Γ(k,T)=λT/(2πN). For wave vectors k<<k_us the damping rate is dominated by hard loop momenta and given by Γ(k,T)=4πT/[3N ln(T/k)]. Analogously, for homogeneous fluctuations in the ultracritical region m_T<<k_us the damping rate is given by Γ₀(m_T,T)=4πT/[3N ln(T/m_T)]. Thus critical slowing down emerges for ultrasoft fluctuations. In such a regime the rate is independent of the coupling λ and both perturbation theory and the classical approximation within the large N limit break down. The strong coupling regime and the shortcomings of the quasiparticle interpretation are discussed.