Dynamics near the critical point: The hot renormalization group in quantum field theory

The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an epsilon expansion in d=5-epsilon space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are omega(p)similar top(z) and Gamma(p)similar to(z-1)omega(p), respectively, and the group velocity of quasiparticles v(g)similar top(z-1) vanishes in the long-wavelength limit at the critical point. Away from the critical point for Tgreater than or similar toT(c) we find omega(p)similar toxi(-z)[1+(pxi)(2z)](1/2) and Gamma(p)similar to(z-1)omega(p)(pxi)(2z)/[1+(pxi)(2z)] with xi the finite temperature correlation length xiproportional to\T-T-c\(-nu). The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with O(N) symmetry we find z=1+epsilon(N+2)/(N+8)(2)+O(epsilon(2)). This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.