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 ɛ expansion in d=5-ɛ 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 ω_p~p^z and Γ_p~(z-1)ω_p, respectively, and the group velocity of quasiparticles v_g~p^z-1 vanishes in the long-wavelength limit at the critical point. Away from the critical point for T>~T_c we find ω_p~ξ^-z[1+(pξ)^2z]^1/2 and Γ_p~(z-1)ω_p(pξ)^2z/[1+(pξ)^2z] with ξ the finite temperature correlation length ξ~\|T-T_c\|^-ν. 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+ɛ(N+2)/(N+8)²+O(ɛ²). 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.