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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 \emph{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 $ε$ expansion in $d=5-ε$ space-time dimensions. The long wavelength physics is determined by a non-trivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum $p$ is $ω_p \sim p^{z}$ and $Γ_p \sim (z-1) ω_p$ respectively, the group velocity of quasiparticles $v_g \sim p^{z-1}$ vanishes in the long wavelength limit at the critical point. Away from the critical point for $T\gtrsim T_c$ we find $ω_p \sim ξ^{-z}[1+(p ξ)^{2z}]^{1/2}$ and $Γ_p \sim (z-1) ω_p \frac{(p ξ)^{2z}}{1+(p ξ)^{2z}}$ with $ξ$ the finite temperature correlation length $ ξ\propto |T-T_c|^{-ν}$. The new \emph{dynamical} exponent $z$ results from anisotropic renormalization in the spatial and time directions. For a theory with O(N) symmetry we find $z=1+ ε \frac{N+2}{(N+8)^2}+\mathcal{O}(ε^2)$. 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.