Universality of pseudo-Goldstone damping near critical points

Recently, in studies of holographic models and hydrodynamics with spontaneous breaking of approximate symmetries, it has been proposed that the damping of pseudo-Goldstone modes at finite temperatures is universally constrained in the way that Ωφ/mφ² Dφ in the broken phase, where Ωφ and mφ are the relaxation rate at zero wavenumber and the mass of pseudo-Goldstones, Dφ is the Goldstone diffusivity in the limit of purely spontaneous breaking. In this paper, we investigate the pseudo-Goldstone damping in a purely relaxational O(N) model by performing the functional renormalization group calculations at the full quantum and stochastic level within the Schwinger-Keldysh formalism. We find that, away from the critical temperature, the proposed relation is always valid. When the temperature is very close to the critical value such that the mass of the Higgs mode is comparable to the mass of the pseudo-Goldstone modes, the pseudo-Goldstone damping displays a novel scaling behavior that follows Ω_φ/m_φ²∝ mφΔ_η with a correction Δ_η controlled by the critical universalities. Moreover, we study how the correction depends on the value of N and show that Δ_η → 0 when fluctuations are infinitely suppressed in the large N limit. In this case, the proposed relation works even in the critical region. Finally, we match our results to the dissipative sector of the pion dynamics near the chiral phase transition.