  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 $\Omega_{\varphi}/m_{\varphi}^2\simeq D_{\varphi}$ in the broken
phase, where $\Omega_{\varphi}$ and $m_{\varphi} $ are the relaxation rate at
zero wavenumber and the mass of pseudo-Goldstones, $D_{\varphi}$ 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 $\Omega_\varphi/m_\varphi^2\propto
m_{\varphi}^{\Delta_\eta}$ with a correction $\Delta_\eta$ controlled by the
critical universalities. Moreover, we study how the correction depends on the
value of $N$ and show that $\Delta_\eta \rightarrow 0$ 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.
