Dynamical Manifestation of the Goldstone Phenomenon at 1-loop

We have calculated the damping rate $\mathrm{\Gamma }(\mathrm{\mid }\mathbf{k}\mathrm{\mid })$ for classical on-shell Goldstone modes of the O(2) symmetric scalar fields propagating in a thermal medium of the broken symmetry phase taking into account the effect of the explicit symmetry breaking. The result of the one-loop analysis can be expanded around $\mathrm{\Gamma }(0)$, which depends non-analytically on the parameter of the explicit symmetry breaking, h. $\mathrm{\Gamma }(0)$ vanishes when $h\to 0$, demonstrating in this way the absence of the restoring force, when the equilibrium direction of the symmetry breaking is modulated homogeneously.