Microscopic identification of dissipative modes in relativistic field theories

We present an argument to support the existence of dissipative modes in relativistic field theories. In an O(N) $\varphi^4$ theory in spatial dimension $d\le 3$, a relaxation constant $\Gamma$ of a two-point function in an infrared region is shown to be finite within the two-particle irreducible (2PI) framework at the next-leading order (NLO) of 1/N expansion. This immediately implies that a slow dissipative mode with a dispersion $p_0\sim i\Gamma \p^2$ is microscopically identified in the two-point function. Contrary, NLO calculation in the one-particle irreducible (1PI) framework fails to yield a finite relaxation constant. Comparing the results in 1PI and 2PI frameworks, one concludes that dissipation emerges from multiple scattering of a particle with a heat bath, which is appropriately treated in the 2PI-NLO calculation through the resummation of secular terms to improve long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent $z$ for the two-point function as $z=2-\eta$. We also discuss possible improvement of the result.