Microscopic identification of dissipative modes in relativistic field theories

We present an argument to support the existence of dissipative modes in relativistic field theories. For an O(N) phi(4) theory in spatial dimension 1 <= d <= 3, the two-point function of phi is shown to develop a pole of the form p(0) similar to i Gamma(-1) (p(2) + m(2)) at small energy p(0) and momentum p, with a nonzero finite coefficient or relaxation constant, Gamma, when evaluated in the two-particle irreducible (2PI) framework at the next-leading order (NLO) of 1/N expansion. In contrast, an NLO calculation in the one-particle irreducible (1PI) framework fails to give a nonzero Gamma. The dissipative mode emerges from multiple scattering of a particle with other particles, which is appropriately treated in the 2PI-NLO calculation through the resummation of secular terms to improve the 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.