On the definition of a unique effective temperature for non-equilibrium critical systems

We consider the problem of the definition of an effective temperature via the long-time limit of the fluctuation-dissipation ratio X-infinity after a quench from the disordered state to the critical point of an O(N) model with dissipative dynamics. The scaling forms of the response and correlation functions of a generic observable O(t) are derived from the solutions of the corresponding renormalization group equations. We show that within the Gaussian approximation all the local observables have the same X-O(infinity), allowing for a definition of a unique effective temperature. This is no longer the case when fluctuations are taken into account beyond that approximation, as shown by a computation up to the first order in the epsilon-expansion for two quadratic observables. This implies that, contrarily to what is often conjectured, a unique effective temperature cannot be defined for this class of models.