  We consider the problem of the definition of an effective temperature via the
long-time limit of the fluctuation-dissipation ratio (FDR) 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 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 FDR, 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 often conjectured, a unique effective
temperature can not be defined for this class of models.
