# Is there a multiscale analysis for field theories?

Consider a (zero dimensional) Gaussian field theory described by the dynamical action

$$S = \int_t \tilde{\phi}(t) \left[\partial_t \phi(t) + M(t) \phi(t)\right] - \gamma \tilde{\phi}(t)^2\, .$$

$$\phi(t)$$ and $$\tilde{\phi}(t)$$ are scalar fields. The mass, $$M(t)$$ is slowly time-dependent,

$$M(t) = M_0 + M_1(\Omega t) \, .$$

$$M_0$$ does not depend on time. $$M_1(\tau)$$ is not small, but $$\Omega$$ is very small so that $$M_1(\Omega t)$$ has a slow time-dependence. Then there are two time scales $$t_1 = 1/M_0$$ and $$t_2 = 1/\Omega>>t_1$$, which are very different. How do I construct an expansion in powers of $$\Omega/M_0$$ for the Green functions (two-point correlation functions) of my field theory?

I know about multiscale analysis for differential equations (see also this), which looks very much like my problem. Presently I do not know how to apply this type of analysis to a field theory. Do you?