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?


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