Integrating out high momentum modes in $\phi^4$ theory

Question

I'm trying to follow section 12.1 of Peskin & Schroeder, which describes how integrating out the high momentum modes of the field in $\phi^4$ theory transforms the Lagrangian both by changing the values of $m$ and $\lambda$ and by introducing new interaction terms such as $\phi^6$, $\phi^8$ etc. I get the idea, but I'm a bit fuzzy on some of the math. In equation 12.5 Peskin separates the field into low momentum modes $\phi$ and high momentum modes $\hat{\phi}$. Rewriting the Lagrangian in terms of these fields gives terms like $\hat{\phi}^2\phi^2$ and $\hat{\phi}\phi^3$, which generate the new interactions, but also terms which depend only on the high momentum field - $m\hat{\phi}^2$ and $\lambda\hat{\phi}^4$. What do these terms do when integrated? Do they just add constants to the new Lagrangian?

Answer 1

Yes, terms like $\lambda \hat \phi^4$ only depend on the high-energy modes, so they're constant functions as functions of the low-energy modes. From the low-energy modes' viewpoint, i.e. when it comes to the dynamics of the low-energy effective theory, they just combine to a constant term $C$ in the Lagrangian, a vacuum energy density that has no impact (unless one considers gravity or compares two situations with different $C$).