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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?

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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 $C$).

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I've thought about it some more, and I'm no longer sure this is right. Shouldn't the $\lambda\hat{\phi}^4$ term have an affect on the low momentum modes due to intermediate states? For example, the following diagram represents a correction to the low momentum mass: (Single lines are low momentum, double lines are high momentum) Shouldn't there also be diagrams like which further modify the low momentum mass and wouldn't exist if it weren't for the $\lambda\hat{\phi}^4$ term? – Ergil Jun 7 '13 at 12:26

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