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I am studying Renormalization but I don't understand why theories which have a coupling constant with negative dimension of mass requires more and more counterterms going up with the perturbative order.

Let we use for simplicity the quartic Fermi Theory in four dimensions: $\mathcal{L}=G(\bar{\psi}\psi)(\bar{\psi}\psi)$, since $[\psi]=[M]^{3/2}$ and $[\mathcal{L}]=[M]^4$ we have $[G]=[M]^{-2}$, so far so good.

From what I understand is that the first loop correction has two vertecies, and so 1-Loop-Correction ~ $ G^{2} $ ~ $ [M]^{-4}$, this means that we could also add a counterterm with one vertex which has a coupling constant $\delta G$ with $\delta G=[M]^{-4}$.

This is my first problem, because the interaction $\delta G(\bar{\psi}\psi)(\bar{\psi}\psi)$ has not the right dimensions of $[M]^{4}$ but instead of $[M]^{2}$, so what I am doing wrong?

Secondly my professor said that we could also add another term: $\tilde{G}(\partial_{\mu}\bar{\psi}\partial^{\mu}\psi)(\partial_{\nu}\bar{\psi}\partial^{\nu}\psi)$ ($\tilde G=[M]^{0}$? I am not sure) because it has the same dimension of the one loop divergence; but again the thing that I don't understand is that $[(\partial_{\mu}\bar{\psi}\partial^{\mu}\psi)(\partial_{\nu}\bar{\psi}\partial^{\nu}\psi)]=[M]^{10}$.

In the ends this should demonstrate that we need all the possible counterterms (compatibles with the symmetries of our theory), but I don't understand why these countermens have different dimensions of mass from the Lagrangian.

I checked some books (Weinberg, Peskin, Maggiore) but I didn't find these explanations in detail, if someone can point a book and chapter where this (or any non renormalizable theory) is explained better I would be more than happy!

Qmechanic
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Aleph12345
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1 Answers1

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  1. The interaction term $G\psi^2\bar{\psi}^2$ has a coupling constant $G$ of mass-dimension $[G]=-2$, and the interaction term $\tilde{G}(\partial\psi)^2(\partial\bar{\psi})^2$ has a coupling constant $\tilde{G}$ of mass-dimension $[\tilde{G}]=-6$, i.e. both coupling constants are irrelevant/non-renormalizable, cf. e.g. this Phys.SE post.

  2. The 1-loop diagram $>\!\!O\!\!<$ with 4 external fermions contributes to the interaction term $G\psi^2\bar{\psi}^2$, and hence to the running of the coupling constant $G$. OP's mismatch in mass dimension seems caused by the omission of 2 internal fermion propagators and various integrations in the 1-loop diagram.

Qmechanic
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