Renormalizable Lagrangians?

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Hello everyone,

my teacher asked in last day class as a curiosity to be discussed: As a function of the space-time dimension "d", which Lagrangians containing an scalar and a fermion field are renormalisable?. Then he encouraged us to think in the interaction vertex of the form: $\phi^n(\bar{\psi}\psi)^m$.

And now I am curious. Answer will be much appreciated.

 

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Where could I find the renormalization of the $\phi^4$ theory using dimensional regularisation? As I only found for $\phi^3$ theory, I am interested in compare both cases to learn better the process.

 

[vanhees71]

Concerning your first question, I guess, you should carefully think yourself about it before we tell you the answer. It's not that difficult as soon as you have understood what's behind the power-counting argument for (superficial) renormalizability, which of course works in any space-time dimension.

For renormalization and $\phi^4$ theory I like to advertize my QFT manuscript. There it's treated in various ways: using cutoff regularization, dimensional regularization, and last but not least direct renormalization without an intermediate step of regularization (called BPHZ renormalization, named after Bogoliubov, Parasiuk, Hepp, and Zimmermann).

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

 

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So it is totally justified to say that as scalar field has dimension 1, fermion field has d= 3/2 we have 5/2 "filled" so as "d" must be an integer we can set another fermion field plus a coupling constant of dim= d-4 ? A question now is, a coupling constant can be a half-integer dimensional?.

 

[vanhees71]

Be careful: The energy dimension of the fields depends on the space-time dimension! You get the counting by looking at the kinetic terms of the field. E.g., for a scalar field, it's $\partial_{\mu} \phi \partial^{\mu} \phi$. The action as a whole is dimensionless (when setting $\hbar=1$, using natural units). $\partial_{\mu}$ has energy dimension 1 (or space-time dimension -1, which is equivalent when also $c=1$ is used). The space-time four-volume element $\mathrm{d}^d x$ has energy dimension $-d$ Thus the energy dimension of the scalar field must fulfill $2d_{\phi}+2-d=0$, i.e., you have $d_{\phi}=(d-2)/2$, which is of course 1 for $d=4$.

Now think about the Dirac field. The kinetic term in the Lagrangian contains $\overline{\psi} \gamma_{\mu} \psi$. So now you can easily do the dimensional analysis yourself.

Now think about $\phi^4$ theory. The interaction Lagrangian is $\lambda \phi^4$. Now you can check, which dimension this coupling must have in $d$ space-time dimensions (only for $d=4$ the coupling is dimensionless). That explains, why it's necessary to introduce a energy scale into the theory in order to keep $\lambda$ dimensionless in any dimension. This is a rather indirect way, how the renormalization scale enters the very formal dimensional regularization, which has its merits through its convenience with regard to gauge symmetries, not as a tool to understand the physical meaning of regularization and renormalization, which is given by the Wilsonian point of view on renormalization. This is quite well treated in principle in Peskin & Schroeder. Unfortunately this otherwise very good book, ironically messes up the proper treatment of the renormalization scale, and (horribile dictu) you find dimensionful quantities as arguments of logarithms in the chapter on the Wilsonian treatment of renormalization, which is a real pity!).

 

[Breo]

Thank you for the response.

So if we have several dimensions... we can set several different terms with fermion fields and scalar fields?