I'm trying to find the RG flow to lowest order in $\epsilon = 3 -d $ for the energy functional:

$$ f=\frac{1}{2} \phi ^2 +u \phi ^6 +\frac{c}{2} (\nabla \phi ) ^2 $$

where $\ d$ is the dimension we're interested in and I've previously found that 3 is the critical dimension.

As you can see it's the standard Gaussian Model with an additional $\ \phi ^6 $ term. Now, I've seen the approach used to deal with an interaction of the form $\ \phi ^4 $, but I'm struggling to apply that here. When I perform a diagrammatic perturbation expansion, I invariably end up with terms $\ \propto \phi ^4 $ which don't appear in the original energy functional and can't be eliminated by the usual rescaling of the coefficients and fields.

Am I going about this completely the wrong way, or is there some trick for dealing with situations like this? I'd be very grateful if somebody could point me in the right direction. My apologies if this is a "standard problem", but I haven't been able to find any similar examples in textbooks or online and I have only received a very rudimentary introduction to RG.

  • 3
    $\begingroup$ Does RG = renormalization group? In general, try to avoid too many abbreviations to help readers understand your question. $\endgroup$ Jan 11, 2015 at 20:23
  • $\begingroup$ Yes, my apologies, RG = renormalization group. $\endgroup$ Jan 12, 2015 at 8:46
  • $\begingroup$ As you probably know, the $\phi^4$ term is relevant near the Gaussian when $d<4$, so the flow will always generate this term in $3-\epsilon$ dimensions. $\endgroup$
    – Adam
    May 26, 2016 at 7:01

1 Answer 1


I think what you are asking is whether your theory is renormalizable. Renormalizable means: all divergences can be absorbed by terms in the original energy functional by rescaling of the coefficients and fields. If you cannot, then your theory is not renormalizable and can therefore not exist as a quantum field theory at all energies.

If you persevere and demand that your theory be the fundamental theory of some toy model universe, then the low-energy effective theory describing that universe will contain a term $\propto \phi^4$, with the coefficient depending on the cuf-off of your universe.

  • $\begingroup$ Thanks for your answer David. I also reached the conclusion that the given energy functional is not renormalizable. The difficulty is that this is a homework assignment, rather than just a model I plucked out of thin air, so I assumed that there must be some method that one can use to find the RG flow. $\endgroup$ Jan 14, 2015 at 8:58
  • $\begingroup$ I would assume that who gave you this as homework would want you to think about the fact that strictly speaking the model is not renormalisable, because as you found out you need a $\propto \phi^4$ counterterm. However, they probably wanted to think about what happens once you include that. $\endgroup$
    – fqq
    Aug 18, 2015 at 11:18

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