Dimensional transmutation in Gross-Neveu vs others

Question

• Firstly I don't know how generic is dimensional transmutation and if it has any general model independent definition.

Is dimensional transmutation in Gross-Neveau somehow fundamentally different than in say massless-scalar QED or QCD?

I mean in the later two cases it shows up in doing perturbation theory - as in QCD, you calculate the 1-loop beta function and try to integrate it to get the flow of the coupling and in the answer you see that its possible to tradeoff the bare-coupling and the renormalization scale in terms of a fictitious mass scale.

• But doesn't it happen in a fundamentally different way in Gross-Neveau?

In here because of chiral symmetry isn't it guaranteed that nothing will diverge in perturbation theory and no mass scale will appear perturbatively? So the dimensional transmutation happens only when you try to exactly try to do the path-integral (..which only works may be in the limit of large number of Fermions..)

I mean - in general shouldn't one be surprised if no scale appears in doing perturbation theory?

• Also if the above is true then what is exactly so special about Gross-Neveu comapared to say QED with massless Fermions? Why is it that in the former the masslessness is perturbatively stable but so is not true for massless QED?

  How well do we know or can prove that RG flow will not break any symmetries of the classical Lagrangian?

Answer 1

Dimensional transmutation is of course not generic and is shown by few models, Gross-Neveu being one of those.

I guess fundamentals are the same, the theory has dependence on a dimensionless parameter which transforms into a dimensionful parameter.

All you have to find is the dependence of coupling constant on renormalization scale, which is also the case in Gross-Neveu (if you are reading Coleman).