78. A talk on scalar QFT, exact renormalization group and RG fixed points

Oliver Rosten who, as I gather,  now works in the U. of Sussex with Daniel Litim, recently gave a talk on exact renormalization group in Perimeter. The main conclusion is that there are non non-trivial RG fixed points for a scalar QFT in D=4 (that is, no fixed points except the gaussian one corresponding to free field theory), and the theory is trivial in the IR.

What is this all about? Exact renormalization group differs from the poor old Wilsonian RG by taking irrelevant couplings into account. Than, when we are calculating a beta-function, it is not given by the sum of contributions from all loops as in old Wilsonian RG, but is completely defined by the 1-loop expression (that is why this RG is called exact). On the other hand, now, instead of having just a single beta-function for the relevant coupling, we have infinite number of couplings, with corresponding RG equations coupled in turn to each other.

No, the question is: having RG equations rewritten in such a way, can we get any new information about regimes of the theory where perturbative expansion breaks down? I doubt it.

First issue is that, as Jacques Distler explains, nobody is able to properly treat this infinite set of coupled equations completely, so people usually cut it at some point. It is not quite clear why infinite set of neglected interactions is than unimportant.

The second issue mentioned by Jacques is that ERG is not truly perturbative since the effective UV cutoff in the ERG is the cutoff for loop integrals appearing in the Feynman perturbation theory. It therefore does not know anything about nonperturbative physics – one can recall  examples of theories with SUSY, where coupling remains unrenormalized in all orders of perturbation theory but acquires corrections from non-perturbative effects (instantons, for example).

The third issue that came on my mind is the following. Suppose I start with a renormalizable theory that is incomplete.  For example, I take QED which is theory with zero charge. Does ERG say me anything about correct extension of QED that might cure the zero charge problem? (As we know, the solution is electroweak theory.) I think the answer is negative – I shall just blankly perform RG analysis to find that my beta-function diverges at some k.

If you know something about ERG (or even better – know how to deal with these three issues), please feel free to explain it to me, and I shall be limitlessly thankful.