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Weinberg, non-renormalizable theories and asym safety

  1. Jun 12, 2010 #1
    Until recently, I thought that any theory that contains non-renormalizable interactions in the power-counting sense (i.e. those whose couplings have negative mass dimension) must be an 'effective' theory that necessarily breaks down at some energy. However, I've been looking at Weinberg's QFT textbook and according to him it might be possible for a perturabatively non-renormalizable theory to be asymptotically safe. Does anyone know if this is just a conjecture, or whether there actually exist successful and convincing physical models of a perturbatively non-renormalizable theory that is asymptotically safe? Any info would be most appreciated.
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  3. Jun 12, 2010 #2


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  4. Jun 14, 2010 #3
    Power counting does not yield the correct scaling if the relevant fixed point is not the trivial Gaussian one. In many theories close to the upper critical dimension the anomalous scaling is not too great and power counting yields reasonable estimates. This is not necessarily the case in general.
  5. Jun 14, 2010 #4
    Thanks so much for that. I've read that the Gaussian fixed point is the kind of fixed point you get if you employ a perturbative framework. Does this mean that if we ARE using a perturbative framework to do QFT, it's still necessary that the theory is power-counting renormalizable if it is to be valid to arbitrary energy (though of course it's also necessary that it have such a fixed point and hence be asymptotically free?)
  6. Jun 14, 2010 #5
    To the best of my knowledge, d=4 N=8 SUGRA naively appears to be a nonrenormalizable theory. Nonetheless, to several (seven?) orders in perturbation theory, the theory turns out to be either renormalizable or finite; I believe it's the latter. N=4 d=4 SYM is finite, and I believe that statement is true to all orders. If I'm correct in my thinking, this would imply that there are theories with magical cancellations that take place that power counting couldn't tell us about.
  7. Jun 14, 2010 #6
    You need to distinguish between the statements "this theory is UV complete (well defined at arbitrary energies)" and "this theory is renormalisable". The former is what we really want, the latter is a formal statement that the (possibly Gaussian) fixed point has no non-renormalisable terms.

    Trying to connect the two can run into problems of several kinds.

    1. As already noted, if the fixed point is actually not Gaussian, then one gets "anomalous" scaling, and power counting is not sufficient to determine the actual flow under renormalisation. The fundamental problem here is that we cannot (generically) do any calculation with anything other than a Gaussian theory (exceptions include conformal theories in 2D, exactly integrable systems, etc.), and then the only option available is perturbation theory about a Gaussian one. Note that just because we can only do calculations with such a theory doesn't make it right.

    2. A non-renormalisable/irrelevant term may, under RG flow, cause a relevant term to grow, which then flows away from the RG fixed point (I'm thinking of running the cutoff downwards here...).

    Fundamentally, one should not get too hung up on this issue. Asking for theories to be effective up to arbitrary energies/length scales is always going to get you into trouble. It's just a shame that particle physics textbooks are still written with that arrogance built-in, and students have to unlearn it themselves.
  8. Jun 15, 2010 #7
    Thank you, that is really useful. Just to finish up, can I ask if the following sort of thought is broadly right.

    When Weinberg first raises non-renormalizable theories in volume 1 of his textbooks, he says that "there seem to be just two possibilities" for what happens at high energies: "one is that the growing strength of the effects somehow 'saturates', avoiding any conflict with unitarity [thus achieving asymptotic safety]. The other is that new physics enters at the scale M". Would I be right in thinking that it takes a very special conspiracy of couplings (one of the contributors to this post has dubbed it 'magical') in order for a power-counting non-renormalizable theory to manage to pull off asymptotic safety? In other words, is it really just overwhelmingly unlikely that an arbitrary non-renormalizable theory would be UV stable, while that's not the case for super-renormalizable theories?

    Thanks so much for your help up to here - the clarifications have really helped me cut through what seemed to be conflicting stories.
  9. Jun 15, 2010 #8
    I think it's fair to say that no-one really knows. It's hard to say "unlikely" if reality does indeed turn out that way...
  10. Jun 15, 2010 #9


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    This is already known for GR w/o matter. Afaik the 1-loop divergence disappears due to an "extra symmetry". Afaik in d=4 N=8 SUGRA one necessary ingredient for finiteness is the restriction to on-shell amplitudes; so divergencies may still appear off-shell.
  11. Jun 20, 2010 #10


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