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metroplex021

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In summary, the conversation discusses the concept of non-renormalizable interactions in theories and their implications for the validity of the theory at high energies. The possibility of a perturbatively non-renormalizable theory being asymptotically safe is mentioned, and the question is posed if there are any successful physical models of such a theory. The conversation also touches on the difference between a theory being UV complete and being renormalizable, and the potential for magical cancellations in non-renormalizable theories.

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metroplex021

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atyy

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genneth

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metroplex021

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genneth said:

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?)

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chrispb

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genneth

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metroplex021 said: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?)

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.

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metroplex021

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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.

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genneth

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tom.stoer

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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.chrispb said: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. ... 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.

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Excellent link!atyy said:

The Weinberg angle, also known as the weak mixing angle, is a parameter that describes the strength of the interaction between the electromagnetic and weak forces. It is important because it helps explain the origin of mass and is a key component of the Standard Model of particle physics.

A non-renormalizable theory is a theoretical framework that cannot be made mathematically consistent through the process of renormalization, which is a mathematical technique used to remove infinities from quantum field theories. These theories are often considered to be incomplete or lacking in predictive power.

Asymptotic safety is a proposed solution to the problem of non-renormalizable theories. It suggests that at very high energies, the strength of the interactions in these theories approaches a fixed value, making them mathematically consistent and predictive. This idea is still being explored and is not yet fully understood.

Some examples of non-renormalizable theories include quantum gravity, which attempts to reconcile general relativity with quantum mechanics, and some extensions to the Standard Model of particle physics, such as grand unified theories and supersymmetry. These theories have not yet been proven to be correct and are still being studied by scientists.

While non-renormalizable theories may not be fully consistent or predictive on their own, they are still important in the quest for a more complete understanding of the universe. They serve as starting points for further research and can provide insights into new theoretical frameworks and experimental techniques. Additionally, the concept of asymptotic safety is still being explored and could potentially lead to a breakthrough in understanding these theories.

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