# On the asymptotics of interacting correlation functions

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Consider an interacting QFT (for example, in the context of the Wightman axioms). Let $G_2(x)$ be the two-point function of some field $\phi(x)$: $$G_2(x)=\langle \phi(x)\phi(0)\rangle$$

Question: What is known about the behaviour of $G_2^{-1}(p)$ at $p\to\infty$? Is there any bound to its growth rate?

It would be nice to have some (non-perturbative) theorem for general spin, but in case this is not possible, you may assume that $\phi(x)$ is scalar. Any reference is also welcome.

Some examples:

A free scalar field has $$G_2^{-1}(p)=p^2+\mathcal O(1)$$ while an interacting one, to first order in perturbation theory, has $$G_2^{-1}(p)=cp^2+\mathcal O(\log p^2)$$ for some $c>0$. Of course, there are large logs at all orders in perturbation theory, and so this result doesn't represent the true $p\to\infty$ behaviour of $G_2(p)$. One could in principle sum the leading logs to all orders but the result, being perturbative, is not what I'm looking for.

Similarly, a free spinor field has $$G_2^{-1}(p)=\not p+\mathcal O(1)$$ while an interacting one, to first order in perturbation theory, has $$G_2^{-1}(p)=c\not p+\mathcal O(\log p^2)$$ as before.

Finally, a free massive vector field has $$G_2^{-1}(p)=\mathcal O(1)$$ while preturbative interactions introduce logs, as usual. It seems natural for me to expect that, non-perturbatively, the leading behaviour is given by the free theory (which has $G_2=p^{2(s-1)}$ for spin $s$), but I'd like to known about the sub-leading behaviour, in a non-perturbative setting.

Update: unitarity

User Andrew has suggested that one can use the optical theorem to put bounds on the rate of decrease of the two-point function: for example, in the case of a scalar field we have $$G_2^{-1}(p^2)\overset{p\to\infty}\ge \frac{c}{p^2}$$ for some constant $c$ (see Andrew's link in the comments for the source).

I'm not sure that this qualifies as an asymptotic for $G_2$ because it doesn't rely on the properties of $G_2(x)$ (nor $\phi(x)$), but it is just a consequence of $SS^\dagger=1$. In other words, we are not really using the axiomatics of the fields, but the physical requirement of a unitary $S$ matrix. As far as I know, in AQFT there is little reference to unitarity. Maybe I'm asking too much, but I have the feeling that one can say a lot about the $n$-point function of the theory using only a few axioms, à la Wightman.

As a matter of fact, I believe that it is possible to use Froissart's theorem to obtain tighter bounds on the decay of the two-point functions, bounds more restrictive than those of the optical theorem alone. But I haven't explored this alternative in detail for the same reasons as above.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user AccidentalFourierTransform

edited Jan 9, 2017
In my understanding (which might be wrong), a theory obeying the Wightman axioms is "already renormalized", i.e. you have no notion of "bare propagators" in it, so how are you defining the self-energy in a Wightman theory? The self-energy is a "perturbative object".

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user ACuriousMind
@ACuriousMind there are non-perturbative definitions of irreducible (aka, fully connected) correlation functions (obtained by taking functional derivatives of the Legendre transform of the partition function). In practice, the irreducible two-point function is just the inverse of the full two point function: $\Pi(p)=G_2(p)^{-1}$, where $G_2(x)=\langle\phi(x)\phi(0)\rangle$. In other words, and to be clear: I am asking about the behaviour of the two-point function, in momentum space, at $p\to\infty$.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user AccidentalFourierTransform

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user ved
@ved thanks, but that post is discussing the propagator (ie, the free correlation function). What I'd like to know is the behaviour of the interacting correlation function.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user AccidentalFourierTransform
Interaction terms will basically modify the mass term of propagator and the propagator will involve a physical mass for on-shell regularization scheme (or at some scale for others), so If you consider $p\to\infty$ limit then behaviour of propagator would stay same.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user ved
For the bounty: I'm looking for the asymptotics of two-point functions in a non-perturbative setting. Thanks!

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user AccidentalFourierTransform
Not sure if this is what you had in mind, but there is a non-perturbative bound which follows from unitarity that the propagator cannot fall off faster than $1/p^2$ as $p\rightarrow \infty$ (for example see equations 84 and 85 of the notes my Matt Schwartz isites.harvard.edu/fs/docs/icb.topic1146665.files/…). He shows it for spin 0 but I believe this result holds for general spins.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user Andrew
@Andrew thanks! I have edited the question to discuss your comment.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user AccidentalFourierTransform
@Andrew The unitarity fall off $1/p^2$ that you quote actually applies only for correlation functions that are assumed to vanish at infinity. There are perfectly healthy CFT where $\phi$ is a primary operator with dimension $\Delta>2$ where this is not the case. One has to perform the so-called subtractions in order to do the Fourier transforms, and this implies the presence of a finite polynomial in the propagator, on top of the decreasing contribution when $p\rightarrow\infty$. Just try to Fourier transform the 2pt-function of a field $\phi$ with dimension $\Delta>2$, and you see the point.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user TwoBs
beside unitarity, see my comment above to Andrew though, one `requires' polynomially boundedness that comes from the tempered distribution nature of the Wightman functions. It is believe that string theory and other very peculiar theories (such as the Galileon) seem to violate this condition as they have some degree of non-locality built-in. As for your last comment about the Froissart bound, there exist plenty of interesting and well defined theories (e.g. all gapless ones, CFTs, gravity,...) where it is violated.

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user TwoBs
@TwoBs huh, that's really interesting and I hadn't thought of that. Maybe the question is more complicated than I thought, and I should narrow it down a bit? Anyway, you gave me a couple of topics to think about, thanks!

This post imported from StackExchange Physics at 2017-01-09 20:52 (UTC), posted by SE-user AccidentalFourierTransform

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