quantum mechanics - On the asymptotics of interacting correlation functions

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.

Answer

Terrific question, OP! I don't have a definitive answer yet, but for lack of a better one, let me mention that the book Quantum fields and strings, by Deligne P., Kazhdan D. and Etingof P. study the asymtotics of Wightman functions in several occasions. Perhaps the most obvious one is section 1.6 Asymptotics of Wightman functions (page 384), where we can read

$$\begin{equation} W_2(x^2)\overset{x^2\to-\infty}\sim\exp\left[-m\sqrt{-x^2}\right] \end{equation}$$ where $W_2$ is $G_2$ in the OP, and $m$ is the lowest eigenvalue of $H$. They don't seem to mention how this generalises to higher spin theories. Perhaps this result is sufficient for your purposes. Please let me know.