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