# 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.

• 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". – ACuriousMind Dec 4 '16 at 13:39
• @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 $p\to\infty$. – AccidentalFourierTransform Dec 4 '16 at 13:42
• This may be helpful.physics.stackexchange.com/questions/274265/… – ved Dec 5 '16 at 8:54
• @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. – TwoBs Jan 5 '17 at 20:36
• 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. – TwoBs Jan 5 '17 at 20:44

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 $$W_2(x^2)\overset{x^2\to-\infty}\sim\exp\left[-m\sqrt{-x^2}\right]$$ 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.