# Kallen-Lehmann representation and branch cuts at threshold masses

Let us consider the Kallen-Lehmann representation for the two-point function of scalar fields

$$\langle \Omega | T\left\{\phi(x) \phi(y)\right\}|\Omega\rangle = \int \frac{d^4 p}{(2\pi)^4} e^{ip\cdot (x-y) }i \Pi(p^2)$$

where $$\Pi(p^2)$$ is expressed in terms of a spectral density function $$\rho(p^2)$$

$$\Pi(p^2) \equiv \int_0^{\infty}dq^2 \frac{\rho(q^2)}{p^2-q^2+i\epsilon}.$$

The density $$\rho(p^2)$$ is real and positive, and hence $$\rho(q^2)=-\frac{1}{\pi}\text{Im}\left[\Pi(q^2)\right]$$.

Is there a general non-perturbative argument and not relying on the specific dynamics of the theory, to conclude that $$\Pi(p^2)$$ has branch-cuts starting at $$p^2=m_{th}^2$$ where $$m_{th}$$ is possibly a threshold mass corresponding to production of $$n$$-particles state?

I am not aware of any such general non-perturbative result. For instance, this statement is usually shown in explicit simple examples at one-loop level and perturbation theory (e.g. see Eq.24.77 of [1]), so I think this result is perturbative. Any comment on this?

References

[1] : M. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013)