In QFT, for instance in $\phi^3$ theory, the scattering amplitudes are said to be constrained to feature so called "physical poles" only.

Consider generalized Mandelstam variables $$s_{ij},s_{ijk},s_{ijkl},...$$

defined as

$$s_{i_1i_2,...,i_m}=\left(\sum_{j=1}^m p_{i_j}\right)^2$$ where each $p_{i_j}^\mu$ is a four momentum corresponding to kinematics of external particles in the scattering process.

In case of $\phi^3$ theory the set of physical poles for $n$ point scattering is given by the generalized Mandelstam variables with indices strictly neighboring - i.e. $s_{12},s_{2,3,4}$ or wrapping around as $s_{n-1,n,1,2,3}$. What tells us that these indices should be neighboring?

Or, more generally:

I wonder where does the information come from for how exactly physical poles are supposed to look like? How would we go about finding a constraint for the shape of physical poles in a generic amplitude in a different theory?