Analytical structure of S-matrix

I wonder whether a reader will have an answer - or even a complete answer.

Imagine the S-matrix of a quantum field theory. It may have poles. The resonances are always above the real positive semi-axis in the "k_0" plane - by causality - and there are images below the negative semi-axis of "k_0". It would seem very bad if a pole - or a branch cut - were located just below the positive semi-axis of "k_0". Such a pole would look like an exponentially growing resonance, and it does not seem to agree with the "i.epsilon" prescription which reflects the causality of Feynman's propagators, as Nima has also emphasized to me right now.

But what about the poles that are further from the real axis? Is there some constraint that there can't be any poles in the whole two quadrants of the "k_0" plane with different signs of "Im(k_0)" and "Re(k_0)"? Or is the condition simpler in the s-plane where "s" is the Mandelstam variable? What is exactly the condition and how can one prove it? The main reason for this question is that what I've believed is the analytical structure of the M-theoretical S-matrix in 11 dimensions - with a Z_3 symmetry - seems to violate these rules. If you know the answer, thanks in advance.