In studying coupled-channel of $\pi \eta$ and $K\bar{K}$ systems, when calculating $\pi \eta \rightarrow K\bar{K}$ scattering amplitude, the threshold energy is at $s=2 m_{K}$, where the center of mass momentum is: $q=\frac{1}{2}\sqrt{s^2-4m_K^2}$.
My question is that when we go to the complex plane ($s \in \mathbb C$) via unitarizing the amplitude (by the use of analytic continuation) in order to determine pole positions, what should be the condition for this scattering to take place:
$Re(s)>2m_K$ or
$|s|>2m_K$?
I fear the question is not properly phrased: The physical scattering process happens for real values of s only - accrodingly here s needs to be larger than 4m_K^2. On the other hand is the scattering amplitude an analytic function in the whole complex plane (besides cuts and poles). In your example it has 4 sheets and the most relavant pole is that of the a_0(980) which is very close to the bar KK threshold either on the physical sheet for Kbar K or on the corresponding unphysical sheet. The up to date best study for this system is described in
M.~Albaladejo and B.~Moussallam, %``Form factors of the isovector scalar current and the $\eta \pi$ scattering phase shifts,'' Eur.\ Phys.\ J.\ C {\bf 75} (2015) no.10, 488 doi:10.1140/epjc/s10052-015-3715-z [arXiv:1507.04526 [hep-ph]].
There is also a lattice QCD study of the system: See
J.~J.~Dudek {\it et al.} [Hadron Spectrum Collaboration], %``An $a_0$ resonance in strongly coupled $\pi \eta$, $K\overline{K}$ scattering from lattice QCD,'' Phys.\ Rev.\ D {\bf 93} (2016) no.9, 094506 doi:10.1103/PhysRevD.93.094506 [arXiv:1602.05122 [hep-ph]].
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