On the existence of bound states in SIMP dark sectors

In strongly interacting massive particle (SIMP) scenarios, dark matter is comprised of stable dark pions whose $3\to 2$ or $4\to 2$ reactions set the dark matter relic abundance. Recent work has shown that shallow two-pion bound states significantly affect the freeze-out, but did not establish whether such states actually form. In this work we demonstrate that a scalar isosinglet bound state does exist in a well-defined region of parameter space by solving an on-shell Lippmann--Schwinger equation in a chiral-unitary framework and analyzing the $S$-wave $\pi\pi$ amplitude in the complex energy plane. We determine the range of $m_\pi/f_\pi$ for which a pole appears below the two-pion threshold, extract the corresponding residue, and, in the non-relativistic limit, obtain the bound-state wave function at the origin, $|\Psi(0)|$, which controls bound-state-assisted annihilation and decay rates relevant for catalyzed freeze-out. Comparing this T-matrix based result with variational estimates using simple finite-range potentials, we find agreement within order-one factors for shallow binding. For binding energies of order the freeze-out temperature, $E_B \sim m_\pi/20$, we obtain $|\Psi(0)|\sim \mathcal{O}(0.1)\,m_\pi^{3/2}$, thereby supporting the parametric assumptions used in previous phenomenological analyses.