The spin parity of $Z_c^-$(4100), $Z_1^+$(4050) and $Z_2^+$(4250)

Abstract

We conjecture that $Z_c^-$(4100) found by LHCb group from a Dalitz plot analysis of $B^0\to \eta_c K^+\pi^-$ decay is the charge conjugate of $Z_1^+$(4050) observed in $\chi_{c1}\pi^+$ distribution from Belle collaboration. Some interesting conclusions are inferred from this assumption. The $Z_2$(4250) would be assigned to be a $J^P = 1^+$ or $1^-$ state because of its absence in $\eta_{c}\pi^-$ invariant mass distribution, while $Z_1^+$(4050)/$Z_c^-$(4100) could be a $0^+$ or $1^-$ state but $2^+$ is unfavored because it would be coupled to $\eta_{c}\pi$ in $D$-wave. The null observation of $Z_1 Z_2$, $Z_1 Z_1$ and $Z_2 Z_2$ production in $e^+ e^-$ annihilation and $\Upsilon(1S,2S)$ decay by Belle collaboration would further allocate the spin parity combination of $Z_1^+$(4050)/$Z_c^-$(4100) and $Z_2$(4250). Our deductions can be used to exclude a set of proposed models and could be further tested by future experiment, e.g. in $\gamma \gamma$ collisions.