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 $B^0\to η_c K^+π^-$ decay is the charge conjugate of $Z_1^+$(4050) observed in $χ_{c1}π^+$ 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 $η_{c}π^-$ 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 $η_{c}π$ 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 $Υ(1S,2S)$ 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 $γγ$ collisions.