SU(3) breaking effect in the $Z_c$ and $Z_{cs}$ states

Based on the hadronic resonance picture, we propose a possible framework to simultaneously describe the resonance parameters of the observed $Z_{cs}$ and $Z_{c}$ states that are close to the thresholds of the $D^{(*)}\bar{D}^{(*)}_s$ and $D^{(*)}\bar{D}^{(*)}$ systems, respectively. We construct the effective potentials of the $Z_{cs}$ and $Z_{c}$ states by analogy with the effective potentials of the leading order (LO) and next-to-leading-order (NLO) $N-\bar{N}$ interactions. Then we introduce an SU(3) breaking factor $g_x$ to identify the differences between the effective potentials in the $Z_{cs}$ and $Z_{c}$ states. We perform two calculations to discuss the differences and similarities of the $Z_{cs}$ and $Z_{c}$ states. In the first calculation, we adopt the LECs extracted from the experimental $Z_{cs}$ states to calculate the $Z_c$ states, and show that if the $Z_{cs}(4000)$ is the SU(3) partner of the $Z_c(3900)$, then our framework can reproduce the large width difference between the $Z_{cs}(4000)$ and $Z_{c}(3900)$ by adjusting the SU(3) breaking factor in a reasonable region. Besides, this SU(3) breaking effect also accounts for the absence of a $Z_{c}$ state with $J^{PC}=1^{++}$, which should be the SU(3) partner of the $Z_{cs}(3985)$. In the second calculation, we separately fit the LECs from the experimental $Z_{cs}$ and $Z_{c}$ states. We show that these two sets of LECs are very similar to each other, indicating a unified set of LECs that could describe the effective potentials of the $Z_{cs}$ and $Z_{c}$ states simultaneously. Then we proceed to systematically predict the other possible $Z_{cs}$ and $Z_{c}$ states that are close to the $D^{(*)}\bar{D}^{(*)}_s$ and $D^{(*)}\bar{D}^{(*)}$ systems, respectively. Further explorations on the $Z_{cs}$ states would be crucial to test our theory.