Z_c and Z_cs systems with operator mixing at NLO in QCD sum rules

We study the mass spectra of hidden-charm tetraquark systems with quantum numbers (I^G)J^P = (1⁺)1⁺ (and their I = $\frac{1}{2}$ \( \frac{1}{2} \) partners) using QCD sum rules. The analysis incorporates the complete next-to-leading order (NLO) contribution to the perturbative QCD part of the operator product expansions, with particular attention to operator mixing effects due to renormalization group evolution. We find that both the parametric dependence and the perturbative convergence are significantly improved for the two mixed operators $J_{1,5}^{\text{Mixed}}$ \( {J}_{1,5}^{\textrm{Mixed}} \) and $J_{2,6}^{\text{Mixed}}$ \( {J}_{2,6}^{\textrm{Mixed}} \) , compared with those for the unmixed meson-meson or diquark-antidiquark type ones. For the $\overline{d}c\overline{c}u$ \( \overline{d}c\overline{c}u \) system, the masses of $J_{1,5}^{\text{Mixed}}$ \( {J}_{1,5}^{\textrm{Mixed}} \) and $J_{2,6}^{\text{Mixed}}$ \( {J}_{2,6}^{\textrm{Mixed}} \) are determined to be ${3.89}_{-0.12}^{+0.18}$ \( {3.89}_{-0.12}^{+0.18} \) GeV and ${4.03}_{-0.07}^{+0.06}$ \( {4.03}_{-0.07}^{+0.06} \) GeV, respectively, closely matching those of Z_c(3900) and Z_c(4020). Similarly, for the $\overline{s}c\overline{c}u$ \( \overline{s}c\overline{c}u \) states, the masses of $J_{1,5}^{\text{Mixed}}$ \( {J}_{1,5}^{\textrm{Mixed}} \) and $J_{2,6}^{\text{Mixed}}$ \( {J}_{2,6}^{\textrm{Mixed}} \) are found to be ${4.02}_{-0.09}^{+0.17}$ \( {4.02}_{-0.09}^{+0.17} \) GeV and ${4.21}_{-0.07}^{+0.08}$ \( {4.21}_{-0.07}^{+0.08} \) GeV, respectively, in close proximity to Z_cs(3985)/Z_cs(4000) and Z_cs(4220), consistent with the expectation that they are the partners of Z_c(3900) and Z_c(4020). Our results highlight the crucial role of operator mixing, an inevitable effect in a complete NLO calculation, in achieving a robust phenomenological description for the tetraquark system.