NLO effects for $\Omega_{QQQ}$ Baryons in QCD Sum Rules

We study the triply heavy baryons $\Omega_{QQQ}$ $(Q=c, b)$ in the QCD sum rules by performing the first calculation of the next-to-leading order (NLO) contribution to the perturbative QCD part of the correlation functions. Compared with the leading order (LO) result, the NLO contribution is found to be very important to the $\Omega_{QQQ}$. This is because the NLO not only results in a large correction, but also reduces the parameter dependence, making the Borel platform more distinct, especially for the $\Omega_{bbb}$ in the $\overline{\rm{MS}}$ scheme, where the platform appears only at NLO but not at LO. Particularly, owing to the inclusion of the NLO contribution, the renormalization schemes ($\bar {MS}$ and On-Shell) dependence and the scale dependence are significantly reduced. Consequently, after including the NLO contribution to the perturbative part in the QCD sum rules, the masses are estimated to be $4.53^{+0.26}_{-0.11}$ GeV for $\Omega_{ccc}$ and $14.27^{+0.33}_{-0.32}$ GeV for $\Omega_{bbb}$, where the results are obtained at $\mu=M_B$ with errors including those from the variation of the renormalization scale $\mu$ in the range $(0.8-1.2) M_B$. A careful study of the $\mu$ dependence in a wide range is further performed, which shows that the LO results are very sensitive to the choice of $\mu$ whereas the NLO results are considerably better. In addition to the $\mu=M_B$ result, a more stable value, (4.75-4.80) GeV, for the $\Omega_{ccc}$ mass is found in the range of $\mu=(1.2-2.0) M_B$, which should be viewed as a more relevant prediction in our NLO approach because of $\mu$ dependence.