Next-to-leading-order QCD corrections to the decay of $Z$ boson into ${\chi }_c({\chi }_b)$

Abstract

Based on the framework of nonrelativistic Quantum Chromodynamics (NRQCD), we carry out next-to-leading order (NLO) QCD corrections to the decay of $Z$ boson into ${\chi }_c$ and ${\chi }_b$, respectively. The branching ratio of $Z\to {\chi }_c({\chi }_b)+X$ is about $10^{-5}(10^{-6})$. It is found that, for $Z\to {\chi }_c({\chi }_b)+X$, the single gluon fragmentation diagrams of ${}^3{S}_1^{[8]}$, which first appear at the NLO level, can provide significant contributions, leading to a great enhancement on the leading-order results. Consequently the contributions from the color octet (CO) channels will account for a large proportion of the total decay widths. Moreover, the introduction of the CO processes will thoroughly change the color singlet (CS) predictions on the ratios of ${\Gamma }_{{\chi }_{c1}}/{\Gamma }_{{\chi }_{c0}}$, ${\Gamma }_{{\chi }_{c2}}/{\Gamma }_{{\chi }_{c0}}$, ${\Gamma }_{{\chi }_{b1}}/{\Gamma }_{{\chi }_{b0}}$ and ${\Gamma }_{{\chi }_{b2}}/{\Gamma }_{{\chi }_{b0}}$, which can be regarded as an outstanding probe to distinguish the CO and CS mechanism. With regard to the CS (${}^3{P}_J^{[1]}$) channels, the heavy quark pair associated processes serve as the leading role, however, in the case of ${\chi }_b$, $Z\to b\bar{b}{[}^3{P}_J^{[1]}]+g+g$ can also contribute significantly. Summing over all the feeddown contributions from ${\chi }_{cJ}$ and ${\chi }_{bJ}$, respectively, we find $\Gamma (Z\to J/\psi +X){|}_{{\chi }_c-\text{feeddown}}=(0.28-2.4)\times 10^{-5}$ and $\Gamma (Z\to {\rm Y}(1S)+X){|}_{{\chi }_b-\text{feeddown}}=(0.15-0.49)\times 10^{-6}$.

Comments: 18 pages, 6 figures and 6 tables. Accepted for publication in PRD