Revisiting $B_c^{(\ast )}$ production in $Z$-boson decays at NLO in QCD

We study $B_c^{(\ast )}$ production in $Z$-boson decays at next-to-leading order (NLO) in QCD within the color-singlet nonrelativistic QCD framework. The analysis covers both the $1S$ and $2S$ pseudoscalar and vector states, extending the phenomenological study to radially excited $B_c$ mesons. We calculate direct production widths and energy-fraction and invariant-mass distributions, including both charm- and bottom-quark production topologies. For the central inputs and ${\mu }_R=2m_b$, the $K$ factors are $1.52$ for $B_c$ and $1.33$ for $B_c^{\ast }$. We examine uncertainties from the renormalization scale, heavy-quark masses, and wave functions at the origin, and apply the principle of maximum conformality to fix the renormalization scale from the ${\beta }_0$-dependent NLO contribution. The NLO corrections enhance the small- and intermediate-energy-fraction regions and modify the invariant-mass spectra. We discuss these features in relation to heavy-quark fragmentation. The results provide production-rate and kinematic inputs for studies of ground-state and radially excited $B_c$ mesons at high-statistics $Z$ factories.