The $B_c^{+}$ family of $(c\overline{b})$ mesons with beauty and charm is of special interest among heavy quarkonium systems. The $B_c^{+}$ mesons are intermediate between $(c\overline{c})$ and $(b\overline{b})$ states both in mass and size, so many features of the $(c\overline{b})$ spectrum can be inferred from what we know of the charmonium and bottomonium systems. The unequal quark masses mean that the dynamics may be richer than a simple interpolation would imply, in part because the charmed quark moves faster in $B_c$ than in the $J/\psi$. Close examination of the $B_c^{+}$ spectrum can test our understanding of the interactions between heavy quarks and antiquarks and may reveal where approximations break down. Whereas the $J/\psi$ and $\mathrm{\Upsilon }$ levels that lie below the flavor threshold are metastable with respect to strong decays, the $B_c$ ground state is absolutely stable against strong or electromagnetic decays. Its dominant weak decays arise from $\overline{b}\to \overline{c}{W}^{\star +}$, $c\to s{W}^{\star +}$, and $c\overline{b}\to W^{\star +}$ transitions, where $W^{\star }$ designates a virtual weak boson. Prominent examples of the first category are quarkonium transmutations such as $B_c^{+}\to J/\psi {\pi }^{+}$ and $B_c^{+}\to J/\psi {\ell }^{+}{\nu }_{\ell }$, where $J/\psi$ designates the $(c\overline{c})$ $1S$ level. The high data rates and extraordinarily capable detectors at the Large Hadron Collider give renewed impetus to the study of mesons with beauty and charm. Motivated by the recent experimental searches for the radially excited $B_c$ states, we update the expectations for the low-lying spectrum of the $B_c$ system. We make use of lattice QCD results, a novel treatment of spin splittings, and an improved quarkonium potential to obtain detailed predictions for masses and decays. We suggest promising modes in which to observe excited states at the LHC. The $3P$ and $3S$ states, which lie close to or just above the threshold for strong decays, may provide new insights into the mixing between quarkonium bound states and nearby two-body open-flavor channels. Searches in the $B^{(*)}{D}^{(*)}$ final states could well reveal narrow resonances in the $J^P=0^{-}$, $1^{-}$, and $2^{+}$ channels and possibly in the $J^P=0^{+}$ and $1^{+}$ channels at threshold. Looking further ahead, the prospect of very-high-luminosity $e^{+}{e}^{-}$ colliders capable of producing tera-$Z$ samples raises the possibility of investigating $B_c$ spectroscopy and rare decays in a controlled environment.

• Received 8 March 2019

DOI:https://doi.org/10.1103/PhysRevD.99.054025

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Published by the American Physical Society