Mesons with Beauty and Charm: New Horizons in Spectroscopy

The $B_c ^+$ family of $(c\bar{b})$ mesons with beauty and charm is of special interest among heavy quarkonium systems. The $B_c ^+$ mesons are intermediate between $(c\bar{c})$ and $(b\bar{b})$ states both in mass and size, so many features of the $(c\bar{b})$ 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. ...


I. INTRODUCTION
Although the lowest-lying (cb) meson has long been established, the spectrum of excited states is little explored. The ATLAS experiment at CERN's Large Hadron Collider reported the observation of a radially excited B c state [1], but this sighting was not confirmed by the LHCb experiment [2]. The unsettled experimental situation and the large data sets now available for analysis make it timely for us to provide up-to-date theoretical expectations for the spectrum and decay patterns of narrow (cb) states, and for their production in hadron colliders [3]. New work from the CMS Collaboration [4] shows the way toward exploiting the potential of (cb) spectroscopy.

A. What we know of the Bc mesons
The possibility of a spectrum of narrow B c states was first suggested by Eichten and Feinberg [5]. Anticipating the copious production of b-quarks at Fermilab's Tevatron Collider and CERN's Large Electron-Positron Collider (LEP), we presented a comprehensive portrait of the spectroscopy of the B c meson and its long-lived excited states [6], based on then-current knowledge of the interaction between heavy quarks derived from (cc) and (bb) bound states, within the framework of nonrelativistic quantum mechanics [7]. Surveying four representative potentials, we characterized the mass of the J P = 0 − ground state as M (B c ) ≈ 6258 ± 20 MeV. A small number of B c candidates appeared in hadronic Z 0 decays at LEP. The CDF Collaboration observed the decay B ± c → J/ψ ± ν in 1. 8-TeVpp collisions at the Fermilab Tevatron [8], estimating the mass as M (B c ) ≈ 6400 ± 411 MeV. (The generic lepton represents an electron or muon.) Subsequent work by the CDF [9], D0 [10], and LHCb [11] Collaborations has refined the mass to M (B c ) = 6274.9 ± 0.8 MeV [12], with the most precise determinations coming from fully reconstructed final states such as J/ψ π + .
Investigations based on the spacetime lattice formulation of QCD aim to provide ab initio calculations that incorporate the full dynamical content of the theory of strong interactions. Before the nonleptonic B c decays had been observed, a first unquenched lattice QCD prediction, incorporating 2 + 1 dynamical quark flavors (u/d, s) found M (B c ) = 6304 ± 12 +18 −0 MeV [13], where the first error bar represents statistical and systematic uncertainties and the second characterizes heavyquark discretization effects. Calculations incorporating 2 + 1 + 1 dynamical quark flavors (u/d, s, c) [14] yield M (1 1 S 0 ) = 6278 ± 9 MeV, in impressive agreement with the measured B c mass, and predict M (2 1 S 0 ) = 6894 ± 19 ± 8 MeV [15].
Until recently, the only evidence reported for a (cb) excited state was presented by the ATLAS Collaboration [1] in pp collisions at 7 and 8 TeV, in samples of 4.9 and 19.2 fb −1 . They observed a new state at 6842 ± 7 MeV in the M (B ± c π + π − ) − M (B ± c ) − 2M (π ± ) mass difference, with B ± c detected in the J/ψ π ± mode. The mass (527 ± 7 MeV above M (1S) ) and decay of this state are broadly in line with expectations for the second s-wave state, B ± c (2S). In addition to the nonrelativistic potential-model calculations cited above, the HPQCD Collaboration has presented preliminary results from a lattice calculation using 2 + 1 + 1 dynamical fermion flavors and highly improved staggered quark correlators [19]. They report M (2 1 S 0 ) = 6892 ± 41 MeV, which is 576.5 ± 41 MeV above M (1S) ). This result and the NRQCD prediction [14] lie above the ATLAS report by one and two standard deviations, respectively. The significance of the discrepancy is limited for the moment by lattice uncertainties. A plausible interpretation has been that ATLAS might have observed the transition B * c (2S) → B * c (1S)π + π − , missing the low-energy photon from the subsequent B * c → B c γ decay, and that the signal is an unresolved combination of 2 3 S 1 and 2 1 S 0 peaks. A search by the LHCb collaboration in 2 fb −1 of 8-TeV pp data yielded no evidence for either B c (2S) state [2]. As we prepared this article for publication, the CMS Collaboration provided striking evidence for both B c (2S) levels, in the form of well-separated peaks in the B c π + π − invariant mass distribution, closely matching the theoretical template [4]. We incorporate these new observations into the discussion that follows in §V A. B. Analyzing the (cb) bound states The nonrelativistic potential picture, motivated by the asymptotic freedom of QCD [20], gave early insight into the nature of charmonium and generated a template for the spectrum of excited states [21]. For more than four decades, it has served as a reliable guide to quarkonium spectroscopy, including the states lying near or just above flavor threshold for fission into two heavy-light mesons that are significantly influenced by coupled-channel effects [22,23].
We view the nonrelativistic potential-model treatment as a steppingstone, not a final answer, however impressive its record of utility. Potential theory does not capture the full dynamics of the strong interaction, and while the standard coupled-channel treatment is built on a plausible physical picture, it is not derived from first principles. Moreover, relativistic effects may be more important for (cb) than for (cc). The c-quark moves faster in the B c meson than in the J/ψ, because it must balance the momentum of a more massive b-quark. One developing area of theoretical research has been to explore methods more robust than nonrelativistic quantum mechanics [17,24,25].
Nonperturbative calculations on a spacetime lattice in principle embody the full content of QCD. This approach is yielding increasingly precise predictions for the masses of (cb) levels up through B * c (2 3 S 1 ) state. It is not yet possible to extract reliable signals for higher-lying states from the lattice, so we rely on potential-model methods to construct a template for the B c spectrum through the 4 3 S 1 level. If experiments should uncover systematic deviations from the expectations we present, they may be taken as evidence of dynamical features absent from the nonrelativistic potential-model paradigm, including-of course-coupling to states above flavor threshold, which we neglect our calculations of the spectrum.
In the following §II, we develop the theoretical tools required to compute the (cb) spectrum. In earlier work [6], we examined the Cornell Coulomb-plus-linear potential [22], a power-law potential [26], Richardson's QCDinspired potential [27], and a second QCD-inspired potential due to Buchmueller and Tye [28], which we took as our reference model. We used a perturbation-theory treatment of spin splittings. Using insights from lattice QCD and higher-order perturbative calculations, we construct a new potential that differs in detail from those explored in earlier work. We also use lattice results and rich experimental information on the (cc) and (bb) spectra to refine the treatment of spin splittings. We present our expectations for the spectrum of narrow states in Section III. We consider decays of the narrow states in section IV, updating the results we gave in Ref. [6]. We compute differential and integrated cross sections for the narrow B c levels in proton-proton collisions at the Large Hadron Collider in §V. Putting all these elements together, we show how to unravel the 2S levels and explore how higher levels might be observed. Prospects for a future e + e − → Tera-Z machine appear in §VI. We draw some conclusions and look ahead in Section VII.

II. THEORETICAL PRELIMINARIES
We take as our starting point a Coulomb-plus-linear potential (the "Cornell potential" [22]), where κ ≡ 4α s /3 = 0.52 and a = 2. 34 GeV −1 were chosen to fit the quarkonium spectra. Analysis of the J/ψ and Υ families led to the choices This simple form has been modified to incorporate running of the strong coupling constant in Refs. [27,28], among others, using the perturbative-QCD evolution equation at leading order and beyond. At distances relevant for confinement, perturbation theory ceases to be a reliable guide. It is now widely held, following Gribov [29], that as a result of quantum screening α s approaches a critical, or frozen, value at long distances (low energy scales). In a light (qq) system, Gribov estimated We incorporate the spirit of this insight into a new version of the Coulomb-plus-linear form that we call the frozenα s potential.
The long-range part is the standard Cornell linear term. To obtain the Coulomb piece, we convert the fourloop running of α s (q) in momentum space [30] to the behavior in position space using the method of [31], with an important modification. We set α s (q = 1.6 GeV) = 0.338 and evolve with three active quark flavors. To enforce saturation of α s (r) at long distances, we alter the recipe of Ref. [31], replacing the identification q = 1/r exp(γ E ), where γ E = 0.57721 . . . is Euler's constant, with the damped form q = 1/[(r exp(γ E ) 2 + µ 2 ] 1/2 . For our reference potential, we have chosen the damping parameter µ = 1.2 GeV. The consequent evolution of α s (r) is plotted as the solid red curve in Figure 1, where we also show an alternative choice of µ = 0.8 GeV (dashed gold curve), the constant α s of the original Cornell potential (dotted green curve) and α s (r) corresponding to the Richardson potential (dot-dashed blue curve).
We plot in Figure 2 the frozen-α s potential for both our chosen example, µ = 1.2 GeV, and the alternative, µ = 0.8 GeV. There we also show the Richardson and Cornell potentials. All coincide at large distances. The Cornell potential is deeper at short distances than any of the potentials that take account of the evolution of α s . For the convenience of others who may wish to apply the new potential, we present values of α s (r) suitable for interpolation in an Appendix. We presented the general formalism for spin-dependent interactions as laid out by Eichten & Feinberg [5] and Gromes [32] in § II B of Ref. [6], where we took a perturbative approach to the spin-orbit and tensor interactions. In the intervening time, the charmonium and bottomonium spectra have been mapped in detail, as summarized in Table I. This wealth of information leads us now to choose a more phenomenological approach.
We write the spin-dependent contributions to the (cb) masses as where the individual terms are s i and s j are the heavy-quark spins, S = s i + s j is the total spin, L is the orbital angular momentum of quark and antiquark in the bound state, S ij = 4 [3( s i ·n)( s j ·n) − s i · s j ] is the tensor operator, andn is an arbitrary unit vector. We will deal with the hyperfine interaction T 3 momentarily. We express the other T k as where we have introduced the phenomenological coefficientsc 2 andc 4 , which take the value unity in the perturbative approach.
We extract values of T 2 and T 4 for the observed levels that appear in Table I. These are shown as the underlined entries in Table II. Then, we combine the definitions in Eq. (6) with our calculated values of α s /r 3 to determinẽ c 2 andc 4 in the (cc) and (bb) families. The geometric mean of these values is our estimates for the coefficients in the cb system. We insert these back into Eq. (6) to estimate the values of T 2 and T 4 for the B c family. For completeness, we include our evaluations of (1/r)dV /dr in the Table.  For the J/ψ and Υ families, composed of equal-mass heavy quarks, the familiar LS coupling scheme, in which states are labeled by n 2S+1 L J , is apt. When the quark masses are unequal, as in the case at hand, spindependent terms in the Hamiltonian mix the spin-singlet and spin-triplet J = L states. We define with and Then our calculations of the T k defined in Eq. (6) lead to these values for the mixing angle: The masses of the mixed states are At lowest order, the hyperfine splitting between s-wave states, arising from T 3 , is given by which is susceptible to significant quantum corrections. Rather than make a priori calculations of the hyperfine splitting, we adopt the lattice QCD result for the ground state and scale the splittings of excited states according to  (18)(1) [14] 551 2 3 S1 580 601 (19)(1) [14] 582 1276 -1280

III. THE Bc SPECTRUM
The vector meson B * c , the 1 3 S 1 hyperfine partner of B c and analogue of J/ψ and Υ, has not yet been observed. Modern lattice calculations [14,34,35] give consistent values for the hyperfine splitting M (B * c ) − M (B c ) = (53 ± 7, 54 ± 7, 55 ± 3 MeV), so we take the mass of the vector state to be M (B * c ) = 6329 MeV and fix the centroid M (1S) of the ground-state s-wave doublet at 6315.5 MeV for the lattice.
We summarize in Table III predictions for the spectrum of mesons with beauty and charm from our 1994 article [6], lattice QCD calculations, and the present work, expressed as excitations with respect to the 1S centroid. Other potential-model calculations, some incorporating relativistic effects, may be found in the works cited in Ref. [7].
Our expectations for the spectrum of states are shown in the Grotrian diagram, Figure 3, along with several of the lowest-lying open-flavor thresholds. The thresholds for strong decays of excited (cb) levels are known experimentally to high accuracy, as shown in Table IV. Comparing with the model calculations summarized in Table III, we conclude that two sets of narrow s-wave (cb)  Table IV. 1212 levels will lie below the beauty+charm flavor threshold, in agreement with general arguments [36]. All of the potential models cited in Ref. [7] predict 3 3 S 1 masses well above the 829-MeV BD threshold. For the 3 1 S 0 level, only the Ebert et al. prediction does not lie significantly above B * D threshold. Lattice QCD calculations do not yet exist for states beyond the 2S levels.

A. Electromagnetic transitions
The only significant decay mode for the 1 3 S 1 (B * c ) state is the magnetic dipole (spin-flip) transition to the ground state, B c . The M1 rate for transitions between s-wave levels is given by where the magnetic dipole moment is and k is the photon energy. Apart from that M1 transition, only the electric dipole transitions are important for mapping the (cb) spectrum. The strength of the electric-dipole transitions is governed by the size of the radiator and the charges of the constituent quarks. The E1 transition rate is given by where the mean charge is k is the photon energy, and the statistical factor S if = S f i is as defined by Eichten and Gottfried [37]. S if = 1 for 3 S 1 → 3 P J transitions and S if = 3 for allowed E1 transitions between spin-singlet states. The statistical factors for d-wave to p-wave transitions are reproduced in Table V. The significant M1 and E1 electromagnetic transition rates and the ππ cascade rates are given in Table VI, along with the total widths in the absence of strong decays.

B. Hadronic transitions
We evaluate the rates for hadronic transitions between (cb) levels according to the prescription we detailed in §IIIB of Ref. [6]. The results are included in Table VI. Dipion cascades to the ground-state doublet are the dominant decay modes of 2 3 S 1 and 2 1 S 0 , and will be key to characterizing those states, as we shall discuss in §V A.
As observed long ago by Brown and Cahn [38], an amplitude zero imposed by chiral symmetry pushes the π + π − invariant mass distribution to higher invariant masses than phase-space alone would predict. In its simplest form, this analysis yields a universal form for the normalized dipion invariant mass distribution in quarkonium cascades Φ → Φ π + π − , where x = M/2m π and K is the three-momentum carried by the pion pair. The soft-pion expression (18) describes the depletion of the dipion spectrum at low invariant masses observed in the transitions ψ(2S) → ψ(1S)ππ, Υ(2S) → Υ(1S)ππ, and Υ(3S) → Υ(2S)ππ, but fails to account for structures in the Υ(3S) → Υ(1S)ππ spectrum [39]. We expect the 3S levels to lie above flavor threshold in the (cb) system, and so to have very small branching fractions for cascade decays (but see the final paragraph of §V A.

C. Properties of (cb) wave functions at the origin
For quarks bound in a central potential, it is convenient to separate the Schrödinger wave function into radial and angular pieces, as Ψ n m ( r) = R n (r)Y m (θ, φ), where n is the principal quantum number, and m are the orbital angular momentum and its projection, R n (r) is the radial wave function, and Y m (θ, φ) is a spherical harmonic [40]. The Schrödinger wave function is normalized, d 3 r |Ψ n m ( r)| 2 = 1, so that ∞ 0 r 2 dr|R n (r)| = 1. The value of the radial wave function, or its first nonvanishing derivative, at the origin, is required to evaluate pseudoscalar decay constants and production rates through heavy-quark fragmentation.
Our calculated values of |R ( ) n (0)| 2 are given in Table VII.
The pseudoscalar decay constant f Bc , which enters the calculations of annihilation decays such as cb → W + → τ + + ν τ , is defined by where A µ is the axial-vector part of the charged weak current, V cb is an element of the Cabibbo-Kobayashi-Maskawa quark-mixing matrix, and q µ is the fourmomentum of the B c . Its counterpart for the vector state is where V µ is the vector part of the charged weak current and ε * µ is the polarization vector of the B * c . The groundstate pseudoscalar and vector decay constants are given in terms of the wave function at the origin by the Van Royen-Weisskopf formula [41], generically where the leading-order QCD correction is given by [42] and δ P = 2; δ V = 8/3.
Choosing the representative value α s = 0.38, and using the quark masses given in Eq.
(2), we find Consequently, we estimate the ground-state meson decay constants as so that f B * c /f Bc = 0.945. The compact size of the (cb) system enhances the pseudoscalar decay constant relative to f π and f K . This is to be compared to a state-or-the-art lattice evaluation [43], f Bc = 434 ± 15 MeV, which entails improved NonRelativistic QCD for the valence b quark and the Highly Improved Staggered Quark (HISQ) action for the lighter quarks on gluon field configurations that include the effect of u/d, s and c quarks in the sea with the u/d quark masses going down to physical values.
The same calculation yields f B * c /f Bc = 0.988 ± 0.027. A calculation in the framework of QCD sum rules gives f Bc = 528 ± 19 MeV [44].

V. PRODUCTION OF (cb) STATES AT THE LARGE HADRON COLLIDER
We present in Table VIII cross sections for the production of B c states at the Large Hadron Collider, calculated using the framework of the BCVEGPY2.2 generator [45], which we have extended to include the production of 3P states. Cross sections for the physical (2, 3)P ( ) 1 states are appropriately weighted mixtures of the 3 P 1 and 1 P 1 cross sections.
TABLE VIII. Production rates (in nb) for (cb) states in pp collisions at the LHC. The production rates were calculated using the BCVEGPY2.2 generator of Ref. [45], extended to include the production of 3P states. Color-octet contributions to s-wave production are small; we show them (following |) only for the 1S states.  The rapidity distributions (for B * c production, Figure 4) and transverse-momentum distributions (shown for B c production, Figure 5) are similar in character for √ s = 8, 13, and 14 TeV. The rapidity distributions for low-lying (cb) states are shown in Figure 6. The acceptance of the CMS and ATLAS detectors covers central pseudorapidity |η| ≤ 2.5, whereas the geometrical acceptance of the LHCb detector is characterized by 2 ≤ η ≤ 5.  6. Rapidity distributions for the production of low-lying (cb) states in pp collisions at √ s = 13 TeV, calculated using BCVEGPY2.2 [45]. From highest to lowest, the histograms refer to production of the 1 3 S1, 1 1 S0, 2 3 S1, 2 1 S0, 2 3 P2, 2P 1 ( ) , 2 3 P0 levels.
For comparison, approximately 68% of the B * c cross section lies within |y| ≤ 2.5, and approximately 22% is produced at forward rapidities y > 2. Similar fractions hold for all the (cb) levels.

A. Dipion cascades
The path to establishing excited states will proceed by resolving two separate peaks in the invariant mass distributions associated with the cascades B c → B c π + π − and B * c → B * c + π + π − , B * c → B c + / γ (gamma unobserved). The splitting between the peaks is set by the difference of mass differences, generically expected to be negative [46]. The corresponding quantity is approximately −64 MeV in the (cc) family and −37 MeV in the (bb) family [12]. For the (cb) system, a modern lattice simulation [14] gives ∆ 21 = −15 MeV, whereas the result of our potential-model calculation is −23 MeV. In these circumstances, the undetected fourmomentum of the photon means that the reconstructed "B * c " mass should correspond to the lower peak.
We show an example of what is to be expected in Figure 7, taking the direct production cross sections (with no rapidity cuts) from Table VIII and the branching fractions from Table VI. The (relative heights of, relative number of events in) the peaks measures the ratio  Table VIII and branching fractions in Table VI. We assume that the photon in the transition B * c → Bc + / γ is not included in the reconstruction. Rates confined to rapidity |y| ≤ 2.5 are 0.68× those shown.

At
√ s = 13 TeV, the ratio of cross sections is nearly 2. 5. Taking account of the branching fractions, we estimate R ≈ 2. If B * c and B c were produced with equal frequency, we would find R ≈ 0. 8 MeV. An unbinned extended maximum-likelihood fit to the CMS data returns 66 ± 10 events for the lower peak and 51 ± 10 for the upper. These yields are not yet corrected for detection efficiencies and acceptances, so they cannot be used to infer ratios of production cross sections times branching fractions. We look forward to the final result and to studies of the π + π − invariant mass distribution as next steps in B c spectroscopy.
Our calculations indicate that the 3S levels will lie above flavor threshold (see §V C, especially the discussion surrounding Figures 9 and 10), but it is conceivable that coupled-channel effects might push one or both states lower in mass. For that reason, it is worth examining the B c π + π − mass spectrum up through 7200 MeV for indications of 3 1 S 0 → B c π + π − and 3 3 S 1 → B * c π + π − lines.  According to our estimate of the 3S hyperfine splitting, the 3 3 S 1 line would lie about 28 MeV below the 3 1 S 0 line (36 MeV if we reset the 1S splitting to 68 MeV). For orientation, note that B(Υ(3S) → Υ(1S)π + π − ) = 4.37±0.08%, while 36% of Υ(3S) decays proceed through the ggg channel, which is not available to the (cb) states. According to Table VIII, the 3S states are produced at approximately 44% of the rate for their 2S counterparts.

B. Electromagnetic transitions
It may in time become possible for experiments to detect some of the more energetic E1-transition photons that appear in Table VI. As an incentive for the search, we show in Figure 8 the spectrum of E1 photons in decays of the 2 3 S 1 and 2 1 S 0 levels as well as the 2P → 2S transitions, assuming as always a missing B * c → B c / γ photon in the reconstruction. Here we include direct production of the 2P states as well as feed-down from 2S → 2P transitions. The strong B * c → B c line arising from direct production of B * c , for which we calculate σ · B ≈ 225 nb at √ s = 13 TeV, is probably too low in energy to be observed. More promising are the 2P levels, which might show themselves in B c + γ invariant mass distributions. These lines make up the right-hand group (black lines) in Figure 8. The 2 3 P 2 (6750) → B * c γ line is a particularly attractive target for experiment, because of the favorable production cross section, branching fraction, and 409-MeV photon energy. The 2P masses inferred from transitions to B * c will be shifted downward because of the unobserved M1 photon. It is not possible to produce enriched samples of the 2S levels by tuning the energy of e + e − collisions, as is done for J/ψ and Υ, so reconstruction of the left-hand group of 2S → 2P transitions (blue lines in Figure 8) will be problematic.
In the far future, combining the photon transition energies and relative rates with expectations for production and decay may eventually make it possible to disentangle mixing of the spin-singlet and spin-triplet J = L states.

C. States above open-flavor threshold
We estimate the strong decay rates for (cb) states that lie above flavor threshold using the Cornell coupledchannel formalism [22] that we elaborated and applied to charmonium states in [23].
We expect both the 3 1 S 0 and 3 3 S 1 states to lie above threshold for strong decays. The 3 1 S 0 state can decay into the final state B * D and the 3 3 S 1 level has decays into both the BD and B * D final states. The open decay channels as a function of the masses of these states is shown in Figures 9 and 10.
The 3 3 P 2 state might be observed as a very narrow (d-wave) BD line near open-flavor threshold. Its decay width as a function of mass for the 2P states are given in Figure 11.
In the phenomenological models the remaining 3P states lie just below the thresholds for strong decays. However they are near enough to these thresholds that there might be interesting behavior at the threshold for B * D in the 3P ( ) 1 cases and for the BD threshold in the case of the 3 3 P 0 state. Figure 12 shows that the 3 3 P 0 width grows rapidly just above threshold. The strong decay widths as a function of mass for the 3P 1 and 3P 1 states have a common behavior, displayed in Figure 13.
It is worth keeping in mind that while narrow BD peaks may signal excited (cb) levels, narrowBD peaks could indicate nearly bound bcq kql tetraquark states [47].
The largest existing e + e − → Z 0 → hadrons data sets were recorded by experiments at CERN's Large Electron-Positron collider (LEP) during the 1990s. In samples of (3.02, 3.9, and 4.2) million hadronic Z 0 decays, the DELPHI, ALEPH, and OPAL Collaborations [52] found a small number of candidates for the decays B c → J/ψπ + , J/ψ + ν, and J/ψ3π. Those few specimens were not sufficient to establish a discovery, but the experiments were able to bound combinations of branching fractions B as at 90% confidence level, where X denotes anything. The relative simplicity of e + e − → Z 0 events and the boosted kinematics of resulting B c mesons suggest that a Tera-Z factory might be a felicitous choice to investigate 2P → 1S + γ lines.

VII. CONCLUSIONS AND OUTLOOK
In this article, we have presented a new analysis of the spectrum of mesons with beauty and charm. First, we modified the traditional Coulomb-plus-linear form of the quarkonium potential to incorporate running of the strong coupling constant α s that saturates at a fixed value at long distances. The new frozen-α s potential incorporates both perturbative and nonperturbative aspects of quantum chromodynamics. Second, we have set aside the perturbative treatment of spin splittings, instead incorporating lessons from Lattice QCD and observations of the (cc) and (bb) spectra.
We look forward to additional experimental progress, first by confirming and elaborating the characteristics of the 2S levels reported by the CMS Collaboration [4]. Key observables are the mass of the 2 1 S 0 state, the splitting between the two lines, and the ratio of peak heights corrected for efficiencies and acceptance. It is also of interest to test whether the dipion mass spectra in the cascade decays B c → B c π + π − and B * c → B * c π + π − follow the pattern seen in ψ(2S) → J/ψπ + π − and Υ(2S) → Υ(1S)π + π − decays. Although we expect the 3S levels to lie above flavor threshold, exploring the B c π + π − mass spectrum up through 7200 MeV might yield indications of 3 1 S 0 → B c π + π − and 3 3 S 1 → B * c π + π − lines. The presence of one or the other of these could signal interactions of bound states with open channels. Prospecting for narrow B ( * ) D ( * ) peaks near threshold could yield evidence of B c states beyond the 2S levels.
The next frontier is the search for radiative transitions among (cb) levels. The most promising candidate for first light is the 2 3 P 2 (6750) → B * c γ transition. Determining the B * c mass, perhaps by reconstructing B * c → B c γ, would provide an important check on lattice QCD calculations and a key input to future calculations.
Detecting the B c → τ ν τ and B c → ppπ + decays would be impressive experimental feats, and would provide another test of the short-distance behavior of the groundstate wave function, complementing what will be learned from the B * c -B c splitting.

Appendix: Strong coupling evolution
To make calculations with the frozen-α s potential, one must combine a linear term with a Coulomb term, −4α s (r)/3r, for which α s (r) is characterized by the solid red curve of Figure 1. We present in Table IX numerical values of the strong coupling over the relevant range of distances, 0 ≤ r ≤ 0.8 fm. The entries advance in steps of δ ln r = 0.1.