PHYSICAL REVIEW D 99, 054025 (2019)
Editors' Suggestion
Mesons with beauty and charm: New horizons in spectroscopy
† Estia J. Eichten* and Chris Quigg Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510, USA
(Received 8 March 2019; published 28 March 2019)
þ ¯ The Bc family of ðcbÞ mesons with beauty and charm is of special interest among heavy quarkonium þ ¯ systems. The Bc mesons are intermediate between ðcc¯Þ and ðbbÞ states both in mass and size, so many features of the ðcb¯Þ 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 Bc than in the J=ψ. Close examination of þ the Bc spectrum can test our understanding of the interactions between heavy quarks and antiquarks and may reveal where approximations break down. Whereas the J=ψ and ϒ levels that lie below the flavor threshold are metastable with respect to strong decays, the Bc ground state is absolutely stable against strong or electromagnetic decays. Its dominant weak decays arise from b¯ → cW¯ ⋆þ, c → sW⋆þ, and cb¯ → W⋆þ transitions, where W⋆ designates a virtual weak boson. Prominent examples of the first þ þ þ þ category are quarkonium transmutations such as Bc → J=ψπ and Bc → J=ψl νl, where J=ψ designates the ðcc¯Þ 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 Bc states, we update the expectations for the low-lying spectrum of the Bc 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 Bð ÞDð Þ final states could well reveal narrow resonances in the JP ¼ 0−, 1−, and 2þ channels and possibly in the JP ¼ 0þ and 1þ channels at threshold. Looking further ahead, the prospect of very-high-luminosity eþe− colliders capable of producing tera-Z samples raises the possibility of investigating Bc spectroscopy and rare decays in a controlled environment.
DOI: 10.1103/PhysRevD.99.054025
I. INTRODUCTION shows the way toward exploiting the potential of ðcb¯Þ spectroscopy. Although the lowest-lying ðcb¯Þ meson has long been established, the spectrum of excited states is little B explored. The ATLAS experiment at CERN’s Large A. What we know of the c mesons Hadron Collider reported the observation of a radially The possibility of a spectrum of narrow Bc states was excited Bc state [1], but this sighting was not confirmed first suggested by Eichten and Feinberg [5]. Anticipating by the LHCb experiment [2]. The unsettled experimental the copious production of b-quarks at Fermilab’s Tevatron situation and the large datasets now available for analysis Collider and CERN’s Large Electron-Positron Collider make it timely for us to provide up-to-date theoretical (LEP), we presented a comprehensive portrait of the expectations for the spectrum and decay patterns of spectroscopy of the Bc meson and its long-lived excited narrow ðcb¯Þ states and for their production in hadron states [6], based on then-current knowledge of the inter- colliders [3]. New work from the CMS Collaboration [4] action between heavy quarks derived from ðcc¯Þ and ðbb¯Þ bound states, within the framework of nonrelativistic quan- *[email protected] tum mechanics [7]. Surveying four representative poten- † P − [email protected] tials, we characterized the mass of the J ¼ 0 ground state as MðBcÞ ≈ 6258 20 MeV. A small number of Bc can- Published by the American Physical Society under the terms of 0 the Creative Commons Attribution 4.0 International license. didates appeared in hadronic Z decays at LEP. The CDF → ψl ν Further distribution of this work must maintain attribution to Collaboration observed the decay Bc J= in 1.8- the author(s) and the published article’s title, journal citation, TeV pp¯ collisions at the Fermilab Tevatron [8], estima- 3 and DOI. Funded by SCOAP . ting the mass as MðBcÞ ≈ 6400 411 MeV. (The generic
2470-0010=2019=99(5)=054025(15) 054025-1 Published by the American Physical Society ESTIA J. EICHTEN and CHRIS QUIGG PHYS. REV. D 99, 054025 (2019)
lepton l represents an electron or muon.) Subsequent work the low-energy photon from the subsequent Bc → Bcγ by the CDF [9],D0[10], and LHCb [11] Collaborations decay, and that the signal is an unresolved combination 3 1 has refined the mass to MðBcÞ¼6274.9 0.8 MeV [12], of 2 S1 and 2 S0 peaks. A search by the LHCb collabo- with the most precise determinations coming from fully ration in 2 fb−1 of 8-TeV pp data yielded no evidence for ψπþ reconstructed final states such as J= . either Bcð2SÞ state [2]. As we prepared this article for Investigations based on the spacetime lattice formulation publication, the CMS Collaboration provided striking ab initio of QCD aim to provide calculations that incor- evidence for both Bcð2SÞ levels, in the form of well- þ − porate the full dynamical content of the theory of strong separated peaks in the Bcπ π invariant mass distribution, interactions. Before the nonleptonic Bc decays had been closely matching the theoretical template [4]. We incorpo- observed, a first unquenched lattice QCD prediction, incor- rate these new observations into the discussion that follows porating 2 þ 1 dynamical quark flavors ðu=d; sÞ found in Sec. VA. þ18 MðBcÞ¼6304 12−0 MeV [13], where the first error bar represents statistical and systematic uncertainties and the B. Analyzing the (cb¯) bound states second characterizes heavy-quark discretization effects. The nonrelativistic potential picture, motivated by the 2 1 1 Calculations incorporating þ þ dynamical quark fla- asymptotic freedom of QCD [20], gave early insight into 11 6278 9 vors ðu=d; s; cÞ [14] yield Mð S0Þ¼ MeV, in the nature of charmonium and generated a template for impressive agreement with the measured Bc mass, and 1 the spectrum of excited states [21]. For more than four predict Mð2 S0Þ¼6894 19 8 MeV [15]. decades, it has served as a reliable guide to quarkonium Three distinct elementary processes contribute to the spectroscopy, including the states lying near or just above ¯ ¯ ⋆þ decay of Bc: the individual decays b → cW and c → flavor threshold for fission into two heavy-light mesons sW⋆þ of the two heavy constituents, and the annihilation that are significantly influenced by coupled-channel cb¯ → W⋆þ through a virtual W-boson. Several examples effects [22,23]. of the b¯ → c¯ transition have been observed, including We view the nonrelativistic potential-model treatment as þ þ þ þ þ − the final states J=ψl νl, J=ψπ , J=ψK , J=ψπ π π , a stepping stone, not a final answer, however impressive its þ þ − þ þ þ − − þ ⋆þ record of utility. Potential theory does not capture the full J=ψπ K K , J=ψπ π π π π , J=ψDs , J=ψDs , and þ þ 0 þ dynamics of the strong interaction, and while the standard J=ψπ pp¯ ; and ψð2SÞπ . A single channel, Bsπ , repre- senting the c → s transition is known. The annihilation coupled-channel treatment is built on a plausible physical þ picture, it is not derived from first principles. Moreover, mechanism, which would lead to final states such as τ ντ ¯ ¯ πþ relativistic effects may be more important for ðcbÞ than for and pp , has not yet been established. The observed ¯ τ 0 507 0 009 ðccÞ. The c-quark moves faster in the Bc meson than in the lifetime, ðBcÞ¼ð . . Þ ps [12], is consistent J=ψ, because it must balance the momentum of a more with theoretical expectations [16,17]. Predictions for partial massive b-quark. One developing area of theoretical decay rates (or relative branching fractions) await exper- research has been to explore methods more robust than imental tests. Some recent theoretical works explore the nonrelativistic quantum mechanics [17,24,25]. potential of rare B decays [18]. c ¯ Nonperturbative calculations on a spacetime lattice in Until recently, the only evidence reported for a ðcbÞ principle embody the full content of QCD. This approach is excited state was presented by the ATLAS Collaboration yielding increasingly precise predictions for the masses of [1] in pp collisions at 7 and 8 TeV, in samples of 4.9 ¯ 0 23 −1 ðcbÞ levels up through the Bc ð S1Þ state. It is not yet and 19.2 fb . They observed a new state at 6842 7 MeV possible to extract reliable signals for higher-lying states πþπ− − − 2 π in the MðBc Þ MðBc Þ Mð Þ mass difference, from the lattice, so we rely on potential-model methods to ψπ 3 with Bc detected in the J= mode. The mass construct a template for the Bc spectrum through the 4 S1 (527 7 MeV above hMð1SÞi) and decay of this state level. If experiments should uncover systematic deviations are broadly in line with expectations for the second s-wave from the expectations we present, they may be taken as state, Bc ð2SÞ. In addition to the nonrelativistic potential- evidence of dynamical features absent from the nonrela- model calculations cited above, the HPQCD Collaboration tivistic potential-model paradigm, including—of course— has presented preliminary results from a lattice calculation coupling to states above flavor threshold, which we neglect using 2 þ 1 þ 1 dynamical fermion flavors and highly in our calculations of the spectrum. improved staggered quark correlators [19]. They report In the following Sec. II, we develop the theoretical tools 1 ¯ Mð2 S0Þ¼6892 41 MeV, which is 576.5 41 MeV required to compute the ðcbÞ spectrum. In earlier work [6], above hMð1SÞi). This result and the NRQCD prediction we examined the Cornell Coulomb-plus-linear potential [14] lie above the ATLAS report by 1 and 2 standard [22], a power-law potential [26], Richardson’s QCD- deviations, respectively. The significance of the discrep- inspired potential [27], and a second QCD-inspired poten- ancy is limited for the moment by lattice uncertainties. tial due to Buchmüller and Tye [28], which we took as our A plausible interpretation has been that ATLAS might have reference model. We used a perturbation-theory treatment þ − observed the transition Bcð2SÞ → Bcð1SÞπ π , missing of spin splittings. Using insights from lattice QCD and
054025-2 MESONS WITH BEAUTY AND CHARM: NEW HORIZONS IN … PHYS. REV. D 99, 054025 (2019) 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 ðcc¯Þ and ðbb¯Þ spectra to refine the treatment of spin splittings. We present our expectations for the spectrum of narrow states in Sec. III. We consider decays ) r of the narrow states in Sec. IV, updating the results we gave ( s in Ref. [6]. We compute differential and integrated cross α sections for the narrow Bc levels in proton-proton collisions at the Large Hadron Collider in Sec. 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 eþe− → tera-Z machine appear in Sec. VI. We draw some conclusions and look ahead in Sec. VII. r [fm]
II. THEORETICAL PRELIMINARIES α FIG. 1. Dependence of the running coupling sðrÞ on the We take as our starting point a Coulomb-plus-linear interquark separation r. The strong coupling for our chosen potential (the “Cornell potential” [22]), potential is shown in the solid red curve. Those corresponding to the Cornell potential (green dots) [22], Richardson potential (blue κ r dash-dotted) [27], and an alternative version of the new potential VðrÞ¼− þ ; ð1Þ μ 0 8 r a2 with ¼ . GeV (gold dashes) are shown for comparison. where κ ≡ 4α =3 ¼ 0.52 and a ¼ 2.34 GeV−1 were s The consequent evolution of α ðrÞ is plotted as the solid chosen to fit the quarkonium spectra. Analysis of the s ψ ϒ red curve in Fig. 1, where we also show an alternative choice J= and families led to the choices μ 0 8 α of ¼ . GeV (dashed gold curve), the constant s of the original Cornell potential (dotted green curve), and α ðrÞ m ¼ 1.84 GeV m ¼ 5.18 GeV: ð2Þ s c b corresponding to the Richardson potential (dot-dashed blue curve). This simple form has been modified to incorporate running We plot in Fig. 2 the frozen-α potential for both our of the strong coupling constant in Refs. [27,28], among s chosen example, μ ¼ 1.2 GeV, and the alternative, others, using the perturbative-QCD evolution equation at μ ¼ 0.8 GeV. There we also show the Richardson and leading order and beyond. At distances relevant for con- finement, perturbation theory ceases to be a reliable guide. It is now widely held, following Gribov [29], that as a result α of quantum screening s approaches a critical, or frozen, value at long distances (low-energy scales). In a light ðqq¯Þ system, Gribov estimated 3π pffiffiffiffiffiffiffiffi α → α¯ 1 − 2 3 ≈ 0 14π 0 44 s s ¼ 4 ð = Þ . ¼ . : ð3Þ ) [GeV] r (
We incorporate the spirit of this insight into a new version V “ α of the Coulomb-plus-linear form that we call the frozen- s potential.” The long-range part is the standard Cornell linear term. To obtain the Coulomb piece, we convert the four-loop α running of sðqÞ in momentum space [30] to the behavior in position space using the method of [31], with an important α 1 6 0 338 r [fm] modification. We set sðq ¼ . GeVÞ¼ . and evolve with three active quark flavors. To enforce saturation of α FIG. 2. Dependence of quarkonium potentials VðrÞ on the sðrÞ at long distances, we alter the recipe of Ref. [31], α 1 γ γ interquark separation r. Our frozen- s potential is shown in the replacing the identification q ¼ =r expð EÞ, where E ¼ 0 57721… ’ solid red curve. The Cornell potential (green dots) [22], Richard- . is Euler s constant, with the damped form son potential (blue dash-dotted) [27], and an alternative version of 1 γ 2 μ2 1=2 q ¼ =½ðr expð EÞ þ . For our reference potential, the new potential with μ ¼ 0.8 GeV (gold dashes) are shown for we have chosen the damping parameter μ ¼ 1.2 GeV. comparison.
054025-3 ESTIA J. EICHTEN and CHRIS QUIGG PHYS. REV. D 99, 054025 (2019)
TABLE I. P-state masses [12] and splittings, in MeV.
State 2Pðcc¯Þ 2Pðbb¯Þ 3Pðbb¯Þ 3 χ0ð P0Þ 3414.71 0.30 9859.44 0.52 10232.5 0.64 3 χ1ð P1Þ 3510.67 0.05 9892.78 0.4 10255.46 0.55 1 hð P1Þ 3525.38 0.11 9899.3 0.8 10259.8 1.12 3 χ2ð P2Þ 3556.17 0.07 9912.21 0.4 10268.85 0.55 3 3525 29 0 01 9899 87 0 17 10260 35 0 31 PJ centroid, hχi . . . . . . h − hχi 0.09 0.11 −0.57 0.82 −0.55 1.25 3 3 χ1ð P1Þ − χ0ð P0Þ 95.96 0.30 33.34 0.66 22.96 0.84 3 3 χ2ð P2Þ − χ0ð P2Þ 45.5 0.09 19.43 0.57 13.39 0.78
˜ 1 dV ˜ Cornell potentials. All coincide at large distances. The T1ðm ;m Þ¼− þ 2T2ðm ;m Þ i j r dR i j Cornell potential is deeper at short distances than any of α 4˜ α the potentials that take account of the evolution of s. ˜ c2 sðrÞ T2ðm ;m Þ¼ For the convenience of others who may wish to apply the i j 3 r3 α new potential, we present values of sðrÞ suitable for c˜4 α r interpolation in the Appendix. ˜ sð Þ T4ðmi;mjÞ¼ 3 3 ; ð6Þ We presented the general formalism for spin-dependent r interactions as laid out by Eichten and Feinberg [5] and Gromes [32] in Sec. II B of Ref. [6], where we took a where we have introduced the phenomenological coeffi- ˜ ˜ perturbative approach to the spin-orbit and tensor inter- cients c2 and c4, which take the value unity in the perturbative approach. actions. In the intervening time, the charmonium and ˜ ˜ bottomonium spectra have been mapped in detail, as We extract values of T2 and T4 for the observed levels summarized in Table I. This wealth of information leads that appear in Table I. These are shown as the boldface us now to choose a more phenomenological approach. entries in Table II. Then, we combine the definitions in ¯ Eq. (6) with our calculated values of hα =r3i to determine We write the spin-dependent contributions to the ðcbÞ ¯ s masses as c˜2 and c˜4 in the ðcc¯Þ and ðbbÞ families. The geometric mean of these values is our estimate for the coefficients X4 in the cb¯ system. We insert these back into Eq. (6) to Δ ¼ T ; ð4Þ ˜ ˜ k estimate the values of T2 and T4 for the Bc family. For k¼1 completeness, we include our evaluations of hð1=rÞdV=dri where the individual terms are in the table. For the J=ψ and ϒ families, composed of equal-mass heavy quarks, the familiar LS coupling scheme, in which ⃗ ⃗ ⃗ ⃗ 2 1 hL · sii ˜ hL · sji ˜ Sþ T1 T1 m ;m T1 m ;m states are labeled by n LJ, is apt. When the quark ¼ 2 2 ð i jÞþ 2 2 ð j iÞ mi mj masses are unequal, as in the case at hand, spin-dependent ⃗ ⃗ ⃗ ⃗ terms in the Hamiltonian mix the spin-singlet and spin- hL · sii ˜ hL · sji ˜ T2 ¼ T2ðmi;mjÞþ T2ðmj;miÞ triplet J ¼ L states. We define mimj mimj ⃗ ⃗ hsi · sji ˜ T3 ¼ T3ðmi;mjÞ ˜ ˜ mimj TABLE II. Values of T2 and T4 extracted from data (bold) and the inferred values of the phenomenological coefficients c˜2 and hSiji ˜ c˜4 for the J=ψ and ϒ families, from which the coefficients for the T4 ¼ T4ðmi;mjÞ; ð5Þ ¯ mimj ðcbÞ system are derived.
α sðrÞ 1 dV ⃗ ⃗ ⃗ ⃗ ⃗ h r3 i hr dr i si and sj are the heavy-quark spins, S ¼ si þ sj is the total ˜ ˜ 3 3 System T2 T4 ðGeV Þ c˜2 c˜4 ðGeV Þ spin, L⃗ is the orbital angular momentum of quark and 1Pðcc¯Þ 0.088 0.0308 0.0527 1.25 1.77 0.141 antiquark in the bound state, Sij ¼ 4½3ðs⃗i · nˆÞðs⃗j · nˆÞ − s⃗i · s⃗j 1 ¯ 0.258 0.0835 0.220 0.82 0.99 0.383 is the tensor operator, and nˆ is an arbitrary unit vector. PðbbÞ 2Pðbb¯Þ 0.181 0.0547 0.166 0.82 0.99 0.278 We will deal with the hyperfine interaction T3 momen- ¯ ˜ 1PðcbÞ 0.119 0.0388 0.0885 1.012 1.316 0.198 tarily. We express the other Tk as
054025-4 MESONS WITH BEAUTY AND CHARM: NEW HORIZONS IN … PHYS. REV. D 99, 054025 (2019)
0 1 3 ¯ jnLLi ¼ cos θjn LLiþsin θjn LLi; TABLE III. Calculated excitation energies (in MeV) for ðcbÞ B 1S − θ 1 θ 3 levels with respect to the cð Þ centroid according to potential jnLLi¼ sin jn LLiþcos jn LLi; ð7Þ models and lattice QCD simulations. The potential models have been aligned with the 1S doublet centroid at 6315.5 MeV. where Communication with states above flavor threshold is neglected.
2A Level EQ94 [6] Lattice QCD This work tan θ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð8Þ 2 4 2 1 B þ B þ A 1 S0 −54.8 −40.5 [14,34,35] −40.5 13 with S1 18.2 13.5 [14,34,35] 13.5 23 P0 381 393(17)(7) [35] 377 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2P1 411 417(18)(7) [35] 415 1 − ˜ 0 A ¼ LðL þ Þ 2 2 T1 ð9Þ 2P1 417 446(30) [33] 423 4 m m 3 c b 2 P2 428 464(30) [33] 435 1 2 S0 537 561(18)(1) [14] 551 and 3 2 S1 580 601(19)(1) [14] 582 33 1 1 1 1 1 D1 693 691 ˜ ˜ − 2 ˜ 3D2 693 690 B ¼ 2 þ 2 T1 þ T2 T4: ð10Þ 4 0 mc mb mcmb mcmb 3D2 686 700 3 3 D3 690 695 ˜ 3 Then our calculations of the Tk defined in Eq. (6) 3 P0 789 789 3 lead to these values for the mixing angle: θ2P ¼ 18.7°, P1 823 820 0 3P1 θ3D ¼ −49.2°, θ3P ¼ 21.2°, θ4F ¼ −49.5°, θ4D ¼ −40.3°. 816 828 33 A lattice calculation in quenched QCD [33] gave P2 834 839 1 θ 33 2 3 S0 925 938 2P ¼ °. 3 3 S1 961 964 The masses of the mixed states are 3 4 F2 918 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 1 2 2 F3 906 MðnL0 Þ¼hMðnLÞi − ðB − 4A þ B Þ; 4 0 L 2 F3 922 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 F4 908 1 2 2 3 − 4 4 D1 1031 MðnLLÞ¼hMðnLÞi 2 ðB þ A þ B Þ; ð11Þ 4D2 1033 0 4D2 1040 where hMðnLÞi is the nL centroid. 3 4 D3 1038 At lowest order, the hyperfine splitting (HFS) between s- 3 4 P0 1121 wave states, arising from T3,isgivenby 4P1 1149 4P0 8α 0 2 1 1158 ðnÞ 3 1 sjRn0ð Þj 43 1167 Δ ¼ Mðn S1Þ − Mðn S0Þ¼ ; ð12Þ P2 HFS 9 1 mcmb 4 S0 1243 1257 3 4 S1 1276 1280 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 We summarize in Table III predictions for the spectrum state and scale the splittings of excited states according to of mesons with beauty and charm from our 1994 article [6], lattice QCD calculations, and the present work, expressed ΔðnÞ 0 2 HFS jRn0ð Þj as excitations with respect to the 1S centroid. Other 1 ¼ 2 : ð13Þ Δð Þ jR10ð0Þj potential-model calculations, some incorporating relativis- HFS tic effects, may be found in the works cited in Ref. [7]. Our expectations for the spectrum of states are shown in B III. THE c SPECTRUM the Grotrian diagram, Fig. 3, along with several of the 3 B 1 S1 B lowest-lying open-flavor thresholds. The thresholds for The vector meson c,the hyperfine partner of c ¯ andanalogofJ=ψ and ϒ, has not yet been observed. strong decays of excited ðcbÞ levels are known exper- Modern lattice calculations [14,34,35] give consistent imentally to high accuracy, as shown in Table IV. values for the hyperfine splitting MðB Þ − MðB Þ¼ Comparing with the model calculations summarized in c c ¯ ð53 7; 54 7; 55 3 MeVÞ, so we take the mass of Table III, we conclude that two sets of narrow s-wave ðcbÞ the vector state to be MðBcÞ¼6329 MeV and fix the levels will lie below the beauty þ charm flavor threshold, in centroid hMð1SÞi of the ground-state s-wave doublet at agreement with general arguments [36]. All of the potential 3 6315.5 MeV for the lattice. models cited in Ref. [7] predict 3 S1 masses well above the
054025-5 ESTIA J. EICHTEN and CHRIS QUIGG PHYS. REV. D 99, 054025 (2019)
where the magnetic dipole moment is 7600 − mbec mceb¯ 7400 μ ¼ ð15Þ 4mcmb 7200 and k is the photon energy. 7000 Apart from that M1 transition, only the electric dipole ¯ 6800 transitions are important for mapping the ðcbÞ spectrum. Mass [MeV] The strength of the electric-dipole transitions is governed 6600 by the size of the radiator and the charges of the constituent quarks. The E1 transition rate is given by 6400 4α e 2 6200 Γ → γ h Qi 3 2 1 2S E1ði f þ Þ¼ 27 k ð Jf þ Þjhfjrjiij if; ð16Þ
FIG. 3. Calculated cb¯ spectrum, with (spin-singlet, spin-triplet) where the mean charge is states shown on the left (right) in red (blue) for each orbital- − mbec mceb¯ angular-momentum family S, P, D, F. Dashed lines indicate heQi¼ ; ð17Þ thresholds for decay into two-body open-flavor channels given in mb þ mc Table IV. k is the photon energy, and the statistical factor Sif ¼ Sfi 1 829-MeV BD threshold. For the 3 S0 level, only the Ebert is as defined by Eichten and Gottfried [37]. Sif ¼ 1 for et al. 3 3 prediction does not lie significantly above the B D S1 → PJ transitions and Sif ¼ 3 for allowed E1 transi- threshold. Lattice QCD calculations do not yet exist for tions between spin-singlet states. The statistical factors for states beyond the 2S levels. d-wave to p-wave transitions are reproduced in Table V. The significant M1 and E1 electromagnetic transition ¯ IV. DECAYS OF NARROW ðcbÞ LEVELS rates and the ππ cascade rates are given in Table VI, along with the total widths in the absence of fission decays. A. Electromagnetic transitions 3 The only significant decay mode for the 1 S1 ðBcÞ state is the magnetic dipole (spin-flip) transition to the ground B. Hadronic transitions state, Bc. The M1 rate for transitions between s-wave levels We evaluate the rates for hadronic transitions between is given by ðcb¯Þ levels according to the prescription we detailed in Sec. III B of Ref. [6]. The results are included in Table VI. 16α Γ → γ μ2 3 2 1 2 2 Dipion cascades to the ground-state doublet are the dom- M1ði f þ Þ¼ k ð Jf þ Þjhfjj0ðkr= Þjiij ; 3 1 3 inant decay modes of 2 S1 and 2 S0 and will be key to ð14Þ characterizing those states, as we shall discuss in Sec. VA. As observed long ago by Brown and Cahn [38],an amplitude zero imposed by chiral symmetry pushes the πþπ− TABLE IV. Open-flavor ðcb¯Þ thresholds and excitations above invariant mass distribution to higher invariant masses than phase space alone would predict. In its simplest form, 1S centroid, 6315.5 MeV, for Bc levels, in MeV. this analysis yields a universal form for the normalized State Flavor threshold Excitation energy dipion invariant mass distribution in quarkonium cascades − 0 Φ0 → Φπþπ BþD 7144.15 0.15 829 , B0Dþ 7149.28 0.16 834 0 B þD 7189.48 0.26 874 S S 3 → 3 γ 0 TABLE V. Statistical factor if ¼ fi for PJ DJ0 þ and B Dþ 7194.30 0.26 879 3 3 DJ → PJ0 þ γ transitions. BþD 0 7286.17 0.15 971 0 þ 7289 89 0 16 0 B D . . 974 JJ Sif B þD 0 7331.50 0.26 1016 01 2 0 þ 7334.91 0.26 1019 B D 111=2 0 þ 7335 23 0 20 Bs Ds . . 1020 129=10 0 þ 7479 09 0 44 Bs Ds . . 1164 211=50 0 þ 7383 74þ1.80 Bs Ds . −1.50 1068 229=50 0 þ 7527 60þ1.84 18 25 Bs Ds . −1.55 1212 23 =
054025-6 MESONS WITH BEAUTY AND CHARM: NEW HORIZONS IN … PHYS. REV. D 99, 054025 (2019)
TABLE VI. Total widths Γ and branching fractions B for TABLE VI. (Continued) principal decay modes of ðcb¯Þ states below threshold, updating Table IX of Ref. [6]. Dissociation into BD, etc., will dominate Branching over the tabulated decay modes for states above threshold. Decay mode kγ (keV) fraction (%) 13 ππ Branching S1 þ 12.4 0 Decay mode kγ (keV) fraction (%) 2P1 (6738) 272 37.2 2P1 11 (6730) 279 41.0 S0 (6275): Weak decays 3 3 P0 (7104): Γ 60.9 keV 3 ¼ 1 S1 (6329): Γ ¼ 0.144 keV 3 1 S1 (6329) 733 46.4 1 1 S0 þ γ 54 100 3 2 S1 (6897) 204 45.0 23 Γ 53 1 3 P0 (6692): ¼ . keV 3 D1 (7006) 97 8.44 3 1 S1 (6329) 354 100 3P1 (7135): Γ ¼ 87.1 keV 2 Γ 72 5 3 P1 (6730): ¼ . keV 1 S1 (6329) 761 32.6 13 389 86.2 1 S1 (6329) 1 S0 (6275) 809 6.24 11 440 13.7 3 S0 (6275) 2 S1 (6897) 234 40.4 0 1 2P1 (6738): Γ ¼ 99.9 keV 2 S0 (6866) 264 8.39 1 1 S0 (6275) 448 92.4 3D2 (7005) 129 4.74 3 0 1 S1 (6329) 397 7.51 3D2 (7015) 119 4.55 3 3 3 D1 (7006) 128 2.88 2 P2 (6750): Γ ¼ 79.7 keV 3 3 0 Γ 113 1 S1 (6329) 409 100 P1 (7143): ¼ keV 1 1 1 S0 (6275) 816 32.9 2 S0 Γ 73 1 (6866): ¼ . keV 3 1 1 S1 (6329) 768 3.88 1 S0 ππ 81.1 þ 1 0 2 S0 (6866) 272 47.0 2P1 (6738) 126 16.5 3 2 S1 (6897) 242 5.15 2P1 (6730) 134 2.24 3 0 3 D2 (7015) 127 4.22 2 S1 (6897): Γ ¼ 76.8 keV 3D2 (7005) 137 6.72 13 ππ S1 þ 65.0 3 3 3 P2 (7154): Γ ¼ 100 keV 2 P0 (6692) 201 7.66 13 2 S1 (6329) 777 35.2 P1 (6730) 165 11.5 3 0 2 S1 (6897) 252 49.2 2P1 (6738) 157 1.13 3 3D2 (7005) 147 1.09 2 P2 (6750) 145 14.6 0 3D2 (7015) 137 1.18 3D2 (7005): Γ ¼ 93.7 keV 3 3 D1 (7006) 146 0.16 11 ππ S0 þ 12.2 33 3 D3 (7010) 142 13.0 1 S1 þ ππ 9.0 4F3 (7221): Γ ¼ 77.6 keV 2P1 (6730) 270 29.1 3 0 D2 (7005) 213 52.4 2P1 (6738) 262 42.3 3 0 23 D2 (7015) 203 42.8 P2 (6750) 250 7.24 3 3 D3 (7010) 208 4.71 33 Γ 117 D1 (7006): ¼ keV 43 Γ 79 9 3 F4 (7223): ¼ . keV 1 S1 þ ππ 17.0 3 3 D3 (7010) 210 100 23 306 53.4 P0 (6692) 3 4 F2 (7233): Γ ¼ 95.3 keV 2P1 (6730) 270 25.3 0 3D2 (7005) 225 6.73 2P1 (6738) 262 2.62 0 3 3D2 (7015) 215 7.96 2 P2 (6750) 251 1.51 33 224 84.8 3 D1 (7006) 3 D3 (7010): Γ 87.2 keV ¼ 33 220 0.42 3 D3 (7010) 1 S1 þ ππ 22.9 4F0 (7237): Γ ¼ 89.9 keV 23 255 77.0 3 P2 (6750) 0 3D2 (7015) 218 47.4 3D0 (7015): Γ ¼ 92.1 keV 2 3D2 (7005) 228 52.5 1 1 S0 þ ππ 9.2 (Table continued)
054025-7 ESTIA J. EICHTEN and CHRIS QUIGG PHYS. REV. D 99, 054025 (2019)
⃗ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 dΓ jKj TABLE VII. Squares of radial wave functions at the origin and constant × 2x2 − 1 2 x2 − 1; 18 Γ M ¼ 2 ð Þ ð Þ related quantities [cf. Eq. (19)] for ðcb¯Þ mesons. d MΦ0 l ⃗ Level jRð Þð0Þj2 where x ¼ M=2mπ and K is the three-momentum carried nl by the pion pair. The soft-pion expression (18) describes the 1S 1.994 GeV3 depletion of the dipion spectrum at low invariant masses 2P 0.3083 GeV5 3 observed in the transitions ψð2SÞ→ψð1SÞππ, ϒð2SÞ→ 2S 1.144 GeV 7 ϒð1SÞππ, and ϒð3SÞ → ϒð2SÞππ, but fails to account for 3D 0.0986 GeV 3 5 structures in the ϒð3SÞ → ϒð1SÞππ spectrum [39].We P 0.3939 GeV 3 3 expect the 3S levels to lie above flavor threshold in the S 0.9440 GeV 4F 0 0493 9 ðcb¯Þ system and so to have very small branching fractions . GeV 4D 0 1989 7 for cascade decays (but see the final paragraph of Sec. VA). . GeV 4P 0.4540 GeV5 4S 0.8504 GeV3 ¯ C. Properties of ðcbÞ wave functions at the origin For quarks bound in a central potential, it is convenient to separate the Schrödinger wave function into radial α m − m m C¯ 2ðα Þ¼1 − s δP;V − c b ln c ; ð23Þ and angular pieces, as Ψ l ðr⃗Þ¼R lðrÞYl ðθ; ϕÞ, where s n m n m π mc þ mb mb n is the principal quantum number, l and m are the orbital angular momentum and its projection, RnlðrÞ is and the radial wave function, and Ylmðθ; ϕÞ is a spherical harmonicR [40]. The Schrödinger waveR function is normal- δP ¼ 2; δV ¼ 8=3: ð24Þ 3⃗ ⃗ 2 ∞ 2 ized, d rjΨnlmðrÞj ¼ 1, so that 0 r drjRnlðrÞj ¼ 1. α 0 38 The value of the radial wave function, or its first non- Choosing the representative value s ¼ . , and using the vanishing derivative, at the origin, quark masses given in Eq. (2), we find
l l d R lðrÞ 0.904; P ð Þ 0 ≡ n ¯ α Rnl ð Þ l ; ð19Þ Cð sÞ¼ : ð25Þ dr r¼0 0.858; V is required to evaluate pseudoscalar decay constants and Consequently, we estimate the ground-state meson decay production rates through heavy-quark fragmentation. Our constants as ðlÞ 0 2 calculated values of jRnl ð Þj are given in Table VII. f 498 ; 471 The pseudoscalar decay constant Bc , which enters the fBc ¼ MeV fBc ¼ MeV; ð26Þ calculations of annihilation decays such as cb¯ →Wþ →τþþ ¯ ν 0 945 τ, is defined by so that fBc =fBc ¼ . . The compact size of the ðcbÞ system enhances the pseudoscalar decay constant relative to h0jAμð0ÞjB ðqÞi ¼ if V qμ; ð20Þ c Bc cb fπ and fK. This is to be compared to a state-of-the-art lattice where Aμ is the axial-vector part of the charged weak current, evaluation [43], f ¼ 434 15 MeV, which entails V Bc cb is an element of the Cabibbo-Kobayashi-Maskawa improved nonrelativistic QCD for the valence b-quark quark-mixing matrix, and qμ is the four-momentum of and the highly improved staggered quark action for the the Bc. Its counterpart for the vector state is lighter quarks on gluon field configurations that include the effect of u=d, s, and c quarks in the sea with the u=d 0 0 V ε h jVμð ÞjBcðqÞi ¼ ifBc cb μ; ð21Þ quark masses going down to physical values. The same 0 988 0 027 calculation yields fBc =fBc ¼ . . . A calculation where Vμ is the vector part of the charged weak current and in the framework of QCD sum rules gives fB ¼ 528 ε c μ is the polarization vector of the Bc. The ground-state 19 MeV [44]. pseudoscalar and vector decay constants are given in terms of the wave function at the origin by the Van Royen- V. PRODUCTION OF (cb¯) STATES AT THE Weisskopf formula [41], generically LARGE HADRON COLLIDER 3 0 2 We present in Table VIII cross sections for the produc- 2 jR10ð Þj ¯ 2 α f ð Þ ¼ C ð sÞ; ð22Þ Bc πM tion of Bc states at the Large Hadron Collider, calculated using the framework of the BCVEGPY2.2 generator [45], where the leading-order QCD correction is given by [42] which we have extended to include the production of 3P
054025-8 MESONS WITH BEAUTY AND CHARM: NEW HORIZONS IN … PHYS. REV. D 99, 054025 (2019)
10 TABLE VIII. Production rates (in nanobarn) for ðcb¯Þ states in pp collisions at the LHC. The production rates were calculated using the BCVEGPY2.2 generator of Ref. [45], extended to 5 include the production of 3P states. Color-octet contributions to s-wave production are small; we show them (following j) only for ] the 1S states. 2 2
ðcb¯Þ pffiffiffi pffiffiffi pffiffiffi 1 level σ s 8 TeV σ s 13 TeV σ s 14 TeV ð ¼ Þ ð ¼ Þ ð ¼ Þ [nb/GeV 2 T
1 p 1 S0 46.8j1.01 80.3j1.75 88.0j1.90 3 /d 0.5 1 1 123.0j4.08 219.1j6.97 237.0j7.55 σ
S d 3 2 P0 1.113 1.959 2.108 23 P1 2.676 4.783 5.214 0.2 1 2 P1 3.185 5.702 6.166 3 2 P2 6.570 11.57 12.64 1 0.1 2 S0 9.58 16.94 18.45 0 2 46810 23 S1 23.46 41.72 45.53 pT [GeV] 3 3 P0 0.915 1.642 1.806 33 FIG. 5. Transverse momentum distribution of B produced in P1 2.263 4.082 4.478 pffiffiffi pc ffiffiffi 1 8 13 3 P1 2.695 4.817 5.287 pp collisions at s ¼ TeVffiffiffi (dotted blue curve), s ¼ TeV 3 p 3 P2 5.53 9.98 10.90 (solid black curve), and s ¼ 14 TeV (dashed red curve), 1 3 S0 4.23 7.53 8.08 calculated using BCVEGPY2.2 [45]. Small shape variations 3 3 S1 10.16 18.21 19.83 are statistical fluctuations.
are shown in Fig. 6. The acceptance of the CMS and 2 3 ð0Þ states. Cross sections for the physical ð ; ÞP1 states are ATLAS detectors covers central pseudorapidity jηj ≤ 2.5, 3 1 appropriately weighted mixtures of the P1 and P1 cross whereas the geometrical acceptance of the LHCb detector sections. is characterized by 2 ≤ η ≤ 5. For comparison, approxi- The rapidity distributions (for Bc production, Fig. 4) and mately 68% of the Bc cross section lies within jyj ≤ 2.5, transverse-momentum distributions (shown for B produc- pffiffiffi c and approximately 22% is produced at forward rapidities tion, Fig. 5) are similar in character for s ¼ 8, 13, and y>2. Similar fractions hold for all the ðcb¯Þ levels. 14 TeV. The rapidity distributions for low-lying ðcb¯Þ states
40 102
35
30
101 ]
25 ] y y Δ Δ
20 [nb/ [nb/ y /d /dy σ σ d
d 15 100 10
5
0 10–1 -4 -2 0 2 4 -4 -2 0 2 4 y y
FIG. 4. Rapidity distribution for production of B in pp FIG. 6. Rapidity distributions for the production of low-lying pffiffiffi pffiffiffi c ¯ pffiffiffi collisions at s 8 TeV (dotted blue curve), s 13 TeV ðcbÞ states in pp collisions at s ¼ 13 TeV, calculated using ¼ pffiffiffi ¼ (solid black curve), and s ¼ 14 TeV (dashed red curve), BCVEGPY2.2 [45]. From highest to lowest, the histograms refer 3 1 3 1 3 3 calculated using BCVEGPY2.2 [45]. The bin width is to production of the 1 S1, 1 S0, 2 S1, 2 S0, 2 P2, 2P1ð0Þ , 2 P0 Δy ¼ 0.5. The mild asymmetries are statistical fluctuations. levels.
054025-9 ESTIA J. EICHTEN and CHRIS QUIGG PHYS. REV. D 99, 054025 (2019)
2.0 which they identify as Bcð2SÞ, and a second peak 29.0 1.5 MeV lower in mass (statistical errors only). (The
1.5 observed Bc mass is replaced, event by event, with the world-average value to sharpen resolution.) The putative 1 Bcð2SÞ lies within 5 MeV of our expectation for the 2 S0 1.0 level, and the separation is to be compared with our expectation of 23 MeV. If we impose the scaling relation /dM [nb/MeV] σ
d Eq. (13) for the hyperfine splittings, we reproduce the 0.5 observed 29-MeV separation with MðBcÞ − MðBcÞ¼ 0 0 68 MeV, MðBc Þ − MðBcÞ¼39 MeV. The Bc → Bc þ γ
0 photon momentum would be 68 MeV. 6820 6830 6840 6850 6860 6870 6880 6890 An unbinned extended maximum-likelihood fit to the M(B π+π–) [MeV] c CMS data returns 66 10 events for the lower peak and 51 10 FIG. 7. Calculated positions and relative strengths of the two- for the upper. These yields are not yet corrected for þ − detection efficiencies and acceptances, so they cannot be pion cascades Bcð2SÞ → Bcð1SÞπ π , represented as Gaussian line shapes with standard deviation of 4 MeV. Production rates used to infer ratios of production cross sections times are given in Table VIII and branching fractions in Table VI.We branching fractions. We look forward to the final result and þ − assume that the photon in the transition Bc → Bc þ =γ is not to studies of the π π invariant mass distribution as next ≤ included in the reconstruction. Rates confined to rapidity jyj steps in Bc spectroscopy. 2.5 are 0.68× those shown. Our calculations indicate that the 3S levels will lie above flavor threshold (see Sec. VC, especially the discussion sur- A. Dipion cascades rounding Figs. 9 and 10), but it is conceivable that coupled- The path to establishing excited states will proceed by channel effects might push one or both states lower in mass. πþπ− resolving two separate peaks in the invariant mass distri- For that reason, it is worth examining the Bc mass 0 þ − 31 → butions associated with the cascades Bc → Bcπ π and spectrum up through 7200 MeV for indications of S0 0 þ − πþπ− 33 → πþπ− Bc → Bc þ π π , Bc → Bc þ =γ (gamma unobserved). Bc and S1 Bc lines. According to our estimate 3 The splitting between the peaks is set by the difference of the 3S hyperfine splitting, the 3 S1 line would lie about 1 of mass differences, 28 MeV below the 3 S0 line (36 MeV if we reset the 1S splitting to 68 MeV). For orientation, note that Bðϒð3SÞ → Δ ≡ 0 − 0 − − − 21 ½MðBc Þ MðBcÞ ½MðBcÞ MðBcÞ ; ð27Þ ϒð1SÞπþπ Þ¼4.37 0.08%, while 36% of ϒð3SÞ decays proceed through the ggg channel, which is not available generically expected to be negative [46]. The correspond- cb¯ 3S ing quantity is approximately −64 MeV in the ðcc¯Þ family to the ð Þ states. According to Table VIII,the states are produced at approximately 44% of the rate for their 2S and −37 MeV in the ðbb¯Þ family [12]. For the ðcb¯Þ system, counterparts. a modern lattice simulation [14] gives Δ21 ¼ −15 MeV, whereas the result of our potential-model calculation is −23 MeV. In these circumstances, the undetected four- B. Electromagnetic transitions momentum of the photon means that the reconstructed It may in time become possible for experiments to detect “ ” Bc mass should correspond to the lower peak. some of the more energetic E1-transition photons that We show an example of what is to be expected in Fig. 7, appear in Table VI. As an incentive for the search, we show 3 taking the direct production cross sections (with no rapidity in Fig. 8 the spectrum of E1 photons in decays of the 2 S1 1 cuts) from Table VIII and the branching fractions from and 2 S0 levels as well as the 2P → 2S transitions, Table VI. The relative heights of (and relative number of assuming as always a missing Bc → Bc=γ photon in the events in) the peaks measures the ratio reconstruction. Here we include direct production of the 2P 2 → 2 σðB 0 þ XÞBðB 0 → B þ πþπ−Þ states as well as feed-down from S P transitions. The R ≡ c c c : ð28Þ strong B → B line arising from direct production of B , σðB0 þ XÞBðB0 → B þ πþπ−Þ c c pffiffiffi c c c c for which we calculate σ · B ≈ 225 nb at s ¼ 13 TeV, is pffiffiffi At s ¼ 13 TeV, the ratio of cross sections is nearly 2.5. probably too low in energy to be observed. More promising Taking account of the branching fractions, we estimate are the 2P levels, which might show themselves in Bc þ γ R ≈ 2.IfBc and Bc were produced with equal frequency, invariant mass distributions. These lines make up the right- 3 we would find R ≈ 0.8. hand group (black lines) in Fig. 8. The 2 P2ð6750Þ → Bcγ Now the CMS Collaboration [4] at the Largepffiffiffi Hadron line is a particularly attractive target for experiment because Collider, analyzing 140 fb−1 ofpp collisions at s ¼ 13 TeV, of the favorable production cross section, branching frac- has observed a pattern that closely resembles the template tion, and 409-MeV photon energy. The 2P masses inferred þ − obs of Fig. 7. In the distribution of MðBcπ π Þ − MðBcÞ þ from transitions to Bc will be shifted downward because of MðBcÞ, they reconstruct a peak at 6871.0 1.2 MeV, the unobserved M1 photon. It is not possible to produce
054025-10 MESONS WITH BEAUTY AND CHARM: NEW HORIZONS IN … PHYS. REV. D 99, 054025 (2019)
3.5
3.0
2.5
2.0 [nb/MeV] σ 1.5
1.0
0.5
0.0 100 150 200 250 300 350 400 450 500 3 ¯ k [MeV] FIG. 10. Strong decay widths of the 3 S1 ðcbÞ level near open- flavor threshold. The shaded band on the mass axis indicates FIG. 8. Photon energies k and relative strengths of E1 tran- 20 MeV around our nominal value for the mass of this state, sitions from 2S → 2P (left group, blue curves) and 2P → 1S 7279 MeV. (right group, black curves) ðcb¯) states. Production rates are taken from Table VIII and branching fractions from Table VI.We → γ suppose that the photon transition Bc Bc þ = goes unobserved C. States above open-flavor threshold in the cascade transitions. We assume Gaussian line shapes with ¯ standard deviation 2 MeV. We estimate the strong decay rates for ðcbÞ states that lie above flavor threshold using the Cornell coupled-channel formalism [22] that we elaborated and applied to charmo- enriched samples of the 2S levels by tuning the energy of nium states in [23]. − 1 3 eþe collisions, as is done for J=ψ and ϒ, so reconstruction We expect both the 3 S0 and 3 S1 states to lie above 1 of the left-hand group of 2S → 2P transitions (blue lines in threshold for strong decays. The 3 S0 state can decay into 3 Fig. 8) will be problematic. the final state B D and the 3 S1 level can decay into both In the far future, combining the photon transition the BD and B D final states. The open decay channels as a energies and relative rates with expectations for production function of the masses of these states is shown in Figs. 9 and decay may eventually make it possible to disentangle and 10. mixing of the spin-singlet and spin-triplet J ¼ L states.
1 ¯ 3 ¯ FIG. 9. Strong decay widths of the 3 S0 ðcbÞ level near open- FIG. 11. Strong decay widths of the 3 P2 ðcbÞ level near open- flavor threshold. The shaded band on the mass axis indicates flavor threshold. The shaded band on the mass axis indicates 20 MeV around our nominal value for the mass of this state, 20 MeV around our nominal value for the mass of this state, 7253 MeV. 7154 MeV.
054025-11 ESTIA J. EICHTEN and CHRIS QUIGG PHYS. REV. D 99, 054025 (2019)
It is worth keeping in mind that while narrow BD peaks may signal excited ðcb¯Þ levels, narrow BD¯ peaks could ¯ ¯ indicate nearly bound bcqkql tetraquark states [47].
VI. TERA-Z PROSPECTS In response to the discovery of the 125-GeV Higgs boson, Hð125Þ [48], plans for large circular electron- positron colliders (FCC-ee [49] and CEPC [50]) are being developed as eþe− → HZ0 “Higgs factories” to run at c.m. pffiffiffi energy s ≈ 240 GeV. As now envisioned, these machines would haveffiffiffi the added capability of high-luminosity run- p 12 ning at s ¼ MZ that would accumulate 10 examples of the reaction eþe− → Z0. With the observed branching fraction, BðZ0 → bb¯Þ¼ð15.12 0.05Þ% [12], the tera- Z mode would produce some 3 × 1011 boosted b-quarks, 33 ¯ FIG. 12. Estimated strong decay widths of the P0 ðcbÞ level which would enable high-sensitivity searches for ðcb¯Þ near open-flavor threshold. The shaded band on the mass axis states in a variety of decay channels. A recent computation indicates 20 MeV around our nominal value for the mass of this B 0 → ¯ ≈ 6 10−4 state, 7104 MeV. suggests that ðZ ðcbÞþXÞ × [51]. The largest existing eþe− → Z0 → hadrons datasets were recorded by experiments at CERN’s Large 33 The P2 state might be observed as a very narrow Electron-Positron Collider during the 1990s. In samples (d-wave) BD line near open-flavor threshold. Its decay of (3.02, 3.9, and 4.2) million hadronic Z0 decays, the 2 width as a function of mass for the P states is given DELPHI, ALEPH, and OPAL Collaborations [52] found a in Fig. 11. small number of candidates for the decays B → J=ψπþ, 3 c In the phenomenological models the remaining P states J=ψlþν and J=ψ3π. Those few specimens were not lie just below the thresholds for strong decays. However, sufficient to establish a discovery, but the experiments they are near enough to these thresholds that there might be were able to bound combinations of branching fractions ð0Þ interesting behavior at the threshold for B D in the 3P1 B as 3 cases and for the BD threshold in the case of the 3 P0 state. 0 8 91 3 ψπþ Figure 12 shows that the 3 P0 width grows rapidly just <> J= => B Z0 → B X B C above threshold. The strong decay widths as a function of ð c þ Þ B → ψlþν ≲ 10−4 0 @Bc J= A few × ; 0 B → > > mass for the 3P1 and 3P states have a common behavior, ðZ hadronsÞ : ; 1 J=ψ3π displayed in Fig. 13. ð29Þ
at 90% confidence level, where X denotes anything. The relative simplicity of eþe− → Z0 events and the boosted kinematics of resulting Bc mesons suggest that a tera-Z factory might be a felicitous choice to investigate 2P → 1S þ 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 α coupling constant s that saturates at a fixed value at long α distances. The new frozen- s potential incorporates both perturbative and nonperturbative aspects of quantum chromodynamics. Second, we have set aside the perturba- ð0Þ tive treatment of spin splittings, instead incorporating FIG. 13. Strong decay widths of the 3P1 or 3P1 . The shaded band on the mass axis indicates 20 MeV around our nominal lessons from lattice QCD and observations of the ðcc¯Þ ¯ values for the masses of these state, 7135 and 7143 MeV. and ðbbÞ spectra.
054025-12 MESONS WITH BEAUTY AND CHARM: NEW HORIZONS IN … PHYS. REV. D 99, 054025 (2019)
We look forward to additional experimental progress, TABLE IX. Evolution of the strong coupling. first by confirming and elaborating the characteristics of α the 2S levels reported by the CMS Collaboration [4]. r (fm) sðrÞ 1 Key observables are the mass of the 2 S0 state, the 0.0080 0.1706 splitting between the two lines, and the ratio of peak 0.0088 0.1742 heights corrected for efficiencies and acceptance. It is 0.0097 0.1780 also of interest to test whether the dipion mass spectra 0.0108 0.1819 0 þ − 0 þ − 0.0119 0.1862 in the cascade decays Bc → Bcπ π and Bc → Bcπ π ψ 2 → ψπþπ− 0.0132 0.1908 follow the pattern seen in ð SÞ J= and 0.0145 0.1957 ϒ 2 → ϒ 1 πþπ− ð SÞ ð SÞ decays. Although we expect the 0.0161 0.2007 3S levels to lie above flavor threshold, exploring the 0.0178 0.2061 þ − Bcπ π mass spectrum up through 7200 MeV might 0.0196 0.2116 1 þ − 3 yield indications of 3 S0 → Bcπ π and 3 S1 → 0.0217 0.2174 þ − Bcπ π lines. The presence of one or the other of 0.0240 0.2235 these could signal interactions of bound states with open 0.0265 0.2299 channels. Prospecting for narrow Bð ÞDð Þ peaks near 0.0293 0.2365 0.0323 0.2434 threshold could yield evidence of Bc states beyond the 2 0.0358 0.2505 S levels. 0.0395 0.2579 The next frontier is the search for radiative transitions ¯ 0.0437 0.2659 among ðcbÞ levels. The most promising candidate for first 0.0483 0.2743 3 light is the 2 P2ð6750Þ → Bcγ transition. Determining the 0.0533 0.2829 Bc mass, perhaps by reconstructing Bc → Bcγ, would 0.0589 0.2915 provide an important check on lattice QCD calculations 0.0651 0.3001 and a key input to future calculations. 0.0720 0.3087 ¯ þ 0.0796 0.3171 Detecting the Bc → τντ and Bc → ppπ decays would be impressive experimental feats and would provide 0.0879 0.3252 another test of the short-distance behavior of the ground- 0.0972 0.3330 0.1074 0.3403 state wave function, complementing what will be learned 0.1187 0.3471 – from the Bc Bc splitting. 0.1312 0.3533 0.1450 0.3590 ACKNOWLEDGMENTS 0.1602 0.3640 This work was supported by Fermi Research Alliance, 0.1771 0.3685 0.1957 0.3723 LLC under Award No. DE-AC02-07CH11359 with the 0.2163 0.3757 U.S. Department of Energy, Office of Science, Office of 0.2390 0.3786 High Energy Physics. 0.2642 0.3811 0.2920 0.3832 APPENDIX: STRONG COUPLING EVOLUTION 0.3227 0.3849 α 0.3566 0.3864 To make calculations with the frozen- s potential, one 0.3941 0.3876 must combine a linear term with a Coulomb term, 0.4355 0.3886 −4α 3 α sðrÞ= r, for which sðrÞ is characterized by the solid 0.4813 0.3895 red curve of Fig. 1. We present in Table IX numerical 0.5320 0.3902 values of the strong coupling over the relevant range of 0.5879 0.3908 distances, 0 ≤ r ≤ 0.8 fm. The entries advance in steps 0.6497 0.3913 of δ ln r ¼ 0.1. 0.7181 0.3917 0.7936 0.3920
054025-13 ESTIA J. EICHTEN and CHRIS QUIGG PHYS. REV. D 99, 054025 (2019)
et al. þ [1] G. Aad (ATLAS Collaboration), Observation of an First Observation of a Baryonic Bc Decay, Phys. Rev. Lett. Excited Bc Meson State with the ATLAS Detector, Phys. 113, 152003 (2014). Rev. Lett. 113, 212004 (2014). [12] M. Tanabashi et al. (Particle Data Group), Review of þ [2] R. Aaij et al. (LHCb Collaboration), Search for excited Bc particle physics, Phys. Rev. D 98, 030001 (2018). states, J. High Energy Phys. 01 (2018) 138. [13] I. F. Allison, C. T. H. Davies, A. Gray, A. S. Kronfeld, P. B. [3] For a recent assessment, see contributions to the Mackenzie, and J. N. Simone (Fermilab Lattice, HPQCD, þ Micro Workshop on Bc physics at LHCb, 13 July and UKQCD Collaborations), Mass of the Bc Meson in 2016, https://indico.cern.ch/event/549155/, especially Z. Three-Flavor Lattice QCD, Phys. Rev. Lett. 94, 172001 þ Yang, Experiment—Recent history of the Bc meson, (2005). https://indico.cern.ch/event/549155/contributions/2226387/ [14] R. J. Dowdall, C. T. H. Davies, T. C. Hammant, and R. R. attachments/1308369/1957602/Bc_results20160713.pdf; Horgan, Precise heavy-light meson masses and hyperfine A. Luchinsky, Theory–Status of the theoretical description splittings from lattice QCD including charm quarks in the þ of the Bc meson, https://indico.cern.ch/event/549155/ sea, Phys. Rev. D 86, 094510 (2012). 2Sþ1 contributions/2226440/attachments/1308367/1956579/bc_ [15] We use spectroscopic notation n LJ, where n is the Luchinsky.pdf; R. Oldeman, Experiment—Perspectives principal quantum number, S is total spin, and L ¼ for future studies, https://indico.cern.ch/event/549155/ S; P; D; … represents the angular momentum 0; 1; 2; …. contributions/2226443/attachments/1308858/1957460/ [16] M. Beneke and G. Buchalla, The Bc meson lifetime, Phys. þ 20160713_Bc.pdf; A. V. Berezhnoy, Theory–Bc produc- Rev. D 53, 4991 (1996); A. Yu. Anisimov, I. M. Narodetsky, tion and spectroscopy, https://indico.cern.ch/event/549155/ C. Semay, and B. Silvestre-Brac, The Bc meson lifetime in contributions/2226438/attachments/1309000/1957694/ the light front constituent quark model, Phys. Lett. B 452, Berezhnoy_Bc.pdf. 129 (1999); V. V. Kiselev, A. E. Kovalsky, and A. K. et al. [4] A. M. Sirunyan (CMS Collaboration), Observation of Likhoded, Bc decays and lifetime in QCD sum rules, Nucl. two excited Bþ states and measurement of the Bþ (2S) mass Phys. B585, 353 (2000); C.-H. Chang, S.-L. Chen, T.-F. c pffiffiffi c in pp collisions at s ¼ 13 TeV, arXiv:1902.00571. Feng, and X.-Q. Li, The lifetime of Bc meson and some [5] E. Eichten and F. Feinberg, Spin dependent forces in QCD, relevant problems, Phys. Rev. D 64, 014003 (2001). Phys. Rev. D 23, 2724 (1981). [17] N. Brambilla et al. (Quarkonium Working Group), Heavy [6] E. J. Eichten and C. Quigg, Mesons with beauty and charm: quarkonium physics, CERN Yellow Report No. CERN- Spectroscopy, Phys. Rev. D 49, 5845 (1994). 2005-005, 2005, DOI: 10.5170/CERN-2005-005. [7] For other work in a similar spirit, see V. V. Kiselev, A. K. [18] A. Ali, L. Maiani, A. D. Polosa, and V. Riquer, Bc decays Likhoded, and A. V. Tkabladze, Bc spectroscopy, Phys. Rev. into tetraquarks, Phys. Rev. D 94, 034036 (2016);A. D 51, 3613 (1995); L. P. Fulcher, Phenomenological pre- Esposito, M. Papinutto, A. Pilloni, A. D. Polosa, and N. Ξ dictions of the properties of the Bc system, Phys. Rev. D 60, Tantalo, Doubly charmed tetraquarks in Bc and bc decays, 074006 (1999); S. Godfrey and N. Isgur, Mesons in a Phys. Rev. D 88, 054029 (2013); W. Wang, Y.-L. Shen, and − → 3872 π− − relativized quark model with chromodynamics, Phys. Rev. C.-D. Lu, The study of Bc Xð Þ ðK Þ decays in the covariant light-front approach, Eur. Phys. J. C 51, 841 D 32, 189 (1985); S. Godfrey, Spectroscopy of Bc mesons in the relativized quark model, Phys. Rev. D 70, 054017 (2007); W. Wang and Q. Zhao, Decipher the short-distance 3872 (2004); D. Ebert, R. N. Faustov, and V. O. Galkin, Properties component of Xð Þ in Bc decays, Phys. Lett. B 755, 261 (2016). of heavy quarkonia and B mesons in the relativistic quark c [19] A. Lytle, B. Colquhoun, C. Davies, and J. Koponen, B model, Phys. Rev. D 67, 014027 (2003); A. V. Berezhnoy, c spectroscopy using highly improved staggered quarks, Proc. V. V. Kiselev, A. K. Likhoded, and A. I. Onishchenko, B c Sci. LATTICE2018 (2018) 083. meson at LHC, Yad. Fiz. 60N10, 1889 (1997) [Phys. At. [20] T. Appelquist and H. D. Politzer, Heavy Quarks and eþe− Nucl. 60, 1729 (1997)]; N. R. Soni, B. R. Joshi, R. P. Shah, ¯ Annihilation, Phys. Rev. Lett. 34, 43 (1975). H. R. Chauhan, and J. N. Pandya, QQ (Q ∈ fb; cg) spec- [21] T. Appelquist, A. De Rujula, H. D. Politzer, and S. L. troscopy using the Cornell potential, Eur. Phys. J. C 78, 592 Glashow, Spectroscopy of the New Mesons, Phys. Rev. (2018). Lett. 34, 365 (1975); E. Eichten, K. Gottfried, T. Kinoshita, [8] F. Abe et al. (CDF Collaboration), Observation of the B pffiffiffi c J. B. Kogut, K. D. Lane, and T.-M. Yan, Spectrum of Meson in pp¯ Collisions at s ¼ 1.8 TeV, Phys. Rev. Lett. Charmed Quark-Antiquark Bound States, Phys. Rev. Lett. 81, 2432 (1998). 34, 369 (1975); Erratum, Phys. Rev. Lett. 36, 1276(E) [9] T. Aaltonen et al. (CDF Collaboration), Observation of the (1976). þ þ Decay Bc → J=ψπ and Measurement of the Bc Mass, [22] E. Eichten, K. Gottfried, T. Kinoshita, K. D. Lane, and T.-M. Phys. Rev. Lett. 100, 182002 (2008). Yan, Charmonium: The Model, Phys. Rev. D 17, 3090 [10] V. M. Abazov et al. (D0 Collaboration), Observation of the (1978); , Charmonium: The Model, Phys. Rev. D 21, 313(E) Bc Meson in the Exclusive Decay Bc → J=ψπ, Phys. Rev. (1980); Charmonium: Comparison with experiment, Phys. Lett. 101, 012001 (2008). Rev. D 21, 203 (1980). þ [11] R. Aaij et al. (LHCb Collaboration), Measurements of Bc [23] E. J. Eichten, K. Lane, and C. Quigg, Charmonium levels þ þ þ − Production and Mass with the Bc → J=ψπ Decay, Phys. near threshold and the narrow state Xð3872Þ → π π J=ψ, þ Rev. Lett. 109, 232001 (2012); Observation of Bc → Phys. Rev. D 69, 094019 (2004); New states above charm þ þ þ J=ψDs and Bc → J=ψDs decays, Phys. Rev. D 87, threshold, Phys. Rev. D 73, 014014 (2006); Erratum, Phys. 112012 (2013); Publisher’s Note 89, 019901(A) (2014); Rev. D 73, 079903(E) (2006).
054025-14 MESONS WITH BEAUTY AND CHARM: NEW HORIZONS IN … PHYS. REV. D 99, 054025 (2019)
[24] N. Brambilla et al., Heavy quarkonium: Progress, puzzles, Photonic production of S- and P wave Bc states and doubly and opportunities, Eur. Phys. J. C 71, 1534 (2011). heavy baryons, Z. Phys. A 356, 89 (1996). [25] Y. Sumino, Computation of heavy quarkonium spectrum in [43] B. Colquhoun, C. T. H. Davies, R. J. Dowdall, J. Kettle, J. perturbative QCD, Proc. Sci. LL2016 (2016) 011. Koponen, G. P. Lepage, and A. T. Lytle (HPQCD Collabo- [26] A. Martin, A fit of upsilon and charmonium spectra, Phys. ration), B-meson decay constants: A more complete picture Lett. 93B, 338 (1980). from full lattice QCD, Phys. Rev. D 91, 114509 (2015); For [27] J. L. Richardson, The heavy quark potential and the ϒ, J=ψ further work on semileptonic decays, see A. Lytle, B. systems, Phys. Lett. 82B, 272 (1979). Colquhoun, C. Davies, J. Koponen, and C. McNeile, [28] W. Buchmüller and S. H. H. Tye, Quarkonia and quantum Semileptonic Bc decays from full lattice QCD, Proc. Sci., chromodynamics, Phys. Rev. D 24, 132 (1981). BEAUTY2016 (2016) 069; A. Lytle, Theory—Semilep- þ [29] V. N. Gribov, The theory of quark confinement, Eur. Phys. J. tonic Bc decays and Lattice QCD, in Ref. [3], https://indico C 10, 91 (1999). .cern.ch/event/549155/contributions/2226442/attachments/ [30] K. G. Chetyrkin, Four-loop renormalization of QCD: Full 1308725/1957203/Bc_lattice.pdf. set of renormalization constants and anomalous dimensions, [44] M. J. Baker, J. Bordes, C. A. Dominguez, J. Penarrocha, and
Nucl. Phys. B710, 499 (2005); M. Czakon, The Four-loop K. Schilcher, B meson decay constants fBc , fBs and fB from QCD beta-function and anomalous dimensions, Nucl. Phys. QCD sum rules, J. High Energy Phys. 07 (2014) 032. B710, 485 (2005). [45] C.-H. Chang, J.-X. Wang, and X.-G. Wu, BCVEGPY2.0: An [31] M. Jeżabek, M. Peter, and Y. Sumino, On the relation upgraded version of the generator BCVEGPY with the between QCD potentials in momentum and position space, addition of hadroproduction of the P-wave Bc states, Phys. Lett. B 428, 352 (1998). Comput. Phys. Commun. 174, 241 (2006). We use [32] D. Gromes, Spin dependent potentials in QCD and the (derivatives of) wave functions at the origin derived from correct long range spin orbit term, Z. Phys. C 26, 401 our current work. The quark mass parameters in this program (1984). vary with the produced state, to reproduce its mass. 1 ∶ 5 000 1 275 2 ∶ 5 234 1 633 [33] C. T. H. Davies, K. Hornbostel, G. P. Lepage, A. J. Lidsey, J. S mb ¼ . , mc ¼ . ; S mb ¼ . , mc ¼ . ; 2 ∶ 5 184 1 573 3 ∶ 5 447 1 825 Shigemitsu, and J. H. Sloan, B spectroscopy from lattice P mb ¼ . , mc ¼ . ; S mb ¼ . , mc ¼ . ; c 3 ∶ 5 502 1 633 QCD, Phys. Lett. B 382, 131 (1996). P mb ¼ . , mc ¼ . , all in GeV. λ ν Δ 0 [34] E. B. Gregory, C. T. H. Davies, E. Follana, E. Gamiz, I. D. [46] In an effective power-law potential VðrÞ¼ r , 21 < so ν 1 Kendall, G. P. Lepage, H. Na, J. Shigemitsu, and K. Y. long as < . See Secs. 4.1.1 and 5.3.2 of C. Quigg and Wong, A Prediction of the B Mass in Full Lattice QCD, J. L. Rosner, Quantum mechanics with applications to c 56 Phys. Rev. Lett. 104, 022001 (2010). quarkonium, Phys. Rep. , 167 (1979), particularly [35] N. Mathur, M. Padmanath, and S. Mondal, Precise Pre- Eqs. (4.21) and (4.22). [47] E. J. Eichten and C. Quigg, Heavy-Quark Symmetry Implies dictions of Charmed-Bottom Hadrons from Lattice QCD, Stable Heavy Tetraquark Mesons Q Q q¯ q¯ , Phys. Rev. Phys. Rev. Lett. 121, 202002 (2018). i j k l Lett. 119, 202002 (2017). [36] C. Quigg and J. L. Rosner, Counting narrow levels of [48] G. Aad et al. (ATLAS Collaboration), Observation of a new quarkonium, Phys. Lett. 72B, 462 (1978). particle in the search for the Standard Model Higgs boson [37] E. Eichten and K. Gottfried, Heavy quarks in eþe− with the ATLAS detector at the LHC, Phys. Lett. B 716,1 annihilation, Phys. Lett. 66B, 286 (1977). (2012); S. Chatrchyan et al. (CMS Collaboration), Obser- [38] L. S. Brown and R. N. Cahn, Chiral Symmetry and ψ 0 → vation of a new boson at a mass of 125 GeV with the CMS ψππ Decay, Phys. Rev. Lett. 35, 1 (1975). experiment at the LHC, Phys. Lett. B 716, 30 (2012). [39] See Sec. 7 of Ref. [17] and Sec. 3.3 of Ref. [24] for surveys [49] FCC-ee, the electron-positron option of the future circular of cascade decays. R colliders design study, http://tlep.web.cern.ch. [40] We adopt the conventional normalization, dΩ∶ [50] CEPC, Circular electron-positron collider, http://cepc.ihep Y θ; ϕ Yl0 0 θ; ϕ δll0 δ 0 lmð Þ m ð Þ¼ mm . See, e.g., the Appendix .ac.cn. Quantum Mechanics of of H. A. Bethe and E. E. Salpeter, [51] Q.-L. Liao, Y. Yu, Y. Deng, G.-Y. Xie, and G.-C. Wang, One- and Two-Electron Atoms (Springer-Verlag, Berlin, Excited heavy quarkonium production via Z0 decays at a 1957). high luminosity collider, Phys. Rev. D 91, 114030 (2015). [41] R. Van Royen and V. F. Weisskopf, Hadron decay processes [52] P. Abreu et al. (DELPHI Collaboration), Search for the Bc and the quark model, Nuovo Cimento A 50, 617 (1967); meson, Phys. Lett. B 398, 207 (1997); R. Barate et al. Erratum, Nuovo Cimento A 51, 583(E) (1967). The factor of (ALEPH Collaboration), Search for the Bc meson in 3 accounts for quark color. hadronic Z decays, Phys. Lett. B 402, 213 (1997);K. [42] E. Braaten and S. Fleming, QCD radiative corrections to the Ackerstaff et al. (OPAL Collaboration), Search for the Bc 0 leptonic decay rate of the Bc meson, Phys. Rev. D 52, 181 meson in hadronic Z decays, Phys. Lett. B 420, 157 (1998). (1995); A. V. Berezhnoy, V. V. Kiselev, and A. K. Likhoded,
054025-15