Bottomonium Spectrum Research Articles

Spectroscopy and annihilation decay widths of charmonium and bottomonium excitations

We study the charmonium ( cc¯ ) and bottomonium ( bb¯ ) spectra, using Cornell potential with additional constant potential term in a non-relativistic framework, with spin-dependent corrections corresponding to the spin–spin, spin–orbit, and tensor interactions added perturbatively. We predict the masses of low-lying and excited states of charmonium and bottomonium up to n = 6 and L = 2. We analyze the radial and orbital Regge trajectories of both the systems and investigate their departure from linearity. Further, we estimate the wave function at the origin and, consequently, predict the decay widths of charmonium and bottomonium states annihilating to leptons and photons. Also, we investigate the effect of scale dependence of the wave function at the origin and the strong coupling constant on the predictions of annihilation decay widths. We compare our predictions of heavy quarkonium spectra and decay widths with experimental results and other theoretical models.

Bottomonium suppression from the three-loop QCD potential

We compute the suppression of bottomonium in the quark-gluon plasma using the three-loop QCD static potential. The potential describes the spin-averaged bottomonium spectrum below threshold with a less than 1% error. Within potential nonrelativistic quantum chromodynamics and an open quantum systems framework, we compute the evolution of the bottomonium density matrix. The values of the quarkonium transport coefficients are obtained from lattice QCD measurements of the bottomonium in-medium width and thermal mass shift; we additionally include for the first time a vacuum contribution to the dispersive coefficient γ. Using the three-loop potential and the values of the heavy quarkonium transport coefficients, we find that the resulting bottomonium nuclear modification factor is consistent with experimental observations, while at the same time reproducing the lattice measurements of the in-medium width. Published by the American Physical Society 2024

Do near-threshold molecular states mix with neighboring Q¯Q states?

The last two decades are marked by a renaissance in hadronic spectroscopy caused by the arrival of vast experimental information on exotic states in the spectrum of charmonium and bottomonium. Most of such states have properties at odds with the predictions of the quark model and reside very close to strong hadronic thresholds. Prominent examples are provided by the glorious $X(3872)$ charmonium-like state and the doubly charmed tetraquark $T_{cc}^+$ with the masses within less than 1 MeV from the $D\bar{D}^*$ and $DD^*$ open-charm thresholds, respectively. The universality of this feature hints towards the existence of a general pattern for such exotic states. In this work we discuss a possible generic mechanism for the formation of near-threshold molecular states as a result of the strong coupling of compact quark states with a hadronic continuum channel. The compact states that survive the strong coupling limit decouple from the continuum channel and therefore also from the formed hadronic molecule - if realised this scenario would provide a justification to treat hadronic molecules isolated, ignoring the possible influence from surrounding, compact quark-model states. We confront the phenomenology of the $D_{s1}(2460)$ and $D_{s1}(2536)$ with this picture and find consistency, although other explanations remain possible for those states.

Bottomonium spectroscopy using Coulomb plus linear (Cornell) potential

The Coulomb plus linear (Cornell) potential is used to investigate the mass spectrum of bottomonium. Gaussian wave function is used in position and momentum space to estimate values of potential and kinetic energies, respectively. Based on our calculations, we study newly observed $${\varUpsilon {(10860)}}$$ as an admixture of $$4^{3}S_{1}$$ with $$4^{3}D_{1}$$ and $${\varUpsilon {(10753)}}$$ as an admixture of $$6^{3}S_{1}$$ with $$4^{3}D_{1}$$ ; also, we try to assign $${{\varvec{\Upsilon }}{(11020)}}$$ as a pure $$4^{3}D_{1}$$ bottomonium state. We also study the Regge trajectories in the $$(J,M^{2})$$ and $$(n_r,M^{2})$$ planes to help prove our association. We estimate the pseudoscalar and vector decay constants, the radiative (electric and magnetic dipole) transition rates, and the annihilation decay width for bottomonium states.

Computation of the quarkonium and meson-meson composition of the ϒ(nS) states and of the new ϒ(10753) Belle resonance from lattice QCD static potentials

We compute the composition of the bottomonium $\mathrm{\ensuremath{\Upsilon}}(nS)$ states [including $\mathrm{\ensuremath{\Upsilon}}(10860)$] and the new $\mathrm{\ensuremath{\Upsilon}}(10753)$ resonance reported by Belle in terms of quarkonium and meson-meson components. We use a recently developed novel approach utilizing lattice QCD string breaking potentials for the study of resonances. This approach is based on the diabatic extension of the Born-Oppenheimer approximation and the unitary emergent wave method and allows us to compute the poles of the $\mathrm{S}$ matrix. We focus on $I=0$ bottomonium $S$ wave bound states and resonances, where the Schr\"odinger equation is a set of coupled differential equations. One of the channels corresponds to a confined heavy quark-antiquark pair $\overline{b}b$, and the others correspond to pairs of heavy-light mesons. In a previous study, only one meson-meson channel ${\overline{B}}^{(*)}{B}^{(*)}$ was considered. Now, we also include the closed strangeness channel ${\overline{B}}_{s}^{(*)}{B}_{s}^{(*)}$, extending our formalism significantly to have a more realistic description of bottomonium. We confirm the new Belle resonance $\mathrm{\ensuremath{\Upsilon}}(10753)$ as a dynamical meson-meson resonance with around 76% meson-meson content. Moreover, we identify $\mathrm{\ensuremath{\Upsilon}}(4S)$ and $\mathrm{\ensuremath{\Upsilon}}(10860)$ as states with both sizable quarkonium and meson-meson contents. With these results, we contribute to the clarification of ongoing controversies in the vector bottomonium spectrum.

Light-by-light scattering sum rule for radiative transitions of bottomonia

We generalize a forward light-by-light scattering sum rule to the case of heavy quarkonium radiative transitions. We apply such sum rule to the bottomonium states, and use available data on radiative transitions in its evaluation. For the transitions that are not known experimentally, we provide theoretical estimates within a potential model, and consider the spread between similar approaches in the literature as an estimate for the model error. For the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ states we observe that, due to a cancellation between transitions involving $\chi_{b0}, \chi_{b1}$, and $\chi_{b2}$ states, the sum rule is satisfied within experimental and theoretical error estimates. Having tested this sum rule for the low-lying bottomonium states, it may be used as a tool to investigate the nature of exotic states in the charmonium and bottomonium spectrum through the corresponding radiative transitions.

Bottomonium spectrum with a Dirac potential model in the momentum space

We study the bottomonium spectrum using a relativistic potential model in the momentum space. This model is based on a complete one gluon exchange interaction with a momentum dependent screening factor to account for the effects due to virtual pair creation that appear close to the decay thresholds. The overall model does not make use of nonrelativistic approximations. We fit well established bottomonium states below the open bottom threshold and predict the rest of the spectrum up to approx 11200 MeV and J^{PC}=3^{--}. Uncertainties are treated rigorously and propagated in full to the parameters of the model using a Monte Carlo to identify if which deviations from experimental data can be absorbed into the statistical uncertainties of the models and which can be related to physics beyond the bbar{b} picture, guiding future research. We get a good description of the spectrum, in particular the Belle measurement of the eta _b(2S) state and the varUpsilon (10860) and chi _b(3P) resonances.

Triply charmed and bottom baryons in a constituent quark model

In this work, we study the mass spectrum of the $\Omega_{ccc}$ and $\Omega_{bbb}$ baryons up to the $N=2$ shell within a nonrelativistic constituent quark model (NRCQM). The model parameters are adopted from the determinations by fitting the charmonium and bottomonium spectra in our previous works. The masses of the $\Omega_{ccc}$ and $\Omega_{bbb}$ baryon states predicted in present work reasonably agree with the results obtained with the Lattice QCD calculations. Furthermore, to provide more knowledge of the $\Omega_{ccc}$ and $\Omega_{bbb}$ states, we evaluate their radiative decays with the available masses and wave functions from the potential model.

Bottomonium spectrum in the relativistic flux tube model

The bottomonium spectrum is far from being established. The structures of higher vector states, including the $\Upsilon(10580)$, $\Upsilon(10860)$, and $\Upsilon(11020)$ states, are still in dispute. In addition, whether the $\Upsilon(10750)$ signal which was recently observed by the Belle Collaboration is a normal $b\bar{b}$ state or not should be examined. Faced with such a situation, we carried out a systematic investigation of the bottomonium spectrum in the scheme of the relativistic flux tube (RFT) model. A Chew-Frautschi like formula was derived analytically for the spin average mass of bottomonium states. We further incorporated the spin-dependent interactions and obtained a complete bottomonium spectrum. We found that the most established bottomonium states can be explained in the RFT scheme. The $\Upsilon(10750)$, $\Upsilon(10860)$, and $\Upsilon(11020)$ could be predominantly the $3^3D_1$, $5^3S_1$, and $4^3D_1$ states, respectively. Our predicted masses of $1F$ and $1G$ $b\bar{b}$ states are in agreement with the results given by the method of lattice QCD, which can be tested by experiments in future. We also compared the RFT model with the quark potential model in detail. The differences of these two kinds of models were discussed.

Heavy quarkonia: the beauty and the beasts

New enhancements in the charmonium and bottomonium spectra observed since 2003 are very briefly reviewed. Special attention is paid to χc1(3872) (formerly X(3872)) owing to its remarkable proximity to the threshold, which allows modelling as a quasibound axial-vector state with a large admixture. In contrast, the interpretation of many other charmonium-like and bottomonium-like states is still very controversial and some may not even correspond to genuine resonances. Accordingly, several entries in the PDG tables have been wildly changing over the years. Three representative states are reviewed here as non-resonant enhancements due to threshold effects, viz. ψ(4260), ψ(4660), and ϒ(10580).

Determination of ΛMS¯(nf=2) and analytic parametrization of the static quark-antiquark potential

While lattice QCD allows for reliable results at small momentum transfers (large quark separations), perturbative QCD is restricted to large momentum transfers (small quark separations). The latter is determined up to a reference momentum scale $\Lambda$, which is to be provided from outside, e.g. from experiment or lattice QCD simulations. In this article, we extract $\Lambda_{\overline{\textrm{MS}}}$ for QCD with $n_f=2$ dynamical quark flavors by matching the perturbative static quark-antiquark potential in momentum space to lattice results in the intermediate momentum regime, where both approaches are expected to be applicable. In a second step, we combine the lattice and the perturbative results to provide a complete analytic parameterization of the static quark-antiquark potential in position space up to the string breaking scale. As an exemplary phenomenological application of our all-distances potential we compute the bottomonium spectrum in the static limit.

Dissociation of 1P states in hot QCD Medium Using Quasi-Particle Model

We extend the analysis of a very recent work [1] to study the dissociation phenomenon of 1P states of the charmonium and bottomonium spectra (χc and χb) in a hot QCD medium using Quasi-Particle Model. This study employed a medium modified heavy quark potential which has quite different form in the sense that it has a lomg range Coulombic tail in addition to the Yukawa term even above the deconfinement temperature. Then we study the flavor dependence of their binding energies and explore the nature of dissociation temperatures by employing the Quasi-Particle debye mass for pure gluonic and full QCD case. Interestingly, the dissociation temperatures obtained by employing EoS1 and EoS2 with the Γ criterion, is closer to the upper bound of the dissociation temperatures which are obtained by the dissolution of a given quarkonia state by the mean thermal energy of the quasi-partons in the hot QCD/QGP medium.

Bottom and charm mass determinations from global fits to Qoverline{Q} bound states at N3LO

The bottomonium spectrum up to n = 3 is studied within Non-Relativistic Quantum Chromodynamics up to N3LO. We consider finite charm quark mass effects both in the QCD potential and the overline{mathrm{MS}} -pole mass relation up to third order in the Y-scheme counting. The u = 1/2 renormalon of the static potential is canceled by expressing the bottom quark pole mass in terms of the MSR mass. A careful investigation of scale variation reveals that, while n = 1, 2 states are well behaved within perturbation theory, n = 3 bound states are no longer reliable. We carry out our analysis in the nℓ = 3 and nℓ = 4 schemes and conclude that, as long as finite mc effects are smoothly incorporated in the MSR mass definition, the difference between the two schemes is rather small. Performing a fit to boverline{b} bound states we find {overline{m}}_bleft({overline{m}}_bright) = 4.216 ± 0.039 GeV. We extend our analysis to the lowest lying charmonium states finding {overline{m}}_cleft({overline{m}}_cright) = 1.273 ± 0.054 GeV. Finally, we perform simultaneous fits for {overline{m}}_b and αs finding {alpha}_s^{left({n}_f=5right)}left({m}_Zright)=0.1178pm 0.0051 . Additionally, using a modified version of the MSR mass with lighter massive quarks we are able to predict the uncalculated mathcal{O}left({alpha}_s^4right) virtual massive quark corrections to the relation between the overline{mathrm{MS}} and pole masses.

Spin partners of the Zb(10610) and Zb(10650) revisited

We study the implications of the heavy-quark spin symmetry for the possible spin partners of the exotic states Zb(10610) and Zb(10650) in the spectrum of bottomonium. We formulate and solve numerically the coupled-channel equations for the Zb states that allow for a dynamical generation of these states as hadronic molecules. The force includes short-range contact terms and the one-pion exchange potential, both treated full nonperturbatively. The strength of the potential at leading order is fixed completely by the pole positions of the Zb states so that the mass and the most prominent contributions to the width of the isovector heavy-quark spin partner states WbJ with the quantum numbers J++ (J = 0, 1, 2) come out as predictions. In particular, we predict the existence of an isovector 2++ tensor state lying a few MeV below the {B}^{ast }{overline{B}}^{ast } threshold which should be detectable in the experiment. Since the accuracy of the present experimental data does not allow one to fix the pole positions of the Zb’s reliably enough, we also study the pole trajectories of their spin partner states as functions of the Zb binding energies. It is shown that, once the heavy-quark spin symmetry is broken by the physical B and B∗ mass difference, especially the pion tensor force has a significant impact on the location of the partner states clearly demonstrating the need of a coupled-channel treatment of pion dynamics to understand the spin multiplet pattern of hadronic molecules.

Spectrum and electromagnetic transitions of bottomonium

Stimulated by the exciting progress in the observation of new bottomonium states, we study the bottomonium spectrum. To calculate the mass spectrum, we adopt a nonrelativistic screened potential model. The radial Schr\"{o}dinger equation is solved with the three-point difference central method, where the spin-dependent potentials are dealt with non-perturbatively. With this treatment, the corrections of the spin-dependent potentials to the wave functions can be included successfully. Furthermore, we calculate the electromagnetic transitions of the $nS$ ($n\leq 4$), $nP$ ($n\leq 3$), and $nD$ ($n\leq 2$) bottomonium states with a nonrelativistic electromagnetic transition operator widely applied to meson photoproduction reactions. Our predicted masses, hyperfine and fine splittings, electromagnetic transition widths and branching ratios of the bottomonium states are in good agreement with the available experimental data. Especially, the EM transitions of $\Upsilon(3S)\to \chi_{b1,2}(1P)\gamma$, which were not well understood in previous studies, can be reasonably explained by considering the corrections of the spin-dependent interactions to the wave functions. We also discuss the observations of the missing bottomonium states by using radiative transitions. Some important radiative decay chains involving the missing bottomonium states are suggested to be observed. We hope our study can provide some useful references to observe and measure the properties of bottomonium mesons in forthcoming experiments.

Heavy Hybrids: decay to and mixing with Heavy Quarkonium

We report on a recent QCD based research on hybrid mesons containing $c\bar c$ or $b\bar b$ quarks. We present results for the spectrum, the decay widths to heavy quarkonium, and the role of mixing with the latter. We point out that mixing with heavy quarkonium provides a potentially large source of spin symmetry breaking. We identify candidates to hybrid mesons among the so called XYZ states in the charmonium and bottomonium spectrum.

Phenomenology of near-threshold states: a practical parametrisation for the line shapes

In the last decade many states in the spectrum of charmonium and bottomonium have been observed experimentally above the lowest open-flavour threshold. Most of these states reside in the vicinity of strong thresholds and show properties that cannot be captured by simple quark models. Description and understanding of such exotic states is a challenge for the phenomenology of strong interactions, since it requires building adequate theoretical tools and approaches. In this work, a practical parametrisation for the line shapes of near threshold resonance(s) is derived in the framework of a coupled-channel model which includes an arbitrary number of elastic and inelastic channels as well as of bare pole terms. Parameters of the distribution have a direct relation to phenomenology and the resulting analytical parametrisation is therefore ideally suited to harvest the full information content provided by the measurements and to establish a link between the experimental data and their theoretical interpretation.

QCD string in excited heavy-light mesons and heavy-quark hybrids

The QCD string model is employed to evaluate the masses of orbitally and radially excited heavy-light mesons and lightest hybrids in the spectrum of charmonium and bottomonium. The number of parameters of the model is reduced to only seven which are the string tension, the two values of the strong coupling constant (one for heavy-light and $\overline{c}c$ mesons and one for $\overline{b}b$ mesons), and the four overall spectrum shift constants which depend on the quark contents of the particular meson or hybrid family. A few well-established states in the spectrum of heavy-light and heavy-heavy mesons are used to fix these parameters, and then the masses of other mesons and hybrids come out as predictions of the model which are confronted with the existing experimental data, and a few suggestions are made concerning yet not measured quantum numbers of some states in the spectrum of charmonium and bottomonium.

Hadronic resonances enhanced by thresholds

We present a neat example of a meson--baryon system where the vicinity of two different thresholds enhances the binding of a hadronic resonance, a pentaquark. As a consequence the pattern of states may change when moving among different flavor sectors, what poses a warning on naive extrapolations to heavy flavor sectors based on systematic expansions. For this purpose we simultaneously analyze the $N\bar D$ and $NB$ two-hadron systems looking for possible bound states or resonances. When a resonance is controlled by a coupled-channel effect, going to a different flavor sector may enhance or diminish the binding. This effect may, for example, generate significant differences between the charmonium and bottomonium spectra above open-flavor thresholds or pentaquark states in the open-charm and open-bottom sectors.

Bottomonium spectrum revisited

We revisit the bottomonium spectrum motivated by the recently exciting experimental progress in the observation of new bottomonium states, both conventional and unconventional. Our framework is a nonrelativistic constituent quark model which has been applied to a wide range of hadronic observables from the light to the heavy quark sector and thus the model parameters are completely constrained. Beyond the spectrum, we provide a large number of electromagnetic, strong and hadronic decays in order to discuss the quark content of the bottomonium states and give more insights about the better way to determine their properties experimentally.