Bottomonium Spectrum Research Articles

Bottomonium mesons and strategies for their observation

The $B$-factories and Large Hadron Collider experiments have demonstrated the ability to observe and measure the properties of bottomonium mesons. In order to discover missing states it is useful to know their properties to develop a successful search strategy. To this end we calculate the masses and decay properties of excited bottomonium states. We use the relativized quark model to calculate the masses and wavefunctions and the $^3P_0$ quark-pair creation model to calculate decay widths to open bottom. We also summarize results for radiative transitions, annihilation decays, hadronic transitions and production cross sections which are used to develop strategies to find these states. We find that the $b\bar{b}$ system has a rich spectroscopy that we expect to be substantially extended by the LHC and $e^+e^-$ experiments in the near future. Some of the most promising possibilities at the LHC are observing the $\chi_{b(1,2)}(3P)$, $\chi_{b(1,2)}(4P)$ and $\eta_b(3S)$ states in $\gamma \mu^+\mu^-$ final states that proceed via radiative transitions through $\Upsilon(nS)$ intermediate states and $1^3D_J$ and $2^3D_J$ into $\gamma\gamma \mu^+\mu^-$ final states proceeding via $1^3P_J\to 1^3S_1$ and $2^3P_J\to 2^3S_1$ intermediate states respectively. Some of the most interesting possibilities in $e^+e^-$ collisions are studying the $1^3D_J$ states via $4\gamma$ cascades starting with the $\Upsilon (3S)$ and the $3^3P_J$ states in $\gamma\gamma\mu^+ \mu^-$ final states starting with the $\Upsilon (4S)$ and proceeding via $\Upsilon (nS)$ intermediate states. Completing the bottomonium spectrum is an important validation of lattice QCD calculations and a test of our understanding of bottomonium states in the context of the quark model.

The unquenched quark model

In this contribution, we briefly analyze the formalism of the unquenched quark model (UQM) and its application to the description of several observables of hadrons. In the UQM, the effects of charges sea pairs are introduced explicitly into the quark model through a QCD- inspired 3P0 pair-creation mechanism. We present our description of flavour asymmetry and strangeness in the proton when baryon-meson components are included. In the meson sector, the charmonium and bottomonium spectra with self-energy corrections due to the coupling to the meson-meson components.

Charmonium description from a generalized screened potential model

A generalized screened potential model (GSPM), recently developed to study the bottomonium spectrum, is applied to the calculation of charmonium masses and electromagnetic widths. The presence in the GSPM of more quark-antiquark bound states than in conventional non screened potential models, allows for the assignment of GSPM states to cataloged non conventional $J^{++}$ charmonium resonances as well as for the prediction of new (non cataloged) $J^{++}$ states. The results obtained seem to indicate that a reasonable overall description of $J^{++}$ charmonium resonances is feasible.

The unquenched quark model

In this contribution, we briefly discuss the results for charmonium and bottomonium spectra with self-energy corrections in the unquenched quark model, due to the coupling to the meson-meson continuum The UQM formalism can be extended to include also the effects of hybrid mesons, i.e. hybrid loops. Finally, we discuss the results of a calculation of hybrid mesons spectrum in Coulomb Gauge QCD.

Higher mass bottomonia

We show the results of a quark model calculation of the bottomonium spectrum with self energy corrections due to the coupling to the meson-meson continuum. The bare masses used in the calculation are computed within the relativized quark model by Godfrey and Isgur. We also give results for the open-bottom strong decay amplitudes of higher bottomonia in a 3P0 model.

Linear confinement in momentum space: Singularity-free bound-state equations

Relativistic equations of Bethe-Salpeter type for hadron structure are most conveniently formulated in momentum space. The presence of confining interactions causes complications because the corresponding kernels are singular. This occurs not only in the relativistic case but also in the nonrelativistic Schr\"odinger equation where this problem can be studied more easily. For the linear confining interaction the singularity reduces to one of Cauchy principal value form. Although this singularity is integrable, it still makes accurate numerical solutions difficult. We show that this principal value singularity can be eliminated by means of a subtraction method. The resulting equation is much easier to solve and yields accurate and stable solutions. To test the method's numerical efficiency, we performed a three-parameter least-squares fit of a simple linear-plus-Coulomb potential to the bottomonium spectrum.

Unraveling the pattern of the XYZ mesons

We present a plausible mechanism for the origin of the XYZ mesons in the heavy meson spectra within a standard quark-model picture. We discuss the conditions required for the existence of four-quark bound states or resonances contributing to the heavy meson spectra, being either compact or molecular. We concentrate on charmonium and bottomonium spectra, where several new states, difficult to understand as simple quark–antiquark pairs, have been reported by different experimental collaborations. The pivotal role played by entangled meson–meson thresholds is emphasized.

Generalized screened potential model

A new non relativistic quark model to calculate the spectrum of heavy quark mesons is developed. The model is based on an interquark potential interaction that implicitly incorporates screening effects from meson-meson configurations. An analysis of the bottomonium spectrum shows the appearance of extra states as compared to conventional non screened potential models.

The bottomonium spectrum at finite temperature from N f = 2 + 1 lattice QCD

We present results on the bottomonium spectrum at temperatures above and below the deconfinement crossover temperature, $T_c$, from dynamical lattice QCD simulations. The heavy quark is treated with a non-relativistic effective field theory on the lattice and serves as a probe of the hot medium. Ensembles with a finer spatial lattice spacing and a greater range of temperatures below $T_c$ than those previously employed by this collaboration are used. In addition, there are $N_f=2+1$ flavours of Wilson clover quark in the sea with $M_\pi\approx400$ MeV and we perform a more careful tuning of the bottom quark mass in this work. We calculate the spectral functions of S and P wave bottomonium states using the maximum entropy method and confirm earlier findings on the survival of the ground state S wave states up to at least $2T_c$ and the immediate dissociation of the P wave states above $T_c$.

The vector coupling α V((r) and the scales r 0, r 1 in the background perturbation theory

We study the universal static potential V st(r) and the force, which are fully determined by two fundamental parameters: the string tension σ = 0.18 ± 0.02 GeV2 and the QCD constants $$\Lambda _{\overline {MS} } (n_f )$$ , taken from pQCD, while the infrared (IR) regulator M B is taken from the background perturbation theory and expressed via the string tension. The vector couplings α V(r) in the static potential and α F(r) in the static force, as well as the characteristic scales, r 1(n f = 3) and r 0(n f = 3), are calculated and compared to lattice data. The result $$r_0 \Lambda _{\overline {MS} } (n_f = 3) = 0.77 \pm 0.03$$ , which agrees with the lattice data, is obtained for M B = (1.15 ± 0.02) GeV. However, better agreement with the bottomonium spectrum is reached for a smaller $$\Lambda _{\overline {MS} } (n_f = 3) = (325 \pm 15)$$ MeV and the frozen value of α V = 0.57 ± 0.02. The mass splittings $$\bar M(1D) - \bar M(1P)$$ and $$\bar M(2P) - \bar M(1P)$$ are shown to be sensitive to the IR regulator used. The masses M(1 3 D 3) = 10169(2) MeV andM(1 3 D 1) = 10155(3) MeV are predicted.

Perturbative heavy quarkonium spectrum at next-to-next-to-next-to-leading order

We compute the energy levels of some of the lower-lying heavy quarkonium states perturbatively up to O(αs5m) and O(αs5mlogαs). Stability of the predictions depends crucially on the unknown 4-loop pole-MS¯ mass relation. We discuss the current status of the predictions with respect to the observed bottomonium spectrum.

Higher bottomonia in the unquenched quark model

We show our results for the bottomonium spectrum with self energy corrections, due to the coupling to the meson-meson continuum. We also discuss our results for the open bottom strong decays of higher bottomonia in the 3 P 0 pair-creation model.

Bottomonia: open bottom strong decays and spectrum

We present our results for the bottomonium spectrum with self energy corrections. The bare masses used in the calculation are computed within Godfrey and Isgur's relativized quark model. We also discuss our results for the open bottom strong decay widths of higher bottomonia in the 3 P0 pair-creation model.

Renormalized quarkonium

We extend our previous study of the vector charmonium states within a renormalization approach with boundary conditions to the full spectrum of charmonium and bottomonium. In light of the predicted spectrum we comply with assignments suggested in the literature. A comparison with the regularized quark model is also included.

Thermal modification of bottomonium spectra from QCD sum rules with the maximum entropy method

The bottomonium spectral functions at finite temperature are analyzed by employing QCD sum rules with the maximum entropy method. This approach enables us to extract the spectral functions without any phenomenological parametrization, and thus to visualize deformation of the spectral functions due to temperature effects estimated from quenched lattice QCD data. As a result, it is found that \Upsilon and \eta_b survive in hot matter of temperature up to at least 2.3T_c and 2.1T_c, respectively, while \chi_{b0} and \chi_{b1} will disappear at T<2.5T_c. Furthermore, a detailed analysis of the vector channel shows that the spectral function in the region of the lowest peak at T=0 contains contributions from the excited states, \Upsilon(2S) and \Upsilon(3S), as well as the ground states \Upsilon (1S). Our results at finite T are consistent with the picture that the excited states of bottomonia dissociate at lower temperature than that of the ground state. Assuming this picture, we find that \Upsilon(2S) and \Upsilon(3S) disappear at T=1.5-2.0T_c.

Bottomonium spectrum with coupled-channel effects

We study the bottomonium spectrum in the nonrelativistic quark model with the coupled-channel effects. The mass shifts and valence $b\bar{b}$ component are evaluated to be rather large. We find that the hadronic loop effects can be partially absorbed into a reselection of the model parameters. No bottomonium state except $\Upsilon(5\,^1S_0)$ and $\Upsilon(5\,^3S_1)$ with mass around 10890 MeV is found in the quark models both with and without coupled-channel effects, so we suggest that $Y_{b}(10890)$ is an exotic state beyond the quark model, if it is confirmed to be a new resonance. The predictions for the $\chi_b(3P)$ masses are consistent with the ATLAS measurements. If some new bottomonium-like states are observed at LHCb or SuperB in the future, we can determine whether they are conventional bottomonium or exotic states by comparing their masses with the mass spectrum predicted in our work.

χ&amp;lt;sub&amp;gt;c&amp;lt;/sub&amp;gt; and χ&amp;lt;sub&amp;gt;b&amp;lt;/sub&amp;gt; States in Hot Quark-Gluon Plasma

We have studied the dissociation phenomenon of 1p states (χc and χb) of the charmonium and bottomonium spectra in a hot QCD medium. This study employed a medium modified heavy quark potential encoding the medium effects in the dielectric function to the full Cornell potential. The medium modified potential has a quite different form in the sense that it has a long range Coulomb tail in addition to the usual Yukawa term even above the deconfinement temperature. We further study the flavor dependence of their binding energies and explore the nature of dissociation by employing the perturbative, non-perturbative, and the lattice parametrized form of the Debye masses in the medium-modified potential. Interestingly, perturbative result of the Debye mass predicts the dissociation temperatures closer to the results obtained in lattice correlator studies whereas the lattice parametrized form of the Debye masses gives the results closer to the current theoretical works based on potential studies.

Quark-antiquark potential to order1/mand heavy quark masses

An updated heavy quark-antiquark potential is constructed by matching the short-distance perturbative part to long-distance lattice QCD results at an intermediate r scale. The static potential and the order 1/m potential are both analyzed in this way. Effects if order 1/m in charmonium and bottomonium spectra are discussed in comparison. Charm and bottom quark masses are deduced from the spectra and related to the quark masses of other schemes.

Bottomonium spectrum at orderv6from domain-wall lattice QCD: Precise results for hyperfine splittings

The bottomonium spectrum is computed in dynamical 2+1 flavor lattice QCD, using NRQCD for the b quarks. The main calculations in this work are based on gauge field ensembles generated by the RBC and UKQCD collaborations with the Iwasaki action for the gluons and a domain-wall action for the sea quarks. Lattice spacing values of approximately 0.08 fm and 0.11 fm are used, and simultaneous chiral extrapolations to the physical pion mass are performed. As a test for gluon discretization errors, the calculations are repeated on two ensembles generated by the MILC collaboration with the Luscher-Weisz gauge action. Gluon discretization errors are also studied in a lattice potential model using perturbation theory for four different gauge actions. The nonperturbative lattice QCD results for the radial and orbital bottomonium energy splittings obtained from the RBC/UKQCD ensembles are found to be in excellent agreement with experiment. To get accurate results for spin splittings, the spin-dependent order-v^6 terms are included in the NRQCD action, and suitable ratios are calculated such that most of the unknown radiative corrections cancel. The cancellation of radiative corrections is verified explicitly by repeating the calculations with different values of the couplings in the NRQCD action. Using the lattice ratios of the S-wave hyperfine and the 1P tensor splitting, and the experimental result for the 1P tensor splitting, the 1S hyperfine splitting is found to be 60.3+-5.5(stat)+-5.0(syst)+-2.1(exp) MeV, and the 2S hyperfine splitting is predicted to be 23.5+-4.1(stat)+-2.1(syst)+-0.8(exp) MeV.

Quarkonium mass splittings in three-flavor lattice QCD

We report on calculations of the charmonium and bottomonium spectrum in lattice QCD. We use ensembles of gauge fields with three flavors of sea quarks, simulated with the asqtad improved action for staggered fermions. For the heavy quarks we employ the Fermilab interpretation of the clover action for Wilson fermions. These calculations provide a test of lattice QCD, including the theory of discretization errors for heavy quarks. We provide, therefore, a careful discussion of the results in light of the heavy-quark effective Lagrangian. By and large, we find that the computed results are in agreement with experiment, once parametric and discretization errors are taken into account.