Properties of bottomonium in a relativistic quark model

Describing a bottomonium bound state problem, incorporating the elements of non perturbative low energy dynamics is a challenging problem. The two main approaches in this line have been effective field theoriesHG90 (); M97 (); NB05 () and lattice gauge theories. Though the calculations away from threshold have been proven successful, the threshold region remain still tricky. And hence a relativistic potential model would be a befitting one for our study of bottomonium states. Essentially in all phenomenological relativistic QCD base quark models, the Hamiltonian for the quark system consists of a two body confinement potential and Coulomb like potential which takes care of relativistic effects. The Hamiltonian in our model has the confinement potential and a two body confined one gluon exchange potential(COGEP) PC92 (); SB91 (); KB93 (); KB97 ().
The RHM equation is SB83 (); VH04 (); VB09 ():

The Dirac wave function $\psi$ is written as $\mid \psi ⟩=N\left(\begin{array}{c}\varphi \\ \chi \end{array}\right)$,where $\varphi$ is the large componet and $\chi$ is the small component for positive energy state solution in the non-relativistic limit. Using standard matrix form of $\alpha$ and $\beta$,the Dirac equation is expressed in terms of the large and small components.

Hence, solution of the full Dirac equation $\psi$ is written as

where ${\gamma }_0$ is the Dirac matrix, M is a constant mass and $A^2$ is the confinement strength.

with normalization constant

E is the eigenvalue of the single particle Dirac equation with interaction potential given by equation (LABEL:eq:A). We perform a unitary transformation to eliminate the lower component of $\psi$. With this transformation, $\varphi$ satisfies the ’harmonic oscillator’ wave equation

the eigenvalue of which is given by

where $N_{OSC}=2n+l$ is the oscillator quantum number and ${\Omega }_N$ is the energy dependent oscillator size parameter given by

The total energy or the mass of the meson is obtained by adding the individual contributions of the quarks. The spurious centre of mass (CM) is corrected GG57 () by using intrinsic operators for the ${\sum }_i{r}_i^2$ and ${\sum }_i{\nabla }_i^2$ terms appearing in the Hamiltonian. This amounts to just subtracting the CM motion zero contribution from the $E^2$ expression.

NRQM employing OGEP, which takes into consideration the confining effect of quarks on mesonic states, does not shed light on confinement of gluons. Hence one needs to incorporate confining effects of gluons on mesonic states since confined dynamics of gluons plays a decisive role in determining the hadron spectrum and in hadron-hadron interaction. There are various confinement models for the gluons. The effect of the confined gluons on the masses of mesons has been studied using the successful current confinement model (CCM). The confined gluon propagators (CGP) are derived in CCM to obtain the COGEP.

The spin orbit part of COGEP is

where ${\hat{P}}_1$, ${\hat{P}}_2$ are the momentum of quarks. The spin orbit term has been split into the symmetric $({\sigma }_1+{\sigma }_2)$ and anti symmetric $({\sigma }_1-{\sigma }_2)$ spin orbit terms.

The tensor part of the COGEP is,

where

The total potential is then

We then obtain Hamiltonian by adding interaction potential given by equation 15 to the the ’harmonic oscillator’ wave equation 6,

We consider two types of radiative transitions of the $b\overline{b}$ meson:

The harmonic oscillator wave function allows the separation of the motion of the center of mass and has been widely used to classify the spectra of baryons and mesons FD69 (); RP71 () and extending to nucleon-nucleon interaction is straight forward KS89 (); FA83 (); KB93 (). If the basic states are the harmonic oscillator wave functions, then it is straightforward to evaluate the matrix elements of few body systems such as mesons or baryons. Since the basic states are the products of the harmonic oscillator wave functions they can be chosen in a manner that allows the product wave functions to be expanded as a finite sum of the corresponding products for any other set of Jacobi coordinates. It is advantageous to use the Gaussian form since in the annihilation of quark-anti quark into lepton pairs, the amplitude of the emission or absorption processes depends essentially on the overlap of initial and final hadrons and hence the overlap depends only on the intermediate distance region of the spatial wave functions which can extend up to 0.5 fm. This intermediate region can be described by potentials that are similar in this region and hence harmonic oscillator wave functions are expected to reproduce emission and absorption processes quite well.

The five parameters used in our model are the mass of beauty quark $m_b$, the harmonic oscillator size parameter $b$, the confinement strength $A^2$, the CCM parameter $c$ and the quark-gluon coupling constant ${\alpha }_s$. There are several papers in literature where the size parameter $b$ is defined SN86 (); IM92 (). The value of $b$ is fixed by minimizing the expectation value of the Hamiltonian for the vector meson. To start with, we construct the $5\times 5$ Hamiltonian matrix for $b\overline{b}$ states in the harmonic oscillator basis. The confinement strength $A^2$ is fixed by the stability condition for variation of mass of the meson against the size parameter $b$.

The parameter $c$ in CCM PC92 (); SB85 (); SB87 () was obtained by fitting the iota (1440 MeV)$0^{-+}$ as a digluon glue ball.To fit ${\alpha }_s$ , $m_b$ we start with a set of reasonable values and diagonalize the matrix for $b\overline{b}$ meson. Then we tune these parameters to obtain an agreement with the experimental value for the mass of $b\overline{b}$ meson. In literature we find different sets of values for $m_b$, which are listed in Table 1.

The values of strong coupling constant ${\alpha }_s$ in literature are listed in Table 2. The value of strong coupling constant (${\alpha }_s$) used is compatible with the perturbative treatment.

The mass spectrum has been obtained by diagonalizing the Hamiltonian in a large basis of $5\times 5$ matrix. The calculation clearly indicates that masses for both pseudo scalar and vector mesons converge to the experimental values when the diagonalization is carried out in a larger basis. In our earlier work also, we had come to the similar conclusion while investigating light meson spectrumVH04 (); BK05 (). The diagonalization of the Hamiltonian matrix in a larger basis leads to the lowering of the masses and justifies the perturbative technique to calculate the mass spectrum. The calculation clearly indicates that when diagonalization is carried out in a larger basis convergence is achieved both for pseudo-scalar mesons and vector mesons to the respective experimental values.
We use the following set of parameter values.

The calculated masses of the $b\overline{b}$ states after diagonalization are listed in Table  3. Our calculated mass value for ${\eta }_b\left(1S\right)$ is 9399 MeV which agrees with the experimental value 9398.0$\pm 3.2$ MeVPDG () and for ${\rm Y}$(1S) is 9460.30 $\pm 0.26$. $J/\psi$(1S) is heavier than ${\eta }_b\left(1S\right)$ by 61 MeV. This difference is justified by calculating the ${}^3{S}_1-{}^1{S}_0$ splitting of the ground state which is given by

The experimental value of hyperfine mass splitting for the ground state is given by PDG ()

The hyperfine mass splitting calculated in our model is in good agreement with both experimental dataPDG () and lattice QCD result ($60.3\pm 7.7)$ MeVSM10 (). The EFTs have predicted a hyperfine mass splitting of $41\pm {11}_{-8}^{+9}$ MeVBA04 () which is lower than the experimental value.

We predict a mass of 10001.12 MeV for the ${\eta }_b\left(2S\right)$ state which is in good agreement with the experemental value $9999.0\pm 3.5$MeV PDG (). The calculated mass of spin triplet state(${\rm Y}\left(2S\right)$) of ${\eta }_b\left(2S\right)$ is 10024.50 MeV which is found to be in good agreement with the experemntal value $10023.26\pm 0.31$ MeV. The hyperfine mass splitting for the 2S state ${\Delta }_{hf}\left({\eta }_b\right(2S\left)\right)=23.38$ MeV is in good agreement with both the experemental one ${\Delta }_{hf}\left({\eta }_b\right(2S\left)\right)={24.3}_{-4.5}^{+4.0}$ MeVPDG () and with the lattice QCD results ${\Delta }_{hf}\left({\eta }_b\right(2S\left)\right)=(23.5-28.0)$ MeVSM10 (). The predicted mass values for ${\eta }_b\left(3S\right)$ and ${\rm Y}\left(3S\right)$ states in our model are 10375.96 MeV and 10392.00 MeV respectively. The corresponding hyperfine mass splitting is ${\Delta }_{hf}\left({\eta }_b\right(3S\left)\right)=16.04$ MeV. We have also claculated masses of higher radially excited states of ${\eta }_b$ and its spin triplet state ${\rm Y}$. The mass of spin singlet P wave state $h_b\left(1P\right)$ calculated in our model is compatible with the experimental one. The masses of spin triplet states ${\chi }_{bJ}$ are in good agreement with experimental values of masses of these states. Our prediction for masses of higher excited P wave states are 10-20 MeV higher than the experimental values and other model calculations. Some of the higher excited states are 50-100 MeV heavier in our model.

Radiative decays of excited bottomonium states are the powerful tools which can be used to study the internal structure of $b\overline{b}$ states and they provide a good test for the predictions from the various models. Radiative transitions between different bottomonium states can be used to discover new bottomonium states experimentally.

The possible $E1$ decay modes have been listed in Table 4 and the predictions for E1 decay widths are given. Also our predictions have been compared with other theoretical models. Most of the predictions for $E1$ transitions are in qualitative agreement. However, there are some differences in the predictions due to differences in phase space arising from different mass predictions and also from the wave function effects particularly from relativization procedure. We find our results are compatible with other theoretical model values for most of the channels. Relativistic corrections to the wave functions tend to reduce the E1 transition widths for most of the channels. We find that the E1 transitions $3S\to 1P$ are suppressed compared to other E1 transitions and it is interesting to see that the E1 transition rates for $3S\to 1P$ are zero in the nonrelativistic caseEG08 (); GR92 (). We find that these E1 transisitions are very sensitive to relativistic effects.