Spectrum and decay properties of bottomonium mesons

Abstract

We calculate the spectrum and wave functions (WFs) of various states of bottomonium($b\overline{b}$) mesons using a non-relativistic quark potential model (NRQPM). The calculated WFs are used to compute the radiative widths of various states of $b\overline{b}$. The strong decays widths of bottomonium states are also calculated using $^3P_0$ model by choosing simple harmonic oscillator wave functions (SHOWFs). The $\beta$ of SHOWFs for various states of the mesons are measured by fitting the numerical wave functions. The radiative and strong decay widths are used to calculate the branching ratios of $b\overline{b}$ mesons. We also compare our calculated masses and widths with available experimental data. On basis of this comparison, it is suggested that $\Upsilon (10860)$ is the 5S state of $b\overline{b}$ mesons.

1 Introduction

Upsilon($\Upsilon$), a state of bottomonium meson, was observed first time in E288 experiment at Fermilab [1] in 1977. The next newly discovered state of $b\overline{b}$ was the 3P state that was observed in Large Hadron Collider (LHC) in 2011 [2, 3]. Uptill now many states of $b\overline{b}$ mesons have been observed in experiments at BaBar, Belle, CDF, D0, ATLAS, CMS and LHCb with lowest state mass equal to $9.3909 \pm 0.0028$ GeV and highest state mass equal to $11.019\pm 0.008$ GeV. Now a days, physicists are taking more interest in the investigation of the highest $b\overline{b}$ states like $\Upsilon (4S), \Upsilon (10860), \Upsilon (11020)$, and $\Upsilon ((10750)$. $\Upsilon (4S), \Upsilon (10860), \Upsilon (11020)$ are discovered in 1980s at CUSB [4]. Among these highly excited $b\overline{b}$ states, study on the $\Upsilon (4S)$ is particularly most important for the investigation of B meson [5] because of its mass which is equal to $10885.3 \pm 1.5_{-0.9}^{2.2}$ MeV [6]. This value of mass is above the $B\overline{B}$ threshold energy, but it is too small that can not produce exciting $B^{*}$ mesons or any other extra particles. This unique quality is beneficial in the production of B mesons and collision of these mesons can produce pairs of B and $B_s$ mesons. Study on these strong decays is very helpful for the advance particle colliders [7, 8].

For theoretical investigation of the experimentally observed states of $b\overline{b}$ mesons and to predict the new states of $b\overline{b}$, different approaches have been used. Martin-like potential model is used in Ref. [9] to calculate the masses and leptonic widths of $b\overline{b}$ and $c\overline{c}$ mesons. Relativistic quark potential model [10,11,12,13,14] is used in Refs. [15, 16] to calculate the masses and decay properties of $b\overline{b}$ mesons. Constituent quark model with the incorporation of spin dependant interaction is used in Ref. [17] to calculate the masses and leptonic widths of various states of $b\overline{b}$ and $c\overline{c}$ mesons. Non-relativistic quark model [18] is used to calculate the masses and decays of $b\overline{b}$ mesons in Refs. [19,20,21,22].

In Refs. [21, 22], spectrum and electromagnetic transitions of $b\overline{b}$ mesons are calculated using non-relativistic screened potential model. In Ref. [22], hadronic transitions of $b\overline{b}$ are calculated with non-relativistic constituent quark model with the Gaussian Expansion Method for $nS (n \le 3), nP (n \le 3)$, and $nD (n \le 2)$ states. In Ref. [21], Schrodinger equation is solved by three-point difference central method to find spectrum. In comparison to Refs. [21, 22], we use shooting method to solve the Schrodinger equation. At short distances, wavefunction becomes unstable due to $\frac{1}{r^3}$ term in the potential model. To solve this problem, we employed the smearing of position coordinates, as discussed in Ref. [23]. In addition, in Ref. [21], electromagnetic transitions are calculated for $nS (n \le 4), nP (n \le 3)$, and $nD (n \le 2)$, while our predictions are for wide range $nS (n \le 5), nP (n \le 4), nD (n \le 4)$ and 1F.

Fig. 1

Decay diagrams in the $^3P_0$ model

Table 1 Masses of ground and excited states of bottomonium mesons. The SHO $\beta$ values are listed in the last column which are obtained by fitting SHO wave functions to the quark model wavefunctions

In this paper, we calculate the masses, radiative transitions, strong decays and branching ratios of $b\overline{b}$ meson. Since bottom quarks are heavier, so $b\overline{b}$ mesons must be non-relativistic. Due to non-relativistic behaviour of $b\overline{b}$ mesons, spectrum can be well studied through non-relativistic approximations. In Refs. [19,20,21,22, 24], properties of heavy mesons are studied through non-relativistic quark potential models which give well satisfactory results. This proves that the approach of using non-relativistic potential models is successful for the study of heavy systems. Due to this reason, we use non-relativistic quark potential model in the columbic plus linear form alongwith the incorporation of spin-spin and spin-angular momentum interactions to find the masses and WFs of $b\overline{b}$ mesons. Parameters are found by fitting the experimentally available masses of $b\overline{b}$, bottom and bottom-strange mesons with the model calculated masses by taking different values for coupling constant for each sector. The calculated WFs are used to calculate the E1 and M1 radiative widths. Strong decay widths are calculated with simple harmonic oscillator wave function (SHOWF) using $^3 P_0$ model for ground and excited states of $b\overline{b}$ mesons. SHOWF depends on the parameter $\beta$.

Table 2 Masses and SHO $\beta$ values of B and $B_s$ mesons used in calculations of strong decay widths

Table 3 Flavor factors for bottomonium decay, where $|X_b\rangle \equiv |b\bar{b}\rangle$

Table 4 Partial widths of radiative transitions and strong decays for 1 S, 2 S and 3 S bottomonium mesons

Table 5 Partial widths of radiative transitions and strong decays for 4 S bottomonium mesons

Table 6 Partial widths of radiative transitions and strong decays for 5 S bottomonium mesons

Table 7 Partial widths of radiative transitions and strong decays for 1P bottomonium mesons

Table 8 Partial widths of radiative transitions and strong decays for 2P bottomonium mesons

Table 9 Partial widths of radiative transitions and strong decays for 3P bottomonium mesons

Table 10 Partial widths of radiative transitions and strong decays for 1D bottomonium mesons

Table 11 Partial widths of radiative transitions and strong decays for 2D bottomonium mesons

Table 12 Partial widths of radiative transitions and strong decays for 3D bottomonium mesons

Table 13 Partial widths of radiative transitions and strong decays for 4D bottomonium mesons

Table 14 Partial widths of radiative transitions and strong decays for 1F bottomonium mesons

In Ref. [25], strong decays for open charm and open bottom flavour mesons are calculated by taking same value of $\beta$ for different flavoured mesons, but in the present paper, strong decay widths of all angularly excited $b\overline{b}$ states are calculated using different values of $\beta$ for different flavoured states. Authors of Ref. [15] used different values of parameter $\beta$ for different states of $b\overline{b}$ mesons in the calculation of decay properties. They found $\beta$ by fitting the RMS radii of SHOWF to the corresponding WF of relativitic quark potential model. In comparison to Ref. [15], we find $\beta$ by fitting SHOWF with the numerically calculated WFs of non-relativistic potential model, which is more accurate method. These two techniques of finding $\beta$ values are compared in our earlier work  [26, 27]. The Ref. [26] shows that the error is less for the $\beta$ fitted by numerical wavefunction as compared to $\beta$ fitted through RMS radius. Furthermore, the strong decay widths are sensitive to the choice of $\beta$ parameter [26] and we are using the most accurate technique to find the $\beta$ values for initial and final mesons. We combine radiative and strong widths to predict the branching ratios of all possible decay channels of $b\overline{b}$ states. In addition, we are studying the spectrum and decay properties of $b\overline{b}$ mesons by incorporating the uncertainties in our parameters.

The paper is organized as follows. In Sect. 2, the potential model is described which is used to calculate the mass and WF of different states of $b\overline{b}$ mesons. In Sect. 3, the expressions used for E1 and M1 radiative transitions are defined. The methodology for the calculation of the strong decay amplitudes using $^3P_0$ decay model is explained in Sect. 4. Results are discussed in Sect. 5 alongwith the concluding remarks.

2 Potential model for bottomonium, charmed bottom and bottom mesons

Following non-relativistic quark anti-quark potential model [14] is used to find the mass spectrum and WFs of $b\overline{b}$, strange-bottom and bottom mesons.

$$\begin{aligned} V_{q\bar{q}}(r)= & {} \frac{-4\alpha _{s}}{3r}+br+\frac{32\pi \alpha _{s}}{ 9 m_{q} m_{\bar{q}}}\left( \frac{\sigma }{\sqrt{\pi }}\right) ^{3}e^{-\sigma ^{2}r^{2}}\textbf{S}_{q}.\textbf{S}_{\bar{q}} \nonumber \\{} & {} \quad +\frac{1}{m_{q} m_{\bar{q}}}\left[ \left( \frac{2\alpha _{s}}{r^{3}}-\frac{b}{2r}\right) \textbf{L}.\textbf{S}+\frac{4\alpha _{s}}{r^{3}}T\right] . \end{aligned}$$

(1)

$m_{q}$, $m_{\bar{q}}$ are the constituent masses of quark and anti-quark respectively. $\alpha _{s}$ is the strong coupling constant, b is the string tension. Columbic interactions, spin-orbit interactions at short distance, and tensor interactions are the result of one gluon exchange process; while spin-orbit interactions at large distances are the result of Lorentz scalar confinement. The spin-spin $\textbf{S}_{b}.\textbf{S}_{\bar{b}}$, spin-orbit $\textbf{L}.\textbf{S}$, and tensor operators in $\left| J, L, S\right\rangle$ basis are given by

$$\begin{aligned} T&=\left\{ \begin{array}{c} -\frac{1}{6(2L+3)},J=L+1 \\ +\frac{1}{6},J=L \\ -\frac{L+1}{6(2L-1)},J=L-1. \end{array}\right. \end{aligned}$$

(2)

The values of parameters $\alpha _{s}$, b, $\sigma$, $m_{q}$, $m_{\bar{q}}$ are found by fitting the mass spectrum of bottomonium, strange-bottom and bottom mesons to the available experimental data of masses. This available data consists of eighteen states of bottomonium mesons, four states of strange-bottom meson and four states of bottom mesons given in Tables 1 and 3. The best fit values of these parameters are $b=0.1139\text { GeV}^{2}$, $\sigma =0.6$ GeV, $m_{b}=4.825$ GeV, $m_{s}=0.41$ GeV, $m_{u}= m_{d}=0.365$ GeV, $\alpha _{s}(b \overline{b})= 0.3339$, $\alpha _{s}(\textbf{B}_s) = 0.738$, and $\alpha _{s}(\textbf{B})= 0.92$. To calculate the spectrum of various states of $b\overline{b}$ system we numerically solved the radial Schr$\ddot{\text {o}}$dinger equation given by

$$\begin{aligned} U^{\prime \prime }(r)+2\mu \left( E-V(r)-\frac{L(L+1)}{2\mu r^{2}}\right) U(r)=0, \end{aligned}$$

(3)

$\mu$ is the reduce mass of meson. Non-trivial solutions of the above equation, existing only for certain discrete values of energy (E), are found by the shooting method. Mass of a $b \overline{b}$ state is found by following expression:

$$\begin{aligned} m_{b\bar{b}}=2m_{b}+E, \end{aligned}$$

(4)

3 Radiative transitions

Radiative transitions are important to investigate the higher states of $b\overline{b}$ mesons. E1 radiative transitions from a $b\overline{b}$ meson to other $b\overline{b}$ meson state are calculated by using the following expression defined in Ref. [14].

$$\begin{aligned}{} & {} \Gamma _{E1}(n^{2S+1}L_J\rightarrow n'^{2S'+1}L'_{J'}+\gamma )\nonumber \\{} & {} \quad =\frac{4}{3}C_{fi}\delta _{S S'}e_b^2 \alpha \mid < \Psi _f \mid r \mid \Psi _i>\mid ^2 E_\gamma ^3 \frac{E^{(b\overline{b})}_f}{M^{(b \overline{b})}_i}. \end{aligned}$$

(5)

Here $E_\gamma$, $E^{b \overline{b}}_f$, and $M_i$ stand for final photon energy ($E_\gamma = \frac{M_i^2 - M_f^2}{2 M_i}$), energy of the final $b\bar{b}$ meson, and mass of initial state of $b\overline{b}$ meson respectively, and

$$\begin{aligned} C_{fi}=max(L, L')(2 J'+1)\left\{ \begin{array}{ccc} L' &{} J' &{} S \\ J &{} L &{} 1 \\ \end{array} \right\} ^2. \end{aligned}$$

(6)

M1 radiative transitions for a $b\overline{b}$ meson state to other $b\overline{b}$ meson state are calculated by the following expression [14]:

$$\begin{aligned}{} & {} \Gamma _{M1}(n^{2S+1}L_J\rightarrow n'^{2S'+1}L'_{J'}+\gamma )=\frac{4}{3}\frac{2J'+1}{2 L+1}\nonumber \\{} & {} \qquad \delta _{L L'}\delta _{S S'\pm 1}e_b^2 \frac{\alpha }{m^2_b}\mid < \Psi _f \mid \Psi _i>\mid ^2 E_\gamma ^3\frac{E^{(b \overline{b})}_f}{M^{(b \overline{b})}_i}. \end{aligned}$$

(7)

4 Open flavor strong decays

We calculate strong decay widths for the states above $B\overline{B}$ threshold using $^3P_0$ model. In the $^3P_0$ model, the open-flavor strong decay of a meson $(A\rightarrow B+C)$ take place through the production of quark anti-quark pair with vacuum quantum numbers ($J^{PC}=0^{++}$) [29]. The produced quark anti-quark pair combines with the quark anti-quark of initial meson A to gives the final mesons B and C. The interaction Hamiltonian for the $^3P_0$ model in nonrelativistic limit is

$$\begin{aligned} H_I=2 m_q \gamma \int d^3{\textbf {x}}\;\overline{\psi }_q({\textbf {x}}) \psi _q({\textbf {x}}), \end{aligned}$$

(8)

where $\psi$ is the Dirac quark field and $\gamma$ is the pair-production strength parameter. We use $\gamma = 0.33$ that obtained from a fit of experimentally known strong decay widths of bottomonium states. The quark anti-quark pair production takes place through $b^{\dag }d^{\dag }$ term in the Hamiltonian

$$\begin{aligned} H_I=2m_q\gamma \int d^3k[\overline{u}(\textbf{k},s)v(\mathbf {-k},\overline{s})]b^{\dag }({\textbf {k}},s)d^{\dag }(-{\textbf {k}},\overline{s}), \end{aligned}$$

(9)

where $b^{\dag }$ and $d^{\dag }$ are the creation operators for quark and antiquark respectively. This interaction Hamiltonian is used to calculate the matrix element $\langle BC|H_I|A\rangle$ for a process $A\rightarrow B+C$. There are two diagrams contribute in the matrix element, shown in Fig. 1. The flavor factors for each diagram along with multiplicity factor $\mathcal {F}$ for all the processes discussed in this work are reported in Table 3. The combined matrix element of both diagrams gives the decay amplitude

$$\begin{aligned} \mathcal {M}_{LS}=\langle j_A,L_{BC},S_{BC}|BC\rangle \langle BC|H_I|A\rangle /\delta (\textbf{A}-\textbf{B}-\textbf{C}). \end{aligned}$$

(10)

The decay width of the process $A \rightarrow B+C$ can be calculated by combining the decay amplitude $(\mathcal {M}_{LS})$ with a relativistic phase space as [30]

$$\begin{aligned} \Gamma _{A\rightarrow BC}=2\pi \frac{P E_BE_C}{M_A}\sum _{LS} |\mathcal {M}_{LS}|^2, \end{aligned}$$

(11)

where $P=|{\textbf {B}}|=|{\textbf {C}}|$ in the center-of-mass of the initial meson-A, $M_A$ is the mass of this initial meson, $E_B$ and $E_C$ are the energies of the final mesons B and C respectively. We use experimental masses of mesons if available; otherwise our theoretically calculated masses of mesons from Table 1 are used. The masses of the final state mesons B and $B_s$ are reported in Table 2. The detailed formulism to calculate the strong decay amplitude by using the $^3P_0$ model is described in our earlier work [26, 27].

In this work, we have computed strong decay widths of kinematically allowed open-flavor decay modes of all the bottomonium states mentioned in Table 1 using the $^3P_0$ model. We use simple harmonic oscillator (SHO) wavefunctions as wavefunctions of initial and final mesons in the momentum space calculations of matrix element $\langle BC|H_I|A\rangle$. The SHO scale $\beta$ for initial and final mesons is taken as parameter of the $^3P_0$ model. In this paper, we fit $\beta$ parameter of SHO wavefunctions to the numerical wavefunctions obtained by solving radial Schrödinger equation. Our fitted $\beta$ values for the initial $b\overline{b}$ mesons are reported in column-5 of Table 1. The $\beta$ values for the final B and $B_s$ mesons appearing in strong decays of higher states of $b\overline{b}$ mesons are mentioned in Table 2.

5 Results and discussion

We use the non-relativistic quark potential model to calculate the numerical wave functions and masses of bottomonium mesons. The mass spectrum of bottomonium mesons are calculated upto 2F energy states. A comparison of our predicted spectrum with recent theoretical studies and experimental data is reported in Table 1.

Percentage error is calculated in masses by changing the parameters of the potential model ($\alpha$, $m_b$,$\sigma$, b). It is found that the relative error in masses of bottomonium states is (0.93–0.98)% for a change of $\pm 1 \%$ in parameters.

Our theoretical masses of bottomonium states Table 1 show that 1 S, 2 S, 3 S and 4 S lying below the $B\overline{B}$ threshold ($\approx 10.558$ GeV). Our theoretical mass of $4^3S_1$ is $10.437 \text { GeV}$ lying below threshold but its experimental mass is $10.5794\pm 0.0012 \text { GeV}$ which is very close to $B\overline{B}$ threshold. Our predicted width of $4^3S_1$ is $20.645\text { MeV}$ which is in good agreement with experimental width $20.5\pm 2.5\text { MeV}$. Our predicted width is also in good agreement with other theoretical predictions 22 MeV [15] and 24.7 MeV [16].

The $\eta _b(5^1S_0)$ is not an established state and its predict mass is 10.6069GeV which is above threshold. According to spin selection rules and energy conservation $\eta _b(5^1S_0)$ has four open-bottom decay channels: $BB^*$, $B^*B^*$, $B_sB_s^*$ and $B_s^*B_s^*$. The predicted width of $\eta _b(5^1S_0)$ is $52.894\text { MeV}$ which is comparable to other theoretical studies 23 MeV [15] and 28.4 MeV [16].

The $\Upsilon (5^3S_1)$ has six open-bottom decay channels: $B\overline{B}$, $BB^*$, $B^*B^*$, $B_sB_s$ $B_sB_s^*$ and $B_s^*B_s^*$ with predicted width $50.47\text { MeV}$. This is $13\text { MeV}$ greater than the experimental measured width which is equal to $37\pm 4\text { MeV}$. Our predicted width is close to recent theoretical predictions of 45.6 MeV [16]. while Ref. [15] calculated 27.4 MeV.

The 1P and 2P bottomonium states are experimentally established but lying below $B\overline{B}$ threshold, therefore only radiative widths are calculated. The experimental masses of two multiplets of 3P bottomonium states are available whereas the masses of other two are not available experimentally. The masses of three multiplets of 4P states are below threshold whereas the mass of $4^3P_2$ is very close to the $B\overline{B}$ threshold and has very small width of 0.01MeV which is not included in the tables. But according to Ref. [15, 16], the masses of all multiplets of 4P states are above threshold. This difference in the theoretical mass predictions is due to the different techniques used in spectrum calculations.

The theoretical masses of 1D, 2D and 3D bottomonium states show that these states are below the $B\overline{B}$ threshold whereas 4D states are above threshold. The $4^1D_2$ state decays strongly through $BB^*$ decay mode only with total predicted width is $4.839\text { MeV}$. The $4^3D_1$ state has two open-bottom decay modes $B\overline{B}$ and $BB^*$ with total predicted width is $3.2\text { MeV}$. The predicted width of $4^3D_2$ multiplet is $3.41\text { MeV}$ with $BB^*$ decay mode only. The $4^3D_3$ bottomonium state can decay strongly through $B\overline{B}$, $BB^*$ and $B^*B^*$ decay channels with total predicted width is $6.12\text { MeV}$. Our predicted widths of 4D states are not in good agreement with Refs. [15, 16] due to our predicted masses which are small as compared to these studies.

We have also included the theoretical masses of 1F and 2F bottomonium states in Table 1 even that the higher states of bottomonium states are not experimentally established. According to our theoretical predictions 1F and 2F states are lying below $B\overline{B}$ threshold and can decay through E1 and M1 transitions only.

Our predicted widths in Tables 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14 show that the M1 radiative widths are very small, but E1 radiative widths are higher values up to 21.75 keV. The reason of this difference is that M1 radiative widths depend on the factor $(\frac{1}{m^2_b})$ while this factor is not used in the calculation of E1 radiative widths. Tables 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 and 14 show that the branching ratios of radiative widths are high below threshold, while the branching ratios of radiative widths decrease above threshold because of the existence of strong decays. Similar behavior is observed in Refs. [15, 16].

The relative percentage error in radiative widths of bottomonium mesons due to $\pm 1 \%$ change in parameters is calculated. It is found that $S\longrightarrow P$, $P\longrightarrow S$, $P\longrightarrow D$, $D\longrightarrow P$, and $D\longrightarrow F$ E1 transitions have relative percentage errors upto 3.37, 3.07, 9.43, 1.997, and 17.88 respectively; while the percentage error in M1 radiative widths varies from 0.0197 to 0.054. Relative percentage error in strong decay width for S, P, and D meson decays is found to be upto 15.5$\%$, 33$\%$, and $9.3 \%$ due to $\pm 1\%$ change in parameters.

5.1 $\Upsilon (10860)$

$\Upsilon (10860)$ with $J^{PC}=1^{--}$ is the first observed resonance that can decay into B and $B_s$ mesons [31]. The experimental mass and decay widths of this state are $10885.2_{-1.6}^{+2.6}$ MeV [28] and $37\pm 4$ MeV. By observing the masses of $1^{--}$ state mesons (reported in Table 1), it is observed that masses of $\Upsilon (5^3S_1$ and $\Upsilon _1(4^3D_1)$ states are close to the experimental measured mass of $\Upsilon (10860)$. But our calculated decay width of $\Upsilon (5^3S_1)$ is more close to measured width of $\Upsilon (10860)$. On basis of this comparison, we suggest that $\Upsilon (10860)$ is the 5S state of bottomonium meson. Similar prediction is suggested in Refs. [15, 16, 19, 32].