The assignments of the bottom mesons within the screened potential model and ${}^{3}P_{0}$ model

Meson spectroscopy is one of the important subjects in hadron physics, and the heavy-light system offers an excellent laboratory for testing the heavy quark symmetry. As we known, most of the charmed and charmed-strange mesons are well established, although there exists some exotic explanations for some states, for example $D^{*}_{s0}(2317)$ and $D_{s1}(2460)$ Chen:2016spr ; Du:2017zvv . For the family of bottom mesons, the first bottom meson $B$ was observed in 1983 by the CLEO Collaboration CLEO:1983mma , and there are only six excited bottom mesons observed experimentally so far, which are $B^{*}$, $B_{1}(5721)$, $B^{*}_{J}(5732)$, $B_{J}(5840)$, $B_{J}(5970)$, and $B^{*}_{2}(5747)$ PDG2021 . Among those bottom states, only $B$ and $B^{*}$ are well assigned as the $S$-wave doublet ($1^{1}S_{0}$ and $1^{3}S_{1}$).

The $B^{*}_{J}(5732)$, as the first orbitally excited bottom meson, was observed in 1994 by the OPAL detector at LEP OPAL:1994hqv , and this state has not yet confirmed in the last two decadeds PDG2021 . It should be streesed that its signal can be interpreted as stemming from several narrow and broad resonances, as discussed in Review of Particle Physics (RPP) PDG2021 , thus we will not discuss this state. In 2007, the D0 Collaboration reported two narrow orbitally excited ($L=1$) bottom mesons, $B_{1}(5721)$ with $J^{P}=1^{+}$ and $B^{*}_{2}(5747)$ with $J^{P}=2^{+}$ D0:2007vzd , later confirmed by the CDF and LHCb Collaborations CDF:2008qzb ; CDF:2013www ; LHCb:2015aaf . The $B_{J}(5970)$ was first observed in 2013 by the CDF Collaboration CDF:2013www . In 2015 the LHCb Collaboration reproted two states $B_{J}(5840)$ and $B_{J}(5960)$ LHCb:2015aaf , where the latter one should be the same state as the $B_{J}(5970)$ since they have the similar properties. Here we list the masses, widths, and the quantum numbers of $B_{1}(5721)$, $B^{*}_{J}(5732)$, $B_{J}(5840)$, $B_{J}(5970)$, and $B^{*}_{2}(5747)$ in Table 1.

Based on the experimental measurements of the bottom mesons, there are many theoretical discussions in literatures Zeng:1994vj ; Godfrey:1985xj ; Lahde:1999ih ; DiPierro:2001dwf ; Zhong:2008kd ; Ebert:2009ua ; Godfrey:2016nwn ; Lu:2016bbk ; Kher:2017mky ; Asghar:2018tha ; Godfrey:2019cmi ; Yu:2019iwm ; Chen:2022fye ; li:2021hss ; Narison:2020wql . The $B^{*}_{2}(5747)$ is commonly regarded as the $B(1^{3}P_{2})$ state Ebert:2009ua ; Godfrey:2016nwn ; Lu:2016bbk ; Kher:2017mky ; Asghar:2018tha ; Godfrey:2019cmi ; Yu:2019iwm ; Chen:2022fye , and the $B_{1}(5721)$ is explained as the $B^{\prime}(1P_{1})$ in Refs. Lu:2016bbk ; Godfrey:2019cmi ; Yu:2019iwm ; Chen:2022fye , $B(1P_{1})$ in Refs. Kher:2017mky ; Asghar:2018tha .

Although the LHCb Collaboration has suggested that the $B_{J}(5840)$ and $B_{J}(5970)$ could be the $2^{1}S_{0}$ and $2^{3}S_{1}$ states, respectively LHCb:2015aaf , the mass difference between them is about 110 MeV, much larger than the theoretical predictions of the $2S$ mass splitting, which is 39 MeV of Godfrey-Isgur model (Godfrey:1985xj, ), 30 MeV in alternate relativized (AR) model Godfrey:2016nwn , and 28 MeV in nonrelativistic potential quark model Asghar:2018tha . On the other hand, there are different interpretations for the $B_{J}(5840)$ and $B_{J}(5970)$, because their masses and widths can not be simultaneously reasonably reproduced.

As discussed in Ref. bai:2009 , the quenched quark models, incorporating a coulomb term at short distances and the linear confining interaction at large distances, will not be reliable for the high excited mesons. This is because the linear potential, which is expected to be dominant in this mass region, will be screened and softened by the vacuum polarization effects of dynamical fermions. It is shown that the screened potential plays an important role in describing the spectra and decay properties of the charmed, charmed-strange mesons and charmonium Song:2015fha ; Song:2015nia ; Wang:2018rjg . Thus one expect that the screened potential could improve the description of the bottom mesons.

In this work, we will investigate the possible assignments of the $B_{1}(5721)$, $B^{*}_{2}(5747)$, $B_{J}(5840)$, and $B_{J}(5970)$ by employing the modified nonrelativistic quark model with screening effect, and the ${}^{3}P_{0}$ model to calculate the mass spectrum and decay properties of the bottom mesons.

This article is organized as follows. In Sec. II, we give a brief review about the modified nonrelativistic quark model and the ${}^{3}P_{0}$ model. In Sec. III, we present the numerical results and discuss the possible assignments of the bottom mesons. The summary is given in Sec. IV

In the modified nonrelativistic quark model, we determine two parameters, $C_{q\bar{q}}=0.0152$ GeV of Eq. (1) and $\beta=0.0246$ GeV of Eq. (5), by fitting the masses of the bottom mesons $B$ and $B^{*}$, which have already been well established as the $B(1^{1}S_{0})$ and $B(1^{3}S_{1})$, respectively. With these values, we calculated the mass spectrum of the bottom mesons, as shown in Table 2, where we also present the predictions with other models without considering the screening effect Zeng:1994vj ; Lahde:1999ih ; DiPierro:2001dwf ; Ebert:2009ua ; Godfrey:2016nwn ; Lu:2016bbk . It should be stressed that the mixing angles are obtained as $\theta_{1P}=-53.5^{\circ}$, $\theta_{2P}=-54.5^{\circ}$, $\theta_{1D}=-50.5^{\circ}$, $\theta_{2D}=-50.5^{\circ}$ and $\theta_{1F}=-49.0^{\circ}$ by solving the potential model with the Hamiltonian of Eq. (1), which are close to the heavy quark limit mixing angles $\theta_{1P}=\theta_{2P}=-54.7^{\circ}$, $\theta_{1D}=\theta_{2D}=-50.8^{\circ}$ and $\theta_{1F}=\theta_{2F}=-49.1^{\circ}$  Cahn:2003cw .

In Table 2, one can find that our results for excited bottom mesons are lower than those of Godfrey-Isgur (GI) Godfrey:2016nwn and Nonrelativistic quark model (NRQM) Lu:2016bbk , which is due to the screening effect. In the modified nonrelativistic quark model, the $B^{*}_{2}(5747)$ mass is consistent with the predicted mass of $B(1^{3}P_{2})$, and the $B_{1}(5721)$ mass is also in well agreement with the one of $B^{\prime}_{1}(1P)$. On the other hand, the $B_{J}(5840)$ can be considered as the candidate of the $B(2^{1}S_{0})$ taking into account the experimental uncertainties, and the $B_{J}(5970)$ can be assigned as the candidate of the $B_{2}(1D)$ or $B(1^{3}D_{3})$. As we known, only the mass information is not enough to make those assignments, and we will calculate their strong decay widths based on these preliminary assignments for the $B_{1}(5721)$, $B^{*}_{2}(5747)$, $B_{J}(5840)$, and $B_{J}(5970)$ to further examine these assignments.

Before presenting the strong decays, we need to determine the quark pair creation strength $\gamma$ firstly. As we discussed above, $B_{1}(5721)$ may be a mixing state, and the assignments of the $B_{J}(5840)$ and $B_{J}(5970)$ are still in debate due to the unknown quantum numbers and poor decay information. In this work we take $\gamma=0.411$ by fitting to the experimental widths of $B^{*}_{2}(5747)$ which is regarded as the $B(1^{3}P_{2})$ state in previous works Lu:2016bbk ; Kher:2017mky ; Asghar:2018tha ; Godfrey:2019cmi ; Yu:2019iwm . The decay properties of the state $B^{*}_{2}(5747)$ with the assignment of $B(1^{3}P_{2})$ are shown in Table 3. The ratio of the decay modes are calculated as,

which is consistent with the experimental data of $1.10\pm 0.42\pm 0.31$ D0:2007vzd and $0.71\pm 0.14\pm 0.30$ LHCb:2015aaf , and also the prediction of the nonrelativistic quark model Lu:2016bbk .

The decay widths of the $B_{1}(5721)$ as the $B_{1}(1P)$ and $B^{\prime}_{1}(1P)$ with the mixing angle $\theta_{1P}=-53.5^{\circ}$ are listed in Table 4. The total widths of the $B_{1}(5721)$ as the $B_{1}(1P)$ and $B^{\prime}_{1}(1P)$ are predicted to be $193.5$ MeV and $16.5$ MeV, respectively. The dependence of the total decay widths of the $B_{1}(5721)$ as the $B_{1}(1P)$ and $B^{\prime}_{1}(1P)$ on the mixing angle is shown in Fig. 1. One can see that the total width is narrow around $\theta_{1P}=-53.5^{\circ}$. We can safely rule out the $B_{1}(1P)$ assignment since the width of this case is much larger than the experimental data. Within in the experimental uncertainties, the total width of the $B^{\prime}_{1}(1P)$ assignment is in fair agreement with the experimental data, which implies that $B_{1}(5721)$ could be the $B^{\prime}_{1}(1P)$ state.

In the heavy quark limit, the $P$-wave heavy-light mesons could be divided into $j=1/2(0^{+},1^{+})$ doublet and $j=3/2(1^{+},2^{+})$ doublet, with $j(=S_{q}+L)$ is the total angular momentum of the light quark. For the bottom mesons, the decay width is broad for $j=1/2$ doublet, which couples to $B\pi$ in $S$-wave, and narrow for $j=3/2$ doublet, which couples to $B\pi$ in $D$-wave. Thus, the $B_{1}(5721)$ and $B^{*}_{2}(5747)$ should be the $j=3/2(1^{+},2^{+})$ doublet.

The decay widths of the $B_{J}(5840)$ as the $B(2^{1}S_{0})$ is shown in Table 5, and the predicted total decay width is 109.5 MeV, in well agreement with the experimental measurement $127\pm 40$ MeV PDG2021 . In this case, the dominant decay mode is $B^{*}\pi$, and the decay mode $B\pi$ is forbidden for the $B(2^{1}S_{0})$ assignment. It should be pointed out that the decay mode $B\pi$ has not yet finally been confirmed by the LHCb LHCb:2015aaf , which implies that assignments of the $B_{J}(5840)$ as the $B(2^{1}S_{0})$ is acceptable.

The decay widths of the $B_{J}(5970)$ as the $B_{2}(1D)$ and $B(1^{3}D_{3})$ are listed in Table 6. The predicted total width of the $B_{2}(1D)$ assignment is $202.5$ MeV with mixing angle $\theta_{1D}=-50.5^{\circ}$, which is about 100 MeV larger than the experimental data $81\pm 12$ MeV, while the one of the $B(1^{3}D_{3})$ assignment is $40.1$ MeV. The dominant decay modes are $B\pi$ and $B^{*}\pi$ decay modes, supported by the measurements of the CDF CDF:2013www and LHCb Collaborations LHCb:2015aaf . The dependence of the decay widths of the $B_{J}(5970)$ as the $B_{2}(1D)$ on the mixing angle is shown in Fig. 2, one can find that the total width of $B_{2}(1D)$ is about 200 MeV around the mixing angle $\theta_{1D}=-50.5^{\circ}$, and the total width of $B^{\prime}_{2}(1D)$ is predicted to be about 50 MeV. Considering the predictive power of the model and the experimental uncertainties, it is reasonable to regard $B_{J}(5970)$ as the $B(1^{3}D_{3})$.

In this paper, we have calculated the bottom meson spectrum with a modified nonrelativistic quark model involving the screening effect. We present a good description of mass spectrum of the bottom mesons, especially for the excited bottom mesons. Furthermore, we also investigate the strong decay properties of the $B_{1}(5721)$, $B^{*}_{2}(5747)$, $B_{J}(5840)$, and $B_{J}(5970)$ with the ${}^{3}P_{0}$ model.

Based on the mass spectrum and decay properties, the $B_{1}(5721)$ and $B^{*}_{2}(5747)$ can be identified as the $B_{1}^{\prime}(1P)$ and $B(1^{3}P_{2})$, respectively. The $B_{J}(5840)$ could be interpreted as the $B(2^{1}S_{0})$, and the $B_{J}(5970)$ could be explained as the $B(1^{3}D_{3})$. Further experimental information, especially the quantum numbers and decay modes of $B_{J}(5840)$ and $B_{J}(5970)$, are necessary to confirm these assignments.