Bottomonia in quark-antiquark confining potential

Different experimental facilities like LHCb, COMPASS, BESIII, STAR, CMS, ATLAS, BABAR, etc., are continuously trying to produce enormous data in the field of heavy-flavor hadrons. The theoretical studies are also heavily invested in the field to interpret the experimental data and predict the properties of hadrons. The heavy quark hadrons like charmonium and bottomonium have also indeed received remarkable experimental growth. Through the efforts of various experimental collaborations and facilities, significant progress has been made in establishing bottomonium spectra. In this paper, we are focussing on the spectra and properties of bottomonium. The states $\Upsilon(1S)$, $\Upsilon(2S)$ and $\Upsilon(3S)$ were observed in 1977 by the E288 collaboration at the Fermi National Accelerator Laboratory (FNAL) [1, 2] and notated as $\Upsilon,\Upsilon^{{}^{\prime}},\Upsilon^{{}^{\prime\prime}}$ states respectively. After the early detection of systems composing of $b\bar{b}$, many of the properties and new states were discovered as given in PDG[3]. During the year 1982, the first excited state $\chi_{bJ}(2P)$ were identified in E1 transition from $\Upsilon^{{}^{\prime\prime}}$ state [4, 5]. Subsequently, in 1983, an exciting breakthrough came with the discovery of $\chi_{bJ}(2P)$ with J = 0, 1, 2 in the E1 transition from $\Upsilon^{{}^{\prime}}$ state [6, 7]. In the following year, 1984, $\Upsilon(4S),\Upsilon(5S),\Upsilon(6S)$ are announced in $e^{+}e^{-}$ cross section above $B\bar{B}$ threshold [8, 9]. These findings opened the search window to investigate higher excited states of bottomonia spectra. After several experimental attempts, in 2008, the BABAR collaboration [10] announced the observation of $\eta_{b}(1S)$, spin partner of $\Upsilon(1S)$. In 2010, BABAR collaboration observed $\Upsilon(1^{3}D_{J})$ with $J=2$ in decay chain $\Upsilon(3S)\rightarrow\gamma\gamma\Upsilon(1^{3}D_{J})\rightarrow\gamma\gamma\Upsilon(1S)$ [11]. In 2011, the BABAR collaboration announced $h_{b}(1P)$ observed via decay process $\Upsilon(3S)\rightarrow\pi^{0}h_{b}(1P)$ [12]. In 2011, Belle Collaboration reported observation of states $h_{b}(1P)$ and $h_{b}(2P)$ in $e^{+}e^{-}\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ process [13]. In the same year, the ATLAS group reconstructed $\chi_{b}(nP)$ via radiative decay $\chi_{b}(nP)\rightarrow\Upsilon(1S)\gamma$ and $\chi_{b}(nP)\rightarrow\Upsilon(2S)\gamma$. A new state $\chi_{b}(3P)$ is identified in these decay modes [14]. This state $\chi_{b}(3P)$ was also reconfirmed by D0 collaborations [15] and the LHCb group [16]. The latest CMS experiment shows resolved peaks of $\chi_{b1}(3P)$ and $\chi_{b2}(3P)$ with mass difference of $10.60\pm 0.64(stat)\pm 0.17(syst)$ $MeV$ using 80 $fb^{-1}$ in pp collisions at a center-of-mass energy of 13 $TeV$ [17]. In 2012, Belle announced the first evidence of the existence of state $\eta_{b}(2S)$, pseudoscalar partner of $\Upsilon(2S)$ [18]. Some charged candidates like $Z_{b}(10610)$ and $Z_{b}(10650)$ are also announced in $\pi^{\pm}\Upsilon(nS)$(n = 1,2,3) and $\pi^{\pm}h_{b}(mP)$ ($m=1,2$) mass spectra by Belle experiment [3]. But, their nature is still enigmatic. These states have been thoroughly investigated and analyzed in theoretical frameworks and interpreted these states as molecular states, exotic states, and tetraquarks states [19, 20, 21]. Recently, there has been a resurgence of interest in the exploration and investigation of states $\Upsilon(10860)$ and $\Upsilon(11020)$, which was first announced in 1985 by CUSB experiment [9]. A recent study in 2019 by Belle investigated $\Upsilon(10860)$ state in process $e^{+}e^{-}\rightarrow\Upsilon(nS)$ ($n=1,2,3$) with high significance and measured masses are $10752.7\pm 5.9$ $MeV$,and decay widths are $35.5^{+17.6}_{-17.6}$ $MeV$ respectively [22]. Now this vector state is mentioned as $\Upsilon(10753)$ state in PDG. Authors in Ref.[23], investigated $\Upsilon(10753)$ using relativistic flux tube and give assignment of $3^{3}D_{1}$ $b\bar{b}$ state. In Ref.[24], with QCD sum rules, Wang et al. interpreted this state as a hidden bottom tetraquark state. Theoretically, the prevailing consensus posits that $\Upsilon(10860)$ and $\Upsilon(11020)$ states correspond to the S-wave vector $b\bar{b}$ states denoted as $\Upsilon(5S)$ and $\Upsilon(6S)$, respectively [25, 26]. Authors in Ref.[27, 28] explicated these states ($\Upsilon(10860)$ and $\Upsilon(11020)$) with instanton-induced potential obtained from the instanton liquid model for QCD vacuum and elucidated $\Upsilon(10860)$ as an admixture of $5^{3}S_{1}-6^{3}D_{1}$ states and $\Upsilon(11020)$ as an admixture of $5^{3}S_{1}-5^{3}D_{1}$ states respectively. Also state $\Upsilon(11020)$ studied in Ref.[27, 28] and suggested as an admixture of $6^{3}S_{1}-5^{3}D_{1}$ states. Very recently, in March 2023, the BelleII detector examined the decay process $e^{+}e^{-}\rightarrow\omega_{\chi_{bJ}}(1P)$ ($J=0,1,2$) and concluded states $\Upsilon(10860)$ and $\Upsilon(10753)$ may have different internal structures [29], which is considered same in previous studies [22]. All these recent observations of numerous states of heavy mesons developed an interest in theoreticians to explore them in all aspects. Different QCD potential models have studied all the above-observed states. It is believed that QCD potential models are the most successful phenomenological approach in the non-perturbative region. There are various kinds of potentials in literature like Cornell potential [30, 31, 32], Martin potential [33, 34], Logarithmic potential [35], Richardson potential [36], and Song and Lin potential [37], which successfully explore quarkonium and their properties. Another potential is used to study charmonium spectroscopy [38] and its decay properties. In the same context, we explore bottomonium spectra and their decay properties using the same non-relativistic potential model from Ref. [38]. Similar potentials have been harnessed with $n=2$ to study the dynamics of light hadrons in the framework of the Bethe- Salpeter equation under Covariant Instantaneous Ansatz (CIA) [39, 40, 41]. It can extensively explore many processes like digamma decays, radiative decays, decay constants, and the structure of hadrons. This framework can also be explored for various energy scales to have better insights. Using quark - antiquark confining potential form [38], we calculated masses and decay properties of bottomonia - leptonic decays, gamma decays, gluons decays, and decay constants, which are very useful in revealing the non-perturbative domain of QCD.

The paper is summarized as follows: Section 2 gives a brief description of phenomenological quark-antiquark confining potential and extraction of potential parameters from Chi-square fitting. Section 3 gives a description of the decay rates of quarkonium. Section 4 presents the numerical analysis where we predict the mass spectra of bottomonia for $nS,nP,nD,nF,n=1,2,3,4,5$ and their decay properties. Section 5 gives the conclusions of the paper.

Various theoretical approaches describe the spectrum of quarkonium states (charmonium, bottomonium, beauty charmed mesons). Still, the phenomenological potential model is one of the most popular and reliable approaches. The famous form of potential model exploited in heavy quarkonium spectroscopy is coulomb plus linear potential. The Coulomb potential component arises from the one-gluon exchange (Lorentz vector exchange) interaction between the quark and antiquark, which is analogous to the electromagnetic interaction between charged particles. On the other hand, the linear potential arises from the strong force (usually associated with Lorentz scalar exchange) that acts between the quark and the antiquark. This potential increases linearly with the separation between the quark and the antiquark. As discussed in the previous section, many other forms of confining potentials are used in literature. We have adopted the following potential form to study the spectroscopy of bottomonium bound states [38].

where $V_{V}$ is the vector part of the potential (Coulomb), $V_{S}$ is the scalar part of the potential (confining), $\alpha_{s}$ is the running coupling constant. $A$ and $B$ are potential parameters estimated with chi-square fitting of low-lying bottomonium states. Since for charmonium, as mentioned in Ref.[38], the results are not consistent for $p=2$, we have also worked for $p=1$ and $B=1$ $GeV^{p}$. The value of the running coupling constant can be obtained by the expression :

Where $\Lambda$ is the QCD scale taken as 0.12 $GeV$, $n_{f}$ is the number of active flavors, number of flavors lighter than scale $\mu$ ($m_{q}\ll\mu$). $\mu$ is the renormalization scale equal to $\frac{2m_{Q}m_{\bar{Q}}}{m_{Q}+m_{\bar{Q}}}$, for bottomonium $n_{f}=4$, as flavors $u$, $d$, $s$, and $c$ are considered light. A similar potential is used to study the ground states of light-flavoured and heavy-light mesons [42, 43, 44, 40, 41, 45, 46, 47, 48, 49]. The confining term in the potential Eq. (1) is supposed to behave as linear confinement ($r$) for the heavy quark ($c$, $b$) and hadronic form of the ($r^{2}$) for the light quarks. $V_{0}$ is a state-dependent constant potential. In adopted potential in eq 1, we added spin-dependent terms - spin-spin, spin-orbit, and tensor to describe the spectrum’s splitting structure and account for each state’s different quantum numbers. The spin-dependent part $V_{SD}$ [50, 51] is given by:

Where coefficients $V_{SS}$, $V_{LS}$, and $V_{T}$ depend on derivatives of vector $V_{V}$ and scalar $V_{S}$ contributions from the adopted potential in eq 1. The expressions for coefficients are the following [50, 52]:

Where $M_{i}$, $M_{j}$ are quark masses. All these spin-dependent interactions are corrections in total mass, treated as first-order perturbation corrections in heavy quark-bound states. As $V_{LS}$, $V_{SS}$, and $V_{T}$ expressions are proportional to $1/m^{2}$ ($M_{i}=M_{j}=m$ for bottomonium), they justify their treatment as first-order perturbative corrections. The spin-orbit term containing $V_{LS}$ and tensor term containing $V_{T}$ describe the fine structure of the state, whereas the spin-spin term containing $V_{SS}$ describes hyperfine splittings. Mass is the prime property to study the spectroscopy of hadrons. To calculate spectra of bottomonium, we estimated the parameters $V_{0}$, bottom quark $m_{b}$, and parameter $A$ appearing in potential in eq (1) with Chi-square fitting procedure. We considered these parameters as free parameters in the fitting procedure for ground-state bottomonium meson in the following range:

$0<A<1$ $(GeV^{3})$; $0<V_{0}<0.5$ $(GeV)$; $4<m_{b}<5$ $(GeV)$

The results for parameters are: $m_{b}$ = 4.72 $GeV$, $V_{0}$ = 0.10 $GeV$, $A$ = 0.20 $GeV^{2}$. The running coupling constant is $\alpha_{s}=0.2053$. The state dependent potential strength $V_{0}$ is given by relation:

Where $n$ and $l$ are the radial and angular quantum numbers of the states. Also, $b$ and $c$ are unknown parameters estimated by fitting the experimental masses of $2S$, $1P$ states of bottomonium. We find $b$ = -0.105 $GeV$, $c$ = 0.0007 $GeV$. Using all these parameters and adopted potential form, we computed mass spectra of $nS$, $nP$, $nD$, and $nF$ bottomonium states listed in Table 1, Table 2, Table 3, and Table 4.

Apart from the masses of bottomonium meson states, precise predictions of decay rates are crucial properties of any successful model. Recently there are several investigations on various phenomena encompassing strong, radiative, and leptonic decays of heavy quarkonium. Such investigations directly probe the hadron structure and shed light on some aspects of the quark-gluon structure. Leptonic decay constants serve as expedient probes for the short-distance structure of hadron and act as tools for studying quark dynamics in this domain. The extracted model parameters and radial wave functions are used here to compute the di-leptonic, two-photon, and two-gluon annihilation rates. Since these rates are related to the wave- function, they provide a better insight into quark-antiquark dynamics within mesons.