Hidden-charm and bottom tetra- and pentaquarks with strangeness in the hadro-quarkonium and compact tetraquark models

Multiquark states are baryons/mesons which cannot be described in terms of $qqq$/$q\bar{q}$ degrees of freedom only. They include $XYZ$ suspected tetraquarks, like the $X(3872)$ [now $\chi_{\rm c1}(3872)$] Choi:2003ue ; Acosta:2003zx ; Abazov:2004kp and $X(4274)$ [also known as $\chi_{\rm c1}(4274)$] Aaltonen:2011at ; Aaij:2016iza , and pentaquark states. The latter were recently discovered by LHCb in $\Lambda_{\rm b}\rightarrow J/\psi\Lambda^{*}$ and $\Lambda_{\rm b}\rightarrow P_{\rm c}^{+}K^{-}\rightarrow(J/\psi p)K^{-}$ decays Aaij:2015tga ; Aaij:2019vzc . The structure of $XYZ$ tetraquarks and $P_{\rm c}$ pentaquarks is still unclear. This is why there are several alternative models to explain their properties. For a review, see Refs. Chen:2016qju ; Ali:2017jda ; Olsen:2017bmm ; Guo:2017jvc . To distinguish among the different pictures (molecular model, diquark model, unquenched quark model, …) one should compare their theoretical predictions for the spectrum, decay amplitudes, production cross-sections, and so on, with the experimental data.

A clean way to discriminate among the previous theoretical interpretations for suspected $XYZ$ tetraquarks was suggested in Ref. Voloshin:2019 . There, Voloshin pointed out that if $Z_{\rm c}$ resonances exist then, because of the SU(3)_f symmetry, one may also expect the emergence of their strange partners, $Z_{\rm cs}$ Voloshin:2019 . The author also argued that the one-pion-exchange interaction of the meson-meson molecular model is impossible between strange and nonstrange heavy mesons, like $B$ and $B_{\rm s}$ Voloshin:2019 . Hidden-charm and bottom mesons with strangeness are also forbidden in the context of the Unquenched Quark Model (UQM) formalism. Indeed, one cannot dress heavy quarkonium $Q\bar{Q}$ states with $Q\bar{s}-n\bar{Q}$ or $Q\bar{n}-s\bar{Q}$ higher Fock components (where $n=u$ or $d$) by creating a light $n\bar{n}$ or $s\bar{s}$ pair with vacuum quantum numbers. Therefore, hidden-charm and bottom tetraquark states with non-null strangeness content cannot take place neither in the UQM Heikkila:1983wd ; Pennington:2007xr ; Danilkin:2010cc ; Ortega:2012rs ; Ferretti:2013faa ; Ferretti:2014xqa ; Ferretti:2013vua ; Achasov:2015oia ; Kang:2016jxw ; Lu:2016mbb ; Ferretti:2018tco nor in the molecular model Tornqvist:1993ng ; Hanhart:2007yq ; Baru:2011rs ; Valderrama:2012jv ; Aceti:2012cb ; Guo:2013sya interpretations.

On the contrary, these exotic configurations are expected (if above threshold) both in the compact tetraquark Jaffe:1976ih ; SilvestreBrac:1993ss ; Brink:1998as ; Maiani:2004vq ; Barnea:2006sd ; Santopinto:2006my ; Ebert:2008wm ; Deng:2014gqa ; Zhao:2014qva ; Lu:2016cwr ; Anwar:2017toa ; Anwar:2018sol ; Esposito:2018cwh ; Bedolla:2019zwg ; Yang:2019itm and hadro-quarkonium Eides:2015dtr ; Perevalova:2016dln ; Alberti:2016dru ; Anwar:2018bpu ; Dubynskiy:2008mq ; Guo:2008zg ; Voloshin:2013dpa ; Wang:2013kra ; Brambilla:2015rqa ; Ferretti:2018kzy ; Panteleeva:2018ijz ; Voloshin:2019 models. See Fig. 1. In light of this, the experimental observation of $XYZ$ states with non-null strangeness content would make it possible to rule out a few possible theoretical interpretations for tetraquarks. Voloshin did not compute the spectrum of $Z_{\rm cs}$ states, but only discussed phenomenological indications for the emergence of those states Voloshin:2019 . The study of their spectrum and that of their pentaquark counterparts is thus the subject of the present manuscript.

Here, we extend the hadro-quarkonium model findings of Refs. Ferretti:2018kzy ; Anwar:2018bpu and calculate the spectrum of hidden-charm and bottom tetraquarks and pentaquarks with strangeness. The hadro-quarkonium picture was developed to explain the experimental observation of heavy-light tetraquark candidates characterized by peculiar properties Dubynskiy:2008mq ; Voloshin:2007dx . Firstly, these exotics are supposed not to be particularly close to a specific heavy-light meson-meson threshold, unlike $D^{0}\bar{D}^{*0}$ in the $X(3872)$ case. Secondly, such states may decay into heavy quarkonia plus one or more light mesons, like $\eta_{\rm c}+\eta$. Even though it was meant for the description of tetraquarks, the hadro-quarkonium model can be easily extended to the baryon sector to study pentaquarks Anwar:2018bpu ; Eides:2015dtr .

We also compute the masses of heavy-light tetraquarks with non-null strangeness content in the compact tetraquark model of Refs. Anwar:2017toa ; Anwar:2018sol ; Bedolla:2019zwg . In the compact tetraquark model, heavy-light $qQ\bar{q}\bar{Q}$ states are modeled as the bound states of a diquark, $qQ$, antidiquark, $\bar{q}\bar{Q}$, pair. The diquark constituents are treated as inert against internal spatial excitations. Their binding is the consequence of one-gluon-exchange forces and their relative dynamics can be described in terms of a relative coordinate $\bf r_{\rm rel}$. The calculation of the spectrum of compact pentaquark configurations in the diquark model is more difficult than that of compact tetraquarks because one has to deal with a three-body problem instead of a two-body one; moreover, one also has to consider both diquark-diquark and diquark-antiquark interactions. This is why here we do not provide results for compact (diquark-diquark-antiquark) pentaquarks, which will be the subject of a subsequent paper.

Our predictions for strange hidden-charm and bottom tetraquarks and, especially, those for $P_{\rm c}$ and $P_{\rm b}$ pentaquarks with non-null strangeness content may soon be tested by LHCb.

The possible existence of binding mechanisms of charmonium states in light-quark matter was discussed long ago Brodsky:1989jd ; Kaidalov:1992hd ; Sibirtsev:2005ex in terms of the interaction of charmonium inside nuclei. The idea of hadro-charmonium (hadro-quarkonium) bound states resembles the previous one.

Hadro-quarkonia are heavy-light tetra- or pentaquark configurations, where a compact $Q\bar{Q}$ state (with $Q=c$ or $b$), labelled as $\psi$ in the following, is embedded in light hadronic matter, $\mathcal{H}=$ $qqq$ or $q\bar{q}$ (where $q=u,d$ or $s$) Eides:2015dtr ; Perevalova:2016dln ; Alberti:2016dru ; Anwar:2018bpu ; Dubynskiy:2008mq ; Guo:2008zg ; Voloshin:2013dpa ; Wang:2013kra ; Brambilla:2015rqa ; Ferretti:2018kzy ; Panteleeva:2018ijz ; Voloshin:2019 . The heavy and light constituents, $\psi$ and $\mathcal{H}$, develop an attractive force, which is the result of multiple-gluon exchange between them. Such interaction, $H_{\rm eff}$, can be written in terms of the multipole expansion in QCD QCDME . In particular, if one considers as leading term the $E1$ interaction with chromo-electric fields ${\bf E}$ and ${\bf E}^{\prime}$ Dubynskiy:2008mq ; Kaidalov:1992hd , one gets the effective Hamiltonian

$$H_{\rm eff}=-\frac{1}{2}\alpha_{\psi\psi^{\prime}}{\bf E}\cdot{\bf E}^{\prime}\mbox{ },$$

(1)

where $\alpha_{\psi\psi^{\prime}}$ is the so-called heavy quarkonium chromo-electric polarizability. By making use of additional approximations, $H_{\rm eff}$ can be further reduced to a simple square-well potential Dubynskiy:2008mq ; Ferretti:2018kzy ; Anwar:2018bpu ,

$$V_{\rm hq}(r)=\left\{\begin{array}[]{ccc}-\frac{2\pi\alpha_{\psi\psi}M_{\mathcal{H}}}{3R_{\mathcal{H}}^{3}}&\mbox{for}&r<R_{\mathcal{H}}\\ 0&\mbox{for}&r>R_{\mathcal{H}}\end{array}\right.\mbox{ },$$

(2)

where $R_{\mathcal{H}}=R_{\mathcal{B}}$ or $R_{\mathcal{M}}$ is the light baryon/meson radius. Eq. (2) can be plugged into a Schrödinger equation and solved for light hadron-heavy quarkonium systems.

There are four quantities to be given as input in the calculation. They are the masses $M_{\psi}$ and $M_{\mathcal{H}}$, the radius $R_{\mathcal{H}}$, and the diagonal chromo-electric polarizability, $\alpha_{\psi\psi}$. See Table 1.

The values of $M_{\psi}$ and $M_{\mathcal{H}}$ are extracted from the PDG Tanabashi:2018oca .

In principle, non-diagonal quarkonium chromo-electric polarizabilities, $\alpha_{\psi\psi^{\prime}}$, can be fitted to the data by considering $\psi\rightarrow\psi^{\prime}+h$ hadronic transitions Voloshin:2007dx ; Chen:2019gty ; see Fig. 2. However, no experimental information can be used to estimate the $\alpha_{\psi\psi}$’s. Therefore, the diagonal chromo-electric polarizabilities, $\alpha_{\psi\psi}(n\ell)$, where $n$ and $\ell$ are the radial quantum number and orbital angular momentum of $\psi$, respectively, have to be extracted from the phenomenology. In the case of charmonia, we consider Anwar:2018bpu : $\alpha_{\psi\psi}(1P)_{c\bar{c}}=11\mbox{ GeV}^{-3}$ and $\alpha_{\psi\psi}(2S)_{c\bar{c}}=18\mbox{ GeV}^{-3}$. In the case of bottomonia, we make use of two sets of values for the chromo-electric polarizabilities. They are Anwar:2018bpu : $\alpha_{\psi\psi}^{(1)}(1P)_{b\bar{b}}=14\mbox{ GeV}^{-3}$ and $\alpha_{\psi\psi}^{(1)}(2S)_{b\bar{b}}=23\mbox{ GeV}^{-3}$; $\alpha_{\psi\psi}^{(2)}(1P)_{b\bar{b}}=21\mbox{ GeV}^{-3}$ and $\alpha_{\psi\psi}^{(2)}(2S)_{b\bar{b}}=33\mbox{ GeV}^{-3}$. We also need the strange mesons’ and hyperons’ radii. While for the kaon we can use the well-established value of the $K^{\pm}$ charge radius reported on the PDG Tanabashi:2018oca , $R_{K}=0.560\pm 0.031$ fm, in the $\Sigma$, $\Xi$ and $K^{*}$ cases the situation is different¹¹1The values of the proton and kaon radii reported by the PDG Tanabashi:2018oca can be regarded as reliable, because they are the result of the average over several measurements. On the contrary, the value of the $\Sigma^{-}$ radius from the PDG is the outcome of a single experiment; moreover, there is no available data for the charge radius of the $\Sigma^{+}$. This is why here we do not extract $R_{\Sigma}$ from the PDG.. Indeed, due to the lack of well-established experimental data, we are forced to extract $R_{\Sigma}$, $R_{\Xi}$ and $R_{K^{*}}$ from phenomenological estimates. For example, see Refs. Ledwig:2006gw ; Kubis:1999xb ; Buchmann:2002et ; Carrillo-Serrano:2016igi ; Godfrey:1985xj . Following Ref. Ledwig:2006gw , we have: $R_{\Sigma}=\frac{1}{2}\left(R_{\Sigma^{+}}+R_{\Sigma^{-}}\right)=0.863$ fm; $R_{\Xi}=0.841$ fm. The $K^{*}(892)$’s radius is calculated in the relativized quark model for mesons of Ref. Godfrey:1985xj : $R_{K^{*}}=0.729$ fm.

Finally, the hadro-quarkonium quantum numbers are obtained by combining those of the hadrons $\mathcal{\psi}$ and $\mathcal{H}$,

$$\left|\Phi_{\rm hq}\right\rangle=\left|(L_{\psi},S_{\psi})J_{\psi};(L_{\mathcal{H}},S_{\mathcal{H}})J_{\mathcal{H}};(J_{\rm hq},\ell_{\rm hq})J_{\rm tot}^{P}\right\rangle\mbox{ }.$$

(3)

Here, ${\bf J}_{\rm hq}={\bf J}_{\psi}+{\bf J}_{\mathcal{H}}$, the hadro-quarkonium parity is $P=(-1)^{\ell_{\rm hq}}$ $P_{\psi}P_{\mathcal{H}}$, and $\ell_{\rm hq}$ is the relative angular momentum between $\psi$ and $\mathcal{H}$. From now on, unless explicitly indicated, we assume that $\ell_{\rm hq}=0$.

In this section, we discuss our results for the spectrum of heavy quarkonium-strange hadron bound states.

The binding energies are computed in the hadro-quarkonium model of Sec. II and Refs. Anwar:2018bpu ; Ferretti:2018kzy ; Dubynskiy:2008mq by solving the two-body eigenvalue problem of Eq. (2) via a finite differences algorithm (Feynman-Lectures, , Vol. 3, Sec. 16-6). As a check, the same results are also obtained by means of a numerical code based on the Multhopp method; see (Richard, , Sec. 2.4). The values of the heavy quarkonium chromo-electric polarizabilities and light hadron radii used here are given in Table 1.