Doubly heavy $QQ$ tetraquarks

TL;DR

This paper explores theoretical models for doubly heavy tetraquarks, ${ m T}_{QQ}$, analyzing their masses and properties, including a soliton-based approach with a distinct large $N_c$ limit, contributing to understanding these exotic states.

Contribution

It introduces a novel soliton-based model for ${ m T}_{QQ}$ tetraquarks and compares its large $N_c$ behavior with other approaches.

Findings

Model estimates suggest possible bound states of ${ m T}_{QQ}$

The soliton approach exhibits a different large $N_c$ limit

Mass predictions vary across models

Abstract

With the discovery of a doubly charmed $\Xi_{cc}$ baryon a somewhat forgotten issue of tetraquarks containing two heavy and two light (anti) quarks, ${\cal T}_{QQ}$, triggered theorist's interest. We discuss quark model estimates of ${\cal T}_{QQ}$ masses and such objects including a model where the light sector is treated as a soliton. We show that this model has different large $N_c$ limit than other approaches.

Equations

$$M_{{\mathcal{T}}_{QQ}}$$

$$M_{J/\Psi }$$

$$M_{{\Lambda }_Q}$$

$$M_{{\mathcal{T}}_{QQ}}\le M_{{\Lambda }_Q}+\frac{1}{2}{M}_{J/\Psi }+\frac{1}{2}⟨T_{QQ}⟩.$$

$$M_{{\mathcal{T}}_{cc}}-M_{\mathrm{thr}}^c\le -55\mathrm{MeV}+\frac{1}{2}⟨T_{cc}⟩$$

$$M_{{\mathcal{T}}_{bb}}-M_{\mathrm{thr}}^b\le -262\mathrm{MeV}+\frac{1}{2}⟨T_{bb}⟩.$$

$$M_{D_Q}$$

$$M_{{\Xi }_{QQ}}$$

$$M_{{\mathcal{T}}_{QQ}}$$

$$M_{{\mathcal{T}}_{cc}}\le 3935\mathrm{MeV},$$

$$M_{{\mathcal{T}}_{bb}}-M_{\mathrm{thr}}^b\sim -224\mathrm{MeV}$$

$$M_{\mathrm{baryon}}^{Q,\overline{3}}$$

$$M_{\overline{3}}^b-M_{\overline{3}}^c=3327\text{MeV,}$$

$$m_c/m_b=0.29-0.31,$$

$$m_c=1357÷1495\text{MeV,}$$

Full text

1 Introduction

Recent discovery of a doubly charmed $\Xi_{cc}^{++}(3621)$ baryon by the LHCb Collaboration at CERN [1] renewed interest in $\bar{Q}\bar{Q}q_{1}q_{2}$ tetraquarks or their antiparticles, essentially for two reasons. Firstly, the LHCb result shows that it is possible to create a $cc$ pair in an attractive channel that can form a bound state with a light quark. Secondly, on theoretical side, heavy $cc$ pair can be described to a good approximation as a pointlike color $\overline{\bf 3}$ source that can form a bound state with two light antiquarks. In the heavy quark limit the $\overline{\bf 3}$ source acts as a heavy antiquark, and therefore the underlying dynamics is identical to the heavy antibaryon case (neglecting spin effects). In this paper we shall consider heavy $\bar{Q}\bar{Q}$ pairs acting as ${\bf 3}$ color source.

Similar excitement arose approximately 17 years ago when SELEX experiment at FermiLab announced a discovery of $\Xi_{cc}^{+}(3519)$ [2], which is today (despite later report [3]) considered as unconfirmed [4]. Phenomenological attempt to estimate the $\bar{c}\bar{c}q_{1}q_{2}$ mass based on SELEX result can be found e.g. in Ref.[5], where also large $N_{c}$ limt for such states is discussed.

In 1993 Manohar and Wise [6] showed within heavy quark symmetry approach that doubly heavy tetraquarks are bound in the limit $m_{Q}\rightarrow\infty$. These arguments were reanalized in 2006 and also very recently, at the time of this conference, by Cohen and collaborators in Refs. [7, 8].

Asymptotic theorems, however, do not provide any hint at what scale they become operative. The aim of this paper is to recall some simple quark model estimates of the doubly heavy tetraquark mass and then apply a phenomenological model based on a soliton description of the light sector to tetraquarks. At the end we discuss the difference between the soliton model and regular quark models in the large $N_{c}$ limit.

2 Quark model estimates of the tetraquark mass

To the best of our knowledge the first phenomenological attempt to estimate doubly heavy $QQ$ tetraquark mass was published by Lipkin in 1986 [9] (although the fourfold heavy tetraquarks were discussed even earlier in 1982 [10]). He used a variational method in the nonrelativistic quark model and one, rather natural assumption, that light quarks see heavy (anti)quark pair as a single object. Lipkin tried to use experimental data available at that time to derive (almost) model-independent estimate of the tetraquark mass. This is schematically shown in Fig. 1 and leads to the following mass formulae for tetraquark ($\mathcal{T}_{{Q}{Q}}$), $J/\Psi$ and $\Lambda_{Q}$:

[TABLE]

where for simplicity we have suppressed bars over heavy quark symbol $Q$. Notation confronted with Fig. 1 is self-explanatory. Note that quark-quark interaction in color $\bar{{\bf 3}}$ is two times weaker than antiquark-quark in ${\bf 1}$.

Formulae (1) lead to the following upper bound for the tetraquark mass:

[TABLE]

With the data available in 1986 one could not eliminate the unknown average $\langle T_{QQ}\rangle$. Nevertheless an important lesson can be drawn from Eq. (2) when plugging in numerical values with a spin averaged mass $M_{\bar{c}c}=(3M_{J/\Psi}+M_{\eta_{c}})/4$ rather than $M_{J/\Psi}$:

[TABLE]

with $M_{\mathrm{thr}}^{c}=M_{D}+M_{D^{*}}$. We see from Eq. (3) that the very existence of the bound heavy tetraquark depends on the subtle balance between $-55$ MeV and $\langle T_{cc}\rangle$. It is easy to convince oneself that adding the D meson does not eliminate $\langle T_{cc}\rangle$. Today we can repeat the same estimate for the $b$ case:

[TABLE]

Now, with the discovery of $\Xi_{cc}$ [1] one can form a linear combination where the troublesome $\langle T_{QQ}\rangle$ term drops out [5]. To this end we have

[TABLE]

With this new input we have one new relation:

[TABLE]

where again we use $M_{D_{c}}=\left(3M_{D^{\ast}}+M_{D}\right)/4$ (for a more accurate choice see [5]) and obtain numerically

[TABLE]

which is 60 MeV above the threshold (as e.g in [11]). This implies that, if the inequality (2) were saturated, $\langle T_{cc}\rangle\sim 230$ MeV. Since $m_{b}/m_{c}\sim 3$, one can reasonably assume that $\langle T_{bb}\rangle\sim\langle T_{cc}\rangle/3$ and we arrive at a conclusion

[TABLE]

in a surprising agreement with much sophisticated quark model of Ref. [12]. In the literature one finds predictions for (8) ranging from $-60$ [13] through $-100$ [11], $-120$ [14, 15] to $-400$ MeV [16] (see Tab. V in [11] for other predictions). Our simple analysis confirms that the binding of a putative doubly heavy $QQ$ tetraquark increases with increasing $m_{Q}$. This is true also in the case when the structure of a heavy diquark can be resolved by the light quarks and repulsive ${\bf 6}$ channel is included [17].

3 Soliton model for tetraquarks

While the nonrelativistic approach may be applicable to heavy quarks, its credibility for the light sector is certainly not at the same footing. In recent papers [18] we have proposed a mean-filed description of heavy baryons as a light quark-soliton and a heavy quark, where the soliton is constructed from $N_{c}-1$ rather $N_{c}$ quarks. The model passes phenomenological tests.

In the limit $m_{q}\rightarrow\infty$ the soliton should not be sensitive to the properties of an object it is interacting with. Moreover, since the ground state soliton is in this case in flavour antitriplet of spin zero [18], the spin of the heavy ”nucleus” is irrelevant. Mass formula for the nonstrange heavy baryons in flavor antitriplet is very simple

[TABLE]

where $\delta_{\bm{\overline{3}}}$ can be extracted from the light hyperon spectrum, and $M_{\mathbf{\overline{3}}}$ is an average mass of the $\bm{\overline{3}}$ flavor multiplet including heavy quark mass $m_{Q}$, classical soliton mass $M_{\rm sol}$ and soliton rotational energy [18]. For strange heavy baryons the coefficient in front of $\delta_{\bm{\overline{3}}}$ is equal $-1/3$. In [18] we never needed $m_{Q}$ and $M_{\rm sol}$ separately. For nonstrange tetraquarks we therefore naturally have

[TABLE]

and it is clear that now we need not only $m_{Q}$ but also $m_{\bar{Q}\bar{Q}}$. For a rough estimate we can approximate $m_{\bar{Q}\bar{Q}}-m_{Q}\sim m_{Q}$, or we may assume a few percent binding following e.g. Ref. [19].

In order to estimate effective $m_{Q}$ we first observe that differences of mean multiplet values, both for flavor $\bm{\overline{3}}$ and ${\bf 6}$, for bottom and charm, should be equal: $m_{b}-m_{c}=M_{\mathbf{\overline{3}}}^{b}-M_{\mathbf{\overline{3}}}^{c}=M_{\mathbf{6}}^{b}-M_{\mathbf{6}}^{c}$. Numerically we have:

[TABLE]

with $M_{\mathbf{\overline{3}},\mathbf{6}}^{Q}$ taken from [18]. We see perfect agreement between both flavor multiplets. Another piece of information comes from the hyperfine splittings in $\mathbf{6}$ that are inversely proportional to the quark masses and have been estimated in [18] yeilding:

[TABLE]

which is compatible with the ratio obtained from the PDG [4]. Now, from Eqs. (11) and (12) we can determine absolute effective masses

[TABLE]

The uncertainty in (13) is due to the uncertainty in the ratio (12). Masses (13) are lower than masses extracted from meson spectra: $m_{c}=1643$ and $m_{b}=4979$ MeV, which are compatible with e.g. [19].

We see from Fig. 2 that for heavy quark masses in the range of Eq. (13) both $cc$ and $bb$ tetraquarks are rather deeply bound. For larger $m_{Q}$, compatible with mesonic spectra, $cc$ tetraquark is most likely unbound and $bb$ is most likely bound.

4 Summary

We have recalled arguments that $QQ$ tetraquarks are bound in the limit $m_{Q}\rightarrow\infty$ and analyzed the mass spectrum of $cc$ and $bb$ tetraquarks with the help of the variational approach of Lipkin [9]. We than employed the quark-soliton model describing light degrees of freedom in the limit $N_{c}\rightarrow\infty$ used previously for heavy baryons with one heavy quark [18] to the problem of $QQ$ tetraquarks. We have argued that the light soliton does not distinguish the nature of the color ${\bf 3}$ heavy source, so that heavy quark can be replaced by a heavy anti-diquark leaving the soliton unaffected. Unfortunately the anti-diquark properties have not been calculated within the soliton model. In fact in the present model the diquark should be considered an $N_{c}-1$ heavy quark system to neutralize the color of the soliton for $N_{c}>3$. This color structure (discussed briefly in [5]) is completely different from the quark model picture where $QQ$ tetraquarks consist from two antiquarks and two quarks for any $N_{c}$. The ”diquark” in the soliton approach is therefore amenable to an effective description, as the light sector that is represented by a soliton, and deserves further studies from this perspective.

Acknowledgments

This work was supported by the Polish NCN grant 2017/27/B/ST2/01314 and NAWA (Polish National Agency for Academic Exchange) Bekker program. The author acknowledges stimulating discussions with H.-C. Kim and M.V. Polyakov and thanks Institute for Nuclear Theory at the University of Washington (where a part of this work has been completed) for hospitality.