Rotation and vibration in tetraquarks

Scientists have detected a new particle, dubbed X(2900), by analyzing all the data collected so far by the LHCb experiment at CERN’s Large Hadron Collider lhcb1 ; lhcb2 . This experiment is renowned for discovering exotic quark combinations, which help scientists study the strong force, one of the four fundamental forces in the universe. LHCb has identified several tetraquarks, made up of four quarks (or two quarks and two antiquarks), including the latest discovery of an entirely new type of tetraquark with a mass of $2.9$ GeV/c², which has only one charm quark. While scientists predicted this particle’s existence in 1964, it is the first observed instance of a tetraquark with only one charm quark. Quarks cannot exist independently; they form composite particles, such as mesons (a quark and an antiquark) or baryons (three quarks or three antiquarks), like the proton. The LHCb detector located at the LHC focuses on studying $B$ mesons, which are composed of a bottom or an anti-bottom quark. These mesons quickly decay into lighter particles shortly after being produced in proton-proton collisions at the LHC. Tetraquarks are believed to be pairs of distinct mesons that are temporarily bound together like a ”molecule,” according to some theoretical models, while others view them as a single cohesive unit of four particles. Identifying and measuring the properties of new kinds of tetraquarks, such as their quantum spin and parity, will provide a better understanding of these strange inhabitants of the subatomic realm. The recently discovered particle, called X(2900), contains an anticharm, an up, a down, and an antistrange quark ($\bar{c}ud\bar{s}]$) and is considered the first open-charm tetraquark, as all previous tetraquark-like states observed by LHCb had a charm-anticharm pair, resulting in a net-zero ”charm flavour.” lhcb1 ; lhcb2

Pairing interactions between fermionic or bosonic systems are common in many physical contexts such as Bose-Einstein Condensation and Superfluidity, airing correlations in nuclei: from microscopic to macroscopic models, high-temperature superconductors,  pit ; fra ; leg ; arim ; oss91 ; oss93 . One example of the application of algebraic methods in hadron physics is the use of such interactions. bar ; kru ; b2 ; b1 ; f1 ; f2 . We establish explicit extensions of duality relations that relate the Hamiltonians and basis classification schemes associated with number-conserving unitary and number-nonconserving quasispin algebras for four-level pairing interactions. The Hamiltonian of the model can be defined using a linear combination of first- and second-order Casimir operators when one- and two-body interactions are present. The four-level pairing model describes a finite system that undergoes a second-order quantum phase transition between the rotation and vibration limits. Recently, we utilized the interacting boson approximation proposed by Arima and Iachello arima75 ; casten to calculate wave functions in an interacting $sl$ many-body boson system pan2002 . It is important to note that, in general, the building blocks of the boson system are associated with both $s$ and $l$ bosons for single and quadrupole angular momentum. The bosonic pairing systems exhibit similarities in their Lie algebraic properties, but the differences are significant in terms of the irreducible representations (irreps) that the eigenstates transform under, which play a critical role in defining the system’s spectroscopy. Finite pairing systems can be described by two complementary algebraic formulations: (1) a unitary algebra consisting of bilinear products of a creation and annihilation operator, and (2) a quasispin algebra that uses creation and annihilation operators for time-reversed pairs of particles hu ; pan2006 ; cap8 ; 27 ; me16 ; ajm17 ; epja .

Tetraquarks are exotic hadrons composed of four quarks that can include two quarks and two antiquarks or four quarks of the same flavor. Despite being first proposed in the 1960s, their existence was only confirmed in 2013 by the Large Hadron Collider experiments. In recent years, the study of tetraquarks has gained increasing interest due to their unique properties and potential implications in particle physics  f1 . In this context, we propose to apply an algebraic framework to investigate the properties of heavy tetraquarks $[QQ][\bar{Q}\bar{Q}]$. Our approach is based on the $SU(1,1)$ algebraic technique  pan2006 ; ajm2021 ; ajmpt and extends the $sl$ boson system. We will derive a new solvable model for hadron physics that takes into account the vector quark pairing strengths and examine the mass spectra of tetraquarks.

In recent years, there has been a growing interest in exploring the properties of fully-heavy tetraquarks. Theoretically, several models have been proposed to describe these states, including the diquark-antidiquark model, the chromomagnetic interaction model Karliner:2016zzc . On the experimental side, various searches have been performed to identify fully-heavy tetraquarks in high-energy experiments. For instance, the LHCb collaboration searched for deeply bound $bb\bar{b}\bar{b}$ tetraquark states, but no significant excess was found in the $\mu^{+}\mu^{-}\Upsilon(1S)$ invariant-mass distribution Aaij:2018zrb . However, the CMS experiment reported a potential candidate of a fully bottom tetraquark $T_{4b}=[bb][\bar{b}\bar{b}]$ around 18-19 GeV dur . Moreover, the LHCb collaboration has recently reported the observation of a narrow peak and a broad structure in the $J/\psi$-pair invariant mass spectrum, which could originate from hadron states consisting of four charm quarks 1804391 . These experimental results provide valuable information for further theoretical investigations of fully-heavy tetraquarks.

Various theoretical models have been developed to study fully-heavy tetraquarks, including phenomenological mass formulae Karliner:2016zzc ; Berezhnoy:2011xn ; Wu:2016vtq , QCD sum rules Chen:2016jxd ; Wang:2017jtz ; Wang:2018poa ; Reinders:1984sr , QCD motivated bag models Heller:1985cb , NR effective field theories Anwar:2017toa ; Esposito:2018cwh , potential models Ader:1981db ; Zouzou:1986qh ; Lloyd:2003yc ; Barnea:2006sd ; Richard:2018yrm ; Richard:2017vry ; Vijande:2009kj ; Debastiani:2017msn ; Liu:2019zuc ; Chen:2019dvd ; Chen:2019vrj ; Chen:2020lgj ; Wang:2019rdo ; Yang:2020rih , non-perturbative functional methods Bedolla:2019zwg , and even some exploratory lattice-QCD calculations Hughes:2017xie . Some models predict that $QQ\bar{Q}\bar{Q}$ ($Q=c$ or $b$) bound states exist and have masses slightly below the respective thresholds of quarkonium pairs (see, for example, Refs.Chen:2016jxd ; Anwar:2017toa ; Karliner:2016zzc ; Berezhnoy:2011xn ; Wang:2017jtz ; Wang:2018poa ; Debastiani:2017msn ; Esposito:2018cwh ). However, other studies suggest that no stable $cc\bar{c}\bar{c}$ and $bb\bar{b}\bar{b}$ tetraquark bound states exist because their masses are larger than two-quarkonium thresholds (see, for example, Refs.Ader:1981db ; Lloyd:2003yc ; Richard:2018yrm ; Wu:2016vtq ; Hughes:2017xie ). A better understanding of the mass locations of fully-heavy tetraquark states is crucial for our comprehension of their underlying dynamics and for experimental studies.

In the context of describing a tetraquark system, diquark clusters play an important role. It is suggested that a tetraquark system, denoted by $T=Q_{1}Q_{2}\bar{Q}_{3}\bar{Q}_{4}$, consists of two point-like diquarks. To consider multi-level pairing in this context, we extend the interacting boson model using algebraic solutions of an $sl$-boson system pan2002 . The dynamical symmetry group in this case is generated by $s$ and $l$ operators, where $l$ represents the configuration of the multiquark states. In the Vibron Model, scalar $s$-bosons with spin and parity $l^{\pi}=0^{+}$ and vector $l$-bosons with spin and parity $l^{\pi}=1^{-}$ represent elementary spatial excitations. The generators in the finite-dimensional $SU(1,1)$ algebra satisfy the following commutation relations.

	$\displaystyle[S^{0}(l),S^{\pm}(l)]$	$\displaystyle=\pm S^{\pm}(l)\,,$		(1a)
	$\displaystyle[S^{+}(l),S^{-}(l)]$	$\displaystyle=-2S^{0}(l).$		(1b)

We can use the $\widehat{SU(1,1)}$ algebra to describe the rotation and vibration transitional Hamiltonian of the $T_{4c}=[cc][\bar{c}\bar{c}]$, $T_{4b}=[bb][\bar{b}\bar{b}]$, and $T_{2bc}=[bc][\bar{b}\bar{c}]$ systems. It is worth mentioning that the quasi-spin algebras have been extensively discussed in previous studies, such as Refs. pan2002 ; ajm17 .

Taking into account the generators of the $SU^{l}(1,1)$-algebra for tetraquarks given by Eqs.(1a) and(1b), we can express the relevant quantities as linear combinations of these generators.

	$\displaystyle S^{+}({l})$	$\displaystyle=\frac{1}{2}\,l^{{\dagger}}\cdot l^{{\dagger}}\,,$		(2a)
	$\displaystyle S^{-}({l})$	$\displaystyle=\frac{1}{2}\,{\tilde{l}}\cdot{\tilde{l}}\,,$		(2b)
	$\displaystyle S^{0}({l})$	$\displaystyle=\frac{1}{2}\,\left(l^{{\dagger}}\cdot{\tilde{l}}+\frac{2l+1}{2}\right)\,,$		(2c)

where $l^{{\dagger}}$ is the creation operator of an l-boson constituting the tetraquark, and $\tilde{l}_{\nu}=(-1)^{\nu}l_{-\nu}$.

A complementary relation for tetraquark states can be expressed by

$\displaystyle|N;n_{l}\,\nu_{l}\,,n_{\Delta}JM\rangle=|N;\kappa_{l}\,\mu_{l}\,,n_{\Delta}JM\rangle\,,$

(3)

with $\kappa_{l}=\frac{1}{2}\nu_{l}+\frac{1}{4}(2l+1)$ and $\mu_{l}=\frac{1}{2}n_{l}+\frac{1}{4}(2l+1)$, where $N$, $n_{l}$, $\nu_{l}$, $J$ and $M$ are quantum numbers of $U(N)$, $U(2l+1)$, $SO(2l+1)$, $SO(3)$ and $SO(2)$, respectively. The quantum number $n_{\Delta}$ is an additional one needed to distinguish different states with the same $J$.

The infinite dimensional $\widehat{SU(1,1)}$ Lie algebra is defined by

	$\displaystyle S_{n}^{\pm}$	$\displaystyle=c_{Q_{1}}^{2n+1}S^{\pm}(l_{1})+c_{Q_{2}}^{2n+1}S^{\pm}(l_{2})+c_{\bar{Q}_{3}}^{2n+1}S^{\pm}(\bar{l}_{3})$
		$\displaystyle+c_{\bar{Q}_{4}}^{2n+1}S^{\pm}(\bar{l}_{4}),$		(4a)
	$\displaystyle S_{n}^{0}$	$\displaystyle=c_{Q_{1}}^{2n}S^{0}(l_{1})+c_{Q_{2}}^{2n}S^{0}(l_{2})+c_{\bar{Q}_{3}}^{2n}S^{0}(\bar{l}_{3})$
		$\displaystyle+c_{\bar{Q}_{4}}^{2n}S^{0}(\bar{l}_{4})\,,$		(4b)

The real-valued control parameters $c_{Q}$ and $c_{\bar{Q}}$ play a crucial role in determining the properties of tetraquarks. Specifically, $l_{1}$ and $l_{2}$ correspond to the first and second tetraquarks, respectively, while $\bar{l}_{3}$ and $\bar{l}_{4}$ correspond to the third and fourth tetraquarks. Additionally, the integer $n$ can take on values of $1$, $2$, $3$, and so on.

To ensure that the fully-heavy tetraquarks satisfy the correct properties, we impose the condition $S^{-}(l)|lw\rangle=0$ on the lowest weight state. The state $|lw\rangle$ can be defined as follows:

$\displaystyle|lw\rangle=|{N};\kappa_{l}\,\mu_{l},n_{\Delta}JM\rangle,$

(5)

where $N=2k+\nu_{Q_{1}}+\nu_{Q_{2}}+\nu_{\bar{Q}_{3}}+\nu_{\bar{Q}_{4}}$. Hence, we have

$$S_{n}^{0}|lw\rangle=\Lambda_{n}^{l}|lw\rangle,\,\,\,\Lambda_{n}^{l}=\sum_{l}c_{l}^{2n}\frac{1}{2}\left(n_{l}+\frac{2l+1}{2}\right).$$

(6)

The system shows vibrational and rotational transitions due to continuous variations of the pairing strengths, $c_{l}$, in the closed interval $[0,1]$. The all-heavy tetraquark pairing model undergoes a quantum phase transition. The vibration limit is reached when ${c_{Q_{1}}=c_{Q_{2}}=c_{\bar{Q}3}=c_{\bar{Q}4}=0}$, while the rotational limit is attained when ${c_{Q_{1}}=c_{Q_{2}}=c_{\bar{Q}3}=c_{\bar{Q}4}=1}$. In our analysis, we obtained diverse values for the control parameters, $c_{Q_{i}}$ and $c_{\bar{Q}_{i}}$, in the interval $[0,1]$ with $i=1,\ldots,4$, between the two limits.

The Hamiltonian of the heavy tetraquark pairing model is expressed in terms of the Casimir operators ${\hat{C}_{2}}$ using branching chains. The first two terms of the Hamiltonian, $S_{0}^{+}S_{0}^{-}$ and $S_{1}^{0}$, are associated with the $SU(1,1)$ algebra, while the remaining terms are constant in terms of the Casimir operators. In the duality relation for tetraquarks, the irreducible representations simplify the quasi-spin algebra chains (4) and (4), and the labels for the chains are related via the duality relations. The Hamiltonian for the heavy tetraquark pairing model is derived by utilizing the generators of the $SU(1,1)$ algebra. However, the pairing models of multi-level are also characterized by an overlaid $U(n_{1}+n_{2}+\ldots)$ algebraic structure with this branching: ${U(10)}_{N}\supset{SO(10)}_{\nu}\supset\mathop{SO(9)}_{\nu}\supset\mathop{SO(3)}_{s}\otimes\mathop{SO(3)}_{Q\,Q}\otimes\mathop{SO(3)}_{\bar{Q}\,\bar{Q}}\otimes\mathop{SO(3)}_{J}$

So, we can define the Hamiltonian with

	$\displaystyle\hat{H}$	$\displaystyle=g\,S_{0}^{+}S_{0}^{-}+\alpha\,S_{1}^{0}+\beta\,\hat{C}_{2}(SO(9))$
		$\displaystyle+\gamma_{1}\,\hat{C}_{2}(SO(3)_{R})+\gamma_{2}\,\hat{C}_{2}(SO(3)_{Q_{1}Q_{2}})$
		$\displaystyle+\gamma_{3}\,\hat{C}_{2}(SO{(3)_{\bar{Q}_{3}\,\bar{Q}_{4}}})+\gamma\,\hat{C}_{2}(SO{(3)_{J}}),$		(7)

where $g$, $\alpha$, $\beta$, $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$, and $\gamma$ are real-valued parameters.

To find the non-zero energy eigenstates with $k$-pairs, we exploit a Fourier Laurent expansion of the eigenstates of Hamiltonians which contain dependences on several quantities in terms of unknown $c$-number parameters $x_{i}$, and thus eigenvectors of the Hamiltonian for excitations can be written as

	$\displaystyle|k;\nu_{Q_{1}}\nu_{Q_{2}}\nu_{\bar{Q}_{3}}\nu_{\bar{Q}_{4}}n_{\Delta}JM\rangle=\sum_{n_{i}\in Z}a_{n_{1}n_{2}\ldots n_{k}}$
	$\displaystyle=x_{1}^{n_{1}}x_{2}^{n_{2}}x_{3}^{n_{3}}\ldots x_{k}^{n_{k}}S_{n_{1}}^{+}S_{n_{2}}^{+}S_{n_{3}}^{+}\ldots S_{n_{k}}^{+}|lw\rangle\,,$		(8)

and

	$\displaystyle S_{n_{i}}^{+}$	$\displaystyle=\frac{c_{Q_{1}}}{1-c_{Q_{1}}^{2}x_{i}}S^{+}(S_{1})+\frac{c_{Q_{2}}}{1-c_{Q_{2}}^{2}x_{i}}S^{+}(S_{2})$
		$\displaystyle+\frac{c_{\bar{Q}_{3}}}{1-c_{\bar{Q}_{3}}^{2}x_{i}}S^{+}(\bar{S}_{3})+\frac{c_{\bar{Q}_{4}}}{1-c_{\bar{Q}_{4}}^{2}x_{i}}S^{+}(\bar{S}_{4})\,.$		(9)

The coefficients $x_{i}$ are determined through the following set of equations

$$\frac{\alpha}{x_{i}}=\sum_{l}\frac{c_{l}^{2}(\nu_{l}+\frac{2l+1}{2})}{1-c_{l}^{2}x_{i}}-{\sum_{j\neq i}{\frac{2}{x_{i}-x_{j}}}}\,.$$

(10)

In the pursuit of finding exact solutions for a spin-spin interaction system, Gaudin utilized a similar structure Gaudin76 as an ansatz, which has now been verified as a consistent operator form in constructing the Bethe ansatz wavefunction for the present tetraquark system. To obtain the energy spectra, the Bethe ansatz equation (BAE), a non-linear equation, is employed for a $k$-pair excitation. The quantum number $k$-pair excitation pertains to the overall number of bosons $N$ and is linked to seniority numbers, specifically the quantum number $\nu_{l}$ of $SO(2l+1)$. As per equation (4), the allowed seniority numbers for a fixed $\nu_{l}$ include $n_{l}=\nu_{l}$, $\nu_{l}+2$, $\nu_{l}+4$, and so on. This information is well-established in the field. Our approach to calculating the masses of heavy tetraquarks follows the procedure outlined in Ref. pan2006 . To account for the bosonic nature of the excitations (vibrations and rotations), we use the totally symmetric representation (II) and define the boson number as the total number of vibrational states in the representation $[N]$.

Pairing in tetraquarks is an interesting phenomenon that affects their rotational and vibrational behavior. The quantum phase transition occurs between the vibrational and rotational limits in the fully-heavy tetraquark pairing model, and the quark (antiquark) configuration can undergo vibrations and rotations described by the quantum numbers $\nu_{Q_{i}}$, $\nu_{\bar{Q}_{i}}$, and $J$. While we will not consider bending and twisting in this analysis due to their higher mass requirements, we must account for the internal degrees of freedom of quarks and antiquarks. To address this complication, we apply the method of pairing strengths, following Refs.f1 ; pan2002 . This scheme illustrates the stringlike configuration of the tetraquark and the vibration-rotation pattern we aim to identify. Within the two-quark configuration, we must follow the operator $Q$ with $\bar{Q}$, but since we are dealing with tetraquarks, we could also have combinations of $QQ$ and $\bar{Q}\bar{Q}$. To determine the appropriate rotation-vibration pattern, we need to ensure that the pairing number $N=2k+\nu_{Q_{1}}+\nu_{Q_{2}}+\nu_{\bar{Q}3}+\nu_{\bar{Q}_{4}}$ is satisfied. We can optimize the control parameters to find the exact symmetry of vibration and rotation that gives us the desired pairing number. By understanding the pairing behavior in tetraquarks, we can better understand their physical properties and potentially make predictions for future experiments. In summary, our work builds on previous research in the field, but we introduce new ideas related to pairing in tetraquarks and optimize control parameters to identify the appropriate symmetry of vibration and rotation.

The determination of tetraquark mass in the diquark–anti-diquark pairing model requires solving the eigenvalue problem of Eq. (II). However, in addition to the spins of diquark and antidiquark clusters, the $J^{PC}$ quantum numbers that define a tetraquark state also include the total spin, spatial inversion symmetry, and charge conjugation of the system. Recent studies have shown that the $J^{PC}$ quantum numbers of a $Q_{1}Q_{2}\bar{Q}_{3}\bar{Q}_{4}$ system can be $0^{++}$, $1^{+-}$, and $2^{++}$, as discussed in Ref. yang . These quantum labels are essential for characterizing the properties of the tetraquark system. The total spin of the tetraquark is determined by the combination of the spins of diquark and antidiquark clusters, and it affects the tetraquark’s stability and decay properties. Spatial inversion symmetry is related to the tetraquark’s mirror image, and it determines whether the system is symmetric or asymmetric with respect to spatial inversion. Charge conjugation, on the other hand, is related to the transformation of particles to their corresponding antiparticles and is a fundamental symmetry of the strong interaction. Understanding the impact of total spin, spatial inversion symmetry, and charge conjugation on tetraquark states is crucial for predicting their properties and behavior. By considering these quantum numbers, we can gain insights into the tetraquark’s internal structure and its interactions with other particles. This knowledge is essential for advancing our understanding of the strong interaction and the behavior of exotic hadrons. For scalar, vector and tensor systems, we have:

1. 1.

   Two states for the scalar system:

   	$\displaystyle|0^{++}\rangle$	$\displaystyle=|0_{Q_{1}Q_{2}},0_{\bar{Q}_{3}\bar{Q}_{4}};J=0\rangle\,,$		(11a)
   	$\displaystyle|0^{++\prime}\rangle$	$\displaystyle=|1_{Q_{1}Q_{2}},1_{\bar{Q}_{3}\bar{Q}_{4}};J=0\rangle\,.$		(11b)

2. 2.

   Three states for the vector system:

   	$\displaystyle|A\rangle$	$\displaystyle=|0_{Q_{1}Q_{2}},1_{\bar{Q}_{3}\bar{Q}_{4}};J=1\rangle\,,$		(12a)
   	$\displaystyle|B\rangle$	$\displaystyle=|1_{Q_{1}Q_{2}},0_{\bar{Q}_{3}\bar{Q}_{4}};J=1\rangle\,,$		(12b)
   	$\displaystyle|C\rangle$	$\displaystyle=|1_{Q_{1}Q_{2}},1_{\bar{Q}_{3}\bar{Q}_{4}};J=1\rangle\,.$		(12c)

   Charge conjugation is a fundamental symmetry of the strong interaction that transforms particles into their corresponding antiparticles. This symmetry leads to different configurations in which $|A\rangle$ and $|B\rangle$ can interchange, while $|C\rangle$ remains odd.

   In the $J^{P}=1^{+}$ configuration, we have one $C$-even and two $C$-odd states. This arrangement plays a crucial role in determining the properties and behavior of the system. Understanding the implications of these configurations is essential for predicting the tetraquark’s stability and decay properties.

   	$\displaystyle|1^{++}\rangle$	$\displaystyle=\frac{1}{\sqrt{2}}(|A\rangle+|B\rangle)\,,$		(13a)
   	$\displaystyle|1^{+-}\rangle$	$\displaystyle=\frac{1}{\sqrt{2}}(|A\rangle-|B\rangle)\,,$		(13b)
   	$\displaystyle|1^{+-\prime}\rangle$	$\displaystyle=|C\rangle\,.$		(13c)

   When considering tetraquarks, it is essential to choose appropriate values for the spin of the quark-antiquark pairs. In particular, the selection of spin states can impact the overall properties and behavior of the system.

   In the case of a tetraquark composed of $Q_{1},Q_{2},\bar{Q}_{3},$ and $\bar{Q}_{4}$, the appropriate spin states depend on the charge conjugation of the system. Specifically, when $C=+$, the only allowed state is one where $Q_{1}\bar{Q}3$ has a spin of $S{Q_{1}\bar{Q}_{3}}=1$.

3. 3.

   One state for the tensor system:

   $$|2^{++}\rangle=|1_{Q_{1}Q_{2}},1_{\bar{Q}_{3}\bar{Q}_{4}};J=2\rangle\,,$$

   (14)

   where this state has also $S_{Q_{1}\bar{Q}_{3}}=1$.