#### Abstract

In this article, we take into account our previous calculations based on the QCD sum rules, and tentatively assign the as the tetraquark molecular state or tetraquark state with the , and assign the and as the 1S and 2S tetraquark states, respectively, with the . Then, we extend our previous works to investigate the LHCb’s new tetraquark candidate as the first radial excited state of the with the QCD sum rules and obtain the mass , which is in very good agreement with the experimental value . Furthermore, we investigate the two-meson scattering state contributions in details and observe that the two-meson scattering states alone cannot saturate the QCD sum rules, the contributions of the tetraquark states play an unsubstitutable role, and we can saturate the QCD sum rules with or without the two-meson scattering states.

#### 1. Introduction

In 2009, the CDF collaboration observed an evidence for the in the mass spectrum for the first time with a significance of larger than [1]. Subsequently, the existence of the was confirmed by the CDF, CMS, and D0 collaborations [2–5]. In 2016, the LHCb collaboration accomplished the first full amplitude analysis of the decays and acquired a good description of the experimental data in the 6D phase space and confirmed the and and determined the spin-parity-charge-conjugation [6, 7]. Furthermore, the LHCb collaboration also observed two new exotic hadrons and in the mass spectrum and determined the quantum numbers [6, 7]. The Breit-Wigner masses and widths are

Recently, the LHCb collaboration accomplished an improved full amplitude analysis of the decays using 6 times larger signal yields than previously analyzed and observed a hidden-charm and hidden-strange tetraquark candidate and () in the mass spectrum of the with a significance of (), the favored assignment of the spin parity is (), and the Breit-Wigner mass and width are () and (), respectively [8]. Furthermore, the LHCb collaboration also observed two new tetraquark (molecular) state candidates and in the mass spectrum of the with the preferred spin parity and updated the experimental values of the masses and widths of the and [8]. The , , , , , and were observed in the mass spectrum of the , and their quantum numbers are , , and for the -wave couplings and , , , and for the -wave couplings. In the present work, we discuss the possible assignments of the , , , and based on the QCD sum rules.

The article is arranged as follows: in Section 2, we discuss the possible assignments of the and based on the QCD sum rules; in Section 3, we get the QCD sum rules for the masses and pole residues of the tetraquark states with the ; in Section 4, we obtain numerical results and give discussions, and Section 5 is aimed to get a conclusion.

#### 2. Possible Assignments of the and Based on the QCD Sum Rules

In this section, we discuss the possible assignments of the and according to our previous calculations with the QCD sum rules.

In Ref. [9], we construct the color-singlet-color-singlet type four-quark current to investigate the molecular state:

The current has definite charge conjugation but has not definite parity, and the components and have positive parity and negative parity, respectively, where the space indexes , , , . The neutral current couples potentially to the two-meson scattering states or tetraquark molecular states with the quantum numbers and , where we use the symbols and to represent the color-neutral clusters with the same quantum numbers as the physical and mesons, respectively. In the QCD sum rules, we choose the local currents, and it is better to call the as the color-singlet-color-singlet type tetraquark state than call it as the tetraquark molecular state. The traditional hidden-flavor mesons, such as the , , and quarkonia, have the normal quantum numbers , , , , , , , , . The components and couple potentially to the and tetraquark molecular states, respectively. We construct projection operators to project out the contribution of the component unambiguously and explore the tetraquark molecular state with the exotic quantum numbers using the QCD sum rules and acquire the prediction [9], which happens to coincide with the mass of the from the LHCb collaboration, [8].

The calculations based on the Bethe-Salpeter equation combined with the heavy meson effective Lagrangian also indicate that there exists such a tetraquark molecular state with the exotic quantum numbers [10, 11]. The predictions in Refs. [9–11] were achieved before the discovery of the . Whether or not the predictions of the QCD sum rules are reliable, the experimental data can reply. After the discovery of the by the LHCb collaboration, Yang et al. study the charmonium-like molecules with hidden-strange via the one-boson exchange mechanism and assign the to be the molecular state with the quantum numbers [12].

As long as the diquark-antidiquark type tetraquark states are concerned, we usually take the scalar (), pseudoscalar (), vector (), axialvector (), and tensor () diquark operators without introducing explicit -waves as the elementary building blocks to construct the interpolating currents. The tensor currents have both vector and axialvector components, and we construct projection operators to project out the spin parity and components explicitly and denote the corresponding operators as and , respectively, to avoid ambiguity. In Ref. [13], we choose the diquark-antidiquark type vector currents and : to interpolate the -type and -type tetraquark states with the quantum numbers and , respectively, and investigate their properties with the QCD sum rules, where the , , , , and are color indexes. We acquire the predictions: which happen to coincide with the masses of the and , respectively [13], and support assigning the and to be the tetraquark states with the symbolic quark constituents and with the quantum numbers and , respectively. The prediction of the mass was achieved long before the discovery of the .

In Refs. [14, 15], we construct the diquark-antidiquark type currents to explore the , , , and tetraquark states with the quantum numbers concordantly via the QCD sum rules, and the numerical results support assigning the and to be the ground state and first radial excited state of the tetraquark states, respectively, with the quantum numbers and assigning the to be the ground state tetraquark states with the quantum numbers . Furthermore, we also obtain the potability that assigning the to be the ground state tetraquark state with the [15]. Our predictions, are in very good agreement with the LHCb improved measurement and [8]. Other assignments of the and , such as the -wave tetraquark states with the , are also possible [16], and more theoretical and experimental works are still needed to obtain definite conclusion.

In summary, according to the (possible) quantum numbers, decay modes, and energy gaps, we can assign the and as the ground state and first radial excited state of the hidden-charm tetraquark states with the [14, 17], assign the and as the ground state and first radial excited state of the hidden-charm tetraquark states with the , respectively [18–20], and assign the and as the ground state and first radial excited state of the hidden-charm tetraquark states with the , respectively [21, 22]. If we assign the to be the first radial excited state of the tentatively, we can get the energy gap , and it is reasonable, see Table 1.

Moreover, in Ref. [17], Lebed and Polosa assign the and to be the and diquark-antidiquark type hidden-charm tetraquark states and , respectively, based on the effective Hamiltonian with the spin-spin and spin-orbit interactions. In Ref. [23], we construct the type and type axialvector currents with the quantum numbers to interpolate the and observe that only the type current can reproduce the mass and width of the in a consistent way.

#### 3. The as the Axialvector Tetraquark States

In this section, we extend our previous work [23] to investigate the as the first radial excitation of the with the QCD sum rules and discuss the possible assignment of the as the tetraquark state having the quantum numbers .

Firstly, we write down the two-point correlation function in the QCD sum rules: where

The current couples potentially to the tetraquark states with the . The tensor diquark operator has both the spin parity and components, and we project out the component via multiplying the tensor diquark operator by the vector antidiquark operator . In Ref. [23], we observe that the current can reproduce the mass and width of the satisfactorily.

At the hadron side, we isolate the ground state () and first radial excited state () contributions, which are supposed to be the pole contributions from the and , respectively, where the pole residues or decay constants are defined by , and the is the polarization vectors of the axialvector tetraquark states .

A hadron, such as the usually called quark-antiquark type meson, tree-quark type baryon, diquark-antidiquark type tetraquark state, and diquark-diquark-antiquark type pentaquark state, has definite quantum numbers and more than one Fock states. Any current operator with the same quantum numbers and same quark structure as a Fock state in the hadron couples potentially to this hadron; in other words, it has nonvanishing coupling to this hadron. Generally speaking, we can construct several current operators to interpolate a hadron or construct a current operator to interpolate several hadrons. Actually, a hadron has one or two main Fock states, and we call a hadron as a tetraquark state if its main Fock component is of the diquark-antidiquark type.

In the present work, the diquark-antidiquark type local four-quark current operator having the quantum numbers couples potentially to the diquark-antidiquark type tetraquark states with the same quantum numbers . On the other hand, this local current can be rearranged into a special superposition of a series of color-singlet-color-singlet type currents through the Fierz transformation both in the Dirac spinor space and color space: which couple potentially to the tetraquark molecular states or two-meson scattering states having the quantum numbers . The diquark-antidiquark type tetraquark states can be viewed as a special superposition of a series of color-singlet-color-singlet molecular states and embody the net effects, and vise versa.

The diquark-antidiquark type tetraquark state can be plausibly described by two diquarks in a double well potential which are separated by a barrier [24, 25], the spatial distance between the diquark and antidiquark leads to smaller wave-function overlap between the quark and antiquark constituents, and the repulsive barrier or spatial distance frustrates the Fierz rearrangements or recombinations between the quarks and antiquarks and therefore suppresses hadronizing to the meson-meson pairs [24–27].

If the color-singlet-color-singlet type components in Eq. (10), such as and , only couple potentially to the two-meson (TM) scattering states, we obtain the correlation function at the hadron side: where

, and we have taken the standard definitions of the decay constants: The is the polarization vectors of the vector and axialvector mesons.

We accomplish the operator product expansion for the correlation function up to the vacuum condensates of dimension 10 consistently [28–30]. In calculations, we assume dominance of the intermediate vacuum state tacitly, just like in previous works [28–30], and insert the intermediate vacuum state alone in all the channels, and vacuum saturation works well in the large limit [31]. Up to now, almost in all the QCD sum rules for the multiquark states, vacuum saturation is assumed for the higher dimensional vacuum condensates, except in some cases, the parameter which parameterizes deviations from the factorization hypothesis, is introduced by hand for the sake of fine-tuning [32], where , , , the , , , and are color indexes, and the , , and are Dirac spinor indexes.

In the original works, Shifman, Vainshtein, and Zakharov took the factorization hypothesis based on two reasons [33, 34]. The first one is the rather large value of the quark condensate , the second one is the duality between the quark and physical states, and they reproduce each other, counting both the quark and physical states (beyond the vacuum states) that maybe lead to a double counting [33, 34].

In the QCD sum rules for the , , and mesons, the are always companied with the fine-structure constant and play a minor important (or tiny) role, and the deviation from , for example, , cannot make much difference in the numerical predictions; though in some cases, the values can lead to better QCD sum rules [35, 36]. However, in the QCD sum rules for the multiquark states, the play an important role, large values, for example, if we take the value in the present case, we can obtain the uncertainties and , which are of the same order of the total uncertainties from other parameters. Sometimes, large values of the can destroy the platforms in the QCD sum rules for the multiquark states [37].

The true values of the higher dimensional vacuum condensates remain unknown or poorly known; if the true values or , the QCD sum rules for the multiquark states have considerably large systematic uncertainties and are less reliable than those of the conventional mesons and baryons [38]. We just make predictions for the multiquark masses with the QCD sum rules based on vacuum saturation and then confront them to the experimental data in the future to examine the theoretical calculations.

After the analytical expression of the QCD spectral density was acquired, we take the quark-hadron duality below the continuum thresholds and by including the contributions of the 1S state and 1S plus 2S states, respectively, and perform Borel transform in regard to the variable to obtain the two QCD sum rules: where . For the explicit expression of the spectral density at the quark and gluon level, one can consult Ref. [23]. We define with , , , and then acquire the QCD sum rules for the masses: where the indexes and . For the technical details in obtaining the QCD sum rules in Eqs. (17) and (18), one can consult Refs. [14, 20, 39].

On the other hand, if we saturate the hadron side of the QCD sum rules with the contributions of the two-meson scattering sates, we obtain the following two QCD sum rules:

In Eqs. (20) and (21), we introduce the parameter to measure the deviations from ; if , we can acquire the conclusion tentatively that the two-meson scattering states can saturate the QCD sum rules.

#### 4. Numerical Results and Discussions

At the QCD side, we choose the conventional values of the vacuum condensates , , , , and