Hidden charm tetraquark states in a diquark model

The purpose of the present study is to explore the mass spectrum of the hidden charm tetraquark states within a diquark model. Proposing that a tetraquark state is composed of a diquark and an antidiquark, the masses of all possible $[qc][\bar{q}\bar{c}]$, $[sc][\bar{s}\bar{c}]$, and $[qc][\bar{s}\bar{c}]$ $\left([sc][\bar{q}\bar{c}]\right)$ hidden charm tetraquark states are systematically calculated by use of an effective Hamiltonian, which contains color, spin, and flavor dependent interactions. Apart from the $X(3872)$, $Z(3900)$, $\chi_{c2}(3930)$, and $X(4350)$ which are taken as input to fix the model parameters, the calculated results support that the $\chi_{c0}(3860)$, $X(4020)$, $X(4050)$ are $[qc][\bar{q}\bar{c}]$ states with $I^GJ^{PC}=0^+0^{++}$, $1^+1^{+-}$, and $1^-2^{++}$, respectively, the $\chi_{c1}(4274)$ is an $[sc][\bar{s}\bar{c}]$ state with $I^GJ^{PC}=0^+1^{++}$, the $X(3940)$ is a $[qc][\bar{q}\bar{c}]$ state with $I^GJ^{PC}=1^-0^{++}$ or $1^-1^{++}$, the $Z_{cs}(3985)^-$ is an $[sc][\bar{q}\bar{c}]$ state with $J^{P}=0^{+}$ or $1^+$, and the $Z_{cs}(4000)^+$ and $Z_{cs}(4220)^+$ are $[qc][\bar{s}\bar{c}]$ states with $J^{P}=1^{+}$. Predictions for other possible tetraquark states are also given.

The purpose of the present study is to explore the mass spectrum of the hidden charm tetraquark states within a diquark model. Proposing that a tetraquark state is composed of a diquark and an antidiquark, the masses of all
In the present work, we use an effective Hamiltonian which contains explicitly the color-spin, spin, flavor-spin, and flavor dependent interactions to investigate the mass spectrum of the hidden charm tetraquark states composed of diquarks and antidiquarks. We systematically calculate the masses of all ¯ ]) hidden charm tetraquark states, and discuss which of the observed states can be accommodated in a diquark scenario and which cannot. We also give predictions for the possible tetraquark states which have not yet been observed so far.
The paper is organized as follows. In Sec. II, we construct the wave functions for the hidden charm tetraquark states. In Sec. III, we present the effective Hamiltonian employed in the present work and give the mass formula for the tetraquark states considered. In Sec. IV, we show the results of the calculated masses of the hidden charm tetraquark states and give discussions for these theoretical results. Finally, in Sec. V, we give a brief summary.

II. WAVE FUNCTIONS OF THE HIDDEN CHARM TETRAQUARK STATES
In color space, each quark (antiquark) belongs to a color 3 (3 ) representation, two quarks can stay in color3 or 6 configurations, and two antiquarks can stay in color 3 or6 configurations. It is well known that the interactions arising from QCD are attractive between a pair of quarks in color3 state or a pair of antiquarks in color 3 state [15]. Thus, the color wave function for a tetraquark state composed of a pair of diquark and antidiquark can be constructed straightforwardly as (1) Thus, similar to the ′¯ ′ -meson octet and singlet multiplets, the hidden charm tetraquark states can decompose into a flavor SU(3) octet multiplet and a flavor SU(3) singlet ′ . Explicitly, the flavor wave functions for the states denoted by the symbols 0 , , + , + , − , 0 , − ,¯ 0 , and ′ can be written as [23] The symmetry states from the flavor SU(3) octet and ′ from the flavor SU(3) singlet have the same isospin 0. They can couple together to result in the physically observed states and¯ ′ : with being the mixing angle. For simplicity, an "ideal mixing" is chosen, i.e. ≈ 54.732°, which gives In short, the above-mentioned flavor wave functions for hidden charm tetraquark states composed of diquarks and antidiquarks could be divided to three types of configurations, i. and [¯ ¯ ] asD andD , respectively, for the sake of simplicity.
In spin space, the wave functions for the three flavor types of configurations, DD, D D , and DD (D D ), can be constructed as follows. We mention that no orbital excitation inside a diquark or between a pair of diquark-antidiquark is considered in the present work.
A. Spin wave functions for DD and D D As can be seen from Eqs. (3)−(11), the isospin triplet − , 0 , and + , and isospin singlet¯ are tetraquark states that have the DD type configurations, and the isospin singlet¯ ′ is the tetraquark state that has the D D type configuration. Their spin wave functions can be constructed as [18,29,30] = 0 ++ : |0, 0 0 , |1, 1 0 ; (12) Here each ket denotes D , D DD or D , D D D , with being the spin of the diquark (antidiquark) or a pair of diquark and antidiquark indicated by the corresponding subscript. Note that for D D states the isospin is = 0 and the parity is given by = (−1) = .
B. Spin wave functions for DD or D D One sees from Eqs. (3)−(11), the isospin doublet 0 and + are tetraquark states that have the DD type configuration, and the isospin doublet − and¯ 0 are tetraquark states that have the D D type configuration. Their spin wave functions can be constructed as Here each ket denotes D , D DD or D , D D D , with being the spin of the diquark (antidiquark) or a pair of diquark and antidiquark indicated by the corresponding subscript. Note that both the DD and D D configuration states are constructed to have particular strangeness quantum numbers [c.f. Eqs. (5) and (6)], and consequently, they are not eigenstates of the charge conjugation operator.

A. The effective Hamiltonian
The phenomenological Hamiltonian of a diquark model is usually parametrized as a sum of the diquark masses and an effective potential composed of color-spin and color-electric interaction terms [15,23,34] which are inspired by one-gluon exchange potential and instant-induced interaction [28]. Such a potential can be written explicitly as [34] where the parameter and the masses and depend on the flavor of constituents and , while the parameter is flavor independent.
represents the Gell-Mann matrix for the color SU(3) group. In order to describe the mass splits of isospin multiplets of the considered tetraquark states, we introduce, analogously to the color dependent potential in Eq. (19), the following flavor dependent potential: where the parameters and are both flavor independent, and represents the Gell-Mann matrix for the flavor SU(3) group. Note that in Eqs. (19) and (20), the summations are performed over all pairs of quarks (antiquarks), either inside a diquark (antidiquark) or between a diquark and an antidiquark for a tetraquark state. The effective Hamiltonian of the model is then written as with being the effective mass of the th constituent which includes also those effects not accounted for by the abovementioned interactions.
The color matrix elements for tetraquark states composed of diquark-antidiquark are The flavor matrix elements are for tetraquark states − , 0 , + , 0 , + , − , and¯ 0 defined in Eqs. (3)−(6), and are for tetraquark states¯ and¯ ′ defined in Eqs. (10)−(11). The spin matrix elements can be calculated directly from the spin wave functions constructed in Eqs. (12)− (18). The details of the calculation are given in Appendix A, and the results are listed in Table I.  Table I, the masses of hidden charm tetraquark states DD, D D , and DD (D D ) can be calculated straightforwardly.
The parameter can be fixed by the mass splits of / and , which gives with / and being the masses of / and , respectively.
All the model parameters needed in the present work are listed in Table II.

IV. NUMERICAL RESULTS
With the values of model parameters listed in Table II, Table III, where the first two columns show the quantum numbers and the calculated mass of each tetraquark configuration, and the last three columns show the particle name, the quantum numbers , and the energy of each state advocated by the Particle Data Group (PDG) [38]. In the third column, the states (3872), (3900), and The masses are in MeV. The first two columns show the quantum numbers and mass of each tetraquark configuration from our theoretical model, and the last three columns show the particle name, quantum numbers, and energy of each state advocated in PDG [38]. In the third column, the states marked with " * " are input used to fix the model parameters, and the states marked with "?" are those that have more than one possible assignment.  The masses are in MeV. The first column shows the quantum numbers of each theoretical tetraquark configuration. The second and third columns show the masses calculated in model I and model II, where the (4350) is assumed to be a tetraquark state with = 0 + 0 ++ and 0 + 2 ++ , respectively. The last three columns show the particle name, quantum numbers, and energy of each state advocated in PDG [38]. In the fourth column, the states marked with " * " and "?" are input used to fix the model parameters and have more than one possible assignment. Note that for [ ] [¯ ¯ ] configurations, = 0 and = (−1) .  . The masses are in MeV. The first column shows the quantum numbers of each theoretical tetraquark configuration. The second and third columns show the masses calculated in model I and model II, respectively. The last three columns show the particle name, quantum numbers, and energy of each state reported in Ref. [8] for (3985) − and in Ref. [9] for (4000) + and (4220) + . In the fourth column, the states marked with "?" are those that have more than one possible assignment.  Table IV. There, the first column shows the quantum numbers of each theoretical tetraquark configuration, the second and third columns show the masses calculated in model I and model II, where the (4350) is assumed to be a tetraquark state with = 0 + 0 ++ and 0 + 2 ++ , respectively. The last three columns show the particle name, quantum numbers, and energy of each state advocated in PDG [38]. In the fourth column, the states marked with " * " and "?" are input used to fix the model parameters and have more than one possible assignment. Note that the quantum numbers are not shown in Table IV Table V. There, the first column shows the quantum numbers of each theoretical tetraquark configuration. The second and third columns show the masses calculated in model I and model II, where the (4350) is assumed to be a tetraquark state with = 0 + 0 ++ and 0 + 2 ++ , respectively. The last three columns show the particle name, quantum numbers, and energy of each state reported in Ref. [8] for (3985) − and in Ref. [9] for (4000) + and (4220) + . In the fourth column, the states marked with "?" are those that have more than one possible assignment.  Thus, we leave such an analysis to future work when more experimental information becomes available.

V. SUMMARY
In the present work, we employ a diquark model to explore the mass spectrum of the hidden charm tetraquark states composed of diquarks and antidiquarks. The effective model Hamiltonian we used contains explicitly the color-spin, spin, flavor-spin, and flavor dependent interactions. We systemati-