Open-charm tetraquark $X_c$ and open-bottom tetraquark $X_b$

Motivated by the LHCb observation of exotic states $X_{0,1}(2900)$ with four open quark flavors in the $D^- K^+$ invariant mass distribution in the decay channel $B^\pm \to D^+ D^- K^\pm$, we study the spectrum and decay properties of the open charm tetraquarks. Using the two-body Coulomb and chromomagnetic interactions diquark configuration model, we find that the two newly observed states can be interpreted as a radial excited tetraquark with $J^P=0^+$ and an orbitally excited tetraquark with $J^P=1^-$, respectively. We then explore the mass and decays of the other flavor-open tetraquarks made of $su \bar d \bar c$ and $ d s \bar u \bar c$, which are in the $\bar 6$ or $15$ representation of the flavor SU(3) group. We point that these two states can be found through the decays: $X^{(\prime)}_{d s\bar{u}\bar{c}}\to (D^- K^-, D_s^- \pi ^-) $, and $X^{(\prime)}_{s u\bar{d}\bar{c}}\to D_s^-\pi ^+ $. We also apply our analysis to open bottom tetraquark $X_b$ and predict their masses. The open-flavored $X_b$ can be discovered through the following decays: $X_{ud\bar s\bar{b}}\to B^0K^+$, $X^{(\prime)}_{d s\bar{u}\bar{b}}\to (B^0 K^-, B_s^0 \pi ^-) $, and $X^{(\prime)}_{s u\bar{d}\bar{b}}\to B_s^0\pi ^+ $.


I. INTRODUCTION
Very recently, the LHCb collaboration has reported an intriguing and important discovery of two exotic structures with open quark flavors in the invariant mass distribution of D − K + of the channel B ± → D + D − K ± [1][2][3]. The relatively narrower one, named as X 0 (2900), has the mass and decay width as [3] m X0(2900) = (2.866 ± 0.007 ± 0.002)GeV, Γ X0(2900) = 57 ± 12 ± 4MeV, while the broader one is called X 1 (2900) and has m X1(2900) = (2.904 ± 0.005 ± 0.001)GeV, Γ X1(2900) = 110 ± 11 ± 4MeV. These two structures are 502MeV and 540MeV higher than the DK threshold, respectively. Both of them can strongly decay into D − K + and thus have the minimum quark content [udsc]. Once that this discovery is confirmed, it is anticipated that our knowledge of QCD color confinement will be greatly deepened.
In Ref. [22], we pointed out the existence of the open charm X c tetraquark states in 2016 and firstly proposed to hunt for the X c states in B and B c decays. Based on the two-body Coulomb and chromomagnetic interactions model, we calculated the masses of the X c tetraquarks. The 0 + and 1 + ground-states composed of [udsc] are predicted to lie in the range 2.4 GeV to 2.6 GeV having a limited phase space for decays into D − K + which cannot be identified with new the X 0,1 (2900) states. But it is interesting to notice that the newly observed X 0,1 (2900) can be attributed to the orbitally and radially excited state. One main focus of this work is to explore this possibility.
In addition, the discovery of the X 0,1 (2900) is of great value to explore other related tetraquark states such as the ones are composed of [usdc] and [dsūc]. In the flavor SU(3) symmetry, the charmed tetraquarks are decomposed as the6 or 15 representation. In the following we will carry out a calculation of the masses for these tetraquarks, and the corresponding open bottom multiplets X b . We will also use flavor SU(3) symmetry to study related strong two body hadronic decays which may provide some guidances for experimental searches.
The rest of this paper is organized as follows. In Sec. II, the heavy tetraquarks are decomposed into different irreducible representations and the spectra of X c and X b tetraquarks is predicted. Using the SU(3) flavor symmetry, decay properties of X c and X b tetraquarks are given in Sec. III. We also discuss the golden channels to hunt for the possible X 0,1 (2900) partners. A brief summary is given in the last section. To start with, we classify heavy tetraquarks with open-charm (bottom) according to SU(3) representations. These tetraquarks can be denoted as X Q (or X qq ′q′′Q when the flavor component is needed), where q, q ′ and q ′′ are light quarks, and Q = c, b is a heavy quark. There are many applications of SU(3) flavor symmetry in Refs. [26][27][28][29][30][31][32][33][34][35][36][37][38]. Considering the fact that the light quarks belong to a triplet 3 representation and the heavy quark Q is a singlet in the flavor SU (3) symmetry, the heavy tetraquarks are classified into different irreducible representations as 3 ⊗ 3 ⊗3 = 3 ⊕ 3 ⊕6 ⊕ 15. When the heavy tetraquarks with four different flavors are involved, one only needs to consider the6 and 15 representations. The observed states may belong to one of these two representations but a specific assignment requests more experimental and theoretical studies.
The6 representation will be denoted as X k [i,j] (i, j, k = 1, 2, 3 corresponding to the u, d, s quark), where the indices i and j are antisymmetric. Their explicit expression are [17] We will use X k {i,j} to abbreviate the 15 representation, where the indices i and j are symmetric [17]: Note that the heavy quark c or b is not explicitly shown in the above. But one can easily add the heavy quark in the following application. The above SU(3) classification is applicable to the ground states, orbitally-excited and radially-excited tetraquarks. In the following we carry out a calculation of their corresponding masses using the two-body Coulomb and chromomagnetic interactions model.
Based on the diquark configuration proposed in Ref. [39], we assume that the open heavy flavor tetraquark is composed of a light diquark, a light quark, and a heavy flavor quark. Their mass spectra can be calculated using the two-body Coulomb and chromomagnetic interactions. Correspondingly, the effective Hamiltonian for a tetraquark state with spin and orbital interaction is written as [40][41][42][43], with the spinal and orbital interactions The parameters in the above equations can be determined from various meson and baryon masses. For example, the quark mass and the color-spin couplings can be determined from the pseudoscalar and vector mesons with According to the previous extractions in Refs [40][41][42][43][44][45][46], we give a collection of the relevant parameters in the following. The quark masses is chosen as m q = 305MeV, m s = 490MeV, m c = 1.670GeV. The diquark mass satisfies the relation m ss − m sq = m sq − m qq and we have m qq = 0.395GeV, m sq = 0.590GeV, and m ss = 0.785GeV. The color-spin coupling constant is used as: and (κ cs ) 0 = 72MeV. We will employ the relation κ ij = 1 4 (κ ij ) 0 for the quark-antiquark coupling, which is derived from one gluon exchange model. The spin-orbit and orbital coupling constants can be extracted from the P-wave meson or baryons. We adopt Asc = A δ = 50MeV and B c = 495MeV [44], Aūb = A δ = 5MeV and B b = 408MeV or Asb = A δ = 3MeV and B b = 423MeV [42,45]. The spectra of S-wave tetraquarks X c (1S) have been given in Ref. [22]. The 0 + [udsc] ground-state was determined to have a mass 2.36GeV, which is much lower than the new LHCb data. Thereby the identification of the observed 0 + and 1 − states is likely to rely on the orbitally or radially excited states.
We now calculate the spectra of X c (1P ) and X c (2S) with different light quark contents from orbital or radial I: Predictions of the masses (GeV) of orbitally excited X c,b (1P ) tetraquarks in both6 and 15 representations. In the table two or more different masses appear in identical J P for some states because of hyperfine splitting from spin-spin or spin-orbital coupling. The same reason also give more than one entries in Table II. excitations, and the results are tabulated in Tab. I and Tab. II, respectively. From the orbitally excited states in Tab. I, one can see that the X udsc in the 15 representation with 1 − has a mass around 2.91GeV and can decay into D − K + . This could be a candidate to explain the newly X 1 (2900) states observed by LHCb collaboration [1]. The J P = 1 − X udsc states with the mass around (2.88, 2.98, 3.00)GeV and the J P = 1 − X ′ udsc states with the mass around (2.81, 2.86)GeV are also interesting and can decay into D − K + , and thus future experiments are likely to discover them. In the table we also listed masses II: Predictions of the masses (GeV) of radially excited X c,b (2S) tetraquarks in both6 and 15 representations. The radially excitation is estimated around 0.5GeV. The ground states of Xc(1S) tetraquarks have been predicted in Ref. [22]. For simplification, the masses of X c,b (1S) tetraquarks can be also obtained through the masses of X c,b (2S) minus the radial excitation 0.5GeV, i.e. m X c,b (1S) ≈ m X c,b (2S) − 0.5GeV. for states with 2 − and 3 − , but other orbitally excited states X c (1P ) either do not have the quark content [udsc] or can not directly decay into D − K + by the spin-parity constraint. We will discuss their decay patterns for experimental searches later.
To explain the X 0 (2900), one needs to find a 0 + state with higher mass than the ground state. We find that X c (2S) has such a possibility. One can estimate the radial excitation using the fact of charmonium radial excitation m J/ψ(2S) − m J/ψ(1S) = (3686.7 − 3096.9)MeV = 589.8MeV [47]. In Tab. II, we give the results for the masses of radially excited X c (2S) tetraquarks. From this table, one can see that the X ′ udsc with 0 + in the6 representation has a mass around 2.86GeV and can decay into D − K + . This state can be identified as the newly X 0 (2900) states observed by LHCb collaboration [1]. The J P = 0 + X ′ udsc state with the mass around 2.99GeV is also interesting for experimental search. Other radially excited states X c (2S) either do not have the quark content [udsc] or can not directly decays into D − K + by spin-parity constraint.
Our analysis can be extended to the Q = b case, and the spectra of X b (1S, 2S) and X b (1P ) tetraquarks are given in Tab. I and Tab. II. In Tab. I, we present the masses of P-wave X b (1P ) tetraquark partners in both 6 and 15 representation. In Tab. II, we present the masses of S-wave X b (2S) tetraquark partners in both 6 and 15 representation, and the masses of S-wave X b (1S) tetraquark partners can be also obtained through We now study the possible strong decays of the X Q (1P ) and X Q (2S) and focus on the Q i + P final states.
The 1 − X Q (1P ) quantum field is labeled as X µ , while the 0 + X Q (2S) is labeled as X. The Q i is one of the heavy meson D i and B i mesons as D i = (D 0 (uc), D − (dc), D − s (sc)) and B i = (B + (ub), B 0 (db), B 0 s (sb)). The P is a pseudo-scalar meson in the octet Using heavy quark effective theory, we find that the interacting termsQv · AX andQA µ X µ are responsible for the leading decays [17]. Here A is the axial-vector field, and v is the heavy quark velocity. Note that all the SU(3) flavor indices are contracted in above equation. Their flavor SU(3) transformation are where ξ † is defined as ξ(x) = Σ(x) and Σ(x) = exp(2iΠ/ √ 2f ). The X c → D i P decay amplitude can then be parameterized as with β and β ′ being the nonperturbative amplitudes to be given latter. Similarly the X b → B i P decay amplitudes can be parameterized as Results for the X c → D i P amplitudes are collected in Tab. III and Tab. IV, while the results for the X b → B i P amplitudes can be obtained using the replacements Tab. III and Tab. IV. It is interesting to note that one can also reconstruct X 0,1 in X 0,1 → D 0 K 0 , whose decay width is the same order of X 0,1 → D − K + . This serves as a confirmation of the model. The other X c tetraquark partners can be searched for using results in Tab. III and IV. Of particular interests are the tetraquarks with four different quarks can be hunted by and the decay width where the dimensionless coupling β ′ c is parameterized as β ′ We have We can estimate the decay width of X 0 as where the SU(3) symmetry breaking effects are neglected.
Using the LHCb measurement m X0 = 2.866GeV and Γ X0 = 57MeV, one can extract the dimensionless coupling as β ′ c ≈ 0.37. For X udsc (1 − ) → D − K + , we have the amplitude and the decay width where the dimensionless coupling β c is parameter- The decay width of X 1 is then given as Using the LHCb measurement m X1 = 2.904GeV and Γ X1 = 110MeV, one can extract the dimensionless coupling β c ≈ 0.30. From the above calculation, one can find that β c ≈ β ′ c .
In the following, we will give some relations of the decay widths of the new decay channels of X 0,1 and their counterparts.
From the flavor SU(3) amplitudes in Tab. III, we have Thus we can estimate the following decay widths for the open charm tetraquarks in6 representation From the flavor SU(3) amplitudes in Tab. IV, we have Thus we can estimate the following decay widths for the open charm tetraquarks in 15 representation As a straightforward extension, one can also investigate the X b tetraquark decays. We explicitly give predictions of the masses and decay widths for X b;0 and X b;1 , which are the partner of X 0 (2900) and X 1 (2900). As discussed before, the X ′ udsc state with 0 + and mass 2.86GeV can be used to explain X 0 (2900) while the X udsc state with 1 − and mass 2.91GeV can be used to explain X 1 (2900). So one can obtain the masses of X b;0 and X b;1 with thec →b replacement from Tab. I and Tab. II. We have m X b;0 = 6.20GeV, m X b;1 = 6.27GeV.

IV. CONCLUSION
In this paper, we have studied the spectra and the decay properties of open-charm tetraquarks X c and openbottom tetraquarks X b . The newly X 0,1 (2900) observed by the LHCb collaboration can be interpreted as a radial excited tetraquark X c composed of [udsc] with J P = 0 + and an orbitally excited tetraquark with J P = 1 − , respectively. Using the flavor SU(3) symmetry, we made a detailed classification of all open charm tetraquarks, and then explored the mass and decays of the other flavoropen tetraquarks made of sudc and dsūc. We pointed that these two states can be found through the decays: X sudb → B 0 s π + . We hope that these theoretical proposals can be carried out in future experimental studies.