Fully open-flavor tetraquark states bcq̄s̄ and scq̄b̄ with J = 0, 1

We have studied the masses for fully open-flavor tetraquark states bcq̄s̄ and scq̄b̄ with quantum numbers J = 0, 1. We systematically construct all diquark-antiquark interpolating currents and calculate the two-point correlation functions and spectral densities in the framework of QCD sum rule method. Our calculations show that the masses are about 7.1−7.2 GeV for the bcq̄s̄ tetraquark states and 7.0 − 7.1 GeV for the scq̄b̄ tetraquarks. The masses of bcq̄s̄ tetraquarks are below the thresholds of B̄sD and B̄∗ sD final states for the scalar and axial-vector channels respectively. The scq̄b̄ tetraquark states with J = 1 lie below the B c K∗ and B∗ sD thresholds. Such low masses for these possible tetraquark states indicate that they can only decay via weak interaction and thus are very narrow and stable.


I. INTRODUCTION
In the conventional quark model [1,2], hadrons generally have two kinds of structures: a meson consisting of a quark and an antiquark, and a baryon consisting of three quarks. However, quantum chromodynamics (QCD) allows the existence of hadrons different from the above two structures, such as the tetraquarks, hadronic molecules, pentaquarks, hybrids and so on [3][4][5][6].
A compact tetraquak is composed of a diquark and an antidiquark, bounding by the color force among quarks and antiquarks. The light tetraquarks have been widely studied via different theoretical methods [7][8][9][10]. For the heavy quark sector, the hidden-charm/bottom QQqq tetraquarks have been extensively investigated to interpret some observed XYZ states in various methods, such as the constituent quark models [11][12][13], meson exchange and scattering methods [14][15][16], QCD sum rules [17][18][19][20], chromomagnetic interaction models [21,22], etc. The doubly heavy tetraquark states QQqq have been studied to investigate the stability of tetraquarks [23,24]. In Ref. [25][26][27], the open-flavor heavy bcqq tetraquark states have also been investigated, the results suggest that their masses may lie below the corresponding two-meson thresholds. In addition, such tetraquarks cannot decay via the annihilation channels and thus they will be very stable with narrow widths.
Comparing to the above several tetraquark configurations, the fully open-flavor tetraquarks bcsq and scqb (q = u, d) are more exotic since they contain four valence quarks with totally different flavors. However, the studies of these tetraquarks have drawn much less interest to date. In Ref. [21], the authors studied the masses of qscb and qcsb tetraquarks by using the color-magnetic interaction with the flavor symmetry breaking corrections. Their results show that the masses of qscb and qcsb tetraquark states are about 7.1 GeV and 7.2 GeV, which are lower than the corresponding two-meson S-wave thresholds. In the heavy quark symmetry, the mass of bcqs tetraquark states with J P = 1 + was also evaluated to be around 7445 MeV [28], which is about 163 MeV above the DB * s threshold and thus allows such decay channel via strong interaction. The above conflicting results from different phenomenological models are inspiring more theoretical studies for the existence of these fully open-flavor tetraquark states. In this paper, we shall study the mass spectra of the fully open-flavor bcqs and scqb tetraquarks in the method of QCD sum rules [29,30]. This paper is organized as follows. In Sec. II, we construct the interpolating tetraquark currents of the bcqs and scqb systems with J P = 0 + , 1 + , respectively. In Sec. III, we evaluate the correlation functions and spectral densities for these interpolating currents. The spectral densities will listed in the appendix because of their complicated form. We extract the masses for these tetraquarks by performing the QCD sum rule analyses in Sec. IV. The last section is a brief summary.

II. INTERPOLATING CURRENTS FOR THE bcqs AND scqb TETRAQUARK SYSTEMS
In this section, we construct the interpolating currents for bcqs and scqb tetraquarks with J P = 0 + , 1 + . In general, there are five independent diquark fields, q T a Cγ 5 q b , q T a Cq b , q T a Cγ µ γ 5 q b , q T a Cγ µ q b , and q T a Cσ µν q b , where q stands for quark field, a, b are the color indices, C denotes the charge conjugate operator, and T represents the transpose of the quark fields. The q T a Cγ 5 q b and q T a Cγ µ q b are S-wave operators while q T a Cq b and q T a Cγ µ γ 5 q b are P-wave operators. The q T a Cσ µν q b contains both S-wave and P-wave operators according to its different components. To study the lowest lying bcqs and scqb tetraquark states, we use only S-wave diquarks and corresponding antidiquark fields to construct the tetraquark interpolating currents with quantum numbers J P = 0 + , 1 + . For the bcqs system, the scalar currents with J P = 0 + can be written as in which q is light quark field (up or down). The color structure for the currents J 1 and J 3 are symmetric [6 c ] bc ⊗ 6 c qs , while for the J 2 and J 4 are antisymmetric 3 c bc ⊗ [3 c ]qs. The axial-vector currents with J P = 1 + can be written as where the currents J 1µ and J 3µ are color symmetric while the J 2µ and J 4µ are color antisymmetric. For the scqb system, the currents with J P = 0 + are where the currents η 1 and η 3 are color symmetric with [6 c ] sc ⊗ 6 c qb , while the η 2 and η 4 are color antisymmetric with 3 c sc ⊗ [3 c ]qb. The currents with J P = 1 + are in which the currents η 1µ and η 3µ are color symmetric while the η 2µ and η 4µ are color antisymmetric.

III. QCD SUM RULES
In this section, we investigate the two-point correlation functions of the above scalar and axial-vector interpolating currents. For the scalar currents, the correlation function can be written as and for the axial-vector current The correlation function Π µν (p 2 ) in Eq. (6) can be expressed as where Π 0 p 2 and Π 1 p 2 are the scalar and vector current polarization functions related to the spin-0 and spin-1 intermediate states, respectively. At the hadron level, the correlation function can be written through the dispersion relation in which b n is the subtraction constant. In QCD sum rules, the imaginary part of the correlation function is defined as the spectral function where the "pole plus continuum parametrization" is adopted. The parameters f H and m H are the coupling constant and mass of the lowest-lying hadronic resonance H respectively with the polarization vector µ . At the quark-gluon level, we can evaluate the correlation function Π(p 2 ) and spectral density ρ(s) using the method of operator product expansion (OPE). To calculate the Wilson coefficients, we use the light quark propagator in coordinate space and heavy quark propagator in momentum space where q represents u, d or s quark and Q represents c or b quark. The superscripts a, b are color indices and x = x µ γ µ ,p = p µ γ µ . In this work, we evaluate the Wilson coefficients up to dimension eight condensates at the leading order in α s . To improve the convergence of the OPE series and suppress the contributions from continuum and higher states region, one can perform the Borel transformation to the correlation function in both hadron and quark-gluon levels. The QCD sum rules are then established as where M B is the Borel mass introduced via the Borel transformation and s 0 the continuum threshold. The lowest-lying hadron mass can be thus extracted via the following expression In this section, we perform the QCD sum rule analyses for the bcqs and scqb tetraquarks. We use the following values of quark masses and condensates [27,[31][32][33][34] in which the masses of u, d, s are the current quark masses in the M S scheme at a scale µ = 2 GeV. We consider the scale dependence of the charm and bottom quark masses at the leading order where is determined by evolution from the τ mass using PDG values.
To obtain a stable sum rule, the working regions should also be determined, i.e, the continuum threshold s 0 and the Borel mass M 2 B . The threshold s 0 can be fixed by minimizing the variation of the hadronic masses m H with the Borel mass M 2 B . The Borel mass M 2 B can be obtained by requiring the OPE convergence, which results in the lower bound of M 2 B , and a sufficient pole contribution, which results in the upper bound of M 2 B . Specific details of these procedures will be shown later. The pole contribution is defined as in which L 0 is defined in Eq. (12).

A. bcqs systems
We firstly perform the QCD sum rule analyses for bcqs tetraquarks. The spectral densities for the interpolating currents in Eqs. (1)-(2) are calculated and listed in the appendix A. For any interpolating current in the bcqs system, contributions from the quark condensate qq and quark-gluon mixed condensate qg s σ·Gq are numerically small since they are proportional to the quark mass m q and m s . The dominant nonperturbative contribution to the correlation function comes from the four-quark condensate qq ss . In Fig.1, we take the scalar interpolating current J 1 as an example to present the contributions to correlation function from the perturbative and various condensate terms. To extract the output parameters, the Borel mass M 2 B should be large enough to guarantee the convergence of OPE series. Here, we require that the four-quark condensate contribution be less than one-fifth of the perturbative term. In Fig.1, we can see that the convergence of OPE series can be ensured while M 2 B ≥ 5.4GeV 2 . . It is shown that the variation of m H with M 2 B minimizes around s 0 ∼ 58 GeV 2 , which will result in the working region 56 ≤ s 0 ≤ 60 GeV 2 for the scalar current J 1 . Using this value of s 0 , the upper bound of M 2 B can be obtained by requiring the pole contribution be larger than 30%. Finally, the working region of the Borel parameter for the scalar current J 1 can be determined to be 5.4 ≤ M 2 B ≤ 5.8GeV 2 . We show the Borel curves in these regions in Fig. 2 and extract the hadron mass to be m H = 7.17 ± 0.11 GeV. The errors come from the continuum threshold s 0 , condensates qq and qg s σ · Gq , the heavy quark masses m b and m c . The errors from the Borel mass and the gluon condensate are small enough to be neglected.
For all other interpolating currents in Eqs. (1)-(2), we perform similar analyses and obtain the suitable working regions for the threshold s 0 , Borel mass M 2 B , output hadron masses, pole contributions. We collect the numerical results in Table I for the scalar bcqs tetraquarks and Table II for the axial-vector bcqs tetraquarks. The last columns are the S-wave two-mesonB s D andB * s D thresholds for these possible tetraquark states. It is shown that both the scalar and axial-vector bcqs tetraquarks lie below the corresponding two-meson thresholds, implying their stabilities against the strong interaction.

B. scqb systems
For the scqb systems, we calculate and list the correlation functions and spectral densities in the appendix A for all interpolating currents in Eqs. (3)-(4). Comparing to the bcqs system, the OPE series behaviors are very different as shown in Fig.3 (for the scalar current η 1 ), where the contributions from the quark condensate qq and quark-gluon  mixed condensate qg s σ · Gq are dominant while the contribution from the four-quark condensate qq 2 is relatively small. Such difference is due to the existence of m Q qq and m Q qg s σ · Gq in the scqb system. These terms are proportional to the heavy quark mass but absent in the OPE series of the bcqs system. For the interpolating current η 1 with J P = 0 + , we perform the numerical analysis and find the suitable working regions for the continuum threshold and Borel parameter are 55 ≤ s 0 ≤ 59 GeV 2 and 7.6 ≤ M 2 B ≤ 7.9 GeV 2 , respectively. We show the variations of m H with threshold s 0 and Borel mass M 2 B in Fig. 4. Accordingly, the hadron mass can be extracted in these parameter regions. For all interpolating currents in the scqb systems, we can only establish reliable mass sum rules for η 1 , η 3 , η 4 , η 1µ and η 3µ . We list the numerical results for the threshold s 0 , Borel mass M 2 B , output masses, pole contributions and the two-meson thresholds in Table III for the scalar scqb system and Table IV for the axial-vector channel. In Table III, the extracted masses for the scalar scqb tetraquarks are slightly below the B s D threshold while higher than the B + c K threshold. However, the numerical results in Table IV show that the axial-vector scqb tetraquarks lie below both the B * s D and B + c K * thresholds.   ----

V. CONCLUSION
We have investigated the mass spectra for the fully open-flavored bcqs and scqb tetraquark states in the framework of QCD sum rules. We construct the interpolating tetraquark currents with J P = 0 + and 1 + and calculate their two-point correlation functions and spectral densities up to dimension eight condensates at the leading order of α s .
For the bcqs tetraquark states, we find that the quark condensate qq and quark-gluon mixed condensate qg s σ ·Gq are proportional to the light quark mass and thus numerically small. The dominant nonperturbative contribution to the correlation function comes from the four-quark condensate qq ss . The OPE series are very different for the scqb tetraquark systems, where the quark condensate and quark-gluon mixed condensate provide more important contribution than the four-quark condensate. Such difference leads to distinct behavior of the mass sum rules between the bcqs and scqb tetraquark systems.
After the numerical analyses, we extract the masses around 7.1 − 7.2 GeV for both the scalar and axial-vector bcqs tetraquark states while 7.0 − 7.1 GeV for the scqb tetraquarks. These results show that the masses of the bcqs tetraquark states are below theB s D andB * s D two-meson S-wave thresholds, which are consistent with the results from the color-magnetic interaction model [21]. For the axial-vector scqb tetraquarks, their masses are also lower than the two-meson thresholds of B + c K * and B * s D modes. Such results indicate that the two-meson strong decay modes are kinematically forbidden for these possible tetraquark states. They can only decay via the weak interaction if they do exist. For the scalar scqb tetraquark state, their decay to the B s D final states is also forbidden, but the B + c K decay mode is allowed due to their slightly higher masses. However, such decay will be difficult since the low production rate of the B + c meson. These stable teraquark states may be found in BelleII and LHCb in future.

ACKNOWLEDGMENTS
This project is supported in part by the Chinese National Youth Thousand Talents Program.

1.
The spectral densities for the bcqs tetraquarks 1.spectral densities for J 1 : δ and H denote the Dirac delta and Heavyside theta function, respectively.