Doubly Heavy Tetraquarks in QCD Sum Rules

In the framework of QCD Sum Rules we investigate the $q q^\prime \bar{Q} \bar{Q}$ tetraquark structure with quantum number $J^P = 1^+$, which are embedded in two types of configurations, the $[8_c]_{q \bar{Q}} \otimes [8_c]_{q^\prime \bar{Q}}$ and $[1_c]_{q \bar{Q}} \otimes [1_c]_{q^\prime \bar{Q}}$ with $Q = b, c$, $q = u$, and $q^\prime = d, s$. Our finding confirms the Lattice QCD result that the bottom tetraquark states could exist and their masses are evaluated. In the calculation, the non-perturbative condensate contributions up to dimension eight in operator product expansion are considered, and those terms linear to the strange quark mass $m_s$ are kept. It is found that for octet-octet configuration the masses of potential tetraquark states are about $11.28$ GeV for the $ud\bar{b}\bar{b}$ system, and $11.31$ and $11.34$ GeV for the $us\bar{b}\bar{b}$ system, which are above the corresponding two-meson thresholds. For molecular configuration, the corresponding masses are found below the thresholds, that is $10.36$ GeV and $10.48$ GeV, respectively. The possible tetraquark decay channels are analyzed and the strong decay rates are evaluated. The mass dependence on the radiative correction and the condensates is estimated. Moreover, the doubly charmed tetraquark states are also considered.


I. INTRODUCTION
In the framework of quantum chromodynamics (QCD) and quark model, various of hadron structures beyond the normal meson and baryon are yet rarely explored, like multiquark state, hybrid, and glueball, etc, which are theoretically legitimate to exist but experimentally hard to identify. With the first observation of X(3872) by Belle Collaboration in 2003 [1], a series of candidates of exotic hadron states were claimed to be measured. It is reasonable to think that we might be now at the dawn of the enormous discovery of novel hadronic structures. In the scope of exotic states, tetraquark state belongs to the simplest extension of the normal hadron.
Doubly heavy tetraquarks composed of two heavy antiquarks (QQ) and two light quarks (qq ′ ), where Q = b, c and q = u, q ′ = d, s, are of peculiar interest in physics. They do not experience quark annihilation process in their decay modes. In the family of tetraquark, the existence of stable doubly heavy tetraquark was first suggested by Jaffe forty years ago [2].
Within the diquark-antidiquark model, the attraction force induced by the color Coulomb potential of two antiquarks in color 3 c configuration enables the existence of stable qq′QQ tetraquark states possible in the m Q → ∞ limit [3]. The properties of such states were explored in depth in a variety of theories, such as the MIT bag model [4], color flux-tube model [5] ,chiral quark model [6,7], constituent quark model [8,9], chiral perturbation [10], and lattice QCD [11][12][13].
In past thirty years, the QCD Sum Rules technique [14], as a model independent approach in the study of hadron physics, especially hadron spectrum, has achieved a great triumph, at least qualitatively. Employing the QCD Sum Rules, in recent years people have performed some remarkable researches on exotic hadrons, including the doubly heavy tetraquark states [15][16][17][18]. In Ref. [15], Navarra, Nielsen and Lee worked with the current of axial heavy diquark and scalar light anti-diquark, and found there possibly exist the J P = 1 + QQ −ūd states.
Ref. [16] calculated the diquark-antidiquark states with various QQqq ′ configurations and J P quantum numbers. Besides, Dias et al. [17] also calculated the tetraquark spectrum but with the QQqq being in molecular structure. Ref. [18] constructed certain (axial vector diquark)-(scalar antidiquark) type currents and investigated the axial vector doubly heavy tetraquark states.
In practice, as noted in Ref. [19], there may exist another type of tetraquark configuration of (Qq)(Qq ′ ), where the diquarks are in color octet as in a hybrid. Though the color-octet diquark itself is loosely bounded relative to the color singlet ones as in the molecular state, the diquark-diquark confining interaction in the former case tends to be stronger than the latter, due to the direct QCD effect. In this work, we analyze in the framework of QCD Sum Rules whether there exists stable J P = 1 + tetraquark state with color-octet-octet configuration or not, and also calculate the doubly heavy molecular tetraquark state by means of currents given in Ref. [12]. The decay properties of the concerned tetraquark states will be evaluated as well.
The paper is organized as follows: after the introduction, we present some primary formulas of the QCD Sum Rules used in our calculation in Sec.II. The numerical analysis and results are given in Sec.III. The last part is left for conclusions and discussion of the results.

II. FORMALISM
The procedure of QCD Sum Rules calculation [14,[20][21][22] starts with the two-point correlation function constructed by hadronic currents. For an axial-vector state considered in this work, the two-point correlation function takes the following form: Here, j µ (x) and j ν (0) are the relevant hadronic currents with J P = 1 + and the Lorentzstructure of the right-hand side is dictated by the conservation of the hadronic current.
For doubly heavy tetraquark in color-octet-octet configuration with quantum number J P = 1 + , the lowest order currents may be constructed exclusively as which are similar to the ones used in the investigation of light tetraquarks [23][24][25][26]. Here, j, k, m, and n denote color indices, t a are SU c (3) color group generators, Q represents the heavy quark b or c, q stands for the up quark u, and q ′ for the down quark d or the strange quark s. Here, the superscripts A, B, C, and D indicate the currents composed of a 0 − color-octet diquark and a 1 − color-octet diquark; a 1 − color-octet diquark and a 0 − color-octet diquark; a 0 + color-octet diquark and a 1 + color-octet diquark; and a 1 + color-octet diquark and a 0 + color-octet diquark, respectively.
For molecular structure, the current goes as [12] j where the subscripts a and b are color indices, and the superscript E means that it is in the molecular configuration.
In order to calculate the spectral density of the operator product expansion (OPE) side, the light quark (q = u, d or s) and heavy quark (Q = c or b) full propagators S q ij (x) and S Q ij (p) are employed, say Here, the vacuum condensates are clearly displayed. For more explanation on above propagators, readers may refer to Refs. [27,28].
In the partonic representation, the dispersion relation may express the correlation function Π(q 2 ) as: where ρ OP E (s) = Im[Π OP E (s)]/π and Here, Π sum (q 2 ) is the sum of those contributions in the correlation function that have no imaginary part but have nontrivial magnitudes after the Borel transformatiom. By applying the Borel transformation to (9), we have To consider the strange quark mass effect, those terms linear to the strange quark mass In the hadronic representation, after isolating the ground state contribution from the tetraquark state, we obtain the correlation function Π(q 2 ) in dispersion integral over the physical region, i.e.
where the superscript X means the lowest lying tetraquark state, M X denotes its mass, ρ(s) is the spectral density that contains the contributions from higher excited states and the continuum states above the threshold s 0 . The coupling constant λ X is defined through By performing the Borel transform on the hadronic side, the Eq.(13), and matching it to Eq.(12), we can then obtain the mass of the tetraqark state where the subscripts i runs from A to E, q ′ = d or s, and the moments L 1 and L 0 are respectively defined as:

A. Inputs
In the QCD Sum Rule calculation, as in any practical theory, to yield meaningful physical results one needs to give certain inputs, such as the magnitudes of condensates and quark masses. Being ascertained in Refs. [14,[20][21][22], for numerical evaluation the inputs we take Moreover, there exist two additional parameters M 2 B and s 0 introduced in establishing the sum rule, which will be fixed in light of the so-called standard procedures abiding by two criteria [14,20,22]. The first one asks for the convergence of the OPE. That is, one needs to compare individual contributions with the total magnitude on the OPE side, and choose a reliable region for M 2 B to retain the convergence. In practice, the degree of convergence may be expressed in fractions of condensates over the total as per where the superscript dim means the dimension of relevant condensate in the OPE of Eq.(15), the subscript i runs from A to E, and q ′ = d or s.
The second criterion requires that the portion of lowest lying pole contribution (PC), the ground state contribution, in the total, pole plus continuum, should be over 50% [22,29], which can be formulated as Here the subscripts i runs from A to E, and q ′ = d or s. Under this prerequisite, the contributions of higher excited and continuum states will be properly suppressed.
In order to find a proper value for continuum threshold s 0 , we perform a similar analysis as in Refs. [30,31]. Notice that the s 0 relates to the mass of the ground state by √ s 0 ∼ (M X + δ) GeV, in which δ lies in the scope of 0.4 ∼ 0.8 GeV, various √ s 0 satisfying this constraint should be taken into account in the numerical analysis. Among these values, one needs to pick up the one which yields an optimal window for Borel parameter M 2 B . That is to say, in the optimal window, the tetraquark mass M X is somehow independent of the Borel parameter M 2 B . In this way, the value of √ s 0 corresponding to the optimal mass curve will be taken as its central value. In practice, we may vary √ s 0 by 0.2 GeV in numerical calculation [32], which set the upper and lower bounds and hence the uncertainties of √ s 0 .

B. The color-octet-octet configuration
For case A, we illustrate the OPE convergence in Figs To determine the √ s 0 , we need to apply the criterion two [29]. Note, for cases C and D, however, we can not find reasonable parameters M 2 B , which implies that those two currents have no strong coupling with the tetraquark states.
After the above evaluation, we can then determine the masses of the tetraquark states with currents A and B, that is where the central values correspond to the results with the optimal stability of M 2 B , and the errors stem from the uncertainties of the condensates, the quark mass, the threshold parameter √ s 0 , and the Borel parameter M 2 B .  Replace m b with m c in the correlation functions and perform the same procedure on justifying the Borel parameter M 2 B and threshold √ s 0 , we may obtain results for doubly charmed systems. That is (4.72±0.13) for the udcc system, and (4.76±0.14) and (4.78±0.14) GeV for the uscc system.
For the charm sector, taking the same procedure as in b sector with m b being replaced by m c in the spectral density, the masses of udcc and uscc in molecular configuration are then obtained as (4.04 ± 0.13), and (4.18 ± 0.13), respectively. Note that Ref. [12] finds that the charm quark mass is not so large as to take the m Q → ∞ limit and the stability of the doubly charmed tetraquark structure is questionable.
For the convenience of reference, a collection of Borel parameters, continuum thresholds, pole contributions and predicted masses for cases A, B and E are tabulated in Table I.  strong interaction, which implies their decay widths might be large. Following we calculate the decay widths by means of the QCD sum rules.
The calculation of the decay vertex starts from the three-point function: where Π µν (x, y) = 0|T [j B * µ (x)j B 0 5 (y)j X † ν (0)]|0 and q = q 1 + q 2 . The interpolating currents of B * and B 0 have the following forms: On the phenomenological side of the QCD sum rules, we insert the intermediate states into Eq.(24), and get Π phen where M X is the mass of the tetraquark state, λ X is its decay constant, and g XB * B 0 (q 2 ) is the form factor. They are defined as follows: On the OPE side, up to dimension five, the only nonzero term is the g sq σ · Gq . As discussed in Ref. [33], the q µ 1 q ν 1 structure would give the dominant contribution, and hence we merely calculate those diagrams shown in Fig.(11) and then extract the coefficient of the q µ 1 q ν 1 term. In the end, we have Performing Borel transforms on both q 2 and q 2 1 , we obtain where Q 2 = −q 2 2 and s 0 is the continuum threshold of the tetraquark state, and A is defined In numerical evaluation, we adopt the mass and decay constant values of the bottom mesons given in Refs. [34,35], i.e., m B * = 5.33 GeV , f B * = 0.180 GeV ; Given the Borel window satisfying the two criteria mentioned above, we can get g XB The decay width takes the form where the subscript A − d stands for case A with q ′ = d, and With the preceding inputs, we then get where the subscript A − s, B − d, and B − s denote case A with q ′ = s, case B with q ′ = d, and q ′ = s, respectively.

E. The Tetraquark mass dependence on radiative corrections
In order to estimate the radiative correction effect on the leading order calculation, we define the ratios of radiative correction over the corresponding leading order ones as Aα s (M 2 B ), , and Eα s (M 2 B ) for the perturbative, two-quark condensate, mixed condensate and four-quark condensate, and dimension-8 condensate composed of the two-quark condensate and mixed condensates in Eqs. (10,11), respectively, as Taking M X E−d as an example, we adjust the parameter A to find the reasonable Borel window and get the valid regions in order to sustain the tetraquark existance. We find that to ensure a reasonable Borel window, while  Fig.(12). one may refer to [12]. For charm sector, we find both color-octet-octet and the color-singletsinglet tetraquark structures are above the corresponding two charm-meson thresholds.
To estimate the calculation uncertainty and reliability, we analyze schematically the mass dependence on radiative corrections and find it may only induce small corrections, -0.043 to 0.161 GeV, to the results.