X(3872) and Its Heavy Quark Spin Symmetry Partners in QCD Sum Rules

X(3872) presents many surprises after its discovery more than ten years ago. Understanding its properties is crucial to understand the spectrum of possible exotic mesons. In this work, X(3872) meson and its heavy quark spin symmetry (HQSS) partners (including the mesons in the bottom sector) are studied within the QCD Sum Rules approach using a current motivated by the molecular picture of X(3872). We predict four heavy partners to X(3872) and bottomonium with the masses and J^PC quantum numbers. Obtained results are in good agreement with the previous studies and available experimental data within errors.


I. INTRODUCTION
X(3872) was first observed by Belle Collaboration in 2003 and the production mode was B + → X(3872)K + → J/ψπ + π − K + [1]. It has been also confirmed by the CDF [2], D0 [3] and BaBar [4] collaborations. The current average mass of X(3872) is 3871.69 ± 0.17M eV and it is only 0.16M eV below the D 0D * 0 threshold with a less then 1.2M eV full width [37]. Its unusual properties presents a puzzle in the meson spectroscopy and up to now there is no concensus about its structure.
One of the puzzles presented by X(3872) is its decays into final states J/ψππ and Jψπππ. In these decays, the pions are produces through the decay of an intermediate ω or ρ mesons. It is observed that their branching ratios are nearly equal [7] B(X(3872) → J/ψρ) B(X(3872) → J/ψω) 1. (1) If one considers the differences in phase space between ρ and ω mesons this ratio turns into [7] A(X(3872) → J/ψρ) A(X(3872) → J/ψω) 0.2 (2) for amplitudes which implies there is a large isospin violation in X(3872) [8]. Such a large isospin violation is hard to reconcile with a quark-anti-quark picture of X(3872), whereas in the molecular picture, it arises naturally due to the mass difference of the neutral and charged D mesons [9]. Another interesting observation about X(3872) is its radiative decays. The branching ratio of X(3872) to J/ψ and ψ(2S) is [10,11] Br(X → ψ(2S)γ) Br(X → J/ψγ) = 2.46 ± 0.64 ± 0.29.
It is a question why X(3872) prefers to decay into ψ(2S)γ even though the phase space is much smaller than its decay into J/ψγ. In the charmonium picture, X → ψ(2S)γ is a ∆L = 1 transition. It is claimed in [12] that this ratio cannot be explained naturally in a pure molecular picture. However it is stated that this ratio can be obtained by adding a charmonium admixture into molecular picture [13,14]. Heavy quarkonium spectra is very rich in its own basis, i.e. XY Z states. Beside X(3872) other examples are Y (4260), Y (3940), Z(3930) and X(3940) [5] and Z b (10610) and Z b (10650) in [41]. These observed states are candidates of so-called exotic states which are not well compatible with quark model interpretation. Actually an old work done by Tornqvist [6] predicted X(3872) using a meson potential. So it is not clear that quark model cannot interpret these exotic states although there are some difficulties. Nonetheless in [42], contrary to general consideration such as molecule or four quark state, they studied X(3872) resonance as cc = χ c1 (2P ) which sits on the D * 0D0 threshold. They also suggest a program for experimental research in order to verify their assumption. In [43], the authors made a threshold parametrization of the Belle and BaBar data on B decays to KJ/ψπ + π − and KDD * 0 . They showed that data can be reproduced with a similar quality for X(3872) being a bound and/or virtual state. They also noted that X(3872) can be a higher order virtual state pole in the limit in which the small D * 0 width vanishes.
Potential models which worked very well in the past failed to obtain such as masses and decay widths of these states. For example, quark model calculations for mass of the χ c1 (2p) state with J P C = 1 ++ quantum number is too large. It is the only state which quark model have the same J P C quantum numbers with X(3872). Relativistic quark model have a mass about 3.95GeV for this state [15]. Among several approaches including tetraquark [16], hybrid [17], glueball [18], charmonium [19][20][21] trying to enlighten this state, the most likely approach is hadronic molecule [22][23][24][25][26][27][28][29]. There are also other models such as Effective Field Theories (EFT) and Heavy Quark Spin Symmetry (HQSS). Recently, a remarkable study was done in [30] by using a new EFT technique to investigate quarkonium contributions to meson molecules. HQSS states that for m Q → ∞ where m Q is the heavy quark mass then Λ QCD m Q → 0 which infers QCD is independent of the heavy quark spin and hadronic states can be classified by light quark's degree of freedom such as flavour, spin, parity etc. In HQSS, physical properties depend on the light quarks spins. When two heavy quarks come together their total spin can be S heavy = 0 or S heavy = 1. If two quarks join this system the total spin changes.
A natural framework to study systems that contain heavy quarks, such as the c or b quarks, is presented by the heavy quark effective theory. In the limit that the masses of heavy quarks are taken to infinity, the spin of the quark decouples from the dynamics. This implies that the states that differ only in the spin of the heavy quark, i.e. states in which the rest of the system has the same total angular momentum, should be degenerate.
In HQSS, physical states that have the same light quark spin S l should have the same mass, i.e. they are degenerate. It is showed in [31,32] under HQSS, if X(3872) is a bound state of DD * then it should have degenerate partners. The same result was achieved in [33] by means of the heavy quark limit of QCD, X(3872) should have degenerate partners independent of its internal structure. In the molecular picture of X(3872), the two charm quarks have a total spin S H = 1, and the light quarks have a total spin S l = 1. The total spin of such a system, assuming L = 0, can be J = 0, J = 1 and J = 2. In the heavy quark limit, all these three states should be degenerate. In addition, the state in which S H = 0 and S l = 1 should also be degenerate with the previous three, forming a heavy quark spin symmetry quartuplet having the quantum numbers J P C = 0 ++ , 1 ++ , 2 ++ and 1 −+ . All these four states have been studied in an effective theory framework in [32,34]. Note also that, in the heavy quark limit for both the c and b quarks, there appears a flavor symmetry between these two quarks. Using this symmetry, it is possible to extract information about the c sector using the b sector, and vice versa.
In the framework of QCD Sum Rules, Sundu studied The mass and current-meson coupling constant of the exotic X(3872) state are computed within the two-point sum rule method using the diquark-antidiquark and molecule interpolating currents [45]. Veliev et al. studied X(3872) meson with thermal QCD sum rules method [46]. Aliev et al. investigated X(3872) via QCD sum rules as a mixing of charmonium and molecular D * D states and found that Y (3940), X(4260) and their orthogonal combinations should exist [47]. Azizi and Er investigated X(3872) in cold nuclear matter using diquark-antidiquark current within the framework of the in-medium two-point QCD sum rule method. They found that the mass, current-meson coupling and vector self energy strongly depend on the density of cold nuclear matter [48].
In [33], currents to be used to study X(3872) and its partners in a QCD sum rules framework has been proposed. In this work, X(3872) and its J P C = 0 ++ and J P C = 2 ++ partners are studied within a QCD sum rules framework using the currents proposed in [33]. Obtained sum rules are also used to study the corresponding mesons in the bottom sector.
The paper is organized as follows. In section II, QCD sum rules is given briefly and the degeneracy of the X(3872) is obtained by QCD Sum Rules method. Section 3 is devoted to numerical results of this degeneracy and in section 4 we summarize our results.

II. QCD SUM RULES AND DEGENERACIES FROM THE CORRELATION FUNCTIONS
QCD Sum Rules are formulated by Shifman, Vainsthein and Zakharov in 1979 [35] for mesons and generalized to baryons by Ioffe [36] in 1981. It is one of the celebrated method among non-perturbative methods such as lattice QCD, AdS/QCD, Chiral Perturbation Theory etc. The method is based on the study of a suitable chosen correlation function in two different kinematical regions.
On one side, it is calculated in the deep Euclidean region where the correlation function receives dominant contribution from short distances. In this case, the correlation function can be calculated using operator product expansion (OPE). On the other side, one calculates the correlation function for positive momentum squared. In this kinematical region, the correlation function can be expressed in terms of the properties of the hadrons. The two expression are matched using spectral representation of the correlation function, and hadronic properties are extracted by this matching.
The fundamental object of the QCD Sum Rule is the correlation function where j(x) is the interpolating current, q is the momentum of the state and T is the time ordering operator. Currents are suitably chosen operators made of quark and gluon fields that can create the studied hadron from vacuum. For q 2 > 0, correlation function can be rewritten as a sum of hadron states by inserting a complete set of eigenstates between the interpolating currents as As can be seen from above equation, singularities (poles) of this expression give the the mass of hadron which is created by the j(x) operator. On the other hand, for q 2 0, the time ordered product can be expanded using OPE as where O d are local operators, d is the mass dimension of the operator O d , and C d (x 2 ) are the Wilson coefficients that can be calculated using perturbation theory. Substituting the OPE into the expression for the correlation function, the correlation function can be expressed as where C d (q 2 ) is the Fourier transform of C d (x 2 ), and O d are the vacuum condensates, which parameterize the properties of the QCD vacuum. Matching Π phen with Π QCD relates the hadronic properties (such as their masses) to the condensates.
The key ingredient in the correlation function is the j(x) operator. If the structure of the operator resembles the structure of the meson, then obtained sum rules are expected to be more reliable. In this paper, the current is used. This current was proposed in [33] to study X(3872) and its partners. For X(3872), Q = c and q = u or q = d and a and b are color indices. As is customary in the QCD sum rules and lattice literature, annihilation diagrams are ignored in this work. Also, the masses of the u and d quarks are taken to be zero The color combi6ion is chosen such that the current can create colorless D and D * states. This current has even charge parity, C = +. An advantage of this current is that, d = 3 term in the OPE, i.e. the quark condensate term, does not contribute to the sum rules. Using this current, the correlation function can be written as Following [33], three projections operators are defined as Using these operators, interpolating currents can be written as the sum of three irreducible representation of the Lorentz group as where In above equations, the superscript denotes the J C quantum numbers of the particle of largest spin that can be created by the corresponding operator. The J P C quantum numbers of the particles that can be created by these operators are as follows: j 2+ µν can create J P C = 0 ++ , J P C = 1 ++ and J P C = 2 ++ , j 1+ µν can create J P C = 1 ++ and J P C = 1 −+ , j 0+ µν can create J P C = 0 ++ from the vacuum. The phenomenological side of the correlation function obtained from j 2+ can be written as where m J P C denotes the mass of the meson whose quantum numbers are J P C , and summations are over the spins of the corresponding meson. The constants λ J P C 2 are defined through the matrix elements where µν and µ are polarization tensors for spin-2 and spin-1 respectively and q is the momentum of the hadron. The polarization tensors satisfy q µ µ = 0, µ µ = −1, q µ µν = 0, µν = νµ , µν g µν = 0 and µν µν = 1. Polarization sum can be done via for spin-2 mesons and s µ * for spin-1 mesons. For simplicity, defining the spin sums as the correlation function can be written as Observing that the contribution of each J P C particle to the correlation function can be extracted as A similar analysis of the phenomenological side of the correlation function made of the j 1+ current can be carried out. By inserting a complete set of states between the interpolating currents, the correlation function can be written as Defining matrix elements as and using polarization sum, spin-1 correlation function becomes Defining the Lorentz structures correlation function can be written in a compact form as Using κ 1+ µναβ κ 1−;µναβ = 0 (38) the contribution of the two particles can be isolated as and Finally, to obtain the phenomenological representation of the correlation function composed of j 0 , first note that j 0 can only create particles with quantum numbers J P C = 0 ++ . With the matrix element defined as the correlation function can be written as follows which can be converted to As can be seen from Eqs. 28-30, 39, 40, and 43, the masses of the hadrons can all be obtained from equations of the form: where P (q 2 ) is a polynomial in q 2 . Note that, the left hand side of Eq. 44 also contains contributions from higher states and the continuum, but only the contribution of the lowest state is explicitly written out.
As is stated earlier, to match Π phen with Π QCD , spectral representation of the correlation function is used: where ρ(s) is the spectral density. To get rid of the unknown polynomials, Borel transformation is carried out. After the Borel transformation, Eq. 44 can be written as: where M 2 is the Borel parameter, and · · · represent the contributions of the higher states and continuum.
To subtract the contributions of the higher states and the continuum, quark hadron duality is used. In quark hadron duality, it is assumed the ρ phen (s) = ρ QCD (s) for s > s 0 , where s 0 is called the continuum threshold. After using quark hadron duality, the sum rules can be obtained as The mass of the relevant meson can be obtained from the sum rules by taking the derivative of the logarithm of both sides with respect to 1/M 2 as: The analytical expression for ρ QCD (s) are presented in the appendix.

III. NUMERICAL ANALYSIS OF MASS SPECTRUM
The numerical values for QCD parameters used in this work are m c = 1.4 GeV 2 , m b = 4.7, m u = m d = 0, 1 4π 2 g 2 s G 2 = 0.012 GeV 4 . There are two additional parameters in QCD Sum Rule calculations. These are the Borel parameter (or Borel mass) and continuum threshold. Borel parameter M 2 , is an auxiliary parameter so physical properties should not depend on it. Due to the approximation made, a residual dependence on M 2 exist. Hence, a range for the Borel parameter in which physical observations are independent of it should be found. The other parameter is continuum threshold, s 0 . In general, this parameter is taken to be s 0 (m + 0.5 GeV ) 2 where m denotes the mass of the studied hadron.
In the present work, the results of the sum rules are studied for the two values of continuum threshold: s 0 = 17 GeV 2 and s 0 = 19 GeV 2 for X(3872) and its partners in the charm sector, and s 0 = 100 GeV 2 or s 0 = 102 GeV 2 for the bottom sector.
In the charm sector, the dependencies of the masses on the Borel parameter for the two values of the continuum threshold are shown in Figures 1, 2, 3, 4, 5, 6. Figures 7, 8, 9, 10, 11 and 12 are the same as Figures 1-6, but for the bottom sector. As can be observed from all the figures, the residual dependence on the Borel parameter is negligible for the chosen continuum thresholds in the chosen Borel range, which is an indication in favor of the chosen ranges.
In Tables I and II we present our predictions for the masses of the particles with J P C quantum numbers 0 ++ , 1 ++ , 1 −+ and 2 ++ in the charm and bottom sector. The error bars in the table are mainly due to the variations of the prediction with the continuum threshold. Note that that 1 −+ particle present in the tables is not the partner of X(3872). X(3872) corresponds to the 1 ++ state, and the predicted mass is higher than the experimental value. Note that in this section, all the obtained masses are almost degenerate with each other.    In this work, we obtained mass spectrum of heavy quark partners of X(3872) and bb in QCD Sum Rule framework. The current we studied has an advantage to study X(3872) partners comparing to for example [38,39]. We obtained m X = 4055 ± 127 M eV which is compatible within the experimental errors for X(3872) and m bb = 9924 ± 43 M eV . The masses for correlation functions can be found in Tables I and II for X(3872) and its partners and bb, respectively. As can be seen from Fig. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 the mass spectrum for the X(3872) andbb and their partners don't depend on the Borel parameter. The degeneracies of X(3872) andbb partners are compatible with the heavy quark spin symmetry prediction. We found that if higher dimensional operators in j(x) are included, the mass plots are not reliable.
In the framework of QCD Sum Rules, in [44] X(3872) is taken as an exotic four quark state with J P C = 1 ++ . They obtained mass as M X = 3.94 ± 0.17 GeV which is in good agreement with our result of J P C = 1 ++ in charm sector. In [45], masses of X(3872) are M X = 3873 ± 127 M eV for diquark picture and M X = 3908 ± 139 M eV for molecular picture. The mass is M X = 3885 ± 85 M eV in [46]. Our result agrees well with these values.
Guo et al. studied X 2 (4012 M eV ) with the quantum numbers J P C = 2 ++ under assumption of being a spin partner of X(3872). They predicted that in a e + e − collision which occurs around 4.4 − 4.5 GeV this spin partner can be produced [40]. Our results for this state with J P C = 2 ++ quantum number is compatible.
Nieves and Valderrama studied X(4012) with the quantum numbers J P C = 2 ++ as a partner of X(3872) [32]. Furthermore they assumed X(3915) with the quantum numbers J P C = 2 ++ as a heavy quark spin symmetry partner and predicted six D * D * molecular states. They used 1 ++ and 0 ++ states as an input for calculation of the masses of molecular states. We obtained these states from our correlation function and our results are in good agreement with their results.
We also predict a parity number for X(3940) which is thought to be heavy spin partner. The quantum numbers for this state has not been determined yet. If this state has a positive parity number, then our mass prediction agrees well within errors.
The other possible heavy spin partner of X(3872) is X(3915). Although, its quantum number is not well determined, J P C = 0 ++ or J P C = 2 ++ , our results fit this possibility.