Exotic , , and states
Abstract
After constructing the possible , and tetraquark interpolating currents in a systematic way, we investigate the twopoint correlation functions and extract the corresponding masses with the QCD sum rule approach. We study the , , and systems with various isospins . Our numerical analysis indicates that the masses of doubly bottomed tetraquark states are below the threshold of the twobottom mesons, twobottom baryons, and one doubly bottomed baryon plus one antinucleon. Very probably these doubly bottomed tetraquark states are stable.
pacs:
12.39.Mk, 12.38.Lg, 14.40.Lb, 14.40.NdI Introduction
In the past decade, many charmonium or charmoniumlike states were observed in the B factories, some of which do not fit in the conventional quark model and are considered as the candidates of the exotic states such as molecular states, tetraquark states, hybrid mesons, baryonium states, etc. For experimental reviews of these new states, see Refs. Swanson (2006); Zhu (2008); Bracko (2009); Yuan (2009); Rosner (2007).
Hadronic molecular states are loosely bound states composed of a pair of mesons. They are probably bound by the longrange, colorsinglet pion exchange. These interesting states generally lie very close to the opencharm/bottom threshold. Some nearthreshold charmoniumlike states such as X(3872) and Z(4430) are very good candidates of the molecular states composed of a pair of charmed and anticharmed mesons. In fact, such a possibility was studied extensively in Refs. Liu et al. (2009, 2008); Swanson (2004a, b); Close and Page (2004); Thomas and Close (2008); FernandezCarames et al. (2009).
Tetraquarks are composed of four quarks. They are bound by colored force between quarks. They decay through rearrangement. Some are charged. Some carry strangeness. There are many states within the same multiplet. The lowlying scalar mesons below 1 GeV have been considered good candidates of the tetraquark states. Some of the recently observed charmoniumlike states were also suggested to be candidates of the hiddencharm tetraquark states Matheus et al. (2007); Maiani et al. (2007); Ebert et al. (2006); Chen and Zhu (2011, 2010); Du et al. (2012); Jiao et al. (2009). We have studied the possible pseudoscalar, scalar, vector, and axialvector hiddencharm/bottom tetraquark states systematically in the framework of the QCD sum rule Chen and Zhu (2011, 2010); Du et al. (2012); Jiao et al. (2009).
Besides the hiddencharm/bottom type tetraquark states, the doublycharmed/bottomed tetraquark states are also very interesting, where Q denotes a heavy quark (beauty or charm) and q denotes one light quark (up, down, strange). Such a fourquark configuration is allowed in QCD. In QED we have the hydrogen atom. One light electron circles around the proton. When two electrons are shared by two protons, a hydrogen molecule is formed. In QCD we have the heavy meson. One light antiquark circles around the heavy quark. If the two light antiquarks were shared by the two heavy quarks in the doubly charmed/bottomed tetraquark system, we would have the QCD analogue of the hydrogen molecule!
On the other hand, if the heavy quark QQ pair is spatially close, it would act as a pointlike antiheavy quark color source and pick up two light quarks to form the bound state . The existence and stability of such systems have been studied in many different models, such as the MIT bag model Carlson et al. (1988), chiral quark model Zhang et al. (2008); Pepin et al. (1997), constituent quark model Vijande et al. (2006, 2009); Brink and Stancu (1998); SilvestreBrac and Semay (1993); Zouzou et al. (1986), and chiral perturbation theory Manohar and Wise (1993). Some other useful references are Navarra et al. (2007); Cui et al. (2007); Gelman and Nussinov (2003); Moinester (1996); Bander and Subbaraman (1994); Ader et al. (1982); Richard (1991); Lipkin (1986, 1973); Carames (2011). However, their existence and stability are model dependent up to now. More theoretical investigations will be helpful in the clarification of the situation.
In this work, we will discuss the systems using the method of QCD sum rule. We first construct the currents with , , , and in a systematic way. These currents have no definite C parity due to their special flavor structures. The isospins of these currents can be with specific quark contents. With the independent currents, we investigate the twopoint correlation functions and spectral densities. After performing the QCD sum rule analysis, we extract the masses of the possible , , , , , and states.
The paper is organized as follows. In Sec. II, we construct the currents with , , , and . In Sec. III, we calculate the correlation functions with the operator product expansion(OPE) method and extract the spectral densities. The results are collected in Appendix A. In Sec. IV, we perform the numerical analysis and extract the masses of these tetraquark states. We discuss the possible decay patterns of these doubly charmed/bottomed tetraquark states in Sec. V. The last section is a brief summary.
Ii tetraquark interpolating currents
In this section, we construct the diquarkantidiquark currents with , and using the same technique in our previous works Chen and Zhu (2011, 2010). One can construct the tetraquark current with the basis or basis, or . However, they can be related by the Fierz transformation. In this work, we consider the first set. Considering the Lorentz structures, there are five independent diquark fields without derivatives: , , , , and , where a, b are the color indices. Since and carry different parity, we consider both operators although they are equivalent. These diquark bilinear can be in the symmetric representation or antisymmetric representation in the color and flavor SU(3) space. Considering the Lorentz structures, we list the properties of these diquark operators in Table II.


States  (Flavor, Color)  

Being composed of two quark (antiquark) fields, the diquark (antidiquark) fields should satisfy Fermi statistics. As shown in Table II, the flavor and color structures are entangled for every diquark operator. For example, the flavor and color structures of the scalar diquark operator are either or . For systems, the heavy quark pair has the symmetric flavor structure . Its flavor and color structures are then fixed as . To construct the color singlet currents, the heavy quark pair and the light antiquark pair should have the same color structures.
According to Ref. Chen and Zhu (2011), there are ten color singlet currents with :
(1) 
where “+” denotes the symmetric color structure and “” denotes the antisymmetric color structure . Due to the symmetry constraint, it’s enough to keep one light diquark piece only in the bracket of Eq. (1) within the calculation. We keep two terms in Eq. (1) to illustrate the color symmetry explicitly.
By considering the symmetric flavor structure for heavy quark pair , only the currents which satisfy the Pauli principle survive. For the pseudoscalar currents in Eq. (1), , and survive and all the other currents vanish. According to Table II, the () operators are isovector currents and are isoscalar currents. Finally, we obtain the following interpolating currents with , , , and :

The tetraquark interpolating currents with are
(2) in which are isovector currents with and , are isoscalar currents with and .

The tetraquark interpolating currents with are
(3) and all the scalar interpolating currents are isovector currents with and .

The tetraquark interpolating currents with are
(4) in which are isovector currents with and , are isoscalar currents with and .

The tetraquark interpolating currents with are
(5) in which are isovector currents with and , are isoscalar currents with and .
For the isovector () currents, we do not differentiate the up and down quarks in our analysis and denote them by . However, they should be differentiated for the isoscalar () currents because the flavor structures of the light antidiquark are antisymmetric. The quark contents are for these currents. For the systems , only the currents with flavor structures survive. The isospins for these systems are . We will also discuss the systems () by using all the currents in Eq. (2)(5). We pick up the interpolating currents with different quark contents in Table. II. To calculate the twopoint correlation functions, the Wick contractions of the currents for and systems are different from those for the and systems.



Quark Content  I  
1  
0  
0  

Iii QCD sum rule
In QCD sum rule Shifman et al. (1979); Reinders et al. (1985); Colangelo (2000), we consider the twopoint correlation functions of the interpolation currents. For the scalar and pseudoscalar currents, the twopoint correlation functions read
(5) 
where is the corresponding interpolating current. The twopoint correlation functions of the vector and axialvector currents are
(6) 
There are two parts of with different Lorentz structures because is not a conserved current. is related to the vector and axialvector meson, while is the scalar and pseudoscalar current polarization function. At the hadron level, we express the correlation function in the form of the dispersion relation with spectral function,
(7) 
where
(8) 
where is the mass of the resonance and is the decay constant of the meson,
(9) 
where is the polarization vector of X ().
One can calculate the correlation functions at the quarkgluon level via the operator product expansion(OPE) method. Using the same technique as in Refs. Chen and Zhu (2011, 2010); Du et al. (2012); Albuquerque and Nielsen (2009); Bracco et al. (2009); Matheus et al. (2007), we calculate the Wilson coefficients while the light quark propagator and heavy quark propagator are adopted as
(10) 
where , , , . The spectral density is obtained with: .
In order to suppress the higherstate contributions and remove the subtraction terms in Eq. (7), we perform the Borel transformation to the correlation function,
(11) 
After performing the Borel transformation and equating the two representations of the correlation function with the quarkhadron duality, we obtain
(12) 
where is the threshold parameter, and is the Borel parameter. We can extract the meson mass ,
(13) 
For all the tetraquark currents in Eq. (2)(5), we collect the spectral densities in Appendix A, respectively. We neglect the threegluon condensate because their contribution is negligible.
Iv Numerical Analysis
In the QCD sum rule analysis, we use the following values of the parameters Shifman et al. (1979); Nakamura et al. (2010); Eidemuller and Jamin (2001); Jamin and Pich (1999); Jamin et al. (2002) in the chiral limit ():
(14) 
The Borel mass and the threshold value are two pivotal parameters. Requiring the convergence of the OPE leads to the lower bound of the Borel parameter. In the present work, we require that the most important condensate contribution be less than one fourth of the perturbative term. We require that the pole contribution be larger than , which determines the upper bound of the Borel parameter. The pole contribution (PC) is defined as
(15) 
which depends on both the Borel mass and the threshold value . is chosen around the region where the variation of with is minimum in the Borel working region. For a genuine hadron state, the extracted mass from the sum rule analysis is expected to be stable with the reasonable variation of the Borel parameter and threshold .
For all the isovector and isoscalar systems, the most important nonperturbative corrections come from the fourquark condensate . Both the quark condensate and the quarkgluon mixed condensate vanish when we let . For the system, only the interpolating currents and lead to a stable mass sum rule after performing the QCD sum rule analysis. In Fig. IV, we show the mass curves of the extracted hadron mass with and for the current with . The variation of with the Borel mass is very weak around GeV. For , the stability of the mass curves is much worse and grows monotonically with and . The situation is very similar for the systems. Now we keep the related terms in the spectral densities. These terms are very important corrections for the OPE series. The dominant nonperturbative contribution is the quark condensate for and . We show the variations of with the Borel mass and threshold parameter for the current in Fig. IV.




With the parameters in Eq. (14), we list the working region of the Borel parameters, threshold value , the extracted masses for the currents with in Table IV. The pole contribution and the masses are extracted using the corresponding threshold values and Borel parameters listed in the table. For the isovector currents , and , one can also investigate the systems besides the and systems. As mentioned in Sec. II, the Wick contractions of the currents for the systems are different from those for the