Mass spectrum of the axial-vector hidden charmed and hidden bottom tetraquark states
aa r X i v : . [ h e p - ph ] A ug Mass spectrum of the axial-vector hidden charmed and hiddenbottom tetraquark states
Zhi-Gang Wang Department of Physics, North China Electric Power University, Baoding 071003, P. R.China
Abstract
In this article, we perform a systematic study of the mass spectrum of the axial-vector hidden charmed and hidden bottom tetraquark states using the QCD sum rules,and identify the Z + (4430) as an axial-vector tetraquark state tentatively. PACS number: 12.39.Mk, 12.38.LgKey words: Tetraquark state, QCD sum rules
The Babar, Belle, CLEO, D0, CDF and FOCUS collaborations have discovered (or con-firmed) a large number of charmonium-like states, such as X (3940), X (3872), Y (4260), Y (4008), Y (3940), Y (4325), Y (4360), Y (4660), etc, and revitalized the interest in thespectroscopy of the charmonium states [1, 2, 3, 4, 5, 6] . Many possible assignments forthose states have been suggested, such as multiquark states (irrespective of the moleculetype and the diquark-antidiquark type), hybrid states, etc [1, 2, 3, 4, 5].The Z + (4430) observed in the decay mode ψ ′ π + ( B → ψ ′ π + K ) by the Belle collab-oration is the most interesting subject [7, 8]. We can distinguish the multiquark statesfrom the hybrids or charmonia with the criterion of non-zero charge. The Z + (4430) can’tbe a pure c ¯ c state due to the positive charge, and may be a c ¯ cu ¯ d tetraquark state. How-ever, the Babar collaboration did not confirm this resonance [9]. Furthermore, the tworesonance-like structures Z (4050) and Z (4250) in the π + χ c invariant mass distributionnear 4 . c ¯ cu ¯ d , just like the Z + (4430), they can’t be the conventionalmesons. There have been several theoretical interpretations for the Z + (4430), such asthe hadro-charmonium resonance [3, 11], the S -wave threshold effect [12], the molecular D ∗ D ( D ′ ) state [13, 14, 15, 16, 17, 18, 19, 20], the tetraquark state [21, 22, 23, 24, 25, 26],the cusp in the D ∗ D channel [27], the radially excited state of the D s [28], the pseudo-resonance structure [29], etc.In Refs.[30, 31], we assume that the hidden charmed mesons Z (4050) and Z (4250) arevector (and scalar) tetraquark states, and study their masses using the QCD sum rules.The numerical results indicate that the mass of the vector hidden charmed tetraquark stateis about M Z = (5 . ± .
15) GeV or (5 . ± .
16) GeV, while the mass of the scalar hiddencharmed tetraquark state is about M Z = (4 . ± .
18) GeV. The resonance-like structure Z (4250) observed by the Belle collaboration in the exclusive decays ¯ B → K − π + χ c can E-mail,[email protected]. There have been many theoretical works on the X , Y , Z hadrons, it is difficult to cite all of them,we prefer the comprehensive review articles [1, 2, 3, 4, 5], where one can find the original literatures.On the other hand, one can consult Ref.[6] for a concise review of the experimental situation of the newcharmonium-like states.
1e tentatively identified as the scalar tetraquark state [31]. In Refs.[32, 33], we study themass spectrum of the scalar hidden charmed and hidden bottom tetraquark states in asystematic way using the QCD sum rules. In Ref.[34], we study the mass spectrum of thevector hidden charmed and hidden bottom tetraquark states systematically. Recently, the0 −− hidden charmed and hidden bottom tetraquark states are studied with the QCD sumrules [35].In this article, we extend our previous works to study the mass spectrum of the axial-vector hidden charmed and hidden bottom tetraquark states in a systematic way withthe QCD sum rules, and make possible explanation for the nature of the Z + (4430). Themass is a fundamental parameter in describing a hadron, whether or not there exist thosehidden charmed or hidden bottom tetraquark configurations is of great importance itself,because it provides a new opportunity for a deeper understanding of the low energy QCD.The axial-vector hidden charmed ( c ¯ c ) and hidden bottom ( b ¯ b ) tetraquark states may beobserved at the LHCb, where the b ¯ b pairs will be copiously produced with the cross sectionabout 500 µb [36].The hidden charmed and hidden bottom tetraquark states (denoted as Z ) have thesymbolic quark structures: Z + = Q ¯ Qu ¯ d ; Z = 1 √ Q ¯ Q ( u ¯ u − d ¯ d ); Z − = Q ¯ Qd ¯ u ; Z + s = Q ¯ Qu ¯ s ; Z − s = Q ¯ Qs ¯ u ; Z s = Q ¯ Qd ¯ s ; Z s = Q ¯ Qs ¯ d ; Z ϕ = 1 √ Q ¯ Q ( u ¯ u + d ¯ d ); Z φ = Q ¯ Qs ¯ s , (1)where the Q denote the heavy quarks c and b .We take the diquarks as the basic constituents following Jaffe and Wilczek [37, 38], andconstruct the axial-vector tetraquark states with the diquark and antidiquark pairs. Thediquarks have five Dirac tensor structures, scalar Cγ , pseudoscalar C , vector Cγ µ γ , axial-vector Cγ µ and tensor Cσ µν , where C is the charge conjunction matrix. The structures Cγ µ and Cσ µν are symmetric, the structures Cγ , C and Cγ µ γ are antisymmetric. Theattractive interactions of one-gluon exchange favor formation of the diquarks in colorantitriplet 3 c , flavor antitriplet 3 f and spin singlet 1 s [39, 40]. In this article, we assume theaxial-vector hidden charmed and hidden bottom tetraquark states Z consist of the Cγ − Cγ µ type rather than C − Cγ µ γ type diquark structures, and construct the interpolatingcurrents J µ ( x ) and η µ ( x ): J µZ + ( x ) = ǫ ijk ǫ imn u Tj ( x ) Cγ Q k ( x ) ¯ Q m ( x ) γ µ C ¯ d Tn ( x ) ,J µZ ( x ) = ǫ ijk ǫ imn √ (cid:2) u Tj ( x ) Cγ Q k ( x ) ¯ Q m ( x ) γ µ C ¯ u Tn ( x ) − ( u → d ) (cid:3) ,J µZ + s ( x ) = ǫ ijk ǫ imn u Tj ( x ) Cγ Q k ( x ) ¯ Q m ( x ) γ µ C ¯ s Tn ( x ) ,J µZ s ( x ) = ǫ ijk ǫ imn d Tj ( x ) Cγ Q k ( x ) ¯ Q m ( x ) γ µ C ¯ s Tn ( x ) ,J µZ ϕ ( x ) = ǫ ijk ǫ imn √ (cid:2) u Tj ( x ) Cγ Q k ( x ) ¯ Q m ( x ) γ µ C ¯ u Tn ( x ) + ( u → d ) (cid:3) ,J µZ φ ( x ) = ǫ ijk ǫ imn s Tj ( x ) Cγ Q k ( x ) ¯ Q m ( x ) γ µ C ¯ s Tn ( x ) , (2)2 µZ + ( x ) = ǫ ijk ǫ imn u Tj ( x ) Cγ µ Q k ( x ) ¯ Q m ( x ) γ C ¯ d Tn ( x ) ,η µZ ( x ) = ǫ ijk ǫ imn √ (cid:2) u Tj ( x ) Cγ µ Q k ( x ) ¯ Q m ( x ) γ C ¯ u Tn ( x ) − ( u → d ) (cid:3) ,η µZ + s ( x ) = ǫ ijk ǫ imn u Tj ( x ) Cγ µ Q k ( x ) ¯ Q m ( x ) γ C ¯ s Tn ( x ) ,η µZ s ( x ) = ǫ ijk ǫ imn d Tj ( x ) Cγ µ Q k ( x ) ¯ Q m ( x ) γ C ¯ s Tn ( x ) ,η µZ ϕ ( x ) = ǫ ijk ǫ imn √ (cid:2) u Tj ( x ) Cγ µ Q k ( x ) ¯ Q m ( x ) γ C ¯ u Tn ( x ) + ( u → d ) (cid:3) ,η µZ φ ( x ) = ǫ ijk ǫ imn s Tj ( x ) Cγ µ Q k ( x ) ¯ Q m ( x ) γ C ¯ s Tn ( x ) , (3)where the i , j , k , · · · are color indexes. In the isospin limit, the interpolating currentsresult in three distinct expressions for the spectral densities, which are characterized bythe number of the s quark they contain. The interpolating currents J µ ( x ) and η µ ( x ) leadto the same expression for the correlation functions Π µν ( p ), for example, J µZ + ∼ η µZ + ; J µZ ∼ η µZ ; J µZ − ∼ η µZ − ; J µZ + s ∼ η µZ + s ; J µZ − s ∼ η µZ − s ; J µZ s ∼ η µZ s ; J µ ¯ Z + s ∼ η µ ¯ Z + s ; J µZ ϕ ∼ η µZ ϕ ; J µZ φ ∼ η µZ φ , (4)where we use ∼ to denote the two interpolating currents lead to the same expression.The special superpositions tJ µ ( x ) + (1 − t ) η µ ( x ) can’t improve the predictions remarkably,where t = 0 −
1. In this article, we take the interpolating currents J µ ( x ) for simplicity,i.e. t = 1.In fact, we can take the colored diquarks as point particles and describe them with thescalar S a , pseudoscalar P a , vector V aµ , axial-vector A aµ and tensor T aµν fields, respectively,where the a is the color index, then introduce the SU (3) color interaction. We constructthe color singlet tetraquark currents with the diquark fields S a , P a , V aµ , A aµ and T aµν ,parameterize the nonpertubative effects with the new vacuum condensates h SS i , h P P i , h V V i , h AA i and h T T i besides the gluon condensate, and perform the standard procedureof the QCD sum rules to study the tetraquark states. The basic parameters such as thediquark masses and the new vacuum condensates can be fitted phenomenally. The nonetscalar mesons below 1 GeV (the f (980) and a (980) especially) are good candidates for thetetraquark states, from those tetraquark candidates, we can obtain the basic parametersand extend the new sum rules to other tetraquark states. As there are many works to do,we prefer another article.The article is arranged as follows: we derive the QCD sum rules for the axial-vectorhidden charmed and hidden bottom tetraquark states Z in Sect.2; in Sect.3, we presentthe numerical results and discussions; and Sect.4 is reserved for our conclusions.3 QCD sum rules for the axial-vector tetraquark states Z In the following, we write down the two-point correlation functions Π µν ( p ) in the QCDsum rules, Π µν ( p ) = i Z d xe ip · x h | T h J µ ( x ) J † ν (0) i | i , (5)where the J µ ( x ) denotes the interpolating currents J µZ + ( x ), J µZ ( x ), J µZ + s ( x ), etc.We can insert a complete set of intermediate hadronic states with the same quantumnumbers as the current operators J µ ( x ) into the correlation functions Π µν ( p ) to obtainthe hadronic representation [41, 42]. After isolating the ground state contribution fromthe pole term of the Z , we get the following result,Π µν ( p ) = λ Z M Z − p (cid:20) − g µν + p µ p ν p (cid:21) + · · · , (6)where the pole residue (or coupling) λ Z is defined by λ Z ǫ µ = h | J µ (0) | Z ( p ) i , (7)the ǫ µ denotes the polarization vector.After performing the standard procedure of the QCD sum rules, we obtain the followingsix sum rules: λ Z e − M ZM = Z s Z ∆ Z dsρ Z ( s ) e − sM , (8)where the Z denote the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s , b ¯ bq ¯ q , b ¯ bq ¯ s and b ¯ bs ¯ s respectively; the s Z are the corresponding continuum threshold parameters, and the M is the Borel parameter.The thresholds ∆ Z can be sorted into three sets, we introduce the q ¯ q , q ¯ s and s ¯ s todenote the light quark constituents in the axial-vector tetraquark states to simplify thenotation, ∆ q ¯ q = 4 m Q , ∆ q ¯ s = (2 m Q + m s ) , ∆ s ¯ s = 4( m Q + m s ) . The explicit expressionsof the spectral densities ρ q ¯ q ( s ), ρ q ¯ s ( s ) and ρ s ¯ s ( s ) are presented in the appendix, where α f = q − m Q /s , α i = − q − m Q /s , β i = αm Q αs − m Q , e m Q = ( α + β ) m Q αβ , ee m Q = m Q α (1 − α ) .We carry out the operator product expansion to the vacuum condensates adding up todimension-10. In calculation, we take vacuum saturation for the high dimension vacuumcondensates, they are always factorized to lower condensates with vacuum saturation inthe QCD sum rules, factorization works well in large N c limit. In reality, N c = 3, someambiguities may come from the vacuum saturation assumption.We take into account the contributions from the quark condensates, mixed conden-sates, and neglect the contributions from the gluon condensate. The gluon condensate h α s GGπ i is of higher order in α s , and its contributions are suppressed by very large denom-inators comparing with the four quark condensate h ¯ qq i (or h ¯ ss i ) and would not play anysignificant role, although the gluon condensate h α s GGπ i has smaller dimension of mass thanthe four quark condensate h ¯ qq i (or h ¯ ss i ). One can consult the sum rules for the lighttetraquark states [43, 44], the heavy tetraquark state [31] and the heavy molecular states445, 46] for example. Furthermore, there are many terms involving the gluon conden-sate for the heavy tetraquark states and heavy molecular states in the operator productexpansion (one can consult Refs.[31, 45]), we neglect the gluon condensate for simplicity.In the special case of the Y (4660) (as a ψ ′ f (980) bound state) and its pseudoscalarpartner η ′ c f (980), the contributions from the gluon condensate h α s GGπ i are rather large[47, 48]. If we take a simple replacement ¯ s ( x ) s ( x ) → h ¯ ss i and (cid:2) ¯ u ( x ) u ( x ) + ¯ d ( x ) d ( x ) (cid:3) → h ¯ qq i in the interpolating currents, the standard heavy quark currents Q ( x ) γ µ Q ( x ) and Q ( x ) iγ Q ( x ) are obtained, where the gluon condensate h α s GGπ i plays an important rulein the QCD sum rules [41]. The interpolating currents constructed from the diquark-antidiquark pairs do not have such feature. There are other interpretations for the Y (4660),for example, the diquark-antidiquark type charmed baryonium [49].We also neglect the terms proportional to the m u and m d , their contributions are ofminor importance due to the small values of the u and d quark masses.Differentiating the Eq.(8) with respect to M , then eliminate the pole residues λ Z , wecan obtain the sum rules for the masses of the Z , M Z = R s Z ∆ Z ds dd ( − /M ) ρ Z ( s ) e − sM R s Z ∆ Z dsρ Z ( s ) e − sM . (9) The input parameters are taken to be the standard values h ¯ qq i = − (0 . ± .
01 GeV) , h ¯ ss i = (0 . ± . h ¯ qq i , h ¯ qg s σGq i = m h ¯ qq i , h ¯ sg s σGs i = m h ¯ ss i , m = (0 . ± .
2) GeV , m s = (0 . ± .
01) GeV, m c = (1 . ± .
10) GeV and m b = (4 . ± .
1) GeV at the energyscale µ = 1 GeV [41, 42, 50].The Q -quark masses appearing in the perturbative terms are usually taken to bethe pole masses in the QCD sum rules, while the choice of the m Q in the leading-ordercoefficients of the higher-dimensional terms is arbitrary [51, 52]. The M S mass m c ( m c )relates with the pole mass ˆ m c through the relation m c ( m c ) = ˆ m c h C F α s ( m c ) π + · · · i − .In this article, we take the approximation m c ( m c ) ≈ ˆ m c without the α s corrections forconsistency. The value listed in the Particle Data Group is m c ( m c ) = 1 . +0 . − . GeV [53],it is reasonable to take ˆ m c = m c (1 GeV ) = (1 . ± .
10) GeV. For the b quark, the M S mass m b ( m b ) = 4 . +0 . − . GeV [53], the gap between the energy scale µ = 4 . m b ≈ m b ( m b ) ≈ m b (1 GeV ) seems rather crude.It would be better to understand the quark masses m c and m b we take at the energy scale µ = 1 GeV as the effective quark masses (or just the mass parameters).In calculation, we also neglect the contributions from the perturbative corrections.Those perturbative corrections can be taken into account in the leading logarithmic ap-proximations through anomalous dimension factors. After the Borel transform, the effectsof those corrections are to multiply each term on the operator product expansion side bythe factor, h α s ( M ) α s ( µ ) i J − Γ O n , where the Γ J is the anomalous dimension of the interpolat-ing current J ( x ) and the Γ O n is the anomalous dimension of the local operator O n (0).We carry out the operator product expansion at a special energy scale µ = 1 GeV , andset the factor h α s ( M ) α s ( µ ) i J − Γ O n ≈
1, such an approximation maybe result in some scale5ependence and weaken the prediction ability. In this article, we study the axial-vectorhidden charmed and hidden bottom tetraquark states systemically, the predictions arestill robust as we take the analogous criteria in those sum rules.In the conventional QCD sum rules [41, 42], there are two criteria (pole dominance andconvergence of the operator product expansion) for choosing the Borel parameter M andthreshold parameter s . We impose the two criteria on the axial-vector heavy tetraquarkstates to choose the Borel parameter M and threshold parameter s .The contributions from the high dimension vacuum condensates in the operator prod-uct expansion are shown in Figs.1-2, where (and thereafter) we use the h ¯ qq i to denote thequark condensates h ¯ qq i , h ¯ ss i and the h ¯ qg s σGq i to denote the mixed condensates h ¯ qg s σGq i , h ¯ sg s σGs i . From the figures, we can see that the contributions from the high dimensioncondensates are very large and change quickly with variation of the Borel parameter atthe values M ≤ . and M ≤ . in the hidden charmed and hidden bot-tom channels respectively, such an unstable behavior cannot lead to stable sum rules, ournumerical results confirm this conjecture, see Fig.4.At the values M ≥ . and s ≥
22 GeV ,
23 GeV ,
23 GeV , the contributionsfrom the h ¯ qq i + h ¯ qq ih ¯ qg s σGq i term are less than 12% , , .
5% in the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s respectively; the contributions from the vacuum condensate of the highest dimension h ¯ qg s σGq i are less than 2 . , , .
5% in the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s respectively; weexpect the operator product expansion is convergent in the hidden charmed channels.At the values M ≥ . and s ≥
136 GeV ,
138 GeV ,
138 GeV , the contri-butions from the h ¯ qq i + h ¯ qq ih ¯ qg s σGq i term are less than 10% , . ,
7% in the channels b ¯ bq ¯ q , b ¯ bq ¯ s , b ¯ bs ¯ s respectively; the contributions from the vacuum condensate of the highestdimension h ¯ qg s σGq i are less than 5 . , ,
3% in the channels b ¯ bq ¯ q , b ¯ bq ¯ s , b ¯ bs ¯ s respec-tively; we expect the operator product expansion is convergent in the hidden bottomchannels.In this article, we take the uniform Borel parameter M min , i.e. M min ≥ . and M min ≥ . in the hidden charmed and hidden bottom channels respectively.In Fig.3, we show the contributions from the pole terms with variation of the Borelparameters and the threshold parameters. The pole contributions are larger than (orequal) 46% , ,
50% at the value M ≤ . and s ≥
22 GeV ,
23 GeV ,
23 GeV in the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s respectively, and larger than (or equal) 48% , , M ≤ . and s ≥
136 GeV ,
138 GeV ,
138 GeV in the channels b ¯ bq ¯ q , b ¯ bq ¯ s , b ¯ bs ¯ s respectively. Again we take the uniform Borel parameter M max , i.e. M max ≤ . and M max ≤ . in the hidden charmed and hidden bottomchannels respectively.In this article, the threshold parameters are taken as s = (23 ±
1) GeV , (24 ±
1) GeV ,(24 ±
1) GeV , (138 ±
2) GeV , (140 ±
2) GeV , (140 ±
2) GeV in the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s , b ¯ bq ¯ q , b ¯ bq ¯ s , b ¯ bs ¯ s respectively; the Borel parameters are taken as M = (2 . − .
2) GeV and (7 . − .
2) GeV in the hidden charmed and hidden bottom channels respectively. Inthose regions, the pole contributions are about (46 − − − − − − c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s , b ¯ bq ¯ q , b ¯ bq ¯ s , b ¯ bs ¯ s respectively; the two criteria of the QCD sum rules are fully satisfied [41, 42].From Fig.3, we can see that the Borel windows M max − M min change with variations ofthe threshold parameters s . In this article, the Borel windows are taken as 0 . and1 . in the hidden charmed and hidden bottom channels respectively; they are small6nough. If we take larger threshold parameters, the Borel windows are larger and theresulting masses are larger, see Fig.4. In this article, we intend to calculate the possiblylowest masses which are supposed to be the ground state masses by imposing the twocriteria of the QCD sum rules.If we take analogous pole contributions, the interpolating current with more s quarksrequires slightly larger threshold parameter due to the SU (3) breaking effects, see Fig.3.In the channels Q ¯ Qq ¯ s and Q ¯ Qs ¯ s , the SU (3) breaking effects on the threshold parametersare tiny, we take uniform threshold parameters in those channels. In Fig.4, we plot theaxial-vector tetraquark state masses M Z with variation of the Borel parameters and thethreshold parameters. Naively, we expect the tetraquark state with more s quarks will havelarger mass. In calculations, we observe that the possibly lowest masses of the axial-vectorheavy tetraquark states Q ¯ Qq ¯ s and Q ¯ Qs ¯ s are almost the same.Taking into account all uncertainties of the relevant parameters, finally we obtain thevalues of the masses and pole resides of the axial-vector tetraquark states Z , which areshown in Figs.5-6 and Table 1. In Table 1, we also present the masses of the scalar hiddencharmed and hidden bottom tetraquark states obtained in our previous works [32, 33].In this article, we calculate the uncertainties δ with the formula δ = vuutX i (cid:18) ∂f∂x i (cid:19) | x i =¯ x i ( x i − ¯ x i ) , (10)where the f denote the hadron mass M Z and the pole residue λ Z , the x i denote the relevantparameters m c , m b , h ¯ qq i , h ¯ ss i , · · · . As the partial derivatives ∂f∂x i are difficult to carry outanalytically, we take the approximation (cid:16) ∂f∂x i (cid:17) ( x i − ¯ x i ) ≈ [ f (¯ x i ± ∆ x i ) − f (¯ x i )] in thenumerical calculations.From Table 1, we can see that the uncertainties of the masses M Z are rather small(about 4% in the hidden charmed channels and 2% in the hidden bottom channels), whilethe uncertainties of the pole residues λ Z are rather large (about 20%). The uncertaintiesof the input parameters ( h ¯ qq i , h ¯ ss i , h ¯ sg s σGs i , h ¯ qg s σGq i , m s , m c and m b ) vary in the range(2 − λ Z are reasonable. We obtain the squaredmasses M Z through a fraction, the uncertainties in the numerator and denominator whichoriginate from a given input parameter (for example, h ¯ ss i , h ¯ sg s σGs i ) cancel out with eachother, and result in small net uncertainty.The SU (3) breaking effects for the masses of the axial-vector hidden charmed and hid-den bottom tetraquark states are buried in the uncertainties. Naively, we expect the axial-vector and vector diquarks have larger masses than the corresponding scalar diquarks,and the masses of the tetraquark states have the hierarchy M Cγ µ − Cγ µ ≥ M Cγ − Cγ µ ≥ M Cγ − Cγ , because the attractive interactions of one-gluon exchange favor formation ofthe diquarks in color antitriplet 3 c , flavor antitriplet 3 f and spin singlet 1 s [39, 40]. FromTable 1, we can see that it is not the case.In the conventional QCD sum rules, we usually consult the experimental data in choos-ing the Borel parameter M and the threshold parameter s . If the mass spectrum of theaxial-vector tetraquark states are well known, we can denote the ground state, the firstexcited state, the second excited state, the third excited state, . . . , as Z , Z ′ , Z ′′ , Z ′′′ , . . . . The critical thresholds for emergence of those excited tetraquark states are T Z ′ ,7 Z ′′ , T Z ′′′ , . . . , respectively. The threshold parameter s should take values in the region( M Z + Γ Z ) ≤ s < T Z ′ . However, the present experimental knowledge about the phe-nomenological hadronic spectral densities of the multiquark states is rather vague, eventhe existence of the multiquark states is not confirmed with confidence, and no knowledgeabout either there are high resonances or not.Taking into account the two criteria (pole dominance and convergence of the operatorproduct expansion) of the QCD sum rules, we can obtain the possibly lowest thresholdparameter s , which is denoted as s min . In this article, we take the value s min and makecrude estimations for the ground state masses.The values of the s min in different channels maybe smaller (or larger) than T Z ′ , oreven smaller than ( M Z + Γ Z ) , the two criteria of the QCD sum rules alone cannot alwayswarrant satisfactory threshold parameters and Borel windows. For example, the nonetscalar mesons below 1 GeV (the f (980) and a (980) especially) are good candidates forthe tetraquark states [38, 54, 55]. The two criteria of the QCD sum rules result in thethreshold parameters s ≫ ( M f /a + Γ f /a ) , the contributions of the excited states arealready included in if there are any, and we have to resort to ”multi-pole + continuumstates” to approximate the phenomenological spectral densities. If we insist on the ”one-pole + continuum states” ansatz, no reasonable Borel window can be obtained, although itis not an indication non-existence of the light tetraquark states (For detailed discussionsabout this subject, one can consult Refs.[31, 56]). The QCD sum rules is just a QCDmodel.In the channel c ¯ cq ¯ q , the threshold parameter s min leads to the mass M c ¯ cq ¯ q = (4 . ± .
18) GeV, which is consistent with the experimental data M Z = (4433 ± ±
2) MeV or4443 +15 − − MeV from the Belle collaboration within uncertainty [7, 8]. The experimentalvalue is ( M Z + Γ Z ) ≤ . , the lower bound of the s min = (22 −
24) GeV issmaller than ( M Z + Γ Z ) , we have to postpone the s min to larger values. If we take s = (26 ±
1) GeV , the prediction M Z = (4 . ± .
19) GeV is in excellent agreement withexperimental data, see Fig.7. In Fig.8, we present the corresponding pole residue, fromthe figure, we can see that larger threshold parameter result in larger pole residue.The predictions of the QCD sum rules favor the scenario of the Z + (4430) as an axial-vector tetraquark state, the Z + (4430) can be tentatively identified as an axial-vectortetraquark state. In other channels, the heavy axial-vector tetraquark states exist innature maybe have larger masses than those theoretical predictions presented in Table1. On the other hand, the upper bound of the threshold parameter s = (25 −
27) GeV maybe larger than the critical threshold T Z ′ , so we identify the Z + (4430) as an axial-vectortetraquark state tentatively, not confidently.In Refs.[31, 32, 33], we observe that the meson Z (4250) may be a scalar tetraquarkstate ( c ¯ cu ¯ d ), irrespective of the Cγ µ − Cγ µ type and the Cγ − Cγ type diquark structures,the decay Z (4250) → π + χ c can take place with the Okubo-Zweig-Iizuka super-allowed”fall-apart” mechanism, which can take into account the large total width naturally. Inthe present case, the decay Z (4430) → ψ ′ π can also take place with the Okubo-Zweig-Iizuka super-allowed ”fall-apart” mechanism, which can take into account the large totalwidth (Γ Z = 45 +18 − − MeV or 107 +86 − − MeV) naturally [7, 8].In this article, we take the simple pole + continuum approximation for the phenomeno-logical spectral densities. In fact, such a simple approximation has shortcomings. In thecase of the non-relativistic harmonic-oscillator potential model, the spectrum of the bound8tates (the masses E n and the wave functions Ψ n ( x )) and the exact correlation functions(and hence its operator product expansion to any order) are known precisely. The non-relativistic harmonic-oscillator potential mω ~r is highly non-perturbative, one supposethe full Green function satisfies the Lippmann-Schwinger operator equation and may besolved perturbatively. We can introduce the Borel parameter dependent effective thresholdparameter z eff ( M ) = ω (cid:2) ¯ z + ¯ z p ωM + ¯ z ωM + · · · (cid:3) and fit the coefficients ¯ z i to reproduceboth the ground energy E and the pole residue R = Ψ ∗ (0)Ψ (0), the phenomenologicalspectrum density can be described by the perturbative contributions well above the effec-tive continuum threshold z eff ( M ), or reproduce the ground energy E only and take thepole residue R as a calculated parameter, there exists a solution for the effective contin-uum threshold z eff ( M ) which precisely reproduces the exact ground energy E for anyvalue of the pole residue R within the range R = (0 . − . R in the limited fiducialBorel window, the value of the pole residue R extracted from the sum rule is determinedto a great extent by the contribution of the hadron continuum [57]. There maybe systemicuncertainties out of control.In the real QCD world, the hadronic spectral densities are not known exactly. In thepresent case, the ground states have not been observed yet, except for the possible axial-vector tetraquark state candidate Z + (4430). So we have no confidence to introduce theBorel parameter dependent effective threshold parameter s eff ( M ) = ¯ s +¯ s M +¯ s M + · · · and approximate the phenomenological spectral densities with the perturbative contribu-tions above the effective continuum threshold s eff ( M ) accurately. Furthermore, the poleresidues (or the couplings of the interpolating currents to the ground state tetraquark) λ Z are not experimentally measurable quantities, and should be calculated by some theoret-ical approaches, the true values are difficult to obtain, which are distinguished from thedecay constants of the pseudoscalar mesons and the vector mesons, the decay constantscan be measured with great precision in the leptonic decays (in some channels).The spectrum of the bound states in the non-relativistic harmonic-oscillator potentialmodel are of the Dirac δ function type, we can choose z eff < E , while in the case ofthe QCD, the situation is rather complex, the effective continuum thresholds s eff ( M )maybe overlap with the first radial excited states, which are usually broad. For example,in the pseudoscalar channels, the widths of the π , π (1300), π (1800), · · · are ∼ . − .
6) GeV, (0 . ± . · · · respectively, while the widths of the K , K (1460), K (1830), · · · are ∼ ∼ (0 . − .
26) GeV, ∼ .
25 GeV, · · · respectively [53]. Inthis article, we prefer (or have to choose) the simple pole + continuum approximation,and cannot estimate the unknown systemic uncertainties of the QCD sum rules beforethe spectral densities in both the QCD and phenomenological sides are known with greataccuracy.In Ref.[58], Lucha, Melikhov and Simula use the correlation function of the pseu-doscalar current J ( x ) = ( m b + m u )¯ q ( x ) iγ b ( x ) to illustrate a Borel-parameter-dependenteffective continuum threshold can reduce considerably the (unphysical) dependence of theextracted bound-state mass and the decay constant on the Borel parameter. In the presentcase, we have no experimental data for the masses and pole residues of the tetraquarkstates to take as a guide and apply the χ minimization by adjusting the effective thresh-old parameters. On the other hand, the Borel-parameter-dependent effective continuumthresholds maybe overlap with the ( M Z + Γ Z ) , T Z ′ , T Z ′′ , · · · , we prefer the Borel-parameter-independent threshold parameters, although the Borel-parameter-dependent9ffective threshold parameters maybe smear some dependence on the Borel parameter.From Figs.5-8, we can see that the dependence of the masses and pole residues on theBorel parameters in the Borel windows are rather mild.The central values of our predictions are much larger than the corresponding onesfrom a relativistic quark model based on a quasipotential approach in QCD [59, 60]. InRefs.[59, 60], Ebert et al take the diquarks as bound states of the light and heavy quarks inthe color antitriplet channel, and calculate their mass spectrum using a Schrodinger typeequation, then take the masses of the diquarks as the basic input parameters, and study themass spectrum of the heavy tetraquark states as bound states of the diquark-antidiquarksystem. In Refs.[61, 62, 63], Maiani et al take the diquarks as the basic constituents,examine the rich spectrum of the diquark-antidiquark states from the constituent diquarkmasses and the spin-spin interactions, and try to accommodate some of the newly ob-served charmonium-like resonances not fitting a pure c ¯ c assignment; furthermore, thecorresponding bottom tetraquark states are also studied with the same method [64]. Thepredictions depend heavily on the assumption that the light scalar mesons a (980) and f (980) are tetraquark states, the basic parameters (constituent diquark masses) are es-timated thereafter. In the conventional quark models, the constituent quark masses aretaken as the basic input parameters, and fitted to reproduce the mass spectra of theconventional mesons and baryons. However, the present experimental knowledge aboutthe phenomenological hadronic spectral densities of the tetraquark states is rather vague,whether or not there exist tetraquark states is not confirmed with confidence, and noknowledge about the high resonances. The predicted constituent diquark masses can notbe confronted with the experimental data.In Refs.[21, 60, 62], the X (3872) and Z (4430) are taken as the ground state axial-vectorand first radially excited axial-vector tetraquark states respectively. In Ref.[65], Matheuset al study the X (3872) as a tetraquark state with J P C = 1 ++ using QCD spectral sumrules, the prediction is consistent with the experimental data within uncertainty. The dis-crepancy between the predictions of Ref.[65] and the present work (analogous interpolatingcurrents are chosen in those works) mainly originates from the high dimensional vacuumcondensates h ¯ qg s σGq i which are neglected in Ref.[65]. The condensates h ¯ qg s σGq i arecounted as O ( m c M ), and the corresponding contributions are greatly enhanced at small M ,and result in rather bad convergent behavior in the operator product expansion, we have tochoose larger Borel parameter M . We insist on taking into account the high dimensionalvacuum condensates, as the interpolating current consists of a light quark-antiquark pairand a heavy quark-antiquark pair, one of the highest dimensional vacuum condensates is h ¯ qq i × h α s GGπ i , we have to take into account the condensates h ¯ qg s σGq i for consistence.The LHCb is a dedicated b and c -physics precision experiment at the LHC (largehadron collider). The LHC will be the world’s most copious source of the b hadrons,and a complete spectrum of the b hadrons will be available through gluon fusion. Inproton-proton collisions at √ s = 14 TeV, the b ¯ b cross section is expected to be ∼ µb producing 10 b ¯ b pairs in a standard year of running at the LHCb operational luminosityof 2 × cm − sec − [36]. The axial-vector tetraquark states predicted in the presentwork may be observed at the LHCb, if they exist indeed. We can search for the axial-vector hidden charm tetraquark states in the D ¯ D ∗ , D ¯ D ∗ s , D s ¯ D ∗ , D s ¯ D ∗ s , J/ψπ , J/ψK , J/ψη , ψ ′ π , ψ ′ K , · · · invariant mass distributions and search for the axial-vector hiddenbottom tetraquark states in the B ¯ B ∗ , B ¯ B ∗ s , B s ¯ B ∗ , B s ¯ B ∗ s , Υ π , Υ K , Υ η , Υ ′ π , Υ ′ K , Υ ′ η ,10 .0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.00.000.020.040.060.080.100.120.140.160.180.20 (I)A f r a c t i on M [GeV ] ; ; ; ; ; . (II)A f r a c t i on M [GeV ] ; ; ; ; ; . (I)B f r a c t i on M [GeV ] ; ; ; ; ; . (II)B f r a c t i on M [GeV ] ; ; ; ; ; . (I)C f r a c t i on M [GeV ] ; ; ; ; ; . (II)C f r a c t i on M [GeV ] ; ; ; ; ; . Figure 1: The contributions from different terms with variation of the Borel parameter M in the operator product expansion. The (I) and (II) denote the contributions fromthe h ¯ qq i + h ¯ qq ih ¯ qg s σGq i term and the h ¯ qg s σGq i term respectively. The A , B and C denote the channels c ¯ cq ¯ q , c ¯ cq ¯ s and c ¯ cs ¯ s respectively. The notations α , β , γ , λ , τ and ρ correspond to the threshold parameters s = 21 GeV , 22 GeV , 23 GeV , 24 GeV ,25 GeV and 26 GeV respectively. 11 .0 6.4 6.8 7.2 7.6 8.0 8.4 8.8 9.2 9.6 10.0-0.20-0.15-0.10-0.050.000.050.100.150.20 (I)A f r a c t i on M [GeV ] ; ; ; ; ; . (II)A f r a c t i on M [GeV ] ; ; ; ; ; . (I)B f r a c t i on M [GeV ] ; ; ; ; ; . (II)B f r a c t i on M [GeV ] ; ; ; ; ; . (I)C f r a c t i on M [GeV ] ; ; ; ; ; . (II)C f r a c t i on M [GeV ] ; ; ; ; ; . Figure 2: The contributions from different terms with variation of the Borel parameter M in the operator product expansion. The (I) and (II) denote the contributions fromthe h ¯ qq i + h ¯ qq ih ¯ qg s σGq i term and the h ¯ qg s σGq i term respectively. The A , B and C denote the channels b ¯ bq ¯ q , b ¯ bq ¯ s and b ¯ bs ¯ s respectively. The notations α , β , γ , λ , τ and ρ correspond to the threshold parameters s = 132 GeV , 134 GeV , 136 GeV , 138 GeV ,140 GeV and 142 GeV respectively. 12 .0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.00.00.10.20.30.40.50.60.70.80.91.0 A po l e M [GeV ] ; ; ; ; ; . B po l e M [GeV ] ; ; ; ; ; . C po l e M [GeV ] ; ; ; ; ; . D po l e M [GeV ] ; ; ; ; ; . E po l e M [GeV ] ; ; ; ; ; . F po l e M [GeV ] ; ; ; ; ; . Figure 3: The contributions of the pole terms with variation of the Borel parameter M . The A , B , C , D , E and F denote the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s , b ¯ bq ¯ q , b ¯ bq ¯ s and b ¯ bs ¯ s respectively. The notations α , β , γ , λ , τ and ρ correspond to the threshold parameters s =21 GeV , 22 GeV , 23 GeV , 24 GeV , 25 GeV and 26 GeV respectively in the hiddencharmed channels; while they correspond to the threshold parameters s = 132 GeV ,134 GeV , 136 GeV , 138 GeV , 140 GeV and 142 GeV respectively in the hidden bottomchannels. 13 .0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.03.63.84.04.24.44.64.85.0 A M Z [ G e V ] M [GeV ] ; ; ; ; ; . B M Z [ G e V ] M [GeV ] ; ; ; ; ; . C M Z [ G e V ] M [GeV ] ; ; ; ; ; . D M Z [ G e V ] M [GeV ] ; ; ; ; ; . E M Z [ G e V ] M [GeV ] ; ; ; ; ; . F M Z [ G e V ] M [GeV ] ; ; ; ; ; . Figure 4: The masses of the axial-vector tetraquark states with variation of the Borelparameter M . The A , B , C , D , E and F denote the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s , b ¯ bq ¯ q , b ¯ bq ¯ s and b ¯ bs ¯ s respectively. The notations α , β , γ , λ , τ and ρ correspond to the thresholdparameters s = 21 GeV , 22 GeV , 23 GeV , 24 GeV , 25 GeV and 26 GeV respectivelyin the hidden charmed channels; while they correspond to the threshold parameters s =132 GeV , 134 GeV , 136 GeV , 138 GeV , 140 GeV and 142 GeV respectively in thehidden bottom channels. 14 .0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.03.63.84.04.24.44.64.85.05.25.45.6 A M Z [ G e V ] M [GeV ] Central value; Upper bound; Lower bound. B M Z [ G e V ] M [GeV ] Central value; Upper bound; Lower bound. C M Z [ G e V ] M [GeV ] Central value; Upper bound; Lower bound. D M Z [ G e V ] M [GeV ] Central value; Upper bound; Lower bound. E M Z [ G e V ] M [GeV ] Central value; Upper bound; Lower bound. F M Z [ G e V ] M [GeV ] Central value; Upper bound; Lower bound.
Figure 5: The masses of the axial-vector tetraquark states with variation of the Borelparameter M . The A , B , C , D , E and F denote the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s , b ¯ bq ¯ q , b ¯ bq ¯ s and b ¯ bs ¯ s respectively. 15 .0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.00123456789 A [ - G e V ] M [GeV ] Central value; Upper bound; Lower bound. B [ - G e V ] M [GeV ] Central value; Upper bound; Lower bound. C [ - G e V ] M [GeV ] Central value; Upper bound; Lower bound. D [ - G e V ] M [GeV ] Central value; Upper bound; Lower bound. E [ - G e V ] M [GeV ] Central value; Upper bound; Lower bound. F [ - G e V ] M [GeV ] Central value; Upper bound; Lower bound.
Figure 6: The pole residues of the axial-vector tetraquark states with variation of theBorel parameter M . The A , B , C , D , E and F denote the channels c ¯ cq ¯ q , c ¯ cq ¯ s , c ¯ cs ¯ s , b ¯ bq ¯ q , b ¯ bq ¯ s and b ¯ bs ¯ s respectively. 16 .0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.03.03.33.63.94.24.54.85.15.4 M Z [ G e V ] M [GeV ] Central value; Upper bound; Lower bound; Experimental data.
Figure 7: The mass of the Z (4430) with variation of the Borel parameter M . [ - G e V ] M [GeV ] Central value; Upper bound; Lower bound.
Figure 8: The pole residue of the Z (4430) with variation of the Borel parameter M .17etraquark states Cγ − Cγ µ λ Z Cγ µ − Cγ µ Cγ − Cγ c ¯ cq ¯ q . ± .
18 2 . ± .
60 4 . ± .
18 4 . ± . c ¯ cq ¯ s . ± .
16 3 . ± .
70 4 . ± . c ¯ cs ¯ s . ± .
16 3 . ± .
70 4 . ± .
16 4 . ± . b ¯ bq ¯ q . ± .
20 1 . ± .
34 11 . ± .
19 11 . ± . b ¯ bq ¯ s . ± .
18 1 . ± .
43 11 . ± . b ¯ bs ¯ s . ± .
16 1 . ± .
41 11 . ± .
16 11 . ± . Z (4430) 4 . ± .
19 3 . ± . − GeV and 10 − GeV in the channels c ¯ c and b ¯ b respectively. · · · invariant mass distributions. In this article, we construct the scalar-diquark-axial-vector-antidiquark type currents tointerpolate the axial-vector tetraquark states, and study the mass spectrum of the axial-vector hidden charmed and hidden bottom tetraquark states with the QCD sum rules in asystematic way. In calculations, we take into account the contributions from the vacuumcondensates adding up to dimension 10 in the operator product expansion, and neglectthe gluon condensates as their contributions are supposed to be very small. The massspectrum are calculated by imposing the two criteria (pole dominance and convergence ofthe operator product expansion) of the QCD sum rules. Our numerical result M Z (4430) =(4 . ± .
19) GeV is in excellent agreement with the experimental data M Z = (4433 ± ±
2) MeV or 4443 +15 − − MeV from the Belle collaboration, which indicates that the Z + (4430) can be identified as the axial-vector tetraquark state tentatively. Considering thelight-flavor SU (3) symmetry and the heavy quark symmetry, we make predictions for otheraxial-vector tetraquark states, the predictions can be confronted with the experimentaldata in the future at the LHCb or the Fermi-lab Tevatron. Appendix
The spectral densities ρ q ¯ q ( s ), ρ q ¯ s ( s ) and ρ s ¯ s ( s ) at the level of the quark-gluon degrees offreedom: 18 q ¯ q ( s ) = 13072 π Z α f α i dα Z − αβ i dβαβ (1 − α − β ) ( s − e m Q ) (35 s − s e m Q + 3 e m Q ) − m Q h ¯ qq i π Z α f α i dα Z − αβ i dβ (1 − α − β )( s − e m Q ) (cid:2) (3 α + 4 β ) s − ( α + 2 β ) e m Q (cid:3) + m Q h ¯ qg s σGq i π Z α f α i dα Z − αβ i dβ (cid:2) (2 α + 3 β ) s − ( α + 2 β ) e m Q ) (cid:3) + m Q h ¯ qq i π Z α f α i dα − m Q h ¯ qq ih ¯ qg s σGq i π Z α f α i dα h sM i δ ( s − ee m Q )+ m Q h ¯ qg s σGq i π M Z α f α i dαs δ ( s − ee m Q ) , (11)19 q ¯ s ( s ) = 13072 π Z α f α i dα Z − αβ i dβαβ (1 − α − β ) ( s − e m Q ) (35 s − s e m Q + 3 e m Q )+ m s m Q π Z α f α i dα Z − αβ i dββ (1 − α − β ) ( s − e m Q ) (5 s − e m Q )+ m s h ¯ ss i π Z α f α i dα Z − αβ i dβαβ (1 − α − β )(15 s − s e m Q + 3 e m Q )+ m Q h ¯ qq i π Z α f α i dα Z − αβ i dβα (1 − α − β )( s − e m Q )( e m Q − s )+ m Q h ¯ ss i π Z α f α i dα Z − αβ i dββ (1 − α − β )( s − e m Q )( e m Q − s )+ m Q h ¯ qg s σGq i π Z α f α i dα Z − αβ i dβα (2 s − e m Q )+ m Q h ¯ sg s σGs i π Z α f α i dα Z − αβ i dββ (3 s − e m Q ) − m s h ¯ sg s σGs i π Z α f α i dα Z − αβ i dβαβ (cid:2) s − e m Q + s δ ( s − e m Q ) (cid:3) − m s m Q h ¯ qq i π Z α f α i dα Z − αβ i dβ ( s − e m Q )+ m Q h ¯ qq ih ¯ ss i π Z α f α i dα + m s m Q h ¯ qg s σGq i π Z α f α i dα − m s m Q h ¯ qq ih ¯ ss i π Z α f α i dαα h sδ ( s − ee m Q ) i − m Q [ h ¯ qq ih ¯ sg s σGs i + h ¯ ss ih ¯ qg s σGq i ]48 π Z α f α i dα h sM i δ ( s − ee m Q )+ m s m Q [2 h ¯ qq ih ¯ sg s σGs i + 3 h ¯ ss ih ¯ qg s σGq i ]288 π M Z α f α i dαα (cid:20) s − s M (cid:21) δ ( s − ee m Q )+ m Q h ¯ qg s σGq ih ¯ sg s σGs i π M Z α f α i dαs δ ( s − ee m Q ) , (12)20 s ¯ s ( s ) = 13072 π Z α f α i dα Z − αβ i dβαβ (1 − α − β ) ( s − e m Q ) (35 s − s e m Q + 3 e m Q )+ m s m Q π Z α f α i dα Z − αβ i dβ (1 − α − β ) ( s − e m Q ) (cid:2) (4 α + 5 β ) s − ( α + 2 β ) e m Q (cid:3) + m s h ¯ ss i π Z α f α i dα Z − αβ i dβαβ (1 − α − β )(15 s − s e m Q + 3 e m Q ) − m Q h ¯ ss i π Z α f α i dα Z − αβ i dβ (1 − α − β )( s − e m Q ) (cid:2) (3 α + 4 β ) s − ( α + 2 β ) e m Q (cid:3) + m Q h ¯ sg s σGs i π Z α f α i dα Z − αβ i dβ (cid:2) (2 α + 3 β ) s − ( α + 2 β ) e m Q (cid:3) − m s h ¯ sg s σGs i π Z α f α i dα Z − αβ i dβαβ (cid:2) s − e m Q + s δ ( s − e m Q ) (cid:3) − m s m Q h ¯ ss i π Z α f α i dα Z − αβ i dβ ( s − e m Q )+ m Q h ¯ ss i π Z α f α i dα + m s m Q h ¯ sg s σGs i π Z α f α i dα − m s m Q h ¯ ss i π Z α f α i dα (1 − α ) h sδ ( s − ee m Q ) i − m s m Q h ¯ ss i π Z α f α i dαα h sδ ( s − ee m Q ) i − m Q h ¯ ss ih ¯ sg s σGs i π Z α f α i dα h sM i δ ( s − ee m Q )+ 5 m s m Q h ¯ ss ih ¯ sg s σGs i π M Z α f α i dαα (cid:20) s − s M (cid:21) δ ( s − ee m Q )+ 5 m s m Q h ¯ ss ih ¯ sg s σGs i π Z α f α i dα (1 − α ) (cid:20) sM + s M (cid:21) δ ( s − ee m Q )+ m Q h ¯ sg s σGs i π M Z α f α i dαs δ ( s − ee m Q ) , (13) Acknowledgements
This work is supported by National Natural Science Foundation of China, Grant Number10775051, and Program for New Century Excellent Talents in University, Grant NumberNCET-07-0282, and the Fundamental Research Funds for the Central Universities.
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