Relativistic description of heavy tetraquarks
aa r X i v : . [ h e p - ph ] F e b Relativistic description of heavy tetraquarks
D. Ebert , R. N. Faustov and V. O. Galkin Institut f¨ur Physik, Humboldt–Universit¨at zu Berlin,Newtonstr. 15, D-12489 Berlin, Germany Dorodnicyn Computing Centre, Russian Academy of Sciences,Vavilov Str. 40, 119991 Moscow, Russia
The masses of the ground state and excited heavy tetraquarks with hidden charmand bottom are calculated within the relativistic diquark-antidiquark picture. Thedynamics of the light quark in a heavy-light diquark is treated completely relativis-tically. The diquark structure is taken into account by calculating the diquark-gluonform factor. New experimental data on charmonium-like states above the opencharm threshold are discussed. The obtained results indicate that X (3872), Y (4260), Y (4360), Z (4433) and Y (4660) can be tetraquark states with hidden charm. Recently the significant experimental progress has been achieved in heavy hadron spec-troscopy. Several new charmonium-like states, such as X (3872), Y (4260), Y (4360), Y (4660), Z (4430), etc., were observed [1] which cannot be simply accommodated in the quark-antiquark ( q ¯ q ) picture. These states can be considered as indications of the possible ex-istence of exotic multiquark states [2, 3]. Here we briefly review our recent results for themasses of heavy tetraquarks in the framework of the relativistic quark model based on thequasipotential approach in quantum chromodynamics. We use the diquark-antidiquark ap-proximation to reduce a complicated relativistic four-body problem to the subsequent moresimple two-body problems. The first step consists in the calculation of the masses, wavefunctions and form factors of the diquarks, composed from light and heavy quarks. At thefinal step, a heavy tetraquark is considered to be a bound diquark-antidiquark system. It isimportant to emphasize that we do not consider a diquark as a point particle but explicitlytake into account its structure by calculating the form factor of the diquark-gluon interactionin terms of the diquark wave functions.In the quasipotential approach the two-particle bound state with the mass M and massesof the constituents m , in momentum representation is described by the wave function Ψ( p )satisfying the quasipotential equation of the Schr¨odinger type b ( M )2 µ R − p µ R ! Ψ d,T ( p ) = Z d q (2 π ) V d,T ( p , q ; M )Ψ d,T ( q ) , (1)where the relativistic reduced mass is µ R = M − ( m − m ) M , and the on-mass-shell relative momentum squared b ( M ) = [ M − ( m + m ) ][ M − ( m − m ) ]4 M . The subscript d refers to the diquark and T refers to the tetraquark composed of a diquarkand antidiquark. The explicit expressions for the corresponding quasipotentials V d,T ( p , q ; M )can be found in Ref. [4]. TABLE I: Masses M and form factor parameters of heavy-light diquarks. S and A denote scalarand axial vector diquarks antisymmetric [ Q, q ] and symmetric { Q, q } in flavour, respectively.Quark Diquark Q = c Q = b content type M (MeV) ξ (GeV) ζ (GeV ) M (MeV) ξ (GeV) ζ (GeV )[ Q, q ] S { Q, q } A Q, s ] S { Q, s } A At the first step, we calculate the masses and form factors of the light and heavy diquarks.As it is well known, the light quarks are highly relativistic, which makes the v/c expansioninapplicable and thus, a completely relativistic treatment of the light quark dynamics isrequired. To achieve this goal we closely follow our consideration of the mass spectra oflight mesons and adopt the same procedure to make the relativistic potential local by re-placing ǫ , ( p ) = q m , + p → E , = ( M − m , + m , ) / M . Solving numerically thequasipotential equation (1) with the complete relativistic potential, which depends on thediquark mass in a complicated highly nonlinear way [5], we get the diquark masses andwave functions. In order to determine the diquark interaction with the gluon field, whichtakes into account the diquark structure, we calculate the corresponding matrix element ofthe quark current between diquark states. Such calculation leads to the emergence of theform factor F ( r ) entering the vertex of the diquark-gluon interaction [5]. This form factoris expressed through the overlap integral of the diquark wave functions. Our estimates showthat it can be approximated with a high accuracy by the expression F ( r ) = 1 − e − ξr − ζr . (2)The values of the masses and parameters ξ and ζ for heavy-light scalar diquark [ Q, q ] andaxial vector diquark { Q, q } ground states are given in Table I.At the final step, we calculate the masses of heavy tetraquarks considered as the boundstates of a heavy-light diquark and antidiquark. In this picture of heavy tetraquarks bothscalar S (asymmetric in flavour [ Qq ] S =0 = [ Qq ]) and axial vector A (symmetric in flavour[ Qq ] S =1 = { Qq } ) diquarks are considered. Therefore we get the following structure of the[ Qq ][ ¯ Q ¯ q ′ ] ground (1 S ) states ( C is defined only for q = q ′ ): • Two states with J P C = 0 ++ : X (0 ++ ) = [ Qq ] S =0 [ ¯ Q ¯ q ′ ] S =0 X (0 ++ ′ ) = [ Qq ] S =1 [ ¯ Q ¯ q ′ ] S =1 • Three states with J = 1: X (1 ++ ) = 1 √ Qq ] S =1 [ ¯ Q ¯ q ′ ] S =0 + [ Qq ] S =0 [ ¯ Q ¯ q ′ ] S =1 ) X (1 + − ) = 1 √ Qq ] S =0 [ ¯ Q ¯ q ′ ] S =1 − [ Qq ] S =1 [ ¯ Q ¯ q ′ ] S =0 ) X (1 + −′ ) = [ Qq ] S =1 [ ¯ Q ¯ q ′ ] S =1 TABLE II: Masses of charm diquark-antidiquark ground (1 S ) states (in MeV). S and A denotescalar and axial vector diquarks.State Diquark Mass J P C content cq ¯ c ¯ q cs ¯ c ¯ s cs ¯ c ¯ q/cq ¯ c ¯ s ++ S ¯ S + ± ( S ¯ A ± ¯ SA ) / √ ++ A ¯ A + − A ¯ A ++ A ¯ A D ¯ D D + s D − s D D ± s D + D − η ′ J/ψ D ± D ∓ s D ¯ D ∗ D ± s D ∗∓ s D ∗ D ± s ρJ/ψ φJ/ψ D D ∗± s D ± D ∗∓ D ∗ + s D ∗− s K ∗± J/ψ ωJ/ψ K ∗ J/ψ D ∗ ¯ D ∗ D ∗ D ∗± s • One state with J P C = 2 ++ : X (2 ++ ) = [ Qq ] S =1 [ ¯ Q ¯ q ′ ] S =1 . The orbitally excited (1 P, D . . . ) states are constructed analogously. As we see a very richspectrum of tetraquarks emerges. However the number of states in the considered diquark-antidiquark picture is significantly less than in the genuine four-quark approach.The diquark-antidiquark model of heavy tetraquarks predicts [6] the existence of theflavour SU (3) nonet of states with hidden charm or beauty ( Q = c, b ): four tetraquarks([ Qq ][ ¯ Q ¯ q ], q = u, d ) with neither open or hidden strangeness, which have electric charges0 or ± Qs ][ ¯ Q ¯ q ] and [ Qq ][ ¯ Q ¯ s ], q = u, d ) with openstrangeness ( S = ± ± ; one tetraquark([ Qs ][ ¯ Q ¯ s ]) with hidden strangeness and zero electric charge. Since in our model we ne-glect the mass difference of u and d quarks and electromagnetic interactions, correspondingtetraquarks will be degenerate in mass. A more detailed analysis [6] predicts that such massdifferences can be of a few MeV so that the isospin invariance is broken for the [ Qq ][ ¯ Q ¯ q ]mass eigenstates and thus in their strong decays. The (non)observation of such states willbe a crucial test of the tetraquark model.The calculated masses of the heavy tetraquark ground (1 S ) states and the correspondingopen charm and bottom thresholds are given in Tables II-V. We find that all S -wavetetraquarks with hidden bottom lie considerably below open bottom thresholds and thus theyshould be narrow states which can be observed experimentally. This prediction significantlydiffers from the molecular picture where bound B − ¯ B ∗ states are expected to lie very close TABLE IV: Masses of bottom diquark-antidiquark ground (1 S ) states (in MeV). S and A denotescalar and axial vector diquarks.State Diquark Mass J P C content bq ¯ b ¯ q bs ¯ b ¯ s bs ¯ b ¯ q/bq ¯ b ¯ s ++ S ¯ S + ± ( S ¯ A ± ¯ SA ) / √ ++ A ¯ A + − A ¯ A ++ A ¯ A B ¯ B B + s B − s BB s B ¯ B ∗ B ± s B ∗∓ s B ∗ B s B ∗ ¯ B ∗ B ∗ + s B ∗− s B ∗ B ∗ s (only few MeV below) to the corresponding thresholds.The situation in the hidden charm sector is considerably more complicated, since most ofthe tetraquark states are predicted to lie either above or only slightly below correspondingopen charm thresholds. This difference is the consequence of the fact that the charm quarkmass is substantially smaller than the bottom quark mass. As a result the binding energiesin the charm sector are significantly smaller than those in the bottom sector.In Table VI we compare our results (EFG [4]) for the masses of the ground and excitedcharm diquark-antidiquark bound states with the predictions of Ref. [6, 7, 8, 9] and withthe masses of the recently observed excited charmonium-like states [1]. We assume that theexcitations occur only inside the diquark-antidiquark bound system. Possible excitationsof diquarks are not considered. Our calculation of the heavy baryon masses supports suchscheme [5]. In this table we give our predictions only for some of the masses of the or-bitally and radially excited states for which possible experimental candidates are available.The differences in some of the presented theoretical mass values can be attributed to thesubstantial distinctions in the used approaches. We describe the diquarks dynamically asquark-quark bound systems and calculate their masses and form factors, while in Ref.[6]they are treated only phenomenologically. Then we consider the tetraquark as purely thediquark-antidiquark bound system. In distinction Maini et al. consider a hyperfine inter-action between all quarks which, e.g., causes the splitting of 1 ++ and 1 + − states arisingfrom the SA diquark-antidiquark compositions. From Table VI we see that our dynamicalcalculation supports the assumption [6] that X (3872) can be the axial vector 1 ++ tetraquarkstate composed from the scalar and axial vector diquark and antidiquark in the relative 1 S state. Recent Belle and BaBar results indicate the existence of a second X (3875) particle afew MeV above X (3872). This state could be naturally identified with the second neutralparticle predicted by the tetraquark model [7]. On the other hand, in our model the lightestscalar 0 ++ tetraquark is predicted to be above the open charm threshold D ¯ D and thus tobe broad, while in the model [6] it lies few MeV below this threshold, and thus is predicted TABLE VI: Comparison of theoretical predictions for the masses of the ground and excited charmdiquark-antidiquark states cq ¯ c ¯ q (in MeV) and possible experimental candidates.State Diquark Theory Experiment J P C content EFG Maiani et al. Maiani et al. ( cs ¯ c ¯ s ) state mass1 S ++ S ¯ S ++ ( S ¯ A + ¯ SA ) / √ † ( X (3872) X (3876) ( . ± . . ± . +1 . − . + − ( S ¯ A − ¯ SA ) / √ ++ A ¯ A + − A ¯ A ++ A ¯ A Y (3943) ( ± ± . +4 . − . P −− S ¯ S ± Y (4260) ( ± +2 − ± +17 − −− ( S ¯ A − ¯ SA ) / √ Y (4260) 4283 +17 − ± −− A ¯ A −− A ¯ A Y (4360) 4361 ± ± S + ± ( S ¯ A ± ¯ SA ) / √ Z (4430) 4433 ± ± ++ A ¯ A + − A ¯ A ∼ P −− S ¯ S Y (4660) 4664 ± ± † input to be narrow. Our 2 ++ state also lies higher than the one in Ref.[6], thus making the inter-pretation of this state as Y (3943) less probable especially if one averages the original Bellemass with the recent BaBar value wich is somewhat lower.The recent discovery of the Y (4260), Y (4360) and Y (4660) indicates an excess of theexpected charmonium 1 −− states [1]. The absence of open charm production is also in-consistent with a conventional c ¯ c explanation. Maini et al. [8] argue that Y (4260) is the1 −− P state of the charm-strange diquark-antidiquark tetraquark. We find that Y (4260)cannot be interpreted in this way, since the mass of such ([ cs ] S =0 [¯ c ¯ s ] S =0 ) tetraquark is foundto be ∼
200 MeV higher. A more natural tetraquark interpretation could be the 1 −− P state ([ cq ] S =0 [¯ c ¯ q ] S =0 ) ( S ¯ S ) which mass is predicted in our model to be close to the mass of Y (4260) (see Table VI). Then the Y (4260) would decay dominantly into D ¯ D pairs. Theother possible interpretations of Y (4260) are the 1 −− P states of ( S ¯ A − ¯ SA ) / √ A ¯ A tetraquarks which predicted masses have close values. These additional tetraquark statescould be responsible for the mass difference of Y (4260) observed in different decay channels.As we see from Table VI the recently discovered resonances Y (4360) and Y (4660) in the e + e − → π + π − ψ ′ cross section can be interpreted as the excited 1 −− P ( A ¯ A ) and 2 P ( S ¯ S )tetraquark states, respectively.Very recently the Belle Collaboration reported observation of a relatively narrow en-hancement in the π + ψ ′ invariant mass distribution in the B → Kπ + ψ ′ decay [1]. This newresonance, Z + (4430), is unique among other exotic meson candidates since it has a non-zeroelectric charge. Different theoretical interpretations were suggested [1]. Maiani et al. [9]give qualitative arguments that the Z + (4430) could be the first radial excitation (2 S ) of adiquark-antidiquark X + u ¯ d (1 + − ; 1 S ) state ( A ¯ A ) with mass 3882 MeV. Our calculations indi-cate that the Z + (4430) can indeed be the 1 + − S [ cu ][¯ c ¯ d ] tetraquark state. It is the firstradial excitation of the ground state ( S ¯ A − ¯ SA ) / √
2, which has the same mass as X (3872).In summary, we calculated the masses of heavy tetraquarks with hidden charm and bot-tom in the diquark-antidiquark picture. In contrast to previous phenomenological treatmentswe used the dynamical approach based on the relativistic quark model. Both diquark andtetraquark masses were obtained by numerical solution of the quasipotential equation withthe corresponding relativistic potentials. The diquark structure was also taken into accountwith the help of the diquark-gluon form factor expressed in terms of diquark wave functions.It is important to emphasize that, in our analysis, we did not introduce any free adjustableparameters but used their fixed values from our previous considerations of heavy and lightmeson properties. It was found that the X (3872), Y (4260), Y (4360), Z (4433) and Y (4660)exotic meson candidates can be tetraquark states with hidden charm. The ground states ofbottom tetraquarks are predicted to have masses below the open bottom threshold and thusshould be narrow.The authors are grateful to A. Badalian, A. Kaidalov, V. Matveev, G. Pakhlova, P.Pakhlov, A. Polosa and V. Savrin for support and discussions. This work was supported inpart by the Deutsche Forschungsgemeinschaft under contract Eb 139/2-4 and by the
RussianFoundation for Basic Research under Grant No.05-02-16243. [1] For recent reviews see e.g. G. V. Pakhlova, talk at the scientific session-conference of NuclearPhysics Department RAS “Physics of fundamental interactions”, 25-30 November 2007, ITEP,Moscow; S. Godfrey and S. L. Olsen, arXiv:0801.3867 [hep-ph].[2] R. L. Jaffe, Phys. Rev. D , 267 (1977); Phys. Rev. Lett. , 195 (1977); V. A. Matveev andP. Sorba, Lett. Nuovo Cim. , 443 (1977).[3] A. M. Badalyan, B. L. Ioffe and A. V. Smilga, Nucl. Phys. B , 85 (1987); A. B. Kaidalov,Surveys in High Energy Physics , 265 (1999).[4] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Lett. B , 214 (2006); D. Ebert, R. N. Faus-tov, V. O. Galkin and W. Lucha, Phys. Rev. D , 114015 (2007)[5] D. Ebert, R. N. Faustov and V. O. Galkin, Phys. Rev. D , 034026 (2005); Phys. Lett. B , 612 (2008).[6] L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D , 014028 (2005).[7] L. Maiani, A. D. Polosa and V. Riquer, Phys. Rev. Lett. , 182003 (2007).[8] L. Maiani, F. Piccinini, A. D. Polosa and V. Riquer, Phys. Rev. D72