Masses of tetraquarks with open charm and bottom

Physics Letters B 696 (2011) 241–245

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Physics Letters B www.elsevier.com/locate/physletb

Masses of tetraquarks with open charm and bottom D. Ebert a , R.N. Faustov a,b , V.O. Galkin a,b,∗ a b

Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany Dorodnicyn Computing Centre, Russian Academy of Sciences, Vavilov Str. 40, 119991 Moscow, Russia

a r t i c l e

i n f o

Article history: Received 12 November 2010 Received in revised form 17 December 2010 Accepted 17 December 2010 Available online 23 December 2010 Editor: A. Ringwald Keywords: Meson spectroscopy Heavy tetraquarks Diquarks Relativistic quark model

a b s t r a c t The masses of the heavy tetraquarks with open charm and bottom are calculated within the diquark– antidiquark picture in the framework of the relativistic quark model. The dynamics of the light quarks and diquarks is treated completely relativistically. The diquark structure is taken into account by calculating the diquark–gluon form factor. New experimental data on charmed and charmed-strange mesons are discussed. Our results indicate that the anomalous scalar D ∗s0 (2317) and axial vector D s1 (2460) mesons could not be considered as diquark–antidiquark bound states. On the other hand, D s (2632) and D ∗s J (2860) could be interpreted as scalar and tensor tetraquarks, respectively. The predictions for masses of the corresponding bottom counterparts of the charmed tetraquarks are given. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In last few years a significant experimental progress has been achieved in charmonium and charmed meson spectroscopy. Many new states, such as X (3872), Y (4260), Y (4360), Y (4660), Z (4248), Z (4430), D ∗s0 (2317), D s1 (2460), D s (2632) and D ∗s J (2860), etc., were observed [1] which cannot be simply accommodated in the quark–antiquark (c c¯ and c s¯ ) picture. These states and especially the charged charmonium-like ones can be considered as indications of the possible existence of exotic multiquark states. The open charm mesons, both with and without open strangeness, represent a special interest. Even seven years after the discovery of the charmed-strange D ∗s0 (2317) and D s1 (2460) mesons their nature remains controversial in the literature. The abnormally light masses of these mesons put them below D K and D ∗ K thresholds, thus making these states narrow, since the only allowed strong de(∗) (∗) cays D s J → D s π violate isospin symmetry. The peculiar feature of these mesons is that they have masses almost equal or even lower than the masses of their charmed counterparts D ∗0 (2400) and D 1 (2427) [2–4]. Most of the theoretical approaches including lattice QCD [5], QCD sum rule [6] and different quark model [7,8] calculations give masses of the 0+ and 1+ P -wave c s¯ states significantly heavier (by 100–200 MeV) than the measured ones. Different theoretical solutions of this problem were proposed in-

*

Corresponding author at: Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany. E-mail address: [email protected] (V.O. Galkin). 0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2010.12.033

cluding consideration of these mesons as chiral partners of 0− and 1− states [9], members of the J P = (0+ , 1+ ) doublet predicted by the heavy quark effective theory [10], c s¯ states which are strongly influenced by the nearby D K thresholds [11], D K or D s π molecules [12], a mixture of c s¯ and tetraquark states [13– 17]. However the universal understanding of their nature is still missing. Therefore it is very important to observe their bottom counterparts. The unquenched lattice calculations of their masses can be found in Ref. [18]. Another unexpectedly narrow charmed-strange meson D s (2632) was discovered by SELEX Collaboration [19]. Its unusual decay properties triggered speculations about its possible exotic origin [1]. However, the status of this state remains controversial since FOCUS [20], BaBar [21] and Belle [22] reported negative results in their search for this state. Three other charmed-strange mesons D s1 (2710), D ∗s J (2860) and D s J (3040) were discovered at B-factories by Belle and BaBar [23,24]. Not all of them could be simply accommodated in the usual c s¯ picture. Their decay pattern implies that D ∗s J (2860) should have natural parity, while D s1 (2710) and D s J (3040) should have unnatural parity. Our recent calculation of the heavy-light quark– antiquark meson spectra [25] have shown that D s1 (2710) and D s J (3040) are good candidates for the 23 S 1 and 2P 1 states, respectively. They nicely fit to the corresponding Regge trajectories, while D ∗s0 (2317), D s1 (2460), D ∗s J (2860) and D s (2632) have anomalously low masses (lower than expected by 100–200 MeV) and do not lie on the respective Regge trajectories. In the charmed sector very recently the BaBar Collaboration discovered four new signal peaks D (2550), D ∗ (2600), D (2750) and

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D. Ebert et al. / Physics Letters B 696 (2011) 241–245

Table 1 Predictions [25] for the masses of charmed (q = u , d) and charmed-strange mesons (in MeV) in comparison with available experimental data [4,26].

JP

Theory mass (cq¯ )

Experiment

n2S +1 L J

State

meson

mass

11 S 0 13 S 1

0− 1−

1871 2010

D D ∗ (2010)

13 P 0

0+

2406

D ∗0 (2400)

1P 1 1P 1 13 P 2 21 S 0

1+ 1+ 2+ 0−

2469 2426 2460 2581

23 S 1

1−

D 1 (2430) D 1 (2420) D ∗2 (2460) D (2550)a ⎧ ⎨ D ∗ (2600)a

2632

⎩ ∗ D (2640) ⎧ ⎨ D ∗ (2760)a

1869.62(16) 2010.25(14)  2403(40)(± ) 2318(29)(0 ) 2427(40) 2423.4(3.1) 2.6 2460.1(+ −3.5 ) 2539.4(8.1)(0 )  2621.3(5.6)(+ ) 2608.7(3.5)(0 ) 2637(6)?

3

1 D1 1D 2 1D 2 13 D 3 23 P 0 2P 1 2P 1 23 P 2 31 S 0 33 S 1 a

1− 2− 2− 3− 0+ 1+ 1+ 2+ 0− 1−

2788

⎩

D (2750)a

Theory mass (c s¯ )

Experiment meson

mass

1969 2111

Ds D ∗s

1968.47(33) 2112.3(5)

2509

D ∗s0 (2317)

2317.8(6)

2574 2536 2571 2688

D s1 (2460) D s1 (2536) D s2 (2573)

2459.5(6) 2535.29(20) 2572.6(9)

2731

D s1 (2710)

9 2709(+ −6 )

D ∗s J (2860)

6 2862(+ −3 )

D s J (3040)

31 3044(+ −9 )



2769.7(4.1)(+ ) 2763.3(3.3)(0 ) 2752.4(3.2)

2913

2850 2806 2863 2919 3021 2932 3012 3062 3096

2961 2931 2971 3054 3154 3067 3142 3219 3242

New states recently observed by BaBar [26].

D ∗ (2760) [26]. The last two signals, observed in D ∗ π and D π modes, have close mass and width values (which differ by 2.6σ and 1.5σ , respectively) and therefore could belong to the same state. The angular analysis shows that these signals can be considered as candidates for the radially excited 21 S 0 , 23 S 1 states and orbitally excited 13 D 1 state, respectively. Their mass values are in a good agreement with the results of our model. We summarize our predictions for the masses of the cq¯ (q = u , d) and c s¯ mesons in Table 1 and confront them with the available experimental data [4, 26]. As it is clearly seen there are indications of the possible existence of the exotic states especially in the charmed-strange sector. In our papers [27,28] we calculated masses of the hidden heavy-flavor tetraquarks and light tetraquarks in the framework of the relativistic quark model based on the quasipotential approach in quantum chromodynamics. Here we extend this analysis to the consideration of heavy tetraquark states with open charm and bottom. This study could help in revealing the nature of the anomalous charmed-strange mesons. As previously [27,28], we use the diquark–antidiquark picture to reduce a complicated relativistic four-body problem to two subsequent more simple two-body problems. The first step consists in the calculation of the masses, wave functions and form factors of the diquarks, composed from light and heavy quarks. At the second step, a tetraquark is considered to be a bound diquark–antidiquark system. It is important to emphasize that we do not consider the diquark as a point particle but explicitly take into account its structure by calculating the form factor of the diquark–gluon interaction in terms of the diquark wave functions. 2. Relativistic model of tetraquarks In the quasipotential approach and diquark–antidiquark picture of heavy tetraquarks the interaction of two quarks in a diquark and the diquark–antidiquark interaction in a tetraquark are described by the diquark wave function (Ψd ) of the bound quark–quark state and by the tetraquark wave function (Ψ T ) of the bound diquark– antidiquark state, respectively. These wave functions satisfy the quasipotential equation of the Schrödinger type [29]



b2 ( M ) 2μ R

−



p2 2μ R

 Ψd,T (p) =

d3 q

(2π )3

V (p, q; M )Ψd, T (q),

(1)

where the relativistic reduced mass is

μR =

E1 E2 E1 + E2

=

M 4 − (m21 − m22 )2 4M 3

(2)

,

and E 1 , E 2 are given by

E1 =

M 2 − m22 + m21 2M

,

E2 =

M 2 − m21 + m22 2M

(3)

.

Here, M = E 1 + E 2 is the bound-state mass (diquark or tetraquark), m1,2 are the masses of quarks (q and Q ) which form the diquark or of the diquark (d) and antiquark (d¯  ) which form the heavy tetraquark (T ), and p is their relative momentum. In the center-ofmass system the relative momentum squared on mass shell reads

b2 ( M ) =

[ M 2 − (m1 + m2 )2 ][ M 2 − (m1 − m2 )2 ] 4M 2

.

(4)

The kernel V (p, q; M ) in Eq. (1) is the quasipotential operator of the quark–quark or diquark–antidiquark interaction. It is constructed with the help of the off-mass-shell scattering amplitude, projected onto the positive-energy states. For the quark–quark interaction in a diquark we use the relation V qq = V qq¯ /2 arising under the assumption of an octet structure of the interaction from the difference in the qq and qq¯ color states. An important role in this construction is played by the Lorentz structure of the confining interaction. In our analysis of mesons, while constructing the quasipotential of the quark–antiquark interaction, we assumed that the effective interaction is the sum of the usual one-gluon exchange term and a mixture of long-range vector and scalar linear confining potentials, where the vector confining potential contains the Pauli term. We use the same conventions for the construction of the quark–quark and diquark–antidiquark interactions in the tetraquark. The quasipotential is then defined as follows [7,27]. (a) For the quark–quark ( Q q) interactions, V (p, q; M ) reads

V (p, q; M ) = u¯ 1 ( p )u¯ 2 (− p )V (p, q; M )u 1 (q)u 2 (−q),

(5)

D. Ebert et al. / Physics Letters B 696 (2011) 241–245

with



V (p, q; M ) =

1 4 2 3

vector in the diquark rest frame. The effective long-range vector vertex of the diquark can be presented in the form μ V αs D μν (k)γ1μ γ2ν + V conf (k)Γ1 (k)Γ2;μ (−k)

S + V conf (k) .

J d;μ =

D

0i

4π

D i j (k) = −

,

k2

4π k2

k

= D = 0,

(6)

and k = p − q; γμ and u ( p ) are the Dirac matrices and spinors,

 u ( p) =

⎛

 ( p) + m ⎝ 2 ( p ) 

⎞

1

⎠ χ λ,

σ ·p

(7)

 ( p) + m

with  ( p ) = p2 + m2 . The effective long-range vector vertex of the quark is defined by [7]

Γμ (k) = γμ +

iκ 2m

σμν k˜ ν , k˜ = (0, k),

(8)

where κ is the Pauli interaction constant characterizing the anomalous chromomagnetic moment of quarks. In configuration space the vector and scalar confining potentials in the nonrelativistic limit [29] reduce to V V conf (r ) S V conf (r )

= ε V conf (r ),

(9)

S V V conf (r ) = V conf (r ) + V conf (r ) = Ar + B ,

(10)

where ε is the mixing coefficient. (b) For the diquark–antidiquark (dd¯  ) interaction, V (p, q; M ) is given by

d( P )| J μ |d( Q ) 4 d ( P  )| J ν |d ( Q  ) V (p, q; M ) = αs D μν (k) √ √ Ed Ed

∗

∗

3

  

+ ψd ( P )ψd P J d;μ   × ψd ( Q )ψd Q  ,

2

μ

V J d V conf (k) +

E d E d



S V conf (k)

(11)

where d( P )| J μ |d( Q ) is the vertex of the diquark–gluon interac

tion which takes into account the finite size of the diquark [ P ( ) =  ( E d( ) , ±p) and Q ( ) = ( E d( ) , ±q), E d = ( M 2 − M d2 + M d2 )/(2M ) and

E d = ( M 2 − M d2 + M d2 )/(2M )]. The diquark state in the confining part of the diquark–antidiquark quasipotential (11) is described by the wave functions



ψd ( p ) =

1

for a scalar diquark,

εd ( p ) for an axial-vector diquark,

(12)

where the four-vector



(εd · p) Md

 (εd · p)p , , εd + Md ( E d ( p) + Md )

μ

εd ( p ) p μ = 0, p2 + M d2 , and

for an axial-vector diquark.

ν is defined by Here, the antisymmetric tensor Σμ

εd (0) = (0, εd ) is the polarization

(15)

and the axial-vector diquark spin Sd is given by ( S d;k )il = −i εkil ; μd is the total chromomagnetic moment of the axial-vector diquark. The constituent quark masses mc = 1.55 GeV, mb = 4.88 GeV, mu = md = 0.33 GeV, ms = 0.5 GeV and the parameters of the linear potential A = 0.18 GeV2 and B = −0.3 GeV have values typical in quark models. The value of the mixing coefficient of vector and scalar confining potentials ε = −1 has been determined from the consideration of charmonium radiative decays [29] and the heavy-quark expansion [30]. The universal Pauli interaction constant κ = −1 has been fixed from the analysis of the fine splitting of heavy quarkonia 3 P J -states [29]. In this case, the long-range chromomagnetic interaction of quarks vanishes in accordance with the flux-tube model. Since we deal with diquarks and tetraquarks containing light quarks and diquarks, respectively, we adopt for the QCD coupling constant αs (μ2 ) the simplest model with freezing [31], namely





αs μ2 =

4π μ2 + M 2B

,

2

β0 = 11 − n f , 3

(16)

Λ2

where the scale is taken √ as μ = 2m1m2 /(m1 + m2 ), the background mass is M B = 2.24 A = 0.95 GeV [31], and the parameter Λ = 413 MeV was fixed from fitting the ρ mass [32]. Note that the other popular parametrization of αs with freezing [33] leads to close values. 3. Diquark and tetraquark masses At the first step, we calculate the masses and form factors of the heavy and light diquarks. Since the light quarks are highly relativistic a completely relativistic treatment of the light quark dynamics is required. To achieve this goal, we closely follow our consideration of diquarks in heavy baryons and adopt the same procedure to make the relativistic potential local by replacing

1,2 ( p ) =

m21,2 + p2 → E 1,2 = ( M 2 − m22,1 + m21,2 )/2M. Solving numerically the quasipotential equation (1) with the complete relativistic potential, which depends on the diquark mass in a complicated highly nonlinear way [34], we get the diquark masses and wave functions. In order to determine the diquark interaction with the gluon field, which takes into account the diquark structure, we calculate the corresponding matrix element of the quark current between diquark states. Such calculation leads to the emergence of the form factor F (r ) entering the vertex of the diquark–gluon interaction [34]. This form factor is expressed through the overlap integral of the diquark wave functions. Our estimates show that this form factor can be approximated with a high accuracy by the expression

F (r ) = 1 − e −ξ r −ζ r . 2

(13)

is the polarization  vector of the axial-vector diquark with momentum p, E d ( p ) =

+

Ed Ed

i μd Σ ν k˜ 2M d μ ν

(14)

= (1 − ε ) V conf (r ),

2

for a scalar diquark,

Ed Ed

⎩ − ( P√+ Q )μ

β0 ln

with

εd ( p ) =

2

  (Σρσ )νμ = −i g μρ δσν − g μσ δρν ,

  ki k j δi j − 2 ,

i0

λ

⎧ (P +Q ) ⎨ √ μ 2

Here, αs is the QCD coupling constant; D μν is the gluon propagator in the Coulomb gauge,

D 00 (k) = −

243

(17)

The values of the masses and parameters ξ and ζ for light and heavy scalar diquark [· · ·] and axial vector diquark {· · ·} ground states were calculated previously [27,34] and are given in Table 2.

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D. Ebert et al. / Physics Letters B 696 (2011) 241–245

Table 2 Masses M and form factor parameters of diquarks. S and A denote scalar and axial vector diquarks which are antisymmetric [· · ·] and symmetric {· · ·} in flavor, respectively.

Table 3 Masses of charmed and bottom diquark–antidiquark ground (1S) states (in MeV) and possible experimental candidates [4]. S and A denote scalar and axial vector diquarks.

Quark content

Diquark type

M (MeV)

ξ

ζ

State

(GeV)

(GeV2 )

JP

Diquark content

Theory mass

[u , d ] {u , d } [u , s ] {u , s } {s, s} [c , q] {c , q} [c , s] {c , s} [b, q] {b, q} [b, s] {b, s}

S A S A A S A S A S A S A

710 909 948 1069 1203 1973 2036 2091 2158 5359 5381 5462 5482

1.09 1.185 1.23 1.15 1.13 2.55 2.51 2.15 2.12 6.10 6.05 5.70 5.65

0.185 0.365 0.225 0.325 0.280 0.63 0.45 1.05 0.99 0.55 0.35 0.35 0.27

0+

S S¯

2399

1+ 1+ 0+ 1+ 2+

¯ SA A S¯ ¯ AA ¯ AA ¯ AA

2558 2473 2503 2580 2698

0+ 1+ 1+ 0+ 1+ 2+

S S¯ ¯ SA A S¯ ¯ AA ¯ AA ¯ AA

cq s¯ q¯ 2619 2723 2678 2689 2757 2863

0+ 1+ 1+ 0+ 1+ 2+

S S¯ ¯ SA A S¯ ¯ AA ¯ AA ¯ AA

cs s¯ q¯ 2753 2870 2830 2839 2901 2998

bs s¯ q¯ 6108 6238 6134 6197 6228 6284

1+ 0+ 1+ 2+

¯ SA ¯ AA ¯ AA ¯ AA

cs s¯ s¯ 3025 3003 3051 3135

bs s¯ s¯ 6383 6353 6372 6411

At the second step, we calculate the masses of heavy tetraquarks considered as the bound states of a heavy-light diquark and light antidiquark. For the potential of the S-wave (L2  = 0) diquark–antidiquark interaction (11) we get [28]

V (r ) = Vˆ Coul (r ) + V conf (r )

+ +



1 E1 E2 2





V p Vˆ Coul (r ) + V conf (r ) p −



3

 Vˆ Coul (r ) +

μd2 E 1 E 2 4 M1 M2

1 4

V  V conf (r )



V  V conf (r ) S1 · S2 ,

(18)

where

Vˆ Coul (r ) = −

4 3

αs

F 1 (r ) F 2 (r ) r

is the Coulomb-like one-gluon exchange potential which takes into account the finite sizes of the diquark and antidiquark through corresponding form factors F 1,2 (r ). Here, S1,2 are the spin operators of diquark and antidiquark. In the following we choose the total chromomagnetic moment of the axial-vector diquark μd = 0. Such a choice appears to be natural, since the long-range chromomagnetic interaction of diquarks proportional to μd then also vanishes in accordance with the flux-tube model. The resulting quasipotential equation with the complete kernel (18) is solved numerically without any approximations. 4. Results and discussion Masses of the heavy tetraquark ground (1S) states calculated in the diquark–antidiquark picture are presented in Table 3. In this table we also give possible experimental candidates for charmed and charmed-strange tetraquarks. In the charmed meson sector the situation is rather complicated. Comparing results presented in Tables 1 and 3 we see that our model predicts very close masses for the scalar 0+ orbitally excited cq¯ meson (13 P 0 ) and the ground state (1S) tetraquark, composed from the scalar [cq] diquark and scalar [¯qq¯ ] antidiquark. The same is true for the masses of the 1+ axial vector cq¯ meson (1P 1 ) and the tetraquark, composed from the axial vector {cq} diquark and scalar [¯qq¯ ] antidiquark. The calculated masses are consistent with the measured masses of the scalar D ∗0 (2400) and axial vector D 1 (2430) mesons. The mixing of the cq¯ and tetraquark states could be responsible for the observed difference in masses of the charged and neutral D ∗0 (2400) mesons. In the charmed-strange sector the 0+ and 1+ tetraquarks are predicted to have masses significantly (by 200–300 MeV)

Experiment meson

Theory mass

mass

cqq¯ q¯ D ∗0 (2400)



2403(40)(± ) 2318(29)(0 )

D 1 (2430)

2427(40)

D s (2632)

2632.5(1.7)

D ∗s J (2860)

6 2862(+ −3 )

bqq¯ q¯ 5758 5950 5782 5896 5937 6007 bq s¯ q¯ 5997 6125 6021 6086 6118 6177

Table 4 Comparison of theoretical predictions for the masses of cq s¯q¯ tetraquarks (in MeV). JP

This Letter

[14]

[15]

[16]

[17]

0+ 1+ 1+ 0+ 1+ 2+

2619 2678 2723 2689 2757 2863

2371 2410 2462 2424 2571 2648

2840 2841 2880 2503 2748 2983

2731

2616

2841 2699 2854

higher than experimentally measured masses of the D ∗s0 (2317) and D s1 (2460) mesons (cf. Tables 1, 3). This excludes the interpretation of these anomalously light D s mesons as the heavy diquark– antidiquark (tetraquark) states in our model. Instead, we find that the lightest scalar 0+ tetraquark, composed from the scalar [cq] diquark and scalar [¯sq¯ ] antidiquark, has a mass consistent with the controversial D s (2632) observed by SELEX [19]. The D ∗s J (2860)

meson, observed by BaBar [24] both in D K and D ∗ K modes,1 has mass coinciding within experimental error bars with the prediction for the mass of the tensor 2+ tetraquark, composed form the axial vector {cq} diquark and axial vector {¯sq¯ } antidiquark. The rather high value of its spin can explain the non-observation of this state by Belle in B decays. In Table 4 we compare our results for the masses of the charmed-strange diquark–antidiquark bound states with the predictions of Refs. [14–17]. From this table we see that only the model of Ref. [14] gives masses of scalar and axial vector tetraquarks compatible with the observed masses of the D ∗s0 (2317) and D s1 (2460) mesons. All other models predict significantly higher mass values for these states. The main difference between

1

This state should therefore have natural parity and total spin J  1.

D. Ebert et al. / Physics Letters B 696 (2011) 241–245

these approaches consists in the substantial distinctions in treating quark dynamics in tetraquarks. The authors of Ref. [14] use a phenomenological approach, determining diquark masses and parameters of hyperfine interactions between quarks from adjusting their predictions to experimental observables. Contrary we describe diquarks and tetraquarks dynamically as quark–quark and diquark–antidiquark bound systems and calculate their masses and form factors in the model where all parameters were previously fixed from considerations of meson properties. Different dynamical approaches were applied in Refs. [15–17]. The authors of Ref. [15] calculate diquark–antidiquark mass spectra in the quark model employing the QCD potential found by means of the AdS/QCD correspondence. Tetraquark masses are calculated in Ref. [16] in the nonrelativistic quark model including both the confining interaction and meson exchanges, while in Ref. [17] the coupled-channel formalism is employed. In the latter two approaches the anomalous charmed-strange mesons could be only accommodated as a mixture of quark–antiquark and tetraquark states with a phenomenologically adjusted mixing interaction. Thus it seems to be unlikely that the D ∗s0 (2317) and D s1 (2460) could be pure diquark– antidiquark bound states. 5. Conclusions We calculated the masses of heavy tetraquarks with open charm and bottom in the diquark–antidiquark picture using the dynamical approach based on the relativistic quark model. Both diquark and tetraquark masses were obtained by the numerical solution of the quasipotential wave equation with the corresponding relativistic potentials. The diquark structure was taken into account in terms of diquark wave functions. It is important to emphasize that, in our analysis, we did not introduce any free adjustable parameters but used their values fixed from our previous considerations of heavy and light hadron properties. It was found that the D ∗0 (2400), D s (2632) and D ∗s J (2860) mesons could be tetraquark states with open charm, while the D ∗s0 (2317) and D s1 (2460) mesons cannot be interpreted as diquark–antidiquark bound states. The masses of the bottom counterparts of charmed tetraquarks were calculated. It is important to search for them in order to help revealing the nature of controversial charmed and charmed-strange mesons. Acknowledgements The authors are grateful to M. Müller-Preussker for support and to V. Matveev, V. Savrin and M. Wagner for discussions. This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) under contract Eb 139/6-1 and the Russian Foundation for Basic Research (RFBR) grants No. 08-02-00582 and No. 10-0291339. References [1] For recent reviews see e.g. N. Brambilla, et al., arXiv:1010.5827 [hep-ph]; N. Drenska, R. Faccini, F. Piccinini, A. Polosa, F. Renga, C. Sabelli, arXiv: 1006.2741 [hep-ph]; S. Godfrey, S.L. Olsen, Annu. Rev. Nucl. Part. Sci. 58 (2008) 51; E.S. Swanson, Phys. Rep. 429 (2006) 243.

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