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New heavy mesons as hadronic molecules
Progress in Particle and Nuclear Physics 61 (2008) 127–132 www.elsevier.com/locate/ppnp
Review
New heavy mesons as hadronic molecules Amand Faessler, Thomas Gutsche, Valery E. Lyubovitskij ∗,1 Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany
Abstract ∗ (2317), D (2460), B ∗ (5725) and B (5778) mesons as hadronic molecules. We discuss a possible interpretation of the Ds0 s1 s1 s0 Using an effective Lagrangian approach we calculate their weak, strong and radiative decays. The new impact of the molecular structure of these states is the presence of u(d) quarks in the K , D (∗) and B (∗) mesons which gives rise to the direct strong isospin∗ (B ∗ ) → D (B ) + π 0 and D (B ) → D ∗ (B ∗ ) + π 0 in addition to the modes generated by η − π 0 violating transitions Ds0 s s s1 s1 s s s0 mixing as was considered before in the literature. c 2007 Elsevier B.V. All rights reserved.
Keywords: Bottom; Charm and strange mesons; Hadronic molecule; Strong and radiative decay; Isospin violation
1. Introduction Nowadays there is much interest to study newly observed mesons and baryons in the context of a hadronic molecule ∗ (2317) and axial D (2460) mesons could be interpretation [1]. As stressed for example in Ref. [2] the scalar Ds0 s1 ∗ candidates for a scalar D K and an axial D K molecule because of a relatively small binding energy of ∼50 MeV. These states were discovered and confirmed just a few years ago by the Collaborations BABAR at SLAC, CLEO at ∗ (2317) and CESR and Belle at KEKB [3]. In the interpretation of these experiments it was suggested that the Ds0 P Ds1 (2460) mesons are the P-wave charm-strange quark states with spin–parity quantum numbers J = 0+ and J P = 1+ , respectively. ∗ (2317) and D (2460) mesons. The simplest The next important question concerns the possible structure of the Ds0 s1 interpretation of these states is that they are the missing js = 1/2 ( js is the angular momentum of the s-quark) members of the c¯s L = 1 multiplet. However, this standard quark model scenario is in disagreement with experimental ∗ (2317) and D (2460) states are narrower and their masses are lower when compared to observation, since the Ds0 s1 theoretical predictions (see e.g. discussion in Ref. [1]). Therefore, in addition to the standard quark–antiquark picture ∗ (2317) and D (2460) mesons have been suggested: four-quark states, mixing of alternative interpretations of the Ds0 s1 two- and four-quark states, two-diquark states and two-meson molecular states. Up to now strong and radiative decays ∗ (2317) and D (2460) mesons have been calculated using different approaches [4–26]: quark models, of the Ds0 s1 effective Lagrangian approaches, QCD sum rules, lattice QCD, etc. ∗ Corresponding author. Tel.: +49 7071 2978637; fax: +49 7071 295388.
E-mail address: [email protected] (V.E. Lyubovitskij). 1 On leave of absence from the Department of Physics, Tomsk State University, 634050 Tomsk, Russia. c 2007 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2007.12.005
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∗ (2317) and D (2460) mesons is that the presence A new feature related to the molecular D (∗) K structure of the Ds0 s1 ∗ → D π 0 and (∗) of u(d) quarks in the D and K mesons gives rise to direct strong isospin-violating transitions Ds0 s ∗ 0 0 Ds1 → Ds π in addition to the decay mechanism induced by η − π mixing, as considered previously. ∗ (2317) and D (2460) In the present paper we will consider the strong, radiative and leptonic decays of the Ds0 s1 ∗ and D are mesons using an effective Lagrangian approach. The approach is based on the hypothesis that the Ds0 s1 ∗ bound states of D, K and D ∗ , K mesons, respectively. In other words, we investigate the position that the Ds0 ∗ and Ds1 are (D K ) and (D K ) hadronic molecules. Their couplings to the constituents are described by effective ∗ D K and g D D ∗ K are determined by the compositeness Lagrangians. The corresponding coupling constants g Ds0 s1 condition Z = 0 [27,28], which implies that the renormalization constant of the hadron wavefunction is set equal to zero. Note, that this condition was originally applied to the study of the deuteron as a bound state of proton and neutron [27]. Then it was extensively used in low-energy hadron phenomenology as the master equation for the treatment of mesons and baryons as bound states of light and heavy constituent quarks (see Refs. [28,29]). Recently the compositeness condition was used to study the light scalar mesons a0 and f 0 as K K¯ molecules [30]. A new ∗ (2317) and D (2460) mesons is that the presence of u(d) quarks in the impact of the molecular structure of the Ds0 s1 ∗ → D π 0 and D ∗ 0 D ∗ and K mesons gives rise to the direct strong isospin-violating transitions Ds0 s s1 → Ds π in 0 addition to the decay induced by η − π mixing considered before in the literature. We show that the direct transition dominates over the η − π 0 mixing transitions. The obtained results for the partial decay widths are consistent with ∗ (5725) and B (5778) previous calculations. We also extend our formalism to the bottom sector, that is to the Bs0 s1 states.
2. Approach: Basic notions and results In this section we briefly discuss the formalism for the study of hadronic molecules. For example, we consider ∗± the Ds0 (2317) meson as a bound state of D and K mesons. Extension to other states is straightforward. First of all ∗± (2317) meson. We use the current results for the quantum numbers of we specify the quantum numbers of the Ds0 ∗ = 2.3173 GeV [3]. Our framework is based on an effective isospin, spin and parity: I (J P ) = 0(0+ ) and mass m Ds0 ∗ (2317) meson and their constituents — D and K interaction Lagrangian describing the coupling between the Ds0 mesons: Z ∗ − 2 ∗ (x) = g ∗ D ∗ (y )D(x + w L Ds0 dyΦ Ds0 (1) K D y)K (x − w D K y) + H.c. s0 (x) D s0
where D and K are the corresponding meson doublets, wi j = m i /(m i + m j ) is a kinematic variable, m D and m K ∗ characterizes the finite size of the D ∗ (2317) are the masses of D and K mesons. The correlation function Φ Ds0 s0 meson as a D K bound state and depends on the relative Jacobi coordinate y with x being the center of mass ∗ . Its Fourier transform reads (CM) coordinate. In the numerical calculations we employ the Gaussian form for Φ Ds0 2 2 2 as Φ˜ D ∗ ( p ) = exp(− p /Λ ∗ ), where p E is the Euclidean Jacobi momentum. Here Λ D ∗ is a size parameter, s0
E
E
Ds0
s0
∗ molecule. The coupling constant g ∗ is which parametrizes the distribution of D and K mesons inside the Ds0 Ds0 determined by the compositeness condition [27,28], which implies that the renormalization constant of the hadron ∗ = 1 − Σ 0 ∗ (m 2 ∗ ) = 0 , where Σ 0 ∗ is the derivative of the D ∗ meson mass wavefunction is set equal to zero: Z Ds0 s0 Ds0 Ds0 Ds0 operator. The effective Lagrangian (1) is the starting point for the study of the decays of hadronic molecules. It defines the transition of the molecule into its constituents. Then we should specify the Lagrangian which describes the interaction of the constituents with external fields (hadrons and gauge bosons) and the diagrams which contribute to the matrix elements of physical processes. All further details can be found in Refs. [21–24].
3. Results ∗ (D ) → D (D ∗ )π ) and radiative D ∗ (D ) → Below, in Tables 1–4, we display our results for the strong Γ (Ds0 s s1 s1 s s0 ∗ → D π ) and R ∗ ∗ Ds (Ds )γ decay widths and their ratios R Ds0 ∗ = Γ (Ds0 → Ds∗ γ )/Γ (Ds0 s Ds1 = Γ (Ds1 → Ds γ )/Γ (Ds1 → Ds∗ π ), including the extension to the bottom sector, and compare them with the predictions of
other approaches.
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A. Faessler et al. / Progress in Particle and Nuclear Physics 61 (2008) 127–132 Table 1 Strong decay widths in keV Approach
∗ → D π) Γ (Ds0 s
Ref. [14] Ref. [7] Ref. [18] Ref. [6] Ref. [8] Ref. [5] Ref. [17] Ref. [13] Ref. [4] Ref. [9] Ref. [10] Ref. [26] Our results [21,23]
6±2 7±1 8.69 10 16 21.5 32 39 ± 5 10–100 155 ± 70 129 ± 43 140 46.7–75
Γ (Ds1 → Ds∗ π ) 7±1 11.41 10 32 21.5 35 43 ± 8 155 ± 70 187 ± 73 140 50.1–79.2
Table 2 Radiative decay widths in keV Approach
∗ → D∗ γ ) Γ (Ds0 s
Γ (Ds1 → Ds γ )
Ref. [8] Ref. [25] Ref. [7] Ref. [12] Ref. [15] Ref. [20] Ref. [10] Ref. [16] Ref. [5] Ref. [6] Ref. [11] Ref. [26] Ref. [9] Our results [21,23]
0.2 0.49 0.85 ± 0.05 1 ≈1.1 1.3–9.9 ≤1.4 1.6 1.74 1.9 4–6 <7 21 0.47–0.63
≤7.3 0.6–2.9 5.5–31.2 ≈2 6.7 5.08 6.2 19–29 '43.6 93 2.37–3.73
Also, we present our results for the leptonic decay constants: f D∗ = 67.1 MeV and f Ds1 = 144.5 MeV. In Table 5 s0 we summarize the present results for f D∗ and f Ds1 obtained in different approaches (either on the basis of hadronic s0 models or from the analysis of experimental data on two-body B-meson decays). Our results are in agreement with the predictions of Refs. [4,31–33], especially with the lower limits derived from an analysis of the branching ratios of ∗ (D ) decays [31,33]. B → D (∗) Ds0 s1 ∗ (D ) decays. Using the predicted decay constants f D∗ and f Ds1 we calculate the branching ratios of B → D (∗) Ds0 s1 s0 For this purpose we use the leptonic decay constants f D∗ , f Ds1 and, in addition, model-independent results for the s0
form factors of B → D (∗) `¯ν` transitions obtained by Caprini, Lellouch and Neubert (CLN) [34]. Latter derivations are based on heavy quark spin symmetry, dispersive constraints, including short-distance and power corrections. In ∗ (D ). For the data Table 6 we present our predictions for the branching ratios of two-body decays B → D (∗) Ds0 s1 we use the averaged lower limits from PDG [3] and in addition for the modes with a Ds1 (2460) meson in the final state the direct results of the BABAR Collaboration. Our predictions are in good agreement with the experimental ∗− ∗+ data except for the marginal situation in the case of the B¯ 0 → Ds0 D decay, where our prediction is slightly ∗ )/Γ (B → D D ∗ ) lower than the experimental limit. In Table 7 we present the results for the ratios Γ (B → D ∗ Ds0 s0 ∗ and Γ (B → D Ds1 )/Γ (B → D Ds1 ). We also use the compilation of experimental data and theoretical results within the covariant light-front (CLF) approach [32] summarized in Table 10 of Ref. [33]. Here, our predictions are in good agreement with the existing experimental data. In comparison with the CLF approach our predictions are lower, although not significantly.
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A. Faessler et al. / Progress in Particle and Nuclear Physics 61 (2008) 127–132
Table 3 Ratios R D ∗ and R Ds1 s0 Approach
R D∗
Ref. [8] Ref. [10] Ref. [26] Ref. [5] Ref. [11] Ref. [6] Ref. [9] Ref. [16] Data [3] Our results [21,23]
0.01 ≤0.02 <0.05 0.08 0.11–0.14 0.19 0.09–0.25 0.16 ≤0.059 '0.01
R Ds1
s0
0.01–0.02 '0.31 0.24 0.62 0.41–1.09 0.67 0.44 ± 0.09 '0.05
Table 4 ∗ (5725) and B (5778) in keV Decay widths of Bs0 s1 Approach
∗ → B π) Γ (Bs0 s
Γ (Bs1 → Bs∗ π )
∗ → B∗γ ) Γ (Bs0 s
Γ (Bs1 → Bs γ )
Refs. [18,19] Our results [24]
7.92 55.2–89.9
10.36 57.4–94.7
3.07–4.06
2.01–2.67
Table 5 Leptonic decay constants f D ∗ and f D s0
s1
f D ∗ (MeV)
fD
225 ± 25 206 ± 120 200±50 170 ± 20 138±16 110±18 >(74 ± 11)/|a1 | 71 60±13 >(58 − 86)/|a1 | 67±13 44 67.1 ± 4.5
225 ± 25
Approach
s0
Ref. [11] Ref. [35] Ref. [36] Ref. [37] Ref. [38] Ref. [39] Ref. [31] Ref. [32] Ref. [32] Ref. [33] Ref. [4] Ref. [40] Our results [22]
s1
(MeV)
247 ± 37 259±13 233±31 >(166 ± 20)/|a1 | 117 150±40 >(90 − 228)/|a1 | 41 144.5 ± 11.1
Table 6 ∗ (D ) decays (in units of 10−3 ) Branching ratios of B → D (∗) Ds0 s1 Mode
Data (averaged) [3]
∗− 0 D B − → Ds0 ∗− + 0 ¯ B → Ds0 D − 0 B − → Ds1 D − + 0 ¯ B → Ds1 D ∗− ∗0 B − → Ds0 D ∗− 0 B¯ → Ds0 D ∗+ − ∗0 B − → Ds1 D − 0 B¯ → Ds1 D ∗+
+0.23 >0.74−0.19 +0.41 >0.97−0.34 +0.6 >1.4−0.5 +0.6 >2.0−0.5 +0.4 >0.9 ± 0.6−0.3 >1.5 ± 0.4+0.5 −0.4 >5.5 ± 1.2+2.2 −1.6 7.6 >7.6 ± 1.7+3.2 −2.4
BABAR [41]
Our results [22] 1.03 ±0.14 0.96 ±0.13
4.3 ±1.6 ±1.3 2.6 ±1.5 ±0.74
2.54 ±0.39 2.36 ±0.36 0.50 ±0.07 0.47 ±0.06
11.2 ±2.6 ±2.0
7.33 ±1.12
8.8 ±2.0 ±1.4
6.85 ±1.05
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A. Faessler et al. / Progress in Particle and Nuclear Physics 61 (2008) 127–132 Table 7 ∗ ,D Ratios M D ∗ /M D = Γ (B → M D ∗ )/Γ (B → M D) for M = Ds0 s1 Mode
Data [33]
CLF [33]
Our results [22]
∗− 0 ∗− ∗0 Ds0 D /Ds0 D ∗− ∗+ ∗− + Ds0 D /Ds0 D − ∗0 − 0 Ds1 D /Ds1 D − ∗+ − + Ds1 D /Ds1 D
0.91±0.73 0.59±0.26 3.4±2.4
0.49 0.49 3.6
0.48 0.48 2.9
2.6±1.5
3.6
2.9
4. Summary ∗ (2317) and D (2460) in the hadronic molecule interpretation, We studied the new charm-strange mesons Ds0 s1 ∗ considering D K and D K bound states, respectively. Using an effective Lagrangian approach we calculated their weak, strong and radiative decays. A new impact of their molecular structure is that the presence of u(d) quarks in the ∗ → D π 0 and D → D ∗ π 0 , in D (∗) and K meson loops gives rise to direct strong isospin-violating transitions Ds0 s s1 s 0 addition to the decay mechanism induced by η −π mixing as was considered before in the literature. Also, we extend ∗ (5725) and B (5778) states. We calculated weak decay properties of our formalism to the bottom sector: that is Bs0 s1 ∗ Ds0 (2317) and Ds1 (2460): the leptonic decay constants f D∗ , f Ds1 and branching ratios of the two-body bottom meson ∗ (D ). decays B → D (∗) Ds0 s1
s0
Acknowledgments This work was supported by the DFG under contracts FA67/31-1 and GRK683. This research is also the part of the EU Integrated Infrastructure Initiative Hadronphysics project under contract number RII3-CT-2004-506078 and the President grant of Russia “Scientific Schools” No. 5103.2006.2. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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Review
New heavy mesons as hadronic molecules Amand Faessler, Thomas Gutsche, Valery E. Lyubovitskij ∗,1 Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen, Auf der Morgenstelle 14, D-72076 T¨ubingen, Germany
Abstract ∗ (2317), D (2460), B ∗ (5725) and B (5778) mesons as hadronic molecules. We discuss a possible interpretation of the Ds0 s1 s1 s0 Using an effective Lagrangian approach we calculate their weak, strong and radiative decays. The new impact of the molecular structure of these states is the presence of u(d) quarks in the K , D (∗) and B (∗) mesons which gives rise to the direct strong isospin∗ (B ∗ ) → D (B ) + π 0 and D (B ) → D ∗ (B ∗ ) + π 0 in addition to the modes generated by η − π 0 violating transitions Ds0 s s s1 s1 s s s0 mixing as was considered before in the literature. c 2007 Elsevier B.V. All rights reserved.
Keywords: Bottom; Charm and strange mesons; Hadronic molecule; Strong and radiative decay; Isospin violation
1. Introduction Nowadays there is much interest to study newly observed mesons and baryons in the context of a hadronic molecule ∗ (2317) and axial D (2460) mesons could be interpretation [1]. As stressed for example in Ref. [2] the scalar Ds0 s1 ∗ candidates for a scalar D K and an axial D K molecule because of a relatively small binding energy of ∼50 MeV. These states were discovered and confirmed just a few years ago by the Collaborations BABAR at SLAC, CLEO at ∗ (2317) and CESR and Belle at KEKB [3]. In the interpretation of these experiments it was suggested that the Ds0 P Ds1 (2460) mesons are the P-wave charm-strange quark states with spin–parity quantum numbers J = 0+ and J P = 1+ , respectively. ∗ (2317) and D (2460) mesons. The simplest The next important question concerns the possible structure of the Ds0 s1 interpretation of these states is that they are the missing js = 1/2 ( js is the angular momentum of the s-quark) members of the c¯s L = 1 multiplet. However, this standard quark model scenario is in disagreement with experimental ∗ (2317) and D (2460) states are narrower and their masses are lower when compared to observation, since the Ds0 s1 theoretical predictions (see e.g. discussion in Ref. [1]). Therefore, in addition to the standard quark–antiquark picture ∗ (2317) and D (2460) mesons have been suggested: four-quark states, mixing of alternative interpretations of the Ds0 s1 two- and four-quark states, two-diquark states and two-meson molecular states. Up to now strong and radiative decays ∗ (2317) and D (2460) mesons have been calculated using different approaches [4–26]: quark models, of the Ds0 s1 effective Lagrangian approaches, QCD sum rules, lattice QCD, etc. ∗ Corresponding author. Tel.: +49 7071 2978637; fax: +49 7071 295388.
E-mail address: [email protected] (V.E. Lyubovitskij). 1 On leave of absence from the Department of Physics, Tomsk State University, 634050 Tomsk, Russia. c 2007 Elsevier B.V. All rights reserved. 0146-6410/$ - see front matter doi:10.1016/j.ppnp.2007.12.005
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A. Faessler et al. / Progress in Particle and Nuclear Physics 61 (2008) 127–132
∗ (2317) and D (2460) mesons is that the presence A new feature related to the molecular D (∗) K structure of the Ds0 s1 ∗ → D π 0 and (∗) of u(d) quarks in the D and K mesons gives rise to direct strong isospin-violating transitions Ds0 s ∗ 0 0 Ds1 → Ds π in addition to the decay mechanism induced by η − π mixing, as considered previously. ∗ (2317) and D (2460) In the present paper we will consider the strong, radiative and leptonic decays of the Ds0 s1 ∗ and D are mesons using an effective Lagrangian approach. The approach is based on the hypothesis that the Ds0 s1 ∗ bound states of D, K and D ∗ , K mesons, respectively. In other words, we investigate the position that the Ds0 ∗ and Ds1 are (D K ) and (D K ) hadronic molecules. Their couplings to the constituents are described by effective ∗ D K and g D D ∗ K are determined by the compositeness Lagrangians. The corresponding coupling constants g Ds0 s1 condition Z = 0 [27,28], which implies that the renormalization constant of the hadron wavefunction is set equal to zero. Note, that this condition was originally applied to the study of the deuteron as a bound state of proton and neutron [27]. Then it was extensively used in low-energy hadron phenomenology as the master equation for the treatment of mesons and baryons as bound states of light and heavy constituent quarks (see Refs. [28,29]). Recently the compositeness condition was used to study the light scalar mesons a0 and f 0 as K K¯ molecules [30]. A new ∗ (2317) and D (2460) mesons is that the presence of u(d) quarks in the impact of the molecular structure of the Ds0 s1 ∗ → D π 0 and D ∗ 0 D ∗ and K mesons gives rise to the direct strong isospin-violating transitions Ds0 s s1 → Ds π in 0 addition to the decay induced by η − π mixing considered before in the literature. We show that the direct transition dominates over the η − π 0 mixing transitions. The obtained results for the partial decay widths are consistent with ∗ (5725) and B (5778) previous calculations. We also extend our formalism to the bottom sector, that is to the Bs0 s1 states.
2. Approach: Basic notions and results In this section we briefly discuss the formalism for the study of hadronic molecules. For example, we consider ∗± the Ds0 (2317) meson as a bound state of D and K mesons. Extension to other states is straightforward. First of all ∗± (2317) meson. We use the current results for the quantum numbers of we specify the quantum numbers of the Ds0 ∗ = 2.3173 GeV [3]. Our framework is based on an effective isospin, spin and parity: I (J P ) = 0(0+ ) and mass m Ds0 ∗ (2317) meson and their constituents — D and K interaction Lagrangian describing the coupling between the Ds0 mesons: Z ∗ − 2 ∗ (x) = g ∗ D ∗ (y )D(x + w L Ds0 dyΦ Ds0 (1) K D y)K (x − w D K y) + H.c. s0 (x) D s0
where D and K are the corresponding meson doublets, wi j = m i /(m i + m j ) is a kinematic variable, m D and m K ∗ characterizes the finite size of the D ∗ (2317) are the masses of D and K mesons. The correlation function Φ Ds0 s0 meson as a D K bound state and depends on the relative Jacobi coordinate y with x being the center of mass ∗ . Its Fourier transform reads (CM) coordinate. In the numerical calculations we employ the Gaussian form for Φ Ds0 2 2 2 as Φ˜ D ∗ ( p ) = exp(− p /Λ ∗ ), where p E is the Euclidean Jacobi momentum. Here Λ D ∗ is a size parameter, s0
E
E
Ds0
s0
∗ molecule. The coupling constant g ∗ is which parametrizes the distribution of D and K mesons inside the Ds0 Ds0 determined by the compositeness condition [27,28], which implies that the renormalization constant of the hadron ∗ = 1 − Σ 0 ∗ (m 2 ∗ ) = 0 , where Σ 0 ∗ is the derivative of the D ∗ meson mass wavefunction is set equal to zero: Z Ds0 s0 Ds0 Ds0 Ds0 operator. The effective Lagrangian (1) is the starting point for the study of the decays of hadronic molecules. It defines the transition of the molecule into its constituents. Then we should specify the Lagrangian which describes the interaction of the constituents with external fields (hadrons and gauge bosons) and the diagrams which contribute to the matrix elements of physical processes. All further details can be found in Refs. [21–24].
3. Results ∗ (D ) → D (D ∗ )π ) and radiative D ∗ (D ) → Below, in Tables 1–4, we display our results for the strong Γ (Ds0 s s1 s1 s s0 ∗ → D π ) and R ∗ ∗ Ds (Ds )γ decay widths and their ratios R Ds0 ∗ = Γ (Ds0 → Ds∗ γ )/Γ (Ds0 s Ds1 = Γ (Ds1 → Ds γ )/Γ (Ds1 → Ds∗ π ), including the extension to the bottom sector, and compare them with the predictions of
other approaches.
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A. Faessler et al. / Progress in Particle and Nuclear Physics 61 (2008) 127–132 Table 1 Strong decay widths in keV Approach
∗ → D π) Γ (Ds0 s
Ref. [14] Ref. [7] Ref. [18] Ref. [6] Ref. [8] Ref. [5] Ref. [17] Ref. [13] Ref. [4] Ref. [9] Ref. [10] Ref. [26] Our results [21,23]
6±2 7±1 8.69 10 16 21.5 32 39 ± 5 10–100 155 ± 70 129 ± 43 140 46.7–75
Γ (Ds1 → Ds∗ π ) 7±1 11.41 10 32 21.5 35 43 ± 8 155 ± 70 187 ± 73 140 50.1–79.2
Table 2 Radiative decay widths in keV Approach
∗ → D∗ γ ) Γ (Ds0 s
Γ (Ds1 → Ds γ )
Ref. [8] Ref. [25] Ref. [7] Ref. [12] Ref. [15] Ref. [20] Ref. [10] Ref. [16] Ref. [5] Ref. [6] Ref. [11] Ref. [26] Ref. [9] Our results [21,23]
0.2 0.49 0.85 ± 0.05 1 ≈1.1 1.3–9.9 ≤1.4 1.6 1.74 1.9 4–6 <7 21 0.47–0.63
≤7.3 0.6–2.9 5.5–31.2 ≈2 6.7 5.08 6.2 19–29 '43.6 93 2.37–3.73
Also, we present our results for the leptonic decay constants: f D∗ = 67.1 MeV and f Ds1 = 144.5 MeV. In Table 5 s0 we summarize the present results for f D∗ and f Ds1 obtained in different approaches (either on the basis of hadronic s0 models or from the analysis of experimental data on two-body B-meson decays). Our results are in agreement with the predictions of Refs. [4,31–33], especially with the lower limits derived from an analysis of the branching ratios of ∗ (D ) decays [31,33]. B → D (∗) Ds0 s1 ∗ (D ) decays. Using the predicted decay constants f D∗ and f Ds1 we calculate the branching ratios of B → D (∗) Ds0 s1 s0 For this purpose we use the leptonic decay constants f D∗ , f Ds1 and, in addition, model-independent results for the s0
form factors of B → D (∗) `¯ν` transitions obtained by Caprini, Lellouch and Neubert (CLN) [34]. Latter derivations are based on heavy quark spin symmetry, dispersive constraints, including short-distance and power corrections. In ∗ (D ). For the data Table 6 we present our predictions for the branching ratios of two-body decays B → D (∗) Ds0 s1 we use the averaged lower limits from PDG [3] and in addition for the modes with a Ds1 (2460) meson in the final state the direct results of the BABAR Collaboration. Our predictions are in good agreement with the experimental ∗− ∗+ data except for the marginal situation in the case of the B¯ 0 → Ds0 D decay, where our prediction is slightly ∗ )/Γ (B → D D ∗ ) lower than the experimental limit. In Table 7 we present the results for the ratios Γ (B → D ∗ Ds0 s0 ∗ and Γ (B → D Ds1 )/Γ (B → D Ds1 ). We also use the compilation of experimental data and theoretical results within the covariant light-front (CLF) approach [32] summarized in Table 10 of Ref. [33]. Here, our predictions are in good agreement with the existing experimental data. In comparison with the CLF approach our predictions are lower, although not significantly.
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Table 3 Ratios R D ∗ and R Ds1 s0 Approach
R D∗
Ref. [8] Ref. [10] Ref. [26] Ref. [5] Ref. [11] Ref. [6] Ref. [9] Ref. [16] Data [3] Our results [21,23]
0.01 ≤0.02 <0.05 0.08 0.11–0.14 0.19 0.09–0.25 0.16 ≤0.059 '0.01
R Ds1
s0
0.01–0.02 '0.31 0.24 0.62 0.41–1.09 0.67 0.44 ± 0.09 '0.05
Table 4 ∗ (5725) and B (5778) in keV Decay widths of Bs0 s1 Approach
∗ → B π) Γ (Bs0 s
Γ (Bs1 → Bs∗ π )
∗ → B∗γ ) Γ (Bs0 s
Γ (Bs1 → Bs γ )
Refs. [18,19] Our results [24]
7.92 55.2–89.9
10.36 57.4–94.7
3.07–4.06
2.01–2.67
Table 5 Leptonic decay constants f D ∗ and f D s0
s1
f D ∗ (MeV)
fD
225 ± 25 206 ± 120 200±50 170 ± 20 138±16 110±18 >(74 ± 11)/|a1 | 71 60±13 >(58 − 86)/|a1 | 67±13 44 67.1 ± 4.5
225 ± 25
Approach
s0
Ref. [11] Ref. [35] Ref. [36] Ref. [37] Ref. [38] Ref. [39] Ref. [31] Ref. [32] Ref. [32] Ref. [33] Ref. [4] Ref. [40] Our results [22]
s1
(MeV)
247 ± 37 259±13 233±31 >(166 ± 20)/|a1 | 117 150±40 >(90 − 228)/|a1 | 41 144.5 ± 11.1
Table 6 ∗ (D ) decays (in units of 10−3 ) Branching ratios of B → D (∗) Ds0 s1 Mode
Data (averaged) [3]
∗− 0 D B − → Ds0 ∗− + 0 ¯ B → Ds0 D − 0 B − → Ds1 D − + 0 ¯ B → Ds1 D ∗− ∗0 B − → Ds0 D ∗− 0 B¯ → Ds0 D ∗+ − ∗0 B − → Ds1 D − 0 B¯ → Ds1 D ∗+
+0.23 >0.74−0.19 +0.41 >0.97−0.34 +0.6 >1.4−0.5 +0.6 >2.0−0.5 +0.4 >0.9 ± 0.6−0.3 >1.5 ± 0.4+0.5 −0.4 >5.5 ± 1.2+2.2 −1.6 7.6 >7.6 ± 1.7+3.2 −2.4
BABAR [41]
Our results [22] 1.03 ±0.14 0.96 ±0.13
4.3 ±1.6 ±1.3 2.6 ±1.5 ±0.74
2.54 ±0.39 2.36 ±0.36 0.50 ±0.07 0.47 ±0.06
11.2 ±2.6 ±2.0
7.33 ±1.12
8.8 ±2.0 ±1.4
6.85 ±1.05
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A. Faessler et al. / Progress in Particle and Nuclear Physics 61 (2008) 127–132 Table 7 ∗ ,D Ratios M D ∗ /M D = Γ (B → M D ∗ )/Γ (B → M D) for M = Ds0 s1 Mode
Data [33]
CLF [33]
Our results [22]
∗− 0 ∗− ∗0 Ds0 D /Ds0 D ∗− ∗+ ∗− + Ds0 D /Ds0 D − ∗0 − 0 Ds1 D /Ds1 D − ∗+ − + Ds1 D /Ds1 D
0.91±0.73 0.59±0.26 3.4±2.4
0.49 0.49 3.6
0.48 0.48 2.9
2.6±1.5
3.6
2.9
4. Summary ∗ (2317) and D (2460) in the hadronic molecule interpretation, We studied the new charm-strange mesons Ds0 s1 ∗ considering D K and D K bound states, respectively. Using an effective Lagrangian approach we calculated their weak, strong and radiative decays. A new impact of their molecular structure is that the presence of u(d) quarks in the ∗ → D π 0 and D → D ∗ π 0 , in D (∗) and K meson loops gives rise to direct strong isospin-violating transitions Ds0 s s1 s 0 addition to the decay mechanism induced by η −π mixing as was considered before in the literature. Also, we extend ∗ (5725) and B (5778) states. We calculated weak decay properties of our formalism to the bottom sector: that is Bs0 s1 ∗ Ds0 (2317) and Ds1 (2460): the leptonic decay constants f D∗ , f Ds1 and branching ratios of the two-body bottom meson ∗ (D ). decays B → D (∗) Ds0 s1
s0
Acknowledgments This work was supported by the DFG under contracts FA67/31-1 and GRK683. This research is also the part of the EU Integrated Infrastructure Initiative Hadronphysics project under contract number RII3-CT-2004-506078 and the President grant of Russia “Scientific Schools” No. 5103.2006.2. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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