cpcChinese Physics CChin. Phys. C1674-1137Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltdcpc_42_12_12310110.1088/1674-1137/42/12/12310142/12/123101PaperParticles and fieldsStrong decays of P-wave mixing heavy-light 1+ states*
Supported by the National Natural Science Foundation of China (NSFC) (11405004, 11405037, 11505039, 11575048), the Key Scientific Research Projects in 2017 at North Minzu University (2017KJ11), Ningxia First-Class Discipline and Scientific Research Projects (Electronic Science and Technology) (NXYLXK2017A07) and the Fundamental Research Funds for the Central Universities, North Minzu University (2018XYZDX08)
Many P-wave mixing heavy-light 1^{+} states have not yet been discovered by experiment, while others have been discovered but without width information, or with large uncertainties on the widths. In this paper, the strong decays of the P-wave mixing heavy-light 1^{+} states D^{0}, D±, Ds±, B^{0}, B± and B_{s} are studied by the improved Bethe-Salpeter (B-S) method with two conditions of mixing angle θ: one is θ = 35.3°; the other is considering a correction to the mixing angle θ=35.3°+θ1. Valuable predictions for the strong decay widths are obtained: Γ(D1′0)=232MeV, Γ(D10)=21.5MeV, Γ(D1′±)=232MeV, Γ(D1±)=21.5MeV, Γ(Ds1′±)=0.0101MeV, Γ(Ds1±)=0.950MeV, Γ(B1′±)=263MeV, Γ(B1±)=16.8MeV, Γ(Bs1′)=0.01987MeV and Γ(Bs1)=0.412MeV. It is found that the decay widths of Ds1± and Bs1 are very sensitive to the mixing angle. The results will provide theoretical assistance to future experiments.
Heavy-light mesons are very important in hadron physics. During the past few years, a lot of interesting processes have been obtained for the heavy-light mesons. Nowadays, many more excited heavy-light mesons have been discovered by experiments. For the P-wave DJ* mesons, D0*(2400)0, D1(2430)0, D1(2420)0, D2*(2460)0 and their charged isospin partners D0*(2400)±, D1(2420)±, D2*(2460)± have been listed in Particle Data Group (PDG) 2016 edition ^{[1]}. Among these, we have little knowledge about D1(2430)0, which has large errors for the decay width, and its charged isospin partner has not been observed. Four P-wave DsJ* mesons, Ds0*(2317)±, Ds1(2460)±, Ds1(2536)± and Ds2*(2573), have been observed in experiments ^{[1]}. The upper bounds on the total decay width of the Ds0*(2317)± and Ds1(2460)± meson are 3.8 MeV at 95% confidence level and 3.5 MeV at 95% confidence level ^{[1]}, respectively. The full width of Ds1(2536)± is rather narrow: Γ=0.92±0.05MeV. In 2007, the D0 Collaboration reported two separate excited B mesons, B1(5721)0 and B2*(5747)0, in fully reconstructed decays to B+(*)π−^{[2]}. The CDF Collaboration observed two orbitally excited narrow B^{0} mesons in 2009 ^{[3]}. They then updated the measurement of the properties of orbitally excited B^{0} and Bs0 mesons in 2014 ^{[4]}. The LHCb Collaboration gave precise measurements for the masses and widths of the B1(5721)0,+ and B2*(5747)0,+ states in 2015 ^{[5]}. The CDF Collaboration reported their observations of Bs1(5830)0 and Bs2*(5840)0 in 2008 ^{[6]}. Later the D0 Collaboration confirmed the existence of Bs2*(5840)0 and indicated that Bs1(5830)0 was not observed with available data ^{[7]}. The LHCb Collaboration confirmed the existence of Bs1(5830)0 and Bs2*(5840)0 in the B(*)+K−^{[8]} channel. The discovery of these excited states not only enriched the spectroscopy of P-wave heavy-light mesons but also provided us an opportunity to research the properties of P-wave heavy-light mesons.
Besides the experimental progress, there have been a lot of theoretical efforts to investigate the properties of the P-wave heavy-light mesons, especially for the newly observed states. In heavy quark effective theory (HQET) ^{[9,10]}, the angular momentum of the light quark jq=sq+L (s_{q} and L are the spin and the orbital angular momentum, respectively, of the light quark) is a good quantum number when the heavy quark is in the mQ→∞ limit, and can be used to label the states, so the physical heavy-light states can be described by HQET. As well as HQET, the mass spectroscopy ^{[11–14]} and strong decays of P-wave heavy-light mesons have also been studied by other methods ^{[15–29]}. The strong decay of P-wave heavy-light mesons can help us understand the properties of these mesons and establish the heavy-light meson spectroscopy.
The mesons can be described by the Bethe-Salpeter (B-S) equation. Reference [30] took the B-S equation to describe the light mesons π and K, then they calculated the mass and decay constant of π by the B-S amplitudes ^{[31]}. The weak decays ^{[32]} and the strong decays ^{[33]} have also been studied, by combining with the Dyson-Schwinger equation. In this paper, however, we describe the properties of heavy mesons and the matrix elements of strong decays by an improved B-S method, which includes two improvements ^{[34]}. One improvement is that relativistic wavefunctions are used which describe bound states with definite quantum numbers, and the relativistic forms of wavefunctions are solutions of the full Salpeter equations. The other is that the matrix elements of strong decays are obtained with relativistic wavefunctions as input. So the improved B-S method is good to describe the properties and decays of heavy mesons with relativistic corrections.
We have previously studied the strong decay of P-wave Bs* mesons by the improved B-S method ^{[35]}, and we have also calculated the production of P-wave mesons in B,Bs and B_{c} mesons ^{[36–38]}. The wavefunctions of mesons are obtained by considering the quantum number J^{P} or JPC for different states. P-wave 1^{+} states are labelled as P13 and P11 in our model. For unequal mass systems, the P13 and P11 states are not physical states, while the two physical states P11/2 and P13/2 are mixtures of P13 and P11^{[9,10]}. In Refs. [35–38], the mixing angle is taken as a definite value θ≈35.3° for the P-wave 1^{+} heavy-light mesons. However, in reality the heavy quark is not infinitely heavy in P-wave 1^{+} states. The mixing angle between P13 and P11 is not a fixed value when considering the correction to the heavy quark limit; there is a shift from the value θ≈35.3°, and the shift is different for the different P-wave mixing 1^{+} heavy-light states ^{[15,19,39]}. In this paper, the strong decays of P-wave mixing 1^{+} heavy-light states (1′+,1+) (just like D1′0 and D10) are studied with two conditions of mixing angle θ. One condition is θ = 35.3°; the other considers the correction to the mixing angle θ=35.3°+θ1. The influence of the shift in the mixing angle on the strong decay of P-wave mixing 1^{+} heavy-light states is discussed afterwards.
The paper is organized as follows. In section 2, we give the formulation and hadronic matrix elements of strong decays. We show the relativistic wavefunctions of initial mesons and final mesons in section 3; We talk about the mixing of P13 and P11 states in section 4, and in section 5, we present the corresponding results and conclusions.
Formulation and hadronic matrix elements of strong decays
In this section, we will show the formulations and the transition matrix elements of P-wave mixing states. The quantum numbers of P-wave mixing states (1′+,1+) are both 1^{+}. Considering the limitations of the phase spaces, the ground P-wave 1^{+} states only have an OZI-allowed dominant strong decay channel: 1+→1−0−. The pseudoscalar 0− state must be the light meson K,π, and the 1− state is a heavy-light meson in the final states.
Strong decays of 1<sup>+</sup> states
In order to calculate the strong decays of the two P-wave mixing states, we take the channel D10→D*+π− as an example in Fig. 1. According to the reduction formula, PCAC relation and low energy theorem, the corresponding amplitude can be written as ^{[40,41]}:
T(D10→D*+π−)=Pf2μfπ〈D*+(Pf1)∣d¯γμγ5u∣D10(P)〉,
where P, Pf1, Pf2 are the momenta of D10 and the final states D*+ and π−, respectively. f_{π} is the decay constant of π−. 〈D*+(Pf1)∣d¯γμγ5u∣D10(P)〉 is the hadronic matrix element.
Strong decay of D10→D*+π−.
With Eq. (1), we obtain the strong decay width formula,
Γ=∣P⃗f1∣24πM2Σ∣T(D10→D*+π−)∣2.M is the mass of the initial meson D10, and P⃗f1 is the three-momentum of the final heavy meson D*+.
With the hadronic matrix element
〈D*+(Pf1)∣d¯γμγ5u∣D10(P)〉,
we will obtain the result of Eq. (1) and Eq. (2). The hadronic matrix element will be discussed in the next subsection.
Hadronic matrix elements of strong decays
In this subsection, we will give the calculation of the hadronic matrix element by the improved B-S method. The improved B-S method, which considers relativistic effects, is a good way to describe bound states. Using the improved B-S method, the instantaneous approach and the Mandelstam formalism ^{[42]}, we can get the hadronic matrix elements. As an example, the hadronic matrix element of D10→D*+π− can be written as ^{[34]},
〈D*+(Pf1)∣d¯γμγ5u∣D10(P)〉=∫dq⃗f(2π)3Trφ¯Pf1++(q⃗f)γμγ5φP++(q⃗)PM,
where P and M are the momentum and mass of the initial state D10, and q⃗ and q⃗f=q⃗−mcmc+mdP⃗f1 are the relative three-momentum of the quark and antiquark in the initial state D10 and final state D*+, respectively. φP++(q⃗) and φPf1++(q⃗f) are the positive energy wavefunctions of D10 and D*+, which are given in the next section.
Relativistic wavefunctions
In the improved B-S method, which is based on the constituent quark model, the forms of wave functions are obtained by considering the quantum numbers J^{P} or JPC for different bound states, which are labelled as S13(1−), P13(1+) and P11(1+) and so on. In this paper, we consider the strong decays of the P-wave mixing states (1′+,1+), which are mixtures of P13(1+) and P11(1+). So we only discuss the relativistic wavefunctions of the P11(1+), P13(1+) and S13(1−) states.
Wavefunctions of the <inline-formula><tex-math>
<?CDATA ${}^{1}P_{1}$?>
</tex-math>
<mml:math overflow="scroll"><mml:mmultiscripts><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>
<inline-graphic xlink:href="cpc_42_12_123101_ieqn112.gif" xlink:type="simple"/>
</inline-formula> state
The general expression for the Salpeter wave function of the P11 state, which has JP=1+ (or JPC=1+− for quarkonium), can be written as ^{[35,43]}:
φ1+−(q⃗)=q⊥·ϵf1(q⃗)+PMf2(q⃗)+q⊥Mf3(q⃗)+Pq⊥M2f4(q⃗)γ5,
where q⊥=q−q·PM is the relative momentum between the quark and anti-quark in the state, P and M are the momentum and mass of the 1^{+} meson, and ϵ is the polarization vector. In the center-of-mass system of the meson, q⊥=(0,q⃗). The wave functions, f_{1}, f_{2}, f_{3} and f_{4}, which are functions of q⊥2, are not independent, but are related because of the constraint equations of the Salpeter equation ^{[35,43]},
f3=−M(w1−w2)m1w2+m2w1f1,f4=−M(w1+w2)m1w2+m2w1f2,
where m1,m2,w1,w2 are the masses and momenta of the quark and anti-quark in the 1+− state, respectively, and w1=m12+q⃗2 and w2=m22+q⃗2. With this wavefunction we can obtain the corresponding positive wavefunction of the P11 state,
φP11++(q⃗)=q⊥·ϵa1+a2PM+a3q⊥M+a4Pq⊥M2γ5,
where the coefficients a1…a4 have been defined in Ref. [35],
a1=12(f1(q⃗)+w1+w2m1+m2f2(q⃗)),a2=m1+m2w1+w2a1,a3=−M(w1−w2)m1w2+w1m2a1,a4=−M(m1+m2)m1w2+w1m2a1.
Wavefunctions of the <inline-formula><tex-math>
<?CDATA ${}^{3}P_{1}$?>
</tex-math>
<mml:math overflow="scroll"><mml:mmultiscripts><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>
<inline-graphic xlink:href="cpc_42_12_123101_ieqn125.gif" xlink:type="simple"/>
</inline-formula> state
The general expression for the Salpeter wave function of the P13 state, for which JP=1+ (or JPC=1++ for quarkonium), can be written as ^{[35,43]}:
φP13(q⃗)=iεμναβPνMq⊥αϵβg1γμ+g2PMγμ+g3q⊥Mγμ+g4Pγμq⊥M2.
According to the relations of the constraint equations of the Salpeter equation ^{[35,43]}, we have:
g3=−M(w1−w2)m1w2+m2w1g1,g4=−M(w1+w2)m1w2+m2w1g2.
Then, we can get the positive energy wavefunction of the P13 state ^{[35]},
φP13++(q⃗)=iεμναβPνMq⊥αϵβγμb1+b2PM+b3q⊥M+b4Pq⊥M2.
The coefficients b1…b4 have been defined in Ref. [35],
b1=12(g1(q⃗)+w1+w2m1+m2g2(q⃗)),b2=−m1+m2w1+w2b1,b3=M(w1−w2)m1w2+w1m2b1,b4=−M(m1+m2)m1w2+w1m2b1.
Wavefunctions of the <inline-formula><tex-math>
<?CDATA ${}^{3}S_{1}$?>
</tex-math>
<mml:math overflow="scroll"><mml:mmultiscripts><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>
<inline-graphic xlink:href="cpc_42_12_123101_ieqn131.gif" xlink:type="simple"/>
</inline-formula> state
The general form for the relativistic wavefunction of the vector state JP=1−(or JPC=1−− for quarkonium) can be written as eight terms, which are constructed by Pf1, q_{f⊥}, ϵ1 and gamma matrices ^{[44]},
φ1−(q⃗f)=qf⊥·ϵ1f1′+Pf1Mf1f2′+qf⊥Mf1f3′+Pf1qf⊥Mf12f4′+Mf1ϵ1f5′+ϵ1Pf1f6′+(qf⊥ϵ1−qf⊥·ϵ1f7′+1Mf1(Pf1ϵ1qf⊥−Pf1qf⊥·ϵ1)f8′,
where ϵ1 is the polarization vector of the vector meson in the final state.
According to the relations of the constraint equations of the Salpeter equation ^{[35,43]}, we have:
f1′=qf⊥2f3′+Mf12f5′(m1′m2′−w1′w2′+qf⊥2)Mf1(m1′+m2′)qf⊥2,f7′=f5′(qf⊥)Mf1(−w1′+w2′)(m1′w2′+m2′w1′),f2′=−qf⊥2f4′+Mf12f6′(qf⊥)(m1′w2′−m2′w1′)Mf1(w1′+w2′)qf⊥2,f8′=f6′Mf1(w1′w2′−m1′m2′−qf⊥2)(m1′+m2′)qf⊥2.
The relativistic positive wavefunction of the S13 state can be written as:
φ1−++(qf⃗)=b1ϵ1+b2ϵ1Pf1+b3(qf⊥ϵ1−qf⊥·ϵ1)+b4(Pf1ϵ1qf⊥−Pf1qf⊥·ϵ1)+qf⊥·ϵ1(b5+b6Pf1+b7qf⊥+b8qf⊥Pf1),
where we first define the parameters n_{i}, which are functions of fi′ (S13 wave functions):
n1=f5′−f6′(w1′+w2′)(m1′+m2′),n2=f5′−f6′(m1′+m2′)(w1′+w2′),n3=f3′+f4′(m1′+m2′)(w1′+w2′),
then we define the parameters b_{i}, which are functions of fi′ and n_{i}:
b1=Mf12n1,b2=−(m1′+m2′)2(w1′+w2′)n1,b3=Mf1(w2′−w1′)2(m1′w2′+m2′w1′)n1,b4=(w1′+w2′)2(w1′w2′+m1′m2′−qf⊥2)n1,b5=12Mf1(m1′+m2′)(Mf12n2+qf⊥2n3)(w1′w2′+m1′m2′+qf⊥2),b6=12Mf12(w1′−w2′)(Mf12n2+qf⊥2n3)(w1′w2′+m1′m2′+qf⊥2),b7=n32Mf1−f6′Mf1(m1′w2′+m2′w1′),b8=12Mf12w1′+w2′m1′+m2′n3−f5′w1′+w2′(m1′+m2′)(w1′w2′+m1′m2′−qf⊥2).
Mixing of <inline-formula><tex-math>
<?CDATA ${}^{3}P_{1}$?>
</tex-math>
<mml:math overflow="scroll"><mml:mmultiscripts><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>
<inline-graphic xlink:href="cpc_42_12_123101_ieqn141.gif" xlink:type="simple"/>
</inline-formula> and <inline-formula><tex-math>
<?CDATA ${}^{1}P_{1}$?>
</tex-math>
<mml:math overflow="scroll"><mml:mmultiscripts><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:none/><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>
<inline-graphic xlink:href="cpc_42_12_123101_ieqn142.gif" xlink:type="simple"/>
</inline-formula> states
The heavy quark is in the mQ→∞ limit in the heavy-light mesons, so the properties of heavy-light mesons are similar to those of the hydrogen atom (which are determined by the orbital electron). The spin of the heavy quark decouples and the properties of heavy-light mesons are determined by light degrees of freedom alone. For the P-wave heavy-light mesons (orbital angular momentum L = 1), the total angular momentum jq=sq+L of the light quark has two conditions (jq=12 and jq=32) with sq=12. The whole angular momentum J=sQ+jq and the spin of the heavy quark SQ=12, so there are two degenerate doublets for P-wave heavy-light mesons: jq=12S doublet (JP=0+,1+) states and jq=32T doublet (JP=1+,2+) states. In this paper, we only talk about the two physical P-wave heavy-light mixing JP=1+ states. In order to distinguish the two 1^{+} states, we use the notation P11/2 (which is from the jq=12S doublet) and P13/2 (which is from the jq=32T doublet) to describe the two physical P-wave heavy-light mixing states.
In our model, we give expressions for the wavefunctions in term of the quantum numbers J^{P} (or JPC), which describe the equal mass systems in heavy mesons very well. There are two physical P-wave states P13(1++) and P11(1+−) for equal mass systems, but they are not physical states when there is no charge conjugation parity for an unequal mass system. In the heavy quark limit, the physical states P11/2 and P13/2 of P-wave heavy-light mesons can be written as ^{[9,10,45]}:
∣P13/2>=23∣P11>+13∣P13>∣P11/2>=−13∣P11>+23∣P13>,
where Eq. (12) is the same as the result of Ref. [35,38] if we take θ = 35.3°. So the wavefunctions for the physical P11/2 and P13/2 states can be obtained by the mixing relations of the P13 and P11 wavefunctions which were shown in section 3. However, when we solve the B-S equation, the mass of the heavy quark is a fixed value which is not infinite. So, considering the correction to the heavy quark limit, the physical states 1′+ and 1^{+} are mixtures of P11/2 and P13/2^{[15,19,39]},
∣1+>=cosθ1∣P13/2>+sinθ1∣P11/2>∣1′+>=−sinθ1∣P13/2>+cosθ1∣P11/2>,
when in the heavy quark limit, e.g: θ1=0°, ∣1′+>=∣P11/2> and ∣1+>=∣P13/2>. For the 1^{+} states D and D_{s}, θ1=−(0.1±0.05) rad=−(5.7±2.9)°^{[15,19]}. For the 1^{+} states B and B_{s}, θ1=−(0.03±0.015) rad=−(1.72±0.86)°^{[19]}. Taking Eq. (12) into Eq. (13), we can get the relation of 1′+,1^{+} and P13, P11,
∣1+>=cosθ∣P11>+sinθ∣P13>∣1′+>=−sinθ∣P11>+cosθ∣P13>,
where θ=35.3°+θ1. According to Eq. (14), we have obtained the mass spectra of the P-wave mixing states 1′+ and 1^{+} by solving the full Salpeter equation in Table 1.
Mass spectra of the P-wave mixing states 1′+ and 1^{+}, in units of MeV. ‘ex.’ means the experimental data from the PDG ^{[1]}, and ‘th.’ means our prediction.
states
th.
ex.
states
th.
ex.
states
th.
ex.
D1′0
2427.0
2427±26±25
Ds1′±
2460.0
2459.6 ± 0.9
B1′0
5710.0
–
D10
2422.0
2421.4 ± 0.6
Ds1±
2536.0
2535.18 ± 0.24
B10
5726.0
5726.0 ± 1.3
D1′±
2427.0
–
Bs1′
5820.0
–
B1′±
5710.0
–
D1±
2422.0
2423.2 ± 2.4
Bs1
5829.0
5828.78 ± 0.35
B1±
5726.0
5726.8−4.0+3.2
Numerical results and discussion
In order to fix the Cornell potentials and masses of quarks, we take these parameters: a = e = 2.7183, λ = 0.21 GeV^{2}, ΛQCD=0.27GeV, α = 0.06 GeV, mu=0.305GeV, m_{d}=0.311 GeV, m_{s}=0.500 GeV, mb=4.96GeV, mc=1.62GeV, etc ^{[46]}, which are best to fit the mass spectra of ground state B and D mesons and other heavy mesons. We get the following masses of the ground states: MD*0=2.007GeV, MD*±=2.010GeV, MDs*±=2.112GeV, MB*=5.325GeV, and MBs*=5.415GeV. For the light mesons, the masses and decay constants are: Mπ=0.140GeV, fπ=0.130GeV, MK=0.494GeV, and fK=0.156GeV^{[1]}.
The D1′0, D10 and D1± states have been listed in the PDG ^{[1]}. D1′0 is a broad state with large uncertainty, and D10 and D1± are narrow states with small uncertainty, but there is no evidence of another state D1′± in experiment. We therefore predict the mass of D1′± to be the same as D1(2430)0 by the improved B-S method in Table 1. The two-body strong decays of these states only happens in the D*π channel, which is OZI-allowed.
We calculate the transition matrix elements by the wavefunctions numerically, and get the strong decay widths of D1′0 and D10. Our predicted results and those of other authors are given in Table 2. The results are shown for the two conditions of the mixing angle θ: θ = 35.3°, and θ=35.3°+θ1. We find that the results of D1′0→D*π are very close for the two conditions. Both of them are smaller than the central experiment value, but if we consider the large uncertainty of the experiment value for D1′0, our results seem reasonable. We also find that our results are consistent with the results of Ref. [21] and Ref. [24], but smaller than the result of Ref. [22]. Though the predicted masses of the P-wave mixing states are similar for different models, the predicted decay widths are much different. The situation is similar in other channels. For example, in our prediction of D10→D*π, when we consider the correction to the heavy quark limit, the decay width is increased, which is consistent with the results of other models and close to the lower limit of the experimental value. In Fig. 2, we plot the relation of mixing angle θ to the strong decay widths of D1′0 and D10. The D1′0 meson is a broad state and the influence of the mixing angle is very small. The width of the D10 meson is at the bottom of the curve, and is sensitive to the mixing angle. We determine the mixing angle θ=35.3°+θ1min=26.7° to be the best description of the experimental value. But for the D1′0 meson, since there are large uncertainties on the experimental value, much stronger experimental confirmation is needed in the future. We give the strong decay widths of D1′± and D1± in Table 3. The strong decay widths of D1± and D1′± are very similar to the results for D1′0 and D10, because of the masses of the light quarks in the P-wave mixing states are very close: mu≈md. The strong decays of D1′± also provide a good way to observe this meson in experiment.
Decay widths of D1′0 and D10.
Decay widths of two-body strong decays of D^{0} mixing states 1′+ and 1^{+}, in units of MeV, with θ1=−(0.1±0.05) rad=−(5.7±2.9)°^{[15,19]}.
mode
θ = 35.3°
θ=35.3°+θ1
^{[17]}
^{[21]}
^{[22]}
^{[24]}
^{[1]}
D1′0→D*π
232
228∼232
–
244
272
220
384−75+107±74
D10→D*π
17.3
17.6∼21.5
11
25
22
21.6
27.4 ± 2.5
Decay widths of two-body strong decays of D± mixing states 1′+ and 1^{+}, in units of MeV, with θ1=−(0.1±0.05) rad=−(5.7±2.9)°^{[15,19]}.
Both Ds1′± and Ds1± have small widths. The Ds1′± meson is below the threshold of D* and K, so the OZI-allowed strong decay is forbidden, and the dominant strong decay channel of the Ds1′± meson is the isospin symmetry violating decay via π0−η mixing, which is Ds1′±→Ds*±η→Ds*±π0^{[47]}. Because of the small value of the mixing parameter tπη=<π0∣∣η>=−0.003GeV2^{[47]}, the decay is heavily suppressed, and the decay width is very narrow. The mass of Ds1± is larger than the threshold of D*K, but the width is also very narrow because of the small kinematic range.
We have given two results of our method in Table 4: (1) θ = 35.3°; (2) considering the correction θ=35.3°+θ1. For the strong decay Ds1′±→Ds*π, the results are very similar for the two conditions and close to the results of Ref. [21] and Ref. [26]. The results for Ds1′±→Ds*π are smaller than the result from Ref. [16], but they are reliable compared with the experimental result. We find that the results for Ds1±→D*K are sensitive to the mixing angle θ. When the mixing angle θ = 35.3°, the decay width of Ds1± is close to the results of Ref. [21] and Ref. [24], but smaller than the result of Ref. [22] and the experimental value in Ref. [1]. With the correction to the mixing angle, we get the decay width of Ds1± as: Γ(Ds1±)=0.950∼5.46MeV. In order to compare with the experimental data, we plot the relation of strong decay width Ds1′± and Ds1± against mixing angle θ in Fig. 3. The results for Ds1± are at the bottom of the curve and close to zero with both conditions, so the result for Ds1± is sensitive to mixing angle. It shows that with θ = 32.5°, the corresponding result Γ(Ds1±)=0.950MeV, which is consistent with the experimental data of Ds1(2536)±: Γ=0.92±0.05 MeV ^{[1]}.
Decay widths of Ds1′± and Ds1±.
Decay widths of two-body strong decays of D_{s} mixing states 1′+ and 1^{+}, in units of MeV, with θ1=−(0.1±0.05) rad=−(5.7±2.9)°^{[15,19]}.
Experiment has only observed B10 and B1±, which are considered to be B1(5721)0 and B1(5721)+^{[2–4]}, and there is no evidence for the B1′0 and B1′±. So we first predict the masses of the B1′0 and B1′±, then calculate the strong decays of these mesons. Because of the large kinematic range, all of these four states have OZI-allowed strong decay channels.
We show the strong decay widths for the P-wave mixing B mesons, under the same two conditions as the P-wave mixing D mesons, in Table 5 and Table 6. For comparison, we list the experimental data and some results from other model predictions, and plot the relations of the decay widths of B1′0 and B10 against the mixing angle θ in Fig. 4. For B1′0 and B1′±, the decay widths with the two conditions are very close, which is consistent with the result of the QCD sum rules: Γ(B1′)≃250MeV^{[20]}. The decay widths of B10 and B1± are close to the result of Ref. [20] and smaller than the results of Ref. [24] and Ref. [27]. The results for B1± are also much smaller than the experimental value, but the results for B10 are very close to the upper limit of the experimental value. Because the experimental results for B1± have large uncertainties, they need to be confirmed experimentally in the future, and our results will provide theoretical assistance.
Decay widths of B1′0 and B10.
Decay widths of two-body strong decays of B^{0} mixing states 1′+ and 1^{+}, in units of MeV, with θ1=−(0.03±0.015)rad=−(1.72±0.86)°^{[19]}.
mode
θ = 35.3°
θ=35.3°+θ1
^{[20]}
^{[24]}
^{[27]}
^{[1]}
B1′0→B*π
262.4
262.1∼262.4
250
219
139
–
B10→B*π
15.6
15.6∼16.0
–
30
20
23 ±3±4
Decay widths of two-body strong decays of B± mixing states 1′+ and 1^{+}, in units of MeV, with θ1=−(0.03±0.015)rad=−(1.72±0.86)°^{[19]}.
In order to compare the results with the two conditions of mixing angle θ for the P-wave Bs* mixing states Bs1′ and Bs1, we also show the corresponding strong decay widths of Bs1′ and Bs1 in Table 7. For the Bs1′ meson, the OZI-allowed strong decay is forbidden, and the dominant strong decay of the Bs1′ meson is the isospin symmetry violating decay via π0−η mixing ^{[47]}. So the decay widths of Bs1′ are very close for both conditions, and the influence of the mixing angle is very small. For the decay width of the Bs1 meson, however, there is a big difference between the two conditions, and the influence of the angle is very large. In our calculation, we plot the decay widths of the mixed states Bs1′ and Bs1 as functions of θ in Fig. 5. The results for the two conditions are at the bottom of the curve and close to zero, which is the same as for Ds1±. If we take the mixing angle θ = 32.7°, we get the decay width Γ(Bs1)=0.412MeV, which is close to the experimental data, within the large uncertainties.
Decay widths of Bs1′ and Bs1.
Decay widths of two-body strong decays of B_{s} mixing states 1′+ and 1^{+}, in units of MeV, with θ1=−(0.03±0.015)rad=−(1.72±0.86)°^{[19]}.
mode
θ = 35.3°
θ=35.3°+θ1
^{[16]}
^{[17]}
^{[24]}
^{[26]}
^{[28]}
^{[29]}
^{[1]}
Bs1′→Bs*π
0.01987
0.01986∼0.01987
0.0215
–
–
0.01036
–
–
–
Bs1→B*K
0.0396
0.0834∼0.412
–
<1
0.4∼1
–
0.7 ± 0.3 ± 0.3
0.098
0.5 ± 0.3 ± 0.3
In conclusion, we have studied the strong decays of the P-wave mixing heavy-light 1^{+} states by the improved B-S method with two conditions of mixing angle θ: θ = 35.3°, and considering a correction to the mixing angle, θ=35.3°+θ1. We find that, for the P-wave 1′+ mesons (D1′0, D1′±, Ds1′±, B1′0, B1′± and Bs1′) and some 1+ states (B10 and B1±), the influence of mixing angle θ between P13 and P11 is very small, and the results with the two conditions are very close. However, for some of the P-wave 1+ mesons (D10, D1±, Ds1± and Bs1), the influence of the mixing angle θ between P13 and P11 is large; especially for the Ds1± and Bs1 mesons, there is a large discrepancy between the two conditions. For the D10 and D1± states, we take the mixing angle θ = 26.7°, which is the best description of the experimental value. For the Ds1± state, the mixing angle θ = 32.5° is the best description of the experimental value. For the Bs1 state, the result for the mixing angle θ = 32.7° is close to the experimental data, within large uncertainties. In this paper, we have studied the strong decays of some special states which have not yet been discovered in experiment, such as D1′±, B1′0, B1′± and Bs1′. This will provide theoretical assistance to future experiments. We have also investigated the strong decays of Ds1′±, B10, B1± and Bs1 with large uncertainties for the experimental data, and given predicted results, which need to be confirmed by future experiments.