Strong decays of P -wave mixing heavy-light 1+ states

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 D0, D± , Ds± , B0, B± and Bs 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.

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 j q = s q +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 m Q →∞ 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][12][13][14] and strong decays of P -wave heavy-light mesons have also been studied by other methods [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. The strong decay of P -wave heavy-light mesons can help us understand the properties of these mesons and establish the heavylight 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 Pwave B * s mesons by the improved B-S method [35], and we have also calculated the production of P -wave mesons in B, B s and B c mesons [36][37][38]. The wavefunctions of mesons are obtained by considering the quantum number J P or J P C for different states. P -wave 1 + states are labelled as 3 P 1 and 1 P 1 in our model. For unequal mass systems, the 3 P 1 and 1 P 1 states are not physical states, while the two physical states P 1/2 1 and P 3/2 1 are mixtures of 3 P 1 and 1 P 1 [9,10]. In Refs. [35][36][37][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 3 P 1 and 1 P 1 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 D ′0 1 and D 0 1 ) 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 • +θ 1 . 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 3 P 1 and 1 P 1 states in Section 4, and in Section 5, we present the corresponding results and conclusions.
2 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 + states
In order to calculate the strong decays of the two Pwave mixing states, we take the channel D 0 1 → 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]: where P , P f 1 , P f 2 are the momenta of D 0 1 and the final states D * + and π − , respectively. f π is the decay constant of π − . D * + (P f 1 )|dγ µ γ 5 u|D 0 1 (P ) is the hadronic matrix element. With Eq. (1), we obtain the strong decay width formula, M is the mass of the initial meson D 0 1 , and P f 1 is the three-momentum of the final heavy meson D * + . With the hadronic matrix element D * + (P f 1 )|dγ µ γ 5 u|D 0 1 (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 D 0 1 →D * + π − can be written as [34], where P and M are the momentum and mass of the initial state D 0 1 , and q and q f = q− mc mc+m d P f 1 are the relative three-momentum of the quark and antiquark in the initial state D 0 1 and final state D * + , respectively. ϕ ++ P ( q) and ϕ ++

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 J P C for different bound states, which are labelled as 3 S 1 (1 − ), 3 P 1 (1 + ) and 1 P 1 (1 + ) and so on. In this paper, we consider the strong decays of the P -wave mixing states (1 ′+ , 1 + ), which are mixtures of 3 P 1 (1 + ) and 1 P 1 (1 + ). So we only discuss the relativistic wavefunctions of the 1 P 1 (1 + ), 3 P 1 (1 + ) and 3 S 1 (1 − ) states.

Wavefunctions of the 1 P 1 state
The general expression for the Salpeter wave function of the 1 P 1 state, which has J P = 1 + (or J P C = 1 +− for quarkonium), can be written as [35,43]: where q ⊥ = q − q·P M 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 ǫ is the polarization vector. In the center-of-mass system of the meson, q ⊥ = (0, q). 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], where m 1 ,m 2 ,w 1 ,w 2 are the masses and momenta of the quark and anti-quark in the 1 +− state, respectively, and w 1 = m 2 1 + q 2 and w 2 = m 2 2 + q 2 . With this wavefunction we can obtain the corresponding positive wavefunction of the 1 P 1 state, where the coefficients a 1 ...a 4 have been defined in Ref. [35],

Wavefunctions of the 3 P 1 state
The general expression for the Salpeter wave function of the 3 P 1 state, for which J P = 1 + (or J P C = 1 ++ for quarkonium), can be written as [35,43]: According to the relations of the constraint equations of the Salpeter equation [35,43], we have: Then, we can get the positive energy wavefunction of the 3 P 1 state [35],

Wavefunctions of the 3 S 1 state
The general form for the relativistic wavefunction of the vector state J P = 1 − (or J P C = 1 −− for quarkonium) can be written as eight terms, which are constructed by 123101-3 P f 1 , q f ⊥ , ǫ 1 and gamma matrices [44], where ǫ 1 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: The relativistic positive wavefunction of the 3 S 1 state can be written as: (11) where we first define the parameters n i , which are functions of f ′ i ( 3 S 1 wave functions): , , , then we define the parameters b i , which are functions of f ′ i and n i : Mixing of 3 P 1 and 1 P 1 states The heavy quark is in the m Q →∞ limit in the heavylight 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 j q =s q +L of the light quark has two conditions (j q = 1 2 and j q = 3 2 ) with s q = 1 2 . The whole angular momentum J = s Q +j q and the spin of the heavy quark S Q = 1 2 , so there are two degenerate doublets for P -wave heavy-light mesons: j q = 1 2 S doublet (J P = 0 + ,1 + ) states and j q = 3 2 T doublet (J P = 1 + ,2 + ) states. In this paper, we only talk about the two physical P -wave heavy-light mixing J P = 1 + states. In order to distinguish the two 1 + states, we use the notation P 1/2 1 (which is from the j q = 1 2 S doublet) and P 3/2 1 (which is from the j q = 3 2 T 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 J P C ), which describe the equal mass systems in heavy mesons very well. There are two physical P -wave states 3 P 1 (1 ++ ) and 1 P 1 (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 P 1/2 1 and P 3/2 1 of P -wave heavy-light mesons can be written as [9,10,45]: 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 P 1/2 when in the heavy quark limit, e.g: θ 1 = 0 • , |1 ′+ >= |P  (13), we can get the relation of 1 ′+ ,1 + and 3 P 1 , 1 P 1 , where θ =35.3 • +θ 1 . 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.

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 The D ′0 1 , D 0 1 and D ± 1 states have been listed in the PDG [1]. D ′0 1 is a broad state with large uncertainty, and D 0 1 and D ± 1 are narrow states with small uncertainty, but there is no evidence of another state D ′± 1 in experiment. We therefore predict the mass of D ′± 1 to be the same as D 1 (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 D ′0 1 and D 0 1 . 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 • +θ 1 . We find that the results of D ′0 1 →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 D ′0 1 , 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 D 0 1 →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 D   are large uncertainties on the experimental value, much stronger experimental confirmation is needed in the future. We give the strong decay widths of D ′± 1 and D ± 1 in Table 3. The strong decay widths of D ± 1 and D ′± 1 are very similar tothe results for D ′0 1 and D 0 1 , because of the masses of the light quarks in the P -wave mixing states are very close: m u ≈ m d . The strong decays of D ′± 1 also provide a good way to observe this meson in experiment.

s1
Both D ′± s1 and D ± s1 have small widths. The D ′± s1 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 D ′± s1 meson is the isospin symmetry violating decay via π 0 −η mixing, which is D ′± s1 →D * ± s η→ D * ± s π 0 [47]. Because of the small value of the mixing parameter t πη =<π 0 |H|η>=−0.003 GeV 2 [47], the decay is heavily suppressed, and the decay width is very narrow. The mass of D ± s1 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 • +θ 1 . For the strong decay D ′± s1 → D * s π, the results are very similar for the two conditions and close to the results of Ref. [21] and Ref. [26]. The results for D ′± s1 → D * s π are smaller than the result from Ref. [16], but they are reli-able compared with the experimental result. We find that the results for D ± s1 → D * K are sensitive to the mixing angle θ. When the mixing angle θ = 35.3 • , the decay width of D ± s1 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 D ± s1 as: Γ(D ± s1 ) = 0.950 ∼ 5.46 MeV. In order to compare with the experimental data, we plot the relation of strong decay width D ′± s1 and D ± s1 against mixing angle θ in Fig. 3. The results for D ± s1 are at the bottom of the curve and close to zero with both conditions, so the result for D ± s1 is sensitive to mixing angle. It shows that with θ = 32.5 • , the corresponding result Γ(D ± s1 ) = 0.950 MeV, which is consistent with the experimental data of D s1 (2536) ± : Γ=0.92±0.05 MeV [1].
Experiment has only observed B 0 1 and B ± 1 , which are considered to be B 1 (5721) 0 and B 1 (5721) + [2][3][4], and there is no evidence for the B ′0 1 and B ′± 1 . So we first predict the masses of the B ′0 1 and B ′± 1 , 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 B ′0 1 and B 0 1 against the mixing angle θ in Fig. 4. For B ′0 1 and B ′± 1 , the decay widths with the two conditions are very close, which is consistent with the result of the QCD sum rules: Γ(B ′ 1 )≃250 MeV [20]. The decay widths of B 0 1 and B ± 1 are close to the result of Ref. [20] and smaller than the results of Ref. [24] and Ref. [27]. The results for B ± 1 are also much smaller than the experimental value, but the results for B 0 1 are very close to the upper limit of the experimental value. Because the experimental results for B ± 1 have large uncertainties, they need to be confirmed experimentally in the future, and our results will provide theoretical assistance.

B ′ s1 and B s1
In order to compare the results with the two conditions of mixing angle θ for the P -wave B * s mixing states B ′ s1 and B s1 , we also show the corresponding strong decay widths of B ′ s1 and B s1 in Table 7. For the B ′ s1 meson, the OZI-allowed strong decay is forbidden, and the dominant strong decay of the B ′ s1 meson is the isospin symmetry violating decay via π 0 −η mixing [47]. So the decay widths of B ′ s1 are very close for both conditions, and the influence of the mixing angle is very small. For the decay width of the B s1 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 B ′ s1 and B s1 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 D ± s1 . If we take the mixing angle θ = 32.7 • , we get the decay width Γ(B s1 ) = 0.412 MeV, which is close to the experimental data, within the large uncertainties.
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 • +θ 1 . We find that, for the P -wave 1 ′+ mesons (D ′0 1 , D ′± 1 , D ′± s1 , B ′0 1 , B ′± 1 and B ′ s1 ) and some 1 + states (B 0 1 and B ± 1 ), the influence of mixing angle θ between 3 P 1 and 1 P 1 is very small, and the results with the two conditions are very close. However, for some of the P -wave 1 + mesons (D 0 1 , D ± 1 , D ± s1 and B s1 ), the influence of the mixing angle θ between 3 P 1 and 1 P 1 is large; especially for the D ± s1 and B s1 mesons, there is a large discrepancy between the two conditions. For the D 0 1 and D ± 1 states, we take the mixing angle θ = 26.7 • , which is the best description of the experimental value. For the D ± s1 state, the mixing angle θ=32.5 • is the best description of the experimental value. For the B s1 state, the result for the mixing angle θ=32.7 • is close to the exper- imental 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 D ′± 1 , B ′0 1 , B ′± 1 and B ′ s1 . This will provide theoretical assistance to future experiments. We have also investi-gated the strong decays of D ′± s1 , B 0 1 , B ± 1 and B s1 with large uncertainties for the experimental data, and given predicted results, which need to be confirmed by future experiments.