Strong decays of excited $ 2^+ $ charmed mesons

The new charmed resonance $ D_2^*(3000) $ was observed by the LHCb Collaboration in $ B $ decays. In this paper, by assigning it as four possible excited states of the $ 2^+ $ family, we use the instantaneous Bethe-Salpeter method to calculate their Okubo-Zweig-Iizuka-allowed two-body strong decays. The results of $ 1^3F_2 $ and $ 3^3P_2 $ states deviate from the present experimental observation while $ 2^3P_2 $ and $ 2^3F_2 $ are in the error range. Our study also reveals that the widths of these states depend strongly on the masses. Due to the large uncertainties and different mass input, variable models get inconsistent conclusions at the moment. The analysis in this work can provide essential assistance for future measurements and investigations.


I. INTRODUCTION
allowed strong decays are dominant for charmed mesons. Some efforts about their decays have been made and several theoretical methods were applied, including heavy quark effective theory (HQET) [8,21], the quark pair creation (QPC) model (also named as the 3P 0 model) [4,9,17,22], and the chiral quark model [20,23]. Significant discrepancies still exist between different works and the comparison will be discussed in Sec. III.
Since the relativistic effects in heavy-light mesons are not negligible, especially for excited states, we use the wave functions obtained by the instantaneous Bethe-Salpeter (BS) approach [24,25] to calculate the OZI allowed two-body strong decays. The BS method has been applied successfully in many previous works [26][27][28][29][30][31][32][33][34][35][36] and the relativistic corrections in the heavy flavor mesons are well considered [37]. When the light pseudoscalar mesons are involved in the final states, the reduction formula, partially conserved axial-vector current (PCAC) relation, and low-energy theorem are used to depict the quark-meson coupling.
Since PCAC is inapplicable when the final light meson is a vector, for instance ρ, ω, and K * , an effective Lagrangian is adopted instead [38,39].
The rest of the paper is organized as follows: In Sec. II, we briefly construct BS wave functions of 2 + state. Then we derive the theoretical formalism of the strong decays with the PCAC relation and the effective Lagrangian method. In Sec. III, the numerical results and detailed discussions are presented. Finally, we summarize our work in Sec. IV.
which are constrained by where m i is the quark/antiquark mass, ω i = m 2 i − q 2 ⊥ , and q 2 ⊥ = −| q| 2 in the rest frame of the meson.
The review of instantaneous approximation and the action kernel adopted in this paper was given in our previous work [41]. Due to the approximation, our previous results showed that the predicted mass spectrum for excited heavy-light mesons may not fit the experimental data very well. Thus we follow our previous works [10,30,35] to use the physical mass as an input parameter to solve the BS wave functions for each state. As we did formerly [11], only the dominant positive-energy part of wave functions ϕ ++ = Λ + 1 / P M ϕ / P M Λ + 2 are kept in the following calculation, which was explained in Ref. [41] in detail.
Feynman diagram of the OZI-allowed two-body strong decay, taking D * 0 2 → D + π − for example.
Taking D * 2 → D + π − as an example, the Feynman diagram of two-body strong decay is shown in Fig. 1. By using the reduction formula, the corresponding transition matrix element has the form in which φ π is the light pseudoscalar meson field. By using the PCAC relation, the field is expressed as where f π is the decay constant of π. Then, by using the low-energy approximation, the transition amplitude in momentum space can be derived as [42] D + (P 1 )π − (P 2 ) D * 0 2 (P ) Besides the PCAC relation, the same transition amplitude can be obtained by introducing an effective Lagrangian [10,38,39] where denotes the chiral field of the light pseudoscalar meson, g is the quark-meson coupling constant and f h is the decay constant.
For further numerical calculation, in the Mandelstam formalism [43], we can write the hadronic transition amplitude as the overlapping integration over the relativistic wave functions of the initial and final mesons [42] where ϕ ++ and ϕ ++ are the positive part of BS wave function and its Dirac adjoint form, and the quark-antiquark relative momenta in the initial and final meson have the relation q 1 = q − mc mc+m d P 1 . When the final light meson is η or η instead of π, we also consider the η − η mixing [1] where η 8 = 1 √ 6 (uū + dd − 2ss), R(θ) or R T (θ) is the mixing matrix defined as We choose the mixing angle θ η = −11.3 • [1] in this work. By extracting the coefficient of mixing, the transition amplitudes with η/η involved are When the final light meson is a vector, such as ρ or ω, the PCAC rule is not valid. Thus we adopt the effective Lagrangian method to derive the transition amplitude. The effective Lagrangian of light vector meson is given by [10,38,39] where m j is the constitute quark mass (we take approximation m j = (m q + mq)/2 in this work), σ µν = i 2 [γ µ , γ ν ]; V µ is the light vector meson field, and the parameters a = −3 and b = 2 denote the vector and tensor coupling strength [39], respectively. Then we use Eq. (14) directly to get the vertex of the light vector and reach the transition amplitude In the possible strong decays of D * 2 (3000), 1 + and 2 − states of D mesons are also involved in the final state. In the heavy quark limit(m Q → ∞), the coupling of spin S and orbital angular momentum L no longer describes the physical states well for these heavy-light states.
The 1 P 1 -3 P 1 and 1 D 2 -3 D 2 mixing are needed. We take the total angular momentum of the light quark in the mesons j l = L + s q (s q is the light-quark spin) to identify the physical doublet. The mixing relations are given by [16,33,44,45]   |J P = 1 + , j l = 3/2 where the ideal mixing angles θ in the heavy-quark limit are adopted. We notice that varying mixing angles could make difference to the decay widths. The dependence of corresponding partial widths on mixing angle will be discussed later.
By performing the integration and trace in Eqs. (9) and (15), the amplitudes of all possible channels within the present study can be simplified as 2β P µ 1 t 6 P ν 1 P α P β + t 7 P ν 1 g αβ + t 8 P α g νβ + t 9 P β g να + t 10 g µα g νβ , in which t i are the form factors achieved by integrating over the wave functions for specific channels, ε αβγδ is the Levi-Civita symbol, , 1 , and 2 are the polarization tensors or vectors of corresponding states. For convenience, we define where P and M are the momentum and mass of the corresponding meson. Then, the completeness relations of polarization vector and tensor are given by The two-body decay width can be achieved by where | P 1 | = λ(M 2 , M 2 1 , M 2 2 )/2M is the momentum of the final charmed meson 1 , J = 2 is the spin quantum number of initial 2 + state.

III. RESULTS AND DISCUSSIONS
In this work, the Cornell potential is taken to solve the BS equations of 2 + states numerically. Within the instantaneous approximation, the Cornell potential in momentum space has the form as follow [41] where V 0 is a free parameter fixed by the physical masses of corresponding mesons; the linear confinement item V s and the one-gluon exchange Coulomb-type item V v are The coupling constant α s is running where e = 2.7183 .
The parameters adopted in the numerical calculation are listed as follows [31]: Before presenting our results, a remark is in order. When we solve the wave functions, the 2 + states actually are the mixture of several partial waves [48]. Within this work, to avoid confusion, we will not discuss this mixing in detail and still use pure P or F wave to mark each 2 + state.
Within our calculation, the wave functions of each state are acquired by fixed on the experimental mass, which also gives the same phase space for every assignment. The total and partial widths of D * 2 (3000) with possible assignments are presented in Table II. In the four possible candidates, D * 2 (3000) as 1 3 F 2 has the largest total width in our prediction, which is about 778.0 MeV and much exceeds the upper limit of present experimental results. The channels of Dπ, D * π, D s K, D 1 (2420)/D 1 (2430)π(η), D s1 (2536)/D s1 (2460)K, and D 2 (2740)/D 2 (2780)π contribute much to the total width. In these dominant channels, the branching fraction of most concerned Dπ mode is about 3%. And the partial width ratio of D * π to Dπ are given by In the case of 2 3 P 2 state, the total width is estimated to be 285.6 MeV, which is larger than the observational value but still in the error range. The dominant channels include Dπ, D * π, Dρ(ω), D * ρ(ω), D * s K * , and D 1 (2420)/D 1 (2430)π. Our predicted branching fractions of Dπ for 2 3 P 2 is about 4%. The partial width ratio between D * π and Dπ is The total widths of 2 3 F 2 and 3 3 P 2 are about 61 MeV and 19.1 MeV, respectively. The width of 2 3 F 2 reaches the lower limit while the result of 3 3 P 2 doesn't. The channels of Dπ, D * π, D 1 (2420)/D 1 (2430)π, and D 2 (2740)/D 2 (2780)π give main contribution to the width of 2 3 F 2 , while 3 3 P 2 dominantly decay into Dπ, D * π, D * ρ(ω), and D * s K * . The branching fractions of Dπ for 2 3 F 2 and 3 3 P 2 are 12% and 10%, respectively. The corresponding partial width ratios are We notice that Dπ mode is appreciable for all four candidates, which is consistent with present experimental observations. The widths of channels involving D 1 (2420)/D 1 (2430), D s1 (2536)/D s1 (2460) and D 2 (2740)/D 2 (2780) are also considerable especially for 2 3 P 2 , 3 3 P 2 and 2 3 F 2 states. In addition, the partial width ratios of Dπ to D * π are different for these candidates, which could be useful for future experimental test.
The results in Table II are  The mixing has been discussed by our previous works [41,49] and ideal mixing is valid for D 1 (2420) and D 1 (2430). Here we only show the sensitive dependence between partial widths and mixing angle. In the following discussion, we keep using the results of ideal mixing for consistency.
Although all four assignments are calculated by fitting the same mass value M = 3214 MeV, which gives equivalent phase space for the same channels, the decay behaviors   D(2 3 P 2 ) → D(1 1 S 0 )π. Both 2 3 P 2 and 2 1 S 0 have one node, which finally enhance the integral. However, in the case of 3 3 P 2 , the structure of two nodes eventually gives the contrary behavior with same two channels. As for 1 3 F 2 and 2 3 F 2 , since the shape of wave functions are different (f 3 , f 4 , and f 5 always have the same sign for n 3 F 2 , while f 3 and f 4 have the same sign for n 3 P 2 ), the widths of D(2 1 S 0 )π are smaller than D(1 1 S 0 )π. In general, 1 3 F 2 and 2 3 P 2 states reach larger widths with less cancellation, while 2 3 F 2 and 3 3 P 2 behave oppositely. It also illustrate that the correction from higher internal momentum q of mesons is non-negligible. At last, we list the full widths of excited 2 + states from some other models in Table  III

IV. SUMMARY
We analyzed the OZI-allowed two-body strong decay behaviors of four possible assignments of 2 + family for the newly observed D * 2 (3000). The wave functions of related charmed mesons was obtained by using the instantaneous Bethe-Salpeter method. In our calculation, D * 2 (3000) as 1 3 F 2 state has an exceeded width of 778 MeV, while the widths of 19.1 MeV for 3 3 P 2 doesn't reach the lower limit. As for 2 3 P 2 and 2 3 F 2 , the total widths are respectively 285.6 MeV and 60.5 MeV, which enter the error range of present experimental results. Dπ channels are important for all four candidates, which shows consistency with the current ex- periment. Additionally, the channels involving D 1 (2420)/D 1 (2430), D s1 (2536)/D s1 (2460), and D 2 (2740)/D 2 (2780) also give significant contribution and the influence of mixing angle was discussed. Our study also indicates a strong dependence between total widths and states masses. Thus different models with varying mass input don't reach a consensus. Considering the large uncertainties of preliminary observation, besides the possible individual states, mixing of several states is also a potential option for the current D * 2 (3000). More accurate measurements and theoretical efforts are expected in the future.