Open-flavor strong decays of open-charm and open-bottom mesons in the $^3P_0$ model

We provide results for the open-flavor strong decays of open-charm ($D$ and $D_{\rm s}$) and open-bottom ($B$, $B_{\rm s}$ and $B_{\rm c}$) mesons. The decays are calculated in a modified version of the $^3P_0$ pair-creation model, assuming harmonic oscillator wave functions. The spectra of open-charm and open-bottom mesons used in the calculations are computed within Godfrey and Isgur's relativized quark model. Quantum number assignments are also provided. Our results are compared with the existing experimental data.

Important information on mesons can be extracted from their possible decay modes, including electromagnetic, weak and strong decays. The possibility to provide a theoretical description of strong (open-and hiddenflavor) decays relies mainly on phenomenological models, because the operators that describe the strong transitions between hadrons, arising from non-perturbative QCD, are essentially unknown. In the open-flavor case, they include "hadrodynamic" models, pair-creation models and elementary meson emission models [19].
In this paper, we focus on the 3 P 0 pair-creation model, in which the decays proceed via the production of qq pairs with vacuum quantum numbers, i.e. J P C = 0 ++ , somewhere in the hadronic medium [20]. An important feature of the 3 P 0 model, apart from its simplicity, is that * Corresponding author: santopinto@ge.infn.it it provides the gross features of several transitions with only one free parameter, the pair-creation strength γ 0 , which is a free constant to be fitted to the experimental data. More recent studies have also discussed the possibility of substituting the constant pair-creation vertex of the model with a more refined one [21][22][23][24][25][26]. Extensively applied to the study of open-flavor strong decays of light mesons [21,27,28] and baryons [29][30][31], the 3 P 0 paircreation model has also been used to compute the decays of charmonia [32][33][34][35], bottomonia [36], open-charm [26,37,38] and open-bottom [39] mesons.
The aim of the present paper is to provide a classification of open-charm and open-bottom mesons in terms of their masses, calculated within Godfrey and Isgur's relativized model [40,41], quantum numbers, and openflavor amplitudes, evaluated within a modified version of the 3 P 0 pair-creation model [34][35][36]42]. As widely shown by previous quark model calculations, we expect to obtain a good overall description of the properties of these mesons, with the possible exception of states close to meson-meson decay thresholds, like D * 0 (2400) and D * s0 (2317) [43,44]. Indeed, it is well known that the quenched approximation may fail for states in the region around the opening of meson-meson decay thresholds, where it is believed that continuum-coupling effects play an important role [34,35,[45][46][47][48]. A study of these particular states in the context of coupled-channel models will be addressed in a future publication.
In the 3 P 0 pair-creation model, the open-flavor strong decay of a hadron A into hadrons B and C takes place in its rest frame, via the creation of an additional qq pair characterized by J P C = 0 ++ quantum numbers [20,29,49]. The decay widths A → BC are calculated as  [20, 28, 29] where the coefficient depending on the relative momentum q 0 and energies of the two decay products, E b = M 2 b + q 2 0 and E c = M 2 c + q 2 0 , is the phase space factor for the decay. We assume harmonic oscillator wave functions, depending on a single oscillator parameter α. The final state is characterized by the relative orbital angular momentum ℓ between B and C and a total angular momentum J = J b + J c + ℓ.
Following Refs. [34][35][36]42], we introduce a few changes into the 3 P 0 model operator, T † . These modifications include the substitution of the pair-creation strength, γ 0 , by an effective one [25,[34][35][36]42], with i = n (i.e. u or d), s, c and b (see Tables II and II), to suppress heavy quark pair-creation. Something similar was also done in Ref. [26], though the authors used a different form for γ eff 0 . We also introduce a Gaussian quark form-factor, because the pair of created quarks has an effective size [34][35][36]42].
The values of the pair-creation model parameters for the SU(4) f and SU(5) f sectors, reported in Table II, are extracted from Refs. [34][35][36]. These are the values we use in our calculations.

B. Godfrey and Isgur's relativized quark model
The relativized quark model [40,41] is based on an effective potential, whose dynamics is governed by a onegluon exchange interaction at short distances plus a longrange linear confining one.
The Hamiltonian of the model is given by [40] where m 1 and m 2 are the masses of the constituent quark and antiquark, q their relative momentum (with conjugate coordinate r), V conf , V hyp and V so the confining, hyperfine and spin-orbit potentials, respectively. The confining potential is the sum of three terms [40], with qq| F 1 · F 2 |qq = − 4 3 . The first term is a constant, the second a spin-independent linear confining one, with parameter b, and the third a Coulomb-like interaction. The hyperfine interaction has the standard form [40] The spin-orbit potential [40], is the sum of two contributions, where is the color-magnetic term and the Thomas-precession one.
In the case of states characterized by quark and antiquark of unequal mass, charge conjugation is not a good  [40]. Columns 4-15 show the decay width contributions (in MeV) from various channels, such as Dπ, D * π, and so on. The values of the 3 P0 model parameters are given in Table II. The symbol -in the table means that a certain decay is forbidden by selection rules or that the decay cannot take place because it is below threshold. The calculated mixing angles are: quantum number. Therefore, states with different spins but the same angular momentum, n 1 L J and n 3 L J , can mix via the spin-orbit interaction. For example, this happens in the case of 1 P 1 and 3 P 1 states, where we consider the linear combinations |nP = cos θ nP n 1 P 1 + sin θ nP n 3 P 1 (9a) and |nP ′ = − sin θ nP n 1 P 1 + cos θ nP n 3 P 1 .
The spectrum of open-charm and open-bottom states, obtained by solving the eigenvalue problem of Eq. (4) with the values of the model paramaters of Ref. [40], is reported in Tables III-VII, third column.  Table II and Refs. [34][35][36]. When available, we calculate the amplitudes by using the experimental values of the meson masses, extracted from the PDG [10]; otherwise, we use the relativized QM predictions reported in the third column of Tables III-VII. Finally, our results for charmed, charmed-strange, bottomed, bottomed-strange and bottomed-charm mesons are reported in Tables III, IV, V, VI and VII, respectively.  See also Table I, which shows the existing experimental data for the total widths of D, D s , B and B s resonances. There are no data available for higher B c resonances [10].
Our theoretical results of Tables III-VII reproduce the global trend of the PDG data [10] (see also Table I), with a few exceptions.
In more detail, starting from the D sector, our result for the open-flavor width of the D * (2007) 0 , Γ th of = 4 keV, is compatible with the total experimental width Γ exp tot < 2.1 MeV [10]. A more refined prediction would require the introduction of coupled-channel effects, the mass of the D * (2007) 0 being very close to Dπ threshold. The same applies to D * 0 (2400), where the presence of higher Fock components in the meson wave function may lower the relativized QM prediction for the mass, 2398 MeV, down to the experimental value, 2318 ± 29 MeV, and also contribute to the open-flavor amplitude. In the D 1 (2420) case, which should mainly decay into D * π with the possible chain D * π → Dππ, our 3 P 0 model prediction is compatible with the data, while this is not true for D 1 (2430), being Γ th of ≪ Γ exp tot . Nevertheless, it is worth noting that, in this second case, the experimental error is still very large; moreover, if D 1 (2420) and D 1 (2430) are mixed by spin-orbit forces, their open-flavor widths are likely to be of the same order of magnitude. Our results for the total open-flavor widths of D * 2 (2460) and D 0 (2550) are compatible with the present experimental data, being Γ th of < Γ exp tot ; there is no experimental information on the partial open-flavor widths. Coupled-channel effects may play an important role in the D * 2 (2460) case, which is very close to Dη and D s K thresholds.
Moving to the D s sector, our predictions for D s1 (2460) are compatible with the data [10], while those for D s1 (2536) are not. The former meson has a narrow width and mainly decays to D * s via photon or π 0 emission, which are normally suppressed decay modes [50]. Because of the large mass difference between D s1 (2460) and D s1 (2536), which cannot be explained in terms of hyperfine or spin-orbit splittings, these mesons may have exotic nature. Our results for the total widths of D * s2 (2573), D * s1 (2700), D * s1 (2860) and D * s3 (2860) are compatible with the experimental data [10], being Γ th of < Γ exp tot . We cannot say much on the single channels, as the PDG only   provides some preliminary results for a few branching fractions, except that, in the D * s2 (2573) case, our predictions are compatible with Γ(D * K)/Γ(DK) < 0.33 [10]. In the D * s3 (2860) case, we also show predictions extracted by using the relativized QM mass for the decaying meson because: I) There is a large difference between experimental and calculated masses; II) The experimental data are not very reliable as, at the moment, the state is excluded from the PDG summary table [10].
Finally, we discuss our predictions for the B and B s sectors.
Our results for the total open-flavor widths of B * 2 (5747) and B * s2 (5840) and for the ratio Γ(B * 2 (5747)→B * π) Γ(B * 2 (5747)→Bπ) are compatible with the experimental data [10,51]. By contrast, our result for the open-flavor width of B s1 (5830) is incompatible with the data. Our prediction is very sensitive to the value of the decaying meson mass -as B s1 (5830) is close to the B * K threshold -and thus a few MeV mass difference can produce large deviations in the calculated decay amplitude.

IV. SUMMARY AND CONCLUSION
We computed the open-flavor strong decays of opencharm and open-bottom mesons within a modified version of the 3 P 0 pair-creation model [20,29].
In the 3 P 0 model, the open-flavor decays take place in the rest frame of the initial state, via the production of a light qq pair (i.e. q = u, d or s) with 3 P 0 quantum numbers. Heavy quark pair production is heavily suppressed, as required by the phenomenology, by substituting the pair-creation strength, γ 0 , with an effective one, γ eff 0 [25,[34][35][36]42]. Moreover, the non-point-like nature of the pair of produced quarks is taken into account by  introducing a quark form-factor [34-36, 42, 52-56] into the model transition operator. The values of the 3 P 0 model parameters in the SU f (4) and SU f (5) sectors were extracted from our previous studies on cc [34,35] and bb [35,36] meson spectroscopy and decays, where they were fitted to the existing experimental data [10].
The open-charm and open-bottom meson spectra we needed in our calculation were predicted within Godfrey and Isgur's relativized quark model [40]. This is one of the most powerful tools for the study of qq meson spectroscopy, and provides a description of the meson spectrum in the light, strange, cc, ..., sectors with a universal set of parameters; moreover, 30 years since its formulation, it still gives a good overall description of the experimental data.
As discussed in our previous papers [34][35][36], there may be substantial deviations between the experimental values of the masses and QM predictions [40] in the case of resonances lying close to meson-meson decay thresholds. In these cases, continuum coupling effects may be important and determine a downward energy shift for the bare meson masses, thus improving the fit to the data; coupled-channel effects may also contribute to the open-flavor amplitudes. Such mesons may have an exotic nature, such as tetraquarks, meson-meson molecules or qq mesons plus continuum components. For example, this may be the case of D(1 3 S 1 ), D * 0 (2400) and D * s0 (2317) [43,44], where QM predictions are incompatible with the present experimental data [10]. The possible interpretations for suspected exotic open-charm and open-bottom mesons will be discussed in a future paper.
In conclusion, we think that our predictions can be   Flavor couplings in the 3 P0 pair-creation model In the following, we show how to calculate the SU f (5) flavor couplings of the 3 P 0 pair-creation model. The SU f (4) couplings can be computed analogously.
We consider the transition A → BC, where A, B and C are quark-antiquark mesons. The SU f (5) flavor couplings can be written as the scalar product between initial, |A(q 1q2 )Φ 0 (q 3q4 ) , and final states, |B(q 1q4 )C(q 3q2 ) , where Φ 0 is the SU f (5) flavor singlet and n f = 5 is the dimension of the SU f flavor group. In general, two different diagrams can contribute to the flavor matrix element BC|AΦ 0 : in the first one, the quark in A ends up in B, while in the second one it ends up in C. In the majority of cases, one of these two diagrams vanishes; however, for some matrix elements, both must be taken into account [34][35][36]42]. Finally, the flavor matrix elements can be calculated as: As an example, we calculate the B 0 → B 0 π 0 flavor coupling. The flavor matrix element can be written as .