Regge trajectories for the mesons consisting of different quarks

By applying the Bohr–Sommerfeld quantization approach to the quadratic form of the spinless Salpeter-type equation (QSSE), we show that the obtained Regge trajectories for the mesons consisting of unequally massive quarks take the form $$M^2=\beta \left( {c_l}l+{\pi }n_r+c_0\right) ^{2/3}+c_1$$M2=βcll+πnr+c02/3+c1, which have the same form as the Regge trajectories for charmonia and bottomonia. Then we apply the obtained Regge trajectories to fit the spectra of the strange mesons, the heavy-light mesons (the D, $$D_s$$Ds, B and $$B_s$$Bs mesons) and the bottom-charmed mesons. The fitted Regge trajectories are in agreement with the experimental data and the theoretical predictions, which demonstrates that the newly proposed Regge trajectories can be applied universally to the light mesons, the heavy-light mesons and the heavy mesons. By fitting the spectra of the mesons composed of different quarks, the concavity of these Regge trajectories are illustrated, which is of cardinal significance for the potential models.


Introduction
The Regge trajectory is one of the effective approaches for studying hadron spectra . For one newly proposed Regge trajectories, checking its universality is an important and necessary task. In Ref. [23], the authors applied the rotational states of the string with massive ends including the spin-orbit correction to describe the Regge trajectories for the light, strange, charmed, bottom mesons, and for the Δ/ /Λ/Λ c baryons. In Ref. [24], the authors used the Regge-like formula (M − m Q ) 2 = πσ L to analyze all the heavy-light mesons and the baryons which can be assumed consisting of one heavy quark and one light cluster of two light quarks.
In Ref. [25], we proposed one new form of the Regge trajectories, a e-mails: chenjk@sxnu.edu.cn;chenjkphy@outlook.com M 2 = β (c l l + π n r + c 0 ) 2/3 + c 1 , where M is the meson mass, l is the orbital angular momentum, n r is the radial quantum number. β and c l are universal parameters. c 0 and c 1 vary with different trajectories. As applying the formula (1) to fit the bottomonia and charmonia, the results are excellent. In the present work, we use the quadratic form of the spinless Salpeter-type equation (QSSE) [26][27][28][29][30][31][32][33] to discuss the Regge trajectories for the mesons consisting of different quarks. We find that the obtained formula can be written in the same form as the Regge trajectories in Eq. (1). The fitted Regge trajectories agree with the experimental data and the theoretical values. By fitting data, we notice that the Regge trajectories for these meson are concave, which is of cardinal significance for the potential models because this property of the Regge trajectories can assist in the choice of the appropriate dynamic equation and potential to describe mesons. This paper is organized as follows. In Sect. 2, the new form of the Regge trajectories for mesons composed of different quarks is obtained from the QSSE, which has the same form as that for heavy quarkonia. In Sect. 3, the obtained Regge trajectory is applied to fit the spectra of the strange mesons, the heavy-light mesons and the bottom-charmed mesons. In Sect. 4, the universality and concavity of the newly proposed Regge trajectories are discussed. We conclude in Sect. 5.

Regge trajectories from the QSSE
In this section, the QSSE is briefly reviewed at first. Then the orbital and radial Regge trajectories for the mesons consisting of unequally massive quarks are obtained from the QSSE by employing the Bohr-Sommerfeld quantization approach [13,34], which have the same form as the Regge trajectories for heavy quarkonia obtained in Ref. [25].

QSSE
It is well known that the Bethe-Salpeter equation [35,36] is an appropriate tool to deal with bound states. In Ref. [27], the authors obtained a first principal Bethe-Salpeter equation, and then reduced it to the eigenvalue equation for the square mass operator [26][27][28][29][30] by means of a three dimensional reduction where In the above equations, M is the bound state mass, p the c.m. momentum of quarks, m 1 and m 2 their constituent masses, is the instantaneous kernel. Neglecting any reference to the spin degrees of freedom of the involved bound-state constituents, Eq. (2) reduces to the QSSE which is written in configuration space as [29][30][31][32][33] where ω i is the square-root operator of the relativistic kinetic energy of constituent Δ is the Laplacian. Let m 1 ≥m 2 , there is the inequality for In case of |p| m 1 , m 2 , there is In case of |p| m 1 , m 2 , there is For simplicity, we assume that U takes the following form which is a variant of the well-known Cornell potential [37].
A and B vary with the discussed mesons. For example, A = 2(m b +m c )α and B = 2(m b +m c )σ for the bottom-charmed mesons, where m c and m b are the charm quark mass and the bottom quark mass, respectively. α = α s 4/3, α s is the strong coupling constant of the color Coulomb interaction. σ is the string tension.

Regge trajectories for the mesons composed of different quarks
Due to the ω 1 ω 2 term, the Regge trajectories can not be obtained directly from Eq. (4). Using Eqs. (4), (6) and (7), we propose two auxiliary equations where Following Refs. [13,25] to employ the Bohr-Sommerfeld quantization approach [34], the orbital Regge trajectories and the radial Regge trajectories can be easily obtained from Eqs. (8), In case of the power-law potential V (r ) = Br a (a > 0), the orbital Regge trajectories for large l are where β l (a) reads The radial Regge trajectories are The Regge slope is where B(x, y) is the beta function [38].

Regge trajectories for the mesons constituting of different quarks
In this section, we employ the Regge trajectory formula M 2 = β (c l l + π n r + c 0 ) 2/3 +c 1 [Eq. (1)] to fit the spectra of the strange mesons, the heavy-light mesons and the bottomcharmed mesons. The parameters of the Regge trajectories are obtained by fitting the experimental data or theoretical values. The fitted radial and orbital Regge trajectories are in agreement with the experimental data or the theoretical predictions. The linear Regge trajectories are also given and we find that the new form of the Regge trajectories are better than the linear ones.

Strange mesons
The universal parameter β is calculated by fitting the radial  Table 1 and they are in good agreement with the experimental data and the theoretical values.

Charmed mesons
By fitting the radial Regge trajectories for D 0 and D * (2007) 0 , the universal parameter β is calculated, β = 1.89 for the charmed mesons. c l is obtained by fitting the orbital Regge trajectories for D 0 , D * (2007) 0 and D * 0 (2400) 0 , c l = 3.02. c 0 and c 1 vary with different Regge trajectories. In Fig. 3 are the fitted radial and orbital Regge trajectories for the charmed mesons. The fitted and predicted masses of the charmed mesons by the Regge trajectories are listed in Table 2 and     Table 2. The well-established states are given by solid dots. The unwell-established states or the states needing confirmation are given by circles. Open squares are predicted masses by the fitted Regge trajectories they are consistent with the experimental data and the theoretical values. D(2550) is taken as a candidate for the 2 1 S 0 state by the helicity distribution analysis [45], and D J (2580) is assumed to be the same state because the resonance parameters of D J (2580) and D(2550) are similar and BaBar assignment [45] is consistent with the LHCb results [46]. D J (3000) is a possible candidate for the 2P state [47], the 3 1 S 0 state [48], or the 3 + state [49]. D * (2600) can be assigned to be the 2 3 S 1 state [48,50], see Ref. [51] for more discussions. D * J (3000) 0 can be the candidate of the 3 3 S 1 state [51]. D * 2 (3000) is possibly the 3 3 P 2 state while the assignment of the 2 3 F 2 state can not be fully excluded [52]. D(2750) 0 is assigned as the 1D 2 state [53,54]. D * 1 (2760) and D * 3 (2760) are assumed to be the 1 3 D 1 state and 1 3 D 3 state [53,55-59], respectively.

Charmed-strange mesons
Due to insufficient experimental data, the radial Regge trajectory for the charmed-strange mesons are obtained by fitting the theoretical values in Ref. [60], β = 1.75, see Fig. 4a. The orbital Regge trajectories by fitting the experimental data and by fitting the theoretical values in Ref. [60] are in consistence, see Fig. 4b. The universal parameter is calculated, c l = 2.44. The experimental data and the theoretical predictions are listed in Table 3. Table 2 Masses of the charmed mesons (in MeV). The experimental data are from PDG [39]. The theoretical values are from EFG [16], GM [53], KDR [54] and LNR [60]. ? denotes the possible candidates for the (2700) ± is taken as the 2 1 S 0 state in Ref. [65] or is suggested to be a mixture of the 2 3 S 1 and 1 3 D 1 states [66][67][68]. In Refs. [57,[69][70][71], D * s1 (2860) and D * s3 (2860) are taken as the 1 3 D 1 state and the 1 3 D 3 state, respectively. In Ref. [72], D * s1 (2860) is interpreted as a mixture of D s (2 3 S 1 ) and D s (1 3 D 1 ).  Table 4.
For the bottom-strange mesons, β = 1.75 and c l = 2.44. The radial and orbital Regge trajectories are given by fitting the theoretical predictions in Ref. [60], see Fig. 6. The experimental and theoretical data are in Table 5.  [16] (EFG), [53] (GM), [54] (KDR) and [60] (LNR). The experimental data (Exp) are from Ref. [39]. The dashed lines are the fitted Regge trajectories for the theoretical values in [60] while the solid line is for the experimental data from [39] Table 3 Masses of the charmed-strange mesons (in MeV). The experimental data are from PDG [39]. The theoretical values are from EFG [16], GM [53], KDR [54] and LNR [60]. ? denotes the possible candidates for the unwell-established states or the mesons needing confirmation By fitting the theoretical values in Ref. [79], we obtain the Regge trajectories for the bottom-charmed mesons, β = 3.49 and c l = 1.75, see Fig. 7. In Table 6 are the experimental and theoretical values.

Discussions
In this section, two important properties, the universality and the concavity, of the Regge trajectories are discussed.     [16] (EFG), [89] (GMS), [87] (KDR) and [60] (LNR). The experimental data (Exp) are from Ref. [39]. The dashed lines are the Regge trajectories by fitting the theoretical values in [60] mesons, for example, see Fig. 8. Therefore, we conclude that the new form the Regge trajectories [Eq. (1)] can be appropriate not only for the heavy mesons but also for the light mesons, not only for the mesons composed of the same quarks but also for the mesons composed of different quarks, that is to say, Eq. (1) is expected to be universal for mesons.
The parameters β n r of the radial Regge trajectories and β l of the orbital Regge trajectories are listed in Tables 7 and  8, respectively. For mesons except for the light mesons, β n r Table 5 Masses of the bottom mesons (in MeV). The experimental data are from PDG [39]. The theoretical values are from EFG [16], GMS [89], LNR [60] and KDR [87]. ? denotes the possible candidates for the unwell-established states or the mesons needing confirmation  The theoretical data are from Ref. [90] (EFG), [91] (SJSCP), [92] (DKR) and [79] (LNR). The experimental data (Exp) are from Ref. [39]. The dashed lines are the Regge trajectories by fitting the theoretical values in [79]  and β l increase with the quark mass, and there is the relation β n r > β l . The exception of the light mesons maybe arises from the complexity of the light mesons.

Conclusions
In this paper, we obtain the radial and orbital Regge trajectories for the mesons composed of unequally massive quarks from the QSSE by employing the Bohr-Sommerfeld quantization approach. The obtained Regge trajectories have the same form as the Regge trajectories for heavy quarkonia [25], M 2 = β (c l l + π n r + c 0 ) 2/3 + c 1 . By fitting the spectra of the strange mesons, the heavy-light mesons (the D, D s , B and B s mesons) and the bottom-charmed mesons, we show that the fitted Regge trajectories are in agreement with the experimental data and the theoretical predictions. By combining the results in Ref. [25] and in the present work, we expect that the new form of the Regge trajectories will be universal for mesons.
By fitting the spectra of mesons, the concavity of these Regge trajectories is illustrated. This property is of cardinal significance for the potential models because it can assist in the choice of the appropriate dynamic equation or the appropriate potential to describe mesons. If the confinement potential is linear, the Schrödinger equation arouses the convex Regge trajectories while the Dirac equation, the Klein-Gordon equation and the spinless Salpeter equation will produce the linear Regge trajectories. The QSSE [Eq. (4)] and the eigenvalue equation for the square mass operator [Eq. (2)] can give the concave Regge trajectories. Therefore, the QSSE and the eigenvalue equation for the square mass operator are preferred if the confinement potential is linear. If the confinement potential takes the form r 0.1 or r 1/2 , all the dynamic equations mentioned above can yield the concave Regge trajectories. In this case, more information is needed to choose the appropriate dynamic equation and the corresponding potential.
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