Regge trajectories for heavy quarkonia from the quadratic form of the spinless Salpeter-type equation

In this paper, we present one new form of the Regge trajectories for heavy quarkonia which is obtained from the quadratic form of the spinless Salpeter-type equation (QSSE) by employing the Bohr-Sommerfeld quantization approach. The obtained Regge trajectories take the parameterized form $$M^2={\beta }({c_l}l+{\pi }n_r+c_0)^{2/3}+c_1$$M2=β(cll+πnr+c0)2/3+c1, which are different from the present Regge trajectories. Then we apply the obtained Regge trajectories to fit the spectra of charmonia and bottomonia. The fitted Regge trajectories are in good agreement with the experimental data and the theoretical predictions.


Introduction
The Regge theory [1,2] is based on Lorentz invariance, unitary and analyticity of the scattering matrix and has nothing with quarks and gluons. Due to its generality, the Regge theory is still an indispensable tool in phenomenological studies. The Regge theory is concerned with almost all aspects of strong interactions. One of the most distinctive features of the Regge theory is the Regge trajectory which is one of the effective approaches for studying hadron spectra [3][4][5][6][7].
It is known that the Regge trajectories for the hadrons constituting of light quarks are approximately linear [8][9][10][11][12][13][14][15][16], that is, the mass M is related to the spin J and the principle quantum number n by where α and β are the Regge slopes which are almost equal for light mesons. For heavy mesons, α will be smaller than β [6,[17][18][19]. In Ref. [20], the authors show that the Regge trajectories are nonlinear in case of both fixed and running coupling constants by using the perturbative QCD. In Ref. [21], the authors show by analyzing the data from the UA8 a e-mails: chenjk@sxnu.edu.cn; chenjkphy@yahoo.com and ISR experiments at CERN that the effective Pomeron Regge trajectories require a term quadratic in t. In Ref. [22], the authors scrutinize the hadronic Regge trajectories in the framework of the string model and the potential model, and recognize that the Regge trajectories for mesons and baryons are not straight and parallel lines in general in the current resonance region both experimentally and theoretically. In Ref. [23], the authors show that meson trajectories are nonlinear and intersecting by fitting experimental data. In Ref. [6], the authors argue that the hadronic Regge trajectories are essentially nonlinear and can be well approximated by a specific square-root form. In Refs. [24,25], the quasilinear Regge trajectories for heavy mesons are discussed. In Refs. [26][27][28], the authors obtain an interpolating mass formula which gives the nonlinear Regge trajectories. The idea of nonlinear Regge trajectories is not new [29][30][31][32][33][34][35]. Different models and different potentials will effect the Regge trajectories [6,36,37]. Different from the massless string model which produces the linear Regge trajectories, the relativistic string with massive ends will result in the complicated behavior of the Regge trajectories [38][39][40][41]. In Ref. [42], the authors obtain the nonlinear Regge trajectories from the gauge/string correspondence. In Ref. [6], the authors present a phenomenological string model for squareroot Regge trajectories. In Refs. [43][44][45], the square-root trajectory is obtained by combining both Regge poles and cuts. The Schrödinger equation with the Cornell potential produces the Regge trajectories M 2 ∼βl 4/3 , and the linear Regge trajectories demands the confining potential rising as r 2/3 [46]. In Ref. [47], the authors show that the Regge trajectories become linear for the Dirac equation with the linear confinement for large orbital angular momenta. The Regge trajectories for the Klein-Gordon equation are considered in Refs. [48][49][50]. In Ref. [51], the authors obtained the linear, non-intersecting and parallel Regge trajectories by calculating the numerical solutions of the relativistic Thompson equation. In Ref. [52], the authors obtain the nonlinear Regge trajectories by including spin dependent terms in a three-dimensional reduction of the Bethe-Salpeter equation. In case of the spinless Salpeter equation, the Cornell potential or the linear potential can yield linear Regge trajectories [53,54]. In Ref. [55], the authors find straight Regge trajectories in case of light-light quarkonia starting from a first principle Salpeter equation. In Ref. [56], the authors discuss the Regge trajectories by solving the eigenvalue equation with the quadratic mass operator. In this paper, we obtain one new form of the Regge trajectories from the quadratic form of the spinless Salpeter-type equation (QSSE) [57][58][59][60] with the Cornell potential or the linear potential. As the obtained new Regge trajectories employed to the spectra of charmonia and bottomonia, the fitted data are in very good agreement with the experimental data and theoretical predictions.
This paper is organized as follows. In Sect. 2, the new form of the Regge trajectories is obtained from the quadratic form of the spinless Salpeter-type equation with the Cornell potential and the power-law potentials. In Sect. 3, the obtained Regge trajectory is applied to fit the spectra of bottomonia and charmonia. We conclude in Sect. 4.

Regge trajectories from the quadratic form of the spinless Salpeter-type equation
In this section, we review briefly the QSSE at first. Then we present the orbital and radial Regge trajectory formulas obtained from the QSSE by employing the Bohr-Sommerfeld quantization approach [5,61], which are different from the known Regge trajectories.

QSSE
The Bethe-Salpeter equation [62,63] is based on the relativistic field theory and is an appropriate tool to deal with bound states. In Ref. [64], a first principal Bethe-Salpeter equation is obtained by using the only assumption that ln W (W being the Wilson loop correlator) can be written in QCD as the sum of its perturbative expression and an area term. By means of a three dimensional reduction, the Bethe-Salpeter equation becomes the eigenvalue equation for the square mass operator [56][57][58]64,65] where where ω i is the square-root operator of the relativistic kinetic energy of constituent m i are effective masses in the phenomenological model. The relativistic corrections with U are small for heavy constituents and will make calculations complicated. Just like the case of the spinless Salpeter equation [66][67][68][69][70], we assume   that U in Eq. (4) can be taken to be nonrelativistic for simplicity although the kinetic energies of the constituents are taken to be relativistic. In the nonrelativistic limit, the configuration-space U reduces to where V (r ) is the usual nonrelativistic potential. In this paper, the Cornell potential [71,72] and the power-law potentials are considered, 9982 9990 9976 9990 9990 10,804 10,790 In Ref. [55], the authors find straight Regge trajectories for the light-light case starting from a first principle Salpeter equation. As a well-defined standard approximation to the Bethe-Salpeter equation, the spinless Salpeter equation [73][74][75] yields the linear Regge trajectories for the linear potential. In Ref. [56], the Regge trajectories for the light-light quarkonium systems are calculated by using both the linear mass operator and the quadratic mass operator, respectively. The authors notice the differences between two kinds of Regge trajectories. In the following, we will propose one kind of nonlinear Regge trajectories from the QSSE [Eq. (4)] which has a quadratic mass operator.

Bohr-Sommerfeld quantization approach
In this subsection, we follow Ref. [5] to employ the Bohr-Sommerfeld quantization approach [61] to obtain the orbital Regge trajectories and the radial Regge trajectories from the QSSE.
In case of equally massive constituents (m 1 = m 2 = m), the quadratic mass operator of the QSSE with the Cornell potential reads from Eq. (4)  where Due to the central interaction, the orbital angular momentum p φ = L is a constant of the motion. From the conservation of the total energy M of the bound state, the radial momentum p r can be written as Solving g(r ) = 0 gives the three roots of g(r ). r j ( j = 0, 1, 2) are where As r 1 ≤ 0 ≤ r 2 ≤ r 0 , the turning points are r − = r 2 and r + = r 0 . The radial motion takes place between two turning points, r − and r + . The action variables in the Bohr-Sommerfeld quantization approach [61] read where s is the degrees of freedom of the bound state, q s and p s are the coordinates and conjugate momenta. The integral is performed over one cycle of the motion. The action variables should be quantized, where h is the Planck constant, n s (≥ 0) is an integral quantum number. c s is a real constant, c s = 1/2 [76]. Using Eqs. (13) and (14) to quantize J φ gives L = l + c φ , c φ = 1/2. Quantization of J r yields where n = n r + c r = n r + In the above equation, n r is the radial quantum number. K (x), E(x) and Π(π/2, x, y) are the complete elliptic integrals of the first, second and third kind, respectively [77]. As angular momenta becomes large, l n r , r − ≈ r + and η ≈ 0. Then, θ = π , K (η) = E(η) = Π(π/2, γ, η) = π/2. Keeping the leading terms in Eq. (15), the Regge trajectories can be obtained As the radial quantum number becomes large, i.e., n r l, In case of the power-law potential V (r ) = σ r a (a > 0), the orbital Regge trajectories for large l are where β l (a) reads In Ref. [52], the Regge trajectories similar to Eq. (19) are obtained under the semiclassical limit, however, the authors did not discuss in detail the linear potential case. As n r becomes large, we have the radial Regge trajectories from Eq. (13) The Regge slope is where B(x, y) is the beta function [77].
The new form of the Regge trajectories [Eqs. (17) and (18)] are obtained from the QSSE (4) by taking the nonrelativistic approximation (6) and neglecting the spin dependent interactions. If higher corrections are considered, the form of the Regge trajectories will become complicated. Moreover, the employed Bohr-Sommerfeld quantization approach is a semiclassical method and is used as an approximation method. Therefore, the obtained Regge trajectories are where And the orbital Regge trajectories (17) can be extended to be By considering Eqs. (23) and (25) and fitting experimental data, we assume that the heavy quarkonium Regge trajectory take the parameterized form In Eq. (26), β is an universal parameter which has a theoretical value according to Eq.

Regge trajectories from the Schrödinger equation, the spinless Salpeter equation and the QSSE
There are many discussions on the Regge trajectories in the framework of the Bethe-Salpeter equation and many reductions of it (see the Introduction), however, most of these discussions rely on the numerical solutions and the explicit formula of the Regge trajectories is not given due to the complexity of these equations. In this subsection, we list the explicit formulas of the Regge trajectories for the equations which can be handled easily.
For simplicity, the power-law potentials are considered in this subsection, see Eq. (7). The nonrelativistic Schrödinger equation reads The radial and orbital Regge trajectories obtained from Eq. (27) read [5,46], respectively, M 2 ∼l 4a/(a+2) , for large l, where M = m 1 + m 2 + E. As a relativistic expansion of the Schrödinger equation and a well-defined standard approximation to the Bethe-Salpeter equation, the semirelativistic spinless Salpeter equation reads The radial and orbital Regge trajectories obtained from Eq. (29) read [5,53,54], respectively, In case of the QSSE, the radial and orbital Regge trajectories read from Eqs. (19) and (21), respectively, For the Schrödinger equation, the linear Regge trajectories need the confining potential rising as r 2/3 ; for the spinless Salpeter equation, the linear Regge trajectories demand the linear confining potential; for the QSSE, the linear Regge trajectories need the harmonic-oscillator-type potential. From Eqs. (28), (30) and (31), we can see that the exponents in the radial and orbital Regge trajectories diminish as the bound-state equations vary from the Schrödinger equation to the spinless Salpeter equation and then to the QSSE. Eqs. (28), (30) and (31) show that different dynamic equations will result in different Regge trajectories although both Eqs. (8) and (29) are reduced from the Bethe-Salpeter equation and the Schrödinger equation can be obtained from Eqs. (8) and (29) by nonrelativistic approximation. The good agreement of the fitted data with the experimental data (see Sect. 3) suggests that the QSSE and the eigenvalue equation for the square mass operator [Eq. (2)] will be better than the Schrödinger equation and the spinless Salpeter equation for heavy quarkonia.

Regge trajectories and mass spectra of heavy quarkonia
In this section, we employ the obtained new form of the Regge trajectories to fit the spectra of charmonium and bottomonium. Experimental data are used as inputs to determine the parameters of the Regge trajectories. Then the fitted Regge trajectories can predict the masses of the unobserved states. In this work, the linear Regge trajectories are also fitted by the experimental data and we find that the new form of the Regge trajectories are better than the linear ones.

Nonlinear Regge trajectories
For light mesons, the Regge trajectories are suggested to be linear although the parameters take different values [78][79][80][81][82][83]. By fitting the experimental data, we find that the new Regge trajectory is better than the linear one even for π mesons, see Fig. 1 [84]. For heavy quarkonia especially for bottomonia, the Regge trajectories deviate significantly from the linearity. The differences between the results obtained by the linear and the results obtained by the nonlinear Regge trajectories are shown in Figs 3, 4, 6 and 7.
For the baryon Regge trajectories within the quarkdiquark models [85], the radial Regge trajectories for the heavy, light vector mesons [86], the Regge-like relation for the heavy-light systems [87], and the light meson Regge trajectories [8,9,88], the linear confining potential plays dominant role. Similarly, the linear confining potential also dominates the new form of the nonlinear Regge trajectories (26), which can be seen from the derivation in the subsection 2.2. The good fitting of the experimental data by the Regge trajectory (26) confirms this observation and suggests that the contributions of the color Coulomb potential to the heavy quarkonium Regge trajectories is subordinate.

Masses of Bottomonia
Using the experimental data of the well-established states, the parameters of the radial Regge trajectories, the orbital Regge trajectories and the linear Regge trajectories are determined. Using the masses of pseudoscalar mesons η b (nS) and the masses of the vector mesons Υ (nS), the universal parameter β is calculated, β = 5.1 GeV 2 for bottomonia which is consistent with the theoretical value. According to Eq. (24), there is a relation between the heavy quark mass m and the string tension σ . The chosen value of β for bottomonium is reasonable because the fitted β can be obtained by reasonable m b and σ , see  Tables 1 and 2 and they are in good agreement with other theoretical predictions.

Masses of charmonia
The universal parameter β for the radial charmonium  Tables 3 and 4 and they are in agreement with other theoretical predictions.
X (3915) is usually identified as the χ c0 (2 p) of charmonium based on its mass and likely J PC [130,145] and this assignment has some problems [146].
In Fig 6c and

Conclusions
In this paper, one new form of the Regge trajectories is proposed based on the quadratic form of the spinless Salpeter-type equation and the Bohr-Sommerfeld quantization method. The obtained Regge trajectories for the Cornell potential or for the linear potential take the parameterized form M 2 = β (c l l + π n r + c 0 ) 2/3 + c 1 . The parameters β and c l and the coefficient π before n r are universal for bottomonia and charmoniua, respectively. It is shown that the experimental data are very well described by the radial Regge trajectories and the orbital Regge trajectories for both bottomonia and charmonia. The good agreement of the fitted data with the experimental data suggests that the QSSE and the eigenvalue equation for the square mass operator [Eq.
One distinctive feature of the new form of the Regge trajectories is that the new form of the heavy quarkonium Regge trajectories and the curvature are determined by the quadratic form of mass operator and the linear potential for long distances which has been validated by lattice QCD calculations, while the contributions of the color Coulomb potential for short distances to the Regge trajectories for heavy quarkonia will be subordinate to that of the linear confining potential.
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