The Regge Trajectories and Leptonic Widths of the Vector ss¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{s}\bar{\varvec{s}}$$\end{document} Mesons

The spectrum of the ss¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\bar{s}$$\end{document} mesons is studied performing a phenomenological analysis of the Regge trajectories defined for the excitation energies. For the ϕ(33S1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (3\,^3S_1)$$\end{document} state the mass M(ϕ(3S))=2100(20)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\phi (3S))=2100(20)$$\end{document} MeV and the leptonic width Γee(ϕ(3S))=0.27(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma _{ee}(\phi (3S))=0.27(2)$$\end{document} keV are obtained, while the mass of the 23D1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\,^3D_1$$\end{document} state, M(ϕ(23D1))=2180(5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\phi (2\,^3D_1))=2180(5)$$\end{document} MeV, appears to be in agreement with the mass of the ϕ(2170)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (2170)$$\end{document} resonance, and its leptonic width, Γee(23D1)=0.20±0.10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varGamma _{ee}(2\,^3D_1)=0.20\pm 0.10$$\end{document} keV, has a large theoretical uncertainty, depending on the parameters of the flattened confining potential.

The masses of the φ(nS) and φ(n 3 D 1 ) states (in MeV) State GI [7] EFG [8] M G I [ 2] Experiment [16]  in Ref. [2] and given in Table 1, where MGI refers to the modified Godfrey-Isgur model with the screened confining potential [2]. From Table 1 one can see that in all RPMs with constituent quark masses, the mass M(φ(3S)) appears to be by (120-200) MeV larger than the mass of X (2000) with J P = 1 − in experiment. Also in RPMs the first excitation φ(2D) has a mass larger than the mass of φ(2170) by ∼ 100 MeV, although in the MGI model, where a screened confining potential (CP) is used, the masses of the φ(3S) and φ(2D) states are about 50 MeV smaller than those in the GI model [7], where the purely linear CP is used. Note that for φ(4S) and φ(3D) the GI model predicts values of the masses, which are already by ∼ 120 MeV larger, which means that within the same model the choice of the parameters of the confining potential at large distances is crucially important to describe higher excitations. The situation is different, if an analysis of the spectrum is performed with the help of Regge trajectories, defined for excitation energies, denoted as ERT [17], where the parameters of the ERT can be extracted from experiment and the predicted masses of high excitations appear to be smaller than those in RPMs. In our paper we perform a phenomenological analysis of the φ(nS) and φ(n D) resonances, using the ERT, introduced by Afonin and Pusenkov [13][14][15], as it was done in the analysis of the heavy-quarkonia spectra in Ref. [17]. We will also discuss how the parameters of the ERT depend on the mass of the s-quark.

The Radial ERT of φ(nS) Mesons
In heavy quarkonia, as well as in the ss system, the ERT are defined for the excitation energies E(n J ) [13][14][15], First, we consider the φ(n 3 S 1 ) radial trajectory, which needs special consideration since its radial slope is larger than that of the ERT for the states with l = 0 [17]; this effect is seen in the RT of light mesons [9][10][11][12][18][19][20][21], as well as in heavy quarkonia [17]. The reason why the radial slope is larger in the S-wave mesons is explained by the stronger gluon-exchange (GE) interaction in the states with l = 0 than in those with l = 0 [22]. The radial ERT of φ(nS) can be presented as, where n r is the radial quantum number. The s-quark mass m s , the intercept a S , and the slope b S can be extracted from experiment, if there are enough experimental data on the φ(nS) excitations, measured with great accuracy. However, the existing experimental data do not allow to extract m s at low scale and here we take m s at low scale, using the relation for the running mass in pQCD [23,24] and the conventional value of m s (q = 2 GeV/c) = 96(5) MeV [16], defined at the scale q = 2 GeV/c. The sizes of high ss mesons are large, > 1.0 fm, and therefore their dynamics are determined by small characteristic momenta, q < ∼ 1 GeV. The following mass relations, m s (q = 1 GeV/c) = 1.  Table 2). In Table 1 for comparison we give also the masses M(φ(nS)) from Table 2 The masses of φ(nS) (in GeV) from the ERT and the parameters a S , b S (in GeV 2 ) for m s = 0.125 GeV and m s = 0.180 GeV and the masses from Ref. [2] and from experiment [16]  Ref. [2], where the MGI model with screened confining potential is used, giving the masses of the 3S and 4S excitations by ∼ 50 MeV larger than those in our analysis of the ERT. From Table 1 one can see that the masses, defined by the ERT with m s = 125 MeV and m s = 180 MeV, coincide within ∼ 10 MeV accuracy for φ(1S) and φ(2S) and differ only by ∼ 20 MeV for the higher 3S and 4S excitations, although their intercepts and the radial slopes are different. Notice that for a smaller m s = 125 MeV the values of the intercept and the radial slope are practically equal to those for light mesons [12,[20][21][22], while for m s = 180 MeV they are closer to the values in heavy quarkonia [17].
Notice that in our analysis, where M(φ(3S)) = 2102 MeV, as well as in RPMs (see Table 1), the mass of φ(3 3 S 1 ) is larger than the experimental mass of X (2000) [1]. However, one cannot exclude a large hadronic shift-down of the 3 3 S 1 state due to the P−wave φφ threshold (with M(thresh.) = 2039 MeV) and then this state could be a candidate to be the X (2000) resonance, as it is assumed in Ref. [2].

The Generalized ERT of the ss Resonances
The experimental masses of the ground states φ(1S), f 2 (1525) and φ 3 (1850) (J = 1, 2, 3) allow to define the orbital parameter b J of the leading ERT with J = l + 1, n r = 0: First, we take m s = 180 MeV and by definition of the ERT the mass differences can be written as giving the following intercept and orbital slope of the leading ERT, a = − 0.4644 GeV 2 ; b J = 0.905 GeV 2 , (m s = 0.180 GeV, J = l + 1).
From here the masses of the ss states with J = l + 1 are following, To describe the radial ss excitations with l = 0, we assume that f 2 (1950) is an ss state, but not assuming a priori that φ(2170) is the first excitation of φ 3 , since its quark structure is still discussed [25,26]. Then the Table 3 The masses (in MeV) of the ss states with l = 0 from ERT Eq. (7) State n r = 0 n r = 1 n r = 2 radial slope of the ERT (for the states with l = 0) can be extracted from experiment (the parameters a, b J are given in Eq. (5)), using the mass difference, From here the radial slope b n = (1.15 ± 0.05) GeV 2 is extracted and the generalized ERT, gives the masses, presented in Table 3, where besides the masses of the n 3 D 3 , the masses of the n 3 D 1 states are given, which are by ∼ 15 (5) MeV smaller due to the fine-structure splitting. In our calculations the mass M(2 3 D 1 ) coincides with that of φ(2070) and this fact indicates that φ(2070) can have large ss component, as it was assumed in Ref. [26].

The Leptonic Widths
To define the leptonic widths of the φ(1020), φ(1680), φ(1 3 D 1 ), φ(2 3 D 1 ) their wave functions (w.f.s) are calculated, using the relativistic string Hamiltonian (RSH) [18,19,22,27,28] without fitting parameters, where the centroid mass M cog (nl) is defined by the eigenvalues M 0 (nl) of the equation, and by two negative corrections, the self-energy and the string corrections [18,19,22,29,30]. Here m s = 180 MeV and the potential V 0 (r ) is taken as the sum of the confining potential and the gluon-exchange (GE) term, V GE = − 4α V (r ) 3r , with the parameters from Ref. [22]. Since the ground states have relatively small sizes, their dynamics is defined by the linear CP, V c (r ) = σ r = 0.18 r GeV, while for high excitations, which have large sizes, a flattened CP V f (r ), has to be taken [22]. We assume that the self-energy and the string correction do not affect the w.f.s and the radial w.f.s at the origin R nS (0) and R n D (0), which are defined by the solutions ϕ nl (r = 0) of Eq. (8), where for the n 3 D 1 states the w.f. R n D (0) is expressed via the derivative R n D (0) [31,32], Here the flattened CP V f (r ) is chosen as in Ref. [22] with the string tension, σ f (r ) = σ (1 − γ f (r )) and the function f (r ), defined by the following parameters,  (12) If the flattened CP+GE term is used, then the sizes of the nS and n D excitations increase, see Table 4, where also the w.f.s R nS (0), R n D (0) and the kinetic energies ω(nL) of the s−quark, entering R n D (0), are given. We have observed an interesting effect: if a flattened CP is used, then the kinetic energies ω(nl) and the w.f. R nS (0) (n r ≥ 1) decrease, while the second derivatives R n D (0) increases. Consequently, the leptonic width of the φ(n D) also increases. The leptonic width of a vector ss meson with the mass M V (nl) (l = 0, 2) (the charge squared e 2 s = 1/9, α = (137) −1 ) is given by the expression [24], where the factor β rel ∼ = (0.72−0.74) takes into account the relativistic effects [24], and β QCD = 1 − 16α s (μ) 3π = 0.40 is the QCD correction, where the strong coupling α s (μ) = 0.353 at the scale μ ∼ 1.4 GeV is taken.
In the GE potential we use the vector coupling constant, which does not contain fitting parameters and takes into account the asymptotic freedom behavior [33], so that the effective coupling of the ground state α V (eff.) = 0.39 is relatively small, while α V = 0.54 is larger for excited states with n r ≥ 2. Details can be found in Ref. [22], where the vector coupling α v (n f = 3) is shown to be defined via the QCD vector constant Λ V (n f = 3) = 0.455 GeV, which corresponds to the QCD constant Λ M S (n f = 3) = 330 MeV from Ref. [34].

Conclusions
The spectrum of the ss mesons was studied with the use of the ERT trajectories, defined for the excitation energies, E(n J ) = M(n J ) − 2m s [13][14][15]. It is shown that the parameters of the ERT depend on the value of the s-quark mass at a low scale. Two values, m s = 125 MeV and m s = 180 MeV, are considered. In both cases the calculated masses coincide within (10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) MeV accuracy, although for m s = 125 MeV the slope b(nS) = 1.465 GeV 2 and the intercept a(nS) = 0.593 GeV 2 are larger than those for m s = 180 MeV, and equal to the parameters of the ρ(nS) RT. If m s = 180 MeV is taken, the values b n (nS) = 1.30 GeV 2 and a(nS) = 0.4356 GeV 2 are smaller and close to those in heavy quarkonia [17]. For φ(3S) the leptonic width Γ ee = 0.42(2) keV is obtained.
With the use of the ERT the predicted masses of the high excitations appear to be smaller than those calculated in potential models with a constituent s-quark mass. For φ(3S) the calculated mass M(φ(3S)) = 2100 (20) MeV is larger than that of the X (2000) resonance, recently observed by BES III, but a large hadronic shift down of this resonance is not excluded. For the states with l = 0 the generalized ERT, which includes the orbital and radial excitations, has the orbital slope b J = 0.905 GeV 2 and the radial slope b n = 1.15(5) GeV 2 . This ERT gives the mass M( f 2 (2P)) = 1938 MeV in agreement with the mass of f 2 (1950) and M( f 2 (3P)) = 2268 MeV, while the mass M(2 3 D 1 ) = 2.180(5) GeV agrees with the mass of the φ(2170) resonance, and therefore φ(2170) could be either the 2 3 D 1 state or contain a large ss component. The leptonic width of φ(2D), Γ ee = 0.20(10) keV, has a large theoretical uncertainty, which occurs because of the strong sensitivity of the radial w.f. at the origin R 2D (0) to the parameters of the flattened (screened) potential.