Strong decays of the higher isovector scalar mesons

Under the assignment of $a_0(1450)$ as the ground isovector scalar meson, the strong decays of $a_0(1950)$ and $a_0(2020)$ are evaluated in the $^3P_0$ model. Our calculations suggest that $a_0(1950)$ and $a_0(2020)$ can be regarded as the same resonance referring to $a_0(3^3P_0)$. The masses and strong decays of $a_0(2^3P_0)$ and $a_0(4^3P_0)$ are also predicted, which can be useful in the search for radially excited scalar mesons in the future.


I. INTRODUCTION
In the framework of quantum chromodynamics (QCD), apart from the ordinary qq states, other exotic states such as glueballs, hybrids, and tetraquarks are permitted to exist in meson spectra. To identify these exotic states, one needs to distinguish them from the background of ordinary qq states, which requires one to understand well the conventional qq meson spectroscopy both theoretically and experimentally.
At present, above the a 0 (980) mass, three higher isovector scalar states, a 0 (1450), a 0 (2020), and a 0 (1950), have been reported experimentally. a 0 (1450) was observed in pp annihilation experiments [8,9], D ± → K + K − π ± [10], and D 0 → K 0 S K ± π ∓ [11]. a 0 (2020) with two alternative solutions of similar masses and widths was found by the Crystal Barrel Collaboration in the partial wave analysis of the data onpp → π 0 η and π 0 η ′ [12], and a 0 (1950) was observed by the BABAR Collaboration in the processes γγ → K 0 s K ± π ∓ and γγ → K + K − π 0 [13]. The masses and widths of the three isovector scalar states are listed in Table I. The lattice QCD calculations support that the lowest isovector scalar qq state corresponds to a 0 (1450) rather than a 0 (980) [14][15][16]. It is widely accepted that a 0 (1450) is the isovector member of the 1 3 P 0 qq nonet [1]. The na-tures of a 0 (2020) and a 0 (1950) are unclear. To be able to understand the nature of a newly observed state, it is natural and necessary to exhaust the possible qq description before restoring to more exotic assignments. Therefore, with the assignment of a 0 (1450) as the ground qq state, one naturally asks whether the higher isovector scalar states, a 0 (2020) and a 0 (1950), can be identified as the radial excitations of a 0 (1450).  [13] Theoretical efforts on the quark model assignments for a 0 (2020) and a 0 (1950) have been carried out. It is suggested that a 0 (1950)/a 0 (2020) can be assigned as the 2 3 P 0 state based on the extended linear sigma model in Ref. [17], where a 0 (2020) is considered earlier evidence for a 0 (1950). In addition, a 0 (1950)/a 0 (2020) is assigned as the 3 3 P 0 state based on the relativistic quark model in Ref. [18], where the predicted a 0 (3 3 P 0 ) mass is about 1993 MeV, in agreement with both the a 0 (1950) and a 0 (2020) masses within errors. Obviously, further studies on the quark model assignments for a 0 (1950) and a 0 (2020) in other approaches are needed. Also, from Table I, one can see that the resonance parameters of a 0 (2020) are close to those of a 0 (1950). The observed mass difference between a 0 (1950) and a 0 (2020) is less than 100 MeV; in such a small mass interval, it would be very difficult to accommodate two radial excitations of a 0 (1450) in practically all the quark models. We therefore conclude that if both a 0 (2020) and a 0 (1950) can be explained as qq states, they should correspond to the same resonance. In this work, we shall discuss the possible quark model assignments of a 0 (2020) and a 0 (1950) by investigating their strong decays in the 3 P 0 model and check whether a 0 (2020) and a 0 (1950) can be identified as the same scalar meson.
The organization of this paper is as follows. In Sec. II, we give a brief review of the 3 P 0 model. In Sec. III, the calculations and discussion are presented, and the summary and conclusion are given in Sec. IV.
Following the conventions in Ref. [31], the transition operator T of the decay A → BC in the 3 P 0 model is given by where the γ is a dimensionless parameter denoting the probability of the quark-antiquark pair q 3q4 with quantum number J P C = 0 ++ . p 3 and p 4 are the momenta of the created quark q 3 and antiquarkq 4 , respectively. χ 34 1,−m , φ 34 0 , and ω 34 0 are the spin, flavor, and color wave functions of q 3q4 , respectively. The solid harmonic poly- The partial wave amplitude M LS (P ) of the decay A → BC can be given by [41], where M MJ A MJ B MJ C (P ) is the helicity amplitude and defined as, |A , |B , and |C denote the mock meson states defined in Ref. [42]. Due to different choices of the pair-production vertex, phase space convention, and employed meson space wave function, various 3 P 0 models exist in the literature. In this work, we employ the simplest vertex as introduced originally by Micu, who assumes a spatially constant pair-production strength γ [19], relativistic phase space, and simple harmonic oscillator (SHO) wave functions. With the relativistic phase space, the decay width Γ(A → BC) can be expressed in terms of the partial wave amplitude, where , and M A , M B , and M C are the masses of the mesons A, B, and C, respectively. The explicit expressions for M LS (P ) can be found in Refs. [31][32][33].
Under the SHO approximation, the meson space wave function in the momentum space is where the radial wave function is given by Here β is the SHO wave function scale parameter, and L is an associated Laguerre polynomial.
The decay widths of a 0 (1450) as the 1 3 P 0 state are listed in Table II. The dominant decay modes of the 1 3 P 0 isovector state are πη, πη ′ , and KK, consistent with observations of a 0 (1450) [8,9,50].
The decay widths of a 0 (1950) as the 2 3 P 0 and 3 3 P 0 states are shown in Table III. If a 0 (1950) is the 2 3 P 0  state, its total width is expected to be about 771 MeV, much larger than the observed a 0 (1950) width of 271 ± 22 ± 29 MeV [13]. The possibility of a 0 (1950) being the 2 3 P 0 state can be ruled out. If a 0 (1950) is the 3 3 P 0 state, its total width is about 207 MeV, reasonably close to the measurement within errors. The dependence of the total width of a 0 (3 3 P 0 ) on the initial state mass is shown in Fig. 1. Within the a 0 (1950) mass errors, the total width does not change too much. The assignment of a 0 (1950) as the 4 3 P 0 state can also be ruled out because the predicted width for a 0 (4 3 P 0 ) with a mass of 1931 MeV is about 37.3 MeV (see also Fig. 3), much smaller than the a 0 (1950) width. Therefore, the measured mass and width for a 0 (1950) are in favor of it being the 3 3 P 0 state. As shown in Fig. 1, a 0 1: The dependence of the total width of a0(3 3 P0) on the initial state mass. The dashed line with a green band denotes the BABAR experimental data [13].
MeV smaller than the lower limit of Crystal Barrel's solution I for the a 0 (2020) width of 330±75 MeV [12], and the predicted width for a 0 (3 3 P 0 ) with a mass of 1980 MeV is about 218 MeV, in agreement with Crystal Barrel's solution II for the a 0 (2020) width of 225 +120 −32 MeV [12]. The possibility of a 0 (2020) being the 2 3 P 0 state can be ruled out because the expected width for a 0 (2 3 P 0 ) with a mass of 1980 (2025) MeV is about 895 (995) MeV, much larger than the observed width of a 0 (2020), as shown in Table I. The predicted width for a 0 (4 3 P 0 ) with a mass of 1980 (2025) MeV is about 36.8 (34.6) MeV (see also Fig. 3), much smaller than the a 0 (2020) width, which makes a 0 (2020) unlikely to be the 4 3 P 0 state. So, the measured mass and width for a 0 (2020) are consistent with an assignment of the 3 3 P 0 state.
The experimental evidence for both a 0 (1950) and a 0 (2020) turns out to be consistent with the presence of the same resonance corresponding to a 0 (3 3 P 0 ). This naturally establishes 1.9 GeV as the approximate mass for the nn members of the 3P nonets, which could be useful to search for the nn members of the 3P nonets experimentally. The dominant decay modes of a 0 (3 3 P 0 ) are π(1300)η, πη(1475), πb 1 (1235), KK 1 (1270), and ρω. a 1 (1640) and a 2 (1700) as the 2P radial excitations have been established [25,51], which also fixes the natural mass scale for the nn members of the 2P multiplets as about 1.7 GeV. One can expect to find a 0 (2 3 P 0 ) near 1.7 GeV. At present, no candidate for the isovector scalar state around 1.7 GeV is reported experimentally. An a 0 -like pole associated to a resonance with a mass of about 1760 MeV is found by investigating the meson-meson interaction in Refs. [52,53]. The a 0 (2 3 P 0 ) mass in the extended linear sigma model is expected to be 1790 ± 35 MeV [17]. Systematic studies on the meson spectra in the relativistic quark models show that the expected a 0 (2 3 P 0 ) mass is about 1679 ∼ 1780 MeV [18,47]. Phenomenologically, it is suggested that the light mesons could be grouped into the following Regge trajectories [54], where M 0 is the lowest-lying meson mass, n is the radial quantum number, and µ 2 is the slope parameter of the corresponding trajectory. In the presence of a 0 (1450) and a 0 (1950)/a 0 (2020) being the 1 3 P 0 and 3 3 P 0 states, respectively, the a 0 (2 3 P 0 ) mass can be determined to be about 1744 MeV based on Eq. (7), 1 consistent with the extended linear sigma model prediction [17] and the quark model predictions [18,47].
The strong decays of a 0 (2 3 P 0 ) with a mass of 1744 MeV are presented in Table IV. The total width of a 0 (2 3 P 0 ) is expected to be about 364 MeV. The dominant decay modes of a 0 (2 3 P 0 ) include πη(1475), πη(1295), πb 1 (1235), πf 1 (1285), and ρω. The dependence of the total width of a 0 (2 3 P 0 ) on the initial state mass is shown in Fig. 2. When the initial state mass varies from 1700 to 1800 MeV, the total width of the a 0 (2 3 P 0 ) varies from about 298 to 460 MeV. With the initial state mass of 1700 MeV, our predicted width of 298 MeV is in agreement with the width of 293 MeV expected by Ref. [25] for a 0 (2 3 P 0 ). 1 We take M a 0 (1450) =1474 MeV, M a 0 (1950)/a 0 (2020) = (1931 + 2025)/2=1978 MeV, the average value of the a 0 (1950) mass reported by the BABAR Collaboration [13] and the favoured solution for the a 0 (2020) mass [12].   MeV based on Eq. (7), consistent with 2250 MeV, the expected mass for a 0 (4 3 P 0 ) in the quark model [18]. The strong decays of a 0 (4 3 P 0 ) with a mass of 2187 MeV are listed in Table V. The dependence of the total width of the 4 3 P 0 isovector state on the initial state mass is shown in Fig. 3. A narrow width for a 0 (4 3 P 0 ) is predicted. The πη(1295), KK 1 (1270), ρω, and πη 2 (1645) channels are the dominant decay modes for a 0 (4 3 P 0 ). As we can see in Figs. 1 and 3, the width derivatives are a discontinuity around 1950 MeV, which is because the decay channel KK(1460) is open above this energy.

IV. SUMMARY AND CONCLUSION
Observations of the state a 0 (1950) by the BABAR Collaboration have enlarged the family of the isovector scalar mesons. In this work, we discuss the possible quark model assignments of a 0 (1950) and a 0 (2020) by calculating their strong decays in the 3 P 0 model. We suggest that a 0 (1950) and a 0 (2020) can be regarded as the same resonance referring to a 0 (3 3 P 0 ). The confirmation of a 0 (1950)/a 0 (2020) as the 3 3 P 0 state thereby estab-lishes about 1.9 GeV as a natural mass scale for the nn members of the 3P nonets.
In the presence of a 0 (1450) and a 0 (1950)/a 0 (2020) being the 1 3 P 0 and 3 3 P 0 states, respectively, in Regge phenomenology, the masses of a 0 (2 3 P 0 ) and a 0 (4 3 P 0 ) are predicted to be about 1744 MeV and 2187 MeV, respectively. The predicted masses for a 0 (2 3 P 0 ) and a 0 (4 3 P 0 ) are consistent with some other theoretical expectations. The total widths of a 0 (2 3 P 0 ) and a 0 (4 3 P 0 ) are expected to be about 364 MeV and 36 MeV, respectively. Our predictions could be useful to study the higher isovector scalar mesons experimentally.