Prediction for $5^{++}$ mesons

In this paper, we study the spectrum and decay behavior of $5^{++}$ meson family which is still missing in experiment. By the modified Godfrey-Isgur model with a color screening effect, we obtain the mass spectrum of $a_5$, $f_5$ and $f_5^\prime$ mesons. And we predict their two-body strong decays by means of a phenomenology quark pair creation model. This study is crucial to establish $J^{PC}=5^{++}$ meson family and it is also helpful to search for these states in the future.

In this paper, we study the spectrum and decay behavior of the 5 ++ meson family which is still missing in experiment. By the modified Godfrey-Isgur model with a color screening effect, we obtain the mass spectrum of a 5 , f 5 and f ′ 5 mesons. And we predict their two-body strong decays by means of a phenomenology quark pair creation model. This study is crucial to establishing the J PC = 5 ++ light meson family and helpful for searching for these states in the future.

I. INTRODUCTION
As an important part of hadron, light meson family is phenomenological studied by many works, such as vector mesons associated with Y(2175) [1, 2], ρ and ρ 3 mesons [3], axial vector mesons [4], tensor mesons [5,6], pseudotensor mesons [7], kaons [8], and higher spin mesons [9]. When checking the experimental status of light mesons [10], we notice an interesting phenomenon, i.e., the light meson family with J PC = 5 ++ are still absent, yet the other families for a and f meson (such as a 0 , a 1 , a 2 , a 3 , a 4 , and a 6 ) are all reported by PDG [10]. This phenomenon stimulates our interest in exploring where are the light mesons with J PC = 5 ++ .
According to the isospin, the light mesons with J PC = 5 ++ can be categorized into three groups, which are isovector a 5 meson family, and isoscalar a 5 , f 5 (nn) and f ′ 5 (ss) meson families. In this work, we firstly study the properties of these states of a 5 , f 5 and f ′ 5 meson families. Since they are missing in experiment, we predict the spectrum and the Okuba-Zweig-Iizuka (OZI) allowed two-body strong decay behavior, which is a key information of experimental search for them.
We hope that our effort will be helpful to establish a 5 , f 5 and f ′ 5 meson families. This paper is organized as follows. In Sec.II, the mass spectrum analysis of the light meson family with J PC = 5 ++ will be performed. In Sec.III, we further study the two-body OZIallowed strong decay behavior of these discussed states. The paper ends with a discussion and conclusion in Sec.IV.

II. THE MASS SPECTRUM ANALYSIS
In this work, the modified GI quark model is utilized to calculate the mass spectrum and wave functions of the light meson family, and relativistic effects is considered.
In the following, this model will illustrated in details.
where m 1 and m 2 denote the mass of quark and antiquark respectively, and effective potentialṼ eff contains two ingredients, a short-range γ µ ⊗γ µ one-gluon-exchange interaction and a 1 ⊗ 1 linear confinement interaction. The meaning of tilde will explain later.
In the nonrelativistic limit, effective potential has familiar format [11,12] V eff (r) = H conf + H hyp + H so , where H conf includes the spin-independent linear confinement piece S (r) and Coulomb-like potential from one-gluonexchange G(r), H hyp denotes the color-hyperfine interaction consists tensor and contact terms, and H SO is the spin-orbit interaction with colour magnetic term causing of one-gluon-exchange and the Thomas precession term. For above formulas, S 1 /S 2 indicates the spin of quark/antiquark and L the orbital momentum between them. F is relevant to the Gell-Mann matrix, i.e., F 1 = λ 1 /2 and F 1 = −λ * 2 /2, and for a meson, F 1 · F 2 = −4/3. Now relativistic effects of distinguish influence must be considered especially in light meson system, which is embedded in two ways. Firstly, based on the nonlocal interactions and new r dependence, a smearing functions is introduced for a meson qq which is applied to S (r) and G(r) to obtain smeared potentials S (r) andG(r) byf where σ 0 and s are defined as [11]. Secondly, owning to relativistic effects, a general potential should rely on the mass-of-center of interacting quarks. Momentum-dependent factors which will be unity in the nonrelativistic limit are applied as (2.12) whereṼ i (r) delegate the contact, tensor, vector spin-orbit and scalar spin-orbit terms, and ǫ i the relevant modification parameters.
Although the GI model has obtained great success in describing the meson spectrum, there are still exist differences between the predicted values of GI models and the experimental observations. In the previous work [13], a modified GI model was proposed, and the prediction results for the charmstrange mesons are consistent with the experimental data. For higher excitation states, the screen effect is considered to be very important by the authors of Ref. [13] . It could be introduced by the transformation br + c → b(1−e −µr ) µ + c, where µ is screened parameter whose particular value is need to be fixed by the comparisons between theory and experiment. Modified confinement potential also requires similar relativistic correction, which has been mentioned in the GI model. Then, we further write V scr (r) as the way given in Eq. (2. 16), By inserting the form of ρ(r − r ′ ) in Eq. (2.9) into the above expression and finishing this integration, the concrete expression forṼ scr (r) is given bỹ Notability, except converting the confinement potential to the screened potential, the other processing contents and the Hamiltonian matrix elements contained in the original GI model are calculated. In our calculation, we need the spatial wave functions of the discussed the light meson family with J PC = 5 ++ which can be numerically obtained by the modified GI model.

B. Mass spectrum analysis
Although the GI model has succeeded in describing the mass of ground states of the light mesons, it does not well describe the excited states. Since unquenched effects are important for a heavy-light system, it is better to adopt the modified GI model (MGI) [13,14] which uses a screening potential with a new parameter µ. The parameter µ describes inverse of the size of screening. In our previous work [8], we calculate the kaon family spectra use MGI model. In this work we will use this MGI model to obtain the mass spetrum of the light meson with J PC = 5 ++ . Beforehand, we need to adjust the parameters of MGI model by fitting with the experiments data. So we fix the following ten parameters listed in Tab. I by fitting forty one experimental data which is listed in Tab. II. In Tab. II, we select forty one experimental data of light meson listed in PDG and optimize these light meson masses to determine ten parameters in Tab. I. This optimization has χ 2 /n = 374 which is smaller than 2638 for the GI model as shown in Tab. II. Of course, besides the mass spectrum light mesons with J PC = 5 ++ was calculated by the GI model, there is a light meson spectrum with a Nambu-Goldstone Pion [16]. But the mass values calculated by A. Le Yaouanc etc are larger than those observed in experiments, which are greater than the values of MGI models, so the accuracy is not very high. Finally, we can obtain the mass spectrum of these four 5 ++ states by the MGI mode list in Tab. III.
So we can conclude that 1. The ground states of the 5 ++ states are still missing in experiments, and the predicted mass are 2.469 GeV for a 5 / f 5 , which are smaller than Ref. [11] and close to the result of Ref. [11]. f ′ 5 (1H) has the mass 2.660 GeV which is smaller than Ref. [11] and Ref. [11].
2. The first exited states of a 5 / f 5 and f ′ 5 have the mass 2.686 GeV and 2.882 GeV, respectively. For the second exited states, a 5 / f 5 (3H) and f ′ 5 (3H) have the mass 2.885 GeV, 3.08 GeV, respectively, which are also smaller than Ref. [11].
The above conclusions are only from the point of mass spectra view and we will study their strong decays in the next section.

A. QPC Model
The structure information of experimentally observed mesons and predicted family members, the QPC model is used , where This work, Exp, and Error represent the theoretical value, experimental results, and experimental error, respectively, and n is the number of the experiment data. We select some established kaon states in PDG [15] for our fitting. The unit of the mass is GeV.
Component State This work GI [11] Exp error nn where P B(C) is a three-momentum of a meson B(C) in the rest frame of a meson A. A superscript M J i (i = A, B, C) denotes an orbital magnetic momentum. The transition operator T is introduced to describe a quark-antiquark pair creation from vacuum, which has the quantum number J PC = 0 ++ , i.e., T can be expressed as 1m; 1 − m|00 dp 3 dp 4 δ 3 (p 3 + p 4 ) This is completely constructed in the form of a visual representation to reflect the creation of a quark-antiquark pair from vacuum, where the quark and antiquark are denoted by indices 3 and 4, respectively. A dimensionless parameter γ depicts the strength of the creation of qq from vacuum, where the concrete values of the parameter R which will be discussed in the later section. Y ℓm (p) = |p| ℓ Y ℓm (p) are the solid harmonics. χ, φ, and ω denote the spin, flavor, and color wave functions respectively, which can be treated separately. Subindices i and j denote the color of a qq pair.
By the Jacob-Wick formula [41], the decay amplitude is where m A is the mass of an initial state A. In our calculation, we need the spatial wave functions of the discussed light mesons. which can be numerically obtained by the modified GI model. In the previous section, we calculate the mass spectrum of the light meson and obtain the wave functions of the light meson. At the same time, we can use QPC model to study the strong decay of the J PC = 5 ++ meson families. The parameter in the QPC model is determined by fitting with the experiment data. Thus, there is no free parameter in the QPC model. We obtain γ = 11.6 as shown in Tab. IV. Next, we will anylise the strong decay behvior of these 5 ++ states.

B. The ground states
The ground state a 5 which is not observed in experiment is predicted in this work, with the mass 2469 MeV (a 5 (2469)), and the total width is 367 MeV. ρ 3 π is its dominant decay channel, the width is about 130 MeV, and the branch ratio is 0.36. a 2 ρ, ωρ, ρπ and h 1 ρ, are its important decay channels which have the branch ratio about 0.08 each one, just as shown in Tab. V . The final states b 1 ω, f 2 π, f 4 (2050)π, ρ(1450)π and a 1 ρ, also have sizable decay widths, in which b 1 ω and f 2 π almost has same width about 20 MeV.
As the iso-spin partner of a 5 (1H), we predict f 5 (1H) will have the mass 2.47 GeV and width 300 MeV, respectively. In the final decay channels of f 5 (1H), ρρ and b 1 ρ will be the most important final states which have the width 67 MeV and 66 MeV respectively, and their branch ratio are about 0.22. a 2 π and a 1 π are the important decay channels too, with the widths 58 MeV and 49 MeV, respectively. In addition, h 1 ω and π 2 π also have visible widths 28 MeV and 25 MeV which is presented in Tab   29.2 f 2 π 2.87 π 2 π 1.71 .6 a 2 f 1 1.08 a 2 a 2 1.20 K * K * 2 (1430) 20.5 a 1 a 1 1.05 3.05 K * (1410)K * 2 (1430) 1.67

C. The first exited states
In this section, we will anylise the strong decay behavior of the first exited states of 5 ++ family. a 5 (2H) has the mass 2686 MeV and narrow width 133 MeV in our theory result. According to Tab. VI, ρ 3 π is the dominant decay channel of a 5 (2H) which is similar with a 5 (1H), the branch ratio is about 0.56. ρ(1450)π and a 0 (1450)ρ are its important decay channels which have the branch ratio about 0.1 and 0.06 respectively, just as Tab. VI shown. The final decay modes π 2 ρ, ω 3 ρ, a 1 b 1 , ωρ and ρ 3 ω, also have sizable decay widths, with the width 4-6 MeV. The other decay information is shown in Tab. VI.
f 5 (2H) as the iso-spin partner(I=0) of a 5 (2H) has the mass 2.69 GeV and width 91 MeV, respectively. f 5 (2H) main decays to ρ(1450)ρ which has the width 33 MeV and the branch ratio 0.36. ω(1420)ω, ρρ and a 1 π are the important final states which have the width 13.7 MeV, 9.64 MeV and 9.23 MeV, respectively, and their branch ratio are 0.15, 0.11 and 0.10, respectively. a 2 π, ω 3 ω a 1 a 2 and ωω also have visible width from 2.3 MeV to 7 MeV which are presented in Tab. VI. The widths of other channels are very small(see Tab. VI) which branch ratios are less than 0.02.

D. The second exited states
We also calculate the two body strong decays of the second exited states of 5 ++ family.
As the iso-vector meson of 5 ++ family, a 5 (3H) has the mass 2.88 GeV, the total width 95 MeV which is very narrow. ρ 3 π also is its dominant decay channel as shown in Tab. VII, the width is about 37.2 MeV, and the branch ratio is 0.4. ρ(1450)π has a large ratio (0.13) in its decay final channels. ρπ, ωρ, a 1 b 1 , π 2 ρ, ω 3 ρ, ω(1420)ρ and ρ(1450)ω also have sizable contribution in the total widths. f 5 (3H) state is the second radial excited state of f 5 , which with the mass 2.88 GeV and width 50 MeV, f 5 (3H) main decays into ρ 3 ρ, ρρ and ρ(1450)ρ, whose decay widths are 11.7, 11.4 and 9.44 MeV, respectively, and each channel almost has the branch ratio 0.2. a 1 a 2 and ω(1420)ω modes are the important decay channels too, with the widths about 5 MeV. In addition, a 2 a 2 , a 0 (1450)a 1 , and π 2 π also have visible widths which are presented in Tab. VII. The width of other modes are very small(Tab. VII).

IV. CONCLUSION
In this paper, we study the spectrum and two body strong decay of the family with J PC = 5 ++ which is still missing in experiment. By the modified Godfrey-Isgur model with a color screening effect, we anylise the mass spectrum of a 5 and f 5 mesons, in which we find that the ground states of the 5 ++ states, a 5 , f 5 and f ′ 5 have the mass 2.469 GeV, 2.469 GeV and 2.66 GeV and the width 367 MeV, 300 MeV, and 813 MeV, respectively. The first exited states of a 5 / f 5 and f ′ 5 have the mass 2.686 GeV and 2.882 GeV and a 5 (3H)/ f 5 (3H) and f ′ 5 (3H) have the mass 2.686 GeV, 2.882 GeV. The total widths are predicted to be 131 MeV(a 5 (2H)), 91 MeV( f 5 (2H)), and 578 MeV( f ′ 5 (2H)) for the first states. For the second exited states of 5 ++ , a 5 (3H), f 5 (3H) and f ′ 5 (3H) have the widths 95 MeV(a 5 ), 50 MeV( f 5 ) and 320 MeV ( f ′ 5 ), respectively. We also predict the detail decay information of 5 ++ family using QPC model which can be helpful for searching the mesons in the future experiments just as BESIII and COM-PASS.