# Prediction for $5^{++}$ mesons

Cheng-Qun Pang1111Corresponding author<EMAIL_ADDRESS>Ya-Rong Wang1 Chao-Hui
Wang1 1College of Physics and Electronic Information Engineering, Qinghai
Normal University, Xining 810000, China

###### Abstract

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.

###### pacs:

14.40.Be, 12.38.Lg, 13.25.Jx

##  I introduction

In the past decades, the quark model has made great achievements. There are
six quarks, up($u$), down($d$), strange($s$), charm($c$), bottom($b$), and
top($t$), in the quark model. Except $t$ quark, the other quarks can form
mesons, baryons ,and other hadrons. As an important part of hadron, the meson
family is phenomenologically studied by many works Wang  _et al._ (2012);
Piotrowska  _et al._ (2017); He  _et al._ (2013); Chen  _et al._ (2015); Ye
_et al._ (2012); Pang  _et al._ (2014); Wang  _et al._ (2015); Pang  _et al._
(2017, 2015). When checking the experimental status of mesons Patrignani  _et
al._ (2016), we notice an interesting phenomenon, the whole meson family with
$J^{PC}=5^{++}$ (with $q\bar{q}$ component, here $q$ defines $u$, $d$, or $s$
quark) is still missing, yet the other families for $a$ and $f$ meson (such as
$a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, and $a_{6}$) have been reported
by Particle Data Group(PDG) Patrignani  _et al._ (2016). This phenomenon
stimulates our interest in exploring where the mesons with $J^{PC}=5^{++}$ are
and their properties.

The mesons with $J^{PC}=5^{++}$ have two families, which are isovector $a_{5}$
meson family, and isoscalar meson family[($f_{5}$($n\bar{n}$, here $n$ defines
$u$ or $d$ quark) and $f_{5}^{\prime}$($s\bar{s}$))]. Due to the present
experimental progress on mesons, it is a suitable time to systematically carry
out phenomenological study of missing $5^{++}$ states. This study is not only
crucial to establish $J^{PC}=5^{++}$ meson family and helpful to search for
these states in the future, but also important for verifying quark model.

As $5^{++}$ mesons have a higher spin, the screening effect will be strong for
the large angular momentum and larger average distance between quark pair. So
we need introduce the screening effect into the quark model in this work when
we deal with the spectrum. In this work, we calculate the mass spectra of the
$5^{++}$ meson family by using the unquenched Godfrey-Isgur(GI) model Song
_et al._ (2015a, b), which contains the screening effect. According to the
former studies Godfrey and Isgur (1985); Barnes  _et al._ (2005); Sun  _et
al._ (2014); Song  _et al._ (2015a, b); Godfrey and Moats (2015); Godfrey  _et
al._ (2016); Pang  _et al._ (2017), the GI model was tested by different
systems, which shows that the GI model works well for describing hadron
spectroscopy. In this work, we continue to apply this model to explore high
spin states, especially estimate the mass spectrum of high spin mesons. Since
the experimental information of high spin states is absent, we hope that these
predicted states can be accessible at future experiment, which can provide
further test of the GI model to high spin states. In this paper, we fix the
parameters in the model by fitting some well-established meson states, which
are adopted when calculating the mass of $5^{++}$ meson states. Then, for
further study the properties of $5^{++}$ mesons, we study their Okubo-Zweig-
Iizuka (OZI)-allowed two-body strong decays taking input with the spatial wave
functions obtaining in mass spectrum numerically calculation. Their partial
and total decay widths are predicted by using a quark pair creation (QPC)
model which was proposed by Micu Micu (1969) and extensively applied to
studies of strong decay of other hadrons Le Yaouanc  _et al._ (1973); van
Beveren  _et al._ (1980, 1983); Le Yaouanc  _et al._ (1988); Roberts and
Silvestre-Brac (1992); Capstick and Roberts (1994); Blundell and Godfrey
(1996); Ackleh  _et al._ (1996); Capstick and Keister ; Bonnaz  _et al._
(2002); Close and Swanson (2005); Zhang  _et al._ (2007); Lu  _et al._ (2006);
Sun and Liu (2009); Liu  _et al._ (2010); Sun  _et al._ (2010); Rijken  _et
al._ (2010); Yu  _et al._ (2011); Zhou and Xiao (2011); Ye  _et al._ (2012);
Wang  _et al._ (2012); Sun  _et al._ (2013); He  _et al._ (2013); Sun  _et
al._ (2014); Pang  _et al._ (2014); Wang  _et al._ (2015); Chen  _et al._
(2015). We hope that our effort will be helpful to establish $a_{5}$, $f_{5}$
and $f_{5}^{\prime}$ meson families.

This paper is organized as follows. In Sec. II, the mass spectrum analysis of
the meson family with $J^{PC}=5^{++}$ will be performed. In Sec. III, we
further study the two-body OZI-allowed strong decay behavior of these
discussed states. The paper ends with a 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 meson family. In the following, this model
will be illustrated in detail.

###  II.1 The modified GI model

In 1985, Godfrey and Isgur raise the GI model for describing relativistic
meson spectra with great success, exactly in low-lying mesonsGodfrey and Isgur
(1985). As for the excited states, the screening potential must be taken into
account for coupled-channel effect.

The interaction between quark and antiquark is depicted by the Hamiltonian of
potential model including kinetic energy pieces and effective potential piece,

$\tilde{H}=\sqrt{m_{1}^{2}+\mathbf{p}^{2}}+\sqrt{m_{2}^{2}+\mathbf{p}^{2}}+\tilde{V}_{\mathrm{eff}}(\mathbf{p,r}),$
(2.1)

where $m_{1}$ and $m_{2}$ denote the mass of quark and antiquark respectively,
and effective potential $\tilde{V}_{\mathrm{eff}}$ contains two ingredients, a
short-range $\gamma^{\mu}\otimes\gamma_{\mu}$ one-gluon-exchange interaction
and a $1\otimes 1$ linear confinement interaction. The meaning of tilde will
be explained later.

In the nonrelativistic limit, effective potential has a familiar formatGodfrey
and Isgur (1985); Lucha  _et al._ (1991)

$\displaystyle
V_{\mathrm{eff}}(r)=H^{\mathrm{conf}}+H^{\mathrm{hyp}}+H^{\mathrm{so}},$ (2.2)

with

$\displaystyle H^{\mathrm{conf}}$
$\displaystyle=\Big{[}-\frac{3}{4}(c+br)+\frac{\alpha_{s}(r)}{r}\Big{]}(\bm{F}_{1}\cdot\bm{F}_{2}),$
$\displaystyle=S(r)+G(r)$ (2.3) $\displaystyle H^{\mathrm{hyp}}$
$\displaystyle=-\frac{\alpha_{s}(r)}{m_{1}m_{2}}\Bigg{[}\frac{8\pi}{3}\bm{S}_{1}\cdot\bm{S}_{2}\delta^{3}(\bm{r})+\frac{1}{r^{3}}\Big{(}\frac{3\bm{S}_{1}\cdot\bm{r}\bm{S}_{2}\cdot\bm{r}}{r^{2}}$
$\displaystyle\quad-\bm{S}_{1}\cdot\bm{S}_{2}\Big{)}\Bigg{]}(\bm{F}_{1}\cdot\bm{F}_{2}),$
(2.4) $\displaystyle H^{\mathrm{so}}=$ $\displaystyle
H^{\mathrm{so(cm)}}+H^{\mathrm{so(tp)}},$ (2.5)

where $H^{\mathrm{conf}}$ includes the spin-independent linear confinement
piece $S(r)$ and Coulomb-like potential from one-gluon-exchange $G(r)$,
$H^{\mathrm{hyp}}$ denotes the color-hyperfine interaction consists tensor and
contact terms, and $H^{\mathrm{SO}}$ is the spin-orbit interaction with

$\displaystyle
H^{\mathrm{so(cm)}}=\frac{-\alpha_{s}(r)}{r^{3}}\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)\left(\frac{\bm{S}_{1}}{m_{1}}+\frac{\bm{S}_{2}}{m_{2}}\right)\cdot\bm{L}(\bm{F}_{1}\cdot\bm{F}_{2}),$
(2.6)

colour magnetic term causing of one-gluon-exchange and

$\displaystyle H^{\mathrm{so(tp)}}=-\frac{1}{2r}\frac{\partial
H^{\mathrm{conf}}}{\partial
r}\Bigg{(}\frac{\bm{S}_{1}}{m^{2}_{1}}+\frac{\bm{S}_{2}}{m^{2}_{2}}\Bigg{)}\cdot\bm{L},$
(2.7)

the Thomas precession term.

For above formulas, $\bm{S}_{1}/\bm{S}_{2}$ indicates the spin of
quark/antiquark and $\bm{L}$ the orbital momentum between them. $\bm{F}$ is
relevant to the Gell-Mann matrix, i.e., $\bm{F}_{1}=\bm{\lambda}_{1}/2$ and
$\bm{F}_{1}=-\bm{\lambda}^{*}_{2}/2$, and for a meson,
$\langle\bm{F}_{1}\cdot\bm{F}_{2}\rangle=-4/3$.

Now relativistic effects of distinguish influence must be considered
especially in meson system, which is embedded in two ways. First, based on the
nonlocal interactions and new $\mathbf{r}$ dependence, a smearing function is
introduced for a meson $q\bar{q}$

$\rho\left(\mathbf{r}-\mathbf{r^{\prime}}\right)=\frac{\sigma^{3}}{\pi^{3/2}}e^{-\sigma^{2}\left(\mathbf{r}-\mathbf{r^{\prime}}\right)^{2}},$
(2.8)

which is applied to $S(r)$ and $G(r)$ to obtain smeared potentials
$\tilde{S}(r)$ and $\tilde{G}(r)$ by

$\tilde{f}(r)=\int
d^{3}r^{\prime}\rho(\mathbf{r}-\mathbf{r^{\prime}})f(r^{\prime}),$ (2.9)

with

$\displaystyle\sigma_{12}^{2}=\sigma_{0}^{2}\Bigg{[}\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{1}m_{2}}{(m_{1}+m_{2})^{2}}\right)^{4}\Bigg{]}+s^{2}\left(\frac{2m_{1}m_{2}}{m_{1}+m_{2}}\right)^{2},$
(2.10)

where the values of $\sigma_{0}$ and $s$ are defined later.

Second, 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

$\tilde{G}(r)\to\left(1+\frac{p^{2}}{E_{1}E_{2}}\right)^{1/2}\tilde{G}(r)\left(1+\frac{p^{2}}{E_{1}E_{2}}\right)^{1/2},$
(2.11)

and

$\frac{\tilde{V}_{i}(r)}{m_{1}m_{2}}\to\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{i}}\frac{\tilde{V}_{i}(r)}{m_{1}m_{2}}\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{i}},$
(2.12)

where $\tilde{V}_{i}(r)$ delegates the contact, tensor, vector spin-orbit and
scalar spin-orbit terms, and $\epsilon_{i}$ the relevant modification
parameters.

Diagonalizing and solving the Hamiltonian in Eq.(2.1) by exploiting a simple
harmonic oscillator (SHO) basis, we will obtain the mass spectrum and wave
functions. As we know, a series of SHO wave function with different radial
quantum number $n$ can be regarded as a complete basis to expand the exact
radial wave function of meson state. The base can produce all kind of states,
even spurious ones(if the motion of the meson is taken into account Wang and
Wong (1988)), However, this model is calculated in center-mass frame, so there
are no spurious states in our numerical result.

In configuration and momentum space, SHO wave functions have explicit forms
respectively,

$\displaystyle\Psi_{nLM_{L}}(\mathbf{r})=R_{nL}(r,\beta)Y_{LM_{L}}(\Omega_{r}),$
$\displaystyle\Psi_{nLM_{L}}(\mathbf{p})=R_{nL}(p,\beta)Y_{LM_{L}}(\Omega_{p}),$
(2.13)

with

$\displaystyle
R_{nL}(r,\beta)=\beta^{3/2}\sqrt{\frac{2n!}{\Gamma(n+L+3/2)}}(\beta
r)^{L}e^{\frac{-r^{2}\beta^{2}}{2}}$ $\displaystyle\times
L_{n}^{L+1/2}(\beta^{2}r^{2}),$ (2.14) $\displaystyle
R_{nL}(p,\beta)=\frac{(-1)^{n}(-i)^{L}}{\beta^{3/2}}e^{-\frac{p^{2}}{2\beta^{2}}}\sqrt{\frac{2n!}{\Gamma(n+L+3/2)}}{(\frac{p}{\beta})}^{L}$
$\displaystyle\times L_{n}^{L+1/2}(\frac{p^{2}}{\beta^{2}}),$ (2.15)

where $Y_{LM_{L}}(\mathrm{\Omega})$ is spherical harmonic function, and
$L_{n-1}^{L+1/2}(x)$ is the associated Laguerre polynomial, and
$\beta=0.4~{}\mathrm{GeV}$ for our calculation.

The space-spin wave function $R_{nL}(r,\beta)\phi_{LSJM}$ with total angular
quantum number $\it J$ can be constructed by coupling $L\otimes S$

$\displaystyle\phi_{LSJM}=\sum_{M_{L}M_{S}}C(LM_{L}SM_{S};JM)Y_{LM_{L}}(\Omega_{r})\chi_{SM_{S}},$
(2.16)

here $C(LM_{L}SM_{S};JM)$ is Clebsch-Gordan coefficient. For the matrix
element $\langle\alpha|\hat{V}(r,\hat{p})|\beta\rangle$ where $|\alpha\rangle$
and $|\beta\rangle$ are arbitrary SHO basis with quantum number
$\\{n,J,L,S\\}$ and $\\{n^{\prime},J^{\prime},L^{\prime},S^{\prime}\\}$. The
matrix element can be calculated conveniently by using SHO base as follows.

$\displaystyle\langle\alpha|\hat{V}(r,\hat{p})|\beta\rangle$
$\displaystyle={\langle\alpha|f(p)g(r)|\beta\rangle}$
$\displaystyle=\sum_{n}{\langle\alpha|f(p)|n\rangle\langle
n|g(r)|\beta\rangle}.$ (2.17)

After diagonalizing the Hamiltonian matrix, we can obtain the mass and wave
function of meson which are available to the following strong decay process.

$5^{++}$ mesons have a higher spin, so the quarks and antiquarks will have
large angular momentum and larger average distance which is greater than about
1 fm. In this circumstance, light quark antiquark pairs will be spontaneously
created and the screening effect will be strong. So we introduce the screening
effect into GI model in this work when we deal with the spectrum. In the
previous work Song  _et al._ (2015a), the modified GI model was proposed, and
the prediction results for the charm-strange 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. Song  _et al._ (2015a)
. It could be introduced by the transformation
$br+c\rightarrow\frac{b(1-e^{-\mu r})}{\mu}+c$, where $\mu$ 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 $V^{\mathrm{scr}}(r)$ as the way given in Eq. (2.18),

$\displaystyle\tilde{V}^{\mathrm{scr}}(r)$ $\displaystyle=$ $\displaystyle\int
d^{3}\bm{r}^{\prime}\rho(\bm{r-r^{\prime}})\frac{b(1-e^{-\mu
r^{\prime}})}{\mu}.$ (2.18)

By inserting the form of $\rho(\bm{r-r^{\prime}})$ in Eq. (2.9) into the above
expression and finishing this integration, the concrete expression for
$\tilde{V}^{\mathrm{scr}}(r)$ is given by

$\displaystyle\tilde{V}^{\mathrm{scr}}(r)$ $\displaystyle=$
$\displaystyle\frac{b}{\mu r}\Bigg{[}r+e^{\frac{\mu^{2}}{4\sigma^{2}}+\mu
r}\frac{\mu+2r\sigma^{2}}{2\sigma^{2}}\Bigg{(}\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{\mu+2r\sigma^{2}}{2\sigma}}e^{-x^{2}}dx-\frac{1}{2}\Bigg{)}$
(2.19) $\displaystyle-e^{\frac{\mu^{2}}{4\sigma^{2}}-\mu
r}\frac{\mu-2r\sigma^{2}}{2\sigma^{2}}\Bigg{(}\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{\mu-2r\sigma^{2}}{2\sigma}}e^{-x^{2}}dx-\frac{1}{2}\Bigg{)}\Bigg{]}.$

Notably, except for 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 meson
family with $J^{PC}=5^{++}$ which can be numerically obtained by the modified
GI model.

###  II.2 Mass spectrum analysis

GI model can describe the mass of ground states of the mesons successfully,
yet it does not describe the excited states well. Since unquenched effects are
important for a heavy-light system, it is better to adopt the modified GI
model (MGI) Song  _et al._ (2015a, b) which uses a screening potential with a
new parameter $\mu$. The parameter $\mu$ describes inverse of the size of
screening. In our previous work Pang  _et al._ (2017), we calculate the kaon
family spectra by using the MGI model. In this work we will use this MGI model
to obtain the mass spectrum of meson with $J^{PC}=5^{++}$. Beforehand, we need
to adjust the parameters of MGI model by fitting with the experimental data.
So we fix the following twelve parameters listed in Table 1.

Table 1: Parameters and their values in this work and GI models.  Parameter | This work | GI Godfrey and Isgur (1985) |
---|---|---|---
$m_{u}$(GeV) | 0.163 | 0.22 |
$m_{d}$(GeV) | 0.163 | 0.22 |
$m_{s}$(GeV) | 0.387 | 0.419 |
$b$(GeV2) | 0.221 | 0.18 |
$c$(GeV) | -0.240 | -0.253 |
$\sigma_{0}$(GeV) | 1.799 | 1.80 |
$s$(GeV) | 1.497 | 1.55 |
$\mu$(GeV) | 0.0635 | 0 |
$\epsilon_{c}$ | -0.138 | -0.168 |
$\epsilon_{sov}$ | 0.157 | -0.035 |
$\epsilon_{sos}$ | 0.9726 | 0.055 |
$\epsilon_{t}$ | 0.893 | 0.025 |

In Table 2, we select forty one experimental data of meson listed in PDG and
optimize these meson masses to determine twelve parameters in Table 1. This
optimization has $\chi^{2}/n=82$ which is smaller than 2638 for the GI model
as shown in Table 2.

Table 2: The experimental data Olive  _et al._ (2014) fitted in our work.
$\chi^{2}=\frac{({\rm{(Th-Exp)}/{Error}})^{2}}{D.O.F}$, where This work, Exp,
and Error represent the theoretical value, experimental results, and
experimental error, respectively. We select some established $q\bar{q}$ meson
states in PDG Olive  _et al._ (2014) for our fitting. The unit of the mass is
GeV.
$\begin{array}[]{cccccccc}\hline\cr\hline\cr\text{Component}&\text{State}&\text{This
work}&\text{GI\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Godfrey:1985xj}{\@@citephrase{(}}{\@@citephrase{)}}}}&\text{Exp}&\text{error}\\\
\hline\cr\hbox{\multirowsetup\it{n}$\bar{n}$}&1^{1}\text{S}_{0}&{0.1397}&0.1524&0.13957&0.0001\\\
&2^{1}\text{S}_{0}&{1.294}&1.293&1.3&0.1\\\
&3^{1}\text{S}_{0}&{1.806}&1.874&1.812&0.012\\\
&1^{1}\text{P}_{1}&{1.228}&1.219&1.2295&0.0032\\\
&2^{1}\text{P}_{1}&{1.736}&1.777&1.96&0.03\\\
&3^{1}\text{P}_{1}&{2.117}&2.236&2.24&0.035\\\
&1^{1}\text{D}_{2}&{1.677}&1.68&1.6722&0.00025\\\
&2^{1}\text{D}_{2}&{2.056}&2.135&2.005&0.025\\\
&3^{1}\text{D}_{2}&{2.366}&2.534&2.285&0.03\\\
&1^{1}\text{F}_{3}&{2.002}&2.033&2.032&0.04\\\
&2^{1}\text{F}_{3}&{2.312}&2.431&2.245&0.019\\\
&1^{3}\text{S}_{1}&{0.7744}&0.7713&0.77526&0.026\\\
&2^{3}\text{S}_{1}&{1.424}&1.456&1.465&0.04\\\
&3^{3}\text{S}_{1}&{1.907}&1.998&1.9&0.022\\\
&4^{3}\text{S}_{1}&{2.258}&2.435&2.265&0.0005\\\
&1^{3}\text{P}_{0}&{1.191}&1.087&1.474&0.016\\\
&1^{3}\text{P}_{1}&{1.235}&1.238&1.23&0.015\\\
&2^{3}\text{P}_{1}&{1.756}&1.818&1.647&0.032\\\
&1^{3}\text{P}_{2}&{1.317}&1.307&1.3183&0.012\\\
&2^{3}\text{P}_{2}&{1.777}&1.823&1.732&0.05\\\
&1^{3}\text{D}_{1}&{1.646}&1.664&1.72&0.04\\\
&2^{3}\text{D}_{1}&{2.048}&2.153&2.0&0.035\\\
&3^{3}\text{D}_{1}&{2.364}&2.557&2.265&0.02\\\
&2^{3}\text{D}_{2}&{2.058}&2.155&1.94&0.03\\\
&3^{3}\text{D}_{2}&{2.370}&2.553&2.225&0.04\\\
&1^{3}\text{D}_{3}&{1.708}&1.683&1.6888&0.0021\\\
&2^{3}\text{D}_{3}&{2.074}&2.131&1.982&0.014\\\
&1^{3}\text{F}_{4}&{2.019}&2.008&1.996&0.035\\\
&2^{3}\text{F}_{4}&{2.322}&2.407&2.237&0.01\\\
&1^{3}\text{G}_{5}&{2.278}&2.296&2.33&0.005\\\
&1^{3}\text{H}_{6}&{2.501}&2.558&2.45&0.13\\\
\hline\cr\hbox{\multirowsetup{\it{n}}$\bar{s}$ or
s$\bar{n}$}&1^{1}\text{S}_{0}&{0.4953}&0.4625&0.4976&0.0004\\\
&2^{1}\text{S}_{0}&{1.471}&1.454&1.46&0.02\\\
&1^{3}\text{S}_{1}&{0.916}&0.9028&0.8958&0.0008\\\
&2^{3}\text{S}_{1}&{1.567}&1.579&1.414&0.015\\\
&1^{3}\text{P}_{0}&{1.325}&1.234&1.425&0.05\\\
&1^{3}\text{P}_{2}&{1.450}&1.428&1.4324&0.0013\\\
&1^{3}\text{D}_{1}&{1.765}&1.776&1.717&0.027\\\
&1^{3}\text{D}_{3}&{1.826}&1.794&1.776&0.007\\\
&1^{3}\text{F}_{4}&{2.126}&2.108&2.045&0.009\\\
&1^{3}\text{G}_{5}&{2.378}&2.388&2.382&0.024\\\
\hline\cr&\chi^{2}&{82}&2638&&\\\ \hline\cr\hline\cr\end{array}$

Of course, besides the mass spectrum mesons with $J^{PC}=5^{++}$ was
calculated by the GI model, Ref. Ebert  _et al._ (2009) also gives a spectrum
for $5^{++}$ meson and we will compare them later. Finally, we can obtain the
mass spectrum of these four $5^{++}$ states by the MGI mode list in Table 3
and compare our numerical result with GI model Godfrey and Isgur (1985) and
Ebert  _et al._ (2009).

Table 3: The mass spectrum of $5^{++}$ states. The unit of the mass is GeV.
$\begin{array}[]{cccccccc}\hline\cr\hline\cr\text{State}&\text{This
work}&\text{GI\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Godfrey:1985xj}{\@@citephrase{(}}{\@@citephrase{)}}}}&\text{Ebert\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ebert:2009ub}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr a_{5}(1H)&{2.492}&2.610&2.359\\\ a_{5}(2H)&{2.719}&2.941&-\\\
a_{5}(3H)&{2.922}&3.255&-\\\ f_{5}(1H)&{2.492}&2.610&2.359\\\
f_{5}(2H)&{2.719}&2.941&-\\\ f_{5}(3H)&{2.922}&3.255&-\\\
f_{5}^{\prime}(1H)&{2.679}&2.771&2.720\\\
f_{5}^{\prime}(2H)&{2.914}&3.097&-\\\ f_{5}^{\prime}(3H)&{3.118}&3.406&-\\\
\hline\cr\hline\cr\end{array}$

So we can conclude that

  1. 1.

The ground states of the $5^{++}$ states are still missing in experiments, and
the predicted mass is 2.492 GeV for $a_{5}/f_{5}$, which are smaller than Ref.
Godfrey and Isgur (1985) and close to the result of Ref. Ebert  _et al._
(2009). $f_{5}^{\prime}(1H)$ has the mass of 2.771 GeV which is smaller than
Ref. Godfrey and Isgur (1985) and Ref. Ebert  _et al._ (2009).

  2. 2.

The first excited states of $a_{5}/f_{5}$ and $f_{5}^{\prime}$ have the mass
of 2.719 GeV and 2.922GeV, respectively. For the second excited states,
$a_{5}/f_{5}$(3H) and $f_{5}^{\prime}$(3H) have the mass of 2.914 GeV and
3.118 GeV, respectively, which are also smaller than Ref. Godfrey and Isgur
(1985).

The above conclusions are only from the point of mass spectra view and we will
study their strong decays in the next section.

##  III THE DECAY BEHAVIOR ANALYSIS

###  III.1 QPC model

The QPC model is used to obtain Okubo-Zweig-Iizuka (OZI) allowed hadronic
strong decays. The QPC model is firstly proposed by Micu Micu (1969), which is
further developed by Orsay group.Le Yaouanc  _et al._ (1973, 1974, 1975,
1977a, 1977b). This model was widely applied to the OZI-allowed two-body
strong decay of hadrons in Refs. van Beveren  _et al._ (1980, 1983); Capstick
and Roberts (1994); Page (1995); Titov  _et al._ (1996); Ackleh  _et al._
(1996); Blundell (1996); Bonnaz  _et al._ (2002); Zhou  _et al._ (2005); Lu
_et al._ (2006); Zhang  _et al._ (2007); Luo  _et al._ (2009); Sun and Liu
(2009); Liu  _et al._ (2010); Sun  _et al._ (2010); Rijken  _et al._ (2010);
Ye  _et al._ (2012); Wang  _et al._ (2012); He  _et al._ (2013); Sun  _et al._
(2013); Pang  _et al._ (2014); Wang  _et al._ (2015).

For a decay process $A\to B+C$, we can write

$\displaystyle\langle
BC|\mathcal{T}|A\rangle=\delta^{3}(\mathbf{P}_{B}+\mathbf{P}_{C})\mathcal{M}^{{M}_{J_{A}}M_{J_{B}}M_{J_{C}}},$
(3.1)

where $\mathbf{P}_{B(C)}$ 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 $\mathcal{T}$ is introduced to
describe a quark-antiquark pair creation from vacuum, which has the quantum
number $J^{PC}=0^{++}$, i.e., $\mathcal{T}$ can be expressed as

$\displaystyle\mathcal{T}$ $\displaystyle=$
$\displaystyle-3\gamma\sum_{m}\langle 1m;1~{}-m|00\rangle\int
d\mathbf{p}_{3}d\mathbf{p}_{4}\delta^{3}(\mathbf{p}_{3}+\mathbf{p}_{4})$ (3.2)
$\displaystyle\times\mathcal{Y}_{1m}\left(\frac{\textbf{p}_{3}-\mathbf{p}_{4}}{2}\right)\chi_{1,-m}^{34}\phi_{0}^{34}\left(\omega_{0}^{34}\right)_{ij}b_{3i}^{{\dagger}}(\mathbf{p}_{3})d_{4j}^{{\dagger}}(\mathbf{p}_{4}).$

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 $\gamma$ depicts the strength of the creation of
$q\bar{q}$ from vacuum, where the concrete values of the parameter $R$ which
will be discussed in the later section. $\mathcal{Y}_{\ell
m}(\mathbf{p})={|\mathbf{p}|^{\ell}}Y_{\ell m}(\mathbf{p})$ are the solid
harmonics. $\chi$, $\phi$, and $\omega$ denote the spin, flavor, and color
wave functions respectively, which can be treated separately. Subindices $i$
and $j$ denote the color of a $q\bar{q}$ pair.

By the Jacob-Wick formula Jacob and Wick (1959), the decay amplitude is
expressed as

$\displaystyle\mathcal{M}^{JL}(\mathbf{P})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{4\pi(2L+1)}}{2J_{A}+1}\sum_{M_{J_{B}}M_{J_{C}}}\langle
L0;JM_{J_{A}}|J_{A}M_{J_{A}}\rangle$ (3.3) $\displaystyle\times\langle
J_{B}M_{J_{B}};J_{C}M_{J_{C}}|{J_{A}}M_{J_{A}}\rangle\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}},$

and the general decay width reads

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{\pi}{4}\frac{|\mathbf{P}|}{m_{A}^{2}}\sum_{J,L}|\mathcal{M}^{JL}(\mathbf{P})|^{2},$
(3.4)

where $m_{A}$ is the mass of an initial state $A$. In our calculation, we need
the spatial wave functions of the discussed mesons. which can be numerically
obtained by the modified GI model.

In the previous section, we obtain the mass spectrum and wave functions of the
mesons. At the same time, we can use QPC model to study the strong decay of
the $J^{PC}=5^{++}$ meson families by the means of these wave functions.

As a phenomenological model of calculating strong decays of hadron, Quark pair
creation (QPC) model was employed to estimate the decay behaviors of hadron.
However, QPC model cannot very precisely describe experimental data. To some
extent, $\rho\to\pi\pi$ cannot be reproduced well as shown in Table 4. In
fact, it is a long-standing question not only for QPC model but also for other
quark models like flux-tube model Wang  _et al._ (2017). Just considering this
point, usually we selected more typical channels to fix $\gamma$ value in QPC
model, where a global fit is adopted. And then, this fixed $\gamma$ value is
applied to calculate other decays. In this work we obtain $\gamma=11.6$ by
fitting the partial decay widths of 30 decay channels as shown in Table 4.

Table 4: The measured partial decay widths of 30 decay channels and the comparison with theoretical calculation (the third column). Here, the minimum of $\chi^{2}$ is $636$.  Decay channel | Exp(MeV) Ye  _et al._ (2012); Pang  _et al._ (2017) | This work
---|---|---
$\rho\to\pi\pi$ | $151.2\pm 1.2$ | 68.9
$b_{1}\rightarrow\omega\pi$ | 142$\pm$8 | 191
$\phi\rightarrow KK$ | 2.08$\pm$0.02 | 1.53
$a_{2}\rightarrow\eta\pi$ | 15.5$\pm$0.7 | 2.65
$a_{2}\to\rho\pi$ | $75.0\pm 4.5$ | 66.2
$a_{2}\rightarrow KK$ | 5.2$\pm$0.2 | 4.53
$\pi_{2}\rightarrow f_{2}(1270)\pi$ | 145.8$\pm$5.1 | 48.6
$\pi_{2}\rightarrow\rho\pi$ | 80.3$\pm$2.8 | 220
$\pi_{2}\to K^{*}{K}$ | $10.1\pm 3.4$ | 26.7
$\rho_{3}\rightarrow\pi\pi$ | 38$\pm$2.4 | 61.8
$\rho_{3}\rightarrow\omega\pi$ | 25.8$\pm$1.6 | 44.3
$\rho_{3}\rightarrow K\overline{K}$ | 2.5$\pm$0.2 | 2.43
$f_{2}\to\pi\pi$ | $156.8\pm 3.2$ | 136
$f_{2}\to K\bar{K}$ |  $8.6\pm 0.8$ s | 8.31
$f_{4}(2050)\to\omega\omega$ | $54\pm 13$ | 70.1
$f_{4}(2050)\to\pi\pi$ | $35.4\pm 3.8$ | 79
$f_{4}(2050)\to KK$ | $1.4\pm 0.7$ | 0.941
$f_{2}^{\prime}(1525)\to K\bar{K}$ | $61\pm 5$ | 53.7
$K^{*}(892)\rightarrow K\pi$ | 48.7$\pm$0.8 | 18.6
$K^{*}(1410)\rightarrow K\pi$ | 15.3$\pm$1.4 | 62.2
$K^{*}_{0}(1430)\rightarrow K\pi$ | 251$\pm$74 | 291
$K^{*}_{2}(1430)\rightarrow K\pi$ | 54.4$\pm$2.5 | 50.2
$K^{*}_{2}(1430)\rightarrow K^{*}\pi$ | 26.9$\pm$1.2 | 19.9
$K^{*}_{2}(1430)\rightarrow K\rho$ | 9.5$\pm$0.4 | 7.18
$K^{*}_{2}(1430)\rightarrow K\omega$ | 3.16$\pm$0.15 | 2.13
$K_{3}^{*}(1780)\to K\rho$ | $74\pm 10$ | 25.8
$K_{3}^{*}(1780)\to K^{*}\pi$ | $45\pm 7$ | 28.3
$K_{3}^{*}(1780)\to K\pi$ | $31.7\pm 3.7$ | 38.1
$K_{4}^{*}(2045)\to K\pi$ | $19.6\pm 3.8$ | 22
$K_{4}^{*}(2045)\to K^{*}\phi$ | $2.8\pm 1.4$ | 34.8
$\gamma=11.6$ |  | $\chi^{2}=636$

Next, we will analyze the strong decay behavior of these $5^{++}$ states.

###  III.2  The ground states

The ground state $a_{5}$ which is not observed in experiment is predicted in
this work, with the mass of 2492 MeV ($a_{5}(2492)$), and the total width is
396 MeV. $\rho_{3}\pi$ is its dominant decay channel, the width is about 137
MeV, and the branch ratio is $0.36$. $a_{2}\rho$, $\omega\rho$, $\rho\pi$ ,and
$h_{1}\rho$, are its important decay channels which have the branch ratio
about 0.08 each one, just as shown in Table 5. The final states $b_{1}\omega$,
$f_{2}\pi$, $f_{4}(2050)\pi$, $a_{1}\rho$ and $\rho(1450)\pi$, also have
sizable decay widths, in which $b_{1}\omega$ and $f_{2}\pi$ almost have the
same width about 20 MeV.

As the isospin partner of $a_{5}(1H)$, we predict $f_{5}(1H)$ will have the
mass of 2.49 GeV and the width of 327 MeV, respectively. In the final decay
channels of $f_{5}(1H)$, $\rho\rho$, and $b_{1}\rho$ will be the most
important final states which have the widths of 73 MeV, and their branch
ratios are about 0.22. $a_{2}\pi$ and $a_{1}\pi$ are the important decay
channels too, with the widths of 58 MeV and 54 MeV, respectively. In addition,
$h_{1}\omega$ and $\pi_{2}\pi$ also have visible widths of 30 MeV and 28 MeV
which are presented in Table 5. The widths of $KK_{3}(1780)$ and
$\rho(1450)\rho$ are very small(see Table 5), their branch ratios are about
0.017.

$f_{5}^{\prime}(1H)$ is the $s\bar{s}$ partner of $f_{5}(1H)$, has the mass of
2.68 GeV and the width more than 850 MeV in our prediction. As shown in Table
5, $f_{5}^{\prime}(1H)$ mainly decays to two strange mesons for its $s\bar{s}$
component. $KK_{3}^{*}(1780)$ is the dominant decay mode with the width 161
MeV. $K^{*}K_{2}^{*}(1430)$, $K_{1}K^{*}$ are also the important decay
channels whose widths are over 100 MeV. Besides, $K^{*}K^{*}$,
$KK_{2}^{*}(1430)$, $KK^{*}(1410)$, $KK^{*}$ are its sizable final channels
with the branch ratios about 0.08. $KK_{1}$, $KK_{4}^{*}(2045)$,
$K^{*}K_{1}^{\prime}$, $KK^{*}(1680)$, $KK_{0}^{*}(1430)$ are the visible
decay channels of $f_{5}^{\prime}(1H)$. Here, we do not consider the mix of
the flavor between $f_{5}(1H)$ and $f_{5}^{\prime}(1H)$.

Table 5: The partial decay widths of the ground states for $5^{++}$ family,
the unit of widths is MeV.
$\begin{array}[]{cccccc}\hline\cr\hline\cr\lx@intercol\hfil{a_{5}\text{(1H)}}\hfil\lx@intercol&\lx@intercol\hfil
f_{5}\text{(1H)}\hfil\lx@intercol&\lx@intercol\hfil
f_{5}^{\prime}\text{(1H)}\hfil\lx@intercol\\\ \\\ \hline\cr
Total&{396}&Total&{327}&Total&{851}\\\ \hline\cr
Channel&Value&Channel&Value&Channel&Value\\\
\rho_{3}\pi&{137}&\rho\rho&{72.7}&KK_{3}^{*}(1780)&{161}\\\
a_{2}\rho&{54.2}&b_{1}\rho&{72.9}&K^{*}K_{2}^{*}(1430)&{144}\\\
\omega\rho&{35.6}&a_{2}\pi&{58.4}&K_{1}K^{*}&{111}\\\
\rho\pi&{31.5}&a_{1}\pi&{54.2}&K^{*}K^{*}&{85.6}\\\
h_{1}\rho&{31.0}&h_{1}\omega&{30.3}&KK_{2}^{*}(1430)&{76.3}\\\
b_{1}\omega&{23.6}&\pi_{2}\pi&{27.6}&KK^{*}(1410)&{73.0}\\\
f_{2}\pi&{21.4}&KK_{3}^{*}(1780)&{5.63}&KK^{*}&{60.9}\\\
f_{4}(2050)\pi&{19.0}&\rho(1450)\rho&{5.38}&KK_{1}&{48.9}\\\
a_{1}\rho&{18.1}&&&KK_{4}^{*}(2045)&{49.4}\\\
\rho(1450)\pi&{16.7}&&&K_{1}^{\prime}K^{*}&{26.2}\\\
KK_{3}^{*}(1780)&{5.63}&&&KK^{*}(1680)&{13.4}\\\
\omega(1420)\rho&{2.05}&&&f_{1}(1425)\eta^{\prime}&{1.31}\\\
\hline\cr\hline\cr\end{array}$

###  III.3 The first excited states

Table 6: The partial decay widths of the first excited states for $5^{++}$
family, the unit of widths is MeV.
$\begin{array}[]{cccccc}\hline\cr\hline\cr\lx@intercol\hfil{a_{5}\text{(2H)}}\hfil\lx@intercol&\lx@intercol\hfil
f_{5}\text{(2H)}\hfil\lx@intercol&\lx@intercol\hfil
f_{5}^{\prime}\text{(2H)}\hfil\lx@intercol\\\ \hline\cr
Channel&Value&Channel&Value&Channel&Value\\\
Total&{159}&Total&{113}&Total&{637}\\\
\rho_{3}\pi&{76.9}&\rho(1450)\rho&{43.5}&K^{*}K^{*}(1680)&{111}\\\
a_{0}(1450)\rho&{17.9}&\omega(1420)\omega&{18.0}&KK_{3}^{*}(1780)&{96.0}\\\
\rho(1450)\pi&{16.9}&\rho\rho&{9.66}&KK^{*}(1680)&{54.5}\\\
\pi_{2}\rho&{9.81}&a_{1}\pi&{8.19}&KK^{*}&{51.6}\\\
a_{1}b_{1}&{8.97}&\omega_{3}\omega&{7.01}&K^{*}K^{*}&{46.7}\\\
\omega_{3}\rho&{7.51}&a_{2}\pi&{6.91}&K^{*}K^{*}(1410)&{43.3}\\\
\rho_{3}\omega&{5.48}&a_{1}a_{2}&{5.88}&K_{1}K_{2}^{*}(1430)&{38.8}\\\
\omega\rho&{4.76}&a_{2}a_{2}&{2.51}&KK_{1}&{37.6}\\\
f_{2}a_{1}&{2.90}&\omega\omega&{2.34}&K^{*}K_{3}^{*}(1780)&{28.1}\\\
f_{2}\pi&{2.82}&\pi_{2}\pi&{2.01}&K_{1}K^{*}&{27.8}\\\
a_{2}f_{2}&{2.08}&f_{1}f_{2}&{1.75}&KK_{2}^{*}(1430)&{26.8}\\\
a_{2}b_{1}&{1.72}&f_{2}f_{2}&{1.63}&K^{*}K_{2}^{*}(1430)&{22.7}\\\
a_{2}f_{1}&{1.65}&b_{1}b_{1}&{1.63}&K_{1}^{\prime}K^{*}(1410)&{16.7}\\\
&&a_{1}a_{1}&{1.45}&K_{1}K^{*}(1410)&{15.8}\\\
&&&&K^{*}(1410)K_{2}^{*}(1430)&{7.32}\\\ &&&&K_{1}K_{1}^{\prime}&{5.67}\\\
&&&&KK_{1}^{\prime}&{3.97}\\\ &&&&KK^{*}(1410)&{2.72}\\\
\hline\cr\hline\cr\end{array}$

In this section, we will analyze the strong decay behavior of the first
excited states of $5^{++}$ family.

$a_{5}(2H)$ has the mass of 2719 MeV and narrow width of 159 MeV in our theory
result. According to Table 6, $\rho_{3}\pi$ is the dominant decay channel of
$a_{5}(2H)$ which is similar to $a_{5}(1H)$, the branch ratio is about 0.49.
$\rho(1450)\pi$ and $a_{0}(1450)\rho$ are its important decay channels which
have the branch ratios about 0.1, just as Table 6 shown. The final decay modes
$\pi_{2}\rho$, $\omega_{3}\rho$, $a_{1}b_{1}$, $\omega\rho$ and
$\rho_{3}\omega$, also have sizable decay widths, with the width of5-10 MeV.
The other decay information is shown in Table 6.

$f_{5}(2H)$ as the isospin partner(I=0) of $a_{5}(2H)$ has the mass of 2.72
GeV and width of 113 MeV, respectively. $f_{5}(2H)$ mainly decays to
$\rho(1450)\rho$ which has the width of 44 MeV and the branch ratio 0.25.
$\omega(1420)\omega$, $\rho\rho$ and $a_{1}\pi$ are the important final states
which have the widths of 18 MeV, 9.7 MeV and 8.2 MeV, respectively, and their
branch ratios are $0.15$, $0.11$ ,and $0.10$, respectively. $a_{2}\pi$,
$\omega_{3}\omega$ $a_{1}a_{2}$, and $\omega\omega$ also have visible widths
from 2.3 MeV to 7 MeV which are presented in Table 6. The widths of other
channels are very small(see Table 6) and their branch ratios are less than
0.02.

As the $s\bar{s}$ partner of $f_{5}(2H)$, $f_{5}^{\prime}(2H)$ has the mass of
2.91 GeV and the width of 637 MeV in our prediction. According to Tab 6,
$f_{5}^{\prime}(2H)$ also mainly decays into two kaon mesons.
$K^{*}K^{*}(1680)$ and $KK_{3}^{*}(1780)$ are the most important decay modes
with the widths of 111 MeV and 96 MeV, respectively. $KK^{*}(1680)$, $KK^{*}$,
$K^{*}K^{*}$ and $K^{*}K^{*}(1410)$ are also the more important decay channels
whose widths are in the range of 43-54 MeV. In addition,
$K_{1}K_{2}^{*}(1430)$, $KK_{1}$, $K^{*}K_{3}^{*}(1780)$, $K_{1}K^{*}$,
$KK_{2}^{*}(1430)$ and $K^{*}K_{2}^{*}(1430)$, are its sizable final channels
whose branch ratios in the range of 0.04-0.07.

###  III.4  The second excited states

Table 7: The partial decay widths of the second excfited states for $5^{++}$
family, the unit of widths is MeV.
$\begin{array}[]{cccccc}\hline\cr\hline\cr\lx@intercol\hfil{a_{5}\text{(3H)}}\hfil\lx@intercol&\lx@intercol\hfil
f_{5}\text{(3H)}\hfil\lx@intercol&\lx@intercol\hfil
f_{5}^{\prime}\text{(3H)}\hfil\lx@intercol\\\ \hline\cr
Channel&Value&Channel&Value&Channel&Value\\\
Total&{103}&Total&{57.2}&Total&{522}\\\
\rho_{3}\pi&{37.7}&\rho\rho&{13.6}&K_{1}K_{2}^{*}(1430)&{101}\\\
\rho(1450)\pi&{12.8}&\rho(1450)\rho&{10.1}&K^{*}K^{*}(1410)&{94.8}\\\
a_{1}b_{1}&{6.97}&a_{1}a_{2}&{6.23}&KK_{3}^{*}(1780)&{58.0}\\\
\omega\rho&{6.74}&\rho_{3}\rho&{5.8\par}&KK^{*}(1680)&{52.3}\\\
\rho\pi&{6.68}&a_{0}(1450)a_{1}&{5.1}&K^{*}K_{3}^{*}(1780)&{48.5}\\\
\pi_{2}\rho&{6.58}&\omega(1420)\omega&{4.48}&KK^{*}&{41.5}\\\
\omega_{3}\rho&{5.82}&a_{2}a_{2}&{3.82}&K^{*}K^{*}(1680)&{30.9}\\\
\omega(1420)\rho&{4.70}&\pi_{2}\pi&{2.76}&KK_{1}&{20.4}\\\
\omega\rho(1450)&{3.23}&a_{1}a_{1}&{2.76}&K_{1}K^{*}(1410)&{11.9}\\\
a_{2}\rho&{2.92}&b_{1}b_{1}&{1.87}&K^{*}K_{1}&{11.3}\\\
f_{2}a_{2}&{2.57}&a_{1}\pi&{0.674}&K_{1}^{\prime}K^{*}(1410)&{10.7}\\\
a_{2}b_{1}&{2.14}&&&K^{*}K_{2}^{*}(1430)&{8.55}\\\
f_{2}a_{1}&{2.90}&&&K^{*}(1410)K^{*}(1410)&{8.03}\\\
b_{1}h_{1}&{1.43}&&&KK^{*}(1410)&{7.72}\\\ &&&&K_{1}K_{1}&{7.37}\\\
&&&&K_{1}K_{1}^{\prime}&{6.78}\\\ &&&&K_{1}K^{*}(1680)&{2.61}\\\
\hline\cr\hline\cr\end{array}$

We also calculate the two body strong decays of the second excited states of
$5^{++}$ family.

As the isovector meson of $5^{++}$ family, $a_{5}(3H)$ has the mass of 2.92
GeV, the total width of 103 MeV which is very narrow. $\rho_{3}\pi$ also is
its dominant decay channel as shown in Table 7, the width is about 37.7 MeV,
and the branch ratio is $0.4$. $\rho(1450)\pi$ has a large ratio (0.13) in its
decay final channels. $\rho\pi$, $\omega\rho$, $a_{1}b_{1}$, $\pi_{2}\rho$,
$\omega_{3}\rho$, $\omega(1420)\rho$, and $\rho(1450)\omega$ also have sizable
contribution in the total widths.

$f_{5}(3H)$ state is the second radial excited state of $f_{5}$, with the mass
of 2.92 GeV and width of 57 MeV. $f_{5}(3H)$ mainly decays into $\rho\rho$ and
$\rho(1450)\rho$, whose decay widths are 13.6 MeV and of 10.1 MeV,
respectively, and each channel almost has the branch ratio 0.2. $a_{1}a_{2}$,
$\rho_{3}\rho$, and $\omega(1420)\omega$ 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 $\pi_{2}\pi$ also have visible widths which are
presented in Table 7. The width of other modes are very small(see Table 7).

$f_{5}^{\prime}(3H)$ has the $s\bar{s}$ component as the partner of
$f_{5}(3H)$ which has the mass of 3.12 GeV. $f_{5}^{\prime}(3H)$ has the total
width of 522 MeV in our calculation. Just as shown in Table 7,
$f_{5}^{\prime}(3H)$ mainly decays to $K_{1}K_{2}^{*}(1430)$ and
$KK^{*}(1410)$ with the widths of 101 MeV and 95 MeV. $KK_{3}^{*}(1780)$,
$KK^{*}(1680)$, $K^{*}K_{3}^{*}(1780)$, and $KK^{*}$ are its important decay
channels with the widths of 58 MeV, 52 MeV, 49 MeV, and 42 MeV, respectively.

$K^{*}K^{*}(1680)$, $KK_{1}$, $K_{1}K^{*}(1410)$, $K^{*}K_{1}$, and
$K_{1}^{\prime}K^{*}(1410)$ are its sizable final channels with the branch
ratios of 0.06, 0.04, 0.023, 0.022, and 0.02, respectively. Besides,
$K^{*}K_{2}^{*}(1430)$, $K^{*}(1410)K^{*}(1410)$, $KK^{*}(1410)$,
$K_{1}K_{1}$, and $K_{1}K_{1}^{\prime})$ have the visible contribution to the
total width too. The other modes have very small widths in the final states of
$f_{5}^{\prime}(3H)$.

##  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 analyze 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 $f_{5}^{\prime}$ have the
mass of 2.492 GeV, 2.492 GeV, and 2.68 GeV and the widths of 400 MeV, 330 MeV,
and 850 MeV , respectively. The first excited states of $a_{5}/f_{5}$ and
$f_{5}^{\prime}$ have the mass of 2.719 GeV and 2.922 GeV and
$a_{5}(3H)/f_{5}(3H)$ and $f_{5}^{\prime}(3H)$ have the mass of 2.914 GeV and
3.118 GeV, respectively. The total widths are predicted to be 160
MeV($a_{5}$(2H)), 110 MeV($f_{5}$(2H)), and 640 MeV($f_{5}^{\prime}$(2H)) for
the first excited states. For the second excited states of $5^{++}$,
$a_{5}$(3H), $f_{5}$ (3H), and $f_{5}^{\prime}(3H)$ have the widths of 100
MeV($a_{5}$), 60 MeV($f_{5}$), and 430 MeV($f_{5}^{\prime}$), respectively.

We also predict the detail of decay information of $5^{++}$ family using QPC
model which can be helpful to search the mesons in the future experiments just
as BESIII and COMPASS.

##  V ACKNOWLEDGMENTS

We thanks Xiang Liu, Jia-lun Ping, and Zhi-feng Sun for helpful communications
and discussions. This work is supported in part by the Nature Science
Foundation Projects of Qinghai Office of Science and Technology, No.
2017-ZJ-748, the Chunhui Plan of China’s Ministry of Education, No. Z2017054.

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