# Anisotropy in Dijet Production in Exclusive and Inclusive Processes

Yoshitaka Hatta  Physics Department, Building 510A, Brookhaven National
Laboratory, Upton, NY 11973, USA  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, New York 11973, USA  Bo-Wen Xiao  School of
Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172,
China  Feng Yuan  Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Jian Zhou  Key Laboratory of Particle
Physics and Particle Irradiation (MOE),Institute of Frontier and
Interdisciplinary Science, Shandong University (QingDao), Shandong 266237,
China

###### Abstract

We investigate the effect of soft gluon radiations on the azimuthal angle
correlation between the total and relative momenta of two jets in inclusive
and exclusive dijet processes. We show that the final state effect induces a
sizable $\cos(2\phi)$ anisotropy due to gluon emissions near the jet cones.
The phenomenological consequences of this observation are discussed for
various collider experiments, including diffractive processes in
ultraperipheral $pA$ and $AA$ collisions, inclusive and diffractive dijet
production at the EIC, and inclusive dijet in $pp$ and $AA$ collisions at the
LHC.

1\. Introduction. Dijet productions are the most abundant events in hadronic
collisions and have been under intensive investigations from both experiment
and theory sides. Typically the two final state jets are produced in the so-
called correlation configuration, namely, back-to-back in the transverse plane
with nearly balanced transverse momenta Abazov:2004hm ; Abelev:2007ii ;
Khachatryan:2011zj ; daCosta:2011ni ; Aad:2010bu ; Chatrchyan:2011sx ;
Adamczyk:2013jei ; Aaboud:2019oop . Deviations from the perfect back-to-back
configuration inform important strong interaction physics, and, in particular,
the non-perturbative structure of the nucleon and nucleus Boer:2003tx ;
Marquet:2007vb ; Dominguez:2010xd ; Dominguez:2011wm ; Boer:2010zf ;
Metz:2011wb ; Dumitru:2015gaa ; Boer:2017xpy ; Boer:2016fqd ; Xing:2020hwh ;
Altinoluk:2015dpi ; Hatta:2016dxp ; Mantysaari:2019csc ; Mantysaari:2019hkq .
It is expected to reveal the medium property by studying de-correlation of
dijet in heavy ion collisions as well Mueller:2016gko ; Mueller:2016xoc ;
Chen:2016vem ; Chen:2016cof ; Chen:2018fqu ; Tannenbaum:2017afg ;
Adamczyk:2017yhe ; Gyulassy:2018qhr .

Let $\vec{q}_{\perp}=\vec{k}_{1\perp}+\vec{k}_{2\perp}$ be the total
transverse momentum of the two jets. We also define
$\vec{P}_{\perp}=(\vec{k}_{1\perp}-\vec{k}_{2\perp})/2$ as the leading jet
transverse momentum, see Fig. 1. The correlation limit is defined by the
condition $q_{\perp}\ll P_{\perp}\sim k_{1\perp}\sim k_{2\perp}$. In general,
the azimuthal angular correlation between $\vec{q}_{\perp}$ and
$\vec{P}_{\perp}$ is isotropic. However, anisotropic distribution can be
generated from nontrivial correlations in the transverse momentum dependent
parton distribution (TMD) associated with the incoming hadrons. This provides
a unique probe to the novel tomography imaging of the nucleon and nucleus in
dijet production.

Among the proposed observables, of particular interest is the $\cos(2\phi)$
anisotropic correlation in exclusive diffractive dijet production
Hatta:2016dxp  where $\phi$ is the angle between $\vec{P}_{\perp}$ and the
target recoil momentum $\vec{\Delta}_{\perp}$. Such a correlation provides a
unique access to the so-called elliptic gluon Wigner distribution of the
target Hatta:2016dxp ; Hagiwara:2016kam ; Zhou:2016rnt . Partly motivated by
this observation (see, also, Hagiwara:2017fye ), very recently, the CMS
collaboration has reported a preliminary measurement of this anisotropy in
ultraperipheral collisions (UPC) at the LHC CMS:2020ekd . Using $q_{\perp}$ as
a proxy for $\Delta_{\perp}$, CMS reported a significant $\cos(2\phi)$
asymmetry. Intrigued by this result, in this paper, we will perform a detailed
analysis of this anisotropic distribution, focusing mainly on the soft gluon
radiation contributions.

Figure 1: Dijet in transverse plane perpendicular to the beam direction at
hadron colliders. Their total transverse momentum
$\vec{q}_{\perp}=\vec{k}_{1\perp}+\vec{k}_{2\perp}$ is much smaller than the
individual jet momentum
$\vec{P}_{\perp}=(\vec{k}_{1\perp}-\vec{k}_{2\perp})/2$. Angular distribution
between $q_{\perp}$ and $P_{\perp}$ has an anisotropy due to soft gluon
radiation associated with the final state jet with a non-zero
$\langle\cos(2\phi)\rangle$.

Soft gluon radiation contributions to the azimuthal angular anisotropy of
$\cos(2\phi)$ and their resummation for various processes have been
investigated in Refs. Boer:2006eq ; Berger:2007si ; Bacchetta:2008xw ;
Bacchetta:2019qkv ; Nadolsky:2007ba ; Catani:2010pd ; Catani:2014qha ;
Catani:2017tuc . For the jet-related processes, they were first studied in
Refs. Catani:2014qha ; Catani:2017tuc  and it was found that their
contributions are not power suppressed and can be resummed to all orders in
perturbation theory. In this paper, we will follow these developments and
apply to dijet production processes. For azimuthally symmetric distributions,
the soft gluon resummation has been derived in Refs. Banfi:2003jj ;
Banfi:2008qs ; Hautmann:2008vd ; Mueller:2013wwa ; Sun:2014gfa ; Sun:2015doa ;
Hatta:2019ixj ; Liu:2018trl ; Liu:2020dct .

The physical picture is as follows. Soft gluons emitted from the final state
jets tend to be aligned with the jet directions. Since gluons emitted too
close to the jet axis become part of the jet, what matters is the emissions
slightly outside the jet cones, and $\vec{q}_{\perp}$ is essentially the
recoil momentum against these gluons. This means that $\vec{q}_{\perp}$ also
tends to point to jet directions on average, resulting in a positive
$\langle\cos(2\phi)\rangle$. When $q_{\perp}\ll P_{\perp}$, one needs to
perform the resummation of logarithms $\alpha_{s}^{n}(\ln
P_{\perp}/q_{\perp})^{m}$ which can be done in the TMD framework. We shall
demonstrate that this simple picture can explain, at least partly, the
observation by the CMS collaboration.

Our result in this paper will have a broad impact on the tomographic study of
nucleons and nuclei at the future electron-ion collider (EIC) Accardi:2012qut
; Proceedings:2020eah . Various anisotropy observables have been proposed to
study the novel structure of nucleon and nucleus Boer:2010zf ; Metz:2011wb ;
Dumitru:2015gaa ; Boer:2017xpy ; Boer:2016fqd ; Xing:2020hwh ;
Altinoluk:2015dpi ; Hatta:2016dxp ; Mantysaari:2019csc ; Mantysaari:2019hkq .
The results in these previous works have to be re-examined.

2\. Diffractive Photoproduction of Dijet. We first study the diffractive
photoproduction of dijets, $\gamma A\to q\bar{q}+A$. The photon fluctuates
into a quark-antiquark pair which then scatters off the nucleon and nucleus
target and forms a final state dijet with momentum $k_{1}$ and $k_{2}$. This
process can be studied, for example, in ultraperipheral $pA$ and $AA$
collisions and the planned EIC in the future.

Let us define the soft factor $S_{J}(q_{\perp})$ as the probability for the
dijet to emit total transverse momentum $q_{\perp}$ outside the jet cones.
Implicitly, it should also depend on the dijet relative transverse momentum
$P_{\perp}=\frac{1}{2}(k_{1\perp}-k_{2\perp})$. The dijet cross section then
reads

$\frac{d\sigma}{d^{2}P_{\perp}d^{2}q_{\perp}}=\int
d^{2}q^{\prime}_{\perp}\frac{d\sigma_{0}}{d^{2}P_{\perp}d^{2}q^{\prime}_{\perp}}S_{J}(q_{\perp}-q^{\prime}_{\perp}),$
(1)

where $\sigma_{0}$ is the cross section without soft radiations. We assume
that the two jets are back-to-back in azimuth, and consider the soft regime
$|q_{\perp}|\ll|P_{\perp}|$. To lowest nontrivial order, $S_{J}$ is given by
the standard eikonal formula

$S_{J}(q_{\perp})=\delta(q_{\perp})+\frac{\alpha_{s}}{2\pi^{2}}\int
dy_{g}\left(\frac{k_{1}\cdot k_{2}}{k_{1}\cdot k_{g}k_{2}\cdot
k_{g}}\right)_{\vec{q}_{\perp}=-\vec{k}_{g\perp}},$ (2)

where $k_{g}$ is the soft gluon momentum emitted from the dijet and $y_{g}$ is
its rapidity. A power-counting argument shows that $S_{J}(q_{\perp})\sim
1/q_{\perp}^{2}$ at small $q_{\perp}$, and we neglect power corrections
$(q_{\perp}/P_{\perp})^{n}$ to this leading behavior such as coming from gluon
emissions from the $t$-channel Pomeron (see, e.g., Boussarie:2014lxa ). We
shall be particularly interested in the $\phi$-distribution where $\phi$ is
the azimuthal angle between $q_{\perp}$ and $P_{\perp}$. In general one can
expand

$\displaystyle
S_{J}(q_{\perp})=S_{J0}(|q_{\perp}|)+2\cos(2\phi)S_{J2}(|q_{\perp}|)+\cdots\
,$ (3)

where for simplicity higher harmonics are neglected in this work. To ${\cal
O}(\alpha_{s})$, $S_{J0,J2}$ can be obtained by calculating the Fourier
coefficients

$\int_{0}^{2\pi}d\phi\int_{-\infty}^{\infty}dy_{g}\frac{k_{1}\cdot
k_{2}}{k_{1}\cdot k_{g}k_{2}\cdot k_{g}}\\{1,\cos(2\phi)\\}.$ (4)

There are collinear divergences when $k_{g}$ is parallel to either $k_{1}$ or
$k_{2}$. They are factorized into the jet functions associated with the two
final state jets. In practice, we introduce the theta-function constraints

$\displaystyle\Theta(2\cosh(y_{1}-y_{g})-2\cos\phi-R^{2})$
$\displaystyle\approx 1-\Theta(R^{2}-(y_{1}-y_{g})^{2}-\phi^{2}),$ (5)

where $R$ is the jet radius, and similarly for the second jet. In the narrow
jet approximation $R\ll 1$ Aversa:1988vb ; Mukherjee:2012uz , we obtain

$S_{J0}(q_{\perp})=\delta(q_{\perp})+\frac{\alpha_{0}}{\pi}\frac{1}{q_{\perp}^{2}}\
,~{}~{}~{}S_{J2}(q_{\perp})=\frac{\alpha_{2}}{\pi}\frac{1}{q_{\perp}^{2}}\ ,$
(6)

where

$\displaystyle\alpha_{0}=\frac{\alpha_{s}C_{F}}{2\pi}2\ln\frac{a_{0}}{R^{2}}\
,~{}~{}~{}\alpha_{2}=\frac{\alpha_{s}C_{F}}{2\pi}2\ln\frac{a_{2}}{R^{2}}\ .$
(7)

$a_{0}$ depends on the rapidity difference $\Delta y_{12}=|y_{1}-y_{2}|$ as
$a_{0}=2+2\cosh(\Delta y_{12})$. A closed analytic expression is not available
for $a_{2}$ except in the two limiting cases $\Delta y_{12}=0,\infty$, but it
has very mild dependence on $\Delta y_{12}$. It increases slightly from
$a_{2}=1/4$ for $\Delta y_{12}=0$ to $a_{2}=1/e$ for $\Delta y_{12}\to\infty$.

We observe that the $\cos(2\phi)$ term is not power suppressed, which is
consistent with the finding in Ref. Catani:2017tuc . In the small-$R$ limit,
the ratio $\alpha_{2}/\alpha_{0}$ is essentially unity, and the $\phi$
distribution reduces to two delta functions around $0$ and $\pi$. However, for
realistic values of $R$, typically $\alpha_{2}/\alpha_{0}\lesssim 0.2$ because
$a_{0}\gg a_{2}$.

To have a reliable prediction, we need to perform the resummation of higher
order contributions to $S_{J}$ Hatta:2019ixj . The constraint
$q_{\perp}=-k_{1g\perp}-k_{2g\perp}-\cdots$ from multiple gluon emissions in
momentum space can be conveniently deconvoluted by Fourier-transforming to
$b_{\perp}$-space,

$\displaystyle\widetilde{S}_{J}(b_{\perp})$ $\displaystyle=$
$\displaystyle\int
d^{2}q_{\perp}e^{i\vec{q}_{\perp}\cdot\vec{b}_{\perp}}S_{J}(q_{\perp})$ (8)
$\displaystyle=$
$\displaystyle\widetilde{S}_{J0}(|b_{\perp}|)-2\cos(2\phi_{b})\widetilde{S}_{J2}(|b_{\perp}|)+\cdots\
,$

where $\phi_{b}$ represents the angle between $\vec{b}_{\perp}$ and
$\vec{P}_{\perp}$. Again we have only kept the leading two harmonics. To one-
loop, we find

$\displaystyle\widetilde{S}_{J0}(b_{\perp})=1+\alpha_{0}\ln(\mu_{b}^{2}/P_{\perp}^{2})\
,~{}~{}\widetilde{S}_{J2}(b_{\perp})=\alpha_{2}\ ,$ (9)

where $\mu_{b}=2e^{-\gamma_{E}}/b_{\perp}$ with $\gamma_{E}$ being the Euler
constant. In the above equation, we have taken into account the virtual
diagram contribution to $S_{J0}$ whose natural scale is the jet momentum
$P_{\perp}$. Note that in the $S_{J2}$ term, the Fourier transformation does
not introduce IR divergence. We extend (9) to all orders by exponentiating to
the Sudakov form factor,

$\displaystyle\widetilde{S}_{J0}(b_{\perp})=e^{-\Gamma_{0}(b_{\perp})}\
,~{}~{}\widetilde{S}_{J2}(b_{\perp})=\alpha_{2}e^{-\Gamma_{0}(b_{\perp})}\ ,$
(10)

where
$\Gamma_{0}(b_{\perp})=\int_{\mu_{b}^{2}}^{P_{\perp}^{2}}\frac{d\mu^{2}}{\mu^{2}}\alpha_{0}$.
Here and in what follows, we neglect the non-global logarithms (NGLs)
Dasgupta:2002bw ; Banfi:2008qs  which are not significant in the kinematics of
our interest.

Figure 2: Anisotropy in diffractive dijet production $\gamma+A\to q\bar{q}+A$
in ultra peripheral heavy ion collisions at the LHC. The kinematics correspond
to the CMS measurements CMS:2020ekd  with the leading jet $P_{\perp}=35\rm
GeV$, the two jets are at the same rapidity $\Delta y_{12}=0$ and the jet size
$R=0.4$. The figure shows $\langle\cos(2\phi)\rangle$ as a function of
$q_{\perp}$, where $\phi$ is the azimuthal angle between $q_{\perp}$ and
$P_{\perp}$. The leading order result (LO) represents the ratio
$\alpha_{2}/\alpha_{0}$ of Eq. (7), whereas full resummation depends on a non-
perturbative parameter $g_{\Lambda}$ (see the main text for the description).

In the fixed coupling case, the Fourier transform of (10) can be obtained
analytically, and we find that
$\langle\cos(2\phi)\rangle=S_{J2}(q_{\perp})/S_{J0}(q_{\perp})$ remains a
constant independent of $q_{\perp}$, but its value is reduced by a factor of
$(1-\alpha_{0})$ compared to the leading order result $\alpha_{2}/\alpha_{0}$.
In the running coupling case, we proceed numerically and adopt the
$b_{*}$-prescription Collins:1984kg :
$\Gamma_{0}(b_{\perp})\Longrightarrow\Gamma_{0}(b_{*})+g_{\Lambda}b_{\perp}^{2}$,
where $b_{*}=b_{\perp}/\sqrt{1+b_{\perp}^{2}/b_{\rm max}^{2}}$ with $b_{\rm
max}=1.5$ GeV-1. The parameter $g_{\Lambda}$ is applied to include non-
perturbative effects at large $b_{\perp}$, which should be order $\Lambda_{\rm
QCD}^{2}$. In the numerical calculations, we take three different values
$g_{\Lambda}=0.05$, $0.1$, $0.2~{}\rm GeV^{2}$. In Fig. 2, we show our
prediction for $\langle\cos(2\phi)\rangle$ for diffractive dijet production
$\gamma A\to q\bar{q}+A$, with $P_{\perp}=35\,\rm GeV$, $R=0.4$ and $\Delta
y_{12}=0$. For this jet size, the narrow jet approximation used in the above
derivations is reasonably good Aversa:1988vb ; Mukherjee:2012uz . We have
explicitly verified this by comparing Eq. (7) with the exact numerical
results. In the figure, we have also plotted the leading order result
$\alpha_{2}/\alpha_{0}\approx 0.14$ for comparison. We see that the anisotropy
does not change dramatically with $q_{\perp}$ in most kinematics except in the
very small $q_{\perp}$ region, where the prediction is sensitive to the non-
perturbative contributions as expected.

In the above result, we have neglected the possible contribution from the
elliptic gluon Wigner distribution Hatta:2016dxp . The anisotropy
$\langle\cos(2\phi)\rangle$ due to the elliptic Wigner is at most a few
percent Hagiwara:2016kam ; Mantysaari:2019csc , which is further suppressed
after convoluted with the symmetric part $S_{J0}$ Hatta:2019ixj . It is thus
negligible compared to the result in Fig. 2.

In Ref. Mantysaari:2019hkq , a significant $\cos(2\phi)$ anisotropy was also
found from a leading order color-glass-condensate calculation in the
kinematics of $q_{\perp}\gg P_{\perp}$, which is opposite to the correlation
limit considered in this paper. Comparison of these $\cos(2\phi)$ anisotropies
will help us to understand the underlying physics in dijet processes.

3\. Inclusive Dijet in $\gamma p$ Collisions. Next we turn to inclusive dijet
photoproduction $\gamma p\to jjX$. We shall show that, in contrast to the
diffractive case, the resummation of initial state radiations in the TMD of
the incoming proton strongly affects the azimuthal distribution of dijets. In
experiments such as at the EIC, photoproduction contains both direct photon
and resolved photon contributions. Here we focus on the direct photon
contribution where the leading order partonic subprocesses are $\gamma g\to
q\bar{q}$ and $\gamma q\to qg$. We take the example of the $\gamma g\to
q\bar{q}$ channel. The other channel $\gamma q\to qg$ can be treated
similarly.

In $\gamma g\to q\bar{q}$, we will have contributions from the soft gluon
radiation associated with the incoming gluon. To lowest nontrivial order, the
differential cross section can be written as,

$\frac{d^{4}\sigma}{d\Omega}=\sigma_{0}x_{g}f_{g}(x_{g})\frac{1}{q_{\perp}^{2}}\left[\frac{\alpha_{0}^{\gamma
g}}{\pi}+\frac{\alpha_{2}^{\gamma g}}{\pi}2\cos(2\phi)\right]\ ,$ (11)

where $\sigma_{0}$ represents the leading order cross section,
$d\Omega=dy_{1}dy_{2}dP_{\perp}^{2}d^{2}q_{\perp}$ for the phase space, and
$f_{g}(x_{g})$ is the gluon distribution with
$x_{g}=P_{\perp}\left(e^{y_{1}}+e^{y_{2}}\right)/\sqrt{S_{\gamma p}}$ is
momentum fraction of the nucleon carried by the gluon. The coefficient
$\alpha_{0,2}$ can be derived from the soft gluon radiation amplitude of
Mueller:2013wwa ,

$\displaystyle\alpha_{0}^{\gamma g}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{2\pi}\left[C_{A}\ln\frac{P_{\perp}^{2}}{q_{\perp}^{2}}+2C_{F}\ln\frac{a_{0}}{R^{2}}\right]\
,$ $\displaystyle\alpha_{2}^{\gamma g}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{2\pi}\left[C_{A}\ln\frac{a_{1}}{R^{2}}-\frac{1}{N_{c}}\ln\frac{a_{2}}{R^{2}}\right]\
,$ (12)

where $a_{1}=1/e$ and $a_{2}$ is the same as in the previous section. The
emission from the incoming gluon is isotropic, and gives rise to the double
logarithmic term $\frac{1}{q_{\perp}^{2}}\ln q_{\perp}^{2}$.

We now perform the resummation of double logarithms in the standard TMD
framework,

$\displaystyle\frac{d^{4}\sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle\sum_{ab}\sigma_{0}\int\frac{d^{2}\vec{b}_{\perp}}{(2\pi)^{2}}e^{-i\vec{q}_{\perp}\cdot\vec{b}_{\perp}}\left[\widetilde{W}_{0}^{\gamma
p}(|b_{\perp}|)\right.$ (13)
$\displaystyle\left.~{}~{}-2\cos(2\phi_{b})\widetilde{W}_{2}^{\gamma
p}(|b_{\perp}|)\right]\ ,$

where the azimuthal symmetric term can be written as

$\displaystyle\widetilde{W}_{0}^{\gamma
p}(b_{\perp})=x_{g}\,f_{g}(x_{g},\mu_{b})e^{-S^{\gamma
p}(P_{\perp}^{2},b_{\perp})}\ .$ (14)

We separate the Sudakov form factor $S(P_{\perp},b_{\perp})$ into perturbative
and non-perturbative parts: $S(P_{\perp},b_{\perp})=S_{\rm
pert.}(P_{\perp},b_{\perp})+S_{\rm NP}(P_{\perp},b_{\perp})$ with the
perturbative part at one-loop order is defined,

$S_{\rm pert.}^{\gamma
p}=\int^{P_{\perp}^{2}}_{\mu_{b}^{2}}\frac{d\mu^{2}}{\mu^{2}}\frac{\alpha_{s}C_{A}}{2\pi}\left[\ln\frac{P_{\perp}^{2}}{\mu^{2}}-2\beta_{0}+\frac{2C_{F}}{C_{A}}\ln\frac{a_{0}}{R^{2}}\right]\
,$ (15)

where $\beta_{0}=11/12-N_{f}/18$. For the non-perturbative part, we follow
those for the TMD quark distributions in Refs. Su:2014wpa ; Prokudin:2015ysa
with an appropriate color factor,

$S_{\rm NP}^{\gamma
p}=\frac{C_{A}}{C_{F}}\left[0.106\,b_{\perp}^{2}+0.42\ln\frac{P_{\perp}}{Q_{0}}\ln\frac{b_{\perp}}{b_{*}}\right]\
,$ (16)

with $Q_{0}^{2}=2.4$ GeV2. For the $\cos(2\phi)$ term, we have

$\displaystyle\widetilde{W}_{2}^{\gamma p}(b_{\perp})=\alpha_{2}^{\gamma
g}\widetilde{W}_{0}^{\gamma p}(b_{\perp})\ .$ (17)

Figure 3: Anisotropy of inclusive dijet production in $\gamma p$ collisions at
the future EIC for the typical kinematics: the leading jet $P_{\perp}=15~{}\rm
GeV$ and both jets are at the same rapidity. Here we plot
$\langle\cos(2\phi)\rangle$ as function and $q_{\perp}$, where $\phi$ is the
azimuthal angle between $q_{\perp}$ and $P_{\perp}$. The leading order (LO)
and full results are calculated from (12) and (13), respectively.

In Fig. 3, we show our numerical results for $\langle\cos(2\phi)\rangle$ as a
function of $q_{\perp}$ in the leading order (from (12)) and resummed (from
(13)) cases. We take the typical kinematics of inclusive dijet production at
the EIC with $\sqrt{S_{\gamma p}}=100\rm GeV$, $P_{\perp}\sim 15~{}\rm GeV$
and the two jets are at the same rapidity. Though it is not manifest in the
plot, the leading order (red) curve goes to zero as $q_{\perp}\to 0$, as it
should. Compared to the results in the diffractive case, the impact of
resummation is more pronounced. This is because the TMD resummation suppresses
the region $q_{\perp}\sim 0$ and shifts the distribution of $q_{\perp}$ to
larger values. This affects the angular dependent and independent parts of the
cross section differently, and therefore their ratio $\langle\cos
2\phi\rangle$ has a stronger $q_{\perp}$-dependence. In contrast, in the
diffractive case where the resummation is single-logarithmic, the effect
largely cancels in the ratio.

The $\cos(2\phi)$ asymmetry in inclusive dijet production in deep-inelastic
scattering ($Q^{2}\neq 0)$ has been proposed to study the so-called linearly
polarized gluon distribution in the nucleon Boer:2010zf ; Metz:2011wb ;
Dumitru:2015gaa ; Boer:2016fqd . As we have seen, the final state radiation
from the dijet system contributes to exactly the same $\cos(2\phi)$ asymmetry
already at leading order. Moreover, at higher orders the effect becomes
sensitive to the initial state radiation through the TMD resummation. It is
important to carefully subtract these radiative contributions in order to
unambiguously extract the linearly polarized gluon distribution from the
experimental data. We will publish the relevant results in a separate paper.

4\. Inclusive Dijet in $pp$ Collisions. We can also extend the above studies
to inclusive dijet production in $pp$ collisions. Again we have to take into
account the soft gluon radiation associated with incoming partons. In the
following, we take the example of $gg\to gg$ channel, which is also the
dominant contribution for dijet production in the typical kinematics at the
LHC. The differential cross section and the resummation formula follow the
similar expressions as Eqs. (11,13,14). For the soft gluon radiation
contribution, we find the following results,

$\displaystyle\alpha_{2}^{gg}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}C_{A}}{2\pi}\left[2\ln\frac{a_{1}}{R^{2}}+\frac{\hat{t}^{2}+\hat{u}^{2}}{2(\hat{s}^{2}-\hat{t}\hat{u})}\ln\frac{a_{2}}{a_{1}}\right]\
,$ (18)

and the associated $\alpha_{0}^{gg}$ has been derived in Ref. Sun:2015doa ,
where $\hat{s}$, $\hat{t}$ and $\hat{u}$ are usual Mandelstam variables in the
partonic process. For simplicity, we take the leading logarithmic
approximation, for example, for the azimuthal symmetric one,

$\widetilde{W}_{0}^{gg}(b_{\perp})=x_{1}\,f_{g}(x_{1},\mu_{b})x_{2}\,f_{g}(x_{2},\mu_{b})e^{-S^{gg}(P_{\perp}^{2},b_{\perp})}\
,$ (19)

where $f_{a,b}(x,\mu_{b})$ are parton distributions for the incoming partons
$a$ and $b$, $x_{1,2}=P_{\perp}\left(e^{\pm y_{1}}+e^{\pm
y_{2}}\right)/\sqrt{S}$ are momentum fractions of the incoming hadrons carried
by the partons. The perturbative part is defined as one-loop order,

$S_{\rm
pert.}^{gg}=\int^{P_{\perp}^{2}}_{\mu_{b}^{2}}\frac{d\mu^{2}}{\mu^{2}}\frac{\alpha_{s}C_{A}}{\pi}\left[\ln\left(\frac{P_{\perp}^{2}}{\mu^{2}}\right)-2\beta_{0}+\ln\frac{a_{0}}{R^{2}}\right]\
.$ (20)

For the $\cos(2\phi)$ term, we have

$\displaystyle\widetilde{W}_{2}^{gg}(b_{\perp})=\alpha_{2}^{gg}\widetilde{W}_{0}^{gg}(b_{\perp})\
.$ (21)

Figure 4: Anisotropy of inclusive dijet production from $gg\to gg$ channel in
$pp$ collisions at the LHC at $\sqrt{S_{pp}}=7~{}\rm TeV$ for the typical
kinematics: the leading jet $P_{\perp}=100\rm GeV$ and both jets are at mid-
rapidity. Here we plot $\langle\cos(2\phi)\rangle$ as function and
$q_{\perp}$, where $\phi$ is the azimuthal angle between $q_{\perp}$ and
$P_{\perp}$.

In Fig. 4, we show the $\langle\cos(2\phi)\rangle$ as function of $q_{\perp}$
for dijet production through $gg\to gg$ channel. In the numeric calculations,
we take the typical kinematics at the LHC with $\sqrt{S_{pp}}=7~{}\rm TeV$,
leading jet $P_{\perp}\sim 100\rm GeV$ and the two jets are both at mid-
rapidity.

In heavy ion collisions, the final state jet suffers multiple interactions
with the hot QCD matter when it traverses through the medium. It also
generates medium induced soft gluon radiations. Taking into account of the
above two effects, we have the following modifications on the Fourier
transform $\widetilde{W}$,

$\displaystyle\widetilde{W}_{0}$ $\displaystyle\Longrightarrow$
$\displaystyle\widetilde{W}_{0}^{med.}=\widetilde{W}_{0}+\
Q_{s}^{2}b_{\perp}^{2}/4,$ (22) $\displaystyle\widetilde{W}_{2}$
$\displaystyle\Longrightarrow$
$\displaystyle\left(\alpha_{2}+\alpha_{2}^{m}\right)\widetilde{W}_{0}^{med.}\
,$ (23)

where $Q_{s}^{2}=\langle\hat{q}L\rangle$ represents the medium
$p_{T}$-broadening effects with the medium transport coefficient $\hat{q}$ and
jet traverse length $L$ and $\alpha_{2}^{m}$ for the medium induced soft gluon
radiation contribution to the $\cos(2\phi)$ anisotropy. Because the medium
induced radiation also has a collimation associated with the jet Baier:1999ds
, this contribution will be similar to what was discussed in previous
sections. Therefore, we may utilize the modification of the $\cos(2\phi)$
anisotropy as a measure of the medium induced soft gluon radiation in quark-
gluon plasma in heavy-ion collisions.

5\. Conclusions. In summary, we have studied the soft gluon radiation
contribution to the dijet production in various hadronic collisions, focusing
on the anisotropy of the total transverse momentum with respect to the leading
jet. Our results have shown that these soft gluon radiation contributions lead
to characteristic behaviors of $\langle\cos(2\phi)\rangle$ as functions of the
total transverse momentum $q_{\perp}$. Experimental studies, as shown in a
preliminary analysis of the diffractive dijet production in ultraperipheral
$AA$ collisions at the LHC CMS:2020ekd , will provide a unique opportunity to
investigate the final state radiation associated with the jet. Our finding has
far-reaching consequences in extracting the linearly polarized gluon
distribution and the elliptic Wigner distribution from the measurement of
angular anisotropies in inclusive and exclusive dijet processes.

A number of further developments shall follow. In this paper we focused on the
$\cos(2\phi)$ anisotropy. Following the examples of Ref. Catani:2017tuc ,
other harmonics such as the $\cos(4\phi)$ anisotropy can be derived as well.
Second, we have made the narrow jet approximation in the derivations. The
finite-$R$ corrections, together with the sub-eikonal corrections which might
be relevant at large $q_{\perp}$, should be taken into account when we compare
to the precise experimental data from the LHC and future EIC. Third, more
rigorous derivation of the resummation formulas needs to be carried out along
the line of Refs. Catani:2014qha ; Catani:2017tuc ; Kang:2016mcy .

Acknowledgments: We thank Daniel Tapia Takaki for discussions which partly
motivated this work. We thank Stefano Catani and Massimiliano Grazzini for
bringing our attention to their publications Catani:2010pd ; Catani:2014qha ;
Catani:2017tuc , which are very relevant for the calculations in this paper.
We also thank Felix Ringer for comments and discussions. Y. H. thanks Yukawa
institute, Kyoto university for hospitality. This material is based upon work
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, under contract numbers DE-AC02-05CH11231 and DE- SC0012704.
This work is also supported by the National Natural Science Foundations of
China under Grant No. 11675093, No. 11575070 and Laboratory Directed Research
and Development (LDRD) funds from Brookhaven Science Associates.

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# Mass spectra and decay properties of the higher excited $\rho$ mesons

Xue-Chao Feng  College of Physics and Electronic Engineering, Zhengzhou
University of Light Industry, Zhengzhou 450002, China  Zheng-Ya Li  School of
Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001,
China  De-Min Li<EMAIL_ADDRESS>School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou, Henan 450001, China  Qin-Tao Song
<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou
University, Zhengzhou, Henan 450001, China  En Wang<EMAIL_ADDRESS>School
of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan
450001, China  Wen-Cheng Yan<EMAIL_ADDRESS>School of Physics and
Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China

###### Abstract

Although there are some experimental hints for the higher excited $\rho$
mesons, our knowledge of their properties is much poor theoretically. Based on
our recent work about excited $\rho$ mesons [Phys.Rev.D104(2021)034013], we
present the mass spectra and decay properties of the higher excited $\rho$
mesons with the modified Godfrey-Isgur quark model and the ${}^{3}P_{0}$
strong decay model, and compare our predictions with the experimental hints,
which should be helpful to search for these higher excited $\rho$ mesons.

###### pacs:

14.40.Be, 13.25.Jx

##  I Introduction

In the last decades, a large number of light vector mesons with mass below 2.4
GeV have been reported experimentally [1, 4, 5, 6, 7, 2, 3], and a lot of
research works have been conducted theoretically based on these experimental
data [10, 8, 9, 12, 11, 13]. These experimental and theoretical studies are
crucial to deepen our understanding of the light vector mesons. In addition,
the higher excited light vector mesons above 2.4 GeV play an important role in
the processes involving the baryon anti-baryon interactions [14]. However,
there are few theoretical and experimental studies on the higher excited light
vector mesons, so it is necessary to study their properties, which should be
helpful to search for them experimentally.

Recently, we have studied the mass spectra and strong decay properties of the
excited $\rho$ mesons $Y(2040)/\rho(2000)$, $\rho(1900)$, and $\rho(2150)$
with the modified Godfrey-Isgur quark model (MGI) including the screening
effects and the ${}^{3}P_{0}$ strong decay model, and found that they can be
assigned as $\rho(2^{3}D_{1})$, $\rho(3^{3}S_{1})$, and $\rho(4^{3}S_{1})$,
respectively [15]. We have also shown the screening effects are crucial to
promote theoretical descriptions of the masses of the observed excited $\rho$
mesons, as shown in Fig. 2 of Ref. [15]. We predict that the $Y(2040)$ mainly
decays to $\pi\pi$ and $\pi\omega$, which is in good agreement with the
measurements of the processes $p\bar{p}\to\pi\pi$ and $e^{+}e^{-}\to\omega\pi$
[16, 5]. For $\rho(1900)$, the predicted main decay modes are $\pi\omega$ and
$a_{2}(1320)\pi$, which is also in consistent with the experimental
measurements [3]. For $\rho(2150)$, its main decay modes are expected to be
$\pi\pi$, $\pi\omega(1420)(\to 6\pi)$, and $\pi a_{2}(1320)(\to 6\pi)$,
consistent with the experimental results [3]. Based on the successful
description of the excited mesons $Y(2040)/\rho(2000)$, $\rho(1900)$, and
$\rho(2150)$, we would like to extend our study to the higher excited $\rho$
mesons with masses above 2.4 GeV in this work.

Although the higher excited $\rho$ mesons above 2.4 GeV have not been reported
experimentally, there exist some hints of them in the experimental
measurements. For instance, in 2002, Anisovich et. al analyzed experimental
data of the process $p\bar{p}\rightarrow\omega\eta\pi^{0}$ measured by the
Crystal Barrel Collaboration, and showed that there exists a peak structure
with $J^{PC}=1^{--}$ around 2.4 GeV [17]. Later in 2006, the BABAR
Collaboration reported the $\omega\pi^{+}\pi^{-}\pi^{0}$ energy-dependent
reaction cross section, as shown in Fig. 18 of Ref. [18], where the blue data
points correspond to the cross section data of the process
$e^{+}e^{-}\rightarrow\omega\pi^{+}\pi^{-}\pi^{0}$. From Fig. 18 of Ref. [18],
one can easily find a peak around 2.6 GeV, which may be the higher excited
$\rho$ meson according to the $G$ parity conservation.

In 2007, the BABAR Collaboration presented the cross section of the process
$e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$, where there is a peak
structure around 2.5 GeV [19]. The analysis of the BABAR measurements within
the Vector Meson Dominance model shows the existences of the $\rho(1900)$ and
another resonance with mass of $M=2550\pm 13$ MeV and width of $\Gamma=209\pm
26$ MeV [20], where the latter one is expected to have the same quantum
numbers as $\rho(1900)$. In addition, the cross sections of
$e^{+}e^{-}\rightarrow\eta^{\prime}\pi^{+}\pi^{-}$ and $e^{+}e^{-}\rightarrow
f_{1}(1285)\pi^{+}\pi^{-}$ processes reported by BABAR indicate that there
exist some structures around 2.5 GeV, and it is difficult to claim the
existences of new resonances due to the insufficient data statistics [19].

In a word, there exist some hints of the higher excited $\rho$ mesons
according to the above experimental measurements, as pointed out in Ref. [21],
and the existences of the higher excited $\rho$ mesons above 2.4 GeV need to
be confirmed by more precise experimental measurements in future. The
theoretical predictions for their mass spectra and decay properties are
crucial to search for them experimentally. Thus, we will study the mass
spectra and decay properties of the higher excited $\rho$ mesons above 2.4 GeV
with the MGI model and ${}^{3}P_{0}$ strong decay model, as used in our
previous work [15]

The organization of this paper is as follows. First we introduce the MGI model
and the ${}^{3}P_{0}$ model in Sec. II and Sec. III, respectively. Then we
present the numerical results of the mass spectra and strong decays for the
higher excited $\rho$ mesons in Sec. IV. Finally a short summary is given in
Sec. V.

##  II Relativistic Quark Model

The Godfrey-Isgure relativistic quark model (GI model) was proposed in 1985 by
Godfrey and Isgur [22], and is widely used in describing the mass spectra of
mesons [23, 24, 25], especially for the lower excited mesons. However, the
masses of the higher excited mesons measured by experiments are much lower
than the predictions of the GI model. Many studies show that the screening
effects play an important role in studying the higher radial and orbital
excited mesons [26, 27]. First we present a brief introduction of the GI
model, and the MGI model that includes the screening effects.

###  II.1 GI model

Within the GI model, the Hamiltonian of the meson’s internal interaction can
be written as [22],

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

where $m_{1}$ and $m_{2}$ denote the masses of quark and antiquark in the
meson, respectively, $\mathbf{p}=\mathbf{p}_{1}=-\mathbf{p}_{2}$ is the
center-of-mass momentum, and $\mathbf{r}$ corresponds to the spatial
coordinate. The $\tilde{V}_{\mathrm{eff}}(\mathbf{p,r})$ is the effective
potential between quark and antiquark, and can be expressed as follows in the
non-relativistic limit,

$V_{\mathrm{eff}}(r)=H^{\mathrm{conf}}+H^{\mathrm{hyp}}+H^{\mathrm{so}},$ (2)

where $H^{\mathrm{conf}}$ is spin-independent potential, includes the spin-
independent linear confinement and Coulomb-type interaction,

$\displaystyle H^{\mathrm{conf}}$
$\displaystyle=\Big{[}-\frac{3}{4}(c+br)+\frac{\alpha_{s}(r)}{r}\Big{]}\bm{F}_{1}\cdot\bm{F}_{2}$
$\displaystyle=S(r)+G(r),$ (3)

where $\langle\bm{F}_{1}\cdot\bm{F}_{2}\rangle=-4/3$ for a meson.
$\alpha_{s}(Q^{2})$ is the running coupling constant of QCD, which depends on
the energy scale $Q$. $\alpha_{s}(Q^{2})$ is divergent at low $Q$ region. The
authors of Ref. [22] assume that $\alpha_{s}(Q^{2})$ saturates as
$\alpha_{s}(Q^{2}=0)=\alpha_{s}^{\mathrm{critital}}$, which is regarded as a
parameter to be determined by fitting to experimental data. $\alpha_{s}(r)$ is
obtained from $\alpha_{s}(Q^{2})$ by using Fourier transform, where $r$ is the
relative distance between quark and antiquark. $H^{\mathrm{hyp}}$ denotes the
color-hyperfine interaction,

$\displaystyle H^{\mathrm{hyp}}$
$\displaystyle=-\frac{\alpha_{s}(r)}{m_{1}m_{2}}\Bigg{[}\frac{1}{r^{3}}\Big{(}\frac{3\bm{S}_{1}\cdot\bm{r}\bm{S}_{2}\cdot\bm{r}}{r^{2}}-\bm{S}_{1}\cdot\bm{S}_{2}\Big{)}$
$\displaystyle+\frac{8\pi}{3}\bm{S}_{1}\cdot\bm{S}_{2}\delta^{3}(\bm{r})\Bigg{]}\bm{F}_{1}\cdot\bm{F}_{2}.$
(4)

$H^{\mathrm{so}}$ is the spin-orbit interaction which contains the color-
magnetic term $H^{\mathrm{so(cm)}}$ and the Thomas-precession term
$H^{\mathrm{so(tp)}}$,

$\displaystyle H^{\mathrm{so}}=H^{\mathrm{so(cm)}}+H^{\mathrm{so(tp)}},$ (5)
$\displaystyle
H^{\mathrm{so(cm)}}=\frac{-\alpha_{s}(r)}{r^{3}}(\frac{1}{m_{2}}+\frac{1}{m_{2}})(\frac{\bm{S}_{1}}{m_{1}}+\frac{\bm{S}_{2}}{m_{2}})\cdot\bm{L}(\bm{F}_{1}\cdot\bm{F}_{2})$
(6) $\displaystyle H^{\mathrm{so(tp)}}=-\frac{1}{2r}\frac{\partial
H^{\mathrm{conf}}}{\partial
r}(\frac{\bm{S}_{1}}{m_{1}^{2}}+\frac{\bm{S}_{2}}{m_{2}^{2}})\cdot\bm{L}.$ (7)

The spin-orbit interaction will give rise to the mixing between spin singlet
${}^{1}L_{J}$ and spin triplet ${}^{3}L_{J}$ if $m_{1}\neq m_{2}$.

In GI model, the relativistic effects are introduced by two main ways.
Firstly, in the quark-antiquark scattering, the interactions should depend on
both quark momentum $\vec{p}_{1}$ and antiquark momentum $\vec{p}_{2}$, or a
linear combination of them as $\vec{p}_{1}-\vec{p}_{2}$ and
$\vec{p}_{1}+\vec{p}_{2}$, so they must be nonlocal interaction potentials as
pointed out in Ref. [22]. In order to take this effect into account, a
smearing function $\rho_{12}\left(\mathbf{r}-\mathbf{r}^{\prime}\right)$ is
used to transform the basic potentials $G(r)$ and $S(r)$ into the smeared ones
$\tilde{G}(r)$ and $\tilde{S}(r)$, the specific form is

$\displaystyle\tilde{f}(r)=\int
d^{3}r^{\prime}\rho_{12}(\mathbf{r}-\mathbf{r^{\prime}})f(r^{\prime}),$ (8)

and the smearing function $\rho_{12}(\mathbf{r}-\mathbf{r^{\prime}})$ is
defined as

$\displaystyle\rho_{12}\left(\mathbf{r}-\mathbf{r^{\prime}}\right)=\frac{\sigma_{12}^{3}}{\pi^{3/2}}e^{-\sigma_{12}^{2}\left(\mathbf{r}-\mathbf{r^{\prime}}\right)^{2}},$
(9)
$\displaystyle\sigma_{12}^{2}=\sigma_{0}^{2}\Bigg{[}\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{1}m_{2}}{(m_{1}+m_{2})^{2}}\right)^{4}\Bigg{]}+s^{2}\left(\frac{2m_{1}m_{2}}{m_{1}+m_{2}}\right)^{2}.$
(10)

For a heavy-heavy $Q\bar{Q}$ meson system,
$\rho_{12}(\mathbf{r}-\mathbf{r^{\prime}})$ will turn into delta function
$\delta^{3}(\mathbf{r}-\mathbf{r^{\prime}})$ as one increases the quark mass
$m_{Q}$, in this case, one can obtain $\tilde{f}(r)=f(r)$, which indicates
that the relativistic effects can be neglected for a heavy-heavy $Q\bar{Q}$
meson. However, the relativistic effects are important for heavy-light mesons
and light mesons, therefore, it is necessary to consider relativistic effects
in the study of excited $\rho$ mesons.

Secondly, the momentum-dependent factors are introduced to modify the
effective potentials, as shown in the following formula,

$\displaystyle\tilde{G}(r)$
$\displaystyle\to\left(1+\frac{p^{2}}{E_{1}E_{2}}\right)^{1/2}\tilde{G}(r)\left(1+\frac{p^{2}}{E_{1}E_{2}}\right)^{1/2},$
(11) $\displaystyle\frac{\tilde{V}_{i}(r)}{m_{1}m_{2}}$
$\displaystyle\to\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{i}}\frac{\tilde{V}_{i}(r)}{m_{1}m_{2}}\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2+\epsilon_{i}},$
(12)

where $\tilde{G}(r)$ is the Coulomb-type potential, and $\tilde{V}_{i}(r)$
represents the contact, tensor, vector spin-orbit and scalar spin-orbit terms
as explained in Ref. [22]. In the non-relativistic limit, those momentum-
dependent factors will become unity.

With the help of the detailed expressions of Eq. (LABEL:hami), we obtain the
mass spectrum and wave functions of mesons by solving the Schrödinger equation
with the Gaussian expansion method, the meson wave functions are used as
inputs to investigate the subsequent strong decays for mesons.

###  II.2 MGI model involving the screening effects

In GI model, linear potential $br$ tends to infinity with the increasing of
the distance between quark and antiquark, however for the large distance, the
quark-antiquark pairs are generated in the vacuum, and the colors of quark and
antiquark are screened in a meson, which implies that the simple linear
potential $br$ cannot accurately describe the interaction between quarks for
higher excited mesons [26, 27, 28, 29]. The study of the lattice QCD shows
that the effective potential of mesons deviates from the traditional linear
potential when the distance between quark and antiquark is greater than 1 fm
[30]. Therefore, it is very important to introduce the screening effects when
studying the masses of higher radial and orbital excited mesons.

The screening effects are introduced to the GI model by substituting linear
potential $br$ with the following form [26, 27, 28],

$\displaystyle V(r)=br\rightarrow V^{\mathrm{scr}}(r)=\frac{b(1-e^{-\mu
r})}{\mu},$ (13)

when $r$ is small enough, we can have $V^{\mathrm{scr}}(r)=V(r)$, therefore,
this replacement will not affect the lower-lying meson states. The parameter
$\mu$ is related to the strength of the screening effects and can be
determined by fitting to the experimental data.

Furthermore, the smeared potential of Eq.(8) can be rewritten as [31, 28],

$\displaystyle\tilde{V}^{\mathrm{scr}}(r)$ $\displaystyle=$
$\displaystyle\frac{b}{\mu r}\Bigg{[}e^{\frac{\mu^{2}}{4\sigma^{2}}+\mu
r}\Bigg{(}\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{\mu+2r\sigma^{2}}{2\sigma}}e^{-x^{2}}dx-\frac{1}{2}\Bigg{)}$
(14)
$\displaystyle\times\frac{\mu+2r\sigma^{2}}{2\sigma^{2}}+r-e^{\frac{\mu^{2}}{4\sigma^{2}}-\mu
r}\frac{\mu-2r\sigma^{2}}{2\sigma^{2}}$
$\displaystyle\times\Bigg{(}\frac{1}{\sqrt{\pi}}\int_{0}^{\frac{\mu-2r\sigma^{2}}{2\sigma}}e^{-x^{2}}dx-\frac{1}{2}\Bigg{)}\Bigg{]},$

where $\sigma=\sigma_{12}$ defined in Eq. (10).

The MGI model is widely used in the studies of mass spectrum of heavy-heavy
mesons, heavy-light meson and light mesons [37, 15, 23, 12, 28, 31, 32, 33,
34, 35, 36]. The parameters of the MGI model are taken from Ref. [23], as
tabulated in Table 1. The values of other parameters can be found in Ref.
[22].

Table 1:  Parameter values in the MGI model [23].  Parameter | Value | Parameter | Value
---|---|---|---
$m_{u}$ (GeV) | 0.163 | $s$ | 1.497
$m_{d}$ (GeV) | 0.163 |  $\mu$ (GeV) | 0.0635
$m_{s}$ (GeV) | 0.387 | $\epsilon_{\rm c}$ | -0.138
$b$ (GeV2) | 0.221 | $\epsilon_{\rm{sov}}$ | 0.157
$c$ (GeV) | -0.240 | $\epsilon_{\rm sos}$ | 0.9726
$\sigma_{0}$ (GeV) | 1.799 | $\epsilon_{\rm t}$ | 0.893

##  III ${}^{3}P_{0}$ strong decay model

In addition to the mass spectrum, the strong decay properties are crucial for
experiments to search for the higher excited $\rho$ mesons. Here, we give a
brief introduction of the ${}^{3}P_{0}$ model which is widely used in studying
two-body OZI-allowed strong decays of mesons [40, 39, 38, 34, 8].

The ${}^{3}P_{0}$ model was originally proposed by Micu [41] in 1968, and
later it was further developed by Le Yaouanc $et$ $al$ [43, 42]. ${}^{3}P_{0}$
model has been considered as an effective tool to study two-body OZI-allowed
strong decays of hadrons. For the process $A\rightarrow B+C$, the main idea of
${}^{3}P_{0}$ model is that a flavor-singlet and color-singlet quark-antiquark
pair with $J^{PC}=0^{++}$ is created from the vacuum firstly, then, the
created antiquark (quark) combines with the quark (antiquark) in meson $A$ to
form meson $B$($C$) as shown in Fig. 1(a) and Fig. 1(b).

Figure 1: Two possible diagrams contributing to the process $A\rightarrow B+C$
in the ${}^{3}P_{0}$ model.

The transition operator $T$ of the decay $A\rightarrow B+C$ in the
${}^{3}P_{0}$ model is given by [44],

$\displaystyle T$ $\displaystyle=-3\gamma\sum\limits_{m}\langle
1m1-m|00\rangle\int
d^{3}\bm{p}_{3}d^{3}\bm{p}_{4}\delta^{3}(\bm{p}_{3}+\bm{p}_{4})$ (15)
$\displaystyle\times{\cal{Y}}^{m}_{1}\left(\frac{\bm{p}_{3}-\bm{p}_{4}}{2}\right)\chi^{34}_{1,-m}\phi^{34}_{0}\omega^{34}_{0}b^{\dagger}_{3}(\bm{p}_{3})d^{\dagger}_{4}(\bm{p}_{4}),$

where $\bm{p}_{3}(\bm{p}_{4}$) is the momentum of the created quark
(antiquark). The dimensionless parameter $\gamma$ stands for the strength of
the quark-antiquark $q_{3}\bar{q}_{4}$ pair created from the vacuum, and is
usually determined by fitting to the experimental data. $\chi^{34}_{1,-m}$,
$\phi^{34}_{0}$ and $\omega^{34}_{0}$ are spin, flavor, and color wave
functions of the created quark-antiquark pair, respectively.

With the transition operator $T$, the helicity amplitude
${\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\bm{P})$ can be written as,

$\displaystyle\langle BC|T|A\rangle$
$\displaystyle=\delta^{3}(\bm{P}_{A}-\bm{P}_{B}-\bm{P}_{C}){\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\bm{P}),$
(16)

where $|A\rangle$, $|B\rangle$ and $|C\rangle$ denote the mock meson states
defined in Ref. [45], and $\bm{P}$ is the momentum of meson $B$ in the center
of mass frame. Then the helicity amplitude is obtained as,

$\displaystyle{\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\bm{P})$
$\displaystyle=$
$\displaystyle\gamma\sqrt{8E_{A}E_{B}E_{C}}\sum_{\begin{array}[t]{l}\scriptstyle
M_{L_{A}},M_{S_{A}},\\\ \scriptstyle M_{L_{B}},M_{S_{B}},\\\ \scriptstyle
M_{L_{C}},M_{S_{C}},m\end{array}}\\!\\!\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle$ (20)
$\displaystyle\times\langle
L_{B}M_{L_{B}}S_{B}M_{S_{B}}|J_{B}M_{J_{B}}\rangle\langle
L_{C}M_{L_{C}}S_{C}M_{S_{C}}|J_{C}M_{J_{C}}\rangle$
$\displaystyle\times\langle
1m1-m|00\rangle\langle\chi^{14}_{S_{B}M_{S_{B}}}\chi^{32}_{S_{C}M_{S_{C}}}|\chi^{12}_{S_{A}M_{S_{A}}}\chi^{34}_{1-m}\rangle$
$\displaystyle\times[f_{1}I(\bm{P},m_{1},m_{2},m_{3})+(-1)^{1+S_{A}+S_{B}+S_{C}}$
$\displaystyle\times f_{2}I(-\bm{P},m_{2},m_{1},m_{3})],$ (21)

where
$f_{1}=\langle\phi^{14}_{B}\phi^{32}_{C}|\phi^{12}_{A}\phi^{34}_{0}\rangle$
and
$f_{2}=\langle\phi^{32}_{B}\phi^{14}_{C}|\phi^{12}_{A}\phi^{34}_{0}\rangle$
correspond to the flavor superposition of the two combinations depicted in
Fig. 1(a) and Fig. 1(b), respectively. The momentum space integral
$I(\bm{P},m_{1},m_{2},m_{3})$ is given by

$\displaystyle I(\bm{P},m_{1},m_{2},m_{3})=$ $\displaystyle\int
d^{3}\bm{p}\psi^{*}_{n_{B}L_{B}M_{L_{B}}}(\frac{m_{3}}{m_{1}+m_{3}}\bm{P_{B}}+\bm{p})$
$\displaystyle\times\psi^{*}_{n_{C}L_{C}M_{L_{C}}}(\frac{m_{3}}{m_{2}+m_{3}}\bm{P_{B}}+\bm{p})$
$\displaystyle\times\psi_{n_{A}L_{A}M_{L_{A}}}(\bm{P_{B}}+\bm{p})\mathcal{Y}_{1}^{m}(\bm{p}).$
(22)

Furthermore, the partial wave amplitude ${\cal{M}}^{LS}(\bm{P})$ for the decay
$A\rightarrow B+C$ can be expressed as [46],

$\displaystyle{\cal{M}}^{LS}(\bm{P})=$
$\displaystyle\sum_{\begin{array}[t]{l}\scriptstyle M_{J_{B}},M_{J_{C}},\\\
\scriptstyle M_{S},M_{L}\end{array}}\\!\\!\langle
LM_{L}SM_{S}|J_{A}M_{J_{A}}\rangle$ (25) $\displaystyle\times\langle
J_{B}M_{J_{B}}J_{C}M_{J_{C}}|SM_{S}\rangle$ $\displaystyle\times\int
d\Omega\,\mbox{}Y^{\ast}_{LM_{L}}{\cal{M}}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\bm{P}).$
(26)

Finally, within the relativistic phase space, the total width
$\Gamma(A\rightarrow B+C)$ can be expressed in terms of the partial wave
amplitude squared [44],

$\displaystyle\Gamma(A\rightarrow
B+C)=\frac{\pi|\bm{P}|}{4M^{2}_{A}}\sum_{LS}|{\cal{M}}^{LS}(\bm{P})|^{2},$
(27)

where
$|\bm{P}|=\frac{\sqrt{[M^{2}_{A}-(M_{B}+M_{C})^{2}][M^{2}_{A}-(M_{B}-M_{C})^{2}]}}{2M_{A}}$,
$M_{A}$, $M_{B}$, and $M_{C}$ are the masses of the mesons $A$, $B$ and $C$,
respectively.

##  IV Numerical RESULTS

###  IV.1 Mass spectrum analysis

First, we have calculated the masses of the higher excited $\rho$ mesons
within the MGI and GI models, which are shown in Table 2. There are six higher
excited $\rho$ mesons in the region of $2.4\sim 3.0$ GeV, which are
$\rho(5^{3}S_{1})$, $\rho(6^{3}S_{1})$, $\rho(7^{3}S_{1})$,
$\rho(4^{3}D_{1})$, $\rho(5^{3}D_{1})$, and $\rho(6^{3}D_{1})$. The deviations
between the predictions of MGI and GI models are about $300\sim 500$ MeV,
which implies the screening potential plays an important role in studying the
masses of the higher excited $\rho$ mesons.

Table 2: Masses of the higher excited $\rho$ mesons predicted by the GI and MGI models. State | MGI (MeV) | GI (MeV) |
---|---|---|---
$\rho(5^{3}S_{1})$ | 2542 | 2817 |
$\rho(6^{3}S_{1})$ | 2774 | 3160 |
$\rho(7^{3}S_{1})$ | 2967 | 3470 |
$\rho(4^{3}D_{1})$ | 2624 | 2915 |
$\rho(5^{3}D_{1})$ | 2840 | 3239 |
$\rho(6^{3}D_{1})$ | 3020 | 3554 |
Figure 2: Comparison of the masses of the higher excited $\rho$ mesons
predicted by MGI model with the cross section of the process
$e^{+}e^{-}\rightarrow\omega\pi^{+}\pi^{-}\pi^{0}$ reported by BABAR. The red
dotted lines are masses predicted with MGI model, and the blue data points are
the BABAR experimental data.

The recent analyses of the BABAR measurements about the process
$e^{+}e^{-}\rightarrow K^{+}K^{-}\pi^{+}\pi^{-}\pi^{0}$ indicate the existence
of one resonance with mass $M=(2550\pm 13)$ MeV [20], which is in good
agreement with our predicted mass of $\rho(5^{3}S_{1})$. In addition, the
cross section of the process
$e^{+}e^{-}\rightarrow\omega\pi^{+}\pi^{-}\pi^{0}$ measured by BABAR indicates
that there are hints of the higher excited $\rho$ mesons. From Fig. 2, one can
find one datum point around 2.54 GeV, very close to our predicted mass $2542$
MeV of $\rho(5^{3}S_{1})$, and an enhancement structure around 2.64 GeV, which
could be related to state $\rho(4^{3}D_{1})$ with mass of $2624$ MeV. Another
datum point appears around 2.76 GeV, close to the predicted mass of
$\rho(6^{3}S_{1})$. Although some hints of the higher excited $\rho$ mesons
exist, one can not claim those resonances due to the poor quality of the
present data, and the more accurate measurements in future are crucial to
confirm the existence of these resonances.

In addition, the light mesons with different radial excitation could be well
fit to the following quasi-linear $(n,M^{2})$ Regge trajectories [47],

$M^{2}_{n}=M^{2}_{0}+(n-1)\mu^{2},$ (28)

where $M_{n}$ stands for the mass of the meson with radial quantum number $n$,
and $M^{2}_{0}$ and $\mu^{2}$ are the paprameters of the trajectories. With
the assignments of $\rho(770)$ as $\rho(1^{3}S_{1})$, $\rho(1450)$ as
$\rho(2^{3}S_{1})$, $\rho(1900)$ as $\rho(3^{3}S_{1})$, $\rho(1700)$ as
$\rho(1^{1}D_{1})$, and $Y(2040)$ as $\rho(2^{3}D_{1})$, respectively, we plot
the Regge trajectories in Fig. 3, where one can find that the masses of
$\rho(5^{3}S_{1})$, $\rho(6^{3}S_{1})$, $\rho(7^{3}S_{1})$,
$\rho(4^{3}D_{1})$, $\rho(5^{3}D_{1})$, and $\rho(6^{3}D_{1})$ predicted by
MGI model are consistent with the estimations of Regge trajectories.

Figure 3: The Regge trajectories for the $\rho(n^{3}S_{1})$ and
$\rho(n^{3}D_{1})$ masses.

###  IV.2 Decay behavior analysis

Table 3:  The decay widths of $\rho(5^{3}S_{1})$ (in MeV), the initial state mass is set to be 2542 MeV and the masses of the final states are taken from RPP [3]. Channel | Mode | $\rho(5^{3}S_{1})$ | Mode | $\rho(5^{3}S_{1})$
---|---|---|---|---
$1^{-}\rightarrow 0^{-}0^{-}$ | $\pi\pi$ | 2.52 | $KK$ | $\textless 0.01$
| $\pi\pi(1300)$ | 12.71 | $KK(1460)$ | 0.01
| $\pi\pi(1800)$ | 9.28 |  |
$1^{-}\rightarrow 0^{-}1^{-}$ | $\pi\omega$ | 0.81 | $KK^{*}(1680)$ | 0.01
| $\rho\eta$ | 0.03 | $\rho(1700)\eta$ | 0.09
| $\omega(1420)\pi$ | 11.07 | $\rho\eta(1475)$ | 0.17
| $\omega(1650)\pi$ | 0.11 | $\omega\pi(1300)$ | 1.02
| $KK^{*}$ | 0.02 | $\rho\eta(1295)$ | 0.67
| $KK^{*}(1410)$ | $\textless 0.01$ | $\rho(1450)\eta$ | 0.65
| $\rho\eta^{\prime}$ | 0.14 |  |
$1^{-}\rightarrow 1^{-}1^{-}$ | $\rho\rho$ | 5.64 | $K^{*}K^{*}$ | 0.15
| $\rho\rho(1450)$ | 12.93 | $K^{*}K^{*}(1410)$ | 0.49
$1^{-}\rightarrow 0^{-}1^{+}$ | $a_{1}(1260)\pi$ | 3.26 | $b_{1}(1235)\eta^{\prime}$ | 0.17
| $h_{1}(1170)\pi$ | 4.22 | $\pi a_{1}(1640)$ | 9.81
| $KK_{1}(1400)$ | 0.10 | $b_{1}(1235)\eta$ | 0.49
| $KK_{1}(1270)$ | 0.10 |  |
$1^{-}\rightarrow 0^{-}2^{+}$ | $a_{2}(1320)\pi$ | 4.83 | $a_{2}(1700)\pi$ | 27.80
| $KK_{2}^{*}(1430)$ | 0.01 |  |
$1^{-}\rightarrow 0^{-}2^{-}$ | $\pi\pi_{2}(1670)$ | 3.65 | $\pi\eta_{2}(1645)$ | 1.94
| $KK_{2}(1770)$ | 0.02 | $KK_{2}(1820)$ | 0.03
$1^{-}\rightarrow 0^{-}3^{-}$ | $\pi\omega_{3}(1670)$ | 2.46 | $\eta\rho_{3}(1690)$ | 0.02
| $KK_{3}(1780)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 1^{-}1^{+}$ | $b_{1}(1235)\rho$ | 3.38 | $K_{1}(1400)K^{*}$ | 0.02
| $a_{1}(1260)\omega$ | 3.14 | $K_{1}(1270)K^{*}$ | 0.09
| $\rho f_{1}(1285)$ | 3.00 | $\rho f_{1}(1420)$ | 0.16
| $\rho h_{1}(1170)$ | 2.50 |  |
$1^{-}\rightarrow 2^{+}1^{-}$ | $\rho f_{2}(1270)$ | 3.93 | $K^{*}K_{2}^{*}(1430)$ | 0.03
| $\omega a_{2}(1320)$ | 5.10 |  |
$1^{-}\rightarrow 0^{-}4^{+}$ | $a_{4}(1970)\pi$ | 0.91 |  |
$1^{-}\rightarrow 1^{-}0^{+}$ | $a_{0}(1450)\omega$ | 0.52 |  |
Total width | 143.01
Table 4:  The decay widths of $\rho(4^{3}D_{1})$ (in MeV), the initial state mass is set to be 2624 MeV and the masses of all the final states are taken from PRR [3]. Channel | Mode | $\rho(4^{3}D_{1})$ | Mode | $\rho(4^{3}D_{1})$
---|---|---|---|---
$1^{-}\rightarrow 0^{-}0^{-}$ | $\pi\pi$ | 5.55 | $KK$ | $\textless 0.01$
| $\pi\pi(1300)$ | 11.16 | $KK(1460)$ | $\textless 0.01$
| $\pi\pi(1800)$ | 7.07 |  |
$1^{-}\rightarrow 0^{-}1^{-}$ | $\pi\omega$ | 0.48 | $KK^{*}(1680)$ | 0.01
| $\rho\eta$ | 0.07 | $\rho(1700)\eta$ | 0.07
| $\omega(1420)\pi$ | 2.23 | $\rho\eta(1475)$ | $\textless 0.01$
| $\omega(1650)\pi$ | 0.23 | $\omega\pi(1300)$ | 0.61
| $KK^{*}$ | 0.01 | $\rho\eta(1295)$ | 0.36
| $KK^{*}(1410)$ | $\textless 0.01$ | $\rho(1450)\eta$ | 0.37
| $\rho\eta^{\prime}$ | 0.14 |  |
$1^{-}\rightarrow 1^{-}1^{-}$ | $\rho\rho$ | 4.81 | $K^{*}K^{*}$ | 0.02
| $\rho\rho(1450)$ | 14.42 | $K^{*}K^{*}(1410)$ | 0.06
$1^{-}\rightarrow 0^{-}1^{+}$ | $a_{1}(1260)\pi$ | 4.00 | $b_{1}(1235)\eta^{\prime}$ | 0.10
| $h_{1}(1170)\pi$ | 6.90 | $\pi a_{1}(1640)$ | 10.54
| $KK_{1}(1400)$ | 0.04 | $b_{1}(1235)\eta$ | 0.82
| $KK_{1}(1270)$ | 0.05 |  |
$1^{-}\rightarrow 0^{-}2^{+}$ | $a_{2}(1320)\pi$ | 0.96 | $a_{2}(1700)\pi$ | 7.09
| $KK_{2}^{*}(1430)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 0^{-}2^{-}$ | $\pi\pi_{2}(1670)$ | 6.02 | $\pi\eta_{2}(1645)$ | 4.48
| $KK_{2}(1770)$ | 0.05 | $KK_{2}(1820)$ | 0.01
$1^{-}\rightarrow 0^{-}3^{-}$ | $\pi\omega_{3}(1670)$ | 1.35 | $\eta\rho_{3}(1690)$ | 0.10
| $KK_{3}(1780)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 1^{-}1^{+}$ | $b_{1}(1235)\rho$ | 1.15 | $K_{1}(1400)K^{*}$ | 0.04
| $a_{1}(1260)\omega$ | 0.39 | $K_{1}(1270)K^{*}$ | 0.08
| $\rho f_{1}(1285)$ | 0.52 | $\rho f_{1}(1420)$ | 0.14
| $\rho h_{1}(1170)$ | 1.14 |  |
$1^{-}\rightarrow 2^{+}1^{-}$ | $\rho f_{2}(1270)$ | 5.71 | $K^{*}K_{2}^{*}(1430)$ | 0.01
| $\omega a_{2}(1320)$ | 3.58 |  |
$1^{-}\rightarrow 0^{-}4^{+}$ | $a_{4}(1970)\pi$ | 0.40 |  |
$1^{-}\rightarrow 1^{-}0^{+}$ | $a_{0}(1450)\omega$ | 0.60 |  |
Total width | 105.82
Table 5:  The decay widths of $\rho(6^{3}S_{1})$ (in MeV), the initial state mass is set to be 2774 MeV and the masses of all the final states are taken from RPP [3]. Channel | Mode | $\rho(6^{3}S_{1})$ | Mode | $\rho(6^{3}S_{1})$
---|---|---|---|---
$1^{-}\rightarrow 0^{-}0^{-}$ | $\pi\pi$ | 0.51 | $KK$ | $\textless 0.01$
| $\pi\pi(1300)$ | 4.83 | $KK(1460)$ | $\textless 0.01$
| $\pi\pi(1800)$ | 5.36 |  |
$1^{-}\rightarrow 0^{-}1^{-}$ | $\pi\omega$ | $\textless 0.01$ | $KK^{*}(1680)$ | $\textless 0.01$
| $\rho\eta$ | 0.02 | $\rho(1700)\eta$ | 0.04
| $\omega(1420)\pi$ | 3.29 | $\rho\eta(1475)$ | 0.12
| $\omega(1650)\pi$ | 0.03 | $\omega\pi(1300)$ | 0.21
| $KK^{*}$ | $\textless 0.01$ | $\rho\eta(1295)$ | 0.14
| $KK^{*}(1410)$ | 0.01 | $\rho(1450)\eta$ | 0.10
| $\rho\eta^{\prime}$ | 0.08 |  |
$1^{-}\rightarrow 1^{-}1^{-}$ | $\rho\rho$ | 3.67 | $K^{*}K^{*}$ | $\textless 0.01$
| $\rho\rho(1450)$ | 7.26 | $K^{*}K^{*}(1410)$ | 0.10
$1^{-}\rightarrow 0^{-}1^{+}$ | $a_{1}(1260)\pi$ | 0.59 | $b_{1}(1235)\eta^{\prime}$ | 0.09
| $h_{1}(1170)\pi$ | 0.79 | $\pi a_{1}(1640)$ | 5.44
| $KK_{1}(1400)$ | 0.02 | $b_{1}(1235)\eta$ | 0.08
| $KK_{1}(1270)$ | 0.02 |  |
$1^{-}\rightarrow 0^{-}2^{+}$ | $a_{2}(1320)\pi$ | 0.36 | $a_{2}(1700)\pi$ | 13.82
| $KK_{2}^{*}(1430)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 0^{-}2^{-}$ | $\pi\pi_{2}(1670)$ | 0.95 | $\pi\eta_{2}(1645)$ | 0.59
| $KK_{2}(1770)$ | $\textless 0.01$ | $KK_{2}(1820)$ | $\textless 0.01$
$1^{-}\rightarrow 0^{-}3^{-}$ | $\pi\omega_{3}(1670)$ | 0.63 | $\eta\rho_{3}(1690)$ | $\textless 0.01$
| $KK_{3}(1780)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 1^{-}1^{+}$ | $b_{1}(1235)\rho$ | 2.57 | $K_{1}(1400)K^{*}$ | $\textless 0.01$
| $a_{1}(1260)\omega$ | 1.78 | $K_{1}(1270)K^{*}$ | 0.01
| $\rho f_{1}(1285)$ | 1.68 | $\rho f_{1}(1420)$ | 0.10
| $\rho h_{1}(1170)$ | 1.94 |  |
$1^{-}\rightarrow 2^{+}1^{-}$ | $\rho f_{2}(1270)$ | 4.12 | $K^{*}K_{2}^{*}(1430)$ | $\textless 0.01$
| $\omega a_{2}(1320)$ | 4.36 |  |
$1^{-}\rightarrow 0^{-}4^{+}$ | $a_{4}(1970)\pi$ | 1.31 |  |
$1^{-}\rightarrow 1^{-}0^{+}$ | $a_{0}(1450)\omega$ | 0.36 |  |
Total width | 67.46
Table 6:  The decay widths of $\rho(5^{3}D_{1})$ (in MeV), the initial state mass is set to be 2840 MeV and the masses of the final states are taken from RPP [3]. Channel | Mode | $\rho(5^{3}D_{1})$ | Mode | $\rho(5^{3}D_{1})$
---|---|---|---|---
$1^{-}\rightarrow 0^{-}0^{-}$ | $\pi\pi$ | 3.46 | $KK$ | $\textless 0.01$
| $\pi\pi(1300)$ | 10.25 | $KK(1460)$ | $\textless 0.01$
| $\pi\pi(1800)$ | 7.90 |  |
$1^{-}\rightarrow 0^{-}1^{-}$ | $\pi\omega$ | 0.04 | $KK^{*}(1680)$ | 0.10
| $\rho\eta$ | $\textless 0.01$ | $\rho(1700)\eta$ | 0.07
| $\omega(1420)\pi$ | 1.19 | $\rho\eta(1475)$ | 0.01
| $\omega(1650)\pi$ | 0.20 | $\omega\pi(1300)$ | 0.17
| $KK^{*}$ | $\textless 0.01$ | $\rho\eta(1295)$ | 0.11
| $KK^{*}(1410)$ | 0.02 | $\rho(1450)\eta$ | 0.10
| $\rho\eta^{\prime}$ | 0.04 |  |
$1^{-}\rightarrow 1^{-}1^{-}$ | $\rho\rho$ | 0.58 | $K^{*}K^{*}$ | $\textless 0.01$
| $\rho\rho(1450)$ | 5.44 | $K^{*}K^{*}(1410)$ | 0.04
$1^{-}\rightarrow 0^{-}1^{+}$ | $a_{1}(1260)\pi$ | 1.34 | $b_{1}(1235)\eta^{\prime}$ | 0.07
| $h_{1}(1170)\pi$ | 2.64 | $\pi a_{1}(1640)$ | 7.35
| $KK_{1}(1400)$ | 0.02 | $b_{1}(1235)\eta$ | 0.16
| $KK_{1}(1270)$ | 0.02 |  |
$1^{-}\rightarrow 0^{-}2^{+}$ | $a_{2}(1320)\pi$ | 0.84 | $a_{2}(1700)\pi$ | 5.81
| $KK_{2}^{*}(1430)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 0^{-}2^{-}$ | $\pi\pi_{2}(1670)$ | 1.58 | $\pi\eta_{2}(1645)$ | 1.68
| $KK_{2}(1770)$ | 0.01 | $KK_{2}(1820)$ | $\textless 0.01$
$1^{-}\rightarrow 0^{-}3^{-}$ | $\pi\omega_{3}(1670)$ | 0.62 | $\eta\rho_{3}(1690)$ | 0.03
| $KK_{3}(1780)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 1^{-}1^{+}$ | $b_{1}(1235)\rho$ | 1.22 | $K_{1}(1400)K^{*}$ | 0.01
| $a_{1}(1260)\omega$ | 0.69 | $K_{1}(1270)K^{*}$ | 0.03
| $\rho f_{1}(1285)$ | 0.91 | $\rho f_{1}(1420)$ | 0.12
| $\rho h_{1}(1170)$ | 0.93 |  |
$1^{-}\rightarrow 2^{+}1^{-}$ | $\rho f_{2}(1270)$ | 1.41 | $K^{*}K_{2}^{*}(1430)$ | $\textless 0.01$
| $\omega a_{2}(1320)$ | 0.84 |  |
$1^{-}\rightarrow 0^{-}4^{+}$ | $a_{4}(1970)\pi$ | 0.52 |  |
$1^{-}\rightarrow 1^{-}0^{+}$ | $a_{0}(1450)\omega$ | 0.46 |  |
Total width | 59.00
Table 7:  The decay widths of $\rho(7^{3}S_{1})$ (in MeV), the initial state mass is set to be 2967 MeV and the masses of the final states are taken from RPP [3]. Channel | Mode | $\rho(7^{3}S_{1})$ | Mode | $\rho(7^{3}S_{1})$
---|---|---|---|---
$1^{-}\rightarrow 0^{-}0^{-}$ | $\pi\pi$ | $\textless 0.01$ | $KK$ | $\textless 0.01$
| $\pi\pi(1300)$ | 4.24 | $KK(1460)$ | 0.01
| $\pi\pi(1800)$ | 5.77 |  |
$1^{-}\rightarrow 0^{-}1^{-}$ | $\pi\omega$ | 0.16 | $KK^{*}(1680)$ | $\textless 0.01$
| $\rho\eta$ | 0.16 | $\rho(1700)\eta$ | 0.09
| $\omega(1420)\pi$ | 1.14 | $\rho\eta(1475)$ | 0.12
| $\omega(1650)\pi$ | 0.16 | $\omega\pi(1300)$ | $\textless 0.01$
| $KK^{*}$ | $\textless 0.01$ | $\rho\eta(1295)$ | $\textless 0.01$
| $KK^{*}(1410)$ | $\textless 0.01$ | $\rho(1450)\eta$ | $\textless 0.01$
| $\rho\eta^{\prime}$ | 0.11 |  |
$1^{-}\rightarrow 1^{-}1^{-}$ | $\rho\rho$ | 6.86 | $K^{*}K^{*}$ | 0.04
| $\rho\rho(1450)$ | 9.09 | $K^{*}K^{*}(1410)$ | $\textless 0.01$
$1^{-}\rightarrow 0^{-}1^{+}$ | $a_{1}(1260)\pi$ | 1.21 | $b_{1}(1235)\eta^{\prime}$ | 0.08
| $h_{1}(1170)\pi$ | 0.42 | $\pi a_{1}(1640)$ | 4.28
| $KK_{1}(1400)$ | 0.01 | $b_{1}(1235)\eta$ | 0.15
| $KK_{1}(1270)$ | 0.01 |  |
$1^{-}\rightarrow 0^{-}2^{+}$ | $a_{2}(1320)\pi$ | $\textless 0.01$ | $a_{2}(1700)\pi$ | 6.86
| $KK_{2}^{*}(1430)$ | 0.02 |  |
$1^{-}\rightarrow 0^{-}2^{-}$ | $\pi\pi_{2}(1670)$ | 0.70 | $\pi\eta_{2}(1645)$ | 0.55
| $KK_{2}(1770)$ | 0.01 | $KK_{2}(1820)$ |
$1^{-}\rightarrow 0^{-}3^{-}$ | $\pi\omega_{3}(1670)$ | 0.04 | $\eta\rho_{3}(1690)$ | 0.05
| $KK_{3}(1780)$ | <0.01 |  |
$1^{-}\rightarrow 1^{-}1^{+}$ | $b_{1}(1235)\rho$ | 3.09 | $K_{1}(1400)K^{*}$ | $\textless 0.01$
| $a_{1}(1260)\omega$ | 1.92 | $K_{1}(1270)K^{*}$ | $\textless 0.01$
| $\rho f_{1}(1285)$ | 1.57 | $\rho f_{1}(1420)$ | 0.06
| $\rho h_{1}(1170)$ | 3.21 |  |
$1^{-}\rightarrow 2^{+}1^{-}$ | $\rho f_{2}(1270)$ | 4.21 | $K^{*}K_{2}^{*}(1430)$ | 0.02
| $\omega a_{2}(1320)$ | 3.72 |  |
$1^{-}\rightarrow 0^{-}4^{+}$ | $a_{4}(1970)\pi$ | 0.53 |  |
$1^{-}\rightarrow 1^{-}0^{+}$ | $a_{0}(1450)\omega$ | 0.24 |  |
Total width | 61.37
Table 8:  The decay widths of $\rho(6^{3}D_{1})$ (in MeV), the initial state mass is set to be 3020 MeV and the masses of the final states are taken from RPP [3]. Channel | Mode | $\rho(6^{3}D_{1})$ | Mode | $\rho(6^{3}D_{1})$
---|---|---|---|---
$1^{-}\rightarrow 0^{-}0^{-}$ | $\pi\pi$ | 0.84 | $KK$ | $\textless 0.01$
| $\pi\pi(1300)$ | 4.29 | $KK(1460)$ | $\textless 0.01$
| $\pi\pi(1800)$ | 5.22 |  |
$1^{-}\rightarrow 0^{-}1^{-}$ | $\pi\omega$ | $\textless 0.01$ | $KK^{*}(1680)$ | $\textless 0.01$
| $\rho\eta$ | 0.02 | $\rho(1700)\eta$ | 0.03
| $\omega(1420)\pi$ | 0.40 | $\rho\eta(1475)$ | 0.03
| $\omega(1650)\pi$ | 0.06 | $\omega\pi(1300)$ | 0.02
| $KK^{*}$ | $\textless 0.01$ | $\rho\eta(1295)$ | 0.01
| $KK^{*}(1410)$ | 0.01 | $\rho(1450)\eta$ | $\textless 0.01$
| $\rho\eta^{\prime}$ | 0.03 |  |
$1^{-}\rightarrow 1^{-}1^{-}$ | $\rho\rho$ | 0.37 | $K^{*}K^{*}$ | $\textless 0.01$
| $\rho\rho(1450)$ | 0.47 | $K^{*}K^{*}(1410)$ | 0.03
$1^{-}\rightarrow 0^{-}1^{+}$ | $a_{1}(1260)\pi$ | 0.16 | $b_{1}(1235)\eta^{\prime}$ | 0.05
| $h_{1}(1170)\pi$ | 0.40 | $\pi a_{1}(1640)$ | 2.90
| $KK_{1}(1400)$ | $\textless 0.01$ | $b_{1}(1235)\eta$ | $\textless 0.01$
| $KK_{1}(1270)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 0^{-}2^{+}$ | $a_{2}(1320)\pi$ | 0.05 | $a_{2}(1700)\pi$ | 2.93
| $KK_{2}^{*}(1430)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 0^{-}2^{-}$ | $\pi\pi_{2}(1670)$ | 0.14 | $\pi\eta_{2}(1645)$ | 0.16
| $KK_{2}(1770)$ | $\textless 0.01$ | $KK_{2}(1820)$ | $\textless 0.01$
$1^{-}\rightarrow 0^{-}3^{-}$ | $\pi\omega_{3}(1670)$ | 0.10 | $\eta\rho_{3}(1690)$ | $\textless 0.01$
| $KK_{3}(1780)$ | $\textless 0.01$ |  |
$1^{-}\rightarrow 1^{-}1^{+}$ | $b_{1}(1235)\rho$ | 0.91 | $K_{1}(1400)K^{*}$ | $\textless 0.01$
| $a_{1}(1260)\omega$ | 0.97 | $K_{1}(1270)K^{*}$ | $\textless 0.01$
| $\rho f_{1}(1285)$ | 0.98 | $\rho f_{1}(1420)$ | 0.07
| $\rho h_{1}(1170)$ | 0.77 |  |
$1^{-}\rightarrow 2^{+}1^{-}$ | $\rho f_{2}(1270)$ | 0.37 | $K^{*}K_{2}^{*}(1430)$ | $\textless 0.01$
| $\omega a_{2}(1320)$ | 0.05 |  |
$1^{-}\rightarrow 0^{-}4^{+}$ | $a_{4}(1970)\pi$ | 0.31 |  |
$1^{-}\rightarrow 1^{-}0^{+}$ | $a_{0}(1450)\omega$ | 0.27 |  |
Total width | 23.88

In this work, we employ the ${}^{3}P_{0}$ model with the realistic meson wave
functions obtained from the MGI model to evaluate the decay widths of
$\rho(5^{3}S_{1})$, $\rho(4^{3}D_{1})$, $\rho(6^{3}S_{1})$,
$\rho(5^{3}D_{1})$, $\rho(7^{3}S_{1})$, and $\rho(6^{3}D_{1})$. The value of
parameter $\gamma$ is taken from our previous work [15], where it was obtained
by fitting to the decay widths of the light mesons with same quantum numbers
as $\rho$ mesons. $\gamma=6.57$ is used for the $u\bar{u}$/$d\bar{d}$ pair
creation, and the $\gamma$ value of $s\bar{s}$ pair creation is suppressed by
an additional factor $m_{u}/m_{s}$. The predicted masses by the MGI model are
used for the unobserved excited $\rho$ mesons in the calculations of the
strong decay widths, and the masses of the other mesons are taken from Review
of Particle Physics (RPP) [3].

The partial widths and total width of $\rho(5^{3}S_{1})$ are listed in Table
3. The total decay width of $\rho(5^{3}S_{1})$ is expected to be $\Gamma=143$
MeV, which is close to the lower limit of the width $\Gamma=(209\pm 26)$ MeV
obtained in Ref. [20]. The main decay modes of $\rho(5^{3}S_{1})$ are
$\pi\pi(1300)(\to 4\pi)$, $\pi\pi(1800)(\to 4\pi)$, $\pi\omega(1420)(\to
4\pi,6\pi)$, $\rho\rho(1450)(\to 4\pi,6\pi)$, $\pi a_{1}(1640)(\to 4\pi)$, and
$\pi a_{2}(1700)(\to\eta\pi\pi,\pi K\bar{K})$, thus one can search for the
$\rho(5^{3}S_{1})$ in the channels of $4\pi$, $6\pi$, $\eta\pi\pi$, and $\pi
K\bar{K}$.

The partial widths and total width of $\rho(4^{3}D_{1})$ are listed in Table
4. The total decay width of $\rho(4^{3}D_{1})$ is expected to be
$\Gamma=105.82$ MeV, consistent with the broad structure around 2.64 GeV in
Fig. 2. The predicted main decay modes are $\pi\pi(1300)(\to 4\pi)$,
$\pi\pi(1800)(\to 4\pi)$, $\rho\rho(1450)(\to 4\pi)$, $\pi h_{1}(1170)(\to
4\pi)$, $\pi a_{1}(1640)(\to 4\pi)$, and $\pi\pi_{1}(1670)(\to 4\pi)$, thus
one can search for $\rho(4^{3}D_{1})$ in the $4\pi$ channel.

The partial widths and total width of $\rho(6^{3}S_{1})$ are listed in Table
5. The total decay width of $\rho(6^{3}S_{1})$ is expected to be
$\Gamma=67.46$ MeV, and the main decay modes are predicted to be
$\pi\pi(1300)(\to 4\pi)$, $\pi\pi(1800)(\to 4\pi)$, $\rho\rho(1450)(\to
4\pi,6\pi)$, $\pi a_{2}(1700)(\to\eta\pi\pi,\pi K\bar{K})$, $\rho
f_{2}(1270)(\to 4\pi)$, and $\omega a_{2}(1320)$. Considering the width of the
intermediate state $a_{2}(1320)$ is $107\pm 5$ MeV, the $\rho(6^{3}S_{1})$
state is expected to be observed in the $\omega a_{2}(1320)$ channel.

The partial widths and total width of $\rho(5^{3}D_{1})$ are listed in Table
6. The total decay width of $\rho(5^{3}D_{1})$ is expected to be
$\Gamma=59.00$ MeV, and the predicted main decay modes are $\pi\pi(1300)(\to
4\pi)$, $\pi\pi(1800)(\to 4\pi)$, $\rho\rho(1450)(\to 4\pi)$, and $\pi
a_{1}(1640)(\to 4\pi)$. Thus $\rho(5^{3}D_{1})$ is expected to be observed in
$4\pi$ channel.

The partial widths and total width of $\rho(7^{3}S_{1})$ are listed in Table
7. The total decay width of $\rho(7^{3}S_{1})$ is expected to be
$\Gamma=61.37$ MeV, and the predicted main decay modes are $\pi\pi(1300)(\to
4\pi)$, $\pi\pi(1800)(\to 4\pi)$, $\rho\rho(\to 4\pi)$, $\rho\rho(1450)(\to
4\pi,6\pi)$, $\pi a_{2}(1700)(\to 4\pi)$, $\pi a_{1}(1640)(\to 4\pi)$, and
$\omega a_{2}(1320)$. Thus one can search for the $\rho(7^{3}S_{1})$ state in
the $\omega a_{2}(1320)$ and $4\pi$ channels.

The partial widths and total width of $\rho(6^{3}D_{1})$ are listed in Table
8. The total decay width of $\rho(6^{3}D_{1})$ is expected to be
$\Gamma=23.88$ MeV, and the predicted main decay modes are $\pi\pi(1300)(\to
4\pi)$, $\pi\pi(1800)(\to 4\pi)$, $\pi a_{1}(1640)(\to 4\pi)$, and $\pi
a_{2}(1700)(\to 4\pi)$. The state $\rho(6^{3}D_{1})$ is expected to be
observed in the $4\pi$ channel.

##  V SUMMARY AND CONCLUSION

In this work, we present the masses and strong decay widths of the higher
excited $\rho$ mesons $\rho(5^{3}S_{1})$, $\rho(4^{3}D_{1})$,
$\rho(6^{3}S_{1})$, $\rho(5^{3}D_{1})$, $\rho(7^{3}S_{1})$, and
$\rho(6^{3}D_{1})$. The deviations between the theoretical masses predicted
with MGI and GI models are about $300\sim 500$ MeV, which implies the
screening effects are important for the higher excited $\rho$ mesons. We also
show some experimental hints for the existences of the higher excited $\rho$
mesons, and one can not claim these resonances based on the poor measurements
at present.

In addition to the masses of the higher exited $\rho$ mesons, the strong decay
widths are also calculated, which are $\Gamma_{\rho(5S)}=143$ MeV,
$\Gamma_{\rho(4D)}=106$ MeV, $\Gamma_{\rho(6S)}=67$ MeV,
$\Gamma_{\rho(5D)}=59$ MeV, $\Gamma_{\rho(7S)}=61$ MeV, and
$\Gamma_{\rho(6D)}=24$ MeV, respectively. According to our theoretical
predictions, those states are expected to be observed in the final states of
$4\pi$, $6\pi$, $\eta\pi\pi$, $\pi K\bar{K}$, $\omega a_{2}(1320)$. Our
theoretical study on their masses and decay properties should be useful to
search for these resonances experimentally.

## Acknowledgements

This work is supported by the Natural Science Foundation of Henan under Grand
No. 222300420554. It is also supported by the Key Research Projects of Henan
Higher Education Institutions under No. 20A140027, the Project of Youth
Backbone Teachers of Colleges and Universities of Henan Province
(2020GGJS017), the Youth Talent Support Project of Henan (2021HYTP002), the
Fundamental Research Cultivation Fund for Young Teachers of Zhengzhou
University (JC202041042), and the Open Project of Guangxi Key Laboratory of
Nuclear Physics and Nuclear Technology, No.NLK2021-08.

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