# MOVESTAR: An Open-Source Vehicle Fuel and Emission Model based on USEPA
MOVES

Ziran Wang, Guoyuan Wu, and George Scora
Center for Environmental Research and Technology
University of California, Riverside
<EMAIL_ADDRESS>{gywu<EMAIL_ADDRESS>

###### Abstract

In this paper, we introduce an open-source model “MOVESTAR” to calculate the
fuel consumption and pollutant emissions of motor vehicles. This model is
developed based on U.S. Environmental Protection Agency’s (EPA) Motor Vehicle
Emission Simulator (MOVES), which provides an accurate estimate of vehicle
fuel consumption and pollutant emissions under a wide range of user-defined
conditions. Originally, MOVES requires users to specify many parameters
through its software graphical user interface (GUI), including vehicle types,
time periods, geographical areas, pollutants, vehicle operating
characteristics, and road types. In this paper, MOVESTAR is developed as a
lite version of MOVES, which only takes the second-by-second vehicle speed
data and vehicle type as inputs111MOVESTAR source code:
https://github.com/ziranw/MOVESTAR-Fuel-and-Emission-Model. To enable easy
integration of this model, its source code is provided in different languages,
including Python, MATLAB and C++. A case study is introduced in this paper to
illustrate how this MOVESTAR model can be utilized in the development of
advanced vehicle technology.

> Keywords: Energy consumption, fuel consumption, emission, open source

##  1 Introduction

Environmental sustainability has been a crucial factor in the development of
transportation systems nowadays. Transportation sources emit greenhouse gases
(GHGs) that contribute to climate change. As the total number of motor
vehicles around the globe increases over time, transportation sector becomes
one of the largest sources of GHG emissions [1]. Specifically in the United
States, transportation sector was in charge of 28.2% of 2018 GHGs, where over
90% of the fuel used for transportation is petroleum based, including
primarily gasoline and diesel. Therefore, the research on reducing fuel
consumption and pollutant emissions of our transportation system (especially
motor vehicles) becomes substantially important.

During the past two decades, a significant amount of “eco-driving” studies has
been proposed, focusing on the policy [2, 3], methodology [4, 5, 6], as well
as field implementation [7, 8, 9] of this emerging concept. By adopting
advanced vehicular technologies, motor vehicles can be driven in an economic
or ecologic style, so that their impacts on the environment can be reduced,
making our transportation systems more sustainable.

In order to conduct environment-oriented research for motor vehicles, a proper
model is needed to calculate their fuel consumption and pollutant emissions.
To this purpose, the United States Environmental Protection Agency (USEPA)
developed MOtor Vehicle Emission Simulator (MOVES) [10]. MOVES is a state-of-
the-science emission modeling system that estimates emissions for mobile
sources at the national, county, and project level for criteria air
pollutants, greenhouse gases, and air toxics. In the modeling process, the
user specifies vehicle types, time periods, geographical areas, pollutants,
vehicle operating characteristics, and road types to be modeled. The model
then performs a series of calculations, which have been carefully developed to
accurately reflect vehicle operating processes, such as running, starts, or
hoteling, and provide estimates of total emissions or emission rates per
vehicle or unit of activity [11].

Although USEPA encourages using MOVES to estimate on-road GHG emissions or
energy consumption regardless of the user’s experience level, it is a
sophisticated model that requires the user to read through extensive
documentations to get familiar with its graphical user interface (GUI) and
data inputs [12]. MOVES is certainly a well-developed and well-maintained
model for experts in the field of fuel or emissions. However, for users that
are not in this field of study, who just want to get an estimate of the fuel
consumption and pollutant emissions of their vehicles (in simulations or field
implementations), MOVES might be relatively difficult to implement.

In this paper, we develop an open-source fuel and emission model for on-road
vehicles based on MOVES, which allows users of all levels to easily satisfy
their research needs. This “MOVESTAR” model is developed in multiple platforms
and programming languages, which enables both real-time calculation and post-
processing evaluation. The model is introduced in section 2, and its
implementations in different platforms are introduced in section 3. Section 4
conducts a case study of MOVESTAR model to show how it can be utilized in the
development of advanced vehicle technology, while section 5 concludes the
paper.

##  2 MOVESTAR Model

Figure 1: Flowchart of the proposed MOVESTAR model as a lite version of USEPA
MOVES

In this section, MOVESTAR model is introduced step by step, which is
summarized by the flowchart shown in Figure 1. The very first step is to
define the vehicle type of the target vehicle that needs to be considered.
Originally, the user of MOVES has to select from 13 “source use types”, which
is the terminology defined by MOVES to describe vehicles. Additionally, the
user also needs to select from six different fuel types, including gasoline,
diesel, ethanol E-85, compressed natural gas (CNG), electricity, and liquefied
petroleum gas. Both steps are supposed to be conducted in the GUI of MOVES.
However, in order to simplify this first process of MOVESTAR, we only provide
limited options of vehicle types, which will be discussed in the
implementation details in section 3.

Actually, before this vehicle type selection, MOVES originally has four
additional steps in its GUI, including specifying “description”, “scale”
(including model type, analysis scale, and calculation type selections), “time
spans” (including time aggregation level, years, months, days, and hours
selections), and “geographic bounds” (including region, states, counties, and
project selections). For the sake of simplicity, these steps are all removed
in our proposed MOVESTAR model.

Once the vehicle type is selected, the related parameters of each vehicle type
can be loaded for further calculation of the vehicle specific power (VSP). The
variables $A,B,C,M$ and $f$ needed in the VSP calculation are derived from the
following Table 1 based on different vehicle types.

Table 1: Coefficients for each vehicle source type

In order to calculate the VSP, we need to also obtain the vehicle dynamic
states, such as speed, acceleration, and instantaneous road elevation. VSP is
calculated by normalizing the continuous power output for each vehicle to its
own weight. Vehicles are tested on full chassis dynamometers, and emission
standards are in units of grams per mile [13].

As mentioned earlier, the proposed MOVESTAR model only requires second-by-
second vehicle speed data as model input, so the user needs to pre-process the
data if its frequency is not 1Hz. Additionally, MOVESTAR model has an
additional function that can convert the second-by-second speed data into the
second-by-second acceleration data, with the assumption that the road grade is
zero.

Then, the VSP or STP (Scaled Tractive Power) can be calculated below

$VSP=\frac{A\cdot v+B\cdot v^{2}+C\cdot
v^{3}+M\cdot(a+g\cdot\sin{\theta})\cdot v}{f}$ (1)

where $A$ is the rolling term; $B$ is the rotating term; $C$ is the drag term;
$M$ is the source mass (in metric tons); $f$ is the fixed mass factor (in
metric tons); $g$ is the acceleration due to gravity (9.8 $m/s^{2}$); $v$ is
the vehicle speed in $m/s$; $a$ is the vehicle acceleration in $m/s^{2}$; and
$\theta$ is the (fractional) road grade.

Once the VSP is calculated, MOVESTAR model will be proceeded to the operating
mode (OpMode) calculation. Operating mode is directly used to estimate fuel
consumption and pollutant emissions in MOVESTAR model. It is calculated on a
second-by-second basis, based on the VSP class, speed class, and/or
acceleration of the vehicle. The mapping relationship from VSP class, speed
class, and/or acceleration to the operating mode is illustrated in Table 2
[14].

Table 2: Operating modes for running exhaust emissions

As can be seen from the table, the values in the center of the table stand for
the operating modes of a vehicle, which are mapped from VSP and speed class.
Specifically, 21 operating modes (all expect for “11” and “21”) represent the
“cruise/acceleration” scenario, where VSP is larger than zero. Two operating
modes (“11” and “21”) represent the “coasting” scenario, where VSP is smaller
than zero. Additionally, two scenarios are not shown in this table since they
do not require the calculation of VSP. One is “deceleration/braking” scenario
which has a mode of “0”; and the other is “idle” scenario which has a mode of
“1”.

Once the operating mode calculation is completed, MOVESTAR model can be
proceeded to the last step. The emission rate associated with the operating
modes is defined in USEPA MOVES, where we only extract the “base rate” values
of specific vehicle types as defined earlier. Therefore, based on the second-
by-second operating modes of a vehicle, second-by-second fuel and emission
outputs of our MOVESTAR model can be generated. A list of all outputs are
shown in Table 3.

Table 3: Outputs of MOVESTAR model

##  3 Open-Source Implementation in Various Platforms

In this section, we introduce the implementation of MOVESTAR in various
platforms as open-source codes. The detailed integration frameworks and
guidelines are introduced with respect to Python, Mathworks MATLAB and PTV
Vissim. Specifically, MOVESTAR is implemented in MATLAB as post-processing
code, and in Vissim as real-time processing code.

###  3.1 Python

MOVESTAR model is implemented in Python as a standalone “.py” file that can
process the speed data generated by vehicles in advance222MOVESTAR for Python
source code: https://github.com/ziranw/MOVESTAR-Fuel-and-Emission-
Model/tree/master/MOVESTAR_Python_v1.0.

In the current Python version of MOVESTAR model, there are only two different
vehicle types that the user needs to choose from. If the user assigns “1” to
the variable “veh_type” in the main function of “MOVESTAR.py” (towards the end
of the script), then the tested vehicle is assumed as a light-duty vehicle;
Otherwise, if “2” is assigned, then the tested vehicle is assumed as a light-
duty truck.

In terms of the other input of MOVESTAR model besides the vehicle type, the
vehicle speed data needs to be processed by the user in the second-by-second
format with a unit of m/s. The Python version of MOVESTAR model provides two
output formats within the script, one is the total fuel and emission results,
while the other is the normalized results based on the travel distance.

###  3.2 Mathworks MATLAB

MOVESTAR model is implemented in Mathworks MATLAB [15] as “.m” files that can
process the speed data generated by vehicles in advance333MOVESTAR for MATLAB
source code: https://github.com/ziranw/MOVESTAR-Fuel-and-Emission-
Model/tree/master/MOVESTAR_MATLAB_v1.1. There are five files in the MATLAB
version of MOVESTAR model, where their features, inputs and outputs are listed
in Table 4.

Table 4: MOVESTAR model implementation in Mathworks MATLAB

In the current MATLAB version of MOVESTAR model, there are only two different
vehicle types that the user needs to choose from. If the user specifies “1” in
the main function “MOVESTAR.m”, then the tested vehicle is assumed as a light-
duty vehicle (e.g., sedan); Otherwise, if “2” is entered, then the tested
vehicle is assumed as a light-duty truck (e.g., SUV).

In terms of the other input of MOVESTAR model besides the vehicle type, the
vehicle speed data needs to be processed by the user in the second-by-second
format. If the user generates the data by MATLAB Simulink, MOVESTAR model
provides a “sbsSpd.m” file that can convert sub-second speed data (in “.mat”
format) into second-by-second speed data.

The MATLAB version of MOVESTAR model provides two output files in “.dat”
format, where the one with an “ER” suffix means the “emission rate” results,
as shown in Table 3. Additionally, MOVESTAR also provides the normalized fuel
and emission results based on the travel distance, which are included in the
file with a “EF” suffix (i.e., “emission factor”).

###  3.3 Microscopic Traffic Simulator Vissim with C++ API

As a widely used microscopic traffic simulator, PTV Vissim [16] provides C++
API for users to load external emission models. MOVESTAR model also provides a
C++ project that can be integrated in Vissim to calculate fuel and emissions
in real simulation time444MOVESTAR for Vissim source code:
https://github.com/ziranw/MOVESTAR-Fuel-and-Emission-
Model/tree/master/MOVESTAR_VISSIM_v1.0.

This C++ project includes an “EmissionModel.cpp” file and an “EmissionModel.h”
file. The “.cpp” file is the main script that includes the calculations of
VSP, operating mode, and emission rate. The “.h” file is defined by Vissim to
allow the user to interact with its API. This “EmissionModel.vcxproj” project
can be compiled to a “.dll” file, which can be further loaded in Vissim GUI.
The high-level integration framework of MOVESTAR model in Vissim is
illustrated as Figure 2, while the detailed integration steps should be
referred to the Vissim manual.

Figure 2: Integration framework of MOVESTAR model in PTV Vissim

##  4 Case Study and Results Evaluation

In this section, we perform a case study of MOVESTAR model based on a
cooperative eco-driving (CED) system with connected and automated vehicles
(CAVs) [17]. This case study develops a system to improve the energy
efficiency along a corridor with signalized intersections. To study the effect
of penetration rate of CAVs, two different types of vehicles are defined in
the system as conventional vehicles and CED vehicles. The high-level concept
of this CED system can be illustrated in Figure 3.

Figure 3: Illustration of the CED system in MOVESTAR case study

A microscopic traffic simulation network based on the real-world map in
Riverside, CA, USA is modeled in Vissim as figure 4, with real-world signal
phase and timing (SPaT) data and traffic count data. Different vehicle
longitudinal control models and their relevant logic (e.g., role transition)
are integrated into the simulation network to simulate vehicles’ behaviors.
MOVESTAR fuel and emission model is implemented to analyze the environmental
impacts of the proposed CED system. This integration architecture is
illustrated in Figure 5.

Figure 4: Vissim simulation traffic network in MOVESTAR case study Figure 5:
Vissim simulation system architecture in MOVESTAR case study

The simulation is conducted under different penetration rates of CED vehicles
and conventional vehicles. The simulation results are compared with two
baseline scenarios, where the first scenario contains 100% conventional
vehicles, and the second scenario contains 100% eco-approach and departure
(EAD) vehicles. EAD vehicles are CAVs that conduct eco-driving maneuvers
solely with the traffic signals, which do not cooperate with other CAVs [7].

As can be seen from Table 5, the results of energy consumption, NOx emissions,
HC emissions, CO emissions, and CO2 emissions can all be calculated by
MOVESTAR model in different scenarios, which are eventually shown as relative
values with respect to two baseline scenarios. All results validate the
effectiveness of the CED system, because the fuel and emission results improve
as the penetration rate of CED vehicles increases. This case study gives an
example on how MOVESTAR model can be applied to the development of advanced
vehicle technology.

Table 5: Simulation results in MOVESTAR case study

##  5 Conclusion

In this paper, we introduced an open-source model “MOVESTAR” for users to
easily generate the fuel and emission results in their research studies. USEPA
MOVES model originally requires users to specify many parameters through its
software GUI. MOVESTAR model, developed as a lite version of MOVES, only takes
second-by-second vehicle speed and vehicle type as inputs of the model without
the consideration of geographical location, road grade, and others. It
provides a much easier access for users of all levels to roughly estimate fuel
and emission results. The open-source implementations of MOVESTAR model were
demonstrated in MATLAB and Vissim C++ API, and a case study with an innovative
CAV technology was conducted to illustrate the utilization of the model.

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#  Mass and decay of the $s\bar{s}$ member of the $1^{3}F_{4}$ meson nonet

Xue-Chao Feng 1<EMAIL_ADDRESS>Ke-Wei Wei 2<EMAIL_ADDRESS>1 College of
Physics and Electronic Engineering, Zhengzhou University of Light Industry,
450002 Zhengzhou, China
2 School of Science, Henan University of Engineering, 451191 Zhengzhou, China

###### Abstract

The mass and decay of the $s\bar{s}$ member of the $1^{3}F_{4}$ meson nonet
are investigated in the framework of the Regge phenomenology and the
${}^{3}P_{0}$ model. We propose, based on the results, that the assignment of
the $s\bar{s}$ member of the $1^{3}F_{4}$ meson nonet will require additional
testing in the future. Our results also provide information for future studies
of the $1^{3}F_{4}$ meson nonet.

11.55.Jy, 12.40.Yx, 14.40.Be

##  1 Introduction

In the 1970s, quantum chromodynamics was developed as a theory to describe
strong interactions. With the discovery of asymptotic freedom in 1973, it was
largely accepted since it satisfactorily explained some of the perplexing
experimental results of the time. Nevertheless, there are still difficulties
with the properties of low-energy QCD that require additional investigation,
since the perturbation theory has been shown to be effective in the high-
energy region, but is inapplicable to the low-energy scale. To investigate the
properties of hadrons, such as mass and decay, various phenomenological models
that embody the most essential properties of strong QCD have been constructed.
The results derived from the theoretical model have dual applicability. On the
one hand, it may be used to assess the validity of the model, and on the
other, it can utilize these estimated properties to map the hadron spectrum.
Inspired by this, we investigate the mass spectrum and decay of $1^{3}F_{4}$
nonet in this work. The investigation into the $1^{3}F_{4}$ nonet began more
than 10 years ago. In this work, we believe it is vital to re-evaluate this
issue because experimental data is continually updated throughout time. In
Table 1, we contrast PDG data from 2002, 2012, and 2022 editions. From Table
1, $a_{4}(1970)$, $f_{4}(2050)$, and $K^{\ast}_{4}(2045)$

Table 1. Masses and decay widths of $J^{PC}=4^{++}$ meson states (in units of
MeV). † The states listed as “further states” in the PDG [1].  PDG State Mass
Width $2022$ [1] $a_{4}(1970)$ $1967\pm 16$ $334^{+15}_{-18}$ $f_{4}(2050)$
$2018\pm 11$ $237\pm 18$ $K^{\ast}_{4}(2045)$ $2048^{+8}_{-9}$
$199^{+27}_{-19}$ $f_{4}(2300)^{{\dagger}}$ $2320\pm 60$ $250\pm 80$
$X(2000)^{{\dagger}}$ $1998\pm 3\pm 5$ $<15$ $a_{4}(2255)^{{\dagger}}$
$2237\pm 5$ $291\pm 12$ $2012$[2] $a_{4}(2040)$ $1996^{+10}_{-9}$
$255^{+28}_{-24}$ $f_{4}(2050)$ $2018\pm 11$ $237\pm 18$ $K^{\ast}_{4}(2045)$
$2045\pm 9$ $198\pm 30$ $f_{J}(2220)^{{\dagger}}$ $2231.1\pm 3.5$
$23^{+8}_{-7}$ $f_{4}(2300)^{{\dagger}}$ $2320\pm 60$ $250\pm 80$
$X(2000)^{{\dagger}}$ $1998\pm 3\pm 5$ $<15$ $2002$ [3] $a_{4}(2040)$ $2011\pm
13$ $360\pm 40$ $f_{4}(2050)$ $2025\pm 8$ $194\pm 13$ $K^{\ast}_{4}(2045)$
$2045\pm 8$ $198\pm 30$ $f_{J}(2200)^{{\dagger}}$ $2231.1\pm 3.5$
$23^{+8}_{-7}$ $f_{4}(2300)^{{\dagger}}$ $2332\pm 15$ $260\pm 57$

are well established as members of the $1^{3}F_{4}$ meson nonet, despite the
fact that the mass of $a_{4}(1970)$ has decreased by approximately 50 MeV in
20 years. The $a_{4}(2225)$ was listed as “further states” and reported in
$\bar{p}p\rightarrow\pi^{0}\eta$, $3\pi^{0}$, $\pi^{0}\eta^{\prime}$ [4], and
$\pi^{-}p\rightarrow\eta\eta\pi^{0}$ [5]. Compared to other states, the
$s\bar{s}$ member of $1^{3}F_{4}$ have not been clearly established yet. In
previous years, the $f_{J}(2220)$ has been thoroughly investigated as a
candidate for $s\bar{s}$ member. However, considering that its width is only
23 MeV, it is too small for traditional mesons. Over the years, people have
also made other speculations, such as Higgs boson [6], bound state of colored
scalars [7], four quark state [8, 9], $\Lambda\bar{\Lambda}$ bound state [10],
hybrid or glueball state [11, 12]. The $f_{4}(2300)$ also has $J^{PC}=4^{++}$
quantum number, although it is omitted from the summery table in the PDG [1].
The $f_{4}(2300)$ was first discovered in Ref. [13] and was subsequently
observed in the reactions, $\bar{p}p\rightarrow K^{-}K^{+}$
[13],$\eta\pi^{0}\pi^{0}$ [14],$\pi\pi$ [15, 16], and $\pi^{-}p\rightarrow
K^{+}K^{-}$ [17]. There are some theoretical studies on the properties of the
observed $f_{4}(2300)$ state. In Ref. [18], Masjuan et al. show that
$f_{4}(2300)$ may be a $s\bar{s}$ member of $1^{3}F_{4}$. The paper by Pang et
al. shows that $f_{4}(2300)$ may be the first radial state of $f_{4}(2050)$
[19]. In a word, the understanding of $s\bar{s}$ member of the $1^{3}F_{4}$
meson nonet is still unclear. In this work, considering the present research
situation, we will systematically analyze the mass and decays of $s\bar{s}$
member of the $1^{3}F_{4}$ meson nonet.

This paper is organized as follows. In Section 2, a brief summary of the Regge
phenomenology and the ${}^{3}P_{0}$ model will be presented. The numerical
results of the $1^{3}F_{4}$ meson nonet are shown in Section 3, while the
results are described in Section 4.

##  2 Theoretical models

###  2.1 Regge phenomenology

In this section, we will review the Regge phenomenology theory. In the 1960s,
the Regge theory was developed, which connects the high-energy behavior of the
scattering amplitude with singularities in the complex angular momentum plane
of the partial wave amplitudes [20]. According to Regge theory, mesons have
poles that shift in the plane of complex angular momentum as a function of
their energy. Regge trajectories of hadrons are often shown on the $(J,M^{2})$
plane, and these plots are known as Chew-Frautschi plots (where $J$ and $M$
are the total spins and masses of the hadrons, respectively)[21]. Recent years
have seen a resurgence in interest in the Regge theory due to the fact that it
may be applied to the prediction of meson masses as well as the determination
of the quantum numbers of newly detected states in experiments [18, 22, 23,
24, 25]. In the past two decades, the quasilinear Regge trajectory was used
for studying hadron spectra and result in reasonable description for the
hadron spectroscopy [23, 26, 27, 28].

Based on the assumption that the hadrons with identical $J^{PC}$ quantum
numbers obey quasi-linear form of Regge trajectories, one have the following
relations

$J=\alpha_{n\bar{n}}(0)+\alpha^{\prime}_{n\bar{n}}M^{2}_{n\bar{n}},$ (1)
$J=\alpha_{n\bar{s}}(0)+\alpha^{\prime}_{n\bar{s}}M^{2}_{n\bar{s}},$ (2)
$J=\alpha_{s\bar{s}}(0)+\alpha^{\prime}_{s\bar{s}}M^{2}_{s\bar{s}},$ (3)

The $\alpha^{\prime}$ and $\alpha$ are the slope and intercept of the Regge
trajectory, respectively. In this work, the intercept and slope can be
expressed as

$\alpha_{n\bar{n}}(0)+\alpha_{s\bar{s}}(0)=2\alpha_{n\bar{s}}(0),$ (4)
$\frac{1}{\alpha^{\prime}_{n\bar{n}}}+\frac{1}{\alpha^{\prime}_{s\bar{s}}}=\frac{2}{\alpha^{\prime}_{n\bar{s}}}.$
(5)

The intercept relation was derived from the dual-resonance model [29], and is
satisfied in two-dimensional QCD [30], the dual-analytic model [31], the quark
bremsstrahlung model [32]. The slope relation (5) was obtained in the
framework of topological expansion and the $q\bar{q}$-string picture of
hadrons [33].

From relations (1)-(5), one has

$M^{2}_{n\bar{n}}\alpha^{\prime}_{n\bar{n}}+M^{2}_{s\bar{s}}\alpha^{\prime}_{s\bar{s}}=2M^{2}_{n\bar{s}}\alpha^{\prime}_{n\bar{s}}.$
(6)

Taking into account the fact that it is assumed that the partners’
trajectories would always have parallel slopes,[22, 28], that is to say,

$\left\\{\begin{array}[]{c}\alpha^{\prime}_{n\bar{n}}(1^{3}P_{2})=\alpha^{\prime}_{n\bar{n}}(1^{3}F_{4})\\\
\\\
\alpha^{\prime}_{n\bar{s}}(1^{3}P_{2})=\alpha^{\prime}_{n\bar{s}}(1^{3}F_{4})\\\
\\\
\alpha^{\prime}_{s\bar{s}}(1^{3}P_{2})=\alpha^{\prime}_{s\bar{s}}(1^{3}F_{4})\\\
\end{array}\right..$ (7)

The following relations, which are connected to meson multiplets with the same
regge slopes, can be obtained by eliminating the slopes of the equations:

$\frac{4M^{2}_{n\bar{s}(1^{3}P_{2})}M^{2}_{n\bar{n}(1^{3}F_{4})}-4M^{2}_{n\bar{n}(1^{3}P_{2})}M^{2}_{n\bar{s}(1^{3}F_{4})}}{M^{2}_{n\bar{n}(1^{3}P_{2})}M^{2}_{s\bar{s}(1^{3}F_{4})}-M^{2}_{s\bar{s}(1^{3}P_{2})}M^{2}_{n\bar{n}(1^{3}F_{4})}}$
$=\frac{M^{2}_{n\bar{s}(1^{3}P_{2})}(M^{2}_{n\bar{n}(1^{3}F_{4})}-M^{2}_{s\bar{s}(1^{3}F_{4})})-M^{2}_{n\bar{s}(1^{3}F_{4})}(M^{2}_{n\bar{n}(1^{3}P_{2})}-M^{2}_{s\bar{s}(1^{3}P_{2})})}{M^{2}_{n\bar{s}(1^{3}P_{2})}M^{2}_{s\bar{s}(1^{3}F_{4})}-M^{2}_{s\bar{s}(1^{3}P_{2})}M^{2}_{n\bar{s}(1^{3}F_{4})}},$
(8)

where $M_{n\bar{n}}$, $M_{n\bar{s}}$, and $M_{s\bar{s}}$ denote the mass
corresponding to the meson nonet member.

###  2.2 ${}^{3}P_{0}$ model

The ${}^{3}P_{0}$ model, also known as the quark-pair creation (QPC) model,
was first proposed by Micu, and developed in the 1970s by LeYaouanc et al [34,
35, 36, 37]. To this day, the model is widely used to calculate the decay
amplitude and decay branch ratios of hadrons and has achieved very good
results [38, 39, 40, 41, 42, 43, 44, 45]. In this model, the decay process
$A\rightarrow BC$ occurs when the quark-antiquark pair produces a state
suitable for quark rearrangement in the vacuum. The transition operator $T$ of
the decay $A\rightarrow BC$ in the ${}^{3}P_{0}$ model is denoted by

$T=-3\gamma\sum_{m}\langle 1m1-m\mid 00\rangle\int
d^{3}\boldsymbol{p}_{3}d^{3}\boldsymbol{p}_{4}\delta^{3}\left(\boldsymbol{p}_{3}+\boldsymbol{p}_{4}\right)$

$\times\mathcal{Y}_{1}^{m}\left(\frac{\boldsymbol{p}_{3}-\boldsymbol{p}_{4}}{2}\right)\chi_{1,-m}^{34}\phi_{0}^{34}\omega_{0}^{34}b_{3}^{\dagger}\left(\boldsymbol{p}_{3}\right)d_{4}^{\dagger}\left(\boldsymbol{p}_{4}\right)$
(9)

where $\boldsymbol{p}_{3}$ and $\boldsymbol{p}_{4}$ are the momentum of the
created quark (antiquark). The dimensionless parameter $\gamma$ represents the
strength of the quark-antiquark pair created from the vacuum.
$\chi_{1,-m}^{34},\phi_{0}^{34}$, and $\omega_{0}^{34}$ are spin, flavor, and
color wave functions of the created quark-antiquark pair, respectively. The
partial wave amplitude $\mathcal{M}^{LS}(\boldsymbol{P})$ for the decay
$A\rightarrow B+C$ may be written as[46]

$\mathcal{M}^{LS}(\boldsymbol{P})=\sum_{\begin{subarray}{c}M_{J_{B},M},M_{J_{C}}\\\
M_{S}M_{L}\end{subarray}}\left\langle LM_{L}SM_{S}\mid
J_{A}M_{J_{A}}\right\rangle\left\langle J_{B}M_{J_{B}}J_{C}M_{J_{C}}\mid
SM_{S}\right\rangle$

$\int d\Omega
Y_{LM_{L}}^{*}\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}(\boldsymbol{P})}$ (10)

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

$\langle
BC|T|A\rangle=\delta^{3}\left(\boldsymbol{P}_{A}-\boldsymbol{P}_{B}-\boldsymbol{P}_{C}\right)\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\boldsymbol{P})$
(11)

The detailed analysis process can be referred to in Refs. [47, 48].

In Refs. [41], Ackleh et al. developed a diagrammatic, momentum space
formulation of the ${}^{3}P_{0}$ model to evaluate the partial width
$\Gamma_{A\rightarrow BC}$

$\Gamma_{A\rightarrow
BC}=2\pi\frac{PE_{B}E_{C}}{M_{A}}\sum_{LS}\left(M_{LS}\right)^{2}$ (12)

where $P$ is the decay momentum, $E_{B}$ and $E_{C}$ are the energies of
mesons $B$ and $C$, in the rest frame of A,

$P=\frac{\left[\left(M_{A}^{2}-\left(M_{B}+M_{C}\right)^{2}\right)\left(M_{A}^{2}-\left(M_{B}-M_{C}\right)^{2}\right)\right]^{1/2}}{2M_{A}}$

$E_{B}=\frac{M_{A}^{2}-M_{C}^{2}+M_{B}^{2}}{2M_{A}}$

$E_{B}=\frac{M_{A}^{2}+M_{C}^{2}-M_{B}^{2}}{2M_{A}}$

$M_{A}$, $M_{B}$, and $M_{C}$ are the masses of mesons $A$, $B$, and $C$.
$M_{LS}$ are proportional to an overall Gaussian in $\frac{P}{\beta}$ times a
channel-dependent polynomial $\xi_{LS}(\frac{P}{\beta})$,

$M_{LS}=\frac{\gamma}{\pi^{1/4}\beta^{1/2}}\xi_{LS}(\frac{P}{\beta})e^{-P^{2}/12\beta^{2}}$

In this work, we take $\beta=0.4GeV$ and $\gamma=0.4$ as input, which is used
in Refs.[40, 41].

##  3 NUMERICAL RESULTS

As presented in Eq. (8), the masses of $n\bar{n}$, $n\bar{s}$, $s\bar{s}$ of
the $1^{3}P_{2}$ state are related to the masses of $n\bar{n}$, $n\bar{s}$,
$s\bar{s}$ of the $1^{3}F_{4}$ state, inserting the values
$M_{n\bar{n}(1^{3}P_{2})}$, $M_{n\bar{s}(1^{3}P_{2})}$,
$M_{s\bar{s}(1^{3}P_{2})}$, $M_{n\bar{n}(1^{3}F_{4})}$ , and
$M_{n\bar{s}(1^{3}F_{4})}$ , the mass of the member of $1^{3}F_{4}$ is
determined to be 2129$\pm$20 MeV. In this work, the values
$M_{n\bar{n}(1^{3}P_{2})}$ , $M_{n\bar{s}(1^{3}P_{2})}$ ,
$M_{n\bar{n}(1^{3}F_{4})}$, and $M_{n\bar{s}(1^{3}F_{4})}$ are taken from PDG,
and the $M_{s\bar{s}(1^{3}P_{2})}$ is calculated using the average values in
Table 2.

Table 2 The mass of $s\bar{s}$ member of the $1^{3}P_{2}$ in different
theoretical models (in units of MeV).  [22] [28] [49] [50] [51] [52] [53]
average $M_{1^{3}P_{2}(s\bar{s})}$ 1544 1546 1530 1529 1539 1513 1550
1536$\pm$11MeV

Based on the previously determined mass value, we used the ${}^{3}P_{0}$ model
to investigate the decays of the $1^{3}F_{4}(s\bar{s})$ state. By inserting
the decay polynomials $\xi_{LS}(\frac{P}{\beta})$ listed in Appendix A, we
obtain the decay results from Eq. (12). The results are shown in Table 3.

Table 3. Strong decay properties for the $1^{3}F_{4}(s\bar{s})$ state(in units
of MeV). The initial state mass is set to be 2129 $\pm$ 20 MeV, and the masses
of all the final states are taken from PDG.  Decay mode Present work Decay
mode Present work $1^{3}F_{4}(s\bar{s})\rightarrow KK$ 25.9 $\pm$ 2.5
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{(}1460)$ 0
$1^{3}F_{4}(s\bar{s})\rightarrow KK^{*}$ 17.9 $\pm$ 2.5
$1^{3}F_{4}(s\bar{s})\rightarrow\eta\eta^{\prime}$ 2.1 $\pm$ 0.4
$1^{3}F_{4}(s\bar{s})\rightarrow K^{*}K^{*}$ 37.6 $\pm$ 4.2
$1^{3}F_{4}(s\bar{s})\rightarrow\eta^{\prime}\eta^{\prime}$ 0
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{1}(1270)$ 11 $\pm$ 2.2
$1^{3}F_{4}(s\bar{s})\rightarrow\eta f_{1}(1420)$ 0.2 $\pm$ 0.01
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{1}(1400)$ 0.6 $\pm$ 0.2
$1^{3}F_{4}(s\bar{s})\rightarrow\eta f_{2}(1525)$ 0
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{2}^{*}(1430)$ 1.6 $\pm$ 0.6
$1^{3}F_{4}(s\bar{s})\rightarrow\eta\eta(1475)$ 0
$1^{3}F_{4}(s\bar{s})\rightarrow KK^{*}(1414)$ 0.2 $\pm$ 0.05
$1^{3}F_{4}(s\bar{s})\rightarrow\phi\phi$ 2.7 $\pm$ 1.5

##  4 Conclusion

According to the introduction in Section 1, it is important to study the mass
spectrum and decay of mesons using phenomenological models. In the present
work, we investigated the mass spectrum and decay of the $s\bar{s}$ member of
the $1^{3}F_{4}$ meson nonet. In the framework of Regge phenomenology, the
mass of $s\bar{s}$ member is determined to be 2129$\pm$ 20 MeV, which is
consistent with prediction with the covariant oscillator quark model [54],
while it is about 100 MeV smaller than the prediction with relativized quark
model [49, 51]. In the new edition of PDG, the $f_{4}(2300)$ has a definite
quantum numbers $4^{++}$, although it is omitted from the summary table [1].
The Ref. [18] suggest that this state may be a $s\bar{s}$ member of
$1^{3}F_{4}$ meson nonet. In our work, considering the fact that mass is about
200 MeV smaller than the measured mass of $f_{4}(2300)$, our analysis does not
support this assignment. Apart from the mass spectrum, the decay was analyzed
using the ${}^{3}P_{0}$ model, and results indicate that $KK$ and $K^{*}K^{*}$
are the two primary decay modes. Due to limited understanding of the
$s\bar{s}$ member of the $1^{3}F_{4}$ meson nonet, we hope that these analyses
can provide some meaningful guidance in understanding the $1^{3}F_{4}$ meson
nonet.

##  Appendix A the polynomial $\xi_{LS}(\frac{P}{\beta})$ for the $1^{3}F_{4}$
meson nonet decay in ${}^{3}P_{0}$ model.

Decay mode | $\xi_{LS}(\frac{P}{\beta})$
---|---
$1^{3}F_{4}(s\bar{s})\rightarrow KK$ | $\xi_{{40}{(^{3}F_{4}\rightarrow^{1}S_{0}+^{1}S_{0})}}=4.29\times 10^{-1}\qquad{}^{1}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow KK^{*}$ | $\xi_{{41}{(^{3}F_{4}\rightarrow^{1}S_{0}+^{3}S_{1})}}=-1.43\times 10^{-1}\qquad{}^{3}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow K^{*}K^{*}$ | $\xi_{{LS}{(^{3}F_{4}\rightarrow^{3}S_{1}+^{3}S_{1})}}=\left\\{\begin{array}[]{cc}-4.87\times 10^{-1}&\qquad{}^{5}{D}_{4}\\\ 3.61\times 10^{-2}&\qquad{}^{1}{G}_{4}\\\ -7.14\times 10^{-1}&\qquad{}^{5}{G}_{4}\end{array}\right.$
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{1}(1270)$ | $\xi_{{LS}{(^{3}F_{4}\rightarrow^{1}P_{1}+^{1}S_{0})}}=\left\\{\begin{array}[]{cc}-1.98\times 10^{-1}&\qquad{}^{3}{F}_{4}\\\ -1.40\times 10^{-1}&\qquad{}^{3}{H}_{4}\\\ \end{array}\right.$
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{1}(1400)$ | $\xi_{{LS}{(^{3}F_{4}\rightarrow^{3}P_{1}+^{1}S_{0})}}=\left\\{\begin{array}[]{cc}5.11\times 10^{-2}&\qquad{}^{3}{F}_{4}\\\ 2.65\times 10^{-3}&\qquad{}^{3}{H}_{4}\\\ \end{array}\right.$
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{2}^{*}(1430)$ | $\xi_{{LS}{(^{3}F_{4}\rightarrow^{3}P_{2}+^{1}S_{0})}}=\left\\{\begin{array}[]{cc}8.53\times 10^{-2}&\qquad{}^{5}{F}_{4}\\\ 2.38\times 10^{-2}&\qquad{}^{5}{H}_{4}\\\ \end{array}\right.$
$1^{3}F_{4}(s\bar{s})\rightarrow KK^{*}(1414)$ | $\xi_{{41}{(^{3}F_{4}\rightarrow^{1}S_{0}+2^{3}S_{1})}}=-2.13\times 10^{-2}\qquad{}^{3}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow KK_{(}1460)$ | $\xi_{{40}{(^{3}F_{4}\rightarrow^{1}S_{0}+2^{1}S_{0})}}=1.90\times 10^{-2}\qquad{}^{1}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow\eta\eta$ | $\xi_{{40}{(^{3}F_{4}\rightarrow^{1}S_{0}+^{1}S_{0})}}=3.79\times 10^{-1}\qquad{}^{1}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow\eta\eta^{\prime}$ | $\xi_{{40}{(^{3}F_{4}\rightarrow^{1}S_{0}+^{1}S_{0})}}=1.63\times 10^{-1}\qquad{}^{1}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow\eta^{\prime}\eta^{\prime}$ | $\xi_{{40}{(^{3}F_{4}\rightarrow^{1}S_{0}+S_{0})}}=2.55\times 10^{-2}\qquad{}^{1}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow\eta f_{1}(1420)$ | $\xi_{{LS}{(^{3}F_{4}\rightarrow^{1}S_{0}+^{3}P_{0})}}=\left\\{\begin{array}[]{cc}4.31\times 10^{-2}&\qquad{}^{3}{F}_{4}\\\ -1.97\times 10^{-3}&\qquad{}^{3}{H}_{4}\\\ \end{array}\right.$
$1^{3}F_{4}(s\bar{s})\rightarrow\eta f_{2}(1525)$ | $\xi_{{LS}{(^{3}F_{4}\rightarrow^{1}S_{0}+^{3}P_{2})}}=\left\\{\begin{array}[]{cc}1.54\times 10^{-2}&\qquad{}^{5}{F}_{4}\\\ 1.34\times 10^{-4}&\qquad{}^{5}{H}_{4}\\\ \end{array}\right.$
$1^{3}F_{4}(s\bar{s})\rightarrow\eta\eta(1475)$ | $\xi_{{40}{(^{3}F_{4}\rightarrow^{1}S_{0}+2^{1}S_{0})}}=4.69\times 10^{-3}\qquad{}^{1}{G}_{4}$
$1^{3}F_{4}(s\bar{s})\rightarrow\phi\phi$ | $\xi_{{LS}{(^{3}F_{4}\rightarrow^{3}S_{1}+^{3}S_{1})}}=\left\\{\begin{array}[]{cc}-1.60\times 10^{-1}&\qquad{}^{5}{D}_{4}\\\ 2.78\times 10^{-3}&\qquad{}^{1}{G}_{4}\\\ -5.51\times 10^{-3}&\qquad{}^{5}{G}_{4}\\\ \end{array}\right.$

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