Excitation functions of related temperatures of
η and η ^0 emission sources from squared momentum transfer spectra
in high-energy collisions

Qi<EMAIL_ADDRESS><EMAIL_ADDRESS>Fu-Hu<EMAIL_ADDRESS><EMAIL_ADDRESS>Khusniddin K<EMAIL_ADDRESS>
$^1$Department of Basic Sciences, Shanxi Institute of
Energy, Jinzhong 030600, China

$^2$Institute of Theoretical Physics, State Key Laboratory of
Quantum Optics and Quantum Optics Devices

& Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan 030006, China

$^3$Laboratory of High Energy Physics, Physical-Technical
Institute of SPA “Physics-Sun" of Uzbek Academy of Sciences,

Chingiz Aytmatov str. $2^b$, Tashkent 100084, Uzbekistan

Abstract: The squared momentum transfer spectra of
$\eta$ and $\eta ^0$, produced in high-energy photon-proton ($\gamma p$)
$\rightarrow {\rm meson + proton}$ process in electron-proton ($ep$)
collisions performed at CEBAF, NINA, CEA, SLAC, DESY, and WLS are
analyzed. The Monte Carlo calculations are used in the
analysis of the squared momentum transfer spectra, where the transfer
undergoes from the incident $\gamma$ to emitted meson or equivalently
from the target proton to emitted proton. In the calculations, the
Erlang distribution and Tsallis-Levy function are used to describe
the transverse momentum spectra of emitted particles. Our results
show that the average transverse momentum ($\langle p_T\rangle$),
the initial-state temperature ($T_i$), and the final-state temperature
($T_0$) roughly decrease from lower center-of-mass energy ($W$) to
higher one, and at several special $W$, this tendency is broken.

Keywords: Initial-state temperature, final-state temperature, squared momentum transfer, Erlang distribution, Tsallis-Levy function

12.40.Ee, 14.40.-n, 24.10.Pa, 25.75.Ag

§ INTRODUCTION

Abundant experimental results produced at LHC and RHIC are helpful
to study the related issues of production of Quark-Gluon Plasma (QGP)
and evolution of collision system. In the process of high energy
heavy-ion collisions, the time evolution of collision system roughly
consists of five stages which are incoming of nuclei, beginning of
collisions, strongly-coupled QGP (sQGP), mixed phase, and hadron
gas [1] respectively. In the stages of incoming of nuclei and beginning of
collisions, two nuclei with the shape of pancake due to the Lorentz
contraction move toward each other and collide. Because of the
transformation from kinetic energy of particles to the huge amount
of thermal energy of system, after a short period of time (about 1 $fm/c$),
QGP is produced which is the extremely hot and dense matter [2, 3, 4, 5].
In the stages of mixed phase and hadron gas, due to the inflation
and cooling down of the system, the hadron matter appears until
the system is hadronic. In high energy collisions, the excitation
and equilibrium degrees of system are considered as the important
characteristics which can help us to study the mechanism of nuclear
reaction and the property of system evolution [6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

In the whole process of high energy collisions, one can use different
temperatures to describe the excitation degree of system or emission
source at different stages [16, 17, 18, 19, 20, 21, 22, 23]. At the first
place, one can choose the initial-state temperature ($T_i$) to describe
the excitation degree of system at the beginning of collisions. At the
second place, one can use the critical temperature ($T_c$) and the
chemical freeze-out temperature ($T_{ch}$) to describe the excitation
degree of system in which the hadron matter appears and at chemical
freeze-out separately. At the last place, one can use the kinetic
freeze-out or final-state temperature ($T_{kin}$ or $T_0$) and the
effective temperature ($T_{eff}$) to describe the excitation degree
of system at kinetic freeze-out. $T_{eff}$ in which the influence
of flow effect is included compared with $T_{kin}$ or $T_0$.

$T_i$, a useful tool for describing the excitation degree of system,
represents the temperature of system or emission source at the
initial-stage of collisions [24]. This initial-stage refers
to a very short stage after thermalization at the beginning of
collisions. To obtain $T_i$, we have several methods. The first
method is to solve the state equation of QGP with fluid model [19].
The second method is to solve the equation for isentropic expansion
in relativistic fluid mechanics [16]. The last method is to
use the transverse momentum ($p_T$) spectra directly, or use various
distributions or functions to fit $p_T$ spectra. The last method
has advantages of accuracy and efficiency. This is because there is
no need to study the concrete evolution process from QGP or sQGP to
hadron phase, but analyze the $p_T$ spectra. Usually, we can use the
Erlang distribution [25, 26, 27], Hagedorn function [28],
and Tsallis-Levy function [29] fitting $p_T$ spectra to
obtain $T_i$, but in this paper, only the Erlang distribution is selected
due to its origin of multiple sources in the multi-source thermal
model [25, 26, 27]. In the process of fitting $p_T$ spectra,
$\langle p_T\rangle$ can be abstracted naturally.

The final-state temperature ($T_0$) known as the kinetic freeze-out
temperature represents the temperature of system or emission source
at the kinetic freeze-out stage. At this stage, interactions between
particles are very week, and there is no elastic collision in
system. $T_0$ obtained by using certain distribution or function to
fit $p_T$ spectra is used to describe the excitation degree of
system. $T_{eff}$ is similar to $T_0$, but there is no influence of
flow effect in $T_0$. In previous works, we use the Tsallis blast-wave
model [30] and Tsallis-Levy function [29] fitting $p_T$
spectra to abstract $T_0$ and $T$ as $T_{eff}$ respectively. Generally,
$T_0$ and $T$ obtained from fit are different due to the influence
of flow effect. In some small systems such as $\gamma p$, $\gamma^* p$,
and $p p$ collisions $T_0 \approx T$, because the flow effect are small
to be neglected. In the $\gamma p$ collision discussed in this work,
we use the Tsallis-Levy function [29] fitting $p_T$ spectra
to abstract $T$ as $T_0$ and study the characteristics of system at
the kinetic freeze-out stage. As for other temperatures, although
they are also important tools to describe the excitation degree of
system, we don't discuss them in the following part.

One can use some parameters to describe the equilibrium degree of system.
By fitting $p_T$ spectra with the Tsallis distribution [31, 32],
the entropy index $q$ is abstracted. The entropy index $q$ is closer to 1,
system is closer to equilibrium, or the equilibrium degree of system is
higher. In the absence of the Tsallis distribution [31, 32], one can
use the Hagedorn function [28] or Tsallis-Levy function [29]
alternatively. In the fitting process, $n$ can be abstracted, which is
used to describe the equilibrium degree of system indirectly due to
$n=1/(q-1)$. In this work, let $n$ be fixed, because we consider that
$T_i$ and $T_0$ are enough to describe the equilibrium degree of system.

To obtain above-mentioned different temperatures and $\langle p_T\rangle$,
one can use various distributions and functions fitting $p_T$ spectra.
In the absence of $p_T$ spectra, the squared momentum transfer ($|t|$)
spectra is alternative. The squared momentum transfer is one of the Mandelstam
variables which consist of four-momentum of particles [33]. In the fitting
process, the squared momentum transfer spectra can not be fitted by those
distributions and functions directly. To fit the squared momentum transfer
spectra, at first we need to obtain many concrete $p_T$ satisfying certain
distribution, then we can obtain many concrete values of the squared momentum
transfer with the Monte Carlo calculation, at last, we obtain the distribution
of the squared momentum transfer.

In this paper, the squared momentum transfer spectra of $\eta$ and $\eta^0$,
produced in high-energy $\gamma p$ collisions performed at CEBAF [34],
NINA [35], CEA [36], SLAC [37], DESY [38], and WLS [39]
are fitted by the results calculating with the Monte Carlo method. These
experimental data are measured at different center-of-mass energy ($W$)
and incident photon energy ($E_\gamma$). In section 4, we give our summary
and conclusions.

§ FORMALISM AND METHOD

i) The Erlang distribution

The Erlang distribution is a type of distribution originating from
the multi-source thermal model [25, 26, 27]. Theory of multi-source
assumes that multiple sources are formed and contribute to $p_T$ of
considered particles in collision process. These sources are considered
as nucleons or partons if we study the formation of nucleon clusters
(nuclear fragments) or particles. Generally speaking, it is enough to
use one or two-component Erlang distribution fitting $p_T$ spectra.

The form of the Erlang distribution is the convolution of multiple
exponential distributions [25, 26, 27]. Every exponential distribution
represents the transverse momentum ($p_t$) distribution of a parton,
and can be regarded as

\begin{align}
f(p_{tj})=\frac{1}{\langle p_t \rangle} \exp\bigg(-\frac{p_{tj}}{\langle
p_t \rangle}\bigg).
\end{align}

Here, $j$, $p_{tj}$, and $\langle p_t\rangle$ refer to the order of
participant partons, the variable which depends on $j$, and the average
contribution of participant partons to $\langle p_T \rangle$ of the
considered particles respectively.

During use, it assumes that $n_s$ partons (partons-like) contribute to
$p_T$ of a given particle. We have the Erlang $p_T$ distribution to be

\begin{align}
p_t \rangle}^{n_s}} \exp\bigg(-\frac{p_T}{{\langle p_t
\rangle}}\bigg).
\end{align}

In Eq. (2), the $p_T$ of a given particle consists of $p_{t1}$, $p_{t2}$,
..., $p_{tn_s}$ of $n_s$ partons. Here $n_s$ is not large and about $2-5$.
This is because that $n_s$ is not determined by the collision system, but
the number of contributor partons. As for $N$, it is the number of particles
and depends on the collision system.

ii) The Tsallis-Levy function

The Tsallis-Levy function is one of applications of the Tsallis statistics [31]
in high energy collisions. We have $p_T$ distribution to be the Tsallis-Levy function [29]

\begin{align}
f_2(p_T)=& \frac{1}{N}\frac{dN}{dp_T} \nonumber\\
=& Cp_T\bigg(1+\frac{\sqrt{p_T^2+m_0^2}-m_0}{nT}\bigg)^{-n}.
\end{align}

Here $T$ and $n$ are free parameters, $\sqrt{p_T^2+m_0^2} \equiv m_T$
is the transverse mass, $m_0$ is the rest mass of the considered particle,
and $C$ is the normalized constant which is related to $T$, $n$, and $m_0$
to make $\int_0^{\infty} f_2(p_T) dp_T=1$. Due to particle mass $m_0$
appearing in $\sqrt{p_T^2+m_0^2}-m_0$ in Eq. (3), $f_2(p_T)$ is related
to $m_0$ and this relation is not strong.

$T$ abstracted from the fitting process of $p_T$ spectra with the Tsallis-Levy
function [29] is the effective temperature. $T$ which includes
the influence of flow effect compared with $T_0$ is used to describe
the excitation degree of system at the kinetic freeze-out stage. Generally
speaking $T \textgreater T_0$, but in the $\gamma p$ collision discussed
in this work, due to the flow effect being small and considered negligible,
we are of the opinion that $T \approx T_0$ roughly. To obtain the more
accurate excitation function of $T$, we set $n$ as a fixed value in the
fitting with the Tsallis-Levy function [29].

iii) Average transverse momentum and initial-state

In the process of fitting $p_T$ spectra with the Erlang distribution [25, 26, 27],
$\langle p_T\rangle$ and $T$ which are used to describe the excitation degree
of system can be abstracted. In the first place, $\langle p_T\rangle$ can be
obtained by

\begin{align}
\langle p_T\rangle =\int_0^{\infty} p_T f(p_T) dp_T =n_s\langle
\end{align}

where $f(p_T)$ is the normalized Erlang distribution. Actually, the Erlang
distribution is normalized by itself. The above-mentioned $\langle p_t\rangle$
and $n_S$ are parameters related to partons.

In the second place, we need to introduce a color string percolation method [40, 41, 42]
to obtain $T_i$. With this method, $T_i$ can be expressed as

\begin{align}
T_i =\sqrt{\frac{\langle p_T^2 \rangle}{2F(\xi)}},
\end{align}

\begin{align}
\langle p_T^2\rangle =\int_0^{\infty} p^2_T f(p_T) dp_T
\end{align}

and $F(\xi)$ is the color suppression factor. In addition, $\sqrt{\langle p_T^2\rangle}$
is the root-mean-square of $p_T$ because $f(p_T)$ is normalized to 1.

In the last place, there is a need to discuss the application of color string percolation
method. In the process of using this method, we can determine the number of string used.
For instance, only one string is used in present work, that is to say $F(\xi)=1$ [43].
If we consider other strings, there will be the minimum $F(\xi)\approx0.6$ which results in
the maximum increase of 29.1% in $T_i$ [43]. Although it is possible to have any other
strings in this work, they do not have a great influence on $T_i$. This is because that one
string accounts for a large proportion, but two and multiple strings account for a small
one in our discussion.

iv) The squared momentum transfer

In the center-of-mass reference frame, in two-body reaction $2+1\rightarrow 4+3$
or two-body-like reaction, three Mandelstam variables [33], $s$, $t$, and $u$ are
defined. They are composed of four-momentum of participated particles and
their forms are

\begin{align}
\end{align}

\begin{align}
\end{align}

\begin{align}
\end{align}

respectively. Here, $P_{1}$, $P_{2}$, $P_{3}$, and $P_{4}$ are four-momenta of
particles 1, 2, 3, and 4 separately. Particle 1 is the target proton which is
supposed to be incident along the $Oz$ direction, and particle 2 is the incident
$\gamma$ which is supposed to be incident along the opposite direction. After
collisions, particle 3 is the emitted proton which is emitted with angle
$\theta$ relative to the $Oz$ direction, and particle 4 is the emitted meson
which is emitted along the opposite direction.

Due to different forms of Mandelstam variables, physical meaning of $s$,
$t$, and $u$ is different. $\sqrt{s}$ is supposed to be the center-of-mass
energy, both $-u$ and $-t$ refer to the squared momentum transfer between
particles. In this work, we choose variable $-t$ to research and its form is

\begin{align}
|t|=& |({E_1}-{E_3})^{2}-({\vec{p}_{1}}-{\vec{p}_{3}})^{2}| \nonumber\\
=& \bigg|m_1^2+m_3^2-2{E_1}\sqrt{\bigg({\frac{p_{3T}}{\sin\theta}}\bigg)^{2}+m_3^2} \nonumber\\
& +2\sqrt{E_1^2-m_1^2}\frac{p_{3T}}{\tan\theta}\bigg|,
\end{align}

where $E_1$ and $E_3$, $\vec{p}_1$ and $\vec{p}_3$, as well as $m_1$ and $m_3$
are the energy, momentum, and rest mass of particles 1 and 3 respectively.
Besides, $p_{3T}$ is the transverse momentum of particle 3 which obeys Equation
(2) or (3).

In this paper, we select the squared momentum transfer spectra at different
center-of-mass energy $W$ and incident photon energy $E_\gamma$ to analyze.
Based on the literature [24], we learned that $W$ = $\sqrt{s}$ = $\sqrt{-(P_1 +P_2)^2}$.
Let $Q^2$ and $x$ be the squared photon virtuality and Bjorken scaling variable,
we have $W \simeq Q^2 /x$ [44, 45, 46, 47, 48, 49, 50, 51].

v) The process of Monte Carlo calculations

Although we can use the Eq. (10) to obtain the single squared momentum
transfer, its distribution is difficult to obtain. To obtain the squared
momentum transfer distribution, we can execute the following steps. At
first, we produce many concrete $p_{3T}$ satisfied
with Eq. (2) or (3) and $\theta$. At second, we can obtain many concrete
squared momentum transfer by calculating with the Eq. (10) repeatedly.
At last, the squared momentum transfer distribution is obtained with the
statistical method.

To produce many concrete $p_{3T}$ and $\theta$, we may use the Monte Carlo
method. Let $R_{1,2}$ and $r_{1,2,3,...,n_s}$ be random numbers distributed
evenly in [0,1]. Then, we obtain many concrete $p_{3T}$ by solving this equation

\begin{align}
\int_0^{p_T}f\left(p'_T\right)dp'_T<R_1<\int_0^{p_T+\delta
\end{align}

where $\delta p_T$ is a small shift relative to $p_T$, and $f(p'_T)$ represents
the Eq. (2) or (3). As for the Eq. (2), there is a simpler expression
of $p_T$. We can solve the equation

\begin{align}
\int_0^{p_{tj}}f\left(p'_{tj}\right)dp'_{tj}=r_j\qquad(j=1,2,3,...,n_s),
\end{align}

then we have

\begin{align}
p_{tj}=-\langle p_t\rangle \ln r_j\qquad(j=1,2,3,...,n_s).
\end{align}

In this way, the simpler expression is written as

\begin{align}
p_{T}=\sum_{j=1}^{n_s} p_{tj} =-\langle p_t\rangle
\sum_{j=1}^{n_s} \ln r_j=-\langle p_t\rangle
\ln\bigg(\prod_{j=1}^{n_s} r_j\bigg).
\end{align}

The distribution of $\theta$ satisfies with

\begin{align}
f_\theta\left(\theta\right)= \frac{1}{2}\sin \theta
\end{align}

which is the half-sine function. In the source's rest frame, it is obtained
under the assumption of isotropic emission. Solving the equation

\begin{align}
\int_0^{\theta}f_\theta\left(\theta'\right)d\theta'=R_2,
\end{align}

we have

\begin{align}
\theta=2\arcsin \left(\sqrt{R_2}\right)
\end{align}

which is needed in the calculations.

The squared momentum transfer distribution obtained through the above steps
is used to fit the experimental data measured at different $W$ and $E_\gamma$.
In the fitting process, parameters $\langle p_t\rangle$, $n_s$, and $T_0$ are
abstracted with the method of least squares. Then, we can obtain $\langle p_T\rangle$
from Eq. (4), $\langle p_T^2\rangle$ from  Eq. (6), and $T_i$ from Eq. (5).
The errors of parameters are obtained by the general method of statistical analysis.

§ RESULTS AND DISCUSSION

Figure 1 shows the differential cross-section, $d\sigma/dt$, in squared
momentum transfer $|t|$ of $\gamma p \rightarrow \eta p$ produced in
different center-of-mass energy ranges $2.52<W<2.56$, $2.56<W<2.60$,
$2.60<W<2.64$, $2.64<W<2.68$, $2.68<W<2.72$, $2.72<W<2.76$, $2.76<W<2.80$,
$2.80<W<2.86$, $2.86<W<2.92$, $2.92<W<2.96$, $2.96<W<3.00$, $3.00<W<3.04$,
$3.04<W<3.08$, and $3.08<W<3.12$ GeV, corresponding to incident photon
energy range $E_\gamma \in \left[ 2.91, 4.72\right]$ GeV. The black squares
represent the experimental
data performed at CEBAF and measured by the CLAS Collaboration [34].
The green solid curves and red dashed and dotted curves are the statistical
results of $|t|$ in which $p_T$ satisfies the Erlang distribution and
Tsallis-Levy function respectively.

Fig. 1. The differential cross-section
$d\sigma/dt$ in $|t|$ of $\gamma p \rightarrow\eta p$ process produced
in $ep$ collisions at energy ranges shown in the panels. The symbols
represent the experimental data measured by the CLAS Collaboration [34].
The green solid curves and red dashed and dotted curves are the statistical
results of $|t|$ in which $p_T$ satisfies the Erlang distribution and
Tsallis-Levy function respectively.

Fig. 2. The differential cross-section
$d\sigma/d|t|$ in $|t|$ of (a, d, e) $\gamma p \rightarrow \eta p$
and (b, c) $\gamma p \rightarrow \eta^0 p$ process produced at (a)
NINA [35], (b) CEA [36], (c) SLAC [37], (d) DESY [38],
and (e) WLS [39] at different $W$ and $E_\gamma$ shown in the
panels. The symbols represent the experimental data. The black solid
curves and black dashed curves are the statistical results of $|t|$
in which $p_T$ satisfies the Erlang distribution and Tsallis-Levy
function respectively.

Table 1. Values of
$\langle p_t\rangle$, $n_s$, $\langle p_T\rangle$, $T_i$, $T_0$,
and the first and last $\chi^2$/ndof corresponding to the statistical
results of $|t|$ in which $p_T$ satisfies the Erlang distribution
and Tsallis-Levy function respectively in Figures 1 and 2.

Fig     $W$ (GeV)     $\langle p_t\rangle$ (GeV/$c$)     $n_s$     $\langle p_T\rangle$ (GeV/$c$)     $T_i$ (GeV)     $\chi^2$/ndof     $T_0$ (GeV)    $\chi^2$/ndof

Fig 1        $(2.52,2.56)$     $0.231\pm0.010$     $3$     $0.693\pm0.030$     $0.566\pm0.024$     $32.76/17$     $0.204\pm0.015$     $33.62/18$

$(2.60,2.64)$     $0.233\pm0.013$     $3$     $0.699\pm0.039$     $0.571\pm0.032$     $22.93/16$     $0.208\pm0.015$     $23.63/17$

$(2.64,2.68)$     $0.212\pm0.014$     $3$     $0.636\pm0.042$     $0.519\pm0.035$     $27.79/16$     $0.189\pm0.014$     $26.76/17$

$(2.68,2.72)$     $0.185\pm0.009$     $3$     $0.555\pm0.027$     $0.453\pm0.022$     $20.99/16$     $0.154\pm0.011$     $21.47/17$

$(2.72,2.76)$     $0.192\pm0.008$     $3$     $0.576\pm0.024$     $0.470\pm0.019$     $14.69/16$     $0.174\pm0.012$     $14.21/17$

$(2.76,2.80)$     $0.197\pm0.011$     $3$     $0.591\pm0.033$     $0.483\pm0.026$     $9.59/16$      $0.178\pm0.009$     $8.83/17$

$(2.80,2.84)$     $0.193\pm0.010$     $3$     $0.579\pm0.030$     $0.473\pm0.024$     $14.40/16$     $0.156\pm0.015$     $14.42/17$

$(2.84,2.88)$     $0.180\pm0.010$     $3$     $0.540\pm0.030$     $0.441\pm0.025$     $20.97/15$     $0.145\pm0.014$     $20.68/16$

$(2.88,2.92)$     $0.178\pm0.008$     $3$     $0.534\pm0.024$     $0.436\pm0.020$     $2.63/13$      $0.143\pm0.009$     $2.67/14$

$(2.92,2.96)$     $0.169\pm0.009$     $3$     $0.507\pm0.027$     $0.414\pm0.022$     $6.19/15$      $0.126\pm0.010$     $5.54/16$

$(2.96,3.00)$     $0.139\pm0.007$     $3$     $0.417\pm0.021$     $0.340\pm0.017$     $8.64/11$      $0.104\pm0.010$     $8.52/12$

$(3.04,3.08)$     $0.138\pm0.010$     $3$     $0.414\pm0.030$     $0.338\pm0.024$     $7.28/11$      $0.097\pm0.013$     $7.14/12$

$(3.08,3.12)$     $0.133\pm0.006$     $3$     $0.399\pm0.018$     $0.326\pm0.015$     $1.78/3$       $0.092\pm0.010$     $1.75/4$

Fig 2(a)     $2.360$             $0.172\pm0.003$     $4$     $0.688\pm0.012$     $0.544\pm0.010$     $0.96/3$      $0.199\pm0.006$     $1.78/4$

$2.551$             $0.165\pm0.002$     $4$     $0.660\pm0.008$     $0.522\pm0.007$     $0.70/3$      $0.188\pm0.002$     $1.39/4$

Fig 2(b)     $(2.694,3.084)$     $0.165\pm0.008$     $4$     $0.660\pm0.032$     $0.522\pm0.026$     $5.87/3$      $0.300\pm0.040$     $9.46/4$

Fig 2(d)     $2.895$             $0.137\pm0.005$     $4$     $0.548\pm0.020$     $0.433\pm0.016$     $9.78/9$      $0.190\pm0.020$     $21.75/10$

$3.484$             $0.129\pm0.006$     $4$     $0.516\pm0.024$     $0.408\pm0.019$     $8.42/9$      $0.182\pm0.013$     $6.37/10$

Fig 2(e)     $2.895$             $0.175\pm0.007$     $4$     $0.700\pm0.028$     $0.553\pm0.022$     $12.21/2$     $0.220\pm0.025$     $13.92/3$

$3.986$             $0.152\pm0.005$     $4$     $0.608\pm0.020$     $0.481\pm0.015$     $2.61/2$      $0.158\pm0.013$     $2.99/3$

Fig. 3. The dependences of (a) $\langle p_T\rangle$,
(b) $T_i$, and (c) $T_0$ on center-of-mass energy ($W$) in $ep$ collisions
produced at different devices.

Table 2. Same as Table

Fig     $E_\gamma$ (GeV)     $\langle p_t\rangle$ (GeV/$c$)     $n_s$     $\langle p_T\rangle$ (GeV/$c$)     $T_i$ (GeV)     $\chi^2$/ndof     $T_0$ (GeV)    $\chi^2$/ndof

Fig 2(c)     $6$     $0.118\pm0.005$     $4$     $0.472\pm0.020$     $0.373\pm0.016$     $1.47/2$     $0.097\pm0.007$     $1.47/3$

In the fitting process, the average transverse momentum $\langle p_t\rangle$,
the number $n_s$ of participant partons, and the final-state temperature
$T_0=T$ of emission source are abstracted. With the values of $\langle p_t\rangle$
and $n_s$, the average transverse momentum $\langle p_T\rangle$ and initial-state
temperature $T_i$ are obtained naturally. To obtain an obvious trend of $T_0$,
let $n$ in the Tsallis-Levy function [29] be fixed. In addition, $E_1$
in Eq. (10) is abstracted as a parameter. In Table 1, we list
the values of these parameters and $\chi^2$/ndof. One can see that the
fitting results are in agreement with the experimental data.

Similar to Figure 1, Figure 2 presents the differential cross-section,
$d\sigma/d|t|$, in $|t|$ of (a, d, e) $\gamma p \rightarrow \eta p$ and
(b, c) $\gamma p \rightarrow \eta^0 p$ produced at (a) NINA [35],
(b) CEA [36], (c) SLAC [37], (d) DESY [38], and (e)
WLS [39] at (a) $W=$ 2.36, 2.551 GeV, in (b) 2.694 GeV $< W <$ 3.084 GeV,
at (c) $E_\gamma=$ 6 GeV, at (d) $W=$ 2.895, 3.484 GeV, and at (e)
$W=$ 2.895, 3.986 GeV. The symbols in Figure 2 represent the experimental
data. The black solid curves and black dashed curves are the statistical
results of $|t|$ in which $p_T$ satisfies the Erlang distribution and
Tsallis-Levy function respectively. The values of parameters and $\chi^2$/ndof
are listed in Table 1. It is clear that the statistical results are in agreement
with the experimental data roughly.

The dependences of (a) $\langle p_T\rangle$, (b) $T_i$, and (c) $T_0$ on
center-of-mass energy ($W$) are given in Figure 3. The different symbols
represent the parameter values derived from free parameters extracted from
Figures 1 and 2. One can see that $\langle p_T\rangle$, $T_i$, and $T_0$
decrease roughly with an increase in $W$. It is worth noting that there are
several abnormal $\langle p_T\rangle$, $T_i$ at $W=$ 2.62, 2.70, 2.74, and
2.98 GeV, and $T_0$ at $W=$ 2.62, 2.70 and 2.74 GeV. In the fitting process
of experimental data produced at CEA and DESY, we consider that the ranges of
$|t|$ are wider than others. To fit better, there is a big difference between
the statistical results of $|t|$ in which $p_T$ satisfies the Erlang distribution
and Tsallis-Levy function, and it results in a higher $T_0$ abstracted from
the statistical results of $|t|$ in which $p_T$ satisfies the Tsallis-Levy function.

Generally speaking, $\langle p_T\rangle$, $T_i$, and $T_0$ increase with
an increase in $W$ in high energy collisions.
It is because that a higher $W$ represents a higher excitation degree
of the system. In $\gamma p$ reaction, there is a reverse situation.
Under the condition of higher $W$, although the energy of emitted $\eta$
or $\eta_0$ is higher, it is more possible to have a small angle
scattering. It is natural that the transverse momentum of $\eta$ or
$\eta_0$ is smaller. As a result, $T_i$ and $T_0$ obtained from the transverse
momentum are smaller. As to those abnormal $\langle p_T\rangle$, $T_i$,
and $T_0$, we consider that there is a high possibility of the large angle
scattering at corresponding $W$.

§ SUMMARY AND CONCLUSIONS

Overall, the squared momentum transfer spectra of $\eta$ and $\eta^0$
produced in $\gamma p\rightarrow{\rm meson} + p$ process have been
analyzed by the statistical results of $|t|$ in which $p_T$ satisfies
the Erlang distribution and Tsallis-Levy function respectively.
The squared momentum transfer undergoes from the incident $\gamma$
to emitted $\eta$ or $\eta^0$, or also equivalently from the
target proton to emitted proton. The statistical results are
in agreement with the experimental data produced at different
devices. In the fitting process, free parameters $\langle p_t\rangle$,
$n_s$, and $T_0$ are extracted. Then, we obtain the dependences of
$\langle p_T\rangle$, $T_i$, and $T_0$ on center-of-mass energy $W$.

The parameters $\langle p_T\rangle$, $T_i$, and $T_0$ decrease with
increasing of $W$ roughly. $W$ is defined as the center-of-mass energy
which effectively reflects the excitation degree of the system. As a rule,
a higher $W$ indicates that the system has a higher excitation degree,
and there should be a larger $\langle p_T\rangle$, $T_i$, and $T_0$.
In fact, it is more possible to have a small angle scattering at
higher $W$. At $W$ corresponding to abnormal $\langle p_T\rangle$, $T_i$,
and $T_0$, there is a high possibility of the large angle scattering.

Data Availability

The data used to support the findings of this study are included
within the article and are cited at relevant places within the
text as references.

Ethical Approval

The authors declare that they are in compliance with ethical
standards regarding the content of this paper.

The funding agencies have no role in the design of the study; in
the collection, analysis, or interpretation of the data; in the
writing of the manuscript; or in the decision to publish the

Conflicts of Interest

The authors declare that there are no conflicts of interest
regarding the publication of this paper.

The work of Q.W. was supported by the Innovative Foundation for
Graduate Education in Shanxi University. The work of Shanxi Group
was supported by the National Natural Science Foundation of China
under Grant Nos. 12147215, 12047571, and 11575103, the Shanxi
Provincial Natural Science Foundation under Grant Nos.
202103021224036 and 201901D111043, the Scientific and
Technological Innovation Programs of Higher Education Institutions
in Shanxi (STIP) under Grant No. 201802017, and the Fund for
Shanxi “1331 Project" Key Subjects Construction. The work of
Kh.K.O. was supported by the Ministry of Innovative Development of
the Republic of Uzbekistan within the fundamental project No.
F3-20200929146 on analysis of open data on heavy-ion collisions at
RHIC and LHC.

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\quantity(m_{i}(t)-2)\quantity(m_{j}(t)-2)&\frac{1}{4}\hat{a}_{i}\delta_{i,j}\differential{t}\\\
\quantity(m_{i}(t)-2)\quantity(m_{j}(t)+2)&\frac{1}{4}\hat{a}_{i}\delta_{i+1,j}\differential{t}\\\
\quantity(m_{i}(t)+2)\quantity(m_{j}(t)-2)&\frac{1}{4}\hat{a}_{i-1}\delta_{i-1,j}\differential{t}\\\
\quantity(m_{i}(t)+2)\quantity(m_{j}(t)+2)&\frac{1}{4}\hat{a}_{i-1}\delta_{i,j}\differential{t}\\\
\quantity(m_{i}(t)-2)m_{j}(t)&\frac{1}{4}\hat{a}_{i}\quantity(1-\delta_{i,j}-\delta_{i+1,j})\differential{t}\\\
\quantity(m_{i}(t)+2)m_{j}(t)&\frac{1}{4}\hat{a}_{i-1}\quantity(1-\delta_{i-1,j}-\delta_{i,j})\differential{t}\\\
m_{i}(t)\quantity(m_{j}(t)+2)&\frac{1}{4}\hat{a}_{j-1}\quantity(1-\delta_{i,j}-\delta_{i,j-1})\differential{t}\\\
m_{i}(t)\quantity(m_{j}(t)-2)&\frac{1}{4}\hat{a}_{j}\quantity(1-\delta_{i,j+1}-\delta_{i,j})\differential{t}\end{cases}$
(150)

From these update rules, we can write the evolution equation of
$C^{mm}_{r}(t,t)$ as

$\displaystyle\derivative{t}\left\langle
m_{i}(t)m_{j}(t)\right\rangle=\sum\limits_{k}\left\langle
m_{i}\Delta_{j,k}\hat{a}_{k}+\Delta_{i,k}\hat{a}_{k}m_{j}\right\rangle_{c}+B_{i,j}.$
(151)

where

$\displaystyle B_{i,j}=$ $\displaystyle\frac{1}{2}\left\langle
3\hat{a}_{i-1}+8\hat{a}_{i}+3\hat{a}_{i+1}\right\rangle\delta_{i,j}$
$\displaystyle-\frac{1}{2}\left\langle
4\hat{a}_{i+1}+4\hat{a}_{i}\right\rangle\delta_{i+1,j}-\frac{1}{2}\left\langle
4\hat{a}_{i-1}+4\hat{a}_{i}\right\rangle\delta_{i+1,j}$
$\displaystyle+\frac{1}{2}\left\langle\hat{a}_{i+1}\right\rangle\delta_{i+2,j}+\frac{1}{2}\left\langle\hat{a}_{i-1}\right\rangle\delta_{i-2,j},$
(152)

is the source of the equal-time mass correlation. In the steady state, where
$\left\langle\hat{a}_{i}(t)\right\rangle=a(\bar{\rho})$, we can write
$B_{i,j}$ as a translationally invariant form, $B_{i,j}\equiv B_{r}$ as

$\displaystyle
B_{r}=7a(\bar{\rho})\delta_{0,r}-4a(\bar{\rho})\quantity(\delta_{0,r+1}+\delta_{0,r-1})+\frac{a(\bar{\rho})}{2}\quantity(\delta_{0,r+2}+\delta_{0,r-2}).$
(153)

Equation (151) has appeared in eq.(36) in the main-text. Using the steady
state condition, we must have $d\left\langle
m_{i}m_{j}\right\rangle_{c}/dt=0$, which implies,

$\displaystyle
2\quantity[C^{m\hat{a}}_{r-1}-2C^{m\hat{a}}_{r}+C^{m\hat{a}}_{r+1}]+B_{r}=0.$
(154)

Eq.(154) can be solved by considering the following generating function,

$\displaystyle G\quantity(z)=\sum\limits_{r=0}^{\infty}C^{m\hat{a}}_{r}z^{r}.$
(155)

We multiply both side of eq.(154) with $z^{r}$ and sum over $r$ to get,

$\displaystyle
G(z)=\frac{4C^{m\hat{a}}_{0}-4zC^{m\hat{a}}_{1}-za\quantity[(z-8)z+14]}{4(1-z)^{2}},$
(156)

where we use the identities,

$\displaystyle\sum\limits_{r=0}^{\infty}C^{m\hat{a}}_{r-1}z^{r}$
$\displaystyle=C^{m\hat{a}}_{1}+zG\quantity(z),$
$\displaystyle\sum\limits_{r=0}^{\infty}C^{m\hat{a}}_{r+1}z^{r}$
$\displaystyle=\frac{G(z)-C^{m\hat{a}}_{0}}{z}.$ (157)

As we are dealing with truncated correlation functions, in the limit
$z\rightarrow 1$, we must have $\lim_{z\rightarrow 1}G(z)<\infty$. Using a new
variable $w\rightarrow 1-z$, we write eq.(156) as

$\displaystyle
G(w)=\frac{1}{4w^{2}}\quantity[a(\bar{\rho})w^{3}+5a(\bar{\rho})w^{2}+(a(\bar{\rho})+4C^{m\hat{a}}_{1})w+-7a(\bar{\rho})+4C^{m\hat{a}}_{0}-4C^{m\hat{a}}_{1}]$
(158)

For the convergence of $G(w)$ in the limit $w\rightarrow 0$, we set

$\displaystyle
a(\bar{\rho})+4C^{m\hat{a}}_{1}=0;-7a(\bar{\rho})+4C^{m\hat{a}}_{0}-4C^{m\hat{a}}_{1}=0,$
(159)

leading to the following exact relations,

$\displaystyle C^{m\hat{a}}_{0}=\frac{3a(\bar{\rho})}{2},$ (160)
$\displaystyle C^{m\hat{a}}_{1}=-\frac{a(\bar{\rho})}{4}.$ (161)

Finally, putting eq.(160) in eq.(156) we get the generating function

$\displaystyle
G\quantity(z)=\frac{3a(\bar{\rho})}{2}-\frac{a(\bar{\rho})}{4}z.$ (162)


[width=]line-topics.png

Figure : Total cost for TAR alternatives to identify 80% of positive documents
for Wikipedia Attack, Other Attack, and Recipient Attack, and ASKfm Curse
Exclusion, General Insult, and Sexism classifications. Values are averaged
over 20 replicates, and a 99% confidence interval on costs is shown as shading
around each curve. Horizontal line is cost to review a random 80% of the data
set.

##  1 Results and Analysis

Our core finding was that, as in eDiscovery, active selection of which
documents to review reduces costs over random selection. Figure  shows mean
cost to reach 80% recall over 20 replications (different seed sets and random
samples) for six representative categories. On all six categories, all TAR
workflows within a few iterations beat the baseline of reviewing a random 80%
of the data set (horizontal line labeled Manual Review). The Wikipedia Attack
category is typical of low to moderate prevalence categories ($p=0.1344$).
Uncertainty sampling strongly dominates both random sampling (too few
positives chosen) and relevance feedback (too many redundant positives chosen
for good training). Costs decrease uniformly with additional training. We plot
99% confidence intervals under the assumption that costs are normally
distributed across replicates. Costs are not only higher for relevance
feedback, but less predictable. The ASKfm Curse Exclusion ($p=0.0169$) and
Wikipedia Other attack ($p=0.0019$) category are typical low prevalence
categories. Uncertainty sampling and relevance feedback act similarly in such
circumstances: even top scoring documents are at best uncertainly positive.
Average cost across replicates levels off and starts to increase after 44
iterations for uncertainty sampling and 45 iterations for relevance feedback.
This is the point at which additional training no longer pays for itself by
improving the ranking of documents. For this category (and typically) this
occurs shortly before 80% recall is reached on the training data alone
(iteration 48 for uncertainty sampling and iteration 52 for relevance
feedback). Task such as the ASKfm Sexism category ($p=0.0030$) that deals with
nuances in human languages requires more training data to produce a stable
classifier. While obtaining training data by random sampling stops reducing
the cost after the first iteration, uncertainty sampling and relevance
feedback continue to take advantage of additional training data to minimize
the cost and become more predictable. Note that the general relationship
between the prevalence of the task and the cost of reaching a certain recall
target using TAR workflows is discussed cost-structure-paper.

Table : Total review cost to reach 80% recall. Values are averaged over all topics for a data set and 20 replicates. Percentages show relative cost reduction over the random sample training baseline. A * indicates that the difference is statistically significant over the random sample training baseline with 99% confidence by conducting paired t-test with Bonferroni correction.  |  | ASKfm | Wikipedia
---|---|---|---
batch | # Train | Random | Relevance | Uncertainty | Random | Relevance | Uncertainty
100 | 202 | 47685.53 | *49833.73 (-4.50) | *50273.21 (-5.43) | 52948.45 | *60751.69 (-14.74) | 52210.00 (01.39)
| 1002 | 46327.93 | *43329.31 (06.47) | *42723.12 (07.78) | 49010.71 | 52931.28 (0-8.00) | *39879.78 (18.63)
| 2002 | 45139.15 | *38179.79 (15.42) | *37938.19 (15.95) | 47805.25 | 46673.34 (002.37) | *29387.06 (38.53)
| 3002 | 44148.28 | *34909.72 (20.93) | *34719.50 (21.36) | 47065.66 | *38964.91 (017.21) | *25676.82 (45.44)
| 4002 | 43731.25 | *33439.69 (23.53) | *32795.05 (25.01) | 47234.75 | *34408.14 (027.16) | *24202.29 (48.76)
| 5002 | 43469.91 | *32261.33 (25.78) | *31957.57 (26.48) | 47125.79 | *31267.88 (033.65) | *22746.94 (51.73)
| 6002 | 42973.85 | *31767.73 (26.08) | *31384.51 (26.97) | 47300.02 | *28945.59 (038.80) | *21922.42 (53.65)
| 7002 | 42563.09 | *30567.00 (28.18) | *30502.95 (28.33) | 47086.42 | *27356.89 (041.90) | *21301.92 (54.76)
| 8002 | 42385.43 | *30708.85 (27.55) | *30441.77 (28.18) | 47106.34 | *25949.51 (044.91) | *21144.28 (55.11)
200 | 202 | 47685.53 | *49302.36 (-3.39) | *49339.93 (-3.47) | 52948.45 | *58866.41 (-11.18) | 55747.35 (-5.29)
| 1002 | 46327.93 | 45014.51 (02.84) | 44733.10 (03.44) | 49010.71 | *55302.14 (-12.84) | *42896.71 (12.47)
| 2002 | 45139.15 | *40473.12 (10.34) | *39894.98 (11.62) | 47805.25 | 49968.88 (0-4.53) | *33981.56 (28.92)
| 3002 | 44148.28 | *37050.02 (16.08) | *36902.63 (16.41) | 47065.66 | 42521.55 (009.65) | *28332.55 (39.80)
| 4002 | 43731.25 | *35310.13 (19.26) | *34888.22 (20.22) | 47234.75 | *37492.98 (020.62) | *25667.95 (45.66)
| 5002 | 43469.91 | *33690.33 (22.50) | *33519.15 (22.89) | 47125.79 | *34933.90 (025.87) | *24070.44 (48.92)
| 6002 | 42973.85 | *32425.25 (24.55) | *32612.13 (24.11) | 47300.02 | *33004.90 (030.22) | *22839.39 (51.71)
| 7002 | 42563.09 | *31488.77 (26.02) | *31813.08 (25.26) | 47086.42 | *31664.04 (032.75) | *22084.88 (53.10)
| 8002 | 42385.43 | *31198.75 (26.39) | *31171.80 (26.46) | 47106.34 | *29346.76 (037.70) | *21837.84 (53.64)
|  |  |  |  |  |  |

Table 1 looks more broadly at the two datasets, averaging costs both over all
topics and over 20 replicate runs for each topic for batch sizes of both 100
and 200 . By 20 iterations with batch size of 100 (2002 training documents),
TAR workflows with both relevance feedback and uncertainty sampling
significantly reduce costs versus TAR with random sampling. (Significance is
based on paired t-tests assuming non-identical variances and making a
Bonferroni correction for 72 tests.) All three TAR methods in turn dominate
reviewing a random 80% of the dataset, which costs 92,590 for Wikipedia and
90,958 for ASKfm. The improvement over cost plateaued after the training sets
reached 5000 documents for ASKfm but continue for Wikipedia. Categories in
Wikipedia ($p=0.1344$ to $0.0018$) are generally more frequent comparing to
ASKfm ($p=0.0463$ to $0.001$), providing more advantage for training to
identify more positive documents. Larger batch size slightly reduce the
improvement as the underlying classifiers are retrained less frequently. In
practice, the sizes are depending on the cost structure of reviewing and
specific workflows in each organization. However, as the classifiers are
frequently updated with more coded documents, the total cost would be reduced
over the iterations.

[width=]bar.png

Figure : Precision in each batch for TAR workflows on Wikipedia
Attack($p=0.1344$) and ASKfm Curse Exclusion($p=0.0169$) classifications. The
x-axis shows the iteration number. A lighter color in an iteration block
indicates higher precision.

Besides the overall cost reduction, Figure 1 shows a heatmap of mean precision
across 20 replicates for batches 1 to 81 with batch size of 100, to give
insight into the moderator experience of TAR workflows. Precision for
relevance feedback starts high and declines very gradually. Uncertainty
sampling maintains relatively constant precision. For the very low prevalence
category Curse Exclusion we cut off the heatmap at 52 iterations for relevance
feedback and 48 iterations for uncertainty sampling since on average 80%
recall is obtained on training data alone by those iterations. For both
categories, even applying uncertainty sampling that is intended to improve the
quality of the classifier improves the batch precision over the random
sampling be a significant amount.