# Mass radius and mechanical properties of the proton via strange $\phi$ meson
photoproduction

Xiao-Yun Wang  Corresponding author<EMAIL_ADDRESS>Department of physics,
Lanzhou University of Technology, Lanzhou 730050, China  Lanzhou Center for
Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province,
Lanzhou University, Lanzhou, Gansu 730000, China  Chen Dong
<EMAIL_ADDRESS>Department of physics, Lanzhou University of Technology,
Lanzhou 730050, China  Quanjin Wang  Department of physics, Lanzhou University
of Technology, Lanzhou 730050, China

###### Abstract

In this work, the cross sections of $\phi$ photoproduction are systematically
investigated with the two gluon exchange model and the pomeron model. The
results obtained show that the theoretical values of the two models agree well
with the experimental data. Since the mass radius and mechanical properties of
the proton are encoded in the scalar gravitational form factor of the momentum
energy tensor, based on the differential cross sections of $\phi$
photoproduction at near-threshold predicted by the two gluon exchange model,
the average mass radius of the proton is derived as $\sqrt{\left\langle
r^{2}\right\rangle_{\mathrm{m}}}=0.78\pm 0.06$ fm. As a comparison, we
directly extract the proton mass radius from the experimental data of $\phi$
photoproduction to obtain an average value of $0.80\pm 0.05$ fm, which is very
close to the result given by the two gluon exchange model. Taking a similar
approach, we extracted the average value of $m_{D}$ to obtain the distribution
of the pressure and shear force contributed within the proton, and compared
the results with other groups. The results of this paper may provide necessary
theoretical information for the subsequent in-depth study of the internal
structure and properties of proton.

##  I Introduction

From the quark model and QCD theory Gross:1973id ; Gross:1973ju ; Gross:1974cs
, it is clear that proton can be divided into quarks and gluons. However, the
mass of three of the valence quarks is only 9.6 MeV/$c^{2}$, which is just
about one percent of the mass of a proton, and we are still exploring other
sources of mass of proton. In addition, quark color confinement and asymptotic
freedom are also problems that have always plagued us. These questions suggest
that our understanding of the internal structure and properties of nucleons is
very limited. Until recent years, it has been found that analyzing the mass
radius and mechanical properties of nucleons may be an effective way to study
and determine the internal structure of nucleons Kharzeev:2021qkd ;
Pefkou:2021fni ; Wang:2021dis ; Mamo:2021krl ; Mamo:2022eui ; Burkert:2021ith
; Pefkou:2021fni . The mass radius is one of the essential characteristics of
the proton structure, which has been proved to be strictly defined by the
Gravitational Form Factors (GFF) of Energy Momentum Tensor (EMT) under the
weak gravitational approximation Kharzeev:2021qkd . In addition, EMT also
contains information related to the variation of the spatial component of the
space-time metric: D-term Polyakov:1999gs ; Polyakov:2018zvc . Since it can be
expressed in terms of the stress tensor, which is also defined as mechanical
properties Polyakov:2002yz ; Burkert:2018bqq . In Ref. Polyakov:2018zvc , a
systematic study of the internal mechanical properties of the proton through
EMT was carried out. At present, the study of mass radius and mechanical
properties is the focus of nuclear physics.

Since the interaction between gravitons and protons is very weak, it is
difficult to directly measure the mass radius and mechanical properties of the
proton through experiments. In Ref. Ji:1996ek ; Ji:1996nm , X. Ji proposed
that the internal quark structure of the proton can be revealed by deeply
virtual Compton scattering (DVCS). In 2018, V. D. Burkert et al. used DVCS
data to extract the pressure distribution inside the proton for the first time
Burkert:2018bqq . The strong repulsive force near the center of the proton and
binding force far away from the center were perceived. Subsequently, the shear
force distribution was derived with the same method Burkert:2021ith , which
discovered that most confinement forces might be around $0.6$ fm from the
center of the proton. In addition to extracting mechanical properties from
DVCS data, the Lattice Quantum Chromodynamics (LQCD) and holographic QCD were
developed for analyzing the mass radius and mechanical properties of the
proton systematically. For example, researchers derived the GFF from LQCD and
the pressure distribution inside the nucleon Shanahan:2018pib  was calculated
with the light quark mass corresponding to $m_{\pi}$. In their subsequent work
Pefkou:2021fni , the mechanical properties of the proton were obtained by the
scalar gravitational tripole fit, and z-expansion was involved to modify the
mechanical distribution. In addition, the mass radius was calculated as
$0.747\pm 0.033$ fm. A novel method was developed for extracting the mass
radius as $0.682$ fm by Holographic QCD in Ref. Mamo:2022eui , and mechanical
properties contained in D-term were described in detail. Besides, in 2019, the
gravitational form factor of valence quarks in nucleons was calculated based
on the light-cone sum rules, and the result agrees well with experimental data
Anikin:2019kwi . In 2021, the light front quark-diquark model was also
proposed to deduce the scalar gravitational form factor and show the pressure
and shear force distribution of the quark inside the proton
Chakrabarti:2021mfd .

Underneath a hypothesis of the scalar gravitational dipole form factor, the
mass radius of the proton can be extracted with the vector quarkoniums
photoproduction at the near-threshold. As well as, the scalar gravitational
tripole form factor is utilized to extract the mechanical properties of the
proton. In 2021, based on associating the GFF with the cross sections of
$J/\psi$ photoproduction from GlueX Collaboration data GlueX:2019mkq , the
mass radius was obtained as $0.55\pm 0.03$ fm Kharzeev:2021qkd  for the first
time. In Ref. Wang:2021dis , the mass radius of $0.67\pm 0.03$ fm was
extracted by utilizing the experimental cross sections of vector mesons
$\phi$, $\omega$ and $J/\psi$ photoproduction. Similarly, the mass radius
$0.68$ fm was derived with the identical method in Ref. Guo:2021ibg . However,
due to the large variation of the cross section of vector mesons
photoproduction at the threshold energy, and the limited accuracy and quantity
of the existing experimental data, there will be some uncertainty if the mass
radius is directly extracted from the photoproduction data of vector mesons.
Therefore, it is necessary to introduce relevant theoretical models to
calculate the cross sections of vector mesons photoproduction, and to further
give the distribution of proton mass radius at the threshold energy. This
allows us to compare and verify the results given by the theoretical model and
the results obtained by direct extraction, so as to give a relatively reliable
value of the proton mass radius. In our previous work Wang:2022vhr , models
such as two gluon exchange was used to predict the photoproduction cross
sections of heavy charmonium $J/\psi$ and $\psi(2S)$. The average proton mass
radius was calculated as $0.67\pm 0.11$ fm, which is basically consistent with
the values given by other groups when errors are considered. However, in
further discussion, it is found that the proton mass radius value extracted
from the near-threshold differential cross section of heavy quarkoniums is
affected by the momentum transfer $|t|_{\min}$, so that the proton mass radius
value obtained around the threshold varies greatly and unstable. Thus, the
lighter vector meson $\phi$ acquired our attention.

As a traditional vector meson of $s\bar{s}$ system, $\phi$ is not only a
bridge to explore the QCD from light to heavy quarks Li:2020xzs , but also its
photoproduction process is suitable for studying the mass radius and
mechanical properties of the proton. As high-energy experiments progress, some
experimental data of $\phi$ photoproduction near-threshold have been
accumulated in recent years by CLAS, SLAC, FERMLAB, etc. Strakovsky:2020uqs ;
Barber:1981fj ; Ballam:1972eq ; Egloff:1979mg ; Busenitz:1989gq ; Owens:2012bv
; Dey:2014tfa ; LEPS:2005hax , which provide the necessary foundation for our
research. In this paper, the two gluon exchange model Ensign:1983jc ;
Baranov:2010zzb ; Suwara:1988uz  is established based on the existing
experiment data of $\phi$ photoproduction, and the pomeron model developed by
Larget Laget:1994ba  is also introduced. Since this pomeron model contains
almost no free parameters and is in good agreement with the previous
experiments, it can be used to compare and verify the validity of the two
gluon exchange model. Based on the cross sections of $\phi$ photoproduction
predicted by the both models, the mass radius $\sqrt{\left\langle
r^{2}\right\rangle_{\mathrm{m}}}$ can be derived by GFF Guo:2021ibg ;
Kharzeev:2021qkd ; Kou:2021qdc ; Ji:1997pf ; Teryaev:2016edw , and proton
mechanical properties including pressure and shear force distributions will be
analyzed. Moreover, the results of this paper can provide theoretical
reference for further study of the proton structure with the Electron-Ion
Collider (EIC) facilities Anderle:2021wcy ; Accardi:2012qut .

The outline of this work is organized as follows. The formalism of the two
gluon exchange and the pomeron models are introduced, and the mass radius and
mechanical properties of the proton are described in Sec. II. The results of
fitting and proton coherence are presented in Sec. III. A general summary in
Sec. IV.

##  II Formalism

###  II.1 Two gluon exchange model

In the two gluon exchange model, the photon fluctuation into the quark-
antiquark pair ($\gamma\rightarrow q+\bar{q}$) is described explicitly in Fig.
1. To begin with, the photon splits into a dipole ($s\bar{s}$). Furthermore,
the dipole exchange two gluons to scatter off the initial proton. Eventually,
the dipole formation of the final state $\phi$, as well as the initial proton,
preserves the proton interaction as two gluons are colorless. In brief, the
two gluon exchange process is dominated by $\phi$ production near the
threshold.

Figure 1:  The Feynman diagram for $\phi$ production based on the two gluon
exchange model.

This work considers the $\phi p$ sattering process $\gamma p\rightarrow\phi p$
with two gluon exchange model. In lowest order perturbative QCD, the amplitude
of $\phi$ photoproduction can be derived from Ref. Brodsky:1994kf ;
Ryskin:1992ui ; Ryskin:1995hz ,

$\displaystyle{\cal T}=$
$\displaystyle\frac{i2\sqrt{2}\pi^{2}}{3}m_{s}\alpha_{s}e_{s}f_{V}F_{2g}(t)$
(1) $\displaystyle\int dl^{2}P_{g}^{2}(l)\left[P_{+}(l)-P_{-}(l)\right]G(l).$

where $m_{s}=0.486$ GeV and $e_{s}=-1/3$ are the mass Kou:2021bez  and charge
of quark, respectively. $f_{v}$ is the pion decay constant, and it can be
extracted from the leptonic decay width, which is given by,

$\Gamma_{e^{+}e^{-}}=\frac{4\pi\alpha^{2}}{3m_{\phi}}f_{V}^{2}.$ (2)

The $\Gamma_{e^{+}e^{-}}=1.27$ keV is the decay width of $\phi$ which can be
obtained from Ref. Sibirtsev:2004ca , and $\alpha=1/137$. With this we can
calculate $f_{V}=0.0761$ GeV. In Eq.(1), $P_{g}(l)$ is the gluon propagator
and it can be taken as $1/l^{2}$. $P_{+}(l)$ is the propagator of the off-
shell quark that two gluons coupled to the same quarks of the vector meson is
written as,

$P_{+}(l)=\left(-2m_{s}^{2}\right)^{-1},$ (3)

and $P_{-}(l)$ is the propagator of the off-shell quark that two gluons
coupled to different quarks of the vector meson is given by,

$P_{-}(l)=\left(-2m_{s}^{2}-2l^{2}\right)^{-1}.$ (4)

The factor $F_{2g}(t)$ in Eq. (1) is interpreted as the dependence of the
amplitude dependence of the two gluons in the proton is expressed as
Ryskin:1992ui ; Xu:2020uaa ,

$F_{2g}(t)=\frac{4m_{p}^{2}-2.8t}{4m_{p}^{2}-t}\frac{1}{\left(1-t/t_{0}\right)^{2}}.$
(5)

where $t_{0}=0.71$ GeV2 and $m_{p}$ is the proton mass. The probability that a
dipole captures a gluon with momentum $l$ from a proton is defined by $G(l)$.
The integral of $G(l)$ is affiliated with the gluon distribution function
$xg(x,Q^{2})$ Ryskin:1992ui ; Brodsky:1994kf ,

$xg\left(x,Q^{2}\right)=\int dl^{2}\frac{G(l)}{l^{2}}.$ (6)

From the above discussion, the amplitude of the two gluon exchange can be
constructed as,

$\displaystyle{\cal T}=$
$\displaystyle\frac{i\sqrt{2}\pi^{2}}{3}m_{s}\alpha_{s}e_{s}f_{V}F_{2g}(t)\left[\frac{xg\left(x,Q_{0}^{2}\right)}{m_{s}^{4}}\right.$
(7)
$\displaystyle\left.+\int_{Q_{0}^{2}}^{+\infty}\frac{dl^{2}}{m_{s}^{2}\left(m_{s}^{2}+l^{2}\right)}\frac{\partial
xg\left(x,l^{2}\right)}{\partial l^{2}}\right].$

$d\sigma/dt=\alpha|{\cal T}|^{2}$ is the differential cross section with the
normalized amplitude. Then, the differential cross section, in the lowest
order is denoted as Xu:2020uaa ,

$\frac{d\sigma}{dt}=\frac{\pi^{3}\Gamma_{e^{+}e^{-}}\alpha_{s}}{6\alpha
m_{s}^{5}}\left[xg\left(x,m_{\phi}^{2}\right)\right]^{2}\exp(b_{0}t),$ (8)

where $x=m^{2}_{\phi}/W^{2}$, $\alpha_{s}=0.701$ is the QCD coupling constant
from Ref. Kou:2021bez . The $t$-dependence of $\phi$ photoproduction can be
exponentially parameterized and $b_{0}$ is constant slope. In addition, the
$|t|$-dependence of exponential slope $b_{0}$ also can be calculated via
$b_{0}=\dfrac{d}{dt}\ln[\dfrac{d\sigma}{dt}]$.

One note that the differential and total cross sections formula obtained
primarily depends on the gluon distribution function $xg(x,m^{2}_{\phi})$. For
example, in Ref. Goloskokov:2007zb , the gluon distribution function given by
CTEQ6M Pumplin:2002vw  can achieve a good description of the cross section of
light vector mesons electroproduction. However, for the $\phi$ meson
photoproduction near threshold, it is difficult to interpret the experimental
data well using the gluon distribution functions given in the current
literature Ball:2011uy ; Owens:2012bv ; Accardi:2016qay ; Wang:2016sfq . In
this work, we simplified parameterized gluon distribute function
$xg(x,m^{2}_{\phi})=A_{0}x^{A_{1}}(1-x)^{A_{2}}$ with three parameters
$A_{0},A_{1},A_{2}$ which can be obtained by the experimental data of $\phi$
photoproduction.

To get the total cross section, one needs to integrate the Eq. (8) from
$t_{min}(W)$ to $t_{max}(W)$ over the allowed kinematical range,

$t_{min}(W)(t_{max}(W))=[\dfrac{m_{1}^{2}-m_{3}^{2}-m_{2}^{2}+m_{4}^{2}}{2W}]^{2}-(P_{1cm}\mp
P_{3cm})^{2},$ (9)

with

$P_{icm}=\sqrt{E_{icm}^{2}-m_{i}^{2}}(i=1,3),$ (10)
$E_{1cm}=\dfrac{W^{2}+m_{1}^{2}-m_{2}^{2}}{2W},E_{3cm}=\dfrac{W^{2}+m_{3}^{2}-m_{4}^{2}}{2W}.$
(11)

Then, the total cross section of $\phi$ photoproduction as,

$\sigma=\frac{\pi^{3}\Gamma_{e^{+}e^{-}\alpha_{s}}}{6\alpha
m_{s}^{5}}\left[xg\left(x,m_{\phi}^{2}\right)\right]^{2}\int_{t_{min}(W)}^{t_{max}(W)}e^{b(W)t}d|t|.$
(12)

The exponential slope $b(W)$ is proportional to the centre of mass energy $W$.
A study by S. Chekanov shows that in the high-energy region the form of $b(W)$
can be derived as Close:2002ky ,

$b(W)=b_{0}+0.46\cdot\operatorname{In}\left(W/W_{0}\right).$ (13)

where $W_{0}=2.29$ GeV is a modified value to calculate $b(W)$ at near-
threshold energy, and $b_{0}$ will be obtained by fitting experimental data of
$\phi$ photoproduction.

###  II.2 Pomeron model

The pomeron model Laget:1994ba  also describes the photoproduction cross
section of vector mesons in attribute, scattering $q\bar{q}$ state by protons
and finally transforming the $q\bar{q}$ state into vector mesons. The
interaction between photon and proton in the pomeron model is elaborated in
Fig. 2, and the differential cross section of meson $\phi$ can be written as,

$\frac{\mathrm{d}\sigma}{\mathrm{d}t}=\frac{81m_{\mathrm{\phi}}^{3}\beta^{4}\mu_{0}^{4}\Gamma_{\mathrm{e}^{+}\mathrm{e}^{-}}}{\pi\alpha}\left(\frac{s}{s_{0}}\right)^{2\alpha(t)-2}\left(\frac{F(t)}{\left(m_{\mathrm{\phi}}^{2}-t\right)\left(2\mu_{0}^{2}+m_{\mathrm{\phi}}^{2}-t\right)}\right)^{2},$
(14)

with the form factor $F(t)$ is,

$F(t)=\frac{4m_{p}^{2}-2.8t}{(4m_{p}^{2}-t)(1-\frac{t}{0.7})^{2}}.$ (15)

and the Regge trajectory $\alpha(t)=1.08+0.25t$ is derived from Ref.
Landshoff:1990kj , where coefficient $\beta=2$ GeV-1, $\mu_{0}^{2}=1.1$ GeV2
and $s_{0}=4$ GeV2.

Figure 2:  The Feynman diagram for $\phi$ production based on the pomeron
model.

###  II.3 Mass radius of proton

The mass radius of the proton describes the mass distribution within the
proton, which can be summarized as the mass density distribution. Generally,
with the weak field approximation, the mass distribution, D-term, and spin we
are interested can be obtained by GFF Ji:1996nm ; Ji:1997pf ; Kharzeev:2021qkd
; Wang:2021dis ; Guo:2021ibg ; Teryaev:2016edw . The mass radius is defined as
the derivative of momentum transfer to GFF, which can be written as
Kharzeev:2021qkd ,

$\left\langle
R_{\mathrm{m}}^{2}\right\rangle=\left.\frac{6}{M}\frac{dG(t)}{dt}\right|_{t=0},$
(16)

where $G(t)=M/(1-t/m_{r}^{2})^{2}$ is the dipole form of scalar GFF and is
normalized. Therefore, we can summarize the form of mass radius as,

$\left\langle R_{\mathrm{m}}^{2}\right\rangle=\frac{12}{m_{r}^{2}}.$ (17)

where $m_{r}$ is a free parameter can be obtained by fitting the quarkonium
photoproduction data. The photoproduction of quarkonium near the threshold is
susceptible to scalar GFF and mass distribution. In the small momentum range,
the photoproduction differential cross section of quarkonium can be described
by the scalar GFF, which can be written as $d\sigma/dt\propto G^{2}(t)$
Kharzeev:2021qkd ; Wang:2021dis . Consequently, it is available to extract the
mass radius of the proton from the cross section of quarkonium
photoproduction.

###  II.4 Mechanical properties of proton

The mechanical properties of the proton, including the pressure and shear
force distribution, are also the direction of proton internal structure
research. These properties are encoded in Quantum Chromodynamics Energy
Momentum Tensor (EMT). The EMT satisfies the conservation law and can be
described as the space-time symmetric translation related conservation current
based on Noether’s Theorem, which can be represented as Kou:2021qdc ,

$\displaystyle\partial^{\mu}\hat{T}_{\mu\nu}=0$ (18)
$\displaystyle\hat{T}_{\mu\nu}=\sum_{q}\hat{T}_{\mu\nu}^{q}+\hat{T}_{\mu\nu}^{g}$

where $\hat{T}_{\mu\nu}^{q}$ and $\hat{T}_{\mu\nu}^{g}$ represent the
contribution of quarks and gluons, respectively. The main object of this work
is the proton related problem. For the proton with spin-$\frac{1}{2}$, the one
particle state can be represented by $|\bm{p}\rangle$, and the normalization
condition is $\left\langle p^{\prime}\mid
p\right\rangle=2p^{0}(2\pi)^{3}\delta^{(3)}\left(\bm{p}^{\prime}-\bm{p}\right)$.
So, the matrix element of EMT for proton can be introduced as Polyakov:2018zvc
,

$\displaystyle\left\langle
p^{\prime},s^{\prime}\left|\hat{T}_{\mu\nu}^{a}(x)\right|p,s\right\rangle=\bar{u}^{\prime}\left[A^{a}(t)\frac{\gamma_{\\{\mu}P_{\nu\\}}}{2}\right.$
(19)
$\displaystyle+B^{a}(t)\frac{iP_{\\{\mu}\sigma_{\nu\\}\rho}\Delta^{\rho}}{4m}+D^{a}(t)\frac{\Delta_{\mu}\Delta_{\nu}-g_{\mu\nu}\Delta^{2}}{4m}$
$\displaystyle\left.+m\bar{c}^{a}(t)g_{\mu\nu}\right]ue^{i\left(p^{\prime}-p\right)x},$

with the notation,

$a_{\\{\mu}b_{\nu\\}}=a_{\mu}b_{\nu}+a_{\nu}b_{\mu},$ (20)

Where $\bar{u}(p,s)u(p,s)=2m$ is spin normalization, $A^{a}(t)$, $B^{a}(t)$,
$D^{a}(t)$, and $\bar{c}^{a}(t)$ represent the form factors of individual
quarks and gluons, and these four form factors depend on renormalization
scale. The $B^{a}(t)=0$ for gluons by calculation of lattice QCD
Alexandrou:2017oeh . The tripole parametrization for D-term from perturbation
theory is,

$D(t)=\frac{D(0)}{\left(1-t/m_{D}^{2}\right)^{3}}.$ (21)

where $m_{D}$ and $D(0)$ are free parameters.

The $D$-term contains information about the distribution of internal pressure
$p(r)$ and shear force $s(r)$ of the proton. Using the Eq. (21) and making
$t=-q^{2}$, then the tripole parameterization $D(t)$ can be rewritten as
$D(-q^{2})=D(0)/(1+q^{2}/m_{D}^{2})^{3}$. The D-term of the three-dimensional
coordinates can be obtained from the Fourier transform of $D(q)$
Polyakov:2018zvc  with $E=m_{N}$,

$\displaystyle\tilde{D}(r)$
$\displaystyle=D(0)\int\frac{d^{3}q}{2m_{N}(2\pi)^{3}}\frac{e^{-iq\cdot
r}}{\left(1+\frac{q^{2}}{m_{D}^{2}}\right)^{3}}$ (22)
$\displaystyle=\int_{0}^{\pi}\int_{0}^{2\pi}\int_{0}^{+\infty}\frac{q^{2}\sin(\theta)}{(2\pi)^{3}2m_{N}}\frac{e^{-iqr\cos(\theta)}D(0)}{\left(1+q^{2}/m_{D}^{2}\right)^{3}}d\theta
d\phi dq$
$\displaystyle=\frac{D(0)}{4\pi^{2}rm_{N}}\int_{0}^{+\infty}q\frac{\sin(qr)}{\left(1+q^{2}/m_{D}^{2}\right)^{3}}dq$
$\displaystyle=\frac{D(0)}{64\pi
m_{N}}e^{-m_{D}r}m_{D}^{4}\left(\frac{1}{m_{D}}+r\right)$

Therefore, $p(r)$ and $s(r)$ can be calculated by Eq. (22)

$\displaystyle p(r)$
$\displaystyle=\frac{1}{3}\frac{1}{r^{2}}\frac{d}{dr}r^{2}\frac{d}{dr}\bar{D}(r)$
(23) $\displaystyle=\frac{D(0)}{192\pi
m_{N}}\left(m_{D}r-3\right)m_{D}^{5}e^{-m_{D}r}$

$\displaystyle s(r)$
$\displaystyle=-\frac{1}{2}\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\bar{D}(r)$ (24)
$\displaystyle=-\frac{D(0)}{128\pi m_{N}}m_{D}^{5}re^{-m_{D}r}$

##  III RESULTS

The parameterized gluon distribution function
$xg(x,m_{\phi}^{2})=A_{0}x^{A_{1}}(1-x)^{A_{2}}$ has been introduce in the
previous section. $A_{0},A_{1},A_{2}$ and $b_{0}$ are derived by combining a
global analysis of the total cross section of $\phi$ photoproduction from Ref.
Strakovsky:2020uqs ; Barber:1981fj ; Ballam:1972eq ; Egloff:1979mg ;
Busenitz:1989gq ; Owens:2012bv  and the differential cross section
($W=1.98\sim 2.30$ GeV) by CLAS Dey:2014tfa  and LEPS LEPS:2005hax
collaboration which are listed in Tab. 1. The fitted $b_{0}=3.60\pm 0.04$
GeV-2 is near to $3.4$ GeV-2 obtained by exponential slope $I\sim
e^{-b_{0}(t-t_{min})}$ based on the pomeron exchange model mentioned in the
Ref. CLAS:2013jlg ; Titov:2003bk , which adequately demonstrates the
effectiveness of the two gluon exchange model. The total and differential
cross sections of the two gluon exchange model can be rewritten with the gluon
distribution function $xg(x,m_{\phi}^{2})=0.36x^{-0.055}(1-x)^{0.12}$.

Table 1:  The slope $b_{0}$ and parameters $A_{0},A_{1},A_{2}$ of the gluon distribution function are obtained by fitting the experimental data and $\chi^{2}/$d.o.f value. $A_{0}$ | $A_{1}$ | $A_{2}$ |  $b_{0}$ (GeV-2) |  $\chi^{2}/$d.o.f
---|---|---|---|---
$0.36\pm 0.04$ | $-0.055\pm 0.003$ | $0.12\pm 0.03$ | $3.60\pm 0.04$ | $2.87$
Figure 3:  The total cross section of the channel $\gamma p\rightarrow\phi p$
as a function of center of mass energy $W$. The line (blue) and dashed (black)
lines are for the two gluon exchange model and pomeron model, respectively.
Here, taking the error of $A_{0}$ as the error band.

The cross sections of $\phi$ photoproduction at the near-threshold are
predicted as shown in Fig. 3 and Fig. 4. One can figure the two gluon exchange
model shows satisfactory agreement with experimental data. And
$\chi^{2}/$d.o.f is calculated to be $2.87$, indicating that the fit is in
line with expectations. But $\chi^{2}/$d.o.f is not tiny enough, mainly
because the error bar measured in the experiment is large. To demonstrate the
availability of $\phi$ photoproduction based on the two gluon exchange model,
the pomeron model without any fitting parameters is introduced which was
developed by Larget in Ref. Laget:1994ba . One can conclude that the two gluon
exchange model and the pomeron model almost overlap at the location nearest to
the threshold in Fig. 3. However, the differential cross section of $\phi$
fitted by the two gluon exchange model is more efficient than the pomeron
model as shown in Fig. 4. Combined with our previous work Wang:2022vhr ;
Zeng:2020coc , the total cross section of $J/\psi$ and $\psi(2S)$ were
predicted based on the two gluon exchange model, which is in agreement with
the experimental data. Therefore, the current results can sufficiently
illustrate the validity of the cross sections of vector mesons photoproduction
at the near-threshold predicted via the two gluon exchange model.

Figure 4:  The differential cross sections of the channel $\gamma
p\rightarrow\phi p$ as a function of $-t$ at different $W$ values. Here, The
blue solid line and the black dashed line correspond to the two gluon exchange
model and the pomeron model, respectively.

Ensure that the consequences are persuasive, the mass radius is obtained via
the differential cross sections of the $\phi$ photoproduction predicted at
$W\in[1.96,2.36]$ with $\Delta W=0.08$ GeV, and the range of $|t|$ from
$t=|t_{min}|$ select ten points with $\Delta|t|=0.05$ GeV2. Then, the average
mass radius is derived as $0.78\pm 0.06$ fm and $1.02\pm 0.01$ fm,
corresponding to the two gluon exchange model and the pomeron model,
respectively. The trend of mass radius with center of mass energy based on the
two models as shown in Fig. 5 and Fig. 6 shows the fitting of $d\sigma/dt$ and
$G(t)$. A noticeable decline for both models around the threshold $W=1.96$
GeV, while the trend gradually decreases with the increase of energy. After
$W=2.04$ GeV, the trend tends to be flat. The condition is primarily due to
the influence of momentum transfer $|t_{min}|$, and a similar decline also
appears in the heavy charmoniums $J/\psi$ and $\psi(2S)$. However, the light
meson $\phi$ is less affected by $|t_{min}|$, so the trend decreases smaller
than the heavy vector meson at the threshold.

Figure 5:  The mass radius of the proton from differential cross sections of
$\phi$ photoproduction at the center of mass energy $W\in[1.96,2.36]$ GeV,
$\Delta W=0.08$ GeV. The average is $0.78\pm 0.06$ fm and $1.02\pm 0.01$ fm,
corresponding to the two gluon exchange and pomeron models, respectively.
Figure 6: The fitting of $d\sigma/dt$ and $G(t)$ based on the two gluon
exchange model (a) and pomeron model (b).

Although $|t_{min}|$ has less influence on light vector mesons than on heavy
vector mesons, there still exists some variation with the increase of center
of mass energy $W$, as shown in Fig. 7. With an increase of energy, the impact
of $|t_{min}|$ on the differential cross section decreases. Away from the
threshold, it can be approximated as $|t_{min}|\rightarrow 0$. If the points
at the threshold are not considered, the average mass radius is calculated at
$W\in[2.04,2.36]$ GeV as $\sqrt{\left\langle
r^{2}\right\rangle_{\mathrm{m}}}=0.75$ and $\sqrt{\left\langle
r^{2}\right\rangle_{\mathrm{m}}}=1.00$ fm, corresponding to the two gluon
exchange model and the pomeron model, respectively. By comparing the results
obtained above, the average radius of ignoring the threshold is only $0.02\sim
0.03$ fm smaller, which means that $d\sigma_{\gamma p\rightarrow\phi p}/dt$ at
the threshold has little effect on the average radius. This situation is in
intense contrast to the heavy charmoniums $J/\psi$ and $\psi(2S)$ Wang:2022vhr
, which proves that the mass radius from the cross sections of $\phi$
photoproduction at the near-threshold is more stable.

As a comparison, we directly extract the proton mass radius from the CLAS
Dey:2014tfa  and LEPS LEPS:2005hax  experimental data as shown in Tab. 2. One
noticed that the extracted proton mass radius values showed irregular
fluctuations with increasing energy. Especially at the first energy point
$W=1.98$ GeV, the extracted mass radius value has a large error bar,
reflecting the large error of the experimental data at this point. The fitting
results and the differential cross sections of $\phi$ photoproduction are
presented in Fig. 8, from which it can be seen that the data points at
$W=1.98$ GeV are discretely distributed with large uncertainties. So the mass
radius obtained at $W=1.98$ GeV is indeterminate. The phenomenon might be
related to the difficulty of the experiment in which the energy closer to the
threshold is more complicated, while $W=1.98$ GeV is very near to the
threshold. Therefore, the average mass radius is $\sqrt{\left\langle
r^{2}\right\rangle_{\mathrm{m}}}=0.80\pm 0.05$ fm with $W\in[2.02,2.29]$ GeV,
which is close to the result of the two gluon exchange model. Taking into
account the error bar, we summarize the mass radius of the proton from this
work and other groups Kharzeev:2021qkd ; Wang:2022vhr ; Pefkou:2021fni ;
Mamo:2022eui ; Guo:2021ibg ; Wang:2021dis ; Wang:2022zwz  generally exist in
$r_{m}\in[0.52,0.94]$ fm as shown in Fig. 9.

It should be noticed that the closed interval does not consider the mass
radius from the pomeron model. Because the result of the pomeron model is more
than $1.00$ fm and significantly larger than that of the two gluon exchange
model and other groups. The primary explanation for the consequence is that
the slope of the curve of the differential cross sections obtained via the
pomeron model in Fig. 4 is massive, which leads to a larger mass radius. So
the result obtained by the pomeron model is not considered in the subsequent
discussion of this paper. However, the pomeron model is still powerful for
predicting the cross section of vector mesons photoproduction. In future
studies, the pomeron model will be upgraded and modified to better simulate
the process of vector mesons photoproduction.

Figure 7: The change of $|t|$ with center of mass energy $W$ is near the photoproduction threshold of $\phi$. Table 2:  Mass radius is derived by the differentiation experiment cross section from CLAS Dey:2014tfa  and LEPS LEPS:2005hax  collaboration, the average is $0.80\pm 0.05$ fm with $W\in[2.02,2.29]$ GeV. $W$ GeV | $1.98$ | $2.02$ | $2.07$ | $2.12$
---|---|---|---|---
$\sqrt{\left\langle r^{2}\right\rangle_{\mathrm{m}}}$ fm | $0.30\pm 0.36$ | $0.80\pm 0.07$ | $0.85\pm 0.04$ | $0.95\pm 0.04$
$W$ GeV | $2.16$ | $2.20$ | $2.25$ | $2.29$
$\sqrt{\left\langle r^{2}\right\rangle_{\mathrm{m}}}$ fm | $0.80\pm 0.05$ | $0.79\pm 0.04$ | $0.68\pm 0.05$ | $0.70\pm 0.04$

Figure 8:  The fitting of $d\sigma/dt$ and $G(t)$ based Based on the
differentiation experiment cross section from CLAS Dey:2014tfa  and LEPS
LEPS:2005hax  collaboration. Figure 9:  Comparison of the results of proton
mass radius $\sqrt{\left\langle r^{2}\right\rangle_{\mathrm{m}}}$ with several
groups. The average based on the two gluon exchange model is $0.78\pm 0.06$
fm. The average value of proton mass radius extracted directly from the near-
threshold cross section of $\phi$ photoproduction is $0.80\pm 0.05$ fm. The
result of $0.55\pm 0.03$ fm is derived from Ref. Kharzeev:2021qkd  and
$0.67\pm 0.03$ fm is the results of Ref. Wang:2021dis . The results of
$0.747\pm 0.033$ fm and $0.682\pm 0.012$ fm are developed by Lattice QCD
Pefkou:2021fni  and Holographic QCD Mamo:2022eui . The result of $0.68$ fm is
obtained from Ref. Guo:2021ibg . The result of $0.67\pm 0.11$ fm and $0.86\pm
0.08$ fm are extraction from charmoniums photoproduction Wang:2022vhr  and
$\rho$ photoproduction Wang:2022zwz , respectively.

The differential cross section of vector meson photoproduction is connected
with the D-term Hatta:2021can , which is $d\sigma/dt\propto D^{2}(t)$. So
mechanical properties of the proton can be analyzed with the same data. Then,
Eq. (23) and Eq. (24) are used to calculate the pressure and shear force
distributions, respectively. $D(0)$ and $m_{D}$ in $D(t)$ are free parameters
that affect the size and shape of the two distributions, respectively. The
shape $m_{D}$ primarily controls the distribution structure of pressure and
shear force, while the size $D(0)$ only affects the distribution intensity.
Therefore, exclusively $m_{D}^{2}$ is involved in the subsequent discussion.
$m_{D}^{2}$ extracted by the two gluon exchange model and from CLAS
Dey:2014tfa  and LEPS LEPS:2005hax  experiment data are calculated with the
analogous way of the mass radius and the result of relevant models as shown in
Tab. 3. The mechanical properties based on the two gluon exchange model and
the experimental data are shown in Fig. 10.

Figure 10: The internal pressure (a) and shear force (b) distribution of the
proton. Here, the blue solid line is the result based on the two gluon
exchange model, and the green dash-dotted line represents the result extracted
from CLAS Dey:2014tfa  and LEPS LEPS:2005hax  experimental data. Here, taking
the error of $m_{D}$ as the error band.

Figure 11: The internal pressure (a) and shear force (b) distribution of the proton from two gluon exchange model (blue solid line ) and other groups Pefkou:2021fni ; Mamo:2022eui ; Burkert:2018bqq ; Burkert:2021ith ; Chakrabarti:2021mfd . The pale magenta dash line is the result of LQCD from Ref. Pefkou:2021fni . The dark yellow dashed line and purple dash-dotted line are the result of Holographic QCD Mamo:2022eui  and DVCS Burkert:2018bqq ; Burkert:2021ith  , respectively. The orange double dot dash line is the result of the light front quark-diquark model Chakrabarti:2021mfd . Table 3:  The average $\sqrt{\left\langle m_{D}^{2}\right\rangle}$ from this work and other groups Pefkou:2021fni ; Mamo:2022eui ; Burkert:2021ith ; Chakrabarti:2021mfd . Method | $\sqrt{\left\langle m_{D}^{2}\right\rangle}$
---|---
Two gluon exchange model (this work) | $1.15\pm 0.05$
Extraction from CLAS and LEPS data (this work) | $1.09\pm 0.07$
Lattice QCD Pefkou:2021fni  | $1.07$
Holographic QCD Mamo:2022eui  | $0.963$
DVCS Burkert:2021ith  | $1.01\pm 0.13$
Light-front quark-diquark model Chakrabarti:2021mfd  | $1.37$

Initiating from the center of the proton $r=0$ fm, the radial distribution of
pressure $r^{2}p(r)$ is divided into positive and negative parts by zero-
crossing, dominated by repulsive force and binding force. It can be observed
from Fig. 10 that the zero-crossing based on the experiment data has a bit
larger than that on the two gluon exchange model, and the positive and
negative peaks are also more sizeable. What is striking is that the zero-
crossing are significantly adjacent, which is similar to the result of mass
radius. The peak of the radial distribution of shear force $r^{2}s(r)$ has the
same trend as that of pressure. The result developed by other groups
Pefkou:2021fni ; Mamo:2022eui ; Burkert:2018bqq ; Burkert:2021ith ;
Chakrabarti:2021mfd  are compared with that of the two gluon exchange model,
as shown in Fig. 11. It is found that the distribution results based on the
two gluon exchange model are roughly in line with those of the other groups,
but there are some differences. In fact, the current research on the internal
mechanical properties of the proton is still in the early stage, including the
different contributions of quarks and gluons to the distribution of mechanical
properties given by different models or experiments. Therefore, we need to
give the results of the corresponding proton mechanics properties from
different models or experiments, so as to facilitate the in-depth study of
this problem in the future.

In addition, it can be seen from Tab. 3 that the zero-crossing point decreases
with the increase of $\sqrt{\left\langle m_{D}^{2}\right\rangle}$. The
stability condition of the mechanical system argued in Ref. Polyakov:2018zvc
is $(2/3)s(r)+p(r)>0$. That is, the corresponding force must be outward. As
shown in Fig. 12, the mechanical system based on the two gluon exchange model
and the the result extracted from the $\phi$ photoproduction data satisfies
the inequality. However, although the mechanical properties satisfy the
mechanical stability condition of the system, and the radial distribution
agrees with other models, there still exist uncertain factors in the proton
mechanical properties, among which $D(0)$ is one.

2D displays of pressure and shear force are established based on the two gluon
exchange model in Fig. 13, which describes the pressure and shear force
distribution from the proton center. The colorbar on the Fig.13 (a) describes
the areas above $0$ where repulsive forces dominate and below $0$ where
binding forces dominate. With $0$ as the node, the intensity increases first
and then decreases as the distance from the center of mass increases. The
color bar on Fig.13 (b) describes the variation of the strength of the shear
force distribution as the distance from the center of mass increases.

Figure 12: Schematic diagram of $(2/3)s(r)+p(r)$ as a function of $r$ for a
mechanical system satisfies the stability conditions. Here, the notations are
the same as in Fig. 10.

Figure 13: 2D display of the pressure $r^{2}p(r)$ (a) and shear force
$r^{2}s(r)$ (b) distribution in proton from the proton center based on the two
gluon exchange model.

##  IV Summary

This work reproduces the differential and total cross sections of $\gamma
p\rightarrow\phi p$ at the near-threshold based on the two gluon exchange
model with the gluon distribution function and pomeron model developed by
larget Laget:1994ba . The parameters $A_{0},A_{1},A_{2}$ as well as $b_{0}$
are obtained by fitting differential cross sections from CLAS Dey:2014tfa  and
LEPS LEPS:2005hax  and total cross section from CLAS Barber:1981fj ;
Strakovsky:2020uqs , SLAC Egloff:1979mg , FERMI Busenitz:1989gq ; Owens:2012bv
, LAMP2 Ballam:1972eq . The average mass radius from the two gluon exchange
and pomeron models are $0.78\pm 0.06$ fm and $1.02\pm 0.01$ fm, respectively.
In addition, the mass radius is analyzed with the CLAS Dey:2014tfa  and LEPS
LEPS:2005hax  experimental data to be $0.80\pm 0.05$ fm which is close to the
result of the two gluon exchange model. Based on the mass radius obtained by
this work, and ignoring the conclusions of the pomeron model, we concluded
that the mass radius of the proton given by the $\phi$ meson photoproduction
process should be attributed in $r_{m}\in[0.78,0.80]$ fm. At present, the
proton mass radius value extracted by the cross section of heavy charmoniums
photoproduction is roughly between 0.55 and 0.68 fm. In our recent work
Wang:2022zwz , the proton mass radius was $0.86\pm 0.08\text{ fm }$ extracted
from the lighter $\rho$ meson photoproduction data. Interestingly, the mass of
$\phi$ is between the charmoniums and the $\rho$ meson, and the corresponding
proton mass radius value is also roughly in the middle. In fact, if we assume
that for lighter vector mesons, the contribution from the gluon operator in
the production amplitude is gradually diminishing, while the contribution from
the light quark operator dominates, resulting in an increasing proton mass
radius extracted and is close to the value of the proton charge radius. If
this assumption holds, it means that the proton mass radius extracted by the
vector meson photoproduction have some rationality. However, we must note that
the proton mass radius values obtained so far are with error bars, and if we
take the error into account, it is difficult to give a clear size
relationship. Therefore, high-precision experimental measurement data and more
complete theoretical models are very much needed.

Afterwards, taking a similar approach, we systematically investigated the
mechanical properties of the proton and compared the obtained results with
those of other groups. The results show that the proton internal pressure and
shear force distributions given by us are close to the results of light-front
quark-diquark model Chakrabarti:2021mfd  and LQCD Pefkou:2021fni , but there
is a little difference with DVCS Burkert:2018bqq ; Burkert:2021ith  and
holographic QCD Mamo:2022eui . In fact, the results of proton mechanics
properties given by different models or experiments are not the same.
Therefore, our study may provide necessary reference or supplementary
information for subsequent in-depth research on the internal properties of
protons.

The Electron-Ion Collider (EIC) is a powerful facility for studying nucleons’
structure and strong interactions in non-perturbation energy regions. At
present, EIC facilities Accardi:2012qut ; Anderle:2021wcy  in China and the
United States take the study of nuclear structure and properties as an
important scientific goal, and our theoretical results on the photoproduction
of $\phi$ meson may provide an important reference and basis for further
accurate measurements of $\phi$ meson electroproduction at EIC facilities
Accardi:2012qut ; Anderle:2021wcy .

###### Acknowledgements.

X.-Y. Wang would like to acknowledge Prof. Dmitri E. Kharzeev for useful
discussions about proton mass radius. This project is supported by the
National Natural Science Foundation of China under Grant Nos. 12065014 and
12047501, and by the West Light Foundation of The Chinese Academy of Sciences,
Grant No. 21JR7RA201.

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