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##  Appendix A Appendix

###  A.1 Trap Frequency of an ODT

We often wish to approximate the potential of an optical dipole potential by a
harmonic oscillator. To do so we find the second derivative in potential as a
function of R and Z. The optical dipole potential due to a Gaussian beam is
given by [40]

$U_{dip}=-\frac{1}{2\epsilon_{0}c}Re(\alpha)I_{0}\left(\frac{w_{0}}{w(z)}\right)^{2}e^{-\frac{2r^{2}}{w(z)^{2}}}.$
(A.1)

where the waist as a function of axial position is defined as

$w(z)=w_{0}\sqrt[\textbullet]{1+(\frac{z\lambda}{\pi w_{0}^{2}})^{2}}$ (A.2)

we then take the second derivative in both x and z about the origin.

$\frac{\partial U_{dip}}{\partial
r}=\frac{1}{2\epsilon_{0}c}Re(\alpha)I_{0}\left(\frac{w_{0}}{w(z)}\right)^{2}\frac{4r}{w(z)^{2}}e^{-\frac{2r^{2}}{w(z)^{2}}}$
(A.3)

$\frac{\partial^{2}U_{dip}}{\partial
r^{2}}=\frac{1}{2\epsilon_{0}c}Re(\alpha)I_{0}\left(\frac{w_{0}}{w(z)}\right)^{2}\frac{4}{w(z)^{2}}e^{-\frac{2r^{2}}{w(z)^{2}}}+\frac{1}{2\epsilon_{0}c}Re(\alpha)I_{0}\left(\frac{w_{0}}{w(z)}\right)^{2}\frac{8r^{2}}{w(z)^{4}}e^{-\frac{2r^{2}}{w(z)^{2}}}$
(A.4)

$\frac{\partial^{2}U}{\partial
r^{2}}\biggr{\rvert}_{x=0,z=0}=\frac{1}{2\epsilon_{0}c}Re(\alpha)I_{0}\frac{4}{w_{0}^{2}}$
(A.5)

$\frac{\partial^{2}U}{\partial
z^{2}}\biggr{\rvert}_{x=0,z=0}=\frac{1}{2\epsilon_{0}c}Re(\alpha)I_{0}\frac{2}{z_{R}^{2}}$
(A.6)

The trap frequency can then be found from this second derivative as

$\omega=\sqrt[\textbullet]{\frac{\frac{\partial^{2}U}{\partial
x^{2}}\biggr{\rvert}_{\frac{\partial U}{\partial x}=0}}{m}}.$ (A.7)

For r and z this then evaluates to

$\omega_{r}=\sqrt[\textbullet]{\frac{1}{2m\epsilon_{0}c}Re(\alpha)I_{0}\frac{4}{w_{0}^{2}}}$
(A.8)

$\omega_{z}=\sqrt[\textbullet]{\frac{1}{2m\epsilon_{0}c}Re(\alpha)I_{0}\frac{2}{z_{R}^{2}}}$
(A.9)

The sum of two trapping frequencies can be seen from Equation 4.12 to be

$\omega_{Net}=\sqrt[\textbullet]{\omega_{Trap}^{2}+\omega_{Probe}^{2}}.$
(A.10)

###  A.2 Polarizability Conversion

To convert the expression for the dynamic polarizability near the tune out
wavelength as given in [1] as
$\frac{\partial\alpha}{\partial\lambda}=-1.913a_{0}^{3}(CGS)$ to SI. We first
convert into units of volume polarizability using the value for $a_{0}$ [89].
To convert this to si units we then use [38]

$4\pi\varepsilon\alpha^{\prime}=\alpha.$

With

$\varepsilon=8.85\dot{1}0^{-12}$ $a_{0}=0.529\cdot 10^{-10}$

To give the SI value of

$\frac{\partial\alpha}{\partial\lambda}=3.15\cdot 10^{-50}\>C\cdot m\cdot
v^{-1}.$ (A.11)

The resultant potential can then be given by

$U_{dipole}=\frac{-1}{2\varepsilon_{0}c}\frac{\partial\alpha}{\partial\lambda}\Delta\lambda(m)\;I$

and is evaluated with the value above to

$\frac{U(\omega)}{\Delta\lambda}=5.9*10^{-39}\frac{Jm^{2}}{nm_{TO}W}=2.8*10^{-16}\frac{Km^{2}}{nm_{TO}W}.$

###  A.3 Outcoupling Calculations

To calculate the outcoupling surface integral we used analytical calculations
in Mathematica. The code is below:

This gives the outcoupling density $\eta$ as


# Systematic investigation of trace anomaly contribution in nucleon mass

Xiao-Yun Wang<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, and Key Laboratory of
Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou, Gansu
730000, China  Jingxuan Bu  Department of physics, Lanzhou University of
Technology, Lanzhou 730050, China

###### Abstract

In this work, under the framework of vector meson dominance model, the trace
anomaly contribution value inside neutrons are extracted for the first time
based on vector meson photoproduction data. Furthermore, we systematically
compare and analyze the trace anomaly contributions of protons and neutrons.
The results show that the trace anomaly contributions of protons and neutrons
are close, which indirectly confirms that their internal structures and
dynamic properties may have certain similarities. In addition, the main
factors affecting the extraction of the trace anomaly contribution of nucleons
are discussed in detail. This study not only provides a theoretical basis for
us to better understand the source of nucleon mass, but also makes a useful
exploration and discussion on how to extract the trace anomaly contribution of
nucleon more accurately in the future.

##  I introduction

As the elementary particle of most visible matter, the study on the internal
properties of nucleon is a valuable subject of quantum chromodynamics (QCD),
where the source of the nucleon mass has always been a mystery. The Higgs
mechanism explains only 2% of it, most of the rest comes from the complex
strong interactions. However, due to the weak gravitational interactions
acting on the single nucleon, the source of the nucleon mass cannot be
directly measured experimentally. Nevertheless, Ji et al. ji94 ; ji95 ; lorce
found a way to represent the nucleon mass in terms of quark and gluon energy
theoretically, the nucleon mass is divided into four parts under their
separation.

The separation starts from the QCD energy-momentum tensor (EMT)
${T}^{\mu\nu}$, which can be decomposed into the trace and traceless parts
ji95

$\displaystyle{T}^{\mu\nu}=\hat{T}^{\mu\nu}+\bar{T}^{\mu\nu}$ (1)

Further, the trace parts is a sum of the quark mass and trace anomaly
contributions, the traceless part is consist of the quark and gluon energy
contributions, respectively.

$\displaystyle\hat{T}^{\mu\nu}=\hat{T}^{\mu\nu}_{a}(\mu^{2})+\hat{T}^{\mu\nu}_{m}(\mu^{2})$
(2)
$\displaystyle\bar{T}^{\mu\nu}=\bar{T}^{\mu\nu}_{q}(\mu^{2})+\bar{T}^{\mu\nu}_{g}(\mu^{2})$

And the tensor defines the QCD Hamiltonian operator as ji95

$H_{\mathrm{QCD}}=\int d^{3}\vec{x}\ T^{00}(0,\vec{x})$ (3)

According to the above definition and decomposition of EMT, the corresponding
four parts separations of the QCD Hamiltonian operator are obtained as ji95

$\displaystyle H_{QCD}=H_{q}+H_{g}+H_{m}+H_{a}$ (4)

In Reference ji95 , hadron mass is defined as the expectation value of the
Hamiltonian operator in the rest frame,

$\displaystyle
M=\frac{\left<P|H_{QCD}|P\right>}{\left<P|P\right>}\Bigg{|}_{\rm rest\ frame}$
(5)

Thus, based on Eq. 4 and 5 the hadron mass decomposition is derived as ji95

$\displaystyle M_{q}=\frac{3}{4}\ (a-\frac{b}{1+\gamma_{m}})M$ (6)
$\displaystyle M_{g}=\frac{3}{4}\ (1-a)M$ $\displaystyle
M_{m}=\frac{4+\gamma_{m}}{4(1+\gamma_{m})}bM$ $\displaystyle
M_{a}=\frac{1}{4}(1-b)M$

Noted from the above, the specific value of the mass decomposition are
determined by parameters $a$ and $b$, which is the fraction of the nucleon
momentum carried by quarks and the trace anomaly parameter, respectively,
$\gamma_{m}$ represents the anomalous dimensions ji94 ; ji95 . Notice that
there has a new source of mass that appears in the last term, which is the so-
called trace anomaly contribution, it depends only on the parameter $b$. Work
Ji:2021pys  argues that the trace anomaly comes from the scale symmetry
breaking among the regulation of the UV divergences, with the scheme
independent. It contributes the nucleon mass by a Higgs-like mechanism and
sets the scale of the other part of the mass separations. Besides, one can
carry out the related mechanism of the quark confinement by studying the
anomalous energy.

Currently, one of the feasible research methods on the trace anomaly is
through the direct lattice QCD calculation latice . In addition, the trace
anomaly contribution can be calculated from the scattering between the
quarkonium and the nucleon I ; light ; vmd ; nature ; Hatta:2018sqd ;
Hatta:2019lxo ; Hatta:2018ina ; wr . This implies that we can study the trace
anomaly based on the existing photoproduction experimental data by combining
the above process with the vector meson photoproduction process under the
vector meson dominance (VMD) model vmd . There have been some studies on the
trace anomaly contribution of proton I ; light ; nature ; wr ; latice , but
not yet on neutron. One of the purposes of this work is to extract the trace
anomaly contribution of neutron mass under the VMD model for the first time.
As another purpose, we will further analyze the factors affecting the anomaly
contribution. In our previous work I , by observing the trace anomaly
distribution extracted from the experimental $J/\psi$ photoproduction cross
section, we discovered the energy dependence problem of the VMD method and
solved it by remoulding the model. We further extended the work to other
vector meson photoproduction processes light , showing that the trace anomaly
contributions from different processes are not the same, although they are all
relatively small. Such results bring the nucleon anomaly contribution a large
error, these inspire us to further analyze the affecting factors in order to
give a more accurate proportion of the trace anomaly part. A comprehensive
analysis will be discussed in detail in summary, and the research direction of
obtaining higher precision nucleon trace anomaly contribution is pointed out.
In part before the summary, the influence of one of the parameter
$\alpha_{s}$, which has a sizable inaccurate precision QCD 2016 ; 2018 ; 1992
; effective 2007 ; 2008 ; charge 2022 ; low 2022 ; HERMES 1997 ; HERMES 1998 ;
g1(p) 1998 ; HERMES 2003 ; HERMES 2007 ; Novel 2022 , on the nucleon anomaly
contribution is first studied numerically. To achieve this, we adopt two sets
of $\alpha_{s}$ on the extraction named $\alpha_{A}$ and $\alpha_{B}$, one set
of $\alpha_{s}$ ($\alpha_{A}$) comes from some previous prediction and fitting
result phi 0.770 ; OR alpha s , another group ($\alpha_{B}$) comes from a
recent work machine  on predicting $\alpha_{s}$ based on machine learning
methods and large amounts of experimental data.

The structure of the paper is as follows. After the introduction, the review
of the calculation method VMD model is in Sec. II, the corresponding numerical
results are in Sec. III, and finally Sec. IV gives a summary.

##  II FORMALISM

In the VMD model, process $\gamma N\to VMN$ is related with process $VMN\to
VMN$ and the forward differential cross section of $\gamma N\to VMN$ reaction
is expressed as vmd

$\displaystyle\frac{d\sigma_{\gamma N\to
VMN}}{dt}\Bigg{|}_{t=t_{min}}=\frac{3\Gamma_{e^{+}e^{-}}}{\alpha
m_{V}}\left(\frac{p_{VMN}}{p_{\gamma N}}\right)^{2}\frac{d\sigma_{VMN\to
VMN}}{dt}\Bigg{|}_{t=t_{min}}$ (7)

where $\Gamma_{e^{+}e^{-}}$ is the radiative decay width, the value of the
electromagnetic coupling constant $\alpha$ is equal to 1/137 and $m_{V}$ is
the vector meson mass.
${p}_{ab}=\frac{1}{2W}\sqrt{W^{4}-2\left(m_{a}^{2}+m_{b}^{2}\right)W^{2}+\left(m_{a}^{2}-m_{b}^{2}\right)^{2}}$
denotes the center of mass momentum of the photon and vector meson,
respectively. The differential cross section part of $VMN\to VMN$ process in
the formula is given as

$\displaystyle\frac{d\sigma_{VMN\to
VMN}}{dt}\Bigg{|}_{t=t_{min}}=\frac{1}{64}\frac{1}{m_{V}^{2}(\lambda^{2}-M_{N}^{2})}|F_{VMN}|^{2}$
(8)

where the nucleon energy is $\lambda=(W^{2}-m^{2}_{V}-M^{2}_{N})/(2m_{V})$ vmd
, and at the low energy region, the elastic scattering amplitude of $VMN\to
VMN$ process is taken as antip

$\displaystyle F_{VMN}$ $\displaystyle\simeq
r_{0}^{3}d_{2}\frac{2\pi^{2}}{27}\left(2M_{N}^{2}-\left\langle
N\left|\sum\limits_{h=u,d,s}m_{h}\bar{q}_{h}q_{h}\right|N\right\rangle\right)$
(9) $\displaystyle=r_{0}^{3}d_{2}\frac{2\pi^{2}}{27}2M_{N}^{2}(1-b)$

where the Bohr radius and the Wilson coefficient are antip ; d2

$\displaystyle r_{0}=\frac{4}{3\alpha_{s}m_{q}}$ (10) $\displaystyle
d_{n}=\left(\frac{32}{N_{c}}\right)^{2}\sqrt{\pi}\frac{\Gamma(n+5/2)}{\Gamma(n+5)}$
(11)

respectively. Where $\alpha_{s}$ is the strong coupling constant and $m_{q}$
is the mass of the constituent quark, the relevant parameters are listed in
Table 1.

As we can see, the above formula relates the vector meson photoproduction
cross section to the parameter $b$, besides the trace anomaly can be
represented by $\frac{1}{4}(1-b)M$ when away from the chiral limit. Thus the
trace anomaly contribution can be studied through VMD model. However, the
four-momentum transfers $t_{min}$ varies significantly with $W$ while
$t_{thr}=m_{V}^{2}M_{N}/(m_{V}+M_{N})$ stay steady igor . The varies of
$t_{min}$ leads to an energy dependence of $\frac{d\sigma_{\gamma N\to
VMN}}{dt}\Bigg{|}_{t=t_{min}}$, ends up resulting the energy dependence of the
trace anomaly. In Reference thr ; igor , the relationship between the
differential and total cross section at the near-threshold is given as

$\displaystyle\frac{d\sigma_{\gamma N\to
VMN}}{dt}\Bigg{|}_{t=t_{min},W=W_{thr}}$ $\displaystyle=\frac{d\sigma_{\gamma
N\to VMN}}{dt}\Bigg{|}_{t=t_{thr},W=W_{thr}}$ (12)
$\displaystyle=\frac{\sigma_{\gamma N\to
VMN}(W_{thr})}{4|p_{\gamma}|\cdot|p_{V}|}$

In order to decrease the energy dependence, we would apply the last two
methods to study the trace anomaly contribution.

Table 1: The relevant parameters of $\rho,\omega$ and $\phi$ pdg ; 0.330 ; Kou 2021 ; phi 0.770 ; OR alpha s ; machine . Meson |  $\Gamma_{e^{+}e^{-}}$(keV) |  $m_{q}$(GeV) |  $m_{V}$(GeV) | $\alpha_{A}$ | $\alpha_{B}$
---|---|---|---|---|---
$\rho$ | $7.04$ | $0.330$ | $0.770$ | $0.439$ | $0.601$
$\omega$ | $0.60$ | $0.330$ | $0.782$ | $0.460$ | $0.595$
$\phi$ | $1.27$ | $0.486$ | $1.019$ | $0.770$ | $0.506$

##  III result and discussion

The $\alpha_{A}$ of the three vector mesons come from several previous work,
where the $\alpha_{A}$ for $\omega$ and $\rho$ are taken from the estimates
based on a relativistic quantum-field model OR alpha s , and for $\phi$, the
$\alpha_{A}$ is taken from the calculations based on a quark potential model
phi 0.770 . $\alpha_{B}$ are all derived from the machine learning prediction
machine , which varies with the energy scale $Q$. Here the respective meson
mass is taken as the corresponding energy scale meson .

Figure 1: The neutron trace anomaly contributions $M_{a}/M_{N}$ as a function
of $R$ extracted from the experimental $\omega$ differential and total cross
section at $\alpha_{A}$ OR alpha s ; omega data . The solid red and black
circles represent the result from the differential and total cross section,
respectively. Bands in the corresponding colour are the error bar.

After determining the relevant parameters, the calculations are carried out.
Firstly, based on the experimental differential and total cross section of
$\omega$ at $W\in[1.75,2.15]$ measured by ELSA omega data , the neutron trace
anomaly contributions were extracted under the framework of VMD model. Results
at $\alpha_{A}$ were listed in Table 2 and shown in Fig. 1 as a function of
$R$. The results from the differential and the total section are slightly
different due to the numerical difference between the two sections. In order
to consider the outcomes in both cases, we process all of the results in the
way of root-mean-square, which is $1.43^{+0.58}_{-0.33}\ \%$.

Table 2:  The neutron trace anomaly contribution extracted from the experimental $\omega$ photoproduction cross section at $\alpha_{A}$ OR alpha s ; omega data . The average is $1.43^{+0.58}_{-0.33}\ \%$. W (GeV) |  $M_{a}/M_{N}$ ($\%$) | W (GeV) |  $M_{a}/M_{N}$ ($\%$)
---|---|---|---
|  (from $d\sigma/dt)$ |  |  (from $\sigma$)
$1.76$ | $1.41\pm 0.18$ | $1.76$ | $1.31\pm 0.07$
$1.78$ | $1.39\pm 0.21$ | $1.78$ | $1.39\pm 0.09$
$1.79$ | $1.31\pm 0.21$ | $1.79$ | $1.46\pm 0.07$
$1.81$ | $1.28\pm 0.12$ | $1.81$ | $1.37\pm 0.07$
$1.84$ | $1.43\pm 0.13$ | $1.84$ | $1.34\pm 0.05$
$1.86$ | $1.40\pm 0.12$ | $1.86$ | $1.36\pm 0.06$
$1.90$ | $1.59\pm 0.08$ | $1.90$ | $1.38\pm 0.08$
$1.95$ | $1.49\pm 0.06$ | $1.95$ | $1.30\pm 0.04$
$2.00$ | $1.49\pm 0.07$ | $2.00$ | $1.27\pm 0.05$
$2.04$ | $1.70\pm 0.06$ | $2.04$ | $1.23\pm 0.06$
$2.09$ | $1.73\pm 0.06$ | $2.09$ | $1.27\pm 0.05$
$2.13$ | $1.93\pm 0.08$ | $2.13$ | $1.28\pm 0.05$

Using the same way of $\omega$, the neutron trace anomaly contributions were
extracted from the SLAC differential cross section rho data  of $\rho$
photoproduction at W=3.87GeV as $1.50\pm 0.22$% under the condition of
$\alpha_{A}$. In addition, the neutron trace anomaly contributions also
extracted from the predicted $\phi$ differential cross section phi data  at
different energies under $\alpha_{A}$, shown in Table 3 and Fig. 2 as a
function of $R$, the root mean square of the results is
$7.89^{+1.74}_{-4.61}$%. However, the results from $\phi$ photoproduction
process vary greatly under different energies, which is caused by the
uncertainty of the prediction. So this set of results is for reference only.

Figure 2: The neutron trace anomaly contributions from the predicted $\phi$ differential cross section at $\alpha_{A}$ as a function of $R$ phi 0.770 ; phi data , which is represented by the solid blue square, meanwhile the blue band is the error bar. Table 3: The neutron trace anomaly contribution extracted from the predicted $\phi$ photoproduction differential cross section at $\alpha_{A}$ phi 0.770 ; phi data . The average is $7.89^{+1.74}_{-4.61}\ \%$. $W$(GeV) |  2.01 |  2.12 |  2.15
---|---|---|---
$M_{a}/M_{N}(\%)$ |  9.12 |  9.63 |  3.27
Table 4: The trace anomaly contribution RMS of neutron at two sets of $\alpha_{s}$ phi 0.770 ; OR alpha s ; machine . Meson | $\rho$ | $\omega$ | $\phi$
---|---|---|---
$M_{a}/M_{N}(\%)$ | $1.50\pm 0.22$ | $1.43^{+0.58}_{-0.33}$ | $7.89^{+1.74}_{-4.61}$
(from $\alpha_{A}$) |  |  |
$M_{a}/M_{N}(\%)$ | $3.85\pm 0.56$ | $3.10^{+1.26}_{-0.72}$ | $2.24^{+0.50}_{-1.31}$
(from $\alpha_{B}$) |  |  |

Under the VMD model, the trace anomaly contribution of the neutron has been
extracted on the above, where $\alpha_{s}$ are derived from several different
physical models. Another purpose is to discuss the effect of $\alpha_{s}$
variation on the anomaly contribution. We extract the anomaly contribution at
$\alpha_{B}$ with the same method through three vector meson photoproduction
processes, $\alpha_{B}$ comes from the pure prediction by data learning and
has no model dependence machine . The trace anomaly contributions RMS of the
neutron mass at different $\alpha_{s}$ are compared in Table 4. Combined with
the value of $\alpha_{s}$ shown in Table 1, one finds that the trace anomaly
contribution is very sensitive to it. That indicates $\alpha_{s}$ is a crucial
uncertain physical quantity for itself and for calculating the nucleon anomaly
contribution.

In works I  and light , the trace anomaly contributions of the proton mass
were extracted through the heavy and light vector meson photoproduction
processes. We display the comparison of the proton and neutron results at the
same set of $\alpha_{s}\ (\alpha_{A})$ in Table 5, results indicating that the
trace anomaly contributions between neutron and proton from the same
photoproduction are close considering the error bar, although the results from
the different process are still different from each other.

As for the percentage of the nucleon anomaly contribution, although the
results obtained with the present calculation accuracy from our works show
that it is small, it cannot be taken as the final conclusion yet, the specific
reasons will be discussed in the next section.

Table 5: The trace anomaly contribution RMS of nucleons at $\alpha_{A}$ light ; phi 0.770 ; OR alpha s . Meson | $\rho$ | $\omega$ | $\phi$
---|---|---|---
$M_{a}/M_{N}(\%)$ | $0.53\pm 0.15$ | $1.47\pm 0.48$ | $3.63\pm 0.64$
(proton) |  |  |
$M_{a}/M_{N}(\%)$ | $1.50\pm 0.22$ | $1.43^{+0.58}_{-0.33}$ | $7.89^{+1.74}_{-4.61}$
(neutron) |  |  |

##  IV SUMMARY

In this work, we calculate the trace anomaly contribution of the neutron mass
through the light vector meson photoproduction processes for the first time
based on the VMD model. The numerical results obtained from the three
processes are all relatively small, which are $1.43^{+0.58}_{-0.33}\ \%$ from
the $\omega$ photoproduction, $1.50\pm 0.22$% from the $\rho$ photoproduction
and $7.89^{+1.74}_{-4.61}$% from the $\phi$ photoproduction. At the same time,
results show that under the same parameter, the trace anomaly contribution of
proton and neutron are very close, indicating that the mass distribution and
quark-gluon structure inside them may be similar. And we also study the
influence of parameter $\alpha_{s}$ on the nucleon anomaly contribution and
proved that $\alpha_{s}$ has a significant affection on it. In addition, the
accuracy of the experimental data of photoproduction cross sections and the
differences of vector meson production mechanisms also affect the extraction
of trace anomaly contribution of nucleon.

In terms of experimental data, the current shortcoming is that the neutron
target photoproduction data is insufficient. However, the proton target
photoproduction data is relatively abundant, we consider that there is a
possibility of using machine learning algorithms to predict the cross section
of $J/\psi$ photoproduction off neutron. Besides, the physical model could be
effective in predicting the neutron target photoproduction data due to the
exclusive reaction channels of the $J/\psi$ production. We will explore the
above aspects in our follow-up work. At the same time, more high-precision
experimental measurements for the photoproduction are expected.

As for the differences in vector meson production mechanisms, we found that
the difference in mesons’ related properties visibly affects the nucleon trace
anomaly contribution by analyzing the results from different photoproduction
processes under the VMD model. For example, the decay width of mesons has a
significant effect on the proton anomaly contribution. The $\omega$ meson and
the $\rho$ meson have a similar mass, corresponding $\alpha_{s}$, and
component quark mass, but there is a large gap between the anomaly
contribution results calculated from the two processes, which comes from the
large decay width of $\rho$. These results bring the anomaly contribution of
proton mass a notable error, if the partial width $\Gamma_{\rho\to
e^{+}e^{-}}$ of $\rho$ is set as $\Gamma_{\omega\to e^{+}e^{-}}$, the results
of the proton anomaly contribution will be very close light . And based on the
verification in Sec. III, we have learned that $\alpha_{s}$ is an important
parameter noticeably affecting the nucleon trace anomaly contribution. At the
same time, $\alpha_{s}$ also has a large uncertainty in the low energy range,
this is due to the complexity of the non-perturbative QCD. Meanwhile, the
corresponding scales of $\alpha_{s}$, which are taken as the meson mass, are
included within this range exactly. The appreciable error of $\alpha_{s}$ and
its significant effect on the nucleon trace anomaly contribution leads to the
uncertainty of the anomaly contribution. We will follow on using the machine
learning algorithms to study $\alpha_{s}$, to explore the physics involved and
try to correct its accuracy. In fact, $\alpha_{s}$ has great accuracy both
experimentally and theoretically in the high $Q$ range. If photoproduction
experimental data of heavier mesons are available, the uncertainty from
$\alpha_{s}$ will be eliminated through the calculation from these processes.

The scattering length of vector meson-nucleon(VMN) can also be calculated
under the VMD model, and the related studies are abundant igor ; thr ; 41 ; 42
; 43 ; 44 . Meanwhile, the uncertainty of the involved parameters in the
calculation is small. Therefore, we will consider exploring the possibility of
associating the VMN scattering length with the nucleon trace anomaly
contribution under the VMD model, trying to find another way to reveal the
nucleon trace anomaly contribution.

In a word, although the current numerical results of the nucleon trace anomaly
contribution are small, this can only be considered a partial conclusion due
to a series of uncertainties. We will conduct our follow-up based on the above
points, aiming to extract a more accurate nucleon trace anomaly contribution.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
under Grants No. 12065014 and No. 12247101, and by the Natural Science
Foundation of Gansu province under Grant No. 22JR5RA266. We acknowledge the
West Light Foundation of The Chinese Academy of Sciences, Grant No.
21JR7RA201.

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