# Dynamical Model of $J/\Psi$ photo-production on the nucleon

S. Sakinah<EMAIL_ADDRESS>Department of Physics, Kyungpook National
University, Daegu 41566, South Korea  T.-S. H. Lee<EMAIL_ADDRESS>Physics
Division, Argonne National Laboratory, Argonne, Illinois 60439, USA  H. M.
Choi<EMAIL_ADDRESS>Department of Physics, Kyungpook National University,
Daegu 41566, South Korea

###### Abstract

A dynamical model based on a phenomenological charm quark-nucleon ($c$-N)
potential $v_{cN}$ and the Pomeron-exchange mechanism is constructed to
investigate the $J/\Psi$ photo-production on the nucleon from threshold to
invariant mass $W=300$ GeV. The $J/\Psi$-N potential, $V_{J/\Psi N}(r)$, is
constructed by folding $v_{cN}$ into the wavefunction
$\Phi_{J/\Psi}(c\bar{c})$ of $J/\Psi$ within a Constituent Quark Model (CQM)
of Ref. SEFH13 . A photo-production amplitude is also generated by $v_{cN}$ by
a $c\bar{c}$-loop integration over the $\gamma\rightarrow c\bar{c}$ vertex
function and $\Phi_{J/\Psi}(c\bar{c})$. No commonly used Vector Meson
Dominance assumption is used to define this photo-production amplitude which
is needed to describe the data near the threshold. The $c$-N potential
$v_{cN}(r)$ is parameterized in a form such that the predicted $V_{J/\Psi
N}(r)$ at large distances has the same Yukawa potential form extracted from a
Lattice QCD (LQCD) calculation of Ref. KS10b  . The parameters of $v_{cN}$ are
determined by fitting the total cross section data of JLab by performing
calculations that include $J/\Psi$-N final state interactions (FSI). The
resulting differential cross sections $d\sigma/dt$ are found in good
agreements with the data. It is shown that the FSI effects dominate the cross
section in the very near-threshold region, allowing for sensitive testing of
the predicted $J/\Psi$-N scattering amplitudes. By imposing the constraints of
$J/\Psi$-N potential extracted from the LQCD calculation of Ref. KS10b , we
have obtained three $J/\Psi$-N potentials which fit the JLab data equally
well. The resulting $J/\Psi$-N scattering lengths are in the range of
$a=(-0.05$ fm $\sim$ $-0.25$ fm). With the determined $v_{cN}(r)$ and the
wavefunctions generated from the same CQM, the constructed model is used to
predict the cross sections of photo-production of $\eta_{c}(1S)$ and
$\Psi(2S)$ mesons for future experimental tests.

###### pacs:

13.60.Le, 14.20.Gk

††preprint: KNU-ANL-02/2024

##  I Introduction

It is well recognized lso23  that the information on the interactions between
the $J/\Psi$ meson and the nucleon(N) can improve our understanding of the
roles of gluons ($g$) in determining the structure of hadrons and hadron-
hadron interactions. In addition, a model of the $J/\Psi$-N interaction is
needed to understand the nucleon resonances $N^{*}(P_{c})$ reported by the
LHCb collaboration LHCb-15 ; LHCb-16a ; LHCb-19 ; LHCb-21a . It is also needed
to extract the gluonic distributions in nuclei, and to study the existence of
nuclei with hidden charms BD88a ; BSD90 ; GLM00 ; BSFS06 ; WL12 .

The leading $J/\Psi$-N interaction is the two-gluon exchange mechanism, as
illustrated in Fig. 1. Higher order multi-gluon exchange effects can not be
neglected in the non-perturbative region. By using continuum and lattice
studies at low energies and the heavy quark effective field theory and
perturbative QCD (PQCD) at high energies, the $J/\Psi$-N interaction can be
estimated.

Figure 1: Two gluon exchange mechanism of $J/\psi$-N reaction in QCD.

Several attempts had been made to determine the $J/\Psi$-N interactions.
Peskin Peskin79  applied the operator product expansion to evaluate the
strength of the color field emitted by heavy $q\bar{q}$ systems, and suggested
BP79  that the van der Waals force induced by the color field of $J/\Psi$ on
nucleons can generate an attractive short-range $J/\Psi$-N interaction. The
results of Peskin were used by Luke, Manohar, and Savage LMS92  to predict,
using the effective field theory method, the $J/\Psi$-N forward scattering
amplitude which was then used to get an estimation that $J/\Psi$ can have a
few MeV/nucleon attraction in nuclear matter. The $J/\Psi$-N forward
scattering amplitude of Ref. LMS92  was further investigated by Brodsky and
Miller BM97a  to derive a $J/\Psi$-N potential which gives a $J/\Psi$-N
scattering length of $-0.24$ fm. The result of Peskin was also used by
Kaidalov and Volkovitsky KV92 , who differed from Ref. BM97a  in evaluating
the gluon content in the nucleon, to give a much smaller scattering length of
$-0.05$ fm. In LQCD calculations using the approach of Refs. IAH06 ; AHI09 ,
Kawanai and Sasaki KS10b ; KS11  obtained an attractive $J/\Psi$-N potential
of the Yukawa form $V_{J/\Psi N,J/\Psi N}=-\alpha e^{-\mu r}/{r}$ with
$\alpha=0.1$ and $\mu=0.6$ GeV, which gives a scattering length of $-0.09$ fm.

Figure 2: $J/\Psi$-N interaction within the VMD assumption.  Figure 3: Quark-
antiquark loop mechanism within assuming the $v_{cN}$ as quark interaction in
$J/\Psi$-N reaction.

To make progress, it is necessary to have experimental information on the
$J/\Psi$-N scattering to test the theoretical results described above and
future LQCD calculations. One can adopt the traditional approach for
determining the vector meson-nucleon (V-N) interaction the Vector Meson
Dominance (VMD) assumption to extract $J/\Psi$-N cross sections from the data
of $J/\Psi$ photo-production reactions. In this approach, the incoming photon
is converted into a vector meson $V$ which is then scattered from the nucleon,
as illustrated in Fig. 2. However the approach based on VMD is not valid for
$J/\Psi$ because the VMD coupling constant is determined by the
$J/\Psi\to\gamma\to e^{+}e^{-}$ decay width at $q^{2}=m^{2}_{J/\Psi}\sim
9~{}\mbox{GeV}^{2}$ which is far from $q^{2}=0$ of the $\gamma+p\to J/\Psi+p$
reaction. Furthermore, the use of VMD for $J/\Psi$ is questionable as
discussed in Ref. XCYBCR21 . In addition, the $J/\Psi+N\rightarrow J/\Psi+N$
transition amplitude $t_{J/\Psi N,J/\Psi N}(k,q,W)$ near threshold is far off-
shell; for example, at $W=(m_{N}+m_{J/\Psi})+0.5$ GeV, the incoming $\gamma N$
relative momentum is $q=0.8$ GeV which is much larger than the outgoing
$J/\Psi$-N relative momentum $k=0.1$ GeV in the center of mass (CM) system.
Thus the VMD approach is clearly not directly applicable for describing
$J/\Psi+N\rightarrow J/\Psi+N$ in the near threshold energy region.

In this paper, we present a reaction model for extracting the $J/\Psi$
scattering amplitudes from the data of $\gamma+N\rightarrow J/\Psi+N$
reactions of Jefferson Laboratory (JLab) GlueX-19 ; GlueX-23 . In the
meantime, we will obtain phenomenological $J/\Psi$-N potentials, $V_{J/\Psi
N,J/\Psi N}$, for investigating nuclear reactions involving $J/\Psi$ meson. No
VMD is assumed by taking the $c\bar{c}$ structure of $J/\Psi$ into account to
define the model Hamiltonian. For simplicity in this exploring work, we will
follow the Pomeron-exchange model of Donnachie and Landshoff (DL) DL84  to
neglect the quark sub-structure of the nucleon and assume that that the
interactions between the charm-anticharm ($c\bar{c}$) quarks in $J/\Psi$ and
the nucleon can be defined by a phenomenological quark-N potential $v_{cN}$.
It follows that the $\gamma+N\rightarrow J/\Psi+N$ transition amplitude,
$B_{\gamma N,J/\Psi N}$, and the $J/\Psi+N\rightarrow J/\Psi+N$ potential,
$V_{J/\Psi N}$, are defined by $c\bar{c}$-loop mechanisms, as illustrated in
Fig. 3. Following the dynamical formulation SL96 ; MSL06 ; JLMS07 ; KNLS13
within which the unitarity condition requires that $J/\Psi$-N final state
interaction (FSI) effects must be included and the total amplitude of
$\gamma+N\rightarrow J/\psi+N$ is of the form

$\displaystyle T^{\rm D}_{\gamma N,J/\Psi N}=B_{\gamma N,J/\Psi
N}+T^{(fsi)}_{\gamma N,J/\Psi N},$ (1)

with

$\displaystyle T^{(fsi)}_{\gamma N,J/\Psi N}=B_{\gamma N,J/\Psi
N}(W)\,G_{J/\Psi N}\,T_{J/\Psi N,J/\Psi N},$ (2)

where $G_{J/\Psi N}$ is the $J/\Psi$-N propagator, and $T_{J/\Psi N,J/\Psi N}$
is the $J/\Psi$-N scattering amplitude calculated from the $J/\Psi$-N
potential, $V_{J/\Psi N}(r)$, by solving the following Lippman-Schwinger
Equation

$\displaystyle T_{J/\Psi N,J/\Psi N}$ $\displaystyle=$ $\displaystyle
V_{J/\Psi N,J/\Psi N}$ $\displaystyle+V_{J/\Psi N,J/\Psi N}\,G_{J/\Psi
N}\,T_{J/\Psi N,J/\Psi N}.$

To also describe data up to 300 GeV GlueX-19 ; GlueX-23 ; ZEUS-95b ; H1-00b ,
we add the Pomeron-exchange amplitude $T^{\rm Pom}_{\gamma N,J/\Psi N}$ of DL,
reviewed in Ref. lso23 , such that the total amplitude can be used to
investigate nuclear reactions involving $J/\Psi$ at all energies. To be
consistent, the Pomeron-exchange amplitude should also be defined by the
similar $c\bar{c}$-loop mechanism of Fig. 3. By using a hadron model RW94 ;
Roberts94  based on Dyson-Schwinger equation (DSE) of QCD, such a quark-loop
Pomeron-exchange model was explored in Refs. PL96 ; PL97 . It will be
interesting to use the recent DSE models MR97a ; CLR10 ; CCRST13 ; QR20 ;
YBCR22 ; AV00 ; SEVA11 ; EF11  to improve the results of Refs. PL96 ; PL97 .
Within the Hamiltonian formulation of this work, it requires a realistic
Constituent Quark Model (CQM) to generate the $J/\Psi$ wavefunction and
careful treatments of relativistic kinematic effects within Dirac’s
formulation of relativistic Quantum Mechanics KP91 . We therefore will not
pursue this here. Rather we focus on the near the threshold region, and any
effects from Pomeron-exchange which is very weak in the near threshold region,
can be considered as an estimate of uncertainties of the results presented in
this paper.

Since the Pomeron-exchange amplitude $T^{\rm Pom}_{\gamma N,J/\Psi N}$ has
been determined in Ref. lso23 , our task is to develop a model for calculating
the ampllitude $T^{\rm D}_{\gamma N,J/\Psi N}$ defined by Eqs.
(1)-(LABEL:eq:totamp-02). As defined by the loop-mechanism, Fig. 3, the
$J/\Psi$-N potential, $V_{J/\Psi N}(r)$, is constructed by folding $v_{cN}(r)$
into the wavefunction $\Phi_{J/\Psi}$. By using the $\Phi_{J/\Psi}$ from the
CQM of Ref. SEFH13 , the amplitude $T^{D}_{\gamma N,J/\Psi N}$ is completely
determined by the $c$-N potential $v_{cN}(r)$. To establish correspondence
with the LQCD calculations, the parametrization is chosen such that the
predicted $V_{J/\Psi N}(r)$ at large distances exhibits the same Yukawa
potential form extracted from a LQCD calculation of KS10b .

We determine the parameters of $v_{cN}$ by fitting the total cross section
data of JLab’s Hall-D GlueX-19 . As will be presented later, the resulting
differential cross sections $d\sigma/dt$ are in reasonably good agreements
with the data from JLab’s Hall-C GlueX-23 . More importantly, it is shown that
the FSI effects dominate the cross section in the very near-threshold region,
allowing for sensitive testing of the predicted $J/\Psi$-N scattering
amplitudes. Within the experimental uncertainties, this procedure allow us to
obtain four $J/\Psi$-N potentials which all fit the available JLab data
reasonably well. They however predict rather different cross sections near
threshold and the resulting $J/\Psi$-N scattering lengths. More extensive and
precise data in the very near threhold region are needed for making further
progress.

By using the determined $c$-N potential $v_{cN}(r)$ and the wavefunctions
generated from the same CQM of Ref. SEFH13 , we can apply the constructed
dynamical model to predict the cross sections of photo-production of the other
Charmonnim states. The results for the production of $\eta_{c}(1S)$ and
$\Psi(2S)$ mesons are presented for future experimental tests at JLab and the
future Electron-ion colliders (EIC).

Here we note that the JLab data of $J/\Psi$ photo-production had also been
investigated by using models based on two- and three-gluons exchange
mechanisms BCHL01 , Generalized Parton Distribution (GPD) of the nucleon GJL21
, and the Holographic QCD MZ19 . All of these approaches have rather different
assumptions in treating the quark sub-structure of $J/\Psi$. They are
distinctively different from our approach which accounts for the
$c\bar{c}$-loop mechanisms in calculating both the $J/\Psi$ photo-production
amplitudes and $J/\Psi$-N final-state interactions. Thus their objective is
not to extract $J/\Psi$-N interactions at low energies, as we are tring to
achieve in this work.

In section II, we present our formulation. The results are presented in
section III. Section IV is for giving a summary and discussing the possible
future improvements.

##  II Formulation

We follow Ref. marvin-gold-ken  to use the normalization $\langle{\bf k}|{\bf
k}^{\prime}\rangle=\delta({\bf k}-{\bf k}^{\prime})$ for plane wave state
$|{\bf k}\rangle$ and
$\langle\phi_{\alpha}|\phi_{\beta}\rangle=\delta_{\alpha,\beta}$ for bound
state $|\phi_{\alpha}\rangle$. The $J/\Psi$ meson will be denoted as $V$ in
the rest of the paper.

The differential cross section of vector meson photo-production reaction,
$\gamma({\bf q},\lambda_{\gamma})+N({\bf p},m_{s})\rightarrow V({\bf
k},\lambda_{V})+N({\bf p}^{\prime},m_{s}^{\prime})$, is calculated from lso23

$\displaystyle\frac{d\sigma_{VN,\gamma N}}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{(2\pi)^{4}}{|{\bf q}|^{2}}\frac{|{\bf k}|\,E_{V}({\bf
k})E_{N}({\bf p})}{W}\frac{|{\bf q}|^{2}E_{N}({\bf q})}{W}$ (4)
$\displaystyle\hskip-51.21504pt\mbox{}\times\frac{1}{4}\sum_{\lambda_{V},m^{\prime}_{s}}\sum_{\lambda_{\gamma},m_{s}}\left|\braket{\mathbf{k},\lambda_{V}m^{\prime}_{s}}{T_{VN,\gamma
N}}{\mathbf{q},\lambda_{\gamma}m_{s}}\right|^{2},$

where $m^{(\prime)}_{s}$ denotes the $z$-component of the initial (final)
state nucleon spin, and $\lambda_{V}$ and $\lambda_{\gamma}$ are the
helicities of vector meson $V$ and photon $\gamma$, respectively. In the CM
frame, the magnitudes of $|\mathbf{p}|=-|\mathbf{q}|=-q$ and
$|\mathbf{p}^{\prime}|=-|\mathbf{k}|=-k$ are defined by the invariant mass
$W=q+E_{N}(q)=E_{V}({\bf k})+E_{N}({\bf p}^{\prime})$.

The reaction amplitude $T_{VN,\gamma N}=T_{VN,\gamma N}(W)$ can be decomposed
into the sum of the dynamical scattering amplitude $T^{\rm D}$ and the
Pomeron-exchange amplitude $T^{\rm Pom}$ as lso23

$\displaystyle T_{VN,\gamma N}=T^{\rm D}_{VN,\gamma N}+T^{\rm Pom}_{VN,\gamma
N}.$ (5)

We shall describe each amplitude in the following subsections.

###  II.1 Dynamical model for $T^{\rm D}$

Following the dynamical approach of Refs. SL96 ; MSL06 ; JLMS07 ; KNLS13 , the
amplitude $T^{\rm D}$ is calculated from using the following Hamiltonian

$\displaystyle H=H_{0}+\Gamma_{\gamma,c\bar{c}}+v_{c\bar{c}}+v_{cN},$ (6)

where $H_{0}$ is the free Hamiltonian, $v_{cN}$ is a phenomenological quark-
nucleon potential to be determined, $v_{c\bar{c}}$ is the $c{\bar{c}}$
potential of CQM, and $\Gamma_{\gamma,c\bar{c}}$ is the electromagnetic
coupling of $\gamma\rightarrow{c\bar{c}}$ defined by

$\displaystyle\braket{{\bf q}}{\Gamma_{\gamma,c\bar{c}}}{{\bf k}_{1},{\bf
k}_{2}}$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2|{\bf
q}|}}\frac{1}{\sqrt{2E_{c}({\bf k}_{1})}}\frac{1}{\sqrt{2E_{c}({\bf k}_{2})}}$
(7)
$\displaystyle\hskip-14.22636pt\mbox{}\times\frac{e_{c}}{(2\pi)^{3/2}}[\bar{u}({\bf
k}_{1})\gamma^{\mu}\epsilon_{\mu}(q)v({\bf k}_{2})].$

Here $e_{c}$ is the charge of the charmed quark, $\epsilon_{\mu}(q)$ is the
photon polarization vector, $\bar{u}({\bf k}_{1})$ and $v({\bf k}_{2})$ are
the Dirac spinors with the normalization $\bar{u}({\bf k})u({\bf
k})=\bar{v}({\bf k})v({\bf k})=1$.

Using the potential $v_{c\bar{c}}(r)$ in the Hamiltonian, the wavefunction
$\ket{\phi_{V}}$ of the $J/\Psi$ is obtained by solving the bound-state
equation within a CQM developed by Segovia et al. SEFH13 , expressed as:

$\displaystyle(H_{0}+v_{c\bar{c}})\ket{\phi_{V}}=E_{V}\ket{\phi_{V}}.$ (8)

Here, we assume a simple $s$-wave wavefunction $\phi_{{\bf p}_{V}}({\bf
k},{\bf k^{\prime}})\equiv\braket{\phi^{J_{V}m_{V}}_{V,{\bf p}_{V}}}{{\bf
k}m_{s_{c}},{\bf k^{\prime}}m^{\prime}_{s_{\bar{c}}}}$ and have the following
matrix elements

$\phi_{{\bf p}_{V}}({\bf k},{\bf k^{\prime}})=\delta({\bf p}_{V}-{\bf k}-{\bf
k^{\prime}})\braket{J_{V}m_{V}}{\frac{1}{2}\frac{1}{2}m_{s_{c}}m^{\prime}_{s_{\bar{c}}}}\phi({\bf\bar{k}}),$
(9)

where ${\bf p}_{V}$ is the momentum of $J/\Psi$, ${\bf k^{(\prime)}}$ is the
momentum of $c({\bar{c}})$, ${\bf\bar{k}}=({\bf k}-{\bf k^{\prime}})/2$,
$J_{V}m_{V}$ is the total angular momentum of $J/\Psi$, $m_{s_{c}}$ and
$m_{s_{\bar{c}}}$ are the spins of $c$ and ${\bar{c}}$, respectively.

With the Hamiltonian Eq. (6) and neglecting the quark-quark scattering, the
scattering amplitude $T(E)$ for $\gamma N\rightarrow VN$ process is defined by
the following Lippmann-Schwinger equation,

$\displaystyle T(E)=H^{\prime}+T(E)\frac{1}{E-H_{0}+i\epsilon}H^{\prime}.$
(10)

where

$\displaystyle H^{\prime}=\Gamma_{\gamma,c\bar{c}}+v_{cN}.$ (11)

Inserting the intermediate states of $\ket{VN}$,$\ket{c\bar{c}N}$ and keeping
only the first order in electromagnetic coupling $e$ in Eq. (10), we obtain

$\displaystyle T^{\rm D}_{VN,\gamma N}(E)=B_{VN,\gamma
N}(E)+T^{\rm(fsi)}_{VN,\gamma N}(E),$ (12)

where $B_{VN,\gamma N}(E)$ is the Born term of $J/\psi$ photoproduction and
the FSI term $T^{\rm(fsi)}_{VN,\gamma N}(E)$ is required by the unitary
condition and is defined by

$\displaystyle T^{\rm(fsi)}_{VN,\gamma
N}(E)=T_{VN,VN}(E)\frac{1}{E-H_{0}+i\epsilon}B_{VN,\gamma N}.$ (13)

In this work, we assume that the $VN$ potential can be constructed by the
Folding model using the quark-N interaction $v_{cN}$ and the wavefunction
$\phi_{V}$ generated from Eq. (8)

$\displaystyle V_{VN,VN}=\braket{\phi_{V},N}{\sum_{c}v_{cN}}{\phi_{V},N}.$
(14)

The wavefunction and $v_{cN}$ potential are also used to construct the
$J/\Psi$ photo-production process with the following form

$\displaystyle B_{VN,\gamma N}$ $\displaystyle=$
$\displaystyle\bra{\phi_{V},N}\left[\sum_{c}v_{cN}\frac{\ket{c\bar{c}}\bra{c\bar{c}}}{E_{c\bar{c}}-H_{0}}\,\Gamma_{\gamma,c\bar{c}}\right]\,\ket{\gamma,N},$

where $E_{c\bar{c}}$ is the energy available to the propagation of $c\bar{c}$.

In the following, we give explicit expressions of the matrix elements of
$V_{VN,VN}$, $B_{VN,\gamma N}$, and $T^{(\rm fsi)}_{VN,\gamma N}$.

####  II.1.1 Matrix elements of $V_{VN,VN}$

Figure 4: $J/\Psi$-N scattering indicate to Eq. (20) with $v_{cN}$ potential
as quark-N interaction in the scattering.

To evaluate Eq. (14), we assume for simplicity that quark-N interaction,
$v_{cN}$, is independent of spin variables,

$\displaystyle\bra{{\bf k}\,m_{s_{c}}\,,{\bf p}\,m_{s_{N}}}v_{cN}\ket{{\bf
k}^{\prime}\,m^{\prime}_{s_{c}}\,,{\bf p}^{\prime}\,m^{\prime}_{s_{N}}}$
$\displaystyle\hskip
5.69046pt=\delta_{m_{s_{c}},m^{\prime}_{s_{c}}}\delta_{m_{s_{N}},m^{\prime}_{s_{N}}}\delta({\bf
k+p-k^{\prime}-p^{\prime}})\braket{{\bf q}}{v_{cN}}{{\bf q^{\prime}}},$ (16)

where the relative momenta of quark and nucleon are defined by

$\displaystyle{\bf q}$ $\displaystyle=$ $\displaystyle\frac{m_{N}{\bf
k}-m_{c}{\bf p}}{m_{N}+m_{c}},$ (17) $\displaystyle{\bf q}^{\prime}$
$\displaystyle=$ $\displaystyle\frac{m_{N}{\bf k}^{\prime}-m_{c}{\bf
p}^{\prime}}{m_{N}+m_{c}}.$ (18)

Here $m_{c}$ and $m_{N}$ are the masses of the quark $c$ and the nucleon,
respectively.

With the $J/\Psi$ wavefunction given in Eq. (9) and the spin independent
quark-N potential defined by Eq. (16), we define the matrix element of the
potential $V_{VN,VN}$ given by Eq. (14) in the CM frame, where ${\bf
p}_{V}=-{\bf p}$ and ${\bf p}^{\prime}_{V}=-{\bf p}^{\prime}$ as illustrated
in Fig. 4. In this frame, the matrix element of Eq. (14) can be written as

$\displaystyle\braket{{\bf p}_{V}m_{V},{\bf p}m_{s}}{V_{VN,VN}}{{\bf
p}^{\prime}_{V}m^{\prime}_{V},{\bf p}^{\prime}m^{\prime}_{s}}$
$\displaystyle\hskip
5.69046pt=\delta_{m_{V},m^{\prime}_{V}}\delta_{m_{s},m^{\prime}_{s}}\delta({\bf
p}_{V}+{\bf p}-{\bf p}^{\prime}_{V}-{\bf p}^{\prime})\braket{{\bf
p}}{V_{VN}}{{\bf p}^{\prime}},$ (19)

where

$\displaystyle\braket{{\bf p}}{V_{VN}}{{\bf p}^{\prime}}$ $\displaystyle=$
$\displaystyle 2\int d{\bf k}\,\textstyle\phi^{*}\left({\bf k}-\frac{{\bf
p}}{2}\right)$ (20) $\displaystyle\hskip-14.22636pt\mbox{}\times\Braket{{\bf
p}-\frac{m_{N}}{m_{N}+m_{c}}{\bf k}}{v_{cN}}{{\bf
p}^{\prime}-\frac{m_{N}}{m_{N}+m_{c}}{\bf k}}$
$\displaystyle\hskip-14.22636pt\mbox{}\times\textstyle\phi\left({\bf
k}-\frac{{\bf p}^{\prime}}{2}\right).$

Here the factor 2 arises from summation of contributions from the two quarks
within the $J/\psi$ meson and we have used the definitions of Eqs. (17) and
(18).

For a potential $v_{cN}(r)$ depending only on the relative distance $r$
between $c$ and $N$, we have

$\displaystyle\braket{{\bf q}}{v_{cN}}{{\bf q^{\prime}}}$ $\displaystyle=$
$\displaystyle v_{cN}({\bf q}-{\bf q}^{\prime})$ (21) $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\int\,d{\bf r}\,e^{i{\bf({\bf q}-{\bf
q}^{\prime})}\cdot{\bf r}}v_{cN}(r).$

The matrix element of $v_{cN}$ in Eq. (20) can then be written as

$\displaystyle\Braket{{\bf p}-\frac{m_{N}}{m_{N}+m_{c}}{\bf k}}{v_{cN}}{{\bf
p}^{\prime}-\frac{m_{N}}{m_{N}+m_{c}}{\bf k}}=v_{cN}({\bf p}-{\bf
p}^{\prime}).$ (22)

For later calculations, we note here that for a Yukawa form
$v_{cN}=\alpha\frac{e^{-\mu r}}{r}$, Eq. (21) leads to

$\displaystyle v_{cN}({\bf p}-{\bf p}^{\prime})$ $\displaystyle=$
$\displaystyle\alpha\frac{1}{(2\pi)^{2}}\frac{1}{({\bf p}-{\bf
p}^{\prime})^{2}+\mu^{2}}.$ (23)

Using Eq. (22), Eq. (20) can now be expressed in the following factorized
form:

$\displaystyle\braket{{\bf p}}{V_{VN}}{{\bf p}^{\prime}}$ $\displaystyle=$
$\displaystyle F_{V}({\bf t})\left[2v_{cN}({\bf t})\right],$ (24)

where ${\bf t}={\bf p}-{\bf p}^{\prime}$, and

$\displaystyle F_{V}({\bf t})$ $\displaystyle=$ $\displaystyle\int d{\bf
k}\,\textstyle\phi^{*}\left({\bf k}-\frac{{\bf p}}{2}\right)\phi\left({\bf
k}-\frac{{\bf p}^{\prime}}{2}\right)$ (25) $\displaystyle=$ $\displaystyle\int
d{\bf k}\,\textstyle\phi^{*}\left({\bf k}-\frac{{\bf
t}}{2}\right)\phi\left({\bf k}\right)$

is the form factor of the vector meson $V$ and $\phi({\bf k})$ is the
wavefunction of $J/\Psi$ in momentum space.

####  II.1.2 Matrix element of $B_{VN,\gamma N}$

Figure 5: J/$\Psi$ photo-production on the proton target with the momentum
variable indicate to Eq. (LABEL:eq:tmx-b-2).

By using the $J/\Psi$ wavefunction and Eq. (16) for quark-N potential, the
matrix element of photo-production of Eq. (LABEL:eq:photo-b) can be
calculated. With the variables in the CM system, as illustrated in Fig. 5, we
obtain

$\displaystyle\Braket{{\bf p}^{\prime}m_{V}m^{\prime}_{s}}{B_{VN,\gamma
N}}{{\bf q}\lambda m_{s}}$
$\displaystyle=\sum_{m_{c},m_{\bar{c}}}\frac{1}{(2\pi)^{3}}\frac{e_{c}}{\sqrt{2|{\bf
q}|}}\int d{\bf
k}\textstyle\braket{J_{V}m_{V}}{\frac{1}{2}\frac{1}{2}m_{c}m_{\bar{c}}}\phi\left({\bf
k}-\frac{1}{2}{\bf p}^{\prime}\right)$ $\displaystyle\hskip
11.38092pt\mbox{}\times\delta_{m_{s},m^{\prime}_{s}}\Braket{{\bf
p}^{\prime}-\frac{m_{N}}{m_{N}+m_{c}}{\bf k}}{v_{cN}}{{\bf
q}-\frac{m_{N}}{m_{N}+m_{c}}{\bf k}}$ $\displaystyle\hskip
11.38092pt\mbox{}\times\frac{1}{W-E_{N}({\bf q})-E_{c}({\bf q-k})-E_{c}({\bf
k})+i\epsilon}$ $\displaystyle\hskip
11.38092pt\mbox{}\times\bar{u}_{m_{c}}({\bf
k})[{\bf\epsilon_{\lambda}}\cdot{\bf\gamma}]v_{m_{\bar{c}}}({\bf q-k}).$ (26)

If one chooses the same Yukawa form of $v_{cN}(r)$, one obtains the following
factorized form:

$\displaystyle\Braket{{\bf p}^{\prime}m_{V}m^{\prime}_{s}}{B_{VN,\gamma
N}}{{\bf q}\lambda m_{s}}$ $\displaystyle\hskip
5.69046pt=C_{\lambda,m_{V}}\delta_{m_{s},m_{s}^{\prime}}B({\bf
p}^{\prime},{\bf q})\left[2v_{cN}({\bf q}-{\bf p}^{\prime})\right],$ (27)

where

$C_{\lambda,m_{V}}=\sum_{m_{c},m_{\bar{c}}}\textstyle\braket{J_{V}m_{V}}{\frac{1}{2}\frac{1}{2}m_{c}m_{\bar{c}}}\braket{m_{\bar{c}}}{{\bf\sigma}\cdot{\bf\epsilon_{\lambda}}}{m_{c}},$
(28)

and

$\displaystyle B({\bf p}^{\prime},{\bf q})$
$\displaystyle=\frac{1}{(2\pi)^{3}}\frac{e_{c}}{\sqrt{2|{\bf q}|}}\int d{\bf
k}\,\phi\left({\bf k}-{\textstyle\frac{1}{2}}{\bf p}^{\prime}\right)$
$\displaystyle\times\frac{1}{W-E_{N}({\bf q})-E_{c}({\bf q-k})-E_{c}({\bf
k})+i\epsilon}$ $\displaystyle\mbox{}\times\sqrt{\frac{E_{c}({\bf
k})+m_{c}}{2E_{c}({\bf k})}}\sqrt{\frac{E_{c}({\bf q-k})+m_{c}}{2E_{c}({\bf
q-k})}}$ $\displaystyle\mbox{}\times\left(1-\frac{{\bf
k}\cdot{\bf(q-k)}}{[E_{c}({\bf k})+m_{c}][E_{c}({\bf q-k})+m_{c}]}\right).$

####  II.1.3 Final State Interactions

Figure 6: J/$\Psi$ photo-production on the proton target with final state
interaction indicate to Eq. (LABEL:eq:pom-cqm-t0). Figure 7: The VN scattering
indicate to Eq. (LABEL:eq:lseq-00).

Including the final state interaction, as illustrated in Fig. 6, the matrix
element of the total amplitude in Eq. (12) is

$\displaystyle\Braket{{\bf p}^{\prime}m_{V}m^{\prime}_{s}}{T_{VN,\gamma
N}}{{\bf q}\lambda m_{s}}$ $\displaystyle=$ $\displaystyle\Braket{{\bf
p}^{\prime}m_{V}m^{\prime}_{s}}{B_{VN,\gamma N}}{{\bf q}\lambda m_{s}}$
$\displaystyle+$ $\displaystyle\braket{{\bf
p}^{\prime}m_{V}m^{\prime}_{s}}{T^{\rm(fsi)}_{VN,\gamma N}}{{\bf q}\lambda
m_{s}},$

with

$\displaystyle\Braket{{\bf
p}^{\prime}m_{V}m^{\prime}_{s}}{T^{\rm(fsi)}_{VN,\gamma N}}{{\bf q}\lambda
m_{s}}$ $\displaystyle\hskip
5.69046pt=\sum_{m^{\prime\prime}_{V},m^{\prime\prime}_{s}}\int d{\bf
p}^{\prime\prime}\Braket{{\bf p}\,m_{V}m^{\prime}_{s}}{T_{VN,VN}}{{\bf
p}^{\prime\prime}m^{\prime\prime}_{V},m^{\prime\prime}_{s}}$
$\displaystyle\hskip
34.14322pt\mbox{}\times\frac{1}{W-E_{N}(p^{\prime\prime})-E_{V}(p^{\prime\prime})+i\epsilon}$
$\displaystyle\hskip 34.14322pt\mbox{}\times\Braket{{\bf
p}^{\prime\prime}m^{\prime\prime}_{V}m^{\prime\prime}_{s}}{B_{VN,\gamma
N}}{{\bf q}\lambda,m_{s}}.$ (31)

In the above equation, the spin independent amplitude of $J/\Psi$-N scattering
is

$\displaystyle\Braket{{\bf p}\,m_{V}m_{s}}{T_{VN,VN}(W)}{{\bf
p}^{\prime}\,m^{\prime}_{V}m^{\prime}_{s}}$ $\displaystyle\hskip
5.69046pt=\delta_{m_{V},m^{\prime}_{V}}\delta_{m_{s},m^{\prime}_{s}}\Braket{{\bf
p}^{\prime}}{T_{VN}(W)}{{\bf p}},$ (32)

where $\braket{{\bf p}^{\prime}}{T_{VN}((W)}{{\bf p}}$ is defined by the
following Lippmann-Schwinger Equation, as illustrated in Fig. 7:

$\displaystyle\braket{{\bf p}^{\prime}}{T_{VN}(W)}{{\bf p}}$ $\displaystyle=$
$\displaystyle\braket{{\bf p}^{\prime}}{V_{VN}}{{\bf p}}$ $\displaystyle+\int
d{\bf p^{\prime\prime}}\frac{\braket{{\bf p}^{\prime}}{V_{VN}}{{\bf
p}^{\prime\prime}}\braket{{\bf p}^{\prime\prime}}{T_{VN}(W)}{{\bf
p}}}{W-E_{N}(p^{\prime\prime})-E_{V}(p^{\prime\prime})+i\epsilon}.$

Here $\braket{{\bf p}^{\prime}}{V_{VN}}{{\bf p}}$ has been defined by Eq.
(24).

We solve Eq. (LABEL:eq:lseq-00) in the partial-wave representation. For a spin
independent $c$-N potential $v_{cN}(r)$, which gives also an spin-independent
$V_{VN}$, we have

$\braket{{\bf p}^{\prime}}{V_{VN}(W)}{{\bf
p}}=\sum_{L}\frac{2L+1}{4\pi}V_{L}(p^{\prime},p,W)P_{L}(x),$ (34)

and

$\braket{{\bf p}^{\prime}}{T_{VN}(W)}{{\bf
p}}=\sum_{L}\frac{2L+1}{4\pi}T_{L}(p^{\prime},p,W)P_{L}(x),$ (35)

where $x={\bf\hat{p}\cdot\hat{p}^{\prime}}$ and $P_{L}(x)$ is the Legendre
function of the first kind. With Eq. (34), the partial-wave matrix element of
potential can be calculated by

$\displaystyle V_{L}(p^{\prime},p,W)$ $\displaystyle=$
$\displaystyle(2\pi)\int_{-1}^{+1}dxP_{L}(x)\braket{{\bf
p}^{\prime}}{V_{VN}(W)}{{\bf p}},$

where $\braket{{\bf p}^{\prime}}{V_{VN}(W)}{{\bf p}}$ is defined by Eq. (24).
If we set $F({\bf t})=1$ in Eq. (24) and use Eq. (23) for a Yukawa form for
$c$-N potential $v_{cN}(r)=\alpha\frac{e^{-\mu r}}{r}$, one finds

$\displaystyle
V_{L}(p^{\prime},p,W)=\frac{2}{\pi}\left(\frac{\alpha}{2pp^{\prime}}\right)Q_{L}(Z),$
(37)

where $Z=\frac{p^{2}+p^{\prime 2}+\mu^{2}}{2pp^{\prime}}$ and $Q_{L}(Z)$ is
the Legendre function of the second kind.

By using Eqs. (34)-(35), Eq. (LABEL:eq:lseq-00) then leads to

$\displaystyle T_{L}(p^{\prime},p,W)=V_{L}(p^{\prime},p,W)$
$\displaystyle\hskip 62.59596pt+\int dp^{\prime\prime}p^{\prime\prime
2}\biggl{[}V_{L}(p^{\prime},p^{\prime\prime},W)$ $\displaystyle\hskip
85.35826pt\times\;\frac{1}{W-E_{N}(p^{\prime\prime})+E_{V}(p^{\prime\prime})+i\epsilon}$
$\displaystyle\hskip 85.35826pt\times\;T_{L}(p^{\prime\prime},p,W)\biggr{]}.$
(38)

We solve Eq. (38) by using the standard numerical method described in Ref.
Hftel-frank . The scattering phase shifts $\delta_{L}$ are calculated from the
resulting $T_{L}(p^{\prime\prime},p,W)$

$\displaystyle
e^{i\delta_{L}}\sin\delta_{L}=-\pi\frac{p_{0}E_{N}(p_{0})E_{V}(p_{0})}{E_{N}(p_{0})+E_{V}(p_{0})}T_{L}(p_{0},p_{0},W),$
(39)

where the on-shell momentum defined by

$\displaystyle W=E_{N}(p_{0})+E_{V}(p_{0})\,.$ (40)

We will also calculate the scattering length $a$ which is defined by $L=0$
partial-wave at $p_{0}\rightarrow 0$ by

$\displaystyle p_{0}\,\cot\delta_{0}=\frac{-1}{a}\,.$ (41)

###  II.2 Pomeron-exchange amplitude

Following the approach of Donnachie and Landshoff in Refs. DL84 ; DL92 ; DL95
; DL98 , $T^{\rm Pom}$ was constructed within Regge Phenomenology and is given
by

$\displaystyle\braket{\mathbf{k},m_{V}m^{\prime}_{s}}{T^{\rm Pom}_{VN,\gamma
N}(W)}{\mathbf{q},\lambda_{\gamma}m_{s}}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\sqrt{\frac{m_{N}m_{N}}{4E_{V}(\mathbf{k})E_{N}(\mathbf{p}^{\prime})|\mathbf{q}|E_{N}(\mathbf{p})}}$
$\displaystyle\mbox{}\times[\bar{u}(p^{\prime},m^{\prime}_{s})\epsilon^{*}_{\mu}(k,\lambda_{V})\mathcal{M}^{\mu\nu}_{\mathbb{P}}(k,p^{\prime},q,p)\epsilon_{\nu}(q,\lambda_{\gamma})u(p,m_{s})].$

In this approach, the incoming photon will be converted to vector meson, then
will be scattered from nucleon by Pomeron exchange, as illustrated by Feynman
diagram in Fig. 8.

Figure 8: The Feynman diagram of $J/\Psi$ potoproduction with DS-pom approach

The Pomeron-exchange amplitude
$\mathcal{M}^{\mu\nu}_{\mathbb{P}}(k,p^{\prime},q,p)$ is given by

$\mathcal{M}^{\mu\nu}_{\mathbb{P}}(k,p^{\prime},q,p)=G_{\mathbb{P}}(s,t)\mathcal{T}^{\mu\nu}_{\mathbb{P}}(k,p^{\prime},q,p),$
(43)

and

$\displaystyle\mathcal{T}^{\mu\nu}_{\mathbb{P}}(k,p^{\prime},q,p)$
$\displaystyle=$ $\displaystyle
i\,2\frac{e\,m_{V}^{2}}{f_{V}}[2\beta_{q_{V}}F_{V}(t)][3\beta_{u/d}F_{1}(t)]$
(44) $\displaystyle\mbox{}\\{\not{q}g^{\mu\nu}-q^{\mu}\gamma^{\nu}\\}\,,$

where $m_{V}$ is the mass of the vector meson, and $f_{V}=5.3$, $15.2$,
$13.4$, $11.2$, $40.53$ for $V=\rho,\omega,\phi,J/\Psi,\Upsilon$ are
traditionally determined by calculating the $V\to\gamma\to e^{+}e^{-}$ decay
width. The parameters $\beta_{q_{V}}$ ($\beta_{u/d}$) define the coupling of
the Pomeron with the quark $q_{V}$ ($u$ or $d$) in the vector meson $V$
(nucleon $N$). In Eq. (44), a form factor for the Pomeron-vector meson vertex
is also introduced with

$\displaystyle
F_{V}(t)=\frac{1}{m_{V}^{2}-t}\left(\frac{2\mu_{0}^{2}}{2\mu_{0}^{2}+m_{V}^{2}-t}\right),$
(45)

where $t=(q-k)^{2}=(p_{f}-p_{i})^{2}$. By using the Pomeron-photon analogy,
the form factor for the Pomeron-nucleon vertex is defined by the isoscalar
electromagnetic form factor of the nucleon as

$F_{1}(t)=\frac{4m_{N}^{2}-2.8t}{(4m_{N}^{2}-t)(1-t/0.71)^{2}}.$ (46)

Here $t$ is in the unit of GeV2, and $m_{N}$ is the proton mass.

Figure 9: Total cross sections from Pomeron-exchange amplitude for $4\leq
W\leq 300$ GeV (top) and for $4\leq W\leq 6$ GeV (bottom).

The propagator $G_{\mathbb{P}}$ of the Pomeron in Eq. (43) follows the Regge
phenomenology form:

$G_{\mathbb{P}}=\left(\frac{s}{s_{0}}\right)^{\alpha_{P}(t)-1}\exp\left\\{-\frac{i\pi}{2}\left[\alpha_{P}(t)-1\right]\right\\}\,,$
(47)

where $s=(q+p_{i})^{2}=W^{2}$,
$\alpha_{P}(t)=\alpha_{0}+\alpha^{\prime}_{P}t$, and
$s_{0}=1/\alpha^{\prime}_{P}$. We use the value of $s_{0}=0.25$ GeV from the
works DL84 ; DL92 ; DL95 ; DL98  of Donnachie and Landshoff.

$T^{\rm Pom}$ has been determined in Ref. WL13 ; Lee20 ; lso23  by fitting the
data of total cross sections up to 300 GeV. The resulting parameters for
$\rho^{0}$, $\omega$, $\phi$ photo-production OL02  have been determined as
follows: $\mu_{0}=1.1$ GeV2, $\beta_{u/d}=2.07$ GeV-1, $\beta_{s}=1.38$ GeV-1,
$\alpha_{0}=1.08$ for $\rho$ and $\omega$, $\alpha_{0}=1.12$ for $\phi$. For
the heavy quark systems, we find that using the same $\mu_{0}^{2}$,
$\beta_{u/d}$, and $\alpha^{\prime}_{P}$ values, the photo-production data for
$J/\Psi$ and $\Upsilon$ can be fitted by setting $\beta_{c}=0.32$ GeV-1 and
$\beta_{b}=0.45$ GeV-1, along with a larger $\alpha_{0}=1.25$.

Figure 9 depicts the total cross section for the $J/\psi$ photo-production on
the nucleon obtained solely from the contribution of the Pomeron-exchange
amplitude, and compares it with the data GlueX-19 ; ZEUS-95b ; H1-00b . One
can see from Fig. 9 that while the Pomeron-exchange mechanism effectively
describes the data (black circles) ZEUS-95b ; H1-00b  at very high energies,
it falls short in accurately representing the JLab data (blue squares)
GlueX-19 .

##  III Results

###  III.1 Determination of quark-Nucleon potential $v_{cN}$

We are guided by the Yukawa form of $J/\Psi$-N potential extracted from the
LQCD calculation of Ref. KS10b . We observe that the factorized form of
$V_{VN,VN}$ of Eq. (24) suggests that if $v_{cN}(r)$ is also of the Yukawa
form, the resulting $V_{VN,VN}$ can approach the $J/\Psi$-N potential of Ref.
KS10b  at large distance $r$. We therefore consider the following
parameterization

$\displaystyle v_{cN}(r)=\alpha\left(\frac{{}^{-\mu
r}}{r}-c_{s}\frac{e^{-\mu_{1}r}}{r}\right).$ (48)

We first consider a model (1Y) with $c_{s}=0$, meaning that the resulting
$J/\Psi$-N potential takes on the Yukawa form extracted from LQCD. The
calculation then only has two free parameters $\alpha$ and $\mu$. We find that
the JLab data GlueX-19  for small $W$ region can be best fitted by choosing
$\alpha=-0.067$ and $\mu=0.3$ GeV if the $J/\Psi$ wavefunction is taken from
the CQM model of Ref. SEFH13 .

Figure 10: Total cross sections for $4\leq W\leq 5$ GeV from $1Y$ model with
$\alpha=-0.067$.

We show in Fig. 10 the total cross sections for the $J/\psi$ photo-production
in the small $4\leq W\leq 5$ GeV region obtained from the 1Y model. The
dotted, dot-dashed, and dashed lines represent the results obtained from the
Pomeron-exchange, Born term, and the sum of Pomeron-exchange and Born term
contributions, respectively. The solid line represents the full result
including the final state interaction in addition to the Pomeron-exchange and
Born terms. As seen in Fig. 10, the Pomeron-exchange mechanism alone is
insufficient, particularly for low-energy regimes. However, the contribution
from the $c$-N interaction $v_{cN}$ is essential for fitting the data in this
1Y model. In particular, the cross sections in the very near threshold region
are largely determined by the FSI term. These results demonstrate that
$J/\Psi$-N interactions can be extracted rather clearly from the $J/\Psi$
photo-production data within this model which does not use the VMD assumption.
More importantly, the calculations properly account for off-shell effects and
satisfy the unitarity condition, a feature that is not considered in most, if
not all, previous approaches discussed in Ref. lso23 .

Figure 11: The effect of the momentum cut-off $k_{\rm max}=500$ MeV (top), and
the effect of the form factor (bottom), on the total cross section.

To see the importance of using a realistic $J/\Psi$ wavefunction, we compare
the cross section from the full calculation and that from using a wavefunction
that employs a momentum cut-off in evaluating $B_{VN,\gamma N}$ amplitude.
Specifically, we replace the integral $B=\int_{0}^{\infty}\phi(k)(\cdots)dk$
with $\int_{0}^{k_{\rm max}}\phi(k)(\cdots)dk$ to evaluate the cut-off effect.

In the top panel of Fig. 11, we illustrate the impact of the momentum cut-off
$k_{\rm max}$ on the total cross section. The solid line represents the result
obtained without the cut-off (i.e. $k_{\rm max}\to\infty$), while the dashed
line corresponds to the result obtained with the cut-off value of $k_{\rm
max}=500$ MeV. Clearly, the high momentum tail of the $J/\psi$ wavefunction is
crucial for fitting the data, particularly in the low-energy region. We also
note that using the Gaussian wavefunction determined by $J/\Psi\rightarrow
e^{+}e^{-}$ of Ref. lso23  results in much larger cross sections, and fitting
to the JLab data requires a much smaller value of $\alpha$.

As one can see from Eq. (24), the $J/\Psi$-N potential is determined by the
form factor $F_{V}(t)$, which is obtained from the convolution of the initial
and final state $J/\Psi$ wavefunctions. In the bottom panel of Fig. 11, we
illustrate the impact of $F_{V}(t)$ on the total cross section by comparing
the momentum-dependent form factor $F_{V}(t)$ (solid line) with the constant
form factor $F_{V}(t)=1$ (dashed line). It is evident that the effect of
$F_{V}(t)$ on the total cross section is significant, underscoring the
importance of using a realistic $J/\Psi$ wavefunction in determining the FSI
effects.

###  III.2 The 2Y model

Figure 12:  (Top) Total cross sections from 1Y model with $\alpha=-0.067$ and
2Y model with $\alpha=-0.145$ and $N=5$. (Bottom) Short-range behavior of
$v_{cN}(r)$ for the 1Y and 2Y models.

We now consider the model ($2Y$) by incorporating the second term with
$c_{s}=1$ in addition to the first term in Eq. (48). The cut-off of the short-
range part is set as $\mu_{1}=N\mu$. We observe that for large values of
$N>50$, the 2Y model closely approximates the 1Y model, and the JLab data can
be fitted with any $N<20$ by adjusting the coupling constant $\alpha$. The
result of the total cross section from the 2Y model with $N=5$ and
$\alpha=-0.145$ is comparable to that of 1Y model with $\alpha=-0.067$, as
shown in the top panel of Fig. 12. The two models fit the JLab data equally
well, but have very large difference at very near threshold which is dominated
completely by FSI. The origin of it is that they have very different short
range behaviors of the potentials $v_{cN}(r)$, as shown in the bottom panel of
Fig. 12. To compare the potential for 1Y model (black line), we evaluate for
the 2Y model with $N=2$ (red line) and $N=5$ (blue line), using the same
$\alpha=-0.145$ for both $N$ values. This ensures clear comparisons of their
$r$-dependence at short distances.

We thus expect that their difference in differential cross section must be
large at large $t$.

Figure 13: Differential cross sections from 1Y model with $\alpha=-0.067$ and
2Y model with $\alpha=-0.145$ and $N=5$.

This is shown in Fig. 13 where the differential cross sections as a function
of $t$ from the 1Y model with $\alpha=-0.067$ and the 2Y model with
$\alpha=-0.145$ and $N=5$ are compared. Both models describe reasonably well
the JLab data from the GlueX Collaboration GlueX-23  at $E_{\gamma}=8.8-10.53$
GeV. We observed that notable differences between the predictions of the 1Y
and 2Y models at large values of $t$. More precise data are clearly needed for
making further progress, while 2Y model appears to be better. For future
experiment at very near threshold, we present in Fig. 14 the differential
cross sections at $W=4.055$ GeV. We compare the predictions of the 2Y model
with (solid line) and without (dashed line) including FSI.

Figure 14: Prediction of the differential cross sections near threshold at
$W=4.055$ GeV by using 2Y model.

###  III.3 LQCD constraints

Figure 15: The $J/\Psi$-N potential $V_{J/\Psi N,J/\Psi
N}(r)=\alpha_{L}\frac{e^{-\mu_{L}r}}{r}$ extracted from the LQCD calculations
KS10b ; sasaki-1 .

We have found that the $J/\Psi$-N potentials resulted from fitting the JLab
data in the previous section give $J/\Psi$-N scattering lengths, $-a\sim
0.3-0.7$ fm, which are rather different from those extracted from the LQCD of
Ref. KS10b , as discussed in section I. Thus it raises the question on our
parameterization of quark-N potential $v_{cN}(r)$. At low energies, the
$J/\Psi$-N system has a small relative momentum and the multi-gluon exchange
involving two quarks in $J/\Psi$ may play important roles, and we need to
extend our parametrization of $v_{cN}$ to include two-body operator. We,
however, will not explore this possibility in this exploratory work because
the model will then have too many parameters which can not be determined by
the still very limited data in the near threshold region. Instead, we next
consider the models imposing LQCD constraints on FSI and just use their LQCD
data of $V_{J/\Psi N}(r)=\alpha_{L}\frac{e^{-\mu_{L}r}}{r}$, where
$\alpha_{L}=-0.06\sim-0.11$ and $\mu_{L}=0.3\sim 0.5$ (in unit of GeV) KS10b ;
sasaki-1  to determine the parameters of our 1Y model with $F_{V}(t)=1$, i.e.
$V^{1Y}_{J/\Psi N}(r)=2v_{cN}(r)=2\alpha\frac{e^{-\mu r}}{r}$ as one can see
from Eqs. (24) and (48).

We have obtained three $V_{J/\Psi N}(r)$ which all fit LQCD data at large $r$,
with different short-range parts, as showing Fig. 15.

The potential fits “pot-1” and “pot-2” are obtained from the two limiting sets
for $\alpha_{L}$, i.e. $(\alpha_{L},\mu_{L})=(-0.06,0.3)$ and $(-0.11,0.5)$,
respectively. Since the parameter $\alpha_{L}$ corresponds to $2\alpha$ in our
model, we utilize the parameter sets $(\alpha\equiv\alpha_{\rm
FSI},\mu)=(-0.03,0.3)$ fixed from pot-1 and $(-0.055,0.5)$ fixed from pot-2.
The potential labeled as “Fit-1” in Fig. 15 corresponds to the 2Y model fit to
LQCD data. For this fit, we utilized the parameter set $(\alpha_{\rm
FSI},\mu)=(-0.1,0.3)$ with $N=2$ to accurately describe the LQCD constraints
on FSI. Consequently, each potential’s parameter set for the contribution of
the $T^{({\rm fsi})}_{J/\Psi N,J/\Psi N}$ term was fixed in this manner.

For the potential parameter of the Born term $B_{J/\Psi N,\gamma N}$, we first
tried the fixed $\alpha_{\rm B}=-0.145$ obtained from the 2Y model. Combining
both Born- and FSI-terms, we now construct the three models from the three
potentials (pot-1, pot-2, Fit-1), i.e. model (a) with $(\alpha_{\rm
FSI},\alpha_{\rm B},\mu)=(-0.03,-0.145,0.3)$ and model (b) with $(\alpha_{\rm
FSI},\alpha_{\rm B},\mu)=(-0.055,-0.145,0.5)$, and model (c) with
$(\alpha_{\rm FSI},\alpha_{\rm B},\mu,N)=(-0.1,-0.145,0.3,5)$. We note that
these three models share a common parameter $\alpha_{\rm B}$.

Figure 16:  Total cross-sections of $J/\Psi$ obtained from the three models
(a), (b), (c) constrained by the LQCD data with the common $\alpha_{\rm
B}=-0.145$ for $4\leq W\leq 5$ GeV (top) and for the near threshold region
(bottom).

Using those three models, we plot in Fig. 16 the total cross sections for
$4\leq W\leq 5$ (GeV) region (top) and for the near threshold region (bottom),
respectively. For comparison, we also plot the result obtained from the pure 2
Y model with $(\alpha_{\rm B},\mu,N)=(-0.145,0.3,5)$. As depicted in the top
panel of Fig. 16, no significant difference is observed between the models
imposing LQCD constraints on FSI and the pure 2Y model. However, as
illustrated in the bottom panel, the models imposing LQCD constraints on FSI
also exhibit a notable effect at $W\leq 4.1$ GeV compared to the pure 2Y
model. This observation indicates that the models imposing LQCD constraints on
FSI improve the threshold behavior and predict the cross-section closer to the
JLab data than the pure 2Y model. Additionally, we observe that it is possible
to adjust the model (c) to closely match the predictions of the model (a).

To enhance our results in comparison to the JLab data GlueX-19 , we attempted
to adjust the parameters $\alpha_{\rm B}$ of the Born term for each model
while keeping the parameters of the FSI term fixed. The best-fit parameters
for each model are summarized in Table 1. The scattering length $a$ is
predicted from the $T^{\rm D}$ matrix, with the other listed model parameters
set accordingly.

Using the model parameters listed in Table 1, we present a comparison of cross
sections among the models in Fig. 17. Especially, the bottom panel of Fig. 17
displays a linear plot of the cross section at very near threshold energies
($W\leq 4.1$ GeV). It illustrates the notable improvement achieved by the
$J/\Psi$-N potentials extracted from LQCD calculations, especially in the
region close to the threshold.

Figure 17: Total cross-sections of $J/\Psi$ obtained from the three models (a), (b), (c) listed in Table 1 for $4\leq W\leq 5$ GeV (top) and for the near threshold region (bottom). Table 1:  The parameters for the models (a),(b),(c) imposing LQCD constraints on FSI. Model | $\alpha_{\rm FSI}$ |  $\mu$ (GeV) |  $\mu_{1}$ (GeV) |  $a$(fm) | $\alpha_{B}$
---|---|---|---|---|---
(a) | -0.03 | 0.3 | - | -0.15 | -0.162
(b) | -0.055 | 0.5 | - | -0.233 | -0.152
(c) | -0.1 | 0.9 | 1.8 | -0.057 | -0.163

The comparison of predicted differential cross sections as a function of $t$
is shown in Fig. 18. We compared the models’ predictions at the energy closest
to the JLab data GlueX-23  threshold. We observe that modifying the FSI with
the LQCD potential, using a fixed parameter, yields predictions that closely
match the data compared to the pure 2Y prediction. By observing the cross
section and differential cross section predictions, the LQCD constraint can
improve the 2Y model to have good behavior at very near threshold and at a
large $t$.

Figure 18: Differential cross sections for each model listed in Table 1.

###  III.4 Predictions of $\eta_{c}(1S)$ and $\Psi(2S)$

With the parameters of $v_{cN}$ determined from fitting the JLab data, we
proceed to predict the photo-production of $\eta_{c}(1S)$ and $\Psi(2s)$
mesons, in addition to $J/\psi$. Because $v_{cN}$ is spin independent and the
wavefunctions for $\eta_{c}(1S)$ and $\Psi(2S)$ are also of the $s$-wave form
Eq. (9), the calculations can be done by using the same formula presented in
section III by simply changing the wavefunctions and the kinemtics associated
with the masses of mesons. The predicted total cross sections for the photo-
productions of $\eta_{c}(1S)$, $\Psi(2s)$, and $J/\psi$ are presented in the
top panel of Fig. 19, along with the comparison of their wavefunctions in the
bottom panel. Using the pure 2Y model with $(\alpha,\mu,N)=(-0.145,0.3,5)$, we
predicted the differential cross sections for $\eta_{c}(1S)$ at 20 MeV (upper
left) and 500 MeV (upper right), and for $\Psi(2S)$ at 20 MeV (lower left) and
500 MeV (lower right) in Fig. 20.

As a validation test for our model, it would be intriguing to compare our
predictions with forthcoming data expected from experiments at JLab and the
future EIC.

Figure 19: The prediction of $\eta_{c}(1S)$ and $\Psi(2S)$ total cross
sections (left), and the wavefunctions of $J/\Psi$, $\eta_{c}(1S)$ and
$\Psi(2S)$ (right).

Figure 20: (Upper) Prediction of $\eta_{c}(1S)$ differential cross sections at
20 MeV (left) and 500MeV (right) from threshold. (Lower) Prediction of
$\Psi(2S)$ differential cross sections at 20 MeV (left) and 500MeV (right)
from threshold.

##  IV Summary and Future improvements

Within a Hamiltonian formulation SL96 ; MSL06 ; JLMS07 ; KNLS13 , a dynamical
model based on the Constituent Quark Model (CQM) and a phenomenological charm
quark-nucleon potential $v_{cN}(r)$ is constructed to investigate the $J/\Psi$
photo-production on the nucleon at energies near threshold. The main feature
of the model lies in the quark-N potential, $v_{cN}$, which generates a photo-
production amplitude defined by a $c\bar{c}$-loop integration over the
$\gamma\rightarrow c\bar{c}$ vertex function and the $J/\Psi$ wavefunction,
$\Phi_{J/\Psi}(c\bar{c})$, from the CQM described in Ref. SEFH13 . In
addition, the $J/\Psi$-N final state interaction is calculated from a
$J/\Psi$-nucleon potential ($V_{J/\Psi N}(r)$) which is constructed by folding
$v_{cN}(r)$ into the same wavefunction $\Phi_{J/\Psi}(c\bar{c})$. By also
incorporating the Pomeron-exchange amplitudes determined in Refs. WL13 ; lso23
, the constructed model can describe the available data from threshold to high
energies, up to the invariant mass $W=300$ GeV.

The parametrization of the $c$-N potential $v_{cN}(r)$ is chosen such that the
predicted $V_{J/\Psi N}(r)$ at large distances has the same Yukawa potential
form extracted from a LQCD calculation. The parameters of $v_{cN}$ are
determined by fitting the total cross section data of JLab’s Hall-D by
performing calculations that include $J/\Psi$-N final state interactions, as
required by the unitarity condition. The predicted differential cross sections
$d\sigma/dt$ are in reasonably good agreements with the data from JLab’s
Hall-C. Furthermore, it is shown that the FSI effects dominate the cross
section in the very near-threshold region. This indicates that the low energy
$J/\Psi$-N scattering amplitudes and the associated models of $J/\Psi$-N
potentials $V_{J/\Psi N}(r)$ can be extracted from the $J/\Psi$ photo-
production data with high sensitivity. The determined $V_{J/\Psi N}(r)$ can be
used to understand the nucleon resonances $N^{*}(P_{c})$ reported by the LHCb
collaboration LHCb-15 ; LHCb-16a ; LHCb-19 ; LHCb-21a , to extract the gluonic
distributions in nuclei, $J/\Psi$ production in relativistic heavy-ion
collisions, and to study the existence of nuclei with hidden charms BD88a ;
BSD90 ; GLM00 ; BSFS06 ; WL12 .

By imposing the constraints of $J/\Psi$-N potential extracted from the LQCD
calculation of Ref. KS10b , we have obtained three $J/\Psi$-N potentials which
fit the JLab data equally well. The resulting $J/\Psi$-N scattering lengths
are in the range of $a=(-0.05$ fm $\sim$ $-0.25$ fm), which rule out the
existence of free $J/\Psi$-N bound states. On the other hand, the available
data near threshold is far from sufficient to draw a conclusion. Clearly, more
extensive and precise experimental data are needed to make further progress.

With the determined $c$-N potential $v_{cN}(r)$ and the wavefunctions
generated from the same CQM of Ref. SEFH13 , the constructed dynamical model
has been used to predict the cross sections of photo-production of
$\eta_{c}(1S)$ and $\Psi(2S)$ mesons. It will be interesting to have data from
experiments at JLab and EIC to test our predictions.

The model presented here needs to be improved in the future. Our
parametrization of quark-nucleon potential $v_{cN}$ is guided by the Yukawa
form extracted from a LQCD calculation of Ref. KS10b . This must be improved
by using more advanced LQCD calculations of $J/\Psi$-N scattering, in
particular the short-range part of the potential. A recent LQCD calculation of
$\phi$-N interaction leads to a more complex form than the simple Yukawa form.

It is necessary to improve our model by developing $c\bar{c}$-loop
calculations of Pomeron-exchange mechanism which is needed to fit high energy
data. This requires an extension of our approach to include relativistic
effects. This can be done straightforwardly within the Instant Form of
relativistic Quantum Mechanics of Dirac KP91 , while the Light-front form is
also possible with much more works.

## Acknowledgement

We would like to thank J. Segovia for providing us with the CQM wavefunctions
used in this work. The works of S.S and H.-M.C. were supported by the National
Research Foundation of Korea (NRF) under Grant No. NRF- 2023R1A2C1004098. The
work of T.-S.H.L. was supported by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics, under Contract No.AC02-06CH11357.

## References

  * (1) T.-S. H. Lee, S. Sakinah, Yongseok Oh,  Models of J/$\Psi$ photo-production reactions on the nucleon, Eur. Phys. J.A 58, 12 252 (2023).
  * (2) R. Aaij et al., LHCb Collaboration,  Observation of $J/\psi p$ resonances consistent with pentaquark states in $\Lambda_{b}^{0}\to J/\psi K^{-}p$ decays, Phys. Rev. Lett. 115, 072001 (2015).
  * (3) R. Aaij et al., LHCb Collaboration,  Evidence for exotic hadron contributions to $\Lambda_{b}^{0}\to J/\psi p\pi^{-}$ decays, Phys. Rev. Lett. 117, 082003 (2016).
  * (4) R. Aaij et al.,  Observation of a narrow pentaquark state, $P_{c}(4312)^{+}$, and of the two-peak structure of the $P_{c}(4450)^{+}$, Phys. Rev. Lett. 122, 222001 (2019).
  * (5) R. Aaij et al., LHCb Collaboration,  Evidence for a new structure in the $J/\psi p$ and $J/\psi\bar{p}$ systems in $B_{s}^{0}\to J/\psi p\bar{p}$ decays, Phys. Rev. Lett. 128, 062001 (2022).
  * (6) S. J. Brodsky, G. F. de Teramond,  Spin correlations, QCD color transparency, and heavy-quark thresholds in proton-proton scattering, Phys. Rev. Lett. 60, 1924 (1988).
  * (7) S. J. Brodsky, I. Schmidt, G. F. de Teramond,  Nuclear-bound quarkonium,  Phys. Rev. Lett. 64, 1011 (1990).
  * (8) H. Gao, T.-S. H. Lee, V. Marinov,  $\phi$-$N$ bound state,  Phys. Rev. C 63, 022201(R) (2001).
  * (9) V. B. Belyaev, N. V. Shevchenko, A. Fix, W. Sandhas,  Binding of charmonium with two- and three-body nuclei,  Nucl. Phys. A 780, 100 (2006).
  * (10) J.-J. Wu, T.-S. H. Lee,  Photoproduction of bound states with hidden charm, Phys. Rev. C 86, 065203 (2012).
  * (11) M. E. Peskin,  Short-distance analysis for heavy-quark systems (I). Diagrammatics,  Nucl. Phys. B 156, 365 (1979).
  * (12) G. Bhanot, M. E. Peskin,  Short-distance analysis for heavy-quark systems (II). Applications,  Nucl. Phys. B 156, 391 (1979).
  * (13) M. Luke, A. V. Manohar, M. J. Savage,  A QCD calculation of the interaction of quarkonium with nuclei,  Phys. Lett. B 288, 355 (1992).
  * (14) S. J. Brodsky, G. A. Miller,  Is $J/\psi$-nucleon scattering dominated by the gluonic van der Waals interaction?,  Phys. Lett. B 412, 125 (1997).
  * (15) A. B. Kaidalov, P. E. Volkovitsky,  Heavy-quark interactions with nucleons and nuclei,  Phys. Rev. Lett. 69, 3155 (1992).
  * (16) N. Ishii, S. Aoki, T. Hatsuda,  Nuclear force from lattice QCD,  Phys. Rev. Lett. 99, 022001 (2007).
  * (17) S. Aoki, T. Hatsuda, N. Ishii,  Theoretical foundation of the nuclear force in QCD and its applications to central and tensor forces in quenched lattice QCD simulations,  Prog. Theor. Phys. 123, 89 (2010).
  * (18) T. Kawanai, S. Sasaki,  Charmonium-nucleon potential from lattice QCD,  Phys. Rev. D 82, 091501(R) (2010).
  * (19) T. Kawanai, S. Sasaki,  Charmonium-nucleon interaction from lattice QCD with $2+1$ flavors of dynamical quarks,  AIP Conf. Proc. 1388, 640 (2011).
  * (20) Y.-Z. Xu, S.-Y. Chen, Z.-Q. Yao, D. Binosi, Z.-F. Cui, C. D. Roberts,  Vector-meson production and vector meson dominance,  Eur. Phys. J. C 81, 895 (2021).
  * (21) A. Ali et al., GlueX Collaboration,  First measurement of near-threshold $J/\psi$ exclusive photoproduction off the proton,  Phys. Rev. Lett. 123, 072001 (2019).
  * (22) S. Adhikari et al., GlueX Collaboration,  Measurement of the $J/\psi$ photoproduction cross section over the full near-threshold kinematic region,  Phys. Rev. C 108, 025201 (2023).
  * (23) A. Donnachie, P. V. Landshoff,  Elastic scattering and diffraction dissociation,  Nucl. Phys. B 244, 322 (1984).
  * (24) T. Sato, T.-S. H. Lee,  Meson-exchange model for $\pi N$ scattering and $\gamma N\to\pi N$ reaction,  Phys. Rev. C 54, 2660 (1996).
  * (25) A. Matsuyama, T. Sato, T.-S. H. Lee,  Dynamical coupled-channel model of meson production reactions in the nucleon resonance region,  Phys. Rep. 439, 193 (2007).
  * (26) B. Juliá-Díaz, T.-S. H. Lee, A. Matsuyama, T. Sato,  Dynamical coupled-channels model of $\pi N$ scattering in the $W\leq 2$ GeV nucleon resonance region, Phys. Rev. C 76, 065201 (2007).
  * (27) H. Kamano, S. X. Nakamura, T.-S. H. Lee, T. Sato,  Nucleon resonances within a dynamical coupled-channels model of $\pi N$ and $\gamma N$ reactions,  Phys. Rev. C 88, 035209 (2013).
  * (28) M. Derrick et al., ZEUS Collaboration,  Measurement of the cross section for the reaction $\gamma p\to J/\psi p$ with the ZEUS detector at HERA,  Phys. Lett. B 350, 120 (1995).
  * (29) C. Adloff et al., H1 Collaboration,  Elastic photoproduction of $J/\psi$ and $\Upsilon$ mesons at HERA, Phys. Lett. B 483, 23 (2000).
  * (30) C. D. Roberts and A. G. Williams,  Dyson–Schwinger equations and their application to hadronic physics,  Prog. Part. Nucl. Phys. 33, 477 (1994).
  * (31) C. D. Roberts,  Electromagnetic pion form factor and neutral pion decay width,  Nucl. Phys. A 605, 475 (1996).
  * (32) M. A. Pichowsky, T.-S. H. Lee,  Pomeron-exchange and exclusive electroproduction of $\rho$-mesons in QCD,  Phys. Lett. B 379, 1 (1996).
  * (33) M. A. Pichowsky, T.-S. H. Lee,  Exclusive diffractive processes and the quark substructure of mesons,  Phys. Rev. D 56, 1644 (1997).
  * (34) B. D. Keister and W. N. Polyzou  Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics, Adv. Nucl. Phys. 20, 225 (1991).
  * (35) P. Maris, C. D. Roberts,  $\pi$\- and $K$-meson Bethe-Salpeter amplitudes,  Phys. Rev. C 56, 3369 (1997).
  * (36) L. Chang, Y.-X. Liu, C. D. Roberts,  Dressed-quark anomalous magnetic moments,  Phys. Rev. Lett. 106, 072001 (2011).
  * (37) L. Chang, I. C. Cloët, C. D. Roberts, S. M. Schmidt, P. C. Tandy,  Pion electromagnetic form factor at spacelike momenta,  Phys. Rev. Lett. 111, 141802 (2013).
  * (38) S.-X. Qin, C. D. Roberts,  Impressions of the continuum bound state problem in QCD,  Chinese Phys. Lett. 37, 121201 (2020).
  * (39) Z.-Q. Yao, D. Binosi, Z.-F. Cui, C. D. Roberts,  Semileptonic transitions: $B_{(s)}\to\pi(K)$; $D_{s}\to K$; $D\to\pi,K$; and $K\to\pi$,  Phys. Lett. B 824, 136793 (2022).
  * (40) R. Alkofer, L. von Smekal,  The infrared behavior of QCD Green’s functions. Confinement, dynamical symmetry breaking, and hadrons as relativistic bound states,  Phys. Rep. 353, 281 (2001).
  * (41) H. Sanchis-Alepuz, G. Eichmann, S. Villalba-Chávez, R. Alkofer,  Delta and Omega masses in a three-quark covariant Faddeev approach,  Phys. Rev. D 84, 096003 (2011).
  * (42) G. Eichmann, C. S. Fischer,  Unified description of hadron-photon and hadron-meson scattering in the Dyson-Schwinger approach,  Phys. Rev. D 85, 034015 (2012).
  * (43) J. Segovia, D. R. Entem, F. Fernandez, E. Hernandez,  Constituent quark model description of charmonium phenomenology,  Int. J. Mod. Phys. E 22, 1330026 (2013).
  * (44) S. J. Brodsky, E. Chudakov, P. Hoyer, J. M. Laget,  Photoproduction of charm near threshold,  Phys. Lett. B 498, 23 (2001).
  * (45) Y. Guo, X. Ji, Y. Liu,  QCD analysis of near-threshold photon-proton production of heavy quarkonium,  Phys. Rev. D 103, 096010 (2021).
  * (46) K. A. Mamo, I. Zahed,  Diffractive photoproduction of $J/\psi$ and $\Upsilon$ using holographic QCD: Gravitational form factors and GPD of gluons in the proton,  Phys. Rev. D 101, 086003 (2020).
  * (47) Marvin L. Goldberger and Kenneth M. Watson,  ”Collision Theory”, Robert E. Krieger Pub- lishing Company, Huntington, New York (1975).
  * (48) Michael I. Haftel and Frank Tabakin,  Nuclear Saturation and the smoothness off Nucleon-Nucleon Potential., Nucl. Phys. A 158, 1 (1970).
  * (49) A. Donnachie, P. V. Landshoff,  Total cross sections,  Phys. Lett. B 296, 227 (1992).
  * (50) A. Donnachie, P. V. Landshoff,  Exclusive vector meson production at HERA,  Phys. Lett. B 348, 213 (1995).
  * (51) A. Donnachie, P. V. Landshoff,  Small $x$: two pomerons!,  Phys. Lett. B 437, 408 (1998).
  * (52) T.-S. H. Lee,  Pomeron-LQCD model of $J/\psi$ photo-production on the nucleon, arXiv:2004.13934.
  * (53) J.-J. Wu, T.-S. H. Lee,  Production of $J/\Psi$ on the nucleon and on deuteron targets,  Phys. Rev. C 88, 015205 (2013).
  * (54) Y. Oh, T.-S. H. Lee,  One-loop corrections to $\omega$ photoproduction near threshold,  Phys. Rev. C 66, 045201 (2002).
  * (55) S. Sasaki,  private communications,  (2020).


# Computer Vision Aided Beam Tracking in
A Real-World Millimeter Wave Deployment

Shuaifeng Jiang and Ahmed Alkhateeb
Arizona State University - Emails: {s.jiang<EMAIL_ADDRESS>The authors
are with the School of Electrical, Computer and Energy Engineering, Arizona
State University, Tempe, AZ, 85281 USA (Email: s.jiang, alkhateeb@asu.edu).
This work is supported by the National Science Foundation under Grant No.
2048021.

###### Abstract

Millimeter-wave (mmWave) and terahertz (THz) communications require
beamforming to acquire adequate receive signal-to-noise ratio (SNR). To find
the optimal beam, current beam management solutions perform beam training over
a large number of beams in pre-defined codebooks. The beam training overhead
increases the access latency and can become infeasible for high-mobility
applications. To reduce or even eliminate this beam training overhead, we
propose to utilize the visual data, captured for example by cameras at the
base stations, to guide the beam tracking/refining process. We propose a
machine learning (ML) framework, based on an encoder-decoder architecture,
that can predict the future beams using the previously obtained visual sensing
information. Our proposed approach is evaluated on a large-scale real-world
dataset, where it achieves an accuracy of $64.47\%$ (and a normalized receive
power of $97.66\%$ ) in predicting the future beam. This is achieved while
requiring less than $1\%$ of the beam training overhead of a corresponding
baseline solution that uses a sequence of previous beams to predict the future
one. This high performance and low overhead obtained on the real-world dataset
demonstrate the potential of the proposed vision-aided beam tracking approach
in real-world applications.

###### Index Terms:

beam tracking, vision, sensing, machine learning, DeepSense 6G, real-world
data

##  I Introduction

The millimeter-wave(mmWave) and terahertz (THz) have been considered as key
enabler for the high data rate communication in future wireless networks [1].
The high carrier frequencies provide an order of magnitude more bandwidth
compared with existing wireless communication systems. The move to higher
frequencies, however, brings new challenges such as the higher path-loss. To
overcome that and ensure sufficient receive power, mmWave/sub-THz
communication systems need to deploy large antenna arrays at the
transmitters/receivers and use narrow beams. Nevertheless, obtaining the
optimal narrow beams often requires large beam training overhead, which
occupies wireless resources and decreases spectral efficiency. This becomes
more significant for high-mobility applications such as autonomous vehicles
and vehicle-to-everything (V2X) communications [2], which are considered key
applications for future wireless communication systems [3]. All that motivates
the need to develop novel approaches that can find the optimal beams with low
or negligible beam training overhead.

An important observation is that the use of narrow beams at mmWave/sub-THz
networks and the reliance on line-of-sight (LoS) links give a special
importance to the knowledge of the physical location of the
transmitters/receivers and the geometry of the environment around the
communication systems. This motivates the use of position/environment sensing
devices (such as position sensors, cameras, etc.) at the communication
terminals to guide the different link establishment/resource allocation tasks.
Prior works have studied improving mmWave/THz beam selection, blockage
prediction, and hand-over based on sensing information of different modalities
[4, 5, 6, 7, 8, 9, 10]. In [4] the author proposes that the sub-6 GHz channel
contains adequate information of mmWave channel, therefore it can be used to
predict mmWave beam and blockage status. The vison/camera sensing modality has
been increasingly studied [5, 7, 8]. [5] leverages the camera signals to
proactively predict dynamic link blockages and handoff for mmWave systems. [7]
employs cameras at the mmWave base stations and leverages the visual data to
help overcome the beam training overhead. In [6, 10, 9], other data modalities
such as the radar senory information and the UE position are utilized to guide
the beam selection process. However, the above work focused mainly on the
current beam prediction task instead of the beam tracking which aims to
predict future beams. Beam tracking is particularly important for the sensing-
aided mmWave communicatons systems as the latency of capturing and processing
the sensory data will unlikely enable current beam prediction, but could be
useful for predicting future beams.

In this paper, we propose to utilize visual sensing information to enable fast
and low-overhead mmWave/THz beam tracking. The main contribution can be
summarized as follows.

  * •

We propose a universal problem formulation for the auxiliary data-aided beam
tracking, which could be used for different auxiliary data modalities, such as
leveraging a sequence of beams, or a sequence of RGB images.

  * •

We propose a machine learning (ML) framework for sensing information aided
mmWave/THz beam tracking exploiting an encoder-decoder architecture.

  * •

We evaluate the proposed vision-aided beam tracking on the real-world
DeepSense 6G dataset [11], which comprises co-existing visual and wireless
beam data.

Simulation results demonstrate the capability of the proposed vision-aided
beam tracking approach in achieving high accuracy and receive power. This
highlights the potential gains of incorporating visual sensors (cameras) in
real-world mmWave/THz communication systems.

##  II System and Problem Formulation

In Section II, we first introduce the considered system model for mmWave
communications. Then, we formulate beam tracking into an optimization problem.
After that, we clearly define the vision-aided beam tracking machine learning
task. Lastly, we also present a baseline beam tracking ML task using the
previous optimal beam sequence.

###  II-A System Model

Fig. 1 shows the considered system model for mmWave communications, where the
base station (BS) is serving a mobile user equipment (UE). The BS is equipped
with an antenna array of $N$ elements and an RGB camera (visual data sensor).
Using the antenna array, the BS performs beamforming to achieve adequate
receive power. We assume that the BS has a pre-defined beamforming codebook
${\bm{\mathcal{F}}}=\\{{\mathbf{f}}_{1},\ldots,{\mathbf{f}}_{|{\bm{\mathcal{F}}}|}\\}$
containing $|{\bm{\mathcal{F}}}|$ beams
${\mathbf{f}}_{m}\in{\mathbb{C}}^{N\times 1}$ [12]. For the sake of
simplicity, the UE is assumed to have a single antenna. At time step $t$, the
BS transmits a compelex symbol $s[t]\in{\mathbb{C}}$. We assume the downlink
signal $s[t]$ satiesfies the power constraint
${\mathbb{E}}\left[s^{H}s\right]=P$ with $P$ denoting the transmit power.
Then, the corresponding downlink receive signal $y[t]$ can be written as

$\displaystyle y[t]={\mathbf{h}}^{H}[t]{\mathbf{f}}[t]s[t]+n[t],$ (1)

where ${\mathbf{h}}[t]\in{\mathbb{C}}^{N\times 1}$ denotes the channel between
the BS and the UE at time step $t$. ${\mathbf{f}}[t]\in{\bm{\mathcal{F}}}$ is
the transmit beamforming vector used at the BS at time step $t$. $n[t]$ is the
receive noise which satisfies ${\mathbb{E}}[n[t]n^{H}[t]]=\sigma_{n}^{2}$, and
$\sigma_{n}^{2}$ denotes the receive noise power.

Figure 1: This figure illustrates the considered system model: the BS senses
the environment and the moving UE with an RGB camera. The obtained sensing
information is then utilized for the BS beam management.

###  II-B Problem Formulation

This paper focuses on the beam tracking problem at the base station, which is
defined as follows: Given the available sensing inforamtion up to time $t-1$,
the BS attempts to determine the optimal beams of $\xi\in{\mathbb{Z}}^{+}$
future time steps, that is, optimal beams of $t,\ldots,(t+\xi-1)$. First, we
define the optimal beam at time step $t$ as the one which gives the highest
beamforming gain. The optimal beam at time step $t$ is then represented by

$\displaystyle{\mathbf{f}}^{\star}[t]$
$\displaystyle=\underset{{\mathbf{f}}[t]\in{\bm{\mathcal{F}}}}{\arg\max}\
\big{|}{\mathbf{h}}^{H}[t]{\mathbf{f}}[t]\big{|}^{2}.$ (2)

With this pre-defined codebook constraint, the optimal beam
${\mathbf{f}}^{\star}[t]$ can be uniquely represented by its beam index in the
codebook. The optimal beam index at time step $t$ satisfies

$\displaystyle
p^{\star}[t]=\underset{p[t]\in[1,|{\bm{\mathcal{F}}}|]}{\arg\max}\
\big{|}{\mathbf{h}}^{H}[t]{\mathbf{f}}_{p[t]}\big{|}^{2}.$ (3)

Note that, under the codebook constraint, obtaining the optimal beam is
equivalent to obtaining the optimal beam index. With the definition of the
optimal beam index in (3), we formulate the beam tracking problem as follows:

$\displaystyle\max_{\hat{p}[t]}$
$\displaystyle{\mathbb{P}}\left\\{\hat{p}[t]=p^{\star}[t]\mid\mathcal{O}_{t-\xi}\right\\}$
(4) s.t. $\displaystyle\hat{p}[t]\in[1,|{\bm{\mathcal{F}}}|],$

where $\hat{p}[t]$ is the predicted optimal beam index for time step $t$.
$\mathcal{O}_{t-\xi}$ is any auxiliary obtained before time step $(t-\xi+1)$
that contains the partial information of the optimal beam at time step $t$.

Figure 2: This figure shows the block diagram of the proposed ML framework for
sensing aided beam tracking. The ML framework adopts an encoder-decoder
architecture, and incorporates the feature extraction block, the embedding
block, the RNN block, and the classifier block.

###  II-C vision-aided Beam Tracking

In this paper, We propose exploiting the visual sensing inforamtion to achieve
accurate beam tracking. The BS uses its RGB camera to capture the visual
sensing information of the mobile UE and the surrounding environment. Let
${\mathbf{X}}[t]\in{\mathbb{R}}^{H\times W\times 3}$ denote the visual sensing
information (RGB image) captured at time step $t$. $H$ and $W$ are the height
and width of the captured image, and the last dimension represnts the $3$ RGB
channels. To utilize this visual sensing information for beam tracking, our
objective then becomes obtaining a function which can predict the optimal
future beams starting from time step $t$ based on the visual sensing
information obtained up to time step $t-1$. Let
$X_{t,i}=\left\\{{\mathbf{X}}[t-i+1],\ldots,{\mathbf{X}}[t]\right\\}$ denote a
sequence of visual sensing information with $i$ representing the observation
time window. Then, from (4), the vision-aided beam tracking optimization
problem can be wirtten as

$\displaystyle\max_{\hat{p}[t]}$
$\displaystyle{\mathbb{P}}\left\\{\hat{p}[t]=p^{\star}[t]\mid
X_{t-\xi,i}\right\\}$ (5) s.t.
$\displaystyle\hat{p}[t]\in[1,|{\bm{\mathcal{F}}}|]$

Since the precise joint probability distribution of $p[t]^{\star}$ and
$X_{t-\xi,i}$ is difficult to model, we propose to leverage the powerful
learning capabilitites of ML models to solve (5) in an data-driven approach.
Let $f(;\theta)$ denote an ML model with $\theta$ representing its trainable
parameters. To solve the vision-aided beam tracking in (5), the ML model aims
at predicting the optimal beam index $p^{\star}[t]$ using the side information
$X_{t-\xi,i}$. Therefore the optimal ML model for vision-aided beam tracking
can be mathmatically represented by

$\displaystyle
f_{v}^{\star}(;\theta_{v}^{\star})=\underset{f_{v}(;\theta_{v})}{\arg\max}\
{\mathbb{P}}\left\\{f_{v}\left(X_{t-\xi,i};\theta\right)=p^{\star}[t]\right\\},$
(6)

where $\theta_{v}^{\star}$ is the associated optimal parameters of
$f^{\star}$.

###  II-D Baseline Beam Tracking

The sequence of beams resulting from exhaustive search beam training in the
previous time steps may also carry information of the mobile UE’s future
optimal beam. Therefore, it can be exploited as the side information
$\mathcal{O}_{t-\xi}$ in (4) to predict the future beam (in the beam tracking
problem). In this paper, we employ this approach as a baseline for the beam
tracking task. If this approach is implemented using machine learning, then we
define this ML task as:

$\displaystyle
f_{b}^{\star}(;\theta_{b}^{\star})=\underset{f_{b}(;\theta_{b})}{\arg\max}\
{\mathbb{P}}\left\\{f_{b}\left(F^{\star}_{t-\xi,i};\theta_{b}\right)=p^{\star}[t]\right\\},$
(7)

where
$F^{\star}_{t,i}=\left\\{{\mathbf{f}}^{\star}[t-i+1],\ldots,{\mathbf{f}}^{\star}[t]\right\\}$
denotes the optimal beam sequence from time step $(t-i+1)$ to time step $t$.
$f_{b}^{\star}$ and $\theta_{b}^{\star}$ are the optimal ML model and
trainable parameters associated to this baseline beam tracking task which uses
the previous optimal beam sequence as input.

In Section III, we will explain in detail the proposed ML models for the
vision-aided beam tracking and the baseline beam tracking tasks.

##  III Vision-Aided Beam Tracking

In this section, we first elaborate on the motivation of exploiting sensing to
overcome the difficulty of mmWave communications. Then we propose an ML
framework that solves the vision beam tracking and the baseline beamforming
tasks formulated in Section II.

###  III-A Enabling Highly-Mobile mmWave Systems with Cameras

mmWave/sub-THz communication is considered as a key component of wireless
systems in 5G and beyond. These systems can take advantage of their larger
transmission bandwidth to achieve high data rates. The use of high-frequency
carriers, however, brings new challenges that originated from the propagation
characteristics at these bands. For example, these systems need to use large
antenna arrays and narrow beams to achieve sufficient receive signal power.
Adjusting these beams, however, involves high beam training overhead [13].
This training overhead becomes even more challenging for high-mobility
scenarios such as autonomous vehicles and V2X communications. These are
considered as the key applications of advanced communication systems [3].

Further, the high propagation loss at mmWave/sub-THz makes these systems
highly rely on line-of-sight (LoS) links and dominant non-line-of-sight (NLOS)
propagation paths corresponding to the main reflectors. Therefore, the optimal
beam for mmWave/THz systems is highly dependent on the direction/position of
the transmission targets and the geometry and layout of the surrounding
environment. This motivates leveraging sensors, such as cameras [5, 8, 7], to
obtain awareness about the user location and the surrounding environment.
Using cameras has the following advantages.

  * •

Camera provides rich and fine-grain information on the potential transmission
target and the surroundings.

  * •

Camera does not require any wireless resource in its operation (compared to
radars), and can also work without feedback signals (compared with wireless
communication based sensing).

  * •

Cameras are mature and well-commercialized sensors. They have the advantage in
terms of price, and are also relatively easy for implementations and
deployments.

  * •

It is possible to utilize the advanced models and algorithms developed for
computer vision applications.

Next, we will present the proposed ML framework for the sensing aided beam
tracking task, and explain in detail the DL model proposed for the vision-
aided beam tracking.

###  III-B Developed Deep Learning Model

Fig. 2 shows the block diagram of the proposed ML framework for the sensing-
aided beam tracking task. The ML framework adopts an encoder-decoder
architecture featuring four components: the feature extraction block, the
embedding block, the RNN block, and the classifier block. The encoder
processes the previously obtained information, and passes the information to
the decoder. The decoder predicts the future beams based on the information it
receives from the encoder.

Fearture Extraction Block: The first component of the proposed ML framework is
the feature extraction block which directly processes the raw input data. The
raw input data often contain extra information which does not contribute to
the beam tracking tasks. This irrelevant information can be detrimental since
the ML model can overfit on them while neglecting the useful information.
Moreover, dropping this unnecessary information also results in a smaller
feature space, thus, results in a more stable training process and lower
computational overhead. Therefore, the feature extraction block is designed to
filter out the unwanted information and extract the features that are
informative for the downstream task.

Figure 3: System setup of the DeepSense 6G Secnerio 8. The mmWave BS antenna
arrays opterateing at 60GHz receives the signal transmitted by the moving UE.
A Camera is placed under the BS to obtain sensing information.

To process the raw RGB data for the vision-aided beam tracking task, we detect
the potential transmission target (UE) in the RGB image. We take advantage of
the advanced computer vision ML models and employ the YOLOv4 [14] object
detector. YOLOv4 is a state-of-the-art convolutional neural network (CNN)
based ML model designed to detect thousands of classes of objects from real-
world RGB images. The YOLOv4 object detector can achieve real-time prediction
and high accuracy which is suitable for the vision-aided beam tracking task.
Given an input RGB image, the YOLOv4 object detector predicts a class index
$c\in{\mathbb{Z}}$, a confidence score $s_{c}\in[0,1]$, and a bounding box
vector ${\mathbf{b}}\in{\mathbb{R}}^{4\times 1}$. Note that the bounding box
vector ${\mathbf{b}}=[x_{c},y_{c},w,h]^{T}$ consists of the $x$-center,
$y$-center, width, and height of the detected object in the RGB image. For
more detailed information related to the YOLOv4 model, we refer the readers to
[14]. As discussed in Section III-A, the optimal beam is highly dependent on
the direction/position of the transmission target. Therefore, we exploit the
bounding boxes as the extracted feature for the successive blocks in the
framework.

The feature extraction block is skipped in the baseline approach since the
inputs are the optimal beam indices.

Embedding Block: The embedding layer transforms the input feature into a
different vector space. The embedded vector ideally captures some of the
semantics of the input vector such that semantically similar input vectors
form clusters in the embedding space. For the vision feature (the bounding
boxes) embedding, we employ a fully connected layer which linearly transforms
the bounding box vector ${\mathbf{b}}\in{\mathbb{R}}^{4\times 1}$ into
$\tilde{{\mathbf{b}}}\in{\mathbb{R}}^{E_{v}\times 1}$. To embed the previous
beam indices, we apply the same approach as the natural language processing
(NLP) ML models embed the word token. For the $|{\bm{\mathcal{F}}}|$ beam
indices in the codebook ${\bm{\mathcal{F}}}$, we employ a look-up table of
$|{\bm{\mathcal{F}}}|$ trainable embedding vectors
$\left\\{\tilde{{\mathbf{p}}}_{1},\ldots,\tilde{{\mathbf{p}}}_{|{\bm{\mathcal{F}}}|}\right\\}$.
In the simulations, we set $E_{v}$ and $E_{b}$ to $64$.

RNN Block: The third component of the proposed ML model is the recurrent
neural network (RNN) block. The recurrent neural networks (RNNs) have been
extensively studied for processing sequential signal and data such as the NLP
and speech recognition tasks. Empirical results have shown that RNNs can
effectively capture and process sequential features. Due to the sequential
nature of the beam tracking task, we employ the RNN architecture to process
the sequential input $\mathcal{O}_{t-i}$ and predict the future beams. We
adopt single-layer gated recurrent units (GRUs) with $\xi$ units to process
the sequence of visual information or the previous optimal beam index. The
hidden state size of the GRU is set to 64.

Classifier Block: A fully connected layer is used as the classifier block.
This block predicts the future optimal beam index from the high-level features
obtained by the RNN block. The softmax activation function is applied to this
fully connected layer to output a confidence score vector
$\hat{{\mathbf{s}}}=[s_{1},\ldots,{\mathbf{s}}_{|{\bm{\mathcal{F}}}|}]^{T}$ of
the beam indices in ${\bm{\mathcal{F}}}$. The beam index with the highest
score is predicted as the optimal future beam

$\displaystyle\hat{p}=\underset{p\in[1,|{\bm{\mathcal{F}}}|]}{\arg\max}\
{\mathbf{s}}_{p}.$ (8)

Learning Phase: The encoder of the proposed ML framework processes the
previously obtained information. The input sequence to the encoder is
$X_{t-\xi,i}$ for the vision-aided beam tracking task or $F_{t-\xi,i}$ for the
baseline beam tracking task. Based on the information received from the
encoder, the decoder predicts the future optimal beams. A padding sequence of
$\xi$ zero vectors $Z_{\xi}$ is input to the decoder as a placeholder.

Since the ML framework is designed to solve a classification problem, we
employ the cross-entropy loss. The loss function can be written as

$\displaystyle
J=\sum_{j=t-\xi+1}^{t}\sum_{m=1}^{|{\bm{\mathcal{F}}}|}p^{\star}_{m}[j]\log_{2}\left(\hat{s}_{m}[j]\right),$
(9)

where $p^{\star}_{m}[j]$ is the $m$-th element of the one-hot coded vector of
${\mathbf{p}}^{\star}$ at time step $j$. $\hat{s}_{m}[j]$ is the $m$-th
element of the output vector $\hat{{\mathbf{s}}}[j]$ at time step $j$.

##  IV Experimental Setup

Our proposed vision-aided beam tracking approach and the ML framework are
designed to manage real-world mmWave beam tracking. Therefore, we need a high-
quality real-world dataset consisting of co-existing RGB images and beam data
to evaluate our proposed approach. In this paper, we adopt the DeepSense 6G
dataset [11] in our simulation and performance evaluation. The DeepSense 6G is
a multi-modal dataset comprising real-world measurements. The DeepSence 6G
dataset incorporates co-existing including wireless beam data, visual sensing
data, among other modalities.

###  IV-A DeepSense 6G Secnerio 8

We adopt Scenario 8 of the DeepSense 6G dataset for our simulation. The system
setup of Scenario 8 is shown by Fig. 3. In Scenario 8, a fixed BS receives
signals from a moving UE. The BS is equipped with a square uniform planar
array (UPA) of 64 ($8\times 8$) elements and an RGB camera installed under the
UPA. The BS adopted a beamforming codebook consisting of $64$ pre-defined
beams. The codebook is horizontal-only and $8$-time oversampled. The UE is a
moving vehicle equipped with the same UPA and transmits mmWave signal on the
60 GHz band using an omni-directional beam. During the data collection
process, the UE passes by the BS multiple times. At each time step, the BS
measures the receive power of all beams in the codebook by beam sweeping, and
captures the UE with the RGB camera.

Figure 4: This figure compares the accuracy and normalized receive power of
the top-$k$ predictions of future beam 1 and 3. For both the vision-aided and
baseline beam tracking, the accuracy increases as $k$ increases. However, the
normalized receive power saturates to the optimum even when $k=1$.

###  IV-B Devolop Dataset Generation

We evaluated our proposed vision-aided beam tracking approach on the official
development dataset split of DeepSense 6G Scenario 8. The training dataset and
the validation dataset consist of $70\%$ and $20\%$ of the raw dataset. The
DeepSence 6G Scenario 8 dataset consists of multiple data sequences. In a data
sequence, the vehicle completes its path passing by the BS for one time. Each
data sequence consists of co-existing RGB images and beam receive powers of
multiple time steps. We follow the official development dataset split of
DeepSense 6G Scenario 8. The training dataset and the validation dataset
consist of $70\%$ and $20\%$ of the raw dataset data sequences. Note that the
data in two datasets come from different vehicle passes to assure there is no
data leakage. For each data sequence, we break it into data samples using a
sliding window size of $13$. One data sample consists of $13$ time steps and
can be written as
$\left\\{({\mathbf{X}}[1],p^{\star}[1]),\ldots,({\mathbf{X}}[13],p^{\star}[13])\right\\}$.

In the training process, we use an observation window size $i=8$, and we train
the models to predict the future beam $1\sim 5$ ($\xi\in[1,5]$). Therefore,
the input to the encoder is
$\left\\{{\mathbf{X}}[1],\ldots,{\mathbf{X}}[8]\right\\}$ for the vision-aided
beam trakcing model, and $\left\\{p^{\star}[1],\ldots,p^{\star}[8]\right\\}$
for the baseline beam trakcing model. In both two beam tracking approaches,
the decoder is expected to output
$\left\\{\hat{{\mathbf{s}}}[9],\ldots,\hat{{\mathbf{s}}}[13]\right\\}$.

##  V Evaluation Results

In this section, we evaluate the proposed vision-aided beam tracking approach
and compares its performance with the baseline beam tracking’s performance.
The metrics adopted in the evaluation are the following.

  * •

Top-$k$ accuracy: the percentage of the time steps of all validation samples
where the beam corresponding to the top-$k$ confidence scores include the
optimal beam.

  * •

Normalized receive power: the ratio between the highest receive power achieved
by the top-$k$ predicted beams and the receive power of the optimal beam. This
metric is averaged over all time steps and all validation samples.

###  V-A Do the ML Models Learn to Predict Future Beams?

Figure 5: This fugure compares the training overhead required in nine times
steps of the vision-aided and the baseline approaches. The vision-aided
approach only consumes $1\%$ beam training overhead compared to the baseline.

In Fig. 4, we present the top-$k$ accuracy and normalized receive power of the
future beam 1 and future beam 3. The top-$k$ accuracy improves significantly
as $k$ increases for both the vision-aided and baseline beam tracking
approaches. It can also be observed that accuracy decreases when predicting
beams in the further future. The top-$5$ accuracy of the vision-aided beam
tracking for future beam 1 is $99.37\%$. This accuracy implies that the
proposed vision beam tracking approach can find the optimal future beam 1 with
$99.37\%$ probability by testing $5$ beams in the beam sweeping process.
However, in terms of receive power, testing more beams (increasing $k$) may
not be worth the price of the training overhead. For the vision-aided beam
training, $\bf 97.66\%$ receive power can already be obtained for future beam
1 even without any beam training ($k=1$), leaving little room for testing 5
beams for improvement. Despite the relatively low top-$1$ accuracy obtained by
both beam tracking approaches, the near-optimal receive power highlights that
the ML models effectively learn to predict future beams, and most of the
mistakes occur at the sub-optimal beam with near-optimal receive power.
Overall, the vision-aided beam tracking approach can achieve comparable
performance to the baseline approach in terms of the two metrics. Note that
the baseline beam tracking approach is a strong baseline since it inputs the
optimal beams of the $8$ previous time steps. This highlights the capability
of the proposed vision-aided approach in accurate beam tracking. It is worth
mentioning that the adopted DeepSense 6G scenario 8 mainly consists of LoS
data. However, the baseline beam tracking approach could be more sensitive to
NLoS scenarios since the previous optimal beam may not contain enough
information on the blockages, reflectors, and scatterers. The baseline beam
tracking is also expected to further degrade when the surrounding environment
becomes more dynamic. On the contrary, The visual data obtained by the camera
captures rich information on the surrounding object and the dynamics.
Moreover, the baseline beam tracking approach requires information on the
optimal previous beams, which may not always be applicable in practice. The
baseline model can instead rely on the beams it previously predicted. This may
cause the baseline approach to deviate more from the optimal beam as the beam
tracking goes on without calibration. The vision-aided approach, however,
keeps capturing the latest information on the environment. Therefore, it is
not likely to suffer from this deviation.

###  V-B Vision-aided vs. Baseline: Beam Training Overhead

In Section V-A, we analyzed the accuracy and received power performance of the
two beam tracking approaches. However, in our simulation, the baseline
approach requires knowing the previous eight optimal beams to predict the
optimal beam of the future time step and match the performance of the vision-
aided approach. Therefore, Fig. 5 studies the beam training overhead required
by the vision-aided and the baseline approaches in these nine time steps when
top-$k$ beams are predicted. It is assumed that both approaches will conduct
beam sweeping over the predicted top-$k$ beams at the future time step when
$k\geq 2$. For the 5 cases shown in Fig. 5, the beam training overhead
required by the vision-aided approach is less than $1\%$ of the baseline
approach. Furthermore, when top-$1$ beam is predicted the vision-aided
approach completely eliminates the beam training overhead.

###  V-C What is the effect of the Beamforming Codebook Size

In this section, we study the effect of the codebook size on the performance
of the vision-aided and baseline beam tracking approaches. Fig. 6 shows the
top-1 accuracy and normalized receive power of the future beam 1. It can be
seen that, as codebook size increases, the top-$k$ accuracies of both beam
tracking approaches decrease as can be expected. However, the normalized
receive power increases as a larger beam codebook size is adopted. Using the
vision-aided beam tracking approach with 16 pre-defined beams, the normalized
receive power of the top-$1$ prediction is $92.17\%$ for future beam 1.
Exploiting the codebook with $64$ beams, a normalized receive power of
$97.66\%$ can be achieved, which is a $6\%$ relative improvement. This
demonstrates that reasonable receive power improvement can be achieved by
using oversampling codebook at a price of slightly more computational
complexity.

Figure 6: This figure shows the accuracy and receive power performance of the
vision-aided and baseline beam tracking approaches with different codebook
sizes of 16, 32, and 64.

##  VI Conclusion

This paper proposes a machine learning based vision-aided beam tracking
framework. Exploiting this framework, we also develop an efficient baseline
beam tracking approach that utilizes the previous optimal beams. The proposed
approaches are evaluated using a large-scale real-world dataset comprising co-
existing visual and wireless mmWave data. Evaluation results demonstrate that
the proposed vision-aided beam tracking approach can learn to accurately
predict future beams and achieve comparable performance to the baseline
solution. It achieves a top-1 accuracy of $64.47\%$ and a top-5 accuracy of
$98.95\%$ in predicting the future beam. The robustness of the proposed
vision-aided beam tracking is illustrated by the $97.66\%$ normalized receive
power of the top-$1$ prediction. Moreover, the proposed vision-aided beam
tracking only requires $1\%$ of the beam tracking overhead of the baseline
approach. These results highlight the potential of leveraging visual sensors
in improving mmWave communications.

## References

  * [1] T. S. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal, A. Alkhateeb, and G. C. Trichopoulos, “Wireless communications and applications above 100 ghz: Opportunities and challenges for 6g and beyond,” _IEEE access_ , vol. 7, pp. 78 729–78 757, 2019.
  * [2] Z. MacHardy, A. Khan, K. Obana, and S. Iwashina, “V2x access technologies: Regulation, research, and remaining challenges,” _IEEE Communications Surveys Tutorials_ , vol. 20, no. 3, pp. 1858–1877, 2018.
  * [3] J. He, K. Yang, and H.-H. Chen, “6g cellular networks and connected autonomous vehicles,” _IEEE Network_ , vol. 35, no. 4, pp. 255–261, 2021.
  * [4] M. Alrabeiah and A. Alkhateeb, “Deep learning for mmwave beam and blockage prediction using sub-6 ghz channels,” _IEEE Transactions on Communications_ , vol. 68, no. 9, pp. 5504–5518, 2020.
  * [5] G. Charan, M. Alrabeiah, and A. Alkhateeb, “Vision-aided 6G wireless communications: Blockage prediction and proactive handoff,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 10, pp. 10 193–10 208, 2021\.
  * [6] G. Charan, T. Osman, A. Hredzak, N. Thawdar, and A. Alkhateeb, “Vision-position multi-modal beam prediction using real millimeter wave datasets,” _arXiv preprint arXiv:2111.07574_ , 2021.
  * [7] M. Alrabeiah, A. Hredzak, and A. Alkhateeb, “Millimeter wave base stations with cameras: Vision-aided beam and blockage prediction,” in _2020 IEEE 91st Vehicular Technology Conference (VTC2020-Spring)_ , 2020, pp. 1–5.
  * [8] M. Alrabeiah, U. Demirhan, A. Hredzak, and A. Alkhateeb, “Vision aided URLL communications: Proactive service identification and coexistence,” in _2020 54th Asilomar Conference on Signals, Systems, and Computers_ , 2020, pp. 174–178.
  * [9] U. Demirhan and A. Alkhateeb, “Radar aided 6G beam prediction: Deep learning algorithms and real-world demonstration,” _arXiv preprint arXiv:2111.09676_ , 2021.
  * [10] T.-H. Chou, N. Michelusi, D. J. Love, and J. V. Krogmeier, “Fast position-aided mimo beam training via noisy tensor completion,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 15, no. 3, pp. 774–788, 2021.
  * [11] A. Alkhateeb, G. Charan, M. Alrabeiah, T. Osman, A. Hredzak, N. Srinivas, and M. Seth, “DeepSense 6G: A large-scale real-world multi-modal sensing and communication dataset,” _available on arXiv_ , 2021. [Online]. Available: https://www.DeepSense6G.net
  * [12] Y. Zhang, M. Alrabeiah, and A. Alkhateeb, “Reinforcement learning of beam codebooks in millimeter wave and terahertz MIMO systems,” _IEEE Transactions on Communications_ , pp. 1–1, 2021.
  * [13] 3GPP TR 38.802 version 14.1.0 Release 14, “Study on New Radio Access Technology—Physical Layer Aspects,” Tech. Rep., Jun 2017. [Online]. Available: http://ftp.3gpp.org
  * [14] “YOLOv4: Optimal speed and accuracy of object detection, author=Bochkovskiy, Alexey and Wang, Chien-Yao and Liao, Hong-Yuan Mark, journal=arXiv preprint arXiv:2004.10934, year=2020.”