# Study on the $\phi$-meson photoproduction off the proton target
with the pentaquark-like $K^{*}\Sigma$ bound state $P_{s}$

Sang-in Shim  0000-0003-3161-1018<EMAIL_ADDRESS>Center for
Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Korea
Department of Physics, Pukyong National University (PKNU), Busan 48513, Korea
Yongsun Kim  0000-0002-9025-0489<EMAIL_ADDRESS>Department of Physics,
Sejong University, Seoul 05006, Korea  Center for Extreme Nuclear Matters
(CENuM), Korea University, Seoul 02841, Korea  Seung-il Nam
0000-0001-9603-9775<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics, Pukyong National University (PKNU), Busan 48513, Korea  Center for
Extreme Nuclear Matters (CENuM), Korea University, Seoul 02841, Korea  Asia
Pacific Center for Theoretical Physics (APCTP), Pohang 37673, Korea

###### Abstract

We utilize the effective Lagrangian method within the tree-level Born
approximation to explore $\phi$-meson photoproduction, i.e., $\gamma p\to\phi
p$. Our analysis encompasses contributions from various sources, including the
Pomeron, $f_{1}$-Regge, pseudoscalar particles ($\pi$, $\eta$), scalar
particles ($a_{0}$, $f_{0}$), protons, and three-nucleon resonance states. In
addition, we consider a possible pentaquark-like $K^{*}\Sigma$-bound state
$P_{s}$. The findings indicate that, apart from the region near the threshold,
contributions other than the Pomeron generally have a limited impact on the
total cross section. However, at specific angles, alternative contributions
become crucial, particularly at smaller values of ${\rm cos}\,\theta$. The
incorporation of $P_{s}$ and other nucleon resonances proves essential to
elucidate the bump observed near $W\sim 2.15$ GeV at very forward angles and
behaviors within the range of $W=(2.0-2.3)$ GeV. Furthermore, in the region
with $W\geq 2.5$ GeV, where nucleon resonances become negligible,
contributions from the t-channel mesons become pivotal. Our calculations for
spin density matrix components, examined in various frames, exhibit
improvement when considering all contributions. This comprehensive approach
successfully reproduces the observed bump by including $P_{s}$. We also
briefly estimate the $P_{s}$ production via $\phi$-meson photoproduction in
the future Electron-Ion Collider (EIC), resulting in the luminosity of 10 fb-1
per month.

$\phi$-meson photoproduction, effective Lagrangian method, Born approximation,
Regge approach, $P_{s}$ pentaquark-like state, spin-density matrix element
(SDME), Electron-Ion Collider (EIC), luminosity.

††preprint: PKNU-NuHaTh-2023

##  I Introduction

Research on $\phi$-meson photoproduction has been extensive, encompassing both
theoretical investigations and experimental inquiries. This phenomenon serves
as a valuable tool for gaining insights into the properties of the strong
interaction and structure of hadrons. Early theoretical explorations of
$\phi$-meson photoproduction focused on the Pomeron exchange mechanism, a non-
resonant process primarily driven by the exchange of colorless gluons. While
this mechanism is believed to account for the overall energy dependence of the
cross section, it falls short of explaining the observed bump structure near
the threshold. In 2005, the LEPS Collaboration made a significant observation,
reporting a distinctive bump structure in the total cross section for
$\phi$-meson photoproduction near the threshold LEPS:2005hax .

Despite consistent observations of this structure in subsequent experiments
CLAS:2013jlg ; Dey:2014tfa  and theoretical studies Ozaki:2009mj ;
Kiswandhi:2010ub ; Kiswandhi:2011cq ; Ryu:2012tw , its origin remains a
perplexing question. A clue to this puzzle may be found in exotic hadrons.
After the recent observation of the pentaquarks $P_{c}(4312)$, $P_{c}(4440)$,
$P_{c}(4450)$ at the Large Hadron Collider (LHC) LHCb:2019kea , additional
theoretical and experimental investigations are underway. Among them, there
are investigations into $J/\psi$-photoproduction involving the consideration
of $P_{c}(4450)$ with a mass similar to the threshold of $D^{*}\Sigma_{c}$
Wang:2015jsa ; HillerBlin:2016odx ; Wang:2019krd ; GlueX:2019mkq .
Substituting charm contents with strange ones in this reaction leads to
$\phi$-photoproduction, considering pentaquarks with a mass similar to the
$K^{*}\Sigma$ threshold. Theoretical studies on $\phi$-photoproduction,
including nucleon resonances with masses similar to the $K^{*}\Sigma$
threshold and having the same spin-parity as their bound state, have been
conducted Kiswandhi:2010ub ; Kiswandhi:2011cq . However, in these works, the
couplings between the resonance states, photon, and $\phi$-meson, as well as
the widths of the resonances, are treated as free parameters. The
$K^{*}\Sigma$ molecular bound-state was already suggested using the coupled-
channel method including the mass, coupling with proton and $\phi$-meson, and
the width Khemchandani:2011et . Based on this, an investigation on hidden-
strangeness production, such as $K^{+}p\to K^{+}\phi p$, has also been
conducted Nam:2021ayk .

In the present work, we investigate the role of the $K^{*}\Sigma$-bound state
(we denote it as $P_{s}$) in the $\phi$-meson photoproduction. $P_{s}$ is a
poorly known state with $S=0$ and $J^{P}=3/2^{-}$ suggested by the unitarized
coupled-channel meson-baryon interaction model Khemchandani:2011et . If proven
to exist this structure would play a role in various hadronic reactions,
including the process of $\phi$-meson photoproduction. To explain the total
(Refs. Ballam:1972eq ; Barber:1981fj ; Egloff:1979mg ; Busenitz:1989gq ) and
differential (Ref. Dey:2014tfa ) cross section of $\phi$-meson
photoproduction, we employ a comprehensive theoretical framework that
considers contributions from various components, including the Pomeron
exchange, t-channel meson exchanges, and the direct $\phi$-radiation via the
proton and nucleon resonances including $P_{s}$. We determine the relevant
parameters by fitting the model to the experimental data. In addition to the
total and differential cross sections, our analysis also discusses the spin-
density matrix elements (SDMEs).

The numerical results indicate that contributions aside from the Pomeron are
typically insignificant regarding the total cross section, except especially
near the threshold. As energy levels increase, contributions other than the
Pomeron and $f_{1}$ Regge are considered insignificant. However, when
examining production reactions at specific angles rather than the total cross
section, relying solely on the Pomeron contribution proves insufficient.
Hence, it becomes essential to analyze additional contributions. We calculate
$d\sigma/d{\rm cos}\,\theta$ to evaluate various angles and center-of-mass
(cm) energies, comparing them with CLAS data. It is clear that the Pomeron
contribution adequately explains the experimental data in the forward-
scattering region. However, as ${\rm cos}\,\theta$ increases, the deviation
from the data increases, indicating the need for contributions other than the
Pomeron, especially at larger ${\rm cos}\,\theta$ values. Importantly,
including $P_{s}$ and other nucleon resonances is crucial for accurately
describing the bump observed near $W\sim 2.15$ GeV at very forward angles, as
well as for capturing behaviors within the range of $(2.0-2.3)$ GeV. Here, $W$
denotes the total energy in the center-of-mass (cm) frame. Additionally, in
the region for $W\geq 2.5$ GeV, where nucleon resonances are scarce,
contributions from $t$-channel mesons, separate from the Pomeron, become
necessary.

We also compute the SDME $\rho^{0}_{00}$ at ${\rm cos}\,\theta=0.7$ in the
Gottfried-Jackson, Adair, and Helicity frames, and compare them with CLAS
data. When considering only the Pomeron, the results consistently
underestimate the data across all frames. However, by including all
contributions, this disparity is significantly reduced, resulting in a better
alignment between the data and the model. An important achievement of this
study is the successful replication of the observed bump at $W\sim 2.15$ GeV
by incorporating $P_{s}$. Our findings suggest that the upcoming Electron-Ion
Collider (EIC) experiment will facilitate precise measurements of the $P_{s}$
resonance. We estimate the $P_{s}$ production via $\phi$-meson photoproduction
in the EIC.

The remainder of this paper is organized as follows. In Section II, we outline
the framework and provide numerical inputs for the reaction. Section III is
devoted to comparing our numerical results with existing data and discussing
them, while the final section offers a summary and conclusion.

##  II Theoretical Framework

In this Section, we would like to describe the theoretical framework briefly
for the $\phi$-photoproduction:

$\gamma(k_{1})+p(p_{1})\rightarrow\phi(k_{2})+p(p_{2}).$ (1)

In the cm frame, the momenta of the particles can be defined as follows:

$\displaystyle k_{1}$ $\displaystyle=$ $\displaystyle(k,0,0,k),$ (2)
$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle(\sqrt{k^{2}+M_{N}^{2}},0,0,-k),$ (3) $\displaystyle k_{2}$
$\displaystyle=$ $\displaystyle(\sqrt{k^{\prime
2}+M_{\phi}^{2}},k^{\prime}{\rm sin}\,\theta,0,k^{\prime}{\rm cos}\,\theta),$
(4) $\displaystyle p_{2}$ $\displaystyle=$ $\displaystyle(\sqrt{k^{\prime
2}+M_{N}^{2}},-k^{\prime}{\rm sin}\,\theta,0,-k^{\prime}{\rm cos}\,\theta).$
(5)

Here, the $z$-axis is set to be parallel to the direction of the photon and
the $y$-axis is normal to the reaction plane. The masses of the proton and the
$\phi$-meson are denoted by $M_{N}$ and $M_{\phi}$, respectively. The angle
between the directions of the initial photon and the final $\phi$-meson is
given by $\theta$. The magnitudes of the initial and final three momenta $k$
and $k^{\prime}$ are defined as

$\displaystyle k$ $\displaystyle=$ $\displaystyle(W^{2}-M_{N}^{2})/(2W),$ (6)
$\displaystyle k^{\prime}$ $\displaystyle=$
$\displaystyle\sqrt{(M_{\phi}^{2})^{2}+(M_{N}^{2})^{2}+(W^{2})^{2}-2(M_{\phi}^{2}M_{N}^{2}+M_{\phi}^{2}W^{2}+M_{N}^{2}W^{2})}/(2W),$
(7)

where $W$ is the total energy of the system in the cm frame.

The invariant amplitude can be written as

$\mathcal{M}=\varepsilon_{\nu}^{*}(k_{2},\lambda^{\prime})\bar{u}(p_{2},s^{\prime})\mathcal{M}^{\mu\nu}(k_{1},p_{1},k_{2},p_{2})u(p_{1},s)\epsilon_{\mu}(k_{1},\lambda)$
(8)

where $\epsilon_{\mu}(k_{1},\lambda)$ and
$\varepsilon_{\nu}^{*}(k_{2},\lambda^{\prime})$ are the polarization vector of
the photon and $\phi$-meson, respectively. $u(p_{1},s)$ and
$u(p_{2},s^{\prime})$ are the spinors of the initial- and final-state protons,
respectively.

The invariant amplitude of Eq. (8) consists of the contributions from
$t$-channel Pomeron exchange, $t$-channel pseudo-scalar ($\pi$, $\eta$),
scalar ($a_{0}(980)$, $f_{0}(980)$), axial-vector ($f_{1}(1285)$) meson
exchange, $s$\- and $u$-channel nucleon $N$, and nucleon resonance $N^{*}$
exchange as follows:

$\mathcal{M}=\mathcal{M}_{\mathbb{P}}+\mathcal{M}_{\rm PS}+\mathcal{M}_{\rm
S}+\mathcal{M}_{\rm AV}+\mathcal{M}_{N}+\mathcal{M}_{N^{*}}$ (9)

In Fig. 1, Feynman diagrams for each contribution are drawn. For the nucleon
resonance $N^{*}$, we consider only three resonace states,
$N^{*}(2000,5/2^{+})$, $N^{*}(2300,1/2^{+})$, and the pentaquark-like
$K^{*}\Sigma$-bound state $P_{s}$, which is assigned by $J^{P}=3/2^{-}$ from
the coupled-channel $V$-$B$ interaction model, in which $V$ and $B$ denote the
vector meson and baryon, respectively Khemchandani:2011et .

Figure 1: Relevant contributions from (a) $t$-channel Pomeron exchange, (b)
$t$-channel pseudo-scalar ($\pi$, $\eta$), scalar ($a_{0}(980)$,
$f_{0}(980)$), axial-vector ($f_{1}(1285)$) meson exchange, (c, d) $s$\- and
$u$-channel nucleon $N$, and nucleon resonance $N^{*}$ exchange

###  II.1 Pomeron exchange

In what follows, we introduce the effective Lagrangians of the various
contributions for the $\phi$ photoproduction. First, for the calculation of
Pomeron-exchange contribution, we consider the $C=+1$ vector-like Pomeron
($\mathbb{P}$), following Refs. Titov:1999eu ; Titov:2003bk ; Kim:2019kef ;
Kim:2020wrd . Its amplitude is depicted in Fig. 1(a). Taking into account the
VMD model, we can construct the following effective Lagrangians:

$\displaystyle\mathcal{L}_{\gamma\mathbb{P}\phi}$ $\displaystyle=$
$\displaystyle
ig_{\gamma\mathbb{P}\phi}F_{\phi}(t)\left[(\phi_{\nu}\partial^{\mu}\mathbb{P}^{\nu}-\mathbb{P}_{\nu}\partial^{\mu}\phi^{\nu})A_{\mu}-(A_{\nu}\partial^{\mu}\mathbb{P}^{\nu}-\mathbb{P}_{\nu}\partial^{\mu}A^{\nu})\phi_{\mu}-(\phi_{\nu}\partial^{\mu}A^{\nu}-A_{\nu}\partial^{\mu}\phi^{\nu})\mathbb{P}_{\mu}\right],$
(10) $\displaystyle\mathcal{L}_{\mathbb{P}NN}$ $\displaystyle=$ $\displaystyle
g_{\mathbb{P}NN}F_{N}(t)\bar{N}\hbox to0.0pt{/\hss}{\mathbb{P}}N,$ (11)

where $N$, $\mathbb{P}^{\mu}$, $A^{\mu}$, and $\phi^{\mu}$. The invariant
amplitude derived from Eq. (11) can be written as

$\displaystyle\mathcal{M}_{\mathbb{P}}$ $\displaystyle=$
$\displaystyle\mathcal{\varepsilon^{*}_{\nu}(\lambda^{\prime})}\bar{u}_{N}(\lambda_{f})\mathcal{M}^{\mu\nu}_{\mathbb{P}}u_{N}(\lambda_{i})\epsilon_{\mu}(\lambda),$
(12) $\displaystyle\mathcal{M}_{\mathbb{P}}^{\mu\nu}$ $\displaystyle=$
$\displaystyle-
M_{\mathbb{P}}(s,t)\left[\left(g^{\mu\nu}-\frac{k_{2}^{\mu}k_{2}^{\nu}}{k_{2}^{2}}\right)\hbox
to0.0pt{/\hss}{k}_{1}-\left(k_{1}^{\nu}-\frac{k_{2}^{\nu}k_{1}\cdot
k_{2}}{k_{2}^{2}}\right)\gamma^{\mu}-\left(\gamma^{\nu}-\frac{\hbox
to0.0pt{/\hss}{k}_{2}k_{2}^{\nu}}{k_{2}^{2}}\right)\left(k_{2}^{\mu}-\frac{\bar{p}^{\mu}k_{1}\cdot
k_{2}}{\bar{p}\cdot k_{1}}\right)\right].$ (13)

Here, $\bar{p}=(p_{1}+p_{2})/2$. The last term $\propto\bar{p}$ in the square
bracket is added to restore the gauge invariance Titov:2003bk . The scalar
factor $M_{\mathbb{P}}(s,t)$ in Eq. (13) is given by

$M_{\mathbb{P}}(s,t)=C_{\mathbb{P}}F_{\phi}(t)F_{N}(t)\frac{1}{s}\left(\frac{s}{s_{\mathbb{P}}}\right)^{\alpha_{\mathbb{P}}(t)}\exp\left[-\frac{i\pi}{2}\alpha_{\mathbb{P}}(t)\right].$
(14)

Here, we define the strength factor
$C_{\mathbb{P}}=g_{\gamma\mathbb{P}\phi}g_{\mathbb{P}NN}=3.6$ Titov:2003bk ;
Kim:2020wrd  for convenience and the energy-scale factor
$s_{\mathbb{P}}=(M_{N}+M_{\phi})^{2}$. The Pomeron trajectory is known to be
$\alpha_{\mathbb{P}}(t)=1.08+0.25t$. The form factor for the
$\gamma\mathbb{P}\phi$ vertex $F_{\phi}(t)$ Jaroszkiewicz:1974ep ;
Donnachie:1983hf  and the form factor for the nucleon $F_{N}(t)$ Laget:1994ba
; Donnachie:1988nj  take the forms

$\displaystyle F_{\phi}(t)$ $\displaystyle=$
$\displaystyle\frac{2\mu_{0}^{2}\Lambda_{\phi}^{2}}{(\Lambda_{\phi}^{2}-t)(2\mu_{0}^{2}+\Lambda_{\phi}^{2}-t)},$
(15) $\displaystyle F_{N}(t)$ $\displaystyle=$
$\displaystyle\frac{4M_{N}^{2}-2.8t}{(4M_{N}^{2}-t)(1-t/t_{0})^{2}}.$ (16)

where the scale parameters and the cutoff are given by $\mu_{0}^{2}=1.1\,{\rm
GeV}^{2}$, $t_{0}=0.71\,{\rm GeV}^{2}$, and $\Lambda_{\phi}=4M_{\phi}^{2}$.

###  II.2 Meson exchange

For the meson-exchange contributions, shown in Fig. 1(b), we consider the
Reggeized axial-vector meson ($f_{1}$-Regge) related to $f_{1}(1285)$ which
has the quantum number $I^{G}(J^{PC})=0^{+}(1^{++})$ Donnachie:2002en ;
Kochelev:1999zf , pseudoscalar mesons, and scalar mesons. The effective
Lagrangians for the mesonic vertices are written as follows:

$\displaystyle\mathcal{L}_{f_{1}\gamma\phi}$ $\displaystyle=$ $\displaystyle
g_{f_{1}\gamma\phi}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}A_{\nu}\partial^{2}\phi_{\alpha}f_{1\beta},$
(17) $\displaystyle\mathcal{L}_{\Phi\gamma\phi}$ $\displaystyle=$
$\displaystyle\frac{eg_{\Phi\gamma\phi}}{M_{\phi}}\epsilon^{\mu\nu\alpha\beta}\partial_{\mu}A_{\nu}\partial_{\alpha}\phi_{\beta}\Phi,$
(18) $\displaystyle\mathcal{L}_{S\gamma\phi}$ $\displaystyle=$
$\displaystyle\frac{eg_{S\gamma\phi}}{M_{\phi}}F^{\mu\nu}\phi_{\mu\nu}S,$ (19)

and, for the nucleon vertices,

$\displaystyle\mathcal{L}_{f_{1}NN}$ $\displaystyle=$ $\displaystyle-
g_{f_{1}NN}\bar{N}\left[\gamma_{\mu}-i\frac{\kappa_{f_{1}NN}}{2M_{N}}\gamma_{\nu}\gamma_{\mu}\partial^{\nu}\right]f^{\mu}_{1}\gamma_{5}N,$
(20) $\displaystyle\mathcal{L}_{\Phi NN}$ $\displaystyle=$ $\displaystyle
ig_{\Phi NN}\bar{N}\Phi\gamma_{5}N,$ (21) $\displaystyle\mathcal{L}_{SNN}$
$\displaystyle=$ $\displaystyle-g_{SNN}\bar{N}SN,$ (22)

where $f_{1}$, $P$, and $S$ stand $f_{1}(1285)$, a pseudosclar meson
($\pi(135,0^{-})$, $\eta(548,0^{-})$), and a scalar meson ($a_{0}(980,0^{+})$,
$f_{0}(980,0^{+})$), and their fields, respectively. The coupling constant
$g_{{\rm M}\gamma\phi}$ can be calculated from the meson decays as follows.

$\displaystyle\Gamma_{f_{1}\rightarrow\gamma\phi}$ $\displaystyle=$
$\displaystyle\frac{k_{\gamma}^{3}}{12\pi}\frac{M_{\phi}^{2}}{M_{f_{1}}^{2}}(M_{f_{1}}^{2}+M_{\phi}^{2})g_{f_{1}\gamma\phi}^{2},$
(23) $\displaystyle\Gamma_{\phi\rightarrow\gamma\Phi}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{3}\frac{q_{\gamma}^{3}}{M_{\phi}^{2}}g_{\Phi\gamma\phi}^{2},$
(24) $\displaystyle\Gamma_{\phi\rightarrow\gamma S}$ $\displaystyle=$
$\displaystyle\frac{4\alpha}{3}\frac{q_{\gamma}^{3}}{M_{\phi}^{2}}g_{S\gamma\phi}^{2}.$
(25)

Here, $\alpha=e^{2}/(4\pi)$ and
$q_{\gamma}=(M_{\phi}^{2}-M_{\Phi,S}^{2})/(2M_{\phi})$. By using the values
given in the Particle Data Group (PDG) for the branching ratios
ParticleDataGroup:2018ovx , the coupling constants become

$g_{f_{1}\gamma\phi}=0.17\,{\rm
GeV}^{-2},\,\,\,\,g_{\pi\gamma\phi}=-0.14,\,\,\,\,g_{\eta\gamma\phi}=-0.71,\,\,\,\,g_{a_{0}\gamma\phi}=-0.77,\,\,\,\,g_{f_{0}\gamma\phi}=-2.44.$
(26)

For the nucleon vertices, from $g_{f_{1}NN}=2.5\pm 0.5$ suggested in Ref.
Birkel:1995ct  and the values obtained using the Nijmegen potentials
Stoks:1999bz ; Rijken:1998yy , we use following coupling constants.

$g_{f_{1}NN}=3.0,\,\,\,\,g_{\pi NN}=3.0,\,\,\,\,g_{\eta
NN}=6.34,\,\,\,\,g_{a_{0}NN}=4.95,\,\,\,\,g_{f_{0}NN}=-0.51.$ (27)

Form factors for both the mesonic and the nucleon vertices are given in the
same form as follows.

$F_{\rm M}(t)=\frac{\Lambda_{\rm M}^{2}-M_{\rm M}^{2}}{\Lambda_{\rm M}^{2}-t}$
(28)

To include not only the exchange of $f_{1}(1285)$ but also exchanges of
higher-spin mesons, we consider $f_{1}$-Regge which is the Reggeized axial-
vector Donnachie:2002en ; Kochelev:1999zf . Then, the invariant amplitude of
$f_{1}$ from the effective Lagrangians in Eq. (19) and (22) needs to change
the form. Specifically, the Feynman propagator must be substituted with the
Regge propagator Donnachie:2002en  as follows:

$P_{f_{1}}^{\rm
Feyn}(t)=\frac{1}{t-M_{f_{1}}^{2}}\,\,\,\,\rightarrow\,\,\,\,P_{f_{1}}^{\rm
Regge}(t)=\left(\frac{s}{s_{f1}}\right)^{\alpha_{f_{1}}(t)-1}\frac{\pi\alpha_{f_{1}}^{\prime}}{{\rm
sin}[\pi\alpha_{f_{1}}(t)]}\frac{1}{\Gamma[\alpha_{f_{1}}(t)]}D_{f_{1}}(t),$
(29)

where the energy-scale factor $s_{f_{1}}=1\,{\rm GeV}^{2}$ and the odd
signature factor $D_{f_{1}}$ is given as

$D_{f_{1}}(t)=\frac{{\rm exp}\big{[}-i\pi\alpha_{f_{1}}(t)\big{]}-1}{2}.$ (30)

The $f_{1}$-Regge trajectory is
$\alpha_{f_{1}}(t)=0.99+\alpha_{f_{1}}^{\prime}t$ and the slope is chosen as
$\alpha_{f_{1}}^{\prime}=0.028\,{\rm GeV^{-2}}$ Kochelev:1999zf .

###  II.3 $P_{s}$ contribution

Here, we define the effective Lagrangians for the $P_{s}$ for the Feynman
diagram Fig. 1(c). Employing the Rarita-Schwinger formalism for the
$J^{P}=\frac{3}{2}^{-}$ baryon, we can write the following Wang:2019krd ;
Park:2022nza .

$\displaystyle\mathcal{L}_{\gamma NP_{s}}^{3/2-}$ $\displaystyle=$
$\displaystyle
e\left[\frac{ih_{1}}{2m_{N}}\bar{N}\gamma^{\nu}-\frac{h_{2}}{4m^{2}_{N}}\partial^{\nu}\bar{N}\right]F_{\mu\nu}P_{s}^{\mu}+\mathrm{h.c},$
(31) $\displaystyle\mathcal{L}_{P_{s}\phi N}^{3/2-}$ $\displaystyle=$
$\displaystyle\,\frac{~{}ig_{P_{s}\phi
N}^{3/2-}}{2m_{N}}\bar{N}\gamma^{\nu}\phi_{\mu\nu}P_{s}^{\mu}-\frac{g_{2}}{4m^{2}_{N}}\partial^{\nu}\bar{N}\phi_{\mu\nu}P_{s}^{\mu}+\frac{g_{2}}{4m^{2}_{N}}\bar{N}\partial^{\nu}\phi_{\mu\nu}P_{s}^{\mu}+\mathrm{h.c},$
(32)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and $N$, $A$,
$P_{s}$, and $\phi$ represent the nucleon, photon, $P_{s}$, and $\phi$ meson
fields, respectively. In the present work, we consider only the leading terms
of $\mathcal{L}_{\gamma NP_{s}}^{3/2-}$ and $\mathcal{L}_{P_{s}\phi N}^{3/2-}$
to avoid additional ambiguities. We adopt $g_{P_{s}\phi N}=0.14+i,0.2$ from
Ref. Khemchandani:2011et . For the mass and total width, to reproduce the
currently existing data, we use slightly larger mass values, $M_{P_{s}}=2.21$
GeV and $\Gamma_{P_{s}}=28$ MeV, compared to the original values of 2.071 GeV
and 14 MeV, respectively. The electromagnetic (EM) coupling constant $eh_{1}$
in $\mathcal{L}_{\gamma NP_{s}}$ can be obtained from the strong coupling
constant $g_{P_{s}\phi N}$ by using the vector meson dominance (VMD) approach
similarly to Ref. Wang:2019krd ; Park:2022nza .

$\displaystyle eh_{1}$ $\displaystyle=$ $\displaystyle g_{P_{s}\phi
N}^{3/2-}\frac{e}{f_{\phi}}\frac{2m_{N}(m_{N}+m_{P_{s}})}{(m_{P_{s}}^{2}-m_{N}^{2})m_{\phi}}\sqrt{\frac{6m_{\phi}^{2}m_{P_{s}}^{2}+m_{N}^{4}-2m_{N}^{2}m_{P_{s}}^{2}+m_{P_{s}}^{4}}{3m_{P_{s}}^{2}+m_{N}^{2}}}.$
(33)

Here, for the decay constant of the $\phi$ meson $f_{\phi}$, in VMD approach,
photons can interact with $P_{s}$ through $\phi$ vector mesons. The
interaction Lagrangian for the $\phi$ and the photon can be written as

$\mathcal{L}_{\phi\gamma}=-\frac{em_{\phi}^{2}}{f_{\phi}}\phi_{\mu}A^{\mu},$
(34)

where $f_{\phi}$ is the decay constant of the $\phi$ meson. Then, one can
determine $e/f_{\phi}$ from the $\phi\rightarrow e^{+}e^{-}$ decay.

$\Gamma_{\phi\rightarrow
e^{+}e^{-}}=\left(\frac{e}{f_{\phi}}\right)^{2}\frac{8\alpha|\vec{p}_{e}^{\,cm}|^{3}}{3m_{\phi}^{2}},$
(35)

where $\vec{p}_{e}^{\,cm}$ denotes the momentum of an electron in the rest
frame of the $\phi$ meson. The $\alpha=e^{2}/4\pi$ is the fine-structure
constant. Using the partial decay width $\Gamma_{\phi\rightarrow
e^{+}e^{-}}=4.249\,\mathrm{MeV}\times 2.979\times 10^{-4}=1.265\times
10^{-3}\,\mathrm{MeV}$,

$\left(\frac{e}{f_{\phi}}\right)^{2}=\frac{3m_{\phi}^{2}\Gamma_{\phi\rightarrow
e^{+}e^{-}}}{\alpha(m_{\phi}^{2}-4m_{e}^{2})^{3/2}}\simeq 0.5101\times
10^{-4},\,\,\,\,\frac{e}{f_{\phi}}\simeq 0.02259.$ (36)

Thus, $eh_{1}$ can be determined as $eh_{1}=0.00858+i\,0.01226$.

###  II.4 Other nucleon resonances

Though, there are various resonance states beyond the $\phi N$ threshold
$W\sim 1.96$ GeV and their helicity amplitudes of $N^{*}\rightarrow N\gamma$
decay are well known, the information of $N^{*}\rightarrow\phi N$ process is
very limited. Therefore, to describe the present reaction process, we select
the two nucleon-resonance states $N^{*}(2000,5/2^{+})$ and
$N^{*}(2300,1/2^{+})$ based on recent CLAS experimental data CLAS:2013jlg  and
prior theoretical studies Kim:2019kef ; Kim:2021adl . The effective
Lagrangians of $J^{P}=\frac{1}{2}^{\pm}$ and $\frac{5}{2}^{\pm}$ for $\gamma
NN^{*}$ vertex for Fig. 1(c) can be written as

$\displaystyle\mathcal{L}_{\gamma NN^{*}}^{1/2\pm}$ $\displaystyle=$
$\displaystyle\frac{eh}{2M_{N}}\bar{N}\Gamma^{(\mp)}\sigma_{\mu\nu}\partial^{\nu}A^{\mu}N^{*}+\mathrm{h.c},$
(37) $\displaystyle\mathcal{L}_{\gamma NN^{*}}^{5/2\pm}$ $\displaystyle=$
$\displaystyle
e\left[\frac{h_{1}}{(2M_{N})^{2}}\bar{N}\Gamma^{(\mp)}_{\nu}-\frac{ih_{2}}{(2M_{N})^{3}}\partial_{\nu}\bar{N}\Gamma^{(\mp)}\right]\partial^{\alpha}F^{\mu\nu}N^{*}_{\mu\alpha}+\mathrm{h.c},$
(38)

and for $\phi NN^{*}$ vertex,

$\displaystyle\mathcal{L}_{\phi NN^{*}}^{1/2\pm}$ $\displaystyle=$
$\displaystyle\pm\frac{1}{2M_{N}}\bar{N}\left[\frac{g_{1}M_{\phi}^{2}}{M_{N^{*}}\mp
M_{N}}\Gamma^{(\mp)}_{\mu}\mp
g_{2}\Gamma^{(\mp)}_{\mu}\sigma_{\mu\nu}\partial^{\nu}\right]\phi^{\mu}N^{*}+\mathrm{h.c},$
(39) $\displaystyle\mathcal{L}_{\phi NN^{*}}^{5/2\pm}$ $\displaystyle=$
$\displaystyle\left[\frac{g_{1}}{(2M_{N})^{2}}\bar{N}\Gamma^{(\mp)}_{\nu}-\frac{ig_{2}}{(2M_{N})^{3}}\partial_{\nu}\bar{N}\Gamma^{(\mp)}+\frac{ig_{3}}{(2M_{N})^{3}}\bar{N}\Gamma^{(-\mp)}\partial\right]\partial^{\alpha}\phi^{\mu\nu}N^{*}_{\mu\alpha}+\mathrm{h.c},$
(40)

where the notations for $\Gamma^{(\pm)}$ and $\Gamma^{(\pm)}_{\mu}$ are
defined as

$\Gamma^{(\pm)}=\left(\begin{array}[]{c}\gamma_{5}\\\ I_{4\times
4}\end{array}\right),\quad\Gamma^{(\pm)}_{\mu}=\left(\begin{array}[]{c}\gamma_{\mu}\gamma_{5}\\\
\gamma_{\mu}\end{array}\right).$ (41)

Here, we consider only $g_{1}$ to avoid ambiguities due to the lack of
information on $N^{*}\rightarrow\phi N$, and the values of other $g_{2,3}$ are
set to be zero for simplicity. From the values of the Breit-Wigner helicities
given in PDG data ParticleDataGroup:2018ovx  for
$N^{*}(2000,\frac{5}{2}^{+})$, the coupling constants of the electromagnetic
transition become $h_{1}^{N^{*}(2000)}=-4.24$ and $h_{2}^{N^{*}(2000)}=4.00$.
For other couplings and the decay widths, we follow the values given in Ref.
Kim:2019kef .

$h^{N^{*}(2300)}=1.0,\quad g^{N^{*}(2000)}=4.0,\quad
g^{N^{*}(2300)}=0.1,\,\,\,\,\Gamma_{N^{*}(2000)}=200\,{\rm
MeV},\,\,\,\,\Gamma_{N^{*}(2300)}=300\,{\rm MeV}.$ (42)

To avoid unreasonably increasing cross sections with respect to $W$ for the
nucleon resonance contributions and consider the finite spatial distributions
of the hadrons, we use the following Gaussian form factor for $P_{s}$,
$N^{*}(2000,5/2^{+})$, and $N^{*}(2300,1/2^{+})$ Kim:2017nxg ; Kim:2018qfu ;
Suh:2018yiu ; Corthals:2005ce ; DeCruz:2012bv .

$F_{N^{*}}(x)={\rm
exp}\left[-\frac{(x-M_{N^{*}}^{2})^{2}}{\Lambda_{N^{*}}^{4}}\right],\quad
x=(s,u).$ (43)

###  II.5 Direct $\phi$-meson radiation

The effective Lagrangians for the direct $\phi$-meson radiation contributions,
depicted by Fig. 1(d), can be written as

$\displaystyle\mathcal{L}_{\gamma NN}$ $\displaystyle=$
$\displaystyle-e\bar{N}\left[\gamma_{\mu}-\frac{\kappa_{N}}{2M_{N}}\sigma_{\mu\nu}\partial^{\nu}\right]NA^{\mu},$
(44) $\displaystyle\mathcal{L}_{\phi NN}$ $\displaystyle=$ $\displaystyle-
g_{\phi NN}\bar{N}\left[\gamma_{\mu}-\frac{\kappa_{\phi
NN}}{2M_{N}}\sigma_{\mu\nu}\partial^{\nu}\right]N\phi^{\mu},$ (45)

where $\kappa_{N}=1.79$. The $\phi NN$ coupling constants are chosen to be
$g_{\phi NN}=-0.24$ and $\kappa_{\phi NN}=0.2$ following the values given in
Ref. Meissner:1997qt . The individual $s$\- and $u$-channel invariant
amplitude from Eqs.(45) does not satisfy the Ward-Takahashi identity (WTI).
This problem can be addressed by taking an appropriate sum of those invariant
amplitudes with correct form factors Ohta:1989ji ; Haberzettl:1997jg ;
Haberzettl:1998aqi ; Davidson:2001rk ; Kim:2020wrd . For the magnetic parts of
the $s$\- and $u$-channel amplitudes, which include the second term of
$\mathcal{L}_{\gamma NN}$ in Eq. (45), they satisfy the WTI by themselves.
Thus, we can use the following form factor:

$F_{N}(x)=\frac{\Lambda_{N}^{4}}{\Lambda_{N}^{4}+(x-M_{N}^{2})^{2}},\quad
x=(s,u).$ (46)

For the electric parts, we choose the following common form factor which
conserves the on-shell condition and the crossing symmetry Davidson:2001rk .

$F_{c}(s,u)=1-[1-F_{N}(s)][1-F_{N}(u)].$ (47)

##  III Numerical results and Discussions

In this section, we present the numerical results of the cross sections,
angular distributions, SDMEs, and so on, and discuss them in detail. The free
parameters in our model are determined as follows. Initially, we set
$C_{\mathbb{P}}=6.5$ for the Pomeron, attributing the largest contribution, to
account for the cross section behavior at high energies Ballam:1972eq ;
Barber:1981fj . This value was also employed in a prior study in which one of
the present authors participated Kim:2019kef . Following this, we examine the
relative phase factors $e^{i\pi\beta}$ for all contributions, with Pomeron’s
$\beta$ fixed to zero for brevity. The relative phases $\beta$ and the cutoffs
$\Lambda$ are determined as in Table 1, taking into account the CLAS data for
differential cross sections and SDMEs. The differential cross section
$d\sigma/d{\rm cos}\,\theta$ is obtained using the following equation.

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^{2}s}\frac{k^{\prime}}{k}\frac{1}{4}\sum_{\textrm{spins}}\left|\mathcal{M}_{\lambda,\lambda^{\prime},s,s^{\prime}}\right|^{2}$
(48)

As mentioned in Sec. II, the invariant amplitude $\mathcal{M}$ consists of
various t-channel, proton, and nucleon-resonance parts.

|  $f_{1}$ |  PS |  S |  $N$ | $N^{*}(2000)$ | $N^{*}(2300)$ |  $P_{s}$
---|---|---|---|---|---|---|---
Phase $\beta$ | 1 | 0 | 3/2 | 1 | 1 | 1/2 | 1
Cutoff $\Lambda$ [GeV] | 1.5 | 0.87 | 1.35 | 1.0 | 1.0 | 1.0 | 1.0
Table 1: The relative phases $\beta$ and the cutoffs $\Lambda$ for the
numerical calculations. Figure 2: The total cross sections for the total and
each contribution are plotted in the left panel. The experimental data are
taken from Refs. Ballam:1972eq ; Barber:1981fj ; Egloff:1979mg ;
Busenitz:1989gq . The total cross section for all contributions is plotted in
the logarithmic scale in the right panel for clarity.

The numerical results for the total cross section, which is the integral of
Eq. 48 are shown in Fig. 2. The left panel shows the results of the Pomeron
(green dotted), without resonances (blue dashed), $N^{*}(2000,5/2^{+})$
(orange dash-dot-dot), $N^{*}(2300,1/2^{+})$ (pink dash-dot), $P_{s}$ (violet
dot-dash-dash), and the total (red solid) contributions, along with the
experimental data Ballam:1972eq ; Barber:1981fj ; Egloff:1979mg ;
Busenitz:1989gq . The total result of the present work well explains the
experimental data up to $10$ GeV Ballam:1972eq ; Barber:1981fj , but shows
slightly overestimated values than the experimental data in the high-energy
region Egloff:1979mg ; Busenitz:1989gq . It can also be seen that most of the
experimental data for the cross section can be explained by the Pomeron
contribution alone. The right panel shows all contributions in the entire
region on the logarithmic scale. One can see a bump and peak at
E${}_{\gamma}^{\textrm{ lab}}\sim 2$ GeV caused by the nucleon resonances and
$P_{s}$. In the right panel, the cross sections of $\pi$ and $a_{0}$ are
comparable to those of the resonances; however, this effect is not very
prominent in the total result due to their destructive interference with each
other. The direct $\phi$-radiation via the proton has almost no contribution
across the entire region. It can also be seen that most of the contributions
other than Pomeron disappear in the high-energy region, but the contribution
of $f_{1}$-Regge remains.

Figure 3: The differential cross sections $d\sigma/d{\rm cos}\,\theta$ as a
function of $W$. The green dotted line represents the Pomeron contribution,
the yellow dashed line represents the resonance-free contribution, the blue
dash-dotted line represents the resonance contribution, the red solid line
represents the total contribution, and the error bars on triangles and orange
circles represent the CLAS experiment data for the charged and neutral
$K\bar{K}$ decay modes of $\phi$, respectively.

In Fig. 3, the results of differential cross section
$d\sigma/d\,{\mathrm{c}os}\,\theta$ as a function of $W$ measured by CLAS
experiment Dey:2014tfa  are compared with the theory in different
${\mathrm{c}os}\,\theta$ values. The data are obtained for the charged
($\phi\rightarrow K^{+}K^{-}$) and neutral ($\phi\rightarrow
K^{0}_{S}K^{0}_{L}$) decay modes. They are measured at uniform angular
intervals of 0.05 for $0.375\leq{\rm cos}\,\theta\leq 0.925$, and 0.1 for
$-0.8\leq{\rm cos}\,\theta\leq 0.3$. As recommended in Ref. Dey:2014tfa , we
compare our numerical result to the experimental data by treating the data of
the charged and neutral modes as a unified dataset rather than independent
measurements. In Fig. 3, it can be seen that the contribution of resonances,
including $P_{s}$, plays a crucial role at all angles in reproducing the peak-
like structures in the data. In particular, at the most forward angle, ${\rm
cos}\,\theta=0.925$, our result is somewhat overestimated for $W>2.2$ GeV.
Nevertheless, it partially accounts for the bump observed near $W=2.15$ GeV,
attributed to the contribution from resonances, especially $P_{s}$. On the
other hand, the contribution of the direct $\phi$-radiation via proton is
almost negligible at all angles as mentioned previously. At angles of ${\rm
cos}\,\theta\geq 0$, the contribution of the Pomeron is dominant while other
t-channel contributions are very small. However, explaining the experimental
data with the Pomeron alone at the angles of ${\rm cos}\,\theta<0$ is
difficult because its contribution becomes smaller and smaller. Therefore, the
contributions of other $t$ channels and resonances are essential.

Figure 4: The differential cross sections $d\sigma/d{\rm cos}\,\theta$ as a
function of ${\rm cos}\,\theta$. The notation of the lines and error bars are
the same as Fig. 3.

Fig. 4 shows the numerical calculation results for $d\sigma/d{\rm
cos}\,\theta$ as a function of ${\rm cos}\,\theta$, compared to the CLAS
charged-decay experimental data. As shown in the results for $W\approx 2.5$
GeV or higher, the Pomeron contribution explains the results near the forward
region (${\rm cos}\,\theta\sim 1$) well, but the difference between the
experimental results and the Pomeron contribution increases as ${\rm
cos}\,\theta$ decreases. This difference is compensated by other contributions
other than the Pomeron and nucleon resonances. It can be seen that the
resonances do not contribute to this region because the result without
resonances shows no difference from the result of the total contribution.
However, in the lower $W$ region, the contribution of resonances is very
important. In particular, in the vicinity of $W\approx 2.2$ GeV, it is
difficult to explain the experimental data without resonance contributions,
being crucial in the vicinity of cos $\theta<0$.

Figure 5: Spin-density matrix elements (SDMEs) for the Pomeron only (left) and
total (right) in the Gottfried Jackson (red solid lines), Adair (blue dot-
dashed lines), and Helicity (green dashed) frames.

Fig. 5 shows the numerical result for the SDME $\rho^{0}_{00}$ for the Pomeron
only (left) and total (right) as a function of $W$ at the angle of ${\rm
cos}\,\theta=0.7$. The results are plotted in the Gottfried-Jackson (red solid
lines), Adair (blue dot-dashed lines), and Helicity (green dashed) frames, and
compared to the CLAS data Dey:2014tfa . The definition of $\rho^{0}_{00}$ is

$\rho^{0}_{00}\propto|\mathcal{M}_{\lambda=1,\lambda^{\prime}=0}|^{2}+|\mathcal{M}_{\lambda=-1,\lambda^{\prime}=0}|^{2}$
(49)

where $\lambda$ and $\lambda^{\prime}$ are the polarizations of the incoming
photon and outgoing $\phi$-meson, respectively. For more details about SDMEs
and the reference frames, one can refer to Appendix and Ref. Schilling:1969um
. From this definition, the nonzero value of $\rho^{0}_{00}$ implies that
helicity flip occurs between the photon and the $\phi$-meson. Because of this,
$\rho^{0}_{00}$ was often considered when discussing s-channel helicity
conservation (SCHC) and $t$-channel helicity conservation (TCHC) along with
the properties of the Pomeron Kim:2019kef . In the left panel of Fig. 5, the
numerical results of the Pomeron for $\rho^{0}_{00}$ are underestimated
compared to the experimental data in all three frames. This shows that other
contributions besides Pomeron are required for the $\phi$-photoproduction. In
addition, the experimental data shows bump structures with increasing $W$ in
each coordinate system. By analyzing the difference between the Pomeron
results and these bump structures, we can find additional contributions to be
considered. In the right panel of Fig. 5 the results for the total
contribution are improved and describe the experimental data better. In
particular, the bump structures at $W\sim 1.5$ GeV in each frame are
reproduced to some extent and this is due to the contribution of $P_{s}$. It
is worth mentioning that the former research without the $P_{s}$ cannot
explain this bump structure. Thus, the present SDME data can be experimental
evidence for the pentaquark-like state $P_{s}$.

As mentioned above, to explain the behavior in the cross sections, including
the bump structure in forward angles, and the SDMEs in the present work,
contributions from the processes of $N^{*}(2000,\frac{5}{2}^{+})$,
$N^{*}(2300,\frac{1}{2}^{+})$, and $P_{s}(2210,\frac{3}{2}^{-})$ are
essential. However, in reality, additional resonance states not currently
considered may contribute. Moreover, the fitted parameters such as the
relative phases, couplings, widths, masses, and the spin-parity of the nucleon
resonances can vary, because of interferences with the additional resonances.
Therefore, further investigation is required to eliminate such uncertainties
together with more reliable experimental data. For example, an experiment
using the EIC to be built at the Brookhaven National Laboratory (BNL)
AbdulKhalek:2021gbh  will be the best to investigate the electromagnetic
production of $\phi$ meson, exploring the existence of $P_{s}$.

Theoretical research on a similar reaction, particularly electroproduction of
the hidden-flavor vector meson $J/\Psi$, has already been conducted in Ref.
Park:2022nza . In Ref. Park:2022nza , the differential cross section for the
production of $J/\psi$ meson via $P_{c}$ was evaluated as a function of the
pseudorapidity and the transverse momentum in the collider kinematics. Using
the same formalism, the total cross section for $P_{s}$ production turns out
to be 4.09 pb in $e$+$p$ collisions at the EIC’s peak energy ($W=126$ GeV).
Given the luminosity of 10 fb-1 per month, this translates to the potential
measurement of about 40k $P_{s}$ particles annually. In this photo-production
process, $P_{s}$ has similar rapidity with the proton beam, necessitating a
dedicated detector positioned at the very forward (proton-going) direction to
effectively measure these. Furthermore, as observed in Ref. Park:2022nza ,
considering polarized cross sections, the spin-parity of $P_{c}$ can be
distinguished. Similarly, we can theoretically predict polarized cross
sections for the nucleon resonances including $P_{s}$, and comparing them with
future experimental results, is anticipated to enhance our understanding of
the $\phi$-photoproduction with the exotic baryon $P_{s}$.

##  IV Summary

We explore the photoproduction reaction of $\phi$-mesons utilizing the
effective Lagrangian method within the tree-level Born approximation. Our
investigation encompasses contributions from various contributions, including
the Pomeron, $f_{1}$-Regge, pseudoscalar particles ($\pi$, $\eta$), scalar
particles ($a_{0}$, $f_{0}$), protons, and three-nucleon resonance states. In
addition to the $N^{*}(2000,5/2^{+})$ and $N^{*}(2300,1/2^{+})$ states
considered in prior theoretical studies Kim:2019kef , we introduce the
contribution of the pentaquark-like nucleon-resonance state $P_{s}$,
anticipated to be a $K^{*}\Sigma$-bound state, as suggested in Ref.
Khemchandani:2011et . To refine our model, we determine unknown parameters,
including the Pomeron parameter $C_{\mathbb{P}}$, relative phases between
contributions denoted as $\beta$, and cutoffs $\Lambda$. This determination is
achieved through a comparison with early total cross section data
Ballam:1972eq ; Barber:1981fj ; Egloff:1979mg ; Busenitz:1989gq  and recent
data from the CLAS experiment Dey:2014tfa .

The numerical findings from this investigation reveal that, in terms of the
total cross section, contributions other than the Pomeron are generally minor,
particularly except for behavior near the threshold. At higher energies,
contributions other than the Pomeron and $f_{1}$ Regge are deemed negligible.
When focusing on the production reaction at specific angles rather than the
total cross section, relying solely on the Pomeron contribution becomes
inadequate. In such cases, analyzing additional contributions becomes
necessary. Consequently, we compute $d\sigma/d{\rm cos}\,\theta$ to assess
various angles and center-of-mass energies, providing a comparison with CLAS
data. From this analysis, it becomes evident that the Pomeron contribution
alone effectively accounts for the experimental data in the vicinity of ${\rm
cos}\,\theta\sim 1$. However, as ${\rm cos}\,\theta$ decreases, the deviation
from the data increases, highlighting the necessity of contributions other
than the Pomeron, particularly at lower ${\rm cos}\,\theta$ values. Crucially,
the inclusion of $P_{s}$ and other nucleon resonances proves essential for
accurately describing the observed bump near $W\sim 2.15$ GeV in very forward
angles, as well as capturing the behaviors within the range of $(2.0-2.3)$
GeV. Additionally, in the region for $W\geq 2.5$ GeV, where nucleon resonances
are nearly absent, contributions from t-channel mesons, distinct from the
Pomeron, become necessary.

In conclusion, we compute $\rho^{0}_{00}$ at ${\rm cos}\,\theta=0.7$ in the
Gottfried-Jackson, Adair, and Helicity frames, subsequently comparing these
calculations with CLAS data. When solely accounting for the Pomeron, the
results consistently fall below the data in all frames. However, with the
inclusion of all contributions, this discrepancy is notably diminished, and
the data aligns well with the model. Notably, a significant accomplishment of
this study is the successful reproduction of the observed bump at $W\sim 2.15$
GeV through the incorporation of $P_{s}$. Our study demonstrates that the
upcoming EIC experiment will enable high-precision measurements of the $P_{s}$
resonance with the luminosity of 10 fb-1 per month. More detailed works on
$P_{s}$ production in EIC are in progress and will appear elsewhere.

## Acknowledgment

The authors are grateful for the fruitful discussions with Sang-Ho Kim
(Soongsil University), Kanchan Pradeepkumar Khemchandani (Federal University
of São Paulo), and Alberto Martinez Torres (University of São Paulo). This
work is supported by the National Research Foundation of Korea (NRF) grants
funded by the Korean government (MSIT) (2018R1A5A1025563). The work of Y.S.K.
was supported by the NRF grant (2022R1A2C1011549). The work of S.i.N. was also
supported by the NRF grants (2022R1A2C1003964 and 2022K2A9A1A06091761).

## Appendix

Here we briefly discuss the SDMEs and helicity conservation. The spin density
matrix element (SDMEs) is defined in terms of the Helicity amplitude as
follows Schilling:1969um :

$\displaystyle\rho^{0}_{\lambda^{\prime}_{1}\lambda^{\prime}_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{s,s^{\prime},\lambda}\mathcal{M}_{s,s^{\prime},\lambda,\lambda^{\prime}_{1}}\mathcal{M}^{*}_{s,s^{\prime},\lambda,\lambda^{\prime}_{2}},$
(50) $\displaystyle\rho^{1}_{\lambda^{\prime}_{1}\lambda^{\prime}_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{s,s^{\prime},\lambda}\mathcal{M}_{s,s^{\prime},\lambda,\lambda^{\prime}_{1}}\mathcal{M}^{*}_{s,s^{\prime},\lambda,\lambda^{\prime}_{2}},$
(51) $\displaystyle\rho^{2}_{\lambda^{\prime}_{1}\lambda^{\prime}_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{s,s^{\prime},\lambda}\mathcal{M}_{s,s^{\prime},\lambda,\lambda^{\prime}_{1}}\mathcal{M}^{*}_{s,s^{\prime},\lambda,\lambda^{\prime}_{2}},$
(52) $\displaystyle\rho^{3}_{\lambda^{\prime}_{1}\lambda^{\prime}_{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{s,s^{\prime},\lambda}\mathcal{M}_{s,s^{\prime},\lambda,\lambda^{\prime}_{1}}\mathcal{M}^{*}_{s,s^{\prime},\lambda,\lambda^{\prime}_{2}},$
(53)

where the normalization factor is
$N=\sum|\mathcal{M}_{s,s^{\prime},\lambda,\lambda^{\prime}}|^{2}$. Because the
SDMEs are not Lorentz invariant quantities, the values change depending on the
choice of the spin-quantization direction for the $\phi$ meson. There are
three reference frames often used, the Adair frame, the helicity frame, and
the Gottfried-Jackson frame, as shown in Fig. 6.

Figure 6: The spin-quantization axes for the helicity ($z_{\textrm{Hel}}$, in
green), Adair ($z_{\textrm{Ad}}$, in red), and Gottfried-Jackson
($z_{\textrm{GJ}}$, in blue) frames, in the c.m. frame.

The spin-quantization axis $z_{\textrm{Ad}}$ coincides with the z-axis for the
c.m. frame. Since $z_{\textrm{Hel}}$ points along the direction of the $\phi$
meson in the c.m. frame, the angle between the helicity and Adair frames is
$\theta$. In the Gottfried-Jackson frame, $z_{\textrm{GJ}}$ is equal to the
direction of the incoming photon in the $\phi$-meson rest frame. The
$z_{\textrm{Hel}}$ and $z_{\textrm{GJ}}$ are favored for discussion of the
s-channel and t-channel helicity conservations, respectively. The SDMEs in one
reference frame can be transformed to them in other frames by a Wigner
rotation. The rotation angle $\alpha$’s are expressed as in Ref.
Schilling:1969um

$\displaystyle\alpha_{\textrm{Ad}\rightarrow\textrm{Hel}}$ $\displaystyle=$
$\displaystyle\theta,$ (54)
$\displaystyle\alpha_{\textrm{Hel}\rightarrow\textrm{GJ}}$ $\displaystyle=$
$\displaystyle-{\rm cos}^{-1}\left(\frac{v-{\rm cos}\,\theta}{v\,{\rm
cos}\,\theta-1}\right),$ (55)
$\displaystyle\alpha_{\textrm{Ad}\rightarrow\textrm{GJ}}$ $\displaystyle=$
$\displaystyle\alpha_{\textrm{Ad}\rightarrow\textrm{Hel}}+\alpha_{\textrm{Hel}\rightarrow\textrm{GJ}},$
(56)

where $v$ is the velocity of the $\phi$ meson in the c.m. frame.

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