# Analysis of the data for $\gamma p\to f_{1}(1285)p$ photoproduction

Ai-Chao Wang  School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 101408, China  Neng-Chang Wei  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
101408, China  Fei Huang<EMAIL_ADDRESS>School of Nuclear Science and
Technology, University of Chinese Academy of Sciences, Beijing 101408, China

###### Abstract

The photoproduction of $f_{1}(1285)$ meson off proton is investigated within
an effective Lagrangian approach. The $t$-channel $\rho$\- and
$\omega$-exchange diagrams, $u$-channel nucleon-exchange diagram, generalized
contact term, and $s$-channel pole diagrams of nucleon and a minimal number of
nucleon resonances are taken into account in constructing the reaction
amplitudes to describe the experimental data. Three different models, i.e.,
the Feynman model, the Regge model, and the interpolated Regge model, are
employed where the $t$-channel reaction amplitudes are constructed in Feynman
type, Regge type, and interpolated Regge type, respectively. The results show
that in neither Feynman model with two nucleon resonances nor interpolated
Regge model with one nucleon resonance can the available data for $\gamma p\to
f_{1}(1285)p$ be satisfactorily reproduced. Nevertheless, in the Regge model,
when any one of the $N(1990){7/2}^{+}$, $N(2000){5/2}^{+}$,
$N(2040){3/2}^{+}$, $N(2060){5/2}^{-}$, $N(2100){1/2}^{+}$,
$N(2120){3/2}^{-}$, $N(2190){7/2}^{-}$, $N(2300){1/2}^{+}$, and
$N(2570){5/2}^{-}$ resonances is considered, the data can be well described.
The resulted resonance parameters are consistent with those advocated in
Particle Data Group (PDG) review. Further analysis shows that in high-energy
region, the peaks of $\gamma p\to f_{1}(1285)p$ differential cross sections at
forward angles are dominated by the contributions from $t$-channel $\rho$\-
and $\omega$-exchange diagrams, while in low-energy region, the $s$-channel
pole diagrams of resonances also provide significant contributions to the
$\gamma p\to f_{1}(1285)p$ cross sections.

###### pacs:

25.20.Lj, 13.60.Le, 14.20.Gk

##  I Introduction

The study of nucleon resonances ($N^{\ast}$’s) and $\Delta$ resonances
($\Delta^{\ast}$’s) has always been of great interest in hadron physics
community, since a thorough understanding of the structure and properties of
$N$ and $\Delta$ resonances is essential to understand clearly the
nonperturbative behavior of quantum chromodynamics (QCD), the fundamental
theory of strong interactions. It is well known nowadays that the experimental
and theoretical studies of $\pi N$ scattering and $\pi$ photoproduction
reactions provide us the most knowledge about the $N^{\ast}$’s and
$\Delta^{\ast}$’s. Nevertheless, quark models Isgur:1978 ; Capstick:1986 ;
Loring:2001  predicated much more $N^{\ast}$’s and $\Delta^{\ast}$’s than
experimentally observed. The possible reason might be that some of the
unobserved $N^{\ast}$’s or $\Delta^{\ast}$’s couple weakly to $\pi N$ channel
and thus escaped from the observation. On the other hand, there are many one-
star and two-star $N^{\ast}$’s and $\Delta^{\ast}$’s whose parameters
especially the decay branching ratios to various final states are not yet
known in the most recent Particle Data Group (PDG) review Workman:2022 . These
situations force us to investigate the $N^{\ast}$’s and $\Delta^{\ast}$’s in
reaction channels other than $\pi$ hadro- and photoproductions. In the past
few years, lots of experimental and theoretical efforts have been devoted to
the study of $\eta N$, $\eta^{\prime}N$, $KY$, $\omega N$, $K^{\ast}Y$, and
$KY^{\ast}$ $(Y=\Lambda,\Sigma)$ photoproduction reactions, and valuable
achievements have been obtained Kashevarov:2017 ; Anisovich:2018 ; Zhangy:2021
; Bradford:2006 ; Mart:2019 ; CLAS-beam ; Zachariou:2020kkb ; Wei:2022 ;
Wang:2022 ; Anisovich:2017rpe ; Wei:2020 ; Moriya:2013 ; Wangx:2020 ;
Zhangyx:2021 ; Wein:2021 ; Roy:2018 ; Weinc:2019 . In the present work, we
concentrate on the photoproduction reaction of $f_{1}(1285)$ meson off the
proton target. The $f_{1}(1285)$ is an axial-vector meson with quantum numbers
$I^{G}J^{PC}=0^{+}1^{++}$, mass $M=1281.9\pm 0.5$ MeV and width
$\Gamma=22.7\pm 1.1$ MeV Workman:2022 . As the $f_{1}(1285)p$ threshold is
above $2.2$ GeV, this reaction is more suitable than $\pi$ production
reactions to investigate the $N^{\ast}$’s with higher mass in the less-
explored energy region. Furthermore, the $f_{1}(1285)p$ photoproduction
reaction acts as an “isospin filter”, isolating the nucleon resonances with
isospin $I=1/2$ and eliminating the interference of the $\Delta$ resonances
which have isospin $I=3/2$.

Experimentally, the differential cross-section data for $\gamma p\to
f_{1}(1285)p$ with $f_{1}(1285)\to\eta\pi^{+}\pi^{-}$ in the energy range from
the $f_{1}(1285)p$ threshold up to the center-of-mass energy $W\approx 2.8$
GeV were released in $2016$ by the CLAS Collaboration at the Thomas Jefferson
National Accelerator Facility (JLab) Dickson:2016gwc . Compared with the
simultaneously released cross-section data for $\gamma
p\to\eta^{\prime}(958)p$ with $\eta^{\prime}(958)\to\eta\pi^{+}\pi^{-}$, the
$f_{1}(1285)$ cross sections are much flatter in $\cos\theta$ angular
dependence, indicating less important $t$\- and $u$-channel contributions and
possible $s$-channel nucleon resonance contributions.

Theoretically, several works have been committed to the study of $\gamma p\to
f_{1}(1285)p$ photoproduction reaction. Before the publication of the CLAS
data, predictions of the differential cross sections for $\gamma p\to
f_{1}(1285)p$ from various theoretical models were available in Refs.
Kochelev:2009 ; Domokos:2009 ; HuangY:2014 . Nevertheless, none of them seemed
to be able to even qualitatively describe the CLAS data reported in 2016
Dickson:2016gwc . Since the CLAS differential cross-section data for $\gamma
p\to f_{1}(1285)p$ became available, there were two theoretical works
analyzing them, both in the framework of the so-called interpolated Regge
model Wangxiaoyun:2017 ; Wangyanyan:2017 . However, the conclusions drawn from
these two analyses are quite different. In Ref. Wangxiaoyun:2017 , all
contributions from resonance-pole diagrams, $NN\gamma$ vector coupling
($\propto{\bar{N}}\gamma_{\mu}A^{\mu}N$) in $s$-channel $N$-pole diagram and
$u$-channel $N$-exchange diagram, and contact term were omitted, and only the
contributions from $t$-channel $\rho$\- and $\omega$-exchange diagrams
together with those from the $NN\gamma$ tensor coupling
($\propto{\bar{N}}\sigma^{\mu\nu}\partial_{\nu}A_{\mu}N$) in $s$-channel
$N$-pole diagram and $u$-channel $N$-exchange diagram were considered in the
calculation. It was concluded that the CLAS differential cross-section data
can be reproduced without considering any nucleon resonances. In Ref.
Wangyanyan:2017 , besides the $t$-channel $\rho$\- and $\omega$-exchange
diagrams, $s$-channel $N$-pole diagram, $u$-channel $N$-exchange diagram, and
contact term contributions, the $N(2300){1/2}^{+}$-pole diagram was further
considered in constructing the reaction amplitudes. It was reported that the
contribution from the $N(2300){1/2}^{+}$-pole diagram plays an important role
in reproducing the CLAS differential cross-section data.

As both Refs. Wangxiaoyun:2017  and Wangyanyan:2017  used the interpolated
Regge model and they analyzed the same set of data, one is confused whether
the differences of their conclusions about the $\gamma p\to f_{1}(1285)p$
reaction mechanisms and the extracted resonance contents are due to the fact
that the contributions from contact term and $NN\gamma$ vector coupling
($\propto{\bar{N}}\gamma_{\mu}A^{\mu}N$) were omitted in Ref. Wangxiaoyun:2017
. Furthermore, one is curious whether the $N(2300){1/2}^{+}$ resonance is the
only one that is needed to describe the CLAS differential cross-section data
for $\gamma p\to f_{1}(1285)p$ in an interpolated Regge model as done in Ref.
Wangyanyan:2017 . Besides, one wants to make clear whether the traditional
Feynman model and/or Regge model are capable of describing the available
cross-section data from the CLAS Collaboration for the $\gamma p\to
f_{1}(1285)p$ reaction.

To eliminate the above-mentioned confusions, in the present work, we present
an independent and comprehensive analysis of the CLAS data Dickson:2016gwc  on
differential cross sections for $f_{1}(1285)p$ photoproduction in an effective
Lagrangian approach. We build three different reaction models, i.e., the
Feynman model, the Regge model, and the interpolated Regge model, where the
reaction amplitudes of $t$-channel $\rho$ and $\omega$ exchanges are
constructed in Feynman type, Regge type, and interpolated Regge type,
respectively. In each of these three models, the $t$-channel $\rho$\- and
$\omega$-exchange diagrams, the $s$-channel $N$-pole diagram, the $u$-channel
$N$-exchange diagram, and the generalized interaction current are considered
as background ingredients, and besides, the $s$-channel pole diagrams of a
minimum number of nucleon resonances are considered in order to get a
satisfactory description of the available data. The gauge invariance of the
photoproduction amplitude is guaranteed by introducing an auxiliary current
which ensures that the full reaction amplitude satisfies the generalized Ward-
Takahashi identity and thus is fully gauge invariant, independent of any
specific type of form factors introduced in hadronic vertices Haberzettl:1997
; Haberzettl:2006 ; Huang:2012 ; Huang:2013 .

The present paper is organized as follows. In Sec. II, we introduce the
framework of our theoretical model. In Sec. III, we present our theoretical
results and a discussion of them. Finally, a brief summary and conclusions are
given in Sec. IV.

##  II Formalism

The full photoproduction amplitudes for $\gamma N\to f_{1}(1285)N$ can be
expressed as

$M^{\nu\mu}=M^{\nu\mu}_{s}+M^{\nu\mu}_{t}+M^{\nu\mu}_{u}+M^{\nu\mu}_{\rm
int},$ (1)

with $\nu$ and $\mu$ being Lorentz indices of the $f_{1}(1285)$ meson and the
photon, respectively. The first three terms $M^{\nu\mu}_{s}$,
$M^{\nu\mu}_{t}$, and $M^{\nu\mu}_{u}$ stand for the $s$-, $t$-, and
$u$-channel amplitudes, respectively, with $s$, $t$, and $u$ being the
Mandelstam variables of the internally exchanged particles. They arise from
the photon attaching to the external particles in the underlying $NNf_{1}$
interaction vertex. The last term, $M^{\nu\mu}_{\rm int}$, stands for the
interaction current that arises from the photon attaching to the internal
structure of the $NNf_{1}$ interaction vertex. All four terms in Eq. (1) are
diagrammatically depicted in Fig. 1.

(a) $s$ channel (b) $t$ channel

(c) $u$ channel (d) Interaction current

Figure 1: Generic structure of the amplitude for $\gamma N\to f_{1}(1285)N$.
Time proceeds from left to right. The symbols $f_{1}$ denote $f_{1}(1285)$.

As shown in Fig. 1, in the present work, contributions from the following
interaction diagrams are considered in constructing the $s$-, $t$-, and
$u$-channel reaction amplitudes: (i) $s$-channel $N$\- and $N^{\ast}$-pole
diagrams, (ii) $t$-channel $\rho$\- and $\omega$-exchange diagrams, and (iii)
$u$-channel $N$-exchange diagram. Note that the $u$-channel interaction is
expected to contribute significantly at high energy backward angles. However,
the available cross section data for $\gamma p\to f_{1}(1285)p$ are sparse in
this energy and angle regime. In this situation, we simply ignore the
$u$-channel resonance exchanges in the present work, as the additional
parameters introduced with these interactions cannot be well determined by the
available data. The expressions for all the $s$-, $t$-, and $u$-channel
amplitudes can be obtained straightforwardly by evaluating the corresponding
Feynman diagrams, except that in Regge model and interpolated Regge model
appropriate substitutions are made for the propagators of intermediate $\rho$
and $\omega$ mesons in the $t$-channel Feynman amplitudes (cf. Sec. II.4). The
interaction current $M^{\nu\mu}_{\rm int}$ in Eq. (1) cannot be calculated
directly, and we follow Refs. Haberzettl:1997 ; Haberzettl:2006 ; Huang:2012 ;
Huang:2013  to model this term by a generalized contact current,

$M^{\nu\mu}_{\rm int}=\Gamma^{\nu}_{NNf_{1}}(q)\,C^{\mu}.$ (2)

Here $\Gamma^{\nu}_{NNf_{1}}(q)$ is the vertex function of $NNf_{1}$ coupling
given by the Lagrangian of Eq. (12),

$\Gamma^{\nu}_{NNf_{1}}(q)=ig_{NNf_{1}}\left(1+\frac{\kappa_{f_{1}}}{2m_{N}}{q\\!\\!\\!/}\right)\gamma^{\nu}\gamma^{5},$
(3)

with $q$ being the four-momentum of the outgoing $f_{1}$ meson; $C^{\mu}$ is
an auxiliary current, which is nonsingular and is introduced to ensure that
the full photoproduction amplitudes of Eq. (1) satisfy the generalized Ward-
Takahashi identity and thus are fully gauge invariant. Following Refs.
Haberzettl:2006 ; Huang:2012 , we choose $C^{\mu}$ for $\gamma p\to
f_{1}(1285)p$ as

$C^{\mu}=-Q_{N_{u}}\frac{f_{u}-\hat{F}}{u-p^{\prime
2}}(2p^{\prime}-k)^{\mu}-Q_{N_{s}}\frac{f_{s}-\hat{F}}{s-p^{2}}(2p+k)^{\mu},$
(4)

with

$\hat{F}=1-\hat{h}\left(1-f_{u}\right)\left(1-f_{s}\right).$ (5)

Here $p$, $p^{\prime}$, and $k$ are the four-momenta for the incoming $N$,
outgoing $N$, and incoming photon, respectively; $Q_{N_{u}}$ and $Q_{N_{s}}$
are electric charges of $u$-channel $N$ and $s$-channel $N$, respectively;
$f_{u}$ and $f_{s}$ are the phenomenological form factors attached to the
amplitudes of $u$-channel $N$-exchange diagram and $s$-channel $N$-pole
diagram, respectively; $\hat{h}$ is an arbitrary function that goes to unity
in the high-energy limit to prevent the “violation of scaling behavior”
Drell:1972 . In the present work, we choose $\hat{h}=1$ for the sake of
simplicity.

###  II.1 Effective Lagrangians

The effective interaction Lagrangians used in the present work for
constructing the production amplitudes are given below. For further
convenience, we define the operators

$\Gamma^{(+)}=\gamma_{5}\qquad\text{and}\qquad\Gamma^{(-)}=1,$ (6)

and the field-strength tensors

$\displaystyle f_{1}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial^{\mu}f_{1}^{\nu}-\partial^{\nu}f_{1}^{\mu},$ (7)
$\displaystyle F^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu},$ (8)

with $f_{1}^{\nu}$ and $A^{\mu}$ denoting the $f_{1}$ vector-meson field and
electromagnetic field, respectively.

The electromagnetic interaction Lagrangians for $s$-channel $N$-pole diagram,
$u$-channel $N$-exchange diagram, and $t$-channel $\rho$\- and
$\omega$-exchange diagrams read

$\displaystyle{\cal L}_{NN\gamma}$
$\displaystyle=-e\bar{N}\left[\left(\hat{e}\gamma^{\mu}-\frac{\hat{\kappa}_{N}}{2M_{N}}\sigma^{\mu\nu}\partial_{\nu}\right)A_{\mu}\right]N,$
(9) $\displaystyle{\cal L}_{Vf_{1}\gamma}$
$\displaystyle=ig_{Vf_{1}\gamma}\varepsilon_{\mu\nu\alpha\beta}\left(\partial^{\mu}A^{\alpha}\right)\left(\partial^{2}V^{\nu}\right)f_{1}^{\beta},$
(10)

where $e$ is the elementary charge unit and $\hat{e}$ stands for the charge
operator;
$\hat{\kappa}_{N}=\kappa_{p}\left(1+\tau_{3}\right)/2+\kappa_{n}\left(1-\tau_{3}\right)/2$,
with the anomalous magnetic moments $\kappa_{p}=1.793$ and
$\kappa_{n}=-1.913$; $M_{N}$ stands for the masse of $N$;
$\varepsilon_{\mu\nu\alpha\beta}$ is the totally antisymmetric Levi-Civita
tensor with $\varepsilon_{0123}=1$; $V$ represents the vector meson $\rho$ or
$\omega$. The coupling constant $g_{\rho f_{1}\gamma}$ can be determined by
the decay width of $f_{1}\to\rho\gamma$,

$\Gamma_{f_{1}\to\rho\gamma}=\frac{g_{\rho f_{1}\gamma}^{2}m_{\rho}^{2}}{96\pi
m_{f_{1}}^{5}}\left(m_{f_{1}}^{2}+m_{\rho}^{2}\right)\left(m_{f_{1}}^{2}-m_{\rho}^{2}\right)^{3}.$
(11)

With the value $\Gamma_{f_{1}\to\rho\gamma}\simeq 0.45$ MeV from the CLAS
experiment Dickson:2016gwc , one gets $g_{\rho f_{1}\gamma}=0.94$ GeV-2. The
coupling constant $g_{\omega f_{1}\gamma}$ can be estimated within a quark
model assuming that $f_{1}$ consists of two-flavor $u,d$ quarks, which results
in $g_{\omega f_{1}\gamma}\approx g_{\rho f_{1}\gamma}/3$ Kochelev:2009 .

The hadronic interaction Lagrangians for $s$-channel $N$-pole diagram,
$u$-channel $N$-exchange diagram, and $t$-channel $\rho$\- and
$\omega$-exchange diagrams read

$\displaystyle{\cal L}_{NNf_{1}}$
$\displaystyle=g_{NNf_{1}}\bar{N}\left(f_{1}^{\mu}-i\frac{\kappa_{f_{1}}}{2m_{N}}\gamma^{\nu}\partial_{\nu}f_{1}^{\mu}\right)\gamma_{\mu}\gamma^{5}N,$
(12) $\displaystyle{\cal L}_{NNV}$
$\displaystyle=-g_{NNV}\bar{N}\left[\gamma_{\mu}V^{\mu}-\frac{\kappa_{V}}{2m_{N}}\sigma_{\mu\nu}\partial^{\nu}V^{\mu}\right]N.$
(13)

The value of coupling constant $g_{NNf_{1}}=2.5$ is obtained according to the
relation between the reduced elements $F$ and $D$ from a study of the axial
vector currents in Ref. Birkel:1996 . The coupling $\kappa_{f_{1}}$ is treated
as a parameter to be fixed by fitting the CLAS experimental data
Dickson:2016gwc . For nucleon and vector meson couplings, the empirical values
$g_{NN\rho}=3.25$, $g_{NN\omega}=11.76$, $\kappa_{\rho}=6.1$ and
$\kappa_{\omega}=0$ from Refs. Huang:2012 ; Ronchen:2013  are quoted.

The Lagrangians for resonance-nucleon-photon transition read

$\displaystyle{\cal L}_{RN\gamma}^{1/2\pm}$ $\displaystyle=$ $\displaystyle
e\frac{g_{RN\gamma}^{(1)}}{2M_{N}}\bar{R}\Gamma^{(\mp)}\sigma_{\mu\nu}\left(\partial^{\nu}A^{\mu}\right)N+\text{H.\,c.},$
(14) $\displaystyle{\cal L}_{RN\gamma}^{3/2\pm}$ $\displaystyle=$
$\displaystyle-\,ie\frac{g_{RN\gamma}^{(1)}}{2M_{N}}\bar{R}_{\mu}\gamma_{\nu}\Gamma^{(\pm)}F^{\mu\nu}N$
(15)
$\displaystyle+\,e\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{2}}\bar{R}_{\mu}\Gamma^{(\pm)}F^{\mu\nu}\partial_{\nu}N+\text{H.\,c.},$
$\displaystyle{\cal L}_{RN\gamma}^{5/2\pm}$ $\displaystyle=$ $\displaystyle
e\frac{g_{RN\gamma}^{(1)}}{\left(2M_{N}\right)^{2}}\bar{R}_{\mu\alpha}\gamma_{\nu}\Gamma^{(\mp)}\left(\partial^{\alpha}F^{\mu\nu}\right)N$
(16)
$\displaystyle\pm\,ie\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{3}}\bar{R}_{\mu\alpha}\Gamma^{(\mp)}\left(\partial^{\alpha}F^{\mu\nu}\right)\partial_{\nu}N$
$\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{RN\gamma}^{7/2\pm}$
$\displaystyle=$ $\displaystyle
ie\frac{g_{RN\gamma}^{(1)}}{\left(2M_{N}\right)^{3}}\bar{R}_{\mu\alpha\beta}\gamma_{\nu}\Gamma^{(\pm)}\left(\partial^{\alpha}\partial^{\beta}F^{\mu\nu}\right)N$
(17)
$\displaystyle-\,e\frac{g_{RN\gamma}^{(2)}}{\left(2M_{N}\right)^{4}}\bar{R}_{\mu\alpha\beta}\Gamma^{(\pm)}\left(\partial^{\alpha}\partial^{\beta}F^{\mu\nu}\right)\partial_{\nu}N$
$\displaystyle+\,\text{H.\,c.},$

where $R$ designates the nucleon resonance, and the superscript of ${\cal
L}_{RN\gamma}$ denotes the spin and parity of the resonance $R$. The coupling
constants $g_{RN\gamma}^{(i)}$ $(i=1,2)$ are treated as fit parameters.

The effective Lagrangians for hadronic vertices including nucleon resonances
read

$\displaystyle{\cal L}_{RNf_{1}}^{1/2\pm}$ $\displaystyle=$
$\displaystyle-\frac{g_{RNf_{1}}}{2M_{N}}\bar{R}\Gamma^{(\pm)}\left[\left(\frac{\gamma_{\mu}\partial^{2}}{M_{R}\mp
M_{N}}\pm i\partial_{\mu}\right)f_{1}^{\mu}\right]N$ (18)
$\displaystyle+\,\text{H.\,c.},$ $\displaystyle{\cal L}_{RNf_{1}}^{3/2\pm}$
$\displaystyle=$ $\displaystyle
i\frac{g_{RNf_{1}}}{2M_{N}}\bar{R}_{\mu}\gamma_{\nu}\Gamma^{(\mp)}f_{1}^{\mu\nu}N+\text{H.\,c.},$
(19) $\displaystyle{\cal L}_{RNf_{1}}^{5/2\pm}$ $\displaystyle=$
$\displaystyle-\frac{g_{RNf_{1}}}{\left(2M_{N}\right)^{2}}\bar{R}_{\mu\alpha}\gamma_{\nu}\Gamma^{(\pm)}\left(\partial^{\alpha}f_{1}^{\mu\nu}\right)N+\text{H.\,c.},$
(20) $\displaystyle{\cal L}_{RNf_{1}}^{7/2\pm}$ $\displaystyle=$
$\displaystyle-i\frac{g_{RNf_{1}}}{\left(2M_{N}\right)^{3}}\bar{R}_{\mu\alpha\beta}\gamma_{\nu}\Gamma^{(\mp)}\left(\partial^{\alpha}\partial^{\beta}f_{1}^{\mu\nu}\right)N$
(21) $\displaystyle+\,\text{H.\,c.}$

The couplings $g_{RNf_{1}}$ are fit parameters. Actually, in a single-channel
calculation as the one performed in the present work, only the products of the
electromagnetic coupling constants and hadronic coupling constants are
relevant to the reaction amplitudes, and in practice we fit these products
directly instead of fitting the hadronic and electromagnetic couplings
separately.

###  II.2 Resonance propagators

In the present work, contributions from $s$-channel pole diagrams of a minimal
number of nucleon resonances with various spin and parity are included to
describe the data. For spin-$1/2$ resonance, the propagator reads

$S_{1/2}(p)=\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma/2},$ (22)

where $M_{R}$, $\Gamma$, and $p$ are mass, width, and four-momentum of
resonance $R$, respectively.

Following Refs. Behrends:1957 ; Fronsdal:1958 ; Zhu:1999 , the prescriptions
of the propagators for resonances with spin-$3/2$, -$5/2$, and -$7/2$ read

$\displaystyle S_{3/2}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma/2}\left(\tilde{g}_{\mu\nu}+\frac{1}{3}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\nu}\right),$
(23) $\displaystyle S_{5/2}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma/2}\,\bigg{[}\,\frac{1}{2}\big{(}\tilde{g}_{\mu\alpha}\tilde{g}_{\nu\beta}+\tilde{g}_{\mu\beta}\tilde{g}_{\nu\alpha}\big{)}$
(24)
$\displaystyle-\,\frac{1}{5}\tilde{g}_{\mu\nu}\tilde{g}_{\alpha\beta}+\frac{1}{10}\big{(}\tilde{g}_{\mu\alpha}\tilde{\gamma}_{\nu}\tilde{\gamma}_{\beta}+\tilde{g}_{\mu\beta}\tilde{\gamma}_{\nu}\tilde{\gamma}_{\alpha}$
$\displaystyle+\,\tilde{g}_{\nu\alpha}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\beta}+\tilde{g}_{\nu\beta}\tilde{\gamma}_{\mu}\tilde{\gamma}_{\alpha}\big{)}\bigg{]},$
$\displaystyle S_{7/2}(p)$ $\displaystyle=$
$\displaystyle\frac{i}{{p\\!\\!\\!/}-M_{R}+i\Gamma/2}\,\frac{1}{36}\sum_{P_{\mu}P_{\nu}}\bigg{(}\tilde{g}_{\mu_{1}\nu_{1}}\tilde{g}_{\mu_{2}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}$
(25)
$\displaystyle-\,\frac{3}{7}\tilde{g}_{\mu_{1}\mu_{2}}\tilde{g}_{\nu_{1}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}+\frac{3}{7}\tilde{\gamma}_{\mu_{1}}\tilde{\gamma}_{\nu_{1}}\tilde{g}_{\mu_{2}\nu_{2}}\tilde{g}_{\mu_{3}\nu_{3}}$
$\displaystyle-\,\frac{3}{35}\tilde{\gamma}_{\mu_{1}}\tilde{\gamma}_{\nu_{1}}\tilde{g}_{\mu_{2}\mu_{3}}\tilde{g}_{\nu_{2}\nu_{3}}\bigg{)},$

where

$\displaystyle\tilde{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\,g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{M_{R}^{2}},$ (26)
$\displaystyle\tilde{\gamma}_{\mu}$ $\displaystyle=$
$\displaystyle\gamma^{\nu}\tilde{g}_{\nu\mu}=-\gamma_{\mu}+\frac{p_{\mu}{p\\!\\!\\!/}}{M_{R}^{2}}.$
(27)

###  II.3 Form factors

Each hadronic vertex obtained from the Lagrangians given in Sec. II.1 is
accompanied with a phenomenological form factor to parametrize the structure
of the hadrons and to normalize the behavior of the production amplitude.
Following Refs. Wang:2017 ; Wang:2018 , for intermediate baryon exchange we
take the form factor as

$f_{B}(p^{2})=\left[\frac{\Lambda_{B}^{4}}{\Lambda_{B}^{4}+\left(p^{2}-M_{B}^{2}\right)^{2}}\right]^{2},$
(28)

where $p$ denotes the four-momentum of the intermediate baryon and $M_{B}$ is
the mass for exchanged baryon $B$. For intermediate meson exchange, we take
the form factor as

$f_{M}(q^{2})=\frac{\Lambda_{M}^{2}-M_{M}^{2}}{\Lambda_{M}^{2}-q^{2}},$ (29)

where $q$ represents the four-momentum of the intermediate meson, and $M_{M}$
is the mass of exchanged meson $M$. The cutoffs $\Lambda_{B(M)}$ for each
exchanged baryon (meson) in Eqs. (28) and (29) are treated as fit parameters.

###  II.4 Treatments of $t$-channel reaction amplitudes

The $t$-channel reaction amplitudes are usually constructed in three different
types, i.e., the Feynman type, the Regge type, and the interpolated Regge type
Wang:2020 . In the present work, we try all these three possibilities to
explore how the reaction mechanisms of $\gamma p\to f_{1}(1285)p$ and the
extracted resonance contents and parameters depend on the choices of different
types of $t$-channel interactions, and what can we really learn from the
available cross-section data for $\gamma p\to f_{1}(1285)p$.

####  II.4.1 Feynman model

In Feynman model, the reaction amplitudes from $t$-channel $\rho$ and $\omega$
exchanges are constructed directly by evaluating the Feynman diagram of Fig.
1(b). The electromagnetic and hadronic vertices can be obtained directly by
use of the Lagrangians given in Sec. II.1. The amplitudes for $\rho$ and
$\omega$ exchanges read

$\displaystyle{\cal M}_{V}^{\nu\mu}=$
$\displaystyle-i\,g_{NNV}\,g_{Vf_{1}\gamma}\,\epsilon^{\alpha\beta\mu\nu}\,k_{\alpha}\,t\,\frac{-g^{\beta\eta}+q^{\beta}q^{\eta}/m_{V}^{2}}{t-m_{V}^{2}}$
$\displaystyle\times\left[\gamma^{\eta}-i\frac{\kappa_{V}}{2M_{N}}\sigma^{\eta\delta}q_{\delta}\right],$
(30)

where $V$ represents $\rho$ or $\omega$, and $q$ denotes the four-momentum of
the exchanged vector meson.

####  II.4.2 Regge model

In high energy region, the differential cross sections are dominated at
forward angles, where the effects of high-spin meson exchanges matter. An
economic way to describe the $t$\- and $s$-dependence of the cross sections in
forward direction is Regge phenomenology Huang:2009 ; Sibirtsev:2009 ;
Huang:2010 . The standard Reggeization of the $t$-channel Feynman amplitudes
of $\rho$ and $\omega$ exchanges corresponds to the following replacements of
the propagators:

$\displaystyle\frac{1}{t-m^{2}_{V}}\quad\Longrightarrow\quad{\cal P}^{V}_{\rm
R}=$
$\displaystyle\;\left(\frac{s}{s_{0}}\right)^{\alpha_{V}(t)-1}\frac{\pi\alpha^{\prime}_{V}}{\sin[\pi\alpha_{V}(t)]}$
$\displaystyle\;\times\frac{1}{\Gamma[\alpha_{V}(t)]}.$ (31)

Here $s_{0}$ is a mass scale which is conventionally taken as $s_{0}=1$ GeV2,
and $\alpha^{\prime}_{V}$ is the slope of the Regge trajectory
$\alpha_{V}(t)$. For $V=\rho$ and $\omega$, the trajectories are parametrized
as Guidal:1997

$\displaystyle\alpha_{\rho}(t)$ $\displaystyle=0.55+0.8~{}{\rm GeV}^{-2}t,$
(32) $\displaystyle\alpha_{\omega}(t)$ $\displaystyle=0.44+0.9~{}{\rm
GeV}^{-2}t.$ (33)

Note that in Eq. (31) the degenerate trajectories are employed for $\rho$ and
$\omega$ exchanges, thus the corresponding signature factors reduce to $1$.
Such a choice is preferred by data, which has been tested by our numerical
calculation.

####  II.4.3 Interpolated Regge model

It is believed that Regge model works properly in the large-$s$ and
small-$|t|$ region, and Feynman model works properly in the low-energy region.
In literature, the so-called interpolated Regge model is widely used Toki:2008
; Nam:2010 ; He:2012 ; Wang:2020 . In this model, an interpolating form factor
is introduced to parametrize the smooth transition from Feynman amplitudes to
Regge amplitudes. Instead of the replacement given in Eq. (31), in the
interpolated Regge model, one has the following replacement of the propagators
of $t$-channel $\rho$ and $\omega$ exchanges in Feynman amplitudes:

$\displaystyle\frac{1}{t-m^{2}_{V}}\quad\Longrightarrow\quad{\cal P}^{V}_{\rm
IR}={\cal P}^{V}_{\rm R}F+\frac{1}{t-m^{2}_{V}}\left(1-F\right),$ (34)

where $F=F_{s}F_{t}$ with

$\displaystyle F_{s}$
$\displaystyle=\frac{1}{1+e^{-\left(s-s_{R}\right)/s_{0}}},$ (35)
$\displaystyle F_{t}$
$\displaystyle=\frac{1}{1+e^{-\left(t+t_{R}\right)/t_{0}}}.$ (36)

Here $s_{R}$, $s_{0}$, $t_{R}$, and $t_{0}$ are parameters to be fixed by
fitting the data. The parameters $s_{R}$ and $t_{R}$ denote where do the
amplitudes transit from Feynman type to Regge type, and $s_{0}$ and $t_{0}$
indicate how fast do this transit occur.

It is seen that by making the replacement of Eq. (34), the $t$-channel
amplitude is a combination of Regge amplitude and Feynman amplitude, with a
weight factor $F$ for the former and $(1-F)$ for the latter. In the low-energy
$(s\ll s_{R})$ region, the factor $F$ tends towards $0$, ensuring that one has
almost pure Feynman amplitude. In the high-energy $(s\gg s_{R})$ and
small-$|t|$ $(|t|\ll t_{R})$ region, the factor $(1-F)$ tends towards $0$,
ensuring that one has almost pure Regge amplitude. In the intermediate energy
region, the amplitude is constructed as a mixture of Feynman amplitude and
Regge amplitude, which transits smoothly to Feynman amplitude in low-energy
region and Regge amplitude in high-energy region.

##  III Results and discussion

In the present work, we perform a comprehensive investigation of the $\gamma
p\to f_{1}(1285)p$ reaction within an effective Lagrangian approach. The
contributions from the $s$-channel $N$-pole diagram, $u$-channel $N$-exchange
diagram, $t$-channel $\rho$\- and $\omega$-exchange diagrams, and the
interaction current are considered as background ingredients in constructing
the reaction amplitudes. We build three reaction models, i.e., the Feynman
model, the Regge model, and the interpolated Regge model, where the
$t$-channel reaction amplitudes of $\rho$ and $\omega$ exchanges are built in
Feynman type, Regge type, and interpolated Regge type, respectively (cf. Sec.
II.4). In each model, we consider pole diagrams in the $s$ channel of as few
as possible nucleon resonances in order to achieve satisfactory descriptions
of the available differential cross-section data.

In the most recent PDG review Workman:2022 , there are nine nucleon resonances
with spin $J=1/2\sim 7/2$ in the energy region considered in the present work,
namely the $N(1990){7/2}^{+}$, $N(2000){5/2}^{+}$, $N(2040){3/2}^{+}$,
$N(2060){5/2}^{-}$, $N(2100){1/2}^{+}$, $N(2120){3/2}^{-}$,
$N(2190){7/2}^{-}$, $N(2300){1/2}^{+}$, and $N(2570){5/2}^{-}$ resonances. As
it is not clear how many nucleon resonances and what nucleon resonances are
really needed in the $\gamma p\to f_{1}(1285)p$ reaction, we follow the
strategy employed in Refs. Zhangy:2021 ; Wei:2022 ; Wang:2022 ; Wei:2020 ;
Wangx:2020 ; Zhangyx:2021 ; Wein:2021 ; Weinc:2019  to choose nucleon
resonances, i.e., we strive to describe the available data by introducing as
few as possible nucleon resonances in our theoretical models. In practice, we
make numerous trials of various number and different combination of these nine
nucleon resonances in our calculations. Firstly, we try to reproduce the data
without considering any nucleon resonance. If fail, we introduce one nucleon
resonance to the model by testing the above-mentioned nine nucleon resonances
one by one. If the data still cannot be well reproduced, various combinations
of two nucleon resonances will be further considered in the model. In
principle, more nucleon resonances should be further introduced in the
calculations unless the data can be satisfactorily described. Nevertheless,
for the $\gamma p\to f_{1}(1285)p$ reaction, so far, we only have differential
cross-section data with relatively large error bars at $5$ energy points.
Adding more than two nucleon resonances to the model will bring in too many
parameters (masses, widths, cutoffs, and coupling constants, et al.) to the
model which are difficult to be well determined by the available (very
limited) data. Therefore, in each model we stop introducing more than two
nucleon resonances if the data cannot be satisfactorily described.

In the following, we show and discuss the results from each model separately.

###  III.1 Results in Feynman model

Figure 2: Differential cross sections of $\gamma p\to f_{1}(1285)p$ as a function of cosine of scattering angle $\theta$ in center-of-mass frame obtained in the Feynman model with the contributions from the $N(2000){5/2}^{+}$ and $N(2060){5/2}^{-}$ resonances being considered. The black solid lines represent the results from the full calculation. The blue dash-double-dotted, green dashed, and red dash-dotted lines represent the individual contributions from $t$-channel $\rho$\- and $\omega$-exchange diagrams, $s$-channel $N(2060){5/2}^{-}$-pole diagram, and $s$-channel $N(2000){5/2}^{+}$-pole diagram, respectively. The scattered symbols denote the CLAS data in Ref. Dickson:2016gwc . The numbers in parentheses denote the centroid value of the photon laboratory incident energy (left number) and the corresponding total center-of-mass energy of the system (right number), in MeV. Table 1:  Fitted values of parameters in Feynman model with the $N(2000){5/2}^{+}$ and $N(2060){5/2}^{-}$ resonances being considered. The asterisks below resonance names denote the overall status of these resonances evaluated by the PDG review Workman:2022 . The numbers in brackets below resonance mass $M_{R}$ and width $\Gamma_{R}$ represent the range of the corresponding quantities given by the PDG review Workman:2022 . $\kappa_{f_{1}}$ | $-5.96\pm 5.0$ |
---|---|---
$\Lambda_{N}$ [MeV] | $578\pm 8$ |
$\Lambda_{\rho}$ [MeV] | $769\pm 1$ |
$\Lambda_{\omega}$ [MeV] | $834\pm 3$ |
| $N(2000){5/2}^{+}$ | $N(2060){5/2}^{-}$
|  $\ast$$\ast$ |  $\ast$$\ast$$\ast$
$M_{R}$ [MeV] | $2250\pm 3$ | $2030\pm 7$
| [$\sim 2000$] | [$2030\sim 2200$]
$\Gamma_{R}$ [MeV] | $126\pm 30$ | $450\pm 52$
| [$\sim 300$] | [$\sim 400$]
$\Lambda_{R}$ [MeV] | $1396\pm 28$ | $1399\pm 59$
$g_{RN\gamma}^{(1)}g_{RNf_{1}}$ | $0.013\pm 0.002$ | $-18.5\pm 0.8$
$g_{RN\gamma}^{(2)}g_{RNf_{1}}$ | $16.8\pm 0.7$ | $-61.0\pm 2.0$

In the Feynman model, it is expected that the near threshold structures
exhibited in the angular distributions of $\gamma p\to f_{1}(1285)p$ are
dominated by contributions from nucleon resonances, and the cross-section
peaks in the high-energy region at forward angles are governed by
contributions from $t$-channel $\rho$ and $\omega$ exchanges. However, after
numerous tests, we found that in the cases when no nucleon resonance or any
one or any two of the above-mentioned nine nucleon resonances are introduced
in the calculations, the available differential cross-section data for $\gamma
p\to f_{1}(1285)p$ from CLAS Collaboration Dickson:2016gwc  cannot be well
reproduced in the Feynman model.

In Fig. 2, we show the results of differential cross sections of $\gamma p\to
f_{1}(1285)p$ as a function of cosine of scattering angle $\theta$ in center-
of-mass frame obtained in the Feynman model with the $N(2000){5/2}^{+}$ and
$N(2060){5/2}^{-}$ contributions being considered. The fit with these two
nucleon resonances being considered is the best one as the corresponding
$\chi^{2}/N$ ($\chi^{2}$ per data point) is the smallest among all the fits
with no nucleon resonance, one nucleon resonance, or two nucleon resonances
being considered. In Fig. 2, the black solid lines represent the results from
the full calculation. The blue dash-double-dotted, green dashed, and red dash-
dotted lines represent the individual contributions from $t$-channel $\rho$\-
and $\omega$-exchange diagrams, $s$-channel $N(2060){5/2}^{-}$-pole diagram,
and $s$-channel $N(2000){5/2}^{+}$-pole diagram, respectively. The scattered
symbols denote the CLAS data from Ref. Dickson:2016gwc . The numbers in
parentheses denote the centroid value of the photon laboratory incident energy
(left number) and the corresponding total center-of-mass energy of the system
(right number), in MeV. One sees from Fig. 2 that the differential cross-
section data in low-energy region can be well described, while the data in
high-energy region near the forward angles are too much underestimated.
Therefore, this set of results is not considered as acceptable. The model
parameters corresponding to this set of theoretical results are listed in
Table 1. One sees that the fitted mass and width of $N(2000){5/2}^{+}$ are a
little bit far away from the corresponding values advocated in the PDG review
Workman:2022 . If we manually restrict these parameters in a range as
advocated in the PDG review Workman:2022  in the fitting procedure, the
agreement of the fitted differential cross sections with the corresponding
data will be even worse than those shown in Fig. 2.

The reason that the differential cross-section data of $\gamma p\to
f_{1}(1285)p$ in high-energy region near forward angles cannot be well
reproduced in the Feynman model is possibly that the Feynman type amplitudes
of $t$-channel $\rho$ and $\omega$ exchanges do not result in proper angular
dependence of the high-energy differential cross sections as demonstrated by
the data. As so far we only have differential cross-section data at $5$ energy
points, introducing more nucleon resonances to this model to improve the
quality of the theoretical description of the data does not make too much
sense. We thus conclude that the available differential cross-section data of
$\gamma p\to f_{1}(1285)p$ cannot be well reproduced in Feynman model.

###  III.2 Results in Regge model

Table 2:  $\chi^{2}/N$ for differential cross sections fitted by including one nucleon resonance in Regge model. The asterisks below resonance names denote the overall status of these resonances rated by the PDG review Workman:2022 . $N^{\ast}$ | $N(1990){7/2}^{+}$ | $N(2000){5/2}^{+}$ | $N(2040){3/2}^{+}$
---|---|---|---
|  $\ast$$\ast$ |  $\ast$$\ast$ | $\ast$
$\chi^{2}/N$ | $6.5$ | $4.0$ | $4.1$
$N^{\ast}$ | $N(2060){5/2}^{-}$ | $N(2100){1/2}^{+}$ | $N(2120){3/2}^{-}$
|  $\ast$$\ast$$\ast$ |  $\ast$$\ast$$\ast$ |  $\ast$$\ast$$\ast$
$\chi^{2}/N$ | $5.1$ | $6.5$ | $4.9$
$N^{\ast}$ | $N(2190){7/2}^{-}$ | $N(2300){1/2}^{+}$ | $N(2570){5/2}^{-}$
|  $\ast$$\ast$$\ast$$\ast$ |  $\ast$$\ast$ |  $\ast$$\ast$
$\chi^{2}/N$ | $6.2$ | $4.7$ | $4.6$
Figure 3: Differential cross sections of $\gamma p\to f_{1}(1285)p$ as a function of cosine of scattering angle $\theta$ in center-of-mass frame obtained in the Regge model with the contribution from $s$-channel $N(2000)5/2^{+}$-pole diagram being considered. Notations are the same as in Fig. 2. Table 3:  Fitted values of parameters in Regge model (second column) and interpolated Regge model (third column) with the $N(2000){5/2}^{+}$ resonance being considered. The asterisks below resonance names denote the overall status of these resonances evaluated by the PDG review Workman:2022 . The numbers in brackets below resonance mass $M_{R}$ and width $\Gamma_{R}$ represent the range of the corresponding quantities given by the PDG review Workman:2022 . | Regge | interpolated Regge
---|---|---
$\kappa_{f_{1}}$ | $14.4\pm 7.0$ | $14.6\pm 5.0$
$\Lambda_{N}$ [MeV] | $500\pm 103$ | $500\pm 124$
$\Lambda_{\rho}$ [MeV] | $1083\pm 17$ | $1079\pm 5$
$\Lambda_{\omega}$ [MeV] | $745\pm 83$ | $749\pm 51$
| $N(2000){5/2}^{+}$ | $N(2000){5/2}^{+}$
|  $\ast$$\ast$ |  $\ast$$\ast$
$M_{R}$ [MeV] | $2042\pm 33$ | $2042\pm 11$
| [$\sim 2000$] | [$\sim 2000$]
$\Gamma_{R}$ [MeV] | $450\pm 57$ | $450\pm 72$
| [$\sim 300$] | [$\sim 300$]
$\Lambda_{R}$ [MeV] | $1167\pm 17$ | $1172\pm 3$
$g_{RN\gamma}^{(1)}g_{RNf_{1}}$ | $-64.7\pm 0.5$ | $-62.8\pm 0.5$
$g_{RN\gamma}^{(2)}g_{RNf_{1}}$ | $27.9\pm 1.5$ | $28.6\pm 1.5$
Figure 4: Predicted total cross sections with dominant individual
contributions for $\gamma p\to f_{1}(1285)p$ in Regge model with the
contribution from $s$-channel $N(2000){5/2}^{+}$-pole diagram being
considered. Notations are the same as in Fig. 3. Figure 5: Predicted target
nucleon asymmetries as functions of $\cos\theta$ for $\gamma p\to
f_{1}(1285)p$ at two selected energies. The numbers in parentheses denote the
photon laboratory incident energy (left number) and the total center-of-mass
energy of the system (right number), in MeV. The black solid, blue dotted,
green dashed, red long-dashed, cyan dot-dashed, brown dot-long-dashed, magenta
dash-double-dotted, violet dot-double-dashed, and orange solid curves
represent the results in Regge model with consideration of the contribution
from $s$-channel $N(2100)1/2^{+}$, $N(2300)1/2^{+}$, $N(2040)3/2^{+}$,
$N(2120)3/2^{-}$, $N(2000)5/2^{+}$, $N(2060)5/2^{-}$, $N(1990)7/2^{+}$,
$N(2190)7/2^{-}$, and $N(2570){5/2}^{-}$, respectively.

As indicated in Sec. II.4.2, the Regge type amplitudes of $t$-channel $\rho$
and $\omega$ exchanges have rather different angular dependence than the
Feynman type amplitudes. Thus, although the Feynman model fails, the Regge
model may still have the opportunity to describe the available differential
cross-section data for $\gamma p\to f_{1}(1285)p$.

We made numerous tests and found that if no contribution from $s$-channel
$N^{\ast}$-pole diagram is considered in the Regge model, the resulted
$\chi^{2}/N$ will be greater than $10$, indicating significantly poor fitting
quality of the obtained results. This means that in Regge model when no
nucleon resonance is considered, the shapes of angular distributions in low-
energy region and high-energy region cannot be simultaneously described by
non-resonant contributions, which are mainly the $t$-channel $\rho$ and
$\omega$ exchanges, only. We then tried to add one of those nine nucleon
resonances as mentioned in the text above Sec. III.1 to this model. We tested
one by one, and it was found that by adding any one of those nine nucleon
resonances in the Regge model, the available differential cross-section data
of $\gamma p\to f_{1}(1285)p$ can always be satisfactorily described with
similar fitting qualities, $4.0<\chi^{2}/N<6.5$, as listed in Table 2.

In Fig. 3, we show the theoretical results, compared with the corresponding
data, for differential cross sections of $\gamma p\to f_{1}(1285)p$ as a
function of cosine of scattering angle $\theta$ in center-of-mass frame
obtained in the Regge model with the contribution from $s$-channel
$N(2000)5/2^{+}$-pole diagram being considered. The notations of this figure
are the same as in Fig. 2. The fitted values of model parameters correspond to
this set of results are listed in the second column of Table 3. It is seen
that the fitted resonance mass and width are consistent with the values
advocated in the most recent PDG review Workman:2022 . Note that the resonance
electromagnetic and hadronic couplings are not shown separately, since in a
single-channel calculation as the one performed in the present work, the
reaction amplitudes depend only on the products of them. In PDG review
Workman:2022 , no electromagnetic decay amplitudes or $f_{1}(1285)p$ hadronic
branching ratio for the two-star resonance $N(2000){5/2}^{+}$ are advocated.
If we use the helicity amplitudes $A_{1/2}=0.031$ GeV-1/2 and $A_{3/2}=-0.043$
GeV-1/2 from a BnGa partial wave analysis Sokhoyan:2015  to fix the
electromagnetic couplings, we get $g_{RN\gamma}^{(1)}=-3.9$,
$g_{RN\gamma}^{(2)}=3.6$. A refit of the hadronic coupling results in
$g_{RNf_{1}}=19.8$, while the cross-section results change very little.

One sees from Fig. 3 that the angular distribution data for $\gamma p\to
f_{1}(1285)p$ in the whole energy region considered are quite well described
in the Regge model with the $N(2000)5/2^{+}$ resonance being considered. In
the high-energy region, the peaks at forward angles are dominated by the
$t$-channel $\rho$ and $\omega$ Regge exchanges. In the low-energy region, the
resonance contribution also provides significant contributions to the observed
cross sections. Note that compared with the contributions from the $t$-channel
$\rho$\- and $\omega$-exchange diagrams in the Feynman model (cf. Fig. 2), now
the contributions from the $t$-channel $\rho$\- and $\omega$-trajectory
exchanges in the Regge model provide much steeper uplifts of the angular
distributions at forward angles, which is the main reason why the Regge model
succeeds in reproducing the data while the Feynman model fails. Considering
that a meson trajectory exchange is an economic treatment of the exchanges of
all mesons lying on the trajectory, such a fact may indicate that for the
$\gamma p\to f_{1}(1285)p$ reaction, contributions from the $t$-channel
exchanges of high-spin mesons are important. The $\chi^{2}/N$ is $4.0$ for
this set of solution. For the other eight sets of solutions, each with one of
the other eight nucleon resonances being considered, the resulted $\chi^{2}/N$
are between $4.0$ and $6.5$. They all reproduce the available differential
cross-section data for $\gamma p\to f_{1}(1285)p$ satisfactorily. The reaction
mechanisms in those sets of solutions are shown to be quite similar — the
$t$-channel $\rho$ and $\omega$ exchanges dominate the high-energy peaks at
forward angles, and the $s$-channel resonance contribution provides
considerable contributions in low-energy region. Nevertheless, the available
differential cross-section data are not sufficient to distinguish those
different sets of solutions in the Regge model.

The predicted total cross sections with dominant individual contributions for
$\gamma p\to f_{1}(1285)p$ in Regge model with the $N(2000){5/2}^{+}$
resonance being considered are shown in Fig. 4. One sees that in the high-
energy region, it is the $t$-channel interaction that dominates the cross
sections, while in the low-energy region, the $N(2000){5/2}^{+}$ resonance
also contributes significantly. This observation is consistent with the
behavior of the differential cross sections as shown in Fig. 3. The results
for models with other nucleon resonances are very close to those with the
$N(2000){5/2}^{+}$ resonance.

In Fig. 5 we show the predictions of target nucleon asymmetries at two
selected energies in Regge model with each of the $N(2100){1/2}^{+}$,
$N(2300){1/2}^{+}$, $N(2040){3/2}^{+}$, $N(2120){3/2}^{-}$,
$N(2000){5/2}^{+}$, $N(2060){5/2}^{-}$, $N(1990){7/2}^{+}$,
$N(2190){7/2}^{-}$, and $N(2570){5/2}^{-}$ resonances. One sees that the signs
and magnitudes of the target nucleon asymmetries predicted in Regge model with
various nucleon resonances are different. Future data on this quantity are
expected to be used to distinguish the Regge models with various nucleon
resonances.

###  III.3 Results in interpolated Regge model

Figure 6: Differential cross sections of $\gamma p\to f_{1}(1285)p$ as a function of cosine of scattering angle $\theta$ in center-of-mass frame obtained in the interpolated Regge model with no resonance being considered. Notations are the same as in Fig. 2. Table 4:  Fitted values of auxiliary parameters introduced in interpolated Regge model. See Sec. II.4.3 for the definitions of these parameters. All numbers are in GeV2. $s_{0}$ | $s_{R}$ | $t_{0}$ | $t_{R}$
---|---|---|---
$0.26\pm 0.35$ | $0.80\pm 2.30$ | $0.05\pm 0.70$ | $10.20\pm 6.25$

As mentioned in the introduction section, two works in literature
Wangxiaoyun:2017 ; Wangyanyan:2017  have already been devoted to the study of
$\gamma p\to f_{1}(1285)p$ photoproduction reaction in the interpolated Regge
model. Although these two theoretical works used the same reaction model,
their results and conclusions are quite different. In Ref. Wangxiaoyun:2017 ,
it was claimed that the available differential cross-section data for $\gamma
p\to f_{1}(1285)p$ can be reproduced without considering any nucleon
resonances. However, in Ref. Wangyanyan:2017 , it was argued that the
$N(2300){1/2}^{+}$ resonance plays an important role in describing the
available differential cross-section data for $\gamma p\to f_{1}(1285)p$. In
the present work, we also perform a comprehensive investigation of $\gamma
p\to f_{1}(1285)p$ in interpolated Regge model, with special attention paying
to whether the available data for $\gamma p\to f_{1}(1285)p$ can be reproduced
in interpolated Regge model without or with the contributions from any nucleon
resonances.

As the interpolated Regge model contains four more parameters compared with
the Regge model [cf. Eqs. (35) and (36) in Sec. II.4.3], naively, one may
expect that the interpolated Regge model can describe the data better than the
Regge model. However, this is not the truth.

Firstly, we varied the parameters in non-resonant contributions by use of
MINUIT to fit the data in the interpolated Regge model without considering any
nucleon resonances. After numerous trials, it was found that the fitting
quality of the results is poor, as the $\chi^{2}/N$ is $7.9$. The fitted
results are illustrated in Fig. 6, from which one sees clearly that the
available differential cross-section data for $\gamma p\to f_{1}(1285)p$
cannot be reasonably described in the interpolated Regge model when no
$s$-channel $N^{\ast}$-pole diagrams are considered. Such a conclusion
contradicts with Ref. Wangxiaoyun:2017  but is in agreement with Ref.
Wangyanyan:2017 . Note that in Ref. Wangxiaoyun:2017 , the contributions from
the $NN\gamma$ vector coupling [$\propto{\bar{N}}\gamma_{\mu}A^{\mu}N$, cf.
Eq. (9)] in $s$-channel $N$-pole diagram and $u$-channel $N$-exchange diagram,
and the contact term [cf. Eqs. (1) and (2)] were totally omitted, which sounds
need further justification.

We then tried to add one of those nine nucleon resonances as mentioned in the
text above Sec. III.1 to the interpolated Regge model to reproduce the data.
Surprisingly, for each set of results, the obtained $\chi^{2}/N$ is very close
to that of the solution in the Regge model with the same resonance content.
Actually, the corresponding values of $\chi^{2}/N$ in these two models only
differ in the second digits (see Table 2 for $\chi^{2}/N$ in Regge model).
Moreover, the fitted values of the model parameters in the interpolated Regge
model are also very close to those in the Regge model when the same nucleon
resonance is considered. As an example, the fitted values of the model
parameters with the $N(2000){5/2}^{+}$ resonance considered in the
interpolated Regge model are listed in the third column of Table 3. One sees
that they are very close to those listed in the second column of this table
for the Regge model.

This unusual phenomenon is not difficult to be understood from the fitted
values of those four auxiliary parameters introduced in the interpolated Regge
model, as listed in Table 4. The parameter $s_{R}$ indicates that around which
energy point does the Feynman amplitude transit to the Regge amplitude, and
the parameter $s_{0}$ demonstrates how fast does this transition occur. From
Table 4 one sees that for the $\gamma p\to f_{1}(1285)p$ reaction, the
$t$-channel amplitudes transit from Feynman type to Regge type around
$s_{R}=0.8$ GeV2, and the transition occurs very fast as $s_{0}$ is a very
small number, $0.26$ GeV2. This means that above the $f_{1}(1285)p$ threshold,
i.e., $s\sim 4.95$ GeV2, the $t$-channel amplitudes are almost purely in Regge
type. In other words, the so-called interpolated Regge model is in substance
the Regge model for the $\gamma p\to f_{1}(1285)p$ reaction. We observe that
in Ref. Wangyanyan:2017 , the values of these two parameters also manifest
that the interpolated Regge model claimed by the authors is essentially the
Regge model.

If we manually force the parameter $s_{R}$ greater than the $f_{1}(1285)p$
threshold value $\sim 4.95$ GeV2 and $s_{0}$ not that small in the fitting
procedure, so that the reaction amplitudes indeed have the spirits of
interpolated Regge model, i.e., they transit smoothly from Feynman type in
low-energy region to Regge type in high-energy region, the resulted
$\chi^{2}/N$ will be greater than $10.0$, and consequently the fitting quality
is too inferior to be accepted. We thus conclude that the available angular
distribution data for $\gamma p\to f_{1}(1285)p$ cannot be reproduced in
interpolated Regge model.

##  IV Summary and conclusion

The CLAS differential cross-section data for $\gamma p\to f_{1}(1285)p$
reaction Dickson:2016gwc  have so far been analyzed in Refs. Wangxiaoyun:2017
; Wangyanyan:2017 . Although both of these two theoretical works used the
interpolated Regge model, their conclusions are quite different. In Ref.
Wangxiaoyun:2017 , it was claimed that the available data for $\gamma p\to
f_{1}(1285)p$ can be reproduced in the interpolated Regge model without
considering any nucleon resonances. However, in Ref. Wangyanyan:2017 , it was
argued that the $N(2300)1/2^{+}$ resonance is needed in the interpolated Regge
model to describe the data.

In the present work, we performed a comprehensive investigation of the $\gamma
p\to f_{1}(1285)p$ reaction in an effective Lagrangian approach. We considered
the contributions from $s$-channel $N$-pole diagram, $u$-channel $N$-exchange
diagram, $t$-channel $\rho$\- and $\omega$-exchange diagrams, and generalized
interaction current in constructing the non-resonant reaction amplitudes. We
built three reaction models, i.e., the Feynman model, the Regge model, and the
interpolated Regge model, where the $t$-channel amplitudes are constructed in
Feynman type, Regge type, and interpolated Regge type, respectively. In each
model, we introduced as few as possible nucleon resonances to reproduce the
data. All nine nucleon resonances listed in the most recent PDG review
Workman:2022  with spin $J=1/2\sim 7/2$ in the energy region considered in the
present work, namely the $N(1990){7/2}^{+}$, $N(2000){5/2}^{+}$,
$N(2040){3/2}^{+}$, $N(2060){5/2}^{-}$, $N(2100){1/2}^{+}$,
$N(2120){3/2}^{-}$, $N(2190){7/2}^{-}$, $N(2300){1/2}^{+}$, and
$N(2570){5/2}^{-}$ resonances, are allowed in practice. The purpose of the
present work was to find concise and clear answers of the following questions:
i) Whether the available differential cross-section data for $\gamma p\to
f_{1}(1285)p$ can be reproduced in Feynman model, Regge model, and
interpolated Regge model, respectively? ii) If yes, how many nucleon
resonances are at least needed and what are the resonance contents and
associated parameters in these three models?

Our results showed that the available angular distribution data for $\gamma
p\to f_{1}(1285)p$ from the CLAS Collaboration Dickson:2016gwc  cannot be well
reproduced in either Feynman model or interpolated Regge model. In the Regge
model, when no nucleon resonance is considered, the data cannot be
satisfactorily described either. However, when any one of the
$N(1990){7/2}^{+}$, $N(2000){5/2}^{+}$, $N(2040){3/2}^{+}$,
$N(2060){5/2}^{-}$, $N(2100){1/2}^{+}$, $N(2120){3/2}^{-}$,
$N(2190){7/2}^{-}$, $N(2300){1/2}^{+}$, and $N(2570){5/2}^{-}$ resonances is
taken into account in the Regge model, the experimental data can be well
described. The resulted resonance masses and widths are consistent with those
advocated in PDG review Workman:2022 . In each set of results in the Regge
model, it was found that in the high-energy region, the peaks of differential
cross sections at forward angles are dominated by $t$-channel $\rho$ and
$\omega$ exchanges, and in the low-energy region, the resonance contribution
provides significant contributions also.

The fact that the available differential cross-section data for $\gamma p\to
f_{1}(1285)p$ can only be reproduced in Regge model but not Feynman model or
interpolated Regge model reveals that the angular and energy dependence of the
$f_{1}(1285)$ photoproduction amplitudes is rather unusual. This indicates
that the $f_{1}(1285)$ meson might have an unconventional $q{\bar{q}}$ content
which deserves further investigations.

###### Acknowledgements.

This work is partially supported by the National Natural Science Foundation of
China under Grants No. 12175240, No. 12147153, and No. 11635009, the
Fundamental Research Funds for the Central Universities, and the China
Postdoctoral Science Foundation under Grants No. 2021M693141 and No.
2021M693142.

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