Study on the reaction of $\gamma p \to f_1(1285) p$ in Regge-effective Lagrangian approach

TL;DR

This study investigates the photoproduction of the $f_1(1285)$ resonance on protons using a Regge-effective Lagrangian approach, highlighting the significance of the $N(2300)$ resonance in the reaction dynamics.

Contribution

It introduces a comprehensive model including various channels and fits to experimental data, emphasizing the role of the $N(2300)$ resonance in $f_1(1285)$ production.

Findings

The $s$-channel $N(2300)$ term is crucial for explaining the data.

A bump in the cross section around $W=2.3$ GeV indicates the $N(2300)$ contribution.

The reaction can serve as a probe for studying the $N(2300)$ resonance.

Abstract

The production of the $f_1(1285)$ resonance in the reaction of $\gamma p \rightarrow f_1(1285) p$ is investigated within a Regge-effective Lagrangian approach. Besides the contributions of the $t$-channel $\rho$ and $\omega$ trajectories exchanges, we also take into account the contributions of $s/u$-channel $N(2300)$ terms, $s/u$-channel nucleon terms, and the contact term. By fitting to the CLAS data, we find that the $s$-channel $N(2300)$ term plays an important role in this reaction. We predict the total cross section for this reaction, and find a clear bump structure around $W=2.3$ GeV, which is associated with the $N(2300)$ state. The reaction of $\gamma p \to f_1(1285) p$ could be useful to further study of the $N(2300)$ experimentally.

Tables

Table 1. Table 1: The fitted parameters in this work.

	Fit A	Fit B
$g_{N^{\ast }}$ (GeV-1)	-0.052 $\pm$ 0.006	$-$
$g_t$ (GeV${}_{}{}^{-2})$	0.335 $\pm$ 0.068	-0.443 $\pm$ 0.117
$g_{f_{1}NN}$	0.347 $\pm$ 0.258	0.110 $\pm$ 0.016
${\kappa }_N$	-4.034 $\pm$ 2.459	9.999 $\pm$ 9.801
${\Lambda }_{N^{\ast }}$ (GeV)	1.354 $\pm$ 0.269	$-$
${\Lambda }_N$ (GeV)	1.285 $\pm$ 0.216	1.540 $\pm$ 0.157
${\Lambda }_t$ (GeV)	0.582 $\pm$ 0.219	0.610 $\pm$ 0.187
$t_0$ (GeV${}_{}{}^2)$	2.153 $\pm$ 0.445	1.612 $\pm$ 0.507
$\Delta t$ (GeV${}_{}{}^2)$	0.736 $\pm$ 0.433	0.850 $\pm$ 0.344
${\chi }^2$/dof	1.05	1.89

Equations

$${\mathcal{L}}_{VNN}=g_{VNN}{\bar{\psi }}_N{\gamma }_{\mu }{\psi }_N{V}^{\mu }+\frac{g_V^T}{2M_N}{\bar{\psi }}_N{\sigma }_{\mu \nu }{\psi }_N{V}^{\mu \nu },$$

$${\mathcal{L}}_{\gamma V{f}_1}=g_{\gamma V{f}_1}{q}_V^2{ϵ}_{\mu \nu \alpha \beta }{ϵ}_V^{\nu }{\xi }^{\beta }{ϵ}_{\gamma }^{\alpha }{p}_1^{\mu },$$

$${\mathcal{L}}_{\gamma N{N}^{\ast }}=\frac{e{g}_{\gamma N{N}^{\ast }}}{2M_N}{\bar{\psi }}_{N^{\ast }}{\sigma }_{\mu \nu }{\partial }^{\nu }{A}^{\mu }{\psi }_N+H.c.,$$

$${\mathcal{L}}_{f_{1}N{N}^{\ast }}=g_{f_{1}N{N}^{\ast }}{\bar{\psi }}_{N^{\ast }}{\gamma }_5{\sigma }_{\mu \nu }{\partial }^{\nu }{f}_1^{\mu }{\psi }_N+H.c.,$$

$${\mathcal{L}}_{\gamma NN}=-e{\bar{\psi }}_{N^{\ast }}[{\gamma }^{\mu }{A}_{\mu }-\frac{{\kappa }_N}{2M_N}{\sigma }_{\mu \nu }{\partial }^{\nu }{A}^{\mu }]{\psi }_N+H.c.,$$

$${\mathcal{L}}_{f_{1}NN}=g_{f_{1}NN}{\bar{\psi }}_{N^{\ast }}{\gamma }^{\mu }{\gamma }_5{f}_{1\mu }{\psi }_N+H.c.,$$

$$g_{\omega }^T=0,$$

$$g_{\rho }^T/g_{\rho NN}={\kappa }_{\rho }=\mathrm{6.1.}$$

$${\mathcal{F}}_{N^{\ast }}(q^2)=\frac{{\Lambda }_{N^{\ast }}^4}{{\Lambda }_{N^{\ast }}^4+(q^2-M_{N^{\ast }}^2{)}^2},$$

$${\mathcal{F}}_N(q^2)=\frac{{\Lambda }_N^4}{{\Lambda }_N^4+(q^2-M_N^2{)}^2},$$

$${\mathcal{F}}_{VNN}(q^2)=\frac{{\Lambda }_t^2-M_V^2}{{\Lambda }_t^2-q^2},$$

$${\mathcal{F}}_{V{f}_1\gamma }(q^2)={\left(\frac{{\Lambda }_t^2-M_V^2}{{\Lambda }_t^2-q^2}\right)}^2,$$

$${\mathcal{M}}_{N^{\ast }}^s$$

$${\mathcal{M}}_u^{N^{\ast }}$$

$${\mathcal{M}}_s^N$$

$${\mathcal{M}}_u^N$$

$${\mathcal{M}}_t^V$$

$${\mathcal{M}}_c$$

$$G_{N^{\ast }}(q^2)=i\frac{/q+M_{N^{\ast }}}{q^2-M_{N^{\ast }}^2+i{M}_{N^{\ast }}\Gamma },$$

$$G_N(q^2)=i\frac{/q+M_N}{q^2-M_N^2+i{M}_N\Gamma },$$

$$G_V^{\mu \nu }(q^2)=-i\frac{g^{\mu \nu }-q^{\mu }{q}^{\nu }/M_V^2}{q^2-M_V^2},$$

$$\mathcal{M}$$

$$\frac{\mathrm{d}\sigma }{\mathrm{d}\Omega }=\frac{M_N^2}{16{\pi }^{2}s}\frac{|{\vec{p}}_3^{\mathrm{cm}}|}{|{\vec{p}}_1^{\mathrm{cm}}|}\sum ^{ˉ}|\mathcal{M}{|}^2,$$

$$\frac{1}{q^2-M_V^2}\to {\left(\frac{s}{s_0}\right)}^{{\alpha }_V-1}\frac{\pi {\alpha }_V^{{}^{\prime }}}{\sin (\pi {\alpha }_V)\Gamma ({\alpha }_V)}{D}_V,$$

$${\alpha }_{\omega }(t)$$

$${\alpha }_{\rho }(t)$$

$$D_{\omega }(t)$$

$$D_{\rho }(t)$$

$${\mathcal{M}}_t^{\mathrm{H}}={\mathcal{M}}_V^{\mathrm{Regge}}\times \mathcal{R}+{\mathcal{M}}_t^V\times (1-\mathcal{R}),$$

$$\mathcal{R}$$

$${\mathcal{R}}_s$$

Full text

I Introduction

It has been known that the nucleon is a bound state of three valence quarks since 1970s. Many nucleon resonances, referred as $N^{*}$, have been observed Olive:2016xmw , the properties of nucleon resonances are the important issues in hadron physics, and attract lots of attentions Lutz:2015ejy ; Aznauryan:2012ba ; Klempt:2009pi . For the nucleon resonances with masses below 2 GeV, their properties have been widely investigated in literature. However, the current knowledge on the properties of excited nucleon states with masses above 2 GeV is scarce. On the other hand, many missing $N^{*}$s, predicted by the constituent quark models are not yet found Capstick:2000qj .

Recently, the CLAS Collaboration has measured the $f_{1}(1285)$ meson for the first time in photoproduction from a proton target, and presented the $f_{1}(1285)$ differential photoproduction cross section into $\eta\pi^{+}\pi^{-}$ final states from the threshold up to a center-of-mass (c.m.) energy of $W=2.8$ GeV Dickson:2016gwc . A cross section comparison for $\gamma p\to\eta^{\prime}(958)p\to\eta\pi^{+}\pi^{-}p$ and $\gamma p\to f_{1}(1285)p\to\eta\pi^{+}\pi^{-}p$ at $W=2.55$ GeV in Fig. 10 of Ref. Dickson:2016gwc shows that the $\eta^{\prime}(958)$ cross section exhibits much stronger $t$- and $u$-channel signatures in the angle dependence than dose the one of $f_{1}(1285)$, which is quite flat. This may imply that the $f_{1}(1285)$ photoproduction mechanism is not dominated alone by $t$-channel production processes.

Before the experimental study on $\gamma p\to f_{1}(1285)p$ Dickson:2016gwc , there are several theoretical works on this reaction. Within the Regge-model, considering the exchanges of $t$-channel $\rho$ and $\omega$ trajectories, Kochelev et al. calculated the differential cross sections of $\gamma p\to f_{1}(1285)p$ Kochelev:2009xz . Comparison of the Regge-model calculations and the CLAS data shows the $t$-channel production process alone does not reproduce the CLAS measurements. Within a model motivated by Chern-Simons-term-induced interactions in holographic QCD, Domokos et al. Domokos:2009cq predicted the differential cross sections of $\gamma p\to f_{1}(1285)p$. The predictions of Ref. Domokos:2009cq are much smaller than the CLAS data, even in the most forward region. Based on the effective-Lagrangian approach with tree-level $\rho$ and $\omega$ exchanges in $t$-channel Huang:2013jda , Huang et al. presented the differential cross sections as shown in Fig. 12 of Ref. Dickson:2016gwc . The results of Huang et al. are also much smaller than the CLAS data. In order to describe the CLAS data, the further model calculations are needed.

The differences between these model predictions and the CLAS data suggest that the $s$-channel intermediate baryon resonances may play an important role in the reaction of $\gamma p\to f_{1}(1285)p$. That is to say, the decay of the excited $N^{*}$ intermediate states may be important, as pointed out by the CLAS Collaboration Dickson:2016gwc . The reaction of $\gamma p\to f_{1}(1285)p$ filters the nucleon resonances with isospin $I=1/2$, and provides a natural mode to investigate the higher excited nucleon resonance with a mass above 2.2 GeV and a sizeable coupling to the final states $f_{1}(1285)p$.

Among the possible nucleon resonances $N^{*}$ [$N(2220)$ $9/2^{+}$, $N(2250)$ $9/2^{-}$, $N(2300)$ $1/2^{+}$] 333According to the PDG Olive:2016xmw , there are three $N^{*}$ [$N(2220)$ $9/2^{+}$, $N(2250)$ $9/2^{-}$, $N(2300)$ $1/2^{+}$] in the mass range of 2.2-2.5 GeV. The differential cross sections of $\gamma p\to f_{1}(1285)p$ above $W=2.5$ GeV are much forward, which should be dominated by the $t$-channel mesons exchanges, as shown in Fig. 12 of Ref. Dickson:2016gwc , the $s$-channel nucleon resonances with masses above 2.55 GeV are not expected to give the dominant contributions., the $N(2300)$ can couple to the $f_{1}(1285)p$ in the $S$ wave, while the other two states $N(2220)$ and $N(2250)$ couple to the $f_{1}(1285)p$ in the $F$ and $E$ waves, respectively. It would be expected that the contributions of $E$ and $F$ waves are strongly suppressed. Thus, we will consider the state $N(2300)$ as the intermediate state in the $\gamma p\to f_{1}(1285)p$ reaction.

In the present work, we shall study the reaction of $\gamma p\to f_{1}(1285)p$ within the Regge-effective Lagrangian approach by considering the $t$-channel $\rho$ and $\omega$ trajectories exchanges, the $s/u$-channel $N(2300)$ resonance mechanisms, the $s/u$-channel nucleon terms, and contact term.

The experimental information of the two-star444According the PDG Olive:2016xmw , the existence evidence of the baryon states with two stars is only fair. $N(2300)$ is very scarceOlive:2016xmw . Until now, it was observed only in the decay of $\psi(3686)\to p\bar{N}^{*}(\bar{p}N^{*})\to p\bar{p}\pi^{0}(\bar{p}p\pi^{0})$ by the BESIII Collaboration, and its mass and width are determined to be $2300^{+40}_{-30}$ MeV and $340\pm 30$ MeV, respectively Ablikim:2012zk . Searching for the $N(2300)$ state in other processes, for instance the photoproduction, could be useful to provide more information about the properties of $N(2300)$ state. As an isospin $1/2$ filter process, the $\gamma p\to f_{1}(1285)p$ is a potential mode to study the $N(2300)$ state.

This paper is organized as follows. In Sec. II, we discuss the formalism and the main ingredients of the Regge-effective Lagrangian approach. In Sec. III, the results and discussions are presented. Finally, a short summary is given in Sec. IV.

II FORMALISM AND INGREDIENTS

II.1 Feynman amplitudes

For the process $\gamma p\rightarrow f_{1}(1285)p$, we will take into account the basic tree level Feynman diagrams depicted in Fig. 1, where the $t$-channel $\rho$ and $\omega$ exchanges, the $s$- and $u$-channel $N(2300)$ terms, the $s$- and $u$-channel nucleon terms, and contact term are considered. The relevant effective Lagrangians of the vertices are given as Kochelev:2009xz ; Domokos:2009cq ; Kochelev:1999zf ; Kim:2014hha ,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $V^{\mu\nu}=\partial^{\mu}V^{\nu}-\partial^{\nu}V^{\mu}$, $f_{1}^{\mu}$ is axial-vector meson $f_{1}(1285)$ field, $A^{\mu}$ and $V^{\mu}$ are electromagnetic field and the vector meson ($\rho$ or $\omega$) field, respectively. $q_{V}$ is the momentum of the exchanged vector meson, and $p_{i}(i=1,2,3,4)$ are the four-momentum of the initial or final states, as shown in Fig. 1. $\epsilon_{V}$, $\xi$, $\epsilon_{\gamma}$ are the polarizations of the vector meson in $t$-channel, $f_{1}(1285)$, and the photon, respectively.

The numerical values of the coupling constants are taken from Ref. Sibirtsev:2004 :

[TABLE]

[TABLE]

Since the hadrons are not point-like particles, we need to include the form factors to describe the off-shell effects. We adopt here the form factors used in many previous works,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

These form factors are similar to those used in Refs. Xie:2013mua ; Xie:2010yk , and the same cut-off $\Lambda_{t}$ is used for the vertices of $VNN$ and $Vf_{1}\gamma$.

Then, according to the Feynman rules, the scattering amplitudes for the $\gamma p\rightarrow f_{1}(1285)p$ reaction can be obtained straightforwardly with the above effective Lagrangians,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $q_{s}=p_{1}+p_{2}$, $q_{u}=p_{2}-p_{3}$, and $q_{t}=p_{1}-p_{3}$. $G_{N^{*}}$ and $G_{N}$ are the propagators for the $N^{*}$ and proton, and $G^{\sigma\nu}$ is the propagators for the $\rho$ or $\omega$ meson. We also define $g_{N^{*}}\equiv\sqrt{4\pi}g_{\gamma NN^{*}}\times g_{f_{1}NN^{*}}$ and $g_{t}\equiv g_{VNN}\times g_{\gamma Vf_{1}}$ for convenience. With the SU(3) invariant Lagrangians and flavor symmetry, one can have $g_{\omega NN}=3g_{\rho NN}$. On the other hand, we have $g_{f_{1}\omega\gamma}\approx\frac{e_{u}+e_{d}}{e_{u}-e_{d}}g_{f_{1}\rho\gamma}=\frac{1}{3}\times g_{f_{1}\rho\gamma}$ Kochelev:2009xz , thus the $g_{t}$ is same for the $\rho$ and $\omega$ mesons.

The contact term is required to keep the full amplitude gauge invariant, and can be written as

[TABLE]

The propagator for the $N^{*}$ and proton term can be written as

[TABLE]

[TABLE]

and the one for vector meson $\rho$ or $\omega$ is

[TABLE]

The total amplitude for the process $\gamma p\rightarrow f_{1}(1285)p$ is the coherent sum of $\mathcal{M}_{s}^{N^{*}}$, $\mathcal{M}_{u}^{N^{*}}$, $\mathcal{M}_{t}^{\rho}$, $\mathcal{M}_{t}^{\omega}$, $\mathcal{M}_{s}^{N}$, $\mathcal{M}_{u}^{N}$, and $\mathcal{M}_{c}$,

[TABLE]

The unpolarized differential cross section in the c.m. frame for the $\gamma p\rightarrow f_{1}(1285)p$ reaction is given as,

[TABLE]

where $s$ is the invariant mass square of the $\gamma p$ system, ${\rm d}\Omega=2\pi{\rm dcos}\theta$, $\theta$ denotes the angle of the outgoing meson $f_{1}(1285)$ relative to the beam direction in the c.m. frame, while $\vec{p}^{\,\rm~{}c.m.}_{1}$ and $\vec{p}^{\,\rm~{}c.m.}_{3}$ are the 3-momentum of the initial photon and final $f_{1}(1285)$ in the c.m. frame.

II.2 $\rho$ and $\omega$ trajectories contributions

At high energies and forward angles, Reggeon exchange mechanism plays a crucial role Grishina:2005cy ; Donnachie:1987pu . Therefore, in modeling the reaction amplitude for the $\gamma p\to f_{1}(1285)p$ reaction at high energies, instead of considering the exchange of a finite selection of individual particles, the exchange of entire Regge trajectories is taken into account, and this exchange can take place in the $t$-channel $\rho$ and $\omega$ trajectories Wang:2014jxb .

One can obtain the amplitude of the $\rho$ or $\omega$ trajectory exchange $\mathcal{M}_{V}^{\rm Regge}$ from the Feynman amplitude $\mathcal{M}_{t}^{V}$ of Eq. (17) by replacing the usual vector meson propagator with a so-called Regge propagator Laget:2005be ,

[TABLE]

where the mass scale constant $s_{0}=1$ GeV, and $\alpha^{{}^{\prime}}_{V}$ is the slope of the trajectory. The $\rho$ and $\omega$ trajectories are taken from Ref. Laget:2005be ,

[TABLE]

and the signature factor $D_{V}(t)$ is taken from Refs. Kochelev:2009xz ; Collins:1977 ; Guidal:1997hy

[TABLE]

In this work, we adopt a hybrid approach to describe the contributions of $t$-channel $\rho$ and $\omega$ exchanges in the range of laboratory photon energies explored by the CLAS data. In the hybrid approach, the amplitude $\mathcal{M}_{t}^{V}$ ($\mathcal{M}_{t}^{\rho}$ and $\mathcal{M}_{t}^{\omega}$) in Eq. 22 is replaced by $\mathcal{M}_{t}^{\rm H}$ Wang:2014jxb ,

[TABLE]

with

[TABLE]

[TABLE]

[TABLE]

where we consider $t_{0}$, $\Delta t$ as free parameters that will be fitted to experimental data, and $W_{0}=2.1$ GeV,$\Delta W=0.08$ GeV from the qualitative comparison of the predictions of Ref. Kochelev:2009xz with the CLAS measurement Dickson:2016gwc , and from the findings of Ref. Wang:2017plf .

From the Eq. (29), we can see that for the region of high energies ($W\equiv\sqrt{s}>W_{0})$ and forward angles ($|t|>t_{0}$), the Regge mechanism is dominant .

III NUMERICAL RESULTS AND DISCUSSIONS

There are nine parameters in our model, (a) four relevant couplings $g_{N^{*}}=\sqrt{4\pi}g_{\gamma NN^{*}}\times g_{f_{1}NN^{*}}$ for the $N(2300)$ term, the $g_{t}=g_{VNN}\times g_{\gamma Vf_{1}}$ for the $t$-channel $\rho$ and $\omega$ trajectories exchanges, $g_{f_{1}NN}$ and $\kappa_{N}$, (b) three cut-off parameters $\Lambda_{N^{*}}$, $\Lambda_{N}$ and $\Lambda_{t}$, (c) $t_{0}$, $\Delta t$. We will obtain these parameters by fitting to the recent differential cross sections data from the CLAS experiment. Since the CLAS Collaboration presents the differential cross sections for $\gamma p\to f_{1}(1285)p\to\eta\pi^{+}\pi^{-}p$, our results for the total cross section and differential cross sections have been scaled by the PDG branching fraction $\Gamma[f_{1}(1285)\to\eta\pi^{+}\pi^{-}]$ in the fit: $0.52\times(2/3)$, which was used by CLAS Collaboration Dickson:2016gwc . In our fit, $M_{N}=0.938$ Gev, $M_{N^{*}}=2.30$ GeV, $\Gamma_{N^{*}}=0.34$ GeV, $M_{\rho}=0.775$ GeV, and $M_{\omega}=0.783$ GeV Olive:2016xmw .

There are a total of 45 CLAS experimental data, and the statistical and systematic uncertainties are taken into account. Considering all the contributions depicted in Fig. 1, we perform a fit to the CLAS data, and the corresponding results are shown in Table 1 (Fit A). With these parameters in Fit A, the calculated differential cross sections from $W=2.35$ to $W=2.75$ GeV as well as the 45 available data are depicted in Fig. 2, where the blue dash-dot-dotted and magenta dotted lines correspond to the contributions of the $s$-channel $N(2300)$ and $t$-channel exchanges, respectively, the black dash-dotted is the contribution of the $u$-channel proton, and other contributions are very small, and can be neglected. The red solid lines stand for the total contributions. Only the statistical errors are shown in Fig. 2.

From Fig. 2, we can see that our model gives an overall reasonable description of the data in the range of $W=2.35\sim 2.75$ GeV. The $N(2300)$ provides a flat contribution for the differential cross sections, since the $N(2300)$ couples to the final states $f_{1}(1285)p$ in the $S$-wave. Near the threshold, the $s$-channel $N(2300)$ gives a large contribution. At higher energies, the contributions of the $t$-channel $\rho$ and $\omega$ trajectories exchanges are responsible for the shapes of the differential cross sections. The contribution of $u$-channel nucleon term is dominant at the backward angles, especially in the region of high energies. Other contributions are very small and can be neglected.

In order to study the role of the $N(2300)$ resonance in the $\gamma p\to f_{1}(1285)p$ reaction, we also perform a fit by excluding the $s/u$-channel $N(2300)$ terms and remaining the other terms. The corresponding results are listed in Table 1(Fit B), and the differential cross sections are shown in Fig. 3. From the Table 1, one can see that the $\chi^{2}$/dof=1.89 in Fit B is larger than $\chi^{2}$/dof=1.05 in Fit A, which shows that the model including the $N(2300)$ contributions can better describe the CLAS data.

Finally, we present the total cross section of the $\gamma p\to f_{1}(1285)p$ reaction with and without $N(2300)$ terms, respectively in Fig. 4 and Fig. 5. From Fig. 4, it can be seen that the $s$-channel $N(2300)$ term gives a clear peak structure around $W=2.3$ GeV with the magnitude of order $30$ nb, but there is no such structure for the case of excluding the $N(2300)$ contributions, which can be used to test our model.

IV Summary

In this work, we have performed the study of $\gamma p\rightarrow f_{1}(1285)p$ reaction within the Regge-effective Lagrangian approach. Besides the contributions from the $t$-channel $\rho$ and $\omega$ trajectories exchanges, we also consider the $s/u$-channel $N(2300)$ terms, the $s/u$ channel of nucleon terms, and the contact term.

We extract the information about the intermediate states by fitting to the CLAS data. We find that the model including the $N(2300)$ contributions can better describe the CLAS data.

Our results indicate that the contributions of the $u$-channel $N(2300)$ term, the $s$-channel nucleon term and contact term, are very small and can be neglected. However, the contribution of $u$-channel nucleon term is dominant at the backward angles, especially in the region of high energies. With the preferred parameters (Fit A), we predict the total cross section. There is a clear bump structure around $W=2.3$ GeV, which is associated with the $N(2300)$ state. Thus, the reaction of $\gamma p\to f_{1}(1285)p$ could be useful to further study of the $N(2300)$ experimentally.

It should be noted that after we submitted this work to arXiv, Wang and He also discussed the $\gamma p\to f_{1}(1285)p$ reaction within a similar way Wang:2017plf , where the $t$-channel $\rho$ and $\omega$ trajectories exchanges, the $s$- and $u$-channel nucleon term are considered, and the $s$-channel nucleon resonances are not included. They suggested that the $s$-channel nucleon resonances is not very large. Our calculations show that the $s$-channel N(2300) term plays an important role in the $\gamma p\to f_{1}(1285)p$ reaction, and a clear bump structure in the total cross section around $\sqrt{s}=2.3$ GeV is predicted. The current information about this reaction is not enough to distinguish our model and the one of Ref. Wang:2017plf . To shed light on the relevant mechanisms of the $\gamma p\to f_{1}(1285)p$ reaction, the further measurement of the total cross sections is called for.

Acknowledgements

We would like to thank Dr. Ju-Jun Xie and Dr. Qi-Fang Lü for valuable discussions. This work is partly supported by the National Natural Science Foundation of China under Grant No. 11505158, the China Postdoctoral Science Foundation under Grant No. 2015M582197, the Postdoctoral Research Sponsorship in Henan Province under Grant No. 2015023, and the Startup Research Fund of Zhengzhou University under Grants No. 1511317001 and No. 1511317002.