# Pion photoproduction off nucleon with Hamiltonian effective field theory

Dan Guo  Zhan-Wei Liu<EMAIL_ADDRESS>1School of Physical Science and
Technology, Lanzhou University, Lanzhou 730000, China
2Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China

###### Abstract

We analyze the $\gamma N\to\pi N$ process in the negative parity channel with
Hamiltonian effective field theory which is successful in studying the nucleon
resonances associated with the meson-nucleon scattering and lattice QCD
simulations. We provide the scattering amplitude of pion photoproduction and
extract the electric dipole amplitudes $E_{0+}$. The bare $N^{*}(1535)$ is
important for improving the consistency between our results and the
experimental data.

##  I Introduction

An important challenge in hadron physics is to understand the nature of
excited nucleon resonances. These excitations are coupled to some meson-baryon
and $\gamma N$ channels Crede:2013kia (1, 2). Extensive data are accumulated
from photoproduction, electroproduction, and meson-nucleon scattering for
decades of efforts, which can help us disclose the structure of these
resonances and furthermore the properties of QCD in the nonperturbative
region.

The $\pi N\to\pi N$ scattering is a powerful measured process to study the
nucleon excitations Klempt:2009pi (3, 1), and it can determine the baryon
masses, widths, pole positions and decay branching ratios
ParticleDataGroup:2020ssz (4). The first negative-parity nucleon $N^{*}(1535)$
is closely related to this process and has been discussed in many works
Doring:2009yv (5, 6, 7, 8, 9, 10, 11, 12, 13). Its mass is larger than the
Roper state $N^{*}(1440)$ with positive parity, which is controversial with
the prediction of conventional quark model Capstick:1986ter (14). Thus the
pentaquark component is expected in $N^{*}(1535)$, coupling with the
conventional triquark kernel. Moreover, it is also suggested that the
resonance may be dynamically generated from meson-nucleon scattering
Kaiser:1995cy (15, 8).

The lattice QCD simulation, from the first principle of QCD, has achieved
remarkable successes. It can reproduce the masses and other properties of
hadrons on unphysical pion masses in finite volumes Fodor:2012gf (16, 17, 18).
The well-known Lüscher formulation and extended models have been used to
extract phase shifts from lattice QCD simulation Luscher:1985dn (19, 20).
There are also some other phenomenological approaches which are set in the
finite volumes and used to extract the useful physical information from the
lattice QCD calculations.

Hamiltonian effective field theory (HEFT) also enables an interpretation
linking the experimental data with the lattice QCD energy levels in a
consistent way Liu:2015ktc (21, 22, 23, 24, 25, 26, 27, 28). Additionally, one
can use the eigenvectors of the discretized Hamiltonian to analyze the
structures of finite-volume states, which can further predict what the energy
levels can be possibly observed with different interpolation operators in
lattice QCD.

In another aspect, photoproduction of meson is also an excellent tool to
extract information about the mechanism of the strong interaction at low
energies Xing:2018axn (29, 30, 9, 31, 32, 33, 7, 34, 35, 36, 37, 38). It is
helpful to determine the basic properties of nucleon resonances, such as the
spin, pole positions, beam asymmetries, anomalous magnetic moments, and
electromagnetic couplings Workman:2012jf (39, 40, 41, 42, 10, 43, 44, 45, 46).
The electromagnetic properties of hadrons, such as magnetic moments, are
essential for constructing the picture of hadrons and widely discussed with
various approaches Li:2021ryu (47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59). In addition, the relevant studies are the necessities for the photon-
nucleus investigation which can detect the structure and electromagnetic
properties of nuclei. For example, meson photoproductions on
${}^{4}\mathrm{He}$ are investigated based on the description of scattering on
nucleon Kim:2021adl (45, 46).

At low energies, the elegant chiral perturbation theory is successful in
exploring photoproduction processes $\gamma N\to\pi N$, concerning the near
threshold region and low partial waves, which often give a precise accord in
this region Ma:2020hpe (60, 61, 62, 63, 64, 65). References Ma:2020hpe (60,
66) studied $S_{11}$ partial-wave multipole amplitude $E_{0}^{+}$ and pion
electroproduction process from threshold to below the $\Delta(1232)$ energy
region. The K-matrix and its variations are widely used to reproduce
experimental data of photoproduction process Arndt:2002xv (67, 68, 69, 70, 71,
72, 73, 74). The $\gamma N\to\pi N$ and $\pi N\to\pi N$ reactions are
simultaneously analyzed with $N^{*}(1535)$ and $N^{*}(1650)$ both taken into
account in Ref. Doring:2009uc (75).

Dynamical coupled-channel models, like the EBAC model Matsuyama:2006rp (11,
12, 40), $\mathrm{J\ddot{u}lich}$ model Doring:2009uc (75, 76, 5) and other
approaches Pascalutsa:2004pk (77, 78, 79), are proposed to give a
comprehensive view about plenty of partial waves with energy up to 2 GeV.
Usually the effective Lagrangians are employed to obtain the Bethe-Salpeter
equation kernel, and these models can well interpret the data from SAID
partial-wave analysis group gws (80).

The photoproduction can ideally inspect the potential meson-nucleon
configuration in nucleon resonances since the electromagnetic properties of
the nucleon are measured pretty well. Thus, the comprehensive analysis,
combining empirical meson-nucleon scattering, the lattice QCD simulation, and
photoproduction on nucleon, can be an ideal platform to promote our
understanding of the properties of nucleon resonances.

HEFT uses the potentials as the bridge to connect the infinite-volume and
finite-volume results. By solving the dynamical equations like Bethe-Salpeter
equation with the potentials, one obtains scattering T matrix and thus other
information. We solve the eigenstates and eigenvalues of finite-volume matrix
Hamiltonian with discretizing the potentials, which can be used to analyze the
lattice QCD simulations. HEFT has been successful in dealing with the systems
involving $N^{*}(1535)$, $N^{*}(1440)$, $\Lambda(1405)$, kaonic deuteron, and
so on Liu:2015ktc (21, 22, 23, 24, 25, 26, 27, 28).

In this work we use HEFT to further analyze the $\gamma N\to\pi N$ processes
in the negative parity channels based on our previous work in which the pion-
nucleon scattering and lattice QCD results have been investigated. With the
well-performed analysis of $\pi N\to\pi N$ scattering by HEFT Liu:2015ktc
(21), a set of couplings about meson and baryons has been obtained. Having
been examined by the lattice QCD results, this set of parameters will not be
adjusted in this work. Combining the data of pion photoproduction, we further
discuss the structure of $N^{*}(1535)$ and the electromagnetic properties and
strong interactions of relevant hadrons.

We organize this work as follows. In Sec. II we extend HEFT to deal with the
$\gamma N\to\pi N$ scattering T matrix and corresponding multipole amplitudes.
The numerical results and discussion are given in Sec. III, followed by a
brief summary in Sec. IV.

##  II Framework

We present the scattering amplitudes of $\gamma N\to\pi N$ and extract the
corresponding electromagnetic multipole amplitudes in this section. The
scattering amplitudes are split into two parts. One is the pure
electromagnetic potential without considering the finite state interactions
(FSI), and the other is FSI correction.

We focus on the finite state $\pi N$ in $S_{11}$ partial wave at low energies,
and the coupled channel effects from $\eta N$ are also included. The
electromagnetic amplitudes of $\gamma N\overset{\rm EM}{\to}\,\pi N/\eta N$
are provided in Sec. II.1 and the corresponding potentials are listed in Sec.
II.2. The FSI corrections are dealt within HEFT in Sec. II.3. We combine these
two contributions $\gamma N\overset{\rm EM}{\to}\,\pi N/\eta N\,\overset{\rm
FSI}{\to}\pi N$ to obtain the final results in Sec. II.4.

###  II.1 Electromagnetic amplitudes without FSI

Figure 1: Tree-level diagrams for pure electromagnetic amplitude of $\gamma
N\to\pi N/\eta N$ process without FSI: (a) s channel, (b) u channel, (c) t
channel (exchanging $\pi$ and $\rho$), (d) the contact term. The solid,
wiggly, and dashed lines represent the baryons, photons, and mesons,
respectively.

Near threshold the nucleon resonances do not affect the $\gamma N\to\pi N$
process very much. The Lagrangians for $\gamma N\overset{\rm EM}{\to}\,\pi N$
can be written as Matsuyama:2006rp (11)

$\displaystyle\mathcal{L}_{\pi NN}$ $\displaystyle=$
$\displaystyle-\frac{f_{\pi
NN}}{m_{\pi}}\bar{N}\gamma_{\mu}\gamma_{5}\vec{\tau}\cdot\partial^{\mu}\vec{\pi}N,$
(1) $\displaystyle\mathcal{L}_{\gamma NN}$ $\displaystyle=$ $\displaystyle
e\bar{N}\left[\hat{e}_{N}\gamma^{\mu}A_{\mu}+\frac{\hat{\kappa}_{N}}{4m_{N}}\sigma^{\mu\nu}F_{\mu\nu}\right]N,$
(2) $\displaystyle\mathcal{L}_{\gamma\pi\pi}$ $\displaystyle=$ $\displaystyle
e\left[\vec{\pi}\times\partial^{\mu}\vec{\pi}\right]_{3}A_{\mu},$ (3)
$\displaystyle\mathcal{L}_{\gamma N\pi N}$ $\displaystyle=$ $\displaystyle
e\frac{f_{\pi
NN}}{m_{\pi}}\bar{N}\gamma^{\mu}\gamma_{5}\left[\vec{\tau}\,\times\vec{\pi}\right]_{3}NA_{\mu},$
(4) $\displaystyle\mathcal{L}_{\gamma\rho\pi}$ $\displaystyle=$ $\displaystyle
e\frac{g_{\gamma\rho\pi}}{m_{\pi}}\varepsilon^{\mu\nu\alpha\beta}\vec{\pi}\cdot\partial_{\mu}\vec{\rho}_{\nu}\partial_{\alpha}A_{\beta},$
(5) $\displaystyle\mathcal{L}_{\rho NN}$ $\displaystyle=$ $\displaystyle
g_{\rho
NN}\bar{N}\left[\gamma_{\mu}-\frac{\kappa_{\rho}}{2m_{N}}\sigma_{\mu\nu}\partial^{\nu}\right]\vec{\rho}\,^{\mu}\cdot\frac{\vec{\tau}}{2}N,$
(6)

where $\hat{e}_{N}\equiv{\rm diag}\\{+1,0\\}$, the electromagnetic field
tensor $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and the
nucleon isospinor $N=(p,n)^{T}$. $\hat{\kappa}_{N}\equiv{\rm
diag}\\{\kappa_{p},\kappa_{n}\\}$, where $\kappa_{p}=\mu_{p}-1=1.79$ and
$\kappa_{n}=\mu_{n}=-1.91$ with $\mu_{p}$ and $\mu_{n}$ being the magnetic
moments of proton and neutron in units of nuclear magneton $\mu_{N}$. These
well-defined Lagrangians are constrained by various symmetry properties, such
as the invariance under isospin, parity and gauge transformation. We list the
relevant Lagrangians for $\gamma N\overset{\rm EM}{\to}\,\eta N$
Matsuyama:2006rp (11)

$\displaystyle\mathcal{L}_{\eta NN}$ $\displaystyle=$
$\displaystyle-\frac{f_{\eta
NN}}{m_{\eta}}\bar{N}\gamma_{\mu}\gamma_{5}N\partial^{\mu}\eta,$ (7)
$\displaystyle\mathcal{L}_{\gamma\rho\eta}$ $\displaystyle=$ $\displaystyle
e\frac{g_{\gamma\rho\eta}}{m_{\rho}}\varepsilon^{\mu\nu\alpha\beta}\partial_{\mu}\rho^{0}_{\nu}\partial_{\alpha}A_{\beta}\eta.$
(8)

In this work, the $\pi N$ final state with the total isospin $I$ being
$\frac{1}{2}$ is mainly concerned, and thus for concise expressions we define

$\hat{\mathcal{M}}_{\pi
N}\equiv\left(\begin{array}[]{cc}\hat{\mathcal{M}}_{\pi N,\gamma p}&0\\\
0&\hat{\mathcal{M}}_{\pi N,\gamma n}\end{array}\right),$ (9)

where $\hat{\mathcal{M}}_{\pi N,\gamma p}$ specially means the amplitude of
$\gamma p\to(\pi N)_{I=\frac{1}{2},I_{3}=+\frac{1}{2}}$, and
$\hat{\mathcal{M}}_{\pi N,\gamma n}$ refers to that of $\gamma n\to(\pi
N)_{I=\frac{1}{2},I_{3}=-\frac{1}{2}}$.

With the above Lagragians, we can write out the electromagnetic amplitude of
$\gamma(\vec{q})\,N(\vec{p})\to[\pi(\vec{k})\,N(\vec{p}\,^{\prime})]_{I=1/2}$
from Fig. 1

$\displaystyle\hat{\mathcal{M}}_{\pi
N}^{(0)}=\frac{\sqrt{3}e}{m_{\pi}}\left\\{f_{\pi
NN}\left[+\frac{2}{3}\tau^{3}\frac{\tilde{\not{k}}\gamma_{5}}{\tilde{k}^{2}-m^{2}_{\pi}+i\epsilon}(\tilde{k}+k)\cdot\varepsilon_{\gamma}-\frac{2}{3}\tau^{3}\not{\varepsilon}_{\gamma}\gamma_{5}\right.\right.$
$\displaystyle\left.\left.\qquad-\not{k}\gamma_{5}\frac{1}{\not{q}+\not{p}-m_{N}+i\epsilon}\Gamma_{N}-\Gamma_{N}\frac{1}{\not{p}-\not{k}-m_{N}+i\epsilon}\not{k}\gamma_{5}\right]\right.$
$\displaystyle\left.\qquad-\frac{g_{\rho
NN}g_{\gamma\rho\pi}}{2}\frac{\Gamma_{\rho}}{\tilde{k}^{2}-m^{2}_{\rho}+i\epsilon}\right\\}.\qquad$
(10)

where $\varepsilon_{\gamma}$ is the photon polarization vector,
$\Gamma_{N}=\hat{e}_{N}\not{\varepsilon}_{\gamma}+\frac{\hat{\kappa}_{N}}{4m_{N}}[\not{q}\not{\varepsilon}_{\gamma}-\not{\varepsilon}_{\gamma}\not{q}]$,
$\Gamma_{\rho}=i\varepsilon_{\mu\nu\alpha\beta}\tilde{k}^{\mu}[\gamma^{\nu}+\frac{\kappa_{\rho}}{4m_{N}}(\gamma^{\nu}\tilde{\not{k}}-\tilde{\not{k}}\gamma^{\nu})]q^{\alpha}\varepsilon^{\beta}_{\gamma}$,
and $\tilde{k}^{\mu}=k^{\mu}-q^{\mu}$.

From the diagrams (a), (b), and (c) in Fig. 1, the electromagnetic amplitude
of $\gamma(\vec{q})\,N(\vec{p})\to\eta(\vec{k})\,N(\vec{p}\,^{\prime})$ is

$\displaystyle\hat{\mathcal{M}}_{\eta N}^{(0)}=e\frac{f_{\eta
NN}}{m_{\pi}}\left[-\not{k}\gamma_{5}\frac{1}{\not{q}+\not{p}-m_{N}+i\epsilon}\Gamma_{N}\right.$
$\displaystyle\left.-\Gamma_{N}\frac{1}{\not{p}-\not{k}-m_{N}+i\epsilon}\not{k}\gamma_{5}\right]-\,e\frac{g_{\rho
NN}g_{\gamma\rho\eta}}{m_{\eta}}\frac{\tau^{3}}{2}\frac{\Gamma_{\rho}}{\tilde{k}^{2}-m^{2}_{\rho}+i\epsilon}.\qquad$
(11)

In addition, the contribution from the resonance $N^{*}(1535)$ should be taken
into account especially as the energy approaches the mass since it has the
same quantum number with the $S_{11}$ $\pi N$ channel. We use the following
Lagrangians Shyam:2008fr (71, 81, 10, 82)

$\mathcal{L}_{\gamma NN^{*}}=e\bar{N}^{*}\frac{\hat{g}_{N\gamma
N^{*}}}{4m_{N}}\gamma_{5}\sigma_{\mu\nu}F^{\mu\nu}N+\mathrm{H.c.},$ (12)
$\mathcal{L}_{\pi NN^{*}}=\frac{g_{\pi
NN^{*}}}{f_{\pi}}\bar{N}^{*}\gamma^{\mu}\vec{\tau}\cdot\partial_{\mu}\vec{\pi}N+\mathrm{H.c.},$
(13) $\mathcal{L}_{\eta NN^{*}}=\frac{\sqrt{3}g_{\eta
NN^{*}}}{f_{\pi}}\bar{N}^{*}\gamma^{\mu}\partial_{\mu}\eta N+\mathrm{H.c.}.$
(14)

The electromagnetic amplitudes of $\gamma N\to\pi N/\eta N$ involving
$N^{*}(1535)$ as intermediate state through s and u channels are

$\displaystyle\hat{\mathcal{M}}_{\alpha}^{(N^{*})}=e\frac{\sqrt{3}g_{\alpha
N^{*}}}{f_{\pi}}\frac{\hat{g}_{N\gamma
N^{*}}}{4m_{N}}\left\\{\not{k}\frac{1}{\not{q}+\not{p}-m_{N^{*}}^{0}+i\epsilon}\gamma_{5}(\not{q}\not{\varepsilon}_{\gamma}-\not{\varepsilon}_{\gamma}\not{q})\right.$
$\displaystyle\left.\qquad\qquad\quad+\gamma_{5}(\not{q}\not{\varepsilon}_{\gamma}-\not{\varepsilon}_{\gamma}\not{q})\frac{1}{\not{p}-\not{k}-m_{N^{*}}^{0}+i\epsilon}\not{k}\right\\}u(k),$
(15)

where $\alpha$ refers to the channel $\pi N$ or $\eta N$, and $m_{N^{*}}^{0}$
is the $N^{*}$ bare mass. The dipole form factor,
$u(k)=(1+k^{2}/\Lambda^{2})^{-2}$, regulates the high momentum divergence with
cutoff parameter $\Lambda=0.8$ GeV. $\hat{g}_{N\gamma N^{*}}\equiv{\rm
diag}\\{g_{p\gamma N^{*+}},g_{n\gamma N^{*0}}\\}$ is the bare
$N^{*}$-$N$-$\gamma$ coupling.

One obtains the final scattering amplitude $\hat{\mathcal{M}}_{\alpha}$ by
summing the above two contributions
$\hat{\mathcal{M}}_{\alpha}=\hat{\mathcal{M}}_{\alpha}^{(0)}+\hat{\mathcal{M}}_{\alpha}^{(N^{*})}$.
In the general case, the Lorentz and gauge invariant amplitude
$\hat{\mathcal{M}}_{\alpha;\gamma N}$ can be decomposed into Chew-Goldberger-
Low-Nambu (CGLN) amplitudes after preforming the nonrelativistic reduction
Chew:1957tf (83),

$\displaystyle\hat{\mathcal{M}}_{\alpha,\gamma N}$ $\displaystyle=$
$\displaystyle\vec{\sigma}\cdot\vec{\varepsilon}\,t_{\alpha;\gamma
N}^{(1)}+i\vec{\sigma}\cdot\hat{k}\,\vec{\sigma}\cdot\hat{q}\times\vec{\varepsilon}\,t_{\alpha;\gamma
N}^{(2)}$ (16)
$\displaystyle+\vec{\sigma}\cdot\hat{q}\,\hat{k}\cdot\vec{\varepsilon}\,t_{\alpha;\gamma
N}^{(3)}+\vec{\sigma}\cdot\hat{k}\,\hat{k}\cdot\vec{\varepsilon}\,t_{\alpha;\gamma
N}^{(4)},$

where $\hat{k}$ and $\hat{q}$ mean the unit vectors of $\vec{k}$ and
$\vec{q}$, respectively. $t_{\alpha;\gamma N}^{(i)}$ is the function of
$\vec{k}$ and $\vec{q}$ in the center-of-mass reference frame, that is,
$t_{\alpha;\gamma N}^{(i)}=t_{\alpha;\gamma N}^{(i)}(\vec{k},\vec{q})$.

For the process
$\gamma(\vec{q})\,N(\vec{p})\to\pi/\eta(\vec{k})\,N(\vec{p}\,^{\prime})$,
there are $\vec{p}\,^{\prime}=-\vec{k}$ and $\vec{p}=-\vec{q}$ in the center-
of-mass reference frame, and we can choose the direction of photon momentum
$\vec{q}$ along the $z$ axis. The $z$-component of the initial nucleon spin
equals to its negative helicity $-\lambda_{N}$. We show the complete
scattering amplitude of $\gamma N\to\pi/\eta\,N$ with explicit spin directions

$\displaystyle\mathcal{M}_{\alpha,\gamma N}(s_{z}^{\prime
N},\lambda_{N},\lambda_{\gamma};\vec{k},\vec{q})$ (17) $\displaystyle\equiv$
$\displaystyle\langle\pi/\eta\,(\vec{k}),\,N(\vec{p}\,^{\prime},s_{z}^{\prime
N})|\,H^{\rm
EM}\,|\gamma(\vec{q},\lambda_{\gamma}),\,N(\vec{p},-\lambda_{N})\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega_{\pi/\eta}(k)}}\frac{1}{\sqrt{2|\vec{q}|}}\,\,\bar{u}_{N}(s_{z}^{\prime
N})\,\hat{\mathcal{M}}_{\alpha,\gamma N}\,u_{N}(-\lambda_{N}),\qquad$

where $\omega_{\pi/\eta}(k)=\sqrt{m_{\pi/\eta}^{2}+k^{2}}$ is the energy of
$\pi$ or $\eta$ meson, $u_{N}$ and $\bar{u}_{N}$ are the Pauli spinors for
nucleons, and the photon polarization vector with helicity
$\lambda_{\gamma}=\pm 1$ is expressed as
$\varepsilon_{\gamma}^{\pm}=\mp\frac{1}{\sqrt{2}}(\hat{x}\pm i\hat{y})$.

###  II.2 Electromagnetic potentials without FSI

The electromagnetic potentials are directly related to the scattering
amplitudes in Eq. (17). However, Eq. (17) gives the potentials about $|\gamma
N\rangle\to|\pi/\eta\,(\vec{k}),\,N(-\vec{k},s_{z}^{\prime N})\rangle$ while
we need those about $|\gamma N\rangle\to|\pi/\eta\,N;k,J,J_{z},L\rangle$ at
last. Here $J$ and $L$ are total angular momentum and the orbital angular
momentum of $\pi/\eta\,N$, respectively. For the $z$-component of $J$, there
is $J_{z}=\lambda_{\gamma}-\lambda_{N}$.

Before obtaining the potentials of $|\gamma
N\rangle\to|\pi/\eta\,N;k,J,J_{z},L\rangle$, we first introduce those for
$|\gamma N\rangle\to|\pi/\eta\,N;k,J,J_{z},\lambda_{N}^{\prime}\rangle$ with
$\lambda_{N}^{\prime}$ being the helicity of the outgoing nucleon

$\displaystyle V_{\alpha,\gamma
N}(J,\lambda_{N}^{\prime},\lambda_{\gamma},\lambda_{N};k,q)=2\pi\int_{-1}^{1}\mathrm{d}(\cos\theta)\;\sum_{s_{z}^{\prime
N}}$ $\displaystyle\qquad
d^{J}_{\lambda_{\gamma}-\lambda_{N},-\lambda_{N}^{\prime}}(\theta)d^{1/2}_{s_{z}^{\prime
N},-\lambda_{N}^{\prime}}(\theta)^{*}\mathcal{M}_{\alpha,\gamma
N}(s_{z}^{\prime N},\lambda_{N},\lambda_{\gamma};\vec{k},\vec{q}),\qquad$ (18)

where $\theta$ is the angle between $\vec{k}$ and $\vec{q}$, and
$d^{j}_{m,m^{\prime}}(\theta)$ is the Wigner rotation matrix. About the
transformations between different representations one can see Refs.
Jacob:1959at (84, 85). With the intermediate potentials in Eq. (II.2), we can
provide the final potentials we concern

$\displaystyle V^{JLS;\lambda_{\gamma}\lambda_{N}}_{\alpha,\gamma N}(k,q)$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2L+1}{2J+1}}\sum_{\lambda_{N}^{\prime}}\langle
L,S,0,-\lambda_{N}^{\prime}|J,-\lambda_{N}^{\prime}\rangle$ (19)
$\displaystyle\qquad\times V_{\alpha,\gamma
N}(J,\lambda_{N}^{\prime},\lambda_{\gamma},\lambda_{N};k,q).\qquad$

Here $S$ is the total spin of $\pi/\eta\,N$ and equals to $\frac{1}{2}$ since
the spin of $\pi$ or $\eta$ is 0.

We introduce the short notation $V_{\alpha,\gamma
N}^{\lambda_{\gamma},\lambda_{N}}$ for the final state $\pi N$ with
$J=\frac{1}{2},L=0,S=\frac{1}{2}$

$\displaystyle V_{\alpha,\gamma N}^{\lambda_{\gamma},\lambda_{N}}(k,q)$
$\displaystyle\equiv$ $\displaystyle
V^{J=\frac{1}{2},L=0,S=\frac{1}{2};\lambda_{\gamma},\lambda_{N}}_{\alpha,\gamma
N}(k,q).$ (20)

From Eqs. (16)-(19) we can obtain the relation

$\displaystyle V_{\alpha,\gamma
N}^{\lambda_{\gamma}=1,\lambda_{N}=\frac{1}{2}}(k,q)=\frac{1}{8\pi^{2}}\frac{-1}{\sqrt{\omega_{\pi/\eta}(k)q}}\int_{0}^{\pi}\mathrm{d}\theta\;\Big{[}2\sin\theta
t_{\alpha,\gamma N}^{(1)}$ $\displaystyle\qquad+2\sin\theta\cos\theta
t_{\alpha,\gamma N}^{(2)}+0\cdot t_{\alpha,\gamma N}^{(3)}+\sin^{3}\theta
t_{\alpha,\gamma N}^{(4)}\Big{]}.$ (21)

###  II.3 FSI within HEFT

In our previous work Liu:2015ktc (21), HEFT is successful in studying both
$\pi N\to\pi N$ scattering and the related nucleon resonance spectra in finite
volume. Hence, the same formalism and parameters are now also adopted this
work for the FSI effects. We briefly repeat the relevant formalism in this
subsection.

The interacting Hamiltonian includes two parts, $H_{I}=g+v$. The $g$ describes
the vertex interaction between bare state $N^{*}$ and coupled channel
$\alpha$,

$g=\sum_{\alpha}\int\mathrm{d^{3}}\vec{k}\,\left\\{|\alpha(\vec{k})\rangle\,G_{\alpha}^{\dagger}(k)\,\langle
N^{*}|+|N^{*}\rangle\,G_{\alpha}(k)\,\langle\alpha(\vec{k})|\right\\},$ (22)

where $G_{\alpha}(k)$ depicts the ordinary S-wave coupling and can be derived
from the Lagrangians Eqs. (13)-(14)

$G_{\alpha}(k)=\frac{\sqrt{3}g_{\alpha N^{*}}}{2\pi
f_{\pi}}\sqrt{\omega_{\pi/\eta}(k)}\,u(k).$ (23)

The other interacting part, $v$, is phenomenologically describing direct two-
to-two particle interaction by

$v=\sum_{\alpha,\beta}\int\mathrm{d^{3}}\vec{k}\mathrm{d^{3}}\vec{k}^{\prime}\,|\alpha(\vec{k})\rangle\,V^{S}_{\alpha,\beta}(k,k^{\prime})\,\langle\beta(\vec{k}^{\prime})|.$
(24)

Here, the separable potentials, $V^{S}_{\alpha,\beta}(k,k^{\prime})$, is
introduced. For instance, the potential for $\pi N$ channel direct scattering
is

$V^{S}_{\pi N,\pi N}(k,k^{\prime})=\frac{g^{S}_{\pi
N}\tilde{u}(k)\tilde{u}(k^{\prime})}{4\pi^{2}f_{\pi}^{2}}.$ (25)

where $\tilde{u}(k)=u(k)[m_{\pi}+\omega_{\pi}(k)]/\omega_{\pi}(k)$.

Then solving the $\pi N\to\pi N$ coupled-channel Bethe-Salpeter equation, one
can get the corresponding partial wave T-matrix,

$\displaystyle T_{\pi N,\pi N}(k,k^{\prime};E)=V_{\pi N,\pi
N}(k,k^{\prime};E)+\sum_{\alpha}\int\mathrm{d}k^{\prime\prime}\,k^{\prime\prime
2}$ $\displaystyle\qquad\times V_{\alpha,\pi
N}(k^{\prime\prime},k;E)\frac{1}{E-\omega_{\alpha}(k^{\prime\prime})+i\epsilon}T_{\pi
N,\alpha}(k^{\prime},k^{\prime\prime};E),\qquad$ (26)

where
$\omega_{\alpha}(k)=\omega_{\pi/\eta}+\omega_{N}=\sqrt{m_{\pi/\eta}^{2}+k^{2}}+\sqrt{m_{N}^{2}+k^{2}}$
is the total energy of the channel $\alpha$, and the coupled-channel potential
are calculated from interacting Hamiltonian

$V_{\alpha,\beta}(k,k^{\prime};E)=G_{\alpha,N^{*}}^{\dagger}(k)\frac{1}{E-m_{N^{*}}^{0}}G_{\beta,N^{*}}(k^{\prime})+V_{\alpha,\beta}^{S}(k,k^{\prime}).$
(27)

###  II.4 Pion photoproduction off nucleon with FSI

With the electromagnetic potentials
$V^{\lambda_{\gamma},\lambda_{N}}_{\alpha,\gamma N}$ in Eq. (II.2) and the
scattering T-matrix $T_{\beta,\alpha}$ in Eq. (II.3), one can obtain the
scattering T-matrix for $\gamma N\to\pi N$ including FSI effects
Matsuyama:2006rp (11, 12, 40)

$\displaystyle T^{\lambda_{\gamma},\lambda_{N}}_{\pi N,\gamma
N}(k,q;E)=V^{\lambda_{\gamma},\lambda_{N}}_{\pi N,\gamma
N}(k,q)+\sum_{\alpha}\int\mathrm{d}k^{\prime}\,k^{\prime 2}$
$\displaystyle\qquad\times V^{\lambda_{\gamma},\lambda_{N}}_{\alpha,\gamma
N}(k^{\prime},q)\frac{1}{E-\omega_{\alpha}(k^{\prime})+i\epsilon}T_{\pi
N,\alpha}(k,k^{\prime};E).\quad$ (28)

Here we mainly concern the outgoing $\pi N$ with the quantum number $S_{11}$,
and such T-matrix is related to the electric dipole $E_{0+}$ Matsuyama:2006rp
(11)

$\displaystyle E_{0+}(E_{\rm cm})$ $\displaystyle=$ $\displaystyle\frac{\pi
m_{N}\sqrt{\omega_{\pi}(k_{\rm on})\,q_{\rm on}}}{E_{\rm
cm}}T^{\lambda_{\gamma}=1,\lambda_{N}=\frac{1}{2}}_{\pi N,\gamma N}(k_{\rm
on},q_{\rm on};E_{\rm cm}),\qquad$ (29)

where $k_{\rm on}$ and $q_{\rm on}$ are the 3-momenta of $\pi N$ and $\gamma
N$ corresponding to the total energy $E_{\rm cm}$ in the center-of-mass frame,
respectively.

More information of electromagnetic multipoles for meson photoproduction off
nucleon can refer to Refs. Chew:1957tf (83, 86, 41).

##  III Numerical Results and Discussion

|
---|---
(a) | (b)
Figure 2: The electric dipole amplitudes $E_{0+}$ with and without the
contributions from the $\rho$ meson or the bare $N^{*}$ resonance. The solid,
dashed and dashed-dotted lines refer to the full, $\hat{g}_{N\gamma N^{*}}=0$,
and $g_{\rho NN}=0$ cases, respectively.

We can now obtain the numerical results for the electric dipole amplitudes
$E_{0^{+}}$. We take the couplings and bare resonance mass from our previous
HEFT for the FSI part and do not adjust them, that is, $g_{\pi NN^{*}}=0.186$,
$g_{\eta NN^{*}}=0.185$, $g^{S}_{\pi N}=-0.0608$, and $m_{N^{*}}^{0}=1601$ MeV
Liu:2015ktc (21). The common couplings are set as in Refs. Matsuyama:2006rp
(11, 12), $f_{\pi NN}=1.00$, $g_{\gamma\rho\pi}=0.13$,
$g_{\gamma\rho\eta}=1.15$, $\kappa_{\rho}=1.82$, $g_{\rho NN}=6.20$,
$f_{\pi}=92$ MeV. We do not consider the contribution from $\eta^{\prime}$
mesons in this work, which may be partly absorbed by $f_{\eta NN}$. We fit the
experimental data of $E_{0^{+}}$ and obtain $f_{\eta NN}=-1.42$, $g_{p\gamma
N^{*+}}=0.27$ and $g_{n\gamma N^{*0}}=-0.25$. The masses of nucleon and mesons
are taken from PDG ParticleDataGroup:2020ssz (4). We can provide the electric
dipole amplitudes $E_{0^{+}}$ using our approach in Fig. 2, along with the
experimental data gws (80).

As shown in Fig. 2, the electric dipoles $E_{0+}$ in our framework are
compatible with experimental data for both proton and neutron targets up to
1.6 GeV. Our results can describe the data both at low energies close to $\pi
N$ threshold and at $N^{*}$ resonance energies.

The basic contributions $\hat{\mathcal{M}}_{\pi N}^{(0)}$ and
$\hat{\mathcal{M}}_{\eta N}^{(0)}$ describe the nonresonant meson-baryon
interaction at tree level with the effective Lagrangian approach. The
amplitude $\hat{\mathcal{M}}_{\pi N}^{(0)}$ consists of s- and u-, and
t-channel term, and the contact term which depicts the short-range heavy
particle exchanging effects and ensure the gauge invariance. Among these terms
of tree-level amplitude, the contact term occupies the main share, the
t-channel term has moderate contribution, and the s- and u-channel terms are
usually smallest. It is that only the antinucleon can propagate in the s
channel and u channel because of the parity conservation that makes the
corresponding suppression in this case. Since these basic contributions
contain none resonance, they are smooth without bumplike, cusplike or diplike
structures.

With the center-of-mass energy $E_{\rm cm}$ increasing, the absolute
magnitudes of the contact and the pion-exchanging t-channel terms in
$\hat{\mathcal{M}}_{\pi N}^{(0)}$ are continuously declining, the u-channel
contribution also declines and changes the sign, and the s-channel part varies
little. The $\rho$-exchanging t-channel term is added in this work, and its
contribution is increasing as $E_{\rm cm}$ grows. Thus, the contact term
dominates the amplitude near $\pi N$ threshold, and the $\rho$-exchanging term
takes more contribution at larger $E_{\rm cm}$.

We give $E_{0+}$ with $g_{\rho NN}=0$ in Fig. 2, and one can estimate the
$\rho$ meson contribution by comparing with full $E_{0+}$. From the figure,
the $\rho$ exchanges are indispensable in both $I_{3}=\frac{1}{2}$ and
$I_{3}=-\frac{1}{2}$ channels around 1200$\sim$1400 MeV.

Here, the $\omega$-exchanging channel is not taken into account, since it
behaves similarly like the $\rho$-exchanging one and thus can be combined into
$\rho$-exchanging contribution. The related coupling parameters about the
$\omega$ meson are also set 0 in Ref. Matsuyama:2006rp (11).

The main part of amplitude in Eq. (II.1), the contact term, is with isospin
factor $\tau_{3}$ being opposite for $I_{3}=\frac{1}{2}$ and
$I_{3}=-\frac{1}{2}$. This makes the electric dipole $E_{0+}$ amplitude having
opposite signs between $p$ and $n$ target around $\pi N$ threshold.

Similar trends of $\hat{\mathcal{M}}_{\eta N}^{(0)}$ for each channel are
retained, but begins from $\eta N$ threshold. The contact term is discarded in
$\hat{\mathcal{M}}_{\eta N}^{(0)}$ due to flavor forbidden, and hence the
magnitude of $\hat{\mathcal{M}}_{\eta N}^{(0)}$ contribution is minor to that
of $\hat{\mathcal{M}}_{\pi N}^{(0)}$.

There are two terms in Eq. (II.4). The first one
$V^{\lambda_{\gamma},\lambda_{N}}_{\pi N,\gamma N}$ is directly from the
$\hat{\mathcal{M}}_{\pi N}^{(0)}$ and contributes to more than 80% of ${\rm
Re}(E_{0+})$ up to 1400 MeV in our framework. The second term receives the
corrections from the hadron rescattering $T_{\pi N,\alpha}$ and produces the
imaginary part of $E_{0+}$.

When considering the contributions of $N^{*}(1535)$, we first use the bare
propagator of the resonance to obtain the pure electromagnetic potentials, and
the FSI effects recover its physical mass and generate the decay width
naturally. As the energy approaches the bare mass $m_{N^{*}}^{0}$, the two
terms in Eq. (II.4) are both divergent due to the lack of a finite imaginary
part in the bare propagator. However, the two divergences cancel each other if
the couplings $g_{\pi NN^{*}}$ and $g_{\eta NN^{*}}$ are the same for the pure
electromagnetic potentials and FSI effects. This subtle cancellation indicates
the consistence of different parts in our framework.

Another simple treatment for the $N^{*}(1535)$ contribution can be that the
resonance propagator uses the Breit-Wigner form depending on the physical mass
and width without FSI effects. Some dynamics would be missed in that way.

If $N(1535)$ is purely dynamically generated by the $\pi N$, $\eta N$
interactions and so on without the bare triquark kernel, $\hat{g}_{N\gamma
N^{*}}$ would be 0 and the $N^{*}(1535)$ contribution would only appear in the
FSI effects. We provide $E_{0+}$ with $\hat{g}_{N\gamma N^{*}}=0$ in Fig. 2,
and one can clearly see that $\hat{g}_{N\gamma N^{*}}$ term is important
around 1.5 GeV. The bare triquark kernel of $N^{*}(1535)$ makes the real part
of $E_{0+}$ on $p$ target lower and imaginary part larger, which improves the
consistence between our results and the empirical data, especially for the
imaginary parts. Similar improvement can be found in the $n$ target case
though not very obvious as in the $p$ target case.

While the present description are well agreed with SAID electric dipoles,
there are some residual discrepancy near 1.6 GeV that could be traced back to
the absence of $N^{*}(1650)$ resonance and its interference, and $K\Lambda$,
and $K\Sigma$ coupled-channel contribution as mentioned in Refs. Doring:2009uc
(75, 76, 5). Also the $\pi\pi N$ contribution may improve the consistence
better at low energies.

Now we focus on the differences from other studies of photoproduction
processes and therein one can notice the improvements in this work. The
partial-wave analysis, K-matrix methods and their variations are widely used
to reproduce large amounts of experimental data from photoproduction and pion
production processes Arndt:2002xv (67, 8, 68, 69, 70, 71, 72, 73, 74, 39).
When implementing the K-matrix approximation, the real parts of meson-baryon
propagator in coupled-channel integral equations are omitted for technical
simplicity, and thus the integral equations are reduced to algebraic
equations. Such methods sometimes cannot show the essential dynamics.

The chiral perturbation theory (CHPT) can give a nice description of
photoproduction observables near threshold where the spontaneously chiral
symmetry breaking plays a central role Ma:2020hpe (60, 61, 62, 63, 64, 65).
CHPT can improve its results order by order. But when moving to the resonance
region, one is still facing some challenges, for example, the convergence may
not be good. Comparing with relevant ChPT works, our results also give a good
consistence at low energies though not as precise as Refs. Ma:2020hpe (60, 63)
and can extend well to the larger energies.

Dynamical coupled-channel models and some equivalent models are used to
analyze the extensive and precise meson- or photon-production data and then
extract the resonance parameters Matsuyama:2006rp (11, 12, 40, 75, 76, 5, 77,
78, 79). In these models, plenty of resonances and coupled channels are
considered, where exhaustedly include the almost all experimental data, such
as total cross section, differential cross sections, polarization observables,
and helicity amplitudes. As a consequence, though with more disagreement
appearing in some specific partial-wave amplitude or polarization observables
due to huge amounts of data taken into account, these models provide good
interpretations to the experimental data and complement to partial-wave
analysis group SAID gws (80). Apparently, these models often aim at describing
the raw experimental data as precise as possible, determining the partial-wave
amplitude, then extracting the resonances parameters like mass, widths,
quantum numbers, couplings, magnetic moments, helicity amplitudes, and
transition form factors, and so on.

|
---|---
Figure 3: The electric dipole amplitudes $E_{0+}$ from the SAID gws (80), EBAC
DCC model (88), and our results.

The well-performed dynamical coupled-channel model, EBAC Matsuyama:2006rp (11,
12, 40), performing a dynamical coupled-channel analysis of worldwide data
about total cross section, differential cross sections, polarization
observables from pion- and photon-induced reactions, also provides the
electromagnetic multipoles online DCC model (88). We plot the EBAC, SAID and
our results for $E_{0+}$ on proton target in Fig. 3, and noticed that EBAC and
SAID extractions are consistent below 1600 MeV. From the figure, our real
(imaginary) part is closer to the EABC data near threshold (at larger
energies).

A dynamical coupled channel framework and semiphenomenological approach
Huang:2011as (43, 44), both with the FSI effects provided by the
$\mathrm{J\ddot{u}lich}$ model, have studied the cross section, differential
cross section, polarization observable and helicity amplitude plots for the
photoproduction processes at different c.m. energies, and do exhibit a well
description. The analyticity is guaranteed in Ref. Ronchen:2014cna (44) and
the $\chi^{2}$ of $pE_{0+}$ is 1574 (${\rm am}^{2}$) there and smaller than
2136 in this work below 1600 MeV. Reference Huang:2011as (43) also suggests
that the fitting results of multipole amplitude is very difficult and
uncertain.

We do not include extra form factors for the electromagnetic vertices since
the energy does not change too widely, which can largely reduce the manmade
arbitrariness and uncertainties. However, the adjustable cutoffs in the
electromagnetic form factors are critical for the photoproduction studies with
most dynamical coupled-channel models Huang:2011as (43, 44).

Another interesting discovery, we find the $\rho$-exchanging t-channel term is
generally considered in Refs. Huang:2011as (43, 44, 75, 11, 12, 40), but is
not discussed very much. However, as seen in Fig. 2, the $\rho$ exchanging
channel plays an important role in this work to regulate the magnitudes of $p$
and $n$ target multipoles, simultaneously making the modulus of $p$ target
higher and $n$ target lower around 1.5 GeV, since the contributions of $\rho$
exchanging have same signs for the $I_{3}=\frac{1}{2}$ and
$I_{3}=-\frac{1}{2}$ processes. The $\rho$ exchange provide the medium and
short range force, which makes it become bigger around 1.5 GeV than threshold.
However, the role of $\rho$ exchange in the interactions among the photon,
meson, and nucleon need be checked in other works and may be more obvious in
other channels.

To explain the mass inverse problem between $N^{*}(1535)$ and $N^{*}(1440)$
and sizable couplings of $N^{*}(1535)$ to $K\Lambda$, $N\eta^{\prime}$ and
$N\phi$ channels, it is proposed that the $N^{*}$ resonances contain 5-quark
components Zou:2010tc (6, 7, 59). The interference of $N^{*}(1535)$ from
$N^{*}(1650)$ are significant, and strong couplings to $K\Lambda$ and
$K\Sigma$ also suggest dynamical generation nature Doring:2009uc (75).
Reference Inoue:2001ip (8, 15, 75) discuss the dynamical generation mechanism
of $N^{*}(1535)$, generated from the pure meson and baryon interactions, but
the predicted amplitudes appear not to be in good agreement with experiment.
The $N^{*}(1535)$ structure is still a well-known unsolved question. From Fig.
2, the imaginary part of the S11 multipole deviates from the experimental data
a little bigger than the real part starting at $E_{cm}\sim$1450 MeV. This may
be improved by including the contributions from $\pi\pi N$, $K\Lambda$
$K\Sigma$, $N^{*}(1650)$, and their interference, which would disclose the
fine structure of the $N^{*}(1535)$.

These dynamical coupled-channel models can extract the parameters for the
resonances and help us understand the structure, but hadron structure models
or lattice QCD calculations can provide other useful information as mentioned
in Crede:2013kia (1, 87). Lattice QCD starts from the first principles of QCD
and is a model-independent approach. Usually it gives the results depending on
unphysical quark masses with finite volume effects. Thus a comprehensive
analysis, interpreting the results of lattice QCD simulation, meson-nucleon
scattering, and photon-induced production, is appealing.

In our previous work $N^{*}(1535)$ is interpreted as a primary three-quark
state, meanwhile with some five-quark components Liu:2015ktc (21). It contains
about 50% of the bare triquark part near physical pion mass, within lattice
boxes $L\simeq$ 2, 3 fm, by comparing the predicted energy levels of HEFT and
the positions of finite-volume lattice QCD simulations. Moreover, as
analytically continuing the scattering amplitude to complex plane, the
resonance pole locating on the unphysical sheet are discovered as
1531$\pm$29-88$i\pm$2 MeV, which is consistent with the PDG value
ParticleDataGroup:2020ssz (4).

Looking to the bare electromagnetic coupling constants, they are fitted as
$g_{p\gamma N^{*+}}=0.27$ and $g_{n\gamma N^{*0}}=-0.25$, and originate from
the contribution of the triquark component in $N^{*}(1535)$. As shown in Fig.
2, a nonvanishing bare $\hat{g}_{N\gamma N^{*}}$ must exist. If comparing the
physical electromagnetic couplings extracted from the electromagnetic decay
widths of $N(1535)$ in PDG ParticleDataGroup:2020ssz (4), $|g^{\rm
phy}_{p\gamma N^{*+}}|=0.613^{+0.095}_{-0.112}$ and $|g^{\rm phy}_{n\gamma
N^{*0}}|=0.466^{+0.180}_{-0.337}$, one can see that the bare couplings are not
small. The bare state cannot be absent in this work before considering more
coupled channels and resonances.

---
Figure 4: The imaginary electric dipole amplitudes $E_{0+}$ on the proton
target with the consideration of $N^{*}(1650)$.

In Ref. Doring:2009uc (75), the simultaneous study of the $\pi N\to\pi N$ and
$\gamma N\to\pi N$ reactions examines the structure of $N^{*}(1535)$ and the
interference effect from $N^{*}(1650)$. The poles of $N^{*}(1535)$ and
$N^{*}(1650)$ can be found on different Riemann sheets intuitively and their
roles are carefully analyzed. The interference of $N^{*}(1650)$ is important
for the properties of $N^{*}(1535)$ therein.

The biggest deviations of our results from the SAID data are the imaginary
parts of the proton target. $K\Lambda$ and $K\Sigma$ thresholds are larger and
thus they may contribute little to the imaginary parts below 1.6 GeV. We would
focus on the estimation of $N^{*}(1650)$ resonance for the imaginary parts.

It is natural to expect the inclusion of $N^{*}(1650)$ with a sizable width
will promote the present status of the imaginary amplitude near 1.6 GeV. For a
rough estimation, we add a Breit-Wigner form for the $N^{*}(1650)$
contribution with physical mass and width from PDG, and explicitly,

$\displaystyle\hat{\mathcal{M}}^{(N^{*}(1650))}=e\frac{\sqrt{3}g^{\rm
phy}_{\pi NN^{*}(1650)}}{f_{\pi}}\frac{\hat{g}^{\rm phy}_{N\gamma
N^{*}(1650)}}{4m_{N}}$
$\displaystyle\left\\{\not{k}\frac{\not{q}+\not{p}+m_{N^{*}(1650)}}{(\not{q}+\not{p})^{2}-(m_{N^{*}(1650)})^{2}+im_{N^{*}(1650)}\Gamma_{N^{*}(1650)}}\gamma_{5}(\not{q}\not{\varepsilon}_{\gamma}-\not{\varepsilon}_{\gamma}\not{q})\right\\}.\qquad$

Then using the Eqs. (17)-(19) we can obtain $V_{\pi N,\gamma
N}^{\lambda_{\gamma}=1,\lambda_{N}=\frac{1}{2},(N^{*}(1650))}(k,q)$, where the
FSI contributions are included in the width of $N^{*}(1650)$ effectively.

From central values of PDG decay information, we can obtain $|g^{\rm
phy}_{p\gamma N^{*+}(1650)}|=0.336$, $|g^{\rm phy}_{n\gamma
N^{*+}(1650)}|=0.285$ and $g^{\rm phy}_{\pi NN^{*}(1650)}=0.092$. We plot the
imaginary electric dipole amplitudes $E_{0+}$ on the proton target with these
couplings for the effect of $N^{*}(1650)$ in Fig. 4 as the Scenario I. The
imaginary amplitude is promoted about 0.50 am at $E_{cm}=$1550 MeV. When
taking the couplings from the upper limit of PDG widths, the scenario II in
Fig. 4 shows a growth of 0.87 am at $E_{cm}=$1550 MeV.

Unfortunately, our photoproduction amplitudes are no longer gauge invariant
for either $\hat{\mathcal{M}}_{\alpha}^{(0)}$ or
$\hat{\mathcal{M}}_{\alpha}^{(N^{*})}$ after taking into account the final
state interaction. One can refer to Refs. Haberzettl:2011zr (89, 90) for the
discussion about gauge invariance and off-shell condition when taking account
of final-state interaction. The guiding principles of construction of gauge
invariant amplitude are the consistent and complete implementation to all
levels of reaction mechanisms Haberzettl:2011zr (89, 90). In the processes
$\gamma p\to\pi^{0}\eta p$ and $\gamma p\to\pi^{0}K^{0}\Sigma^{+}$, when
considering the FSI effect and using Bethe-Salpeter equation with the kernel
of the Weinberg-Tomozawa term, the gauge invariance is automatically satisfied
as long as the photon not only attached to external legs and vertexes, but
also to intermediate propagators and vertexes Doring:2005bx (91). We should
also consider these effects in the future.

In short, a consistent understanding of $N^{*}(1535)$ is obtained with
simultaneously interpreting QCD simulation, meson-nucleon scattering
amplitudes and photon-induced dipole amplitudes.

##  IV Summary

The photoproduction information of nucleon resonance is particularly
interesting to reflect the internal structure, where both hadronic and
electromagnetic coupling are accessible. Electromagnetic properties afford
some additional information for the structure of strong interaction systems.
The multipoles are directly tied to the scattering amplitudes and one can gain
the insight into the nature of resonances more sensitively.

We analyze the role of $N^{*}(1535)$ and hadron interactions in the $\gamma
N\to\pi N$ and $\pi N\to\pi N$ reactions and lattice QCD simulation
simultaneously together with our previous work Liu:2015ktc (21). We extend
HEFT for the $\gamma N\to\pi N$ process, and the parameters for the FSI part
are the same as previous ones. First we give the general $\gamma N\to\pi N$
potentials without FSI and transform them among different representations, and
then provide the corresponding scattering T matrix with FSI and associate
electric dipole amplitudes.

From the numerical results, the contact term dominates the region near the
threshold, and the $\rho$-exchanging contributions become important around
1200$\sim$1400 MeV. The $N^{*}(1535)$ resonance is dynamically generated among
the bare triquark kernel and $\pi N$ and $\eta N$ channels in our framework
Liu:2015ktc (21). The coupling of the bare $N^{*}(1535)$ with $\gamma N$ is
indispensable for the electric dipole amplitudes around 1500 MeV.

Combined with out previous work Liu:2015ktc (21), an excellent picture of
simultaneous descriptions is revealed for pion photoproduction, $\pi N$
scattering, resonance pole position, and the finite-volume energy level and
structure analysis associated with lattice QCD simulations, in spite of
certain residual discrepancy in the end of considered energy range, near 1.6
GeV. The $S_{11}$ partial wave meson-nucleon channel with negative parity is
complicated, since there are many open thresholds such as $K\Lambda$ and
$K\Sigma$ and also two interfering resonances $N^{*}(1535)$ and $N^{*}(1650)$.
Including these effects sequentially in future will provide a better
deciphering of nucleon resonances and understanding of photo- and hadroinduced
production.

##  V Acknowledgment

We would like to thank Dr. Hiroyuki Kamano for useful discussion. This project
is supported by the National Natural Science Foundation of China under Grants
No. 12175091, No. 11965016, and No. 12047501, and CAS Interdisciplinary
Innovation Team.

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