# Neutral pion photoproduction on the nucleon in a chiral quark model

Li-Ye Xiao1, Xu Cao2 and Xian-Hui Zhong1 111E-mail<EMAIL_ADDRESS>1)
Department of Physics, Hunan Normal University, and Key Laboratory of Low-
Dimensional Quantum Structures and Quantum Control of Ministry of Education,
Changsha 410081, China  2) Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China

###### Abstract

A chiral quark-model approach is adopted to study the $\gamma
p\rightarrow\pi^{0}p$ and $\gamma n\rightarrow\pi^{0}n$ reactions. Good
descriptions of the total and differential cross sections and single-
polarization observables are obtained from the pion production threshold up to
the second resonance region. It is found that (i) the $n=0$ shell resonance
$\Delta(1232)P_{33}$, the $n=1$ shell resonances $N(1535)S_{11}$ and
$N(1520)D_{13}$, and the $n=2$ shell resonance $N(1720)P_{13}$ play crucial
roles in these two processes. They are responsible for the first, second and
third bump structures in the cross sections, respectively. (ii) Furthermore,
obvious evidences of $N(1650)S_{11}$ and $\Delta(1620)S_{31}$ are also found
in the reactions. They notably affect the cross sections and the polarization
observables from the second resonance region to the third resonance region.
(iii) The $u$-channel background plays a crucial role in the reactions. It has
strong interferences with the $s$-channel resonances. (iv) The $t$-channel
background seems to be needed in the reactions. Including the $t$-channel
vector-meson exchange contribution, the descriptions in the energy region
$E_{\gamma}=600\sim 900$ MeV are improved significantly. The helicity
amplitudes of the main resonances, $\Delta(1232)P_{33}$, $N(1535)S_{11}$,
$N(1520)D_{13}$, $N(1720)P_{13}$, $N(1650)S_{11}$, and $\Delta(1620)S_{31}$,
are extracted and compared with the results from other groups.

###### pacs:

13.60.Le, 14.20.Gk, 12.39.Jh, 12.39.Fe

##  I Introduction

Understanding of the baryon spectrum and searching for the missing nucleon
resonances and new exotic states are hot topics in hadronic physics
Klempt:2009pi . Photoproduction of mesons is an ideal tool for the study of
nucleon and $\Delta(1232)$ spectroscopies in experiments Krusche:2003ik .
Neutral pion photoproduction reactions are of special interest because the
neutral pions do not couple directly to photons so that nonresonant background
contributions are suppressed (i.e., no contact term contribution)
Dieterle:2014blj . In the past few years, great progress has been achieved in
experiments studying of the $\gamma p\rightarrow\pi^{0}p$ reaction at JLab,
CB-ELSA, MAMI, and GRAAL. These experimental groups have carried out precise
measurements of the differential cross sections and single-polarization
observables with a large solid angle coverage and a wide photon energy range
Fuchs:1996ja ; Beck:2006ye ; Sparks:2010vb ; Crede:2011dc ; vanPee:2007tw ;
Elsner:2008sn ; Adamian:2000yi ; Dugger:2007bt ; Bartalini:2005wx ;
Hartmann:2014mya ; Bartholomy:2004uz ; Anisovich:2005tf . Recently they also
have finished some measurements of the double-polarization observables
Hartmann:2014mya ; Thiel:2012yj ; Sikora:2013vfa ; Gottschall:2013uha .
Furthermore, in recent years significant progress has been achieved in
experiments measuring the $\gamma n\rightarrow\pi^{0}n$ reaction as well. In
2009, some measurements of the beam asymmetries for the $\gamma
n\rightarrow\pi^{0}n$ process were obtained by the GRAAL experiment in the
second and third resonances region DiSalvo:2009zz . Very recently, the quasi-
free differential and total cross sections in the second and third resonances
region for this reaction were also measured by the Crystal Ball/TAPS
experiment at MAMI Dieterle:2014blj . Thus, improvement of the experimental
situations gives us a good opportunity to study the excitation spectroscopies
of the nucleon and $\Delta(1232)$.

Stimulated by these new measurements, many partial-wave analysis groups, such
as BnGa Anisovich:2009zy ; Anisovich:2011fc ; Anisovich:2013jya , SAID
Chen:2012yv ; Workman:2011vb ; Workman:2012jf , MAID Drechsel:2007if , Kent
Shrestha:2012ep , Jülich Huang:2011as ; Ronchen:2014cna  and ANL-Osaka
Kamano:2013iva , have updated their analysis in recent years. For the $\gamma
p\rightarrow\pi^{0}p$ reaction, good descriptions of the data up to the second
and third resonances region have been obtained by different groups. However,
the explanations of the reaction data and the extracted resonance properties
from the reaction still exhibit strong model dependencies. For example, the
$\gamma p$ couplings for some well-established resonances, such as
$N(1535)S_{11}$, $N(1650)S_{11}$ and $N(1520)D_{13}$, extracted by various
groups differ rather notably from each other. For the $\gamma
n\rightarrow\pi^{0}n$ reaction, consistent predictions from different
approaches can only be obtained in the first resonance region Dieterle:2014blj
. Because of the lack of data, the predictions from different models in the
second and third resonances region are very different. Fortunately, in this
energy region some new measurements of the cross section for the $\gamma
n\rightarrow\pi^{0}n$ reaction at MAMI Dieterle:2014blj  were reported about
one year ago.

These new data for the $\gamma n\rightarrow\pi^{0}n$ reaction not only provide
us a good opportunity to extract more knowledge of the neutron resonances, but
also shed light on the puzzle of the narrow structure around $W=1.68$ GeV
observed in the excitation function of $\eta$ production off quasi-free
neutrons by several experimental groups Kuznetsov:2007gr ; Jaegle:2008ux ;
Jaegle:2011sw . This narrow structure has been listed by the Particle Data
Group (PDG) as a new nucleon resonance $N(1685)$ Agashe:2014kda . However,
many controversial explanations about this narrow structure, such as the
$N(1650)S_{11}$ and $N(1710)P_{11}$ coupled-channel effects, interference
effects between $N(1650)S_{11}$, $N(1710)P_{11}$ and $N(1720)P_{13}$, and
effects from strangeness threshold openings, can be found in the literature
Shyam:2008fr ; Shklyar:2006xw ; Doring:2009qr . In our quark model study, we
find that the narrow structure around $W=1.68$ GeV can be explained by the
constructive interferences between $N(1535)S_{11}$ and $N(1650)S_{11}$
Zhong:2011ti . Our conclusion is consistent with the analysis from the BnGa
group Anisovich:2015tla ; Anisovich:2008wd . It should be mentioned that, the
$\gamma n$ coupling for $N(1650)S_{11}$ extracted by us and BnGa group has a
positive sign, which is opposite to that of PDG Agashe:2014kda . Now, two
questions arise naturally: (i) Can some clues about the controversially
discussed $N(1685)$ be found in the $\gamma n\rightarrow\pi^{0}n$ reaction?
(ii) Are the properties of $N(1535)S_{11}$ and $N(1650)S_{11}$ extracted from
the $\eta N$ channel consistent with those extracted from the $\pi^{0}N$
channel? To better understand these questions, a systematic analysis of the
recent data for the neutral pion production off nucleons is urgently needed.

In this work, we carry out a combined study of the $\gamma
p\rightarrow\pi^{0}p$ and $\gamma n\rightarrow\pi^{0}n$ reactions in a chiral
quark model. By systematically analyzing the new data for neutral pion
photoproduction on the nucleons, we attempt to uncover some puzzles existing
in the photoproduction reactions and obtain a better understanding of the
excitation spectra of the nucleon and $\Delta(1232)$. It should be mentioned
that there are interesting differences between $\gamma p\to\pi^{0}p$ and
$\gamma n\to\pi^{0}n$. In the $\gamma p$ reactions, contributions from the
nucleon resonances of representation $[70,^{4}8]$ will be suppressed by the
Moorhouse selection rule Moorhouse:1966jn ; Zhao:2006an . In contrast, all the
octet states can contribute to the $\gamma n$ reactions. In other words, more
states will be present in the $\gamma n$ reactions. Therefore, by studying
neutral pion photoproduction on nucleons, we expect that the role played by
intermediate baryon resonances can be highlighted.

In the chiral quark model, an effective chiral Lagrangian is introduced to
account for the quark-pseudoscalar-meson coupling. Since the quark-meson
coupling is invariant under the chiral transformation, some of the low-energy
properties of QCD are retained. The chiral quark model has been well developed
and widely applied to meson photoproduction reactions Zhong:2011ti ;
Zhong:2011ht ; Li:1994cy ; Li:1995si ; Li:1997gd ; Zhao:2002id ; Zhao:1998fn ;
Zhao:2001jw ; Zhao:1998rt ; Li:1998ni ; Saghai:2001yd ; He:2008ty ; He:2009zzi
. Recently, this model has been successfully extended to $\pi N$ and $KN$
reactions as well Zhong:2007fx ; Zhong:2008km ; Zhong:2013oqa ; Xiao:2013hca .

The paper is organized as follows. In Sec. II, a brief review of the chiral
quark model approach is given. The numerical results are presented and
discussed in Sec. III. Finally, a summary is given in Sec. IV.

##  II The model

In this section, we give a brief review of the chiral quark model. In this
model, the $s$\- and $u$-channel transition amplitudes are determined by
Li:1997gd ; Li:1995si

$\displaystyle\mathcal{M}_{s}=\sum_{j}\langle N_{f}|H_{m}|N_{j}\rangle\langle
N_{j}|\frac{1}{E_{i}+\omega_{\gamma}-E_{j}}H_{e}|N_{i}\rangle,$ (1)
$\displaystyle\mathcal{M}_{u}=\sum_{j}\langle
N_{f}|H_{e}\frac{1}{E_{i}-\omega_{m}-E_{j}}|N_{j}\rangle\langle
N_{j}|H_{m}|N_{i}\rangle,$ (2)

where $H_{m}$ and $H_{e}$ stand for the quark-pseudoscalar-meson and
electromagnetic couplings at the tree level, respectively. They are described
by Li:1995si ; Li:1997gd ; Zhao:2002id

$\displaystyle H_{m}$ $\displaystyle=$
$\displaystyle\sum_{j}\frac{1}{f_{m}}\bar{\psi}_{j}\gamma^{j}_{\mu}\gamma^{j}_{5}\psi_{j}\vec{\tau}\cdot\partial^{\mu}\vec{\phi}_{m},$
(3) $\displaystyle H_{e}$ $\displaystyle=$
$\displaystyle-\sum_{j}e_{j}\gamma^{j}_{\mu}A^{\mu}(\mathbf{k},\mathbf{r}),$
(4)

where $\psi_{j}$ represents the $j$-th quark field in a hadron, $\phi_{m}$ is
the field of the pseudoscalar-meson octet, and $f_{m}$ is the meson’s decay
constant. The $\omega_{\gamma}$ is the energy of the incoming photons. The
$|N_{i}\rangle$, $|N_{j}\rangle$ and $|N_{f}\rangle$ stand for the initial,
intermediate and final states, respectively, and their corresponding energies
are $E_{i}$, $E_{j}$ and $E_{f}$, which are the eigenvalues of the non-
relativistic constituent quark model Hamiltonian $\hat{H}$ Isgur:1978xj ;
Isgur:1977ef ; Capstick:1986bm . The $s$\- and $u$-channel transition
amplitudes have been worked out in the harmonic oscillator basis in Refs.
Li:1995si ; Li:1997gd ; Zhao:2002id .

The $t$-channel contributions of vector meson exchange are included in this
work. The effective Lagrangians for the vector meson exchange for the
$\gamma\pi V$ and $Vqq$ couplings are adopted as Zhao:2002id

$\displaystyle\mathcal{L}_{\gamma\pi V}$ $\displaystyle=$ $\displaystyle
e\frac{g_{V\pi\gamma}}{m_{\pi}}\varepsilon_{\alpha\beta\gamma\delta}\partial^{\alpha}A^{\beta}\partial^{\gamma}V^{\delta}\pi,$
(5) $\displaystyle\mathcal{L}_{Vqq}$ $\displaystyle=$ $\displaystyle
g_{Vqq}\bar{\psi}_{j}(\gamma_{\mu}+\frac{\kappa_{q}}{2m_{q}}\sigma_{\mu\nu}\partial^{\nu})V^{\mu}\psi_{j},$
(6)

where $A$ and $V$ denote the photon and vector-meson fields, respectively;
$\pi$ stands for the $\pi$-meson field; $g_{V\pi\gamma}$ and $g_{Vqq}$ are the
coupling constants. The $t$-channel transition amplitude has been given in the
harmonic oscillator basis in Refs. Zhao:2002id .

It should be remarked that the amplitudes in terms of the harmonic oscillator
principle quantum number $n$ are the sum of a set of SU(6) multiplets with the
same $n$. To obtain the contributions of individual resonances, we need to
separate out the single-resonance-excitation amplitudes within each principle
number $n$ in the $s$-channel. Taking into account the width effects of the
resonances, the resonance transition amplitudes of the $s$-channel can be
generally expressed as Li:1997gd

$\displaystyle\mathcal{M}^{s}_{R}=\frac{2M_{R}}{s-M^{2}_{R}+iM_{R}\Gamma_{R}}\mathcal{O}_{R}e^{-(\textbf{k}^{2}+\textbf{q}^{2})/6\alpha^{2}},$
(7)

where $\sqrt{s}=E_{i}+\omega_{\gamma}$ is the total energy of the system,
$\alpha$ is the harmonic oscillator strength, $M_{R}$ is the mass of the
$s$-channel resonance with a width $\Gamma_{R}$, and $\mathcal{O}_{R}$ is the
separated operators for individual resonances in the $s$-channel. In the Chew-
Goldberger-Low-Nambu (CGLN) parameterization, the transition amplitude can be
written in a standard form Chew:1957tf :

$\displaystyle\mathcal{O}_{R}$ $\displaystyle=$ $\displaystyle
if^{R}_{1}\mbox{\boldmath$\sigma$\unboldmath}\cdot\mbox{\boldmath$\epsilon$\unboldmath}+f^{R}_{2}\frac{(\mbox{\boldmath$\sigma$\unboldmath}\cdot\mathbf{q})\mbox{\boldmath$\sigma$\unboldmath}\cdot(\mathbf{k}\times\mbox{\boldmath$\epsilon$\unboldmath})}{|\mathbf{q}||\mathbf{k}|}$
(8)
$\displaystyle+if^{R}_{3}\frac{(\mbox{\boldmath$\sigma$\unboldmath}\cdot\mathbf{k})(\mathbf{q}\cdot\mbox{\boldmath$\epsilon$\unboldmath})}{|\mathbf{q}||\mathbf{k}|}+if^{R}_{4}\frac{(\mbox{\boldmath$\sigma$\unboldmath}\cdot\mathbf{q})(\mathbf{q}\cdot\mbox{\boldmath$\epsilon$\unboldmath})}{|\mathbf{q}|^{2}},$

where $\sigma$ is the spin operator of the nucleon, $\epsilon$ is the
polarization vector of the photon, and $\mathbf{k}$ and $\mathbf{q}$ are
incoming photon and outgoing meson momenta, respectively. In the
SU(6)$\otimes$O(3) symmetry limit, we have extracted the CGLN amplitudes for
the $s$-channel resonances in the $n\leq 2$ shell for the $\gamma
p\rightarrow\pi^{0}p$ and $\gamma n\rightarrow\pi^{0}n$ processes, which have
been listed in Tables 1 and 2, respectively. Comparing the CGLN amplitudes of
different resonances with each other, one can easily find which states are the
main contributors to the reactions in the SU(6)$\otimes$O(3) symmetry limit.

Table 1: The CGLN amplitudes of $s$-channel resonances in the $n\leq 2$ shell for the $\gamma p\rightarrow\pi^{0}p$ process in the SU(6)$\otimes$O(3) symmetry limit. We have defined $A\equiv-(\frac{\omega_{m}}{E_{f}+M_{N}}+1)|\mathbf{q}|$, $B\equiv\frac{\omega_{m}}{\mu_{q}}+\frac{2|\mathbf{q}|}{3\alpha^{2}}A$, $C\equiv\frac{\omega_{m}}{\mu_{q}}+\frac{|\mathbf{q}|}{\alpha^{2}}A$, $D\equiv\frac{\omega_{m}}{\mu_{q}}+\frac{2|\mathbf{q}|}{5\alpha^{2}}A$, $x\equiv\frac{|\mathbf{k}||\mathbf{q}|}{3\alpha^{2}}$, $P_{l}^{\prime}(z)\equiv\frac{\partial P_{l}(z)}{\partial z}$, $P_{l}^{\prime\prime}(z)\equiv\frac{\partial^{2}P_{l}(z)}{\partial z^{2}}$. The $\omega_{\gamma}$, $\omega_{m}$ and $E_{f}$ stand for the energies of the incoming photon, outgoing meson and final nucleon, respectively. The $m_{q}$ is the constituent $u$ or $d$ quark mass. $1/\mu_{q}$ is a factor defined by $1/\mu_{q}=2/m_{q}$. $P_{l}(z)$ is the Legendre function with $z=\cos\theta$. resonance | $[N_{6},^{2S+1}N_{3},n,l]$ | $f^{R}_{1}$ | $f^{R}_{2}$ | $f^{R}_{3}$ | $f^{R}_{4}$
---|---|---|---|---|---
$N(938)P_{11}$ | $[56,^{2}8,0,0]$ | 0 | $+i\frac{5\sqrt{2}}{2}\frac{k}{6m_{q}}A$ | 0 | 0
$\Delta(1232)P_{33}$ | $[56,^{4}10,0,0]$ | $+i\frac{4\sqrt{2}}{3}\frac{k}{6m_{q}}AP_{2}^{\prime}(z)$ | $+i\frac{8\sqrt{2}}{3}\frac{k}{6m_{q}}A$ | $-i\frac{4\sqrt{2}}{3}\frac{k}{6m_{q}}AP_{2}^{\prime\prime}(z)$ | 0
$N(1535)S_{11}$ | $[70,^{2}8,1,1]$ | $-i\frac{\sqrt{2}}{18}k(1+\frac{k}{2m_{q}})B$ | 0 | 0 | 0
$\Delta(1620)S_{31}$ | $[70,^{2}10,1,1]$ | $+i\frac{\sqrt{2}}{36}k(1-\frac{k}{6m_{q}})B$ | 0 | 0 | 0
$N(1520)D_{13}$ | $[70,^{2}8,1,1]$ | $+i\frac{\sqrt{2}}{27}k(1+\frac{k}{2m_{q}})\frac{|\mathbf{q}|}{\alpha^{2}}A$ | $+i\frac{\sqrt{2}}{54}k\frac{k}{m_{q}}\frac{|\mathbf{q}|}{\alpha^{2}}AP_{2}^{\prime}(z)$ | 0 | $-i\frac{\sqrt{2}}{27}k\frac{|\mathbf{q}|}{\alpha^{2}}AP_{2}^{\prime\prime}(z)$
$\Delta(1700)D_{33}$ | $[70,^{2}10,1,1]$ | $-i\frac{\sqrt{2}}{54}k(1-\frac{k}{6m_{q}})\frac{|\mathbf{q}|}{\alpha^{2}}A$ | $+i\frac{\sqrt{2}}{54}k\frac{k}{6m_{q}}\frac{|\mathbf{q}|}{\alpha^{2}}AP_{2}^{\prime}(z)$ | 0 | $+i\frac{\sqrt{2}}{54}k\frac{|\mathbf{q}|}{\alpha^{2}}AP_{2}^{\prime\prime}(z)$
$N(1440)P_{11}$ | $[56,^{2}8,2,0]$ | 0 | $+i\frac{11\sqrt{2}}{36\times 18}\frac{15}{19}k\frac{k}{m_{q}}Cx$ | 0 | 0
$N(1710)P_{11}$ | $[70,^{2}8,2,0]$ | 0 | $+i\frac{11\sqrt{2}}{36\times 18}\frac{6}{19}k\frac{k}{m_{q}}Cx$ | 0 | 0
$\Delta(1750)P_{31}$ | $[70,^{2}10,2,0]$ | 0 | $-i\frac{11\sqrt{2}}{36\times 18}\frac{2}{19}k\frac{k}{m_{q}}Cx$ | 0 | 0
$N(1720)P_{13}$ | $[56,^{2}8,2,2]$ | $-i\frac{\sqrt{2}}{90}\frac{25}{12}k(1+\frac{k}{2m_{q}})DP_{2}^{\prime}(z)x$ | $-i\frac{\sqrt{2}}{90}\frac{25}{12}k\frac{k}{2m_{q}}Dx$ | $-i\frac{\sqrt{2}}{90}\frac{25}{12}kDP_{2}^{\prime\prime}(z)x$ | 0
$N(1900)P_{13}$ | $[70,^{2}8,2,2]$ | $-i\frac{\sqrt{2}}{90}\frac{10}{12}k(1+\frac{k}{2m_{q}})DP_{2}^{\prime}(z)x$ | $-i\frac{\sqrt{2}}{90}\frac{10}{12}k\frac{k}{2m_{q}}Dx$ | $-i\frac{\sqrt{2}}{90}\frac{10}{12}kDP_{2}^{\prime\prime}(z)x$ | 0
$\Delta(1985?)P_{33}$ | $[70,^{2}10,2,2]$ | $+i\frac{\sqrt{2}}{90}\frac{5}{12}k(1-\frac{k}{6m_{q}})DP_{2}^{\prime}(z)x$ | $-i\frac{\sqrt{2}}{90}\frac{5}{12}k\frac{k}{6m_{q}}Dx$ | $+i\frac{\sqrt{2}}{90}\frac{5}{12}kDP_{2}^{\prime\prime}(z)x$ | 0
$\Delta(1920)P_{33}$ | $[56,^{4}10,2,2]$ | 0 | $-i\frac{\sqrt{2}}{90}\frac{10}{9}k\frac{k}{2m_{q}}Dx$ | $+i\frac{\sqrt{2}}{90}\frac{10}{9}k\frac{k}{2m_{q}}DP_{2}^{\prime\prime}(z)x$ | 0
$\Delta(1600)P_{33}$ | $[56,^{4}10,2,0]$ | $+i\frac{\sqrt{2}}{90}\frac{10}{9}k\frac{k}{2m_{q}}CP_{2}^{\prime}(z)x$ | $+i\frac{\sqrt{2}}{90}\frac{20}{9}k\frac{k}{2m_{q}}Cx$ | $-i\frac{\sqrt{2}}{90}\frac{10}{9}k\frac{k}{2m_{q}}CP_{2}^{\prime\prime}(z)x$ | 0
$\Delta(1905)F_{35}$ | $[56,^{4}10,2,2]$ | $+i\frac{2\sqrt{2}}{3}\frac{5k}{630m_{q}}AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{2\sqrt{2}}{3}\frac{2k}{630m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $+i\frac{2\sqrt{2}}{3}\frac{3k}{630m_{q}}AP_{2}^{\prime\prime}(z)x^{2}$ | $-i\frac{2\sqrt{2}}{3}\frac{3k}{630m_{q}}AP_{3}^{\prime\prime}(z)x^{2}$
$\Delta(?)F_{35}$ | $[70,^{2}10,2,2]$ | $-i\frac{\sqrt{2}}{180}(1-\frac{k}{6m_{q}})AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{180}\frac{k}{6m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $-i\frac{\sqrt{2}}{180}AP_{2}^{\prime\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{180}AP_{3}^{\prime\prime}(z)x^{2}$
$N(1680)F_{15}$ | $[56,^{2}8,2,2]$ | $+i\frac{5\sqrt{2}}{180}(1+\frac{k}{2m_{q}})AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{5\sqrt{2}}{180}\frac{k}{2m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $+i\frac{5\sqrt{2}}{180}AP_{2}^{\prime\prime}(z)x^{2}$ | $-i\frac{5\sqrt{2}}{180}AP_{3}^{\prime\prime}(z)x^{2}$
$N(?)F_{15}$ | $[70,^{2}8,2,2]$ | $+i\frac{2\sqrt{2}}{180}(1+\frac{k}{2m_{q}})AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{2\sqrt{2}}{180}\frac{k}{2m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $+i\frac{2\sqrt{2}}{180}AP_{2}^{\prime\prime}(z)x^{2}$ | $-i\frac{2\sqrt{2}}{180}AP_{3}^{\prime\prime}(z)x^{2}$
$\Delta(1950)F_{37}$ | $[56,^{4}10,2,2]$ | $+i\frac{2\sqrt{2}}{3}\frac{k}{70m_{q}}AP_{4}^{\prime}(z)x^{2}$ | $+i\frac{2\sqrt{2}}{3}\frac{2k}{105m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $-i\frac{2\sqrt{2}}{3}\frac{k}{210m_{q}}AP_{4}^{\prime\prime}(z)x^{2}$ | $+i\frac{2\sqrt{2}}{3}\frac{k}{210m_{q}}AP_{3}^{\prime\prime}(z)x^{2}$
Table 2:  The CGLN amplitudes of $s$-channel resonances in the $n\leq 2$ shell for the $\gamma n\rightarrow\pi^{0}n$ process in the SU(6)$\otimes$O(3) symmetry limit.  resonance | $[N_{6},^{2S+1}N_{3},n,l]$ | $f^{R}_{1}$ | $f^{R}_{2}$ | $f^{R}_{3}$ | $f^{R}_{4}$
---|---|---|---|---|---
$N(940)P_{11}$ | $[56,^{2}8,0,0]$ | 0 | $+i\frac{5\sqrt{2}}{3}\frac{k}{6m_{q}}A$ | 0 | 0
$\Delta(1232)P_{33}$ | $[56,^{4}10,0,0]$ | $+i\frac{4\sqrt{2}}{3}\frac{k}{6m_{q}}AP_{2}^{\prime}(z)$ | $+i\frac{8\sqrt{2}}{3}\frac{k}{6m_{q}}A$ | $-i\frac{4\sqrt{2}}{3}\frac{k}{6m_{q}}AP_{2}^{\prime\prime}(z)$ | 0
$N(1535)S_{11}$ | $[70,^{2}8,1,1]$ | $-i\frac{\sqrt{2}}{18}k(1+\frac{k}{6m_{q}})B$ | 0 | 0 | 0
$N(1650)S_{11}$ | $[70,^{4}8,1,1]$ | $+i\frac{\sqrt{2}}{36}k\frac{k}{6m_{q}}B$ | 0 | 0 | 0
$\Delta(1620)S_{31}$ | $[70,^{2}10,1,1]$ | $+i\frac{\sqrt{2}}{36}k(1-\frac{k}{6m_{q}})B$ | 0 | 0 | 0
$N(1520)D_{13}$ | $[70,^{2}8,1,1]$ | $+i\frac{\sqrt{2}}{9}(1+\frac{k}{6m_{q}})Ax$ | $+i\frac{\sqrt{2}}{9}\frac{k}{6m_{q}}AxP_{2}^{\prime}(z)$ | 0 | $-i\frac{\sqrt{2}}{9}AxP_{2}^{\prime\prime}(z)$
$N(1700)D_{13}$ | $[70,^{4}8,1,1]$ | $+i\frac{\sqrt{2}}{18}\frac{4}{5}\frac{k}{6m_{q}}Ax$ | $+i\frac{\sqrt{2}}{18}\frac{1}{5}\frac{k}{6m_{q}}AxP_{2}^{\prime}(z)$ | 0 | $-i\frac{\sqrt{2}}{18}\frac{3}{5}\frac{k}{6m_{q}}AxP_{2}^{\prime\prime}(z)$
$\Delta(1700)D_{33}$ | $[70,^{2}10,1,1]$ | $-i\frac{\sqrt{2}}{18}(1-\frac{k}{6m_{q}})Ax$ | $+i\frac{\sqrt{2}}{18}\frac{k}{6m_{q}}AxP_{2}^{\prime}(z)$ | 0 | $+i\frac{\sqrt{2}}{18}AxP_{2}^{\prime\prime}(z)$
$N(1675)D_{15}$ | $[70,^{4}8,1,1]$ | $+i\frac{\sqrt{2}}{6}\frac{k}{15m_{q}}AxP_{3}^{\prime}(z)$ | $+i\frac{\sqrt{2}}{6}\frac{k}{10m_{q}}AxP_{2}^{\prime}(z)$ | $-i\frac{\sqrt{2}}{6}\frac{k}{2m_{q}}Axz$ | $+i\frac{\sqrt{2}}{6}\frac{k}{30m_{q}}AxP_{2}^{\prime\prime}(z)$
$N(1440)P_{11}$ | $[56,^{2}8,2,0]$ | 0 | $+i\frac{47\sqrt{2}}{36\times 108}\frac{10}{11}k\frac{k}{m_{q}}Cx$ | 0 | 0
$N(1710)P_{11}$ | $[70,^{2}8,2,0]$ | 0 | $+i\frac{47\sqrt{2}}{36\times 108}\frac{2}{11}k\frac{k}{m_{q}}Cx$ | 0 | 0
$\Delta(1750)P_{31}$ | $[70,^{2}10,2,0]$ | 0 | $-i\frac{47\sqrt{2}}{36\times 108}\frac{1}{11}k\frac{k}{m_{q}}Cx$ | 0 | 0
$N(?)P_{11}$ | $[70,^{4}8,2,2]$ | 0 | $-i\frac{47\sqrt{2}}{36\times 108}\frac{1}{9}k\frac{k}{m_{q}}Dx$ | 0 | 0
$\Delta(1910)P_{31}$ | $[56,^{4}10,2,2]$ | 0 | $-i\frac{47\sqrt{2}}{36\times 108}\frac{8}{9}k\frac{k}{m_{q}}Dx$ | 0 | 0
$N(1720)P_{13}$ | $[56,^{2}8,2,2]$ | $-i\frac{\sqrt{2}}{108}\frac{10}{2}k\frac{k}{6m_{q}}DP_{2}^{\prime}(z)x$ | $-i\frac{\sqrt{2}}{108}\frac{10}{2}k\frac{k}{6m_{q}}Dx$ | 0 | 0
$N(1900)P_{13}$ | $[70,^{2}8,2,2]$ | $-i\frac{\sqrt{2}}{108}k(1+\frac{k}{6m_{q}})DP_{2}^{\prime}(z)x$ | $-i\frac{\sqrt{2}}{108}k\frac{k}{6m_{q}}Dx$ | $-i\frac{\sqrt{2}}{108}kDP_{2}^{\prime\prime}(z)x$ | 0
$N(?)P_{13}$ | $[70,^{4}8,2,2]$ | 0 | $-i\frac{\sqrt{2}}{108}\frac{1}{2}k\frac{k}{6m_{q}}Dx$ | $+i\frac{\sqrt{2}}{108}\frac{1}{2}k\frac{k}{6m_{q}}DP_{2}^{\prime\prime}(z)x$ | 0
$\Delta(1985?)P_{33}$ | $[70,^{2}10,2,2]$ | $+i\frac{\sqrt{2}}{108}\frac{1}{2}k(1-\frac{k}{6m_{q}})DP_{2}^{\prime}(z)x$ | $-i\frac{\sqrt{2}}{108}\frac{1}{2}k\frac{k}{6m_{q}}Dx$ | $+i\frac{\sqrt{2}}{108}\frac{1}{2}kDP_{2}^{\prime\prime}(z)x$ | 0
$\Delta(1920)P_{33}$ | $[56,^{4}10,2,2]$ | 0 | $-i\frac{\sqrt{2}}{108}\frac{8}{2}k\frac{k}{6m_{q}}Dx$ | $+i\frac{\sqrt{2}}{108}\frac{8}{2}k\frac{k}{6m_{q}}DP_{2}^{\prime\prime}(z)x$ | 0
$\Delta(1600)P_{33}$ | $[56,^{4}10,2,0]$ | $+i\frac{\sqrt{2}}{108}\frac{8}{2}k\frac{k}{6m_{q}}CP_{2}^{\prime}(z)x$ | $+i\frac{\sqrt{2}}{108}\frac{24}{2}k\frac{k}{6m_{q}}Cx$ | $-i\frac{\sqrt{2}}{108}\frac{16}{2}k\frac{k}{6m_{q}}CP_{2}^{\prime\prime}(z)x$ | 0
$N(?)P_{13}$ | $[70,^{4}8,2,0]$ | $+i\frac{\sqrt{2}}{108}\frac{1}{2}k\frac{k}{6m_{q}}CP_{2}^{\prime}(z)x$ | $+i\frac{\sqrt{2}}{108}\frac{3}{2}k\frac{k}{6m_{q}}Cx$ | $-i\frac{\sqrt{2}}{108}\frac{2}{2}k\frac{k}{6m_{q}}CP_{2}^{\prime\prime}(z)x$ | 0
$N(1680)F_{15}$ | $[56,^{2}8,2,2]$ | $+i\frac{\sqrt{2}}{18}\frac{k}{6m_{q}}AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{k}{6m_{q}}AP_{3}^{\prime}(z)x^{2}$ | 0 | 0
$N(?)F_{15}$ | $[70,^{2}8,2,2]$ | $+i\frac{\sqrt{2}}{18}\frac{1}{5}(1+\frac{k}{6m_{q}})AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{1}{5}\frac{k}{6m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{1}{5}AP_{2}^{\prime\prime}(z)x^{2}$ | $-i\frac{\sqrt{2}}{18}\frac{1}{5}AP_{3}^{\prime\prime}(z)x^{2}$
$N(?)F_{15}$ | $[70,^{4}8,2,2]$ | $+i\frac{\sqrt{2}}{18}\frac{1}{14}\frac{k}{6m_{q}}AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{1}{35}\frac{k}{6m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{3}{70}\frac{k}{6m_{q}}AP_{2}^{\prime\prime}(z)x^{2}$ | $-i\frac{\sqrt{2}}{18}\frac{3}{70}\frac{k}{6m_{q}}AP_{3}^{\prime\prime}(z)x^{2}$
$\Delta(?)F_{35}$ | $[70,^{2}10,2,2]$ | $-i\frac{\sqrt{2}}{18}\frac{1}{10}(1-\frac{k}{6m_{q}})AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{1}{10}\frac{k}{6m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $-i\frac{\sqrt{2}}{18}\frac{1}{10}AP_{2}^{\prime\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{1}{10}AP_{3}^{\prime\prime}(z)x^{2}$
$\Delta(1905)F_{35}$ | $[56,^{4}10,2,2]$ | $i\frac{\sqrt{2}}{18}\frac{4}{7}\frac{k}{6m_{q}}AP_{2}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{8}{35}\frac{k}{6m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $+i\frac{\sqrt{2}}{18}\frac{12}{35}\frac{k}{6m_{q}}AP_{2}^{\prime\prime}(z)x^{2}$ | $-i\frac{\sqrt{2}}{18}\frac{12}{35}\frac{k}{6m_{q}}AP_{3}^{\prime\prime}(z)x^{2}$
$\Delta(1950)F_{37}$ | $[56,^{4}10,2,2]$ | $+i\frac{3\sqrt{2}}{4}\frac{8}{9}\frac{k}{70m_{q}}AP_{4}^{\prime}(z)x^{2}$ | $+i\frac{3\sqrt{2}}{4}\frac{8}{9}\frac{2k}{105m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $-i\frac{3\sqrt{2}}{4}\frac{8}{9}\frac{k}{210m_{q}}AP_{4}^{\prime\prime}(z)x^{2}$ | $+i\frac{3\sqrt{2}}{4}\frac{8}{9}\frac{k}{210m_{q}}AP_{3}^{\prime\prime}(z)x^{2}$
$N(1990)F_{17}$ | $[70,^{4}8,2,2]$ | $+i\frac{3\sqrt{2}}{4}\frac{1}{9}\frac{k}{70m_{q}}AP_{4}^{\prime}(z)x^{2}$ | $+i\frac{3\sqrt{2}}{4}\frac{1}{9}\frac{2k}{105m_{q}}AP_{3}^{\prime}(z)x^{2}$ | $-i\frac{3\sqrt{2}}{4}\frac{1}{9}\frac{k}{210m_{q}}AP_{4}^{\prime\prime}(z)x^{2}$ | $+i\frac{3\sqrt{2}}{4}\frac{1}{9}\frac{k}{210m_{q}}AP_{3}^{\prime\prime}(z)x^{2}$

Finally, the differential cross section $d\sigma/d\Omega$, photon beam
asymmetry $\Sigma$, polarization of recoil protons $P$, and target asymmetry
$T$ are given by the following standard expressions Walker:1968xu ;
Krusche:2003ik ; Fasano:1992es :

$\displaystyle\frac{d\sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{e}\alpha_{\pi}(E_{i}+M_{N})(E_{f}+M_{N})}{16sM_{N}^{2}}\frac{1}{2}\frac{|\mathbf{q}|}{|\mathbf{k}|}\sum^{4}_{i=1}|H_{i}|^{2},$
(9) $\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
2\mathrm{Re}(H_{4}^{*}H_{1}-H_{3}^{*}H_{2})/\sum^{4}_{i=1}|H_{i}|^{2},$ (10)
$\displaystyle P$ $\displaystyle=$
$\displaystyle-2\mathrm{Im}(H_{4}^{*}H_{2}+H_{3}^{*}H_{1})/\sum^{4}_{i=1}|H_{i}|^{2},$
(11) $\displaystyle T$ $\displaystyle=$ $\displaystyle
2\mathrm{Im}(H_{2}^{*}H_{1}+H_{4}^{*}H_{3})/\sum^{4}_{i=1}|H_{i}|^{2},$ (12)

where the transition amplitudes $H_{i}$ in the helicity space can be expressed
by the CGLN amplitudes $f_{i}$ Walker:1968xu :

$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2}}\sin\theta\cos\frac{\theta}{2}(f_{3}+f_{4}),$
(13) $\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\sqrt{2}\cos\frac{\theta}{2}[(f_{2}-f_{1})+\sin^{2}\frac{\theta}{2}(f_{3}-f_{4})],$
(14) $\displaystyle H_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\sin\theta\sin\frac{\theta}{2}(f_{3}-f_{4}),$
(15) $\displaystyle H_{4}$ $\displaystyle=$
$\displaystyle\sqrt{2}\sin\frac{\theta}{2}[(f_{2}+f_{1})+\cos^{2}\frac{\theta}{2}(f_{3}-f_{4})].$
(16)

In Eq.(9), the fine-structure constant $\alpha_{e}$ is well determined, and
the $\pi NN$ coupling constant $\alpha_{\pi}$ is related to the axial vector
coupling $g_{A}$ by the generalized Goldberg-Treiman relation

$\displaystyle\alpha_{\pi}=\frac{1}{4\pi}\left(\frac{g_{A}M_{N}}{f_{\pi}}\right)^{2}\equiv\frac{g_{\pi
NN}^{2}}{4\pi}.$ (17)

However, the quark model predicts rather large values $g_{A}=5/3$ for charged
pions and $g_{A}=5\sqrt{2}/6$ for neutral pions. In our paper, the coupling
$\alpha_{\pi}$ is determined by fitting the data.

##  III CALCULATION AND ANALYSIS

###  III.1 Parameters

In our framework, the $s$-channel resonance transition amplitude,
$\mathcal{O}_{R}$, is derived in the SU(6)$\otimes$O(3) symmetry limit. In
reality, the SU(6)$\otimes$O(3) symmetry is generally broken due to, e.g.,
spin-dependent forces in the quark-quark interaction. As a consequence,
configuration mixings would occur. The configuration mixings break the
SU(6)$\otimes$O(3) symmetry, which can change our theoretical predictions.
Furthermore, the helicity couplings and strong decay couplings of some
resonances might be over/under-estimated with the simple quark model. To
accommodate the uncertainties in the symmetric quark model framework, we
introduce a set of coupling strength parameters, $C_{R}$, for each resonance
amplitude by an empirical method Li:1998ni ; Saghai:2001yd ; He:2008ty ;
He:2009zzi :

$\displaystyle\mathcal{O}_{R}\rightarrow C_{R}\mathcal{O}_{R},$ (18)

where $C_{R}$ can be determined by fitting the experimental observables. In
the SU(6)$\otimes$O(3) symmetry limit one finds $C_{R}=1$, while deviations of
$C_{R}$ from unity imply the SU(6)$\otimes$O(3) symmetry breaking.

In our previous study of the $\eta$ photoproduction on the nucleons, we found
the configuration mixings seem to be inevitable for the low-lying $S$-wave
nucleon resonances $N(1535)S_{11}$ and $N(1650)S_{11}$, and $D$-wave nucleon
resonances $N(1520)D_{13}$ and $N(1700)D_{13}$. By including configuration
mixing effects in the $S$\- and $D$-wave states, we explicitly express their
transition amplitudes as follows:

$\displaystyle\mathcal{O}_{R}\rightarrow
C^{[70,^{2}8]}_{R}\mathcal{O}_{[70,^{2}8,J]}+C^{[70,^{4}8]}_{R}\mathcal{O}_{[70,^{4}8,J]}.$
(19)

The coefficients $C^{[70,^{2}8]}_{R}$ and $C^{[70,^{4}8]}_{R}$ can be related
to the mixing angles. We adopt the same mixing scheme as in our previous work
Zhong:2011ti ,

$\left(\begin{array}[]{c}S_{11}(1535)\cr
S_{11}(1650)\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{S}&-\sin\theta_{S}\cr\sin\theta_{S}&\cos\theta_{S}\end{array}\right)\left(\begin{array}[]{c}\left|70,^{2}8,1/2^{-}\right\rangle\cr\left|70,^{4}8,1/2^{-}\right\rangle\end{array}\right),$
(20)

and

$\left(\begin{array}[]{c}D_{13}(1520)\cr
D_{13}(1700)\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{D}&-\sin\theta_{D}\cr\sin\theta_{D}&\cos\theta_{D}\end{array}\right)\left(\begin{array}[]{c}\left|70,^{2}8,3/2^{-}\right\rangle\cr\left|70,^{4}8,3/2^{-}\right\rangle\end{array}\right).$
(21)

Then, the coefficients defined in Eq.(19) are given by

$\displaystyle C^{[70,^{2}8]}_{S_{11}(1535)}$ $\displaystyle=$ $\displaystyle
R_{2}^{S}\cos\theta_{S}(\cos\theta_{S}-\sin\theta_{S}/2),$ (22) $\displaystyle
C^{[70,^{4}8]}_{S_{11}(1535)}$ $\displaystyle=$ $\displaystyle
R_{4}^{S}\sin\theta_{S}(\sin\theta_{S}-2\cos\theta_{S}),$ (23) $\displaystyle
C^{[70,^{2}8]}_{S_{11}(1650)}$ $\displaystyle=$ $\displaystyle
R_{2}^{S}\sin\theta_{S}(\sin\theta_{S}+\cos\theta_{S}/2),$ (24) $\displaystyle
C^{[70,^{4}8]}_{S_{11}(1650)}$ $\displaystyle=$ $\displaystyle
R_{4}^{S}\cos\theta_{S}(\cos\theta_{S}+2\sin\theta_{S}),$ (25) $\displaystyle
C^{[70,^{2}8]}_{D_{13}(1520)}$ $\displaystyle=$ $\displaystyle
R_{2}^{D}\cos\theta_{D}(\cos\theta_{D}-\frac{1}{2\sqrt{10}}\sin\theta_{D}),$
(26) $\displaystyle C^{[70,^{4}8]}_{D_{13}(1520)}$ $\displaystyle=$
$\displaystyle
R_{4}^{D}\sin\theta_{D}(\sin\theta_{D}-2\sqrt{10}\cos\theta_{D}),$ (27)
$\displaystyle C^{[70,^{2}8]}_{D_{13}(1700)}$ $\displaystyle=$ $\displaystyle
R_{2}^{D}\sin\theta_{D}(\sin\theta_{D}+\frac{1}{2\sqrt{10}}\cos\theta_{D}),$
(28) $\displaystyle C^{[70,^{4}8]}_{D_{13}(1700)}$ $\displaystyle=$
$\displaystyle
R_{4}^{D}\cos\theta_{D}(\cos\theta_{D}+2\sqrt{10}\sin\theta_{D}).$ (29)

The parameters $R_{2}$ and $R_{4}$ are introduced to adjust the overall
strength of the partial wave amplitudes of $[70,^{2}8]$ and $[70,^{4}8]$,
respectively, which may be overestimated or underestimated in the naive quark
model Zhong:2011ti . If $R=1$, one finds that the $C_{R}$ parameters of $S$\-
and $D$-wave states can be explained with configuration mixings only. In the
calculation, the mixing angle between $N(1535)S_{11}$ and $N(1650)S_{11}$ is
adopted to be $\theta_{S}=26^{\circ}$, determined in our previous work
Zhong:2011ti . Notice that in our work, we have adopted Isgur’s later
conventions Koniuk:1979vy  where wave functions are in line with the SU(3)
conventions of de Swart deSwart:1963gc . In this frame, we obtain a positive
mixing angle $\theta_{S}$. However, in line with the old conventions of the
SU(3) wave functions from Isgur and Karl’s early works Isgur:1978xj ;
Isgur:1977ef , one obtains a negative mixing angle $\theta_{S}$ Li:1998ni ;
Saghai:2001yd ; He:2008ty ; He:2009zzi ; Isgur:1978xj ; Hey:1974nc . This
question has been clarified in Refs. An:2011sb ; Zhong:2011ti . Furthermore,
the mixing angle between $N(1520)D_{13}$ and $N(1700)D_{13}$ is adopted to be
$\theta_{D}\simeq 10^{\circ}$ as widely suggested in the literature Hey:1974nc
; Capstick:2004tb ; Isgur:1978xj ; Saghai:2001yd ; He:2008ty ; He:2009zzi .

For the $\gamma p\to\pi^{0}p$ reaction, we obtain $R_{2}^{S}\simeq 1.0$ and
$R_{2}^{D}\simeq 1.5$ by fitting the 450 data points of differential cross
section, and the 53 data points of total cross section collected in Tab. 3.
While for the $\gamma n\to\pi^{0}n$ reaction, we obtain $R_{2}^{S}\simeq
R_{4}^{S}\simeq 0.7$, $R_{2}^{D}\simeq 1.4$ and $R_{4}^{D}\simeq 1.0$ by
fitting the 36 data points of total cross section around the second resonance
energy region $1.30\leq W\leq 1.72$ GeV recently measured at MAMI
Dieterle:2014blj . With these determined $R$ parameters, from Eqs. 22-29 one
can obtain the overall strength parameters $C^{[70,^{2}8]}_{R}$ and
$C^{[70,^{4}8]}_{R}$ for the $S$\- and $D$-wave resonances.

The determined $C_{R}$ values for these low-lying resonances are listed in
Tab. 4. From the table, we find that to reproduce the data we need to
introduce two large coupling strength parameters $C_{P_{33}(1232)}\simeq 1.83$
and $C_{S_{31}(1620)}\simeq 1.8$ for $\Delta(1232)P_{33}$ and
$\Delta(1620)S_{31}$, respectively. The reason may be the well-known
underestimation of their photocouplings in the constituent quark model
Sato:1996gk ; Sato:2000jf . We also need to enhance the contributions of
$N(1520)D_{13}$ by a factor of $C_{D_{13}(1520)}\simeq 1.4$, which can not be
explained with configuration mixings only. The underestimation of the
resonance amplitude of $N(1520)D_{13}$ is also found in the $\gamma N\to\eta
N$ processes within the quark model framework Zhong:2011ti , which is due to
the underestimation of the photocoupling of $N(1520)D_{13}$ in the constituent
quark model. In the $\pi^{0}$ photoproduction processes, the data favor a
smaller contribution of $N(1535)S_{11}$ than that in the SU(6)$\otimes$O(3)
symmetry limit. In the $\gamma p\to\pi^{0}p$ reaction, the strength parameter
$C^{[70,^{2}8]}_{S_{11}(1535)}\simeq 0.61$ can be naturally explained with the
configuration mixings between two $S$-wave nucleon resonances $N(1535)S_{11}$
and $N(1650)S_{11}$ with a mixing angle $\theta_{S}\simeq 26^{\circ}$.
However, in the $\gamma n\to\pi^{0}n$ reaction, the strength parameter
$C^{[70,^{2}8]}_{S_{11}(1535)}\simeq 0.43$ can not be well explained with the
configuration mixing effects only, we should introduce a parameter
$R_{2}^{S}\simeq 0.7$ to slightly decrease the transition amplitude of
$[70,^{2}8,1/2^{-}]$. Furthermore, we find that in the $\gamma n\to\pi^{0}n$
reaction, the enhancement of the contributions of $N(1720)P_{13}$ might
significantly improve the descriptions of the experimental data.

Table 3: 450 data points of differential cross section, and 53 data points of total cross section of $\gamma p\rightarrow\pi^{0}p$ included in our fits. The $\chi^{2}$ datum point is about $\chi^{2}/N_{data}=4.3$.  Data Refs. | Obser. | $E_{\gamma}$ | $N_{data}$ | $\chi^{2}_{i}$ | $\chi^{2}_{i}/N_{data}$
---|---|---|---|---|---
Fuchs:1996ja  MAMI | $d\sigma/d\Omega$ | $240,260,278$ | 27 | 357 | 13.2
Beck:2006ye  MAMI | $d\sigma/d\Omega$ | $300\sim 400$ | 112 | 540 | 4.8
Bartholomy:2004uz  CB-ELSA | $d\sigma/d\Omega$ | $438\sim 862$ | 311 | 1204 | 3.9
Schumann:2010js  MAMI | $\sigma$ | $240\sim 335$ | 20 | 10 | 0.5
Bartholomy:2004uz  CB-ELSA | $\sigma$ | $342\sim 1138$ | 33 | 37 | 1.1
Table 4: The strength parameters $C_{R}$ determined by the experimental data. $C_{R}$ parameter |  $\gamma p\rightarrow\pi^{0}p$ |  $\gamma n\rightarrow\pi^{0}n$
---|---|---
$C^{[70,^{2}8]}_{S_{11}(1535)}$ | $0.61^{+0.06}_{-0.04}$ | $0.43^{+0.09}_{-0.09}$
$C^{[70,^{4}8]}_{S_{11}(1535)}$ | … | $-0.42$
$C^{[70,^{2}8]}_{S_{11}(1650)}$ | $0.39^{+0.04}_{-0.06}$ | $0.27^{+0.06}_{-0.06}$
$C^{[70,^{4}8]}_{S_{11}(1650)}$ | … | 1.12
$C^{[70,^{2}8]}_{D_{13}(1520)}$ | $1.41^{+0.15}_{-0.09}$ | $1.32^{+0.09}_{-0.09}$
$C^{[70,^{4}8]}_{D_{13}(1520)}$ | … | $-1.05$
$C^{[70,^{2}8]}_{D_{13}(1700)}$ | 0.09 | 0.08
$C^{[70,^{4}8]}_{D_{13}(1700)}$ | … | 2.05
$C_{P_{33}(1232)}$ | $1.83^{+0.02}_{-0.04}$ | $1.83$
$C_{S_{31}(1620)}$ | $1.80^{+0.50}_{-0.20}$ | $1.80$
$C_{D_{15}(1675)}$ | 1.00 | 1.00
$C_{P_{13}(1720)}$ | 1.00 | $3.20^{+0.1}_{-0.2}$

To take into account relativistic effects, the commonly applied Lorentz boost
factor is introduced in the resonance amplitude for the spatial integrals
Li:1995si , which is

$\displaystyle\mathcal{O}_{R}(\textbf{k},\textbf{q})\rightarrow\mathcal{O}_{R}(\gamma_{k}\textbf{k},\gamma_{q}\textbf{q}),$
(30)

where $\gamma_{k}=M_{N}/E_{i}$ and $\gamma_{q}=M_{N}/E_{f}$.

The $\pi NN$ coupling $\alpha_{\pi}$ and the coupling
$g_{\omega\pi\gamma}\cdot g_{\omega qq}$ from $\omega$-meson exchange in the
$t$ channel are considered as free parameters in the present calculations. By
fitting the experimental data of $\gamma p\rightarrow\pi^{0}p$ reaction (see
Tab. 3), we get $g_{\pi NN}\simeq 13.2$ (i.e., $\alpha_{\pi}\equiv g^{2}_{\pi
NN}/4\pi\simeq 13.8$) and $g_{\omega\pi\gamma}\cdot g_{\omega qq}\simeq 1.37$.
The $\pi NN$ coupling determined in this work is compatible with the value
$g_{\pi NN}\simeq 13.5$ adopted in other literature Huang:2011as ;
Ronchen:2014cna . According to the decay of $\omega\to\pi\gamma$, one obtains
$g_{\omega\pi\gamma}\simeq 0.32$ Zhao:2002id . Then the $\omega qq$ coupling
extracted by us is $g_{\omega qq}\simeq 4.28$, which is consistent with the
value $g_{\omega qq}\simeq 3$ suggested in Ref. Riska:2000gd .

There are another two parameters, the constituent quark mass $m_{q}$ and the
harmonic oscillator strength $\alpha$, from the transition amplitudes. In the
calculation we adopt their standard values in the the quark model, $m_{q}=330$
MeV and $\alpha^{2}=0.16$ GeV2.

In the calculations, the $n=3$ shell resonances are treated as degeneration;
their degenerate mass and width are taken as $M=2080$ MeV and $\Gamma=200$
MeV, since in the low energy region the contributions from the $n=3$ shell are
not significant. In the $u$ channel, the intermediate states are the nucleon
and $\Delta(1232)$ and their resonances. It is found that contributions from
the $n\geq 1$ shell are negligibly small and insensitive to the degenerate
masses for these shells. In this work, we take $M_{1}=1650$ MeV ($M_{2}=1750$
MeV) for the degenerate mass of $n=1$ ($n=2$) shell resonances. In the $s$
channel, the masses and widths of the resonances are taken from the PDG
Agashe:2014kda , or the constituent quark model predictions Capstick:1986bm
if no experimental data are available. For the main resonances, we allow their
masses and widths to change around the values from PDG Agashe:2014kda  in
order to better describe the data. The determined values are listed in Tab. 5.
As a comparison, the resonance masses and widths of both pole and Breit-Wigner
parametrizations from the PDG Agashe:2014kda  are listed in Tab. 5 as well. It
is found that the resonance masses and widths extracted by us are in good
agreement with the values of pole parametrization. The reason is that when we
fit the data, a momentum independent width $\Gamma_{R}$ is used, which is
similar to the pole parametrization. It should be pointed out that
$N(1720)P_{13}$ seems to be a narrow state with a width of $120$ MeV in our
model, which is about one half of the average value from the PDG
Agashe:2014kda . However, our result is in good agreement with that extracted
from the $\pi^{-}p\to K^{0}\Lambda$ reaction by D. H. Saxon et al.
Saxon:1979xu . The strong decay properties of $N(1720)P_{13}$ will be
discussed in detail in our another work.

To know some uncertainties of a main parameter ($C_{R}$, $M_{R}$,
$\Gamma_{R}$) we vary it around its central value until the predictions are
inconsistent with the data within their uncertainties. The obtained
uncertainties for the main parameters have been given in Tabs. 4 and 5.

Table 5: The masses $M_{R}$ (MeV) and widths $\Gamma_{R}$ (MeV) for the $s$-channel resonances in present work compared with the values of Breit-Wigner (BW) and pole parametrizations from PDG14 Agashe:2014kda .  resonance | $(M_{R},\Gamma_{R})_{\mathrm{Ours}}$ | $(M_{R},\Gamma_{R})_{\mathrm{Pole}}$ | $(M_{R},\Gamma_{R})_{\mathrm{BW}}$
---|---|---|---
$\Delta(1232)P_{33}$ | $1210^{+2}_{-2},98^{+2}_{-2}$ | $1210^{+1}_{-1},100^{+2}_{-2}$ | $1232^{+2}_{-2},117^{+3}_{-3}$
$N(1535)S_{11}$ | $1515^{+5}_{-7},115^{+10}_{-15}$ | $1510^{+20}_{-20},170^{+80}_{-80}$ | $1535^{+10}_{-10},150^{+25}_{-25}$
$N(1650)S_{11}$ | $1660^{+10}_{-20},150^{+30}_{-20}$ | $1655^{+15}_{-15},135^{+35}_{-35}$ | $1655^{+15}_{-10},140^{+30}_{-30}$
$\Delta(1630)S_{31}$ | $1600^{+10}_{-10},135^{+25}_{-15}$ | $1600^{+10}_{-10},130^{+10}_{-10}$ | $1630^{+30}_{-30},140^{+10}_{-10}$
$N(1520)D_{13}$ | $1518^{+5}_{-5},105^{+5}_{-10}$ | $1510^{+5}_{-5},110^{+10}_{-5}$ | $1515^{+5}_{-5},115^{+10}_{-15}$
$N(1720)P_{13}$ | $1685^{+10}_{-5},120^{+5}_{-10}$ | $1675^{+15}_{-15},250^{+150}_{-100}$ | $1720^{+30}_{-20},250^{+150}_{-100}$

###  III.2 $\gamma p\rightarrow\pi^{0}p$

The chiral quark model studies of $\gamma p\rightarrow\pi^{0}p$ were carried
out in Refs. Li:1994cy ; Li:1997gd ; Zhao:2002id  about twenty years ago.
During the past two decades, great progress has been achieved for pion
photoproduction at JLab, CB-ELSA, MAMI, and GRAAL. The new data sets are more
accuracy and have larger solid angle coverage and wider photon energy range.
The improvement of the experimental situations gives us a good opportunity to
test our model and study the excitation spectra of nucleon and $\Delta(1232)$
at the same time. All the intermediate states in the $s$ channel classified in
the quark model with $n\leq 2$ are listed in Tab. 1. It should be pointed out
that in this reaction the contributions from the nucleon excitations with the
representation $[70,^{4}8]$ are forbidden by the Moorhouse selection rule
Moorhouse:1966jn ; Zhao:2006an . In the $n=0$ shell, both nucleon pole and
$\Delta(1232)P_{33}$ contribute to the reaction. Comparing their CGLN
amplitudes listed in Tab. 1, we can obviously see that $\Delta(1232)P_{33}$
plays a dominant role for its larger amplitudes. In the $n=1$ shell, two
$S$-wave states $N(1535)S_{11}$ and $\Delta(1620)S_{31}$ and two $D$-wave
states $N(1520)D_{13}$ and $\Delta(1700)D_{33}$ contribute to the reaction.
Considering configuration mixing effects, we find that $N(1650)S_{11}$ and
$N(1700)D_{13}$ can also contribute to the reaction. Similarly, from Tab. 1 we
can find that $N(1535)S_{11}$ and $N(1520)D_{13}$ play a dominant role in the
$n=1$ shell $S$-wave and $D$-wave resonances, respectively. In the $n=2$ shell
eight $P$-wave resonances and five $F$-wave resonances contribute to the
reaction. Comparing their CGLN amplitudes we find that $N(1720)P_{13}$ and
$N(1680)F_{15}$ play a dominant role in the $n=2$ shell $P$-wave resonances
and $F$-wave resonances, respectively.

In present work, we have calculated the differential cross sections, total
cross section, beam asymmetry, target asymmetry, and polarization of recoil
protons from pion production threshold up to the second resonance region for
the $\gamma p\rightarrow\pi^{0}p$ reaction. The model parameters are
determined by fitting the 450 data points of differential cross section from
MAMI Fuchs:1996ja ; Beck:2006ye  and CB-ELSA Bartholomy:2004uz  in the beam
energy region 240 MeV$\leq E_{\gamma}\leq 862$ MeV, and the 53 data points of
total cross section from MAMI Schumann:2010js  and CB-ELSA Bartholomy:2004uz
in the beam energy region $240\leq E_{\gamma}\leq 1138$ MeV (see Tab. 3). The
$\chi^{2}$ datum point is about $\chi^{2}/N_{data}=4.3$. Our results are
compared with the data in Figs 1-10.

Figure 1: (Color online) Differential cross section of the $\gamma
p\rightarrow\pi^{0}p$ reaction as a function of scattering angle. Data are
taken from Fuchs:1996ja  (solid circles), Beck:2006ye  (solid squares),
Beck:1997ew  (solid triangles), and Bartholomy:2004uz  (open triangles). The
first and second numbers in each figure correspond to the photon energy
$E_{\gamma}$ (MeV) and the $\pi N$ center-of-mass (c.m.) energy $W$ (MeV),
respectively. Figure 2: (Color online) Total cross section as a function of
c.m. energy $W$ for the $\gamma p\rightarrow\pi^{0}p$ reaction. Data are taken
from Bartholomy:2004uz  (open circles), Mazzucato:1986dz  (solid circles), and
Schumann:2010js  (triangles). The results for switching off the contributions
from $N(1650)S_{11}$, $\Delta(1620)S_{31}$, and $t$ channel; and the partial
cross sections for $\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$,
$N(1720)P_{13}$ and the $u$ channel are indicated explicitly by different
legends in the figure.  Figure 3: (Color online) Photon energy dependent
differential cross section of the $\gamma p\rightarrow\pi^{0}p$ reaction. Data
are taken from Beck:2006ye  (left triangles), Govorkov:1966  (open squares),
Fischer:1970dn  (solid stars), DeStaebler:1965sf  (open triangles),
Hemmi:1973nf  (open circles), Yoshioka:1980vu  (solid squares), Bacci:1967zz
(solid circles), Bartholomy:2004uz  (down triangles), Bartalini:2005wx  (up
triangles), Dugger:2007bt  (diamonds). The partial cross sections for
$\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$ and $N(1720)P_{13}$ are
indicated explicitly by different legends in the figure.

The differential and total cross sections are shown in Figs. 1 and 2,
respectively. It is seen that the chiral quark model can obtain a reasonable
description of the data in a wide energy region $E_{\gamma}=200\sim 900$ MeV.
To clearly see the contributions from different resonances, we also plot the
energy dependent differential cross sections in Fig. 3. One can clearly see
three bump structures in both the energy dependent differential cross sections
and the total cross section. According to our calculations, we find that
$\Delta(1232)P_{33}$ is responsible for first bump at $E_{\gamma}\simeq 300$
MeV. It governs the reaction in the first resonance region. Both
$N(1535)S_{11}$ and $N(1520)D_{13}$ together dominate the resonance
contributions in the second resonance region. They give approximately equal
contributions to the second bump at $E_{\gamma}\simeq 700$ MeV. The
$N(1720)P_{13}$ resonance might be responsible for the third bump at
$E_{\gamma}\simeq 1000$ MeV. It should be mentioned that although
$\Delta(1620)S_{31}$ and $N(1650)S_{11}$ do not give obvious structures in the
cross sections, they are crucial to give a correct shape of the differential
cross sections from the second resonance region to the third resonance region
(see Fig. 4). Switching off their contributions one can see that the total
cross sections around $E_{\gamma}=700\sim 1000$ MeV are overestimated slightly
(see Figs. 2). The $u$-channel background plays a crucial role in the
reaction, it has strong destructive interferences with $\Delta(1232)P_{33}$,
$N(1535)S_{11}$ and $N(1720)P_{13}$. By including the $t$-channel vector-meson
exchange contribution, we find that the descriptions of the cross sections in
the energy region $E_{\gamma}=600\sim 900$ MeV are improved notably, while
without the $t$-channel contributions, the cross sections are underestimated
obviously (see Figs. 2 and 4). Finally, it should be mentioned that our quark
model explanation of the first and second bump structures in the cross
sections are consistent with that of the isobar model vanPee:2007tw ;
Anisovich:2005tf ; Bartholomy:2004uz . However, our quark model explanation of
the third bump structure differs from that of the isobar model vanPee:2007tw ;
Anisovich:2005tf ; Bartholomy:2004uz . In Ref. vanPee:2007tw ;
Anisovich:2005tf ; Bartholomy:2004uz , the authors predicted that the third
bump might be due to three major contributions: $\Delta(1700)D_{33}$,
$N(1680)F_{15}$ and $N(1650)S_{11}$, rather than $N(1720)P_{13}$. Thus, to
clarify the puzzle about the third bump structure in the cross section more
studies of the reaction $\gamma p\rightarrow\pi^{0}p$ are needed.

Figure 4: (Color online) Effects of $N(1650)S_{11}$, $\Delta(1620)S_{31}$ and
$t$ channel on the differential cross sections of the $\gamma
p\rightarrow\pi^{0}p$ process. Data are taken from Bartholomy:2004uz  (open
squares) and Bartalini:2005wx  (solid squares). The results by switching off
the contributions from $N(1650)S_{11}$, $\Delta(1620)S_{31}$, and $t$ channel
are indicated explicitly by different legends in the figure. The first and
second number in each figure correspond to the photon energy $E_{\gamma}$
(MeV) and the $\pi N$ center-of-mass energy $W$ (MeV), respectively. Figure 5:
(Color online) Beam asymmetry of the $\gamma p\rightarrow\pi^{0}p$ process as
a function of scattering angle. The data are taken from Beck:2006ye  (solid
left triangles), Blanpied:2001ae  (open squares), Ganenko:1974zp  (open up
triangles), Belyaev:1983xf  (solid circles), Bartalini:2005wx  (solid
squares), Adamian:2000yi  (diamonds), Knies:1974zx  (solid down triangles),
Elsner:2008sn  (stars). The first and second number in each figure correspond
to the photon energy $E_{\gamma}$ (MeV) and the $\pi N$ center-of-mass energy
$W$ (MeV), respectively. Figure 6: (Color online) Photon energy dependent beam
asymmetry of the $\gamma p\rightarrow\pi^{0}p$ reaction. The data are taken
from Ganenko:1974zp  (right triangles), Blanpied:2001ae  (open up triangles),
Belyaev:1983xf  (open diamonds), Bartalini:2005wx  (solid diamonds),
Adamian:2000yi  (down triangles), Knies:1974zx  (open circles), Elsner:2008sn
(solid squares), Barbiellini:1970qu  (solid circles), and Zdarko:1972di  (open
squares). The results by switching off the contributions from various partial
waves are indicated explicitly by different legends in the figure.

The beam asymmetries $\Sigma$ in the energy region $E_{\gamma}=220\sim 900$
MeV are shown in Fig. 5. In this energy region, the polarized data are not as
abundant as those of differential cross sections. In the low energy region
$E_{\gamma}<600$ MeV, until now no data on $\Sigma$ at the forward and
backward angles had been obtained. From Fig. 5, it is seen that the chiral
quark model has achieved good descriptions of the measured beam asymmetries
$\Sigma$ in the energy region $E_{\gamma}=220\sim 800$ MeV. In the higher
energy region $E_{\gamma}>800$ MeV, it is found that the chiral quark model
poorly describes the data at forward angles. To clearly see contributions from
different resonances, the energy dependent beam asymmetries at six angles
$\theta_{c.m.}=20^{\circ},60^{\circ},90^{\circ},125^{\circ},150^{\circ},170^{\circ}$
are shown in Fig. 6 as well. From the figure, it is found that the beam
asymmetry $\Sigma$ is sensitive to $\Delta(1232)P_{33}$. Its strong effects
not only exist in the first resonance region, but also extend to the second
resonance region. If we switch off the contributions of $\Delta(1232)P_{33}$,
the beam asymmetry $\Sigma$ changes drastically. Furthermore, we find that
both $N(1520)D_{13}$ and $N(1535)S_{11}$ have strong effects on the beam
asymmetry $\Sigma$ around the second resonance region (i.e., $E_{\gamma}\simeq
700$ MeV), and without their contributions, the beam asymmetry $\Sigma$ in
this energy region changes notably. In the higher energy region
$E_{\gamma}>800$ MeV, it is found that the resonances $\Delta(1232)P_{33}$,
$N(1520)D_{13}$, $N(1535)S_{11}$, $N(1650)S_{11}$, $\Delta(1620)S_{31}$ and
$N(1720)P_{13}$ together with the $u$-channel background have equally
important contributions to the beam asymmetry $\Sigma$. It should be mentioned
that when the beam energy $E_{\gamma}>800$ MeV, many $P$\- and $F$-wave states
in the $n=2$ shell begin to have obvious effects on the beam asymmetry
$\Sigma$ as well. Thus, so many equal contributors in this higher energy
region make difficult descriptions of the beam asymmetry $\Sigma$.

Figure 7: (Color online) Polarization of recoil protons $P$ of the $\gamma
p\rightarrow\pi^{0}p$ reaction as a function of scattering angle. Data are
taken from Belyaev:1983xf  (circles), Bratashevsky:1980pu  (triangles), and
Hartmann:2014mya  (squares). The first and second number in each figure
correspond to the photon energy $E_{\gamma}$ (MeV) and the $\pi N$ center-of-
mass energy $W$ (MeV), respectively. Figure 8: (Color online) Photon energy
dependent polarization of recoil protons for the $\gamma p\rightarrow\pi^{0}p$
reaction. Data are taken from Lundquist:1968zz  (open circles), Bloom:1967tn
(solid circles), Belyaev:1982zv  (diamonds), Querzoli:1961zya  (solid down
triangles), Althoff:1966  (solid stars), Goncharov:1973we  (open stars),
Bratashevsky:1985pk  (solid squares), Belyaev:1983xf  (solid up triangles),
Kato:1979br  (solid left triangles), Blum:1976mw  (open down triangles), and
Blum:1976ud  (solid right triangles). The results by switching off the
contributions from various partial waves are indicated explicitly by different
legends in the figure.

The polarizations of recoil protons $P$ are shown in Fig. 7. In the low energy
region $E_{\gamma}<650$ MeV, only a few old data with limited angle coverage
were obtained. Recently, some precise new data in the higher energy region
$E_{\gamma}\simeq 700\sim 900$ MeV were reported by the CBELSA/TAPS
Collaboration Hartmann:2014mya . From Fig. 7, it is found that our quark model
descriptions are in reasonable agreement with the measurements in a fairly
wide energy region $E_{\gamma}=280\sim 800$ MeV. Above the photon energy
$E_{\gamma}\simeq 800$ MeV, the quark model descriptions at both forward and
backward angles become worse compared with the data. To clearly see
contributions from different resonances, the energy dependent $P$ at six
angles
$\theta_{c.m.}=40^{\circ},60^{\circ},90^{\circ},110^{\circ},130^{\circ},150^{\circ}$
are shown in Fig. 8 as well. It is found that an obvious dip structure appears
around $E_{\gamma}=700$ MeV, which can be well described in the chiral quark
model. The dip structure is due to the strong effects of $\Delta(1232)P_{33}$.
When we switch off its contribution, we find that the dip structure
disappears. Furthermore, from Fig. 8 it is obviously seen that the
polarization of recoil protons $P$ is sensitive to $N(1520)D_{13}$ and
$N(1535)S_{11}$ around the second resonance region (i.e., $E_{\gamma}\simeq
700$ MeV). In the higher energy region $E_{\gamma}>800$ MeV,
$\Delta(1232)P_{33}$, $N(1520)D_{13}$, $N(1535)S_{11}$, $N(1650)S_{11}$, the
$u$-channel background and other higher partial waves have approximately equal
contributions to $P$, which leads to a complicated description of the higher
energy data.

Figure 9: (Color online) Target asymmetry of the $\gamma p\to\pi^{0}p$
reaction as a function of scattering angle. The data are taken from
Belyaev:1983xf  (solid circles), Fukushima:1977xj  (solid triangles),
Booth:1976es  (open circles), and Hartmann:2014mya  (solid squares). The first
and second number in each figure correspond to the photon energy $E_{\gamma}$
(MeV) and the $\pi N$ center-of-mass energy $W$ (MeV), respectively.  Figure
10: (Color online) Photon energy dependent target asymmetry of the $\gamma
p\rightarrow\pi^{0}p$ reaction. The data are taken from Fukushima:1977xj
(solid squares), Booth:1976es  (open squares), Feller:1976ta  (solid circles),
and Belyaev:1983xf  (open triangles). The results by switching off the
contributions from various partial waves are indicated explicitly by different
legends in the figure.

The target asymmetries $T$ are shown in Fig. 9. Below the photon energy
$E_{\gamma}\simeq 700$ MeV, only a few old data with a very small angle
coverage were obtained. Recently, some precise data with larger angle coverage
in the higher energy region $E_{\gamma}\simeq 700\sim 900$ MeV were published
by CBELSA/TAPS Collaboration Hartmann:2014mya . By comparing with the new data
we find that our chiral quark model calculation obviously underestimates the
target asymmetry $T$ in the higher energy region $E_{\gamma}>700$ MeV, but the
predicted tendency is in rough agreement with the data. In the low energy
region $E_{\gamma}\simeq 280\sim 450$ MeV, the data can be well described in
the chiral quark model, though the data at forward and backward angles are
still absent. In the energy region $E_{\gamma}\simeq 450\sim 660$ MeV, our
quark model results are obviously smaller than the data at the forward angle.
To test our model, we expect that more precise measurements with large angle
coverage can be carried out in the energy region $E_{\gamma}<700$ MeV in the
future. To clearly see contributions from different resonances, the energy
dependent target asymmetries $T$ at six angles
$\theta_{c.m.}=27^{\circ},50^{\circ},83^{\circ},100^{\circ},120^{\circ},145^{\circ}$
are shown in Fig. 10 as well. The data show that there is a dip structure at
the angle $\theta_{c.m.}\simeq 80^{\circ}\sim 100^{\circ}$ around the second
resonance region $E_{\gamma}=700$ MeV. This structure can be explained by the
strong interferences between $\Delta(1232)P_{33}$ and $N(1535)S_{11}$.
Switching off the contributions of either $\Delta(1232)P_{33}$ or
$N(1535)S_{11}$, no obvious dip structure around $E_{\gamma}=700$ MeV can be
found in the target asymmetry $T$. According to the chiral quark model
predictions, the dip structure should be found at the forward and backward
angles as well. In the higher energy region $E_{\gamma}>700$ MeV, it is found
that many contributors, such as $\Delta(1232)P_{33}$, $N(1535)S_{11}$,
$N(1650)S_{11}$, $N(1520)D_{13}$, $\Delta(1620)S_{31}$, $N(1720)P_{13}$ and
the $u$-channel background have obvious effects on the target asymmetry $T$.

In brief, obvious roles of the $\Delta(1232)P_{33}$, $N(1535)S_{11}$,
$N(1650)S_{11}$, $\Delta(1620)S_{31}$, $N(1520)D_{13}$ and $N(1720)P_{13}$
have been found in the $\gamma p\rightarrow\pi^{0}p$ process. (i)
$\Delta(1232)P_{33}$ not only plays a dominant role around the first resonance
region, its strong contributions also extend up to the third resonance region,
which can be obviously seen in the cross section, beam asymmetry, target
asymmetry and polarization of recoil protons. (ii) Both $N(1520)D_{13}$ and
$N(1535)S_{11}$ play a dominant role around the second resonance region. They
are the main contributors of the second bump structure in the energy dependent
differential cross section and total cross section. Their strong effects on
the polarization observables can be seen obviously as well. (iii)
$N(1720)P_{13}$ might play a crucial role in the third resonance region. It
might be responsible for the third bump structure in the energy dependent
differential cross section and total cross section. However, no dominant role
of $N(1720)P_{13}$ is found in the polarization observables. It should be
pointed out that the evidence of $N(1720)P_{13}$ around the third resonance
region should be further confirmed due to our poor descriptions of the
polarization observables in the higher energy region. (iv)
$\Delta(1620)S_{31}$ and $N(1650)S_{11}$ are crucial to give the correct shape
of the differential cross sections in the second resonance region, although
they do not contribute obvious structures in the cross sections. (v)
Furthermore, the $u$\- and $t$-channel backgrounds play crucial roles in the
reaction as well. The $u$ channel has a strong interference with the
resonances, such as $\Delta(1232)P_{33}$, $N(1520)D_{13}$ and $N(1535)S_{11}$.
Including the $t$-channel vector-meson exchange contribution, we find that the
descriptions in the energy region $E_{\gamma}=600\sim 900$ MeV are improved
obviously. (vi) No obvious contributions of the other resonances, such as
$N(1700)D_{13}$, $\Delta(1700)D_{33}$ and $N(1680)F_{15}$, are found in the
$\gamma p\rightarrow\pi^{0}p$ process.

Figure 11: (Color online) Differential cross sections of the $\gamma
n\rightarrow\pi^{0}n$ reaction as a function of scattering angle. Data are
taken from Dieterle:2014blj  (open circles), Ando:1977  (solid circles),
Hemmi:1973ii  (solid squares), and Bacci:1972bh  (solid triangles). The first
and second number in each figure correspond to the photon energy $E_{\gamma}$
(MeV) and the $\pi N$ center-of-mass energy $W$ (MeV), respectively. Figure
12: (Color online) Photon energy dependent differential cross sections of the
$\gamma n\rightarrow\pi^{0}n$ reaction. Data are taken from Dieterle:2014blj
(solid triangles), Kossert:2003zf  (solid stars), Hemmi:1973ii  (solid
squares), and Bacci:1972bh  (solid circles). The partial cross sections for
$\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$ and $N(1720)P_{13}$ are
indicated explicitly by different legends in the figure.  Figure 13: (Color
online) Total cross section as a function of the c.m. energy $W$ for the
$\gamma n\rightarrow\pi^{0}n$ reaction. Data are taken from Dieterle:2014blj .
The results by switching off the contributions from $N(1650)S_{11}$,
$\Delta(1620)S_{31}$, and $t$ channel; and the partial cross sections for
$\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$, $N(1720)P_{13}$ and
$u$ channel are indicated explicitly by different legends in the figure.
Figure 14: (Color online) Cross section ratio $\sigma_{n}/\sigma_{p}$ between
the reactions $\gamma n\rightarrow\pi^{0}n$ and $\gamma p\rightarrow\pi^{0}p$
as a function of the center-of-mass energy $W$. Data are taken from Ref.
Dieterle:2014blj .  Figure 15: (Color online) Beam asymmetry of the $\gamma
n\rightarrow\pi^{0}n$ reaction as a function of scattering angle. Data are
taken from DiSalvo:2009zz . The first and second numbers in each figure
correspond to the photon energy $E_{\gamma}$ (MeV) and the $\pi N$ center-of-
mass energy $W$ (MeV), respectively. Figure 16: (Color online) Photon energy
dependent beam asymmetry of the $\gamma n\rightarrow\pi^{0}n$ reaction. Data
are taken from DiSalvo:2009zz . The results by switching off the contributions
from various partial waves are indicated explicitly by different legends in
the figure.

###  III.3 $\gamma n\rightarrow\pi^{0}n$

The chiral quark model studies of $\gamma n\rightarrow\pi^{0}n$ were carried
out in Refs. Li:1994cy ; Li:1997gd ; Zhao:2002id  about twenty years ago.
However, the model studies were limited in the first resonance region, because
only a few scattered data were obtained from the old measurements in the early
1970s. Fortunately, obvious progress has been achieved in experiments in
recent years. In 2009, some measurements of the beam asymmetries for the
$\gamma n\rightarrow\pi^{0}n$ process were obtained by the GRAAL experiment in
the second and third resonances region DiSalvo:2009zz . In this energy region,
recently the quasi-free differential and total cross sections for this
reaction were also measured by the Crystal Ball/TAPS experiment at MAMI
Dieterle:2014blj . Thus, these new measurements in the higher resonances
region provide us a good opportunity to extend the chiral quark model to study
these high-lying resonances.

The contributors of the $s$-channel intermediate states classified in the
quark model with $n\leq 2$ have been listed in Tab. 2. In the $n=0$ shell, the
dominant contribution to the reaction comes from the $\Delta(1232)P_{33}$,
which has much larger CGLN amplitudes than the nucleon pole. In the $n=1$
shell, three $S$-wave states $N(1535)S_{11}$, $N(1650)S_{11}$ and
$\Delta(1620)S_{31}[70,^{2}10]$, and four $D$-wave states $N(1520)D_{13}$,
$N(1700)D_{13}$, $N(1675)D_{15}$ and $\Delta(1700)D_{33}$ contribute to the
reaction. By comparing their CGLN amplitudes listed in Tab. 2, we find that
$N(1535)S_{11}$ and $N(1520)D_{13}$ play dominant roles in these $S$\- and
$D$-wave resonances. In the $n=2$ shell, twelve $P$-wave resonance and seven
$F$-wave resonances contribute to the reaction. Most of the $P$-wave and
$F$-wave resonances in the $n=2$ shell have comparable amplitudes.
$N(1720)P_{13}$, $N(1900)P_{13}$ and $\Delta(1600)P_{33}$ have relatively
larger CGLN amplitudes in the $n=2$ shell $P$-wave resonances, while
$N(1680)F_{15}$ and $\Delta(1905)F_{35}$ have relatively bigger CGLN
amplitudes among the $n=2$ shell $F$-wave resonances.

In this work, we have carried out a chiral quark model study of the $\gamma
n\rightarrow\pi^{0}n$ reaction up to the second and third resonances region.
In the SU(6)$\otimes$O(3) symmetry limit, the parameters from the $u$\- and
$t$-channel backgrounds and the $\Delta$ resonances $\Delta(1232)P_{33}$ and
$\Delta(1620)S_{31}$ for the $\pi^{0}n$ channel should be the same as those
for the $\pi^{0}p$ channel, which have been well determined by the $\gamma p$
data. Thus, in the $\gamma n\rightarrow\pi^{0}n$ reaction these parameters are
taken to have the same values as in the $\gamma p\rightarrow\pi^{0}p$ process.
The other strength parameters, $C_{R}$, for the main resonances
$N(1535)S_{11}$, $N(1650)S_{11}$, $N(1520)D_{13}$ and $N(1720)P_{13}$ for the
$\gamma n$ reaction can not be well constrained by the $\gamma p$ data for
their different photocouplings, thus, we determine them by fitting the 36
$\gamma n$ data points of total cross section around the second resonance
energy region 1.30 GeV$\leq W\leq 1.72$ GeV recently measured at MAMI
Dieterle:2014blj . The $\chi^{2}$ datum point is about
$\chi^{2}/N_{data}=2.8$. Our results compared with the data have been shown in
Figs 11-16.

The differential cross sections compared with the data are shown in Fig. 11.
In the energy region what we consider, only a few data can be obtained.
Fortunately, the abundant data for the $\gamma p\to\pi^{0}p$ process help us
well constrain some important model parameters, as we pointed out above. From
Fig. 11, one can see that the data of the $\gamma n\rightarrow\pi^{0}n$
reaction are reasonably reproduced. To clearly see the contributions from
different partial waves, we plot the energy dependent differential cross
sections in Fig. 12 as well. Our results obviously show three bump structures
in the forward angle region. It is found that the resonances
$\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$ and $N(1720)P_{13}$
play crucial roles in the $\gamma n\rightarrow\pi^{0}n$ reaction. The
$\Delta(1232)P_{33}$ resonance is responsible for the first bump structure
around $E_{\gamma}\simeq 300$ MeV. Both $N(1535)S_{11}$ and $N(1520)D_{13}$
are the main contributors to the second bump around $E_{\gamma}\simeq 700$
MeV. The $N(1720)P_{13}$ resonance is most likely responsible for the third
bump around $E_{\gamma}\simeq 1000$ MeV.

The total cross sections compared with the data are shown in Fig. 13. Obvious
roles of $\Delta(1232)P_{33}$, $N(1535)S_{11}$ and $N(1720)P_{13}$ in the
$\gamma n\rightarrow\pi^{0}n$ reaction can be found in the total cross section
as well. Recently, the total cross section was measured by the Crystal
Ball/TAPS experiment at MAMI Dieterle:2014blj . There are two obvious bump
structures in the cross section in the second and third resonances region (see
Fig. 13). The bump structure around the second resonance region receives
approximately equal contributions from $N(1535)S_{11}$ and $N(1520)D_{13}$,
while the bump structure around the third resonance region might be due to the
contributions of $N(1720)P_{13}$. There are no measurements of the total cross
section in the first resonance region. In this energy region, we predict that
the ratio of total cross section between the $\pi^{0}n$ channel and the
$\pi^{0}p$ channel $\sigma_{n}/\sigma_{p}$ is around $1$ (see Fig. 14).

Furthermore, by analyzing the data of differential and total cross sections,
we find that $\Delta(1620)S_{31}$ and $N(1650)S_{11}$ play obvious roles
around their mass threshold. If we switch off them, the cross sections around
their mass threshold are overestimated significantly. It should be mentioned
that, the role of $N(1650)S_{11}$ should be confirmed by more accurate data in
the future, which will be further discussed in Sec. III.4. Finally, it should
be pointed out that the backgrounds play a crucial role in the reaction. The
$u$-channel background has strong destructive interferences with
$\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$ and $N(1720)P_{13}$.
Including the $t$-channel vector-meson exchange contribution, we find that the
descriptions of the cross sections in the energy region $E_{\gamma}=600\sim
900$ MeV are improved significantly.

The polarization observations for the $\gamma n\rightarrow\pi^{0}n$ reaction
are very sparse. In the year of 2009, the beam asymmetry $\Sigma$ in the
second and third resonances region was measured by the GRAAL Collaboration for
the first time DiSalvo:2009zz . Our chiral quark model results are shown in
Fig. 15. From the figure, it is seen that the model results are in rough
agreement with the data. Our results are notably smaller than the data at the
intermediate angles. To clearly see the contributions from different partial
waves, the energy dependent beam asymmetries $\Sigma$ at six angles
$\theta_{c.m.}=20^{\circ},52^{\circ},91^{\circ},123^{\circ},144^{\circ},163^{\circ}$
are shown in Fig. 16 as well. From the figure, one can find that below the
photon energy $E_{\gamma}\simeq 500$ MeV, the beam asymmetry is sensitive to
$\Delta(1232)P_{33}$ and the $u$-channel background. By turning off one of
them, the beam asymmetry changes drastically in this energy region. Similarly,
we can obviously find that around the second resonance region, i.e.,
$E_{\gamma}\sim 700$ MeV, the $N(1535)S_{11}$, $N(1650)S_{11}$,
$N(1520)D_{13}$, $\Delta(1232)P_{33}$, $\Delta(1620)S_{31}$ and the
$u$-channel background have strong effects on the beam asymmetry. Up to the
second resonance region, the higher partial wave states, such as
$N(1720)P_{13}$ begin to contribute to beam asymmetry. Many resonances
together with the backgrounds have approximately equal contributions to the
beam asymmetry, leading to a very complicated description of the data.

As a whole, a reasonable chiral quark model description of the $\gamma
n\rightarrow\pi^{0}n$ reaction is obtained from the pion production threshold
up to the second resonance region. Obvious evidences of the
$\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$ and $N(1720)P_{13}$ are
also found in the $\gamma n\rightarrow\pi^{0}n$ reaction. (i) The ground state
$\Delta(1232)P_{33}$, the $S$-wave state $N(1535)S_{11}$ together with the
$D$-wave state $N(1520)D_{13}$, and the $P$-wave state $N(1720)P_{13}$ are
responsible for the first, second, and third bump structures in the cross
sections, respectively. (ii) Furthermore, another two $S$-wave states
$\Delta(1620)S_{31}$ and $N(1650)S_{11}$ have obvious effects on the
differential cross section around their mass threshold, although they do not
give any structure in the cross sections. It should be pointed out that the
role of $N(1650)S_{11}$ should be further confirmed in future experiments.
(iii) The backgrounds play a crucial role in the reaction. The $u$ channel
background has a strong constructive interference with the $s$-channel
resonances $\Delta(1232)P_{33}$, $N(1535)S_{11}$ and $N(1520)D_{13}$. By
including the $t$-channel vector-meson exchange contribution, we find that the
descriptions in the energy region $E_{\gamma}=600\sim 900$ MeV are slightly
improved. (vi) No obvious evidence of the other resonances, such as
$N(1700)D_{13}$, $N(1675)D_{15}$, $\Delta(1700)D_{33}$ and $N(1680)F_{15}$, is
found in the $\gamma n\rightarrow\pi^{0}n$ process.

Table 6: The expressions of $\xi$ in Eq. 32 for various resonances. Here we have defined $\mathcal{K}\equiv\sqrt{\frac{\alpha_{e}\alpha_{\pi}(E_{f}+M_{N})\pi}{2M_{R}^{2}M_{N}}}\frac{1}{\Gamma_{R}}$, $\mathcal{A}\equiv\left[\frac{2\omega_{\gamma}}{m_{q}}-\frac{2q^{2}}{3\alpha^{2}}(1+\frac{\omega_{\pi}}{E_{f}+M_{N}})\right]e^{-\frac{k^{2}+q^{2}}{6\alpha^{2}}}$, $\mathcal{B}\equiv\frac{2q^{2}}{3\alpha^{2}}(1+\frac{\omega_{\pi}}{E_{f}+M_{N}})e^{-\frac{k^{2}+q^{2}}{6\alpha^{2}}}$, $\mathcal{D}\equiv\left[\frac{2\omega_{\gamma}}{m_{q}}-\frac{2q^{2}}{5\alpha^{2}}(1+\frac{\omega_{\pi}}{E_{f}+M_{N}})\right]e^{-\frac{k^{2}+q^{2}}{6\alpha^{2}}}$.  $\Delta(1232)P_{33}$ | $\xi_{1/2}$ | $-\mathcal{K}\sqrt{\frac{1}{2}}\frac{4\omega_{\gamma}}{9m_{q}}(1+\frac{\omega_{\pi}}{E_{f}+M_{N}})|\mathbf{q}|C_{P_{33}(1232)}$
---|---|---
| $\xi_{3/2}$ | $-\mathcal{K}\sqrt{\frac{3}{2}}\frac{4\omega_{\gamma}}{9m_{q}}(1+\frac{\omega_{\pi}}{E_{f}+M_{N}})|\mathbf{q}|C_{P_{33}(1232)}$
$\Delta(1620)S_{31}$ | $\xi_{1/2}$ | $\mathcal{K}\frac{\omega_{\gamma}}{18}(1-\frac{\omega_{\gamma}}{6m_{q}})\mathcal{A}C_{S_{31}(1620)}$
$N(1535)S_{11}$ | $\xi^{p}_{1/2}$ | $\mathcal{K}\frac{\omega_{\gamma}}{9}(1+\frac{\omega_{\gamma}}{2m_{q}})\mathcal{A}C^{[70,^{2}8]}_{S_{11}(1535)}$
| $\xi^{n}_{1/2}$ | $-\mathcal{K}\frac{\omega_{\gamma}}{9}\left[(1+\frac{\omega_{\gamma}}{6m_{q}})+\frac{\tan\theta_{S}\omega_{\gamma}}{6m_{q}}\right]\mathcal{A}C^{[70,^{2}8]}_{S_{11}(1535)}$
$N(1650)S_{11}$ | $\xi^{p}_{1/2}$ | $\mathcal{K}\frac{\omega_{\gamma}}{9}(1+\frac{\omega_{\gamma}}{2m_{q}})\mathcal{A}C^{[70,^{2}8]}_{S_{11}(1650)}$
| $\xi^{n}_{1/2}$ | $-\mathcal{K}\frac{\omega_{\gamma}}{9}\left[(1+\frac{\omega_{\gamma}}{6m_{q}})-\frac{\cot\theta_{S}\omega_{\gamma}}{6m_{q}}\right]\mathcal{A}C^{[70,^{2}8]}_{S_{11}(1650)}$
$N(1520)D_{13}$ | $\xi^{p}_{1/2}$ | $\mathcal{K}\frac{\omega_{\gamma}}{9\sqrt{2}}(1-\frac{\omega_{\gamma}}{m_{q}})\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1520)}$
| $\xi^{p}_{3/2}$ | $\mathcal{K}\sqrt{\frac{3}{2}}\frac{\omega_{\gamma}}{9}\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1520)}$
| $\xi^{n}_{1/2}$ | $-\mathcal{K}\frac{\omega_{\gamma}}{9\sqrt{2}}\left[(1-\frac{\omega_{\gamma}}{3m_{q}})-\frac{\tan\theta_{D}\omega_{\gamma}}{3\sqrt{10}m_{q}}\right]\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1520)}$
| $\xi^{n}_{3/2}$ | $-\sqrt{\frac{3}{2}}\mathcal{K}\frac{\omega_{\gamma}}{9}\left[1-\frac{\tan\theta_{D}\omega_{\gamma}}{\sqrt{10}m_{q}}\right]\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1520)}$
$N(1700)D_{13}$ | $\xi^{n}_{1/2}$ | $-\frac{\omega_{\gamma}}{9\sqrt{2}}\mathcal{K}\left[(1-\frac{\omega_{\gamma}}{3m_{q}})+\frac{\cot\theta_{D}\omega_{\gamma}}{3\sqrt{10}m_{q}}\right]\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1700)}$
| $\xi^{n}_{3/2}$ | $-\sqrt{\frac{3}{2}}\mathcal{K}\frac{\omega_{\gamma}}{9}\left[1+\frac{\cot\theta_{D}\omega_{\gamma}}{\sqrt{10}m_{q}}\right]\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1700)}$
| $\xi^{p}_{1/2}$ | $\mathcal{K}\sqrt{\frac{1}{2}}\frac{\omega_{\gamma}}{9}(1-\frac{\omega_{\gamma}}{m_{q}})\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1700)}$
| $\xi^{p}_{3/2}$ | $\mathcal{K}\sqrt{\frac{3}{2}}\frac{\omega_{\gamma}}{9}\mathcal{B}C^{[70,^{2}8]}_{D_{13}(1700)}$
$N(1675)D_{15}$ | $\xi^{n}_{1/2}$ | $-\mathcal{K}\frac{\omega_{\gamma}^{2}}{40m_{q}}\mathcal{B}C_{D_{15}(1675)}$
| $\xi^{n}_{3/2}$ | $-\mathcal{K}\frac{\omega_{\gamma}^{2}}{20\sqrt{2}m_{q}}\mathcal{B}C_{D_{15}(1675)}$
$N(1720)P_{13}$ | $\xi^{p}_{1/2}$ | $\mathcal{K}\sqrt{\frac{1}{2}}\frac{5}{108}\frac{\omega_{\gamma}^{2}}{\alpha^{2}}(1+\frac{k}{3m_{q}})\mathcal{D}|\mathbf{q}|C^{p}_{P_{13}(1720)}$
| $\xi^{p}_{3/2}$ | $-\mathcal{K}\sqrt{\frac{1}{6}}\frac{5}{108}\frac{\omega_{\gamma}^{2}}{\alpha^{2}}\mathcal{D}|\mathbf{q}|C^{p}_{P_{13}(1720)}$
| $\xi^{n}_{1/2}$ | $-\mathcal{K}\sqrt{\frac{1}{2}}\frac{5}{108}\frac{\omega_{\gamma}^{2}}{\alpha^{2}}\frac{2k}{27m_{q}}\mathcal{D}|\mathbf{q}|C^{n}_{P_{13}(1720)}$
| $\xi^{n}_{3/2}$ | 0

###  III.4 Helicity amplitudes

The accurate data for the $\gamma n\rightarrow\pi^{0}n$ and $\gamma
p\rightarrow\pi^{0}p$ processes provide us a good platform to extract the
helicity amplitudes of the dominant resonances in these reactions.
Theoretically, the helicity amplitudes $A_{\lambda}$ for a baryon resonance
$N^{*}$ photoexcitation on a nucleon are defined by

$\displaystyle A_{\lambda}=\sqrt{2\pi/k}\langle
N^{*};J_{z}=\lambda|H_{e}|N;J_{z}=\lambda-1\rangle,$ (31)

where $\lambda=1/2$ and $3/2$. As we know, the helicity amplitudes of a
resonance are related to the transition amplitudes of the photoproduction
reactions. Thus, we can extract the helicity amplitudes from the neutral pion
photoproduction processes by the relation

$\displaystyle
A^{n,p}_{1/2,3/2}=\sqrt{\frac{|\mathbf{q}|M_{R}\Gamma_{R}}{|\mathbf{k}|M_{N}b_{\pi^{0}N}}}\xi^{n,p}_{1/2,3/2},$
(32)

where $b_{\pi^{0}N}\equiv\Gamma_{\pi^{0}N}/\Gamma_{R}$ is the branching ratio
of the resonance. The quantity $\xi$ for different resonances can be
analytically expressed from their CGLN amplitudes. We have given the
expressions of the $\xi$ for several low-lying nucleon and $\Delta$ resonances
in Tab. 6. We estimate the helicity amplitudes for these main contributing
resonances: $\Delta(1232)P_{33}$, $\Delta(1620)S_{31}$, $N(1535)S_{11}$,
$N(1650)S_{11}$, $N(1520)D_{13}$ and $N(1720)P_{13}$. The branching ratios
$b_{\pi N}$ for $N(1720)P_{13}$ are adopted from our quark model prediction,
and the branching ratios for other resonances are taken from PDG14
Agashe:2014kda  (see table 7). Our extracted helicity amplitudes are listed in
Tab. 8. As a comparison, in the same table we also show our previous solution
extracted from the $\eta$ photoproduction processes Zhong:2011ti , the recent
analysis of the $\gamma N$ data from SAID Workman:2012jf ; Chen:2012yv ;
Workman:2011vb , Kent Shrestha:2012ep  and BnGa Anisovich:2009zy ;
Anisovich:2013jya , the average values from PDG14 Agashe:2014kda , and the
theoretical predictions from different quark models Li:1990qu ;
Capstick:1992uc .

Table 7: Branching ratio $b_{\pi N}$ of the resonances used in the calculation. Resonance | $\Delta(1232)P_{33}$ | $\Delta(1620)S_{31}$ | $N(1535)S_{11}$ | $N(1650)S_{11}$ | $N(1520)D_{13}$ | $N(1720)P_{13}$
---|---|---|---|---|---|---
$b_{\pi N}$ | 1.0 | $20\sim 30\%$ | $35\sim 55\%$ | $50\sim 90\%$ | $55\sim 65\%$ | $60\sim 90\%$
Table 8: Extracted helicity amplitudes for the main nucleon and $\Delta(1232)$ resonances from the neutral pion photoproduction reactions (in units of $10^{-3}$GeV-1/2). resonance | helicity | this work | ZZ11Zhong:2011ti  | PDG14 Agashe:2014kda  | Kent12 Shrestha:2012ep  | BnGa Anisovich:2009zy ; Anisovich:2013jya  | SAID12 Workman:2012jf ; Chen:2012yv  | SAID11Workman:2011vb  | ZF Li:1990qu  | C92 Capstick:1992uc
---|---|---|---|---|---|---|---|---|---|---
$\Delta(1232)P_{33}$ | $A_{1/2}$ | $-133$ | … | $-135\pm 6$ | $-137\pm 1$ | $-136\pm 5$ | $-139\pm 2$ | $-138\pm 3$ | $-94$ | $-108$
| $A_{3/2}$ | $-230$ | … | $-255\pm 5$ | $-251\pm 1$ | $-267\pm 8$ | $-262\pm 3$ | $-259\pm 5$ | $-162$ | $-186$
$N(1535)S_{11}$ | $A_{1/2}^{p}$ | $137\pm 15$ | $60\pm 5$ | $115\pm 15$ | $59\pm 3$ | $90\pm 15$ | $128\pm 4$ | $99\pm 2$ | 142 | 76
| $A_{1/2}^{n}$ | $-77\pm 9$ | $-68\pm 5$ | $-75\pm 20$ | $-49\pm 3$ | $-93\pm 11$ | $-58\pm 6$ | $-60\pm 3$ | $-77$ | $-63$
$N(1650)S_{11}$ | $A_{1/2}^{p}$ | $61\pm 9$ | $41\pm 13$ | $45\pm 10$ | $30\pm 3$ | $60\pm 20$ | $55\pm 30$ | $65\pm 25$ | $78$ | $54$
| $A_{1/2}^{n}$ | $-18\pm 3$ | $24\pm 7$ | $-50\pm 20$ | $11\pm 2$ | $25\pm 20$ | $-40\pm 10$ | $-26\pm 8$ | $-47$ | $-35$
$\Delta(1620)S_{31}$ | $A_{1/2}$ | $80\pm 8$ |  | $40\pm 15$ | $-3\pm 3$ | $63\pm 12$ | $29\pm 3$ | $64\pm 2$ | $72$ | 81
$N(1520)D_{13}$ | $A_{1/2}^{p}$ | $-17\pm 1$ | $-32\pm 7$ | $-20\pm 5$ | $-34\pm 1$ | $-32\pm 6$ | $-19\pm 2$ | $-16\pm 2$ | $-47$ | $-15$
| $A_{3/2}^{p}$ | $109\pm 5$ | $113\pm 23$ | $140\pm 10$ | $127\pm 3$ | $138\pm 8$ | $141\pm 2$ | $156\pm 2$ | 117 | $134$
| $A_{1/2}^{n}$ | $-30\pm 1$ | $-40\pm 8$ | $-50\pm 10$ | $-38\pm 3$ | $-49\pm 8$ | $-46\pm 6$ | $-47\pm 2$ | $-75$ | $-38$
| $A_{3/2}^{n}$ | $-90\pm 4$ | $-126\pm 26$ | $-115\pm 10$ | $-101\pm 4$ | $-113\pm 12$ | $-115\pm 5$ | $-125\pm 2$ | $-127$ | $-114$
$N(1720)P_{13}$ | $A_{1/2}^{p}$ | $-89\pm 9$ | … | $100\pm 20$ | $57\pm 3$ | $130\pm 50$ | $95\pm 2$ | $99\pm 3$ | $-68$ | $-11$
| $A_{3/2}^{p}$ | $34\pm 4$ | … | … | $-19\pm 2$ | $100\pm 50$ | $-48\pm 2$ | $-43\pm 2$ | $53$ | $-31$
| $A_{1/2}^{n}$ | $18\pm 2$ | … | … | $-2\pm 1$ | $-80\pm 50$ | … | $-21\pm 4$ | $-4$ | $4$
| $A_{3/2}^{n}$ | 0 | … | … | $-1\pm 2$ | $-140\pm 65$ | … | $-38\pm 7$ | $-33$ | $11$

From Tab. 8, it is found that the helicity amplitudes of $\Delta(1232)P_{33}$
extracted in present work are in good agreement with the values from PDG14
Agashe:2014kda  and other partial wave analysis groups Shrestha:2012ep ;
Anisovich:2009zy ; Anisovich:2013jya ; Workman:2012jf ; Chen:2012yv ;
Workman:2011vb ; Cao:2013psa .

The $A^{p}_{1/2}$ and $A^{n}_{1/2}$ of $N(1535)S_{11}$ extracted in this work
are compatible with the PDG average values and the latest analysis of the
$\gamma N$ data from the SAID Workman:2012jf ; Chen:2012yv ; Workman:2011vb .
It should be pointed out that with the same model we found a smaller $\gamma
p$ coupling $A^{p}_{1/2}\simeq 60\times 10^{-3}$GeV-1/2 for the
$N(1535)S_{11}$ by the analysis of the $\gamma p\to\eta p$ process
Zhong:2011ti , and similar solution was also obtained in He:2008ty ;
Shklyar:2012js . The reason for the different $\gamma p$ couplings for
$N(1535)S_{11}$ in the $\pi^{0}p$ and $\eta p$ channels should be clarified in
future studies.

Figure 17: (Color online) Effects of $N(1650)S_{11}$ on the differential cross
sections and the total cross section around its mass threshold. Data are taken
from Ref. Dieterle:2014blj . The solid and dashed curves are for the results
with negative and positive $\gamma n$ couplings for $N(1650)S_{11}$,
respectively.

All the partial wave analysis groups have extracted similar $\gamma p$
coupling $A^{p}_{1/2}$ for $N(1650)S_{11}$ from the data, which is also
consistent with the theoretical predictions in quark models Li:1990qu ;
Capstick:1992uc . However, contradictory solutions for the $\gamma n$ coupling
$A^{n}_{1/2}$ of $N(1650)S_{11}$ are obtained by different groups. Our
previous analysis of the $\gamma n\to\eta n$ reaction indicates a positive
helicity coupling $A^{n}_{1/2}\simeq 24\times 10^{-3}$GeV-1/2 for
$N(1650)S_{11}$ Zhong:2011ti , which is supported by the latest analysis of
the same reaction from the BnGa Anisovich:2013jya ; Anisovich:2015tla  and
Kent Shrestha:2012ep  groups. However, in present work by analyzing the recent
the final-state-interaction (FSI) corrected data of the $\gamma n\to\pi^{0}n$
reaction from the A2 Collaboration Dieterle:2014blj , a negative helicity
coupling $A^{n}_{1/2}\simeq-18\times 10^{-3}$GeV-1/2 is obtained, which is
compatible with the values from the PDG14 Agashe:2014kda  and the recent SAID
analysis Workman:2012jf ; Chen:2012yv ; Workman:2011vb . Contradictory results
for the $\gamma n$ coupling $A^{n}_{1/2}$ of $N(1650)S_{11}$ obtained from two
different reactions with the same model indicate that the $N(1650)S_{11}$
state found in the $\gamma n\to\pi^{0}n$ is possibly not the same state found
in the $\gamma n\to\eta n$ if the data are accurate enough. It should be noted
that the FSI is rather rough correction that assumes identical effects on the
proton and the neutron, which certainly does not have to be the case
Dieterle:2014blj . Thus, considering that the data from the A2 Collaboration
might bear large uncertainties in the second resonance region, with a small
positive helicity amplitude, $A^{n}_{1/2}\simeq 20\times 10^{-3}$GeV-1/2 for
$N(1650)S_{11}$, we predict the differential and total cross sections around
the second resonance region ( see Fig. 17). If $N(1650)S_{11}$ has a positive
helicity amplitude, it is found that i) the differential cross section and the
total cross section around the second resonance region should be significantly
larger than the present data; and ii) $N(1650)S_{11}$ has obviously
constructive interference with $N(1535)S_{11}$ and $N(1520)D_{13}$, which can
be tested in future experiments. It was pointed out in Ref. Boika:2014aha
that the positive $A^{n}_{1/2}$ would imply $N(1650)S_{11}$ should have a
large $s\bar{s}$ component in its wave function. To clarify the sign problem
of the $\gamma n$ coupling for $N(1650)S_{11}$, more accurate data are needed.

We find a large helicity amplitude for $\Delta(1620)S_{31}$, which is about a
factor 2 larger than the PDG average value Agashe:2014kda , and $30\%$ larger
than the recent results from the BnGa Anisovich:2009zy ; Anisovich:2013jya
and SAID Workman:2011vb  groups. However, we find that our result is very
close to the theoretical predictions in quark models Li:1990qu ;
Capstick:1992uc .

In our previous work Zhong:2011ti , we gave our estimations of the helicity
amplitudes for $N(1520)D_{13}$ by the analysis of the $\eta$ photoproduction
data. However, the large uncertainties of the branching ratio $b_{\eta N}$
lead to a weak conclusion of these helicity amplitudes. In this work, the
accurate branching ratio $b_{\pi N}$ should let us extract the helicity
amplitudes for $N(1520)D_{13}$ more reliably. It is found that the
$A^{p}_{1/2}$ extracted by us is in good agreement with the results from SAID
group Workman:2011vb  and the PDG average value Agashe:2014kda . However, the
$A^{p}_{3/2}$ extracted in present work are about 30% smaller than the PDG
average value Agashe:2014kda  and the results from other groups. It should be
mentioned that recently the CBELSA/TAPS Collaboration also found a small
helicity amplitude $A^{p}_{3/2}\simeq 118\times 10^{-3}$GeV-1/2 from an
energy-independent multipole analysis based on new polarization data on
photoproduction of neutral pions Hartmann:2014mya . The $\gamma n$ couplings
for the $N(1520)D_{13}$ extracted in this work are compatible with the PDG
values within 30% uncertainties. Our results are slightly smaller than the
results from other partial wave analysis groups.

For $N(1720)P_{13}$, we have noted that the absolute values of the
$A^{p}_{1/2}$ and $A^{p}_{3/2}$ extracted by us are compatible with the
results from the BnGa Anisovich:2009zy ; Anisovich:2013jya  and Kent
Shrestha:2012ep  groups. However, their solutions have opposite signs to our
results. It is interesting to find that our results are consistent with the
quark model predictions by Z. Li and F. Close Li:1990qu , and the partial wave
analysis of the $\gamma n\to\eta n$ reaction from the Giessen group
Shklyar:2012js . Knowledge about the $\gamma n$ couplings, $A^{n}_{1/2}$ and
$A^{n}_{3/2}$, for the $N(1720)P_{13}$ is very poor, and different groups have
given very different predictions. In the SU(6)$\otimes$O(3) symmetry limit, we
predict the $A^{n}_{3/2}$ should be zero, which is compatible with the
analysis of the Kent group Shrestha:2012ep . More studies are needed to
clarify these puzzles about the $N(1720)P_{13}$.

##  IV Summary

In this work, we have studied neutral pion photoproduction on nucleons within
a chiral quark model. We have achieved reasonable descriptions of the data
from the pion production threshold up to the second resonance region.

The roles of the low-lying resonances in the reactions were carefully
analyzed. We found that: (i) $\Delta(1232)P_{33}$, $N(1535)S_{11}$,
$N(1520)D_{13}$, and $N(1720)P_{13}$ play crucial roles in both $\gamma
p\to\pi^{0}p$ and $\gamma n\to\pi^{0}n$ reactions. The $\Delta(1232)P_{33}$
resonance not only plays a dominant role around the first resonance region,
but also contributes up to the third resonance region. Both $N(1535)S_{11}$
and $N(1520)D_{13}$ paly crucial roles around the second resonance region. The
second bump structure around $E_{\gamma}=700$ MeV in the cross section
receives approximately equal contributions from these two resonances.
$N(1720)P_{13}$ might play a crucial role in the third resonance region. It
might be responsible for the third bump structure in cross section, which
should be further investigated due to our relatively poor descriptions of the
polarization observables in this energy region. (ii) Furthermore, obvious
evidence of $N(1650)S_{11}$ and $\Delta(1620)S_{31}$ is also found in the
reactions. They notably affect the cross sections and the polarization
observables from the second resonance region to the third resonance region.
(iii) The $u$\- and $t$-channel backgrounds play a crucial role in the
reaction as well. The $u$ channel has strong interferences with the
resonances, such as $\Delta(1232)P_{33}$, $N(1535)S_{11}$ and $N(1520)D_{13}$.
By including the $t$-channel vector-meson exchange contribution, the
descriptions of the data in the energy region $E_{\gamma}=600\sim 900$ MeV are
improved notably. (iv) No obvious evidence of the other resonances, e.g.,
$N(1700)D_{13}$, $N(1675)D_{15}$, $\Delta(1700)D_{33}$ and $N(1680)F_{15}$,
was found in the reactions.

Furthermore, the helicity couplings for the main resonances,
$\Delta(1232)P_{33}$, $N(1535)S_{11}$, $N(1520)D_{13}$, $N(1720)P_{13}$,
$N(1650)S_{11}$ and $\Delta(1620)S_{31}$, were extracted from the reactions.
We found that: (i) Our extracted helicity amplitudes of $\Delta(1232)P_{33}$
and $N(1535)S_{11}$ are in good agreement with the PDG average values and the
results of other groups. (ii) The $\gamma p$ coupling for $N(1650)S_{11}$
extracted by us is in good agreement with the results from the SAID
Workman:2011vb ; Workman:2012jf ; Chen:2012yv  and BnGa Anisovich:2009zy ;
Anisovich:2013jya . However, properties of the $\gamma n$ coupling for
$N(1650)S_{11}$ are still controversial. Our analysis of the recent data of
the $\gamma n\to\pi^{0}n$ reaction indicates a small negative $\gamma n$
coupling for $N(1650)S_{11}$. Its sign is opposite to that of other analyses
of the $\gamma n\to\eta n$ data Zhong:2011ti ; Anisovich:2013jya ;
Anisovich:2015tla . (iii) We obtain a large helicity coupling for
$\Delta(1620)S_{31}$, but it is very close to the recent analysis from the
BnGa group Anisovich:2009zy ; Anisovich:2013jya . (iv) We give smaller
helicity couplings for $N(1520)D_{13}$, which are compatible with the PDG
values at the 30% level. (v) The helicity couplings $A^{p}_{1/2}$ and
$A^{p}_{3/2}$ for $N(1720)P_{13}$ extracted by us are consistent with the
quark model predictions by Li and Close Li:1990qu ; Capstick:1992uc  and the
analysis of the Giessen group Shklyar:2012js . We find a small positive
helicity coupling $A^{n}_{1/2}$ for $N(1720)P_{13}$, and the $A^{n}_{3/2}$
should be zero in the SU(6)$\otimes$O(3) symmetry limit.

Finally, it should be pointed out that (i) the width of $N(1720)P_{13}$
extracted by us is notably narrower than the estimated values from the PDG,
however, our result is in good agreement with those extracted from the
$\pi^{-}p\to K^{0}\Lambda$ reaction by Saxon et al. Saxon:1979xu . To confirm
the properties of $N(1720)P_{13}$, a study of the $\pi^{-}p\to K^{0}\Lambda$
reaction is needed. (ii) Furthermore, a more realistic correction of the FSI
for neutral pion photoproduction on quasi-free neutrons hopefully will be
obtained in future. Then the sign problem of the $\gamma n$ coupling
$A^{n}_{1/2}$ of $N(1650)S_{11}$ could be clarified in the $\gamma
n\rightarrow\pi^{0}n$ reaction, which seems to be crucial to uncover the
puzzle of the narrow structure around $W=1.68$ GeV observed in the excitation
function of $\eta$ production off quasi-free neutrons. If the $\gamma n$
coupling $A^{n}_{1/2}$ of the $N(1650)S_{11}$ is negative, then the narrow
structure in the $\gamma n\to\eta n$ reaction would no longer be explained by
the interference effects between $N(1535)S_{11}$ and $N(1650)S_{11}$.

##  Acknowledgements

The authors would like to thank Qiang Zhao for useful discussions. We also
thank B. Krusche and M. Dieterle for providing us the data of neutral pion
photoproduction on the nucleons, Jan Hartmann for providing us the new data of
polarization observables $T$, $P$ and $H$ for the $\gamma
p\rightarrow\pi^{0}p$ reaction, Paolo Levi Sandri for providing us the data on
beam asymmetry for the $\gamma n\rightarrow\pi^{0}n$ reaction. This work is
supported, in part, by the National Natural Science Foundation of China
(Grants No. 11075051, No. 11375061, and No. 11405222), and the Hunan
Provincial Natural Science Foundation (Grant No. 13JJ1018).

## References

  * (1) E. Klempt and J. M. Richard, Rev. Mod. Phys. 82, 1095 (2010).
  * (2) B. Krusche and S. Schadmand, Prog. Part. Nucl. Phys. 51, 399 (2003).
  * (3) M. Dieterle et al. [A2 Collaboration], Phys. Rev. Lett. 112, 142001 (2014).
  * (4) M. Fuchs et al., Phys. Lett. B 368, 20 (1996).
  * (5) R. Beck, Eur. Phys. J. A 28S1, 173 (2006).
  * (6) N. Sparks et al. [CBELSA/TAPS Collaboration], Phys. Rev. C 81, 065210 (2010).
  * (7) H. van Pee et al. [CB-ELSA Collaboration], Eur. Phys. J. A 31, 61 (2007).
  * (8) A. V. Anisovich, A. Sarantsev, O. Bartholomy, E. Klempt, V. A. Nikonov and U. Thoma, Eur. Phys. J. A 25, 427 (2005).
  * (9) O. Bartholomy et al. [CB-ELSA Collaboration], Phys. Rev. Lett. 94, 012003 (2005).
  * (10) J. Hartmann, H. Dutz, A. V. Anisovich, D. Bayadilov, R. Beck, M. Becker, Y. Beloglazov and A. Berlin et al., Phys. Rev. Lett. 113, 062001 (2014).
  * (11) O. Bartalini et al. [GRAAL Collaboration], Eur. Phys. J. A 26, 399 (2005).
  * (12) M. Dugger, B. G. Ritchie, J. P. Ball, P. Collins, E. Pasyuk, R. A. Arndt, W. J. Briscoe and I. I. Strakovsky et al., Phys. Rev. C 76, 025211 (2007).
  * (13) F. V. Adamian, A. Y. Bunyatyan, G. S. Frangulian, P. I. Galumian, V. H. Grabsky, A. V. Airapetian, H. H. Hakopian and V. K. Hoktanian et al., Phys. Rev. C 63, 054606 (2001).
  * (14) D. Elsner et al. [CBELSA and TAPS Collaborations], Eur. Phys. J. A 39, 373 (2009).
  * (15) V. Crede et al. [CBELSA/TAPS Collaboration], Phys. Rev. C 84, 055203 (2011).
  * (16) A. Thiel, A. V. Anisovich, D. Bayadilov, B. Bantes, R. Beck, Y. Beloglazov, M. Bichow and S. Bose et al., Phys. Rev. Lett. 109, 102001 (2012).
  * (17) M. H. Sikora, D. P. Watts, D. I. Glazier, P. Aguar-Bartolome, L. K. Akasoy, J. R. M. Annand, H. J. Arends and K. Bantawa et al., Phys. Rev. Lett. 112, 022501 (2014).
  * (18) M. Gottschall et al. [CBELSA/TAPS Collaboration], Phys. Rev. Lett. 112, 012003 (2014).
  * (19) R. Di Salvo, A. Fantini, G. Mandaglio, F. Mammoliti, O. Bartalini, V. Bellini, J. P. Bocquet and L. Casano et al., Eur. Phys. J. A 42, 151 (2009).
  * (20) A. V. Anisovich, E. Klempt, V. A. Nikonov, M. A. Matveev, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A 44, 203 (2010).
  * (21) A. V. Anisovich, R. Beck, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A 48, 15 (2012) [arXiv:1112.4937 [hep-ph]].
  * (22) A. V. Anisovich, V. Burkert, E. Klempt, V. A. Nikonov, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A 49, 67 (2013).
  * (23) W. Chen, H. Gao, W. J. Briscoe, D. Dutta, A. E. Kudryavtsev, M. Mirazita, M. W. Paris and P. Rossi et al., Phys. Rev. C 86, 015206 (2012).
  * (24) R. L. Workman, M. W. Paris, W. J. Briscoe and I. I. Strakovsky, Phys. Rev. C 86, 015202 (2012).
  * (25) R. L. Workman, W. J. Briscoe, M. W. Paris and I. I. Strakovsky, Phys. Rev. C 85, 025201 (2012).
  * (26) D. Drechsel, S. S. Kamalov and L. Tiator, Eur. Phys. J. A 34, 69 (2007).
  * (27) M. Shrestha and D. M. Manley, Phys. Rev. C 86, 055203 (2012).
  * (28) F. Huang, M. Döring, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Krewald, U. G. Meißner and K. Nakayama, Phys. Rev. C 85, 054003 (2012).
  * (29) D. Rönchen, M. Döring, F. Huang, H. Haberzettl, J. Haidenbauer, C. Hanhart, S. Krewald and U.-G. Meißner et al., Eur. Phys. J. A 50, 101 (2014).
  * (30) H. Kamano, S. X. Nakamura, T.-S. H. Lee and T. Sato, Phys. Rev. C 88, 035209 (2013).
  * (31) V. Kuznetsov et al. [GRAAL Collaboration], Phys. Lett. B 647, 23 (2007).
  * (32) I. Jaegle et al. [CBELSA Collaboration and TAPS Collaboration], Phys. Rev. Lett. 100, 252002 (2008).
  * (33) I. Jaegle, B. Krusche, A. V. Anisovich, J. C. S. Bacelar, B. Bantes, O. Bartholomy, D. E. Bayadilov and R. Beck et al., Eur. Phys. J. A 47, 89 (2011).
  * (34) K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C 38, 090001 (2014).
  * (35) R. Shyam and O. Scholten, Phys. Rev. C 78, 065201 (2008).
  * (36) V. Shklyar, H. Lenske and U. Mosel, Phys. Lett. B 650, 172 (2007).
  * (37) M. Döring and K. Nakayama, Phys. Lett. B 683, 145 (2010).
  * (38) X. H. Zhong and Q. Zhao, Phys. Rev. C 84, 045207 (2011).
  * (39) A. V. Anisovich, E. Klempt, B. Krusche, V. A. Nikonov, A. V. Sarantsev, U. Thoma and D. Werthm ller, Eur. Phys. J. A 51, 72 (2015). Anisovich:2008wd
  * (40) A. V. Anisovich, I. Jaegle, E. Klempt, B. Krusche, V. A. Nikonov, A. V. Sarantsev and U. Thoma, Eur. Phys. J. A 41, 13 (2009).
  * (41) R. G. Moorhouse, Phys. Rev. Lett. 16, 772 (1966).
  * (42) Q. Zhao and F. E. Close, Phys. Rev. D 74, 094014 (2006).
  * (43) Z. P. Li, Phys. Rev. D 50, 5639 (1994).
  * (44) Z. P. Li, Phys. Rev. C 52, 1648 (1995).
  * (45) Z. P. Li, H. X. Ye and M. H. Lu, Phys. Rev. C 56, 1099 (1997).
  * (46) Q. Zhao, J. S. Al-Khalili, Z. P. Li and R. L. Workman, Phys. Rev. C 65, 065204 (2002).
  * (47) Q. Zhao, Z. P. Li and C. Bennhold, Phys. Rev. C 58, 2393 (1998).
  * (48) Q. Zhao, J. S. Al-Khalili and C. Bennhold, Phys. Rev. C 64, 052201 (2001).
  * (49) Q. Zhao, Z. P. Li and C. Bennhold, Phys. Lett. B 436, 42 (1998).
  * (50) Z. P. Li and B. Saghai, Nucl. Phys. A 644, 345 (1998).
  * (51) B. Saghai and Z. P. Li, Eur. Phys. J. A 11, 217 (2001).
  * (52) J. He, B. Saghai and Z. P. Li, Phys. Rev. C 78, 035204 (2008).
  * (53) J. He and B. Saghai, Phys. Rev. C 80, 015207 (2009).
  * (54) X. H. Zhong and Q. Zhao, Phys. Rev. C 84, 065204 (2011).
  * (55) X. H. Zhong, Q. Zhao, J. He and B. Saghai, Phys. Rev. C 76, 065205 (2007).
  * (56) X. H. Zhong and Q. Zhao, Phys. Rev. C 79, 045202 (2009).
  * (57) X. H. Zhong and Q. Zhao, Phys. Rev. C 88, 015208 (2013).
  * (58) L. Y. Xiao and X. H. Zhong, Phys. Rev. C 88, 065201 (2013).
  * (59) N. Isgur and G. Karl, Phys. Rev. D 18, 4187 (1978).
  * (60) N. Isgur and G. Karl, Phys. Lett. B 72, 109 (1977).
  * (61) S. Capstick and N. Isgur, Phys. Rev. D 34, 2809 (1986).
  * (62) G. F. Chew, M. L. Goldberger, F. E. Low and Y. Nambu, Phys. Rev. 106, 1345 (1957).
  * (63) R. L. Walker, Phys. Rev. 182, 1729 (1969).
  * (64) C. G. Fasano, F. Tabakin and B. Saghai, Phys. Rev. C 46, 2430 (1992).
  * (65) R. Koniuk and N. Isgur, Phys. Rev. D 21, 1868 (1980) [Phys. Rev. D 23, 818 (1981)].
  * (66) J. J. de Swart, Rev. Mod. Phys. 35, 916 (1963) [Rev. Mod. Phys. 37, 326 (1965)].
  * (67) C. An and B. Saghai, Phys. Rev. C 84, 045204 (2011) [arXiv:1108.3282 [nucl-th]].
  * (68) A. J. G. Hey, P. J. Litchfield and R. J. Cashmore, Nucl. Phys. B 95, 516 (1975).
  * (69) S. Capstick and W. Roberts, Fizika B 13, 271 (2004).
  * (70) T. Sato and T. S. H. Lee, Phys. Rev. C 54, 2660 (1996) [nucl-th/9606009].
  * (71) T. Sato and T. S. H. Lee, Phys. Rev. C 63, 055201 (2001) [nucl-th/0010025].
  * (72) S. Schumann, B. Boillat, E. J. Downie, P. Aguar-Bartolome, J. Ahrens, J. R. M. Annand, H. J. Arends and R. Beck et al., Eur. Phys. J. A 43, 269 (2010).
  * (73) D. O. Riska and G. E. Brown, Nucl. Phys. A 679, 577 (2001).
  * (74) D. H. Saxon, R. D. Baker, K. W. Bell, J. A. Blissett, I. J. Bloodworth, T. A. Broome, J. C. Hart and A. L. Lintern et al., Nucl. Phys. B 162, 522 (1980).
  * (75) R. Beck et al., Phys. Rev. Lett. 78, 606 (1997).
  * (76) E. Mazzucato, P. Argan, G. Audit, A. Bloch, N. de Botton, N. d’Hose, J. L. Faure and M. L. Ghedira et al., Phys. Rev. Lett. 57, 3144 (1986).
  * (77) B. B. Govorkov et al., Yad. Fiz. 4, 507 (1966).
  * (78) G. Fischer, H. Fischer, G. Von Holtey, H. Kaempgen, G. Knop, P. Schulz, H. Wessels and W. Braunschweig et al., Nucl. Phys. B 16, 93 (1970).
  * (79) H. C. DeStaebler, E. F. Erickson, A. C. Hearn and C. Schaerf, Phys. Rev. 140, B336 (1965).
  * (80) Y. Hemmi, Y. Inagaki, T. Inagaki, A. Maki, K. Miyake, T. Nakamura, Tamura and J. Tsukamoto et al., Phys. Lett. B 43, 79 (1973).
  * (81) M. Yoshioka, A. Noda, M. Daigo, Y. Hemmi, R. Kikuchi, M. Minowa, K. Miyake and T. Nakamura et al., Nucl. Phys. B 168, 222 (1980).
  * (82) C. Bacci, G. Penso, G. Salvini, C. Mencuccini, A. Reale, V. Silvestrini, M. Spinetti and B. Stella, Phys. Rev. 159, 1124 (1967).
  * (83) G. Blanpied, M. Blecher, A. Caracappa, R. Deininger, C. Djalali, G. Giordano, K. Hicks and S. Hoblit et al., Phys. Rev. C 64, 025203 (2001).
  * (84) V. B. Ganenko, V. G. Gorbenko, Y. V. Zhebrovskii, L. Y. Kolesnikov, I. I. Miroshnichenko, A. L. Rubashkin, V. M. Sanin and P. V. Sorokin et al., Yad. Fiz. 20, 356 (1974).
  * (85) A. a. Belyaev, V. a. Getman, V. g. Gorbenko, V. a. Gushchin, A. y. Derkach, Y. v. Zhebrovsky, I. m. Karnaukhov and L. y. Kolesnikov et al., Nucl. Phys. B 213, 201 (1983).
  * (86) G. Knies, H. Oberlack, A. Rittenberg, A. H. Rosenfeld, M. Bogdanski and G. Smadja, Phys. Rev. D 10, 2778 (1974).
  * (87) G. Barbiellini, G. Bologna, G. Capon, G. De Zorzi, F. L. Fabbri, G. P. Murtas, G. Diambrini and G. Sette et al., Phys. Rev. 184, 1402 (1969).
  * (88) R. W. Zdarko and E. B. Dally, Nuovo Cim. A 10, 10 (1972).
  * (89) A. S. Bratashevsky, A. I. Derebchinsky, A. A. Zybalov, O. G. Konovalov, A. S. Omelaenko and A. E. Tenishev, Yad. Fiz. 32, 667 (1980).
  * (90) D. E. Lundquist, R. L. Anderson, J. V. Allaby and D. M. Ritson, Phys. Rev. 168, 1527 (1968).
  * (91) E. D. Bloom, C. A. Heusch, C. Y. Prescott and L. S. Rochester, Phys. Rev. Lett. 19, 671 (1967).
  * (92) A. A. Belyaev, V. A. Getman, V. G. Gorbenko, V. A. Gushchin, A. Y. Derkach, Y. V. Zhebrovsky, I. M. Karnaukhov and L. Y. Kolesnikov et al., Yad. Fiz. 35, 693 (1982).
  * (93) R. Querzoli, G. Salvini and A. Silverman, Nuovo Cim. 19, 53 (1961).
  * (94) K. H. Althoff, D. Finken, N. Minatti, H. Piel, D. Trines, M. Unger, Phys. Lett. B 26, 677 (1968).
  * (95) N. V. Goncharov, A. I. Derebchinskii, A. A. Zybalov, O. G. Konovalov, A. E. Tenishev, S. G. Tonapetyan and V. M. Khvorostyan, Zh. Eksp. Teor. Fiz. 64, 401 (1973).
  * (96) A. S. Bratashevsky, A. A. Zybalov, S. P. Karasev, O. G. Konovalov, E. E. Korobova, A. S. Omelaenko, P. V. Sorokin and Y. O. Storozhenko et al., Yad. Fiz. 42, 658 (1985).
  * (97) S. Kato, T. Miyachi, K. Sugano, K. Toshioka, K. Ukai, M. Chiba, K. Egawa and T. Ishii et al., Nucl. Phys. B 168, 1 (1980).
  * (98) P. Blum, P. Brinckmann, R. Brockmann, P. Lutter, W. Mohr and R. Sauerwein, Z. Phys. A 277, 311 (1976).
  * (99) P. Blum, R. Brockmann and W. Mohr, Z. Phys. A 278, 275 (1976).
  * (100) M. Fukushima, N. Horikawa, R. Kajikawa, H. Kobayakawa, K. Mori, T. Nakanishi, C. O. Pak and S. Suzuki et al., Nucl. Phys. B 136, 189 (1978).
  * (101) P. S. L. Booth, L. J. Carroll, G. R. Court, P. R. Daniel, R. Gamet, C. J. Hardwick, P. J. Hayman and J. R. Holt et al., Nucl. Phys. B 121, 45 (1977).
  * (102) P. Feller, M. Fukushima, N. Horikawa, R. Kajikawa, K. Mori, T. Nakanishi, T. Ohshima and C. O. Pak et al., Nucl. Phys. B 110, 397 (1976).
  * (103) A. Ando et al., Physik Daten, 1977 (unpublished).
  * (104) Y. Hemmi, T. Inagaki, R. Kikuchi, A. Maki, K. Miyake, T. Nakamura, A. Sasaki and N. Tamura et al., Nucl. Phys. B 55, 333 (1973).
  * (105) C. Bacci, R. Baldini-Celio, B. Esposito, C. Mencuccini, A. Reale, G. Sciacca, M. Spinetti and A. Zallo, Phys. Lett. B 39, 559 (1972).
  * (106) K. Kossert, M. Camen, F. Wissmann, J. Ahrens, J. R. M. Annand, H. J. Arends, R. Beck and G. Caselotti et al., Eur. Phys. J. A 19, 391 (2004).
  * (107) Z. P. Li and F. E. Close, Phys. Rev. D 42, 2207 (1990).
  * (108) S. Capstick, Phys. Rev. D 46, 2864 (1992).
  * (109) X. Cao, V. Shklyar and H. Lenske, Phys. Rev. C 88, 055204 (2013).
  * (110) V. Shklyar, H. Lenske and U. Mosel, Phys. Rev. C 87, 015201 (2013).
  * (111) T. Boika, V. Kuznetsov and M. V. Polyakov, arXiv:1411.4375 [nucl-th].