11institutetext:  The University of Texas at Austin, Austin TX 78705, USA
11email: {blevac, sidharth.kumar, sofia.kardonik<EMAIL_ADDRESS>
# FSE Compensated Motion Correction for MRI Using Data Driven Methods

Brett Levac Equal contribution Sidharth Kumar⋆ Sofia Kardonik  Jonathan I.
Tamir

###### Abstract

Magnetic Resonance Imaging (MRI) is a widely used medical imaging modality
boasting great soft tissue contrast without ionizing radiation, but
unfortunately suffers from long acquisition times. Long scan times can lead to
motion artifacts, for example due to bulk patient motion such as head movement
and periodic motion produced by the heart or lungs. Motion artifacts can
degrade image quality and in some cases render the scans nondiagnostic. To
combat this problem, prospective and retrospective motion correction
techniques have been introduced. More recently, data driven methods using deep
neural networks have been proposed. As a large number of publicly available
MRI datasets are based on Fast Spin Echo (FSE) sequences, methods that use
them for training should incorporate the correct FSE acquisition dynamics.
Unfortunately, when simulating training data, many approaches fail to generate
accurate motion-corrupt images by neglecting the effects of the temporal
ordering of the k-space lines as well as neglecting the signal decay
throughout the FSE echo train. In this work, we highlight this consequence and
demonstrate a training method which correctly simulates the data acquisition
process of FSE sequences with higher fidelity by including sample ordering and
signal decay dynamics. Through numerical experiments, we show that accounting
for the FSE acquisition leads to better motion correction performance during
inference.

###### Keywords:

Motion Correction Deep Learning Magnetic Resonance Imaging Fast Spin Echo.

##  1 Introduction

MRI is a powerful medical imaging modality due to its superb soft tissue
contrast which can help in diagnosing various types of pathology. However,
unlike imaging methods that use ionizing radiation such as CT, MRI is
predominantly slow, and each MRI scan can take several minutes to acquire.
This is because the MRI acquisition involves repeated application of signal
excitation through radio-frequency (RF) pulses and signal reception using
spatially varying magnetic field gradients. Due to physical limitations
including signal decay, only a small number of measurements in k-space can be
acquired in a single excitation, and a duration (called the repetition time,
TR) must pass before the MRI signal resets to equilibrium or steady state. As
a result, multiple excitations are needed to fully sample k-space. Patient
motion during the scan therefore commonly occurs, and leads to degraded image
quality [22]. If the degradation is not too severe the image might still be
useful for clinical diagnosis. However, oftentimes, the motion is so severe
that the image is nondiagnostic, and the scan needs to be repeated. It has
been reported that motion-corrupted scans lead hospitals to incur nearly $365K
in additional costs each year [22].

Due to the high associated cost, many methods have been proposed to mitigate
and correct for motion [8]. A common approach to reduce motion is to modify
the acquisition and change the sampling pattern and ordering so that the
motion artifacts are incoherent, as is the case for non-Cartesian radial
sampling [26]. Prospective motion correction during the scan is also possible
using external sources such as respiratory bellows, nuclear magnetic resonance
probes [7], optical and electromagnetic tracking [3], and wireless RF tones
[16], or through self-navigators [4]. However, prospective motion correction
in some cases reduces scan efficiency, as motion-corrupt measurements must be
re-acquired with additional RF excitations. Retrospective motion correction is
also popular [13, 5], as it relies on computational processing during
reconstruction in lieu of reducing scan efficiency.

Recently, data-driven approaches to retrospective motion correction have
increased in popularity due to the rising success of deep learning and the
availability of public MRI datasets with raw k-space [27]. These approaches
are attractive as they do not require sequence modification, external
hardware, or explicit knowledge of the motion information, and instead use a
large corpus of training data to learn motion artifact removal. The motion
removal can be posed as either supervised or self-supervised learning, in
which a motion-corrupt image is “denoised” based on a motion-free reference.
Many works that follow this paradigm simulate motion effects by corrupting
k-space lines through linear phase augmentation and rotation [12].

When simulating motion artifacts for data driven motion correction, the ground
truth k-space is divided into non-overlapping regions which are taken from
different motion states of the underlying image. This structure of motion-
corruption reflects the fact that in FSE scans the time between phase encode
lines in different repetition times (TR) (inter-TR time) is much longer than
the time between echoes in a single TR (intra-TR time) and thus motion is
roughly simulated as being inter-TR and not occurring between lines in the
same echo train. Many of the publicly available datasets are acquired using
fast spin-echo (FSE) imaging [27, 17, 1, 2], in which multiple k-space lines
are rapidly acquired per RF excitation through the use of RF refocusing
pulses. Therefore, the specific phase encode lines, their ordering, and signal
decay modulation are all pre-determined by the acquisition parameters and the
tissue relaxation parameters. In this setting it is often the case that when
retrospectively applying motion to different lines of k-space the phase encode
lines are grouped into non-overlapping subsets which are agnostic to the
sampling pattern that was used to collect the data. This approach is
fundamentally incorrect as it neglects the fact that signal decay is occurring
within each echo train of the ground truth FSE image and thus training on such
data will lead to inconsistent results when applying data driven methods to
actual FSE data.

In this work, we propose a method for simulating synthetic motion artifacts
that accounts for the echo ordering and signal decay present in FSE
acquisitions, and we highlight the disparity in performance that occurs when
training on simulated data that is agnostic to the acquisition parameters.

###  1.1 Related Works

In the case of retrospective motion correction with unknown motion, both
physics and learning based methods have been proposed. Authors in [6] showed
that by modeling the forward model of in-plane motion, an alternating
minimization approach can be used to solve for both the parameterized motion
model and the motion-free image. Similarly, authors in [10] proposed an
algorithm to estimate the rotation and translation operator and used the
estimated operator to remove the motion artifacts. Improving on that, the same
authors used a convolutional network to first estimate the motion operator and
then reconstruct the image with information and data consistency [9]. The work
in [15] used image alignment to remove the motion artifacts by inputting
multiple images to the network among which one has motion artifacts.

Authors in [19] used an Inception-ResNet based encoder-decoder network to
correct motion artifacts, the motion generation method only uses two randomly
generated motion states. Whereas [21] proposes a joint image and frequency
space inter connected network to remove motion artifacts, the motion artifacts
are simulated by drawing translation and rotation parameters from a uniform
distribution. Authors in [12] used a Conditional Generative Adversarial
Network (GAN) to correct motion-corrupt images which were generated using
simulating translation and rotational motions. The considered motion model
only used $2$ or $3$ motion events and the k-space data was acquired without
considering the underlying scanning protocol.

##  2 Methods

####  2.0.1 MR Forward Model with Rigid Body Motion:

Neglecting multiple coils and assuming Nyquist sampling, the forward model is
as follows:

$y=Fx+w,\quad w\sim\mathcal{N}(0,\sigma^{2}),$ (1)

where $x$ is the image to be reconstructed, $F$ is the Fourier transform
operator, $y$ represents the k-space measurements for a particular scan, and
$w$ is additive white Gaussian noise. This model assumes that there is no
motion during the scan. With the inclusion of motion, the forward model is as
follows:

$y=PFM(x)+w,\quad w\sim\mathcal{N}(0,\sigma^{2}).$ (2)

Here, $P$ is a sampling operator which takes non-overlapping k-space samples
from each TR, and $M(\cdot)$ contains the linear motion operation for each TR.
If the inverse problem is solved assuming no motion ($M=I$), then the solution
will be a motion-corrupt.

####  2.0.2 MR Forward Model with Rigid Body Motion and Signal Decay:

The forward model in (2) assumes that k-space is acquired at the same echo
time which is only true for steady-state and spin-echo sequences where a
single k-space line is collected in each excitation. However, nearly all
clinical MRI exams employ FSE sequences in which multiple echoes are acquired
per excitation, and each echo has a decay factor depending on T2 relaxation
and the time when the echo is acquired. This gives rise to the well-known
T2-blurring seen in FSE imaging, and can be interpreted as k-space filtering
leading to spatially varying convolutions in image-space [23]. To account for
this decay, the forward model is modified as follows:

$y=PFM(h(TE,T2,PD))+w,\quad w\sim\mathcal{N}(0,\sigma^{2}),$ (3)

where

$h(TE,T2,PD)=PDe^{\frac{-TE}{T2}}.$ (4)

Here $TE$ contains all the echo times at which the signal is being acquired,
T2 is transverse relaxation time, and PD is the proton density. $h(\cdot)$
represents the exponential decay in the transverse magnetization as TE
increases. For simplicity we neglect the effects of imperfect RF refocusing
pulses and we assume complete T1 recovery, i.e. long TR, though these effects
are easy to incorporate into the signal model [23] due to the
differentiability of the extended phase graph algorithm [25].

###  2.1 Image Simulations

####  2.1.1 Ground Truth FSE Image:

To correctly simulate the FSE sequence acquisition we first estimated T2 and
Proton Density (PD) maps from multi-contrast brain MRI scanned at our
institution with institutional board approval and informed consent (details
about the protocol can be found in the supplementary materials section).
Following the FSE example in [14], we simulated echo train lengths (ETL) of 16
and 8, and echo spacing of $12$ms, resulting in $16$ (resp. 8) different echo
images (Fig. 1) which depict how the signal changes during an FSE acquisition.
To form our ground truth FSE image we used a center-out k-space acquisition
ordering along the echo train axis, corresponding to PD contrast (Fig. 1). The
non-overlapping k-space from each echo was combined into one k-space which was
inverse Fourier transformed to make the ground truth image. A total of 18 TRs
were acquired, corresponding to Nyquist rate sampling.

Figure 1: Example of how ground truth FSE images were simulated using PD and
T2 maps. Top row contains images generated using Eq. 3 at 16 different echo
times with an echo spacing of 12ms. The bottom row shows the respective
k-space measurements collected and the sampling pattern used for each echo
image to create the FSE image.

####  2.1.2 FSE Compensated Motion Simulation:

To demonstrate the importance of accounting for the FSE effects, as proof-of-
principle we simulated nine motion events evenly spaced every 2TR seconds.
Each rotation angle was randomly chosen from a Normal distribution
$\mathcal{N}\left(0,2\right)$ based on [12]. We then sampled k-space according
to the particular motion state and echo time across the 18 TRs. The IFFT of
this k-space resulted in a motion-corrupt image (See Supplementary Material
for Simulation Outline). We emphasize that because this motion model used the
same phase encode lines for each TR as was used to generate the ground truth
image, the simulation correctly takes into account the relationship between
the signal decay during the echo trains and the motion state for each TR.

####  2.1.3 FSE Agnostic Motion Simulation

To generate the FSE agnostic motion-corrupted images we simply took the FSE
image (rightmost in Fig. 1), linearly segmented it into 9 disjoint regions of
k-space (corresponding to two TRs each), and applied the same rotations we
used previously (shown in Fig. 2). Although the angles of rotation remain the
same in this method there is no acknowledgment of the temporal relationship
found between lines captured in the same echo train for the true FSE
acquisition and this is what ultimately causes this method of simulation to
diverge from actual FSE scan data.

Figure 2: Example naive motion-corrupted FSE images. The top row contains
images which are rotated version of the GT FSE image. The bottom row depicts
the sampling pattern in k-space for each TR. Note that there is no notion of
signal decay or echo train ordering.

####  2.1.4 Motion Correction Network Training

To test the effect of the different motion models we trained two conditional
GANs: the first model trained with our proposed FSE motion simulation, and the
second trained using the FSE agnostic simulation. The input to the conditional
GAN was the motion-corrupted image and the output was the motion-free FSE
image. Additional information about the training can be found in the
supporting information. Our conditional GAN approach was chosen as it is
commonly used as the backbone for deep learning based motion correction
methods [12]. We anticipate that more sophisticated methods would lead to
similar overall conclusions.

We implemented image reconstruction using the BART toolbox [24] and motion
correction networks using PyTorch; our code is publicly
available111https://github.com/utcsilab/GAN_motion_correction. Both models
were trained using 819 paired images from the respective motion simulation
method. We tested both models on data generated by properly simulating an FSE
acquisition with motion and tracked both structural similarity index (SSIM)
and normalized root mean square error (NRMSE) values to evaluate image quality
and performance. The results for two different ETL values are summarized in
Table 1.

##  3 Results and Discussion

####  3.0.1 Results

Table 1 shows the performance of both models when tested on simulated motion-
corrupt FSE acquisitions. By training on simulated data which is agnostic to
the actual acquisition parameters of an FSE sequence there is a significant
drop in quantitative performance when compared to models trained and tested on
FSE-aware models for motion-corruption and this was observed for different ETL
values. Fig. 3 shows multiple motion correction examples chosen from the test
set. All cases show a significant improvement in both quantitative and
qualitative metrics when accounting for the FSE acquisition. The traditional
simulation method does not properly account for the correct sampling of
k-space, and therefore at test time it is not able to correct the motion
artifacts.

Table 1: Performance comparison of FSE agnostic vs. FSE aware motion correction on the test set for 2 different ETL values. For the ETL = $16$ case, there are 18 repetition times and time between each echo was $12$ms. Whereas for the ETL = $8$ case, there are $36$ repetition times and time between each echo was $20$ms. For consistency, we simulated $9$ motion states for both the cases.  ETL | 16 | 16 | 8 | 8
---|---|---|---|---
Error Metric | SSIM | NRMSE | SSIM | NRMSE
Input | 0.554 $\pm 0.071$ | 0.268 $\pm 0.058$ | 0.659 $\pm 0.053$ | 0.185 $\pm 0.047$
FSE Agnostic | 0.730 $\pm 0.060$ | 0.287 $\pm 0.040$ | 0.885 $\pm 0.058$ | 0.120 $\pm 0.025$
FSE Aware | 0.851 $\pm 0.043$ | 0.215 $\pm 0.039$ | 0.939 $\pm 0.022$ | 0.119 $\pm 0.016$
Figure 3: Example motion artifact correction for 3 different slices. The first
column is the ground truth, second column is for the motion-corrupt images,
third column is for motion correction network trained on traditional motion
simulation methods and the fourth column shows result for correct physics
guided motion simulation methods.

####  3.0.2 Discussion

Many recent works that use deep learning for retrospective motion correction
are trained with simulated motion-corruption. While spin-echo and steady-state
sequences are only weakly dependent on the acquisition ordering, FSE
acquisitions strongly depend on the data ordering. When this ordering is not
correctly accounted for in simulation, there is an expected appreciable drop
in performance at test time. This is important to consider as FSE is the
backbone of clinical MRI and most publicly available raw MRI datasets are all
FSE based. This bias is strongly related to other implicit data crimes arising
from the misuse of public data for MRI reconstruction [20]. Fortunately, this
problem is easily mitigated by accounting for the data ordering [15]. Another
approach to circumventing the issue include unpaired training with
experimentally acquired motion-corrupt data [11, 18].

Further work is required to investigate the disparity in network performances
on prospective motion corrupt data. While our model does not account for
through-plane motion in 2D imaging, the model could be extended to 3D FSE
acquisitions which have significantly longer ETLs and no through-plane motion.

##  4 Conclusion

When applying data driven methods to retrospective motion correction,
simulation parameters are crucial. In particular, when training motion
correction networks for FSE sequences the choice of sampling masks across echo
trains and repetition times is key to ensuring that the correct artifact
distribution is learned by the network to ensure best results at inference on
experimentally acquired motion-corrupt data.

####  4.0.1 Acknowledgements

This work was supported by NIH Grant U24EB029240 and NSF IFML 2019844 Award.

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## Supplementary Material

### Data Acquisition Protocol

#### Base Multicontrast Sequence:

The $T_{2}$ and PD maps used to simulate ground truth images for motion were
estimated using a multi delay multi echo (MDME) sequence protocol at our
institution with proper IRB approval and informed consent from all volunteers.
The protocol consisted of four different inversion times and two different
echo times to encode the $T_{1}$ and $T_{2}$ information respectively
resulting in eight images. For parameter mapping, we simulated a dictionary of
signal evolutions matching the sequence parameters at multiple $T_{1}$ and
$T_{2}$ values and employed dictionary matched filtering by selecting the
entry with the highest inner product with the experimentally observed signal.
The dictionary was generated by changing $T_{1}$ values from $100$ to $6000$ms
and $T_{2}$ values from $10$ to $1000$ms with step sizes of $20$ and $2$ms
respectively. The MRI scans were conducted on a Siemens 3T Vida scanner
(Siemens Healthineers, Erlangen, Germany) and 53 slices were acquired per
subject. The $20$ central slices per subject were used for training of the
neural network and the remaining slices were discarded. The total duration of
the MDME sequence was $8$ minutes with a repetition time (TR) of $7680$ms. The
employed echo time values were $27$, $90$ms while the delay times of $7562$,
$3504$, $1041$ and $171$ms were used. The acquired frequency matrix was of
size $320\times 288$ with a field of view of $22.8$cm. The undersampled
k-space data was acquired with an acceleration factor of $3.2$ and
reconstructed using the BART toolbox [24].

### Motion Simulation

Figure 4 shows how the same rigid body motion can induce different motion
artifacts based on the acquisition simulation. Figure 5 outlines the procedure
followed to generate FSE aware motion artifacts using PD and $T_{2}$ maps.

Figure 4: Ground truth FSE image (left) with example motion artifacts from FSE
Aware simulation (center) and FSE Agnostic simulation (right). Figure 5:
Proposed motion simulation for FSE sequences. T2 and PD masks are used to
generate 16 images with varying TE. Each echo image is randomly rotated for
each TR to simulate inter-TR motion (all echos in the same TR are rotated the
same). These images are then transformed into k-space and sampled once (non-
overlapping) based on their TR and echo number. All sampled k-space lines are
then merged into a single k-space as they would be in a typical FSE
acquisition. The IFFT is taken to obtain the motion-corrupt image.


# Eta-meson production in the resonance energy region. 111Supported by
Transregio SFB/TR16, project B.7

V. Shklyar<EMAIL_ADDRESS>H. Lenske  U. Mosel  Institut
für Theoretische Physik, Universität Giessen, D-35392 Giessen, Germany

###### Abstract

We perform an updated coupled-channel analysis of eta-meson production
including all recent photoproduction data on the proton. The dip observed in
the differential cross sections at c.m. energies W=1.68 GeV is explained by
destructive interference between the $S_{11}(1535)$ and $S_{11}(1560)$ states.
The effect from $P_{11}(1710)$ is found to be small but still important to
reproduce the correct shape of the differential cross section. For the
$\pi^{-}N\to\eta N$ scattering we suggest a reaction mechanism in terms of the
$S_{11}(1535)$, $S_{11}(1560)$, and $P_{11}(1710)$ states. Our conclusion on
the importance of the $S_{11}(1535)$, $S_{11}(1560)$, and $P_{11}(1710)$
resonances in the eta-production reactions is in line with our previous
results. No strong indication for a narrow state with a width of 15 MeV and
the mass of 1680 MeV is found in the analysis. $\eta N$ scattering length is
extracted and discussed.

###### pacs:

11.80.-m,13.75.Gx,14.20.Gk,13.30.Gk

##  I Introduction

The discovery of nucleon resonances in the first pion-nucleon scattering
experiments provided first indications for a complicated intrinsic structure
of the nucleon. With establishing the quark picture of hadrons and
developments of the constituent quark models the interest in the study of the
nucleon excitation spectra was renewed. The major question was the number of
the excited states and their properties. This problem was attacked both
experimentally and theoretically. On the theory side constituent quark (CQM)
models, lattice QCD and Dyson-Schwinger approaches have been developed to
describe and predict the nucleon resonance spectra (see e.g. Aznauryan:2009da
for a review). The main problem remains, however, a serious disagreement
between the theoretical calculations and the experimentally observed baryon
spectra. This concerns both the number and the properties of excited states.

On the experimental side pion-induced reactions have been studied to establish
resonance spectra. However, due to difficulties in detecting neutral particles
most experiments were limited to pion-nucleon elastic scattering with charged
particles in the final state. Being the lightest non-strange particle next to
the pion the $\eta$-meson also becomes an interesting probe to study nucleon
excitations. A few experiments have been made in the past to investigate
$\eta$-production. The first near-threshold measurements Jones:1966zza ;
Richards:1970cy ; Bulos:1970zk  demonstrated that the reaction proceeds
through a strong $S$-wave resonance excitation which was later identified with
$S_{11}(1535)$. An extensive study of the $\pi^{-}p\to\eta n$ reaction above
W$>$1.7 GeV has been made in Brown:1979ii ; Baker:1979aw . Both differential
cross section and asymmetry data have been obtained. However, due to possible
problems with the energy-momentum calibration Clajus:1992dh  the use of these
data might lead to wrong conclusions on the reaction mechanism. Note that
these problems are present not only in the $\eta$-measurements Brown:1979ii ;
Baker:1979aw  but also in the charge-exchange data obtained in the same
experiment.

Presently the development of the high-duty electron facilities (ELSA, JLAB,
MAMI, SPring) offers new possibilities to study the $\eta$-photoproduction
both on the proton ($\eta p$) and on the neutron ($\eta n$). The first
measurement of the $\eta$-photoproduction on the neutron reported an
indication for a resonance-like structure in the reaction cross section at
W=1.68 GeV Kuznetsov:2006 ; Kuznetsov:2006kt . Independent experimental
studies Jaegle:2008ux ; Jaegle:2011sw  confirmed the existence of this effect
in the $\gamma n$ reaction. This phenomenon was predicted in Azimov:2005jj  as
a signal from a narrow state - a possible non-strange partner of the
pentaquark Diakonov:1997mm . Another explanation has been suggested in
Shklyar:2006xw  where the observed effect was described by the contributions
from the $S_{11}(1650)$ or $P_{11}(1710)$ states. Due to the lack of knowledge
of the $S_{11}(1650)$ and $P_{11}(1710)$ resonance couplings to $\gamma n$ a
clean separation of the relative contributions from these states is difficult.
The general conclusion made in Shklyar:2006xw  is that both states might be
good candidates to explain the observed structure.

By fitting to the $\eta n$ cross sections and beam asymmetry the Bonn-Gatchina
group provided an explanation Anisovich:2008wd  for the second peak in terms
of the $S_{11}(1650)$ state. Another contribution to the field has been made
by the authors of Doring:2009qr . There the peak in the
$\sigma_{p}/\sigma_{n}$ cross section ratio was explained by a cusp effect
from the $K\Sigma$ and $K\Lambda$ rescattering channel. All these studies have
been done assuming scattering on a quasi-free nucleon. At the same time a
realistic analysis of meson photoproduction on the quasi-free neutron should
include the nucleon-nucleon and meson-nucleon correlations (FSI-effect) which
were shown to be very important Tarasov:2011ec  and take into account
corresponding experimental cuts applied by the extraction of the quasi-free
neutron data from $\gamma D$-scattering. The later issue might be crucial for
the unambiguous identification of the narrow resonance contribution as
discussed in MartinezTorres:2010zzb .

If it is granted that the signal observed in the $\gamma n$ scattering
Kuznetsov:2006 ; Kuznetsov:2006kt ; Jaegle:2008ux ; Jaegle:2011sw  is due to
the narrow (exotic) state one may expect to observe a similar effect in other
eta-production reactions at the same energies, e.g in gamma-proton scattering.
The experimental investigations of the $\eta$-production on the proton made by
the CLAS, GRAAL, and CB-ELSA/TAPS collaborations Dugger:2002ft ; Crede:2009zzb
; Bartholomy:2007zz ; Bartalini:2007fg  have found an indication of the dip
structure around W=1.68 GeV in the differential cross section but not a
resonance-like structure. This effect was also accompanied by the change in
the angular distribution of the differential cross section. However, despite
of extensive theoretical studies of the $\eta$ -production the reaction
mechanism is still under discussion PhysRevC.84.045207 ; Chiang:2002vq ;
Bartholomy:2007zz ; Feuster:1998b ; An:2011sb ; Anisovich:2011ka ;
Shyam:2008fr ; Nakayama:2008tg ; Choi:2007gy ; Zhong:2007fx ; Fix:2007st ;
Gasparyan:2003fp ; Ruic:2011wf .

Recently the $\eta$-photoproduction on the proton has been measured with high-
precision by the Crystal Ball collaboration at MAMI McNicoll:2010qk . These
high-resolution data provides a new step forward in understanding the reaction
dynamics and in the search for a signal from the ’weak’ resonance states. The
main result reported in McNicoll:2010qk  is a very clean signal of a dip
structure around W=1.68 GeV. It is interesting to note that the old
measurements of the $\pi N\to\eta N$ reaction Richards:1970cy  also give an
indication for the second structure in the differential cross section at W=1.7
GeV. This raises a question whether the dip reported in the $\eta p$ reaction,
the resonance-like signal observed in $\eta n$ and the possible structure in
the $\pi N\to\eta N$ cross section are originating from the same degrees of
freedom or not. The second question is whether one of these phenomena can be
attributed to the signal from a narrow (exotic) resonance state as discussed
in Azimov2 ; Azimov:2005jj ; Azimov:2005hv .

In our previous coupled-channel PWA study Shklyar:2006xw  we proposed an
explanation of the possible dip in the $\eta$-proton cross section in terms of
the destructive interference of the $S_{11}(1535)$ and $S_{11}(1650)$ states.
The result was based on the $\eta p$ photoproduction data taken before 2006
Dugger:2002ft ; Crede:2003ax . The aim of the present study is to extend our
previous coupled-channel analysis of the $\gamma p\to\eta p$ reaction by
including the data from the high-precision measurements McNicoll:2010qk . The
main question is whether the $\eta p$ reaction dynamics can be understood in
terms of the established resonance states. We emphasize that for reliable
identification of the resonance contributions the calculations should maintain
unitarity. Another complication comes from the fact that the most
contributions to the resonance self-energy (total decay width) is driven by
its hadronic couplings. Therefore the analysis of the photoproduction data
requires the knowledge of the hadronic transition amplitudes. Hence the
simultaneous analysis of all open channels (both hadronic and electromagnetic
) is inevitable for the identification of the resonances and extraction of
their properties. In the present study we concentrate on the combined
description of the $(\gamma/\pi)p\to\eta p$ scattering taking also the
$(\gamma/\pi)\to\pi N$, $2\pi N$, $\omega N$, $K\Lambda$ channels into
account. The results on the $\eta n$ reaction will be reported elsewhere.

First, we corroborate our previous findings Penner:2002a ; Penner:2002b ;
Shklyar:2006xw  where the important contributions from the $S_{11}(1535)$,
$S_{11}(1650)$ and $P_{11}(1710)$ resonances to the $\pi N\to\eta N$ reaction
have been found. The major effect comes from the $S_{11}$ and $P_{11}$ partial
waves. The interference between the $S_{11}(1535)$ and $S_{11}(1650)$ states
produces a dip in the $S_{11}$ amplitude. The $P_{11}$ amplitude is influenced
by the contributions from the $P_{11}(1710)$ state. The interference between
the $S_{11}$ and $P_{11}$ partial waves leads to the forward peak in the
differential cross section around W=1.7 GeV. We stress that the interference
between two nearby states also includes rescattering and coupled-channel
effects which are hard to simulate by the simple sum of two Breit-Wigner
forms.

We also confirm our previous finding that the interference between
$S_{11}(1535)$ and $S_{11}(1650)$ is responsible for the dip seen in the $\eta
p$ data. The effect from the $\omega N$ threshold is found to be relatively
small which is also in line with the conclusion of Shklyar:2006xw . Opposite
to Anisovich:2011ka  we do not find any strong indications for a narrow state
in the Crystal Ball/Taps data around W=1.68 GeV. We have also checked our
results for the $\eta p$ reaction above W=2 GeV where a number of new
experimental data are available. Note that we do not use Reggezied $t-$channel
exchange but include all $t$-channel contributions consistently into our
unitarization procedure. Because of the normalization problem Sibirtsev:2010yj
; Dey:2011rh  between the CLAS Williams:2009yj  and the CB-ELSA Crede:2009zzb
datasets the simultaneous description of these data is not possible. Above W=2
GeV our calculations are found to be in closer agreement with the CLAS
measurements Williams:2009yj . The CB-ELSA data Crede:2009zzb  demonstrates a
step rise around W=1.925 GeV for the scattering angles
$\cos\theta=0.85...0.95$. It is not clear whether this phenomenon could be
related to a threshold effect (e.g. $\phi N$, $a_{0}(980)N$, $f_{0}(980)$, or
$\eta^{\prime}N$) or attributed to other reaction mechanisms.

We conclude that further progress in understanding of the $\eta$-meson
production dynamics would be hardly possible without new measurements of the
$\pi N\to\eta N$ reaction.

##  II Database

Here we present a short overview of the experimental database relevant for the
present calculations. The details on the $K\Lambda$, $K\Sigma$, $\omega N$
channels will be given elsewhere.

$\pi N\to\eta N$: The thorough overview of the $\pi N\to\eta N$ experimental
data (except the recently published Crystal Ball measurements Prakhov ), is
given in Clajus:1992dh . As already mentioned in Introduction only few
measurements of the $\eta$-production have been made with pion beams: except
for Danburg:1971  where the eta-meson was produced in $\pi^{+}D$ collisions,
all the data have been taken from the $\pi^{-}p$ scattering Prakhov ;
Baker:1979aw ; Brown:1979ii ; Bulos:1970zk ; Richards:1970cy ; Jones:1966zza ;
Debenham ; Deinet:1969cd . Unfortunately due to numerous problems with the
experimental data from Baker:1979aw ; Brown:1979ii  (see discussion in
Clajus:1992dh  and references therein) the use of these measurements in the
analysis might lead to wrong conclusions for the reaction mechanism.
Therefore, opposite to Shrestha:2012va  we do not include these data in the
analysis. Another measurement available above W=1.65 GeV is the data from
Richards et al Richards:1970cy . In the first resonance energy region this
cross section tends to be lower than results from other experiments. Since the
old measurements quote only statistical uncertainties the reason for these
differences is unclear. In their study the authors of Batinic:1995  added
systematical errors to all differential cross sections. We do not follow this
procedure and include only quoted uncertainties in the analysis.

$\gamma p\to\eta p$: a number of experimental studies have been performed in
the resonance energy region McNicoll:2010qk ; Williams:2009yj ; GRAAL:2002 ;
Dugger:2002ft ; Krusche:1995nv ; Bartalini:2007fg ; Nakabayashi:2006ut ;
Crede:2009zzb ; Crede:2003ax ; Bartholomy:2007zz ; Elsner:2007hm ; Bock:1998rk
; Ajaka:1998zi . Most of these measurement are differential cross sections.
The target asymmetry has been studied in Bock:1998rk . It has been observed
that close to the $\eta N$ production threshold the asymmetry changes the sign
at moderate scattering angles. The previous calculations of the Giessen Model
Penner:2002a ; Shklyar:2006xw  and the Mainz group PhysRevC.60.035210  could
not explain this feature. The description of this data would require an
unexpected phase shift between the $S_{11}$ and $D_{13}$ resonances as noted
in PhysRevC.60.035210 . One may hope that the upcoming new measurements of the
target asymmetry at the ELSA facility will solve this puzzle Hartmann:2011uv .

For the beam asymmetry we use the recent data from the GRAAL Bartalini:2007fg
and CB-ELSA/TAPS Elsner:2007hm  collaborations which cover the energy region
up to W=1.91 GeV. For the differential cross section we use the recent high-
quality Crystal Ball data McNicoll:2010qk . Above W=1.89 GeV our calculations
are constrained by the amalgamated data set from experiments Williams:2009yj ;
Crede:2009zzb ; Crede:2003ax ; Bartholomy:2007zz . Since the experimental
uncertainties of the data Williams:2009yj ; Crede:2009zzb ; Crede:2003ax ;
Bartholomy:2007zz  are much larger than those in McNicoll:2010qk  we reduce
them by factor of 2.

In the $(\pi/\gamma)N\to\pi N$ channels our calculations are constrained by
the single-energy solutions from the GWU (former SAID) analysis Workman:2011vb
; Arndt:2006bf ; Arndt:2008zz . For the $\pi N\to 2\pi N$ transitions we
follow the procedure described in Penner:2002a ; Penner:2002b ; Feuster:1998a
; Feuster:1998b . We continue to parameterize the $2\pi N$ channel in terms of
the effective $\zeta N$ state, where $\zeta$ is an isovector scalar meson of
two pion mass: $m_{\zeta}=2m_{\pi}$. The final $\zeta N$ state is only allowed
to couple to nucleon resonances. Therefore the decay $N^{*}\to\zeta N$ stands
for the sum of transitions $N^{*}\to\Delta\pi$, $\sigma N$, $\rho N$ etc. This
procedure allows for the good description of the $\pi N\to 2\pi N$ partial
wave cross sections extracted in Manley:1984 . However of case of the $\gamma
p\to 2\pi N$ the same agreement cannot be expected. This is because of the
enhanced role of the background contributions (due to e.g. the contact
$\gamma\rho NN$ interaction in the $\gamma N\to\rho N$ transitions). After
fixing the database a $\chi^{2}$ minimization is performed to fix the model
parameters.

##  III Giessen Model

Here we briefly outline the main ingredients of the model. More details can be
found in Feuster:1998a ; Feuster:1998b ; Penner:2002a ; Penner:2002b ;
shklyar:2004a ; shklyar:2005c . The Bethe-Salpeter equation is solved in the
$K$-matrix approximation to obtain multi-channel scattering $T$-matrix:

$\displaystyle
T(\sqrt{s},p,p^{\prime})=K(\sqrt{s},p,p^{\prime})+\int\frac{d^{4}q}{(2\pi)^{4}}K(\sqrt{s},p,q)G_{BS}(\sqrt{s},q)T(\sqrt{s},q,p^{\prime}),$
(1)

where $p$ ($k$) and $p^{\prime}$ ($k^{\prime}$) are the incoming and outgoing
baryon (meson) four-momenta, $T(\sqrt{s},p,p^{\prime})$ is a coupled-channel
scattering amplitude, $G_{BS}$ is a meson-nucleon propagator and
$K(\sqrt{s},p,p^{\prime})$ is an interaction kernel. The quantities
$T(\sqrt{s},p,p^{\prime})$, $G_{BS}$, and $K(\sqrt{s},p,p^{\prime})$ are in
fact multidimensional matrices where the elements of the matrix stand for the
different scattering reactions.

To solve the coupled-channel scattering problem with a large number of
inelastic channels, we apply the so-called K-matrix approximation by
neglecting the real part of the BSE propagator $G_{BS}$. After the integration
over the relative energy, Eq. (1) reduces to

$\displaystyle
T^{\lambda_{f}\lambda_{i}}_{fi}=K^{\lambda_{f}\lambda_{i}}_{fi}+i\int
d\Omega_{n}\sum_{n}\sum_{\lambda_{n}}T^{\lambda_{f}\lambda_{n}}_{fn}K^{\lambda_{n}\lambda_{i}}_{ni},$
(2)

where $T_{fi}$ is a scattering matrix and $\lambda_{i}$($\lambda_{f}$) stands
for the quantum numbers of initial(final) states $f,i,n=$ $\gamma N$, $\pi N$,
$2\pi N$, $\eta N$, $\omega N$, $K\Lambda$, $K\Sigma$. Using the partial-wave
decomposition of $T$, $K$ in terms of Wigner d-functions the angular
integration can be easily carried out and the equation is further simplified
to the algebraic form

$\displaystyle
T^{J\pm,I}_{fi}=\left[\frac{K^{J\pm,I}}{1-iK^{J\pm,I}}\right]_{fi}.$ (3)

The validity of this approximation was demonstrated by Pearce and Jennings in
Pearce:1990uj  by studying different approximations to the BSE for $\pi N$
scattering. Considering different BSE propagators they concluded that an
important feature of the reduced intermediate two particle propagator is the
on-shell part of $G_{BS}$. It has been argued that there is no much difference
between physical parameters obtained using the $K$-matrix approximation and
other schemes. It has also been shown in Goudsmit:1993cp ; Oset:1997it  that
for $\pi N$ and $\bar{K}N$ scattering the main effect from the off-shell part
is a renormalization of the couplings and the masses.

Due to the smallness of the electromagnetic coupling the dominant
contributions to the self energy stem from the hadronic part. Therefore we
treat the photoproduction reactions perturbatively. This is equivalent to
neglecting $\gamma N$ in the sum over intermediate states $n$ in Eq. (2).
Thus, for a photoproduction process the equation (3) can be rewritten as
follows Penner:2002b ; Feuster:1998b

$\displaystyle
T^{J\pm,I}_{f\gamma}=K^{J\pm,I}_{f\gamma}+i\sum_{n}T^{J\pm,I}_{fn}K^{J\pm,I}_{n\gamma},$
(4)

where the summation in Eq.(4) is done over all hadronic intermediate states.
Here the matrix $T^{J\pm,I}_{fn}$ stems only from the hadronic transitions:
indices $f$ and $n$ run over $\pi N$, $2\pi N$, $\eta N$, $K\Lambda$,
$K\Sigma$, $\omega N$ channels. The sum in Eq. (4) reflects the importance of
the hadronic part of the transition amplitude in the description of
photoproduction reactions. In other words, the amplitudes for the $\pi N\to\pi
N$, $\eta N$, $\omega N$ etc. transitions should always be included in the
calculation of the photoproduction amplitudes.

###  III.1 Interaction kernel and resonance parameters

Here we present the main ingredients of the interaction kernel to the BSE
Eq.(1) relevant for $\eta$-production. More details on other reactions can be
found in shklyar:2004a ; Penner:2002a ; Penner:2002b ; Feuster:1998a ;
Feuster:1998b ; shklyar:2004b . The interaction potential ($K$-matrix) of the
BSE is built up as a sum of $s$-, $u$-, and $t$-channel contributions
corresponding to the tree level Feynman diagrams shown in Fig. (1).

Figure 1: $s$-,$u$-, and $t$\- channel contributions to the interaction
potential. $i$ and $f$ stand for the initial and final $\gamma N$, $\pi N$,
$2\pi N$, $\eta N$, $\omega N$, $K\Lambda$, $K\Sigma$ states. $m$ denotes
intermediate $t$-channel meson.

In the isospin $I=\frac{1}{2}$ channel we checked for the contributions from
the $S_{11}(1535)$, $S_{11}(1650)$, $P_{11}(1440)$, $P_{11}(1710)$,
$P_{13}(1720)$, $P_{13}(1900)$, $D_{13}(1520)$ $D_{13}(1900)$, $D_{15}(1675)$,
$F_{15}(1680)$ , $F_{15}(2000)$ resonances. The resonance and background
contributions are consistently generated from the same effective interaction.
The Lagrangian densities are given in shklyar:2004a ; Penner:2002a ;
Penner:2002b ; Feuster:1998a ; Feuster:1998b ; shklyar:2004b  and respect the
chiral symmetry in low-energy regime. The properties of the $t$-channel mesons
important for $\eta$ production are given in Table 1.

| mass [GeV] | $J^{P}$ | $I$ | final state
---|---|---|---|---
$\omega$ | 0.783 | $1^{-}$ | $0$ | $(\gamma,\eta)$
$\rho$ | 0.769 | $1^{-}$ | $1$ | $(\pi,\eta)(\gamma,\eta)$
$a_{0}$ | 0.983 | $0^{+}$ | $1$ | $(\pi,\eta)$
$\phi$ | 1.02 | $1^{-}$ | $0$ | $(\gamma,\eta)$
Table 1: Properties of mesons which give contributions to the $\eta N$ final
state via the $t$-channel exchange. The notation $(\gamma,\eta)$ means $\gamma
N\to\eta N$ etc.

Using the interaction Lagrangians and values of the corresponding meson decay
widths taken from the PDG pdg  the following coupling constants are obtained:

$\displaystyle\begin{array}[]{lcrclcr}g_{a_{0}\eta\pi}&=&-2.100\;,&&g_{\omega\eta\gamma}&=&-0.27\;,\\\
g_{\rho\eta\gamma}&=&-0.64\;,&&g_{\phi\eta\gamma}&=&-0.385\;.\\\ \end{array}$
(7)

All other coupling constants were allowed to be varied during the fit. The
obtained values are given in Table 2. For the $\eta NN$ interaction we use
pseudoscalar coupling , which has been also utilized in our previous studies
Feuster:1998b ; Feuster:1998a ; Penner:2002a ; Penner:2002b ; Shklyar:2006xw .
The derived $g_{\eta NN}$ constant is found to be small which is in line with
our previous results Shklyar:2006xw ; Penner:2002b . To check the dependence
of our results on the choice of the $\eta NN$ interaction we have also
performed calculations with the pseudovector coupling. However also in the
latter case only a small $g_{\eta NN}$ coupling constant has been found.

Since the PDG gives only the upper limit for the decay branching ratio
R$(\rho\to\pi\eta)<6\times 10^{-3}$ we allowed this constant to be varied
during fit. However due to lack of experimental constraints this coupling
cannot be fully fixed in the present calculation. We find a small overall
contribution from the $t$-channel $\rho$-meson exchange to the
$\pi^{-}p\to\eta n$ reaction. The $g_{\phi NN}$ coupling is calculated from
$g_{\omega NN}$ using the relation

$\frac{g_{\phi NN}}{g_{\omega NN}}=-\tan\Delta\theta_{\phi/\omega},$

where $\Delta\theta_{\phi/\omega}$ is a deviation from the ideal
$\phi$-$\omega$ mixing angle. Taking $\Delta\theta_{\phi/\omega}=3.7^{0}$ from
pdg  one gets for the ratio $g_{\phi NN}/g_{\omega NN}\approx-1/15$. Using
this value a very small contribution from the $t$\- channel $\phi$-meson
exchange to the $\eta$-photoproduction has been found.

$g_{\pi NN}$ | 12.85 | $g_{\rho\eta\pi}$ | 0.133 | $g_{\rho NN}$ | 4.98 | $\kappa_{\rho}$ | 2.18
---|---|---|---|---|---|---|---
$g_{\eta NN}$ | 0.31 | $g_{a_{0}NN}$ | -44.37 | $g_{\omega NN}$ | 7.23 | $\kappa_{\omega}$ | -1.50

Table 2: Nucleon and $t$-channel couplings obtained in the present study.

To take into account the finite size of mesons and baryons each vertex is
dressed by a corresponding form factor:

$\displaystyle F_{p}(q^{2},m^{2})$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{4}}{\Lambda^{4}+(q^{2}-m^{2})^{2}},$ (8)

where $q$ is a c.m. four-momentum of an intermediate particle and $\Lambda$ is
a cutoff parameter. The cutoffs $\Lambda$ in Eq. (8) are treated as free
parameters being varied during the calculation. However, we keep the same
cutoffs in all channels for a given resonance spin $J$ : $\Lambda^{J}_{\pi
N}=\Lambda^{J}_{\pi\pi N}=\Lambda^{J}_{\eta N}=...$ etc.,
($J=\frac{1}{2},~{}\frac{3}{2},~{}\frac{5}{2}$). This significantly reduces
the number of free parameters; i.e. for all spin-$\frac{5}{2}$ resonances
there is only one cutoff $\Lambda=\Lambda_{\frac{5}{2}}$ for all decay
channels. However for the photoproduction reactions we use different cutoffs
at the $s$\- and $u$-channel electromagnetic vertices. All values are given in
Table 3. Except for the spin-$\frac{3}{2}$ states, the $s$\- and $u$-channel
cutoffs almost coincide.

$\Lambda_{N}$ [GeV] |  $\Lambda_{\frac{1}{2}}^{h}$ [GeV] |  $\Lambda_{\frac{3}{2}}^{h}$ [GeV] |  $\Lambda_{\frac{5}{2}}^{h}$ [GeV] |  $\Lambda_{\frac{1}{2}}^{\gamma}$ [GeV] |  $\Lambda_{\frac{3}{2}}^{\gamma}$ [GeV] |  $\Lambda_{\frac{5}{2}}^{\gamma}$ [GeV] |  $\Lambda_{t}^{h,\gamma}$[GeV]
---|---|---|---|---|---|---|---
0.952 | 3.0 | 0.97 | 1.13 | 1.69 (1.69) | 4.20 (2.9) | 1.17 (1.25) | 0.7
Table 3: Cutoff values for the form factors. The lower index denotes an
intermediate particle, i.e. $N$: nucleon, $\frac{1}{2}$: spin-$\frac{1}{2}$
resonance, $\frac{3}{2}$: spin-$\frac{3}{2}$, $\frac{5}{2}$:
spin-$\frac{5}{2}$ resonance, $t$: $t$-channel meson. The upper index
$h$($\gamma$) denotes whether the value is applied to a hadronic or
electromagnetic vertex. The cutoff values used at electromagnetic $u$-channel
vertices are given in brackets.

The use of vertex form factors requires special care for maintaining the
current conservation when the Born contributions to photoproduction reactions
are considered. Since the resonance and intermediate meson vertices are
constructed from gauge invariant Lagrangians they can be independently
multiplied by the corresponding form factors. For the nucleon contributions to
meson photoproduction we apply the suggestion of Davidson and Workman
Davidson:2001rk  and use the crossing symmetric common form factor:

$\displaystyle\tilde{F}(s,u,t)=F(s)+F(u)+F(t)-F(s)F(u)-F(s)F(t)-F(u)F(t)+F(s)F(u)F(t).$
(9)

The extracted resonance parameters given in Table 4 are very close to the
values deduced in our previous calculations shklyar:2004b ; Shklyar:2006xw
which indicates the stability of the obtained solution. However some values
changed upon inclusion of the new MAMI data McNicoll:2010qk . The total width
of $S_{11}(1650)$ tends to be larger than that deduced in our previous
calculations shklyar:2004b . The helicity amplitude is also modified but still
is in good agreement with the parameter range provided by PDG pdg . The
opposite effect is found for the $P_{11}(1710)$ state where the total width is
reduced once the data of McNicoll:2010qk  are included. The remaining
resonance parameters are only slightly modified as compared to our previous
results.

The mass and width of the Roper resonance is found to be larger than deduced
in other analyses pdg . However the authors of Vrana:2000  give $490\pm 120$
MeV for the total width. The large decay width $545\pm 170$ MeV has also been
deduced by Cutkosky and Wang Cutkosky:1990 . Note that properties of this
state are strongly influenced by its decay into the $2\pi N$ final state.
Arndt et al Arndt:1990bp  found a second pole structure for the Roper
resonance which might be attributed to the coupling to the $\pi\Delta$
subchannel. Since we use a simplified prescription for the $2\pi N$ reaction
this effect cannot be properly described in the present calculations.

The recent GWU(SAID) study of the $\pi N$ data shows no evidence for the
$P_{11}(1710)$ resonance. An indirect indication for the existence of this
state can be concluded from the analysis of the $\pi N$ inelasticity and $2\pi
N$ cross section in the $P_{11}$ partial wave, see discussion in Section IV.
We find a small coupling of this resonance to the $\pi N$ final state. Since a
clear signal from this state is not seen in the recent GWU solution, the
determination of the total width turns out to be difficult. In our
calculations we assume that this resonance has a large decay branching ratio
to the $\eta N$. However the quality of the $\pi^{-}p\to\eta n$ data does not
allow for an unambiguous determination of the properties of this state.

The mass and width of the $D_{13}(1520)$ is more close to the values obtained
by Arndt et al Arndt:1995ak : $1516\pm 10$ MeV and $106\pm 4$ MeV
respectively. It is interesting to note that the mass of this resonance
deduced from the pion photoproduction tends to be 10 MeV lower that the values
derived from the pion-induced reactions pdg . The second $D_{13}(1900)$ has a
very large decay width. We associate this state with $D_{13}(2080)$ as
suggested in PDG. This resonance is rated with two stars and its existence is
still under discussion. In our updated coupled-channel calculation of the
$\omega$-production shklyar:2004b  a large $\omega N$ and $2\pi N$ decay
branching ratios have been obtained.

The properties of other resonances are very close to the values given in PDG.
Except for $S_{1}(1535)$ and $P_{11}(1710)$ we find only small resonance
couplings to $\eta N$ which is in accordance with our previous conclusions.
One needs to stress that the smallness of the resonance coupling does not
necessarily mean that the contribution from the state is negligible. The
$S_{11}(1650)$ state produces for example a sizable effect in the eta-
production due to overlapping with $S_{11}(1535)$. Another example is the
effect from the $D_{13}(1520)$ state in $\eta$-photoproduction on the proton.
Here the smallness of the $\eta N$ branching ratio is compensated by the
strong electromagnetic coupling of this resonance. Therefore the effect from
this state could be seen in the $E_{2-}$ and $M_{2-}$ multipoles, see Section
IV.5. However in most cases the resonance contributions with small branching
ratios to the eta are hard to resolve unambiguously.

###  III.2 Pole parameters

It is interesting to compare the poles positions and elastic residues with the
results from other studies, see Table 5. The calculated pole masses are very
close to the values obtained in other analyses, see pdg . The agreement
between imaginary parts and elastics residues is also good, though some
differences exist between the present values and the results from other
groups.

For the $S_{11}(1535)$ state we obtain a smaller elastic residue (for
definition of $|R|$ see pdg ) $|R|=15$ MeV which is almost identical to the
result of the GWU group $|R|$=16 MeV Arndt:2006bf . Both values seem to be out
of the range given in PDG pdg  50$\pm$20 MeV. It is interesting to note that
the elastic residue from Arndt:2006bf  is included into the estimation made in
pdg  but still does not fit to the provided range. The value $\Gamma_{\rm
pole}=89$ MeV for the $S_{11}(1650)$ state is also comparable with the result
from Arndt:2006bf : $\Gamma_{\rm pole}=80$ MeV which are again less than the
lower bound given in pdg .

Though the derived pole mass of $P_{11}(1440)$ is very close to the values
deduced in other calculations we obtain a significantly larger pole width. As
a result the elastic pole residue turns out to be also large $|R|$=126 MeV. We
note, that the extraction of the properties of $P_{11}(1440)$ in the complex
energy plane might require a proper treatment of the
$P_{11}(1440)\to\pi\Delta(1232)\to 2\pi N$ isobar decay channel where the
overlap of the self-energies of the $P_{11}(1440)$ and $\Delta(1232)$ states
might be important for the determination of the properties of $P_{11}(1440)$.
This question will be addressed in shklyar:2012 .

As we already mentioned the results for $P_{11}(1710)$ are controversial. We
find 159 MeV for the pole width. Somewhat greater value of 189 MeV has been
obtained in Tiator:2010rp ; Batinic:2010zz . The recent issue of PDG pdg
summarizes results for the pole parameters taken from four different analyses.
Whereas the calculations Anisovich:2011fc ; Hoehler:1993  give 200 MeV for the
pole width, Cutkosky obtains a significantly lower value $\Gamma$=80 MeV
Cutkosky:1990zh ; Cutkosky:1979fy . This results in a large spread of the
resonance width given by PDG, see Table 5. The elastic residue is found to be
small which is in accordance with the small decay branching ratio to $\pi N$.
The similar conclusion has also been drawn in Tiator:2010rp .

Investigation of the $P_{13}$\- wave inelasticity Arndt:2006bf  shows that the
$P_{13}(1720)$ state could have a strong decay flux into the $3\pi N$ channel
Manley:1984 . Therefore the calculation of its pole width might be affected by
deficiencies in description of this channel. PDG estimations are based on
several studies where $\Gamma_{\rm pole}=120\pm 40$ by Cutkosky
Cutkosky:1979fy  is the lower limit. The upper bound $\Gamma_{\rm pole}=450\pm
100$ MeV is given by the recent Bonn-Gatchina analysis Anisovich:2011fc .
Neither of these calculations includes the $3\pi N$ channel explicitely.

The situation with the second $P_{13}(1900)$ state is even more complicated.
This resonance is rated by two stars in PDG and supposed to be rather broad.
The latest GWU analysis Arndt:2006bf  does not find any indication for this
state. The present information about the pole parameters in PDG is based
solely on the result of the Bonn-Gatchina calculations Anisovich:2011fc  which
deduce the pole mass $1900\pm 30$ MeV and the pole width $200^{+100}_{-60}$
MeV. These values are very close to those derived in the present work.

The pole width of the $D_{13}(1520)$ state ( 94 MeV) turns out to be 10 MeV
less than the lower limit given in PDGpdg . The similar value of $\Gamma_{\rm
pole}=$ 95 MeV has also been obtained in the Jülich model Doring:2009yv . Some
analyses find additional poles associated with the $D_{13}(1700)$ and
$D_{13}(1875)$ states pdg . We do not find any indication for $D_{13}(1700)$.
The pole position for the second resonance is close to the results of other
calculation pdg .

Though the elastic residues for the $D_{15}(1675)$ and $F_{15}(1680)$ states
are comparable with the values given in PDG their pole widths are somewhat
lower than those obtained in other studies pdg . We also find an indication
for the second state N(2000) with the pole mass of 1900 MeV and the width of
$123$ MeV, see Table 5. This resonance has a small coupling to the $\pi N$
final state what is in agreement with results from other calculations.

$N^{*}$ | mass (MeV) | $\Gamma_{tot}(MeV)$ | RπN | R2πN | RηN | RωN | $A^{p}_{\frac{1}{2}}$ | $A^{p}_{\frac{3}{2}}$
---|---|---|---|---|---|---|---|---
$S_{11}$(1535) | 1526(2) | 131(12) | 35(3) | $8(2)$ | $58(4)$ | — | 91(4) | —
| 1526 | 136 | 34.4 | $9.5$ | $56.1$ | — | 92 | —
| 1536(10) | 150(25) | 45(10) | $5(5)$ | $42(10)$ | — | 90(30) | —
$S_{11}$(1650) | 1665(2) | 147(14) | 74(3) | $23(2)$ | $1(2)$ | — | 63(6) | —
| 1664 | 131 | 72.4 | $23.1$ | $1.4$ | — | 57 | —
| 1657(13) | 150(30) | 70(20) | $15(5)$ | $10(5)$ | — | 53(16) | —
$P_{11}$(1440) | 1515(15) | 605(90) | 56(2) | $44(2)$ | — | — | -85(3) | —
| 1517 | 608 | 56.0 | $44.0$ | — | — | -84 | —
| 1445(25) | 300(150) | 65(10) | $35(5)$ | — | — | -60(4) | —
$P_{11}$(1710) | 1737(17) | 368(120) | 2(2) | $49(3)$ | $45(4)$ | 3(2) | -50(1) | —
| 1723 | 408 | 1.7 | $49.8$ | $43.0$ | 0.2 | -50 | —
| 1710(30) | 150(100) | 13(7) | $65(25)$ | $20(10)$ | 13(2) | 24(10) | —
$P_{13}$(1720) | 1700(10) | 152(2) | 17(2) | $79(2)$ | $0(1)$ | — | -65(2) | 35(2)
| 1700 | 152 | 17.1 | $78.7$ | $0.2$ | — | -65 | 35
| 1725(24) | 225(125) | 11(3) | $>70$ | $4(1)$ | — | 50(60) | -19(20)
$P_{13}$(1900) | 1998(3) | 359(10) | 25(1) | $61(2)$ | $2(2)$ | 10(3) | -8(1) | 0(1)
| 1998 | 404 | 22.2 | $59.4$ | $2.5$ | 14.9 | -8 | 0
| 1900(-) | 250(-) | 10(-) | — | $12(-)$ | 39(-) | 26(15) | -65(30)
$D_{13}$(1520) | 1505(4) | 100(2) | 57(2) | $44(2)$ | $0(1)$ | — | -15(1) | 146(1)
| 1505 | 100 | 56.6 | $43.4$ | 1.2 | — | -13 | 145
| 1520(5) | 112(12) | 60(5) | $25(5)$ | 2.3$\pm 10^{-3}$ | — | -24(8) | 150(15)
$D_{13}$(1875) | 1934(10) | 857(100) | 11(1) | $69(2)$ | $0(1)$ | 20(5) | 11(1) | 26(1)
| 1934 | 859 | 10.5 | $68.7$ | $0.5$ | 20.1 | 11 | 26
| 1875(45) | 220(100) | 12(10) | $70(20)$ | $3.5(3.5)$ | 21(7) | 18(10) | -9(5)
$D_{15}$(1675) | 1666(2) | 148(1) | 41(1) | $58(1)$ | $0(1)$ | — | 9(1) | 21(1)
| 1666 | 148 | 41.1 | $58.5$ | $0.3$ | — | 9 | 20
| 1675(5) | 150(15) | 40(5) | $55(5)$ | $0(1)$ | — | 19(8) | 15(9)
$F_{15}$(1680) | 1676(2) | 115(1) | 68(1) | $32(1)$ | $0(1)$ | — | 3(1) | 116(1)
| 1676 | 115 | 68.3 | $31.6$ | $0.0$ | — | 3 | 115
| 1685(5) | 130(10) | 67(3) | $35(5)$ | $0(1)$ | — | -15(6) | 132(13)
$F_{15}$(2000) | 1946(4) | 198(2) | 10(1) | $87(1)$ | $2(2)$ | 1(1) | 11(1) | 25(1)
| 1946 | 198 | 9.9 | $87.2$ | $2.0$ | 0.4 | 10 | 25
| 2050(100) | 350(200) | 15(7) | — | — | — | 35(15) | 50(14)

Table 4: Resonance parameters extracted in the present study. The uncertainties are given in brackets. Helicity decay amplitudes are given in $10^{-3}$GeV${}^{-\frac{1}{2}}$. 1st line: present study; 2nd line: shklyar:2004b , 3th line: pdg . (-): the validity range is not given.  | Re $z_{0}$(GeV) | -2Im $z_{0}$(MeV) |  $|$R$|$(MeV) | $\theta^{0}$
---|---|---|---|---
$S_{11}(1535)$ | 1.49 | 100 | 15 | -51
| 1.49-1.53 | 90-250 | 30…70 | -1…-30
$S_{11}(1650)$ | 1.65 | 89 | 19 | -46
| 1.64-1.67 | 100-170 | 20-50 | -50…-80
$P_{11}(1440)$ | 1.386 | 277 | 126 | -60
| 1.35-1.38 | 160-220 | 40-52 | -75…-100
$P_{11}(1710)$ | 1.67 | 159 | 11 | 9
| 1.67-1.77 | 80-380 | 2-15 | -160…+190
$P_{13}(1720)$ | 1.67 | 118 | 12 | -45
| 1.66-1.69 | 150-400 | 7-23 | -90…-160
$P_{13}(1900)$ | 1.91 | 173 | 10 | -64
| 1.870-1.93 | 140-300 | 1-5 | 45…-25
$D_{13}(1520)$ | 1.492 | 94 | 27 | -35
| 1.505-1.515 | 105-120 | 32-38 | -5…-15
$D_{13}(1875)$ | 1.81 | 98 | 3 | -76
| 1.8-1.95 | 150-250 | 2-10 | 20…180
$D_{15}(1675)$ | 1.64 | 108 | 20 | -49
| 1.655-1.665 | 125-150 | 22-32 | -21…40
$D_{15}(1680)$ | 1.66 | 98 | 33 | -32
| 1.665-1.68 | 110-135 | 35-45 | 0…-30
$F_{15}(2000)$ | 1.90 | 123 | 11 | -6
| 1.92-2.15 | 380-580 | 20-115 | -60…-140

Table 5: Pole positions and elastic pole residues. First line: present study,
second line: values from PDG pdg .

##  IV Results and discussion

The lack of the experimental data for the pion-induced reactions does not
provide enough constraints on the resonance parameters. Also the discrepancy
among various measurements (see Section (II)) does not allow for a consistent
description of the data in a full kinematical region. While the contribution
from the $S_{11}(1535)$ state is well established the reaction dynamics above
W=1.6 GeV is still under discussion. One of the early Giessen coupled-channel
calculations Penner:2002a ; Penner:2002b ; Shklyar:2006xw  found a destructive
interference between $S_{11}(1535)$ and $S_{11}(1650)$ states. The second
suggestion is a strong contribution from the $P_{11}(1710)$-resonance
excitation above W=1.68 GeV. This resonance was established in the early
single-channel Karlsruhe-Helsinki and Carnegie Mellon-Berkeley analyses (see
PDG pdg  and references therein). The independent study of the $\pi
N\to(\pi/\eta)N$ reactions by the Zagreb group Ceci:2006ra  provides an
additional evidence for the existence of $P_{11}(1710)$. The result of
Ceci:2006ra  confirm the assumption made in Penner:2002a ; Penner:2002b  on
the important contribution from this state to the $\eta$-production. However
the recent analysis from the GWU group Arndt:2006bf  finds no evidence for
this state. The absence of a clear signal in the $P_{11}$ partial wave of the
elastic $\pi N$ scattering does not necessarily mean that this state does not
exist. If the coupling to the final $\pi N$ state is small, the effect from
this state might not be seen in $\pi N$ scattering. The evidence for the
signal from the $P_{11}(1710)$ resonance has also been reported from the study
of the $\pi N\to K\Lambda$ reaction Ceci:2005vf . On the other hand the result
of the Bayestian analysis performed by the Gent group DeCruz:2012bv
demonstrates that $P_{11}(1710)$ is not needed to describe the $K\Lambda$
photoproduction. An opposite conclusion was drawn by the Bonn-Gatchina group
which finds decay branching ratio of $23\pm 7$% of this state to $K\Lambda$
Anisovich:2011fc .

Figure 2: (Color online) Calculated $\pi N$ inelasticity and $\pi N\to 2\pi N$
cross section in the $P_{11}$ partial wave in comparison with the results from
Arndt:2006bf  (GWU 2006) and Manley:1984 (Manley 1984).

Another indication for this state comes from the analysis of an inelastic flux
in the $P_{11}$ partial wave. In Fig. (2) the total inelasticity from the GWU
analysis vs. the total $2\pi$ cross section extracted in Manley:1984  is
compared. The difference between the total $\pi N$ inelasticity and the total
$2\pi N$ cross section at W=1.7 GeV in the $P_{11}$-wave can be attributed to
the sum of inelastic channels like $3\pi N$, $\eta N$, $\eta\pi N$ etc. We
assume here that the observed difference is due to the $\eta N$ production
channel dominated by the $P_{11}(1710)$ state. As $g_{\pi NN^{*}(1710)}$ is
assumed to be small this raises the question about the magnitude of the
$P_{11}(1710)$ contribution in the $\pi N\to\eta N$ reaction. However the
situation in $\eta$-production is different from the $\pi N$ elastic
scattering. Here the contribution from $P_{11}(1710)$ is proportional to the
product $g_{\pi NN^{*}(1710)}g_{\eta NN^{*}(1710)}$, where $g_{\pi
NN^{*}(1710)}$ is the coupling constant at the $N(1710)\to\eta N$ transition
vertex. It follows that the contribution from the $P_{11}(1710)$ can be
significant provided that $g_{\eta NN^{*}(1710)}$ is large enough. The
interplay with background and coupled-channel rescattering would further
increase this effect.

###  IV.1 $\pi N\to\eta N$

The results of our calculations are presented in Fig. (3) in comparison with
the world data. The first peak at W=1.54 GeV is related to the well
established $S_{11}(1535)$ resonance contribution.

Figure 3: (Color online) Calculated differential $\pi^{-}p\to\eta n$ cross
section in comparison with the experimental data from: Prakhov 2005:Prakhov ,
Deinet 1969:Deinet:1969cd , Richards 1970:Richards:1970cy , Morrison
2000:Morrison:2000kx .

Though the effect from the $S_{11}(1650)$ state is hardly visible in the
differential cross section this state plays an important role leading to the
destructive interference between $S_{11}(1535)$ and $S_{11}(1650)$ as it has
been pointed out in our previous calculations Penner:2002a ; Penner:2002b .

The second rise is due to the $P_{11}(1710)$ resonance. This state has a small
branching ratio to the $\pi N$ system but due to the large $\eta$-coupling
this resonance affects the production cross section at W=1.7 GeV. The coupled-
channel effects and interference with other partial waves further enlarge the
overall contribution from this state.

The total partial wave cross sections are shown in Fig. 4. The destructive
interference between the $S_{11}(1535)$ and $S_{11}(1650)$ leads to the dip in
the total $S_{11}$-partial wave cross section around W=1.64 GeV (dotted line).
The effect from the $P_{11}(1710)$ state is shown by the dashed line, Fig. 4.
The contributions from other partial waves are found to be small. We also
corroborate our previous results shklyar:2004a  where only minor contributions
from spin $J\geq\frac{3}{2}$ resonance states were obtained. Both t-channel
$a_{0}$ and $\rho$ meson exchange and $u-$channel graphs give small effects.
The inclusion of the higher spin state $D_{13}(1520)$ into the calculations is
still important to reproduce the correct shape of the cross section. This
feature is also found in many other calculations, e.g.Penner:2002a ;
Batinic:1995 . It is interesting to note that importance of the $P_{11}(1710)$
resonance contribution has recently been found in Shrestha:2012va  which is in
line with our previous results Shklyar:2006xw ; Penner:2002b .

Since the main contributions in our calculations come mainly from the $S_{11}$
and $P_{11}$ partial waves it is interesting to trace back the interference
effect between them. Neglecting the higher partial waves the differential
cross section can be written in the form

$\displaystyle\frac{d\sigma}{d\cos\,(\theta)}~{}\sim~{}1+\alpha\,\sin^{2}\left(\frac{\theta}{2}\right),$
(10)

where $\theta$ is a scattering angle and
$\alpha=\left(\frac{|S_{11}-P_{11}|^{2}}{|S_{11}+P_{11}|^{2}}-1\right)$ only
depends on the c.m. energy. Then the angular distribution should have a
maximum (minimum) at forward angles depending on the relative phase between
the nonvanishing $S_{11}$ and $P_{11}$ amplitudes. In our calculation the
interference between $S_{11}$ and $P_{11}$ partial waves produces a peak at
forward scattering angles and energies above W=1.67 GeV, see Fig. (3). As a
result the signal from the $P_{11}(1710)$ resonance becomes more transparent
for forward scattering. This is in line with the data of Richards et al
Richards:1970cy  confirming our guess about the production mechanism. The
inclusion of higher partial waves would modify Eq. (10). However these
contributions are relatively small (see Fig.(4)) thus producing only minor
deviations from the distribution Eq. (10).

Note, that due to numerous problems with the experimental data our
calculations above W=1.6 GeV are only partly constrained by experiment.
Indeed, once the data Baker:1979aw ; Brown:1979ii  are neglected there are
only 30 datapoints from experiment Richards:1970cy . This data has relatively
large error bars and seems not to be fully consistent with other measurements
Clajus:1992dh . Therefore, the results for the differential cross section
might be regarded as a prediction rather than an outcome of the fit. This
demonstrates an urgent need for new measurements of the $\pi^{-}N\to\eta N$
reactions above W=1.6 GeV. This would be a challenge for the the upcoming
pion-beams experiment carried out by the HADES collaboration at GSI.

Figure 4: (Color online) Total partial wave cross section $\pi^{-}p\to\eta n$
vs. experimental data.

###  IV.2 $\eta N\to\eta N$ amplitude and $\eta N$ scattering lengths

The result for the $\eta N\to\eta N$ transition amplitude in the $S_{11}$
partial wave is presented in Fig. 5. Close to threshold the elastic $\eta N$
scattering is completely determined by the contribution of the $S_{11}(1535)$
resonance. At higher energies the excitation of $S_{11}(1650)$ also becomes
important. The interference between those two $S_{11}$-states produces an
excess structure in the imaginary part of the amplitude at W=1.65 GeV.

The rapid variation of the $S_{11}$-amplitude close to threshold indicates
that this energy dependence should be taken into account when the $\eta N$
scattering length is calculated. Here we use the definition for the effective
range expansion from PhysRevC.55.R2167 :

$\displaystyle\frac{q_{\rm c.m.}}{S_{11}^{\,\eta N}}+iq_{\rm
c.m.}=\frac{1}{a_{\eta N}}+\frac{r_{0}}{2}q_{\rm c.m.}^{2}+s\,q^{4}_{\rm
c.m.},$ (11)

where $S_{11}^{\,\eta N}$ is an elastic partial S-wave amplitude, and $a_{\eta
N}$, $r_{0}$ and $s$ are scattering length, effective range, and effective
volume respectively. The results are shown in Table 6 in comparison with
values deduced from other coupled-channel calculations ( results published
before 1997 are discussed in PhysRevC.55.R2167  ). The obtained value of
$a_{\eta N}$ is very close to our previous results Penner:2002a . The values
for the real part deduced in Batinic:1996me  and PhysRevC.55.R2167  are lower
than in this work. The study Batinic:1996me  gives 1.550 GeV for the mass and
204 MeV for the width of the $S_{11}(1535)$ state which are somewhat greater
than in the present calculation. This could be one of the reasons for the
differences in $Re\,a_{\eta N}$.

In PhysRevC.55.R2167  only the $S_{11}(1535)$ state is taken into account to
calculate transition amplitudes to the $\eta N$ channel. Since the parameters
of $S_{11}(1535)$ in PhysRevC.55.R2167  are close to the values obtained in
the present study the observed difference in ${\rm Re}\,a_{\eta N}$ might be
attributed to the different treatment of background contributions which have
been assumed in PhysRevC.55.R2167  to be energy-independent. The second piece
of uncertainty is related to the quality of the world data of $\pi N\to\eta N$
scattering. Hence, precise measurements of this reaction would provide an
additional constraint on $\eta N$ scattering length.

The non-vanishing imaginary part of $a_{\eta N}$ is mostly driven by
rescattering in the $\pi N$ channel. Since the largest contributions to the
scattering length are produced by the $S_{11}(1535)$ state the imaginary part
of $a_{\eta N}$ is strongly influenced by the decay branching ratio of this
resonance to $\pi N$. Only a minor effect is found from the rescattering
induced by background contributions and inelastic flux to the $2\pi N$
channel. Since the $\pi NN^{*}(1535)$ coupling is well fixed an agreement in
${\rm Im}(a_{\eta N})$ between various model calculations can be expected
provided that unitarity is maintained.

The obtained value of the scattering length should be taken with care when in-
medium properties of the $\eta$-meson are considered. As it has already been
pointed out in PhysRevC.55.R2167  the $S_{11}$ amplitude has a strong energy
dependence - a feature which might affect the $\eta$-potential. The second
reason is that properties of the $S_{11}(1535)$ resonance might also be
subjected to in-medium modifications Lehr:2003km . Both effects should be
taken into account when $\eta$-meson properties in nuclei are studied.

Figure 5: Calculated $S_{11}$ partial wave amplitude of the elastic $\eta N$ scattering.  Reference |  $a_{\eta N}$(fm) |  $r_{0}$(fm)
---|---|---
present work | 0.99$\pm$0.08 + i0.25$\pm$0.06 | -1.98$\pm$0.1 - i0.43$\pm$0.15
Penner:2002a  | 0.99 + i0.34 | -2.08 - i0.81
Batinic:1996me  | 0.734$\pm$0.026 + i0.269$\pm 0.019$ |
PhysRevC.55.R2167  | 0.75$\pm 0.04$ \+ i0.27$\pm 0.03$ | -1.5$\pm$0.13 - i0.24$\pm$0.04
Lutz:2001  | 0.43+ i0.21 |
Table 6:  Calculated scattering length and effective range in comparison with
results from other works.

###  IV.3 $\gamma N\to\eta N$ below 1.89 GeV

The results of our calculation of the differential cross section in comparison
with the recent Crystal Ball/MAMI measurements are shown in Fig. (6). Our
calculations demonstrate a nice agreement with the experimental data in the
whole kinematical region. The first peak is related to the $S_{11}(1535)$
resonance contribution. Similar to the $\pi^{-}p\to\eta n$ reaction the
$S_{11}(1650)$ and $S_{11}(1650)$ states interfere destructively producing a
dip around W=1.68 GeV. Though the effect from the $P_{11}(1710)$ state is only
minor, the contribution from this resonance produces a rapid change in the
$M_{1-}$ photoproduction multipole, see Section IV.5. The coherent sum of all
partial waves leads to the more pronounced effect from the dip at forward
angles. Note that the resonance contribution to the photoproduction reaction
stems from two sources: the first is related to the direct electromagnetic
excitation of the nucleon resonance and the second comes from rescattering
e.g. $\gamma p\to\pi N\to\eta N$, Eq. (4). At this stage the hadronic
transition amplitudes e.g. $T_{\pi N\to\eta N}$ become an important part of
the production mechanism. The sum of these contributions in the $P_{11}$ wave
turns out to be destructive which reduces the overall contribution from the
$P_{11}(1710)$ state. We also corroborate our previous findings Shklyar:2006xw
where a small effect from the $\omega N$ threshold was found.

Figure 6:  Differential $\eta p$ cross section vs. recent MAMI data
McNicoll:2010qk .

We also do not find any strong indication for contributions from a hypothetic
narrow $P_{11}$ state with a width of 15-20 MeV around W=1.68 GeV. It is
natural to assume that the contribution from this state would induce a strong
modification of the beam asymmetry for energies close to the mass of this
state. This is because the beam asymmetry is less sensitive to the absolute
magnitude of the various partial wave contributions but strongly affected by
the relative phases between different partial waves. Thus even a small
admixture of a contribution from a narrow state might result into a strong
modification of the beam asymmetry in the energy region of W=1.68 GeV.

Figure 7: (Color online) $\gamma p\to\eta p$ partial wave cross sections vs.
measurements Crede:2003ax ; Crede:2009zzb ; Bartholomy:2007zz .

In Fig. (8) we show the calculation of the photon-beam asymmetry in comparison
with the GRAAL measurements Bartalini:2007fg . One can see that even close to
the $\eta N$ threshold where our calculations exhibit a dominant $S_{11}$
production mechanism (see Fig. (7) ) the beam asymmetry is nonvanishing for
angles $\cos(\theta)\geq-0.2$. This shows that this observable is very
sensitive to very small contributions from higher partial waves. At W=1.68 GeV
and forward angles the GRAAL measurements show a rapid change of the asymmetry
behavior. We explain this effect by a destructive interference between the
$S_{11}(1535)$ and $S_{11}(1650)$ resonances which induces the dip at W=1.68
GeV in the $S_{11}$ partial wave. The strong drop in the $S_{11}$ partial wave
modifies the interference between $S_{11}$ and other partial waves and changes
the asymmetry behavior. Note that the interference between $S_{11}(1535)$ and
$S_{11}(1650)$ and the interference between different partial waves are of
different nature. The overlapping of the $S_{11}(1535)$ and $S_{11}(1650)$
resonances does not simply mean a coherent sum of two independent
contributions, but also includes rescattering (coupled-channel effects). Such
interplay is hard to simulate by the simple sum of two Breit-Wigner forms
since it does not take into account rescattering due to the coupled-channel
treatment.

The GRAAL collaboration finds no evidence for a narrow state around W=1.68
GeV. We also find no strong need for the narrow $P_{11}$ resonance
contribution to describe the asymmetry data. Taking contributions from the
established states into account our results are in close agreement with the
experimental data Bartalini:2007fg .

Figure 8: Calculated beam asymmetry. Experimental data are taken from
Bartalini:2007fg (GRAAL07).

###  IV.4 $\gamma N\to\eta N$ above 1.89 GeV

Figure 9: (Color online) Differential $\eta p$ cross section as a function of
the scattering angle. The data are taken from CLAS 2009:Williams:2009yj  and
CB-ELSA:Crede:2009zzb .  Figure 10: (Color online) Differential $\eta p$ cross
section as a function of c.m. energy at fixed forward angles. Data are taken
from CLAS 2009:Williams:2009yj , CB-ELSA:Crede:2009zzb , and
MAMI2010:McNicoll:2010qk .

Since the MAMI measurements are available up to W=1.89 GeV the calculations in
the region W=1.89 …2.GeV are constrained by the combined data set constructed
out of the recent CLAS and CB-ELSA/TAPS Williams:2009yj ; Crede:2009zzb  data.
Due to some inconsistencies between these two experiments Dey:2011rh ;
Sibirtsev:2010yj  we did not try to fit the data above W=2.GeV but instead
extrapolate our calculation into the higher energies. In this region the
$t$-channel exchange starts to play a dominant role. One of the accepted
prescriptions is to use a Reggeized $t$-channel meson exchange as suggested in
Chiang:2002vq . We do not follow this procedure here but include all
$t$-channel exchanges into the interaction kernel. This allows for a
consistent unitary treatment of resonance and background contributions. The
calculated differential cross section is presented in Fig. (9) as a function
of the scattering angle. Except for the energy bin W = 2.097 GeV our results
are found to be in close agreement with the CLAS measurements. The major
contribution to the differential cross section at forward angles comes from
$\rho$\- and $\omega$-meson exchanges. The effect from the $\phi$-meson is
small due to the weakness of the $\phi NN$ coupling as dictated by the OZI
rule Okubo:1963fa ; Zweig:1964jf ; Iizuka:1966fk . We also checked for the
contributions from the Primakoff effect which is found to be negligible at
these energies.

It is interesting to compare our calculations with the data Williams:2009yj ;
Crede:2009zzb  at forward angles plotted as a function of the c.m. energy, see
Fig. (10). The cusp due to the $\omega N$ production threshold is clearly seen
in our calculations around W=1.72 GeV. The quality of the data is still not
good enough to unambiguously resolve the cusp induced by the $\omega N$
threshold in the experimental data. Note, that the calculations are done
assuming a stable $\omega$-meson. Taking into account the final $\omega$-width
would smear out this effect. Since the $\omega N$ threshold lies 45 MeV above
the dip position ( W=1.68 GeV) we conclude that this effect cannot explain the
dip in the differential cross section. This conclusion is opposite to that
drawn in Anisovich:2011ka .

The discrepancy between the CLAS Williams:2009yj  and CB-ELSA/TAPS data is
better seen at $\cos(\theta)=0.75$ whereas for $\cos(\theta)=0.85$ the
measurements are found to be in better agreement. One of the interesting
features observed in the recent CB-ELSA data is a sudden rise of the
differential cross section at W=1.92 GeV. The effect is more pronounced at
$\cos(\theta)=0.85...0.95$ and is absent at other scattering angles. This
phenomena might be attributed to sidefeeding from of one of the inelastic
channels (e.g. $\phi N$, $a_{0}(980)N$, $f_{0}(980)$, or $\eta^{\prime}N$).
However the problem with normalization inconsistencies between the CLAS and
CB-ELSA data should be solved first before any physical interpretation can be
given.

###  IV.5 eta-photoproduction multipoles

The extracted $\gamma p\to\eta p$ multipoles are presented in Fig. 11. The
major contribution to the $E_{0+}$ multipole comes from the $S_{11}(1535)$
resonance. The second $S_{11}(1650)$ plays an important role in the region
W=1.6…1.7 GeV. We corroborate our previous results Shklyar:2006xw  where only
a small effect from the spin-$\frac{5}{2}$ states has been found. A very small
signal from the $F_{15}(1680)$ resonance is seen in the $E_{3-}$ and $M_{3-}$
amplitudes at W=1.68 GeV.

It is interesting to note that the effect of $D_{13}(1520)$ is clearly seen in
the $E_{2-}$ and $M_{2-}$ though the overall contribution from this state
turns out to be small. The $M_{1-}$ multipole is affected by the Roper and
$P_{11}(1710)$ resonances leading to the rapid change in both real and
imaginary parts of the amplitude at W=1.7 GeV. In the region W=1.48…1.6 GeV
both the imaginary and the real parts of all multipoles with $l\neq 0$ are of
the order of magnitude smaller than $E_{0+}$ due to the strong dominant
contribution from $S_{11}(1535)$. However for higher energies the influence of
amplitudes with $l\neq 0$ becomes also important.

Figure 11: (Color online) $\gamma p\to\eta p$ multipoles extracted in the
present study.

##  V Conclusion

We have performed a coupled-channel analysis of pion- and photon-induced
reactions including the recent eta-photoproduction data from the Crystal
Ball/MAMI collaboration. In the region W=1.89…2.0 GeV our solution is
constrained by the combined dataset built from the recent CLAS and CB-
ELSA/TAPS measurements. The dip in the differential cross-sections at W=1.68
GeV reported in McNicoll:2010qk  is described in terms of an interference of
the $S_{11}(1535)$ and $S_{11}(1650)$ states. We stress that such an
interference also includes coupled-channel effects and rescattering which is
hard to simulate by a simple sum of two Breit-Wigner contributions. The
additional contribution at W=1.68 GeV comes from the $M_{1-}$ multipole where
the excitation of the $P_{11}(1710)$ leads to a rapid change of the real and
imaginary parts of the amplitude. We conclude that the cusp due to the $\omega
N$ threshold seen at 1.72 GeV is not important for the explanation of the dip
at W=1.68 GeV. However the quality of the data is still not sufficient to
resolve the threshold effect completely.

Above W=1.9 GeV the $t$-channel $\rho$\- and $\omega$-exchanges start to play
a dominant role in the calculations. The effect from the $\phi$-meson exchange
is less important because of the smallness of the $\phi NN$ coupling. We have
also checked for the contribution from the Primakoff-effect which is found to
be negligible. In the region W=1.9…2.2 GeV our calculations tend to be in
closer agreement with the CLAS data.

It is interesting to note that above W=1.92 GeV the cross sections of the CB-
ELSA/TAPS collaboration indicate a sudden rise from 0.2 $\mu$b up to 0.3
$\mu$b. The effect is observed only for scattering angles
$\cos(\theta)=0.85...0.95$. This phenomenon might be attributed to sidefeeding
from of one of the inelastic channels (e.g. $\phi N$, $a_{0}(980)N$,
$f_{0}(980)$, or $\eta^{\prime}N$). However the origin of the normalization
discrepancies between the CLAS and CB-ELSA/TAPS data should first be
understood before any physical interpretation can be given.

In the $\pi^{-}p\to\eta n$ reaction the main effect comes from three
resonances $S_{11}(1535)$, $S_{11}(1650)$, and $P_{11}(1710)$. Similar to eta-
photoproduction on the proton the overlap of the $S_{11}(1535)$ and
$S_{11}(1650)$ states produces a dip around W=1.68 GeV. For energies $W>1.68$
GeV the contribution from $P_{11}(1710)$ is found to be important. The above
reaction mechanism for the $(\gamma/\pi)N\to\eta N$ reaction is in line with
our early findings Shklyar:2006xw  where the resonance like-structure in
$\eta$-photoproduction at W=1.68 GeV on the neutron was explained by the
excitations of the $S_{11}(1650)$, and $P_{11}(1710)$ resonances.

We conclude that further progress in understanding of $\eta$-meson production
would be hardly possible without new measurements of the $\pi N\to\eta N$
reaction. The experimental investigation of this reaction would help to
establish the resonance contributions to the $\eta$-photoproduction above
$W>1.6$ GeV. Finally, the study of the $\eta N$-channel with pion beams would
solve the question whether the observed phenomena in $\eta$ photoproduction
have their counterparts in $\pi N\to\eta N$ scattering.

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