# Application of the single-channel, single-energy AA/PWA method to
$K^{+}\Lambda$ photoproduction

A. Švarc<EMAIL_ADDRESS>Rudjer Bošković Institute, Bijenička cesta 54, P.O. Box
180, 10002 Zagreb, Croatia,Tesla Biotech d.o.o., Mandlova 7, 10000 Zagreb,
Croatia  Y. Wunderlich  Helmholtz-Institut für Strahlen- und Kernphysik,
Universität Bonn, D-53115 Bonn, Germany, Germany  L. Tiator  Institut für
Kernphysik, Universität Mainz, D-55099 Mainz, Germany

###### Abstract

The new single-channel, single-energy partial wave analysis method based on a
simultaneous use of amplitude and partial wave analysis called AA/PWA,
developed and tested on $\eta$ photoproduction in ref. [1], is applied to the
$K^{+}\Lambda$ photoproduction for the center-of-mass energy range of 1625 MeV
$<W<$ 2296 MeV. A complete set of multipoles has been created. The advantages
of the method have been confirmed, and a comparison with the only existing
single-energy partial wave analysis of $K^{+}\Lambda$ photoproduction given in
refs. [2, 3] is presented. We confirm the size and shape of Bonn-Gatchina
multipoles, but we do not confirm the unambiguous interpretation of the
structure in the $M_{1-}$ multipole as a $N(1880)\frac{1}{2}^{+}$ resonance.
The decisive role of the self-consistency of the world database is emphasized.

##  I Introduction

Single-channel, singe-energy partial wave analysis (SC-SE-PWA) has always been
of particular interest primarily for experimentalists, but also for theorists.
For experimentalists it seemed to be the most direct way to convert measured
data to physically interpretable partial waves, and for theorists it seemed to
be the most direct way to test the validity of their theoretical approach.
Many attempts have been made to prove the uniqueness of SC-SE-PWA [4] even in
the case of a single elastic channel, and they culminated with research by Ion
Sabba Stefanescu, who has formulated necessary conditions for the uniqueness
of SC-SE-PWA in the elastic domain111The proof of uniqueness requires the
multivariate analyticity of the amplitude as a function of two Mandelstam
variables. [5]. However, as the search for nucleon resonances basically takes
place in the inelastic region, the continuum-ambiguity problem became of utter
importance [6]. This discussion has started a long time ago, but has never
been completely finished. Recently, this problem was reopened by our group,
and culminated with the conclusion that each single-channel PWA in the
inelastic region is inherently model dependent, as it depends on the free
energy- and angle-dependent continuum ambiguity phase which leaves all
observables invariant, but the angular-dependent part of the ambiguity mixes
partial waves. Hence this makes the quantum numbers of resonances
unidentifiable without additional information [7]. Since the full information
on the phase can be obtained only from all possible inelastic channels, it
remains undetermined in single-channel models, and the only way to make a
single-channel PWA unique is to fix this phase to some known value. Here we
have to distinguish the following two cases: single-channel energy-dependent
PWA (SC-ED-PWA) and single-channel energy-independent PWA (SC-SE-PWA). In SC-
ED-PWA the phase is automatically determined by the analyticity of the
continuous ED model, and in SC-SE-PWA it is absolutely free, so we have to
take it over from some theoretical calculation. Observe that the common
denominator of both cases is that the analysis has to be extended to multiple
channels, as fixing the continuum-ambiguity phase is only possible by
restoring multi-channel unitarity, and that can only be done in analyzing all
available channels for this reaction. So, in both cases, we have to use
coupled-channels models, but even there the missing phase can only be at least
approximately determined as all possible, open two-body channels are never
known. The problem totally collapses when three-body channels are involved, so
we have to face and live with the fact that each single-channel PWA is
inherently model dependent. This automatically means that SC-SE-PWA is also
inherently model dependent. So, when we compare different PWAs, we have to
match the reaction-amplitude phases first.

Constraining SC-SE-PWA has for decades been done by either fixing some partial
waves to values from some theoretical model, or penalizing some or all partial
waves to the particular constraining theoretical model. This is the
traditional way. A strong step forward was done by the Karlsruhe-Helsinki
group in the 1980s [8] when the problem was raised to the level of reaction
amplitudes where phase ambiguities appear. Following the work of Ion
Stefanescu [5] on the importance of analyticity in two Mandelstam variables,
fixed-$t$ analyticity was introduced in $\pi N$ elastic scattering.
Analyticity in $t$ was achieved using the manifestly analytic Pietarinen
decomposition of invariant amplitudes and fitting the free parameters of the
decomposition to the data base transformed from [Wfixed, $\theta$] coordinates
into [$t_{\rm fixed}$, W] coordinates222Observables are traditionally analyzed
in [Wfixed, $\theta$] space., while analyticity in Mandelstam $s$ is enforced
by using the traditional partial wave decomposition. The free continuum-
ambiguity phase was predetermined by the choice of starting values in the
$t$-variable minimization. As the reaction-amplitude phase is fairly well
known for the elastic scattering, this method resulted in the KH80- and KH84
solutions for elastic $\pi N$ partial waves [8], which have been accepted and
used for decades. The same method was recently revived and applied with great
success to pion photoproduction [9, 10]. Luckily, in pion photoproduction this
phase is for sufficiently low energies linked to the well-determined phase of
elastic $\pi N$ scattering due to Watson’s theorem [11]. Therefore, in refs.
[9, 10] the authors also do not face the problem of the unknown phase.
However, for all other reactions where Watson’s theorem breaks down (like
$\eta$ or $K\Lambda$ photoproduction) this is not true, so the phase stays
poorly determined, and SC-SE-PWA stays model dependent. Triggered by the fact
that the continuum-ambiguity phase in SC method in the inelastic domain has to
be constrained in particular for reactions where Watson’s theorem breaks down,
a new method based on only one, i.e. the Mandelstam $s$ variable has been
developed in ref. [1]. However, in this model we openly acknowledge the
problem that the continuum-ambiguity phase is unknown, and from the very start
constrain it to a phase of some chosen theoretical coupled-channels model. The
rest is very similar to the fixed-$t$ analyticity procedure, but restricted to
the $s$ channel only. The procedure is a two-step process applied to the same
database: the first step consists of an amplitude analysis of the database
where the moduli of the reaction amplitudes are fitted while the reaction-
amplitude phases are fixed to the values of a particular ED coupled-channels
model (this eliminates the continuum ambiguity); the second step is a standard
truncated partial wave analysis (TPWA), where the reaction amplitudes are
forced to be close to the reaction amplitudes of the first step using
penalization techniques. In this way the continuity in energy is ensured
through the continuous phase, and continuity in angle is ensured by the TPWA.
Let us observe that the proposed method relies on using minimal theory
dependence, which is given by fixing the phase only. Unfortunately, this works
perfectly well only for the ideal case when all observables are self-
consistent. This has been shown in ref. [12] for a complete set of pseudo-
observables (numeric data) for $\eta$ photoproduction generated by the ETA-
MAID model [13] which are by definition self-consistent. The free phase in
unconstrained SC-SE-PWA is replaced by the original ETA-MAID phase, and the
continuous generating multipoles were exactly reproduced. Unfortunately, real
data are never self-consistent, so when the method is applied to real data
some discontinuities in partial waves might appear. Therefore, all scatter of
the result is the consequence of experimental errors and data inconsistencies.
The method has been developed and tested on a world database of $\eta$
photoproduction, and presented in ref. [1]. As the results for $\eta$
photoproduction were stable [1], and the pole content of the obtained solution
looked very reasonable [7], in this paper we used this method for the
$K\Lambda$ photoproduction reaction where the isospin structure is identical.
However, the main advantage of $K\Lambda$ photoproduction is that we have
access to results from complementary analyses, to which our results can be
compared. First of all, there exists a very confident theoretical coupled-
channels ED model by the Bonn-Gatchina group to start with [14]. Second,
results of SC-SE-PWA made by the same group are published [2, 3], so we have
direct numbers to compare our results with. The situation is even more
favorable. The Bonn-Gatchina SC-SE-PWA was done in the standard way: only
lower partial waves were left free, while all higher partial waves were fixed
to the Bonn-Gatchina ED model, while we offer the simultaneous variation of
all multipoles within the framework of AA/PWA method. So, results and
advantages/disadvantages of both approaches can be directly compared. Third,
but not the least important is that we had access to a full $K\Lambda$
photoproduction database in numeric form from the Bonn-Gatchina web page [14].
So, as all input can be made identical, the benefits of the new approach could
be clearly detected. We discuss similarities and differences, and point out to
the reasons why this is so.

##  II The AA/PWA method

In the AA/PWA method, from the very start we openly accept the fact that the
overall continuum-ambiguity phase of the reaction amplitudes in any inelastic
SC analysis is by default undetermined as it depends on other channels [7],
and we fix it to a phase of some chosen theoretical coupled-channels model.
However, let us stress that the way it is done in our paper is only an
approximation. The amplitude phases contain two parts: the first part are
relative phases which are determined by single- and double-polarization
observables, and can be uniquely determined in the SC model333Let us remember
that in pseudoscalar meson photoproduction, the four observables
$d\sigma/d\Omega$, $\Sigma$, $T$ and $P$ determine the absolute values of the
four transversity amplitudes(see Table 4 in appendix A), and the remaining
four observables in a complete set determine the three remaining relative
phases., and the second part is the continuum-ambiguity phase which is unknown
in any SC approach [1]. In principle, we should only fix the unknown continuum
ambiguity phase to a theoretical model. However, as the separation of each
reaction-amplitudes phase in relative- and continuum ambiguity part is
unknown, we opted to fully fix the phases of all 4 reaction amplitudes. So, we
expect that the model we use will be fairly good, and fit also the non-
measured spin observables. The rest is very similar to procedures implementing
fixed-$t$ analyticity [8, 15, 9, 10], but restricted to the $s$ channel only.
The procedure is a two-step process applied to the same data base: the first
step is an amplitude analysis of the database, where the moduli of reaction
amplitudes are fitted while the reaction-amplitude phases are fixed to the
values of a particular ED coupled-channels model (this eliminates the
continuum ambiguity); and the second step is a standard truncated partial wave
analysis (TPWA) where the reaction amplitudes are forced to be close to the
reaction amplitudes resulting from the first step, using a penalization
technique. In this way the continuity in energy is ensured through the
continuous phase, and continuity in angle is ensured via the TPWA. All scatter
of the result is, hence, the consequence of experimental errors and data
inconsistencies, as in ref. [12] it has been shown that fixing only the phase
results in a smooth and unique solution in the case where a self-consistent
database has been generated in the form of pseudo observables from a known
model. The AA/PWA method has been developed and tested on a world database for
$\eta$ photoproduction, and in details presented in ref. [1].

For the convenience of the reader, we summarize the essence of the method
mostly relying on the text in ref. [7].

In ref. [1] we have formulated a single-channel, single-energy partial wave
analysis (SC-SE-PWA) procedure of determining reaction amplitude via fitting
scattering data when the number of equations may be less than number of
unknown quantities, which combines amplitude- and partial wave analyses into
one logical sequence, and directly from the data generates a set of continuous
partial waves using a minimally model-dependent input (AA/PWA). We have
demonstrated that by controlling the reaction-amplitude phase, and freely
varying the reaction-amplitude partial waves, we obtain a continuous solution
with far better agreement with the used data base than the original energy
dependent (ED) model.

The most standard, classic approach is the one where one penalizes partial
waves by requiring that fitted partial waves reproduce the observable $\cal O$
and are at the same time close to some partial waves taken from a theoretical
model:

$\displaystyle\chi^{2}(W)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{data}}w^{i}\left[{\cal
O}^{exp}_{i}(W,\Theta_{i})-{\cal O}^{th}_{i}({\cal
M}^{fit}(W),\Theta_{i})\right]^{2}+\lambda_{pen}\sum_{j=1}^{N_{mult}}\left|{\cal
M}^{fit}_{j}(W)-{\cal M}^{th}_{j}(W)\right|^{2}$ (1)

where

$\displaystyle{\cal M}$ $\displaystyle\stackrel{{\scriptstyle def}}{{=}}$
$\displaystyle\left\\{{\cal M}_{0},{\cal M}_{1},{\cal M}_{2},...,{\cal
M}_{N_{mult}}\right\\}$

$w_{i}$ is the statistical weight and $N_{mult}$ is the number of partial
waves (multipoles), ${\cal M}^{fit}$ are fitting parameters and ${\cal
M}^{th}$ are continuous functions taken from a particular theoretical model
(For a detailed outline of how the observables ${\cal O}_{i}$ are composed in
terms of multipoles for pseudoscalar meson photoproduction, see appendix A.).
Instead, we use the possibility to make the penalization function independent
of a particular model as first formulated in the Karlsruhe-Helsinki $\pi N$
elastic PWA by G. Höhler in the mid-80es [8]. Partial waves which are
inherently model dependent are replaced with a penalization function which was
constructed from reaction amplitudes which can be in principle directly linked
to experimental data without any model in the amplitude-reconstruction
procedure. So, the equation (1) was changed to:

$\displaystyle\chi^{2}(W)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N_{data}}w^{i}\left[{\cal
O}^{exp}_{i}(W,\Theta_{i})-{\cal O}^{th}_{i}({\cal
M}^{fit}(W),\Theta_{i})\right]^{2}+{\cal P}(W)$ (2) $\displaystyle{\cal P}(W)$
$\displaystyle=$
$\displaystyle\lambda_{pen}\sum_{i=1}^{N_{data}}\sum_{k=1}^{N_{amp}}\left|{\cal
A}_{k}({\cal M}^{fit}(W),\Theta_{i})-{\cal
A}_{k}^{pen}(W,\Theta_{i})\right|^{2}\,,$

where ${\cal A}_{k}$ is the generic name for any of reaction amplitudes
(invariant, helicity, transversity $\ldots$). ${\cal A}_{k}({\cal
M}^{fit}(W),\Theta_{i})$ is the reaction-amplitude value generated by the
fitted multipoles, and ${\cal A}_{k}^{pen}(W,\Theta_{i})$ is the penalizing
function coming from the amplitude analysis. However, one is now facing two
challenges: to get reaction amplitudes which fit the data, and also to make
them continuous. In the Karlsruhe-Helsinki case [8, 15, 9, 10], this was
accomplished by implementing fixed-$t$ analyticity and fitting the data base
for fixed $t$. So, the first step of the KH fixed-$t$ approach was to create
the data base ${\cal O}(W)|_{t=fixed}$ using the measured base ${\cal
O}(\cos\,\theta)|_{W=fixed}$, and then to fit them with a manifestly analytic
representation of the reaction amplitudes for a fixed $t$. Manifest
analyticity was implemented by using the Pietarinen decomposition of reaction
amplitudes. Then the second step was to perform a penalized PWA defined by Eq.
(2) in the fixed-$W$ channel where the penalizing factor ${\cal A}_{k}({\cal
M}^{pen}(W,\Theta_{i}))$ was obtained in the first step in a fixed-$t$
channel. In that way a stabilized SE PWA was performed. This approach was
revived recently for SE PWA of $\eta p$ and $\pi^{0}p$ photoproduction and
very recently also for pion photoproduction in full isospin by the Mainz-
Tuzla-Zagreb collaboration, and analyzed in details in refs. [15, 9, 10].

We propose an alternative.

We also use Eq. (2), but the penalizing factor ${\cal P}(W)$ is generated by
an amplitude analysis performed in the same fixed-$W$ representation, and not
in the fixed-$t$ one. The phase is in our approach openly recognized as
undeterminable, and taken over from the chosen coupled-channels ED model. This
simplifies the procedure significantly, and avoids quite some theoretical
assumptions on the behavior in the fixed-$t$ representation.

We also propose a 2-step process as in ref. [8, 15, 7]:

  * _Step 1:_

Amplitude analysis of experimental data in fixed-$W$ system to generate
penalizing factor ${\cal P}(W)$

  * _Step 2:_

Penalized PWA using Eq. (2) with the penalization factor from _Step 1_.

And now we are bound to say something about the importance of the reaction-
amplitudes phase. The continuum ambiguity forbids to conclude onto the correct
phase in any single-channel analysis because the loss of probability flux to
other channels starts after the first inelastic threshold opens. The only way
to solve the continuum ambiguity problem is to reintroduce the unitarity
constraint in the context of a coupled-channels formalism. If we pick the
phase in a single-channel analysis arbitrarily by hand, we are departing from
the genuine ’true’ phase, the phase in which partial waves do not mix, and we
introduce a pole-shift from one partial wave into another via the angular
dependent part of the continuum ambiguity (see refs. [12, 16]). However, each
coupled-channels model by construction results in the non-pole-mixing
solution. Namely, some form of interaction introducing poles is formulated,
and the background contribution is included. Then, the data in all channels
are simultaneously fitted forcing the phase to be the correct one, and the
non-pole-mixing situation is established. Background contributions
automatically enforce the phase to be a non-mixing one. It is needless to say
that all coupled-channels models should end up with the same phase in the
ideal case, but incompleteness of the data forbids that to happen. Therefore,
phases of different models [17, 18, 20, 19] are somewhat different, and we
cannot avoid this. However, fixing the phase to the phase of a particular
model ensures to obtain the non-mixing pole solution; departure from it
automatically enforce pole mixing, so the analytic structure of such a
solution is spoiled. So, we can chose a different phase, a phase coming from
any model, but it has to be the proper phase originating from that model. A
free, uncontrolled departure from ED model phase is not allowed.

Of course the question is purely quantitative: How much can we depart from the
’true’ phase in an uncontrolled way to maintain the correct analytic
properties. In other words, the question is how much we are allowed to reduce
the importance of the penalty function and maintain the correct analyticity.

All relevant formulae and more details on the photoproduction formalism are
given in appendix A of this paper.

##  III Application of AA/PWA to $K^{+}\Lambda$ photoproduction data

As the AA/PWA method worked so well on $\eta$ photoproduction, we have decided
to test it on the next natural candidate reaction, and that is $K^{+}\Lambda$
photoproduction. This reaction has the same isospin structure as $\eta$
photoproduction, it has a rich database, so the technical effort involved in
adapting the AA/PWA scheme was minimal. However, there is one big advantage:
we have results from other complementary analyses to compare with. Namely,
four years ago the Bonn-Gatchina group made a classic SC-SE-PWA analysis of
$K^{+}\Lambda$ photoproduction. In ref. [2], the first four multipoles
($E_{0^{+}}$, $M_{1^{-}}$, $E_{1^{+}}$, and $M_{1^{+}}$) were let free, while
all higher multipoles were forced by a penalty function to stay close to the
Bonn-Gatchina ED theoretical coupled-channels model, while in the forthcoming
reference [3] the next three multipoles ($E_{2^{-}}$, $M_{2^{-}}$, and
$M_{2^{+}}$) were released in addition. In this paper we focus on comparing
our results with the results of ref. [2].

However, to do so we have to use the identical data base, and identical Bonn-
Gatchina (BG) ED model constraining partial waves. This turned out not to be a
problem, as the data base is in numerical form given on two very nice web
pages [14, 21]. Unfortunately, choosing BG ED multipoles turned out to be much
more difficult. For some reason, the particular BG ED solution used for both
BG publications [2, 3] is not given on the BG web page, so we obtained these
numbers via private communication [22]. As the reader will see later, this
turned out to be extremely important as this solution was especially tuned to
fit $K^{+}\Lambda$ data, and the absolute normalization of all multipoles is
somewhat different. This is trivially visible for the dominant $E_{0^{+}}$
multipole, where the value of used BG ED solution was notably larger than
either of the solutions BG2014-2 or BG2019 given on their web page.

###  III. A Description of the database

In Table 1 we give our data base which is in numeric form taken over from the
Bonn-Gatchina and George-Washington-University web pages [14, 21]:

Table 1:  Experimental data from CLAS, and GRAAL used in our PWA. Note that
the observables $C_{x}$ and $C_{z}$ are measured in a rotated coordinate frame
[23]. They are related to the standard observables $C_{x^{\prime}}$ and
$C_{z^{\prime}}$ in the $c.m.$ frame by an angular rotation:
$C_{x}=C_{z^{\prime}}\sin(\theta)+C_{x^{\prime}}\cos(\theta)$, and
$C_{z}=C_{z^{\prime}}\cos(\theta)-C_{x^{\prime}}\sin(\theta)$, see ref. [3].
Obs. | $N$ |  $E_{c.m.}$ [MeV] | $N_{E}$ |  $\theta_{cm}$ [deg] | $N_{\theta}$ | Reference
---|---|---|---|---|---|---
$d\sigma/d\Omega\equiv\sigma_{0}$ | $3615$ | $1625-2295$ | $268$ | $28-152$ | $5-19$ | CLAS(2007) [23], CLAS(2010) [24]
$\Sigma$ | $400$ | $1649-2179$ | $34$ | $35-143$ | $6-16$ | GRAAL(2007) [25], CLAS(2016) [26]
$T$ | $408$ | $1645-2179$ | $34$ | $31-142$ | $6-16$ | GRAAL(2007) [25],CLAS(2016) [26]
$P$ | $1597$ | $1625-2295$ | $78$ | $28-143$ | $6-18$ | CLAS(2010) [24], GRAAL(2007) [25]
$O_{x^{\prime}}$ | $415$ | $1645-2179$ | $34$ | $31-143$ | $6-16$ | GRAAL(2007) [25], CLAS(2016) [26]
$O_{z^{\prime}}$ | $415$ | $1645-2179$ | $34$ | $31-143$ | $6-16$ | GRAAL(2007) [25], CLAS(2016) [26]
$C_{x}$ | $138$ | $1678-2296$ | $14$ | $31-139$ | $9$ | CLAS(2007) [23]
$C_{z}$ | $138$ | $1678-2296$ | $14$ | $31-139$ | $9$ | CLAS(2007) [23]

As we see from the Table 1, we have a situation at hand which is very similar
to $\eta$ photoproduction:

  1. 1.

We have eight measured observables at our disposal, and unfortunately,
identically as in $\eta$ photoproduction, this is still not a complete set of
observables (some observables from either the beam-target or the target-recoil
categories are missing). For details, see Appendix B.II. A.

  2. 2.

We have a strong dominance of $d\sigma/d\Omega$ data over all other
observables.

  3. 3.

Only four observables out of eight are given in the full, analyzed energy
range of 1625 MeV $<W_{c.m.}<$ 2296 MeV, and these are $d\sigma/d\Omega$, $P$,
$C_{x}$ and $C_{z}$. The remaining four observables $\Sigma$, $T$,
$O_{x^{\prime}}$, and $O_{z^{\prime}}$ are measured only up to $\approx 2180$
MeV. This might create unwanted discontinuities at this energy444Eight
observables at lower energies might create slightly different multipoles than
only four at energies above $\approx 2180$ MeV. So, the transition may not be
smooth for all multipoles at this energy.. For details see Appendix B.II. B.

Even a superficial glimpse at Table 1 tells us that the measured data are
given at different energies and different angles, so some data re-binning is
in order. Standardly, data binning consists of using the data not at the exact
energy where they were taken, but in the energy interval
$W_{exact}-\Delta/2<W_{exact}<W_{exact}+\Delta/2$ where $\Delta$ is the energy
bin-width. However, in the case of scarce data this procedure might introduce
an unwanted dissipation of data, resulting in possible discontinuities between
energy bins. Therefore, we have adopted and used an altogether different
method. Instead, we rely on a two-dimensional (2D) data interpolation. We
simultaneously interpolate experimental data and their corresponding
experimental error in energy $W$ and angle $\Theta$ using a standard
Mathematica routine [27], and use interpolated values instead of binned ones.
Our interpolation strategy is as follows:

  * •

Energy grid:
The analysis is performed on a collection of energies where at least one
polarization observable apart from the cross section $\sigma_{0}$ was measured
(141 energy points).

  * •

Angular grid:
The analysis is done on the following pre-chosen fixed values of 16 points:
cos($\Theta$) = {-0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0., 0.1, 0.2, 0.3,
0.4, 0.5, 0.6, 0.7, 0.80 } .

So, let us summarize. The AA/PWA method is not done on the realistic, measured
energy and angular values, but at interpolated values of all observables
instead. Instead, some interpolation has to be done since we are doing a
single-energy analysis, i.e. at least energies between analyzed observables
have to match (which they don’t do for the measured data sets). This has some
advantages and possibly some drawbacks. The main advantage is that the number
of analyzed points is increased. In the standard binning method, the number of
analyzed energy and angular points is directly limited by the number of
measured points for the least known observable. So, when we perform the
analysis with the energy-binning technique, we can make an analysis using a
maximum number of observables only on a small number of points, on points
where the least known observable is measured. We can never benefit from the
vast amount of energy- and angular points where all other energies are
measured. On the rest of energies and angles the number of used observables is
smaller, and the uncertainty introduced into the analysis hence grows.
However, when we use interpolation technique, we have all measured observables
at all analyzing points as interpolated values, and the confidence into our
analysis depends on the quality of the interpolation. So, it is of utter
importance to have a confident interpolation for the ”worst” observable where
the separation between measured points is the farthest. In our case the
”worst” observables are $C_{x}$ and $C_{z}$ (see Table 1), and in Fig. 1 we
show the quality of the corresponding interpolations. We are of the opinion
that the interpolation of these two observables is fairly good.

Figure 1:  Quality of the 2D interpolation for $C_{x}$ and $C_{z}$
observables. The red symbols are measured values, and the orange surface shows
the interpolated values. (Color online)

Of course, the natural drawback is that we introduce additional unmeasured
points, so the contribution of poorly measured observables to the overall
$\chi^{2}$ grows. However, this effect of overstressing the statistical
importance of poorly measured observables is present in the binning technique,
but is introduced differently (by increasing the weighting factor for these
observables). So, the idea is similar, but for the interpolation method
implemented in a more confident way.

Our fitting strategy reads as follows:

  * •

The AA step is done on interpolated energies (141 points) and interpolated
angles (16 points).

  * •

The TPWA minimization step for obtaining multipoles is also done on
interpolated energies and pre-chosen angular values. However, the statistical
data analysis to obtain $\chi^{2}$ is done at exact energies and exact number
of angles (energy dependent) for each observable (cf. Table 1).

###  III. B Choosing the energy-dependent constraining model

Our first intention was to take the solution BG2019 from the Bonn-Gatchina web
page [14] as a constraining solution. To our surprise, this solution fits the
polarization observables $\Sigma$, $T$, $O_{x^{\prime}}$ and $O_{z^{\prime}}$
rather poorly at higher energies. We show the discrepancy of that solution
with the above-mentioned four observables at one randomly chosen higher energy
in Fig. 2. However, when the SE-PWA has been performed in refs. [2, 3],
another Bonn-Gatchina ED model was used to constrain the higher partial waves,
and this solution was different from BG2019. We call this solution the BG2017
model. As one of the goals of the present paper is to compare the results of
our AA/PWA method with results from Bonn-Gatchina publications, it is natural
to take the same constraining input, but to be used on the level of reaction-
amplitude phases. As seen in refs. [2, 3] the agreement of the BG2017 model
with polarization observables is very good, and this is very important for
choosing the phase as we fix relative phases in addition to the continuum-
ambiguity phase. Unfortunately, we realized that this solution is never given
anywhere in numbers explicitly, and we got it only via private communication
[22]. As our model requires fairly good agreement with all data, the natural
model of choice for the constraining phase had to be the officially
unpublished BG2017 model, in spite of the fact that it is older, and actually
never published.

Figure 2:  The quality of the BG2017 and BG2019 solutions for the polarization
observables $\Sigma$, $T$, $O_{x^{\prime}}$ and $O_{z^{\prime}}$ is
illustrated at one randomly chosen energy. Red symbols are measured data, cyan
symbols are interpolated data, the dashed blue line is the result of the
BG2019 and BG2017 model respectively, and the red line is the result of our
fit.

###  III. C Results

In Figs. 3 and 4 we show our final results for multipoles on the full set of
energies. The penalty factor $\lambda$ is picked by hand, and is set to
$\lambda=250$. Red symbols give the values of multipoles for our AA/PWA
solution, and the black full line gives the prediction of the BG2017 solution
[14]. In Fig. 5 we give the $\chi^{2}$ per data point for each observable
calculated on measured values of energies and angles as red symbols, and the
same quantity for the ED BG2017 solution [14] as black symbols. In Figs. 6 \-
13 we give the fits to measured observables resulting from our AA/PWA method
(red full line) as well as predictions from the ED BG2017 solution [14]
(dashed blue line) at representative energies only. All further energies are
available upon demand.

Figure 3: (Color online) The multipoles for the $L=0$, $1$ and $2$ partial
waves of our AA/PWA solution are shown. Red discrete symbols correspond to our
solution, and the black full line gives the BG2017 ED solution for comparison.
The thin vertical black line marks the energy where only 4 observables are
measured instead of 8 (cf. Table 1).

Figure 4: (Color online) The multipoles for the $L=2$ and $3$ partial waves of
our AA/PWA solution are shown. Red discrete symbols correspond to our
solution, and the black full line gives the BG2017 ED solution for comparison.
The thin vertical black line marks the energy where only 4 observables are
measured instead of 8 (cf. Table 1).

Figure 5: (Color online) The $\chi^{2}/N_{\text{data}}$ for individual
observables, calculated on measured values of energies and angles, is shown.
Our AA/PWA solution is given with red symbols, and the same quantity evaluated
for the ED BG2017 solution [14] with black symbols.

Figure 6: (Color online) Comparison of experimental data for $\sigma_{0}$
(discrete symbols) with our results from AA/PWA (red full line) and the BG2017
fit (blue dashed line) at representative energies.

Figure 7: (Color online) Comparison of experimental data for $\Sigma$
(discrete symbols) and interpolated values (cyan symbols) with our results
from AA/PWA (red full line) and the BG2017 fit (blue dashed line) at
representative energies. Figure 8: (Color online) Comparison of experimental
data for $T$ (discrete symbols) and interpolated values (cyan symbols) with
our results from AA/PWA (red full line) and the BG2017 fit (blue dashed line)
at representative energies.  Figure 9: (Color online) Comparison of
experimental data for $P$ (discrete symbols) and interpolated values (cyan
symbols) with our results from AA/PWA (red full line) and the BG2017 fit (blue
dashed line) at representative energies.  Figure 10: (Color online) Comparison
of experimental data for $O_{x^{\prime}}$ (discrete symbols) and interpolated
values (cyan symbols) with our results from AA/PWA (red full line) and the
BG2017 fit (blue dashed line) at representative energies. Figure 11: (Color
online) Comparison of experimental data for $O_{z^{\prime}}$ (discrete
symbols) and interpolated values (cyan symbols) with our results from AA/PWA
(red full line) and the BG2017 fit (blue dashed line) at representative
energies. Figure 12: (Color online) Comparison of experimental data for
$C_{x}$ (discrete symbols) and interpolated values (cyan symbols) with our
results from AA/PWA (red full line) and the BG2017 fit (blue dashed line) at
representative energies. Figure 13: (Color online) Comparison of experimental
data for $C_{z}$ (discrete symbols) and interpolated values (cyan symbols)
with our results from AA/PWA (red full line) and the BG2017 fit (blue dashed
line) at representative energies.

We also give predictions resulting from our AA/PWA method for the unmeasured
${\cal BT}$ polarization observables $E$, $F$, $G$, and $H$ at certain,
representative energies in the full energy range.

Figure 14:  Prediction of our AA/PWA method (full red line) and the BG2017 fit
(blue dashed line) for the $E$ spin observable at representative energies in
the full energy range.  Figure 15:  Prediction of our AA/PWA method (full red
line) and the BG2017 fit (blue dashed line) for the $F$ spin observable at
representative energies in the full energy range.  Figure 16:  Prediction of
our AA/PWA method (full red line) and the BG2017 fit (blue dashed line) for
the $G$ spin observable at representative energies in the full energy range.
Figure 17:  Prediction of our AA/PWA method (full red line) and the BG2017 fit
(blue dashed line) for the $H$ spin observable at representative energies in
the full energy range.

We have also given predictions for unmeasured ${\cal BT}$ observables $E$,
$F$, $G$, and $H$. They show good agreement at low energies and larger spread
at the highest energies. This is, however, also already seen in the ${\cal
BR}$ observables that are fitted. Many observables shown in the paper, like
$T$, show sharp structures near 0 and 180 degrees, especially for higher
energies. We consider those structures as natural, since most observables must
vanish at those extreme angles, either as $\sin(\Theta)$ or
$\sin^{2}(\Theta)$. And at higher energies, when lots of multipoles can
contribute, the bending towards zero becomes quite sharp. In the experiment
such structures are hard to see because of the small solid angles.

Up to about $2$ GeV the fitted data is practically complete, and further
additional polarization observables will hardly improve the PWA. Above $2$ GeV
and even more above $2.2$ GeV, the number of observables and also the quality
of the data is more limited and additional measurements of ${\cal BT}$
observables can very well improve the PWA.

###  III. D Discussion

The obtained multipoles given in Figs. 3 and 4 are very close to the values of
the chosen theoretical model BG2017. This demonstrates the stability of that
model, however some additional resonant structures in all multipoles are made
more visible which is to expected as the AA/PWA is significantly improving the
ED BG2017 fit for all observables in the particular channel of $K\Lambda$
photoproduction, see Fig. 5.

The AA/PWA multipoles are fairly smooth. However, we would like to warn the
reader that we basically have two distinct energy ranges: the lower one where
8 observables have been measured (1625 MeV $<W<$ 2179 MeV), and the higher one
where only 4 observables have been measured (2179 MeV $<W<$ 2296 MeV). As fits
are in principle done on individual energies one by one, they are correlated
only through the penalty function, so a change in multipoles due to the change
of the number of observables might be expected (cf. the solution theory
discussed in appendix B). The crucial energy where the transition happens is
indicated by the vertical black line at 2179 MeV in all figures. The fits in
the lower energy range should tend to be smoother, and more constrained, while
some visible changes might occur at higher energies. This produces
discontinuities. As the AA/PWA method gives a set of smooth multipoles for a
self-consistent and complete dataset by forcing the reaction-amplitude phases
to be smooth (this has been shown in ref. [12]), this indicates that the
remaining discontinuities are the result of an inconsistency of the database.
So, the AA/PWA method offers a possibility to test the self-consistency of the
experimental database. However, this discussion can be reliably performed only
when a confident method for pole detection is used, so one should in principle
get some answers with the use of Laurent plus Pietarinen (L+P) formalism [28,
29, 30, 31].

###  III. E Comparison of AA/PWA with BG SC-SE-PWA of ref. [2]

For relatively new methods such as AA/PWA, a comparison with old, double-
checked and worldly accepted models is crucial. Such an opportunity is offered
to us by the Bonn-Gatchina group. They have performed SE-PWA and used the
standard constrained PWA method (letting lower multipoles free, and strongly
penalizing higher ones to theoretical ED values, cf. section II). This enabled
us to make a direct comparison of our results with theirs. As we mentioned
before, their constraining ED solution was, unfortunately, not made public,
but differed notably from the values of the published old BG2014-02 and new
BG2019 solutions [14] (we show the difference for the $E_{0+}$ multipole in
Fig. 18). However, upon request, they provided us with the exact numbers [22]
for all multipoles. So, as our AA/PWA method required exactly the same input
(data base and theoretical ED multipoles), and we have achieved that, we used
the chance to compare the results directly.

Figure 18: (Color online) A comparison of $E_{0+}$ BG ED solution BG2017 from
ref. [2] (red line) which was used in this publication, and the BG2019
solution given in BG2019 [14] (black line).

We show in Fig. 19 the result for the four lower multipoles, where the SE
results of ref. [2] only exist.

Figure 19: (Color online) Comparison of $E_{0+}$, $M_{1-}$, $E_{1+}$ and
$M_{1+}$ SC-SE-BG PWA solutions presented in ref. [2] (red circles) and the
same multipoles obtained using the AA/PWA method in this publication (blue
triangles). The red line is the BG2017 ED solution used in the SC-SE-BG PWA
publication, see ref. [14]. Black arrows on the horizontal axis mark the pole
masses of nucleon resonances as given by PDG [32].

We see a very good agreement between both solutions regarding the overall
absolute values and functional shape. Both solutions are also quite close to
the constraining BG2017 ED solution.

We see that the AA/PWA method results in many more points. This is a result of
the data-preparation philosophy as the AA/PWA method uses data interpolation
instead of data binning as the Bonn-Gatchina SE-PWA does.

We see some discrepancy in the absolute values of some multipoles at lower
energies, but our values are closer to the BG2017 values. We conclude that
both SE methods are in full agreement.

The advantage of the AA/PWA method is obvious: first we generate much many
more data points in reconstruction multipoles, and second we generate all
multipoles, and not only the lowest ones.

The main issue is to answer the question whether we reproduce the structures
in the $M_{1-}$ multipole around 1890 MeV, which were interpreted in ref. [2]
as a confirmation of the $N(1880)\frac{1}{2}^{+}$ resonance. The answer is
definitely affirmative: Yes, we do. However, the ”size” of the structure is
not so pronounced, so we need a detailed L+P analysis to confirm or dismiss
this interpretation. This will be done in the forthcoming chapter of this
paper.

##  IV Extraction of resonance parameters from the single-energy multipoles

Extracting poles directly from SE partial-wave solutions is very difficult,
and the only method which showed quite some flexibility and confidence is the
L+P method [32, 28, 29, 30, 31]. So, we use this method to analyze the crucial
multipole $M_{1-}$. However, for the convenience of the reader, we repeat the
essence of the method in section IV. A.

###  IV. A The L+P method

The driving concept behind the Laurent-Pietarinen expansion (L+P) was the aim
to replace elaborate theoretical models by a local power-series representation
of partial wave amplitudes [29]. The complexity of any reaction-theoretical
model which leads to partial waves is thus replaced by a much simpler model-
independent expansion which just exploits analyticity. The L+P approach
separates the pole- and regular parts in the form of a Laurent expansion, and
instead of modeling the regular part by some physical model it uses a
conformal mapping to expand it into a rapidly converging power series with
well defined analytic properties. In such an approach the model dependence is
minimized, and is reduced to the choice of the number and location of branch
points used in the L+P expansion.

The L+P expansion is based on the Pietarinen expansion used in some older
papers in the analysis of pion-nucleon scattering data [33, 34, 35, 36], but
for the L+P model the Pietarinen expansion is applied in a different manner.
It exploits the Mittag-Leffler theorem555Mittag-Leffler theorem [38]: This
theorem provides the generalization of a Laurent expansion to a more-than-one
pole situation. For simplicity, we will simply refer to this as a Laurent
expansion. of partial wave amplitudes near the real energy axis, representing
the regular, but unknown, background term by a conformal-mapping-generated,
rapidly converging power series called a Pietarinen expansion666A conformal-
mapping expansion of this particular type was introduced by Ciulli and Fisher
[33, 34]. It was described in detail and used in pion-nucleon scattering by
Esco Pietarinen [35, 36]. The procedure was denoted as a Pietarinen expansion
by G. Höhler in [8].. The method was used successfully in several few-body
reactions [30, 31, 28], and recently generalized to the multi-channel case
[37]. The formulae used in the L+P approach are collected in Table 2.

Table 2: Formulae defining the Laurent plus Pietarinen (L+P) expansion (see
ref. [3]). $\displaystyle T^{a}(W)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{{N}_{pole}}\frac{x^{a}_{j}+\imath\,\,y^{a}_{j}}{W_{j}-W}+\sum_{k=0}^{K^{a}}c^{a}_{k}\,X^{a}(W)^{k}+\sum_{l=0}^{L^{a}}d_{l}^{a}\,Y^{a}(W)^{l}+\sum_{m=0}^{M^{a}}e_{m}^{a}\,Z^{a}(W)^{m}\hskip
142.26378pt$ $\displaystyle X^{a}(W)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{a}-\sqrt{x_{P}^{a}-W}}{\alpha^{a}+\sqrt{x_{P}^{a}-W}};\,\,\,\,\,Y^{a}(W)=\frac{\beta^{a}-\sqrt{x_{Q}^{a}-W}}{\beta^{a}+\sqrt{x_{Q}^{a}-W}};\,\,\,\,\,Z^{a}(W)=\frac{\gamma^{a}-\sqrt{x_{R}^{a}-W}}{\gamma^{a}+\sqrt{x_{R}^{a}-W}}\hskip
119.50157pt$ $\displaystyle D_{dp}^{a}\ \ \ \ \ $ $\displaystyle=$
$\displaystyle\frac{1}{2\,N_{W}^{a}-N_{par}^{a}}\,\,\sum_{i=1}^{N_{W}^{a}}\left\\{\left[\frac{{\rm
Re}\,T^{a}(W^{(i)})-{\rm Re}\,T^{a,exp}(W^{(i)})}{Err_{i,a}^{\rm
Re}}\right]^{2}+\right.\left.\left[\frac{{\rm Im}\,T^{a}(W^{(i)})-{\rm
Im}\,T^{a,exp}(W^{(i)})}{Err_{i,a}^{\rm Im}}\right]^{2}\right\\}+{\cal
P}^{a}\hskip 8.53581pt$ $\displaystyle{\cal P}^{a}\ \ \ \ \ $ $\displaystyle=$
$\displaystyle\lambda_{c}^{a}\,\sum_{k=1}^{K^{a}}(c^{a}_{k})^{2}\,k^{3}+\lambda_{d}^{a}\,\sum_{l=1}^{L^{a}}(d^{a}_{l})^{2}\,l^{3}+\lambda_{e}^{a}\,\sum_{m=1}^{M^{a}}(e^{a}_{m})^{2}\,m^{3}\qquad\qquad
D_{dp}=\sum_{a}^{all}D_{dp}^{a}\hskip 113.81102pt$ $\displaystyle
a\,\,.....\,\,{\rm channel\,\,index}\qquad\qquad N_{pole}\;.....\;{\rm
number\;of\;poles}\qquad\qquad W_{j},W\in\mathbb{C}\hskip 119.50157pt$
$\displaystyle
x_{i}^{a},\,y_{i}^{a},\,c_{k}^{a},\,d_{l}^{a},\,e_{m}^{a},\,\alpha^{a},\,\beta^{a},\,\gamma^{a}...\in\mathbb{R}\,\,\hskip
273.14645pt$ $\displaystyle
K^{a},\,L^{a},\,M^{a}\,...\,\in\mathbb{N}\,\,\,{\rm
number\,\,of\,\,Pietarinen\,\,coefficients\,\,in\,\,channel\,\,\mathit{a}}.\hskip
150.79959pt$ $\displaystyle D_{dp}^{a}\;.....\;{\rm
discrepancy\;function\;in\;channel\;}a\qquad\qquad\hskip
34.1433ptN_{W}^{a}\;.....\;{\rm number\;of\;energies\;in\;channel\;}a\hskip
22.76219pt$ $\displaystyle N_{par}^{a}\;.....\;{\rm
number\;of\;fitting\;parameters\;in\;channel\;}a\qquad\qquad{\cal
P}^{a}\,\,.....\,\,{\rm Pietarinen\,\,penalty\,\,function}\hskip 39.83385pt$
$\displaystyle\lambda_{c}^{a},\,\lambda_{d}^{a},\,\lambda_{e}^{a}\,\,\,.....\,\,{\rm
Pietarinen\,\,weighting\,\,factors}\qquad\qquad\hskip
28.45274ptx_{P}^{a},\,x_{Q}^{a},\,x_{R}^{a}\in\mathbb{R}\,\,\,\,({\rm
or}\,\,\in\mathbb{C}).\hskip 73.97716pt$ $\displaystyle Err_{i,a}^{\rm
Re,\,Im}.....{\rm\,\,minimization\,\,error\,\,of\,\,real\,\,and\,\,imaginary\,\,part\,\,respectively.}\hskip
128.0374pt$

In the fits, the regular background part is represented by three Pietarinen
series and all free parameters are fitted. The first Pietarinen expansion with
branch point $x_{P}$ is restricted to an unphysical energy range and
collectively represents all left-hand singularities. The next two Pietarinen
expansions describe the background in the physical energy-range with branch
points $x_{Q}$ and $x_{R}$ respecting the analytic properties of the analyzed
partial wave. The second branch point is in most cases fixed to the elastic
channel branch point, the third one is either fixed to the dominant inelastic
threshold, or left free. Thus, only rather general physical assumptions about
the analytic properties are made like the number of poles and the number and
the position of branch points, and the simplest analytic function with the
correct set of poles and branch points is constructed. The method is
applicable to both, theoretical and experimental input, and represents the
first reliable procedure to extract pole positions from experimental data,
with minimal model bias.

The generalization of the L+P method to a multichannel L+P method is performed
in the following way: i) separate Laurent expansions are made for each
channel; ii) pole positions are fixed for all channels, iii) residues and
Pietarinen coefficients are varied freely; iv) branch points are chosen as for
the single-channel model; v) the single-channel discrepancy function
$D_{dp}^{a}$ (see Eq. (5) in ref. [28]) which quantifies the deviation of the
fitted function from the input is generalized to a multi-channel quantity
$D_{dp}$ by summing up all single-channel contributions, and vi) the
minimization is performed for all channels in order to obtain the final
solution.

###  IV. B Detailed analysis of the $M_{1-}$ multipole using the L+P method

The only reliable way to establish whether the structure seen in the $M_{1-}$
multipole corresponds to a resonance pole or not is to use the L+P formalism,
and try to find any analytic function with an explicit pole and realistic
background which fits the data. If we find it, then we can claim that the
observed structure is at least consistent with a function having a pole. Very
often, especially for narrow or small resonances, such a function cannot be
found, so this is an indication that the observed structure is originating not
from a pole, but from some other effect possible in the method.

In ref. [2] the L+P formalism was used to analyze the obtained result, and it
was found that a function containing a pole of mass $M=1876$ MeV and a width
of $\Gamma=33$ MeV can describe the data well, so in spite of the fact that
the width was rather narrow for the formerly found state
$N(1880)\frac{1}{2}^{+}$ (33(19) MeV corresponding to previously established
230(50) MeV) it was interpreted as a signal of a new resonant state.
Unfortunately, in the present paper we do not confirm this finding.

In this paper we have performed an L+P analysis of our AA/PWA solution.
Instead of using the whole energy range up to 2295 MeV, from our analysis we
have omitted the high-energy part above 2179 MeV, where four spin observables
$\Sigma$, $T$, $O_{x^{\prime}}$ and $O_{z^{\prime}}$ are not measured (cf.
Table 1), as we anticipate discontinuities in our solution in that energy
range. Observe that this is also the energy range analyzed in refs. [2, 3].
The fit has been performed with two and three poles, and our solutions are
documented in Fig. 20 and Table 3.

Figure 20: (Color online) The L+P fit of the $M_{1-}$ multipole. In Fig. (a)
we show the result of the single-channel fit of BG-SE data, in Fig. (b) we
show the result of the 2-pole fit of our AA/PWA solution, and in Fig. (c) we
show the result of its 3-pole fit. Red and blue symbols are single-energy BG-
SE and AA/PWA solutions, the thin dashed red line is the resonant
contribution, thin dashed black line is the background part, and thick red
line is the result of the full fit. Table 3:  The pole parameters for the
$M_{1-}$ multipole extracted using the single-channel L+P formalism of ref.
[28, 29, 30, 31] are given. BG-SE denotes the solution of refs. [2, 3] with
one pole, and AA/PWA denotes two solutions for AA/PWA determined in this
paper, with two and three poles respectively. $M_{i}$, $\Gamma_{i}$, $r_{i}$
and $\Theta_{i}$, $i=1,\ldots,3$ are pole masses, widths as well as absolute
values of the residue and its phase, while $\chi^{2}$ is the value of the
total chi-squared. Values given by the PDG [32] for these resonances are given
in bold letters for comparison.

Model | $M_{1}$ | $\Gamma_{1}$ | $|a_{1}|$ | $\Theta_{1}$ | $M_{2}$ | $\Gamma_{2}$ | $|a_{2}|$ | $\Theta_{2}$ | $M_{3}$ | $\Gamma_{3}$ | $|a_{3}|$ | $\Theta_{3}$ | $\chi^{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
PDG | 1700(40) | 120(40) | - | - | 1860(40) | 230(50) | - | - | 2100(50) | 290(50) | - | - |
BG-SE | - | - | - | - | 1876(11) | 31(19) | 6(2) | 57(14) | - | - |  | - | 36
AA/PWA2poles | 1715(12) | 118(31) | 117(62) | 12454(35) | - | - | - | - | 2002(59) | 201(100) | 84(103) | -116(92) | 32.7
AA/PWA3poles | 1714(19) | 120(33) | 122(98) | 121(45) | 1882(3) | 6(6) | 0.6(0.6) | 123(54) | 2007(70) | 187(102) | 71(120) | -113(108) | 29.2

In spite of obtaining a suspiciously narrow width of 33 MeV, the BG-SE
solution is completely consistent with a $N(1880)\frac{1}{2}^{+}$ resonant
state, and is in refs. [2, 3] it has been interpreted as such. However, both
our fits depicted in Fig. (20) (b) and (c) do not support this conclusion.
While the BG-SE fit identifies only a $N(1880)\frac{1}{2}^{+}$ state, our
model clearly confirms the existence of $N(1710)\frac{1}{2}^{+}$ and
$N(2100)\frac{1}{2}^{+}$ states too. In Fig. (20) (b) we show the fit of the
data with two poles only. The fit is smooth and reliable, with $\chi^{2}=32$,
and covers all data in the whole energy range very reliably. Some deviation
from the data is only seen in the energy range of the $N(1880)\frac{1}{2}^{+}$
resonance, but it is small. In Fig. (20) (c) we show the three-pole fit to the
data. The $\chi^{2}$ is slightly improved from 32 to 29.2, but the obtained
resonance, in spite of being in the right place at 1882 MeV, has an extremely
narrow width of 6 MeV.

As such a narrow state of 6 MeV is extremely unlikely, and we do not see the
mechanism how a wider state could influence data in such a narrow energy
range, we do not confirm the existence of $N(1880)\frac{1}{2}^{+}$ state
contrary to Bonn-Gatchina result. We confirm that we do have a ”one- or two-
points structure”, but as the energy step in BG-SE case is much wider (20 MeV)
the width of the disturbance raises to ca 30 MeV. In our case our energy step
is much lower (ca 2-5 MeV), a ”two-point” structure remains, but the width of
a corresponding resonance becomes much narrower.

We conclude that L+P analysis indicates that the narrow disturbance in this
energy range is more consistent with the instability due to violation of data
consistency at this narrow energy range than to the existence of
$N(1880)\frac{1}{2}^{+}$ resonance.

##  V Conclusions and Outlook

The new single-channel, single-energy data analysis method AA/PWA, fully
explained and demonstrated for $\eta$ photoproduction in ref. [1], has been
applied to the world collection of data for $K\Lambda$ photoproduction. It is
shown that a the precise set of multipoles which improves the agreement with
the data compared to the theoretical ED Bonn-Gatchina model [22] was obtained.
Some discontinuities in the obtained multipoles are observed, and this is
explained by the violation of self-consistency of measured data, as well as
the incompleteness of existing data sets. In order to overcome these
difficulties a stronger constraint on the penalizing functions is required.
Let us observe that after Step 1, the constraining amplitudes can be
discontinuous because their continuity is in the present AA/PWA method
guaranteed only by data consistency and completeness of the data set. As both
requirements are not met in most measured processes, additional conditions for
the achievement of continuity are needed. One natural way is to impose
fixed-$t$ analyticity [8, 9, 10].

The obtained results were compared to the already published classic SE-PWA
made in refs. [2, 3], and it has been shown that our results qualitatively and
quantitatively agree with the results of these references. The mutual
agreement of both approaches improves the probability that they are correct.

We confirm that the $M_{1-}$ multipole in our model reproduces a narrow
structure around 1880 MeV, but our L+P analysis fails to confirm that this
structure is the confirmation of a $N(1880)\frac{1}{2}^{+}$ resonant state.
The pole which would explain this structure in our model is much narrower than
the anyways very narrow pole given in refs. [2, 3] (1876 MeV - $i$ 31 MeV), so
we interpret it as a discontinuity due to data inconsistency rather than a
resonance signal. Preliminary tests have confirmed that if we more strongly
constrain the TPWA of the second step of the AA/PWA method to the smooth
amplitudes of the ED BG2017 model, this structure disappears. However, this
has to be elaborated in forthcoming publications.

As the method shows a strong sensitivity to the self-consistency of the data,
we advocate it as a reliable method to perform data self-consistency testing.

###### Acknowledgements.

Y.W. was supported financially by the Transdisciplinary Research Area -
Building Blocks of Matter and Fundamental Interactions (TRA Matter) during the
completion of this work.

##  Appendix A General photoproduction formalism

In this appendix, we collect essential parts of the general formalism for
pseudoscalar meson photoproduction, in order to keep the present work self-
contained. We consider the following $2\rightarrow 2$ reaction:

$\gamma\left(p_{\gamma};m_{\gamma}\right)+N\left(P_{i};m_{s_{i}}\right)\longrightarrow
K\left(p_{K}\right)+\Lambda\left(P_{f};m_{s_{f}}\right)\mathrm{.}$ (3)

The $4$-momenta as well as the variables necessary to label the spin-states
have been indicated for each particle.

The photoproduction process is conventionally described using the well-known
Mandelstam variables $s$, $t$ and $u$. Since $4$-momentum conservation holds,
i.e. $p_{\gamma}+P_{i}=p_{K}+P_{f}$, each of the Mandelstam variables can be
written in two equivalent forms:

$\displaystyle s$
$\displaystyle=(p_{\gamma}+P_{i})^{2}=(p_{K}+P_{f})^{2}\mathrm{,}$ (4)
$\displaystyle t$
$\displaystyle=(p_{\gamma}-p_{K})^{2}=(P_{f}-P_{i})^{2}\mathrm{,}$ (5)
$\displaystyle u$
$\displaystyle=(p_{\gamma}-P_{f})^{2}=(P_{i}-p_{K})^{2}\mathrm{.}$ (6)

Since all particles in the initial- and final state of the reaction (3) are
assumed to be on the mass-shell, the whole reaction can be described by two
independent kinematic invariants. The latter are often chosen to be the pair
$(s,t)$.

In this work, center-of-mass (CMS) coordinates are adopted. The following
relations are valid between $(s,t)$ and the center-of-mass energy $W$ and
scattering angle $\theta$ of the reaction:

$\displaystyle s$ $\displaystyle=W^{2},$ (7) $\displaystyle t$
$\displaystyle=m_{K}^{2}-2k\sqrt{m_{K}^{2}+q^{2}}+2kq\cos\theta.$ (8)

In these expressions, $k$ and $q$ are the absolute values of the CMS
$3$-momenta for the photon and the kaon, respectively. Both of these variables
can be expressed in terms of $W$ and the masses of the particles in the
initial- and final states. One can therefore describe the reaction
equivalently in terms of $(W,\theta)$. The phase-space factor for the
considered reaction (3) is defined as $\rho=q/k$.

The general decomposition of the reaction amplitude into contributions of
individual spin amplitudes has been found by Chew, Goldberger, Low and Nambu
(CGLN) [39] and it reads as follows

$\mathcal{F}=\chi_{m_{s_{f}}}^{\dagger}\left(i\vec{\sigma}\cdot\hat{\epsilon}\;F_{1}+\vec{\sigma}\cdot\hat{q}\;\vec{\sigma}\cdot\hat{k}\times\hat{\epsilon}\;F_{2}+i\vec{\sigma}\cdot\hat{k}\;\hat{q}\cdot\hat{\epsilon}\;F_{3}+i\vec{\sigma}\cdot\hat{q}\;\hat{q}\cdot\hat{\epsilon}\;F_{4}\right)\hskip
2.0pt\chi_{m_{s_{i}}}\mathrm{.}$ (9)

In this expression, $\hat{k}$ and $\hat{q}$ are normalized CMS 3-momenta,
$\hat{\epsilon}$ is the normalized photon polarization vector and
$\chi_{m_{s_{i}}}$, $\chi_{m_{s_{f}}}$ are Pauli spinors for the baryons in
the initial- and final states. The complex amplitudes $F_{1},\ldots,F_{4}$ are
called CGLN amplitudes and they depend on $(W,\theta)$. This set of $4$
amplitudes contains the full information on the dynamics of the considered
process (3).

The so-called transversity amplitudes $b_{1},\ldots,b_{4}$ are defined by
rotating the spin-quantization axis away from the $\hat{z}$-axis of the CMS
frame, which has been inherent to equation (9), to the direction normal to the
so-called reaction plane. The latter plane is spanned by the CMS $3$-momenta
$\vec{k}$ and $\vec{q}$. Using the conventions employed implicitly in the work
of Chiang and Tabakin [40], one obtains the following set of linear and
invertible relations between transversity- and CGLN amplitudes (see also
reference [41]):

$\displaystyle b_{1}\left(W,\theta\right)$
$\displaystyle=-b_{3}\left(W,\theta\right)-\frac{1}{\sqrt{2}}\sin\theta\left[F_{3}\left(W,\theta\right)e^{-i\frac{\theta}{2}}+F_{4}\left(W,\theta\right)e^{i\frac{\theta}{2}}\right]\mathrm{,}$
(10) $\displaystyle b_{2}\left(W,\theta\right)$
$\displaystyle=-b_{4}\left(W,\theta\right)+\frac{1}{\sqrt{2}}\sin\theta\left[F_{3}\left(W,\theta\right)e^{i\frac{\theta}{2}}+F_{4}\left(W,\theta\right)e^{-i\frac{\theta}{2}}\right]\mathrm{,}$
(11) $\displaystyle b_{3}\left(W,\theta\right)$
$\displaystyle=\frac{i}{\sqrt{2}}\left[F_{1}\left(W,\theta\right)e^{-i\frac{\theta}{2}}-F_{2}\left(W,\theta\right)e^{i\frac{\theta}{2}}\right]\mathrm{,}$
(12) $\displaystyle b_{4}\left(W,\theta\right)$
$\displaystyle=\frac{i}{\sqrt{2}}\left[F_{1}\left(W,\theta\right)e^{i\frac{\theta}{2}}-F_{2}\left(W,\theta\right)e^{-i\frac{\theta}{2}}\right]\mathrm{.}$
(13)

The transversity basis greatly simplifies the mathematical form of the
definitions of the polarization observables (see Table 4 and the discussion
further below). For this reason, this basis is also used in the analysis
performed in the present work (cf. section III).

In order to access information on individual resonances, partial waves have to
be analyzed. In the present work, we adopt the well-known expansion of the
CGLN amplitudes into electric and magnetic multipoles
$\left\\{E_{\ell\pm},M_{\ell\pm}\right\\}$, i.e. [39, 42]:

$\displaystyle F_{1}\left(W,\theta\right)$
$\displaystyle=\sum\limits_{\ell=0}^{\infty}\Big{\\{}\left[\ell
M_{\ell+}\left(W\right)+E_{\ell+}\left(W\right)\right]P_{\ell+1}^{{}^{\prime}}\left(\cos\theta\right)$
$\displaystyle\quad\quad\quad+\left[\left(\ell+1\right)M_{\ell-}\left(W\right)+E_{\ell-}\left(W\right)\right]P_{\ell-1}^{{}^{\prime}}\left(\cos\theta\right)\Big{\\}}\mathrm{,}$
(14) $\displaystyle F_{2}\left(W,\theta\right)$
$\displaystyle=\sum\limits_{\ell=1}^{\infty}\left[\left(\ell+1\right)M_{\ell+}\left(W\right)+\ell
M_{\ell-}\left(W\right)\right]P_{\ell}^{{}^{\prime}}\left(\cos\theta\right)\mathrm{,}$
(15) $\displaystyle F_{3}\left(W,\theta\right)$
$\displaystyle=\sum\limits_{\ell=1}^{\infty}\Big{\\{}\left[E_{\ell+}\left(W\right)-M_{\ell+}\left(W\right)\right]P_{\ell+1}^{{}^{\prime\prime}}\left(\cos\theta\right)$
$\displaystyle\quad\quad\quad+\left[E_{\ell-}\left(W\right)+M_{\ell-}\left(W\right)\right]P_{\ell-1}^{{}^{\prime\prime}}\left(\cos\theta\right)\big{\\}}\mathrm{,}$
(16) $\displaystyle F_{4}\left(W,\theta\right)$
$\displaystyle=\sum\limits_{\ell=2}^{\infty}\left[M_{\ell+}\left(W\right)-E_{\ell+}\left(W\right)-M_{\ell-}\left(W\right)-E_{\ell-}\left(W\right)\right]P_{\ell}^{{}^{\prime\prime}}\left(\cos\theta\right)\mathrm{.}$
(17)

The multipoles can be assigned to definite conserved spin-parity quantum
numbers $J^{P}$ (resonances with spin $J=\left|\ell\pm\frac{1}{2}\right|$
couple to the multipoles $E_{\ell\pm}$ and $M_{\ell\pm}$). The multipole
expansion of the $F_{i}$ can be formally inverted using a set of known
projection integrals [43, 44].

The polarization observables accessible in pseudoscalar meson photoproduction
are dimensionless asymmetries among differential cross sections for different
beam-, target- and recoil polarization states:

$\mathcal{O}=\frac{\beta\left[\left(\frac{d\sigma}{d\Omega}\right)^{\left(B_{1},T_{1},R_{1}\right)}-\left(\frac{d\sigma}{d\Omega}\right)^{\left(B_{2},T_{2},R_{2}\right)}\right]}{\sigma_{0}}\mathrm{.}$
(18)

The factor $\beta$ is a consistency factor introduced in reference [42]. It
takes the value $\beta=\frac{1}{2}$ for observables which involve only beam-
and target polarization and $\beta=1$ for quantities with recoil polarization.
The unpolarized cross section $\sigma_{0}$ is always the sum of the two
polarization configurations present in equation (18):
$\sigma_{0}=\beta\left[\left(\frac{d\sigma}{d\Omega}\right)^{\left(B_{1},T_{1},R_{1}\right)}+\left(\frac{d\sigma}{d\Omega}\right)^{\left(B_{2},T_{2},R_{2}\right)}\right]$.
The dimensioned asymmetry $\sigma_{0}\mathcal{O}$ is often called a profile
function [40, 44] and it is distinguished by a hat-mark on the $\mathcal{O}$:

$\hat{\mathcal{O}}=\beta\left[\left(\frac{d\sigma}{d\Omega}\right)^{\left(B_{1},T_{1},R_{1}\right)}-\left(\frac{d\sigma}{d\Omega}\right)^{\left(B_{2},T_{2},R_{2}\right)}\right]\mathrm{.}$
(19)

For single-meson photoproduction, there exist in total $16$ polarization
observables [40, 42]. They include the unpolarized cross section $\sigma_{0}$
and $15$ further single- and double-polarization observables. The full set of
$16$ observables can be divided into the four groups of single-spin
observables ($\mathcal{S}$), beam-target- ($\mathcal{BT}$), beam-recoil-
($\mathcal{BR}$) and target-recoil ($\mathcal{TR}$) observables [40]. Each
group is composed of $4$ quantities. The definitions of the $16$ observables
in terms of transversity amplitudes are given in Table 4.

Observable | Group
---|---
$\sigma_{0}=\frac{1}{2}\left(\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}+\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right)$ |
$\hat{\Sigma}=\frac{1}{2}\left(-\left|b_{1}\right|^{2}-\left|b_{2}\right|^{2}+\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right)$ | $\mathcal{S}$
$\hat{T}=\frac{1}{2}\left(\left|b_{1}\right|^{2}-\left|b_{2}\right|^{2}-\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right)$ |
$\hat{P}=\frac{1}{2}\left(-\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}-\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right)$ |
$\hat{E}=\mathrm{Re}\left[-b_{3}^{\ast}b_{1}-b_{4}^{\ast}b_{2}\right]=-\left|b_{1}\right|\left|b_{3}\right|\cos\phi_{13}-\left|b_{2}\right|\left|b_{4}\right|\cos\phi_{24}$ |
$\hat{F}=\mathrm{Im}\left[b_{3}^{\ast}b_{1}-b_{4}^{\ast}b_{2}\right]=\left|b_{1}\right|\left|b_{3}\right|\sin\phi_{13}-\left|b_{2}\right|\left|b_{4}\right|\sin\phi_{24}$ | $\mathcal{BT}$
$\hat{G}=\mathrm{Im}\left[-b_{3}^{\ast}b_{1}-b_{4}^{\ast}b_{2}\right]=-\left|b_{1}\right|\left|b_{3}\right|\sin\phi_{13}-\left|b_{2}\right|\left|b_{4}\right|\sin\phi_{24}$ |
$\hat{H}=\mathrm{Re}\left[b_{3}^{\ast}b_{1}-b_{4}^{\ast}b_{2}\right]=\left|b_{1}\right|\left|b_{3}\right|\cos\phi_{13}-\left|b_{2}\right|\left|b_{4}\right|\cos\phi_{24}$ |
$\hat{C}_{x^{\prime}}=\mathrm{Im}\left[-b_{4}^{\ast}b_{1}+b_{3}^{\ast}b_{2}\right]=-\left|b_{1}\right|\left|b_{4}\right|\sin\phi_{14}+\left|b_{2}\right|\left|b_{3}\right|\sin\phi_{23}$ |
$\hat{C}_{z^{\prime}}=\mathrm{Re}\left[-b_{4}^{\ast}b_{1}-b_{3}^{\ast}b_{2}\right]=-\left|b_{1}\right|\left|b_{4}\right|\cos\phi_{14}-\left|b_{2}\right|\left|b_{3}\right|\cos\phi_{23}$ | $\mathcal{BR}$
$\hat{O}_{x^{\prime}}=\mathrm{Re}\left[-b_{4}^{\ast}b_{1}+b_{3}^{\ast}b_{2}\right]=-\left|b_{1}\right|\left|b_{4}\right|\cos\phi_{14}+\left|b_{2}\right|\left|b_{3}\right|\cos\phi_{23}$ |
$\hat{O}_{z^{\prime}}=\mathrm{Im}\left[b_{4}^{\ast}b_{1}+b_{3}^{\ast}b_{2}\right]=\left|b_{1}\right|\left|b_{4}\right|\sin\phi_{14}+\left|b_{2}\right|\left|b_{3}\right|\sin\phi_{23}$ |
$\hat{L}_{x^{\prime}}=\mathrm{Im}\left[-b_{2}^{\ast}b_{1}-b_{4}^{\ast}b_{3}\right]=-\left|b_{1}\right|\left|b_{2}\right|\sin\phi_{12}-\left|b_{3}\right|\left|b_{4}\right|\sin\phi_{34}$ |
$\hat{L}_{z^{\prime}}=\mathrm{Re}\left[-b_{2}^{\ast}b_{1}-b_{4}^{\ast}b_{3}\right]=-\left|b_{1}\right|\left|b_{2}\right|\cos\phi_{12}-\left|b_{3}\right|\left|b_{4}\right|\cos\phi_{34}$ | $\mathcal{TR}$
$\hat{T}_{x^{\prime}}=\mathrm{Re}\left[b_{2}^{\ast}b_{1}-b_{4}^{\ast}b_{3}\right]=\left|b_{1}\right|\left|b_{2}\right|\cos\phi_{12}-\left|b_{3}\right|\left|b_{4}\right|\cos\phi_{34}$ |
$\hat{T}_{z^{\prime}}=\mathrm{Im}\left[-b_{2}^{\ast}b_{1}+b_{4}^{\ast}b_{3}\right]=-\left|b_{1}\right|\left|b_{2}\right|\sin\phi_{12}+\left|b_{3}\right|\left|b_{4}\right|\sin\phi_{34}$ |
Table 4: The definitions of the $16$ polarization observables of pseudoscalar
meson photoproduction are given here in terms of transversity amplitudes
$b_{1},\ldots,b_{4}$ (cf. ref. [40]; sign conventions are consistent with
[44]). Expressions are given both in terms of real- and imaginary parts of
bilinear products of amplitudes and in terms of moduli and relative phases of
the amplitudes. Furthermore, the phase-space factor $\rho$ has been suppressed
in the given expressions (i.e. we have set $\rho=1$). The four different
groups of four observables each are indicated as well.

##  Appendix B Solution-theory for the complete-experiment analysis (CEA) of
the investigated database

In the following, we give some more mathematical details on the possible
ambiguities of the CEA when it is applied to the database analyzed in this
work (cf. section III. A). We are well aware that the facts given below can be
extracted from the well-known mathematical treatments in the CEA for
photoproduction [40, 45]. Still, we hope that the details given in the
following can provide some intuition on the mathematical ambiguities one has
to be careful about when analyzing the present database.

###  II. A Lower-energy region ($1625$ \- $2179$ MeV): observables
$\left\\{\sigma_{0},\hat{\Sigma},\hat{T},\hat{P},\hat{O}_{x^{\prime}},\hat{O}_{z^{\prime}},\hat{C}_{x^{\prime}},\hat{C}_{z^{\prime}}\right\\}$

We consider the definitions (cf. Table 4) of the eight analyzed observables
(in the following, we set $\rho=q/k\equiv 1$):

$\displaystyle\sigma_{0}$
$\displaystyle=\frac{1}{2}\left(\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}+\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right),$
(20) $\displaystyle\hat{\Sigma}$
$\displaystyle=\frac{1}{2}\left(-\left|b_{1}\right|^{2}-\left|b_{2}\right|^{2}+\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right),$
(21) $\displaystyle\hat{T}$
$\displaystyle=\frac{1}{2}\left(\left|b_{1}\right|^{2}-\left|b_{2}\right|^{2}-\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right),$
(22) $\displaystyle\hat{P}$
$\displaystyle=\frac{1}{2}\left(-\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}-\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right),$
(23) $\displaystyle\hat{O}_{x^{\prime}}$
$\displaystyle=\mathrm{Re}\left[-b_{1}b_{4}^{\ast}+b_{2}b_{3}^{\ast}\right],$
(24) $\displaystyle\hat{O}_{z^{\prime}}$
$\displaystyle=\mathrm{Im}\left[b_{1}b_{4}^{\ast}+b_{2}b_{3}^{\ast}\right],$
(25) $\displaystyle\hat{C}_{x^{\prime}}$
$\displaystyle=\mathrm{Im}\left[-b_{1}b_{4}^{\ast}+b_{2}b_{3}^{\ast}\right],$
(26) $\displaystyle\hat{C}_{z^{\prime}}$
$\displaystyle=\mathrm{Re}\left[-b_{1}b_{4}^{\ast}-b_{2}b_{3}^{\ast}\right].$
(27)

In case the phases of the transversity amplitudes
$b_{j}=\left|b_{j}\right|e^{i\phi_{j}}$ are fixed to values coming from an
energy-dependent, unitary PWA model (e.g. BnGa), one can see very quickly that
the system of equations composed of the eight observables (20) to (27) is in
principle capable of fixing the four moduli $\left|b_{i}\right|$ uniquely. In
fact, the group-$\mathcal{S}$ observables
$\left\\{\sigma_{0},\hat{\Sigma},\hat{T},\hat{P}\right\\}$ alone are already
capable of that feat and the four additional observables only should make the
solution more stable.

In case we also wish to determine the phases of the four transversity
amplitudes, the situation is as follows: the four moduli $\left|b_{i}\right|$
are fixed uniquely by the group-$\mathcal{S}$ observables and the four
additional observables
$\left\\{\hat{O}_{x^{\prime}},\hat{O}_{z^{\prime}},\hat{C}_{x^{\prime}},\hat{C}_{z^{\prime}}\right\\}$
can uniquely pin down the relative phases $\phi_{14}$ and $\phi_{23}$. The
latter statement is true due to the inverse relations:

$\displaystyle e^{i\phi_{14}}$
$\displaystyle=\frac{\text{Re}\left[b_{1}b_{4}^{\ast}\right]+i\text{Im}\left[b_{1}b_{4}^{\ast}\right]}{\left|b_{1}\right|\left|b_{4}\right|}=\frac{\left(-\hat{O}_{x^{\prime}}-\hat{C}_{z^{\prime}}\right)+i\left(\hat{O}_{z^{\prime}}-\hat{C}_{x^{\prime}}\right)}{2\left|b_{1}\right|\left|b_{4}\right|},$
(28) $\displaystyle e^{i\phi_{23}}$
$\displaystyle=\frac{\text{Re}\left[b_{2}b_{3}^{\ast}\right]+i\text{Im}\left[b_{2}b_{3}^{\ast}\right]}{\left|b_{2}\right|\left|b_{3}\right|}=\frac{\left(\hat{O}_{x^{\prime}}-\hat{C}_{z^{\prime}}\right)+i\left(\hat{O}_{z^{\prime}}+\hat{C}_{x^{\prime}}\right)}{2\left|b_{2}\right|\left|b_{3}\right|}.$
(29)

The complex exponential functions $e^{i\phi_{jk}}$ can be inverted uniquely
for phases on the interval $\phi_{jk}\in[0,2\pi)$ (via the arctan2 function).
The amplitude-arrangement of four transversity amplitudes is however not
uniquely fixed777I.e., uniquely up to one overall phase for all $4$
amplitudes.. One additional ’connecting’ relative phase is missing, for
instance $\phi_{12}$ or $\phi_{34}$. In other words, the two subsets of
amplitudes $\left\\{b_{1},b_{4}\right\\}$ and $\left\\{b_{2},b_{3}\right\\}$
can be rotated relative to each other in a completely free way and the
observables
$\left\\{\hat{O}_{x^{\prime}},\hat{O}_{z^{\prime}},\hat{C}_{x^{\prime}},\hat{C}_{z^{\prime}}\right\\}$
are completely blind to such a rotation. As an example, consider a rotation of
only the two amplitudes $b_{2}$ and $b_{3}$ by the same phase
$\tilde{\varphi}$, which can have any dependence on energy and angle:

$b_{2}(W,\theta)\longrightarrow
e^{i\tilde{\varphi}(W,\theta)}b_{2}(W,\theta)\text{ and
}b_{3}(W,\theta)\longrightarrow
e^{i\tilde{\varphi}(W,\theta)}b_{3}(W,\theta).$ (30)

This rotation leaves both relative phases $\phi_{14}$ and $\phi_{23}$
invariant and therefore also all four observables
$\left\\{\hat{O}_{x^{\prime}},\hat{O}_{z^{\prime}},\hat{C}_{x^{\prime}},\hat{C}_{z^{\prime}}\right\\}$.
However, a rotation like (30) generally leads to a new set of amplitudes with
a very different partial wave decomposition, since it changes the unknown
continuum ambiguity phase (i.e. one overall phase for all $4$ amplitudes) as
well as the ’connecting’ relative phases $\phi_{12}$ and $\phi_{34}$. One has
to be careful about such effects when analyzing the data although as mentioned
above, in our analysis this ambiguity is removed by fixing the phases of all
$4$ transversity amplitudes to a known ED model (cf. sections II and III).

###  II. B Higher-energy region ($2179$ \- $2296$ MeV): observables
$\left\\{\sigma_{0},\hat{P},\hat{C}_{x^{\prime}},\hat{C}_{z^{\prime}}\right\\}$

We start by considering the definitions (cf. Table 4) of the four analyzed
observables (again setting $\rho=q/k\equiv 1$):

$\displaystyle\sigma_{0}$
$\displaystyle=\frac{1}{2}\left(\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}+\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right),$
(31) $\displaystyle\hat{P}$
$\displaystyle=\frac{1}{2}\left(-\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}-\left|b_{3}\right|^{2}+\left|b_{4}\right|^{2}\right),$
(32) $\displaystyle\hat{C}_{x^{\prime}}$
$\displaystyle=\mathrm{Im}\left[-b_{1}b_{4}^{\ast}+b_{2}b_{3}^{\ast}\right],$
(33) $\displaystyle\hat{C}_{z^{\prime}}$
$\displaystyle=\mathrm{Re}\left[-b_{1}b_{4}^{\ast}-b_{2}b_{3}^{\ast}\right].$
(34)

We assume that the phases of the four transversity amplitudes are fixed to a
model and define:

$c_{ij}:=\cos\phi_{ij}\text{, and }s_{ij}:=\sin\phi_{ij}.$ (35)

We re-consider the equations for $\hat{C}_{x^{\prime}}$ and
$\hat{C}_{z^{\prime}}$:

$\displaystyle\hat{C}_{x^{\prime}}$
$\displaystyle=-\left|b_{1}\right|\left|b_{4}\right|s_{14}+\left|b_{2}\right|\left|b_{3}\right|s_{23},$
(36) $\displaystyle\hat{C}_{z^{\prime}}$
$\displaystyle=-\left|b_{1}\right|\left|b_{4}\right|c_{14}-\left|b_{2}\right|\left|b_{3}\right|c_{23},$
(37)

and recognize that these equations can be inverted for the following products
of moduli:

$\displaystyle\left|b_{1}\right|\left|b_{4}\right|$
$\displaystyle=\frac{-c_{23}\hat{C}_{x^{\prime}}-s_{23}\hat{C}_{z^{\prime}}}{s_{14}c_{23}+c_{14}s_{23}}=:\xi_{1},$
(38) $\displaystyle\left|b_{2}\right|\left|b_{3}\right|$
$\displaystyle=\frac{c_{14}\hat{C}_{x^{\prime}}-s_{14}\hat{C}_{z^{\prime}}}{s_{14}c_{23}+c_{14}s_{23}}=:\xi_{2}.$
(39)

The quantities $\xi_{1}$ and $\xi_{2}$ are uniquely fixed from the observables
and the employed model phases.

We could now choose to eliminate, for instance, the quantities
$\left|b_{3}\right|$ and $\left|b_{4}\right|$ in the equations for the cross
section (31) and the observable $\hat{P}$ (32) and thus obtain:

$\displaystyle\sigma_{0}$
$\displaystyle=\frac{1}{2}\left[\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}+\left(\frac{\xi_{2}}{\left|b_{2}\right|}\right)^{2}+\left(\frac{\xi_{1}}{\left|b_{1}\right|}\right)^{2}\right],$
(40) $\displaystyle\hat{P}$
$\displaystyle=\frac{1}{2}\left[-\left|b_{1}\right|^{2}+\left|b_{2}\right|^{2}-\left(\frac{\xi_{2}}{\left|b_{2}\right|}\right)^{2}+\left(\frac{\xi_{1}}{\left|b_{1}\right|}\right)^{2}\right],$
(41)

These are two quadratic equations for the two remaining unknowns
$\left|b_{1}\right|$ and $\left|b_{2}\right|$. One can make the following
attempt at solving them. We multiply the equation (41) by
$\left|b_{2}\right|^{2}$ in order to obtain:

$\left|b_{2}\right|^{4}+\left[\left(\frac{\xi_{1}}{\left|b_{1}\right|}\right)^{2}-\left|b_{1}\right|^{2}-2\hat{P}\right]\left|b_{2}\right|^{2}-\left(\xi_{2}\right)^{2}=0.$
(42)

This is a quadratic equation for $\left|b_{2}\right|^{2}$ and thus allows for
the following two solutions:

$\left|b_{2}\right|^{2}_{\text{I,II}}=\frac{1}{2}\left[2\hat{P}+\left|b_{1}\right|^{2}-\left(\frac{\xi_{1}}{\left|b_{1}\right|}\right)^{2}\right]\pm\sqrt{\frac{1}{4}\left[2\hat{P}+\left|b_{1}\right|^{2}-\left(\frac{\xi_{1}}{\left|b_{1}\right|}\right)^{2}\right]^{2}+\left(\xi_{2}\right)^{2}}.$
(43)

Irrespective of whether solution ’I’ or ’II’ is the correct one, the ’+’
branch of the square root has to be taken in order to arrive at a positive
modulus $\left|b_{2}\right|$. In case both solutions ’I’ or ’II’ are allowed
(i.e. larger than zero) in equation (43), one obtains two permissible moduli
$\left|b_{2}\right|_{\text{I,II}}$. Then, one has to substitute these
solutions into the equation for the cross section (40) and solve for
$\left|b_{1}\right|$. In this way, at most a set of four discrete ambiguities
can emerge and all continuous ambiguities are resolved for the four moduli
$\left|b_{1}\right|,\ldots,\left|b_{4}\right|$.
Therefore, in case one would attempt to let all four moduli
$\left|b_{1}\right|,\ldots,\left|b_{4}\right|$ run freely in the AA step (i.e.
our step ’1’), we expect the found solution to lie on a well-defined
(approximately) parabolic minimum where the derivative of the minimized
(’$\chi^{2}$-like’) function exactly vanishes. Data for the four observables
$\left\\{\sigma_{0},\hat{P},\hat{C}_{x^{\prime}},\hat{C}_{z^{\prime}}\right\\}$
are in principle only capable to distinguish solutions up to the above-
mentioned discrete ambiguity. However, in case the initial conditions are
well-chosen for the minimization procedure, we are confident that the correct
minimum can be found, i.e. that the moduli extraction is sufficiently stable.

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