#  Testing chiral unitary models for the $\Lambda(1405)$ in $K^{+}\pi\Sigma$
photoproduction

P. C. Bruns<EMAIL_ADDRESS>Nuclear Physics Institute of the Czech Academy
of Sciences, 250 68 Řež, Czechia  A. Cieplý<EMAIL_ADDRESS>Nuclear Physics
Institute of the Czech Academy of Sciences, 250 68 Řež, Czechia  M. Mai
<EMAIL_ADDRESS>Helmholtz-Institut für Strahlen- und Kernphysik
(Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, D–53115
Bonn, Germany  The George Washington University, Washington, DC 20052, USA

###### Abstract

We adopt a novel formalism for the low-energy analysis of the $\gamma p\to
K^{+}\pi\Sigma$ photoproduction reaction to calculate the $\pi\Sigma$
invariant mass distributions in the $\Lambda(1405)$ resonance region. The
approach adheres to constraints arising from unitarity, gauge invariance and
chiral perturbation theory, and is used without adjusting any model
parameters. It is found that the meson-baryon rescattering in the final state
has a major impact on the magnitude and structure of the generated spectra
that are compared with the experimental data from the CLAS collaboration. We
demonstrate a large sensitivity of the theoretical predictions to the choice
of the coupled-channel $\pi\Sigma-\bar{K}N$ model amplitudes which should
enable one to constrain the parameter space of these models.

(a) Weinberg-Tomozawa (WT) graphs.

(b) Born term (BT) graphs of two types, B1 and B2.

(c) The anomalous (AN) graph.

Figure 1: Three classes of tree graphs for the two-meson photoproduction
amplitude. Directed lines: baryons, dashed lines: pseudoscalar mesons, crossed
circles: possible photon insertions. Working at order $\mathcal{O}(e)$, the
first figure (a) represents five graphs, each with the photon attached to a
different crossed vertex. The same notation system applies to figures (b) and
(c) giving altogether rise to $5+(2\times 7)+1=20$ tree graphs.

##  I Introduction

Unraveling the general patterns and the nature of individual (excited) states
in the hadron spectrum is essential to test our understanding of the strong
interaction. In that, only few other states have such a long-lasting history
as the enigmatic $\Lambda(1405)$ resonance, becoming a benchmark for our
understanding of the $\rm SU(3)$ hadron dynamics. Starting with its
theoretical prediction Dalitz and Tuan (1959) and later experimental
verification Alston  _et al._ (1961), until the most recent debates on the
existence of the second broad pole, it emerged as a fruitful research area
sparking many theoretical and experimental developments. For a deeper recap of
the character and history of $\Lambda(1405)$, established to be an
$I(J^{P})=0\,(1/2^{-})$ resonance in the strangeness $S=-1$ sector, we refer
the reader to the recent dedicated reviews Hyodo and Jido (2012); Mai (2021).
More general reviews of our present understanding of excited hadrons can be
found in Refs. Hyodo and Niiyama (2021); Mai  _et al._ (2022).

Broader impact of the research related to the antikaon-nucleon systems
includes: (1) Application to the $\Lambda_{b}\to J/\psi\,\Lambda(1405)$ decay
Roca  _et al._ (2015) and similar processes with the final-state interaction
dominated by the non-perturbative meson-baryon dynamics Oset  _et al._ (2016);
(2) Investigation of and search for $\bar{K}$-nuclei ($\bar{K}NN$,
$\bar{K}KN$, $\bar{K}NNN$, etc.), being part of large experimental programs at
DA$\rm\Phi$NE Filippini  _et al._ (1998); Agnello  _et al._ (2005), Saclay
Yamazaki  _et al._ (2009); Maggiora  _et al._ (2010); Yamazaki  _et al._
(2010) or at J-PARC Sada  _et al._ (2016); Yamaga  _et al._ (2020) to name
just a few; (3) The exploration of the in-medium properties of anti-kaons and
strange nuclear matter Cieplý  _et al._ (2011); Gal  _et al._ (2014);
Hrtánková  _et al._ (2018). One natural application is also represented by
studies of the equation of state of neutron stars in relation to the
strangeness, see the comprehensive review Gal  _et al._ (2016) on many aspects
related to strangeness in nuclear physics.

The state-of-the-art theoretical approaches to the universal parameters of the
sub-threshold $\Lambda(1405)$ resonance include various forms of coupled-
channel models implementing $S$-matrix unitarity, while also adding
constraints from chiral symmetry of QCD to various degrees. Parameters of such
models are typically fixed by fitting $K^{-}p$ reactions and threshold data,
and the chirally motivated coupled-channel dynamics generates two poles in the
complex energy plane Oller and Meißner (2001); Jido  _et al._ (2003); Mai and
Meißner (2013); Cieplý  _et al._ (2016). The theoretical predictions Ikeda
_et al._ (2012); Guo and Oller (2013); Mai and Meißner (2015); Feijoo  _et
al._ (2019); Bruns and Cieplý (2022) tend to agree on the position of the
narrow pole located around ${1425~{}{\rm MeV}}$, coupling strongly to the
$\bar{K}N$ channel and mostly interpreted as a meson-baryon molecular state.
The position of the second pole is not determined so well, is broader and at
significantly lower energies. This so-called _double-pole_ structure of
$\Lambda(1405)$ is what makes this state particularly curious, while we know
now that such a structure is common to many other states in the hadron
spectrum Meißner (2020); Mai  _et al._ (2022).

One particular complication in accessing the parameters of both
$\Lambda(1405)$ states directly lies in the fact that there is no $\Sigma$
target to perform $\pi\Sigma$ scattering experiments. This hurdle can be
overcome by analyzing processes with $\pi\Sigma$ appearing in the final state.
This is the case, e.g., for the _two-meson photoproduction_ reaction $\gamma
p\to K^{+}\pi\Sigma$ in which the $K^{+}$ meson takes away momentum, enabling
a scan in the invariant mass of the $\pi\Sigma$ system down to its production
threshold. Such determined line-shape of the $\pi\Sigma$ system covers the
relevant energy and, in principle, allows one to extract the parameters of the
$S=-1$ meson-baryon system more directly. The corresponding experiment was
performed by the CLAS collaboration several years ago, taking very precise,
high-resolution data Moriya  _et al._ (2013a); Schumacher and Moriya (2013);
Moriya  _et al._ (2013b). These data have already made a significant impact on
the field, see Refs. Roca and Oset (2013a); Mai and Meißner (2015); Roca and
Oset (2013b); Nakamura and Jido (2014); Wang  _et al._ (2017); Nam (2017);
Anisovich  _et al._ (2020); Anisovich (2021), where various photoproduction
models have been tested and the generated $\pi\Sigma$ mass distributions
compared with the data. Our work aims to contribute to these efforts by
implementing a recently proposed formalism Bruns (2020) for the $\gamma p\to
K^{+}\pi\Sigma$ transition and discussing the sensitivity of the calculated
cross sections to ambiguities inherent in extending the $\pi\Sigma-\bar{K}N$
coupled-channel models to energies below the $\bar{K}N$ threshold. In the
future, this should allow us to reduce the model dependence and shrink at the
same time the parameter space.

At this point, it might be appropriate to mention other contemporary and
ongoing experimental studies aiming at shedding more light on this enigmatic
part of the hadron spectrum. First of all, the $\pi\Sigma$ mass spectra were
also measured in other reactions, some of them compatible with the CLAS data
Zychor  _et al._ (2008); Scheluchin  _et al._ (2021), some other not
Agakishiev  _et al._ (2013). Completely new experimental data on the isoscalar
$\pi^{0}\Sigma^{0}$ photoproduction should also be reported soon by the GlueX
collaboration Wickramaarachchi . Other efforts aim at determining the
isovector part of the $\bar{K}N$ interaction via precise measurements of the
kaonic hydrogen and kaonic deuterium characteristics Bazzi  _et al._ (2013);
Curceanu  _et al._ (2013); Marton  _et al._ (2017) or by studying the related
$K^{0}_{L}p\to K^{+}\Xi^{0}$ reaction Amaryan  _et al._ (2020). Finally,
exciting new prospects in exploring strangeness in the hadron spectrum may
open with the PANDA experiment Lutz  _et al._ (2009); Barucca  _et al._
(2021).

The article is organized as follows: In section II we introduce the formalism
adopted to study the $K^{+}\pi\Sigma$ photoproduction on proton targets. In
section III we discuss the predictions made using this formalism, and compare
the resulting $\pi\Sigma$ mass distributions with the CLAS data. Finally, in
section IV we conclude summarizing our findings and provide an outlook for
future directions.

##  II Formalism

We consider the two-meson photoproduction reaction
$\gamma(k)p(p_{N})\,\rightarrow\,K(q_{K})\pi(q_{\pi})\Sigma(p_{\Sigma})$ ,
where the symbol in brackets denotes the four-momentum of the indicated
particle. For a process with five external particles, we can form five
independent Mandelstam variables, which we choose here as

$\displaystyle s$
$\displaystyle=(p_{N}+k)^{2}=(q_{K}+q_{\pi}+p_{\Sigma})^{2}\,,$ $\displaystyle
M_{\pi\Sigma}^{2}$ $\displaystyle=(q_{\pi}+p_{\Sigma})^{2}\,,\quad
t_{\Sigma}=(p_{\Sigma}-p_{N})^{2}\,,$ $\displaystyle u_{\Sigma}$
$\displaystyle=(p_{\Sigma}-k)^{2}\,,\quad\;t_{K}=(q_{K}-k)^{2}\,.$ (1)

The two-meson photoproduction amplitude can be decomposed as

$\displaystyle\mathcal{M}^{\mu}=$
$\displaystyle\,\gamma^{\mu}\mathcal{M}_{1}+p_{N}^{\mu}\mathcal{M}_{2}+p_{\Sigma}^{\mu}\mathcal{M}_{3}+q_{K}^{\mu}\mathcal{M}_{4}$
(2)
$\displaystyle+\not{k}\left(\gamma^{\mu}\mathcal{M}_{5}+p_{N}^{\mu}\mathcal{M}_{6}+p_{\Sigma}^{\mu}\mathcal{M}_{7}+q_{K}^{\mu}\mathcal{M}_{8}\right)$
$\displaystyle+\not{q}_{K}\left(\gamma^{\mu}\mathcal{M}_{9}+p_{N}^{\mu}\mathcal{M}_{10}+p_{\Sigma}^{\mu}\mathcal{M}_{11}+q_{K}^{\mu}\mathcal{M}_{12}\right)$
$\displaystyle+\not{q}_{K}\not{k}\left(\gamma^{\mu}\mathcal{M}_{13}+p_{N}^{\mu}\mathcal{M}_{14}+p_{\Sigma}^{\mu}\mathcal{M}_{15}+q_{K}^{\mu}\mathcal{M}_{16}\right)\,,$

where we have suppressed structures $\sim k^{\mu}$ that do not contribute to
the photoproduction cross section. The structures $\mathcal{M}$ can be
obtained diagrammatically as it is shown explicitly in App. C of Ref. Bruns
(2020). At the leading order, there are 20 tree level diagrams contributing to
these structures, which can be arranged in three classes: (WT) graphs with one
Weinberg-Tomozawa contact term with a photon attached to one of five possible
places; (B1/B2) Born graphs with the pion/kaon emitted before the kaon/pion,
respectively, including seven possible photon coupling possibilities; (AN)
graphs with a three-meson-photon vertex from the anomalous Wess-Zumino-Witten
Lagrangian Wess and Zumino (1971); Witten (1983). Thus, $5+(2\times 7)+1=20$
tree diagrams comprise our photoproduction kernel. We also remark that the
latter class of graphs yields contributions of sub-leading chiral order in the
low-energy power counting, but we include it here as a test of higher-order
contributions, without introducing new adjustable parameters.

Our notation for kinematics is such that Lorentz-non-invariant quantities
(like angles and three-momenta) are marked with a $(\ast)$ if they are
evaluated in the $\pi\Sigma$ c.m. frame (where
$\vec{p}_{\Sigma}^{\,\ast}+\vec{q}_{\pi}^{\,\ast}=\vec{0}$), and otherwise
refer to the overall c.m. frame where $\vec{p}_{N}+\vec{k}=\vec{0}$. For
example, we have

$\displaystyle|\vec{q}_{K}|$
$\displaystyle=\frac{\sqrt{\lambda(s,M_{\pi\Sigma}^{2},M_{K}^{2})}}{2\sqrt{s}}\,,$
(3) $\displaystyle|\vec{p}_{\Sigma}^{\,\ast}|$
$\displaystyle=\frac{\sqrt{\lambda(M_{\pi\Sigma}^{2},m_{\Sigma}^{2},M_{\pi}^{2})}}{2M_{\pi\Sigma}}\,,$
(4)

employing $\lambda(x,y,z):=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$ .

In the coupled-channel formalism of _Unitarized ChPT_ , one constructs meson-
baryon partial-wave scattering amplitudes
$f_{\ell\pm}^{c^{\prime},c}(M_{\pi\Sigma})$, which aim to describe the
scattering from channel $c$ to channel $c^{\prime}$ (here,
$c,c^{\prime}=\pi\Sigma,\,\bar{K}N,\,\eta\Lambda,\ldots$) for total angular
momentum $\ell\pm\frac{1}{2}$ and orbital angular momentum
$\ell=0,1,2,\ldots\,$. These amplitudes are designed in such a way to be
consistent with ChPT up to a fixed order of the low-energy expansion (in
practice, usually on tree level, i.e., $\mathcal{O}(p)$ or
$\mathcal{O}(p^{2})$). At the same time, they fulfill the requirement of
coupled-channel unitarity,

$\mathrm{Im}(f_{\ell\pm})=(f_{\ell\pm})^{\dagger}(|\vec{p}^{\,\ast}|)(f_{\ell\pm})\;,$
(5)

above the lowest reaction threshold. Here, $f_{\ell\pm}$ denotes a complex-
valued matrix in the space of the considered meson-baryon channels
$c,c^{\prime}$, with entries $f_{\ell\pm}^{c^{\prime},c}(M_{\pi\Sigma})$,
while $(|\vec{p}^{\,\ast}|)$ is a diagonal matrix in this space, the entries
of which are given by the moduli of the three-momenta (for each channel $c$)
in the meson-baryon c.m. frame (see eq. 4) above the threshold of channel $c$,
and zero below it.

Figure 2: Representation of the final-state interaction (FSI) of the $S=-1$
meson-baryon pair $MB$. $\mathcal{M}$ is the amplitude for $\gamma
p\rightarrow K^{+}MB$ without FSI, and $\mathcal{T}$ is the $S=-1$ meson-
baryon scattering amplitude, which can be decomposed into partial waves
$f_{\ell\pm}$ Chew  _et al._ (1957a).

It is our aim to implement the scattering amplitudes $f_{0+}$ in the
photoproduction formalism in such a way, that it describes the final-state
s-wave interaction of the $S=-1$ meson-baryon pair $(MB)$ produced in the
$\gamma p\rightarrow K^{+}MB$ reaction as illustrated in Fig. 2. Clearly, we
have to find the combinations of (partial-wave projections of) the structure
functions $\mathcal{M}_{i=1,\ldots,16}$ which project on the $\ell=0$ state of
this meson-baryon pair. In fact, one can construct projected photoproduction
amplitudes $\mathcal{A}^{i=1,\ldots,4}_{0+}(s,M_{\pi\Sigma}^{2},t_{K})$ from
the $\mathcal{M}_{i}$ which have simple unitarity relations with the pertinent
s-wave scattering amplitudes,

$\mathrm{Im}(\mathcal{A}^{i}_{0+})=(f_{0+})^{\dagger}(|\vec{p}^{\,\ast}|)(\mathcal{A}^{i}_{0+})\,,\quad
i=1,\ldots,4\,.$ (6)

Note that there are _four_ amplitudes for the production of a state with
$\pi\Sigma$ being in an s-wave, corresponding to the four invariant amplitudes
for the photoproduction process $\gamma p\rightarrow K^{+}\Lambda^{\ast}$, see
Ref. Bruns (2020) for more details. The projections $\mathcal{A}^{i}_{0+}$ can
be found, e.g., by applying the Cutkosky rules Cutkosky (1960) to the $MB$
loop in Fig. 2, and studying the consequences of unitarity for the various
invariant structures in $\mathcal{M}^{\mu}$, or equivalently by the methods
used in Ref. Chew  _et al._ (1957b); Pearlstein and Klein (1957); Berends  _et
al._ (1967) to extract the multipole amplitudes for single-meson
photoproduction. Explicit expressions are provided in Appendix A. This
analysis is somewhat simplified employing an approximation which is motivated
by the observations made in Sec. 4 of Ref. Bruns (2020). In that, confining
ourselves to a low-energy analysis in the $\Lambda(1405)$ energy region, it is
reasonable to neglect higher partial waves
$\sim\mathcal{Y}_{\ell>0,m}(\theta_{\Sigma}^{\ast},\phi_{\Sigma}^{\ast})$ in
the decomposition of the $\mathcal{M}_{i}$ into spherical harmonics, and
express the projections solely through

$\overline{\mathcal{M}}_{i}(s,M_{\pi\Sigma}^{2},t_{K}):=\int\frac{d\Omega_{\Sigma}^{\ast}}{4\pi}\mathcal{M}_{i}(s,M_{\pi\Sigma}^{2},t_{K},t_{\Sigma},u_{\Sigma})\,.$
(7)

The s-wave resonance pole terms in the photoproduction amplitude are not
affected by this approximation. We also note that this approximation can be
used for any c.m. energy $\sqrt{s}$, as long as $M_{\pi\Sigma}$ stays
sufficiently close to the $MB$ threshold region. Therefore, kinematics
involving high-energy kaons can, in principle, be treated within the framework
of this section. Of course, the ChPT tree graphs might not be sufficient for
this purpose, and the elementary photoproduction amplitude would have to be
amended.

Neglecting contributions due to the $\pi\Sigma$ states with $\ell>0$, the
double-differential cross section for the $\gamma p\rightarrow K^{+}\pi\Sigma$
reaction can be expressed through the $\mathcal{A}^{i}_{0+}$ amplitudes as
follows,

$\frac{d^{2}\sigma}{d\Omega_{K}dM_{\pi\Sigma}}=\frac{|\vec{q}_{K}||\vec{p}_{\Sigma}^{\,\ast}|}{(4\pi)^{4}s|\vec{k}|}|\mathcal{A}|^{2}\,,$
(8)

where we introduced

$\displaystyle
4|\mathcal{A}|^{2}=(1\\!-\\!z_{K})\left|\mathcal{A}_{0+}^{1}+\mathcal{A}_{0+}^{2}\right|^{2}+(1\\!+\\!z_{K})\left|\mathcal{A}_{0+}^{1}-\mathcal{A}_{0+}^{2}\right|^{2}$
$\displaystyle\hskip
23.68048pt+(1\\!-\\!z_{K})\,\biggl{|}\mathcal{A}_{0+}^{1}+\mathcal{A}_{0+}^{2}$
$\displaystyle+\frac{2|\vec{q}_{K}|(1+z_{K})}{M_{K}^{2}-t_{K}}\left((\sqrt{s}+m_{N})\mathcal{A}_{0+}^{3}+(\sqrt{s}-m_{N})\mathcal{A}_{0+}^{4}\right)\biggr{|}^{2}$
$\displaystyle\hskip
23.68048pt+(1\\!+\\!z_{K})\,\biggl{|}\mathcal{A}_{0+}^{1}-\mathcal{A}_{0+}^{2}$
$\displaystyle-\frac{2|\vec{q}_{K}|(1\\!-\\!z_{K})}{M_{K}^{2}-t_{K}}\left((\sqrt{s}+m_{N})\mathcal{A}_{0+}^{3}-(\sqrt{s}-m_{N})\mathcal{A}_{0+}^{4}\right)\biggr{|}^{2}\,,$

with $z_{K}\equiv\cos\theta_{K}$, $\theta_{K}$ being the angle between
$\vec{q}_{K}$ and $\vec{k}$ in the overall c.m. frame.

Returning to the partial-wave unitarity statement in eq. 6, the proposed
coupled-channel formalism is now easily explained. We have to construct an
elementary photoproduction amplitude, e.g., from the tree graphs discussed
above, computing the according projected amplitudes
$\mathcal{A}^{i(\mathrm{tree})}_{0+}$, as detailed. Since we treat the
outgoing $K^{+}$ effectively as a spectator particle in our unitarization
procedure, as was also done in previous studies Nacher  _et al._ (1999); Lutz
and Soyeur (2005); Nakamura and Jido (2014); Roca and Oset (2013a, b); Mai and
Meißner (2015), the final-state interaction in our model is restricted to the
$S=-1$ meson-baryon subspace, which is the relevant sector for the formation
of the $\Lambda(1405)$. Therefore, unitarized amplitudes for $\gamma
p\rightarrow K^{+}MB$ will be taken as the coupled-channel vector

$(\mathcal{A}^{i}_{0+})=(\mathcal{A}^{i(\mathrm{tree})}_{0+})+(f_{0+})\left(8\pi
M_{\pi\Sigma}G(M_{\pi\Sigma})\right)(\mathcal{A}^{i(\mathrm{tree})}_{0+})\,.$
(9)

Here, $G(M_{\pi\Sigma})$ is a diagonal channel-space matrix, with entries
given by suitably regularized loop integrals

$\displaystyle\mathrm{i}\,$ $\displaystyle G^{c=MB}(M_{\pi\Sigma})=$ (10)
$\displaystyle\int_{\mathrm{reg.}}\frac{d^{4}l}{(2\pi)^{4}}\frac{1}{((p_{\Sigma}+q_{\pi}-l)^{2}-m_{B}^{2}+i\epsilon)(l^{2}-M_{M}^{2}+i\epsilon)}\,.$

The $MB=\pi\Sigma$ entries of the vector $\mathcal{A}^{i}_{0+}$ can then be
inserted in formula eq. 8 to obtain the required s-wave cross sections. Using
the fact that

$8\pi
M_{\pi\Sigma}\,\mathrm{Im}\,G^{MB}(M_{\pi\Sigma})=|\vec{p}_{B}^{\,\ast}|\Theta(M_{\pi\Sigma}-m_{B}-M_{M})\,,$
(11)

where $\Theta(\cdot)$ denotes the Heaviside step function, together with Eq.
(5), it is straightforward to show that the ansatz (9) solves the unitarity
requirement (6).

In the actual calculations presented in the next section we will use $f_{0+}$
amplitudes generated by two different approaches to $MB$ coupled-channel
interactions. Both of them derive the interaction kernel from the $MB$ chiral
Lagrangian taken up to the NLO order, but differ (among other details) in the
methods adopted to regularize the loop function $G$. In order to stay
consistent with the construction of the amplitudes, we will use in each case
the appropriate regulation procedure that is employed in the respective model.
Thus, in the case of the Bonn models presented in Refs. Mai and Meißner (2015)
and Sadasivan  _et al._ (2019), we take the dimensionally regularized integral
$G=-I_{MB}$ (see App. B of the first reference), while in the case of the
Prague model Bruns and Cieplý (2022), we take the loop integral of Eq. (8) in
Bruns and Cieplý (2019), divided by $8\pi M_{\pi\Sigma}\,g^{2}_{jb}$, to
conform with Eq. (11). Requiring that both versions of the loop function $G$
agree at the threshold energy $M_{MB}=m_{B}+M_{M}$ yields a matching relation
between the regulator scale $\alpha$ used in Refs. Bruns and Cieplý (2022,
2019) and the mass scale $\mu$ adopted in Refs. Mai and Meißner (2015);
Sadasivan  _et al._ (2019):

$\pi\alpha\overset{!}{=}m_{B}+M_{M}-m_{B}\log\left(\frac{m_{B}^{2}}{\mu^{2}}\right)-M_{M}\log\left(\frac{M_{M}^{2}}{\mu^{2}}\right)\,.$
(12)

For a _natural_ scale of $\mu\approx 1~{}{\rm GeV}$ used in dimensional
regularization this somewhat ad-hoc matching procedure provides
$\alpha_{\pi\Sigma}\approx 460$ MeV and $\alpha_{\bar{K}N}\approx 715$ MeV,
both values obtained via Eq. (12) at the respective channel thresholds.

After ensuring two-body unitarity in the final state we turn to another
aspect, the gauge invariance of the photoproduction amplitude. Could there be
a conflict between Eqs. (8)-(9) and gauge invariance? The answer is not
trivial. Even a gauge-invariant tree-level amplitude
$\mathcal{M}^{\mu}_{(\mathrm{tree})}$ does not necessarily generate a gauge-
invariant unitarized amplitude when it is just plugged into a loop integral to
couple it to the final-state interaction, since it will usually depend on the
loop momentum that is integrated over. We refer here to the discussions in van
Antwerpen and Afnan (1995); Borasoy  _et al._ (2005, 2007); Mai  _et al._
(2012), see also Refs. Haberzettl  _et al._ (2019); Haberzettl (2021). In our
present framework, the issue is resolved as follows. Given the functions
$\mathcal{A}^{i}_{0+}(s,M_{\pi\Sigma}^{2},t_{K})$, we can find a set of
invariant amplitudes $\mathcal{M}_{i}^{\mathcal{C}}$ (provided in App. A)
which form a gauge-invariant amplitude by construction, and yield the same
projections $\mathcal{A}^{i}_{0+}$. The s-wave cross-section calculated
directly from this gauge-invariant amplitude exactly equals the one in Eq.
(8). The difference between the invariant amplitudes, thus, resides in the
higher meson-baryon partial waves. This strategy works for every gauge-
invariant set of $\mathcal{A}^{i(\mathrm{tree})}_{0+}$ in (9), which allows
for some flexibility in the construction of models for these functions. For
example, we could add higher-order contact terms to the ChPT tree graphs. All
these models will yield cross-sections that are in accord with s-wave coupled-
channel unitarity, gauge invariance and the chiral low-energy theorems Bruns
(2020) (as long as the model used for the $f_{0+}$ does not spoil the proper
low-energy behaviour) at the same time.

Figure 3: The calculated $\pi\Sigma$ mass distributions are compared with
experimental data taken from Moriya  _et al._ (2013a). The lines demonstrate
the contributions from WT and other (BT+AN) graphs, and the impact of the $MB$
rescattering in the final state (FS).

##  III Comparison with data

We use the formalism described in the previous section to calculate the
$\pi\Sigma$ mass distributions observed in the CLAS experiment Moriya  _et
al._ (2013a) for the photoproduction reactions on a proton target.

First, we look at the relevance of various contributions to the s-wave
amplitude of the photoproduction process. In Fig. 3 we present the results
obtained for the mass spectra at $W=\sqrt{s}=2.0~{}{\rm GeV}$ when only the
tree level graphs contribute to the amplitude, and demonstrate the additional
impact of including the $MB$ rescattering in the final state. For the latter,
we have adopted only the recent version of the $\bar{K}N$ Prague model Bruns
and Cieplý (2022) to generate the rescattering amplitudes $f_{0+}$ required in
Eq. (9), but will present results including more models below. We observe that
when only the WT tree graph is considered (dotted red lines in the figure),
without the rescattering second term of Eq. (9), the $\pi\Sigma$ cross
sections are either exactly zero (for $\pi^{+}\Sigma^{-}$)111This result is
caused by a structure of the WT graph in this case., quite negligible (for
$\pi^{0}\Sigma^{0}$) or remain very small (for $\pi^{-}\Sigma^{+}$). The
addition of Born and anomalous terms leads to larger cross sections (shown by
dot-dashed blue lines in the figure) and two of the predicted $\pi\Sigma$
spectra compare reasonably well with those observed in the CLAS experiment
Moriya  _et al._ (2013a). The exception here is the $\pi^{-}\Sigma^{+}$
distribution that does not match the data neither in magnitude, nor in its
shape. It should be noted that all three $\pi\Sigma$ mass distributions
generated with photoproduction amplitudes constructed from tree level graphs
are very flat. The peak structure appears only when the $MB$ rescattering is
taken into account in the final state, as the dashed red (for only WT graph)
and continuous blue lines (for all graphs) show. In general, the inclusion of
the Born terms moves the peak structure to lower energies. We have also
checked that the contribution of the anomalous graphs is relatively small. For
this reason, we do not show their impact on the spectra separately to avoid
overcrowding the figures with too many lines. We also conclude that at least
at the c.m. energy $W=2.0~{}{\rm GeV}$ the description of the
$\pi^{0}\Sigma^{0}$ and $\pi^{+}\Sigma^{-}$ cross sections is quite
reasonable. Although we cannot say the same about the $\pi^{-}\Sigma^{+}$ mass
spectrum, one should bear in mind that our theoretical predictions are
provided without any adjustment of the $MB$ rescattering amplitudes that have
a major impact on the calculated spectra but are generated by a model fitted
to a completely different sector of experimental data and for much higher
energies from the $\bar{K}N$ threshold up.

It is also well known that the chirally motivated $\bar{K}N$ models provide
very different predictions for the energies below the $\bar{K}N$ threshold as
well as in the isovector sector Cieplý  _et al._ (2016). Thus, we felt it
necessary to check the sensitivity of the calculated $\pi\Sigma$ mass spectra
to variations of the $MB$ re-scattering amplitudes. In Fig. 4 we show our
results obtained at two sample energies, $W=2.0$ (left figures) and
$2.4~{}{\rm GeV}$ (right figures), employing four different $\bar{K}N$ models
that provide the final state rescattering amplitudes $f_{0+}$ required in Eq.
(9). Besides the already mentioned Prague amplitudes Bruns and Cieplý (2022),
used to calculate the spectra shown in Fig. 3 and tagged as P model here, we
also show results obtained with three versions of the Bonn model amplitudes:
B2, B4 Mai and Meißner (2015), and BW Sadasivan  _et al._ (2019). For
$W=2.0~{}{\rm GeV}$, the continuous blue lines generated by the P model are
exactly the same as those shown in Fig. 3.

Figure 4: Comparison of $\pi\Sigma$ mass distributions calculated at two c.m.
energies, $W=2.0$ and $2.4~{}{\rm GeV}$, while employing $MB$ amplitudes
generated by four different coupled-channel models tagged as P Bruns and
Cieplý (2022), B2, B4 Mai and Meißner (2015) and BW Sadasivan  _et al._
(2019). The experimental data are taken from Moriya  _et al._ (2013a).

The first impression one gets from Fig. 4 is that, taking aside the
$\pi^{0}\Sigma^{0}$ and $\pi^{+}\Sigma^{-}$ results at $W=2.0~{}{\rm GeV}$,
the theoretical predictions provide much higher cross sections than those
observed experimentally. We remark that similarly large cross sections of
$d\sigma/dM_{\pi\Sigma}\approx 5-10\;\mu{\rm b/GeV}$ at the peak mass were
predicted in the early Refs. Nacher  _et al._ (1999); Lutz and Soyeur (2005)
as well. As we will illustrate below, however, there is a hope that the
magnitude of the $\pi\Sigma$ mass spectra can be reduced by tuning the
rescattering contribution to the photoproduction amplitude. We also note that
our formalism is based on LO ChPT which is expected to work much better at low
energies. Specifically, a kaon momentum of
$|\vec{q}_{K}|(W=2\,\mathrm{GeV},M_{\pi\Sigma}=1.35\,\mathrm{GeV})\approx
350\,\mathrm{MeV}$ is still reasonable for three-flavor ChPT. However, at
$W=2.4~{}{\rm GeV}$ the kaon momentum is much larger, $|\vec{q}_{K}|\approx
717$ MeV, probably too large for the LO ChPT photo-kernel. Therefore, the
worse description of the observed mass distributions at $W=2.4~{}{\rm GeV}$
can be anticipated.

It is interesting to note the close proximity of the theoretical predictions
made with the P and BW model amplitudes. Despite the Prague and Bonn
approaches to $\pi\Sigma-\bar{K}N$ coupled-channel interactions do differ
significantly in several aspects including the sets of experimental data their
model parameters were fitted to, the P and BW models generate quite similar
$\pi\Sigma$ photoproduction mass spectra up to about $1400$ MeV. This can be
understood by comparing the respective rescattering amplitudes which we
demonstrate in Fig. 5. There, one can see that both models generate very
similar $MB$ amplitudes for energies up to the $\bar{K}N$ threshold.

Figure 5: The real (left) and imaginary (right) parts of s-wave scattering
amplitudes for reactions $\pi^{0}\Sigma^{0}\rightarrow\pi^{0}\Sigma^{0}$,
$K^{-}p\rightarrow\pi^{0}\Sigma^{0}$, and $K^{-}p\rightarrow K^{-}p$. The
amplitudes were generated by four different $\bar{K}N$ coupled-channel models
specified in the text. The P model amplitude of the inelastic
$K^{-}p\rightarrow\pi^{0}\Sigma^{0}$ process is shown with an opposite sign
for an easier comparison with the other models.

The other two Bonn models, B2 and B4, were in fact used in fits that included
the $\pi\Sigma$ photoproduction data from the CLAS experiment Mai and Meißner
(2015). However, in those fits a simplified phenomenological approach was
adopted to model the photoproduction part (see also Ref. Roca and Oset
(2013b)) of the process in terms of makeshift energy-dependent constants used
to multiply the amplitudes responsible for the $MB$ rescattering in the final
state. When compared with our present formulation of the photoproduction
amplitude given in Eq. (9) the B2 and B4 models correspond to setting the tree
graphs in the first term on the r.h.s. to zero and (at the same time)
replacing them by energy-dependent constants in the second term. It is obvious
that such an ad-hoc treatment cannot reflect fully the complexity of the
photoproduction process and the resulting $MB$ amplitudes may not be quite
reliable. However, the B2 and B4 amplitudes still describe the data well for
$K^{-}p$ reactions and represent a good option to test the model dependence of
our theoretical predictions. Looking at Fig. 4 it seems that the available
$\bar{K}N$ models are flexible enough to generate a peak in the $\pi\Sigma$
mass spectra at varied energies, and at least in the $\pi^{-}\Sigma^{+}$
channel provide even two peaks in case the data would support such option. The
BW model predicts two peaks also in the $\pi^{0}\Sigma^{0}$ mass spectra, the
one at higher mass quite close to the $\bar{K}N$ threshold where the chirally
motivated approaches predict a resonance interpreted as a $K^{-}p$ molecular
state.

Coming back to the the $\pi\Sigma$ mass spectra, we recall that generating the
peak structures is only possible including the $MB$ rescattering in the final
state as shown in fig. 3. Algebraically, this is represented by the second
term in Eq. (9) and one may consider modifications of either the rescattering
amplitude or of the loop function that connects it to the tree level
photoproduction graphs. The first option would require a completely new fit to
the experimental data including both, the $K^{-}p$ reactions data (at
threshold and higher energies) as well as the $\pi\Sigma$ mass spectra
discussed here. The complexity of such fits goes well beyond the scope of our
current work, but we wish to illustrate how the magnitude of the
photoproduction cross sections can be tuned by modifying the loop function
$G(M_{\pi\Sigma})$ that enters Eq. (9). This is demonstrated in Fig. 6 where
we present the results obtained with different choices of the regularization
scales – the inverse range $\alpha$ for the P model and the mass scale $\mu$
in case of the BW model. For simplicity, the same scale $\alpha$ or $\mu$ is
used for all ten $MB$ channels. The original mass spectra generated by the P
and BW models (employing different regularization scales in different
channels, matching those used to generate the rescattering amplitudes) are
presented in the figure for comparison as well.

Figure 6: A demonstration of the impact caused by modifying the regularization
scales adopted in the loop functions connecting the tree level photoproduction
graphs with the meson-baryon rescattering amplitudes. The top (bottom) figures
present the $\pi^{0}\Sigma^{0}$ mass spectra generated with the modified P
model (BW model), the results obtained with the unmodified loop functions are
shown for comparison as well.

The calculated spectra shown in Fig. 6 illustrate that lower values of the
regularization scales $\alpha$ or $\mu$ lead to smaller cross sections. It
seems that for $\alpha\approx 400$ MeV or for $\mu\approx 1.0~{}{\rm GeV}$ the
magnitude of the generated $\pi\Sigma$ distributions is comparable to the one
found in the CLAS experiment. Of course, the peak position and structure of
the generated mass spectra require additional modifications that are driven by
the energy dependence of the adopted $f_{0+}$ amplitudes. As already stated,
there is a room for such an adjustment since different $MB$ coupled-channel
models provide varied energy dependence of the amplitudes in between the
$\pi\Sigma$ and $\bar{K}N$ thresholds.

Finally, we would like to comment on the feasibility of altering only the
first $MB$ loop functions that appears in Eq. (9) while keeping unchanged the
rescattering amplitudes and the loop functions used in coupled-channel
approaches that generate them. Since the vertices connecting the
photoproduction tree graphs with the $MB$ rescattering part have a different
structure than the $MB\rightarrow M^{\prime}B^{\prime}$ vertices one can
consider altering the first, e.g., by imposing additional form factors on
them, while keeping the latter intact. Thus, the adopted modifications of the
first loop functions that we used to tune the magnitude of the $\pi\Sigma$
cross sections can effectively relate to modifications of the $\cal{M}$ vertex
in Fig. 2. We note that a good reproduction of the CLAS data presented in Ref.
Nakamura  _et al._ (2015) required a combined impact of modifications of the
first $MB$ loop functions, addition of phenomenological contact terms
contributing to the $\cal{M}$ vertex as well as addition of form factors
applied to it.

##  IV Summary and Conclusion

In this article, we have outlined a methodology to incorporate the meson-
baryon final-state interaction into the two-meson photoproduction amplitude
through partial-wave amplitudes $f_{\ell\pm}$ constructed within the framework
of _Unitarized ChPT_. This formalism can be used to implement coupled-channel
unitarity, low-energy theorems from ChPT and gauge invariance in the
description of the photoproduction process. Our results obtained with two
versions of modern $\bar{K}N$ models, adopted to generate the $MB$ amplitudes,
demonstrate the crucial role played by the final-state interaction in the
$\pi\Sigma$ photoproduction process. When the $MB$ rescattering is omitted,
only the considered tree graphs contribute to the photoproduction amplitude,
and the generated cross sections are rather small and flat. The peak structure
observed in the mass spectra reported by the CLAS collaboration and related to
a formation of the $\Lambda(1405)$ resonance appears only when the $MB$
rescattering in the final state is accounted for in our formalism.

Our results show that the formalism is capable of reproducing the experimental
data at c.m. energies $W\approx 2.0~{}{\rm GeV}$ where the pertinent kaon
momenta are relatively small, complying with the limitations of three-flavor
ChPT. The situation is worse at higher energies where our predictions provide
too large cross sections when compared with the CLAS data. However, we would
like to emphasize once again that these predictions are made without
introducing any adjustment to the $\pi\Sigma-\bar{K}N$ coupled-channel models
that were fitted to describe the data on $K^{-}p$ reactions, at a completely
different sector of kinematics and energies. We have also refrained from
introducing any additional mechanisms in the tree level photoproduction graphs
that would go beyond the standard ChPT approach, nor applied any ad-hoc energy
dependent factors to moderate the generated mass spectra. In this sense our
current predictions are completely parameter-free. A much better agreement
with the experimental data can be achieved by a combination of including the
$\pi\Sigma$ photoproduction data in the fits of the $\bar{K}N$ models and
maybe by applying an additional form factor to the $\mathcal{M}$ vertex
depicted in Fig. 2. Moreover, we have shown that adopting different $\bar{K}N$
models for the $f_{0+}$ amplitudes leads to varied structure of the computed
$\pi\Sigma$ mass distributions that can accommodate spectra with either one or
two peaks. At the same time the magnitude of the calculated cross sections can
be tuned by modifying the first $MB$ loop function or the $\mathcal{M}$
vertex.

Of course, it is also natural to ask about a role played by other kinds of
final-state interaction inherent in the process, as, e.g., the pion-kaon
interaction, irreducible three-body interactions, or even triangle-graph
mechanisms as studied in Ref. Wang  _et al._ (2017). In this respect we note
that effects due to an enhanced final-state interaction in the other channels
have already been subtracted in the CLAS data with which we compare our
predictions Moriya  _et al._ (2013a). There, it was also reported that these
effects have only a moderate impact on the $\pi\Sigma$ invariant mass spectra.
Thus, it seems that a direct application of the approach proposed here is a
reasonable strategy. In the future, one should also make efforts for a more
complete description of the photoproduction process, along the lines explained
so far. This would enable us to compare our predictions with a more complete
subset of the provided data, as, e.g., the kaon angular distributions, which
are presumably sensitive to final-state interaction in sectors not considered
in this work.

Acknowledgements

The work of P.C.B. and A.C. was supported by the Grant Agency of the Czech
Republic under Grant No. 19-19640S. M.M. acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) and the NSFC through
the funds provided to the Sino-German Collaborative Research Center CRC 110
“Symmetries and the Emergence of Structure in QCD” (DFG Project-ID 196253076 -
TRR 110, NSFC Grant No. 12070131001).

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##  Appendix A Projection on $\mathbf{\pi\Sigma}$ partial waves

Here we provide explicit expressions for the s-wave projections of the
photoproduction amplitude for $MB=\pi\Sigma$, but it is obvious that analogous
expressions for the other meson-baryon-channels can be found by simple
replacements of the appropriate masses. As a first step, we define the
combinations

$\displaystyle\mathcal{C}_{0+}^{1}=\mathcal{M}_{1}^{\prime}+(\sqrt{s}+m_{N})\mathcal{M}_{5}^{\prime}+(\sqrt{s}-M_{\pi\Sigma})\left(\mathcal{M}_{9}^{\prime}+(\sqrt{s}+m_{N})\mathcal{M}_{13}^{\prime}\right)\,,$
(A.1)
$\displaystyle\mathcal{C}_{0+}^{2}=\mathcal{M}_{1}^{\prime}-(\sqrt{s}-m_{N})\mathcal{M}_{5}^{\prime}-(\sqrt{s}+M_{\pi\Sigma})\left(\mathcal{M}_{9}^{\prime}-(\sqrt{s}-m_{N})\mathcal{M}_{13}^{\prime}\right)\,,$
(A.2)
$\displaystyle\mathcal{C}_{0+}^{3}=\mathcal{M}_{1}^{\prime}+(\sqrt{s}-M_{\pi\Sigma})\mathcal{M}_{9}^{\prime}+\frac{1}{2}(\sqrt{s}+m_{N})\bigl{(}\mathcal{M}_{2}^{\prime}+(\sqrt{s}-m_{N})\mathcal{M}_{6}^{\prime}+(\sqrt{s}-M_{\pi\Sigma})\left(\mathcal{M}_{10}^{\prime}+(\sqrt{s}-m_{N})\mathcal{M}_{14}^{\prime}\right)\bigr{)}\,,$
(A.3)
$\displaystyle\mathcal{C}_{0+}^{4}=\mathcal{M}_{1}^{\prime}-(\sqrt{s}+M_{\pi\Sigma})\mathcal{M}_{9}^{\prime}-\frac{1}{2}(\sqrt{s}-m_{N})\bigl{(}\mathcal{M}_{2}^{\prime}-(\sqrt{s}+m_{N})\mathcal{M}_{6}^{\prime}-(\sqrt{s}+M_{\pi\Sigma})\left(\mathcal{M}_{10}^{\prime}-(\sqrt{s}+m_{N})\mathcal{M}_{14}^{\prime}\right)\bigr{)}\,,$
(A.4)

employing the following abbreviations:

$\displaystyle\mathcal{M}_{1}^{\prime}$ $\displaystyle:=$
$\displaystyle\overline{\mathcal{M}}_{1}-\frac{1}{3}(E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{3}\,,\qquad\mathcal{M}_{5}^{\prime}:=\overline{\mathcal{M}}_{5}+\frac{1}{3}(E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{7}\,,$
(A.5) $\displaystyle\mathcal{M}_{9}^{\prime}$ $\displaystyle:=$
$\displaystyle\overline{\mathcal{M}}_{9}+\frac{1}{3}(E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{11}\,,\qquad\mathcal{M}_{13}^{\prime}:=\overline{\mathcal{M}}_{13}-\frac{1}{3}(E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{15}\,,$
(A.6) $\displaystyle\mathcal{M}_{2}^{\prime}$ $\displaystyle:=$
$\displaystyle\overline{\mathcal{M}}_{2}+\frac{1}{3M_{\pi\Sigma}}(4E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{3}\,,\qquad\mathcal{M}_{6}^{\prime}:=\overline{\mathcal{M}}_{6}+\frac{1}{3M_{\pi\Sigma}}(4E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{7}\,,$
(A.7) $\displaystyle\mathcal{M}_{10}^{\prime}$ $\displaystyle:=$
$\displaystyle\overline{\mathcal{M}}_{10}+\frac{1}{3M_{\pi\Sigma}}(4E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{11}\,,\qquad\mathcal{M}_{14}^{\prime}:=\overline{\mathcal{M}}_{14}+\frac{1}{3M_{\pi\Sigma}}(4E_{\Sigma}^{\ast}-m_{\Sigma})\overline{\mathcal{M}}_{15}\,.$
(A.8)

See Eq. (2) for the definition of the $\mathcal{M}_{i}$, and Eq. (7) for the
definition of the $\overline{\mathcal{M}}_{i}$. We point out that the
structure functions $\mathcal{M}_{4,8,12,16}$ are always eliminated via the
gauge-invariance constraints

$\displaystyle(s-m_{N}^{2})\mathcal{M}_{2}\,$ $\displaystyle\overset{!}{=}$
$\displaystyle\,(u_{\Sigma}-m_{\Sigma}^{2})\mathcal{M}_{3}+(t_{K}-M_{K}^{2})\mathcal{M}_{4}\,,$
$\displaystyle 2\mathcal{M}_{1}+(s-m_{N}^{2})\mathcal{M}_{6}\,$
$\displaystyle\overset{!}{=}$
$\displaystyle\,(u_{\Sigma}-m_{\Sigma}^{2})\mathcal{M}_{7}+(t_{K}-M_{K}^{2})\mathcal{M}_{8}\,,$
$\displaystyle(s-m_{N}^{2})\mathcal{M}_{10}\,$ $\displaystyle\overset{!}{=}$
$\displaystyle\,(u_{\Sigma}-m_{\Sigma}^{2})\mathcal{M}_{11}+(t_{K}-M_{K}^{2})\mathcal{M}_{12}\,,$
$\displaystyle 2\mathcal{M}_{9}+(s-m_{N}^{2})\mathcal{M}_{14}\,$
$\displaystyle\overset{!}{=}$
$\displaystyle\,(u_{\Sigma}-m_{\Sigma}^{2})\mathcal{M}_{15}+(t_{K}-M_{K}^{2})\mathcal{M}_{16}\,.$
(A.9)

In the simple case of structure functions $\mathcal{M}_{i}$ independent of
$t_{\Sigma},\,u_{\Sigma}$, vanishing for $i=3,7,11,15$, we have
$\mathcal{M}_{i}^{\prime}=\mathcal{M}_{i}$ . In fact, we can easily construct
a gauge-invariant amplitude $\mathcal{M}^{\mu}$ of such a simplified form,
which yields a set of prescribed
$\mathcal{C}_{0+}^{i}(s,M_{\pi\Sigma}^{2},t_{K})$,

$\displaystyle\mathcal{M}_{1}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{s}M_{\pi\Sigma}}\biggl{(}(\sqrt{s}+M_{\pi\Sigma})(\sqrt{s}-m_{N})\mathcal{C}_{0+}^{1}-(\sqrt{s}-M_{\pi\Sigma})(\sqrt{s}+m_{N})\mathcal{C}_{0+}^{2}\biggr{)}\,,$
(A.10) $\displaystyle\mathcal{M}_{2}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s}M_{\pi\Sigma}}\biggl{(}(\sqrt{s}+M_{\pi\Sigma})\mathcal{C}_{0+}^{3}+(\sqrt{s}-M_{\pi\Sigma})\mathcal{C}_{0+}^{4}\biggr{)}\,,$
(A.11) $\displaystyle\mathcal{M}_{3}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle 0\,,$ (A.12) $\displaystyle\mathcal{M}_{5}^{\mathcal{C}}$
$\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{s}M_{\pi\Sigma}}\biggl{(}(\sqrt{s}+M_{\pi\Sigma})\mathcal{C}_{0+}^{1}+(\sqrt{s}-M_{\pi\Sigma})\mathcal{C}_{0+}^{2}\biggr{)}\,,$
(A.13) $\displaystyle\mathcal{M}_{6}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s}M_{\pi\Sigma}}\biggl{(}\frac{\sqrt{s}+M_{\pi\Sigma}}{\sqrt{s}+m_{N}}\left(\mathcal{C}_{0+}^{3}-\mathcal{C}_{0+}^{1}\right)+\frac{\sqrt{s}-M_{\pi\Sigma}}{\sqrt{s}-m_{N}}\left(\mathcal{C}_{0+}^{2}-\mathcal{C}_{0+}^{4}\right)\biggr{)}\,,$
(A.14) $\displaystyle\mathcal{M}_{7}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle 0\,,$ (A.15) $\displaystyle\mathcal{M}_{9}^{\mathcal{C}}$
$\displaystyle=$
$\displaystyle-\frac{1}{4\sqrt{s}M_{\pi\Sigma}}\biggl{(}(\sqrt{s}-m_{N})\mathcal{C}_{0+}^{1}+(\sqrt{s}+m_{N})\mathcal{C}_{0+}^{2}\biggr{)}\,,$
(A.16) $\displaystyle\mathcal{M}_{10}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s}M_{\pi\Sigma}}\left(\mathcal{C}_{0+}^{4}-\mathcal{C}_{0+}^{3}\right)\,,$
(A.17) $\displaystyle\mathcal{M}_{11}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle 0\,,$ (A.18) $\displaystyle\mathcal{M}_{13}^{\mathcal{C}}$
$\displaystyle=$
$\displaystyle\frac{1}{4\sqrt{s}M_{\pi\Sigma}}\left(\mathcal{C}_{0+}^{2}-\mathcal{C}_{0+}^{1}\right)\,,$
(A.19) $\displaystyle\mathcal{M}_{14}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{s}M_{\pi\Sigma}}\left(\frac{\mathcal{C}_{0+}^{1}-\mathcal{C}_{0+}^{3}}{\sqrt{s}+m_{N}}+\frac{\mathcal{C}_{0+}^{2}-\mathcal{C}_{0+}^{4}}{\sqrt{s}-m_{N}}\right)\,,$
(A.20) $\displaystyle\mathcal{M}_{15}^{\mathcal{C}}$ $\displaystyle=$
$\displaystyle 0\,,$ (A.21)

with $\mathcal{M}_{4,8,12,16}^{\mathcal{C}}$ accordingly fixed from the gauge-
invariance constraints. Finally, we define

$\displaystyle\mathcal{A}_{0+}^{1}$ $\displaystyle:=$
$\displaystyle\sqrt{E_{\Sigma}^{\ast}+m_{\Sigma}}\sqrt{E_{\pi\Sigma}+M_{\pi\Sigma}}\,\left(\mathcal{C}_{0+}^{1}\right)\sqrt{E_{N}-m_{N}}\,/\sqrt{2M_{\pi\Sigma}}\,,$
$\displaystyle\mathcal{A}_{0+}^{2}$ $\displaystyle:=$
$\displaystyle\sqrt{E_{\Sigma}^{\ast}+m_{\Sigma}}\sqrt{E_{\pi\Sigma}-M_{\pi\Sigma}}\,\left(\mathcal{C}_{0+}^{2}\right)\sqrt{E_{N}+m_{N}}\,/\sqrt{2M_{\pi\Sigma}}\,,$
$\displaystyle\mathcal{A}_{0+}^{3}$ $\displaystyle:=$
$\displaystyle\sqrt{E_{\Sigma}^{\ast}+m_{\Sigma}}\sqrt{E_{\pi\Sigma}+M_{\pi\Sigma}}\,\left(\mathcal{C}_{0+}^{3}\right)\sqrt{E_{N}-m_{N}}\,/\sqrt{2M_{\pi\Sigma}}\,,$
$\displaystyle\mathcal{A}_{0+}^{4}$ $\displaystyle:=$
$\displaystyle\sqrt{E_{\Sigma}^{\ast}+m_{\Sigma}}\sqrt{E_{\pi\Sigma}-M_{\pi\Sigma}}\,\left(\mathcal{C}_{0+}^{4}\right)\sqrt{E_{N}+m_{N}}\,/\sqrt{2M_{\pi\Sigma}}\,,$

where $E_{\pi\Sigma}$ is the c.m. energy of the $\pi\Sigma$ pair,
$E_{\pi\Sigma}=\sqrt{s}-E_{K}=(s+M_{\pi\Sigma}^{2}-M_{K}^{2})/(2\sqrt{s})$. We
also note that $|\vec{q}_{K}|=\sqrt{E_{\pi\Sigma}^{2}-M_{\pi\Sigma}^{2}}\,$
and $E_{N}=(s+m_{N}^{2})/(2\sqrt{s})$ ,
$E_{\Sigma}^{\,\ast}=(M_{\pi\Sigma}^{2}+m_{\Sigma}^{2}-M_{\pi}^{2})/(2M_{\pi\Sigma})$
.


Further author information: (Send correspondence to K.A.)
K.A<EMAIL_ADDRESS>+49 551 384 979-186
J.H<EMAIL_ADDRESS>
D.B<EMAIL_ADDRESS>
# Autonomous on-board data processing and instrument calibration software for
the SO/PHI

K. Albert  Max Planck Institute for Solar System Research, Justus-von-Liebig-
Weg 3, Göttingen, Germany  J. Hirzberger  Max Planck Institute for Solar
System Research, Justus-von-Liebig-Weg 3, Göttingen, Germany  D. Busse  Max
Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3,
Göttingen, Germany  T. Lange  Institute of Computer and Network Engineering at
the TU Braunschweig, Hans-Sommer-Stra$\ss$e 66, Braunschweig, Germany  M.
Kolleck  Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg
3, Göttingen, Germany  B. Fiethe  Institute of Computer and Network
Engineering at the TU Braunschweig, Hans-Sommer-Stra$\ss$e 66, Braunschweig,
Germany  D. Orozco Suárez  Instituto de Astrofísica de Andalucía (IAA - CSIC),
Apartado 3004, Granada, Spain  J. Woch  Max Planck Institute for Solar System
Research, Justus-von-Liebig-Weg 3, Göttingen, Germany  J. Schou  Max Planck
Institute for Solar System Research, Justus-von-Liebig-Weg 3, Göttingen,
Germany  J. Blanco Rodríguez  Universidad de Valencia, Catedrático José
Beltrán 2, Paterna (Valencia), Spain  A. Gandorfer  Max Planck Institute for
Solar System Research, Justus-von-Liebig-Weg 3, Göttingen, Germany  Y. Guan
Institute of Computer and Network Engineering at the TU Braunschweig, Hans-
Sommer-Stra$\ss$e 66, Braunschweig, Germany  J.P. Cobos Carrascosa  Instituto
de Astrofísica de Andalucía (IAA - CSIC), Apartado 3004, Granada, Spain  D.
Hernández Expósito  Universidad de Valencia, Catedrático José Beltrán 2,
Paterna (Valencia), Spain  J.C. del Toro Iniesta  Instituto de Astrofísica de
Andalucía (IAA - CSIC), Apartado 3004, Granada, Spain  S. K. Solanki  Max
Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3,
Göttingen, Germany  H. Michalik  Institute of Computer and Network Engineering
at the TU Braunschweig, Hans-Sommer-Stra$\ss$e 66, Braunschweig, Germany

###### Abstract

The extension of on-board data processing capabilities is an attractive option
to reduce telemetry for scientific instruments on deep space missions. The
challenges that this presents, however, require a comprehensive software
system, which operates on the limited resources a data processing unit in
space allows.
We implemented such a system for the Polarimetric and Helioseismic Imager
(PHI) on-board the Solar Orbiter (SO) spacecraft. It ensures autonomous
operation to handle long command-response times, easy changing of the
processes after new lessons have been learned and meticulous book-keeping of
all operations to ensure scientific accuracy. This contribution presents the
requirements and main aspects of the software implementation, followed by an
example of a task implemented in the software frame, and results from running
it on SO/PHI.
The presented example shows that the different parts of the software framework
work well together, and that the system processes data as we expect. The
flexibility of the framework makes it possible to use it as a baseline for
future applications with similar needs and limitations as SO/PHI.

##  1 Introduction

Many state of the art scientific space instruments produce more data than what
is possible to download to ground. This discrepancy is especially significant
when, due to the orbit design, the mission can only ensure a low amount of
telemetry. One way to reduce the necessary telemetry is to extend the on-board
processing capabilities of the instrument.
This strategy is adapted for the Polarimetric and Helioseismic Imager
(PHI)[1]. PHI is the first imaging solar spectropolarimeter on-board a deep
space mission: the Solar Orbiter (SO). It will image the solar photosphere in
the light of the Fe I 617.3 nm absorption line, at six wavelengths and in four
polarisation states of light. This spectral line is sensitive to the magnetic
field (through the Zeeman effect) and to the line of sight (LOS) velocities in
the photosphere (due to the Doppler effect). Through the measurements
described we can determine characteristics of the magnetic field and the LOS
velocity at the average of the formation height for the absorption line. To
retrieve these quantities the polarised radiative transfer equation (RTE) must
be inverted[2].
SO/PHI implements on-board autonomous instrument calibration and on-board
autonomous science data analysis including RTE inversion. This means that the
calibration data can be directly applied on-board to the science observables
without adding to the telemetry volume. Therefore the necessary telemetry is
reduced to the continuum intensity, the magnetic field vector and the LOS
velocity instead of the raw science observables. These functions are
implemented for the first time on this type of instrument with the limited
computational resources a space instrument offers. Comparable systems have
been implemented on the AMPTE IRM 3D Plasma Instrument, the Giotto RPA
Experiment[3] and the SOHO/MDI[4]. However, all these cases are considerably
more limited in capability than the system implemented for SO/PHI.
The hardware available for data processing[5],[6] integrates a system
controller microprocessor, a system supervisor Field Programmable Gate Array
(FPGA), and two dynamically reconfigurable FPGAs (RFPGAs). See fig. 1. The
processing is aided with a 1 GiB processing memory connected to one of the
RFPGAs, and a much smaller, 256 MiB memory connected to the microprocessor, of
which only a part is available for image processing. A non-volatile storage of
512 GiB is available for storing data in the instrument. The in flight
reconfigurability of the two Xilinx FPGAs enables us to take a time-sharing
approach, saving volume, mass and energy.

Figure 1: The data processing of SO/PHI is distributed between a Leon-3FT
microprocessor and Xilinx Virtex-4 RFPGAs. These units are aided by memories
of different capacities. Data is stored in the non-volatile memory.

SO/PHI captures science images during dedicated observation windows along the
orbits, each lasting for 10 days. An in-orbit calibration campaign is
associated with each observation window with the aim of characterising the
instrument in the observational conditions. The data processing is done after
the observation window, using the data collected in the calibration campaign.
The planning of the operations and the data processing is done together and on
long term, well ahead of the time of their execution. This includes the
specification of all input and output addresses, as well as the selection of
the correct calibration data for the reduction of the dataset.
In the following we describe the software frame for the on-board data
calibration and on-board data pre-processing steps performed before the
inversion of the RTE.

##  2 Requirements of the data processing system

The data processing system of SO/PHI performs three different tasks. It:

  * •

determines the optimal operational parameters for the instrument (e.g. focus
position) from dedicated datasets, refining the result in a second
observation,

  * •

calculates the calibration data from datasets obtained during dedicated
observations,

  * •

processes the raw science observables by removing the instrumental effects,
inverting the RTE and compressing the data.

The processing tasks involve computationally demanding image processing steps.
To shorten run time, these functions are implemented in one of the two RFPGAs.
However, a back-up solution, implemented in the microprocessor is also
required to facilitate early testing during development, as well as to
contribute to fault tolerance. This results in the need of ensuring that the
same task can run on two platforms with different amount of processing memory.
To reduce the implementation complexity of the RFPGA functions, a fixed point
notation has been adapted for the processing system. To maintain the
requirements for scientific accuracy the images are rescaled before the
processing steps are applied to them.
Due to the novelty of the mission a number of questions related to the data
processing steps may only be answered after first light. This requires the
possibility for easy changes in the task implementations.
Due to the autonomous execution and no access to the intermediate results, the
only record of the operations is kept in a processing log file, as part of the
metadata of each dataset. The metadata recording therefore must be integral
and secure, as well as provide data that facilitates error search and
improvement of the algorithms.

##  3 Implementation of the data processing system

We take a unified approach to the tasks, implementing them in the form of
pipelines. A pipeline is defined to be a series of operations executed on the
same target (a dataset or a subset of a dataset). All pipelines start with
loading the target from the non-volatile memory into the processing memory,
and end by either storing the result dataset in the non-volatile memory, or by
returning a calculated parameter.
To implement the computationally demanding functions on the RFPGAs we define
several FPGA configurations, each of them containing dedicated hardware logic
that implements parts of the on-board processing (e.g. RTE inversion, or a
group of different image processing functions), and load them into the RFPGAs
as needed.[7]
To fulfil the requirement on easily changing the pipelines a block approach is
taken. The blocks access the image processing functions running on the RFPGAs,
or the microprocessor in back-up implementation, and combine them into useful
steps. To facilitate changing their order, they have a unified interface.
Within the pipeline all blocks have the same target, loaded as the first step
of the pipeline into the processing memory. The blocks ensure that the target
at their interface is scaled to be represented on the maximum number of
available bits. All blocks may load additional data necessary for the
processing steps, which are discarded at their termination. Each block must
also determine the dataset history, to facilitate their free ordering. The
parameters necessary for this are not passed as variables in-between the
blocks, instead they are transferred through the metadata. All blocks are
executed sequentially, and a pipeline is manipulating the target only through
blocks.
Due to the requirement on executing the pipeline both with and without the
RFPGAs, the pipeline must run with different amounts of processing memory. In
the back-up solution, the memory cannot hold a full dataset. Therefore, the
pipeline is parametrised and can be configured to run on a subset of the full
dataset. In many steps the pipeline needs one image of the dataset. However,
there are exceptions in which corresponding pixels from multiple images are
necessary. Consequently, we split the pipeline into an image linear and image
parallel part, each running on a part of the full dataset: a number of images
or a number of rows from all images.
Error handling is done with two error levels: errors and warnings, based on
their severity. Errors interrupt the processing, while warnings are checked on
ground to evaluate the correctness of the results. Both are marked in the
return value of all operations, blocks and pipelines. In order to log multiple
warnings and errors detected, the return values are defined as bit masks.
The metadata of a dataset contains all relevant information and is recorded at
different times: at image acquisition, and at data processing. The information
is organised by entries, containing parameters that form a logical unit (e.g.
the return value and parameters of the image processing functions), and the
metadata file is treated as a log.

##  4 Example pipeline and its run

To demonstrate how the system works, we take an example of a science data pre-
processing pipeline, consisting of 3 steps: dark field subtraction, flat field
division, and polarimetric demodulation. This is the minimum that is needed
for pre-processing, and would be followed by RTE inversion for full on-board
analysis.
Each step becomes a block, and the whole pipeline is split into the image
linear and image parallel part. Each of these starts with the loading of the
target dataset, and ends with the storage of the result (see fig. 2). Each
block scales the datasets to adapt to the defined scaling interface between
them, and to calculate the results at the required accuracy. The blocks also
check the history of the target dataset and adapt the calibration data applied
to them. Additional verifications generate warnings to determine possible
problem-sources on ground. Each pipeline is inside a loop and is executed the
number of times necessary to process the whole dataset, when this is split
into sub-parts.

(a) Image linear part.

(b) Image parallel part.

Figure 2: Example science data processing pipeline. In (a): the parameter ni
denotes the number of images processed at a given time, while nl is the number
of times the image linear portion of the pipeline has to run to process the
whole dataset. In (b): n$\lambda$ denotes the number of wavelengths processed
at a given time, while np denotes the iterations of the image parallel part of
the pipeline necessary for the full processing of the dataset.

(a) $I$

(b) $Q$

(c) $U$

(d) $V$

(e) I1

(f) I2

(g) I3

(h) I4

Figure 3: Test dataset at line centre. (a) - (d): Stokes images from
SOPHISM[8]. (e) - (h): Input to the pipeline, Stokes images modified according
to eq. 1. The input images show the dark field (the stripes), and the flat
field (smudges and dust grains). These effects are compensated by applying the
calibration data shown in fig. 4, then demodulated to arrive at the Stokes
images once again.

(a) Dark field.

(b) Flat field.

(c) DM Uniformity across FOV.

Figure 4: The calibration data used in the test run is recorded in the
laboratory. (a): The dark field shows the characteristic sensor pattern: four
distinct vertical stripes and row-to-row horizontal variations. (b): The flat
field used in the tests is the same for all wavelengths and modulation states.
(c): The variation of the 4$\times$4 demodulation matrix elements across the
FOV, treated as a dataset of 16 images.

(a) I1

(b) I2

(c) I3

(d) I4

(e) $I$

(f) $Q$

(g) $U$

(h) $V$

(i) $\delta I$

(j) $\delta Q$

(k) $\delta U$

(l) $\delta V$

Figure 5: Result dataset at line centre. (a) - (d): Input images corrected for
the effects of dark and flat field. (e) - (h): Result of the demodulation of
the images in fig. 5(a) to 5(d). (i) - (l): The difference after demodulation
when executing the pipeline in floating point representation on a desktop, and
executing it in fixed point representation on SO/PHI. The differences occur
inside the solar disk in form of quantisation noise. (The units are in fixed
point DN-s.)

The input data for the demonstration is generated from images taken by the
SDO/HMI instrument[9], transformed into Stokes images with the SO/PHI
instrument simulator, SOPHISM[8]. See fig. 3(a) to 3(d).

These images are further manipulated to represent the raw observables,
according to the following equation:

$\bm{I}^{\text{input}}_{\lambda}\unboldmath(x,y)=c_{1}c_{2}[\bm{M}(x,y)\cdot\bm{I}^{\text{Stokes}}_{\lambda}(x,y)]I^{\text{Flat}}(x,y)+I^{\text{Dark}}(x,y),$
(1)

where ”$\cdot$” stands for matrix multiplication, $\lambda$ denotes the
wavelength, $x$ and $y$ denote the image row and column size,
$\bm{I}^{\text{input}}_{\lambda}$ is the created test dataset, a $1\times 4$
matrix of images at $6$ different wavelengths,
$\bm{I}^{\text{Stokes}}_{\lambda}$ is the $1\times 4$ matrix of Stokes images
created with SOPHISM at 6 wavelengths, which can be expressed for any of the
wavelengths as:

$\bm{I}^{\text{Stokes}}=[I,Q,U,V]^{T},$

$\bm{M}$ is the instrument Modulation Matrix, a $4\times 4$ matrix of images,
describing how the instrument transforms the sought Stokes images into
practically measurable light levels for each pixel of its field of view (FOV),
applied for all wavelengths, $c_{1}$ is the constant that scales the
normalised images to represent the number of incident photons in 20 ms
exposure time, $c_{2}$ is the constant that converts the pixels from number of
photons accumulated on the detector to Digital Numbers (DN-s),
${I}_{\text{Flat}}$ is one image, the normalised flat field of the telescope,
applied to all wavelengths and polarisation states, and ${I}_{\text{Dark}}$ is
the dark field image of the sensor at 20 ms integration time in DN-s, the same
for all wavelengths and polarisation states.

Table 1: The metadata from the image linear pipeline portion shows all executed blocks, ending with the entry by the pipeline. Parameter 1 shows the ID of the Target Dataset (TD), Dark Field (DF), Flat Field (FF) and Result Dataset (RD). The dataset is processed in the RFPGA, treating all 24 images (see image start (S) and end (E)), and the full field of view (see row and column start (S) and end (E)) at once. The flat correction block generates a warning about produced NaN-s, which is also shown in the pipeline return. Return value | Function ID | param. 1 | param. 2 | image S | image E | row S | row E | col. S | col. E
---|---|---|---|---|---|---|---|---|---
Correct | Load | TD ID | 0 | 0 | 23 | 0 | 2047 | 0 | 2047
Correct | Correct Dark | DF ID | 0 | 0 | 23 | 0 | 2047 | 0 | 2047
W: NaNs | Correct Flat | FF ID | 0 | 0 | 23 | 0 | 2047 | 0 | 2047
Correct | Store | RD ID | 0 | 0 | 23 | 0 | 2047 | 0 | 2047
W: NaNs | Linear ppln. | RD ID | 0 | 0 | 23 | 0 | 2047 | 0 | 2047

The pipeline removes the dark field and the flat field from the input images,
then demodulates them to arrive to the Stokes images. The calibration data
applied by the pipeline is recorded in the laboratory, using the SO/PHI Flight
Model (see fig. 4). The dark field is the offset of the image detector, while
the flat field shows the gain variations across the FOV caused by the optics.
The Demodulation Matrix is the inverse of the Modulation Matrix used in eq. 1.
The test is run on a model of SO/PHI, which is representative in its data
processing unit.
We compare the output of the pipeline run on SO/PHI to results produced on the
desktop with floating point calculations on the same input data. We expect
these to give similar results with differences within an error margin, coming
from the pixel quantisation in fixed point notation. The differences are as we
expect them, appearing as quantisation noise over the solar disk. The standard
deviation of the differences is 0.05% for the Stokes $I/I_{r}$ image, and 6%,
5.3% and 1.8% for $Q/Q_{r}$, $U/U_{r}$ and $V/V_{r}$, respectively, where
$I_{r}$, $Q_{r}$, $U_{r}$ and $V_{r}$ denote the resulting Stokes images from
the reference pipeline. The polarimetric errors coming from the fixed point
representation are 0.017%, 0.015% and 0.016% for $Q/I_{c}$, $U/I_{c}$ and
$V/I_{c}$, respectively, where $I_{c}$ is the Stokes $I$ at continuum
wavelength. These values are within the SO/PHI scientific requirements, and
may be further optimised by adjusting the scaling parameters.
The metadata entries, recorded to summarise the blocks and pipelines executed
on the dataset, contain their ID, given parameters and return values. See
table 1. To keep a complete record, in addition to these, entries from other
sources are also logged (e.g. entries from the image processing operations, or
describing the dataset).

##  5 Conclusions

SO/PHI is the first solar imaging spectropolarimeter to perform autonomous on-
board instrument calibration and autonomous data reduction. The data
processing involved with these tasks is implemented in a software framework
distributed between a microprocessor and two RFPGAs. The microprocessor
controls the RFPGAs, which implement functions with long computational times
to accelerate them (e.g. image processing functions). The framework is
designed to ensure the implementation of a large variety of algorithms, and
their easy modification when necessary, while operating with limited memory
resources. To ensure that the autonomous process can be fully recovered on
ground, a metadata log is kept about all operations performed, their
parameters and return values. A warning system is implemented for supporting
the scientists on ground to find possible errors in the processing. The
presented results demonstrate the integrity of the system and its ability to
process the data.
The data processing software frame has the following key features:

  * •

block approach,

  * •

pipeline implementation of tasks,

  * •

parametrised pipelines for processing data in sub-sets,

  * •

wide error detection and handling,

  * •

a system for maintaining required accuracy during fixed point operations,

  * •

extensive metadata logging.

The software system was developed for the specific needs of the SO/PHI
instrument on-board the Solar Orbiter, to cope with the telemetry limitations.
However, due to the adaptability of the system, it is planned to serve as
basis for on-board data processing systems for future instruments in similarly
challenging orbits. A more detailed description of the SO/PHI data processing
system will be given in a paper currently in preparation.

## Acknowledgements

This work was carried out in the framework of the International Max Planck
Research School (IMPRS) for Solar System Science at the Max Planck Institute
for Solar System Research (MPS). Solar Orbiter is a mission lead by the
European Space Agency (ESA) with significant contribution from National
Aeronautics and Space Administration (NASA). The SO/PHI instrument is
supported by the German Aerospace Center (DLR) through Grant 50 OT 1201. The
Spanish contribution has been partly funded by the Spanish Research Agency
under project ESP2016-77548-C5, partially including European FEDER funds. The
solar data used in the test are the courtesy of NASA/SDO HMI science team.

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