###### Abstract

We present a chiral ${\bar{K}}N$ interaction model that has been developed and
optimized in order to account for the experimental data of inelastic
${\bar{K}}N$ reaction channels that open at higher energies. In particular, we
study the effect of the higher partial waves which originate directly from the
chiral Lagrangian, as they could supersede the role of high-spin resonances
employed in earlier phenomenological models to describe meson–baryon cross
sections in the 2 GeV region. We present a detailed derivation of the partial
wave amplitudes that emerge from the chiral SU(3) meson-baryon Lagrangian up
to the d-waves and next-to-leading order in the chiral expansion. We implement
a nonperturbative unitarization in coupled channels and optimize the model
parameters to a large pool of experimental data in the relevant energy range
where these new contributions are expected to be important. The obtained
results are encouraging. They indicate the ability of the chiral higher
partial waves to extend the description of the scattering data to higher
energies and to account for structures in the reaction cross sections that
cannot be accommodated by theoretical models limited to the s-waves.

###### keywords:

chiral lagrangian; unitarization; resonances; ${\bar{K}}N$ interaction

xx 1 5 Received: date; Accepted: date; Published: date The ${\bar{K}}N$
interaction in higher partial waves Albert Feijoo1,2, Daniel Gazda2, Volodymyr
Magas3 and Àngels Ramos3 Albert Feijoo, Daniel Gazda, Volodymyr Magas and
Àngels Ramos

##  1 Introduction

Over the last forty years, strong interaction processes in the low-energy
domain have been described with great success by chiral perturbation theory
($\chi$PT), an effective field theory with hadron degrees of freedom which
respects all the symmetries of the underlying theory of the strong
interaction, Quantum Chromodynamics (QCD), particularly its spontaneously
broken chiral symmetry Gasser:1983yg .

In the early years of $\chi$PT many studies were focused on the low-energy
pion interactions with pions and nuclei, although extensions to the
strangeness sector, invoking flavour SU(3) symmetry, were also explored. In
particular, the ${\bar{K}}N$ interaction was especially challenging because
the description of the scattering amplitude cannot be properly achieved within
a perturbative treatment, owing to the presence of a resonance, the
$\Lambda(1405)$, located 27 MeV below the ${\bar{K}}N$ threshold.

Another challenge in hadron physics over the past few decades has been to
disentangle the nature of some hadron resonances that do not fit well in a
conventional quark model description, according to which baryons are composed
by three quarks and mesons by a quark-antiquark pair Guo:2017jvc . For many
years, a large amount of theoretical and experimental activity has been
devoted to the discovery and characterization of exotic mesons and baryons.
One of the early evidences was provided, precisely, by the $\Lambda(1405)$
resonance, whose mass was systematically predicted to be too high by quark
models but it found a better explanation if it was described as a $\bar{K}N$
quasi-bound state. This was already pointed out in the late fifties
Dalitz:1959dn ; Dalitz:1960du , and was later corroborated by models that
built the meson–baryon interaction from a chiral effective Lagrangian, derived
respecting the symmetries of QCD, and implementing unitarization Kaiser:1995eg
; Oset:1997it ; Oller:2000fj ; Lutz:2001yb ; GarciaRecio:2002td ;
Borasoy:2005ie . The confirmation of the $\Lambda(1405)$ being essentially a
meson-baryon bound state came after establishing its double-pole nature
Oller:2000fj ; Jido:2003cb  from comparing different experimental line shapes
Magas:2005vu , which unambiguously indicates the different coupling strength
to the meson-baryon components of the $\Lambda(1405)$ wave-function. See the
recent overview on this issue in Ref. Meissner:2020khl .

Several experiments aiming at establishing the shape of the $\Lambda(1405)$,
as those employing $pp$ reactions by the COSY Zychor:2007gf  and HADES
HADES:2012csk  collaborations or photo- or electro-production processes at
LEPS Niiyama:2008rt  and CLAS CLAS:2013rjt ; CLAS:2013rxx ; CLAS:2013zie ,
together with the precise determination of the energy shift and width of the
1s state in kaonic hydrogen measured by SIDDHARTA SIDD , triggered a renewed
interest in improving the chiral unitary theories GO ; IHW ; Roca:2013cca ;
Mai:2014xna ; Cieply:2016jby ; Feijoo:2015yja ; Ramos:2016odk ; Feijoo:2018den
for a better description of the ${\bar{K}}N$ interaction and related
phenomenology. In our previous works Feijoo:2015yja ; Ramos:2016odk ;
Feijoo:2018den , the relevance of the terms next in the hierarchy after the
lowest order Weinberg–Tomozawa (WT) term was studied, in connection with the
inclusion of higher-energy experimental data. More explicitly, we first
focused on the $K\Xi$ production reactions because they do not proceed
directly via the lowest order WT contribution Feijoo:2015yja . Our further
studies Ramos:2016odk ; Feijoo:2018den  indeed demonstrated that the so-called
Born diagrams as well as the next-to-leading order (NLO) terms are far from
being mere corrections when it comes to the reproduction of the
$K^{-}p\to\eta\Lambda,\eta\Sigma^{0},K^{+}\Xi^{-},K^{0}\Xi^{0}$ reaction cross
sections. In particular, the inclusion of isospin filtering reactions in
Feijoo:2018den ; Feijoo:2015cca  served to emphasize their relevance for
avoiding potential ambiguities in the isospin components of the scattering
amplitude. All this translates into stronger constraints on the models by
means of which one can derive more reliable values of the low-energy constants
of the chiral Lagrangian.

In the same spirit, the inclusion of new ingredients that are expected to be
specially relevant at higher energies could reveal more information about the
physics behind the NLO terms of the chiral Lagrangian. In the present work, we
explore the relevance of including partial waves higher than the $L=0$, which
is usually the only component considered in the literature to study the
${\bar{K}}N$ scattering phenomenology. In particular, we focus on the p-wave
contribution, the effect of which is expected to be non-negligible, as we aim
at obtaining the ${\bar{K}}N$ scattering amplitudes at higher energies,
necessary to describe the $\eta\Lambda$, $\eta\Sigma^{0}$, $K^{0}\Xi^{0}$ and
$K^{+}\Xi^{-}$ production reactions. Extending the ${\bar{K}}N$ interaction to
p-wave components is also relevant for studies of bound ${\bar{K}}$ mesons in
nuclei Garcia-Recio:2000seh ; Cieply:2011yz ; Hrtankova:2017zxw ;
Hrtankova:2019jky , since their local momentum can acquire sizable values.
Finally, from a formal perspective, it is interesting to explore whether the
unitarized interaction in higher partial waves might give rise to dynamically
generated states, similarly to the case of the $\Lambda(1405)$ resonance in
s-wave.

Former studies of the effect of p-wave contributions derived from chiral
Lagrangians have been done in the strangeness $S=0$ sector CaroRamon:1999jf
and, more recently, in the $S=+1$ sector Aoki:2018wug . As for the $S=-1$
sector considered in the present work, the study of Ref. Jido:2002zk , limited
to the lowest-order chiral Lagrangian, found the traditional low-energy data
not to be sensitive to the new p-wave components. These findings were
corroborated by a later study Cieply:2015pwa  that followed similar ideas. The
recent work of Ref. Sadasivan:2018jig  obtains the s- and p-wave scattering
amplitudes from the leading-order (LO) and NLO Lagrangians, the parameters of
which are fitted to the low-energy data, also including the invariant
$\pi\Sigma$ mass distributions from photo-production reactions at CLAS
CLAS:2013rjt . A dynamically generated p-wave state with isospin one and
$J^{P}=1/2^{+}$ is found in that study, which in fact mimics the absence of
the $\Sigma^{*}(1385)$ resonance that plays a relevant role in the $\pi\Sigma$
distributions.

In the present work we consider the s- and p-wave contributions of the
scattering amplitudes but, differently to Ref. Sadasivan:2018jig , we also
employ the data of inelastic reactions opening at higher energies to constrain
our models. This implies that the NLO Lagrangian, which we take from Ref.
Aoki:2018wug , must also consider terms that have usually been disregarded
owing to their negligible effect at lower energies. We will show that the new
terms of the NLO Lagrangian are relevant, being in fact strongly intertwined
with the p-wave components of the ${\bar{K}}N$ interaction.

Our paper is organized as follows. In Sec. 2 we develop the chiral unitary
formalism, show the LO and NLO contributions of the chiral Lagrangian, give
the explicit expressions of the corresponding interaction kernels—with their
complete momentum dependence structure—and, finally, describe the procedure
that allows one to obtain the partial-wave components of the scattering
amplitude. The models studied in this work are presented in Sec. 3, together
with the data employed in the fits. Sec. 4 is devoted to the discussion of our
results, with a special focus on the role played by the p-wave amplitudes and
the new terms of the NLO Lagrangian. A few concluding remarks are given in
Sec. 5.

##  2 Formalism

Chiral unitary approaches (UChPT) have shown to be a powerful tool to treat
the meson–baryon scattering at energies around resonances. These
nonperturbative schemes prevent plain chiral perturbation theory from non
converging and guarantee the unitarity and analyticity of the scattering
amplitude. In the present work, unitarity is implemented by solving the
Bethe–Salpeter (BS) equation with coupled channels. Following Oset:1997it ;
Hyodo:2011ur , the interaction kernel is conveniently split into its on-shell
contribution and the corresponding off-shell one. The off-shell part gives
rise to a tadpole-type diagram which can be reabsorbed into renormalization of
couplings and masses and, hence, be omitted from the calculation. This method
permits factorizing the interaction kernel and the scattering amplitude out of
the integral equation, transforming a complex system of coupled integral
equations into a simple system of algebraic equations which, in matrix form,
reads:

$T_{ij}={(1-V_{il}G_{l})}^{-1}V_{lj},$ (1)

where $V_{ij}$ is the driving kernel derived from the chiral Lagrangian,
$T_{ij}$ is the corresponding scattering amplitude for the transition from an
$i$ channel to a $j$ one, and $G_{l}$ is the loop function of the intermediate
channel $l$, which reads:

$G_{l}={\rm
i}\int\frac{d^{4}q_{l}}{{(2\pi)}^{4}}\frac{2M_{l}}{{(P-q_{l})}^{2}-M_{l}^{2}+{\rm
i}\epsilon}\frac{1}{q_{l}^{2}-m_{l}^{2}+{\rm i}\epsilon},$ (2)

with $M_{l}$ and $m_{l}$ being the baryon and meson masses of the channel. As
this function diverges logarithmically, a dimensional regularization scheme is
applied which leads to the expression:

$G_{l}=\frac{2M_{l}}{(4\pi)^{2}}\Bigg{\\{}a_{l}(\mu)+\ln\frac{M_{l}^{2}}{\mu^{2}}+\frac{m_{l}^{2}-M_{l}^{2}+s}{2s}\ln\frac{m_{l}^{2}}{M_{l}^{2}}+\frac{q_{\rm
cm}}{\sqrt{s}}\ln\left[\frac{(s+2\sqrt{s}q_{\rm
cm})^{2}-(M_{l}^{2}-m_{l}^{2})^{2}}{(s-2\sqrt{s}q_{\rm
cm})^{2}-(M_{l}^{2}-m_{l}^{2})^{2}}\right]\Bigg{\\}}.$ (3)

The subtraction constants $a_{l}$ replace the divergence for a given
dimensional regularization scale $\mu$, which is taken to be $1$ GeV in the
present work. These constants are unknown parameters to be fitted to the
experimental data. For the $S=-1$ meson–baryon interaction they amount to ten,
one per channel, although it is quite common to reduce them to six taking into
account isospin symmetry arguments, as we shall also do.

The kernel employed in the chiral unitary approaches is derived from the SU(3)
effective chiral Lagrangian, which provides the fundamental blocks of the
interaction that preserve the symmetries of QCD employing hadron fields as the
relevant degrees of freedom. Following a power counting scheme, these
constituent pieces are arranged in the expansion by order of relevance (see
for instance Scherer:2002tk  for a more detailed explanation). At leading
order (LO), the most general Lagrangian can be expressed as:

$\mathcal{L}_{\phi B}^{(1)}=i\langle\bar{B}\gamma_{\mu}[D^{\mu},B]\rangle-
M_{0}\langle\bar{B}B\rangle-\frac{1}{2}D\langle\bar{B}\gamma_{\mu}\gamma_{5}\\{u^{\mu},B\\}\rangle\\\
-\frac{1}{2}F\langle\bar{B}\gamma_{\mu}\gamma_{5}[u^{\mu},B]\rangle,$ (4)

where $M_{0}$ is the common baryon octet mass in the chiral limit, the
constants $D$, $F$ denote the axial vector couplings of the baryons to the
mesons, and the symbol $\langle\cdot\rangle$ stands for the trace in flavour
space. The baryon octet field ($N,\Lambda,\Sigma,\Xi$) is denoted by $B$,
while the pseudoscalar meson octet field $\phi$ ($\pi,K,\eta$) enters in
$u_{\mu}=iu^{\dagger}\partial_{\mu}Uu^{\dagger}$, where
$U(\phi)=u^{2}(\phi)=\exp{\left(\sqrt{2}i\phi/f\right)}$ (see Feijoo:2015yja )
with $f$ being the pseudoscalar decay constant that acts as typifying scale
factor in the expansion in powers of momentum. Finally, $[D_{\mu},B]$ contains
the covariant derivative that accounts for the local character of the chiral
transformation of $u$ and it is defined as:

$[D_{\mu},B]=\partial_{\mu}B+[\Gamma_{\mu},B]\,$ (5)

with $\Gamma_{\mu}=[u^{\dagger},\partial_{\mu}u]/2$ being the chiral
connection. The next-to-leading order (NLO) contributions are given by:

$\begin{split}\mathcal{L}_{\phi
B}^{(2)}&=b_{D}\langle\bar{B}\\{\chi_{+},B\\}\rangle+b_{F}\langle\bar{B}[\chi_{+},B]\rangle+b_{0}\langle\bar{B}B\rangle\langle\chi_{+}\rangle+d_{1}\langle\bar{B}\\{u_{\mu},[u^{\mu},B]\\}\rangle\\\
&+d_{2}\langle\bar{B}[u_{\mu},[u^{\mu},B]]\rangle+d_{3}\langle\bar{B}u_{\mu}\rangle\langle
u^{\mu}B\rangle+d_{4}\langle\bar{B}B\rangle\langle u^{\mu}u_{\mu}\rangle\\\
&-\frac{g_{1}}{8M_{N}^{2}}\langle\bar{B}\\{u_{\mu},[u_{\nu},\\{D^{\mu},D^{\nu}\\}B]\\}\rangle-\frac{g_{2}}{8M_{N}^{2}}\langle\bar{B}[u_{\mu},[u_{\nu},\\{D^{\mu},D^{\nu}\\}B]]\rangle\\\
&-\frac{g_{3}}{8M_{N}^{2}}\langle\bar{B}u_{\mu}\rangle\langle[u_{\nu},\\{D^{\mu},D^{\nu}\\}B]\rangle-\frac{g_{4}}{8M_{N}^{2}}\langle\bar{B}\\{D^{\mu},D^{\nu}\\}B\rangle\langle
u_{\mu}u_{\nu}\rangle\\\
&-\frac{h_{1}}{4}\langle\bar{B}[\gamma^{\mu},\gamma^{\nu}]Bu_{\mu}u_{\nu}\rangle-\frac{h_{2}}{4}\langle\bar{B}[\gamma^{\mu},\gamma^{\nu}]u_{\mu}[u_{\nu},B]\rangle-\frac{h_{3}}{4}\langle\bar{B}[\gamma^{\mu},\gamma^{\nu}]u_{\mu}\\{u_{\nu},B\\}\rangle\\\
&-\frac{h_{4}}{4}\langle\bar{B}[\gamma^{\mu},\gamma^{\nu}]u_{\mu}\rangle\langle
u_{\nu},B\rangle+\text{h.c.},\end{split}$ (6)

where $M_{N}$ stands for the nucleon mass, while the coefficients $b_{D}$,
$b_{F}$, $b_{0}$, $d_{i}$ $(i=1,\ldots,4)$, $g_{i}$ $(i=1,\ldots,4)$ and
$h_{i}$ $(i=1,\ldots,4)$ are the low-energy constants (LECs) at this order.
Despite the symmetries of the underlying theory cannot fix these constants,
some of them can be constrained by the mass splitting of baryons, the pion-
nucleon sigma term and from the strangeness content of the proton
Gasser:1990ce , and some others using data from low-energy meson–baryon
interactions, such as the isospin even $\pi N$ s-wave scattering length
Koch:1985bn  and the isospin zero kaon–nucleon s-wave scattering length
Dover:1982zh . However, we treat them as free parameters in the fitting
procedure as it is usually done in the literature. The reason lies in the
coupled-channel unitarization that produces amplitudes that go beyond tree
level. The quantity
$\chi_{+}=2B_{0}(u^{\dagger}\mathcal{M}u^{\dagger}+u\mathcal{M}u)$ explicitly
breaks chiral symmetry via the quark mass matrix $\mathcal{M}={\rm
diag}(m_{u},m_{d},m_{s})$, while $B_{0}=-\braket{0}{\bar{q}q}{0}/f^{2}$
relates to the order parameter of spontaneously broken chiral symmetry.

Before proceeding further, we would like to make a few remarks on the NLO
Lagrangian of Eq. (6), taken from Aoki:2018wug , where the authors study the
$KN$ scattering amplitude in a chiral unitary approach looking for a possible
broad resonance in the $S=+1$ sector. Formally speaking, the expression
differs from the one we employed in our previous works Feijoo:2015yja ;
Ramos:2016odk ; Feijoo:2018den  in the terms with coefficients $g_{i}$
$(i=1,\ldots,4)$ and $h_{i}$ $(i=1,\ldots,4)$ (from now on $g$-terms and
$h$-terms), which naturally appear from an extension of the SU(2) chiral
Lagrangian introduced in Bernard:1995dp ; Fettes:2000gb ; Lutz:2001yb , but
have been commonly discarded in the literature studying $S=-1$ meson–baryon
scattering within the UChPT framework owing to their non significant role at
low energies. Nevertheless, as our models explore higher energies, it is
reasonable to consider these additional contributions. We will show that they
play a relevant role mainly in the $\bar{K}N$ transitions to the
$\eta\Lambda$, $\eta\Sigma^{0}$, and $K\Xi$ channels. We also note that the
$b_{D}$, $b_{F}$, $b_{0}$, $d_{i}$ and $h$-terms are invariant under the
hermitian conjugation (h.c.) transformation in Eq. (6), hence their h.c.
contributions can be reabsorbed in the LECs of the corresponding original
terms. In contrast, the h.c. of the $g$-terms produces new structures in the
Lagrangian, which have to be considered explicitly. Specially remarkable is
the cancellation of the $g_{3}$ monomial, which implies the reduction of the
parameter space in the derived models. Finally, it should be pointed out that
other versions for SU(3) Lagrangians at $O(q^{2})$ are presented in
Frink:2004ic ; Oller:2006yh .

Figure 1:  Diagrammatic representation of the meson-baryon interaction
kernels: Weinberg–Tomozawa term (i), direct and crossed Born terms (ii) and
(iii), and NLO terms (iv). Dashed (solid) lines represent the pseudoscalar
octet mesons (octet baryons).

Not without a tedious calculation, one can derive the amplitudes for the
$\phi_{i}B_{i}^{s}\to\phi_{j}B_{j}^{s^{\prime}}$ processes (with the incoming
and outgoing spins $s,s^{\prime}$, respectively) which are schematically
represented in Fig. 1. Diagram (i) corresponds to the Weinberg–Tomozawa (WT)
term that can be written in momentum space as:

$V_{ij}^{WT}=-\frac{N_{i}N_{j}}{4f^{2}}C_{ij}\left\\{(2\sqrt{s}{-}M_{i}{-}M_{j})\chi_{j}^{\dagger
s^{\prime}}\chi_{i}^{s}+\frac{2\sqrt{s}{+}M_{i}{+}M_{j}}{(E_{i}{+}M_{i})(E_{j}{+}M_{j})}\chi_{j}^{\dagger
s^{\prime}}\left[\vec{q}_{j}\cdot\vec{q}_{i}+i(\vec{q}_{j}\times\vec{q}_{i})\cdot\vec{\sigma}\right]\chi_{i}^{s}\right\\}.$
(7)

The indices $(i,j)$ cover all possible initial and final channels in the
$S=-1,Q=0$ sector. The matrix of coefficients $C_{ij}$ can be found in Table
VII of Ref. Feijoo:2015yja . The normalization factor $N$ is defined as
$N=\sqrt{(M+E)/(2M)}$, with $M$ and $E$ being, respectively, the mass and
energy of the baryon in the corresponding channel. The two-component Pauli
spinor of a baryon with spin projection $s$ is denoted by $\chi^{s}$. The
symbol for the spin, only present in the spinors, should not be confused with
$\sqrt{s}$ that represents the total energy of the meson-baryon system in the
center-of-mass (CM) frame. The CM three-momentum of the meson (baryon) is
given by $\vec{q}$ ($-\vec{q}$).

In Fig. 1, the Born contributions are represented by diagrams (ii) (direct
Born term) and (iii) (crossed Born term), the vertices of which are obtained
from the $D$ and $F$ terms of Eq. (4). The analytical form of the direct Born
term is given by:

$\begin{split}V_{ij}^{D}&=\frac{N_{i}N_{j}}{12f^{2}}\sum_{k}\frac{C^{(\text{Born})}_{\bar{i}i,k}C^{(\text{Born})}_{\bar{j}j,k}}{s-M_{k}^{2}}\left\\{\vphantom{\frac{1}{(E_{i}+M_{i})(E_{j}+M_{j})}}(\sqrt{s}-M_{k})(s+M_{i}M_{j}-\sqrt{s}(M_{i}+M_{j}))\chi_{j}^{\dagger
s^{\prime}}\chi_{i}^{s}\right.\\\
&\left.+\frac{(s+\sqrt{s}(M_{i}+M_{j})+M_{i}M_{j})(\sqrt{s}+M_{k})}{(E_{i}+M_{i})(E_{j}+M_{j})}\chi_{j}^{\dagger
s^{\prime}}\left[\vec{q}_{j}\cdot\vec{q}_{i}+i(\vec{q}_{j}\times\vec{q}_{i})\cdot\vec{\sigma}\right]\chi_{i}^{s}\right\\}\end{split}$
(8)

and, similarly, the crossed Born term reads:

$\begin{split}V^{C}_{ij}&=-\frac{N_{i}N_{j}}{12f^{2}}\sum_{k}\frac{C^{(\text{Born})}_{\bar{j}k,i}C^{(\text{Born})}_{\bar{i}k,j}}{u-M_{k}^{2}}\left\\{\vphantom{\frac{\vec{q}_{j}\cdot\vec{q}_{i}+i(\vec{q}_{j}\times\vec{q}_{i})\cdot\vec{\sigma}}{(E_{i}+M_{i})(E_{j}+M_{j})}}\left[\vphantom{M_{i}^{2}}u(\sqrt{s}+M_{k})+\sqrt{s}(M_{j}(M_{i}+M_{k})+M_{i}M_{k})\right.\right.\\\
&\left.-M_{j}(M_{i}+M_{k})(M_{i}+M_{j})-M_{i}^{2}M_{k}\right]\chi_{j}^{\dagger
s^{\prime}}\chi_{i}^{s}+\left[u(\sqrt{s}-M_{k})+\sqrt{s}(M_{j}(M_{i}+M_{k})+M_{i}M_{k})\right.\\\
&\left.\left.+M_{j}(M_{i}+M_{k})(M_{i}+M_{j})+M_{i}^{2}M_{k}\right]\chi_{j}^{\dagger
s^{\prime}}\frac{\vec{q}_{j}\cdot\vec{q}_{i}+i(\vec{q}_{j}\times\vec{q}_{i})\cdot\vec{\sigma}}{(E_{i}+M_{i})(E_{j}+M_{j})}\chi_{i}^{s}\right\\},\end{split}$
(9)

where the label $k$ refers to the intermediate baryon involved in the process.
Here, $u$ stands for the Mandelstam variable defined by
$u=({p_{i}}^{\mu}-{q_{j}}^{\mu})^{2}$, with ${q_{j}}^{\mu}$ being the four-
momentum of the final meson and ${p_{i}}^{\mu}$ of the initial baryon. The
coefficients $C^{(\text{Born})}_{\bar{x}y,z}$, which depend on the axial
vector constants $D$ and $F$, can be found in Apendix A of Ref. Borasoy:2005ie
.

Taking into account only the tree-level contributions at NLO (diagram (iv) in
Fig. 1), one obtains:

$\begin{split}V^{NLO}_{ij}&=\frac{N_{i}N_{j}}{f^{2}}\left[D_{ij}-2L_{ij}{q_{j}}^{\mu}{q_{i}}_{\mu}+\frac{1}{2M_{N}^{2}}g_{ij}({p_{i}}^{\mu}{q_{j}}_{\mu}{p_{i}}^{\nu}{q_{i}}_{\nu}+{p_{j}}^{\mu}{q_{j}}_{\mu}{p_{j}}^{\nu}{q_{i}}_{\nu})\right]\left(\vphantom{\frac{1}{(E_{i}+M_{i})(E_{j}+M_{j})}}\chi_{j}^{\dagger
s^{\prime}}\chi_{i}^{s}\right.\\\ &-\left.\chi_{j}^{\dagger
s^{\prime}}\frac{\vec{q}_{j}\cdot\vec{q}_{i}+i(\vec{q}_{j}\times\vec{q}_{i})\cdot\vec{\sigma}}{(E_{i}+M_{i})(E_{j}+M_{j})}\chi_{i}^{s}\right)+\frac{N_{i}N_{j}}{f^{2}}h_{ij}\left[-\left(\frac{{q_{j}}_{0}{q_{i}}^{2}}{E_{i}+M_{i}}+\frac{{q_{i}}_{0}{q_{j}}^{2}}{E_{j}+M_{j}}\right.\right.\\\
&+\left.\left.\frac{{q_{j}}^{2}{q_{i}}^{2}}{(E_{i}+M_{i})(E_{j}+M_{j})}+\frac{(\vec{q}_{j}\cdot\vec{q}_{i})^{2}}{(E_{i}+M_{i})(E_{j}+M_{j})}\right)\chi_{j}^{\dagger
s^{\prime}}\chi_{i}^{s}\right.\\\
&+\left(\frac{{q_{i}}_{0}}{E_{i}+M_{i}}+\frac{{q_{j}}_{0}}{E_{j}+M_{j}}\right)\chi_{j}^{\dagger
s^{\prime}}\vec{q}_{j}\cdot\vec{q}_{i}\chi_{i}^{s}+\left(\frac{{q_{i}}_{0}}{E_{i}+M_{i}}+\frac{{q_{j}}_{0}}{E_{j}+M_{j}}\right.\\\
&+\left.\left.\frac{\vec{q}_{j}\cdot\vec{q}_{i}}{(E_{i}+M_{i})(E_{j}+M_{j})}-1\right)i\chi_{j}^{\dagger
s^{\prime}}(\vec{q}_{j}\times\vec{q}_{i})\cdot\vec{\sigma}\chi_{i}^{s}\right],\end{split}$
(10)

where the $D_{ij}$ and $L_{ij}$ coefficients are constructed from combinations
of the NLO parameters $b_{0}$, $b_{D}$, $b_{F}$, $d_{1}$, $d_{2}$, $d_{3}$ and
$d_{4}$ and are given in Table VIII of Ref. Feijoo:2015yja , while the
$g_{ij}$ and $h_{ij}$ matrices are listed in Appendix A.

One should bear in mind that the direct incorporation of the amplitudes given
by equations (7)–(10) into Eq. (1) is simply not possible since such
interaction kernels are composed of a mixture of contributions with different
angular momenta. Thus, it is convenient to express the $T$-matrix in terms of
the spin-nonflip and spin-flip parts.

$T_{ij}(s,s^{\prime})=\chi_{j}^{\dagger
s^{\prime}}[f(\sqrt{s},\theta)-i(\vec{\sigma}\cdot\hat{n})g(\sqrt{s},\theta)]\chi_{i}^{s},$
(11)

where $\theta$ is the CM angle between the inital and final meson momenta and
$\hat{n}=\vec{q}_{j}{\times}\vec{q}_{i}/\lvert\vec{q}_{j}{\times}\vec{q}_{i}\rvert$
is the normal vector to the scattering plane. The functions
$f(\sqrt{s},\theta)$ and $g(\sqrt{s},\theta)$ can be expanded in Legendre
polynomials as:

$\begin{split}f(\sqrt{s},\theta)&=\sum_{l=0}^{\infty}f_{l}(\sqrt{s})\,P_{l}(\cos\theta),\\\
g(\sqrt{s},\theta)&=\sum_{l=1}^{\infty}g_{l}(\sqrt{s})\,\sin\theta\,\frac{\mathrm{d}P_{l}(\cos\theta)}{\mathrm{d}\cos\theta},\end{split}$
(12)

with $f_{l}$ and $g_{l}$ being the projections of the sum of all the above
defined kernels onto $P_{l}(\cos\theta)$ and $\sin\theta\,\frac{{\rm
d}P_{l}(\cos\theta)}{{\rm d}\cos\theta}$, respectively.

The amplitudes can be redefined as:

$\begin{split}f_{l+}(\sqrt{s})&=\frac{1}{2l+1}\left(f_{l}(\sqrt{s})+l\,g_{l}(\sqrt{s})\right)\text{for~{}}J=l+\frac{1}{2},\\\
f_{l-}(\sqrt{s})&=\frac{1}{2l+1}\left(f_{l}(\sqrt{s})-(l+1)\,g_{l}(\sqrt{s})\right)\text{for~{}}J=l-\frac{1}{2},\end{split}$
(13)

to ensure that the quantum numbers of spin ($\frac{1}{2}$), orbital angular
momentum $(l)$, and total angular momentum $(J)$ are preserved in the
unitarization procedure implemented by the BS equations. Following the
notation of Ref. Jido:2002zk , each unitarized $J$-scattering amplitude should
be calculated by a new version of Eq. (1), which in matrix form reads:

$f_{l\pm}=\left[1-f^{\rm tree}_{l\pm}G\right]^{-1}f^{\rm tree}_{l\pm},$ (14)

where the amplitudes $f^{\rm tree}_{l\pm}$ are obtained from equations
(11)–(13) but employing, for the amplitude $T_{ij}(s,s^{\prime})$, the sum of
the interaction kernels shown in equations (7)–(10).

The aim of this work is the study of the effects caused by the inclusion of
higher partial waves on the physical observables in this sector, particularly
on the $K^{-}p$ cross sections. The most general expression of the
differential cross sections for a given
$\phi_{i}B_{i}^{s}\to\phi_{j}B_{j}^{s^{\prime}}$ process is

$\frac{d\sigma_{ij}}{d\Omega}=\frac{M_{i}\,M_{j}\,q_{j}}{16\,\pi^{2}\,s\,q_{i}}\,S_{ij},$
(15)

with

$S_{ij}=\frac{1}{2s+1}\sum_{s,s^{\prime}}\,\lvert
T_{ij}(s,s^{\prime})\rvert^{2}=\lvert f(w,\theta)\rvert^{2}+\lvert
g(w,\theta)\rvert^{2},$

where the first factor averages over the initial baryon spin projections,
giving $1/2$ for this particular case, and where we have also summed over all
possible final baryon spin projections.

Since the scope of the present study does not go beyond d-wave contributions,
focusing mostly on p-wave effects, the expression for the total cross section
incorporating such partial waves can be written as:

$\sigma_{ij}=\frac{M_{i}\,M_{j}\,q_{j}}{4\,\pi\,s\,q_{i}}\,\left[\lvert
f_{0}\rvert^{2}+2\lvert f_{1+}\rvert^{2}+\lvert f_{1-}\rvert^{2}+3\lvert
f_{2+}\rvert^{2}+2\lvert f_{2-}\rvert^{2}\right].$ (16)

##  3 Models and Data Treatment

The availability of a large collection of experimental data makes the $S=-1$
sector an interesting benchmark to test meson-baryon EFTs, not only for
checking their predictive power but also to extract information about the
LECs, especially those beyond LO. Constructing a model with higher partial
waves capable to describe the phenomenology of the $S=-1$ meson–baryon
interaction provides, indeed, a tool to explore other sectors lacking
experimental data, like $S=-2$, $S=-3$, and even to check the validity of
SU(3) symmetry upon comparing the resulting parameters to those of existing
chiral models that describe the meson–baryon interaction in the $S=0$ sector.

###  3.1 Models

In this section, we introduce the models derived for this study and discuss
the fitting procedure. Since the present study is the natural extension of the
one presented in Feijoo:2018den , we include a description of the old model as
a reminder and to provide a better comprehension of the p-wave effects.

  * •

s-wave (old): This first model corresponds to the fit called WT+Born+NLO
carried out in Feijoo:2018den . It was constructed by adding the interaction
kernels derived from the Lagrangian up to NLO and neglecting the $h$\- and
$g$-terms. We limited this model to the s-wave contribution.

  * •

s-wave: The second model improves upon the first one by incorporating the
novel $h$\- and $g$-terms that come from the NLO interaction kernel and, as in
the first model, only the s-wave contribution is taken into account.

  * •

s+p-waves: This model employs the same Lagrangian as the s-wave fit, but it
also incorporates the p-wave contributions.

###  3.2 Data Treatment and Fitting Procedure

The parameters in our models originate from the chiral Lagrangian as well as
from the employed regularization scheme. One finds 17 LECs in accordance with
equations (7)–(10), namely the meson decay constant $f$, the axial vector
couplings $D$ and $F$, and the NLO coefficients $b_{0}$, $b_{D}$, $b_{F}$,
$d_{1}$, $d_{2}$, $d_{3}$, $d_{4}$, $g_{1}$, $g_{2}$, $g_{4}$, $h_{1}$,
$h_{2}$, $h_{3}$, and $h_{4}$. It is a well-known fact that, in the UChPT
models, the $f$ parameter effectively takes larger values than the
experimental one. Literature offers plenty of miscellaneous choices but, in
our particular case, in order to allow for a sort of average over all mesons
involved in the various coupled channels, this parameter is constrained to
vary from $f=f_{\pi}$ to $f=1.3f_{\pi}$. The axial vector couplings $D$ and
$F$ are allowed to vary within 15% of their canonical values in order to
accommodate the spread of values found in the literature Ramos:2016odk . The
NLO LECs are treated as completely free parameters in the fit. Additionally,
the dimensional regularization introduces six subtraction constants,
$a_{\pi\Sigma}$, $a_{\bar{K}N}$, $a_{\pi\Lambda}$, $a_{\eta\Sigma}$,
$a_{\eta\Lambda}$, and $a_{K\Xi}$ that are also parameters of the fit.
However, these parameters are restricted to their natural-sized values, which
should, as discussed in Ref. Feijoo:2018den , lie in the range $[-10,10]\times
10^{-3}$ for $\mu=1$ GeV.

Table 1: Number of experimental points used in our fits, which are extracted from exp_1 ; exp_2 ; exp_3 ; exp_4 ; exp_5 ; exp_6 ; exp_7 ; exp_8 ; exp_9 ; exp_10 ; exp_11 ; br_1 ; br_2 ; SIDD ; Starostin ; Baxter ; Jones ; Berthon ; Jones:1974si , distributed per observable. Observable |  Points | Observable |  Points
---|---|---|---
$\sigma_{K^{-}p\to K^{-}p}$ | 23 | $\sigma_{K^{-}p\to\bar{K}^{0}n}$ | 9
$\sigma_{K^{-}p\to\pi^{0}\Lambda}$ | 3 | $\sigma_{K^{-}p\to\pi^{0}\Sigma^{0}}$ | 3
$\sigma_{K^{-}p\to\pi^{-}\Sigma^{+}}$ | 20 | $\sigma_{K^{-}p\to\pi^{+}\Sigma^{-}}$ | 28
$\sigma_{K^{-}p\to\eta\Sigma^{0}}$ | 9 | $\sigma_{K^{-}p\to\eta\Lambda}$ | 49
$\sigma_{K^{-}p\to K^{+}\Xi^{-}}$ | 46 | $\sigma_{K^{-}p\to K^{0}\Xi^{0}}$ | 29
$\gamma$ | 1 | $\Delta E_{1s}$ | 1
$R_{n}$ | 1 | $\Gamma_{1s}$ | 1
$R_{c}$ | 1 |  |

Table 1 displays the distribution per observable of the 224 experimental
points employed in the current fitting procedure (the same as in
Feijoo:2018den ), most of them coming from the total cross section for
$K^{-}p$ scattering into different final channels exp_1 ; exp_2 ; exp_3 ;
exp_4 ; exp_5 ; exp_6 ; exp_7 ; exp_8 ; exp_9 ; exp_10 ; exp_11 ; Starostin ;
Baxter ; Jones ; Berthon ; Jones:1974si . Apart from this, the measured
branching ratios of cross section yields br_1 ; br_2  are also fitted. Such
yields can be defined from the elastic and inelastic $K^{-}p$ cross sections
evaluated at threshold:

$\begin{split}\gamma&=\frac{\Gamma(K^{-}p\rightarrow\pi^{+}\Sigma^{-})}{\Gamma(K^{-}p\rightarrow\pi^{-}\Sigma^{+})}=2.36\pm
0.04,\\\
R_{c}&=\frac{\Gamma(K^{-}p\rightarrow\pi^{+}\Sigma^{-},\pi^{-}\Sigma^{+})}{\Gamma(K^{-}p\rightarrow\text{inelastic
channels})}=0.189\pm 0.015,\\\
R_{n}&=\frac{\Gamma(K^{-}p\rightarrow\pi^{0}\Lambda)}{\Gamma(K^{-}p\rightarrow\text{neutral
states})}=0.664\pm 0.011.\end{split}$ (17)

The precise measurement of the energy level shift and width of the atomic $1s$
state in kaonic hydrogen by the SIDDHARTA Collaboration SIDD  implied a
substantial change in constraining the theoretical models. We incorporate this
input in our fits via the $K^{-}p$ scattering length, which is obtained from
the $K^{-}p$ scattering amplitude at threshold as:

$a_{\scriptscriptstyle
K^{-}p}=-\frac{1}{4\pi}\frac{M_{p}}{M_{p}+M_{\bar{K}}}T_{K^{-}p\to K^{-}p}.$
(18)

The scattering length is connected to the shift and width of kaonic hydrogen
by means of the second order corrected Deser-type formula Meissner:2004jr :

$\Delta E-i\frac{\Gamma}{2}=-2\alpha^{3}\mu_{r}^{2}a_{\scriptscriptstyle
K^{-}p}\Big{[}1+2a_{\scriptscriptstyle
K^{-}p}\,\alpha\,\mu_{r}\,(1-\ln\alpha)\Big{]},$ (19)

where $\alpha$ is the fine-structure constant and $\mu_{r}$ the reduced mass
of the $K^{-}p$ system.

The minimizing criteria taken for the fitting procedure is based on the
$\chi^{2}$ per degree of freedom ($\chi^{2}_{\rm d.o.f.}$). We avoid the use
of the the standard definition of $\chi^{2}_{\rm d.o.f.}$ since it favors
observables with a larger number of experimental points over those with a
smaller amount. To get around this misleading effect, we adopt the method used
in GarciaRecio:2002td ; GO ; IHW ; Mai:2014xna ; Feijoo:2015yja ;
Ramos:2016odk ; Feijoo:2018den , consisting of the assignment of equal weights
to the different measurements by defining a renormalized $\chi^{2}_{\rm
d.o.f.}$ as:

$\chi^{2}_{\rm
d.o.f}=\frac{\sum_{k=1}^{K}n_{k}}{\sum_{k=1}^{K}n_{k}-p}\frac{1}{K}\sum_{k=1}^{K}\frac{1}{n_{k}}\chi^{2}_{k},$
(20)

with

$\chi^{2}_{k}=\sum_{i=1}^{n_{k}}\frac{\left(y_{i,k}^{\rm th}-y_{i,k}^{\rm
exp}\right)^{2}}{\sigma_{i,k}^{2}}.$

Here, $y_{i,k}^{\rm exp}$, $y_{i,k}^{\rm th}$, and $\sigma_{i,k}$ are the
experimental value, the theoretical prediction, and the experimental
uncertainty of the $i^{\text{th}}$ point of the $k^{\text{th}}$ observable,
respectively. Each observable has a total of $n_{k}$ points; $K$ is the total
number of observables, and $p$ denotes the number of free fit parameters. In
Eq. (20), the renormalization is incorporated by averaging the $\chi^{2}$ per
degree of freedom over the different experiments.

##  4 Results and Discussion

As mentioned in the introduction, our previous works Feijoo:2015yja ;
Ramos:2016odk ; Feijoo:2018den  studied the influence of the Born and NLO
contributions, especially on the inelastic processes that open up at higher
energies, namely the
$K^{-}p\to\eta\Lambda,\eta\Sigma^{0},K^{+}\Xi^{-},K^{0}\Xi^{0}$ reactions. In
the present work we include new ingredients, such as the $h$\- and $g$-terms
of the NLO Lagrangian and the p-wave contributions, which are expected to play
a relevant role at these energies. The effects of these new contributions on
the model parameters are displayed in Table 2, where the parameters of the
s-wave (old) model, obtained in Ref. Feijoo:2018den , are also included for
comparison.

Table 2:  Values of the parameters and the corresponding $\chi^{2}_{\rm d.o.f.}$, defined in Eq. (20), for the different models described in the text. The subtraction constants are taken at a regularization scale $\mu=1$ GeV. The error bars in the parameters of the s-wave (old) are determined as explained in Feijoo:2018den , while those of s+p-waves and s-wave are directly provided by the MINUIT James:1975dr  minimization procedure. | s+p-waves | s-wave | s-wave (old)
---|---|---|---
$a_{\bar{K}N}\ (10^{-3})$ | $1.322\pm 0.643$ | $2.105\pm 0.378$ | $1.268^{+0.096}_{-0.096}$
$a_{\pi\Lambda}\ (10^{-3})$ | $10.000\pm 19.962$ | $9.999\pm 18.190$ | $-6.114^{+0.045}_{-0.055}$
$a_{\pi\Sigma}\ (10^{-3})$ | $1.247\pm 1.913$ | $3.413\pm 1.609$ | $0.684^{+0.429}_{-0.572}$
$a_{\eta\Lambda}\ (10^{-3})$ | $-3.682\pm 6.541$ | $-4.585\pm 1.322$ | $-0.666^{+0.080}_{-0.140}$
$a_{\eta\Sigma}\ (10^{-3})$ | $5.528\pm 2.571$ | $3.017\pm 0.027$ | $8.004^{+2.282}_{-0.978}$
$a_{K\Xi}\ (10^{-3})$ | $-2.077\pm 0.931$ | $0.997\pm 0.038$ | $-2.508^{+0.396}_{-0.297}$
$f/f_{\pi}$ | $1.110\pm 0.068$ | $1.042\pm 0.003$ | $1.196^{+0.013}_{-0.007}$
$b_{0}\ (GeV^{-1})$ | $0.394\pm 0.087$ | $-0.079\pm 0.005$ | $0.129^{+0.032}_{-0.032}$
$b_{D}\ (GeV^{-1})$ | $0.206\pm 0.064$ | $0.112\pm 0.008$ | $0.120^{+0.010}_{-0.009}$
$b_{F}\ (GeV^{-1})$ | $0.303\pm 0.058$ | $0.117\pm 0.011$ | $0.209^{+0.022}_{-0.026}$
$d_{1}\ (GeV^{-1})$ | $0.246\pm 0.077$ | $-0.848\pm 0.039$ | $0.151^{+0.021}_{-0.027}$
$d_{2}\ (GeV^{-1})$ | $0.120\pm 0.057$ | $0.634\pm 0.036$ | $0.126^{+0.012}_{-0.009}$
$d_{3}\ (GeV^{-1})$ | $0.270\pm 0.082$ | $0.463\pm 0.047$ | $0.299^{+0.020}_{-0.024}$
$d_{4}\ (GeV^{-1})$ | $0.723\pm 0.085$ | $-0.678\pm 0.032$ | $0.249^{+0.027}_{-0.033}$
$g_{1}\ (GeV^{-1})$ | $0.105\pm 0.114$ | $-1.211\pm 0.049$ | -
$g_{2}\ (GeV^{-1})$ | $-0.024\pm 0.056$ | $0.764\pm 0.041$ | -
$g_{4}\ (GeV^{-1})$ | $0.301\pm 0.097$ | $-1.030\pm 0.036$ | -
$h_{1}\ (GeV^{-1})$ | $0.540\pm 1.070$ | $-0.533\pm 0.373$ | -
$h_{2}\ (GeV^{-1})$ | $0.387\pm 0.483$ | $-1.979\pm 0.229$ | -
$h_{3}\ (GeV^{-1})$ | $0.472\pm 0.821$ | $7.452\pm 0.159$ | -
$h_{4}\ (GeV^{-1})$ | $-0.291\pm 0.832$ | $-2.547\pm 0.319$ | -
$D$ | $0.701\pm 0.102$ | $0.899\pm 0.004$ | $0.700^{+0.064}_{-0.144}$
$F$ | $0.510\pm 0.056$ | $0.510\pm 0.017$ | $0.510^{+0.060}_{-0.050}$
$\chi^{2}_{\rm d.o.f.}$ | $0.86$ | $0.77$ | $1.14$

The parameters of the new s-wave model are displayed in the second column of
Table 2. Although this fit has the lowest $\chi^{2}_{\text{d.o.f.}}$ of
$0.77$, we observe an overall discrepancy in the values of the parameters and
in their accuracy compared to those obtained in our previous s-wave (old)
study. It is important to stress that the error bars can not be directly
compared because of the different methods employed to calculate them, as
mentioned in the caption of Table 2. Despite this fact, some qualitative
comparison can be made, as we discuss below.

Although we get natural-sized values for all the subtraction constants
obtained with the two s-wave models, the parameter sets are very different.
Moreover, the s-wave model provides larger values of the $d_{i}$ parameters
compared to those of the s-wave (old) model, quite high values of the new
$g_{i}$’s and $h_{i}$’s, as well as a small value of $f/f_{\pi}$, almost at
the lower edge, together with values of the axial vector constants at their
upper edges. A plausible explanation is that the $g$\- and $h$-terms of the
new model can play a role in accommodating the data at higher energies, where
these terms are more important, and the fitting procedure forces them to be
sizable at the expense of producing extreme values for the other parameters.

It is therefore expected that the incorporation of the p-waves in the fit,
acting also more importantly at higher energies CaroRamon:1999jf , might bring
the parameters of the $g$\- and $h$-terms within a more natural range. This is
indeed what happens for the s+p-waves model, whose parameters are compiled in
the first column of Table 2. The parameters $b_{i}$’s and $d_{i}$’s become
also more moderate in size, except for the value of $d_{4}$ which shows big
differences among the different models. We note that the volatility of this
NLO LEC was already discussed in Feijoo:2018den  and it was tied to the fact
that this parameter only plays a role in the elastic transitions of the
sector. Finally, we observe that the fit of the new s+p-waves model produces a
value of $f/f_{\pi}$ very close to that of the old model, together with
exactly the same values for $D$ and $F$.

Table 3: Threshold observables obtained from our fits. Experimental data is taken from br_1 ; br_2 ; SIDD .  | $\gamma$ | $R_{n}$ | $R_{c}$ | $a_{p}(K^{-}p\rightarrow K^{-}p)$ | $\Delta E_{1s}$ | $\Gamma_{1s}$
---|---|---|---|---|---|---
s+p-waves | 2.36 | 0.188 | 0.662 | $-0.70+{\rm i\,}0.81$ | 297 | 532
s-wave | 2.40 | 0.179 | 0.665 | $-0.64+{\rm i\,}0.83$ | 280 | 560
s-wave (old) | $2.36^{+0.03}_{-0.03}$ | $0.188^{+0.010}_{-0.011}$ | $0.659^{+0.005}_{-0.002}$ | $-0.65^{+0.02}_{-0.08}+{\rm i\,}0.88^{+0.02}_{-0.05}$ | $288^{+23}_{-8}$ | $588^{+9}_{-40}$
Exp. | $2.36{\pm}0.04$ | $0.189{\pm}0.015$ | $0.664{\pm}0.011$ | $(-0.66{\pm}0.07){+}{\rm i\,}(0.81{\pm}0.15)$ | $283{\pm}36$ | $541{\pm}92$

The results of the threshold observables for the previous models are collected
in Table 3, where they can be compared to the corresponding experimental
values also included there. There is no substantial change in the reproduction
of the experimental values when comparing the new models to the s-wave (old)
one.

Figure 2: Total cross sections of the $K^{-}p\to
K^{-}p,\bar{K}^{0}n,\pi^{-}\Sigma^{+},\pi^{+}\Sigma^{-},\pi^{0}\Sigma^{0},\pi^{0}\Lambda,\eta\Lambda,\eta\Sigma^{0},K^{+}\Xi^{-},K^{0}\Xi^{0}$
reactions obtained for the s+p-waves (solid black line), s-wave (old) (dashed
green line), with the corresponding estimation of the error bands (grey), and
for s-wave (dotted red line) models (dashed line). Experimental data has been
taken from exp_1 ; exp_2 ; exp_3 ; exp_4 ; exp_5 ; exp_6 ; exp_7 ; exp_8 ;
exp_9 ; exp_10 ; exp_11 ; Starostin ; Baxter ; Jones ; Berthon ; Jones:1974si
. See the text for a detailed description of the models.

In Fig. 2, the calculated total cross sections of $K^{-}p$ scattering to all
channels of the $S=-1$ sector are compared to experimental data. As can be
seen from the first three panel rows, the agreement of all models in
reproducing the threshold observables is also reflected on the total cross
sections of the classical processes, namely $K^{-}p\to K^{-}p$,
$\bar{K}^{0}n$, $\pi^{-}\Sigma^{+}$, $\pi^{+}\Sigma^{-}$, $\pi^{0}\Sigma^{0}$,
and $\pi^{0}\Lambda$, which are studied at energies close to the $K^{-}p$
threshold. It is interesting to comment on the little enhancement of strength
seen for the s+p-waves model at CM energies around $1500$ MeV in the
$K^{-}p\to\pi^{0}\Lambda$ cross section. In this case, since the corresponding
threshold is located several tens of MeV below the other channels, the
momentum of the outgoing particles can reach higher values thereby providing a
more relevant role to the p-wave contribution.

In contrast to what we see in the conventional channels, larger differences
among the models are observed in the cross sections of the
$K^{-}p\to\eta\Lambda,\eta\Sigma^{0},K^{+}\Xi^{-},K^{0}\Xi^{0}$ processes,
shown in the two bottom panel rows of Fig. 2. In the case of the $\eta$
channels, one can clearly see that all models describe qualitatively well the
data, exhibiting minor discrepancies at energies below $1800$ MeV. The
differences among the models become appreciable at higher energies. From the
study in GO ; Feijoo:2018den , we know that the $\eta\Lambda$ and
$\eta\Sigma^{0}$ reactions are sensitive to the NLO (and Born) terms up to the
extent that one cannot reproduce properly the experimental data in the case of
$\eta\Lambda$ channel at this energy range unless such terms are taken into
account. Consequently, the disagreement observed between the s-wave (old) and
s-wave models can be directly attributed to effects induced by the
incorporation of the new $g$\- and $h$-terms. The additional p-wave
contribution in the s+p-waves model, which gains significance as the momenta
of particles involved increase, seems to provide more strength to the
$K^{-}p\to\eta\Lambda$ cross section around $1900$ MeV and to the
$K^{-}p\to\eta\Sigma^{0}$ one from $2000$ MeV on.

In the case of the $K\Xi$ channels, there are appreciable differences between
the models in the whole energy range. In particular, the s+p-waves model for
the $K^{0}\Xi^{0}$ cross section slightly overestimates the experimental data
below $2000$ MeV while remaining somewhat below the experimental structure
that appears at $2100$ MeV. On the contrary, this model offers the best
description for the experimental points located at higher energies for both
the $K^{0}\Xi^{0}$ and $K^{+}\Xi^{-}$ total cross sections. This is in
opposition to the s-wave and s-wave (old) models that, in such a region,
accommodate the experimental points of the $K^{0}\Xi^{0}$ and $K^{+}\Xi^{-}$
cross sections, respectively, at the edges of their errors bars. In general,
the s+p-waves model is smoothly averaging over the whole set of points and the
s-wave models are able to provide a more marked structure around 2100 MeV.

Figure 3: Total cross sections of the $K^{-}p\to
K^{-}p,\bar{K}^{0}n,\pi^{-}\Sigma^{+},\pi^{+}\Sigma^{-},\pi^{0}\Sigma^{0},\pi^{0}\Lambda,\eta\Lambda,\eta\Sigma^{0},K^{+}\Xi^{-},K^{0}\Xi^{0}$
reactions obtained for s+p-waves (solid black line), and the corresponding
contributions for $J^{P}={\frac{1}{2}}^{-}$ (dotted red line),
$J^{P}={\frac{1}{2}}^{+}$ (dashed green line) and $J^{P}={\frac{3}{2}}^{+}$
(dash-dotted violet line).

For a deeper understanding of the role of the p-waves, in Fig. 3 we present
the individual partial-wave contributions to the $K^{-}p$ processes included
in the s+p-waves model, namely the $J^{P}={\frac{1}{2}}^{-}$ (s-wave),
${\frac{1}{2}}^{+}$ and ${\frac{3}{2}}^{+}$ (p-wave) channels. Turning first
to the classical processes (top six panels), we notice that p-wave effects are
only noticeable in the $\pi^{0}\Lambda$ production process at the upper edge
of the energy range, as discussed in the previous paragraph. The other cross
sections are essentially built from s-wave contributions at the energies
explored here.

The situation becomes much more interesting for those reaction channels that
open up at higher energies. Examining the $K^{-}p\to\eta\Lambda$ cross
section, one can observe that the s-wave $J^{P}={\frac{1}{2}}^{-}$
contribution (dotted red line) is dominant just above threshold due to the
$\Lambda(1670)$ resonance, generated dynamically in this partial wave. Then,
as the energy increases, its role is moderately losing relevance against the
$J^{P}={\frac{3}{2}}^{+}$ component (dash-dotted violet line) which is the
main contribution in the region ranging from $1750$ to $2100$ MeV. As a matter
of fact, the predominant character of the $J^{P}={\frac{3}{2}}^{+}$ component
is another proof of the need to incorporate NLO terms in the kernel, as it can
be seen that the WT and the Born s-channel terms do not contribute to this
partial wave. Moreover, it is interesting to note that the present
$J^{P}={\frac{3}{2}}^{+}$ contribution may replace the role of the explicit
resonance $\Lambda(1890)$ with these quantum numbers that was incorporated, in
addition to the s-wave chiral amplitudes, in one of the models studied in
Feijoo:2018den  producing a similar description of the $\eta\Lambda$ cross
section above $1900$ MeV. Concerning the $J^{P}={\frac{1}{2}}^{+}$ partial-
wave (dashed green line), it is hardly contributing to the cross section which
is in contrast to the coupled channel partial wave analysis carried out in
Ref. Matveev:2019igl . There, the corresponding cross section contains a
contribution coming from the $J^{P}={\frac{1}{2}}^{+}$ channel comparable in
size to that of the $J^{P}={\frac{3}{2}}^{+}$ one, the latter having a similar
strength as ours. As a last comment, we emphasize that the information encoded
in this process can be very valuable as it acts as an $I=0$ filter, thereby
helping to reduce the potential ambiguities in the amplitudes Feijoo:2018den ;
Feijoo:2015cca .

Next, we focus on another isospin filtering process, namely the
$K^{-}p\to\eta\Sigma^{0}$ reaction which proceeds in the $I=1$ channel. In
this case, the s-wave $J^{P}={\frac{1}{2}}^{-}$ contribution is clearly the
dominant one over the whole range of energies explored. The p-wave
contributions start being noticeable from $1850$ MeV, representing 20% of the
total cross section at most. However, one should be very careful about drawing
conclusions because of the lack of experimental data in this channel beyond
$1800$ MeV.

The partial-wave decomposition of the $K\Xi$ total cross section provides very
valuable information about the relevance of higher partial waves in our model.
On the one hand, the $K^{0}\Xi^{0}$ production discloses the fundamental role
played by the $J^{P}={\frac{1}{2}}^{+}$ contribution above $2200$ MeV and how
the low-energy regime is dominated by the $J^{P}={\frac{1}{2}}^{-}$ wave. The
$J^{P}={\frac{3}{2}}^{+}$ component provides a nonnegligible strength around
2100 MeV, which is however not enough to reproduce the sizable experimental
structure. On the other hand, in the $K^{-}p\to K^{+}\Xi^{-}$ process, the
$J^{P}={\frac{1}{2}}^{-}$ and $J^{P}={\frac{1}{2}}^{+}$ contributions are
moderate and roughly constant, being the $J^{P}={\frac{3}{2}}^{+}$ component
the one that governs the description of the experimental data. A subdominance
of the s-wave component was also obtained for the $\pi^{+}p\to
K^{+}\Sigma^{+}$ process in Ref. CaroRamon:1999jf , where a similar chiral
scheme up to NLO for the $S=0$ sector was employed. The resonant model of Ref.
Matveev:2019igl  produces $K^{0}\Xi^{0}$ and $K^{+}\Xi^{-}$ cross section that
are dominated by the ${\frac{3}{2}}^{+}$ contribution with a sizable influence
of the ${\frac{5}{2}}^{-}$ one. The s-wave $J^{P}={\frac{1}{2}}^{-}$ component
is weak and the ${\frac{1}{2}}^{+}$ contribution practically nonexistent, in
strong opposition to the present results.

Figure 4: Total cross sections of the
$K^{-}p\to\eta\Lambda,\eta\Sigma^{0},K^{+}\Xi^{-},K^{0}\Xi^{0}$ reactions
obtained for s+p-waves (solid black line) compared to the contributions for
$J^{P}={\frac{3}{2}}^{-}$ (dashed red line) and $J^{P}={\frac{5}{2}}^{-}$
(dotted green line) one obtains employing the same parametrization (first
column in Table 2). See text for a detailed explanation of the content.

The incorporation of the d-wave components of the interaction is the natural
next step It can be easily verified that these partial waves emerge from the
crossed Born and the NLO kernels. It seems that a model containing the s-, p-
and d-wave components simultaneously would provide the most realistic
information about the NLO LECs of the Lagrangian (6). However, in practice,
reaching a proper parametrization of the LECs when the d-waves are included is
highly non-trivial, given the interdependence of the parameters contributing
to the various partial waves at tree level, together with the complications of
the coupled-channel scheme. Preliminary attempts have led us to the conclusion
that a proper and controlled inclusion of higher partial waves in the fit
requires the extension of the data set to include the available total and
differential cross sections at higher energies, an enormous task that is
beyond the scope of the present work.

However, in order to illustrate the relevance of the d-waves, we take the
parameters of the s+p-waves model and present, in Fig. 4, the
$J^{P}={\frac{3}{2}}^{-}$ and $J^{P}={\frac{5}{2}}^{-}$ contributions to the
cross sections of the
$K^{-}p\to\eta\Lambda,\eta\Sigma^{0},K^{+}\Xi^{-},K^{0}\Xi^{0}$ reactions,
which are the ones expected to be more substantially affected by the higher
partial waves. As can be noticed from all panels in Fig. 4, the d-wave
contributions start to have a non-negligible effect about 100 MeV above the
threshold of the corresponding reaction. In the cases of the $\eta\Lambda$ and
$K^{+}\Xi^{-}$ cross sections these partial waves contribute in a very
remarkable way. This demonstrates that the d-wave terms seem to be an
essential ingredient to describe the $K^{-}p$ inelastic cross sections at
higher energies and, consequently, a fitting procedure including the effect of
the d-wave terms needs to be performed. From a purely phenomenological point
of view, it seems that the $J^{P}={\frac{3}{2}}^{-}$ and
$J^{P}={\frac{5}{2}}^{-}$ contributions could indeed provide the additional
structures needed to accommodate the experimental data. But the most important
consequence of the inclusion of such contributions is the possibility to get
more reliable values for the NLO LECs. This prospect stems from the fact that
the d-waves receive contributions mostly from the NLO terms of the Lagrangian,
whereas the d-wave contributions of the LO terms come exclusively from the
crossed Born diagram.

Table 4:  Comparison of the pole positions between the models: s+p-waves,
s-wave (old) and s-wave (in MeV) and the corresponding modulus of the
couplings $\lvert g_{i}\rvert$ found in $I=0$ and $I=1$ channels. s+p-waves
---
$J^{P}=\frac{1}{2}^{-}$ | $(I,S)=(0,-1)$ | $(I,S)=(1,-1)$
| $\Lambda(1405)$ | $\Lambda(1670)$ |  | $\Sigma^{*}$
$M\;\rm[MeV]$ | $1399.71$ | $1423.30$ | $1674.05$ | $M\;\rm[MeV]$ | $1590.97$
$\Gamma\;\rm[MeV]$ | $118.50$ | $58.02$ | $31.08$ | $\Gamma\;\rm[MeV]$ | $480.14$
| $\lvert g_{i}\rvert$ | $\lvert g_{i}\rvert$ | $\lvert g_{i}\rvert$ |  | $\lvert g_{i}\rvert$
$\pi\Sigma$ | $3.45$ | $2.57$ | $0.37$ | $\pi\Lambda$ | $1.24$
$\bar{K}N$ | $3.19$ | $3.70$ | $0.40$ | $\pi\Sigma$ | $1.36$
$\eta\Lambda$ | $0.66$ | $0.85$ | $1.33$ | $\bar{K}N$ | $1.79$
$K\Xi$ | $0.72$ | $0.52$ | $3.65$ | $\eta\Sigma$ | $0.37$
|  |  |  | $K\Xi$ | $0.57$
s-wave
$J^{P}=\frac{1}{2}^{-}$ | $(I,S)=(0,-1)$ | $(I,S)=(1,-1)$
| $\Lambda(1405)$ | $\Lambda(1670)$ |  | $\Sigma^{*}$
$M\;\rm[MeV]$ | $1364.13$ | $1419.54$ | $1679.16$ | $M\;\rm[MeV]$ | $-$
$\Gamma\;\rm[MeV]$ | $190.58$ | $39.14$ | $62.36$ | $\Gamma\;\rm[MeV]$ | $-$
| $\lvert g_{i}\rvert$ | $\lvert g_{i}\rvert$ | $\lvert g_{i}\rvert$ |  | $\lvert g_{i}\rvert$
$\pi\Sigma$ | $3.00$ | $1.59$ | $0.26$ | $\pi\Lambda$ | $-$
$\bar{K}N$ | $2.41$ | $3.14$ | $0.64$ | $\pi\Sigma$ | $-$
$\eta\Lambda$ | $0.71$ | $1.21$ | $1.78$ | $\bar{K}N$ | $-$
$K\Xi$ | $1.86$ | $2.14$ | $4.50$ | $\eta\Sigma$ | $-$
|  |  |  | $K\Xi$ | $-$
s-wave (old)
$J^{P}=\frac{1}{2}^{-}$ | $(I,S)=(0,-1)$ | $(I,S)=(1,-1)$
| $\Lambda(1405)$ | $\Lambda(1670)$ |  | $\Sigma^{*}$
$M\;\rm[MeV]$ | $1419^{+16}_{-22}$ | $1420^{+15}_{-21}$ | $1675^{+10}_{-11}$ | $M\;\rm[MeV]$ | $1701^{+16}_{-1}$
$\Gamma\;\rm[MeV]$ | $142^{+48}_{-62}$ | $54^{+36}_{-22}$ | $62^{+8}_{-14}$ | $\Gamma\;\rm[MeV]$ | $340^{+4}_{-14}$
| $\lvert g_{i}\rvert$ | $\lvert g_{i}\rvert$ | $\lvert g_{i}\rvert$ |  | $\lvert g_{i}\rvert$
$\pi\Sigma$ | $3.40$ | $2.31$ | $0.47$ | $\pi\Lambda$ | $1.96$
$\bar{K}N$ | $2.98$ | $3.51$ | $0.59$ | $\pi\Sigma$ | $0.47$
$\eta\Lambda$ | $1.10$ | $1.26$ | $1.74$ | $\bar{K}N$ | $1.21$
$K\Xi$ | $0.65$ | $0.36$ | $3.71$ | $\eta\Sigma$ | $0.36$
|  |  |  | $K\Xi$ | $0.98$

Finally, we analyze the pole content of the scattering amplitudes derived from
the s+p-waves and s-wave models and compare with the corresponding pole
content from the s-wave (old) model. The first thing to be mentioned is that
we cannot dynamically generate any state neither with
$J^{P}={\frac{1}{2}}^{+}$ nor with $J^{P}={\frac{3}{2}}^{+}$ from the
s+p-waves model, in spite of the existence of some $\Lambda^{*}$ and
$\Sigma^{*}$ resonances with these quantum numbers in the range covered by the
higher-energy inelastic channels. According to the PDG compilation PDG , these
resonances decay mostly into $\bar{K}N$, $\pi\Sigma$, $\pi\Lambda$ or to more
exotic two- and three-body products. Thus, the inclusion of higher-energy
experimental data from the $K^{-}p\to\bar{K}N$, $\pi\Sigma$, and $\pi\Lambda$
processes in future fits could favor the appearance of poles in the scattering
amplitude, since there are clear structures reflecting resonant signals in the
corresponding cross sections.

The results of the pole position of the resonances with
$J^{P}={\frac{1}{2}}^{-}$, together with their couplings to the different
channels, are compiled in Table 4 for the s+p-waves (top), the s-wave (middle)
and the s-wave (old) (bottom) models.

In the $I=0$ channel, all models generate the double-pole structure of
$\Lambda(1405)$ and a pole corresponding to $\Lambda(1670)$. As can be
appreciated from the panel on the top, the broader pole of $\Lambda(1405)$ has
been shifted $20$ MeV towards lower energies compared to the corresponding
resonance found in the s-wave (old) amplitudes. Despite the location of this
first pole is not well established, given the scattered positions found in the
literature, the new position of this pole is in better agreement with most of
the studies devoted to this topic (see e.g. Cieply:2016jby ; Feijoo:2018den ).
This is also the case for the corresponding pole in the s-wave model of the
middle panel, but in this case it is shifted downwards by 55 MeV. The widths
of these broad poles behave in a contrary way, while that of s+p-waves model
gets narrower, compared to that of the s-wave (old) model, the one obtained
from the s-wave model is even larger. All these broad poles couple most
notably to $\pi\Sigma$ and $\bar{K}N$ states, the coupling to $\pi\Sigma$
being slightly higher. All models produce a higher energy and narrower
$\Lambda(1405)$ state. While the position in energy is similar for the three
models, the pole found by the s-wave model is narrower by more than $10$ MeV
compared to those of found by the two other models. The pattern of the
couplings for the narrow $\Lambda(1405)$ in the three models is in accordance
to most of the works in literature, meaning that this pole couples mostly to
the $\bar{K}N$ channel but has a sizable coupling to the $\pi\Sigma$ one. As
for the $\Lambda(1670)$ resonance, the most remarkable fact is that the
s+p-waves model produces a width that is half of that obtained by the other
two models and in much better agreement with the value quoted by the PDG PDG .

Concerning the $I=1$ sector, the s-wave model does not generate any state,
whereas the s+p-waves model produces a wide resonance about 100 MeV below in
energy and substantially wider than that found by the s-wave (old) model.
These states can hardly be identified with any of the $\Sigma$ resonances
listed in the PDG with $J^{P}=1/2^{-}$ PDG . For instance, the observed 1-star
$\Sigma(1620)$ resonance is substantially narrower and has been seen to decay
into $\pi\Sigma$ states with twice as much probability than into $\pi\Lambda$
states, a decay pattern that cannot be reproduced by neither of the $I=1$
states listed in Table 4 according to the size of their couplings to the
different meson-baryon channels. The identification with the $\Sigma(1750)$
resonance is also discarded, as it has been seen to decay more strongly into
$\eta\Sigma$ states and, with a somewhat smaller branching ratio, into
$\pi\Lambda$ and $\pi\Sigma$. We note, however, that the large width of the
$I=1$ states found here hinders their experimental identification in reactions
where they might be produced.

##  5 Conclusions

We have performed a new study of the meson–baryon interaction in the $S=-1$
sector including both s- and p-waves, aimed at improving our knowledge about
the NLO terms of the chiral SU(3) Lagrangian. We pay special attention to the
processes that are very sensitive to these subleading terms of the Lagrangian,
such as the $K^{-}p\to K^{+}\Xi^{-},K^{0}\Xi^{0},\eta\Lambda,\eta\Sigma^{0}$
reactions. This work presents a detailed derivation of the formalism needed to
obtain the higher partial waves of the scattering amplitudes, beyond the
s-wave component usually considered in the literature.

Our previous models evolved in the direction of implementing systematic
improvement to obtain a more reliable determination of the NLO terms of the
Lagrangian, namely the $b_{i}$ and $d_{i}$ LECs, culminating into the s-wave
(old) set obtained in Feijoo:2018den . In the present work we have first
improved upon the later model by incorporating the novel $g$\- and $h$-terms
from the NLO interaction kernel. This has led to the new s-wave set of
parameters, which reproduces experimental data very well and achieves very low
$\chi^{2}_{d.o.f.}$ of 0.77. However, we have observed that the LECs of this
s-wave model, both the LO and NLO ones, are qualitatively very different from
our earlier estimations.

We have then checked how stable is this new s-wave set by incorporating into
the model the p-wave contributions, which strongly depend on the novel $g$\-
and $h$-terms but also on the rest of NLO parameters. With respect to the
s-wave results, the s+p-waves model also produces a very good fit, although
with a slightly worse $\chi^{2}_{d.o.f.}$ value of 0.86 and a rather different
set of LECs. The latter fact is an indication of the clear interconnection
between the higher partial waves and the higher orders of the chiral
expansion, as both contributions acquire importance at higher energies. It is
interesting to note, however, that the size of the LECs obtained with the
s+p-waves model is smaller and closer to that of our earlier s-wave (old)
model. This makes us believe that the s+p-waves model developed in the present
work constitutes a very good starting point for subsequent implementations of
higher partial waves. In this work, we have shown that a notable fraction of
the cross sections of the $K^{-}p\to
K^{+}\Xi^{-},K^{0}\Xi^{0},\eta\Lambda,\eta\Sigma^{0}$ reactions comes indeed
from the novel p-wave contributions, thereby making it clear that higher
partial waves cannot be considered as minor corrections in these processes.

The main conclusion of this study is that one does need to incorporate higher
partial-wave contributions to the scattering amplitudes in order to properly
describe the $K^{-}p$ inelastic reactions opening up at higher energies. This
is in contrast to our previous works, where the experimental data of such
processes was reasonably reproduced employing pure s-wave scattering
amplitudes. In other words, our previous models have effectively overestimated
the lowest partial-wave contributions, thereby masking the physics of higher
partial waves behind the values of the NLO parameters. As for the relevance of
the new $g$\- and $h$-terms considered in this work, compared to the rest of
NLO contributions, we cannot say anything conclusive, as we have obtained very
different values of the parameters in the two new fits presented here. What
seems to be clear is that they provide a non-negligible contribution to the
scattering amplitudes, especially far enough from thresholds, because of their
strong dependence on momenta. Therefore, it is natural to consider the $g$\-
and $h$-terms when implementing higher partial waves in the scattering
amplitudes, since both contributions acquire a relevant role at higher
energies.

As a final remark we would like to comment that the s+p-waves model is the
natural continuation of our previous ones Feijoo:2015yja ; Ramos:2016odk ;
Feijoo:2018den . Since it employs the same set of data points in the fit, it
provides a clear quantification of the effect of the new ingredients, namely
the $g$\- and $h$-terms and the p-waves. This model should then be considered
as a promising reference point upon which additional partial-wave
contributions, in particular the d-waves, need to be added. However, this
challenging study, which is already in progress, requires including more
experimental data points of total and differential cross sections within a
wider range of energies in the fits. As can be clearly seen in Table 2, the
determination of LECs becomes an extensive nonlinear optimization problem
where the relatively large number of parameters makes it challenging to
determine their optimal values. Thus, it is also crucial to establish a
credible program to quantify the magnitude of statistical and systematic
uncertainties of the LECs and to study their propagation into experimental
observables. This can be achieved by employing advanced mathematical
optimization tools, as introduced e.g. in NavarroPerez:2012qf ;
Carlsson:2015vda . We believe that this is the proper procedure to have a
realistic determination of the NLO parameters of the chiral SU(3) Lagrangian.

All authors contributed on an equal footing to the research presented in this
work. All authors have read and agreed to the published version of the
manuscript.

V.M. and A.R. acknowledge support from the Ministerio de Ciencia, Innovación y
Universidades under the project CEX2019-000918-M of ICCUB (Unidad de
Excelencia “María de Maeztu”), and from the Ministerio de Economía y
Competitividad, with additional European Regional Development Funds, under the
contract FIS2017-87534-P. A.F. acknowledges support from the from the Spanish
Ministerio de Ciencia, Innovación y Universidades under the project
PID2020-112777GB-I00, from the Ministerio de Economía y Competitividad and the
European Regional Development Fund, under contract FIS2017-84038-C2-1-P, and
from the Generalitat Valenciana under contract PROMETEO/2020/023. V.M., A.R.
and A.F. also acknowledge support from the EU STRONG-2020 project under the
program H2020-INFRAIA-2018-1, grant agreement no. 824093. The work of D.G. was
supported by the Czech Science Foundation, GAČR grant no. 19-19640S.

The authors declare no conflict of interest.

no

## Appendix A

Table 5: The $g_{ij}$ and $h_{ij}$ coefficients of Eq. (10).

$g_{ij}$ coefficients
---
| $K^{-}p$ | $\bar{K}^{0}n$ | $\pi^{0}\Lambda$ | $\pi^{0}\Sigma^{0}$ | $\eta\Lambda$ | $\eta\Sigma^{0}$ | $\pi^{+}\Sigma^{-}$ | $\pi^{-}\Sigma^{+}$ | $K^{+}\Xi^{-}$ | $K^{0}\Xi^{0}$
$K^{-}p$ | $2(g_{2}+g_{4})$ | $g_{1}+g_{2}$ | $-\frac{\sqrt{3}}{2}(g_{1}+g_{2})$ | $-\frac{1}{2}(g_{1}+g_{2})$ | $\frac{1}{2}(g_{1}-3g_{2})$ | $\frac{1}{2\sqrt{3}}(g_{1}-3g_{2})$ | $-2g_{2}$ | $g_{2}-g_{1}$ | $-4g_{2}$ | $-2g_{2}$
$\bar{K}^{0}n$ |  | $2(g_{2}+g_{4})$ | $\frac{\sqrt{3}}{2}(g_{1}+g_{2})$ | $-\frac{1}{2}(g_{1}+g_{2})$ | $\frac{1}{2}(g_{1}-3g_{2})$ | $-\frac{1}{2\sqrt{3}}(g_{1}-3g_{2})$ | $g_{2}-g_{1}$ | $-2g_{2}$ | $-2g_{2}$ | $-4g_{2}$
$\pi^{0}\Lambda$ |  |  | $2g_{4}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $\frac{\sqrt{3}}{2}(g_{1}-g_{2})$ | $\frac{\sqrt{3}}{2}(g_{2}-g_{1})$
$\pi^{0}\Sigma^{0}$ |  |  |  | $2g_{4}$ | $0$ | $0$ | $-2g_{2}$ | $-2g_{2}$ | $\frac{1}{2}(g_{1}-g_{2})$ | $\frac{1}{2}(g_{1}-g_{2})$
$\eta\Lambda$ |  |  |  |  | $2g_{4}$ | $0$ | $0$ | $0$ | $-\frac{1}{2}(g_{1}+3g_{2})$ | $-\frac{1}{2}(g_{1}+3g_{2})$
$\eta\Sigma^{0}$ |  |  |  |  |  | $2g_{4}$ | $\frac{2}{\sqrt{3}}g_{1}$ | $-\frac{2}{\sqrt{3}}g_{1}$ | $-\frac{1}{2\sqrt{3}}(g_{1}+3g_{2})$ | $\frac{1}{2\sqrt{3}}(g_{1}+3g_{2})$
$\pi^{+}\Sigma^{-}$ |  |  |  |  |  |  | $2(g_{2}+g_{4})$ | $-4g_{2}$ | $g_{1}+g_{2}$ | $-2g_{2}$
$\pi^{-}\Sigma^{+}$ |  |  |  |  |  |  |  | $2(g_{2}+g_{4})$ | $-2g_{2}$ | $g_{1}+g_{2}$
$K^{+}\Xi^{-}$ |  |  |  |  |  |  |  |  | $2(g_{2}+g_{4})$ | $g_{2}-g_{1}$
$K^{0}\Xi^{0}$ |  |  |  |  |  |  |  |  |  | $2(g_{2}+g_{4})$
$h_{ij}$ coefficients
| $K^{-}p$ | $\bar{K}^{0}n$ | $\pi^{0}\Lambda$ | $\pi^{0}\Sigma^{0}$ | $\eta\Lambda$ | $\eta\Sigma^{0}$ | $\pi^{+}\Sigma^{-}$ | $\pi^{-}\Sigma^{+}$ | $K^{+}\Xi^{-}$ | $K^{0}\Xi^{0}$
$K^{-}p$ | $-h_{1}-h_{2}-h_{3}-h_{4}$ | $-h_{2}-h_{3}-h_{4}$ | $\frac{1}{2\sqrt{3}}(-h_{1}+3h_{2}+h_{3})$ | $\frac{1}{2}(-h_{1}+h_{2}-h_{3}-2h_{4})$ | $\frac{1}{6}(h_{1}+3h_{2}-7h_{3}-6h_{4})$ | $\frac{1}{2\sqrt{3}}(h_{1}+h_{2}-h_{3})$ | $h_{2}-h_{3}-h_{4}$ | $-h_{1}-h_{4}$ | $2(h_{2}-h_{3}-h_{4})$ | $h_{2}-h_{3}-h_{4}$
$\bar{K}^{0}n$ |  | $-h_{1}-h_{2}-h_{3}-h_{4}$ | $\frac{1}{2\sqrt{3}}(h_{1}-3h_{2}-h_{3})$ | $\frac{1}{2}(-h_{1}+h_{2}-h_{3}-2h_{4})$ | $\frac{1}{6}(h_{1}+3h_{2}-7h_{3}-6h_{4})$ | $\frac{1}{2\sqrt{3}}(-h_{1}-h_{2}+h_{3})$ | $-h_{1}-h_{4}$ | $h_{2}-h_{3}-h_{4}$ | $h_{2}-h_{3}-h_{4}$ | $2(h_{2}-h_{3}-h_{4})$
$\pi^{0}\Lambda$ |  |  | $-\frac{1}{3}(h_{1}+2h_{3})$ | $0$ | $0$ | $-\frac{1}{3}(h_{1}+2h_{3}+3h_{4})$ | $0$ | $0$ | $\frac{1}{\sqrt{3}}(h_{1}-h_{3})$ | $\frac{1}{\sqrt{3}}(h_{3}-h_{1})$
$\pi^{0}\Sigma^{0}$ |  |  |  | $-h_{1}-2h_{3}-2h_{4}$ | $-\frac{1}{3}(h_{1}+2h_{3}+3h_{4})$ | $0$ | $h_{2}-h_{3}-h_{4}$ | $h_{2}-h_{3}-h_{4}$ | $-h_{3}-h_{4}$ | $-h_{3}-h_{4}$
$\eta\Lambda$ |  |  |  |  | $-h_{1}-2h_{3}-2h_{4}$ | $0$ | $-\frac{1}{3}(h_{1}+2h_{3}+3h_{4})$ | $-\frac{1}{3}(h_{1}+2h_{3}+3h_{4})$ | $\frac{1}{3}(-h_{1}+3h_{2}-2h_{3}-3h_{4})$ | $\frac{1}{3}(-h_{1}+3h_{2}-2h_{3}-3h_{4})$
$\eta\Sigma^{0}$ |  |  |  |  |  | $\frac{1}{3}(-h_{1}-2h_{3})$ | $\frac{1}{\sqrt{3}}(h_{1}-h_{2}-h_{3})$ | $-\frac{1}{\sqrt{3}}(h_{1}-h_{2}-h_{3})$ | $\frac{1}{\sqrt{3}}h_{2}$ | $-\frac{1}{\sqrt{3}}h_{2}$
$\pi^{+}\Sigma^{-}$ |  |  |  |  |  |  | $-h_{1}-h_{2}-h_{3}-h_{4}$ | $2(h_{2}-h_{3}-h_{4})$ | $-h_{2}-h_{3}-h_{4}$ | $h_{2}-h_{3}-h_{4}$
$\pi^{-}\Sigma^{+}$ |  |  |  |  |  |  |  | $-h_{1}-h_{2}-h_{3}-h_{4}$ | $h_{2}-h_{3}-h_{4}$ | $-h_{2}-h_{3}-h_{4}$
$K^{+}\Xi^{-}$ |  |  |  |  |  |  |  |  | $-h_{1}-h_{2}-h_{3}-h_{4}$ | $-h_{1}-h_{4}$
$K^{0}\Xi^{0}$ |  |  |  |  |  |  |  |  |  | $-h_{1}-h_{2}-h_{3}-h_{4}$

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