# First application of a microscopic $K^{-}NN$ absorption model in
calculations of kaonic atoms

J. Óbertová<EMAIL_ADDRESS>Nuclear Physics Institute of the
Czech Academy of Sciences, 25068 Řež, Czech Republic  Faculty of Nuclear
Sciences and Physical Engineering, Czech Technical University in Prague,
Břehová 7, 11519 Prague, Czech Republic  E. Friedman  Racah Institute of
Physics, The Hebrew University, Jerusalem 91904, Israel  J. Mareš  Nuclear
Physics Institute of the Czech Academy of Sciences, 25068 Řež, Czech Republic

###### Abstract

Strong interaction energy shifts and widths in kaonic atoms are calculated for
the first time using microscopic $K^{-}N$ \+ $K^{-}NN$ potentials derived from
$K^{-}N$ scattering amplitudes constructed within SU(3) chiral coupled-
channels models of meson-baryon interactions. The in-medium modifications of
the free-space amplitudes due to the Pauli correlations are taken into
account. The $K^{-}N+K^{-}NN$ potentials evaluated for 23 nuclear species are
confronted with kaonic atoms data. The description of the data significantly
improves when the $K^{-}NN$ absorption is included. To get $\chi^{2}$ as low
as for the $K^{-}N+$phenomenological multi-nucleon potential an additional
phenomenological term, accounting for $K^{-}-3N(4N)$ processes, is still
needed. However, density dependence of this phenomenological term points out
some deficiencies in the microscopic potentials and further improvements of
the applied model are thus desirable. The calculated branching ratios for
$K^{-}N$ and $K^{-}NN$ absorption channels in the 12C$+K^{-}$ atom are in
reasonable agreement with the old bubble chamber data, as well as with the
latest data from the AMADEUS Collaboration.

##  I Introduction

The $K^{-}N$ interaction near threshold has recently been described in the
framework of SU(3) chiral coupled-channels models of meson-baryon
interactions. Above threshold, the models are tuned to reproduce low-energy
$K^{-}p$ scattering and reaction data [1, 2, 3]. At threshold, important
constraints on the $K^{-}p$ interaction are provided by three known threshold
branching ratios [4, 5], and particularly by precisely measured strong-
interaction level shift and width of the $1s$ state in the kaonic hydrogen by
the SIDDHARTA Collaboration [6]. On the other hand, the $K^{-}n$ interaction
is poorly determined due to the absence of sufficiently accurate data.
Moreover, the $K^{-}N$ interaction is known substantially less below
threshold, where chiral models differ considerably in their predictions.
Information about the subthreshold $K^{-}$ interaction is provided by the
analyses of $\pi\Sigma$ spectra in the region of the $\Lambda(1405)$ resonance
[7, 8, 9, 10, 11, 12, 13], dynamically generated in the SU(3) chiral coupled-
channels models, and particularly by kaonic atom data throughout the periodic
table. The database of 65 points involves strong-interaction energy shifts,
widths and yields (upper level widths) from CERN, Argonne, RAL, and BNL (see
[14] and references therein).

The above chiral models of the $K^{-}N$ interaction include only the
$K^{-}N\rightarrow\pi Y\;\;(Y=\Lambda,\;\Sigma)$ decay channel. However, in
the nuclear medium, $K^{-}$ interactions with two and more nucleons should be
included as well, e.g., $K^{-}NN\rightarrow YN$. In fact, the $K^{-}$
absorption on two and more nucleons amounts to about 20% of the total $K^{-}$
absorption at the surface of atomic nuclei and the role of multi-nucleon
absorption increases rapidly with density. The multi-nucleon absorption ratios
were first measured in bubble chamber experiments for $K^{-}$ capture on a
mixture of C, F, Br [15], on Ne [16], and C [17]. The $K^{-}$ two-nucleon
absorption fractions for all possible final states on 4He were published in
Ref. [18]. More recently, $K^{-}$ three- and four-nucleon absorption fractions
on 4He for channels with $\Lambda$ in the final state were measured in the
E549 experiment at KEK [19], and the FINUDA Collaboration studied the
$\Sigma^{-}p$ emission rate in reactions of low-energy $K^{-}$ with light
nuclei [20]. Finally, the AMADEUS Collaboration reported the measured $K^{-}$
two-nucleon branching ratios with $\Lambda p$ and $\Sigma^{0}p$ in the final
state for low-energy antikaons absorbed in 12C [21, 22]. The above experiments
provided valuable information on the $K^{-}$ multi-nucleon absorption in the
nuclear medium.

A recent study of kaonic atoms performed by Friedman and Gal [23] revealed
that $K^{-}$ optical potentials based on the $K^{-}N$ scattering amplitudes
derived within SU(3) chiral models fail to reproduce experimental data.
However, once a phenomenological optical potential accounting for the $K^{-}$
multi-nucleon processes in nuclear matter was added, a very good global fit of
kaonic atoms was achieved. Moreover, when extra constraint to reproduce
simultaneously the $K^{-}$ single-nucleon absorption fractions from bubble
chamber experiments [15, 16, 17] was applied, only the Kyoto-Munich [24],
Prague [25], and Barcelona [26] models were found acceptable.

The Prague and Kyoto-Munich models supplemented by the phenomenological
$K^{-}$ multi-nucleon potential were applied in calculations of $K^{-}$
nuclear quasi-bound states [27, 28]. The $K^{-}$ multi-nucleon absorption was
found to contribute considerably to the total widths of these $K^{-}$ nuclear
states, which then substantially exceeded the corresponding binding energies.
However, since analyses of Friedman and Gal [23] have shown that kaonic atom
data probe reliably the real part of the $K^{-}$ optical potential up to
$\approx$ 30% and its imaginary part up to $\approx$ 50% of nuclear density
$\rho_{0}$, the shape of the phenomenological $K^{-}$ multi-nucleon potential
in the nuclear interior is a matter of extrapolation to higher densities.
Therefore, a proper microscopic model for $K^{-}$ absorption on two and more
nucleons in nuclear matter is needed for a reliable description of $K^{-}$
absorption in atomic nuclei.

Sekihara et al. [29] developed a microscopic model for the $K^{-}NN$
absorption in nuclear matter employing a chiral unitary approach to a free-
space $\bar{K}N$ interaction and evaluated the branching ratios of mesonic and
nonmesonic $K^{-}$ absorption in the nuclear medium.

Inspired by the evaluation of an $\eta^{\prime}$-nucleus optical potential
including $2N$ absorption using an $\eta^{\prime}$ self-energy constructed
within a meson exchange formalism [30], Hrtánková and Ramos [31] developed a
microscopic model for the $K^{-}NN$ absorption in symmetric nuclear matter.
The absorption was described within a meson-exchange picture and the primary
$K^{-}N$ interaction strength was derived from chiral interaction models. The
medium modification of the $K^{-}N$ scattering amplitudes due to the Pauli
correlations was taken into account, which appeared crucial. The derived
$K^{-}N$ and $K^{-}NN$ optical potentials were applied in calculations of the
$K^{-}$ single- and two-nucleon absorption fractions and branching ratios for
various mesonic and nonmesonic channels.

In the present work, we apply for the first time the microscopic
$K^{-}N+K^{-}NN$ potentials derived from chiral $K^{-}N$ scattering amplitudes
in calculations of the strong interaction energy shifts and widths in kaonic
atoms. The chiral amplitudes are constructed within the Barcelona and Prague
models and the in-medium modifications of the free-space amplitudes due to the
Pauli principle are taken into account. In addition, we calculate branching
ratios for all $K^{-}N$ and $K^{-}NN$ absorption channels in the 12C+$K^{-}$
atom.

The paper is organized as follows. Section II provides a brief description of
the formalism used to derive the microscopic $K^{-}N$ and $K^{-}NN$ optical
potentials, followed by a construction of underlying in-medium $K^{-}N$
amplitudes from free-space amplitudes obtained within chiral interaction
models. A discussion of subthreshold kinematics applied to kaonic atoms is
also presented. In Section III, the microscopic $K^{-}N$ and $K^{-}NN$ optical
potentials are confronted with kaonic atoms data. It is demonstrated that the
description of the data improves considerably when the $K^{-}NN$ potential is
included. An additional phenomenological term, introduced to incorporate
missing $K^{-}-3N$($4N$) processes, is discussed. Finally, branching ratios
for all $K^{-}N$ and $K^{-}NN$ absorption channels in 12C+$K^{-}$, calculated
using microscopic $K^{-}N+K^{-}NN$ potentials are compared with old bubble
chamber data and with the branching ratios reported recently by the AMADEUS
Collaboration. A brief summary is given in Section IV.

##  II Model

This section provides a brief introduction to the kaonic atom methodology and
to our microscopic $K^{-}NN$ absorption model applied in the present
calculations. For more details see Refs. [23, 31].

The binding energies $B_{K^{-}}$ and widths $\Gamma_{K^{-}}$ of $K^{-}$ atomic
states are determined by solution of the Klein-Gordon equation

$\left[\vec{\nabla}^{2}+\tilde{\omega}_{K^{-}}^{2}-m_{K^{-}}^{2}-\Pi_{K^{-}}(\omega_{K^{-}},\rho)\right]\phi_{K^{-}}=0~{},$
(1)

where $\tilde{\omega}_{K^{-}}=m_{K^{-}}-B_{K^{-}}-{\rm
i}\Gamma_{K^{-}}/2-V_{C}=\omega_{K^{-}}-V_{C}$, $m_{K^{-}}$ is the $K^{-}$
mass, $\omega_{K^{-}}$ stands for a complex kaon energy, $V_{C}$ is the
Coulomb potential introduced via the minimal substitution [32], and $\rho$ is
the nuclear density distribution. The $K^{-}$ interaction with the nuclear
medium is described by the energy- and density-dependent kaon self-energy
operator

$\Pi_{K^{-}}=2{\mu}_{K^{-}}(V_{K^{-}N}+V_{K^{-}NN})~{},$ (2)

where $V_{K^{-}N}$ denotes the $K^{-}$ single-nucleon potential, $V_{K^{-}NN}$
is the $K^{-}$ two-nucleon potential, and $\mu_{K^{-}}$ is the $K^{-}$-nucleus
reduced mass. The $K^{-}N$ potential is taken in a $t\rho$ form

$2\mu_{K^{-}}V_{K^{-}N}=-4\pi\left(1+\frac{A-1}{A}\frac{\mu_{K^{-}}}{m_{N}}\right)\left(F_{0}\frac{1}{2}\rho_{p}+F_{1}\left(\frac{1}{2}\rho_{p}+\rho_{n}\right)\right)~{},$
(3)

where $F_{0}$ and $F_{1}$ are the isospin 0 and 1 $s$-wave in-medium
amplitudes in the $K^{-}N\rightarrow K^{-}N$ channel, respectively, and
$m_{N}$ is the nucleon mass. The symbols $\rho_{p}$ and $\rho_{n}$ denote
proton and neutron density distributions, respectively, calculated within the
relativistic mean field model TM2 for light and medium mass nuclei ($A<40$)
and TM1 for heavier nuclei ($A\geq 40$) [33]. It is to be noted that very
similar results were obtained for other density distributions (two-parameter
Fermi distributions and the NL-SH parametrization [34]).

Apart from that, the imaginary part of the $K^{-}$ single-nucleon potential
can be evaluated as the self-energy of the Feynman diagram shown in Fig. 1
[31]. Here, the shaded circles denote the $K^{-}N\rightarrow\pi
Y,~{}(Y=\Lambda,~{}\Sigma)$ t-matrices derived from a chiral coupled-channels
meson-baryon interaction model. The total imaginary $K^{-}N$ potential is then
built as a sum of the contributions from each absorption channel listed on the
left-hand side of Table 1,

$\text{Im}V_{K^{-}N}=\sum_{\rm channels}\text{Im}V_{K^{-}N\rightarrow\pi
Y}~{}.$ (4)

We checked that ImV${}_{K^{-}N}$ from Eq. (4) and the imaginary part of the
’$t\rho$’ potential [Eq. (3)] yield identical results for kaonic atoms.

Table 1: Channels considered for $K^{-}$ single-nucleon (left) and two-nucleon (right) absorption in nuclear matter.  $K^{-}N$ | $\rightarrow\pi Y$ | $K^{-}N_{1}N_{2}$ | $\rightarrow YN$
---|---|---|---
$K^{-}p$ | $\rightarrow\pi^{0}\Lambda$ | $K^{-}pp$ | $\rightarrow\Lambda p$
| $\rightarrow\pi^{0}\Sigma^{0}$ |  | $\rightarrow\Sigma^{0}p$
| $\rightarrow\pi^{+}\Sigma^{-}$ |  | $\rightarrow\Sigma^{+}n$
| $\rightarrow\pi^{-}\Sigma^{+}$ | $K^{-}pn(np)$ | $\rightarrow\Lambda n$
$K^{-}n$ | $\rightarrow\pi^{-}\Lambda$ |  | $\rightarrow\Sigma^{0}n$
| $\rightarrow\pi^{-}\Sigma^{0}$ |  | $\rightarrow\Sigma^{-}p$
| $\rightarrow\pi^{0}\Sigma^{-}$ | $K^{-}nn$ | $\rightarrow\Sigma^{-}n$
Figure 1:  Feynman diagram for $K^{-}$ single-nucleon absorption in nuclear
matter. The shaded circles denote the $K^{-}N\rightarrow\pi
Y,~{}(Y=\Lambda,~{}\Sigma)$ t-matrices derived from a chiral coupled-channels
meson-baryon interaction model. Figure adapted from Ref. [31].

The $K^{-}NN$ potential is constructed within our recently developed
microscopic model [31]. The $K^{-}NN$ absorption is described as a process
with different intermediate virtual mesons exchanged
($\overline{K},~{}\pi,~{}\eta$) as represented by Feynman diagrams in Figs. 2
and 3. Diagrams in Fig. 3 are obtained by antisymmetrizing the initial
$N_{1}N_{2}$ system and exchanging the $N$ and $Y$ lines in the final state.
The channels considered for $K^{-}NN$ absorption in nuclear matter are listed
on the right-hand side of Table 1. Each channel can proceed via direct (Fig.
2) and exchange (Fig. 3) diagrams with the corresponding intermediate mesons.

Figure 2:  Direct Feynman diagrams for $K^{-}$ absorption on two nucleons
$N_{1},~{}N_{2}$ in nuclear matter. The shaded circles denote the $K^{-}N$
t-matrices derived from a chiral coupled-channels meson-baryon interaction
model. Figure adapted from Ref. [31]. Figure 3:  Exchange Feynman diagrams for
$K^{-}$ absorption on two nucleons $N_{1},~{}N_{2}$ in nuclear matter. The
shaded circles denote the $K^{-}N$ t-matrices derived from a chiral coupled-
channels meson-baryon interaction model. Figure adapted from Ref. [31].

The total $K^{-}NN$ potential is obtained as a sum of contributions coming
from the direct and exchange diagrams for all considered channels

$V_{K^{-}NN}=\sum_{\rm channels}V_{K^{-}NN}^{\rm direct}+V_{K^{-}NN}^{\rm
exchange}~{}.$ (5)

The underlying chiral $K^{-}N$ scattering amplitudes are derived within the
Barcelona (BCN) [26] and Prague (P) [25] models. These models supplemented by
a phenomenological multinucleon term were found to describe simultaneously the
$K^{-}$ atoms data and the $K^{-}$ single-nucleon absorption fraction [23]. In
the nuclear medium, the $K^{-}N$ interaction is modified due to the Pauli
principle [35, 36] and hadron self-energies [37, 38, 39, 40]. We incorporated
the in-medium modifications due to the Pauli blocking using two different
approaches in this work. First, the Pauli blocking effect was included
directly in the chiral amplitudes by restricting the nucleon momentum in the
intermediate meson-nucleon loops of the uniterized amplitude to be larger than
the Fermi momentum (denoted further by ‘Pauli’). The second method is based on
the multiple scattering approach by T. Wass, M. Rho and W. Weise (denoted
further by ’WRW’) [41]. Here, the in-medium isospin 0 and 1 amplitudes $F_{0}$
and $F_{1}$, respectively, for the diagonal channels $K^{-}N\rightarrow
K^{-}N$ are evaluated from the free-space amplitudes $f_{K^{-}N\rightarrow
K^{-}N}$ using the following formulas:

$F_{1}=\frac{f_{K^{-}n\rightarrow
K^{-}n}(\sqrt{s})}{1+\frac{1}{4}\xi_{k}c_{lab}f_{K^{-}n\rightarrow
K^{-}n}(\sqrt{s})\rho}~{},\quad F_{0}=\frac{[2f_{K^{-}p\rightarrow
K^{-}p}(\sqrt{s})-f_{K^{-}n\rightarrow
K^{-}n}(\sqrt{s})]}{1+\frac{1}{4}\xi_{k}c_{lab}[2f_{K^{-}p\rightarrow
K^{-}p}(\sqrt{s})-f_{K^{-}n\rightarrow K^{-}n}(\sqrt{s})]\rho}~{}.$ (6)

Amplitudes for the non-diagonal channels $K^{-}N\rightarrow\pi/\eta
Y~{}(Y=\Lambda,\Sigma)$ are modified as follows

$F_{1(0)}^{\pi/\eta Y}=\frac{f_{1(0)}^{\pi/\eta
Y}(\sqrt{s})}{1+\frac{1}{4}\xi_{k}c_{\rm lab}f_{1(0)}(\sqrt{s})\rho}$ (7)

where $f_{1(0)}^{\pi/\eta Y}$ is the free-space isospin 0 or 1 amplitude in
the channel $K^{-}N\rightarrow\pi/\eta Y$, and $f_{1(0)}$ denotes
corresponding free-space isospin amplitude in the channel $K^{-}N\rightarrow
K^{-}N$. Here,

$c_{\rm lab}=1+\frac{A-1}{A}\frac{\mu_{K^{-}}}{m_{N}}$ (8)

and

$\xi_{k}=\frac{9\pi}{p_{\rm
F}^{2}}\,4I,\;\;\;\;\;I=\int_{0}^{\infty}\frac{dr}{r}\exp(ikr)j_{1}^{2}(p_{F}r),$
(9)

$p_{\rm F}$ is the Fermi momentum corresponding to density $\rho=2p_{\rm
F}^{3}/(3\pi^{2})$, $j_{1}$ is the spherical Bessel function and $k$ is the
kaon momentum

$k=\sqrt{\omega_{K^{-}}^{2}-m_{K^{-}}^{2}}~{}.$ (10)

The integral $I$ in Eq. (9) can be evaluated analytically [23]

$4I(q)=1-\frac{q^{2}}{6}+\frac{q^{2}}{4}\left(2+\frac{q^{2}}{6}\right)\ln\left(1+\frac{4}{q^{2}}\right)-\frac{4}{3}q\left(\frac{\pi}{2}-\arctan(q/2)\right)~{},$
(11)

where $q=-ik/p_{F}$.

Figure 4:  Comparison of the Pauli (green) and WRW (red) modified $K^{-}p$
(left) and $K^{-}n$ (right) BCN amplitudes, printed from the 12C$+K^{-}$ atom
calculation with the $K^{-}N+K^{-}NN$ potential. ’WRW2’ (blue) denotes WRW
amplitudes evaluated using a cms kaon momentum $k_{\rm cms}$ (see text for
details). The vertical line denotes the $K^{-}N$ threshold.

In Fig. 4, we compare the Pauli (green) and WRW (red) modified $K^{-}p$ and
$K^{-}n$ BCN amplitudes in the 12C$+K^{-}$ atom, calculated with the
$K^{-}N+K^{-}NN$ potential as functions of $\sqrt{s}$, where $s$ is the
Mandelstam variable. The free-space amplitudes (black) are also shown for
comparison. The in-medium modification causes the peak of the $K^{-}p$
amplitude to shift towards higher $\sqrt{s}$. There is a noticeable difference
between the Pauli and WRW modified $K^{-}p$ (left panel) and $K^{-}n$ (right
panel) amplitudes. The WRW method yields smaller values of the real and
imaginary parts of the $K^{-}p$ and $K^{-}n$ amplitudes than the Pauli
approach in most of the subthreshold region. The reason for the discrepancy
between the Pauli and WRW amplitudes lies in the definition of momentum $k$
employed in the two methods. While in the WRW method the ansatz for $k$ is
given by Eq. (10), in the evaluation of the Pauli blocked amplitudes a center-
of-mass system (cms) formula is used

$k_{\rm
cms}=\sqrt{\frac{[s-(m_{N}+m_{K^{-}})^{2}][s-(m_{N}-m_{K^{-}})^{2}]}{4s}}~{}.$
(12)

If the expression for $k$ from Eq. (12) is used in the WRW method then the
resulting in-medium amplitudes get considerably closer to the Pauli amplitudes
(see blue lines in Fig. 4 denoted by ’WRW2’). Nonetheless, we used the WRW
method with $k$ defined by Eq. (10) in the present calculations, following
previous kaonic atoms studies [23, 42].

The in-medium amplitudes in Eqs. (6) and (7) (as well as the Pauli amplitudes)
are functions of energy $\sqrt{s}$ given by the Mandelstam variable

$s=(E_{N}+E_{K^{-}})^{2}-(\vec{p}_{N}+\vec{p}_{K^{-}})^{2}~{},$ (13)

where $E_{N}=m_{N}-B_{N}$, $E_{K^{-}}=m_{K^{-}}-B_{K^{-}}$ and
$\vec{p}_{N(K^{-})}$ is the nucleon (kaon) momentum. The momentum dependent
term $(\vec{p}_{N}+\vec{p}_{K^{-}})^{2}\neq 0$ in the $K^{-}$-nucleus cm frame
and generates additional substantial downward energy shift [40]. The $K^{-}N$
amplitudes can then be expressed as a function of energy $\sqrt{s}=E_{\rm
th}+\delta\sqrt{s}$ where $E_{\rm th}=m_{N}+m_{K^{-}}$. The relative energy
$\delta\sqrt{s}$ is expanded near threshold in terms of binding and kinetic
energies (to leading order) and specific forms of density dependence are
introduced ensuring that $\delta\sqrt{s}\rightarrow 0$ as $\rho\rightarrow 0$
(for details see Refs. [40, 23, 27, 28]):

$\delta\sqrt{s}=-B_{N}\frac{\rho}{\bar{\rho}}\,-\beta_{N}\\!\left[B_{K^{-}}\frac{\rho}{\rho_{\rm
max}}+T_{N}\left(\frac{\rho}{\bar{\rho}}\right)^{2/3}\\!\\!\\!\\!+V_{C}\left(\frac{\rho}{\rho_{\rm
max}}\right)^{1/3}\right]+\beta_{K^{-}}{\rm Re}V_{K^{-}}(r)~{},$ (14)

where $\beta_{N(K^{-})}={m_{N(K^{-})}}/(m_{N}+m_{K^{-}})$, $B_{N}=8.5$ MeV is
the average binding energy per nucleon, $\rho_{\rm max}$ and $\bar{\rho}$ are
the maximal and average value of the nuclear density, respectively. Since
$\delta\sqrt{s}$ depends on Re$V_{K^{-}}$ [ and thus $f_{K^{-}N}(\sqrt{s})$]
and $B_{K^{-}}$ which by themselves depend on $\sqrt{s}$, it is clear that for
a given value of $B_{K^{-}}$, $f_{K^{-}N}(\sqrt{s})$ has to be determined
self-consistently by iterations.

##  III Results

Using the method described in Section II, we performed calculations for 23
atomic species from lithium up to uranium. For the first time, we evaluated
energy shifts and widths for lower and upper states (65 data points) using
microscopic $K^{-}N+K^{-}NN$ potentials based on chiral amplitudes and
compared them with available kaonic atom data. Three cases were considered.
First, the data were compared with predictions of the $K^{-}$ single-nucleon
potential. Next, the $K^{-}$ two nucleon potential was added to the $K^{-}$
single-nucleon potential ($K^{-}N+K^{-}NN$) and finally, the $K^{-}N+K^{-}NN$
potential was supplemented by an additional phenomenological term (+phen.) in
order to cover $3N(4N)$ processes not included in our microscopic model:

$V_{\rm phen}=-4\pi B\left(\frac{\rho}{\rho_{0}}\right)^{\alpha}\rho~{},$ (15)

where $B$ is a complex amplitude, $\alpha$ is a positive number and
$\rho_{0}=0.17$ fm-3 is the saturation density.

Table 2: Values of $\chi^{2}(65)$ resulting from comparison of predictions of $K^{-}N$, $K^{-}N+K^{-}NN$, and $K^{-}N+K^{-}NN+$phen. multi-N potentials with kaonic atom data. Microscopic potentials are based on the Pauli and WRW modified BCN amplitudes. Values of the complex amplitude $B$ and parameter $\alpha$ for the additional phenomenological term are presented as well. | $K^{-}N$ | $K^{-}N+K^{-}NN$ | \+ phen. | Re$B$ (fm) | Im$B$ (fm) | $\alpha$
---|---|---|---|---|---|---
Pauli | 825 | 565 | 105 | -1.97(13) | -0.93(11) | 1.4
WRW | 2378 | 1123 | 116 | -0.90(9) | 0.72(10) | 0.6

The results of calculations using the Pauli and WRW modified BCN amplitudes
are presented in Table 2. The description of the data improves significantly
when the $K^{-}NN$ absorption is taken into account. The value of
$\chi^{2}(65)$ decreases to about one half with respect to the case of pure
$K^{-}N$ potentials, however, it still remains considerable. This suggests
that some additional processes of $3N(4N)$ absorption might be missing. When
the additional phenomenological term is added to the $K^{-}N+K^{-}NN$
potentials, values of $\chi^{2}(65)$ further decrease to $\approx 100$, which
is comparable with the best fit $K^{-}N$ \+ phenomenological multi-nucleon
potential, Re$B$ = -1.3 fm, Im$B$ = 1.9 fm, $\alpha=1$, $\chi^{2}(65)=112.3$
[43]. In general, the value of $\chi^{2}(65)$ is much lower for the potentials
based on the Pauli amplitudes than for the potentials derived from the WRW
amplitudes. On the other hand, the value of Im$B$ of the additional
phenomenological term is negative for the Pauli amplitudes which means
negative absorption. In other words, the $K^{-}N+K^{-}NN$ potentials based on
the Pauli BCN amplitudes seem to be too absorptive in the relevant density
region and there is no space for additional absorption from $3N(4N)$
processes. On the contrary, the resulting best fit value of Im$B=0.72\pm 0.10$
fm is positive for potentials based on the WRW modified amplitudes. It is
about half of the value obtained in the fit with $K^{-}N$ \+ phenomenological
multi-nucleon optical potential, Im$B=1.9$ fm, which seems reasonable.

The parameter $\alpha$ controls density dependence of the additional
phenomenological term. It is expected to be $\alpha\geq 2$ for the $3N(4N)$
processes. However, the fit yields values of $\alpha$ lower than 2 in both
cases, which indicates certain deficiencies in the microscopic $K^{-}$
potentials. Although the WRW amplitudes seem to yield reasonable fit with a
positive value of Im$B$, the unexpected density dependence ($\alpha<2$) of the
additional phenomenological term may suggest that the $K^{-}N+K^{-}NN$
microscopic potential should be more absorptive in the relevant density
region.

One of the possible improvements is the proper inclusion of self-energy
insertions in terms of hadron-nucleon potentials for the intermediate hadrons
in the amplitudes employed here. The self-energy effects are expected to
partly compensate for the upward Pauli shift of the scattering amplitudes [37,
38, 39, 40].

Complete results of the fit, i.e. the values of $\chi^{2}(65)$, Re$B$, and
Im$B$, scanned for values of $\alpha$ from 0 to 2 are presented in Figs. 5 and
6. Fig. 5 contains results obtained with $K^{-}$ potentials based on the WRW
in-medium BCN amplitudes and Fig. 6 those obtained with the Pauli BCN
amplitudes. The Prague model, not presented here, yields similar results.

Figure 5: Values of $\chi^{2}(65)$ (upper panel) and the complex amplitude $B$
(lower panel) as functions of the parameter $\alpha$ from best fits based on
$K^{-}N+K^{-}NN$ potentials using the WRW BCN amplitudes. Figure 6: Values of
$\chi^{2}(65)$ (upper panel) and the complex amplitude $B$ (lower panel) as
functions of parameter $\alpha$ obtained in the fit with $K^{-}N+K^{-}NN$
potentials based on the Pauli BCN amplitudes.

Figure 7: Top panel presents total $K^{-}N$, $K^{-}N+K^{-}NN$, and
$K^{-}N+K^{-}NN+$phen. potentials in 31P$+K^{-}$ calculated with the WRW (left
panel) and Pauli (right panel) BCN amplitudes. The $K^{-}N$ \+
phenomenological multi-nucleon optical potential (denoted by ’$K^{-}N+{\rm
multi}N$’) is also shown for comparison. Lower panel shows overlap of the
$K^{-}$ wave function squared, $|\psi|^{2}$, with Im$V_{K^{-}}$ for all
considered potentials.

Top panel of Fig. 7 demonstrates the depth of $K^{-}$ potentials in the
31P$+K^{-}$ atom calculated with the WRW (left panel) and Pauli (right panel)
BCN amplitudes. In the lower panel, the overlaps of $K^{-}$ imaginary
potential with $|\psi|^{2}$, where $\psi$ is the kaon radial wave function,
are plotted for all considered potentials (note that the width of an atomic
level is proportional to the overlap integral of Im$V_{K^{-}}$ and
$|\psi|^{2}$). The relative density $\rho/\rho_{max}$ is shown for
illustration. The pure $K^{-}$ single-nucleon potential (black lines) based on
the WRW amplitudes differs considerably from the potential based on the Pauli
amplitudes. Adding the microscopic $K^{-}NN$ potential (red lines) increases
the depth of the imaginary potential in both cases and slightly decreases the
depth of the real $K^{-}$ potential. The best fit $K^{-}N$\+ phenomenological
multi-nucleon potential (’$K^{-}N+{\rm multi}N$’, blue lines) is presented for
comparison. In the relevant density region, the $K^{-}N+K^{-}NN$ potential
based on the Pauli amplitudes has considerably deeper imaginary part than the
$K^{-}N$\+ multi-N potential. This is compensated by the negative value of
Im$B$ for the additional phenomenological potential in the
$K^{-}N+K^{-}NN+$phen. case (green line). The WRW modified amplitudes generate
less absorptive $K^{-}N+K^{-}NN$ potential, leaving space for additional
$3N(4N,...)$ absorption. The overall $K^{-}N+K^{-}NN+$phen. potential
evaluated using the WRW amplitudes is then very close to the $K^{-}N+$phen.
multi-N potential in the relevant region. It is worth mentioning that the
right balance between the real and imaginary parts of the $K^{-}$ potential is
crucial. The $K^{-}N+$multi-N as well as $K^{-}N+K^{-}NN+$phen. potentials are
less attractive, even repulsive, and very absorptive in the central region of
the nucleus whereas the $K^{-}N$ and $K^{-}N+K^{-}NN$ potentials are strongly
attractive and less absorptive inside the nucleus. As a consequence, the
overlap of Im$V_{K^{-}N}$ and Im$V_{K^{-}N+K^{-}NN}$ with the kaon wave
function spans deeper inside the nucleus, towards higher densities, than in
the $K^{-}N+K^{-}NN+$phen. and $K^{-}N+$multi-N cases.

Table 3: Values of $\chi^{2}$ for shifts $\Delta(\epsilon)$, widths $\Gamma$ for lower states and width $\Gamma^{*}$ for upper states in selected $K^{-}$ atoms, calculated with the $K^{-}N$, $K^{-}N+K^{-}NN$ and $K^{-}N+$phen. multi-N potentials based on the BCN Pauli and WRW modified amplitudes. Experimental values of $\Delta(\epsilon)$ and $\Gamma$ (in keV), and $\Gamma^{*}$ (in eV) including errors are shown in the last column for completeness. BCN | WRW | Pauli | phen. | Exp. [14]
---|---|---|---|---
$\bm{\chi^{2}}$ | $K^{-}N$ | $+K^{-}NN$ | $K^{-}N$ | $+K^{-}NN$ |  $K^{-}N$ \+ phen. multiN |
| $\Delta(\epsilon)$ | 101.52 | 34.35 | 25.13 | 11.48 | 1.76 | -0.59 (0.08)
12C | $\Gamma$ | 44.80 | 27.45 | 17.00 | 9.44 | 0.70 |  1.73 (0.15)
|  $\Gamma^{*}$ | 1.71 | 1.47 | 0.15 | 0.67 | 2.74 |  0.99 (0.20)
| $\Delta(\epsilon)$ | 41.04 | 15.13 | 10.46 | 6.35 | 0.03 | -0.33 (0.08)
31P | $\Gamma$ | 13.72 | 10.34 | 11.43 | 6.42 | 0.24 |  1.44 (0.12)
|  $\Gamma^{*}$ | 5.17 | 4.70 | 5.98 | 1.87 | 0.30 |  1.89 (0.30)
| $\Delta(\epsilon)$ | 475.71 | 209.40 | 90.77 | 80.82 | 1.24 | -0.494 (0.038)
32S | $\Gamma$ | 0.76 | 2.83 | 67.35 | 43.29 | 9.24 |  2.19 (0.10)
|  $\Gamma^{*}$ | 13.32 | 10.85 | 9.45 | 2.78 | 0.47 |  3.03 (0.44)
| $\Delta(\epsilon)$ | 38.27 | 17.69 | 4.23 | 4.62 | 2.10 | -0.99 (0.17)
35Cl | $\Gamma$ | 5.94 | 2.56 | 10.94 | 5.39 | 0.00 |  2.91 (0.24)
|  $\Gamma^{*}$ | 7.92 | 4.53 | 2.27 | 0.74 | 0.15 |  5.8 (1.70)
| $\Delta(\epsilon)$ | 33.50 | 8.93 | 1.54 | 2.71 | 3.19 | -0.370 (0.047)
63Cu | $\Gamma$ | 0.31 | 0.02 | 4.90 | 3.57 | 2.25 |  1.37 (0.17)
|  $\Gamma^{*}$ | 0.98 | 0.13 | 0.24 | 0.73 | 1.52 |  5.2 (1.1)
| $\Delta(\epsilon)$ | 9.00 | 8.81 | 6.57 | 8.50 | 2.15 | -0.41 (0.18)
118Sn | $\Gamma$ | 0.42 | 0.03 | 0.35 | 0.71 | 0.29 |  3.18 (0.64)
|  $\Gamma^{*}$ | 24.53 | 15.08 | 5.04 | 4.80 | 4.09 |  15.1 (4.4)
| $\Delta(\epsilon)$ | 7.52 | 3.67 | 3.24 | 4.84 | 0.34 | -0.02 (0.012)
208Pb | $\Gamma$ | 0.12 | 0.10 | 0.31 | 0.38 | 0.39 |  0.37 (0.15)
|  $\Gamma^{*}$ | 0.06 | 0.18 | 0.35 | 0.41 | 0.52 |  4.1 (2)
$\bm{\chi^{2}}$ | total | 820.37 | 378.24 | 277.69 | 200.54 | 33.71 |
|  32S out | 330.58 | 155.16 | 110.13 | 73.65 | 22.76 |

In Table 3, we present values of $\chi^{2}$ for shifts and widths for lower
state and width for upper state in selected $K^{-}$ atoms (21 data points),
calculated using the $K^{-}N$, $K^{-}N+K^{-}NN$ and $K^{-}N+$phen. multi-N
potentials based on the Pauli and WRW modified BCN amplitudes. When the
microscopic $K^{-}NN$ absorption is taken into account the description of the
atomic data improves considerably – $\chi^{2}(21)$ decreases from $\approx
800$ to $\approx 400$ for the WRW amplitudes and from $\approx 300$ to
$\approx 200$ for the Pauli amplitudes. Note that the most pronounced
contribution to the total $\chi^{2}(21)$ comes from 32S due to a very small
experimental error in energy shift. This may imply that the microscopic
potentials are not able to describe the energy shift reasonably well. However,
the experimental data on $\Delta(\epsilon)$ and $\Gamma$ (in keV), and
$\Gamma^{*}$ (in eV) presented with corresponding errors in the last column of
Table 3 for completeness, were compiled by Batty (private communication) and
[14] as weighted averages. In the case of sulfur, the data come from three
different experiments (see Ref. [14]) and the values of energy shifts and
widths span over a quite large range. For instance, if only the data from Ref.
[44] are considered, the total value of $\chi^{2}$ for sulfur drops down
significantly, e.g., from $\chi^{2}=127$ to $\chi^{2}=42$ for $K^{-}N+K^{-}NN$
Pauli potentials! Moreover, when the 32S data are excluded from the fit, the
total $\chi^{2}$ drops by more than a half (see the last two rows in Table 3).
This indicates that a new measurement of kaonic sulfur is needed. It would be
desirable to remeasure kaonic sulfur within current experiments.

Next, we calculated branching ratios for $K^{-}N$ and $K^{-}NN$ absorption
channels in 12C$+K^{-}$ using the WRW and Pauli BCN and P amplitudes. The
branching ratios were evaluated as fractions of partial width in the
respective channel (see Table 1) over the total width

$\text{BR}=\frac{\Gamma_{\rm channel}}{\Gamma_{\rm
total}}=\frac{\int\text{Im}V_{\rm
channel}(r)|\psi(r)|^{2}dr}{\int\text{Im}V_{K^{-}}(r)|\psi(r)|^{2}dr}~{},$
(16)

where $\psi(r)$ is the $K^{-}$ radial wave function. The branching ratios for
the lower (l=1) and upper (l=2) states are presented in Table 4 and Table 5,
respectively.

Table 4: Primary-interaction branching ratios (in $\%$) for mesonic ($K^{-}N\rightarrow Y\pi$, $Y=\Lambda,\Sigma$) and non-mesonic ($K^{-}NN\rightarrow YN$) absorption of $K^{-}$ from the l=1 state in 12C$+K^{-}$, calculated with $K^{-}N+K^{-}NN$ potentials based on the WRW and Pauli BCN and P amplitudes. The experimental data for primary-interaction branching ratios are shown for comparison. 12C + $K^{-}$ (l=1) | BCN | P | Exp. [17]
---|---|---|---
mesonic ratio |  WRW |  Pauli |  WRW |  Pauli |  12C
$\Sigma^{+}\pi^{-}$ | 25.3 | 21.7 | 24.5 | 20.0 | 29.4 $\pm$ 1.0
$\Sigma^{-}\pi^{0}$ | 7.7 | 6.7 | 7.3 | 5.3 | 2.6 $\pm$ 0.6
$\Sigma^{-}\pi^{+}$ | 7.6 | 12.9 | 7.7 | 14.7 | 13.1 $\pm$ 0.4
$\Sigma^{0}\pi^{-}$ | 7.8 | 6.8 | 7.4 | 5.3 | 2.6 $\pm$ 0.6
$\Sigma^{0}\pi^{0}$ | 12.7 | 14.2 | 12.5 | 15.0 | 20.0 $\pm$ 0.7
$\Lambda\pi^{0}$ | 6.0 | 5.0 | 5.2 | 3.8 | 3.4 $\pm$ 0.2
$\Lambda\pi^{-}$ | 11.8 | 10.1 | 10.3 | 7.4 | 6.8 $\pm$ 0.3
total 1N ratio | 79.0 | 77.4 | 75.6 | 74.6 | 77.9 $\pm$ 1.6
$R_{\pm}=\frac{(\Sigma^{+}\pi^{-})}{(\Sigma^{-}\pi^{+})}$ | 3.3 | 1.7 | 3.2 | 1.4 | 2.24 $\pm$ 0.12
$R_{pn}=\frac{(\Sigma^{+}\pi^{-})+(\Sigma^{-}\pi^{+})}{(\Sigma^{-}\pi^{0})}$ | 4.3 | 5.2 | 4.4 | 6.6 | 16.3 $\pm$ 4.0
non-mesonic ratio |  WRW |  Pauli |  WRW |  Pauli | 76% CF3Br + 24% C3H8 [15]
$\Lambda p+\Lambda n+\Sigma^{0}p+\Sigma^{0}n$ | 11.4 | 11.8 | 12.8 | 13.3 | 14.1 $\pm$ 2.5$~{}^{\rm{a}}$
$\Sigma^{-}p+\Sigma^{-}n$ | 3.8 | 4.4 | 5.0 | 5.2 | 7.3 $\pm$ 1.3$~{}^{\rm{a}}$
$\Sigma^{+}n$ | 5.7 | 6.4 | 6.6 | 6.9 | 4.3 $\pm$ 1.2$~{}^{\rm{a}}$
$\Sigma^{0}p+\Sigma^{0}n$ | 4.6 | 5.2 | 5.6 | 5.8 | -
total 2N ratio | 21.0 | 22.6 | 24.4 | 25.4 | 25.7 $\pm$ 3.1 [15] 111multinucleon capture rate
$16\pm 3(\text{stat.})^{+4}_{-5}(\text{syst.})$ [22]
Table 5: Primary-interaction branching ratios (in $\%$) for mesonic ($K^{-}N\rightarrow Y\pi$, $Y=\Lambda,\Sigma$) and non-mesonic ($K^{-}NN\rightarrow YN$) absorption of $K^{-}$ from the l=2 state in 12C$+K^{-}$, calculated with $K^{-}N+K^{-}NN$ potentials based on the WRW and Pauli blocked BCN and P amplitudes. The experimental data for primary- interaction branching ratios are shown for comparison. 12C + $K^{-}$ (l=2) | BCN | P | Exp. [17]
---|---|---|---
mesonic ratio |  WRW |  Pauli |  WRW |  Pauli |  12C
$\Sigma^{+}\pi^{-}$ | 26.9 | 22.4 | 28.1 | 22.1 | 29.4 $\pm$ 1.0
$\Sigma^{-}\pi^{0}$ | 8.3 | 7.7 | 7.2 | 5.9 | 2.6 $\pm$ 0.6
$\Sigma^{-}\pi^{+}$ | 15.5 | 17.5 | 17.1 | 17.6 | 13.1 $\pm$ 0.4
$\Sigma^{0}\pi^{-}$ | 8.4 | 7.9 | 7.3 | 5.9 | 2.6 $\pm$ 0.6
$\Sigma^{0}\pi^{0}$ | 17.2 | 16.4 | 19.3 | 17.3 | 20.0 $\pm$ 0.7
$\Lambda\pi^{0}$ | 5.2 | 5.0 | 4.2 | 3.7 | 3.4 $\pm$ 0.2
$\Lambda\pi^{-}$ | 10.4 | 9.9 | 8.3 | 7.2 | 6.8 $\pm$ 0.3
total 1N ratio | 91.9 | 87.0 | 90.7 | 82.0 | 77.9 $\pm$ 1.6
$R_{\pm}=\frac{(\Sigma^{+}\pi^{-})}{(\Sigma^{-}\pi^{+})}$ | 1.7 | 1.3 | 1.6 | 1.3 | 2.24 $\pm$ 0.12
$R_{pn}=\frac{(\Sigma^{+}\pi^{-})+(\Sigma^{-}\pi^{+})}{(\Sigma^{-}\pi^{0})}$ | 5.1 | 5.2 | 6.3 | 6.7 | 16.3 $\pm$ 4.0
non-mesonic ratio |  WRW |  Pauli |  WRW |  Pauli | 76% CF3Br + 24% C3H8 [15]
$\Lambda p+\Lambda n+\Sigma^{0}p+\Sigma^{0}n$ | 4.2 | 6.7 | 4.6 | 9.0 | 14.1 $\pm$ 2.5$~{}^{\rm{a}}$
$\Sigma^{-}p+\Sigma^{-}n$ | 1.7 | 3.1 | 2.1 | 4.2 | 7.3 $\pm$ 1.3$~{}^{\rm{a}}$
$\Sigma^{+}n$ | 2.2 | 3.5 | 2.6 | 4.8 | 4.3 $\pm$ 1.2$~{}^{\rm{a}}$
$\Sigma^{0}p+\Sigma^{0}n$ | 1.9 | 3.1 | 2.2 | 4.2 | -
total 2N ratio | 8.1 | 13.0 | 9.3 | 18.0 | 25.7 $\pm$ 3.1 [15] 222multinucleon capture rate
$16\pm 3(\text{stat.})^{+4}_{-5}(\text{syst.})$ [22]

Theoretical values are compared with experimental data on primary-interaction
branching ratios, i.e., corrected for secondary interactions of particles
created in the absorbing nucleus, measured in old bubble chamber experiments
[17, 15]. Note that the branching ratios are evaluated only for $K^{-}$
single-nucleon and $K^{-}$ two-nucleon absorptive potentials and the effect of
$3N(4N)$ absorption is not taken into account. Both BCN and P models yield the
values of branching ratios in reasonable agreement with experimental data,
except the branching ratios for $\Sigma^{-}\pi^{0}$ and $\Sigma^{0}\pi^{-}$
production, which differ from the experimental values for all considered
amplitudes. The WRW and Pauli in-medium amplitudes in both interaction models
yield comparable branching ratios for the total 1N and 2N absorption in the
l=1 case (see Table 4). In the l=2 case (see Table 5), there are evident
differences between the total branching ratios calculated using various in-
medium amplitudes under consideration.

Valuable information about absorption of an antikaon in the nuclear medium has
been provided recently by the AMADEUS Collaboration which measured branching
fractions for the $K^{-}$ two-nucleon absorption in reactions of low-energy
$K^{-}$ with a carbon target [21, 22]. The total 2N absorption ratios
calculated within the BCN model (presented in Tables 4 and 5) could be
considered consistent with the value measured by the AMADEUS Collaboration,
BR($K^{-}$2N$\rightarrow$YN) = $(16\pm
3(\text{stat.})^{+4}_{-5}(\text{syst.}))\%$ [22]. The P model yields branching
ratios for the $K^{-}$ two-nucleon absorption from the l=1 state slightly
above the error bars.

Table 6: Branching ratio (in %) for $\Lambda N$ and $\Sigma^{0}N$ production in $K^{-}NN$ absorption at rest ($p_{K^{-}}=0$ MeV/c) in the 12C$+K^{-}$ atom for the lower (l=1) and upper (l=2) state, calculated with $K^{-}N+K^{-}NN$ potentials based on the WRW modified and Pauli amplitudes derived from the BCN and P models. Theoretical values are compared with the AMADEUS data.  BCN | WRW | Pauli |  P | WRW | Pauli |
---|---|---|---|---|---|---
BR |  l=1 |  l=2 |  l=1 |  l=2 |  |  l=1 |  l=2 |  l=1 |  l=2 | Exp. [22]
$\Lambda N$ | 5.45 | 2.32 | 4.23 | 3.02 |  | 5.27 | 2.17 | 4.30 | 3.19 | 6.45 $\pm~{}1.41$(stat.)${}^{+0.5}_{-0.6}$(syst.)
$\Sigma^{0}N$ | 4.44 | 2.09 | 3.99 | 2.93 |  | 5.11 | 2.37 | 4.53 | 3.47 | 7.55 $\pm~{}2.2$(stat.)${}^{+4.2}_{-5.4}$(syst.)

The AMADEUS Collaboration determined among others branching ratios for quasi-
free (QF) production of $\Lambda p$ and $\Sigma^{0}p$ pairs from the $K^{-}$
two-nucleon absorption in 12C, without final state interaction (FSI) with a
residual nucleus, and branching ratios for processes where the primary created
$\Lambda(\Sigma^{0})$’s undergo elastic FSI (see Table 1 in Ref. [22]). The
FSI ratios include also channels $K^{-}pn\rightarrow\Lambda n$ and
$K^{-}pn\rightarrow\Sigma^{0}n$. It is to be noted that the AMADEUS data do
not include QF production of $\Lambda n$ and $\Sigma^{0}n$ pairs. In our
present calculations of kaonic atoms, we evaluated branching ratios for the
total $\Lambda N$ and $\Sigma^{0}N$ production. The branching ratios
calculated within our microscopic model are compared with experimental data on
the QF+FSI branching ratios in Table 6. The theoretical values of branching
ratios for both $\Lambda N$ and $\Sigma^{0}N$ production in the $K^{-}NN$
absorption from the l=1 state in the ${}^{12}{\rm C}+K$ atom are in agreement
with the experimental data for both the BCN and P model. As for the l=2 state,
the branching ratios for $\Lambda N$ and $\Sigma^{0}N$ production are
significantly lower than the ratios for $K^{-}NN$ absorption from the l=1
state. Nonetheless, the calculated $\Sigma^{0}N$ branching ratios could be
still considered consistent with experiment due to rather large experimental
errors. Following the finding that 75% of $K^{-}$ absorption in 12C takes
place from the upper $l=2$ state [23], we evaluated weighted average of the
l=1 (25 %) and l=2 (75%) values of the $\Lambda N$ branching ratios within the
BCN model for completeness. The resulting values BR($\Lambda N)_{\rm WRW}$ =
3.1 % and BR($\Lambda N)_{\rm Pauli}$ = 3.3 % are well below the experimental
value and again out of the experimental error.

Finally, the AMADEUS Collaboration reported the ratio of branching ratios [22]

$R=\frac{{\rm BR}(K^{-}pp\rightarrow\Lambda p)}{{\rm
BR}(K^{-}pp\rightarrow\Sigma^{0}p)}=0.7\pm~{}0.2(\text{stat.})^{+0.2}_{-0.3}(\text{syst.})~{}.$
(17)

We calculated this ratio in 12C$+K^{-}$ atom, using the $K^{-}N+K^{-}NN$
microscopic potentials based on the BCN and P amplitudes. The value of $R$ is
about 1 for both the WRW and Pauli in-medium amplitudes and both interaction
models, which is within error bars of the experimental value, as demonstrated
in Table 7.

Table 7: Ratio $R$ in the 12C$+K^{-}$ atom for lower (l=1) and upper (l=2) state and $K^{-}NN$ absorption at rest ($p_{K^{-}}=0$ MeV/c), calculated with $K^{-}N+K^{-}NN$ potentials based on the WRW modified and Pauli amplitudes from the BCN and P models. Theoretical values are compared with the AMADEUS data.  | WRW | Pauli |
---|---|---|---
$R$ |  l=1 |  l=2 |  l=1 |  l=2 | Exp. [22]
BCN | 1.1 | 1.0 | 1.1 | 1.0 | 0.7 $\pm~{}0.2$(stat.)${}^{+0.2}_{-0.3}$(syst.)
P | 1.1 | 1.0 | 1.0 | 1.0

##  IV Conclusions

In this work, we calculated strong-interaction energy shifts and widths in
various kaonic atoms, from lithium up to uranium. For the first time, the
calculations were performed using microscopic $K^{-}N+K^{-}NN$ potentials
based on scattering amplitudes derived from two chiral coupled-channels meson-
baryon interaction models – the Barcelona and Prague model.

The $K^{-}NN$ potentials were constructed within our recently formulated
microscopic $K^{-}NN$ absorption model [31]. We took into account medium
modifications of the free-space amplitudes due to the Pauli principle. They
were incorporated by two different methods – i) Pauli blocking included
directly in the chiral amplitudes and ii) WRW procedure based on multiple
scattering approach.

The $K^{-}N+K^{-}NN$ potentials, based on in-medium chiral amplitudes,
calculated for 23 nuclear species were confronted with kaonic atom data. The
value of $\chi^{2}(65)$ significantly improves when the $K^{-}$ two-nucleon
potentials are included for both the Pauli and WRW in-medium amplitudes. It
drops approximately to one half of the value corresponding to just $K^{-}N$
chiral potential, nonetheless, it remains still sizable. Next, we added a
phenomenological term to the microscopic $K^{-}N+K^{-}NN$ potentials in order
to incorporate and quantify missing $K^{-}-3N(4N)$ processes. After adding
this term the description of the data further improved and the resulting
$\chi^{2}(65)\approx 100$ was comparable with the best fit $K^{-}N$ \+
phenomenological multi-nucleon potential. However, the Pauli in-medium
amplitudes yielded negative imaginary amplitude $B$ for the additional
phenomenological term. This implies that the $K^{-}N+K^{-}NN$ potentials based
on the Pauli amplitudes are too absorptive and there is no space for
additional absorption from $3N(4N)$ processes, which is in contrast to
experimental measurements of the AMADEUS Collaboration [22].

Unanticipated values of the fitted parameters of the additional
phenomenological term indicate certain deficiencies in microscopic potentials
in the region where $K^{-}$ absorption takes place, i.e., in their density
dependence. Our microscopic model could be further improved by introducing
hadron self-energies as another component of in-medium modifications of
$K^{-}N$ scattering amplitudes in the construction of $K^{-}N$ and $K^{-}NN$
potentials. This extension of our microscopic model is currently under
investigation and will be published elsewhere.

On the other hand, the analyses of kaonic atom data including microscopic as
well as phenomenological $K^{-}$ optical potentials hint at certain
inconsistencies in the available experimental data which were compiled from
different experiments, performed in the 1970s. This indicates that it would be
opportune to repeat some of these measurements.

Finally, we calculated branching ratios for all $K^{-}N$ and $K^{-}NN$
absorption channels in the 12C$+K^{-}$ atom, using microscopic
$K^{-}N+K^{-}NN$ potentials based on the in-medium BCN and P amplitudes. Our
results are in reasonable agreement with old bubble chamber data on primary-
interaction branching ratios. The total ratio for the $K^{-}$ two-nucleon
absorption was found to be in accordance with the latest measurement by the
AMADEUS Collaboration, BR($K^{-}$2N$\rightarrow$YN) = $(16\pm
3(\text{stat.})^{+4}_{-5}(\text{syst.}))\%$. Moreover, the newly measured
ratio $R=({\rm BR}(K^{-}pp\rightarrow\Lambda p))/{\rm
BR}(K^{-}pp\rightarrow\Sigma^{0}p))$ was found to be consistent with the one
calculated in this work. The AMADEUS Collaboration reported also branching
ratios for the $\Lambda N$ and $\Sigma^{0}N$ production in $K^{-}$ two-nucleon
absorption. Here, only the results for the $K^{-}$ absorption from the lower
l=1 state are in agreement with the data, while the branching ratios for
$K^{-}$ absorption from the upper l=2 state are too low. This is in
contradiction with the empirical fact that $75\%$ of $K^{-}$ absorption takes
place from the upper level in 12C.

To summarize, although some positive results have been presented in this work,
confrontation of our microscopic model with all available kaonic atom data
revealed need for a further theoretical as well as experimental study in order
to get a consistent description of $K^{-}$ absorption in the nuclear medium.

## Acknowledgements

We thank Àngels Ramos for providing us with the BCN model amplitudes and for
careful reading of the manuscript. J.O. and J.M. acknowledge support from the
Czech Science Foundation GACR Grant No. 19-19640S. The present work is part of
a project funded by the European Union’s Horizon 2020 research & innovation
programme, Grant Agreement No. 824093

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$\displaystyle\begin{aligned}
&{\dot{h}}_{ij}=\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}g_{ik}(h_{kj}-h_{ik}h_{ij})+g_{jk}(h_{ik}-h_{kj}h_{ij})\Big{)},\\\
&{\dot{G}}_{ij}=\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}h_{ik}(G_{kj}-G_{ik}G_{ij})+h_{jk}(G_{ik}-G_{ij}G_{kj})\Big{)}.\end{aligned}$
(6.6)

Since a solution to (6.3) is analytic and we assume (6.5), there exists a
finite time $T_{1}>0$ such that

$\mathcal{A}(\mathcal{X}(t))>0,\quad\mathcal{D}(\mathcal{U}(t))<\sqrt{2},\quad
t\in[0,T_{1}).$

Moreover, there also exists a finite time $T_{2}>0$ such that

$\mathcal{A}(\mathcal{X}(t))=h_{i_{t}j_{t}}(t),\quad t\in[0,T_{2}).$

We set

$T_{*}:=\min\\{T_{1},T_{2}\\}.$

Since the following relation holds:

$\|U_{i}-U_{j}\|_{\textup{F}}^{2}=2(d-g_{ij})<2,\quad\textup{or
equivalently,}\quad g_{ij}>d-1,\quad t\in[0,T_{*}),$

one has

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{A}(\mathcal{X})&\geq\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}g_{i_{t}k}(h_{kj_{t}}-h_{i_{t}k}\mathcal{A}(\mathcal{X}))+g_{j_{t}k}(h_{i_{t}k}-h_{kj_{t}}\mathcal{A}(\mathcal{X}))\Big{)}\\\
&>\frac{\kappa(n-1)}{N}\mathcal{A}(\mathcal{X})\sum_{k=1}^{N}(1-h_{i_{t}k}+1-h_{kj_{t}})>0,\quad
t\in[0,T_{*}).\end{aligned}$

On the other hand, we rewrite $\eqref{E-5}_{2}$ as

$\displaystyle{\dot{g}}_{ij}$
$\displaystyle=\frac{\kappa}{N}\sum_{k=1}^{N}h_{ik}\Big{(}\|I_{n}-G_{ij}\|_{\textup{F}}^{2}-\|I_{n}-G_{kj}\|_{\textup{F}}^{2})+h_{jk}(\|I_{n}-G_{ij}\|_{\textup{F}}^{2}-\|I_{n}-G_{kj}\|_{\textup{F}}^{2}\Big{)}$
$\displaystyle\hskip
14.22636pt+(2-\|I_{n}-G_{ij}\|_{\textup{F}}^{2})\cdot\frac{\kappa}{2N}\sum_{k=1}^{N}\Big{(}h_{ik}\|I_{n}-G_{ik}\|_{\textup{F}}^{2}+h_{jk}\|I_{n}-G_{jk}\|_{\textup{F}}^{2}\Big{)}.$

By the same argument, one has

$\displaystyle\begin{aligned}
\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})^{2}&=-\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}h_{i_{t}k}(\mathcal{D}(\mathcal{U})^{2}-\|I_{n}-G_{kj_{t}}\|_{\textup{F}}^{2})+h_{j_{t}k}(\mathcal{D}(\mathcal{U})^{2}-\|I_{n}-G_{ki_{t}}\|_{\textup{F}}^{2})\Big{)}\\\
&\hskip
14.22636pt-(2-\mathcal{D}(\mathcal{U})^{2})\cdot\frac{\kappa}{2N}\sum_{k=1}^{N}\Big{(}h_{i_{t}k}\|I_{n}-G_{ki_{t}}\|_{\textup{F}}^{2}+h_{j_{t}k}\|I_{n}-G_{kj_{t}}\|_{\textup{F}}^{2}\Big{)}\\\
&<0,\quad t\in[0,T_{*}).\end{aligned}$

Hence, if we continue the process above successively, then we obtain the
desired assertion. ∎

Now, we are ready to provide a sufficient condition leading to complete
aggregation for (LABEL:E-1).

######  Theorem 6.1.

Suppose the initial data $(\mathcal{X}^{0},\mathcal{U}^{0})$ satisfy

$\mathcal{A}(\mathcal{X}^{0})>0,\quad\mathcal{D}(\mathcal{U}^{0})<\sqrt{2},$

and let $(\mathcal{X},\mathcal{U})$ be a solution to (6.3). Then, system (6.3)
exhibits complete aggregation:

$\lim_{t\to\infty}\Big{(}\mathcal{D}(\mathcal{X}(t))+\mathcal{D}(\mathcal{U}(t))\Big{)}=0.$

Moreover, the convergence rate is exponential.

###### Proof.

Since we assume (5.21), we can apply Lemma 5.1:

$h_{ij}(t)>0,\quad g_{ij}(t)>d-1,\quad t>0,\quad i\neq j.$

Note that

$\mathcal{D}(\mathcal{X}(t))^{2}=\max_{1\leq i\neq j\leq
N}|x_{i}(t)-x_{j}(t)|^{2}=2(1-\mathcal{A}(\mathcal{X}(t))).$

Now we choose extremal indices $(i_{t},j_{t})$ such that

$\mathcal{A}(\mathcal{X}(t))=\langle x_{i_{t}},x_{j_{t}}\rangle.$

Then, one has

$\displaystyle 1-h_{i_{t}k}+1-h_{j_{t}k}$
$\displaystyle=\frac{1}{2}(|x_{i_{t}}-x_{k}|^{2}+|x_{j_{t}}-x_{k}|^{2})>\frac{1}{4}(|x_{i_{t}}-x_{k}|+|x_{j_{t}}-x_{k}|)^{2}$
$\displaystyle>\frac{1}{4}|x_{i_{t}}-x_{j_{t}}|^{2}=\frac{1}{4}\mathcal{D}(\mathcal{X})^{2}=\frac{1}{2}(1-\mathcal{A}(\mathcal{X})).$

It follows from (5.8) and (5.12) that

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{A}(\mathcal{X})>\frac{\kappa(n-1)}{N}\mathcal{A}(\mathcal{X})\sum_{k=1}^{N}(1-h_{i_{t}k}+1-h_{j_{t}k})>\frac{\kappa(n-1)}{2}\mathcal{A}(\mathcal{X})(1-\mathcal{A}(\mathcal{X})).$

Thus, we obtain the desired exponential convergence of
$\mathcal{D}(\mathcal{X})$. On the other hand for $\mathcal{D}(\mathcal{U})$,
we use (LABEL:E-12) to find

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})^{2}<-\frac{\kappa}{4}\mathcal{A}(\mathcal{X}^{0})(2-\mathcal{D}(\mathcal{U})^{2})\mathcal{D}(\mathcal{U})^{2}.$

This shows the desired assertion. ∎

######  Remark 6.2.

The initial conditions in Theorem 6.1 leading to complete aggregation appear
in [7] and [21] for decoupled systems, respectively. To be more specific, by
letting $U_{i}\equiv U$ in $\eqref{E-1}_{1}$, the model reduces to the model
on $\mathbb{S}^{d-1}$:

$\dot{x}_{j}=\frac{\kappa}{N}\sum_{k=1}^{N}(x_{k}-\langle x_{i},x_{k}\rangle
x_{i}).$ (6.7)

Then, the authors in [7] showed that if the initial data satisfy
$\mathcal{A}(\mathcal{X}^{0})>0$, then system (6.7) displays complete
aggregation. Similarly by setting $x_{j}\equiv x$ in $\eqref{E-1}_{2}$, it
reduces to the model on ${\mathbf{SO}}(d)$:

$\dot{U}_{j}=\frac{\kappa}{2N}\sum_{k=1}^{N}(U_{k}-U_{j}U_{k}^{\top}U_{j}).$
(6.8)

Although the model in [21] is defined on the unitary group $\mathbf{U}(d)$, a
similar calculation leads to the same result, i.e., if the initial data
satisfy $\mathcal{D}(\mathcal{U}^{0})<\sqrt{2}$, then (6.8) exhibits complete
synchronization.

###  6.2. Heterogeneous ensemble

In this subsection, we consider a heterogeneous ensemble:

$\Omega_{i}\neq\Omega_{j},\quad A_{i}\neq A_{j}\quad\textup{for some $i\neq
j\in[N]$.}$ (6.9)

Due to the dissimilarity between natural frequencies, we might not expect the
emergence of complete aggregation. Instead, our goal is to provide a
sufficient framework leading to practical aggregation.

######  Theorem 6.2.

Suppose initial data and system parameters satisfy

$\mathcal{A}(\mathcal{X}^{0})>\frac{1}{2},\quad\mathcal{D}(\mathcal{U}^{0})^{2}<1+\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}},\quad\kappa>\min\left\\{\frac{4\mathcal{D}(\mathcal{A})}{\mathcal{A}(\mathcal{X}^{0})},\frac{8\mathcal{D}(\Omega)}{\mathcal{A}(\mathcal{U}^{0})}\right\\},$
(6.10)

and let $(\mathcal{X},\mathcal{U})$ be a solution to (6.2) with (6.9). Then,
practical aggregation emerges asymptotically:

$\limsup_{t\to\infty}\mathcal{D}(\mathcal{X}(t))\leq
1-\sqrt{1-\frac{8\mathcal{D}(\Omega)}{\kappa\mathcal{A}(\mathcal{U}^{0})}}\quad\textup{and}\quad\limsup_{t\to\infty}\mathcal{D}(\mathcal{U}(t))\leq
1-\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}}.$

###### Proof.

By slightly modifying (6.6), one has

$\displaystyle\begin{aligned} &{\dot{h}}_{ij}=\langle
x_{i},(\Omega_{j}-\Omega_{i})x_{j}\rangle+\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}g_{ik}(h_{kj}-h_{ik}h_{ij})+g_{jk}(h_{ik}-h_{kj}h_{ij})\Big{)},\\\
&{\dot{G}}_{ij}=A_{i}G_{ij}-G_{ij}A_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}h_{ik}(G_{kj}-G_{ik}G_{ij})+h_{jk}(G_{ik}-G_{ij}G_{kj})\Big{)}.\end{aligned}$
(6.11)

Now, we use the conditions on initial data:

$\mathcal{A}(\mathcal{X}^{0})>\frac{1}{2},\quad\mathcal{A}(\mathcal{U}^{0})>d-\frac{1}{2}\left(1+\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}}\right)>0,$

and the continuity of solutions to see that there exists a finite time
$T_{*}>0$ such that

$\mathcal{A}(\mathcal{X}(t))>0,\quad\mathcal{A}(\mathcal{U}(t))>0,\quad
t\in[0,T_{*}).$

It follows from (5.12) and (6.11) that

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{A}(\mathcal{X})>\frac{\kappa}{2}\mathcal{A}(\mathcal{U})\mathcal{A}(\mathcal{X})(1-\mathcal{A}(\mathcal{X}))-\mathcal{D}(\Omega),\quad
t\in[0,T_{*}).$

At $t=0$, since $\mathcal{A}(\mathcal{X}^{0})>\frac{1}{2}$, one has

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{A}(\mathcal{X})\biggl{|}_{t=0}>\frac{\kappa}{2}\mathcal{A}(\mathcal{U})\mathcal{A}(\mathcal{X})(1-\mathcal{A}(\mathcal{X}))-\mathcal{D}(\Omega)\biggl{|}_{t=0}=\frac{\kappa\mathcal{A}(\mathcal{U}^{0})}{2}\mathcal{A}(\mathcal{X}^{0})(1-\mathcal{A}(\mathcal{X}^{0}))-\mathcal{D}(\Omega)>0.$

Thus, we see that $\mathcal{A}(\mathcal{X})$ begins to increase

$\mathcal{A}(\mathcal{X}(t))>\mathcal{A}(\mathcal{X}^{0}),\quad
t\in[0,T_{**}).$

For $t\in[0,T_{**})$, the diameter $\mathcal{D}(\mathcal{U})$ satisfies

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})^{2}=-\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}h_{i_{t}k}(\mathcal{D}(\mathcal{U})^{2}-\|I_{n}-G_{kj_{t}}\|_{\textup{F}}^{2})+h_{j_{t}k}(\mathcal{D}(\mathcal{U})^{2}-\|I_{n}-G_{ki_{t}}\|_{\textup{F}}^{2})\Big{)}\\\
&-(2-\mathcal{D}(\mathcal{U})^{2})\cdot\frac{\kappa}{2N}\sum_{k=1}^{N}h_{i_{t}k}\|I_{n}-G_{ki_{t}}\|_{\textup{F}}^{2}+h_{j_{t}k}\|I_{n}-G_{kj_{t}}\|_{\textup{F}}^{2})+\textup{tr}(G_{i_{t}j_{t}}(A_{i}-A_{j}))\\\
&<-\mathcal{A}(\mathcal{X}^{0})(2-\mathcal{D}(\mathcal{U})^{2})\cdot\frac{\kappa}{2N}\sum_{k=1}^{N}\Big{(}\|I_{n}-G_{ki_{t}}\|_{\textup{F}}^{2}+\|I_{n}-G_{kj_{t}}\|_{\textup{F}}^{2}\Big{)}+\mathcal{D}(\mathcal{A})\\\
&<-\frac{\kappa\mathcal{A}(\mathcal{X}^{0})}{2}(2-\mathcal{D}(\mathcal{U})^{2})D(\mathcal{U})^{2}+\mathcal{D}(\mathcal{A}).\end{aligned}$

Since $\mathcal{D}({\mathcal{U}}^{0})$ satisfies (6.10), then we have

$\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})^{2}\biggl{|}_{t=0}<0,$

which implies that $\mathcal{D}(\mathcal{U})$ starts to increase and
$\mathcal{A}(\mathcal{U})$ start to decrease. Thus, if we continue this
process, then we conclude

$\mathcal{A}(\mathcal{X}(t))>\frac{1}{2},\quad\mathcal{D}(\mathcal{U}(t))^{2}<1+\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}},\quad
t>0.$

In addition, since $\mathcal{A}(\mathcal{X})$ and $\mathcal{D}(\mathcal{U})$
satisfy for $t>0$,

$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{A}(\mathcal{X})>\frac{\kappa}{2}\mathcal{A}(\mathcal{U}^{0})\mathcal{A}(\mathcal{X})(1-\mathcal{A}(\mathcal{X}))-\mathcal{D}(\Omega),$
$\displaystyle\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})^{2}<-\frac{\kappa\mathcal{A}(\mathcal{X}^{0})}{2}(2-\mathcal{D}(\mathcal{U})^{2})D(\mathcal{U})^{2}+\mathcal{D}(\mathcal{A}).$

By dynamical systems theory, there exists a finite entrance time $T_{1}>0$
such that

$\displaystyle\mathcal{A}(\mathcal{X}(t))>\frac{1}{2}\left(1+\sqrt{1-\frac{8\mathcal{D}(\Omega)}{\kappa\mathcal{A}(\mathcal{U}^{0})}}\right)\quad\textup{or}\quad\mathcal{D}(\mathcal{X}(t))^{2}<1-\sqrt{1-\frac{8\mathcal{D}(\Omega)}{\kappa\mathcal{A}(\mathcal{U}^{0})}},$
$\displaystyle\mathcal{D}(\mathcal{U}(t))^{2}<1-\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}},\quad
t>T_{1}.$

This shows the desired assertion. ∎

###  6.3. Partially homogeneous ensemble

In this subsection, we consider a partially homogeneous ensemble:

$\Omega_{\ell}\equiv\Omega,\quad\ell=1,\cdots,N,\quad A_{i}\neq
A_{j}\quad\textup{for some $i\neq j$.}$ (6.12)

Although the natural frequencies $\\{\Omega_{j}\\}$ are the same, complete
aggregation for $\\{x_{j}\\}$ can emerge. However, since $\\{A_{j}\\}$ are
different in general, $\\{U_{j}\\}$ tends to locked state. Thus, our goal in
this subsection is to find a sufficient framework such that the following
relation holds:

$\lim_{t\to\infty}|x_{i}(t)-x_{j}(t)|=0\quad\textup{and}\quad\exists~{}~{}\lim_{t\to\infty}U_{i}(t)U_{j}^{\top}(t).$

First, we show that relative distances for $\\{x_{j}\\}$ tend to zero and the
maximal diameter $\mathcal{D}(\mathcal{U})$ can be controlled in terms of the
coupling strength $\kappa$.

######  Lemma 6.2.

Suppose initial data and system parameters satisfy

$\displaystyle\mathcal{A}(\mathcal{X}^{0})>0,\quad\mathcal{D}(\mathcal{U}^{0})^{2}<1+\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}},\quad\kappa>\frac{4\mathcal{D}(\mathcal{A})}{\mathcal{A}(\mathcal{X}^{0})},$

and let $(\mathcal{X},\mathcal{U})$ be a solution to (6.2) with (6.12). Then,
there exists $T_{*}>0$ such that

$\lim_{t\to\infty}\mathcal{D}(\mathcal{X}(t))=0,\quad\mathcal{D}(\mathcal{U}(t)^{2}<1-\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}},\quad
t>T_{*}.$

###### Proof.

We closely follow Theorem 6.3 to conclude that

$\displaystyle\begin{aligned}
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{A}(\mathcal{X})>\frac{\kappa}{2}\mathcal{A}(\mathcal{U}^{0})\mathcal{A}(\mathcal{X})(1-\mathcal{A}(\mathcal{X})),\\\
&\frac{{\textup{d}}}{{\textup{d}t}}\mathcal{D}(\mathcal{U})^{2}<-\frac{\kappa\mathcal{A}(\mathcal{X}^{0})}{2}(2-\mathcal{D}(\mathcal{U})^{2})D(\mathcal{U})^{2}+\mathcal{D}(\mathcal{A}).\end{aligned}$
(6.13)

Thus, inequalities (6.13) are verified. For the convergence of
$\mathcal{D}(\mathcal{X})$, we use $\eqref{F-20}_{1}$ to show that
$\mathcal{A}(\mathcal{X})$ tends to 1 and recall the relation
$\mathcal{D}(\mathcal{X})^{2}=2(1-\mathcal{A}(\mathcal{X}))$. ∎

In order to establish the desired convergence of $G_{ij}=U_{i}U_{j}^{\top}$,
we adopt the strategy in [21]. Recall that $G_{ij}$ satisfies

$\frac{{\textup{d}}}{{\textup{d}t}}G_{ij}=A_{i}G_{ij}-G_{ij}A_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}h_{ik}(G_{kj}-G_{ik}G_{ij})+h_{jk}(G_{ik}-G_{ij}G_{kj})\Big{)},$
(6.14)

and it follows from Lemma 6.2 that all $h_{ij}$ converges to 1. Thus, we
rewrite (6.14) as

$\frac{{\textup{d}}}{{\textup{d}t}}G_{ij}=A_{i}G_{ij}-G_{ij}A_{j}+\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}(G_{kj}-G_{ik}G_{ij})+(G_{ik}-G_{ij}G_{kj})\Big{)}+\kappa\varepsilon_{ij},$

where the auxiliary functional $\varepsilon_{ij}$ is defined as

$\varepsilon_{ij}:=\frac{1}{N}\sum_{k=1}^{N}\Big{(}(h_{ik}-1)(G_{kj}-G_{ik}G_{ij})+(h_{jk}-1)(G_{ik}-G_{ij}G_{kj})\Big{)}.$

Let $\\{(\tilde{x}_{j},\tilde{U}_{j})\\}$ be any solution to (6.2) with
(6.12). Then, we write

$\displaystyle\tilde{G}_{ij}:=\tilde{U}_{i}\tilde{U}_{j}^{\top},\quad\tilde{h}_{ij}:=\langle\tilde{x}_{i},\tilde{x}_{j}\rangle,$
$\displaystyle\tilde{\varepsilon}_{ij}:=\frac{1}{N}\sum_{k=1}^{N}\Big{(}(\tilde{h}_{ik}-1)(\tilde{G}_{kj}-\tilde{G}_{ik}\tilde{G}_{ij})+(\tilde{h}_{jk}-1)(\tilde{G}_{ik}-\tilde{G}_{ij}\tilde{G}_{kj})\Big{)}.$

Then, we observe

$\displaystyle(G_{ij}-\tilde{G}_{ij})^{\prime}$
$\displaystyle=A_{i}(G_{ij}-\tilde{G}_{ij})-(G_{ij}-\tilde{G}_{ij})A_{j}+\varepsilon_{ij}-\tilde{\varepsilon}_{ij}$
$\displaystyle\hskip
14.22636pt+\frac{\kappa}{N}\sum_{k=1}^{N}\Big{(}(G_{kj}-G_{ik}G_{ij})-(\tilde{G}_{kj}-\tilde{G}_{ik}\tilde{G}_{ij})$
$\displaystyle\hskip
85.35826pt+(G_{ik}-G_{ij}G_{kj})-(\tilde{G}_{ik}-\tilde{G}_{ij}\tilde{G}_{kj})\Big{)}.$

Define the diameter between $G_{ij}$:

$d(\mathcal{U},\tilde{\mathcal{U}}):=\max_{1\leq i,j\leq
N}\|G_{ij}-\tilde{G}_{ij}\|_{\textup{F}}=\max_{1\leq i,j\leq
N}\|U_{i}U_{j}^{\top}-\tilde{U}_{i}\tilde{U}_{j}^{\top}\|_{\textup{F}}.$

Then, after algebraic manipulation, one finds a differential inequality for
$d(\mathcal{U},\tilde{\mathcal{U}})$:

$\frac{{\textup{d}}}{{\textup{d}t}}d(\mathcal{U},\tilde{\mathcal{U}})\leq-2\kappa(1-2\mathcal{D}(\mathcal{U}))d(\mathcal{U},\tilde{\mathcal{U}})+\|\varepsilon_{ij}-\tilde{\varepsilon}_{ij}\|_{\textup{F}}.$

Below, we recall an elementary lemma whose proof can be found in Lemma A.1 of
[6].

######  Lemma 6.3.

Suppose that $f=f(t)\in(L^{1}\cap L^{\infty})(\mathbb{R}_{+})$ is a positive
and bounded function whose argument tends to zero, as $t\to\infty$, and let
$y=y(t)$ be a nonnegative $\mathcal{C}^{1}$-function satisfying

$\begin{cases}\dot{y}\leq-ay+f,\quad t>0,\\\ y(0)=y_{0},\end{cases}$

where $a$ is a positive constant. Then, $y=y(t)$ tends to zero with an
integrable convergence rate.

$\lim_{t\to\infty}y(t)=0.$

Now, we are ready to state the main theorem.

######  Theorem 6.3.

Suppose initial data and system parameters satisfy

$\mathcal{A}(\mathcal{X}^{0})>0,\quad\mathcal{D}(\mathcal{U}^{0})^{2}<1+\sqrt{1-\frac{4\mathcal{D}(\mathcal{A})}{\kappa\mathcal{A}(\mathcal{X}^{0})}},\quad\kappa>\frac{4\mathcal{D}(\mathcal{A})}{\mathcal{A}(\mathcal{X}^{0})},$

and let $(\mathcal{X},\mathcal{U})$ be a solution to (6.2) with (6.12). Then,
a solution to system (6.2) tends to the partially locked state.

###### Proof.

It follows from Lemma 6.2 that relative distances for $\mathcal{X}$ tend to
zero. Thus, it suffices to consider the dynamics of $\mathcal{U}$. From the
arguments in the above, we show that $d(\mathcal{U},\tilde{\mathcal{U}})$
tends to zero. As a next step, since our system is autonomous, for any $T>0$,
we choose $\tilde{U}_{j}$ and $\tilde{V}_{j}$ as

$\tilde{U}_{j}(t)=U_{j}(t+T).$

By discretizing the time $t\in\mathbb{R}_{+}$ as $n\in\mathbb{Z}_{+}$ and
setting $T=m\in\mathbb{Z}_{+}$, we conclude that
$\\{U_{i}(n)U_{j}^{\top}(n)\\}_{n\in\mathbb{Z}_{+}}$ becomes a Cauchy sequence
in the complete spaces $\mathbf{SO}(n)$. Hence, there exists constant unitary
matrices $U_{ij}^{\infty}\in\mathbf{SO}(n)$ such that

$\lim_{t\to\infty}\|U_{i}(t)U_{j}^{\top}(t)-U_{ij}^{\infty}\|_{\textup{F}}=0.$

This establishes the convergence of $U_{i}U_{j}^{\top}$. ∎

##  7\. Conclusion

In this paper, we have provided a systematic algebraic approach for the weak
coupling of the Lohe tensor models with possibly different ranks and sizes.
The Lohe tensor model incorporates well-studied first-order aggregation models
such as the Kuramoto model, the swarm sphere model, the Lohe matrix model that
have been extensively studied in previous literature. For the systematic
modeling of the weak coupling of two models, we introduced a characteristic
symbol corresponding to a given Cauchy problem to the Lohe tensor model, and
it consists of four components, namely a size vector, a coupling strength
tensor, set of natural frequency tensors and initial configuration, and then
we also introduce a fusion operation $\star$ which provides a binary operation
on the space of all characteristic symbols. It is worthwhile to mention that
such fusion operation can form a monoid structure. In this way, we can derive
a weakly coupled model for multiple Lohe tensor models. As a concrete example,
we presented a weakly coupled model for the swarm sphere model and the Lohe
matrix model. For this weakly coupled model, we also provided several
sufficient frameworks leading to collective emergent dynamics. Of course,
there are many untouched issues in this work. For example, we have not studied
the emergent dynamics of the multiple tensor model and did not investigate the
solution structure of the proposed model. These interesting issues will be
left for future work.

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