# In-medium effects in strangeness production in heavy-ion collisions at
(sub-)threshold energies

Taesoo Song<EMAIL_ADDRESS>GSI Helmholtzzentrum für Schwerionenforschung GmbH,
Planckstrasse 1, 64291 Darmstadt, Germany  Laura Tolos<EMAIL_ADDRESS>Institute of Space Sciences (ICE, CSIC), Campus Universitat Autonoma de
Barcelona, Carrer de Can Magrans, 08193 Bellaterra, Spain  Institut d’Estudis
Espacials de Catalunya (IEEC), 08034 Barcelona, Spain  Frankfurt Institute for
Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany
Joana Wirth<EMAIL_ADDRESS>Excellence Cluster ’Origin and Structure of
the Universe’, 85748 Garching, Germany  Physik Department E62, Technische
Universität München, 85748 Garching, Germany  Joerg Aichelin
<EMAIL_ADDRESS>SUBATECH UMR 6457 (IMT Atlantique, Université de
Nantes, IN2P3/CNRS), 4 Rue Alfred Kastler, F-44307 Nantes, France  Frankfurt
Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt am
Main, Germany  Elena Bratkovskaya<EMAIL_ADDRESS>GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany
Institute for Theoretical Physics, Johann Wolfgang Goethe Universität,
Frankfurt am Main, Germany

###### Abstract

We study the in-medium effects in strangeness production in heavy-ion
collisions at (sub-)threshold energies of 1 - 2 A GeV based on the microscopic
Parton-Hadron-String Dynamics (PHSD) transport approach. The in-medium
modifications of the antikaon $(\bar{K}=K^{-},\bar{K}^{0})$ properties are
described via the self-consistent coupled-channel unitarized scheme based on a
SU(3) chiral Lagrangian which incorporates explicitly the $s-$ and $p-$ waves
of the kaon-nucleon interaction. This scheme provides the antikaon potential,
spectral functions and reaction cross sections as well as their dependence on
baryon density, temperature and antikaon momentum in the nuclear medium, which
are incorporated in the off-shell dynamics of the PHSD. The in-medium
modification of kaons $(K=K^{+},K^{0})$ are accounted via the kaon-nuclear
potential, which is assumed to be proportional to the local baryon density.
The manifestation of the medium effects in observables is investigated for the
$K$ and ${\bar{K}}$ rapidity distributions, $p_{T}$-spectra as well as the
polar and azimuthal angular distributions, directed ($v_{1}$) and elliptic
($v_{2}$) flow in C+C, Ni+Ni, and Au+Au collisions. We find - by comparison to
experimental data from the KaoS, FOPI and HADES Collaborations - that the
modifications of (anti)kaon properties in nuclear matter are necessary to
explain the data in a consistent manner. Moreover, we demonstrate the
sensitivity of kaon observables to the equation-of-state of nuclear matter.

##  I Introduction

The modification of hadron properties at finite temperatures and densities due
to the onset of chiral symmetry restoration is an open problem in present
strong interaction physics. This question is of primary importance for the
field of heavy-ion collisions (HICs) and it has also implications for
astrophysics, in particular for the evolution of the early universe and
neutron stars. As suggested early by Kaplan and Nelson kaplan , the attractive
kaon-nucleon interaction might lead to kaon condensation in the interior of
neutron stars. Experimental data FOPI ; Laue99 ; Menzel:2000vv ; Sturm01 ;
Forster02 ; Zinyuk:2014zor ; Agakishiev:2010rs ; Adamczewski-Musch:2018xwg
and theoretical studies Aichelin:1986ss ; Ko1 ; Ko4 ; lix ; cmko ; Li2001 ;
Cass97 ; brat97 ; CB99 ; laura03 ; Effenber00 ; Aichelin ; Fuchs ;
Mishra:2004te ; Hartnack:2011cn ; Kolomeitsev:2004np  on $K^{\pm}$ production
from A+A collisions at SIS energies of 1-2 A$\cdot$GeV have shown that in-
medium modifications of the properties of (anti)kaons have been seen in the
collective flow pattern of $K^{+}$ mesons Ko2  as well as in the abundance and
spectra of kaons and antikaons brat97 ; laura03 ; Ko2 ; Hartnack:2011cn .
Moreover, as advocated early in Ref. Aichelin:1986ss , the strangeness
production at threshold energies is sensitive to the equation-of-state (EoS)
of nuclear matter and thus can provide a comprehensive information about the
compressibility of nuclear matter Fuchs ; Hartnack:2011cn .

The theoretical study of in-medium properties of hadrons has been launched in
part by the early suggestion of Brown and Rho brown , that the modifications
of hadron masses should scale with the scalar quark condensate
$\langle\bar{q}q\rangle$ at finite baryon density and temperature. In this
scenario the kaon masses ’grow’ with the density while antikaon masses ”drop”.
The mass modification at normal nuclear matter density $\rho_{0}$ could be
attributed to the (anti)kaon-nuclear potential. The first theoretical attempts
on the extraction of the antikaon-nucleus potential from the analysis of
kaonic-atom data were in favour of very strong attractive potentials of the
order of -150 to -200 MeV at normal nuclear matter density $\rho_{0}$ FGB94 ;
Gal ; Friedman:2007zza . However, later self-consistent calculations based on
a chiral Lagrangian Lutz:1997wt ; Waas:1996xh ; Waas:1996fy ; Lutz021 ; Lutz02
; Ramos:1999ku ; Mishra:2003tr  or the coupled-channel unitarized scheme (or
G-matrix approach) using meson-exchange potentials Tolos:2000fj ; Tolos:2002ud
predicted a moderate attractive depths of -40 to -60 MeV at density
$\rho_{0}$.

The complexity in the determination of the antikaon potential at finite baryon
density is related to the fact that the antikaon-nucleon amplitude in the
isospin channel $I=0$ is dominated by the $\Lambda(1405)$ resonant structure
Koch:1994mj , which in free space is only $\simeq$ 27 MeV below the
${\bar{K}}N$ threshold. In the vacuum the antikaon-nucleon interaction is
repulsive at low energies, however, in the medium it becomes attractive. The
onset of an attractive $\bar{K}N$ interaction at low densities is a
consequence of an up-shift of the $\Lambda(1405)$ resonance mass induced by
Pauli blocking on the intermediate nucleon states Koch:1994mj ; Waas:1996xh ;
Waas:1996fy ; Lutz:1997wt . Additional medium effects such as the self-energy
of mesons in related coupled channels and the binding of hyperons in the
nuclear environment bring a smoothed $\Lambda(1405)$ back to its vacuum
position Ramos:1999ku , while keeping the attractive character of the
$\bar{K}N$ interaction in the medium. We refer the reader to Ref.
Tolos:2020aln  for a recent review on strangeness.

Another problem related to the complexity of the description of strangeness
dynamics in relativistic heavy-ion reactions is that their in-medium complex
self-energies and scattering amplitudes, partly far from the mass shell, have
to be considered Tolos:2002ud ; laura03 , because the antikaon couples
strongly to the baryons and achieves a nontrivial spectral shape in the
medium. The propagation of such broad states which change their properties
dynamically depending on the local environment, i.e. a temperature $T$ and a
baryon density $\rho_{B}$ (or baryon chemical potential $\mu_{B}$), requires
to go beyond the standard on-shell BUU type of approaches and to employ a
transport theory for strongly interacting systems. This became possible after
a sizeable progress in the understanding of the propagation of strongly
interacting off-shell particles in phase-space configurations has been
obtained in the end of the last millennium: in Refs. Cassing:1999mh ;
Cassing:1999wx  an off-shell transport approach has been developed by Cassing
and Juchem on the basis of the Kadanoff-Baym equations that include the
propagation of particles with dynamical spectral functions (cf. also the
review Cassing:2008nn ). They derived relativistic off-shell generalized
transport equations from the Wigner transformed Kadanoff-Baym equations in the
limit of first order gradients in phase space and realized them - within an
extended test-particle representation - in the Hadron-String Dynamics (HSD)
transport approach Ehehalt:1996uq ; CB99  extended later to the Parton-Hadron-
String Dynamics (PHSD) approach Cassing:2008sv ; Cassing:2008nn ;
Cassing:2009vt ; Bratkovskaya:2011wp ; Linnyk:2015rco ; Moreau:2019vhw . The
PHSD incorporates also partonic degrees-of-freedom in terms of strongly
interacting dynamical quasiparticles, whose properties are evaluated within
the Dynamical Quasi-Particle Model (DQPM) Peshier:2005pp ; Cassing:2007nb ;
Cassing:2007yg  in line with the lattice QCD EoS.

The formulation of off-shell transport found an immediate application for the
description of the strangeness dynamics: whereas in the early calculations lix
; cmko ; Li2001 ; Cass97 ; brat97 ; CB99  the in-medium cross sections have
been simply extrapolated from on-shell cross sections in vacuum, in Ref.
laura03  the full off-shell dynamics - based on coupled channel G-matrix
calculations Tolos:2002ud  \- has been incorporated for the first time in the
HSD transport approach for the description of strangeness production at SIS
energies. It has been shown that the antikaon observables (multiplicities,
rapidity and $p_{T}$-spectra, angular distributions, flow harmonics
$v_{1},v_{2}$ etc.) are very sensitive to the in-medium interactions, i.e. to
the details of the G-matrix approach. A summary of the research on strangeness
dynamics at SIS energy can be found in the review Hartnack:2011cn .

The results obtained with the G-matrix approach in Ref. laura03  are based on
the Jülich meson-exchange model Tolos:2000fj ; Tolos:2002ud  as the effective
$\bar{K}N$ interaction in matter. During the last decade the understanding of
the $\bar{K}N$ interaction has been further improved: in Ref. Tolos:2006ny  a
chiral unitarity approach in coupled channels has been developed by
incorporating $s$\- and $p$\- waves of the kaon-nucleon interaction at finite
density and zero temperature in a self-consistent manner. In Ref. Tolos:2008di
finite temperature effects have been also implemented, however, a full self-
consistent solution has been only achieved for the $s$\- wave effective
$\bar{K}N$ interaction, while the $p$-wave contribution was treated by means
of hyperon-nucleon insertions. Later on, the chiral effective scheme in dense
(and hot) matter developed in Refs. Tolos:2006ny ; Tolos:2008di  has been
substantially improved in Ref. Cabrera:2014lca . There the full self-
consistency in $s$\- and $p$-waves at finite density and temperature has been
achieved. In this way, it became possible to generate in-medium antistrange
meson-baryon cross sections (amplitudes) at finite density and temperature as
well as to determine the single-particle properties of hyperons, such as the
$\Lambda(1115)$, $\Sigma(1195)$ and $\Sigma^{*}(1385)$, at finite momentum
with respect to the medium at rest, and finite density and temperature. The
latter are important for calculations of the in-medium antikaon scattering
cross sections.

While antikaons follow strongly attractive interactions in the medium, which
broadens their spectral function substantially, the kaon spectral function
stays narrow since there are no baryonic resonances that couple to kaons.
Nevertheless, they are affected by the repulsive interaction with nucleons,
which can be approximated by the kaon-nuclear potential proportional to the
local baryon density. This scenario has been widely explored in different
transport approaches Ko1 ; Ko4 ; lix ; cmko ; Li2001 ; Cass97 ; brat97 ; CB99
; laura03 ; Effenber00 ; Aichelin ; Fuchs ; Mishra:2004te  (although there are
models which includes the (anti)kaon on-shell production via the coupling to
heavy baryonic resonances with vacuum properties UrQMDstrange ;
Steinberg:2018jvv ). It has been found that the kaon repulsive potential of
+20-30 MeV at normal nuclear density $\rho_{0}$ is mostly consistent with a
variety of experimental data on $p+A$, $\pi+A$ and $A+A$ collisions. We refer
the reader to the review Hartnack:2011cn  and references therein.

The goal of this work is to study the in-medium effects in the strangeness
production in heavy-ion collisions at (sub-)threshold energies employing the
microscopic transport approach PHSD which incorporates the in-medium
description of the antikaon-nucleon interactions based on ’state of the art’
many-body theory realized by the G-matrix formalism Cabrera:2014lca . This
chiral unitary approach in coupled channels incorporates the $s$\- and
$p$-waves of the kaon-nucleon interaction, its modification in the hot and
dense medium to account for Pauli blocking effects, mean-field binding for
baryons, as well as pion and kaon self-energies. Moreover, it implements
unitarization and self-consistency for both the $s$\- and $p$-wave
interactions at finite temperature and density. This provides access to in-
medium amplitudes in several elastic and inelastic coupled channels with
strangeness content $S=-1$. For the in-medium scenario of the kaon-nucleon
interaction we will adopt here a repulsive potential linear in the baryon
density. As an other novel development, we mention also the implementation of
detailed balance on the level of $2\leftrightarrow 3$ reactions for the main
channels for strangeness production/absorption by baryons ($B=N,\Delta$) and
pions: $B+B\leftrightarrow N+Y+K$ and $B+\pi\leftrightarrow N+N+\bar{K}$, as
well as for the non-resonant reactions $N+N\leftrightarrow N+N+\pi$ and
$\pi+N\leftrightarrow N+\pi+\pi$. We will confront our theoretical results
with available experimental data on (anti)kaon production at SIS energies and
analyse the consequences of chiral symmetry restoration in strangeness
observables.

The outline of the paper is as follows: In section II we shall briefly recall
the basic concepts of the chiral SU(3) model and G-matrix method used in the
present investigation and describe the medium modifications of the
$K(\bar{K}$) mesons in this effective many-body model. In Section III we
remind the basic ideas of the PHSD approach. Sections IV and V are,
respectively, dedicated to the production and absorption of $K$ and $\bar{K}$
mesons and their modifications in the nuclear medium. In section VI we
investigate the properties of $K(\bar{K}$) mesons in heavy-ion collisions and
the results are compared to experimental data from different collaborations in
section VII. Section VIII summarizes the findings of the present investigation
and discusses future extensions.

##  II Chiral unitarized model for $\bar{K}N$ in hot nuclear matter

Here we summarize the main features of the self-consistent unitarized coupled-
channel model in dense and hot matter (or G-matrix approach) based on the
$SU(3)$ meson-baryon chiral Lagrangian, which incorporates the $s$\- and $p$\-
waves of the antikaon-nucleon interaction, as developed in Ref.
Cabrera:2014lca .

The lowest-order chiral Lagrangian which couples the octet of light
pseudoscalar mesons to the octet of $1/2^{+}$ baryons is given by

$\displaystyle
L=\langle\bar{B}i\gamma^{\mu}\nabla_{\mu}B\rangle-M\langle\bar{B}B\rangle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\frac{1}{2}D\langle\bar{B}\gamma^{\mu}\gamma_{5}\\{u_{\mu},B\\}\rangle+\frac{1}{2}F\langle\bar{B}\gamma^{\mu}\gamma_{5}[u_{\mu},B]\rangle,$
(1)

where $\langle..\rangle$ denotes the trace over SU(3) flavor matrices, $M$ is
the baryon mass, $D$ and $F$ stand for the vector and axial-vector coupling
constants, and $\nabla_{\mu}$ is the covariant derivative that couples the
baryon fields to the pseudoscalar meson axial vector current $\Gamma_{\mu}$,

$\displaystyle\nabla_{\mu}B$ $\displaystyle=$
$\displaystyle\partial_{\mu}B+[\Gamma_{\mu},B]\ ,$ $\displaystyle\Gamma_{\mu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}(u^{\dagger}\partial_{\mu}u+u\,\partial_{\mu}u^{\dagger})\
,$ $\displaystyle U$ $\displaystyle=$ $\displaystyle u^{2}={\rm
exp}(i\sqrt{2}\Phi/f)\ ,$ (2) $\displaystyle u_{\mu}$ $\displaystyle=$
$\displaystyle iu^{\dagger}\partial_{\mu}Uu^{\dagger}\ ,$

with $f$ the meson weak decay constant, and $\Phi$ and $B$ the standard SU(3)
meson and baryon field matrices.

The $s$-wave meson-baryon interaction results from the covariant derivative
term in Eq. (1). Keeping terms up to two meson fields, one gets

$\displaystyle
L\sim\frac{i}{4f^{2}}\bigg{\langle}\bar{B}\gamma^{\mu}[\Phi\partial_{\mu}\Phi-\partial_{\mu}\Phi\Phi,B]\bigg{\rangle}.$
(3)

from which one derives the meson-baryon (tree-level) $s$\- amplitudes

$\displaystyle V_{ij}^{s}=-C_{ij}{1\over
4f^{2}}\bar{u}(p^{\prime})\gamma^{\mu}u(p)(k_{\mu}+k^{\prime}_{\mu}),$ (4)

where $k,k^{\prime}~{}(p,p^{\prime})$ are the initial and final meson (baryon)
momenta, respectively, and the coefficients $C_{ij}$, with $i$, $j$ indicating
the particular meson-baryon channel, can be found explicitly in Ref.
Oset:1997it . For low-energy scattering, the $s$-wave meson-baryon interaction
can be written as

$\displaystyle V_{ij}^{s}$ $\displaystyle=$ $\displaystyle-
C_{ij}\,\frac{1}{4f^{2}}\,(2\,\sqrt{s}-M_{B_{i}}-M_{B_{j}})$ (5)
$\displaystyle\times$
$\displaystyle\left(\frac{M_{B_{i}}+E_{i}}{2\,M_{B_{i}}}\right)^{1/2}\,\left(\frac{M_{B_{j}}+E_{j}}{2\,M_{B_{j}}}\right)^{1/2}$
$\displaystyle\simeq$ $\displaystyle-C_{ij}\,{1\over
4f^{2}}(k^{0}_{i}+k^{0}_{j})\ ,$

where $\sqrt{s}$ is the center-of-mass (c.m.) energy, $M_{B_{i(j)}}$ and
$E_{i(j)}$ are the mass and energy of the baryon in the $i(j)$ channel,
respectively. The second equation is satisfied to a good approximation for
practical purposes. In this study, we consider the following channels which
couple to $K^{-}p$ :

$\displaystyle
K^{-}p,~{}\bar{K}^{0}n,~{}\pi^{0}\Lambda,~{}\pi^{0}\Sigma^{0},~{}\eta\Lambda,~{}\eta\Sigma^{0},\hfill\phantom{Aa}$
$\displaystyle\pi^{+}\Sigma^{-},~{}\pi^{-}\Sigma^{+},K^{+}\Xi^{-},~{}K^{0}\Xi^{0}~{},$
(6)

and the following channels which couple to $K^{-}n$ :

$\displaystyle
K^{-}n,~{}\pi^{0}\Sigma^{-},~{}\pi^{-}\Sigma^{0},~{}\pi^{-}\Lambda,~{}\eta\Sigma^{-},K^{0}\Xi^{-}.\hfill$
(7)

The main contribution to the $p$-wave meson-baryon interactions comes from the
$\Lambda$ and $\Sigma$ pole terms, which result from the $D$ and $F$ terms of
the lowest-order meson-baryon chiral Lagrangian of Eq. (1). The
$\Sigma^{*}(1385)$ is also taken into account explicitly, as done in Ref.
Oset:2000eg . Moreover, the Lagrangian in Eq. (3) also provides a small part
of the $p$-wave.

Following Refs. Tolos:2006ny ; Cabrera:2014lca , the $p$-wave meson-baryon
interaction is given by

$\displaystyle\tilde{V}_{ij}^{p}=3\,[f_{ij}(\sqrt{s})\hat{q}^{\prime}\cdot\hat{q}-ig_{ij}(\sqrt{s})(\hat{q}^{\prime}\times\hat{q})\cdot\vec{\sigma}]\
,$ (8)

where $\vec{q}\,(\vec{q}\,^{\prime})$ are the on-shell c.m. three-momentum of
incoming (outgoing) mesons, and $f_{ij}(\sqrt{s})$ and $g_{ij}(\sqrt{s})$
correspond to the spin-non-flip and spin-flip amplitudes, respectively, that
read

$\displaystyle f_{ij}(\sqrt{s})$ $\displaystyle=$ $\displaystyle{1\over
3}\left\\{-C_{ij}{1\over 4f^{2}}\,a_{i}\,a_{j}\left({1\over b_{i}}+{1\over
b_{j}}\right)\right.$ (9)
$\displaystyle+\left.{D^{\Lambda}_{i}D^{\Lambda}_{j}\left(1+{q_{i}^{0}\over
M_{i}}\right)\left(1+{q_{j}^{0}\over
M_{j}}\right)\over\sqrt{s}-\tilde{M}_{\Lambda}}\right.$
$\displaystyle\left.+{D^{\Sigma}_{i}D^{\Sigma}_{j}\left(1+{q_{i}^{0}\over
M_{i}}\right)\left(1+{q_{j}^{0}\over
M_{j}}\right)\over\sqrt{s}-\tilde{M}_{\Sigma}}\right.$
$\displaystyle\left.+{2\over
3}{D^{\Sigma^{*}}_{i}D^{\Sigma^{*}}_{j}\over\sqrt{s}-\tilde{M}_{\Sigma}^{*}}\right\\}q_{i}q_{j}$
$\displaystyle g_{ij}(\sqrt{s})$ $\displaystyle=$ $\displaystyle{1\over
3}\left\\{C_{ij}{1\over 4f^{2}}\,a_{i}\,a_{j}\left({1\over b_{i}}+{1\over
b_{j}}\right)\right.$ (10)
$\displaystyle\left.-{D^{\Lambda}_{i}D^{\Lambda}_{j}\left(1+{q_{i}^{0}\over
M_{i}}\right)\left(1+{q_{j}^{0}\over
M_{j}}\right)\over\sqrt{s}-\tilde{M}_{\Lambda}}\right.$
$\displaystyle\left.-{D^{\Sigma}_{i}D^{\Sigma}_{j}\left(1+{q_{i}^{0}\over
M_{i}}\right)\left(1+{q_{j}^{0}\over
M_{j}}\right)\over\sqrt{s}-\tilde{M}_{\Sigma}}\right.$
$\displaystyle\left.+{1\over
3}{D^{\Sigma^{*}}_{i}D^{\Sigma^{*}}_{j}\over\sqrt{s}-\tilde{M}_{\Sigma}^{*}}\right\\}q_{i}q_{j}\
,$

where $q_{i(j)}\equiv|\vec{q}_{i(j)}|$. The first term in both, $f_{ij}$ and
$g_{ij}$, comes from the small $p$-wave component in the meson-baryon
amplitudes from the lowest order chiral Lagrangian in Eq. (1) Tolos:2006ny ;
Cabrera:2014lca , with

$a_{i}=\sqrt{E_{i}+M_{i}\over 2M_{i}}\ ,\hskip 19.91684ptb_{i}=E_{i}+M_{i}\
,\hskip 19.91684ptE_{i}=\sqrt{M_{i}^{\,2}+\vec{q}_{i}\,^{2}}\ ,$ (11)

given in the c.m. frame. Moreover, $D^{Y}_{i}$ are the couplings of $\Lambda$,
$\Sigma$ and $\Sigma^{*}$ to a given meson-baryon pair:

$\displaystyle D^{\Lambda}_{i}$ $\displaystyle=$ $\displaystyle
c_{i}^{D,\Lambda}\sqrt{20\over 3}{D\over 2f}-c_{i}^{F,\Lambda}\sqrt{12}{F\over
2f}\ ,$ $\displaystyle D^{\Sigma}_{i}$ $\displaystyle=$ $\displaystyle
c_{i}^{D,\Sigma}\sqrt{20\over 3}{D\over 2f}-c_{i}^{F,\Sigma}\sqrt{12}{F\over
2f}\ ,$ (12) $\displaystyle D^{\Sigma^{*}}_{i}$ $\displaystyle=$
$\displaystyle c_{i}^{S,\Sigma^{*}}{12\over 5}{D+F\over 2f}\ .$

The constants $c^{D}$, $c^{F}$, $c^{S}$ are the $SU(3)$ Clebsch-Gordan
coefficients (see Table I of Ref. Jido:2002zk ), whereas the couplings $D$ and
$F$ are chosen as $D=0.85$ and $F=0.52$. The masses $\tilde{M}_{\Lambda}$,
$\tilde{M}_{\Sigma}$, $\tilde{M}_{\Sigma^{*}}$ are bare masses of the hyperons
($\tilde{M}_{\Lambda}$$=$1030 MeV, $\tilde{M}_{\Sigma}$$=$1120 MeV,
$\tilde{M}_{\Sigma^{*}}$$=$1371 MeV), which will turn into physical masses
upon the unitarization procedure described below.

Due to its spin structure, the $p$-wave contribution mixes different total
angular momenta ($J=1/2,3/2$). For unitarization, it is convenient to rewrite
the $p$-wave amplitudes according to the total angular momentum $J$, since
these amplitudes can be unitarized independently. Thus, one can then define
the $p$-wave tree-level amplitudes according to the total angular momentum,
$V^{p}_{ij-}$ ($L=1$, $J=1/2$) and $V^{p}_{ij+}$ ($L=1$, $J=3/2$), as

$\displaystyle V^{p}_{ij-}$ $\displaystyle=$ $\displaystyle f_{ij}-2g_{ij}$
$\displaystyle V^{p}_{ij+}$ $\displaystyle=$ $\displaystyle f_{ij}+g_{ij}.$
(13)

Note that the $\Sigma^{*}$ pole is contained in the $V^{p}_{+}$ amplitude,
whereas the $V^{p}_{-}$ amplitude includes the $\Lambda$ and $\Sigma$ poles.

The transition amplitudes can be obtained by means of a unitarization coupled-
channel procedure based on solving the Bethe-Salpeter equation by using the
tree level contributions to the $s$\- and $p$-wave meson-baryon scattering as
the kernel of the equation. Within the on-shell factorization Oset:1997it ;
Oller:2000fj , the Bethe-Salpeter equation is given in matrix notation as

$\displaystyle T$ $\displaystyle=$ $\displaystyle V+VGT,$ $\displaystyle T$
$\displaystyle=$ $\displaystyle[1-VG]^{-1}V\ ,$ (14)

where $V$ is the kernel (potential), and the $G$ is a diagonal matrix
accounting for the loop function of a meson-baryon $G_{l}$ propagator,

$\displaystyle G_{l}(\sqrt{s})$ $\displaystyle=$ $\displaystyle
i\int\frac{d^{4}q}{(2\pi)^{4}}\frac{M_{l}}{E_{l}(\vec{P}-\vec{q})}\frac{1}{\sqrt{s}-q_{0}-E_{l}(\vec{P}-\vec{q})+i\varepsilon}$
(15) $\displaystyle\times$
$\displaystyle\frac{1}{q_{0}^{2}-\vec{q}\,^{2}-m_{l}^{2}+i\varepsilon},$

with $(P_{0},\vec{P})$ being the total four-momentum of the meson-baryon pair
and $s=P_{0}^{2}-\vec{P}\,^{2}$. The loop is divergent and needs to be
regularized. This can be done by adopting either a cutoff method or
dimensional regularization. We will adopt the cut-off method as it is easier
and more transparent when dealing with particles in the medium. Within this
method, and taking advantage of Lorentz invariance to calculate in the c.m.
frame, the loop function reads

$\displaystyle G_{l}(\sqrt{s})$ $\displaystyle=$
$\displaystyle\int_{\mid{\vec{q}}\,\mid<q_{\rm
max}}\,\frac{d^{3}q}{(2\pi)^{3}}\,\frac{1}{2\omega_{l}(\vec{q}\,)}\frac{M_{l}}{E_{l}(-\vec{q}\,)}$
(16) $\displaystyle\times$
$\displaystyle\frac{1}{\sqrt{s}-\omega_{l}(\vec{q}\,)-E_{l}(-\vec{q}\,)+i\varepsilon}\,,$

with $\omega_{l}$ ($E_{l}$) being the energy of the meson (baryon) in the
intermediate state in the c.m. frame, respectively, and $q_{\rm max}=630$ MeV,
which has been fixed in this scheme to reproduce the $\Lambda(1405)$
properties and several threshold branching ratios Oset:1997it .

Whereas the $s$-wave amplitude of Eq. (5) can be used directly to solve Eq.
(14) to obtain the transition amplitude in the $s$-wave ($T^{s}$), for the
$p$-wave one proceeds as

$\displaystyle T^{p}_{+}$ $\displaystyle=$
$\displaystyle[1-V^{p}_{+}G]^{-1}V^{p}_{+}\ ,$ $\displaystyle T^{p}_{-}$
$\displaystyle=$ $\displaystyle[1-V^{p}_{-}G]^{-1}V^{p}_{-}\ .$ (17)

Nuclear matter effects at finite temperature are introduced through the
modification of the meson-baryon propagators, as described in Refs.
Tolos:2008di ; Cabrera:2014lca . On the one hand, one of the main sources of
medium modifications comes from the Pauli principle. On the other hand, all
mesons and baryons in the intermediate loops interact with the nucleons of the
Fermi sea, and their properties are modified with respect to those in vacuum.
All these changes are straightforward implemented within the Imaginary Time
Formalism, as shown in Refs. Tolos:2008di ; Cabrera:2014lca . Here we
summarize the most important findings.

As found in Ref. Cabrera:2014lca , the in-medium loop function $\bar{K}N$
states reads

$\displaystyle{G}_{\bar{K}N}(P_{0},\vec{P};T)=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{M_{N}}{E_{N}(\vec{P}-\vec{q},T)}$
(18) $\displaystyle\times$ $\displaystyle\left[\int_{0}^{\infty}d\omega
S_{\bar{K}}(\omega,{\vec{q}};T)\frac{1-n_{N}(\vec{P}-\vec{q},T)}{P_{0}+{\rm
i}\varepsilon-\omega-E_{N}(\vec{P}-\vec{q},T)}\right.$ $\displaystyle+$
$\displaystyle\left.\int_{0}^{\infty}d\omega
S_{K}(\omega,{\vec{q}};T)\frac{n_{N}(\vec{P}-\vec{q},T)}{P_{0}+{\rm
i}\varepsilon+\omega-E_{N}(\vec{P}-\vec{q},T)}\right].\ \ \ \ \ $

The Pauli blocking of nucleons is given by the 1-$n_{N}(\vec{P}-\vec{q},T)$
term, where
$n_{N}(\vec{P}-\vec{q},T)=[1+\exp(E_{N}(\vec{p},T)-\mu_{B})/T)]^{-1}$ is the
nucleon Fermi-Dirac distribution. The medium modification on kaons and
antikaons appears through the corresponding spectral functions, defined as

$\displaystyle S_{M}(\omega,{\vec{q}};T)=-\frac{1}{\pi}\frac{{\rm
Im}\,\Sigma_{M}(\omega,\vec{q};T)}{\mid\omega^{2}-\vec{q}\,^{2}-m_{M}^{2}-\Sigma_{M}(\omega,\vec{q};T)\mid^{2}}\
,\ \ $ (19)

where $\Sigma_{M}$ is the meson self-energy that will be described in more
detail below for the $\bar{K}$ case. The $K$ spectral function, $S_{K}$, can
be replaced by a free-space delta function, as the interaction of kaons with
matter is rather weak Tolos:2008di . Note also that the second term in the
$\bar{K}N$ loop function typically provides a small, real contribution for the
studied energy range in $P_{0}$.

In the case of $\pi Y(=\Lambda,\Sigma)$ one finds

$\displaystyle{G}_{\pi
Y}(P_{0},\vec{P};T)=\int\frac{d^{3}q}{(2\pi)^{3}}\frac{M_{Y}}{E_{Y}(\vec{P}-\vec{q},T)}$
(20) $\displaystyle\times$ $\displaystyle\int_{0}^{\infty}d\omega
S_{\pi}(\omega,{\vec{q}},T)\left[\frac{1+f(\omega,T)}{P_{0}+{\rm
i}\varepsilon-\omega-E_{Y}(\vec{P}-\vec{q},T)}\right.$ $\displaystyle+$
$\displaystyle\left.\frac{f(\omega,T)}{P_{0}+{\rm i}\varepsilon+\omega-
E_{Y}(\vec{P}-\vec{q},T)}\right]\ ,$

where the $\pi Y$ loop function incorporates the pion spectral function
$S_{\pi}$. The first term is the dominant one for the studied energy range in
$P_{0}$ and carries the $1+f(\omega,T)$ enhancement factor, which accounts for
the contribution from thermal pions at finite temperature. For hyperons, we
neglect the fermion distribution given the low population for the studied
densities and temperatures.

As for $r=\eta\Lambda$, $\eta\Sigma$ and $K\Xi$ intermediate states, we simply
consider the meson propagator in vacuum and include only the effective baryon
energies modified by the mean-field binding potential for $\Lambda$ and
$\Sigma$ hyperons Cabrera:2014lca , i.e.

$\displaystyle G_{r}(P_{0},\vec{P};T)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\frac{1}{2\omega_{r}(\vec{q}\,)}\,\frac{M_{r}}{E_{r}(\vec{P}-\vec{q},T)}$
(21) $\displaystyle\times$ $\displaystyle\frac{1}{P_{0}+{\rm
i}\varepsilon-\omega_{r}(\vec{q}\,)-E_{r}(\vec{P}-\vec{q},T)}\,.\ \ $

The in-medium $s$\- and $p$-wave transition amplitudes for $\bar{K}N$ at
finite density and temperature are obtained solving Eq. (14) in matter. The
on-shell factorization of the amplitudes in the Bethe-Salpeter equation can be
maintained in the case of the in-medium calculation for $s$-wave scattering
Tolos:2006ny . The solution of the Bethe-Salpeter equation with on-shell
amplitudes can be kept for the $p$-waves with a simple modification of the
meson-baryon loop function, as shown in Ref. Tolos:2006ny . If we denote by
$G_{l}^{L}(P_{0},\vec{P};T)$ the in-medium meson-baryon propagator for $s$\-
($L=0$) and $p$-wave ($L=1$), one has

$\displaystyle G_{l}^{s}(P_{0},\vec{P};T)$ $\displaystyle=$ $\displaystyle
G_{l}(P_{0},\vec{P};T)\ ,$ $\displaystyle G_{l}^{p}(P_{0},\vec{P};T)$
$\displaystyle=$ $\displaystyle G_{l}(s)+\frac{1}{\vec{q}\,^{2}_{\rm
on}}[\tilde{G}_{l}(P_{0},\vec{P};T)-\tilde{G}_{l}(s)],\ \ \ \ \ $ (22)

where the $\tilde{G}$ functions carry an extra $\vec{q}\,^{2}$ factor in the
integrand, corresponding to the off-shell $p$-wave vertex. Moreover, nuclear
short-range correlations have to be taken into account, and the $p$-wave
amplitudes need to be corrected to incorporate, in the external states, the
proper off-shell momentum Tolos:2006ny .

Figure 1: (Color online) The $K^{-}p$ cross sections to the final states
$\pi^{0}\Sigma^{0},~{}\pi^{-}\Sigma^{+},~{}\pi^{+}\Sigma^{-},$ and
$\pi^{0}\Lambda$ according to the G-matrix calculations at
$\rho=\rho_{0},~{}2\rho_{0}$ and $T=0,~{}100$ MeV in comparison with the
experimental data 62 .

Once the transition amplitudes in free space and in matter are evaluated, the
corresponding cross sections can be obtained by

$\displaystyle\frac{d\sigma_{ij}}{d\Omega}(\sqrt{s})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi^{2}}\frac{M_{i}M_{j}}{s}\frac{q_{j}}{q_{i}}$ (23)
$\displaystyle\times$
$\displaystyle\\{|T^{s}_{ij}+(2T^{p}_{ij+}+T^{p}_{ij-})\cos\theta|^{2}$
$\displaystyle+$ $\displaystyle|T^{p}_{ij+}-T^{p}_{ij-}|^{2}\sin^{2}\theta\\}\
,$

where $q_{i}(q_{j})$ is the modulus of the on-shell c.m. three-momentum of the
incoming (outgoing) meson-baryon pair, and $\theta$ is the scattering angle in
that frame.

Figure 1 shows the $K^{-}p$ cross sections to the final states
$\pi^{0}\Sigma^{0},~{}\pi^{-}\Sigma^{+},~{}\pi^{+}\Sigma^{-},$ and
$\pi^{0}\Lambda$ at $T=0$ and $T=100$ MeV for several baryon densities. The
total three-momentum of $K^{-}p$ is taken to be zero, while $k_{K^{-}}$ is
calculated in the proton rest frame. One can see that the cross section in
each channel is close to the experimental data at low densities, while it
decreases with increasing baryon density and temperature. As density and
temperature increase, the rapid fall of the cross section close to threshold
is softened and the strength is distributed over a wide range of energies, as
the $\Lambda(1405)$ melts in matter.

Figure 2: (Color online) (left) The real part and (right) the imaginary part
of antikaon self-energy as a function of the baryon density for antikaon
momenta of 0, 150, 300, and 450 MeV in the rest frame of nuclear matter at
$T=$ 50 MeV (upper) and 100 MeV (lower).

Once the transition amplitudes are computed, the self-energy of the antikaon
in either $s$\- or $p$-wave are obtained by integrating over the nucleon Fermi
distribution at a given temperature,

$\displaystyle\Sigma_{\bar{K}}^{L}(k_{0},\vec{k};T)=4\int\frac{d^{3}p}{(2\pi)^{3}}n_{N}(\vec{p},T)\bar{T}^{L}_{\bar{K}N\rightarrow\bar{K}N}(P_{0},\vec{P};T),$

where $P_{0}=k_{0}+E_{N}(\vec{p},T)$ and $\vec{P}=\vec{k}+\vec{p}$ are the
total energy and momentum of the $\bar{K}N$ pair in the nuclear medium rest
frame, $k$ stands for the momentum of the $\bar{K}$ meson also in this frame,
and $\bar{T}^{L}$ indicates the spin and isospin averaged scattering amplitude
for a given partial wave ($L=0$ or $L=1$). For example, for $K^{-}$ we have

$\displaystyle\Sigma_{K^{-}}^{L}(k_{0},\vec{k};T)=2\int\frac{d^{3}p}{(2\pi)^{3}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\times\bigg{[}n_{p}(\vec{p},T)\bar{T}^{L}_{K^{-}p\rightarrow
K^{-}p}(P_{0},\vec{P};T)~{}~{}~{}$
$\displaystyle+n_{n}(\vec{p},T)\bar{T}^{L}_{K^{-}n\rightarrow
K^{-}n}(P_{0},\vec{P};T)\bigg{]},$ (25)

where the $L=1$ transition amplitude reads $T^{p}=3\,[T^{p}_{-}+2T^{p}_{+}]$,
with $T^{p}_{-}$ and $T^{p}_{+}$ given in Eqs. (II). The calculation of the
antikaon self-energy is a self-consistent process, since the self-energy is
obtained from the in-medium transition amplitude,
$T_{\bar{K}N\rightarrow\bar{K}N}$, which requires the evaluation of the
$\bar{K}N$ loop function, $G_{\bar{K}N}$, and the latter itself is a function
of $\Sigma_{\bar{K}}(k_{0},\vec{k};T)$ through the antikaon spectral function
$S_{\bar{K}}(k_{0},\vec{k};T)$,

$S_{\bar{K}}(k_{0},{\vec{k}};T)=-\frac{1}{\pi}\frac{{\rm
Im}\,\Sigma_{\bar{K}}(k_{0},\vec{k};T)}{\mid
k_{0}^{2}-\vec{k}\,^{2}-m_{\bar{K}}^{2}-\Sigma_{\bar{K}}(k_{0},\vec{k};T)\mid^{2}}\
.$ (26)

Figure 2 shows the real and imaginary parts of antikaon self-energy as a
function of the baryon density for a couple of antikaon momenta in the rest
frame of nuclear matter at $T=$ 50 and 100 MeV. As the baryon density
increases, the imaginary part of the self-energy decreases continuously due to
the increase of phase space, whereas the real part decreases up to twice
saturation density for momenta up to 0.3 GeV. With increasing antikaon
momentum, the changes in the real part of the self-energy become smaller,
while the imaginary part is much less sensitive to the momentum.

Figure 3: (Color online) Spectral function of $\bar{K}$ as a function of
invariant mass at $\rho=0.5\rho_{0}$, $\rho_{0}$, $1.5\rho_{0}$ and
$2\rho_{0}$ with $\rho_{0}$ being saturation density for antikaon momentum
$p=$ 0, 150, 300 and 450 MeV in the rest frame of nuclear matter at $T=$ 50
MeV (left) and 100 MeV (right).

The spectral function can be then reconstructed as

$\displaystyle A_{\bar{K}}(\omega,{\bf k})=\frac{-2~{}{\rm
Im}\Sigma_{\bar{K}}}{(\omega^{2}-{\bf k}^{2}-m_{\bar{K}}^{2}-{\rm
Re}\Sigma_{\bar{K}})^{2}+({\rm Im}\Sigma_{\bar{K}})^{2}},$ (27)

where $m_{\bar{K}}$ is the antikaon mass in vacuum and the spectral function
is normalized as

$\displaystyle\int_{0}^{\infty}\frac{d\omega}{2\pi}\omega
A_{\bar{K}}(\omega,{\bf k})=\frac{1}{2}.$ (28)

From now on we denote $A_{\bar{K}}$ as the spectral function, as shown in the
literature laura03 , which is related to $S_{\bar{K}}$ in Eq. (26) by
$S_{\bar{K}}=A_{\bar{K}}/(2\pi)$.

One can change the integral variable from the energy $\omega$ to the invariant
mass of the $\bar{K}$,

$\displaystyle A_{\bar{K}}(m^{2})=\frac{-2~{}{\rm
Im}\Sigma_{\bar{K}}}{(m^{2}-m_{\bar{K}}^{2}-{\rm
Re}\Sigma_{\bar{K}})^{2}+({\rm Im}\Sigma_{\bar{K}})^{2}},$ (29)

where $m^{2}=\omega^{2}-k_{\bar{K}}^{2}$ and

$\displaystyle\int_{0}^{\infty}\frac{dm^{2}}{2\pi}A_{\bar{K}}(m^{2})=1.$ (30)

Figure 3 shows the spectral function of the $\bar{K}$ as a function of
invariant mass at four baryon densities for antikaon momenta $p=$ 0, 150, 300,
and 450 MeV in the rest frame of nuclear matter at $T=$50 MeV (left panels)
and $T=$ 100 MeV (right panels). The quasiparticle peak appears at
$m^{2}-m_{\bar{K}}^{2}-{\rm Re}\Sigma_{\bar{K}}=0$, being its height
proportional to $-1/{\rm Im}\Sigma_{\bar{K}}$.

##  III Basic concepts of the PHSD

We start with a reminder of the basic ideas of the PHSD transport approach.
The Parton–Hadron–String Dynamics transport approach Cassing:2008sv ;
Cassing:2008nn ; Cassing:2009vt ; Bratkovskaya:2011wp ; Linnyk:2015rco  is a
microscopic off-shell transport approach for the description of strongly
interacting hadronic and partonic matter in and out-of equilibrium. It is
based on the solution of Kadanoff–Baym equations in first-order gradient
expansion in phase space Cassing:2008nn . The approach consistently describes
the full evolution of a relativistic heavy-ion collision from the initial hard
scatterings and string formation through the dynamical deconfinement phase
transition to the strongly-interacting quark-gluon plasma (sQGP) as well as
hadronization and the subsequent interactions in the expanding hadronic phase
as in the Hadron-String-Dynamics (HSD) transport approach Ehehalt:1996uq ;
CB99 . The description of partonic degrees-of-freedom and their interactions
in the PHSD is based on the Dynamical Quasi-Particle Model (DQPM)
Peshier:2005pp ; Cassing:2007nb ; Cassing:2007yg  that is constructed to
reproduce lQCD results for a quark-gluon plasma in thermodynamic equilibrium
Cassing:2008nn  at finite temperature $T$ and baryon (or quark) chemical
potential $\mu_{q}$ on the basis of effective propagators for quarks and
gluons. Correspondingly, the EoS in the PHSD is the lQCD EoS of crossover
type.

The dynamics of heavy-ion collisions within PHSD proceed as follows:
$\bullet$ Initially, the penetrating nuclei are initialized according to Wood-
Saxon density distributions in coordinate space and by local Thomas-Fermi
distributions in momentum space. The initial phase of nucleus-nucleus
collisions starts with primary nucleon-nucleon scatterings from the impinging
nuclei. The further dynamics strongly depends on the collision energy. For low
energy heavy-ion collisions the dynamics is driven by hadronic degrees of
freedom with binary collisions, possible resonance excitations and decays.
However, with increasing bombarding energy the color-neutral strings are
produced which decay to multi-particle states. In the PHSD the description of
multi-particle production in elementary baryon-baryon ($BB$), meson-baryon
($mB$) and meson-meson ($mm$) reactions is realized within the Lund model LUND
in terms of the event generators FRITIOF 7.02 FRITIOF  and PYTHIA 6.4
Sjostrand:2006za . We note that in the PHSD the Lund event generators (FRITIOF
7.02 and PYTHIA 6.4) are ”tuned”, i.e. adjusted, to get a better agreement
with experimental data on elementary $p+p$ collisions, especially at low
energies (cf. Ref. Kireyeu:2020wou ).

$\bullet$ If the local energy density is below the critical one for
deconfinement ($\epsilon_{c}\simeq 0.5$ GeV/fm3) the excited strings are
dissolved into ’pre-hadrons’ (the string decay products) with a formation time
of $\sim$ 0.8 fm/c in their rest frame, except for the ’leading hadrons’, i.e.
the fastest residues of the string ends (quarks $q$ or diquarks $qq$), which
can re-interact (practically instantly) with hadrons with a reduced cross
sections in line with the constituent quark contents.

We note that in Refs. PHSD_CSR ; Alessia  the chiral symmetry restoration
effect has been incorporated for the string decay in the dense hadronic medium
that is created in the early stage of the reaction during the penetration of
the colliding nuclei. The restoration of chiral symmetry is reflected in the
dropping of the scalar quark condensate $<\bar{q}q>$, which can be evaluated
in the transport approach in each local cell within the non-linear
$\sigma-\omega$ model from the local scalar baryon and meson density according
to the Hellman-Feynmann theorem. The dropping of the scalar quark condensate
leads to a modification of the constituent quark masses for light and strange
quarks bound in strings and thus affects the ”chemistry” of decaying strings
via the Schwinger mechanism. This leads to an enhancement of strangeness
production in the dense baryonic medium before the deconfined phase may set in
PHSD_CSR ; Alessia .

$\bullet$ If the local energy density in the cell is above the critical value
of $\epsilon_{c}\sim 0.5\,$GeV/fm3 LQCDx , the ’deconfinement’ (i.e. a
transition of hadronic to partonic degrees-of-freedom) is implemented by
dissolving the ’pre-hadrons’ (in this cell) into the massive colored
quarks/antiquarks and mean-field energy, keeping the ’leading hadrons’ out of
dissolution (cf. Refs. Cassing:2008sv ; Cassing:2009vt  for details). This
procedure allows to keep the microscopic description of a changing degrees-of-
freedom by conserving energy-momentum, charge, flavour etc. The propagation of
the off-shell partons in the hot and dense environment in the self-generated
repulsive mean-field potential is realized by solving the Cassing-Juchem off-
shell transport equations for testparticles Cassing:2008nn .

$\bullet$ The cross sections for the elastic and inelastic partonic
interactions in the QGP phase are determined in the DQPM model as well as the
covariant transition rates for the hadronization to colorless off-shell
hadrons when the local energy density decreases and become close to or lower
than $\epsilon_{c}$ due to the expansion of the system. In PHSD 4.0 the
partonic cross sections, evaluated from the widths in the quark/gluon
propagators, depend only on $T$ Ozvenchuk:2012fn , while in the recent version
PHSD 5.0 the cross sections are evaluated from the leading order scattering
diagrams using the ’resummed’ propagators of the DQPM, and they have an
explicit $(T,\mu_{B})$\- dependence Moreau:2019vhw . However, for the present
study we will use the PHSD 4.0 since we concentrate on low energies where the
partonic phase is not the relevance.

$\bullet$ The final stage of a heavy-ion collision at relativistic energies is
dominated by hadronic interactions. On the other hand, if the initial
bombarding energy is too low for the formation of the QGP, as in the present
study, the whole dynamics of the colliding system proceeds by interactions of
hadronic degrees-of-freedom.

The hadronic degrees-of-freedom in the PHSD are the baryon octet and
decouplet, the ${0}^{-}$ and ${1}^{-}$ meson nonets and higher resonances. The
hadronic interactions include elastic and inelastic collisions between
baryons, mesons and resonances (including the backward reactions through
detailed balance) in line with the HSD approach Ehehalt:1996uq ; CB99 . We
note that in the PHSD the multi-meson fusion reactions to baryon-antibaryon
pairs and backward reactions ($n\ {\rm mesons}\leftrightarrow B+\bar{B}$) are
included, too Cassing:2001ds ; Seifert:2017oyb . Following the formulation of
the collision term for $n\leftrightarrow m$ reactions in Cassing:2001ds ;
Seifert:2017oyb , some other $3\leftrightarrow 2$ reactions have been
implemented recently in the PHSD as the non-resonant reactions
$N+N\leftrightarrow N+N+\pi$ and $\pi+N\leftrightarrow N+\pi+\pi$ and for the
dominant channel in strangeness production/absortion at the threshold by
baryons ($B=N,\Delta$) and pions: $B+B\leftrightarrow N+Y+K$ and
$\pi+B\leftrightarrow K+\bar{K}+N$ (as will be discussed in the next Section).

We mention that at low energies a density dependent Skyrme nucleon-nucleon
potential is used for the mean-field propagation of nucleons, which
corresponds to a ’middle’-soft EoS (default in the PHSD) with the compression
modulus $K\simeq 300$ MeV (cf. Section VII.E). The non-strange baryon
resonances (such as $\Delta$’s) are propagated in the same manner as nucleons
with the same $NN$ potential while strange baryon resonances (such as
$\Lambda$’s, $\Sigma$’s) feel only 2/3 of the nucleon-nucleon potential.

The PHSD incorporates also in-medium effects related to the changes of
hadronic properties in the dense and hot environment during the time evolution
of the system, such as a collisional broadening of the spectral functions of
the light vector mesons ($\rho,\omega,\phi,a_{1}$) Bratkovskaya:2007jk  and
strange vector mesons $K^{*},\bar{K}^{*}$ Ilner:2016xqr .

As mentioned in the Introduction, in Ref. laura03  the in-medium modification
of the strange mesons $K,\bar{K}$ has been implemented in HSD 1.5 using an
early G-matrix approach. The propagation of broad states - described by the
complex self-energies and medium-dependent cross sections - is only possible
in a consistent way by using the off-shell transport theory as reviewed in
Cassing:2008nn . This provides a solid ground for the implementation of the
novel G-matrix described in Section II in the PHSD approach. The PHSD version,
which incorporates the novel G-matrix Cabrera:2014lca , we will denote by PHSD
4.5.

##  IV On-shell kaon, antikaon production/absorption

We start with the description of the on-shell strangeness production channels
in elementary baryon-baryon ($BB$), meson-baryon ($mB$) and meson-meson ($mm$)
reactions (where $B=p,n,\Delta,...$ and $m=\pi,\rho,\omega,...$) as realized
in the PHSD 4.5. Due to the strangeness conservation $s,\bar{s}$ quarks can be
produced only in pairs, which implies the production of kaon-antikaon
$K\bar{K}$ or kaon-hyperon pairs $KY$, where $Y=(\Lambda,\Sigma)$. We mention
that the production mechanism strongly depends on the bombarding energy of the
heavy-ion collision. At low energies only one or two hadrons are associated to
the production of the strange pair, while with increasing energy multi-hadron
states are energetically possible. The latter are realized within string
excitation and decay.

###  IV.1 Baryon+baryon scattering

There are only two channels for strangeness production in baryon+baryon
scattering at low energy:
$\bullet$ The channel with the lowest threshold energy is $N+N\rightarrow
N+Y+K$ with nucleon $N$, hyperon $Y$ and kaon $K$ in the final state
($\sqrt{s_{0}}=m_{N}+m_{\Lambda}+m_{K}=2.546$ GeV). We take into account the
inverse reaction $N+Y+K\rightarrow N+N$ as well, which is relevant in a dense
baryonic matter and required by detailed balance. The implementation of the
$N+Y+K\rightarrow N+N$ channel is realized by a calculation of the transition
rate following Refs. Cassing:2001ds ; Seifert:2017oyb . The details of the
numerical realization of $3\rightarrow 2$ reactions in the PHSD are presented
in Appendix B.
$\bullet$ With increasing energy the strange pair can be produced in the
channel $N+N\rightarrow N+N+K+\bar{K}$ ($\sqrt{s_{0}}=2(m_{N}+m_{K})=2.862$
GeV).

The interactions of nucleons with $\Delta$’s (as well as $\Delta+\Delta$
interactions) are treated in the same way as $N+N$ interactions. We mention
that the contribution of $\Delta N$ reactions (especially $\Delta+N\to N+Y+K$)
is very important for strangeness production at (sub-)threshold energies since
$\Delta N$ channel has a larger $\sqrt{s}$ compared to the scattering of two
nucleons with the same kinetic energy as $\Delta$ and $N$. Moreover, since the
density of $\Delta$ at SIS energies is high, the $\Delta N$ channels dominate
$NN$ channels. The quantitative results on this issue will be presented in
Section VI.

The cross sections for both channels and the inverse reactions of the first
channel $N+Y+K\rightarrow N+N$ are, respectively, described in Appendices A
and B.
$\bullet$ With further increasing collision energy the string formation and
fragmentation becomes the main source of strangeness production. The threshold
for the formation of baryon-baryon strings is taken in the PHSD as
$\sqrt{s_{th}}(BB\ str)=$2.65 GeV.

###  IV.2 Meson+baryon scattering

$\bullet$ Dominant channels for kaon production in meson+baryon scattering are
$\pi+N\rightarrow K+Y$ and $\pi+N\rightarrow K+\bar{K}+N$ with cross sections
given in Appendix A. We also account for the inverse process $N+K+\bar{K}\to
N+\pi$ by detailed balance (see Appendix B).

$\bullet$ As has been found first in Ref. Barz:1985xc  and supported by other
transport calculations Cass97 ; Hartnack:2011cn , the most important channel
for antikaon production in nucleus-nucleus collisions is $\pi+Y\rightarrow
N+\bar{K}$ which involves two secondary particles since the hyperons and pions
have to be first produced by initial $NN$ collisions. The $\pi+Y\rightarrow
N+\bar{K}$ cross section is related to the inverse reaction
$N+\bar{K}\rightarrow\pi+Y$ by detailed balance:

$\displaystyle\sigma_{\pi
Y\rightarrow\bar{K}N}(s)=\frac{D_{\bar{K}}D_{N}}{D_{\pi}D_{Y}}\bigg{(}\frac{p_{\bar{K}}}{p_{\pi}}\bigg{)}^{2}\sigma_{\bar{K}N\rightarrow\pi
Y}(s)~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=\frac{D_{\bar{K}}D_{N}}{D_{\pi}D_{Y}}\frac{\\{s-(m_{\bar{K}}+m_{N})^{2}\\}\\{s-(m_{\bar{K}}-m_{N})^{2}\\}}{\\{s-(m_{\pi}+m_{Y})^{2}\\}\\{s-(m_{\pi}-m_{Y})^{2}\\}}$
$\displaystyle\times\sigma_{\bar{K}N\rightarrow\pi Y}(s),~{}~{}~{}$ (31)

where $D_{i}$ is the degeneracy factor of particle $i$, and $p_{\pi}$ and
$p_{\bar{K}}$ are, respectively, the momenta of pion and antikaon in the c.m.
frame. The cross section for $N+\bar{K}\rightarrow\pi+Y$ is provided by the
G-matrix approach described in Section II, if the initial and final states are
included in Eqs. (6) and (7).
The reactions of mesons with $\Delta$’s are taken into account, too. For
example, for the interactions with $\Delta$’s such as
$\Delta+\bar{K}\rightarrow\pi+Y$, the cross sections are given in Appendix A.
$\bullet$ At energies above the threshold for the string formation in
baryon+meson collisions, which is taken to be $\sqrt{s_{th}}(mB\ str)=$2.4 GeV
in the PHSD, strangeness can be produced in an associated multi-hadron
environment.

###  IV.3 Meson+meson scattering

$\bullet$ The most dominant channel for kaon and antikaon production from
meson scattering is $\pi+\pi\rightarrow K+\bar{K}$ which is given by $\simeq
3$ mb Protopopescu:1973sh . If the total electric charge is +1 or -1, there is
only one channel: $\pi^{+}+\pi^{0}\rightarrow K^{+}+\bar{K}^{0}$ or
$\pi^{0}+\pi^{-}\rightarrow K^{0}+K^{-}$. On the other hand, $\pi^{+}+\pi^{-}$
and $\pi^{0}+\pi^{0}$ can produce either $K^{+}+K^{-}$ or $K^{0}+\bar{K}^{0}$.
We assume that

$\displaystyle\sigma_{\pi^{+}\pi^{-}\rightarrow
K^{+}K^{-}}=\sigma_{\pi^{+}\pi^{-}\rightarrow K^{0}\bar{K}^{0}}$ (32)
$\displaystyle=\sigma_{\pi^{0}\pi^{0}\rightarrow
K^{+}K^{-}}=\sigma_{\pi^{0}\pi^{0}\rightarrow K^{0}\bar{K}^{0}}=1.5~{}[\rm
mb].$ $\displaystyle\sigma_{\pi^{+}\pi^{0}\rightarrow
K^{+}\bar{K}^{0}}=\sigma_{\pi^{-}\pi^{0}\rightarrow K^{-}K^{0}}=3~{}[\rm mb].$

The annihilation of $K$ and $\bar{K}$ is taken into account by the detailed
balance consistently.

$\bullet$ Again, at relativistic energies the strangeness can be produced by
the excitation and decay of meson-meson strings above the threshold, which is
taken to be $\sqrt{s_{th}}(mm\ str)=$1.3 GeV.

###  IV.4 Resonance decays

The decay of $K^{*}(\bar{K}^{*}$) and $\phi$ mesons produces a (anti)kaon and
antikaon-kaon pairs, respectively. $K^{*}(\bar{K}^{*}$) decays into
$K(\bar{K})\pi$ with almost 100 % probability, while $\phi$ meson decays into
$K^{+}K^{-}$ with 49.2 % and $K_{L}^{0}K_{S}^{0}$ with 34 % Tanabashi:2018oca
. The total decay widths of $K^{*}(\bar{K}^{*})$ and $\phi$ are respectively
about 50 MeV and 4 MeV in vacuum Tanabashi:2018oca  (and getting broader in
the medium).

##  V Off-shell kaon, antikaon production/absorption

The properties of (anti)kaons are modified in nuclear matter with respect to
the vacuum. The energy or the rest mass of the kaon increases in nuclear
matter while that of the antikaon decreases. This effectively shifts the
threshold energy for kaon production up and that of antikaon production down.

The modification of the reaction thresholds in the nuclear medium has
important consequences for the strangeness production: the reduction of the
antikaon threshold leads to an enhancement of antikaon production in the
medium. Moreover, it opens sub-threshold, i.e. below the vacuum threshold,
production of antikaons. This sub-threshold antistrangeness production in
$A+A$ reactions - with respect to the elementary $N+N$ reactions - has been an
exciting experimental discovery and extensively studied by transport models in
the past (cf. the review for details Hartnack:2011cn ). Contrary to antikaons,
the kaon threshold in the nuclear medium is shifted to larger energies, i.e.
above the vacuum threshold for corresponding elementary $BB$ and $mB$
($B=N,\Delta$) reactions for kaon production, which suppresses the production
of kaons in $A+A$ collisions when accounting for the in-medium effects
compared to the calculations without medium effects.

In this Section we present our modelling of the in-medium strangeness
production and propagation which accounts for the modification of the
(anti)kaon properties in the nuclear environment. We stress that the antikaon
scattering in the medium is evaluated following the coupled-channel G-matrix
approach as described in Section II.

###  V.1 Antikaon production in the nuclear medium

The antikaon production cross section in the medium is evaluated by folding
the corresponding vacuum production cross section with the in-medium spectral
function in Eq. (29):

$\displaystyle\sigma_{\bar{K}}^{*}(\sqrt{s})=\int_{0}^{(\sqrt{s}-m_{4})^{2}}\frac{dm^{2}}{2\pi}A(m^{2})~{}\sigma_{\bar{K}}(\sqrt{s}-\Delta
m_{\bar{K}}),$ (33)

where $m_{4}$ is the invariant mass of the final particles different from
antikaons (i.e. for $\bar{K}$ production via $1+2\to\bar{K}+X$, $m_{4}$ is the
mass of all particles $X$), $m$ is the off-shell mass of the $\bar{K}$ meson
in the medium defined by the spectral function $A(m^{2})$ according to Eq.
(29); $m_{\bar{K}}$ is the vacuum mass of the antikaon in the final state and
$\Delta m_{\bar{K}}=m-m_{\bar{K}}$ describes the deviation from the vacuum
mass. In Eq. (33) we have assumed that the mass shift of the antikaon is
realized in the scattering cross section by a shift of the incoming invariant
energy $\sqrt{s}$. An explicit derivation is given in Appendix C. The mass of
a produced antikaon is determined by a Monte Carlo procedure on the basis of
the probability distribution function

$\displaystyle\frac{dP(m^{2})}{dm^{2}}=\frac{A(m^{2})~{}\sigma_{\bar{K}}(\sqrt{s}-\Delta
m_{\bar{K}})}{2\pi\sigma_{\bar{K}}^{*}(\sqrt{s})}.$ (34)

The self-energy of the antikaon, which fully determines the spectral function,
explicitly depends on the antikaon three momentum in the rest frame of the
nuclear medium. In the present study we take the velocity of the c.m. frame of
the scattering particles in the medium as the antikaon velocity assuming that
the kinetic energy of the antikaon is small in the c.m. frame. In other words,
the three-momentum of the antikaon in the nuclear medium - on which self-
energy depends - is given by $m_{K}\gamma\vec{\beta}$ with $m_{K}$ and
$\vec{\beta}$ being the antikaon mass in vacuum and the velocity of the c.m.
frame in the medium (with $\gamma=1/\sqrt{1-\beta^{2}}$). Considering that the
collision energy of heavy-ions in the present study is close to the threshold
energy for antikaon production, this should be a reasonable approximation.

###  V.2 Antikaon propagation in the nuclear medium

The equations of motion for off-shell particles are given by the Cassing-
Juchem off-shell transport equations for testparticles Cassing:1999mh  used
also for the propagation of antikaons in the early HSD study laura03 :

$\displaystyle\frac{dr_{i}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{1-C}\frac{1}{2E}\bigg{[}2p_{i}+\nabla_{p}{\rm
Re}\Sigma+\frac{M^{2}-M_{0}^{2}}{{\rm Im}\Sigma}\nabla_{p}{\rm
Im}\Sigma\bigg{]},$ $\displaystyle\frac{dp_{i}}{dt}$ $\displaystyle=$
$\displaystyle\frac{-1}{1-C}\frac{1}{2E}\bigg{[}\nabla_{r}{\rm
Re}\Sigma+\frac{M^{2}-M_{0}^{2}}{{\rm Im}\Sigma}\nabla_{r}{\rm
Im}\Sigma\bigg{]},$ (36) $\displaystyle\frac{dE}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{1-C}\frac{1}{2E}\bigg{[}\partial_{t}{\rm
Re}\Sigma+\frac{M^{2}-M_{0}^{2}}{{\rm Im}\Sigma}\partial_{t}{\rm
Im}\Sigma\bigg{]},$ (37)

where $M_{0}$ is the pole mass in vacuum, $M^{2}=E^{2}-p_{i}^{2}-{\rm
Re}\Sigma$, and

$\displaystyle C=\frac{1}{2E}\bigg{[}\frac{\partial}{\partial E}{\rm
Re}\Sigma+\frac{M^{2}-M_{0}^{2}}{{\rm Im}\Sigma}\frac{\partial}{\partial
E}{\rm Im}\Sigma\bigg{]}.$ (38)

However, as shown in Appendix D, the time-evolution of the off-shell particle
mass is more practical for actual calculations than the time-evolution of the
energy, and Eq. (37) is substituted by

$\displaystyle\frac{dM^{2}}{dt}=\frac{M^{2}-M_{0}^{2}}{{\rm
Im}\Sigma}\frac{d{\rm Im}\Sigma}{dt},$ (39)

which manifests directly that the off-shell mass $M$ approaches $M_{0}$ as the
imaginary part of the self-energy converges to zero when propagating in the
vacuum where the antikaon is on-shell, i.e. stable with respect to strong
interaction.

###  V.3 Kaons in nuclear matter

In this study we assume that the kaon-nuclear potential increases linearly
with baryon density $\rho$ as

$\displaystyle V_{K}=25~{}{\rm MeV}~{}(\rho/\rho_{0}),$ (40)

where $\rho_{0}$ is the nuclear saturation density (taken to be $0.16/{\rm
fm^{3}}$). Since the single-particle energy of the kaon, $\mathcal{E}$, in
nuclear matter is approximated in the nonrelativistic limit by

$\displaystyle\mathcal{E}=\sqrt{m_{K}^{2}+p^{2}+{\rm Re}\Sigma}\simeq
E_{K}+\frac{{\rm Re}\Sigma}{2E_{K}}=E_{K}+V_{K},$ (41)

where $m_{K}$ is the kaon mass in vacuum, ${\rm Re}\Sigma$ is the real part of
the kaon self-energy and $E_{K}=\sqrt{m_{K}^{2}+p^{2}}$. The in-medium kaon
mass $m_{K}^{*}$ is expressed in terms of $V_{K}$ as follows:

$\displaystyle m_{K}^{*}=\sqrt{m_{K}^{2}+{\rm
Re}\Sigma}=\sqrt{m_{K}^{2}+2E_{K}V_{K}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\simeq m_{K}\bigg{(}1+\frac{E_{K}V_{K}}{m_{K}^{2}}\bigg{)}\simeq
m_{K}\bigg{(}1+\frac{25~{}{\rm MeV}}{m_{K}}\frac{\rho}{\rho_{0}}\bigg{)}.$
(42)

As in case of antikaon production, the cross section is shifted by the mass
difference:

$\displaystyle\sigma_{NN\rightarrow
NYK}(\sqrt{s})\rightarrow\sigma_{NN\rightarrow NYK^{*}}(\sqrt{s}-\Delta
m_{K})$ (43)

where $\Delta m_{K}=m_{K}^{*}-m_{K}$. Since $\Delta m_{K}$ is positive, the
mass shift suppresses $K$ production in heavy-ion collisions.

In some scattering channels such as $N+N\rightarrow N+N+K+\bar{K}$ and
$\pi+N\rightarrow N+K+\bar{K}$, the kaon and antikaon are simultaneously
produced. In this case the cross section is expressed as the combination of
Eqs. (33) and (43):

$\displaystyle\sigma_{K\bar{K}}^{*}(\sqrt{s})=\int_{0}^{(\sqrt{s}-m_{4})^{2}}\frac{dm^{2}}{2\pi}A(m^{2})$
$\displaystyle\times\sigma_{K\bar{K}}(\sqrt{s}-\Delta m_{K}-\Delta
m_{\bar{K}}).$ (44)

The production of kaon and antikaon through the decay of $K^{*}$ and $\phi$ is
also modified in nuclear matter. Assuming that the transition amplitude does
not depend on the mass of the daughter particle, as shown in Appendix C, the
mass distribution of the antikaon - produced from the decay - follows

$\displaystyle\frac{dP(m^{2})}{dm^{2}}=\frac{A(m^{2})p^{c.m.}(m^{2})}{8\pi^{2}M^{2}\Gamma}=A(m^{2})~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\times\frac{\sqrt{\\{M^{2}-(m+m_{2})^{2}\\}\\{M^{2}-(m-m_{2})^{3}\\}}}{(4\pi)^{2}M^{3}\Gamma},$
(45)

where $p^{c.m.}$ is the three-momentum of the antikaon in the c.m. frame and
$M$ and $m_{2}$ are, respectively, the masses of the mother particle and the
other daughter particle different from the antikaon.

The position and momentum of the kaon are updated in nuclear matter according
to the equations of motion given by

$\displaystyle\frac{dr_{i}}{dt}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial p_{i}}=\frac{p_{i}}{E},$ $\displaystyle\frac{dp_{i}}{dt}$
$\displaystyle=$ $\displaystyle-\frac{\partial H}{\partial r_{i}}=-\nabla
V_{K}(r),$ (46)

where $i=1,2,3$. The kaon energy is defined by Eq. (41).

##  VI $K$/$\bar{K}$ production in heavy-ion collisions

We step now to the investigation of strangeness dynamics in heavy-ion
collisions within the PHSD 4.5 which incorporates the strangeness cross
sections and their modifications in the medium (defined in Sec. V) and the
off-shell propagation of antikaons according to Eqs. (LABEL:x-update)-(39).
The kaon cross sections and potential depend on the baryon density while the
antikaon properties, defined by self-energies of the G-matrix, depend on the
baryon density, the temperature and the three-momentum of the antikaon in the
rest frame of the medium.

Figure 4: (Color online) The distributions in baryon density $\rho/\rho_{0}$
(upper), temperature $T$ (middle) and momentum $p_{K}$ in the matter-rest
frame (lower) for $K^{+}$’s (left) and $K^{-}$’s (right) at their production
points in central Au+Au collisions ($b=0$ fm) at 1.5 A GeV. The lines show the
individual production channels involving non-strange baryons $B$, hyperons
$Y=(\Lambda,\Sigma)$ and non-strange mesons $m$. The PHSD calculations include
the in-medium effects, i.e. a repulsive potential for $K^{+}$ and the self-
energy from the G-matrix approach for $K^{-}$.

Figure 4 shows the distributions of baryon density, temperature and momentum
in the matter-rest frame of $K^{+}$ and $K^{-}$ at their production points for
individual production channels and for their sum (solid black lines) in
central Au+Au collisions at 1.5 A GeV. The PHSD calculations include the in-
medium effects, i.e. a repulsive potential for $K^{+}$ and the self-energy
from the G-matrix approach for $K^{-}$. The channel decomposition involves
non-strange baryons $B$, hyperons $Y=(\Lambda,\Sigma)$ and non-strange mesons
$m$. From the upper row of Fig. 4 one can see that the density at $K^{+}$
production is larger than at $K^{-}$ production. That is due to the fact that
the main production channels of $K^{+}$’s is $B+B\rightarrow B+Y+K$ which
occurs promptly in heavy-ion collisions while $K^{-}$’s are produced not
directly but in two steps, first $B+B\rightarrow B+Y+K$ and then
$Y+m\rightarrow B+\bar{K}$. We note that we consider here central Au+Au
collisions at $b=$ 0 fm, where the pion density is larger than for the minimal
bias selection. This enhances the contribution from $B+\pi\rightarrow Y+K$
compared to that from $B+B\rightarrow B+Y+K$ for the $K^{+}$ production. There
are many production channels which are not explicitly shown in the figure. For
example, $\pi+\pi\rightarrow K+\bar{K}$ and $B+m\rightarrow B+K+\bar{K}$ are
only shown for $K^{-}$ production, though they contribute to both, $K^{+}$ and
$K^{-}$ production, because their relative contribution to $K^{+}$ production
is small. The lines ’other’ channels in Fig. 4 for $K^{+}$ and $K^{-}$
production include $\phi$ decay, $B+B\rightarrow B+B+K+\bar{K}$, and string
decay of $B+B$, $B+m$ and $m+m$ (if the final state is different from the
channels listed in the figure explicitly). The channels $Y+m\rightarrow\Xi+K$,
$\Xi^{0}+K^{0}\rightarrow\Xi^{-}K^{+}$ contribute only to $K^{+}$ production
and $\Xi+K\rightarrow B+\bar{K}$ and $\Xi+m\rightarrow Y+\bar{K}$ only to
$K^{-}$ production are also included in ’the others.’

The strangeness production in heavy-ion collisions at threshold energies is
dominated by secondary reactions in the hadronic medium, i.e. through multi-
step processes. For example, the primary nucleon+nucleon scattering excites
either one or both nucleons into a $\Delta$, a process which is more
favourable than $NN$ to produce strangeness via $\Delta+N\rightarrow N+Y+K$,
because the $\Delta$ is heavier than the nucleon. Or the $\Delta$ decays into
a nucleon and a pion which produces strangeness via $B+\pi\rightarrow Y+K$.

channel | produced $K^{+}$ per event
---|---
$N+m\rightarrow Y+K$ | 8.79$\times 10^{-2}$
$\Delta+m\rightarrow Y+K$ | 2.30$\times 10^{-2}$
$N+N\rightarrow B+Y+K$ | 1.87$\times 10^{-2}$
$N+\Delta\rightarrow B+Y+K$ | 3.39$\times 10^{-2}$
$\Delta+\Delta\rightarrow B+Y+K$ | 1.22$\times 10^{-2}$
$N+m\rightarrow B+K+\bar{K}$ | 8.79$\times 10^{-4}$
$\Delta+m\rightarrow B+K+\bar{K}$ | 8.79$\times 10^{-4}$
Table 1: The number of produced $K^{+}$’s (with in-medium effects) per event
from several channels in central Au+Au collisions ($b=0$ fm) at 1.5 A GeV.

In order to quantify the importance of secondary processes involving
$\Delta$’s and pions, we present in Table 1 the number of produced $K^{+}$’s
per event in subchannels of the channels $B+m\rightarrow Y+K$, $B+B\rightarrow
B+Y+K$ and $B+m\rightarrow B+K+\bar{K}$, depending on whether the $\Delta$
baryon is involved in strangeness production or not. In the case of
$B+m\rightarrow Y+K$ about 20 $\%$ of $K^{+}$ production is induced by a
$\Delta$ baryon. On the other hand, $\Delta+N\rightarrow B+Y+K$ is about 80
$\%$ more frequent than $N+N\rightarrow B+Y+K$. However,
$\Delta+\Delta\rightarrow B+Y+K$ is kinematically more favourable for $K^{+}$
production, but it is less frequent since the possibility that two $\Delta$’s
scatter is lower. In the case of $B+m\rightarrow B+K+\bar{K}$ the contribution
from $N+m$ and that from $\Delta+m$ are similar. We note that the contribution
of each subchannel depends on the collision system and energy as well as on
centrality. It is worth noting that the contribution from $K^{*}(\bar{K}^{*})$
decay is not negligible for $K^{+}(K^{-})$ production. Thus up to 15 % for
$K^{+}$ and 3.7 % for $K^{-}$ come from their decays. Since the decay width of
the $K^{*}(\bar{K}^{*})$ is about 50 MeV, their decays ($K^{*}\rightarrow
K+\pi$ or $\bar{K}^{*}\rightarrow\bar{K}+\pi$) happen on avarage 4 fm/c after
$K^{*}(\bar{K}^{*})$ production. Therefore the baryon density at $K(\bar{K})$
production through $K^{*}(\bar{K}^{*})$ decays is quite low as follows from
Fig. 4. On the other hand, the $\phi$ decay barely contributes to (anti)kaon
production: 0.07 % for $K^{+}$ and 0.67 % for $K^{-}$.

The middle panels in Fig. 4 show the temperature distributions at the $K^{+}$
and $K^{-}$ production points. The temperature is extracted from the local
energy density and baryon density by using the hadron resonance gas model
(HRG) which includes all mesons up to 1.5 GeV of mass and all (anti)baryons up
to 2 GeV, i.e. in line with the hadronic degrees of freedom of the PHSD. A
similar HRG model has been used in the past to study the thermal equilibration
in infinite hadron-string matter as well as in HICs within the BUU
Bratkovskaya:2000qy , however, here we consider all hadrons ”on-shell”, i.e
neglect their spectral functions (as realized in most of the statistical
models, cf. Cleymans:1992zc ), which might slightly influence the final
temperature Bratkovskaya:2000qy . Although the produced nuclear matter is not
in complete thermal equilibrium, we calculate local energy density and baryon
density in the rest frame of the local cell and map them into those from the
HRG model to obtain a corresponding temperature, which is a commonly used
method in coarse graining. Indeed, such mapping of non-equilibrium
distributions to its equilibrium values can be considered only as a leading
order approximation. However, we note that the temperature dependence of the
antikaon spectral function is relatively weak compared to its density
dependence which washes out the uncertainties related to the extraction of
local temperatures within our HRG model.

One can see that the temperature distribution is highly peaked around $T=$
90-100 MeV for $K^{+}$ while in the case of $K^{-}$ it is less sharply peaked
around $T=$ 80-90 MeV. Considering that $K^{+}$ is earlier produced than
$K^{-}$, this is expected. In addition, the temperature distribution of
$K^{*}(\bar{K}^{*})$ decay is lower than for the other channels, since it
happens later when the matter is cooling due to the expansion.

The distributions in three-momentum of the (anti)kaon at its production time
in the local rest frame of the baryon matter are displayed in the lower panels
of Fig. 4. The figure shows that the three-momentum of $K^{+}$ is peaked
around 300 MeV$/c$ while for $K^{-}$ it is a bit lower. In general the three-
momentum of $K^{+}$ is slightly larger than that of $K^{-}$, though the
distributions are quite similar. One can also find that the momentum of
$K^{+}$ from $B+m\rightarrow Y+K$ is larger than that from other channels and
$K^{-}$ from $Y+m\rightarrow B+\bar{K}$ has a larger momentum than that from
$\pi+\pi\rightarrow K+\bar{K}$.

Figure 5: (Color online) The mass distribution of $K^{+}$’s (upper) and
$K^{-}$’s (lower) at their production points in central Au+Au collisions
($b=0$ fm) at 1.5 A GeV. The lines show the individual production channels.
The PHSD calculations include the in-medium effects for $K^{+}$ and $K^{-}$.

Figure 5 shows the mass distributions of $K^{+}$’s (upper plot) and $K^{-}$’s
(lower plot) at their production points in central Au+Au collisions ($b=0$ fm)
at 1.5 A GeV by summing up all production contributions discussed above with
respect to Fig. 4. The $K^{+}$ mass in the medium mainly depends on the baryon
density as follows from Eq. (42). Considering that the baryon density at
$K^{+}$ production is peaked around $3\rho_{0}$ (as follows from Fig. 4), it
is consistent that the $K^{+}$ mass is distributed around 560-570 MeV in Fig.
5. We note that the $K^{+}$ mass from the $K^{*}$ decay is relatively low
because this decay happens later at low baryon densities.

On the other hand, the $K^{-}$ mass depends on baryon density, temperature and
the three-momentum of the antikaon in the rest frame of the baryonic medium
through the self-energy of the antikaon. Figure 4 shows that the $K^{-}$ is
produced in a wide range of baryon densities from 0 to $3\rho_{0}$ and at
temperatures between 50 MeV and 100 MeV with the three-momentum peaking at
200-400 MeV$/c$. As follows from Fig. 2, the absolute value of the real part
of the $K^{-}$ self-energy rapidly decreases with increasing antikaon momentum
and is less than 20 MeV for $p=$ 300 MeV$/c$, while the imaginary part is not
much affected by the change of momentum. Therefore, the pole mass of the
$K^{-}$ spectral function changes only slightly and the spectral width
broadens as shown in Fig. 5.

Figure 6: (Color online) The contour plot for the mass distributions of
$K^{+}$’s (upper) and $K^{-}$’s (lower) as a function of time in central Au+Au
collisions ($b=0$ fm) at 1.5 A GeV. The PHSD calculations include the in-
medium effects for $K^{+}$ and $K^{-}$.

Figure 6 shows the contour plot for the mass distributions of $K^{+}$’s (upper
plot) and $K^{-}$’s (lower plot) as a function of time for central Au+Au
collisions ($b=0$ fm) at 1.5 A GeV. While at the production times the masses
of $K^{+}$ and $K^{-}$ are highly off-shell they approach dynamically
(according to the off-shell propagation) to the on-shell masses following the
decrease of the baryon density and temperature with time according to Eqs.
(39) and (42), respectively. As displayed in Fig. 5, the $K^{+}$ mass is
initially larger than the vacuum mass while the pole position of the $K^{-}$
mass is close to the vacuum mass with a large spreading in spectral width
towards lower masses. The decrease of the baryon density with time leads to a
reduction of the off-shell $K^{+}$ mass to the vacuum mass and shrinks the
$K^{-}$ spectral function such that the width vanishes in vacuum.

Figure 7: (Color online) The channel decomposition for $K^{+}$ (upper) and
$K^{-}$ (lower) production as a function of time in central Au+Au collisions
($b=0$ fm) at 1.5 A GeV. The PHSD calculations include the in-medium effects
for $K^{+}$ and $K^{-}$.

We show in Fig. 7 the channel decomposition of the $K^{+}$ and $K^{-}$
production as a function of time in central Au+Au collisions ($b=0$ fm) at 1.5
A GeV where the color lines indicate the dominant production channels. One can
see that the $K^{+}$ production via the $B+B\rightarrow B+Y+K$ reaction takes
place early, dominantly due to initial $N+N$ scattering. The channels
involving secondary particles - mesons or baryons - come with some delay,
since their production also needs some time. The $B+m\rightarrow Y+K$ channel
involves produced mesons; the isospin exchange reactions ($p+K^{0}\rightarrow
n+K^{+}$) can happen only after kaons ($K^{0}$) are produced; the $K^{*}$
decays come even with a larger delay due to their later production.

In the case of $K^{-}$’s, the reactions $Y+m\rightarrow B+\bar{K}$,
$\pi+\pi\rightarrow K+\bar{K}$, and $B+m\rightarrow B+K+\bar{K}$ take place at
a similar time after meson production, dominantly pions from $\Delta$ decays.
After that the isospin exchanges ($n+\bar{K}^{0}\rightarrow p+K^{-}$) and then
$K^{*}$ decays are involved. As follows from Fig. 7, the dominant channel for
$K^{-}$ production is a meson-hyperon reaction which is in line with previous
findings Barz:1985xc ; Cass97 ; Hartnack:2011cn . We note that many antikaons
are absorbed after production because the absorbtion cross section of
antikaons by nucleons is very large (cf. Fig. 1). That is why the $K^{-}$
masses in Fig. 6 bent down after initial strong rise.

channel | produced | survived
---|---|---
total $K^{+}$ | 2.57$\times 10^{-1}$ | 1.68$\times 10^{-1}$
$B+B\rightarrow B+Y+K$ | 6.48$\times 10^{-2}$ | 3.92$\times 10^{-2}$
$B+m\rightarrow Y+K$ | 1.13$\times 10^{-1}$ | 6.90$\times 10^{-2}$
$p+K^{0}\rightarrow n+K^{+}$ | 3.15$\times 10^{-2}$ | 2.25$\times 10^{-2}$
$K^{*}\to\pi+K$ decay | 3.89$\times 10^{-2}$ | 3.14$\times 10^{-2}$
total $K^{-}$ | 4.10$\times 10^{-2}$ | 4.13$\times 10^{-3}$
$B+m\rightarrow B+K+\bar{K}$ | 4.57$\times 10^{-3}$ | 1.85$\times 10^{-4}$
$n+\bar{K}^{0}\rightarrow p+K^{-}$ | 4.65$\times 10^{-3}$ | 6.19$\times 10^{-4}$
$Y+m\rightarrow B+\bar{K}$ | 2.16$\times 10^{-2}$ | 2.61$\times 10^{-3}$
$\pi+\pi\rightarrow K+\bar{K}$ | 7.42$\times 10^{-3}$ | 3.69$\times 10^{-4}$
$\bar{K}^{*}\to\pi+\bar{K}$ decay | 1.41$\times 10^{-3}$ | 2.0$\times 10^{-4}$
Table 2: The numbers of produced and survived $K^{+}$ and $K^{-}$ mesons per
event from several channels in central Au+Au collisions ($b=0$ fm) at 1.5 A
GeV. The PHSD calculations are done including the in-medium effects for
$K^{+}$ and $K^{-}$.

Figure 8: (Color online) The rapidity distributions (upper) and
$m_{T}$-spectra (lower) of $K^{+}$ (left) and $K^{-}$ (right) mesons at
freeze-out (i.e. survived) in central Au+Au collisions ($b=0$ fm) at 1.5 A
GeV. The colored lines show the different production channels. The black
dashed line show the rapidity distribution of $K^{+}$ and $K^{-}$ at the
production point normalized to the total number of survived (anti)kaons (solid
black lines). The PHSD calculations include the in-medium effects for $K^{+}$
and $K^{-}$.

Table 2 lists the number of produced and surviving $K^{+}$’s and $K^{-}$’s for
each production channel per event in central Au+Au collisions at 1.5 A GeV.
One can see that about 65 % of produced $K^{+}$ mesons survive, while only 10
% of $K^{-}$ remain and the other $K^{-}$ are absorbed by nucleons or switched
to $\bar{K}^{0}$ by isospin exchange. The survival probability of $K^{+}$ from
$K^{*}$ decay is relatively large and amounts to 81 %, because the $K^{*}$
decays happen later (at low baryon density) than for other production
channels.

We note that (anti)kaon production in Table 2 as well as in Figs. 4 and 5 does
not only include primary production but also secondary and third productions,
e.g. a $K^{+}$ turns to a $K^{0}$ by isospin exchange and then returns to
$K^{+}$ by another isospin exchange. In this case $K^{+}$ production is
counted twice.

Figure 8 shows the rapidity distribution and $p_{T}$-spectra within the
rapidity window $|y|<0.5$ of $K^{+}$ and $K^{-}$ mesons at freeze-out in
central Au+Au collisions at 1.5 A GeV. The integrated yields correspond to the
numbers in the right column (’survived’) of Table 2. The black dashed lines
show the rapidity distribution of $K^{+}$ and $K^{-}$ at the production point
normalized to the total number of surviving (anti)kaons (solid black lines).
The colored lines indicate the different production channels of the surviving
$K^{+},K^{-}$ mesons. One can see that the rapidity distribution of surviving
$K^{+}$ mesons is broader than that of produced $K^{+}$ while the rapidity
distribution of surviving $K^{-}$ has practically the same shape as at
production.

The lower panels of Fig. 8 show the $m_{T}$-spectra of $K^{+}$ and of $K^{-}$.
Again the colored lines indicate the different production channels of
surviving $K^{+},K^{-}$ mesons. One can see that the production channels have
slightly different shapes, however, the final shape is related to the dominant
channels: $B+m\to YK$ for $K^{+}$ mesons and $Y+m\to B+\bar{K}$ for $K^{-}$
mesons. The final $m_{T}$-spectrum (black solid line) of $K^{+}$ mesons is
harder than that at the production point (black dashed line), while the final
$K^{-}$ spectrum becomes slightly softer.

We note (without showing the results explicitly) that the channel
decomposition depends on the system size: the lighter the colliding system,
the larger is the role of primary baryon-baryon collisions for $K^{+}$
production. For a light system as C+C, the production of $K^{+}$ by the $BB\to
BYK$ reactions exceeds that by the $Bm\to YK$ reactions which is dominant
channel for Au+Au collisions at the same energy of 1.5 A GeV - cf. Fig. 8. The
$K^{-}$ mesons are dominantly produced by secondary reactions involving mesons
and hyperons. Such the $Ym\to B\bar{K}$ channel is dominant for the C+C
collisions similar to the Au+Au collisions. The total yield is, however,
substantially reduced due to the low abundances of produced hyperons and
mesons in C+C collisions as compared to more heavy systems.

Figure 9: (Color online) Comparison of initial (dashed lines) and final (solid
lines) $p_{T}$-spectra of $K^{+}$ (upper blue lines) and $K^{-}$ (lower red
lines) mesons without (upper plot) and with (lower plot) in-medium effects
(repulsive potential for $K^{+}$ and the self-energy within the G-matrix
approach for $K^{-}$) in central Au+Au collisions ($b<3$ fm) at 1.5 A GeV
including the angular cut $|\theta-90^{o}|<10^{o}$. Here the number of
produced (anti)kaons is normalized to the number of surviving (anti)kaons in
each plot.

The modification of the final rapidity and $m_{T}$-distributions of the
$K^{+}$ and $K^{-}$ mesons has two origins Hartnack:2011cn  \- the
rescattering in the hadronic medium and the in-medium effects in terms of a
repulsive potential for $K^{+}$ and self-energies within the G-matrix approach
for $K^{-}$. The influence of each effect is quantified in Fig. 9: The upper
plot displays the $p_{T}$\- distributions of produced (initial) i) and
surviving (final) $K^{+}$ and $K^{-}$ mesons without in-medium effects in
central Au+Au collisions at 1.5 A GeV at mid-rapidity (which corresponds to
the angular cut $|\theta-90^{o}|<10^{o}$). Since the number of produced
(anti)kaons is different than for surviving (anti)kaons, the initial
distribution is normalized to the number of surviving (anti)kaons for easy
comparison. One can see that both $p_{T}$\- distributions of the produced
$K^{+}$ and $K^{-}$ are shifted to larger transverse momenta due to
scattering/absorption in the medium with baryons and mesons. We note that the
$K^{-}$ absorption is stronger for slow $K^{-}$. ii) The lower panel of Fig. 9
shows the PHSD results including the kaon potential and the self-energy within
the G-matrix approach for $K^{-}$. One sees that the $p_{T}$\- distribution of
the surviving $K^{+}$ is further shifted to larger $p_{T}$ due to the
repulsive forces. On the other hand, the $p_{T}$\- distribution of surviving
$K^{-}$ is shifted back to small $p_{T}$ because of the attractive potential
related to the real part of antikaon self-energy and partially due to the
broadening of the spectral function related to the imaginary part of the self-
energy (as will be also discussed in Section VII.A). As a result, the final
$p_{T}$\- distribution turns out to be not far from the $p_{T}$\- distribution
of the produced $K^{-}$, as has been already seen in the $m_{T}$-spectra of
$K^{-}$ in Fig. 8 (low, right). We note also that -as demonstrated in Fig. 8
\- the shape of the $K^{-}$ $m_{T}$\- (or $p_{T}$-) distribution follows the
shape of the dominant production channel $Y+m\to B+\bar{K}$ (cf. also Fig. 7)
which involves the secondary particles, i.e. such processes happen with a time
delay when the density of the system is decreasing due to the expansion.
Consequently, the $K^{-}$ mesons feel a relatively lower density by
propagation in the medium - cf. Fig. 4. Then, one should expect larger medium
effects for $K^{-}$ for heavy systems where the baryon density and the size of
the fireball are larger. This will be demonstrated in the next section.

##  VII Comparison with experimental data

In this Section we compare our results on strangeness production with
experimental data from the KaoS, FOPI and HADES Collaborations at SIS energies
and demonstrate the influence of different effects on the observables such as
the in-medium modifications of (anti)kaon properties in terms of the kaon-
nuclear potential for kaons and the self-energy within the G-matrix approach
for antikaons - as well as scattering effects and the influence of the
equation-of-state.

We compare here two scenarios: with and without medium effects in the
following sense:

  * •

with medium effects:
i) for kaons ($K^{+},K^{0}$) we include a repulsive density-dependent
potential which leads to a shift of the kaon mass to a larger value and thus
to a shift of the production threshold, which leads to a modification of kaon
production and its interaction cross sections - cf. Section V.C.
ii) for antikaons ($K^{-},\bar{K}^{0}$) the off-shell dynamics includes the
full density and temperature dependent self-energy within the G-matrix
approach: the modification of the antikaon masses in line with the in-medium
spectral function, off-shell dispersion relations including complex self-
energies, in-medium cross sections for the production and interactions as well
as $\bar{K}$ off-shell propagation \- cf. Sections II and V.A,B.

  * •

without medium effects:
i) for kaons ($K^{+},K^{0}$) we employ on-shell dynamics without a kaon
potential and use free cross sections - cf. Section IV.
ii) for antikaons ($K^{-},\bar{K}^{0}$) we employ on-shell dynamics using the
free production cross sections in the coupled channels - cf. Sections II and
IV.

Since strangeness is produced always in pairs of $(s,\bar{s})$, the kaon and
antikaon (and hyperon) dynamics is coupled. The in-medium effects are applied
for kaons and antikaons simultaneously.

###  VII.1 $y$-distributions

Figure 10: (Color online) Upper row: the PHSD results for the rapidity
distributions of $K^{+}$ (left), $K^{0}_{s}$ (middle) and $K^{-}$ (right)
mesons in 35% central Ar+KCl collisions at 1.76 A GeV in comparison to the
experimental data of the HADES Collaboration Agakishiev:2009ar ;
Agakishiev:2010zw . Middle row: the PHSD results for the rapidity
distributions of $K^{+}$ (left), $K^{0}$ (middle) and $K^{-}$ (right) mesons
in central Ni+Ni collisions at 1.93 A GeV in comparison to the experimental
data of the KaoS Collaboration Menzel:2000vv  for $K^{+},K^{-}$ and the FOPI
Collaboration Merschmeyer:2007zz  for $K^{0}_{s}$. Lower row: the PHSD results
for the rapidity distributions of $K^{+}$ (left) and $K^{-}$ (right) mesons in
40% central Au+Au collisions at 1.23 A GeV in comparison to experimental data
of the HADES Collaboration Adamczewski-Musch:2017rtf  and of $K^{0}_{s}$
mesons (where $K^{0}_{s}=(K^{0}+\bar{K}^{0})/2$) (middle) in 10% central and
40% central (scaled by a factor of 1.8) Au+Au collisions at 1.23 A GeV in
comparison to experimental data of the HADES Collaboration Adamczewski-
Musch:2018xwg . The dashed lines indicate the PHSD results without medium
effects for (anti)kaons, while the solid lines display the results with the
medium effects: a repulsive potential for kaons and the self-energy within the
G-matrix approach for antikaons.

Figure 10 shows the rapidity distributions of $K^{+}$, $K^{0}$,
$K^{0}_{s}=(K^{0}+\bar{K}^{0})/2$ and $K^{-}$ in central Ar+KCl collisions at
1.76 A GeV Ni+Ni collisions at 1.93 A GeV and Au+Au collisions at 1.23 A GeV.
The dashed lines indicate the results without medium effects for (anti)kaons,
while the solid lines denote the results with the medium effects: a repulsive
potential for kaons and the self-energy within the G-matrix approach for
antikaons. One can see that the kaon potential suppresses $K^{+}$, $K^{0}$ (or
$K^{0}_{s}$) production since it increases the kaon mass in the dense medium
and thus increases the threshold for their production. Oppositely, the
inclusion of in-medium effects within the G-matrix approach for $K^{-}$ leads
to an enhancement of their production due to a reduction of the threshold in
the medium. The latter is caused dominantly by a broadening of the $K^{-}$
spectral function in the hadronic medium rather than by a shift of the pole
mass of the spectral function as discussed in the previous Section. As seen
from Fig. 10, the rapidity distributions of $K^{+}$ and $K^{0}$ with potential
are broader than those without potential due to the repulsive forces. On the
other hand, the rapidity distribution of $K^{-}$ becomes slightly narrower
with the medium effects although the broadening of the antikaon spectral
function in the medium leads to the increase in mass for some fraction of
antikaons \- cf. Fig. 5. The narrowing of the final rapidity distributions
occurs due to the following reasons: i) Due to the attractive nature of the
antikaon potential, related to the real part of the antikaon self-energy
$Re\Sigma$ as seen in Fig.2. ii) In addition to $Re\Sigma$, the imaginary part
of the antikaon self-energy $Im\Sigma$ plays an important role in the off-
shell propagation since it is entering the equations-of-motion Eqs.
(LABEL:x-update,36,37) and influence the forces. The effect of $Im\Sigma$ can
be illustrated in the following example: let’s assume ${\rm Re}\Sigma\approx
0$, i.e. that the pole mass does not change, then Eq. (36) is approximated by

$\displaystyle\frac{dp_{i}}{dt}\approx-\frac{1}{2E}\frac{M^{2}-M_{0}^{2}}{{\rm
Im}\Sigma}\nabla_{r}{\rm Im}\Sigma,$ (47)

where $M_{0}$ is the $K^{-}$ mass in vacuum. Since ${\rm Im}\Sigma$ is
negative in dense nuclear matter, $\nabla_{r}{\rm Im}\Sigma$ is directed
outward in heavy-ion collisions. Therefore $K^{-}$ mesons lighter than $M_{0}$
feel an attractive force while heavier $K^{-}$ mesons feel a repulsive force.
As shown in Fig. 5, much more $K^{-}$ mesons are lighter than $M_{0}$ and thus
are attracted inwards or to low rapidity.

One can see from Fig. 10 that in-medium effects are necessary to explain the
experimental data on the rapidity distributions of $K^{+}$, $K^{0}$ and
$K^{-}$ mesons in Ni+Ni collisions at 1.93 A GeV from the KaoS and FOPI
Collaborations as well as $K^{+}$, $K^{0}_{s}$ and $K^{-}$ mesons in Ar+KCl
collisions at 1.76 A GeV from the HADES collaboration. However, there is a
tension between the PHSD results on $K^{+}$, $K^{0}_{s}$ and $K^{-}$ meson
rapidity distributions in Au+Au Collisions at 1.23 A GeV and the HADES data
(which are slightly lower in energy) - without as well as with medium effects.
The overestimation for $K^{+}$ and $K^{0}_{s}$ (including in-medium effects)
is about 20%, while it is about of factor 2 for $K^{-}$ with medium effects.
Thus, within the same model we do not find a consistent description of all
three experiments. The origin for this contradictory message from the
comparison of our results to the experimental data of the HADES Collaboration,
and to the FOPI and the Kaos Collaborations requires further investigation.

Figure 11: (Color online) The PHSD results for the rapidity distributions of
neutral hyperons $\Lambda+\Sigma^{0}$ (blue solid lines) in central (upper)
Ni+Ni collisions at 1.93 A GeV in comparison to the experimental data from the
FOPI Collaboration Merschmeyer:2007zz , (midle) in central Ar+KCl collisions
at 1.76 GeV and (lower) in central Au+Au collisions at 1.5 GeV in comparison
to the experimental data from the HADES Collaboration Agakishiev:2010rs ;
Adamczewski-Musch:2018xwg . The red dashed lines show the contribution from
$\Lambda$ and the green dot-dashed lines present the contribution from
$\Sigma$ hyperons.

Since strangeness is not present in the initial nuclei, strangeness
conservation holds strictly , i.e. the number of strange hadrons is always
equal to that of antistrange hadrons. Therefore the number of produced kaons
is the same as that of hyperons and antikaons (the production of multi-strange
states at low energies is negligible in the total strangeness balance). Thus,
the dynamics of hyperons and antikaons is closely linked since the
$\pi+Y\to\bar{K}+N$ reaction is the dominant channel for antikaon production.
Since the PHSD results for kaon and antikaon rapidity and the $p_{T}$-spectra
for Ni+Ni are in a good agreement with experimental data, we confront our
results for the hyperon distribution with the experimental data, too. We
recall that in the PHSD the strangeness is strictly conserved.

In the upper part of Fig. 11 we show the rapidity distribution of neutral
hyperons $\Lambda+\Sigma^{0}$ in central Ni+Ni collisions at 1.93 A GeV in
comparison to the experimental data from the FOPI Collaboration
Merschmeyer:2007zz . A small deviation is seen at midrapidity where our
results underestimate the FOPI data, and the theoretical distribution is
slightly broader than the data. A similar disagreement is found in the middle
part of Fig. 11 which shows the comparison with the experimental data from the
HADES Collaboration Agakishiev:2010rs  in central Ar+KCl collisions at 1.76 A
GeV. However, the PHSD result for the $y$\- distribution of
$\Lambda+\Sigma^{0}$ agrees very well with the HADES data for Au+Au collisions
at 1.23 A GeV as seen from the lower part of Fig. 11.

###  VII.2 $p_{T}$-distributions

We note that the rapidity distributions are not measured directly in the
experiments. They are extrapolated from the measured $p_{T}$-spectra assuming
a thermal distribution. In this respect it is very important to compare the
model results with directly measured quantities such as $p_{T}$\-
distributions which we present in this subsection.

Figure 12: (Color online) The PHSD results for the $m_{T}$-spectra of $K^{+}$
(upper) and $K^{-}$ (lower) as a function of the transverse kinetic energy
$m_{T}-m_{K}$ in central Ni+Ni collisions at 1.93 A GeV in comparison to the
experimental data of the KaoS Collaboration Menzel:2000vv . The dashed lines
indicate the results without medium effects for (anti)kaons, while the solid
lines display the results with the medium effects: a repulsive potential for
kaons and the self-energy within the G-matrix approach for antikaons.

Figure 13: (Color online) The PHSD results (including in-medium effects) for
the $m_{T}$-spectra of $K^{+}$ (upper), $K^{0}_{S}$ (midle) and $K^{-}$
(lower) as a function of the transverse kinetic energy $m_{T}-m_{K}$ for
different rapidity bins in 35% central Ar+KCl collisions at 1.76 A GeV in
comparison to the experimental data of the HADES Collaboration
Agakishiev:2009ar ; Agakishiev:2010zw .

Figure 14: (Color online) The PHSD results for the midrapidity $m_{T}$-spectra
of $K^{+}$ (blue line) and $K^{-}$(green line) as a function of the transverse
kinetic energy $m_{T}-m_{K}$ in 40% central Au+Au collisions at 1.23 A GeV in
comparison to the experimental data of the HADES Collaboration Adamczewski-
Musch:2017rtf . The dashed lines indicate the results without medium effects
for (anti)kaons, while the solid lines display the results with the medium
effects: a repulsive potential for kaons and the self-energy within the
G-matrix approach for antikaons.

Figure 15: (Color online) The PHSD results (including in-medium effects) for
the $m_{T}$-spectra of $K^{+}$ (upper) and $K^{-}$ (lower) as a function of
the transverse kinetic energy $m_{T}-m_{K}$ for different rapidity bins in 40%
central Au+Au collisions at 1.23 A GeV in comparison to the experimental data
of the HADES Collaboration Adamczewski-Musch:2017rtf .

In Fig. 13 we show the PHSD results (including in-medium effects) for the
$m_{T}$-spectra of $K^{+}$ (upper), $K^{0}_{S}$ (midle) and $K^{-}$ (lower) as
a function of the transverse kinetic energy $m_{T}-m_{K}$ for different
rapidity bins in 35% central Ar+KCl collisions at 1.76 A GeV in comparison to
the experimental data of the HADES Collaboration Agakishiev:2009ar ;
Agakishiev:2010zw . One can see that the PHSD agrees with HADES data for
Ar+KCl very well for all rapidity bins.

Figure 12 shows the PHSD results for the $m_{T}$-spectra versus the transverse
kinetic energy $m_{T}-m_{K}$ of $K^{+}$ and $K^{-}$ at midrapidity in central
Ni+Ni collisions at 1.93 A GeV in comparison to the experimental data of the
KaoS Collaboration Menzel:2000vv . We note that here the invariant
$p_{T}$-spectra are divided by transverse mass $m_{T}$ to extract the
effective temperature which will be discussed later in this section; therefore
the spectra are not Lorentz-invariant. As for the rapidity distribution the
kaon-nuclear potential suppresses the $K^{+}$ yield and pushes the $K^{+}$
spectrum to larger transverse momenta. On the other hand, the antikaon
potential enhances the $K^{-}$ yield and pulls the $p_{T}$-spectrum toward
smaller transverse momenta. Again the kaon potential is necessary to reproduce
the experimental data on the $p_{T}$\- spectra of $K^{+}$; the data for
$K^{-}$ are also in favour of the inclusion of the self-energy within the
G-matrix approach, similar to their yields and rapidity distributions, but, as
discussed in the last section, scattering of antikaons in the medium and the
in-medium effect within the G-matrix for antikaons partially compensate each
other.

Figure 16: (Color online) The PHSD results for the inclusive invariant cross
sections at midrapidity as a function of the kinetic energy $E_{c.m.}-m_{K}$
for $K^{+}$ (left) and $K^{-}$ (right) mesons in minimum-bias Au+Au collisions
at 1.5 A GeV (upper), Ni+Ni collisions at 1.93 A GeV (middle) and C+C
collisions at 1.8 A GeV (lower) with (solid lines) and without (dashed lines)
medium effects, compared to the experimental data from the KaoS Collaboration
Forster:2007qk . The midrapidity condition is a selection of
$\theta_{c.m.}=90^{0}\pm 10^{0}$.

Figure 17: (Color online) The PHSD results for the inclusive invariant cross
section at midrapidity as a function of the kinetic energy $E_{c.m.}-m_{K}$
for $K^{+}$ mesons in minimum-bias Au+Au collisions at 1.0 A GeV without (red
dashed line) and with (blue solid line) medium effects compared to the
experimental data from the KaoS Collaboration Forster:2007qk . The midrapidity
condition is a selection of $\theta_{c.m.}=90^{0}\pm 10^{0}$.

Now we advance to a heavy system and show in Fig. 14 the PHSD results for the
midrapidity $m_{T}$-spectra of $K^{+}$ (blue line) and $K^{-}$(green line) as
a function of the transverse kinetic energy $m_{T}-m_{K}$ in 40% central Au+Au
collisions at 1.23 A GeV in comparison to the HADES data Adamczewski-
Musch:2017rtf . The HADES data for $K^{+}$ are overestimated by about 20-30%
at large $m_{T}$ and agree with the data at small $m_{T}$ for the in-medium
scenario (solid blue line). Without medium effects (dashed blue line) we over-
predict the data by a factor of 2. The discrepancies for the $K^{-}$ with
medium effects are about factor of 2 at low $m_{T}$ (solid green line) and by
about 30% (within statistical fluctuations) at larger $m_{T}$ for the free
scenario (dashed green line).

It is interesting to compare our results for the $m_{T}$-distributions not
only to the mid-rapidity data as shown in Fig. 14, but also to the non-central
rapidity bins which have been used for the extrapolation of the HADES results
for $dN/dy$ shown in Fig. 10. In Fig. 15 we display the PHSD $m_{T}$-spectra
of $K^{+}$ (upper) and $K^{-}$ (lower) (including in-medium effects) for
different rapidity bins in 40% central Au+Au collisions at 1.23 A GeV in
comparison to the experimental data of the HADES Collaboration Adamczewski-
Musch:2017rtf . As one can see, the slope of the $m_{T}$-spectra of $K^{+}$ is
getting harder when coming to midrapidity. The $m_{T}$-spectra of $K^{+}$ for
rapidity bins $|y|>0.45$ are very well described, however, the agreement is
getting worse (up to 30% overestimation for large $m_{T}$) when coming to the
midrapidity bin, as discussed above for Fig. 14. For $K^{-}$ mesons the slopes
of the $m_{T}$-spectra for each $y$-bin are approximately in line with the
HADES data, however, the absolute yield for each $y$-bin is overestimated by
about of a factor of 2. This is reflected in the $y$\- distribution of $K^{+}$
and $K^{-}$ of Fig. 10, too.

Figure 16 shows the inclusive invariant cross sections at midrapidity as a
function of the kinetic energy $E_{c.m.}-m_{K}$ for $K^{+}$ and $K^{-}$ mesons
in minimum-bias Au+Au collisions at 1.5 A GeV, Ni+Ni collisions at 1.93 A GeV
and C+C collisions at 1.8 A GeV with and without medium effects, which are
compared to the experimental data from the KaoS Collaboration Forster:2007qk .
Contrary to the kaon $p_{T}$-spectra in central Ni+Ni collisions (shown in
Fig. 12), the yields of $K^{+}$ mesons in minimum-bias are slightly
underestimated as compared to the experimental data, which implies that our
$K^{+}$ production in non-central collisions is smaller than in the
experimental data. However, the slopes of the $K^{+}$ spectra are close to the
experimental data with the kaon potential switched on. As seen from Fig. 12,
the repulsive force from the kaon potential hardens the spectra relative to
those without kaon potential. The effects of the kaon potential are stronger
in Au+Au collisions than in C+C collisions, since the baryon density achieved
in Au+Au collisions is higher than in C+C collisions.

The same applies for the $K^{-}$ mesons (right panel of Fig. 16): the maximal
softening of the $p_{T}$-spectrum of $K^{-}$ occurs in Au+Au collisions. The
softening of the $p_{T}$-spectrum is also well visible in minimum-bias Ni+Ni
collisions, however, the effect is smaller than in central Ni+Ni collisions in
Fig. 12, since the baryon density is larger for central collisions than in
minimum-bias Ni+Ni collisions. For C+C collisions the softening is invisible
(with the statistics achieved in the calculations), moreover, the absolute
yield is underestimated for C+C even with the inclusion of in-medium effects.

Additionally, in Fig. 17 we show the inclusive invariant cross section at
midrapidity as a function of the kinetic energy $E_{c.m.}-m_{K}$ for the
$K^{+}$ mesons in minimum-bias Au+Au collisions at strongly sub-threshold
energy of 1.0 A GeV without (red dashed line) and with (blue solid line)
medium effects compared to the experimental data from the KaoS Collaboration
Forster:2007qk . One can see that the PHSD describes the $K^{+}$ spectra very
well with in-medium effects and substantially overestimates the data without
the repulsive potential.

Figure 18: (Color online) The PHSD results for the effective temperatures (or
the inverse slopes) of $K^{+}$ and $K^{-}$mesons as a function of the mass
number of projectile and target nuclei in minimum-bias collisions of C+C at
1.5 and 1.8 A GeV, Ni+Ni at 1.93 A GeV and Au+Au at 1.5 A GeV in comparison to
the experimental data from the KaoS Collaboration Forster:2007qk . We shift
the PHSD results for C+C at 1.5 A GeV to a lower mass number for better
visibility.

The transverse momentum spectra can be characterized by an effective
temperature $T_{eff}$ which can be obtained from a fit of the (anti)kaon
spectra with a thermal distribution function:

$\displaystyle E\frac{d\sigma}{d^{3}p}\sim E\exp\bigg{(}-\frac{E}{T}\bigg{)}.$
(48)

Since radial flow, which hardens the spectrum, contributes to the slope, $T$
is called an effective temperature $T_{eff}$. Fig. 18 shows the effective
temperatures for $K^{+}$ and $K^{-}$ mesons in minimum-bias C+C at energies of
1.5 and 1.8 A GeV; Ni+Ni and Au+Au collisions, respectively, at energies of
1.93 and 1.5 A GeV as a function of the mass number of target and projectile
nuclei. The effective temperature depends on the range of transverse momenta
considered and our results are fitted to the spectra between
$0<E-m_{K}<0.5~{}{\rm GeV}$. Because of collective flow the effective
temperature in Au+Au collisions is higher than that in C+C collisions,
although the collision energy is higher in the latter case.

One can see that the effective temperature for $K^{+}$ mesons is higher than
for $K^{-}$ mesons, because the kaon potential - which generates a repulsive
force - hardens the $K^{+}$ spectra, while the inclusion of the self-energy
within the G-matrix approach softens the $K^{-}$ spectra. This effect is
stronger for the heavy systems: the splitting of the effective temperatures
for $K^{+}$ and $K^{-}$ is small in C+C collisions, where the size and density
of nuclear matter is small, and the splitting is larger for Au+Au collisions
which produce a much larger and denser fireball. Moreover, the larger density
achieved in Au+Au collisions compared to C+C collisions lead to the relative
increase of the $T_{eff}$ of $K^{+}$ and $K^{-}$ with system size. Our results
are close to the experimental data of the KaoS Collaboration Forster:2007qk
and qualitatively consistent.

The difference of the effective temperatures of (anti)kaons for C+C and Au+Au
at 1.5 A GeV is mostly attributed to in-medium effects. As demonstrated in
Fig. 9 (upper plot), the elastic scattering increases the effective
temperature for both, kaons and antikaons, considerably (since the (anti)kaons
are produced near the thresholds, they are not energetic due to phase space
limitation). The interaction with the medium is repulsive for kaons and mostly
attractive for antikaons, and this leads to the separation of their effective
temperatures in the Au+Au reaction. Such separation is not visible for the
light C+C system due to the small volume of the medium and lower baryon
densities.

We note that the different slopes of the $K^{+}$ and $K^{-}$ $m_{T}$-spectra
for Au+Au collisions at 1.23 A GeV have been interpreted by HADES
collaboration Agakishiev:2009ar ; Adamczewski-Musch:2017rtf  by the feed-down
from $\phi$-meson decay, which substantially softens the spectra of $K^{-}$
mesons compared to the ’thermal’ hard component - cf. Fig. 2 in Ref.
Adamczewski-Musch:2017rtf . However, according to the present PHSD
calculations, the contribution to the $K^{-}$ abundances for Au+Au collisions
at 1.5 GeV from the $\phi$ decay plays a sub-leading role - cf. Fig. 7 and the
different slopes of $K^{+}$ and $K^{-}$ is attributed to the in-medium effects
(as discussed above). This result is in line with the experimental data of the
KaoS Collaboration which show that the splitting $T_{eff}$ of $K^{+}$ and
$K^{-}$ spectra increases with the system size. One can emphasize that if the
antikaons are dominantly produced by $\phi$ decay one would expect a different
behaviour: since the $\phi$-nucleon interaction cross section is small (cf.
Ref. joos ) and the $\phi$-meson life time is long, i.e. the most of the
$\phi$ mesons decay to $K\bar{K}$ outside of the fireball at almost zero
baryon density, one would expect that the slope of the spectra of antikaons do
not show the prominent dependence on the system size. Thus, we stress, that by
a measurement of the effective temperature of $K^{-}$ for different system
sizes for the same beam energy one can shed more light on the origin of the
antikaon production and the role of the in-medium effects.

We note that our results for the rapidity and $p_{T}$-spectra are consistent
with previous findings within the HSD Cass97 ; brat97 ; CB99 ; laura03  and
other groups Hartnack:2011cn ; Kolomeitsev:2004np  about the necessity to
include medium effects for the proper description of the experimental data on
strangeness production at low energies.

We stress again that the comparison to the HADES data on $m_{T}$-spectra for
Au+Au collisions at 1.23 A GeV falls out of this systematics, although the
PHSD results with in-medium effects (as well as the results of other transport
models Hartnack:2011cn ; Kolomeitsev:2004np ) agree well with the KaoS and
FOPI data even for Au+Au collisions at 1.0 and 1.5 A GeV. Moreover, the PHSD
reproduces well the KaoS and FOPI data for more light system as Ni+Ni at 1.93
A GeV and the HADES data for Ar+KCl at 1.76 A GeV. This tension is presently
unsolved and requires further investigations from theoretical and experimental
sides.

In this respect, we note that in the present calculation the $s-$wave and
$p-$wave of the antikaon-nucleon interaction have been obtained from the
leading-order contribution together with the $\Lambda,\Sigma$ and $\Sigma^{*}$
pole terms. In order to improve the agreement with data for large momenta,
future work should include the analysis of the effect of next-to-leading-order
corrections to the antikaon-nucleon interaction as well as the inclusion of
strange baryonic resonances located at energies between 1.89 and 2.35 GeV,
well above the antikaon-nucleon threshold, as recently discussed in Ref.
Feijoo:2018den .

###  VII.3 Polar distributions

Figure 19: (Color online) The polar angular distributions of $K^{+}$ (left)
and $K^{-}$ (right) mesons in central (upper) and noncentral (lower) Au+Au
collisions at 1.5 A GeV compared to the experimental data from the KaoS
Collaboration Forster:2007qk . The distributions are normalized to unity at
$\cos(\theta_{c.m.})=0$, i.e. presented as the ratios
$\sigma_{inv}(\theta_{c.m.})/\sigma_{inv}(\theta_{c.m.}=90^{0})$. The solid
blue and dashed red lines show the (anti)kaon angular distributions with and
without the medium effects, respectively. The dot-dashed green lines show
their distribution at the production point (with medium effects). The black
dashed lines indicate the nucleon angular distributions.

Figure 20: (Color online) The polar angle distributions of $K^{+}$ (left) and
$K^{-}$ (right) mesons in central (upper) and noncentral (lower) Ni+Ni
collisions at 1.93 A GeV compared to the experimental data from the KaoS
Collaboration Forster:2007qk . The line description is the same as in Fig. 19.

Figure 19 shows the polar angle distributions of $K^{+}$ and $K^{-}$ in
central (upper row: $(0-18.1)\%$ of total reaction cross section $\sigma$) and
in noncentral (lower row: $(18.1-100)\%\cdot\sigma$) Au+Au collisions at 1.5 A
GeV in comparison to the experimental data from the KaoS Collaboration
Forster:2007qk . The polar angle is defined as the angle between the beam axis
and the particle, i.e. $\cos(\theta_{c.m.})=\pm 1$ corresponds to the beam
direction and $\cos(\theta_{c.m.})=0$ to midrapidity. The distributions are
normalized to unity at $\cos(\theta_{c.m.})=0$, i.e. presented as the ratios
$\sigma_{inv}(\theta_{c.m.})/\sigma_{inv}(\theta_{c.m.}=90^{0})$, where
$\sigma_{inv}(\theta_{c.m.})$ is the invariant particle production cross
section measured at a polar angle $\theta_{c.m.}$.

As seen from Fig. 19, the polar distribution of $K^{+}$ mesons in noncentral
collisions is strongly peaked in beam direction, while the distribution in
central collisions is rather flat. This results from a strong correlation of
$K^{+}$ mesons with nucleons through the production of $K^{+}$ from
nucleon+nucleon or nucleon+pion scattering and the interactions of $K^{+}$
mesons with nucleons. One can see that the polar distribution of $K^{+}$
follows that of nucleons which is shown by black dashed lines both in central
and noncentral Au+Au collisions.

The dot-dashed green lines display the polar distributions of produced
$K^{+}$. We note that the distribution of produced $K^{+}$ is only slightly
affected when including the kaon potential. For central collisions the angular
distribution of kaons is also rather flat, while for noncentral it grows
towards beam direction. The differences between the dot-dashed green line and
the dashed red line are caused by scattering and those between the dashed red
line and the blue solid line is the effect of the kaon potential. Since the
kaon potential is repulsive, it pushes the polar distributions of $K^{+}$
mesons towards forward and backward directions. One can see that the effects
of scattering on the polar distribution of $K^{+}$ mesons are stronger than
those of the kaon potential.

The right two panels of Fig. 19 show the polar distributions of $K^{-}$ mesons
in central and noncentral Au+Au collisions, respectively. Though the
statistics is limited in noncentral collisions, one can see that the
scattering pushes $K^{-}$ mesons backward and forward as in the case of
$K^{+}$. However, the effects of the antikaon potential are opposite such that
it pulls $K^{-}$ mesons into the middle of the fireball, since the antikaon
feels attraction.

In Fig. 20 we show the same polar distributions for Ni+Ni collisions at 1.93 A
GeV. One can see a rather similar behaviour of $K^{+}$ and $K^{-}$ polar
distributions as for Au+Au at 1.5 A GeV, although the effects of scattering
and the potential are slightly weaker than in Au+Au collisions since the size
and density of the produced fireball is smaller.

The minor influence of the in-medium effects on the polar distribution for
Ni+Ni and Au+Au collisions might seem to be in conflict with the enhancement
of the $K^{+}$ rapidity distribution shown in Fig. 10. However, both results
are consistent due to the following reasons: one has to keep in mind that the
polar distribution is normalized to $\sigma_{inv}(\theta=90^{0})$, i.e. at
$\cos\theta_{c.m.}=0$. For kaons with a moderate transverse momentum in the
range of $0.2~{}{\rm GeV}<p_{T}<0.8~{}{\rm GeV}$ (to which most kaons belong
to), the rapidity $y=0.5$ corresponds to $0.8<\cos\theta_{c.m.}<0.9$.
Therefore, a wide range of $\cos\theta_{c.m.}$ in Fig. 20 corresponds to a
narrow range close to midrapidity in Fig. 10. If $dN/dy$ with and without
medium effects would be rescaled (i.e. assuming the same number of kaon
production with and without medium effects), the shapes of the $y-$
distribution near mid-rapidity $(-0.5<y<0.5)$ would be similar to each other.
That is why the polar distributions with and without medium effects are
similar to each other in Figs. 19 and 20, too.

###  VII.4 Azimuthal distributions

The azimuthal distribution of particles is parameterized in a Fourier series
as:

$\displaystyle\frac{dN(p_{T},y)}{d\phi}=C[1+2v_{1}(p_{T},y)\cos\phi$
$\displaystyle+2v_{2}(p_{T},y)\cos(2\phi)+\cdots],$ (49)

where the coefficients $v_{1}$ and $v_{2}$ are, respectively, denoted as the
directed and elliptic flows obtained by

$\displaystyle v_{1}(p_{T},y)$ $\displaystyle=$ $\displaystyle\frac{\int
d\phi\frac{dN(p_{T},y)}{d\phi}\cos\phi}{\int d\phi\frac{dN(p_{T},y)}{d\phi}},$
(50) $\displaystyle v_{2}(p_{T},y)$ $\displaystyle=$ $\displaystyle\frac{\int
d\phi\frac{dN(p_{T},y)}{d\phi}\cos(2\phi)}{\int
d\phi\frac{dN(p_{T},y)}{d\phi}}.$ (51)

Figure 21: (Color online) (Upper) The $\left<p_{x}\right>/m$ of $\Lambda$ (red
solid line) and proton (blue solid line) as a function of normalized rapidity
and (middle) the $v_{1}$ of $K^{+}$ mesons as a function of $p_{T}$ in central
Ni+Ni collisions at 1.93 A GeV in comparison to the experimental data from the
FOPI Collaboration Ritman:1995tn ; Crochet:2000fz . (Lower) The $v_{1}$ of
$K^{+}$ and $K^{-}$ with and without medium effects as a function of rapidity
in the same collisions. The blue solid line in the lower plot shows the PHSD
results for the $v_{1}$ of final $K^{+}$ mesons with medium effects, the green
dotted line indicates the $v_{1}$ of kaons at the production point for the in-
medium scenario, while the red dashed line shows the $v_{1}$ of kaons without
medium effects.

Since the collective flow of kaons is related to the flow of nucleons involved
in the production and interactions of kaons, we first show in Fig. 21 (upper
plot) the directed flows of protons and $\Lambda$’s (which are the associated
partner for the kaon production) as a function of the normalized rapidity in
central Ni+Ni collisions at 1.93 A GeV. In our initialization of A+A
collisions, the projectile nucleus moving in $z$-direction is located at
$x=b/2$ and the target nucleus at $x=-b/2$, where $b$ is the impact parameter.
One can see that the average $p_{x}$ of the protons and $\Lambda$’s is
antisymmetric with respect to $y=0$ and shows a linear increase with rapidity
in line with the FOPI data.

The middle panel of Fig. 21 shows the directed flow $v_{1}$ of $K^{+}$ mesons
as a function of $p_{T}$ in central Ni+Ni collisions at the same energy.
Considering $y_{\rm lab}/y_{\rm c.m.}-1=(y/y_{\rm beam})_{\rm c.m.}$, where
$y_{c.m.}$ is the c.m. rapidity of two colliding heavy-ions, the directed flow
of nucleons within $-1.2<(y/y_{\rm beam})_{\rm c.m.}<-0.65$ is negative as
seen from the upper panel of Fig. 21. In fact, the directed flow $v_{1}$ of
the produced $K^{+}$ mesons - which is shown by the dotted line - is negative,
since they are produced by the scattering of nucleons. The dashed line
represents the $v_{1}$ of $K^{+}$ mesons after freeze-out (’survived’) without
kaon potential. The $v_{1}$ of $K^{+}$ mesons is only slightly changed due to
the interaction with nucleons which have a negative $v_{1}$. Including the
kaon potential, however, the $v_{1}$ of $K^{+}$ drastically changes and
becomes positive. This happens because the nucleons from the target nucleus,
which are dominantly located in $-x$ direction, push the $K^{+}$ in $+x$
direction. Similar phenomena occur at forward rapidity, where the nucleons
from the projectile nucleus push the $K^{+}$ in $-x$ direction. The effects of
the kaon potential are stronger for $K^{+}$ mesons with small transverse
momentum and weaker for those with large transverse momentum, since it is
harder to change the direction of $K^{+}$ mesons which have large momentum.
One can see that the experimental data are well explained if one includes the
kaon-nuclear potential in the calculations. The lower panel shows the $v_{1}$
of $K^{+}$ and $K^{-}$ as a function of rapidity. It demonstrates more clearly
the effect of the (anti)kaon potential on the directed flow. As explained
above the repulsive kaon potential pushes the directed flow of $K^{+}$ away
from that of nucleons while the attractive antikaon potential pulls the
directed flow of $K^{-}$ closer.

Figure 22: (Color online) Upper and middle plots: The flow coefficient $v_{2}$
of $K^{+}$ (upper) and $K^{-}$ (middle) mesons as a function of $p_{T}$ in
semi-central Au+Au collisions at 1.5 A GeV in comparison to the experimental
data from the KaoS Collaboration Ploskon:2005qr . The solid blue and dashed
red solid lines show the $v_{2}$ with and without medium effects,
respectively, while the green dotted lines represent $v_{2}$ at the production
point for the in-medium scenario. Lower plot: The $v_{2}$ of $K^{+}$ and
$K^{-}$ with and without medium effects as a function of rapidity in the same
collisions.

Figure 23: (Color online) the integrated $v_{2}$ of $K^{+}$ (blue lines) and
$K^{-}$ (red lines) as a function of time with (solid lines) and without
(dashed lines) medium effects in the same collisions as in figure 22. The
green dot-dashed line corresponds to the $v_{2}$ of nucleons.

Figure 22 shows the elliptic flow $v_{2}$ of $K^{+}$ (upper plot) and $K^{-}$
(middle plot) mesons as a function of $p_{T}$ and $v_{2}$ of $K^{+}$ and
$K^{-}$ as a function of rapidity $y$ (lower plot) in semi-central Au+Au
collisions at 1.5 A GeV in comparison to the experimental data from the KaoS
Collaboration Ploskon:2005qr . Since $0.3<y/y_{\rm beam}<0.7$ is equivalent to
$-0.2<(y/y_{\rm beam})_{c.m.}<0.2$, the shown (in the upper and middle plots)
$v_{2}$ corresponds to midrapidity. For a better understanding of the results
we display in Fig. 23 the integrated $v_{2}$ of $K^{+}$, $K^{-}$ mesons and
also nucleons as a function of time in the same rapidity range as well as the
same collision system. The dotted lines in the upper and middle panels are,
respectively, the $v_{2}$ of produced $K^{+}$ and $K^{-}$ mesons. Since
(anti)kaons are produced through the scattering of nucleons with other
nucleons or mesons, their $v_{2}$ is closely related to the $v_{2}$ of
nucleons. Considering that the baryon density is peaked around $t=10~{}{\rm
fm/c}$ for this centrality/energy and that most of the $K^{+}$ and $K^{-}$
mesons are produced before $t=15~{}{\rm fm/c}$, one can find that the $v_{2}$
of nucleons at the production time of (anti)kaons is positive. This explains
the positive $v_{2}$ of the produced $K^{+}$ and $K^{-}$ mesons. After that
the $v_{2}$ of nucleons decreases and changes sign due to the attractive
nuclear potential. Since the produced $K^{+}$ and $K^{-}$ interact with the
nucleons, their $v_{2}$ follows the $v_{2}$ of nucleons as shown by the blue
and red dashed lines in the lower panel, which, respectively, indicate the
$v_{2}$ of $K^{+}$ and of $K^{-}$ mesons considering scattering turned on, but
(anti)kaon-nuclear potential turned off. As a result, the final $v_{2}$ of
$K^{+}$ and of $K^{-}$ without the potential turn out negative as seen from
the upper and middle panels of Fig. 22. This is also well seen on the lower
panel of Fig. 22 which shows that $v_{2}$ of $K^{+}$ and $K^{-}$ without
medium effects is negative. An inclusion of the repulsive kaon potential leads
to a further reduction of the $K^{+}$ elliptic flow, while the in-medium
effects for $K^{-}$ lead to an enhancement of $v_{2}$, which fluctuates around
zero with the presently achieved numerical statistics.

The difference between the dashed line and solid line shows the effects of the
(anti)kaon-nuclear potential on the $v_{2}$ of $K^{+}$ and $K^{-}$ mesons. One
can see that the kaon potential - which is repulsive - shifts the $v_{2}$ of
$K^{+}$ mesons to a more negative value, while the antikaon potential - which
is attractive - moves the $v_{2}$ of $K^{-}$ mesons towards less negative
values. We note that the kaon-nuclear potential is necessary to reproduce the
experimental data from the KaoS Collaboration, but antikaon medium effects do
not help to explain the KaoS experimental data for the $v_{2}$ of $K^{-}$
mesons.

Figure 24: (Color online) The azimuthal angular distributions of $K^{+}$ in
semi-central Au+Au collisions at 1.5 A GeV (upper) and in semi-central Ni+Ni
collisions at 1.93 A GeV (lower) in comparison to the experimental data from
the KaoS Collaboration Uhlig:2004ue . The solid lines show the calculations
with medium effects, while the dashed lines display results without medium
effects.

Finally we show in Fig. 24 the azimuthal angular distribution of $K^{+}$ in
semi-central Au+Au collisions at 1.5 A GeV (upper plot) and in semi-central
Ni+Ni collisions at 1.93 A GeV (lower plot). Since the rapidity range is the
same as in Fig. 22, the contributions from $v_{1}$ are small and the
distributions are dominated by $v_{2}$. One can see that the $v_{2}$ of
$K^{+}$ mesons is negative both in Au+Au and Ni+Ni collisions and turning on
the kaon-nuclear potential enhances the negative flow, which is consistent
with the results for $v_{2}$ in Fig. 22. We note that an increase of the kaon-
nuclear potential (Eq. (40)) from +25 MeV at $\rho_{0}$ to a large value (e.g.
+35 MeV) leads to some small increase of the anisotropy in the azimuthal
angular distribution. However, the transverse momentum spectra of $K^{+}$
mesons become harder which is not supported by the experimental data. Though
the $v_{2}$ of $K^{+}$ mesons - with the default kaon-nuclear potential of +25
MeV at $\rho_{0}$ \- is smaller than with larger repulsive potential, our
results qualitatively agree with the experimental data from the KaoS
Collaboration Uhlig:2004ue . However, as seen from Fig. 24, the azimuthal
angular distribution depends only very modestly on the inclusion of the medium
effects for (anti)kaons.

###  VII.5 Multiplicity dependence on $A_{part}$ and on the EoS

The threshold kaon production in heavy-ion collisions is one of the sensitive
probes for the nuclear equation-of-state Aichelin:1986ss ; Fuchs ;
Hartnack:2011cn . In the present study we use a static Skyrme potential Skyrme
parameterized by

$\displaystyle
U(\rho)=a\bigg{(}\frac{\rho}{\rho_{0}}\bigg{)}+b\bigg{(}\frac{\rho}{\rho_{0}}\bigg{)}^{\gamma},$
(52)

where $a=-153$ MeV, $b=$ 98.8 MeV, $\gamma=$ 1.63. The parameters a,b,$\gamma$
are chosen to assure that the energy per nucleon has a minimum of
$\frac{E}{A}(\rho_{0})=-16$ MeV at $\rho_{0}$. The third condition to fix the
parameters is the choice of the nuclear compressibility
$\chi=\frac{1}{V}\frac{dV}{dP}$, where $P$ is the pressure and $V$ is the
volume of the system. Usually its inverse quantity - the compression modulus
$K$ \- is quoted Song:2015hua  which is defined as

$\displaystyle
K=-V\frac{dP}{dV}=9\rho^{2}\frac{\partial^{2}(E/A)}{\partial\rho^{2}}\bigg{|}_{\rho_{0}}$
(53)

A large compression modulus $K$ reflects that the nuclear matter can be hardly
compressed, thus the equation-of-state is called a ’hard’ EoS. Oppositely, a
small $K$ stands for a ’soft’ EoS. The soft EoS enhances the production of
kaons in heavy-ion collisions for two reasons: i) it allows for the formation
of dense nuclear matter where nucleons have more chances to collide and
produce kaons; ii) the nucleons loose less compression energy and the ’saved’
energy can be used for the production of kaons. Our set of parameters gives a
compression modulus $K\simeq 300$ MeV at saturation density, which is in
between the soft and hard equations-of-state, i.e. a ’middle-soft’ EoS. This
’middle-soft’ EoS provides an optimal description of the different observables
at SIS energies and, thus is used as ’default’ value in the PHSD for low
energy heavy-ion collisions. In order to investigate the sensitivity of the
$K^{+}$, $K^{-}$ production on the EoS, we have changed the EoS in PHSD to a
softer with $K=210$ MeV and to a harder with $K=380$ MeV. We found that the
softening of the EoS leads to an increase of the $K^{+}$ and $K^{-}$ yields by
$\sim 13$ % and $\sim 24$ %, respectively, in central Ni+Ni collisions at 1.93
GeV, while a harder EoS decreases them, respectively, by $\sim 14$ % and $\sim
19$ %. We mention that also momentum dependent Skyrme interaction have been
successfully employed to study the in medium properties of kaon Aichelin ;
Hartnack:2011cn .

The stiffness of the nuclear equation-of-state affects also the dependence of
kaon production on the centrality of heavy-ion collisions. The softer the
equation-of-state is, the more kaons are produced in central collisions
compared to semi-central and peripheral collisions Hartnack:2011cn . The
centrality dependence might be converted to the dependence on $A_{\rm part}$
being the number of participants. The multiplicity of (anti)kaons in heavy-ion
collisions is proportional to $M\sim(A_{\rm part})^{\alpha}$ where $\alpha$ is
a fitting parameter. The deviation from a scaling with $A_{part}$ ($\alpha=1$)
points towards ’collective production effects’: e.g. a formation of $K^{+}$ by
scattering of resonances such as $\Delta N\to N\Lambda K^{+}$ collisions. The
probability that a $\Delta$ interacts with nucleons and produces a kaon
(before it disintegrates to a pion and nucleon) increases with baryon density,
which is larger for the most central collisions. For pion production the
deviation from the linear $A_{part}$ dependence is not expected since, in
spite of multiple $\Delta$ decay and regeneration by pion-nucleon scattering,
the number of final pions is not much affected. Indeed, as found by the KaoS
Collaboration Forster:2007qk , in the case of pion production, the
experimental value $\alpha_{\pi}$ is compatible with unity,
($\alpha_{\pi}(Au)=0.96\pm 0.05$, $\alpha_{\pi}(Ni)=1.0\pm 0.05$). while for
kaons $\alpha_{K}(Au)=1.34\pm 0.16$, i.e. the kaon multiplicity rises stronger
than linear with $A_{part}$. A similar behaviour has been observed for
antikaons ($\alpha_{\bar{K}}(Au)=1.22\pm 0.27$) despite the different
production thresholds. This leads to constant ratio of $K-/K+$ versus
centrality. We note that the PHSD reproduces the KaoS results on the pion
value of $\alpha_{\pi}$ rather well.

Figure 25: (Color online) Multiplicities per mean number of participants
$A_{part}$ of $K^{+}$ (upper), $K^{-}$ (middle) and $\Lambda+\Sigma^{0}$
(lower) in Au+Au collisions at 1.5 A GeV. The solid red lines correspond to
the PHSD results (including in-medium effects) with default EoS, the blue
dashed line and green dotted show the results with a hard and soft EoS,
respectively. The grey dot-dashed lines indicate the fit of the PHSD results
with the default EoS (see text). The solid squares show the KaoS data for
$K^{+}$ taken from Ref. Forster:2007qk .

Figure 25 shows the PHSD results (including the medium effects) for the
multiplicities per mean number of participants $A_{part}$ of $K^{+}$ (upper),
$K^{-}$ (middle) and $\Lambda+\Sigma^{0}$ (lower) in Au+Au collisions at 1.5 A
GeV in comparison to the KaoS data for $K^{+}$ taken from Ref. Forster:2007qk
. The solid red lines represent the PHSD results with the default EoS, the
blue dashed line and green dotted show the results with a hard and soft EoS,
respectively. One can see a strong sensitivity of the multiplicity of kaons
and $\Lambda+\Sigma^{0}$ on the compression modulus of the EoS - the hard EoS
leads to a strong reduction of the multiplicity in central collisions while
the soft EoS to an enhancement of kaons and $\Lambda+\Sigma^{0}$. We note that
we obtain the best description of the experimental data with the default
’middle-soft’ EoS in the PHSD.

The grey dot-dashed lines indicate a fit of the PHSD results (with default
EoS) for the multiplicities as $M\sim(A_{\rm part})^{\alpha}$: we find for
kaons $\alpha_{K}^{PHSD}(\rm{Au})\simeq 1.34$ in agreement with the KaoS data.
Moreover, for $\Lambda+\Sigma^{0}$ –
$\alpha_{\Lambda+\Sigma^{0}}^{PHSD}(Au)\simeq 1.32$ which is almost the same
as for kaons. For the antikaons $\alpha_{K}^{PHSD}(\rm{Au})\simeq 1.4$ which
is a bit larger than for kaons, but still in the range of experimental
errorbars Forster:2007qk .

We stress that the change of the equation-of-state leads to a modification of
the whole dynamics of the heavy-ion collisions at low energies which is seen
in the ’bulk’ observables such as proton and pion rapidity and
$p_{T}$-spectra, angular distributions, flow harmonics $v_{1},v_{2}$ etc. The
strange hadrons are reflecting these changes since their production mechanisms
are tightly linked to the dynamics of non-strange baryons and pions,
especially at subthreshold energies considered in this study. Thus, in order
to pin down a robust information on the nuclear EoS from low energy
experimental data, one has to analyse of all observables - for strange and for
non-strange hadrons - simultaneously.

Figure 26: (Color online) The PHSD results for the $A_{part}$ dependence of
the multiplicity over averaged $A_{part}$ of $\Lambda+\Sigma^{0}$ (blue solid
line), $K^{+}$ (green dot-dot-dashed line), $K^{0}_{s}=(K^{0}+\bar{K}^{0})/2$
(red dot-dashed line) and $K^{-}$ (violet dashed line, multiplied by 40) in
comparison to the experimental data from the HADES Collaboration Adamczewski-
Musch:2018xwg .

As follows from Fig. 25 the PHSD reproduces well the $A_{part}$ dependence
$K^{+}$ for the Au+Au at 1.5 A GeV. Now we step down in energy and come to
sub-threshold $K^{+}$ production in Au+Au collisions at 1.23 A GeV. In Fig. 26
we show the PHSD results for the $A_{part}$ dependence of the multiplicity
over averaged $A_{part}$ of $\Lambda+\Sigma^{0}$ (blue solid line), $K^{+}$
(green dot-dot-dashed line), $K^{0}_{s}=(K^{0}+\bar{K}^{0})/2$ (red dot-dashed
line) and $K^{-}$ (violet dashed line, multiplied by 40) in comparison to the
experimental data from the HADES Collaboration Adamczewski-Musch:2018xwg . One
can see that the $\Lambda+\Sigma^{0}$ and $K^{+}$ data are reasonable well
described by PHSD. However, as expected, we see the same tension of the PHSD
results for the $K^{0}_{s}$ and $K^{-}$ with the HADES data as in Figs. 10 and
14: the $K^{0}_{S}$ data are overestimated by 20% and $K^{-}$ data are