# Medium modification of pion production in low energy Au+Au collisions

C. Kummer<EMAIL_ADDRESS>Institut für Theoretische
Physik, Universität Giessen, Giessen, Germany  K. Gallmeister  Institut für
Theoretische Physik, Universität Giessen, Giessen, Germany  Helmholtz Research
Academy Hesse for FAIR (HFHF), Campus Giessen, Giessen, Germany  L. von Smekal
Institut für Theoretische Physik, Universität Giessen, Giessen, Germany
Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Giessen, Giessen,
Germany

###### Abstract

There is a major mismatch between the charged pion yields in Au+Au collisions
at low energies calculated by various transport models and the experimental
measured values from the HADES collaboration. In this work, reasonable
improvements on the equation of state, in-medium modification of cross
sections, and the influence of the nuclear potential for $\Delta$ resonances
will be investigated in the framework of the GiBUU transport model. As a
result, we demonstrate that theoretical calculations can indeed describe the
charged pion yields measured by HADES for Au$+$Au collisions rather well, but
that a mismatch then remains between calculations and data for the yields of
neutral pions extracted from dileptons within the same experiment.

##  I Introduction

In the paper by Adamczewski-Musch _et al._ Adamczewski-Musch  _et al._ (2020)
the HADES collaboration showed that transport models systematically
overpredict measured pion yields. Every transport code was overshooting the
rapidity and $p_{t}$ spectra by nearly a factor of 2. Such a failure is
already known since almost 30 years Bass  _et al._ (1995); Teis  _et al._
(1997); Uma Maheswari  _et al._ (1998); Pelte  _et al._ (1997); Larionov  _et
al._ (2001); Larionov and Mosel (2003). This is disturbing for a few reasons.
Due to their low mass, pions play an important role in many theoretical models
Ericson and Weise (1988). Assuming two flavors of massless quarks, for
example, QCD exhibits a chiral symmetry which, when spontaneously broken,
gives rise to three massless Nambu-Goldstone bosons. These are typically
identified with the pions to explain why they are the lightest hadrons, and
hence are commonly produced more abundantly than others in heavy-ion
collisions. Thus pion production is important from an experimental as well as
a theoretical point of view. For the system considered by HADES in Ref.
Adamczewski-Musch  _et al._ (2020), Au+Au at an incident energy of $E_{\rm
kin}=1.23\,A\,{\rm Ge\kern-0.6458ptV}$, a mismatch between measured and
calculated pion multiplicities by not quite a factor of two but on the
50$\,{\rm\%}$ level was reported in Ref. Larionov  _et al._ (2021).
Nevertheless, one has to require that transport models reproduce the most
frequent particles better than that. It is therefore most important to develop
a framework with some theoretical footing that can describe the HADES
findings.

As the conventional assumptions and modes typically employed in transport
codes failed to reproduce the data, new and creative ways have to be adapted.
Indeed, Godbey _et al._ Godbey  _et al._ (2022) found a prescription to
reproduce the correct pion numbers, whereby a density-dependent suppression
was applied on the cross section for each charged pion. While this method is
without a doubt successful, some questions still remain open. For once, it is
unclear why an isospin dependent factor needs to be introduced in an otherwise
isosymmetric theory. Furthermore the exponential suppression itself is an
instrument to artificially lower hadron numbers. Its theoretical foundations
need to be studied, but for now its phenomenological effectiveness will be
appreciated by us as well.

Another approach lies in the modification of potentials and cross sections.
The argument for these implementations comes from the assumption, that the
emergence of scalar and vector fields in bulk matter changes the behaviour of
hadrons compared to vacuum. Admittedly some of the in-medium modifications
performed are controversial throughout the community, but they have some
backing in literature.

Furthermore, we argue in favor of using the Liu equation of state instead of
the commonly used NL2 by Lang. This claim is not just supported by our
results, but also by the astronomic constraints obtained from neutron star
mergers.

The structure of this letter is the following. In section II the relativistic
mean field method of the Giessen Boltzmann-Uehling-Uhlenbeck (GiBUU) model is
explained. Most information about it can be found in the review paper Ref.
Buss  _et al._ (2012), however, it seems appropriate to elaborate on the
assumptions and implementations made in this project. A focus is then given to
the equation of state by Liu _et al._ Liu  _et al._ (2002) and on the in-
medium modifications. How the modifications affect particle production and
baryon densities is explored in section III. Then in section IV a summary of
the results shows how different modifications compare to experiments. These
are the FOPI experiment for proton and pion data, and HADES, where the pions
and dielectrons are being looked at. A summary of our findings is given in
section V.

##  II Model

###  II.1 RMF

In the present work, the propagation of particles is carried out using a
relativistic mean field (RMF) for the equations of motion. Its workings are
elaborated here, as it is less common than non-relativistic Skyrme-like
potentials typically used in heavy-ion collisions. More details can be found
in the GiBUU review paper Buss  _et al._ (2012). The Lagrangian is given by

$\mathcal{L}=\overline{\psi}[\gamma_{\mu}(i\partial^{\mu}-g_{\omega}\omega^{\mu}-g_{\rho}\boldsymbol{\tau}\boldsymbol{\rho}^{\,\mu}-\frac{e}{2}(1+\tau^{3})A^{\mu})-m_{N}-g_{\sigma}\sigma]\psi+\frac{1}{2}\partial_{\mu}\sigma\partial^{\mu}\sigma-U(\sigma)+\frac{1}{2}m_{\omega}^{2}\omega^{2}+\frac{1}{2}m_{\rho}^{2}\boldsymbol{\rho}^{\,2}-\frac{1}{16\pi}F_{\mu\nu}F^{\mu\nu}~{},$
(1)

with the Dirac-spinor $\psi$, and the fields $\sigma$, $\omega$, and
$\boldsymbol{\rho}$. The meson-nucleon couplings
$g_{\omega},g_{\rho},g_{\sigma}$ and meson mass $m_{\sigma}$ in the Lagrangian
eq. 1 are determined by the equation of state (EOS) at hand, while the masses
$m_{\rho},m_{\omega},m_{N}$ for rho-meson, omega-meson and nucleon are fixed
at their conventional value. In usual notations $\boldsymbol{\tau}$ are the
Pauli matrices, $A^{\mu}$ is the electromagnetic field and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ the corresponding
field tensor. The scalar field self interaction is given by

$U(\sigma)=\frac{1}{2}m_{\sigma}^{2}\sigma^{2}+\frac{1}{3}g_{2}\sigma^{3}+\frac{1}{4}g_{3}\sigma^{4}\,,$
(2)

with additional coefficients $g_{2}$ and $g_{3}$. The Lagrangian eq. 1 leads
to the equations of motion for the Dirac-spinor $\psi$,

$\displaystyle\Big{[}\gamma_{\mu}\Big{(}i\partial^{\mu}-g_{\omega}\omega^{\mu}-g_{\rho}\boldsymbol{\tau}\boldsymbol{\rho}^{\,\mu}-\frac{e}{2}(1+\tau^{3})A^{\mu}\Big{)}$
$\displaystyle-m_{N}-g_{\sigma}\sigma\Big{]}\psi$ $\displaystyle=0\ ,$ (3)

for the isoscalar-scalar field $\sigma$,

$\displaystyle\partial_{\mu}\partial^{\mu}\sigma+\frac{\partial
U(\sigma)}{\partial\sigma}$ $\displaystyle=-g_{\sigma}\rho_{S}\ ,$ (4) for the
isoscalar-vector field $\omega$, $\displaystyle m_{\omega}^{2}\omega^{\nu}$
$\displaystyle=\ g_{\omega}j^{\nu}_{b}\ ,$ (5) for the isovector-vector field
$\boldsymbol{\rho}$, $\displaystyle m_{\rho}^{2}\boldsymbol{\rho}^{\nu}$
$\displaystyle=\ g_{\rho}\boldsymbol{j}_{I}^{\nu}\ ,$ (6) and for the
electromagnetic potential $A$,
$\displaystyle\partial_{\mu}\partial^{\mu}A^{\nu}$ $\displaystyle=\ 4\pi
ej_{c}^{\nu}\ .$ (7)

To be clear, there is no spinor implemented in GiBUU as in eq. 3. Instead,
GiBUU uses the particle distribution functions $f_{i}(x,\boldsymbol{p})$ for
this evaluation. These $f$ further define the r.h.s’s of the equations of
motion which are the scalar density (for $p^{*}$ and $m^{*}$ see below),

$\displaystyle\rho_{S}$
$\displaystyle=\frac{g}{(2\pi)^{3}}\sum_{i=p,n,\bar{p},\bar{n}}\int\,\frac{\mathrm{d}^{3}p}{p_{i}^{*\,0}}\,m_{N}^{*}\,f_{i}(x,\boldsymbol{p})\
,$ (8) the baryon density, $\displaystyle j^{\mu}_{b}$
$\displaystyle=\frac{g}{(2\pi)^{3}}\Bigg{(}\sum_{i=p,n}-\sum_{i=\bar{p},\bar{n}}\Bigg{)}\int\,\frac{\mathrm{d}^{3}p}{p_{i}^{*\,0}}\,p_{i}^{*\mu}\,f_{i}(x,\boldsymbol{p})\,$
(9) the isospin current, $\displaystyle\boldsymbol{j}_{I}^{\mu}$
$\displaystyle=\frac{g}{(2\pi)^{3}}\sum_{i=p,n,\bar{p},\bar{n}}\int\,\frac{\mathrm{d}^{3}p}{p_{i}^{*\,0}}\,p_{i}^{*\mu}\,\boldsymbol{\tau}\,f_{i}(x,\boldsymbol{p})~{},$
(10) and the charge current, $\displaystyle j_{c}^{\mu}$
$\displaystyle=\frac{1}{2}(j_{b}^{\mu}+j_{I}^{3,\mu})\ .$ (11)

Here $g$ accounts for the spin-degeneracy and has a numerical value of 2. In
principle there should also be partial derivatives of $\omega$ in eq. 5, but
in practice they are neglected, as it is a short-ranged field. Thus the
$\omega$-field is proportional to the baryon current just as stated in eq. 5.
Derivatives in eq. 6 are neglected for the same reason as well, in addition to
the isospin-mixed nucleon states. Hence the first two components are set to
zero, i.e. $j_{I}^{1,\nu}=j_{I}^{2,\nu}=\rho^{1,\nu}=\rho^{2,\nu}=0$, and only
$\rho^{3,\nu}$ and $j_{I}^{3,\nu}$ need to be calculated. Like $\omega$, the
$\boldsymbol{\rho}$-field is proportional to a current. Finally in the last
equation the electromagnetic current $j_{c}^{\nu}$ is related to the
electromagnetic field. Useful kinematic quantities are the kinetic four-
momentum

$\displaystyle p^{*}$ $\displaystyle=p-V$ (12) and the effective nucleon mass
$\displaystyle m_{N}^{*}$ $\displaystyle=m_{N}+S\ .$ (13)

The scalar and vector fields are given by

$\displaystyle V^{\nu}$
$\displaystyle=\,g_{\omega}\omega^{\nu}+g_{\rho}\tau^{3}\rho^{3,\nu}+\frac{e}{2}(1+\tau^{3})A^{\nu}\,,$
(14) $\displaystyle S$ $\displaystyle=\,g_{\sigma}\sigma\,.$ (15)

By assuming a plane wave solution to the Dirac equation, eq. 3, one obtains
the dispersion relation for the nucleons in the form

$(p^{*})^{2}\,-\,(m_{N}^{*})^{2}\,=\,0\,.$ (16)

Albeit the above discussion was only for nucleons, this RMF prescription will
be applied to all baryons. The Coulomb potential is effective for all particle
species.

###  II.2 Liu EOS

An EOS relates observables to the nuclear density. They are always fixed at a
binding energy per nucleon of $E/A=-16\,{\rm Me\kern-0.6458ptV}$ at the
saturation density $\rho_{0}=0.16\,{\rm fm}^{-3}$. In other regards they can
and do differ significantly. This can be explained by the theoretical
backgrounds of the researchers responsible for them. For example, to study
heavy-ion collisions A. Lang _et al._ Lang  _et al._ (1992) devised parameter
sets suitable for this task, while G.A. Lalazissis _et al._ Lalazissis  _et
al._ (1997) needed a different set to work on nuclear structure models and B.
Liu _et al._ Liu  _et al._ (2002) investigated the influence of isovector
scalar fields with another EOS. Hence the EOSs by Lang are typically more
applicable for a dynamic description of heavy-ion collisions, while Lalazissis
deals better with nuclear ground states. The EOS of Liu _et al._ is also used
by Godbey _et al._ Godbey  _et al._ (2022).

Astronomic observations on neutron star mergers give some constraints on the
EOS, an extensive review which summarizes those was done by Li _et al._ Li
_et al._ (2021). For our purposes the most relevant work is a study by Xie _et
al._ Xie and Li (2020), where restraints for possible potentials bounded by
those astronomical constraints is provided. This is shown in fig. 1, where the
binding energy per nucleon and the effective mass of the nucleons are given as
a function for the normalized density for various EOSs.

Figure 1: The binding energy per nucleon and the effective mass as a function
of nuclear density for various EoSs Lalazissis  _et al._ (1997); Lang  _et
al._ (1992); Liu  _et al._ (2002). The red band in the plot of the binding
energies shows the estimates of the study by Xie _et al._ Xie and Li (2020).

Constraints apply to the binding energy, where a band shows the possible
behaviour of the EOS as allowed by astronomical observations. Right in the
middle lies the EOS by Liu, which we will consider from now on. NL2 by Lang,
commonly used in transport codes, has a similar binding energy and lies on the
very edge of the band, but the ratio $m^{*}/m$ is dropping off a lot weaker
than Liu EOS. In case there is a considerable disagreement between results
obtained with Liu and NL2 by Lang, we will showcase it.

The coefficients from eq. 1 required to reproduce the EOSs by Liu and Lang,
are given in table 1.

EOS | K | $m_{N}^{*}/m_{N}$ | $g_{\sigma}$ | $g_{\omega}$ | $g_{\rho}$ | $g_{2}$ | $g_{3}$ | $m_{\sigma}$
---|---|---|---|---|---|---|---|---
| $\,{\rm Me\kern-0.6458ptV}$ |  |  |  |  | $\,{\rm Ge\kern-0.6458ptV}$ |  | $\,{\rm Me\kern-0.6458ptV}$
Liu | 240 | 0.75 | 8.958 | 9.238 | 3.769 | -4.681 | -30.909 | 550
Lang, NL2 | 210 | 0.83 | 8.5 | 7.54 | 0.0 | -9.939 | -6.26 | 550.5
Table 1:  Parameter sets for Liu and NL2 by Lang EOS’s and their saturation
properties in terms of the compression modulus K and the effective mass
$m_{N}^{*}$ in units of the bare nucleon mass $m_{N}$.

###  II.3 Cross Section Modification

As mentioned above, tranport theoretical calculations overestimate the
experimental measured pion yields in heavy ion collisions. There are multiple
handles to change this.

####  II.3.1 Exponential Suppression

One way to reduce pion yields is the exponential suppression of cross sections
in the spirit of Refs Song and Ko (2015); Godbey  _et al._ (2022). Here the
$NN\to N\Delta$ and $NN\to NN\pi$ cross sections are multiplied by a factor

$f=\exp\bigg{(}-\alpha\,\bigg{(}\frac{\rho}{\rho_{0}}\bigg{)}^{\beta}\
\bigg{)}\ ,$ (17)

with constants $\alpha$ and $\beta$, where the factor $\beta$ was introduced
here to have a more general expression. The underlying concept is that pion
distributions can be expressed with this density dependent suppression. In our
investigation we saw that leaving $\beta$ at the default value 1 is
sufficient, the data can be described fine just with the factor $\alpha$. To
maintain detailed balance, the factor $f$ is also included in the cross
section of the back reaction $N\Delta\to NN$ and in the rate of pion
absorption by two nucleons $\pi NN\to NN$. Hereby two dominant channels of the
pion production get affected. This method of medium modification is not
perfect, as pions get produced by other processes, mainly $NN\to NR$, where
$R$ is a resonance, at increased rates, but it is an effective method to
reduce the overall pion numbers.

####  II.3.2 Effective Mass

A second possibility for medium modifications is given by the effective mass.

To calculate heavy-ion collisions effectively, one needs to incorporate cross
sections for the myriad of different events. One can derive from QFT that the
differential cross sections in vacuum are given by

$\mathrm{d}\sigma_{12\to 1^{\prime}2^{\prime}\dots
N^{\prime}}=(2\pi)^{4}\delta^{(4)}\left(p_{1}+p_{2}-\sum_{i=1^{\prime}}^{N^{\prime}}p_{i}\right)\frac{n_{1}n_{2}\prod_{i=1^{\prime}}^{N^{\prime}}n_{i}}{4I_{12}}\overline{|\mathfrak{M}_{12\to
1^{\prime}2^{\prime}\dots N^{\prime}}|^{2}}\
\mathcal{S}_{1^{\prime}2^{\prime}\dots
N^{\prime}}\prod_{i=1^{\prime}}^{N^{\prime}}A_{i}(p_{i})\frac{\mathrm{d}^{4}p_{i}}{(2\pi)^{3}2p_{i}^{0}}~{},$
(18)

where the symmetry factor

$\mathcal{S}_{ab}=\begin{cases}1&\text{if $a$ and $b$ are not identical},\\\
\frac{1}{2}&\text{if $a$ and $b$ are identical}\,,\end{cases}$ (19)

takes into account that one cannot differentiate between two elementary
particles of the same kind and

$I_{12}\,=\,\sqrt{(p_{1}p_{2})^{2}-(m_{1}m_{2})^{2}}$ (20)

is the flux factor. The spin-averaged spectral function is defined as

$A(x,p)=-\frac{1}{g\pi}\text{tr}\Big{[}\text{Im}(\tilde{S}^{\text{ret}}(x,p))\gamma^{0}\Big{]}$
(21)

with Dirac matrix $\gamma^{0}$ and the retarded Green’s function
$\tilde{S}^{\text{ret}}(x,p)$. $\mathfrak{M}$ is the matrix element used in
the Bjorken-Drell convention, which is related to the PDG convention by

$\mathcal{M}_{if}=\mathfrak{M}_{if}\prod_{j}\sqrt{n_{j}}\ ,\quad
n_{j}=\begin{cases}1&\text{$j$ is boson},\\\ 2m_{j}&\text{$j$ is
fermion}~{}.\end{cases}$ (22)

Having said that, it is important to keep in mind, that not every cross
section can be calculated like that from first principles. Many have to be
taken from experiment and many arise from educated guesses as there is not
enough data or underlying theory to proceed otherwise.

For processes under investigation, particle number and density are not close
to vacuum anymore. Hence it is plausible to allow some in-medium
modifications. There are many options to apply this modifications, the
prescription described here is the cross section modification by the effective
mass. The in-medium cross section can be given by Buss  _et al._ (2012)

$\mathrm{d}\sigma^{*}_{12\to 1^{\prime}2^{\prime}\dots
N^{\prime}}=(2\pi)^{4}\delta^{(4)}\left(p_{1}+p_{2}-\sum_{i=1^{\prime}}^{N^{\prime}}p_{i}\right)\frac{n_{1}^{*}n_{2}^{*}\prod_{i=1^{\prime}}^{N^{\prime}}n_{i}^{*}}{4I_{12}^{*}}\overline{|\mathfrak{M}_{12\to
1^{\prime}2^{\prime}\dots N^{\prime}}|^{2}}\
\mathcal{S}_{1^{\prime}2^{\prime}\dots
N^{\prime}}\prod_{i=1^{\prime}}^{N^{\prime}}A_{i}(p_{i})\frac{\mathrm{d}^{4}p_{i}}{(2\pi)^{3}2p_{i}^{*0}}~{},$
(23)

with the in-medium flux factor

$I^{*}_{12}\,=\,\sqrt{(p^{*}_{1}p^{*}_{2})^{2}-(m_{1}^{*}m^{*}_{2})^{2}}\,.$
(24)

As one can see, eq. 23 is almost identical to eq. 18, except that effective
masses and momenta appear. ($n_{j}^{*}$ is defined as $n_{j}$, but also with
the effective mass.) In fact, this is why the Bjorken-Drell convention is
used: The matrix element does not need to be in-medium modified, only the
other factors are changed. However the delta function has to have the actual
four momentum of the particles, as that is required by energy and momentum
conservation.

The Bjorken-Drell convention of the matrix element leads to another problem,
as it depends on the free c.m. energy of the system and thus the free four-
momenta of colliding particles, while only the kinetic four-momenta are
available in the medium. In other words the existing potentials have to be
taken into account when calculating the available energy of the system. Thus
the c.m. energy has to be modified as well. One option, which many transport
codes (including GiBUU in Skyrme-like mode) use for hadron-hadron reactions,
is with the so-called free c.m. energy

$s_{\text{free}}\,=\,(p_{1,\text{free}}+p_{2,\text{free}})^{2}$ (25)

with free momenta

$p_{\text{free}}=(\sqrt{m^{2}+\boldsymbol{p}^{2}},\boldsymbol{p})\,.$ (26)

This prescription assumes that the potential acts as a background field, which
does not affect reaction rates. A drawback of this method is the disregard of
the in-medium threshold for particle production. Consequently a different
prescription is used, namely

$\sqrt{s_{\text{free}}}\,=\,\sqrt{s^{*}}-(m_{1}^{*}-m_{1})-(m^{*}_{2}-m_{2})$
(27)

with $s^{*}=(p_{1}^{*}+p_{2}^{*})^{2}$. To show that this prescription
conserves in-medium threshold, one has to start with the assumption, that the
sum of vector fields $V$ stays constant pre- and post collision, meaning
$V_{1}+V_{2}=\sum_{i=1^{\prime}}^{N^{\prime}}V_{i}$. Then one can replace the
four momenta in the delta-function of eq. 23 by the kinetic momenta. This is
done in the GiBUU code mainly due to technical reasons. Accordingly the cross
section becomes proportional to the $N$-body phase-space volume element,

$\mathrm{d}\sigma^{*}_{12\to 1^{\prime}\dots
N^{\prime}}\propto\mathrm{d}\Phi_{N}(p_{1}^{*}+p_{2}^{*};p_{1^{\prime}}^{*},\dots,p_{N^{\prime}}^{*})\,,$
(28)

when we assume the outgoing particles are on their Dirac mass shell defined in
eq. 16. The phase space volume element is defined by

$\displaystyle\mathrm{d}\Phi_{N}(P;p_{1},\dots,p_{N})$
$\displaystyle=\frac{\mathrm{d}^{3}\boldsymbol{p}_{1}}{(2\pi)^{3}2p_{1}^{0}}\dots\frac{\mathrm{d}^{3}\boldsymbol{p}_{N}}{(2\pi)^{3}2p_{N}^{0}}$
(29) $\displaystyle\quad\times\ \delta^{(4)}(P-p_{1}-\dots-p_{N})\,.$

Now one can define the in-medium excess energy
$Q^{*}=\sqrt{s^{*}}-\sum_{i=1^{\prime}}^{N^{\prime}}m_{i}^{*}$ and the in-
medium threshold condition

$Q^{*}>0$ (30)

follows immediately from eq. 28. Hence it makes sense to define the in-medium
invariant energy as

$\sqrt{s_{\text{free}}}=Q^{*}+\sum_{i=1^{\prime}}^{N^{\prime}}m_{i}=\sqrt{s^{*}}-\sum_{i=1^{\prime}}^{N^{\prime}}(m_{i}^{*}-m_{i})\,.$
(31)

Nevertheless this last equation is hard to compute, as the final particles are
not known initially, but produced during the collision. To overcome this
problem another assumption is made, namely that the sum of initial scalar
fields equals the sum of final scalar fields, e.g.
$S_{1}+S_{2}=\sum_{i=1^{\prime}}^{N^{\prime}}S_{i}$. This allows to rewrite
eq. 31 into eq. 27. The assumptions made in this derivation, namely
$V_{1}+V_{2}=\sum_{i=1^{\prime}}^{N^{\prime}}V_{i}$ and
$S_{1}+S_{2}=\sum_{i=1^{\prime}}^{N^{\prime}}S_{i}$, are usually fulfilled in
present simulations, as all baryon-meson couplings are set equal to the
nucleon-meson coupling and the action of the mean-field on mesons is
neglected. Cases when the assumptions do not hold are always explicitly given
and explained below.

All in all it is now possible to calculate the in-medium cross sections for
different final states of the baryon-baryon collisions. If the final particles
are assumed to be on their mass shell, the relation between in-medium and
vacuum cross sections is given by

$\sigma^{*}_{12\to 1^{\prime}2^{\prime}\dots
N^{\prime}}(\sqrt{s^{*}})=\mathcal{F}\,\sigma^{\text{vac}}_{12\to
1^{\prime}2^{\prime}\dots N^{\prime}}(\sqrt{s_{\text{free}}})$ (32)

with free center of mass (c.m.) energy given by equation eq. 27 and the
modification factor

$\mathcal{F}=\frac{n_{1}^{*}n_{2}^{*}n_{1^{\prime}}^{*}\dots
n_{N^{\prime}}^{*}}{n_{1}n_{2}n_{1^{\prime}}\dots
n_{N^{\prime}}}\frac{I_{12}}{I_{12}^{*}}\frac{\Phi_{N^{\prime}}(\sqrt{s^{*}};m_{1^{\prime}}^{*},\dots,m_{N^{\prime}}^{*})}{\Phi_{N^{\prime}}(\sqrt{s_{\text{free}}};m_{1^{\prime}},\dots,m_{N^{\prime}})}\,.$
(33)

It contains all terms of eq. 23 which do depend on the kinetic momenta and
effective mass. The $N$-body phase space volume is the integration of the
infinitesimal element eq. 29,

$\Phi_{N}(M;m_{1},...,m_{N})=\int\mathrm{d}\Phi_{N}(P;p_{1},...,p_{N})\ ,$
(34)

with the mass shell condition $p_{i}^{2}=m_{i}^{2}$ and $P^{2}=M^{2}$. In the
case of baryon-baryon scattering, eq. 33 takes the form

$\mathcal{F}\propto\frac{m_{1}^{*}m_{2}^{*}}{m_{1}m_{2}}\
\prod_{i=1^{\prime}}^{N^{\prime}}\frac{m_{i}^{*}}{m_{i}}$ (35)

which shows, that the ratio of the effective (Dirac) mass to the bare mass is
of importance for the cross section modification.

Not all cross sections need to be modified. There is some theoretical backing
Ter Haar and Malfliet (1987); Li and Machleidt (1993, 1994) in only modifying
inelastic cross sections and leaving the elastic channels unmodified. Assuming
that $NN\leftrightarrow N\Delta$ and $NN\leftrightarrow NN\pi$ reactions would
dominate the inelastic collisions, these cross sections were modified, as in
section II.3.1. Cross sections involving higher resonances can be left
unmodified, as it can be assumed, that they are governed by a short-range
interaction (heavy meson exchange), and, thus, in-medium effects can be
neglected.

####  II.3.3 Delta Potential Modifications

Another aspect that possibly needs to be refined is the baryon potential. The
free c.m. energy of eq. 27 can be rewritten into

$\sqrt{s_{\text{free}}}=\sqrt{s^{*}}-S_{\text{fin}}\ ,$ (36)

where $S_{\text{fin}}$ is the sum of scalar potentials acting on the final
particles. $\sqrt{s^{*}}$ is the invariant energy of the colliding particles
including their vector potentials. According to phenomenological observation
Mosel (2020) the potential energy of the delta is

$U_{\Delta}\simeq-30\,{\rm Me\kern-0.6458ptV}\simeq\frac{2}{3}\,U_{N}\,.$ (37)

Thus by reducing the scalar and vector potentials of the $\Delta(1232)$ to two
thirds the nucleons potential, the production of deltas is penalized compared
to the default of using the nucleons potential. In the spirit of Kosov _et
al._ Kosov  _et al._ (1998), these modifications are applied by changing the
couplings to the scalar and vector fields. Subsequently the ratios

$r_{S}=\frac{g_{S}^{\prime}}{g_{S}}\ \text{ and }\
r_{V}=\frac{g_{V}^{\prime}}{g_{V}}$ (38)

are defined, where $g_{S}$ is the nucleons coupling to the scalar field $S$,
$g_{V}$ the nucleons coupling to the vector field $V$, and $g_{S}^{\prime}$
and $g_{V}^{\prime}$ are the corresponding couplings of the $\Delta$.

Unfortunately, there is an issue with this prescription. As mentioned above,
the cross sections are not computed really by eq. 23, instead the Dirac delta
function evaluates the kinetic 4-momentum $p^{*}$ due to technical reasons.
This causes no problems, when $r_{V}=1$, but changing this coupling does
violate momentum conservation for the $\Delta$. The question of how to
quantify this effect is left for future studies. Having said that, it is also
possible to only modify the scalar potential, but then the condition eq. 37
requires $r_{S}=0.878$ to keep $U_{\Delta}\simeq-30\,{\rm Me\kern-0.6458ptV}$.
We will not discuss this possibility in this work.

##  III Model Predictions

In our simulations we consider the EOS by Liu with vacuum cross sections and
in-medium modifications. In the following shown figures, denoted with
"$\alpha=2.0$" is the exponential suppression with the factor $\alpha=2.0$
from eq. 17. Modifications via Dirac mass with the prescription defined in
section II.3.2 for the processes $NN\leftrightarrow N\Delta$ and
$NN\leftrightarrow NN\pi$ are denoted by "$\sigma^{*}_{\Delta,\pi}$". The
label "$r_{S,V}$" stands for modified coupling to scalar and vector field as
described in section II.3.3 with the default value $r_{S,V}=\frac{2}{3}$. We
are interested in how medium modifications alter nuclear densities and
collision rates. Thus a few simulations at the HADES incident energy of
$1.23\,A{\rm Ge\kern-0.6458ptV}$ with zero impact parameter were run. Here we
expect the largest impact of the modifications, as less central events have
less participants and more spectators and thus water down results.

Figure 2: Particle numbers as a function of time. Note the different scales
for different particles and how the $y$-axis for the nucleons does not start
at zero.

The total number of nucleons, deltas and pions is shown in fig. 2. The number
of nucleons decreases through collisions, as other hadrons are being produced
by inelastic processes, before back reactions and particle decays lead again
to an increase of nucleon numbers. Deltas are the resonances with the lowest
mass and are produced more abundantly than others. Further they are of great
importance to pion production, so they deserve some investigation. As they are
not present in the nuclei at the beginning, they have to be created during
collisions. Thus their numbers increase starting from zero, until decays and
back reactions terminate almost all $\Delta$ at the end of the simulation. The
medium modifications are effective and lower production and consequently the
yield of deltas. The nucleon numbers also reflect this behaviour: If less
$\Delta$ are being produced, which after all is the most dominant resonance,
more nucleons remain and the dip in fig. 2 is less pronounced. However, the
difference gets somewhat mitigated by the enhanced production of higher
resonances. For the time frames considered in heavy ion collisions pions can
be assumed to be stable particles, which can only disappear by inelastic
collisions. Their production vastly outweighs such processes, so it is no
surprise that pion numbers only increase over time. Just as for the deltas,
the modifications lower yields. Furthermore they change the timing of delta
and pion formation. Compared to the modification via effective mass and
potential the exponential suppression has its delta peak delayed by $1.6\,{\rm
fm}/c$. This is due to the pion capture $\pi N\to\Delta$, so the pion numbers
are decreased in favor of the deltas. The modification of the potential
coupling makes it less likely for deltas to form, so the pion capture is
slightly hindered. In the end deltas decay into pions, $\Delta\to\pi N$, and
the exponential suppression gives a $2.6\,{\rm fm}/c$ delay for pion numbers
compared to the other modification.

The central baryon density is calculated at the center of mass of the system.
So at the start of the simulations the nuclei are separated and the density is
far below the nuclear density of $\rho_{0}=0.16\,{\rm fm}^{-3}$. As the nuclei
collide the density increases rapidly up to a maximum at $t=12\,{\rm fm}/c$,
when the centers of mass from both nuclei overlap. Then the density decreases
until it asymptotically reaches zero. This process is depicted in fig. 3.

Figure 3: Central baryon density in units of the nuclear density
$\rho_{0}=0.16\,{\rm fm}^{-3}$ throughout the collision. The vertical line at
$t=12\,{\rm fm}/c$ corresponds to the moment of overlap of both nuclei, thus
the moment of maximum density.

Note that density is not symmetric to the maximum due to stopping which slows
particles down and causes the expansion to be slower than the compression. The
medium modifications lower the central density significantly from
$2.82\,\rho/\rho_{0}$ without modifications to $2.54\,\rho/\rho_{0}$ with
exponential suppression, a $10\,{\rm\%}$ decrease. An explanation lies in the
reduced cross sections due to the modifications, which causes less stopping to
occur. Less stopping means that the nuclei become more transparent for the
nucleons in the reaction so they move past each other instead of accumulating
in the center region and increasing the density. The effects of the medium
modifications on stopping are discussed in the section below as well.

At the beginning of the collision only nucleons are present, all other hadrons
are being produced in the reaction by collisions. So studying the collision
rates is of utmost importance for understanding the results. The net collision
rates, i.e. "reaction - back reaction" for the modified processes is shown in
fig. 4.

Figure 4: Net collision rates for $NN\to N\Delta$ and $NN\to NN\pi$.

Compared to vacuum cross sections medium modifications, both effective mass
and exponential suppression, lower the production of $\Delta$ and $\pi$
significantly. The exponential suppression is in fact almost completely
ceasing these processes. Now the question arises how total pion numbers can be
similar between both modification scenarios, if the exponential suppression is
so much stronger. The answer lies in the production of (heavier) resonances.
As fig. 5 shows, both modifications increase the production of resonances
higher than $\Delta$, which are named $R$.

Figure 5: Net collision rates for $NN\to NR$ and $NN\to\Delta\Delta$.

This is a direct result of the modifications, as more nucleons are present to
react and the production of resonances becomes more likely if the previously
dominant reaction channel diminishes. The double delta production is more
interesting. The effective mass and coupling modification hinders this
reaction again, but this is entirely due to the altered coupling. The
effective mass modification enlarges the production rate slightly, but not
nearly as much as the exponential suppression. The reason for the increase is
the same as for the resonance production, the lowered collision rate in the
channel $NN\to N\Delta$ leads to an increase in $NN\to\Delta\Delta$. Of course
this is unphysical as the double delta production must be lower than the
single delta production, so this is a point where the code must be improved in
the future. But for our purposes in this paper this is sufficient.

##  IV Comparison with Experimental Data

We compare our calculations to two experiments, namely HADES Adamczewski-Musch
_et al._ (2020) and FOPI Reisdorf  _et al._ (2004, 2010). The pion yields by
HADES started our investigation and their dielectrons are also a good
benchmark test for our model. The FOPI collaboration measured pions and
protons, the latter provide an excellent method to calibrate mean-field
potentials.

###  IV.1 Nucleons and Pions from FOPI

In the papers by Reisdorf _et al._ Reisdorf  _et al._ (2004, 2010) data for
nucleons and pions obtained by the FOPI collaboration are given. In the
experiment ${{}^{197}\text{Au}}$ projectiles were collided with stationary
${{}^{197}\text{Au}}$ targets with a kinetic energy of 400$\,{\rm
Me\kern-0.6458ptV}$ or 1.5$\,{\rm Ge\kern-0.6458ptV}$ per nucleon,
respectively. The FOPI measurements are very valuable, because they include
cumulated protons instead of clusters like deuterons or helium-nuclei, which
GiBUU cannot produce. Hence the predictions from our model are directly
comparable to data. The FOPI collaboration decided to present their data as
function of a logitudinal, $y_{z}$, and a transversal rapidity, $y_{x}$. The
first one corresponds to the usual definition oriented along the beam axis,
while the second one measures the rapidity in direction to the transversal
$x$-direction (in the laboratory)Reisdorf  _et al._ (2010),

$y\equiv y_{z}=\frac{1}{2}\log\bigg{(}\frac{E+p_{z}}{E-p_{z}}\bigg{)}\ ,\quad
y_{x}=\frac{1}{2}\log\bigg{(}\frac{E+p_{x}}{E-p_{x}}\bigg{)}\ .$ (39)

Figure 6: Proton rapidity spectra for Liu EOS with in-medium modifications.
Experimental data taken from Reisdorf  _et al._ (2004). A kinetic energy of
$0.4\,A{\rm Ge\kern-0.6458ptV}$ is present.

The notion of the transversal rapidity might be misleading, as it is neither
about the transverse momentum $p_{t}$ nor about the transverse reaction plane,
but solely refers to a fixed laboratory $x$ direction which is perpendicular
to the beam direction. In addition, published data Reisdorf  _et al._ (2004)
is in respect of the normalized rapidity
$y_{(z,x)}^{0}=y_{(z,x)}/y_{\text{proj}}$, which is normalized by the
projectile’s rapidity in the c.m. frame. Only central events with a scaled
impact parameter up to $b^{0}=b/b_{\text{max}}=0.15$ were considered,
equivalent to an impact parameter of up to $b=2.0\,{\rm fm}$ with
$b_{\text{max}}=1.15(A_{P}+A_{T})^{\frac{1}{3}}\,{\rm fm}$ being the
conventional maximum impact parameter.

The rapidity spectra for protons are shown in fig. 6 and fig. 7.

Figure 7: Same as fig. 6, but at $1.5\,A{\rm Ge\kern-0.6458ptV}$.

The description of the experimental data using Liu EOS is fine, and the cross
section modifications according the two different prescriptions do both not
destroy the agreement. In fact, at midrapidity the agreement to data according
the longitudinal axis gets better. Furthermore the curves are less peaked at
mid-rapidity. This implies that less stopping occurs in the collisions which
is to be expected as the cross sections are reduced by medium modifications.
Transversal spectra are (nearly) unaffected by the used medium modifications.

A more complex observable is the scaled directed flow, also called sideflow,
as it was introduced by FOPI Reisdorf  _et al._ (2004). It is given by

$p_{xdir}^{0}=\frac{p_{xdir}}{u_{1cm}}\quad,\qquad
p_{xdir}=\frac{\sum\text{sign}(y)Zu_{x}}{\sum Z}\ ,$ (40)

where the sums run over all charged particles with $Z<10$, excluding pions.
Here $u_{1cm}$ is the spatial part of the center of mass 4-velocity of the
projectile, $y$ the (longitudinal) rapidity and $Z$ the charge of a fragment.
$u_{x}=\beta_{x}\gamma$ is the 4-velocity projected onto the reaction plane.
In fig. 8 the sideflow is plotted against the scaled impact parameter and
compared to the data.

Figure 8: Sideflow for Liu EOS with in-medium modifications. Experimental data
taken from Reisdorf  _et al._ (2004). MUL and ERAT are the two prescriptions
used by FOPI to determine the impact parameter.

Figure 9: Same as fig. 8, but for EOS NL2 by Lang.

For a kinetic energy of $400\,{\rm Me\kern-0.6458ptV}$ per nucleon the
agreement is fine and both methods of medium modifications do not break it. At
the higher energy of $1.5\,A{\rm Ge\kern-0.6458ptV}$ GiBUU slightly
overpredicts the sideflow in vacuum and the modifications correct towards the
data and are acceptable. The shift to more inelastic collisions explains why
the modifications are more influential for altering the flow at higher
energies. Consequently the mean field can be studied more clearly at the lower
energy as inelastic collisions are less pronounced. Thus it is reassuring,
that Liu with and without modifications agrees well with the data at
$400\,A{\rm Me\kern-0.6458ptV}$. On the other hand, NL2 by Lang does not agree
with the data at lower energies, and stays significantly underneath Liu EOS as
can be seen in fig. 9. This is supported by earlier work Blaettel  _et al._
(1993), where a strong influence of the incompressibility on transverse flow
is disproved. Nara _et al._ Nara and Stoecker (2019) also find that the
effective mass has a stronger effect than the incompressibility, which is
almost identical between Liu and NL2 by Lang. They argue it is because flow is
mostly governed by the stiffness at larger densities, which itself is closely
related to effective mass and barely dependent on the incompressibility Boguta
and Stoecker (1983). Physically the smaller effective mass requires a larger
$\omega$ meson coupling and hence more repulsion is generated. This is not the
case in our model, but we also include the $\boldsymbol{\rho}$ coupling in our
EOS, so it is reasonable that meson interactions lead to more repulsion.

The FOPI collaboration also published the transverse momentum spectra for
pions Reisdorf  _et al._ (2010) for $1.5\,A{\rm Ge\kern-0.6458ptV}$ and
$b^{0}\leq 0.15$. Comparisons of calculations with these data are shown in
fig. 10.

Figure 10: Pion transverse momentum spectra compared to data from FOPI at
$1.5\,A{\rm Ge\kern-0.6458ptV}$ Reisdorf  _et al._ (2010) for Liu EOS with in-
medium modifications. Data points for momenta less than $100\,{\rm
Me\kern-0.6458ptV}$ were extrapolated, not measured and are shown as hollow
dots.

Calculations using the Liu EOS agree reasonably well with the data at
$p_{t}=100\text{-}150\,{\rm Me\kern-0.6458ptV}$, which are the lowest measured
data points, since for lower $p_{t}$, published data is extrapolated. The
calculated spectra show a slope, which is too hard compared to the
experimental results. The medium modification methods even worse the picture
by reducing the yield of pions at low momenta. While integrated pion yields
(even with the uncertainty of the extrapolation to low transverse momenta)
with the default setup seem to be corectly described, the medium modifications
yield to some underprediction of the experimental yields.

###  IV.2 Pions from HADES

For the HADES experiment, Au+Au heavy-ion collisions were performed at an
incident energy of $1.23\,A{\rm Ge\kern-0.6458ptV}$ and pion spectra were
published by Adamczewski-Musch _et al._ Adamczewski-Musch  _et al._ (2020).
Right off the bat, the pion multiplicities are significantly lower than
measured by FOPI, which was also discussed by both collaborations Adamczewski-
Musch  _et al._ (2020); Reisdorf  _et al._ (2007). The following comparisons
of calculations to experimental spectra are for the innermost 10$\,{\rm\%}$
centrality class of events. As shown in fig. 11 one observes, that both
methods of the in-medium modifications reduce the pion yield and the rapidity
curves agree perfectly with data.

Figure 11: Rapidity spectra of pions for Liu EOS with in-medium modifications.
Experimental data taken from Adamczewski-Musch  _et al._ (2020).

Figure 12: Rapidity spectra of pions with medium modifications according
effective mass for both pion species as in fig. 11. Calculations with and
without modification of the $\Delta$ potential are shown. Top lines are for
$\pi^{-}$, bottom lines for $\pi^{+}$.

Figure 13: Same as fig. 13, but for NL2 by Lang EOS.

In the case of the medium modification via the effective mass it is important
to observe, that only the interplay with the modification of the $\Delta$
potential yields to similar reduction of both pion species. Theoretical
calculations without any medium modifications yield a ratio $\pi^{-}/\pi^{+}$
very similar to the experimental value, while the absolute numbers are off.
Thankfully this ratio is robust and the medium modifications do not break it.
This is illustrated in fig. 13, where rapidity spectra with medium
modifications, but with and without modification of the $\Delta$ potential are
shown.

Figure 14: Transverse momentum spectra of pions for Liu EOS with in-medium
modifications. Experimental data taken from Adamczewski-Musch  _et al._
(2020).

It is noteworthy that the modification via the effective mass or the $\Delta$
potential lower the rapidity spectra only slightly, whereas their combined
effect results in a significant reduction. Especially the $m^{*}/m$
prescription has an almost negligible effect on its own. For comparison the
same curves with NL2 by Lang EOS are shown in fig. 13. One can see that both
the effective mass and the $\Delta$ potential lower pion yields with more
intense than with NL2, even though the rapidity curves with vacuum cross
sections are almost identical. This is to be expected, as Liu has a stronger
drop of $m^{*}/m$. Consequently Liu lies perfectly on the data with the
modifications, while NL2 by Lang still overpredicts the yields significantly.
Similarly the prescription with Liu EOS and exponential suppression with
$\alpha=2.0$ works for both species and even slightly better with NL2 by Lang
EOS. The $\pi^{-}/\pi^{+}$ ratio is within the data. (Please recall that the
Coulomb potential is in effect for all particle species in our calculations.)
This has to be considered an improvement to the results of Godbey _et al._
Godbey  _et al._ (2022) who needed the factor $\alpha$ to be isospin-dependent
in order to overcome the burden of starting with a wrong ratio already without
modification.

Unfortunately the good agreement of Liu EOS with the rapidity spectra does not
hold for the $p_{t}$ spectra, as shown in fig. 14: The in-medium modifications
are too hard to reproduce the curves. This is in agreement with the above
comparison to the pion spectra measured by FOPI. Another problem is that
$\pi^{-}$ at low $p_{t}$ are not well described. But overall this is a
significant improvement to earlier transport calculations.

Figure 15: Invariant mass spectra of dielectrons for Liu EOS with and without
in-medium modifications. Data taken from Adamczewski-Musch  _et al._ (2019).
Bremsstrahlung-enhancement as in Larionov  _et al._ (2021).

###  IV.3 Dileptons from HADES

Figure 16: Same as fig. 15, but zoomed in on the $\pi^{0}$ Dalitz decay. Other
contributions than the Dalitz decay are not shown separately.

Another observable measured by HADES are dielectron spectra. For the situation
at hand, gold on gold at $1.23\,A{\rm Ge\kern-0.6458ptV}$, the invariant mass
spectra for the innermost 40$\,{\rm\%}$ of events were published by
Adamczewski-Musch _et al._ Adamczewski-Musch  _et al._ (2019). The effect of
the in-medium modification on the full spectrum can be seen in fig. 15. Slight
relative differences in the various yields may be visible, while the sum of
all contributions is nearly unaffected. Some mystery is the behaviour of the
$\pi N$-bremsstrahlung. The in-medium modifications increase the yield of this
channel by more than a factor 3. More research is needed to find the source of
this enhancement. Details about the dilepton calculations may be found in ref.
Larionov  _et al._ (2021).

The region of the Dalitz peak is emphasized in fig. 16. The $\pi^{0}$ Dalitz
decay is troublesome: If total pion numbers agree well with experiment, too
few dielectrons are computed. Bringing the $\pi^{0}$ Dalitz decay closer to
the experiment on the other hand gives too many pions in total, which is
consistent with earlier works Larionov  _et al._ (2021). For now it seems
impossible to get the correct number of dielectrons from the $\pi^{0}$ Dalitz
decay and the correct number of charged pions at the same time.

##  V Conclusions

As shown in Adamczewski-Musch  _et al._ (2020), transport models drastically
overpredict experimental measured pion yields. The goal of this letter was to
solve this pion puzzle by adjusting the underlying physics of the collisions
to in-medium conditions.

There are three methods considered throughout this report which work in
reducing pion production. One option is by applying in-medium modifications on
the cross section via the Dirac mass, or more precisely the ratio of Dirac
mass to bare mass $m^{*}/m$. This prescription is not uncontroversial, but it
has some backing in literature. Another method of reducing pion numbers is the
modification of the coupling of the $\Delta$ to scalar and vector field and
thus changing the potential for these baryons. Whether this couplings are
stronger or weaker than the nucleons’, and what its precise values are, is
still an unanswered question with active research. Further it is unclear,
whether this prescription is implemented in a physical manner in GiBUU, but
for now it is assumed that this is the case. Another way of lowering the
abundance of pions is by suppressing the $NN\leftrightarrow N\Delta$ and
$NN\leftrightarrow NN\pi$ reaction with an ad-hoc density dependent factor.
Similar work was done by Godbey _et al._ Godbey  _et al._ (2022), but they
required this factor to also be isospin dependent.

In order to give robust results, it is demanded that GiBUU can accurately
describe proton spectra. Data on cumulated protons was published by the FOPI
collaboration Reisdorf  _et al._ (2004), which was needed, as GiBUU does not
produce clusters, but only protons. From the proton rapidity spectra it
follows, that a soft EoS has to be considered the way to go. This becomes
blatantly apparent at lower energies ($400\,A{\rm Me\kern-0.6458ptV}$) where
the mean-field can be studied most clearly. Another interesting observable is
the sideflow, which proves to be highly sensible on the EoS. There a stiff EoS
immediately has to be ruled out. In agreement with astronomical constraints Li
_et al._ (2021) the EoS by Liu seems to be most appropriate to describe
nuclear matter.

The pion rapidity spectra obtained by HADES is well described by the EoS by
Liu with some in-medium modifications. Using the exponential suppression good
agreement is possible without introducing different factors for positively and
negatively charged pions. In this way this study is an improvement on earlier
works on this topic. Similar agreement to data can be achieved by modifying
the cross sections for $NN\leftrightarrow N\Delta$ and $NN\leftrightarrow
NN\pi$ and by simultaneously changing the coupling to scalar and vector
fields, $r_{S,V}$. Especially the coupling $r_{S,V}$ turns out to be important
for the ratio of $\pi^{-}/\pi^{+}$. This is exciting as this formalism would
offer a better theoretical description on the underlying physical situation.
However, both exponential suppression and effective mass mechanisms yield
$p_{t}$ spectra which are too stiff. Too many high momentum pions are produced
and for now it is unclear how to solve this problem. The comparison to the
pion spectra from FOPI is ambivalent, nothing can be excluded. Furthermore,
FOPI yields are significantly higher than HADES’s, which is also discussed by
the HADES collaboration Adamczewski-Musch  _et al._ (2020).

One remaining problem lies in the $\pi^{0}$ Dalitz decay. There is the
paradoxical situation, that the Dalitz decay is under-predicted, if the
multiplicites of the charged pions match the experiment. However, if the pion
numbers are over-predicted, the $\pi^{0}$ Dalitz decay lines up with the data.
For now there is also no solution for this conundrum.

###### Acknowledgements.

We are grateful for close collaboration with Alexei Larionov in the early
stages of this project. We also gratefully acknowledge helpful discussions
with Ulrich Mosel and Jan-Hendrik Otto. This work was supported by BMBF grant
no. 05P21RGFCA.

## References

  * Adamczewski-Musch  _et al._ (2020) J. Adamczewski-Musch _et al._ (HADES), Eur. Phys. J. A 56, 259 (2020), arXiv:2005.08774 [nucl-ex] .
  * Bass  _et al._ (1995) S. A. Bass, C. Hartnack, H. Stoecker, and W. Greiner, Phys. Rev. C 51, 3343 (1995), arXiv:nucl-th/9501002 .
  * Teis  _et al._ (1997) S. Teis, W. Cassing, M. Effenberger, A. Hombach, U. Mosel, and G. Wolf, Z. Phys. A 356, 421 (1997), arXiv:nucl-th/9609009 .
  * Uma Maheswari  _et al._ (1998) V. S. Uma Maheswari, C. Fuchs, A. Faessler, L. Sehn, D. S. Kosov, and Z. Wang, Nucl. Phys. A 628, 669 (1998), arXiv:nucl-th/9706004 .
  * Pelte  _et al._ (1997) D. Pelte _et al._ (FOPI), Z. Phys. A 357, 215 (1997).
  * Larionov  _et al._ (2001) A. B. Larionov, W. Cassing, S. Leupold, and U. Mosel, Nucl. Phys. A 696, 747 (2001), arXiv:nucl-th/0103019 .
  * Larionov and Mosel (2003) A. B. Larionov and U. Mosel, Nucl. Phys. A 728, 135 (2003), arXiv:nucl-th/0307035 .
  * Ericson and Weise (1988) T. E. O. Ericson and W. Weise,  _Pions and Nuclei_ (Clarendon Press, Oxford, UK, 1988).
  * Larionov  _et al._ (2021) A. B. Larionov, U. Mosel, and L. von Smekal, Phys. Rev. C 102, 064913 (2021), arXiv:2009.11702 [nucl-th] .
  * Godbey  _et al._ (2022) K. Godbey, Z. Zhang, J. W. Holt, and C. M. Ko, Phys. Lett. B 829, 137134 (2022), arXiv:2107.13384 [nucl-th] .
  * Buss  _et al._ (2012) O. Buss, T. Gaitanos, K. Gallmeister, H. van Hees, M. Kaskulov, O. Lalakulich, A. B. Larionov, T. Leitner, J. Weil, and U. Mosel, Phys. Rept. 512, 1 (2012), arXiv:1106.1344 [hep-ph] .
  * Liu  _et al._ (2002) B. Liu, V. Greco, V. Baran, M. Colonna, and M. Di Toro, Phys. Rev. C 65, 045201 (2002), arXiv:nucl-th/0112034 .
  * Lang  _et al._ (1992) A. Lang, W. Cassing, U. Mosel, and K. Weber, Nucl. Phys. A 541, 507 (1992).
  * Lalazissis  _et al._ (1997) G. A. Lalazissis, J. Konig, and P. Ring, Phys. Rev. C 55, 540 (1997), arXiv:nucl-th/9607039 .
  * Li  _et al._ (2021) B.-A. Li, B.-J. Cai, W.-J. Xie, and N.-B. Zhang, Universe 7, 182 (2021), arXiv:2105.04629 [nucl-th] .
  * Xie and Li (2020) W.-J. Xie and B.-A. Li, Astrophys. J. 899, 4 (2020), arXiv:2005.07216 [astro-ph.HE] .
  * Song and Ko (2015) T. Song and C. M. Ko, Phys. Rev. C 91, 014901 (2015).
  * Ter Haar and Malfliet (1987) B. Ter Haar and R. Malfliet, Phys. Rev. C 36, 1611 (1987).
  * Li and Machleidt (1993) G.-Q. Li and R. Machleidt, Phys. Rev. C 48, 1702 (1993), arXiv:nucl-th/9307028 .
  * Li and Machleidt (1994) G.-Q. Li and R. Machleidt, Phys. Rev. C 49, 566 (1994), arXiv:nucl-th/9308016 .
  * Mosel (2020) U. Mosel, (2020), arXiv:2007.10260 [nucl-th] .
  * Kosov  _et al._ (1998) D. S. Kosov, C. Fuchs, B. V. Martemyanov, and A. Faessler, Phys. Lett. B 421, 37 (1998), arXiv:nucl-th/9704012 .
  * Reisdorf  _et al._ (2004) W. Reisdorf _et al._ (FOPI), Phys. Rev. Lett. 92, 232301 (2004), arXiv:nucl-ex/0404037 .
  * Reisdorf  _et al._ (2010) W. Reisdorf _et al._ (FOPI), Nucl. Phys. A 848, 366 (2010), arXiv:1005.3418 [nucl-ex] .
  * Blaettel  _et al._ (1993) B. Blaettel, V. Koch, and U. Mosel, Rept. Prog. Phys. 56, 1 (1993).
  * Nara and Stoecker (2019) Y. Nara and H. Stoecker, Phys. Rev. C 100, 054902 (2019), arXiv:1906.03537 [nucl-th] .
  * Boguta and Stoecker (1983) J. Boguta and H. Stoecker, Phys. Lett. B 120, 289 (1983).
  * Reisdorf  _et al._ (2007) W. Reisdorf _et al._ (FOPI), Nucl. Phys. A 781, 459 (2007), arXiv:nucl-ex/0610025 .
  * Adamczewski-Musch  _et al._ (2019) J. Adamczewski-Musch _et al._ (HADES), Nature Phys. 15, 1040 (2019).