# Temperature and net baryochemical potential dependence of $\eta/s$ in a
hybrid approach

Niklas Götz1,2 and Hannah Elfner3,1,2,4 1Institute for Theoretical Physics,
Goethe University, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
2Frankfurt Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438
Frankfurt am Main, Germany  3GSI Helmholtzzentrum für Schwerionenforschung,
Planckstr. 1, 64291 Darmstadt, Germany  4Helmholtz Research Academy Hesse for
FAIR (HFHF), GSI Helmholtz Center, Campus Frankfurt, Max-von-Laue-Straße 12,
60438 Frankfurt am Main, Germany

###### Abstract

In this work, the qualitative impact of the net baryochemical potential
dependence of the shear viscosity to entropy density ratio $\eta/s$ in
hydrodynamical simulations is studied. The effect of a predicted non-constant
$\eta/s$($\mu_{B}$) is largely unexplored in hydrodynamic simulations.
Previous studies focus only on a temperature dependence or even only a
constant effective shear viscosity. This work addresses this issue by studying
qualitatively the effect of a generalized $\eta/s$($T,\mu_{B}$) in the hybrid
approach SMASH-vHLLE, composed of the hadronic transport approach SMASH and
the (3+1)d viscous hydrodynamic code vHLLE. In order to reduce the bias of the
result on the equation of state used in the hydrodynamic part of the model,
$\eta/s$ is parameterized directly in the energy density and net baryon number
density. The parameterization takes into account the constraints of matching
to the transport coefficients in the hadronic phase, as well as pQCD results.
This work compares the impact of the density dependence for different system
sizes and energies and compares the observables with experimental results in
the RHIC - BES region $\sqrt{s_{NN}}=7.7-39.0$ GeV, as the effect of this
generalization is especially relevant for intermediate collision energies, for
which the system is in equilibrium for a relevant amount of time, but the net
baryochemical potential does not vanish. It is shown that the effect of an
explicit net baryon number dependence on the elliptic flow is negligible and
only relevant in the early stages of the collision. Additionally, we find that
the proposed parameterization could be a good proxy for the shear viscosity in
the non-equilibrium hadronic transport stage.

##  I Introduction

Collisions of atomic nuclei at relativistic velocities are one of the most
important means to study and understand the fundamental properties of matter
Busza  _et al._ (2018). Different experimental facilities exist to conduct
such heavy-ion collisions, such as the Large Hadron Collider (LHC) at CERN,
SIS-18 at GSI and the Relativistic Heavy-Ion Collider (RHIC) at BNL, which
allow to study the different regions of the QCD diagram. While at high beam
energies high temperatures are reached at vanishing net baryon chemical
potential, the region accessed by collisions at intermediate energy is of
special interest as signatures of of a first order phase transition and a
critical end point are expected there Stephanov  _et al._ (1998); Kumar
(2013). The theoretical understanding of the dynamics at such intermediate
beam energies requires the treatment of finite net baryon chemical potential.

Hybrid approaches based on relativistic viscous fluid dynamics for the hot and
dense stage and hadronic transport for the non-equilibrium rescattering have
been established as a great tool to describe the dynamics of high-energy
heavy-ion reactions. Recent progress has been made in the theoretical
description of heavy-ion collisions at finite baryon densities Schäfer  _et
al._ (2021); Karpenko  _et al._ (2015); Petersen  _et al._ (2008); Wu  _et
al._ (2022); Shen  _et al._ (2017); Akamatsu  _et al._ (2018); Du and Heinz
(2020); Nandi  _et al._ (2020) employing such hybrid approaches, which combine
the successful description of hadronic transport at low energies, where
hadronic interactions prevail, with the high energy description of
relativistic hydrodynamic calculations. The hadronic transport typically
serves as a hadronic afterburner, which has shown to successfully reproduce
experimental observables Petersen (2014), but it can also serve to create the
initial conditions. Besides the initial conditions, relativistic viscous
hydrodynamic calculations need as an input an equation of state as well as the
transport coefficients. This indicates also the main advantage of
hydrodynamics, namely direct access to the properties of the medium and a
potential phase transition.

Regarding the transport coefficients, extensive comparisons of theoretical
calculations with experimental flow measurements have shown that there is
evidence that the shear and bulk viscosity are not vanishing in the quark-
gluon plasma Shen (2021). From a theoretical perspective, the transport
coefficients are currently not easily accessible from first principle lattice
QCD calculations due to numerical challenges Petreczky and Teaney (2006);
Aarts and Resco (2002). Nevertheless, many theoretical predictions support a
non-vanishing shear viscosity over entropy ratio $\eta/s$ Kovtun  _et al._
(2005); Auvinen  _et al._ (2018); Nakamura and Sakai (2005).

All known fluids in nature have a temperature dependent $\eta/s$, with a
minimum close to the phase transition Chapman  _et al._ (1990). A similiar
behavior is also predicted for the shear viscosity of nuclear matterEverett
_et al._ (2021); JETSCAPE collaboration (2021); Ghiglieri  _et al._ (2018);
Auvinen  _et al._ (2020); Nijs  _et al._ (2021); G. Nijs, W. van der Schee, U.
Gürsoy, and R. Snellings (2021); Greiner  _et al._ (2011); Gorenstein  _et
al._ (2008); Csernai  _et al._ (2006). Apart from this, there is also
theoretical support for a dependence on the net baryochemical potential
Denicol  _et al._ (2013); Itakura  _et al._ (2008); Kadam and Mishra (2015);
Demir and Bass (2009). The effect of including such a non-constant
$\eta/s$($T,\mu_{B}$) is largely unexplored in hydrodynamic simulations.
Previous studies focus on a temperature dependence or even a constant
effective shear viscosity Petersen  _et al._ (2008); Petersen (2014); Everett
_et al._ (2021); JETSCAPE collaboration (2021); G. Nijs, W. van der Schee, U.
Gürsoy, and R. Snellings (2021); Auvinen  _et al._ (2018); Magdy  _et al._
(2021); Niemi  _et al._ (2016). Therefore, in the following the effects of
including both a temperature and a net baryochemical potential dependence of
the shear viscosity on the evolution and observables will be investigated.

This work is structured as follows: In Sec. II, the hybrid approach SMASH-
vHLLE-hybrid, within which this study is performed, is briefly summarized. In
the following, a parameterization for $\eta/s(T,\mu_{B})$ is proposed and the
choice of parameters as well as parameterizations chosen for comparison are
explained. Additionally, the setup of the computations to investigate the
qualitative impact of the $\mu_{B}$ dependence is illustrated in the same
section. In Sec. III, this qualitative impact is studied both at the example
of the midrapidity yield and mean transverse momentum as well as the elliptic
flow. This is accompanied by a comparison to data as well as a study of the
effect of changing the switching energy density, which shows that the proposed
parameterization approximates the shear viscosity in the hadronic phase. To
conclude, a brief summary and outlook can be found in Section IV.

##  II Model Description

The theoretical calculations presented in this work are performed using the
SMASH-vHLLE-hybrid hybrid approach hyb , which is publicly available and
suited for the theoretical description of heavy-ion collisions between
$\sqrt{s_{NN}}$ = 4.3 GeV and $\sqrt{s_{NN}}$ = 5.02 TeV. SMASH-vHLLE-hybrid
has been shown to reproduce experimental data well across a wide range of
collision energies and conserves all charges ($B,Q,S$). It is especially
successful in reproducing the longitudinal baryon dynamics at intermediate
collision energies Schäfer  _et al._ (2021).

In the following, a short overview of its components is given (a more detailed
description can be found in Schäfer  _et al._ (2021)), as well as the
description and motivation of the proposed parameterization for the shear
viscosity over entropy density ratio.

###  II.1 Hybrid approach

Hybrid approaches combine viscous fluid dynamics for the hot and dense phase
of heavy ion collisions with microscopic non-equilibrium transport approaches
for the cold and dilute stage. The SMASH-vHLLE-hybrid employs the same
hadronic transport approach SMASH Weil  _et al._ (2016); Oliinychenko  _et
al._ (2021); sma  for both the initial conditions and the afterburner. This
allows for direct comparisons between hydrodynamic and non-equilibrium
evolution as well as a gradual turn on of the fluid dynamic stage when moving
to higher beam energies.

####  II.1.1 Hadronic transport

SMASH effectively solves the relativistic Boltzmann equation by simulating the
collision integral through formations, decays and elastic scatterings of
hadronic resonances, for which all hadrons listed by the PDG up to a mass of
2.35 GeV are included Tanabashi  _et al._ (2018). For hadronic interactions at
high energies, hard scatterings are carried out within Pythia 8 Sjöstrand  _et
al._ (2008) and a soft string model is employed.

The initial conditions of the hydrodynamic evolution are generated by SMASH on
a hypersurface of constant proper time $\tau_{0}$, which corresponds to the
passing time of the two nuclei Karpenko  _et al._ (2015), following the
assumption that at this time local equilibrium is reached and the hydrodynamic
description becomes applicable Oliinychenko and Petersen (2016); Inghirami and
Elfner (2022).

In the last stage of the hybrid evolution, SMASH is employed for hadron
rescattering. The hadrons obtained from particlization on the Cooper-Frye
hypersurface are propagated back to the earliest time and appear subsequently
in the hadronic transport evolution and scatter or decay. The calculation is
terminated when the medium becomes sufficiently dilute.

####  II.1.2 Hydrodynamic evolution

vHLLE Karpenko  _et al._ (2014) is a 3+1D viscous hydrodynamics code and used
to model the evolution of the hot and dense fireball. It solves the
hydrodynamic equations

$\partial_{\nu}T^{\mu\nu}=0\quad,$ (1)
$\partial_{\nu}j^{\nu}_{B}=0\quad\partial_{\nu}j_{Q}^{\nu}=0\quad\partial_{\nu}j_{S}^{\nu}=0\quad.$
(2)

These equations represent the conservation of net-baryon, net-charge and net-
strangeness number currents as well as the conservation of energy and
momentum. The energy-momentum tensor is decomposed as

$T^{\mu\nu}=\epsilon u^{\mu}u^{\nu}-\Delta^{\mu\nu}(p+\Pi)+\pi^{\mu\nu}$ (3)

with $\epsilon$ the local rest frame energy density, $p$ and $\Pi$ the
equilibrium and bulk pressure and $\pi^{\mu\nu}$ is the shear stress tensor.
These equations are solved in the second order Israel-Stewart framework
Denicol  _et al._ (2014); Ryu  _et al._ (2015).

In order to initialise the hydrodynamic evolution with the iso-$\tau$ particle
list generated from SMASH, as outlined in the last section, some smoothing has
to be performed in order to prevent shock waves. For this purpose, a Gaussian
smearing kernel Karpenko  _et al._ (2015) with the parameters taken from
Schäfer  _et al._ (2021) is applied. At this step, particles have been
converted into fluid elements which are evolved using a chiral model equation
of state Steinheimer  _et al._ (2011) matched to a hadron gas equation of
state. This evolution is performed until a switching energy density is
reached, which is set in this work to a default value of
$\epsilon_{\text{switch}}$=0.3 GeV/fm3, if not mentioned differently. At this
point, the freezeout hypersurface is constructed using the CORNELIUS
subroutine Huovinen and Petersen (2012) and the thermodynamical properties of
the freezeout elements are calculated according to the SMASH hadron resonance
gas equation of state Schäfer  _et al._ (2021) to prevent discontinuities.
$\epsilon_{\text{switch}}$ is a free parameter and different choices are
possible. This relies on the assumption that there is a region in the QCD
phase diagram where both hydrodynamic evolution as well as hadronic transport
are equivalently applicable, which in turn validates the application of hybrid
approaches. It is important to notice that $\epsilon_{\text{switch}}$ is a
technical parameter and does only control the application of the hybrid and
transport approach, but does not determine which degrees of freedom are
realized in the QCD matter.

####  II.1.3 Particle sampler

As SMASH evolves particles and not fluid elements, particlization has to be
performed, for which the SMASH-vhLLE hybrid approach applies the SMASH-hadron-
sampler sam . The SMASH-hadron-sampler employs the grand-canonical ensemble in
order to particlize each surface element independently. Hadrons are sampled
according to a Poisson distribution with the mean at the thermal mulitplicity,
and momenta are sampled according to the Cooper-Frye formula Cooper  _et al._
(1975). The corrections to the distribution function $\delta f_{\rm shear}$
associated to a finite shear viscosity are considered. This procedure provides
a particle list that can be evolved by SMASH. Because of the grand-canonical
sampling procedure, quantum numbers are not conserved event-by-event but only
on average. This is sufficient for this work as we are only interested in
averaged observables. It is crucial to match the hadronic degrees of freedom
in the equation of state to the ones that are active in SMASH to maintain
energy conservation at the transition even for the average over events. A more
thorough introduction into the sampling procedure can be found in Karpenko
_et al._ (2015).

####  II.1.4 Configuration details

For each simulation presented, 100 event-by-event viscous hydrodynamics events
were generated, each initialized from one single SMASH initial condition. The
smearing parameters for the transition from transport to hydrodynamic
evolution are taken from Table 1 of Schäfer  _et al._ (2021), as these have
been shown to reproduce experimental data well. From the resulting freezeout
hypersurface 1000 events are sampled for the hadronic afterburner evolution,
in order to allow for on-average quantum number conservation. Whereas the bulk
viscosity is set to zero for all runs, the choice of the shear viscosity will
be discussed in the following. Unless stated otherwise, the code versions
SMASH-vHLLE-hybrid:03232b2, SMASH-2.1.4, vhlle-params:99ef7b4 and SMASH-
hadron-sampler-1.0 are used in this work. The modifications to the
hydrodynamic code in order to allow for different parameterisations are based
on vHLLE:efa9e28.

###  II.2 Parametrization of $\eta/s$

Figure 1: $\eta/s(\epsilon,\rho)$ for $S_{\rho}=0.05$ fm3 (top) and the same
parametrization mapped into $(T,\mu_{B})$ using a chiral equation of state
(bottom).

This work aims at investigating both a temperature and net baryochemical
potential dependence of the shear viscosity in the fireball. It is a common
approach to parametrize $\eta/s$ directly in $T$ and it would be possible to
add additional terms proportional to $\mu_{B}$ as well. However, parametrizing
instead in the local rest frame energy density $\epsilon$ and the net baryon
number $\rho$ has the advantage that these are the quantities which are
evolved throughout the hydrodynamic evolution, and therefore reduces the
dependency of results on the choice of the equation of state. Additionally, as
outlined before, a minimum is expected close to the cross-over transition
between the quark-gluon plasma and the hadron gas phase. As the transition
line is expected to follow approximately a line of constant energy density,
this behavior is easier to replicate with a parameterization in the energy
density itself.

For this qualitative study, several simplifications are made. First of all,
although at finite $\mu_{B}$ not $\eta/s$ but $\eta T/w$, with the enthalpy
$w=\epsilon+p$, is the more correct measure of fluidity in the system Liao and
Koch (2010), we use $\eta/s$ as an approximation, as both terms have the same
limit for small $\mu_{B}$. Next, as pointed out above, we expect a minimum of
$\eta/s$ near the transition. In the limit of vanishing $\mu_{B}$, we want to
replicate this behavior of $\eta/s(T)$ also in $\eta/s(\epsilon)$, as
$\epsilon\sim T^{4}$. As observables are more sensitive to the effective shear
viscosity than to the functional form of the shear viscosity Gardim and
Ollitrault (2021), we restrict ourselves to a linear dependence of the shear
viscosity both in the region of high and low energy densities. Up to now, this
already implies a $\mu_{B}$-dependence, since our parameterization
$\eta/s(\epsilon)$ is dependent on $\mu_{B}$, whereas $\eta/s(T)$ is not. For
example, the minimum of $\eta/s(\epsilon)$ moves to lower temperatures with
increasing $\mu_{B}$. Nevertheless, we want to be able to study the effects of
a $\mu_{B}$-dependence more in detail, which is why we include a linear term
proportion to the net baryon number density $\rho$. Together with the
condition that the shear viscosity always stays positive, this leads us to the
following functional form:

$\eta/s(\epsilon,\rho)=\max\left(0,(\eta/s)_{\text{kink}}+\begin{cases}S_{\epsilon,H}(\epsilon-\epsilon_{\text{kink}})+S_{\rho}\rho,&\epsilon<\epsilon_{\text{kink}}\\\
S_{\epsilon,Q}(\epsilon-\epsilon_{\text{kink}})+S_{\rho}\rho&\epsilon>\epsilon_{\text{kink}}\end{cases}\right)$
(4)

This form of the parameterization results in five parameters, out of which all
but $S_{\rho}$ will be fixed by constraints. We set $\epsilon_{\text{kink}}$,
the position of the minimum and vanishing net baryochemical potential, to 1
GeV/fm3 according to lattice QCD Borsányi  _et al._ (2014) and experimental
results Cleymans  _et al._ (2006). The value of the shear viscosity at this
minimum is set to the KSS-bound Kovtun  _et al._ (2005). The steepness for the
high energy density region is motivated by matching pQCD results Ghiglieri
_et al._ (2018) at a temperature of 400 MeV and vanishing net baryochemical
potential. The steepness in the low energy region is set to match the shear
viscosity extracted from box calculations in SMASH at vanishing net
baryochemical potential and at the particlization temperature in order to
reduce discontinuities. It is however important to note that this is only an
approximation of the shear viscosity in hadronic transport, as box
calculations differ substantially from the expanding medium in the
rescattering phase.

This leaves one free parameter, $S_{\rho}$, which is going to be varied in
order to investigate its effect on the observables. The parameterization for
the choice of $S_{\rho}=0.05$ fm3 can be see in in Fig. 1, both in
($\epsilon$,$\rho$) as well as ($T$,$\mu_{B}$), for which the mapping is
performed by using the same equation of state as employed in vHLLE. Both show
the typical kink structure. The temperature dependence of the minimum in
($T$,$\mu_{B}$) is present also for vanishing $S_{\rho}$, whereas the value of
the shear viscosity over entropy density at the minimum changes only due to
non-vanishing $S_{\rho}$.

In the following, we want to compare our parameterization with other existing
choices for constant or temperature-dependent $\eta/s$. The first comparison
choice is a constant value for $\eta/s$, with different values at different
collision energies (see table 1 in Schäfer  _et al._ (2021)). These are the
default values in SMASH-vHLLE-hybrid, and were originally taken from
UrQMD+vHLLE in order to be in optimal agreement with experimental data. Such a
choice of $\eta/s$ is often very successful, as many observables are mostly
sensitive to the effective shear viscosity Gardim and Ollitrault (2021), but
rather insensitive to different values of the viscosities in different regions
and stages of the evolution of the fireball.

The second parameterization chosen for comparison is representative for the
increasing efforts of extracting $\eta/s(T)$ from experimental data by means
of Bayesian analysis. We use the parameterization in JETSCAPE collaboration
(2021) and choose parameters which lie central in the 60% confidence interval
of the posterior. Though being tuned to be in good agreement with experimental
data at high collision energies and providing a minimum of the shear viscosity
close to the transition temperature, it does not include any dependence on the
net baryochemical potential.

##  III Results

In the following, we investigate the qualitative effects that the proposed
parameterization and different choices of $S_{\rho}$ have on the evolution and
observables in heavy-ion collisions. Since the region of non-vanishing net
baryochemical potential is of special interest, the chosen energies and
systems are AuAu collisions for $\sqrt{s_{NN}}$= 7.7 and 39 GeV as well as
PbPb collisions at $\sqrt{s_{NN}}$= 17.3 GeV. For observables, the focus will
lie on midrapidity yield and mean transverse momentum of central collisions,
which are only weakly sensitive to the shear viscosity but can be computed
with precision even at low statistics, and the integrated elliptic flow
$v_{2}$ at centrality ranges 20-30%, which is highly sensitive to the shear
viscosity. Centrality is for this study defined by the impact parameter. For
the collision setups investigated in this study, the temperature reaches
throughout the hydrodynamic evolution values between 108 and 407 MeV, whereas
the net baryochemical potential reaches values between 0 and 583 MeV.

These observables will be compared for two sets of parameterizations: One
which compares $\eta/s(\epsilon)$ (with $S_{\rho}$ equal to zero) to both the
constant and temperature dependent parameterization mentioned in the last
section for comparison to prior choices and another set where the impact of an
explicit net baryon number dependence by comparing the cases $S_{\rho}=$ 0,
0.05 and -0.05 fm3 is studied.

###  III.1 Effective $\eta/s$ and time evolution

Figure 2: Energy-density weighted mean shear viscosity over entropy ratio
throughout the whole hydrodynamic evolution. From top to bottom: Comparison
between different parametrization choices for central and off-central
collisions, impact of net baryon number density dependence for central and
off-central collisions.

An important mean to compare different, non-constant parameterizations of the
shear viscosity is the effective shear viscosity, which is a weighted average
throughout the hydrodynamic evolution. Different ways to weight the
contribution of the shear viscosity in a single fluid cell at a specific time
step exist, but for this work, we restrict ourselves to choose the energy
density as a weight. The result of this comparison can be seen in Fig. 2.

We see that the parameterization in the energy densitiy reaches in general
higher values for the shear viscosity than the parameterizations chosen for
comparison, especially in peripheral collisions. The origin of this lies in
the matching to the shear viscosity in SMASH which is larger than the one in
the purely temperature dependent parameterization. This hints at the fact that
many cells close to the switching energy density provide a significant
contribution to the averaged shear viscosity. The temperature dependent shear
viscosity parameterization shows only a small dependence on the collision
energy, as the posterior of the Bayesian inference is centered around
relatively small steepnesses in comparison to the proposed parameterization,
while also having a similar value of the minimum. The constant values for the
shear viscosity, on the other hand, agree with the effective values for low
collision energies but have a much faster decline.

Figure 3: Time evolution of the energy density weighted shear viscosity over
entropy ratio at $\sqrt{s_{NN}}=39$ GeV central collisions. The net baryon
number density dependence is only relevant for the early stage of the
collision.

For the comparison of $\rho$ dependence, it becomes clear that the effect is
rather small and reduces with the collision energies as well as with
decreasing centrality. In both cases either the density of the medium is
reduced or the net baryochemical potential. A closer insight is given in Fig.
3, which shows the effective shear viscosity over entropy ratio during each
timestep for central collisions at $\sqrt{s_{NN}}$= 39 GeV. The strongest
effect can be seen in the very early stages of the hydrodynamic evolution,
where densities are high. In general, the parameterization in $\epsilon$ shows
a strong time dependent behavior. At the observed energies, many fluid
elements are initialised close or slightly above $\epsilon_{\text{kink}}$ and
have small shear viscosities, which quickly decreases when the cells cool down
and become more dilute. Let us emphasize here again, that this does not imply
that the finite net baryochemical potential is irrelevant, since our
$\eta/s(\epsilon)$ incorporates this dependence implicitly.

###  III.2 Bulk observables

Figure 4: Excitation function of mean transverse momentum of charged hadrons
at midrapidity ($|y|<0.5$) for different parametrization strategies (top) and
different values of the net baryon number density dependence (bottom).

A first insight into the effect of net baryochemical potential dependence can
be gained by looking at the bulk observables in central collisions. In Fig. 4,
the mean transverse momentum at midrapidity for charged hadrons is compared
for the different parameterizations. The parameterization in $\epsilon$
results in smaller mean transverse momentum for low collisions energies, but
approaches the value of other parameterizations at higher values. Introducing
a $\rho$ dependence, both positive and negative, increases the transverse
momentum, which hints to non-linear effects to the evolution of the fireball.

Figure 5: Excitation function of midrapidity yield ($|y|<0.5$) of charged
hadrons for different parametrization strategies (top) and different values of
the net baryon number density dependence (bottom).

In Fig. 5, the midrapditiy yield for charged hadrons is shown as a function of
beam energy. The differences between the different parametrization strategies
become only significant at higher collision energies, where the energy density
dependent parameterization increases the yield. With respect to the explicit
net baryon number dependence, the addition of $S_{\rho}$ is also more
important at higher collision energies. The effect is however not linear -
adding a term which increases the shear viscosity with increasing net baryon
number density has virtually no effect on the yield, whereas decreasing the
shear viscosity with increasing net baryon number density significantly
reduces the yield.

The increasing importance of $S_{\rho}$ with increasing collision energy seems
surprising, as $\mu_{B}$ decreases with collision energy. However, as observed
in Schäfer  _et al._ (2021), the lifetime of the fireball evolved within
hydrodynamics strongly increases in this energy range, resulting in an
increasing importance of the choice of parameterization.

Figure 6: Comparison of proton mean transverse momentum and midrapidity yield
to data from NA49 et al. (2005) and STAR et al. (2017).

At this point, it is also interesting to take a look at the available
experimental data, even though we are aiming only at a qualitative study of
the impact of an net baryon chemical potential dependence in the transport
coefficients. In Fig. 6, the mean transverse momentum and midrapidity yield
for protons are compared between the parameterization in the energy density
with vanishing $S_{\rho}$, the original, constant shear viscosity and
experimental values. The choice of protons for this comparison is motivated by
the available data. In both cases, the agreement is fairly similar and
experimental values are reproduced in a reasonable fashion.

###  III.3 Elliptic flow

Figure 7: Integrated event plane elliptic flow of charged hadrons at
midrapidity ($|y|<0.5$) for different parametrization strategies (left) and
different values of the net baryon number density dependence (right).

The anisotropic flow coefficients are very sensitive to the shear viscosity
Magdy  _et al._ (2021), especially higher order flow coefficients. These are
however only accessible with high statistics, which is the reason why we
restrict ourselves to the integrated $v_{2}$ in collisions at 20%-30% only.
The results can be found in Fig. 7 for the excitation function of $\langle
v_{2}\rangle$. We notice first of all that the energy density dependent
parameterization leads to a significantly reduced elliptic flow in comparison
to the investigated alternatives, which can be explained by the higher value
of the effective shear viscosity. In contrast the different values of
$S_{\rho}$ have only minimal effects on the flow which are mostly restricted
to low collision energies. This might be a hint that the late phases of the
hydrodynamic evolution are more important for the flow than the early stages.

Figure 8: Comparison of charged hadron integrated event plane elliptic flow at
midrapidity to STAR data et al. (2018).

Comparing the integrated flow of charged hadrons with experimental data, the
reduction in flow introduced with $\eta/s(\epsilon)$ increases disagreement
with experimental data. Except for low energies, where also the original
configuration of SMASH-vhlle-hybrid is missing the data point due to the short
hydrodynamic evolution, the agreement in the original configuration is better
than with $\eta/s(\epsilon)$. This again does not mean that the net baryon
chemical potential dependence can be neglected, but is rather a result of our
qualitative approach, where we did not attempt to fit other parameters of the
approach at the same time.

#### Impact of switching energy density

Figure 9: Contribution to the elliptic flow from the different stages of the
hybrid simulation, for different values of the $\epsilon_{\text{switch}}$.
Top: 0.1 GeV/fm3, middle: 0.3 GeV/fm3, bottom 0.5 GeV/fm3.

As pointed out before, the duration of the hydrodynamic evolution has an
important effect on the role the parameterization of $\eta/s$ for the
observables, as the viscous corrections are applied to the system for a longer
period. The time spent during the hydrodynamic evolution is defined by the
point, when the last fluid element reaches an energy density smaller than
$\epsilon_{\text{switch}}$. Depending on the value of
$\epsilon_{\text{switch}}$, the system spends more time in the hydrodynamic
evolution and less in the transport calculation, or vice versa. This, in turn,
also determines in which part of the hybrid approach elliptic flow is
generated. Fig. 9 builds on an investigation performed in Auvinen and Petersen
(2013) and compares the elliptic flow in the initial condition with the
elliptic flow without rescattering, which approximates the flow at the end of
the hydrodynamic evolution, and the flow of the full evolution. The
calculations were done choosing the default constant shear viscosity and
$\epsilon_{\text{switch}}$=0.5, 0.3 and 0.1 GeV/fm3.

The comparison of the flow contribution of the different phases of the hybrid
approach shows that for increasing $\epsilon_{\text{switch}}$ and decreasing
collision energy, more and more of the flow is generated in the hadronic
transport evolution. For $\epsilon_{\text{switch}}$=0.5 GeV/fm3, the
contribution of the hydrodynamic evolution is only of significant importance
from $\sqrt{s_{NN}}$=39 GeV on, whereas for a value of 0.1 GeV/fm3, the
afterburner stage does not contribute anymore to the observed elliptic flow.

Figure 10: Integrated event plane elliptic flow of charged hadrons at
midrapidity ($|y|<0.5$) for different parametrization strategies (top) and
different values of the net baryon number density dependence (bottom), for
$\epsilon_{\text{switch}}$ decreased to 0.1 GeV/fm3 (left) and increased to
0.5 GeV/fm3 (right). Figure 11: Integrated event plane elliptic flow of
charged hadrons at midrapidity ($|y|<0.5$) for different parametrization
strategies and values of $\epsilon_{\text{switch}}$.

In light of this insight, Fig. 10 shows the integrated charged hadron elliptic
flow for the cases $\epsilon_{\text{switch}}$=0.5 GeV/fm3 and 0.1 GeV/fm3 and
should be compared with 7. Independent of $\epsilon_{\text{switch}}$, the
impact of $S_{\rho}$ is virtually independent of the switching energy density.
The elliptic flow for $\eta/s(T)$ and constant $\eta/s$ changes significantly
when a different value of $\epsilon_{\text{switch}}$ is chosen. On the other
hand, this is not the case for $\eta/s(\epsilon)$.

This becomes more clear in Fig. 11. Here, the integrated elliptic flow of
charged hadrons is plotted at all three investigated values for
$\epsilon_{\text{switch}}$ for the default choice of $\eta/s$ as well as the
parameterizations in $T$ and $\epsilon$. We see that the lines for constant
$\eta/s$ and $\eta/s(T)$ differ strongly for changing values of
$\epsilon_{\text{switch}}$ \- with reducing $\epsilon_{\text{switch}}$, the
flow strongly increases. In contrast, for $\eta/s(\epsilon)$, the lines stay
virtually on top of each other, which in turn means that the flow is, for this
range of $\epsilon_{\text{switch}}$, independent of the switching energy
density for $\eta/s(\epsilon)$.

As mentioned, the elliptic flow is strongly sensitive to the shear viscosity.
When increasing $\epsilon_{\text{switch}}$, parts of the late stage evolution
which were before performed in the hydrodynamic evolution, where the shear
viscosity is defined by the input parameterization, are then performed in the
hadronic transport, where the shear viscosity is not directly accessible.

This, in turn, means that the independence of the integrated flow from the
value of $\epsilon_{\text{switch}}$ for $\eta/s(\epsilon)$ is a strong sign
that $\eta/s(\epsilon)$ approximates the shear viscosity in the non-
equilibrium hadronic transport stage. To phrase this important finding
differently, the independence of the elliptic flow on the value of the
switching transition energy density gives confidence that the viscous
hydrodynamic and transport descriptions are equivalent over a broad region in
the phase diagram.

##  IV Conclusions and Outlook

In this work, a new parameterization of the shear viscosity over entropy ratio
as a function of energy density and net baryon number density,
$\eta/s(\epsilon,\rho)$ based on known constraints is tested within the hybrid
approach SMASH-vHLLE-hybrid. The parameterization is compared to a temperature
dependent parameterization extracted from Bayesian inference as well as with
the default setting of SMASH-vHLLE-hybrid, a constant value of $\eta/s$.
Additionally, the impact of a $\rho$ dependence was investigated. For this
comparison, the midrapdity yield and midrapidity mean transverse momentum of
central collisions and the integrated elliptic flow of collisions at 20% – 30%
centrality at $\sqrt{s_{NN}}$= 7.7 GeV, 17.3 GeV and 39.0 GeV were studied.
The dependence on the net baryon number number does have no significant effect
on the flow as its effect on the shear viscosity is limited to the early
stages of the hydrodynamic evolution. Nevertheless, it introduces a
significant difference in the yield and mean transverse momentum. In
comparison with alternative parameterizations, $\eta/s(\epsilon)$ reproduces
proton midrapidity yield and mean transverse momentum well, but underestimates
the elliptic flow. In a further study, it is shown that significant amounts of
the elliptic flow originate from the hadronic transport in the initial
condition and the rescattering phase. This is dependent on the collision
energy, as this increases the lifetime of the fireball in the hydrodynamic
evolution, as well as on the switching energy density. The flow is sensitive
to the switching energy density when using constant or temperature dependent
$\eta/s$, but stays at the same values for $\eta/s(\epsilon)$. This shows that
the parameterization in the energy density approximates the shear viscosity in
non-equilibrium hadronic transport within the region of $\epsilon$ = 0.1–0.5
GeV/fm3.

For this study, bulk viscosity is neglected, but the effect of bulk viscosity
on observables is expected to be significant. Since it is expected to peak
around the phase transition, a parameterization of the bulk viscosity in the
energy density is worthwhile to explore, too. Additionally, the introduction
of non-constant viscosities affects the evolution of the fireball and can lead
to a change in the effective shear viscosity. A higher bulk viscosity could,
for example, slow down the expansion of the medium, which leads to higher
energy densities and reduced shear viscosity. Additionally, the range of
collision energies has been restricted to up to 39.0 GeV in order to study
regions where $\mu_{B}$ is high enough to see significant effects. The
presented results show that, although the difference in the effective shear
viscosity due to a non-vanishing $S_{\rho}$ are decreasing, the difference in
midrapidity yield and mean transverse momentum are still significant due to
the increased fireball lifetime. Therefore, it might be promising to include
higher beam energies in the future. Additionally, increased statistics could
give access to $v_{3}$, which has a considerably increases sensitivity to
$\eta/s$.

## Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project number 315477589 – TRR 211. N.G. acknowledges
support by the Stiftung Polytechnische Gesellschaft Frankfurt am Main as well
as the Studienstiftung des Deutschen Volkes. Computational resources have been
provided by the GreenCube at GSI.

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