# Extraction of the microscopic properties of quasi-particles using deep
neural networks

Olga Soloveva<EMAIL_ADDRESS>Helmholtz Research Academy Hesse
for FAIR (HFHF), GSI Helmholtz Center for Heavy Ion Physics, Campus Frankfurt,
60438 Frankfurt, Germany  Institut für Theoretische Physik, Johann Wolfgang
Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
Andrea Palermo<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794–3800, USA  Elena Bratkovskaya  Helmholtz Research Academy Hesse for
FAIR (HFHF), GSI Helmholtz Center for Heavy Ion Physics, Campus Frankfurt,
60438 Frankfurt, Germany  Institut für Theoretische Physik, Johann Wolfgang
Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
GSI Helmholtzzentrum für Schwerionenforschung GmbH,Planckstrasse 1, D-64291
Darmstadt, Germany

###### Abstract

We use deep neural networks (DNN) to obtain the microscopic characteristics of
partons in terms of dynamical degrees of freedom on the basis of an off-shell
quasiparticle description. We aim to infer masses and widths of quasi-gluons,
up/down, and strange quarks using constraints on the macroscopic thermodynamic
observables obtained by the first-principles calculations lattice QCD. In this
work, we use 3 independent dimensionless thermodynamic observables from lQCD
for minimization. First, we train our DNN using the DQPM (Dynamical
QuasiParticle Model) Ansatz for the masses and widths. Furthermore, we use the
DNN capabilities to generalize this Ansatz, to evaluate which quasiparticle
characteristics are desirable to describe different thermodynamic functions
simultaneously. To evaluate consistently the microscopic properties obtained
by the DNN in the case of off-shell quarks and gluons, we compute transport
coefficients using the spectral function within Kubo-Zubarev formalism in
different setups. In particular, we make a comprehensive comparison in the
case of the dimensionless ratios of shear viscosity over entropy density
$\eta/s$ and electric conductivity over temperature $\sigma_{Q}/T$, which
provide additional constraints for the parameter generalization of the
considered models.

relativistic heavy ion collisions, transport coefficients, quark-gluon plasma

##  I Introduction

In the exploration of quark-gluon plasma (QGP) matter produced during heavy-
ion collisions, a critical step is establishing the equation of state (EoS)
and transport coefficients of the matter corresponding to a specific range of
parameters, such as energy density, which is associated with temperature, and
baryon density, associated with baryon chemical potential. Moreover, for the
fully dynamical evolution of the system, it is required to consider
microscopic characteristics for the partonic degrees of freedom. Although
first-principles Lattice Quantum Chromodynamics (lQCD) simulations provide
vital predictions, especially for temperatures above the critical temperature
$T_{c}$, they are complex and challenging to execute. While lQCD has
successfully outlined the main thermodynamic quantities, finer details of
microscopic characteristics and dynamic properties, such as transport
coefficients, remain somewhat elusive. These properties are crucial as they
depend on the microscopic interactions among quarks and gluons. However,
assessing the microscopic properties of QGP matter at finite temperatures and
baryon chemical potentials from first principles is a challenging task. To
bridge this gap, it is worthwhile to investigate new methodologies that can
further elucidate the established lQCD predictions, thereby enhancing our
understanding of the QGP matter in these extreme conditions. In the past
decade, methods of machine learning (ML) have been developed as a powerful
computational tool and novel problem-solving perspective for physics, which
offers new avenues for exploring strongly interacting QCD matter under extreme
conditions such as finite $T$ and baryon chemical potential [1, 2, 3, 4, 5, 6,
7, 8, 9].

Generally, ML stands apart from conventional minimization methods by aiming
for predictions instead of just fitting, thus offering enhanced adaptability
for applications in physics [10, 11]. In the context of Relativistic Heavy-Ion
Collisions (HICs) and phenomenology of the QGP, this includes the use of novel
techniques such as active learning [12], transfer learning [13], among other
methods [14, 15, 16, 17]. In this work, we use machine learning techniques, in
particular, Deep Neural Networks (DNNs) [18, 19], to gain insights into the
phenomenological description of deconfined QCD matter and test the
applicability of these methods to phenomenology. We confine ourselves to the
case of $2+1$ flavours and colours QCD, $N_{f}=N_{c}=3$, with degenerate light
quarks at vanishing quark chemical potential, $\mu_{q}=0$. To train the DNNs
used in this work we use lQCD data provided by the WB Collaboration, in
particular results on the entropy density $s(T)$ and baryon susceptibilities
$\chi_{2}^{B}(T),\chi_{2}^{S}(T)$ [20, 21] $(T\leq 3T_{c})$ and latest
estimates for the strange susceptibility $\chi_{2}^{S}(T)$ [22] $(T\leq
1.7T_{c})$. We choose these thermodynamic quantities in order to fix
consistently the strange quark, the light quark, and gluon sectors
independently. The functional form of the masses and widths of quarks and
gluons is inferred by multi-layer Feed-forward neural networks (FFNNs), which
are trained on the entropy density, baryon, and strange susceptibilities. The
feasibility of this approach is grounded in the universal approximation
theorem [23, 24]. This theorem states that multi-layer feed-forward neural
networks (FFNNs), owing to their non-linear characteristics and given a
sufficient number of hidden neurons, can effectively approximate any function.

To evaluate the thermodynamic quantities required for the training of DNNs we
consider an off-shell quasi-particle description of the QGP, where quasi-
quarks and gluons acquire a thermal width as in the original Dynamical Quasi-
Particle Model (DQPM) [25, 26, 27, 28, 29, 30, 31].

The DQPM is based on a propagator representation with effective (anti-)quarks
and gluons whose properties are defined by complex self-energies and off-shell
spectral functions. The advantage of the DQPM compared to other QPM is that it
is by construction a 2PI (two particle irreducible) model while the other on-
shell quasiparticle models are 1PI (one particle irreducible) in nature. This
avoids the introduction of an extra “bag constant” often employed for on-shell
quasi-particle models in order to describe the thermodynamic properties of the
QGP.

Quasiparticle models are highly favored for their ease of integration into
transport frameworks, which are crucial for simulating the evolution of QGP
matter. Specifically, the DQPM has been incorporated into the Parton-Hadron-
String Dynamics (PHSD) transport approach [32, 29]. Meanwhile, the
Quasiparticle Model (QPM) has been adopted in the Catania transport approach
[33] and [34]. Additionally, recent developments have seen the implementation
of approximate QCD models, like the Nambu–Jona-Lasinio (NJL) and
Polyakov–Nambu–Jona-Lasinio (PNJL) models, within the A Multi-Phase Transport
(AMPT) model [35]. These implementations utilize scalar and vector potentials
to approximate QCD dynamics.

In addition to analyzing thermodynamic quantities, we also explore transport
coefficients, using the resulting microscopic properties, i.e. spectral
functions. Our previous systematic comparison reveals that these transport
coefficients are significantly influenced by the properties of the constituent
degrees of freedom [36]. This leads to the observation that different
theoretical models, despite having an almost identical EoS, can yield markedly
different transport coefficients. The goal of this study is to explore the
properties of strongly interacting quasi-particles, i.e. the $T$\- dependence
of masses and widths which are consistent with the lQCD thermodynamics at
vanishing chemical potential, by using DNN methods. The techniques used here
can be also applied at finite $\mu_{b}$. Starting from the DQPM Ansatz we
gradually generalize it, aiming at a simultaneous description of both the
entropy density $s(T)$ and the susceptibilities $\chi_{2}^{B,S}(T)$.

In general, a substantial amount of data is essential to effectively train a
neural network to uncover established physics. In the phenomenology of the
QGP, these get even worse since the only available lQCD data represents a
small amount of points. Therefore the simple architecture and small number of
hyperparameters of FFNNs compared to CNNs and ResNets is beneficial for this
task.

The structure of the paper is as follows. In Sec. II we briefly describe the
framework of the off-shell Quasi-particle model including the evaluation of
thermodynamic observables and transport coefficients. In Sec. III we describe
our method to determine the microscopic quantities using DNNs, starting from
the DQPM and generalizing it further in Sec. IV. Finally, our results and
conclusions are summarized in Sec. V.

##  II  Framework of the off-shell quasiparticle model

In the quasiparticle approach, the degrees of freedom are strongly interacting
dynamical quasiparticles: quarks and gluons with a broad spectral function,
whose ‘thermal’ masses and widths increase with temperature. The microscopic
and macroscopic properties of the off-shell quasiparticles are defined by the
complex self-energies and spectral functions. The spectral functions can be
taken in various forms. Here we consider the most successful physically
motivated Lorentzian form [26]:

$\displaystyle\rho_{i}(\omega,{\bf p})=$
$\displaystyle\frac{\gamma_{i}}{\tilde{E}_{i,\mathbf{p}}}\left(\frac{1}{(\omega-\tilde{E}_{i,\mathbf{p}})^{2}+\gamma_{i}^{2}}-\frac{1}{(\omega+\tilde{E}_{i,\mathbf{p}})^{2}+\gamma_{i}^{2}}\right)\
$
$\displaystyle=\frac{4\omega\gamma_{i}}{\left(\omega^{2}-\mathbf{p}^{2}-m^{2}_{i}\right)^{2}+4\gamma^{2}_{i}\omega^{2}},$
(1)

where $i$ runs over the particle species ($i=q,{\bar{q}},g$). Here, we
introduced the off-shell energy $\tilde{E}_{i,\mathbf{p}}=\sqrt{{\bf
p}^{2}+m_{i}^{2}-\gamma_{i}^{2}}$, $m_{i}$ and $\gamma_{i}$ being the
particle’s pole mass and width. $\rho_{i}$ is a real function, odd in $\omega$
and for all ${\bf p}$ it fulfills the sum rule

$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\omega\rho_{i}(\omega,{\bf p})=1.$
(2)

Using the spectral function, the quasiparticle (retarded) propagators can be
expressed in the Lehmann representation as:

$\Delta_{i}(\omega,{\bf
p})=\int_{-\infty}^{\infty}\frac{d\omega^{\prime}}{2\pi}\frac{\rho_{i}(\omega^{\prime},{\bf
p})}{\omega-\omega^{\prime}}=\frac{1}{\omega^{2}-{\bf
p}^{2}-m_{i}^{2}+2i\gamma_{i}\omega}.$ (3)

In (3), we are considering all the (retarded) quasiparticle self-energies to
be equal, $\Pi=\Sigma=\Sigma_{q}\approx\Sigma_{\bar{q}}$, and they are
expressed via dynamical masses and widths as:

$\Pi_{i}=m_{i}^{2}-2i\gamma_{i}\omega.$ (4)

In the off-shell case, $\omega$ is an independent off-shell energy.

###  II.1 DQPM Ansatz

In the DQPM, the microscopic quantities, i.e. effective masses and thermal
widths, depend on the effective coupling constant which, in turn, acquires an
explicit temperature and chemical potential dependence [29]. This is the main
characteristic of the DQPM parametrization used for our first model study.

Now we briefly recall the main details of the parametrization of the DQPM,
which has been studied in many variations in Refs [25, 26, 27, 28, 29, 30,
31]:
$\bullet$ In the DQPM, the coupling constant is adjusted by fixing the
quasiparticle entropy density to reproduce the entropy density
$s(T,\mu_{\mathrm{B}}=0)$ from lQCD, e.g. [21, 20] (see a parametrization
method introduced in Ref. [37]). It has been shown that, for a given value of
$g^{2}$, the ratio $s(T,g^{2})/T^{3}$ is almost constant for different
temperatures but identical $g^{2}$, i.e. ${\frac{\partial}{\partial
T}}(s(T,g^{2})/T^{3})=0$. Therefore the entropy density $s$ and the EoS in the
DQPM is a function of $g^{2}$ only, i.e. $s(T,g^{2})/s_{SB}(T)=f(g^{2})$ where
$s^{QCD}_{SB}=19/9\pi^{2}T^{3}$ corresponds to the Stefan-Boltzmann limit of
the entropy density for massless quarks and gluons. Thus, by inverting the
$f(g^{2})$ function, the coupling constant $g^{2}$ can be directly obtained
from the parametrization of lQCD data for the entropy density
$s(T,\mu_{B}=0)$. The resulting parametrization for the coupling constant
reads [28]:

$g^{2}(T,\mu_{\mathrm{B}}=0)=d\cdot\Big{[}\left(s(T)/s^{\mathrm{QCD}}_{\mathrm{SB}}\right)^{e}-1\Big{]}^{f}.$
(5)

Here the parameters $d=169.934$, $e=-0.178434$, and $f=1.14631$ are fixed in
accordance with the $s(T)$ calculations by the WB Collaboration from Refs.
[21, 20].

$\bullet$ The exact form of pole masses, in the DQPM, depends on the effective
coupling constant as predicted by Hard Thermal Loop (HTL) calculations in the
high-temperature regime [38, 39]. Considering three flavors and colors,
$N_{f}=N_{c}=3$, and for vanishing chemical potential, $\mu_{q}=0$, the gluon
($g$) and light quarks ($u$ and $d$, both labeled with $l$) masses read [38,
39]:

$\displaystyle m^{2}_{g}(T)$
$\displaystyle=C_{a}\frac{g^{2}(T)}{6}T^{2}\left(1+\frac{N_{f}}{2N_{c}}\right)=\frac{3}{4}g^{2}(T)T^{2},$
(6a) $\displaystyle m^{2}_{l(\bar{l})}(T)$
$\displaystyle=C_{f}\frac{g^{2}(T)}{4}T^{2}=\frac{1}{3}g^{2}(T)T^{2},$ (6b)

where $C_{F}=\dfrac{N_{c}^{2}-1}{2N_{c}}=4/3$ and $C_{A}=N_{c}=3$ are the QCD
color factors for quarks and gluons, respectively. It is important to note
that here $g^{2}(T)$ is the effective temperature-dependent coupling constant.

$\bullet$ The strange quark mass, whose scaling has not been studied in detail
within the HTL framework, is given by the phenomenological equation:

$m_{s(\bar{s})}(T)=m_{l(\bar{l})}(T)+\Delta m_{ls},$ (7)

where $\Delta m_{ls}=0.03$ GeV is a constant mass shift. Later on, we will
consider how the change of this parameter affect transport coefficients.
However, this simple shift is justified by the larger bare mass of the strange
quark, which enhances its dynamic mass. Previously, the value of $\Delta
m_{ls}$ has been fixed by comparing experimental data for strange hadrons
abundances and the $K^{+}/\pi^{+}$ ratio in relativistic heavy-ion collisions
using the PHSD, a microscopic covariant transport approach [29]. There, the
microscopic properties of the partonic degrees of freedom are described by the
DQPM (see latest results and discusions in [30, 29]).

$\bullet$ In contrast to the on-shell quasi-particle models [33, 40], where
thermal widths are absent and a bag constant must be introduced, in the DQPM
the thermal widths are chosen to follow the smooth increase of the interaction
rate $\Gamma$ with T. Therefore, the thermal widths have a physical meaning,
as they reflect the collision frequency of particles at finite temperature T.
In the DQPM parametrization, the thermal widths read [29, 28, 39]:

$\gamma_{i}(T)=\frac{1}{3}C_{i}\frac{g^{2}(T)T}{8\pi}\ln\left(\frac{2c_{m}}{g^{2}(T)}+1\right).$
(8)

The constant parameter $c_{m}=14.4$ was fixed in Ref. [26] and is related to a
magnetic cut-off. The finite width reflects the dynamical modification of the
spectral function of quasiparticles during their propagation in the sQGP
medium.

Furthermore, the thermal widths of all (anti-)quarks are assumed to be equal
and completely fixed by $g^{2}$: $\gamma_{u}=\gamma_{d}=\gamma_{s}$. This
assumption has been checked in Ref. [29], where it was shown that the
interaction rates of strange and light quarks coincide, apart from the small
difference in the vicinity of the phase transition $T<1.5T_{c}$. In the next
section, we will discuss possible generalizations of this model using DNNs.

###  II.2 Thermodynamics

In this subsection, we detail the precise method by which thermodynamic
observables are derived from the microscopic characteristics of off-shell
quasiparticles. For the analysis of the main thermodynamic properties within
thermal QCD, employing the $\Phi$-functional approach is beneficial. This
technique represents the thermodynamic potential $\Omega$ via the dressed
propagators $\Delta_{i}$ [41, 42]. Following this representation, the
derivatives of $\Omega$ can be efficiently calculated. In this formalism, the
entropy density $s^{\mathrm{dqp}}=-\dfrac{\partial\Omega/V}{\partial T}$ for
off-shell quasiparticles can be written as:

$s^{dqp}=2\int_{0}^{+\infty}\dfrac{d\omega dp\
p^{2}}{4\pi^{3}}F_{s}(\omega,p,T,\mu),$ (9)

with

$\displaystyle F_{s}(\omega,p,T,\mu)=-d_{g}\frac{\partial
f_{g}(\omega)}{\partial
T}\left(\mathrm{Im}(\ln\Delta^{-1})-\mathrm{Im}\Pi\mathrm{Re}\Delta\right)$
$\displaystyle-d_{q}\frac{\partial f_{q}(\omega-\mu_{q})}{\partial
T}\left(\mathrm{Im}(\ln
S_{q}^{-1})-\mathrm{Im}\Sigma_{q}\mathrm{Re}S_{q}\right)$ $\displaystyle-
d_{\bar{q}}\frac{\partial f_{\bar{q}}(\omega+\mu_{q})}{\partial
T}\left(\mathrm{Im}(\ln
S_{\bar{q}}^{-1})-\mathrm{Im}\Sigma_{\bar{q}}\mathrm{Re}S_{\bar{q}}\right),$

where $\Delta_{i}=(p^{2}-\Pi_{i})^{-1}$, $S_{q}=(p^{2}-\Sigma_{q})^{-1}$ and
$S_{\bar{q}}=(p^{2}-\Sigma_{\bar{q}})^{-1}$ stand for the full (scalar)
quasiparticle propagator of gluons $g$, quarks $q$, and antiquarks
${\bar{q}}$.
Similarly, the quark density of the off-shell quasi-particles
$n^{\mathrm{dqp}}=-\dfrac{\partial\Omega/V}{\partial\mu}$ reads:

$n^{dqp}=2\int_{0}^{+\infty}\dfrac{d\omega dp\
p^{2}}{4\pi^{3}}F_{n}(\omega,p,T,\mu),$ (10)

with

$\displaystyle F_{n}(\omega,p,T,\mu)=-d_{q}\frac{\partial
f_{q}(\omega-\mu_{q})}{\partial\mu_{q}}\left(\mathrm{Im}(\ln
S_{q}^{-1})-\mathrm{Im}\Sigma_{q}\mathrm{Re}S_{q}\right)$ $\displaystyle-
d_{\bar{q}}\frac{\partial
f_{\bar{q}}(\omega+\mu_{q})}{\partial\mu_{q}}\left(\mathrm{Im}(\ln
S_{\bar{q}}^{-1})-\mathrm{Im}\Sigma_{\bar{q}}\mathrm{Re}S_{\bar{q}}\right).$

In the above formulae, $d_{g}=2\times(N_{c}^{2}-1)$ is the number of
transverse gluonic degrees of freedom while $d_{q}=d_{\bar{q}}=2\times N_{c}$
is the fermionic one. Furthermore, $f_{g,q}=f_{B,F}$ is the Bose or Fermi
distribution for gluon or quark respectively:

$f_{B,F}=\left[\exp\left((E_{i}-\mu_{i})/T\right)\pm 1\right]^{-1},$ (11)

where $\mu_{i}$ is the quark chemical potential. The quasi-particle energy is
$E_{i}=\sqrt{\mathbf{p}_{i}^{2}+m_{i}^{2}}$ for the on-shell case or
$\omega_{i}$ for the considered off-shell case.

Here we emphasize that the entropy and quark number density functionals are
not restricted to narrow quasiparticles, i.e. with spectral widths
$\gamma_{i}$ much smaller than the typical energy.

Baryon and strange charge densities follow from the quark densities:

$\displaystyle\langle n_{B}\rangle=n^{\mathrm{dqp}}/3=1/3(\langle
n_{u}\rangle+\langle n_{d}\rangle+\langle n_{s}\rangle)$ $\displaystyle\langle
n_{S}\rangle=-\langle n_{s}\rangle.$

At vanishing chemical potential, the properties of the quasi-particles are
explored using quark-number susceptibilities instead of the densities:

$\chi_{i}(T,m_{i},\gamma_{i})=\dfrac{\partial
n_{i}}{\partial\mu_{i}}\Biggr{|}_{\mu_{i}=0}.$ (12)

Quark-number susceptibilities are related to the pressure by

$\displaystyle\frac{\chi_{q}(T,\mu_{q})}{T^{2}}=\frac{\partial^{2}(P/T^{4})}{\partial(\mu_{q}/T)^{2}},$

and therefore can be related to the conventional second-order susceptibilities
used in the Taylor expansion $\chi_{2}^{ij}$ at vanishing quark chemical
potential as [21]

$\displaystyle\frac{P(T,{\mu_{i}})}{T^{4}}=\frac{P(T,{0})}{T^{4}}+\frac{1}{2}\sum_{i,j}\frac{\mu_{i}\mu_{j}}{T^{2}}\chi_{2}^{ij},$
$\displaystyle\textrm{with}\ \ \chi_{2}^{ij}=\frac{1}{T^{2}}\frac{\partial
n_{j}(T,{\mu_{i}})}{\partial\mu_{i}}\Biggr{|}_{\mu_{i}=\mu_{j}=0}.$ (13)

For the $N_{f}=3$ case and transverse gluons, the total entropy and the baryon
and strange susceptibilities read:

$\displaystyle s(T)$
$\displaystyle=-d_{g}I^{B}_{g}(T)-d_{q}\sum_{i=q,s}I^{F}_{i}(T),$ (14a)
$\displaystyle\chi_{2}^{B}(T)$
$\displaystyle=\frac{1}{T^{2}}\frac{d_{q}}{9}(2\chi_{2}^{l}(T)+\chi_{2}^{s}(T)),$
(14b) $\displaystyle\chi_{2}^{S}(T)$
$\displaystyle=\frac{1}{T^{2}}d_{q}\chi_{2}^{s}(T),$ (14c)

where we have introduced the integrals:

$\displaystyle I^{B,F}_{i}(T,m,\gamma)=$
$\displaystyle\frac{1}{2\pi^{2}T}\int\mathrm{d}\,p^{2}\frac{4p^{2}+3m_{i}^{2}}{3\sqrt{\omega^{2}+p^{2}}}f_{B,F}(\omega,T)+2\int_{0}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\partial
f_{B,F}(\omega,T)}{\partial T}h(\omega,p,m_{i},\gamma_{i}),$ (15a)
$\displaystyle\chi^{i}_{2}(T,m,\gamma)=$
$\displaystyle\frac{1}{2\pi^{2}T}\int_{0}^{\infty}\mathrm{d}p\,\frac{p^{2}}{1+\cosh\left(\sqrt{m_{i}^{2}+p^{2}}/T\right)}+2\int_{0}^{\infty}\frac{\mathrm{d}\omega}{2\pi}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{\sinh(\omega/T)}{T^{2}(1+\cosh(\omega/T))^{2}}h(\omega,p,m_{i},\gamma_{i}),$
(15b)

where the function $h$ is the auxiliary function:

$h(\omega,p,m_{i},\gamma_{i})=2\gamma_{i}\omega\frac{\omega^{2}-\bm{p}^{2}-m_{i}^{2}}{(\omega^{2}-\bm{p}^{2}-m_{i}^{2})^{2}+4\gamma_{i}^{2}\omega^{2}}-\arctan\left(\frac{2\gamma_{i}\omega}{\omega^{2}-\bm{p}^{2}-m_{i}^{2}}\right).$
(16)

Notice that all thermodynamical functions implicitly depend on $g^{2}$ through
the dressed masses and widths if we consider a specific parametric ansatz as
in the DQPM. An alternative approach involves a traditional perturbative
series expansion of the entropy with respect to the coupling constant $g$ as
in references [43, 44]:

$s/T^{3}=c_{0}+c_{2}g^{2}+c_{3}g^{3}+\dots$ (17)

Here, $c_{0}=\frac{\pi^{2}}{45}(4(N_{c}^{2}-1)+7N_{c}N_{f})$ corresponds to
the Stefan-Boltzmann limit. Moreover, there are different resummations
techniques for thermal QCD such as hard-thermal-loop perturbation theory
(HTLpt), which are not discussed here (for further reading we refer to Ref.
[45]). To enhance the convergence of the HTL resummation, a self-consistent
quasi-particle expansion has been adopted, as described in Ref. [42].

From the entropy density and quark densities, other thermodynamic quantities
follow. The pressure at vanishing baryon chemical potential $\mu_{B}=0$ is
defined employing the entropy density as

$\displaystyle p(T)=p^{lqcd}(T_{0})+\int\limits_{T_{0}}^{T}s(T^{\prime})\
dT^{\prime},$ (18)

where $p^{lqcd}(T_{0})$ is taken from lQCD after fixing $T_{0}$.

The energy density $\epsilon$ then follows from the Euler relation

$\epsilon=Ts-p.$ (19)

Another important thermodynamic observable is the trace of the energy-momentum
tensor, also known as interaction measure or trace anomaly:

$I=\epsilon-3p=Ts-4p.$ (20)

From lQCD calculations [46, 20, 22, 47], the trace anomaly $I$ is expected to
be sizeable in the vicinity of the cross-over transition, indicating a strong
interaction of the medium. Consequently, one can expect the shear viscosity
close to $T_{c}$ to be correspondingly smaller than for a weakly interacting
medium, as considered in the pQCD limit.

###  II.3 Transport coefficients

As complementary observables, transport coefficients can reveal how well the
considered models can microscopically describe the dense QGP medium.
Therefore, we aim to evaluate transport coefficients using the Kubo-Zubarev
formalism, where we employ the parton spectral functions without relying on
the relaxation time approximation or effective coupling constant. Since
quasiparticles masses and widths - obtained from the DNNs - can be larger than
those in the original DQPM, this method is better suited.

The shear viscosity is evaluated from the slope of the Fourier transform of
the spectral function for the spatial traceless part of the stress tensor
$\langle[\pi_{ij}(x),\pi_{ij}(0)]\rangle$ in the limit $\omega\rightarrow 0$.
Here we employ the following formula for the shear viscosity:

$\displaystyle\eta^{\text{Kubo}}(T)$
$\displaystyle=-\int\frac{d^{4}p}{(2\pi)^{4}}\ \sum_{i=q,\bar{q},g}d_{i}\
\frac{\partial f_{i}(\omega)}{\partial\omega}\
\rho_{i}(\omega,\mathbf{p})^{2}\Pi_{i}$ $\displaystyle=\frac{1}{15T}\int$
$\displaystyle\frac{d^{4}p}{(2\pi)^{4}}\ \sum_{i=q,\bar{q},g}d_{i}\left[(1\pm
f_{i}(\omega))f_{i}(\omega)\right]\rho_{i}(\omega,\mathbf{p})^{2}\Pi_{i},$
(21)

where the notation $f_{i}(\omega)=f_{i}(\omega,T,\mu_{q})=f_{B,F}$ is used for
the distribution functions. The corresponding derivative of the distribution
function accounts for the Pauli-blocking ($-$) and Bose-enhancement ($+$)
factors, and $\rho_{i}$ denotes the spectral functions from Eq. (1). Using the
notation $\Pi_{i=q,g}$ we differentiate the contribution from transverse
gluons [48, 25]:

$\Pi_{g}=7\omega^{4}-10(\omega\mathbf{p})^{2}+7\mathbf{p}^{4},$ (22)

and from quarks [48, 49]:

$\Pi_{q}=p_{x}^{2}p_{y}^{2}.$ (23)

We note that (for weak coupling) it is common to derive $\eta$ from a
Boltzmann equation to next-to-leading log (NLL) order [50]:

$\eta^{NLL}\approx\dfrac{T^{3}}{g^{2}\ln(1/g)}.$ (24)

This approach is suited for on-shell or narrow quasi-particles, whereas we
don’t assume small $g$ and employ the more rigorous Kubo-formalism. It is
expected that, near $T_{c}\approx 158$ MeV, the results of Eq. (24) are not
applicable.

We consider here another important transport coefficient, i.e. the electric
conductivity for stationary electric fields $\sigma_{Q}(T)$, which describes
the response of the system to an external electric field. In the case of the
electrical conductivity only quark degrees of freedom contribute, resulting in
an analogous expression [51]:

$\displaystyle\sigma_{Q}^{\text{Kubo}}(T)$
$\displaystyle=-\int\frac{d^{4}p}{(2\pi)^{2}}\ \sum_{i=q,\bar{q}}d_{i}\
\frac{\partial f_{i}(\omega)}{\partial\omega}\rho_{i}(\omega,\mathbf{p})^{2}$
$\displaystyle=\frac{1}{3T}\int$ $\displaystyle\frac{d^{4}p}{(2\pi)^{4}}\
\mathbf{p}^{2}\sum_{i=q,\bar{q}}d_{i}\left[(1\pm
f_{i}(\omega))f_{i}(\omega)\right]\rho_{i}(\omega,\mathbf{p})^{2},$ (25)

for $q=u,d,s$. By separating the contributions of the strange flavor one can
also single out the strange quark’s contribution to the conductivity, which
gives us a complementary observable to infer the properties of the strange
quarks.

##  III  Neural networks for the regression applied to the quasi-particle
model

In this section, we describe the technical details of our framework, covering
aspects such as the input and output observables, the architecture of the
DNNs, the training process, and the evaluation methods. The code of the neural
network is written in Python and it is implemented using the Keras Deep
Learning API v2.13.1 [52] together with the Tensorflow v2.13.0 library [53].

For the regression task, we utilize Feed-forward neural networks (FFNNs),
which are a preferred choice over CNN especially for simple, non-spatially-
distributed datasets, due to their efficiency and the smaller number of
hyperparameters, allowing a faster training. The number of layers of the DNNs
is chosen heuristically, balancing between the quality of the fit, the smaller
number of parameters, and the training speed. Three hidden layers, with 24,
12, and 12 neurons respectively, turn out to be sufficient for the purpose of
this study. For all layers, the activation function is _sigmoid_ :

$\sigma(x)=\frac{1}{e^{x}+1}.$ (26)

For the minimization procedure, we employ the Adam algorithm [54]. For all of
the presented results, the learning rate is initially set to 0.005 and decays
by 90% every 1000 epochs of the training. The number of epochs used in the
training is $4\times 10^{3}$, and we observed that increasing this number by a
factor of three doesn’t change our results and leads to overfitting.

In this study, we use machine learning to infer the functional form of
microscopic quasiparticles properties (masses, widths and coupling constant)
as a function of temperature, such that the formulae obtained with the
$\Phi$-functional approach - detailed in Sec. II.2 \- are in agreement with
lattice QCD data. Therefore, we consider a DNN-based DQPM model, which we will
refer to as DQPMnn. The input of the DQPMnn is the temperature, expressed in
GeV, in the range $T>T_{c}$, $T_{c}=0.158$ GeV in accordance with the employed
lQCD data from the WB collaboration.

We will explore two distinct regression tasks. As outputs for these tasks, we
consider (i) the squared coupling constant $g^{2}(T)$; and (ii) a combination
of $g^{2}(T)$, the masses $m_{i}(T)$, and the widths $\gamma_{i}(T)$ of the
quasi-particles. The next subsections describe the details of the loss
function, and the extraction of the physical properties of the quasiparticles.

###  III.1 DQPMnn: $\mathcal{L}_{0}$ \- extraction of $g^{2}(T)$

The original idea of DQPM relies on adjusting the effective coupling to fit
the entropy density, employing an HTL-based parametrization for the effective
masses and widths. In this section, as proof of principle, we employ the DQPM
parametrization, Eqs. (6), (7) and (8), and compare the outputs of the DQPMnn
to different lattice observables, such as the EoS and the transport
coefficients. In what follows, and throughout the paper, in order to
distinguish quantities from the the DQPMnn, we use an underline to identify
them. For example, $\underline{g}^{2}(T)$ is the function associated with the
running coupling output by the DQPMnn, whereas $g^{2}(T)$ is the one used in
the original DQPM.

To extract the coupling constant $\underline{g}^{2}(T)$ with the DQPMnn, we
use a loss function:

$\displaystyle\mathcal{L}_{0}=\beta_{G}\left[\frac{s(T)/T^{3}-s_{\text{lQCD}}/T^{3}}{\Delta
s_{lQCD}/T^{3}}\right]^{2}$
$\displaystyle+\beta_{L}\left[\frac{\chi_{2}^{B}(T)-{\chi^{B}_{2}}_{\text{lQCD}}}{\Delta{\chi^{B}_{2}}_{lQCD}}\right]^{2}$
$\displaystyle+\beta_{S}\left[\frac{\chi_{2}^{S}(T)-{\chi^{S}_{2}}_{\text{lQCD}}}{\Delta{\chi^{S}_{2}}_{lQCD}}\right]^{2}.$
(27)

The minimization of the above loss function is analogous to the minimization
of the $\chi^{2}$ in the standard fitting procedures, but the parameters
$\beta_{G},\beta_{L},\beta_{S}$ allow to regulate the contribution of each
thermodynamic quantity to the loss function. Notice that the loss function in
eq. (27) is not only a minimum squared error, as it was used for example in
[40], but also takes into account the uncertainties associated with the
lattice measurements.

The case $\beta_{G}=1$ and $\beta_{L}=\beta_{S}=0$, where microscopic
quantities are inferred from the entropy density only, should reproduce the
results of the DQPM. The thermodynamic functions are computed from Eqs. (14)
and the subscript “lQCD” labels the lattice data, which have been taken from
[46, 20, 22]. All the thermodynamic functions appearing in the loss have an
implicit dependence on $\underline{g}(T)$ through the mass and width of the
different quasi-particles. To be more explicit, one should write, for example,
$s(T)\rightarrow s(T,\underline{g}(T)$), but we choose to simplify the
notation and drop the implicit dependence of the thermodynamic functions.

Notice that, to train the DQPMnn, we have chosen thermodynamic quantities that
are dimensionless. The main advantage of the use of dimensionless observables
such as $s/T^{3}$, $\chi_{2}^{B}$, $\chi_{2}^{S}$ is that they mitigate the
impact of discrepancies between lattice EoS results obtained by different
groups: HotQCD [47] or WB Collaboration [20, 21, 22] results influence less
the final results for the desired microscopic quantities: masses, widths. In
contrast, if we consider $s(T)$ in $\text{GeV}^{3}$ as an input for the
training, the difference between the two EoS would be larger. This is in
contrast with a recent work, Ref. [40], where DNNs were used for an on-shell
quasi-particle model with a bag constant and $s(T)$ and $\Delta=I(T)$ were
used for training.

However, the loss function in Eq. (27) requires the evaluation of
thermodynamic integrals at each step during the training process. While the
computation of one of the integrals in Eqs. (15) is rather fast, the frequency
at which they must be calculated renders the use of Eq. (27) not optimal, as
it would significantly prolong the training duration. To overcome this
complication, we use other 2 neural networks as _surrogate models_ to
approximate the integrals in Eqs. (15), and hence $s(T)$ and
$\chi_{2}^{B,S}(T)$. These neural networks take $T$, $m/T$ and $\gamma/T$ as
input and are trained to reproduce the values of the integrals $I^{B,F}$ and
$\chi^{i}_{2}(i=l,s)$ in Eqs. (15). A schematic representation of the
surrogate models is given in Fig. 2. Denoting the functions resulting from the
surrogate model with a tilde ($\tilde{I}^{F,B}$ and $\tilde{\chi_{2}}$) to
distinguish them from the actual values of Eqs. (15), we can use a loss
function with the substitutions $s\rightarrow\tilde{s}$ and
$\chi_{2}^{B,S}\rightarrow\tilde{\chi}_{2}^{B,S}$, where the tilded
thermodynamic functions are computed from Eqs. (14) using $\tilde{I}^{F,B}$
and $\tilde{\chi}_{2}$ in place of $I^{F,B}$ and $\chi_{2}$:

$\displaystyle\tilde{s}(T)$
$\displaystyle=-d_{g}\tilde{I}^{B}_{g}(T)-d_{q}\sum_{i=q,s}\tilde{I}^{F}_{i}(T),$
(28a) $\displaystyle\tilde{\chi}_{2}^{B}(T)$
$\displaystyle=\frac{1}{T^{2}}\frac{d_{q}}{9}(2\tilde{\chi}_{2}^{l}(T)+\tilde{\chi}_{2}^{s}(T)),$
(28b) $\displaystyle\tilde{\chi}_{2}^{S}(T)$
$\displaystyle=\frac{1}{T^{2}}d_{q}\tilde{\chi}_{2}^{s}(T).$ (28c)

The use of the surrogate model leads to a significant improvement in the speed
of the training. The actual loss function used in the training therefore
becomes:

$\displaystyle\mathcal{L}_{0}=\beta_{G}\left[\frac{\tilde{s}(T)/T^{3}-s_{\text{lQCD}}/T^{3}}{\Delta
s_{lQCD}/T^{3}}\right]^{2}$
$\displaystyle+\beta_{L}\left[\frac{\tilde{\chi}_{2}^{B}(T)-{\chi^{B}_{2}}_{\text{lQCD}}}{\Delta{\chi^{B}_{2}}_{lQCD}}\right]^{2}$
$\displaystyle+\beta_{S}\left[\frac{\tilde{\chi}_{2}^{S}(T)-{\chi^{S}_{2}}_{\text{lQCD}}}{\Delta{\chi_{2}^{S}}_{lQCD}}\right]^{2}.$
(29)

Further details about the surrogate model are reported in Appendix Appendix A:
Surrogate model approach.

The structure of the DQPMnn is depicted schematically in Figure 1. However, in
this section, we are only considering $\underline{g}$ as an output of the
DQPMnn, whereas the masses and the width are taken from the DQPM
parametrization, Eqs. (6), (7) and (8), with the substitution
$g\rightarrow\underline{g}$. The consequences of relaxing the DQPM
parametrization will be explored in the forthcoming sections.

Figure 1: The representation of the quasiparticle DNN model, DQPMnn. An input
temperature is used to generate as output the expected coupling constant $g$,
as well as two key parameters: the dimensionless masses $m_{i}/T$ and widths
$\gamma_{i}/T$ for each particle species. The latter two parameters are then
fed into the surrogate model, in light blue in the figure, to obtain the
functions $\tilde{I}$ and $\tilde{\chi_{2}}$. These functions are used to
compute the entropy density and the susceptibilities as described in the text.
The loss function minimizes the squared error between the neural network’s
guess and the lattice data. Figure 2: A schematic representation of the neural
network used for the surrogate model. The input data, $T$, $m/T$, and
$\gamma/T$ are used to evaluate the functions $\tilde{I}$ and
$\tilde{\chi}_{2}$ in Eqs. (15). The training uses the mean squared error as a
loss function, such that $\tilde{I}^{F,B}$ and $\tilde{\chi}_{2}$ reproduce
$I^{F,B}$ and $\chi_{2}^{i}$ evaluated numerically from Eqs. (15). The
complete procedure is described in Appendix Appendix A: Surrogate model
approach.

We have examined different values of the weights
$\beta_{G},\beta_{L},\beta_{S}$ in the loss function $\mathcal{L}_{0}$ to
understand how these adjustments influence the microscopic quantities, such as
masses, widths, and the coupling constant, based on the selected thermodynamic
observable. In particular, we have tested the three setups resulting from
setting one of the $\beta_{i}$ to one and the others to zero. These setups
called “I”, “II” and “III”, are explicitly summarized in Table 1. They
correspond to

  * •

setup “I” – best DNN fit of $\mathcal{L}_{0}$ to $s/T^{3}$ (notations in the
figures “$\mathcal{L}_{0}:s/T^{3}$”),

  * •

setup “II” – best DNN fit of $\mathcal{L}_{0}$ to $\chi_{2}^{B}$ (notations in
the figures “$\mathcal{L}_{0}:\chi_{2}^{B}$”),

  * •

setup “III” – best DNN fit of $\mathcal{L}_{0}$ to $\chi_{2}^{S}$ (notations
in the figures“$\mathcal{L}_{0}:\chi_{2}^{S}$”).

setups |  $\beta_{G}$ : $\beta_{L}$ : $\beta_{S}$
---|---
I |  $1$ : $0$ : $0$
II |  $0$ : $1$ : $0$
III |  $0$ : $0$ : $1$
Table 1: The three setups used to test the DQPM model parametrization on
different observables. The weights refer to the loss $\mathcal{L}_{0}$ in Eq.
(29).

We begin with the comparison of the thermodynamic observables used for the
minimization of the loss function $\mathcal{L}_{0}$ to the true values - the
lQCD estimates. Fig. 3 shows the dimensionless entropy ($s/T^{3}$) in the top
panel, baryon susceptibility ($\chi_{2}^{B}$) in the middle panel, and
strangeness susceptibility ($\chi_{2}^{S}$) in the bottom panel, all as
functions of the scaled temperature ($T/T_{c}$). The lines represent
predictions from the DQPMnn with $\mathcal{L}_{0}$ in setups I (red solid
lines), II (blue solid lines), and III (green solid lines). The symbols
correspond to the true values - estimates from lattice QCD by the WB
Collaboration [46, 20, 22]. We observe that the setup I underestimates the
susceptibility, similar to the DQPM model. Setups II and III are very similar
to each other and overestimate the entropy.

###  III.2 Results: $g^{2},m,\gamma$ and transport coefficients.

Now, let’s delve into the microscopic characteristics, starting with the
output of DQPMnn - $g^{2}$. For a comparison with the previous results, the
running coupling $\alpha_{S}=g^{2}/(4\pi)$ is displayed in Fig 4 as a function
of the scaled temperature $T/T_{c}$ in the setups I-III and is compared to the
DQPM result.

Figure 3: The dimensionless entropy $s/T^{3}$ (top panel), baryon
susceptibility $\chi_{2}^{B}$ (middle panel) and strangeness susceptibility
$\chi_{2}^{S}$ (bottom panel) as a function of the scaled temperature
$T/T_{c}$. Lines correspond to the predictions of the DQPMnn with
$\mathcal{L}_{0}$ in setups I (red solid lines), II (blue solid lines), and
III (green solid lines). The symbols correspond to the true values - lQCD
results from the WB Collaboration [46, 20, 22].  Figure 4: The running
coupling $\alpha_{S}=g^{2}/(4\pi)$ as a function of the scaled temperature
$T/T_{c}$ constrained from the DQPMnn with $\mathcal{L}_{0}$ in the setups
I-III is compared to the DQPM parametrization (black dashed lines). The color
coding of the different setups is the same as in Fig. 3. The lattice results
for quenched QCD, $N_{f}=0$, (grey circles) are taken from Ref. [55],
$N_{f}=2$ (black triangles) from Ref. [56], and for $N_{f}=2+1$ (brown square)
from Ref. [57].

Figure 5: Masses [GeV] as a function of the scaled temperature $T/T_{c}$ for
gluons (top panel) and light quarks (bottom panel) from the DQPMnn with the
loss function $\mathcal{L}_{0}$ in the setups I-III in comparison to the
standard DQPM parametrization (black dashed lines). The color coding for
different setups is the same as in Fig. 3.

Figure 6: Widths [GeV] as a function of the scaled temperature $T/T_{c}$ for
gluons (top panel) and light quarks (bottom panel) from the DQPMnn with loss
function $\mathcal{L}_{0}$ in the setups I-III in comparison to the standard
DQPM parametrization (black dash-dotted line). The color coding for different
setups is the same as in Fig. 3.

Figure 7: Specific shear viscosity $\eta/s$ as a function of the scaled
temperature $T/T_{c}$ constrained from the DNN with the loss function
$\mathcal{L}_{0}$ in the setups I-III estimated from Eq. (21). The color
coding for different setups is the same as in Fig. 3. The black and cyan
dashed lines correspond to the DQPM estimates for Kubo (Eq. (21)) [29] and RTA
with the interaction rate formalism [30], respectively. The dashed gray line
demonstrates the Kovtun-Son-Starinets bound [58, 59]
$(\eta/s)_{\text{KSS}}=1/4\pi$. The symbols show lQCD data for pure SU(3)
gauge theory taken from Refs. [60] (black squares), [61] (blue line), [62]
(green triangles), [63] (magenta circles). The grey area corresponds to the
latest estimates from a Bayesian analysis by the JETSCAPE Collaboration [64].

Figure 8: Scaled electric conductivity $\sigma_{Q}/T$ for all flavors (top
panel) and strange quarks (bottom panel) as a function of the scaled
temperature $T/T_{c}$ constrained from the DQPMnn with the loss function
$\mathcal{L}_{0}$ in the setups I-III estimated from Eq. (25). The color
coding for different setups is the same as in Fig. 3. Black solid and cyan
lines correspond to the DQPM estimates for Kubo (Eq. (25)) and RTA with the
interaction rate formalism [30], respectively. The symbols correspond to the
lQCD results. In the top panel, black squares represent $N_{f}=2$ data from
Ref. [65], whereas $N_{f}=2+1$ data are blue triangles and stars from Ref.
[66], and brown circles from Ref. [67]. The latter reference is also used for
the strangeness contribution to the conductivity in the bottom panel, where
lattice data are brown circles.

Additionally, we show lQCD estimates: quenched QCD $N_{f}=0$ results (grey
circles) are taken from Ref. [55], $N_{f}=2$ (black inverted triangles) from
Ref. [56] and $N_{f}=2+1$ (brown squares) from Ref. [57] (from Fig. 7). It is
important to note that the coupling constant obtained by lQCD strongly depends
on the definition of $\alpha_{s}$ extracted from the static potential [55,
57]. The main feature of the resulting coupling constants is a significant
increase approaching $T_{c}$, comparable to the predictions from lQCD, showing
the importance of nonperturbative effects for $T<3T_{c}$.

One can see that the initial choice of fitting $s/T^{3}$ in the case of setup
I aligns with the DQPM parametrization. On the other hand, setups II and III
lead to an unrealistically low coupling.

In Figs. 5, 6 we show the masses and widths for gluons (top panels) and light
quarks (bottom panels) as a function of $T/T_{c}$ for different setups and
compare them to the benchmark DQPM results (black dash-dotted line). As seen
from the figures, in the DQPM and DQPMnn $\mathcal{L}_{0}$ setup I, the masses
and widths of quarks and gluons increase with $T$. For all presented setups
the results respect the colour factors
$M_{q}=\frac{2}{3}M_{g},\gamma_{q}=\frac{4}{9}\gamma_{g}$. The temperature
dependence of the masses and widths are rather different for setups II and III
since they follow the extracted $g^{2}$. Similarly the $m_{i},\gamma_{i}$ for
setups II and III are smaller than the DQPM one. Notable small differences are
observed between setups II and III, particularly near the critical temperature
$T_{c}$ and for temperatures exceeding $2.5T_{c}$. This observation aligns
well with our initial expectations, which are based on the comparative
analysis of thermodynamic observables used for training and prediction, as
depicted in Fig. 3.

It’s important to highlight that fitting all three thermodynamic observables
using the basic DQPM Ansatz is not feasible: as we have just shown fitting the
entropy density underestimates the susceptibilities, whereas fitting the
latter overestimates the entropy density. Thus, in the subsequent section, we
will modify the DQPM parametrization to potentially extend the model for a
more comprehensive representation of all thermodynamic observables.

Similarly to the thermodynamic observables, we compare the transport
coefficients from the Kubo-Zubarev method for three setups to the estimates of
the original DQPM spectral functions.

To ensure an accurate comparison we focus on dimensionless transport
coefficients. These include the shear viscosity to entropy density ratio,
denoted as $\eta/s$, and the ratio of electric conductivity to temperature,
expressed as $\sigma_{Q}/T$. Fig. 7 displays the specific shear viscosity
$\eta/s$ as a function of the scaled temperature $T/T_{c}$ in comparison to
various results from the literature. Multiple colored lines represent the
DQPMnn estimates from Eq. (21) using three different setups for the weights.
The black and cyan dashed lines correspond to the DQPM estimates for Kubo (Eq.
(21)) $\eta^{\text{Kubo}}/s$ [29] and RTA with the interaction rate
$\eta^{\text{RTA}}/s$ [30], respectively. The RTA estimate of the shear
viscosity is found to be very close to the one from the Kubo formalism [29]
indicating that the quasiparticle limit ($\gamma\ll M$) holds in the DQPM
Ansatz.

We also present the primary estimates obtained from well-established methods
such as lQCD calculations, AdS/CFT correspondence, and Bayesian analysis,
which can serve as valuable guidance for further improvements, although these
estimates still rely on certain assumptions about the QGP. A dashed gray line
highlights the well-known Kovtun-Son-Starinets bound
$(\eta/s)_{\text{KSS}}=1/(4\pi)$ [58, 59]. Symbols on the plot represent lQCD
data for pure SU(3) gauge theory from various references: black squares from
[60], a blue line from [61], green triangles from [62], and magenta circles
from [63]. The grey area encompasses estimates from the Bayesian analysis
conducted by the JETSCAPE Collaboration [64]. We see that the DQPM results are
in agreement with the lattice estimations and close to the range provided by
JETSCAPE. For a reasonably good description of the QGP it is important to have
$\eta/s>(\eta/s)_{\text{KSS}}=1/(4\pi)$.

Comparing results for different setups we see that estimates from setup I are
in agreement with the standard DQPM result and with the lattice results, while
in the case of setups II and III the values obtained are higher. This can be
explained by the smaller masses and widths, resulting from unrealistically
small coupling. The setup I, as well as the original DQPM, is in good
agreement with the lattice data for gluodynamics, however, the
phenomenologically constrained results from state-of-the-art Bayesian
statistical analyses for dynamical, i.e. out of equilibrium, full QCD medium
indicate smaller values of $\eta/s$ [68, 69, 70]. In this respect, we expect
that an improved description of the QGP matter should correspond to
$\eta/s>(\eta/s)_{\text{KSS}}=1/(4\pi)$ and within the range of Bayesian
estimates.

Fig. 8 presents the scaled electric conductivity, $\sigma_{Q}/T$, for all
flavors, versus the scaled temperature, $T/T_{c}$. The DQPMnn results derived
from Eq. (25) are characterized by red solid lines (representing calculations
using $\mathcal{L}_{0}$ in the setup I), blue solid lines (setup II), and
green dashed lines (setup III). Lattice QCD data points are represented using
various symbols: for $N_{f}=2$, black squares denote the results from Ref.
[65], while for $N_{f}=2+1$, brown circles represent estimates from Ref. [67],
and blue triangles and stars from Ref. [66].

The bottom panel of Fig. 8 focuses specifically on the scaled electric
conductivity of strange quarks, maintaining a similar structure and
representation as the upper plot. One can see the same tendency in the case of
different setups, i.e. setups II and III predict a larger value of the
conductivities due to the smaller effective coupling.

The consistent overestimation of $\sigma_{Q}(\text{strange})/T$ across all
setups compellingly suggests the need for revising the masses and widths of
strange quarks to achieve closer alignment with the lQCD data. This
observation is not merely an anomaly but a clear indication that our current
understanding and modeling of the strange quark behavior requires refinement.

Furthermore, the simultaneous pursuit of a more accurate agreement with the
lQCD data for both $\sigma_{Q}(\text{full})/T$ and
$\sigma_{Q}(\text{strange})/T$ demands a thoughtful reevaluation and potential
modification of the Ansatz employed in the DQPM. In the forthcoming two
subsections we will explore possible generalizations of the model, which can
shed light on which direction microscopic quantities can be adjusted to
improve $\chi_{2}^{S}$ and $\sigma_{Q}(\text{strange})/T$.

In summary, we have found that setups II and III, which do not fit the lQCD
entropy density, provide a better description of
$\chi^{B}_{2}(T),\chi^{S}_{2}(T)$. However, because of the DQPM Ansatz, these
two setups yield an unrealistically small value compared to the estimates from
lQCD and the analytical 2/1 loop running constant [71, 72] for
$\alpha_{s}=\underline{g}^{2}(T)/(4\pi)$ at $T>1.5T_{c}$. In particular, one
can see that the effective coupling $\underline{g}^{2}(T)\rightarrow 0$
already at $T\approx 2.5T_{c}$.

In the next section we will relax the form of the Ansatz aiming at a
consistent simultaneous description of $s/T^{3}$, $\chi^{B}_{2}$ and
$\chi^{S}_{2}$ in a quasiparticle model.

##  IV Modified quasiparticle model DNN $\mathcal{L}_{1}$: extraction of
$m_{i}(T),\gamma_{i}(T)$, and $g^{2}(T)$

Now we explore a generalization of the DQPM ansatz considered in the previous
section. The properties of each quasiparticle are the output of the fully
connected NN model depicted in Fig. 1. For this purpose we consider 6
quantities as output of the DQPMnn: the coupling constant
$\underline{g}^{2}(T)$, gluon and light quark masses,
$\underline{M}_{g}$,$\underline{M}_{l}$, and three independent quasiparticle
widths $\underline{\gamma}_{g}$, $\underline{\gamma}_{l}$,
$\underline{\gamma}_{s}$. The mass of the strange quark is still assumed to be
$\underline{M}_{s}=\underline{M}_{l}+0.03\text{GeV}$.

In order to make the extraction of the effective coupling $\underline{g}(T)$
possible, we follow the idea proposed in [56] of absorbing the nonperturbative
corrections to the masses out of the effective coupling constant. In the case
of gluodynamics, the screening masses were generalized taking into account
non-perturbative corrections in the partonic phase in the following way:

$\dfrac{m_{D}(T)}{T}=A(T)(1+N_{f}/6)^{1/2}g(T),$ (30)

where $A(T)$ describes non-perturbative corrections. As in Ref. [56], $A(T)$
turns out to be larger than the perturbative limit even at high T. Based on
these observations we modify the effective masses using the following ansatz:

$\dfrac{\underline{m}_{q/g}(T)}{T}=\underline{A}_{i}(T)\underline{g}(T),$ (31)

where $\underline{A}_{i}$ and $\underline{g}$ are computed from the Neural
Network. In the DQPM $A_{g}=3/4$, $A_{q}=1/3$.

It is important to note that, using the above parametrization,
$\underline{g}(T)$ remains relatively similar to the original effective
coupling $g^{DQPM}(T)$ whereas the masses can differ. This allows to obtain a
better description of thermodynamical quantities while preserving transport
coefficients and cross-sections from dramatic changes. The widths
$\underline{\gamma}_{i}(T)$ are left completely free, and no parametrization
in terms of $\underline{g}(T)$ is assumed. This differentiation in non-
perturbative corrections within the masses emphasizes that gluons and quarks
may exhibit distinct temperature dependence in their non-perturbative
adjustments. Additionally, this distinction leaves more freedom to the DQPMnn
to learn and adapt the T-dependence of masses and widths, thus achieving a
better fit for the susceptibilities $\chi_{2}^{B,S}$.

In this case study we are relaxing the DQPM assumptions in order to achieve a
better agreement with thermodynamic functions. However, to meet physical
requirements such as the asymptotic HTL scaling and the hierarchy between
masses and widths, we need to modify the loss function by adding a
regularization term.

The loss function used in this section reads:

$\displaystyle\mathcal{L}_{1}=\mathcal{L}_{0}+\beta_{reg}\mathcal{L}_{DQPM},$
(32)

where the regularization term is:

$\displaystyle\mathcal{L}_{DQPM}=\sum_{i=g,l,s}\left[\underline{\gamma}_{i}(T)-\gamma_{i}^{DQPM}(T)\right]^{2}+$
$\displaystyle\left[\underline{A}_{g}(T)-3/4\right]^{2}+\left[\underline{A}_{q}(T)-1/3\right]^{2}.$
(33)

The inclusion of $\mathcal{L}_{DQPM}$ acts as a constraint on the neural
network, ensuring that the produced outputs stay relatively close to the
expectations derived from the DQPM. In order to regulate the influence of
$\mathcal{L}_{0}$ on the predictions, one has to fix $\beta_{reg}$ taking into
account the relative contribution for this term in the loss function. In this
section, we present two distinct setups for $\beta_{i}$, as given in Table 2.

setups | $\beta_{G}:\beta_{L}:\beta_{S}:\beta_{reg}$
---|---
A |  $100$ : $1$ : $1$ : $10^{3}$
B |  $500$ : $1$ : $1$ : $10^{4}$
Table 2: Considered setups for the modified loss function $\mathcal{L}_{1}$

  * •

In the setup “A” the weights were chosen such that all the contributions to
the loss function are of the same order of magnitude. The use of a larger
$\beta_{G}$ compared to $\beta_{L,S}$ is due to the larger errors $\Delta
s_{lQCD}/T^{3}$ associated with the lattice data for $s/T^{3}$. Choosing a
$\beta_{G}$ of order one would result in the NN regarding the entropy as an
irrelevant feature and learning only from the susceptibilities. Furthermore,
the regularization loss is not enhanced by dividing by the variance, hence
using a smaller $\beta_{reg}$ would effectively remove any sizeable effect
from the regularization loss.

  * •

The setup “B”, on the other hand, is such that the entropy and the DQPM
regularization term have a bigger role compared to the susceptibilities,
yielding an intermediate result between the setup “I” of the previous section
and the setup “A”.

Furthermore, in this section we train the DQPMnn multiple times randomizing
the initial weight distribution each time. The results of these variations are
plotted as shaded areas.

Figure 9: The dimensionless entropy $s/T^{3}$ (top panel), baryon
susceptibility $\chi_{2}^{B}$ (middle panel) and strangeness susceptibility
$\chi_{2}^{S}$ (bottom panel) as a function of the scaled temperature
$T/T_{c}$. Colored areas correspond to the predictions from the DQPMnn with
the modified loss function $\mathcal{L}_{1}$ within the setups A (blue areas),
and B (orange areas). The red dashed lines depict predictions generated by the
DQPMnn with loss function $\mathcal{L}_{0}:s/T^{3}$, as discussed in Section
III.1 and trained in setup I. The symbols correspond to the lQCD results from
the WB Collaboration [46, 20, 22].

Fig. 9 shows the resulting thermodynamic observables for the setups A (blue
areas), and B (orange areas) as a function of the scaled temperature
$T/T_{c}$: dimensionless entropy $s/T^{3}$ (top panel), baryon susceptibility
$\chi_{2}^{B}$ (middle panel), and strangeness susceptibility $\chi_{2}^{S}$
(bottom panel). The red dashed lines depict predictions generated by the
DQPMnn with the loss function $\mathcal{L}_{0}:s/T^{3}$ (Eq. (29)), as
discussed in Section III.1 and trained in setup I. The symbols correspond to
the lQCD results from the WB Collaboration [46, 20, 22]. The dimensionless
entropy $s/T^{3}$ is described well for both setups, while in the case of
setup A, the DQPMnn can describe $\chi_{2}^{B}$ and $\chi_{2}^{S}$ better than
the original DQPM or the DQPMnn $\mathcal{L}_{0}:s/T^{3}$. This is expected,
as the setup B is closer to the standard DQPM due to the larger weight of the
entropy and regularization terms $\beta_{g}$ and $\beta_{reg}$ in the loss
function. In both setups, due to the regularization term in the loss function,
$\chi_{2}^{B}$ and $\chi_{2}^{S}$ start to approach DQPM asymptotics for
$T>2.5-3T_{c}$, as the absence of lattice data leads the DQPMnn to learn only
from the regularization loss term. In general, one can introduce other
asymptotic terms for high $T$, such as the Stefan-Boltzmann limit. The main
objective here is to obtain physically motivated values of microscopic
observables - non-vanishing effective masses and widths, therefore we choose
the DQPM form.

Figure 10: The effective coupling $g^{2}$ as a function of the scaled
temperature $T/T_{c}$ constrained from the DQPMnn with the modified loss
function $\mathcal{L}_{1}$ (colored areas) in the setups A-B in comparison to
the DQPMnn $\mathcal{L}_{0}:s/T^{3}$, as discussed in Section III.1 and
trained in the setup I (red dashed line). The color coding for different
setups is the same as in Fig. 9. Figure 11: The dimensionless quantities
$A_{q}$ and $A_{g}$ as a function of the scaled temperature $T/T_{c}$
constrained from DQPMnn with the modified loss function $\mathcal{L}_{1}$
(colored areas) in the setups A-B in comparison to the DQPM values (black
dashed lines). The color coding for different setups is the same as in Fig. 9.

Figure 12: Masses [GeV] as a function of the scaled temperature $T/T_{c}$ for
gluons (top panel) and light quarks (bottom panel) from the DQPMnn with the
modified loss function $\mathcal{L}_{1}$ (colored areas) in the setups A-B in
comparison to the DQPM values (black dashed lines). The color coding for
different setups is the same as in Fig. 9.

Figure 13: Widths [GeV] as a function of the scaled temperature $T/T_{c}$ for
gluons (top panel), light quarks (middle panel) and strange quarks (bottom
panel) from the DQPMnn with the modified loss function $\mathcal{L}_{1}$
(colored areas) in the setups A-B in comparison to the DQPM values (black
dashed lines). The color coding for different setups is the same as in Fig. 9.

Figure 14: Specific shear viscosity $\eta/s$ as a function of the scaled
temperature $T/T_{c}$. Colored areas correspond to estimates from the DQPMnn
with the modified loss function $\mathcal{L}_{1}$ within the setups A (blue
areas), and B (orange areas) from Eq. (21). The red dashed line depicts
predictions generated by the DQPMnn with loss function
$\mathcal{L}_{0}:s/T^{3}$, as discussed in Section III.1 and trained in setup
I. The dashed gray line demonstrates the Kovtun-Son-Starinets bound [58, 59]
$(\eta/s)_{\text{KSS}}=1/4\pi$. The symbols show lQCD data for pure SU(3)
gauge theory taken from Refs. [60] (black squares), [61] (blue line), [62]
(green triangles), [63] (magenta circles). The grey area corresponds to the
latest estimates from a Bayesian analysis by the JETSCAPE Collaboration [64].

Figure 15: Scaled electric conductivity $\sigma_{Q}/T$ for all flavors (top
panel) and strange quarks (bottom panel) as a function of the scaled
temperature $T/T_{c}$. Colored areas correspond to estimates from the DQPMnn
with the modified loss function $\mathcal{L}_{1}$ within the setups A (blue
areas), and B (orange areas) from Eq. (25). The symbols correspond to the lQCD
results. In the top panel black squares represent $N_{f}=2$ data from Ref.
[65], whereas $N_{f}=2+1$ data are blue triangles and stars from Ref. [66],
and brown circles from Ref. [67]. The latter reference is also used for the
strangeness contribution to the conductivity in the bottom panel, where
lattice data are brown circles.

###  IV.1 Results: $g^{2},m,\gamma$ and transport coefficients.

Now we look closely into the resulting microscopic observables and compare
them to the original DQPM parametrization. Fig. 10 depicts the effective
coupling $g^{2}$ as a function of the scaled temperature $T/T_{c}$ constrained
in the setups $\mathcal{L}_{1}$: A-B in comparison to the DQPMnn with
$\mathcal{L}_{0}:s/T^{3}$ (setup I, red dashed line). In setup A, the coupling
exhibits higher values within the $T<3T_{c}$ range. Unlike setups II and III
discussed in the previous section, the improved description of $\chi_{2}^{B}$
and $\chi_{2}^{S}$ in setup A does not lead to unphysically small values of
the effective coupling. This observation is less surprising considering that,
in the setups we are using in this section, macroscopic quantities do not
depend solely on $g^{2}(T)$ but also to the additional $A_{i}(T)$ parameters,
and the widths $\gamma(T)$, which we are leaving completely free. Allowing a
general form of the widths, we can keep the masses of quasi-particles in the
physical range of $1-1.4$ GeV for gluons and $0.3-0.7$ GeV for
$T=T_{c}-3T_{c}$. Examining the dimensionless factors $A_{q}$ and $A_{g}$ as
functions of the scaled temperature $T/T_{c}$ (as depicted in Fig. 11) in
setups A-B (colored areas), in comparison to the DQPM values (indicated by the
black dashed lines) we can quantify the difference between these scenarios.
Specifically for gluons the observed $A_{g}$ values are higher than those
suggested by the DQPM, in agreement with the results of Ref. [56]. The
convergence of these values between the different setups becomes more
pronounced for temperatures near $3T_{c}$, eventually aligning with the DQPM
values. This trend is dominantly attributed to the regularization term in the
loss function $\mathcal{L}_{1}$. On the contrary, for quarks, the variation
among the setups is more pronounced, and the overall values remain
consistently lower than those predicted by the DQPM. The smaller value of
$A_{q}$ can be understood from the results about the susceptibilities obtained
in the previous section. Indeed, the setups II and III show that, in order to
fit $\chi_{2}^{B,S}$, a smaller coupling is needed. In the previous section,
this can also be rephrased by saying that a smaller mass is needed. However,
in the present section, the mass is given by the product between
$\underline{A}(T)$ and $\underline{g}^{2}$, therefore a favorable condition to
fit the susceptibilities can also be obtained by decreasing the value of
$\underline{A}$, as observed in Fig. 11. This underlines a fundamental
difference in the behavior of quarks when compared to gluons within the
framework provided by the DQPMnn with a relaxed DQPM Ansatz.

Fig. 12 illustrates masses in GeV as functions of the scaled temperature
$T/T_{c}$ for gluons (top panel) and light quarks (bottom panel). Here
predictions generated by the DQPMnn with $\mathcal{L}_{1}$ are depicted by
colored areas within setups A-B contrasted with the DQPM values represented by
the black dashed lines. As expected from the results on $g^{2}$ and $A_{q/g}$
the gluon masses are higher, while quark masses are smaller than the DQPM
parametrization. Not imposing the DQPM asymptomatic behavior for $T>3T_{c}$ we
would expect smaller masses for light quarks and less-pronounced T-dependence
for gluons.

Fig. 13 presents the widths in GeV as a function of the scaled temperature
$T/T_{c}$ across three panels, covering gluons (top panel), light quarks
(middle panel), and strange quarks (bottom panel). Similarly to the masses, we
show the comparison to the DQPM values represented by the black dashed lines.
This comparative analysis reveals the behavior of widths as a function of
temperature within these distinct setups depicted by colored areas. For gluon
widths, both setup A and B are close to the DQPM, with setup B showing smaller
values. In contrast, for light quarks the overall values are higher than the
DQPM, while setup B yields smaller widths. Similarly, for the strange quarks,
both setups predict higher values of $\gamma$. Therefore we expect that
transport properties will change for setup A and B compared to the DQPM
predictions.

Now we look closely into the influence of the change in microscopic quantities
on the transport coefficients. The comparison between DQPMnn results with
different loss functions in the case of $\eta/s$ vs $T/T_{c}$ is depicted in
Fig. 14. The red dashed line shows the predictions generated by the DQPMnn
with the loss function $\mathcal{L}_{0}:s/T^{3}$, as discussed in Section
III.1 and trained in the setup I. One can see that for both setups predictions
by DQPMnn with the modified description are smaller than in the case of
$\mathcal{L}_{0}:s/T^{3}$. Since the microscopic properties of the setup B are
closer to the DQPM parametrization (as have shown above), the $\eta/s$
approaches the value of $\mathcal{L}_{0}:s/T^{3}$ at $T\approx 2.5T_{c}$. The
main difference is due to the difference in quark and gluon masses compared to
the DQPM values: smaller values of quark masses and increased quark width. In
the case of setup A the specific shear viscosity is even smaller due to the
larger widths and gluon mass.

More interesting is to look into the quark sector and compare predictions of
DQPMnn with different setups while considering the full and strange electric
conductivity. We show the resulting $\sigma_{Q}/T$, for all flavors (top
panel) and strange quarks (bottom panel) as a function of the scaled
temperature $T/T_{c}$ in Fig. 15. For the total conductivity, we see that
predictions from the DQPMnn with $\mathcal{L}_{1}$ in the setup B are larger
than setup A, but overall values are smaller than in the case of DQPMnn with
$\mathcal{L}_{0}:s/T^{3}$. In the strange sector, both setups perform better
than the standard DQPM model as compared to the lattice data.

###  IV.2 Strange sector refinements: DNN $\mathcal{L}_{1}$ set A -
modifications $\Delta m_{ls}$

Figure 16: The dimensionless entropy $s/T^{3}$ (top panel), baryon
susceptibility $\chi_{2}^{B}$ (middle panel) and strangeness susceptibility
$\chi_{2}^{S}$ (bottom panel) as a function of the scaled temperature
$T/T_{c}$. Colored areas correspond to the predictions from the DQPMnn with
the modified loss function $\mathcal{L}_{1}$ (set A) with $\Delta m_{ls}$=0.0
GeV (blue areas), $\Delta m_{ls}$=0.03 GeV (orange areas), and $\Delta
m_{ls}$=0.09 GeV (green areas), $\Delta m_{ls}$=0.15 GeV (magenta areas). The
red dashed lines depict predictions generated by the DQPMnn with loss function
$\mathcal{L}_{0}:s/T^{3}$, as discussed in Section III.1 and trained in setup
I. The symbols correspond to the lQCD results from the WB Collaboration [46,
20, 22].

Figure 17: Masses [GeV] as a function of the scaled temperature $T/T_{c}$ for
gluons (top panel) and light quarks (bottom panel) from the DQPMnn with the
modified loss function $\mathcal{L}_{1}$ (set A) with $\Delta m_{ls}$=0.0 GeV
(blue areas), $\Delta m_{ls}$=0.03 GeV (orange areas), $\Delta m_{ls}$=0.09
GeV (green areas) and $\Delta m_{ls}$=0.15 GeV (magenta areas) in comparison
to the DQPM values (black dashed lines).

Figure 18: Widths [GeV] as a function of the scaled temperature $T/T_{c}$ for
gluons (top panel), light quarks (middle panel), and strange quarks (bottom
panel) from the DQPMnn with the modified loss function $\mathcal{L}_{1}$ (set
A) with $\Delta m_{ls}$=0.0 GeV (blue areas), $\Delta m_{ls}$=0.03 GeV (orange
areas), $\Delta m_{ls}$=0.09 GeV (green areas) and $\Delta m_{ls}$=0.15 GeV
(magenta areas) in comparison to the DQPM values (black dashed lines).

Figure 19: Specific shear viscosity $\eta/s$ as a function of the scaled
temperature $T/T_{c}$. Colored areas correspond to estimates from the DQPMnn
with the modified loss function $\mathcal{L}_{1}$ (set A) with $\Delta
m_{ls}$=0.0 GeV (blue areas), $\Delta m_{ls}$=0.03 GeV (orange areas), $\Delta
m_{ls}$=0.09 GeV (green areas) and $\Delta m_{ls}$=0.15 GeV (magenta areas)
from Eq. (21). The red dashed lines depict predictions generated by the DQPMnn
with the loss function $\mathcal{L}_{0}:s/T^{3}$. The dashed gray line
demonstrates the Kovtun-Son-Starinets bound [58, 59]
$(\eta/s)_{\text{KSS}}=1/4\pi$. The symbols show lQCD data for pure SU(3)
gauge theory taken from Refs. [60] (black squares), [61] (blue line), [62]
(green triangles), [63] (magenta circles). The grey area corresponds to the
latest estimates from a Bayesian analysis by the JETSCAPE Collaboration [64].

Figure 20: Scaled electric conductivity $\sigma_{Q}/T$ for all flavors (top
panel) and strange quarks (bottom panel) as a function of the scaled
temperature $T/T_{c}$. Colored areas correspond to estimates from the DQPMnn
with the modified loss function $\mathcal{L}_{1}$ (set A) with $\Delta
m_{ls}$=0.0 GeV (blue areas), $\Delta m_{ls}$=0.03 GeV (orange areas), $\Delta
m_{ls}$=0.09 GeV (green areas) and $\Delta m_{ls}$=0.15 GeV (magenta areas)
from Eq. (25). The red dashed lines depict predictions generated by the DQPMnn
with loss function $\mathcal{L}_{0}:s/T^{3}$. The symbols correspond to the
lQCD results. In the top panel, black squares represent $N_{f}=2$ data from
Ref. [65], whereas $N_{f}=2+1$ data are blue triangles and stars from Ref.
[66], and brown circles from Ref. [67]. The latter reference is also used for
the strangeness contribution to the conductivity in the bottom panel, where
lattice data are brown circles.

Here we present results on the influence of $\Delta m_{ls}$ \- the difference
between the masses of strange and light quarks/antiquarks:

$\Delta m_{ls}=m_{s(\bar{s})}-m_{l(\bar{l})}.$

For this purpose, we train the DQPMnn with the loss function $\mathcal{L}_{1}$
in the setup A for different values of $\Delta m_{ls}$. Now comparing to the
standard DQPM Ansatz we have the following modifications for the strange
quarks: the value of $\gamma_{s}$ is not equal to $\gamma_{l}$, and different
values of $\Delta m_{ls}$. In this section, we present results from four
different setups, characterized by $\Delta m_{ls}$ values of $0.0$, $0.03$,
$0.09$, and $0.15$ GeV. With these values, we obtain a good fit for the EoS
and reasonable values of the transport coefficients.

If we look into the training observables, $s/T^{3}$, $\chi_{2}^{B}$,
$\chi_{2}^{S}$ depicted in Fig. 16 in three panels from top to bottom,
respectively, we find that a higher value of $\Delta m_{ls}$ slightly improves
the fit of $\chi_{2}^{B}$, however $\chi_{2}^{S}$ deviates from the data for
$T>2T_{C}$.

Now we look at the microscopic properties predicted by the DQPMnn when we
tweak the mass shift $\Delta m_{ls}$. In Fig. 17, we show the masses for the
four scenarios, utilizing the $\mathcal{L}_{1}$ loss function (set A) with
$\Delta m_{ls}$ set to 0.0 GeV (blue areas), 0.03 GeV (orange areas), 0.09 GeV
(green areas) and 0.15 GeV (magenta areas), compared to the DQPM values
(depicted by the black dashed line). We see that the light-quarks masses are
most affected by the mass-shift, becoming smaller as $\Delta m_{ls}$
increases, especially at $T<2T_{c}$. The gluon mass, on the other hand, stays
approximately the same.

Similarly, for the widths shown in Fig. 18 we see that the main deviations
between the different scenarios are seen for the quarks. The plot displays
widths [GeV] as functions of the scaled temperature $T/T_{c}$ for gluons (top
panel), light quarks (middle panel), and strange quarks (bottom panel),
derived from the DQPMnn. We can see that, as $\Delta m_{ls}$ increases, the
light-quarks width becomes smaller, whereas the opposite trend is found in the
strange-quark width. For $\Delta m_{ls}\geq 0.09$ GeV the widths of the
strange quarks become larger than the light quarks, and the difference is more
prominent for the larger strange mass quark in the region of $T<2.5T_{c}$,
where lattice data for $\chi_{2}^{S}$ are present. It is noticeable that the
asymptotic behaviour of $\gamma_{l}$ has also changed, and shows different
asymptotics for $T>2T_{c}$ compared to the standard DQPM parametrization and
predictions from set A.

Now let’s look how the mass shift $\Delta m_{ls}$ changes the transport
properties. We start with the shear viscosity, which is less affected by the
strange quark. Fig . 19 depicts the comparison of the specific shear viscosity
($\eta/s$) as a function the scaled temperature ($T/T_{c}$). Colored areas
represent estimates from the DQPMnn with the loss function $\mathcal{L}_{1}$
(set A) for four values of $\Delta m_{ls}$ (0.0 GeV in blue, 0.03 GeV in
orange, 0.09 GeV in green and 0.15 GeV in magenta) from Eq. (21). Only a
subtle difference can be seen in the vicinity of $T_{c}$, where the widths and
the masses change more drastically.

In Fig. 20 the scaled electric conductivity ($\sigma_{Q}/T$) is shown for all
flavors (top panel) and strange quarks (bottom panel), plotted against the
scaled temperature ($T/T_{c}$) computed from Eq. (25). A more precise
description is obtained when using the larger $\Delta m_{ls}$ value,
particularly within the temperature range of $T<1.5T_{c}$, for both full and
strange conductivities. Nevertheless, there is a decrease in the value of the
full conductivity observed in the $T>2T_{c}$ region, which can be attributed
to the temperature-dependent behavior of $\gamma_{l}$.

We find that for larger masses of the strange quarks the agreement for the
electric conductivity from strange quarks becomes better, however, the full
conductivity suffers in that case.

At this stage, it is unclear which specific value within the range of $\Delta
m_{ls}$ = 0.0 - 0.15 GeV would be optimal. It would be more beneficial to
consider $\Delta m_{ls}=f(T)$ as an additional output of the DQPMnn, however,
in the current framework and without additional input from lattice data or HTL
calculations, the results obtained are challenging to interpret physically. If
this approach is to be pursued, which would be desirable according to the
findings discussed, more data are needed. We expect a $T$ dependent mass shift
for the strange quark to be larger than $\Delta m_{ls}=0.03$ GeV (DQPM value)
in the region $T<1.5T_{c}$ and to be decreasing with T.

##  V Conclusions and outlook

In this study, we have addressed the phenomenological problem of extraction of
the microscopic properties of the off-shell quasi-particles in the QGP using
machine learning techniques based on macroscopic observables measured in
lattice QCD.

DNNs are used to learn the relationship between $T$, which serves as an input
to the network, and dependent variables, i.e. microscopic observables, that
are designed as outputs of the network. In particular, we have addressed a few
scenarios for temperature-dependent masses, widths, and effective coupling
constants inferred from the DNNs. In our framework, we trained our DNNs using
three independent thermodynamic quantities, the entropy density $s/T^{3}$ and
the baryon and strange susceptibilities $\chi_{2}^{B,S}$ using lQCD data
provided by the WB Collaboration. The key ingredient is that our simple
formulation can provide a playground for fixing the strange quark and light
quark sectors and studying the interplay between different sectors by
utilizing both thermodynamic and transport properties. In addition, it has the
flexibility to preserve a chosen scaling by adding additional terms.

Our results cover the temperature range $T_{c}<T<3.5T_{c}$. To assess the
quality of our model description of the QGP we have incorporated the Kubo-
Zubarev formalism into our framework to compute transport coefficients (the
dimensionless specific shear viscosity $\eta/s$ and the ratio of electric
conductivity to the temperature $\sigma_{Q}/T$ ) and compared them to results
from first principle lQCD calculations and phenomenologically constrained
Bayesian estimations by the JETSCAPE Collaboration. In particular, for the
comparison, we consider various lQCD results for the gluodynamics in the case
of $\eta/s$, while in the case of the $\sigma_{Q}/T$ there are recent lattice
results for $(2+1)$-flavor QCD that separates additionally the strange quark
contribution.

I. Firstly, we have considered a DNN with one output microscopic quantity, the
effective coupling constant $g^{2}$, and used the DQPM parametrization for
quasiparticle’s masses and widths. In this framework, we have tested three
different setups, labeled I-III, and explored how model parameters can affect
microscopic quantities and transport coefficients.

We found that setups II and III, despite not fitting the lQCD entropy density,
offer a more accurate description of $\chi_{2}^{B}(T)$ and $\chi_{2}^{S}(T)$.
This is possible by using a smaller coupling constant and consequently quark
masses are smaller compared to the ones employed in the DQPM. Furthermore, in
these setups, the thermal widths are found to be smaller than in the DQPM,
which generally translates into a smaller interaction rate and larger
transport coefficients (notice however that the interaction rate was not
employed in this work, but instead we use the more general Kubo-Zubarev
formalism). These observations show that in order to achieve a good
simultaneous description of both $s/T^{3}$ and $\chi_{2}^{B,S}$, as well as of
the transport coefficients, one should reconsider the parameterization for the
DQPM.

II. Such a generalization has been performed in the second part of this work,
where we have considered a DNN with 6 output microscopic quantities:
temperature-dependent masses, widths, and effective coupling constants. In
this case, a regularization term has been introduced in the loss function in
order to preserve a HTL-predicted asymptotic behaviour.

The modifications introduced in this section, which have been explored in two
setups A and B, allow to fit simultaneously both the entropy density and the
susceptibilities. This less constrained model favors heavier gluons and
lighter quarks, which is in agreement with the findings of the first part of
this paper. For what concerns the thermal widths, the gluon sector is very
similar to the DQPM parametrization, whereas the quark widths are larger than
in the DQPM. We also observe that the DNN favors different values for the
widths of the strange and the light quarks.

These modifications decrease the value of the transport coefficients. The
specific shear viscosity $\eta/s$ now lies within the JETSCAPE predictions.
Concerning the conductivities, the description of the strange quark
contribution to the conductivity now is in better agreement with the lattice
data, however, the total conductivity is found to be smaller than the lQCD
results.

III. We further tested another assumption commonly used in the quasi-particle
models, namely that the light and strange quark masses are related by a
temperature-independent mass shift $\Delta m_{ls}$. We have explored the value
of $\Delta m_{ls}$ in the range $[0,0.15]$ GeV and repeated the previous
study, finding that this mass shift also has an impact on transport
coefficients. We demonstrate that the strange quark electric conductivity can
be better aligned with the lQCD results for a larger $\Delta m_{ls}$. Our
study suggests that to achieve a more realistic quark sector description and a
better fit for the scaled conductivities considering a temperature-dependent
mass shift $\Delta m_{ls}(T)$ with a higher value ($\Delta m_{ls}$>0.03 GeV)
near $T_{c}$ is essential. However, the lack of physical constraints on its
value and on its asymptotic scaling, prevents us from making any physically
sensible assessment. In this sense, first-principle-based studies, such as
those employing functional methods in the case of Dyson–Schwinger equations
(DSE) [73] or functional renormalisation group (FRG) [74], constraining the
strange quark mass would be highly desirable.

Overall, our work identifies machine learning as a promising flexible
framework, which can provide hints for the improvements of a model description
in the phenomenology of HICs. The parameterization of the quasiparticle
properties inferred in this work is useful for transport approaches that
incorporate a partonic phase, such as the Parton-Hadron-String Dynamics
(PHSD), based on Kadanoff-Baym off-shell dynamics (cf. the reviews [75, 39,
76]). Furthermore, a similar scheme to the one proposed here can be used to
study systems at finite baryon density, with a modest chemical potential. In
this context, reliable lattice results are being produced, and the
generalization capabilities of neural networks may provide yet another tool to
investigate the QGP at large $\mu_{B}$.

###### Acknowledgements.

The authors acknowledge inspiring discussions with L. Wang, J. Aichelin, E.
Grossi, M. Ruggieri, O. Kaczmarek, and R. Pisarski. A.P. would also like to
thank L. Anderlini and P. Braccia for useful suggestions. O.S. would like to
thank V. Dusiak for the valuable comments. Also we thank to W. Cassing for the
critical reading of our manuscript. Furthermore, we acknowledge support by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the
grant CRC-TR 211 “Strong-interaction matter under extreme conditions” –
project number 315477589 – TRR 211 as well as for the European Union’s Horizon
2020 research and innovation program under grant agreement STRONG–2020 – No
824093. The computational resources have been provided by the Center for
Scientific Computing (CSC) of Goethe University.

## Appendix A: Surrogate model approach

This appendix is devoted to the description of the surrogate model used in
this work. As described in the main text, the training process of the neural
networks involves the calculation of the integrals in Eqs. (15) appearing in
Eq. (14). A direct numerical calculation of these integrals during training
would be too computationally intensive and time-consuming, making the overall
training process prohibitively lengthy. To circumvent this challenge, we
employ a surrogate model.

The primary role of the surrogate is to approximate the results of the
aforementioned integrals in a computationally efficient way, allowing the
neural network training to proceed faster. In practice, we have computed
numerically the functions $I_{B},I_{F}$, and $\chi_{2}^{i}$ appearing in Eqs.
(15) for different values of temperature $T$, mass over temperature $m/T$ and
width over temperature $\gamma/T$. The range of these parameters has been
chosen to include the physical values of $T$, $m/T$, and $\gamma/T$ for all
quasiparticles as a subset. Therefore, we have computed the integrals in the
range $155\text{ MeV}\leq T\leq 510\text{ MeV}$, $0.2\leq m/T\leq 10$ and
$0.02\leq\gamma/T\leq 6.2$. In terms of the critical temperature $T_{c}=158$
MeV, the temperature range explored corresponds to $T\in[0.98T_{c},3.2T_{c}]$
approximately. For the numerical integration, we have used Mathematica [77],
and generated tables, which were then employed in the training of the
surrogate models. These tables are formatted to represent data for $s/T^{3}$:

$T[\text{GeV}]\ \ |\ \ m/T\ \ |\ \ \gamma/T\ \ |\ \ I_{B}[\text{GeV}^{3}]\ \
|\ \ I_{F}[\text{GeV}^{3}]$

and for $\chi_{2}$:

$T[\text{GeV}]\ \ |\ \ m/T\ \ |\ \ \gamma/T\ \ |\ \ \chi^{q}_{2}.$

We use the tables to train two neural networks that take as input $T$, $m/T$,
and $\gamma/T$ and give as output the functions $\tilde{I}^{F,B}$ and
$\tilde{\chi}_{2}$, respectively. These neural networks are represented
schematically in Figure 2. The loss function used to train these DNNs is the
mean squared error, explicitly:

$\displaystyle\mathcal{L}^{surr}_{s}=$
$\displaystyle\sum_{j=I,F}(\tilde{I}^{j}(T,m/T,\gamma/T)-I^{j}(T,m/T,\gamma/T))^{2},$
$\displaystyle\mathcal{L}^{surr}_{\chi_{2}}=$
$\displaystyle(\tilde{\chi}_{2}^{q}(T,m/T,\gamma/T)-\chi_{2}^{q}(T,m/T,\gamma/T))^{2}.$

The surrogates are trained for $6\times 10^{4}$ epochs and at the end of the
training the loss function is $\mathcal{L}^{surr}\sim 10^{-7}-10^{-8}$ for
both training and validation samples in both models. The small value of the
loss function testifies to the good accuracy achieved by the surrogates in
approximating the numerical values of the integrals in Eqs. (15).

The surrogate models effectively perform a non-linear interpolation of the
data collected in the training table. We prefer to use this approach with
respect to standard interpolation procedures because of its speed, its non-
linearity, and because of its ability to generalize outside of the range of
the table.

The surrogate model is used in the training process as described in the main
text. Given the accuracy of the training of the surrogate model and the fact
that the physical values of $m/T$ and $\gamma/T$ are well within the range of
values used in the training of the surrogate, we expect that numerical errors
and artifacts caused by the use of the surrogate will be much smaller than the
errors on the lattice data. Therefore, we conclude that the surrogate model
can be applied safely to the case studies. Notice that, despite the ability of
the neural network to generalize outside of its range of training, any result
out of the training intervals of the surrogate models mentioned above is, in
principle, less reliable.

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