# Inelastic and elastic parton scatterings in the strongly interacting quark-
gluon plasma

Ilia Grishmanovskii<EMAIL_ADDRESS>Institut für Theoretische
Physik, Johann Wolfgang Goethe-Universität,Max-von-Laue-Str. 1, D-60438
Frankfurt am Main, Germany  Olga Soloveva  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
Taesoo Song  GSI Helmholtzzentrum für Schwerionenforschung GmbH,Planckstrasse
1, D-64291 Darmstadt, Germany  Carsten Greiner  Institut für Theoretische
Physik, Johann Wolfgang Goethe-Universität,Max-von-Laue-Str. 1, D-60438
Frankfurt am Main, Germany  Helmholtz Research Academy Hesse for FAIR (HFHF),
GSI Helmholtz Center for Heavy Ion Physics, Campus Frankfurt, 60438 Frankfurt,
Germany  Elena Bratkovskaya  GSI Helmholtzzentrum für Schwerionenforschung
GmbH,Planckstrasse 1, D-64291 Darmstadt, Germany  Institut für Theoretische
Physik, Johann Wolfgang Goethe-Universität,Max-von-Laue-Str. 1, D-60438
Frankfurt am Main, Germany  Helmholtz Research Academy Hesse for FAIR (HFHF),
GSI Helmholtz Center for Heavy Ion Physics, Campus Frankfurt, 60438 Frankfurt,
Germany

###### Abstract

We investigate the role of inelastic processes in the strongly interacting
quark-gluon plasma (sQGP) based on the effective dynamical quasi-particle
model (DQPM). In the DQPM the non-perturbative properties of the sQGP at
finite temperature $T$ and baryon chemical potential $\mu_{B}$ are described
in terms of strongly interacting off-shell partons (quarks and gluons) with
dynamically generated spectral functions whose properties are adjusted to
reproduce the lQCD EoS for the QGP in thermodynamic equilibrium. For the first
time the massive gluon radiation processes from the off-shell quark-quark
($q+q$) and quark-gluon ($q+g$) scatterings are calculated explicitly within
leading order Feynman diagrams with effective propagators and vertices from
the DQPM without any further approximations. We present the results for the
energy and temperature dependencies of the total and differential radiative
cross sections and compare them to the corresponding elastic cross sections.
We show that our results reproduce the pQCD calculations in the limit of zero
masses and widths of quasiparticles. Also we study the $\mu_{B}$ dependence of
the inelastic cross sections. Moreover, we present estimates for the
transition rate and relaxation time of radiative versus elastic scatterings in
the sQGP.

##  I Introduction

Heavy-ion collisions provide a unique possibility to explore the structure of
the QCD phase diagram and study the properties of the quark-gluon plasma under
extreme conditions. The experimental observations Arsene  _et al._ (2005);
Adcox  _et al._ (2005); Back  _et al._ (2005); Adams  _et al._ (2005) suggest
that the matter created in central heavy-ion collisions shows the properties
of a strongly-coupled liquid Shuryak (2005); Gyulassy and McLerran (2005)
rather than a weakly-coupled gas of quark and gluons as expected initially.
The theoretical description of such a strongly interacting QGP is a
challenging task since perturbative methods of QCD do not work in the vicinity
of the critical temperature $T_{c}$ where the strong coupling constant is
large Ghiglieri  _et al._ (2018). Moreover, a solution of QCD on the lattice
(lQCD) can provide presently only a limited knowledge about the phase diagram
at moderate baryon $\mu_{B}$ (or quark $\mu_{q}=\mu_{B}/3$) chemical potential
where $\mu_{B}\simeq 3\mu_{q}$.

In order to overcome these difficulties and obtain information about the
degrees-of-freedom of the sQGP and their properties effective models are
applied such as quasiparticle models (QPM) Blaizot  _et al._ (2001); Peshier
_et al._ (1996); Sambataro  _et al._ (2023); Liu  _et al._ (2022). One of the
prominent quasiparticle models is the QCD-inspired effective field-theoretical
Dynamical QuasiParticle Model (DQPM) introduced in Ref. Peshier and Cassing
(2005) and worked out in Refs. Cassing (2007a, b); Berrehrah  _et al._
(2016a); Moreau  _et al._ (2019); Soloveva  _et al._ (2020a) and others. The
main difference and advantage of the DQPM with respect to other QPM is that it
is by construction a 2PI model while the other quasiparticle models are 1PI in
nature. The DQPM is based on a propagator representation with effective
(anti-)quarks and gluons which properties are defined by complex self-
energies. Thus, additionally to the temperature (and $\mu_{B}$ in DQPM)
dependence of quasiparticle masses – as in other effective QPM – the DQPM
contains the information about their interactions in the sQGP via dynamically
generated widths, i.e. the imaginary part of self-energies, which also depend
on temperature $T$ (and $\mu_{B}$). Consequently, the DQPM quasiparticles are
off-shell broad states with dynamically generated spectral functions whose
properties (masses and widths) are adjusted by fitting the DQPM entropy
density to the lattice QCD (lQCD) data at zero $\mu_{B}$ Cassing (2007a, b).
This avoids an introduction of a Debye mass in the propagator for the
regularization of infrared divergences when calculating cross sections as well
as an extra bag constant when computing the thermodynamic properties of the
QGP.

The effective off-shell strongly interacting partonic quasiparticles of the
DQPM allow for a microscopic interpretation of the lattice QCD results in the
$T,\mu_{B}$ plane for thermodynamic quantities – such as entropy density,
pressure, energy density – as well as for transport coefficients which measure
the interactions of the QGP degrees-of-freedom in thermodynamic equilibrium.
The transport coefficients are evaluated by computing the off-shell partonic
elastic interaction cross sections based on the leading order scattering
Feynman diagrams with effective propagators and vertices and depend on
$T,\mu_{B}$, the invariant energy of the colliding partons $\sqrt{s}$ as well
as the scattering angle Moreau  _et al._ (2019). The DQPM results for the
shear and bulk viscosity, electric conductivity, magnetic susceptibility as
well as the full diffusion matrix show a good agreement with existing lQCD
results as well as with a Bayesian analysis based on hydrodynamic description
of the heavy-ion data Ozvenchuk  _et al._ (2013); Cassing  _et al._ (2013);
Steinert and Cassing (2014); Moreau  _et al._ (2019); Fotakis  _et al._
(2021); Soloveva  _et al._ (2021). We note that the DQPM quasiparticles have
been incorporated in the PHSD approach, based on Kadanoff-Baym off-shell
dynamics, for the description of the partonic phase and hadronization (cf. the
reviews Cassing (2009); Linnyk  _et al._ (2016); Bleicher and Bratkovskaya
(2022)), and has led to a successful description of a large variety of
experimental data.

Recently, in Ref. Grishmanovskii  _et al._ (2022) the DQPM has been applied
for studying the elastic interactions of leading jet partons in the sQGP and
for the calculation of the jet transport coefficients, i.e. the transverse
momentum transfer squared per unit length $\hat{q}$ as well as the energy loss
per unit length $\Delta E=dE/dx$. However, for a realistic description of jet
attenuation in heavy-ion collisions, additionally to elastic scattering, the
gluon radiation processes have to be accounted for, which contribution grows
with increasing the energy of jet partons Djordjevic (2006). The role of
radiative processes for a jet suppression has been extensively studied by a
variety of groups – cf. the review and refs. therein Cao and Wang (2021).
Moreover, the radiative processes are also important for an understanding of
the energy loss of heavy quarks in heavy-ion collisions: while at low and
intermediate $p_{T}$ the elastic processes are dominant, at high $p_{T}$ the
gluon radiation can be sizeable Cao  _et al._ (2019).

When considering a thermalized sQGP, one expects that the inelastic processes
of gluon radiation from quark-quark ($q+q$) and quark-gluon ($q+g$)
scatterings ($q+q\to q+q+g$ and $q+g\to q+g+g$) should be strongly suppressed
relative to the elastic scattering due to the low average collision energy of
thermal partons (related to the thermal distribution of partons). In the sQGP
– consisting of thermal quasiparticles – such radiative processes are expected
to be even more suppressed relative to the pQCD plasma due to the large mass
of the emitted gluons and final off-shell partons. However, such an
expectation has to be quantified.

Thus, the goal of this study is to extend the DQPM to radiative processes for
the $q+q$ and $q+g$ scatterings, to calculate the inelastic $q+q\to q+q+g$ and
$q+g\to q+g+g$ cross sections, to estimate the inelastic rate relatively to
the elastic one and find the energy and temperature regimes where radiative
processes become visible and even dominant in the strongly interacting QGP.
The main difference from previous studies is that now we avoid any kind of
approximations (on the gluon energy, transverse momentum transfer and etc.)
and calculate all possible $2\to 3$ diagrams explicitly. This approach will
allow us to explore the entire radiative processes from massive partons with
spectral functions and study different limits and applications.

We recall that a study of radiative processes in QCD has a long history.
Explicit analytical expressions for the $2\to 3$ amplitudes in the case of
massless pQCD quarks have been derived in Refs. Berends  _et al._ (1981);
Ellis and Sexton (1986). However, the actual calculations of cross-sections of
radiative processes in pQCD with massless quarks and gluons require regulating
an infrared divergence of the amplitudes realized by the introduction of a
Debye mass (which is a model parameter) in propagators. Moreover, big efforts
were related to the accounting of the Landau-Pomeranchuk-Migdal effect (LPM)
Landau and Pomeranchuk (1953); Migdal (1956), which suppresses the soft gluon
radiation Wang  _et al._ (1995).

In order to simplify the calculations different approximations are used. A
commonly used approximation to the leading order pQCD matrix element for
partonic $2\to 3$ processes is a formula derived by Gunion and Bertsch (GB) in
1981 Gunion and Bertsch (1982). The original GB approximation is strictly
valid at mid-rapidity and implies that the radiated gluon and the momentum
transfer of the process are soft and all particles are massless. In Ref. Das
and Alam (2010); Abir  _et al._ (2011); Bhattacharyya  _et al._ (2012) efforts
are made to go beyond the soft GB approximation while still obtaining a
relatively compact form. In Ref. Fochler  _et al._ (2013) the GB approximation
was extended to be valid in the entire region of rapidity and in Ref. Uphoff
(2014) the latter was generalized to the case of heavy quarks. These
approximations have been widely used when solving transport problems, for
example via rate equations or via microscopic transport approaches, most
notably the study of heavy quark and jet suppression within the microscopic
transport model Boltzmann Approach to MultiParton Scatterings (BAMPS) Uphoff
_et al._ (2015); Senzel (2020); Fochler  _et al._ (2013); Senzel (2020).

The influence of radiative processed on jet attenuation beyond pQCD has been
studied in the framework of finite temperature field theory in the hard
thermal loop (HTL) approximation Djordjevic (2006); Djordjevic and Heinz
(2008); Djordjevic (2009); Djordjevic and Djordjevic (2012); Caron-Huot
(2009), in non-perturbative calculations within EQCD Moore and Schlusser
(2020) and in AdS/CFT models Liu  _et al._ (2006); Grefa  _et al._ (2022),
which show the importance of non-perturbative effects for an understanding of
the transverse jet momentum broadening and jet quenching Moore  _et al._
(2022).

The emission of massless and massive gluons by heavy quark scattering with
massless quarks and gluons from the QGP medium has been studied within scalar
pQCD in Refs. Gossiaux  _et al._ (2010); Aichelin  _et al._ (2014); Gossiaux
_et al._ (2014). It has been shown that the emission rate decreases with
increasing gluon mass.

Recently, in Ref. Song  _et al._ (2023) the soft gluon approximation (SGA) has
been applied to evaluate the massive gluon emission from the non-perturbative
sQGP medium described by the DQPM. The SGA simplifies the calculations making
the scattering amplitude factorizable into the elastic scattering and the
emission of a soft gluon. In the present study we depart from this
approximation which makes our results applicable for the emission of gluons of
any energy and angular direction. However, we mention (without showing
explicitly) that the SGA qualitatively agrees with our exact results for gluon
radiation cross sections in the range of validity of the SGA.

The paper is organized as follows: we start with the Introduction in Section
I. We recall the basis of the DQPM model in Section II. In Section III we
present the framework for the calculation of radiative processes from quark-
quark and quark-gluon scattering in the DQPM. In Section IV we continue with
the results for the inelastic and elastic total and differential cross
sections as a function of temperature and scattering energy, transition rates
and relaxation times for quarks and gluons in the sQGP. We finish the paper
with a Summary in Section V. Moreover, the details of our calculations for the
cross sections are presented in the Appendices.

##  II Dynamical Quasiparticle Model

We start with a brief reminder of the basis of the DQPM and refer the reader
to Ref. Soloveva  _et al._ (2021) for a more detailed description of the
latest version of the DQPM.

The Dynamical Quasiparticle Model (DQPM) Peshier and Cassing (2005); Cassing
(2007a, b); Berrehrah  _et al._ (2016a); Moreau  _et al._ (2019); Soloveva
_et al._ (2021) is an effective model which describes the Quark-Gluon Plasma
(QGP) in terms of strongly interacting quarks and gluons. This model is based
on fitting the properties of these particles to reproduce the results of
lattice QCD calculations in thermal equilibrium and at vanishing chemical
potential.

The quasi-particles in the DQPM are characterized by:
$\bullet$ ”dressed” propagators, i.e. single-particle (two-point) Green’s
functions, which have the form

$G^{R}_{j}(\omega,{\bf p})=\frac{1}{\omega^{2}-{\bf
p}^{2}-M_{j}^{2}+2i\gamma_{j}\omega}$ (1)

for quarks, antiquarks, and gluons ($j=q,\bar{q},g$), using $\omega=p_{0}$ for
energy, the widths $\gamma_{j}$ and the masses $M_{j}$.
$\bullet$ complex self-energies for gluons $\Pi=M_{g}^{2}-2i\omega\gamma_{g}$
and for (anti)quarks $\Sigma_{q}=M_{q}^{2}-2i\omega\gamma_{q}$, where the real
part of the self-energies is associated with dynamically generated thermal
masses, while the imaginary part provides information about the lifetime and
reaction rates of the particles.

The spectral functions in the DQPM are no longer $\delta$-functions, but have
a finite width Linnyk  _et al._ (2016).

$\displaystyle\rho_{j}(\omega,{\bf p})$
$\displaystyle=\frac{\gamma_{j}}{\tilde{E}_{j}}\left(\frac{1}{(\omega-\tilde{E}_{j})^{2}+\gamma^{2}_{j}}-\frac{1}{(\omega+\tilde{E}_{j})^{2}+\gamma^{2}_{j}}\right)$
$\displaystyle\equiv\frac{4\omega\gamma_{j}}{\left(\omega^{2}-\vec{p}^{2}-M^{2}_{j}\right)^{2}+4\gamma^{2}_{j}\omega^{2}}$
(2)

Here, $\tilde{E}_{j}^{2}({\bf p})={\bf p}^{2}+M_{j}^{2}-\gamma_{j}^{2}$. The
spectral function is antisymmetric in $\omega$ and normalized as

$\int\limits_{-\infty}^{\infty}\frac{d\omega}{2\pi}\ \omega\
\rho_{j}(\omega,{\bf p})=\int\limits_{0}^{\infty}d\omega\frac{\omega}{\pi}\
\rho_{j}(\omega,{\bf p})=1.$ (3)

$\bullet$ a model ansatz for the masses $M_{j}(T,\mu_{q})$ and widths
$\gamma_{j}(T,\mu_{q})$ as functions of the temperature $T$ and the quark
chemical potential $\mu_{q}$.

With the quasiparticle properties (or propagators) fixed as described above,
one can evaluate thermodynamic quantities such as the entropy density
$s(T,\mu_{B})$, the pressure $P(T,\mu_{B})$ and energy density
$\epsilon(T,\mu_{B})$ in a straight forward manner by starting with the
entropy density and number density in the propagator representation from Baym
Vanderheyden and Baym (1998); Blaizot  _et al._ (2001).

By comparison of the entropy density – computed within the DQPM framework – to
the lQCD data, one can fix the few parameters used in the ansatz for
quasiparticle masses and widths.

###  II.1 Quasiparticle properties

The following ansatz is used in the DQPM for the definition of the
quasiparticle properties (masses and widths) as a functions of $T$ and
$\mu_{q}$:
$\bullet$ The dynamical quasiparticle pole masses are given by the HTL thermal
mass in the asymptotic high-momentum regime, i.e. for gluons by Bellac (2011);
Linnyk  _et al._ (2016),

$M^{2}_{g}(T,\mu_{q})=\frac{g^{2}(T,\mu_{q})}{6}\left(\left(N_{c}+\frac{1}{2}N_{f}\right)T^{2}+\frac{N_{c}}{2}\sum_{q}\frac{\mu^{2}_{q}}{\pi^{2}}\right),$
(4)

and for quarks (antiquarks) by

$M^{2}_{q(\bar{q})}(T,\mu_{q})=\frac{N^{2}_{c}-1}{8N_{c}}g^{2}(T,\mu_{q})\left(T^{2}+\frac{\mu^{2}_{q}}{\pi^{2}}\right),$
(5)

where $N_{c}\ (=3)$ stands for the number of colors and $N_{f}\ (=3)$ denotes
the number of flavors. Eq. (5) determines masses for the ($u,d$) quarks, the
strange quark has a larger bare mass for controlling the strangeness ratio in
the QGP. Empirically, we find $M_{s}(T,\mu_{B})=M_{u/d}(T,\mu_{B})+\Delta M$,
where $\Delta M$ = 30 MeV, which has been fixed once in comparison to
experimental data Moreau  _et al._ (2019)

$\bullet$ The effective quarks, antiquarks, and gluons in the DQPM acquire
sizable widths $\gamma_{j}$, which are taken in the form Linnyk  _et al._
(2016)

$\gamma_{j}(T,\mu_{\mathrm{B}})=\frac{1}{3}C_{j}\frac{g^{2}(T,\mu_{\mathrm{B}})T}{8\pi}\ln\left(\frac{2c_{m}}{g^{2}(T,\mu_{\mathrm{B}})}+1\right).$
(6)

Here $c_{m}=14.4$ is related to a magnetic cutoff, which is an additional
parameter of the DQPM, and $C_{q}=\dfrac{N_{c}^{2}-1}{2N_{c}}=4/3$ and
$C_{g}=N_{c}=3$ are the QCD color factors for quarks and for gluons,
respectively. We also assume that all (anti-)quarks have the same width.

$\bullet$ As follows from Eqs. (5), (6) the masses and widths depend on the
coupling constant (squared) $g^{2}=4\pi\alpha_{s}$ which defines the strength
of the interaction. This parameter is a critical quantity that determines all
the microscopic properties and transport coefficients of the Dynamical
Quasiparticle Model (DQPM), including the thermal masses and widths.

In the DQPM, the value of $g^{2}$ is extracted from lattice Quantum
Chromodynamics (lQCD) by utilizing a parametrization method introduced in Ref.
Berrehrah  _et al._ (2016b), where 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 and identical to $g^{2}$, i.e. ${\frac{\partial}{\partial
T}}(s(T,g^{2})/T^{3})=0$. Therefore the entropy density $s$ and the
dimensionless equation of state in the DQPM is a function of the effective
coupling only, i.e. $s(T,g^{2})/s_{SB}(T)=f(g^{2})$ where
$s^{QCD}_{SB}=19/9\pi^{2}T^{3}$ is the Stefan-Boltzmann entropy density. 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 $s(T,\mu_{B}=0)$ at zero baryon chemical potential:

$g^{2}(T,\mu_{B}=0)=d\left((s(T,0)/s^{QCD}_{SB})^{e}-1\right)^{f}.$ (7)

Here $d=169.934,e=-0.178434$ and $f=1.14631$ are the dimensionless parameters
obtained by adjusting quasiparticle entropy density $s(T,\mu_{B}=0)$ to the
lQCD data provided by the BMW Collaboration Borsanyi  _et al._ (2012, 2014).

Figure 1: Running coupling constant $\alpha_{s}$ as a function of $T/T_{c}$
for $\mu_{B}=0$. Black solid line corresponds to the DQPM running coupling
defined by Eq. (8), gray dashed line indicates a constant value of 0.3. The
lattice results for quenched QCD, $N_{f}=0$, (blue circles) are taken from
Ref. Kaczmarek  _et al._ (2004), for $N_{f}=2$ (orange inverted triangles) –
from Ref. Kaczmarek and Zantow (2005) and for $N_{f}=2+1$ (red triangles) –
from Ref. Kaczmarek (2007).

Fig. 1 displays the running coupling constant $\alpha_{s}$ for $\mu_{B}=0$ as
a function of $T/T_{c}$, represented by the black solid line in accordance
with Eq. (7). The lattice results for quenched QCD, $N_{f}=0$, (blue circles)
are taken from Ref. Kaczmarek  _et al._ (2004), for $N_{f}=2$ (orange inverted
triangles) – from Ref. Kaczmarek and Zantow (2005) and for $N_{f}=2+1$ (red
triangles) – from Ref. Kaczmarek (2007). The fit obtained from the entropy
density for the DQPM $\alpha_{s}$ shows a significant increase approaching
$T_{c}$ comparable to the predictions from lQCD.

It is important to note that the value of the coupling constant obtained by
lQCD (i.e. lQCD ”data” in Fig. 1) depends on the definition of $\alpha_{s}$
extracted from the static potential Kaczmarek  _et al._ (2004); Kaczmarek
(2007). Moreover, the DQPM $\alpha_{s}$ accounts for nonperturbative effects
which make it larger compared to the analytical 2 or 1 loop running constant
Caswell (1974) approaching low temperatures. The gray dashed line in the
figure represents a constant value of $\alpha_{s}=0.3$ frequently used in
perturbative QCD models, particularly in the BAMPS calculations.

The extension of the coupling constant to finite baryon chemical potential
$\mu_{B}$ is realized using a scaling hypothesis Cassing (2009), which works
up to $\mu_{B}\approx 500$ MeV. It assumes that $g^{2}$ is a function of the
ratio of the effective temperature $T^{*}=\sqrt{T^{2}+\mu_{q}^{2}/\pi^{2}}$
and the $\mu_{B}$-dependent critical temperature $T_{c}(\mu_{B})$ as

$g^{2}(T/T_{c},\mu_{B})=g^{2}\left(\frac{T^{*}}{T_{c}(\mu_{B})},\mu_{B}=0\right)$
(8)

with $T_{c}(\mu_{B})=T_{c}\sqrt{1-\alpha\mu_{B}^{2}}$, where $T_{c}$ is the
critical temperature at vanishing chemical potential ($\approx$=0.158 GeV) and
$\alpha\simeq 0.974$ GeV-2 Berrehrah  _et al._ (2016a). This extension of the
coupling constant for finite $\mu_{B}$ increase the $\mu_{B}$ \- dependence of
pole masses and widths, especially in the vicinity of $T_{c}$ Soloveva  _et
al._ (2020b).

The actual pole masses of quarks $M_{q}$ and gluons $M_{g}$ as well as their
widths – $\gamma_{q}$ and $\gamma_{g}$ – are depicted in Fig. 2 as a function
of $T$ for different baryon chemical potentials $\mu_{B}=0,0.2,0.4$ and 0.6
GeV. As seen from the figure, the masses grow with $T$ and decrease with
increasing $\mu_{B}$ and move to the low $T$ following the $\mu_{B}$ dependent
critical temperature $T_{c}(\mu_{B})$ (depicted as dashed vertical lines). The
widths of quarks and gluons increase with $T$ and show only a weak $\mu_{B}$
dependence. For $\mu_{B}=0$ the DQPM gives
$M_{q}=\frac{2}{3}M_{g},\gamma_{q}=\frac{4}{9}\gamma_{g}$.

Figure 2: The effective quark (upper) and gluon (lower plot) masses $M$ and
widths $\gamma$ (from the parametrizations - Eqs. (5), (4) and (6)) as a
function of the temperature $T$ for different $\mu_{B}$. The vertical dashed
lines correspond to the DQPM $\mu_{B}$ dependent critical temperature
$T_{c}(\mu_{B})$.

Having the quasiparticle properties, dressed propagators, and coupling
constant as given by the DQPM, one can evaluate the scattering amplitudes as
well as the cross sections and the transport coefficients of quarks and gluons
in the QGP as a function of the temperature and the chemical potential
Berrehrah  _et al._ (2014); Moreau  _et al._ (2019).

###  II.2 Matrix elements in the DQPM

In order to calculate the matrix elements corresponding to a scattering of
DQPM partons, the scalar propagator has to be replaced by the following
propagators – with full Lorentz structure – to describe a massive vector gluon
and massive (spin $1/2$) fermion with a finite width Berrehrah  _et al._
(2014):

$\includegraphics[width=43.36133pt]{propg.pdf}=-i\delta_{ab}\frac{g^{\mu\nu}-q^{\mu}q^{\nu}/M^{2}_{g}}{q^{2}-M^{2}_{g}+2i\gamma_{g}q_{0}},$
(9)

$\includegraphics[width=43.36133pt]{propq.pdf}=i\delta_{ij}\frac{\not{q}+M_{q}}{q^{2}-M^{2}_{q}+2i\gamma_{q}q_{0}},$
(10)

where $q$ is the 4-momentum of the exchanged particle. The delta functions
ensure that the exchanged quark or gluon is connected with other parts of the
diagram with the same color ($a,b$ for the gluon and $i,j$ for the quark),
i.e. color and spin during the propagation is conserved.

In the DQPM, only transverse gluons are included in the calculation of
thermodynamic quantities referring that the contribution of hard longitudinal
gluons is found to be negligible Blaizot  _et al._ (2001); Peshier (2001). The
details on the summation over gluon polarization for massive gluons are given
in Appendix A. We note that we use the covariant gauge for the evaluation of
cross sections.

For the calculation of elastic quark-quark and quark-gluon scatterings we
refer the reader to Refs. Moreau  _et al._ (2019); Soloveva  _et al._ (2021);
Grishmanovskii  _et al._ (2022).

##  III Gluon radiation processes in the DQPM

Now we step to the calculation of the inelastic processes for gluon radiation
from thermal quark-quark ($q+q$) and quark-gluon ($q+g$) scatterings: $q+q\to
q+q+g$ and $q+g\to q+g+g$ within leading order Feynman diagrams with effective
propagators and vertices from the DQPM as specified in Section II.B. We note
that here and further on when noting reactions with ”quark” we also imply the
corresponding reactions for antiquarks (if not specified explicitly).

###  III.1 Matrix elements for $2\to 3$ reactions

The inelastic scattering is accounted by the leading order Feynman diagrams
for $2\to 3$ processes considering $t-$, $u-$ and $s-$ channels. All diagrams
are calculated explicitly without assumptions, accounting for all possible
interferences between different channels.

All processes are computed for the ”on-shell” case, when the masses of final
scattering partons as well as radiative gluon are taken at the pole position
of their spectral functions since ”off-shell” calculations including the
integration over their spectral functions are much more computationally
expensive. We note that on-shell elastic cross sections differ from off-shell
ones by less than 20% Grishmanovskii  _et al._ (2022).

####  III.1.1 Gluon radiation from quark + quark scattering

We start with gluon radiation from quark + quark scattering $q+q\to q+q+g$.
The corresponding Feynman diagrams for $t$-channels are presented in Fig. 3.
Here $p_{a},p_{b}$ and $p_{1},p_{2}$ stand for the 4-momenta of initial and
final quarks, respectively, while $p_{3}$ corresponds to the 4-momentum of the
emitted gluon.

Figure 3: Feynman diagrams for the $t$-channel of the $q+q\to q+q+g$
scattering.

Denoting

$\displaystyle\Pi_{\mu\nu}(p)=\left[-i\frac{g_{\mu\nu}-(p_{\mu}p_{\nu})/M_{g}^{2}}{p^{2}-M_{g}^{2}+2i\gamma_{g}\omega_{p}}\right]\quad\text{(gluon
propagator)},$
$\displaystyle\Lambda(p)=\left[i\frac{\not{p}+M_{q}}{p^{2}-M_{q}^{2}+2i\gamma_{q}\omega_{p}}\right]\quad\text{(quark
propagator)},$ $\displaystyle
V^{\nu,a}_{ik}=\left(-ig\gamma^{\nu}T^{a}_{ik}\right)\quad\text{(quark-gluon
vertex)},$
$\displaystyle\Omega^{abc,\mu\nu\tau}(p_{1},p_{2},p_{3})=-gf^{abc}C^{\mu\nu\tau}(p_{1},p_{2},p_{3})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{(three
gluon vertex)},$ (11)

where $\mu,\nu$ stand for the Lorentz indices, $i,k=1..3$ for the quark color
indices and $a,c,b=1..8$ for the gluon color indices, one can write the
invariant amplitudes $i\mathcal{M}_{i}=i\mathcal{M}_{i}(q^{i}q^{k}\to
q^{j}q^{l}g^{b})$ – corresponding to the Feynman diagrams in Fig. 3 \- as:

$\displaystyle i\mathcal{M}_{1}$
$\displaystyle=\bar{u}^{l}(p_{2})V^{\nu,a}_{lk}u^{k}(p_{b})\Pi_{\mu\nu}(p_{b}-p_{2})\bar{u}^{j}(p_{1})\varepsilon_{\tau}^{*}(p_{3})V^{\tau,b}_{jm}\Lambda(p_{1}+p_{3})V^{\mu,a}_{mi}u^{i}(p_{a})$
(12) $\displaystyle i\mathcal{M}_{2}$
$\displaystyle=\bar{u}^{j}(p_{1})V^{\mu,a}_{ji}u^{i}(p_{a})\Pi_{\mu\nu}(p_{a}-p_{1})\bar{u}^{l}(p_{2})\varepsilon_{\tau}^{*}(p_{3})V^{\tau,b}_{lm}\Lambda(p_{2}+p_{3})V^{\nu,a}_{mk}u^{k}(p_{b})$
$\displaystyle i\mathcal{M}_{3}$
$\displaystyle=\bar{u}^{l}(p_{2})V^{\nu,a}_{lk}u^{k}(p_{b})\Pi_{\mu\nu}(p_{b}-p_{2})\bar{u}^{j}(p_{1})V^{\mu,a}_{jm}\Lambda(p_{a}-p_{3})\varepsilon_{\tau}^{*}(p_{3})V^{\tau,b}_{mi}u^{i}(p_{a})$
$\displaystyle i\mathcal{M}_{4}$
$\displaystyle=\bar{u}^{j}(p_{1})V^{\mu,a}_{ji}u^{i}(p_{a})\Pi_{\mu\nu}(p_{a}-p_{1})\bar{u}^{l}(p_{2})V^{\nu,a}_{lm}\Lambda(p_{b}-p_{3})\varepsilon_{\tau}^{*}(p_{3})V^{\tau,b}_{mk}u^{k}(p_{b})$
$\displaystyle i\mathcal{M}_{5}$
$\displaystyle=\bar{u}^{j}(p_{1})V^{\mu,a}_{ji}u^{i}(p_{a})\;\bar{u}^{l}(p_{2})V^{\lambda,c}_{lk}u^{k}(p_{b})\Pi_{\mu\nu}(p_{a}-p_{1})\Pi_{\lambda\sigma}(p_{b}-p_{2})\varepsilon^{*}_{\tau}(p_{3})\Omega^{abc,\sigma\tau\nu}(p_{b}-p_{2},-p_{3},p_{2}-p_{b}+p_{3}).$

The invariant amplitude squared - averaged over initial states of spin and
color and summed over final states - are:

$\displaystyle|\bar{\mathcal{M}}_{q+q\prime\to
q+q\prime+g}|^{2}=\frac{1}{N_{c}^{2}}\sum_{\text{color}}\frac{1}{(2s_{q}+1)(2s_{q\prime}+1)}$
$\displaystyle\times\sum_{\text{spin}}|\mathcal{M}_{1}+\mathcal{M}_{2}+\mathcal{M}_{3}+\mathcal{M}_{4}+\mathcal{M}_{5}|^{2},$
(13)

where $q^{\prime}$ denotes a quark with a possible different flavours than the
$q$ quark.

There are $5\times 5=25$ possible combinations of multiplications, which, in
turn, can be reduced to 15 using symmetry relations:
$|\bar{M}_{ij}|=|\bar{M}_{ji}|$, where $|\bar{M}_{ij}|\sim M_{i}^{*}M_{j}$
(cf. Eq. (13)). An explicit expression for the squared amplitude can not be
written in a compact form and therefore the analysis of this expression can
only be performed numerically. In this study the final analytical expression
was evaluated using the FeynCalc package Mertig  _et al._ (1991); Shtabovenko
_et al._ (2016).

In general, the $q+q$ reaction implies different channels depending on the
flavors of the interacting quarks. The only channel that is common for all
$q+q$ reaction is the $t$-channel.
The $u$-channel is relevant only for quarks with the same flavors, and even
though the angular distribution for this reaction is different, the integrated
cross section is of the same value as for the quarks with different flavors.
The $s$-channel is relevant for the $q+\bar{q}$ reaction and for the $2\to 3$
reaction is expected to be strongly suppressed for any value of $\sqrt{s}$.
Therefore in this study, we consider only the $t$-channel for any $q+q$
reaction noting that the corrections given by the other channels are
negligible.

####  III.1.2 Gluon radiation from quark + gluon scattering

We continue with gluon radiation from quark + gluon scattering $q+g\to q+g+g$.
Contrary to $q+q$ reactions, $q+g$ reactions assume all $t$-, $u$\- and
$s$-channels regardless of the quark flavor. However, we will show explicitly
that, as in case of the $q+q$ reaction, the $t$-channel plays the dominant
role in the $q+g$ and contributions from the other channels could as well be
omitted. The corresponding Feynman diagrams for $t$-, $u$-, $s$\- channels are
presented in Fig. 4.

Figure 4: Feynman diagrams for $t$-channels (upper 6 diagrams), $u$-channels
(lower left 5 diagrams), $s$-channels (lower right 5 diagrams) for the $q+g\to
q+g+g$ process.

For the $t$-channel of the $q+g$ reaction the matrix amplitudes (according to
the numeration in the upper 2 rows in Fig. 4) read as:

$\displaystyle i\mathcal{M}_{1}$
$\displaystyle=\bar{u}(p_{1})V^{\mu,d}_{kn}\Lambda(p_{1}+p_{3},m_{q})V^{\lambda,f}_{ni}u(p_{a})\Omega^{bef,\sigma\tau\nu}(p_{b}-p_{2},-p_{b},p_{2})\Pi_{\lambda\sigma}(p_{2}-p_{b})\varepsilon^{*}_{\mu}(p_{3})\varepsilon^{*}_{\nu}(p_{2})\varepsilon_{\tau}(p_{b})$
(14) $\displaystyle i\mathcal{M}_{2}$
$\displaystyle=\bar{u}(p_{1})V^{\sigma,f}_{kn}\Lambda(p_{1}+p_{2}-p_{b},m_{q})V^{\mu,d}_{ni}u(p_{a})\Omega^{bef,\lambda\tau\nu}(p_{b}-p_{2},-p_{b},p_{2})\Pi_{\lambda\sigma}(p_{2}-p_{b})\varepsilon^{*}_{\mu}(p_{3})\varepsilon^{*}_{\nu}(p_{2})\varepsilon_{\tau}(p_{b})$
$\displaystyle i\mathcal{M}_{3}$
$\displaystyle=\bar{u}(p_{1})V^{\lambda,f}_{ki}u(p_{a})\Omega^{bfn,\nu\mu\xi}(-p_{2},-p_{3},p_{2}+p_{3})\Omega^{deh,\sigma\tau\rho}(-p_{b}+p_{2}+p_{3},p_{b},-p_{2}-p_{3})$
$\displaystyle\times\Pi_{\rho\xi}(p_{2}-p_{b})\Pi_{\lambda\sigma}(p_{2}-p_{b})\varepsilon^{*}_{\mu}(p_{3})\varepsilon^{*}_{\nu}(p_{2})\varepsilon_{\tau}(p_{b})$
$\displaystyle i\mathcal{M}_{4}$
$\displaystyle=\bar{u}(p_{1})V^{\lambda,f}_{ki}u(p_{a})\Omega^{efh,\tau\mu\rho}(-p_{b},p_{3},p_{b}-p_{3})\Omega^{bdh,\nu\sigma\xi}(p_{2},p_{b}-p_{2}-p_{3},-p_{b}+p_{3})$
$\displaystyle\times\Pi_{\rho\xi}(p_{2}-p_{b})\Pi_{\lambda\sigma}(p_{2}-p_{b})\varepsilon^{*}_{\mu}(p_{3})\varepsilon^{*}_{\nu}(p_{2})\varepsilon_{\tau}(p_{b})$
$\displaystyle i\mathcal{M}_{5}$
$\displaystyle=\bar{u}(p_{1})V^{\lambda,f}_{ki}u(p_{a})\Omega^{dfh,\sigma\mu\xi}(-p_{b}+p_{2}+p_{3},-p_{3},-p_{2}+p_{b})\Omega^{beh,\nu\tau\rho}(-p_{2},p_{b},-p_{b}+p_{2})$
$\displaystyle\times\Pi_{\rho\xi}(p_{2}-p_{b})\Pi_{\lambda\sigma}(p_{2}-p_{b})\varepsilon^{*}_{\mu}(p_{3})\varepsilon^{*}_{\nu}(p_{2})\varepsilon_{\tau}(p_{b})$
$\displaystyle i\mathcal{M}_{6}$
$\displaystyle=\bar{u}(p_{1})V^{\lambda,f}_{ki}u(p_{a})\Gamma^{fbde}_{\mu\nu\sigma\tau}\Pi_{\lambda\sigma}(p_{2}-p_{b})\varepsilon^{*}_{\mu}(p_{3})\varepsilon^{*}_{\nu}(p_{2})\varepsilon_{\tau}(p_{b}).$

We omit showing explicitly the expressions for $u$\- and $s$-channels noting
that they can be expressed in a similar way.

###  III.2 Cross section for $2\to 3$ reactions

The cross section for a $2\to 3$ process $(a+b\to 1+2+3)$ is given by

$\displaystyle\sigma_{2\to 3}$
$\displaystyle=\frac{1}{F}\int\frac{d^{3}p_{1}}{(2\pi)^{3}2E_{1}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\int\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{a}+p_{b}-p_{1}-p_{2}-p_{3})|\bar{\mathcal{M}}_{2\to
3}|^{2},$ (15)

where the flux is $F=4p_{C}\sqrt{s}$ with $p_{C}$ being the initial three
momentum in the center-of-mass (CMS) frame.

This integration implies, in general, 9-dimensional integration. However, it
is numerically more efficient to analytically integrate out the delta function
to end up with a five dimensional integration. The further choice of
independent phase space variables depends on the observables we are interested
in. The first possible choice to perform the integration is to consider the
$2\to 3$ reaction as a sum $(2\to 2)+(1\to 2)$ processes by introducing the
intermediate ”state” $p_{x}=p_{2}+p_{3}$ (or $p_{x}=p_{1}+p_{2}$). In this
case the set of independent variables can be chose as
$\\{m_{x},\theta,\varphi,\xi,\psi\\}$, where $\\{\theta,\varphi\\}$ are polar
and azimuthal angles between $\vec{p}_{1}$ and $\vec{p}_{x}$ (or $\vec{p}_{3}$
and $\vec{p}_{x}$) in the CMS frame, $\\{\xi,\psi\\}$ are polar and azimuthal
angles of $\vec{p}_{2}$ and $\vec{p_{3}}$ (or $\vec{p_{1}}$ and $\vec{p_{2}}$)
in $p_{x}$ rest frame and $m_{x}=\sqrt{p_{x}^{2}}$. The integration over
$\varphi$ can be done analytically due to the symmetry of the system leaving
with the final expression for the total cross section (see B.1 for details):

$\displaystyle\sigma_{2\to 3}$
$\displaystyle=\frac{1}{8(2\pi)^{5}\sqrt{s}F}\int_{0}^{1}d\cos\theta\int_{0}^{1}d\cos\xi\int_{0}^{2\pi}d\psi$
$\displaystyle\times\int_{m_{2}+m_{3}(m_{1}+m_{2})}^{\sqrt{s}-m_{1}(\sqrt{s}-m_{3})}dm_{x}\
p_{x,C}\ p_{2,X}|\bar{\mathcal{M}}_{2\to 3}|^{2}.$ (16)

This choice of variables allows to investigate the angular distribution of the
inelastic cross section, in particular $d\sigma/d\cos\theta$ with $\theta$
being the scattering angle of $p_{1}$ (leading quark) or $p_{3}$ (emitted
gluon) in the CMS frame. Also this set of variables is convenient when one
wants to implement the cross sections into transport simulations.

The second choice of variables (in the CMS) implies switching to GB
coordinates $(q_{t},k_{t},y,\varphi)$, where $q_{t}$ is the transverse
momentum transfer, $k_{t}$ is transverse momentum of the emitted gluon, $y$ is
the rapidity of the emitted gluon and $\varphi$ is the angle between $k_{t}$
and $q_{t}$. In these variables the $2\to 3$ cross section can be expressed as

$\displaystyle\sigma_{2\to 3}$
$\displaystyle=\frac{1}{128\pi^{4}F}\int_{0}^{p_{C}^{2}}dq_{t}^{2}\int_{0}^{p_{C}^{2}}dk_{t}^{2}\int_{y_{\min}}^{y_{\max}}dy\int_{0}^{\pi}d\varphi$
$\displaystyle\times|\bar{\mathcal{M}}_{2\to
3}|^{2}\sum\left(\left.\frac{\partial\mathcal{F}}{\partial
y_{1}}\right|_{\mathcal{F}=0}\right)^{-1},$ (17)

where $y_{1}$ denotes the rapidity of particle 1 and $\varphi$ is the angle
between $\vec{q}_{t}$ and $\vec{k}_{t}$. The available phase space limits the
rapidity to
$y_{\max/\min}=\pm\text{arccosh}(\frac{\sqrt{p_{C}^{2}+m_{3}^{2}}}{m_{3,t}})$.
The function $\mathcal{F}$ comes from the transformation of the delta function
(see B.2).

We have numerically checked that both Eqs. (III.2) and (III.2) give the same
results.

##  IV Results

We proceed with showing the results from numerical calculations of cross
sections for radiative processes $q+q\to q+q+g$ and $q+g\to q+g+g$. We note
that all calculations here and below are done for $\mu_{B}=0$. If not stated
otherwise, all parton masses/widths are chosen to be thermal DQPM
masses/widths (Moreau  _et al._ (2019)) which depend of temperature and
$\mu_{B}$. The parton masses are taken at the pole position of the spectral
functions (cf. Fig. 2), i.e.

$\displaystyle m_{a}$
$\displaystyle=m_{b}=m_{1}=m_{2}=M_{q}=M^{DQPM}_{q}(T,\mu_{B}),$
$\displaystyle m_{3}$ $\displaystyle=M_{g}=M^{DQPM}_{g}(T,\mu_{B}),$
$\displaystyle\gamma_{q}$ $\displaystyle=\gamma^{DQPM}_{q}(T,\mu_{B}),$
$\displaystyle\gamma_{g}$ $\displaystyle=\gamma^{DQPM}_{g}(T,\mu_{B}).$ (18)

We mention that when indicating the processes $q+q\to q+q$, $q+q\to q+q+g$ we
imply the reactions of light quarks of different flavours, too. The same
convention holds for the scatterings including anti-quark.

###  IV.1 pQCD limit of the DQPM

We start with showing explicitly that the DQPM amplitudes reproduce the pQCD
amplitudes Berends  _et al._ (1981); Ellis and Sexton (1986) in the ”pQCD”
limit of massless partons with zero widths.

Figure 5: Squared scattering amplitude (scaled by $g^{6}$) for the $q+q\to
q+q+g$ process for $\sqrt{s}=5$ GeV as a function of the transverse momentum
of the emitted gluon $k_{t}$ in case of the full DQPM (calculated with pole
masses $M_{q},M_{g}$) for T=0.4 GeV (blue line), pQCD result from Berends  _et
al._ (1981) (orange line) and DQPM in the massless limit (black dotted line).
The dashed green and red lines stand for the DQPM amplitudes calculated with
reduced pole masses of quarks and gluons $M_{q}/n,M_{g}/n,\ n=2,4$.

Fig. 5 shows the squared amplitude for the inelastic squared amplitude (scaled
by $g^{6}$) for the $q+q\to q+q+g$ process as a function of the transverse
momentum of the emitted gluon $k_{t}$ in case of the full DQPM (blue line),
pQCD result from Berends  _et al._ (1981) (orange line) and DQPM in the
massless limit (black dotted line). The quantity $|M_{qq}|^{2}/g^{6}$ is
computed for $q_{t}=0.4$ GeV, $y=0$, $\varphi=0$.

The convergence of the ”massless DQPM” and ”pQCD” results illustrates the
correspondence between the pQCD and DQPM model in the massless limit as it
follows from the form of the quark, Eq. (10), and gluon, Eq. (9), propagators.
One can also see that the pQCD amplitude diverges in the infrared ($k_{t}\to
0$) and collinear ($\vec{k}_{t}=\vec{q}_{t}$) limits, which makes it
impossible to calculate the integrated cross section without introducing the
corresponding cut-offs. The result from the full DQPM model calculated with
the pole masses of quarks and gluons (blue line) shows the absence of these
divergencies and therefore allows for the calculation of the integrated cross
section. The dashed green and red lines show the DQPM amplitudes calculated
with reduced masses of quarks and gluons $M_{q}/n,M_{g}/n,\ n=2,4$. One can
see that the reduction of the DQPM masses leads to an enhancement of the DQPM
amplitudes that approach the pQCD limit.

###  IV.2 Angular distributions

Figure 6: Inelastic differential cross sections for $q+q\to q+q+g$ scattering
of light quarks of different flavours for $T=3T_{c}$ in the CMS as a function
of the angle $\theta_{q}$ between the initial and final momentum of the quark.
The different lines correspond to different collision energies $\sqrt{s}$ from
3 to 10 GeV (see legend). Figure 7: Inelastic differential cross sections for
$q+q\to q+q+g$ processes for $T=3T_{c}$ as a function of the angle
$\theta_{g}$ between the momentum of the emitted gluon relative to the
collision axis in the CMS. The different lines correspond to different
collision energies $\sqrt{s}$ from 3 to 10 (see legend).

Fig. 6 shows the inelastic differential cross sections for $q+q\to q+q+g$
scattering for light quarks of different flavours for $T=3T_{c}$ as a function
of the angle $\theta_{q}$ between the outgoing quark after emission of the
gluon relative to the collision axis in the CMS for different collision
energies $\sqrt{s}$. As one can see the angular distribution becomes steeper
with increasing collision energy, so for high-energy scatterings a forward
scattering to small angles dominates.

Fig. 7 shows the inelastic differential cross sections for $q+q\to q+q+g$
processes for $T=3T_{c}$ GeV as a function of the angle $\theta_{g}$ between
the momentum of the emitted gluon relative to the collision axis in the CMS
for different collision energies $\sqrt{s}$. All lines are symmetric with
respect to the angle $\theta_{g}$ meaning that forward and backward directions
for the emitted gluon are equal. At the same time the distribution shows a
rapid growth of the cross section for small $\theta_{g}$ with increasing
energy $\sqrt{s}$ indicating that small angles are dominant for the gluons as
well as for the quarks.

###  IV.3 Transverse momentum distributions, comparison of DQPM and BAMPS

Figure 8: Differential cross section $d\sigma/dk_{t}$ as a function of $k_{t}$
(upper) and $d\sigma/dq_{t}$ as a function of $q_{t}$ (lower) for the $q+q\to
q+q+g$ scattering of light quarks of different flavours calculated within the
DQPM (blue lines) and BAMPS model (orange lines) Fochler  _et al._ (2013).

Fig. 8 shows the differential cross section $d\sigma/dk_{t}$ as a function of
transverse momentum of the emitted gluon $k_{t}$ (upper plot) and
$d\sigma/dq_{t}$ (lower plot) versus transverse momentum transfer $q_{t}$
(lower plot), respectively, for the $q+q\to q+q+g$ process for light quarks of
different flavours calculated with the DQPM (solid blue lines) matrix
elements.

We compare the DQPM transverse momentum distributions in $k_{t}$ and $q_{t}$
to the distributions calculated based on the pQCD BAMPS model (orange lines)
for the improved Gunion-Bertsch description following Ref. Fochler  _et al._
(2013). We recall that in the BAMPS model Fochler  _et al._ (2013) the strong
coupling constant is fixed as $\alpha_{s}=0.3$; the exchange parton has a
Debye mass in case of gluon
$m_{D}^{2}=\dfrac{8\alpha_{s}}{\pi}(N_{c}+N_{f})T^{2}$, while scattered
partons are assumed to be massless. The squared matrix element Fochler  _et
al._ (2013) is taken as

${\left|\overline{\mathcal{M}_{qq\to
qqg}}\right|}^{2}=\frac{32}{3}g^{6}\frac{s^{2}}{(q_{t}^{2}+m_{D}^{2})^{2}}(1-\bar{x})^{2}\\\
\times\left[\frac{{\bf k}_{t}}{k_{t}^{2}+m_{D}^{2}}+\frac{{\bf q}_{t}-{\bf
k}_{t}}{({\bf q}_{t}-{\bf k}_{t})^{2}+m_{D}^{2}}\right]^{2},$ (19)

where ${\bf q}_{t},{\bf k}_{t}$ are the 2-vectors of the transverse momentum
of the quark and emitted gluon, respectively. Here the quantity $x$
characterizes the fraction of the light cone momentum carried away by the
radiated gluon. It can be related to the rapidity of the emitted gluon by
$x=\frac{k_{t}}{\sqrt{s}}{\rm e}^{y}$. In Eq. (19) the $k_{t}\to 0$ divergence
(as well as $q_{t}\to 0$ divergence) of the pQCD amplitude is regularized by
an extra Debye mass – also for the emitted gluon – which leads to the final
cross section.

As seen from Fig. 8 the DQPM distributions are enhanced at low $k_{t}$ and
$q_{t}$ momenta compared to the BAMPS results which show broader
distributions. Since we selected for comparison a large $T=0.4$ GeV where the
DQPM $\alpha_{s}(T)\sim 0.3$ (cf. Fig. 1), i.e. similar to $\alpha_{s}=0.3$ in
BAMPS, the main differences between the DQPM and BAMPS results can be related
to the thermal masses in the DQPM versus the zero masses of partons in the
pQCD and the different propagators.

###  IV.4 Channel decomposition for the total cross sections

We step on to present the DQPM results for the energy and temperature
dependence of the total inelastic cross sections. However, firstly we
investigate which channels and which diagrams play a dominant role for the
gluon radiation from $q+q$ and $q+g$ scattering.

In Fig. 9 we show the contributions of non-interference terms corresponding to
the squared amplitude $|\bar{M}_{ii}|$ (where amplitudes $M_{i}$ are defined
by Eq. (12) and shown by the diagrams ”1-5” in Fig. 3) (dashed lines of
different colors) to the total $q+q\to q+q+g$ cross section (solid black line)
calculated including all interference terms as a function of the collision
energy $\sqrt{s}$ calculated at $T=3T_{c}$.

One can see that the dominant contribution comes from the term
$|\bar{M}_{55}|$ corresponding to the squared amplitude of diagrams ”5” in
Fig. 3 with the emission of a gluon from the gluon exchange line. It is larger
than the contribution of each of the other diagrams ”1-4” at all $\sqrt{s}$.
The diagrams ”1”, ”2”, ”3” and 4” contribute equally, i.e.
$|\bar{M}_{11}|=|\bar{M}_{22}|=|\bar{M}_{33}|=|\bar{M}_{44}|$. As follows from
Fig. 9 the total cross section (solid black line) – calculated including all
terms and their interference – is smaller than the $|\bar{M}_{55}|$ term due
to destructive interference.

Figure 9: Contributions of the squared amplitudes $|\bar{M}_{ii}|$ (i.e. non-
interference terms, shown by the dashed lines of different colors - cf. the
legend) to the total $q+q\to q+q+g$ cross section with inclusion of all
interference terms (solid black line) as a function of the collision energy
$\sqrt{s}$ at $T=3T_{c}$. Figure 10: The contribution to the DQPM inelastic
total cross sections for the $q+g\to q+g+g$ process at $T=1.2T_{c}$ from
$t$-channel (blue line), $u$-channel (orange line), $s$-channel (green line)
and from all channels (black line).

Fig. 10 illustrates the contribution to the DQPM inelastic total cross
sections for the $q+g\to q+g+g$ process from different interaction channels:
$t$-channel (blue line), $u$-channel (orange line), $s$-channel (green line)
and from all channels (black line). As it is seen from the figure, the
contribution from the $u$\- and $s$-channels are almost two orders of
magnitude lower than for the $t$-channel and do not give a significant
contribution to the cross section for the entire energy region. This let us
neglect the contributions from the $u$\- and $s$-channels channels in further
calculations (similar to the $q+q\to q+q+g$ process).

###  IV.5 Total inelastic cross section versus $\sqrt{s}$ and $T$

####  IV.5.1 $q+q\to q+q+g$ process

Figure 11: Total inelastic $q+q\to q+q+g$ cross sections (solid lines) as a
function of the collision energy $\sqrt{s}$ at different temperatures
$T=1.2T_{c}$ (blue), $T=3T_{c}$ (orange) $T=6T_{c}$ (green) in comparison to
the elastic $q+q\to q+q$ cross sections (dashed lines). Figure 12: Total
inelastic $q+q\to q+q+g$ cross sections (solid lines) as a function of
temperature $T$ at different collision energies $\sqrt{s}=2$ GeV (blue),
$\sqrt{s}=5$ GeV (orange) and $\sqrt{s}=9$ GeV (green) in comparison to the
elastic $q+q\to q+q$ cross sections (dashed lines).

In Fig. 11 we present the DQPM results for the total cross sections for gluon
radiation from $q+q\to q+q+g$ process (solid lines) as a function of the
collision energy $\sqrt{s}$ at different temperatures $T=1.2~{}T_{c}$ (blue),
$T=3~{}T_{c}$ (orange) $T=6~{}T_{c}$ (green). The inelastic cross sections are
compared to the elastic $q+q\to q+q$ cross sections (dashed lines) at the same
$T$. Both processes are calculated for on-shell partons where their masses are
taken as the pole masses of the spectral functions at a given temperature $T$.
In Fig. 12 the total inelastic cross sections (solid lines) are shown as a
function of temperature $T$ at different the collision energies $\sqrt{s}=2$
GeV (blue), $\sqrt{s}=5$ GeV (orange) and $\sqrt{s}=9$ GeV (green) in
comparison to the elastic cross sections (dashed lines).

As one can see from the figures, both elastic and inelastic cross sections
increase monotonically with energy and decrease with temperature. However, the
form of these dependencies is rather different. For low energies the inelastic
cross section is suppressed for the entire range of temperatures. At the same
time for high temperatures the inelastic cross section is also suppressed for
the range of energies up to 15 GeV. However, in the low-temperature regime the
inelastic cross can become significant. For example, for $T=1.2~{}T_{c}$ the
inelastic cross section is of the same order as for the elastic one already at
$\sqrt{s}\approx 3.5$ GeV and $T=3~{}T_{c}$ at $\sqrt{s}\approx 12$ GeV.

The reason for such a behavior of the inelastic cross section lies in the form
of the scattering amplitudes, which for the inelastic reactions (Eq. (12)) are
proportional to the coupling cubed $(|\bar{\mathcal{M}}_{2\to
3}|^{2}\propto\alpha_{s}^{3})$ while for the elastic reaction the amplitudes
are proportional to the coupling squared $(|\bar{\mathcal{M}}_{2\to
2}|^{2}\propto\alpha_{s}^{2})$. Since in the DQPM the coupling constant is
temperature dependent $\alpha_{s}(T)=g^{2}(T)/(4\pi)$ and it grows above one
in the vicinity of the critical temperature (cf. Fig. 1), it leads to a rapid
increase of the inelastic amplitude as the temperature goes down. Thus, the
relative contribution of elastic and inelastic cross sections depends strongly
on the temperature dependence of the coupling constant $\alpha_{s}(T)$.
However, if $\alpha_{s}$ is taken as a fixed constant below 1, e.g.
$\alpha_{s}=0.3$ as in many pQCD calculations or effective models, the
inelastic cross sections are suppressed.

Figure 13: Total inelastic (solid lines) and elastic (dashed lines) cross
sections scaled by the $g^{4}$ and $g^{6}$, respectively, versus the invariant
energy $\sqrt{s}$.

This is demonstrated in Fig. 13 which shows the total inelastic (solid lines)
and elastic (dashed lines) cross sections scaled by $g^{4}(T)$ and $g^{6}(T)$,
respectively. One can see a strong suppression of inelastic cross sections at
low $\sqrt{s}$ compared to elastic ones since the energy is not sufficient to
emit a heavy thermal gluon. While $2\to 3$ cross section grows with increasing
energy, it is below the elastic $2\to 2$ cross sections by about of factor of
10 even at $\sqrt{s}=15$ GeV for all $T$.

We note that the estimated averaged temperature of the QGP created in central
Au+Au collisions at $\sqrt{s}=200$ GeV at RHIC is on a level of $2~{}T_{c}$
and in central Pb+Pb collisions at $\sqrt{s}=2.76$ TeV at LHC is about
$3~{}T_{c}$. For these temperatures the mean values of $\sqrt{s}$ of colliding
partons (for the DQPM) are less than 2 GeV, which according to the above
figures implies that the processes of gluon radiation by thermal partons are
expected to be negligible. However, these processes may play an important role
in case of jets (fast light partons or heavy quarks) as they can have a very
large momentum (above 10 GeV/c) and therefore produce collisions with
significant values of $\sqrt{s}$.

####  IV.5.2 $q+g\to q+g+g$ process

Figure 14: Comparison of the total inelastic cross sections of the $q+g\to
q+g+g$ reaction with inelastic $q+q\to q+q+g$ reaction (scaled by the factor
of 9/4) as a function of the collision energy $\sqrt{s}$ for different
temperatures $T=1.2~{}T_{c},\ 3~{}T_{c},\ 6~{}T_{c}$ (upper plot) and as a
function of the temperature for different collision energies $\sqrt{s}=2,\ 5,\
9$ GeV (lower plot).

Fig. 14 shows the total inelastic cross sections for $q+g\to q+g+g$ reactions
– calculated by summing the diagrams in Fig. 4 according to the amplitudes
(14) – as a function of the collision energy $\sqrt{s}$ for different medium
temperatures $T=1.2~{}T_{c},\ 3~{}T_{c},\ 6~{}T_{c}$ (upper plot) and as a
function of $T$ for different $\sqrt{s}=2,\ 5,\ 9$ GeV (lower plot). The
$q+g\to q+g+g$ cross-sections are compared to the $q+q\to q+q+g$ cross-section
scaled by the factor of 9/4. The ratio for the $q+q$ and $q+g$ cross-sections
reaches the expected pQCD color factor limit $4/9$ Berends  _et al._ (1981) at
high energies. At low energies the inelastic $q+g$ cross sections are smaller
than the $q+q$ ones due to the different thresholds for $q+q$ and $q+g$
reactions, i.e.
$\sqrt{s}_{th}^{q+q}=2m_{q}+m_{g}<\sqrt{s}_{th}^{q+g}=m_{q}+2m_{g}$: since the
calculations are done for the pole masses of initial partons and the pole
gluon mass is larger than the pole quark mass.

###  IV.6 $\mu_{B}$-dependence of the gluon radiative cross sections

Figure 15: Total inelastic $q+q\to q+q+g$ cross section as a function of the
baryon chemical potential $\mu_{B}$ for $T=1.5~{}T_{c}$ (upper) and
$T=3~{}T_{c}$ (lower) at different collision energies $\sqrt{s}=2$ GeV (blue),
$\sqrt{s}=5$ GeV (orange) and $\sqrt{s}=9$ GeV (green). Here $T_{c}=0.158$
GeV.

Fig. 15 shows the total inelastic $q+q\to q+q+g$ cross section as a function
of the baryon chemical potential $\mu_{B}$ for $T=3~{}T_{c}$ at different
collision energies $\sqrt{s}=2$ GeV (blue), $\sqrt{s}=5$ GeV (orange) and
$\sqrt{s}=9$ GeV (green). One can see that the $\mu_{B}$ dependence of the
cross section $\sigma_{qq}$ is not strong, however, non-trivial and changes
with $T$ and $\sqrt{s}$. We note that in the DQPM the quark and gluon masses
as well as $\alpha_{s}$ decrease with increasing $\mu_{B}$ Moreau  _et al._
(2019); Soloveva  _et al._ (2021). At low temperature $T=1.5~{}T_{c}$ the
cross section $\sigma_{qq}$ increases for $\sqrt{s}=2$ GeV due to the lowering
of the threshold $\sqrt{s}_{th}^{q+q}=m_{q}+m_{q}+m_{g}$ for gluon production
in the $2\to 3$ reaction which increases the open phase space. With increasing
energy ($\sqrt{s}=5,\ 9$ GeV) the threshold effect doesn’t play a role anymore
since the open phase space is large anyhow and the cross section decreases
with increasing $\mu_{B}$ partially due to the reduction of
$\alpha_{s}(\mu_{B})$. Similar at $T=3~{}T_{c}$ the cross section (which is
substantially decreases with $T$ \- cf. Fig. 12) decreases with increasing
$\mu_{B}$.

###  IV.7 Interaction rates and relaxation times

The thermal interaction rate and relaxation time of the on-shell and off-shell
partons within the DQPM have been studies in Refs. Moreau  _et al._ (2019);
Soloveva  _et al._ (2021) by accounting for the elastic $q+q$ and $q+g$
reactions. Here we investigate the contribution of gluon radiative reactions
to the interaction rate and relaxation time. We note that in this study we
consider the on-shell case for elastic and inelastic scattering – since we
concentrate on relative contributions of these reactions – to reduce the
numerical complexity of the full off-shell calculations for $2\to 3$
processes. However, we point out that the off-shell elastic rate evaluated
earlier in Refs Moreau  _et al._ (2019); Soloveva  _et al._ (2021) is smaller
than the on-shell rate here due to the averaging over the spectral functions
of final partons.

We recall that the interaction rate for the corresponding on-shell partons
(i.e. taken at their pole masses) is given by

$\displaystyle\Gamma_{j}^{\text{inel}}(p_{j},T,\mu_{q})=\frac{1}{2E_{j}}\sum_{i=q,\bar{q},g}\int\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}d_{i}f_{i}\int\frac{d^{3}p_{1}}{(2\pi)^{3}2E_{1}}$
$\displaystyle\times\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\int\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}(1\pm
f_{1})(1\pm f_{2})(1\pm f_{3})$ $\displaystyle\times\ |\bar{\mathcal{M}}_{\
}|^{2}(2\pi)^{4}\delta^{(4)}(p_{j}+p_{i}-p_{1}-p_{2}-p_{3}),$ (20)

where $d_{i}$ is the degeneracy factor for spin and color ($2N_{c}$ for quarks
and $2(N_{c}^{2}-1)$ for gluons); $f_{i}=f_{i}(E_{i},T,\mu_{q})$ are the Fermi
distribution functions for quarks and $f_{i}=f_{i}(E_{i},T)$ are the Bose
distribution functions for gluons. Here
$v_{\text{rel}}=F/(4E_{\text{j}}E_{i})$ with $F$ being the flux of the
incident particles. The Pauli-blocking $(-)$ and Bose-enhancement $(+)$
factors account for the available density of final states.

To evaluate the average width of the parton $j$, we finally have to average
its interaction rate (IV.7) over its momentum distribution,

$\Gamma_{j}^{\text{inel}}(T,\mu_{q})=\frac{d_{j}}{n_{j}(T,\mu_{q})}\int\frac{d^{3}p_{j}}{(2\pi)^{3}}f_{j}(E_{j},T,\mu_{q})\Gamma_{j}(p_{j},T,\mu_{q})$
(21)

with the on-shell density of partons $j$ at $T$ and $\mu_{q}$ given by

$n_{j}(T,\mu_{q})=d_{j}\int\frac{d^{3}p}{(2\pi)^{3}}f_{j}(E_{j},T,\mu_{q}).$
(22)

Figure 16: On-shell interaction rate $\Gamma$ of a light quark (upper) and a
gluon (lower) as a function of the temperature $T$ for $\mu_{q}=0$ for elastic
(blue lines) and for inelastic (orange lines) processes.

Fig. 16 illustrates the on-shell interaction rate of a light quark (upper
plot) and for the gluon (lower plot) as a function of the temperature for
elastic (blue line) and inelastic (orange line) processes. Here, similar to
Refs. Moreau  _et al._ (2019); Soloveva  _et al._ (2021), for the elastic
light quark scattering (e.g. $u+u$) we account for the contributions for all
the possible interaction channels such as $uu\to uu$, $u\bar{u}\to u\bar{u}$,
$u\bar{u}\to d\bar{d}$, $u\bar{u}\to s\bar{s}$, $ud\to ud$, $u\bar{d}\to
u\bar{d}$, $us\to us$, $u\bar{s}\to u\bar{s}$ and $ug\to ug$. For the
inelastic scatterings we account for the same channels as for elastic (with an
additional emitted gluon in the final state), however, we omit the
contributions from the annihilation processes such as $u\bar{u}\to d\bar{d}$
and $u\bar{u}\to s\bar{s}$ and assume identical contributions from all $uq\to
uq$ processes for the reason described in Sec. III.1.1.

For the gluon elastic scatterings the following reactions are considered
(similar to Refs. Moreau  _et al._ (2019); Soloveva  _et al._ (2021)): $gu\to
gu$, $g\bar{u}\to g\bar{u}$, $gd\to gd$, $d\bar{d}\to g\bar{d}$, $gs\to gs$,
$g\bar{s}\to g\bar{s}$, and $gg\to gg$. For the inelastic scatterings we
account the same channels (with an additional emitted gluon in the final
state). We note that the $gg\to ggg$ channel has been evaluated using the
relation $\sigma_{gg}=\frac{9}{4}\sigma_{gq}$. We note that the evaluated on-
shell interaction rate here is the same as in previous DQPM studies in Refs.
Moreau  _et al._ (2019); Soloveva  _et al._ (2021).

Figure 17: Distribution of on-shell $u-$quark scatterings as a function of the
invariant energy $\sqrt{s}$ for different temperatures at $\mu_{q}=0$. The
vertical dashed lines correspond to the mean $\sqrt{s}$ for each temperature.

As follows from Fig. 16, the inelastic rates of thermal light quarks and
gluons are strongly suppressed at all temperatures compared to the elastic
rates in the QGP medium of quarks and gluons with thermal masses.

For a better understanding of this result we present in Fig. 17 the
distribution of on-shell $u-$quark scatterings (assuming the scattering
probability equal to one) as a function of the invariant energy $\sqrt{s}$ for
different temperatures at $\mu_{B}=0$. The vertical dashed lines correspond to
the mean $\sqrt{s}$ for each temperature $T$. The sharp threshold is due to
the large pole masses of scattered quarks - cf. Fig. 2. We note that in the
off-shell case – as considered in Refs. Moreau  _et al._ (2019); Soloveva  _et
al._ (2021) – the distribution $dN/d\sqrt{s}$ goes to zero by accounting for
all possible masses of the initial quarks according to their spectral
functions.

As seen from Fig. 17, the $dN/d\sqrt{s}$ distribution of scattered
quasiparticles favours low values of $\sqrt{s}$ due to the Fermi-Dirac
distribution function $f_{j}(E_{j},T,\mu_{q})$ which suppresses collisions at
large $\sqrt{s}$ in the thermal QCD medium. Thus, only the low $\sqrt{s}$
region of elastic and inelastic cross sections dominantly contributes to the
interaction rates $\Gamma$. As follows from Fig. 12 at all achievable
$\sqrt{s}$ (cf. Fig. 17) the inelastic cross sections are much smaller than
the elastic ones at all $T$ which leads to smaller inelastic rates compared to
the elastic rates at all $T$. Thus, the contribution of radiative processes –
i.e. the emission of massive thermal gluons by collisions of massive partons –
is negligible for thermal properties of the sQGP within the quasiparticle
picture. The situation will change when considering the collisions of fast jet
quarks with the thermal medium since the available $\sqrt{s}$ is large in this
case (cf. Fig. 11), thus the radiative contribution becomes important
Grishmanovskii  _et al._ .

Figure 18: Relaxation time $\tau$ of a light quark (upper) and gluon (lower)
as a function of the temperature $T$ for $\mu_{q}=0$ evaluated by the average
parton interaction rate for elastic contribution only (blue lines) and for the
sum of elastic and inelastic contributions (orange lines).

The interaction rate can be directly related to the relaxation time
$\tau=\hbar c/\Gamma$ which is shown in Fig. 18. There $\tau$ for a light
quark (upper plot) and gluons (lower plot) is presented as a function of the
temperature $T$ evaluated by the average parton interaction rate for elastic
contributions only (blue line) and for the sum of elastic and inelastic
contributions (orange line) from Fig. 16. One can see that by accounting for
inelastic processes the relaxation time of thermal sQGP partons is only
slightly shortened. As discussed above this is related to the fact that the
thermal radiation of heavy gluons from $q+q$ and $q+g$ scattering is
suppressed compared to their elastic scattering.

Since the inelastic reactions only slightly contribute to the relaxation times
of quarks and gluons, the DQPM transport coefficients of the thermal sQGP –
such as the specific shear and bulk viscosities $\eta/s$, $\zeta/s$, electric
conductivity $\sigma_{Q}/T$ as well as the diffusion coefficient matrix –
evaluated for the elastic reactions within the relaxation time approximation
(RTA) Moreau  _et al._ (2019); Soloveva  _et al._ (2020a); Fotakis  _et al._
(2021) will change insignificantly. Only in the high-temperature region
transport coefficients from the RTA including the inelastic contributions are
expected to be smaller when compared to previous estimations that solely
considered elastic ones.

##  V Summary

We have studied the inelastic versus elastic scatterings in the strongly
interacting quark-gluon plasma (sQGP) based on the effective dynamical quasi-
particle model (DQPM), which describes the non-perturbative nature of the sQGP
at finite temperature $T$ and baryon chemical potential $\mu_{B}$ in terms of
off-shell quarks and gluons with properties (masses and widths) adjusted in
order to reproduce the lQCD EoS for the QGP in thermodynamic equilibrium. We
stress that contrary to the pQCD based models with massless weakly interacting
quarks and gluons, the concept of massive strongly interacting quasiparticles
in the DQPM allows not only to reproduce the lQCD EoS at zero and finite
$\mu_{B}$, but to explore QCD in the non-perturbative regime of ($T,\mu_{B})$
phase diagram where the lQCD results are not yet available, however, this
region is relevant for heavy-ion physics. While in previous DQPM studies we
focused on elastic scattering of quasiparticles (which are assumed to be
dominant in the quasiparticle picture due to the large thermal masses) by
evaluating the Born type scattering diagrams with effective propagators and
vertices, here we extend our study to the evaluation of inelastic processes
with the goal to quantify explicitly the relative contribution of the emission
of massive gluons from scatterings of thermal quarks and gluons to the thermal
properties of the QGP.

We summarize our study as follows:

  * •

We have calculated the thermal gluon radiation processes from the massive
quark-quark ($q+q\to q+q+g$) and quark-gluon ($q+g\to q+g+g$) scatterings
(similar for the scatterings including anti-quarks) by evaluating explicitly
the leading order Feynman diagrams for $2\to 3$ processes with effective
propagators and vertices from the DQPM for the first time without any further
approximations and simplifications. The calculations of $2\to 3$ cross
sections have been performed for the ”on-shell” case, where the masses of
scattered initial as well as final partons, including the emitted gluon, have
been taken at their pole masses, thus omitting the integrations over the
spectral functions of final partons as for the off-shell calculation of
elastic scattering in earlier DQPM studies Moreau  _et al._ (2019); Soloveva
_et al._ (2021).

  * •

We have shown that for $q+q$ and $q+g$ radiative cross sections the
$t$-channel is the dominant channel compared to $u$\- and $s$-channels.
Moreover, the large contribution to the $q+q\to q+q+g$ cross sections stem
from the diagram ”5” in Fig. 3 with the emission of a gluon from the gluon
exchange lines, this contribution is larger than from each of the other
diagrams ”1-4” (they give about equal contributions) at all $\sqrt{s}$.
However, the total cross section is smaller than the sum of the squared
amplitudes and even the squared amplitude of diagram ”5” due to a strong
destructive interference between different amplitudes, which shows the
importance of an explicit evaluation of all diagrams without approximations.

  * •

We have found that the DQPM radiative cross sections for $q+q$ and $q+g$
reactions scale with a factor $4/9$ at high collision energies $\sqrt{s}$ \-
in line with the pQCD calculations. However, at low $\sqrt{s}$ the cross
sections for $q+g\to q+g+g$ reactions are smaller than for $q+q\to q+q+g$
reactions due to the larger threshold (i.e. the pole mass of thermal gluons is
larger than the pole mass of quarks).

  * •

We have proven that the DQPM results for $2\to 3$ processes reproduces the
pQCD calculations in the limit of zero masses and widths of quasiparticles in
the DQPM. In particular we have shown the identity of the DQPM and pQCD
amplitudes Berends  _et al._ (1981); Ellis and Sexton (1986) in the ”pQCD”
limit.

  * •

Moreover, we have compared our results for the angular distributions of the
emitted gluons and final scattered quark from $q+q\to q+q+g$ processes to the
pQCD-based BAMPS model calculations and have found a qualitative agreement.
The quantitative differences between the DQPM and the pQCD BAMPS results can
be attributed to the following reasons: i) the thermal masses of scattered
quarks and emitted gluons in the DQPM versus zero mass partons in BAMPS; ii)
differences in the propagators: the necessity to introduce a Debye mass (i.e.
an extra parameter in the model) to regularize the infrared divergence in the
gluon pQCD propagator while the ”resummed” DQPM propagator has no divergence
due to the self-consistent dynamical widths of the 2PI approach; iii)
different strong coupling constant $\alpha_{s}$ used in the calculations:
while the pQCD calculations assume a small $\alpha_{s}$ (e.g. in our study
$\alpha_{s}=0.3$), independent on temperature, the DQPM calculations involve
$T$-dependent $\alpha_{s}(T)$ which grows above one for $T\to T_{c}$ in line
with the lQCD data (cf. Fig. 1).

  * •

We have calculated the energy and temperature dependencies of the total and
differential radiative cross sections for the thermal quark-quark and quark-
gluon scatterings and compared them to the corresponding elastic cross
sections. We have found that the elastic cross sections dominate the radiative
ones at low $\sqrt{s}$ energies for $T\geq 2T_{C}$, however, at low $T\to
T_{c}$ the radiative cross sections become larger. The reason is related to
the fact that the inelastic cross sections are proportional to the coupling
cubed $(\sigma_{2\to 3}\propto\alpha_{s}^{3}(T))$ while elastic ones to the
coupling squared $(\sigma_{2\to 2}\propto\alpha_{s}^{2}(T))$.

  * •

We have studied the $\mu_{B}$ dependence of the radiative cross sections and
found a relatively weak dependence on $\mu_{B}$: at larger $\sqrt{s}$, where
the phase space is open sufficiently, the total cross sections decreases with
increasing $\mu_{B}$.

  * •

We have estimated the interaction rate $\Gamma$ and relaxation time $\tau$ of
quarks and gluons in the sQGP versus the temperature $T$. We have found that
the transition rate defined by elastic scattering is strongly dominates the
inelastic transition rate. Thus, the relaxation time $\tau$ calculated within
the elastic plus inelastic transition rates $\Gamma$ is only slightly shorter
than that with elastic $\Gamma$ only. It implies that the radiative processes
– with the emission of massive gluon – are strongly suppressed in the non-
perturbative sQGP medium within the dynamical quasiparticle model.

We mention that our study of thermal radiative processes can be extended for
the investigation of jet (highly energetic partons) attenuation in the sQGP
and jet transport properties. This work is in progress Grishmanovskii  _et
al._ .

## ACKNOWLEDGEMENTS

The authors acknowledge inspiring discussions with J. Aichelin, W. Cassing, M.
Djordjevic, P.-B. Gossiaux and O. Kaczmarek. 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. I.G. also
acknowledges support from the ”Helmholtz Graduate School for Heavy Ion
research”. This work is supported by the European Union’s Horizon 2020
research and innovation program under grant agreement No 824093 (STRONG-2020).
The computational resources have been provided by the Goethe-HLR Center for
Scientific Computing.

##  Appendix A Gluon polarization

In case of a massive gluon, the sum over polarizations is given by:

$\sum_{\text{pol.}}(\epsilon_{i})_{\mu}(\epsilon^{\star}_{i})_{\mu^{\prime}}=-g_{\mu\mu^{\prime}}+\frac{(k_{i})_{\mu}(k_{i})_{\mu^{\prime}}}{(M_{i})^{2}_{g}},$
(23)

where additionally to the first ”simple” term - as for massless gluons, there
is the second term which is over proportional to the gluon thermal mass.

When calculating the cross sections of gluon radiation, we have found that the
first term gives the dominant contribution such that the total cross section
calculated including both terms differs only very little from those calculated
with the first term only. This is illustrated in Fig. 19 which shows the total
$q+q\to q+q+g$ cross section as a function of the collision energy for the
”full” (blue line) and ”simple” (orange line) gluon polarizations. The
comparison shows that the second term in the gluon polarization sum (23) gives
an almost negligible contribution to the cross section and therefore can be
omitted in all further calculations.

Figure 19: Total $q+q\to q+q+g$ cross section at $T=3T_{c}$ as a function of
the collision energy for the ”full” (blue line) and ”simple” (orange line)
gluon polarizations.

##  Appendix B Details on the calculation of the cross section

The differential cross section for a $2\to 3$ process for on-shell particles
$(a+b\to 1+2+3)$ is given by

$d\sigma_{2\to 3}=\frac{|\bar{\mathcal{M}}|^{2}}{F}d\Phi_{3},$ (24)

where $d\Phi_{3}$ denotes the three-body Lorentz invariant phase-space:

$d\Phi_{3}=(2\pi)^{4}\frac{d^{3}p_{1}}{(2\pi)^{3}2E_{1}}\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\delta^{4}(p_{a}+p_{b}-p_{1}-p_{2}-p_{3}).$
(25)

Note, that in general the total cross section is not only a function of the
system energy, but also a function of all parton masses and widths, i.e.

$\sigma_{2\to 3}=\sigma_{2\to
3}(s,m_{a},m_{b},m_{1},m_{2},m_{3},M_{q},M_{g},\gamma_{q},\gamma_{g})$ (26)

Obtaining this total cross section implies, in general, a 9-dimensional
integration, where 4 integrations can be done analytically using the
properties of the delta function. There are several ways to perform this
integration depending on the choice of integration variables.

###  B.1 Variant 1

Figure 20: Schematic view of a $2\to 3$ reaction.

The first way to perform the integration over the delta function is to
consider $2\to 3$ reaction as $(2\to 2)+(1\to 2)$ by introducing the
intermediate state $p_{x}$ (Fig. 20), where

$p_{x}=p_{2}+p_{3},\quad p_{x}^{2}=m_{x}^{2},\quad
m_{2}+m_{3}<m_{x}<\sqrt{s}-m_{1}.$ (27)

Employing for $p_{3}$ the general relation:

$\int\frac{d^{3}p}{2E}=\int d^{4}p\;\delta(p^{2}-m^{2})\,\theta(E)$ (28)

that stems from the property of the delta function,

$\delta(f(x))=\sum_{i}\frac{1}{|f^{\prime}(x_{i})|}\delta(x-x_{i}),$ (29)

where $x_{i}$ are the roots of $f(x)$. Then after substituting

$1=\int d^{4}p_{x}\delta^{4}(p_{x}-p_{2}-p_{3})$ (30)

into (25) and integrating over $p_{3}$, we obtain

$d\Phi_{3}=(2\pi)^{-5}\frac{d^{3}p_{1}}{2E_{1}}\frac{d^{3}p_{2}}{2E_{2}}\;d^{4}p_{x}\;\delta\left((p_{x}-p_{2})^{2}-m_{3}^{2}\right)\;\delta^{4}(p_{a}+p_{b}-p_{1}-p_{x})$
(31)

Introducing the same procedure for $p_{1}$ we arrive at

$d\Phi_{3}=(2\pi)^{-5}\frac{d^{3}p_{2}}{2E_{2}}d^{4}p_{x}\;\delta\left((p_{a}+p_{b}-p_{x})^{2}-m_{1}^{2}\right)\;\delta\left((p_{x}-p_{2})^{2}-m_{3}^{2}\right).$
(32)

In the CMS we now can simplify an expression inside the first delta function:

$(p_{a}+p_{b}-p_{x})^{2}=(p_{a}+p_{b})^{2}-2(p_{a}+p_{b})p_{x}+p_{x}^{2}=s-2\sqrt{s}E^{C}_{x}+m_{x}^{2},$
(33)

where the upper index $C$ denotes the CMS frame. Having in mind that

$d^{4}p=d^{3}pdE=p^{2}dpd\Omega dE=dm^{2}\frac{p}{2}dEd\Omega,$ (34)

we now can write

$d^{4}p_{x}\delta\left((p_{a}+p_{b}-p_{x})^{2}-m_{1}^{2}\right)=dp_{x}^{2}\frac{p_{x}^{C}}{2}dE_{x}^{C}d\Omega_{x}^{C}\;\delta(s-2\sqrt{s}E^{C}_{x}+m_{x}^{2}-m_{1}^{2})=\frac{p_{x}^{C}}{2}\frac{dm_{x}d\Omega_{x}^{C}}{2\sqrt{s}}dE_{x}^{C}\delta(E_{x}^{C}-E_{x0}^{C}).$
(35)

For the second delta function in the rest frame of (2+3) we have

$(p_{x}-p_{2})=p_{x}^{2}-2p_{x}p_{2}+p_{2}^{2}=m_{x}^{2}-2m_{x}^{X}E_{2}^{X}+m_{2}^{2},$
(36)

where the upper index $X$ denotes the rest frame of (2+3). Substituting this
into (32) we have

$\frac{d^{3}p_{2}}{2E_{2}}\delta\left((p_{x}-p_{x})^{2}-m_{3}^{2}\right)=\frac{p_{2}^{X}}{2}dE_{2}^{X}d\Omega_{2}^{X}\delta(p_{x}^{2}-2E_{x}^{X}E_{2}^{X}+m_{2}^{2}-m_{3}^{2})=\frac{p_{2}^{x}}{2}\frac{d\Omega_{2}^{X}}{2E_{x}^{X}}dE_{2}^{X}\delta(E_{2}^{X}-E_{20}^{X})$
(37)

Integrating over the delta functions we end up with

$d\Phi_{3}=(2\pi)^{-5}\left(\frac{p_{2}^{X}}{4E_{x}^{X}}d\Omega_{2}^{X}\right)\left(\frac{p_{x}^{C}}{4\sqrt{s}}d\Omega_{x}^{C}\right)dm_{x}^{2},$
(38)

and the expression for the differential cross section now reads

$d\sigma_{2\to 3}=\frac{|\bar{\mathcal{M}}_{2\to
3}|^{2}}{F}(2\pi)^{-5}\left(\frac{p_{2}^{X}}{4E_{x}^{X}}d\Omega_{2}^{X}\right)\left(\frac{p_{x}^{C}}{4\sqrt{s}}d\Omega_{x}^{C}\right)dm_{x}^{2}.$
(39)

Noting

$d\Omega_{x}^{C}=d\cos\theta d\varphi,\quad d\Omega_{2}^{X}=d\cos\xi
d\psi,\quad E_{x}^{X}=m_{x},$ (40)

we can now express the final integral for the total cross section:

$\sigma_{2\to
3}=\frac{1}{8(2\pi)^{5}\sqrt{s}F}\int_{0}^{1}d\cos\theta\int_{0}^{2\pi}d\varphi\int_{0}^{1}d\cos\xi\int_{0}^{2\pi}d\psi\int_{m_{2}+m_{3}}^{\sqrt{s}-m_{1}}dm_{x}p_{x}^{C}p_{2}^{X}|\bar{\mathcal{M}}_{2\to
3}|^{2}.$ (41)

The integration over the polar angle (of $p_{x}^{C}$) can be done
analytically, giving a factor $2\pi$. In order to obtain
$|\bar{\mathcal{M}}|^{2}$ one needs to reconstruct a full set of 4-momentums
according to $\theta,\varphi,\xi,\psi,m_{x}$. Since $p_{2}^{X}$ and
$p_{x}^{C}$ are expressed in different systems, one should perform the Lorenz
transformation to one chosen system. For example, one can define the following
set of momenta:

$\displaystyle\not{p}_{a}^{C}$ $\displaystyle=(E_{a},0,0,p_{a}^{C}),\quad
p_{a}^{C}=\lambda(\sqrt{s},m_{a},m_{b})/(2\sqrt{s}),$
$\displaystyle\not{p}_{b}^{C}$ $\displaystyle=(E_{b},0,0,-p_{a}^{C}),$
$\displaystyle\not{p}_{1}^{C}$
$\displaystyle=(E_{1},p_{x}^{C}\sin\theta\cos\varphi,p_{x}^{C}\sin\theta\sin\varphi,p_{x}^{C}\cos\theta),\quad
p_{x}^{C}=\lambda(\sqrt{s},m_{1},m_{x})/(2\sqrt{s}),$
$\displaystyle\not{p}_{x}^{C}$ $\displaystyle=(E_{x},-\vec{p}_{1}),$
$\displaystyle\not{p}_{2}^{X}$
$\displaystyle=(E_{2}^{X},p_{2}^{X}\sin\xi\cos\psi,p_{2}^{X}\sin\xi\sin\psi,p_{2}^{X}\cos\xi),\quad
p_{2}^{X}=\lambda(m_{x},m_{2},m_{3})/(2m_{x}),$ $\displaystyle\not{p}_{3}^{X}$
$\displaystyle=(E_{3}^{X},-\vec{p}_{2}^{X}),$ $\displaystyle\not{p}_{2}^{C}$
$\displaystyle=L_{-\vec{v}_{x}}(p_{2}^{X}),$ $\displaystyle\not{p}_{3}^{C}$
$\displaystyle=L_{-\vec{v}_{x}}(p_{3}^{X}),$ (42)

where $L_{\vec{v}}(p)$ denotes a Lorentz transformation of the 4-vector $p$
with the velocity $\vec{v}$ and

$\lambda(a,b,c)=\sqrt{\left(a^{2}-(b+c)^{2}\right)\left(a^{2}-(b-c)^{2}\right)},\quad\vec{v}=\vec{p}_{x}/E_{x}$
(43)

###  B.2 Variant 2

In order to properly compare DQPM results with well known Gunion-Bertsch
approximation, we should express the integration in terms of another set of
variables such as $(q_{t},k_{t},y,\phi)$, where $q_{t}$ is the transverse
momentum transfer, $k_{t}$ is the transverse momentum of the emitted gluon,
$y$ is the rapidity of the emitted gluon and $\phi$ is the angle between
$k_{t}$ and $q_{t}$. For a given particle with four-momentum
$(E,p_{x},p_{y},p_{z})$ we can express the components in terms of transverse
momentum, rapidity, and azimuthal angle using the following relation:

$(E,p_{x},p_{y},p_{z})=(m_{t}\cosh{y},p_{t}\cos{\varphi},p_{t}\sin{\varphi},m_{t}\sinh{y}),\quad
m_{t}=\sqrt{m^{2}+p_{t}^{2}}.$ (44)

The scalar product of 2 four vectors is expressed now as

$p_{i}p_{j}=m_{i,t}m_{j,t}(\cosh{y_{i}}\cosh{y_{2}}-\sinh{y_{1}}\sinh{y_{2}})-p_{i,t}p_{j,t}\cos\varphi.$
(45)

Starting again from the definition of the phase-space from (25) and
integration over $p_{2}$ using relations (28) and (29) we obtain

$d\Phi_{3}=\frac{1}{128\pi^{5}}\frac{d^{3}p_{1}}{E_{1}}\frac{d^{3}p_{3}}{E_{3}}\delta((p_{a}+p_{b}-p_{1}-p_{3})^{2}-m_{2}^{2}).$
(46)

Denoting the argument of the delta function
$F=(p_{a}+p_{b}-p_{1}-p_{3})^{2}-m_{2}^{2}$ we have

$\displaystyle F$
$\displaystyle=\left((p_{a}+p_{b})-(p_{1}+p_{3})\right)^{2}-m_{2}^{2}=(p_{a}+p_{b})^{2}-2(p_{a}+p_{b})(p_{1}+p_{3})+(p_{1}+p_{3})^{2}-m_{2}^{2}$
$\displaystyle=s-2(E_{a}+E_{b})(E_{1}+E_{3})+2(\vec{p}_{a}+\vec{p}_{b})(\vec{p}_{1}+\vec{p}_{3})+m_{1}^{2}+m_{3}^{2}+2p_{1}p_{3}-m_{2}^{2}.$
(47)

In the CMS $\vec{p}_{a}+\vec{p}_{b}=0$ and $E_{a}+E_{b}=\sqrt{s}$, so the
above expression can be simplified to

$F=s-2\sqrt{s}(E_{1}+E_{3})+m_{1}^{2}+m_{3}^{2}+2p_{1}p_{3}-m_{2}^{2}.$ (48)

Now we are going to express it in terms of new variables:

$\displaystyle F$
$\displaystyle=s-2\sqrt{s}(m_{1,t}\cosh{y_{1}}+m_{3,t}\cosh{y_{3}})+2m_{1,t}m_{3,t}(\cosh{y_{1}}\cosh{y_{3}}-\sinh{y_{1}}\sinh{y_{3}})$
$\displaystyle+2p_{1,t}p_{3,t}\cos\varphi+m_{1}^{2}+m_{3}^{2}-m_{2}^{2}=s-2\sqrt{s}(\sqrt{p_{1,t}^{2}+m_{1}^{2}+p_{1,z}^{2}}+E_{3})$
$\displaystyle+2E_{3}\sqrt{p_{1,t}^{2}+m_{1}^{2}+p_{1,z}^{2}}-2p_{1,z}p_{3,z}+2p_{1,t}p_{3,t}\cos\varphi+m_{1}^{2}+m_{3}^{2}-m_{2}^{2}.$
(49)

The derivative of $F$ reads

$\displaystyle\frac{\partial F}{\partial y_{1}}$
$\displaystyle=-2\sqrt{s}(m_{1,t}\sinh{y_{1}})+2m_{1,t}m_{3,t}(\sinh{y_{1}}\cosh{y_{3}}-\cosh{y_{1}}\sinh{y_{3}})=$
$\displaystyle-2\left[p_{1,z}(\sqrt{s}-m_{3,t}\cosh{y_{3}})+\sqrt{p_{1,t}^{2}+m_{1}^{2}+p_{1,z}^{2}}m_{3,t}\sinh{y_{3}}\right].$
(50)

The solutions for $F=0$ are

$p_{1,z}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a},$ (51)

with

$\displaystyle a$ $\displaystyle=(\sqrt{s}-E_{3})^{2}-p_{3,z}^{2},$
$\displaystyle b$ $\displaystyle=p_{3,z}\,\eta,$ $\displaystyle c$
$\displaystyle=(\sqrt{s}-E_{3})^{2}(p_{1,t}^{2}+m_{1}^{2})-\frac{1}{4}\eta^{2},$
$\displaystyle\eta$
$\displaystyle=s-2\sqrt{s}E_{3}+2p_{1,t}p_{3,t}\cos\varphi+m_{1}^{2}+m_{3}^{2}-m_{2}^{2}.$
(52)

The solutions have to satisfy the additional constraint

$s-\sqrt{s}E_{3}+2p_{1,t}p_{3,t}\cos{\varphi}-2p_{1,z}p_{3,z}+m_{1}^{2}+m_{3}^{2}-m_{2}^{2}.$
(53)

By performing the analytical integration over the delta function, we obtain
the final expression for the phase space:

$d\Phi_{3}=\frac{1}{128\pi^{4}}dp_{1,t}^{2}dp_{3,t}^{2}dyd\varphi\sum\left(\left.\frac{\partial
F}{\partial y_{1}}\right|_{F=0}\right)^{-1},$ (54)

and the total cross section reads

$\sigma_{2\to
3}=\frac{1}{128\pi^{4}F}\int_{0}^{p_{C}}dp_{1,t}^{2}\int_{0}^{p_{C}}dp_{3,t}^{2}\int_{y_{\min}}^{y_{\max}}dy\int_{0}^{\pi}d\varphi|\bar{\mathcal{M}}_{2\to
3}|^{2}\sum\left(\left.\frac{\partial F}{\partial
y_{1}}\right|_{F=0}\right)^{-1},$ (55)

where $p_{C}=\lambda(\sqrt{s},m_{1},m_{2})/(2\sqrt{s})$ and
$y_{\max/\min}=\pm\text{arccosh}(\frac{\sqrt{p_{C}^{2}+m_{3}^{2}}}{m_{3,t}})$.

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