compat=1.0.0

# Soft gluon emission from heavy quark scattering in strongly interacting
quark-gluon plasma

Taesoo Song<EMAIL_ADDRESS>GSI Helmholtzzentrum für Schwerionenforschung GmbH,
Planckstrasse 1, 64291 Darmstadt, Germany  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<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

###### Abstract

We apply the Low’s theorem to soft gluon emission from heavy quark scattering
in the nonperturbative strongly interacting quark-gluon plasma (sQGP). The
sQGP is described in terms of the dynamical quasi-particles and adjusted to
reproduce the EoS from lQCD at finite temperature and chemical potential.
Since the emitted gluon is soft and of long wavelength, it does not provide
information on the detailed structure of the scattering, and only the emission
from incoming and outgoing partons is enough. It simplifies the calculations
making the scattering amplitude factorizable into the elastic scattering and
the emission of soft gluon. Imposing a proper upper limit on the emitted gluon
energy, we obtain the guage-invariant scattering cross sections of heavy
quarks with the massive partons of the medium as well as their transport
coefficients (momentum drag and diffusion) in the QGP and compare with those
from the elastic scattering without gluon emission.

##  I Introduction

Heavy flavor is one of the important probes for the properties of the quark-
gluon plasma (QGP) produced in ultra-relativistic heavy-ion collisions Uphoff
_et al._ (2012); He  _et al._ (2014); Cao  _et al._ (2013); Gossiaux  _et al._
(2010); Das  _et al._ (2015); Song  _et al._ (2015, 2016); Plumari  _et al._
(2018). The production of heavy flavor is reliably described by perturbative
Quantum Chromodynamics (pQCD), since a large energy-momentum transfer is
required. However, the hadronization of heavy quark to a heavy meson or heavy
baryon is a soft process whose realization depends on model. If the heavy
quark has a large momentum, phenomenological models such as heavy quark
fragmentation functions work well Peterson  _et al._ (1983). On the other
hand, the hadronization of soft heavy quarks often adopts the coalescence
model where the heavy quark combines with an anti-light quark or with a di-
quark to form a heavy meson or a heavy baryon, respectively Song  _et al._
(2015, 2016).

The production and hadronization processes of heavy flavor are common in p+p
and heavy-ion collisions. The difference between the two collisions is the
presence or absence of a hot dense nuclear matter with which the heavy quark
interacts and changes energy-momentum. A heavy quark with a small momentum is
shifted towards a larger momentum by collective flows, while the one with a
large momentum is suppressed due to energy loss in the QGP. They are expressed
by the nuclear modification factor which is the heavy flavor distribution in
heavy-ion collisions scaled by that in p+p collisions and the number of
nucleon+nucleon binary collisions.

A heavy quark interacts with matter through elastic scattering and inelastic
scattering. The former brings about the collisional energy loss of heavy
quark, while the latter the radiative energy loss because it induces gluon
emission. The collisional energy loss is dominant at low or intermediate
momentum of a heavy quark, which is taken over by the radiative energy loss at
high momentum of heavy quark Uphoff  _et al._ (2012); Cao  _et al._ (2013);
Aichelin  _et al._ (2014).

The Parton-Hadron-String Dynamics (PHSD) adopts the Dynamical QuasiParticle
Model (DQPM) to describe the strongly interacting partonic matter as well as
partonic interactions with massive off-shell quasiparticles, contrary to the
massless pQCD partons, whose properties are described by the complex self-
energies and spectral functions. The real part of self-energy is related to
the pole mass and the imaginary part to the spectral width of partons which
are taken in the form of the Hard Thermal Loop (HTL) calculations. The DQPM is
adjusted to reproduce the lattice equation-of-state (EoS) through the strong
coupling which depends on temperature and baryon chemical potential Cassing
and Bratkovskaya (2009); Bratkovskaya  _et al._ (2011); Voronyuk  _et al._
(2011); Linnyk  _et al._ (2014, 2016); Moreau  _et al._ (2019). It has been
found that the DQPM which is extended to heavy quark interactions in the QGP
reproduces the heavy quark transport coefficients from lattice calculations as
well as the experimental data on heavy flavor production in heavy-ion
collisions Berrehrah  _et al._ (2014a, b); Song  _et al._ (2015, 2016). One
limitation of the DQPM for heavy quarks is the absence of radiative energy
loss. Though it can be justified at low and intermediate energies of heavy
quarks due to the large gluon mass in the DQPM, the radiative energy loss
cannot be neglected at large momenta of heavy quarks Grishmanovskii  _et al._
(2022).

The radiative processes play an important role in quantum electrodynamics
(QED). Bremsstrahlung photons are emitted from charged particles which are
accelerated or decelerated by scattering (interaction). According to Refs. Low
(1958); Song and Moreau (2018) a low-energy photon is emitted from the
external charged particles in Feynman diagrams. In other words, the
complicated inner structure of scattering can be ignored in the limit of low
energy photon emission, as shown in Fig. 1. Then the Feynman diagram can be
factorized into an elastic scattering part and photon emission part. The
reason for ignoring the inner structure of the scattering in the soft photon
limit is found in Ref. Song and Moreau (2018) where the Bremsstrahlung photon
spectrum becomes soft with increasing stopping time of the charged particle,
while a low-frequency (low-energy) photon is not affected by the stopping
time, because the low-frequency photon cannot provide microscopic information
on the scattering but only the macroscopic information, for example, incoming
and outgoing momenta of the charged particles before and after scattering.

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$

Figure 1:  According to Refs. Low (1958); Song and Moreau (2018) a complicated
scattering process on the left hand side is simplified to the right side
Feynman diagram, if the emitted photon is soft.

In this study we extend the soft photon approximation to QCD, i.e. to the soft
gluon emission from strong interactions. This extension is reasonable because
(anti)quarks and gluons have color charge. A difference from QED is that the
color charge is in SU(3) and non-commutative. We prove that the soft gluon
approximation satisfies the Slavnov-Taylor identities in quark-quark and
quark-gluon scatterings, as the soft photon approximation satisfies the Ward-
Takahashi identity. Then it is applied to the heavy quark scatterings with
light quark or gluon with a soft gluon emission in the sQGP as described by
the DQPM.

There have been several studies which deal with the soft gluon emission from
partonic scatterings Gunion and Bertsch (1982); Djordjevic and Gyulassy
(2004); Lushozi  _et al._ (2020). Most of them are focused on gluon emission
from the scattering of energetic partons such as jet. In this case forward
scattering is dominant and light-cone coordinate system is convenient, because
$\sqrt{s}$ is the largest energy scale, compared to the energy-momentum
transfer for the scattering and the emitted gluon energy. The present study,
however, does not assume close to the forward scattering but deals with all
possible scattering angles, and treats soft gluon emission systematically up
to the leading order of $(q/p)$ with $q$ and $p$ being the energy-momenta of
emitted gluon and scattered parton, respectively, such that the Slavnov-Taylor
identities are satisfied and the results are explicitly gauge-invariant.

This paper is organized as follows: in section II the soft photon
approximation is rederived up to the leading order for both boson and fermion
scatterings, which is extended to QCD in section III. Then the soft gluon
emission is applied in section IV to partonic interactions in the QGP where
partons are dressed and thus massive. The final state phase space and the
scattering cross sections with the soft gluon emission are discussed in
section V, with which the transport coefficients of heavy quarks are
calculated in section VI. A summary is given in Section VII.

##  II soft photon approximation

In this section we rederive the formalism for the soft photon emission from
both boson and fermion scatterings, which corresponds to the first dominant
term in Low’s calculations Low (1958).

###  II.1 Emission from pseudoscalar particles

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{1}-q$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{2}-q$

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{3}+q$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{4}+q$

Figure 2: Photon emission from the four external legs of 2-to-2 scattering

The transition amplitude for photon emission from the scattering of two
pseudoscalar particles, as shown in figure 2, is given by

$\displaystyle M_{2\rightarrow
2+\gamma}(p_{1},p_{2};p_{3},p_{4},q)=\varepsilon_{\mu}^{*}(q)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\times\\{M_{2\rightarrow
2}(p_{1}-q,p_{2};p_{3},p_{4})G(p_{1}-q)V^{\mu}(p_{1};p_{1}-q)$
$\displaystyle+M_{2\rightarrow
2}(p_{1},p_{2}-q;p_{3},p_{4})G(p_{2}-q)V^{\mu}(p_{2};p_{2}-q)$
$\displaystyle+V^{\mu}(p_{3}+q;p_{3})G(p_{3}+q)M_{2\rightarrow
2}(p_{1},p_{2};p_{3}+q,p_{4})$
$\displaystyle+V^{\mu}(p_{4}+q;p_{4})G(p_{4}+q)M_{2\rightarrow
2}(p_{1},p_{2};p_{3},p_{4}+q)\\},$ (1)

where $\varepsilon_{\mu}^{*}(q)$ is the polarization vector of the emitted
photon, and $G(p)$ and $V^{\mu}(p+q;p)$ are, respectively, the propagator of
the photon-emitting particle and the electromagnetic vertex with photon
momentum $q$, which are expressed for the pseudoscalar particle (or pion) as
Kapusta  _et al._ (1991),

$\displaystyle G(p)$ $\displaystyle=$
$\displaystyle\frac{i}{p^{2}-m^{2}+i\varepsilon},$ $\displaystyle
V^{\mu}(p+q,p)$ $\displaystyle=$ $\displaystyle-iQ(2p+q)^{\mu}.$ (2)

Adopting the soft-photon approximation ($q\ll p_{1}$, $p_{2}$, $p_{3}$,
$p_{4}$), Eq. (1) is simplified into Haglin  _et al._ (1993)

$\displaystyle M_{2\rightarrow 2+\gamma(p_{1},p_{2};p_{3},p_{4},q)}$
$\displaystyle=\varepsilon_{\mu}^{*}(q)\bigg{\\{}-\frac{Q_{1}p_{1}^{\mu}}{p_{1}\cdot
q}-\frac{Q_{2}p_{2}^{\mu}}{p_{2}\cdot q}+\frac{Q_{3}p_{3}^{\mu}}{p_{3}\cdot
q}+\frac{Q_{4}p_{4}^{\mu}}{p_{4}\cdot q}\bigg{\\}}$
$\displaystyle~{}~{}~{}\times M_{2\rightarrow 2}(p_{1},p_{2};p_{3},p_{4}),$
(3)

which satisfies the Ward-Takahashi identity: $q^{\mu}(M_{2\rightarrow
2+\gamma})_{\mu}=$0 where $M_{2\rightarrow
2+\gamma}=\varepsilon^{\mu*}(q)(M_{2\rightarrow 2+\gamma})_{\mu}$.

###  II.2 Emission from fermions

The photon emission from a fermion is more complicated than the emission from
a boson due to the spin of fermion. For example, both propagator and vertex
include a gamma matrix:

$\displaystyle G(p)$ $\displaystyle=$ $\displaystyle
i\frac{\not{p}+m}{p^{2}-m^{2}+i\varepsilon},$ $\displaystyle V^{\mu}(p+q,p)$
$\displaystyle=$ $\displaystyle-iQ\gamma^{\mu}.$ (4)

If a photon goes out from $p_{3}$ as in the lower left diagram of figure 2,
the spinor of $p_{3}$ is substituted by

$\displaystyle\bar{u}^{s}(p_{3})$ $\displaystyle\rightarrow$
$\displaystyle-\bar{u}^{s}(p_{3})iQ_{3}\gamma^{\mu}i\frac{\not{p_{3}}+\not{q}+m}{(p_{3}+q)^{2}-m^{2}+i\varepsilon}\varepsilon_{\mu}^{*}(q)$
(5) $\displaystyle=$
$\displaystyle-\bar{u}^{s}(p_{3})iQ_{3}\gamma^{\mu}i\frac{u^{r}(p_{3}+q)\bar{u}^{r}(p_{3}+q)}{(p_{3}+q)^{2}-m^{2}+i\varepsilon}\varepsilon_{\mu}^{*}(q)$
$\displaystyle=$ $\displaystyle
Q_{3}\frac{\bar{u}^{s}(p_{3})\gamma^{\mu}u^{r}(p_{3}+q)}{2p_{3}\cdot
q}\varepsilon_{\mu}^{*}(q)\bar{u}^{r}(p_{3}+q).$

Making use of the Gordon decomposition Halzen and Martin (1984),

$\displaystyle\bar{u}^{s}(p_{3})\gamma^{\mu}u^{r}(p_{3}+q)$
$\displaystyle=\frac{1}{2m}\bar{u}^{s}(p_{3})\\{(2p_{3}+q)^{\mu}-i\sigma^{\mu\nu}q_{\nu}\\}u^{r}(p_{3}+q)$
$\displaystyle\approx\frac{p_{3}^{\mu}}{m}\bar{u}^{s}(p_{3})u^{r}(p_{3})=2p_{3}^{\mu}\delta_{sr}$
(6)

in the limit $q\ll p_{3}$, where the superscripts $s$ and $r$ are spin indices
and

$\displaystyle\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}],$ (7)

one finds that the modification of the transition amplitude in Eq. (5) is the
same as that for a pion in Eq. (3):

$\displaystyle\bar{u}^{s}(p_{3})\rightarrow\varepsilon_{\mu}^{*}(q)\frac{Q_{3}p_{3}^{\mu}}{p_{3}\cdot
q}\bar{u}^{s}(p_{3}+q)\approx\varepsilon_{\mu}^{*}(q)\frac{Q_{3}p_{3}^{\mu}}{p_{3}\cdot
q}\bar{u}^{s}(p_{3}).$ (8)

Therefore the soft photon approximation of Eq. (3) is applied not only to
pseudoscalar particle scattering but also to fermion scattering.

##  III Soft gluon emission

Now we apply the same approach to gluon emission, assuming that the emitted
gluon is soft and has a long wavelength.

###  III.1 Emission from (anti)quarks

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{1}-q$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{2}-q$

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{3}+q$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{4}+q$

Figure 3: Gluon emission from $q+q\rightarrow q+q$ scattering

As shown in Fig. 3 gluon emission is the same as photon emission except for a
color factor. The quark propagator and gluon vertex are given by

$\displaystyle G_{ij}(p)$ $\displaystyle=$ $\displaystyle
i\frac{\not{p}+m}{p^{2}-m^{2}+i\varepsilon}\delta_{ij},$ $\displaystyle
V_{ij}^{\mu,a}(p+q,p)$ $\displaystyle=$ $\displaystyle
ig\gamma^{\mu}T_{ij}^{a},$ (9)

where $i,j$ and $a$ are, respectively, the color indices of quark and gluon.
Making the same substitution as in Eq. (5),

$\displaystyle\bar{u}_{i}^{s}(p_{3})M_{2\rightarrow 2,i}$ (10)
$\displaystyle\rightarrow$
$\displaystyle-g\bar{u}_{i}^{s}(p_{3})\gamma^{\mu}T_{ij}^{a}\varepsilon_{\mu}^{a*}(q)\frac{\not{p_{3}}+\not{q}+m}{(p_{3}+q)^{2}-m^{2}+i\varepsilon}M_{2\rightarrow
2,j}$ $\displaystyle=$
$\displaystyle-g\frac{\bar{u}_{i}^{s}(p_{3})\gamma^{\mu}T_{ij}^{a}u_{j}^{r}(p_{3}+q)}{2p_{3}\cdot
q}\varepsilon_{\mu}^{a*}(q)$
$\displaystyle\times\bar{u}_{j}^{r}(p_{3}+q)M_{2\rightarrow 2,j},$

one can see that Eq. (10) is very similar to Eq. (5) except the color factor
$T_{ij}^{a}$. The transition amplitude turns out as

$\displaystyle M_{2q\rightarrow
2q+g}^{kl;ij}(p_{1},p_{2};p_{3},p_{4},q)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=g\varepsilon_{\mu}^{a*}(q)\bigg{\\{}\frac{p_{1}^{\mu}}{p_{1}\cdot
q}M_{2q\rightarrow 2q}^{kl;mj}T^{a}_{mi}+\frac{p_{2}^{\mu}}{p_{2}\cdot
q}M_{2q\rightarrow 2q}^{kl;im}T^{a}_{mj}$
$\displaystyle-\frac{p_{3}^{\mu}}{p_{3}\cdot q}T^{a}_{km}M_{2q\rightarrow
2q}^{ml;ij}-\frac{p_{4}^{\mu}}{p_{4}\cdot q}T^{a}_{lm}M_{2q\rightarrow
2q}^{km;ij}\bigg{\\}},$ (11)

where $i,j$ are the color indices of the incoming quarks and $k,l$ those of
the outgoing quarks.

The simplest color structure of $M_{2q\rightarrow 2q}^{kl;ij}$ is from the
one-gluon exchange:

$\displaystyle M_{2q\rightarrow 2q}^{kl;ij}\sim
T^{b}_{ki}T^{b}_{lj}=\frac{1}{2}\bigg{(}\delta_{jk}\delta_{il}-\frac{1}{N_{c}}\delta_{ik}\delta_{jl}\bigg{)}.$
(12)

Substituting Eq. (12) into Eq. (11) one finds that the transition amplitude
satisfies the Ward-Takahashi identity:

$\displaystyle q_{\mu}M_{2q\rightarrow
2q+g}^{\mu,a,kl;ij}(p_{1},p_{2};p_{3},p_{4},q)=0,$ (13)

where

$\displaystyle M_{2q\rightarrow
2q+g}^{kl;ij}=\varepsilon_{\mu}^{a*}(q)M_{2\rightarrow 2+g}^{\mu,a,kl;ij}.$
(14)

We note that the Ward-Takahashi identity is equivalent to the Slavnov-Taylor
identities in the case of on-shell external gluons.

The transition amplitude squared is given by

$\displaystyle|M_{2q\rightarrow
2q+g}|^{2}=-\frac{g^{2}}{2N_{c}}\bigg{[}(N_{c}^{2}-1)\bigg{\\{}\frac{m_{1}^{2}}{(p_{1}\cdot
q)^{2}}+\frac{m_{2}^{2}}{(p_{2}\cdot q)^{2}}$
$\displaystyle+\frac{m_{3}^{2}}{(p_{3}\cdot
q)^{2}}+\frac{m_{4}^{2}}{(p_{4}\cdot q)^{2}}\bigg{\\}}-\frac{4p_{1}\cdot
p_{2}}{(p_{1}\cdot q)(p_{2}\cdot q)}$ $\displaystyle-\frac{4p_{3}\cdot
p_{4}}{(p_{3}\cdot q)(p_{4}\cdot q)}+\frac{2p_{1}\cdot p_{3}}{(p_{1}\cdot
q)(p_{3}\cdot q)}+\frac{2p_{2}\cdot p_{4}}{(p_{2}\cdot q)(p_{4}\cdot q)}$
$\displaystyle-2(N_{c}^{2}-2)\bigg{\\{}\frac{p_{1}\cdot p_{4}}{(p_{1}\cdot
q)(p_{4}\cdot q)}+\frac{p_{2}\cdot p_{3}}{(p_{2}\cdot q)(p_{3}\cdot
q)}\bigg{\\}}\bigg{]}$ $\displaystyle\times|M_{2q\rightarrow
2q}|^{2}.~{}~{}~{}~{}~{}$ (15)

The detailed derivations of Eqs. (13) and (15) are presented in Appendix A.

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$ Figure 4: Soft gluon emission from the
exchanged gluon in quark-quark scattering

One may think of soft gluon emission from the exchanged gluon as in figure 4,
which is expressed as

$\displaystyle\varepsilon_{\mu}^{a*}(q)gf^{abc}[g^{\nu\mu}(-p_{1}+p_{3}-2q)^{\lambda}+g^{\lambda\mu}(-p_{1}+p_{3}+q)^{\nu}$
$\displaystyle+g^{\nu\lambda}(2p_{1}-2p_{3}+q)^{\mu}]\frac{-i}{(p_{1}-p_{3}+q)^{2}}\bigg{(}M_{2q\rightarrow
2q}^{kl;ij}\bigg{)}_{\nu\lambda}^{bc},~{}~{}~{}$ (16)

considering that the one gluon propagator and a three gluon vertex are
attached to the original 2-to-2 Feynman diagram. Since

$\displaystyle\bar{u}(p_{3})(\not{p_{3}}-\not{p_{1}})u(p_{1})$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\bar{u}(p_{4})(\not{p_{3}}-\not{p_{1}})u(p_{2})$
$\displaystyle=$
$\displaystyle\bar{u}(p_{4})(\not{p_{2}}-\not{p_{4}})u(p_{2})=0,$

Eq. (16) is simplified in the limit $q\rightarrow 0$ into

$\displaystyle-
ig\varepsilon_{\mu}^{a*}(q)\frac{2(p_{1}-p_{3})^{\mu}}{(p_{1}-p_{3})^{2}+2(p_{1}-p_{3})\cdot
q}f^{abc}\bigg{(}M_{2q\rightarrow 2q}^{kl;ij}\bigg{)}^{bc}.$ (17)

Removing $(p_{1}-p_{3})^{2}$ in the denominator, it looks similar to the terms
in Eq. (11). Comparing Eqs. (11) and (17) without $M_{2q\rightarrow 2q}$, the
former is of the order of $1/q$ and the latter of the order of $1/p=1/q(q/p)$
where $q$ and $p$ are, respectively, the momenta of the soft gluon and of the
scattering parton. Since $q/p\ll 1$, the diagram in figure 4 is of higher
order than those in figure 3, if the transition amplitude is expanded in term
of $q/p$:

$\displaystyle{\rm Fig.~{}\ref{feynman-quark}}$ $\displaystyle\sim$
$\displaystyle\frac{1}{q}\bigg{\\{}A_{0}+A_{1}\bigg{(}\frac{q}{p}\bigg{)}+...\bigg{\\}},$
$\displaystyle{\rm Fig.~{}\ref{exchange-fig}}$ $\displaystyle\sim$
$\displaystyle\frac{1}{q}\bigg{\\{}B_{1}\bigg{(}\frac{q}{p}\bigg{)}+...\bigg{\\}}.$
(18)

If the diagram of Fig. 4 is taken into account, one should also include the
next-to-leading order term in figure 3, that is, $A_{1}(q/p)$ in the above
equation, for the final results to be gauge-invariant. There is one more
important advantage in taking only the leading order in $(q/p)$. The soft
gluon emission from the exchanged virtual gluon is rather simple in
$q+q\rightarrow q+q+g$, but much more complicated in $q+g\rightarrow q+g+g$
which will be explained in the next section. That is why the soft gluon
emission is studied only in quark-quark scattering in Ref. Gunion and Bertsch
(1982); Lushozi  _et al._ (2020). Some works Uphoff  _et al._ (2012); Aichelin
_et al._ (2014) include $q+g\rightarrow q+g+g$ to study the radiative energy
loss of heavy quark in QGP. However, they consider only the gluon emission
from t-channel which is dominant in high-energy scattering, though it is not
gauge-invariant without $s-$ and $u-$ channels.

In spite of the order counting in Eq. (18), $(p_{1}-p_{3})^{2}$ in the
denominator of Eq. (17) can be smaller than the second term,
$(p_{1}-p_{3})\cdot q$, near forward scattering. Therefore, Eq. (11) is not a
good approximation for nearly forward scattering and the valid kinematic
region must properly be restricted. It will be discussed in the next section.

$-p_{1}$$-p_{2}$$-p_{3}$$-p_{4}$$q$$-p_{1}+q$$-p_{1}$$-p_{2}$$-p_{3}$$-p_{4}$$q$$-p_{2}+q$

$-p_{1}$$-p_{2}$$-p_{3}$$-p_{4}$$q$$-p_{3}-q$$-p_{1}$$-p_{2}$$-p_{3}$$-p_{4}$$q$$-p_{4}-q$

Figure 5: Gluon emission from $\bar{q}+\bar{q}\rightarrow\bar{q}+\bar{q}$
scattering

One can think about gluon emission from antiquark scattering
($\bar{q}+\bar{q}\rightarrow\bar{q}+\bar{q}$) as in figure 5. Then Eq. (10)
changes into

$\displaystyle M_{2\bar{q}\rightarrow 2\bar{q},i}v_{i}^{s}(p_{3})$ (19)
$\displaystyle\rightarrow$ $\displaystyle gM_{2\rightarrow
2,j}\frac{\not{p_{3}}+\not{q}-m}{(p_{3}+q)^{2}-m^{2}+i\varepsilon}\varepsilon_{\mu}^{a*}(q)\gamma^{\mu}T_{ji}^{a}v_{i}^{s}(p_{3})$
$\displaystyle=$ $\displaystyle gM_{2\rightarrow 2,k}v_{k}^{r}(p_{3}+q)$
$\displaystyle\times\frac{\bar{v}_{j}^{r}(p_{3}+q)\gamma^{\mu}T_{ji}^{a}v_{i}^{s}(p_{3})}{2p_{3}\cdot
q}\varepsilon_{\mu}^{a*}(q).$

The Gordon decomposition for the antifermion current is the same as in Eq. (6)
except for an overall minus sign:

$\displaystyle\bar{v}_{j}^{r}(p_{3}+q)\gamma^{\mu}v_{i}^{s}(p_{3})$
$\displaystyle=-\frac{1}{2m}\bar{v}_{j}^{r}(p_{3}+q)\\{(2p_{3}+q)^{\mu}+i\sigma^{\mu\nu}q_{\nu}\\}v_{i}^{s}(p_{3})$
$\displaystyle\approx-\frac{p_{3}^{\mu}}{m}\bar{v}_{j}^{r}(p_{3})v_{i}^{s}(p_{3})=2p_{3}^{\mu}\delta_{sr}$
(20)

Therefore, the expression of Eq. (11) is valid not only for $qq$ but also for
$\bar{q}\bar{q}$ and $q\bar{q}$ elastic scatterings.

###  III.2 Emission from gluon

Now we turn to the soft gluon emission from $q+g\rightarrow q+g$ scattering in
pQCD as shown in Fig. 6. The gluon propagator and three-gluon vertex are,
respectively, given by

$\displaystyle
G^{\mu\nu,ab}(p)=\frac{-ig^{\mu\nu}}{p^{2}+i\varepsilon}\delta_{ab},$
$\displaystyle
V^{\mu\nu\lambda,abc}(p+q,p)=gf^{abc}[g^{\nu\mu}(p+2q)^{\lambda}$
$\displaystyle+g^{\lambda\mu}(-2p-q)^{\nu}+g^{\nu\lambda}(p-q)^{\mu}]$ (21)

where $\mu,~{}\nu,~{}\lambda$ are Lorentz indices and $a,~{}b,~{}c$ are color
indices of the emitted, incoming and outgoing gluons, respectively.

Making the similar substitution as in Eq. (5),

$\displaystyle\varepsilon_{\lambda}^{b*}(p_{4})M_{2\rightarrow
2}^{\lambda,b}\rightarrow-
igf^{bdc}\varepsilon_{\lambda}^{b*}(p_{4})\varepsilon_{\mu}^{c*}(q)\bigg{[}g^{\mu}_{\nu}(p_{4}+2q)^{\lambda}$
$\displaystyle+g^{\lambda\mu}(p_{4}-q)_{\nu}+g_{\nu}^{\lambda}(-2p_{4}-q)^{\mu}\bigg{]}\frac{1}{2p_{4}\cdot
q}M_{2\rightarrow 2}^{\nu,d}$ $\displaystyle\approx
igf^{bdc}\varepsilon_{\mu}^{c*}(q)\frac{p_{4}^{\mu}}{p_{4}\cdot
q}\times\varepsilon_{\nu}^{b*}(p_{4})M_{2\rightarrow 2}^{\nu,d},~{}~{}~{}$
(22)

because $p_{4}^{\lambda}\varepsilon_{\lambda}^{c*}(p_{4})=0$ and
$p_{4\nu}M_{2\rightarrow 2}^{\nu,b}=0$. Similarly,

$\displaystyle\varepsilon_{\lambda}^{a}(p_{2})M_{2\rightarrow
2}^{\lambda,a}\rightarrow
igf^{adc}\varepsilon_{\mu}^{c*}(q)\frac{p_{2}^{\mu}}{p_{2}\cdot
q}\times\varepsilon_{\nu}^{a}(p_{2})M_{2\rightarrow 2}^{\nu,d}.~{}~{}~{}$ (23)

From Eqs. (11), (22) and (23) one gets

$\displaystyle M_{q+g\rightarrow
q+g+g}^{\mu,jbc;ia}(p_{1},p_{2};p_{3},p_{4},q)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle=g\bigg{\\{}\frac{p_{1}^{\mu}}{p_{1}\cdot q}M_{q+g\rightarrow
q+g}^{jb;ma}T^{c}_{mi}-\frac{p_{3}^{\mu}}{p_{3}\cdot
q}T^{c}_{jm}M_{q+g\rightarrow q+g}^{mb;ia}$
$\displaystyle+i\frac{p_{2}^{\mu}}{p_{2}\cdot q}f^{adc}M_{q+g\rightarrow
q+g}^{jb;id}+i\frac{p_{4}^{\mu}}{p_{4}\cdot q}f^{bdc}M_{q+g\rightarrow
q+g}^{jd;ia}\bigg{\\}},$ (24)

where $a$ and $b$ are, respectively, the colors of the incoming and outgoing
gluons, and $i$ and $j$ are the colors of the incoming and outgoing quarks,
respectively.

Considering that the color structure of $M_{q+g\rightarrow q+g}^{jb;ia}$ is
given by $[T^{a},T^{b}]_{ji}$ or $if^{abc}T_{ji}^{c}$, the matrix element in
Eq. (24) satisfies current conservation:

$\displaystyle q_{\mu}M_{q+g\rightarrow
q+g+g}^{\mu,jbc;ia}(p_{1},p_{2};p_{3},p_{4},q)=0,$ (25)

where

$\displaystyle M_{q+g\rightarrow
q+g+g}^{jb;ia}=\varepsilon_{\mu}^{c*}(q)M_{q+g\rightarrow
q+g+g}^{\mu,jbc;ia},$ (26)

and the transition amplitude squared is given by

$\displaystyle|M_{q+g\rightarrow
q+g+g}|^{2}=-g^{2}\bigg{[}\frac{N_{c}^{2}-1}{2N_{c}}\bigg{(}\frac{m_{1}^{2}}{(p_{1}\cdot
q)^{2}}+\frac{m_{3}^{2}}{(p_{3}\cdot q)^{2}}\bigg{)}$
$\displaystyle+\frac{1}{N_{c}}\frac{p_{1}\cdot p_{3}}{(p_{1}\cdot
q)(p_{3}\cdot q)}-\frac{N_{c}}{2}\bigg{(}\frac{2p_{2}\cdot p_{4}}{(p_{2}\cdot
q)(p_{4}\cdot q)}$ $\displaystyle+\frac{p_{1}\cdot p_{2}}{(p_{1}\cdot
q)(p_{2}\cdot q)}+\frac{p_{3}\cdot p_{4}}{(p_{3}\cdot q)(p_{4}\cdot
q)}+\frac{p_{1}\cdot p_{4}}{(p_{1}\cdot q)(p_{4}\cdot q)}$
$\displaystyle+\frac{p_{2}\cdot p_{3}}{(p_{2}\cdot q)(p_{3}\cdot
q)}\bigg{)}\bigg{]}\times|M_{q+g\rightarrow q+g}|^{2}.$ (27)

We note that the transition amplitude, in principle, must be divided by 2, for
the two gluons in the final state are indistinguishable. Since it is assumed
that one gluon is the hard gluon involved in the elastic scattering and the
other gluon the soft gluon emitted from the scattering, 2 is not divided in
Eq. (27). The proof and the derivation are presented in Appendix B.

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{1}-q$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{2}-q$

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{3}+q$$p_{1}$$p_{2}$$p_{3}$$p_{4}$$q$$p_{4}+q$

Figure 6: Gluon emission from $q+g\rightarrow q+g$ scattering

##  IV Soft gluon emission in the sQGP within the DQPM

In the Dynamical Quasi-Particle Model (DQPM), quark and gluon are dressed in
the sQGP and gain effective masses ($m_{q,g}$) and spectral widths
($\Gamma_{q,g}$) which depend on temperature and quark chemical potential
Moreau  _et al._ (2019). Main properties and parameters of the DQPM is
described in Appendix C. The finite width is reflecting the dynamical
modification of spectral function of quasiparticles during their propagation
in the sQGP medium. In the DQPM the propagators of quark and gluon are
modified, respectively, into

$\displaystyle G_{ij}(p)$ $\displaystyle=$ $\displaystyle
i\frac{\not{p}+m_{q}(T,\mu)}{p^{2}-m_{q}^{2}(T,\mu)+i|p_{0}|\Gamma_{q}(T,\mu)}\delta_{ij},$
$\displaystyle G^{\mu\nu,ab}(p)$ $\displaystyle=$
$\displaystyle\frac{-ig^{\mu\nu}}{p^{2}-m_{g}^{2}(T,\mu)+i|p_{0}|\Gamma_{g}(T,\mu)}\delta_{ab}.$
(28)

Then the scalar products of the external momentum $p_{i}$ and the soft gluon
momentum $q$ in the denominators of Eq. (15) are modified as

$\displaystyle p_{1}\cdot q\rightarrow p_{1}\cdot
q-m_{g}^{2}/2-i|p_{1}^{0}-q^{0}|\Gamma_{q}/2,$ $\displaystyle p_{2}\cdot
q\rightarrow p_{2}\cdot q-m_{g}^{2}/2-i|p_{2}^{0}-q^{0}|\Gamma_{q}/2,$
$\displaystyle p_{3}\cdot q\rightarrow p_{3}\cdot
q+m_{g}^{2}/2+i|p_{3}^{0}+q^{0}|\Gamma_{q}/2,$ $\displaystyle p_{4}\cdot
q\rightarrow p_{4}\cdot q+m_{g}^{2}/2+i|p_{4}^{0}+q^{0}|\Gamma_{q}/2,$

and in Eq. (27) $\Gamma_{g}(T,\mu)$ is introduced for $p_{2}\cdot q$ and
$p_{4}\cdot q$:

$\displaystyle p_{2}\cdot q\rightarrow p_{2}\cdot
q-m_{g}^{2}/2-i|p_{2}^{0}-q^{0}|\Gamma_{g}/2,$ $\displaystyle p_{4}\cdot
q\rightarrow p_{4}\cdot q+m_{g}^{2}/2+i|p_{4}^{0}+q^{0}|\Gamma_{g}/2.$

Since the product is now replaced with a complex function due to the imaginary
term, the following substitution must be made:

$\displaystyle\frac{1}{(p_{i}\cdot q)(p_{j}\cdot q)}\rightarrow{\rm
Re}\bigg{[}\frac{1}{(p_{i}\cdot q)(p_{j}\cdot q)}\bigg{]}$
$\displaystyle~{}~{}~{}~{}=\frac{1}{2}\bigg{\\{}\frac{1}{(p_{i}\cdot
q)(p_{j}\cdot q)^{*}}+\frac{1}{(p_{i}\cdot q)^{*}(p_{j}\cdot q)}\bigg{\\}}.$
(29)

For example,

$\displaystyle\frac{1}{(p_{1}\cdot q)(p_{3}\cdot q)}\rightarrow$
$\displaystyle\frac{1}{2}\bigg{\\{}\frac{1}{p_{1}\cdot
q-m_{g}^{2}/2-i|p_{1}^{0}-q^{0}|\Gamma_{q}/2}$
$\displaystyle~{}~{}~{}\times\frac{1}{p_{3}\cdot
q+m_{g}^{2}/2-i|p_{3}^{0}+q^{0}|\Gamma_{q}/2}$
$\displaystyle~{}+\frac{1}{p_{1}\cdot
q-m_{g}^{2}/2+i|p_{1}^{0}-q^{0}|\Gamma_{q}/2}$
$\displaystyle~{}~{}~{}\times\frac{1}{p_{3}\cdot
q+m_{g}^{2}/2+i|p_{3}^{0}+q^{0}|\Gamma_{q}/2}\bigg{\\}}.$ (30)

Furthermore, the first term in the square bracket of Eq. (27) has additional
terms which vanish for massless gluons:

$\displaystyle-\frac{N_{c}^{2}-1}{2N_{c}}\bigg{(}\frac{m^{2}}{(p_{1}\cdot
q)^{2}}+\frac{m^{2}}{(p_{3}\cdot q)^{2}}\bigg{)}\rightarrow$
$\displaystyle-\frac{N_{c}^{2}-1}{2N_{c}}\bigg{\\{}\frac{m_{q}^{2}}{\bigg{|}p_{1}\cdot
q-m_{g}^{2}/2-i|p_{1}^{0}-q^{0}|\Gamma_{q}/2\bigg{|}^{2}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{m_{q}^{2}}{\bigg{|}p_{3}\cdot
q+m_{g}^{2}/2+i|p_{3}^{0}+q^{0}|\Gamma_{q}/2\bigg{|}^{2}}\bigg{\\}}$
$\displaystyle-N_{c}\bigg{\\{}\frac{m_{g}^{2}}{\bigg{|}p_{2}\cdot
q-m_{g}^{2}/2-i|p_{2}^{0}-q^{0}|\Gamma_{g}/2\bigg{|}^{2}}$
$\displaystyle~{}~{}~{}~{}+\frac{m_{g}^{2}}{\bigg{|}p_{4}\cdot
q+m_{g}^{2}/2+i|p_{4}^{0}+q^{0}|\Gamma_{g}/2\bigg{|}^{2}}\bigg{\\}}.$ (31)

##  V Phase space and cross section

The cross section for 2-to-3 scattering is given by

$\displaystyle\sigma_{2\rightarrow
3}=\frac{1}{4p_{i}\sqrt{s}}\int\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\int\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}\int\frac{d^{3}q}{(2\pi)^{3}2E_{g}}$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p_{3}-p_{4}-q)\overline{|M_{2\rightarrow
3}|}^{2},~{}~{}~{}$ (32)

where $p_{i}$ is the initial three momentum in the center-of-mass frame and
the line over the transition amplitude squared implies spin-color average.
Introducing the variable $p=p_{3}+p_{4}$,

$\displaystyle\sigma_{2\rightarrow 3}=\frac{1}{4p_{i}\sqrt{s}}\int
d^{4}p\int\frac{d^{3}p_{3}d^{3}p_{4}}{(2\pi)^{6}4E_{3}E_{4}}\delta^{(4)}(p-p_{3}-p_{4})$
$\displaystyle\times\int\frac{d^{3}q}{(2\pi)^{3}2E_{g}}(2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}-p-q)\overline{|M_{2\rightarrow
3}|}^{2}$ $\displaystyle=\frac{1}{32\pi
p_{i}\sqrt{s}}\int\frac{d^{3}q}{(2\pi)^{3}2E_{g}}\frac{|{\bf
p}_{3}|}{\sqrt{s_{2}}}\int d\cos\theta\overline{|M_{2\rightarrow
3}}|^{2}~{}~{}~{}~{}$ (33)

with the constraint of energy-momentum conservation in the second equation. In
the above equation $s_{2}=(p_{3}+p_{4})^{2}$, and $|{\bf p}_{3}|$ and $\theta$
are, respectively, the three-momentum and scattering angle of $p_{3}$ in the
center-of-mass frame of $p_{3}+p_{4}$:

$\displaystyle|{\bf
p}_{3}|=\sqrt{\frac{\\{s_{2}-(m_{3}+m_{4})^{2}\\}\\{s_{2}-(m_{3}-m_{4})^{2}\\}}{4s_{2}}}.$
(34)

Assuming that the emitted gluon is soft, the center-of-mass frame of
$p_{3}+p_{4}$ is similar to that of $p_{1}+p_{2}$ and the differential cross
section is approximated as

$\displaystyle\frac{d\sigma_{2\rightarrow
3}}{d\cos\theta}\approx\frac{1}{32\pi
p_{i}\sqrt{s}}\int\frac{d^{3}q}{(2\pi)^{3}2E_{g}}\frac{|{\bf
p}_{3}|}{\sqrt{s_{2}}}\overline{|M_{2\rightarrow 3}}|^{2}$
$\displaystyle\approx\frac{d\sigma_{2\rightarrow
2}}{d\cos\theta}\int\frac{d^{3}q}{(2\pi)^{3}2E_{g}}|\epsilon\cdot
J|^{2}\frac{|{\bf p}_{3}|\sqrt{s}}{p_{f}\sqrt{s_{2}}},$ (35)

where

$\displaystyle\overline{|M_{2\rightarrow 3}}|^{2}$ $\displaystyle\equiv$
$\displaystyle|\epsilon\cdot J|^{2}~{}\overline{|M_{2\rightarrow 2}}|^{2}$
(36) $\displaystyle=$ $\displaystyle 32\pi s|\epsilon\cdot
J|^{2}\frac{p_{i}}{p_{f}}\frac{d\sigma_{2\rightarrow 2}}{d\cos\theta}$

with

$\displaystyle
p_{f}=\sqrt{\frac{\\{s-(m_{3}+m_{4})^{2}\\}\\{s-(m_{3}-m_{4})^{2}\\}}{4s}}.$
(37)

We note that $|{\bf p}_{3}|\sqrt{s}/(p_{f}\sqrt{s_{2}})$ in Eq. (35) is
responsible for the reduction of phase space of $p_{3}+p_{4}$ in 2-to-3
process, compared to 2-to-2 process, because $s_{2}$ is always smaller than
$s$ Linnyk  _et al._ (2016).

For simplicity we patch gluon energy-momentum to 2-to-2 elastic scattering,
ignoring energy-momentum conservation:

$\displaystyle p_{1}$ $\displaystyle=$
$\displaystyle(E_{1},~{}0,~{}0,~{}p_{1}),$ (38) $\displaystyle p_{2}$
$\displaystyle=$ $\displaystyle(E_{2},~{}0,~{}0,~{}-p_{1}),$ $\displaystyle
p_{3}$ $\displaystyle=$
$\displaystyle(E_{3},~{}0,~{}p_{1}\sin\theta,~{}p_{1}\cos\theta),$
$\displaystyle p_{4}$ $\displaystyle=$
$\displaystyle(E_{4},~{}0,~{}-p_{1}\sin\theta,~{}-p_{1}\cos\theta),$
$\displaystyle q$ $\displaystyle=$ $\displaystyle(E_{g},~{}{\bf
q}\sin\theta^{\prime}\cos\phi^{\prime},~{}{\bf
q}\sin\theta^{\prime}\sin\phi^{\prime},~{}{\bf q}\cos\theta^{\prime}),$

which will be used to calculate $|\epsilon\cdot J|^{2}$ in Eq. (36). The
integration in Eq. (35) is then expressed as

$\displaystyle\int\frac{d^{3}q}{E_{g}}=\int_{m_{g}}^{E_{max}}\sqrt{E_{g}^{2}-m_{q}^{2}}\
dE_{g}d\cos\theta^{\prime}d\phi^{\prime},$ (39)

where $E_{max}$ is the maximum energy of soft gluon. From energy conservation
it is given by

$\displaystyle
q_{max}^{2}=\frac{\\{s-(m_{3}+m_{4}+m_{g})^{2}\\}\\{s-(m_{3}+m_{4}-m_{g})^{2}\\}}{4s}$
(40)

with $E_{max}=\sqrt{m_{g}^{2}+q_{max}^{2}}$.

However, we have neglected gluon emission from the interaction region which is
figured as a blob in the Feynman diagrams, because the wavelength of the
emitted gluon is assumed larger than the scattering scale which is roughly
$1/\sqrt{-t}$ with $t=(p_{1}-p_{3})^{2}$. Therefore, more reasonable limit for
our calculations to be valid will be

$\displaystyle E_{max}={\rm min}\bigg{[}E_{max}~{}{\rm
from~{}Eq.}~{}(\ref{limit1}),~{}\sqrt{-t}\bigg{]},$ (41)

which also justifies the ignoring of figure 4. From here on the upper limit
from Eq. (40) will be denoted the kinematic upper limit and Eq. (41) the
realistic upper limit.

Figure 7: (Color online) Differential and integrated cross sections for
$c+q(g)\rightarrow c+q(g)$ and $c+q(g)\rightarrow c+q(g)+g$ with the kinematic
and realistic upper limits of soft gluon energy, which are, respectively,
displayed by dashed, dotted and solid lines as a function of (upper) the
scattering angle of the charm quark, (middle) scattering energy and (lower)
temperature.

Now we apply the formula, which has been derived, to the soft gluon emission
from charm quark elastic scattering in the QGP with charm quark mass being 1.5
GeV.

Figure 7 shows the differential and integrated cross sections for
$c+q(g)\rightarrow c+q(g)$ and $c+q(g)\rightarrow c+q(g)+g$ scattering with
the kinematic and realistic upper limits of gluon energy, which are,
respectively, displayed by dashed, dotted and solid lines, as a function of
the scattering angle of the charm quark, scattering energy and temperature.

One can see that the differential cross sections for gluon emission with the
two different upper limits of the gluon energy are almost same except near
$\cos\theta=1$, i.e. near forward scattering where $t$ is small and thus the
realistic upper limit of the soft gluon energy is low.

The middle panels show that the integrated cross sections for gluon emission
are suppressed at low energy, because the collision energy is not enough or
hard to produce a massive gluon in the final state. The integrated cross
section for the kinematic upper limit is much larger than for the realistic
upper limit, since the forward scattering cross section is very large which,
however, is suppressed in the latter case.

Finally, comparing 2-to-2 scattering cross sections and 2-to-3 scattering
cross sections in the lower panel, the temperature dependence is stronger for
the latter. The reason is that the former is proportional to $\alpha_{s}^{2}$
while the latter to $\alpha_{s}^{3}$, and $\alpha_{s}$ is very large near
$T_{c}$. Therefore, charm quark scattering with gluon emission is more
suppressed than the elastic scattering with increasing temperature.

##  VI Transport coefficients of heavy quark in the sQGP

Charm quarks change their energy-momentum in the QGP with time through
scattering, which is expressed in a Fokker-Planck equation as Svetitsky
(1988):

$\displaystyle\frac{\partial f(p)}{\partial t}=\frac{\partial}{\partial
p_{i}}\bigg{[}A_{i}(p)f(p)+\frac{\partial}{\partial
p_{i}}[B_{ij}(p)f(p)]\bigg{]},$ (42)

where $f(p)$ is the charm distribution function and the drag,
transverse/longitudinal diffusion coefficients and $\hat{q}$ of the charm
quark in the medium, supposing that the charm quark moves in $z-$direction,
are defined as

$\displaystyle A(p)$ $\displaystyle=$
$\displaystyle-\frac{\vec{A}(p)\cdot\vec{p}}{|\vec{p}|}=-\frac{d\langle\Delta
p\rangle}{dt}=\langle\langle(p-p^{\prime})_{z}\rangle\rangle,$ (43)
$\displaystyle B_{L}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{p_{i}p_{j}}{|\vec{p}|^{2}}B_{ij}(p)$ (44)
$\displaystyle=$ $\displaystyle\frac{1}{2}\frac{d\langle(\Delta
p_{L})^{2}\rangle}{dt}=\frac{1}{2}\langle\langle(p-p^{\prime})_{z}^{2}\rangle\rangle,$
$\displaystyle B_{T}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\bigg{(}\delta_{ij}-\frac{p_{i}p_{j}}{|\vec{p}|^{2}}\bigg{)}B_{ij}(p)$
(45) $\displaystyle=$ $\displaystyle\frac{1}{4}\frac{d\langle(\Delta
p_{T})^{2}\rangle}{dt}=\frac{1}{4}\langle\langle p_{x}^{\prime
2}+p_{y}^{\prime 2}\rangle\rangle,$ $\displaystyle\hat{q}(p)$ $\displaystyle=$
$\displaystyle\frac{d\langle(\Delta
p_{T})^{2}\rangle}{dz}=\frac{4E}{p_{L}}B_{T}(p),$ (46)

where $p_{L}$ and $p_{T}$ are respectively the longitudinal and transverse
momentum of charm quark.

The double bracket in Eqs. (43) to (46) implies Berrehrah  _et al._ (2014b);
Song  _et al._ (2020)

$\displaystyle\langle\langle O^{*}\rangle\rangle\equiv\sum_{i=q,\bar{q},g}\int
dm_{i}dm_{f}dm_{g}A_{i}(m_{i})A_{i}(m_{f})A_{g}(m_{g})$
$\displaystyle\times\int\frac{d^{3}k}{(2\pi)^{3}}f_{i}(k)~{}O^{*}~{}v_{ic}\sigma_{ic},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(47)

for the scattering of offshell partons, where $m_{i}$, $m_{f}$ and $m_{g}$ are
respectively the incoming and outgoing parton masses and the emitted gluon
mass, and $A_{i}$, $A_{f}$ and $A_{g}$ their spectral functions whose pole
masses and spectral widths are presented in Appendix C; $f_{i}(k)$ is a
distribution function of parton $i$, and $v_{ic}$ and $\sigma_{ic}$ are the
relative velocity and the scattering cross section of the charm quark and
parton $i$, respectively. We note that the scattering cross section in Eq.
(47) is multiplied by 2 in order to reproduce the lattice data on the spatial
diffusion coefficient and the experimental data on heavy flavors in heavy-ion
collisions Song  _et al._ (2017).

Figure 8: (Color online) (upper) Drag coefficients and (lower) $\hat{q}/T^{3}$
of the charm quark at T=200 MeV (left) and 400 MeV (right) without and with
$c+q(g)\rightarrow c+q(g)$ and $c+q(g)\rightarrow c+q(g)+g$ for the kinematic
and realistic upper limits of soft gluon energy, which are, respectively,
displayed by dashed, dotted and solid lines.

Figure 8 shows the drag coefficients and $\hat{q}/T^{3}$ of the charm quark
without and with soft gluon emission for the kinematic and realistic upper
limits of soft gluon energy at T=200 MeV and 400 MeV. One can see that 2-to-3
scattering hardly changes the transport coefficients at low momentum, because
the scattering energy of a slow charm quark with a thermal parton is not large
enough to produce a massive gluon. As a result, the spatial diffusion
coefficient of a heavy quark in the QGP, which is obtained from the drag
coefficient in the static limit of charm quark, is not much affected by the
2-to-3 scattering:

$\displaystyle D_{s}=\lim_{p_{c}\rightarrow 0}\frac{T}{m_{c}(A/p_{c})},$ (48)

where $A$ is the drag coefficient shown in the upper panels of Fig. 8.

The spatial diffusion coefficient is presently available in lattice
calculations Banerjee  _et al._ (2012) and it is well reproduced within the
DQPM Berrehrah  _et al._ (2014b); Song and Berrehrah (2016). Therefore, this
good reproduction will not change even after including the 2-to-3 processes in
the DQPM, as shown in Fig. 9.

Figure 9: (Color online) spatial diffusion coefficient of charm quark for only
elastic scattering and both elastic and radiative scatterings in comparison
with lattice data Banerjee  _et al._ (2012).

On the other hand, the radiative energy loss enhances the drag and $\hat{q}$
of the charm quark at large momentum, which is not consistent with the PHSD
results presented in Refs. Song  _et al._ (2015, 2016), because the $R_{\rm
AA}$ of D mesons in heavy-ion collisions at RHIC and LHC are well reproduced
only with elastic scattering. Recently we have found in Ref. Grishmanovskii
_et al._ (2022) that the strong coupling $\alpha_{s}(T)$ \- extracted from the
lattice EoS - seems to overestimate jet quenching and the mixture of
$\alpha_{s}(T)$ and a constant strong coupling may be more realistic, because
an energetic parton is far off thermal equilibrium. This kind of change of the
strong coupling at large momentum will help the PHSD to reproduce experimental
data with the radiative energy loss, too.

We note that the transport coefficients from the kinematic upper limit and
those from the realistic upper limits are not much different from each other,
while the difference of the scattering cross sections is huge in Fig. 7. The
reason is that the two cross sections mostly differ near the forward
scattering, which is not so effective to increase the transport coefficients.

Figure 10: (Color online) same as figure 8 but for bottom quark.

Figure 10 is same as the figure 8 except that charm quark is substituted with
bottom quark whose mass is taken to be 4.8 GeV. Compared with figure 8, the
transport coefficients of bottom quark seem to be less affected by the gluon
emission. In fact, it is attributed to the large mass of bottom quark which is
about three times larger than charm quark mass. A fair comparison between
charm and bottom will be made when $p_{T}$ of bottom quark is rescaled by
three times, because interaction rate is proportional to heavy quark velocity.
We also note that $\hat{q}$ of bottom quark is a bit rising at low momentum,
because $\hat{q}$ is defined as $d\langle(\Delta p_{T})^{2}\rangle/dz$ rather
than $d\langle(\Delta p_{T})^{2}\rangle/dt$ in Eq. (46). The transport
coefficients of bottom quark are in general smaller than those of charm quark,
because the differential cross section of bottom quark is more highly peaked
in forward direction due to the large mass Song  _et al._ (2017).

##  VII Summary

In this study we have extended the formalism for soft photon emission from the
scattering of electric-charged particles to soft gluon emission from the
scattering of color-charged particles, i.e. of partons. We have found that the
soft gluon is emitted from incoming or outgoiong partons as the emission of
soft photon, and the scattering amplitude with the approximation satisfies the
Slavnov-Taylor identities. This is so because the Slavnov-Taylor identities
are satisfied order by order, if the scattering amplitude is expanded in terms
of the emitted gluon energy divided by the energy of the scattering particles.
It enables the factorization of radiative scattering into elastic scattering
and gluon/photon emission and guarantees gauge-invariance.

Heavy flavor is an important probe particle searching for the properties of an
extremely hot and dense matter produced in ultra-relativistic heavy-ion
collisions. We have applied the formalism for the soft gluon emission to the
heavy quark scattering off massive quarks or gluons in the sQGP. For the soft
gluon approximation to be valid, emitted gluon energy is restricted up to the
energy-momentum transfer of 2-to-2 scattering.

Comparing the integrated cross sections for elastic scattering and for soft
gluon emission, we have found that the latter is strongly suppressed at low
momentum, because the scattering energy is not large enough to produce a
massive gluon, and that the scattering for gluon emission is more strongly
suppressed with increasing temperature than the elastic scattering, because
the former is proportional to $\alpha_{s}^{3}(T)$ while the latter to
$\alpha_{s}^{2}(T)$.

The results have been extended to the calculations of the transport
coefficients of heavy quarks in the sQGP. For the same reason as for the cross
section the transport coefficients little change at small momentum of heavy
quark in spite of incluing gluon emission, which means that the spatial
diffusion coefficient of heavy quark is not affected by the radiative
scattering. However, the transport coefficients are enhanced by it with
increasing charm quark momentum. Since the energetic parton is far off thermal
equilibrium, it is doubtable that the thermal strong coupling extracted from
the lattice EoS, $\alpha_{s}(T,\mu)$, can be applied to the scattering of the
energetic parton. Instead a perturbative strong coupling or a mixture of them
may be more reasonable, which will be our next study.

## Acknowledgements

The authors acknowledge helpful discussions with E. Bratkovskaya, J. Aichelin
and W. Cassing. 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”. The computational resources have been provided by the
LOEWE-Center for Scientific Computing and the ”Green Cube” at GSI, Darmstadt.

## Appendix A

We first show the current conservation of Eq. (13). From Eqs. (11) and (14) we
get

$\displaystyle q_{\mu}M_{2\rightarrow 2+g}^{\mu,a,kl;ij}$ $\displaystyle=$
$\displaystyle g\bigg{\\{}M_{2q\rightarrow
2q}^{kl;mj}T^{a}_{mi}+M_{2q\rightarrow 2q}^{kl;im}T^{a}_{mj}$ (49)
$\displaystyle-T^{a}_{km}M_{2q\rightarrow
2q}^{ml;ij}-T^{a}_{lm}M_{2q\rightarrow 2q}^{km;ij}\bigg{\\}}$
$\displaystyle\sim$ $\displaystyle
T^{b}_{km}T^{b}_{lj}T^{a}_{mi}+T^{b}_{ki}T^{b}_{lm}T^{a}_{mj}$
$\displaystyle-T^{a}_{km}T^{b}_{mi}T^{b}_{lj}-T^{a}_{lm}T^{b}_{ki}T^{b}_{mj}.$

Using Eq. (12) for $M_{2q\rightarrow 2q}^{kl;ij}$,

$\displaystyle q_{\mu}M_{2\rightarrow
2+g}^{\mu,a,kl;ij}\sim\bigg{(}T^{a}_{li}\delta_{kj}-\frac{1}{N_{c}}T^{a}_{ki}\delta_{lj}\bigg{)}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle+\bigg{(}T^{a}_{kj}\delta_{il}-\frac{1}{N_{c}}T^{a}_{lj}\delta_{ki}\bigg{)}-\bigg{(}T^{a}_{kj}\delta_{il}-\frac{1}{N_{c}}T^{a}_{ki}\delta_{lj}\bigg{)}$
$\displaystyle-\bigg{(}T^{a}_{li}\delta_{kj}-\frac{1}{N_{c}}T^{a}_{lj}\delta_{ki}\bigg{)}\equiv
T_{1}+T_{2}-T_{3}-T_{4}=0.~{}~{}$ (50)

Now we turn to the transition amplitude squared, for which the following
combinations of color structure are needed:

$\displaystyle|T_{1}|^{2}=|T_{2}|^{2}=|T_{3}|^{2}=|T_{4}|^{2}=\frac{N_{c}^{2}-1}{4}C_{F},$
$\displaystyle T_{1}T_{2}^{*}=T_{3}T_{4}^{*}=-\frac{1}{2}C_{F},$
$\displaystyle T_{1}T_{3}^{*}=T_{2}T_{4}^{*}=-\frac{1}{4}C_{F},$
$\displaystyle T_{1}T_{4}^{*}=T_{2}T_{3}^{*}=\frac{N_{c}^{2}-2}{4}C_{F},$ (51)

where $(T^{a}_{ij})^{*}=T^{a}_{ji}$ and $C_{F}=(N_{c}^{2}-1)/(2N_{c})$.

Taking into account the color factors in Eq. (51), the transition amplitude
squared turns to

$\displaystyle|M_{2q\rightarrow
2q+g}|^{2}=-\frac{g^{2}}{2N_{c}}\bigg{[}(N_{c}^{2}-1)\bigg{\\{}\frac{m_{1}^{2}}{(p_{1}\cdot
q)^{2}}+\frac{m_{2}^{2}}{(p_{2}\cdot q)^{2}}$
$\displaystyle+\frac{m_{3}^{2}}{(p_{3}\cdot
q)^{2}}+\frac{m_{4}^{2}}{(p_{4}\cdot q)^{2}}\bigg{\\}}-4\frac{p_{1}\cdot
p_{2}}{(p_{1}\cdot q)(p_{2}\cdot q)}$ $\displaystyle-4\frac{p_{3}\cdot
p_{4}}{(p_{3}\cdot q)(p_{4}\cdot q)}+2\frac{p_{1}\cdot p_{3}}{(p_{1}\cdot
q)(p_{3}\cdot q)}+2\frac{p_{2}\cdot p_{4}}{(p_{2}\cdot q)(p_{4}\cdot q)}$
$\displaystyle-2(N_{c}^{2}-2)\bigg{\\{}\frac{p_{1}\cdot p_{4}}{(p_{1}\cdot
q)(p_{4}\cdot q)}+\frac{p_{2}\cdot p_{3}}{(p_{2}\cdot q)(p_{3}\cdot
q)}\bigg{\\}}\bigg{]}$ $\displaystyle\times|M_{2q\rightarrow
2q}|^{2},~{}~{}~{}~{}~{}$ (52)

where $N_{c}C_{F}/2$ is absorbed into $|M_{2q\rightarrow 2q}|^{2}$.

## Appendix B

Substituting $M_{q+g\rightarrow q+g}^{jb;ia}$ with $if^{abe}T_{ji}^{e}$ in Eq.
(24), one can show the current conservation as follows:

$\displaystyle q_{\mu}M_{q+g\rightarrow
q+g+g}^{\mu,jbc;ia}(p_{1},p_{2};p_{3},p_{4},q)\sim
if^{abe}T_{jm}^{e}T^{c}_{mi}$ $\displaystyle-
if^{abe}T^{c}_{jm}T_{mi}^{e}-f^{adc}f^{dbe}T_{ji}^{e}-f^{bdc}f^{ade}T_{ji}^{e}$
$\displaystyle=-f^{abe}f^{ecd}T^{d}_{ji}-f^{adc}f^{dbe}T_{ji}^{e}-f^{bdc}f^{ade}T_{ji}^{e}$
$\displaystyle=(-f^{abd}f^{ced}-f^{cbd}f^{ead}-f^{ebd}f^{acd})T_{ji}^{e}=0,$
(53)

which needs the cyclic property of $f^{abc}$ and
$[T^{e},T^{c}]_{ji}=if^{ecd}T_{ji}^{d}$ Muta (1987).

Now we turn to the scattering amplitude squared. Considering only color
factors,

$\displaystyle|M_{q+g\rightarrow
q+g+g}^{jbc;ia}|^{2}(p_{1},p_{2};p_{3},p_{4},q)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(54) $\displaystyle\sim$
$\displaystyle-\bigg{|}if^{abe}(T^{e}T^{c})_{ji}\frac{p_{1}^{\mu}}{p_{1}\cdot
q}-if^{abe}(T^{c}T^{e})_{ji}\frac{p_{3}^{\mu}}{p_{3}\cdot q}$
$\displaystyle-f^{adc}f^{dbe}T^{e}_{ji}\frac{p_{2}^{\mu}}{p_{2}\cdot
q}-f^{bdc}f^{ade}T^{e}_{ji}\frac{p_{4}^{\mu}}{p_{4}\cdot q}\bigg{|}^{2}$
$\displaystyle=$
$\displaystyle\bigg{[}-\frac{N_{c}^{2}-1}{2N_{c}}\bigg{(}\frac{m^{2}}{(p_{1}\cdot
q)^{2}}+\frac{m^{2}}{(p_{3}\cdot q)^{2}}\bigg{)}$
$\displaystyle-\frac{1}{N_{c}}\frac{p_{1}\cdot p_{3}}{(p_{1}\cdot
q)(p_{3}\cdot q)}+\frac{N_{c}}{2}\bigg{(}\frac{p_{2}\cdot p_{4}}{(p_{2}\cdot
q)(p_{4}\cdot q)}$ $\displaystyle+\frac{p_{1}\cdot p_{2}}{(p_{1}\cdot
q)(p_{2}\cdot q)}+\frac{p_{3}\cdot p_{4}}{(p_{3}\cdot q)(p_{4}\cdot
q)}+\frac{p_{1}\cdot p_{4}}{(p_{1}\cdot q)(p_{4}\cdot q)}$
$\displaystyle+\frac{p_{2}\cdot p_{3}}{(p_{2}\cdot q)(p_{3}\cdot
q)}\bigg{)}\bigg{]}\times N_{c}{\rm Tr}[T^{e}T^{e}],$

for which the followings are useful:

$\displaystyle f^{ade}f^{bef}f^{cfd}$ $\displaystyle=$
$\displaystyle\frac{N_{c}}{2}f^{abc},$ $\displaystyle f^{abc}T^{b}T^{c}$
$\displaystyle=$ $\displaystyle\frac{i}{2}N_{c}T^{a},$ $\displaystyle
T^{b}T^{a}T^{b}$ $\displaystyle=$ $\displaystyle-\frac{1}{2N_{c}}T^{a}.$ (55)

Since the color structure for $q+g\rightarrow q+g$ is given by

$\displaystyle|M_{q+g\rightarrow
q+g}^{jb;ia}|^{2}\sim|if^{abe}T_{ji}^{e}|^{2}=N_{c}{\rm Tr}[T^{e}T^{e}]$ (56)

from $f^{abd}f^{abe}=N_{c}\delta_{de}$, the scattering amplitude squared for
$q+g\rightarrow q+g+g$ turns to

$\displaystyle|M_{q+g\rightarrow
q+g+g}|^{2}=-g^{2}\bigg{[}\frac{N_{c}^{2}-1}{2N_{c}}\bigg{(}\frac{m_{1}^{2}}{(p_{1}\cdot
q)^{2}}+\frac{m_{3}^{2}}{(p_{3}\cdot q)^{2}}\bigg{)}$
$\displaystyle+\frac{1}{N_{c}}\frac{p_{1}\cdot p_{3}}{(p_{1}\cdot
q)(p_{3}\cdot q)}-\frac{N_{c}}{2}\bigg{(}\frac{2p_{2}\cdot p_{4}}{(p_{2}\cdot
q)(p_{4}\cdot q)}$ $\displaystyle+\frac{p_{1}\cdot p_{2}}{(p_{1}\cdot
q)(p_{2}\cdot q)}+\frac{p_{3}\cdot p_{4}}{(p_{3}\cdot q)(p_{4}\cdot
q)}+\frac{p_{1}\cdot p_{4}}{(p_{1}\cdot q)(p_{4}\cdot q)}$
$\displaystyle+\frac{p_{2}\cdot p_{3}}{(p_{2}\cdot q)(p_{3}\cdot
q)}\bigg{)}\bigg{]}\times|M_{q+g\rightarrow q+g}|^{2}.$ (57)

## Appendix C

Here we recall main properties and parameters of the DQPM. We fix the strength
of the quasiparticle interaction by the adjusting the coupling constant at
$\mu_{B}=0$ so that quasiparticle entropy density reproduces the entropy
density $s(T,\mu_{\mathrm{B}}=0)$ from the lattice QCD calculations provided
by the BMW collaboration Borsanyi  _et al._ (2012, 2014) in the following way
Berrehrah  _et al._ (2016):

$g^{2}(T,\mu_{\mathrm{B}}=0)=d\cdot\Big{[}\left(s(T,0)/s^{\mathrm{QCD}}_{\mathrm{SB}}\right)^{e}-1\Big{]}^{f},$
(58)

where $s_{\mathrm{SB}}^{\mathrm{QCD}}/T^{3}=19\pi^{2}/9$ is the Stefan-
Boltzmann limit of entropy density for massless quarks and gluons and the
dimensionless parameters $d=169.934$, $e=-0.178434$ and $f=1.14631$.

At finite $\mu_{\mathrm{B}}$ the $g^{2}$ is obtained by employing the ‘scaling
hypothesis’ introduced in Ref. Cassing (2007). It assumes that $g^{2}$ is a
function of the ratio of the effective temperature

$T^{*}=\sqrt{T^{2}+\mu^{2}_{q}/\pi^{2}}$ (59)

(where the quark chemical potential is defined as
$\mu_{q}=\mu_{u}=\mu_{s}=\mu_{B}/3$ ) and the $\mu_{\mathrm{B}}$-dependent
critical temperature $T_{c}(\mu_{\mathrm{B}})$ defined as in Ref. Berrehrah
_et al._ (2016):

$T_{c}(\mu_{B})=T_{c}(0)\sqrt{1-\alpha\mu_{B}^{2}},$ (60)

where $T_{c}(0)$ is the critical temperature at vanishing chemical potential
($T_{c}(0)\approx 0.158$ GeV) and $\alpha=0.974$ GeV-2. Thus, the DQPM
effective coupling constant $g_{DQPM}^{2}(T,\mu_{\mathrm{B}})$ reads

$\displaystyle
g_{DQPM}^{2}(T,\mu_{\mathrm{B}})\equiv\begin{cases}&\mu_{\mathrm{B}}=0:\
g^{2}(T,\mu_{\mathrm{B}}=0)\\\ &\mu_{\mathrm{B}}>0:\
g^{2}(T_{scale}(T,\mu_{\mathrm{B}}))\end{cases}$ (61)

with $T_{scale}=T^{*}/(T_{c}(\mu_{\mathrm{B}})/T_{c}(0))$.

In the DQPM the quasiparticle pole masses are adopted in the form of
asymptotic quark or gluon masses respectively $m_{\infty}\sim m_{D}/2$ or
$\sqrt{2}m_{f}$, where $m_{D}$ is the HTL Debye mass, and $m_{f}$ is the HTL
thermal fermion mass Bellac (2011); Linnyk  _et al._ (2016):

$m^{2}_{g}(T,\mu_{\mathrm{B}})=C_{g}\frac{g^{2}(T,\mu_{\mathrm{B}})}{6}T^{2}\left(1+\frac{N_{f}}{2N_{c}}+\frac{1}{2}\frac{\sum_{q}\mu^{2}_{q}}{T^{2}\pi^{2}}\right),$
(62)
$m^{2}_{q(\bar{q})}(T,\mu_{\mathrm{B}})=C_{q}\frac{g^{2}(T,\mu_{\mathrm{B}})}{4}T^{2}\left(1+\frac{\mu^{2}_{q}}{T^{2}\pi^{2}}\right).$
(63)

In Eqs. (62),(63) $N_{c}=3$ and $N_{f}=3$ denote the number of colors and the
number of flavors respectively, $C_{q}=\dfrac{N_{c}^{2}-1}{2N_{c}}=4/3$ and
$C_{g}=N_{c}=3$ are the QCD color factors for quarks and for gluons,
respectively. The strange quark has a larger bare mass which enhances its
dynamical mass. This essentially suppresses the channel $g\rightarrow
s+{\bar{s}}$ relative to the channel $g\rightarrow u+{\bar{u}}$ or
$d+{\bar{d}}$ and controls the strangeness ratio in the QGP. Empirically
$m_{s}(T,\mu_{B})=m_{u}(T,\mu_{B})+\Delta m=m_{d}(T,\mu_{B})+\Delta m$ where
$\Delta m$ =30 MeV has been used Moreau  _et al._ (2019). This model parameter
has been fixed in an empirical way by comparing to experimental data for
strange hadron abundances and the $K^{+}/\pi^{+}$ ratio from heavy-ion
collisions at relativistic energies obtained within – the PHSD approach – a
microscopic covariant transport approach.

Furthermore, thermal widths in the DQPM are adopted in the following form
Berrehrah  _et al._ (2016); Linnyk  _et al._ (2016):

$\gamma_{j}(T,\mu_{\mathrm{B}})=\frac{1}{3}C_{j}\frac{g^{2}(T,\mu_{\mathrm{B}})T}{8\pi}\ln\left(\frac{2c_{m}}{g^{2}(T,\mu_{\mathrm{B}})}+1\right),$
(64)

where the parameter $c_{m}=14.4$ was fixed in Cassing (2007), which is related
to a magnetic cut-off. Furthermore, we assume that all (anti-)quarks have the
same thermal width: $\gamma_{u}=\gamma_{d}=\gamma_{s}$

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Gaël Poux-Médard, Julien Velcin, and Sabine Loudcher 11institutetext:
Université de Lyon, Lyon 2, ERIC UR 3083, Bron 69676, France
11email<EMAIL_ADDRESS>
WWW home page: https://gaelpouxmedard.github.io

# Properties of Reddit News Topical Interactions

Gaël Poux-Médard 11 Julien Velcin 11 Sabine Loudcher 11

###### Abstract

Most models of information diffusion online rely on the assumption that pieces
of information spread independently from each other. However, several works
pointed out the necessity of investigating the role of interactions in real-
world processes, and highlighted possible difficulties in doing so:
interactions are sparse and brief. As an answer, recent advances developed
models to account for interactions in underlying publication dynamics. In this
article, we propose to extend and apply one such model to determine whether
interactions between news headlines on Reddit play a significant role in their
underlying publication mechanisms. After conducting an in-depth case study on
100,000 news headline from 2019, we retrieve state-of-the-art conclusions
about interactions and conclude that they play a minor role in this dataset.

###### keywords:

Dirichlet Hawkes process, Topic modeling, Interaction, Information spread

##  1 Introduction

As the volume of data available on the Internet keeps on growing
exponentially, so does the need for efficient data-processing tools and
methods. In particular, the user-generated content produced on online social
platforms provides a detailed snapshot of the world population’s thoughts and
interests. This kind of data can be used for many different applications (e.g.
in marketing, opinion mining, fake news control, summary generation, etc.).
However, we need a fine understanding of the underlying data-generation
mechanisms at stake to refine the results of these possible applications.

Figure 1: An output of the proposed approach – (Top) A set of inferred topics
along with the vocabulary of their documents. (Bottom) Instantaneous
probability for one observation of a topic to trigger other observations in
other topics.

Early works on information spread consider that a user publishing a piece of
information (or meme) did so due to an earlier exposure to this same meme
[11]. Users are represented as a network on which edges pieces of information
flow independently from each other. This model has seen several refinements,
that consider additional information about nodes and edges to model the way
information spreads. Nodes participate the spread when exposed a certain
number of times [15, 17], have a different influence on their neighbours
depending on their position in the network [12], can publish exogenous memes
(e.g. that were not flowing on the network beforehand) [16, 10], etc.. Edges
between nodes represent the likeliness of a meme flowing from one node to the
other; later works consider edges whose intensity depends on the meme
considered [6, 1], on the temporal distance to the last exposure [8, 9], etc.

However, a core assumption of most of these models and extensions is that
memes spread independently from each other. It has been highlighted in some
occasions that modeling the interaction between pieces of information might
bring interesting insights in the way data gets generated online [3, 15, 7].
For instance, we expect memes about two politically opposed candidates to have
some interaction (the publication of one would inhibit the publication chances
of the other one); memes raising climate change awareness might be more likely
to spread when coexisting with memes about ecologic disasters, etc.

In this work, we conduct an in-depth study of such interaction mechanisms on a
large-scale Reddit dataset –gathered from 9 news subreddits over 2019. We
propose to use a model that answers the challenges of interaction modeling
raised in recent works on interactions –interactions between memes are likely
sparse and brief. This model, the Multivariate Powered Dirichlet-Hawkes
Process (MPDHP), groups textual memes into topics. A possible output is
represented in Fig. 1. Topics interact in pairs to yield a probability for a
new meme about a given topic to get published, in a continuous-time setting.
The output of MPDHP is a time-dependent topical interaction network. Once this
network has been inferred, we can analyse the properties of topical
interactions. We conduct experiments using several iterations of MPDHP
(accounting for different timescales and topical modeling), so that we get an
exhaustive panorama of interactions at stake. Overall, our conclusions hint
that interactions play a minor role on news subreddits; most information
indeed spreads independently in this context. From a broader perspective, we
introduce a methodology to assess the role of temporal topical interactions
textual datasets.

##  2 Background

####  2.0.1 Modeling interactions

In this section, we briefly review previous works on interaction modeling in
spreading processes. To the best of our knowledge, the first work tackling the
problem proposes a SIR-based model that comprises an interaction term $\beta$
between two co-existing viruses –in their case, the adoption of either Firefox
or Chrome web browser [3]. Interaction is modeled as a hyper-parameter, that
has to be tuned manually to retrieve global shares of each virus over time.
Building on this work, [15] later proposed to model interaction between memes
at the agent level: given a user is exposed to tweet A at time $t_{A}$, what
is the probability to retweet tweet B at a time $t_{B}$ later. This model
attempts to learn the interaction parameter $\beta$ from the data instead of
fine-tuning it. Their conclusion is that interactions can have a large overall
effect of the data-generation processes (here retweeting mechanisms), but that
most interactions have little influence. Later works tackling the problem from
a similar perspective on several real-world datasets find similar results:
significant interactions between clusters of memes are sparse [19, 21]. This
highlight the need to cluster memes so that it becomes possible to retrieve
meaningful interaction terms.

Some other works tackled the problem of pair-interaction modeling from a
temporal perspective. Simple considerations show that interactions cannot
remain constant over time; a user that read a meme 5min earlier and another
meme 5 days earlier is likely to be much more influenced the first one than
the second one. Interactions are likely to be brief. This temporal dependence
is explicitly modeled in [18], where the interaction strength is shown to
typically decay exponentially, which correlated the findings of [4], and of
[15] to a certain extent. It highlights the need to consider time to
relevantly model interactions. Thus the need for a model able to handle large
piles of data, performing topic inference, allowing interactions between these
topics, and accounting for time.

####  2.0.2 Dirichlet-Hawkes processes

The Dirichlet-Hawkes process [5] seem to qualify for these requirements. In
particular, it has been extended as the Powered Dirichlet-Hawkes process [20]
to allows for extended modeling flexibility –to which extend should we favor
the textual information over the temporal information. However, both these
models do not account for pair-interaction between the inferred topics.
Instead, topics can only trigger new observations from themselves. In this
work, we consider a Multivariate extension of the Powered Dirichlet-Hawkes
process (MPDHP). This extension boils down to substituting the Hawkes process
described in [5, 20] by a multivariate Hawkes process [13], so that topics can
influence the probability of new observations to belong to either other
topics.

The principle of the Dirichlet-Hawkes approaches relies on Bayesian inference.
An online language model accounts for the textual content of documents (Fig.
1-top). This model, typically a Dirichlet-Multinomial bag of words as we will
use later, expresses the likelihood for a new document to belong to any
existing topic. This language model is coupled to a Dirichlet-Hawkes prior,
that can be expressed as a sequential process. The prior assigns to a new
observation a prior probability to belong to either topic based on its
temporal interaction with earlier publications (Fig. 1-middle). In the
remaining of this section, we present the modified formulation of the base
models [5, 20] so that it accounts for topical pair-interactions.

We first rewrite and detail the expression of the Dirichlet-Hawkes process as
introduced in [5, 20]:

$\begin{split}P&(C_{i}=c|n_{i},N_{<i,c},t_{i},\lambda(t_{i}),\mathcal{H},\theta_{0},\lambda_{0},r)\\\
&\propto\underbrace{P(n_{i}|C_{i}=c,N_{<i,c},\theta_{0})}_{\begin{subarray}{c}\text{Textual
likelihood}\\\ \text{(Dirichlet-
Multinomial)})\end{subarray}}\underbrace{P(C_{i}=c|t_{i},\lambda(t_{i}),\mathcal{H},\lambda_{0},r)}_{\begin{subarray}{c}\text{Temporal
prior}\\\ \text{(MPDHP)}\end{subarray}}\\\
&=\frac{\Gamma(N_{c}+\theta_{0})}{\Gamma(N_{c}+n_{i}+\theta_{0})}\prod_{v}\frac{\Gamma(N_{c,v}+n_{i,v}+\theta_{0,v})}{\Gamma(N_{c,v}+\theta_{0})}\begin{cases}\frac{\lambda_{c}^{r}(t_{i})}{\lambda_{0}+\sum_{c^{\prime}}\lambda_{c^{\prime}}^{r}(t_{i})}\text{
if c$\leq$K}\\\
\frac{\lambda_{0}}{\lambda_{0}+\sum_{c^{\prime}}\lambda_{c^{\prime}}^{r}(t_{i})}\text{
if c=K+1}\end{cases}\end{split}$ (1)

where $C_{i}$ represents the cluster chosen by the $i^{\text{th}}$ document,
$c$ is the random variable accounting for this allocation, $n_{i}$ is a vector
that whose $v^{\text{th}}$ entry represents the count of word $v$ in document
$i$, $N_{<i,c}$ is the vector whose entry $v$ represents the total count of
word $v$ in all documents up to $i$ that belong to cluster $c$, $t_{i}$ the
arrival time of document $i$, $\lambda(t_{i})$ the vector of intensity
functions at time $t_{i}$ whose $c^{\text{th}}$ entry corresponds to cluster
$c$, and $\mathcal{H}$ the history of all previous documents that appeared
before time $t_{i}$. The three last symbols are hyperparameters: $\theta_{0}$
is the concentration parameter of the Dirichlet-Multinomial language model [5,
22], $\lambda_{0}$ is the concentration parameter of the MPDHP temporal prior
[5], and $r$ controls the extent to which we rely of either textual of
temporal information [20]. Data is processed sequentially using a Sequential
Monte Carlo (SMC) algorithm similar to [5, 20, 14]. For each new observation,
we get from Eq. 1 the posterior probability that it belongs to either of $K$
existing clusters, or to start a cluster of its own ($K+1$). The SMC algorithm
accounts for several allocation hypotheses at once, and discard the most
unlikely ones every other iteration.

Now, the extension of the Dirichlet-Hawkes process to the multivariate case
boils down to giving a new definition for the vector $\lambda(t)$. We express
it as:

$\lambda_{c}(t)=\sum_{t_{i}^{c^{\prime}}<t}\vec{\alpha}^{T}_{c,c^{\prime}}\cdot\vec{\kappa}(t-t_{i}^{c^{\prime}})\,\,\,\,\,\,\,\text{where}\,\,\,\,\,\,\,\kappa_{l}(\Delta
t)=\frac{1}{\sqrt{2\pi\sigma_{l}^{2}}}e^{-\frac{(\Delta
t-\mu_{l})^{2}}{2\sigma_{l}^{2}}}$ (2)

where $\vec{\alpha}_{c,c^{\prime}}$ is a vector of $L$ parameters to infer,
and $\vec{\kappa}(t-t_{i}^{c^{\prime}})$ is a vector of $L$ given temporal
kernel functions depending only on the time difference between two events. We
consider $\vec{\kappa}(\Delta t)$ to be a Gaussian RBF kernel with fixed mean
$\vec{\mu}$ and deviation $\vec{\sigma}$, which allows us to model a range of
different intensity functions. Each parameter $\alpha_{c,c^{\prime},l}$
accounts for the influence of $c^{\prime}$ on $c$ according to the $l^{th}$
entry of the temporal kernel. The dot product of $\alpha$ with $\kappa$ yields
the intensity function vector $\lambda$, which represents the topical
interactions’ temporal adjacency matrix (represented in an alternative way in
Fig. 1-middle).

##  3 Experimental setup

Figure 2: Characteristics of the News dataset — For $\sim$100,000 headlines
and $\sim$13,000 different words: (Top-Left) Distribution of the words count.
(Top-Right) Distribution of headlines popularity. (Bottom-Left) Distribution
of headlines over subreddits. (Bottom-Right) Distribution of headlines over
time.

####  3.0.1 Dataset

The dataset used in this study has been gathered from the Pushshift Reddit
repository [2], which contains archives of the entirety of Reddit posts and
comments up to June 2021. For each Reddit post, we can retrieve the subreddit
it came from, the title of the publication, its publication date and its score
(number of upvotes minus number of downvotes).

We choose to only consider popular English news subreddits. Namely, we select
only posts from 2019 published on the following subreddits: inthenews,
neutralnews, news, nottheonion, offbeat, open news, qualitynews, truenews,
worldnews. This leaves us with a corpus of 867,328 headlines, which makes a
total of 1,111,955 words drawn from a vocabulary of size 36,284.

Finally, we discard uninformative words and documents from the dataset.
Explicitly, we remove the stopwords, punctuation signs, web addresses, words
whose length is lesser than 4 characters, and words that appear less than 3
times in the whole dataset. Then, we remove publications that carry lesser
textual or temporal information. Firstly, we choose not to consider the
publications that have a popularity lesser than 20 – meaning that they
received less than 20 positive votes more than negative votes. We make this
choice so that we consider publications that are visible enough to have any
influence on the data generation process. Secondly, we remove publications
that comprise less than 3 words. The semantic information so-carried is
expected to be poor and is not considered in our analysis.

After curating the dataset in the way described above, we are left with
102,045 news headlines (one-eighth of the original data), which makes a total
of 875,334 tokens (named entities, verbs, numbers, etc.) drawn from a
vocabulary of size 13,241 (one-third of the original vocabulary). The
characteristics of this dataset are shown in Fig. 2.

####  3.0.2 Temporal kernel

We run our experiments using three different RBF kernels, which account for
publication dynamics at three different timescales: minute, hour, and day. The
“Minute” RBF kernel is made of Gaussian functions centered at the following
times: $\left[0,10,20,30,40,05,60,70,80\right]$ minutes; each entry shares a
same standard deviation $\sigma$ of 5 minutes, and $\lambda_{0}=0.01$. The
“Hour” RBF kernel has Gaussians centered around $\left[0,2,4,6,8\right]$
hours, with a standard deviation $\sigma$ of 1 hour, and $\lambda_{0}=0.001$.
The “Day” RBF kernel is centered around $\left[0,1,2,3,4,5,6\right]$ days,
with a standard deviation $\sigma$ of 0.5 days, and $\lambda_{0}=0.0001$. For
each of these kernels, we set the concentration parameter $\lambda_{0}$ so
that it reaches roughly the value of one Gaussian function evaluated at
$2\sigma$. It means that an event which is $2\sigma$ away from the center of
the Gaussian kernel of a single observation has 50% chances of getting
associated with this Gaussian kernel entry, and 50% chances of opening a new
cluster.

####  3.0.3 Hyper-parameters

We consider two values for the concentration parameter of the language model:
$\theta_{0}=0.001$ and $\theta_{0}=0.01$. The choice of this range is standard
in the literature [5] and supported by our own observations. A larger value of
$\theta_{0}$ makes the inferred clusters cover a broader range of document
types, whereas a small value makes the inferred clusters more specific to a
topic.

The value of $r$ is chosen to be either $0$ (no use of the temporal
information), $0.5$, $1$ and $1.5$. The larger $r$, the more the inference
relies of the temporal dynamics instead of the textual content of the
documents.

The SMC algorithm described in [5, 20] is run using 8 particles and 100,000
samples used to estimate the matrix of parameters $\alpha$.

##  4 Results

###  4.1 Overview of the experiments

Table 1: Results for each experiment — We characterize each run in terms of clusters (number, size, textual entropy, subreddits entropy). The standard deviation on the last digits is given between parenthesis. $\vec{\kappa}(t)$ |  $\theta_{0}$ |  $r$ |  $K$ |  $<N>$ |  $S_{text}^{(20)}$ |  $S_{sub}^{(20)}$
---|---|---|---|---|---|---
Minute |  0.01 |  0.0 |  16150 |  6 |  0.744(68) |  0.400(36)
0.5 |  8498 |  12 |  0.796(55) |  0.441(24)
1.0 |  5069 |  20 |  0.790(49) |  0.476(53)
1.5 |  2730 |  37 |  0.808(43) |  0.490(52)
2-7 |  0.001 |  0.0 |  46304 |  2 |  0.475(48) |  0.239(56)
|  0.5 |  37277 |  3 |  0.485(40) |  0.256(60)
|  1.0 |  26858 |  4 |  0.501(37) |  0.266(72)
|  1.5 |  19275 |  5 |  0.506(39) |  0.280(58)
Hour |  0.01 |  0.0 |  3792 |  27 |  0.798(52) |  0.474(106)
0.5 |  1735 |  59 |  0.791(45) |  0.469(77)
1.0 |  825 |  124 |  0.803(47) |  0.484(75)
1.5 |  426 |  240 |  0.795(37) |  0.489(69)
2-7 |  0.001 |  0.0 |  18012 |  6 |  0.760(86) |  0.397(63)
|  0.5 |  11923 |  9 |  0.784(72) |  0.425(38)
|  1.0 |  4837 |  21 |  0.821(50) |  0.497(30)
|  1.5 |  2368 |  43 |  0.814(42) |  0.481(91)
Day |  0.01 |  0.0 |  609 |  168 |  0.713(34) |  0.413(103)
0.5 |  326 |  313 |  0.728(33) |  0.429(98)
1.0 |  172 |  593 |  0.743(31) |  0.461(94)
1.5 |  96 |  1063 |  0.755(36) |  0.464(86)
2-7 |  0.001 |  0.0 |  4349 |  23 |  0.705(49) |  0.396(103)
|  0.5 |  2654 |  38 |  0.721(59) |  0.404(104)
|  1.0 |  1399 |  73 |  0.734(57) |  0.431(106)
|  1.5 |  764 |  134 |  0.741(54) |  0.442(98)

In Table 1, we represent the main characteristics of each run in terms of
number of inferred clusters $K$, the average cluster population $<N>$ (where
$<\cdot>$ denotes the average), the average normalized entropy of the
vocabulary of the top 20 clusters $S_{text}^{(20)}$, the average normalized
entropy of the subreddits partition of the most populated 20 clusters
$S_{sub}^{(20)}$. The normalized entropy is bounded between 0 and 1. It is
defined so that a low entropy $S_{text}^{(20)}$ (resp. $S_{sub}^{(20)}$) means
that the top 20 clusters contain documents that are concentrated around a
reduced set of words (resp. of subreddits); conversely, a large entropy means
that these top 20 clusters do not account for documents concentrated around a
specific vocabulary (resp. set of subreddits). We can make several
observations from Table 1:

  * •

The number of inferred clusters decreases with $r$, and their average
population increases.

  * •

The number of clusters grows large for the “Minute” kernel. This is because
the short time range considered does not allow for clusters to last in time. A
cluster that does not replicate within 1h30 is forgotten.

  * •

We recover the fact that textual clusters have a lower entropy for small
values of $r$ [20]; this is because their creation is based more on textual
coherence than on temporal coherence.

  * •

The subreddit entropy seems to increase as $r$ grows. A possible
interpretation is that favouring the temporal information for cluster creation
results in larger clusters (see $<N>$). They would be too large to account for
subreddit-specific dynamics. However, the entropy remains lower than the
entropy of the distribution Fig. 2-top-left, equal to 0.51.

###  4.2 Quantifying interactions

####  4.2.1 Effective interaction

We introduce the parameters we are going to use in follow-up analyses. The
output of MPDHP consists of a list of clusters comprising timestamped bags of
words –news headlines. Between each pair of clusters, MPDHP inferred a
temporal influence function $\lambda(t)$, that represents the probability for
one cluster to trigger publications from another. Therefore, our model yields
an adjacency matrix $A\in\mathbb{R}^{K\times K\times L}$, where $K$ is the
number of clusters and $L$ the size of the RBF kernel $\vec{\kappa}(t)$. One
entry $a_{i,j,l}$ represents the strength of the influence of $j$ in $i$ due
to the $l^{th}$ entry of $\vec{\kappa}(t)$.

However, we must consider the effective number of interactions, that is to
which extent the intensity function has effectively had a role in the
inference. To do so, we simply consider a weight matrix
$W\in\mathbb{R}^{K\times K\times L}$, whose entries $w_{i,j,l}$ are the
average of the intensity of $i$ above $\lambda_{0}$ due to $j$ from the kernel
entry $l$ for all observations. Explicitly:

$w_{i,j,l}=\frac{1}{|\mathcal{H}_{i}|}\sum_{t_{i}\in\mathcal{H}_{i}}\sum_{t_{j}<t_{i}}max(a_{i,j,l}\kappa_{l}(t_{i}-t_{j})-\lambda_{0},0)$
(3)

The notations are the same as in Eq. 1 and Eq. 2. Note that we retract
$\lambda_{0}$ from the intensity term, because it is considered as a
background probability for a publication to happen. Therefore, $W$ can also be
interpreted as the instantaneous increase in probability due to interactions.

####  4.2.2 Interactions strength

In Table 2, we investigate the effective impact of interactions in the
dataset. We consider the following metrics:

  * •

$<A>$: the average value of the whole adjacency matrix; to which extent topics
interact with each other according to MPDHP.

  * •

$<W>$: the average value of the effective interactions; the extent to which
the interactions (encoded in $A$) effectively happen in the dataset.

  * •

$<A>_{W}$: the average of the inferred interaction matrix $A$ weighted by the
effective interactions $W$. In this case, $W$ can be interpreted as our
confidence in the corresponding entries of $A$.

  * •

$\frac{<W^{intra}>}{<W^{extra}>}$: ratio of the intra-cluster effective
interactions with the extra-cluster effective interactions; how much clusters
self-interact versus how much they interact with other ones.

Table 2: Interaction strength — Overall, interaction between clusters is weak. The standard deviation on the last digits are given between parenthesis. The large standard deviations suggest that there is a large variety of interacting behaviours. Interactions tend to happen within a cluster (self-interactions).  $\vec{\kappa}(t)$ | $\theta_{0}$ | $r$ |  $<A>$ ($10^{-3}$) |  $<W>$ ($10^{-5}$) |  $<A>_{W}$ ($10^{-3}$) |  $\frac{<W^{intra}>}{<W^{extra}>}$
---|---|---|---|---|---|---
Minute |  0.01 | 0.5 |  49(21) |  342(889) |  66(17) |  1.8(62)
1.0 |  48(20) |  478(1124) |  60(17) |  1.4(43)
1.5 |  48(20) |  746(1901) |  60(17) |  1.0(33)
2-7 |  0.001 | 0.5 |  50(22) |  316(882) |  66(17) |  3.1(138)
| 1.0 |  50(21) |  279(752) |  67(16) |  2.6(105)
| 1.5 |  50(22) |  268(665) |  67(16) |  2.3(84)
Hour |  0.01 | 0.5 |  49(18) |  389(843) |  56(17) |  0.5(13)
1.0 |  49(18) |  478(1187) |  56(17) |  0.6(15)
1.5 |  48(17) |  471(789) |  52(15) |  0.7(13)
2-7 |  0.001 | 0.5 |  50(21) |  110(398) |  61(17) |  1.7(67)
| 1.0 |  50(18) |  133(506) |  57(17) |  1.4(60)
| 1.5 |  49(17) |  183(554) |  55(17) |  1.1(37)
Day |  0.01 | 0.5 |  49(18) |  41(97) |  55(17) |  1.2(34)
1.0 |  49(19) |  63(131) |  54(17) |  1.2(31)
1.5 |  49(19) |  91(187) |  53(18) |  1.2(31)
2-7 |  0.001 | 0.5 |  50(20) |  18(90) |  60(19) |  1.1(59)
| 1.0 |  50(19) |  23(101) |  58(17) |  1.0(50)
| 1.5 |  50(19) |  37(111) |  56(18) |  1.0(36)
Figure 3: Distribution of interaction strength – A large effective interaction
strength means that the publication of a new document can be significantly
explained by the presence of previously published news. We see that most
interactions are weak.

The main conclusion of the results Table 2 is that most interactions are weak.
The average value of $A$ tells us that the average value of the inferred
parameters is around $0.05$, which is few given entries of $A$ are bounded
between 0 and 1. The metric ${<W>}$ tells us that on all events, the
interaction between clusters rose the probability of publication by 0.1%-1% on
average. We can also note that the values of ${<W>}$ are of the same order of
magnitude as $\lambda_{0}$ (0.01 for the “Minute” kernel, 0.001 for “Hour”,
and 0.0001 for “Day”). We can interpret this as the probability for a new
document belonging to a cluster or being from a new cluster is roughly the
same from a temporal perspective. The metric $<A>_{W}$ tells us that when
weighting the average of $A$ with the effective interaction, the values of $A$
are slightly higher than $0.05$; we can now be confident in this value, given
it has been inferred on a statistically significant number of observations.
Still, only some interactions seem to be significant, which correlates with
[15, 19]. Finally, the last metric $\frac{<W^{intra}>}{<W^{extra}>}$ finds
that most effective interactions take place more within the same cluster;
clusters tend to self-replicate. Further studies of this phenomena would
involve an extension of the MPDHP that considers Nested Dirichlet Processes
instead of Powered Dirichlet Processes. Clusters would then be broken down
into smaller ones, whose interactions could be analyzed.

Another major observation from Table 2 is that standard deviations of
effective interactions are large: it hints that some of interactions may play
a more significant role in the dataset. In Fig. 3, we plot the distribution of
effective interactions for one specific run (“Hour” kernel, $\theta_{0}=0.01$,
$r=1$; we recover the same trend in all other experiments). The results of
this figure are similar to the ones of previous studies [15, 19].

####  4.2.3 Interactions range

Finally, in Table 3, we investigate the range of effective interactions. We
compute the effective interaction for each entry of the temporal kernel
individually and average it over all existing clusters. Importantly, the
effective interaction of $\kappa_{1}$ is smaller than others. This is induced
by our kernel choice: because $\kappa_{1}$ is centered around $t=0$, half of
the associated Gaussian function accounts negative time differences, which
never happens by design. Therefore, only other kernels entries can contribute
on both sides of their mean.

Table 3: Interaction range — All the values for effective interaction are given in ten-thousandth ($10^{-5}$). Influence tends to decrease over time for all kernels. $\vec{\kappa}(t)$ | $\theta_{0}$ | $r$ |  $\kappa_{1}$ |  $\kappa_{2}$ |  $\kappa_{3}$ |  $\kappa_{4}$ |  $\kappa_{5}$ |  $\kappa_{6}$ |  $\kappa_{7}$ |  $\kappa_{8}$ |  $\kappa_{9}$
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  0m |  10m |  20m |  30m |  40m |  50m |  60m |  70m |  80m
Minute (m) |  0.01 | 0.5 |  133 |  421 |  407 |  451 |  428 |  403 |  395 |  345 |  121
1 |  198 |  591 |  580 |  607 |  532 |  575 |  521 |  507 |  224
1.5 |  308 |  937 |  893 |  914 |  955 |  840 |  808 |  810 |  304
2-12 |  0.001 | 0.5 |  218 |  509 |  457 |  424 |  371 |  340 |  313 |  178 |  52
| 1 |  142 |  435 |  396 |  388 |  343 |  327 |  272 |  187 |  45
| 1.5 |  104 |  388 |  366 |  353 |  326 |  333 |  290 |  215 |  61
|  |  |  0h |  2h |  4h |  6h |  8h |  - |  - |  - |  -
Hour (h) |  0.01 | 0.5 |  247 |  430 |  502 |  456 |  324 |  |  |  |
1 |  329 |  538 |  549 |  542 |  451 |  |  |  |
1.5 |  229 |  615 |  532 |  526 |  411 |  |  |  |
2-8 |  0.001 | 0.5 |  62 |  149 |  119 |  137 |  92 |  |  |  |
| 1 |  77 |  164 |  172 |  149 |  111 |  |  |  |
| 1.5 |  104 |  244 |  223 |  197 |  156 |  |  |  |
|  |  |  0d |  1d |  2d |  3d |  4d |  5d |  6d |  - |  -
Day (d) |  0.01 | 0.5 |  22 |  45 |  47 |  46 |  47 |  47 |  37 |  |
1 |  35 |  71 |  72 |  68 |  68 |  70 |  61 |  |
1.5 |  51 |  100 |  101 |  105 |  98 |  105 |  82 |  |
2-10 |  0.001 | 0.5 |  9 |  20 |  21 |  21 |  21 |  21 |  17 |  |
| 1 |  12 |  26 |  24 |  27 |  28 |  26 |  22 |  |
| 1.5 |  11 |  41 |  42 |  41 |  41 |  41 |  35 |  |

We see in Table 2 that influence tends to decrease over time for all the
kernels considered. Overall, the interaction between documents still plays a
marginal role. We did not plot the standard deviation for visualization
purposes, but they are similar as in Table 2; most interactions do not play a
significant role in the publication of subsequent documents over time.
Overall, the increase in probability for a new document to belong to a cluster
due to interactions is within 0.1%-1%.

##  5 Conclusion

In this work, we conducted extensive experiments on a real-world large-scale
dataset from Reddit. We conducted 24 different experiments, each accounting
for a given combination of parameters, that determine the timescale considered
($\kappa(\Delta t)$), the sparsity of the language modeling ($\theta_{0}$),
and the extent on which we rely on text or time during the inference ($r$).

Our experiments hint that interactions do not play a significant role in this
dataset. We proposed several ways to assess the role of interactions in the
dataset. In particular, we introduced the notion of effective interaction as a
way to evaluate how confident we are in MPDHP’s output. On this basis, we
analysed the importance of interactions in general, as well as from a temporal
perspective. We recovered the conclusions of prior works: interactions are
sparse and decay over time. By looking at the global effective interaction
average, we conclude that interactions play a minor role this dataset.
Overall, they only increase the instantaneous probability for a new
observation to appear by 1%. Even the most extreme interactions seem to only
increase this probability by 12% top.

However, despite intending our study as exhaustive, there is room for
improvement in interaction modelling using MPDHP. In particular, there are two
biases that we could not explore in this work. Firstly, the parameter
$\lambda_{0}$ has been set according to a heuristic (so that a new cluster is
opened with fifty percent chances when we are 95% sure that it does not match
the existing one). Its direct inference would robustify the approach.
Nevertheless, this inclusion sounds challenging: $\lambda_{0}$ does not
account for individual events realizations, but for Hawkes processes starts,
whose inference is not a trivial extension. Another improvement would be to
allow clusters to passively replicate –without the need for an interaction. We
expect that this would boil down to adding a time-independent kernel entry to
$\vec{\kappa}$. However, other questions may arise from such modification:
when to consider a cluster as extinct given a non-fading kernel? How should
this kernel relate to the temporal concentration parameter $\lambda_{0}$? We
believe such improvements would make MPDHP more robust and interpretable, and
find applications beyond interaction modelling.

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