# Accretion disks properties around regular black hole solutions obtained from
non-linear electrodynamics

Yergali Kurmanov<EMAIL_ADDRESS>National Nanotechnology Laboratory
of Open Type, Almaty 050040, Kazakhstan.  Al-Farabi Kazakh National
University, Al-Farabi av. 71, 050040 Almaty, Kazakhstan.  International
Engineering Technological University, Al-Farabi av. 93G/5, 050060 Almaty,
Kazakhstan.  Kuantay Boshkayev<EMAIL_ADDRESS>National Nanotechnology
Laboratory of Open Type, Almaty 050040, Kazakhstan.  Al-Farabi Kazakh National
University, Al-Farabi av. 71, 050040 Almaty, Kazakhstan.  Institute of Nuclear
Physics, Ibragimova, 1, 050032 Almaty, Kazakhstan.  Talgar Konysbayev
<EMAIL_ADDRESS>National Nanotechnology Laboratory of Open Type, Almaty
050040, Kazakhstan.  Al-Farabi Kazakh National University, Al-Farabi av. 71,
050040 Almaty, Kazakhstan.  Orlando Luongo<EMAIL_ADDRESS>Al-Farabi
Kazakh National University, Al-Farabi av. 71, 050040 Almaty, Kazakhstan.
Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino, Italy.
SUNY Polytechnic Institute, 13502 Utica, New York, USA.  Istituto Nazionale di
Fisica Nucleare, Sezione di Perugia, 06123, Perugia, Italy.  INAF -
Osservatorio Astronomico di Brera, Milano, Italy.  Nazym Saiyp
<EMAIL_ADDRESS>Al-Farabi Kazakh National University, Al-Farabi av. 71,
050040 Almaty, Kazakhstan.  Ainur Urazalina<EMAIL_ADDRESS>National
Nanotechnology Laboratory of Open Type, Almaty 050040, Kazakhstan.  Al-Farabi
Kazakh National University, Al-Farabi av. 71, 050040 Almaty, Kazakhstan.
Institute of Nuclear Physics, Ibragimova, 1, 050032 Almaty, Kazakhstan.
Gulfeiruz Ikhsan<EMAIL_ADDRESS>National Nanotechnology Laboratory
of Open Type, Almaty 050040, Kazakhstan.  Gulnara Suliyeva
<EMAIL_ADDRESS>National Nanotechnology Laboratory of Open Type,
Almaty 050040, Kazakhstan.  Al-Farabi Kazakh National University, Al-Farabi
av. 71, 050040 Almaty, Kazakhstan.  Fesenkov Astrophysical Institute,
Observatory 23, 050020 Almaty, Kazakhstan.

###### Abstract

We investigate a family of spherically symmetric, static, charged regular
black hole solutions derived within the framework of Einstein-nonlinear
electrodynamics. Our study focuses on examining the characteristics of
accretion disks in the spacetimes described by the Dymnikova and Fan-Wang
solutions. We explore circular geodesics of test particles and calculate
various properties, including the radius of the innermost stable circular
orbit, radiant energy, temperature, and conversion efficiency of accretion
mass into radiation. We employ the Novikov-Thorne-Page thin accretion disk
model as a background. By comparing our findings with those obtained in the
Schwarzschild black hole case, we reveal significant modifications in the
overall spectral properties. Specifically, we observe an increase in the
energy emitted from the disk surface, resulting in higher temperatures for the
accretion disks under certain values of the free parameters. Consequently, we
note an enhanced efficiency of mass conversion into radiation compared to the
Schwarzschild spacetime.

Regular black holes, accretion disk luminosity, non-linear electrodynamics

##  I Introduction

Unlike black holes (BHs), non-singular or regular black holes (RBHs) have been
predicted based on specific energy-momentum tensors, introducing additional
effects. Recent findings indicate that the violation of the strong energy
condition (SEC) is a key feature of RBHs. It’s generally observed that in any
static region of the event horizon, SEC is violated, resulting in a negative
Tolman mass Zaslavskii (2010). Conversely, in the non-static case, this
property leads to a violation of the dominant energy condition.

The search for RBHs originated from contributions by Sakharov and Gliner
Sakharov (1966); Gliner (1966), who proposed that introducing a medium with a
de Sitter metric instead of a vacuum could potentially circumvent the
existence of singularities. This idea was further developed by Dymnikova,
Gurevich, and Starobinsky Gliner and Dymnikova (1975); Gurevich (1975);
Starobinskii (1979), who proposed an intricate interplay between gravitational
and electromagnetic fields.

Specifically, non-linear electrodynamics (NLED) can produce fields that
interact with extremely strong gravitational sources, particularly in the
vicinity of massive objects such as BHs. In such conditions, the gravitational
field itself acts as a source of electromagnetic fields, amplifying them and,
in turn, influencing the curvature of spacetime.

As a consequence, exact solutions for RBHs can be constructed if electric or
magnetic charges associated with NLED are non-zero Bronnikov (2001); Fan and
Wang (2016).

In this regard, various approaches have been discussed in the literature. One
prototype involves the utilization of the Born-Infeld Lagrangian, aimed at
eliminating the central singularity of a point charge’s electromagnetic field
as well as its energy divergence Born and Infeld (1933). Following a similar
approach, RBH solutions in NLED include those proposed by Bardeen, Hayward,
Dymnikova, Fan-Wang, among others. Bardeen introduced the first model of RBHs,
now known as the Bardeen BH Bardeen (1968) 111Subsequently, Ayon-Beato-Garcia
and Bronnikov Bronnikov (2001) demonstrated that these solutions arise from
NLED as a source. They proposed a magnetic monopole as a source within the
framework of NLED, leading to the Bardeen BH solution from the Einstein field
equations Ayón-Beato and García (1998, 2000); Bronnikov (2001)..

Interestingly, Hayward constructed a RBH by assuming a particular mass
function Hayward (2006) where vacuum energy avoids the singularity.
Introducing the Lagrangian for NLED to the first order, Bronnikov Bronnikov
(2001), Berej et al. Berej et al. (2006) initially constructed the RBH,
whereas in Ref. Dymnikova (2004), using the concepts put forth by Bronnikov
Bronnikov (2001), Berej et al. Berej et al. (2006), Dymnikova presented an
exact, regular, spherically symmetric, charged BH solution222This solution was
constructed using a NLED theory with a Hamiltonian-like function. Balart and
Vagenas Balart and Vagenas (2014) constructed charged RBHs using Einstein-NLED
theory. They developed the general lapse function for mass distribution
functions based on continuous probability distributions.. Additionally, Fan
and Wang constructed a new class of magnetic solutions with spherical symmetry
Fan and Wang (2016).

Interestingly, Hayward constructed a RBH by assuming a particular mass
function Hayward (2006), where vacuum energy prevents the singularity.
Introducing the Lagrangian for NLED to the first order, Bronnikov and Berej et
al. initially constructed RBHs Bronnikov (2001); Berej et al. (2006). In a
subsequent work, Dymnikova presented an exact, regular, spherically symmetric,
charged BH solution, building upon the concepts proposed by Bronnikov and
Berej et al. Dymnikova (2004). 333This solution was derived using an NLED
theory with a Hamiltonian-like function. Balart and Vagenas also constructed
charged RBHs using Einstein-NLED theory, developing the general lapse function
for mass distribution functions based on continuous probability distributions
Balart and Vagenas (2014).. Additionally, Fan and Wang constructed a new class
of magnetic solutions with spherical symmetry Fan and Wang (2016).

When NLED is connected to General Relativity and complies with the Weak Energy
Condition (WEC), the presence of electrically charged regular structures and a
limitation on self-energy that becomes infinite for a point charge are
ensured. It is important to note that a field need not be weak to be regular;
instead, one should disregard only the requirement of Maxwell’s weak field
limit at the center, which forms the basis of non-existence theorems. It was
demonstrated in Dymnikova (2004) that there are electrically charged
structures with regular centers that exhibit regular geometry, fields, and
stress-energy tensors without a Maxwell limit as $r\to 0$.

Moreover, the study of BHs holds significant importance in modern
astrophysics. A hot accretion disk, typically formed around a BH by gaseous
material Churilova et al. (2021), generates a distinct spectrum of
electromagnetic radiation. The processes occurring within these accretion
disks result in the radiation they produce. Hence, the physics of BHs
captivates researchers in testing theories of gravity in regions of strong
gravitational fields. Analyzing images of BH accretion provides an opportunity
to bridge the gap between theoretical predictions and observational evidence.

This encourages an investigation of the luminosity properties of the accretion
disk surrounded by NLED sources for RBHs. Relevant clues have recently been
carried forward. For example, in Ref. Rezaei Akbarieh et al. (2023), the
authors investigated the non-rotating Bardeen and Hayward BHs. They examined
the physical properties of non-rotating RBHs, differential luminosity,
temperature, and efficiency of converting accretion mass into
radiation444Remarkably, the authors showed that the accretion disk around the
non-rotating Bardeen and Hayward RBHs emits more energy compared to the
Schwarzschild BH. Furthermore, the authors demonstrated that Bardeen and
Hayward RBHs are more efficient than Schwarzschild BHs at converting aggregate
mass into radiation.. Thin accretion disks in the gravitational field of a
class of rotating RBHs were examined by the authors in Ref. Boshkayev et al.
(2024a). Important accretion disk parameters were determined: disk radiation
flux, differential and spectral luminosity, and the radius of the innermost
stable circular orbit ($r_{\text{ISCO}}$). Additionally, the quantification of
the mass-to-energy conversion efficiency in accretion disks revealed distinct
departures from the Kerr solution predictions.

In this article, following the standard theory of BH accretion developed in
Refs. Novikov and Thorne (1973); Page and Thorne (1974), we work with a
theoretical model for determining the key features of accretion disks. First,
the motion of particles on circular orbits within the Dymnikova and Fan-Wang
metrics is analyzed. After that, we determine the metrics’ kinematic
properties and the innermost stable circular orbits (ISCOs). Additionally, we
compute the radiative flux, temperature, and differential luminosity of the
accretion disk for both metrics. These computations provide valuable insights
into the observable aspects of accretion processes and the radiative emissions
from surrounding materials. We compare our results in view of recent
developments related to BHs and regular solutions previously investigated.

The paper is organized as follows. In Sec. II, we contemplate the class of
NLED RBHs. In Sec. III, we provide a comprehensive review of the thin
accretion disk formalism. In Sec. IV, we explore the conversion efficiency of
mass to radiation and, finally, in Sec. V we present conclusions and
perspectives555Throughout the paper, we use natural units, setting $G=c=1$..

##  II Regular solutions from NLED: The Dymnikova and Fan-Wang spacetimes

Investigation the structures of RBHs Zaslavskii (2010); Bonanno et al. (2021);
Carballo-Rubio et al. (2018); Borde (1997), in analogy to singular BHs, can
include dynamics Li and Miao (2022, 2021); Guo et al. (2022), thermodynamics
Maluf and Neves (2018); Wei (2018); Huang et al. (2023); Sharif and Saba Nawaz
(2020); Myung et al. (2009); Guo and Miao (2022); Lan et al. (2021); Bouhmadi-
López et al. (2021), shadows Olmo et al. (2023); Tsukamoto (2018); Sau and
Moffat (2023); Ling and Wu (2022); Stuchlík and Schee (2019); Ahmed et al.
(2022); Ghosh et al. (2020a, b); Kumar et al. (2019); Dymnikova and Kraav
(2019); Li and Bambi (2014); Abdujabbarov et al. (2016), the properties of the
accretion disk around RBHs Boshkayev et al. (2024a); Guo et al. (2023);
Narzilloev et al. (2022); De Martino et al. (2023); Zeng et al. (2023); Rezaei
Akbarieh et al. (2023), as well as quasinormal modes Nomura and Tamaki (2005);
Toshmatov et al. (2015); Flachi and Lemos (2013); Rincón and Santos (2020);
Panotopoulos and Rincón (2019); Lan and Wang (2023), among others.

Focusing on spherically-symmetric spacetime, we have

$\displaystyle
ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}),$
(1)

where for NLED, we can first single out the Dymnikova BH Dymnikova (2004); Wan
and Wu (2022):

$\displaystyle f_{D}(r)=1-\frac{4m}{\pi
r}\left(\arctan\Big{(}\frac{r}{l_{D}}\Big{)}-\frac{rl_{D}}{r^{2}+l_{D}^{2}}\right),$
(2)

where $l_{D}$ is the length scale, $l_{D}=\frac{\pi q_{D}^{2}}{8m}$, $m$ is
the total mass and $q_{D}$ is the electric charge for the Dymnikova BH.

Additionally, another possible solution within the realm of NLED is offered by
the Fan-Wang spacetime, having Fan and Wang (2016):

$\displaystyle f_{FW}(r)=1-\frac{2mr^{2}}{(r+l_{FW})^{3}},$ (3)

where $l_{FW}$ is the magnetic charge parameter for the Fan-Wang BH.
Hereafter, the indices “D” and “FW” will stand for Dymnikova and Fan-Wang,
respectively.

The Schwarzschild solution is recovered in the limit of either
$l_{D},l_{FW}\rightarrow 0$, whereas for the Dymnikova and Fan-Wang BHs the
event horizons imply the conditions

$\displaystyle\frac{\pi
r}{4m}-\arctan\Big{(}\frac{r}{m\bar{l}_{D}}\Big{)}-\frac{mr\bar{l}_{D}}{r^{2}+m^{2}\bar{l}_{D}^{2}}=0,$
(4)

and

$\displaystyle(r+m\bar{l}_{FW})^{3}-2mr^{2}=0,$ (5)

where $\bar{l}_{D}=l_{D}/m$ and $\bar{l}_{FW}=l_{FW}/m$ are the normalized
charges. For both Dymnikova and Fan-Wang BHs, the event horizons exist only
when $\bar{l}_{D}\leq 0.452$ and $\bar{l}_{FW}\leq 8/27$, respectively.

Figure 1: The inner and outer horizons radii normalized by the mass $r_{H}/m$
versus $l/m$. The dashed black line indicates the boundary between inner and
outer horizons. on the left, we have the Dymnikova BH, where at the border
$r_{H}/m=0.82532$ and $l/m=0.452167$. On the right, we have the Fan-Wang BH,
the boundary is at $r_{H}/m=16/27\approx 0.59259$ and $l/m=8/27\approx
0.29629$.

###  II.1 Numerical analysis of the solutions

In Fig. 1, we present the radii of inner and outer horizons normalized by mass
versus the parameters of the model also normalized by mass. The left panel
illustrates the horizon radii for the Dymnikova BH, while the right panel
depicts the horizon radii for the Fan-Wang BH. The dashed black line in both
panels marks the boundary between the inner and outer horizons. At this
boundary, the values of $r_{H}/m$ and $l/m$ correspond to the extreme BH
cases. As $l/m$ approaches zero, the inner horizon $r_{H}/m$ tends to 0, while
the outer horizon tends to 2 in both models, as expected.

Figure 2: Left panel: The metric coefficient $f(r)$ for the Dymnikova and Fan-
Wang BHs with different values of the free parameter $l_{D}$ and $l_{FW}$.
Right panel: The metric coefficient $f(r)$ plot for Dymnikova and Fan-Wang BHs
with varying mass $m$ and constant free parameter values $l_{D}=0.452$ and
$l_{FW}=8/27$.

In Fig. 2 (left panel), we plotted $f(r)$ against $r$ for the Dymnikova and
Fan-Wang BHs with varying free parameter values. The Dymnikova BH is depicted
in blue, and the Fan-Wang BH is represented in red. The blue and red solid
curves indicate critical values of the free parameters, while the blue and red
dashed and dotted curves correspond to values below the critical threshold.
For comparison, the function $f(r)$ for the Schwarzschild metric with $m=1$ is
shown as the black solid curve. Fig. 2 (left panel) shows that, in comparison
to the Schwarzschild BH, the event horizons of RBHs are shifted to lower
values. Both the Dymnikova and Fan-Wang BHs exhibit one event horizon at
critical values ${l}_{D}=0.452$ and ${l}_{FW}=8/27$. The blue (Dymnikova
solution) and red (Fan-Wang solution) dashed curves represent $f(r)$ for the
free parameters ${l}_{D}=0.30$ and ${l}_{FW}=6/27$, which are below the
critical values. Additionally, the blue and red dotted curves correspond to
the values of ${l}_{D}=0.17$ and ${l}_{FW}=4/27$, which are also below the
critical values. Furthermore, similar to the Hayward and Bardeen BHs Rezaei
Akbarieh et al. (2023), the function $f(r)$ tends towards $1$ as $r\rightarrow
0$. Additionally, for $r\rightarrow\infty$, the behavior of Dymnikova and Fan-
Wang BHs approaches that of a Schwarzschild BH (see Fig. 2 left panel).

In Fig.2 (right panel), we plot $f(r)$ as a function of $r$ for Dymnikova and
Fan-Wang BHs with varying mass $m$ and constant values of the free parameter
$\bar{l}_{D}=0.45$, $\bar{l}_{FW}=8/27$.

###  II.2 Morphological structures of the solutions

In order to make our computation more compact, we can take into account an
effective mass representation, $M(r)$ as a function of the radial coordinate,
rewriting Eqs. (2) and (3) as

$\displaystyle
ds^{2}=-\left(1-\frac{2M(r)}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M(r)}{r}\right)}+r^{2}d\Omega^{2},$
(6)

as inspired by Ref. Dutta Roy and Kar (2022), following the Damour-Solodukhin
description and, so, having for the Dymnikova and Fan-Wang, respectively

$\displaystyle M_{D}(r)$ $\displaystyle=$
$\displaystyle\frac{2m}{\pi}\left(\arctan\Big{(}\frac{r}{l_{D}}\Big{)}-\frac{r{l}_{D}}{r^{2}+l_{D}^{2}}\right),$
(7) $\displaystyle M_{FW}(r)$ $\displaystyle=$
$\displaystyle\frac{mr^{3}}{(r+l_{FW})^{3}}.$ (8)

Lying on the equatorial plane, $\theta=\pi/2$, the Lagrangian reads
$2\mathcal{L}=-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+r^{2}\dot{\phi}^{2}$,
where dot represents the derivative with respect to the affine parameter.
Accordingly, the generalized momenta are

$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{t}}=-\Big{(}1-2\frac{M(r)}{r}\Big{)}\dot{t}=-E,$
(9) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{r}}=\Big{(}1-2\frac{M(r)}{r}\Big{)}^{-1}\dot{r},$
(10) $\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=r^{2}\dot{\phi}=L.$
(11)

where, $E$ and $L$ are the energy and angular momentum per unit of the
particle’s rest mass, respectively. The Hamiltonian is

$\displaystyle
2H=-E\dot{t}+\Big{(}1-2\frac{M(r)}{r}\Big{)}^{-1}\dot{r}^{2}+L\dot{\phi}=-1.$
(12)

The energy equation can be obtained by solving Eqs. (9) and (11) for
$\dot{t}$, $\dot{\phi}$ and then combining the results within Eq. (12),

$\displaystyle\frac{1}{2}\dot{r}^{2}+V_{eff}(r)=\frac{1}{2}\Big{(}E^{2}-1\Big{)},$
(13)

where the effective potential per unit mass acquires the form,

$\displaystyle
V_{eff}(r)=-\frac{M(r)}{r}+\frac{L^{2}}{2r^{2}}-\frac{M(r)L^{2}}{r^{3}}.$ (14)

It is helpful to write $r=r(\phi)$ Itin (2008), thus having from Eq. 11,

$\displaystyle\dot{r}=\frac{dr}{d\tau}=\frac{dr}{d\phi}\frac{d\phi}{d\tau}=\frac{L}{r^{2}}\frac{dr}{d\phi},$
(15)

and so, substituting Eq. (15) in Eq. (13), we obtain

$\displaystyle\Big{(}\frac{du}{d\phi}\Big{)}^{2}+u^{2}=\frac{E^{2}-1}{L^{2}}+\frac{2uM(u)}{L^{2}}+2u^{3}M(u),$
(16)

where $r=1/u$.

Further, replacing the mass profiles for the two solutions, and then taking
the derivative with respect to $\phi$ of Eq. (16), we infer

$\displaystyle\frac{d^{2}u}{d\phi^{2}}+u$ $\displaystyle=$
$\displaystyle\frac{2m}{\pi\Big{(}1+u^{2}l_{D}^{2}\Big{)}^{2}}\Bigg{[}\frac{1}{L_{D}^{2}}\Bigg{(}\Big{(}1+u^{2}l_{D}^{2}\Big{)}^{2}\arctan\Big{(}\frac{1}{ul_{D}}\Big{)}-ul_{D}\Big{(}3+u^{2}l_{D}^{2}\Big{)}\Bigg{)}+$
(17) $\displaystyle+$ $\displaystyle
3u^{2}\Big{(}1+u^{2}l_{D}^{2}\Big{)}^{2}\arctan\Big{(}\frac{1}{ul_{D}}\Big{)}-u^{3}l_{D}\Big{(}5+3u^{2}l_{D}^{2}\Big{)}\Bigg{]}.$

and

$\displaystyle\frac{d^{2}u}{d\phi^{2}}+u=\frac{m}{\Big{(}1+ul_{FW}\Big{)}^{4}}\Bigg{[}\frac{1}{L_{FW}^{2}}\Big{(}1-2ul_{FW}\Big{)}+3u^{2}\Bigg{]},$
(18)

for the Dymnikova and Fan-Wang spacetimes, respectively.

For circular orbits in the equatorial plane, $\dot{r}=0$, i.e., showing that
$r$ and $u=1/r$ are constant. As a byproduct, we calculate the angular momenta
using Eqs. (17) and (18), having

$\displaystyle L_{D}$ $\displaystyle=$
$\displaystyle\sqrt{2}mx\sqrt{\frac{\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)-x\bar{l}_{D}\Big{(}3x^{2}+\bar{l}_{D}^{2}\Big{)}}{x\Big{(}6\bar{l}_{D}^{3}+\pi\bar{l}_{D}^{4}+10x^{2}\bar{l}_{D}+2\pi
x^{2}\bar{l}_{D}^{2}+\pi
x^{4}\Big{)}-6\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)}},$
(19)

and

$\displaystyle L_{FW}$ $\displaystyle=$ $\displaystyle
mx^{2}\sqrt{\frac{x-2\bar{l}_{FW}}{\Big{(}x+\bar{l}_{FW}\Big{)}^{4}-3x^{3}}},$
(20)

again for the Dymnikova and Fan-Wang spacetimes, being conveniently calculated
in dimensionless values $u=1/mx$. Additionally, the specific energies for the
two cases read

$\displaystyle E_{D}$ $\displaystyle=$
$\displaystyle\frac{x\Big{(}4\bar{l}_{D}+\pi\bar{l}_{D}^{2}+\pi
x^{2}\Big{)}-4\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}\arctan\left(\frac{x}{\bar{l}_{D}}\right)}{\sqrt{{\pi
x^{2}\Big{(}6\bar{l}_{D}^{3}+\pi\bar{l}_{D}^{4}+10x^{2}\bar{l}_{D}+2\pi
x^{2}\bar{l}_{D}^{2}+\pi x^{4}\Big{)}-6\pi
x\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)}}},$
(21)

and

$\displaystyle
E_{FW}=\frac{\Big{(}x+\bar{l}_{FW}\Big{)}^{3}-2x^{2}}{\Big{(}x+\bar{l}_{FW}\Big{)}\sqrt{\Big{(}x+\bar{l}_{FW}\Big{)}^{4}-3x^{3}}},$
(22)

The angular velocity for the Dymnikova and Fan-Wang BHs acquires the form

$\displaystyle\Omega_{D}=\frac{\dot{\phi}}{\dot{t}}=\frac{1}{mx}\sqrt{\frac{2}{\pi
x}\arctan\left(\frac{x}{\bar{l}_{D}}\right)-\frac{2\bar{l}_{D}\Big{(}3x^{2}+\bar{l}_{D}^{2}\Big{)}}{\pi\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}}},$
(23)

and

$\displaystyle\Omega_{FW}$ $\displaystyle=$
$\displaystyle\frac{\dot{\phi}}{\dot{t}}=\frac{\sqrt{x-2\bar{l}_{FW}}}{m\Big{(}x+\bar{l}_{FW}\Big{)}^{2}}.$
(24)

The effective potentials for Dymnikova and Fan-Wang BHs can be found by
substituting Eqs. (19) and (20) into Eq. (14), respectively.

$\displaystyle
V_{eff-D}=\frac{x^{2}\bar{l}_{D}\Big{(}8\bar{l}_{D}+\pi\bar{l}_{D}^{2}-\pi
x^{2}\Big{)}-x\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}\Big{(}16\bar{l}_{D}+\pi\bar{l}_{D}^{2}+\pi
x^{2}\Big{)}\arctan\left(\frac{x}{\bar{l}_{D}}\right)+8\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)^{2}}{{\pi
x^{2}\Big{(}6\bar{l}_{D}^{3}+\pi\bar{l}_{D}^{4}+10x^{2}\bar{l}_{D}+2\pi
x^{2}\bar{l}_{D}^{2}+\pi x^{4}\Big{)}-6\pi
x\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)}},$
(25)

and

$\displaystyle V_{eff-FW}$ $\displaystyle=$
$\displaystyle\frac{x^{2}\Big{[}4x^{2}-\Big{(}x+4\bar{l}_{FW}\Big{)}\Big{(}x+\bar{l}_{FW}\Big{)}^{2}\Big{]}}{2\Big{(}x+\bar{l}_{FW}\Big{)}^{2}\Big{[}\Big{(}x+\bar{l}_{FW}\Big{)}^{4}-3x^{3}\Big{]}}.$
(26)

At this stage, the dimensionless ISCO radii, $x_{ISCO}$, may be computed from
Page and Thorne (1974), through the definition,
$\frac{d^{2}V_{eff}}{dx^{2}}=0$.

Thus, the derivative of the effective potential $V_{eff}$ is computed Harko et
al. (2009); Kovács and Harko (2010)

$\displaystyle\frac{d^{2}V_{eff}}{dx^{2}}$ $\displaystyle=$
$\displaystyle\frac{4x\bar{l}_{D}\Big{(}\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}+4x^{4}\Big{)}-4\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{3}\arctan\left(\frac{x}{\bar{l}_{D}}\right)}{\pi
x^{3}\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{3}}+$ (27) $\displaystyle+$
$\displaystyle\bar{L^{2}}_{D}\Bigg{(}\frac{3}{x^{4}}+\frac{x\bar{l}_{D}\Big{(}\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}+16\Big{(}2x^{2}+\bar{l}_{D}^{2}\Big{)}\Big{)}-24\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{3}\arctan\left(\frac{x}{\bar{l}_{D}}\right)}{\pi
x^{5}\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{3}}\Bigg{)},$

and

$\begin{split}\frac{d^{2}V_{eff}}{dx^{2}}=&\frac{12x\bar{l}_{FW}-2(x+\bar{l}_{FW})^{2}}{(x+\bar{l}_{FW})^{5}}\\\
&+3\bar{L^{2}}_{FW}\Big{(}\frac{1}{x^{4}}-\frac{4}{(x+\bar{l}_{FW})^{5}}\Big{)},\end{split}$
(28)

where for brevity $\bar{L}\equiv L/m$. So, by inserting Eqs. (19) and (20) in
Eqs.(27) and (28), we get

$\displaystyle\frac{d^{2}V_{eff}}{dx^{2}}$ $\displaystyle=$
$\displaystyle\frac{2x\Big{(}24\bar{l}_{D}\Big{(}2x^{4}+\bar{l}_{D}^{4}\Big{)}+104x^{2}\bar{l}_{D}^{3}+\pi\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{3}\Big{)}\arctan\left(\frac{x}{\bar{l}_{D}}\right)-2x^{2}\bar{l}_{D}\Big{(}12\bar{l}_{D}^{3}+\pi\bar{l}_{D}^{4}+68x^{2}\bar{l}_{D}+8\pi
x^{2}\bar{l}_{D}^{2}-\pi x^{4}\Big{)}}{\pi
x^{3}\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}\Big{(}x\Big{(}6\bar{l}_{D}^{3}+\pi\bar{l}_{D}^{4}+10x^{2}\bar{l}_{D}+2\pi
x^{2}\bar{l}_{D}^{2}+\pi
x^{4}\Big{)}-6\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)\Big{)}}-$
(29) $\displaystyle-$
$\displaystyle\frac{24\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)^{2}}{\pi
x^{3}\Big{(}x\Big{(}6\bar{l}_{D}^{3}+\pi\bar{l}_{D}^{4}+10x^{2}\bar{l}_{D}+2\pi
x^{2}\bar{l}_{D}^{2}+\pi
x^{4}\Big{)}-6\Big{(}x^{2}+\bar{l}_{D}^{2}\Big{)}^{2}\arctan\left(\frac{x}{\bar{l}_{D}}\right)\Big{)}},$

and

$\begin{split}\frac{d^{2}V_{eff}}{dx^{2}}=&\frac{12x\bar{l}_{FW}-2\Big{(}x+\bar{l}_{FW}\Big{)}^{2}}{\Big{(}x+\bar{l}_{FW}\Big{)}^{5}}\\\
&+3\frac{\Big{(}x-2\bar{l}_{FW}\Big{)}\Big{[}\Big{(}x+\bar{l}_{FW}\Big{)}^{5}-4x^{4}\Big{]}}{\Big{(}x+\bar{l}_{FW}\Big{)}^{5}\Big{[}\Big{(}x+\bar{l}_{FW}\Big{)}^{4}-3x^{3}\Big{]}},\end{split}$
(30)

The $x_{ISCO}$ value for the free parameters’ critical values
$\bar{l}_{D}=0.45$, $\bar{l}_{FW}=\frac{8}{27}$ one can to compute for the
Dymnikova and Fan-Wang BHs using Eqs. (29) and (30).

$\displaystyle\frac{dL}{dx}=\frac{dE}{dx}=0.$ (31)

The $x_{ISCO}$ dependence on the free parameter $\bar{l}$ for the Dymnikova BH
(left panel) and Fan-Wang BH (right panel) is shown in Fig. 3.

The effective potential for the constant value of $\bar{L}=L/m$ as determined
in the ISCO for the free parameters of the Dymnikova (blue) and Fan-Wang (red)
is shown in Figure 4 (left panel). The locations of the ISCO for the
Schwarzschild, Dymnikova, and Fan-Wang BHs are shown by the black, magenta,
and green dots, respectively.

Figure 3: Dependence of the $x_{ISCO}$ on the free parameter $l/m$ . Left
panel: for the Dymnikova RBH. Right panel: for the Fan-Wang RBH.

Figure 4: Left panel: The Effective potential for the values of $\bar{L}=L/m$
when evaluated at the ISCO versus of $x$ for the Dymnikova BH with
$\bar{l}_{D}=0.45$ and the Fan-Wang BH with $\bar{l}_{FW}=8/27$. Right panel:
The effective potential as a function of $x$ for the Dymnikova and Fan-Wang
BHs.

Additionally, we compared the Dymnikova and Fan-Wang BHs with the
Schwarzschild BH by plotting the effective potential $V_{eff}$ as a function
of $x$ in Fig.4 (right panel). As seen from Fig. 4 for both panels the
$x_{ISCO}$ and the minimum values of the effective potential move to smaller
values. The values of $x_{ISCO}$ for the Schwarzschild, Dymnikova, and Fan-
Wang BHs are shown by the black, red, and blue dots, respectively.

Figure 5: Left panel: Angular velocity of test particles versus $x$ for
Dymnikova space-time. Right panel: Angular velocity of test particles versus
$x$ for Fan-Wang space-time.

In Fig. 5 we show the orbital angular velocity $\Omega$ of test particles as a
function of $x$ Dymnikova (left panel) and Fan-Wang (right panel) spacetimes.
In both panels, we see that for different values of
$\bar{l}_{D},\bar{l}_{FW}$, the angular velocity is always less than the
Schwarzschild BH (black solid curve).

Figure 6: Left panel:Angular momentum $\bar{L}=L/m$ as a function of $x$.
Right panel: Specific energy $E$ as a function of $x$.

In Fig. 6 (left panel), the angular momentum $\bar{L}$ is depicted as a
function of the $x$ for Dymnikova and Fan-Wang spacetimes and compared with
the Schwarzschild BH. It is evident from Fig. 6 that the angular momentum has
also been marginally moved to lower values in the $x_{isco}$ region.

The same pattern as the angular momentum is replicated in the curve of
specific energy $E$ in terms of $x$, as seen in Fig. 6,(right panel). The
specific energy $E$ of RBHs is somewhat skewed towards lesser values when
compared with the Schwarzschild BH, as seen in Fig. 6,(right panel), which is
presented for the critical values of the free parameters $\bar{l}_{D}$ and
$\bar{l}_{FW}$ (See Tab. 1).

In Fig. 7 we compare the mass curve of the Dymnikova and the Fang-Wang BHs as
a function of $x$ with respect to the Schwarzschild BH (horizontal black
line). The vertical lines show the ISCO for the Dymnikova and Fan-Wang BHs
with the free parameters $\bar{l}_{D}$ and $\bar{l}_{FW}$. The horizontal
solid line corresponds to the Schwarzschild space-time. As can be seen from
Fig. 7, for a given value of the free parameters $\bar{l}_{D}$ and
$\bar{l}_{FW}$, the particle falling into the BH experiences less
gravitational pull due to the decreasing mass profile as the $x$ decreases
(See. Fig. 7).

Figure 7: The mass curves as a function of $x$ for the Dymnikova and Fan-Wang
BHs.

##  III Relativistic Characteristics of Thin Accretion Disks

The derivative of the luminosity at infinity $\mathcal{L}_{\infty}$ is
obtained by combining the laws of energy conservation and angular momentum
Page and Thorne (1974); Joshi et al. (2014)

$\displaystyle\frac{d\mathcal{L}_{\infty}}{dlnr}=4\pi
r\sqrt{-g}E\mathcal{F}(r),$ (32)

The radiant energy flux $\mathcal{F}$ emitted from the disk’s surface is
determined by

$\displaystyle\mathcal{F}(r)=-\frac{\dot{{\rm
m}}}{4\pi\sqrt{-g}}\frac{\Omega_{,r}}{\left(E-\Omega
L\right)^{2}}\int^{r}_{r_{i}}\left(E-\Omega L\right)L_{,\tilde{r}}d\tilde{r},$
(33)

where $\dot{m}$ is the mass accretion rate of the disk, which is assumed to be
constant, $r_{i}=r_{ISCO}$ and $\sqrt{-g}=r$ applies to the Dymnikova, Fan-
Wang, and Schwarzschild spacetimes.

If the accretion disk is assumed to be in thermodynamic equilibrium, then its
surface radiation can be considered blackbody radiation with a temperature
determined by Zuluaga and Sánchez (2021)

$\displaystyle{T}(r)=\sigma^{-\frac{1}{4}}\mathcal{F}(r)^{\frac{1}{4}},$ (34)

where $\sigma$ is the Stefan-Boltzmann constant.

In Figs. 8 and 9 we plotted the flux, the temperature of the accretion disk in
the accretion rate unit, and the differential curve of luminosity for the
Dymnikova, Fan-Wang and Schwarzschild BHs.

Figure 8: Left panel: Radiative flux $\mathcal{F}^{*}$ of the accretion disk
multiplied by $10^{5}$ versus $x$. Right panel: The temperature $T^{*}$ in the
accretion rate unit of the disk around the RBHs.

Now we compare the flux from the accretion disk of RBHs with the one of a
Schwarzschild BH (Fig. 8 (left panel)). The accretion disk around RBHs emits
more energy than the Schwarzschild BH. This occurs as a result of the RBH’s
inner edge being moved toward smaller values of $x$ in contrast to the
Schwarzschild BH.

The temperature of accretion disks in the unit of disk accretion rate is
displayed in Fig 8 (right panel). Here we compare the temperature curve of the
accretion disk around the Schwarzschild BH (black solid curve) with the one
around the Dymnikova and Fan-Wang BHs (blue and red solid curves). As seen in
the figure, the temperature of the Fan-Wang BHs is greater than that of the
Dymnikova and Schwarzschild BHs due to the shift of the inner edge of the
accretion disk toward the smaller values of $x$. This is due to the ISCO of
the Fan-Wang BH being less than that of the Dymnikova BH (See Tab 1).

A comparison between the luminosity derivative of the classical BH and the one
of the Dymnikova and Fan-Wang BHs is shown in Fig. 9. As seen in Fig. 9 the
derivative of the luminosity of the accretion disk of the Dymnikova and Fan-
Wang BHs differs from that of a Schwarzschild BH. The luminosity derivative of
the Dymnikova and Fan-Wang BHs is higher than the Schwarzschild BH.

Figure 9: Differential luminosity of the thin accretion disk around RBHs.

##  IV Efficiency of Mass Conversion to Radiation

We compute the efficiency of converting mass to radiation, symbolized as
$\epsilon$, by analyzing the energy dissipation of a test particle moving from
infinity to the disk’s inner edge. It is assumed that all photons emitted from
the disk’s surface can reach infinity. Thus, the particle’s energy tends to
unity, or $E_{\infty}\approx 1$, as the radius gets closer to infinity. So, we
write

$\displaystyle\epsilon=\frac{E_{\infty}-E_{isco}}{E_{\infty}}\approx
1-E_{isco},$ (35)

For the critical values of Dymnikova and Fan-Wang’s free parameters, we
calculate the appropriate bounds on specific energy in ISCOs, $E_{isco}$, and
the conversion efficiency of the accreting mass into the corresponding
radiation, $\epsilon$, see Tab. 1.

This trend seems to be the opposite of what the Schwarzschild BH predicted,
and it is especially noticeable for the Dymnikova and Fan-Wang BHs.

Table 1: Specific energy in the innermost stable circular orbit $E_{isco}$ and mass-to-radiation conversion efficiency $\epsilon$ for the Dymnikova and Fan-Wang RBHs with different values of free parameters, and comparison with the Schwarzschild-space-time. $Free\,\,Parameter$ | $x_{isco}$ | $E_{isco}$ | $\epsilon(\%)$
---|---|---|---
$\bar{l}_{D}=0.45$ | 3.60 | 0.9112 | 8.8767
$\bar{l}_{D}=0.35$ | 4.54 | 0.9238 | 7.6209
$\bar{l}_{D}=0.26$ | 4.85 | 0.9308 | 6.9197
$\bar{l}_{D}=0.17$ | 5.29 | 0.9359 | 6.4081
$\bar{l}_{FW}=8/27$ | 2.67 | 0.8856 | 11.4314
$\bar{l}_{FW}=6/27$ | 3.74 | 0.9130 | 8.6982
$\bar{l}_{FW}=4/27$ | 4.57 | 0.9270 | 7.3036
$\bar{l}_{FW}=2/27$ | 5.31 | 0.9361 | 6.3874
$\bar{l}_{D},\bar{l}_{FW}=0$ | $6$ | 0.9428 | 5.7191

Furthermore, increases in flux, temperature, and differential luminosity, as
well as the conversion efficiency of accreting mass to radiation, are directly
related to the displacement of the ISCO to lower values.

It turns out that the key quantity for describing the physical characteristics
(both optical and structural) of an accretion disk is the $r_{ISCO}$, which
is, in turn, defined by the parameters of the gravitational source. In the
case of the Dymnikova and Fan-Wang solutions, in addition to the mass of the
source, additional length parameters related to electric and magnetic charges
emerging from NLED substantially alter the geometry around RBHs in a similar
manner to an electric charge in the Reissner–Nordström BH or color charges in
the charged dilatonic BH solutions Pugliese et al. (2011); Boshkayev et al.
(2024b). Hence, any BH with any form of charge induces additional gravity.

##  V Final outlooks and perspectives

As a consequence of the modern impressive interest around RBHs, we here
explored the thermodynamic properties of thin accretion disks around precise
regular spacetimes that can be obtained from NLED Lagrangian scenarios and may
turn to be interesting in modelling compact objects without singularities.

Recently, the study of accretion disks surrounding RBHs has assumed
considerable importance in astrophysics, as this research avenue provides
chances to deepen our understanding of RBHs as distinct entities from
conventional BHs.

As regular solutions, we here considered the Fan-Wang and Dymnikova RBHs,
evaluating for them the simplest accretion disk model proposed by Novikov-
Thorne and Page-Thorne.

In so doing, we calculated temperature, differential luminosity, and mass
conversion efficiency into radiation to compare the actual findings with
previous literature.

Our findings indicated that introducing non-zero free parameters, distinct for
Fan-Wang and Dymnikova metrics and here denoted as ${l}_{D}$ and ${l}_{FW}$,
resulted in a reduction in the corresponding ISCOs. Consequently, we
interpreted that non linear couplings of gravity with electrodynamics can
affect the measurements of the expected overall spectral properties.

Nevertheless, we uncovered another noteworthy result due to the non-zero free
parameters, ${l}_{D}$ and ${l}_{FW}$, namely, an increase in energy emitted
from the disk surface around the RBH solution.

Remarkably, utilizing specific values for the free parameters, and precisely
${l}_{D}=0.452$ and ${l}_{FW}=8/27$, we found an increase in the temperature
of the accretion disk surrounding RBHs. Additionally, incorporating non-zero
values for these parameters resulted in an enhancement in the differential
luminosity of the accretion disk around Dymnikova and Fan-Wang BHs.

Additionally, we observed that non-zero values of the free parameters
increased the efficiency of mass conversion into radiation compared to the
Schwarzschild spacetime.

Thus, the here-involved particular solutions obtained from NLED may be seen to
be more efficient in converting mass into radiation than Schwarzschild BHs,
suggesting how future explorations can check whether a given compact object
can be described by such solutions or not.

In view of our findings, future investigations will go through diverse
scenarios involving rotating RBHs, encompassing the impact of magnetic fields
and dark matter on the physical characteristics of the accretion disk.
Furthermore, embracing alternative and potentially more intricate accretion
disk models will be crucial to probing potential deviations from our current
approach. By scrutinizing these alternative models, we aim to elucidate
differences arising as we move away from the assumption of spherical symmetry
within the employed spacetimes. This proposal will clearly shed light on the
expected complexities of accretion disk dynamics, providing valuable insights
into the behavior of RBH accretion disk systems.

###### Acknowledgements.

YeK acknowledges Grant No. AP19575366, TK acknowledges the Grant No.
AP19174979, and GS acknowledges Grant No. AP19680128, and KB and AU
acknowledge the Grant No. BR21881941 from the Science Committee of the
Ministry of Science and Higher Education of the Republic of Kazakhstan.

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# Cross-Domain Image Classification through Neural-Style
Transfer Data Augmentation

Yijie Xu
University College London
<EMAIL_ADDRESS>Arushi Goel
University of Edinburgh
<EMAIL_ADDRESS>
###### Abstract

In particular, the lack of sufficient amounts of domain-specific data can
reduce the accuracy of a classifier. In this paper, we explore the effects of
style transfer-based data transformation on the accuracy of a convolutional
neural network classifiers in the context of automobile detection under
adverse winter weather conditions. The detection of automobiles under highly
adverse weather conditions is a difficult task as such conditions present
large amounts of noise in each image. The InceptionV2 architecture is trained
on a composite dataset, consisting of either normal car image dataset , a
mixture of normal and style transferred car images, or a mixture of normal car
images and those taken at blizzard conditions, at a ratio of 80:20. All three
classifiers are then tested on a dataset of car images taken at blizzard
conditions and on vehicle-free snow landscape images. We evaluate and contrast
the effectiveness of each classifier upon each dataset, and discuss the
strengths and weaknesses of style-transfer based approaches to data
augmentation.

##  1 Introduction

It is commonly accepted that large amounts of data is needed to train deep
neural networks which yields high performance. When the number of training
examples is reduced in size, such networks become more prone to overfitting
which reduces their generalizable capability [9]. To improve generalization
and performance, several data augmentation and regularization techniques have
been explored in the research community [7, 11]. In this paper, we present a
data augmentation technique for improving the performance of neural networks.
The availability of data is known to improve performance as more
discriminative features are extracted by the model . However, the resource
intensive nature of data acquisition can render it unfeasible, necessitating a
need to generate synthetic data.

In this paper, we introduce a novel neural style transfer-based data
transformation and augmentation method for cross-domain classification
applications, where an existing data domain (cars) is shifted to target a
separate cross-domain application (cars obscured with snow). Due to lack of
suitable conditions required to collect data specific to a domain, style-
transfer based data augmentation may become a necessary aspect of data
collection. In other words, while many applications have standardized data
generation procedures, others are subject to time, temperature, or cost-
sensitive conditions, greatly restricting the data collection time.

One practical example to consider is that of car detection under adverse
winter conditions. The acquisition of such data may only be done in
geographically suitable areas under suitable weather conditions, which is
subject to change without notice. With increasing global temperatures due to
climate change, tracking suitable time for data collection is becoming
increasingly difficult.

Our style-transfer based data transformation tackles this problem by applying
appropriate low-level transformations to existing vanilla car image datasets
[4]. As the degree and type of style transfer is controlled by user-defined
weights and reference images respectively, the various kinds of synthetic
composite outputs could be generated to match a particular feature domain of
interest. To the best of our knowledge, we are the first to use neural style
transfer for data augmentation to improve cross-domain performance using
domain-specific noise.

The rest of the paper discusses related work, our proposed method,
experimental results and conclusion.

##  2 Related Work

###  2.1 Data Augmentation

Data augmentation improves model robustness and sample size [11, 8].
Traditional augmentation approaches for image classification applications
include rotations, translations, zooms, Gaussian noise addition, and mirror
flips. More advanced methods to data augmentation have been investigated by Wu
_et al_. [19], who argues that data augmentation itself is invaluable in
preventing the learning of image-specific artifacts in training datasets and
to improve learning performance features that are invariant for specific
classes. Besides traditional augmentation methods, they incorporated
techniques such as adaptive resolution scaling, color changes, lens distortion
effects, and global or localized brightness changes to augment their data, and
achieve a performance surpassing the top-scoring models from the ILSVRC
classification tasks of the preceding three years [19].

The generation of completely synthetic data as a data augmentation approach
has also been investigated for image classification applications, with varying
success. A study by Alhajia _et al_. [1] on automobile classification in
traffic situations concluded that a dataset consisting of real images
augmented with 3D-modelled vehicles outperformed both a purely 3D-modelled
dataset as well as the original dataset [1]. Similarly, augmenting clear
facial data with synthetically modelled accessories has also been shown to
improve recognition performance in surveillance industry [18]. More recently,
adversarial architectures such as GANs have shown promise in being able to
generate new synthetic data for applications such as in medical imaging and
facial recognition by combining feature elements from existing data. However,
these often suffer from restrictions in feature domains, being subject to
domain bias [2, 16]. In our method, we target the problem of domain bias by
augmenting the training dataset using neural style transfer, which has not yet
been explored in existing works.

###  2.2 Domain Bias

The subject of cross-domain classification, or the use of a classifier trained
on a particular distribution of data to generalize on other previously unseen
datasets, has been investigated in literature. Previous approaches in
literature suffered from poor cross-domain discriminative performance, with
the phenomenon termed as domain bias, where a model will not generalize well
to data not observed during training [17, 3]. The differences between such
datasets may be ascribed to factors such as variations in camera pose,
illumination, lens properties, noise, background, and the presence of foreign
artifacts [6].

Transfer learning addresses domain bias by fine-tuning an existing network
trained for one domain to classify samples from the other domain [10, 14, 20].
The approach relies on the understanding that convolutional features generated
from larger datasets are shared across multiple domains, allowing for a faster
and resource-efficient training process to be pursued [13]. This has been
extensively demonstrated for medical applications, where access to large
amounts of domain-specific data is limited, the use of network pre-trained on
the ImageNet dataset has shown to improve classification accuracy for the
detection of mammographic tumours, lung disease, and abnormal lymph nodes in
test images [15, 5]. However, the lack of specifity of such approaches to the
target domain has made the acquisition of domain-specific data a preferable
course of action. In contrast, our approach utilizes transfer learning for
cross-domain classification using a composite dataset, where the approach
consisted of a complete retraining of the network using combination of
examples from both the original and target domains, wherein the data for the
target domain was syntheticially derived from the original dataset using
neural style transfer. As such, we aimed for the generation of synthetic data
representative of the target domain to improve classifier performance.

Figure 1:  Workflow pipeline of our network architecture. A target and
reference image are fed into the style-transfer architecture to generate a
composite output. Style-transferred outputs are combined with original car
images to form a composite dataset in order to evaluate test set inputs via a
InceptionV2-based classifier.

###  2.3 Style Transfer

First defined by Gatys _et al_. [4] in the context of artwork generation,
style transfer refers to the altering of low-level features within an image
while retaining its semantic content [4, 6]. Conceptually, style transfer
relies on the underlying fact that different levels of a convolutional neural
network capture different aspects of the level. Gatys _et al_. [4] defined the
convolutional features captured by lower levels within the network, such as
the local texture or patterns, as the “style” of an image, while those
captured by higher levels as the semantic content of an image. This separate
capture process allows for the independent manipulation and transfer of
individual components into a generated composite. Gatys _et al_.’s original
paper demonstrated how landscape images could be altered to resemble modern
artworks while retaining high-level semantic content.

The exact mechanism of style transfer has been discussed extensively in
literature [4, 8]. Briefly, a VGG network [9] stripped of any fully connected
layers serves as the model for style transfer. To extract high level semantic
content, a target image is fed into this network to be encoded by successive
convolutional layers. To replicate the features captured at a layer of
interest, gradient descent is used on a white-noise image matching feature
responses are observed, measured by squared loss error.

$L_{content}(\vec{p},\vec{x},l)=\frac{1}{2}\sum_{i,j}(F_{ij}^{l}-P_{ij}^{l})^{2}$
(1)

Where P, F, and l represent the original image, the generated image, and l the
layer of interest, respectively. To extract the low-level “style” content of
an image, a Gram matrix is constructed to capture the correlations between
different filter responses from layer of interest, defined as:

$G_{ij}^{l}=\sum_{k}F_{ik}^{l}F_{jk}^{l}$ (2)

Or the inner product of the vectorized feature maps i and j within a layer l.
Similar to above, gradient descent is then used on a white-noise image to
create a matching representation, with the loss defined as the mean-squared
distance between the matrices of the original and the generated counterpart.

$E_{l}=\frac{1}{4N_{l}^{2}M_{l}^{2}}\sum_{i,j}(G_{ij}^{l}-A_{ij}^{l})^{2}$ (3)

Finally, an additional loss function is defined that weighs the contribution
of each layer to the total loss.

$L_{style}(\vec{a},\vec{x})=\sum_{l=0}^{L}w_{l}E_{l}$ (4)

The generated representations can then be combined to form a composite image.
Note that the ratio between the two is controlled by manually defined weights,
allowing for a degree of control over the outputs. In both the original work
and current work, this was done in an iterative manner, so as to evaluate the
effect of iterative evolutions of the output images on the classification
accuracy.

While style transfer has been extensively demonstrated for generative
applications, studies on its usage in a data augmentation context for
discriminative applications are limited. Jackson _et al_. [6] demonstrated the
capability of neural style transfer in randomizing dataset color, texture and
contrast whilst preserving semantic geometry, and improved classification
accuracy of vanilla traffic scenes by roughly 1.4% when compared to
traditional data augmentation strategies. Similarly, Zheng _et al_. [21]
observed a 2% and 1.3% increases in performance by vanilla VGG16 and VGG19
networks featuring style transfer-based data augmentation on the Caltech 101
and 256 datasets, when compared to models utilizing traditional data
augmentation strategies [21]. These works primarily focus on intra-domain
discriminative performance improvements, the capability of neural style
transfer to improve cross-domain discriminative performance under a real-world
context is a topic of interest for research.

##  3 Proposed Method

In this section, we describe our method to achieve data augmentation using
neural-style transfer.

###  3.1 Data Augmentation using Style Transfer

Our approach involves the utilization of style transfer to introduce
appropriate domain-specific noise to our vanilla car images in order to create
images more reminiscent of adverse winter conditions. Figure 1 depicts the
overview of our methodology.

Initially, image samples from the original car dataset are fed together with a
winter reference image into the VGG19-based neural style transfer architecture
to generate synthetic outputs. These outputs are then grouped together with
original car images to form a composite dataset representing the generic “car”
class, at a ratio of 20:80. This dataset was then used to train an
InceptionV2-based architecture on the car class and four auxiliary classes.
The resulting model is evaluated on a variety of test datasets, and its
performance is measured by the true positive (TP) and false positive rates
(FP).Figure 2 illustrates the style transfer composite generation process in
detail.

Figure 2:  Reference, target, and output images at iterations 1, 3, 5, and 7,
used in our experiments.

As described in Section 2.3, a target image is fed into a VGG-19-based
convolutional neural network together with a reference winter image. By
extracting low-level features from the reference together with high level
semantic features from the target image with specific weights, successive
style transfer iterations are applied to generate the desired composite
output. With an increasing number of iterations, the output becomes
correspondingly blurry, as the proportion of low-level features is
accumulated. To evaluate the effect of increased iterations, multiple models
trained using composite datasets featuring style-transferred images of a
specific iteration were systematically evaluated.

##  4 Results and Experiments

###  4.1 Datasets

We evaluated our method using a composite dataset, based on the natural-images
dataset. The natural-images dataset was built by Roy _et al_. [12], and
consists of 8 classes of images from various published datasets [12]. For this
implementation, only 4 classes of images were considered – car, flower, cat,
and dog.

As a test dataset for our class of interest, Google Image Search was used to
acquire 200 images of cars under adverse winter conditions, 100 of which were
used for training purposes. This dataset (henceforth known as “Blizzard”)
consisted of images selected as being human-identifiable as cars: this meant
that instances where the vehicle was covered in snow past human legibility
were excluded. Additionally, 100 images of empty snow landscapes (henceforth
known as “Landscape”) were acquired to evaluate the misclassification rates of
our models, along with a dataset consisting of 100 style-transferred car
images (henceforth known as “Styled”) produced using our approach. Examples of
all three datasets are displayed below in Figure 3

Figure 3: Example images of the “Blizzard” (a,d),“Styled” (b,e), and
“Landscape” (c) test datasets used to evaluate our trained models.

###  4.2 Implementation Details

The generation of our images was done on a style-transfer convolutional neural
network based on the VGG19 architecture as defined by Gatys _et al_. [4], with
a style weight of 1, content weight of 0.0003, and a total variational weight
of 0.00001. For classification, InceptionV2 ResNet34 architecture [8] was pre-
trained for ImageNet classification task. The architecture of the model was
modified by removing the output layer prior to the classification layer and
include a dropout layer with a dropout rate of 0.5 to limit overfitting during
the training process. A small learning rate of 5x$10^{-4}$ was used to fine-
tune our model, and the network was retrained on our composite datasets for a
maximum of 10 epochs with ADAM optimizer. Traditional augmentation approaches
such as image rotation and flipping were also used consistently across all
models. Each model was run 20 times to generate statistically significant
results. Our code is implemented in Python, using the Keras and Tensorflow
packages.

###  4.3 Results

TASK | Vanilla(100%) / A | Styled (20%) / B | Blizzard (20%) /C
---|---|---|---
Blizzard Images | 0.803$\pm$0.085 | 0.888$\pm$0.035 | 0.967$\pm$0.013
Landscape Images | 0.022$\pm$0.023 | 0.086$\pm$0.060 | 0.236$\pm$0.064
Styled Images | 0.102$\pm$0.086 | - | -

Table 1: The classification accuracy across our different training domain
combinations. Model A was trained with the baseline original car image
dataset, while Model B was trained with a augmented dataset, where 20% of the
images in the original dataset were replaced with style-transferred images.
Model C was also trained with an augmented dataset, where 20% of the images
were replaced with images from the “Blizzard” dataset. All three models were
then evaluated on the “Blizzard” and “Landscape” datasets, respectively, with
model A being further tested on the “Styled” dataset.

Table 1 shows the classification accuracy across our different training domain
combinations. Model A was trained with the baseline original car image
dataset, while Model B was trained with a augmented dataset, where 20% of the
images in the original dataset were replaced with style-transferred images .
Model C was also trained with an augmented dataset, where 20% of the images
were replaced with images from the “Blizzard” dataset. All three models were
then evaluated on the “Blizzard” and “Landscape” datasets, respectively, with
model A being further tested on the “Styled” dataset.

The baseline test accuracy utilizing the original car dataset is described by
Model A, at $80.3\text{\,}\mathrm{\char 37\relax}$ ±
$8.5\text{\,}\mathrm{\char 37\relax}$. The model possesses the smallest
misclassification rate when presented with the landscape data, at
$2.2\text{\,}\mathrm{\char 37\relax}$ ± $2.3\text{\,}\mathrm{\char 37\relax}$.
The addition of either style transferred or blizzard car images into the
training dataset increased the blizzard car detection accuracy while reducing
the model uncertainty (from 8.5 to 3.5 and 1.3%, for vanilla , styled, and
blizzard car augmented datasets, respectively). This is supported by the low
detection rate ($10.2\text{\,}\mathrm{\char 37\relax}$ ±
$8.6\text{\,}\mathrm{\char 37\relax}$) of Model A on the “Styled” images test
dataset, which suggest that the generated data has shifted beyond the original
domain. As Model A captures features specific to the original domain, it is
difficult for such a model to generalize on out-of-domain data which results
in low detection accuracy.

Interestingly, while Model C certainly possesses the highest detection
accuracy and confidence interval among all models, it suffers from high false
detection rate when applied to the “Landscape” dataset, where an increase in
rate is 10-fold over Model A (2.2% to 23.6%), accompanied by a near 3-fold
increase in model-to-model uncertainty (2.3% to 6.4%) was observed. The high
misclassification rate of Model C may be linked to the high variation found in
the real-world dataset images, whether it be difference in background terrain,
the weather conditions, or even the amount of snow (noise) observed on the
vehicles differing for each image. This is to be expected and remains
exceptionally difficult to compensate for due to the levels of entropy
observed in weather conditions. It may be possible to improve upon the
misclassification rate by the incorporation of a larger dataset, but this may
not be feasible.

In contrast, the car images in the “Styled” dataset are all similar in their
level of noise, resulting in a more consistent dataset capable of improving
detection accuracy while suffering from only a small increase in
misclassification rates, as was observed with Model B. This small increase may
be attributed to the randomness observed in the degree of adversity in the
conditions of the blizzard car images test set.

Our results are in line with by previous cross-domain work by Jackson _et al_.
[6], who observed a maximum of 11% increase in classification accuracy for in
cross domain applications of the Office dataset when utilizing a combined
style-transfer and traditional data augmentation approach compared to purely
traditional approach. Notably, the magnitude of the accuracy increase was
subject to domain-specific differences between the training data and target
data: an accuracy increase of 6.2% was observed for networks tested on the
DSLR domain when trained on the Amazon and Webcam domains.10

Interestingly, when our testing was repeated on models trained on style-
transferred outputs generated after 5, 7, or 10, iterations, a progressively
worse performance in the composite datasets could be observed, with all models
exhibiting notably worse classification accuracy and standard deviation over
multiple attempts in comparison to the reference model. We hypothesize the
strong stylistic abstraction observed across multiple iterations shifts the
domain further from that of the both the vanilla dataset and that of vehicles
under adverse conditions, with a detrimental effect on accuracy as a
consequence.

##  5 Conclusion

In this paper we proposed and evaluated style transfer based data
transformation to improve classification accuracy of cars under adverse winter
weather conditions. Such augmentation improved the classification accuracy on
a foreign domain with only minimal impact on the false detection rate. We
compared the performance of our method with the original dataset augmented
with real-world blizzard car images. This successfully evaluated our approach
to use neural style transfer as being superior to utilizing a purely real-
world blizzard dataset, as the latter exhibited high false detection rates due
to large variations across its real-world data samples.

Excessive number of iterations for style transfer may have a detrimental
effect on classifier performance. As a rule of thumb, we propose that the
appropriate number of iterations of style transfer can be judged visually, and
must bear superficial resemblance to the domain of the original dataset.

Given the blur-like qualities of our synthetic generated examples, future
studies should focus on the generation of more realistic snow effects to
evaluate both the ability of generative models to mimic true reality, and to
discern if generated realistic examples of the target domain also suffer from
the detrimental effects of high data variation.

Thus, in this paper we evaluate the superiority of a consistent, artificially
generated dataset, and show improvements over directly using real-world data
given a constrained dataset size. Hence, style transfer is one of the
successful ways for synthetic data generation on other foreign domain
applications.

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###### Abstract

In this talk the answer to the question: what neutrinos have to do with QCD,
the present theory of strong interactions? is presented. The answer is
positive as some strong arguments on this assertion that are presented bring
out.

NEUTRINOS AND STRONG INTERACTIONS: QCD (and BEYOND?)111Invited talk at LISHEP
2021, Rio de Janeiro, July 2021

C.A. García Canal

IFLP/CONICET and Departamento de Física
Universidad Nacional de La Plata, C.C.67, 1900, La Plata, Argentina

##  1 Introduction

The first question to pose and to answer seems to be: what has to do a
neutrino that only participates of the weak interaction with QCD the theory of
strong interactions? Through this talk we hope that it will became clear that,
no doubt, the neutrino has a lot to do with QCD.

The history (or perhaps I should write story…) starts in the seventies when
the quark structure of nucleons, the proton in particular, was confirmed by
the experimental data obtained in the Gargamelle bubble chamber at CERN acted
by neutrino beams and it continues nowadays, mainly in connection with the
IceCube detected events initiated by very energetic cosmic neutrinos in this
case.

The experimental study of electron inelastic scattering at SLAC [1] and the
results obtained at CERN [2] in the bubble chamber Gargamelle, in the
seventies, detecting neutrino reactions, gave strong support to the image of
the nucleon containing particular constituents, called partons, that
afterwards were identified with the quarks of the $SU(3)_{flavor}$
spectroscopy hypothesis. Initially, data confirm the scaling behavior of deep
inelastic cross sections, and moreover, that the electric charge assigned to
quarks was confirmed.

Gargamelle apparatus owes its name to the mother of Gargantua, the character
of Rabelais’ series of bookas: ”Gargantua et Pantagruel”, mainly due to the
big size of the bubble chamber. It measured $4.8\,m$ long and $2\,m$ in
diameter, weighed $1000\,T$ and was filler with $12m^{3}$ of heavy-liquid
freon ($CF_{3}Br$). In July 1973, the Gargamelle collaboration announced the
detection of weak neutral currents [3] being the first experimental indication
of the existence of the $Z_{0}$ weak boson, and for that reason a verification
of the electroweak theory based upon the $SU(2)\otimes U(1)$ gauge symmetry.
Unfortunately, in 1979 an irreparable crack appeared in the bubble chamber,
and Gargamelle was decommissioned. It is at present part of the ”Microcosm”
exhibition at CERN. (At that time, the cracking was particularly impressive
because the tank of Gargamelle was builded by the same industry that
constructed nuclear reactor tanks…). This cracking is directly connected with
my participation in the physics of Gargamelle. Together with the Brazilian
colleagues María Beatriz Gay and Jose Martins Sim$\tilde{o}$es we were
contacted in Strasbourg by one of the experimental collaborations working in
Gargamelle, the Aachen-Bergen-Brussels-London-Strasbourg one, that wanted us
to compute the expected number of same sign dimuon events in neutrino-nucleon
scattering to be expected taking into account the small time of recording of
data due to the decommission of the bubble chamber. By using the QCD improved
parton model we concluded that the collaboration could have registered at most
$5$ dimuon events [4]. If they were democratic, each member of the
collaboration should get one event…

In 1961 Gell-Mann, based on the symmetry group $SU(3)_{flavor}$, introduced
the Eightfold Way scheme, which classified the hadrons into representations of
that group. Mathematically, Gell-Mann’s $SU(3)$ model has a fundamental
representation of dimension $3$ that perhaps would imply the existence of
three more elementary particles, that were called ”quarks”. At the same time,
Zweig in a paper of 1964, proposed that mesons and baryons are formed from a
set of three fundamental particles that he called aces.Both Gell-Mann’s quarks
and Zweig’s aces had exotic electrical charges equal to $-1/3$ and $+2/3$ of
the electron charge. The situation, at that time (1966) was such that Gell-
Mann himself stated: ”..the idea that mesons and baryons are made primarily of
quarks is hard to believe…”

As it was previously mentioned, in 1968, electron-proton scattering
experiments by the MIT-SLAC collaboration at SLAC revealed the inner structure
of protons. The experiment was based on electrons colliding protons and the
detection of the electrons that emerge. The scattering results allows one to
conclude that the interaction were caused by point-like, incoherent
constituents inside the protons. In the subsequent years, these results were
confirmed and combined with those from neutrino-scattering in the Gargamelle
bubble chamber at CERN, showing conclusively that these point constituents
have charges of $+2/3\,e$ and $-1/3\,e$.

A brief reminder of the background for this result [5]: In the case of
electromagnetic deep inelastic scattering (DIS) one has for the structure
function in the quark-parton model

$F_{2}(x)=\sum_{i}\,e_{i}^{2}\,q_{i}(x)$

where $e_{i}$ stands for the electric charge of the $i$ quark involved with
quark distribution function $q_{i}$. Consequently, for a proton target

$F_{2}^{\gamma
p}=\frac{4}{9}\,(u+\bar{u})+\frac{1}{9}\,(d+\bar{d})+\frac{1}{9}\,(s+\bar{s})$

and a similar one for $F_{2}^{\gamma n}$ with the replacement
$u\leftrightarrow d$ in the case of a neutron.

On the other hand, when the DIS is initiated by neutrinos, the interactions
present: $\nu\,d\rightarrow\ell^{-}\,u$ and
$\nu\,\bar{u}\rightarrow\ell^{-}\,\bar{d}$ give rise to

$F_{2}^{\nu p}=2(d+\bar{u})$

and again the replacement $u\leftrightarrow d$ allows one to write $F_{2}^{\nu
n}$ For the case of an isoscalar target $N$, the previous expressions ends in
the relation

$\frac{F_{2}^{\nu N}}{F_{2}^{\gamma
N}}=\frac{\frac{5}{9}\,(u+\bar{u}+d+\bar{d})+\frac{2}{9}\,(s+\bar{s})}{2\,(u+\bar{u}+d+\bar{d})}\geq\frac{5}{18}$

This number $5/18$ that comes from the electric charges of quarks, was
experimentally obtained comparing Gargamelle data with the SLAC data, as it is
shown in Fig.1

Figure 1: Comparison of SLAC-Gargamelle DIS data

Consequently, neutrinos gave physical sense to the fractional charge of
quarks, now identified with the nucleon components This also allows stating
that the electron and neutrino scattering measurements found the same
structure. These detected components are the basic degrees of freedom for
building up a quantum field theory of strong interactions based upon the gauge
symmetry $SU(3)_{c}$, QCD.

Much water has gone under the bridge since Gargamelle’s successes and we are
not trying here to summarize the recent history of the neutrino physics [6].
We are only interested in presenting one of the most recent approaches of
using very high energy neutrino data to eventually confirm at these energies
QCD or detect new physics beyond the Standard Model. It is important to
realize that even if neutrinos participate directly only of the weak
interactions (leaving aside gravity of course) they are very useful as probes
of the structure of hadrons and in particular of proton structure, as it was
remarked above.

##  2 Neutrino-Nucleon Interaction

In order to prepare the presentation of the impact of very high energy
neutrinos detected by IceCube, we summarize here the most relevant equations
connected with the neutrino-nucleon interaction, mainly in the deep inelastic
(DIS) regime.

The kinematics of neutrino-nucleon scattering is described in terms of the
momentum transferred measured by $Q^{2}$, the Bjorken $x=Q^{2}/2\,m_{N}\,\nu$,
and the inelasticity $y=Q^{2}/sx=\nu/E$ that measures the energy transfer
$\nu=E_{\nu,f}-E_{\nu,i}$ between the neutrino and the nucleon and with $s$
being the square of the center-of-mass energy. The cross-section for CC
neutrino (and antineutrino) scattering on isoscalar nucleon targets is given
by [7]

$\sigma_{{\rm
CC},0}=\int_{0}^{1}dx\int_{0}^{xs}dQ^{2}{d^{2}\sigma^{\nu(\bar{\nu})N}\over
dx\ dQ^{2}}\,,$ (1)

where

$\displaystyle{d^{2}\sigma^{\nu(\bar{\nu})N}\over dx\ dQ^{2}}$
$\displaystyle=$ $\displaystyle{G_{\mathrm{F}}^{2}\over 2\pi
x}\bigg{(}{m^{2}_{W}\over
Q^{2}+m^{2}_{W}}\bigg{)}^{2}\bigg{[}Y_{+}\,F_{2}^{\nu(\bar{\nu})}(x,Q^{2})$
(2) $\displaystyle-y\,F_{\rm
L}^{\nu(\bar{\nu})}(x,Q^{2})+Y_{-}\,xF_{3}^{\nu(\nu)}(x,Q^{2})\bigg{]}$

is the differential cross-section given in terms of the structure functions
$F_{2}^{\nu(\bar{\nu})},$ $F_{\rm L}^{\nu(\bar{\nu})}$ and
$xF_{3}^{\nu(\bar{\nu})}$, and $Y_{+}=1+(1-y)^{2}$, $Y_{-}=1-(1-y)^{2}$. Here,
$G_{\rm F}$ is the Fermi constant and $m_{W}$ is the $W$-boson mass. At
leading order in perturbative QCD (LO), the structure functions can be given
in terms of parton distributions

$F_{2}^{\nu(\bar{\nu})}=x\big{[}\sum_{i}\alpha_{i}q_{i}(x,Q^{2})+\sum_{j}\alpha_{j}\bar{q}_{j}(x,Q^{2})\big{]}$
$xF_{3}^{\nu(\bar{\nu})}=x\big{[}\sum_{i}\beta_{i}q_{i}(x,Q^{2})+\sum_{j}\beta_{j}\bar{q}_{j}(x,Q^{2})\big{]}$

and, at LO

$F_{\rm L}^{\nu(\bar{\nu})}=0$

For neutrinos, $i=u,d,s,b$ and $j=u,d,c$, with
$\alpha_{i}=\alpha_{j}=\beta_{i}=1$ for $u,d$;
$\alpha_{i}=\alpha_{j}=\beta_{i}=2$ for $s,b$; $\beta_{j}=-1$ for $u,d$;
$\beta_{j}=-2$ for $c$ quarks. For antineutrinos, $i=u,d,c$ and $j=u,d,s,b$,
with $\alpha_{i}=\alpha_{j}=\beta_{i}=1$ for $u,d$;
$\alpha_{i}=\alpha_{j}=\beta_{i}=2$ for $c$; $\beta_{j}=-1$ for $u,d$;
$\beta_{j}=-2$ for $s,b$ quarks.

The NC cross sections on isoscalar targets via $Z$ are given by expressions
similar to (1) and (2), with the $W$ propagator replaced by the $Z$ one. For
NC interactions the LO expressions for the structure functions are given by

$F_{2}^{\nu(\bar{\nu})}=x\big{[}\sum_{i}\alpha_{i}[q_{i}(x,Q^{2})+\bar{q}_{i}(x,Q^{2})]+\sum_{j}\alpha_{j}[q_{j}(x,Q^{2})+\bar{q}_{j}(x,Q^{2})]$
$+\sum_{k}\alpha_{k}[q_{k}(x,Q^{2})+\bar{q}_{k}(x,Q^{2})]\big{]}$

and

$xF_{3}^{\nu(\bar{\nu})}=\sum_{i}x(v_{u}a_{u}+v_{d}a_{d})[q_{i}(x,Q^{2})-\bar{q}_{i}(x,Q^{2})]$

where $i=u,d$, $j=s,b$, $k=c$,
$\alpha_{i}=(a_{u}^{2}+v_{u}^{2}+a_{d}^{2}+v_{d}^{2})/2$,
$\alpha_{j}=a_{d}^{2}+v_{d}^{2}$, and $\alpha_{k}=a_{u}^{2}+v_{u}^{2}$, with
$v_{u}$, $v_{d}$, $a_{u}$, $a_{d}$ the NC vector and axial couplings of $u-$
and $d$-type quarks.

##  3 Experimental Information

High-energy neutrinos are unique messengers of information of very far-away
physics phenomena. For this reason, they could be eventually considered as an
informant of physics beyond the Standard Model. The event rates that one could
registered are a combination of neutrino flux and the corresponding cross
section. In order to avoid the astrophysical uncertainties implied, the
concomitant consideration of up-going and down-going events is in order. By
using this combination one can disentangle physics from astrophysics [8]. This
is precisely the job that IceCube collaboration [9], the neutrino telescope
installed in the Amundsen-Scott station of the South Pole, is doing for us.
This is precisely a counting experiment.

We have investigated the sensitivity of the present and future South Pole
neutrino-detection experiments, in particular on the neutrino-nucleon cross-
section [10]. This was triggered by the detection in IceCube of sensible
neutrino event rates at PeV energies [11, 12]. In fact, the IceCube Neutrino
Observatory at the South Pole, which detects Cherenkov light from charged
particles produced in neutrino interactions, firmly established the existence
of an astrophysical high-energy neutrino component. They particularly refer to
the so-called $HESE$-events (High Energy Starting Events), those that start
inside the detector and in this way the $\mu$-background is substantially
reduced. For energies above $60\,TeV$ the measured flux is

$\phi_{0}(E_{\nu})=2.2\times 10^{-18}~{}\left(\frac{E_{\nu}}{100~{}{\rm
TeV}}\right)^{-2.58}~{}(\mathrm{GeV\,s\,sr\,cm^{2}})^{-1}$

Now is mandatory to go ”up to the South” (see Fig.2) to learn about IceCube as
presented in Fig.3

Figure 2: Our South pictured by Joaquín Torres García Figure 3: Sketch of the
IceCube facility

The IceCube Collaboration recent measurement of the neutrino-nucleon cross
section [13] in the energy region $6.3<E_{\nu}/{\rm TeV}<980$ is

$\sigma_{\nu N}=\sigma_{\rm
SM}\times\left[1.30^{+0.21}_{-0.19}{\rm(stat.)}^{+0.39}_{-0.43}{\rm(syst.)}\right]\,,$

where $\sigma_{\rm SM}$ stands for the Standard Model prediction [14]. The
energy dependence of the cross section was also determined [15]. It has
already been proposed the upgrade to the IceCube-Gen2 [16] to work
technologically at the level of IceCube. A mere scaling of the corresponding
dimensions allows one to estimate that IceCube-Gen2 will have an order of
magnitude larger aperture than IceCube, providing an expected improvement in
the precision of measurements of the neutrino-nucleon cross-section to be
comparable to the perturbative QCD related to collider data. Consequently, our
main goal was to quantitatively infer the sensitivity of IceCube, present and
future, for $\sigma_{\nu N}$ in an energy region well beyond that available
for neutrino beams.

The three neutrino flavors produces distinctive signal when they interact in
ice producing the Cherenkov light detected by the IceCube. The charged current
interaction of $\nu_{e}$ triggers an electromagnetic cascade (or shower) $S$
producing a quite spherically symmetric signal, and therefore exhibits a low
angular resolution but a relatively precise measurement of the $\nu_{e}$
energy. The situation is different for CC interaction $\nu_{\mu}$ induced
events. In this case, the secondary muons travel relatively unhurt through the
ice leaving tracks $T$ that point nearly in the direction of the original
$\nu_{\mu}$, allowing a high angular resolution but the energy deposited
represents only a lower bound of the $\nu_{\mu}$ energy. Certainly,
misclassification is possible. The topology of events produced by different
neutrino flavors and interactions, are indicated in Table 1.

Table 1: Event topology for each neutrino flavor. Interaction type |  $e$ |  $\mu$ |  $\tau$
---|---|---|---
CC | $S$ | $T$ | $S$
NC | $S$ | $S$ | $S$

The rates at IceCube for down- and up-going events have been found [17] to
scale respectively as $\Gamma_{\rm down}\propto\phi\ \sigma_{i}$ and
$\Gamma_{\rm up}\propto\phi\ \sigma_{i}/\sigma_{a}$, where $\phi$ is the
neutrino flux, $\sigma_{i}$ is the cross section for the interaction that
produces the event ($i\in\\{{\rm CC},{\rm NC}\\}$), and $\sigma_{a}$ is the
attenuation cross section, including all the effects due to the fact that
neutrinos go through the Earth.

For a given flux $\phi$ and cross sections $\sigma_{i}$ and $\sigma_{a}$, the
expected number of up-going events of a flavor $\alpha$ produced by a charged
or neutral current interaction may be expressed as

$N_{u}^{i,\alpha}\equiv\tilde{N}_{u}^{i,\alpha}\frac{\phi}{\phi_{0}}\frac{\sigma_{i}/\sigma_{a}^{\alpha}}{\sigma_{i,0}/\sigma_{a,0}^{\alpha}},$
(3a) and for down going events,
$N_{d}^{i,\alpha}\equiv\tilde{N}_{d}^{i,\alpha}\frac{\phi}{\phi_{0}}\frac{\sigma_{i}}{\sigma_{i,0}},$
(3b)

with $i\in\\{{\rm CC},{\rm NC}\\}$ and where the $\tilde{N}$-constants are
obtained assuming that the flux and cross sections are equal to the reference
values, $\sigma_{i,0}$ and $\sigma_{a,0}$.

##  4 Likelihood Analysis

The tool for the analysis we have proposed is the Likelihood estimation.
Likelihood states how probable a set of observations are given particular
values of statistical parameters. To go ahead one introduces the set
$\theta\doteq\\{F,H\\}$ of parameters involved in the data analysis defined by

$\phi=F\,\phi_{0}\,\,\,\,;\,\,\,\,\sigma_{tot}=H\,\sigma_{tot;0}$ (4)

that are relevant to vary $\sigma_{CC;0}$ and $\sigma_{NC;0}$.

Now, maximizing $L(\theta)$, the likelihood, allows one to estimate the most
likely values of the parameters $F$ and $H$. This is obtained for the pair of
values

$\left\\{\begin{array}[]{rcl}H&=&1.16^{+0.51}_{-0.34}\,(1\sigma\,\mbox{C.L.}),\\\\[5.69046pt]
F&=&0.86^{+0.27}_{-0.22}\,(1\sigma\,\mbox{C.L.}).\end{array}\right.$ (5)

In Fig. 4 we show the confidence contours and the associated curves in the
$F-H$ plane for each event type that would produce the observed number of
events of each type.

Figure 4: 1, 3, and $5\sigma$ confidence contours for $(F,H)$ for scaled total
cross section $\sigma_{\rm tot}$ and flux $\phi$ with respect to their
reference values $\sigma_{{\rm tot},0}$, $\phi_{0}$.

Therefore, the cross-section is consistent at $1\sigma$ with the value from
QCD guided by HERA data with some statistics limited by an order of $37\%$.

The next step in our analysis includes $\sigma_{CC}/\sigma_{NC}$ as a free
parameter. This is guided by the fact that in general, all new physics
processes increases the neutral current cross section via, for example,
gravitons, strings, sphalerons, etc. [18], i.e, as non-perturbative SM
effects.

We have used the complete set of ($S\,+\,T$) HESE data to find a bound to the
rise of $\sigma_{\rm NC}$. Because the low number of data and the large
uncertainties in arrival direction of shower events we have integrated over
the angular distribution. Note that the analysis presented herein is
complementary to those reported in [13, 15] as it test a different region of
the neutrino-nucleon cross section parameter space. Indeed, the likelihood fit
given in (6) provides the first clear constraint coming from IceCube data on
beyond the Standard Model phenomena.

We have done again our analysis but keeping the ratio $\sigma_{\rm
CC}/\sigma_{\rm NC}$ as a free parameter in the likelihood function instead to
mantain it fixed to the value $\sigma_{\rm CC}/\sigma_{\rm NC}=3$ that is
expected in the Standard Model.

We write the total neutrino-nucleon cross section as $\sigma_{\rm
tot}=\sigma_{\rm CC,0}+\sigma_{\rm NC}$. Instead of considering the the
parameter $H$ as the one of interest, we perform the analysis to constrain the
ratio $H_{\rm NC}\equiv\sigma_{\rm NC}/\sigma_{\rm NC,0}$. Following a process
similar to that previously used, maximizing the likelihood for the parameters
$\boldsymbol{\theta}=\\{F,S_{\rm NC}\\}$ provides the values

$\left\\{\begin{array}[]{rcl}S_{\rm
NC}&=&0.00^{+0.27}_{-0.00}\,(1\sigma\,\mbox{C.L.}),\\\\[5.69046pt]
F&=&1.16^{+0.20}_{-0.18}\,(1\sigma\,\mbox{C.L.}).\end{array}\right.$ (6)

Figure 5: 1, 3, and $5\sigma$ confidence contours in the $(F,H_{\rm SN})$
plane.

In Fig. 5 the confidence contours and the associated curves in the $F-S_{\rm
NC}$ plane for each event type that would produce the observed number of
events of each type is presented.

We can conclude that $H_{\rm NC}>1$ is excluded at $2\sigma$ level.
Consequently, there is a consistency with the value of QCD coming from HERA
data. This result also is a constraint from IceCube data on non-perturbative
SM phenomena, such as sphaleron transitions, that remain almost unconstrained
by LHC data [19].

##  5 Looking Ahead

Studies for the upgrade to IceCube-Gen2 high-energy array are going on [16].
The planned instrumented volume is around $10~{}{\rm km}^{3}$ and will lead to
larger neutrino detection rates, in all neutrino flavor and detection
channels. A rapid estimate indicates about an order of magnitude increase per
year in the exposure. The sample size of events can be obtained by simply
scaling the instrumented volume.

To get information on the sensitivity of IceCube-Gen2 to probe strong
dynamics, we generate random samples of events following the Poisson
distribution, with the parameters for a scaled total cross section found in
the IceCube data analysis. In 10 years of observation IceCube-Gen2 will
collect about 500 neutrino events in the energy range $0.1\lesssim
E_{\nu}/{\rm PeV}\lesssim 2$, and about 1000 events in 20 years. Thus we adopt
20 and 40 as the representative multiplicative factors associated with these
data samples. Using the high-energy and high-statistics sample to be collected
by IceCube-Gen2, we perform the same likelihood analysis as with the real
data. The precision on the cross section determination would be 7.9% and 5.5%
for $\sim 500$ and $\sim 1000$ events, respectively. This precision is
comparable to that obtained in perturbative QCD calculations guided by HERA
data. Detailed evolution of the uncertainty with sample sizes is illustrated
in Fig. 6.

Figure 6: Evolution of the cross section precision measurement.

We can also dream an IceCube-like detector of $100~{}{\rm km}^{3}$,
specifically designed to probe strong dynamics. In this case the $1\sigma$
contour regions could reach a precision of less than 2% level.

Before ending, we refer to the FASER$\nu$ collaboration at the CERN Large
Hadron Collider. This is an experiment designed to detect collider neutrinos
directly, in order to analyze the neutrino cross-section at the TeV energies
of LHC, a region of energies not yet studied. On May 13, the first evidence of
neutrino interactions at LHC was announced and the event display is shown in
Fig. 7.

Figure 7: Event displays of two of the neutral vertices in the y-z projection
longitudinal to the beam direction (left) and in the view transverse to the
beam direction (right), taken from Ref.[20]

##  6 Conclusions

Neutrino had, have and will have a lot to say about the strong dynamics. We
have recalled the importance of neutrino data in giving physics reality to the
fractional charge of quarks and recently, the IceCube observations informed
about strong dynamics with the neutrino telescope in the Antarctic. By
comparing the rate for up-going and down-going neutrino events detected by
IceCube, one can disentangle effects from the unknown flux and those from
strong (QCD) dynamics. Current experimental results from IceCube provide
constraints on the flux cross-section parameter space. The IceCube HESE data
provide a measurement of the neutrino-nucleon cross section at $\sqrt{s}\sim
1~{}{\rm TeV}$ that is consistent within $1\sigma$ with perturbative QCD
calculations constrained by HERA measurements. The data have also constrained
contributions from non-perturbative processes to the neutrino-nucleon cross
section. In fact, the contributions to the NC interaction at $\sqrt{s}\sim
1~{}{\rm TeV}$ from electroweak sphaleron transitions are excluded at the
$2\sigma$ level.

The potential of future neutrino-detection facilities as IceCube-Gen2 for
measuring the neutrino-nucleon cross section shows a clear improvements to
determine both astrophysical neutrino fluxes and cross section that can arrive
to a precision of about a $6\%$, comparable to perturbative QCD informed by
HERA data. Moreover, a $100~{}{\rm km}^{3}$ detector would reach the precision
of less than a 2% level.

## Acknowledgements

I would like to warmly acknowledge Alberto Santoro for his kind invitation to
participate in LISHEP 2021. I also thanks to all the organizers of this
enjoyable meeting. My gratitude to my colleagues M.B. Gay, J.A. Martins
Simoes, L. Anchordoqui and J. Soriano for their generous attitude in sharing
with me their scientific findings and particularly to L. Anchordoqui for
helping me in improving the manuscript.

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