#  Exact black holes in string-inspired Euler-Heisenberg theory

Athanasios Bakopoulos  0000-0002-3012-6144<EMAIL_ADDRESS>Physics Division,
School of Applied Mathematical and Physical Sciences, National Technical
University of Athens, Zografou Campus, Athens 15780, Greece  Thanasis
Karakasis  0000-0002-5479-6513<EMAIL_ADDRESS>Physics
Division, School of Applied Mathematical and Physical Sciences, National
Technical University of Athens, Zografou Campus, Athens 15780, Greece  Nick E.
Mavromatos  0000-0001-7683-8625<EMAIL_ADDRESS>Physics Division, School
of Applied Mathematical and Physical Sciences, National Technical University
of Athens, Zografou Campus, Athens 15780, Greece  Theoretical Particle Physics
and Cosmology group, Department of Physics, King’s College London, London WC2R
2LS, UK.  Theodoros Nakas  0000-0002-3522-5803<EMAIL_ADDRESS>Physics Division, School of Applied Mathematical and Physical Sciences,
National Technical University of Athens, Zografou Campus, Athens 15780, Greece
Eleftherios Papantonopoulos  0000-0003-1244-922X<EMAIL_ADDRESS>Physics
Division, School of Applied Mathematical and Physical Sciences, National
Technical University of Athens, Zografou Campus, Athens 15780, Greece

###### Abstract

We consider higher-order derivative gauge field corrections that arise in the
fundamental context of dimensional reduction of String Theory and Lovelock-
inspired gravities and obtain an exact and asymptotically flat black-hole
solution, in the presence of non-trivial dilaton configurations. Specifically,
by considering the gravitational theory of Euler-Heisenberg non-linear
electrodynamics coupled to a dilaton field with specific coupling functions,
we perform an extensive analysis of the characteristics of the black hole,
including its geodesics for massive particles, the energy conditions,
thermodynamical and stability analysis. The inclusion of a dilaton scalar
potential in the action can also give rise to asymptotically (A)dS spacetimes
and an effective cosmological constant. Moreover, we find that the black hole
can be thermodynamically favored when compared to the Gibbons-Maeda-Garfinkle-
Horowitz-Strominger (GMGHS) black hole for those parameters of the model that
lead to a larger black-hole horizon for the same mass. Finally, it is observed
that the energy conditions of the obtained black hole are indeed satisfied,
further validating the robustness of the solution within the theoretical
framework, but also implying that this self-gravitating dilaton-non-linear-
electrodynamics system constitutes another explicit example of bypassing
modern versions of the no-hair theorem without any violation of the energy
conditions.

††preprint:  KCL-TH-PH/2024-03

###### Contents

  1. I Introduction
  2. II String inspired non-linear electrodynamics
  3. III Theoretical framework and black-hole solutions
  4. IV Geodesics and energy conditions
    1. IV.1 Geodesics
    2. IV.2 Energy Conditions
  5. V Thermodynamic analysis
    1. V.1 GMGHS black hole
    2. V.2 Black hole with non-linear electrodynamics
  6. VI Linear Perturbations
    1. VI.1 Radial Stability
    2. VI.2 Scalar Quasi-Normal Modes
  7. VII Other Solutions
    1. VII.1 Asymptotically (A)dS spacetimes
    2. VII.2 Solutions for general $\gamma$
  8. VIII Conclusions

##  I Introduction

In the pursuit of a comprehensive understanding of gravitational phenomena in
the cosmos as well as gravity itself, the theoretical examination of black
holes stands as an essential frontier. The General theory of Relativity (GR),
while highly successful in describing the macroscopic behavior of these
celestial entities, becomes subject to scrutiny under extreme conditions. This
investigation prompts the exploration of modified gravitational theories and
theories with extra dimensions. Among the theories attempting to unify
fundamental interactions, String Theory stands as the leading contender. In
particular, the heterotic string theory stands as an essential branch within
the broader scope of String Theory, distinguished by its capability to unify
gravitational interactions with other fundamental forces. Notably, it excels
in synthesizing these interactions into a cohesive framework. A focal point of
interest lies in the derivation of an effective four-dimensional theory,
offering insights into quantum corrections that modify Einstein’s theory of
gravity. These corrections can potentially incorporate terms ranging from the
Gauss-Bonnet, quadratic-curvature term [1, 2, 3, 4] to non-linear
electromagnetic corrections (see e.g. [5, 6, 7, 8] and references within).

Drawing therefore inspiration only from the aforementioned corrections
introduced by string/brane theory, in this article, we aim to elucidate the
implications of departing from the conventional electromagnetic framework and
embracing the intricacies of non-linear electrodynamics (NED) and scalar
fields within the context of black hole solutions. In addition to the above,
it is important to mention that four-dimensional scalar-vector-tensor theories
can be also obtained via an appropriate reduction from a higher-dimensional
Lovelock theory [9]. Under this perspective, scalar-tensor-vector theories can
be understood as natural extensions of the scalar-tensor theories that have
been extensively studied in the last decades [4, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]. Such scalar-
vector-tensor theories offer a very fruitful framework for finding novel
compact-object solutions, eluding the constraints imposed by the “no-hair”
theorems [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68,
69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81]. This departure from
traditional limitations is attributed to the dilaton field, which, notably,
introduces no additional independent free parameter into the resulting
solution [82]. Instead, these solutions manifest a distinct feature known as
“secondary hair”, intricately determined by the compact object’s mass, charge,
and angular momentum.

Despite the motivation coming from higher-dimensional theories, there are also
additional reasons that lead us to explore non-linear electrodynamics. First
and foremost, in regions with strong gravitational fields, such as those near
black holes, traditional linear theories may break down. Non-linear
electrodynamics becomes important in these strong field regimes, where the
intensity of electromagnetic fields can become comparable to the strength of
gravitational fields. Studying how non-linearities affect the behavior of
electromagnetic fields in these regimes is crucial for understanding the
physics of objects like black holes, neutron stars, and other astrophysical
phenomena. Moreover, non-linear electrodynamics is expected to lead to
phenomena that are absent in linear theories. In the early universe for
example, when energy densities were extremely high, the interplay between
gravitational and electromagnetic fields was significant. NED can be crucial
in modeling the behavior of these fields during cosmological evolution.
Consequently, NED allows us to investigate how electromagnetic interactions
influenced the dynamics of the early universe and whether non-linear effects
played a role in the formation of cosmic structures. Understanding these
cosmological implications helps build a more complete picture of the evolution
of the universe. For a review on non-linear electrodynamics and its
applications, see [83] and references therein.

In this paper, we embark on a comprehensive exploration of a gravitational
theory extending the classical Euler-Heisenberg (EH) electrodynamics coupled
to a non-trivial dilaton field. Our motivation for this study stems from the
rich theoretical landscape it promises, building upon the established
framework of self-gravitating dilaton-linear-electrodynamics. This extension
allows us to delve into intriguing phenomena, notably exemplified by the
Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole [84, 85], a
significant exact solution within this domain. In our investigation, we
examine the intricacies of our proposed model and unravel its associated
black-hole solution in detail. One of our key insights lies in the strategic
assumption of a specific profile governing the dilaton coupling to the Euler-
Heisenberg terms. This choice results in an exact analytic black-hole
solution, facilitating a straightforward examination of its physical
characteristics.

Having the solution at hand, we then commence a rigorous analysis encompassing
various facets of our model’s implications. This includes a thorough
examination of the geodesics of massive test particles within the black-hole
spacetime, followed by a meticulous scrutiny of the energy conditions.
Subsequently, we delve into the thermodynamic aspects of the black hole,
computing the relevant thermodynamic quantities, such as the temperature, the
entropy, and the magnetic potential ($\Phi_{m}$), to demonstrate the validity
of the first law of thermodynamics. Moreover, within the parameter space of
solutions, we unveil the existence of pairs consisting of two distinct black
holes characterized by different ratios $Q_{m}/M$, both more compact than the
Schwarzschild solution yet sharing identical horizon radii. Intriguingly,
despite their geometric similarity, a thermodynamic analysis reveals clear
distinctions, with one black hole exhibiting thermodynamic stability while its
_doppelgänger_ proves to be thermodynamically unstable. Additionally, we
explore the radial stability of the black-hole solution under linear
perturbations and also its scalar quasi-normal modes, shedding light on its
potential as an astrophysical entity. Furthermore, we extend our discussions
to encompass other solutions and extensions of our model theory, including
asymptotically (Anti-)de Sitter (AdS) spacetimes and more general dilaton
couplings, providing a comprehensive overview of the theoretical landscape. In
conclusion, our work offers a thorough investigation into the gravitational
theory of non-linear EH electrodynamics coupled to a non-trivial dilaton
field, unraveling a plethora of intriguing phenomena and paving the way for
further exploration and theoretical advancements in this domain.

The structure of the current article is the following: In the next section II
we motivate our study by discussing a gravitational theory of non-linear
Euler-Heisenberg (EH) electrodynamics coupled to a non-trivial dilaton field.
This generalises the corresponding self-gravitating dilaton-linear-
electrodynamics case, known to admit the GMGHS black hole [84, 85] as an exact
solution. In the subsequent section III, we discuss our model and its
associated black hole solution. By assuming a specific profile for the dilaton
coupling to the EH terms, in such a way that, additionally to non-trivial
dilaton couplings, one has also dilaton-independent EH terms, we demonstrate
the possibility of studying analytically the corresponding black-hole
solution. In section IV we first discuss the geodesics of test particles in
such black hole spacetimes, and then demonstrate the satisfaction of the
energy conditions for appropriate sets of the parameters of the solution. In
section V, we study the thermodynamics of the black hole, and show explicitly,
by computing the relevant thermodynamical quantities, that the first law of
thermodynamics is satisfied in a coordinate-independent way, as should have
been expected. In the parameter space of solutions, it is possible to obtain
two distinct black holes with different ratios $Q_{m}/M$ that are more compact
than the Schwarzschild solution and share the same horizon radius. However,
these black holes even though they have the same horizon radius, from a
thermodynamic point of view, are quite distinguishable, since the solution
with a greater value for the ratio $Q_{m}/M$ is thermodynamically stable,
while its doppelgänger with a lower value for the ratio $Q_{m}/M$ is
thermodynamically unstable. In section VI, we demonstrate the radial stability
of the black-hole solution under linear perturbations, and study its scalar
quasi-normal models, which provide insights into its properties as a potential
astrophysical object. Other solutions of (extensions of) our model theory
(15), including asymptotically (Anti-)de Sitter (AdS) spacetimes, as well as
solutions corresponding to more general couplings $\exp(-2\gamma\phi)$, of the
dilaton to the Maxwell term in the action, rather than the $\gamma=1$ in
closed strings, are discussed in section VII. Finally, conclusions and outlook
are given in section VIII.

##  II String inspired non-linear electrodynamics

One of the particular aspects of string/brane-induced non-linear
electrodynamics effects is that the higher order in the Maxwell tensor can be
combined into an all-order expression, the so-called Born-Infeld (BI)
Lagrangian [86, 87, 88, 89, 90, 91], as a result of re-summation of open
string excitations (attached to, e.g., 3-brane worlds in the D-brane extension
of string theory, in which case the world-volume of ($d=3$)-brane leads to the
DiracBI (DBI) action (see [92, 93, 94] and references therein). In such
models, the BI electrodynamics in four spacetime dimensions originates from
the higher ($d=10$)-dimensional superstring action upon either
compactification or appropriate restriction on a $3d$-brane volume. It is
important to note that in all such string-inspired models the BI Lagrangian
couples to the inverse of the open-string coupling $g_{s}=e^{\phi}$, where
$\phi$ is the (dimensionless) dilaton field, so the corresponding four-
dimensional action in a curved four-dimensional background metric (in the
Jordan or $\sigma$-model frame), $g^{J}_{\mu\nu}$, reads

$\displaystyle\mathcal{S}^{J}_{\rm BI}=-\mathcal{T}_{4}^{2}\int
d^{4}x\,e^{-\phi}\sqrt{{\rm
Det}\Big{(}-g^{J}_{\mu\nu}+\mathcal{T}_{4}^{-1}\,\mathcal{F}_{\mu\nu}\Big{)}}\,,$
(1)

where $\mathcal{F}_{\mu\nu}$ is the Maxwell tensor
$\mathcal{F}_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}$,
and $\mathcal{T}_{4}=\frac{1}{2\pi\alpha^{\prime}}=\frac{M_{s}^{2}}{2\pi}\,,$
is the (open) string tension, with $\alpha^{\prime}=M_{s}^{-2}$ the Regge
slope ($M_{s}$ the string mass scale, which in general is different from the
four-dimensional Planck scale). One may go away from string/brane theory and
define the BI action as a starting point of an effective modified
electrodynamics field theory. In such a case the string tension
$\mathcal{T}_{4}$ may be considered as an arbitrary phenomenological
dimensionful parameter which is not related to the Regge slope
$\alpha^{\prime}$. We term such a parameter the BI parameter.

We next remark that, in four spacetime dimensions, the determinant in the
argument of the square root in the BI action (1) can be expanded exactly to
yield:

$\displaystyle\mathcal{S}_{\rm BI}=-\mathcal{T}_{4}^{2}\int
d^{4}x\,\sqrt{-g^{J}}\,e^{-\phi}\sqrt{1+\frac{1}{2\,\mathcal{T}_{4}^{2}}\mathcal{F}_{\mu\nu}\,\mathcal{F}^{\mu\nu}-\frac{1}{16\,\mathcal{T}_{4}^{4}}\,\Big{(}\mathcal{F}_{\mu\nu}\,\widetilde{\mathcal{F}}^{\mu\nu}\Big{)}^{2}}\,$
(2)

where
$\widetilde{\mathcal{F}}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}\,\mathcal{F}^{\rho\sigma}$
is the dual of the Maxwell tensor, with $\varepsilon_{\mu\nu\rho\sigma}$ the
Levi-Civita fully antisymmetric symbol in curved spacetime with metric
$g^{J}_{\mu\nu}$. Expanding the (square root in the) four-dimensional BI
action (2) in inverse powers of the BI parameter $\mathcal{T}_{4}$, leads to
effective dimension $8$ (and higher) operators in the effective field theory,
which make contact with the generic Euler-Heisenberg (EH) non-linear
electrodynamics (NED) [91, 95, 96]:

$\displaystyle\mathcal{S}_{\rm BI}$ $\displaystyle=\int
d^{4}x\sqrt{-g^{J}}\,e^{-\phi}\Big{[}-\mathcal{T}_{4}^{2}\,I_{2}-\mathcal{T}_{4}^{4}\,I_{4}\Big{(}1+\mathcal{O}(\mathcal{F}^{2})\Big{)}\Big{]}\,,$
$\displaystyle I_{2}$
$\displaystyle=\frac{1}{4\mathcal{T}_{4}^{2}}\,\mathcal{F}_{\mu\nu}\,\mathcal{F}^{\mu\nu}\,,\quad
I_{4}=-\frac{1}{8\,\mathcal{T}_{4}^{4}}\mathcal{F}_{\mu\nu}\,\mathcal{F}^{\nu\rho}\,\mathcal{F}_{\rho\lambda}\,\mathcal{F}^{\lambda\mu}+\frac{1}{32\,\mathcal{T}_{4}^{4}}\,\Big{(}\mathcal{F}_{\mu\nu}\,\mathcal{F}^{\mu\nu}\Big{)}^{2}\,.$
(3)

Hence, ignoring for the moment the dilaton, to fourth order in the field
strength $\mathcal{F}_{\mu\nu}$ one obtains (up to the dilaton factors) a
special case of the generic EH NED with dimension $8$ operators, with
Lagrangian:

$\displaystyle\mathcal{L}_{\rm
EH}=c_{1}\,\Big{(}\mathcal{F}_{\mu\nu}\,\mathcal{F}^{\mu\nu}\Big{)}^{2}+c_{2}\,\mathcal{F}_{\mu\nu}\,\mathcal{F}^{\nu\rho}\,\mathcal{F}_{\rho\lambda}\,\mathcal{F}^{\lambda\mu}\,,$
(4)

where the BI Lagrangian corresponds to [95, 96]

$\displaystyle c_{1}=-\frac{1}{32\mathcal{T}_{4}^{2}}\,,\quad
c_{2}=\frac{1}{8\,\mathcal{T}_{4}^{2}}\,.$ (5)

The reader should notice that the ratio of $c_{2}/c_{1}=-4$ exactly, which is
a characteristic prediction of the BI theory.

Phenomenologically, assuming a constant dilaton and flat Minkowski spacetime,
the BI parameter $\mathcal{T}_{4}^{2}$ can be constrained in collider physics,
via light-by-light scattering, for which there is clear experimental evidence
these days at LHC experiments (see [97, 98, 99]). Such light-by-light
scattering studies [95] can place a lower bound on the BI parameter
$\mathcal{T}_{4}\gtrsim 100~{}{\rm GeV}$. In the case of string theory, this
would lead to a (weak) lower bound of the string mass scale $M_{s}\gtrsim
0.25~{}{\rm TeV}$. Notably, extra dimension collider (LHC) searches place
currently this bound to $M_{s}\gtrsim\mathcal{O}(10)$ TeV. Forecasts for much
larger values of the lower bounds of the BI parameter in future colliders, in
particular FCC, have been given in [96]. Embedding the BI (or more generally
Euler-Heisenberg) theory into curved spacetime, and fully incorporating the
dilaton effects, leads to a whole new area of tests of NED by employing the
entire machinery of modern gravitational experiments technology.

The BI action $\mathcal{S}_{\rm BI}$ (1) and (2) in curved background metrics
can be augmented, at an effective field theory level, by including the
dynamics of the gravitational ($g^{J}_{\mu\nu}$) and dilaton ($\phi$) fields.
In this respect, we recall that the D-brane action is by construction in the
so-called Jordan (or $\sigma$-model) frame. Passing into the Einstein frame in
four dimensions, via the transformation of the metric: $g^{J}_{\mu\nu}\to
g_{\mu\nu}=e^{-2\phi}g^{J}_{\mu\nu}$, we write for the pertinent gravitational
action (in geometrized units $c=G=1$, in which we work from now on):

$\displaystyle\mathcal{S}=\frac{1}{16\pi}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\Big{[}\mathcal{R}-2\nabla^{\mu}\phi\nabla_{\mu}\phi\Big{]}-\int
d^{4}x\,\sqrt{-g}\,e^{-\phi}\Big{[}\mathcal{T}_{4}^{2}\,I_{2}^{\rm
E}+\mathcal{T}_{4}^{4}\,e^{-4\phi}\,I_{4}^{\rm E}\Big{]}+\dots\,,$ (6)

where the quantities $I_{i}^{\rm E},\,i=2,4$ are given by the corresponding
ones in (II), but the indices contraction is made by the Einstein-frame metric
$g_{\mu\nu}$.

Departing from the case of the brane DBI action (1), one may consider higher-
order (in derivatives, that is in a Regge slope $\alpha^{\prime}$ expression)
electromagnetic terms in effective low energy field theories stemming only
from closed strings, e.g. the heterotic string [1]. In such theories, unlike
the DBI brane or open-string case, there is no resummation in closed form of
the gauge terms. Nonetheless, some authors have generalised the BI effective
action in a curved $(3+1)$-dimensional spacetime, by considering the following
form of dilaton couplings to the electromagnetic fields in a BI NED setting
[100, 101, 102]:

$\displaystyle\mathcal{S}$
$\displaystyle=\frac{1}{16\pi}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\Big{[}\mathcal{R}-2\nabla^{\mu}\phi\nabla_{\mu}\phi+\mathcal{L}_{\rm
BI}\Big{]}~{},$ $\displaystyle\mathcal{L}_{\rm BI}$ $\displaystyle=4\beta_{\rm
BI}\,e^{2\gamma\phi}\,\Big{(}1-\sqrt{1+\frac{e^{-4\gamma\phi}}{2\beta_{\rm
BI}}\mathcal{F}^{2}-\frac{e^{-8\gamma\phi}}{16\,\beta_{\rm
BI}^{2}}(\mathcal{F}\widetilde{\mathcal{F}})^{2}}\,\,\,\Big{)}$ (7)

where the notation has been defined above, $\gamma$ defines the dilaton
coupling, and now $\beta_{\rm BI}$ plays the role of the generalised BI
parameter, with mass dimensions $+2$ (which is identified with
$\mathcal{T}_{4}^{2}$ in the case of strings, in which case, to match with the
corresponding $\mathcal{O}(\alpha^{\prime})$ Maxwell terms of the heterotic-
string effective action [1], $e^{-2\phi}\mathcal{F}^{2}$, one should fix
$\gamma=1$).

The above considerations deal with tree-level in string loops, that is first
quantised actions on world-sheet with trivial topology ($2d$ sphere for closed
string sectors, and disc for open one). In general, string loop effective
actions are not known in closed form. In simplified phenomenological scenarios
such effective actions can be expressed in the generic form, e.g. in the
closed string sector in the string (or $\sigma$-model frame with metric
$\widehat{g}_{\mu\nu}$ in ($3+1$)-dimensions, after string compactification)
[103]:

$\displaystyle S=\int
d^{4}x\,\sqrt{-\widehat{g}}\,\Big{(}\frac{1}{\alpha^{\prime}}\,B_{g}(\Phi)\,\widehat{\mathcal{R}}+\frac{1}{\alpha^{\prime}}\,B_{\Phi}(\Phi)\,\Big{[}\widehat{\Box}\Phi-4\widehat{\nabla}_{\mu}\Phi\,\widehat{\nabla}^{\mu}\Phi\Big{]}-\frac{B_{F}(\Phi)}{4}\widehat{F}_{\mu\nu}\,\widehat{F}^{\mu\nu}-B_{\psi}(\Phi)\overline{\widehat{\psi}}\,\not{\widehat{D}}\widehat{\psi}+\dots\Big{)}\,,$
(8)

where the $\widehat{(...)}$ symbol above a tensorial quantity implies
contraction of the world-indices with the string-frame metric
$\widehat{g}_{\mu\nu}$, $F_{\mu\nu}$ denotes the field strength of the gauge
field, $D_{\mu}$ is the gauge covariant derivative, $\psi$ are fermionic
matter fields and the $\dots$ denote other matter fields as well as (an
infinity of) higher-derivative (higher order in $\alpha^{\prime}$) terms, The
quantities $B_{i}(\Phi),i=g,\Phi,F,\psi$ are non-derivative functions of the
dilaton which arise from summing over (closed) world-sheet topologies, that is
these functions involve powers of the string coupling $g_{s}=\exp(\Phi)$ of
the form $g_{s}^{-\chi}$, where $\chi=2-2N$ where $N$ denbotes the number of
handles, is the genus of the world-sheet surface (sphere has $N=0$, torus (one
string loop) $=0$ etc. Thus,

$\displaystyle
B_{i}(\Phi)=e^{-2\Phi}+c_{0}^{(i)}+c_{1}^{(i)}e^{2\Phi}+\dots+c_{2n}^{(i)}\,e^{2n\Phi}+\dots\,,$
(9)

where the constant quantities $c_{i}$ pertain to effects of string loops, so
that the expressions (9) involve a power series in the square of the string
coupling $g_{s}^{2}=\exp(2\Phi)$. The first term on the right-hand side of (9)
leads to the standard closed string expression for the gauge field Maxwell
terms in the low-energy effective action, $e^{-2\phi}\mathcal{F}^{2}$ for
standard dilaton kinetic term normalization in the Einstein frame [1, 2,
3]).111We note for completion that a similar factor accompanies the quadratic
gravitational curvature (Gauss-Bonnet (GB)) terms in the action at string-loop
tree level. This is a remnant of the corresponding situation of the ten-
dimensional target-spacetrime heterotic-string effective field theory action,
which in the extra (compact) dumensional sector leads to the celebrated
anomaly cancellation by equating the extra-dimensional (non-Abelian) gauge
with the corresponding quadratic-curvature gravitational GB terms $\int
d^{6}x\sqrt{-G^{(6)}}e^{-2\Phi}\Big{(}{\rm Tr}\mathbf{F}^{2}-\mathbf{R}_{\rm
GB}^{2}\Big{)}\to 0$ (with the Tr being a group-index trace), which leads to
the Heterotic string selecting the $E_{8}\times E_{8}$ gauge group as the
unique target-space group before compactification to (3+1)-dimensions [2, 3].

Passing to the Einstein frame, via appropriate redefinitions of [103]: the
metric $\widehat{g}_{\mu\nu}\rightarrow
g_{\mu\nu}=C\,B_{g}(\Phi)\widehat{g}_{\mu\nu}$, where $C$ are numerical
normalization constants, the dilaton $\Phi\rightarrow\phi=\int
d\Phi\,\sqrt{\frac{3}{4}\Big{(}\frac{B_{g}^{\prime}}{B_{g}}\Big{)}^{2}+2(\frac{B_{\Phi}^{\prime}}{B_{\Phi}}+\frac{B_{\Phi}}{B_{g}})}$,
where the prime denotes $d/d\Phi$, and the fermionic matter fields,
$\widehat{\psi}\rightarrow\psi=C^{-3/4}\,B_{g}^{-3/4}\,B_{\psi}^{1/2}\,\widehat{\psi}$,
leaving the gauge fields as they are, yields the effective action:

$\displaystyle S$ $\displaystyle=\int
d^{4}x\,\sqrt{-g}\frac{1}{16\pi\overline{\rm G}}\,R-\frac{1}{8\pi\overline{\rm
G}}+S_{\rm matter}\,,$ $\displaystyle S_{\rm matter}$ $\displaystyle=\int
d^{4}x\,\sqrt{-g}\Big{[}-\overline{\psi}\not{D}\psi-\frac{1}{4}B_{F}(\Phi)F_{\mu\nu}\,F^{\mu\nu}+\dots\Big{]}\,,$
(10)

where the reader should notice the potential existence (depending on the
specific type of string theory considered) of constant (dilaton-independent)
terms involving $F^{2}$ terms (cf. the $c_{0}^{(F)}$ terms on the right-hand
side of (9)).

In case we consider more general theories involving a combination of closed
and open strings (the latter attached, e.g. to brane universe)s, for which one
obtains effective actions in the Einstein frame that include both closed- and
open-string sectors (the latter leading to DBI terms of the form appearing in
the second integral on the right-hand side of (6)), then, the inclusion of
string loops can lead, following similar arguments to the closed-string case
(II), to generalised situations, in which the (string-loop corrected)
effective action acquires the form in the Einstein frame [104]: 222Indeed, if
only Abelian gauge fields are considered then only open world-sheet surfaces
are taken into account, in order to evaluate the pertinent contribution to the
effective action, as discussed explicitly in [104] where it was shown that the
loop corrected effective action acquires the form in the sigma-model frame
(ignoring antisymmetric tensor fields contributions, which are of no interest
in the present discussion):  $\displaystyle S=\int
d^{4}x\sqrt{-\widehat{g}}\,{\alpha^{\prime}}^{-2}e^{-2\phi}\,\Big{[}-\frac{3}{2}\alpha^{\prime}\,(\widehat{\mathcal{R}}+4(\partial\phi)^{2}+\dots]+d_{1}\,e^{-\phi}\,\sqrt{{\rm
det}\Big{(}\widehat{g}_{\mu\nu}+2\pi\alpha^{\prime}F_{\mu\nu}\Big{)}}+d_{2}+d_{3}+d_{4}\,e^{\phi}+\dots\,,$
(11) where $d_{i}$, $i=1,2,\dots,$ denote finite parts of the dilaton
tadpoles, and the dots denote contributions from higher derivative
corrections, as well as higher string loops (that is higher powers of the
string coupling $g_{s}=\exp(\phi)$. Passing onto the appropriate Einstein
frame leads to actions of the form (12).

$\displaystyle\mathcal{S}=\frac{1}{16\pi}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\Big{[}\mathcal{R}-2\nabla^{\mu}\phi\nabla_{\mu}\phi\Big{]}-\int
d^{4}x\,\sqrt{-g}\,B_{F^{2}}(\phi)\,\Big{[}\mathcal{T}_{4}^{2}\,I_{2}^{\rm
E}\Big{]}-\int
d^{4}x\,\sqrt{-g}\,\mathcal{T}_{4}^{4}\,B_{F^{4}}(\phi)\,I_{4}^{\rm
E}\Big{]}+\dots\,,$ (12)

where the functions $B_{F^{i}}(\phi),\,i=2,4$ admit a power series expansion
in the string coupling, summing up terms of the generic form

$\displaystyle
B_{F^{i}}(\phi)=\sum_{\chi}g_{s}^{-\chi}c_{\chi}^{(F^{i})}\,,\qquad
i=1,2\,,\qquad g_{s}=\exp(\phi)\,,$ (13)

where $\chi=2-2N-N_{H}$, with $N_{H}$ the number of holes (or boundaries) (eg
disc has genus $\chi=1$, since $N=0$, $N_{H}=1$ etc), where finite parts of
dilaton tadpoles contribute to the coefficients $c_{\chi}^{(F^{i})}$.

In heterotic strings, which do not involve branes, the higher derivative EH
electrodynamics terms do not appear in a closed BDI form. In that case a more
general action, involving EH terms, after summation over string loops, might
then be considered, in the Einstein frame:

$\displaystyle\mathcal{S}=\frac{1}{16\pi}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\Big{[}\mathcal{R}-2\nabla^{\mu}\phi\nabla_{\mu}\phi\Big{]}-\int
d^{4}x\,\sqrt{-g}\,B_{F^{2}}(\phi)\,\Big{[}\mathcal{T}_{4}^{2}\,I_{2}^{\rm
E}\Big{]}-\int d^{4}x\sqrt{-g}\,\mathcal{T}_{4}^{4}B_{F^{4}}(\phi)\,L_{\rm
EH}+\dots\,,$ (14)

where the ellipsis ($\dots$) includes possible string-loop generated dilaton-
potential terms, whose precise form is not known at present, as this is a
highly string-model-dependent issue, and the functions $B_{F^{2}}(\phi)$,
$B_{F^{4}}(\phi)$ in this case are given by power series expansions of even
powers of the string coupling, of the form (9), as only closed world-sheet
surfaces are involved. The Euler-Heisenberg Lagrangian $L_{\rm EH}$ is given
by (4), but the coefficients $c_{i},i=1,2$ no longer satisfy (5), given that
the DBI action no longer describes the electromagnetic self-interactions in
closed form. In this work we shall use the framework (14) to discuss black
hole solutions in a phenomenological manner, keeping the coefficients $c_{i}$
of the EH Lagrangian as arbitrary. As already mentioned, in such a framework,
for some sufficiently high string-loop order one can conjecture that there
would exist a dilaton independent term in the function $B_{F^{4}}(\phi)$. As
we shall discuss in the next section, such a constant term plays a crucial
rôle in our phenomenological analysis in this paper in yielding analytic black
hole solutions of a dilaton-EH theory that could be the low energy limit of an
appropriate underlying string theory. However in our work, we shall be more
general, and our analysis will be presented independent of strings.

##  III Theoretical framework and black-hole solutions

In the geometrised unit system ($c=G=1$), the Einstein-frame action functional
that we will occupy us in this article is a simplified version of (14) and
reads

$\mathcal{S}=\frac{1}{16\pi}\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\Big{[}\mathcal{R}-2\nabla^{\mu}\phi\nabla_{\mu}\phi-e^{-2\phi}\mathcal{F}^{2}-f(\phi)\big{(}2\alpha\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}-\beta\mathcal{F}^{4}\big{)}\Big{]}~{}.$
(15)

Such a field theoretic gravitational actions also arises as part of a non-
diagonal reduction of the Gauss-Bonnet action [9] and admits the GMGHS black
hole [84, 85] as an exact solution when $f(\phi)=0$. In (15), $\mathcal{R}$ is
the Ricci scalar,
$\mathcal{F}^{2}\equiv\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}\sim\bf{E}^{2}-\bf{B}^{2}$
is the usual Faraday scalar, and
$\mathcal{F}^{4}\equiv\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu}\mathcal{F}_{\alpha\beta}\mathcal{F}^{\alpha\beta}$,
where $\mathcal{F}_{\mu\nu}$ stands for the usual field strength
$\mathcal{F}_{\mu\nu}=\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}$
and $\alpha,\beta$ are coupling constants of the theory, with dimensions
(length)2, which in our discussion are treated phenomenologically. The scalar
field $\phi$ and the associated scalar function $f(\phi)$ are both
dimensionless.333The reader should be reminded at this stage that in the
special case of (open)string/brane-inspired BI theory at tree-level in string
loops, the function $f(\phi)\sim e^{-5\phi}$ (cf. (6)), however in such a case
the Maxwell term $\mathcal{F}^{2}$ in (15) should be accompanied by the
inverse of the open string coupling, ie. $e^{-\phi}$, instead of $e^{-2\phi}$
that appears in (15). On the other hand, in the heterotic-string-inspired
model (15) can be mapped to the model (II), upon choosing $\gamma=1$, and
$\mathcal{F}\widetilde{\mathcal{F}}=0$, that is concentrating on magnetically
charged black holes only (in which case, the function $f(\phi)\sim
e^{-6\phi}$). However, as we have already stressed, and we shall argue below,
it is crucial for an analytic treatment of the black-hole solution to have a
dilaton-independent term in $f(\phi)$, which, as we have argued in the
previous section, can be induced by considering higher-order string loop
corrections in the underlying string-theory model. For the moment we do not
consider a potential for the dilaton, but only its non-linear interactions
with the EH terms. The addition of a pure dilaton potential
$\mathfrak{V}(\phi)$ can lead to interesting alternative solutions, including
a cosmological constant, which we discuss in section VII.

The field equations emanating from (15) are of the following form

$\displaystyle G_{\mu\nu}=$ $\displaystyle\
2\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\partial^{\alpha}\phi\partial_{\alpha}\phi+2e^{-2\phi}\left(\mathcal{F}_{\mu}^{~{}\alpha}\mathcal{F}_{\nu\alpha}-\frac{1}{4}g_{\mu\nu}\mathcal{F}^{2}\right)+$
$\displaystyle
f(\phi)\left\\{8\alpha\mathcal{F}_{\mu}^{~{}\alpha}\mathcal{F}_{\nu}^{~{}\beta}\mathcal{F}_{\alpha}^{~{}\eta}\mathcal{F}_{\beta\eta}-\alpha
g_{\mu\nu}\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}-4\beta\mathcal{F}_{\mu}^{~{}\xi}\mathcal{F}_{\nu\xi}\mathcal{F}^{2}+\frac{1}{2}g_{\mu\nu}\beta\mathcal{F}^{4}\right\\}~{},$
(16)
$4\square\phi=-2e^{-2\phi}\mathcal{F}^{2}+\frac{df(\phi)}{d\phi}\left(2\alpha\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}-\beta\mathcal{F}^{4}\right)~{},$
(17) $\partial_{\mu}\Big{\\{}\sqrt{-g}\Big{[}4\mathcal{F}^{\mu\nu}\left(2\beta
f(\phi)\mathcal{F}^{2}-e^{-2\phi}\right)-16\alpha\mathcal{F}^{\mu}{}_{\kappa}\mathcal{F}^{\kappa}{}_{\lambda}\mathcal{F}^{\nu\lambda}\Big{]}\Big{\\}}=0\,.$
(18)

By taking into account the higher-order electromagnetic invariants
$\mathcal{F}^{4}$ and
$\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}$,
we are interested in extending the GMGHS solution [84, 85]. To do so, we
introduce the most general spherically symmetric metric ansatz in the form

$\mathop{}\\!\mathrm{d}s^{2}=-B(r)\mathop{}\\!\mathrm{d}t^{2}+\frac{\mathop{}\\!\mathrm{d}r^{2}}{B(r)}+[R(r)]^{2}\mathop{}\\!\mathrm{d}\Omega^{2}~{},$
(19)

where $B(r),R(r)$ are two unknown functions to be determined from the field
equations, while
$\mathop{}\\!\mathrm{d}\Omega^{2}=\mathop{}\\!\mathrm{d}\theta^{2}+\sin^{2}\theta\mathop{}\\!\mathrm{d}\varphi^{2}$.
444Note that throughout this article, $\varphi$ will always denote the
azimuthal coordinate, while $\phi$ will always denote the scalar field.
Moreover, we consider both electric and magnetic charges, via the following
four-vector, which is compatible with spherical symmetry,

$\mathcal{A}_{\mu}=(V(r),0,0,Q_{m}\cos\theta)~{},$ (20)

where $Q_{m}$ stands for the magnetic charge carried by the black hole. This
ansatz for the electromagnetic field solves by construction the $\varphi$
component of the Maxwell equations iff one considers that the scalar field
inherits the spacetime symmetries, namely $\phi\equiv\phi(r)$. Interestingly,
one can see that the combination

$2\alpha\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}-\beta\mathcal{F}^{4}=\frac{4(\alpha-\beta)Q_{m}^{4}}{[R(r)]^{8}}+\frac{8\beta
Q_{m}^{2}[V^{\prime}(r)]^{2}}{[R(r)]^{4}}+4(\alpha-\beta)[V^{\prime}(r)]^{4}~{},$
(21)

will vanish if one does not consider both electric and magnetic configurations
in the case of $\alpha=\beta$. In the above, prime denotes derivation with
respect to $r$. Maxwell’s equation is very difficult to be integrated for the
dyonic case and as a result we will consider pure magnetic fields, that is
$V(r)=0$. Consequently, both these non-linear electrodynamics terms will
contribute iff $\alpha\neq\beta$. We will begin our analysis for the scalar
free scenario $\phi=0,f(\phi=0)=1$, for which the solution reads

$B(r)=1-\frac{2M}{r}+\frac{Q_{m}^{2}}{r^{2}}+\frac{2(\alpha-\beta)Q_{m}^{4}}{5r^{6}}~{},$
(22)

and $R(r)=r$. This solution resembles the Einstein-Euler-Heisenberg black hole
[105]. The interesting thing to notice in (22) is that the non-linear
electromagnetic terms
$\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}$
and $\mathcal{F}^{4}$ affect the spacetime geometry in a similar way. It is
solely the values of the coupling constants $\alpha$ and $\beta$ that
determine whether this contribution survives or not. Note that in the case of
$\alpha=\beta$ the higher-order electromagnetic term does not contribute at
all. However, in the case where $\alpha\neq\beta$, we notice that depending on
the signs of the parameters $\alpha$ and $\beta$, the non-linear
electromagnetic terms can act either attractively or repulsively. Black holes
with a scalar hair in the Euler-Heisenberg theory have been discussed in [64],
and it was found that the scalar hair results in a more compact black hole
(having a smaller radius for the event horizon) when compared to the non-hairy
Einstein-Euler-Heisenberg black hole.

Let us now assume a non-trivial profile for the coupling function $f(\phi)$.
In particular, we consider

$f(\phi)=-[3\cosh(2\phi)+2]\equiv-\frac{1}{2}\left(3e^{-2\phi}+3e^{2\phi}+4\right)~{}.$
(23)

Notice here that the coupling function $f(\phi)$ contains the dilatonic
coupling $e^{2\xi\phi}$ with $\xi=\pm 1$ as well as a constant (dilaton
independent) term. At this point the reader is invited, for completion, to
compare such couplings with the string-loop corrected coupling functions
$B_{F^{4}}(\phi)$, in the framework of string-inspired models (14), discussed
in the previous section II. In such a stringy context, the exponential dilaton
terms in the coupling function (23) can be written as
$f(\phi)=-\frac{3}{2}(g_{s}^{-2}+g_{s}^{2})-2$, where $g_{s}=\exp(\phi)$ is
the string coupling. As discussed in section II, the $g_{s}^{-2}$ is the
standard tree-level dilaton-Maxwell term coupling [1, 2, 3], while the
$g_{s}^{2}$ indicates two-string-loop corrections (genus-$\chi=2$ world-sheet
surfaces). The crucial, for our subsequent discussion, dilaton-independent
term in $f(\phi)$ might be the result of appropriate combinations of higher-
string-loop corrections in the Einstein-frame effective action.

It is now straightforward to solve the field equations of (15), with (23), in
order to determine the geometry of the spacetime and the functional expression
for the scalar field. By doing so, one obtains a simple exact, magnetically
charged black-hole solution, for which it holds that

$\displaystyle
B(r)=1-\frac{2M}{r}-\frac{2(\alpha-\beta)Q_{m}^{4}}{r^{3}\left(r-\frac{Q_{m}^{2}}{M}\right)^{3}}~{},\hskip
15.00002pt[R(r)]^{2}=r\left(r-\frac{Q_{m}^{2}}{M}\right)~{},$ (24)
$\displaystyle\phi(r)=-\frac{1}{2}\ln\left(1-\frac{Q_{m}^{2}}{Mr}\right)~{},\hskip
15.00002pt\mathcal{A}_{\mu}=(0,0,0,Q_{m}\cos\theta)~{}.$ (25)

We observe that in this case, for $\alpha=\beta$ we obtain the GHS solution
[85], while the radial coordinate $r\in(Q_{m}^{2}/M,+\infty)$ in order to have
$R\in(0,+\infty)$. In this case, it is also intriguing to observe that the
sign of the combination $\alpha-\beta$ among the coupling constants determines
whether the higher-order electromagnetic terms in the theory will contribute
attractively or not.

To obtain a better understanding of the spacetime geometry, one may express
the line element (19) in terms of the physical coordinate system with $R$
playing the role of the radial coordinate. By doing so, one finds that

$\displaystyle
ds^{2}=-B(R)dt^{2}+\frac{[W(R)]^{2}dR^{2}}{B(R)}+R^{2}d\Omega^{2}\,,$ (26)

with functions $B(R)$, $W(R)$, and $\phi(R)$ being given by

$\displaystyle
B(R)=1-\frac{4M^{2}}{Q_{m}^{2}+\sqrt{Q^{4}_{m}+4M^{2}R^{2}}}-\frac{2(\alpha-\beta)Q_{m}^{4}}{R^{6}}\,,$
(27) $\displaystyle[W(R)]^{2}=\frac{4M^{2}R^{2}}{Q^{4}_{m}+4M^{2}R^{2}}\,,$
(28)
$\displaystyle\phi(R)=-\frac{1}{2}\ln\left(\frac{\sqrt{Q_{m}^{4}+4M^{2}R^{2}}-Q_{m}^{2}}{\sqrt{Q_{m}^{4}+4M^{2}R^{2}}+Q_{m}^{2}}\right)\,.$
(29)

In the physical coordinate system $(t,R,\theta,\varphi)$, one can verify that
the curvature invariant quantities $\mathcal{R}$,
$\mathcal{R}_{\mu\nu}\mathcal{R}^{\mu\nu}$, and
$\mathcal{R}_{\alpha\beta\gamma\delta}\mathcal{R}^{\alpha\beta\gamma\delta}$
possess a single spacetime singularity residing at $R=0$, while the function
$B(R)$ satisfies the following expansions

$\displaystyle
B(R\rightarrow+\infty)=1-\frac{2M}{R}+\frac{Q_{m}^{2}}{R^{2}}-\frac{Q_{m}^{4}}{4MR^{3}}+\frac{Q_{m}^{8}}{64M^{3}R^{5}}-\frac{2(\alpha-\beta)Q^{4}_{m}}{R^{6}}+\mathcal{O}(1/R^{7})\,,$
(30) $\displaystyle B(R\rightarrow
0)=-\frac{2(\alpha-\beta)Q_{m}^{4}}{R^{6}}+\left(1-\frac{2M^{2}}{Q_{m}^{2}}\right)+\mathcal{O}(R^{2})\,.$
(31)

From (30), it becomes apparent that the spacetime (26) is practically
indistinguishable from that of a magnetically charged Reissner-Nordström black
hole for an observer at infinity, with the parameter $M$ corresponding to the
ADM mass of the solution. However, an observer much closer to the black hole
(26) would perceive a completely different picture. Indeed these quantum-
gravity corrections are important near the singularity, since the geometry
there is determined by their behavior.

(a)

(b)

Figure 1: The metric function $B(R)$ in terms of $R/(2M)$ for various values
of the parameter $Q_{m}/M$ and (a) $(\alpha-\beta)/M^{2}=1$, (b)
$(\alpha-\beta)/M^{2}=-1$. All parameters are dimensionless, and the
horizontal axis in both figures is logarithmic.

The radial null-trajectories for the spacetime (26), lead to the relation

$\frac{dt}{dR}=\pm\frac{2MR}{\sqrt{Q^{4}_{m}+4M^{2}R^{2}}}\frac{1}{|B(R)|}\,,$
(32)

which by its turn means that the roots of the function $B(R_{h})$ correspond
to black-hole horizons. In Fig. 1, one can observe the behavior of the metric
function $B(R)$ in terms of the dimensionless quantity $R/(2M)$. We see that
the solution (26) describes a black hole with a single horizon when
$(\alpha-\beta)/M^{2}=1$, while for $(\alpha-\beta)/M^{2}=-1$ the black-hole
horizons can range from two to none. It is essential to note that the previous
assertion holds in general for $(\alpha-\beta)/M^{2}$ being either greater or
lower than zero. Analysis of Fig. 1(a) reveals that a positive value for the
combination $(\alpha-\beta)/M^{2}$ results in black-hole solutions featuring a
single horizon. To facilitate comparison, we have also included the
Schwarzschild solution which can be obtained by simply setting $Q_{m}=0$. One
can readily observe that within our theory’s solution spectrum, black holes
can exhibit either greater compactness or sparsity relative to the
Schwarzschild solution. In astrophysical scenarios where $Q_{m}$ is relatively
small compared to the mass, our solution appears more compact. Conversely,
when the fraction $(\alpha-\beta)/M^{2}$ takes a negative value, the solutions
range from black holes with two horizons to naked singularities. The
transition from one class of solutions to the other occurs continuously as the
magnetic charge $Q_{m}$ increases, as depicted in Fig. 1(b). Consequently, in
this scenario, there always exists a specific value for the ratio $Q_{m}/M$
that renders the black hole extremal, meaning the inner and outer horizons
coincide.

Figure 2: The ratio $R_{h}/(2M)$ in terms of the ratio $Q_{m}/M$ for various
values of the dimensionless parameter $(\alpha-\beta)/M^{2}$. Both axes are
logarithmic.

It is crucial to highlight here the intriguing behavior observed in the realm
of single-horizon black-hole solutions, for which $\alpha-\beta>0$.
Specifically, there exists a minimum value for the ratio $R_{h}/(2M)$, which
is below unity, resulting in more compact black holes compared to the
Schwarzschild solution. Starting from $Q_{m}=0$ (Schwarzschild) and increasing
the magnetic charge, the resulting black holes become progressively more
compact until reaching the point where $R_{h}/(2M)$ attains its minimum value.
Beyond this point, further increase in the ratio $Q_{m}/M$ causes $R_{h}/(2M)$
to rise again, eventually reaching $R_{h}/(2M)=1$, albeit now with $Q_{m}\neq
0$. Subsequently, any additional increase in the ratio $Q_{m}/M$ yields a
solution more sparse than the Schwarzschild counterpart. This particular
behavior is elucidated by analyzing Fig. 2, where the relationship between the
ratio of the black hole horizon ($R_{h}$) to twice the black hole mass ($2M$)
and the ratio $Q_{m}/M$ is depicted for various values of the dimensionless
parameters $(\alpha-\beta)/M^{2}$. Conversely, it is observed that when
$\alpha-\beta<0$, the outer horizon radius of the resulting black holes is
consistently smaller than that of the corresponding Schwarzschild black hole
with the same mass. Furthermore, it is important to note that in this
scenario, the graph reaches a termination point. This occurs because, beyond a
certain threshold of the ratio $Q_{m}/M$ (which is less than unity), there is
a significant transition in the nature of the compact object. Specifically,
the object transitions from being an extremal black hole to a naked
singularity. Consequently, for this particular choice of parameters, there is
no horizon to be depicted. These observations are further corroborated by the
findings depicted in Fig. 1(b). Returning now to the case $\alpha-\beta>0$,
the discovery of black-hole solutions sharing identical horizon radii yet
varying in the ratios $Q_{m}/M$ unveils a realm of _doppelgänger black holes_
within the framework of theory (15). While it is typical to find black holes
stemming from different theoretical paradigms with shared horizon radii but
differing physical attributes such as mass, electric charge, or secondary
scalar hair, such occurrences are notably rare when considering black holes
that arise from the same theory. Even more remarkable is the fact that these
two doppelgänger black holes, despite having identical horizon radii, exhibit
distinguishable thermodynamic behaviors. One is thermodynamically stable while
the other is unstable. This distinctive feature is thoroughly explored in
Section V.

##  IV Geodesics and energy conditions

###  IV.1 Geodesics

In this subsection, we will examine the geodesic curves of massive particles
and the effective gravitational potential generated by the spacetime geometry
given by eqs. (19) and (24). We choose to work with the $(t,r,\theta,\varphi)$
coordinate system, as it facilitates a straightforward derivation of the
effective gravitational potential $V_{\text{eff}}$ through a well-established
procedure. This will help us to better comprehend the geometry of the
aforementioned black hole solutions. To do so, we introduce the effective
Lagrangian

$\displaystyle
2L_{\text{eff}}=g_{\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=-B(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{B(r)}+[R(r)]^{2}\left(\dot{\theta}^{2}+\sin^{2}\theta\,\dot{\varphi}^{2}\right)\,,$
(33)

the Euler-Lagrange equations of which yield the geodesic equations. In the
above, $\tau$ is an affine parameter of motion which can be identified with
the proper time of a particle, dot denotes derivation with respect to $\tau$,
while $2L_{\text{eff}}=-1$ corresponds to massive particles which follow a
timelike path. Note that massless particles will not follow the geodesics
induced by the geometry $g_{\mu\nu}$, instead they will follow the geodesics
induced by an effective geometry that accounts for photon-photon interactions,
introduced by the non-linear electromagnetic terms in our action. Upon
inspecting the Lagrangian (33), it becomes evident that there is no explicit
dependence on the coordinates $(t,\varphi)$. As a result, the Euler-Lagrange
equations for $t$ and $\varphi$ yield two conserved quantities: the energy $E$
and the angular momentum $J$ of the particle under consideration,
respectively. Hence, we have

$\displaystyle E=B(r)\dot{t}~{},$ (34) $\displaystyle
J=[R(r)]^{2}\sin^{2}\theta\,\dot{\varphi}~{}.$ (35)

The equation of motion for $\theta$ reads

$[R(r)]^{2}\,\ddot{\theta}+2R(r)R^{\prime}(r)\,\dot{r}\dot{\theta}-J\,\frac{\cos\theta}{\sin\theta}=0~{},$
(36)

and by choosing $\theta=\pi/2$ ($\dot{\theta}=0$), the particles stay fixed at
the equatorial plane. Now plugging these results back to (33) we obtain the
radial equation of motion

$\frac{1}{2}\dot{r}^{2}+V_{\text{eff}}(r)=\frac{1}{2}E^{2},$ (37)

with the effective potential induced by the geometry being

$V_{\text{eff}}(r)=\frac{B(r)}{2}\left(1+\frac{J^{2}}{[R(r)]^{2}}\right)~{},$
(38)

and the functions $B(r)$ and $R(r)$ given by (24) . As we have already
mentioned in the previous section, the radial coordinate $r$ ranges from
$Q_{m}^{2}/M$ to plus infinity because the physical radial coordinate
$R\in(0,+\infty)$.

Figure 3: The effective potential $V_{\text{eff}}$ in terms of the quantity
$r/M$ for $Q_{m}/M=0.5$, $J/M=5$, and $(\alpha-\beta)/M^{2}=\\{-2,0,2\\}$. All
parameters are dimensionless, and the horizontal axis is logarithmic.

In Fig. 3, we depict the behavior of the effective potential $V_{\text{eff}}$
in terms of the dimensionless parameter $r/M$, considering three distinct
values for the fraction $(\alpha-\beta)/M^{2}$. Upon close examination, it
becomes evident that the scenarios where $\alpha=\beta$ and
$(\alpha-\beta)/M^{2}=2$ share a strikingly similar pattern in the effective
potential. In both cases, the potential curve features one maximum and one
minimum value, corresponding to unstable and stable circular orbits,
respectively. On the other hand, in the case where $(\alpha-\beta)/M^{2}=-2$,
an additional minimum emerges, exhibiting local behavior that closely
resembles the Newtonian potential. To understand the origin of this
difference, we have to examine the expansion of the potential $V_{\text{eff}}$
in the limit $r\rightarrow Q_{m}^{2}/M$, where one can verify that

$V_{\text{eff}}(r\rightarrow
Q_{m}^{2}/M)=-\frac{J^{2}M^{4}(\alpha-\beta)}{Q_{m}^{4}\left(r-\frac{Q_{m}^{2}}{M}\right)^{4}}+\frac{M^{3}(\alpha-\beta)\left(4J^{2}M^{2}-Q_{m}^{4}\right)}{Q_{m}^{6}\left(r-\frac{Q_{m}^{2}}{M}\right)^{3}}+\mathcal{O}\Bigg{[}\left(r-\frac{Q_{m}^{2}}{M}\right)^{-2}\Bigg{]}\,.$
(39)

We observe that the first term, which dominates in this particular regime,
depends explicitly on the sign of the quantity $\alpha-\beta$. When
$\alpha-\beta>0$, the potential tends toward negative infinity, whereas for
$\alpha-\beta<0$, the potential tends toward positive infinity. This alignment
precisely mirrors our observations in Fig. 3. Finally, from Fig. 3, it is also
clear that for $r/M>2$, the effective potential in all cases exhibits the same
profile, independently of the relative values of the coupling constants
$\alpha$ and $\beta$. This can be naively understood through the expansion of
the potential at infinity, which is of the following form

$V_{\text{eff}}(r\rightarrow+\infty)=\frac{1}{2}-\frac{M}{r}+\frac{J^{2}}{2r^{2}}+\left(\frac{Q_{m}^{2}}{2M^{2}}-1\right)\left[\frac{J^{2}M}{r^{3}}+\frac{J^{2}Q_{m}^{2}}{r^{4}}+\frac{J^{2}Q_{m}^{4}}{M\,r^{4}}+\frac{J^{2}Q_{m}^{6}}{M^{2}r^{4}}\right]-\frac{(\alpha-\beta)Q_{m}^{4}}{r^{6}}+\mathcal{O}\left(\frac{1}{r^{7}}\right)\,.$
(40)

It is obvious that in the asymptotic regime, the coupling constants $\alpha$
and $\beta$ cease to influence the potential profile, as their first
contribution comes into play only in the seventh term of the expansion.
Consequently, even at medium distances, we anticipate that beyond a certain
point, the coupling constants will have negligible impact on the potential’s
behavior.

###  IV.2 Energy Conditions

We will now turn our attention to the energy conditions associated with the
stress-energy tensor of our theory. In the physical coordinate system
$(t,R,\theta,\varphi)$ the stress-energy tensor is described by an anisotropic
fluid which in a covariant form can be written as

$T^{\mu\nu}=(\rho_{E}+p_{\theta})u^{\mu}u^{\nu}+(p_{R}-p_{\theta})n^{\mu}n^{\nu}+p_{\theta}g^{\mu\nu}\,.$
(41)

(a)

(b)

Figure 4: The energy conditions for $Q_{m}/M=0.5$ with (a) a positive and (b)
a negative value assigned to the dimensionless quantity
$(\alpha-\beta)/M^{2}$. The vertical lines correspond to the horizon of the
black hole determined by these parameters.

In the above, $\rho_{E}$ is the energy density of the fluid measured by a
comoving observer with the fluid, $p_{R}$ is its radial pressure, $p_{\theta}$
is its tangential pressure, while $u^{\mu}$ and $n^{\mu}$ are its timelike
four-velocity and a spacelike unit vector orthogonal to $u^{\mu}$ and also to
both angular directions. The four-vectors $u^{\mu}$ and $n^{\mu}$ satisfy the
following relations:

$\displaystyle u^{\mu}=u(R)\,\delta^{\mu}_{0}\,,\hskip
10.00002ptu^{\mu}u^{\nu}g_{\mu\nu}=-1\,,$ (42) $\displaystyle
n^{\mu}=n(R)\,\delta^{\mu}_{1}\,,\hskip
10.00002ptn^{\mu}n^{\nu}g_{\mu\nu}=1\,.$ (43)

Given eqs. (III), (23), (26)-(29), and (41)-(43), one can readily compute that

$\displaystyle\rho_{E}=-T^{t}{}_{t}=\frac{B(R)}{[W(R)]^{2}}\left(\frac{\mathop{}\\!\mathrm{d}\phi}{\mathop{}\\!\mathrm{d}R}\right)^{2}+\frac{Q_{m}^{2}}{R^{4}}\,e^{-2\phi}+\frac{2(\alpha-\beta)Q_{m}^{4}}{R^{8}}\,f(\phi)\,,$
(44) $\displaystyle
p_{R}=T^{R}{}_{R}=\frac{B(R)}{[W(R)]^{2}}\left(\frac{\mathop{}\\!\mathrm{d}\phi}{\mathop{}\\!\mathrm{d}R}\right)^{2}-\frac{Q_{m}^{2}}{R^{4}}\,e^{-2\phi}-\frac{2(\alpha-\beta)Q_{m}^{4}}{R^{8}}\,f(\phi)\,,$
(45) $\displaystyle
p_{\theta}=T^{\theta}{}_{\theta}=T^{\varphi}{}_{\varphi}=-\frac{B(R)}{[W(R)]^{2}}\left(\frac{\mathop{}\\!\mathrm{d}\phi}{\mathop{}\\!\mathrm{d}R}\right)^{2}+\frac{Q_{m}^{2}}{R^{4}}\,e^{-2\phi}+\frac{6(\alpha-\beta)Q_{m}^{4}}{R^{8}}\,f(\phi)\,.$
(46)

For the anisotropic fluid of (41), the energy conditions take the following
expression:

  * •

Null Energy Conditions (NEC): $\hskip 16.99998pt\rho_{E}+p_{R}\geq 0\hskip
10.00002pt\&\hskip 10.00002pt\rho_{E}+p_{\theta}\geq 0\,,$

  * •

Weak Energy Conditions (WEC): $\hskip 10.00002pt\text{NEC}\hskip
10.00002pt\&\hskip 10.00002pt\rho_{E}\geq 0\,,$

  * •

Strong Energy Conditions (SEC): $\hskip 10.00002pt\text{NEC}\hskip
10.00002pt\&\hskip 10.00002pt\rho_{E}+p_{R}+2p_{\theta}\geq 0\,.$

In Figs. 4(a) and 4(b), we illustrate the graphs of the quantities $\rho_{E}$,
$\rho_{E}+p_{R}$, $\rho_{E}+p_{\theta}$, and $\rho_{E}+p_{R}+2p_{\theta}$ each
plotted against the dimensionless parameter $R/M$. The free parameters of our
model and solution have been chosen to be $Q_{m}/M=0.5$, while the combination
$(\alpha-\beta)/M^{2}$ takes values of $1$ and $-1$, respectively. It is
evident from Fig. 4 that all the aforementioned quantities maintain positive
values within the causal region of spacetime and as a result, all energy
conditions are satisfied.

These results imply, therefore, the existence of a dilaton hair in the black
hole’s exterior, while the energy conditions are satisfied, thereby leading to
a bypass of the pertinent (modern version of the) no-hair theorems [106, 107]
in the spirit of [108]. The situation can be understood as a consequence of
the fact that the stress-energy tensor of our theory (15), with (23), is such
that the tangential component of the pressure
($p_{\theta}=T^{\theta}_{\,\,\theta}$) dominates over its radial one
($p_{R}=T^{R}_{\,\,R})$ (in the $(t,R,\theta,\phi)$ coordinate system),
outside the horizon. That is, the following quantity is positive in the
exterior region of the black hole,

$\displaystyle\mathcal{G}-\mathcal{J}=T^{\theta}_{\,\,\theta}-T^{R}_{\,\,R}>0\,,$
(47)

where $\mathcal{G}=\rho_{E}+T^{\theta}_{\,\,\theta}$ and
$\mathcal{J}\equiv\rho_{E}+T^{R}_{\,\,R}$. Note that the condition (47)
follows from NEC. As discussed in detail in [108], the quantity
$2\mathcal{G}/R$ is the effective gradient pressure force, and its positivity
(i.e. that of $\mathcal{G}$, since $R>0$) explains in a physical way the
existence of scalar hair in the black-hole’s exterior, without any violation
of the energy conditions. The validity of the condition (47) can also be
explicitly checked in our model from Eqs. (45) and (46). Thus, the exact black
hole solution of the self-gravitating scalar-EH (non-linear) electrodynamics
examined in this paper constitutes another explicit example of the general
considerations of [108] for bypassing the no-hair theorem without any
violation of the energy conditions.

##  V Thermodynamic analysis

In this section, we will discuss the thermodynamics of both the GMGHS and our
black-hole solution by considering their Euclidean actions. We will consider
the Grand Canonical Ensemble and enclose the black hole spacetime in a cavity
with a large radius $r_{c}$. In the Grand Canonical Ensemble, the black hole
is allowed to exchange energy/mass and charge with its environment, so these
two quantities are allowed to flow in and out through the boundary keeping the
temperature and the magnetostatic potential of the boundary fixed. This
effectively means that $T(r_{h})=T(r_{c})$ and
$\Phi_{m}(r_{h})=\Phi_{m}(r_{c})$ and the system black hole-cavity is in
thermodynamic equilibrium. Note that $T$ is the black-hole temperature and
$\Phi_{m}$ is the magnetostatic potential. The quantum partition function for
the system is then given by

$\mathcal{Z}=\int\mathop{}\\!\mathrm{d}[g^{(L)}_{\mu\nu},\psi]e^{i\mathcal{S}(g^{(L)}_{\mu\nu},\psi)}=\int\mathop{}\\!\mathrm{d}[g^{(E)}_{\mu\nu},\psi]e^{-\mathcal{I}_{E}(g^{(E)}_{\mu\nu},\psi)}~{},$
(48)

where $\mathcal{S}$ is the Lorentzian action, $\mathcal{I}_{E}$ is the
Euclidean action and $\psi$ denotes all other possible fields included besides
the metric tensor. The two actions are related via
$\mathcal{I}_{E}=-i\mathcal{S}$ [109]. The quantity $g_{\mu\nu}^{(L)}$ is the
Lorentzian metric with signature $(-+++)$, which corresponds to a
$\mathbb{R}^{3,1}$ spacetime, while $g_{\mu\nu}^{(E)}$ is the Euclidean metric
with signature $(++++)$, which is obtained from the Lorentzian one by
performing a Wick rotation [110] of the time coordinate ($\tau=it$). In the
standard Matsubara formalism of finite-temperature systems, the Euclidean
metric corresponds to a space $\mathbb{R}^{3}\times S_{\beta_{\tau}}^{1}$,
where the radius $\beta_{\tau}$ of the $S^{1}$ is the inverse temperature
$T^{-1}$ in units of the Boltzman factor $k_{B}=1$. Hence, the second integral
in (48) is evaluated over all possible field configurations that have an
imaginary time $\tau$ with period $\beta_{\tau}$. From the partition function,
using standard thermodynamic relations one can obtain the Free Energy
$\mathcal{G}$ of the system as

$\mathcal{G}=-\frac{1}{\beta_{\tau}}\ln\mathcal{Z}~{}.$ (49)

By using the saddle point approximation (Laplace’s method) we will consider
that the classical action contributes the most and as a result we may drop the
integral in the partition function $\mathcal{Z}$. Then the Euclidean action
$\mathcal{I}_{E}$ can be related to the free energy evaluated on shell through
the following relation

$\mathcal{I}_{E}=\mathcal{G}\beta_{\tau}~{}.$ (50)

Having the expression of the free energy for the black hole solution, we will
compare it with the free energy of the grand canonical ensemble in order to
extract the mass (internal energy), the entropy, and the magnetostatic
potential of the black hole. For more information in the discussion that
follows, we refer the reader to the original work of Gibbons and Hawking
[111].

###  V.1 GMGHS black hole

We start our analysis with the thermodynamics of the GMGHS solution. The
Euclidean action, including the appropriate boundary terms, is given by

$\mathcal{I}_{E}=-\frac{1}{16\pi}\int_{\Sigma}\mathop{}\\!\mathrm{d}^{4}x\sqrt{g}\left(\mathcal{R}-2\nabla_{\alpha}\phi\nabla^{\alpha}\phi-e^{-2\phi}F_{\mu\nu}F^{\mu\nu}\right)-\frac{1}{8\pi}\int_{\partial\Sigma}\left(K-K_{0}\right)d^{3}x\sqrt{h}~{}.$
(51)

In the Euclidean signature, the GMGHS black hole is described by the following
metric:

$\displaystyle\mathop{}\\!\mathrm{d}s^{2}=B(r)\mathop{}\\!\mathrm{d}\tau^{2}+\frac{\mathop{}\\!\mathrm{d}r^{2}}{B(r)}+[R(r)]^{2}\mathop{}\\!\mathrm{d}\Omega^{2}~{},$
(52)

where $B(r)=1-2M/r$, while $R(r)$ has the same form as in eq. (24). In this
coordinate system, the Euclidean time coordinate is periodic and takes values
in the range $0\leq\tau\leq\beta_{\tau}$. For the derivation of the
thermodynamic quantities, we assume that we have enclosed the black hole in a
large cavity with radius $r_{c}$. Therefore, the radial coordinate takes
values in $r_{h}\leq r<r_{c}$. Finally, the two angular coordinates take their
usual values. The boundary term $K$ represents the trace of the extrinsic
curvature, which in our case reads

$K=\nabla^{\alpha}n_{\alpha}=\frac{2R^{\prime}(r)\sqrt{B(r)}}{R(r)}+\frac{B^{\prime}(r)}{2\sqrt{B(r)}}\,,$
(53)

where $n_{\alpha}=\sqrt{1/B(r)}\,\delta^{r}_{\alpha}$ is a normalized
spacelike vector field. The $K$ term in the above hypersurface integral
represents the Gibbons-Hawking-York boundary term, ensuring a well-defined
variational principle. The second boundary term $K_{0}$ serves as a
subtraction term to render the action finite for flat space (in the absence of
the black hole). For flat space, $K_{0}$ equals $2/r$, obtained by setting
$B(r)=1$ and $R(r)=r$ in the above relation. Utilizing these relations, one
can readily compute the Euclidean action (51) to be

$\mathcal{I}_{E}=\frac{\beta_{\tau}M}{2}-\frac{\beta_{\tau}Q_{m}^{2}}{4M}~{}.$
(54)

In the above, we have used that the horizon radius is given by $r_{h}=2M$. In
the Grand Canonical Ensemble, the Euclidean action is identified with the free
energy of the thermodynamic system as
$\mathcal{I}_{E}=\beta_{\tau}\mathcal{G}$, thus, we can rewrite (54) as

$\mathcal{I}_{E}=\beta_{\tau}M-\beta_{\tau}\Phi_{m}Q_{m}-S~{},$ (55)

where $S$ is the entropy and $\Phi_{m}$ is the magnetostatic potential,
$\Phi_{m}=Q_{m}/r_{h}$. For the derivation of the above equation, we have used
the fact that $\beta_{\tau}=8\pi M\equiv 1/T$ with $T$ being the temperature
of the black hole. By combining now eqs. (54) and (55) we can evaluate the
black-hole entropy $S$, which is given by the following relation

$S=2\pi M\left(2M-\frac{Q_{m}^{2}}{M}\right)=\pi[R(r_{h})]^{2}=\frac{A}{4},$
(56)

where $A$ denotes the horizon area. It is evident that in this case, the
entropy function has the well-known form of the Bekenstein-Hawking entropy.
For validation, the same result may also be obtained using Wald’s formula or
even using the Arnowitt-Deser-Misner (ADM) formalism [112]. For a
comprehensive analysis of the ADM formalism, readers are directed to [113].
Additionally, for its explicit application in black-hole solutions, we refer
the interested reader to [114]. In the subsequent subsection, we will utilize
the ADM formalism for the thermodynamic analysis of our black-hole solution.

The inclusion of the Gibbons-Hawking-York boundary term ensures that the
Euclidean action attains an extremum within the class of fields considered
here, $\delta\mathcal{I}_{E}=0~{}.$ As a result, it is evident that the first
law of thermodynamics in the Grand Canonical Ensemble (keeping the temperature
and the magnetic potential fixed) takes the form

$\delta M=T\delta S+\Phi_{m}\delta Q_{m}~{},$ (57)

derived from (55), and holds by construction. The first law is also evident by
taking the variation of the entropy with respect to the primary black-hole
charges. The temperature of this black hole is the same as that of the
Schwarzschild black hole, as pointed out in [115], since the Euclidean
continuation does not care about the angular part. Consequently, the heat
capacity $C$ for constant charge will also be negative, $C=-1/(8\pi T^{2})$;
hence, these types of black holes cannot reach thermal equilibrium.

###  V.2 Black hole with non-linear electrodynamics

We will now focus on our black-hole solution, emanating from the action (15)
and characterized by the line element (26)-(28). The scalar and the gauge
fields are of the form $\phi=\phi(R)$—with $R$ being the physical radial
coordinate—and $\mathcal{A}_{\mu}=(0,0,0,A(\theta))$, respectively. In this
case, to determine the thermodynamic quantities associated with the resulting
black-hole solution, we will make use of the Euclidean signature and also
utilize the ADM formalism [112, 113]. Hence, we consider the line element of
the form

$\mathop{}\\!\mathrm{d}s^{2}=[N(R)]^{2}B(R)\mathop{}\\!\mathrm{d}\tau^{2}+\frac{[W(R)]^{2}\mathop{}\\!\mathrm{d}R^{2}}{B(R)}+R^{2}\mathop{}\\!\mathrm{d}\Omega^{2}~{},$
(58)

where the Euclidean time takes values in the range
$0\leq\tau\leq\beta_{\tau}$, while the radial coordinate
$R\in[R_{h},+\infty)$. To obtain the temperature, that is the period of the
Matsubara frequency $\tau$, in our case, we follow the calculation of [37]. To
this end, we first ignore the angular part of the line element and perform a
series expansion near the horizon. Thus, we are left with a two-dimensional
line element which is compared with the line element of two-dimensional space
expressed in polar coordinates
$\mathop{}\\!\mathrm{d}S=\mathop{}\\!\mathrm{d}\hat{R}^{2}+\hat{R}^{2}\mathop{}\\!\mathrm{d}\Theta^{2}$.
By doing so, we obtain

$\displaystyle\mathop{}\\!\mathrm{d}\hat{R}^{2}=\frac{W(R_{h})^{2}}{B^{\prime}(R_{h})(R-R_{h})}\mathop{}\\!\mathrm{d}R^{2}~{},$
(59) $\displaystyle
B^{\prime}(R_{h})(R-R_{h})\mathop{}\\!\mathrm{d}\tau^{2}=\hat{R}^{2}\mathop{}\\!\mathrm{d}\Theta^{2}~{}.$
(60)

The coordinate $\Theta$ is periodic with a period $2\pi$ which implies that
$\tau$ is also a periodic coordinate with a period $\beta_{\tau}$ given by:

$\beta_{\tau}=\frac{1}{T}=\frac{4\pi
W(R)}{N(R)B^{\prime}(R)}\Bigg{|}_{R_{h}}~{},$ (61)

where $T$ is the temperature of the black hole. For completeness, we also
remark at this point that we have also checked that, as expected, the
temperature will take on the same values at the event horizon regardless of
the coordinate system we are using ($r$ or $R$).

The Euclidean action is related to the Lorentzian action via
$\mathcal{I}_{E}=-i\mathcal{S}$ and we will consider the following variational
problem which basically consists of the theory (15) alongside a boundary term
denoted by $\mathcal{B}_{E}$ which we will consider in order to have a well-
defined variational principle $\delta\mathcal{I}_{E}=0$. Thus, we have

$\mathcal{I}_{E}=\frac{2\pi\beta_{\tau}}{16\pi}\int^{\pi}_{0}\mathop{}\\!\mathrm{d}\theta\int_{R_{h}}^{\infty}\mathop{}\\!\mathrm{d}R\left[-NR^{2}W\sin\theta\,\mathcal{L}(R,\theta)\right]+\mathcal{B}_{E}~{}.$
(62)

Here $\mathcal{L}$ denotes the Lagrangian of the theory which is a function of
$R,\theta$ coordinates. After canceling total derivatives, the Euclidean
action reads

$\displaystyle\mathcal{I}_{E}=\beta_{\tau}\int^{\pi}_{0}\mathop{}\\!\mathrm{d}\theta\int_{R_{h}}^{\infty}\mathop{}\\!\mathrm{d}R\,\hat{\mathcal{L}}(Q^{i},\partial_{\mu}Q^{i})+\mathcal{B}_{E}~{}\,,$
(63)

with $Q^{i}=\\{N(R),W(R),B(R),\phi(R),A(\theta)\\}$ and
$\hat{\mathcal{L}}(Q^{i},\partial_{\mu}Q^{i})$ given by

$\displaystyle\hat{\mathcal{L}}(Q^{i},\partial_{\mu}Q^{i})=\frac{N\sin\theta}{4W^{2}R^{6}}\bigg{[}WR^{7}B^{\prime}$
$\displaystyle+W^{3}\left(2f(\phi)(\alpha-\beta)\frac{(\partial_{\theta}A)^{4}}{\sin^{4}\theta}+e^{-2\phi}R^{4}\frac{(\partial_{\theta}A)^{2}}{\sin^{2}\theta}-R^{6}\right)$
$\displaystyle+BR^{6}\left(WR^{2}\phi^{\prime
2}-2RW^{\prime}+W\right)\bigg{]}\,.$ (64)

Following the ADM formalism, we have to vary the above Euclidean action with
respect to each one of the dynamical fields $Q^{i}$ to obtain the field
equations. By doing so, we obtain none other than the well-known Euler-
Lagrange equations, namely

$\frac{\partial\hat{\mathcal{L}}}{\partial
Q^{i}}-\partial_{\mu}\left(\frac{\partial\hat{\mathcal{L}}}{\partial(\partial_{\mu}Q^{i})}\right)=0\,.$
(65)

Let us now apply the above equation for the dynamical field $Q^{1}=N(R)$. Upon
substituting the expression of $\hat{\mathcal{L}}$ from (64) into (65), we
find that the second term vanishes identically, while
$\partial\hat{\mathcal{L}}/\partial N=\hat{\mathcal{L}}/N$. As a result, the
equation for $N(R)$ indicates that $\hat{\mathcal{L}}=0$, which in turn
implies $\mathcal{I}_{E}=\mathcal{B}_{E}$. This outcome is anticipated in the
ADM formalism, where the metric construction (58) is specifically tailored to
yield this result. Additionally, by solving the field equations (65) for all
dynamical fields $Q^{i}$, one can determine the unknown functions and verify
that the resulting solution is the one obtained in Sec. III with line element
(26)-(28), alongside a constant $N$ which without loss of generality we may
set equal to $1$. It is important to mention at this point, that during the
derivation of the Euler-Lagrange equations, certain boundary terms were
omitted. These terms are of the following form

$\beta_{\tau}\left(\frac{R}{2W}\delta
B+\frac{2BR^{2}\phi^{\prime}}{W}\,\delta\phi-\frac{BR}{W^{2}}\,\delta
W\right)\Bigg{|}_{R_{h}}^{\infty}~{},$ (66)

and

$\beta_{\tau}\int_{R_{h}}^{\infty}\mathop{}\\!\mathrm{d}R\left(\frac{We^{-2\phi}(\partial_{\theta}A)}{2R^{2}\sin\theta}+\frac{2(\alpha-\beta)Wf(\phi)(\partial_{\theta}A)^{3}}{R^{6}\sin^{3}\theta}\right)\delta
A\Bigg{|}_{\theta=0}^{\theta=\pi}~{}.$ (67)

The variation of the boundary term $\delta\mathcal{B}_{E}$ will account for
the neglected boundary terms, ensuring the attainment of a well-defined
variational principle $\delta\mathcal{I}_{E}=0$. Utilizing the fact that the
variation of $A$ yields $\delta A=(\delta Q_{m})\cos\theta$, and substituting
the expressions for the functions in (67), one can integrate and derive the
following expression:

$\beta_{\tau}\,\frac{Q_{m}\left\\{\sqrt{4M^{2}R_{h}^{2}+Q_{m}^{4}}\left[R_{h}^{4}-4(\alpha-\beta)Q_{m}^{2}\right]-R_{h}^{4}Q_{m}^{2}\right\\}}{2MR_{h}^{6}}\,\delta
Q_{m}~{}.$ (68)

Now, the variation of the dynamical fields at infinity yield

$\displaystyle\delta B=-\frac{2\delta
M}{R}+\mathcal{O}\left(\frac{1}{R^{2}}\right)~{},$ (69)
$\displaystyle\delta\phi=\frac{Q_{m}}{MR}\delta
Q_{m}-\frac{Q_{m}^{2}}{2M^{2}R}\delta
M+\mathcal{O}\left(\frac{1}{R^{3}}\right)~{},$ (70) $\displaystyle\delta
W=\frac{Q_{m}^{4}}{4M^{3}R^{2}}\delta M-\frac{Q_{m}^{3}}{2M^{2}R^{2}}\delta
Q_{m}+\mathcal{O}\left(\frac{1}{R^{3}}\right)~{},$ (71)

while at the horizon we have that

$\displaystyle\delta B|_{R_{h}}=-B^{\prime}(R_{h})\delta R_{h}~{},$ (72)
$\displaystyle\delta\phi|_{R_{h}}=\delta\phi(R_{h})-\phi^{\prime}(R_{h})\delta(R_{h})~{},$
(73) $\displaystyle\delta W|_{R_{h}}=\delta
W(R_{h})-W^{\prime}(R_{h})\delta(R_{h})~{}.$ (74)

Note that the parameters $\alpha,\beta$ are fixed by the theory and thus not
allowed to vary, while $M$ and $Q_{m}$ are pure integration constants allowed
in the variation.

As previously mentioned, to ensure a well-defined variational procedure, it is
desirable to have $\delta\mathcal{I}_{E}=0$. For clarity and convenience, we
will partition the variation of the boundary term $\delta\mathcal{B}_{E}$ into
two components: one at infinity and another at the event horizon, expressed
as:

$\delta\mathcal{B}_{E}=\delta\mathcal{B}_{E}(\infty)+\delta\mathcal{B}_{E}(R_{h})~{}.$
(75)

Evaluating now (66) at infinity and considering the variation of the boundary
term at infinity we find that a zeroth order contribution survives, which
according to the variations of the fields leads to

$\displaystyle\frac{\beta_{\tau}}{2}R\left(\delta B-2\delta
W\right)+\delta\mathcal{B}_{E}(\infty)=0\Rightarrow$ (76)
$\displaystyle\delta\mathcal{B}_{E}(\infty)=\beta_{\tau}\,\delta M~{}.$ (77)

On the other hand, eq. (66) at the horizon, alongside the variation of the
boundary term at the horizon and (68) results in

$2\pi R_{h}\delta
R_{h}+\beta_{\tau}\,\frac{Q_{m}\left\\{\sqrt{4M^{2}R_{h}^{2}+Q_{m}^{4}}\left[R_{h}^{4}-4(\alpha-\beta)Q_{m}^{2}\right]-R_{h}^{4}Q_{m}^{2}\right\\}}{2MR_{h}^{6}}\,\delta
Q_{m}+\delta\mathcal{B}_{E}(R_{h})=0~{},$ (78)

which might be written equivalently as

$\displaystyle\frac{\delta A}{4}+\beta_{\tau}\Phi_{m}\delta
Q_{m}+\delta\mathcal{B}_{E}(R_{h})=0~{},$ (79)

where we have used the fact that the area of the black hole is given by
$A=4\pi R_{h}^{2}$ and we have defined the magnetic potential as

$\Phi_{m}=\frac{Q_{m}\left\\{\sqrt{4M^{2}R_{h}^{2}+Q_{m}^{4}}\left[R_{h}^{4}-4(\alpha-\beta)Q_{m}^{2}\right]-R_{h}^{4}Q_{m}^{2}\right\\}}{2MR_{h}^{6}}~{}.$
(80)

Considering now that we are dealing with the Grand Canonical Ensemble, we keep
the temperature and the magnetic potential of the system fixed and as a result
we can drop the variations to obtain

$\displaystyle\mathcal{B}_{E}(\infty)=\beta_{\tau}M~{},$ (81)
$\displaystyle\mathcal{B}_{E}(R_{h})=-\frac{A}{4}-\beta_{\tau}\Phi_{m}Q_{m}~{}.$
(82)

Therefore, the value of the Euclidean action is given by

$\mathcal{I}_{E}=\beta_{\tau}M-\frac{A}{4}-\beta_{\tau}\Phi_{m}Q_{m}~{},$ (83)

and since the Euclidean action is related to the free energy $\mathcal{G}$ of
the system via
$\mathcal{I}_{E}=\beta_{\tau}\mathcal{G}=\beta_{\tau}\mathcal{M}-S-\beta_{\tau}\Phi_{m}Q_{m}$
we can identify, by comparison the conserved black hole mass and the entropy
of the black hole as

$\displaystyle\mathcal{M}=M~{},$ (84) $\displaystyle S=A/4~{}.$ (85)

Finally, the First Law of Thermodynamics (57) holds by construction as in the
GMGHS black hole.

(a)

(b)

Figure 5: The black-hole temperature for (a) attractive and (b) repulsive
higher-order electromagnetic contributions, with varying values of the
magnetic charge ($Q_{m}$), while keeping the mass ($M$) the same. The axes in
both figures are logarithmic.

With the confirmation that the black-hole thermodynamic quantities in our case
adhere to the standard relations, we can now proceed to analyze the black
hole’s temperature. In Fig. 5, we depict the black-hole temperature as a
function of the dimensionless quantity $R_{h}/(2M)$. Notice that the
temperature is scaled by the temperature of the Schwarzschild black-hole to
form a dimensionless quantity, ensuring its independence from the chosen unit
system. In Figs. 5(a) and 5(b), we explore the effects of the higher-order
electromagnetic contributions on black-hole temperature, considering fixed
(yet distinct) values for the coupling constants $\alpha$ and $\beta$, along
with varying magnetic charge ($Q_{m}$) values, but maintaining the same value
for the black-hole mass. Both Figs. 5(a) and 5(b) were generated using the
following procedure: For each value $(\alpha-\beta)/M^{2}$ and the ratio
$Q_{m}/M$, we numerically evaluate the value of the ratio $R_{h}/(2M)$ using
eq. (27). Subsequently, employing equation (61), we calculate the temperature
of the black hole for each parameter pair. Finally, for each
$(\alpha-\beta)/M^{2}$ we plot the points from the list
$\\{R_{h}/(2M),T(R_{h})/T_{sch}\\}$. Note that we use the same mass parameter
$M$ for the temperature calculation of the Schwarzschild black hole $T_{sch}$.
In both figures, we have also incorporated a distinctive dot symbolizing a
constant value for the quantity $T(R_{h})/T_{sch}$, irrespective of the ratio
$R_{h}/(2M)$. Apparently, this is not coincidental, as it mirrors the
characteristics of the Schwarzschild black hole, where the horizon radius
precisely equals $2M$ and its temperature is determined by the established
formula $T=1/(8\pi M)$.

Focusing now on the thermodynamical characteristics of our solution, we
observe that regardless of the value $(\alpha-\beta)/M^{2}$, for $Q_{m}=0$,
our solution reduces to the Schwarzschild black hole and therefore all graphs
in Fig. 5 have as a starting point the Schwarzschild point. However, when we
depart from this limit, we notice that for $\alpha-\beta>0$, as illustrated in
Fig. 5(a), the temperature of the resulting black holes consistently surpasses
that of the Schwarzschild black hole, whereas for $\alpha-\beta<0$ (Fig.
5(b)), the opposite effect occurs. Furthermore, this temperature increase, in
the $\alpha-\beta>0$ scenario, is independent of whether the black hole under
examination possesses a smaller or larger horizon radius compared to the
corresponding Schwarzschild black hole. As previously observed in Fig. 2 and
discussed in Sec. III, it becomes evident in Fig. 5(a) that there are
consistently pairs of black-hole solutions, more compact than the
Schwarzschild solution, that share the same horizon radius $R_{h}$ but with
different ratios $Q_{m}/M$. However, now we see that although these solutions
possess the same horizon radius, their temperatures differ significantly. This
can be understood through the relation (61) where it is evident that the
formula determining the temperature of a black hole depends on the first
derivative of the function $B(R)$. This means that black-hole solutions which
for different ratios $Q_{m}/M$ result in the same horizon radius $R_{h}$
through the equation $B(R_{h})=0$, their temperatures are not necessarily the
same since $B^{\prime}(R_{h})$ could differ in these two cases.

Moreover, we can deduce the thermodynamic stability of these black holes by
examining how the temperature changes with a change in the mass. In Fig. 5(a)
it is evident that there are two distinct branches of black-hole solutions. In
the first branch we have black-hole solutions that get colder as the mass is
decreasing, while in the second branch we have black holes that are getting
hotter as the mass is getting smaller. This implies that the heat capacity
$C\equiv dM/dT$ for the first branch is positive since both $dM,dT$ are
negative and the black holes are thermally stable, while the second branch,
for which the temperature rises with the decrease of mass, exhibits negative
heat capacity and are thermally unstable. Notice also the fact that the
Schwarzchild black hole lies in the second (unstable) branch which is a well-
known result. Furthermore, the parameter space of these black holes exhibits a
point where $dT=0$ indicating the divergence of the heat capacity and as a
result a phase transition from hot to cold black holes. In Fig. 5(b), we can
see that as the black holes lose mass they get colder which implies that they
are thermally stable since they possess positive heat capacity. These results
are in agreement with the studies in [116], where the black holes are viewed
as defects in the thermodynamical spacetime [117].

In the next section, we proceed to study the stability of the black holes from
a linear-perturbation point of view, which, in general, is distinct from the
thermodynamic stability. Indeed, as we shall demonstrate explicitly below,
such a linearised stability analysis does not necessarily imply
thermodynamical stability, in the sense that the thermodynamically unstable
branches found above exhibit stability under linear perturbations.

##  VI Linear Perturbations

###  VI.1 Radial Stability

In this section, we investigate the stability of our solution under radial
perturbations. For simplicity, we focus on linear and radial perturbations.
Therefore, we use the following ansatz:

$ds^{2}=-P(R,t)dt^{2}+Q(R,t)dR^{2}+R^{2}d\Omega^{2},\qquad
A=a_{0}(R,t)dt+Q_{m}\cos\theta\,d\varphi,\qquad\phi=\phi(R,t),$ (86)

where

$\displaystyle P(R,t)=B(R)\left[1-\epsilon\,e^{-i\omega
t}h_{1}(R)\right],\qquad Q(R,t)=\frac{1}{H(R)}+\epsilon\,e^{-i\omega
t}h_{2}(R),$ (87) $\displaystyle\phi(R,t)=\phi(R)+\epsilon\,e^{-i\omega
t}\phi_{1}(R),\qquad a_{0}(R,t)=\epsilon\,e^{-i\omega t}a_{0}(R).$ (88)

For the stability analysis, it is more convenient to work within the physical
coordinate system; hence, in the above equations $B(R)$, $\phi(R)$ and
$H(R)=B(R)/W^{2}(R)$ are given in eqs. (27–29), and correspond to the
background/unperturbed spacetime. Note that the radial perturbations are
associated with the $L=0$ perturbation 555$L$ is the “angular momentum” index
in the spherical harmonics function $Y_{L}^{M_{L}}(\theta,\varphi)$. For more
information about the decomposition of the perturbations in spherical
harmonics, see [118, 119, 120, 121]. in the even sector of the gravitational
perturbations. Therefore, in the electromagnetic part, only the electric-type
perturbations contribute, as the magnetic-type corresponds to the odd sector
[121, 122]. The dimensionless constant $\epsilon$ determines the order of the
perturbation. Finally, $\omega$ specifies the decomposition of the modes with
fixed energy.

(a)

(b)

Figure 6: The graph of the effective potential $\mathcal{V}$ in terms of
$r^{*}$ for various values of the parameter $Q_{m}/M$ and (a)
$(\alpha-\beta)/M^{2}=1$, (b) $(\alpha-\beta)/M^{2}=-1$.

A direct calculation reveals that both the spacetime and the matter field
perturbations are determined from the function $\phi_{1}$. Consequently, the
investigation of the system is simplified to a single equation for the
perturbation of the scalar field. This specific equation can be expressed in
the conventional Schrödinger form:

$\frac{d^{2}\Psi(r^{*})}{{dr^{*}}^{2}}+\left[\omega^{2}-\mathcal{V}(R)\right]\Psi(r^{*})=0,$
(89)

where we have defined a new perturbation function as $\Psi\equiv R\phi_{1}$
and the tortoise coordinate is $dr^{*}\equiv\frac{dR}{\sqrt{B(R)H(R)}}$. The
potential of the Schrödinger equation has the following form

$\displaystyle\mathcal{V}(R)=$
$\displaystyle\,\frac{HB^{\prime}+2BR\phi^{\prime}\left[\phi^{\prime}\left(R\left(H^{\prime}+HR\phi^{\prime
2}\right)+4H\right)+3HR\phi^{\prime\prime}\right]+BH^{\prime}}{2R}$
$\displaystyle+\frac{B(\alpha-\beta)Q_{m}^{4}\left(R\dot{f}\phi^{\prime}+\ddot{f}\right)}{R^{8}}+\frac{B\gamma
e^{-2\phi}Q_{m}^{2}\left(2-R\phi^{\prime}\right)}{R^{4}}\,.$ (90)

Considering our emphasis on the stability of black-hole solutions, there is no
need to solve eq. (89). The time evolution factor, $\exp{(-i\omega t)}$,
simplifies the task, requiring us only to ascertain whether the frequency,
$\omega$, is purely imaginary or not. In the scenario where the frequency
$\omega$ is purely imaginary, the mode undergoes exponential growth due to the
presence of the term $\exp{(-i\omega t)}$ making the solution unstable.
Therefore, a negative eigenvalue, $\omega^{2}<0$, that signifies an unstable
mode, corresponds to a bound state in the Schrödinger equation (89). A general
result in quantum physics is that for a potential that vanishes in both
asymptotic regions, has a barrier form, and is positive definite. Therefore,
eq. (89) does not exhibit bound states. In Fig. 6, we depict the potential of
the Schrödinger equation for two families of solutions. The first one
corresponds to $(\alpha-\beta)/M^{2}=1$, while the second one corresponds to
$(\alpha-\beta)/M^{2}=-1$. By carefully examining the parametric space of the
solutions, we deduce that the potential always takes a form similar to the
potentials in Fig. 6. Therefore, we conclude that our solutions are stable
under radial perturbations.

Although radial stability is a strong indication regarding the stability of a
particular solution, a more careful and general perturbation analysis has to
be performed to extract a stronger result, which, however, lies beyond the
scope of this work. Moreover, as shown in the previous section V, linear
stability does not necessarily imply thermodynamical stability for the black
hole, in the sense that the latter, although linearly stable, nonetheless it
possesses thermodynamically unstable branches.

###  VI.2 Scalar Quasi-Normal Modes

Quasi-normal modes (QNMs) play a crucial role in the study of black holes and
other astrophysical objects [123, 124, 125]. These modes represent the
characteristic vibrations or oscillations of a black hole after a
perturbation, such as a gravitational wave or a scattering event. QNMs are
characterized by complex frequencies, i.e. eigenvalues of the Schrödinger
equation, consisting of a real part and an imaginary part. The real part
corresponds to the oscillation frequency, while the imaginary part reflects
the damping or decay of the mode. The study of QNMs provides valuable insights
into the nature and properties of black holes, offering a unique window into
their internal dynamics.

For simplicity, we will consider the propagation of a test scalar field $\Phi$
in the background of our black hole and extract its QNMs. We begin our
analysis from the following action functional for the scalar field,

$S=\int
d^{4}x\sqrt{-g}\left[\nabla^{\mu}\Phi\nabla_{\mu}\Phi+m^{2}\Phi^{2}\right],$
(91)

where $m$ is the mass of the test scalar field. The variation of the above
action with respect to the scalar field yields the Klein-Gordon equation in
the black hole background

$\left(\square-m^{2}\right)\Phi=0\,.$ (92)

Note that the test scalar field $\Phi$, as a perturbation field, does not
back-react on the spacetime metric and is a function of all spacetime
coordinates $\Phi=\Phi(t,R,\theta,\varphi)$. For clarity, we choose to work in
the physical coordinate system. Therefore, the background metric is given by
eq. (26). We can apply the separation of variables as follows

$\Phi(t,R,\theta,\varphi)=e^{-i\omega
t}Y_{L}^{M_{L}}(\theta,\varphi)\frac{\Uppsi(R)}{R}~{},$ (93)

where $Y_{L}^{M_{L}}(\theta,\varphi)$ represents the spherical harmonics
function. By using the tortoise coordinate,
$dr^{*}=\frac{dR}{\sqrt{B(R)H(R)}}$, one can rewrite the perturbation equation
in a Schrödinger form as:

$\frac{d^{2}\Uppsi(r^{*})}{{dr^{*}}^{2}}+\left[\omega^{2}-V(R)\right]\Uppsi(r^{*})=0,$
(94)

where $V(R)$ is the effective potential of the Schrödinger equation and is
given by

$V=\frac{HB^{\prime}+BH^{\prime}}{2R}+\frac{B}{R^{2}}L(L+1)+m^{2}B.$ (95)

The background metric functions $B$ and $H=B/W^{2}$ are given in eqs. (27–28).

In the pursuit of calculating the QNMs, the WKB (Wentzel-Kramers-Brillouin)
approximation stands as a valuable method. Particularly useful in the context
of wave-like phenomena, the WKB approximation provides an efficient and semi-
classical approach to estimating the complex frequencies associated with QNMs.
By treating the Schrödinger-like equation governing the perturbations as a
semiclassical wave equation, the WKB method allows for the determination of
QNM frequencies without the need for an exact solution. The WKB method was
initially developed for quantum mechanical problems; however, Schutz and Will
were the first to apply this method to the problem of scattering around black
holes [126]. Later, Iyer and Will extended this approach to the third WKB
order beyond the eikonal approximation [127], and Konoplya further advanced it
to the sixth order [128, 125]. Interestingly, the sixth order yields a
relative error of approximately two orders of magnitude lower than that of the
third WKB order [128, 125]. However, for simplicity, in this work, we will
employ the first-order WKB approximation, in which the QNM frequencies are
obtained from the solution of the following equation

$n+\frac{1}{2}=-i\left[\frac{\omega^{2}-V(r^{*})}{\sqrt{-2V^{\prime\prime}(r^{*})}}\right]_{r*=r^{*}_{\rm
max}},\qquad n\in\mathbb{Z}^{\geq}.$ (96)

The expression in the right-hand part of the above equation is evaluated at
the maximum of the potential $r^{*}_{\rm max}$ while $n$ is the overtone
number of the QNMs.

| $(\alpha-\beta)/M^{2}=0.01$ | $(\alpha-\beta)/M^{2}=-1$ | $(\alpha-\beta)/M^{2}=1$
---|---|---|---
$Q_{m}/M=0.01$ | $0.329438-0.096255\,i$ | $0.329438-0.096255\,i$ | $0.329438-0.096255\,i$
$Q_{m}/M=0.3\,\,$ | $0.333767-0.096712\,i$ | $0.333767-0.096728\,i$ | $0.333767-0.096696\,i$
$Q_{m}/M=0.6\,\,$ | $0.348235-0.098171\,i$ | $0.348226-0.098595\,i$ | $0.348243-0.097727\,i$
Table 1: The dimensionless $L=1$, $n=0$ quasinormal modes ($M\omega$) for $m=0$. | $(\alpha-\beta)/M^{2}=0.01$ | $(\alpha-\beta)/M^{2}=-1$ | $(\alpha-\beta)/M^{2}=1$
---|---|---|---
$Q_{m}/M=0.01$ | $0.401632-0.050195\,i$ | $0.401632-0.050195\,i$ | $0.401632-0.050195\,i$
$Q_{m}/M=0.3\,\,$ | $0.404334-0.052296\,i$ | $0.404335-0.052294\,i$ | $0.404332-0.052298\,i$
$Q_{m}/M=0.6\,\,$ | $0.413700-0.058602\,i$ | $0.413733-0.058522\,i$ | $0.413667-0.058679\,i$
Table 2: The $L=1$, $n=0$ dimensionless quasinormal modes ($M\omega$) for
$m/M=0.4$.

In Tables 1 and 2, we present the dimensionless QNM frequencies, denoted by
$(M\omega)$, for two distinct scenarios: when $m=0$ and $m/M=0.4$,
respectively. Notably, as $Q_{m}$ approaches 0, our solution converges to the
Schwarzschild black hole, irrespective of the $(\alpha-\beta)/M^{2}$
parameter. This convergence is evident in the first row of both tables, where
$Q_{m}/M=0.01$, as the QNM values remain constant across varying
$(\alpha-\beta)/M^{2}$. Furthermore, as $(\alpha-\beta)/M^{2}$ tends toward 0,
our solution adopts the characteristics of the GMGHS black hole. Consequently,
the QNMs in the first row of both tables, specifically when
$(\alpha-\beta)/M^{2}=0.01$, align with those of the GMGHS black hole. The
subsequent rows in the tables provide additional insights into the
characteristics of our solution. For instance, in the second and third rows,
where $Q_{m}/M=0.3$ and $0.6$, respectively, we observe a systematic variation
in the QNM values with changes in both $Q_{m}/M$ and $(\alpha-\beta)/M^{2}$.
This behavior highlights the sensitivity of the QNM frequencies to the
parameters characterizing the black hole solution. Furthermore, by comparing
these results to the Schwarzschild and GMGHS cases, we discern how our
solution deviates from these benchmark scenarios. Additionally, the tables
reveal intriguing patterns in the imaginary parts of the QNMs. For varying
$Q_{m}/M$ and $(\alpha-\beta)/M^{2}$, the imaginary parts exhibit non-trivial
changes, reflecting the impact of the black hole’s charge and the parameter
$(\alpha-\beta)/M^{2}$ on the damping behavior of the perturbations.

##  VII Other Solutions

###  VII.1 Asymptotically (A)dS spacetimes

Let us now, briefly discuss asymptotically (A)dS spacetimes. Following [129],
introducing a scalar potential $\mathfrak{V}(\phi)$ in the action and
considering

$\mathcal{S}=\int\mathop{}\\!\mathrm{d}^{4}x\sqrt{-g}\Big{(}\mathcal{R}-2\nabla^{\mu}\phi\nabla_{\mu}\phi-e^{-2\phi}\mathcal{F}^{2}+f(\phi)\big{(}-2\alpha\mathcal{F}^{\alpha}_{~{}\beta}\mathcal{F}^{\beta}_{~{}\gamma}\mathcal{F}^{\gamma}_{~{}\delta}\mathcal{F}^{\delta}_{~{}\alpha}+\beta\mathcal{F}^{4}\big{)}-\mathfrak{V}(\phi)\Big{)}~{},$
(97)

with a $\mathfrak{V}(\phi)$ of the form

$\mathfrak{V}(\phi)=\frac{1}{3}\Lambda e^{-2\phi}+\frac{1}{3}\Lambda
e^{2\phi}+\frac{4\Lambda}{3}=\frac{2}{3}\Lambda(\cosh(2\phi)+2)~{},$ (98)

we can obtain $B(r)$ as

$B(r)=1-\frac{2M}{r}-\frac{2(\alpha-\beta)Q_{m}^{4}}{r^{3}\left(r-\frac{Q_{m}^{2}}{M}\right){}^{3}}-\frac{1}{3}\Lambda
r\left(r-\frac{Q_{m}^{2}}{M}\right)~{},$ (99)

while $\phi(r),R(r)$ will remain the same. Note here that the potentials
$\mathfrak{V}(\phi)$ and $f(\phi)$ are almost identical, and they are both
Liouville-type potentials [70].666Nonetheless, we should notice that the
presence of a positive cosmological constant term in the gravitational
effective actions is problematic within the context of string/brane-inspired
quantum gravity theories, due to the so-called swampland conjecture [130, 131,
132]. This issue is open at present, and its study goes beyond the purpose of
the current article.

###  VII.2 Solutions for general $\gamma$

Assuming that the coupling term between the dilaton and the Maxwell term is of
the form $e^{-2\gamma\phi}$ we can obtain the same geometry with the
$\gamma=1$ case with the coupling function $f(\phi)$ now being given by

$f(\phi)=-3\cosh(2\phi)-2-\frac{e^{2\phi}\left(\left(e^{-2\phi}\right)^{\gamma-1}-1\right)Q_{m}^{6}}{2M^{4}\left(e^{2\phi}-1\right)^{4}(\alpha-\beta)}~{}.$
(100)

In this case, the charge-to-mass ratio is fixed by the theory. As a result,
such black holes are described by a constrained phase space of free
parameters, since this situation reduces the number of primary black hole
hairs from two to one. A more physical result would be to let the form of the
dilaton field to be affected by the change of the coupling function with the
Maxwell term, however, we were not able to derive exact results in this case,
so one has to employ numerical techniques. Such endeavors may be undertaken in
subsequent works.

##  VIII Conclusions

In the quest to comprehend gravitational phenomena and the nature of gravity
itself, the theoretical exploration of black holes stands as a pivotal
frontier. The predictions of General Relativity (GR) are in good agreement
with current observations related to black holes. This is attributed to the
large mass of the observed objects and therefore their large horizon radius
and small horizon curvature. Additionally, a plethora of cosmological
observations indicates instances where GR exhibits limitations, with the most
notable challenges being the Dark Energy Problem and GR’s inability to account
for the inflationary epoch in our universe. Therefore, the validity of General
Relativity is expected to come under scrutiny in extreme conditions. General
Relativity is commonly acknowledged as an effective theory applicable only
within the realm of low energies. Consequently, such observations motivate us
to explore modified gravitational theories, especially in extreme conditions
where GR’s validity may be compromised. Among these theoretical frameworks,
modifications originating from String Theory, particularly the heterotic
string theory, emerge as leading contenders. Notably, String Theory offers
insights into high-order corrections, ranging from the Gauss-Bonnet term to
non-linear electromagnetic effects, and provides a rich avenue for exploring
the behavior of black holes under diverse conditions.

One intriguing aspect of string/brane-induced non-linear electrodynamics is
the emergence of the Born-Infeld (BI) Lagrangian, which encapsulates higher-
order corrections to Maxwell’s theory. This Lagrangian arises from the
resummation of open string excitations, particularly in the context of D-brane
worlds in string theory. The coupling of the BI Lagrangian to the dilaton
field in curved spacetime leads to an effective four-dimensional action,
offering a novel perspective on electromagnetic interactions in the presence
of gravity. Furthermore, considerations of higher-order electromagnetic terms,
originating from closed string sectors, broaden the theoretical landscape. The
inclusion of string loops leads to generalized effective actions,
incorporating both closed and open string contributions, and potentially
revealing novel phenomena beyond conventional electromagnetic frameworks.
Departing from traditional electromagnetic theories, the exploration of non-
linear electrodynamics within the context of black hole solutions offers a
rich avenue for understanding strong-field regimes and cosmological
implications. Non-linear effects become crucial in regions with intense
gravitational fields, such as those near black holes, shedding light on
phenomena absent in linear theories. Moreover, non-linear electrodynamics
holds relevance for early universe cosmology, where the interplay between
gravitational and electromagnetic fields played a significant role.

In this work, we considered a string-inspired theory that involves a scalar
field $\phi$ coupled to the electromagnetic field via a non-linear function
$f(\phi)$. The action encompasses higher-order electromagnetic invariants,
contributing to the field equations and leading to novel black hole solutions.
Furthermore, we investigated the impact of a non-trivial coupling function
$f(\phi)$, considering a specific functional form motivated by string-inspired
models. The resulting exact, magnetically charged black hole solution revealed
significant departures from the classical General Relativity predictions, with
the scalar field and the electromagnetic field configurations exhibiting non-
trivial behavior. We explored the implications of different coupling constants
$\alpha$ and $\beta$ on the spacetime geometry and electromagnetic field
configurations. The solutions obtained exhibit intriguing features, including
dependence on the sign of $\alpha-\beta$ which determines whether the higher-
order electromagnetic terms contribute attractively or repulsively to the
spacetime geometry. Additionally, we examined the horizon structure of the
black hole solutions, observing transitions from single to multiple horizons
and even to naked singularities as the parameters varied. Notably, the
compactness of the black holes relative to the Schwarzschild solution depended
on the the magnetic charge to mass ratio. Our findings suggest a rich
interplay between the scalar field, electromagnetic field, and spacetime
geometry, highlighting the potential implications of such theories in
astrophysical contexts, and the search for potential signatures of string
theory in black hole physics, at least those signatures that can be manifested
through effective string-inspired field theory models. It goes without saying,
however, that the present work does not deal with a detailed experimental
sensitivity analysis of such objects, which still remains to be done.

The examination of geodesics and energy conditions delves into the intricate
dynamics of particles moving within the spacetime geometry described by the
black hole solutions under investigation. By analyzing the geodesic equations
we unveil the behavior of massive particles in the vicinity of these black
holes, elucidating the role of the effective gravitational potential. Notably,
the effective potential exhibits distinct features depending on the relative
values of the coupling constants, offering insights into the stability and
nature of orbits around the black holes. Additionally, the examination of
energy conditions associated with the effective stress-energy tensor reveals
intriguing properties of the spacetime, indicating the existence of a dilaton
hair in the black hole’s exterior while satisfying all energy conditions. This
observation challenges the traditional no-hair theorems, underscoring the
nuanced interplay between gravitational theories, non-linear electrodynamics,
and scalar fields in modified theories. Our thermodynamic analysis provided
valuable insights into the properties of black holes in both the GMGHS
solution and our black-hole solution with non-linear electrodynamics. Our
analysis allows for the extraction of important thermodynamic quantities such
as mass, entropy, magnetostatic potential, and the extraction of the first Law
of Thermodynamics. Notably, the entropy of both black hole is consistent with
the Bekenstein-Hawking entropy formula. By examining the behavior of the
temperature, we concluded that when the non-linear electrodynamics terms act
attractively, there exist two distinct branches of black holes, one that is
getting colder as the mass is decreasing and therefore is thermally stable,
and another one that is getting hotter as the mass is decreasing which is
thermally unstable. On the other hand, when the non-linear electrodynamics
terms have a repulsive effect, the black holes are getting colder as the mass
is decreasing and as a result are thermally stable.

Finally, our analysis of linear perturbations and scalar quasi-normal modes
(QNMs) provides valuable insights into the stability and dynamic behavior of
black hole solutions with non-linear electrodynamics. Through a rigorous
investigation of radial stability, we demonstrated that our black hole
solutions remain stable under linear and radial perturbations. This finding
underscores the robustness of our black hole solutions against radial
perturbations, supporting their viability as physically meaningful
configurations within the framework of non-linear electrodynamics.
Furthermore, our examination of scalar QNMs yielded intriguing results
regarding the characteristic vibrations and oscillations of the black hole
spacetime. Moreover, the analysis of scalar QNMs revealed the intricate
interplay between the black hole parameters, such as charge and
$(\alpha-\beta)/M^{2}$, and the frequency and damping behavior of
perturbations. By systematically varying these parameters, we observed
distinct patterns in the QNM frequencies, indicating the sensitivity of the
black hole’s dynamic properties to its intrinsic characteristics. Notably, our
results exhibited convergence to the Schwarzschild black hole in the limit of
vanishing charge and alignment with the GMGHS black hole in specific parameter
regimes. These observations highlight the rich phenomenology associated with
black holes in our string-inspired theory.

## Acknowledgements

This work was supported by the Hellenic Foundation for Research and Innovation
(H.F.R.I.) under the “3rd Call for H.F.R.I. Research Projects to support Post-
Doctoral Researchers” (Project Number: 7212). The work of NEM was supported in
part by the UK Science and Technology Facilities research Council (STFC) and
UK Engineering and Physical Sciences Research Council (EPSRC) under the
research grants ST/X000753/1 and EP/V002821/1, respectively. NEM also
acknowledges participation in the COST Association Action CA21136 “Addressing
observational tensions in cosmology with systematics and fundamental physics
(CosmoVerse)”.

## References

  * [1] D. J. Gross and J. H. Sloan, “The Quartic Effective Action for the Heterotic String,” Nucl. Phys. B 291 (1987) 41–89.
  * [2] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory Vol. 1: 25th Anniversary Edition.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 11, 2012.
  * [3] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory Vol. 2: 25th Anniversary Edition.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 11, 2012.
  * [4] P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, “Dilatonic black holes in higher curvature string gravity,” Phys. Rev. D 54 (1996) 5049–5058, arXiv:hep-th/9511071.
  * [5] M. Natsuume, “Higher order correction to the GHS string black hole,” Phys. Rev. D 50 (1994) 3949–3953, arXiv:hep-th/9406079.
  * [6] Y. Kats, L. Motl, and M. Padi, “Higher-order corrections to mass-charge relation of extremal black holes,” JHEP 12 (2007) 068, arXiv:hep-th/0606100.
  * [7] D. Anninos and G. Pastras, “Thermodynamics of the Maxwell-Gauss-Bonnet anti-de Sitter Black Hole with Higher Derivative Gauge Corrections,” JHEP 07 (2009) 030, arXiv:0807.3478 [hep-th].
  * [8] J. T. Liu and P. Szepietowski, “Higher derivative corrections to R-charged AdS(5) black holes and field redefinitions,” Phys. Rev. D 79 (2009) 084042, arXiv:0806.1026 [hep-th].
  * [9] C. Charmousis, B. Gouteraux, and E. Kiritsis, “Higher-derivative scalar-vector-tensor theories: black holes, Galileons, singularity cloaking and holography,” JHEP 09 (2012) 011, arXiv:1206.1499 [hep-th].
  * [10] E. Babichev and C. Charmousis, “Dressing a black hole with a time-dependent Galileon,” JHEP 08 (2014) 106, arXiv:1312.3204 [gr-qc].
  * [11] E. Babichev, C. Charmousis, and A. Lehébel, “Black holes and stars in Horndeski theory,” Class. Quant. Grav. 33 no. 15, (2016) 154002, arXiv:1604.06402 [gr-qc].
  * [12] E. Babichev, C. Charmousis, and A. Lehébel, “Asymptotically flat black holes in Horndeski theory and beyond,” JCAP 04 (2017) 027, arXiv:1702.01938 [gr-qc].
  * [13] C. Charmousis, E. J. Copeland, A. Padilla, and P. M. Saffin, “General second order scalar-tensor theory, self tuning, and the Fab Four,” Phys. Rev. Lett. 108 (2012) 051101, arXiv:1106.2000 [hep-th].
  * [14] C. A. R. Herdeiro and E. Radu, “Asymptotically flat black holes with scalar hair: a review,” Int. J. Mod. Phys. D 24 no. 09, (2015) 1542014, arXiv:1504.08209 [gr-qc].
  * [15] O. Baake, A. Cisterna, M. Hassaine, and U. Hernandez-Vera, “Endorsing black holes with beyond Horndeski primary hair: An exact solution framework for scalarizing in every dimension,” arXiv:2312.05207 [hep-th].
  * [16] G. Antoniou, A. Bakopoulos, and P. Kanti, “Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories,” Phys. Rev. Lett. 120 no. 13, (2018) 131102, arXiv:1711.03390 [hep-th].
  * [17] G. Antoniou, A. Bakopoulos, and P. Kanti, “Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories,” Phys. Rev. D 97 no. 8, (2018) 084037, arXiv:1711.07431 [hep-th].
  * [18] G. Antoniou, A. Bakopoulos, P. Kanti, B. Kleihaus, and J. Kunz, “Novel Einstein–scalar-Gauss-Bonnet wormholes without exotic matter,” Phys. Rev. D 101 no. 2, (2020) 024033, arXiv:1904.13091 [hep-th].
  * [19] A. Bakopoulos, G. Antoniou, and P. Kanti, “Novel Black-Hole Solutions in Einstein-Scalar-Gauss-Bonnet Theories with a Cosmological Constant,” Phys. Rev. D 99 no. 6, (2019) 064003, arXiv:1812.06941 [hep-th].
  * [20] A. Bakopoulos, P. Kanti, and N. Pappas, “Existence of solutions with a horizon in pure scalar-Gauss-Bonnet theories,” Phys. Rev. D 101 no. 4, (2020) 044026, arXiv:1910.14637 [hep-th].
  * [21] A. Bakopoulos, P. Kanti, and N. Pappas, “Large and ultracompact Gauss-Bonnet black holes with a self-interacting scalar field,” Phys. Rev. D 101 no. 8, (2020) 084059, arXiv:2003.02473 [hep-th].
  * [22] A. Bakopoulos and T. Nakas, “Analytic and asymptotically flat hairy (ultra-compact) black-hole solutions and their axial perturbations,” JHEP 04 (2022) 096, arXiv:2107.05656 [gr-qc].
  * [23] A. Bakopoulos, C. Charmousis, and P. Kanti, “Traversable wormholes in beyond Horndeski theories,” JCAP 05 no. 05, (2022) 022, arXiv:2111.09857 [gr-qc].
  * [24] A. Bakopoulos, C. Charmousis, P. Kanti, and N. Lecoeur, “Compact objects of spherical symmetry in beyond Horndeski theories,” JHEP 08 (2022) 055, arXiv:2203.14595 [gr-qc].
  * [25] A. Bakopoulos, C. Charmousis, P. Kanti, N. Lecoeur, and T. Nakas, “Black holes with primary scalar hair,” Phys. Rev. D 109 no. 2, (2024) 024032, arXiv:2310.11919 [gr-qc].
  * [26] A. Bakopoulos and T. Nakas, “Novel exact ultracompact and ultrasparse hairy black holes emanating from regular and phantom scalar fields,” Phys. Rev. D 107 no. 12, (2023) 124035, arXiv:2303.09116 [gr-qc].
  * [27] A. Bakopoulos, N. Chatzifotis, and T. Nakas, “Compact objects with primary hair in shift and parity symmetric beyond Horndeski gravities,” arXiv:2312.17198 [gr-qc].
  * [28] A. Bakopoulos, N. Chatzifotis, C. Erices, and E. Papantonopoulos, “Stealth Ellis wormholes in Horndeski theories,” JCAP 11 (2023) 055, arXiv:2306.16768 [hep-th].
  * [29] T. Nakas, T. D. Pappas, and Z. Stuchlík, “Bridging dimensions: General embedding algorithm and field-theory reconstruction in 5D braneworld models,” Phys. Rev. D 109 no. 4, (2024) L041501, arXiv:2309.00873 [gr-qc].
  * [30] N. Chatzifotis, E. Papantonopoulos, and C. Vlachos, “Disformal transition of a black hole to a wormhole in scalar-tensor Horndeski theory,” Phys. Rev. D 105 no. 6, (2022) 064025, arXiv:2111.08773 [gr-qc].
  * [31] N. Chatzifotis, P. Dorlis, N. E. Mavromatos, and E. Papantonopoulos, “Scalarization of Chern-Simons-Kerr black hole solutions and wormholes,” Phys. Rev. D 105 no. 8, (2022) 084051, arXiv:2202.03496 [gr-qc].
  * [32] N. Chatzifotis, N. E. Mavromatos, and D. P. Theodosopoulos, “Global monopoles in the extended Gauss-Bonnet gravity,” Phys. Rev. D 107 no. 8, (2023) 085014, arXiv:2212.09467 [gr-qc].
  * [33] T. Karakasis, E. Papantonopoulos, Z.-Y. Tang, and B. Wang, “Black holes of (2+1)-dimensional $f(R)$ gravity coupled to a scalar field,” Phys. Rev. D 103 no. 6, (2021) 064063, arXiv:2101.06410 [gr-qc].
  * [34] T. Karakasis, E. Papantonopoulos, Z.-Y. Tang, and B. Wang, “Exact black hole solutions with a conformally coupled scalar field and dynamic Ricci curvature in f(R) gravity theories,” Eur. Phys. J. C 81 no. 10, (2021) 897, arXiv:2103.14141 [gr-qc].
  * [35] T. Karakasis, E. Papantonopoulos, and C. Vlachos, “f(R) gravity wormholes sourced by a phantom scalar field,” Phys. Rev. D 105 no. 2, (2022) 024006, arXiv:2107.09713 [gr-qc].
  * [36] T. Karakasis, E. Papantonopoulos, Z.-Y. Tang, and B. Wang, “(2+1)-dimensional black holes in f(R,$\phi$) gravity,” Phys. Rev. D 105 no. 4, (2022) 044038, arXiv:2201.00035 [gr-qc].
  * [37] T. Karakasis, N. E. Mavromatos, and E. Papantonopoulos, “Regular compact objects with scalar hair,” Phys. Rev. D 108 no. 2, (2023) 024001, arXiv:2305.00058 [gr-qc].
  * [38] T. Karakasis, G. Koutsoumbas, and E. Papantonopoulos, “Black holes with scalar hair in three dimensions,” Phys. Rev. D 107 no. 12, (2023) 124047, arXiv:2305.00686 [gr-qc].
  * [39] Z.-Y. Tang, Y. C. Ong, B. Wang, and E. Papantonopoulos, “General black hole solutions in ( 2+1 )-dimensions with a scalar field nonminimally coupled to gravity,” Phys. Rev. D 100 no. 2, (2019) 024003, arXiv:1901.07310 [gr-qc].
  * [40] Z.-Y. Tang, B. Wang, T. Karakasis, and E. Papantonopoulos, “Curvature scalarization of black holes in f(R) gravity,” Phys. Rev. D 104 no. 6, (2021) 064017, arXiv:2008.13318 [gr-qc].
  * [41] H.-S. Liu, H. Lu, Z.-Y. Tang, and B. Wang, “Black hole scalarization in Gauss-Bonnet extended Starobinsky gravity,” Phys. Rev. D 103 no. 8, (2021) 084043, arXiv:2004.14395 [gr-qc].
  * [42] E. Babichev, C. Charmousis, M. Hassaine, and N. Lecoeur, “Conformally coupled theories and their deformed compact objects: From black holes, radiating spacetimes to eternal wormholes,” Phys. Rev. D 106 no. 6, (2022) 064039, arXiv:2206.11013 [gr-qc].
  * [43] E. Babichev, C. Charmousis, M. Hassaine, and N. Lecoeur, “Selecting Horndeski theories without apparent symmetries and their black hole solutions,” Phys. Rev. D 108 no. 2, (2023) 024019, arXiv:2303.04126 [gr-qc].
  * [44] E. Babichev, C. Charmousis, and N. Lecoeur, “Rotating black holes embedded in a cosmological background for scalar-tensor theories,” JCAP 08 (2023) 022, arXiv:2305.17129 [gr-qc].
  * [45] E. Babichev, C. Charmousis, and N. Lecoeur, “Exact black hole solutions in higher-order scalar-tensor theories,” arXiv:2309.12229 [gr-qc].
  * [46] E. Babichev, C. Charmousis, M. Hassaine, and N. Lecoeur, “Conformally coupled scalar in Lovelock theory,” Phys. Rev. D 107 no. 8, (2023) 084050, arXiv:2302.02920 [gr-qc].
  * [47] C. Charmousis, A. Lehébel, E. Smyrniotis, and N. Stergioulas, “Astrophysical constraints on compact objects in 4D Einstein-Gauss-Bonnet gravity,” JCAP 02 no. 02, (2022) 033, arXiv:2109.01149 [gr-qc].
  * [48] G. Antoniou, A. Lehébel, G. Ventagli, and T. P. Sotiriou, “Black hole scalarization with Gauss-Bonnet and Ricci scalar couplings,” Phys. Rev. D 104 no. 4, (2021) 044002, arXiv:2105.04479 [gr-qc].
  * [49] N. Andreou, N. Franchini, G. Ventagli, and T. P. Sotiriou, “Spontaneous scalarization in generalised scalar-tensor theory,” Phys. Rev. D 99 no. 12, (2019) 124022, arXiv:1904.06365 [gr-qc]. [Erratum: Phys.Rev.D 101, 109903 (2020)].
  * [50] G. Ventagli, A. Lehébel, and T. P. Sotiriou, “Onset of spontaneous scalarization in generalized scalar-tensor theories,” Phys. Rev. D 102 no. 2, (2020) 024050, arXiv:2006.01153 [gr-qc].
  * [51] S. Mahapatra, S. Priyadarshinee, G. N. Reddy, and B. Shukla, “Exact topological charged hairy black holes in AdS Space in $D$-dimensions,” Phys. Rev. D 102 no. 2, (2020) 024042, arXiv:2004.00921 [hep-th].
  * [52] J. D. Bekenstein, “Nonexistence of baryon number for static black holes,” Phys. Rev. D 5 (1972) 1239–1246.
  * [53] J. D. Bekenstein, “Transcendence of the law of baryon-number conservation in black hole physics,” Phys. Rev. Lett. 28 (1972) 452–455.
  * [54] C. Teitelboim, “Nonmeasurability of the lepton number of a black hole,” Lett. Nuovo Cim. 3S2 (1972) 397–400.
  * [55] J. D. Bekenstein, “Novel “no-scalar-hair” theorem for black holes,” Phys. Rev. D 51 (Jun, 1995) R6608–R6611. https://link.aps.org/doi/10.1103/PhysRevD.51.R6608.
  * [56] A. E. Mayo and J. D. Bekenstein, “No hair for spherical black holes: Charged and nonminimally coupled scalar field with selfinteraction,” Phys. Rev. D 54 (1996) 5059–5069, arXiv:gr-qc/9602057.
  * [57] P. O. Mazur, “Black hole uniqueness theorems,” arXiv:hep-th/0101012.
  * [58] P. T. Chrusciel, J. Lopes Costa, and M. Heusler, “Stationary Black Holes: Uniqueness and Beyond,” Living Rev. Rel. 15 (2012) 7, arXiv:1205.6112 [gr-qc].
  * [59] S. Priyadarshinee, S. Mahapatra, and I. Banerjee, “Analytic topological hairy dyonic black holes and thermodynamics,” Phys. Rev. D 104 no. 8, (2021) 084023, arXiv:2108.02514 [hep-th].
  * [60] S. Priyadarshinee and S. Mahapatra, “Analytic three-dimensional primary hair charged black holes and thermodynamics,” Phys. Rev. D 108 no. 4, (2023) 044017, arXiv:2305.09172 [gr-qc].
  * [61] A. Daripa and S. Mahapatra, “Analytic three-dimensional primary hair charged black holes with Coulomb-like electrodynamics and their thermodynamics,” arXiv:2401.04561 [gr-qc].
  * [62] D. P. Theodosopoulos, T. Karakasis, G. Koutsoumbas, and E. Papantonopoulos, “Motion of particles around a magnetically charged Euler-Heisenberg black hole with scalar hair and the Event Horizon Telescope,” arXiv:2311.02740 [gr-qc].
  * [63] T. Karakasis, E. Papantonopoulos, Z.-Y. Tang, and B. Wang, “Rotating (2+1)-dimensional black holes in Einstein-Maxwell-dilaton theory,” Phys. Rev. D 107 no. 2, (2023) 024043, arXiv:2210.15704 [gr-qc].
  * [64] T. Karakasis, G. Koutsoumbas, A. Machattou, and E. Papantonopoulos, “Magnetically charged Euler-Heisenberg black holes with scalar hair,” Phys. Rev. D 106 no. 10, (2022) 104006, arXiv:2207.13146 [gr-qc].
  * [65] M. Kord Zangeneh, B. Wang, A. Sheykhi, and Z. Y. Tang, “Charged scalar quasi-normal modes for linearly charged dilaton-Lifshitz solutions,” Phys. Lett. B 771 (2017) 257–263, arXiv:1701.03644 [hep-th].
  * [66] Z.-Y. Tang, C.-Y. Zhang, M. Kord Zangeneh, B. Wang, and J. Saavedra, “Thermodynamical and dynamical properties of Charged BTZ Black Holes,” Eur. Phys. J. C 77 no. 6, (2017) 390, arXiv:1610.01744 [hep-th].
  * [67] N. Sanchis-Gual, J. Calderón Bustillo, C. Herdeiro, E. Radu, J. A. Font, S. H. W. Leong, and A. Torres-Forné, “Impact of the wavelike nature of Proca stars on their gravitational-wave emission,” Phys. Rev. D 106 no. 12, (2022) 124011, arXiv:2208.11717 [gr-qc].
  * [68] C. Herdeiro, E. Radu, and H. Rúnarsson, “Kerr black holes with Proca hair,” Class. Quant. Grav. 33 no. 15, (2016) 154001, arXiv:1603.02687 [gr-qc].
  * [69] C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Font, “Spontaneous Scalarization of Charged Black Holes,” Phys. Rev. Lett. 121 no. 10, (2018) 101102, arXiv:1806.05190 [gr-qc].
  * [70] C. Charmousis, B. Gouteraux, and J. Soda, “Einstein-Maxwell-Dilaton theories with a Liouville potential,” Phys. Rev. D 80 (2009) 024028, arXiv:0905.3337 [gr-qc].
  * [71] E. Babichev, C. Charmousis, and M. Hassaine, “Black holes and solitons in an extended Proca theory,” JHEP 05 (2017) 114, arXiv:1703.07676 [gr-qc].
  * [72] E. Babichev, C. Charmousis, and M. Hassaine, “Charged Galileon black holes,” JCAP 05 (2015) 031, arXiv:1503.02545 [gr-qc].
  * [73] J. Barrientos, A. Cisterna, D. Kubiznak, and J. Oliva, “Accelerated black holes beyond Maxwell’s electrodynamics,” Phys. Lett. B 834 (2022) 137447, arXiv:2205.15777 [gr-qc].
  * [74] A. Cisterna, M. Hassaine, J. Oliva, and M. Rinaldi, “Static and rotating solutions for Vector-Galileon theories,” Phys. Rev. D 94 no. 10, (2016) 104039, arXiv:1609.03430 [gr-qc].
  * [75] A. Cisterna, G. Giribet, J. Oliva, and K. Pallikaris, “Quasitopological electromagnetism and black holes,” Phys. Rev. D 101 no. 12, (2020) 124041, arXiv:2004.05474 [hep-th].
  * [76] A. Cisterna, C. Henríquez-Báez, N. Mora, and L. Sanhueza, “Quasitopological electromagnetism: Reissner-Nordström black strings in Einstein and Lovelock gravities,” Phys. Rev. D 104 no. 6, (2021) 064055, arXiv:2105.04239 [gr-qc].
  * [77] H. Rehman and G. Abbas, “Accretion around a hairy black hole in the framework of gravitational decoupling theory,” Chin. Phys. C 47 no. 12, (2023) 125106.
  * [78] H. Rehman, G. Abbas, T. Zhu, and G. Mustafa, “Matter accretion onto the magnetically charged Euler–Heisenberg black hole with scalar hair,” Eur. Phys. J. C 83 no. 9, (2023) 856, arXiv:2307.16155 [gr-qc].
  * [79] D. Magos, N. Bretón, and A. Macías, “Orbits in static magnetically and dyonically charged Einstein-Euler-Heisenberg black hole spacetimes,” Phys. Rev. D 108 no. 6, (2023) 064014.
  * [80] M. Dernek, C. Tekincay, G. Gecim, Y. Kucukakca, and Y. Sucu, “Hawking radiation of Euler–Heisenberg-adS black hole under the GUP effect,” Eur. Phys. J. Plus 138 no. 4, (2023) 369.
  * [81] R. H. Ali and G. Abbas, “Thermodynamics under the impact of thermal fluctuations and quasi-normal modes of Euler-Heisenberg AdS BH in the framework of NLED,” Chin. Phys. C 47 no. 11, (2023) 115106.
  * [82] S. Kiorpelidi, T. Karakasis, G. Koutsoumbas, and E. Papantonopoulos, “Scalarization of the Reissner-Nordström black hole with higher derivative gauge field corrections,” Phys. Rev. D 109 no. 2, (2024) 024033, arXiv:2311.10858 [gr-qc].
  * [83] D. P. Sorokin, “Introductory Notes on Non-linear Electrodynamics and its Applications,” Fortsch. Phys. 70 no. 7-8, (2022) 2200092, arXiv:2112.12118 [hep-th].
  * [84] G. W. Gibbons and K.-i. Maeda, “Black Holes and Membranes in Higher Dimensional Theories with Dilaton Fields,” Nucl. Phys. B 298 (1988) 741–775.
  * [85] D. Garfinkle, G. T. Horowitz, and A. Strominger, “Charged black holes in string theory,” Phys. Rev. D 43 (1991) 3140. [Erratum: Phys.Rev.D 45, 3888 (1992)].
  * [86] M. Born, “Modified field equations with a finite radius of the electron,” Nature 132 no. 3329, (1933) 282.1.
  * [87] M. Born, “On the quantum theory of the electromagnetic field,” Proc. Roy. Soc. Lond. A 143 no. 849, (1934) 410–437.
  * [88] M. Born and L. Infeld, “Foundations of the new field theory,” Proc. Roy. Soc. Lond. A 144 no. 852, (1934) 425–451.
  * [89] R. R. Metsaev, M. Rakhmanov, and A. A. Tseytlin, “The Born-Infeld Action as the Effective Action in the Open Superstring Theory,” Phys. Lett. B 193 (1987) 207–212.
  * [90] O. D. Andreev and A. A. Tseytlin, “Partition Function Representation for the Open Superstring Effective Action: Cancellation of Mobius Infinities and Derivative Corrections to Born-Infeld Lagrangian,” Nucl. Phys. B 311 (1988) 205–252.
  * [91] A. A. Tseytlin, “Born-Infeld action, supersymmetry and string theory,” arXiv:hep-th/9908105.
  * [92] R. G. Leigh, “Dirac-Born-Infeld Action from Dirichlet Sigma Model,” Mod. Phys. Lett. A 4 (1989) 2767.
  * [93] J. Dai, R. G. Leigh, and J. Polchinski, “New Connections Between String Theories,” Mod. Phys. Lett. A 4 (1989) 2073–2083.
  * [94] J. Polchinski, “Tasi lectures on D-branes,” in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, pp. 293–356.  11, 1996.  arXiv:hep-th/9611050.
  * [95] J. Ellis, N. E. Mavromatos, and T. You, “Light-by-Light Scattering Constraint on Born-Infeld Theory,” Phys. Rev. Lett. 118 no. 26, (2017) 261802, arXiv:1703.08450 [hep-ph].
  * [96] J. Ellis, N. E. Mavromatos, P. Roloff, and T. You, “Light-by-light scattering at future $e^{+}e^{-}$ colliders,” Eur. Phys. J. C 82 no. 7, (2022) 634, arXiv:2203.17111 [hep-ph].
  * [97] D. d’Enterria and G. G. da Silveira, “Observing light-by-light scattering at the Large Hadron Collider,” Phys. Rev. Lett. 111 (2013) 080405, arXiv:1305.7142 [hep-ph]. [Erratum: Phys.Rev.Lett. 116, 129901 (2016)].
  * [98] ATLAS Collaboration, M. Aaboud et al., “Evidence for light-by-light scattering in heavy-ion collisions with the ATLAS detector at the LHC,” Nature Phys. 13 no. 9, (2017) 852–858, arXiv:1702.01625 [hep-ex].
  * [99] CMS Collaboration, A. M. Sirunyan et al., “Evidence for light-by-light scattering and searches for axion-like particles in ultraperipheral PbPb collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV,” Phys. Lett. B 797 (2019) 134826, arXiv:1810.04602 [hep-ex].
  * [100] S. S. Yazadjiev, “Einstein-Born-Infeld-dilaton black holes in non-asymptotically flat spacetimes,” Phys. Rev. D 72 (2005) 044006, arXiv:hep-th/0504152.
  * [101] M. H. Dehghani, S. H. Hendi, A. Sheykhi, and H. Rastegar Sedehi, “Thermodynamics of rotating black branes in (n+1)-dimensional Einstein-Born-Infeld-dilaton gravity,” JCAP 02 (2007) 020, arXiv:hep-th/0611288.
  * [102] A. Sheykhi and N. Riazi, “Thermodynamics of black holes in (n+1)-dimensional Einstein-Born-Infeld dilaton gravity,” Phys. Rev. D 75 (2007) 024021, arXiv:hep-th/0610085.
  * [103] T. Damour and A. M. Polyakov, “The String dilaton and a least coupling principle,” Nucl. Phys. B 423 (1994) 532–558, arXiv:hep-th/9401069.
  * [104] R. R. Metsaev and A. A. Tseytlin, “ON LOOP CORRECTIONS TO STRING THEORY EFFECTIVE ACTIONS,” Nucl. Phys. B 298 (1988) 109–132.
  * [105] H. Yajima and T. Tamaki, “Black hole solutions in Euler-Heisenberg theory,” Phys. Rev. D 63 (2001) 064007, arXiv:gr-qc/0005016.
  * [106] J. D. Bekenstein, “Novel “no-scalar-hair” theorem for black holes,” Phys. Rev. D 51 no. 12, (1995) R6608.
  * [107] T. P. Sotiriou and S.-Y. Zhou, “Black hole hair in generalized scalar-tensor gravity,” Phys. Rev. Lett. 112 (2014) 251102, arXiv:1312.3622 [gr-qc].
  * [108] P. Dorlis, N. E. Mavromatos, and S.-N. Vlachos, “Bypassing Bekenstein’s no-scalar-hair theorem without violating the energy conditions,” Phys. Rev. D 108 no. 6, (2023) 064004, arXiv:2305.18031 [gr-qc].
  * [109] M. Bravo-Gaete and M. Hassaine, “Thermodynamics of a BTZ black hole solution with an Horndeski source,” Phys. Rev. D 90 no. 2, (2014) 024008, arXiv:1405.4935 [hep-th].
  * [110] S. Fulling and S. Ruijsenaars, “Temperature, periodicity and horizons,” Physics Reports 152 no. 3, (1987) 135–176. https://www.sciencedirect.com/science/article/pii/0370157387901360.
  * [111] G. W. Gibbons and S. W. Hawking, “Action Integrals and Partition Functions in Quantum Gravity,” Phys. Rev. D 15 (1977) 2752–2756.
  * [112] R. Arnowitt, S. Deser, and C. W. Misner, “Dynamical structure and definition of energy in general relativity,” Phys. Rev. 116 (Dec, 1959) 1322–1330. https://link.aps.org/doi/10.1103/PhysRev.116.1322.
  * [113] A. Corichi and D. Núñez, “Introduction to the ADM formalism,” Rev. Mex. Fis. 37 (1991) 720–747, arXiv:2210.10103 [gr-qc].
  * [114] C. Martinez, R. Troncoso, and J. Zanelli, “Exact black hole solution with a minimally coupled scalar field,” Phys. Rev. D 70 (2004) 084035, arXiv:hep-th/0406111.
  * [115] G. T. Horowitz, “The dark side of string theory: Black holes and black strings.,”  10, 1992.  arXiv:hep-th/9210119.
  * [116] N. Chatzifotis, P. Dorlis, N. E. Mavromatos, and E. Papantonopoulos, “Thermal stability of hairy black holes,” Phys. Rev. D 107 no. 8, (2023) 084053, arXiv:2302.03980 [gr-qc].
  * [117] S.-W. Wei, Y.-X. Liu, and R. B. Mann, “Black Hole Solutions as Topological Thermodynamic Defects,” Phys. Rev. Lett. 129 no. 19, (2022) 191101, arXiv:2208.01932 [gr-qc].
  * [118] T. Regge and J. A. Wheeler, “Stability of a Schwarzschild singularity,” Phys. Rev. 108 (1957) 1063–1069.
  * [119] F. J. Zerilli, “Effective potential for even parity Regge-Wheeler gravitational perturbation equations,” Phys. Rev. Lett. 24 (1970) 737–738.
  * [120] F. J. Zerilli, “Gravitational field of a particle falling in a schwarzschild geometry analyzed in tensor harmonics,” Phys. Rev. D 2 (1970) 2141–2160.
  * [121] F. J. Zerilli, “Perturbation analysis for gravitational and electromagnetic radiation in a reissner-nordstroem geometry,” Phys. Rev. D 9 (1974) 860–868.
  * [122] D. Astefanesei, J. L. Blázquez-Salcedo, C. Herdeiro, E. Radu, and N. Sanchis-Gual, “Dynamically and thermodynamically stable black holes in Einstein-Maxwell-dilaton gravity,” JHEP 07 (2020) 063, arXiv:1912.02192 [gr-qc].
  * [123] K. D. Kokkotas and B. G. Schmidt, “Quasinormal modes of stars and black holes,” Living Rev. Rel. 2 (1999) 2, arXiv:gr-qc/9909058.
  * [124] E. Berti, V. Cardoso, and A. O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant. Grav. 26 (2009) 163001, arXiv:0905.2975 [gr-qc].
  * [125] R. A. Konoplya and A. Zhidenko, “Quasinormal modes of black holes: From astrophysics to string theory,” Rev. Mod. Phys. 83 (2011) 793–836, arXiv:1102.4014 [gr-qc].
  * [126] B. F. Schutz and C. M. Will, “BLACK HOLE NORMAL MODES: A SEMIANALYTIC APPROACH,” Astrophys. J. Lett. 291 (1985) L33–L36.
  * [127] S. Iyer and C. M. Will, “Black Hole Normal Modes: A WKB Approach. 1. Foundations and Application of a Higher Order WKB Analysis of Potential Barrier Scattering,” Phys. Rev. D 35 (1987) 3621.
  * [128] R. A. Konoplya, “Gravitational quasinormal radiation of higher dimensional black holes,” Phys. Rev. D 68 (2003) 124017, arXiv:hep-th/0309030.
  * [129] C. J. Gao and S. N. Zhang, “Dilaton black holes in de Sitter or Anti-de Sitter universe,” Phys. Rev. D 70 (2004) 124019, arXiv:hep-th/0411104.
  * [130] G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, “De Sitter Space and the Swampland,” arXiv:1806.08362 [hep-th].
  * [131] E. Palti, “The Swampland: Introduction and Review,” Fortsch. Phys. 67 no. 6, (2019) 1900037, arXiv:1903.06239 [hep-th].
  * [132] H. Ooguri, E. Palti, G. Shiu, and C. Vafa, “Distance and de Sitter Conjectures on the Swampland,” Phys. Lett. B 788 (2019) 180–184, arXiv:1810.05506 [hep-th].