# De-singularizing the extremal GMGHS black hole
via higher derivatives corrections

Carlos Herdeiro†, Eugen Radu† and Kunihito Uzawa‡

† Departamento de Matemática da Universidade de Aveiro and
Centre for Research and Development in Mathematics and Applications (CIDMA),
Campus de Santiago, 3810-183 Aveiro, Portugal
‡Department of Physics, School of Science and Technology,
Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan

(February 2021)

###### Abstract

The Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole is an
influential solution of the low energy heterotic string theory. As it is well
known, it presents a singular extremal limit. We construct a regular extension
of the GMGHS extremal black hole in a model with
$\mathcal{O}(\alpha^{\prime})$ corrections in the action, by solving the fully
non-linear equations of motion. The de-singularization is supported by the
$\mathcal{O}(\alpha^{\prime})$-terms. The regularised extremal GMGHS BHs are
asymptotically flat, possess a regular (non-zero size) horizon of spherical
topology, with an $AdS_{2}\times S^{2}$ near horizon geometry, and their
entropy is proportional to the electric charge. The near horizon solution is
obtained analytically and some illustrative bulk solutions are constructed
numerically.

##  1 Introduction

Low energy string theory compactified to four spacetime dimensions admits a
famous black hole (BH) solution, found by Gibbons and Maeda [1] and,
independently, by Garfinkle, Horowitz and Strominger [2] – from now on dubbed
the GMGHS BH. This solution can be described either in the Einstein frame or
in the conformally related string frame. In the former case (which is the one
considered in this work), the model’s action is the sum of an Einstein term, a
kinetic term for a scalar field (the dilaton) and a Maxwell term with an
exponential coupling to the dilaton - see Eq. (2.3) below. The GMGHS BH is the
simplest solution of this model, representing a charged, static and
spherically symmetric horizon surrounded by scalar hair. This stringy
extension of the Reissner-Nordström (RN) BH has attracted an enormous
attention, finding a variety of interesting applications, $e.g.$ [2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30].

An awkward property of the GMGHS solution is that its extremal limit is
singular: when taking this limit the area of the spatial sections of the
horizon shrinks to zero and the Kretschmann scalar blows up at the (would be)
horizon. Since neither Ref. [1] nor [2] take into account the possible stringy
$\alpha^{\prime}$-corrections to the Einstein-Maxwell-dilaton action one may
ask if such corrections could de-singularize the extremal solution.

A perturbative extension of the extremal magnetic GMGHS BH has been
constructed by Natsuume in Ref. [9] (to first order order in
$\alpha^{\prime}$). As found therein [15, 21, 20, 27, 26, 28], the corrected
solution inherits all basic properties of the extremal GMGHS BH; in
particular, the horizon area still vanishes. On the other hand, to the best of
our knowledge, the task of constructing the fully non-linear BH solutions of
the $\mathcal{O}(\alpha^{\prime})$ corrected action has not yet been
considered in the literature. This is presumably due to the complexity of the
field equations. Yet, such construction can reply to the key question whether
such corrections can desingularise the extremal GMGHS solution. The main
purpose of this work is to report results in this direction. Starting with a
general model for the $\mathcal{O}(\alpha^{\prime})$ corrections to the
Einstein-Maxwell-dilaton action (which is essentially the one in Ref. [9]), we
find that the field equations of the full model possess an exact solution
describing a Robinson-Bertotti-type vacuum, with an $AdS_{2}\times S^{2}$
metric, an electric field and a constant dilaton. On the other hand, we also
find there is no counterpart of this solution with a magnetic charge.

The supergravity action includes, in general, a tower of corrections of all
powers in $\alpha^{\prime}$ because of the recursive definition of the Kalb-
Ramond 3-form field strength, which breaks the supersymmetry in the
supergravity theory. The term of quadratic order in curvature is obtained
imposing supersymmetry of the theory at first order in $\alpha^{\prime}$ if we
consider the Chern-Simons term in the field strength. There are further higher
power corrections in the curvature of the torsionful spin connection, which
are required to preserve supersymmetry order by order [31]. Additional higher-
curvature corrections unrelated to the supersymmetrization of the Kalb- Ramond
kinetic term are also present, although those appear first at cubic order in
$\alpha^{\prime}$. These additional higher-order (like
${\alpha^{\prime}}^{2}$, ${\alpha^{\prime}}^{3}$ $\cdots$) corrections may
drastically modify the non-perturbative result obtained from our setup. The
properties of the solutions with higher curvature corrections, within the
heterotic string theory, is still largely an uncharted territory. Although the
conditions imposed by supersymmetry have been studied in detail, the solution
of the Einstein equations are not well known yet, especially for the
interesting case of the heterotic string with fluxes.

As for the extremal RN solution in the Einstein-Maxwell model, we expect this
“attractor” to describe the near-horizon geometry of an extremal BH with a
regular horizon. That is, the $\mathcal{O}(\alpha^{\prime})$-corrections de-
singularize the extremal limit of the GMGHS BH, leading to a non-zero size,
regular horizon111 Finding the local solutions in the vicinity of the
(extremal) horizon ($i.e.$ the attractor) does not guarantee the existence of
global solutions with the right asymptotics. Progress in this direction
requires an explicit construction of the bulk extremal BHs.. It is well known
that such near horizon geometry is a key feature of static supersymmetric BHs,
providing the attractor mechanism by which horizon scalar fields values are
determined by the charges carried by the BH and insensitive to the asymptotic
values of the scalar fields [32, 33, 34, 35]. The entropy of these BHs is
consistent with the microscopic states counting of the associated $D$-brane
system. These are described by the inclusion of higher derivative corrections
in the generalized prepotential together with the supergravity low energy
description [36, 37, 38].

In this work we shall provide numerical evidence for the existence of non-
perturbative extensions of the GMGHS BHs which are extremal, asymptotically
flat and regular, on and outside the horizon. These solutions represent global
(bulk) extensions of the aforementioned attractors found analytically and
possess a variety of interesting properties. For example, as for the
attractors, the entropy of the extremal BHs is proportional to the electric
charge. Also, their charge to mass ratio is always greater than one.

This paper is organised as follows. In Section 2 we describe the full model,
including $\alpha^{\prime}$ corrections and describe briefly the GMGHS
solution. In Section 3 we consider the $\alpha^{\prime}$ corrected solutions;
first we construct (analytically) the attractors and then we give
(numerically) examples of global bulk solutions. Section 4 gives some
concluding remarks. In the appendix, we summarize some formulas, including the
equations of motion and our static spherically symmetric ansatz, which are
needed for the results in section 3.

##  2 The model

###  2.1 The action

The starting point is the string-frame action including order
$\alpha^{\prime}$ terms

$\displaystyle S_{s}=\int d^{4}x\sqrt{-\tilde{g}}{\rm
e}^{-2\phi}\bigg{[}\tilde{R}+4\left(\tilde{\nabla}\phi\right)^{2}-F^{2}+\alpha^{\prime}\bigg{\\{}a\left(\tilde{R}_{\mu\nu\rho\sigma}\tilde{R}^{\mu\nu\rho\sigma}-4\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}+\tilde{R}^{2}\right)$
(2.1)
$\displaystyle+b\left(F^{2}\right)^{2}+cF^{2}\left(\tilde{\nabla}\phi\right)^{2}+h\tilde{R}^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}\bigg{\\}}\bigg{]},$

where $\tilde{g}$ , $\tilde{R}$ , $\tilde{R}_{\mu\nu}$ ,
$\tilde{R}_{\mu\nu\rho\sigma}$ , $\tilde{\nabla}$ denote determinant, Ricci
scalar, Ricci tensor, Riemann curvature tensor, covariant derivative
constructed from the string frame metric $\tilde{g}_{\mu\nu}$ , respectively.
Also, $F=dA$ is the $U(1)$ field strength tensor and $\phi$ is a real scalar
field: the dilaton. $a,b,c,h$ are constant coefficients.

The corresponding expression of the action in the Einstein frame (which is the
case of interest in this work) is found via a conformal transformation, with

$\tilde{g}_{\mu\nu}={\rm e}^{2\phi}g_{\mu\nu}\,.$ (2.2)

This results in ($g_{\mu\nu}$ being the Einstein frame metric):

$\displaystyle S_{E}=\int d^{4}x\sqrt{-g}{\cal L},\qquad~{}~{}{\rm
where}~{}~{}{\cal L}=R-2\left(\nabla\phi\right)^{2}-{\rm
e}^{-\phi}F^{2}+\alpha^{\prime}{\cal L}^{\prime}~{},$ (2.3)

with the leading order corrections in $\alpha^{\prime}$

$\displaystyle{\cal L}^{\prime}$ $\displaystyle=$ $\displaystyle
a~{}e^{-\phi}\left[R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}-2R{\triangle}\phi+4\nabla_{\mu}\nabla_{\nu}\phi\nabla^{\mu}\nabla^{\nu}\phi-\left({\triangle}\phi\right)^{2}\right.$
(2.4)
$\displaystyle\left.+3{\triangle}\phi\left(\nabla\phi\right)^{2}+\frac{3}{4}\left(\nabla\phi\right)^{4}+2R_{\mu\nu}\left(2\nabla^{\mu}\nabla^{\nu}\phi-\nabla^{\mu}\phi\nabla^{\nu}\phi\right)\right]$
$\displaystyle+b~{}{\rm e}^{-3\phi}\left(F^{2}\right)^{2}+\frac{c}{4}\,{\rm
e}^{-2\phi}F^{2}\left(\nabla\phi\right)^{2}+h~{}e^{-2\phi}F_{\mu\nu}F_{\rho\sigma}\left[R^{\mu\nu\rho\sigma}+g^{\sigma\left[\mu\right.}\nabla^{\left.\nu\right]}\nabla^{\rho}\phi-g^{\rho\left[\mu\right.}\nabla^{\left.\nu\right]}\nabla^{\sigma}\phi\right.$
$\displaystyle\left.+\frac{1}{2}\left\\{\left(\nabla^{\left[\mu\right.}\phi\right)g^{\left|\sigma\right|\left.\nu\right]}\nabla^{\rho}\phi-\left(\nabla^{\left[\mu\right.}\phi\right)g^{\left|\rho\right|\left.\nu\right]}\nabla^{\sigma}\phi-g^{\rho\left[\mu\right.}g^{\left|\sigma\right|\left.\nu\right]}g^{\rho^{\prime}\sigma^{\prime}}\nabla_{\rho^{\prime}}\phi\nabla_{\sigma^{\prime}}\phi\right\\}\right].$

The variation of (2.3) with respect to $g_{\mu\nu}$, $\phi$ and $A_{\mu}$
leads to the equations of motion of the model. However, their expression is
too complicated to include here.

Let us briefly comment on the string theory origin of the above action. The
$U(1)$ gauge field arises as a subgroup of the ${\rm E}_{8}\times{\rm E}_{8}$
or $SO(32)$ gauge group in the low-energy effective theory of the heterotic
string [3]. We set the remaining gauge fields and the antisymmetric field
strength $H_{\mu\nu\rho}$ to zero. Then the four-dimensional action emerges
from heterotic string theory compactified on a six-dimensional torus. This
includes corrections due to the next order terms such as $R^{2}$ , $F^{4}$ ,
$F^{2}\left(\nabla\phi\right)^{2}$ [9, 15] . After eliminating terms in the
effective action by field redefinitions, the corrections to the leading order
in $\alpha^{\prime}$ can be expressed as (2.4).

The higher-order ($\alpha^{\prime}$) terms in the four-dimensional effective
action can be considered corrections due to quantum gravity [15, 21, 39]. Let
us estimate the mass scaling of these terms, for extremal BHs, near the
horizon $r\sim M$. The GMGHS BH solution implies that any derivative
contributes a factor of order $M^{-1}$; then, the curvature tensors are $R\sim
M^{-2}$ , $R^{2}\sim M^{-4}$ and the gauge field strength gives $F\sim M^{-1}$
. Hence we have, $e.g.$ $(F^{2})^{2}\sim M^{-4}$ , $F^{2}(\nabla\phi)^{2}\sim
M^{-4}$ , $F^{2}\nabla^{2}\phi\sim M^{-4}$ and $F^{2}R\sim M^{-4}$. We assume
the classical mass-charge relation for extremal BHs approximately holds and
they are sufficiently large, $Q\simeq M\gg 1$, in units of the Planck mass
$M_{\rm Pl}$ . We conclude that the $\alpha^{\prime}$ corrections are
suppressed by powers of $(\alpha^{\prime}/M)^{2}$ . Thus, we consider the
leading-order $\alpha^{\prime}$ corrections throughout only, assuming the BHs
are sufficiently macroscopic $\alpha^{\prime}/M\ll 1$.

Moreover, the leading order terms in the Lagrangian ${\cal L}$ have $M^{-2}$
while the leading order correction in the Einstein frame are given by (2.4)
[9, 15]; all terms involving $\nabla_{\rho}F_{\mu\nu}$ are dropped, without
loss of generality, since they are equivalent, via the Bianchi identities, to
terms already accounted for or terms involving $\nabla_{\mu}F^{\mu\nu}$ [21,
39] . Such terms vanish in the absence of charged matter sources, which is the
case considered in this paper.

We do not consider terms that are proportional to the Kalb-Ramond three-form
field strength $H$,

$H=dB+\frac{1}{4}\left(\omega_{\rm L}-\omega_{\rm Y}\right)\,,$ (2.5)

where $B$ is 2-form gauge potential, $\omega_{\rm L}$, $\omega_{\rm Y}$ denote
the Lorentz and Yang-Mills Chern-Simons forms, respectively. The presence of
$H$ in the heterotic string is required due to $\alpha^{\prime}$ corrections
in the Bianchi identity which are needed for anomaly cancellation. Since the
Chern-Simons forms act as sources for $H$ , we are in general not able to
simply set $H=0$ . However, the Lorentz Chern-Simons form can be written by
the exterior derivative of a three form for the spherically symmetric metrics.
Hence, we can absorb it into the definition of 2-form gauge potential $B$
[40]. Moreover, the Yang-Mills Chern-Simons form vanishes for the purely
magnetic case [9].

In the expression (2.3), $a,b,c$ and $h$ are constants which are not fixed a
priori. The coefficient $a$ is arbitrary (and nonzero), since it can be
changed by scaling $\alpha^{\prime}$. The choice in Ref. [9] (employed also
here) is $a=1/8$. That reference gives also an argument that $h=0$.

As for $b$ and $c$, to the best of our knowledge, the only concrete
computation is in Ref. [9]. The idea there was to construct the (leading
order) $\alpha^{\prime}$-correction to the magnetic GMGHS BH, in the extremal
limit. The corrected solution is supposed to possess the same near-throat
behaviour and the same far field asymptotics as the original solution (with
$\alpha^{\prime}=0$). This argument implies

$\displaystyle c=2\ ,$ (2.6)

(for the conventions here), while the value of $b\geqslant 0$ is
arbitrary222Since the four-derivative terms can be fixed by supersymmetry, we
can obtain the value of $b$ [31]. In this letter, however, we do not focus on
the supersymmetric solutions. .

Generalizing the discussion, $\alpha^{\prime}-$ (or quantum gravity) corrected
Einstein-Maxwell-dilaton theories have attracted much attention [15, 26, 27,
28] even beyond the stringy models. Within string theory and its low-energy
effective field theories, the coefficients of each terms in the string
effective action are invariant under field redefinitions [9, 15], which were
then taken from the heterotic string calculations [3]: $a=1/8$ and $h=0$ . The
parameter $c$ can be fixed by a requirement of consistency with exact results
that were obtained for the GMGHS BH [7] while the value of $b$ does not affect
the correction to the BH mass. On the other hand, there are studies of BHs
based on the most general collection of four-derivative terms for
4-dimensional Einstein-Maxwell-dilaton theories [26]. We will here construct
the $\alpha^{\prime}$ corrected GMGHS solution within the framework of the
low-energy effective theory of the heterotic string. But our choice here is to
work with some slight more generality. Thus we take:

$\displaystyle a=\frac{1}{8}\ ,\qquad b,c~{}~{}{\rm arbitrary}\ ,\qquad h=0\
.$ (2.7)

###  2.2 Ansatz and entropy

In this work we are interested in static spherically symmetric solutions with
a purely electric $U(1)$ potential. An Ansatz suitable to address both the
(generic) BH solutions and the Robinson-Bertotti ones (with an $AdS_{2}\times
S^{2}$ near horizon geometry) reads

$\displaystyle
ds=-a(r)^{2}dt^{2}+c(r)^{2}dr^{2}+b(r)^{2}d\Omega^{2},\qquad\phi\equiv\phi(r),\qquad
A=V(r)dt~{},$ (2.8)

with $(r,t)$ the radial and time coordinates, respectively, while
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the usual metric on
$S^{2}$.

The entropy of generic solutions (extremal or not) is fully accounted for by
the Wald formula [41]

$\displaystyle S$ $\displaystyle=$ $\displaystyle-2\pi
A_{H}\,\frac{\delta{\cal L}}{\delta
R_{\mu\nu\rho\sigma}}\varepsilon_{\mu\nu}\varepsilon_{\rho\sigma}$ (2.9)
$\displaystyle=$ $\displaystyle 2\pi
A_{H}\left[-2g^{\rho\mu}g^{\nu\sigma}+2\alpha^{\prime}a\,{\rm
e}^{-\phi}\left\\{-R^{\mu\nu\rho\sigma}+4g^{\rho\mu}R^{\nu\sigma}+\left(-R+\Delta\phi\right)g^{\rho\mu}g^{\nu\sigma}\right.\right.$
$\displaystyle\left.\left.-g^{\rho\mu}\left(2\nabla^{\nu}\nabla^{\sigma}\phi-\nabla^{\nu}\phi\nabla^{\sigma}\phi\right)\right\\}\right]\varepsilon_{\mu\nu}\varepsilon_{\rho\sigma}\,,$

where $A_{H}$ is the event horizon area, $\varepsilon_{\mu\nu}$ is the
binormal to the horizon of the BH, normalized so that
$\varepsilon_{\mu\nu}\varepsilon^{\mu\nu}=-2$ and relation (2.9) is evaluated
at the horizon.

###  2.3 The $\alpha^{\prime}=0$ limit: the GMGHS solution

The GMGHS solution can be written in the form (2.8), with the metric functions

$a(r)^{2}=\frac{1}{c(r)^{2}}=\left(1-\frac{r_{+}}{r}\right)\ \ ,\qquad
b(r)^{2}=r^{2}\left(1-\frac{r_{-}}{r}\right)\ ,$ (2.10)

and the Maxwell potential and dilaton field

$A=\frac{Q}{r}dt\ ,\qquad
e^{2\phi}=\frac{1}{2}\left(1-\frac{r_{-}}{r}\right).$ (2.11)

The two free parameters $r_{+}$, $r_{-}$ (with $r_{-}<r_{+}$), corresponding
to outer and inner horizon radius, respectively They are related to the ADM
mass, $M$, and (total) electric charge, $Q$, by

$\displaystyle M=\frac{r_{+}}{2},\qquad
Q=\left(\frac{r_{-}r_{+}}{2}\right)^{\frac{1}{2}}\ .$ (2.12)

Other quantities of interest are the horizon area $A_{H}$ and the Hawking
temperature $T_{H}$

$\displaystyle A_{H}=4\pi r_{+}^{2}\left(1-\frac{r_{-}}{r_{+}}\right),\qquad
T_{H}=\frac{1}{4\pi r_{+}}\ .$ (2.13)

The extremal limit, which corresponds to the coincidence limit $r_{-}=r_{+}$,
results in a singular solution (as can be seen $e.g.$ by evaluating the
Kretschmann scalar). In this limit, the area of the event horizon goes to
zero. The Hawking temperature, however, approaches a constant.

For completeness, let us mention the existence of a magnetic version of the
GMGHS solution, which possesses the same metric, while the $U(1)$ potential
and the dilaton are

$\displaystyle A=Q\cos\theta d\varphi,\qquad
e^{-2\phi}=\frac{1}{2}\left(1-\frac{r_{-}}{r}\right)~{}.$ (2.14)

However, this solution also becomes singular as $r_{-}\to r_{+}$. A full
duality orbit of solutions that interpolate between these two in Einstein-
Maxwell-scalar models was recently discussed in [42] (See also [43, 44]).

##  3 Non-perturbative electrically charged solutions

###  3.1 The attractors

Taking into account $\alpha^{\prime}$ corrections, the possible existence of
non-perturbative generalizations of the extremal GMGHS BHs with a nonzero
horizon size is suggested by the presence of a Robinson-Bertotti-type exact
solution, which we shall now discuss. The existence of such near-horizon
geometry is, moreover, closely connected with the attractor mechanism [32, 33,
34, 35], as discussed above. We consider the following Ansatz, which is a
particular case of (2.8),

$\displaystyle
ds^{2}=v_{0}^{2}\left(\frac{dr^{2}}{r^{2}}-r^{2}dt^{2}\right)+v_{1}^{2}d\Omega_{(2)}^{2},\qquad\phi(r)=\phi_{0},\qquad
V=qr,$ (3.15)

where $v_{0}$, $v_{1}$, $\phi_{0}$ and $q$ are parameters which satisfy a set
of algebraic relations which result from the field equations. The ansatz
(3.15) was discussed by [34, 45, 46] as the most general near horizon field
configuration consistent with the SO(2, 1)$\times$SO(3) symmetry of
AdS${}_{2}\times$S2. Instead of following this route, however, in what follows
we choose to determine the unknown parameters by using the formalism proposed
in Refs. [34, 45, 46], thus by extremizing an entropy function333 We have
verified that the same solution is recovered when solving the full set of
covariant field equations. . This alternative approach allows us to also
compute the entropy of these BHs, and to show that the solutions exhibit
attractor behaviour.

The entropy function is defined as

$\displaystyle
F(v_{0},v_{1},q,Q,\phi_{0})=2\pi\left(qQ-f(v_{0},v_{1},q,\phi_{0})\right)\ ,$
(3.16)

where $Q$ is the electric charge of the solutions, while
$f(v_{0},v_{1},q,\phi_{0})$ is the Lagrangian density of the model (2.3)
evaluated for the Ansatz (3.15) and integrated over the angular coordinates,

$\displaystyle f(v_{0},v_{1},q,\phi_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int d\theta d\varphi\sqrt{-g}{\cal L}$
$\displaystyle=$
$\displaystyle\frac{v_{0}^{2}-v_{1}^{2}}{2}+e^{-2\phi_{0}}q^{2}\frac{v_{1}^{2}}{2v_{0}^{2}}-\frac{1}{4}\alpha^{\prime}e^{-2\phi_{0}}\left(1-4b\frac{e^{-4\phi_{0}}q^{4}v_{1}^{2}}{v_{0}^{6}}\right)~{}.$

The attractor equations are:

$\displaystyle\frac{\partial F}{\partial v_{0}}$ $\displaystyle=$
$\displaystyle
0\,\,\,\Rightarrow~{}~{}~{}1=\frac{e^{-2\phi_{0}}q^{2}v_{1}^{2}}{v_{0}^{4}}\left(1+\frac{6\alpha^{\prime}be^{-4\phi_{0}}q^{2}}{v_{0}^{4}}\right),$
(3.18) $\displaystyle\frac{\partial F}{\partial v_{1}}$ $\displaystyle=$
$\displaystyle
0\,\,\,\Rightarrow~{}~{}~{}1=\frac{e^{-2\phi_{0}}q^{2}}{v_{0}^{2}}\left(1+\frac{2\alpha^{\prime}be^{-4\phi_{0}}q^{2}}{v_{0}^{4}}\right),$
(3.19) $\displaystyle\frac{\partial F}{\partial\phi_{0}}$ $\displaystyle=$
$\displaystyle
0\,\,\,\Rightarrow~{}~{}~{}q^{2}=\frac{\alpha^{\prime}v_{0}^{2}}{2v_{1}^{2}}\left(1-\frac{12be^{-4\phi_{0}}q^{4}v_{1}^{2}}{v_{0}^{6}}\right),$
(3.20)

and

$\displaystyle\frac{\partial F}{\partial q}$ $\displaystyle=$ $\displaystyle
0\,\,\,\Rightarrow~{}~{}~{}Q=\frac{e^{-2\phi_{0}}qv_{1}^{2}}{v_{0}^{2}}\left(1+\frac{4\alpha^{\prime}be^{-4\phi_{0}}q^{2}}{v_{0}^{4}}\right),$
(3.21)

The unique solution of the eqs. (3.18)-(3.21) reads

$\displaystyle
v_{0}=\frac{\sqrt{\alpha^{\prime}}e^{-\phi_{0}}}{\sqrt{2}},~{}~{}~{}v_{1}=\sqrt{\alpha^{\prime}}e^{-\phi_{0}}\frac{\sqrt{2b}}{\sqrt{1+12b-\sqrt{1+16b}}},~{}~{}~{}q=\sqrt{\alpha^{\prime}}\frac{\sqrt{\sqrt{1+16b}}-1}{4\sqrt{b}}~{},$
(3.22)

and

$\displaystyle
Q=\sqrt{\alpha^{\prime}}e^{-2\phi_{0}}\frac{\sqrt{b(1+16b)(\sqrt{1+16b}-1)}}{1+12b-\sqrt{1+16b}}~{}.$
(3.23)

Replacing the above expression and back in the entropy function we obtain the
remarkable simple expression of the entropy of an extremal BH:

$\displaystyle S_{\rm
extremal}=\alpha^{\prime}~{}e^{-2\phi_{0}}\frac{2\pi\sqrt{1+16b}}{3\sqrt{1+16b}-1}~{},$
(3.24)

which can also be expressed in terms of the electric charge,

$\displaystyle S_{\rm extremal}=Q\sqrt{\alpha^{\prime}}s_{0}\,\qquad{\rm
with}\qquad s_{0}=\frac{\pi\sqrt{\sqrt{1+16b}-1}}{2\sqrt{b}}\ .$ (3.25)

As a check, we note that the result agrees with Wald’s form (3.11) evaluated
for the near horizon geometry. Let us also remark that the constant $c$ which
enters (2.3) does not enter the above expressions444 This is a consequence of
the fact the corresponding term in the Lagrangian density vanishes for the
ansatz (3.15). . Moreover, one can easily see that (3.22) solves indeed the
field equations, being a consistent solution.

As a final remark, we mention that there is no magnetic counterpart of the
above solution. That is, when employing the same metric ansatz as in (3.15), a
constant scalar and a magnetic $U(1)$ form $A=Q\cos\theta d\varphi$, the field
equations imply the following relation

$\displaystyle 4a\alpha^{\prime}+v_{1}^{2}=0\ ,$ (3.26)

which does not possess any physical solution.

###  3.2 The bulk extremal BHs

On general grounds, we expect that the above attractor solution describes the
neighbourhood of the event horizon of a bulk extremal BH. In what follows, we
give numerical evidence for the existence of these configurations.

In the numerical study of the solutions, it is convenient to work in
Schwarzschild-like coordinates, with the following choice in (2.8): $b(r)=r$,
$a(r)^{2}=e^{-2\delta(r)}N(r)$ and $c(r)^{2}=1/N(r)$, which results in the
line element

$ds^{2}=-e^{-2\delta(r)}N(r)dt^{2}+\frac{dr^{2}}{N(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\ .$ (3.27)

To construct RN-like extremal BH solutions, we assume the existence of a
horizon located at $r=r_{H}>0$. In its exterior neighbourhood, one finds the
following approximate solution (which holds for an extremal BH only):

$\displaystyle N(r)=N_{2}(r-r_{H})^{2}+\dots\
,\qquad\delta(r)=\delta_{0}+\delta_{1}(r-r_{H})+\dots\ ,$ (3.28)
$\displaystyle\phi(r)=\phi_{0}+\phi_{1}(r-r_{H})+\dots\ ,\qquad
V(r)=v_{1}(r-r_{H})+\dots\ ,$

with

$\displaystyle N_{2}=\frac{2}{\alpha^{\prime}}\ ,\qquad
r_{H}=\frac{e^{-\phi_{0}}\sqrt{\alpha^{\prime}}\sqrt{2b}}{\sqrt{1+12b-\sqrt{1+16b}}}>0\
,\qquad
v_{1}=\frac{e^{-\delta_{0}}\sqrt{\sqrt{1+16b}-1}}{2\sqrt{\alpha^{\prime}}\sqrt{b}}\
,$ (3.29)

as imposed by the field equations. Thus it turns out that only the parameters
$\phi_{0}$ and $\delta_{0}$ in the above near-horizon expansion are essential,
the coefficients $\phi_{1}$ and $\delta_{1}$ being determined in terms of
these. However, their expression is too complicated to include here.

For large $r$, one finds the following asymptotic form of the solution:

$N(r)=1-\frac{Q^{2}+Q_{s}^{2}}{r}+\dots\ ,\qquad\phi(r)=\frac{Q_{s}}{r}+\dots\
,\qquad
V(r)=V_{0}+\frac{Q}{r}+\dots,\qquad\delta(r)=\frac{Q_{s}^{2}}{2r^{2}}+\dots\
.$ (3.30)

The essential parameters in the above expansion are the mass $M$, electric
charges $Q$, electrostatic potential at infinity $V_{0}$ and scalar ’charge’
$Q_{s}$.

These extremal BHs have finite global charges $M,Q$ as well as a finite scalar
’charge’ $Q_{s}$. Their Hawking temperature vanishes, while the entropy takes
a very simple form, as resulting from (2.9)

$\displaystyle S=S_{E}+S_{GB}\ ,\qquad{\rm with}\qquad S_{E}=\frac{1}{4}A_{H}\
,\qquad
S_{GB}=\frac{1}{2}\alpha^{\prime}e^{-\phi_{0}}\int_{H}d^{2}x\sqrt{h}{\rm
R}~{},$ (3.31)

with ${\rm R}$ the Ricci scalar of the induced horizon metric:

$d\sigma^{2}=h_{ab}dx^{a}dx^{b}=r_{H}^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\ .$

After replacing the near horizon expression of the solution ($e.g.$
$A_{H}=4\pi r_{H}^{2}$, ${\rm R}=2/r_{H}^{2}$), the relation (3.24) is
recovered555The expression of entropy becomes more complicated for non-
extremal configurations, with an extra-contribution to (3.31)..

We also note that the equations of the model are invariant under the
transformation

$\displaystyle r\to\lambda
r,~{}~{}~{}\alpha^{\prime}\to\lambda^{2}\alpha^{\prime},$ (3.32)

(with $\lambda>0$ an arbitrary positive constant), various quantities of
interest scaling accordingly, $e.g.$

$\displaystyle M\to\lambda M,~{}~{}~{}(Q,Q_{s})\to\lambda~{}(Q,Q_{s})~{},$
(3.33)

such that only quantities invariant under (3.32) (like $Q/M$) have a physical
meaning. In what follow, we use this symmetry to study solutions with
$\alpha^{\prime}=0.5$ only, without any loss of generality.

The metric functions $N(r)$, $\delta(r)$, the scalar field $\phi(r)$ and the
electric potential $V(r)$ are found by solving a set of four ordinary
differential equations. However, these equations are too long to display here,
with hundreds of independent terms. However, we mention that we have been able
to find a suitable combination of the field equations such that the functions
$N,\delta$ still solve first order equations, while the functions $\phi$ and
$V$ satisfy second order equations. Moreover, the 2nd order equation for the
electric potential $V(r)$ possesses a 1st integral which is used in practice
to check the accuracy of the numerical results.

The solutions which smoothly connect the asymptotics (3.28) and (3.30) are
constructed numerically. The only input parameters are

$\displaystyle\\{\alpha^{\prime},~{}b,~{}c\\},$ (3.34)

while the constants $\phi_{0},$ $\delta_{0}$ and $M$, $V_{0}$, $Q$, $Q_{s}$
result from the numerical output, with the electric charge $Q$ given by
(3.23), a relation which provides an extra-test of the numerical accuracy.

We follow the usual approach in such problems and, by using a standard
ordinary differential equation solver, we evaluate the initial conditions at
$r=r_{H}(1+10^{-5})$ for global tolerance $10^{-14}$, adjusting for ’shooting’
parameters $\phi_{0}$ and $\delta_{0}$, and integrating towards $r\to\infty$,
looking for solutions with proper asymptotics. The profile of a typical BH
solution is shown in Figure 1. There, apart from the metric and matter
functions, we display also the Ricci and Kretschmann scalars, which show that
the geometry is regular on and outside the horizon.

Figure 1: The profile functions (left) of an illustrative extremal BH solution
are shown as functions of the compactified coordinate $1-r_{H}/r$. The Ricci
($R$) and the Kretschmann ($K$) scalars are also shown (right).

Figure 2: The charge to mass ratio (left) and the value of the dilaton field
at the horizon (right) are shown as a function of the parameter $c$ for
several values of $b$; $b,c$ are the two parameters which enter the (first
order) $\alpha^{\prime}$-corrections to the action.

Figure 3: The BH mass $M$ and electric charge $Q$ are shown to decrease as the
parameter $c$ increases, for $b=0.01$ . If we compare it with the figure 2, we
observe that the charge-to-mass ratio $Q/M$ decreases as $M$ decreases.

The basic properties of the extremal solutions can be summarized as follows.
First, given the input constants $(\alpha^{\prime},~{}b,~{}c)$, regular BH
solution appears to exist for a single set of the ’shooting’ parameters
$(\phi_{0},\delta_{0})$, while the profile of the scalar field $\phi(r)$ is
nodeless. Thus it is natural to conjecture that, similar to the
$\alpha^{\prime}=0$ case in [1, 2], only nodeless solutions exist also for the
model in this work.

Second, for a given $b$, the existence of solutions depends on the value of
input constant $c$, which enters the $\alpha^{\prime}$-term in the action
(2.3) (we recall that $c$ is absent in the attractor solution). Taking
$\alpha^{\prime}=1/2$, we have considered various values of the constant $b$
and varied the value of the parameter $c$. Then no upper bound on $c$ was
found. However, for a given $b$, the solutions stop to exist for $c<c_{0}$,
with $c_{0}$ increasing as $b$ increases (see Figure 2, both panels). As $c\to
c_{0}$, the value of the scalar field at the horizon appears to increase (in
modulus) without bound, and the solution becomes singular, a feature which
cannot be predicted by the attractor analysis in Section 3.1. A detailed study
of the critical behaviour in this limit may be of interest, but it is outside
the scope of this work.

Given the above remark, the existence of regular solutions with $c=2$ (as
implied by the perturbative results in Ref. [9]) seems to require very small
values of the parameter $b$, of the order $10^{-5}$ or smaller. Unfortunately,
the numerical accuracy deteriorates in that region of the parameter space and
we could not explore this case.

Finally, perhaps the most interesting feature of the solutions found so far is
that the ratio $Q/M$ is always greater than one - see Figure 2, left panel.
But the charge-to-mass ratio $Q/M$ decreases when $c$ increases for fixed $b$.
On the other hand, from Figure 3, we find that as the parameter $c$ grows, the
BH mass $M$ decreases for fixed value of $b$ . This feature seems to be at
tension with the rationale of the weak gravity conjecture.

##  4 Discussion

In this work we have confirmed that $\alpha^{\prime}$ corrections can de-
singularize the extremal GMGHS solution, an influential stringy BH whose
extremal limit is long known to be singular. This gives an illustrative
example of how higher order corrections motivated by quantum gravity can be
key (and non-negligible) to understand the BH geometry on and outside a
horizon. These higher-curvature terms also contribute to the global charges
and energy of the system [20, 47, 48].

The BH solutions with $\alpha^{\prime}$ corrections in string theory,
moreover, shows the importance of corrections in the Riemann tensor (rather
than, say, just in the Ricci scalar as in $f(R)$ models); taking them into
account is fundamental because the curvature scalars do not capture all the
possible terms in the equations of motion at higher orders in
$\alpha^{\prime}$ corrections [49]. Here we have focused on static BHs but
there are also studies with rotating BHs with first order correction in
$\alpha^{\prime}$ [50, 40, 51] \- see also the related studies in [52, 53, 54,
55, 56].

We observe that the charge-to-mass ratio $Q/M$ decreases when the BH mass $M$
decreases for fixed value of $b$ , which is different from previous examples
[21, 24]. Whether the charge-to-mass ratio increases or decreases depends on
the particular structure of the higher derivative terms. If it increases then
BHs can decay to smaller BHs, while such decays are forbidden in the other
case. Although we get $Q/M>1$, the charge-to-mass ratio decreases as the mass
decreases. Hence, our results do not assure that an extremal BH is always able
to decay to smaller extremal BHs of marginally higher charge-to-mass ratio.

Let also remark that the results reported here are exploratory, but establish
a proof of concept that such higher order corrections lead to BH solution that
are non-singular on the horizon and can be extended throughout the whole
spacetime. We have also confirmed the existence of non-extremal BH solutions
of the same model. These BHs possess very similar far field and near horizon
expressions, the main difference being that the function $N(r)$ possesses a
single zero as $r\to r_{H}$,

$\displaystyle N(r)=N_{1}(r-r_{H})+\dots~{},$ (4.35)

(also the value $r_{H}$ is not fixed in this case). Moreover, we have found
clear numerical evidence that the non-extremal BH solutions exist for a large
part of the $(b,c)$-plane. We hope to return elsewhere with a detailed study
of these configurations.

As a final intriguing remark, it would be interesting to clarify the absence
of a magnetic counterpart to the electric attractor in Section 3.1, and its
implication in the contex electric/magnetic duality for the $\alpha^{\prime}$
corrected theory.

Acknowledgements
The work of E.R. is supported by the Center for Research and Development in
Mathematics and Applications (CIDMA) through the Portuguese Foundation for
Science and Technology (FCT - Fundacao para a Ciência e a Tecnologia),
references UIDB/04106/2020 and UIDP/04106/2020, and by national funds (OE),
through FCT, I.P., in the scope of the framework contract foreseen in the
numbers 4, 5 and 6 of the article 23,of the Decree-Law 57/2016, of August 29,
changed by Law 57/2017, of July 19. We acknowledge support from the projects
PTDC/FIS-OUT/28407/2017, CERN/FIS-PAR/0027/2019 and PTDC/FIS-AST/3041/2020.
This work has further been supported by the European Union’s Horizon 2020
research and innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No.
FunFiCO-777740. The authors would like to acknowledge networking support by
the COST Action CA16104. The work of K. U. is supported by Grants-in-Aid from
the Scientific Research Fund of the Japan Society for the Promotion of
Science, under Contract No. 16K05364.

## Appendix

In this appendix, we work out the explicit form of the components of the
Einstein equations for a metric (2.8) which is compatible with the expected
form of the sought for attractor solution. We refer to section 3 for the
physical motivation leading to a metric of the form. Using again the
expressions (3.27), Einstein equations become

$\displaystyle rN^{\prime}+N-1+r^{2}N\phi^{\prime
2}+e^{2(\delta-\phi)}r^{2}V^{\prime
2}+\alpha^{\prime}e^{-2\phi}N\bigg{(}4aP_{N}$ (.36)
$\displaystyle{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}+6be^{4(\delta-\phi)}\frac{r^{2}V^{\prime
4}}{N}-ce^{2(\delta-\phi)}r^{2}\phi^{\prime 2}V^{\prime 2}\bigg{)}=0,$
$\displaystyle\delta^{\prime}+r\phi^{\prime
2}+\alpha^{\prime}e^{-2\phi}\left(4P_{\delta}a+ce^{2(\delta-\phi)}r\phi^{\prime
2}V^{\prime 2}\right)=0,$ (.37)
$\displaystyle\delta^{\prime\prime}-\frac{N^{\prime\prime}}{2N}+\delta^{\prime}\left(\frac{1}{r}-\delta^{\prime}\right)+\frac{N^{\prime}}{N}\left(\frac{3\delta^{\prime}}{2}-\frac{1}{r}\right)-\phi^{\prime
2}+\frac{e^{2(\delta-\phi)}V^{\prime 2}}{N}$ (.38)
$\displaystyle{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}+\alpha^{\prime}e^{-2\phi}\left(\frac{4a}{r}P_{c}+\frac{2be^{4(\delta-\phi)}V^{\prime
4}}{N}-ce^{2(\delta-\phi)}\phi^{\prime 2}V^{\prime 2}\right)=0\,,$

while the equations of motion for scalar and gauge fields are

$\displaystyle\phi^{\prime\prime}+\left(\frac{2}{r}+\frac{N^{\prime}}{N}-\delta^{\prime}\right)\phi^{\prime}-\frac{e^{2(\delta-\phi)}V^{\prime
2}}{N}+\alpha^{\prime}e^{-2\phi}\bigg{(}2aP_{\phi}-b\frac{6e^{4(\delta-\phi)}V^{\prime
4}}{N}$ (.39)
$\displaystyle{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}+ce^{2(\delta-\phi)}V^{\prime}((\frac{2}{r}+\delta^{\prime}+\frac{N^{\prime}}{N}-2\phi^{\prime})\phi^{\prime}V^{\prime}+V^{\prime}\phi^{\prime\prime}+2\phi^{\prime}V^{\prime})\bigg{)}=0,$
$\displaystyle\left(e^{\delta-2\phi}r^{2}V^{\prime}(1+\alpha^{\prime}e^{-2\phi}(4be^{2(\delta-\phi)}V^{\prime
2}-cN\phi^{\prime 2}))\right)^{\prime}=0,$ (.40)

where a prime ”′” states a derivative with respect to the radial coordinate
$r$ . The equation (.40) for the electric potential $V(r)$ consists of a first
integral. Also, in order to simplify the relations, we have defined the
following quantities

$\displaystyle P_{N}=N^{\prime}\phi^{\prime}\left(3(1+r^{2}\phi^{\prime
2})-\frac{1}{N}\right)+2\phi^{\prime 2}\bigg{(}3(1+rN^{\prime})$
$\displaystyle{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}-N(1+r\phi^{\prime}(4+3r\phi^{\prime}))\bigg{)}+2\phi^{\prime\prime}(N-1+rN\phi^{\prime}(4+3r\phi^{\prime})),$
$\displaystyle
P_{\delta}=\left(\frac{3N-1}{r}+3N\phi^{\prime}(2+r\phi^{\prime})\right)\delta^{\prime}\phi^{\prime}+2\phi^{\prime
2}\bigg{(}\frac{2(1-N)}{r}-N\phi^{\prime}(5+3r\phi^{\prime})\bigg{)}$
$\displaystyle{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}+\phi^{\prime\prime}\left(\frac{N-1}{r}+N\phi^{\prime}(4+3r\phi)\right),$
$\displaystyle
P_{\phi}=\frac{1}{r^{2}}\bigg{(}\left(2(1-N)\delta^{\prime}+5N^{\prime}-4(1-3N)\phi^{\prime}\right)\delta^{\prime}-12N^{\prime}\phi^{\prime}-\left(3\delta^{\prime}+N^{\prime}-4\phi^{\prime}\right)\frac{N^{\prime}}{N}\bigg{)}$
$\displaystyle-\frac{4}{r}\bigg{(}\delta^{\prime}(2N\delta^{\prime}-5N^{\prime})+\frac{N^{\prime
2}}{N}\bigg{)}\phi^{\prime}+\bigg{(}-\frac{4(1+2N)}{r^{2}}+(2N(\frac{8}{r}-3\delta^{\prime})+15N^{\prime})\delta^{\prime}-(\frac{26}{r}+\frac{3N^{\prime}}{N})N^{\prime}$
$\displaystyle+12\phi^{\prime}(N\phi^{\prime}-N^{\prime})\bigg{)}\phi^{\prime
2}+2\bigg{(}\frac{N-1}{r^{2}}+\frac{4N\phi^{\prime}}{r}+3N\phi^{\prime
2}\bigg{)}\delta^{\prime\prime}+\bigg{(}\frac{1-N}{r^{2}N}-\frac{4\phi^{\prime}}{r}-3\phi^{\prime
2}\bigg{)}N^{\prime\prime}$
$\displaystyle+\frac{2}{r}\left(\frac{2(1-N)}{r}+4N\delta^{\prime}-2N^{\prime}-3(rN^{\prime}+2N(2-r\delta^{\prime}+2r\phi^{\prime}))\phi^{\prime})\right)\phi^{\prime\prime},$
$\displaystyle
P_{c}=(5N^{\prime}-2N\delta^{\prime})\delta^{\prime}\phi^{\prime}-\left(N^{\prime}(2-5r\delta^{\prime}+r\phi^{\prime})+2N(r\delta^{\prime
2}+\delta^{\prime}(1+2r\phi^{\prime})-\phi^{\prime}(2+3r\phi^{\prime}))\right)\phi^{\prime
2}$
$\displaystyle+2N\phi^{\prime}(1+r\phi^{\prime})\delta^{\prime\prime}-\phi^{\prime}(1+r\phi^{\prime})N^{\prime\prime}+\bigg{(}-N^{\prime}(1+2r\phi^{\prime})+2N(\delta^{\prime}+2(-1+r\delta^{\prime})\phi^{\prime}$
$\displaystyle{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}-3r\phi^{\prime
2})\bigg{)}\phi^{\prime\prime}-\frac{N^{\prime
2}\phi^{\prime}(1+r\phi^{\prime})}{N}.$

The component of Einstein equations displayed in (.38) is the constraint which
will be used when searching for black hole solutions, including the extremal
configuration. Here the functions $N,\delta$ solve first order equations,
while the functions $\phi$ and $V$ satisfy second order equations, which was
used in our numerical treatment of the problem.

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