# The dilation and Scalar Weak Gravity Conjecture

###### Abstract

Einstein-Maxwell-dilaton theory with non-trivial dilaton potential is known to
admit asymptotically flat and (Anti-)de Sitter charged black hole solutions.
We investigate the conditions for the presence of horizons as function of the
parameters mass $M$, charge $Q$ and dilaton coupling strength $\alpha$. We
observe that there is a value of $\alpha$ which separate two regions, one
where the black hole is Reissner-Nordström-like from a region where it is
Schwarzschild-like. We find that for de Sitter and small non-vanishing
$\alpha$, the extremal case is not reached by the solution. We also discuss
the attractive or repulsive nature of the leading long distance interaction
between two such black holes, or a test particle and one black hole, from a
world-line effective field theory point of view. Finally, we discuss possible
modifications of the Weak Gravity Conjecture in the presence of both a
dilatonic coupling and a cosmological constant.

Dilatonic (Anti-)de Sitter Black Holes and Weak Gravity Conjecture

Karim Benakli♠<EMAIL_ADDRESS>Carlo Branchina
<EMAIL_ADDRESS>and Gaëtan Lafforgue-Marmet♣
<EMAIL_ADDRESS>

Laboratoire de Physique Théorique et Hautes Energies (LPTHE),
UMR 7589, Sorbonne Université et CNRS, 4 place Jussieu, 75252 Paris Cedex 05,
France.

###### Contents

  1. 1 Introduction
  2. 2 Einstein-Maxwell-dilaton Black Holes
  3. 3 Asymptotically Flat Black Holes: $\Lambda=0$
  4. 4 Dilatonic de Sitter Black Holes: $\Lambda>0$
    1. 4.1 $\alpha=1$
    2. 4.2 $\alpha>1$
    3. 4.3 $\alpha<1$
      1. 4.3.1 $\frac{1}{\sqrt{3}}<\alpha<1$
      2. 4.3.2 $\alpha=\frac{1}{\sqrt{3}}$
      3. 4.3.3 $\alpha<\frac{1}{\sqrt{3}}$
  5. 5 Dilatonic Anti-de Sitter Black Holes
  6. 6 Thermodynamics
  7. 7 Test particles in charged dilatonic black hole metric
    1. 7.1 Large distance action of the dilatonic black holes on a test particle
    2. 7.2 Forces between two point-like states with the black holes charges
  8. 8 Summary and conclusions

##  1 Introduction

While global symmetries seem fine in Quantum Field Theory (QFT), their
existence is a no-go for any Quantum Theory of Gravity: global charges are not
conserved when falling in black holes (see [1]). This illustrates the fact
that not all model builder’s ingredients are allowed in Nature: some
consistent QFTs will never be derived from an ultraviolet (UV) theory that
includes quantum gravity, such as String Theory. They fall in the Swampland,
contrary to those that form the Landscape (see [2]). Maybe the best-tested
condition that discriminates between the two sets of theories is the Weak
Gravity Conjecture [3]. This requires, for an abelian $U(1)$ gauge symmetry,
the presence of at least one state carrying a charge $Q$ bigger than its mass
$M$, measured in Planck length units: ${M^{2}}<Q^{2}$. Arguments based on
black hole (BH) physics have allowed to extend this conjecture to either
Einstein-Maxwell-dilaton theory in flat space-time [4] or to de Sitter space-
time [5]. It was the aim of this work to put the two together: Einstein-
Maxwell-dilaton theory with (Anti-) de Sitter ((A)dS) backgrounds.

The present work began then when we asked ourselves what happens to the de
Sitter Weak Gravity Conjecture in the case of a dilatonic gauge coupling. In
[5], the loci of black hole horizons were interpreted as the result of the
competition between a repulsive electromagnetic energy density on one side,
and gravity on the other, with attractive and repulsive contributions from the
black hole mass and the cosmological constant, respectively. It was suggested
that the parameter region where the electromagnetic contribution dominates
defines simultaneously the absence of a black hole solution and the WGC
conditions. It was comforting to see that in the limit of vanishing
cosmological constant one recovers the flat space-time result, in contrast
with previous attempts [6, 7] that provide complementary criteria for the
consistency of the theory and therefore different proposals for a WGC. We were
also interested in the dilaton as an example of scalar field that allows to
probe the Scalar WGC [8, 9, 13, 11, 10, 12, 14, 15, 16].

An extension of the Reissner-Nordström de Sitter black hole solution [17, 18,
19] to the case of Einstein-Maxwell-dilaton theory was constructed in [20, 21,
22]. This is a generalisation of the well known flat space-time solution of
Gibbons-Maeda [23] and Garfinkle-Horowitz-Strominger [24]. For the de Sitter
background, we were not able to find in the literature a discussion on the
conditions for the existence of horizons with a dilatonic coupling $\alpha\neq
0$. Some aspects of the asymptotically Anti-de Sitter metric were discussed
for the case of AdS5 in [21] and for AdS4 in some limits by [25]. Some
properties of these solutions, as the photon spheres, were considered in [26].
It is the main goal of this work to provide the missing comprehensive
analysis. In the different cases, the WGC states will then be considered to be
contained within the parametric regions complementary to those where black
holes exist.

The paper is organized as follows. Section 2 presents the black hole solution
and some formulae generic to all values of $\alpha$. A very brief review of
the asymptotically flat space-time is given in Section 3 for completeness and
comparison. The horizons of charged dilatonic de Sitter black hole are
described in Section 4. The Anti-de Sitter case is studied in Section 5. Some
thermodynamic quantities are computed in Section 6. The issue of attractive
and repulsive forces, in the case of asymptotically flat space-time, are
analyzed in Section 7. For the convenience of the readers, our results are
summarized in Section 8 with our conclusions.

##  2 Einstein-Maxwell-dilaton Black Holes

The Reissner-Nordström Black Holes are parametrized by their charge
$\tilde{Q}$ and mass $\tilde{M}$. It is useful to define the analog of such
quantities in the so-called geometrized units:

$M=\frac{\kappa^{2}\,\tilde{M}}{8\pi},\qquad
Q^{2}=\frac{\kappa^{2}\tilde{Q}^{2}}{32\pi^{2}}\qquad\Rightarrow\qquad\frac{M^{2}}{Q^{2}}=\frac{\kappa^{2}}{2}\frac{\tilde{M}^{2}}{\tilde{Q}^{2}},$
(1)

with $\kappa^{2}={1}/{M_{P}^{2}}=8\pi G\equiv 8\pi$ and $G$ Newton’s constant.
The absence of a naked singularity requires $Q\leq M$.

In the following we consider the extension provided by the Einstein-Maxwell-
dilaton action

$\mathcal{S}=\int
d^{4}x\sqrt{-g}\frac{1}{2\kappa^{2}}\left(R-2\left(\partial\phi\right)^{2}-e^{-2\alpha\phi}F^{2}-V(\phi)\right),$
(2)

where $F_{\mu\nu}$ is the field strength tensor related to the massless gauge
field $A_{\mu}$. For $V(\phi)=0$, the values $\alpha=1$ and $\alpha=\sqrt{3}$
are those obtained from string theory and Kaluza-Klein compactifications,
respectively. Note that $\phi$ and $A_{\mu}$ here are dimensionless. The
dimensionful physical fields are

$\tilde{\phi}=\sqrt{2}M_{P}\,\phi;\,\,\,\,\,\,\,\,\tilde{A_{\mu}}=\sqrt{2}M_{P}\,A_{\mu}.$
(3)

In the following, for notation simplicity, we will use $\phi$ for both the
dimensionful and the dimensionless fields.

A static, spherically symmetric solution to the Einstein’s equations was given
in [20] for $V(\phi)$ of the form

$V(\phi)=\frac{2}{3}\frac{\Lambda}{1+\alpha^{2}}\left(\alpha^{2}(3\alpha^{2}-1)e^{-2\frac{\phi-\phi_{0}}{\alpha}}+(3-\alpha^{2})e^{2\alpha(\phi-\phi_{0})}+8\alpha^{2}e^{\alpha(\phi-\phi_{0})-\frac{\phi-\phi_{0}}{\alpha}}\right),$
(4)

where $\Lambda$ is the cosmological constant and $\phi_{0}$ the asymptotic
value of $\phi(r)$ for $r\to\infty$. It reads

$\begin{cases}\mathrm{d}s^{2}=&-\left[\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}\mp
H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}\right]\mathrm{d}t^{2}\\\
&+\left[\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}\mp
H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}\right]^{-1}\mathrm{d}r^{2}\\\
&+r^{2}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}\mathrm{d}\Omega_{2}^{2},\\\
e^{2\alpha\phi}=&e^{2\alpha\phi_{0}}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}},\\\
F=&\frac{1}{\sqrt{4\pi
G}}\frac{Qe^{2\alpha\phi_{0}}}{r}\,\mathrm{d}t\wedge\mathrm{d}r.\end{cases}$
(5)

Here $H^{2}$ is the Hubble parameter $H^{2}={|\Lambda|}/{3}$. When
$\Lambda=0$, this reproduces the asymptotically flat black hole solutions of
[23, 24]. Otherwise, the solution is either an asymptotically dS (upper sign)
or AdS (lower sign) space-time. The relation between the integration constants
$r_{+}$, $r_{-}$ and the mass and charge is

$\begin{cases}2M=r_{+}+\frac{1-\alpha^{2}}{1+\alpha^{2}}r_{-},\\\
Q^{2}e^{2\alpha\phi_{0}}=\frac{r_{+}r_{-}}{1+\alpha^{2}},\\\
D=\frac{\alpha}{1+\alpha^{2}}r_{-},\end{cases}$ (6)

where $D$ is the scalar charge of the black hole defined as the integral over
a two sphere at infinity, $D=\frac{1}{4\pi}{\underset{r\to\infty}{\lim}}\int
d^{2}\Sigma^{\mu}\nabla_{\mu}\phi$, or, equivalently, through the expansion
$\phi=\phi_{0}-\frac{D}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)$ at large
$r$. This family of solutions have only two independent parameters (in
addition to the constant asymptotic value $\phi_{0}$): $r_{+},r_{-}$ or $Q,M$.
Inverting the relations, we obtain $r_{+}$, $r_{-}$ from $Q$ and $M$ as

$\begin{cases}r_{+}=M\pm\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}\\\
r_{-}=\frac{(1+\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}{M\pm\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}},\end{cases}$
(7)

and $D$

$D=\alpha\frac{Q^{2}e^{2\alpha\phi_{0}}}{M\pm\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}},$
(8)

as long as $(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}>{M^{2}}$. In order to map
the uncharged $Q=0$ case to the Schwarzschild metric, we choose this limit to
correspond exclusively to $r_{-}=0$. Contrary to the choice $r_{+}=0$, this
allows the metric (5) to truly take the desired form. Thus, we will consider
in the following only solutions with a ’$+$’ sign in (7) and (8).

As a consequence of the above relations between $(M,Q)$ and the integration
constants $(r_{+},r_{-})$, some peculiarities arise.

  * •

When $\alpha\geq 1$, probing the $(r_{+},r_{-})$ plane allows to sweep the
entire $(M,Q)$ one. The region
$r_{+}<\left[{(\alpha^{2}-1)}/{(\alpha^{2}+1)}\right]r_{-}$ defines negative
masses $M<0$ and is unphysical. A bijection is then defined between the
$r_{+}\geq\left[{(\alpha^{2}-1)}/{(\alpha^{2}+1)}\right]r_{-}$ portion of the
$(r_{+},r_{-})$ plane and the whole $(M,Q)$ one.

  * •

When $0<\alpha<1$, for $M^{2}<(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}$ both the
constants $r_{+},r_{-}$ and the metric become complex. A part of the $(M,Q)$
plane is inaccessible to the solution, a manifestation of the fact that

$M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}=\left(\frac{r_{+}}{2}-\frac{1-\alpha^{2}}{1+\alpha^{2}}\frac{r_{-}}{2}\right)^{2}$

is always positive in the parametric coordinates system $(r_{+},r_{-})$.
There, writing $r_{-}=r_{+}\tan\theta$, the charge-to-mass ratio

$\frac{Q^{2}e^{2\alpha\phi_{0}}}{M^{2}}=\frac{4}{1+\alpha^{2}}\frac{\tan\theta}{\left(1+\frac{1-\alpha^{2}}{1+\alpha^{2}}\tan\theta\right)^{2}}$

monotonically increases from $0$ to ${1}/{(1-\alpha^{2})}$ for
$\theta\in\left[0,\arctan\frac{1+\alpha^{2}}{1-\alpha^{2}}\right]$, reaches
its maximal value and then monotonically decreases to 0 for
$\theta\in\left[\arctan\frac{1+\alpha^{2}}{1-\alpha^{2}},\frac{\pi}{2}\right]$.
In this second copy of the
$\left(1-\alpha^{2}\right)Q^{2}e^{2\alpha\phi_{0}}<M^{2}$ parametric region,
$Q$ vanishes for $r_{+}=0$ and, following the discussion below (8), we discard
it. The bijection is now defined between the
$r_{+}\geq\left[{(1-\alpha^{2})}/{(1+\alpha^{2})}\right]r_{-}$ and the
$M^{2}\geq(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}$ portions of the planes.

Combining them, these observations show that
$r_{+}<\left|{(1-\alpha^{2})}/{(1+\alpha^{2})}\right|r_{-}$ defines a non
physical region for all $\alpha\neq 0$. The Reissner-Nordström solution
($\alpha=0$) does not suffer from the same issue, as will be discussed in the
next section.

##  3 Asymptotically Flat Black Holes: $\Lambda=0$

For $\alpha=0$,
$g_{00}=-\left(1-\frac{r_{+}+r_{-}}{r}+\frac{r_{+}r_{-}}{r^{2}}\right)$
$=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)$, we recover the Reissner-
Norsdtröm solution. The $r_{+}$ and $r_{-}$ constants do not enter separately
into the metric anymore but only through the combinations $r_{+}+r_{-}$ and
$r_{+}r_{-}$. It is thanks to this property that Reissner-Nordström solutions
(either flat or asymptotically (A)dS) do not suffer from the complex valued
region discussed above.

When $\alpha\neq 0$, $r_{-}$ is the location of a singular surface while
$r_{+}$ is the only event horizon of the black hole. The condition for the
singularity to be shielded by the horizon is simply $r_{+}>r_{-}$, that is:

$Q^{2}e^{2\alpha\phi_{0}}<\left(1+\alpha^{2}\right)M^{2}$ (9)

In this case of asymptotically flat black holes, the complex valued region is
beyond the reach of the black hole solution.

##  4 Dilatonic de Sitter Black Holes: $\Lambda>0$

When $\alpha=0$, the dilaton decouples and we recover the Reissner-Nordström-
de Sitter solution studied in [5]. For $\alpha\neq 0$, one needs to
distinguish between several cases, corresponding to different behaviours of
$g_{00}$. Here, $r_{+}$ does not determine the location of the horizon
anymore, while $r_{-}$ still indicates the coordinate of a singular surface.

###  4.1 $\alpha=1$

The $\alpha=1$ case allows for explicit expressions of the black hole
horizons. The metric can be written as

$\mathrm{ds^{2}}=-\left(1-\frac{2M}{r}-H^{2}r(r-2D)\right)\mathrm{d}t^{2}+\left(1-\frac{2M}{r}-H^{2}r(r-2D)\right)^{-1}\mathrm{d}r^{2}+r(r-2D)\mathrm{d}\Omega_{2}^{2}.$
(10)

$D$ is related to $M$ and $Q$ through

$D=\frac{Q^{2}e^{2\phi_{0}}}{2M}$ (11)

and $r=r_{-}=2D$ is a singular surface. The horizons correspond to the loci of
the roots of the polynomial of degree 3 in $r$:

$P(r)=H^{2}r^{3}-2DH^{2}r^{2}-r+2M$ (12)

Their explicit expression is not very illuminating. We find more instructive,
in particular for discussing below $\alpha\neq 1$, to provide a description of
the behaviour of the roots as functions of $M$ and $D$.

First, note that $P(r)\underset{r\rightarrow+\infty}{\rightarrow}+\infty$, and
can have two extrema $R_{-}<R_{+}$ given by the roots of
$P^{\prime}(r)=3H^{2}r^{2}-4DH^{2}r-1$. As $R_{-}R_{+}=-1$, $R_{-}<0$ while
$R_{+}=\frac{2}{3}D+\frac{1}{6}\sqrt{16D^{2}+\frac{12}{H^{2}}}>0$. We are
interested only in solutions of $P(r)=0$ in the region $r>2D$ outside the
singularity. Therefore, we will discuss the signs of $P(2D)$ and, when
$R_{+}>2D$, $P(R_{+})$.

The case of $R_{+}<2D$, i.e. $D^{2}H^{2}>\frac{1}{4}$, corresponds to
$r_{-}^{2}H^{2}>1$, which means that the radius of the singular surface is
greater than the Hubble’s one. No black hole solutions can arise there: when
$P(2D)<0$ one root is present, otherwise the polynomial is always positive for
all $r>2D$.

We restrict from now on to $R_{+}>2D$. When $P(2D)=2(M-D)\leq 0$, $P$ only has
one root. If $P(2D)>0$, there can be 0,1 or 2 roots, depending on the sign of
$P$ at the minimum $R_{+}$.

Studying this case, i.e. $M>D$, we have

$P(R_{+})=-\frac{16}{27}D^{3}H^{2}-\frac{2}{3}D+2M-\sqrt{4D^{2}+\frac{3}{H^{2}}}\left(\frac{8}{27}D^{2}H^{2}+\frac{2}{9}\right).$
(13)

If $-\frac{16}{27}D^{3}H^{2}-\frac{2}{3}D+2M$ is negative, the sign of
$P(R_{+})$ is fixed to be negative, and there are two zeros for $P$. In order
to further investigate the sign of $P(R_{+})$, it is helpful to consider the
function

$\begin{cases}&U(D)\equiv-\frac{16}{27}D^{3}H^{2}-\frac{2}{3}D+2M\\\
&D<D_{1}\rightarrow U(D)>0\\\ &D=D_{1}\rightarrow U(D)=0\\\
&D>D_{1}\rightarrow U(D)<0\end{cases}$ (14)

We have $U(0)=2M>0$ and $U$ is decreasing with $D$. There is so one solution
$D_{1}$ such that $U(D_{1})=0$. For $D>D_{1}$, $U$ is negative and $P$ has two
zeros. The region $D<D_{1}$, where $U(D)$ is positive, needs further
investigation.

It is easier for the rest of the computation, with $D<D_{1}$, to reformulate
the zeros of $P(R_{+})$ as the zeros of a simpler function:

$\displaystyle P(R_{+})=0$
$\displaystyle\Leftrightarrow\left(-\frac{16}{27}D^{3}H^{2}-\frac{2}{3}D+2M\right)^{2}=\left(4D^{2}+\frac{3}{H^{2}}\right)\left(\frac{8}{27}D^{2}H^{2}+\frac{2}{9}\right)^{2}$
$\displaystyle\Leftrightarrow-\left(\frac{4}{3}\right)^{3}H^{2}MQ(D)=0,$

where $Q$ is a function of $D$ defined as

$Q(D)=D^{3}+\frac{1}{16H^{2}M}D^{2}+\frac{9}{8H^{2}}D-\frac{27M}{16H^{2}}+\frac{1}{16H^{4}M},$
(15)

so that $P(R_{+})<0$ when $Q(D)>0$. $Q$ is an increasing function for positive
$D$. The sign of $Q(0)=-\frac{27M}{16H^{2}}+\frac{1}{16H^{4}M}$ discriminates
between two cases. If $Q(0)>0$, $Q(D)$ is positive for all positive $D$. If
$Q(0)<0$, there is one $D_{0}$ such that $Q(D_{0})=0$:

$\begin{cases}&D<D_{0}\rightarrow Q(D)<0\Rightarrow P(R_{+})>0\Rightarrow
P(r)\neq 0,\,\,\forall r\in{\mathbb{R}^{+}}\\\ &D=D_{0}\rightarrow Q(D)=0\\\
&D>D_{0}\rightarrow Q(D)>0\Rightarrow P(R_{+})<0.\end{cases}$

Imposing the necessary condition $D^{2}H^{2}<\frac{1}{4}$, we shall now group
all cases. There are three possibilities corresponding to 0, 1 or 2 roots.

  * •

For $P$ to have 2 roots, the first condition to be satisfied is $P(2D)>0$,
i.e. $D<M$. If $D>D_{1}$, $P$ has two roots and there is no need for further
investigations. If $D<D_{1}$, $P$ is also assured to have two roots when
$M^{2}H^{2}<\frac{1}{27}$. On the contrary, when $M^{2}H^{2}\geq\frac{1}{27}$,
$P$ has two roots when the additional condition $Q(D)>0\Leftrightarrow
D>D_{0}$ is met.

  * •

There are two scenarios where $P$ has one root. The first is realized when
$P(2D)\leq 0$, corresponding to $D\geq M$. As $P(0)>0$, this happens when one
of the two roots above is behind the singularity. The second scenario is met
when $P(2D)\geq 0$ and $P(R_{+})=0$, corresponding to $M\geq D$ and $D=D_{0}$
with $D<D_{1}$. This latter case is found when the two horizons discussed in
the previous point coincide.

  * •

Finally, the case where $P$ does not have roots corresponds to $D<M$,
$D<D_{1}$, $D<D_{0}$ and $M^{2}H^{2}>\frac{1}{27}$.

All the cases listed above depend on the values of $D_{0}$ and $D_{1}$. Those
are given in terms of $M$ as roots of the polynomials $Q$ and $U$. More
compact expressions, that we present below, can be given using the variables
$Y=DH$ and $X=MH$. This gives:

$\displaystyle Y_{0}=$
$\displaystyle-\frac{1}{48X}+\frac{1}{48}\left(-\frac{1}{X^{3}}-\frac{2^{4}.3^{3}.5}{X}+2^{7}.3^{6}X+\frac{48\sqrt{3}}{X^{2}}\sqrt{1+2^{2}.3^{4}X^{2}+2^{4}.3^{7}X^{4}+2^{6}.3^{9}X^{6}}\right)^{\frac{1}{3}}$
$\displaystyle-\frac{16}{3}\\!\\!\left(\frac{27}{8}-\\!\\!\frac{1}{256X^{2}}\\!\\!\right)\\!\\!\left(-\frac{1}{X^{3}}-\frac{2^{4}.3^{3}.5}{X}+2^{7}.3^{6}X+\frac{48\sqrt{3}}{X^{2}}\sqrt{1+2^{2}.3^{4}X^{2}+2^{4}.3^{7}X^{4}+2^{6}.3^{9}X^{6}}\right)^{-\frac{1}{3}}$
$\displaystyle Y_{1}=$
$\displaystyle\frac{3}{2\sqrt{6}}\left(-3\sqrt{6}X+\sqrt{1+54X^{2}}\right)^{-\frac{1}{3}}-\frac{3}{2\sqrt{6}}\left(-3\sqrt{6}X+\sqrt{1+54X^{2}}\right)^{\frac{1}{3}}$
(16)

Actually, the value of $Y_{0}$ presented just above is complex for
$X<\frac{1}{12\sqrt{6}}$. In that range of parameters, of the three roots of
$Q(D)$, it is another one which is real, corresponding to $Y_{0}$ with an
absolute value taken on the factors elevated to the $\pm\frac{1}{3}$ power and
on the factor $\frac{27}{8}-\frac{1}{256X^{2}}$. However, as we are only
interested in $D>0$ and $D_{0}$ is positive only for $X>\frac{1}{\sqrt{27}}$,
the expression for $Y_{0}$ given above is real in the whole range of interest
for $D$ and the absolute values are of no use.

The different cases for the black hole horizons are represented graphically in
figure 1. Instead of $D$, we used the electric charge $Q$ (actually, $Q$
really is $Qe^{\phi_{0}}$) to define $x$-axis.

Figure 1: Number of horizons of the $\alpha=1$ de Sitter black hole as a
function of $MH$ and $QH$. The green curve represents $HD_{0}$, the yellow one
the limit $\sqrt{2}MH=Qe^{\alpha\phi_{0}}H$, and the red one
$D^{2}H^{2}=\frac{1}{4}$. Dotted lines are for intermediate steps and
discussions in the text.

The green curve represents $D=D_{0}$, while the yellow one is $M=D$. The
function $D_{1}$, represented by the dashed blue curve, is below $M=D$ for
$D^{2}H^{2}<\frac{1}{4}$ so that, according to our previous findings, it plays
no role in the separation of the different regimes. In the region between the
green and the yellow curves, black hole solutions with two horizons are found.
For $Q=0$, the discriminant between solutions with two and zero horizons is
$M=\frac{1}{\sqrt{27}H}$, as it should. Solutions describing a naked
singularity with a cosmological horizon are found below the yellow curve.
Finally, the red curve is defined by $2DH=1$. On its right, the radius of the
singularity is greater than the Hubble’s.

To illustrate the solution, we now follow two straight horizontal lines, like
the green and yellow dashed ones, in figure 1, with $MH<\frac{1}{\sqrt{27}}$
in one case and $\frac{1}{\sqrt{27}}<MH\leq\frac{1}{2}$ in the other.

  * •

$\boldsymbol{MH<\frac{1}{\sqrt{27}}}$. At $Q=0$ there are two horizons: the
event and the cosmological horizon. As $Q$ grows, the radius of the
cosmological horizon and of the singularity increase while that of the event
horizon decreases until the value $Qe^{\phi_{0}}=\sqrt{2}M$ is reached. Here,
the event horizon coincides with the singularity. For
$Qe^{\phi_{0}}>\sqrt{2}M$, only the cosmological horizon surrounds
$r_{-}=\frac{Q^{2}e^{2\alpha\phi_{0}}}{M}$. The radius of the singular surface
increases with $Q$ until it meets the Hubble radius when
$Q^{2}e^{2\alpha\phi_{0}}=\frac{M}{H}$.

  * •

$\boldsymbol{\frac{1}{\sqrt{27}}<MH\leq\frac{1}{2}}$. With $MH<\frac{1}{2}$,
at $Q=0$ no horizons are present. This remains true until the condition
$\frac{Q^{2}e^{2\alpha\phi_{0}}}{2M}=D_{0}$ is met: at this point, one horizon
appears. Here, two roots of $g_{00}$ coincide, meaning that the event and
cosmological horizons have the same size. As $Q$ further grows the two
horizons disentangle, the radius of the event horizon shrinks, while that of
the cosmological horizon increases. From now on, the analysis is the same as
in the previous point: when the condition $Qe^{\alpha\phi_{0}}=\sqrt{2}M$ is
reached, the singularity merge with the event horizon, and for greater charges
the solutions only show a cosmological horizon.
When $MH=\frac{1}{2}$, the region with two horizons disappears. At the point
$MH=\frac{1}{2}$, $Qe^{\alpha\phi_{0}}H=\sqrt{2}MH=\frac{1}{\sqrt{2}}$, the
green, yellow and red curves meet. Here, both the two roots of $g_{00}$
coincide with the singularity that coincides, in turn, with the Hubble
horizon. Put simply, the locations of the singularity, the event, the
cosmological and the Hubble horizons all coincide. In terms of the previously
defined quantities, this corresponds to the case where $P(2D)=P(R_{+})=0$ with
$R_{+}=2D$. This point defines the maximal mass and charge for which a black
hole solution exists. Larger charges allow for the presence of a cosmological
horizon, with the singularity bigger than the Hubble surface.

For $MH>\frac{1}{2}$, no black hole solution is possible: the singularity is
either naked, when $Qe^{\alpha\phi_{0}}<\sqrt{2}M$, or shielded by a
cosmological horizon when $Qe^{\alpha\phi_{0}}>\sqrt{2}M$, with the latter
coinciding with the singularity when the equality is verified. The condition
for the singularity to be bigger than the Hubble horizon is now met before
this last one.

If we follow the arguments of [5] to infer the WGC condition from the absence
of event horizons shielding the singularity, the WGC would require the
existence of a state with mass $m$ and charge $q$, in geometrized units,
satisfying $qe^{\phi_{0}}>\sqrt{2}m$. This corresponds to the dilatonic WGC
bound in asymptotically flat space-time, as discussed above. Thus, for
$\alpha=1$, the dilatonic WGC seems to be insensitive to the presence of a
cosmological constant.

###  4.2 $\alpha>1$

We first study the $\alpha\to\infty$ limit, where one should recover the
Schwarzschild-de Sitter solution, and then look at the generic $\alpha>1$
case.

In the $\alpha\to\infty$ limit the metric reads

$\mathrm{d}s^{2}=-\left[\frac{1-\frac{r_{+}}{r}}{1-\frac{r_{-}}{r}}-H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{2}\right]\mathrm{d}t^{2}+\left[\frac{1-\frac{r_{+}}{r}}{1-\frac{r_{-}}{r}}-H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{2}\right]^{-1}\mathrm{d}r^{2}+r^{2}\left(1-\frac{r_{-}}{r}\right)^{2}\mathrm{d}\Omega_{2}^{2}.$
(17)

To study the horizons of the above metric, we need to find the roots of the
polynomial

$G(r)\equiv H^{2}(r-r_{-})^{3}-(r-r_{+})=0.$ (18)

$G$ has two extrema: a minimum at $r=r_{-}+\frac{1}{\sqrt{3}H}$ and a maximum
at $r=r_{-}-\frac{1}{\sqrt{3}H}$. The latter is inside the singular surface.
The knowledge of the values on the singular surface, $G(r_{-})$, and at its
minimum, $G(r_{-}+\frac{1}{\sqrt{3}H})$, allows to find the number of roots of
$G$. We have

$\begin{cases}G(r_{-})=r_{+}-r_{-}\\\
G(r_{-}+\frac{1}{\sqrt{3}H})=r_{+}-r_{-}-\frac{2}{\sqrt{27}H}.\end{cases}$
(19)

For $0<r_{+}-r_{-}<\frac{2}{\sqrt{27}H}$, the singularity is protected by two
horizons: the event and the cosmological horizons. When
$r_{+}-r_{-}=\frac{2}{\sqrt{27}H}$, the two horizons merge. Above, neither the
event nor the cosmological horizon are present. At $r_{+}=r_{-}$, the event
horizon coincides with the singularity. Using (7) one obtains, for
$\alpha\to\infty$, $r_{+}-r_{-}=2M$. Thus, we discard the $r_{+}<r_{-}$ region
as corresponding to negative masses. In the $\alpha\to\infty$ limit of (5) the
Schwarzschild-de Sitter solution is thus recovered: the discriminant between a
naked and a shielded singularity is the sign of $M-\frac{1}{\sqrt{27}H}$.

Now, we consider the general metric (5) and define a new function $F$ that
vanishes for the same values of $r$ than $g_{00}$:

$F(r)\equiv
r-r_{+}-H^{2}r^{3}\left(1-\frac{r_{-}}{r}\right)^{\frac{3\alpha^{2}-1}{\alpha^{2}+1}}.$
(20)

To investigate the solutions of $F(r)=0$, we divide $F$ into the sum of two
contributions: $A(r)\equiv r-r_{+}$, and $B(r)\equiv
H^{2}r^{3}\left(1-\frac{r_{-}}{r}\right)^{\frac{3\alpha^{2}-1}{\alpha^{2}+1}}$.
The intersection points of the two curves defined by $A$ and $B$ give the
zeros of $F$. We carry the analysis in two regions of the parameter space:

  * •

For $r_{+}\leq r_{-}$, $A(r_{-})\geq 0$ and the two curves always cross in one
point. Accordingly there is, in this case, only one zero, corresponding to the
cosmological horizon.

  * •

For $r_{+}>r_{-}$, $A(r_{-})<0$ and there are either two, one or zero
solutions depending on the location of the point $r_{0}$ where
$B^{\prime}(r_{0})=1$. $B(r_{0})\leq A(r_{0})$ corresponds to the case where
the function has two zeros, coalescing into one when the equality is
satisfied. $B(r_{0})>A(r_{0})$ will determine the horizon-less regime where
the dS space-time causal patch has been completely eaten by the black hole.

We consider, from now on, $r_{+}>r_{-}$. To discriminate between the different
regimes we just described, we proceed in the following way.

First, we observe that the limit for the two zeros to collapse into one is
obtained where $A(r)$ and $B(r)$ are tangent, thus $F(r_{0})=0$ and
$F^{\prime}(r_{0})=0\left(B^{\prime}(r_{0})=1\right)$. Consider111The
solutions of the system are always two as the equation $F(r)=0$, with the
prior $F^{\prime}(r)=0$, reduces to a quadratic equation for $r$. $r_{0\,\pm}$
two functions of $r_{\pm}$ given by:

$r_{0\,\pm}=\frac{(3-\alpha^{2})r_{-}+3(1+\alpha^{2})r_{+}}{4(1+\alpha^{2})}\pm\sqrt{\left(\frac{(3-\alpha^{2})r_{-}+3(1+\alpha^{2})r_{+}}{4(1+\alpha^{2})}\right)^{2}-2\frac{r_{+}r_{-}}{1+\alpha^{2}}}.$
(21)

When

$\begin{cases}F(r_{0\pm})=0\\\ F^{\prime}(r_{0\pm})=0\end{cases}$ (22)

the event and cosmological horizons coincide. As $F(r)\rightarrow-\infty$ for
$r\rightarrow\infty$, starting with $F(r_{-})<0$, if $F(r)$ takes a positive
value at some coordinate value this ensures that it crosses twice the abscissa
axis thus allowing the existence of two horizons. Therefore, when $r_{0\pm}$
are both greater than $r_{-}$, $r_{+}>r_{-}$, the black hole solution exists
in the parameter region of $(r_{+},r_{-})$ where $F(r_{0+})>0$ and
$F(r_{0-})<0$.

Next, note that for $\alpha>1$, $r_{0-}<r_{-}$ does not intervene. Using
$F(r_{-})<0$, the region where two horizons are present is defined by
$F(r_{0+})>0$. In terms of $M$ and $Q$, $F(r_{0+})>0$ translates to

$\displaystyle\left((1-2\alpha^{2})M+(\alpha^{2}-2)\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}+\sqrt{P(M,Q,\alpha,\phi_{0})}\right)$
$\displaystyle-H^{2}\left((10+9\alpha^{2})M-(4+3\alpha^{2})\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}+\sqrt{P(M,Q,\alpha,\phi_{0})}\right)^{\frac{3\alpha^{2}-1}{1+\alpha^{2}}}\times$
$\displaystyle\left((4+3\alpha^{2})M-\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}+\sqrt{P(M,Q,\alpha,\phi_{0})}\right)^{\frac{4}{1+\alpha^{2}}}>0,$
(23)

where $P(M,Q,\alpha,\phi_{0})$ is defined by

$P(M,Q,\alpha,\phi_{0})=(17+24\alpha^{2}+9\alpha^{4})M^{2}-(9+15\alpha^{2}+8\alpha^{4})Q^{2}e^{2\alpha\phi_{0}}-(8+6\alpha^{2})M\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}.$
(24)

Figure 2: Number of horizons of the $\alpha>1$ de Sitter black hole as a
function of $MH$ and $QH$. The green curve represents $F(r_{0+})=0$, the
yellow one the limit $Q^{2}=(1+\alpha^{2})M^{2}$, and the red one $r_{-}H=1$.
We chose for the illustration $\alpha=2$ and $\phi_{0}=0$.

Figure 2 presents the results of the discussion above. In analogy to the case
$\alpha=1$, we have added a further constraint for the singularity to be
inside the Hubble horizon, $r_{-}<\frac{1}{H}$. When the black hole charge
vanishes, the equation $F(r_{0+})=0$ reduces to $M=\frac{1}{\sqrt{27}H}$.
Consider in this figure a point in the region corresponding to a black hole
with two horizons and vary the charge or the mass:

  * •

Increasing the mass, the event horizon reaches the cosmological one for the
black hole mass $M$ such that $F(r_{0+})=0$. Beyond this value, the
singularity is naked.

  * •

Increasing the charge, instead, we encounter at some point the line
$Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$, where the event horizon and
the singular surface $r_{-}$ coalesce. Continuing to increase the charge, the
electromagnetic energy density becomes strong enough to prevent the formation
of an event horizon. The WGC states are expected to have mass and charge in
this region of parameters.

The WGC would then require the existence of a state with
$q^{2}e^{2\alpha\phi_{0}}>(1+\alpha^{2})m^{2}$, as in the case $\alpha=1$.
Although the presence of a cosmological constant changes the form of
$g_{00}(r)$, in the $\alpha>1$ case the weak gravity bound would take the same
form as in asymptotically flat space-time.

###  4.3 $\alpha<1$

For $\alpha<1$, both the terms in $g_{00}$ given by:

$g_{00}(r)=-\left[\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}-H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}\right]$

vanish when $r\to r_{-}$. For $\alpha^{2}=\frac{1}{3}$, the $r_{-}$ dependence
factorizes and $g_{00}$ can be written as

$g_{00}(r)\Big{|}_{\alpha^{2}=\frac{1}{3}}=-\left(1-\frac{r_{-}}{r}\right)^{\frac{1}{2}}\left(1-\frac{r_{+}}{r}-H^{2}r^{2}\right),$
(25)

where the second factor takes the form of the $g_{00}$ of a Schwarzschild-de
Sitter metric with mass $M^{*}\equiv\frac{r_{+}}{2}$. The relative importance
of the two terms in $g_{00}$ depends on whether $\alpha^{2}<\frac{1}{3}$ or
$\alpha^{2}>\frac{1}{3}$. A priori, we may expect a dilatonic-like black hole
behaviour for $\frac{1}{\sqrt{3}}<\alpha<1$, similar to the $\alpha>1$, while
a different, de Sitter-like black hole, behaviour for
$\alpha<\frac{1}{\sqrt{3}}$. We thus split the $\alpha<1$ analysis in three
parts: $\frac{1}{\sqrt{3}}<\alpha<1$, $\alpha=\frac{1}{\sqrt{3}}$ and
$\alpha<\frac{1}{\sqrt{3}}$.

####  4.3.1 $\frac{1}{\sqrt{3}}<\alpha<1$

Factorizing the $\left(1-\frac{r_{-}}{r}\right)$ term, we study the zeros of
$F$ defined in (20), with $3\alpha^{2}-1>0$. The only difference with the
$\alpha>1$ case comes from the convergence of the first term in $g_{00}$ when
$r\to r_{-}$. The convergence is to $0^{+}$ when $r_{+}>r_{-}$ and to $0^{-}$
when $r_{+}<r_{-}$. Concretely, this does not affect the zeros of $F$, and
thus the results obtained in the case $\alpha\geq 1$. In the $(Q,M)$ plane,
the boundaries of the region allowing black holes is still given by
$Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$ and $F(r_{0+})=0$. Note that
the most involved part of the analysis in the case $\alpha>1$ was for the
situation $r_{+}>r_{-}$ and used $r_{0-}<r_{-}$. While in the region
$\alpha\in\left]\frac{1}{\sqrt{3}},1\right[$, $r_{0-}$ can become greater than
$r_{-}$, this only happens when $r_{+}<r_{-}$ and therefore does not modify
that analysis. Black hole arguments would again indicate for the WGC the
existence of a particle satisfying
$q^{2}e^{2\alpha\phi_{0}}>(1+\alpha^{2})m^{2}$. As long as the second term in
$g_{00}$ (de Sitter-like) is sub-dominant, the transition between black holes
and naked singularities (with a cosmological horizon) seems to happen in the
same parametric region as in asymptotically flat space-time.

####  4.3.2 $\alpha=\frac{1}{\sqrt{3}}$

The $\alpha=\frac{1}{\sqrt{3}}$ case allows for explicit expressions of the
horizons and can be studied in full details. The second factor in (25) can be
seen as the time component of a Schwarzschild-de Sitter metric with an
effective mass $M^{*}\equiv r_{+}/2$. This factor has two zeros for
$r_{+}<\frac{2}{\sqrt{27}H}$, degenerate for $r_{+}=\frac{2}{\sqrt{27}H}$, and
none for $r_{+}>\frac{2}{\sqrt{27}H}$. The roots of the polynomial $P(r)\equiv
r-r_{+}-H^{2}r^{3}$ are:

$\begin{cases}r_{c}=\frac{1}{H}\left(\frac{\left(\frac{2}{3}\right)^{1/3}}{\left(-9r_{+}H+\sqrt{3}\sqrt{-4+27r_{+}^{2}H^{2}}\right)^{1/3}}+\frac{\left(-9r_{+}H+\sqrt{3}\sqrt{-4+27r_{+}^{2}H^{2}}\right)^{1/3}}{2^{1/3}3^{2/3}}\right)\\\
r_{h}=-\frac{1}{H}\left(\frac{\left(\frac{2}{3}\right)^{1/3}e^{-i\pi/3}}{\left(-9r_{+}H+\sqrt{3}\sqrt{-4+27r_{+}^{2}H^{2}}\right)^{1/3}}+\frac{\left(-9r_{+}H+\sqrt{3}\sqrt{-4+27r_{+}^{2}H^{2}}\right)^{1/3}e^{i\pi/3}}{2^{1/3}3^{2/3}}\right)\\\
r_{--}=-\frac{1}{H}\left(\frac{\left(\frac{2}{3}\right)^{1/3}e^{i\pi/3}}{\left(-9r_{+}H+\sqrt{3}\sqrt{-4+27r_{+}^{2}H^{2}}\right)^{1/3}}+\frac{\left(-9r_{+}H+\sqrt{3}\sqrt{-4+27r_{+}^{2}H^{2}}\right)^{1/3}e^{-i\pi/3}}{2^{1/3}3^{2/3}}\right),\end{cases}$
(26)

where $r_{c}$, $r_{h}$ are the cosmological and the event horizons,
respectively, and $r_{--}$ is negative thus of no physical interest. In fact,
one can see from the coefficients of $P$ that the product of the roots is
$-\frac{r_{+}}{H^{2}}<0$ and their sum is null. So, there are either two real
positive roots and a negative one (corresponding to the case where we have two
horizons) or two complex conjugate and a negative root (corresponding to the
case where no horizon is present). The transition between these two regimes
happens when the horizons coincide, $r_{c}=r_{h}$, i.e. for
$r_{+}=\frac{2}{\sqrt{27}H}$. We also require that these roots are located
outside the singular surface at $r_{-}$. In order to study the behaviour of
the roots of $g_{00}$, we consider the equations:

$\begin{cases}r_{c}-r_{-}=0\\\ r_{h}-r_{-}=0\\\
r_{+}=\frac{2}{\sqrt{27}H}.\end{cases}$ (27)

Figure 3: Number of horizons for $\alpha^{2}=\frac{1}{3}$ in $(r_{+}H,r_{-}H)$
(left) and $(QH,MH)$ coordinates (right).
Left: The region below the blue curve corresponds to $r_{c}>r_{-}$ and the
region below the yellow one to $r_{h}>r_{-}$. The intersection of the two
regions is populated by black hole solutions. On its right, the dS space-time
causal patch is completely eaten. The blue dotted curve is $r_{+}=r_{-}$, the
yellow dotted one is $r_{+}={r_{-}}/{2}$. The portion of plane above the
latter does not have counterpart in $(QH,MH)$. Right: Green and yellow curves
translate, respectively, the blue and yellow ones from left panel. Blue dotted
curves are $Q^{2}=({4}/{3})M^{2}$ and $Q^{2}=({3}/{2})M^{2}$. The union of the
region between the yellow curve and the lower dotted blue one from $(0,0)$ to
$\left(\frac{1}{\sqrt{6}},\frac{7}{12\sqrt{3}}\right)$ and the one between the
green and the same dotted line from
$\left(\frac{1}{\sqrt{6}},\frac{7}{12\sqrt{3}}\right)$ to
$\left(\frac{\sqrt{3}}{4},\frac{1}{2\sqrt{2}}\right)$ give the region with a
singularity surrounded only by a cosmological horizon. The yellow dotted line
represents the maximal black hole mass:
$M_{\mathrm{max}}=\frac{7}{12\sqrt{3}H}$.

We can see in figure 3 (left panel) the different regimes for $g_{00}$ in the
$(r_{+}H,r_{-}H)$ plane. It is instructive to understand the $(r_{+}H,r_{-}H)$
diagram before moving to the physical parameters $M$ and $Q$. The red curve is
$r_{+}=\frac{2}{\sqrt{27}H}$, while the blue and yellow ones represent
$r_{c}-r_{-}$ and $r_{h}-r_{-}$, respectively. They intersect each other at
the point
$(r_{+},r_{-})=\left(\frac{2}{\sqrt{27}H},\frac{1}{\sqrt{3}H}\right)$. As long
as $r_{+}<\frac{2}{\sqrt{27}H}$, $r_{c}>r_{-}$ is realized below the blue
curve and $r_{h}>r_{-}$ below the yellow one. As a consistency check of the
method used for generic $\alpha$, one can verify here, thanks to the explicit
expressions of the horizons, the equivalence of the condition $r_{c}\geq
r_{-},\,r_{+}\leq\frac{2}{\sqrt{27}H}$ with $F(r_{0+})\geq 0$ and of
$r_{h}\geq r_{-},\,r_{+}\leq\frac{2}{\sqrt{27}H}$ with $F(r_{0-})\leq 0$. The
different regimes are related in the following way:

  * •

The transition between a black hole solution and a naked singularity with no
cosmological horizon happens for $r_{+}=\frac{2}{\sqrt{27}H}$ when
$r_{-}\leq\frac{1}{\sqrt{3}H}$.

  * •

A transition from the black hole to a naked singularity with a cosmological
horizon can only happen in the combined interval
$r_{+}\in\left[0,\frac{2}{\sqrt{27}H}\right]$,
$r_{-}\in\left[0,\frac{1}{\sqrt{3}H}\right]$ when we cross the yellow curve
representing $r_{h}=r_{-}$. We find here a new bound compared to the
$r_{+}=r_{-}$ (blue dashed one) present for $\alpha^{2}>\frac{1}{3}$.

  * •

The regions defining a naked singularity with or without a cosmological
horizon meet on the blue curve when $r_{-}>\frac{1}{\sqrt{3}H}$.

The region of existence of the black hole is larger than what it would have
been if it was bounded by $r_{+}=r_{-}$
$\left(Q^{2}e^{2\alpha\phi_{0}}=\frac{4}{3}M^{2}\left(=(1+\alpha^{2})M^{2}\right)\right)$.
The line $r_{+}={r_{-}}/{2}$, corresponding to $Q^{2}=({3}/{2})M^{2}$
$\left(={M^{2}}/{(1-\alpha^{2})}\right)$, is also shown in the diagram (yellow
dashed one). It entirely lies in a region where the singularity is naked.
Along this dotted line, the separation between the regions where the
singularity is surrounded or not by a cosmological horizon is given by its
intersection with the blue curve. The problematic region of a complex valued
metric limiting the definition of the coordinates does not intervene in the
black hole region. It can only be reached after the singularity has been
exposed. Note that each value of $(QH,MH)$ corresponds to two choices of the
coordinates $(r_{+},r_{-})$, one of which is above the line
$r_{+}={r_{-}}/{2}$ and one below it. However, only this last region leads to
black hole solutions.

A similar analysis can be carried in terms of the mass and charge parameters
$(M,Q)$, using the equations
$r_{+}=M+\sqrt{M^{2}-\frac{2}{3}Q^{2}e^{2\alpha\phi_{0}}}$,
$r_{-}=\frac{4}{3}{Q^{2}}/{\left(M+\sqrt{M^{2}-\frac{2}{3}Q^{2}e^{2\alpha\phi_{0}}}\right)}$.
It is convenient to use $\hat{M}\equiv MH$ and $\hat{Q}\equiv
e^{\alpha\phi_{0}}QH$, and the conditions (27) then take the following form

$\displaystyle r_{c}-r_{-}=0\Leftrightarrow$
$\displaystyle\frac{\left(\frac{2}{3}\right)^{1/3}}{\left(-9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{-4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}$
$\displaystyle+\frac{\left(-9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{-4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}{2^{1/3}\,3^{2/3}}$
$\displaystyle=\frac{4\hat{Q}^{2}}{3\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)},$
(28)

$\displaystyle r_{h}-r_{-}=0\Leftrightarrow$
$\displaystyle\frac{\left(\frac{2}{3}\right)^{1/3}e^{-i\pi/3}}{\left(-9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{-4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}$
$\displaystyle+\frac{e^{i\pi/3}\left(-9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{-4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}{2^{1/3}\,3^{2/3}}$
$\displaystyle=-\frac{4\hat{Q}^{2}}{3\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)},$
(29)

and

$r_{+}=\frac{2}{\sqrt{27}H}\Leftrightarrow\hat{M}=\frac{1}{\sqrt{27}}+\frac{\sqrt{3}\hat{Q}^{2}}{2}.$
(30)

It is possible to identify the triple point
$(r_{+},r_{-})=\left(\frac{2}{\sqrt{27}H},\frac{1}{\sqrt{3}H}\right)$ where
the regions with two, one and zero solutions meet with
$(\hat{Q},\hat{M})=\left(\frac{1}{\sqrt{6}},\frac{7}{12\sqrt{3}}\right)$. We
now have all the elements to understand the phase diagram in the
$(\hat{Q},\hat{M})$ plane as displayed in Figure 3 (right panel):

  * •

Restricting to masses below the triple point, thus
$\hat{M}<\frac{7}{12\sqrt{3}}$, the upper bound on the mass allowing the two-
horizons solution is given by $r_{+}=\frac{2}{\sqrt{27}H}$, as shown in the
$(r_{+}H,r_{-}H)$ plane. It is represented by the green curve,
$\hat{M}=\frac{1}{\sqrt{27}}+\frac{\sqrt{3}\hat{Q}^{2}}{2}$, from
$\left(0,\frac{1}{\sqrt{27}}\right)$ up to the triple point. Above it, the
bound from (4.3.2) is stronger and the green curve now corresponds to
$r_{c}=r_{-}$ .

  * •

The lower bound on this black hole region is given by (4.3.2) and it is
represented in yellow in figure 3. It corresponds to the limit where the event
horizon coincides with the singularity. Note that the lower bound yellow curve
starts at $(0,0)$ and crosses the upper bound green curve at the triple point.
After that, the yellow curve runs above the green one and does not bound any
physical region.

The graphical representation of the two bounds reveals that for masses
$M>\frac{7}{12\sqrt{3}H}$ the event horizon cannot form: this is the maximal
mass above which asymptotically de Sitter black hole solutions are no more
possible (yellow dashed line). Accordingly, this point corresponds to a
maximal charge $Q_{\mathrm{max}}=\frac{1}{\sqrt{6}H}$.

  * •

Singularities with only a cosmological horizon are found in the domain given
by the union of: (1) the region between the yellow and the lower dashed blue
curve from the origin up to the triple point, (2) the region between the green
and the same dashed blue curve, now above it. On this dashed curve
$Q^{2}e^{2\alpha\phi_{0}}=({3}/{2})M^{2}$ marks the limit of definition of the
metric.

The ($r_{c}=r_{-}$) green curve crosses the blue one, corresponding to
$Q^{2}e^{2\alpha\phi_{0}}=({3}/{2})M^{2}$ in the point
$\left(Qe^{\alpha\phi_{0}},M\right)=\left(\frac{\sqrt{3}}{4H},\frac{1}{2\sqrt{2}H}\right)$.
This is a point of maximal charge and mass. Above it, the green curve
delimiting singularities with and without cosmological horizon cannot be
drawn: either it is not defined, or it lies inside the inaccessible region
(complex metric).

To confirm that (4.3.2) can be seen as a WGC bound, one can look at its
behaviour when $H\to 0$. To look at this limit, let us rewrite $r_{h}$, in the
region where it is real, as

$r_{h}=\frac{2}{\sqrt{3}H}\sin\left(\frac{\theta}{3}\right),$ (31)

where the angle $\theta$ is defined by $\sin(\theta)=({3\sqrt{3}}/{2})r_{+}H$
and $\cos(\theta)=\sqrt{1-({27}/{4})r_{+}^{2}H^{2}}$. In the limit
$H\ll{1}/{r_{+}}$, one obtains

$r_{h}=r_{+}+H^{2}r_{+}^{3}+\mathcal{O}(r_{+}^{3}H^{4}).$ (32)

Looking at $r_{h}-r_{-}=0$, replacing $r_{+}$ and $r_{-}$ by their definition
in function of $M$ and $Q$ (7), one can write the expansion of $Q$ in powers
of $H$ as

$Q^{2}=\frac{4}{3}M^{2}+\frac{4^{3}}{3^{4}}M^{4}H^{2}+\mathcal{O}(M^{6}H^{4}).$
(33)

In the limit $H\to 0$, the bound given by (4.3.2) reduces to (9).

In conclusion, for $\alpha={1}/{\sqrt{3}}$, the study of horizons of these
dilatonic black holes would rather suggest (4.3.2) as a WGC bound than (9).

####  4.3.3 $\alpha<\frac{1}{\sqrt{3}}$

As shown above, when $\alpha={1}/{\sqrt{3}}$, $r_{+}=r_{-}$ is no longer the
black hole extremality condition, as it was for all cases with
$\alpha>{1}/{\sqrt{3}}$. In the following, we will see that this remains true
for $\alpha<{1}/{\sqrt{3}}$.

The first difference one can observe with respect to previous cases is the
change in the behaviour of the derivative of $g_{00}(r)$ in a neighborhood of
the singularity. Leading terms are given by

$\begin{cases}\partial_{r}(g_{00})\underset{r\to
r_{-}}{\sim}-\frac{1-\alpha^{2}}{1+\alpha^{2}}\frac{r_{-}}{r^{2}}\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{-2\frac{\alpha^{2}}{1+\alpha^{2}}}\qquad\mathrm{for}\;\alpha>\frac{1}{\sqrt{3}}\\\
\partial_{r}(g_{00})\underset{r\to
r_{-}}{\sim}2\frac{\alpha^{2}}{1+\alpha^{2}}H^{2}r_{-}\left(1-\frac{r_{-}}{r}\right)^{-\frac{1-\alpha^{2}}{1+\alpha^{2}}}\qquad\mathrm{for}\;\alpha<\frac{1}{\sqrt{3}}.\end{cases}$
(34)

When $\alpha<{1}/{\sqrt{3}}$, the sign of the derivative at the vicinity of
the singular surface $r=r_{-}$ is independent of $r_{+}$ and is always
positive, with $\underset{r\to r_{-}^{+}}{\lim}g_{00}(r)=0^{+}$: $g_{00}$
always reaches $0$ from above. This, combined with the asymptotic value
$g_{00}\to+\infty$ when $r\to\infty$, implies that the metric exhibits a
horizon only if the parameters in $g_{00}$ are such that the function is
decreasing in an interval to reach a negative minimum. In this situation,
there are two horizons, coincident when the minimum of $g_{00}$ is $0$. This
leads to a first conclusion:

  * •

In the parametric $(QH,MH)$ space, a singularity surrounded only by a
cosmological horizon can only appear on a curve, rather than in a portion of
the plane as happened for all cases with $\alpha\geq\frac{1}{\sqrt{3}}$.

Remember that, in contrast with the asymptotically flat case, here both
$r_{+}>r_{-}$ and $r_{+}\leq r_{-}$ are now allowed. The method used above to
investigate the limits between regions with different behaviours of the
horizons was valid only for $r_{+}>r_{-}$ but can now be extended to all the
situations. We proceed thus by using the function $F$ defined in (20), and its
decomposition into $A$ and $B$. For $\alpha^{2}<{1}/{3}$, $B\to+\infty$ when
$r\to r_{-}$. In a neighborhood of $r_{-}$ we always have $B(r)>A(r)$.

We notice that the condition $F(r_{0-})<0$ is not very illuminating in this
case of $\alpha^{2}<1/3$. Indeed, for $r_{+}>r_{-}$, $F(r_{0-})$ is always
negative while for $r_{+}\leq r_{-}$, expanding (21), we have $r_{0-}\leq
r_{-}$, i.e. $r_{0-}$ lies inside the singular surface and $F(r_{0-})<0$
should not be considered.

One should recall here that, for any $\alpha\neq 0$, the solution is plagued
by the appearance of the complex valued metric for
$Q^{2}e^{2\alpha\phi_{0}}(1-\alpha^{2})>M^{2}$. Sweeping the whole
$r_{+},r_{-}\in\mathbb{R}^{+}$ parametric space, we have no access to that
region. All we can say here is that constraints from $F(r_{0-})$ do not appear
in the region where the metric is defined real. For $\alpha=0$, the metric is
real valued in the whole $(Q,M)$ plane: this allows us to verify that
$F(r_{0-})=0$ matches the condition for the existence of naked singularities
with a cosmological horizon in the RNdS metric found in [5].

Therefore:

  * •

The condition for the existence of black hole solutions is given by
$F(r_{0+})>0$. When $F(r_{0+})=0$, the event and cosmological horizons
coincide. $F(r_{0+})<0$ defines naked singularities with no cosmological
horizon.

There is a maximal mass, above which there is no black hole solution. This
mass corresponds to the point where the curve defined by $F(r_{0+})=0$ crosses
the line defined by $Q^{2}e^{2\alpha\phi_{0}}(1-\alpha^{2})=M^{2}$. It is
given by

$M_{max}=\frac{1}{2\sqrt{2}H}\left(\frac{1-3\alpha^{2}}{2(1-\alpha^{2})}\right)^{\frac{1-3\alpha^{2}}{2(1+\alpha^{2})}}$
(35)

The behaviour of the horizons for the asymptotically de Sitter metric for
$\alpha<1/{\sqrt{3}}$ is described in figure 4 where we have taken, for an
explicit illustrative example, $\alpha=1/2$.

In figure 4, the region of the $(QH,MH)$ plane with two horizons shows an
upper bound represented by the green curve, $F(r_{0+})=0$. On the green curve,
the event and cosmological horizons coincide. The lower bound is given by the
blue curve, where $Q^{2}e^{2\alpha\phi_{0}}(1-\alpha^{2})=M^{2}$ and the
metric is on the verge of becoming complex. Plots of $g_{00}$ reveal that, on
this line, the event horizon and the singularity are still far apart.
Approaching this line from above (the black hole solution region), we see that
the event horizon and the singularity get closer but never touch. Expected
solutions with only the cosmological horizon seem to be hidden inside the
inaccessible region. The maximal mass for the black hole, corresponding to the
crossing point of the green and blue curves are given by
$M_{max}H=2^{\frac{-8}{5}}3^{\frac{-1}{10}}\simeq 0.3$.

Figure 4: Number of horizons for $\alpha<{1}/{\sqrt{3}}$, here illustrated by
the value $\alpha={1}/{2}$. The green curve represents $F(r_{0+})=0$ and gives
an upper bound on the mass. The yellow line represents
$Q^{2}=(1+\alpha^{2})M^{2}$, that does not play anymore the same role for
$\alpha<{1}/{\sqrt{3}}$. The blue one is $Q^{2}={M^{2}}/{(1-\alpha^{2})}$. In
the region between the green and the blue curves, cosmological and event
horizons are present. Below the blue one, $r_{+}$, $r_{-}$ and the metric
become complex valued.

We can compare with the RN-dS black hole solution studied in [19, 5],
corresponding here to $\alpha=0$. In that case the $(QH,MH)$ plane shows a
central region with three horizons surrounded by two regions with one horizon.
One of them is attained, in parametric space, after the event horizon has
reached the cosmological one, and is interpreted as a dS space-time causal
patch eaten by the black hole. The other, related in [5] to dS-WGC states, is
beyond the locus of the coincidence of the inner and event horizons. For
$0<\alpha<1/{\sqrt{3}}$, there are two horizons in a central region and zero
horizons outside. Strictly speaking, $r=r_{-}$ is a zero of $g_{00}$ for all
$\alpha<1$ but the inner horizon is traded for a singularity. The instability
of Cauchy horizons may provide an additional motivation towards their
identification. However, we have observed that for $\alpha\rightarrow 0$,
while the two cosmological and event horizons tend to their corresponding
surfaces in RN-dS, numerically the singularity $r_{-}$ seems not to coincide
exactly with the inner horizon but lies slightly above: it is no more a
singularity neither an horizon. In fact, in this limit, the expression of
$r_{-}=r_{-}(Q,M)$, given in (7), takes the same form as the RN black hole
inner horizon and becomes trivially this surface when $H\rightarrow 0$.

Finally, note that for $\alpha=0$ the parametric equations $F(r_{0\pm})=0$
reproduce the relations separating the regions with different horizons in [5]

$\begin{cases}F(r_{0-})=0\underset{\alpha=0}{\Leftrightarrow}M^{2}_{-}=\frac{1}{54l}[l(l^{2}+36Q^{2})-(l^{2}-12Q^{2})^{\frac{3}{2}}]\\\
F(r_{0+})=0\underset{\alpha=0}{\Leftrightarrow}M^{2}_{+}=\frac{1}{54l}[l(l^{2}+36Q^{2})+(l^{2}-12Q^{2})^{\frac{3}{2}}]\end{cases}$
(36)

with $l=\frac{1}{H}$. It is $M^{2}_{-}$, and thus $F(r_{0-})$, that marks the
transition between black holes and naked singularities with cosmological
horizon. However, the solution of $F(r_{0-})=0$ can not be used for
$0<\alpha<1/{\sqrt{3}}$, as the metric is complex in that region. Note that in
all previous literature, because the asymptotically flat metric always shows a
naked singularity before turning complex, this region was simply ignored.

##  5 Dilatonic Anti-de Sitter Black Holes

Changing the sign of the $H^{2}$ terms in the metric (5) allows to obtain a
particular class of dilatonic asymptotically AdS black hole solutions. For
completeness, we will investigate the phase space exhibiting the behaviour of
the horizons as one varies $\alpha$, $M$ and $Q$ using the same method as for
the de Sitter case.

We start by briefly recalling the Reissner Nordström AdS case as it will
correspond to the $\alpha\to 0$ limit. The time component of the metric is
$g_{00}(r)=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+H^{2}r^{2}\right)$. Its
roots are given by those of the polynomial $G(r)\equiv
H^{2}r^{4}+r^{2}-2Mr+Q^{2}$. They are two (degenerate in the extremal case)
real positive roots as long as

$M^{2}\geq\frac{1}{54}\left(36Q^{2}-\frac{1}{H^{2}}+\frac{\left(1+12H^{2}Q^{2}\right)^{\frac{3}{2}}}{H^{2}}\right).$
(37)

The presence of horizons can be inspected through the study of the zeros of
the function:

$F_{AdS}(r)\equiv
r-r_{+}+H^{2}r^{3}\left(1-\frac{r_{-}}{r}\right)^{\frac{3\alpha^{2}-1}{1+\alpha^{2}}}=-r\left(1-\frac{r_{-}}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}g_{00}(r).$
(38)

This turns out to be much simpler to study than the corresponding dS function
$F$. It is indeed straightforward to see that whenever $r_{+}<r_{-}$,
$F_{AdS}>0$ for all $r\in\left[r_{-},\infty\right[$ and so the function cannot
have zeros. It will prove useful to split $F_{AdS}$ into the sum of the two
contributions $A_{AdS}(r)\equiv r-r_{+}$, a straight line, and
$B_{AdS}(r)\equiv-H^{2}r^{3}\left(1-\frac{r_{-}}{r}\right)^{\frac{3\alpha^{2}-1}{1+\alpha^{2}}}$
which is always negative. In this way, the problem is again recast in terms of
the intersection points of $A_{AdS}$ and $B_{AdS}$. We split the discussion
into three parts depending on the value of $\alpha$.

#####  $\boldsymbol{\alpha^{2}>\frac{1}{3}}$:

When approaching the singularity, $B_{AdS}$ goes to $0^{-}$. As a consequence,
the curves defined by $A_{AdS}$ and $B_{AdS}$ have either one intersection
point when $r_{+}\geq r_{-}\,\left(A_{AdS}(r_{-})\leq 0\right)$, or no
intersections at all when $r_{+}<r_{-}\,\left(A_{AdS}(r_{-})>0\right)$. We
conclude that the discriminant between the black hole regime and the naked
singularity is given by $r_{+}=r_{-}$ i.e.
$Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$.

#####  $\boldsymbol{\alpha^{2}=\frac{1}{3}}$:

$B_{AdS}$ does not depend on $r_{-}$: this is again related to a factorization
in the metric as we have seen in the dS case. As such, $F_{AdS}$ now
corresponds to the Schwarzschild-AdS polynomial
$F_{AdS}(r)=r-r_{+}+H^{2}r^{3}$, whose roots are given by

$\begin{cases}r_{h}=\frac{1}{H}\left(-\frac{\left(\frac{2}{3}\right)^{1/3}}{\left(9r_{+}H+\sqrt{3}\sqrt{4+27r_{+}^{2}H^{2}}\right)^{1/3}}+\frac{\left(9r_{+}H+\sqrt{3}\sqrt{4+27r_{+}^{2}H^{2}}\right)^{1/3}}{2^{1/3}3^{2/3}}\right)\\\
r_{--}=\frac{1}{H}\left(\frac{\left(\frac{2}{3}\right)^{1/3}e^{-i\pi/3}}{\left(9r_{+}H+\sqrt{3}\sqrt{4+27r_{+}^{2}H^{2}}\right)^{1/3}}-\frac{\left(9r_{+}H+\sqrt{3}\sqrt{4+27r_{+}^{2}H^{2}}\right)^{1/3}e^{i\pi/3}}{2^{1/3}3^{2/3}}\right)\\\
r_{--}^{*}=\frac{1}{H}\left(-\frac{\left(\frac{2}{3}\right)^{1/3}e^{i\pi/3}}{\left(9r_{+}H+\sqrt{3}\sqrt{4+27r_{+}^{2}H^{2}}\right)^{1/3}}+\frac{\left(9r_{+}H+\sqrt{3}\sqrt{4+27r_{+}^{2}H^{2}}\right)^{1/3}e^{-i\pi/3}}{2^{1/3}3^{2/3}}\right),\end{cases}$
(39)

where $r_{h}$ is the horizon and $r_{--}$ and $r_{--}^{*}$ are two complex
conjugate (non-physical) roots. Accordingly, the condition for the singularity
to be shielded by the horizon is just $r_{h}>r_{-}$, which reads:

$\displaystyle\frac{\left(9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}{2^{1/3}\,3^{2/3}}$
$\displaystyle-\frac{\left(\frac{2}{3}\right)^{1/3}}{\left(9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}>\frac{4\hat{Q}^{2}}{3\left(M+\sqrt{M^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}.$
(40)

In the $(\hat{Q},\hat{M})$ space this gives a lower bound on the mass which is
a little higher than the asymptotically flat case:
$\hat{Q}^{2}>({4}/{3})\hat{M}^{2}$, as shown in figure 5 (left panel). The
$H\to 0$ limit of (5) is again given by (33) with the change of sign in front
of the $H^{2}$ term.

Figure 5: The dilatonic AdS black hole case. The left panel describes the
phase diagram for the $\alpha^{2}={1}/{3}$, the right panel shows the explicit
example of $\alpha^{2}={1}/{4}$ to illustrate the situation for
$\alpha^{2}<{1}/{3}$. In both cases, the blue region corresponds to black hole
solutions, the yellow line is the flat-space discriminant between shielded and
naked singularities (playing no role here but shown for comparison) and the
red region is the inaccessible region where the metric becomes complex.

#####  $\boldsymbol{\alpha^{2}<\frac{1}{3}}$:

This is again the most intricate parametric region. Here, $B_{AdS}$ diverges
to $-\infty$ when $r\to r_{-}$. For $r_{+}\leq r_{-}$, since $A_{AdS}$ is
positive for all $r\geq r_{-}$, and so is the difference $A_{AdS}-B_{AdS}$, no
horizon can ever be present.

On the other hand, when $r_{+}>r_{-}$ we have $A_{AdS}(r)<0$ for all
$r\in[r_{-},r_{+}[$, so the combination $A_{AdS}-B_{AdS}$ could result to be
negative there. As $B_{AdS}$ is a concave function of $r$, two roots will be
present when $A_{AdS}$ and $B_{AdS}$ intersect, collapsing to one when they
are tangent, and zero otherwise. The region of parameters allowing the
presence of two horizons, the black hole solution region, is obtained as in
the dS case by solving the combined equations $F_{AdS}=0$ and
$F_{AdS}^{\prime}=0$. The solutions to this system are the same $r_{0\pm}$
found in (21)222Writing $F_{AdS}^{\prime}(r)$ with the prior $F_{AdS}(r)=0$,
one sees that it is independent of the sign in front of the $H^{2}r^{3}$ term
and gives the same equations as in the dS case, $F^{\prime}(r)=0$ with the
prior $F(r)=0$.. In the $(r_{+}H,r_{-}H)$ plane, $F_{AdS}(r_{0+})$ is always
null or positive, leading to no constraint in practice. As a consequence:

  * •

The condition for the singularity to be shielded can be simply expressed as
$F_{AdS}(r_{0-})\leq 0$, with the equality being satisfied by extremal
solutions with coincident horizons.

This leads to a lower bound on the mass, that lies above the flat-space one
($Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$), as illustrated in the
example of figure 5 (right panel).

For $\alpha^{2}={1}/{3}$, the lower bound on the mass coincides with the one
obtained above by simply requiring $r_{h}-r_{-}\geq 0$, the set of curves are
continuously connected. The presence of black holes with two horizons is a new
characteristic that was not present for $\alpha\geq{1}/{\sqrt{3}}$.

As in the dS case, we verify again the equivalence between the limit
$\underset{\alpha\to 0}{\lim}F_{AdS}(r_{0-})\leq 0$ and (37).

Note that the singularity at $r=r_{-}$ changes its nature: from a space-like
one, as it happens when it is behind the single $\alpha^{2}>{1}/{3}$ horizon,
to a time-like one. This is dictated by the derivative of $g_{00}$ that
diverges now to $-\infty$ for $r\to r_{-}^{+}$. The $\alpha^{2}<{1}/{3}$ AdS
black holes are the only ones where the singularity can be avoided: $r=r_{-}$
is not in the future light-cone of all the observers that crosses the
horizons, $g_{00}$ becomes time-like again before reaching it, as already
observed by [25]. Of all the setups we studied, this is the only case where
$\underset{r\to r_{-}^{+}}{\lim}g_{00}<0$ in a black hole parametric region.

This is similar to what happens in the Reissner Nordstrom AdS metric for the
$r=0$ singularity. Varying $\alpha$, starting with $\alpha^{2}>{1}/{3}$, we
encounter at $\alpha^{2}={1}/{3}$ a transition from Schwarzschild-AdS like
black holes, with only one horizon and a space-like singularity, to Reissner-
Nordstrom AdS like ones, with two horizons and a time-like singularity. For
all $\alpha\neq 0$, the singularity at $r=r_{-}$ resembles here the
singularity at $r=0$ of $\alpha=0$. Note that we do not encounter the issue of
a complex metric, in contrast with the dS case, as the naked singularity bound
is reached for values $M^{2}>(1-\alpha^{2})Q^{2}$.

It can be interesting to look at the first correction to the flat space-time
condition $r_{-}=r_{+}$ for small $H$. For $H=0$, $F(r_{0-})=0$ reduces to
$r_{0-}=r_{+}$ which is equivalent to $r_{-}=r_{+}$.
In order to find the first term in the expansion in $H$, we set

$r_{-}=r_{+}+cr_{+}^{\gamma+1}H^{\gamma}+o(r_{+}^{\gamma+1}H^{\gamma}),$ (41)

where the constants $c$ and $\gamma$ have to be fixed.

From (41), it is possible to express $r_{0-}$ as
$r_{0-}=r_{+}+\frac{(1+\alpha^{2})}{2(1-\alpha^{2})}cr_{+}^{\gamma+1}H^{\gamma}+o(r_{+}^{\gamma+1}H^{\gamma})$.
Requiring $F(r_{0-})=0$ at first order gives:

$\gamma=\frac{1+\alpha^{2}}{1-\alpha^{2}}\qquad{\rm
and}\qquad\frac{1+\alpha^{2}}{2(1-\alpha^{2})}c+\left[\frac{3\alpha^{2}-1}{2(1-\alpha^{2})}c\right]^{\frac{3\alpha^{2}-1}{1+\alpha^{2}}}=0.$
(42)

For $\alpha<{1}/{\sqrt{3}}$, $c$ is single valued, negative, with the limits
$c\to-2$ when $\alpha\to 0$ and $c\to-1$ for $\alpha\to{1}/{\sqrt{3}}$.
Plugging the relation between $(r_{+},r_{-})$ and $(Q,M)$ given by (7) in
(41), we can look for the corresponding relation
$Q^{2}=(1+\alpha^{2})M^{2}+bM^{2+\delta}H^{\delta}$, that defines the boundary
of the black hole region. It is possible to determine the constants $\delta$
and $b$: $\delta=\gamma$ and
$b=\alpha^{2}(1+\alpha^{2})^{\frac{2}{1-\alpha^{2}}}c$. Thus, the constraint
$F(r_{0-})=0$ can be expanded for $H\to 0$ as

$Q^{2}=(1+\alpha^{2})M^{2}+\alpha^{2}(1+\alpha^{2})^{\frac{2}{1-\alpha^{2}}}c\,M^{\frac{3-\alpha^{2}}{1-\alpha^{2}}}H^{\frac{1+\alpha^{2}}{1-\alpha^{2}}}+o(M^{\frac{3-\alpha^{2}}{1-\alpha^{2}}}H^{\frac{1+\alpha^{2}}{1-\alpha^{2}}}).$
(43)

We see that for $\alpha={1}/{\sqrt{3}}$, it reduces to

$Q^{2}=\frac{4}{3}M^{2}-\frac{4^{3}}{3^{4}}M^{4}H^{2}+o(M^{4}H^{2}).$ (44)

It is the same equation as for the dS case, with a difference of sign. For
$\alpha\to 0$, the power of H tends to $1$, but the coefficient in front
vanishes. This is coherent with [19], since there is no linear term in the
expansion for small $H$.

##  6 Thermodynamics

The analysis of the existence of horizons shows that above and below the value
$\alpha=1/\sqrt{3}$ both the dS and AdS black holes have different properties.
We have seen these differences in the behaviours of $g_{00}$ and its first
derivative $\partial_{r}g_{00}$. In this section, we shall analyse them using
thermodynamic quantities of the black holes 333For the physics of
thermodynamic quantities in de Sitter space-time, we refer the reader to
[27]..

The Hawking temperature of the black holes is given by

$\displaystyle
T=-\frac{\partial_{r}g_{00}}{4\pi}\Bigg{|}_{r=r_{h}}=\frac{1}{4\pi}$
$\displaystyle\Bigg{[}\frac{r_{+}}{r_{h}}\left(1-\frac{r_{-}}{r_{h}^{2}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}+\frac{1-\alpha^{2}}{1+\alpha^{2}}\left(1-\frac{r_{+}}{r_{h}}\right)\left(1-\frac{r_{-}}{r_{h}}\right)^{-\frac{2\alpha^{2}}{1+\alpha^{2}}}\frac{r_{-}}{r_{h}^{2}}$
$\displaystyle\,\,\mp
2H^{2}r_{h}\left(1-\frac{r_{-}}{r_{h}}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}\mp
2\frac{\alpha^{2}}{1+\alpha^{2}}H^{2}r_{-}\left(1-\frac{r_{-}}{r_{h}}\right)^{-\frac{1-\alpha^{2}}{1+\alpha^{2}}}\Bigg{]},$
(45)

where $r_{h}$ is the radial coordinate of the event horizon. The upper (lower)
sign refers to the dS (AdS) solution. The above expression could be simplified
using the relation between $r_{+},r_{-}$ and $r_{h}$ given by
$g_{00}(r_{h})=0$, but it is more useful for our purposes to keep it in this
form. Note that the second term in (6) does not vanish identically as now
$r_{h}\neq r_{+}$ in general. We write the Hawking-Beckenstein black hole
entropy proportional to the horizon area as

$S=\pi
r_{h}^{2}\left(1-\frac{r_{-}}{r_{h}}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}},$
(46)

Note that this is the same as in asymptotically flat space and vanishes when
$r_{h}\to r_{-}$ for any $\alpha\neq 0$.

For $\alpha>1/\sqrt{3}$ extremal solutions are defined to have $r_{h}\to
r_{-}=r_{+}$. The temperature diverges in the limit $r_{h}\to r_{-}$ when
$\alpha\neq 1$ and we distinguish two cases.

  * •

For $\alpha>1$, the first and the second term of (6) lead the divergence. The
temperature goes as

$T\underset{r_{h}\to r_{-}}{\sim}\frac{1}{4\pi
r_{h}}\left(1+\frac{1-\alpha^{2}}{1+\alpha^{2}}\right)\left(1-\frac{r_{-}}{r_{h}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}.$
(47)

  * •

For $1/\sqrt{3}<\alpha<1$, the divergence is lead by the last term ($H\neq 0$)

$T\underset{r_{h}\to
r_{-}}{\sim}\frac{1}{2\pi}\frac{\alpha^{2}}{1+\alpha^{2}}H^{2}r_{-}\left(1-\frac{r_{-}}{r_{h}}\right)^{-\frac{1-\alpha^{2}}{1+\alpha^{2}}}$
(48)

For $\alpha=1$, the temperature is finite and reads

$T=\frac{1}{4\pi}\left(\frac{1}{2M}\mp 2MH^{2}\right)$ (49)

In this case, with the extremality condition $D=M$ and the necessary
requirement for the singularity to be smaller than the Hubble radius in the dS
case, $4D^{2}H^{2}\leq 1$, the expression (49) is always positive (or null).
Actually, this points out a rather interesting property: the extremal solution
with the singularity of the same size as the Hubble horizon (denoted by the
point where the green, yellow and red curve of figure 1 meet) has a null
temperature. It is thus a trivial endpoint of Hawking evaporation. This is a
not the case in the asymptotically flat metric where $T={1}/{8\pi M}$ and one
questioned whether the extremal solutions are endpoints of Hawking evaporation
or not [28]. Note that the finiteness of $T$ comes from the vanishing of the
exponent $\left|{(1-\alpha^{2})}/{(1+\alpha^{2})}\right|$ in the continuous
limit $\alpha\to 1$.

The divergence of the temperature was discussed in [28] for the asymptotically
flat $\alpha>1$ case. The divergence for $1/\sqrt{3}<\alpha<1$ is new and
entirely due to the presence of a non-vanishing cosmological constant. For the
presence of horizons, the $1/\sqrt{3}<\alpha<1$ and $\alpha>1$ black holes
share the same properties and are not much sensitive to the value of $H$ when
they approach extremal solutions. The temperature, however, has a different
form in the two cases and shows a dependence on $H$.

When $0<\alpha\leq 1/\sqrt{3}$, extremal solutions no longer have
$r_{+}=r_{-}$. We distinguish two cases.

  * •

For $\alpha=1/\sqrt{3}$, the extremality condition $r_{h}=r_{-}$ is reached
when $F(r_{0-})=0$ (defined in (20) and (21)) or $F_{AdS}(r_{0-})=0$ (defined
in (38)). The temperature of such black holes is

$T\underset{r_{h}\to
r_{-}}{\sim}\frac{1}{8\pi}\frac{r_{-}}{r_{h}^{2}}\left(1-\frac{r_{+}}{r_{h}}\mp
H^{2}r_{h}^{2}\right)\left(1-\frac{r_{-}}{r_{h}}\right)^{-\frac{1}{2}}=0,$
(50)

thus vanishes for the extremal solution as the factor inside the first
parenthesis corresponds to $g_{00}(r_{h})$ and is identically null. If one
were to blindly take the limit $\alpha\to 1/\sqrt{3}$ of (48), the temperature
of extremal solutions would seem to diverge with an exponent $1/2$. This shows
a discontinuity in the $\alpha$-dependence of such exponent that can be traced
back to the factorization in the metric and the loss of the extremality
condition $r_{+}=r_{-}$.

  * •

When $0<\alpha<1/\sqrt{3}$, we have seen in both the dS and AdS cases that
$r_{h}=r_{-}$ is never reached for different reasons.

In the dS case, extremal solutions could not be defined within the domain of a
real valued metric.

In the AdS case, the black holes on the verge of exposing a naked singularity
have the event horizon coincident with their inner horizon. As these do not
coincide with the singularity, the temperature does not diverge anymore but
vanishes as $\partial_{r}g_{00}(r_{h})=0$.

A simple interpretation of the behaviour of the temperature is as follows. The
$\alpha\to\infty$ and $\alpha\to 0$ limits of such black holes are given by
Schwarzschild and Reissner-Nordström black holes, respectively, with
temperatures given by

$T_{Sc}=\frac{1}{8\pi M},\qquad
T_{RN}=\frac{1}{2\pi}\frac{\sqrt{M^{2}-Q^{2}}}{\left(M+\sqrt{M^{2}-Q^{2}}\right)^{2}}$
(51)

in asymptotically flat space. In that case, the dilatonic black holes have a
temperature

$T=\frac{1}{4\pi
r_{+}}\left(1-\frac{r_{-}}{r_{+}}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}},$
(52)

diverging for $\alpha>1$, finite for $\alpha=1$, and vanishing for
$0<\alpha<1$ extremal solutions. We retrieve a vanishing temperature for small
$\alpha$, here $\alpha=0$, as the extremality condition reads now $M=Q$. For
large values of $\alpha$, the extremality condition obtained by identification
of the horizon $r_{h}(=r_{+})$ with the singularity $r_{-}$, would formally
correspond in the Schwarzschild case to put the horizon at the origin i.e.
formally take the limit $M$ tends to $0$ in the black hole solution which in
turn leads to a divergent temperature.

This remains true also in asymptotically (A)dS space-time. For large $\alpha$,
we consider the temperature of the Schwarzschild (A)dS black hole

$T_{Sc}=\frac{1}{4\pi}\frac{1\mp 3H^{2}r_{h}^{2}}{r_{h}},$ (53)

where $r_{h}$ is the radius of the event horizon 444Note, that in the dS case,
the singularity is shielded as long as $M^{2}H^{2}<1/27$, while in the AdS
case it is for any mass $M>0$. We do not discuss issues related to thermal
equilibrium of observers in the region between the event and cosmological
horizons in asymptotically dS space [29]. We restrict to the small $H$ limit
where the two are far away.. The extremality corresponds again to a formal
$r_{h}\rightarrow 0$ limit, obtained when $M\to 0$ in (53). This leads to a
divergent temperature. For small $\alpha$, we consider instead the Reissner-
Nordström (A)dS black holes, and $T$ has a similar form [19]

$T_{RN}=\frac{1}{4\pi}\frac{1-\frac{Q^{2}}{r_{h}^{2}}\mp
3H^{2}r_{h}^{2}}{r_{h}},$ (54)

where $r_{h}$ indicates again the radius of the event horizon but defined by a
different metric. The region of validity for the dS and AdS solutions are
given in (36) and (37), respectively. Extremality is obtained then by taking
the lower mass bound for which the temperature vanishes.

In this picture, the transition between diverging and vanishing temperatures
of extremal dilatonic black holes could be then seen as the thermal footprint
of a transition from a Schwarzschild-like behaviour to a Reissner-Nordström
like one. Such transition happens for $\alpha=1$ in asymptotically flat space,
where the temperature is equal to $T=\frac{1}{8\pi M}$, and for
$\alpha=1/\sqrt{3}$ in asymptotically (A)dS space, where $T=0$.

For the AdS black holes, one can observe a peculiar behaviour of the entropy
formula applied to the extremal solution. It trivially vanishes above
$\alpha=1/\sqrt{3}$ as one has a naked singularity. It is finite for
$\alpha<1/\sqrt{3}$, increasing as $\alpha\to 0$: there, the extremal
condition corresponds to the coincidence of two (non-singular) horizons.

##  7 Test particles in charged dilatonic black hole metric

The weak gravity conjectures, for abelian gauge symmetries, dilatonic or
scalar interactions, have been formulated as constraints on the non-
relativistic and, often but not always, large distance interactions between
charged states. These states can be elementary in the theory, but also
solitonic as D-branes. Accordingly, we wish to study non-relativistic, large
distance, leading interactions between the charged black holes. The latter,
separated by very large distances and interacting through gravitons, gauge
bosons and scalar fields with large wavelengths compared to their typical
size, i.e. their horizon radius, look like point particles. One challenge for
the point-like particle description of the interactions is to identify here
the expression of the scalar coupling and associated scalar charge of these
states.

It appears instructive to first consider the simpler case of a test particle
submitted to the forces generated by a black hole. Also, taking in our
computations the limit $H=0$, allows to compare with the available results of
explicit amplitude computation.

###  7.1 Large distance action of the dilatonic black holes on a test particle

In our effective theory description, the scalar charge of a point-like
particle with respect to the dilaton $\phi$ appears encoded in the field-
dependent mass $m(\phi)$. This in turn will be translated into a three-point
coupling in a field theory context, as we shall discuss later. The action for
the motion of a test particle of mass $m(\phi)$ and charge $q$ in the black
hole geometry is given by

$S_{m}=\int\mathrm{d}\tau\left(-m(\phi)\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}+\sqrt{4\pi
G}gqA_{\mu}\dot{x}^{\mu}\right),$ (55)

where $x^{\mu}$ represent the particle’s coordinates and the dot indicates a
derivative with respect to the proper time $\tau$. The last term is the
coupling to the abelian gauge field $A_{\mu}$, with a gauge coupling constant
$g$. The mass $m$ and the charge $q$ are in geometrized units for consistency.
The geodesic equations are

$-m(\phi)\left(\ddot{x}^{\mu}+\Gamma^{\mu}_{\nu\rho}\dot{x}^{\nu}\dot{x}^{\rho}\right)+\sqrt{4\pi
G}gqF^{\mu}_{\,\,\rho}\dot{x}^{\rho}-\frac{dm(\phi)}{d\phi}\left(\dot{x}^{\mu}\dot{x}^{\rho}\partial_{\rho}\phi-\dot{x}^{\rho}\dot{x}_{\rho}\partial^{\mu}\phi\right)=0,$
(56)

where the $\Gamma$s denote the Christoffel symbols and $F_{\mu\rho}$ is the
gauge field strength. Here, we are mainly interested in the last term,
interpreted as a scalar force.

To study the motion of the test particle in the space-time defined by the
metric (5), we first rewrite the Lagrangian as

$\mathcal{L}=-m(\phi)\sqrt{f(r)\dot{t}^{2}-\frac{\dot{r}^{2}}{f(r)}-r^{2}g(r)\dot{\theta}^{2}-r^{2}g(r)\sin^{2}{\theta}\,\dot{\varphi}^{2}}-\frac{e^{2\alpha\phi_{0}}q{Q}}{r}\dot{t},$
(57)

where the gauge field was chosen as $A=\left(-\frac{gQ}{\sqrt{4\pi
G}r},0,0,0\right)$ and the gauge coupling is now $g=e^{\alpha\phi_{0}}$. In
(57), we have introduced two functions $f$ and $g$:

$\begin{cases}f(r)\equiv\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{\frac{1-\alpha^{2}}{1+\alpha^{2}}}\mp
H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}\\\
g(r)\equiv\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}},\end{cases}$
(58)

where the $\mp$ signs depend on whether we consider an asymptotically dS or
AdS space-time, respectively.

We shall start by taking $H=0$, corresponding to the asymptotically flat
space-time solution, and discuss the $H\neq 0$ case in a second moment.

The spherical symmetry allows us to restrict the analysis to the equatorial
plane $\theta=\frac{\pi}{2}$. The two Killing vectors $\partial_{t}$ and
$\partial_{\varphi}$ correspond to two constant conserved quantities $E$ and
$L$, proportional to the energy and angular momentum as measured at infinity,
respectively. They are given by:

$\begin{cases}E=-\frac{\partial\mathcal{L}}{\partial\dot{t}}=m(\phi)f(r)\dot{t}+\frac{e^{2\alpha\phi_{0}}qQ}{r}\\\
L=\frac{\partial\mathcal{L}}{\partial\dot{\varphi}}=m(\phi)r^{2}g(r)\dot{\varphi},\end{cases}$
(59)

In (59) we have used the normalization
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$ whose explicit form reads

$-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+r^{2}g(r)\dot{\varphi}^{2}=-1,$
(60)

or, using (59),

$-\frac{1}{m^{2}(\phi)f(r)}\left(E-\frac{e^{2\alpha\phi_{0}}qQ}{r}\right)^{2}+\frac{\dot{r}^{2}}{f(r)}+\frac{L^{2}}{m^{2}(\phi)r^{2}g(r)}=-1.$
(61)

Restricting to radial paths and null angular momentum $L$, this gives

$\left(\frac{\mathrm{d}r}{\mathrm{d}\tau}\right)^{2}=-f(r)+\frac{1}{m^{2}(\phi)}\left(E-\frac{e^{2\alpha\phi_{0}}qQ}{r}\right)^{2}.$
(62)

After putting the equation in the form
$\frac{1}{2}\left(\frac{\mathrm{d}r}{\mathrm{d}\tau}\right)^{2}+V_{\mathrm{eff}}(r)=0$,
one can read the forces from the ${1}/{r}$ coefficient in
$V_{\mathrm{eff}}(r)$, the Newtonian approximation being recovered at large
distances ($r\gg r_{-}$). In this limit, using (5), the leading order of $f$
and $m$ are

$\displaystyle
f(r)=1-\frac{1}{r}\left(r_{+}+\frac{1-\alpha^{2}}{1+\alpha^{2}}r_{-}\right)+\mathcal{O}\left(\frac{1}{r^{2}}\right)$
(63) $\displaystyle
m^{2}(\phi)=m^{2}\left(\phi_{0}-\frac{\alpha}{1+\alpha^{2}}\frac{r_{-}}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)\right)=m^{2}(\phi_{0})-\frac{\mathrm{d}m^{2}}{\mathrm{d}\phi}\Bigg{|}_{\phi_{0}}\frac{\alpha}{1+\alpha^{2}}\frac{r_{-}}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right).$
(64)

Together with the identification in (6), this gives

$\frac{1}{2}\left(\frac{\mathrm{d}r}{\mathrm{d}\tau}\right)^{2}=\frac{\frac{E^{2}}{m_{0}^{2}}-1}{2}+\frac{M}{r}+\frac{E^{2}}{m_{0}^{2}}\frac{\frac{m^{\prime}_{0}}{m_{0}}D}{r}-\frac{E}{m_{0}}\frac{e^{2\alpha\phi_{0}}\frac{q}{m_{0}}Q}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right)$
(65)

where the ′ stands for the derivative with respects to $\phi$ and the
subscripts $0$ denote quantities evaluated at $\phi=\phi_{0}$. In (65), $M$,
$Q$ and $D$ are the mass, the charge and scalar charge of the black hole
expressed in geometrized units. Note that $D$ is a secondary charge related to
$M$ and $Q$ in (8). The ${E}/{m_{0}}$ factors should be seen as relativistic
corrections, intrinsically present in the GR framework, and important for
${v}/{c}\sim 1$. At leading order, the non-relativistic potential $V_{pp}$
felt by a point-particle takes then the form

$V_{pp}(r)\equiv
m_{0}V_{\mathrm{eff}}(r)=-\frac{m_{0}M+m^{\prime}_{0}D-e^{2\alpha\phi_{0}}qQ}{r}+\mathcal{O}\left(\frac{1}{r^{2}}\right).$
(66)

We have therefore shown that the forces felt by a test particle match the
result for the non-relativistic limit of the $2\to 2$ scattering amplitude
between this test particle and a state with the gauge charge, scalar charge
and mass of the black hole.

Before turning to the question of how to extend this picture to the case of
scattering of two point-like dilatonic black holes, we give here the final
expression for the second order term in the expansion of $V_{\mathrm{eff}}(r)$

$\begin{split}V^{(2)}_{\mathrm{eff}}(r)=-\frac{1}{2r^{2}}\Bigg{[}&\frac{e^{4\alpha\phi_{0}}q^{2}}{m_{0}^{2}}Q^{2}-(1-\alpha^{2})\left(e^{2\alpha\phi_{0}}Q^{2}-D^{2}\right)-\frac{1}{2}\frac{E^{2}}{m^{2}_{0}}D^{2}\frac{{{m^{\prime\prime}}_{0}}^{2}}{m^{2}_{0}}\big{.}\\\
&\left.+\frac{E^{2}}{m_{0}^{2}}\frac{1+\alpha^{2}}{\alpha}D^{2}\frac{m^{\prime}_{0}}{m}-4\frac{E}{m_{0}}e^{2\alpha\phi_{0}}\frac{q}{m_{0}}QD\frac{m^{\prime}_{0}}{m_{0}}+4\frac{E^{2}}{m_{0}^{2}}D^{2}\left(\frac{m^{\prime}_{0}}{m_{0}}\right)^{2}\right],\end{split}$
(67)

This generalizes the $1/r^{2}$ term one finds in the Reissner-Nordström case:

$V^{(2)}_{\mathrm{eff}}(r)=-\frac{1}{2r^{2}}\left(\frac{q^{2}}{m^{2}}Q^{2}-Q^{2}\right)$
(68)

that is recovered in the limit $\alpha\to 0$ (thus $D\to 0$), giving a non
vanishing contribution even for purely radial motion. Actually, in the
dilatonic case, the expansion of the various terms formally gives
contributions to the different orders in $1/r$.

For generic paths with $L\neq 0$, the $1/r^{2}$ term gets contributions both
from (67) and from the angular momentum term

$\frac{1}{m^{2}(\phi)}\frac{L^{2}}{r^{2}}\frac{f(r)}{g(r)}\qquad{\rm
with}\qquad\frac{f(r)}{g(r)}=\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)^{\frac{1-3\alpha^{2}}{1+\alpha^{2}}}$
(69)

The explicit expression of the angular momentum-related potential term is

$\displaystyle\frac{1}{m^{2}}\frac{L^{2}}{r^{2}}\frac{f}{g}=\frac{L^{2}}{m_{0}^{2}r^{2}}$
$\displaystyle\Bigg{[}1+\frac{2D\frac{m^{\prime}_{0}}{m_{0}}-2M+2\alpha D}{r}$
$\displaystyle+\frac{1}{r^{2}}\bigg{[}\frac{1+\alpha^{2}}{\alpha}\frac{m^{\prime}_{0}}{m_{0}}D-\frac{{m^{2}}^{\prime\prime}_{0}}{m^{2}_{0}}\frac{D}{2}+2\left(\frac{m^{\prime}_{0}}{m_{0}}\right)^{2}+(1-3\alpha^{2})e^{2\alpha\phi_{0}}Q^{2}+4(1+\alpha^{2})\frac{m^{\prime}_{0}}{m}D^{2}$
$\displaystyle\qquad\,\,\,-2(1+\alpha^{2})(1-3\alpha^{2})D^{2}-2\frac{m^{\prime}_{0}}{m_{0}}DM-2\frac{1-\alpha^{2}}{\alpha}\frac{m^{\prime}_{0}}{m_{0}}D\bigg{]}+\mathcal{O}\left(\frac{1}{r^{3}}\right)\Bigg{]}.$
(70)

In the limit $\alpha\to 0$ this reduces to

$\frac{L^{2}}{m^{2}r^{2}}\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)$ (71)

An important difference for $\alpha\neq 0$, for the overall potential sourced
by both $V_{\mathrm{eff}}$ and the angular momentum, is that the sub-leading
contributions are parts of a formally infinite expansion in powers of
${r_{-}}/{r}$, where $r=r_{-}$ is the location of the singularity. As we
approach $r_{-}$, higher orders become important and the whole expansion needs
to be taken into account. Contrary to Schwarzschild or Reissner-Nordström
solutions, there is no fixed-order dominant term whose sign determines whether
the overall effective potential is attractive or repulsive around the
singularity.

In the presence of a non-zero cosmological constant, the computation leading
to (60) and (61) is unchanged but with the additional $\mp
H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}$
in $f(r)$. At large $r\gg r_{-}$, this can be expanded to give

$\displaystyle\mp
H^{2}r^{2}\left(1-\frac{r_{-}}{r}\right)^{\frac{2\alpha^{2}}{1+\alpha^{2}}}\underset{r\gg
r_{-}}{\sim}\mp$ $\displaystyle\left(H^{2}r^{2}-2\alpha
DH^{2}r+(\alpha^{2}-1)D^{2}H^{2}+\frac{2}{3}\frac{\alpha^{2}-1}{\alpha}\frac{D^{3}H^{2}}{r}\right.$
$\displaystyle\left.\,\,+\frac{1}{6}\frac{(\alpha^{2}-1)(\alpha^{2}+3)}{\alpha^{2}}\frac{D^{4}H^{2}}{r^{2}}+\mathcal{O}\left(\frac{r_{-}^{5}}{r^{5}}\right)\right).$
(72)

The result is an additional contribution to the ${1}/{r}$ potential, trivially
vanishing for $H\to 0$, $\alpha\to 0$ and $\alpha=1$. Finally, the additional
contribution to the angular momentum terms, given by
$({L^{2}}/{m^{2}(\phi)r^{2}})({f(r)}/{g(r)})$ leads to
$\mp{L^{2}H^{2}}/{m^{2}_{0}}$, at leading order.

###  7.2 Forces between two point-like states with the black holes charges

We will investigate now the interaction between two point-like states both
describing the black hole type solutions. These states will be characterized
by their mass, their charge and their coupling to the dilaton $\phi$. We will
pursue this description in region of parameters of the solution even beyond
the extremal solution limit, therefore point-like states not corresponding to
black holes anymore, in an attempt to get some indication of what happens
where the metric becomes complex.

The question of how to associate the parameters of a dilatonic black hole to a
particle state was addressed in [30, 31] for the case of an asymptotically
flat space-time solution.
The black hole parameters (say its ADM mass, gauge charge and scalar charge)
are defined at infinity. As such, for a point particle to effectively describe
this black hole, its charge $q$, mass $m(\phi)$ and first derivative
$m^{\prime}(\phi)$ observed at infinity must satisfy the conditions

$\begin{cases}m(\phi_{0})=M=\frac{1}{2}\left(r_{+}+\frac{1-\alpha^{2}}{1+\alpha^{2}}r_{-}\right)\\\
q=Q=\sqrt{\frac{r_{+}r_{-}}{1+\alpha^{2}}}e^{-\alpha\phi_{0}}\\\
m^{\prime}(\phi)\big{|}_{\phi_{0}}=D=\frac{\alpha}{1+\alpha^{2}}r_{-},\end{cases}$
(73)

where $\phi_{0}$ is the asymptotic value of $\phi$ at infinity.

In order to obtain an explicit expression for the scalar charge/scalar
coupling of the point-like black hole approximation, we express the relation
(8) as:

$D=\frac{\alpha}{1-\alpha^{2}}\left(M-\sqrt{M^{2}-(1-\alpha^{2})Q^{2}e^{2\alpha\phi_{0}}}\right)$
(74)

and consider that the point-like state lives in a region where
$\phi_{0}\simeq\phi$, generated by the other (distant) black hole, and
therefore has a coupling to the dilaton given by:

$\frac{\mathrm{d}m}{\mathrm{d}\phi}=\frac{\alpha}{1-\alpha^{2}}\left(m(\phi)-\sqrt{m^{2}(\phi)-(1-\alpha^{2})q^{2}e^{2\alpha\phi}}\right).$
(75)

As was shown in [30, 31], the useful parameters to describe the scalar
interactions are

$\gamma(\phi)\equiv\frac{\mathrm{d}}{\mathrm{d}\phi}\ln{m(\phi)}\quad\mathrm{and}\quad\beta(\phi)\equiv\frac{\mathrm{d}\gamma(\phi)}{\mathrm{d}\phi},$
(76)

and the mass $m(\phi)$ can be expanded around a background value $\bar{\phi}$
as

$m(\phi)=m(\bar{\phi})\left(1+\gamma(\bar{\phi})(\phi-\bar{\phi})+\frac{1}{2}\left(\gamma^{2}(\bar{\phi})+\beta(\bar{\phi})\right)(\phi-\bar{\phi})^{2}+\mathcal{O}\left((\phi-\bar{\phi})^{3}\right)\right).$
(77)

Using (75) one obtains

$\begin{cases}\gamma(\phi)=\frac{\alpha}{1-\alpha^{2}}\left(1-\sqrt{1-(1-\alpha^{2})\frac{q^{2}}{m^{2}(\phi)}e^{2\alpha\phi}}\right)\\\
\beta(\phi)=\frac{\alpha^{2}}{1-\alpha^{2}}\frac{q^{2}e^{2\alpha\phi}}{m^{2}(\phi)}\left(1-\frac{\alpha^{2}}{\sqrt{1-(1-\alpha^{2})\frac{q^{2}}{m^{2}(\phi)}e^{2\alpha\phi}}}\right).\end{cases}$
(78)

With these formulae at hand, we can now extend the analysis of one black hole
and a test particle case to the present case with two black holes.

For $\alpha=1$, it is easy to see that an explicit solution is
$m(\phi)=\sqrt{\mu^{2}+{q^{2}e^{2\phi}}/{2}}$, where $\mu$ is an integration
constant. It is useful for the discussion below to recall that geometrized
units have been used so far and that the $\phi$ field here is dimensionless
(see (3)). In terms of physical quantities, this translates into

$m(\phi)=\sqrt{\mu^{2}+M_{P}^{2}q^{2}e^{\sqrt{2}\frac{\phi}{M_{P}}}},$ (79)

where, although we have used the same notation for simplicity, the quantities
should now be understood to be the physical ones.

The tree-level $t$-channel contribution to the $2\to 2$ scattering amplitude
of 2 such states with same charge $q$ and mass $m(\phi)$ reads555The two
states being the same, they have the same asymptotic value of $\phi$ thus
$\bar{\phi}=\phi_{0}$. We however keep the bar notation here.:

$\mathcal{A}=\frac{4m^{2}}{t}\left(q^{2}e^{\sqrt{2}\frac{\phi}{M_{P}}}-\left(\partial_{\phi}m\right)^{2}-\frac{1}{2}\frac{m^{2}}{M_{P}^{2}}\right)\Bigg{|}_{\bar{\phi}}=\frac{4m^{2}}{t}\left(q^{2}e^{\sqrt{2}\frac{\phi}{M_{P}}}-\frac{1}{2}\frac{M_{P}^{2}}{m^{2}}q^{4}e^{2\sqrt{2}\frac{\phi}{M_{P}}}-\frac{1}{2}\frac{m^{2}}{M_{P}^{2}}\right)\Bigg{|}_{\bar{\phi}},$
(80)

where the bar from now on indicates quantities evaluated at the background
value $\bar{\phi}$. The amplitude can then be put in the simple form

$\mathcal{A}=-2\frac{M_{P}^{2}}{t}\left(\frac{{\bar{m}}^{2}}{M_{P}^{2}}-q^{2}e^{\sqrt{2}\frac{\bar{\phi}}{M_{P}}}\right)^{2}=-\frac{2\mu^{4}}{M_{P}^{2}\,t}$
(81)

from which it is straightforward to observe that it vanishes for
$q^{2}e^{\sqrt{2}\frac{\bar{\phi}}{M_{P}}}={{\bar{m}}^{2}}/{M_{P}^{2}}\,(\mu=0)$
and is always negative otherwise. The no-force condition is readily seen to
correspond, once we revert again to geometrized units, to the black hole
extremality

$q^{2}e^{2\bar{\phi}}=2{\bar{m}}^{2}.$ (82)

As such, for $\alpha=1$, the leading classical force between two particles
with charge $q$ and mass (79) giving an effective description of a pair of the
same black holes is always attractive and vanishes only for extremal states.

Figure 6: Comparison of the leading overall forces for the scattering of two
point-like states approximating far-away black holes in blue, and one Kaluza-
Klein-like state and one similar to the black hole in yellow. Here,
$\alpha=0.7$, and $m=0.3$ in Planck units.

Turning now to generic values of $\alpha$, $\gamma(\phi)$ can be rewritten in
terms of physical quantities as

$\gamma(\phi)=\frac{\alpha}{1-\alpha^{2}}\left(1-\sqrt{1-2M_{P}^{2}(1-\alpha^{2})\frac{q^{2}}{m^{2}(\phi)}e^{\sqrt{2}\alpha\frac{\phi}{M_{P}}}}\right).$
(83)

The tree-level contribution to the force between two states is given by the
coefficient of the $t$-channel pole is

$\mathcal{A}^{\mathrm{t-pole}}=4e^{\sqrt{2}\alpha\frac{\bar{\phi}}{M_{P}}}q^{2}\bar{m}^{2}-2\frac{\bar{m}^{4}}{M_{P}^{2}}-2\frac{\alpha^{2}}{(1-\alpha^{2})^{2}}\frac{\bar{m}^{4}}{M_{P}^{2}}\left(1-\sqrt{1-2M_{P}^{2}(1-\alpha^{2})\frac{q^{2}}{m^{2}_{b}}e^{\sqrt{2}\alpha\frac{\phi}{M_{P}}}}\right)^{2}$
(84)

The resulting behaviour of $\mathcal{A}$ is represented by the blue curve in
figure 6. We observe again that the overall force between two particles is
always attractive, even beyond the point
$q^{2}e^{\sqrt{2}\alpha\frac{\bar{\phi}}{M_{P}}}=\frac{1+\alpha^{2}}{2}\frac{{\bar{m}}^{2}}{M_{P}^{2}}$
corresponding to extremality, which is again found to be the only point where
the force vanishes. Contrary to the Reissner-Nordstrom case, the dilatonic
coupling does not allow repulsive forces beyond extremality: increasing $q$ at
fixed $m$, the scalar force grows at least as strong as the gauge one. As
observed in [30] and later shown in [32], in the asymptotic $\phi\to\infty$
limit, the solution to (75) takes the form
$\sqrt{{(1+\alpha^{2})}/{2}}\left({m(\phi)}/{M_{P}}\right)=qe^{\frac{\alpha}{\sqrt{2}}\frac{\phi}{M_{P}}}$.
We note here that it coincides with the extremal relation.

We now discuss the force felt by a particle of charge $q$ and Kaluza-Klein-
like mass $m(\phi)=m_{A}e^{\alpha\phi}$ (in geometrized units). Our result for
the effective potential for the motion of such a particle in the
asymptotically flat background metric, when applied for this case, agrees with
[33]. We will address here the attractive or repulsive nature of the leading
interaction, the ${1}/{r}$ contribution, for the case of our point-like
states.

The effective potential takes the form (66), with $m^{\prime}=\alpha m$. If we
choose the two, particle and black hole, states such that they share the same
mass ($m(\bar{\phi})=M$) and charge ($q=Q$), we find that, for $\alpha<1$, the
overall force, proportional to $M^{2}+\alpha MD-e^{2\alpha\bar{\phi}}Q^{2}$,
as shown by the yellow curve in figure 6, vanishes at two points:

  * •

the point defining the extremality condition

$M^{2}=\frac{e^{2\alpha\bar{\phi}}Q^{2}}{1+\alpha^{2}}$ (85)

  * •

and in the point

$M^{2}=(1-\alpha^{2})e^{2\alpha\bar{\phi}}Q^{2}$ (86)

where the metric is on the verge of becoming complex.

This remains valid as long as ${m}/{q}={M}/{Q}$.

Denoting the point-like state approximating a black hole as $S$, we now turn
to the computation of the amplitude $S\to S\phi\phi$ for the emission of a
pair of dilatons due to the couplings in (77). It takes the form

$\mathcal{A}(S\to
S\phi\phi)=-2i\frac{\bar{m}^{4}}{M_{P}^{2}}{\bar{\gamma}}^{2}\left(\frac{1}{t-\bar{m}^{2}}+\frac{1}{u-\bar{m}^{2}}\right)-i\frac{\bar{m}^{2}}{M_{P}^{2}}\left(2{\bar{\gamma}}^{2}+{\bar{\beta}}\right),$
(87)

where ${\bar{\beta}}$ is now written in physical units as the other
quantities. At threshold, $t,u=-\bar{m}^{2}$, and this simplifies to

$\mathcal{A}(S\to
S\phi\phi)=-i\frac{\bar{m}^{2}}{M_{P}^{2}}{\bar{\beta}}=-2i\frac{\alpha^{2}}{1-\alpha^{2}}q^{2}e^{\sqrt{2}\alpha\frac{\bar{\phi}}{M_{P}}}\left(1-\frac{\alpha^{2}}{\sqrt{1-(1-\alpha^{2})2M_{P}^{2}\frac{q^{2}}{{\bar{m}}^{2}}e^{\sqrt{2}\alpha\frac{\bar{\phi}}{M_{P}}}}}\right).$
(88)

For any finite value $\alpha\geq 1$, the amplitude is finite as well and
reduces to $\mathcal{A}=-2iq^{2}e^{\sqrt{2}\alpha\frac{\bar{\phi}}{M_{P}}}$
for $\alpha=1$. However, for $\alpha<1$ the amplitude diverges at

$q^{2}e^{\sqrt{2}\alpha\frac{\phi}{M_{P}}}=\frac{1}{2(1-\alpha^{2})}\frac{{\bar{m}}^{2}}{M_{P}^{2}},$
(89)

corresponding to the largest charge before the metric becomes complex. It is
also easily verified that the amplitude (88) vanishes when the no force
(extremality) condition (85) is met.

##  8 Summary and conclusions

We give here an overview of the results obtained above for the existence of
horizons in the black hole solution (5). We also attempt to infer from them
new bounds for the Dilatonic Weak Gravity Conjecture (DWGC). We will keep
separate the discussions about asymptotically flat, AdS and dS space-time. For
AdS, this extends previous partial results for the dilatonic black hole
solution given in [21, 25]. Also, the authors of [25] focused on the region
$r_{+}\gg r_{-}$. Whereas, our analysis of the horizons in the dS solution is
to our knowledge new. Finally, for the WGC in this cases, we follow [5], where
the dS-WGC bound was conjectured to be set by the boundary, in the $(QH,MH)$
plane, between the black hole solution region exhibiting both an event and a
cosmological horizon and the naked singularity region with only a cosmological
horizon.

We are particularly interested in the behaviour of the black hole solutions as
function of the dilaton coupling $\alpha$. We notice that the
$\alpha\to\infty$ and $\alpha\to 0$ limits of such black holes are
approximated by Schwarzschild and Reissner-Nordström black holes, and we find
a value of $\alpha=1,1/\sqrt{3}$ which separates between the two behaviours in
flat and (A)dS backgrounds, respectively.

The flat space-time BH and DWGC:

We have retrieved the well-known result [23, 24]

$Q^{2}e^{2\alpha\phi_{0}}\leq(1+\alpha^{2})M^{2}$ (90)

As formulated by [4], we assume that the WGC corresponds to the opposite
inequality.

The asymptotically AdS BH and AdS-DWGC:

  * •

For $\alpha>{1}/{\sqrt{3}}$:

The black hole solutions exhibit only one (event) horizon. It is located
outside the singular surface as long as

$Q^{2}e^{2\alpha\phi_{0}}<(1+\alpha^{2})M^{2}$ (91)

and the two surfaces coincide when the inequality turns to equality. The DWGC
condition is the same as in the asymptotically flat-space one.

  * •

When $\alpha={1}/{\sqrt{3}}$:

The black hole solutions still possess only one horizon. The coincidence of
that horizon with the singularity is no more obtained for
$Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$ but for a smaller charge now,
saturating the inequality

$\displaystyle\frac{\left(9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}{2^{1/3}\,3^{2/3}}$
$\displaystyle-\frac{\left(\frac{2}{3}\right)^{1/3}}{\left(9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}\geq\frac{4\hat{Q}^{2}}{3\left(M+\sqrt{M^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}.$
(92)

The expansion for small $H$ (large $L=\frac{1}{H}$, with $L$ the AdS length
scale) gives

$Q^{2}e^{(2/\sqrt{3})\phi_{0}}\leq\frac{4}{3}M^{2}-\frac{4^{3}}{3^{4}}M^{4}H^{2}+o(H^{2}).$
(93)

which reproduces the flat space-time case for $H\to 0$.

  * •

For $0<\alpha<{1}/{\sqrt{3}}$:

The black holes have both an inner and an outer horizon. The extremality
condition is now obtained when the two horizons coincide and is expressed as

$F_{AdS}(r_{0-})=0$ (94)

where $F_{AdS}$ is defined by (38) and $r_{0-}$ by (21). Here the two
coincident horizons are located outside the singularity. The expansion for
small $H$ gives

$Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}+\alpha^{2}(1+\alpha^{2})^{\frac{2}{1-\alpha^{2}}}cM^{\frac{3-\alpha^{2}}{1-\alpha^{2}}}H^{\frac{1+\alpha^{2}}{1-\alpha^{2}}}+o(H^{\frac{1+\alpha^{2}}{1-\alpha^{2}}}),$
(95)

where $c$ is defined in (42). From this expansion one can see that the
condition tends to the flat space one in the limit $H\to 0$. Black hole states
solve $F_{AdS}(r_{0-})<0$.

  * •

For $\alpha=0$:

This is the well studied case of charged AdS without dilaton (see for example
[34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]).

A WGC bound can be identified as the requirement of the presence of a state
with a charge $Q$ and a mass $M$ that verifies an inequality opposite to the
ones above respected by black holes with a horizon.

In [45], the classical decay of (near)-extremal solutions through the charged
superradiance mechanism was used to obtain a WGC bound in asymptotically AdS
space-time with a dilaton. The conjecture requires the existence of a state
with mass $m$ and charge $q$ solving

$q\geq\frac{\Delta}{\mu},\qquad\mathrm{with}\qquad\Delta=\frac{3H}{2}+\sqrt{\frac{9H^{2}}{4}+m^{2}},\qquad\mu=\frac{Q}{r_{+}}.$
(96)

where $\Delta$ is the minimum frequency of a scalar perturbation in AdS and
$\mu$ the difference between the component $A_{t}$ of the gauge field at
infinity and on the horizon for extremal solutions. The condition for the
onset of superradiance was obtained considering the leading order in the
horizon radius for small black hole and the equation
$Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$ to define extremal states. With
these assumptions, eq. (96) reads

$q\geq\Delta\sqrt{1+\alpha^{2}}.$ (97)

Here, for $\alpha\leq 1/\sqrt{3}$, we found that the extremality condition
receives corrections. Accordingly, the bound from (96) becomes

$q\geq\frac{r_{0-}}{\sqrt{r_{+}r_{-}}}\Delta\sqrt{1+\alpha^{2}},$ (98)

where $r_{+}$ and $r_{-}$ are related by the condition $F_{AdS}(r_{0-})=0$,
$r_{0-}=r_{0-}(r_{+},r_{-})$. Again, $F_{AdS}$ is defined by (38) and $r_{0-}$
in (21). For $r_{+}\geq r_{-}$, and thus for extremal solutions,
$r_{0-}<\sqrt{r_{+}r_{-}}$, so that the bound (98) is weaker than (97). Using
(21) and the expansion of the extremality condition for small $H$ (41), this
gives at leading order:

$q\geq\Delta\sqrt{1+\alpha^{2}}\left(1+\frac{\alpha^{2}}{1-\alpha^{2}}cr_{+}^{\gamma}H^{\gamma}+o(r_{+}^{\gamma}H^{\gamma})\right),$
(99)

where $\gamma={(1+\alpha^{2})}/{(1-\alpha^{2})}$ and $c$ is a constant
solution of the equation given in (42). Note that the expression of the
minimum frequency in AdS might also receive corrections which we expect to be
sub-leading (for RN-AdS, we have $\omega=H\Delta+o(r_{h}H^{2})$).

The asymptotically dS-BH and dS-DWGC

  * •

For $\alpha>{1}/{\sqrt{3}}$, the condition for the event horizon to coincide
with the singularity is given by
$Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$. It is again the same as in
both asymptotically flat and AdS space.

  * •

When $\alpha={1}/{\sqrt{3}}$, the extremal solution solutions solve

$\displaystyle\frac{\left(\frac{2}{3}\right)^{1/3}e^{-i\pi/3}}{\left(-9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{-4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}$
$\displaystyle+\frac{e^{i\pi/3}\left(-9\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)+\sqrt{3}\sqrt{-4+27\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}\right)^{1/3}}{2^{1/3}\,3^{2/3}}$
$\displaystyle=-\frac{4\hat{Q}^{2}}{3\left(\hat{M}+\sqrt{\hat{M}^{2}-\frac{2}{3}\hat{Q}^{2}}\right)}.$
(100)

This condition can be expanded for small $H$ as

$Q^{2}e^{(2/\sqrt{3})\phi_{0}}=\frac{4}{3}M^{2}+\frac{4^{3}}{3^{4}}M^{4}H^{2}+\mathcal{O}(M^{6}H^{4}),$
(101)

showing that it allows for slightly greater charges than the corresponding
flat space limit. It goes to the flat space-time condition for $H\to 0$.

  * •

For $0<\alpha<{1}/{\sqrt{3}}:$

For a real valued metric, one has always either two horizons or, trivially for
huge masses, a naked singularity. We have found no region of parameters with a
(real) metric exhibiting only a naked singularity with a cosmological horizon.

In both limits $\alpha\to 0$ and $\alpha\to{1}/{\sqrt{3}}$ limits, we retrieve
the $\alpha=0$ and $\alpha={1}/{\sqrt{3}}$ results, respectively, where a
transition from a black hole to a naked singularity with cosmological horizon
happens for $F(r_{0-})=0$ (defined by (20) and (21)).

We discuss further this case below.

  * •

For $\alpha=0$

The phases of the dS-RN black hole metric are described in details in [19, 5].
The condition for the existence of the black hole is

$Q^{2}\leq M^{2}+M^{4}H^{2}+{\cal O}(M^{6}H^{4})$ (102)

with $\quad M^{2}H^{2}\leq\displaystyle\frac{2}{27}$.

The appearance of a complex valued metric leaves the lower bound on the black
hole mass for the $0<\alpha<{1}/{\sqrt{3}}$ asymptotically dS solutions
undefined. Because previous literature focused on the asymptotically flat
metric which always shows a naked singularity before reaching the complex
valued metric region, it was not investigated. Here, however, the condition
for the metric becoming complex,

$Q^{2}e^{2\alpha\phi_{0}}>\frac{M^{2}}{1-\alpha^{2}}$ (103)

can be reached within the domain of the black hole solution region and
represents a new bound.

To investigate the nature of this bound, we studied, in the asymptotically
flat space, the behaviour of the interaction between point-like particles
approximating at long distances the charged dilatonic black holes. The main
difficulty is then to express the coupling of the dilaton to the point-like
particles as a function of the charge and mass which is non-trivial for the
generic solution. A first approximation was to adopt the relation between
scalar and gauge charges and mass as it appears for the black hole solution at
infinity. We found then that the self interaction between two such states is
always attractive, even in the super-extremal case, and is null only for
extremal solutions with $Q^{2}e^{2\alpha\phi_{0}}=(1+\alpha^{2})M^{2}$.
Increasing further the charge to mass ratio, the production of dilaton pairs
seems to diverge for $\alpha<1$ when we reach
$Q^{2}e^{2\alpha\phi_{0}}=M^{2}/(1-\alpha^{2})$ above which the metric becomes
complex. Considering instead a particle with a mass
$m(\phi)=m_{A}e^{\alpha\phi}$, interacting with a black hole of same mass and
charge at large distance, we found that for $0<\alpha<1$ the leading order
forces between these two states cancel at the extremality but also at the
point where the metric is on the verge of becoming complex. We might consider
that (103) represents a new dilatonic de Sitter WGC bound for this domain of
dilaton couplings ($0<\alpha<1/\sqrt{3}$), but further investigation is needed
to confirm or infirm this. Also, it will be interesting to consider the effect
of higher derivative corrections to the black hole solutions.

## Acknowledgments

K.B. acknowledges the support of the Agence Nationale de Recherche under grant
ANR-15-CE31-0002 “HiggsAutomator”.

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