# Higher dimensional dilaton black holes in the presence of exponential
nonlinear electrodynamics

A<EMAIL_ADDRESS>and A. Kazemi 1 1 Physics Department
and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454,
Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran

###### Abstract

We examine the higher dimensional action in which gravity is coupled to the
exponential nonlinear electrodynamic and a scalar dilaton field. We construct
a new class of $n$-dimensional static and spherically symmetric black hole
solutions of this theory in the presence of the dilaton potential with two
Liouville-type terms. In the presence of two Liouville-type dilaton potential,
the asymptotic behavior of the obtained black holes are neither flat nor
(A)dS. Due to the nonlinear nature of electrodynamic field, the electric field
has finite value near the origin where $r\rightarrow 0$ and goes to zero as
$r\rightarrow\infty$. Interestingly enough, we find that in the absence of the
dilaton field, the electric field has a finite value at $r=0$, while as soon
as the dilaton field is taken into account, the electric field diverges as
$r\rightarrow 0$. This implies that the presence of the dilaton field changes
the behaviour of the electric field near the origin. In the limiting case
where the nonlinear parameter $\beta$ goes to infinity, our solutions reduce
to dilaton black holes of Einstein-Maxwell-dilaton gravity in higher
dimensions. We compute the conserved and thermodynamic quantities of the
solutions and show that these quantities satisfy the first law of black holes
thermodynamics on the horizon.
PACS numbers: 04.70.-s, 04.30.-w

##  I Introduction

There are several motivations for studying black hole solutions in higher
dimensional spacetimes. The first reason comes from string theory which
contains gravity and requires more than four dimensions. In fact, the first
successful statistical counting of black hole entropy in string theory was
performed for a five dimensional black hole Stro . This example provides the
best laboratory for the microscopic string theory of black holes. Another
motivation originates from the AdS/CFT correspondence which relates the
properties of an $n$-dimensional black hole with those of a quantum field
theory in $(n-1)$-dimensions Ah . Besides, the production of higher-
dimensional black holes in future colliders becomes a conceivable possibility
in scenarios involving large extra dimensions and TeV-scale gravity MC . In
addition, as mathematical objects, black hole spacetimes are among the most
important Lorentzian Ricci-flat manifolds in any dimension Emp .

In this paper, we turn to the investigation on higher dimensional black holes
in the presence of nonlinear electrodynamics and dilaton field. The theory of
nonlinear electrodynamics was first introduced in 1930’s by Born and Infeld to
obtain a classical theory of charged particles with finite self-energy BI .
Born-Infeld (BI) theory has received renewed attentions since it turns out to
play an important role in string theory. It arises naturally in open
superstrings and D-branes Frad ; Cal . The low energy effective action for an
open superstring in loop calculations lead to BI type actions. The BI action
including a dilaton and an axion field, appears in the coupling of an open
superstring and an Abelian gauge field theory Frad . This action, describing a
Born-Infeld-dilaton-axion system coupled to Einstein gravity, can be
considered as a nonlinear extension in the Abelian field of Einstein-Maxwell-
dilaton-axion gravity. Nonlinear BI theory in the context of dilaton gravity
have been investigated by many authors. See for e.g Tam1 ; Tam2 ; YI ; yaz ;
Clement ; yazad ; SRM ; Shey1 ; Shey2 ; DHSR  and references therein. In the
framework of Einstein-Maxwell-dilaton (EMd) gravity with a Liouville type
dilaton potential non-degenerate and degenerate (extremal) Killing horizons of
arbitrary geometry and topology were investigated Shohreh . Here we would like
to consider another type of nonlinear electrodynamics, namely exponential form
of the nonlinear electrodynamics in the setup of dilaton gravity. The
lagrangian of the exponential nonlinear (EN) electrodynamics is given by
HendiJHEP ,

$L_{\rm
EN}=4\beta^{2}\left[\exp\left(-\frac{F^{2}}{4\beta^{2}}\right)-1\right],$ (1)

where $\beta$ is called the nonlinear parameter with dimension of mass,
$F^{2}=F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}$ is the electromagnetic field
tensor. The advantages of the EN Lagrangian compared to BI nonlinear
electrodynamics is that, it does not cancel, at least for some parameters, the
divergency of the electric field at $r=0$, however, its singularity is much
weaker than Einstein-Maxwell theory Hendi2 . In our previous paper ShHaj , we
studied dilaton black holes coupled to EN electrodynamics in four dimensional
spacetime. In the present work, we would like to extend our study to all
higher dimensions and construct a new class of $n$-dimensional black holes in
dilaton gravity which is coupled to nonlinear matter field. We shall also
investigate the physical properties of the spacetime and obtain conserved and
thermodynamic quantities of the solutions.

This paper is organized as follows. In the next section, we introduce the
Lagrangian of EN electrodynamics in $n$-dimensions coupled to the dilaton
field in Einstein gravity, and obtain the corresponding field equations by
varying the action. In section III, we find a new class of static and
spherically symmetric black hole solutions of this theory and investigate
their properties. In section IV, we study thermodynamics of higher dimensional
dilaton black holes in the presence of nonlinear electrodynamics. The last
section is devoted to conclusion and discussion.

##  II Basic field equations

We consider the $n$-dimensional $(n\geq 4)$ action in which gravity is coupled
to dilaton and nonlinear electrodynamic fields

$S=\frac{1}{16\pi}\int{d^{n}x\sqrt{-g}\left(\mathcal{R}\text{
}-\frac{4}{n-2}(\nabla\Phi)^{2}-V(\Phi)+L(F,\Phi)\right)},$ (2)

where $\mathcal{R}$ and $\Phi$ are, respectively, the Ricci scalar curvature
and the dilaton filed, and $V(\Phi)$ is a potential for $\Phi$. We further
assume the dilaton potential contain two Liouville terms,

$V(\Phi)=2\Lambda_{0}e^{2\zeta_{0}\Phi}+2\Lambda e^{2\zeta\Phi},$ (3)

where $\Lambda_{0}$, $\Lambda$, $\zeta_{0}$ and $\zeta$ are constants. This
kind of potential was previously investigated in the context of BI-dilaton
(BId) black holes Shey1 ; Shey2  as well as EMd gravity MW ; CHM ; Cai ; yaz2
; Clem ; Shey3 .

We choose the Lagrangian of the exponential nonlinear electrodynamics coupled
to the dilaton field (ENd) in $n$-dimensions as

$L(F,\Phi)=4\beta^{2}e^{4\alpha\Phi/(n-2)}\left[\exp\left(-\frac{e^{-8\alpha\Phi/(n-2)}F^{2}}{4\beta^{2}}\right)-1\right],$
(4)

where $\alpha$ is a constant determining the strength of coupling of the
scalar and electromagnetic field. In order to justify such a choice for the
Lagrangian of ENd field, let us invoke the BId Lagrangian in $n$-dimensions
which is written as Shey1

$L_{\rm
BId}(F,\Phi)=4\beta^{2}e^{4\alpha\Phi/(n-2)}\left(1-\sqrt{1+\frac{e^{-8\alpha\Phi/(n-2)}F^{2}}{2\beta^{2}}}\right).$
(5)

It is worthy to note that Lagrangian (5) originates from open string version
of the BI action coupled to a dilaton field and only valid for the pure
electric case Tam1 . Clearly, this version of the BId action does not enjoy
electric-magnetic duality Clement . This form for the BId term have been
investigated previously by a number of authors Tam1 ; Tam2 ; YI ; yaz ;
Clement ; yazad ; SRM ; Shey1 ; Shey2 ; DHSR . It is easy to check that the
series expansion of both BId Lagrangian (5) and ENd Lagrangian (4), for large
$\beta$, have the same behavior

$L_{\rm
BId}(F,\Phi)=-e^{-4\alpha\Phi/(n-2)}F^{2}+\frac{e^{-12\alpha\Phi/(n-2)}F^{4}}{8\beta^{2}}-\frac{e^{-20\alpha\Phi/(n-2)}F^{6}}{32\beta^{4}}+O\left(\frac{1}{\beta^{6}}\right),$
(6)
$L(F,\Phi)=-e^{-4\alpha\Phi/(n-2)}F^{2}+\frac{e^{-12\alpha\Phi/(n-2)}F^{4}}{8\beta^{2}}-\frac{e^{-20\alpha\Phi/(n-2)}F^{6}}{96\beta^{4}}+O\left(\frac{1}{\beta^{6}}\right).$
(7)

This similarity implies that one can either consider (4) or (5) as the
nonlinear electrodynamic Lagrangian coupled to the dilaton field. Here we
would like to study the new ENd Lagrangian (4), and investigate the effects of
this kind of nonlinear electrodynamics coupled to the dilaton field on the
behavior of the solutions. In the absence of the dilaton field ($\alpha=0$)
and in four dimensions where $n=4$, $L(F,\Phi)$ reduces to EN electrodynamic
Lagrangian presented in Hendi2 . On the other hand, in the limiting case where
$\beta\rightarrow\infty$, both $L(F,\Phi)$ and $L_{\rm BId}(F,\Phi)$ recovers
the standard linear Maxwell lagrangian coupled to the dilaton field in
$n$-dimensions CHM

$L(F,\Phi)=L_{\rm BId}(F,\Phi)=-e^{-4\alpha\Phi/(n-2)}F^{2}.$ (8)

This is an expected result, since in this case the nonlinear electrodynamics
reduces to the linear Maxwell electrodynamics. For latter convenience we
rewrite

$L(F,\Phi)=4\beta^{2}e^{4\alpha\Phi/(n-2)}{\mathcal{L}}(Y),$ (9)

where

$\displaystyle{\mathcal{L}}(Y)$ $\displaystyle=$ $\displaystyle\exp(-Y)-1,$
(10) $\displaystyle Y$ $\displaystyle=$
$\displaystyle\frac{e^{-8\alpha\Phi/(n-2)}F^{2}}{4\beta^{2}}.$ (11)

By varying action (2) with respect to the gravitational field $g_{\mu\nu}$,
the dilaton field $\Phi$ and the gauge field $A_{\mu}$ we obtain the field
equations as

$\displaystyle{\cal R}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{4}{n-2}\left(\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{4}g_{\mu\nu}V(\Phi)\right)-2e^{-4\alpha\Phi/(n-2)}\partial_{Y}{{\cal
L}}(Y)F_{\mu\eta}F_{\nu}^{\text{ }\eta}$ (12)
$\displaystyle+\frac{4\beta^{2}}{n-2}e^{4\alpha\Phi/(n-2)}\left[2Y\partial_{Y}{{\cal
L}}(Y)-{{\cal L}}(Y)\right]g_{\mu\nu},$
$\nabla^{2}\Phi=\frac{n-2}{8}\frac{\partial
V}{\partial\Phi}+2\alpha\beta^{2}e^{4\alpha\Phi/(n-2)}\left[2{Y}\partial_{Y}{{\cal
L}}(Y)-{\cal L}(Y)\right],$ (13)
$\nabla_{\mu}\left(e^{-4\alpha\Phi/(n-2)}\partial_{Y}{{\cal
L}}(Y)F^{\mu\nu}\right)=0.$ (14)

When $\beta\rightarrow\infty$, we have ${\cal L}(Y)=-Y$, and the system of
field equations (12)-(14) restore the well-known equations of EMd gravity CHM
; Cai ; yaz2 ; Clem ; Shey3 .

##  III higher dimensional dilaton black hole

In this section, we would like to find static and spherically symmetric
solutions of the field equations (12)-(14). The metric of such a spacetime can
be written

$ds^{2}=-f(r)dt^{2}+{dr^{2}\over f(r)}+r^{2}R^{2}(r)d\Omega^{2}_{n-2},$ (15)

where $d\Omega^{2}_{n-2}$ denotes the metric of an unit $(n-2)$-sphere and
$f(r)$ and $R(r)$ are functions of $r$ which should be determined. Integrating
the electromagnetic field equation (14) and assuming all components of
$F_{\mu\nu}$ are zero except $F_{tr}$, we find

$F_{tr}=\frac{qe^{4\alpha\Phi/(n-2)}}{\left(rR\right)^{n-2}}\exp\left[-\frac{1}{2}L_{W}\left(\frac{q^{2}}{\beta^{2}\left(rR\right)^{2n-4}}\right)\right],$
(16)

where $q$ is an integration constant which is related to the electric charge
of the black hole. We recall that in BId theory the electric field can be
written as Shey1

$F_{tr}=\frac{qe^{4\alpha\Phi/(n-2)}}{\left(rR\right)^{n-2}}\sqrt{1+\frac{2q^{2}}{\beta^{2}(n-2)\left(rR\right)^{2n-4}}}.$
(17)

Using the Gauss’s law,
$Q=\frac{1}{4\pi}\int\exp\left[{-4\alpha\Phi/(n-2)}\right]\text{
}^{*}Fd{\Omega},$ we can calculate the flux of the electromagnetic field at
infinity to obtain the electric charge of black hole as

${Q}=\frac{q\omega_{n-2}}{4\pi},$ (18)

where $\omega_{n-2}$ represents the volume of an unit $(n-2)$-sphere. Note
that in (16), $L_{W}(x)={LambertW(x)}$ is the Lambert function which satisfies
the identity Lambert

$L_{W}(x)e^{L_{W}(x)}=x,$ (19)

and has the following series expansion

$L_{W}(x)=x-x^{2}+\frac{3}{2}x^{3}-\frac{8}{3}x^{4}+....$ (20)

Clearly, series (20) converges for $|x|<1$. In the absence of the nonlinear
dilaton field ($\alpha=0$) we have $R(r)=1$, and Eq. (16) reduces to

$F_{tr}=\frac{q}{r^{n-2}}\exp\left[-\frac{1}{2}L_{W}\left(\frac{q^{2}}{\beta^{2}r^{2n-4}}\right)\right],$
(21)

while in the limiting case where $\beta\rightarrow\infty$, it reduces to the
electric field of $n$-dimensional EMd black holes Shey3

$F_{tr}=\frac{qe^{4\alpha\Phi/(n-2)}}{(rR)^{n-2}}+O\left(\frac{1}{\beta^{2}}\right).$
(22)

In order to solve the system of equations (12) and (13) for three unknown
functions $f(r)$, $R(r)$ and $\Phi(r)$, we make the ansatz Shey3

$R(r)=e^{2\alpha\Phi/(n-2)}.$ (23)

It is important to note that in the absence of the nontrivial dilaton field
($\alpha=0$), we have $R(r)=1$, as one expected (see Eq. 15). Substituting
(23), the electromagnetic field (16) and the metric (15) into the field
equations (12) and (13), one can obtain the following solutions

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{{(n-3)}{{({\alpha}^{2}+1)}^{2}}}{{({\alpha}^{2}+n-3)}{({\alpha}^{2}-1)}}{b}^{-\gamma}{r}^{\gamma}-\frac{m}{r^{n-3-(n-2)\gamma/2}}+\frac{2{{(\alpha^{2}+1)}^{2}}{(\Lambda+2\beta^{2})}{{b}^{\gamma}}}{{(n-2)}{({\alpha}^{2}-n+1)}}r^{2-\gamma}$
(24) $\displaystyle-\frac{4\beta
q\left(\alpha^{2}+1\right)b^{{(4-n)\gamma}/{2}}}{{(n-2)}{{r}^{n-3-(n-2)\gamma/2}}}\int
r^{-\gamma}\left(\sqrt{L_{W}(\eta)}-\frac{1}{\sqrt{L_{W}(\eta)}}\right){dr},$
$\Phi(r)=\frac{(n-2)\alpha}{2(\alpha^{2}+1)}\ln\left(\frac{b}{r}\right),$ (25)

where $b$ is an arbitrary constant, $\gamma=2\alpha^{2}/(1+\alpha^{2})$, and

$\eta\equiv\frac{q^{2}b^{(2-n)\gamma}}{\beta^{2}r^{(n-2)(2-\gamma)}}.$ (26)

In the above expression, $m$ appears as an integration constant and is related
to the mass of the black hole. The obtained solutions fully satisfy the system
of equations (12) and (13) provided we take

$\zeta_{0}=\frac{2}{\alpha(n-2)},\hskip
22.76228pt\zeta=\frac{2\alpha}{n-2},\hskip
22.76228pt\Lambda_{0}=\frac{(n-2)(n-3)\alpha^{2}}{2b^{2}(\alpha^{2}-1)}.$ (27)

Notice that here $\Lambda$ remains as a free parameter which plays the role of
the cosmological constant. For later convenience, we redefine it as
$\Lambda=-(n-1)(n-2)/2l^{2}$, where $l$ is a constant with dimension of
length. The integration of Eq. (24) can be performed using the Mathematica
software. The resulting solution can be written

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{{(n-3)}{({\alpha}^{2}+1)^{2}}}{{({\alpha}^{2}-1)}{({\alpha}^{2}+n-3)}}{b}^{-\gamma}{r}^{\gamma}-\frac{m}{r^{n-3-(n-2)\gamma/2}}+\frac{2(\Lambda+2\beta^{2})\left({\alpha}^{2}+1\right)^{2}{b}^{\gamma}}{{(n-2)}{(\alpha^{2}-n+1)}}r^{2-\gamma}$
(28) $\displaystyle+\frac{2\beta
q(\alpha^{2}+1)^{4}b^{{(4-n)\gamma}/{2}}}{{(n-2)^{2}}{(\alpha^{2}-1)^{2}}}\left(\frac{\beta^{2}b^{(n-2)\gamma}}{q^{2}}\right)^{\frac{1-\gamma}{(n-2)(\gamma-2)}}r^{{(n-2)\gamma}/{2}-n+3}$
$\displaystyle\times\left(\frac{1-\alpha^{2}}{2n-4}\right)^{\frac{2-2n+\gamma
n}{(\gamma-2)(2n-4)}}\Bigg{\\{}-(n-2)^{2}(\gamma-2)^{2}\left[\Gamma\left(\frac{\alpha^{2}+3n-7}{2n-4},\frac{1-\alpha^{2}}{2n-4}L_{W}(\eta)\right)-\Gamma\left({\frac{\alpha^{2}+3n-7}{2n-4}}\right)\right]$
$\displaystyle+(\gamma-1)^{2}\left[\Gamma\left(\frac{\alpha^{2}-n+1}{2n-4},\frac{1-\alpha^{2}}{2n-4}L_{W}(\eta)\right)-\Gamma\left({\frac{\alpha^{2}-n+1}{2n-4}}\right)\right]\Bigg{\\}}.$

where $\Gamma(a,z)$ and $\Gamma(a)$ are Gamma functions and they are related
to each other via,

$\displaystyle\Gamma(a,z)=\Gamma(a)-\frac{z^{a}}{a}\digamma\left(a,1+a,-z\right).$
(29)

where $\digamma(a,b,z)$ is hypergeometric function Lambert . Using (29),
solution (28) can also be reexpressed in terms of hypergeometric function,

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{(n-3)({\alpha}^{2}+1)^{2}}{({\alpha}^{2}-1)({\alpha}^{2}+n-3)}{b}^{-\gamma}{r}^{\gamma}-\frac{m}{r^{n-3-{(n-2)\gamma}/{2}}}+\frac{2(\Lambda+2\beta^{2})\left({\alpha}^{2}+1\right)^{2}{b}^{\gamma}}{(n-2)(\alpha^{2}+1-n)}r^{2-\gamma}$
(30) $\displaystyle+\frac{4\beta
q{(\alpha^{2}+1)^{2}}b^{(4-n)\gamma/2}}{(n-2)r^{n-3-(n-2)\gamma/2}}{\left(\frac{\beta^{2}b^{(n-2)\gamma}}{q^{2}}\right)}^{\frac{1-\gamma}{(n-2)(\gamma-2)}}{{L_{W}^{\frac{\alpha^{2}-n+1}{2n-4}}}(\eta)}$
$\displaystyle\times\Bigg{\\{}\frac{L_{W}^{2}(\eta)}{\alpha^{2}+3n-7}\digamma\left(\frac{\alpha^{2}+3n-7}{2n-4},\frac{\alpha^{2}+5n-11}{2n-4},\frac{\alpha^{2}-1}{2n-4}L_{W}(\eta)\right)$
$\displaystyle-\frac{1}{\alpha^{2}-n+1}\digamma\left(\frac{\alpha^{2}-n+1}{2n-4},\frac{\alpha^{2}+n-3}{2n-4},\frac{\alpha^{2}-1}{2n-4}L_{W}(\eta)\right)\Bigg{\\}}.$

Using the fact that $L_{W}(x)$ has a convergent series expansion for $|x|<1$
as given in (20), we can expand (28) for large $\beta$. The result is

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{(n-3)(\alpha^{2}+1)^{2}}{(\alpha^{2}+n-3)(\alpha^{2}-1)}b^{-\gamma}r^{\gamma}-\frac{m}{r^{n-3-{(n-2)\gamma}/{2}}}+\frac{2\Lambda(\alpha^{2}+1)^{2}}{(n-2)(\alpha^{2}-n+1)}b^{\gamma}r^{2-\gamma}$
(31)
$\displaystyle+\frac{2q^{2}({\alpha^{2}+1})^{2}b^{-(n-3)\gamma}}{(n-2)(\alpha^{2}+n-3)r^{-(n-3)\gamma+2n-6}}-\frac{q^{4}(\alpha^{2}+1)^{2}b^{-(2n-5)\gamma}}{2\beta^{2}(n-2)(\alpha^{2}+3n-7)r^{(2n-5)(2-\gamma)}}+O\left(\frac{1}{\beta^{4}}\right).$

When $\beta\rightarrow\infty$, solution (31) is exactly the one obtained for
higher dimensional black holes in EMd gravity Shey3 . This is an expected
result, since as we discussed already in this limit the Lagrangian of ENd
theory reduces to EMd gravity. In the absence of the dilaton field
($\alpha=0=\gamma$), solution (31) can be further simplified as

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle
1-\frac{m}{r^{n-3}}+\frac{r^{2}}{l^{2}}+\frac{2q^{2}}{{(n-2)(n-3)r^{2n-6}}}-\frac{1}{2\beta^{2}(n-2)(3n-7)}\frac{q^{4}}{r^{4n-10}}+O\left(\frac{1}{\beta^{4}}\right),$
(32)

which has the form of static spherically symmetric $n$-dimensional RN-AdS
black holes in the limit $\beta\rightarrow\infty$. The last term in the right
hand side of (32) is the leading nonlinear correction term to the RN-AdS black
hole in the large $\beta$ limit.

### Physical properties of the solutions

Now we back to the electric field obtained in (16). In order to study the
behaviour of the electric field, we combine Eqs. (23) and (25) with (16). We
find

$F_{tr}=E(r)=\frac{qb^{(4-n)\gamma/2}}{r^{n-2+2\gamma-n\gamma/2}}\exp\left[-\frac{1}{2}L_{W}\left(\frac{q^{2}b^{-(n-2)\gamma}}{\beta^{2}r^{(n-2)(2-\gamma)}}\right)\right].$
(33)

Expanding for large $\beta$, we arrive at

$E(r)=\frac{qb^{(4-n)\gamma/2}}{r^{n-2+(4-n)\gamma/2}}-\frac{q^{3}}{2\beta^{2}}\frac{b^{\gamma(4-3n/2)}}{r^{3n-6+\gamma(4-3n/2)}}+\frac{5q^{5}}{8\beta^{4}}\frac{b^{\gamma(6-5n/2)}}{r^{5n-10+\gamma(6-5n/2)}}+O\left(\frac{1}{\beta^{6}}\right).$
(34)

In order to analyze the behaviour of $F_{tr}$, we choose $b=1$, $n=6$ and
$\alpha=\sqrt{2}$ ($\gamma=4/3$). For these values of the parameters we have

$E(r)=\frac{q}{r^{8/3}}-\frac{q^{3}}{2\beta^{2}r^{16/3}}+\frac{5q^{5}}{8\beta^{4}r^{8}}+O\left(\frac{1}{\beta^{6}}\right).$
(35)

From Eq. (35) we see that in the presence of the dilaton field, the electric
field diverges as $r\rightarrow 0$. For more details see table A.

Figure 1: The behavior of the electric field $E(r)$ of ENd black holes versus
$r$ for $n=6$, $b=1$ and $q=2$.

$r=$ | $10^{3}$ | $10^{2}$ | $10^{1}$ | $10^{-1}$ | $10^{-10}$ | $10^{-1000}$
---|---|---|---|---|---|---
$\alpha=0.0$ | $10^{-11}$ | $10^{-7}$ | $10^{-4}$ | $10$ | $40$ | $400$
$\alpha=0.4$ | $10^{-10}$ | $10^{-7}$ | $10^{-4}$ | $20$ | $10^{4}$ | $10^{300}$
$\alpha=0.8$ | $10^{-9}$ | $10^{-6}$ | $10^{-3}$ | $50$ | $10^{10}$ | $10^{800}$

Table A: $E(r)$ for ENd versus $r$ for $\beta=3$, $n=6$, $q=2$ and $b=1$ and
different values of $\alpha$.

Figure 2: The behavior of the electric field $E(r)$ of ENd black holes versus
$r$ for $b=1$, $\alpha=0.5$, $\beta=2$ and $q=2$.

Figure 3: The behavior of the electric field $E(r)$ versus $r$ and for EMd,
ENd and BId field. Here we have taken $b=1$, $\beta=3$, $\alpha=0.4$, $n=5$
and $q=1$.

Figure 4: The behavior of the electric field $E(r)$ of ENd black holes versus
$r$ for $b=1$, $n=6$, and $q=1$.

In order to have better understanding of the behavior of the electric field
(33), we plot $E(r)$ versus $r$ for different values of the parameters in
figures 1-4. From these figures and table A we see that the electric field
goes to zero for large $r$ independent of the value of the other parameters.
Figure 1 shows that for ENd black holes, and in the absence of the dilaton
field ($\alpha=0$), the electric field has a finite value near $r=0$, while as
soon as the dilaton field is taken into account ($\alpha>0$), the electric
field diverges as $r\rightarrow 0$. This implies that the presence of the
dilaton field changes the behavior of the electric field near the origin. The
behavior of the electric fields for ENd black holes in different dimensions
are shown in figure 2. From this figure we see that in any dimension as
$r\rightarrow 0$, the electric field goes to infinity. In figures 3 we have
compared the behavior of $E(r)$ for BId, ENd and EMd black holes. From this
figure we see that for ENd case the electric field has a finite value _near_
the origin, while it diverges exactly at $r=0$, however its singularity is
weaker than EMd. This is in contrast to the BId electrodynamics which the
electric field has finite value at $r=0$. Finally, we have plotted in figure 4
the electric field of ENd black holes for different values of the nonlinear
parameter $\beta$. From this figure we see that with increasing $\beta$, the
electric field diverges as $r\rightarrow 0$. This is an expected result, since
for large $\beta$ our theory reduces to the well-known EMd gravity Shey3 .

The next step is to investigate the casual structure of the solutions and
check whether there is or not the curvature singularities and horizons. We
find that Kretschmann scalar $R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$
diverges as $r\rightarrow 0$. This implies that our spacetime has an essential
singularity located at $r=0$.

In order to study the asymptotic behaviour of the solutions, we expand the
metric function $f(r)$ for $r\rightarrow\infty$ limit. We find

$\displaystyle\lim_{r\longrightarrow\infty}f(r)=-\frac{(n-3)(\alpha^{2}+1)^{2}}{(\alpha^{2}+n-3)(\alpha^{2}-1)}b^{-\gamma}r^{\gamma}+\frac{2\Lambda(\alpha^{2}+1)^{2}}{(n-2)(\alpha^{2}-n+1)}b^{\gamma}r^{2-\gamma}.$
(36)

Let us note that in the absence of the dilaton field ($\alpha=0=\gamma$), the
metric function becomes

$\displaystyle\lim_{r\longrightarrow\infty}f(r)=1-\frac{2\Lambda
r^{2}}{(n-1)(n-2)},$ (37)

which describes an asymptotically flat ($\Lambda=0$), AdS ($\Lambda<0$) or dS
($\Lambda>0$) spacetimes. However, as one can see from Eq. (36), in the
presence of the dilaton field the asymptotic behaviour is neither flat nor
(A)dS. For example, taking $\alpha=\sqrt{2}$, $n=6$ and $b=1$, we have

$\displaystyle\lim_{r\longrightarrow\infty}f(r)=-\frac{27}{5}r^{4/3}-\frac{3\Lambda}{2}r^{2/3}.$
(38)

Clearly, the metric function (38) is neither falt nor (A)dS. Indeed, it has
been shown that no dilaton dS or AdS black hole solution exists with the
presence of only one or two Liouville-type dilaton potential MW . In the
presence of one or two Liouville-type potential, black hole spacetimes which
are neither asymptotically flat nor (A)dS have been explored by many authors
(see e.g. CHM ; Cai ; yaz2 ; Clem ; Shey3 ). It is important to note that this
asymptotic behaviour is not due to the nonlinear nature of the electrodynamic
field, since as $r\rightarrow\infty$ the effects of the nonlinearity
disappear. This is due to the fact that, $r\rightarrow\infty$ limit
corresponds to $\beta^{2}\rightarrow\infty$, and in this case $F_{tr}$ as well
as the metric functions $f(r)$ restore the result of EMd with unusual
asymptotic Shey3 .

Furthermore, from the dilaton field (25) we see that as $r\rightarrow\infty$,
the dilaton field does not vanishes, while in case of asymptotic flat or (A)dS
we expect to have $\lim_{r\longrightarrow\infty}\Phi(r)=0.$ Indeed, by solving
the field equation (13) we find

$\Phi(r)=\frac{(n-2)\alpha}{2(\alpha^{2}+1)}\ln\left(a+\frac{b}{r}\right),$
(39)

however, the system of equation (12)-(14) will be fully satisfied provided we
choose $a=0$. From the above arguments we conclude that the asymptotic
behaviour of the obtained solutions is neither flat nor (A)dS.

It is also worthwhile to note that in case of $\alpha=\sqrt{n-1}$ and
$\alpha=1$ the solutions are ill-defined as one can see from Eq. (28). In
order to obtain the location of the horizons of spacetime, we have to find the
roots of $f(r_{+})=0$. However, due to the complexity of $f(r)$ given in (28),
it is not possible to find the roots of $f(r_{+})=0$, analytically.
Nevertheless, we can plot the function $f(r)$ versus $r$ for different model
parameters as in figures 5 and 6. For simplicity, in these figures, we kept
fixed the other parameters $b=l=1$. Figure 5 shows that the obtained solutions
may represent a black hole with two horizons, an extreme black hole or a naked
singularity depending on the metric parameters. It also shows that for fixed
value of the other parameters, the number of horizons decreases with
increasing $\alpha$. On the other hand from figure 6 we see that for fixed
value of $m$, $\alpha$ and $q$, there is a minimum ($\beta_{\rm min}$) and
extreme $(\beta_{\rm ext})$ value for the nonlinear parameter for which we
have black hole with a non-extreme horizon provided $\beta\leq\beta_{\rm
min}$, black hole with two horizons for $\beta_{\rm min}<\beta<\beta_{\rm
ext}$, black hole with an extreme horizon for $\beta=\beta_{\rm ext}$ and
naked singularity for $\beta>\beta_{\rm ext}$. Clearly, $\beta_{\rm min}$ and
$\beta_{\rm ext}$ depend on the other parameters of the model. It is worth
mentioning that $\beta_{\rm ext}$ is the value of $\beta$ in which the two
horizons meet and our black hole has only one horizon. In other words,
$f(r)=0$ has a degenerate solution. This correspond to green curve in figure
6. Besides in this figure, the red curve corresponds to $\beta\leq\beta_{\rm
min}$, the blue curve shows the case $\beta_{\rm min}<\beta<\beta_{\rm ext}$,
and the purple curve indicates the case with $\beta>\beta_{\rm ext}$. On the
other hand, since $\beta_{\rm ext}$ corresponds to the minimum value of
$f(r)$, thus in this case $f^{\prime}(r)\mid_{r=r_{\rm ext}}=0$, which implies
that the surface gravity should be vanished.

Figure 5:  $f(r)$ versus $r$ for $q=2$, $n=6$ and $m=2.5$.

Figure 6: $f(r)$ versus $r$ for $q=2$, $m=2.5$ and $n=7$.

Figure 7: The mass parameter $m$ versus $r_{+}$ for different $\alpha$ and
$n=8$ and $q=1$.

Figure 8: The mass parameter $m$ versus $r_{+}$ for different value of the
charge parameter $q$. Here we have taken $\alpha=0.5$, $n=8$ and $\beta=1$.

Figure 9: The mass parameter $m$ versus $r_{h}$ for $\alpha=0.6$, $n=8$,
$\beta=2$ and $q=1$. $m<m_{\rm ext}$ (red line), $m=m_{\rm ext}$ (blue line),
$m>m_{\rm ext}$ (purple line).

The nature of the horizons can be further understood if we plot the mass
parameter $m$ as a function of the horizon radius $r_{h}$ for different model
parameters. Solving Eq. $f(r_{h})=0$, for the mass parameter yields

$\displaystyle m(r_{h})$ $\displaystyle=$
$\displaystyle-\frac{(n-3)({\alpha}^{2}+1)^{2}}{({\alpha}^{2}+n-3)({\alpha}^{2}-1)}{b}^{-\gamma}{r_{h}}^{2\gamma+n-3-n\gamma/2}+\frac{4(\Lambda+2\beta^{2})\left({\alpha}^{2}+1\right)^{2}{b}^{\gamma}}{(2n-4)(\alpha^{2}-n+1)}r_{h}^{n-1-{n\gamma}/{2}}$
(40) $\displaystyle+\frac{4\beta
q{(\alpha^{2}+1)^{2}}{b^{(4-n)\gamma/2}}}{n-2}{\left(\frac{\beta^{2}b^{(n-2)\gamma}}{q^{2}}\right)}^{\frac{1-\gamma}{(n-2)(\gamma-2)}}{L_{W}^{\frac{\alpha^{2}-n+1}{2n-4}}(\eta_{h})}$
$\displaystyle\times\Bigg{\\{}\frac{L_{W}^{2}(\eta_{h})}{\alpha^{2}+3n-7}\digamma\left(\frac{\alpha^{2}+3n-7}{2n-4},\frac{\alpha^{2}+5n-11}{2n-4},\frac{\alpha^{2}-1}{2n-4}L_{W}(\eta_{h})\right)$
$\displaystyle-\frac{1}{\alpha^{2}-n+1}\digamma\left(\frac{\alpha^{2}-n+1}{2n-4},\frac{\alpha^{2}+n-3}{2n-4},\frac{\alpha^{2}-1}{2n-4}L_{W}(\eta_{h})\right)\Bigg{\\}},$

where $\eta_{h}=\eta(r=r_{h})$. Figures 7, 8 and 9 show that for fixed value
of other parameters, the value of $m$ determines the number of horizons. For
simplicity in these figure we set $l=b=1$. From figure 9 we see that if we
solve Eq. $m=\rm const.$ for $r_{h}$, we can distinguish three cases depending
on the value of $m$. For $m>m_{\rm ext}$, there exist two value for
$r_{h}=r_{\pm}$ and thus we have two horizons, for $m=m_{\rm ext}$ the two
horizons meet. In this case we encounter an extremal black hole with zero
temperature. As we will show in the next section the extremal black hole with
mass $m_{\rm ext}$ and degenerate horizon has zero temperature. Besides, for
$m<m_{\rm ext}$ there is no horizon. Furthermore, from figure 8 we see that in
the limit $r_{h}\rightarrow 0$ we have a nonzero value for the mass parameter
$m$. This is in contrast to the Schwarzschild black holes in which the mass
parameter goes to zero as $r_{h}\rightarrow 0$. This is due to the effect of
the nonlinearity of the electrodynamic field and in case of $q=0$, the mass
parameter $m$ goes to zero as $r_{h}\rightarrow 0$. Physically
$r_{h}\rightarrow 0$ means that the radius of the horizon becomes very small.
Since the horizon radius depends on the solutions parameters such as $m$, thus
we have plotted this behavior in figure 8. For the simple Schwarzschild black
hole $m=\frac{r_{h}}{2}$ and so as $r_{h}\rightarrow 0$ we have $m\rightarrow
0$.

##  IV Thermodynamics of dilaton black holes

The Hawking temperature of the black hole on the outer horizon $r_{+}$, may be
obtained through the use of the definition of surface gravity HendiJHEP

$\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle\frac{\kappa}{2\pi}=\frac{1}{2\pi}\sqrt{-\frac{1}{2}(\nabla_{\mu}\chi_{\nu})(\nabla^{\mu}\chi^{\nu})},$
(41)

where $\kappa$ is the surface gravity and $\chi=\partial/\partial t$ is the
null killing vector of the horizon. Taking $\chi^{\nu}=(-1,0,0,...)$, we have
$\chi_{\nu}=(f(r_{+}),0,0,...)$ and hence
$(\nabla_{\mu}\chi_{\nu})(\nabla^{\mu}\chi^{\nu})=-\frac{1}{2}\left[f^{\prime}(r_{+})\right]^{2}$
which leads to

$\displaystyle\kappa=\sqrt{-\frac{1}{2}(\nabla_{\mu}\chi_{\nu})(\nabla_{\mu}\chi_{\nu})}=\frac{1}{2}\left(\frac{df(r)}{dr}\right)_{r=r_{+}}.$
(42)

Thus, the temperature is obtained as

$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{f^{\text{
}^{\prime}}(r_{+})}{4\pi}=-\frac{(\alpha^{2}+1)}{4\pi}{r_{+}^{1-\gamma}}\Bigg{\\{}\frac{(n-3)b^{-\gamma}r_{+}^{2\gamma-2}}{\alpha^{2}-1}+\frac{2(\Lambda+2\beta^{2})b^{\gamma}}{n-2}$
(43) $\displaystyle-\frac{4\beta
q}{n-2}r_{+}^{{(n-2)(\gamma-2)}/{2}}{b^{{(4-n)\gamma}/{2}}}\left(\frac{1}{\sqrt{L_{W}(\eta_{+})}}-\sqrt{L_{W}(\eta_{+})}\right)\Bigg{\\}},$

where $\eta_{+}=\eta(r=r_{+})$ and we have used $f(r_{+})=0$ for deleting $m$.
The behavior of $T$ versus $r_{+}$ is shown in figures 10 and 11. From these
figures we find out that, for large value of $r_{+}$, the temperature tends to
a constant independent of the model parameters, while for small values of
$r_{+}$, the temperature may be negative ($T<0$). In this case we encounter a
naked singularity. On the other hand, for an extremal black hole the
temperature is zero and the horizon is degenerate. In this case $r_{\rm ext}$
is the positive root of the following equation:

$\displaystyle\frac{(n-3)b^{-\gamma}r_{\rm
ext}^{2\gamma-2}}{\alpha^{2}-1}+\frac{2(\Lambda+2\beta^{2})b^{\gamma}}{n-2}$
$\displaystyle-\frac{4\beta q_{\rm ext}}{n-2}r_{\rm
ext}^{{(n-2)(\gamma-2)}/{2}}{b^{{(4-n)\gamma}/{2}}}\left(\frac{1}{\sqrt{L_{W}(\eta_{\rm
ext})}}-\sqrt{L_{W}(\eta_{\rm ext})}\right)=0.$ (44)

where

$\eta_{\rm ext}\equiv\frac{q_{\rm ext}^{2}r_{\rm
ext}^{(n-2)(\gamma-2)}}{\beta^{2}b^{(n-2)\gamma}}.$ (45)

Figure 10:  $T$ versus $r_{+}$ for different values of $\alpha$ parameter.
Here we take $l=b=1$, $\beta=2$, $n=5$ and $q=1$.

Figure 11: $T$ versus $r_{+}$ for different values of charge parameter $q$.
Here we take $l=b=1$, $\alpha=0.8$, $n=6$ and $\beta=2$.

From figures 10 and 11, we see that $r_{\rm ext}$ decreases as $\alpha$
increases, while $r_{\rm ext}$ increases with increasing $q$. Indeed, the
metric of Eqs. (15) and (28) can describe a nonlinear dilaton black hole with
inner and outer event horizons located at $r_{-}$ and $r_{+}$, provided
$r>r_{\mathrm{ext}}$, an extreme ENd black hole in the case of
$r=r_{\mathrm{ext}}$, and a naked singularity if $r<r_{\mathrm{ext}}$. Note
that in the limiting case where $\beta\rightarrow\infty$, expression (43)
reduces to the temperature of higher dimensional EMd black holes Shey3 ,

$\displaystyle T_{+}$ $\displaystyle=$
$\displaystyle-\frac{b^{-\gamma}(\alpha^{2}+1)(n-3)}{4\pi(\alpha^{2}-1)}r_{+}^{\gamma-1}-\frac{\Lambda(\alpha^{2}+1)b^{\gamma}}{2(n-2)\pi}r_{+}^{1-\gamma}-\frac{q^{2}b^{-\gamma(n-3)}(\alpha^{2}+1)}{2\pi(n-2)}r_{+}^{(\gamma{n}+5-3\gamma-2n)}+O\left(\frac{1}{\beta^{2}}\right).$
(46)

The entropy of the ENd black hole still obeys the so called area law of the
entropy which states that the entropy of the black hole is a quarter of the
event horizon area Beck . This near universal law applies to almost all kinds
of black holes, including dilaton black holes, in Einstein gravity hunt . It
is easy to show

${S}=\frac{b^{(n-2)\gamma/2}r_{+}^{(n-2)(1-\gamma/2)}\omega_{n-2}}{4}.$ (47)

The gauge potential $A_{t}$ corresponding to the electromagnetic field (33)
can be obtained through relation
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Since our solution
is static, thus the gauge potential is only a function of $r$. From
$F_{tr}=\partial_{t}A_{r}-\partial_{r}A_{t}$ with $\partial_{t}A_{r}=0$ we
have

$F_{tr}+\partial_{r}A_{t}(r)=0,$ (48)

and hence the gauge potential $A_{t}$ can be derived as

$A_{t}(r)=-\int{F_{tr}dr}=-\beta
b^{\gamma}\int{drr^{-\gamma}\sqrt{L_{W}(\eta)}}$ (49)

where $\eta$ is defied in (26). Integrating yields

$\displaystyle A_{t}$ $\displaystyle=$ $\displaystyle
b^{\gamma}\beta(\alpha^{2}+1)\left(\frac{\beta^{2}b^{(n-2)\gamma}}{q^{2}}\right)^{\frac{\alpha^{2}-1}{2n-4}}\left(\frac{\alpha^{2}-1}{4-2n}\right)^{\frac{\alpha^{2}+n-3}{4-2n}}$
(50)
$\displaystyle\times\Bigg{\\{}-\frac{1}{2(n-2)}\Gamma\left(\frac{\alpha^{2}+n-3}{2n-4},\frac{1-\alpha^{2}}{2n-4}L_{W}(\eta)\right)$
$\displaystyle+\frac{1}{\alpha^{2}-1}\left[\Gamma\left(\frac{\alpha^{2}+3n-7}{2n-4},\frac{1-\alpha^{2}}{(2n-4)}L_{W}(\eta)\right)-\frac{1}{2}\Gamma\left(\frac{\alpha^{2}+n-3}{2n-4}\right)\right]\Bigg{\\}}.$

The electric potential $U$, measured at infinity with respect to the horizon,
is defined by DHSR

$U=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(51)

where $\chi=\partial_{t}$ is the null generator of the horizon. It is a matter
of calculation to show that

$\displaystyle U$ $\displaystyle=$ $\displaystyle
b^{\gamma}\beta(\alpha^{2}+1)\left(\frac{\beta^{2}b^{(n-2)\gamma}}{q^{2}}\right)^{\frac{\alpha^{2}-1}{2n-4}}\left(\frac{\alpha^{2}-1}{4-2n}\right)^{\frac{\alpha^{2}+n-3}{4-2n}}$
(52)
$\displaystyle\times\Bigg{\\{}-\frac{1}{2(n-2)}\Gamma\left(\frac{\alpha^{2}+n-3}{2n-4},\frac{1-\alpha^{2}}{2n-4}L_{W}(\eta_{+})\right)$
$\displaystyle+\frac{1}{\alpha^{2}-1}\left[\Gamma\left(\frac{\alpha^{2}+3n-7}{2n-4},\frac{1-\alpha^{2}}{2n-4}L_{W}(\eta_{+})\right)-\frac{1}{2}\Gamma\left(\frac{\alpha^{2}+n-3}{2n-4}\right)\right]\Bigg{\\}}.$

Expanding for large value of $\beta$, we get

$\displaystyle
U=\frac{q(\alpha^{2}+1)b^{(4-n)\gamma/2}}{(\alpha^{2}+n-3)r_{+}^{n-3+2\gamma-n\gamma/2}}+O\left(\frac{1}{\beta^{2}}\right).$
(53)

We have shown the the behavior of the electric potential $U$ as a function of
horizon radius $r_{+}$ in figures 12 and 13 for $b=1$. Due to the nature of
the nonlinear electrodynamics, the electric potential can be finite as
$r_{+}\rightarrow 0$, depending on the model parameters, and goes to zero for
large $r_{+}$ independent of the model parameters. From these figures we find
that for fixed value of other parameter, the divergency of $U$, for small
$r_{+}$, increases with increasing $\alpha$ and $\beta$.

Figure 12: The behaviour of $U(r_{+})$ versus $r_{+}$ for $\alpha=0.5$,
$\beta=2$ and $n=6$ and different $q$.

Figure 13:  $U(r_{+})$ versus $r_{+}$ for $\beta=2$, $n=8$ and $q=1$ and
different $\alpha$.

There are several ways for calculating the mass of the black holes. For
example, for asymptotically AdS solution one can use the conterterm method
inspired by (A)dS/CFT correspondence Mal ; Haro . Another way for calculating
the mass is through the use of the substraction method of Brown and York BY .
Such a procedure causes the resulting physical quantities to depend on the
choice of reference background. In our case, due to the presence of the non-
trivial dilaton field, the asymptotic behaviour of the solutions are neither
flat nor (A)dS, therefore we have used the reference background metric and
calculate the mass. According to the substraction method of BY , if we write
the metric of $n$-dimensional static and spherically symmetric spacetime in
the form CHM

$ds^{2}=-W^{2}(r)dt^{2}+{dr^{2}\over V^{2}(r)}+r^{2}d\Omega_{n-2}^{2},$ (54)

and the matter action contains no derivatives of the metric, then the
quasilocal mass is given by CHM

${\cal M}=\frac{n-2}{2}r^{n-3}W(r)\left(V_{0}(r)-V(r)\right).$ (55)

Here $V_{0}(r)$ is an arbitrary function which determines the zero of the
energy for a background spacetime and $r$ is the radius of the spacelike
hypersurface boundary. It was argued that the $ADM$ mass $M$ is the ${\cal M}$
determined in (55) in the limit $r\rightarrow\infty$ CHM . Transforming metric
(15) in the form (54), the mass of the solution is obtained as

${M}=\frac{b^{(n-2)\gamma/2}(n-2)\omega_{n-2}}{16\pi(\alpha^{2}+1)}m.$ (56)

Next, we want to check the first law of thermodynamics for $n$-dimensional ENd
black holes. For this purpose, we first obtain the mass $M$ as a function of
extensive quantities $S$ and $Q$. Combining expressions for the charge, the
mass and the entropy given in Eqs. (18), (56) and (47), and using the fact
that $f(r_{+})=0$, we obtain a Smarr-type formula as

$\displaystyle M(S,Q)$ $\displaystyle=$
$\displaystyle-\frac{b^{-\alpha^{2}}{(n-2)(n-3)(\alpha^{2}+1)}{\left(4S\right)}^{\frac{\alpha^{2}+n-3}{n-2}}}{16\pi(\alpha^{2}+n-3)(\alpha^{2}-1)}+\frac{(\alpha^{2}+1)b^{\alpha^{2}}{(\Lambda+2\beta^{2})}}{8\pi(\alpha^{2}-n+1)}{\left(4S\right)}^{\frac{-\alpha^{2}+n-1}{n-2}}$
(57) $\displaystyle+\frac{b^{\gamma}\beta
Q}{(2n-4)}\frac{(\alpha^{2}+1)^{3}}{(\alpha^{2}-1)^{2}}{\left(\frac{\beta^{2}b^{(n-2)\gamma}}{16\pi^{2}Q^{2}}\right)}^{\frac{\alpha^{2}-1}{2n-4}}\left(\frac{1-\alpha^{2}}{2n-4}\right)^{\frac{\alpha^{2}+1-n}{4-2n}}$
$\displaystyle\times\Bigg{\\{}-(n-2)^{2}(\gamma-2)^{2}\left[\Gamma\left(\frac{\alpha^{2}+3n-7}{2n-4},\frac{1-\alpha^{2}}{2n-4}L_{W}\left(\zeta\right)\right)-\Gamma\left(\frac{\alpha^{2}+3n-7}{2n-4}\right)\right]$
$\displaystyle+(\gamma-1)^{2}\left[\Gamma\left(\frac{\alpha^{2}-n+1}{2n-4},\frac{1-\alpha^{2}}{2n-4}L_{W}\left(\zeta\right)\right)-\Gamma\left({\frac{\alpha^{2}-n+1}{2n-4}}\right)\right]\Bigg{\\}},$

where $\zeta=\frac{\pi^{2}Q^{2}}{S^{2}\beta^{2}}$. If we expand $M(S,Q)$ for
large $\beta$, we arrive at

$\displaystyle M(S,Q)$ $\displaystyle=$
$\displaystyle-\frac{b^{-\alpha^{2}}(n-2)(n-3)(\alpha^{2}+1){\left(4S\right)}^{\frac{\alpha^{2}+n-3}{n-2}}}{16\pi(\alpha^{2}+n-3)(\alpha^{2}-1)}+\frac{\Lambda(\alpha^{2}+1)b^{\alpha^{2}}}{8\pi(\alpha^{2}-n+1)}{\left(4S\right)}^{\frac{-\alpha^{2}+n-1}{n-2}}$
(58)
$\displaystyle+\frac{2\pi(\alpha^{2}+1)Q^{2}b^{\alpha^{2}}}{\alpha^{2}+n-3}(4S)^{\frac{\alpha^{2}+n-3}{2-n}}+O\left(\frac{1}{\beta^{2}}\right),$

which is exactly the Smarr-type formula obtained for EMd black in the limit
$\beta\rightarrow\infty$ Shey3 . Now, if we consider $S$ and $Q$ as a complete
set of extensive parameters for the mass $M(S,Q)$, we can define the intensive
parameters conjugate to $S$ and $Q$ as

$T=\left(\frac{\partial M}{\partial S}\right)_{Q},\ \ U=\left(\frac{\partial
M}{\partial Q}\right)_{S}.$ (59)

Numerical calculations show that the intensive quantities calculated by Eq.
(59) coincide with Eqs. (43) and (52). Thus, these thermodynamic quantities
satisfy the first law of black hole thermodynamics

$dM=TdS+Ud{Q}.$ (60)

The satisfaction of the first law of thermodynamics for the obtained conserved
and thermodynamic quantities, together with the fact that these quantities in
two limiting cases, namely in the absence of the dilaton field
($\alpha=0=\gamma$), and for large values of the nonlinear parameter
($\beta\rightarrow\infty$), reduce to the known results in the literature
Hendi2 ; Shey3 , indicate that the conserved and thermodynamic quantities
obtained in this paper are correct and in agreement with other method such as
Euclidean action method Mis .

##  V conclusion and discussion

In this paper, we generalized the study on the EN electrodynamics by taking
into account the dilaton scalar field in the action. We first proposed the
suitable Lagrangian for EN electrodynamics coupled to the dilaton field and in
the presence of two Liouville-type dilaton potential for the dilaton field in
all higher dimensions. By varying the action we obtained the field equations
of $n$-dimensional EN electrodynamics coupled to dilaton field in Einstein
gravity. Then, we constructed a new class of higher dimensional static and
spherically symmetric black hole solutions of this theory. When
$\beta\rightarrow\infty$, our solutions reduce to higher dimensional EMd black
hole solutions Shey3 , while in the absence of the dilaton field,
($\alpha=0=\gamma$), they restore charged black holes coupled to EN
electrodynamics. Although the behavior of the electric field near the origin
depends on the model parameters, however for large $r$ the asymptotic behavior
of electric field are exactly the same as linear Maxwell field. Interestingly
enough, we found that the electric field of ENd black hole is finite near the
origin and diverges exactly at $r=0$ depending on the model parameters,
however its divergency is much weaker than Maxwell field. Besides, in the
absence of the dilaton field ($\alpha=0$), the electric field has a finite
value near $r=0$, while as soon as the dilaton field is taken into account
($\alpha>0$), the electric field diverges as $r\rightarrow 0$. This implies
that the presence of the dilaton field changes the behaviour of the electric
field near the origin where $r=0$.

We also found that the dilaton field changes the asymptotic behavior of the
solutions to be neither flat nor (A)dS. This is consistent with the fact that
no dilaton dS or AdS black hole solution exists with the presence of only one
or two Liouville-type dilaton potential MW . The obtained solutions can
represent black holes with inner and outer horizons, an extreme black hole or
naked singularity depending on the model parameters. For fixed value of $m$,
$\alpha$ and $q$, we found that there is a minimum (extreme) value for
nonlinear parameter $\beta_{\rm min}$ $(\beta_{\rm ext})$, for which we have
black hole with a non-extreme horizon provided $\beta\leq\beta_{\rm min}$,
black hole with two horizons for $\beta_{\rm min}<\beta<\beta_{\rm ext}$,
black hole with an extreme horizon for $\beta=\beta_{\rm ext}$ and naked
singularity for $\beta>\beta_{\rm ext}$. We computed conserved and
thermodynamic quantities and obtained Smarr-type formula,$M(S,Q)$. We checked
that the conserved and thermodynamic quantities obtained for higher
dimensional dilaton black holes satisfy the first law of black holes
thermodynamics on the horizon.

Let us emphasize that the higher dimensional dilaton black holes coupled to EN
electrodynamics constructed in this paper are static. Thus, it would be
interesting to derive $n$-dimensional rotating black hole/brane solutions of
these field equations. The study can also be applied for other type of
nonlinear electrodynamics such as logarithmic nonlinear electrodynamics with
suitable lagrangian in the presence of dilaton field. It is also intersecting
to study stability of the solutions. These issues are now under investigations
and the results will be appeared elsewhere.

###### Acknowledgements.

A. Sheykhi thanks Shiraz University Research Council. This work has been
supported financially by Center for Excellence in Astronomy and Astrophysics
of IRAN (CEAAI), Maragheh, IRAN.

## References

  * (1) A. Strominger and C. Vafa, Phys. Lett. B 379, 99 (1996).
  * (2) O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, Phys. Rept. 323, 183 (2000).
  * (3) M. Cavaglia, Int. J. Mod. Phys. A 18, 1843 (2003);
P. Kanti, Int. J. Mod. Phys. A 19, 4899 (2004).

  * (4) R. Emparan and H. S. Reall, Living Rev. Rel. bf11, 6 (2008).
  * (5) M. Born and L. Infeld, Proc. R. Soc. A 144, 425 (1934).
  * (6) E. Fradkin and A. Tseytlin, Phys. Lett. B 163, 123 (1985);
R. Matsaev, M. Rahmanov and A. Tseytlin, Phys. Lett. B 193, 205 (1987);
E. Bergshoeff, E. Sezgin, C. Pope and P. Townsend, Phys. Lett. B 188, 70
(1987).

  * (7) C. Callan, C. Lovelace, C. Nappi and S. Yost, Nucl. Phys. B 308, 221 (1988);
O. Andreev and A. Tseytlin, Nucl. Phys. B 311, 221 (1988);
R. Leigh, Mod. Phys. Lett. A 4, 2767 (1989).

  * (8) T. Tamaki and T. Torii, Phys. Rev. D 62, 061501 (R) (2000).
  * (9) T. Tamaki and T. Torii, Phys. Rev. D 64, 024027 (2001).
  * (10) R. Yamazaki and D.Ida, Phys. Rev. D 64, 024009 (2001).
  * (11) S. S. Yazadjiev, Phys.Rev. D 72, 044006 (2005).
  * (12) G. Clement and D. Gal’tsov, Phys. Rev. D 62, 124013 (2000).
  * (13) S. S. Yazadjiev, P. P. Fiziev, T. L. Boyadjiev, M. D. Todorov, Mod. Phys. Lett. A 16, 2143 (2001).
  * (14) A. Sheykhi, N. Riazi and M. H. Mahzoon, Phys. Rev. D 74, 044025 (2006).
  * (15) A. Sheykhi, N. Riazi, Phys. Rev. D 75, 024021 (2007);
A. Sheykhi, Int. J. Mod. Phys. D 18, 25 (2009).

  * (16) A. Sheykhi, Phys. Lett. B 662, 7 (2008).
  * (17) M. H Dehghani, S. H. Hendi, A. Sheykhi and H. Rastegar Sedehi, JCAP 0702 (2007) 020;
M. H. Dehghani, A. Sheykhi, S. H. Hendi, Phys. Lett. B 659, 476 (2008).

  * (18) S. Abdolrahimi and A. A. Shoom, Phys. Rev. D 83, 104023 2011.
  * (19) S. H. Hendi, JHEP 03, 065 (2012);
S. H. Hendi, A. Sheykhi, Phys. Rev D 88 (2013) 044044\.

  * (20) S. H. Hendi, Annals of Phys. 333, 282 (2013);
S. H. Hendi, arXiv:1405.6996

  * (21) A. Sheykhi, S. Hajkhalili, Phys. Rev. D 89, 104019 (2014).
  * (22) S. J. Poletti, D. L. Wiltshire, Phys. Rev. D 50, 7260 (1994);
S. J. Poletti, J. Twamley and D. L. Wiltshire, Phys. Rev. D 51, 5720 (1995);
S. Mignemi and D. L. Wiltshire, Phys. Rev. D 46, 1475 (1992).

  * (23) K. C. K. Chan, J. H. Horne and R. B. Mann, Nucl. Phys. B447, 441 (1995).
  * (24) R. G. Cai, J. Y. Ji and K. S. Soh, Phys. Rev D 57, 6547 (1998);
R. G. Cai and Y. Z. Zhang, ibid. 64, 104015 (2001).

  * (25) S. S. Yazadjiev, Class. Quant. Gravit. 22, 3875(2005).
  * (26) G. Clement, D. Gal’tsov and C. Leygnac, Phys. Rev. D 67, 024012 (2003);
G. Clement and C. Leygnac, ibid. 70, 084018 (2004).

  * (27) A. Sheykhi, Phys. Rev. D 76, 124025 (2007).
  * (28) M. Abramowitz and I. A. Stegun, _Handbook of Mathematical Functions_ , Dover, New York, (1972);
R. M. Corless, etal., Adv. Computational Math. 5, 329 (1996).

  * (29) J. D. Beckenstein, Phys. Rev. D 7, 2333 (1973); S. W. Hawking, Nature (London) 248, 30 (1974); G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).
  * (30) C. J. Hunter, Phys. Rev. D 59, 024009 (1999); S. W. Hawking, C. J Hunter and D. N. Page, ibid. 59, 044033 (1999); R. B. Mann ibid. 60, 104047 (1999);ibid. 61, 084013 (2000).
  * (31) J. Maldacena, Adv. Theor. Math. Phys., 2, 231 (1998); E. Witten, ibid. 2, 253 (1998); O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rep. 323,183 (2000); V. Balasubramanian and P. Kraus, Commun. Math. Phys. 208, 413 (1999).
  * (32) S. de Haro, K. Skenderis and S. N. Solodukhin, Commun. Math. Phys. 217, 595 (2001).
  * (33) J. Brown and J. York, Phys. Rev. D 47, 1407 (1993);
J.D. Brown, J. Creighton, R.B. Mann, Phys. Rev. D 50, 6394 (1994).

  * (34) O. Miskovic and R. Olea, Phys. Rev. D 77, 124048 (2008).