# Charged and radially excited boson stars (in Anti-de Sitter)

Yves Brihaye<EMAIL_ADDRESS>Service de Physique de l’Univers,
Champs et Gravitation, Université de Mons, Mons, Belgium  Felipe Console
<EMAIL_ADDRESS>Instituto de Física de São Carlos, Universidade de São
Paulo, São Carlos, São Paulo 13560-970, Brazil  Betti Hartmann
<EMAIL_ADDRESS>Department of Mathematics, University College London,
Gower Street, London, WC1E 6BT, UK

###### Abstract

We study charged and radially excited boson stars in both asymptotically flat
as well as asymptotically Anti-de Sitter space-time. We demonstrate that two
different types of radially excited solutions exist : one that persists in the
linear limit of small scalar fields, in which analytical arguments suggest the
existence of these solutions, and one that appears only in the highly non-
linear regime of the model. We also demonstrate that the formation of wavy
scalar hair discussed previously for black holes and boson stars in
asymptotically flat space-time persists for asymptotically Anti-de Sitter
space-time.

##  I Introduction

Anti-de Sitter (AdS) – together with Minkowski and de Sitter (dS) – space-time
plays an important role amongst known exact solutions of the Einstein
equation. All three space-times are maximally symmetric, i.e. admit the
largest possible amount of Killing vectors. Minkowski, dS and AdS possess
constant vanishing, positive and negative curvature, respectively, with the
Ricci scalar being proportional to the cosmological constant. Amongst these
three space-times, AdS is special as it possesses a conformal boundary, i.e. a
boundary that is in causal contact with the interior of the space-time. As
such, physics in the interior can be connected to (albeit different) physics
on the boundary. The Anti-de Sitter/Conformal Field Theory (AdS/CFT)
correspondence Maldacena:1997re  (or more generally the gauge-gravity
correspondence gauge_gravity ) as a prediction rooted in String Theory uses
exactly this idea: a gravity theory in $d$-dimensional AdS is dual to a
conformal field theory on the $(d-1)$-dimensional boundary of AdS. The duality
also predicts that there exists a weak-strong coupling duality between the
theories in the bulk and on the boundary. Consequently, solutions to (weakly
coupled) gravitational theories in $d$-dimensional asymptotically AdS (aAdS)
have been used to make predictions about strongly coupled field theories in
$(d-1)$ dimensions. One of the main examples that has been widely studied are
models for high-temperature superconductivity hhh . Using bulk black holes
allows to study the temperature dependence of the phase transition’s order
parameter, while phase transitions can also be modelled using solitonic
objects Horowitz:2010jq . The latter have no temperature ascribed to them, but
phase transitions appear e.g. when varying the chemical potential. Most
discussions of these so-called holographic superconductors have been done
using a scalar field in the bulk and consequently interpreting the value of
this scalar field on the conformal boundary as the expectation value of an
operator in the dual field theory with dimension equal to the power of the
fall-off of the scalar field in aAdS. In the language of condensed matter,
these superconductors would correspond to $s$-wave superconductors.

Scalar fields are well motivated in Particle Physics models as well as
effective descriptions of collective phenomena. $Q$-balls are lumps of complex
scalar field carrying a conserved Noether charge that results from the
invariance of the model under a U(1) gauge transformation Coleman . They
possess a scalar field that oscillates with constant frequency such that the
energy-momentum tensor (and consequently the space-time) stays static. These
objects appear e.g. in supersymmetric extensions of the Standard Model Kusenko
and need specific scalar field potentials (with attractive and repulsive
terms) in order to exist. In Copeland  an effective exponential potential that
appears in gauge-mediated supersymmetric breaking has been discussed in this
context. Boson stars Kaup ; Friedberg ; Jetzer ; Schunck  are the curved
space-time equivalent of $Q$-balls. Other than their flat space-time
counterparts, they do not require a higher order scalar field potential, but
exist for a non-self-interacting and massive scalar field. Boson stars also
exist in aAdS space-time radu , but now – in contrast to the asymptotically
flat space-time case – are no longer exponentially localized, but possess a
power-law fall-off for the scalar field function. The power of this fall-off
is determined by the mass and the angular frequency of the scalar field. The
value of the scalar field on the conformal boundary can then be interpreted as
the expectation value of a dual operator in the corresponding boundary field
theory, e.g. as a glueball condensate Hartmann_Riedel .

Interesting new phenomena appear when gauging the U(1) symmetry, i.e. when
electrically charging the boson star. This has been studied for a 4th-order
scalar field potential cbs1 , a $V$-shaped potential cbs2 ; cbs3 , an
exponential potential cbs4 , a 6th order potential gauged_BS_power_law  as
well as for a massive scalar field with no self-interaction cbs5 ,
respectively. When the gauge interaction becomes strong, charged boson stars
cease to exist, which is related to the fact that the individual constituents,
the scalar bosons, that make up the star are now also electrically charged and
hence the electric repulsive force becomes important.

In this paper, we study charged boson stars in asymptotically flat and AdS
space-time, respectively. In the former case, which has been discussed in the
literature previously, we show that radially excited charged boson stars exist
and that these have very different qualitative properties as compared to the
unexcited boson stars. In particular, we demonstrate that two disconnected
branches of solutions exist.

##  II Set-up

We consider a U(1)-charged complex scalar field, $\Psi$, minimally coupled to
gravity with negative cosmological constant $\Lambda$ and a scalar field
potential $\mathcal{U}$. The action is given by

$S=\int d^{4}x\sqrt{-g}\left(\frac{\mathcal{R}}{16\pi G}-\frac{\Lambda}{8\pi
G}-\left(D_{\mu}\Psi\right)(D^{\mu}\Psi)^{*}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\mathcal{U}\left(|\Psi|^{2}\right)\
\right)$ (1)

where $\mathcal{R}$ is the Ricci scalar, $G$ is the Newton gravitational
constant, $D_{\mu}=\nabla_{\mu}+iqA_{\mu}$ is the gauge covariant derivative,
$A_{\mu}$ the U(1) gauge field,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ the field strenght
tensor and $q$ the scalar charge.

The equations of motion are obtained by varying the action with respect to
metric and the matter fields and read, respectively :

$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}\hskip 5.69046pt,\hskip
28.45274pt\nabla_{\mu}F^{\nu\mu}=iq\left(\Psi^{*}\left(D^{\nu}\Psi\right)-\Psi\left(D^{\nu}\Psi\right)^{*}\right)\equiv
qj^{\nu}\hskip 5.69046pt,\hskip
28.45274ptD_{\mu}D^{\mu}\Psi=\frac{d\mathcal{U}}{d|\Psi|^{2}}\Psi\hskip
5.69046pt.$ (2)

$G_{\mu\nu}$ is the Einstein tensor and the energy-momentum tensor,
$T_{\mu\nu}$, can be written as the sum of the energy momentum-tensor for the
scalar field, $T_{\mu\nu}^{(\Psi)}$, and for the electromagnetic field,
$T_{\mu\nu}^{(EM)}$ as follows :

$T_{\mu\nu}=T_{\mu\nu}^{(\Psi)}+T_{\mu\nu}^{(EM)}$ (3)

with

$T_{\mu\nu}^{(\Psi)}=\left(D_{\mu}\Psi\right)\left(D_{\nu}\Psi\right)^{*}+\left(D_{\mu}\Psi\right)^{*}\left(D_{\nu}\Psi\right)-g_{\mu\nu}\left(\left(D_{\alpha}\Psi\right)\left(D^{\alpha}\Psi\right)^{*}+\mathcal{U}\right)\hskip
5.69046pt,\hskip 28.45274ptT_{\mu\nu}^{(EM)}=F_{\mu\alpha}F_{\nu}^{\hskip
5.69046pt\alpha}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\hskip
5.69046pt.$ (4)

We consider a scalar self-interacting potential of the form :

$\mathcal{U}\left(|\Psi|^{2}\right)=\mu^{2}\eta^{2}\left(1-\exp\left(-\frac{|\Psi|^{2}}{\eta^{2}}\right)\right)\hskip
5.69046pt,$ (5)

where $\mu$ is the mass of the boson and $\eta$ is an energy scale. This
potential has been considered previously in the construction of Q-balls and
boson stars in AdS Copeland ; Hartmann_Riedel .

In the following the space-time is chosen to be static and spherically
symmetric with adapted coordinates $\\{t,r,\theta,\varphi\\}$ such that
$\xi=\partial_{t}$ and $\eta=\partial_{\varphi}$ are a timelike and spacelike
Killing vector, respectively. On the other hand, the Ansatz for the matter
fields is non-static since, although static solutions are not forbidden by
Derrick’s theorem when gauge fields are involved, static gauge fields with
vanishing $A_{t}$ component lead to magnetic fields only and these cannot be
spherically symmetric in Abelian gauge field theory (there are no monopole
terms). However, we will choose an Ansatz that will lead to a static energy-
momentum tensor to be consistent with a static space-time. The Ansatz for the
metric, scalar field $\Psi$ and U(1) gauge field $A_{\mu}$ then reads :

${\rm d}s^{2}=-N(r)\sigma(r)^{2}{\rm d}t^{2}+\frac{1}{N(r)}{\rm
d}r^{2}+r^{2}{\rm d}\theta^{2}+r^{2}\sin^{2}\theta{\rm d}\varphi^{2}\hskip
2.84544pt,\hskip 11.38092pt\Psi=\Psi(t,r)=e^{-i\omega
t}\psi\left(r\right)\hskip 2.84544pt,\hskip 11.38092ptA_{\mu}{\rm
d}x^{\mu}=V(r){\rm d}t\hskip 5.69046pt,$ (6)

where we will use $N(r)=1-2m(r)/r$ with $m(r)$ a mass function. Replacing the
above Ansatz into (2) we obtain a system of four coupled non-linear ordinary
differential equations, which read :

$m^{\prime}=\frac{1}{2}r^{2}\Lambda+4\pi
Gr^{2}\left(\mathcal{U}+\frac{\left(\omega-
qV\right)^{2}}{N\sigma^{2}}\psi^{2}+\frac{(V^{\prime})^{2}}{2\sigma^{2}}+N(\psi^{\prime})^{2}\right)\hskip
28.45274pt\sigma^{\prime}=\frac{8\pi Gr}{N^{2}\sigma}\left(\left(\omega-
qV\right)^{2}\psi^{2}+N^{2}\sigma^{2}\left(\psi^{\prime}\right)^{2}\right)$
(7)
$\psi^{\prime\prime}=-\left(\frac{N^{\prime}}{N}+\frac{\sigma^{\prime}}{\sigma}+\frac{2}{r}\right)\psi^{\prime}-\frac{\left(\omega-
qV\right)^{2}}{N^{2}\sigma^{2}}\psi+\frac{1}{N}\frac{d\mathcal{U}}{d|\Psi|^{2}}\psi\hskip
28.45274ptV^{\prime\prime}=-\left(\frac{2}{r}-\frac{\sigma^{\prime}}{\sigma}\right)V^{\prime}-\frac{2q\left(\omega-
qV\right)}{N}\psi^{2}\ .$ (8)

The above equations obviously depend only on the gauge invariant combination
$\omega-qV$, which also appears in the expression for the globally conserved
Noether charge that reads :

$Q_{N}=\int_{0}^{\infty}{\rm d}r\hskip
1.42271pt\frac{2r^{2}\psi^{2}}{N\sigma}\left(qV-\omega\right)\hskip
5.69046pt.$ (9)

$Q_{N}$ has often been interpreted as the number of bosonic particles of mass
$\mu$ that make up the boson star. While in the ungauged case, next to the
gravitational attraction, only the scalar field potential is responsible for
repulsion between these individual constituents, there is an additional
repulsive electric force acting in the gauged case. Hence, the properties of
electrically charged boson stars are similar to those of the ungauged stars as
long as the electric field is small, but deviate significantly when the gauge
interaction becomes important.

In order to find globally regular solutions in an asymptotically AdS space-
time (aAdS), we need to impose appropriate boundary conditions. Close to the
origin $r=0$ the equations of motion (7) and (8) imply that

$m(0)=0\hskip 2.84544pt,\hskip 28.45274pt\hskip
28.45274pt\psi^{\prime}(0)=0\hskip 2.84544pt,\hskip
28.45274ptV^{\prime}(0)=0\hskip 2.84544pt,$ (10)

while $\psi(0)\equiv\psi_{0}$, $\sigma(0)\equiv\sigma_{0}$ and $V(0)\equiv
V_{0}$ are a priori free parameters. However, we can use the gauge freedom to
fix $V(0)=0$. The values $\psi_{0}$ and $\sigma_{0}$ will, on the other hand,
be determined numerically.

Asymptotically, the leading terms of the metric functions are :

$N(r\rightarrow\infty)\sim 1-\frac{\Lambda}{3}r^{2}\hskip 2.84544pt,\hskip
28.45274pt\sigma(r\rightarrow\infty)\sim 1\hskip 5.69046pt.$ (11)

from which we can derive the behaviour of the matter field functions. This
becomes :

$V(r\rightarrow\infty)\sim\Phi-\frac{Q_{e}}{r}\ \ \ ,\ \
\psi(r\rightarrow\infty)\sim\frac{\psi_{+}}{r^{\Delta_{+}}}+\frac{\psi_{-}}{r^{\Delta_{-}}}\hskip
5.69046pt,\hskip
28.45274pt\Delta_{\pm}=\frac{3}{2}\pm\frac{1}{2}\sqrt{9-\frac{12\mu^{2}}{\Lambda}}\
\ ,$ (12)

where $\Phi$ is a constant that we will often refer to as “the chemical
potential” and $Q_{e}$ is the electric charge of the solution that can be
shown to be related to the Noether charge by $Q_{e}=eQ_{N}$. In fact, with our
choice $V(0)=0$, $\Phi$ denotes the electric potential difference between the
origin and infinity.

Note that, different from the asymptotically flat space-time case, where
$\psi$ falls of exponentially

$\psi(r\rightarrow\infty)\sim\frac{\exp\left(-\sqrt{\mu^{2}-(\omega-q\Phi)^{2}}r\right)}{r}\
\ \ \ \ \ \ \ \ \ \ (\Lambda=0)\ ,$ (13)

the scalar field now has a power-law fall-off as long as $\mu^{2}>3\Lambda/4$.
In the following, we will choose $\mu^{2}>0$, and since $\Lambda<0$, this
implies that $\Delta_{-}$ is negative and the corresponding solution to $\psi$
would diverge as $r\rightarrow\infty$. For this reason we set $\psi_{-}=0$ and
consider only the regular solution at infinity,
$\psi\sim\psi_{+}/r^{\Delta_{+}}$. $\psi_{+}$ is a constant that can then be
interpreted as the expectation value of a “condensate” in the dual description
on the conformal boundary.

Finally, the mass $M$ of the solution can be read of from the asymptotic
behaviour of the mass function $m(r)$, which behaves as radu  :

$m(r\rightarrow\infty)\sim M+\frac{\Lambda r^{3}}{6}-\frac{2\pi
GQ^{2}}{r}+n_{1}r^{-2\Delta_{+}+3}\ ,$ (14)

where $n_{1}$ is a constant depending on $\Lambda$ and we need to subtract the
diverging part for $\Lambda\neq 0$, which - more formally - would be done
using the counterterm approach.

###  II.1 Radially excited solutions and gauged oscillons

Assuming $\psi$ to be small we find the linearized scalar field equation

$\frac{{\rm d}}{{\rm d}z}\left(z^{2}f\frac{{\rm d}\psi}{{\rm
d}z}\right)=\left(\mu^{2}-\frac{\Omega^{2}}{f}\right)\ell^{2}z^{2}\psi\ \ ,\ \
f=1+z^{2}\ ,$ (15)

where we have defined $z=r/\ell$ with $\ell=\sqrt{-3/\Lambda}$ the AdS radius
and $\Omega=\omega-q\Phi$. The solutions to this equation are so-called
oscillons and have been discussed in the ungauged case before (see e.g.
Bizon:2011gg ; Dias:2012tq ; Brihaye:2014bqa ). Since the scalar field sources
the U(1) gauge field only via a current proportional to $\psi^{2}$, the gauge
field equation can be integrated to $V(r)\sim\Phi-Q_{e}/r$ in the limit of
$\psi\ll 1$. The leading term of the gauge field inserted into the scalar
field equation is then $V\sim\Phi$, i.e. $V$ equal to its chemical potential.
Now, we will introduce

$y=\frac{1}{1+z^{2}}\ \ ,\ \ \psi=y^{p}F(y)\ \ ,$ (16)

to rewrite (15). This gives a hypergeometric equation of the form

$y(y-1)\frac{{\rm d}^{2}F}{{\rm d}y^{2}}+((a+b+1)y-c)\frac{{\rm d}F}{{\rm
d}y}+abF=0$ (17)

with

$a=p-\frac{\ell\Omega}{2}\ \ ,\ \ b=p+\frac{\ell\Omega}{2}\ \ ,\ \
c=2p+1-\frac{3}{2}$ (18)

and the parameter $p$ is determined by

$4p^{2}-6p-\ell^{2}\mu^{2}=0\ \ \longrightarrow\ \
p_{\pm}=(3\pm\sqrt{9+4\ell^{2}\mu^{2}})/4\ .$ (19)

The general solution to (17) is given in terms of the hypergeometric function
$\phantom{X}{}_{2}F_{1}(a,b,c;y)$ and reads

$F(y)=C_{1}\phantom{X}{}_{2}F_{1}(a,b,c;y)+C_{2}y^{1-c}\phantom{X}_{2}F_{1}(a-c+1,b-c+12-c;y)\
,$ (20)

where $C_{i}$, $i=1,2$ are integration constants. In general, this solution is
not regular for $y\to 0$ unless $a=-k$, $k\in\mathbb{N}$. This leads to the
following requirement for $\Omega$:

$\Omega=\frac{2(p+k)}{\ell}\ \ \ ,\ \ \ k\in\mathbb{N}\ ,$ (21)

i.e. $\Omega$ is quantised. Moreover, the solutions indexed by $k$ possess $k$
zeros. These provide the gauged oscillon basis that describes the solutions to
the linearized equation (15). This oscillon basis has been discussed
previously in the case of massless scalar fields Bizon:2011gg ; Dias:2012tq
as well as including the mass of the scalar field Brihaye:2014bqa .

We hence expect that solutions with zeros in the scalar field function exist
also in the non-linear case and would interpret them as radially excited
solutions. In the following we will demonstrate that these solutions can,
indeed, be constructed numerically and agree with the analysis done above in
the linear limit. Note that in the asymptotically flat case, $\Lambda=0$,
there is no equivalent linear argument. However, as we will demonstrate below,
excited solutions do exist.

##  III Results

In order to solve the equations numerically it is convenient to rescale as
follows

$m\rightarrow\frac{m}{\mu}\hskip 2.84544pt,\hskip
22.76228pt\Omega\rightarrow\mu\Omega\hskip 2.84544pt,\hskip
22.76228pt\Lambda\rightarrow\Lambda\mu^{2}\hskip 2.84544pt,\hskip
22.76228pt\psi\rightarrow\eta\psi\hskip 2.84544pt,\hskip
22.76228ptV\rightarrow\eta V\hskip 2.84544pt,$ (22)

and define the following dimensionless quantities :

$x=\frac{r}{\mu}\hskip 2.84544pt,\hskip 22.76228pte=\frac{\eta q}{\mu}\hskip
2.84544pt,\hskip 22.76228pt\alpha=4\pi G\eta^{2}\ .$ (23)

Solitonic solutions to the model given here have been previously constructed
in the asymptotically flat case cbs4  (see also gauged_BS_power_law  for a
very similar study, albeit with a 6th order scalar potential) as well as in
the ungauged case in aAdS Hartmann_Riedel . Here, we will be interested in the
influence of the charge on asymptotically Anti-de Sitter (aAdS) boson stars as
well as radially excited solutions in both aAdS ($\Lambda<0$) and
asymptotically flat ($\Lambda=0$) space-time.

###  III.1 Unexcited solutions

As discussed above, we expect the model to possess solutions with nodes in the
scalar field function. Here, we first discuss the solutions with no nodes,
i.e. the solutions that are not radially excited and will frequently refer to
them as $k=0$ solutions.

####  III.1.1 $\Lambda=0$

Very similar to what has been discussed in gauged_BS_power_law  for a power-
law scalar field potential, we find that also for the exponential scalar field
potential three branches of solutions exist when plotting the diverse
quantities versus $\Omega=\omega-e\Phi$ and choosing $\alpha$ and $e$
appropriately. In the following, we will refer to these branches as branches
$A$, $B$ and $C$, respectively, where branch $A$ corresponds to the branch of
lowest mass solutions for a given $\Omega$ and exists in the limit $\Omega\to
1$. In this limit the scalar field $\psi(r)\equiv 0$, while the mass $M$ and
Noether charge $Q_{N}$ both approach a non-vanishing value. In Fig. 1 we show
$M$ and $Q_{N}$ (left) as well as the value of the scalar field function at
the origin, $\psi(r)$, (right) in function of $\Omega$ for $\alpha=0.0012$ and
$e=0.02$. The three branches for the unexcited solution are clearly visible
and we note that on all branches we have $M<Q_{N}$ indicating that the
solutions on all branches are stable with respect to the decay into $Q_{N}$
individual bosons of mass $\mu\equiv 1$. Moreover, we find that $M\rightarrow
Q_{N}$ in the limit $\Omega\rightarrow 1$. The dependence of $\psi(0)$ on
$\Omega$ (see Fig. 1 (right)) demonstrates that the central value of the
scalar field cannot exceed a finite value. While on branch $A$ the increase of
$\psi(0)$ is connected to an increase of the Noether charge $Q_{N}$, i.e. the
number of bosonic particles making up the boson star, this is no longer true
for branch $B$. On this branch, the number of bosonic particles increases, but
the central density decreases. This suggests that more and more bosonic
particles are pushed out from the center due to the increased electromagnetic
repulsion that the bosonic particles feel. On branch $C$ finally, the value of
$\psi(0)$ as well as the Noether charge $Q_{N}$ stay more or less constant.

Figure 1: Left: The dependence of the mass $M$ (dashed) and Noether charge
$Q_{N}$ (solid) on $\Omega$ for unexcited boson stars ($k=0$) as well as boson
stars with one node in the scalar field function ($k=1$), respectively. Here
we have chosen $\alpha=0.0012$, $e=0.02$, $\Lambda=0$. Right: The value of the
scalar field at the origin, $\psi(0)$, in function of $\Omega$ for the same
solutions. $A$, $B$, $C$ indicate the branches for $k=0$ (solid), while $a$,
$b$, $b^{\prime}$ refer to those for the $k=1$ solutions (dashed).

In order to understand this latter feature, let us discuss in more detail what
happens on branch $C$. Again, this has been discussed previously for a power-
law potential in gauged_BS_power_law . Here we show that this feature persists
for the exponential potential and point out that the interpretation of results
becomes easier when using $\omega$ instead of $\Omega$. As argued in
gauged_BS_power_law , the scalar field starts oscillating in regimes where the
square of the effective mass $m_{\rm eff}$ defined by (note that $\mu\equiv 1$
due to our rescalings (22) and (23)) :

$m_{\rm eff}^{2}(x)=\frac{1}{N(x)}-\frac{\left(\omega-
eV(x)\right)^{2}}{N^{2}(x)\sigma^{2}(x)}$ (24)

becomes negative. This is shown in Fig. 2 for $\alpha=0.0012$, $e=0.02$ and
two different values of $\omega$. We observe that when decreasing $\omega$ the
value of the minimum of the metric function $N(x)$ decreases and in the limit
$\omega\rightarrow 0$ tends to zero. We find e.g. when decreasing $\omega$
from $\omega=0.001$ to $\omega=0.0002$ that the value of the minimum of $N(x)$
decreases from $\approx 0.004$ to $\approx 0.0007$. At the same time, the
number of oscillations in the scalar field function stays constant (equal to
ten in our case), but the oscillations increase in amplitude and – at the same
time – are squeezed to a smaller interval in $x$. This is clearly related to
the fact that the effective mass $m^{2}_{\rm eff}$ is more negative.

Figure 2: We show the profiles of the metric function $N(x)$, of the scalar
field function $\psi(x)$, of the electric potential $-V(x)$ (left only) as
well as of the effective mass $m_{\rm eff}^{2}$ (see (24)) for
$\alpha=0.0012$, $e=0.02$ and $\omega=0.001$ (upper row) as well as
$\omega=0.0002$ (lower row). The right figures in each row show a zoom into
the $x$-interval where $\psi(x)$ oscillates.

####  III.1.2 $\Lambda<0$

In order to understand the influence of the charge on the solutions, we have
first set $\alpha=0$ (such that $N=1-\Lambda x^{2}/3$, $\sigma\equiv 1$) and
studied the behaviour of the solutions when varying $e$. This is shown in Fig.
3 for $e=0,\,0.5,\,1,\,1.1,$ corresponding to $\Omega\approx
1.40,\,1.47,\,1.76,\,1.89$, respectively. We observe that the scalar field
function falls off slower when increasing $e$ which means a larger radius for
the solitonic object. This is not surprising noticing that the configuration
can be interpreted as being made of $Q_{N}$ scalar and charged bosons, each
charged with $e$. Increasing $e$ hence increases the repulsion between the
individual constituents and since the space-time does not backreact, there is
no attractive force to compensate for the repulsion. Hence, we cannot make a
gauged scalar field object possess arbitrarily large scalar field values (as
is, e.g. the case in flat space-time in the ungauged case). This is
demonstrated in Fig. 4 (left), where we give the value of the scalar field at
the origin, $\psi(0)$ in dependence of $\Omega$ for $\alpha=0$, $\Lambda=-1/6$
and different values of $e$. For $e=0$, the value of $\psi(0)$ can become
arbitrarily large with the Noether charge $Q_{N}\rightarrow\infty$ (see Fig. 4
(right)). Increasing $e$ from zero, the behaviour is qualitatively different.
Solutions exist only down to a minimal value of $\Omega$, where a second
branch of solutions exists that extends backwards in $\Omega$ with increasing
values of $\psi(0)$ and $Q_{N}$. This means that the frequency $\Omega$ needs
to increase in order for the gauged boson star to be composed of more bosonic
particles. The larger $e$, the smaller is $\psi(0)$, which again can be
explained by the increased electromagnetic repulsion.

Figure 3: _Left_ : We show the scalar field function $\psi(x)$ of the
unexcited solution ($k=0)$ for $\alpha=0$, $\Lambda=-1/6$ and different values
of the gauge coupling $e$. _Right_ : The corresponding gauge field function
$-V(x)$.

Figure 4: _Left_ : We show the value of the scalar field at the origin,
$\psi(0)$, as a function of $\Omega$ for $\alpha=0$, $\Lambda=-1/6$, $k=0$ and
different values of $e$. _Right_ : The Noether charge $Q_{N}$ as function of
$\Omega$ for the same solutions.

To understand the influence of gravity, we give the values of $\psi_{0}$ and
the Noether charge $Q_{N}$ in dependence of $\Omega$ for $\alpha=0.1$ in Fig.
5. Comparing this with the results for the $\alpha=0$ case (see Fig. 4) the
interplay between the electromagnetic repulsion and gravitational attraction
becomes apparent. For $\alpha=0.1$ the curve of $\psi(0)$ as function of
$\Omega$ does not show the above mentioned back-bending behaviour until $e$ is
large enough. This is also seen when considering the Noether charge $Q_{N}$.
For $e=0$, $e=0.25$ and $e=0.4$ the curves show the typical spiraling
behaviour for uncharged boson stars. Only when the electromagnetic interaction
becomes sufficiently strong and dominates the system, does this behaviour
change. We also observe when comparing with the $\alpha=0$ case that $\psi(0)$
is much larger when backreaction of the space-time is not taken into account.
This is related to the fact that when the space-time backreacts a limit for
the mass of any gravitating object exists : $2M$ per Schwarschild radius.

The branches $A$, $B$ and $C$ discussed in the asymptotically flat case exist
as well in the $\Lambda<0$ case, see Fig. 6 (left), where we give $\psi(0)$ in
function of $\omega$ for three different values of $\Lambda$ including the
already discussed asymptotically flat case. Clearly, the branches $A$, $B$ and
$C$ are present for $\Lambda<0$ and the qualitative dependence of $\psi(0)$ on
$\omega$ does not change significantly when changing $\Lambda$. In fact, we
find that the wavy scalar hair solutions discussed above also exist in the
$\Lambda<0$ case. In Fig. 7 we show the scalar field function for
$\alpha=0.0012$, $e=0.02$, $\omega=0.001$ and $\Lambda=10^{-5}$ and compare it
with the corresponding solution for $\Lambda=0$ (see Fig. 2 (upper right)). We
observe that when changing $\Lambda$ that the first three local maxima are
slightly shifted to larger $x$, while the seven remaining maxima are slightly
shifted to smaller values of $x$, i.e. the oscillations appear in a smaller
interval of $x$. More significantly, however, we observe also that the
amplitude of the oscillations decreases when decreasing $\Lambda$ from zero.
We would expect that decreasing $\Lambda$ further, i.e. decreasing the AdS
radius, will suppress the oscillations as soon as the AdS radius becomes
comparable to the $x$-values at which these oscillations appear.

Figure 5: _Left_ : We show the value of the scalar field at the origin,
$\psi(0)$, as a function of $\Omega$ for $\alpha=0.1$, $\Lambda=-1/6$, $k=0$
and different values of $e$. _Right_ : The Noether charge $Q_{N}$ as function
of $\Omega$ for the same solutions.

Figure 6: Left: We show the dependence of $\psi(0)$ on $\omega$ for the
unexcited solution ($k=0$) and three values of $\Lambda$ indicating branches
$A$, $B$ and $C$. Right: We show the dependence of $\psi(0)$ on $\omega$ for
the first radially excited solution ($k=1$) and three values of $\Lambda$
indicating branches $a$, $b$, $b^{\prime}$ and $a^{*}$, $b^{*}$, respectively,
for three different values of $\Lambda$. For both cases we have chosen
$\alpha=0.0012$ and $e=0.02$.  Figure 7: We show the profile of the scalar
field function $\psi(x)$ for $\alpha=0.0012$, $e=0.02$, $\omega=0.001$ and
$\Lambda=10^{-5}$ (solid) and $\Lambda=0$ (dashed), respectively.

In order to make a connection to the AdS/CFT correspondence and the suggested
interpretation of the field values of solitonic solutions in aAdS on the
conformal boundary in terms of glueball condensates (see e.g. Horowitz:2010jq
; Hartmann_Riedel ), we plot the value of $\psi_{+}$ as function of $-\Phi$
for $\Lambda=-1/6$, several values of $e$ and $\alpha=0.01$ (left) and
$\alpha=0.1$ (right) in Fig. 8. This is hence a holographic phase diagram
showing the condensate in function of the chemical potential. Increasing the
backreaction of the space-time, i.e. increasing $\alpha$, changes the
qualitative behaviour of the condensate. At given fixed value of the gauge
coupling $e$, the increase of the potential difference between the origin and
the conformal boundary $V(0)-V(r\rightarrow\infty)=-\Phi$ increases the value
of the condensate up to a maximal value of both the potential difference and
the condensate value. We observe that this is true for small $\alpha$ and $e$
and find that the maximal possible value of the condensate decreases with
increasing $e$ and $\alpha$. For larger values of $\alpha$ and sufficiently
large values of $e$ we observe that at a given value of $-\Phi$ the condensate
reaches a maximum and then decreases again when increasing $-\Phi$ further. In
fact, we find that for $e$ large enough a spiraling behaviour of the
condensate value appears, i.e. we observe that for a given values of $e$ and
$\alpha$ an interval in $-\Phi$ exists on which different values of the
condensate are possible for the same value of $-\Phi$.

Figure 8: _Left_ : The value of the condensate, $\psi_{+}$, as a function of
the chemical potential, $-\Phi$, for several values of $e$, $\alpha=0.01$ and
$\Lambda=-1/6$. _Right_ : Same as left, but for $\alpha=0.1$.

###  III.2 Radially excited solutions

Following our discussion above, we would expect that radially excited boson
stars in aAdS should exist in our model. In fact, they also exist for the
$\Lambda=0$ limit and since these solutions have not been discussed in the
literature so far, we will first present our results for the asymptotically
flat case.

####  III.2.1 $\Lambda=0$

In Fig. 1 we show the mass $M$ and the Noether charge $Q_{N}$ of the radially
excited boson stars with one node in the scalar field function ($k=1$) for
$\alpha=0.0012$ and $e=0.02$ in dependence on $\Omega$. We find that the
unexcited and the one-node solutions show a qualitatively similar behaviour in
the limit $\psi(0)\to 0$, e.g. $\Omega\rightarrow 1$ for both $k=0$ and $k=1$
and the scalar field function tends uniformly to zero while the mass $M$ and
Noether charge $Q_{N}$ tend to non-vanishing values.

The behaviour, however, changes when increasing $\psi(0)$. While we also find
several branches for the radially excited case, these have very different
features as compared to the branches discussed for $k=0$. Rather than finding
branches $A$, $B$ and $C$, the excited solutions form several branches in the
form of a spiral for which we will use the labels $a$, $b$, $b^{\prime}$ (for
the first three branches) in the following. In fact, the second branch of
excited solutions has lower mass and Noether charge, respectively, as the
first, major branch that connects to $\Omega=1$. To understand this result, we
show the scalar field and the associated effective energy density
$\sqrt{-g}T^{0}_{0}=\sqrt{-g}\epsilon$ in Fig.10 (upper row) for $\Omega=0.55$
and the branches $a$ and $b$. We observe that the solutions on branch $b$ have
larger central value of the scalar field, $\psi(0)$, but that the scalar
field’s node is at smaller $x$. Moreover, we find that the effective energy
density possesses two local maxima, one close to the node of the scalar field
function. These maxima are more pronounced for solutions on branch $a$.

In Fig. 10 (middle row) we show the electric potential $V(x)$ and the electric
field $-V^{\prime}(x)$ for the same solutions. Both potential and electric
field are stronger close to the core of the boson star for the solutions on
branch $b$ as compared to those on branch $a$. Consequently, the local space-
time curvature is also stronger there, see the metric functions $N(x)$ and
$\sigma(x)$ in Fig. 10 (bottom row).

Figure 9: We compare the two radially excited solutions on branch $a$ and
branch $b$ for $k=1$, $\alpha=0.0012$, $e=0.02$, $\Lambda=0$ and
$\Omega=0.55$, see Fig.1 (right). Left: Scalar field function $\psi(x)$ and
effective energy density $\sqrt{-g}T_{0}^{0}=\sqrt{-g}\epsilon$. Right: Metric
function $N(x)$ and negative of electric field $V^{\prime}(x)$.

Interestingly, we find that next to the branch of solutions discussed above
another, completely disconnected branch exists that appears only in the highly
non-linear regime. This is shown in Fig. 6 (right) where we give the value of
$\psi(0)$ in dependence on $\omega$. To compare, we have also plotted again
the values for the branches $a$, $b$ and $b^{\prime}$. At sufficiently small
$\omega$ and sufficiently large $\psi(0)$ we find two new branches $a^{*}$ and
$b^{*}$ that are disconnected from the other branches and do not connect to
the $\omega=1$, $\psi(0)=0$ limit. In keeping with the notation used above,
the $b^{*}$ part of these new solutions has higher mass as compared to $a^{*}$
part.

Figure 10: We compare the four radially excited solutions on branches $a$,
$b$, $a^{*}$ and $b^{*}$ for $k=1$, $\alpha=0.0012$, $e=0.02$, $\Lambda=0$ and
$\Omega=0.53$, see Fig.6. Top Left: Scalar field function $\psi(x)$, Top
Right: effective energy density $\sqrt{-g}T_{0}^{0}=\sqrt{-g}\epsilon$. Middle
Left: Electric potential $-V(x)$, Middle Right: Electric field
$-V^{\prime}(x)$, Bottom Left: Metric function $N(x)$, Bottom Right: Metric
function $\sigma(x)$.

In order to understand the difference between these one-node solutions, we
show the scalar field function $\psi(x)$ as well as the effective energy
density $\sqrt{-g}\epsilon$ of the four $k=1$ solutions for $\Omega=0.53$ in
Fig. 10 (upper row), the corresponding electric potential $V(x)$ and field
$-V^{\prime}(x)$, respectively, are shown in Fig. 10 (middle row), while we
give the metric functions $N(x)$ and $\sigma(x)$, respectively, in Fig.10
(bottom row). Interestingly, branches $a$, $b$ and $a^{*}$ have the zero of
the scalar field function roughly at the same value of $x$, while the zero of
the solution on branch $b^{*}$ is at much larger values of $x$. The
corresponding maximum of the effective energy density $\sqrt{-g}\epsilon$ as
well as the minimum of the metric function $N(x)$ consequently also appear at
larger $x$ for branch $b^{*}$ as compared to the remaining branches.

####  III.2.2 $\Lambda<0$

Figure 11: (_Left_): We show the scalar field function $\psi(x)$ of the
unexcited solution ($k=0)$ and the first three radially excited solutions
($k=1$, $2$, $3$) for $\alpha=1$, $e=0.1$ and $\Lambda=-3/4$. (_Right_): The
corresponding gauge field function $-V(x)$.

Figure 12:  We show the metric function $N(x)+\frac{\Lambda}{3}x^{2}$ of the
unexcited solution ($k=0)$ and the first three radially excited solutions
($k=1$, $2$, $3$) for $\alpha=1$, $e=0.1$ and $\Lambda=-3/4$. (_Right_): The
corresponding metric function $\sigma(x)$.

We have constructed these solutions numerically. In Fig. 11 and Fig. 12 we
show the matter and metric functions, respectively, of solutions with
$k=1,2,3$ zeros in the scalar field function corresponding to $\Omega\approx
2.32,\,3.23,\,4.15$, respectively, for $\alpha=1$, $e=0.1$ and $\Lambda=-3/4$.
Increasing $k$, we observe that the minimum of the metric function $N(x)$
decreases in value and moves closer to the origin, while $\sigma(x)$ reaches
its asymptotic value quicker when increasing $k$.

In Fig. 13 (left) we give $\Omega$ as function of the scalar field function at
the origin, $\psi(0)\equiv\psi_{0}$ for solutions with $k=0,1,2,3$ nodes in
the scalar field for $\mu=1$, $\Lambda=-3/4$. Using the discussion on
oscillons (see II.1) we have hence $p_{+}=2$ and consequently $\Omega=2+k$,
$k\in\mathbb{N}$. The data in Fig. 13 clearly demonstrates that for small
values of $\psi_{0}$ we find that $\Omega=2+k$, where $k$ is the number of
nodes of the scalar field function in agreement with the discussion given
above. Increasing $\psi_{0}$ leads to a decrease in the value of $\Omega$
until $\Omega_{\rm min}$ is reached at some intermediate value of $\psi_{0}$.
We find that $\Omega$ reaches its minimal value at decreasing values of
$\psi_{0}$ when increasing the number of nodes in the scalar field.

In Fig. 13 (right) we give the value of the condensate, i.e. the value of the
scalar field on the conformal boundary, $(\psi_{+})^{1/\Delta_{+}}$, in
function of the chemical potential $-\Phi$. For all $k$ we find that the
condensate builds up strongly when increasing $-\Phi$ from zero. It then
reaches a maximal value and this is attained at decreasing values of $-\Phi$
when increasing $k$. The condensate then decreases up to a maximal value of
$-\Phi$ and starts to form a spiral with several branches appearing. We
observe a similar behaviour for all $k$, but note that the interval in $-\Phi$
for which solutions exist decreases with $k$. Interestingly, we also find that
the value of the condensate increases when increasing $k$ at a fixed chemical
potential and that different values of the condensate are possible for
sufficiently large and fixed $-\Phi$.

Figure 13: (_Left_): We show the value of $\Omega$ in dependence of the value
of the scalar field function at the origin, $\psi_{0}$, of the unexcited
solution ($k=0)$ and the first three radially excited solutions ($k=1$, $2$,
$3$) for $\alpha=1$, $e=0.1$ and $\Lambda=-3/4$. (_Right_): The corresponding
condensate $(\psi_{+})^{1/\Delta_{+}}$ as function of the chemical potential
$-\Phi$ for $k=0,1,2$.

Interestingly, we find that the new branches of solutions found in the
$\Lambda=0$ limit exist as well for $\Lambda<0$, at least as long as $\Lambda$
is sufficiently close to zero. This is shown in Fig. 6 (right). The branches
$a$, $b$ and $b^{\prime}$ move to larger values of $\omega$, while the
branches $a^{*}$ and $b^{*}$ move to smaller values of $\omega$. For
$\Lambda=-0.01$, the $a^{*}$ and $b^{*}$ branches have nearly disappeared.
Hence, these branches exist only for $\Lambda$ sufficiently close to zero.

##  IV Conclusions

In this paper, we have studied electrically charged boson stars with an
exponential scalar field interaction in asymptotically flat and Anti-de Sitter
space-time, respectively.

Next to the standard boson stars discussed in the literature previously, we
find several new branches of solutions. For the unexcited solutions ($k=0$),
i.e. solutions without nodes in the scalar field function inside the boson
star core, we observe that the formation of wavy scalar hair on the boson
stars discussed before in asymptotically flat space-time gauged_BS_power_law
also exists in aAdS space-time, at least for small values of $|\Lambda|$. This
hair appears for gravitational interaction sufficiently large and
electromagnetic interaction sufficiently small and involves the splitting of
the space-time into a ”false vacuum” interior (where the scalar field is
constant, but non-zero) and a ”true vacuum” exterior (where $\psi\equiv 0$).
The interior hence corresponds to a de Sitter space-time with the constant
scalar field leading to a constant positive potential energy that can be
interpreted as a positive cosmological constant. The exterior corresponds to a
space-time without scalar field, i.e. an extremal Reissner-Nordström (-AdS)
solution. At the intersection between these space-times, the scalar field
starts oscillating.

When considering radially excited solutions, we find that the wavy scalar hair
is not present, which also confirms that the spatial oscillations interpreted
as wavy scalar hair are not radial excitations of the boson star. In fact, we
observe a qualitatively very different behaviour. When increasing the central
value of the scalar field, instead of reaching a maximal possible value as in
the $k=0$ case, we find that the central value can become much larger and
seems (at least within our numerical analysis) not to be bounded from above as
soon as the boson star is radially excited. Moreover, we have constructed new
branches of radially excited solutions that seem disconnected from the linear
scalar field limit and seem to be a truly non-linear phenomenon. These
branches disappear when decreasing $\Lambda$ to strongly from zero, i.e. exist
only when the AdS radius is sufficiently large.

Finally, within the context of the holographic interpretation, we find that
the radially excited solutions can produce larger values of the condensate,
i.e. the scalar field value on the conformal boundary, as compared to the
unexcited solutions. In order to get a better understanding of these
condensates, it will be interesting in the future to construct the
corresponding black hole solutions and find holographic phase diagrams. This
is currently under investigation.

Acknowledgments F. Console thanks CAPES for financial support.

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