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\begin{document}

\title{Stability of six-dimensional hyperstring braneworlds}

\author{Patrick Peter}
\email{peter@iap.fr}
\affiliation{Institut d'Astrophysique de Paris, \GReCO, FRE
2435-CNRS, 98bis boulevard Arago, 75014 Paris, France}

\author{Christophe Ringeval}
\email{christophe.ringeval@physics.unige.ch}
\affiliation{D\'epartement de Physique Th\'eorique, Universit\'e de
Gen\`eve, 24 quai Ernest Ansermet, 1211 Gen\`eve 4, Switzerland,\\
Institut d'Astrophysique de Paris, \GReCO, FRE
2435-CNRS, 98bis boulevard Arago, 75014 Paris, France}

\author{Jean-Philippe Uzan}
\email{uzan@iap.fr}
\affiliation{Institut d'Astrophysique de Paris, \GReCO, FRE
2435-CNRS, 98bis boulevard Arago, 75014 Paris, France \\ Laboratoire
de Physique Th\'eorique, CNRS-UMR 8627, Universit\'e Paris Sud,
B\^atiment 210, F-91405 Orsay Cedex, France.}

\date{January 22, 2003}

\begin{abstract}
We study a six dimensional braneworld model in the case where the four
dimensional brane is described by a topologically stable vortex of a
U(1) symmetry-breaking Abelian Higgs model. We revisit the range of
the microscopic parameters that leads to gravity confinement on the
brane. In particular, we find that the necessary fine-tuning is much
tighter than previously though, and qualitatively different from that
of a five dimensional braneworld with a brane described by a domain
wall. The investigation of gauge invariant perturbations around this
background solution reveals that, contrary to the five dimensional
case, there exist perturbation modes which can only satisfy the
boundary condition of being bounded far from the brane provided they
are tachyonic inside the brane. The warped space-time structure that
is obtained this way is thus unstable. The genericness of our
conclusions is discussed; this will shed some light on the possibility
of describing our spacetime as a general six dimensional braneworld.
\end{abstract}
\pacs{04.50.+h, 11.10.Kk, 98.80.Cq}
\maketitle

\section{Introduction}\label{Sec:I}

Following the advent of string theory~\cite{string} and its
implication that space may have more than the three dimensions (in
the Kaluza-Klein way) came the suggestion that the extra
dimensions could be much larger than previously expected, would it
be because of a smaller value of the Planck energy in the (finite
size) bulk~\cite{large}, or because of a large curvature in the
(infinite) extra-dimensions~\cite{RS}. A novel idea came into play
with the assumption that we live on a hyper-surface, a
three-spatial dimensional `brane', embedded in a larger
dimensional warped space-time bulk~\cite{warp}.

For any higher dimensional Universe model, it is essential to
confine gravity since gravitation is experimentally tested to be
three dimensional on many different scales, ranging from the
millimeter~\cite{limit} to some Mega-parsec~\cite{largescales}.
For a five dimensional anti-de Sitter bulk, gravity was shown to
be localized on the brane~\cite{RS} and to lead to a viable
cosmological framework~\cite{cosmo5D} provided the brane and bulk
cosmological constants are adjusted by hand. The situation is not
yet settled concerning the cosmological perturbations induced in
the brane~\cite{perturb5D}. In the case the brane is modeled as a
domain wall-like topological defect, this fine-tuning transforms
into a tuning of the underlying parameters (masses, coupling
constant and bulk cosmological constant)~\cite{rpu01}.

Many mechanisms have been proposed to confine the other known
interactions and their associated particles: scalar~\cite{bajc} and
gauge~\cite{gauge} bosons, and fermions~\cite{fermions}. In the latter
case the mechanism relies on a generalization of the cosmic string
case~\cite{jackiw,ringeval}. The fermions are trapped in the brane in
the form of massless zero modes and some can even become massive,
although their mass spectrum is not compatible~\cite{rpu01} with the
observed one~\cite{pdg}. It was also suggested~\cite{higgs} that the
electroweak Higgs field, and thus the origin of electroweak symmetry
breaking, could be understood from the existence of an extra-dimension
in the form of a transverse gauge field component. Most of these works
were based on the simplifying assumption of a reflection symmetry with
respect to the brane, although a more refined treatment, not assuming
such a symmetry, appears possible~\cite{bcjpu1}.

Most of the relevant discussions on braneworld models have been
restricted to the case of one spatial extra-dimension, as
advocated \eg in the framework of the eleven dimensional
realization of $M-$theory proposed by Ho\v{r}ava and
Witten~\cite{HW}. Moreover, the underlying brane model of the
Universe is often assumed to be infinitely thin in the transverse
direction, so that (i) the induced gravity stems essentially from
Darmois~--~Israel junction conditions~\cite{cosmo5D} and (ii) is
mostly independent of the microstructure of the brane, if any. No
such general condition is available in the less restrictive
situation of more than one extra-dimension that is the subject of
the present work.

To study braneworlds with more than one extra-dimension, it is
necessary to specify the microstructure of the brane to fix a model,
taking into account in particular the possibly finite thickness of the
brane~\cite{nostring}. In particular, one needs to regularize long
range self interaction forces (including gravity). In the case the
force derives from a potential (\eg in linearized gravity), the
self-interaction potential is well behaved outside the brane but will
be singular on the brane in the thin brane limit. One needs to
introduce a UV cutoff associated with the underlying
microstructure~\cite{bcjpu2}. The only case where such a procedure can
be avoided is that of hyper-membranes that can be satisfactorily
treated without recourse to regularization.

Much work has been devoted to warped geometries in six
dimensions~\cite{6D}. When considering explicit realizations in terms
of an underlying topological defect forming field
model~\cite{GS,model6D}, it seems that the six dimension case
represents a limiting situation: for more than two extra-dimensions,
it is not possible to confine gravity on the brane~\cite{6+}. At least
two questions arise: are the properties of gravity dependent on the
microscopic structure of the brane and is the chosen microscopic model
consistent with $M-$theory? The second question has started to be
addressed in Ref.~\cite{ACHL}, and it turns out that, for many
purposes, it is possible to consider defect-like realizations of
branes, as in the present article.

Before considering the possibility of trapping~\cite{gauge6D} particle
fields in a hyper-string embedded in an anti-de Sitter six-dimensional
bulk space-time (adS$_{_6}$), it is necessary to determine the
background structure itself within a given field content, and decide
whether it is possible to localize gravity in the Universe thus
obtained, thereby generalizing the five dimensional case. This article
is devoted to this task and accordingly models the brane by a vortex
of a U(1) Abelian gauge Higgs model. Despite the fact that we agree on
the existence of a solution exhibiting a gravity localized on the
vortex, we find that it requires a more restrictive set of parameters
than found in previous analysis~\cite{gms01}. In fact, it differs also
qualitatively from the five dimensional case. We then go on to analyze
the stability of this solution by studying the scalar modes of the
gauge invariant perturbations, as originally suggested in
Ref.~\cite{zero6D}. Restricting our attention to bounded
perturbations, \ie perturbations satisfying the asymptotic condition
of not being divergent away from the brane (a condition necessary in
order to ensure that the bulk is close to anti-de Sitter space), we
find that the physically relevant modes must be tachyonic inside the
vortex-brane, hence initiating instabilities.

The article is organized as follows: after setting the field
theoretic framework both for the particles and gravity in
Sec.~\ref{Sec:II}, we construct the Nielsen-Olesen ansatz for a
three-dimensional vortex configuration, set and discuss the
corresponding Euler-Lagrange field equations in
Sec.~\ref{Sec:III}. We then show how to handle the boundary
conditions in Sec.~\ref{Sec:IV} and solve numerically the field
equations in Sec.~\ref{Sec:V}, insisting in particular on the
numerous technical difficulties, usually not enough emphasized in
previous works. This permits us to obtain the parameter range over
which gravity is localized. We then discuss some arguments leading
to the suggestion that such a defect realization of a brane in six
dimensions may be unstable in Sec.~\ref{Sec:VI}, and explicitly
exhibit the instabilities in Sec.~\ref{sec:pert} before ending by
some concluding remarks.

\section{Vortex configuration in adS$_{_6}$}\label{Sec:II}

We consider the action for a complex scalar field $\Phi$ coupled to
gravity in a six dimensional spacetime
\begin{equation}
\label{eq:action} S=\int\left[\frac{1}{2\kappa_{_6}^2}(R-2\Lambda)
+{\cal L}_{\matter}\right]\sqrt{-g}\,\dd^6 x,
\end{equation}
where $g_{\si{AB}}$ is the six dimensional metric with signature
$(+,-,-,-,-,-)$, $R$ its Ricci scalar, $\Lambda$ the six dimensional
cosmological constant and $\kappa_{_6}^2\equiv16\pi^2G_{_6}/3$,
$G_{_6}$ being the six dimensional gravity constant\footnote{In $D$
dimensions, we relate $\kappa_\si{D}^2$ to the $D$-dimensional
gravitational constant $G_\si{D}$ by
$\kappa_\si{D}^2=(D-2)\Omega^{[\si{D}-2]}G_\si{D}$, where
$\Omega^{[\si{D}-1]}=2\pi^{D/2}/\Gamma(D/2)$ is the surface of the
$(D-1)$-sphere.}. The matter Lagrangian reads
\begin{equation}\label{eq:lag}
{\cal L}_{\matter} =
\frac{1}{2}g^{\si{AB}}\DD_{\si{A}}\Phi\left(\DD_{\si{B}}\Phi\right)^*
-V(\Phi)-\frac{1}{4}\F_{\si{AB}}\F^{\si{AB}},
\end{equation}
in which capital Latin indices $A,B\ldots$ run from 0 to 5,
$\F_{\si{AB}}$ is the electromagnetic-like tensor defined by
\begin{equation}\label{eq:fab}
\F_{\si{AB}}=\partial_{\si{A}} C_{\si{B}} -\partial_{\si{B}}
C_{\si{A}},
\end{equation}
where $C_{\si{B}}$ is the connection 1-form. The U(1) covariant
derivative $\DD_{\si{A}}$ is defined by
\begin{equation}\label{eq:d}
\DD_{\si{A}}\equiv\partial_{\si{A}} -iqC_{\si{A}},
\end{equation}
where $q$ is the charge. The potential of the scalar field $\Phi$ is
chosen to break the underlying U(1) symmetry and thereby allow for
topological vortex (cosmic string like) configurations,
\begin{equation}\label{eq:V}
V(\Phi)=\frac{\lambda}{8}\left(\vert\Phi\vert^2-\eta^2\right)^2,
\end{equation}
where $\lambda$ is a coupling constant and $\eta=\langle
|\Phi|\rangle$ is the magnitude of the scalar field vacuum expectation
values (VEV)\footnote{Note, that, because of the unusual number of
spacetime dimensions, the fields have dimensions given by $[R]=M^2$,
$[\Phi]=M^{2}$, $[\Lambda]=M^2$, $[\lambda]=M^{-2}$, $[\eta]=M^{2}$,
$[\kappa_{_6}^2]=M^{-3}$, $[C_{\si{A}}]=M^2$, $[\F_{\si{AB}}]=M^3$ and
$[q]=M^{-1}$ ($M$ being a unit of mass). This can be further
generalized in the $D-$dimensional case by: $[\Phi]=M^{(D-2)/2}$,
$[\lambda]=M^{4-D}$, $[\eta]=M^{(D-2)/2}$, $[\kappa_{_D}^2]=M^{2-D}$,
$[C_{\si{A}}]=M^{D-2}$, $[\F_{\si{AB}}]=M^{D-3}$ and $[q]=M^{2-D/2}$,
$[R]$ and $[\Lambda]$ being unchanged.}.

Motivated by the brane picture, we choose the metric of the bulk
spacetime to be of the warped static form
\begin{equation}\label{eq:metric}
\dd s^2=g_{\si{AB}} \dd x^{\si{A}} \dd x^{\si{B}} =
\ee^{\sigma(\rdim)}\eta_{\mu\nu} \dd x^\mu\dd x^\nu
-\dd\rdim^2-\rdim^2\ee^{\gamma(\rdim)}\dd\theta^2,
\end{equation}
where $\eta_{\mu\nu}$ is the four dimensional Minkowski metric of
signature ($+,-,-,-$), and $(\rdim,\theta)$ the polar coordinates in
the extra-dimensions. Greek indices $\mu,\nu\ldots$ run from 0 to 3
and describe the brane worldsheet and we set $g_{\mu\nu} \equiv
\exp [{\sigma(\rdim)}] \eta_{\mu\nu}$. The action (\ref{eq:action})
with the ansatz (\ref{eq:metric}) will admit static solutions
depending only on $\rdim$ so that the general covariance along the
four dimensional (physical) spacetime is left unbroken.

The Nielsen-Olesen like~\cite{NO} ansatz for a generalized vortex
configuration is taken to be of the form~\cite{neutral}
\begin{equation}\label{eq:ansatz}
\Phi= \varphi(\rdim)\ee^{i n\theta} = \eta f(\rdim)\ee^{i
n\theta},\qquad C_\theta= \displaystyle
\frac{1}{q}\left[n-Q(\rdim)\right]
\end{equation}
where $n$ is an integer, so that the only non-vanishing component of
the electromagnetic tensor is $\F_{\theta\rdim}=Q'/q$. With such an
ansatz, we shall now derive the relevant field equations and discuss
their solutions.

\section{Equations of motion}\label{Sec:III}

With the metric given by Eq.~(\ref{eq:metric}), the non-vanishing
Einstein tensor components reduce to
\begin{eqnarray}\label{eq:G}
G_{\mu\nu} &=&
\frac{1}{4}g_{\mu\nu}\bigg(6\sigma''+\frac{6}{\rdim}\sigma'
+6\sigma'^2+3\sigma'\gamma'\nonumber \\ & &
+2\gamma''+\gamma'^2+ \frac{4}{\rdim}\gamma'\bigg), \nonumber\\
G_{\rdim\rdim}&=&-\frac{1}{2}\sigma'\left(3\sigma'+\frac{4}{\rdim}+2\gamma'
\right), \nonumber\\
G_{\theta\theta}&=&-\frac{1}{2}\rdim^2\hbox{e}^\gamma\left(4\sigma''
+5\sigma'^2\right),
\end{eqnarray}
where a prime denotes differentiation with respect to
$\rdim$. Similarly, the matter stress-energy tensor,
\begin{equation}
\label{eq:tmunumatt}
T_{\si{AB}} \equiv 2{\delta {\cal L}_{\matter} \over \delta
g^{\si{AB}}} - g_{\si{AB}} {\cal L}_{\matter},
\end{equation}
has non-vanishing components provided by the Nielsen-Olesen
ansatz~(\ref{eq:ansatz}) that are given by
\begin{eqnarray} \label{eq:tmunu}
T_{\mu\nu} &=&
g_{\mu\nu}\left[V+\frac{\eta^2}{2}\left(f'^2+\frac{Q^2f^2}{\rdim^2}
\hbox{e}^{-\gamma}\right)+\frac{1}{2}\frac{Q'^2}{q^2\rdim^2}\hbox{e}^{-\gamma}
\right],\nonumber\\ T_{\rdim\rdim} &=& -V
+\frac{\eta^2}{2}\left(f'^2-\frac{Q^2f^2}{\rdim^2}\hbox{e}^{-\gamma}\right)
+\frac{1}{2}\frac{Q'^2}{q^2\rdim^2}\hbox{e}^{-\gamma} ,\nonumber\\
T_{\theta\theta} &=& \rdim^2\hbox{e}^\gamma\left[ -V
-\frac{\eta^2}{2}\left(f'^2-\frac{Q^2f^2}{\rdim^2}\hbox{e}^{-\gamma}\right)
+\frac{1}{2}\frac{Q'^2}{q^2\rdim^2}\hbox{e}^{-\gamma} \right]
.\nonumber \\ & &
\end{eqnarray}
It follows that the six dimensional Einstein equations, with our
conventions,
\begin{equation}
\label{eq:einstein}
G_{\si{AB}} + \Lambda g_{\si{AB}} + \kappa_{_6}^2 T_{\si{AB}} = 0,
\end{equation}
can be cast in the form
\begin{eqnarray}
\frac{3}{2}\ddot\sigma+\frac{3}{2}\dot\sigma^2+\frac{3}{2\radim}\dot\sigma
+\frac{3}{4}\dot\sigma\dot\gamma+\frac{1}{2}\ddot\gamma
+\frac{1}{4}\dot\gamma^2+\frac{1}{\radim}\dot\gamma=-\frac{\Lambda}{|\Lambda|}
\hfill \nonumber\\ -\alpha\left[\beta\left(f^2-1\right)^2+\dot f^2
+\frac{\hbox{e}^{-\gamma}}{\radim^2}\left(Q^2f^2+\frac{\dot
Q^2}{\varepsilon} \right)\right],\label{eq:einstein1}
\end{eqnarray}

\begin{eqnarray}
\frac{3}{2}\dot\sigma^2+\frac{2}{\radim}\dot\sigma
+\dot\sigma\dot\gamma&=&-\frac{\Lambda}{|\Lambda|}
-\alpha\Bigg[\beta\left(f^2-1\right)^2\nonumber \\ &-& \dot f^2
+\frac{\hbox{e}^{-\gamma}}{\radim^2}\left(Q^2f^2-\frac{\dot
Q^2}{\varepsilon} \right)\Bigg],\label{eq:einstein2}
\end{eqnarray}

\begin{eqnarray}
2\ddot\sigma+\frac{5}{2}\dot\sigma^2&=&-\frac{\Lambda}{|\Lambda|}
-\alpha\Bigg[\beta\left(f^2-1\right)^2\nonumber \\ &+&\dot f^2
-\frac{\hbox{e}^{-\gamma}}{\radim^2}\left(Q^2f^2+\frac{\dot
Q^2}{\varepsilon} \right)\Bigg],\label{eq:einstein3}
\end{eqnarray}
where we have introduced the dimensionless radial coordinate
\begin{equation}\label{eq:dimless1}
\radim\equiv\sqrt{\vert\Lambda\vert}\rdim,
\end{equation}
as well as the dimensionless parameters
\begin{equation}\label{eq:dimless2}
\alpha\equiv\frac{1}{2}\kappa_{_6}^2\eta^2,\qquad
\beta\equiv\frac{1}{4}\frac{\lambda\eta^2}{|\Lambda|},\qquad
\varepsilon\equiv\frac{q^2\eta^2}{|\Lambda|}.
\end{equation}
In Eqs.~(\ref{eq:einstein1}~--~\ref{eq:einstein3}), we have introduced
the convention that a dot refers to a differentiation with respect to
the dimensionless radial coordinate $\radim$.

The scalar field dynamics is given by the Klein-Gordon equation
\begin{eqnarray}\label{eq:KG}
\nabla_{\si{A}}\nabla^{\si{A}}\Phi&=&-\frac{\lambda}{2}
\left(\vert\Phi\vert^2-
\eta^2\right)\Phi +q^2C^2\Phi \\ \nonumber &
&+iqC^{\si{A}}\partial_{\si{A}}\Phi+iq\nabla_{\si{A}}
\left(C^{\si{A}}\Phi\right),
\end{eqnarray}
which takes the reduced form
\begin{equation}\label{eq:KGred}
\ddot f +\left(2\dot\sigma
+\frac{1}{2}\dot\gamma+\frac{1}{\radim}\right)\dot f =
\frac{Q^2}{\radim^2}f\hbox{e}^{-\gamma}+2\beta\left(f^2-1\right)f,
\end{equation}
while the Maxwell equations
\begin{equation}\label{eq:max}
\nabla_{\si{A}}
\F^{\si{AB}}=-q^2C^2\vert\Phi\vert^2+\frac{i}{2}q\left(
\Phi\partial^{\si{B}}\Phi^*-\Phi^*\partial^{\si{B}}\Phi\right),
\end{equation}
provide the single reduced equation for the only non-vanishing
component of the gauge vector field
\begin{equation}\label{eq:maxred}
\ddot Q + \left(2\dot\sigma-\frac{1}{2}\dot\gamma-
\frac{1}{\radim}\right)\dot Q = \varepsilon f^2 Q.
\end{equation}

The set of equations (\ref{eq:einstein1}~--~\ref{eq:einstein3},
\ref{eq:KGred}, \ref{eq:maxred}) is a set of five differential
equations for 4 unknown functions ($\sigma$, $\gamma$, $f$, $Q$).
Indeed, we have a redundant equation due the Bianchi identities. For
numerical practicality, we decide to pick up the following six
first-order independent equations (one can verify that the Higgs field
equation can be recovered from the following ones for $\dot{f} \ne 0$)
\begin{eqnarray}
\label{eq:s}
\dot s & = & -2 s^2 - {1 \over 2} s l - \calF + {1\over 2} \calV, \\
\label{eq:u}
\dot l & = & -{1\over 2} l^2 - 2 s l - \calF -2\calVV -{3\over 2} \calV, \\
\label{eq:m}
\dot m & = & m l,\\
\label{eq:f}
2\alpha \dot f^2 & = & 3 s^2 + 2 s l + 2 \calF + \calVV - \calV, \\
\label{eq:Q}
\dot Q & = & w, \\
\label{eq:w}
\dot w & = & \varepsilon Q f^2 + w \left({1\over 2}l - 2s \right),
\end{eqnarray}
where we have set the new independent functions
\begin{equation}
s \equiv \dot{\sigma}, \qquad u\equiv \dot \gamma, \qquad v\equiv
\ee^{\gamma},
\end{equation}
as well as
\begin{equation}
\label{eq:lmdefs}
l \equiv \frac{2}{\radim} + u, \qquad m \equiv \radim^2 v,
\end{equation}
and
\begin{eqnarray}
\label{eq:funcdefs} {\calF} &\equiv& \alpha\beta
\left(f^2-1\right)^2 +{\Lambda\over|\Lambda|},\\
\calV &\equiv& {2\alpha \over \varepsilon}{w^2 \over m}. \qquad
\calVV \equiv {2\alpha} {f^2 Q^2\over m}.\label{eq:funcdefs2}
\label{eq:defs}\end{eqnarray}

After discussing the behavior of these fields far from the vortex, \ie
far in the bulk, and on the brane itself in the following section, we
shall solve numerically Eqs.~(\ref{eq:s}--\ref{eq:w}) in order to
determine the structure of the spacetime and defect system.

\section{Boundary conditions}\label{Sec:IV}

Before solving any set of differential equations, one must discuss its
boundary conditions. By definition of the topological defect like
configuration, we require that the Higgs field vanishes on the
membrane itself, \ie $\Phi = 0$ for $\radim=0$, while it recovers its
VEV, $\eta$, in the bulk. These requirements translate into the
following boundary conditions for the function $f$:
\begin{equation}
\label{eq:limhiggs}
f(0)=0,\qquad \lim_{\radim\to +\infty}f = 1.
\end{equation}
The corresponding boundary conditions for the 1-form connection are
given by
\begin{equation}
\label{eq:limgauge}
 Q(0)=n,\qquad \dot Q(0)=w(0)=0, \qquad\lim_{\radim\to +\infty}Q = 0.
\end{equation}
Note that only two out of these three conditions are actually
necessary to fully solve the system.

In order to avoid any curvature singularity on the string, the Ricci
scalar stemming from Eq.~(\ref{eq:action}), namely
\begin{equation}
\label{eq:ricci}
R=-\left(\gamma'' + 4 \sigma'' + \frac{1}{2} \gamma'^2 + 5
\sigma'^2 + 2 \gamma ' \sigma' + \frac{2
\gamma'}{\rdim} + \frac{4 \sigma'}{\rdim}\right),
\end{equation}
has to be finite at $\rdim=0$ as the vortex is assumed to represent
our physical four-dimensional space. As a result, the warp functions
$u$ and $s$ have to vanish in the string core,
\begin{equation}
\label{eq:warp0}
u(0)=s(0)=0,
\end{equation}
and the warp function $l$ therefore scales near the string like
\begin{equation}
\label{eq:l0}
l(\radim) \sim \frac{2}{\radim}.
\end{equation}
A coordinate transformation allows to choose $\sigma(0)=0$, while
$\gamma(0)$ and $v_\zero$, defined by
\begin{equation}
\label{eq:v0}
v_\zero\equiv\ee^{\gamma(0)},
\end{equation}
can only be determined by the boundary conditions at infinity. At
this point, we remark that, contrary to what is assumed in
Ref.~\cite{gms01}, $v_\zero$ is not a freely adjustable parameter,
and its value is completely determined as soon as the other
boundary conditions are imposed. Demanding $v_\zero =1$, for
quantum stability reasons for instance, as we shall discuss later,
thus leads to an extra, physically motivated, constraint.

\subsection{Near the string} \label{sect:near}

The field behaviors near the string, to leading order in $\radim$, can
be extracted from the equations of motion (\ref{eq:s}) to
(\ref{eq:w}). Assuming the truncated Taylor expansions
\begin{equation}
w \sim w_\one \radim^{\kappaw}, \qquad Q \sim n +
\frac{w_\one}{1+\kappaw} \radim^{1+\kappaw},
\end{equation}
for the gauge fields, and
\begin{equation}
\label{eq:f1}
f \sim f_\one \radim^{\kappaf},
\end{equation}
for the Higgs field, with $\kappaw \ge 1$ in order to satisfy the
boundary conditions (\ref{eq:limgauge}), and $\kappaf \ge 1$ in
order for $\dot{f}$ to be well defined at the origin,
Eqs.~(\ref{eq:w}), (\ref{eq:warp0}) and (\ref{eq:l0}) yield
$\kappaw=1$. Similarly, the warp factor $s$ can be expanded around
$\radim=0$ as
\begin{equation}
s \sim s_\one \radim^{\kappas},
\end{equation}
with $\kappas \ge 1$ in order for the Ricci scalar to be well defined
[see Eq.~(\ref{eq:ricci})]. By means of Eqs.~(\ref{eq:s}),
(\ref{eq:lmdefs}) and (\ref{eq:v0}), one gets
\begin{equation}
\label{eq:s1}
\kappas=1, \qquad s_\one=\frac{1}{2}\left(1 - \alpha \beta +
\frac{\alpha}{\varepsilon} \frac{w_\one^2}{v_\zero} \right).
\end{equation}
The next order in the Higgs field behavior is given by
Eq.~(\ref{eq:f}) which, making use of
Eqs.~(\ref{eq:funcdefs}~--~\ref{eq:funcdefs2}), reads
\begin{equation}
\alpha \kappaf^2 f_\one^2 \radim^{2 \kappaf-2}=2 s_\one +\left(\alpha
\beta -1\right) - \frac{\alpha}{\varepsilon} \frac{w_\one^2}{v_\zero} + 
\alpha \frac{f_\one^2 n^2}{v_\zero} \radim^{2 \kappaf -2}.
\end{equation}
With $s_\one$ given by the expression (\ref{eq:s1}), this implies
\begin{equation}
\label{eq:powerf}
\kappaf=\frac{n}{\sqrt{v_\zero}}.
\end{equation}
As a result, the behavior of the Higgs field around the string core is
only determined by the asymptotic solutions at infinity through
$v_\zero$, as announced above, and the requirement $\kappaf \ge 1$
only provides the constraint
\begin{equation}
\label{eq:v0cond}
v_\zero \le n^2.
\end{equation}

{}From a purely classical point of view, only $\dot{f}$ has to be
well defined at $\radim=0$. However, in the general situation,
there always exists an integer $p \in \setN$ such that $v_{\zero}
> n^2/p$, in which case all derivatives of the Higgs field,
$f^{(k)}$, with $k\geq p$ are divergent in the core since the
$p^{\mathrm{th}}$ derivative already is. This is so unless the
bound is saturated, \ie the equality $v_\zero=n^2/p$ is strictly
satisfied so that $\kappaf=p$ and then all derivatives of order
larger than $p$ will strictly vanish. It is interesting to note
that in Eqs.~(\ref{eq:s}) to (\ref{eq:w}), the warp function $m$
(and thus $v$) appears only through the ratio $Q^2/m$ and $w^2/m$.
As a result, the solution for a given winding number $n$ and
$v_\zero$, is also solution of a winding number $\tilde{n}=pn$ and
$\tilde{v}_\zero=p^2 v_\zero$. This scaling permits to solve the
equations for a reduced set of parameters and yet obtain the
complete spectrum of solutions.

\subsection{Far from the string}

Asymptotically, the anti-de Sitter spacetime is recovered provided
\begin{equation}
\label{eq:sinf}
\lim_{\radim \to +\infty} \dot{s} =\lim_{\radim \to +\infty} \dot{u} =
0.
\end{equation}
Denoting by an index `{\small F}' (standing for ``fixed'') the value
of the fields at infinity, it follows from Eqs.~(\ref{eq:limhiggs}),
(\ref{eq:limgauge}) and (\ref{eq:sinf}) that the adS$_{_6}$ solution
is a fixed point for the set of Eqs. (\ref{eq:s}~--~\ref{eq:w}) with
$\ffx=1$, $\wfx=\Qfx=0$ for the Higgs and gauge fields, and with the
equations
\begin{eqnarray}
\label{eq:sfix}
2 \sfx^2 + \frac{1}{2} \sfx \lfx + \frac{\Lambda}{|\Lambda|} & = & 0,\\
\label{eq:lfix}
\frac{1}{2} \lfx^2 + 2 \sfx \lfx + \frac{\Lambda}{|\Lambda|} & = &0,\\
\label{eq:mfix}
\mfx \lfx & = & 0, \\
\label{eq:ffix}
3 \sfx^2 + 2 \sfx\lfx + 2 \frac{\Lambda}{|\Lambda|} & = & 0,
\end{eqnarray}
for the warp factors. Eq. (\ref{eq:mfix}) can only be satisfied for
$\mfx \to 0$, since $\lfx=0$ would lead to $\Lambda=0$ through
Eq.~(\ref{eq:lfix}). We can also check that the existence of such a
fixed point requires that the functions $\calV$ and $\calVV$
vanish. In this case, the asymptotic warp factors reduce to
\begin{equation}
\label{eq:warpF}
\lfx^2 = \sfx^2 = -\frac{2}{5} \frac{\Lambda}{|\Lambda|}.
\end{equation}
The anti-de Sitter solution is obtained for $\Lambda <0$ so that
$\lfx=\sfx=-\sqrt{2/5}$ and we can now verify that $\calV$ and
$\calVV$ effectively vanish. Indeed, from Eq.~(\ref{eq:m}), $m$ scales
at infinity as
\begin{equation}
\label{eq:mF}
\lim_{\radim \to + \infty} m \propto \ee^{-\sqrt{2/5} \radim},
\end{equation}
while Eqs.~(\ref{eq:Q}) and (\ref{eq:w}) lead asymptotically to
\begin{eqnarray}
\label{eq:gaugeF}
Q \propto \ee^{-\ell \radim}, \qquad w \sim -\ell Q,
\end{eqnarray}
with
\begin{equation}
\ell=\frac{3}{4} \sqrt{2\over5}\left(\sqrt{\frac{40}{9}
\varepsilon +1} - 1 \right).
\end{equation}
{}From Eq.~(\ref{eq:defs}), the gauge functions $\calV$ and $\calVV$
vanish at infinity provided
\begin{equation}
\label{eq:epsilonmin}
2 \ell > \sqrt{2\over 5} \quad \Longleftrightarrow \quad \varepsilon >
\frac{2}{5}.
\end{equation}
This is the first restriction on the available parameter space.

\section{Numerical approach and problems}\label{Sec:V}

Numerically, it is more convenient to solve the first order
differential system (\ref{eq:s}~--~\ref{eq:w}) where the explicit
dependencies in $1/\radim$ have been absorbed into the warp function
$l=2/\radim + u$. Indeed, would we not have defined the variables $l$
and $m$ through Eq.~(\ref{eq:lmdefs}) and considered instead the
original system in terms of the functions $u(\radim)$ and $v(\radim)$,
we would have had to face the following technical difficulty. The
effect of the $1/\radim$ term in the right hand side is to add flow
turning points where the signs of $\dot{s}$, $\dot{u}$, and $\dot{f}$
change. As a result, growing exponential behaviors appear from these
turning points whatever the initial conditions and the direction of
integration. On the other hand, since there is no such flow inversion
in Eqs.~(\ref{eq:s}~--~\ref{eq:w}), the exponential growth can be
simply suppressed by integrating from the fixed point, at a given
cutoff $\radim_\infty$, toward the string core.

A backward Runge-Kutta numerical method of integration has been
used with the gauge field values starting at a large enough
distance $\radim_\infty$ such that the fields are given by their
asymptotic expansion [see Eq.~(\ref{eq:gaugeF})] with sufficient
accuracy,
\begin{eqnarray}
\label{eq:minf}
m(\radimF) & = & \minf \ee^{-\sqrt{2/5} \radimF},\\
\label{eq:Qinf}
Q(\radimF) & = & \Qsminf \sqrt{\minf} \ee^{-\ell \radimF},\\
\label{eq:winf} w(\radimF) & = & -\ell \Qsminf \sqrt{\minf}
\ee^{-\ell \radimF},
\end{eqnarray}
while the Higgs field and warp factors are given by their values
at the fixed point, namely,
\begin{eqnarray}
\label{eq:warpinf}
s(\radimF) & = & l(\radimF)=-\sqrt{2\over5},\\
f(\radimF) & = & 1.
\end{eqnarray}
In order to recover the expected fields behaviors near the string
core, we demand that the numerical solution verifies
\begin{eqnarray}
\label{eq:f0num}
\lim_{\radim \to 0} f=0, \\
\label{eq:l0num}
\lim_{\radim \to 0} \frac{\radim}{2} l = 1,\\
\label{eq:Q0num}
\lim_{\radim \to 0} Q=n,
\end{eqnarray}
the other conditions being automatically satisfied (see
Sec.~\ref{sect:near}). In general, the fine tuning of $\minf$ and
$\Qsminf$ is not sufficient to obtain the previous constraints, the
integration stopping at a finite distance from the core at which the
square of the Higgs derivative vanishes and would become negative [see
Eq.~(\ref{eq:f})]. As it was the case in five dimensions~\cite{rpu01}
and previously discussed also in six dimensions~\cite{gms01}, a fine
tuning between the parameters $\alpha$, $\beta$ and $\varepsilon$ is
required in order to reach $\radim=0$. The matching with the expected
solution near the string is thus performed in three steps.

First, at fixed $\alpha$ and $\varepsilon$, and for arbitrary values
of $\minf$ and $\Qsminf$, the parameter $\beta$ is adjusted to push
the point where the Higgs field vanishes toward $\radim=0$ in order to
satisfy the constraint (\ref{eq:f0num}). Once this is performed, the
parameter $\Qsminf$ is fine tuned in order to satisfy
Eq.~(\ref{eq:l0num}), while for each value of $\Qsminf$, $\beta$ is
also recomputed to satisfy Eq.~(\ref{eq:f0num}).  Finally, once
Eqs.~(\ref{eq:f0num}) and (\ref{eq:l0num}) are verified, the parameter
$\minf$ allows to normalize the fields according to
Eq.~(\ref{eq:Q0num}). Once all the constraints
(\ref{eq:f0num}~--~\ref{eq:Q0num}) are satisfied, one can check that
the warp factors are well defined in the core, and extract the values
of $v_\zero$. The previous method is then reproduced for other values
of $\alpha$ and $\varepsilon$ in order to compute the parameter
surface in the three dimensional space $(\alpha,\varepsilon,\beta)$
leading to physical solutions.  However, keep in mind that there are
also additional constraints given by Eqs.~(\ref{eq:v0cond}) and
(\ref{eq:epsilonmin}) which limit the surface.

\begin{figure*}
\begin{center}
\includegraphics[width=12cm]{surfparam.eps}
\caption{Parameter surface leading to anti-de Sitter spacetime in the
$(\alpha,\varepsilon,\beta)$ space. The green region refers to
parameters leading to $v_\zero < 1/4$ where the second derivative of
the Higgs field is well defined at $\radim=0$, while the red region
corresponds to $ 1/4< v_\zero < 1$, where only the first derivative is
finite in the string core. The thick curve gives the points leading to
$v_\zero=1$ along which all the derivatives of the Higgs field are
well defined. We have also represented in purple the parameter area
giving $v_\zero=1 \pm 0.1$. The shaded region is associated with
$v_\zero > 1$ and is thus not physical.}
\label{fig:surfparam}
\end{center}
\end{figure*}

Once a warped solution is reached, one still has to demand that the
first derivative of the Higgs field on the brane be well-behaved in
order for the brane energy, hence the cosmological constant, be
finite. This not-so-obvious requirement restricts again the available
parameter space. Moreover, one can also assume, as was done in
Ref.~\cite{gms01}, that the Higgs field expansion around the vortex
core starts as $f\propto\radim$, an assumption that may be justifiable
on the ground that the Higgs field is expected to be an analytic
function of space~\cite{quantum}. In terms of our model, this
translates into $v_\zero =1$. This provides an orthogonal constraint
which restricts even more the parameter space to a line, \ie a one
dimensional subspace of a three dimensional space.

On Fig.~\ref{fig:surfparam}, we have depicted the domains of
parameters leading to $v_\zero < 1/4$, $1/4 < v_\zero < 1$, and
$v_\zero=1 \pm 0.1$. In particular, $v_\zero=1$ can only be obtained
for the parameters $\alpha$, $\beta$ and $\varepsilon$, living on the
curve whose least square power fit gives
\begin{eqnarray}
\left\lbrace\begin{array}{lll}
\ln \beta & \sim & 2.1 - 3.3 \ln{\alpha}, \\
\ln \varepsilon  & \sim & 4.2 - 3.0 \ln{\alpha},
\end{array}\right.
\end{eqnarray}
the standard deviations being less than $2 \%$ all along the space we
studied. The typical field solutions in these three regions have also
been plotted on Fig.~\ref{fig:solgrav}. As expected from the
analytical approach, the second derivative of the Higgs field is only
well defined in $\radim=0$ for $v_\zero < 1/4$.

At this point, our results differ from those of Giovannini
\etal~\cite{gms01} since we find only a curve in the parameters space
leading to $v_\zero=1$. Indeed, in Ref.~\cite{gms01} it was assumed
that the relationship $v_\zero=1$ was always verified, and the authors
have looked for parameters leading asymptotically to an anti-de Sitter
spacetime at infinity without considering the modification of the
fields in the string core. As a result, their relevant parameter space
was artificially enlarged to include the more general solutions
leading to $v_\zero \ne 1$ in our approach, and in particular the
unphysical ones with $v_\zero > 1$ (see Fig.~\ref{fig:surfparam}).

\begin{figure*}
\begin{center}
\includegraphics[height=5cm,width=5cm]{vortex.eps}
\includegraphics[height=5cm,width=5cm]{sgrav.eps}
\includegraphics[height=5cm,width=5cm]{stest.eps}\\
\includegraphics[height=5cm,width=5cm]{vortex2.eps}
\includegraphics[height=5cm,width=5cm]{sgrav2.eps}
\includegraphics[height=5cm,width=5cm]{stest2.eps}\\
\includegraphics[height=5cm,width=5cm]{vortex3.eps}
\includegraphics[height=5cm,width=5cm]{sgrav3.eps}
\includegraphics[height=5cm,width=5cm]{stest3.eps}
\caption{The field solutions leading to anti-de Sitter spacetime in
the regimes where $v_\zero < 1/4$, $ 1/4< v_\zero < 1$ and $v_\zero
\lsim 1$, obtained for $(\alpha,\varepsilon,\beta)$ equals to
$(1,10,2.286)$, $(1.75,15,3.548)$ and $(1.655,14.447,3.398)$
respectively. In each case, the Higgs and gauge fields are plotted on
the left picture, the warp factors are plotted in the middle one,
while the right plot represents the field $\radim l/2$ involved in the
constraint equation (\ref{eq:l0num}), with the warp function
$v(\radim)$ and $f(\radim)/\radim$.} \label{fig:solgrav}
\end{center}
\end{figure*}

\section{Fine-tuning and stability}\label{Sec:VI}

The solution we have just obtained for the fields surrounding a
brane-like vortex requires a fine-tuning of the underlying microscopic
parameters. We would like to compare this fine-tuning to its
equivalent in five dimensions. We argue here that even though the
fine-tuning required in five dimensions appears to be of the same
nature as the one needed in six dimensions, there exists a qualitative
difference. Whereas the five dimensional brane configuration turns out
to be stable with respect to scalar perturbations in the
bulk~\cite{perturb5D}, this will not be the case for the
six-dimensional vortex under scrutiny here. Let us first see
qualitatively why we expect such a conclusion. Our analysis is
analogous to the one developed in Ref.~\cite{GS} in the case of global
vortex and leads to similar conclusions.

In the five dimensional case, there are only two parameters,
essentially equivalent to our $\alpha$ and $\beta$ (see, \eg
Ref.~\cite{rpu01} and references therein) which have to be adjusted so
that the warp function, denoted in what follows by $S_{_5}$ to
distinguish it from its six-dimensional counterpart $s$, vanishes at
the brane location. The corresponding tuning, necessary for the
consistency of a reflection symmetry with respect to the brane,
translates into a relation between $\alpha$ and $\beta$, hence a one
dimensional curve in a two dimensional parameter space.  Let us
recapitulate the essentials of the five dimensional brane. The metric
is chosen as
\begin{equation}
\dd s^2_{_5} = \ee^{\sigma (y)} \eta_{\mu\nu} \dd x^\mu \dd x^\nu -\dd
y^2,
\end{equation}
and we now have $S_{_5}\equiv \dd \sigma / \dd \radim$, with $\radim =
y \sqrt{|\Lambda|}$.

In the limit $\radim\gg 1$, the equivalent of Eq.~(\ref{eq:s}), \ie
the $(\mu\nu)$ component of the 5D Einstein equations,
reads~\cite{rpu01}
\begin{equation}
S_{_5}^2 +\dot S_{_5}={2\over 3},
\label{5Dmunu}
\end{equation}
whose general solution is
\begin{equation}
S_{_5} = c_{_5} \times {A_{_5} \ee^{c_{_5} \radim} -
\ee^{-c_{_5} \radim} \over {A_{_5} \ee^{c_{_5} \radim} + \ee^{-c_{_5}
\radim}}},
\label{sol5}
\end{equation}
with $c_{_5}=\sqrt{2/3}$ and $A_{_5}$ an arbitrary constant of
integration. It is clear than in the general situation for which
$A_{_5}\not= 0$, the solution~(\ref{sol5}) satisfies, recalling that
$\rho$ is not necessary positive in this case,
\begin{equation}
\lim_{\radim\to\pm\infty} S_{_5}(\radim) = \pm \sqrt{2\over 3}, \qquad
\forall A_{_5} \in \setR^*.
\label{inf5}
\end{equation}
This means that the boundary condition leading to the requirement of
anti-de Sitter in the bulk is a repeller (one would need the sign to
be $\mp$ in the previous solution to be an attractor). One would thus
expect instabilities to easily develop. However, in the same limit
$\radim\gg 1$, one should consider as well the $(yy)$ part of the
Einstein field equations. It actually reads $S_{_5}^2=2/3$ which, by
comparison with Eq.~(\ref{5Dmunu}) and its solution~(\ref{sol5}),
implies that one must set $A_{_5}=0$ (we discard here the unphysical
situation, also satisfying the constraint, having $A_{_5}=\infty$ and
corresponding to a de Sitter divergent exponential in the metric).
This relation is therefore the only possible solution for the complete
set of Einstein equations. It is not a choice that can be made, or
fine-tuned, arbitrarily. This stems from the fact that there are only
two independent Einstein equations, one [the $(\mu\nu)$ part] being
dynamical while the other one [the $(yy)$ part] is a constraint. As a
result, there is no arbitrary integration constant that can be chosen,
and the solution, as it turns out, is stable~\cite{perturb5D}.

This situation is to be contrasted with the string-like brane solution
discussed here. In six dimensions, the asymptotic form of
Eq.~(\ref{eq:s}) reads
\begin{equation}
\dot s+{5\over 4} s^2 ={1\over 2},
\label{s6}
\end{equation}
for which one obtains the general solution in the form
\begin{equation}
s = \sqrt{2\over 5} \times {A_{_6} \ee^{c_{_6} \radim} - \ee^{-c_{_6}
\radim} \over {A_{_6} \ee^{c_{_6} \radim} + \ee^{-c_{_6} \radim}}},
\label{sol6}
\end{equation}
where $c_{_6}=\sqrt{5/8}$. As in the previous case, $A_{_6}$ is an
arbitrary constant, to be matched with the vortex interior
solution. Again, if $A_{_6}\not= 0$, one has
\begin{equation}
\lim_{\radim\to +\infty} s(\radim) = \sqrt{2\over 5},
\end{equation}
leading to an exponentially divergent warp factor for the
metric~(\ref{eq:metric}). Only the particular value $A_{_6}=0$ can
smoothly join the interior metric to a six-dimensional anti-de Sitter
asymptotic spacetime. But now, because of the extra degree of freedom
provided by the other function $\gamma$, this is an explicit choice
and not a mandatory consequence of the Einstein equations. In other
words, the solution satisfying $\lim_{\radim\to\infty} s =-\sqrt{2/5}$
is also a point from which all trajectories diverge. This is related
to the fact that in the limit where the field contribution in the
stress-energy tensor is negligible with respect to the bulk
cosmological constant, Eqs.~(\ref{eq:einstein1}) and
(\ref{eq:einstein3}) are two dynamical equations for the warp
functions $\sigma$ and $\gamma$, with Eq.~(\ref{eq:einstein2}) being a
constraint. The two first order (in $\dot\sigma$ and $\dot\gamma$)
dynamical equations require two constants of integration, of which the
constraint fixes only one (together with the requirement of an anti-de
Sitter asymptotic spacetime). The solution with arbitrary
(non-vanishing) $A_{_6}$ is thus a valid solution, contrary to the
five dimensional case. However, any non-zero value of $A_{_6}=0$ does
not correspond to a solution with gravity localized on the vortex.
One is thus led to conclude that the fine-tuning required in the 6D
case is much worse than in 5D since the physically relevant solution
is a set of measure zero in the full set of solutions.  This drives us
to ask whether such a solution is stable with respect to perturbations
in the fields or the metric. Note that the situation is even worse in
higher dimensions for which it is known that there is no warped
solution~\cite{6+} localizing gravity at all.

\section{Gauge invariant perturbations}\label{sec:pert}

\subsection{Gauge invariant variables}

In order to conclude on the stability of the configuration we
obtained, it is necessary to perturb this background solution in a
gauge-invariant way. As vector and tensor perturbations have been
investigated elsewhere~\cite{gauge6D}, we shall concentrate on the
scalar part~\cite{zero6D} of the perturbations\footnote{Here and
in what follows, the Scalar-Vector-Tensor decomposition is
understood to be with respect to the four-dimensional vortex.}. We
thus expand the metric as
\begin{eqnarray}
\label{eq:pertmetric}
\dd s^2 &=& \ee^{\sigma(r)} \left[ \eta_{\mu\nu} \left( 1+\psi \right)
+ \partial_\mu \partial _\nu E \right] \dd x^\mu \dd x^\nu -
(1+\xi)\dd r^2 \nonumber \\ & & - 2 \zeta \dd r\dd \theta
-r^2\ee^{\gamma(r)}\left(1+\omega\right) \dd \theta^2 \nonumber \\ &
&-2 \left( \partial _\mu B \dd r +\partial _\nu C \dd\theta \right)
\dd x^\mu ,
\label{pertScalar}
\end{eqnarray}
where the scalar functions $\psi$, $E$, $\xi$, $\zeta$, $\omega$, $B$ and
$C$ depend on all the coordinates $(x^\alpha ,r,\theta)$ and are
assumed to be small.

A gauge transformation $x^{\si{A}} \to \tilde x^{\si{A}} = x^{\si{A}}
+ \epsilon^{\si{A}}$, with $\epsilon^\mu = \partial^\mu \epsilon$
(scalar transformations only) implies three gauge degrees of freedom,
so we are left with four unknown functions to determine. The scalar
functions transform under a gauge transformation as
\begin{equation}
\left\{
\begin{aligned}
\tilde{\psi} & = \psi - \sigma' \epsilon_r,\\
\tilde E &= E + 2 \ee^{-\sigma} \epsilon ,\\
\tilde\xi &= \xi - 2 \epsilon_r',\\
 \tilde \zeta &= \zeta -
\frac{1}{2} \left[\epsilon_\theta' +\partial_\theta
\epsilon_r - \left(\displaystyle{2\over r}+\gamma'\right)
\epsilon_\theta \right],\\
 \tilde\omega &= \omega
- {2\ee^{-\gamma} \over r^2} \partial_\theta\epsilon_\theta -
\left({2\over r}+\gamma'\right)\epsilon_r,\\
\tilde B &= B -  {1\over 2} \left( \epsilon_r + \epsilon'-\sigma'
\epsilon \right),\\
\tilde C &= C - \displaystyle {1\over 2} \left( \epsilon_\theta +
\partial_\theta \epsilon \right).
\end{aligned}
\right.
\end{equation}
{}From these relations, we can derive the four gauge invariant
variables
\begin{equation}
\left\{
\begin{aligned}
 \Psi & \equiv \psi - \displaystyle {1\over2} \sigma'
    \left(4B + \ee^\sigma E' \right),\\
 \Xi & \equiv \xi
    -\partial_r\left(4B + \ee^\sigma E' \right),\\
 \Upsilon
    & \equiv\zeta - \displaystyle {1\over 4} \partial_r \left(4C +
    \ee^{\sigma} \partial_\theta E\right) -\displaystyle {1\over 4}
    \partial_\theta \left(4B + \ee^{\sigma} E'\right) \\
&  + \displaystyle{1\over4} \left({2\over r} + \gamma'\right) \left(4C
    + \ee^{\sigma} \partial_\theta E \right),\\
 \Omega & \equiv \omega - \displaystyle {\ee^{-\gamma} \over r^2}
    \partial_\theta \left(4C + \ee^\sigma \partial_\theta E \right) \\
 &     - \displaystyle {1\over2} \left({2 \over r} +
    \gamma' \right) \left(4B + \ee^\sigma \partial_r E
    \right).
\end{aligned}
\right.
\end{equation}
As in the usual cosmological case~\cite{mfb}, these variables are
identical to the original variables once the choice of longitudinal
gauge ($E=B=C=0$) is made. The transformation leading to this gauge,
starting from an arbitrary gauge transformation, reads
\begin{equation}
\left\{
\begin{aligned}
\epsilon &= \displaystyle -{1\over 2} \ee^\sigma E,\\
\epsilon_r &= \displaystyle 2B+{1\over 2} \ee^\sigma E',\\
\epsilon_\theta &= \displaystyle 2C+{1\over 2} \ee^\sigma \partial_\theta E,
\end{aligned}
\right.
\end{equation}
and is unique. This gauge choice, which we shall for now on adopt, is
thus complete for metric perturbations, but also for the matter
ones. Indeed, in our model (\ref{eq:lag}), the matter perturbations
concern only the hyperstring forming scalar field $\Phi$ and its
associated gauge field $C_{\si{A}}$. Note that in our framework, the
location of the brane is given by the zeroes of the Higgs field and
thus directly taken into account in its perturbations. Since we are
only interested in scalar perturbations, the perturbed fields can be
expanded as
\begin{eqnarray}
\label{eq:perthiggs}
\delta \Phi & = & \sum_{p\in \setZ} \chi_p \ee^{ip\theta},\\
\label{eq:pertgauge}
\delta
C_{\si A} & = & \left(\partial_\mu \gper,\gperR,\gperT \right),
\end{eqnarray}
where we have decomposed the scalar field perturbations in Fourier
modes around the vortex. Note that, for consistency, all the other
perturbations have also to be invariant by a complete rotation around
the hyperstring, and thus can also be decomposed in this way. We shall
in what follows consider a single excitation mode $p$ and thus drop
the sum in Eq.~(\ref{eq:perthiggs}). Under the gauge transformation
$x^{\si A} \to \tilde x^{\si A} = x^{\si A} + \epsilon^{\si A}$ these
perturbations transform to
\begin{equation}
\left\{
\begin{aligned}
\tilde{\chi_p} & = \displaystyle \chi_p - \varepsilon_r \Phi' - i n
\frac{\ee^{-\gamma}}{r^2} \varepsilon_\theta \Phi,\\ \tilde{\gper} & =
\displaystyle \gper -\frac{\ee^{-\gamma}}{r^2} C_\theta
\varepsilon_\theta,\\ \tilde{\gperR} & = \displaystyle \gperR -
\frac{\ee^{-\gamma}}{r^2} C_\theta \varepsilon_{\theta}' + 2
\frac{\ee^{-\gamma}}{r^2} C_\theta \left(\frac{1}{r} +
\frac{\gamma'}{2} \right) \varepsilon_\theta,\\ \tilde{\gperT} & =
\displaystyle \gperT - C_{\theta}' \varepsilon_r -
\frac{\ee^{-\gamma}}{r^2} C_\theta \partial_\theta \varepsilon_\theta,
\end{aligned}
\right.
\end{equation}
and are therefore not invariant. Similarly to the metric tensor
decomposition, we then define the gauge invariant quantities through
the relations
\begin{equation}
\left\{
\begin{aligned}
\Chip &\equiv\displaystyle \chi_p - \frac{1}{2} \Phi'\left( 4 B +
\ee^\sigma E' \right) -\frac{i n \ee^{-\gamma}}{2 r^2} \Phi
\left(4C + \ee^\sigma \partial_\theta E \right),\\ \Gper & \equiv
\displaystyle \gper - \frac{1}{2} \frac{\ee^{-\gamma}}{r^2} C_\theta
\left(4C+\ee^{\sigma} \partial_\theta E \right),\\ \GperR & \equiv 
\displaystyle \gperR - \frac{1}{2} \frac{\ee^{-\gamma}}{r^2} C_\theta
\left[4C'+ \left(\ee^{\sigma} \partial_\theta E\right)' \right] \\ & +
\displaystyle \frac{\ee^{-\gamma}}{r^2} C_\theta \left(\frac{1}{r} +
\frac{\gamma'}{2} \right) \left(4C + \ee^{\sigma} \partial_\theta E
\right),\\ \GperT & \equiv \displaystyle \gperT - \frac{1}{2} C_\theta'
\left(4B+\ee^{\sigma} E' \right) \\ & - \displaystyle \frac{1}{2}
\frac{\ee^{-\gamma}}{r^2} C_\theta \left(4 \partial_\theta C +
\ee^{\sigma} \partial^2_\theta E \right),
\end{aligned}
\right.
\end{equation}
which also match with the original variables in the longitudinal gauge
($E=B=C=0$).

Note that since we are interested in perturbation theory, we have to
keep in mind that all the perturbed physical quantities involved at
some initial time have to be close to the background solution. This
implies in particular that we must impose on the physically meaningful
perturbations to be asymptotically bounded: of all the possible
solutions of the perturbation equations which we discuss below, we
shall retain only those for which no long-distance divergence
appear. This, as it turns out, is extremely restrictive, and will lead
to the existence of tachyonic, \ie unstable, modes.

\subsection{Perturbed Einstein equations}

The Einstein equations, perturbed at first order, stem from
Eq.~(\ref{eq:einstein}). The perturbed metric tensor $\delta
g_\si{AB}$ is explicitly written in Eq.~(\ref{eq:pertmetric}) and
allows, by means of Eqs.~(\ref{eq:perthiggs}) and
(\ref{eq:pertgauge}), the determination of the scalar part of the
perturbed Einstein and stress-energy tensors. They are derived in the
appendix, and Eq.~(\ref{eq:einstein}) leads, in terms of the gauge
invariant variables, to the following equation of motion
\begin{widetext}
\begin{eqnarray}
\label{eq:einstein_munu}
\left(\partial_\mu \partial_\nu - \eta_{\mu \nu} \Box \right) \left(
\frac{\Xi + \Omega}{2} + \Psi \right) & + & \frac{1}{2} \ee^\sigma
\eta_{\mu \nu} \Bigg\{3 \Psi'' + 3 \gTTi \partial^2_\theta \Psi +
\Omega'' - 2 \gTTi \partial_\theta \Upsilon' + \gTTi \partial_\theta^2
\Xi + 3 \left(2 \sigma' + \lngTTp \right) \Psi' \nonumber \\ & - &
\left. \left(\frac{3}{2} \sigma' + \lngTTp \right) \Xi' +
\left[\frac{3}{2} \sigma' + 2 \left(\lngTTp\right) \right] \Omega' - 3
\gTTi \sigma' \partial_\theta \Upsilon \right. \nonumber \\ & + &
\left[3\sigma'' + 3\sigma'^2 + 3\sigma' \left(\lngTTp\right) +
2\left(\lngTTp\right)' + 2\left(\lngTTp\right)^2 \right] \left(\Psi -
\Xi \right) \Bigg\} \nonumber \\ & + & \kappasix^2 \ee^{\sigma}
\eta_{\mu \nu} \Bigg\{-\gTTi \frac{Q'}{q} \left(\GperT' -
\partial_\theta \GperR \right) + \left[\varphi' \Chip' + \gTTi Q
(Q-n+p) \varphi \Chip + {\dd V \over \dd\varphi} \Chip \right]
\nonumber \\ & \times & \cos\left[(n-p)\theta\right] -
\frac{1}{2}\left( \gTTi \frac{Q'^2}{q^2} + \varphi'^2 \right) \Xi -
\frac{1}{2} \gTTi \left( \frac{Q'^2}{q^2} + \varphi^2 Q^2 \right)
\Omega \nonumber \\ & + & \left[\frac{1}{2} \gTTi \left(
\frac{Q'^2}{q^2} +\varphi^2 Q^2 \right) + \frac{1}{2} \varphi'^2 +
V(\varphi) \right] \Psi - \gTTi q \varphi^2 Q \GperT \Bigg\} = 0,
\end{eqnarray}
for the $(\mu,\nu)$ part of Eq.~(\ref{eq:einstein}). The $(\mu,r)$ and
$(\mu,\theta)$ components lead to the following equations,
respectively,
\begin{eqnarray}
\label{eq:einstein_mur}
3 \Psi' + \Omega' - \gTTi \partial_\theta \Upsilon & -
&\left(\frac{3}{2} \sigma' + \lngTTp \right) \Xi + \left(-\frac{1}{2}
\sigma' + \lngTTp \right) \Omega \nonumber \\ & + & 2 \kappasix^2
\left\{- \gTTi \frac{Q'}{q} \left(\GperT -
\partial_\theta \Gper \right) + \varphi' \Chip
\cos\left[(n-p)\theta\right] \right\} = 0,
\end{eqnarray}
and
\begin{eqnarray}
\label{eq:einstein_mutheta}
\partial_\theta\left(3 \Psi + \Xi \right) - \Upsilon' - \left(\sigma'
+ \lngTTp \right) \Upsilon + 2 \kappasix^2 \left\{ \frac{Q'}{q}
\left(\GperR - \Gper' \right) -\varphi Q \Chip \sin\left[(n-p)\theta
\right] -q \varphi^2 Q \Gper \right\} = 0.
\end{eqnarray}
The purely bulk components of the first order perturbation of
Eq.~(\ref{eq:einstein}) read
\begin{eqnarray}
\label{eq:einstein_rr}
\frac{1}{2} \ee^{-\sigma} \Box\left(3 \Psi + \Omega\right) & - & 2
\gTTi \partial_\theta^2 \Psi + 2 \gTTi \sigma' \partial_\theta
\Upsilon - \left[3 \sigma' + 2 \left(\lngTTp \right) \right] \Psi' -
\sigma' \Omega' \nonumber \\ & + & \kappasix^2 \Bigg\{-\gTTi
\frac{Q'}{q}\left(\GperT' - \partial_\theta \GperR \right) + \left[
\varphi' \Chip' - \gTTi \varphi Q(Q-n+p) \Chip - {\dd V\over
\dd\varphi} \Chip \right] \cos\left[(n-p)\theta\right] \nonumber \\ &
- & \left[\frac{1}{2} \gTTi \varphi^2 Q^2 + V(\varphi) \right] \Xi -
\frac{1}{2} \gTTi \left(\frac{Q'^2}{q^2} - \varphi^2 Q^2 \right)
\Omega + \gTTi q \varphi^2 Q \GperT \Bigg\} - \Lambda \Xi = 0,
\end{eqnarray}
for the $(r,r)$ part,
\begin{eqnarray}
\label{eq:einstein_thetatheta}
\frac{1}{2} \gTT \ee^{-\sigma} \Box \left(3 \Psi + \Xi \right) & - & 2
\gTT \Psi'' + \gTT \sigma' \left(\Xi' - 5 \Psi' \right) + \frac{1}{2}
\gTT \left(4 \sigma'' + 5 \sigma'^2 \right) \left( \Xi - \Omega
\right) \nonumber \\ & + & \kappasix^2 \Bigg\{-\frac{Q'}{q}
\left(\GperT' - \partial_\theta \GperR \right) + \left[-\gTT \varphi'
\Chip' + \varphi Q(Q-n+p) \Chip - \gTT {\dd V\over \dd \varphi} \Chip
\right] \cos\left[(n-p) \theta \right] \nonumber \\ & + & \frac{1}{2}
\left(\gTT \varphi'^2 - \frac{Q'^2}{q^2} \right) \Xi - \gTT \left[
\frac{1}{2} \varphi'^2 + V(\varphi) \right] \Omega - q \varphi^2 Q
\GperT \Bigg\} - \gTT \Lambda \Omega = 0,
\end{eqnarray}
for the $(\theta,\theta)$ component, while the mixed one $(r,\theta)$
leads to the equation
\begin{eqnarray}
\label{eq:einstein_rtheta}
{}-\frac{1}{2} \ee^{-\sigma} \Box \Upsilon & + & 2 \partial_\theta
\Psi' -\sigma' \partial_\theta \Xi + \left[\sigma' - 2
\left(\lngTTp\right) \right] \partial_\theta \Psi -\frac{1}{2} \left(4
\sigma'' + 5 \sigma'^2 \right) \Upsilon \nonumber \\ & + &
\kappasix^2 \Bigg\{\left[-\varphi Q \Chip' + \varphi'(Q-n+p) \Chip
\right] \sin \left[(n-p)\theta\right] + \left[\frac{1}{2} \gTTi\left(
\frac{Q'^2}{q^2} - \varphi^2 Q^2\right) - \frac{1}{2} \varphi'^2
-V(\varphi) \right] \Upsilon \nonumber \\ & - & q \varphi^2 Q \GperR
\Bigg\} - \Lambda \Upsilon = 0,
\end{eqnarray}
\end{widetext}
where $\Box$ stands for the flat four-dimensional d'Alembertian,
\ie
\begin{equation}
\Box = \eta^{\mu \nu} \partial_\mu \partial_\nu = \partial _t^2 -
\nabla^2.\label{nabla}
\end{equation}

\subsection{Perturbed Maxwell equations}

By means of Eqs.~(\ref{eq:perthiggs}) and (\ref{eq:pertgauge}), we can
also derive the perturbed Maxwell equations stemming from
Eq.~(\ref{eq:max}), at first order in the fields. Since the Einstein
equations impose to the stress-energy tensor to be conserved, the perturbed
Maxwell equations are certainly already included in
Eqs.~(\ref{eq:einstein_munu}) to
(\ref{eq:einstein_rtheta}). Nevertheless, they mainly involve the
matter fields and may help to decouple the whole system. The $(\mu)$
component of the perturbed Faraday tensor gives
\begin{widetext}
\begin{eqnarray}
\label{eq:pertmax_mu}
\GperR' & - & \Gper'' + \left(\sigma' + \lngTTp \right) \left(\GperR -
\Gper' \right) + \gTTi \partial_\theta \left(\GperT
 -  \partial_\theta \Gper \right)  + q \varphi \left\{
\Chip \sin \left[(n-p)\theta\right] + q \varphi \Gper \right\} = 0,
\end{eqnarray}
while the $(r)$ and $(\theta)$ bulk parts lead to
\begin{eqnarray}
\label{eq:pertmax_r}
\ee^{-\sigma} \Box \left(\GperR - \Gper' \right) & + & \gTTi
\partial_\theta\left(\GperT' - \partial_\theta \GperR \right) - \gTTi
\frac{Q'}{q} \partial_{\theta} \left[ 2 \Psi - \frac{1}{2} \left(\Xi +
\Omega \right) \right] + \gTTi q \varphi^2 Q \Upsilon + q\left(\varphi
\Chip' - \varphi' \Chip \right) \sin \left[(n-p) \theta \right]
\nonumber \\ & + & q^2 \varphi^2 \GperR = 0,
\end{eqnarray}
and
\begin{eqnarray}
\label{eq:pertmax_theta}
\ee^{-\sigma} \Box \left(\GperT - \partial_\theta \Gper \right) & - &
\left(\GperT'' -  \partial_\theta \GperR' \right) - \left[2 \sigma' -
\left(\lngTTp \right) \right] \left( \GperT' -\partial_\theta \GperR
\right) + \frac{Q'}{q} \left[2 \Psi' -\frac{1}{2} \left( \Xi'+ \Omega'
\right) \right] \nonumber \\ & - & \left\{ \frac{Q''}{q} + \left[2
\sigma'- \left(\lngTTp \right)\right] \frac{Q'}{q} \right\} \Xi - q
\varphi ( 2Q -n +p) \Chip \cos\left[(n-p)\theta\right] + q^2 \varphi^2
\GperT = 0,
\end{eqnarray}
where use has been made of Eq.~(\ref{eq:maxred}) to simplify the term
otherwise proportional to $\Omega$.

\subsection{Perturbed Klein-Gordon equation}

In the same way, the Klein-Gordon equation (\ref{eq:KG}) can also be
perturbed in terms of metric and matter fields, and by means of
Eqs.~(\ref{eq:perthiggs}) and (\ref{eq:pertgauge}), one gets
\begin{eqnarray}
\label{eq:pertKG}
\ee^{ip\theta} \Bigg\{ \ee^{-\sigma} \Box \Chip - \Chip'' & - &
\left(2\sigma' + \lngTTp \right) \Chip' + \left[ \gTTi \left(Q - n + p
\right)^2 + \frac{\lambda}{2} \left(2\varphi^2 - \eta^2 \right)
\right] \Chip \Bigg\} +\ee^{i(2n-p)\theta} \frac{\lambda}{2} \varphi^2
\Chip \nonumber \\ & - & \ee^{in\theta} \Bigg\{ \varphi' \left(2\Psi'
+ \frac{\Omega'- \Xi'}{2} - \gTTi \partial_\theta \Upsilon \right) + i
\varphi \gTTi Q \left[\partial_\theta \left(2 \Psi +
\frac{\Xi-\Omega}{2} \right) - \Upsilon' \right] \nonumber \\ & + & i
q \varphi \left(\ee^{-\sigma} \Box\Theta - \Theta_r' - \gTTi
\partial_\theta\Theta_\theta \right) - \left[ \varphi'' + \left(2
\sigma' + \lngTTp \right) \varphi' \right] \Xi \nonumber \\ & - & i
\gTTi \left[Q \varphi \left(2 \sigma' - {1\over \rdim} -{\gamma'\over
2} \right) + \varphi Q' + 2 \varphi' Q \right] \Upsilon + \gTTi
\varphi Q^2 \Omega + 2 \gTTi q \varphi Q \Theta_\theta \nonumber \\ &
- & i q \left[2 \varphi' + \left(2 \sigma' + \lngTTp \right) \varphi
\right] \Theta_r \Bigg\} = 0.
\end{eqnarray}
\end{widetext}
This equation can obviously be split into three terms with respective
phase factors $\ee^{ip\theta}$, $\ee^{i(2n-p)\theta}$ and
$\ee^{in\theta}$. Since the first two terms are real functions [see
Eq.~(\ref{eq:pertKG})], their angular dependence is completely
determined by their phases, while it is not the case for the last term
since it also involves complex numbers. However, as previously pointed
out, all the perturbations have to be invariant by a complete rotation
around the hyperstring and thus must be decomposable as a Fourier
series. As a result, the \apriori complex function that factorizes in
front of the term $\ee^{i n \theta}$ must have an additional phase of
the form $\ee^{i \bar n \theta}$ where $\bar n$ is an integer.

In order to solve Eq.~(\ref{eq:pertKG}), one has to consider various
possibilities. The first is that for which all the phases are
different. This, in turns, implies that all three amplitudes must
vanish separately: direct inspection of Eq.~(\ref{eq:pertKG}) then
shows that this is the case provided $\Chip=0$, or, in other words,
that the perturbation one considers vanishes identically for $p\not=
n$. Another extreme possibility is that for which all the phases are
equal: whatever the value of $\bar n$, this also imposes
$p=n$. Finally, there is the possibility that they are equal two by
two (the third amplitude vanishing). {}From Eq.~(\ref{eq:pertKG}), this
procedure yields that the only solution having non-vanishing $\Chip$
and $p\not=n$ is then obtained for
\begin{equation}
2n-p = n+\bar n \Longleftrightarrow p=n-\bar n.
\end{equation}
However, in the case where $p=n-\bar n$, Eq.~(\ref{eq:pertKG}) also
demands that
\begin{eqnarray}
\label{eq:chipinfty}
\Chip'' & + & \left(2\sigma' + \lngTTp \right) \Chip' - \left[ \gTTi
\left(Q - n + p \right)^2 \right. \nonumber \\
& + & \left. \frac{\lambda}{2} \left(2\varphi^2 -
\eta^2 \right) \right] \Chip - \ee^{-\sigma} \Box \Chip =0.
\end{eqnarray}
Asymptotically, and in terms of the dimensionless parameters
previously introduced, Eq.~(\ref{eq:chipinfty}) reads
\begin{equation}
\ddot{\Chip} -
\sqrt{5\over 2}
\dot{\Chip} +
\frac{M_\chi^2}{|\Lambda|} \exp\left(\sqrt{2\over 5} \radim\right)
\Chip \underset{\infty}{\sim} 0,
\label{eq:Chipasymp}
\end{equation}
where we have defined the four-dimensional mass of the mode by
\begin{equation}
\label{eq:masschi}
\Box \Chip = -M_\chi^2 \Chip,
\end{equation}
and we have made use of Eqs.~(\ref{eq:warpF}) and (\ref{eq:warpinf}).

Eq.~(\ref{eq:Chipasymp}) can be solved through the change of variable
$z = \exp (\radim/\sqrt{10})$ and function $\Chip =
\exp(\sqrt{5/8}\radim) \tilde f$, leading to
\begin{equation}
{1\over z} {\dd\over \dd z} \left( z {\dd \tilde f\over \dd z}\right)
+ \left( {10 M_\chi^2\over |\Lambda|} - \frac{25}{4 z^2} \right)
\tilde f =0,
\end{equation}
whose solution is known \{see, \eg Eq.~(8.491) in Ref.~\cite{Grad}\},
and gives
\begin{equation}
\label{eq:Chipdiv} \Chip \propto \exp\left( \sqrt{5\over 8} \radim
\right) \times \calZ_{5/2} \left[ \sqrt{10  \over
|\Lambda|} |M_\chi|\exp \left( {\radim \over \sqrt{10}}\right) \right] ,
\end{equation}
in which $\calZ_{5/2}$ is a Bessel function of order $5/2$ and of its
argument in brackets for $M_\chi^2>0$, and a modified Bessel function
for $M_\chi^2<0$.

As a result, for any positive mass squared, $M_\chi^2 >0$, the
solution given in Eq.~(\ref{eq:Chipdiv}) behaves, asymptotically far
from the vortex, as an oscillatory exponentially divergent quantity
whose amplitude scales as
\begin{equation}
|\Chip| \underset{\infty}{\propto} \exp{\left(\sqrt{\frac{2}{5}}
\radim \right)}.
\end{equation}
As previously discussed, such solutions are clearly not physical in
the framework of perturbation theory since they cannot be bounded, far
from the string, close to the background solution, here the string
forming Higgs field. Even worse, they would also lead to an
exponentially divergent energy density contrast! This means that the
solutions which respect physical boundary conditions can only be
obtained for negative mass squared, $M_\chi^2 < 0$. Indeed, in this
case, the Bessel function in Eq.~(\ref{eq:Chipdiv}) is of modified
kind and admits an asymptotically exponential of exponential decaying
behavior
\begin{equation}
\Chip \underset{\infty}{\propto} \exp{\left(\sqrt{\frac{2}{5}}\radim
\right)} \exp{\left[- \sqrt{\frac{10}{|\Lambda|}} \left|M_\chi\right|
\exp{\left(\frac{\radim}{\sqrt{10}}\right)} \right]}.
\end{equation}
As a result, the physical perturbative solution leads to the
development of a tachyonic mode instability inside the brane. We are
led to the conclusion that the modes $p\not= n$ are always
unstable. Since those are modes of the Higgs field, one might however
argue that this does not imply that the vortex system itself is
unstable since some symmetry argument [\eg some residual U(1) gauge]
could protect these modes to ever be excited. In the following, we
will accordingly be interested in the not yet excluded case $p=n$.

\subsection{A subset of metric and cylindrical gauge perturbations}

According to the result of the previous section, we shall concentrate
on the case $p=n$. Then, the perturbation, $\Chi\equiv\Chi_n$, of the
Higgs field respects the cylindrical symmetry of the vortex. In the
following we will be interested in the metric and gauge field
perturbations which are also isotropic in the extra-dimensions, \ie
with not explicit dependence in $\theta$. This case is of particular
interest since it separates the full system into two disjoint pieces
that can be studied separately.

Introducing the new dimensionless parameters
\begin{equation}
\Thetadim = q \Theta, \quad \Thetadim_r = \frac{q}{\sqrt{|\Lambda|}}
\Theta_r, \quad \Thetadim_\theta  = q\Theta_\theta ,
\end{equation}
and
\begin{equation}
\Upsiadim = \sqrt{|\Lambda|} \Upsilon,\quad \Chiadim = \frac{\Chi}{\eta},
\end{equation}
the Einstein equations (\ref{eq:einstein_mutheta}) and
(\ref{eq:einstein_rtheta}), together with the Maxwell system
(\ref{eq:pertmax_mu}-\ref{eq:pertmax_r}), can be recast, as
announced above, into the closed set
\begin{eqnarray}
\label{eq:closegauge_mutheta}
\dot{\Upsiadim} + \left(s + \frac{l}{2} \right) -
\frac{4 \alpha}{\varepsilon} \dot{Q} \left( \Thetadim_r - \dot{\Thetadim}
\right) + 4 \alpha Q \Thetadim & = 0,& \\
\label{eq:closegauge_rtheta}
\left(\ee^{-\sigma} \madim_\upsilon^2 - 4 \alpha \frac{f^2 Q^2}{m}
\right) \Upsiadim - 4 \alpha Q \Thetadim_r & = 0,&\\
\label{eq:closegauge_mu}
\dot{\Thetadim}_r - \ddot{\Thetadim} + \left(s + \frac{l}{2} \right)
\left(\Thetadim_r - \dot{\Thetadim} \right) + \varepsilon f \Thetadim
& = 0,&\\
\label{eq:closegauge_r}
\ee^{-\sigma} \madim_\ug^2 \left(\Thetadim_r -\dot{\Thetadim} \right) -
\varepsilon \frac{f^2 Q}{m} \Upsiadim - \varepsilon f^2
\Thetadim_r & = 0,&
\end{eqnarray}
where the four-dimensional dimensionless squared masses $\madim_\ug^2$
and $\madim_\upsilon^2$ have been defined for the gauge and metric
fields $\Thetadim_{\si{A}}$ and $\Upsiadim$, respectively, as in
Eq.~(\ref{eq:masschi})
\begin{eqnarray}
\Box \Thetadim_{\si{A}} & = & - |\Lambda| \madim_\ug^2
\Thetadim_{\si{A}}, \\ \Box \Upsiadim & = & -|\Lambda|  \madim_\upsilon^2
\Upsiadim.
\end{eqnarray}
There are three variables $\Thetadim_r$, $\Thetadim$ and $\Upsiadim$
for four equations, one of them being thus a constraint equation. By
means of Eq.~(\ref{eq:closegauge_rtheta}), the metric perturbation
$\Upsiadim$ can be expressed in terms of $\Thetadim_r$ only
\begin{equation}
\label{eq:upsitheta_r}
\Upsiadim = \dfrac{4 \alpha Q\Thetadim_r }{\ee^{-\sigma}
\madim_\upsilon^2 - 4 \alpha \dfrac{f^2 Q^2}{m}},
\end{equation}
while by means of Eq.~(\ref{eq:upsitheta_r}), Eq.~(\ref{eq:closegauge_r})
gives the relation
\begin{equation}
\label{eq:theta_r_theta}
\Thetadim_r = \calP \dot{\Thetadim},
\end{equation}
with
\begin{equation}
\calP = \dfrac{\ee^{-\sigma} \madim_\ug^2}{\ee^{-\sigma} \madim_\ug^2
- \varepsilon f^2 \dfrac{\ee^{-\sigma} \madim_\upsilon^2 - 4 \alpha
Q^2/m}{\ee^{-\sigma} \madim_\upsilon^2 - 4 \alpha f^2
Q^2/m}}.
\end{equation}
Finally, plugging the previous expressions for $\Upsiadim$ and
$\Thetadim_r$, given by Eq.~(\ref{eq:upsitheta_r}) and
Eq.~(\ref{eq:theta_r_theta}), in Eq.~(\ref{eq:closegauge_mu}), one
gets a second order differential equation involving only the
function $\Thetadim$, namely
\begin{equation}
\label{eq:evolthetadim}
\left(\calP - 1 \right) \ddot{\Thetadim} + \left[ \dot{\calP} +
\left(s + \dfrac{l}{2} \right) \left(\calP - 1 \right) \right]
\dot{\Thetadim} + \varepsilon f \Thetadim = 0,
\end{equation}
whose asymptotic form is
\begin{equation}
\label{eq:evolthetaasymp}
\ddot{\Thetadim} - \sqrt{\dfrac{5}{2}} \dot{\Thetadim} + \madim_\ug^2
\exp{\left(\sqrt{\dfrac{2}{5}} \radim\right)} \Thetadim
\underset{\infty}{\sim} 0.
\end{equation}
This equation is identical to the perturbed Higgs field one
(\ref{eq:Chipasymp}) in the case where $p \neq n$, and leads to the
same conclusion: the only physical gauge field perturbations are
obtained for negative squared mass, $\madim_\ug^2 < 0$, and represent
tachyonic instabilities generated by the gauge vector field inside the
brane.

Now, we have to verify that the constraint equation
(\ref{eq:closegauge_mutheta}) is consistent with
Eq.~(\ref{eq:evolthetaasymp}). By means of Eqs.~(\ref{eq:upsitheta_r})
and (\ref{eq:theta_r_theta}), Eq.~(\ref{eq:closegauge_mutheta}) can be
expressed in terms of $\Thetadim$ only, whose asymptotic form reads
\begin{equation}
\label{eq:evolthetaconst}
\ddot{\Thetadim} + \left[
l\left(\dfrac{\madim_\upsilon^2}{\madim_\ug^2} - 1 \right) -
\sqrt{\dfrac{5}{2}} \right] \dot{\Thetadim} + \madim_\upsilon^2
\exp{\left(\sqrt{\dfrac{2}{5}} \radim \right)} \underset{\infty}{\sim}
0.
\end{equation}
Eqs.~(\ref{eq:evolthetaasymp}) and (\ref{eq:evolthetaconst}) are
consistent only if
\begin{equation}
\madim_\upsilon^2 = \madim_\ug^2.
\end{equation}
Since the physically acceptable solutions demand that
$\madim_\ug^2<0$, this last relation implies that the metric
perturbations $\Upsiadim$ also develop time instabilities on the
brane.

The other set of equations for the variables
$(\Psi,\Omega,\Xi,\Theta_\theta,\Chi_p)$ has not been studied in
details. But it has the same structure that the previous one and
we expect to reach the similar conclusions, specially once
$\Upsiadim$ has been shown to be unstable on the brane.

\section{Conclusions}\label{Sec:VII}


In the braneworld framework, one issue is to determine how to model
the brane, and in particular to investigate whether its internal
structure influences the properties of gravity and of the other fields
living on the brane. Among other solutions, more interests has been
focused on the possibility for the brane to be realized by a
topological defect~\cite{warp,ACHL}.

In five dimensions, it has been found that there always exists a
domain wall solution that confines gravity, which moreover is
symmetric with respect to both sides of the brane provided the
usual relationship between the bulk and the brane cosmological
constants is satisfied. This relationship translates into a
fine-tuning of the underlying microphysics
parameters~\cite{rpu01}. As far as the gravitational sector is
concerned, the properties of the braneworld are mostly independent
of the internal structure of the brane. It is then possible to
find various confinement mechanisms that lead to the existence of
bosonic~\cite{bajc,gauge} as well as fermionic~\cite{fermions}
zero modes that can be made massive~\cite{rpu01,higgs} (although
with a spectrum not yet compatible with accelerator data). In
short, a five dimensional topological model of our Universe is,
for the time being, an open possibility both from the cosmological
and particles physics points of view.

In more than six large dimensions, the situation is much simpler, as
it was shown~\cite{6+} that there is no warped solution if a
topological defect like description is assumed.

In six dimensions, the general machinery used to study
five-dimensional reflection symmetric braneworld does not apply,
mainly because of the necessity to regularize the long range
gravitational self-interaction~\cite{bcjpu2}. A way round is to
specify a complete model determining the internal structure of the
brane in order to grasp some features of six dimensional
braneworld models. Indeed, one will then need to discuss the
genericness of the conclusions drawn on a particular microphysics.
For instance, there exists a vortex-like brane configuration on
which gravity was shown to be localizable~\cite{zero6D}. However,
and contrary to what has been obtained in Ref.~\cite{zero6D}, we
have shown in this article that the fine-tuning necessary to
obtain this solution is much stronger, \ie a line in a
three-dimensional parameter space, than in five dimensions.

To study the stability of this solution, we have performed a full
gauge invariant perturbation theory around the six dimensional vortex
background solution. Focusing on the scalar perturbations, we showed
that both the Higgs scalar and associated gauge fields lead to
instabilities in the brane if they are constrained to be bounded
asymptotically in the bulk. We also showed that there exists gravity
modes with negative mass squared confined to the brane. Note that we
have exhibited unstable bending and torsion modes. By analogy to the
cosmic string case~\cite{ringeval}, these modes are expected to be
excited as soon as one couples fermions to the vortex.

A priori, these conclusions on the stability of this six dimensional
braneworld model are specific to the case at hand. In particular they
depend on the field content of the underlying theory. However, since
we have exhibited a tachyonic mode in the gravitational sector, it
could be conjectured that six dimensional braneworld with anti-de
Sitter bulk will also be unstable, even \eg in the singular brane
case. This would imply that multi-dimensional braneworld models could
only have one (large) extra-dimension of the warped form.

\appendix*

\section{Perturbed quantities}

In this appendix, we derive all the gauge-invariant parts of the
various tensors necessary for the stability analysis.

\subsection{Metric tensor}
According to Eq.~(\ref{eq:pertmetric}), the scalar perturbed metric
tensor reads, in terms of gauge invariant variables,
\begin{equation}
\begin{aligned}
\delta g_{\mu \nu} & = \ee^\sigma \eta_{\mu \nu} \Psi, & \delta g_{\mu
r} & = 0, & \delta g_{\mu \theta} & = 0, \\ \delta g_{r \theta} & =
-\Upsilon, & \delta g_{rr} & = - \Xi, & \delta g_{\theta \theta} & =
-\gTT \Omega.
\end{aligned}
\end{equation}
By means of
\begin{equation}
\delta g^{\si{AB}}=-g^{\si{AC}} g^{\si{BD}} \delta g_{\si{BD}},
\end{equation}
one can get the inverse perturbed metric tensor
\begin{equation}
\begin{aligned}
\delta g^{\mu \nu} & = - \ee^{-\sigma} \eta^{\mu \nu} \Psi, & \delta
g^{\mu r} & = 0, & \delta g^{\mu \theta} & = 0,\\ \delta g^{r \theta}
& = \gTTi \Upsilon, & \delta g^{rr} & = \Xi, & \delta g^{\theta
\theta} & = \gTTi \Omega.
\end{aligned}
\end{equation}
The perturbed Riemann tensor can also be expressed as a function of
the perturbed metric tensor through the perturbed Christoffel symbols
\begin{equation}
\delta R^{\si A}_{\si{\ BCD}} = -\delta \Gamma^{\si A}_{\si{\ BC};{\si
D}} + \delta \Gamma^{\si A}_{\si{\ BD};{\si C}},
\end{equation}
where the covariant derivatives with respect to the unperturbed metric
have been noted with a semicolon, and the perturbed connections are
given by
\begin{equation}
\delta \Gamma^{\si A}_{\si{\ BC}} = \frac{1}{2} g^{\si{AD}} \left(
\delta g_{\si{DB;C}} + \delta g_{\si{DC};{\si B}} - \delta
g_{\si{BC};{\si D}} \right).
\end{equation}

\subsection{Einstein tensor}

{}From the perturbed Riemann tensor, the perturbed Einstein tensor can
be expressed in terms of gauge invariant variables by means of
\begin{equation}
\delta G_{\si{AB}} = \delta R_{\si{AB}} - \frac{1}{2} R \delta g_{\si{AB}} -
\frac{1}{2} g_{\si{AB}} \delta R,
\end{equation}
where the perturbed Ricci scalar is
\begin{equation}
\delta R = g^{\si{AB}} \delta R_{\si{AB}} + \delta g^{\si{AB}}
R_{\si{BD}}.
\end{equation}
After some (tedious) calculations one gets
\begin{widetext}
\begin{equation}
\begin{aligned}
\delta G_{\mu \nu} & = \left(\partial_\mu \partial_\nu - \eta_{\mu
\nu} \Box \right) \left( \frac{\Xi + \Omega}{2} + \Psi \right) +
\frac{1}{2} \ee^\sigma \eta_{\mu \nu} \Bigg\{3 \Psi'' + 3 \gTTi
\partial^2_\theta \Psi + \Omega'' - 2 \gTTi \partial_\theta \Upsilon' +
\gTTi \partial_\theta^2 \Xi \\
& + 3 \left(2 \sigma' + \lngTTp \right)
\Psi' - \left. \left(\frac{3}{2} \sigma' + \lngTTp
\right) \Xi' + \left[\frac{3}{2} \sigma' + 2 \left(\lngTTp\right)
\right] \Omega' - 3 \gTTi \sigma' \partial_\theta \Upsilon
\right\} + G_{\mu \nu}  \left(\Psi - \Xi \right),
\end{aligned}
\end{equation}
for the purely brane part, while the mixed ones read
\begin{equation}
\begin{aligned}
\delta G_{\mu r} & = \frac{1}{2} \partial_\mu \Bigg\{3 \Psi' + \Omega'
- \gTTi \partial_\theta \Upsilon - \left(\frac{3}{2} \sigma' +
\lngTTp \right) \Xi + \left(-\frac{1}{2} \sigma' + \lngTTp \right)
\Omega \Bigg\},\\
\delta G_{\mu \theta} & = \frac{1}{2} \partial_\mu
\Bigg\{\partial_\theta\left(3 \Psi + \Xi \right) - \Upsilon' -
\left(\sigma' + \lngTTp \right) \Upsilon \Bigg\},
\end{aligned}
\end{equation}
and the purely bulk components are
\begin{equation}
\begin{aligned}
\delta G_{rr} & = \frac{1}{2} \ee^{-\sigma} \Box\left(3 \Psi +
\Omega\right) - 2 \gTTi \partial_\theta^2 \Psi + 2 \gTTi \sigma'
\partial_\theta \Upsilon - \left[3 \sigma' + 2 \left(\lngTTp \right)
\right] \Psi' - \sigma' \Omega',\\
\delta G_{\theta \theta} & = \frac{1}{2} \gTT \ee^{-\sigma} \Box
\left(3 \Psi + \Xi \right) - 2 \gTT \Psi'' + \gTT \sigma'
\left(\Xi' - 5 \Psi' \right) + \frac{1}{2} \gTT \left(4 \sigma'' + 5
\sigma'^2 \right) \left( \Xi - \Omega \right),\\
\delta G_{r \theta} & = -\frac{1}{2} \ee^{-\sigma} \Box \Upsilon +
2 \partial_\theta \Psi' - \sigma' \partial_\theta \Xi + \left[\sigma'
  - 2 \left(\lngTTp\right) \right] \partial_\theta \Psi -\frac{1}{2}
\left(4 \sigma'' + 5 \sigma'^2 \right) \Upsilon.
\end{aligned}
\end{equation}
\end{widetext}
where $\Box$ is the brane d'Alembertian defined above
[Eq.~(\ref{nabla})].

\subsection{Stress-energy tensor}

In terms of the underlying fields, the stress-energy tensor stemming from
Eq.~(\ref{eq:tmunumatt}) reads
\begin{equation}
\begin{aligned}
T_{\si{AB}} & =\frac{1}{4} \left(\DD_{\si A} \Phi \right)^\dag
\left(\DD_{\si B}
\Phi \right) + \frac{1}{4} \left(\DD_{\si B} \Phi \right)^\dag
\left(\DD_{\si A} \Phi \right) \\
& - g^{\si{CD}} \F_{\si{AC}} \F_{\si{BD}} -
g_{\si{AB}} \mathcal{L}_{\matter},
\end{aligned}
\end{equation}
with $\mathcal{L}_\matter$ given by Eq.~(\ref{eq:lag}); to zeroth
order, this is given by
\begin{equation}
\mathcal{L}_\matter = -\frac{1}{2} \gTTi \frac{Q'^2}{q^2} -
\frac{1}{2} \varphi'^2 - \frac{1}{2} \gTTi \varphi^2 Q^2 - V(\varphi),
\end{equation}
and to first order in metric and field perturbations, using
Eqs.~(\ref{eq:perthiggs}) and (\ref{eq:pertgauge}), this perturbed
matter Lagrangian reads
\begin{equation}
\begin{aligned}
\delta \mathcal{L}_\matter & = \gTTi \frac{Q'}{q} \left(\GperT' -
\partial_\theta \GperR \right) + \frac{1}{2} \gTTi \frac{Q'^2}{q^2}
\left(\Xi + \Omega \right) \\ & + \frac{1}{2} \varphi'^2 \Xi +
\frac{1}{2} \gTTi \varphi^2 Q^2 \Omega + \gTTi q \varphi^2 Q \GperT \\
& - \left[ \varphi' \Chip' \gTTi \varphi Q(Q+n-p) \Chip + {\dd V\over
\dd \varphi} \Chip \right] \\ & \times \cos\left[(n-p)\theta\right].
\end{aligned}
\end{equation}
The purely brane part of the perturbed stress-energy tensor is therefore
given by
\begin{widetext}
\begin{equation}
\begin{aligned}
\delta T_{\mu \nu} & = \ee^{\sigma} \eta_{\mu \nu} \Bigg\{-\gTTi
\frac{Q'}{q} \left(\GperT' - \partial_\theta \GperR \right) +
\left[\varphi' \Chip' + \gTTi Q (Q-n+p) \varphi \Chip + {\dd V \over
\varphi} \Chip \right] \cos\left[(n-p)\theta\right] \\ & -
\left(\frac{1}{2} \gTTi \frac{Q'^2}{q^2} + \frac{1}{2} \varphi'^2
\right) \Xi - \left(\frac{1}{2} \gTTi \frac{Q'^2}{q^2} + \frac{1}{2}
\gTTi \varphi^2 Q^2 \right) \Omega + \left[\frac{1}{2} \gTTi
\frac{Q'^2}{q^2} + \frac{1}{2} \varphi'^2 + \frac{1}{2} \gTTi
\varphi^2 Q^2 + V(\varphi) \right] \Psi \\ & - \gTTi q \varphi^2 Q
\GperT \Bigg\},
\end{aligned}
\end{equation}
while the mixed components are
\begin{equation}
\begin{aligned}
\delta T_{\mu r} & = \partial_\mu \Bigg\{- \gTTi \frac{Q'}{q}
\left(\GperT - \partial_\theta \Gper \right) + \varphi' \Chip
\cos\left[(n-p)\theta\right] \Bigg\},\\ \delta T_{\mu \theta} & =
\partial_\mu \Bigg\{ \frac{Q'}{q} \left(\GperR - \Gper' \right)
-\varphi Q \Chip \sin\left[(n-p)\theta \right] -q \varphi^2 Q \Gper
\Bigg\},
\end{aligned}
\end{equation}
and the bulk ones
\begin{equation}
\begin{aligned}
\delta T_{rr} & = -\gTTi \frac{Q'}{q}\left(\GperT' -
\partial_\theta \GperR \right) + \left[ \varphi' \Chip' - \gTTi
\varphi Q(Q-n+p) \Chip - {\dd V\over\dd\varphi} \Chip \right]
\cos\left[(n-p)\theta\right] \\ & - \left[\frac{1}{2} \gTTi \varphi^2
Q^2 + V(\varphi) \right] \Xi - \left(\frac{1}{2} \gTTi
\frac{Q'^2}{q^2} - \frac{1}{2} \gTTi \varphi^2 Q^2 \right) \Omega +
\gTTi q \varphi^2 Q \GperT ,\\ 
\delta T_{\theta \theta} & =
-\frac{Q'}{q} \left(\GperT' - \partial_\theta \GperR \right) +
\left[-\gTT \varphi' \Chip' + \varphi Q(Q-n+p) \Chip - \gTT {\dd
V\over\dd\varphi} \Chip \right] \cos\left[(n-p) \theta \right] \\ & +
\left(\frac{1}{2} \gTT \varphi'^2 - \frac{1}{2}
\frac{Q'^2}{q^2}\right) \Xi - \gTT \left[ \frac{1}{2} \varphi'^2 +
V(\varphi) \right] \Omega - q \varphi^2 Q \GperT,\\ 
\delta T_{r \theta} & = \left[-\varphi Q \Chip' + \varphi'(Q-n+p)
\Chip \right] \sin\left[(n-p)\theta\right] + \left[\frac{1}{2} \gTTi
\frac{Q'^2}{q^2} - \frac{1}{2} \varphi'^2 - \frac{1}{2} \gTTi
\varphi^2 Q^2 - V(\varphi) \right] \Upsilon - q \varphi^2 Q
\GperR.
\end{aligned}
\end{equation}
\end{widetext}

\subsection{Faraday tensor}

In order to directly derive the perturbed Maxwell equations from
Eq.~(\ref{eq:max}), we have used the following perturbed Faraday
tensor whose purely brane components vanish
\begin{equation}
\delta \F_{\mu \nu} = 0 = \delta \F^{\mu \nu},
\end{equation}
and with the mixed parts
\begin{equation}
\begin{aligned}
\delta \F_{\mu r} & = \partial_\mu \left(\GperR - \Gper' \right),\\
\delta \F^{\mu r} & = - \ee^{-\sigma} \eta^{\mu \nu} \partial_\nu
\left(\GperR - \Gper' \right),\\ \delta \F_{\mu \theta} & =
\partial_\mu \left(\GperT - \partial_\theta \Gper \right),\\ \delta
\F^{\mu \theta} & = - \frac{\ee^{-(\sigma+\gamma)}}{\rdim^2} \eta^{\mu
\nu} \partial_\nu \left(\GperT - \partial_\theta \Gper \right).
\end{aligned}
\end{equation}
The only non-vanishing purely bulk components end up being
\begin{equation}
\begin{aligned}
\delta \F_{r \theta} & = \GperT' - \partial_\theta \GperR,\\ \delta
\F^{r \theta} & = \gTTi \left[\frac{Q'}{q} \left(\Xi + \Omega \right)
+ \GperT' - \partial_\theta \GperR \right].
\end{aligned}
\end{equation}
Owing to these formulas, one can calculate the perturbed Faraday
tensor divergence involved in Eq.~(\ref{eq:max}) by means of
\begin{equation}
\delta \F^{\si{AB}}_{\ \ \ ;\si{A}} = \partial_{\si A} \delta \F^{\si{AB}} +
\Gamma^{\si A}_{\ \si{DA}} \delta \F^{\si{DB}},
\end{equation}
where we have used the antisymmetry property of $\delta
\F_{\si{AB}}$. The perturbed left hand side of Eq.~(\ref{eq:max}) can
be, in turn, expressed in terms of the perturbed matter fields by
means of Eqs.~(\ref{eq:d}), (\ref{eq:perthiggs}) and
(\ref{eq:pertgauge}) to give the perturbed Maxwell equations
(\ref{eq:pertmax_mu}) to (\ref{eq:pertmax_theta}).

\newpage

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\end{document}

