text:
\section{Introduction}
The conjecture about that our {\it 3 + 1} dimensional world might be
allocated on a brane (or domain wall) in a multi-dimensional space-time
has recently invoked much activity \ci{R-S}-\ci{RS1}, owing to the new
tools it provides to solve the long standing mass and scale hierarchy
problems \ci{ADD}-\cite{fmasshi2} in particle theory.
New extra dimensional physics could manifest itself in accessible
experiments and observations, when the size of extra dimensions is
relatively large \ci{ADD,ADD2} or even infinite
\ci{R-S}, \ci{infin1} -- \ci{infin4},
in as much to feed
research programs in running and forthcoming collider and non-collider
experiments
\ci{ADD2}, \ci{KKgrav1}--\ci{missch}, \ci{bran3}, \ci{fluct}.
Nowadays the brane world scenarios and their applications are well
summarized in a dozen of review articles
\ci{rev1}--\ci{rev10}.

On the one hand, the brane itself is often considered as an
{\sl elementary} geometrical object of a vanishing thickness along
the extra dimensions, as it is promoted by  superstring
theories (with supersymmetric compactifications, see \ci{rev2} and references therein,
and non-supersymmetric ones, see \ci{koko} and references therein).
In other words, this kind of brane represents just a
boundary for higher dimensional objects. On the other hand, such an approach
gives rise to the question about the origin of branes trapping our
matter world. In addition to, when endowing branes with a tiny thickness,
one allows some more options for a solution of the mass hierarchy
problem \ci{fmasshi1,fmasshi2,masshi1,masshi2} (see also reviews \ci{rev1,rev3,rev9}).

A non-antagonistic alternative to the infinitely thin branes is
provided by an effective multi-dimensional field theory,
owing to the possibility of a spontaneous breaking of the
translational symmetry. The thick (or fat) brane (or domain wall)
formation and the trapping of light particles
in its layer might be obtained \ci{scal1}--\ci{galoc3} by a number of particular background
scalar and/or gravitational fields living in the multi-dimensional bulk,
when their vacuum configuration has a non-trivial topology,
thereby generating zero-energy states localized on the brane.

Respectively, the mechanism of how such background fields might emerge and
further induce the spontaneous breaking of translational symmetry is worthy
to be elaborated and the  domain wall creation, due to the self-interaction
of certain particles in the bulk, may become a conceivable and appealing
possibility \ci{aags}.

In this paper we continue the exploration of a non-compact
{\it 4 + 1} dimensional fermion model \ci{aags} with a local strong
four-fermion interaction, by supplementing it with a partially induced
background gravitational field. Both kinds of interactions will lead coherently
first to the discrete symmetry breaking and, further on, to the breaking of translational
invariance. This can be achieved in terms of a particular domain wall pattern of the
vacuum state \ci{Rajar}-- \ci{Baz},
just allowing the light massive Dirac particles to live essentially in
{\it 3 + 1} dimensions. Those very same interactions generate localized zero-modes for the
composite scalar fields, just completing the fermion matter content on the brane
with a scalar counterpart.

We shall concentrate ourselves upon
the main dynamical origin of the spontaneous symmetry breaking
-- on fermion self-interaction supplemented by gravity -- and yet simplify
the model by neglecting all gauge field
interaction. In this sense, our model may be considered a sector
of the so called {\sl domain wall gravitating standard model}.

The five dimensional fermion model with spin-0 and spin-2 induced
self-interaction is formulated in Section 2. There the fermion
self-interaction {\it via} the spin-0 channel is
restricted to a four-fermion type that will be sufficient to
trigger the localization of massive Dirac fermions. It contains two
dimensional coupling constants expressed in units of the compositeness
scale $\Lambda$. The latter one plays the role of a cut-off for virtual
fermion energies and momenta.

Meanwhile, the interaction {\it via} the spin-2 channel ({\sl extra dimensional gravity})
is introduced in a non-linear way, in order to make this model covariant under the
space-time diffeomorphisms, with a specific five dimensional Einstein-Hilbert bare Action
and a bare cosmological constant to balance the formation
of the physical Newton's and cosmological constants on the brane.
Among different options, we pay  attention to the case when the
five dimensional gravity is fundamental and its related
Einstein-Hilbert term is not too much corrected by fermion induced
effective Action. As well, we shall turn ourselves
to the economical scenario of induced gravity when, contrary to the
previous case,  the bare gravitational constant may be taken negligible
and gravity is principally induced by fermion matter being therefore composite.

In this Section 2, once the
scalar bosonization of the four-fermion interaction has been
implemented, the low-energy effective Action for composite scalar and
gravity fields is obtained in the mean field or large-$N$ approximation,
where $N$ roughly counts the number of fermion species in the Standard Model.
This effective Action, which arises out of fermion one-loop radiative
corrections, does accumulate the radiative contributions of high-energy
fermion virtualities to describe infrared phenomena of spontaneous
symmetry breaking. It already contains the kinetic terms for the scalar
auxiliary fields, endowing them with the structure of composite fields.
In the calculation of that effective Action, the Euclidean space-time
approach for the invariant cut-off and the finite-mode regularization \ci{AnBo}
for separation of high- and low-energy fermion fields are employed.

In Section 3 we search for classical vacuum configurations of
gravity and scalar fields by analyzing the low-energy
effective Action. This search is restricted to the class of conformal-like
metrics (warped geometries) with the flat Minkowski hyperplanes at
each point along the fifth coordinate. The joint solution of
Einstein and non-linear Klein-Fock-Gordon equations is properly found within the
weak-gravity approximation, {\it i.e.} assuming a relatively small
five dimensional gravitational constant, what will be eventually justified
in Section 4 after normalization to the Newton's gravitational constant on
the brane. As expected to the leading order, the equations of
motion for the scalar fields are not affected by gravity and their solutions
coincide with the flat case investigated in \ci{aags}. In this sense,
in the model under discussion, the brane and warped geometry creation
is basically maintained by matter but not by gravity\footnote{The interplay
between
gravity and matter effects in brane world formation and corresponding
Einstein equations have been  subjects
of extensive studies in different models
\ci{R-S,rev1,rev5,rev10,scal1,scal2,scal3,scal5,gravbr}.}.

We select out the magnitudes of our four-fermion coupling constants
in such a way that the dynamical breaking of the so called $\tau$-symmetry
and translational invariance were supported and, moreover, that the
brane located Dirac fermions were supplied  with masses.
The characteristic scale $M$ of
the symmetry breaking -- inverse thickness of a brane -- can be
conceivably much less than the compositeness scale $\Lambda\,,$ keeping
consistently the brane formation and particle localization phenomena
in the low-energy region.

The remaining pair of the Einstein-like field equations can be
rearranged so that one of them does specify the warp factor of the
five dimensional anti-de Sitter (AdS$_5$) geometry, whereas the other
one does represent the integral of motion with the bulk cosmological
constant as an integration constant.
The latter property holds exactly in all order of gravitational
perturbation theory. As a consequence, this integration constant
is found to be firmly fixed by the vacuum (kink-like) configurations
of the scalar fields and of the warped geometry.

In Section 4 we examine the spectral properties of light particles
allocated on the brane, we describe the structure of the ultra-low energy
Lagrange density and finally summarize the results of \ci{aags} for
the coupling constants of light particle interactions,
which are parameterized by the ratio $M/\Lambda$ of
the symmetry breaking and the compositeness scales.
The mass spectrum of light particle together with their interactions are described
up to the leading order in the gravitational constant.
We argue that this leading contribution into the brane world matter interaction is,
in fact, independent of the five dimensional gravity, in spite of the
nonperturbative quantum tunneling of massive states off the brane
\ci{galoc2} in the AdS$_5$ geometry.

Section 5 is devoted to the estimation of all the scales and the
coupling constants we have previously introduced. This can be achieved
by imposing the observed value of the  Newton's constant and
the Newton's gravitational law within the presently available limits
in modern experiments \ci{adelb}. It is adopted that
both the scale of compositeness and the
translational symmetry breaking scale must be naturally very high,
in between several TeV's and the GUT scale $10^{15}$ GeV.
Therefrom it is derived that, if five dimensional gravity is
fundamental and displaying some Planck-like scale much larger
than the compositeness scale, then there is a window for a non-trivial
(but weak) interaction between brane fluctuations (branons \ci{bran3,bran1,bran2}),
Higgs-like scalars and fermions, with a low compositeness scale of the order
tens of TeV's, the trapping barrier for light particles as high as $2\div 3$
TeV and the AdS$_5$ curvature scale $\sim 10^{-3}$ eV, which is comparable with the
existing experimental checks of the Newton's law \ci{adelb}.

On the other hand, if the induced gravity is dominated
at ultra-low energies, then the scalar matter interaction is highly
suppressed, just making branons good sterile candidates
to contribute into dark matter \ci{bran3}.
The cosmological constant induced on the
brane is also calculated and found to vanish exactly,
just making consistently endorsed the {\it ansatz} for the flat
Minkowski's hyperplanes\footnote{This
  phenomenon of exact compensation between gravity and matter
  contributions into the brane cosmological constant has been also found  in
 \ci{R-S}, \ci{cosm1} -- \ci{cosm3} in
different brane-world models.}.

In the concluding Section we mainly summarize our observations on the interplay between
the five dimensional Planck-like mass, the compositeness and the dynamical symmetry
breaking scales and the AdS$_5$ curvature. As a further development,
the possibility of implementing an occasional tiny matter and vacuum energy defect,
located in the forth space direction in the presence of gravity, is
shortly discussed. As a matter of fact, in our previous paper \cite{aags} we
have demonstrated that such a defect may trigger and thereby justify
the dynamical breaking of translational invariance at a particular place in
the extra
dimensions\footnote{Another realization of thin defects in extra
  dimensional world see in \ci{def}.}
and it might supply the branon degrees of freedom with a small mass.

\section{Five dimensional fermion model with scalar and gravity
induced self-interaction}

Let us remind \ci{aags} the domain wall phenomenology
 and introduce the necessary notations. We start from the model of
one four-component fermion bi-spinor field $\psi(X)$
defined on a five dimensional flat Minkowski space-time and
coupled to a scalar field $\Phi(X)$.
The extra-dimension coordinate is assumed to be space-like,
$$
X^A = (x^\mu, z)\ , \quad x^\mu = (x^0, x^1, x^2, x^3)\ , \quad
(\eta^{AA}) = (+,-,-,-,-)
$$
and the subspace of coordinates $x^\mu$ eventually corresponds to the
four dimensional Minkowski space.
The extra-dimension size is supposed to be infinite (or large enough).
The fermion wave function is then described by the Dirac equation
\begin{equation}
[\,i\gamma^A \partial_A - \Phi(X)\,]\psi(X) = 0\ , \quad
\gamma^A = (\gamma^\mu, i\gamma_5)\ ,\quad \{\gamma^A, \gamma^B\}
= 2\eta^{AB}\ , \la{5dir}
\end{equation}
$\gamma^\mu$ being a standard set of four dimensional Dirac matrices in the
chiral (or Weyl) representation with
$\gamma_5\equiv i\gamma^0\gamma^1\gamma^2\gamma^3$.

The trapping of light fermions on a four dimensional hyper-plane
-- the domain wall -- localized in the
fifth dimension at $z = z_0$ can be promoted
by a certain {\sl topological}, $z$-dependent configuration of
the vacuum expectation value of the scalar field
\mbox{$\langle\Phi(X)\rangle_0 = \varphi (z)$} -- for
instance $ \varphi (z) =
M\, \mbox{\rm tanh}(Mz)$ --
due to the appearance of zero-modes with a certain chiralities in the spectrum
of the four dimensional Dirac operator \ci{R-S,rev1}.

If we aim to build up a light Dirac fermion, we need two different
chiralities for the same shape of scalar background. Then the minimal
set of fermions over the five dimensional space-time
has to include \ci{galoc2,aags} two proto-fermions $\psi_1(X), \psi_2(X)$.
In order to generate left- and right-handed
parts of a four dimensional Dirac bi-spinor as zero modes,
those fermions have to couple to the  scalar field $\Phi(X)$
with opposite charges,
\begin{equation}
[\,i \not\!\partial - \tau_3\Phi(X)\,]\Psi(X) = 0\ ,\quad
\not\!\partial \equiv \widehat\gamma^A \partial_A\ ,\quad
\Psi(X) =\left\lgroup\begin{array}{c}\psi_1(X)\\
\psi_2(X)\end{array}\right\rgroup\ ,
\la{2fer}
\end{equation}
where $\widehat\gamma^A \equiv \gamma^A\otimes {\bf 1}_2$ are
Dirac matrices and
$\tau_a \equiv {\bf 1}_4 \otimes \sigma_a,\ a=1,2,3 $
are the generalizations of
the Pauli matrices $\sigma_a$ acting on the bi-spinor components $\psi_i(X)$.

In addition to the trapping scalar field,
a further one is required to supply light domain wall fermions
with a mass. Its coupling must mix left and right chiralities as the
mass term breaks the chiral invariance.
Thus we introduce two types of four-fermion self-interactions
to reveal two composite scalar fields with  a proper coupling to fermions.
These two scalar fields
acquire mass spectra similar to fermions with light counterparts located
on the domain wall. The dynamical scheme of creation of domain wall
particles turns out to be quite economical and few predictions on masses
and decay constants of fermion and boson particles have been derived \ci{aags}.
However the allocation of matter on the domain wall certainly lead to
strong gravitational effects. Moreover, the gravity itself may cause
in turn the localization of the matter fields on a domain wall.

In the present paper we shall extend the dynamical mechanism of the fermion
self-interaction by including, partially or completely,
the gravitational contribution as induced by  the high-energy spinor matter.
Our main interest is focused on the
thick brane formation corresponding to the flat {\it 3+1} dimensional
Minkowski space, whereas the gravity is non-trivial in the
fifth direction orthogonal to the Minkowski's world. It is treated
self-consistently together with the vacuum configurations of composite
scalar fields.

Let us now formulate the fermion model in five
dimensions\footnote{We extend the idea of the Top-mode Standard Model (TmSM)
\ci{Topmode} on extra dimensions. Other extensions of TmSM have been
undertaken in \ci{6top1,6top2}.}, which
implements the mechanism of translational symmetry breaking to
create a domain wall. It is described by the classical Lagrange density
\begin{equation}
{\cal L}^{(5)} (\overline{\Psi},\Psi) = \overline{\Psi}\ i \!\not\!\partial \Psi +
\frac{g_1}{4N \Lambda^3}\left(\overline{\Psi}\tau_3 \Psi\right)^2
+ \frac{g_2}{4N \Lambda^3}\left(\overline{\Psi} \tau_1\Psi\right)^2,
\la{mod}
\end{equation}
where
$\Psi(X)$ is an eight-component five dimensional
fermion field, see eq.\,\gl{2fer} --
either a bi-spinor in a four dimensional theory or a spinor in a
six-dimensional theory --
which may also realize a flavor and color multiplet with the total number
$N=N_f\,N_c$ of spinor degrees of freedom. We can say that, {\it grosso modo},
the number of color and flavor
degrees of freedom of massive fermions in the Standard Model is around twenty,
if all of them are originated from corresponding five dimensional proto-fermions.
The ultraviolet cut-off scale $\Lambda$ bounds fermion momenta,
as the four-fermion interaction is supposed here to be an effective one,
whereas $g_1$ and $g_2$ are suitable dimensionless
and eventually scale dependent effective couplings.

This Lagrange density can be more conveniently represented with the help of a pair of
auxiliary scalar fields $\Phi(X)$ and $H(X)$, which eventually will allow to
trap a light fermion on the domain wall and to supply it with a mass: namely,
\begin{equation}
{\cal L}^{(5)} (\overline{\Psi},\Psi,\Phi, H) =
\overline{\Psi} ( i\!\not\!\partial
- \tau_3 \Phi -\tau_1 H) \Psi - \frac{N \Lambda^3}{g_1}\,\Phi^2
- \frac{N \Lambda^3}{g_2}\,H^2\ . \la{aux}
\end{equation}
For sufficiently strong couplings,
this system undergoes
the phase transition to the state  in which the condensation of
fermion-antifermion pairs does spontaneously  break --
partially or completely -- the so-called $\tau$-symmetry: \mbox{$
\Psi \longrightarrow \tau_1 \Psi;\
\Phi \longrightarrow -\,\Phi$;} and $\Psi \longrightarrow \tau_3 \Psi;\
H \longrightarrow -\,H$.

In order to develop the infrared phenomenon of the $\tau$-symmetry breaking,
the effective Lagrange density containing  essential low-energy
degrees of freedom has to be suitably derived.
To this concern, we proceed
in the transition to the Euclidean space,
where the invariant momentum cut-off can be unambiguously implemented.
Within this framework, the notion
of low-energy is referred to momenta $|p| < \Lambda_0$ as compared with
the cut-off $\Lambda_0 \ll \Lambda$. However, after the elaboration
of the domain wall vacuum, we will search for the fermion states
with masses $m_f$ much lighter than the dynamical scale $\Lambda_0$,
{\it i.e.} for the ultralow-energy physics.
Thus, eventually, there are three scales in the present model
in order to implement the domain wall particle trapping.

Let us decompose the momentum space fermion fields into their high-energy part
$\Psi_h(p)\equiv\Psi(p)\vartheta(|p|-\Lambda_0)\vartheta(\Lambda-|p|)$,
their low-energy part $\Psi_l(p)\equiv\Psi(p)\vartheta(\Lambda_0-|p|)$
and integrate out the high-energy part of the fermion spectrum,
$\vartheta(t)$ being the usual Heaviside's step distribution.

The above decomposition of the fermion spectrum
should be done covariantly, {\it i.e.}
in terms of the full Euclidean Dirac's normal operator,
\begin{equation}
\not\!\!D \equiv i (\not\!\partial + \tau_3 \Phi + \tau_1 H)\ .\la{eudir}
\end{equation}
As we want to investigate  the
$\tau$-symmetry breaking by fermion condensation, in what follows we can
neglect the high-energy components of the auxiliary boson fields,
what is equivalent to adopt the mean-field or large $N$ approximations.
Then the low-energy Euclidean Lagrange density, which accounts for
low-energy fermion and boson fields, can be written as the sum of
the Euclidean counterpart of the classical Lagrange density
\gl{aux} and the induced one-loop contribution, arising by
functional integration of the high-energy fermion field components:
namely,
\begin{equation}
{\cal L}^{(5)}_{\rm low} (\overline{\Psi}_l,\Psi_l,\Phi, H) =
{\cal L}^{(5)}_E (\overline{\Psi}_l,\Psi_l, \Phi, H) +
\Delta {\cal L}^{(5)} (\Phi, H)\ , \la{lowl}
\end{equation}
where the classical Euclidean Lagrange density reads
\begin{equation}
{\cal L}^{(5)}_E (\overline{\Psi}_l,\Psi_l,\Phi, H) =
\overline{\Psi}_l\not\!\!D \Psi_l + \frac{N \Lambda^3}{g_1}\,\Phi^2
+ \frac{N \Lambda^3}{g_2}\,H^2\ .
\la{auxeucl}
\end{equation}
The one-loop contribution of high-energy fermions is given by
\begin{eqnarray}
&&\Delta {\cal L}^{(5)} (\Phi, H)=
-(N/2)\,\mbox{tr}\,\langle X|{\rm A}|X\rangle\la{leff}\ , \\
&&{\rm A} \equiv \vartheta(\Lambda^2 - \not\!\!D^\dagger\!\not\!\!D)
\ln\frac{\not\!\!D^\dagger\!\not\!\!D}{\Lambda^2} - \vartheta(\Lambda_0^2 -
\not\!\!D^\dagger\!\not\!\!D) \ln\frac{\not\!\!D^\dagger\!\not\!\!D}{\Lambda_0^2}\ ,
\nonumber
\end{eqnarray}
where the symbol [ tr ] stands for the trace over
spinor and internal degrees of freedom. In the latter operator [
${\rm A}$ ] we have incorporated the cut-offs which select out the
above defined high-energy region  \ci{AnBo}. For $n=5$ we eventually
found \cite{aags}
\begin{eqnarray} &&\Delta {\cal L}^{(5)} (\Phi,
H)\stackrel{\Lambda\rightarrow\infty}{\sim}\nonumber\\
&&\frac{N\Lambda}{4\pi^3}\left[\,(\partial_A \Phi)^2 + (\partial_A
H)^2\,\right]-\frac{N\Lambda^3}{9\pi^3}\left(\Phi^2 + H^2\right)
+\frac{N\Lambda}{4\pi^3}\left(\Phi^2 + H^2\right)^2\ .\la{a12}
\end{eqnarray}
\medskip
Now, let us switch on gravity as described by the metric field
$g_{AB} (X)$ on a five dimensional Riemannian manifold ${\cal M}_5$
which is called the base space. The capital Latin indexes $A,B,C,\ldots $
are usually called the base indexes or holonomic indexes.
The Action is appropriately normalized
with the help of the determinant of this metric,
\begin{equation}
{S}(\Phi, H, \overline{\Psi}_l, \Psi_l,g) = \int_{{\cal M}_5} d^5 X
\sqrt{g} \ \left[ {\cal L}^{(5)}_{\rm fermion} +
{\cal L}^{(5)}_{\rm boson}\right]\ ;
\quad
g \equiv \mbox{\rm det}(g_{AB})\ , \label{invact}
\end{equation}
where the fermion (spinor) and boson (scalar and gravity) parts of the Lagrange density
will be  defined here below.
Namely, for the Lagrange density \gl{auxeucl} the form invariant under diffeomorphisms
of the part bilinear in fermion fields is given in terms of the pentad--fields
or {\it f\"unfbeine} $e^i_A(X)$,
which connect locally the curved manifold ${\cal M}_5$ with the flat
space with  Euclidean signature, so that
\begin{equation}
e^i_A(X) e^i_B(X)  = g_{AB}(X)\ ;\quad
e_i^A(X) e_i^{B}(X)  = {g}^{AB}(X)\ ;\quad
e^i_A(X) e_j^A(X) = \delta^i_j\ ,
\end{equation}
where the  Euclidean frame Latin small indexes, also called anholonomic,
run from one to five, in such a way that
\begin{equation}
\{\widehat\gamma_j, \widehat\gamma_k\} =
2\delta_{jk}\,\left({\bf 1}_4\otimes{\bf 1}_2\right)\ .
\end{equation}
The invariant spinor Lagrange density then reads
\begin{eqnarray}
{\cal L}^{(5)}_{\rm fermion} (\overline{\Psi}_l,\Psi_l,\Phi, H, g) &=&
i\overline{\Psi}_l\left[\,\widehat\gamma_k e^A_k \left(\partial_A + \omega_A\right)
+ \tau_3 \Phi +\tau_1 H\,\right] \Psi_l \nonumber\\
&=&
i\overline{\Psi}_l\left(\,\!\not\!\nabla
+ \tau_3 \Phi + \tau_1 H\right) \Psi_l\nonumber\\
&\equiv&
\overline{\Psi}_l\not\!\!{\bf D}\Psi_l\ , \la{aux1}
\end{eqnarray}
where the spin connection $\omega_A$ can be represented in terms of the
{\it f\"unfbeine} and the affine connection
\begin{equation}
\Gamma^{\,C}_{AB} = \frac12
g^{CD}\left(\partial_A g_{BD} + \partial_B g_{AD}
- \partial_D g_{AB}\right)\ .
\end{equation}
The spin connection has the following form
\begin{equation}
\omega_A \equiv \frac18\,[\,\widehat\gamma_i, \widehat\gamma_j\,]\,\Gamma^{ij}_A\ ;\quad
\Gamma_A^{ij}=e^{Bj} \Gamma_{AB}^{\,C} e_C^i -
e^{Bj} \partial_A e_B^i\ .\label{spincon}
\end{equation}
As before in eq.~\gl{auxeucl}, the fermion self-interaction is
induced by those Yukawa--like vertexes which are
linear in the auxiliary scalar fields $\Phi, H$. We remind that the
symmetry breaking phase arises when the classical scalar interaction
compensates large contributions $\sim \Lambda^3$ in the low-energy effective
action \gl{a12} induced by high-energy fermions.
When the bare dynamical gravity is added, one is expected
to find similar large contributions $\sim \Lambda^5$ and
$\sim \Lambda^3$
in the one-loop effective Action, which may be tuned to end up with a
gravitational dynamics not
severely suppressed by a very high cosmological constant and a
very small Newton's constant. As we will see the zeroth- and first-order
Seeley--Gilkey
coefficients $a_0, a_2$ contain respectively the invariant measure
$\sqrt{g}$ and the scalar curvature $R\,,$
where the curvature scalar is defined in the
conventional way \cite{pauli,land}. Actually, let us denote with
\begin{eqnarray}
&&{R^A}_{BCD} =
\partial_C \Gamma_{BD}^A -
\partial_D \Gamma_{BC}^A +
\Gamma_{BD}^E \Gamma_{EC}^A -
\Gamma_{BC}^E \Gamma_{ED}^A\ ;
\label{Riemann}\\
&&R_{BD} \equiv {R^A}_{BAD}\ ;\quad
R \equiv g^{BD}R_{BD}
\end{eqnarray}
the Riemann curvature tensor, the Ricci tensor and the scalar curvature
respectively.

Now, if we aim to compensate all the one-loop large
contributions as induced by the fermionic matter,
we must add the classical bosonic Euclidean Lagrange
density
\begin{equation}
{\cal L}^{(5)}_{\rm boson} (\Phi, H, g) = N \Lambda^3
\left(\frac{\Phi^2}{g_1} + \frac{H^2}{g_2}\right) - {\Lambda\over
{\cal G}}\left( \varepsilon \frac{R}{2} - \lambda_0\right)\ , \label{bare}
\end{equation}
We notice that in eq.~(\ref{bare}) the quantity $\lambda_0 \sim \Lambda^2$
stands for the bare cosmological constant of the five dimensional
universe. In turn, the Newton--like constant ${\cal G} \sim
1/\Lambda^2$ does specify the strength of the five dimensional
gravitational interaction and its order of magnitude will be
suitably tuned later on.

Yet, with the help of the factor \mbox{$\varepsilon
= 0, \pm 1$} we reserve ourselves the possibility
to regulate different physical options for bare
gravity: namely, for \mbox{$\varepsilon = \pm 1$} the bare Einstein-Hilbert
Action is admittedly generated from a more fundamental theory.
On the one hand, it turns out that either this bare Action
screens the induced gravity contribution for \mbox{$\varepsilon = -1\,,$}
just like in the spirit of phenomenological supersymmetry,
or, conversely, it enhances the induced gravity effect, for \mbox{$\varepsilon=1\,,$}
so that the five dimensional gravity appears to be very weak.
Both cases do imply some
particular dynamics beyond the compositeness scale responsible for the
formation of a pure gravitational interaction.
On the other hand, the choice of \mbox{$\varepsilon = 0$} does give rise
to another scenario, in which gravity is entirely induced by fermionic matter,
{\it i.e.} by proto-fermions in the present model.
In this latter case, we see how the five dimensional gravitational fields will
certainly result to be very weak as for \mbox{$\varepsilon =1\,.$}
We find this option quite interesting and we shall estimate the related scales
of the five dimensional bulk physics, providing eventually
the usual Newton's gravity in our brane universe.

Now we want to calculate the low-energy effective Action
in the curved five dimensional space owing to the presence of gravity.
To this purpose, let us
define the elliptic second-order operator having the
same spectrum as the original Dirac operator in \gl{aux1}.
We use the conjugation property
$\not\!\!{\bf D}^\dagger = \tau_2\not\!\!{\bf D}\tau_2$ which relates the diffeomorphism
and frame covariant Dirac operator $\not\!\!{\bf D}$ to its hermitian
conjugate with respect to the invariant scalar product, as defined
by the invariant measure of eq.\gl{invact}.
Thus, the diffeomorphism and frame
covariant Euclidean Dirac operator
is still a normal operator, which has to be implemented in order to get
a real effective Action
and to define the spectral cut-offs with the help of the positive operator
\begin{eqnarray}
\not\!\!{\bf D}^\dagger\not\!\!{\bf D} &=& (i\!\not\!\nabla)^2 +
\Phi^2(X)  + H^2(X)   - \tau_3 \!\not\!\partial \Phi(X)
 - \tau_1 \!\not\!\partial H(X)\ . \la{quad}
\end{eqnarray}
This elliptic second-order differential positive operator can be re-expressed in a quite
general form \ci{cogia},
\begin{equation}
\not\!\!{\bf D}^\dagger\not\!\!{\bf D}\ =\ -\ g^{AB}(X)D_A D_B + {\cal M}^2(X)\
\equiv\ -\ D^2 + {\cal M}^2(X)
\label{eq:KG-heat}
\end{equation}
where ${\cal M}^2(X)$ is a matrix--valued multiplicative ({\it i.e.} non--differential)
operator, whereas
the full covariant derivative for a mixed quantity $f$ having anholonomic
(left understood) and holonomic indexes is given by
$D_A f^C = (\partial_A + \omega_A)f^C + \Gamma^C_{AB} f^B\,.$
Notice that the full covariant Laplace operator $D^2$
reduces to the Laplace-Beltrami operator when acting on
a scalar quantity
\begin{equation}
D^2f(X)=g^{AB}(X)\partial_A\partial_B f(X)+
\frac{1}{\sqrt{g(X)}}\,\partial_A
\left[\,g^{AB}(X)\sqrt{g(X)}\,\right]\partial_B f(X)\ .
\end{equation}
Explicit evaluation yields
\begin{equation}
{\cal M}^2(X) =
\frac{R}{4} + \Phi^2(X)  + H^2(X) -
\tau_3 \!\not\!\partial \Phi(X) -
\tau_1 \!\not\!\partial H(X)\ .
\end{equation}
where the identity matrix $\widehat{\bf 1}\equiv {\bf 1}_4\otimes{\bf 1}_2$
is always left understood.
Now, let us calculate the effective low energy Lagrange density
induced by high energy proto--fermions.
One can see that, in fact, the scale anomaly only contributes
into  $\Delta {\cal L}^{(5)}$, {\it i.e.} that part which depends upon the scales.
Thus, equivalently,
\begin{equation}
 \Delta {\cal L}^{(5)} (\Phi, H, g) = N \int^{\Lambda}_{\Lambda_0}
\frac{dQ}{Q}\
\mbox{tr}\langle X|
\vartheta(Q^2 - \not\!\!{\bf D}^\dagger\not\!\!{\bf D})|X\rangle\ .
\la{scan}
\end{equation}
As we assume that the scalar fields carry momenta much smaller than the lower scale
$\Lambda_0$, then the diagonal matrix element
in the RHS of eq.\,\gl{scan} can be
calculated with the help of the derivative expansion of the
representation \ci{AnBo}
\begin{equation}
\mbox{tr}\langle X|\vartheta(Q^2 - \not\!\!{\bf D}^\dagger\not\!\!{\bf D})|X\rangle =
\int_{-\infty}^{+\infty}\frac{d t}{2\pi i}\
\frac{\exp\{i t\}}{t - i\varepsilon}\ \mbox{tr}\langle X|
\ \exp\left\{- i\,\not\!\!{\bf D}^\dagger\not\!\!{\bf D}/{Q^2}\right\}|X\rangle
\label{intrep}
\end{equation}
where $X$ belongs to the base space,
which is supposed to be a Riemannian manifold of dimension $n\,.$
The trace in eq.~(\ref{intrep}) can be calculated by means
of the heat kernel method (see its review \ci{vasil}), as shown in Appendix A.
For $n=5 $, only three heat kernel coefficients  at most
are proportional to non--negative powers of the large parameter $Q$,
\begin{eqnarray}
&&\mbox{tr}\langle X|\vartheta(Q^2 - \not\!\!{\bf D}^\dagger\not\!\!{\bf D})|X\rangle \approx \frac{Q^5}{15\pi^3}
- \frac{Q^3}{3\pi^3}\,\left[\,\Phi^2(X) + H^2(X) + \frac{R(X)}{12}\,\right] \nonumber\\
&& +\ \frac{Q}{4\pi^3}\left\{
\partial_A\Phi(X)\partial^A\Phi(X) +
\partial_A H(X)\partial^A H(X) +
[\,\Phi^2(X)+H^2(X)\,]^2\right.\nonumber\\
&&\left. +\ \frac{1}{6}\,R(X)[\,\Phi^2(X)+H^2(X)\,]
- \frac13\,D^2[\,\Phi^2(X)+H^2(X)\,] - \frac{1}{60}\,D^2 R(X)
\right.\nonumber\\
&&\left. -\ \frac{7}{720}\,
R_{ABCD}(X)R^{ABCD}(X) - \frac{1}{90}\,
R_{AB}(X)R^{AB}(X) + \frac{R^2(X)}{144}\right\}\ ,
\la{elem}
\end{eqnarray}
where, for large scales $\Lambda_0\ll Q < \Lambda$, the neglected terms
rapidly vanish.
Inserting the RHS of eq.\,\gl{elem} in eq.\,\gl{scan} and taking into account
that $\Lambda_0 \ll \Lambda$, one can neglect the $\Lambda_0$--dependence
and find, up to a total penta--divergence,
\begin{eqnarray}
\Delta {\cal L}^{(5)} (\Phi, H, g)\stackrel{\Lambda\rightarrow\infty}{\sim}&&
\frac{N\Lambda^5}{75\pi^3}-\frac{N \Lambda^3}{9\pi^3}
\left(\Phi^2 + H^2 + \frac{R}{12}\right)\nonumber\\
+&&\frac{N \Lambda}{4\pi^3}
\left\lbrace\,\partial_A \Phi\,\partial^A \Phi +
\partial_A H \partial^A H\right.\nonumber\\
+&&\left.\left(\Phi^2 + H^2\right)^2 +
 \frac{R}{6}\,\left(\Phi^2 + H^2\right) +
\frac{ R^2}{144}\right.\nonumber\\
-&&\left.\frac{1}{90}\,
R_{AB}R^{AB}- \frac{7}{720}\,
R_{ABCD}R^{ABCD}\,\right\rbrace\ .
\la{a123}
\end{eqnarray}
Although the actual values of the coefficients might
be regulator--dependent, as already noticed,
the coefficients of the kinetic
and quartic terms of the effective low energy Lagrange density are definitely equal,
no matter how the latter is
obtained from the basic Dirac operator of eq.\,\gl{eudir}.

\section{Scalars and gravity: classical configurations}

The interplay between different operators in the low-energy
Lagrange density \gl{lowl} may lead to different dynamical
regimes, depending of $\tau$-symmetry breaking.
The low-energy Euclidean Lagrange density can be cast in the form
\begin{eqnarray}
{\cal L}_{\rm low}^{(5)} (\overline{\Psi}_l,\Psi_l,\Phi, H, g) &\equiv&
{\cal L}^{(5)}_{\rm fermion} (\overline{\Psi}_l,\Psi_l,\Phi, H, g) +
{\cal L}^{(5)}_{\rm boson} (\Phi, H, g) +
\Delta {\cal L}^{(5)} (\Phi, H, g)\nonumber\\
&=& i\overline{\Psi}_l(X)\left[\,\!\not\!\nabla
+ \tau_3 \Phi(X) + \tau_1 H(X)\,\right]\Psi_l(X)\nonumber\\
&+& \frac{N\Lambda}{4\pi^3}\,\left\{\partial_A \Phi(X)\partial^A \Phi(X)+
\partial_A H(X)\partial^A H(X)\right.\nonumber\\
&-&\left. 2 \Delta_1\Phi^2(X)
- 2\Delta_2 H^2(X)\right\} - \frac{\Lambda}{2 \kappa \cal G}\left\lbrace\,
R(X) - 2\lambda\,\right\rbrace\nonumber\\
&+& \frac{N\Lambda}{4\pi^3}\,\left[\,\Phi^2(X) + H^2(X)\,\right]\left\{
\Phi^2(X) + H^2(X) + {R(X)\over 6}\right\}\nonumber\\
&+&\frac{N\Lambda}{2880\pi^3}\left\lbrace 5R^2(X) -
8R_{AB}(X)R^{AB}(X)\right.\nonumber\\
&-& \left.
7R_{ABCD}(X)R^{ABCD}(X)\right\rbrace
\la{lowmin}
\end{eqnarray}
where the two mass scales $\Delta_i$ characterize the
deviations from the critical point
\begin{equation}
\Delta_i(g_i)
=\frac{2\Lambda^2}{9g_i}\left(g_i-g_i^{\rm cr}\right);\quad g_1^{\rm cr} = g_2^{\rm cr}
= 9\pi^3 \ .
\end{equation}
The {\sl dressed} cosmological  constant $\lambda$ and the
gravitational low-energy dimensionless parameter $\kappa$
do arise as a net effect of the interplay between the classical
and fermion induced contributions\footnote{We remark that the
combination of terms quadratic in the scalar and tensor curvatures
does not form the Gauss-Bonnet gravity interaction \ci{corra} and, as
well, the scalar curvature ``mass'' $R/6$ for scalar matter fields does
not imply an additional
conformal symmetry of Klein-Fock-Gordon equations for the
five-dimensional manifold \ci{tagir} .}.
If we introduce some average compositeness scale
\begin{equation}
\Lambda^3_c \equiv N\Lambda^3/54\pi^3, \la{compo}
\end{equation}
together with a five dimensional counterpart of the Planck scale
\begin{equation}
{\hbar c\Lambda}/{{\cal G}\kappa} \equiv M^3_*\ ,
\la{plfive}
\end{equation}
then we can easily obtain
\begin{equation}
\lambda=\kappa \lambda_0+\frac{N\Lambda^4}{75\pi^3}\,{\cal G} \kappa \ ,
\end{equation}
together with the relationship
\begin{equation}
 {\kappa}
= \frac{1}{\varepsilon + N\Lambda^2{\cal G}/54\pi^3}\,\quad
\varepsilon\kappa=1-\left({\Lambda_c\over M_*}\right)^3
\label{kappa}
\end{equation}
where $\kappa$ must be positive for the gravitational
interaction to be attractive.
We shall be interested in different scenarios which one can
qualitatively specify as follows:\\
\begin{itemize}
\item[(A)]
{\tt Fundamental gravity},\\
when the bare and dressed gravitational couplings are comparable
\begin{equation}
\varepsilon  = 1\ ,\qquad
\kappa \simeq 1\ ,\qquad
\mbox{\rm so that}\qquad
\Lambda_c \ll  M_*\ .
\label{fundgrav}
\end{equation}
\item[(B)]
{\tt Induced gravity},\\
when the bare gravitational Action is
either absent  or irrelevant whilst the fermion induced
gravitational Action is dominant
\begin{equation}
\varepsilon  = 0\ ,\qquad\mbox{and/or}\qquad
\kappa \ll 1\ ,\qquad
\mbox{\rm so that}\qquad
\Lambda_c \simeq  M_*\ .
\label{indgrav}
\end{equation}
\item[(C)]
{\tt Strong gravity},\\
when the bare gravitational Action is dominating
with respect to the fermion induced gravitational Action
\begin{equation}
\varepsilon  = -1\ ,\qquad
\kappa > 1\ ,\qquad
\mbox{\rm so that}\qquad
\Lambda_c >  M_*\ .
 \label{strongrav}
\end{equation}
\end{itemize}
Eventually, we shall demand for all scales to be much less than
the five dimensional Planck-like mass, otherwise
our effective model would go beyond its own essence and
must be replaced by the still unknown genuine theory of quantum gravity.
This criterion rather rules out the scenario $(C)$ with $\kappa \gg 1,\quad
\Lambda_c \gg  M_*$ as the very concept of composite
particles and fields far above the Planck scale certainly needs to engage
quantum gravity. This is why we will not focus on it keeping the side
of a weak quasi--classical gravity.

In what follows we assume that all the matter effective couplings substantially reduce
the cut-off scale $\Lambda$ to a low
energy scale $M \ll \Lambda$ and, to be definite, we take $\Delta_1=M^2\,.$
Thus, in
order to make the different parts of the equations of motion comparable in mass scales,
we shall suitably tune in the sequel the different
couplings of our five dimensional model.
In particular, the bare cosmological constant must be negative
$\lambda_0 <0$ in order to provide
a small constant $\lambda \ll \Lambda^2\,.$

Let us obtain brane-like solutions which describe the localization
along the fifth coordinate $z$. In this paper we are interested in the
static effects of particle trapping on a brane and this is why we shall
consider here only the flat four dimensional space-time, with
Euclidean signature to simplify our calculations.
As a consequence, we restrict ourselves to analyze
the quasi-flat Riemannian metric, the invariant line element
of which can be suitably chosen as follows
\begin{eqnarray}
ds^2 = g_{AB}(X)\,dX^A\,dX^B
= \exp\{-2\rho(z)\}\,dx_\mu dx_\mu + dz^2
 \label{metr}
\end{eqnarray}
with the Euclidean signature. The physical motivations for the above choice will
be clear later on.
The related invariant volume factor is $\sqrt{g} = \exp\{-4\rho(z)\}$.
The scalar and gravity parts of the low-energy
Lagrange density \gl{lowmin} for this metric reduce to
(see the expressions for curvature terms in Appendix B)
\begin{eqnarray}
&&{\cal L}^{(5)}_{\rm boson} (\Phi, H, g) + \Delta {\cal L}^{(5)} (\Phi, H, g)\ =\nonumber\\
&& ({N\Lambda}/{4\pi^3})\,\exp\{2\rho(z)\}\,
\left[\,\partial_\mu \Phi(X)\partial_\mu \Phi(X) + \partial_\mu H(X) \partial_\mu H(X)\right]
+ ({N\Lambda}/{4\pi^3})\ \times\nonumber\\
&& \times \left\lbrace
 \left[\,\partial_z \Phi(X)\,\right]^2 +\left[\,\partial_z H(X)\,\right]^2
- 2\Delta_1\Phi^2(X) - 2\Delta_2 H^2(X)
+ \left[\,\Phi^2(X) + H^2(X)\,\right]^2\right\rbrace\nonumber\\
&& +\ ({N\Lambda}/{3\pi^3})\,\left[\,\Phi^2(X) + H^2(X)\,\right]
\left\{\rho''(z) - (5/2)\,[\,\rho'(z)\,]^2\,\right\}\nonumber\\
&& +\ ({\Lambda/\kappa{\cal G}})\,\left\{- 4\rho''(z)
+\ 10\,[\,\rho'(z)\,]^2\, +\ \lambda\right\}\nonumber\\
&& +\ ({N\Lambda}/{120\pi^3}) \left\lbrace
2[\,\rho''(z)\,]^2  - 36 \rho''(z)[\,\rho'(z)\,]^2 + {45}[\,\rho'(z)\,]^4\right\rbrace
\la{confm}
\end{eqnarray}
where $\rho^\prime(z)\equiv d\rho/dz\,,\  \rho^{\prime\prime}(z)\equiv d^{\,2}\rho/dz^2\,.$
Let us search for classical solutions in the form of a metric \gl{metr} in the
gravitational weak coupling regime in which we assume that all along
the large extra dimension we have
\begin{equation}
|\rho^\prime (z)|/M = \rm O(\kappa)\ ,\qquad
|\rho^{\prime\prime} (z)|/M^2 = \rm O(\kappa)\ .
\end{equation}
This means that the conformal function $\rho(z)$ is
slowly varying over a distance, in the  extra dimension,
of the order of magnitude of the typical Compton wavelength of the low energy particles.
This requirement, together with $0< \kappa\ll 1$ and $M\ll \Lambda$,
yields that the last
line in eq.~\gl{confm}, which corresponds to the terms
quadratic in the curvature scalar and curvature tensors
of the Lagrange density \gl{lowmin},
will be sub--leading and negligible with respect to the other terms.
Nevertheless, just for completeness, we shall display the contribution
of those terms into the equations of motion in Appendix C.

Within the present gravitational
weak coupling approximation,
the leading dynamics is mainly determined by the
Einstein--Hilbert Action, on the five dimensional Riemannian
manifold ${\cal M}_5$,
and its coupling to the scalar matter fields.
As shown in Appendix D, the field equations become
\begin{equation}
R_{AB} -\frac12\,g_{AB}\left(R - 2\lambda\right)\
\equiv G_{AB}\ +\ \lambda\,g_{AB}\ =\
\frac{N\kappa{\cal G}}{2\pi^3}\,t_{AB}
\end{equation}
where the normalized energy--momentum tensor of the scalar matter reads
\begin{eqnarray}
t_{AB}\ &\equiv&\ ({4\pi^3}/{N\Lambda})\,T_{AB}\ \equiv\
\partial_A \Phi\,\partial_B\Phi +
\partial_A H\,\partial_B H\nonumber\\
&-&\ \frac12\,g_{AB}\left[\,
\partial_C \Phi\, \partial^C \Phi +
\partial_C H\,\partial^C H - 2 \Delta_1\Phi^2
- 2\Delta_2 H^2 + \left(\Phi^2 + H^2\right)^2\,\right]\nonumber\\
&+&\  \frac16\left(R_{AB} - \frac12\,g_{AB}\,R
+ g_{AB}\,D^C\partial_C  - D_B\,\partial_A\right)
\left(\Phi^2 + H^2\right)\ .
\end{eqnarray}
The equations of motion for the scalar fields read
\begin{eqnarray}
&&2\left[\Delta_1  - \Phi^2 - H^2\,\right]\Phi\ =\
\left(\frac{R}{6}\ -\ \frac{1}{\sqrt{g}}\,
{\partial}_C \sqrt{g} \ {g}^{CD} {\partial}_D\right) \Phi\ ,\nonumber\\
&&2 \left[\,\Delta_2  - H^2 - \Phi^2\,\right]H\ =\
\left(\frac{R}{6}\ -\ \frac{1}{\sqrt{g}}\,
{\partial}_C \sqrt{g} \ {g}^{CD} {\partial}_D\right) H\ .
\la{phivar1}
\end{eqnarray}
In the case of a quasi--flat metric \gl{metr} and for
kink--like profiles of the vacuum expectation values of the scalar fields
\mbox{$\langle\Phi(X)\rangle_0 = \Phi(z),$} \mbox{$\langle
  H(X)\rangle_0 = H(z)$}
one finds the following equations: namely,
\begin{eqnarray}
G_{\alpha\alpha}(z)\ +\ g_{\alpha\alpha}(z)\,\lambda\ &=&\
\left({N\kappa{\cal G}}/{2\pi^3}\right)\,t_{\alpha\alpha}(z)\ ,\\
G_{55}(z)\ +\ \lambda\ &=&\
\left({N\kappa{\cal G}}/{2\pi^3}\right)\,t_{55}(z)\ ;
\end{eqnarray}
which respectively reduce to
\begin{eqnarray}
\rho^{\,\prime\prime} - 2\rho^{\,\prime\,2} - \frac13\,{\lambda} &=&
\frac{N\kappa{\cal G}}{12\pi^3}\left\lbrace
\Phi^{\,\prime\,2} + H^{\,\prime\,2} - 2 \Delta_1\Phi^2 - 2\Delta_2 H^2
+ \left(\Phi^2 + H^2\right)^2\right.\nonumber\\
&+& \left.\left( \rho^{\,\prime\prime} - 2\rho^{\,\prime\,2}
- \frac13\,\frac{d^{\,2}}{dz^2} + \rho^{\,\prime}\,\frac{d}{dz}\right)
\left(\Phi^2 + H^2\right)\right\rbrace
\label{eqgrav1}
\end{eqnarray}
\begin{eqnarray}
2\rho^{\,\prime\,2} + \frac13\,{\lambda}  &=&
\frac{N\kappa{\cal G}}{12\pi^3}\left\lbrace
\Phi^{\,\prime\,2} + H^{\,\prime\,2} + 2 \Delta_1\Phi^2 + 2\Delta_2 H^2
\right.\nonumber\\
&-& \left.\left(\Phi^2 + H^2\right)^2 + \left(2\rho^{\,\prime\,2}
- \frac43\,\rho^{\,\prime}\,\frac{d}{dz}\right)
\left(\Phi^2 + H^2\right)\right\rbrace\ .
\label{eqgrav2}
\end{eqnarray}
The field equations for the scalar matter become
\begin{equation}
\Phi^{\,\prime\prime} =
2 \Phi\left(\Phi^2 + H^2\right) - 2\Delta_1\Phi
+ 4\rho^\prime\, \Phi^\prime
+ \frac{2}{3}\,\Phi\left(2\rho^{\,\prime\prime}
- 5\rho^{\,\prime\,2}\right)\ ,\label{eqscalar1}
\end{equation}
\begin{equation}
H^{\,\prime\prime} =
2 H\left(\Phi^2 + H^2\right) - 2\Delta_2 H
+ 4\rho^\prime\, H^\prime
+ \frac{2}{3}\,H\left(2\rho^{\,\prime\prime}
- 5\rho^{\,\prime\,2}\right)\ .
\label{eqscalar2}
\end{equation}

Now, it turns out to be convenient to put forward the role
of the low--energy relevant mass scale $\Delta_1=M^2$. This can be better
achieved after a suitable redefinition of the gravitational low--energy
dimensionless parameter
\begin{equation}
{\overline \kappa}\ \equiv\ \frac{N\kappa}{6\pi^3}\,M^2{\cal G}\ \ll\ 1
\label{newcoup}
\end{equation}
and of the cosmological constant
\begin{equation}
\lambda\ \equiv\ 3{\overline \kappa}\,\lambda_{\rm
  eff}\ .
\label{coseff}
\end{equation}
It is worthwhile to notice that the sum of Eqs.~\gl{eqscalar1} and \gl{eqscalar2}
does not include the five dimensional rescaled cosmological constant $\lambda_{\rm eff}$
and reads
\begin{equation}
\rho^{\prime\prime}  = \frac{{\overline \kappa}}{M^2}
\left\{\Phi^{\,\prime\,2} + H^{\,\prime\,2} +
\frac12\left(\rho^{\prime\prime}
- \frac13\,\frac{d^{\,2}}{dz^2} - \frac13\,\rho^{\,\prime}\frac{d}{dz}\right)
\left(\Phi^2 + H^2\right)\right\}\ .
\label{eqgrav05}
\end{equation}
On the other hand, eq.~\gl{eqgrav2} can be rewritten in the form
\begin{eqnarray}
2M^2\lambda_{\rm eff}\  &=&
\Phi^{\prime\,2}
+ H^{\prime\,2}  + 2 \Delta_1\Phi^2 + 2\Delta_2 H^2
- \left(\Phi^2 + H^2\right)^2\nonumber\\
&+& \left(2\rho^{\,\prime\,2} - \frac43\,\rho^\prime\frac{d}{dz}\right)
\left(\Phi^2 + H^2\right) - \frac{4M^2}{{\overline \kappa}}\,\rho^{\,\prime\,2}
\label{intcosm}
\end{eqnarray}
which actually represents the integral of motion with
the rescaled five dimensional cosmological constant playing
the role of an integration constant.
This can be checked by differentiating the above  equation
and taking into account Eqs.~\gl{eqscalar1}, \gl{eqscalar2}
and \gl{eqgrav05}.

We can approximate our field equations
keeping in mind that we still assume
\begin{equation}
\frac{|\rho^\prime(z)|}{M}\ =\ \rm O({\overline \kappa})\ =\
\frac{|\rho^{\prime\prime}(z)|}{M^2}\ ,
\label{approx}
\end{equation}
as previously specified. We emphasize that the validity of
perturbation theory in $\overline \kappa$ is endorsed by the choice
of a reference frame where the metric takes the form \gl{metr}.
For different choices such as, for instance, the conformal metric
\begin{equation}
ds^2 = \exp\{-2\sigma(y)\}\,\Bigl(dx_\mu dx_\mu + dy^2\Bigr)\ ;\quad
 \exp\{-\sigma(y)\}dy \equiv dz\ ,
\end{equation}
the expansion in  $\overline \kappa$
is not uniform and fails for large $z\,.$

On the other hand, the expansion in powers of ${\overline \kappa}$
of the solutions of Eqs.~\gl{eqscalar1}, \gl{eqscalar2},
\gl{eqgrav05} and \gl{intcosm} is well defined on the whole
extra dimension $-\infty < z < \infty\,.$
Let us find solutions by expanding in
${\overline \kappa}\ll 1$.
According to our approximation \gl{approx},
taking into account that $\Phi/M = {\rm O}(1) = H/M\,,$ we obtain in
the leading order
\begin{equation}
\frac{\rho^{\,\prime\prime}}{M^2}  = \frac{{\overline \kappa}}{M^4}
\left\{\Phi^{\prime\,2} + H^{\prime\,2}
- \frac16\,\frac{d^{\,2}}{dz^2}
\left(\Phi^2 + H^2\right)\right\} + \rm O({\overline \kappa}^2)\ ,
\label{grav0}
\end{equation}
whereas the cosmological constant to the lowest order reads
\begin{equation}
\frac{\lambda_{\rm eff}}{M^2}\  =
\frac{1}{2M^4}\left\lbrace\Phi^{\prime\,2}
+ H^{\prime\,2}  + 2 \Delta_1\Phi^2 + 2\Delta_2 H^2
- \left(\Phi^2 + H^2\right)^2\right\rbrace + \rm O({\overline \kappa})\ .
\label{cosm0}
\end{equation}
The latter equation firmly determines the five dimensional
cosmological constant
$\lambda_{\rm eff}$ in terms of the parameters for the kink--like
solutions in the flat space which read
\begin{eqnarray}
&&\Phi^{\,\prime\prime} + 2\,\Phi\left(\Delta_1
- \Phi^2 - H^2\right) = \rm O({\overline \kappa})\ ,\nonumber\\
&&H^{\,\prime\prime} + 2H\left(\Delta_2
- \Phi^2 - H^2\right) = \rm O({\overline \kappa})\ .
\label{scal0}
\end{eqnarray}
To sum up, we have four equations for three functions $\rho(z), \Phi(z), H(z)$
and one integration constant $\lambda_{\rm eff}\,.$
As in \cite{aags} one can
discover two types of kink--like  solutions for
Eqs.~\gl{scal0}: namely,
\begin{eqnarray}
&&(J)\quad \Phi_J \equiv \langle\Phi(X)\rangle_0 = M
\mbox{\rm tanh}(M z)\ , \quad  H_J \equiv\langle H(X)\rangle_0
= 0\ ;\la{Jvacuum}\\
&&(K)\quad \Phi_K \equiv \langle\Phi(X)\rangle_0=  M
\mbox{\rm tanh}(\beta z)\ ,\quad H_K \equiv
\langle H(X)\rangle_0 =\mu\,\mbox{\rm sech}(\beta z)\ ,
\label{Kvacuum}
\end{eqnarray}
where
\begin{equation}
\mu =
\sqrt{2 \Delta_2 - M^2}\ ,\qquad
\beta = \sqrt{M^2 - \mu^2}\ .
\label{mu}
\end{equation}
The solution $(K)$ exists only for $\Delta_2 < M^2 < 2\Delta_2$
and it coincides with the extremum $(J)$ in the limit
$\Delta_2\to M^2/2,\ \mu\to 0,\ \beta\to M$.
Both of them give consistently
\begin{equation}
 \lambda =
\frac{N{\cal G}\kappa M^4}{4\pi^3} = \frac32 \bar\kappa M^2\ .
\label{cosmcon}
\end{equation}
The last term in \gl{grav0} is originated from
the scalar curvature mass--like term $R/6$
in the Lagrange density
\gl{lowmin} for the scalar fields. To unravel its role, we split
the conformal factor in the following way: namely,
\begin{eqnarray}
\rho(z) &=&\rho_1(z) + \rho_2(z)
= \frac{2{\overline \kappa}}{3}
\left(1 +\frac{\mu^2}{2M^2}\right)\ln\mbox{\rm cosh}(\beta z)
+ B z\ ;\label{rhoone}\\
\rho_1^{\,\prime\prime}(z)  &=&
 \frac{{\overline \kappa}}{M^2} \left\lbrace[\,\Phi^\prime(z)\,]^2
+ [\,H^\prime(z)\,]^2\right\rbrace\ ;\quad
\rho_2^{\,\prime\prime}(z)  =
\frac{- {\overline \kappa}}{6M^2}\,{d^2\over dz^2}\,
\left[\,\Phi^2(z) + H^2(z)\,\right]\ ;\nonumber\\
\rho_1(z) &=& \frac{2{\overline \kappa}}{3}
\left(1 +\frac{\mu^2}{2M^2}\right)\ln\mbox{\rm cosh}(\beta z)
+ \frac{{\overline \kappa}}{6}\left(1 -\frac{\mu^2}{M^2}\right)
\mbox{\rm tanh}^2(\beta z) + B z\ ; \nonumber\\
\rho_2(z) &=& \frac{-{\overline \kappa}}{6M^2}
\left[\,\Phi^2(z) + H^2(z) - \mu^2\,\right]\
=\ -\ \frac{{\overline \kappa}}{6}
\left(1 -\frac{\mu^2}{M^2}\right)
\mbox{\rm tanh}^2(\beta z)\ .
\end{eqnarray}
This solution is normalized so that for the vanishing integration constant
\mbox{$B \rightarrow 0$} the function $\rho(z)$ becomes even  and
$\rho (0) = 0$. The latter corresponds to the proper normalization
of the {\it 3+1}
metric on the brane $z = 0\,.$

One can see that the scalar curvature term substantially simplifies the
metric factor $\rho(z)$ around the brane. Evidently, this solution approaches,
when $B=0$, the symmetric Anti-de-Sitter (AdS) metric for large $z$: namely,
\begin{equation}
\rho(z)\stackrel{|z| \rightarrow \infty}{\sim}
 k|z|\ ;\quad k\equiv \frac23\,{\overline \kappa}\beta
\left(1 +\frac{\mu^2}{2M^2}\right)\ \approx\
\frac23\,{\overline \kappa}M\ .
\label{AdSlim}
\end{equation}

\section{Ultra-low energy physics in the matter sector}

Let us summarize the structure \cite{aags} of the spectrum and
of the interaction of the light states trapped on a brane,
in the absence of gravity. The kinetic operators
(second variation of the Action) of the two scalars
$\Phi(X)$ and $H(X)$ and of the spinor field $\Psi(X)$
do exhibit normalizable zero-modes in the extra dimension,
in the vicinity of the vacuum background \gl{Jvacuum} or \gl{Kvacuum},
at the scaling point $M^2 = \Delta_1 = 2 \Delta_2$ or $\mu = 0\,.$
Those zero-modes $\phi_0(z)$, $h_0(z)$ and $\psi_0(z)$, respectively,
are localized at the origin of the $z$-axis, with a localization
width $\sim 1/M$ and, at ultra-low energies, the fluctuations of the
matter fields can be parametrized as follows:  namely,
\begin{eqnarray}
&&\Phi(X) \simeq \langle\Phi(X)\rangle_0 + \phi(x)
\phi_0(z)\ ;\nonumber\\
&& H(X) \simeq\langle H(X)\rangle_0 +  h(x) h_0(z)\ ;\nonumber\\
&& \Psi(X) \simeq
\psi(x)\,\psi_0 (z)\ .
\end{eqnarray}
For these states the ultra-low-energy effective Lagrange density
(still in the Euclidean space) is generated
\begin{eqnarray}
{\cal L}^{(4)} &=&
i\overline{\psi}(x)\left[\,\not\!\partial + g_f\,h(x)\,\right]\psi(x)
+\frac12[\,\partial_\mu\phi(x)\,]^2
+\frac12[\,\partial_\mu h(x)\,]^2\nonumber\\
&+& \lambda_1\phi^4(x) + \lambda_2\phi^2(x) h^2(x) + \lambda_3 h^4(x)\ ,
\la{lelag}
\end{eqnarray}
with the ultra-low energy effective couplings given by
\begin{equation}
g_f = \frac{\pi}{4} \sqrt{\zeta}\ ,\quad \lambda_1 =
\frac{18}{35}\,\zeta\ ,\quad \lambda_2
=\frac{2}{5}\,\zeta\ , \quad \lambda_3
=\frac{1}{3}\,\zeta\ , \qquad \zeta \equiv \frac{M\pi^3}{\Lambda N}\ .
\la{lowcons}
\end{equation}
Once gravity is switched on, it can be shown\footnote{It
has been firstly analyzed in \ci{misse,missch,galoc2}. A detailed analysis more
suitable for our model will be presented elsewhere.}  that the zero-modes
remain localizable -- see below -- and therefore the AdS vacuum solution
does not play any dominant role concerning the determination of the coupling
constants in eq.~\gl{lelag}.

The situation becomes more subtle when light massive states are there.
In the absence of  gravity,
the deviations off the scaling point towards
the $(K)$ vacuum configuration with $\mu \ll M$ do
produce the masses for the Higgs-like particle and the spinor
particles. Furthermore, the scalar self-interaction is induced
in the form
\begin{eqnarray}
&&\Delta {\cal L}_\mu^{(4)} = \frac12\,m^2_h\,h^2(x) + i m_f\,
\overline{\psi}(x)\psi(x) + \lambda_4\,h^3(x)\ ;\nonumber\\
&&m^2_h = \mu \sqrt2\ ;\qquad
 m_f = \frac{\pi}{4}\,\mu\ ;\qquad
\lambda_4 = \mu \sqrt{\zeta}\ .
\la{mumass}
\end{eqnarray}
On the one hand, we see that all interaction vertexes
are governed by the parameter $\zeta \sim M/\Lambda$ and
if $\zeta \ll 1$ the scalar matter essentially
decouples from the fermion sector and
does not interact without gravity. However, the
parameter $\zeta$ is not properly fixed  if gravity is not present
and, in general, it is constrained by experimental bounds.
On the other hand, the masses of Higgs-like scalar and fermions are
controlled by the ultra-low scale $\mu$
independently of $\zeta\,.$ In the spirit of the Top-mode Standard Model
\ci{Topmode} one expects that the heaviest quark mostly
contributes into dynamical symmetry breaking, thereby driving to the scale
$\mu \sim m_{\rm top}\sim 200$ GeV, {\it i.e.} the order of
magnitude of the electroweak symmetry breaking scale \ci{PDG}.

Once the five dimensional gravity is switched on and the background geometry
becomes an anti-de Sitter one, it turns out that light massive scalar and fermion
1-particle states belong, in fact, to the continuous part of the Hamiltonian spectrum,
the latter ones being strongly localized on the brane albeit possessing a tiny
non-vanishing tail for $|z| \gg 1/k\,.$
The origin of this phenomenon can be schematically sketched in terms of
the mass operator for a one dimensional scalar field living on the fifth
dimension (see eq.~\gl{confm} and \ci{aags}): namely,
\begin{eqnarray}
\exp\{-2\rho(z)\} {\bf M}_z &\equiv& -\ \partial_z[\,\exp\{- 4
\rho(z)\}\partial_z\,] + {\bf  V}(z) \exp\{- 4 \rho(z)\}\ ;\nonumber\\
{\bf M}_z\,\varphi(z) &=& m^2 \varphi(z)\ ,
\end{eqnarray}
where ${\bf  V}(z)$ is, in general, some matrix-valued potential
well giving rise to bound states, in the absence of gravity,
and approaching the energy $4M^2$ for large $z\,.$
The mass eigenvalue $m$ stands for the mass of a scalar Higgs-like
1-particle state.
After the suitable redefinition $\varphi(z) = \widetilde\varphi(z)\,
\exp\{2\rho(z)\}\,,$ the eigenvalue problem is transformed into
a zero-mode condition\quad$\widetilde{\bf M}_z\,\widetilde\varphi(z)=0\,,$
where the modified mass operator takes the conventional form,
\begin{equation}
\widetilde{\bf M}_z = -\ \partial_z^2 + \widetilde{\bf  V}(z)
- m^2 \exp\{2\rho(z)\}\ \equiv\ -\ \partial_z^2 + {\bf  W}(z)\ ,
\la{massop2}
\end{equation}
whereas $\widetilde{\bf  V}(z)\simeq {\bf  V}(z)$ uniformly
up to the leading order in ${\overline \kappa}\,.$
Thus we see how the only important difference in the spectral
problem, in comparison with the five dimensional flat space,
does consist in the appearance of the conformal factor,
which is boundless increasingly when $|z| \gg 1/k\,.$
The very last term in eq.~\gl{massop2} becomes now a piece of the
modified potential ${\bf  W}\,,$ whilst the mass eigenvalue $m$
just appears as a coupling constant.

Evidently, this very last piece, being negative,
makes the modified potential unbounded from below: instead of a potential
well,  some finite barrier just arises.
As a consequence, each light massive
state with $0 < m^2 \ll 4M^2$ is at most
quasi-stationary. Nonetheless, let us assign massive scalar particles in the
brane world to be described by normalizable wave packets with a width
of the order $1/M\,.$ The latter ones are not, of course,
exact eigenfunctions of the mass operator \gl{massop2} but may be
taken as close as possible, {\it e.g.} with the help of a variational principle,
to the solutions in the vicinity of the brane $|z| \ll 1/k\,.$
For instance, one can start with bound state wave functions in the absence of
gravity. Then quantum mechanics predicts the decay rates of such a kind of
bound states, due to quantum tunneling.
The quasi-classical
probability of barrier penetration is governed by the change in
the sub-barrier wave function between the two turning points for classical trajectories,
say, $0 < z_0 < z_1\,.$ These turning points can be estimated to be
\begin{equation}
z_0 = \frac{C}{M}\ ,\quad C \simeq 1\ ; \qquad z_1 \simeq \frac{1}{k}
\ln\frac{2M}{m}\ ,
\end{equation}
so that $z_1 \longrightarrow \infty$ for the zero-modes
$m \longrightarrow0\,.$ Taking these relations into account, one finds the suppression
factor for quantum tunneling
\begin{equation}
\exp\left\{- \int^{z_1}_{z_0} dz' \sqrt{{\bf  W}(z')}\,\right\} \simeq
\exp\left\{- z_1 \cdot 2M\right\} \simeq
\exp\left\{- \frac{3}{{\overline \kappa}}\ln\frac{2M}{m}\right\}\ ,
\la{suppr}
\end{equation}
where the limit ${\bf  W} \longrightarrow 4M^2$ for $z_0 \ll z'\ll
z_1$  and the definition \gl{AdSlim} have been used.

As we shall discuss in the next Section, some reasonable order of
magnitude for the weak gravitational constant is
${\overline \kappa}\leq 10^{-8}\,,$
so that it is clear that such a suppression factor \gl{suppr} is
extremely small and spoils in practice any chance for matter to
disappear from the brane. We also notice that the suppression factor
is essentially non-perturbative and cannot be recovered as an expansion
in powers of ${\overline \kappa}$ -- see {\it e.g.} \ci{CGS}  for a
complete discussion of the similar case of the Stark effect.

Now we can outline our further strategy in
calculating gravitational effects on the mass spectrum and coupling
constants of light particles living on the brane. First, particle wave
functions at the lowest order are taken from the flat space limit.
Second, only perturbation theory in powers of ${\overline \kappa}$ is involved
in corrections of particle characteristics due to non-trivial
gravitational background. In this way, our approach is able to
implement the systematic perturbative expansion which provides
light particle wave packets localized upon the brane, to any finite
order in the expansion in powers of ${\overline \kappa}\,.$
A more detailed and rigorous treatment is postponed to the forthcoming paper.

\section{Newton's constant and parameters}

One can find the relation between the five dimensional and brane gravity
constants using the factorized Riemannian metric
\begin{equation}
ds^2 = \exp\{-2\rho(z)\}\,g_{\mu\nu} (x) dx^\mu dx^\nu + dz^2\ .
\label{metr2}
\end{equation}
For this metric, according to eq.~\gl{scalarcur5} of Appendix B,
the effective four dimensional Einstein-Hilbert Action,
at the leading order, becomes
\begin{eqnarray}
S[\,g\,] = &-& \frac{\Lambda}{2\kappa{\cal G}} \int d^5X
\sqrt{g(X)}\left\{ R(X)\ -\ 2\lambda\right\}\nonumber\\
\simeq &-& \frac{\Lambda}{2\kappa{\cal G}} \int d^4 x \sqrt{g(x)}\, R(x)
\int_{-\infty}^{+\infty}dz\ \exp\{-2\rho(z)\}\nonumber\\
&-& \frac{\Lambda}{\kappa{\cal G}} \int d^4 x \sqrt{g(x)}
\int_{-\infty}^{+\infty}dz\ \exp\{-4\rho(z)\}\left\lbrace
6[\,\rho^\prime(z)\,]^2 - \lambda\right\rbrace\nonumber\\
\equiv &-& \frac{1}{16\pi G_N} \int d^4 x \sqrt{g(x)}\left\lbrace
R(x)\ -\ 2\Lambda_{\rm grav}\right\rbrace\ ,
\label{fivefour}
\end{eqnarray}
whence we eventually get the Planck mass
scale $M_{P} \sim 1.22\times 10^{19}$ GeV/c$^2$ which
corresponds to the Newton's gravitational constant
\begin{equation}
M_P^2 = G_N^{-1} \equiv \frac{8\pi\Lambda}{\kappa{\cal G}}
\int_{-\infty}^{+\infty}dz\ \exp\{-2\rho(z)\}
\end{equation}
and the gravitational part of the four dimensional cosmological constant
\begin{equation}
\Lambda_{\rm grav}\equiv
\frac{8\pi G_N\Lambda}{\kappa{\cal G}}
\int_{-\infty}^{+\infty}dz\ \exp\{-4\rho(z)\}\left\lbrace
\lambda - 6[\,\rho^\prime(z)\,]^2\right\rbrace\ .
\end{equation}
Under the assumption of the smallness of the parameter ${\overline \kappa} \ll 1$
and using in eq.~\gl{fivefour} the approximate form for $|z|\to\infty$
of the solution \gl{rhoone} one obtains
\begin{eqnarray}
G_N\ \simeq\ \frac{\pi^2\,{\overline \kappa}^{\,2}}{2N\Lambda M}\ .
\end{eqnarray}
Remarkably, the full value of the  cosmological constant, including the gravitational
as well as the matter vacuum energy densities, does indeed exactly vanish
to all orders in the perturbative expansion in powers of
${\overline \kappa}\,:$ actually,
\begin{eqnarray}
\Lambda_{\rm cosmo}&\equiv&
\frac{2N\Lambda G_N}{\pi^2}
\int_{-\infty}^{+\infty}dz\ \exp\{-4\rho(z)\}\biggl\lbrace
2M^2 \lambda_{\rm eff}  - ({4/{\overline \kappa}})\,{M^2}
\rho^{\prime\,2}(z)\nonumber\\
&+& \Phi^{\prime\,2}(z) + H^{\prime\,2}(z) - 2\Delta_1\Phi^2(z) - 2\Delta_2 H^2(z)
+ \left[\,\Phi^2(z) + H^2(z)\,\right]^2\nonumber\\
&+& \frac23\left[\,\Phi^2(z) + H^2(z)\,\right]
\left[\,2\rho^{\prime\prime}(z) - 5\rho^{\prime\,2}(z)\,\right]\biggr\rbrace\nonumber\\
&=& \frac{4N\Lambda G_N}{\pi^2{\overline \kappa}}
\int_{-\infty}^{+\infty}dz\ \exp\{-4\rho(z)\}\biggl\lbrace
-\ M^2\rho^{\prime\prime}(z) + {\overline \kappa}\left[\,
\Phi^{\prime\,2}(z) + H^{\prime\,2}(z)\,\right]\nonumber\\
&+& \frac23\,{\overline \kappa}\left[\,\Phi^2(z) + H^2(z)\,\right]
\left[\,2\rho^{\prime\prime}(z) - 5\rho^{\prime\,2}(z)\,\right]\biggr\rbrace\
=\ 0\ ,
\label{zerocos}
\end{eqnarray}
where the vacuum expectation values \gl{eqscalar1} and \gl{eqscalar2}
of the scalar fields together with the field equation \gl{eqgrav05}
of the conformal factor have been suitably taken into account.
In order to establish the exact cancellation between the gravitational
and the scalar matter contributions, the five dimensional
cosmological constant has been conveniently substituted from eq.~\gl{intcosm}.
We also notice that the inclusion of the higher-order gravitational interaction
in the last line of eq.~\gl{confm} still keeps equal to zero the value
of the induced cosmological constant $\Lambda_{\rm cosmo}\,,$
as it can be shown with the help of the Appendix C.

Let us find the relations among the AdS curvature scale
$k\simeq 2{\overline \kappa}\,M/3$, the Planck
mass $M_P$ and the spontaneous symmetry breaking scale $M\,.$
We recall that the characteristic parameter of ultra-low energy
dynamics of light fermions and scalar fields has been found
\cite{aags} to be
\begin{equation}
\zeta \equiv \frac{M\pi^3}{N \Lambda}\ ,
\end{equation}
with $N \sim \pi^3$ in the Standard Model (see the summary in the previous Section 4).
Therefore, the relationship among the three scales $M_{P}, k $ and $M$ actually reads
\begin{equation}
k^2 M_{P}^2 = \frac{8\pi}{9\zeta}\,M^4\ ,
\label{scales}
\end{equation}
in accordance with Eqs.~\gl{newcoup} and \gl{AdSlim}.
Correspondingly one obtains that
\begin{equation}
\bar\kappa = \frac{2\pi}{\sqrt{\zeta}}\,(M/M_P)
\la{barkap}
\end{equation}
restricted to $\bar\kappa \ll 1$.

One can now express both the five dimensional gravitational constant and
the five dimensional Planck scale, respectively, just in terms of
the above mentioned scales and parameters, that means
\begin{equation}
\frac{{\cal G}\kappa}{\Lambda} = \frac{6\pi^3{\overline \kappa}}{N\Lambda M^2}
= \frac{9 k\zeta}{M^4}\,=\,\frac{1}{M_*^3}\ ;\quad M_*^{3} = \frac{k M^2_P}{8\pi}\ ,
\la{planck5}
\end{equation}
which leads to the lower bound
for the five dimensional Planck scale $ M_* > 10^8$ GeV
from the experimental bound $ k > 10\
\mbox{\rm mm}^{-1} = 2\cdot 10^{-3}$ eV, the latter one being based on possible deviations
from the Newton's law \ci{adelb}.

Our first scenario -- {\tt fundamental gravity} --
is selected to have a principally detectable dynamics of
scalar fields and of the Higgs-Yukawa
coupling to fermions in the brane world.
Let us therefore analyze the regime where
$M/\Lambda \simeq \zeta \sim 0.1 \div 0.3 \,,$
at least, albeit not much less.
In this case, the experimental bound $ k > 10\
\mbox{\rm mm}^{-1} = 2\cdot 10^{-3}$ eV \ci{adelb} turns out to be
compatible with the lower bound \ci{rev3,rev9} for the localization scale
$M > (2 \div 3)$ TeV, which makes somewhat
challenging to produce new physics related
to the fifth dimension at the next generation of colliders.
The corresponding cut-off is $\Lambda > 10\div 20$ TeV so that,
from eq.~\gl{barkap}, perturbation theory is controlled by
the very tiny constant ${\overline \kappa} > 10^{-15}\,,$
whilst the five dimensional cosmological
constant must be tuned to the value
$\lambda \sim 10^{-2}$ MeV$^2\,.$

Let us find the relationship between the bare value ${\cal G}$ and
dressed value $\kappa {\cal G}$ of the five dimensional gravitational constants.
According to eq.~\gl{kappa}, the latter one is controlled by the ratio
\begin{equation}
\omega =  \frac{N \Lambda^2 {\cal G}}{54\pi^3} =
\frac{{\overline \kappa}\,\pi^6}{9 N^2 \zeta^2} \sim 10\,
{\overline \kappa} \sim 10^{-14}\ll 1
\la{omega}
\end{equation}
for the chosen values of parameters ${\overline \kappa}$ and $\zeta\,.$
Thus we conclude that $\kappa\simeq 1$ and therefore the bare gravitational
constant ${\cal G}$ mostly determines the intensity of the gravitational
attraction in the five dimensional space-time.
But this gravitational force is pretty strong, as its Planck scale
$M_*  \sim 10^8$ GeV is much lower than its four dimensional counterpart
$M_P \sim 10^{19}$ GeV.

A further economical choice might be to identify the AdS curvature
scale  $ k$ with the electroweak symmetry breaking scale $\mu \sim 200$ GeV
with the hope to connect  the top-quark mass formation in the Standard Model
to some extra dimensional gravitational effects.
If $ k \sim \mu  \sim 200$ GeV, and still $\zeta \sim 0.1 \,$
then one finds $M \sim  10^{10}$ GeV and $\Lambda \sim 10^{11}$ GeV so that
the five dimensional cosmological constant must be tuned to $\lambda\sim 10^{12}$ GeV$^2$.

It follows therefrom that the expansion parameter increases up to
${\overline \kappa} \sim 10^{-8}\,.$  Nevertheless,
it does keep to be very small and, consequently, one basically needs the first orders of
perturbation theory in the gravitational interaction to the aim of reaching
a sufficiently good precision. For such a choice, the only signatures of
extra dimensional physics could come from branon detection and
there is no hope to reach sufficiently high energies to overcome
the barrier $M$ towards the fifth dimension.
The estimates coming from eqs.~\gl{omega} and \gl{planck5}, {\it i.e.}
$\omega \sim 10^{-8}\ll 1\,;\quad M_{*} \sim 10^{13}$ GeV,
show that this scenario still corresponds to the fundamental
five dimensional gravity Action with $\kappa \simeq 1\,,$
a relatively strong gravity.

The second scenario under consideration is that one of the
{\tt induced gravity}. Let us adopt the induced gravity relations
of eq.~\gl{indgrav}
\begin{equation}
\kappa\ \simeq\ {54\pi^3}/{N\Lambda^2 {\cal G}}\ ;\qquad
{\overline \kappa}^{\,1/2}\ \simeq\ {3M/\Lambda}\ =\
{3 N\zeta}/{\pi^3} \ll 1\ ,
\end{equation}
in such a way that
\begin{equation}
k M_{P}^2 = {4N\Lambda^3}/{27\pi^3}\ ;\qquad
k^5 M_{P}^4 = {128N^2 M^9}/{27\pi^6}\ .
\end{equation}
In particular, for a lower experimental bound
$k \geq 2\cdot 10^{-12}$ GeV one finds
\begin{eqnarray}
&& M \geq 100\ {\rm GeV}\ ,\qquad
\Lambda \geq 10^{9}\ {\rm GeV}\ ;\nonumber\\
&& \zeta \sim M/\Lambda \sim 10^{-7}\ ,\qquad
{\overline \kappa} \sim 10^{-13}\ .
\end{eqnarray}

This means that the light particle interaction is highly suppressed and
the only particle interaction which is left is the gravitational one.
It is worthwhile to remark that the induced gravity still keeps to be weak
at low energies
${\overline \kappa} \ll 1\,,$ which also entails the equivalence between
the  compositeness scale $\Lambda_c$ defined in eq.~\gl{compo} and
the five dimensional Planck-like scale $M_{*}> 10^8$ GeV (see
eq.~\gl{planck5}).  Evidently the barrier $M \sim 100$ GeV
is too low to be accepted by modern collider experiments, as the
gravity (and gauge bosons) may easily be able to give fermions enough
energy to disappear from our world.

For so far experimentally acceptable barriers of order
$M \sim 1$ TeV, one finds that the AdS curvature scale
$k \sim  10^{-10}$ GeV, which corresponds to distances
of the ${\mu}$m order, becomes  certainly unreachable in a
nearest future experiment hunting for Newton's law deviations
\ci{adelb}. As a matter of fact, the scalar particles essentially decouple
from the fermion world and from each other since
$\Lambda_c \simeq M_{*} \sim 10^{9}$ GeV and
$\zeta \sim M/\Lambda \sim 10^{-6.5}\,;\ {\overline \kappa} \sim 10^{-12}\,.$
Although the Higgs-like particles may be involved into the gauge boson
interaction and be observable by gauge boson mediation, it turns out
that branons, {\it i.e.} the quanta of the field $\phi$,
do represent excitations of a geometrical nature, being related to
the Goldstone bosons of the translational invariance breaking.
As a consequence, their actual decoupling
from any other kind of matter makes them a perfect candidate for the dark
matter/energy, depending on their mass.

For the electroweak breaking scale $k \sim 2\cdot 100$ GeV,
the barrier $M \sim 10^{10}$ GeV is too high to trigger any observable
physics at the Earth laboratories since
$\Lambda \sim 10^{14} {\rm GeV}\,;\ \zeta \sim M/\Lambda \sim 10^{-4}\,,$
whereas branons keep essentially decoupled and thereby belonging to the dark
universe. The low-energy gravity remains weak with
${\overline \kappa} \sim 10^{-8}$ so that
we see that, for matter induced gravity, the parameter
$\zeta$ may not be small only for scales $k \sim  M \sim \Lambda$
approaching the Planck scale, where gravity is actually strong
(${\overline \kappa}\sim 1$) and quantum gravity is in order.

\section{Conclusions, further development}

In this paper we have performed the systematic analysis of how the
brane world can be generated dynamically by matter self-interaction
from a spontaneous breaking of the translational
invariance. The latter breaking is intimately related to the
$\tau$-symmetry breaking in our proto-fermion model.
The pertinent low-energy effective Action for fermions and the
auxiliary scalar fields, as well as for
their gravitational interaction, has been obtained by one-loop
integration of high energy spinor degrees of freedom.

Both the gravitational and the scalar fields turn out to be
responsible for the localization of light matter on the
brane. Nevertheless, even if the scalar kink-like vacuum solution
allows to trap light fermion and scalar particles inside the
brane layer, still the AdS$_5$ gravitational background, in which
the kink-like vacuum configuration are embedded, just induces
the quantum tunneling taking away, in principle, some of those
particles. Meanwhile, some simple WKB estimations prove that such a decay,
eventually due to the quantum tunnel effect,
is extremely well suppressed, so that it becomes practically impossible
to detect the disappearance of any particle in the visible universe during
the universe lifetime.

These circumstances have allowed us to formulate the following principle:\\
{\sl particle lifetimes on the brane must be evaluated by wave packet localization,
which is unambiguously and consistently determined in perturbation theory.}
Within this approach, the mass spectrum of light states on the brane turns out
to be slightly different from the corresponding one in the flat space limit.

We have in turn examined the possible ranges of different scales and coupling
constants in terms of the Newton's constant normalization to its observed value
and in terms of the experimental bounds on the AdS$_5$ curvature and thresholds
on appearance/disappearance of high energy particles in accelerator experiments.
As a summary of these studies, we conclude that:\\
a) on the one hand, fundamental gravity in five dimensions appears to be
more challenging
for future experiments, because the phenomenologically acceptable large values
of AdS$_5$ curvature $k \sim 10^{-3}$ eV together with the relatively low
translational symmetry breaking scales $M \sim 1 \div 3$ TeV turn out to be
compatible with  a weak, but not vanishing, coupling of branons to Higgs-like
scalars and fermions. However, it appears that the lower are the values
of the AdS$_5$ curvature $k$, the higher is the threshold for new physics $M$
and the weaker is the interaction among spinor and scalar matter, in such a
way to move branons to the dark side of the universe.\\
b) On the other hand, induced gravity leads to decoupling of branons
from other matter in a wide range of acceptable scales and coupling
constants, thus putting them straightforwardly to the dark matter realm.\\
c) In any case, the dimensionless parameter which characterizes the
strength of the gravitational interaction is very small, of the order
${\overline \kappa} \leq 10^{-8}$. This feature does justify the use of
the perturbation theory both in the calculation of vacuum field
configurations and gravity background, as well as in the derivation of
mass spectrum of localized particles.

For nearest future developments, we deserve the following interesting
problems:\\
a) interplay in deviation from the scaling point (where only zero-mass
states are localized on the brane) between
manifest driving with the scale $\mu$
and mass generation by the AdS background gravity; \\
b) next-to-leading effects in the gravitational coupling constant
$\bar\kappa$
and, in
particular, scalar gravity (``radion''\ci{rad1,rad2,rad3}) influence
on the matter spectrum;\\
c) soft breaking of the translational symmetry due to scalar field defects,
necessarily accompanied with defects in the cosmological constant and
consequent production of massive branons;\\
d) supplementing our model with gauge bosons;\\
e) search for new experimental perspectives to detect branons and
other signatures of extra dimensions based on the model presented in
our paper.

\acknowledgments
We are grateful to Luca Vecchi for his accurate check of certain formulas.
This work was supported by Grant INFN/IS-PI13. Two of us (A.A.A. and V.A.A.)
were supported by Grants RFBR 04-02-26939 and Grant UR.02.01.299.

redpajama_set_name: RedPajamaArXiv


Many cultures and early medicine in the past clearly defined mind and body as two separate entities. While this categorized approach isn't as widespread nowadays, we tend to separate the two on a subconscious level, often times not fully aware of just how interconnected our mind and body truly are.
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Doreen Virtue – It's super windy here today, so we filmed indoors to have better sound quality.
In Part 1, we work in a meditation with Archangel Michael to lift fear and clear lower energies, and clearly tuning into our divine guidance. It's a time of exploring your options to move forward as God is guiding you with your home, career, spiritual path, and lifestyle.
In Part 2, we work on healing our hearts from post-traumatic experiences, so that we can be peaceful and open to love.
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Pandemonium: Excerpt from Chapter 3
The squad is en-route. It's been decided that they will try a different method this wee, co-led by Thalir. The pressure is on, and everyone seems more nervous. But they can't back down, these raids
The Vanguard must act, it is time. What most consider to be routine just has a different feel to it, and there is a deeper more difficult feel to the task at hand. Odhiin and Calibo struggle to decide whether they continue as always, or try something else.
The scene is set, and the pieces are moving. But it is quiet. Too quiet. It feels like the same, old dreary day, yet that is when one must be must prepared.

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These wholesale loose fit diabetic socks fit size 9-11 and work for both men and women. Each case comes with 120 pairs of black crew socks. These diabetic socks are physician approved and work to help improve circulatory problems, diabetes, edema, sensitive feet and neuropathy.
These wholesale loose fit diabetic socks fit size 9-11 and work for both men and women. Each case comes with 120 pairs of white crew socks. These diabetic socks are physician approved and work to help improve circulatory problems, diabetes, edema, sensitive feet and neuropathy.
These wholesale loose fit diabetic socks fit size 9-11 and work for both men and women. Each case comes with 120 pairs of grey crew socks. These diabetic socks are physician approved and work to help improve circulatory problems, diabetes, edema, sensitive feet and neuropathy.
These wholesale unisex winter gloves, scarves, and beanies come in 5 assorted colors and a bulk case of 144. The perfect winter accessory for your wardrobe, keep your body dry and warm while enduring the cold winter weather. Ideal for charity, donations, giveaways, promotions, non-profits, homeless shelters and more!
These wholesale unisex winter gloves and beanies come in 5 assorted colors and a bulk case of 96. The perfect winter accessory for your wardrobe, keep your body dry and warm while enduring the cold winter weather. Ideal for charity, donations, giveaways, promotions, non-profits, homeless shelters and more!
These wholesale 4 piece woven, breaded, leather, stackable bracelets come in 6 assorted styles. These unique and stylish bracelets are the perfect fashion accessory for your attire.


Mini Fit & Flare Dresses in Teal, Squirrel & Mars Red.
Because it's all about spontaneous online shopping in early the a.m., right? For some reason I've been obsessed with fit and flare dresses lately so I bought…3 of them. Sigh. But I picked out fun colors so I can continue growing the color department of my city-girl wardrobe.
Dahlings, what was your most recent spontaneous buy???


Less than an hour after the Henry Abbott Technical High School's graduating seniors marched into the O'Neill Center Sunday while "Pomp and Circumstance'' played over the public address system, caps, beachballs and confetti were tossed into the air when school Director Robert Sandagata presented the Class of 2004 to the audience.
Commentary, ranging from the address by class valedictorian Justin Paul to remarks from State Sen. David Cappiello, R-Danbury and the keynote speech from commencement speaker Fil Cerminara, was brief and to the point.
Cappiello reflected on his carefree high school days, and advised the new graduates to make good use of the knowledge they had acquired.
"There is so much potential in this room,'' he said, adding "There is nothing worse than wasted potential."
While some people will experience success solely on the basis of good fortune, Cappiello said, hard work and maximizing one's talents are the primary reasons that most people succeed in life.
Cerminara, a member of Abbott Tech's Class of 1979, encouraged the graduates to enjoy the achievement for a few days.
Graduation day is "a day you feel on top of the world,'' he said.
But then, it's time to get serious, and to think about giving back to society, he said.
Of the 119 graduates, Sandagata said, 35 percent are planning to further their education, and five percent will enter the military.
"But 100 percent are prepared to go into the world of work,'' he said.


China Mobile Limited volume of trade today and historical data of China Mobile Limited capitalization since start date of trade of CHL stocks.
Today's market capitalization for China Mobile Limited is 198 105 595 904 US dollar.
Today's China Mobile Limited volume of trade is 242 701 US dollar.
Volume of trade of China Mobile Limited - total sum in US dollars for all deals with CHL stocks bought and sold during selected date.
China Mobile Limited capitalization is 198 105 595 904 $ now. Market capitalization of China Mobile Limited changed by -0.925% since yesterday. Today's volume of trade of CHL stocks was 242 701 $. CHL volume of trade amount changed to -92.632% since yesterday.
China Mobile Limited stock price today, CHL stock quote.
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CHL volume of trade today and historical data of China Mobile Limited capitalization.
CHL dividend dates, history of China Mobile Limited annual dividend yield in 2019.
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Puja Detjen came to Patterson + Sheridan with a unique skillset that she has since put to good use in her intellectual property law practice. As a former member of GE's Operations Management Leadership Program, Puja supported corporate strategy objectives across manufacturing, operations, and supply chain management. Today, she leverages her diverse business experience to lead the way for inventors seeking to align their IP efforts with a strategic plan.
Puja's practice includes litigation, along with preparation and prosecution of both domestic and international patents with an emphasis on wireless communications. Her specialized knowledge in this area encompasses a wide range of evolving technologies – from LTE and WiFi to radio front ends and artificial nervous systems. Puja's ability to focus on the bigger picture without losing sight of the details ensures that her client's inventions have the needed protection to thrive across multiple jurisdictions.
In addition to preparing and prosecuting patents, Puja provides transactional support for clients engaging in mergers, acquisitions, and divestitures. For Puja, this involves performing IP portfolio due diligence and analyzing assets such as trademarks, patents, and licensing agreements. Puja also collaborates with inventors, in-house counsel, and foreign and domestic associates to manage the myriad of working components and competing interests that accompany these complex transactions. Relying upon a cohesive mix of her legal expertise and management skills, Puja is able to deliver exceptional results that meet each party's individual needs.
When she isn't devoting her time to helping clients navigate the patent application process or the intricacies of an M&A deal, Puja enjoys the performing arts, spending time with her husband and children, and pursuing her passion for healthy living.


Legendary (and I don't use that term casually) Southern Gospel pianist, Roger Bennett, passed away early Saturday morning. His death was the result of complications from his continuing battle with cancer.
We have really lost a great performer, but more importantly a great person.
God bless and keep his family at this difficult time.
Here are some courageous words from his wife, Debbie.


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Q: How do I change this code to use File I/O to Save a "High Score" File for my Game I am now using YAML to store the data and it works in creating and storing the file. I am still trying to figure out how to get the table to print to the terminal in the correct format using the text-table gem. Here is the code:
def highscore
if File.exists?('highscore.txt')
    hs = YAML.load_file("highscore.txt")
else
    hs = {
        1 =>  { player: '', score: 0, date: '' },
        2 =>  { player: '', score: 0, date: '' },
        3 =>  { player: '', score: 0, date: '' },
        4 =>  { player: '', score: 0, date: '' },
        5 =>  { player: '', score: 0, date: '' },
        6 =>  { player: '', score: 0, date: '' },
        7 =>  { player: '', score: 0, date: '' },
        8 =>  { player: '', score: 0, date: '' },
        9 =>  { player: '', score: 0, date: '' },
        10 => { player: '', score: 0, date: '' },
        }
end

(1..10).each do |rank|
  t = Time.now
  if @grand_total > hs[rank][:score]
    hs[rank][:score] = @grand_total
    hs[rank][:date] = "#{t.month}/#{t.day}/#{t.year}"
    puts "Congratulations you set a new HIGH SCORE! Enter your initials."
    initials = gets.chomp.upcase
    hs[rank][:player] = initials
    break
  else
    puts "Sorry, you didn't get a high score. Try again!"
  end
end

File.write('highscore.txt', hs.to_yaml)

puts hs.to_table
end

A: I'd recommend you look into using the YAML data serializer. It creates a nicely formatted and readable file that is easily read by other languages.
require 'pp'
require 'yaml'

hs = [["RANK", "PLAYER", "SCORE", "DATE"],
      ["1st", "-", 0, "-"],
      ["2nd", "-", 0, "-"],
      ["3rd", "-", 0, "-"],
      ["4th", "-", 0, "-"],
      ["5th", "-", 0, "-"],
      ["6th", "-", 0, "-"],
      ["7th", "-", 0, "-"],
      ["8th", "-", 0, "-"],
      ["9th", "-", 0, "-"],
      ["10th", "-", 0, "-"]]

puts hs.to_yaml

Which looks like:
---
- - RANK
  - PLAYER
  - SCORE
  - DATE
- - 1st
  - '-'
  - 0
  - '-'
- - 2nd
  - '-'
  - 0
  - '-'
- - 3rd
  - '-'
  - 0
  - '-'
- - 4th
  - '-'
  - 0
  - '-'
- - 5th
  - '-'
  - 0
  - '-'
- - 6th
  - '-'
  - 0
  - '-'
- - 7th
  - '-'
  - 0
  - '-'
- - 8th
  - '-'
  - 0
  - '-'
- - 9th
  - '-'
  - 0
  - '-'
- - 10th
  - '-'
  - 0
  - '-'

Saving to file is easy:
File.write('path/to/file.yaml', hs.to_yaml)

And reading it back in is as easy:
hs = YAML.load_file('path/to/file.yaml')

Read through the "Yaml Cookbook" for more information how Ruby and YAML structures are related.
That said, I'd recommend a different data structure for your high score table. Instead of arrays of arrays, which are a pain to access, use a hash of hashes:
hs = {
  1 =>  { player: '', score: 0, date: '' },
  2 =>  { player: '', score: 0, date: '' },
  3 =>  { player: '', score: 0, date: '' },
  4 =>  { player: '', score: 0, date: '' },
  5 =>  { player: '', score: 0, date: '' },
  6 =>  { player: '', score: 0, date: '' },
  7 =>  { player: '', score: 0, date: '' },
  8 =>  { player: '', score: 0, date: '' },
  9 =>  { player: '', score: 0, date: '' },
  10 => { player: '', score: 0, date: '' },
}

This will make the YAML file easier to understand:
---
1:
  :player: ''
  :score: 0
  :date: ''
2:
  :player: ''
  :score: 0
  :date: ''
3:
  :player: ''
  :score: 0
  :date: ''
4:
  :player: ''
  :score: 0
  :date: ''
5:
  :player: ''
  :score: 0
  :date: ''
6:
  :player: ''
  :score: 0
  :date: ''
7:
  :player: ''
  :score: 0
  :date: ''
8:
  :player: ''
  :score: 0
  :date: ''
9:
  :player: ''
  :score: 0
  :date: ''
10:
  :player: ''
  :score: 0
  :date: ''

And will simplify your accesses to the table in your code.
In general, your code needs a major clean up. A very strong principle in writing code is "DRY" AKA "Don't repeat yourself". You have a huge amount of redundancy in your code, that can be simplified greatly. That needs to be handled in a separate question, but, basically your score-handling code could be reduced to a case statement and a method to store values in the hash.

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What's Up Heathens! Today we are going over why I don't believe in the historicity of Jesus. I go over all the points which leads me to conclude that was most probably a fictional character made up by first century Jews.
There are multiple lines of evidence against the historicity of Jesus. The first line is Paul. He never cements Jesus here on earth. He has only a spiritual Jesus that exists. He emphatically says that he got his information only from a vision of the risen Jesus and scriptures. Basically, he hallucinated the encounters with Jesus and then used scriptures to gain more knowledge about Jesus. This is why Paul never mentions an earthly ministry or birth.
Another bit of evidence is Philo of Alexandria where he describes the messiah of the Jews and gives characteristic things that we attribute to Jesus. This is because the Messiah is a pre-existent figure in Judaism. He was around when God created the earth. In fact, Jesus is described as God's "Agent of Creation. " The Gospel of John is the most blatant example of this theology in the New Testament (John 1:1).
Then there are the supposed extra-biblical references to Jesus. I go through the most popular ones like Tacitus and Josephus, but none of them actually prove Jesus existed on earth. Josephus contains blatantly forged insertions that are defended with bad apologetics. Tacitus only recites what Christians believed at the time and not any kind of first-hand account of Jesus living on earth. These two examples represent the status of the extra-biblical sources because they are either forged (like 2 Peter) or they only relay information about the belief and not eyewitness accounts.


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text:
Home » Your Health
Dispelling Schizophrenia Myths and Stigmas
Published on May 18, 2016 Updated on September 29, 2020
Health Check Certified By: Greg Dorter, RP
Tips on How to Deal with a Narcissist
Schizophrenia may just be one of the most misunderstood health conditions out there. Unfortunately, it carries a remarkable stigma that can result in people fearing or even outright avoiding people dealing with the issue.
Related Topics (Ads):
Mental Health Therapy OptionsTreatments for Mental Health DisordersSymptoms of Mental HealthCommon Mental Health Disorders
That's why it's important to ignore the movies and television shows and do some research about the condition. As with any condition, schizophrenia varies from person to person, which means it won't affect two people in the same way. Here are nine of the most prominent myths about schizophrenia…
Myth 1: Schizophrenics are Lazy
Some people with serious mental health conditions like schizophrenia do struggle to complete the kinds of tasks that make living independently possible. However, this misconception could be said of just about anyone with a serious condition affecting their mental or physical health.
In reality, schizophrenia has little effect on a person's drive to be successful in their personal or professional lives. According to this article published in The Atlantic, with the right treatment, it can be hard to tell the difference between a schizophrenic and someone who has no link to the condition.
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Myth 2: Schizophrenics Belong in Insane Asylums
It's true that, in the past, many people with serious mental health conditions like schizophrenia were sent to insane asylums and mental hospitals, where they were often subjected to painful and inhumane "treatment." Ask any doctor and they will refer to this as a dark time in the history of healthcare and medicine.
However, that was long before the development of advanced pharmaceuticals capable of helping people with schizophrenia manage their condition and lead normal lives. So, by no means do all people with schizophrenia deserve to be institutionalized.
Myth 3: Having Schizophrenia Makes Having a Job Impossible
You'd think that having a serious mental health condition like schizophrenia makes living a normal life difficult, and that holding down a vibrant and growing career would be difficult to maintain. But schizophrenics who receive the right treatment report have little trouble getting, keeping, and flourishing in employment.
It certainly helps that many employers have developed progressive, forward-thinking policies on the hiring and treatment of people with mental health conditions. Simply put, there's no reason someone with schizophrenia can't thrive in a job so long as they're receiving the right treatment.
Myth 4: Children of Schizophrenics Will Get it as Well
There's no denying that genetics plays a role in the development of schizophrenia. But this does not mean that every child of a schizophrenic is bound to develop the disorder. In fact, experts estimate that only about one in ten children of schizophrenics will be affected by the same issue.
Of course, having two parents with schizophrenia does raise the risk substantially. However, before resolving yourself to a lifetime of dealing with schizophrenia, talk to your doctor about the risks.
Signs You're an Introvert
Myth 5: Good Parents Don't Have Schizophrenic Children
Nurture or nature? It's an ongoing debate when it comes to the development of children. That's why parents, and especially mothers, often blame themselves when a child is diagnosed with schizophrenia (or another significant mental health issue).
However, schizophrenia is rarely caused by poor parenting. It may be the result of trauma, drug abuse, or genetics, and rarely do any of these matters have anything to do with the actions of a mother or father.
Myth 6: Schizophrenics are Slow Learners
It's true that some people with mental health conditions have a harder time learning. And some schizophrenics do have difficulty completing exams testing attention memory, and learning ability.
However, this hardly means schizophrenics lack intelligence. In fact, some of history's most remarkable figures (i.e., mathematician, John Nash, whose life was loosely captured in the film A Beautiful Mind) have been diagnosed with schizophrenia. With the right treatment, the true intelligence of people with schizophrenia can be allowed to shine through.
Myth 7: Schizophrenics are Dangerous
Many people believe that schizophrenics should be avoided because they're volatile, highly unpredictable, and probably dangerous. Too often television and movies have led us to believe that this is true.
However, according to research from WebMD, it's the opposite that's most often the case. While schizophrenics can be unpredictable at moments, they're usually quite calm and stable, particularly if they're receiving the right medication for their condition. In other words, there's no reason to fear schizophrenics.
Tips for Dealing with Obsessive Compulsive Disorder
Myth 8: Schizophrenia Results in Multiple Personality Disorder
One prevailing belief about schizophrenia is that it results in people developing multiple personalities, some of which may be highly volatile and dangerous.
The reality is that most people with schizophrenia don't have multiple personality disorder or Dissociative Identity Disorder (DID), which in itself is a separate condition that is always treated as such. As with any physical or mental illnesses, schizophrenia can lead to the development of other issues, but it is by no means destined to be multiple personality disorder.
Emily Lockhart is a weight loss expert who specializes in healthy living. She is dedicated to providing health-conscious individuals with the information they need to make great lifestyle choices that will make them look and feel better. In her spare time, Emily teaches Pilates at a local studio and enjoys activities like hiking, rowing and biking.
Mental Illness in Teens: Common Conditions and Symptoms
Psychopath vs. Sociopath: What's The Difference?
Psychosis: Symptoms of a Psychotic Episode
Telltale Traits of a Sociopath
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Physical Symptoms of Mental Illnesses
Is My Teen Showing Signs of Schizophrenia?
Are You A Narcissist? The Most Common Signs & Symptoms of Narcissistic Personality Disorder
Symptoms and Causes of Schizophrenia
Signs of Hypochondria
Behavioral Indicators of Antisocial Personality Disorder
Symptoms of Obsessive-Compulsive Disorder