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\begin{document}

\begin{center}

\vspace{0.5cm}

{\large
\bf{ Unifying the Strengths of Forces in Higher
Dimensions}}

\bf{A. Dedes}$^1$ and \bf{P. Kanti}$^2$

$^1$ {\it  Physikalisches Institut der Universit\"at Bonn,\\ 
 Nu\ss allee 12, D-53115 Bonn, Germany}\\
$^2$ {\it Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7,
\\ I-56126 Pisa, Italy}

\end{center}

\centerline{\bf Abstract}
{\small We consider the embedding of the Standard Model fields in a
$(4+d)$-dimensional theory while gravitons may propagate in $d'$
extra, compact dimensions. We study the modification of strengths
of the gravitational and gauge interactions and, for various values
of $d$ and $d'$, we determine the energy scale at which these
strengths are unified. We demonstrate that our ``unification''
constraint can be reduced to the Type I string theory or M-theory
constraint and, therefore, the scale $m_U$ where the unification of the
strengths of forces takes place should be interpreted as the true
unification scale in the framework of a more fundamental
theory. We show that our analysis succeeds in reproducing all known
results coming from string theory and in providing a simple
framework for the study of alternative models.}
\vspace*{2mm}
PACS : 11.10.Kk, 04.50.+h \\
\vspace*{3mm}


 
%\baselineskip=13pt

It is widely believed that any fundamental theory capable of
describing our world at higher energy scales always predicts the
existence of extra, spatial, compact dimensions. Being motivated by
attempts to lower the string scale at the gauge unification
scale~\cite{witten} or the TeV scale~\cite{Lykken},
the concept of large~\cite{ADD} or
small~\cite{RS} extra dimensions, that are being felt only by
gravitons, has been used in order to attack the hierarchy problem.
On the other hand, the existence of extra dimensions,
where gauge bosons can propagate, was
proposed in an attempt to explain the size of the supersymmetry
breaking scale~\cite{antoniadis} or to lower the unification scale
of the gauge interactions~\cite{unification1}. 
In both cases, the Kaluza-Klein excitations of either gravitons
or Standard Model particles appear in the framework of the
4-dimensional, effective theory and, in principle, may modify
the low-energy physics. 
The accuracy with which the electroweak and strong interactions
have been probed at low energies demands that any contributions
to SM processes coming from the KK excitations be extremely
small which, in turn, places a lower bound on the energy scale
associated with the extra dimensions.

In this letter, we assume the existence of two different sets of
extra dimensions: one being felt only by gravitons and one where
gauge bosons can also propagate. 
The gravitational sector of the theory is described by a
higher-dimensional version of Einstein's gravity while
a higher-dimensional analog of the Standard Model describes
the self- and cross-interactions of bosons, fermions and
Higgs-like particles. Both theories should be considered
as effective theories valid from the decompactification scales
of the two sets of extra dimensions up to a higher scale where 
a more fundamental theory (string or M-theory) takes over.
The problem of unification of forces, in a higher-dimensional
framework, has been studied before from the string theory point
of view~\cite{unification1}. Here, we address the same problem
by using an alternative point of view, that of the 
higher-dimensional effective theories for the gravitons
and Standard Model particles. 
It is well known that the presence of the extra dimensions
modifies both the strength of the gravitational forces \cite{ADD}
and Standard Model couplings \cite{Peskin}. We take this idea
one step further and we calculate the strength of the
electromagnetic (gauge) interactions in the presence 
of extra dimensions. We search for specific combinations of
the numbers and sizes of the extra dimensions, that lead to the
unification of strengths of forces, in the framework of the
higher-dimensional theory, and determine the corresponding
pattern of decompactification and unification scales. We
demonstrate that the apparently
{\it ad hoc} demand of the unification of strengths of forces,
in the higher-dimensional framework, is justified by the fact
that our unification constraint straightforwadly reduces to
either the Type I string constraint~\cite{unification1}
or the M-theory constraint~\cite{witten}, and, therefore,
our results are relevant to the true unification of forces
in the context of a more fundamental theory.
We conclude our paper by demonstrating that
results presented previously in the literature
can be easily recovered in the context of our analysis and 
that alternative models with new patterns of energy scales can
be easily determined.

We start our analysis by considering a higher-dimensional formulation
of an $SU(N)$ Yang-Mills theory with matter. We assume that the vector
and scalar particles, i.e. the gauge bosons, $\hat A^\alpha_M$, and
the Higgs field, $\hat \phi$, may propagate
in $4+d$ dimensions, while all the fermionic particles, $\Psi$, are
localized on the 4-dimensional boundary\footnote{Our results apply
also in the case where the fermions live in the bulk.}. 
The $(4+d)$-dimensional action functional, that describes the above
theory and preserves Lorentz invariance, may be written as~\cite{Peskin}
%%%%%%%%%%%%%%%%%%
\ba
%\hspace*{-3mm}
S_{4+d} &= & \int d^4x\,d^dz\,\Biggl\{
%
- \frac{1}{4}\,\Bigl(\partial_M \hat{A}^\alpha_N-
\partial_N \hat{A}^\alpha_M+\frac{\hat{g}}{\sqrt{\Lambda^d}}\,
C_{\alpha\beta\gamma}\,\hat{A}_M^\beta\,\hat{A}_N^\gamma\Bigr)^2
%\nonumber \\[1mm]
%
- \bar{\Psi}_{L,R} \gamma^\mu \Bigl(\partial_\mu
-\frac{i\,\hat{g}}{\sqrt{\Lambda^d}}\,t^\alpha \hat{A}^\alpha_\mu
\Bigr)\,\Psi_{L,R}\,\delta(\vec{z}) \nonumber \\[1mm]
%
&-& \Bigl|\Bigl(\partial_M 
-\frac{i\,\hat{g}}{\sqrt{\Lambda^d}}\,t^\alpha \hat{A}^\alpha_M
\Bigr)\,\hat{\phi}\Bigr|^2 - \hat{\mu}^2\hat{\phi}^*\hat{\phi}
-\frac{\hat{\lambda}}{2\Lambda^d}\,
(\hat{\phi}^*\hat{\phi})^2  
%\nonumber\\[1mm]
%
- 
\Bigl(\frac{\hat{Y}_1}{\sqrt{\Lambda^d}}\,\bar{\Psi}_L\,\hat{\phi}\,\Psi_R
+ \frac{\hat{Y}_2}{\sqrt{\Lambda^d}}\,\bar{\Psi}_L\,\hat{\phi}_c\,\Psi_R
+ h.c. \Bigl)\,\delta(\vec{z}) \Biggr\}\,,
\label{daction}
\ea
%%%%%%%%%%%%%%%%
where $\hat{g}$, $C_{\alpha\beta\gamma}$ and $t^a$ are the coupling constant,
the structure constants and the generators, respectively, of the $SU(N)$
gauge group while $\hat \mu^2$ and $\hat \lambda$ are the mass and coupling
constant of the Higgs field. In order to render the coupling constants of
the theory dimensionless in $(4+d)$-dimensions, an arbitrary energy scale
$\Lambda$ has been introduced. Note that, in the above,
$M, N=\{t, x_1, x_2, x_3, z_1, z_2,..., z_d\}$,
$\mu, \nu=\{t, x_1, x_2, x_3\}$ and the hat 
denotes $(4+d)$-dimensional quantities.

Next, we assume that the extra $d$ dimensions are compactified over an
internal manifold with the size of every compact dimension being $2L$.
Then, we can Fourier expand the $(4+d)$-dimensional vector and scalar
fields along the compact dimensions in the following way
%%%%%%%%%%%%%%
\be
\hat{\Phi}(x,z) = \hat{\Phi}^{(0)}(x) + \sum_{\vec{n}=1}^\infty\,
\frac{\hat{\Phi}^{(\vec{n})}(x)}{\sqrt{2}}\,\Bigl(
e^{i\frac{\pi\vec{n}}{L}\vec{z}} + e^{-i \frac{\pi\vec{n}}{L}\vec{z}}
\Bigr)\,, 
\label{kk}
\ee
%%%%%%%%%%%%%%%
where $\vec{n}=\{n_1, n_2,..., n_d\}$. By performing a
Kaluza-Klein compactification, i.e by using the above field expansion
and integrating over the extra dimensions, the action (\ref{daction})
reduces to an effective 4-dimensional theory. A prominent feature of this
effective theory is its complexity due to extra terms involving
the massive Kaluza-Klein (KK) excitations, $\hat{\Phi}^{(\vec{n})}(x)$,
of all the fields propagating in the extra dimensions, apart from the
usual, massless
zero-modes, $\hat{\Phi}^{(0)}(x)$.  Here, we are only interested in the part of the
effective theory that contains the zero modes of the various fields
and, more specifically, in the relations that hold between the $(4+d)$
and 4-dimensional couplings and masses. These are found to be 
%%%%%%%%%%%
\be
g=\frac{\hat{g}}{(2L \Lambda)^{d/2}}\,, \qquad
Y_{1,2}=\frac{\hat{Y}_{1,2}}{(2L \Lambda)^{d/2}}\,,
\label{gauge}
\ee
%%%%%%%%%%%%%
\vspace*{-5mm}
%%%%%%%%%%%%%
\be
\mu^2=\hat{\mu}^2\,, \qquad
\lambda=\frac{\hat{\lambda}}{(2L \Lambda)^d}\,, 
\label{higgs}
\ee
%%%%%%%%%%%%%%%%%%
where we have used the following field redefinitions 
%%%%%%%%%%%%%%%
\be 
A_\mu^a=(2L)^{d/2}\,\hat{A}^a_\mu\,\,, \qquad
\phi=(2L)^{d/2}\,\hat{\phi}
\label{fields}
\ee
%%%%%%%%%%%%%
in order to obtain canonical kinetic terms in 4-dimen\-sions.
Note that the fermionic fields being always localized on the
4-dimensional boundary remain unchanged. 

In the framework of the 4-dimensional effective theory, the gauge bosons
and fermions acquire mass through the spontaneous symmetry breaking of
the gauge group. Their masses are given in terms of the vacuum expectation
value of the Higgs field which is also redefined according to
Eq.~(\ref{fields}). 
The question, then, arises  whether the mass spectrum
of the 4-dimensional effective theory corresponds to a different one
in the framework of the original (4+d)-dimensional theory, 
a point which is going to be of significant importance in our
subsequent analysis. To
answer this question, we consider the masses of the fermions which are
given by the following expression
%%%%%%%%%%%%%%%
\be
m_{\Psi_{L,R}}=Y_{1,2}\,\frac{\upsilon}{\sqrt{2}}=
\frac{\hat{Y}_{1,2}}{\Lambda^{d/2}}\,\frac{\hat{\upsilon}}{\sqrt{2}}
=\hat{m}_{\Psi_{L,R}}\,.
\label{masses}
\ee
%%%%%%%%%%%%%%%%%%%
Similar relations can also be written for the masses of the gauge bosons
and the Higgs field. In all cases, the tree-level masses, that are
generated via the Higgs mechanism in the 4-dimensional theory, remain
unaltered when one embeds this theory in a higher-dimensional one. 

As it is well known~\cite{Peskin},
this is not the case with the coupling constant $\hat g$
that eventually determines the strength of the gauge
interactions of the theory. The $SU(N)$ gauge group of the
$(4+d)$-dimensional theory (\ref{daction}) could be replaced
by the $U(1) \times SU(2) \times SU(3)$ group with the gauge field
$\hat A^\alpha_M$, the generators $t^\alpha$ and the coupling constant
$\hat g$ standing for each one of the corresponding quantities of the
Standard Model group, i.e. $(\hat{B}_M, \hat{W}^a_M,
\hat{G}^\alpha_M)$, $(Y/2,\tau^a/2, \lambda^\alpha/2)$ and
$(\hat{g}_1, \hat{g}_2, \hat{g}_3)$, respectively. In that case,
each coupling constant and gauge boson of the SM group is redefined
according to Eqs.~(\ref{gauge}) and (\ref{fields}), respectively.

By making use of the redefinition (\ref{gauge}) for the gauge coupling
constants, we find that the electric charge $e$ changes as follows
%%%%%%%%%%%%%%
\be
\hat{e} = \hat{g}_1 \cos\hat{\theta}_W = (2L\Lambda)^{d/2}
g_1 \cos\theta_W =(2L\Lambda)^{d/2}\,e\,,
\ee
%%%%%%%%%%%%%%
where we have used the fact that the Weinberg angle $\theta_W$
remains unaltered since its tangent is given by the ratio of the gauge
couplings $g_1$ and $g_2$. The above rescaling of the electric charge
inevitably affects the strength of the electromagnetic interactions in
higher dimensions. The strength of a force is classically defined
\cite{halzen} as the
potential energy of the corresponding interaction between two identical
particles, separated by a distance equal to the particle's Compton
wavelength, $\lambdabar=\hbar/mc$, compared to the energy of the rest
mass of each particle. This is indeed the only way of providing a 
dimensionless measure of the strength of a force that depends on the
distance between the interacting particles.
As is well known, in 4-dimensions and for two
particles with mass $m$ and charge $e$, the above rule gives the result
%%%%%%%%%%%%%
\be
\alpha_{EM}=\frac{E_{int}}{E_m}=\frac{1}{mc^2}\,\frac{e^2}
{4\pi\epsilon_0\,(\hbar/mc)}=\frac{e^2}{4\pi\epsilon_0}
\simeq \frac{1}{137}\,,
\label{electro4D}
\ee
%%%%%%%%%%%%%
where we have set $c=\hbar=1$. 
Generalizing the above definition of the strength of a force in $(4+d)$
dimensions, in the limit $\lambdabar < 2L$, i.e. for energies
larger than the decompactification scale, we obtain
%%%%%%%%%%%%%
\be
\hat{\alpha}_{EM}=\frac{\hat{E}_{int}}{\hat{E}_m}=
\frac{1}{\hat{m}}\,\frac{(\hat{e}^2/\Lambda^d)}{4\pi\epsilon_0\,
(1/\hat{m})^{1+d}}=\alpha_{EM}\,\Bigl(\frac{m}{M_x}\Bigr)^d\,,
\label{electro}
\ee
%%%%%%%%%%%%%
where we have used the fact that the masses of the particles do not
change, according to Eq. (\ref{masses}), and where we have defined
$M_x \equiv (2L)^{-1}$. It is straightforward to check that, in
the limit, $\lambdabar > 2L$, the above formula reduces to
Eq. (\ref{electro4D}) as expected. The above result clearly
reveals that the strength of the electromagnetic force changes as the
gauge bosons start ``feeling" the extra dimensions. Moreover, the
strength of the force strongly depends not only on the number and size 
of extra dimensions but also on the mass of the test particle, i.e.
on the energy scale where the measurement takes place
\footnote{A similar dependence of the strength of electromagnetic
forces on the mass of the test particle arises also in 
lower-dimensional problems, i.e. in the case where the total
number of dimensions is lower than four. We thank M. Shifman
for pointing this out.}. 
It is also worth noting that the auxiliary energy scale $\Lambda$
does not appear in the electromagnetic strength
formula (\ref{electro}).


So far, we have not considered the gravitational interactions. 
We now assume that gravitons may propagate, apart from the usual
4 dimensions, in $d'=\delta + d$ extra dimensions, where $\delta$ is
the number of transverse dimensions, with size $2L'$, felt only by
gravitons. In that case, the $(4+d')$-dimensional action functional  
%%%%%%%%%%%%%%
\be
S_{4+d'}  =  -\frac{M^{2+d'}_{GR}}{16\pi}
\int d^4x\,d^{d'}z\,\sqrt{G_{4+d'}}\,R_{4+d'}
\label{grav}
\ee
%%%%%%%%%%%%%%%%%%%%%
reduces to an effective, 4-dimensional Einstein's theory of gravity 
only when the following relation between the sizes of the
extra dimensions and the energy scales of gravity, $M_P$ and 
$M_{GR}$, in 4 and $(4+d')$ dimensions, respectively, 
%%%%%%%%%%%%%%%
\be
(2L')^\delta\,(2L)^d\,M^{2+d'}_{GR}=M_P^2 
\label{defmgr}
\ee
%%%%%%%%%%%%%%%%%%%%
holds. Here, we restrict ourselves in the case of product
spacetimes where the 4-dimensional subspace does not depend
on the geometry of the internal, compact manifolds.
In this case, the strength of the gravitational interactions also
changes when one introduces extra dimensions for the gravitons
\cite{ADD}. Using the
same rule as above, the strength of the gravitational
interaction between two particles with mass $m$,
in $4+d'$ dimensions, is given by the expression
%%%%%%%%%%%%
\be
\hat{\alpha}_{GR}=\frac{\hat{E}_{int}}{\hat{E}_m}=\frac{1}{\hat{m}}\,
\frac{\hat{G}_N\,\hat{m}^2}{(1/\hat{m})^{1+d'}}=
\Bigl(\frac{m}{M_{GR}}\Bigr)^{2+d'}\,\,,
\label{gravi1}
\ee
%%%%%%%%%%%%%
where $\hat{G}_N=1/M_{GR}^{2+d'}$. By choosing appropriately the
sizes of the transverse and longitudinal, extra 
dimensions, the higher-dimensional
gravity scale, $M_{GR}$, could be much lower than the 4-dimensional one,
according to Eq. (\ref{defmgr}),
a fact which, subsequently, changes the strength of gravity.


The question we would like to address is the following:
can these two effective theories, described by Eqs. (\ref{daction})
and (\ref{grav}), provide useful information on the problem
of unification of forces? Note that, here, we are interested
in bridging the gap between the gravitational
and electromagnetic forces. The effective theory (\ref{daction})
being non-renormalizable can not address the problem of the
unification of gauge couplings. This problem can be studied
independently, through the power-law mechanism \cite{unification1,ross},
in the framework of the 4-dimensional, effective theory.
Here, we ignore the small differences in the strengths of
electromagnetic, weak and strong interactions and we focus on 
the study of the unification of gravitational and gauge
interactions, the latter being represented by the electromagnetic
interactions\footnote{Under
the assumption that the electromagnetic, weak and strong
forces ``feel" the same number $d$ of extra dimensions, we may
assume that their strengths remain comparable at every scale even
in $(4+d)$-dimensions.}.
Having derived the strengths of these forces, in the presence 
of extra dimensions, it seems natural to assume that there is
a scale $m_U$ where the strengths of
gravitational and electromagnetic forces become equal. Combining
Eqs. (\ref{electro}) and (\ref{gravi1}), we obtain the 
``unification'' constraint 
%%%%%%%%%%%%%%%%%
\be
\hat{\alpha}_{EM}=\hat{\alpha}_{GR} \Rightarrow
\alpha_{EM} \biggl (\frac{m_U}{M_x} \biggl )^d = \biggl (\frac{m_U}{M_{GR}}
\biggr )^{2+d'} \;.
\label{unif}
\ee
%%%%%%%%%%%%%%%%
A priori,
there is no reason to believe that such a demand is indeed
relevant to the true unification of forces in the framework
of a more fundamental theory, like string or M-theory. One
could argue that demanding unification of the relevant mass
scales, instead of unification of couplings, would be more
robust. Moreover, any combination of numbers and sizes of
the two sets of extra dimensions, that would lead to the
unification of gravitational and gauge forces, should be
chosen in such a way that they would also lower the scale
of the fundamental theory, in a consistent way, down to
$m_U$ \footnote{We are grateful to Keith Dienes for a
fruitful discussion on this point.}.
We will now demonstrate that this is indeed the case
with the numbers and sizes of the extra dimensions that 
follow from our ``unification'' constraint, Eq. (\ref{unif}). 
As we will see, the above relation reduces to the fundamental
constraint of Type I string theory or M-theory, according
to our assumptions, and that the scale of the unification
of strengths of the forces $m_U$ can be interpreted
as the string scale $M_s$ or the M-theory scale $M_M$, respectively.
To this end, we rewrite Eq. (\ref{unif}), combined with 
Eq. (\ref{defmgr}), in the form
%%%%%%%%%%%%
\be
\Bigl(\frac{m_U}{M_P}\Bigl)^2=\hat \alpha_{EM}\,(m_U\,2L')^{-\delta}\,
(m_U\,2L)^{-d}\,,
\label{string}
\ee
%%%%%%%%%%%%%
which reduces to the Type I string constraint
$M_s^2 \sim (\alpha_{gauge} M_P \sqrt{V})^{-1}$\cite{unification1}
if we identify $m_U$ with $M_s$, and $\hat \alpha_{EM}$
with $\alpha^2_{gauge}$. The total compactified volume is now
$V=(2L)^d (2L')^\delta$ and a T-duality transformation
$m_U\,r \leftrightarrow (m_U\,r)^{-1}$ needs to be performed,
where $r \equiv 2L, 2L'$.
Alternatively, for $\delta=1$ and $d=0$, Eq. (\ref{unif})
reduces to the relation
%%%%%%%%%%%
\be
\frac{1}{M_c}=\hat \alpha_{EM}\,\frac{M_P^2}{m_U^3}
\label{m-theory}
\ee
%%%%%%%%%%%
which is nothing else than the M-theory constraint
$R_{11}=\alpha^2_{gauge}\,(l_M^3/l_P^2)$~\cite{witten}, that relates
the size of the extra dimension for the gravity with the scale
of the M-theory and the Planck length, if we identify
$m_U$ with $M_M$ and $R_{11}$ with $M_c^{-1}$. We may therefore
conclude that the constraint (\ref{unif}) not only is the
right constraint for the unification of forces, in the framework
of the higher-dimensional effective theories (\ref{daction}) and
(\ref{grav}), but also any combination of compactification
radii is directly embedded into a finite framework such as
string or M-theory. 

We will now study the four distinct categories, that are formed
according to the values\footnote{Motivated by M-theory,
we assume that $0 \leq \delta, d \leq 7$.}
of the numbers $\delta$ and $d$ of the extra dimensions, in order
to demonstrate the simplicity with which results, presented previously
in the literature, arise in the framework of our analysis, but also
the fact that new patterns of energy scales, associated with the
extra dimensions, can easily be determined. 
Hereafter, for simplicity, we will take $\alpha_{EM} \simeq 10^{-2}$.

({\it i}) $\delta=d=0$ . In this case, $M_{GR} \equiv M_P$ and {\it only} 
at energies $m_U \simeq 10^{18}$ GeV, i.e. close to the Planck scale, 
the strengths of the forces become equal, as expected.

({\it ii}) $\delta \ne 0$, $d=0$ . Here, we assume that only the 
gravitational fields can feel extra $\delta$ dimensions. Then, 
from Eq. (\ref{unif}, we may easily conclude that 
the unification of strengths takes place at a scale
$m_U=(0.2-0.6) M_{GR}$, for $1 \le \delta \le 7$.
This means that the strength of
the gravitational force becomes comparable to that of the other forces
only when the measurement takes place near the gravity scale,
independently of where $M_{GR}$ lies. 
The results associated with M-theory \cite{witten} can be recovered for
$\delta=1$ and $M_c \equiv (2L')^{-1}=10^{12}$ GeV. Then, the unification
takes place at $m_U \simeq M_{GUT} \simeq M_{GR} \sim 10^{16}$ GeV.
The low-scale gravity model of Ref.~\cite{ADD}, with $M_{GR} \sim 1$
TeV can be also reproduced by choosing appropriately the value of
$\delta$ and the size of the transverse dimensions $M_c$.
In the framework of our analysis, we may easily see that the new
hierarchy problem, between the compactification
scale $M_c$ and the electroweak scale $M_W$, can be resolved
by increasing the number $\delta$ of
extra dimensions accompanied with a small increase in the value of the
higher-dimensional gravity scale $M_{GR}$. For example, allowing
the compactification scale, $M_c$, to be close to the electroweak
one, i.e. $M_c=10^{\pm 1} M_W$, the gravity scale lies in the range
$M_{GR}=(10^5-10^7)$ GeV, for $\delta \ge 6$. In this case, there is no energy
gap between $M_c$ and $M_W$ while the gravity scale is still 
close to the electroweak scale (the ratio $M_{GR}/M_W$, in this case,
is the same as the one of the top quark mass over the down quark
mass, $m_t/m_d \sim 10^4$).


({\it iii}) $\delta=0$ , $d \ne 0$ . In this case, it is the $d$ extra,
longitudinal dimensions that ``open up" for the gauge and scalar fields,
as well as for gravitons, at the scale $M_x$. Substituting
Eq. (\ref{defmgr}) into Eq. (\ref{unif}) and solving for $m_U$, we find
that the unification of strengths takes place only near the string
scale, i.e. $m_U=M_s \simeq 10^{18}$ GeV, independently of the number $d$ of
extra dimensions. If we take the new gravity scale, $M_{GR}$, to be
the string scale, then, from Eq. (\ref{defmgr}), we are led to the
following relation between $M_x$ and $M_{GR}$:
%%%%%%%%%%%%%%
\be
M_x=10^{-2/d} M_{GR} = (0.01 - 0.5) M_{GR} \;,
\label{eq15}
\ee
%%%%%%%%%%%%%%%
for $1 \le d \le 7 $. In this case, the compactification
scale $M_x$ is 1 or 2 orders of magnitude smaller than the gra\-vity scale.
A quite interesting result arises for $d=1$. Then, Eq. (\ref{eq15})
gives the result $M_{x} \simeq 10^{16}$
GeV, which coincides with the scale of the unification of gauge
couplings, $M_{GUT}$, in the minimal supersymmetric standard model
(according to the power-law mechanism for the unification of
gauge couplings \cite{unification1,ross},
the presence of the extra dimensions does not affect the
unification scale $M_{GUT}$ when $M_x \simeq M_{GUT}$).
The compactification scale, $M_x$, being very close to $M_s$
does not introduce any new energy scale in the theory and is
sufficiently high to suppress the ``dangerous'' baryon number violating
operators. From that point of view, the above arrangement of energy
scales resembles the one that arises in the framework of string
unification~\cite{kaplunovsky} where the compactification scale, $M_x$,
is very close to the string scale, $M_s$. This arrangement is indeed
necessary in order to prevent the gauge coupling $\alpha_{EM}$ from
acquiring an unacceptably small value near the unification scale, a goal
which is also accomplished in our case. Although the hierarchy
between the electroweak scale, $M_W$, and the gravity scale, $M_{GR}$,
still remains, the existence of an extra, longitudinal dimension
that ``opens up" just above $M_{GUT}$ may provide a natural explanation
for the energy gap between $M_{GUT}$ and $M_s$. 



({\it iv}) $\delta \ne 0$ , $d \ne 0$ . This is the most general 
case where both gravitational and gauge fields feel a number of extra
dimensions. According to Eq. (\ref{unif}), the unification of strengths,
now, takes place at the energy scale
%%%%%%%%%%%%%%
\be
m_U^{2+\delta}=\alpha_{EM}\,M_P^2\,M_c^\delta \;.
\label{eq16}
\ee
%%%%%%%%%%%%%%%
We make the natural assumption that the ultimate unification of strengths
occurs near the new gravity scale, i.e. $m_U=M_{GR}$. Then, the combination
of Eqs. (\ref{defmgr}) and (\ref{eq16}) leads to the same relation
(\ref{eq15}) between the compactification scale $M_x$ and the gravity 
scale $M_{GR}$. On the other hand, the scale $M_c$, associated with the size
of the transverse dimensions felt only by gravitons, can be found
from the following expression:
%%%%%%%%%%%%%
\be
M_c^\delta= \alpha_{EM}^{-1}\,\frac{M_{GR}^{2+\delta}}{M_P^2}\;.
\label{eq17}
\ee
%%%%%%%%%%%%%
Note that it is {\it only} the number of transverse dimensions $\delta$ that
determines the compactification scale $M_c$, in terms of $M_P$ and $M_{GR}$,
exactly in the same way that it is {\it only} the number of longitudinal
dimensions $d$ that determines the corresponding expression of $M_x$.
If we, now, choose $M_{GR}=M_s \simeq 10^{18}$ GeV and $d=1$, the interesting
result $M_x \simeq M_{GUT}$ of case ({\it iii}) arises once again. However,
in order for this picture to be viable in the existence of extra, transverse
dimensions for gravitons, the compactification scale $M_c$ should lie very
close to the string scale for every value of $\delta$. Since
$M_c, M_x \geq M_{GUT}$, the unification pattern of gauge couplings is not
affected by the presence of the extra dimensions, either transverse or
longitudinal, and the ultimate unification of all forces takes place
exactly at the string scale. Moreover, the compactification scale $M_c$,
being very close to $M_s$, does not introduce any new energy scale in
the theory. The only other case where this problem is also
resolved is when $M_c = 10^{\pm 1} M_W$, according to the analysis
presented in case ({\it ii}). For $\delta=6$ and $M_c=10$ GeV,
the gravity scale lies at the energy scale $2\times 10^5$ GeV.
Then, for $d=1$, Eq. (\ref{eq15}) shows that the unification of strengths
takes place only if $M_x$ is exactly two orders of magnitude
smaller than $M_{GR}$, i.e. $M_x=2$ TeV. Remarkably, this is in
perfect agreement with the proposal of the existence of a single
extra dimension for the gauge bosons with size at the TeV scale
necessary for the explanation of the supersymmetry breaking scale
\cite{antoniadis}. The existence of an extra dimension of this size
would lead to the unification of gauge couplings, through the power-law
mechanism \cite{unification1}, at an energy scale
which is, approximately, one order of magnitude larger than $M_x$
\cite{unification1,ross}. The extra hypothesis, that gravitons may
feel 6 extra, compact dimensions, brings the scale of gravity
down to $200$ TeV, thus, completing the picture of unification
of forces and removing the hierarchy problem. It is worth noting
that, by changing appropriately the values of $\delta$ and $M_c$,
we may easily recover the results of Ref.~\cite{unification1}, where
$M_{GUT}=M_{GR}=M_s=10$ TeV, and which were derived with a completely
different method.


In conclusion, in this letter, we have demonstrated that the addition
of extra, compact dimensions for gravitons and gauge bosons modifies
the strengths of gravitational and gauge interactions. We have
formulated a ``unification'' constraint under the assumption that
the strengths of gravitational and gauge interactions become equal
at a scale $m_U$. This scale is uniquely defined in terms of the numbers
and sizes of the two sets of extra dimensions. We have demonstrated
that the aforementioned ``unification'' constraint reduces either
to the Type I string theory or the M-theory constraint and, therefore,
$m_U$ should be interpreted as the string theory scale $M_s$ or
the M-theory scale $M_M$, respectively. As a result, the scale of the
unification of strengths of forces, in the framework of the 
higher-dimensional effective theory for gravitons and gauge bosons,
signifies the true unification of forces in the framework of a more
fundamental theory. We have shown that previously derived results
can be easily recovered in the framework of our analysis and that
new features of the unification and patterns of compactification
radii can be easily derived. For example,  we have found that,
when gravitons propagate in $4+\delta$ dimensions, the unification
always takes place near the new gravity scale, $M_{GR}$, whereas,
when gauge bosons and gravitons feel the same number $d$ of extra
dimensions, the unification occurs only near the string
scale, $M_s$. In the latter case, the existence of only one extra
dimension, that ``opens up" just above $M_{GUT}$, can justify the
energy gap between $M_{GUT}$ and $M_s$. In the more interesting
case where gravitons and gauge bosons feel a different number of 
extra dimensions, a low-scale unification pattern arises: 6 extra
dimensions ``open up'' only for gravitons at $M_c=10$ GeV while
one extra dimension ``opens up'' for the gauge bosons, as well,
at $M_x=2$ TeV, a value that could explain the size of the
supersymmetry breaking scale. The unification of all forces takes
place at the new gravity scale of the theory, i.e. at 
$M_{GR}=200$ TeV, thus rendering this 11-dimensional theory free
of any hierarchy problem. It is worth noticing that our analysis
can lead to many more alternative models, possibly phenomenologically
viable, and only few of them have been presented above as illustrating
examples.








The authors would like to thank K.R. Dienes, H.P. Nilles, R.G. Roberts,
G.G. Ross, M. Shifman and K. Tamvakis for fruitful discussions and comments.
A.D is supported from Marie Curie Research Training Grants
ERB-FMBI-CT98-3438. P.K. is grateful to the Particle Physics Group
at Rutherford Appleton Laboratory for the warm hospitality and financial
support. P.K. also acknowledges partial financial support from the
DOE Grant No. DE-FG-02-94-ER-40823.


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