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\preprint{\vbox{
\rightline{BUHEP-02-40}
\rightline{FSU-HEP-021202} 
\rightline{IFUAP-HEP-03-02}
}}

{\tiny .} \vspace{1.5cm}

\title{Where is the Higgs boson?}

\author{A. Aranda}
\email[]{aranda@buphy.bu.edu}
\affiliation{Department of Physics, Boston University, Boston, MA 02215 U.S.A}

\author{C. Bal\'azs}
\email[]{balazs@hep.fsu.edu}
\affiliation{Department of Physics, Florida State University, 
Tallahassee, FL 32306 U.S.A.}

\author{J.L. D\'\i az-Cruz}
\email[]{ldiaz@sirio.ifuap.buap.mx}
\affiliation{Instituto de Fisica, BUAP, Puebla, Pue. 72570, M\'exico}

\date{December 2, 2002}

%\maketitle

\begin{abstract}
Electroweak precision measurements indicate that the standard model Higgs boson 
is light and that it could have already been discovered at LEP 2, or might be 
found at the Tevatron Run 2. In the context of a TeV$^{-1}$ size extra 
dimensional model, we argue that the Higgs boson production rates at LEP and the 
Tevatron are suppressed, while they might be greatly enhanced at the LHC or at a 
LC. This is due to the possible mixing between brane and bulk components of the 
Higgs boson, that is, the non-trivial brane-bulk `location' of the lightest 
Higgs.
%
To parametrize this mixing, we consider two Higgs doublets, one confined to the 
usual space dimensions and the other propagating in the bulk. Calculating Higgs 
production and decay rates, we find that compared to the standard model (SM),
the cross section receives a suppression well below but an enhancement close to 
and above the compactification scale $M_c$. This impacts the discovery of the 
lightest (SM like) Higgs boson at colliders. 
%
To find a Higgs signal in this model a higher luminosity would be required at 
the Tevatron Run 2 and the LC with $\sqrt{s}=500$ GeV than in the SM case. 
Meanwhile, at the LHC and a LC with $\sqrt{s}=1.5$ TeV, one might find highly 
enhanced production rates. This will enable the latter experiments to distinguish 
between the extra dimensional and the SM for $M_c$ up to about 6--8 TeV.
\end{abstract}

\maketitle

%\pacs{PACS:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction} 
\label{sec:intro}

% Higgs
The Higgs boson is the missing link connecting the real world with the unified 
electroweak (EW) gauge group by spontaneously breaking the latter. Precision 
measurements of EW observables constrain the Higgs mass below about 200 GeV at 
95\% C.L. \cite{Langacker:2002sy,Villa:2002zt,Chanowitz:2002cd} within the 
standard model (SM). Thus, it is expected that a Higgs particle will be 
discovered at the Run 2 of the Tevatron, provided sufficient luminosity 
\cite{Carena:2000yx}.
%
But it is intriguing to notice that the EW observables strongly prefer a SM like 
Higgs with mass below 114.1 GeV \cite{Langacker:2002sy,Chanowitz:2002cd}, which 
is the present lower limit from LEP 2. The data also indicate that the Higgs boson 
should have already been discovered \cite{Langacker:2002sy}, and the fact that 
it was not can be interpreted as new physics crucially affecting the Higgs sector 
\cite{Chanowitz:2002cd}.
%
In this work we put forward a model in which the presently missing signal of the 
lightest Higgs boson is due to a suppression of the Higgs production cross 
section at LEP and the Tevatron. This suppression arises from the non-trivial 
`location' of the lightest Higgs boson in a five dimensional space. However, the 
same feature promises significant enhancement of the Higgs signal at the CERN 
Large Hadron Collider (LHC) and possibly at the next Linear Collider (LC).

% XD
The five dimensional model that is used in this work arises as follows.
The idea that our universe could be confined to a higher dimensional defect 
\cite{exdimori}, has recently been revived in the context of non-perturbative 
string analyses~\cite{Polchinski:1995mt,Horava:1996ma}, and applied as a 
possible solution to the gauge hierarchy problem \cite{exdimcom}. Such a 
solution relies on the existence of $n > 0$ additional compact space-like 
dimensions.
% hierarchy
In models based on this idea, the four dimensional Planck scale $M_{P\ell}$ 
becomes an effective quantity and is related to the fundamental scale $M$ via 
the relation $M_{P\ell}^2 = M^{n+2} V_n \label{mp}$ by the volume of the extra 
space $V_n$. If one requires $M = {\cal O}$(TeV) then for $n = 2$, the 
compactification radius $R \sim V_n^{1/n}$ is in the order of a millimeter, but 
for $n = 7$ it is less than a fermi, not far from the inverse of a TeV.
It is remarkable to notice that with ${\cal O}$(TeV$^{-1}$) size extra 
dimensions the hierarchy problem is indeed nullified, since the fundamental 
scales $M \sim 1/R$ are close to TeV. 

% TeV scale XD
The original string arguments \cite{Polchinski:1995mt,Horava:1996ma} 
also allow the standard gauge and Higgs sectors to penetrate the bulk. If the 
compactification scale $M_c = 1/R$ is higher than about 1 TeV, experiments do 
not conflict with this scenario either~\cite{quiros,ddg}. This makes the inverse 
TeV size extra dimensional models attractive.
% or
Alternatively, the hierarchy problem can be solved with the use a non-% 
factorizable geometry, which has also been proposed in five dimensions. The 
introduction of an exponential `warp' factor reduces all mass parameters of 
the order of a fundamental $M_{P\ell}$ on a distant brane to TeV's on the brane 
where we live~\cite{exdimran}. In order to explain large hierarchies among 
energy scales one simply has to explain small distances along the extra 
dimension, thus this mechanism requires a Planck scale size extra 
dimension~\cite{rs2}.

These intriguing possibilities have opened a new window for the exploration of 
physics beyond the SM~\cite{Uehara:2002yv}, in particular, the phenomenology of 
the Higgs sector. In Ref.~\cite{edhixgun} it is shown that it is possible to 
obtain electroweak symmetry breaking in an extra dimensional scenario even in 
the absence of tree-level Higgs self interactions. Also, in 
Ref.~\cite{edhixcoll} we find scenarios in which the radion in the 
Randall-Sundrum model is contrasted with the SM Higgs boson.
Studying several 
models that lead to a universal rate suppression of Higgs boson observables, 
Ref.~\cite{wellsupr} concluded that the Tevatron and LHC will have 
difficulty finding evidence for extra dimensional effects. Yet 
another study of universal extra dimensions~\cite{universal} conjectures that a 
suppression of the Higgs rates occurs~\cite{petriello}.

However, just as in the SM and other four dimensional theories, the Higgs sector 
remains the least constrained, since it can live either 
in the brane or the bulk, each choice being phenomenologically consistent. One way to 
parametrize this freedom is to consider an extra dimensional two Higgs doublet model 
(THDM) where one doublet lives in the bulk while the other is confined to the 
brane. The lightest Higgs boson state, which will resemble the SM one, will then be a 
linear combination of the neutral components of the two doublets. 
Constraints from electroweak precision data have been applied to such a model, 
and it was found that the compactification scale is larger than a couple of 
TeVs~\cite{masipomarol}. 

In this paper we study the ability of present and future colliders 
to find the lightest Higgs boson, for simplicity, in a model with a single 
TeV$^{-1}$ size new dimension. We assume that the standard model gauge bosons 
and one of the Higgs doublets propagate in this compact dimension. 
The SM $W^{\pm}$ and $Z$ particles are identified with the zero modes of the 
five dimensional gauge boson fields. There
is a second Higgs field restricted to the brane together with all
the matter fields of the SM. The CP even Higgses may be the combinations, i.e. 
brane-bulk mixed states, of the two Higgs doublets.
We derive the Lagrangian for Higgs interactions and apply it to
calculate the cross section of the associated production of
Higgs with gauge bosons at the LC, as well as the dominant Higgs decays including the
possible contributions from virtual KK states. 
Crucial to our approach is the cumulative effect of the virtual gauge boson KK 
states, $W^{\pm(n)}$ and $Z^{(n)}$, which contribute to the cross
section for the production of the Higgs associated with the $W^\pm$ and $Z$
and to the three-body decay $h\to Vf\bar{f}'$ ($V=W^{\pm(0)},Z^{(0)}$). 
The corresponding reactions at hadron colliders are studied as well. 
%Taking advantage of the large energy that can be achieved at the LHC, one could 
%also consider the production of Higgs associated with $W^{\pm(n)}$ or $Z^{(n)}$ 
%KK states.

The organization of our paper is as follows. In Section~\ref{sec:modeling}, we
present the Higgs model that is used to study the brane-bulk mixing of the
Higgs boson. Then in Section~\ref{production}, we derive formulae for
the Higgs production and decays. These include the evaluation of the
contribution from virtual $Z^{(n)}$ KK states to the associated production at 
linear colliders, i.e. $e^+ e^- \to hZ^{(0)}$, as well as 
to the three-body decay $h\to Vf\bar{f}'$.
The calculation for the hadron collider case, i.e. $pp \to hZ^{(0)}$ 
% and $pp \to hZ^{(n)}$, 
is also included. The discussion of the implications of our 
results for present and future colliders appears in 
Section~\ref{sec:Implications}, while the conclusions are presented in 
Section~\ref{sec:conclusion}.


\section{Modeling the Higgs location} 
\label{sec:modeling}

To describe the brane-bulk Higgs mixing, we work with a five dimensional 
(5D) extension of the standard model that contains two Higgs doublets.
The SM fermions and one Higgs doublet ($\Phi_u$) live on the 4D boundary,
the brane, while the gauge bosons and the second Higgs doublet ($\Phi_d$),
are all allowed to propagate in the bulk. The constraints from electroweak
precision data~\cite{masipomarol}, show that the compactification scale
can be of ${\cal O}$(TeV) (2--3 TeV at 95 \% C.L.).
The relevant terms of the 5D $SU(2)\times U(1)$ gauge and Higgs Lagrangian are 
given by
\be \label{5dlagrangian}
{\cal L}^5 = -\frac{1}{4}\left(F^a_{MN}\right)^2
 -\frac{1}{4}\left(B_{MN}\right)^2
+ |D_M \Phi_d|^2 + |D_{\mu}\Phi_u|^2 \delta(x^5) \, ,
\ee
where the Lorentz indices $M$ and $N$ run from $0$ to $4$, and $\mu$ runs 
from $0$ to $3$. The covariant derivative is given by
\be \label{covariant}
D_M = \partial_M + 
i g_5^{\prime}\frac{Y}{2} B_M + 
i g_5 \frac{\sigma^a}{2} A^a_M \, .
\ee
Given this definition, the mass dimensions of the fields are: 
$\dim(\Phi_d) = 3/2$, $\dim(\Phi_u) = 1$,
$\dim(A_M) = 3/2$, $\dim(B_M) = 3/2$, and the 5D gauge
couplings have a mass dimension of $-1/2$.
Bulk fields are defined to have even parity under $x^5\to -x^5$, 
and are expanded as
\be \label{decomposition}
{\cal S}(x^{\mu},x^5) = \frac{1}{\sqrt{\pi R}}\left(
S^{(0)}(x^{\mu}) + 
\sqrt{2}\sum_{n=1}^{\infty} \cos\left(\frac{nx^5}{R}\right) S^{(n)}(x^{\mu}) 
\right)\, .
\ee
This decomposition, together with Eq.~(\ref{5dlagrangian}), 
guarantees that after compactification we obtain the usual 4D kinetic
terms for all fields. 

Spontaneous symmetry breaking (SSB) of EW 
symmetry occurs when the Higgs doublets acquire vacuum expectation values (vevs). 
After SSB the Higgs fields on the brane can be written as
\bea \label{higgs}
\Phi_u & = & \frac{1}{\sqrt{2}}\left( \begin{array}{c} 
\Phi_u^{0*} \\ \Phi_u^{-} \end{array} \right) =
\frac{1}{\sqrt{2}}\left( \begin{array}{c} 
v_u + h\cos\alpha + H\sin\alpha + i\cos\beta A \\ 
 \cos\beta H^-  \end{array} \right)\, ,
\\
\Phi_d^{(0)} & = & \frac{1}{\sqrt{2}}\left( \begin{array}{c} 
\Phi_d^{+} \\ \Phi_d^{0} \end{array} \right) =
\frac{1}{\sqrt{2}}\left( \begin{array}{c} \sin\beta H^+ \\
v_d - h\sin\alpha + H\cos\alpha + i\sin\beta A \end{array} \right) \, ,
\eea
where the neutral CP-even bosons are denoted by $h$ and $H$, and $h$ is identified 
with the lightest Higgs: $m_h < m_H$.
The mixing angle $\alpha$ is introduced to diagonalize the CP-even mass matrix.
The CP-odd and charged Higgs fields are denoted by $A$ and $H^\pm$, and $v_u$ 
and $v_d$ are the vevs of $\Phi_u$ and $\Phi_d^{(0)}$ respectively. Note that 
the angles $\alpha$ and $\beta = \arctan(v_u/v_d)$ parametrize what we call 
brane-bulk mixing, or higher dimensional `location', of the neutral Higgses.

After performing the KK-mode expansion and identifying the
physical states, one derives the interaction Lagrangian for all 
the vertices of the neutral and charged Higgses. In particular,
the interactions $ZZh$ and $ZZ^{(n)}h$, which are necessary to 
calculate the Higgs production in association with a $Z$, 
as well as the vertices involving the $W^{\pm}$
bosons, are given by the following 4D Lagrangian:
\bea \label{interactions}
\nonumber
{\cal L}^4 & \supset & \frac{g M_Z}{2c_W}\left(h\sin(\beta-\alpha)
+H\cos(\beta-\alpha)\right)Z_{\mu}Z^{\mu} \\ \nonumber
           &    +    &
\sqrt{2} \frac{g M_Z}{c_W} \left( h \sin\beta\cos\alpha
+H\sin\beta\sin\alpha \right)\sum_{n=1}^{\infty} Z_{\mu}^{(n)}Z^{\mu}
\\ \nonumber
           &    +    & g M_W \left(
h\sin(\beta -\alpha) +H\cos(\beta -\alpha)\right)
W^+_{\mu}W^{-\mu} \\  
           &    +    &
\sqrt{2}gM_W\left(h\sin\beta\cos\alpha + H\sin\beta\sin\alpha\right)
\sum_{n=1}^{\infty}\left(W^+_{\mu}W^{-(n)\mu} + 
W^-_{\mu}W^{+(n)\mu}\right) \, .
\eea
%where $\tan\beta = v_u/v_d$.
Thus, the vertices $hZZ$ and $hWW$ have the same form as in the usual 4D THDM, 
i.e. proportional to $\sin(\beta-\alpha)$.
Meanwhile the couplings $hZZ^{(n)}$ and $hW^{\pm}W^{\pm(n)}$ are proportional to 
$\sin\beta\cos\alpha$, vanishing either when $\beta=0$ or $\alpha = \pi/2$, 
i.e. either when EWSB is driven exclusively by the vev of $\Phi_d^{(0)}$ or
when the CP-even Higgs comes entirely from $\Phi_d^{(0)}$. 
%
Similarly, the couplings of the CP-odd Higgs $A$ and the charged Higgs 
resemble the THDM, although new vertices of the type $H^+W^-Z^{(n)}$ or 
$H^+W^{-(n)}Z$ could be induced.

On the other hand, because the fermions are confined to the brane,
the Higgs-fermion couplings could take any of the
THDM I, II or III versions \cite{mythdm}. However, for the THDM III version 
the possible FCNC problems would be ameliorated, as the bulk-brane couplings 
will be suppressed by 
the factor $1 / \sqrt{2\pi R}$ \cite{sakamura}. Thus, for the flavor conserving
and gauge type couplings one can use the formulae of the THDM II vertices that
appears for instance in the ``Higgs Hunters Guide'' \cite{hhunter}.


\section{Extra dimensional contribution to Higgs production and decays} 
\label{production}

\subsection{Associated production $h+Z$ at linear colliders}

In order to study Higgs production at future colliders, first 
we derive the cross section for the Bjorken process, namely for
$e^+e^-\to hZ$. The total amplitude includes the contribution of
virtual $Z=Z^{(0)}$ and $Z^{(n)}$ states in the $s$-channel. 
The sum over all KK modes can be performed analytically, which simplifies
considerably the final expression.
Our result  for the cross section is given by
\bea \label{cross} 
\sigma(e^+e^- \rightarrow Zh) 
& = & \sigma_{SM}
\left[ \sin(\beta-\alpha) + 2\cos\alpha\sin\beta F_{KK} \right]^2  \, , 
\eea
where $\sigma_{SM}$ denotes the SM cross section, given by
\be \label{crossSM}
\sigma_{SM} = \frac{G_F^2 M_Z^4}{3\pi} \left(4s_w^4-2s_w^2+\frac{1}{2} \right)
\frac{|{\bf k}|}{\sqrt{s}}\frac{(3M_Z^2+|{\bf k}|^2)}{(s-M_Z^2)} \, ,
\ee
and
\bea 
|{\bf k}| = 
\frac{1}{\sqrt{s}}\left(\left(\frac{s+M_Z^2-m_h^2}{2}\right)^2 
-s M_Z^2\right)^{1/2}
\eea
is the 3-momentum of the $Z$ boson.

The function $F_{KK}$, which arises after summing over all the virtual KK-modes, 
is given by
\be \label{fkk}
F_{KK}  = 2 \sum_{n=1}^{\infty} \frac{s-M_Z^2}{s-M_n^2} 
        = \pi R A(s) \left[\cot(\pi R A(s)) - 1 \right]\, ,
\ee
where $M_n = \sqrt{n^2/R^2 + M_Z^2}$ is the mass of the $n^{th}$ KK level,
$A(s) = \sqrt{s-M_Z^2}$,  and we neglected 
the widths of the vector boson and its KK modes. It is trivial to modify the 
above for the process $q{\bar q}' \to W^{\pm}h$.


\subsection{Associated production $h+Z, \, h+W^\pm$ at hadron colliders}

When considering Higgs production at hadron colliders, a similar expression 
holds at the parton level for the production cross section of the Higgs in
association with a $W^\pm$ or $Z$. To obtain the hadronic cross section
$h_1 h_2 \to hZ$, the partonic cross section must be convoluted with the parton
distribution functions (PDFs):
\be
\sigma(h_1 h_2 \rightarrow Z h) = 
\sum_{q \bar{q}} \int_{0}^{1} \int_{0}^{1}
f_{q/h_1}(x_1,\hat{s}) 
\sigma(q \bar{q} \rightarrow Z h) %(x_1,x_2,\hat{s})
f_{\bar{q}/h_2} (x_2,\hat{s}) d x_1 d x_2
+ {q \leftrightarrow \bar{q}} \, .
\label{hadrXS}
\ee
Here $f_{q/h_i}(x,{\hat s})$ gives the distribution of a parton $q$ in the 
hadron $h_i$ as a function of the longitudinal momentum fraction $x$ and the 
factorization scale, which is chosen to be the partonic center of mass 
$\hat{s}$. In our numeric study we use CTEQ4M PDFs \cite{cteq4}. The sum in 
Eq.~(\ref{hadrXS}) extends over the quark flavors $q = u,d,s,c,b$.

The large center of mass energy that can be achieved at the LHC also opens up 
the possibility to produce a Higgs boson in association with KK states, for 
instance $hZ^{(1)}$, which will have a very distinctive signature that could 
allow `direct' detection of the first KK modes at the LHC. This possibility is 
studied elsewhere.


\subsection{Higgs decays}

For Higgs bosons lying in the intermediate
mass range, which is in fact favored by the analysis of electroweak 
radiative corrections, the dominant decay is into $b\bar{b}$ pairs.
In our higher dimensional model this decay width is given by the formulae of the
THDM, just as that of the other tree-level two-body modes. On the other hand, for
the 3-body decays $h \to W l \nu_l$ and $h\to Z l^+l^-$, which can play 
a relevant role at Tevatron and LHC, the corresponding decay width could
receive additional contributions from the virtual $KK$ states. 
The inclusion of these KK modes leads to the following
expression for the differential decay width:
\be \label{gamma}
\frac{d\Gamma}{dx} (h\to W l\bar{\nu}_l)= \frac{g^4m_h}{3072 \pi^3}
\frac{(x^2-4r_w)^{1/2}}{1-x} f_V(x)
\left[ \sin(\beta-\alpha) + 2 \cos\alpha\sin\beta F_{KK} \right]^2 \, ,
\ee 
where
$f_V(x)= x^2-12r_w x +8r_w +12r^2_w$, with $r_w=M_W^2/m_h^2$,
and $2r^{1/2}_w < x <1+r_w$. The $F_{KK}$ function
is given as in Eq.~(\ref{fkk}), with the replacements 
$s\to q^2=m_h^2 (1-x)$ and $M_Z \to M_W$. 
A similar expression can be derived for the
decay $h\to  Z l^+l^-$.

To study the effect of the KK modes on the decay $h\to W l\bar{\nu}_l$, 
we have evaluated the ratio of the corresponding decay width in the
extra dimensional scenario over the SM decay width:
\be
R_{hWW^*}=\frac{\Gamma( h\to W l\bar{\nu}_l)_{XD}}
               {\Gamma( h\to W l\bar{\nu}_l)_{SM}} \, .
\label{RhWW}
\ee
Results for this ratio are shown in Table~\ref{tab:decays}, for 
several representative sets of parameters which are chosen as
\begin{itemize}
 \item[{\bf A}]: $M_c = 2$ TeV, $\alpha = \pi/3$, ~~~ 
       {\bf B} : $M_c = 2$ TeV, $\alpha = \pi/1.28$,
 \item[{\bf C}]: $M_c = 2$ TeV, $\alpha = \pi$,   ~~~~~~
       {\bf D} : $M_c = 5$ TeV, $\alpha = \pi$, 
\end{itemize}
while $\beta=\pi/4$ remains fixed. 
One can appreciate that significant deviations
from the SM can appear, though the larger effect comes from the 
deviations for the THDM Higgs couplings from the SM case.

\begin{table}
\begin{tabular}{|c|c|c|c|c|}
\hline
~$m_h$ (GeV)~ & ~$R_{hWW^*}$ (set {\bf A})~ & ~$R_{hWW^*}$ (set {\bf B})~ & 
~$R_{hWW^*}$ (set {\bf C}) ~& $R_{hWW^*}$ (set {\bf D})~ \\ 
\hline
 130   & $6.0\times 10^{-2}$ & 0.999 & 0.51 & 0.50 \\
\hline
 140   & $6.0\times 10^{-2}$ & 0.998 & 0.51 & 0.50 \\
\hline
 150   & $6.1\times 10^{-2}$ & 0.996 & 0.50 & 0.50 \\
\hline
 160   & $6.1\times 10^{-2}$ & 0.993 & 0.50 & 0.50 \\
\hline
\end{tabular}
\caption{The ratio $R_{hWW^*}$, introduced in Eq.~(\ref{RhWW}), for several sets 
of parameters {\bf A}, {\bf B}, {\bf C}, {\bf D} (as defined in the text), and 
with $\beta=\pi/4$.}
\label{tab:decays}
\end{table}

On the other hand, the loop induced decays $h\to \gamma \gamma, Z \gamma$
also receive contributions from the $W^{(n)}$ KK modes. But because the
coupling $hWW^{(n)}$ that appears in the loop is proportional to $M_W$, rather
than $M_{W^{(n)}}$, the KK states contribution will decouple, as there are no
mass factors in the numerator that could cancel the ones that appear in
the numerator. Thus, the KK contribution can be neglected for the 
decay widths of the loop induced decays.

We conclude that in the intermediate Higgs mass range
%, which is in fact the one preferred by the analysis of EW precision measurements, 
the decay $h\to b\bar{b}$ will continue to dominate, even more than in the SM case
for some values of parameters.


\section{Implications for Higgs search at future colliders} 
\label{sec:Implications}

\subsection{The LC case}

\begin{figure}
\vspace{-1cm}
%  \begin{centering}
%  \def\epsfsize#1#2{1.0#2}
\resizebox*{.45\textwidth}{.35\textheight}
%\hfil\hspace{-10em} 
{\includegraphics{fig1a.ps}}
\resizebox*{.45\textwidth}{.35\textheight} 
{\includegraphics{fig1b.ps}}\\
\resizebox*{.45\textwidth}{.35\textheight} 
{\includegraphics{fig1c.ps}}
\resizebox*{.45\textwidth}{.35\textheight} 
{\includegraphics{fig1d.ps}}
%\epsfbox{fig1a.ps} \hfill
\caption{SM, THDM and XD cross sections for $e^+e^- \rightarrow Zh$. Each plot
corresponds to a different set of values for $\alpha$ and $\beta$ all with
$m_h = 120$~GeV and with a compactification scale $M_c = 2$~TeV.}
\label{figure1}
%  \end{centering}
\end{figure}

Because of its simplicity, first we discuss Higgs production at the next linear 
collider. The results for the $e^+e^- \to hZ$ cross section, after the inclusion 
of the virtual $Z$ KK contribution, are shown in Figs.~\ref{figure1} and 
\ref{figure2}.
%
Fig.~\ref{figure1} shows the cross section as a function of the center of mass 
energy ($200 < \sqrt{s} < 2000$ GeV),  for $m_h=120$ GeV and a value of the 
compactification scale $M_c=2$ TeV. We plot the SM cross section $\sigma_{SM}$, the
THDM cross section $\sigma_{THDM}$, as well as the extra dimensional cross section
$\sigma_{XD}$. Four different pairs of $\alpha$ and $\beta$ were chosen to 
compute $\sigma_{THDM}$ and $\sigma_{XD}$. They are given by
\begin{enumerate}
 \item[{\bf a}]: $\beta=\pi/4$, \, $\alpha=-\pi/4$,  ~~~\,
       {\bf b} :  $\beta=\pi/4$, \, $\alpha=0$, 
 \item[{\bf c}]: $\beta=\pi/4$, \, $\alpha=0.79\pi$, ~~~
       {\bf d} :  $\beta=\pi/3$, \, $\alpha=0.32\pi$. 
\end{enumerate}
Choice {\bf a} corresponds to a case in which $\sigma_{THDM} = \sigma_{SM}$; {\bf b} corresponds
to a case where the effect of the mixing due to $\alpha$ in the term involving the
KK sum is maximum; {\bf c} and {\bf d} correspond to values of $\alpha$ and $\beta$ where 
$\sigma_{XD}/\sigma_{THDM}$ at $\sqrt{s} = 500$~GeV is about $1.5$ and $10$ 
respectively.
 
We can see that in all four cases the cross section of the XD model is always
smaller at $\sqrt{s} = 500$~GeV than that of the SM, and that in order to obtain a larger
cross section, one needs energies greater than 
$\sqrt{s} \sim 1$~TeV. This is understood from the fact that the
heavier KK modes, through their propagators, interfere destructively with the SM 
amplitude thus 
reducing the cross section. It is only until one reaches energies close to the
first KK mass that an enhancement is observed due to the dominating KK 
contribution. 

We mention that the four cases presented in Fig.~\ref{figure1} are only representative, 
and that one can find broad regions of parameter space in which $\sigma_{XD} > \sigma_{THDM}$.
This can be seen in Fig.~\ref{figure2}, where we show $\sigma_{XD}$ as a function  of
$\alpha$ for fixed values of $m_h=120$ GeV, $M_c=2$ TeV and
$\beta=\pi/4$. We present two cases for $\sqrt{s} = 500$~GeV and $1$~TeV respectively.
The dips that appear in the figure are related to
cancellations between $Z$ and $Z^{(n)}$ $s$-channel contributions.
The top solid line denotes the SM cross section, and we also have shown 
the minimum cross section needed to get 20 events with an integrated
luminosity of 500 $fb^{-1}$, i.e. $\sigma_{min}= 4 \times 10^{-5}$ pb.
From the figure we can appreciate that only a small range of $\alpha$ is not
covered at the LC with $\sqrt{s}=1$ TeV.
%{\bf CsB- This is great! How much does it depend on $\beta$?}

\begin{figure}
\vspace{-3cm}
%  \begin{centering}
%  \def\epsfsize#1#2{1.0#2}
\resizebox*{.45\textwidth}{.45\textheight}
%\hfil\hspace{-10em} 
{\includegraphics{fig2a.ps}}
\resizebox*{.45\textwidth}{.45\textheight}
{\includegraphics{fig2b.ps}}
\caption{SM, THDM and XD cross sections as a function of $\alpha$ for fixed
$\beta = \pi/4$ and for $M_c = 2$~TeV. The cross sections are evaluated at
$\sqrt{s} = 500$~GeV in frame {\bf a} and $1$~TeV in {\bf b}. The line denoted
by $\sigma_{min}$ is explained in the text.}
\label{figure2}
%  \end{centering}
\end{figure}

However, as Fig.~\ref{figure1} shows, once we start approaching the threshold
for the production of the first KK state, then the cross section
starts to grow, and for instance with $M_c=2$ TeV,  
$\sigma_{SM} \simeq \sigma_{XD} $ for  $\sqrt{s} \simeq 1.3$ TeV,
which can be clearly detectable. One would need higher
energies in order to have a cross section larger than that of the SM.

According to current studies, when  the cross section is 4\% larger
than the SM cross section, with the estimated precision
that could be obtained at the LC \cite{LCp}, it can be possible to distinguish
between the SM and XD Higgs scenarios. We can see that this will
be possible for compactification scales of the order 
2 TeV for a broad range of values of parameters.
%
It is interesting to note that such deviations in the cross section 
from the SM prediction, arise
even when the couplings of the Higgs to the gauge bosons 
are indistinguishable from the SM ones.


\subsection{Implications for the Tevatron}

After the productive but unsuccessful Higgs search at LEP2, the Run 2 of the 
Tevatron continues the search until the LHC starts operating. The luminosity 
that is required to achieve a 5 or 3 $\sigma$ discovery, or a 95\% C.L. 
exclusion limit, was presented by the Run 2 Higgs working group 
\cite{Carena:2000yx}. For instance, with $m_h=120$ GeV, the corresponding 
numbers are about 20, 6 and 2 $fb^{-1}$ respectively.

Assuming $M_c \ge 2$ TeV, the inclusion of the KK modes decreases the 
$h W^\pm$ and $h Z$ associated production cross section at the Tevatron.
%
% If we consider the mass region in which the Higgs is produced in association 
% with a $W^\pm$, followed by its decay into b-quarks, the cross section will 
% scale like $\sigma_{SM} \to Z_1 \sigma_{SM}$. In order to obtain a fixed number 
% of events (above the backgrounds) the luminosity will now scale as ${\cal{L}} 
% \to {\cal{L}}/Z_1$. Therefore, we find that $Z_1=1/2$, i.e. the luminosity will 
% be twice of the numbers quoted before, namely we only need to have: 2, 6 and 20 
% $fb^{-1}$ to obtain  a 5, 3 sigma effect, or 95\% exclusion limits 
% respectively.{\bf )}
%
Depending on the actual values of $M_c$, $\alpha$ and $\beta$ the suppression in 
the parton level cross section may be anything between 1 and 99 percent. For 
example, if $M_c$ is a few TeV then the $s$-channel process $q{\bar q'} \to 
W^{\pm}h$ can receive a considerable suppression, as it can be inferred from 
Figs.~\ref{figure1} and \ref{figure2}. This is so unless the KK contribution 
itself is suppressed by $\cos\alpha$ and/or $\sin\beta$ in Eq.~(\ref{cross}), 
in which case the presented model has little relevance.
%
Thus, as a general prediction of this model, we conclude that more luminosity is 
required to find a light Higgs boson than what is listed above. This
slims the chances of the Tevatron to find the Higgs of this model.

% We note that, the Higgs boson can also be produced via gluon fusion, and for 
% values of the Higgs mass where the decay mode $h\to Wl\nu_l$ can be used, we 
% need to include also the corrections to the decay width  $\Gamma (h \to 
% WW^*)=Z_2 \Gamma_{SM} $. It turns out that the resulting number of events will 
% scale like $N_{SM} \to Z_1 Z_2 N_{SM}$. Thus, the required luminosity to get a 
% fixed number of events will scale like ${\cal{L}} \to {\cal{L}}/(Z_1Z_2)$, which 
% can produce interesting effects...


\subsection{Higgs production at the LHC}

The Higgs discovery potential in this model is more promising at the LHC. We 
illustrate this in Fig.~\ref{figLHC1} showing the $pp\to hZ$ cross section as 
a function of the compactification scale $M_c$. (It is needless to say that
similar results hold for $pp\to hW^\pm$.) The main features of this plot 
can be understood examining Eqs.~(\ref{cross}) and (\ref{fkk}).
%
When $M_c = 1/R$ is close to the average partonic center of mass energy of the 
LHC, then the factor $\cot(\pi R A(s))$ of Eq.~(\ref{fkk}) can dominate, since 
its argument $\pi R A(\hat{s}) \sim \pi \sqrt{\hat{s}}/M_c$ is close to $\pi$. 
This happens around $M_c = 4$ TeV and leads to the large enhancement of the 
extra dimensional cross section over the SM one. 

\begin{figure}
\includegraphics[width=12cm]{figLHC1.eps}
\caption{Higgs production cross section in association with a $Z$ boson
as a function of the compactification scale for selected values of the mixing
parameters. For reference the SM cross section is shown with a 10\% uncertainty 
band around it.}
\label{figLHC1}
\end{figure}

% width
The singularity at $R A(s) = 1$ is regulated by the width of the KK mode, which 
is not included in our calculation. Thus the peak region in Fig.~\ref{figLHC1} is 
only correct in the order of magnitude. Yet, since the width of the $n^{th}$ KK 
mode $\Gamma_n \sim 2 \alpha_{EM} m_n$ is rather small (due to the smallness of 
$\alpha_{EM}$), the large enhancement is expected to prevail after the inclusion 
of the width.

% enhancement
Depending on the particular values of $\alpha$ and $\beta$ the enhancement is 
more or less pronounced. But independently from the values of $\alpha$, $\beta$ 
(and the KK width) the XD production cross section is considerably enhanced 
compared to the SM for $M_c \lesssim 6$ TeV. For certain parameter values there 
is an enhancement even up to $M_c \sim 8$ TeV.
% alpha-beta
For particular $\alpha$ and $\beta$ values it might also happen that the term 
containing the extra dimensional factor $F_{kk}$ in Eq.~(\ref{cross}) (nearly) 
cancels the first term. This cancellation also depends on the center of mass 
energy ${\hat s}$ and $M_c$ and happens close to 8.5 TeV in 
Fig.~\ref{figLHC1}.

\begin{figure}
\includegraphics[width=15cm]{figLHC2.eps}
\caption{Higgs production cross section in association with a $Z$ boson
as a function of the Higgs mass for selected values of the compactification scale
$M_c$.}
\label{figLHC2}
\end{figure}

An alternative way to gauge the XD enhancement is shown in Fig.~\ref{figLHC2}. 
Here the Higgs production cross section is shown as the function of the Higgs 
mass for selected values of $M_c$. The mixing parameters are fixed to values of 
$\alpha = \pi/1.28$ and $\beta = \pi/4$. For these values the enhancement is the 
least around $M_c = 6$ TeV in Fig.~\ref{figLHC1}. Still, Fig.~\ref{figLHC2} 
shows that there is a well pronounced enhancement for $M_c \sim 5$ TeV for the 
entire Higgs mass range considered.

Based on these results, we conclude that in the Bjorken process alone the reach 
of the LHC may extend to $M_c =$ 6--8 TeV, depending on the values of $\alpha$ 
and $\beta$.
%
Finally we note that the XD contribution to the running of the gauge couplings is 
important when the effective center of mass energy of the collider is close to 
$1/R$~\cite{ddg}. Since not included in this work, this contribution is expected to 
change our results quantitatively for $\sqrt{\hat s} \gtsim M_c$ .


\section{Conclusions} 
\label{sec:conclusion}

In this work, we studied the extent to which present and future colliders can 
probe the brane-bulk location of the Higgs boson in a model with a TeV$^{-1}$ 
size extra dimension. In this model one Higgs doublet is located on the brane 
while another one propagates in the bulk.
%
We found that the virtual KK states of the gauge bosons contribute to the 
associated production of the Higgs with $W^{\pm(n)}$ and $Z^{(n)}$, and at low 
energies ($\sqrt{s}\leq 1/R$~GeV) the cross section is suppressed compared to 
the SM case. Meanwhile at higher energies, i.e. $\sqrt{s} \sim 1/R$, the cross 
section can receive an enhancement that has important effects for the discovery 
of the Higgs at future colliders. 

Assuming compactification scales in the order of 2--14 TeV, we concluded that to 
find a Higgs signal in this model, the Tevatron Run 2 and the LC with 
$\sqrt{s}=500$ GeV are required to have a luminosity higher than in the SM case. 
Meanwhile, the LHC and LC with $\sqrt{s}=1.5$ TeV should have greater possibilities 
to find and study a Higgs signal. Depending on the model parameters, these colliders may 
be able to distinguish between the extra dimensional and the SM for 
compactification scales up to about 6--8 TeV. If this model is relevant for weak 
scale physics, the LHC should see large enhancements in the associated 
production rates. Thus, not finding the Higgs at the Tevatron may be good news for 
the XD Higgs search at the LHC.


\section*{Acknowledgments}

A.A. was supported by the U.~S. Department of Energy under grant 
DE-FG02-91ER40676.
C.B. was also supported by the DOE, under contract number 
DE-FG02-97ER41022.
J.L. D.-C.'s was supported by CONACYT and SNI (M\'exico).


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