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\begin{document}
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\title{Implication of Brane fluctuations to 
indirect collider signals}

\author{ Seong Chan Park$^{(a)}$ 
and H.~S.~ Song $^{(b)}$}
\address{
Center for Theoretical physics and
School of Physics, Seoul National University,
Seoul 151-742, Korea \\
$^{(a)}$ schan@mulli.snu.ac.kr \\
$^{(b)}$ hssong@physs.snu.ac.kr
}

%\author{Seong Chan Park}
%
%\author{H. ~S. ~Song} 
%\address{Center for Theoretical physics and
%School of Physics, Seoul National University,
%Seoul 151-742, Korea} 


\date{\today}

\maketitle
                    
\begin{abstract}
We study the effect of brane fluctuation on the
indirect signals of high energy colliders. 
Brane fluctuation could act as a regulator of
divergent expression of infinite tower of Kaluza-Klein
graviton effects. 
The phenomenological parameter $\lambda$,
introduced by Hewett, 
is shown to be determined in our setting, and its dramatic
behaviors depending on the $D=4+\delta$ dimensional
gravitation scale $M_D$, `softening parameter' $\Delta$,
and $\sqrt{s}$ of collider are presented. 
The present exclusion bounds from the processes
$e^+e^-\rightarrow \gamma\gamma$ and
$p\bar{p}\rightarrow e^+ e^-, \gamma\gamma(\gamma)$ are 
considered within the parameter space $(M_D, \Delta)$
with respect to the number of extra dimensions.
\end{abstract}

\pacs{PACS number(s): 11.30.Pb, 11.30.Er}
\vskip2.0pc]    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\it Introduction}---It is 
phenomenologically interesting when the size of extra 
dimension is so large that the Kaluza-Klein excitations of 
bulk fields could  directly affect the low energy phenomena.
If all the standard model(SM) fields are confined on the brane
and only graviton can propagate through the bulk, the 
size of extra dimension can be large enough to nullifying
the hierarchy between the weak scale and the Planck scale \cite{ADD}.
In that case, the Kaluza-Klein (KK) tower of graviton can make
 sizable contributions to the collider physics.
Not only signals with direct production of KK graviton, 
but indirect signals with the KK mediation can provide chances
to detect the effects from the extra dimensions. 
The scattering cross section including KK graviton mediating 
diagrams can be written as
%
\be
\sigma_{\rm Total}= \sigma_{\rm SM} + \eta_{KK} \sigma_{\rm Mix}
                    + \eta_{KK}^2 \sigma_{KK}.
\ee
%
Where 
$\eta_{KK}$ denotes the propagating factor of KK graviton tower 
defined as
%
\be
\eta_{KK}\equiv \frac{i^2}{8 M_P^2}\sum_{KK} \frac{1}{s-m_{KK}^2}
\ee
%
when only s-channel diagrams are involved. Here 
$m_{KK}^2 \equiv \vec{n}\cdot\vec{n}/R^2 $ is the mass of the 
KK graviton in state $\vec{n}=(n_1, n_2 ,...,n_\delta)$, the factor $1/8$ is 
introduced for future convenience and the Planck scale $M_P$
is for gravitational coupling.
The case with t-, u- channel 
KK mediating diagrams
also contributing like $e^+e^-\rightarrow e^+e^-$ can be considered
as a straightforward ways. 
The summation through the whole tower of KK states 
is generally divergent and so we should take
a cutoff scale ($N_\Lambda =\frac{\Lambda}{R}$) to get finite result
and then we simply throw the rests away \cite{Sundrum}
(see, for loop calculations of Kaluza-Klein states \cite{Strumia}). 
The size of extra
dimension $R$ is related to the scales of the gravitation as
$R^\delta M_D^{2+\delta} = M_P^2$, where $\delta$ is number of extra
dimensions.
With small spacing ($\sim \frac{1}{R}$) the summation could be
approximated to the integration, we can obtain $\eta_{KK}$ at the
limit $\Lambda/\sqrt{s} >>1$ as
%
\bea
\eta_{KK}(Rigid)~~
          \approx  \frac{\pi}{2 M_D^4}\ln(\Lambda/\sqrt{s}) 
     ~~~~~~~&&(\delta=2), \\
      \approx  \frac{\Omega_\delta s^{\delta/2-1}}{8 M_D^{\delta+2}}
             (\Lambda/\sqrt{s})^{\delta-2} ~~&&(\delta>2)
\eea
where $\Omega_\delta$ is the solid angle in $\delta$ dimensional
space, e.g., $\Omega_2 = 2\pi$ in 2 dimensional space.

%
By taking $\Lambda \sim M_D$, the factor could be estimated as
$\eta_{KK}\sim \lambda/M_D^4$, where $\lambda$ 
encapsulates all the 
uncertainties from the number of extra dimensions, 
the unknown relation between $\Lambda$ and $M_D$,
and threshold effects coming from the string
theory beyond the cutoff scale \cite{Hewett}. 
Though it is obvious that $\lambda$ is dependent on the number of
extra dimensions and the energy scale of collider $\sqrt{s}$, 
it is usually assumed that the value is ${\cal O}(1)$ and
insensitive to the number of extra dimensions.
With this assumptions, indirect signals of extra dimensions have been 
treated independently of $\delta$ etc. 
\cite{Review}, \cite{Review2}. 
However from a more close study 
including the dynamics of the brane fluctuation,  
$\lambda$ shows dramatic behavior depending $\delta$, $\sqrt{s}$ and
softening scale parameter which is essentially determined
by tension of the brane.   


{\it Brane Fluctuations and determining $\lambda$ parameter}
---In the string theory embedding of large extra dimensional theory,
our world might be a dynamical object carrying finite tension
\cite{String}. It is very natural since any relativistic consideration
does not allow strictly rigid objects. 
Thus the formalism including brane fluctuation 
must ultimately be employed to probe the high energy physics of
extra dimensions with brane\cite{Sundrum},\cite{Bando}. 
The brane fluctuation could be described by introducing Nambu-Goldstone
boson $\vec{\phi}(x)$
which came from the spontaneous translational symmetry breaking
\cite{Kugo}.
The dynamics of the Namby-Goldstone
 boson, inducing the metric on the fluctuating brane, 
is described by the Nambu-Goto action with the induced metric.
%
\bea
{\cal L}&&=-\tau \int d^4 x \sqrt{-g} \\ \nonumber
&&= -\tau \int d^4 x 
(1-\frac{1}{2}\partial_\mu \vec{\phi}(x) \cdot \partial^\mu \vec{\phi}(x) 
 + \cdots) 
\eea
%
where the tension of the brane is denoted as $\tau$.  
After expanding the bulk graviton field around the compact extra
dimensions
and taking normal ordering for the expansion modes in perturbation
framework, the interaction Lagrangian with the KK gravitons and 
the SM particles is shown to carry an exponential
`softening factor' \cite{Murayama},\cite{Park}:
%
\bea 
{\cal L} &&\supset 
-\frac{1}{M_P}g_{\mu\nu}T^{\mu\nu}({\rm SM}) \\ \nonumber
&&\Rightarrow
-\frac{1}{M_P}\sum_{\vec{n}}e^{-\frac{1}{2}m_{KK}^2/\Delta^2 }
g_{\mu\nu}^{\vec{n}}T^{\mu\nu}({\rm SM})
\eea
%
where $\frac{1}{\Delta^2}$ is the free propagator of $\vec{\phi}$,
%
\be
\frac{1}{\Delta^2} \equiv <\phi(x)\phi(y)>|_{x\rightarrow y}
=-\frac{1}{4\pi^2 \tau} (x-y)^{-2}|_{x\rightarrow y} \cite{Bando}.
\ee
%
In principle, the scale $\Delta^2$ could be determined 
by the tension of the 
brane and the cutoff scale of loops of a $\phi(x)$ field. 
In this study we just take the scale $\Delta$ as a free parameter
of the theory which shows the effect of the brane fluctuation.
Note that when the scale is chosen at infinity, the action
reduced to the normal UN-fluctuating case and 
if it is chosen at the same order of
$M_D$ scale it will provide very rich phenomenology of
high energy colliders.
Since the coupling of the higher KK states are quite suppressed
with the exponential softening factor, divergent expression for
the infinite KK graviton contribution could be naturally 
regularized. 

For the indirect collider signals, we can get the regularized 
expression for $\eta_{KK}$ or $\lambda$ parameter as 
%
\be
\eta_{KK}(Fluctuating) \Rightarrow 
\frac{-1}{8 M_P^2} \int d^\delta n 
\frac{e^{-\vec{n}\cdot\vec{n}/(R^2 \Delta^2)}}{s-\vec{n}\cdot\vec{n}/R^2}.
\ee
%
From the above relation we can derive the expression for the parameter
$\lambda$ as following.
%
\be
\lambda(Fluctuating)=\frac{-\Omega_\delta}{8}
(\frac{ \sqrt{s} }{M_D})^{\delta-2} {\cal I}(\delta, s/\Delta^2). 
\ee
% 
The integral function is introduced as
%
\be
{\cal I}(\delta, s/\Delta^2)=\int dx 
\frac{x^{\delta-1}e^{-x^2 (s/\Delta^2)}}{1-x^2}
\ee
%
and we take the principal value for the singular integral
without pole contribution of KK state. 
The value for the integration is very stable with respect to
 the UV cutoff
and we understand the behavior with exponential suppression 
factor for the large $x$ region. This is very general 
behavior from the brane fluctuation working as physical 
regularization factor.  
In Fig.1,2 and 3, we present the numerical results for
the parametric dependence of $\lambda$ parameter.
Note that the sign of $\lambda$ is essentially determined
by the ratio of $(s/\Delta^2)$ for given $\delta$. If the ratio
is large enough that the contribution of the light KK gravitons
are quite suppressed, the integral function surely gives negative 
sign. In that case, $\lambda$ could have positive sign. But
for the case with the large softening parameter, $\lambda$
usually carries negative sign. With our existing colliders
like LEP-II or Tevatron, we can safely assume the sign of $\lambda$
parameter is negative. 
  

(Fig.1) shows the softening parameter dependence of the $\lambda$ 
parameter when $M_D$ and $\sqrt{s}$ are set to be 2 and 0.5 TeV,
respectively.
The $\delta$ dependence is very crucial. 
It should be noted that within the reasonable 
range of the parameter space, $\lambda$ value is well approximated
as ${\cal O}(1)$ value for the case with $\delta=2$ and $4$.
But for the case with $\delta=6$, the absolute value can be much larger.

% --------------------------------------
% Figure 1
% --------------------------------------
\begin{figure}[ht]
\begin{center}
\vspace*{-0.5cm}
\epsfysize=7truecm\epsfbox{lam_f.eps}
\end{center}
\vspace*{-0.5cm}
\caption{\it 
Softening parameter $\Delta$ dependence of $\lambda$. The solid 
line, dotted line and dashed line denote the case for $\delta=2,4$ and
$6$, respectively.(All the following graphes take the same convention
for $\delta$ lines.
$M_D$ and $\sqrt{s}$ are set as 2 and 0.5 TeV, respectively.
}
\end{figure}

In (Fig.2), we plot the $M_D$ dependence when $\Delta$ and $\sqrt{s}$
are set to be 3 and 0.5 TeV, respectively. The absolute value is suppressed
with larger $M_D$. We can understand this behavior from the
inversely depending relation given above. The case for $\delta=2$
is very stable with respect to varying $M_D$ but it is not general
behavior for larger dimensions. We can also see that the absolute value
is ${\cal O}(1)$ for the case with two or four extra dimensions
but  can be very large when $\delta=6$. 
Crossing occurs at $M_D\approx 3$ TeV and which is generally possible
from the non-linear dependence of the parameters. 

% --------------------------------------
% Figure 2
% --------------------------------------
\begin{figure}[ht]
\begin{center}
\vspace*{-0.5cm}
\epsfysize=7truecm\epsfbox{lam_m.eps}
\end{center}
\vspace*{-0.5cm}
\caption{\it 
$M_D$ dependence of $\lambda$ with varying $\delta=2,4$ and 6.
$\Delta$ and $\sqrt{s}$ are set as 3 and 0.5 TeV, respectively.
}
\end{figure}

In (Fig.3), $\sqrt{s}$ dependence is plotted when $M_D$ and $\Delta$
are set to be 2 and 3 TeV, respectively.
We can see the rather smooth dependence for two or four extra dimensional
cases but the slope is a bit steeper with six extra dimensions.
With given parameter set, the $\lambda$ has ${\cal O}(1)$ value
in the parameter space.


% --------------------------------------
% Figure 3
% --------------------------------------
\begin{figure}[ht]
\begin{center}
\vspace*{-0.5cm}
\epsfysize=7truecm\epsfbox{lam_s.eps}
\end{center}
\vspace*{-0.5cm}
\caption{\it 
The collider C.M. energy $\sqrt{s}$ dependence of
$\lambda$ with respect to the varying $\delta=2,4$ and 6. 
$M_D$ and $\Delta$ are set as 2 and 3 TeV,respectively.
}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%             Section 3
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
{\it Collider bounds}
---Now let us consider the experimental bounds for the indirect 
signals of KK gravitons from the fluctuating brane.
Here we consider the dielectron and diphoton production
processes as a concrete example. However, our method is
quite general so that we can extend the study to any indirect
collider signals of extra dimensions with brane fluctuation
regularization. 
Usually experimental bounds for the extra dimensions are given
by $M_D$ assuming $\lambda= \pm 1$.
However,as was seen at the above section,
 the parameter $\lambda$ generally depends on 
not only $M_D$ but also  $\delta$ and $\Delta$ with brane fluctuation 
effects. We could only impose experimental bounds within
the parameter space of three dimensions:($M_D, \Delta, \delta$)
with given center of mass energy of collider.
 
We first consider the LEP-II bound from the 
$e^+e^- \rightarrow \gamma\gamma(\gamma)$ \cite{Mele},\cite{Agashe}
 at center of mass
energy ranging from 189 GeV to 202 GeV by DELPHI.
The bound was given as $M_D>713$ GeV with $\lambda=1$ and
$M_D>691$ GeV with $\lambda=-1$ (95\% C.L.)\cite{DELPHI}. 
As was commented, we  only 
consider the case with negative $\lambda$ case.
In (Fig.4), the exclusion bound is presented. 
For  the case with $\delta=2$, our new bound is 
$M_D >810$ GeV and the bound is
almost independent of $\Delta$ scale. 
Generally the exclusion
bound is larger with larger $\delta$ and increasing with larger $\Delta$.  
% --------------------------------------
% Figure 4
% --------------------------------------
\begin{figure}[ht]
\begin{center}
\vspace*{-0.5cm}
\epsfysize=7truecm\epsfbox{mde.eps}
\end{center}
\vspace*{-0.5cm}
\caption{\it 
95 \% C.L. $M_D$ exclusion  bound 
from the LEPII data with the process
$e^+e^- \rightarrow \gamma\gamma(\gamma)$
with C.M. energies from 189 to 202 GeV.
}
\end{figure}
 
Now let us consider the Tevatron bound from the
diphoton and dielectron pair production processes\cite{Cheung}
obtained by D0.
The center of mass energy is taken about $1.8$ TeV.
The lower limits at 95 \% C.L. on the $M_D$ scale
was given between 1.0 and 1.1 TeV with $\lambda=1$ and $-1$ 
, respectively\cite{D0}. 
In (Fig.5), we see the exclusion bound for $M_D$ scale with
respect to the softening scale and number of extra dimensions
for the data. The bound for $\delta=2$ case is similar with
LEP-II bound but in the cases with larger number of extra dimensions
the bounds are much higher. 
This behavior could be understood by C.M. energy 
dependence of $\lambda$ parameter in Eq.(9) and (10).



% --------------------------------------
% Figure 5
% --------------------------------------
\begin{figure}[ht]
\begin{center}
\vspace*{-0.5cm}
\epsfysize=7truecm\epsfbox{mdp.eps}
\end{center}
\vspace*{-0.5cm}
\caption{\it 
95 \% C.L. exclusion bounds from the Tevatron data with the processes 
$p\bar{p} \rightarrow e^+e^-,\gamma\gamma$ detected by  D0 detector.
C.M. energy is $1.8$ TeV.
}
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%  Summary and Conclusion
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
{\it Summary and conclusion}
---We study the implication of the brane fluctuation to
the indirect signals of collider physics.
From the exponential softening factor the
brane fluctuation provide very natural regularization
scheme for KK states summation. 
With this regularization we can determine 
$\lambda$ parameter with respect to parameters of
$M_D$, $\Delta$ ,$\sqrt{s}$
and $\delta$.  
The experimental exclusion bounds of existing data and
sensitivity bounds for future colliders could be determined
within the parameter space of $(M_D,\Delta,\delta)$. 
As a concrete example, we found the 
exclusion bounds from the LEP-II diphoton production data 
with C.M. energy from 189 to 202 GeV 
and
Tevatron dielectron and diphoton production data with C.M.
energy of 1.8 TeV.

The brane fluctuation provides a new setup for 
phenomenology of extra dimension searches.
The indirect signals of extra dimensions,
the data is shown to be highly dependent on not only $M_D$ scale but
also $\delta$ and $\Delta$.
All the sensitivity bounds should be determined within the 
parameter set of
$(M_D, \Delta, \delta)$ if we consider the brane fluctuation.

Finally, let us briefly comment on the brane fluctuation regularization
within the Randall-Sundrum setting \cite{RS}. 
Unfortunately, it is not possible to introduce the brane fluctuation
regularization in Randall-Sundrum's case.
In the RS case, our TeV brane carries negative tension to make
the warped geometry and so we cannot apply any regularization
of brane fluctuation.
Just sharp cutoff at reasonable range could provide a finite
results of low energy observables by assuming only KK modes
up to the cutoff scale is responsible for calculation 
(see e.g.,\cite{Park_RS}, \cite{Agashe_RS}).
Further study is needed to formulate natural regularization formalism
with the negative tension brane.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The work was supported in part by the BK21 program and in part by
the Korea Research Foundation(KRF-2000-D00077). 
%%%%%%%%%%%%%%%%%% References
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%BBBB%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}


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