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\title{$B \to X_s +$ missing energy in models with large extra dimensions}
\author{Namit Mahajan\thanks{E--mail : nm@ducos.ernet.in, 
nmahajan@physics.du.ac.in}\\
	{\em Department of Physics and Astrophysics,} \\
	 {\em University of Delhi, Delhi-110 007, India.}}
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\begin{abstract}
We study the neutral current flavour changing rare decay mode
$B\rightarrow X_s +$ missing energy within the framework of
theories with large extra spatial dimensions. The corresponding Standard Model
signature is $B\rightarrow X_s + \nu \bar{\nu}$. But in theories with large
extra dimensions, it is possible to have scalars and gravitons in the final
state making it quite distinct from any other scenario where there are 
no gravitons and the scalars are far too heavier than the B-meson to be
present as external particles. We give an estimate of the branching ratio
for such processes for different values of the number of extra dimensions and 
the effective scale of the effective theory.\\
{\bf Keywords}: Extra dimensions, Rare B decay\\
{\bf PACS}: 11.25.Mj, 13.25.Hw
\end{abstract} 

\begin{section}{Introduction}
The investigations of flavour changing neutral current (FCNC) transitions
of b-quark offer excellent opportunities to test the basic structure
of the underlying theory. These processes are quite sensitive to QCD
and possible long distance corrections as well as new contributions from
loops. Consequently, such processes become a very useful tool for testing
new physics as well. The measurement of $B \to X_s \gamma$ (and 
subsequently the exclusive channel $B \to K^* \gamma$) by CLEO \cite{cleo}
has been used to stringently constrain the parameter space of supersymmetric
(SUSY) and various other theories (see foe example \cite{const} and references
therein).\\

The same is true for any other FCNC process. In particular, the quark
level transition $b \to s \nu \bar{\nu}$ can turn out to be a very 
successful place for putting strong and meaningful bounds on the underlying
 theory. The Standard Model (SM) experimental signature in this case is 
the observation
of the decay process $B\to X_s +$ missing energy in the form of neutrinos.   
In any extension of the SM, decay of a b-quark to an s-quark and a neutral,
 ultra light particle can mimic the SM process and will interfere with it
apart from an additional contribution to the SM process.\\

It is well known that SM is plagued with the hierarchy problem and many
possible solutions have been proposed in the past. However, the idea that
the fundamental scale of gravitational interaction being quite distinct
from the Planck scale (${\mathcal{O}}\sim 10^{19}$ GeV) and possibly
as low as TeV has attracted a lot of attention. In the simplest version,
as put forward by Arkani-Hamed et.al. (referred to as ADD model/scenario from
now on) \cite{nima}, the basic idea is 
the existence of n spatially compact large dimensions. The spacetime
is a direct product of the four dimensional Minkowski space and 
the compact space spanned by the n spatial extra dimensions. The SM fields
are all resticted to a 3-brane while gravity is free to propagate in the bulk.
Seen from a four dimensioanl point of view, it simply means that apart from
the usual massless graviton mode, there is a tower of Kaluza-Klein (KK) 
excitations from the gravity sector due to compactification. Also, if $M_{*}$
and $M_{Pl}$ denote the effective scale of gravity and four dimensioal
Planck scale respectively and if $R$ is the radius of compactification
of the extra dimensions (assuming the same radius for all the n dimensions), 
then it turns out that they are related as follows
\begin{equation}
M_{Pl}^2 \sim R^n~M_{*}^{n+2}
\end{equation} 
Therefore, for large values of $R$, the effective scale $M_*$ can be as
low as TeV to satisfy this relation. The individual KK modes couple
to the SM fields on the 3-brane by the ordinary gravitational strength
but the presence of a large number of them makes the effective coupling
appear to be of the order of TeV$^{-1}$ rather than Planck scale inverse.
This means that gravity starts to become a strong force at TeV scales.\\

In the present study, we investigate the inclusive decay $B \to X_s +$ missing
energy in the context of ADD scenario. In the SM or most of its extensions,
the missing energy is in the form of the neutrinos that escape without
being detected and it is only new contributions to the relevant Wilson 
coefficients that are induced by any model beyond SM. However, there is a sharp
contrast in the case at hand. In the present scenario, there can be very light
gravitons and associated scalars (dilatons) that can be present in the
final state. The situation is clearly distinct from SM or its usual 
extensions as in all such cases there are no spin-2 gravitons involved and
the scalars, both neutral and charged, are much much massive than the b-quark
to be present in the external legs. But this is possible in theories with
large compact extra dimensions. It is precisely this advantageous aspect
that we would like to exploit in the present study and see whether we
get any meaningful results.
\end{section}

\begin{section}{$b \to s + KK$ modes}
The effective Hamitonian for $b \to s$ transition in the SM is \cite{buchalla}
\be
{\cal{H}}_{eff} = -\frac{G_F}{\sqrt{2}}V_{ts}^*V_{tb}\sum_i 
                         C_i(\mu)O_i(\mu),
\ee
with
\[
O_1 = (\bar{s}_ic_j)_{V-A}(\bar{c}_jb_i)_{V-A},
\]
\[
O_2 = (\bar{s}_ic_i)_{V-A}(\bar{c}_jb_j)_{V-A},
\]
\[
O_3 = (\bar{s}_ib_i)_{V-A}\sum_q(\bar{q}_jq_j)_{V-A},
\]
\[
O_4 = (\bar{s}_ib_j)_{V-A}\sum_q(\bar{q}_jq_i)_{V-A},
\]
\be
O_5 = (\bar{s}_ib_i)_{V-A}\sum_q(\bar{q}_jq_j)_{V+A},
\ee
\[
O_6 = (\bar{s}_ib_j)_{V-A}\sum_q(\bar{q}_jq_i)_{V+A},
\]
\[
O_7 = \frac{e}{16\pi^2}\bar{s}_i\sigma^{\mu\nu}(m_sP_L +
m_bP_R)b_iF_{\mu\nu},
\]
and
\[
O_8 = \frac{g}{16\pi^2}\bar{s}_i\sigma^{\mu\nu}(m_sP_L + m_bP_R)
           T_{ij}^ab_jG_{\mu\nu}^a.
\]
Apart from these there are semi-leptonic operators as well
\[
O(ee)_V = (\bar{s}_ib_i)_{V-A}(\bar{e}e)_V
\]
\be
O(ee)_A = (\bar{s}_ib_i)_{V-A}(\bar{e}e)_A
\ee
and finally 
\[
O(\nu\bar{\nu}) = (\bar{s}_ib_i)_{V-A}(\bar{\nu}\nu)_{V-A}
\]
It is this last operator that is responsible for the relevant deacy channel in 
SM. In order to calculate any process beyond these known operators, one will 
have to either write down all possible new operators respecting the symmetries
of the low energy theory and then indirectly fix the associated coefficients
or explicitly make the claculation. We follow the latter route.\\

The coupling of the gravitons and the dilatons can be obtained from the
low energy effective action \cite{tao,giudice}. As expected, 
the gravitons couple
to the energy-momentum tensor of the SM fields and the dilaton
couples to the trace of the energy-momentum tensor. It is then
straight forward to get the Feynman rules and compute the individual
contributions. We consider the scalar KK emission first and comment on the
analogous graviton emission later. The diagrams contributing to the
dilaton emission process are shown in Fig.1.
It may be important to mention here that the 
dilaton-gauge boson-fermion-fermion vertex does not exist, although it is there
in \cite{tao}. This is not hard to see. The trace of the energy-momentum tensor
for the fermionic plus gauge bosonic parts (including interaction term)
is nothing but
\be
T^{\mu}_{\mu} = -m_f\bar{f}f + m_V^2A^2
\ee
In obtaining this, one needs to consider the interacting Heisenberg field
equations rather than the free ones. Thus, clearly the gauge boson-fermion-
fermion-dilaton vertex will never be present. But a similar vertex for
the gravitons will be there as in \cite{tao}. This reduces the number
of diagrams in the dilaton emission process by two when compared with the 
graviton emission process. Also, in the limit of neglecting the strange quark
mass, the diagram with dilaton being emitted from the s-quark line also
vaishes.\\

Clearly, all the diagrams are divergent and there is no underlying symmetry
like the one present in SM that ensures cancellations between the individual
diagrams to render a finite result. By naive power counting, one finds that
there are hard divergences which are quadratic, quartic and possibly even 
higher in the case of graviton emission. It now brings the question 
of handling such divergences in the context of effective field
theories. It has been very strongly advocated \cite{london} that in any
sensible effective field theory all the quadratic and higher divergences
are cancelled by counter-terms arising out of the complete underlying 
high-energy theory. At the one loop level, it is thus the 
logarithmic dependence
on the cut-off only that can be extracted from the low energy effective
theory. Thus, it makes more sense to keep only the logarithmic pieces as
far as the one loop calculation within an effective theory is concerned.
We adhere to this approach in our calculations and thus use dimensional
regularization throughout and at the end replace $1 \over{\epsilon}$ by
$\ln\Bigg({\Lambda^2 \over{m_W^2}}\Bigg)$ with the identification $\Lambda = M_*$.\\

As mentioned earlier, it is expected that the presence of the scalars and
gravitons in the external legs makes the situation more interesting as new 
oprators are introduced. The operators are of the form (denoting the dilaton 
and the graviton fields ny $\Phi$ and $h_{\mu\nu}$ respectively)
\be 
O(dilaton) \sim (\bar{s}_i(1\pm \gamma_5)b_i) \Phi
\ee
and
\be 
O(graviton) \sim (\bar{s}_i\gamma_{\mu}p_{\nu}(b)(1-\gamma_5)b_i) h^{\mu\nu} 
\ee 

The invariant matrix element for the quark level process $b \to s \Phi^{(n)}$
for a dilaton of mass $m_n$ is
\bea
{\mathcal{M}}(b \to s \Phi) &=& \Bigg({\imath\over{16\pi^2}}\Bigg)
{G_F\omega\kappa\over{\sqrt{2}}}\Bigg({1\over{\epsilon}}\Bigg)
\Bigg(\sum_i \xi_i m_i^2\Bigg)\Bigg({m_b\over{3}}\Bigg) \\ \nonumber
&&\Bigg[18\Bigg({m_s^2 \over{m_b^2-m_s^2}}\Bigg) - 1\Bigg]\bar{s}(1+\gamma_5)b
\eea
where $\xi_i$ is the CKM factor, $m_i$ is the mass of the up-type quark in the 
loop. We have neglected ${\mathcal{O}}({m_s^2\over{m_W^2}})$ and higher terms
 in obtaining theis expression. Using this matrix amplitude it is now trivial 
to compute the decay rate
for the emission of a single dilaton of mass $m_n$. For the inclusive process,
it suffices to use the quark level amplitude to get the decay rate.
 One should sum over the
tower of these dilatons tilll the b-quark mass scale. Following the summing 
techniques illustrated in \cite{tao}, we get the following expression for the
decay rate 
\bea
\Gamma(B\to X_s \Phi) = &=& \Bigg({1\over{16\pi^2}}\Bigg)^2
{G_F^2\over{2}}\Bigg[\ln\Bigg({M_*^2\over{m_W^2}}\Bigg)\Bigg]^2
\Bigg(\sum_i \xi_i m_i^2\Bigg)^2\Bigg({4m_b\over{27}}\Bigg) \\ \nonumber
&&\Bigg({1\over{n+2}}\Bigg)
\Bigg[{1\over{n}} + {1\over{n+4}} - {2\over{n+2}}\Bigg]
\Bigg({m_b\over{M_*}}\Bigg)^{n+2}
\eea 
where we have made the replacement 
\[
{1\over{\epsilon}} \longrightarrow \ln\Bigg({M_*^2\over{m_W^2}}\Bigg)
\]
The expression for the graviton emission rate will look similar in its final
form but the rate itself is expected to be higher because of two
 reasons. Firstly, there are two extra diagrams (with the graviton being
attached to either of the ferrmion-fermion-gauge boson vertex and secondly,
the graviton coupling does not have the $\omega$ factor suppression present
in the dilaton coupling. Thus, the rate is expected to be
\be
\Gamma(B\to X_s G) \sim {\mathcal{F}}~\Gamma(B\to X_s \Phi)
\ee
where ${\mathcal{F}}$ is a numerical factor of the order $1-10$.     
\end{section}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{section}{Results}
The total contribution to the process $B \rightarrow X_s + KK$ modes is
\[
\Gamma(B\to X_s \Phi) + \Gamma(B\to X_s G)
\]
Below we quote the branching fraction for the dilaton mode for different
values of $n$ and $M_*$.
\begin{table}[ht]
    \caption{${\it{Br}}(B\to X_s + \Phi)$}
    \begin{center}

    \begin{tabular}{|c|c|c|}
	\hline
	$n$ & $M_*$ (TeV) & Branching ratio \\
		\hline
	2	&1 	& $1.28 \times 10^{-3}$\\ 
       3 	& 1     & $2.25 \times 10^{-6}$ 	\\ 
       4        & 1     & $4.93 \times 10^{-9}$ \\ 
\hline
	2	&5 	& $5.50 \times 10^{-6}$\\ 
       3 	& 5     & $1.93 \times 10^{-9}$ 	\\ 
       4        & 5     & $8.45 \times 10^{-13}$ \\ 
\hline
	2	&10	& $4.69 \times 10^{-7}$\\ 
       3 	& 10    & $8.23 \times 10^{-11}$ 	\\ 
       4        & 10    & $1.80 \times 10^{-14}$ \\ 
\hline
	2	&50	& $1.33 \times 10^{-9}$\\ 
       3 	& 50    & $4.68 \times 10^{-14}$ 	\\ 
       4        & 50    & $2.05 \times 10^{-18}$ \\ 
\hline
\end{tabular}
\end{center}
\end{table}
These numbers are to be compared with the SM expectation, ${\it Br}(b\to s
\nu\bar{\nu}) = 5 \times 10^{-5}$ and the experimental limits on the
same, ${\it Br}(b\to s\nu\bar{\nu})_{exp} < 6.2 \times 10^{-4}$ at $90\%$
confidence level \cite{aleph}. 
It is quite evident from the above table that for $M_*=1$ TeV, practically
all the quoted $n$ values stand a chance to compete with the SM predictions 
for the
branching fraction of $B\to X_s +$ missing energy and it is only
 $n=2$ for other
$M_*$ values that seems to be reasonable from the point of view of 
observation. Other values are far tooo small to be observed. Further, it
is apparent that higher the $n$ or $M_*$ value, lower the predicted 
branching ratio, making such choices uninteresting.

It may be useful to mention here that the mode $B\to X_s G$ can contribute to
SM process $B\to X_s \nu\bar{\nu}$ via $G\to \nu\bar{\nu}$ and the sum of
the SM and the extra contribution will form the complete matrix element.
But, in the case of the scalar mode, $B\to X_s \Phi$, there can be no cascade
decay of $\Phi$ to $\nu\bar{\nu}$. Hence, until and unless the produced
$\Phi$ decays into low mass particles and escapes as missing energy,
the energy and angular spectrum of the hadronic junk, $X_s$, will be
markedly different from the corresponding distribution observed in the case of
$B\to X_s \nu\bar{\nu}$ coming from any theory. The same is true for the 
graviton emission process provided it does not decay further into
$\nu\bar{\nu}$ or any other pair of low mass particles.
From the structure of $O(graviton)$ it is evident that the operator
corresponding to $b\to s \nu\bar{\nu}$ will have the form
\[
O(\nu\bar{\nu})_{graviton} \sim  (\bar{s}_ib_i)_{V-A}(\bar{\nu}\nu)_V
\]
which is pure vector current for the $\nu\bar{\nu}$ pair, a feature different
 from the SM again where the intermediate particle responsible is the 
Z-boson, making the angular distribution different.
\\

Also observe that if instead of following the approach advocated in 
\cite{london}, we had retained quadratic and higher divergences, we
would have obtained enormously large, untolerable numbers. This again
justifies the retaining of only logarithmic terms and in turn justifies 
our use of dimensional regularization in carrying out the calculations, which 
otherwise would have to be carried out with a hard ultra-violet cut-off.\\

In conclusion, we can say that for two extra dimensions and with $M_*\sim
 {\mathcal{O}}$(TeV), the desired branching ratio is sizeable and we
ca hope that future observation of the decay mode
$B\to X_s +$ missing energy and also the exclusive modes will put
severe constraints on the number of extra dimensions and the effective
scale of gravitational interactions in these theories.     
 

\end{section}
%\vskip 1cm 

\section*{Acknowledgements} 
NM would like to thank the University Grants Commission,
India, for a fellowship.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak
\vskip 3cm
\begin{figure}[ht]
\vspace*{-1cm}
\centerline{
\epsfxsize=5cm\epsfysize=3.5cm
                    \epsfbox{fd1.eps}
\hskip 1.5cm
\epsfxsize=5cm\epsfysize=3.5cm
                    \epsfbox{fd2.eps}}
\vskip 1cm
\centerline{
\epsfxsize=5cm\epsfysize=3.5cm
                    \epsfbox{fd3.eps}
\hskip 1.5cm
\epsfxsize=5cm\epsfysize=3.5cm
                    \epsfbox{fd4.eps}}
\caption{\em The diagrams contributing to the dilaton emission. For
 the graviton emission there are two more diagrams with the graviton
being hooked to either of the two ends of the loop.}
%	}
%label{fig:fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\end{document}

