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\begin{document}


\title{}

\author{Ulrich Nierste} 


\maketitle


\section{Theory of very rare decays}
The LHC is constructed to reveal the nature of new physics around the
TeV scale, which is postulated to stabilise the electroweak scale.  The
production of new particles at the high energy frontier needs to be
complemented by precision measurements which determine the couplings and
mixing patterns of these new particles. To this end flavour-changing
neutral current (FCNC) processes play a pivotal role, because they are
highly suppressed in the Standard Model (SM) but not in its generic
extensions. Clearly, the more suppression factors pile up in a given SM
FCNC decay rate, the higher is the sensitivity to new physics. A
particularly important class of very rare decays are the leptonic decays
of a $B_d$ or a $B_s$ meson. In addition to the electroweak-loop
suppression the corresponding decay rates are helicity suppressed by a
factor of $m_{\ell}^2/m_B^2$, where $m_\ell$ and $M_B$ are the masses of
lepton and $B$ meson, respectively. The SM decay rates were calculated
in \cite{bb} at the next-to-leading-order (NLO) of QCD. The effective
$|\Delta B|=|\Delta S| =1$ hamiltonian, which describes $b\to s$ decays,
already contains 17 different operators in the Standard Model, in a
generic model-independent analysis of new physics this number will
exceed 100. One virtue of purely leptonic $B_s$ decays is their
dependence on a small number of operators, so that they are accessible
to model-independent studies of new physics. These statements, of
course, equally apply to $b\to d$ transitions and leptonic $B_d$ decays.
While in the Standard Model all six $B_q\to \ell^+ \ell^- $ decays (with
$q=d$ or $s$ and $\ell=e,\mu$ or $\tau$) are related to each other in a
simple way, this is not necessarily so in models of new physics.
Therefore all six decay modes should be studied. 

Other very rare decays to be mentioned in this context are non-hadronic
$B$ decays which have multiple leton pairs in the final state or photons
in addition to one or more lepton pairs. The theoretical information
contained in these decays can be more easily obtained from other decays,
for instance $B\to X_s \gamma$ or $B\to X_s \ell^+\ell^-$. Still they
need to be considered as background processes to the leptonic decays of
interest because of the finite resolution of the invariant mass of the
lepton pair. Another related class of very rare B decays are
lepton-flavour violating decays like $B\to e^- \mu^-$. In the Standard
Model (meant here to include neutrino mass terms) they are suppressed by
two powers of $m_\nu/M_W$, where $m_\nu$ denotes the largest neutrino
mass. However, this suppression factor is absent in certain models of
new physics.
 
\boldmath
\subsection{$B_q \to \ell^+ \ell^-$ in the Standard Model} 
\unboldmath
Photonic penguins do not contribute to $B_q \to \ell^+ \ell^-$, because 
a lepton pair with zero angular momentum has charge conjugation quantum
number C=1, while the photon has C=-1. 
The dominant contribution stems from the Z-penguin diagram shown in
\fig{fig:smp}.
\begin{figure}[t]
\begin{center}
  \includegraphics[width=.3\textwidth]{smpeng.ps}
\caption{Left: Z-penguin contribution to $B_s\to \ell^+\ell^-$.}\label{fig:smp}
\end{center}
\end{figure}
There is also a box diagram with two W bosons, which is suppressed by a
factor of $M_W^2/m_t^2$ with respect to the Z-penguin diagram. These
diagrams determine the Wilson coefficient $C_A$ of the operator 
\begin{eqnarray}
Q_A & = & \ov b_L \gamma^{\mu} q_L \, 
          \ov \ell \gamma_{\mu} \gamma_5 \ell . \label{defqa} 
\end{eqnarray}
We will further need operators with scalar and pseudoscalar couplings 
to the leptons:  
\begin{eqnarray}
Q_S & = & m_b \ov b_R q_L \, 
          \ov \ell \ell , \qquad\qquad
Q_P \; = \; m_b \ov b_R q_L \, 
          \ov \ell \gamma_5 \ell . \label{defsc} 
\end{eqnarray}
Their coefficients $C_S$ and $C_P$ are determined from 
penguin diagrams involving the Higgs or 
the neutral Goldstone boson, respectively. While $C_S$ and $C_P$ are
tiny and can be safely neglected in the Standard Model, the situation
changes dramatically in popular models of new physics discussed below. 
The effective hamiltonian reads 
\begin{eqnarray}
       H & = & \frac{G_F}{\sqrt{2}} \, 
        \frac{\alpha_{\rm QED}}{\pi \sin^2\theta_W} \, V_{tb}^* V_{tq} \, 
        \left[ \, C_S Q_S + C_P Q_P + C_A Q_A \, \right].
        \label{hami}
\end{eqnarray} 
$C_A$ has been determined in the  next-to-leading order (NLO) of QCD
\cite{bb,bb2}. The NLO corrections are in the percent range and
higher-order corrections play no role. $C_A$ is commonly expressed in
terms of the $\ov{\rm MS}$ mass $\ov m_t$ of the top quark. A pole mass
of $m_t^{\rm pole}=171.4 \pm 2.1 \, \gev $ corresponds to
$\ov{m}_t=163.8\pm 2.0\, \gev $.  An excellent approximation to the NLO
result for $C_A$, which holds with an accuracy of $5 \cdot 10^{-4}$ for
$149 \, \gev < \ov m_t < 179 \,\gev$, is  
\begin{eqnarray}% good to 5*10^(-4) for 149 GeV < mt < 179 GeV:   
 C_A (\ov m_t)  & = &  0.9636 
   \lt[ \frac{80.4 \, \gev}{M_W} 
        \frac{\ov m_t}{164 \, \gev}\rt]^{1.52} \label{defca}
\end{eqnarray}
In the literature $C_A(\ov m_t)$ is often called $Y(\ov m_t^2/M_W^2)$. 
The exact expression can be found e.g.\ in Eqs.~(16-18) of \cite{bb2}.  
The branching fraction can be compactly expressed in terms of the Wilson
coefficients $C_A$, $C_S$ and $C_P$: 
\begin{eqnarray}% 
{\cal B} \lt( B_q \to \ell^+\ell^- \rt) &=&
\frac{G_F^2\,\alpha_{\rm QED}^2}{64\, \pi^3 \sin^4\theta_W} \, 
      \lt| V_{tb}^* V_{tq} \rt|^2 \, \frac{\tau_{B_q}}{\hbar} 
     \, M_{B_q}^3 f_{B_q}^2 \,
      \sqrt{1-\frac{4 m_{\ell}^2}{M_{B_q}^2}} \nn
&& \times \lt[ \lt( 1- \frac{4 m_{\ell}^2}{M_{B_q}^2} \rt) 
    \, M_{B_q}^2 \, C_S^2 \; +\; 
    \lt( M_{B_q} C_P - \frac{2 m_{\ell}}{M_{B_q}} C_A \rt)^2 \rt] 
  \label{defbr}. 
\end{eqnarray}
Here $f_{B_q}$ and $\tau_{B_q}$ are the decay constant and the lifetime
of the $B_q$ meson, respectively, $\theta_W$ is the Weinberg angle and  
$\hbar=6.582 \cdot 10^{-13} \,\gev\, \mbox{ps}$. Since $B_q \to \ell^+\ell^-$
is a short-distance process, the appropriate value of $\alpha_{\rm QED}$   
is $\alpha_{\rm QED}(M_Z)=1/128$. With \eq{defca} and $C_S=C_P=0$
\eq{defbr} gives the following Standard Model predictions:
\begin{eqnarray}% 
{\cal B} \lt( B_s \to \tau^+\tau^- \rt) &=& 
  \lt( 8.20\pm 0.31 \rt) \cdot 10^{-7\phantom{4}} \, \times
  \frac{\tau_{B_s}}{1.527\, \mbox{ps}}  \, 
  \lt[ \frac{\lt| V_{ts}\rt|}{0.0408} \rt]^2 \,  
  \lt[ \frac{f_{B_s}}{240\,\mev } \rt]^2 \label{stnum} \\[1mm]
{\cal B} \lt( B_s \to \mu^+\mu^- \rt) &=& 
  \lt( 3.86\pm 0.15 \rt) \cdot 10^{-9\phantom{4}} \, \times
  \frac{\tau_{B_s}}{1.527\, \mbox{ps}}  \, 
  \lt[ \frac{\lt| V_{ts}\rt|}{0.0408} \rt]^2 \, 
  \lt[ \frac{f_{B_s}}{240\,\mev } \rt]^2 \label{smunum} \\[1mm]
{\cal B} \lt( B_s \to e^+ e^- \rt) &=& 
  \lt(9.05 \pm 0.34 \rt) \cdot 10^{-14} \, \times 
    \frac{\tau_{B_s}}{1.527\, \mbox{ps}}  \, 
  \lt[ \frac{\lt| V_{ts}\rt|}{0.0408} \rt]^2 \,  
  \lt[ \frac{f_{B_s}}{240\,\mev } \rt]^2 \label{senum}  \\[1mm]
{\cal B} \lt( B_d \to \tau^+\tau^- \rt) &=& 
  \lt( 2.23 \pm 0.08 \rt) \cdot 10^{-8\phantom{0}} \, \times
  \frac{\tau_{B_d}}{1.527\, \mbox{ps}}  \, 
  \lt[ \frac{\lt| V_{td}\rt|}{0.0082} \rt]^2 \,  
  \lt[ \frac{f_{B_d}}{200\,\mev } \rt]^2 \label{dtnum} \\[1mm]
{\cal B} \lt( B_d \to \mu^+\mu^- \rt) &=& 
  \lt( 1.06\pm 0.04 \rt) \cdot 10^{-10} \, \times
  \frac{\tau_{B_d}}{1.527\, \mbox{ps}}  \, 
  \lt[ \frac{\lt| V_{td}\rt|}{0.0082} \rt]^2 \, 
  \lt[ \frac{f_{B_d}}{200\,\mev } \rt]^2 \label{dmunum} \\[1mm]
{\cal B} \lt( B_d \to e^+ e^- \rt) &=& 
  \lt( 2.49\pm 0.09 \rt) \cdot 10^{-15} \, \times 
    \frac{\tau_{B_d}}{1.527\, \mbox{ps}}  \, 
  \lt[ \frac{\lt| V_{td}\rt|}{0.0082} \rt]^2 \,  
  \lt[ \frac{f_{B_d}}{200\,\mev } \rt]^2 \label{denum} 
\end{eqnarray}
The dependences on the decay constants, which have sizable theoretical
uncertainties, and on the relevant CKM factors have been factored out.
While $|V_{ts}|$ is well-determined through the precisely measured 
$|V_{cb}|$, the determination of $|V_{td}|$ involves the global fit to
the unitarity triangle and suffers from larger uncertainties. The
residual uncertainty in \eqsto{stnum}{denum} stems from the   
2$\,\gev$ error in $\ov{m}_t$. 

\boldmath
\subsection{$B_q \to \ell^+ \ell^-$ and new physics} 
\unboldmath% 
\subsubsection{Additional Higgs bosons}
The helicity suppression factor of $m_\ell/M_{B_q}$ in front of $C_A$ in
\eq{defbr} makes ${\cal B} (B_q \to \ell^+ \ell^-)$ sensitive to physics
with new scalar or pseudoscalar interactions, which contribute to $C_S$
and $C_P$. This feature renders $B_q \to \ell^+ \ell^-$ highly
interesting to probe models with an extended Higgs sector. Practically
all weakly coupled extensions of the Standard Model contain extra Higgs
multiplets, which puts ${\cal B} (B_q \to \ell^+ \ell^-)$ on the center
stage of indirect new physics searches. Higgs bosons couple to fermions
with Yukawa couplings $y_f$. In the Standard Model $y_b\propto m_b/M_W$
and $y_\ell\propto m_\ell/M_W$ are so small that Higgs penguin diagrams,
in which the Z-boson of \fig{fig:smp} is replaced by a Higgs boson, play
no role. In extended Higgs sectors the situation can be dramatically
different. Models with two or more Higgs multiplets can not only
accomodate Yukawa couplings of order one, they also generically contain
tree-level FCNC couplings of neutral Higgs bosons. In simple
two--Higgs--doublet models these unwanted FCNC couplings are usually
switched off in an ad-hoc way by imposing a discrete symmetry on the
Higgs and fermion fields, which leads to the celebrated
two-Higgs-doublet models of type I and type II. Here we only discuss the
latter model, in which one Higgs doublet $H_u$ only couples to up-type
fermions while the other one, $H_d$, solely couples to down-type
fermions \cite{ghkd}. The parameter controling the size of the
down--type Yukawa coupling is $\tan\beta=v_u/v_d$, the ratio of the
vacuum expectation values acquired by $H_u$ and $H_d$. The Yukawa
coupling of $H_d$ to the fermion $f$ is $y_f\sin\beta = m_f \tan\beta/v$
with $v=\sqrt{v_u^2+v_d^2}=174\, \gev$.  Hence $y_b\approx 1$ for $\tan
\beta \approx 50$. The dominant contributions to $C_S$ and $C_P$ for
large $\tan\beta$ involve charged and neutral Higgs bosons, but the
final result can be solely expressed in terms of $\tan\beta$ and the
charged Higgs boson mass $M_{H^+}$ \cite{ln}:
\begin{eqnarray}% 
        C_S \;=\; C_P &=& \frac{m_{\ell}}{2 M_W^2} \tan^2 \beta \,\,
        \frac{\ln r}{r-1}\qquad\qquad\quad \mbox{with}
\quad r\;=\; \frac{M_{H^+}^2}{\ov m_t^2} .
        \label{wc}
\end{eqnarray}
While for very large values of $\tan\beta/M_{H^+}$ the branching
fraction can be enhanced, the contributions in \eq{wc} typically reduce
${\cal B} (B_q \to \ell^+ \ell^-)$ with respect to the Standard Model
value. The decoupling for $M_{H^+}\to \infty$ is slow, e.g.\ for 
$\tan\beta=60$ and $ M_{H^+}=500 \gev$ the new Higgs contributions
reduce ${\cal B} (B_q \to \ell^+ \ell^-)$ by 50\%!  

\subsubsection{Supersymmetry}  
The generic Minimal Supersymmetric Standard Model (MSSM) contains many
new sources of flavour violation in addition to the Yukawa couplings.
These new flavour violating parameters stem from the
supersymmetry--breaking terms and their effects could easily exceed
those of the CKM mechanism. In view of the success of the CKM
description of flavour--changing transitions one usually supplements the
MSSM with the hypothesis of \emph{Minimal Flavour Violation (MFV)},
which can be justified by symmetry arguments \cite{dgi}.  In the
MFV--MSSM the only sources of flavour violation are the Yukawa
couplings, just as in the Standard Model. In this section the MSSM is
always understood to be supplied with the assumption of MFV. While in
MFV scenarios the contributions from virtual supersymmetric particles to
FCNC processes are normally smaller than the Standard Model
contribution, the situation is very different for $B_q \to
\ell^+\ell^-$.

The MSSM has two Higgs doublets.  At tree-level the couplings are as in
the two-Higgs-doublet model of type II, because the holomorphy of the
superpotential forbids the coupling of $H_u$ to down-type fermions and
that of $H_d$ to up-type fermions. At the one-loop level, however, the
situation is different, and both doublets couple to all fermions.  The
loop-induced couplings are proportional to the product of a
supersymmetry-breaking term and the $\mu$ parameter.  If $\tan\beta$ is
large, the loop-induced coupling of $H_u^*$ and the tree-level coupling
of $H_d$ give similar contributions to the masses of the down-type
fermions, because the loop suppression is compensated by a factor of
$\tan \beta$ \cite{hr}. In this scenario the Higgs sector is that of a
\emph{general}\ two-Higgs-doublet model, which involves FCNC Yukawa
couplings of the heavy neutral Higgs bosons $A^0$ and $H^0$ \cite{hpt}.
The Wilson coefficients $C_S$ and $C_P$ differ from those in \eq{wc} in
two important aspects: they involve three rather than two powers of
$\tan \beta$ and they depend on the mass $M_{A^0}\sim M_{H^0}$ instead
of the charged Higgs boson mass. The branching ratios scale as
\begin{eqnarray}% 
Br^{\rm SUSY} (B_q\to \ell^+\ell^-)&\propto &
        \frac{m_b^2 m_\ell^2\, \tan^6\beta}{M_{A^0}^4}\no
\end{eqnarray}
and could, in principle, exceed the Standard Model results in
\eqsto{stnum}{denum} by a factor of 1000 \cite{bk}. Thus the
experimental upper limit on $Br (B_s\to \mu^+\mu^-)$ from the Tevatron,
which is larger than $Br^{\rm SM} (B_s\to \mu^+\mu^-)$ in \eq{smunum} by
a factor of 25, already severly cuts into the parameter space of the
MSSM.  $Br (B_s\to \mu^+\mu^-)$ in MSSM scenarios with large $\tan\beta$
has been studied extensively \cite{bk,ltb,bcrs,ddn}.

Very popular special cases of the MSSM are the minimal supergravity
model (mSUGRA) \cite{n} and its slight variant, the Constrained Minimal
Supersymmetric Standard Model (CMSSM). While the MSSM contains more than
100 parameters, mSUGRA involves only 5 parameters and is therefore much
more predictive. In particular correlations between $Br (B_s\to
\mu^+\mu^-)$ and other observables emerge, for example with the
anomalous magnetic moment of the muon and the mass of the lightest
neutral Higgs boson \cite{ddn}.  Other well-motivated variants of the
MSSM incorporate the parameter constraints from grand unified theories
(GUTs).  $Br (B_s\to \mu^+\mu^-)$ is especially interesting in GUTs
based on the symmetry group SO(10) \cite{ddn,drrr}. In the minimal
SO(10) GUT the top and bottom Yukawa couplings $y_b$ and $y_t$ unify at
a high scale implying that $\tan\beta$ is of order 50. While realistic
SO(10) models contain a non--minimal Higgs sector, any experimental
information on the deviation of $y_b/y_t$ from 1 is very desirable, as
it probes the Higgs sectors of GUT theories. In conjunction with other
observables like the mass difference in the \bbs\ system \cite{bcrs} or
$Br(B^+\to \tau^+ \nu_\tau)$ \cite{ip}, which depend in different ways
on $\tan\beta$ and the masses of the non-Standard Higgs bosons and the
supersymmetric particles, the measurement of $Br (B_s\to \mu^+\mu^-)$ at
the LHC will, within the MSSM, answer the question whether the top and
bottom Yukawa couplings unify at high energies.

\boldmath
\subsection{Other very rare decays}
\unboldmath% 
The decays $B_q \to \ell^+ \ell^- \gamma$ and $B_q \to \ell^+ \ell^-
\ell^{\prime +} \ell^{\prime -}$ are of little interest from a
theoretical point of view. First, they are difficult to calculate, since
they involve photon couplings to quarks and are thereby sensitive to
soft hadron dynamics. Second, they are not helicity--suppressed, because
the (real or virtual) photon can recoil against a lepton pair in a $J=1$
state. This implies that they probe operators of the effective
hamiltonian which can more easily be studied from $B_q\to X \gamma$ and
$B\to X \ell^- \ell^-$ decays. However, the absence of a helicity
suppression makes $B_q \to \ell^+ \ell^- \gamma$ a dangerous background
to $B_q \to \ell^+ \ell^-$, if the experimental resolution of the
invariant mass of the lepton pair is limited. A naive estimate gives $Br
(B_s\to \mu^+\mu^- \gamma)\sim (m_B^2/m_\mu^2)\, \alpha_{\rm QED}/(4
\pi)\sim Br (B_s\to \mu^+\mu^-) $, while a more detailed analysis even
finds $Br (B_s\to \mu^+\mu^- \gamma) > Br (B_s\to \mu^+\mu^-)$
\cite{mn}.

Lepton-flavour violating (LFV) decays like $B_q \to \ell^\pm \mu^\mp$
are negligibly small in the Standard Model. In supersymmetric theories
with R parity (such as the MSSM) their branching ratios are smaller than
those of the corresponding lepton-flavour conserving decay, e.g.\ $B_q
\to \mu^+ \mu^-$. Large effects, however, are possible in models
which contain LFV tree-level couplings or leptoquarks. Here
supersymmetric theories without R parity and the Pati-Salam model should
mentioned. Supersymmetry without R parity involves a plethora of new
couplings, which are different for all combinations of quark and lepton 
flavour involved, so that no other experimental constraints prevent
large effects in $B_q \to \ell^\pm \mu^\mp$. 
Flavour physics in the Pati-Salam model has been studied in \cite{vw}. 
   


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