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\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
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\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}

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\begin{document}

\def\xspace{\hskip 0.000cm}
\def\Phis {\ensuremath{\Phi_s}\xspace}
\def\Bs {\ensuremath{ B_s} \xspace}
\def\Bsbar {\ensuremath{ \bar{B}_s} \xspace}
\def\mubarn {\ensuremath{ \mathrm{\mu b}} \xspace}
\def\BsMuMu {\ensuremath{ B_s \rightarrow \mu^+\mu^-}\xspace}
\def\BsJPsiPhi {\ensuremath{ B_s \rightarrow J / \psi \phi}\xspace}
\def\BdKstarMuMu {\ensuremath{ B^0  \rightarrow K^\ast\mu^+\mu^-}\xspace}
\def\BtoDK {\ensuremath{ B  \rightarrow D K }\xspace}
\def\BsBsbar {\ensuremath{ B_s - \bar{B}_s}\xspace}
\def\JPsiPhi{\ensuremath{J/\psi\phi }\xspace}
\def\Invfb{\ensuremath{\mathrm{fb}^{-1}}\xspace}
\def\Dbar{\ensuremath{\bar{D}}\xspace}
\def\BmDKm{\ensuremath{B^- \rightarrow D^0 K^-}\xspace}
\def\BmDbarKm{\ensuremath{B^- \rightarrow \bar{D}^0 K^-}\xspace}
\def\BpmDKpm{\ensuremath{B^\pm \rightarrow D K^\pm}\xspace}
\def\BzDzKstar{\ensuremath{B^0 \rightarrow D K^\ast}\xspace}
\def\BsDsKpm{\ensuremath{B_s \rightarrow D_s^\mp K^\pm  } \xspace}
\def\BsDsPipm{\ensuremath{B_s \rightarrow D_s^\mp \pi^\pm  } \xspace}
\def\DspmKmp{\ensuremath{D_s^\pm K^\mp } \xspace}
\def\BtoKstarMM{\ensuremath{ B\rightarrow K^\ast \mu^+\mu^- } \xspace}
\def\BstoMM{\ensuremath{ B_s\rightarrow  \mu^+\mu^- } \xspace}
\def\BtoHH{\ensuremath{ B \rightarrow  h^+ h^- } \xspace}





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\vspace*{4cm}

\title{Looking for New Physics: Prospects for B Physics at LHCb}

\author{U. Uwer}

\address{
Physikalisches Institut, 
Universit\"at Heidelberg, Philosophenweg 12,\\
69120 Heidelberg, Germany}

\maketitle\abstracts{
With the startup of LHCb, the dedicated heavy flavor experiment 
at the LHC, the next round of precision B-experiments will be launched. 
LHCb has access to about $10^{12}$ B meson decays per year, allowing
significant measurements  of even very rare B decays and, in particular, 
the precision study of the \Bs\ system. With the  measurement of rates, 
angular distributions and CP asymmetries of loop suppressed B decays, 
LHCb will probe the
quantum corrections predicted by the Standard Model. Many 
observables show a large sensitivity to New Physics contributions.
In the following the expected LHCb physics performance and the potential to
search for New Physics is discussed for a set of key measurements.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

The LHCb experiment~\cite{LHCb} is a dedicated beauty and charm-physics experiment at the
proton-proton collider LHC at CERN. The experiment exploits the copious
production of heavy quarks in proton-proton collisions. For a 
centre-of-mass energy of 14 TeV  a $b\bar{b}$ production cross section  
of more than 500 \mubarn is expected. At LHC, all b-hadron species 
($ B_d $, $ B_u$, $ B_s$, $ B_c$,
$\Lambda_b$, ...) will be produced. 

To unambigiously assign long-lived particles to an interaction point (primary
vertex) and to facilitate the event reconstruction, single interactions per 
beam crossing are prefered. It is therefore planned, to adjust the
beam luminosity at the LHCb interaction point to a value of $2\times 10^{32}$ 
$\mathrm{cm}^{-2}\mathrm{s}^{-1}$. 
At this luminosity, 
% which leads to a rate of ${\cal O}(10^4)$~Hz of $b\bar{b}$ pairs produced 
% in the acceptance of LHCb, 
the number of bunch crossings with a single interaction is 
maximized. 


With the  measurement of rates, angular distributions and CP
asymmetries of loop suppressed B decays LHCb will probe the
quantum corrections predicted by the Standard Model. Many of the
observables show large sensitivities to New Physics contributions.
The unprecedented quantity of b hadrons will allow LHCb to perform
significant measurements  of rare B decays down to branching fractions
of $10^{-9}$. One of the early LHCb key measurements will be the determination
of the \Bs\ mixing phase, \Phis , in the decay \BsJPsiPhi . Data of a few months
of LHC running will allow to cross-check the deviation from the Standard
Model prediction currently observed by the two TEVATRON experiments CDF and 
D0~\cite{Roadmap13-14}. The measurement of the direct CP asymmetry in \BtoDK\
tree-level decays will result in a determination of the CKM angle $\gamma$ with an 
expected final precision better than 3$^\circ$. The comparison with results from loop-suppressed
decays and with the result of the indirect determination of $\gamma$ from the
CKM fits will probe possible new physics contributions. Flavor changing
neutral current (FCNC) decays such as \BsMuMu\ and \BdKstarMuMu\ are very
sensitive  to different extensions of the Stadard Model and might
be the first place to observe effects of New Physics at the LHC.

The LHCb detector construction was completed in summer 2008. The
commissioning of the detector is well advanced. A large sample
of cosmic muon events was taken and was used to perform a first time
and and space alignment of the large area tracking detectors 
and the calorimeters. Data taken during injection tests, 
where the LHC beam was dumped before reaching the LHC main ring upstream 
of the LHCb detector, showed to be very useful for the understanding of the 
silicon detectors. The events with muon fluxes of several 
particles/$\mathrm{cm}^2$ were successfully used to align the silicon detectors.

In the following the expected physics performance of the LHCb experiment 
is discussed  for four key measurements.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Measurement of the \Bs\ mixing phase}

The interference between \BsJPsiPhi\ decays  with or without \BsBsbar\
oscillation gives rise to an observable CP violating phase \Phis. In the Standard Model, this 
phase is predicted to be $\Phis = -2\beta_s = -0.0368 \pm 0.0017$, where $\beta_s = arg(
- \frac{V_{ts} V_{tb}^\ast}{V_{cs}V_{cb}^\ast})$ is the smaller angle of the $bs$ unitary triangle 
of the CKM matrix. 


The value of \Phis\ can be extracted from the mesurement of the time dependent 
CP asymmetry $A_{CP}(t)$,
$$
A_{CP}(t) = \frac{\eta_f \sin \Phis \sin(\Delta m_s t)}{ \cosh(\Delta\Gamma_s
  t / 2 ) -\eta_f\cos\Phis \sinh(\Delta\Gamma_s  t / 2) }\, ,
$$
where $\Delta m_s$ and $\Delta \Gamma_s$ are the differences in mass and width of
the two \Bs\ mass eigenstates, and $\eta_f = \pm 1$ is the CP eigenvalue of
the final state.

The measurement of the phase \Phis\ is a sensitive test of the Standard Model as
 \Phis\ is one of the CP observables with the smallest theoretical uncertainty.
New Physics could  significantly modify this prediction, if, for example, new particles
contribute to the \BsBsbar\ mixing diagram.

As the final state \JPsiPhi\ contains two vector mesons its CP value 
depends on the relative angular momentum of both vector mesons. The relative contribution of
the two CP states has to be determined on a statistical basis using an angular analyis
of the three decay product angles $\Omega = \{ \theta, \phi, \psi  \} $ as depicted in 
Figure~\ref{Fig_JPsiPhi_Angles}. 

\begin{figure}[h]
\begin{center}
\epsfig{file=TransversityBasis.eps, width=0.8\textwidth}
\caption{Definition of the angles $\Omega = \{ \theta, \phi, \psi  \} $ in the
transversity basis.\label{Fig_JPsiPhi_Angles}}
\end{center}
\end{figure}


Both TEVATRON collaborations, CDF and D0, have recently presented measurements
of the time dependent CP asymmetrie for the channel \BsJPsiPhi\ and the phase
\Phis\ \cite{Roadmap13-14}. A combination  by HFAG~\cite{HFAG} results in
the following experimentally prefered 90\% CL intervalls for \Phis,
$$
\Phis \in [-1.47; -0.29] \cup [-2.85; -1.65]~~\mbox{at 90\% CL},
$$ 
These experimental constraints deviate from the Standard Model prediction. An
update by 
CDF~\cite{Roadmap16} using 2.8~\Invfb\ further degrades the consistency with the
Standard Model value.


For the decay \BsJPsiPhi\ the LHCb experiment expects a proper time resolution 
of about 38~ps, a \Bs mass resolution of 16~MeV and an effective tagging power 
($\epsilon D^2$) of 6.2~\%. With a total efficiency of 2.6\% (including the
trigger) the simulation predicts
a yield of 117k selected signal events per 2~\Invfb\ with a background to signal ratio
 of long-lived 
backgrounds of about 0.5. The signal events will allow a determination of the 
phase $\Phis = -2 \beta_s$ with a statistical precision 
$\sigma (\Phis) \sim 0.03$. Systematic effects due to proper time 
and angular resolutions, angular acceptance and flavour tagging have been 
studied and found to be smaller than the statistical uncertainty 
expected for 2~\Invfb. 

Figure~\ref{Fig_JPsiPhi_Error} shows the expected statistical uncertainty for
$\Phis =-2\beta_s$ versus the integrated luminosity ${\cal L}_{int}$. The sensitivity
has been estimated for 0.5 and 2~\Invfb, the values are $0.060 \pm 0.005$ and
$0.030 \pm 0.002$ respectively. The red line is an extrapolation from these
two values. 
The blue lines show the uncertainties related to the $b\bar{b}$ cross-section
and the visible branching ratio of  $ B_s \rightarrow J / \psi(\mu\mu)
\phi(KK)$. The line labelled "CDF+D0" is the combined CDF/D0 uncertainty in 2008 scaled 
to an integrated luminosity of $2\times 9$~\Invfb, as expected for the TEVATRON by 2010. 
It should be noted that LHCb is able to perform a competitive measurement of
\Phis\ already with the very first data.

\begin{figure}[ht]
\begin{center}
\epsfig{file=BetasError.eps, width=0.5\textwidth}
\caption{Expected statistical uncertainty on $\Phis=-2\beta_s$ as function of the 
integrated luminosity.\label{Fig_JPsiPhi_Error}}
\end{center}
\end{figure}



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

\section{Measurement of the CKM angle $\gamma$}

Several methods to directly measure the CKM angle $\gamma$ in \BtoDK\ tree
decays have been pioneered by the BABAR and BELLE experiment. The achieved 
precision so far is limited by statistics and is still much worse than the precision
of $\gamma$ determined indirectly from the CKM fits~\cite{Tisserand_Proceedings}.   

In all cases, the measurement of $\gamma$ in \BtoDK\ tree decays exploits the interference
between the two decay amplitudes, \BmDKm\ and \BmDbarKm , depicted in
Figure~\ref{Fig_Gamma_Feynman}, which appears  in the case that the $D$ and \Dbar\
are observed in the same final state. The interference 
allows the observation of the phase difference between
the two amplitudes,
$$
\frac{ {\cal A}(\BmDKm)}{ {\cal A}(\BmDbarKm)} = r_B e^{\delta_B} e^{-i\gamma} \, ,
$$ 
where $r_B$ is the relative magnitude of the two amplitudes and 
$\delta_B$ is an additional strong phase difference.

\begin{figure}[b]
\begin{center}
\epsfig{file=BtoDK.eps, width=0.8\textwidth}
\caption{Feynman diagrams for the decays \BmDKm\ and \BmDbarKm.\label{Fig_Gamma_Feynman}}
\end{center}
\end{figure}


LHCb has investigated the expected sensitivity to $\gamma$ for the ADS~\cite{ADS}
the GLW~\cite{GLW} and the GGSZ~\cite{GGSZ} method. The different methods result into a similar
sensitivity (between 10 and 13 degrees for a data-sample corresponding to 2~\Invfb).
For the combination, a sensitivity, depending on the strong
phase $\delta_B$, between 
4.3$^\circ$ ($\delta_B =180^\circ$) and 6.1$^\circ$ ($\delta_B =45^\circ$) 
 is expected~\cite{LHCb-2008-031}. 
 
An alternative way to determine $\gamma$, unique to LHCb, is the determination of
the sum of the two phases $\gamma + \Phis$ from the observation of the time
dependent CP asymmetry of the decays \BsDsKpm.  As the \Bs\ and the \Bsbar\ can 
both decay to \DspmKmp, there exists an intereference between amplitudes with and 
without a preceding oscillation. The main issue of this analysis is to separate 
the decay \BsDsKpm\ from the very similar decay \BsDsPipm\ with a $\sim$15 times larger 
branching fraction. Here, the excellent LHCb particle identification based on the two 
RICH counters allows a significant reduction of the background contamination.
For a data-sample equivalent to 2~\Invfb, a signal yield of 6.2$\times 10^3$ \BsDsKpm\ events 
with a background-to-signal ratio of $\sim$0.7 is expected. The resulting sensitivity to 
$\gamma + \Phis$ depends on the strong phase and varies between 9$^\circ$ and 12$^\circ$.

Combining all tree-level measurements of $\gamma$ (\BpmDKpm, \BzDzKstar\ and \BsDsKpm)
one obtains the expected sensitivities listed in
Table~\ref{Tab_Gamma_Sensitivity}~\cite{LHCb-2008-031}. As can be seen, data corresponding to about 
0.5~\Invfb\ would already allow a determination of $\gamma$ with an error better than 10$^\circ$.

\begin{table}[t]
\caption{ The expected combined sensitivity to $\gamma$  from \BtoDK\ and time-dependent
measurements~{\protect\cite{LHCb-2008-031}} for data sets corresponding to
integrated luminosities of 0.5, 2
and 10~\Invfb. \label{Tab_Gamma_Sensitivity}}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{lccccc}
\hline
$\delta_B (^\circ)$                      & 0   & 45  &  90  & 135 & 180 \\ \hline
$\sigma_\gamma$ for 0.5~\Invfb ($^\circ$)& 8.1 & 10.1&  9.3 & 9.5 & 7.8 \\
$\sigma_\gamma$ for 2~\Invfb ($^\circ$)  & 4.1 & 5.1 & 4.8  & 5.1 & 3.9 \\
$\sigma_\gamma$ for 0.5~\Invfb ($^\circ$)&  2.0& 2.7 & 2.4  & 2.6 & 1.9 \\ \hline
\end{tabular}
\end{center}
\end{table}



\section{Search for New Physics in flavor changing neutral current processes}


\subsection{\BtoKstarMM}
In the Standard Model, the flavor changing neutral current  
decay $b\rightarrow s\ell^+\ell^-$ cannot occur at 
tree level but only through electroweak penguin amplitudes with small branching 
fractions, e.g. $BR(\BtoKstarMM)\sim 1.2 \times 10^{-6}$. An angular analysis
of the decay \BtoKstarMM\ is well suited to search for New Physics: most New Physics
scenarios make definite predictions, for the forward-backward asymmetry, $A_{FB}$,    
of the $\mu^+$  relative to the B direction in the $\mu^+\mu^-$ rest frame.
In particular, the value $s_0$ of the $\mu^+\mu^-$ invariant mass, $q^2$, for which the 
asymmetry becomes zero, is predicted with small theoretical uncertainties
and may provide a stringent test of the Standard Model and New Physics scenarios.

Because of the clear di-muon signature 
and the good $K/\pi$ separation provided by the RICH detectors, the 
exclusive decay \BtoKstarMM\ can be triggered and reconstructed in LHCb 
with high efficiency~\cite{Teubert_50}. The selection 
including trigger has an efficiency of $\sim$1\% resulting in an expectation 
of $\sim$7.2k signal events for an integrated luminosity of 2~\Invfb. A simple counting
of forward and backward events together with a linear fit results in a
precision on $s_0$ 
 of 0.5~GeV$^2$. In addition to measuring $A_{FB}$, LHCb
is able to extract other angular observables, such as the longuitudinal 
$K^\star$ polarization $F_L(q^2)$ or the 
transverse asymmetry $A^{(2)}_T(q^2)$~\cite{Egede_LHCb-2007-057}. These quantities
show a high sensitivity to minimal flavor violating MSSM models with small 
$\tan\beta$. In the theoretically favored region away from the photon pole and below
the charm resonances ($1~\mathrm{GeV}^2 < q^2 < 6~\mathrm{GeV}^2$) the sensitivity 
of $F_L$ and $A^{(2)}_T$ is 0.016 and 0.42 respectively, for the above mentionned
data sample. 


\subsection{\BstoMM}

The purely leptonic decays $ B^0_{d,s} \rightarrow \mu^+\mu^-$ are FCNC
processes. In the Standard Model, these decays can only
proceed at a very low rate through higher order diagrams (Z/Higgs-penguins
and W-box diagram - the latter is suppressed). Moreover, the decays are helicity 
suppressed by a factor $(m_\mu/M_{B})^2$.
The decays are therefore very sensitive to any New Physics model with new scalar 
or pseudo-scalar interactions, in particular, they are sensitive to models with
an extended Higgs sector. In the MSSM the branching fractions are known to
increase with $\tan^6\beta$. 

The Standard Model branching fraction for \BstoMM\ is computed~\cite{BsMM} 
to be $BR(\BstoMM)=(3.35\pm 0.32)
\times 10^9$. The branching fraction for $B^0_d \rightarrow \mu^+\mu^-$  is further
suppressed by the CKM ratio $|V_{td}/V_{ts}|^2$ leading to a predicted Standard Model 
branching fraction of $\sim 1 \times 10^{-10}$. The current upper limit of the \BstoMM 
branching fraction from the TEVATRON experiments~\cite{HFAG,Teubert_55+56} is $4.7 
\times 10^{-8}$.

At LHCb, the background expected in the search for the decay \BstoMM\ is dominated
by random combinations of two muons originating from two distinct B decays. This 
background can be kept under control by exploiting the excellent tracking and
vertexing 
capabilities of the detector. Due to a very low muon misidentification rate, 
two-body hadronic decays, \BtoHH ,  where the hadrons are misindentified as muons, 
do not contribute to a significant level compared to the combinatorial background.
The LHCb experiment has the sensitivity to observe a branching fraction of
$1\times 10^{-8}$ with a 3$\sigma$ significance already with 0.1~\Invfb\ of data. 
A data-set of 2(6)~\Invfb will allow the observation of 
the Standard Model branching fraction with a 3(5)$\sigma$ signficance.


\section{Summary and Outlook}

The large $b\bar{b}$ production cross section at the LHC provides a unique opportunity 
to measure rates, angular distributions, and CP asymmetries of even very rare, loop suppressed 
 b decays. Many of the observables show large sensitivities to New Physics contributions.
In particular, the production of \Bs mesons could play a crucial role to identify new
 CP violating effects originating from New Physics. Already with a data-set of 0.5~\Invfb\ 
LHCb expects to perform sensitive tests of the Standard Model and New Physics
scenarios:  The \Bs\ mixing phase \Phis\ can be measured with a precision of 
$\pm 0.06$; the reachable precision of the CKM phase 
$\gamma$ is estimated to be between 8$^\circ$ and 10$^\circ$; a \BstoMM\ branching
fraction of $5\times 10^{-9}$ can be observed.  
  
The LHCb measurements will either limit New Physics contributions to B decays
or, more optimistically, uncover them. The measurements will complement the direct 
searches of New Physics performed by ATLAS and CMS. If New Physics phenomena will be 
observed first by the high-$p_t$ experiments, the LHCb measurements will help to 
understand the flavor structure of the new phenomena.


\section*{References}
\begin{thebibliography}{99}

\bibitem{LHCb} The LHCb Collaboration,
{\em Journal of Instrumentation}, JINST 3 S08005 (2008).

\bibitem{Roadmap13-14}
The CDF collaboration,\Journal{\PRL}{100}{16802}{2008}, \\
The D0 collaboration, {\em hep-ex}, arXiv:0802.2255v1, (2008).

\bibitem{HFAG}
Heavy Flavor Averaging Group, {\tt http://www.slac.stanford.edu/xorg/hfag/}

\bibitem{Roadmap16} The CDF collaboration, CDF/ANAL/BOTTOM/PUBLIC/9458 
(2008).

\bibitem{Tisserand_Proceedings} V.~Tisserand, these proceedings.

\bibitem{ADS} D. Atwood, I. Dunietz and A. Soni, \Journal{\PRL}{78}{3257}{1997},\\
  Dunietz and A. Soni, \Journal{\PRD}{63}{036005}{2001}.

\bibitem{GLW} M. Gronau and D. London, \Journal{\PLB}{253}{483}{1991}, \\
           M. Gronau and D.Wyler, \Journal{\PLB}{265}{172}{1991}.

\bibitem{GGSZ} A. Giri, Yu. Grossman, A. Soffer and J. Zupan, 
             \Journal{\PRD}{68}{054018}{2003}.

\bibitem{LHCb-2008-031} K.~Akiba {\em et al.}, {\em CERN-LHCb-2008-038} (2008).

\bibitem{Teubert_50} J.~Dickens {\em et al.}, {\em CERN-LHCb-2007-038} (2007).
 
\bibitem{Egede_LHCb-2007-057} U.~Egede, {\em CERN-LHCb-2007-057} (2007).

\bibitem{BsMM} M.~Blanke, A.J.~Buras, D.~Guadagnoli and C.~Tarantino 
\Journal{\em JHEP}{0610}{003}{2006}.   

\bibitem{Teubert_55+56} The D0 Collaboration,\\ 
{\tt http://wwwd0.fnal.gov/Run2Physics/WWW/results/bhtm} (2006),\\
The CDF Collaboration, CDF/ANAL/BOTTOM/PUBLIC/8176 (2006).

\end{thebibliography}


\end{document}

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