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\title{Potential for precise Unitarity Triangle angles measurements
in LHC}

\author{M. Musy\address[MCSD]{Physics Dept.
        Universit\`a di Milano--Bicocca \\ 
        Piazza delle Scienze 3 U2, 20126 Milano, Italy}\\
  On Behalf of the LHCb Collaboration
}       
\runtitle{Potential for precise Unitarity Triangle angles measurements in LHC}
\runauthor{M. Musy}
\begin{document}

\begin{abstract}
The Large Hadron Collider (LHC) will represent a very important
opportunity for B physics research. The two multi-purpose experiments,
ATLAS and CMS will have the capability to
realize competitive programmes, while a dedicated experiment
LHCb will have the explicit task to exploit a wide range
of physics decays involving B mesons.  This paper is a review of the main
characteristics of the future measurements of the Unitarity Triangle
angles at these experiments, and the expected achievable precisions.
\vspace{1pc}
\end{abstract}

\maketitle

\section{Present results}

The Standard Model (SM) allows for the CKM unitarity triangle~\cite{cab} only 
a very restricted region of the $(\bar\eta,\bar\rho)$ plane. In
Figure~\ref{utfit} the present knowledge of the angles of Unitarity
Triangle (plus $\varepsilon_K$) is represented, the most stringent
constraint coming from the $\beta$ angle~\cite{utfitpeople}.  The
general consistency of the various available measurements is very
good. Together with the measurements of the sides and $\Delta
m_s$ recent results, it leads to rather precise predictions for all
the three angles:
\begin{equation}
\alpha = 94.6^\circ\pm 4.6^\circ
\end{equation}
\begin{equation}
\beta  = 23.9^\circ\pm 1.0^\circ
\end{equation}
\begin{equation}
\gamma  = 61.3^\circ\pm 4.5^\circ
\end{equation}
\begin{equation}
\phi_s  = 2.1^\circ\pm 0.2^\circ
\end{equation}
New Physics effects are certainly capable of modifying this pattern, but
it will need quite a high statistical and systematic 
precision in order to disentangle them from SM
components effects.\\

\vspace{-9pt}\begin{figure}[htb]
\includegraphics[width=18pc]{CKM_fit_sides.eps}
\caption{The allowed region in the $(\bar\eta,\bar\rho)$ plane
for the vertex of the Unitarity Triangle~\cite{cab}.}
\label{utfit}\end{figure}

\section{LHC Experiments}

Three experiments are foreseen at LHC which may give important
contribution to the CKM Triangle determination.  ATLAS~\cite{atlas}
and CMS~\cite{CMS} will explore B physics mainly through the use of
high $p_T$ muons, and in decay modes involving di--muons.  The
LHCb~\cite{lhcb} experiment is dedicated explicitly to B-physics
dedicated experiment. It consists of a single-arm spectrometer in the
``forward'' region. To that extent, it is complementary to Atlas and
CMS experiments which are sensitive to the central rapidity
region. LHCb will have a $b\bar{b}$ production cross section of
230$\mu$b, more than a factor 2 with respect to ATLAS and CMS.

Trigger and B flavour tagging are very important and challanging
aspects for all the three experiments due to the enormous inelastic
cross section and the fact that the B mesons are produced
incoherently.  ATLAS will have a trigger output rate of 10--15~Hz,
with a tagging effective efficiency of about 4\%. CMS foresees to have
5~Hz of inclusive and about 1~Hz of exclusive trigger output rate and
comparable tagging power.  They will both run a few years at $L=2\cdot
10^{33}$/cm$^2$/s. LHCb will run at ten times less luminosity reducing the
probability of pile-up down to 0.5.  The tagging effective efficiency will range
between 4\%--5\%, and 7\%--9\% for B$_d$ and B$_s$ respectively.  The
LHCb output rate will amount to 200~Hz of exclusive B modes plus
900~Hz of inclusive b (e.g. b$\rightarrow \mu X$).

\section{Unitarity angle measurements}

The potential for the future determination of each of the unitarity
angles is reviewed in the following.

\subsection{Angle $\beta$}
The $\beta$ angle is already well measured at the B factories through the
decay $B_d\rightarrow J/\Psi K_s$. Still it is an important
measurement given the amount of statistics available at LHC.  The
extraction of $\beta$ proceeds through the fit of the time dependance
of the CP asymmetry:
\begin{equation}
A_{CP}(t) = A^{dir}\cos(\Delta m_d t)+A^{mix}\sin(\Delta m_d t)
\end{equation}
the first term being 0 in the SM. In one year LHCb will collect 2/fb
in this channel allowing to reach a precision of
$\sigma(\beta)=0.6^\circ$.  ATLAS will achieve similar sensitivity with
30/fb.  Thanks to the high statistics, it might be feasible to compare
against other channels like $B_d\rightarrow \phi K_s, \eta' K_s$,
where a preliminary estimate of the yield gives a precision on
$\sigma(\sin 2\beta_{eff})\approx 0.4$.

\subsection{Angle $\phi_s$}

The $\phi_s$ angle can be obtained from \\ $B_s\rightarrow J/\Psi\phi,
\eta_c\phi, J/\Psi\eta', D_sD_s$ decay modes. The highest yield being
the decay to the final state $J/\Psi\phi$ (131k/year at LHCb). This mode
for the determination of $\phi_s$
is the B$_s$ counterpart of the $B_d\rightarrow J/\Psi K_s$ for
$\beta$. In the SM it is expected to be a very small number ($\approx
0.04$), and therefore can be sensitive to New Physics effects in the
$B_s$--$\bar{B}_s$ system. The analysis is complicated by the fact
that the final state is a mixture of 2 CP even and 1 CP odd eigenstates, so
that an angular analysis is necessary to disentangle them. The result is
then obtained by a simultaneous fit to $\phi_s$ angle,
$\Delta\Gamma_s$ and CP-odd fraction.  Assuming $\Delta m_s=17.5/ps$,
$\phi_s=-0.04$ and $\Delta\Gamma_s/\Gamma_s=0.15$ one obtaines a
precision of $\sigma(\phi_s)=0.022$~rad (with 2/fb at LHCb).  ATLAS and
CMS will achieve a precision of $\approx 0.08$~rad with 10/fb integrated
luminosity.


\subsection{Angle $\gamma$}
There are various methods which have been considered to extract the
$\gamma$ angle using different channels.

\subsubsection{$\gamma$ from $B_s\rightarrow D_s K$}
This channels involves 4 decay rates used for the measurement of 2
time dependent asymmetries which depend on the angle $\gamma+\phi_s$,
so that $\gamma$ can be derived assuming $\phi_s$ known from
elsewhere.  In this case, contributions from New Physics are very
unlikely because of the large asymmetries induced by the interference of
same-order tree level amplitudes proportional to $\lambda^3$.  In this
channel we will collect 5.4k events/year (at LHCb) with an expected
precision of $\sigma(\gamma)=14^\circ$. Discrete ambiguities on the
measurement of $\gamma$ can be resolved if $\Delta\Gamma_s$ is large
enough or using $B_d\rightarrow D\pi$~\cite{bsdsk}.

\subsubsection{$\gamma$ from $B^0\rightarrow D^0 K^*$}
In the Ref~\cite{dunietz} a method has been proposed to evaluate 
$\gamma$ from these decays. It makes use of the interference between 
two color suppressed diagrams which interfere through the D mixing.
Decay amplitudes depend on the weak phase $\gamma$ plus a strong phase $\Delta$.
At LHCb experiment, the extraction of $\gamma$ can be
 obtained from the measurement of 3 decay rates (plus charge conjugates):\\
$B^0\rightarrow D^0(K^+\pi^-) K^*(K^+\pi^-)$ (3.4k/year),\\
$B^0\rightarrow D^0(K^-\pi^+) K^*(K^+\pi^-)$ (0.5k/year)\\
$B^0\rightarrow D^0_{CP}(K^-K^+) K^*(K^+\pi^-)$ (0.6k/year).\\
The achievable precision is $\sigma(\gamma)\approx 8^\circ$ (assuming
$55^\circ <\gamma <105^\circ$, and $|\Delta| <20^\circ$).
In this channel tagging of the B meson is not necessary 
because of the charge of the kaon in the final state, thus improving
the performance.

\begin{table*}[htb]
\caption{Summary table of the achievable performaces of the
LHCb experiment alone on the Unitarity Triangle angles in one year nominal
data taking corresponding to 2/fb integrated luminosity.}
\label{table:1}
\newcommand{\m}{\hphantom{$-$}}
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\begin{tabular}{@{}llllll}
\hline\hline
Angle &  Channel & Yield & B/S & Exp. Precision & Theoretical Precision \\
\hline\hline
$\beta$ & $B_d\rightarrow J/\Psi K_s$ & 216k & 0.8    & $\sigma(\beta)\approx 0.6^\circ$ & $\sigma(\beta)\approx 0.2^\circ$ \\
        & $B_d\rightarrow \Phi K_s$    & 0.8k & $<$2.4 & $\sigma(\beta)\approx 12^\circ$  & $\sigma(\beta)\approx 2^\circ$ \\
\hline
        & $B_s\rightarrow J/\Psi\phi$ & 125k & 0.3    &  & \\
$\phi_s$& $B_s\rightarrow J/\Psi\eta$ & 12k  & 2--3   & $\sigma(\phi_s)\approx 1.2^\circ$ & $\sigma(\phi_s)\approx 0.2^\circ$ \\
        & $B_s\rightarrow \eta_c\phi$ & 3k   & 0.7    &  & \\
\hline
        & $B_s\rightarrow D_s K$      & 5.4k & $<$1   & $\sigma(\gamma)\approx 13^\circ$ &  $\sigma(\phi_s)<< 1^\circ$\\
        & $B_d\rightarrow \pi\pi$     & 26k  & $<$0.7 &                                   & (if U--spin\\
        & $B_s\rightarrow KK$         & 37k  & $<$0.3 &  $\sigma(\gamma)\approx 5^\circ$ & symmetry holds)\\
        & $B_d\rightarrow D^0(K^-\pi^+)K^*$ & 0.5k  & $<$0.3 & & \\
$\gamma$& $B_d\rightarrow D^0(K^+\pi^-)K^*$ & 2.4k  & $<$2   & $\sigma(\gamma)\approx 8^\circ$ & \cc{ --}\\
        & $B_d\rightarrow D_{CP}(K^+K^-)K^*$ & 0.6k  & $<$0.3   &  & \\
        & $B^-\rightarrow D^0(K^+\pi^-)K^-$ & 60k  & $<$0.5   &  & \\
        & $B^-\rightarrow D^0(K^-\pi^+)K^-$ & 2k   & $<$0.5   & $\sigma(\gamma)\approx 8^\circ$--$13^\circ$ & 
         $\sigma(\gamma)<< 1^\circ$ \\
\hline
$\alpha$& $B_d\rightarrow \rho\pi, \rho\rho$ & 14k  & $<$0.8   & $\sigma(\alpha)\approx 10^\circ$ & 
           $\sigma(\alpha)\approx 1^\circ$ \\
\hline\hline
\end{tabular}\\[2pt]
\end{table*}

\subsubsection{$\gamma$ from $B^\pm\rightarrow D K^\pm$}
Also $B^\pm\rightarrow D K^\pm$ decays involve $b\rightarrow c$ and
$b\rightarrow u$ transitions and are therefore sensitive to $\gamma$
if a common final state is reached for the $D$ and $\bar D$
mesons~\cite{atwood}. In this case there are 2 interfering B diagrams
(one color suppressed), and 2 interfering D diagrams (one doubly suppressed)
leading to large interference effects because of similar final amplitudes. 
From the measurement of these 4 rates one has 2 observables, but still
4 unknown parameters ($\gamma, \delta_B, r_B, \delta_D^{K\pi}$).
To constrain the problem further one can add more decay modes like
$D\rightarrow K\pi\pi\pi$ (4 rates and one new strong phase
$\delta_D^{K3\pi}$) and $D\rightarrow KK$ (CP eigenstate, 2 rates) and
extract all the unknown parameters in a global fit.  The LHCb achievable
precision on $\gamma$ from this method is $\sigma(\gamma)\approx
4^\circ-13^\circ$, depending on the actual value of strong phases
$\delta_D^{K\pi}$ and $\delta_D^{K3\pi}$ (this study assumes 
$|\delta_D^{K\pi}|<25^\circ$ and $|\delta_D^{K3\pi}|<180^\circ$).
The extraction of  $\gamma$ via Dalitz plot study using $D\rightarrow K_s\pi\pi$ 
decays is under investigation in LHCb.

\subsubsection{$\gamma$ from $B_d\rightarrow \pi\pi, B_s\rightarrow KK$}
A further metod for $\gamma$ extraction is from the study of these
decay channels where penguin contributions are possible and for this
reason can be sensitive to New Physics. For both channels one measures
$A^{dir}_{CP}$ and $A^{mix}_{CP}$ which are parameters depending on
$\gamma$, $\phi_{s}, \phi_d$ and penguin over tree complex ratio
$de^{i\theta}$. The U-spin symmetry allows to simplify
the real part $d=d_{\pi\pi}=d_{KK}$ and the phase 
$\theta=\theta_{\pi\pi}=\theta_{KK}$, which
remains as a theoretical assumption.  The precision on $\gamma$ in
this case is $\sigma(\gamma)\approx 5^\circ$ in one year LHCb data
taking.


\subsection{Angle $\alpha$}

The analysis of $B_d\rightarrow \rho\pi$ events allows for the extraction
of $\alpha$. However the simple approach, where the $\rho$ meson is
considered as stable particle, is affected by both higher order discrete
ambiguities and penguin pollution, like for $B_d\rightarrow \pi\pi$.
To solve these problems a method was proposed~\cite{snider} which
analyses the decay $B_d\rightarrow \pi^+\pi^-\pi^0$ in the $\rho$
resonance region, taking into account interference effects between
vector mesons of different charges. The knowledge of the strong decay
$\rho\rightarrow \pi\pi$, parametrized as a Breit-Wigner amplitude,
allows the extraction of $\alpha$ and penguin-to-tree ratio
simultaneously from a multi-dimensional fit.  To evaluate the
precision of the method on $\alpha$ a number of pseudoexperiment were
performed  including all known experimental smearings, finite resolution,
acceptance, and wrong tag fraction.  A background/signal ratio equal
to 1 was also considered.  Under these assumptions the precision of the
determination (LHCb, for 1 year data taking) is of the order of $10^\circ$ with
85\% of the pseudoexperiments converging to the correct solution. The
probability of mirror solution decreases with the increasing
statistics, down to about $0.2$\% for 10/fb.  The inclusion in the
analysis of $B_d\rightarrow \rho^0\rho^0$ decays can give further
contraint to the measurement of $\alpha$. The result of the fit goes
from $98.1^{+14^\circ}_{-8^\circ}$ to $97.0^{+5.9^\circ}_{-3.8^\circ}$
when such events are considered. Figure~\ref{alpha} shows the likelihood
of the result of the combination of  $B_d\rightarrow \rho\pi,\rho\rho$
(with $\rho^0\rho^0$ coming from LHCb)
channels after one year data taking.

\vspace{-9pt}\begin{figure}[htb]
\includegraphics[width=19pc]{fig.ps}
\caption{Likelihood function for the combination of 
$B_d\rightarrow \rho\pi,\rho\rho$ channels for the extraction of
the angle $\alpha$.}
\label{alpha}\end{figure}

\section{Summary}
Table~\ref{table:1} shows the status of the expected precisions for
the Unitarity Triangle angles at LHC. For each angle, the relevant
decay channels are considered together with the annual LHCb yield
after trigger and channel selection, and the expected background over
signal ratio. In the last column the theoretical precision of the
calculation is indicated as well.  In all the measurements the
uncertainty on the CKM angles will be dominated by the experimental
accuracy and not by the thoeretical one.  After some year of
successful running LHC will be able to determine the Unitarity
Triangle very precisely, possibly highlighting New Physics effects.


\begin{thebibliography}{9}
\bibitem{cab} N Cabibbo, Phys. Rev. Lett. {\bf 10},531 (1963).\\
M.Kobayashi and T.Maskawa, Prog.Th. Phys, {\bf 49}. 652 (1973).
\bibitem{atlas} The ATLAS Collaboration, ATLAS Technical Proposal,
CERN/LHCC/94-43,\\
The ATLAS Collaboration, ATLAS Detector and Physics Performance
Technical Design Report, CERN/LHCC/99-14 and CERN/LHCC/99-15.
\bibitem{CMS} The CMS Collaboration, CMS Technical Proposal,CERN/LHCC/98-38.
\bibitem{lhcb} The LHCb Collaboration, LHCb Technical Proposal, CERN/LHCC98-4\\
The LHCb Collaboration, Reoptimized Detector Design and Performance,
CERN/LHCC/2003-030.
\bibitem{bsdsk} R. Fleischer, Nucl. Phys. {\bf B 671} (2003) 459.
\bibitem{snider} A. Snyder, H.R. Quinn, Phys. Rev. {\bf D48}, 2139 (1993). 
\bibitem{dunietz} Gronau and London, Phys. Lett. {\bf B 253}, 483
(1991), Gronau and Wyler, Phys. Lett. {\bf B 265}, 172 (1991) and
Dunietz, Phys. Lett. {\bf B 270}, 75 (1991).
\bibitem{atwood} Atwood, Dunietz and Soni, Phys. Rev. Lett. 78, 3257 (1997).
\bibitem{utfitpeople} M.~Bona {\it et al.}  [UTfit Collaboration], 
hep-ph/0606167, 2006.

\end{thebibliography}

\end{document}

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