\documentclass[fleqn,twoside]{article}
\usepackage[headings]{espcrc2}

\usepackage{graphicx}

\include{babarsym}
\def\rhop  {\ensuremath{\rho^+}}
\def\rhom  {\ensuremath{\rho^+}}
\def\rhoz  {\ensuremath{\rho^0}}
\def\alphaeff {\ensuremath{\alpha_\mathrm{eff}}}
\def\dAlpha {\ensuremath{\Delta\alpha}}

% declarations for front matter
\title{Measurements of $\alpha$ at the B-Factories}

\author{D. Hutchcroft
  \address{Department of Physics, University of Liverpool \\
    Liverpool L69 7ZE, United Kingdom}
}
       
\begin{document}

\begin{abstract}
 Measurements from BaBar and Belle collaborations are presented that
 lead to the constraint of the angle $\alpha$ in the CKM unitarity triangle.
\vspace{1pc}
\end{abstract}

\maketitle

\section{Introduction}

The Cabibbo-Kobayashi-Maskawa matrix ($V_{CKM}$) describes the unitary
rotation between the mass eigenstates and weak Eigen states of the quark
system.  This matrix can be described in terms of three angles and one
non-trivial phase \cite{CKM}. CP violation is possible if the remaining
phase is non-zero, leading to complex terms in $V_{CKM}$. The unitary
relation $V^*_{ub}V_{ud}+V^*_{cb}V_{cd}+V^*_{tb}V_{td}=0$ describes a
unitary triangle, which has three angles $(\alpha, \beta,\gamma)$ or
sometimes $(\phi_2,\phi_1,\phi_3)$.

$\alpha$ (or $\phi_2$) is defined as
$\arg[-V_{td}V^*_{tb}/V_{ud}V^*_{ub}]$. To be sensitive to $\alpha$ two
processes are required that interfere, one that depends on $V_{td}$ and
the other $V_{ub}$. In $\Bz$ decays\footnote{Charge conjugation is
  assumed throughout} the $\Bz$ oscillates with the $\Bzb$ with a
frequency that depends on $V_{tb}$, in the tree decay of $\Bz$ to two
charmless mesons ($\pi$ or $\rho$) the decay amplitude depends on
$V_{ub}$ which means there is a time dependant difference between
$\Bz\to\pi^+\pi^-$ and $\Bzb\to\pi^+\pi^-$. However the gluonic penguin
decays give the same final state but have an amplitude that depends on
$V_{td}$ which dilutes the time dependent interference and adds an
additional phase difference between the strong phases of the final
states. Using analyses of $\Bz\to\rhoz\rhoz$ and $\Bp\to\rhop\rhoz$
provide constraints on the correction required for the gluonic penguins
\cite{gronau}.

The time dependant partial width of \Bz and \Bzb decays are described by
equation \ref{eqn:td}. 
\begin{eqnarray}
f(\Bz\to f,\deltat) &=& {\Gamma \over 4}e^{-\Gamma|\deltat|}
                       [1 + \eta S\sin(\deltamd\deltat) \nonumber\\
                    &&      - \eta C\cos(\deltamd\deltat)]
\label{eqn:td}
\end{eqnarray}
Where $\eta= +1(-1)$ for \Bz(\Bzb) and $S$ and
$C$ are the indirect and direct \CP components of the time dependence.
In the absence of penguin decays the prediction is that $C=0$ and
$S=\sin(2\alpha)$, when gluonic penguin diagrams are also considered
this becomes $C\propto \sin(\delta), S=\sqrt{1-C^2}\sin(2\alphaeff)$ and
$\delta$ is the difference between the tree and penguin strong phases.
The difference between the effective and true values of $\alpha$ is
$\dAlpha = |\alpha - \alphaeff|$.

                        
The branching ratios for charmless \B decays are small, ${\cal
  O}(10^{-5})$, so very large samples of \BB\ events are required to
measure properties of theses decays. There are two $B-$factories one in
the United States called PEP-II and one in Japan called KEKII, these
each have one detector called BaBar\cite{babar} and Belle\cite{belle}
respectively. Both of the accelerators predominately operate on the \Y4S
resonance producing \BB events and non-resonant continuum events. The
two detectors have collected around 1\invab between them, totalling more
than $1\times 10^9$ \BB pairs. Additional data has been taken below the
\Y4S resonance which is used to characterise the backgrounds.

%Two kinematic variables are defined to separate signal from \qqbar\
%backgrounds, the energy substituted mass
%$\mes=\sqrt{(s/2+\mathbf{p}_0\cdot\mathbf{p}_B)^2/E_0^2-\mathbf{p}_B^2}$,
%where $\sqrt{s}$ is the energy of the \epem\ system, $\mathbf{p}_B$ is
%the $B$ momentum and $(E_0,\mathbf{p}_0)$ is the 4-momentum of the \Y4S\
%system, and the difference between the reconstructed energy and half the
%beam energy, $\DeltaE = E_B - \sqrt{s}/2$ where $E_B$ is the
%reconstructed energy of the $B$ meson.

\section{$\Bz\to\pi\pi$}
Both Belle and BaBar have measured the properties of the decays of
$\Bz\to\pip\pim$ \cite{belle:pippim,babar:pippim}. The measured values
of $S$ and $C$, given in Table \ref{tab:pippim}, are taken from fits to
the time dependent decay of $\Bz\to\pip\pim$ shown in Figure
\ref{fig:pipiTD}. The combination of these results provides a
measurement of \alphaeff.

\begin{table}[t]
\caption{\CP violating properties of the $\Bz\to\pip\pim$ decays}
\label{tab:pippim}
\begin{tabular}{rcc}
& Belle & BaBar \\
$S=$&$-0.67\pm0.16\pm0.06$ & $-0.30\pm0.17\pm0.03$\\
$C=$&$-0.56\pm0.12\pm0.06$ & $-0.09\pm0.15\pm0.04$\\
\end{tabular}
\end{table}

\begin{figure*}
  \resizebox{\textwidth}{!}{
    \resizebox{0.5\textwidth}{!}{
      \includegraphics{belle_pipi.eps}  
    }
    \resizebox{0.5\textwidth}{!}{
      \includegraphics{babar_pipi.eps}
    }
  }
  \caption{The time dependant decays of $\Bz\to\pip\pim$ and the
    asymmetry as measured by Belle (left) and BaBar (right). Belle and
    BaBar have separated the \Bz and \Bzb into (a) and (b), and shown
    the asymmetry between them in the lower parts of the figures, (c)
    for BaBar. Belle further divide the sample into poor tagged (c) and
    good tagged (d) samples. }
  \label{fig:pipiTD}
\end{figure*}

The decays of $\Bz\to\piz\piz$ can be used to constrain \dAlpha, a small
branching ratio implying a small penguin contribution to the $\pi\pi$
system and that \dAlpha\ is small. The averaged measurements of BaBar and
Belle \cite{babar:pizpiz,belle:pizpiz,hfag2006} are
$\BR(\Bz\to\piz\piz)=1.45\pm0.29$, which means in the isospin
interpretation $\dAlpha\le35^\circ$ at 90\% confidence level (CL). 

\section{$\B\to\rho\rho$}
The same isospin analysis holds for the $\rho\rho$ system. Starting with
the neutral decays, BaBar has measured
$\BR(\Bz\to\rhoz\rhoz)=(1.16^{+0.37}_{-0.36}\pm0.27)\times10^{-6}$ with
a 3$\sigma$ significance of having observed a signal
\cite{babar:rhozrhoz}. This implies that the tree decays dominate in the
$\rho\rho$ modes. The $\Bp\to\rhop\rhoz$ decay have also been measured
with higher statistics by BaBar \cite{babar:rhoprhoz} with the results
given in Table \ref{tab:rhoprhoz}. The branching ratio is consistent
with the isospin expectation, the polarisation ($f_L$) means the final
state is almost entirely \CP even and the charge asymmetry ($A_{CP}$) is
compatible with zero and the assumption that electro-weak penguin
effects are very small. Using these results the constraint on
$\alphaeff<11^\circ$ at 68\% CL can be derived.

\begin{table}[!h]
\caption{Properties of the $\Bp\to\rhop\rhoz$ decays.}
\label{tab:rhoprhoz}
\begin{tabular}{rc}
 & BaBar \\
\BR= & $(16.8\pm2.2\pm2.3)\times10^{-6}$\\
$f_L=$ & $0.905\pm0.042^{+0.023}_{-0.027}$ \\
$A_{CP}=$ & $-0.12\pm0.13\pm0.10$ \\
\end{tabular}
\end{table}

Both Belle and BaBar have measured the time dependant asymmetry in
$\Bz\to\rhop\rhom$ decays \cite{belle:rhoprhom,babar:rhoprhom}, with the
results listed in Table \ref{tab:rhoprhom}.

\begin{table}[!h]
\caption{Properties of $\Bz\to\rhop\rhom$ decays}
\label{tab:rhoprhom}
\begin{tabular}{rcc}
& Belle & BaBar \\
$\BR=$&$(22.8\pm3.8^{+2.8}_{-2.6})$ & $(23.5\pm 2.2\pm4.1)$ \\
 &$\times10^{-6}$ &$\times10^{-6}$ \\
$f_L=$& $0.941^{+0.034}_{-0.040}\pm0.030$& $0.977\pm0.024^{+0.015}_{-0.013}$\\
$S=$& $0.08\pm0.41\pm0.09$& $-0.19\pm0.21^{+0.05}_{-0.07}$\\
$C=$& $0.00\pm0.30\pm0.09$ & $-0.07\pm0.15\pm0.06$
\end{tabular}
\end{table}

\section{$\Bz\to(\rho\pi)^0$}
The final useful channel is the $\Bz\to(\rho\pi)^0$, which is not CP
Eigenstate, so requires a full Dalitz analysis of the 6 decays
($\Bz(\Bzb)\to\rho^\pm\pi^\mp,\rho^0\pi^0$ to disentangle the results
\cite{snyderQuinn:rhopi}. BaBar and Belle have measured the $\rho^\pm$
parts of the phase space only but were not able to set limits on $S$ and
$C$ using those analyses. BaBar have also measured the full Dalitz plane
\cite{babar:rhopiDalitz}, using a square variant of the Dalitz plot, see
Figure \ref{fig:rhopiDaltiz}. The results are shown in table
\ref{tab:rhopi}.

\begin{table}[!h]
\caption{\CP violating properties of the $\Bz\to(\rho\pi)^0$ decays}
\label{tab:rhopi}
\begin{tabular}{rc}
 & BaBar \\
C=&$-0.154\pm0.090\pm0.037$\\
S=&$0.01\pm0.12\pm0.03$\\
\end{tabular}
\end{table}

\begin{figure*}
  \resizebox{\textwidth}{!}{
    \includegraphics{mprime.eps}  
    \includegraphics{hprime.eps}
  }
  \caption{The projection on the two square Dalitz variables of the
    BaBar $(\rho\pi)^0$ analysis. The grey bands represent the
    continuum, \B and miss reconstructed backgrounds in that order on the
    plot. The points are the data and the solid line is the projection
    of the PDF of the fit to the sample. }
  \label{fig:rhopiDaltiz}
\end{figure*}


\section{Conclusion}
Each of the above decay modes contributes to the constraints on
$\alpha$, the $\rho\rho$ system gives the best measurement and the
$\rho\pi$ unfolds the ambiguity in the sign of $\alphaeff$. The CKM
fitter group combination \cite{CKMFitter2006} of the measurements of
$\alpha$ is shown in Fig. \ref{fig:ckmfitter_alpha}. The HFAG average
\cite{hfag2006} of the measurements of $\alpha$ is
$\alpha=99^{+12}_{-9}$, which is consistent with the fit to all of the
CKM data except that of $\alpha$ which predicts $\alpha=97^{+5}_{-16}$
\cite{CKMFitter2006}.

\begin{figure}
  \begin{center}
    \resizebox{0.5\textwidth}{!}{
      \includegraphics{alpha_all_win06.eps}
    }
    \caption{CKM fitter combination of the light charmless meson decays
      contributions to the measurement of $\alpha$. Also indicated is the
      average of the indirect measurement of $\alpha$ from CKM fits.}
    \label{fig:ckmfitter_alpha}
  \end{center}
\end{figure}

\begin{thebibliography}{9}
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\bibitem{gronau}
M.~Gronau and D.~London, \jprl{65} (1990) 3381 .

\bibitem{babar}
\babar\ Collaboration, B.\ Aubert {\em et al.},
Nucl.\ Instrum.\ Meth.\ A {\bf A479} (2002) 1

\bibitem{belle}
Nucl.\ Instrum.\ Meth.\ A {\bf 479} (2002) 117.

\bibitem{belle:pippim}
Belle Collaboration, \jprl{95} (2005) 101801

\bibitem{babar:pippim}
\babar\ Collaboration, \jprl{95} (2005) 151803

\bibitem{babar:pizpiz}
  B.~Aubert {\it et al.}  [BABAR Collaboration],
  %``Branching fractions and CP asymmetries in $B^0 \to \pi^0 \pi^0$, $B^+ \to
  %\pi^+  \pi^0$ and $B^+ \to K^+ \pi^0$ decays and isospin analysis of the $B
  %\to \pi \pi$  system,''
  Phys.\ Rev.\ Lett.\  {\bf 94} (2005) 181802

\bibitem{belle:pizpiz}
  K.~Abe {\it et al.}  [Belle Collaboration],
  %``Observation of B0 --> pi0 pi0,''
  Phys.\ Rev.\ Lett.\  {\bf 94} (2005) 181803

\bibitem{hfag2006}
  E. Barberio et al., \hepex{0603003}

\bibitem{babar:rhozrhoz}
  \babar\ Collaboration, \hepex{0607097}

\bibitem{babar:rhoprhoz}
  \babar\ Collaboration,  \hepex{0607092} Submitted to PRL.

\bibitem{babar:rhoprhom}
 \babar\ Collaboration, \hepex{0607098}

\bibitem{belle:rhoprhom}
  Belle Collaboration 
  \jprl{96} 171801 (2006)

\bibitem{snyderQuinn:rhopi}
  A.~E.~Snyder and H.~R.~Quinn,
  %``Measuring CP asymmetry in B $\to$ rho pi decays without ambiguities,''
  Phys.\ Rev.\ D {\bf 48} (1993) 2139.

\bibitem{babar:rhopiDalitz}
\babar\ Collaboration, \hepex{0608002}

\bibitem{CKMFitter2006}
 CKMfitter Group (J. Charles et al.),
Eur. Phys. J. {\bf C41} (2005) 1-131 
\end{thebibliography}
\end{document}
