\contentsline {section}{\numberline {1}General Introduction}{2}
\contentsline {subsection}{\numberline {1.1}Extracting $\alpha $}{2}
\contentsline {subsection}{\numberline {1.2}Why $B\rightarrow 3\pi $}{2}
\contentsline {section}{\numberline {2}Theoretical framework}{2}
\contentsline {subsection}{\numberline {2.1}Full Analysis}{4}
\contentsline {subsubsection}{Neutral-$B$ Dalitz plot Analysis: Ideal Case}{4}
\contentsline {subsubsection}{Kinematical Uncertainties}{6}
\contentsline {subsubsection}{Taking into account the Electroweak Penguins}{8}
\contentsline {subsubsection}{Charged-$B$ Decays}{9}
\contentsline {subsection}{\numberline {2.2}Charged-$\rho $ Bands Analysis}{10}
\contentsline {subsection}{\numberline {2.3}Data Simulation}{14}
\contentsline {subsubsection}{Factorization of the Non-Leptonic Amplitudes}{14}
\contentsline {subsubsection}{Heavy-to-Light Form Factors}{15}
\contentsline {subsection}{\numberline {2.4}The main equations}{17}
\contentsline {subsection}{\numberline {2.5}The resonances and their Breit Wigner description}{17}
\contentsline {subsection}{\numberline {2.6}The theoretical scenarios}{17}
\contentsline {subsection}{\numberline {2.7}Interferences Description}{17}
\contentsline {section}{\numberline {3}Description of the fitting procedure}{17}
\contentsline {subsection}{\numberline {3.1}The likelihood function}{17}
\contentsline {subsubsection}{Neutral B decays}{19}
\contentsline {subsubsection}{Adding Charged B decays}{20}
\contentsline {subsection}{\numberline {3.2}Fitting 9 parameters: procedure and mirror solutions}{20}
\contentsline {subsubsection}{Fitting procedure}{20}
\contentsline {subsubsection}{Model independent parameterization}{20}
\contentsline {subsubsection}{Symetries of the likelihood function}{22}
\contentsline {subsubsection}{Neglecting the Electroweak Penguins}{22}
\contentsline {subsubsection}{Advantages of the parameterization}{22}
\contentsline {subsubsection}{Fitting Procedure}{23}
\contentsline {subsection}{\numberline {3.3}The perfect case}{23}
\contentsline {subsubsection}{Illustration of the fit}{23}
\contentsline {subsubsection}{Towards Confidence Levels}{24}
\contentsline {subsubsection}{Sample of events}{24}
\contentsline {subsubsection}{Effect of the available statistics}{24}
\contentsline {section}{\numberline {4}Experimental effects on the fit}{26}
\contentsline {subsection}{\numberline {4.1}Impact of acceptance cuts}{26}
\contentsline {subsubsection}{Description of the cuts}{26}
\contentsline {subsubsection}{Illustration}{26}
\contentsline {subsubsection}{LHC-B like cuts}{27}
\contentsline {subsection}{\numberline {4.2}Impact of the detector resolution}{27}
\contentsline {subsubsection}{Description of the resolution}{27}
\contentsline {subsubsection}{Effect on the $\alpha $ extraction}{28}
\contentsline {subsection}{\numberline {4.3}Impact of our knowledge of selection efficiencies}{28}
\contentsline {subsubsection}{Description}{28}
\contentsline {subsubsection}{Effect on the $\alpha $ extraction}{28}
\contentsline {subsection}{\numberline {4.4}Impact of B-Tagging}{28}
\contentsline {subsubsection}{B-tagging description}{28}
\contentsline {subsubsection}{Effect on the $\alpha $ extraction}{28}
\contentsline {subsection}{\numberline {4.5}Impact of $B\mathaccent "7016\relax B$ and $q\mathaccent "7016\relax q$ backgrounds}{28}
\contentsline {subsubsection}{Description of the expected background}{28}
\contentsline {subsubsection}{Description of backgrounds in our Monte Carlo}{29}
\contentsline {subsubsection}{Another description}{29}
\contentsline {subsection}{\numberline {4.6}Neglecting the $\rho ^0\pi ^0$ band}{29}
\contentsline {subsection}{\numberline {4.7}Non-resonant background}{30}
\contentsline {subsection}{\numberline {4.8}Impact of the $K\pi \pi $ background}{30}
\contentsline {section}{\numberline {5}Recovering information...}{30}
\contentsline {subsection}{\numberline {5.1}Adding the charged B}{30}
\contentsline {subsection}{\numberline {5.2}Summary}{32}
\contentsline {section}{\numberline {6}Theoretical effects}{32}
\contentsline {subsection}{\numberline {6.1}Ignoring penguins}{32}
\contentsline {subsubsection}{Estimation of the shift in the $\alpha $ measurement}{33}
\contentsline {subsubsection}{For different scenarios}{33}
\contentsline {subsection}{\numberline {6.2}Neglecting strong phases}{33}
\contentsline {subsection}{\numberline {6.3}Shape of the Breit Wigner}{33}
\contentsline {subsection}{\numberline {6.4}Effect of our knowledge of the $\rho (1450)$}{33}
\contentsline {subsubsection}{Ignoring the $\rho (1450)$}{33}
\contentsline {subsubsection}{Fitting for the $\rho (1450)$ contribution}{33}
\contentsline {subsubsection}{Fitting for the phase between the $\rho (770)$ and the $\rho (1450)$}{33}
\contentsline {subsection}{\numberline {6.5}Effect of badly known resonances}{33}
\contentsline {subsubsection}{The $\rho (1700)$}{33}
\contentsline {subsubsection}{The $\rho _3$}{33}
\contentsline {subsubsection}{The $f_0$}{33}
\contentsline {subsection}{\numberline {6.6}Electroweak penguins}{33}
