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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
\institute{UZH}
\title[$\PLambda_b \to \PLambda_c^{\ast} \ell \nu$ Zurich update]{$\PLambda_b \to \PLambda_c^{\ast} \ell \nu$ Zurich update}
\date{5 February 2016}


\begin{document}
\tikzstyle{every picture}+=[remember picture]

{
\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
\begin{frame}[c]%{\phantom{title page}} 
\begin{center}
\begin{center}
	\begin{columns}
		\begin{column}{0.75\textwidth}
			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {$\PLambda_b \to \PLambda_c^{\ast} \ell \nu$ Zurich update}
		\end{column}
                \begin{column}{0.02\textwidth}
                  {~}
                  \end{column}
                \begin{column}{0.23\textwidth}
                 % \hspace*{-1.cm}
                  \vspace*{-3mm}
                  \includegraphics[width=0.6\textwidth]{lhcb-logo}
                  \end{column}
	        
	\end{columns}
\end{center}
	\quad
	\vspace{3em}
\begin{columns}
\begin{column}{0.44\textwidth}
\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em} Nicola Serra  \\\vspace{-0.1em} Elena Graverini}

\end{column}
\begin{column}{0.53\textwidth}
\includegraphics[height=1.3cm]{uzh-transp}
\end{column}
\end{columns}

\vspace{1em}
%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
\vspace{0.5em}

	\textcolor{normal text.fg!50!Comment}{$R(\Lambda_c^{\ast})$ meeting, Zurich\\February 17, 2016}
\end{center}
\end{frame}
}


\begin{frame}[c]{Introduction}
\begin{minipage}{\textwidth}

The matrix elements for our $\PLambdab$ decays are depended on $F_{1,..,4}$ and $G_{1,...,4}$ form factors:



\begin{tiny}
%\scalebox{0.8}{
\begin{equation}
\langle   \Lambda_{c}^{1/2^-}(p', s') | V_{\mu}| \Lambda_{b}(p, s)\rangle =
\overline{u}(p', s')\left(F_1(q^2) \gamma_{\mu} + F_2(q^2)
\frac{ p_\mu}{m_{\Lambda_Q}} +  F_3(q^2)\frac{ p'_\mu}{m_{\Lambda_q}}
\right) u(p,s), \nonumber
\end{equation}
\begin{equation}
\langle \Lambda_{c}^{1/2^-}(p', s')| A_{\mu}| \Lambda_{b}(p, s)\rangle =
\overline{u}(p', s')\left(G_1(q^2) \gamma_{\mu} + G_2(q^2)
\frac{ p_\mu}{m_{\Lambda_Q}} +  G_3(q^2)\frac{ p'_\mu}{m_{\Lambda_q}}\right)
\gamma_{5} u(p,s), \nonumber
\end{equation}
\begin{equation}
\langle \Lambda_{c}^{3/2^-}(p', s')| V_{\mu}|  \Lambda_{b}(p, s)\rangle=
\overline{u}^\alpha(p', s')\left[
\frac{p_\alpha}{m_{\Lambda_Q}}\left(F_1\gamma_{\mu} + F_2\frac{
p_\mu}{m_{\Lambda_Q}} +  F_3\frac{ p'_\mu}{m_{\Lambda_q^{3/2}}}\right)+F_4 g_{\alpha\mu}\right]
u(p,s), \nonumber
\end{equation}
\begin{equation}
\langle \Lambda_{c}^{3/2^-}(p', s')| A_{\mu}|  \Lambda_{b}(p, s)\rangle = 
\overline{u}^\alpha(p', s')\left[
\frac{p_\alpha}{m_{\Lambda_Q}}\left(G_1 \gamma_\mu + 
G_2\frac{ p_\mu}{m_{\Lambda_Q}} +  G_3
\frac{ p'_\mu}{m_{\Lambda_q^{3/2}}}\right)+G_4 g_{\alpha\mu}\right]\gamma_{5} u(p,s). \nonumber
\end{equation}
,where $\overline{u}^\alpha(p', s')$ is a nasty Rarita-Swinger spinor.
%}
\end{tiny}


	\end{minipage}
\end{frame}




\begin{frame}[c]{What is in the simulation?}
\begin{minipage}{\textwidth}

$\Rrightarrow$ The simulation that we have uses the form factors calculated in \href{http://arxiv.org/pdf/nucl-th/0503030v1.pdf}{arXiv:nucl-th/0503030}\\
$\Rrightarrow$ In this paper the form factors are calculated in constituent quark model.\\
$\Rrightarrow$ Let me quote a theorists that wants to remain anonymous: ,,it's not even wrong''.\\
$\Rrightarrow$ Never the less this is what Syracuse is using and is now in the simulation $\rightarrow$ we will reweigh our MC.\\
$\Rrightarrow$ To do so, we firstly looked at reproducing the calculations from \texttt{EvtGen}.




	\end{minipage}
		\vspace*{2.cm}
\end{frame}


\begin{frame}[c]{Form factor calculus}
\begin{minipage}{\textwidth}
$\Rrightarrow$ So in \texttt{EvtGen} the calculations of From Factors are done only in the harmonic oscillator basis.\\{~}\\
\begin{tiny}

\begin{tabular}{|l|cccccccc|}
\hline
 model & $m_\sigma$ (GeV)& $m_s$ (GeV) & $m_c$ (GeV) & $m_b$ (GeV) & $b$ (GeV$^2$) 
 & $\alpha_{\rm Coul}$ \,\,\,\,\,\,&
$\alpha_{\rm hyp}$ & $C_{qqq}$ (GeV) \\\hline
 HONR & 0.40& 0.65& 1.89 & 5.28 & 0.14 &0.45 & 0.81 & -1.20 \\
 HOSR & 0.38& 0.59& 1.83 & 5.17 & 0.17 &0.09 & 0.26 & -1.45 \\
 STNR & 0.40& 0.64& 1.87 & 5.28 & 0.13 &0.35 & 0.31 & -1.22 \\
 STSR & 0.34& 0.57& 1.78 & 5.22 & 0.15 &0.19 & 0.11 & -1.23  \\ \hline
\end{tabular}
\end{tiny}
$\Rrightarrow$ On top of this you also have the wave size:
\begin{tiny}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{tabular}{|l|l|cccc|}
\hline
$J^P$ &  model & $\Lambda_b$ & $\Lambda_c$ & $\Lambda$ & $N$  \\ 
&  & $\left(\alpha_\lambda,\,\,\,\,\alpha_\rho\right)$ &
$\left(\alpha_\lambda,\,\,\,\,\alpha_\rho\right)$ &
$\left(\alpha_\lambda,\,\,\,\,\alpha_\rho\right)$ &
$\left(\alpha_\lambda,\,\,\,\,\alpha_\rho\right)$  \\ \hline
$1/2^+$ &HONR &  (0.59, 0.61)& (0.55, 0.58) & (0.49, 0.53) & 0.48\\
$1/2^+$ &HOSR & (0.68, 0.68)& (0.60, 0.61) & (0.52, 0.57)& 0.54\\
$1/2^+$ & STNR &  (0.44, 0.66)& (0.41, 0.69) & (0.35, 0.75)&- \\
$1/2^+$ & STSR &  (0.46, 0.64)& (0.43, 0.67) & (0.38, 0.72) &-\\ \hline
$1/2^-$ & HONR &  -& (0.47, 0.49) & (0.40, 0.47) &0.37\\
$1/2^-$ & HOSR &  -& (0.55, 0.59) & (0.48, 0.54) &0.46 \\
$1/2^-$ &STNR & -& (0.60, 0.50) & (0.55, 0.54) &- \\
$1/2^-$ &STSR & -& (0.61, 0.49) & (0.58, 0.51) &-\\ \hline
$3/2^+$ &HONR & -& - & - & 0.35\\
$3/2^+$ &HOSR & -& -& -& 0.44\\\hline
$5/2^+$ &HONR & -& - & - & 0.35\\
$5/2^+$ &HOSR & -& -& -& 0.46\\\hline
\end{tabular}


\end{tiny}
	\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{Form factor results (example $\PLambda_c^{1/2+}$)}
\begin{minipage}{\textwidth}
\begin{columns}
\column{2.5in}
\includegraphics[width=0.95\textwidth]{images/FF_paper.png}


\column{2.5in}

\includegraphics[angle=-90,width=0.95\textwidth]{images/FF.pdf}

\end{columns}
$\Rrightarrow$ So I check each of the three Form factors with calculations from \texttt{EvtGen} and they are in perfect agreement.\\



\end{minipage}
		\vspace*{2.cm}
\end{frame}


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


\begin{frame}[c]{Reweighing the MC}
\begin{minipage}{\textwidth}
{~}\\
Some maths:
\begin{columns}
\column{0.24\textwidth}
\begin{equation}
u^{(1)}=N\begin{pmatrix}
1\\
0\\
\frac{p_z}{E+m}\\
\frac{p_x+i p_y}{E+m}
\end{pmatrix} \nonumber
\end{equation}
\column{0.24\textwidth}
\begin{equation}
u^{(2)}=N\begin{pmatrix}
0\\
1\\
\frac{p_x-i p_y}{E+m}\\
\frac{-p_z}{E+m}
\end{pmatrix} \nonumber
\end{equation}

\column{0.24\textwidth}
\begin{equation}
\nu^{(1)}=N\begin{pmatrix}
\frac{p_x-i p_y}{E+m}\\
\frac{-p_z}{E+m}\\
0\\
1
\end{pmatrix} \nonumber
\end{equation}

\column{0.24\textwidth}
\begin{equation}
\nu^{(2)}=N\begin{pmatrix}
\frac{p_z}{E+m}\\
\frac{p_x+i p_y}{E+m}\\
1\\
0
\end{pmatrix} \nonumber
\end{equation}







\end{columns}

$\Rightarrow$ Should be the standard representation but in case I missed a minus sign let me know!
Now the gamma matrix:
\begin{tiny}
\begin{columns}



\column{0.19\textwidth}
\begin{equation}
\gamma^0=\begin{pmatrix}
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
0 & 0 & -1 & 0\\
0 & 0 & 0 & -1\\
\end{pmatrix} \nonumber
\end{equation}
\column{0.19\textwidth}
\begin{equation}
\gamma^1=\begin{pmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & -1 & 0 & 0\\
-1 & 0 & 0 & 0\\
\end{pmatrix} \nonumber
\end{equation}

\column{0.19\textwidth}
\begin{equation}
\gamma^2=\begin{pmatrix}
0 & 0 & 0 & -i\\
0 & 0 & i & 0\\
0 & i & 0 & 0\\
-i & 0 & 0 & 0\\
\end{pmatrix} \nonumber
\end{equation}



\end{columns}


\begin{columns}

\column{0.19\textwidth}
\begin{equation}
\gamma^3=\begin{pmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & -1\\
-1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
\end{pmatrix} \nonumber
\end{equation}


\column{0.19\textwidth}

\begin{equation}
\gamma^5=\begin{pmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
1 & 0 & 0 & 0\\
0 & 1 & 0 & 0\\
\end{pmatrix} \nonumber
\end{equation}

\end{columns}


\end{tiny}


\end{minipage}
		\vspace*{2.cm}
\end{frame}






\begin{frame}[c]{Lepton current}
\begin{minipage}{\textwidth}
{~}\\
$\Rrightarrow$ Now the lepton current can be written as:

\begin{equation}
l_{\mu} = \overline{\nu} (\gamma_{\mu} - \gamma_{\mu}\gamma_5) \mu \nonumber
\end{equation}
$\Rrightarrow$ There are two possibilities: Either you use for the lepton $u^{(1)}$ or $u^{(2)}$. Neutrino is always left-handed:
\begin{equation}
\nu=N\begin{pmatrix}
\frac{p_x-i p_y}{E+m}\\
1\\
-\frac{p_x-i p_y}{E+m}\\
-1
\end{pmatrix} \nonumber
\end{equation}
$\Rrightarrow$ In total we sum over two leptonic currents.


\end{minipage}
		\vspace*{2.cm}
\end{frame}






\begin{frame}[c]{Hadronic  current}
\begin{minipage}{\textwidth}
{~}\\
$\Rrightarrow$ Now the hadronic is a more complicated beast:

\begin{equation}
h_{\mu} = \overline{u} [[\gamma_{\mu} - \gamma_{\mu}\gamma_5][F_1/G_1(q^2) \gamma_{\mu} + F_2/G_2(q^2)
\frac{ p_\mu}{m_{\Lambda_Q}} +  F_3/G_3(q^2)] u \nonumber
\end{equation}
$\Rrightarrow$ Here I am showing you the simplest $\PLambda_b^{1/2+} \to \PLambda_c^{1/2+}$ transition.\\
$\Rrightarrow$  Others will heave a $\gamma_5$ and minuses in some places to invert in other cases.
$\Rrightarrow$  There will be 4 harmonic currents for $1/2+ \to 1/2\pm$ transition and 8 for the the $1/2+ \to 3/2-$ one.\\
$\Rrightarrow$  NB. the $1/2+ \to 3/2-$ will be described by the Rarita-Schiwnger spinor which is more complicated and didn't want to fit on the slide, so you will have to thrust me on this one :P



\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{Current-Current part}
\begin{minipage}{\textwidth}
{~}\\
Now we calculate all possible combinations:
\begin{equation}
M = \sum_{i=1}^{4/8} \sum_{j=1}^{2} h_i l_j \nonumber
\end{equation}

$\Rrightarrow$ And the probability:
\begin{equation}
P=M M^{\ast}  \nonumber
\end{equation}
$\Rrightarrow$ Now the main purpose of this study was tu study the impact of the form factors on our analysis, that is why I omitted all the constant as they will drop out in the ratio.



\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{Results of the reweighing}
\begin{minipage}{\textwidth}
{~}\\


\begin{columns}
\column{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{images/q2.png}

\column{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{{images/FF_var}.png}


\end{columns}


\begin{columns}
\column{0.5\textwidth}
\includegraphics[width=0.95\textwidth]{images/weights.png}

\column{0.5\textwidth}
$\Rrightarrow$ No strong dependence on the form factor!

\end{columns}


\end{minipage}
		\vspace*{2.cm}
\end{frame}


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

\begin{frame}[c]{Summary}
\begin{minipage}{\textwidth}
\begin{itemize}
\item Form factors implemented with all the algebraic structure of $V-A$ currents.
\item Working on matrix element computations.
\item Reweighing should follow.
\item Once that is ready we will test with of the discriminating variables are less form factor independent so they can be used in the selection.
\end{itemize}




\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{TO do}
\begin{minipage}{\textwidth}
{~}\\
\begin{itemize}
\item Validate the code:
\begin{itemize}
\item Validate the code as it still fresh.
\item Check the C++ implementation for $1/2-$ and $3/2-$.
\item Write the calculations in mathematica for cross-check.
\item Check with Danny EOS implementation.
\end{itemize}
\end{itemize}


\end{minipage}
		\vspace*{2.cm}
\end{frame}

\begin{frame}[c]{Selection}
\begin{minipage}{\textwidth}
{~}\\
\begin{itemize}
\item On my normal web page I have added the wrong sign combination: $\PLambdab \to \PLambda_c \mu^+$.
\item We will use them for the combinatorial background. 
\end{itemize}


\end{minipage}
		\vspace*{2.cm}
\end{frame}


\iffalse

\begin{frame}[c]{Selection}
\begin{minipage}{\textwidth}
{~}\\


\end{minipage}
		\vspace*{2.cm}
\end{frame}

\fi





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