ratio.tex

\section{\texorpdfstring{$\chi_{b1}$}{chib1} and \texorpdfstring{$\chi_{b2}$}{chib2} yields ratio}
\label{sec:ratio}

The ratio between $\chi_{b2}$ and $\chi_{b1}$ candidates is one of the
parameters in our fit model (Section~\ref{sec:chib:fit}). Nowadays this ratio
is estimated only in theoretical works. Recent \lhcb preliminary results were
obtained at \lhcb and \cms. \lhcb study  shows a good agreement with theory,
but \cms\cite{CMS-PAS-BPH-13-005} results are slightly different. 

In \Cref{sec:review} is shown that the theory
predicts~\cite{Likhoded:2012hw} the ratio $\chi_{c2}/\chi_{c1}$ as a function
of $p_T$, which is in agreement with experimental results (\Cref{fig:frac:ratio}).
Since the result for bottomonium ratio could be obtained by rescaling the
charmonium curve, the  \chibone and \chibtwo ratio can be  measured by the
following formula in specified transverse momentum intervals of $\Upsilon$:
\begin{equation}
    \frac{N_{\chibtwo}^{data}}{N_{\chibone}^{data}} = \frac{\sigma(\chibtwo)}{\sigma(\chibone)}
    \frac{Br(\chibtwo\to\Upsilon\gamma)}{Br(\chibone\to\Upsilon\gamma)}\frac{\eps_{\chibtwo}^{\gamma}}{\eps_{\chibone}^{\gamma}}
\label{eqn:mc_ratio}
\end{equation}
\noindent where $\sigma(\chibtwo) / \sigma(\chibone)$ is a ratio
from~\cite{Likhoded:2012hw}, the branching fractions $Br(\chi_{b1,2} \to \Upsilon \gamma)$ 
are known experimentally~(\Cref{tab:branching}) and reconstruction efficiencies 
$\eps_{\chi_{b1,2}}^{\gamma}$ are obtained in this study~(\Crefrange{tab:mc:eff:chib1_y1}{tab:mc:eff:chib3_y3}).

\begin{table}[H]
\caption{The branching fractions of radiative \chibOneP and \chibTwoP mesons
decays are known experimentally~\cite{PDG2012}}
\centering
\begin{tabular}{l}
$Br(\chiboneOneP \to \Y1S \gamma) = 33.9\% \pm 2.2\%$ \\
$Br(\chibtwoOneP \to \Y1S \gamma) = 19.1\% \pm 1.2\%$ \\
$Br(\chiboneTwoP \to \Y1S \gamma) = 9.2\% \pm 0.8\%$ \\
$Br(\chibtwoTwoP \to \Y1S \gamma)  = 7.0\% \pm 0.7\%$ \\
$Br(\chiboneTwoP \to \Y2S \gamma) = 19.9\% \pm 1.9\%$ \\
$Br(\chibtwoTwoP \to \Y2S \gamma) = 10.6\% \pm 2.6\%$ \\
\end{tabular}
\label{tab:branching}
\end{table}

The $N_{\chibone}^{data}/(N_{\chibone}^{data} + N_{\chibtwo}^{data})$ ratio is
estimated by~\Cref{eqn:mc_ratio} to be in the range between 0.4 and 0.7 for
\chibOneP and \chibTwoP decays. The ratio for \chibThreeP decays could not be
calculated because the branching fraction of these decays are unknown. In this
study this ratio is fixed to 0.5 and systematic uncertainty is assigned to this
decision. 

% \input{tables/ratio/lambda}