chib.tex

\section{\texorpdfstring{$\chi_b$}{xb} signal extraction}
\label{sec:chib}

In this study, the photon in \chib decay is measured by the calorimeter system.
Another approach is to look at photons that convert to an electron-positron
(\epem) pair. Converted photons provide a better invariant mass resolution and would 
allow to separate mass peaks due to close resonances, since the 
\epm momentum 
resolution obtained from the tracking stations is better than the photon energy resolution obtained
by the calorimeter system. However, conversions should be required to happen before the magnet in
order to reconstruct the charged tracks. Furthermore, if the photon converts too
early, the \epm has more chance to radiate energy, which leads to worse
track reconstruction and worse energy resolution. Therefore, only photons
converting before the magnet and after the VELO should be used, which severely limits 
the size of the available sample and the decays which can be analyzed. In this study 
unconverted photons are used, in order to obtain a much larger data sample and 
analyze more decays in a wide range of $\Upsilon$ transverse momentum. 

%% ============================================================================
\subsection{Selection}
\label{sec:chib:selection}

The selected $\Upsilon$ candidates  are combined with photon candidates to form
\chib candidates. Well reconstructed photons are selected by requiring their 
transverse momentum to be greater than 600 \mevc. To further suppress background, 
the cosine of the angle of the photon direction in the center-of-mass of the
$\mumu\gamma$ system with respect to the momentum of this system, is required to
be greater than zero. An additional loose cut on the photon confidence level is
required to be greater than 0.01. This confidence level is computed starting from the distributions 
of calorimetric variables which are sensitive to photons, by computing likelihoods under different particle 
hypotheses, and taking   
the ratio of the likelihood for a photon hypothesis divided by the sum of likelihoods for all hypotheses. 

The criteria for event selection with a reconstructed photon are summarized in
Table~\ref{tab:chib:selection:photons}:

\input{tables/chib/selection_photons}

To separate decays into different $\Upsilon$ channels, cuts on dimuon invariant mass are 
applied as shown in Table~\ref{tab:chib:selection:window}:

\input{tables/chib/selection_window}

To avoid \Y2S and \Y3S mixing contamination, the mass ranges of the \Y2S and \Y3S are
asymmetric with respect to the nominal masses.

The $\Upsilon$ selection cuts
~(Table~\ref{tab:upsilon:selection:study:summary}), the cuts  on $\gamma$
~(Table~\ref{tab:chib:selection:photons}) and dimuon mass  
~(Table~\ref{tab:chib:selection:window}) are used to obtain \chib yields
using a fit model which is described in the next section.


%% ============================================================================
\subsection{Fit model}
\label{sec:chib:fit}

This section describes the common properties of the fit model that is used for
obtaining yields in each of the \chib decays. The results of the fits are
given in the following sections.

The \chib signal yields are obtained by fitting event candidates
in the distribution of invariant mass difference $m(\mumu\gamma) - m(\mumu)$.
In this case any biases and resolution effects from the $\Upsilon$ reconstruction
are cancelled at first order. For clearness, the PDG mass of the corresponding $\Upsilon$
particle is added to the mass difference value in each plot.

The $\chi_b(jP)$ (j=1,2,3) signals are the sum of three contributions, due to $\chi_{b0}(jP)$,
$\chi_{b1}(jP)$, $\chi_{b2}(jP)$. The $\chi_{b0}$ meson is difficult to detect
because it has a low radiative branching ratio in comparison with the other two
mesons. So the $\chi_{b0}$ states were excluded from this study and the fit
model.

To determine the $\chi_b$ signal yields, an unbinned maximum likelihood fit to
$m(\mumu\gamma) - m(\mumu)$ has been performed. The signal has been modeled with
a sum of single-sided Crystal Ball (CB)  functions. The background is parameterized with a
product of an exponential function and a linear combination of basic Bernstein
polynomials~\cite{Phillips:2003} with non-negative coefficients $c_{i}^2$:

\begin{equation}
\label{eq:bernstein}
{\mathscr B}_{n}(x) = e^{-\tau x} \times \sum_{i=0}^{n} c_{i}^2 {\mathscr B}_{n}^{i}(x)
\end{equation}
Such combination results in a smooth and non-negative function that can be used
as a PDF.

The CrystalBall function can be written in the following form:

\begin{equation}
CB(x) = N \times
\begin{cases}
\frac{1}{\sqrt{2\pi\sigma}}\exp(-\frac{{(x-\mu)}^2}{2\sigma^2}) & \text{, if $\frac{x-\mu}{\sigma} > -\alpha$} \\
\frac{1}{\sqrt{2\pi\sigma}}{(\frac{n}{|\alpha|})}^n \exp(-\frac{|\alpha|^2}{2}){(\frac{n}{|\alpha|}-|\alpha|-\frac{x-\mu}{\sigma})}^{-n} & \text{, otherwise}
\end{cases}
\label{eq:cb}
\end{equation}

As already mentioned, the CB is similar to a gaussian distribution, but has an asymmetric tail. This
function has five parameters: N, $\mu$, $\sigma$, $\alpha$ and $n$, where
parameters $\mu$ and $\sigma$ have the same meaning as for gaussian. Parameters
$\alpha$ and $n$ describe the tail behavior: $\alpha$ controls the tail start
and $n$ corresponds to the decreasing power of the tail. 
%%% CB WHAT DOES THIS MEAN?!? All three parameters
%%%                                                             $N$, $\alpha$ and $n$ contribute to the amplitude.

The number of CrystalBall functions and the order of the polynomial depend on
the decay under study and are described in the section corresponding to that
specific decay.


The $\alpha$ and $n$
parameters of CB are fixed to the values obtained from simulation~(\Cref{sec:mc})
and are shown in~\Cref{tab:chib:fit:tail}:


\begin{table}[H]
\caption{\small   The $\alpha$ and $n$ parameters of CB functions.}
\centering
\begin{tabular}{lrr}
\toprule
Signal & $\alpha$ & $n$ \\
\midrule
$\chi_{b1,2}(1,2P)$ & -1.1 & 5 \\
$\chi_{b1,2}(3P)$ & -1.25 & 5 \\
\bottomrule
\end{tabular}
\label{tab:chib:fit:tail}
\end{table}

Due to the small mass difference between $\chi_{b2}(jP)$ and $\chi_{b1}(jP)$
(j=1,2,3) states and the insufficient detector resolution, it is not possible
to fit the $\chi_{b1}$ and $\chi_{b2}$ states by two independent CB functions.
Thus, the mean, width and yield values of  $\chi_{b1}$ and $\chi_{b2}$ signals
are linked together by the following constraints:

\begin{equation}
  \begin{aligned}
\mu_{\chi_{b2}(jP)} = \mu_{\chi_{b1}(jP)} + \Delta m_{\chi_{b1,2}(jP)}^{PDG} \text{, j = (1,2)}\\
\mu_{\chi_{b2}(3P)} = \mu_{\chi_{b1}(3P)} + \Delta m_{\chi_{b1,2}(3P)}^{theory} \\
\sigma_{\chi_{b2}} = k \sigma_{\chi_{b1}}\\
N_{\chi_{b2}} = \frac{(1-\lambda)}{\lambda} N_{\chi_{b1}}
  \end{aligned}
\end{equation}

\noindent where $\Delta m_{\chi_{b1,2}(jP)}^{PDG}$ is the corresponding PDG
mass difference which is fixed in the fit; $\Delta
m_{\chi_{b1,2}(3P)}^{theory}$ is fixed to the theoretical predicted mass
difference in 12\mevcc~\cite{Motyka:1997di}. The $\lambda$ parameter depends on
the $p_T(\Upsilon)$ range and is fixed to the value that is based on the
theoretical prediction discussed in Section~\ref{sec:ratio}. The parameter $k$
is the ratio between the resolution of $\chi_{b1}$ and $\chi_{b2}$ signals.
This parameter is obtained from simulation, and is fixed to $1.05$ for
$\chi_{b1,2}(1P)$ signals and fixed to 1 for $\chi_{b1,2}(2,3P)$ signals.

The width of each CB function ($\sigma$) is fixed to the value obtained from
simulation~(Section~\ref{sec:mc}) in each transverse momentum bin, in order to
improve fit convergence and reduce uncertainties.

As already mentioned in the introduction (\Cref{sec:introduction}), the \chibThreeP  was
recently observed, but the mass of this meson was not precisely measured. 
Section \Cref{sec:chib:ups3s:fit} presents a determination of the \chibThreeP mass, which 
was consequently fixed to the measured value of 10.508\gevcc in these studies. 

%% ============================================================================
\subsection{\texorpdfstring{\chib}{xb} yields in
\texorpdfstring{$\chib \to \Y1S \gamma$}{xb -> Y\(1S\) gamma } decays}
\label{sec:chib:ups1s:fit}

If $\chi_{b0}$ decays are neglected, the \Y1S can be produced in radiative decays
of six \chib particles: $\chi_{bi}(jP) \to \Y1S \gamma$ (i=1,2; j=1,2,3). So
the sum of six CB functions is used to determine \chib signals in these decays.
The mass of \chiboneOneP ($\mu_{\chiboneOneP}$) is taken as free parameter in the fit, and  other
parameters are constrained by:

\begin{equation}
  \begin{aligned}
\mu_{\chiboneTwoP} = \mu_{\chiboneOneP} + \Delta m_{\chi_{b1}(2P)}^{PDG} \\
\mu_{\chiboneThreeP} = \mu_{\chiboneOneP} + \Delta m_{\chi_{b1}(3P)}, \\
  \end{aligned}
\end{equation}
\noindent where $\Delta m_{\chi_{b1}(2P)}^{PDG}$ is the difference between the PDG masses of
\chiboneTwoP and \chiboneOneP. The $\Delta  m_{\chi_{b1}(3P)}$ parameter is the difference
between the masses of \chiboneThreeP and \chiboneOneP, where the mass of
\chiboneThreeP was taken from the measurement performed in this 
thesis~(Section~\ref{sec:chib:ups3s:fit}). The parameters  $\Delta
m_{\chi_{b1}(2P)}^{PDG}$ and $\Delta  m_{\chi_{b1}(3P)}$ are fixed in the fit.

The order of the background polynomial in~\Cref{eq:bernstein} depends on the
$p_{T}^{\OneS}$ interval and is given in~\Cref{tab:chib:ups1s:fit:order}.

\input{tables/chib/ups1s_fit_order}

The fit was performed in the mass interval from  9.77 \gevcc to 10.89 \gevcc.
Figure~\ref{fig:chib:ups1s:fit:nominal} shows the mass distribution along with
the pull distribution in the transverse momentum range $14 < p_T^{\OneS} < 40
\gevc$. In this range the fit has the lowest relative error of signal yields. 
Table~\ref{tab:chib:ups1s:nominal} details the corresponding fit parameters.

The pull is the residual divided by the error:
\begin{equation}
\label{eq:pull}
Pull = \frac{N_{data} - N_{model}}{\sqrt{N_{data}}},
\end{equation}
\noindent where $N_{model}$ is the expected number of events in a bin from
the fit function and $\sqrt{N_{data}}$ is the statistical uncertainty on the
number of events in a bin. Pull values  for good fits are normally
distributed around zero, with a standard deviation of 1. 


\input{pics/chib/ups1s_nominal}
\input{tables/chib/ups1s_nominal}

\Cref{tab:chib:ups1s:nominal} shows that the measured \chiboneOneP mass
nicely agrees with the PDG value $9892.78 \pm 0.26 \pm 0.31 \mevcc$. In
the following, this mass was fixed to 9.887 \gevcc which is the value 
measured on the combined 2011 and 2012 datasets in the range $6<p_T^{\Y1S}<40\gevc$.


\Cref{fig:chib:ups1s:yields} illustrates the number of signal events as
a function of $\Y1S$ transverse momentum. The yields
normalized by bin size and luminosity are shown in
\Cref{fig:chib:ups1s:yields_scaled}. The \chibOneP and \chibThreeP yields
are smoothly decreasing functions of $p_T^{\Y1S}$, as expected. Differences between 7 and 8\tev
data, due to different production cross sections, can be seen for the
$\chi_b(1P)$ state, while they are washed out by statistical fluctuations for
the other states. The unexpected difference for $\chibTwoP$ yields in the lowest
$p_T$ bin is due to insufficient fit in this region (see~\Cref{fig:chib:ups1s:fits2011,fig:chib:ups1s:fits2012}).
\Cref{tab:chib:ups1s:fits} in Appendix summarizes the
obtained results.

\input{pics/chib/ups1s_yields}

Even though a correction on the momentum scale was applied on data, a smooth variation of the
\chiboneOneP mass is observed as a function of transverse momentum (see
Figure~\ref{fig:chib:ups1s:mean}). This effect can be explained by the unknown
ratio between the number of \chiboneOneP and \chiboneTwoP candidates.
Figure~\ref{fig:chib-1s:m1p} shows how the measured mass depends on this ratio
($\lambda$ parameter). A systematic uncertainty is assigned to this effect.

\input{pics/chib/ups1s_mean}
\input{pics/chib/ups1s_m1p}


%% ===========================================================================
\subsection{\texorpdfstring{\chib}{xb} yields in
    \texorpdfstring{$\chib \to \Y2S \gamma$}{xb --> Y(2S) gamma} decays}
\label{sec:chib:ups2s:fit}

The fit was performed in the mass interval from 10.16 \gevcc to 11.04 \gevcc.
The  \chiboneTwoP peak width depends on $p_T^{\Y2S}$ interval and is fixed to
the value obtained from simulation (Section~\ref{sec:mc}) without any scaling.
The \chiboneThreeP peak width is fixed to \chiboneTwoP peak width scaled by
1.65, as observed on simulation. 


A third order polynomial for the background (described in \Cref{eq:bernstein}) is used 
for all intervals of \Y2S transverse momentum.


\Cref{fig:chib:ups2s:nominal} shows the mass distribution in the transverse
momentum range $18 < p_T^{\TwoS} < 40 \gevc$. \Cref{tab:chib:ups2s:nominal}
details the corresponding fit parameters.

\input{pics/chib/ups2s_nominal}
\input{tables/chib/ups2s_nominal}

\Cref{tab:chib:ups2s:nominal} shows that the measured \chiboneTwoP mass is
about 5 \mevcc less than the PDG value $10255.46  \pm 0.22 \pm 0.50 \mevcc$.
The same difference is also observed in the smaller $p_T^{\Y2S}$ ranges
(Figure~\ref{fig:chib:ups2s:mean}).
In the following analysis this mass was fixed to 10.250\gevcc, which was measured
in the $18 < p_T^{\Y2S} < 40 \gevc$ interval on the sum of 2011 and 2012 datasets, and the
systematic uncertainty on the results due to this assumption has been
determined.

\input{pics/chib/ups2s_mean}

\Cref{fig:chib:ups2s:yields} shows the number of signal events as a function
of $p_T^{\Y2S}$. \Cref{tab:chib:ups2s:fits} in Appendix summarizes the
obtained results.

\input{pics/chib/ups2s_yields}

\Cref{fig:chib:ups2s:yields_scaled} shows the yields normalized by the bin
size and the luminosity. Both \chibTwoP and \chibThreeP yields are smoothly
decreasing functions of $p_T^{\Y2S}$, as expected.

The dependence of the $\chi_{b1}(2P)$ fitted mass in bins of $p_T^{\Y2S}$ is
shown in Figure~\ref{fig:chib:ups2s:mean}.


%% ===========================================================================
\subsection{\texorpdfstring{\chib}{xb} yields in
    \texorpdfstring{$\chib \to \Y3S \gamma$}{xb --> Y(3S) gamma} decays}
\label{sec:chib:ups3s:fit}

The fit was performed in the mass interval from  10.440 to 10.760 \gevcc. 
A second order polynomial for the background (described in Equation 5.3) is used.
Due to the large fluctuations in the background, the
parameters of this component were fixed.


\Cref{fig:chib:ups3s:nominal} shows the mass distribution in the
transverse momentum range $27<p_T^{\Y3S}<40\gevc$.
\Cref{tab:chib:ups3s:nominal} details the corresponding fit parameters.

\input{pics/chib/ups3s_nominal}
\input{tables/chib/ups3s_nominal}

A good resolution on the \chiboneThreeP mass is observed, so this decay can be
used for \chiboneThreeP mass estimation. \Cref{fig:chib-3s:m3p} shows how the
measured \chiboneThreeP mass depends on the \chiboneThreeP and \chibtwoThreeP
yields ratio ($\lambda$ parameter), which is unknown~(\Cref{sec:ratio}).
The \chiboneThreeP mass is measured to be $10{,}508 \pm
2\stat \pm 8\syst \mevc$, where the combined 2011 and 2012 datasets are used, 
the central value has been obtained by setting $\lambda = 0.5$ in the fit and the 
systematic error takes into account the uncertainties on  
the $\lambda$ parameter and the mass difference between \chiboneThreeP and \chibtwoThreeP. 
This result is in agreement with a recent unpublished
\lhcb study with converted photons, where the \chiboneThreeP mass is
$10{,}510 \pm 3\stat{}^{+4.4}_{-3.4}\syst$. This result is also compatible within~$\sim{}1.5\sigma$ with 
the $\chi_{b1,2}(3P)$ mass barycenter reported by \atlas~\cite{Aad:2011ih} ($10{,}530 \pm 5\stat
\pm 17\syst \mevcc$) and D0~\cite{Abazov:2012gh}  ($10{,}551 \pm 14\syst \pm 17\stat$)


\input{pics/chib/ups3s_m3p}
% \input{pics/chib/ups3s_nominal}