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 provides a better resolution, since the energy
resolution using the tracking stations is better than the resolution obtained
by the calorimeter system. This is due to the fact that the \epm measurements
provide a better precision on the photon momentum and energy than in the
unconverted case. The conversion is required to happen before the magnet in
order to be reconstructed with tracks. Furthermore, if the photon converts too
early, the \epm has more chance to radiate energy, which leads to the worse
track reconstruction and smaller energy resolution. Therefore, only photons
converting before the magnet and after the VELO are used. So, in this study
unconverted photons are used since it provides much larger statistic and allows
to analyze more decays in wide $\Upsilon$ transverse momentum ranges.

%% ============================================================================
\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 the
transverse momentum greater than 600 \mevc. To further suppress the 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.

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

\input{tables/chib/selection_photons}

To separate decays by different $\Upsilon$ channels the cut on dimuon mass is
applied as shown in Table~\ref{tab:chib:selection:window}:

\input{tables/chib/selection_window}

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

The $\Upsilon$ selection cuts
~(Table~\ref{tab:upsilon:selection:study:summary}), cuts  on $\gamma$
~(Table~\ref{tab:chib:selection:photons}) and limit on dimuon mass
~(Table~\ref{tab:chib:selection:window}) are used for obtaining \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 efficient way to obtain \chib signal yields is by fitting event candidates
in the distribution of invariant mass difference $m(\mumu\gamma) - m(\mumu)$.
In this case the bias and resolution effects from the $\Upsilon$ reconstruction
are suppressed. For clearness, the PDG mass of the corresponding $\Upsilon$
particles is added to the mass difference value in each plot.

The $\chi_b(jP)$ (j=1,2,3) signals are composed of three parts: $\chi_{b0}(jP)$,
$\chi_{b1}(jP)$, $\chi_{b2}(jP)$. The $\chi_{b0}$ signals is hard to detect
because it has a low radiative branching ratio in comparison with the other two
parts. 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}

The CB is similar to 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. 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
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 $k$ 
is the ratio between the resolution of $\chi_{b1}$ and $\chi_{b2}$ signals.
This parameter is equal to $1.05$ for $\chi_{b1,2}(1P)$ signals and equal to 1
for $\chi_{b1,2}(2,3P)$ signals.

To reduce fit
errors, the width of each CB function ($\sigma$) is fixed to the value obtained
from simulation~(Section~\ref{sec:mc}).

As was said in the introduction (\Cref{sec:introduction}) the \chibThreeP  was
recently observed and the mass of this meson is not well known.
So in this study this mass is fixed to 10.508\gevcc, which was measured in
\Cref{sec:chib:ups3s:fit}.

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

If you don't take in account $\chi_{b0}$ decays, the \Y1S could 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 a free parameter 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 a difference between PDG masses of
\chiboneTwoP and \chiboneOneP. The $\Delta  m_{\chi_{b1}(3P)}$ is a difference
between masses of \chiboneThreeP and \chiboneOneP, where the mass of
\chiboneThreeP was taken from measurement conducted in this
study~(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 event in a bin. If a fit has most pull values  are normally
distributed around 0 in the range between -1 and 1, it is indicated as a good
fit.


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

\Cref{tab:chib:ups1s:nominal} shows that the measured \chiboneOneP mass
has a nice agreement with the PDG value $9892.78 \pm 0.26 \pm 0.31 \mevcc$. In
the further analysis this mass was fixed to 9.887 \gevcc which which was
measured on combined 2011 and 2012 dataset in $6<p_T^{\Y1S}<40$ range.


\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 fluctuation for
the other states. \Cref{tab:chib:ups1s:fits} in Appendix summarizes the
obtained results.

\input{pics/chib/ups1s_yields}

The momentum scaling correction was applied, but a smooth variation of the
\chiboneOneP mass is seen 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.


The order of the background polynomial in~\Cref{eq:bernstein} is 3 for all
intervals of \Y2S transverse momentum.


\Cref{fig:chib:ups2s:nominal} shows the mass distribution in 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 further analysis this mass was fixed to 10.250\gevcc, which was measured
in $18 < p_T^{\Y2S} < 40 \gevc$ interval in merged 2011 and 2012 data, 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. 
The order of the background polynomial in \Cref{eq:bernstein} is 2.
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}).
Therefore, it could be stated that the \chiboneThreeP mass is $10{,}508 \pm
2\stat \pm 8\syst \mevc$. This result is in agreement with the recent
unpublished
\lhcb study with converted photons~\cite{edwige} where the \chiboneThreeP mass is
$10{,}509 \pm 3\stat{}^{+5.3}_{-2.9}\syst$. The difference with
$\chi_{b1,2}(3P)$ mass barycenter reported by \atlas~\cite{Aad:2011ih} ($10{,}530 \pm 5\stat
\pm 17\syst \mevcc$) and D0~\cite{Abazov:2012gh}  ($10{,}551 \pm 14\syst \pm 17\stat$)
is~$\sim{}1.5\sigma$.


\input{pics/chib/ups3s_m3p}

In this study the mass was fixed to 10.508 \gevcc which was measured 
on combined 2011 and 2012 data with $\lambda$ fixed to 0.5 in the fit.

\input{pics/chib/ups3s_nominal}