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\begin{document}
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\title{OBSERVATION OF THE $\Sigma_b$ BARYONS AT CDF}
\author{ J. PURSLEY on behalf of the CDF Collaboration }
\address{Department of Physics and Astronomy, 366 Bloomberg Center,\\
3400 N. Charles St., Baltimore, MD 21218, USA}
\maketitle\abstracts{
We present a measurement of four new bottom baryons
in proton-antiproton collisions with a center of
mass energy of $1.96$~TeV.
Using $1.1~\rm{ fb}^{-1}$ of data collected by the CDF II
detector, we observe four $\Lambda_b^0\pi^{\pm}$ resonances
in the fully reconstructed decay mode
$\Lambda_b^0 \to \Lambda_c^+ \pi^-$, where
$\Lambda_c^+ \to p K^- \pi^+$.
The probability for the background to produce a
similar or larger signal is less than $8.3\times 10^{-8}$,
corresponding to a signif\mbox{}icance of greater than
5.2~$\sigma$. We interpret these baryons as the
$\Sigma_b^{(*)\pm}$ states.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction\label{sec:intro}}
The Tevatron at the Fermi National Accelerator Laboratory
collides $p\bar{p}$ with a center of mass energy
of 1.96 TeV. The Collider Detector at Fermilab, or CDF,
experiment employs a general multipurpose detector~\cite{CDF}
to reconstruct particle physics events from these collisions.
With a $b$ hadron cross section of $\approx$ 50 $\mu$b
$\big(|\eta| < 1.0\big)$,~\cite{det_coord} CDF has collected a wealth of
experimental data on $b$ hadrons. Using this data, we
announce the f\mbox{}irst observation of the $\Sigma_b^{(*)\pm}$
baryons.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\Sigma_b^{(*)}$ Theoretical Predictions\label{sec:theory}}
Only one $b$ baryon has been previously
established, the ground state $\Lambda^0_b$, which
contains $b$, $u$, and $d$ quarks with the two
light quarks ($u$ and $d$) in a flavor antisymmetric diquark
state. CDF uses a two displaced track trigger to
select the decay of
$\Lambda^0_b \to \Lambda^+_c \pi^-$, with
$\Lambda^+_c \to p K^- \pi^+$ (inclusion of the respective
charge conjugate modes is assumed throughout this paper).
The two displaced track trigger requires two high $p_{\rm T}$
tracks displaced from the $p\bar{p}$ interaction point; in the
decay of $\Lambda_b^0$, the two tracks which satisfy the
requirements are primarily the pion from the $\Lambda_b^0$
decay and the proton from the $\Lambda_c^+$ decay.
Using 1.1 fb$^{-1}$ of data collected by the CDF II detector
between February 2002 and March 2006, CDF possesses the world's
largest sample of bottom baryons with $3180 \pm 60$ (stat.)
$\Lambda^0_b \to \Lambda^+_c \pi^-$
candidates. The reconstructed $\Lambda_b^0$
invariant mass distribution is shown in Fig.~\ref{fig:lb_mass}.
The next accessible $b$ baryons are the lowest lying
$\Sigma_b^{(*)}$ states, which decay strongly to $\Lambda^0_b$
baryons by emitting pions. The $\Sigma_b^{(*)+}$ baryons contain
one $b$ and two $u$ quarks, the $\Sigma_b^{(*)-}$ baryons contain one
$b$ and two $d$ quarks, and the $\Sigma_b^{(*)0}$ baryons contain
the $b$, $u$, and $d$ quarks. In the $\Sigma_b$ baryons, the
two light quarks are in a flavor symmetric diquark state, leading
to a doublet of baryons with $J^P=\frac{1}{2}^+$ ($\Sigma_b$)
and $J^P=\frac{3}{2}^+$ ($\Sigma^*_b$). Because $\Sigma_b^{(*)0}$
decays to $\Lambda_b^0\pi^0$ and the CDF II detector cannot
reconstruct neutral pions, we expect to observe only the charged
$\Sigma_b^{(*)\pm}$ states.
There is predicted to be a hyperf\mbox{}ine mass splitting
between the doublet states $\Sigma_b$ and $\Sigma_b^*$,
as well as a mass splitting between the
$\Sigma^{(*)-}_b$ and $\Sigma^{(*)+}_b$ states due to strong
isospin violation. Predictions for the $\Sigma_b^{(*)}$ masses
exist from heavy quark effective theories, non-relativistic and
relativistic potential models, $1/{\rm N}_c$ expansion, sum rules,
and lattice Quantum Chromodynamics calculations. These predictions
expect~\cite{ref:sib}
$m\big(\Sigma_b\big)-m\big(\Lambda^0_b\big)\sim 180-210$ MeV/$c^2$,
$m\big(\Sigma^*_b\big)-m\big(\Sigma_b\big)\sim 10-40$ MeV/$c^2$,
and $m\big(\Sigma^-_b\big)-m\big(\Sigma^+_b\big)\sim 5-7$ MeV/$c^2$.
The intrinsic width of $\Sigma_b^{(*)}$ baryons is dominated by
the P-wave one-pion transition, whose partial width depends on
the available phase space.~\cite{Korner:1994nh} For the
predicted range of $\Sigma_b^{(*)}$ masses, the intrinsic width
varies between $2$ and $20$ MeV/$c^2$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analysis Methodology\label{sec:analysis}}
We search for four resonant $\Lambda^0_b \pi^\pm$ states
consistent with theoretical predictions for $\Sigma_b$,
where $\Sigma_b$ now refers to both charged $J=\frac{1}{2}^+$
$\big(\Sigma^{\pm}_b\big)$ and $J=\frac{3}{2}^+$
$\big(\Sigma^{*\pm}_b\big)$ states. To minimize
the contribution of the mass resolution of each
$\Lambda^0_b$ candidate, the search is made for narrow
resonances in the mass difference distribution of
$Q = m\big(\Lambda^0_b \pi\big) - m\big(\Lambda^0_b\big) - m_\pi$.
Events are separated into ``$\Lambda^0_b \pi^-$'' and
``$\Lambda^0_b \pi^+$'' subsamples; $\Lambda^0_b \pi^-$
contains $\Sigma^{(*)-}_b$ and $\overline{\Sigma}^{(*)-}_b$
while $\Lambda^0_b \pi^+$ contains $\Sigma^{(*)+}_b$ and
$\overline{\Sigma}^{(*)+}_b$.
The $\Lambda^0_b$ candidate is combined with a prompt
pion, as the $\Sigma_b$ decays strongly at the primary
vertex of the $p\bar{p}$ collision. To perform an unbiased
optimization of the selection criteria, we use as a
background sample only those tracks far from the
expected $\Sigma_b$ signal region. From theoretical
predictions, the signal region is def\mbox{}ined as $30 < Q < 100$
MeV/$c^2$. The principle sources of background in the
$\Sigma_b$ $Q$ distribution are tracks from the hadronization
of prompt $\Lambda^0_b$ baryons and $B$ mesons reconstructed
as $\Lambda^0_b$ baryons, and combinatorial background.
%The underlying event tracks, the remaining debris of
%the $p\bar{p}$ collision, also contribute; as they cannot
%be separated from the hadronization tracks, the latter
%denotes the sum of the two.
The percentage of each background source in the
$\Sigma_b$ $Q$ distribution is f\mbox{}ixed from the $\Lambda^0_b$
invariant mass f\mbox{}it shown in Fig.~\ref{fig:lb_mass}, which is
89.4\% $\Lambda^0_b$ baryons, 7.3\% $B$ mesons, and
3.3\% combinatorial background.
The $Q$ distribution of each background component is
established before unblinding the signal region.
The high mass region above the $\Lambda^0_b \to \Lambda^+_c \pi^-$
signal in the $\Lambda^0_b$ mass distribution (Fig.~\ref{fig:lb_mass})
determines the combinatorial background.
Reconstructing $\bar{B}^0 \to D^+\pi^-$ data as $\Lambda^0_b \to
\Lambda^+_c \pi^-$ gives the background from $B$ hadronization tracks.
The largest background component, from $\Lambda^0_b$ hadronization tracks,
is obtained from a $\Lambda^0_b$ {\sc PYTHIA}~\cite{Sjostrand:2000wi}
Monte Carlo simulation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\Sigma_b^{(*)}$ Results\label{sec:results}}
After determining the background shape, we observe an
excess of events over the expected background in the
$\Sigma_b$ signal region.
The excess in the $\Lambda^0_b \pi^-$ subsample
is $118$ candidates over $288$ expected background candidates,
while in the $\Lambda^0_b \pi^+$ subsample the excess
is $91$ over $313$ expected background candidates.
The strength of the $\Sigma_b$ hypothesis is evaluated
using as a test statistic the likelihood ratio,
$LR \equiv L_{\rm alt} / L$, where $L$ is the f\mbox{}it
likelihood of the four $\Sigma_b$ signal hypothesis and
$L_{\rm alt}$ is the f\mbox{}it likelihood of an alternate hypothesis
such as the no signal hypothesis. Using simplistic Monte
Carlo samples of background fluctuations, we f\mbox{}ind the
probability of the no signal hypothesis to be less than
$8.3\times 10^{-8}$, corresponding to
a signal signif\mbox{}icance of greater than 5.2~$\sigma$.
The subsamples are modeled with a simultaneous unbinned maximum
likelihood f\mbox{}it comprising a signal for each expected $\Sigma_b$ state
plus the background.
Each signal consists of a non-relativistic Breit-Wigner
distribution convoluted with a double Gaussian model of the detector
resolution. The intrinsic width of the Breit-Wigner
is computed from the available phase space given the central
location of the signal. Due to low statistics, the constraint
$m(\Sigma^{*+}_b)-m(\Sigma^+_b) = m(\Sigma^{*-}_b)-m(\Sigma^-_b)
\equiv \Delta(\Sigma^*_b)$ is added. The $\Sigma_b$ signal f\mbox{}it to
data, which has a $\chi^2$ f\mbox{}it probability of 76\% in the range $Q\in[0, 200]$
MeV/c$^2$, is shown in Fig.~\ref{fig:sigmab}.
%Systematic uncertainties are evaluated for the CDF mass scale and
%for assumptions made in the f\mbox{}it to data.
The majority of the systematic uncertainty on
the yield measurement is due to poor knowledge of the $\Lambda_b$
hadronization background,
while the majority of the systematic uncertainty on the mass
measurement is due to the CDF mass scale uncertainty.
The f\mbox{}inal results for the yields are
$N(\Sigma_b^+) = 32^{+13}_{-12}~\mbox{(stat.)} ^{+5}_{-3}~\mbox{(syst.)}$,
$N(\Sigma_b^-) = 59^{+15}_{-14}~\mbox{(stat.)} ^{+9}_{-4}~\mbox{(syst.)}$,
$N(\Sigma_b^{*+}) = 77^{+17}_{-16}~\mbox{(stat.)} ^{+10}_{-6}~\mbox{(syst.)}$, and
$N(\Sigma_b^{*-}) = 69^{+18}_{-17}~\mbox{(stat.)} ^{+16}_{-5}~\mbox{(syst.)}$.
The signal locations are
%\begin{center}
$Q(\Sigma_b^+) = 48.5^{+2.0}_{-2.2}~\mbox{(stat.)}^{+0.2}_{-0.3}~\mbox{(syst.)}$ MeV/$c^2$,
$Q(\Sigma_b^-) = 55.9 \pm 1.0~\mbox{(stat.)} \pm 0.2~\mbox{(syst.)}$ MeV/$c^2$, and
$\Delta(\Sigma_b^*) = 21.2^{+2.0}_{-1.9}~\mbox{(stat.)} ^{+0.4}_{-0.3}~\mbox{(syst.)}$ MeV/$c^2$.
%\end{center}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary\label{sec:summary}}
The $\Lambda^0_b\pi^\pm$ resonant states observed in
$1.1$~fb$^{-1}$ of CDF II data are consistent with the
lowest lying charged $\Sigma_b$ baryons, and the observed
properties are in agreement with theoretical predictions.
Using the CDF II measurement~\cite{ref:lbmass} of
m($\Lambda^0_b$) = $5619.7 \pm 1.2$ (stat.) $\pm 1.2$ (syst.)
MeV/$c^2$, the masses of each state are
%
\begin{center}
m$(\Sigma_b^+) = 5807.8^{+2.0}_{-2.2}~\mbox{(stat.)} \pm 1.7~\mbox{(syst.)}$ MeV/$c^2$,\\
m$(\Sigma_b^-) = 5815.2 \pm 1.0~\mbox{(stat.)} \pm 1.7~\mbox{(syst.)}$ MeV/$c^2$,\\
m$(\Sigma_b^{*+}) = 5829.0^{+1.6}_{-1.8}~\mbox{(stat.)} ^{+1.7}_{-1.8}~\mbox{(syst.)}$ MeV/$c^2$,\\
m$(\Sigma_b^{*-}) = 5836.4 \pm 2.0~\mbox{(stat.)} ^{+1.8}_{-1.7}~\mbox{(syst.)}$ MeV/$c^2$.
\end{center}
This is the f\mbox{}irst observation of the lowest lying
charged $\Sigma_b$ baryons.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figures
\begin{figure}[p]
\begin{center}
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\vskip 2.5cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
\psfig{figure=BaselineBlindedData_massFit.eps,height=3.0in}
\caption{Unbinned maximum likelihood f\mbox{}it to the reconstructed
invariant mass of $\Lambda^0_b \to \Lambda^+_c \pi^-$
candidates. The fully reconstructed $\Lambda_b^0$
modes (such as $\Lambda^0_b \to \Lambda^+_c \pi^-$
and $\Lambda^0_b \to \Lambda^+_c K^-$) are not shown
separately on the f\mbox{}igure.}
\label{fig:lb_mass}
\end{center}
\end{figure}
\begin{figure}[p]
\begin{center}
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\vskip 2.5cm
%\rule{5cm}{0.2mm}\hfill\rule{5cm}{0.2mm}
%\psfig{figure=SigmaB_Points_SmallRange.eps,height=4.0in}
%\psfig{figure=PRL_combined_fig.eps,height=4.0in}
\psfig{figure=PRL_combined_mev.eps,height=4.0in}
\caption{Unbinned maximum likelihood f\mbox{}it to the $\Sigma_b$
$Q$ distributions. The top plot shows the
$\Lambda^0_b \pi^+$ combinations, while the
bottom plot shows the $\Lambda^0_b \pi^-$ combinations.
The insets show the expected background plotted on the data for
$Q \in$ [0, 500] MeV/$c^2$, while the $\Sigma_b$ signal
f\mbox{}it is shown on a reduced range of $Q \in$ [0, 200] MeV/$c^2$.}
\label{fig:sigmab}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgments}
We thank T.~Becher, A.~Falk, D.~Pirjol, J.~Rosner, and D.~Ebert
for useful discussions.
%
We also thank the Fermilab staf\mbox{}f and the technical staf\mbox{}fs of the
participating institutions for their vital contributions. This work
was supported by the U.S. Department of Energy and National Science
Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the
Ministry of Education, Culture, Sports, Science and Technology of Japan;
the Natural Sciences and Engineering Research Council of Canada; the
National Science Council of the Republic of China; the Swiss National
Science Foundation; the A.P. Sloan Foundation; the Bundesministerium f\"ur
Bildung und Forschung, Germany; the Korean Science and Engineering Foundation
and the Korean Research Foundation; the Particle Physics and Astronomy
Research Council and the Royal Society, UK; the Institut National de
Physique Nucleaire et Physique des Particules/CNRS; the Russian Foundation
for Basic Research; the Comisi\'on Interministerial de Ciencia y
Tecnolog\'{\i}a, Spain; the European Community's Human Potential Programme
under contract HPRN-CT-2002-00292; and the Academy of Finland.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{References}
\begin{thebibliography}{99}
\bibitem{CDF} D.~Acosta {\it et al.}~(CDF Collaboration),
\Journal{\PRD}{71}{032001}{2005}.
\bibitem{det_coord} The CDF II detector uses a cylindrical coordinate
system with the $z$-axis along the proton beam direction.
The pseudorapidity $\eta$ is def\mbox{}ined as $\tanh^{-1}(\cos \theta)$.
Transverse momentum, \mbox{$p_{\rm{T}}$}, is the component
of the particle's momentum in the $(x, y)$ plane.
\bibitem{ref:sib} CDF Collaboration, CDF Public Note 8523 (2006)
and references therein,\\
http://www-cdf.fnal.gov/physics/new/bottom/060921.blessed-sigmab.
\bibitem{Korner:1994nh}
J.~G.~K\"{o}rner, M.~Kr\"{a}mer, and D.~Pirjol,
{\em Prog.\ Part.\ Nucl.\ Phys.\ } {\bf 33}, 787 (1994).
\bibitem{Sjostrand:2000wi}
T.~Sj\"{o}strand, P.~Eden, C.~Friberg, L.~Lonnblad, G.~Miu, S.~Mrenna,
and E.~Norrbin, {\em Comput.\ Phys.\ Commun.\ } {\bf 135}, 238 (2001).
\bibitem{ref:lbmass} D.~Acosta {\it et al.}~(CDF Collaboration),
\Journal{\PRL}{96}{202001}{2006}.
\end{thebibliography}
\end{document}
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