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\newcommand{\qqbar}{q\bar{q}}
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\def\hww{$H\rightarrow W^+ W^-$}
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\begin{document} \title{Standard Model Higgs Searches at the Tevatron} 
\author{Ralf Bernhard %\\ on behalf of the D\O~and CDF collaborations}
}
\institute{Physikalisches Institut, %
Albert-Ludwigs Universit\"at Freiburg\\ 
}
\maketitle
\begin{abstract}
The latest searches for the Standard Model Higgs boson at a centre-of-mass energy of $\sqrt{s}=$ 1.96 TeV with the D\O\ and the CDF detectors at the Fermilab Tevatron collider are presented. For the first time since the LEP experiments the sensitivity for a Standard Model Higgs boson has been reached at a Higgs boson mass of 170 GeV/c$^2$.
\end{abstract}

\section{Introduction}\label{sec:introduction}
In the Standard Model (SM) of particle physics the Higgs mechanism is responsible
for breaking electroweak symmetry, thereby giving mass
to the $W$ and $Z$ bosons. It predicts the existence of a heavy scalar boson, the
Higgs boson, with a mass that can not be predicted by the SM. 
Direct searches for the Higgs Boson were performed at the LEP experiments in the process $e^{+}e^{-}\rightarrow ZH$ with a centre of mass energy of 206.6 GeV. A direct mass limit at $m_{H}$ $>$ 114.4 GeV/c$^2$~\cite{LEPlimit} was set at the 95\% confidence level (CL)\footnote{All limits given
in this paper are at $95\%$ CL}. This limit is slightly below the maximum available kinematic limit due to a small excess observed in the LEP data.

Indirect limits have been placed on the Higgs boson mass by the LEP, SLD and Tevatron experiments from electroweak precision measurements \cite{EWconst}. The main contribution to these indirect constraints
from the Tevatron experiments, D\O\ and CDF, are the measurements of the $W$ Boson and top quark
masses~\cite{}. The dependence of the Higgs
mass on these measurements is shown in Figure~\ref{fig-BBplot} on the left and 
the Higgs mass dependence on the measured  electroweak precision parameters in Figure~\ref{fig-BBplot} on the right. The SM fit yields a best value of $m_H=84^{+34}_{-26}$~GeV/c$^2$~\cite{LEPfit}. The upper limit on the Higgs
mass at $95\%$ CL is $m_H<154$~GeV/c$^2$. If the direct mass limit is also taken into account this limit is increased to $m_{H} < $ 185 GeV/c$^2$.

\begin{figure}[t]
\begin{center}
\includegraphics[width=.48\textwidth]{figures/topW_sum08.eps}
\includegraphics[width=.48\textwidth]{figures/BBplot_sum08.eps}
\end{center}
\caption{ 
Constraints on the Higgs mass from precision top and W mass measurements (left) and 
fit for the Higgs Mass from the W data showing the direct search LEP limit (right)\label{fig-BBplot}}
\end{figure}


\section{Higgs Searches at the Tevatron}
\label{sec:higgsattevatron}

The Tevatron experiments CDF~\cite{cdfrunII} and D\O~\cite{d0runII} search for direct Higgs boson production in the mass range
above the LEP limit using $\ppbar$ collisions at $\sqrt{s}=1.96$~TeV.
The relevant processes at these energies are associated Higgs production 
($qq'\to WH$, $q\bar{q}\to ZH$) and gluon fusion ($gg\to H$). 
Typical cross-sections are $\sigma \simeq 0.7-0.15$~pb for gluon fusion
and $\sigma \simeq 0.2-0.02$~pb for associated production at Higgs masses
in the range $115-200$~GeV/c$^2$. 

The Higgs boson predominantly decays into $b\bar{b}$ quark pairs in the low mass range below $135$~GeV/c$^2$. 
Hence the signal in the $gg\to H$ channel is overwhelmed by multi-jet background.
This makes the  process $gg\to H$  therefore not a viable
search channel at low Higgs boson masses. The $WH$ and $ZH$ channels, where the vector boson
decays into leptons, have much lower cross-sections
but the lepton tag from the decay of the $W\to \ell\nu$ or $Z\to\ell\ell$ and 
selections on missing transverse energy from the neutrino in the
decays $W\to \ell\nu$ or $Z\to\nu\nu$  
help to reduce the background significantly.

At higher masses, around $m_H=165$~GeV/c$^2$, the Higgs boson will predominantly
decay into $WW$ pairs. Leptons from the decays of the $W$ bosons and the missing
transverse energy are used to reject background, making the channel $gg\to
H\to WW$ the most promising search channel in this mass region.
A 'hybrid' channel, the associated production with subsequent Higgs decay into
(virtual) W pairs, $qq'\to WH \to WWW$, also contributes 
in the intermediate mass region.

\subsection{The Tools}
The main tools employed in Higgs searches at the Tevatron
are lepton identification and - especially in the low Higgs mass region - 
jet reconstruction and $b$ jet tagging.
The experiments use $b$ jet tagging algorithms that exploit the long lifetime
of $b$ hadrons. 
These algorithms are applied to each jet, searching for tracks with large transverse 
impact parameters relative to the primary vertex and for secondary vertices formed by tracks in the jet.
%\begin{figure}[htbp] 
%\begin{center} 
%\includegraphics[width=0.45\textwidth]{figures/btag.eps} 
%\end{center} 
%\caption[]{
%Dependence of the $b$ tagging efficiency per jet on the
%rate of misidentified jets (fake rate) for different
%operating points of the NN and the JLIP tagging algorithms
%used by the D\O\ experiment.
%}
%\label{fig-btag} 
%\end{figure} 

To further improve the $b$ jet tagging these variables are used as input to a artificial Neural Network (NN) jet-flavor separator. The NN is trained to separate $b$ quark jets from light flavour jets. By adjusting the minimum requirement
on the NN output variable, a range of increasingly stringent $b$ tagging operating points
is obtained, each with a different signal efficiency and purity. 
Using this tool at D\O\,, $b$ tagging efficiencies
have been improved by 33\% while keeping the rate of falsely identified
light flavor jets (mistags) low. The efficiencies range between 40-70\% for $b$ jets 
at a low mistag rates between 0.5-3\% for light flavor jets.

%The improved performance of a NN tagging algorithm over an algorithm
%using jet lifetime probabilities (JLIP) is shown in Figure~\ref{fig-btag}.

Almost all Higgs searches at the Tevatron employ advanced analysis techniques
like artificial Neural Networks (NN), boosted decision trees (BDT) or matrix
element techniques (ME) to combine kinematic characteristics of signal and
background events into a single discriminant. These techniques improve
the separation of signal to background over the invariant Higgs boson mass distribution
which is the most important single variable. Careful validation of all input variables is mandatory for robust results.

Events with neutrinos in the final state are identified
using missing transverse energy. The reconstruction
of all these variables require excellent performance
of all detector components.

\subsection{Signal and Background}

The Higgs signal is simulated with 
PYTHIA~\cite{bib-pythia}. The signal cross-sections are normalised to 
next-to-next-to-leading order (NNLO) calculations~\cite{bib-nnlo, bib-nnlo1} and branching
ratios from HDECAY~\cite{bib-hdecay}.

There are many types of background to the Higgs search.
An important source of background are multi-jet events (often labeled ``QCD background''). This background
and the instrumental background due to mis-identified leptons or $b$ jets is either simulated with PYTHIA (only for the CDF $ZH\to\nu\nu\bbbar$ analysis)  or is taken directly from data, since it is not very well simulated by Monte Carlo.
Determining this background from data is done using control samples with no signal content.

Electroweak background processes such as di-boson production, 
$\ppbar\to VV (V=W,Z)$, $V$+jets or $\ttbar$ pair production often dominate at the final stages
of the selection; these are simulated using leading order Monte Carlo
programs such as PYTHIA, ALPGEN, HERWIG or COMPHEP.
The normalisation of these processes is obtained either from
data or from NLO calculations.

\subsection{Search for $WH \to \ell \nu \bbbar$}
One of the most sensitive channels for a low Higgs boson mass is the decay $WH \to \ell \nu \bbbar$. This final state 
consists of two $b$ jets from the Higgs boson and a charged lepton $\ell$ and a neutrino from the W boson. All three leptonic decays of the W boson are analysed at D\O\,, with the most sensitive being the decays to electrons and muons. Events are selected with one or two $b$ tagged jets an isolated electron or muon and missing transverse energy. The main backgrounds after selection are $W+$jets and $t\bar{t}$ production. The di-jet invariant mass distribution for events with two b-tags is shown in Figure~\ref{d0wh} on the left side. To improve the separation between the signal and the irreducible background a NN is trained which takes a number of kinematic and topological variables as input. The output of this NN is used to extract limits on Higgs production and is shown in Figure~\ref{d0wh} on the right side. The analysis uses 1.7~fb $^{-1}$ of recorded data and sets an observed (expected) limit on $\sigma_{95}/ \sigma_{SM}$= 9.1(8.5) for a Higgs boson mass $m_{H} = 115$ GeV/c$^2$ (where $\sigma_{SM}$ is the cross section predicted for this process by the Standard Model). A dedicated search for $W^{\pm}H\rightarrow \tau^{\pm}\nu b\bar{b}$ with hadronic $\tau$ decays has been added at D\O. Using the di-jet mass distribution to separate signal from background an observed (expected) limit on $\sigma_{95}/ \sigma_{SM}$= 35.4 (42.1) for a Higgs boson mass $m_{H} = 115$ GeV/c$^2$ has been obtained in that channel. 
At CDF a similar analysis using 2.7~fb$^{-1}$ of data with a NN discriminant and a combined ME+BDT technique is performed. The analysis sets an observed (expected) limit on $\sigma_{95}/ \sigma_{SM}$= 5.0 (5.8) for the NN analysis and $\sigma_{95}/ \sigma_{SM}$= 5.8 (5.6) for the ME+BDT analysis for a Higgs boson mass $m_{H} = 115$ GeV/c$^2$.

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=.48\textwidth]{figures/H38F3d.eps}
\includegraphics[width=.48\textwidth]{figures/H38F4d.eps}
\end{center}
\caption{D\O\ $WH\to\ell \nu \bbbar$ channel: Di-jet invariant mass distribution for events with two b-tags and the NN distribution at the final stage of the selection. \label{d0wh}}
\end{figure}

\subsection{$ZH \to \nu\nu \bbbar$}

The channel $ZH \to \nu\nu \bbbar$ has very good sensitivity since the branching ratios
for $Z\to\nu\nu$ and $H\to\bbbar$ decays are large. With the two b-jets
being boosted in the transverse direction, the signature for the final state are acoplanar di-jets and large missing transverse energy.  Thus is in contrast to most background di-jet events which are expected to be back-to-back in the transverse plane. The main background sources in this search channel are $W$ boson or $Z$ boson production in association
with heavy flavour jets, multi-jet events and $\ttbar$ pairs. 

The basic selection requires at least one (CDF) or two jets (D\O ) with a $b$ tag, large missing transverse energy
($E_T^{\rm miss}>50$~GeV), and a veto on any isolated muon or electron in the event.

\begin{figure}[htbp] 
\begin{center} 
%\includegraphics[width=0.48\textwidth]{figures/H49F4b.eps}
\includegraphics[width=0.48\textwidth]{figures/CDFHnunubbnnQCD.eps}
\includegraphics[width=0.48\textwidth]{figures/CDFHnunubbNN.eps}
\end{center} 
\caption[]{CDF $ZH\to\nu\nu\bbbar$ channel:
NN output distribution to separate against the dominate QCD background (left)
and the NN distribution for the remaining backgrounds (right).
}
\label{fig-cdfmet} 
\end{figure} 

In the CDF analysis, the final sample is divided into three samples,
one sample with exactly one tight secondary vertex $b$ tag, the second sample with one tight secondary vertex $b$ tag and one tag with the JetProb algorithm and a third sample with two tight secondary vertex $b$ tags. 
Two NNs are trained one against the dominant QCD background (see Figure~\ref{fig-cdfmet} on the left side for the second b-tag sample) and one against di-boson and $\ttbar$ background (see Figure~\ref{fig-cdfmet} on the right side for the second sample), which is also used to extract limits on the production cross section.

In the case of D\O, events with two NN $b$ tags are used to
construct a BDT for identifying signal events. Asymmetric operating points, one loose and one tight,
are chosen for the two $b$ tags. The output distributions
of the BDT, retrained for every Higgs mass, is shown in Figure~\ref{fig-d0met} on the right side.

\begin{figure}[htbp] 
\begin{center} 
\includegraphics[width=0.48\textwidth]{figures/H49F3b.eps}
\includegraphics[width=0.48\textwidth]{figures/H49F4b.eps}
\end{center} 
\caption[]{D\O\ $ZH\to\nu\nu\bbbar$ channel:
Invariant dijet Mass Distribution (left) and output distribution of the BDT variable (right).
}
\label{fig-d0met} 
\end{figure} 

To increase the sensitivity of this analysis, $WH$ signal events
where the charged lepton has not been identified are also included
in the signal definition. This search yields a median observed (expected) upper limit on the 
$VH (V=W,Z)$  production cross-section of
$\sigma_{95}/\sigma_{SM}=7.9 (6.3)$ for CDF and $7.5 (8.4)$ for D\O\ at a Higgs mass
of $m_H=115$~GeV/c$^2$. The data set for both experiments corresponds to $2.1$~fb$^{-1}$ of analyzed data.

\subsection{$ZH \to \ell\ell \bbbar$}

In the $ZH \to \ell\ell \bbbar$ channel the $Z$ boson is reconstructed through the decay into 
two high-$p_T$ isolated muons or electrons. The reconstructed $Z$ 
and two b-tagged jets are used to select the Higgs signal. 
The invariant mass of the two leptons is required to be in the
$Z$ mass range $70<m_Z<110$~GeV/c$^2$ (D\O ) or $76<m_Z<106$~GeV/c$^2$ (CDF).
Both experiments require two jets with either one tight $b$ tag or two loose $b$ tags.

The main background sources are $Z$ production in association with heavy jets and $\ttbar$ production.
$ZZ$ production is an irreducible background, apart from the mass
discriminant. CDF trains two separate NNs to reject these two background components.
Slices of the output of these NNs, projected on the two axes, is shown in Figure~\ref{fig-llcdf}.
The di-jet mass resolution is improved by training a different NN
using $E_{T}^{miss}$ and the kinematics of both jets. The data set corresponds
to an integrated luminosity of 2.4~fb$^{-1}$. 
The D\O\ analysis is performed with $2.3$~fb$^{-1}$ of data using a kinematic
NN and two NN $b$ tag samples with one tight $b$ tag and two loose $b$ tags.

\begin{figure}[htb] 
\begin{center} 
\includegraphics[width=0.48\textwidth]{figures/ProjXSlice_singleZH120.eps}
\includegraphics[width=0.48\textwidth]{figures/ProjYSlice_singleZH120.eps}
\end{center} 
\caption[]{$ZH\to\ell\ell\bbbar$ channel:
NN output projection with $y \le 0.1$ in the Z+Jets vs. ZH 
projections and $x \ge 0.9$ in the ZH vs. $\ttbar$ projection.
}
\label{fig-llcdf} 
\end{figure} 

These searches yield a median observed (expected) upper limit on the $ZH$ production 
cross-section of $\sigma_{95}/\sigma_{SM}=11.6 (11.8)$ for CDF and $11.0 (12.3)$ for D\O\ at a Higgs mass
of $m_H=115$~GeV/c$^2$. Even though the limits are less stringent than for the $ZH\to\nu\nu\bbbar$ channel, they still provide an important input to increase the overall sensitivity of the analysis.

\subsection{$W \to WW  \to \ell\nu\ell\nu$}

The dominant decay mode for higher Higgs masses is $H\to WW^{(*)}$.
Leptonic decays of the W bosons are therefore used to suppress the QCD background.
The signature of the $gg\to H \to WW^{(*)}$ channel is two high-$p_T$
opposite signed isolated leptons with a small azimuthal separation, $\Delta\phi_{\ell\ell}$,
due to the spin-correlation between the final-state leptons in
the decay of the spin-0 Higgs boson. In contrast, the lepton pairs from background events, mainly $WW$ events, are predominantly back-to-back in $\Delta\phi_{\ell\ell}$. This is shown in Figure~\ref{fig-dphi} (left) for 
a preselected CDF data sample with zero reconstructed jets. 

\begin{figure}[htbp] 
\begin{center} 
\includegraphics[width=0.48\textwidth]{figures/HistStackdPhiLeptons.eps}
\includegraphics[width=0.48\textwidth]{figures/TemplateStackHWW160_HighSB_Log.eps}
\end{center} 
\caption[]{CDF $WW$ channel:
azimuthal angle between the two leptons in the $H \to WW$ search. Due to spin
correlations, the signal is at low $\Delta\phi_{\ell\ell}$,
whereas the background is at high $\Delta\phi_{\ell\ell}$.
}
\label{fig-dphi} 
\end{figure} 

An additional selection requires $E_T^{\rm miss} > 25$~GeV for CDF and $E_T^{\rm miss} > 20$~GeV for D\O\ to account
for the neutrinos in the final state.
D\O\ defines three final states ($e^+e^-$, $e^\pm \mu^\mp$, and $\mu^+\mu^-$).
CDF separates the \hww\ events into five non-overlapping samples, first by
separating the events by jet multiplicity (0, 1 or 2), then subdviding the 
0 and 1 jet samples in two, one having a low signal/bacgkround (S/B)
ratio, the other having a higher one. In these analyses, 
the final discriminants are neural-network outputs
based on several kinematic variables. These 
include likelihoods constructed from matrix-element probabilities as input
to the neural network for CDF and is shown on the right side of Figure~\ref{fig-dphi}.
The background subtracted NN distribution for D\O\ is shown in Figure\ref{fig-wwd0} on the left side. This distribution has been used to extract median observed (expected) limits on the production 
cross-section of $\sigma_{95}/\sigma_{SM}=1.9\, (2.0)$ for  $m_H=165$~GeV/c$^2$. The obtained limits on the production 
cross-section as a function of the Higgs boson mass are shown in Figure~\ref{fig-wwd0} on the right side.
With the NN distributions CDF obtains $\sigma_{95}/\sigma_{SM}=1.7 (1.6)$ for  $m_H=165$~GeV/c$^2$. The data sets analyzed correspond to an integrated luminosity of 3~fb$^{-1}$ for each experiment.
 
\begin{figure}[htbp] 
\begin{center} 
\includegraphics[width=0.48\textwidth]{figures/H61F4d.eps}
\includegraphics[width=0.48\textwidth]{figures/H61F5a.eps}
\end{center} 
\caption[]{D\O\ $WW$ channel: The background subtracted distribution of the NN (left) and the obtained median observed and expected limits on the production cross-section (right).
}
\label{fig-wwd0} 
\end{figure} 


\subsection{$WH \to WWW^{*}  \to \ell \nu \ell^{'} \nu\qqbar$}
In the process $WH \to WWW^{*}  \to \ell \nu \ell^{'}  \nu\qqbar$ the 
Higgs boson is produced in association with a $W$ boson and subsequently decays
into a $WW$ pair. This process is important in the intermediate mass range.
The signature is at least two isolated leptons from the $W$
decays with $p_T>15$~GeV and identical charge. The associated $W$
and one of the two $W$ bosons from the Higgs decay should have
the same charge. For the final signal selection D\O\ used a two-dimensional
likelihood based on the invariant mass of the two leptons,
the missing transverse energy and their azimuthal angular correlations.
%The two-dimensional likelihood separates the signal from physics background
%and instrumental background, respectively.

This same-sign charge requirement is very powerful in rejecting
background from $Z$ production. The remaining background is either
due to di-boson production or due to charge mis-measurements.
The rate of charge mis-measurements for muons is determined by comparing
the independent charge measurements within the solenoidal and in the toroidal fields
of the D\O\ detector. For electrons the charge mis-measurement rate
is determined by comparing the charge measurement from the solenoid
with the azimuthal offset between the track and the calorimeter cluster
associated to the electron.

%The upper limit on the cross-section $\sigma(\ppbar\to W) \BR(H\to WW^*)$
%is between 3.2~pb and 2.8~pb for Higgs masses from 115~GeV to 175~GeV.
The expected cross-section ratio in the mass range 140~GeV/c$^2$ to 180~GeV/c$^2$
is $\sigma_{95}/\sigma_{SM}\simeq 20$, i.e. this channel makes
a significant contribution at the limit in this mass range. 

\section{Combined Tevatron Limit}
The data of both experiments have been combined using the full
set of analyses with luminosities up to 3.0~fb$^{-1}$. 
To gain confidence that the final result does not depend on the details
of the statistical method applied, several types of combination 
were performed, using both Modified Frequentist (sometimes called the
LEP $CL_s$ method) and Bayesian approaches. The results
agree within about $10\%$. Both methods use Poisson likelihoods and
rely on distributions of the final discriminants, e.g. NN output or di-jet mass distributions, not only on event counting.

Systematic uncertainties enter as uncertainties on the expected number of signal
and background events, as well as on the shape of the discriminant
distributions. The correlations of systematic uncertainties between channels,
different background sources, background and signal and between experiments are taken into account. 
The main sources of systematic uncertainties are, depending on channel,
the luminosity and normalisation, the estimates of the multi-jet backgrounds,
the input cross-sections used for the MC generated background sources,
the higher order corrections ($K$ factors) needed to describe heavy flavour
jet production, the jet energy scale, $b$ tagging and lepton identification.

The combinations of results of each single experiment,
yield the following ratios of 95\% C.L. observed (expected) limits to the SM 
cross section: 
4.2~(3.6) for CDF and 5.3~(4.6) for D\O\ at $m_{H}=115$~GeV/c$^2$, and 
1.8~(1.9) for CDF and 1.7~(2.3) for D\O\ at $m_{H}=170$~GeV/c$^2$.

The ratios of the 95\% C.L. expected and observed  limit to the
SM cross section are shown in Figure~\ref{fig-comb}
for the combined CDF and D\O\ analyses on the left side.  The observed and median expected values are
1.2 (1.2) at $m_{H}=165$~GeV/c$^2$,
1.0 (1.4) at $m_{H}=170$~GeV/c$^2$ and
1.3 (1.7) at $m_{H}=175$~GeV/c$^2$.
On the right side in Figure~\ref{fig-comb} the 1-$CL_S$ distribution as a function of 
the Higgs boson mass, which is directly interpreted
as the level of exclusion of the search. For instance,  both the observed and expected
results exclude a Higgs boson with $m_H= $ 165 GeV/$c^2$ at $\approx$ 92\% C.L.
The green and yellow bands show the one and two
sigma bands for background fluctuations. We exclude at the 95\% C.L. the
production of a standard model  Higgs boson with  mass of 170 GeV/c$^2$.

\begin{figure}[htbp] 
\begin{center} 
\includegraphics[width=0.49\textwidth]{figures/H64F2.eps} 
\includegraphics[width=0.49\textwidth]{figures/H64F3.eps} 
\end{center} 
\caption[]{
Expected and observed $95\%$ CL cross-section ratios for the combined
CDF and D\O\ analyses. (status July 2008).
}
\label{fig-comb} 
\end{figure}  

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