\small
\begin{center}
{\it T. Dorigo,
     S. Kuhlmann,
     R. Snihur,
     G. Watts,
     J. Womersley} \\
\end{center}
\normalsize \nopagebreak

For Higgs masses less than about 135~GeV the most promising channels
in Run~2 at the Tevatron involve the decay of the Higgs to $b\overline
b$, such as the process $\pp\to WH\to \ell\nu\bb$.  The signal is an
excess of events having $\bb$ invariant mass peaked at the Higgs mass.
Obtaining the best possible invariant $\bb$ mass resolution is
therefore crucial in establishing a signal.

This section presents the results of two studies of the dijet mass resolution
in the upgraded Run 2 detectors.  

\newpage
\vspace{0.2in} \large
{\bf CDF Dijet Mass Resolution Studies} \\ \normalsize \nopagebreak

The $b\bar{b}$ decay of the Higgs boson dominates if $M_H <
135~GeV/c^2$. Our ability to extract this particle from the large QCD
background in Run 2 (for instance in the $W b \bar{b}$ final state,
when associated $WH$ production is sought) will therefore depend
critically on the resolution we can attain on the Higgs boson mass,
reconstructed from the measured $b$-quark jet energies: both the
possibility to see a bump in a dijet mass spectrum and the option to
apply a mass window cut as a selection tool will strictly depend on
the attainable $b\bar{b}$ mass resolution.

A study of jet--energy resolution in QCD dijet events with one
electromagnetic jet was carried out by CDF.  This study improved upon
the calorimeter--only jet energy measurement typically used in Run 1
CDF (and D\O) analyses by including information from charged--particle
momenta measurements and shower maximum detectors.
Figure~\ref{f-kuhlfig} shows the jet energy resolution for both the
calorimeter--only jet energy algorithm and the improved algorithm.
This shows roughly a 30\% improvement in energy resolution.

\begin{figure}
  \begin{center}\hbox{
  \parbox{3.25in}{\epsfxsize=\hsize\epsffile{bbmass/jetscale.eps}}
  \parbox{3.25in}{\epsfxsize=\hsize\epsffile{bbmass/escale.eps}}}
  \end{center} 
  \caption{The left-hand panel shows CDF jet energy resolution 
           computed for a calorimeter--only 
           measurement similar to Run 1 results and for an
           improved algorithm which incorporates energy measurements 
           from additional detector systems.  The right-hand panel shows
           the resolution obtained from SHW compared with the two CDF
           resolutions and the standard D\O\ jet energy resolution.  It
           is clear that both experiments have achieved energy resolutions 
           significantly better than that used in this study.}
  \label{f-kuhlfig}
\end{figure}

A study of the dijet mass resolution in $Z \to b \bar{b}$ events was
performed by CDF in the context of a search for that process in events
collected by an inclusive muon trigger.  Two million $Z \to b \bar{b}$
decays were generated with the PYTHIA 5.7 Monte Carlo, and filtered as
the data.
 
After the application of the standard CDF jet correction routine, many
observable quantities were studied as a function of the difference
between measured jet momenta and originating parton momenta, in the
hope of using the measured values of these observables to help
reducing the energy mismeasurement.  The quantities that were found
most useful for this purpose were the following: the muon momentum,
the missing transverse energy projection along each jet, and the jet
charged fraction.

The muon momentum is needed in correcting the energy of jets
originating from the semileptonic decays of $b$-quarks, because the
minimum ionizing muons do not contribute linearly to the energy
measured in the calorimeter. The missing $E_T$, projected along the
direction of the jets in the transverse plane, provides useful
information on the momentum taken away by the neutrino and on a
possible fluctuation of the energy measurement. The charged fraction
of the jets (defined as the ratio between the total momentum of
reconstructed charged tracks inside the jet cone and the jet momentum
measured in the calorimeter) also helps reduce the uncertainty in the
energy measurement.

By properly accounting for the value of these observables, it was
possible to reduce the relative width of the dijet mass distribution,
$\sigma_M / M_{jj}$, by nearly $50\%$ (see Figure~\ref{mass_evol}, left).
  
\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{bbmass/zbb_corr_pyt_mass_evol.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{bbmass/zbb_corr_her_h_evol.eps}}
  \end{center}
  \caption{ The four gaussian fits show the improvement of the mass 
            reconstruction
	    for simulated $Z \to b\bar{b}$ decays (PYTHIA 5.7, left) 
            and simulated $H \to b\bar{b}$ decays (HERWIG 5.6, right) 
            when the observable characteristics of the $b$-quark 
            decays are properly taken into account in the mass 
            reconstruction.  The CDF Run~1 simulation was used in this
            study.}
  \label{mass_evol}
\end{figure}

The validity of these corrections was tested with HERWIG \cite{herwig}
to check the
effect of a different fragmentation model, and no differences were
found.  The corrections were also checked on a simulated sample of $gg
\to H \to b\bar{b}$ decays, with $M_H=120~GeV$, and were found to be
equally useful for a different resonance mass (Figure~\ref{mass_evol},
right).

Finally, it was possible to show that a real sample of $Z \to b\bar{b}$
decays extracted from Run 1 data behaved as expected from the simulation:
the reconstructed excess of events over background predictions indeed
showed an average value and width in agreement with Monte Carlo predictions
both before and after the application of the jet corrections 
(Figure~\ref{excess_evol}).

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{bbmass/zbb_bless_count_10_old_new.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{bbmass/zbb_bless_count_10_new.eps}}
  \end{center}
  \caption{ The comparisons between observed data and background predictions
            (insets) allow the extraction of gaussian excesses due to the 
            $Z \to b \bar{b}$ decay both before (left) and after (right)
            the use of the $b$-quark specific jet corrections. 
            The excess behaves as predicted by Monte 
            Carlo expectations in the two cases. }  
  \label{excess_evol}
\end{figure}

The corrections described above were specifically intended for a pair 
of back-to-back $b$-quarks, when one of the two decays to a muon. 
Their success in improving the significance of the $Z \to b\bar{b}$ signal,
however, demonstrates the usefulness of the chosen approach, 
and spurs new studies aimed at different final state
topologies for the search for new resonances.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{0.2in} \large
{\bf D\O\ Dijet Mass Study} \\ \normalsize \nopagebreak

This study is based on events generated using PYTHIA with a Higgs mass
of 90~GeV simulated in the D\O\ detector.  The $W$ decay was forced to
$\mu \nu$ in order to give a relatively ``clean'' dijet Higgs signal
in the calorimeter.  Events were generated with both initial and final
state radiation (ISR and FSR) as default, but we also generated
samples with neither ISR nor FSR, with only ISR, and with only FSR,
for comparison purposes.  Events were then passed through the D0GEANT
detector simulation which provides a full simulation of the
calorimeter response and resolution.  (This is the Run~1,
fortran-based detector simulation).  The digitized events were then
passed through the Run~1 calorimeter reconstruction and jet finding
packages.

In the analyses presented here, we required the following standard jet
selection cuts:
\[E_T > 15\ {\rm GeV\ and}\ |\eta|< 2.0.\]

The reconstructed mass distributions were characterized using a fit
function formed of a gaussian convoluted with two
exponentials~\cite{hobbsfit}.  The fits typically yield a $\chi^2$ per
degree of freedom between 1 and 2.  The bin containing the peak and 
the bin separation corresponding to the FWHM were found by hand.

Two jet clustering algorithms were investigated.  The first is the
standard D\O\ cone algorithm, with a cone size of $\Delta R =0.7$.
The second is the $k_T$ algorithm, with $D$ parameter settings of 0.5
and 1.0 ($D$ is a close approximation to the $\Delta R$ in the cone
algorithm.)  In the $k_T$ algorithm each hit calorimeter cell forms a
vector.  Vectors are combined with each other to build a jet by
joining the two vectors with the smallest relative $p_T$ to form a new
vector.  Combination stops when all vectors are separated by $\Delta R
> D$ \cite{ktalg}.

Our attempt throughout this study is to demonstrate the factors
governing mass resolution, rather than to present final results on the
resolution that can be attained.  Much work will have to be done in
deriving energy scale corrections before final resolution numbers are
finalized.

\vspace{0.2in}
{\bf Parton, Particle and Calorimeter Level Resolutions} \\ \nopagebreak

To demonstrate the effects of hadronization, radiation, and of the
calorimeter energy measurement, we have applied the same $k_T$
algorithm at the parton, particle and calorimeter cell levels.  We
used the D\O\ Run 1 $k_T$ jet finder and excluded the muon and
neutrino from the $W$ decay from the parton and particle level jets.
Two values of the jet size parameter $D$ were used, 0.5 and 1.0.
Figure~\ref{skip_id_1_page1} shows the mass resolution obtained at the
parton level. Without any gluon radiation, the Higgs mass is
reconstructed perfectly (left hand pane); gluon radiation degrades
this somewhat, as shown in the right hand pane which has both initial
state radiation (ISR) and final state radiation (FSR) included.


\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/skip_id_1_page1.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/skip_id_1_page4.eps}}
  \end{center}
  \caption{The dijet invariant mass using reconstructed parton level $k_T$ 
     jets. The upper two plots are $D=0.5$ and the lower two are $D=1.0$; 
     the left hand column has no initial or final-state radiation and the 
     right hand column is a simulation will both on.  The solid line does 
     not include muons and neutrinos in the jet; the dashed line does.}
  \label{skip_id_1_page1}
\end{figure}

Hadronization introduces a low-side tail to the mass
distributions, as may be seen in Figure~\ref{skip_id_2_page1}; 
it is somewhat washed out when ISR and FSR 
are included.  At this point
the larger $D$ value starts to give a better resolution since it
clusters more of the final state particles into the reconstructed jets.

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/skip_id_2_page1.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/skip_id_2_page4.eps}}
  \end{center}
  \caption{The dijet invariant mass calculated using reconstructed particle 
      level $k_T$ jets. The upper two plots are $D=0.5$ and the lower two 
      are $D=1.0$; the left hand column has no initial or final-state 
      radiation and the right hand column is a simulation with both on. }
  \label{skip_id_2_page1}
\end{figure}

Finally at the calorimeter level we obtain the distributions
in Figure~\ref{skip_id_6_page4} which correspond to what would
be measured in the detector.  We will now investigate the
calorimeter level distributions in more detail.

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/skip_id_6_page4.eps}}
  \end{center}
  \caption{The dijet invariant mass calculated using uncorrected 
     calorimeter $k_T$ jets. The upper plot is $D=0.5$ and the lower is 
     for $D=1.0$. Both initial-state and final-state radiation are simulated.}
  \label{skip_id_6_page4}
\end{figure}

\vspace{0.2in}
{\bf Dijet Mass Definition} \\ \nopagebreak

We compared two different dijet mass estimators:
\[M_{4v}^2 = (E_{1} + E_{2})^{2} - (p_{x1} + p_{x2})^{2} + (p_{y1} + p_{y2})^{2} + (p_{z1} + p_{z2})^{2}\]
and
\[M_{jj}^2 =  2 E_{T1} E_{T2} ( \cosh(\eta_{1} - \eta_{2} )
	- \cos(\phi_{1} - \phi_{2}) ). \]

For the particular case of $R=0.7$ cone jets, and plotting the mass
of the leading two jets in the event,
the two quantities are shown in figure \ref{new_t6_15_c_un_1812}.
They yield very similar fractional widths though they are
not bin-by-bin identical.  In most of what follows we shall use $M_{4v}$.

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/new_t6_15_c_un_1812.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/new_t6_15_c_un_2812.eps}}
  \end{center}
  \caption{
     A comparison of two methods of calculating the dijet mass of the Higgs
     decay. The $M_{4v}$ calculation is on the left, and the $M_{jj}$ is on 
     the right.}
  \label{new_t6_15_c_un_1812}
\end{figure}

\vspace{0.2in}
{\bf Energy Scale Corrections} \\ \nopagebreak

We applied the D\O\ Run~1 jet energy scale corrections to the
jets before forming the invariant mass.  To our 
disappointment, the mass resolution was degraded about 10\% by
these ``corrections''. 
%(figures \ref{new_1512} and \ref{new_t6_15_c_un_2512}).
This is perhaps not too surprising since the corrections were
derived for jets at much higher $E_T$ and were also
not optimized for $b$-jets
in any way.  The D\O\ top mass analysis used a different set of jet 
corrections which would probably be more appropriate here, but these
were not available in our analysis package.  In what follows we shall
use uncorrected jet energies.

\vspace{0.2in}
{\bf Cone and $k_T$ Jets Compared} \\ \nopagebreak

One of the hopes of using the $k_T$ algorithm was that the mass resolutions
would be better.  The $k_T$ algorithm should do a good job of recombining
final state radiation with the parent parton jet because it recombines
particles with a small invariant mass. The $k_T$ jet mass resolution shown in
this paper is worse than the resolutions found by the cone jet algorithm
except in the high luminosity case with $D=0.5$.  The mass resolutions from $k_T$ 
are larger than for the cone, as can be seen in Figure~\ref{new} for
$D=1.0$ and 0.5.
The larger $D$ value gives a better resolution and a 
peak mass closer to the true Higgs mass.  
A complete analysis of the 
$k_T$ jet mass resolution, including jet energy corrections, in progress
at D\O.

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/new.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/new_d5.eps}}
  \end{center}
  \caption{The dijet invariant mass calculated using uncorrected calorimeter 
     $k_T$ jets with $D=1.0$ (left hand plot) and $D=0.5$ (right hand plot). 
     Both initial and final state radiation were simulated.}
  \label{new}
\end{figure}

\vspace{0.2in}
{\bf Initial and Final State Radiation} \\ \nopagebreak

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/nofsr.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/noisr.eps}}
  \end{center}
  \caption{The dijet invariant mass calculated using uncorrected calorimeter 
     $k_T$ jets with $D=1.0$. The left hand plot has no final state radiation 
     and the right hand plot has no initial state radiation. These should be 
     compared to the left hand plot of Figure~\ref{new} which has 
     both final and initial state radiation simulated.}
  \label{nofsr}
\end{figure}

Comparison of Figure~\ref{new} with Figure~\ref{nofsr}(left), which was
generated without final state radiation (FSR), shows  
that FSR contributes a low-end tail, 
because the jet algorithms fail to cluster in all the radiation.
Initial State Radiation (ISR) on the other hand, 
contributes a high-end tail,
because the jet algorithms cluster in extra (unwanted) radiation,
as shown by Figure~\ref{nofsr}(right) which was generated without ISR.
Comparison of the figures will also show that ISR and FSR
bias the peak position by a few GeV as well as contributing to
the resolution.

For $D=1.0$, the ISR component of the mass resolution is roughly equivalent
to the FSR piece; for $D=0.5$, FSR contributes relatively 
more (giving a worse resolution overall).
In principle, the $D$ parameter could be optimized to
accept as much FSR as possible while rejecting ISR.

\vspace{0.2in}
{\bf $b$-quark matching} \\ \nopagebreak

To simulate the effect of requiring a $b$-tag within each of the jets
used for the mass reconstruction, we required that the leading jets
be the closest jets to the $b$ and $\overline b$ parton directions.
This significantly improves the resolution, because it cuts out events
where one of the leading jets comes from FSR (and is hence likely to
match a gluon rather than the  $b$ or $\overline b$).  The improvement
is seen both for cone jets and $k_T$ jets in
Figure~\ref{new_t6_15_c_un_11812}.

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/new_t6_15_c_un_11812.eps}}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/new_d1_11212.eps}}
  \end{center}
  \caption{The dijet invariant mass calculated using uncorrected calorimeter 
    level $\Delta R=0.7$ (left) and $D=1.0$ (right) $k_T$ jets. The two leading jets 
    are required to be the closest jets to the $b$ and $\bar{b}$ directions.}
  \label{new_t6_15_c_un_11812}
\end{figure}

\vspace{0.2in}
{\bf Luminosity Effects} \\ \nopagebreak

We investigated the effects of multiple interactions per crossing
on the dijet resolution. We overlaid real D\O\ zero-bias events from 
the highest luminosity phase of Run 1 on to our simulated Higgs
events.  A sample with instantaneous luminosity of 
${\cal L} = 2 \times 10^{31}$~cm$^{-2}$s$^{-1}$ was used, which
has approximately 3.3 interactions per crossing on average.
In Run 2b, with 100+ bunches and 132~ns between crossings,
this is equivalent to ${\cal L} = 3 \times 10^{32}$~cm$^{-2}$s$^{-1}$. 
The effect is shown in Figure~\ref{new_zf} and, as can be seen, results in
a significant worsening of the resolution.  We have not attempted to
recover the resolution but it might be possible to do better by imposing 
a threshold cut on calorimeter towers to remove some of the low-energy
pileup.

We also investigated how the mass resolution is affected in overlaid events
as a function of $D$. Figure~\ref{watts-new1} shows the Higgs mass resolutions
for both $D=1.0$ and $D=0.5$.  The smaller size $D$ exhibits less of a 
dependency on the luminosity.  The $k_T$ algorithm's $D$ parameter controls,
approximately, the size of the jet, and smaller jet size will cover less
calorimeter cells.  Multiple interaction background can be approximated as a 
pedestal, distributed almost evenly in all towers of the calorimeter; so the
smaller size $D$ parameter jets are affected less. The effects of multiple
interactions have also been studies in the cone jet algorithms, as shown
in Figure~\ref{watts-new2}.  The loss in resolution is similar for the
$k_T$ $D=1.0$ algorithm.

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=\hsize\epsffile{./bbmass/new_zf.eps}}
  \end{center}
  \caption{The effect of multiple interactions on the calculation of the 
     Higgs mass. This is the dijet mass calculated with calorimeter level 
      $k_T$ jets with $D=1.0$. The simulation contains overlapped data: D\O\ 
      zero-bias data recorded at a luminosity of ${\cal L} = 2 \times
      10^{31}$cm$^{-2}$s$^{-1}$. In Run 2 this will be equivalent to ${\cal L} =
      3 \times 10^{32} $cm$^{-2}$s$^{-1}$ at 132 ns bunch spacing.}
  \label{new_zf}
\end{figure}

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=3in\epsffile{./bbmass/new_zf_15_c_un_11112.eps}}
    \parbox{3.0in}{\epsfxsize=3in\epsffile{./bbmass/new_zf_d5_15_c_un_11112.eps}}
  \end{center}
  \caption{The effect of varying the $k_T$ algorithm's $D$ parameter on Higgs 
   mass resolutions in simulated multiple interaction data. The dijet mass of 
   the Higgs is calculated at calorimeter level in both plots, and the leading
   two jets are required to match the b quark direction.}
  \label{watts-new1}
\end{figure}

\begin{figure}
  \begin{center}
    \parbox{3.0in}{\epsfxsize=3in\epsffile{./bbmass/new_zf_15_c_un_1512.eps}}
    \parbox{3.0in}{\epsfxsize=3in\epsffile{./bbmass/new_zf_15_c_un_11512.eps}}
  \end{center}
  \caption{The effect of multiple interactions on the calculation of the Higgs
     mass using cone jets. This is the dijet mass calculated with calorimeter 
     level cone jets in overlapped data. The right hand plot requires the 
     leading jets match the direction of a b-quark jet; the left hand plot 
     does not. The jets have had calibration corrections applied.}
  \label{watts-new2}
\end{figure}

\vspace{0.2in}
{\bf Conclusions} \\ \nopagebreak

We have investigated many effects contributing to the invariant mass
resolution.  We find:

\begin{itemize}
\item The $k_T$ algorithm is no silver bullet (at least as implemented
in the D\O\ Run 1 software). In fact it is hard to do much better than a
standard $R=0.7$ cone algorithm, especially if the jets are required
to match the $b$ quark directions.

\item Jet energy corrections will have to be optimized for these 
relatively low-$E_T$ $b$-quark jets. 

\item Maintaining good resolution in the high luminosity environment of 
Run~2 and beyond will be a challenge.  Pileup from minimum bias events
can seriously degrade the energy resolution.  
\end{itemize}

Table~\ref{massconc} summarizes
the resolution and peak position of all the results.
\begin{table}
 \caption{A summary of the mass resolution results.  For each data
   sample, the cone and $k_T$ peak value FWHM are listed for
   a Higgs mass of 90~GeV/c$^2$.}
\label{massconc}
\begin{center}
   \begin{tabular}{|c|c|c|c|c|c|c|} 
\noalign{\vskip-6pt}  %\hline
 & \multicolumn{2}{c|}{Cone} &  \multicolumn{2}{c|}{$k_T$ $D=1.0$}
            &  \multicolumn{2}{c|}{$k_T$ $D=0.5$} \\ 
\tableline   %\hline
 Data Sample & Peak & Width 
             & Peak & Width 
             & Peak & Width \\
             & (GeV/$c^2$) &       
             & (GeV/$c^2$) &       
             & (GeV/$c^2$) &       \\ 
\tableline   %\hline
  Uncorr., no FSR & & & 80.2 & 15.3\% &  & \\
  Uncorr., no ISR & & & 75.7 & 15.0\% &  & \\
  Uncorr., full rad. 
     & 72.5 & 14.6\% & 73.7 & 20.1\% & 67.25 & 22.7\% \\
  Uncorr., full rad., $b$ matching
     & 72.5 & 14.6\% & 77.5 & 16.5\% &       &       \\
  Mult. int., uncorr., full rad.
     & 78.7 & 20.8\% & 83.2 & 24.7\% &       &       \\
  Mult. int., uncorr., full rad., $b$ matching
     & 79.7 & 17.6\% & 85.7 & 18.2\% & 69.75 & 16.4\% \\
  \end{tabular}
\end{center}
\end{table}












