\small
\begin{center}
{\it G. Landsberg,
     K. Matchev}
\end{center}
\normalsize \nopagebreak

The Standard Model (SM) is very economical in the sense that the Higgs
doublet responsible for electroweak symmetry breaking can also be used
to generate fermion masses.  The Higgs boson couplings to the gauge
bosons, quarks, and leptons are therefore predicted in the Standard
Model, where one expects the Higgs boson to decay mostly to b-jets and
tau pairs (for low Higgs masses, $M_h\lsim 140$ GeV), or to $WW$ or
$ZZ$ pairs, (for higher Higgs masses, $M_h \gsim 140$ GeV).  Since the
Higgs boson is neutral and does not couple to photons at tree level,
the branching ratio ${\rm B}(h\rightarrow \gamma\gamma)$ is predicted
to be very small in the SM, on the order of $10^{-3}-10^{-4}$.

In a more general framework, however, where different sectors of the
theory are responsible for the physics of flavor and electroweak
symmetry breaking, one may expect deviations from the SM predictions,
which may lead to drastic changes in the Higgs boson discovery
signatures.  One such example is the so called ``fermiophobic'' (also
known as ``bosophilic'' or ``bosonic'') Higgs, which has suppressed
couplings to all fermions, and may arise in a variety of
models~\cite{fermiophobe}. A variation on this theme is the Higgs in
certain topcolor models, which may couple to heavy quarks
only~\cite{topmodels}. Some even more exotic possibilities have been
suggested in the context of theories with large extra
dimensions~\cite{LED}.

Finally, in the minimal supersymmetric standard model (MSSM), the
width into $b\bar{b}$ pairs can be suppressed due to 1-loop SUSY
corrections, thus enhancing the branching ratios of a light Higgs into
more exotic signatures \cite{CMW1,Mrenna}.  In all these cases, the
Higgs boson decays to photon pairs are mediated through a $W$ or heavy
quark loop and dominate for $M_h\lsim 100$ GeV
\cite{Stange:1994ya,SMW}. In the range $100\lsim M_h\lsim 160$, they
compete with the $WW^\ast$ mode, while for $M_h\gsim 160$ GeV,
$h\rightarrow WW$ completely takes over.  Current bounds from
LEP~\cite{LEP limits} are limited by the kinematic reach of the
machine. The existing Run 1 analyses at the Tevatron have utilized the
diphoton plus 2 jets~\cite{Lauer,D0,Wilson} and inclusive
diphoton~\cite{Wilson} channels and were limited by statistics.  Since
they only looked for a ``bosonic'' Higgs, they did not consider the
Higgs production mechanism through gluon fusion, which can be a major
additional source of signal in certain models~\cite{topmodels}.  Since
$h\rightarrow\gamma\gamma$ is a very clean signature, it will allow
the Tevatron to extend significantly those limits in its next run.

In this study we shall evaluate the Higgs discovery potential of the
upcoming Tevatron runs for several diphoton channels.  We shall
concentrate on the following two questions. First, what is the
absolute reach in Higgs mass as a function of the
$h\rightarrow\gamma\gamma$ branching ratio? Second, which signature
(inclusive diphotons, diphotons plus one jet, or diphotons plus two
jets) provides the best reach.  We believe that neither of those two
questions has been adequately addressed in the literature previously.

\subsubsection{Fermiophobic Higgs Bosons}

Here we consider the case of a fermiophobic Higgs, {\it i.e.} models where
the Higgs couplings to all fermions are suppressed. Then, the main
Higgs production modes at the Tevatron are associated $Wh/Zh$
production, as well as $WW/ZZ$ fusion.  All of these processes have
comparable rates [see fig.~\ref{fg:4}], so it makes sense to consider an
inclusive signature first~\cite{Wilson}.

\vspace{0.2in}
{\bf Inclusive channel: event selection} \\ \nopagebreak

We use the following cuts for our inclusive study: two photons with
$p_T(\gamma)>20$ GeV and rapidity $|\eta(\gamma)| < 2$, motivated by
the acceptance of the CDF or D\O\ detectors in Run~2. Triggering on
such a signature is trivial; both collaborations will have diphoton
triggers that are nearly fully efficient with such offline cuts.

We assume 80\% diphoton identification efficiency, which we apply to
both the signal and background estimates on top of the kinematic and
geometrical acceptance.  Again, this efficiency is motivated by the
CDF/D\O\ EM ID efficiency in Run 1 and is not likely to change in
Run~2.

\vspace{0.2in}
{\bf Inclusive channel: background} \\ \nopagebreak

The main backgrounds to the inclusive diphoton channel come from the
QCD production of dijets, direct photons, and diphotons.  In the
former two cases a jet mimics a photon by fragmenting into a leading
$\pi^0/\eta$ meson that further decays into a pair of photons, not
resolved in the calorimeter.

\begin{figure}[t]
\epsfysize=3.5in
\epsffile[-250 0 100 560]{matchev/qcd_background2.ps}
\begin{center}
\caption[]
{\small The total background in the inclusive diphoton channel,
as well as the individual contributions from $\gamma\gamma$,
$\gamma j$ and $jj$ production. \label{qcd_background}}
\end{center}
\end{figure}


We used the PYTHIA~\cite{PYTHIA} event generator and the
experimentally measured probability of a jet to fake a
photon~\cite{Lauer} to calculate all three components of the QCD
background. The faking probability depends significantly on the
particular photon ID cuts, especially on the photon isolation
requirement~(see, {\it e.g.}, ref.~\cite{Lauer,diboson,monopole}).
For this study we used an $E_T$-dependent jet-faking-photon
probability of $$ P({\rm jet} \rightarrow \gamma) = \exp\left(-0.01\
{E_T\over\mbox{(1 GeV)}}-7.5\right),$$ which is obtained by taking the
$\eta$-averaged faking probabilities used in the D\O\ Run 1
searches~\cite{Lauer}. The fractional error on $P(\mbox{jet}
\rightarrow \gamma)$ is about 25\% and is dominated by the uncertainty
on the direct photon fraction in the $\mbox{jet}+\gamma$ sample used
for its determination.  (For high photon $E_T$, however, the error is
dominated by the available statistics.)  This probability is expected
to remain approximately the same in Run~2 for both the CDF and D\O\
detectors.  We used 80\% ID efficiency for the pair of photons, and
required the photons to be isolated from possible extra jets in the
event.  We accounted for NLO corrections via a constant $k$-factor of
1.34.


\begin{figure}[t]
\epsfysize=3.5in
\epsffile[-250 0 100 560]{matchev/95cl_et.ps}
\begin{center}
\caption[]
{\small The 95\% CL upper limit on the signal cross section after cuts
as a function of the diphoton invariant mass $M_{\gamma\gamma}$,
for several benchmark total integrated luminosities in Run~2. 
\label{ggX}}
\end{center}
\end{figure}


Adding all background contributions, for the total background in the inclusive
diphoton channel we obtain the following parametrization:
$$
{d\sigma\over dM_{\gamma\gamma}}
= \biggl[p_3+p_4 \left({M_{\gamma\gamma}\over 1\ {\rm GeV}}\right)
      +p_5 \left({M_{\gamma\gamma}\over 1\ {\rm GeV}}\right)^2\biggr]\
\exp\,\biggl\{ p_1+
    p_2 \left( {M_{\gamma\gamma}\over 1\ {\rm GeV}}\right)\biggr\},
$$
where $p_1= 6.45$, $p_2=-0.029$, $p_3= 2.44$, $p_4=-0.011$ and $p_5=
0.00005$.  In the region $M_{\gamma\gamma}>100$ GeV it is dominated by
direct diphoton production and hence is irreducible.  The expected
statistical plus systematic error on this background determination is
at the level of 25\%, based on the jet-faking photon probability
uncertainty.  For larger invariant masses, however, the accuracy is
dominated by the uncertainties in the direct diphoton production cross
section, which will be difficult to measure independently in Run~2, so
one will still have to rely on the NLO predictions.  On the other
hand, for narrow resonance searches one could do self-calibration of
the background by calculating the expected background under the signal
peak via interpolation of the measured diphoton mass spectrum between
the regions just below and just above the assumed resonance
mass. Therefore, in our case the background error will be purely
dominated by the background statistics. A combination of the
interpolation technique and the shape information from the theoretical
NLO calculations of the direct diphoton cross section is expected to
result in significantly smaller background error in Run~2.

The total background, as well as the individual contributions from
$\gamma\gamma$, $\gamma j$ and $jj$ production, are shown in
Fig.~\ref{qcd_background}.  Additional SM background sources to the
inclusive diphoton channel include Drell-Yan production with both
electrons misidentified as photons, $W\gamma\gamma$ production,
etc. and are all negligible compared to the QCD background.  The
absolute normalization of the background obtained by the above method
agrees well with the actual background measured by CDF and D\O\ in the
diphoton mode~\cite{Wilson,monopole}.

In Fig.~\ref{ggX} we show the 95\% CL upper limit on the differential
cross section after cuts 
$d(\varepsilon\times\sigma(\gamma\gamma+X))/dM_{\gamma\gamma}$ as a
function of the diphoton invariant mass $M_{\gamma\gamma}$, given the
above background prediction (here $\varepsilon$ is the product of the
acceptance and all efficiencies).  This limit represents $1.96\sigma$
sensitivity to a narrow signal when doing a counting experiment in 1
GeV diphoton mass bins.  This plot can be used to obtain the
sensitivity to any resonance decaying into two photons as follows. One
first fixes the width of the mass window around the signal peak which
is used in the analysis.  Then one takes the average value of the 95\%
C.L. limit in $d\sigma/dM_{\gamma\gamma}$ across the mass window from
Fig.~\ref{ggX} and multiplies it by $\sqrt{w/\mbox{GeV}}$, where $w$
is the width of the mass window\footnote{The square root enters the
calculation since the significance is proportional to the background
to the $-1/2$ power.}, to obtain the corresponding 95\% CL upper limit
on the signal cross-section after cuts.  Similar scaling could be used
if one is interested in the 3$\sigma$ or 5$\sigma$ reach.


\vspace{0.2in}
{\bf Inclusive channel: optimum mass window} \\ \nopagebreak

\begin{figure}[t]
\epsfysize=2.5in
\epsffile[-150 205 200 565]{matchev/sign.ps}
\begin{center}
\caption[]
{\small Significance $S/\sqrt{B}$ (in arbitrary units),
as a function of the mass window width $w$ (in units of $\Gamma$),
for a Breit-Wigner or Gaussian resonance. \label{fig:significance}}
\end{center}
\end{figure}

When searching for narrow resonances in the presence of large
backgrounds ($B$), the best sensitivity toward signal ($S$) is
achieved by performing an unbinned maximum likelihood fit to the sum
of the expected signal and background shapes.  However, simple
counting experiments give similar sensitivity if the size of the
signal ``window'' is optimized. For narrow resonances the observed
width\footnote{Notice that the width is defined so that the
cross-section at $\pm\Gamma/2$ away from the peak is a factor of 2
smaller than the peak value (FWHM). For a Gaussian resonance the width
is related to the variance $\sigma$ by
$\Gamma=2\sigma\sqrt{\ln4}\simeq 2.35\sigma$.}  $\Gamma$ is dominated
by the instrumental effects, and is often Gaussian.  The background in
a narrow window centered on the assumed position $M_0$ of the peak in
the signal invariant mass distribution could be treated as
linear. Therefore, the Gaussian significance of the signal,
$S/\sqrt{B}$, as a function of the window width, $w$, is given by:
\begin{equation}
{S\over \sqrt{B}} 
\sim {1\over\sqrt{w}}\ 
{1\over \sqrt{2\pi}\sigma}
\int_{M_0-w/2}^{M_0+w/2}d\sqrt{s} 
\exp\left(-{(\sqrt{s}-M_0)^2\over 2\sigma^2}\right)
\sim {1\over\sqrt{w/\Gamma}}\ 
{\rm erf}\left(\sqrt{\ln2}\ {w\over\Gamma}\right),
\label{signgaus}
\end{equation}
where erf$(x)$ is the error function
$$
{\rm erf}(x)={2\over\sqrt{\pi}}\int_0^x e^{-t^2} dt.
$$
The function (\ref{signgaus}) is shown in
Fig.~\ref{fig:significance}, and has a maximum at $w \approx 1.2\Gamma$,
which corresponds to $\pm 1.2(\Gamma/2)$ cut around the resonance maximum.

For resonances significantly wider than the experimental resolution,
the shape is given by the Breit-Wigner function, and in this case
the significance is:
\begin{equation}
{S\over\sqrt{B}}
\sim {1\over \sqrt{w}}
\int_{(M_0-w/2)^2}^{(M_0+w/2)^2}
{ds\over (s-M_0^2)^2+M_0^2\Gamma^2}
\sim {1\over \sqrt{w/\Gamma}}\arctan({w\over\Gamma}).
\end{equation}
This function, also shown in Fig.~\ref{fig:significance},
peaks at a similar value of $w$ ($w \approx 1.4\Gamma$).
We see that for both Gaussian and Breit-Wigner resonances,
the significance does not appreciably change when
using a $w = 1\Gamma-2\Gamma$ cuts. For our analysis
we shall use two representative choices: $w=1.2\Gamma$ 
and $w= 2\Gamma$ for the mass window, which we shall
always center on the actual Higgs mass. 

Clearly, one can do even better in principle, by suitably resizing and
repositioning the mass window around the bump in the combined $S+B$
distribution. Because of the steeply falling parton luminosities,
the signal mass peak is skewed and its maximum will appear somewhat
below the actual physical mass. In our analysis we choose
not to take advantage of these slight improvements,
thus accounting for unknown systematics.

\vspace{0.2in}
{\bf Inclusive channel: results} \\ \nopagebreak

In Tables \ref{1sigma} and \ref{2sigma} we show
the inclusive $\gamma\gamma+X$ background rates in fb
for different Higgs masses, for $w=1.2\Gamma$ and $w=2\Gamma$
mass window cuts, respectively.
%
\begin{table*}[ht!]
\renewcommand{\arraystretch}{1.5}
\caption{ Background rates in fb for $w=1.2\Gamma$
mass cut, and significance ($S/\sqrt{B}$, for 1 fb$^{-1}$ of data, and
assuming ${\rm B}(h\rightarrow\gamma\gamma)=100\%$) as
a function of the Higgs mass $M_h$. The signal consists of 
associated $Wh/Zh$ production and $WW/ZZ$ fusion. \label{1sigma}}
\begin{tabular}{||c||c||c||c|c|c|c||c|c|c|c||}
\noalign{\vskip-6pt}\hline
      & $\gamma\gamma+X$ 
      & \multicolumn{9}{c||}{Significance $S/\sqrt{B}$}  \\ \cline{3-11}
$M_h$ & bknd &  $\gamma\gamma+X$
      & \multicolumn{4}{c||}{$\gamma\gamma+1$ jet}
      & \multicolumn{4}{c||}{$\gamma\gamma+2$ jets}  \\ \cline{4-11}
(GeV) & (fb) & 
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$ 
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$  \\ \hline\hline
100. &271.7 & 16.5 & 31.3 & 34.2 & 36.3 & 35.4 & 31.9 & 36.4 & 35.1 & 31.1 \\ \hline
120. &166.6 & 13.0 & 24.4 & 26.7 & 28.5 & 28.5 & 24.7 & 29.5 & 29.2 & 26.7 \\ \hline
140. &103.0 & 10.7 & 20.1 & 22.4 & 23.9 & 23.9 & 20.6 & 24.0 & 23.7 & 21.0 \\ \hline
160. & 64.3 &  8.9 & 17.0 & 19.1 & 20.1 & 20.4 & 17.2 & 20.2 & 20.8 & 19.2 \\ \hline
180. & 40.5 &  7.3 & 13.5 & 15.0 & 16.2 & 16.5 & 13.6 & 16.3 & 16.7 & 16.2 \\ \hline
200. & 26.1 &  5.8 & 10.6 & 11.9 & 12.5 & 12.7 & 10.4 & 12.3 & 12.7 & 12.0 \\ \hline
250. &  9.4 &  3.7 &  6.7 &  7.5 &  7.9 &  8.2 &  6.6 &  7.9 &  8.7 &  8.4 \\ \hline
300. &  4.8 &  2.2 &  3.8 &  4.3 &  4.7 &  4.7 &  3.6 &  4.2 &  4.9 &  4.7 \\ \hline
350. &  2.3 &  1.5 &  2.8 &  3.1 &  3.3 &  3.3 &  2.4 &  2.8 &  3.0 &  3.4 \\ \hline
400. &  1.2 &  1.0 &  1.8 &  2.0 &  2.0 &  2.1 &  1.7 &  1.7 &  2.1 &  2.4 \\ \hline
\hline
\end{tabular}
\end{table*}

\begin{table*}[h!]
\renewcommand{\arraystretch}{1.5}
\caption{ The same as Table \ref{1sigma}, but for a
$w=2\Gamma$ mass window. \label{2sigma}}
\begin{tabular}{||c||c||c||c|c|c|c||c|c|c|c||}
\noalign{\vskip-6pt}\hline
      & $\gamma\gamma+X$ 
      & \multicolumn{9}{c||}{Significance $S/\sqrt{B}$}  \\ \cline{3-11}
$M_h$ & bknd &  $\gamma\gamma+X$
      & \multicolumn{4}{c||}{$\gamma\gamma+1$ jet}
      & \multicolumn{4}{c||}{$\gamma\gamma+2$ jets}  \\ \cline{4-11}
(GeV) & (fb) & 
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$ 
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$  \\ \hline\hline
100. &453.4 & 14.4 & 27.0 & 29.8 & 31.7 & 30.6 & 27.7 & 31.9 & 30.5 & 26.4  \\ \hline
120. &278.1 & 11.3 & 21.3 & 23.5 & 25.1 & 24.9 & 21.9 & 25.5 & 25.1 & 22.5  \\ \hline
140. &171.9 &  9.3 & 17.5 & 19.5 & 21.0 & 21.0 & 17.7 & 20.7 & 21.0 & 18.5  \\ \hline
160. &107.3 &  8.0 & 15.1 & 16.7 & 18.0 & 18.2 & 15.4 & 17.8 & 18.2 & 17.3  \\ \hline
180. & 67.6 &  6.6 & 12.2 & 13.7 & 14.6 & 14.9 & 12.4 & 14.3 & 15.2 & 14.2  \\ \hline
200. & 43.6 &  5.4 & 10.1 & 11.4 & 12.1 & 12.1 &  9.9 & 11.9 & 12.3 & 11.1  \\ \hline
250. & 15.7 &  3.6 &  6.5 &  7.3 &  7.7 &  7.8 &  6.4 &  7.6 &  8.5 &  8.0  \\ \hline
300. &  8.1 &  2.1 &  3.8 &  4.2 &  4.6 &  4.5 &  3.6 &  4.3 &  4.5 &  4.5  \\ \hline
350. &  3.9 &  1.6 &  2.7 &  3.0 &  3.4 &  3.4 &  2.6 &  3.1 &  3.5 &  3.1  \\ \hline
400. &  2.1 &  1.1 &  1.9 &  2.2 &  2.3 &  2.4 &  1.6 &  2.1 &  2.6 &  2.4  \\ \hline
\hline
\end{tabular}
\end{table*}

Here we have added the intrinsic width $\Gamma_h$ and the experimental
resolution $\Gamma_{\rm exp}=2\sqrt{\ln4}\times\sigma_{\rm exp}\simeq
2.35\times0.15\sqrt{2}\sqrt{E(\gamma)} \simeq 0.35\sqrt{M_h}$ in
quadrature: $\Gamma=\left(\Gamma^2_h+\Gamma^2_{\rm exp}\right)^{1/2}$.
The width $\Gamma$ varies between 3.5 GeV for $M_h=100$ GeV and 29.0
GeV for $M_h=400$ GeV. The two tables also show the significance (for
1 fb$^{-1}$ of data, and assuming ${\rm
B}(h\rightarrow\gamma\gamma)=100\%$) in the inclusive diphoton channel
when only associated $Wh/Zh$ production and $WW/ZZ\rightarrow h$
fusion are included in the signal sample. We see that (as can also be
anticipated from Fig.~\ref{fig:significance}) a $w=1.2\Gamma$ cut
around the Higgs mass typically gives a better statistical
significance, especially for lighter (and therefore more narrow) Higgs
bosons.

\begin{figure}[ht!]
\begin{center}
\epsfysize=2.5in
\epsffile[-150 205 200 565]{matchev/nobj.ps}
\end{center}
\caption[]
{\small The number of ``jets'', which stands for
QCD jets, tau jets and electrons, in associated $Wh$
production, once we require the two photons from the
Higgs to pass the photon ID cuts. \label{nobj}}
\end{figure}

\begin{figure}[h!]
\begin{center}			
\epsfysize=3.5in
\epsffile[-20 250 80 550]{matchev/rapidity_bw.ps}
\end{center}
\caption[]
{\small Pseudorapidity distribution of the two spectator jets
in (a) associated $Wh/Zh$ production and 
(b) $WW/ZZ\rightarrow h$ fusion. Here
$\eta(j_1)$ ($\eta(j_2)$) is the pseudorapidity of
the leading (next-to-leading) jet in the event.
\label{rapidity}}
\end{figure}

\vspace{0.2in}
{\bf Exclusive channels: event selection} \\ \nopagebreak

The next question is whether the sensitivity can be further improved
by requiring additional objects in the event.  The point is that a
significant fraction of the signal events from both associated $Wh/Zh$
production and $WW/ZZ$ fusion will have additional hard objects, most
often QCD jets.  In Fig.~\ref{nobj} we show the ``jet'' multiplicity
in associated $Wh$ production, where for detector simulation we have
used the SHW package with a few modifications as in~\cite{SHWmod}.
Here we treat ``jets'' in a broader context, including electrons and
tau jets as well.

Previous studies \cite{Wilson,D0} have required {\em two} or more
additional QCD jets. Here we shall also consider the signature with at
least {\em one} additional ``jet'', where a ``jet'' is an object with
$|\eta|<2$.  The advantages of not requiring a second ``jet'' are
twofold. First, in this way we can also pick up signal from
$WW/ZZ\rightarrow h$ fusion, whose cross-section does not fall off as
steeply with $M_h$, and in fact for $M_h>200$ GeV is larger than the
cross-section for associated $Wh/Zh$ production\footnote{In the case
of a topcolor Higgs (see the next section) we would also pick up
events with initial state gluon radiation, comprising about 30\% of
the gluon fusion signal, which is the dominant production process for
any Higgs mass.}.  Events from $WW/ZZ\rightarrow h$ fusion typically
contain two very hard forward jets, one of which may easily pass the
jet selection cuts. This is illustrated in Fig.~\ref{rapidity}, where
we show the pseudorapidity distribution of the two spectator jets in
(a) associated $Wh/Zh$ production and (b) $WW/ZZ\rightarrow h$ fusion.
Second, by requiring only one additional jet, we win in signal
acceptance. In order to compensate for the corresponding background
increase, we shall consider several $p_T$ thresholds for the
additional jet, and choose the one giving the largest significance.

For the exclusive channels we need to rescale the background from
Fig.~\ref{qcd_background} as follows.  From Monte Carlo we obtain
reduction factors of $4.6\pm0.5$, $6.2\pm1.0$, $7.6\pm1.4$, and
$8.6\pm1.5$ for the $\gamma\gamma+1$ jet channel, with $p_T(j)>20$,
25, 30 and 35 GeV, respectively. For the $\gamma\gamma+2$ jets channel
the corresponding background reduction is $21\pm5$, $38\pm12$,
$58\pm21$, and $74\pm26$, depending on the jet $p_T$ cuts. These
scaling factors agree well with those from the CDF and D\O\ data from
Run 1.

Notice that we choose not to impose an invariant dijet mass ($M_{jj}$)
cut for the $\gamma\gamma+2$ jets channel.  We do not expect that it
would lead to a gain in significance for several reasons. First, given
the relatively high jet $p_T$ cuts needed for the background
suppression, there will be hardly any background events left with
dijet invariant masses below the (very wide) $W/Z$ mass window.
Second, the signal events from $WW/ZZ$ fusion, which typically
comprise about $25-30\%$ of our signal, will have a dijet invariant
mass distribution very similar to that of the background. Finally, not
imposing the $M_{jj}$ cut allows for a higher signal acceptance
because of the inevitable combinatorial ambiguity for the events with
$>2$ jets.


The significances for the two exclusive channels, with the four
different jet $p_T$ cuts, are also shown in Tables \ref{1sigma} and
\ref{2sigma}.  We see that the exclusive $\gamma\gamma+2$ jets channel
with $p_T(j)>30$ GeV typically gives the largest significance, but our
new exclusive $\gamma\gamma+1$ jet channel is following very close
behind.




\vspace{0.2in}
{\bf Exclusive channels: results} \\ \nopagebreak

We are now ready to present our results for the Run~2 Tevatron reach
for a bosonic Higgs.  In Fig.~\ref{bos} we show the 95\% CL upper
limit on the branching ratio ${\rm B}(h\rightarrow \gamma\gamma)$,
with 0.1 (cyan), 2.0 (green) and 30 ${\rm fb}^{-1}$ (red), as a
function of $M_h$. For each mass point, we compare the significance
for both the inclusive as well as the exclusive channels with all the
different cuts, and for the limit we choose the channel with the set
of cuts providing the best reach.  
\vspace{0.4in}

\begin{figure}[hb!]
\centerline{\psfig{file=matchev/bos.ps,width=9cm}}
\begin{center}
\caption[]
{\small 95\% CL upper limit on the branching ratio
${\rm B}(h\rightarrow \gamma\gamma)$, 
with 0.1 (cyan), 2.0 (green) and 30 ${\rm fb}^{-1}$ (red), 
as a function of $M_h$. For each mass point, we compare the
significance for both the inclusive and the exclusive
channels with different cuts, and for the limit we choose the
set of cuts which provides the best reach, denoted by
o: $2\gamma+2j$, with $p_T(j)>25$ GeV and $w=1.2\,\Gamma$,
$\Box$: $2\gamma+2j$, with $p_T(j)>30$ GeV and $w=1.2\,\Gamma$, and
$\Diamond$: $2\gamma+2j$, with $p_T(j)>30$ GeV and $w=2.0\,\Gamma$.
The solid line is the prediction for 
the branching ratio of a ``bosonic'' Higgs.
\label{bos}}
\end{center}
\end{figure}


It turns out that for this case
the optimal selections are
\begin{itemize}
 \item[o:]          $2\gamma+2j$, with $p_T(j)>25$ GeV and $w=1.2\,\Gamma$, 
 \item[$\Box$:]     $2\gamma+2j$, with $p_T(j)>30$ GeV and $w=1.2\,\Gamma$, and 
 \item[$\Diamond$:] $2\gamma+2j$, with $p_T(j)>30$ GeV and $w=2.0\,\Gamma$. 
\end{itemize} 
In the figure we also show the HDECAY
\cite{hdecay} prediction for ${\rm B}(h\rightarrow \gamma\gamma)$ in
case of a ``bosonic'' Higgs.  The reach shown for 0.1 ${\rm fb}^{-1}$
is intended as a comparison to Run 1, in fact for the 0.1 ${\rm
fb}^{-1}$ curve we scaled down both the signal and background
cross-sections to their values at 1.8 TeV center-of-mass energy,
keeping the efficiencies the same. In other words, the region marked
as Run 1$^\prime$ would have been the hypothetical reach in Run 1, if the
improved Run~2 detectors were available at that time.  As seen from
Fig.~\ref{bos}, the reach for a ``bosonic'' Higgs (at 95\% CL) in
Run~2a and Run~2b is $\sim115$ GeV and $\sim125$ GeV,
correspondingly. This is a significant improvement over the ultimate
reach from LEP \cite{LEP limits} of $\sim 105$ GeV.



\subsubsection{Topcolor Higgs Bosons}

Here we consider the case of a ``topcolor'' bosonic Higgs, where the
Higgs also couples to the top and other heavy quarks~\cite{topmodels}.
We therefore include events from gluon fusion into our signal
sample. We used next-to-leading order cross-sections for gluon
fusion~\cite{hxsec}.

In Tables \ref{1sigmatop} and \ref{2sigmatop} we show the significance
(for 1 fb$^{-1}$ of data, and again assuming ${\rm
B}(h\rightarrow\gamma\gamma)=100\%$) in the inclusive and the two
exclusive channels, for the topcolor Higgs case.  Since gluon fusion,
which rarely has additional hard jets, is the dominant production
process, the inclusive channel typically provides the best reach.
However, the $2\gamma+1j$ channel is again very competitive, since the
additional hard jet requirement manages to suppress the background at
a reasonable signal cost.  We see that our new $2\gamma+1j$ channel
clearly gives a better reach than the $2\gamma+2j$
channel~\cite{Lauer,D0,Wilson}. For Higgs masses above $\sim180$ GeV,
it sometimes becomes marginally better even than the inclusive
diphoton channel. The specific jet $p_T$ cut and mass window size $w$
seem to be less of an issue -- from Tables~\ref{1sigmatop} and
\ref{2sigmatop} we see that $p_T(j)>25$, $p_T(j)>30$ GeV and
$p_T(j)>35$ GeV work almost equally well, and for $M_h\gsim 200$ GeV
both values of $w$ are acceptable.
\vspace{0.4in}

\begin{figure}[h!]
\centerline{\psfig{file=matchev/top.ps,width=9cm}}
\begin{center}
\caption[]
{\small The same as Fig.~\ref{bos}, but for
a topcolor Higgs, {\it i.e.} gluon fusion events are
included in the signal. The channels with the best
$S/\sqrt{B}$ ratio are: 
o: inclusive $2\gamma+X$, and 
$\Box$: $2\gamma+1j$, with $p_T(j)>30$ GeV;
both with $w=1.2\,\Gamma$.
\label{topcolor}}
\end{center}
\end{figure}

\begin{table*}[ht!]
\renewcommand{\arraystretch}{1.5}
\caption{ The same as Table \ref{1sigma}, but for a
topcolor Higgs, {\it i.e.} gluon fusion events are included in the
signal. \label{1sigmatop}}
\begin{tabular}{||c||c||c||c|c|c|c||c|c|c|c||}\noalign{\vskip-6pt}\hline
      & $\gamma\gamma+X$ 
      & \multicolumn{9}{c||}{Significance $S/\sqrt{B}$}  \\ \cline{3-11}
$M_h$ & bknd &  $\gamma\gamma+X$
      & \multicolumn{4}{c||}{$\gamma\gamma+1$ jet}
      & \multicolumn{4}{c||}{$\gamma\gamma+2$ jets}  \\ \cline{4-11}
(GeV) & (fb) & 
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$ 
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$  \\ \hline\hline
%     TOPCOLOR HIGGS,  +/-0.6 SIGMA CUTS   
100. &271.7 & 54.3 & 43.4 & 44.6 & 45.1 & 42.5 & 33.6 & 37.8 & 36.4 & 32.1\\ \hline
120. &166.6 & 42.5 & 35.3 & 36.9 & 37.3 & 35.3 & 26.5 & 30.8 & 30.2 & 27.5\\ \hline
140. &103.0 & 35.5 & 30.4 & 32.1 & 33.0 & 31.6 & 22.3 & 25.6 & 25.1 & 22.0\\ \hline
160. & 64.3 & 30.1 & 27.3 & 28.8 & 29.2 & 28.3 & 19.5 & 22.4 & 22.4 & 20.3\\ \hline
180. & 40.5 & 24.5 & 22.4 & 23.8 & 24.5 & 23.8 & 15.7 & 18.2 & 18.3 & 17.6\\ \hline
200. & 26.1 & 19.9 & 18.6 & 19.9 & 20.3 & 20.0 & 12.7 & 14.4 & 14.3 & 13.3\\ \hline
250. &  9.4 & 12.9 & 12.5 & 13.3 & 13.6 & 13.4 &  8.4 &  9.7 & 10.2 &  9.8\\ \hline
300. &  4.8 &  9.1 &  8.7 &  9.3 &  9.6 &  9.4 &  4.9 &  5.6 &  5.9 &  5.9\\ \hline
350. &  2.3 &  8.4 &  7.9 &  8.4 &  8.4 &  7.9 &  4.0 &  4.1 &  4.5 &  4.5\\ \hline
400. &  1.2 &  6.4 &  6.1 &  6.1 &  6.3 &  6.2 &  3.0 &  3.0 &  3.5 &  3.1\\ \hline
\end{tabular}
\end{table*}

\begin{table*}[h!]
\renewcommand{\arraystretch}{1.5}
\caption{ The same as Table \ref{1sigmatop}, but for a
$w=2\Gamma$ mass window. \label{2sigmatop}}
\begin{tabular}{||c||c||c||c|c|c|c||c|c|c|c||}\noalign{\vskip-6pt}\hline
      & $\gamma\gamma+X$ 
      & \multicolumn{9}{c||}{Significance $S/\sqrt{B}$}  \\ \cline{3-11}
$M_h$ & bknd &  $\gamma\gamma+X$
      & \multicolumn{4}{c||}{$\gamma\gamma+1$ jet}
      & \multicolumn{4}{c||}{$\gamma\gamma+2$ jets}  \\ \cline{4-11}
(GeV) & (fb) & 
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$
      & $p_T>20$ & $p_T>25$ & $p_T>30$ & $p_T>35$ \\ \hline\hline
%     TOPCOLOR HIGGS,  +/-ONE SIGMA CUTS   
100. &453.4 & 47.7 & 37.6 & 38.9 & 39.5 & 36.9 & 29.0 & 33.0 & 31.6 & 27.1\\ \hline
120. &278.1 & 37.7 & 31.2 & 32.6 & 33.1 & 31.2 & 23.5 & 26.6 & 25.9 & 23.1\\ \hline
140. &171.9 & 31.4 & 26.7 & 28.2 & 29.0 & 27.9 & 19.2 & 22.1 & 22.0 & 19.4\\ \hline
160. &107.3 & 26.6 & 24.3 & 25.3 & 26.0 & 25.2 & 17.4 & 19.7 & 19.5 & 18.3\\ \hline
180. & 67.6 & 22.3 & 20.5 & 21.7 & 22.1 & 21.6 & 14.2 & 16.2 & 16.8 & 15.4\\ \hline
200. & 43.6 & 18.8 & 17.7 & 19.1 & 19.4 & 18.9 & 12.0 & 13.7 & 13.8 & 12.5\\ \hline
250. & 15.7 & 12.8 & 12.1 & 13.2 & 13.4 & 13.1 &  8.2 &  9.3 & 10.0 &  9.6\\ \hline
300. &  8.1 &  9.0 &  8.5 &  9.1 &  9.2 &  9.0 &  5.0 &  5.6 &  5.6 &  5.7\\ \hline
350. &  3.9 &  8.3 &  7.6 &  8.1 &  8.1 &  7.9 &  4.0 &  4.7 &  4.6 &  4.4\\ \hline
400. &  2.1 &  6.3 &  6.1 &  6.3 &  6.3 &  6.1 &  2.9 &  3.4 &  3.7 &  3.6\\ \hline
\end{tabular}
\end{table*}



In Fig.~\ref{topcolor} we show the Run~2 reach for the branching ratio
${\rm B}(h\rightarrow \gamma\gamma)$ as a function of the Higgs mass,
for the case of a ``topcolor'' Higgs boson. This time the channels
with the best signal-to-noise ratio are: o: inclusive $2\gamma+X$, and
$\Box$: $2\gamma+1j$, with $p_T(j)>30$ GeV; both with $w=1.2\Gamma$.

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

We have studied the Tevatron reach for Higgs bosons decaying into
photon pairs (for an alternative study with similar results, see
\cite{MW}). For purely ``bosonic'' Higgses, which only couple to gauge
bosons, the $2\gamma+2j$ channel offers the best reach, but the
$2\gamma+1j$ channel is almost as good.  For topcolor Higgs bosons,
which can also be produced via gluon fusion, the inclusive $2\gamma+X$
channel is the best, but the $2\gamma+1j$ channel is again very
competitive.  We see that in both cases the $2\gamma+1j$ channel is a
no-lose option!


