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\begin{document}



\pagestyle{empty}

\noi DESY 97-227

\BS\BS

\noi November 1997
\section*{
\vspace{4cm}
\begin{center}
\LARGE{\bf
 Higgs Physics at a 300 - 500 GeV\\ $e^+e^-$ Linear Collider
          }\\
\end{center}
}

\vspace{2.5cm}
\large
\begin{center}
H. J\"urgen\, Schreiber${}^*$\\
\bigskip \bigskip  

DESY- Zeuthen, Germany \\
\end{center}
\vfill
{$*$ Contribution  to the Proceedings of the Internat. Europhysics Conf.
on High Energy Physics,
~16-26 August 1997, Jerusalem, Israel}

\newpage
\pagestyle{plain}
\pagenumbering{arabic}
%----------------------------------------------------------------------
%----------------------------------------------------------------------
% ABSTRACT
%----------------------------------------------------------------------
%----------------------------------------------------------------------
\section*{Abstract}

We summarize discovery potentials for the Standard Model Higgs boson
produced in $e^+e^-$ collisions and measurements of its detailed
properties. Prospects for Higgs boson detection within the Minimal
Supersymmetric Standard Model are also discussed.
\BS\BS\BS

%----------------------------------------------------------------------
%----------------------------------------------------------------------

\large
%----------------------------------------------------------------------
% INTRODUCTION
%----------------------------------------------------------------------
\section{Introduction}


Electroweak breaking due to the Higgs mechanism \cite{Higgs} implies the
existence of at least one new particle, the Higgs boson. Its discovery
is the most important missing link for the formulation of electroweak
interactions. LEP has established a lower bound of the Higgs mass $M_H
\gess$ 77 GeV  \cite{lowM}. High precision data, interpreted within the
Standard Model (SM), favour a Higgs boson with a mass somewhere
between 100 and 180 GeV.

Future $e^+e^-$ linear colliders are the ideal machines
for a straightforward
Higgs boson discovery, its verification and precision measurements
of the Higgs sector properties.

Recent simulation studies \cite{our} involve, thanks to the effort
of several groups,\\
i) the full matrix elements for 4-fermion final states,
ii) all Higgs decay modes (with SM decay fractions $\gess $ 1\%),
iii) initial state QED and beamstrahlung (TESLA design), 
iv) a detector response \cite{CDR} and
v) all important background expected to contribute.


%----------------------------------------------------------------------
% chapter 1
%----------------------------------------------------------------------
\section{SM Higgs discovery potentials}

The Higgs boson can be produced by the Higgsstrahlung process $e^+e^-$
$\rightarrow Z^* \rightarrow ZH^o$  (1), or by the fusion of WW and
ZZ bosons,   $e^+e^- \rightarrow \nu \bar{\nu} H^o$ (2) and 
$e^+e^- \rightarrow e^+e^- H^0$ (3), respectively,
or by 
  radiation off top quarks, $e^+e^- \rightarrow t \bar{t} H^o$, with
  however a very small cross section.
At e.g.
$\sqrt{s}$ = 360 GeV
and $M_H$ = 140 GeV, the Higgsstrahlung process is about four times
more important than the fusion reactions. Reaction (1) 
admits two strategies for the Higgs search: i) calculation of the recoil mass
against the $Z \rightarrow e^+e^- / \mu^+\mu^-, M^2_{rec} =$
$ s - 2\sqrt{s} (E_{l^+} + E_{l^-}) + M^2_Z$, which is independent of
assumptions about Higgs decay modes, and ii) 
the direct reconstruction of the invariant mass of the Higgs decay
products.

In order to achieve the best experimental resolution, energy-momentum
as well as $M(l^+l^-) = M_Z$ constraints have been imposed when
appropriate, and to make a signal-to-background analysis as meaningful
as possible a consistent evaluation of the signal
and all expected background
rates has been made \cite{our}.

The leptonic channel, $e^+e^- \rightarrow ZH^o \rightarrow $
$(e^+e^- / \mu^+\mu^-) (b\bar{b})$, allows Higgs detection either in
the mass recoiling against the Z or in the hadronic two b-quark jet
mass. Typically, an integrated luminosity of $\sim 10 fb^{-1}$ is
needed  to observe  the Higgs boson with a significance
$S/\sqrt{B} > 5$ after application of appropriate selection
procedures \cite{our}.

The tauon channel,  $e^+e^- \rightarrow ZH^o \rightarrow (q\bar{q})$
$(\tau^+\tau^-)$, requires some more refined selection procedures
\cite{our, Hil} due to missing neutrinos from $\tau$ decays.
Hence, the accumulated luminosity to observe a clear $H^o$ signal
in the recoil resp. $\tau^+\tau^-$  mass should be between 50 to
80 $fb^{-1}$. Energy-momentum constraints are required to determine
the original $\tau$ and jet energies by a fit; without that no $H^o$
signal would be visible.

The missing energy channels, $e^+e^- \rightarrow ZH^o \rightarrow$
$(\nu\bar{\nu})(b\bar{b})$ and $e^+e^- \rightarrow \nu\bar{\nu} H^o
\rightarrow \nu\bar{\nu} (b\bar{b})$ are very important due to their
large discovery potential for the Higgs boson. It has been
demonstrated \cite{our,Janot} that with $\sim 1 fb^{-1}$ of integrated
luminosity a convincing signal over some small remaining background in
the doubly-tagged b-jet mass should be obtained (Fig.1).
%neutrinos
%---------------------------------------------------------------------
\begin{figure*}[h!t]
\begin{center}
\mbox{\epsfxsize=17cm\epsfysize=17.5cm\epsffile{hjsfig1.eps}}
\end{center}
\caption[ ]{\sl 2-jet  mass distributions of the
reaction \ee \into \nn\qq,
~for an integrated luminosity of 1 fb$^{-1}$ at $\sqrt{s} =$ 360 GeV.
Background contributions are
included.}
\label{fig:neutrinos}
\end{figure*}
Thus, few days
of running a 300 GeV $e^+e^-$ linear collider at its nominal
luminosity would suffice to
 discover the SM Higgs in the intermediate mass range.

With increasing accumulated luminosity other $H^o$ decay modes like
$WW^{(*)}$ and $c\bar{c}$ + gg would be measurable, allowing to
 verify the Higgs
interpretation; 50 to 100 $fb^{-1}$ are needed for
significant signals. As an example, Fig.2 shows the signal from
 selected $H^o \rightarrow
c\bar{c}$ + gg decays in presence of a huge background
in the 2-jet missing energy event topology.
%gluon-cc
%---------------------------------------------------------------------
\begin{figure*}[h!t]
\begin{center}
\mbox{\epsfxsize=17cm\epsfysize=17.5cm\epsffile{hjsfig2.eps}}
\end{center}
\caption[ ]{\sl 2-jet  mass distribution of the reaction
\ee \into \nn(\cc + \glgl) for an integrated luminosity of 100 fb$^{-1}$
at $\sqrt{s} =$ 360 GeV.
The shaded histogram represents the reducible background expected.}
\label{fig:gluon-h}
\end{figure*}

Observation of the $H^o \rightarrow \gamma\gamma$ decay requires large
statistics $(> 200 fb^{-1})$ and a fine-grained electromagnetic
calorimeter with very good energy resolution \cite{Gunion}.



%----------------------------------------------------------------------
% chapter 2
%----------------------------------------------------------------------
\section{Higgs properties}


If a Higgs candidate is discovered,
 it is imperative to understand its nature. In
the following we assume an integrated luminosity of $100 fb^{-1}$.

Measurements of the Higgs boson mass are best performed by searching for
$Z \rightarrow l^+l^- (l=e/\mu)$ decay products and reconstructing the recoil
mass peak. The mass resolution  depends on
$\sqrt{s}$ and 
the detector performance, in particular on the momentum resolution of
the tracking system. For $M_H = 140$ GeV and $\sqrt{s}$ = 360 GeV,
 we expect
an error for the Higgs mass of $\sim$ 50 MeV for the detector design
of ref.\cite{CDR}.

When the Higgsstrahlung $ZH^o$ rate is significant, the CP-even
component of the $H^o$ dominates. It is possible to cross check this
by studying the Higgs production and the Z decay angular
distributions. Thus, observation of angular distributions as expected
for a pure CP-even state  implies that the
Higgs has spin-0 and that it is not primarily CP-odd \cite{Kramer}.
Further, studying decay angular correlations between the decay
products of the tauons in the reaction
 $e^+e^- \rightarrow ZH^o \rightarrow Z
\tau^+ \tau^-$ provides a democratic probe of the Higgs CP-even and
CP-odd components \cite{Kramer,JFG}. If the $H^o \rightarrow \gamma
\gamma$ decay is visible (or the Higgs is 
produced in $\gamma \gamma$ collisions)
the Higgs must be a scalar and has a CP-even component. Whether a
CP-odd component also exists can be studied by comparing Higgs
production rates in $\gamma \gamma$ collisions with different photon
polarizations \cite{JFG2}.

The determination of the branching fraction of the
 Higgs into a final state X requires to
compute $BF(H \rightarrow X) = [\sigma(ZH) \cdot BF (H \rightarrow
X)] / \sigma (ZH)$, where $\sigma (ZH)$ is the inclusive Higgs
cross section, see Fig. 3. Its error is expected to be $\pm 5\%$
\cite{our}, while the numerator $\sigma (ZH)\cdot BF (H \rightarrow
X)$ and its precision can be obtained from the  X invariant mass
distribution.
%inclusive
%---------------------------------------------------------------------
\begin{figure*}[h!t]
\begin{center}
\mbox{\epsfxsize=17cm\epsfysize=17.5cm\epsffile{hjsfig3.eps}}
\end{center}
\caption[ ]{\sl Inclusive recoil mass distribution in the process
\ee \into \znull(\into \mm/\ee) + anything for an integrated
luminosity of 100 fb$^{-1}$ and $|cos \Theta_Z| <$ 0.8
at $\sqrt{s} =$ 360 GeV.
The shaded histogram represents the reducible background expected,
 whereas the Higgs contribution is shown cross-hatched.
The irreducible background for the \hnull \into WW channel has been
neglected.}
\label{fig:inlusive}
\end{figure*}

Typical statistical branching fraction uncertainties expected for a
140 GeV SM Higgs boson are \cite{our} \\
\\

\begin{tabular}{|c|c|c|c|}\hline \par
%\vspace*{10mm}
$BF(H\rightarrow b \bar{b})$ & $BF(H\rightarrow \tau^+\tau^-)$ & 
$BF(H\rightarrow WW^*)$ & $BF(H\rightarrow c \bar{c} + gg)$ \\ \hline
$\pm 6.1 \%$ & $\pm 22 \%$ & $\pm 22 \%$ & $\pm 28 \%$  \\ \hline
\end{tabular} \\

\vspace{1.0cm}
As can be seen, many of the Higgs decay modes are accessible to
experiment and the measurements allow a
discrimination between the SM-like and e.g. SUSY-like Higgs
bosons. However, more fundamental than branching fractions are the
Higgs partical widths. If $M_H \gess 140$ GeV, $\Gamma(H \rightarrow WW)$
is measurable from the WW fusion cross section, whereas for $M_H \less 
130$ GeV a $\gamma \gamma$ collider allows to measure $\Gamma(H
\rightarrow \gamma \gamma)$.   
Combined with the corresponding branching fraction (from LHC or NLC),
$\Gamma_{tot}^{Higgs}$ is calculable, which in turn allows the
determination of the remaining  partial widths. In addition, the
$H^o \rightarrow ZZ$ coupling-squared is obtained from 
$\sigma (ZH)$ with an error comparable to that of the inclusive
cross section.

%----------------------------------------------------------------------
% chapter 4
%----------------------------------------------------------------------
\section{MSSM Higgs bosons: $h^o, H^o, A^o, H^{\pm}$}


In the minimal supersymmetric scenario five Higgs particles are
expected, two CP-even states $h^o, H^o$ (with $M_{h^o} < M_{H^o})$, one
CP-odd state $A^o$ and two charged states $H^{\pm}$. An upper bound on
the mass of the $h^o$, including radiative corrections \cite{bound},
is predicted to be $M_{h^o} \less$ 130 GeV.
If $M_{h^o}$
exceeds 130 GeV, the MSSM is ruled out.

In the limit of large $M_{A^o}( \gg M_Z)$ the properties of the $h^o$
are very close to those of the SM Higgs and a discrimination between
the two states is only possible if precise measurements show $h^o
\neq H^o_{SM}$ or further Higgs states are discovered.

Besides Higgsstrahlung and fusion production mechanisms for $h^o$ and
$H^o$, Higgs pair production $e^+e^- \rightarrow h^o A^o,\, H^oA^o,\,
H^+H^-$ is possible, with the relations $\sigma (e^+e^- \rightarrow
h^oZ, h^o\nu\bar{\nu}, H^oA^o) \propto sin^2 (\beta - \alpha)\, (I)$ and
$\sigma (e^+e^- \rightarrow
H^oZ, H^o\nu \bar{\nu}, h^oA^o) \\ \propto
 cos^2 (\beta - \alpha)\, (II)$
$( \alpha, \beta$ are mixing angles). One process in each line (I),
(II) always has a substantial rate and the sum of the $h^o$ or $ H^o$
cross sections is constant and equals  $\sigma (H^o_{SM})$ over the
whole $(M_A, \tan{\beta})$-plane. Therefore, a $h^o$ or $ H^o$ cannot
escape detection if it is kinematically accesible.

Concerning the branching fractions it is usually assumed that the
masses of the supersymmetric particles are so heavy that SUSY-Higgses
cannot decay into s-particles; only decays into standard particles are
possible. Whatever the masses of the Higgs particles and tan$\beta$
values are, we expect i) b-quark decays of $h^o, H^o$ and $A^o$ to occur in
large regions of the parameter space, underlining the importance of
excellent b-tagging, ii) vector boson decays are never significant for
the $A^o$ while strong restrictions exist for the $H^o$ and iii) if 
$M_{H^{\pm}} \gess$ 180 GeV, the $H^{\pm}\rightarrow tb$ decay rate is large
leading in most cases via $ e^+e^- \rightarrow H^+H^-  \rightarrow 
W \,b \bar{b}\, W \,b \bar{b}$ to 8-jet final states involving four b-quark
jets, a very challenging task to recognize and
analyze these events. An example for MSSM Higgs boson detection is
shown in Fig.4 for the process $ e^+e^- \rightarrow H^oA^o \rightarrow
b \bar{b} b \bar{b}$ at $\sqrt{s} =$ 800 GeV \cite{CDR}. 
%eeHA
%---------------------------------------------------------------------
\begin{figure*}[h!t]
\begin{center}
\mbox{\epsfxsize=17cm\epsfysize=17.5cm\epsffile{hjsfig4.eps}}
\end{center}
\caption[ ]{\sl Distribution of the di-jet invariant mass for the
jet pair combination with better compatibility with the
reaction $ e^+e^- \rightarrow H^oA^o \rightarrow b \bar{b} b \bar{b}$
at $\sqrt{s} =$ 800 GeV.}
\label{fig:eeHA}
\end{figure*}


If however
s-particle decays of Higgses are possible, the decays end (in many
SUSY models) with the lightest 'neutrino-like' supersymmetrie
particle. Hence, event signatures are characterized by missing
$p_{\bot} / E_{\bot}$ + leptons + jets which warrant further
experimental simulations.


%-----------------------------------------------------------------------
%CONCLUSIONS
%-----------------------------------------------------------------------
\section{Conclusions}

An  $ e^+e^-$ collider focusing on  $ e^+e^- \rightarrow ZH^o$
production in combination with a $\gamma \gamma$ collider allows to
fully explore the properties of an intermediate SM Higgs boson in the
shortest time.

Within the MSSM, one light CP-even Higgs boson must be found or the
MSSM is ruled out. A 500 GeV collider offers very good prospects to
discover all five MSSM Higgs particles, $h^o, H^o, A^o, H^{\pm}$, if 
$M_A \less $ 220 GeV.
If these particles are discoverd large statistics experiments are needed
to measure their parameters.


%-----------------------------------------------------------------------
% ---- Bibliography ----
%-----------------------------------------------------------------------
\begin{thebibliography}{99}

\bibitem{Higgs} P.W.Higgs, Phys.Rev.Lett. 12 (1964) 132 and
Phys. Rev. 145 (1966) 1156.

\bibitem{lowM} W.Murray, Search for the SM Higgs, presented at the
Int. Europhysics Conf. on High Energy Physics, 16-26 August 1997,
Jerusalem, Israel.

\bibitem{our} M.Sachwitz, H.J.Schreiber and S.Shichanin, Contribution
to the Proceedings of the 'ECFA/DESY Study on Physics and
Detectors for the Linear Collider', DESY 97-123E, ed. by R.Settles,
.

\bibitem{CDR} Conceptual Design of a 500 GeV $e^+e^-$ Linear Collider
with Integrated X-ray Laser Facility, DESY 1997-048, ECFA 1997-182,
editors: R.Brinkmann, G.Materlik, J.Rossbach and A.Wagner.

\bibitem{Hil} H.D.Hildreth, T.L.Barklow and D.L.Burke, Phys.Rev.
D49 (1994) 3441.

\bibitem{Janot} P.Janot, preprint LAL 93-38, July 1993.

\bibitem{Gunion} J.F.Gunion and P.Martin, preprint UCD-96-34, hep-ph/
9607360.

\bibitem{Kramer} M.Kramer et al., Z.Phys. C64 (1994) 21.

\bibitem{JFG} J.F.Gunion and B.Grzadkowski, Phys.Lett. B350 (1995) 218.

\bibitem{JFG2} J.F.Gunion and B.Grzadkowski, Phys.Lett. B294 (1992)
361.

\bibitem{bound} e.g. A.Djouadi, J.Kalinowski and P.M.Zerwas,
Z.Phys. C70 (1996) 435.

\end{thebibliography}




\end{document}
 













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\begin{document}







\pagestyle{empty}

\noi DESY 97-??

\BS\BS

\noi January 1997
\section*{
\vspace{4cm}
\begin{center}
\LARGE{\bf
 The Standard Model Higgs: \\
Discovery Potentials and Branching Ratio Measurements at the NLC
       }\\
\end{center}
}

\vspace{2.5cm}
\large
\begin{center}
 M. Sachwitz, H. J. Schreiber and S. Shichanin\\ 
\bigskip \bigskip  

DESY-Institut f\"{u}r Hochenergiephysik, Zeuthen, FRG \\
\end{center}
\newpage
\pagestyle{plain}
\pagenumbering{arabic}
%----------------------------------------------------------------------
%----------------------------------------------------------------------

%----------------------------------------------------------------------
% ABSTRACT
%----------------------------------------------------------------------
%----------------------------------------------------------------------
\section*{Abstract}
\BS\BS\BS

\large
%----------------------------------------------------------------------
% INTRODUCTION
%----------------------------------------------------------------------
\section{Introduction}

The fundamental particles in the Standard Model (SM) \cite{sm}, 
the gauge bosons, the leptons and quarks, acquire their mass by 
means of the Higgs mechanism \cite{higgs}.
This mechanism require the existence of a real physical particle, the
Higgs boson, and its discovery  is the most important experiment 
for the standard formulation of the electroweak interactions.

However, in the SM the mass of the Higgs particle is unknown.
A lower bound on the Higgs mass of about \mh $\ge$ 66 GeV has been
established so far from LEP I \cite{leplimit}; this limit can be raised
to $\sim$ 95 GeV in the second phase of LEP with a total energy of
\SQRTS = 192 GeV \cite{cavena}.
Beyond the LEP II range, the multi-TeV $pp$ collider LHC can cover the
entire Higgs mass range up to the SM limit of about 800 GeV
\cite{froidevaux}. 
For \mh \, above 150 GeV, the lepton channels \hnull \into 4$l^{\pm}$
will be used, while below that value the rare photon decay \hnull \into
\gaga \, is the sole decay channel, established so far.
Due to the overwhelming QCD background, other \hnull \, decay modes
cannot be explored or are difficult to detect. 
\footnote{In a recent study \[S.Moretti, preprint Cavendish-HEP-95/18,
  Nov. 1996\] it has been emphasized that also the \hnull $\rightarrow
  b\bar{b}$ decay mode is viable for the Higgs detection.}
It is this intermediate mass range below 200 - 180 GeV which is
theoretically the most attractive region for Higgs masses.
A lower bound on the Higgs mass in the SM follows from the requirement
of vacuum stability for large top quark masses.
For a top mass of $\sim$ 180 GeV, the Higgs mass must exceed $\sim$
110 GeV.

Future \ee \, linear colliders operating in the 300 to 500 GeV 
center-of-mass energy range,
 denoted as NLC (for Next Linear Collider) in the following, are
the ideal machines to investigate the Higgs sector in the intermediate
mass range since it can be easily discovered and all major decay modes
can be explored.
The confirmation of the existence of the Higgs boson and the study of
its properties are therefore of primary importance in the physics
program at the NLC.

In order to access the actual capability of such a collider concept 
it is extremely interesting and appealing to apply various tools,
thanks to the effort of several groups, to an analysis of SM Higgs
boson physics. In particular, we have included in our simulations
\begin{itemize}
\item{the full matrix elements for 4-fermion find states beyond the
    usual approximation of computing production error sections times
    branching ratios of the Higgs into decay products;}
\item{initial state QED and beamstrahlung,}
\item{a detector response, with parameters of the detector as designed
    on a series of workshops for the conceptual design for an \ee
    -linear collider;}
\item{all reducible backgrounds expected}
\end{itemize}

In particular, we assume, as an example, an \ee \, center-of-mass (cm)
energy \SQRTS = 360 GeV and a Higgs mass of \mh = 140 GeV.
We first describe machine and detector parameters and event
simulations.
We then present our analysis in detail.
It is shown that the Higgs signal can be observed already with limited
integrated luminosity and that the inclusive cross 
section for Higgs production
can be determined with high accuracy without explicit assumptions on
the Higgs decay modes.
Once the Higgs is found it is of very importance to measure the
(relative) Higgs couplings to gauge bosons and fermions via the
appropriate decay branching rations.
We demonstrate with which accuracy these branching ratios can easily
be measured, allowing a potential discrimination of the SM Higgs
particle from the lightest CP-even MSSM Higgs over a large range in
tan$\beta$. 
According to our concern, both the signal and background
rates for all reactions expected to contribute at \SQRTS = 360 GeV are
simulated. it is assumed that \SQRTS has a value such 
that \toto \, pair production as
additional background is kinematically not possible.



%----------------------------------------------------------------------
% chapter 2
%----------------------------------------------------------------------
\section{The center-of-mass energy spectrum and luminosity}

The extremely high density to which the beam particles must be focused
to produce sufficient luminosity for particle physics at a NLC results
in a significant interaction rate between the particles of one beam
and the collective electromagnetic fields produced by the particles in
the opposite beam.
A major consequence of this is that the cm energy spectrum at which
\ee \, interaction takes place is not at all monochromatic.
Radiation of photons during the beam-beam collision (beamstrahlung)
will result in a \SQRTS-spectrum that depends in detail on the energy
and bunch characteristics of each beam.
In our study we have used, as an example, the expectations of the beam
parameters for the TESLA design \cite{tesla}, which are typical of
most machine designs studied so far.
In particular, the beamstrahlung simulation includes the effects of
multiple radiation and beam-beam disruption \cite{schulte}
 and a luminosity of 5
10$^{23}$ cm$^{-2}$sec$^{-1}$\cite{tesla}.

Besides beamstrahlung, a further reduction of the initial state energy
occurs due to radiation of photons off the beam particles (ISR).
In our simulation, ISR as suggested in \cite{kuraev} has been included
and convoluted with the beamstrahlung spectrum. 
The net result of both effects onto the em energy can be seen for
various reactions in Fig. 1. Here, the hatched histograms reveal the
effect due to beamstrahlung for the TESLA design alone. As expected,
and in contrast to ISR effects, beamstrahlung energy degradation is
reaction-independent. It amounts to typically 1/3 of the total energy
loss during beam-beam collision provided the event-related cm energy
cannot be kinematically close to the $Z^o$pole by photon radiation.


In this study, all cross sections resp event rates 
have been computed by convoluting the luminosity
spectrum with each cross section evaluated at the reduced cm energy.
Event rates of reactions with a logarithmic cross section
s-dependence are only little affected by beamstrahlung and ISR.
However, most of our signal and background rates are more sensitive to
the details of the cm energy distribution.
However, varying this spectrum within reasonable ranges we find that the
conclusions that we draw are not altered.

The luminosity is assumed to be 5 10$^{33}$cm$^{-2}$sec$^{-1}$
\cite{tesla}.
Typically, for a year of running (of 10$^7$sec) we obtain an integrated
luminosity of 100 fb$^{-1}$ within two years.
This is considered as our default luminosity for a comprehensive study of the
Higgs particle.




%----------------------------------------------------------------------
% chapter 3
%----------------------------------------------------------------------
\section{Detector response}
 
The properties of a detector that are necessary to successfully carry
out analysis of  the Higgs boson physics 
at the NLC have been developed within
the course of the 'Conceptual Design for an \ee \, Linear Collider and
Detector' of the European high-energy physics community \cite{cdreport}.
The basic components of the detector are a 
vertex detector, a tracker, an electromagnetic and a hadronic 
calorimeter, a muon detector and a luminosity counter.
The parameters of the detector components  
(resolutions, acceptances and granularities) are summarized in Table 1.
As can be seen, the detector properties assumed do not extend 
any technology beyond that now in existence or are feasible in near future.
We can also anticipate that improvements in the 
techniques and technologies used in design and construction 
of detector components will occur.
The emphasis in our analysis is on the tracker, the vertex 
detector and the calorimeters.


\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|lll|}
\hline

&&& \\
Vertexing & $\sigma_{imp} (R\phi)$ & = & 10 $\mu m \oplus 30 \mu m$
$/psin^{3/2} \Theta  $  \\
&&& \\
 & $ |cos \Theta|$ & = & 0.95  \\ [2mm]
&&& \\

\hline
&&& \\
Tracking & $\frac{\sigma_{p \perp}}{p\perp}$ & = & 1.5 $\cdot 10^{-4}
$
 $ p_{\perp} (GeV)  \oplus 0.01 $ \\ [5mm]
& $|cos \Theta_{TRA}| $ & = & 0.95 \\ [5mm]
&  \multicolumn{3}{l|}{radius: $\:$  1.5 m}  \\ [5mm]
&  length: $ \pm$ & 2.5 m & \\
&&& \\
\hline
&&& \\
Electromagnet Calorimeter & $\frac{\sigma_E}{E}$ & = & 0.10 /
$\sqrt{E(GeV)} \oplus 0.01 $ \\ [5mm]
& \multicolumn{3}{l|}{cell size:  $0.7^o \cdot 7^o$ } \\ [5mm]
& $|cos \Theta|$ & = & 0.98481 $\; (\hat{=} 10^o)$ \\ [2mm]
&&& \\

\hline
&&& \\
Hadronic calorimeter & $\frac{\sigma_E}{E}$ & = & 0.50 /
$\sqrt{E(GeV)} \oplus 0.04 $ \\ [5mm]
& \multicolumn{3}{l|}{cell size:  $2^o \cdot 2^o$} \\ [5mm]
& $|cos \Theta|$ & = & 0.98481 $\; (\hat{=} 10^o)$ \\ [2mm]
&&& \\
\hline

&&& \\
Muon detector & \multicolumn{3}{l|}{Fe yoke instrumented as tail
  catcher} \\
& \multicolumn{3}{l|}{and muon tracker} \\
&&& \\

\hline
&&& \\
Luminosity counter & $ \frac{\sigma_E}{E}$ & = & 0.12 /
$\sqrt{E(GeV)}$
$\oplus 0.02 $ \\ [5mm]
& \multicolumn{3}{l|}{cell size:  $2^o \cdot 2^o$}  \\ [5mm]
& \multicolumn{3}{l|}{$4^o < \Theta  < 8^o;$  \hspace{1cm}
$172^o < \Theta < 176^o$}   \\ [2mm]
&&& \\
\hline
\end{tabular}
\end{center}
\caption[ ] {\sl Detector parameters. The symbol $\oplus$ means
  quadratic sum.}
\end{table}


The detector resonance is obtained by modifying
 the generated particle momenta, energies and nature in the following manner.
At first, particles within the acceptance regions are divided into isolated particles and energetic clusters.
Particles are called isolated if they have a 'large' distance to any other when entering the electromagnetic calorimeter, otherwise they are combined to clusters.
The isolation criteria is based on the cell size of the calorimeters.
For clusters, their electromagnetic and hadronic energy components are smeared by a Gaussian distribution separately according to the parameters of Table 1 and added up to the total cluster energy afterwards.
Their directions are modified according to the granularities. Note
that for charged isolated particles, the smeared momentum from the 
tracking system is used except in the case when a better energy value from the calorimeter exists.
We apply for the transverse momentum smearing the formula

\begin{equation}
\frac{\sigma p_{\perp}}{p_\perp} = A p_\perp + 1\%
\end{equation}

with A = 7 10$^{-5}$. This resolution parameter
 is expected from the combined vertex detector and the central 
tracker and a constrained from the main \ee \, vertex.
The tracker resolution scales, however, inverse proportional with the 
square of the projected track length.
Azimuthal and polar angles of charged particles are smeared by 
Gaussian distributions with standard deviations of 1 mrad, 
respectively, 2 mrad.
Isolated neutral hadrons (photons) are modified according to the resolution parameters of the hadron (electromagnetic) calorimeter.
In addition, charged particles are required to have 
 a \Pt \, > 200 MeV, while an electromagnetic (hadronic) calorimeter 
entry should have at least a deposited energy of 100 (300) MeV.
Otherwise they are removed.
Electrons (muons) are misinterpreted as charged pions with 1\% (2\%) 
probability and  muons should exceed an energy of  3 GeV.
For the magnetic field a solenoid of 3 Tesla is assumed.

The relatively  long lifetime of the $b$-quark, expected preferentially to be produced from Higgs boson decays, gives rise to decay vertices displaced from the primary \ee \, vertex or, equivalently, to tracks with large impact parameters.
We included this possibility by simulating the performance of a vertex detector in our Monte Carlo and using the impact parameter information in the plane perpendicular to the beam direction.
We compute for each track projection its distance of closest approach to the primary vertex, b, and assume that the measurement error on b can be parameterized as 
\begin{equation}
\sigma_b = \sqrt{A^2 + [B/(p \sin^{3/2}\Theta)]^2} \: ,
\end{equation}
where the parameters A = 10 $\mu$m and B = 30 $\mu$m reflect the 
intrinsic resolution of the vertex detector and the effect of 
multiple scattering of particles traversing the beam pipe and the detector.
For the analysis in this paper, a charged particle is defined to 
be a 'high impact parameter track' if it is within the vertex 
detector acceptance region and has a value of 
b$_{norm}$ = b/$\sigma_b$ greater than two.
An upper limit of the impact parameter has also been introduced to suppress contamination from $K_S$ and $\Lambda$ decays.
Our simulation is kept simple and generic in order that the success of the analysis does not depend on specific details of the vertex detector.
As soon as a detailed design of the vertex detector exists, dedicated Monte Carlo simulations are demanded. 


%----------------------------------------------------------------------
% chapter 4
%----------------------------------------------------------------------
\section{Event generation}
 
With the exception of the Higgs particle that we are studying, we
have assumed that the Standard Model with three generations of 
quarks and leptons is a correct description of nature.

The reactions considered in our study are listed in Table 2. For the
2- to 4-body reactions (1) - (6), the full matrix elements are used,
beyond the usual approximation of computing production cross sections
x branching ratios of Higgs, into decay products. In this way, thanks
to the effort of the State University of Moscow group, contributions
of non-leading diagrams, inferences, correct spin structures and
non-zero fermion masses were addressed. Our calculation procedure
consists of two main steps. The generation of Feynman diagrams, 
the
analytical expressions for the matrix elements squared and the
corresponding optimized Fortran code have been obtained 
by means of the computer package CompHEP \cite{comphep}. The
integration over phase space and the generation of the event floor
have been done with the help of the adaptive Monte Carlo package
BASES/SPRING \cite{bases}. One has to point out that because of  
the complicated phase space structure of the 4-fermion final 
states and the occurrence of singularities, a reasonable 
choice of variables and smoothing of singularities were 
mandatory. Today's version of CompHEP offers all features 
needed to overcome these problems.   
So far, the reaction \ee $\rightarrow \mu^+ \mu^- b\bar{b}$ has 
been studied in \cite{boos1}. It turned out that this reaction has the
cleanest signature for the Higgs but the event rate is rather
small. Six times more Higgs events are expected in the reaction \ee
$\rightarrow \nu^+ \nu^- b\bar{b}$, for which corresponding complete
tree-level calculations can be found in \cite{boos2}. Here, some extra
contribution comes from the fusion process \ee $\rightarrow \nu \nu
h^o$. The reaction \ee $\rightarrow$ \ee $b \bar{b}$ turned out to be
the most complicated one from the calculational point of view
\cite{boos3}, and the extraction of the Higgs out of a background two
orders of magnitude larger, requires well designed out. Higgs
production and its detection potential have also been considered in
the reaction \ee $\rightarrow b \bar{b}$ + 2 jets (= $q
\bar{q}$). \[e.g. A. Ballestrero .................\]. These studies
have now been extended such that instead of considering only the
b-quark in the final state, we sum over the u-, d-, s-, c- and b-quark
states. In addition, reaction (4), \ee $\ rightarrow \tau^+ \tau^- q
\bar{q}$, has to our knowledge not been considered in the past. Its
tree-level calculation is however done analogous to that of reaction
(1), \ee $\rightarrow \mu^+ \mu^- q \bar{q}$, with the substitution
$m_{\mu} \rightarrow m_{\tau}$ and $Z^o \rightarrow  q \bar{q}$
decays. 


In order to avoid  unnecessary large  event samples the
following conditions were applied during event generation at the
parton level:

\begin{itemize}
\item m(\qq) $>$ 50 GeV, $|$cos($q,\bar{q})| <$ 0.993 and M(\lele) $>$
  30 GeV \\ 
for reactions (1)-(3);
\item M(\qq) $>$ 50 GeV and $|$cos($q,\bar{q})|<$ 0.993 $\:$ for reaction
  (4);
\item m(\qq) $>$ 10 GeV, $|$cos$\Theta(q,\bar{q})| <$ 0.993, $|$cos$(q)| <
  $ 0.993 and $|$cos($\bar{q})| <$ 0.993 $\:$ for reaction (5).
\end{itemize}
\vspace{1cm}

At cm energies around 300 GeV high $p_t$-background contributions in
\ee collisions are theoretically well understood and accurately
calculable. At these energies, most of the background expected is due
to hard electroweak and QCD processes. They are also listed in Tablet
as reactions (6)-(8) together with their cross sections. For an
integrated luminosity of 100 $fb^{-1}$ at $\sqrt{s}$ = 360 GeV, these
reactions however contribute more than 4 million events to the date
volume. \footnote{It should be emphasized that other SM background
  processes, such is \ee $\rightarrow Z^oZ^o$, are involved in the
  reactions [1]-[5] x as irreducible backgrounds.} Such an enormous
data sample might obscure and/or mimic a Higgs signal, so it is
mandatory to include these channels in our analysis.

The Higgs boson with a mass of 140 GeV as assumed in this study decays
in almost 50 \% of the cases into $WW^*$. In order to ensure large
statistics, only the $W \rightarrow q\bar{q}$ and  $Z^o \rightarrow
q\bar{q}$ decays are taken into account. It would be desirable to
embed this signal channel in a 2- to 6- body reaction  
\ee $ \rightarrow 3q 3\bar{q}$ generator, so that the irreducible
background and inferences are included. Such an event generator does
however not exist, so that we proceed here in the usual approximation
of comparing production cross sections $x$ branching ratio of $H^o
\rightarrow WW^*x$ branching ratio of $W \rightarrow q\bar{q}$ x
branching ratio of $W \rightarrow q\bar{q}$. SM 6-jet topology
processes, such is \ee $\rightarrow WW Z^o$ and $Z^oZ^oZ^o$, are
expected to be the only significant sources of background
\footnote{Remember, this study assumes to be in energy below the top
  pair production reaction, \ee $ \rightarrow 3t 3\bar{t}.$} These
reaction channels are generated in the same manner as the Higgs
channel. Table 2 contains also their cross sections.

Reactions (6) - (11) have been generated by means of the computer
package PYTHIA 5.7 \cite{pythia} and, for all reactions considered,
the JETSET 7.4 package \cite{pythia} has been used for quark and gluon
fragmentation and unstable particles decays. In all cases, the events
generated, characterized by the 4-momenta, charges and lifetimes of
the stable particles and vertex informations, are used as input for
the detector simulation program.

It is worth to emphasize that to a good approximation, dedicated
electroweak calculations of the electroweak content of the processes
such as the package CompHEP have achieved very high accuracy but that
relatively large uncertainties exist in the perturbative parton shower
or non-perturbative hadronization procedures. This rises the question
of the reliability of such a procedure for the study of hadronic and
mixed hadronic-lepton final states. We adopted a pragmatic solution by
including a standard interface with parton-shower and hadronization
programs. We are aware that using exact matrix elements it becomes
desirable to include final state QCD corrections, even when
kinematical cuts are imposed. 
%----------------------------------------------------------------------
% chapter 5
%----------------------------------------------------------------------
\section{Higgs Discovery Potential.}


Basically, the Higgs can be produced by the so called bremsstrahlung process\cite{brems} 

\begin{equation}
e^+e^- \longrightarrow Z^* \longrightarrow Z^0H^0
\end{equation}

\noi or by the fusion of $W^+W^-$ and $Z^0Z^0$ bosons\cite{fusion}

\begin{eqnarray}
e^+e^- \longrightarrow \nu \nu H^0  \\
e^+e^- \longrightarrow e^+e^- H^0
\end{eqnarray}

At \SQRTS = 360 GeV and $M_H$ = 140 GeV, the bremsstrahlung process is
4-5 times more important than the fusion reactions. This is in 
good approximation also true if ISR and beamstrahlung are
included. Therefore we restrict ourself to the bremsstrahlung process 
in the following and try to select it from the overwhelming background.

The bremsstrahlung process offers two main strategies for the Higgs search:
\begin{itemize}
\item calculation of the recoil mass against the $Z^0$, most conveniently determined in the $Z^o$ \into \ee \,and \mm \,decay modes.
But also hadronic $Z^o$ and $Z^o$ \into \tata \,decays can be used.
This method has the unique feature of being independent on assumptions about the Higgs decay modes;
\item direct reconstruction of the invariant mass of the Higgs decay products; here the decays \hnull \into \bb, $W^+W^-$, \tata \,and light \qq \, or \glgl \, are appropriate, with the \bb \,decay mode being the most effective one for the \hnull \,discovery.

\end{itemize}

In the past, various \hnull \,analyses have been developed for practical all possible \hnull \,and \znull \,decays, see e.g. \cite{pgw,janot,bark}.
We follow the guidelines of these studies, but have either improved the selection criteria or added further restrictions or impose energy-momentum as well as $M(l^+l^-) = M(Z^0)$ constraints, if appropriate.
If for particular \hnull \,and \znull \,decay mode a set of cuts and 
constraints had been established, all reactions of Table 2 were also 
studied under these conditions.
Their contributions to either the recoil mass or the appropriate 
invariant mass spectrum were added so that the Higgs signal with
appropriate backgrounds can be explored.

Typical Higgs signal topologies expected are shown in Fig. 2.
The associated production \ee \into \znull \hnull \,is followed by 
the decay of the \znull \into \lele (10\%), \nn (20\%),\qq (70\%) 
and the decay of the \hnull \,mostly into \bb, $W W^*$ and
occasionally into \tata \,and more rarely into light quarks \qq \,and
two gluons. Only $W \rightarrow q\bar{q}$ decays are considered.

In the following we apply so-called standard and improved cuts such
that the background is removed to a negligible or small level while
the \hnull is retained to a large extent. Standard cuts (see below)
are mainly introduced to suppress non-high $p_\perp$ low-multiplicity
background whereas the improved cuts are dedicated ones to remove
effectively W pairs, $q\bar{q}$ pairs, $e\nu W$, WW\znull,
\znull\znull\znull events and possible reflections of the signal
channel (1) - (5), (9) onto that one under study. The number of signal
S and background B events in a window of 8 resp. 12 GeV around the
value of $H_H $= 140 GeV are then counted in order to determine the
significance $S/\sqrt{B}$. The value/values $S/\sqrt{B}$ is then
studied as a function of the integrated luminosity and, if
$S/\sqrt{B}$ is typically 5 or larger, Higgs detection either in the
recoil mass or in the invariant mass of its decay products should be
promptly feasible. It is however important to point out that the
proposed criteria are robust and relatively simple, they are
necessarily not optimized and significant improvements are possible
once the real detector behaviours are known. Our today's concern is to
convince the reader that the proposed analysis demonstrates the
potential of a fast Higgs discovery at NLC.





%----------------------------------------------------------------------
% chapter 5.1
%----------------------------------------------------------------------
\subsection{The leptonic channel: \ee \into \znull\hnull \into 
(\ee; \mm)(\bb)}  


According to the topology in Fig. 2a, the Higgs can be detected either
by 
\begin{itemize}
\item[-] calculating the recoil mass $M^2_{rec}  s - 2 \sqrt{s} E_z +
M_z^2$, once two-opposite charged leptons with invariant mass close to
the \znull mass are identified or by
\item[-] investigation of the hadronic 2-jet mass, M(2-jet).
\end{itemize}

The set of cuts we have adapted consists of

\begin{itemize}
\item [a)] the total transverse energy of the event has to be larger 
than 30 GeV but less than 250 GeV; 

\item [b)] the total momentum along the beam is restricted to be
  within the range $\pm$  120 GeV;

\item [c)] the visible energy of the event should be larger than 230 GeV;
\item [d)] the total number of tracks should exceed 14;
\item [e)] $M$(\mm) = $M_Z \pm$ 10 GeV $\:$ resp. $M$(\ee) = $M_Z \pm$ 6
  GeV,  \\
with $|cos(Z)| <$ 0.8;

\item [f)] if \znull $\rightarrow$ \mm, the energy of isolated 
electrons has to be is smaller than 25 GeV whereas for the $Z_o$ 
\into \ee \, decay, it is required to be between 10 and 140 GeV;

\item [g)] if the hadronic system contains two jets\footnote{The
    LUCLUS\cite{pythia} jet finding algorithm has been used throughout
    this study.}, 
with an opening angle larger than $40^o$; each jet has an energy
larger than 10 GeV, a number of particles  5 and $|cos \Theta_{jet}| <$ 0.8;

\item [h)] the number of tracks with 'large impact parameter' has to
  be larger or equal than 3.
\end{itemize}



%----------------------------------------------------------------------
% chapter 5.2
%----------------------------------------------------------------------
\subsection{The tauon channel: \ee \into \znull\hnull \into (\qq)(\tata)}  

Since one of our interest is to measure the branching ration BF (H
\into \tata] we select from the general final state \tata \qq that
part with \znull \into \qq and \hnull \into \tata decays only, which
is about two times more abundant that the complementary one with \znull
\into \tata and \hnull \into \qq decays respectively. Higgs detection is
possible either by inspection of the \tata invariant mass or, once the
\znull \into \qq decay is established, of the mass recoiling against
the \znull. Hence, the final state of interest here is formed by two
isolated energetic tau's and two jets as indicated in Fig. 2b.

In order to distinguish a possible signal from all relevant background
we apply at first the criteria 

\begin{itemize}
\item the total transverse energy should be in the range 30 GeV to 240
  GeV;
\item the total momentum along the beam is restricted to be within
  $\pm$140 G\
eV;
\item the visible energy of the event is larger than 150 GeV and below
  330 Ge\
V;
\item the number of charged particles should exceed 8, so that most of
  the low \Pt background is removed and all signal events are retained.
\end{itemize}

The two $\tau$'s are selected as the two most isolated oppositely
charged particles\footnote{For simplicity, only 1-prong $\tau$ decays
  are considered.} with


\begin{itemize}
\item a transverse momentum of at least 2.5 GeV;
\item no other charged particles in a cone of 25$^0$ around their direction;
\item not identified as an \ee \,or \mm \,pair;
\item all neutrals within a cone of 25$^0$ around the charged 
particle direction are included.
\end{itemize}

These particles are then subtracted from the event and the remainder 
is considered to be a 2-jet system with an invariant  mass within 
15 GeV of the $Z$ mass.
Since we know the cm energy and momentum, the initial $\tau$ and jet
energies are recomputed by a fit. If such an energy rescaling is not
performed, the \hnull signal would not be visible in the recoil and
2-jet masses.

In order to ensure good event containment, reasonable track
reconstruction capability and well-defined \znull \into 2-jet decays
we require additionally

\begin{itemize}
\item isolated electrons should have an energy less than 125 GeV with a polar angle between 15$^0$ and 165$^0$;
\item $|cos\Theta_Z| <$ 0.8
\item the two jets found should have an opening angle larger than 
40$^o$ and, for each jet, it is demanded to have an energy larger 
than 10 GeV, $|cos \Theta| <$ 0.8 and to involve at least 5 particles.
\end{itemize}

Some 25 \% of the signal events survive all cuts. However,
non-negligible background due to insufficient \znull\hnull particle
association out of the general \tata \qq final state and from $W^+W^-$
pair and $2q2 \bar{q}$ production remains. In addition, a worse mass
resolution exists. Therefore, in order to observe a significant
s$\sqrt{B}$ more statistics is needed. The histograms expected for 40
fb$^{-1}$ are shown in Figs.3b and 3d for the recoil mass and the
\tata invariant mass, respectively. In each distribution, about 22
\znull\hnull events exist on a background of 15 events.
 

%----------------------------------------------------------------------
% chapter 5.3
%----------------------------------------------------------------------
\subsection{The missing energy channels: \ee \into $ZH$ \into
  (\nn)(\bb)and \ee \into \nn (\bb)}

This channel is of great interest because of its discovery potential of
the Higgs boson with limited integrated luminosity, thanks to two
contributing diagrams and large branching ratios BF(\znull \into \nn)
and BF(\hnull \into \bb.

The topology expected for both processes (Gig. 2c) is an (acoplanar]
pair of jets accompanied by large missing energy.
The difference the two production mechanisms consists basically
in the constraint M(\nn) = M$-z$ for the bremsstrahlung case.
The isolation criteria and cuts we have applied to select the signal
from background processes, such as \ee \into $W^+W^-$, \znull\znull,
E$\nu$W, \ee \qq or \qq $\gamma$, are

\begin{itemize}
\item[a)] a total transverse energy larger than 10 GeV but less than
  170 GeV;
\item[b)] the total momentum along the beam line is within the range
  $\pm$ 140 GeV;
\item[c)] the visible event energy is larger than 50 GeV and less than
  210 GeV, to allow for large missing energy;
\item[d)] the number of charged particles is larger than 6;
\item[e)] isolated electrons have an energy smaller than 50 GeV and
  their polar angle should be between $20^o - 160^o$;
\item[f)] the hadronic system involves two jets with an opening angle
  of at least $30^o$;
\item[g)] the invisible mass recoiling against the 2-jet system has to
  be within the range 60 - 250 GeV
\item[h)] each jet should have an energy larger than 10 GeV, at least
  5 particles and a polar angle with $|cos\Theta_{jet}| <$ 0.8;
\item[i)] the number of tracks with significant impact parameter,
  b/$\sigma_B$, is required to be larger than 4.
\end{itemize}

Cuts a) - e) reduce background events to a large extent; they do not
remove significantly signal events. The other conditions we have
adopted should accumulate well-defined b-quark jets. Note that we have
strengthened the definition for the 'significant impact parameter track'
with the result that the surviving $W^+W^-$, e$\nu$W, \qq$(\gamma)$ and
\ee\qq background is reduced to a small level.

After applying all criteria, 36 \% of the signal events survive which
together with the large cross sections times branching ratios for the
bremsstrahlung and fusion processes allows to reduce the integrated
luminosity to 1 $fb^{-1}$ to observe 11 events on practically no
background as shown in Fig. 3e. This result makes the \nn\qq channel
very attractive for a quick intermediate Higgs search in \ee
collisions at cm energies 300 - 360 GeV.

%---------------------------------------------------------
%chapter 6.4
%---------------------------------------------------------------------
\subsection{The 4-jet channel: \ee \into \znull\hnull \into
  (\qq)(\bb)}
The topology for this channel (Fig. 1d) involves 4 jets from the
\znull \into \qq and \hnull \into \bb decays. This channel also offers
two possibilities for a direct \hnull search: i) to calculate the
recoil mass against the \znull and ii) to study the \bb invariant
mass. Since numerous backgrounds are expected to contribute to this
topology a more sophisticated analysis has to be developed. Our
proposed results to the following demands: 

\begin{itemize}
\item[a)] the total transverse energy should be larger than 30 GeV and
  smaller tan 




%--------------------------------------------------------------------------
%       bibliography
%--------------------------------------------------------------------------


\begin{thebibliography}{99}


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\end{thebibliography}


\end{document}
 





















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\begin{document}
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\authorrunning{H. J\"urgen \,Schreiber}
\titlerunning{{\talknumber}: Higgs Physics at $e^+e^-$ LC}
 
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% For plenary talks, the talk number is that of the session
\def\talknumber{1808} 

\title{{\talknumber}: Higgs Physics at a 300 - 500 GeV $e^+e^-$ Linear
  Collider  }
\author{H. J\"urgen\, Schreiber
(schreibe@ifh.de)}
\institute{ DESY IfH Zeuthen}

\maketitle

\begin{abstract}
We summarize discovery potentials for the Standard Model Higgs boson
produced in $e^+e^-$ collisions and measurements of its detailed
properties. Prospects for Higgs boson detections within the Minimal
Supersymmetric Standard Model are also discussed.
\end{abstract}
%
%
%
\section{Introduction}
Electroweak breaking due to the Higgs mechanism \cite{Higgs} implies the
existence of at least one new particle, the Higgs boson. Its discovery
is the most important experiment for the formulation of electroweak
interactions. LEP has established a lower bound of the Higgs mass $M_H
\gess$ 77 GeV  \cite{lowM}. High precision data, interpreted within the
Standard Model (SM), favour a Higgs boson with a mass somewhere
between 100 and 180 GeV.

Future $e^+e^-$ linear colliders are the ideal machine for an easily
Higgs boson discovery, its verification and precision measurements
of the Higgs sector properties.

Recent simulation studies \cite{our} involve, thanks due to the effort
of several groups,
i) the full matrix elements for 4-fermion final states,
ii) all Higgs decay modes (with SM decay fractions $\gess $ 1\%),
iii) initial state QED and beamstrahlung (TESLA design), 
iv) a detector response \cite{CDR} and
v) all important background expected to contribute.


\section{SM Higgs discovery potentials}
The Higgs boson can be produced by the Higgsstrahlung process $e^+e^-$
$\rightarrow Z^* \rightarrow ZH^o$  (1), or by the fusion of WW and
ZZ bosons,   $e^+e^- \rightarrow \nu \bar{\nu} H^o$ (2) and 
$e^+e^- \rightarrow e^+e^- H^0$ (3), respectively,
or by 
  radiation off top quarks, $e^+e^- \rightarrow t \bar{t} H^o$, with
  however a very small cross section.
At e.g.
$\sqrt{s}$ = 360 GeV
and $M_H$ = 140 GeV, the Higgsstrahlung process is about four time
more important than the fusion reactions. Reaction (1) 
admits two strategies for the Higgs search: i) calculation of the recoil mass
against the $Z \rightarrow e^+e^- / \mu^+\mu^-, M^2_{rec} =$
$ s - 2\sqrt{s} (E_{l^+} + E_{l^-}) + M^2_Z$, which is independent on
assumptions about Higgs decay modes, and ii) 
the direct reconstruction of the invariant mass of the Higgs decay
products.

In order to achieve the best experimental resolution, energy-momentum
as well as $M(l^+l^-) = M_Z$ constraints have been imposed when
appropriate, and to make a signal-to-background analysis as meaningful
as possible, a consistently evaluation of the signal and all background
rates expected has been done.

The leptonic channel, $e^+e^- \rightarrow ZH^o \rightarrow $
$(e^+e^- / \mu^+\mu^-) (b\bar{b})$, allows Higgs detection either in
the mass recoiling against the Z or in the hadronic two b-quark jet
mass. Typically, an integrated luminosity of $\sim 10 fb^{-1}$ is
needed  to observe  the Higgs boson with a significance
$S/\sqrt{B} > 5$ after application of appropriate selection
procedures \cite{our}.

The tauon channel,  $e^+e^- \rightarrow ZH^o \rightarrow (q\bar{q})$
$(\tau^+\tau^-)$, requires some more refined selection procedures
\cite{our, Hil} due to missing informations of $\tau$ decay products.
Hence, the luminosity accumulated to observe a clear $H^o$ signal
in the recoil resp. $\tau^+\tau^-$  mass should be between 50 to
80 $fb^{-1}$. Energy-momentum constraints are required to determine
the initial $\tau$ and jet energies by a fit; otherwise no $H^o$
signal would be visible.

The missing energy channels, $e^+e^- \rightarrow ZH^o \rightarrow$
$(\nu\bar{\nu})(b\bar{b})$ and $e^+e^- \rightarrow \nu\bar{\nu} H^o
\rightarrow \nu\bar{\nu} (b\bar{b})$ are very important due to their
large discovery potential for the Higgs boson. It has been
demonstrated \cite{our,Janot} that with $\sim 1 fb^{-1}$ of integrated
luminosity a convincing signal over some small remaining background in
the two-tagged b-jet mass should be obtained (Fig.1). 
%\begin{figure}[h]
%\begin{center}
%\epsfig{file=neutrinos.eps}
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%\end{figure}
Thus, few days
of running a 300 GeV $e^+e^-$ linear collider at its nominal
luminosity would discover the SM Higgs in the intermediate mass range.

With increasing accumulated luminosity other $H^o$ decay modes like
$WW^{(*)}$ and $c\bar{c}$ + gg would verify the Higgs
interpretation. However, 50 to 100 $fb^{-1}$ are necessary for
significant signals. As an example, Fig. 2 shows selected $H^o \rightarrow
c\bar{c}$ + gg decays  despite the presence of a huge background
in the 2-jet missing energy event topology.

Observation of the $H^o \rightarrow \gamma\gamma$ decay requires large
statistics $(> 200 fb^{-1})$ and a fine-grained electromagnetic
calorimeter with very good energy resolution \cite{Gunion}.

\section{Higgs properties}
If a Higgs is discovered, it is imperative to understand its nature. In
the following we assume an integrated luminosity of $100 fb^{-1}$.

Measurements of the Higgs boson mass rely at best by searching for 
$Z \rightarrow l^+l^-$ decay products and reconstructing the recoil
mass peak. The mass resolution  depends on the cm energy
$\sqrt{s}$ and 
the detector performance, in particular on the momentum resolution of
the tracking system. We expect 
an error for the Higgs mass of $\sim$ 50 MeV for the detector design
of ref \cite{CDR}.

When the Higgsstrahlung $ZH^o$ rate is significant, the CP-even
component of the $H^o$ dominates. It is possible to cross check that
by studying the Higgs production and the $Z^o$ decay angular
distributions. Thus, observation of angular distributions as expected
for a pure CP-even state  implies that the
Higgs has spin-0 and that it is not primarily CP-odd \cite{Kramer}.
Further, studying decay angular correlations between the decay
products of the tauons in the reaction
 $e^+e^- \rightarrow ZH^o \rightarrow Z
\tau^+ \tau^-$ provides a democratic probe of the Higgs CP-even and
CP-odd components \cite{Kramer,JFG}. If the $H^o \rightarrow \gamma
\gamma$ decay is visible (or the Higgs is 
produced in $\gamma \gamma$ collisions)
the Higgs must be a scalar and has a CP-even component. Whether a
CP-odd component also exists, can be studied by comparing Higgs
production rates in $\gamma \gamma$ collisions with different photon
helicities \cite{JFG2}.

The branching fraction of the Higgs into a final state X requires to
compute $BF(H \rightarrow X) = [\sigma(ZH) \cdot BF (H \rightarrow
X)] / \sigma (ZH)$, where $\sigma (ZH)$ is the inclusive Higgs
cross section, see Fig. 3. Its error is expected to be $\pm 5\%$
\cite{our}, while the nominator $\sigma (ZH)\cdot BF (H \rightarrow
X)$ and its precision can be obtained from the  X invariant mass
distribution. Typical branching fraction uncertainties expected for a
140 GeV SM Higgs boson are \cite{our} \\


\begin{tabular}{|c|c|c|c|}\hline \par
%\vspace*{10mm}
$BF(H\rightarrow b \bar{b})$ & $BF(H\rightarrow \tau^+\tau^-)$ & 
$BF(H\rightarrow WW^*)$ & $BF(H\rightarrow c \bar{c} + gg)$ \\ \hline
$\pm 6.1 \%$ & $\pm 22 \%$ & $\pm 22 \%$ & $\pm 28 \%$  \\ \hline
\end{tabular} \\

As can be seen, many of the Higgs decay modes are accessible to
experimental considerations and most of the measurements allow a
powerful discrimination between the SM-like  and SUSY-like Higgs
bosons. However, more fundamental than branching fractions are the
Higgs partical widths. If $M_H \gess 140$ GeV, $\Gamma(H \rightarrow WW)$
is measurable from the WW fusion cross section, whereas for $M_H \less 
130$ GeV a $\gamma \gamma$ collider allows to measure $\Gamma(H
\rightarrow \gamma \gamma)$.   
Combined with the corresponding branching fraction (from LHC or NLC),
$\Gamma_{tot}^{Higgs}$ is calculable, which in turn allows the
determination of the remaining  partial widths. In addition, the
$H^o \rightarrow ZZ$ coupling-squared is obtained from 
$\sigma (ZH)$ with an error comparable to that of the inclusive
cross section.

\section{MSSM Higgs bosons: $h^o, H^o, A^o, H^{\pm}$}

In the minimal supersymmetric scenario five Higgs particles are
expected, two CP-even states $h^o, H^o$ (with $M_{h^o} < M_{H^o})$, one
CP-odd state $A^o$ and two charged states $H^{\pm}$. An upper bound on
the mass of the $h^o$ is predicted to be $M_{h^o} \less$ 130
GeV, including radiative corrections \cite{bound}. If $M_{h^o}$
exceeds 130 GeV, the MSSM is ruled out.

In the limit of large $M_{A^o}( \gg M_Z)$ the properties of the $h^o$
are very close to those of the SM Higgs and a discrimination between
both states is only possible if very precision measurements show $h^o
\neq H^o_{SM}$ or further Higgs states are discovered.

Besides Higgsstrahlung and fusion production mechanisms for $h^o$ and
$H^o$, Higgs pair production $e^+e^- \rightarrow h^o A^o,\, H^oA^o,\,
H^+H^-$ is possible, with the relations $\sigma (e^+e^- \rightarrow
h^oZ, h^o\nu\bar{\nu}, H^oA^o) \alpha sin^2 (\beta - \alpha)\, (I)$ and 
$\sigma (e^+e^- \rightarrow
H^oZ, H^o\nu \bar{\nu}, h^oA^o) \alpha cos^2 (\beta - \alpha)\, (II)$
$( \alpha, \beta$ are mixing angles). One process in each line (I),
(II) always has a substantial rate and the sum for the $h^o$ or $ H^o$
cross sections are constant and equals  $\sigma (H^o_{SM})$ over the
whole ($M_A,$ tan $\beta)$-plane.

Concerning the branching fractions it is usually assumed that the
masses of the supersymmetric particles are so heavy that SUSY-Higgses
cannot decay into s-particles; only decays into standard particles are
possible. Whatever the masses of the Higgs particles and tan$\beta$
values are, we expect i) b-quark decays of $h^o, H^o$ and $A^o$ occur in
large regions of the parameter space underlying the importance of
excellent b-tagging, ii) vector boson decays are never significant for
the $A^o$ while strong restrictions exist for the $H^o$ and iii) if 
$M_{H^{\pm}} \gess$ 180 GeV, the $H^{\pm}\rightarrow tb$ decay rate is large
leading in most cases via $ e^+e^- \rightarrow H^+H^-  \rightarrow 
W \,b \bar{b}\, W \,b \bar{b}$ to 8-jet final states involving four b-quark
jets, a very challenging task for experimentalists to recognize and
analyze such events. An example for MSSM Higgs boson detection is
shown in Fig. 4 for the process $ e^+e^- \rightarrow H^oA^o \rightarrow
b \bar{b} b \bar{b}$ at $\sqrt{s} =$ 800 GeV \cite{CDR}. 

If however
s-particle decays of Higgses are possible, the decays end (in many
SUSY models) with the lightest 'neutrino-like' supersymmetrie
particle. Hence, event signatures are characterized by missing
$p_{\bot} / E_{\bot}$ + leptons + jets which warrant further
experimental simulations.

\section{Conclusions}
An  $ e^+e^-$ collider focusing on  $ e^+e^- \rightarrow ZH^o$
production in combination with a $\gamma \gamma$ collider allows to
fully explore the properties of an intermediate SM-like Higgs in the
shortest time.

Within the MSSM, one light CP-even Higgs boson must be found or the
MSSM is ruled out. A 500 GeV collider has very good prospects to
discover all five MSSM Higgs particles, $h^o, H^o, A^o, H^{\pm}$, if 
$M_A \less $ 220 GeV.
If these particles are detected large statistics experiments are needed
to measure their parameters.



\section{Here we include figures }
\begin{figure}[!ht]
%\begin{center}
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\caption{.....}
\end{minipage}
%%%%%%%%
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\begin{minipage}[t]{6cm}
%\begin{picture}{6}
\mbox{\epsfig{file=inclusive.eps,height=5cm}} %,width=5cm}}
%\end{picture}\par
\caption{.....}
\end{minipage}
%%%%%%%%%%%%
\hfill
\begin{minipage}[t]{6cm}
%\begin{picture}{6}
\mbox{\epsfig{file=eeHA.eps,height=5cm}} %,width=5cm}}
%\end{picture}\par
\caption{.....}
\end{minipage}


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%\end{figure}



% ---- Bibliography ----
%
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\end{thebibliography}
\end{document}
 







%\end{document}


%\section{Here we reference equations and quote papers}
%As shown in reference \cite{ref}, this reads:
%\begin{equation}
%m(r)=\int^r_04\pi r^2 \rho \,dr  \enspace .\label{mass}
%\end{equation}
%Here we reference to equation \ref{mass}


\section{Here we include figures }
\begin{figure}[ht]
%\begin{center}
\unitlength1cm
\begin{minipage}[t]{6cm}
\begin{picture}{6}
\mbox{\epsfig{file=neutrinos.eps,height=5cm}} %width=5cm}}
\end{picture}\par
\caption{.....}
\end{minipage}
\hfill
\begin{minipage}[t]{6cm}
\begin{picture}{6}
\mbox{\epsfig{file=neutrinos.eps,height=5cm}} %,width=5cm}}
\end{picture}\par
\caption{.....}
\end{minipage}

%\end{center}
\end{figure}
%\caption{The height of this figure is 2.5\,cm}
%\end{figure}


\end{document} 

