%Paper: 
%From: Ed Stoeffhaas <ed@phenxe.physics.wisc.edu>
%Date: Thu, 9 Sep 1993 15:12:01 -0500

lecture1.
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\begin{document}

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\hskip.5in \raise.1in\hbox{\fortssbx University of Wisconsin - Madison}
\hfill$\vcenter{\hbox{\bf MAD/PH/780}
            \hbox{July 1993}}$ }

\vspace{1in}

\begin{center}
{\LARGE\bf COLLIDER PHYSICS 1993\footnotemark}\\[.4in]
{\large V.~Barger$^a$  and R.J.N.~Phillips$^b$}\\[.2in]
\it
$^a$Physics Department, University of Wisconsin, Madison, WI 53706, USA\\
$^b$Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK
\end{center}

\footnotetext{Lectures presented by V.~Barger at the {\it VII Jorge Andr\'e
Swieca Summer School}, S\~ao Paulo, Brazil, January 1993.}

\renewcommand{\LARGE}{\Large}
\renewcommand{\Huge}{\Large}

\vfill

\abstract{
These lectures survey the present situation and future prospects in
selected areas of particle physics phenomenology: (1)~the top quark,
(2)~the  Higgs boson in the Standard Model, (3)~strong $WW$ scattering,
(4)~supersymmetry, (5)~the Higgs sector in minimal supersymmetry,
(6)~low-energy constraints on supersymmetry.
}

%\vspace{1.2in}
\vfill
\newpage

\section*{GENERAL INTRODUCTION}
%\unskip\smallskip
%{\baselineskip13.5pt
   Our present understanding of elementary particle phenomena is
dominated by the Standard Model (SM), based on three generations of basic
spin-1/2 fermions (quarks and leptons), plus the spin-1 gauge fields of
$\rm SU(3)\times SU(2)_L\times U(1)$ symmetry, plus spontaneous symmetry
breaking by one
doublet of spin-0 scalar fields.  The SM works amazingly well --- all of
its predictions that can be tested have so far been verified to high
precision; no discrepancies have yet been found.  However the predicted
top quark has not yet been confirmed experimentally, nor has the Higgs
boson.  The top quark is not seriously in doubt, since lots of indirect
evidence points to its existence, but it is nevertheless important to
discover it to determine its mass and other properties that enter many
calculations.  The symmetry-breaking sector appears to be much more
arbitrary (one can easily imagine alternatives to the simple Higgs
mechanism) and urgently requires investigation.  This sector can be
probed either by searching for the Higgs boson or by studying the
scattering of longitudinally polarized $W$ or $Z$ bosons; these longitudinal
states arise through symmetry-breaking and their scattering reflects
the underlying mechanism.  These topics are addressed in the first
three lectures: (1)~the top quark, (2)~the standard model Higgs boson,
(3)~strong $WW$ scattering.


   The SM has severe shortcomings however.  It contains
many apparently arbitrary and unrelated parameters; this
arbitrariness might be reduced or explained by some Grand Unified Theory
(GUT), where the different gauge symmetries merge into a single higher
symmetry and the masses may be simply related at a very high energy
scale.  But in plain GUT models associated with electroweak symmetry breaking
an arbitrary fine-tuning of parameters
seems to be needed to prevent the scalar particles from acquiring very large
masses; a search for a solution to this ``hierarchy problem" leads to
Supersymmetry (SUSY), where every fermion has a boson partner and vice versa.
No SUSY partners have yet been discovered but there is encouragement from the
success of SUSY-GUT models, and indeed cosmological Dark Matter
may be due to one of these particles.  SUSY is also a likely ingredient
in an eventual unification of strong and electroweak forces with gravity,
while behind all this there lies perhaps a Superstring theory of everything.
All sorts of new phenomena may arise beyond the SM: extra gauge bosons,
exotic fermions, leptoquarks, nucleon decay, new classes of Yukawa
interactions, etc.  Some of these topics are addressed in the later lectures:
(4)~SUSY and GUTs, (5)~Higgs sector in the minimum SUSY extension of the
SM (MSSM), (6)~low-energy constraints on SUSY.

   We give a selection of references, but make no claim to completeness
and specifically exclude references to standard textbook material.

%}



\chapter*{LECTURE 1:\\ THE TOP QUARK}
\setcounter{chapter}{1}

%\thispagestyle{empty}

\section{Introduction}

   The top quark $t$ is an essential part of the third fermion generation
in the SM, together with the $b$-quark, the $\tau$-lepton and its neutrino
$\nu_\tau$.  Their left and right chiral components have the usual
$\rm SU(2)_L\times U(1)_Y$  weak-isospin and hypercharge quantum numbers (with
electric charge $Q=T_3+{1\over2}Y$):
%
% The reason for putting the single numbers in arrays is that Latex (unlike the
% original TeX & Revtex forms of this equation) apparently puts extra space on
% each side of an array, even with no delimiters. Thus the right-justified
% (used to line up signed quantities with unsigned) arrays and the right-just
% single numbers would not line up until the latter put in arrays also.
\[
\begin{array}{c@{\hspace{3em}}r@{\hspace{3em}}r}
\hline\hline
\nv3
                         &       T_3      &           {1\over2} Y \\
\nv3
\hline
\nv3
\left(\begin{array}{c}
       \nu_\tau    \\
        \tau
\end{array}\right)_{\!\!L} &
\begin{array}{r}
 1\over2    \\
- {1\over2}
\end{array} &
\begin{array}{r}
-{1\over2} \\
-{1\over2}
\end{array}
\\
\nv3
\hline
\nv3
          \tau_R          &
\begin{array}{r} 0 \end{array}      &
\begin{array}{r} - 1 \end{array} \\
\nv3
\hline
\nv3
\left( \begin{array}{c}
         t \\
         b
\end{array} \right)_{\!\!L}   &
\begin{array}{r}
    1\over2   \\
  - {1\over 2}
\end{array}  &
\begin{array}{r}
 1\over 6 \\
 1\over 6
\end{array}
\\
\nv3
\hline
\nv3
           t_R           &
\begin{array}{r}  0 \end{array}       &
\begin{array}{r}  2\over3 \end{array} \\
\nv3
\hline
\nv3
           b_R           &
\begin{array}{r}  0  \end{array}      &
\begin{array}{r} - {1\over3} \end{array} \\
\nv3
\hline\hline
\end{array}
\]
%

   Here $t$ (and $\nu_\tau$) are the only SM fermions to escape direct
detection.  In the case of $t$, this is apparently because it is very
heavy~\cite{cdftop},
%
\begin{equation}
          m_t > 108\mbox{ GeV \qquad (1993 SM CDF limit)}\;, \label{m_t CDF}
\end{equation}
%
compared to other quarks $(m_u = 0.004,\ m_d = 0.007,\ m_s = 0.15,\ m_c = 1.3,\
m_b = 4.8$~GeV).  But we have many indirect indications that top exists:
\begin{enumerate}
\item It is needed to cancel chiral anomalies.

\item It is needed for GIM-suppression of flavor-changing neutral currents.
If $b_L$ had no doublet partner $t_L$, then $BF(B\to \ell^+ \ell^- X) >
0.013$~\cite{barpak}; but the CLEO experimental bound is ${}< 0.0012$ at
90\%~CL~\cite{cleollx}. Also, if $b$ were an SU(2)$_L$ singlet, $B_d^0$-$\bar
B_d^0$  oscillations would be near maximal~\cite{roybb}
but in fact they are not.

\item The $e^+e^- \to \bar bb$ forward-backward asymmetry measures
$T_3 (b_L) - T_3 (b_R)$;  LEP data give a value
$-0.504{+0.018\atop-0.011}$~\cite{lepsm},
confirming the SM value $-1/2$ and showing that $b_L$ belongs to a doublet.

\item With a singlet assignment for $b$, the predicted $Z\to b\bar b$ partial
width is a factor of 15 times smaller than the measured value~\cite{roybb}.
\item Electroweak radiative corrections to all available $Z,\ W$ and deep
inelastic scattering data fit beautifully but require top-quark
contributions with a mass of order~\cite{elfog}
\end{enumerate}
%
\begin{equation}
m_t = 141 {+17\atop-19} {+17\atop-18} \mbox{ GeV \qquad  (SM electroweak)} \;.
\label{m_t SM}   \end{equation}
%

\section{Top production, decay and detection}

   Top decays dominantly by  $t \to b W^+$  in the SM (Fig.~1); other
charged-current decays to $dW$ or $sW$ are suppressed by small KM matrix
elements;
neutral-current decays to $uZ$ or $cZ$ are suppressed by the GIM mechanism.
Here $W$ is now known to be on-shell [see Eq.~(\ref{m_t CDF})] and its
subsequent decays are dominantly  $ W^+ \to u\bar d,\ c \bar s,\ \nu \bar e,\
\nu \bar\mu,\ \nu \bar\tau$ with respective branching
fractions 1/3, 1/3, 1/9, 1/9, 1/9, approximately.  The semileptonic modes
$t\to be\nu,\ b\mu\nu$  provide the cleanest signatures, typically containing
\begin{enumerate}
\item an energetic lepton, isolated from jets,
\item missing energy and momentum,
\item a $b$-jet, possibly tagged by a muon from $b\to c \mu \nu$ decay,
\item a displaced vertex from $b$-decay (long lifetime),
\end{enumerate}
which help to separate top events from backgrounds.

\medskip
%\epsfysize=1.5in
\begin{center}
\hspace*{0in}
%\epsffile{fig1-1.eps}

{\small Fig.~1: Top decay in the SM.}
\end{center}

   If $m_t$ is close to threshold, the mean lifetime  $\tau = 1/\Gamma$  is
long enough to allow the usual fragmentation (formation of a hadron)
before the top quark decays. But if  $m_t \agt 120$~GeV the lifetime is
too short to form any hadron (including toponium states) and top
essentially decays as a free quark; in this regime its width is
%
\begin{equation}
     \Gamma(t \to bW) \sim 0.17 (m_t/M_W)^3 \ \rm GeV  \;, \label{Gamma}
\end{equation}
%
shown in Fig.~2~\cite{fujii}. $\Gamma_t$ may be measured from the linewidth;
there are also interference effects between gluons radiated from $t$ and $b$,
which depend sensitively on the top lifetime~\cite{khoze}.

\begin{center}
%\epsfxsize=5in
\hspace*{0in}
%\epsffile{fig1-2.ai.eps}

{\small Fig.~2: SM top decay width versus $m_t$. From Ref.~\cite{fujii}.}
\end{center}

   The only present machine that can now discover SM top quarks is the
Fermilab Tevatron $p\bar p$ collider, with CM energy $\sqrt s=1.8$~TeV.
The lowest-order ($\alpha_s^2$) QCD production subprocesses are light
quark-antiquark and gluon-gluon fusion (Fig.~3).

%\epsfxsize=5in
\begin{center}
\hspace*{0in}
%\epsffile{fig1-3.eps}

\medskip
{\small Fig.~3: $t\bar t$ production via QCD at the Tevatron.}
\end{center}

\noindent
The total cross section  is
given by a convolution of subprocess cross sections $\hat\sigma$ with parton
distributions of the general form
%
\begin{equation}
    \sigma(s) = \sum_{i,j} \int dx_1 dx_2 \hat\sigma_{ij}(x_1x_2s,\,\mu^2)
f_i^A(x_1,\,\mu) f_j^B(x_2,\,\mu) \;,  \label{sigma(s)}
\end{equation}
%
where $A$ and $B$ denote the incident hadrons, $i$ and $j$ are the initial
partons, $x_1$ and $x_2$ are their longitudinal momentum fractions, and
$\mu$ is the renormalization scale. The $t\bar t$ hadroproduction cross section
has been calculated to the next order ($\alpha_s^3$)~\cite{rkellis}; there are
uncertainties from the parton distributions and also from the choice of scale
(Fig.~4). Electroweak subprocesses such as $W^+ g \to t \bar b$, producing
single top quarks, are also interesting but not competitive
at the Tevatron\cite{yuan}.

\begin{center}
%\epsfxsize=6in
\hspace*{0in}
%\epsffile{fig1-4.eps}

\medskip

\parbox{5in}{\small Fig.~4: Range of predictions for $\sigma(p\bar p\to t\bar
tX)$ at $\sqrt s=1.8$~TeV at order $\alpha_s^3$, with  various scales $\mu$.
{}From Ref.~\cite{berends}.}
\end{center}


   To detect $t\bar t$ production is not easy: it is only a tiny fraction of
the total cross section and fake events (containing leptons and jets)
can arise from relatively copious production of $b$-quarks and $W$ or $Z$.
For $p\bar p$ collisions at $\sqrt s=1.8$~TeV we have~\cite{hmrs}
\[
\vbox{\halign{\tabskip1em
$#\hfil$& \hfil#\tabskip0em& \hfil${}=#\,\rm cm^2$\cr
\sigma(\rm total)& 70 mb& 7\times10^{-26}\cr
\sigma(b\bar b)& 30 $\mu$b& 3\times10^{-29}\cr
\sigma(W)& 20 nb& 2\times10^{-32}\cr
\sigma(Z)& 2 nb& 2\times10^{-33}\cr
\sigma(t\bar t)_{m_t=150}& 10 pb& 1\times10^{-35}\cr}}
\]
The reliability of QCD calculations has been tested by $b$-quark data;
Figure~5 shows that CDF measurements of inclusive $b$-production for
transverse momentum $p_T(b) > p_T^{\rm min}$  approximately agree
with expectations~\cite{hmrs}.

The number of observed events in a given channel is
%
\begin{equation}
N_{\rm events} = \sigma\times{\rm BF}\times \int\L dt \times\rm efficiency\;.
\end{equation}
%
Here $\int{\cal L}dt$ is the integrated luminosity, for which the CDF
detector accumulated about 4~pb$^{-1}$ up to 1989. However the D0 detector
is now working too and the accelerator may deliver
25~pb$^{-1}$ per detector in the 1992--3 running, rising to 75~pb$^{-1}$ in
1994 and eventually (with a new main injector) up to 1000~pb$^{-1}$ in 1997. A
possible increase of energy to $\sqrt s=2$~TeV in 1994 would increase
the $t\bar t$ signal by about 30\% for $m_t = 150$~GeV.


\begin{center}
%\epsfxsize=4in
\hspace*{0in}
%\epsffile{fig1-5.eps}

\parbox{5in}{\small Fig.~5: Inclusive $p\bar p\to bX$ cross sections versus
$p_T^{\rm min}$; preliminary CDF data are compared with QCD expectations. From
Ref.~\cite{hmrs}.}
\end{center}



   Top events have the structure $p\bar p\to t\bar t\to (bW^+)(\bar bW^-)$
where typically the $b$-quarks appear as jets and each $W$-boson appears either
as an isolated lepton (plus invisible neutrino) or as a pair of quark
jets. However, the various partons can also radiate additional gluons or
quarks, and final state partons can overlap, so the net number of jets is
not fixed.  Typical topologies for $t\bar t$ signal and $b\bar b$ background
events are shown in Fig.~6.  The $b\bar b$ background process produces a
lepton in or near a jet (i.e.\ non-isolated) and can be greatly suppressed
by a stringent isolation requirement.  The other major background, from
$W+{}$QCD~jets, can give isolated leptons but usually gives less central jet
activity than the $t\bar t$ signal; furthermore the jets usually do not
contain a $b$-jet.

\begin{center}
\hspace*{0in}
%\epsfxsize=4in
%\epsffile{fig1-6.eps}

{\small Fig.~6: Topologies of typical $t\bar t$ signal and $b\bar b$ background
events.}
\end{center}

\section{Top search strategies}

   Exploiting the leptonic $W$-decays, one usually requires either one
or two isolated leptons, for which $t\bar t$ states have the following
branching fractions:
%
\begin{eqnarray}
     B(t\bar t \to e \mu X)        &=&  0.024 \;, \nonumber\\
     B(t\bar t \to ee \mbox{ or } \mu\mu X)  &=&  0.024 \;,  \label{BFs}\\
     B(t\bar t \to e \mbox{ or } \mu +jets) &=&  0.29  \;. \nonumber
\end{eqnarray}
%
There are then various strategies, based on detecting
\begin{enumerate}
\item Dileptons: this is the smallest but cleanest channel. We require two
isolated opposite-sign leptons, not back-to-back, plus missing-$p_T$ (denoted
$\overlay/p_T$). The $b\bar b$ background is suppressed (mostly non-isolated
and back-to-back in azimuth);
$W+{}$jets does not contribute; Drell-Yan and $Z+{}$jets is suppressed by
$\overlay/p_T$
(and additionally if we restrict to $e\mu$ cases). A small background from
direct $WW$ and $WZ$ production remains, further suppressed by requiring
extra jets.

\item Single lepton plus jets: this is bigger but dirtier. We require one
isolated lepton plus large $\overlay/p_T$ plus several jets, of which two have
invariant mass $m(jj)\sim M_W$.  Then $b\bar b$ is suppressed but $W+{}$jets is
only partly suppressed.  This signal is bigger [see Eq.~(\ref{BFs})] and also
offers a complete
reconstruction of top from 3 of the jets  (unlike (a) where there is always
a missing neutrino), but the $W+{}$jets background remains problematical.

\item $b$-tags.  The presence of a muon near a jet, with $p_T > 1$~GeV
relative to the jet axis, tags it as probably a $b$-jet.  Also the
presence of a displaced vertex (with suitable conditions on the
emerging tracks) can give an efficient $b$-tag.  Adding such a tag would
purify both signals (a) and (b) above, but at some cost to the signal
event rate --- perhaps a factor 10 for muon-tagging or a factor 3 for
vertex-tagging.   CDF have used muon-based $b$-tagging to sharpen up
their top search in the single-isolated-lepton channel; a vertex-tagger
is now in use too. A recent study of $b$-tagging in the heavy-top search is
given in Ref.~\cite{han}
\end{enumerate}

   We can illustrate this discussion with some numbers, assuming
integrated luminosity 100~pb$^{-1}$, acceptance cuts $p_T > 15$~GeV on each
lepton and jet and $\overlay/p_T$, plus further detection efficiency
factors 50\% for each lepton. Predicted numbers of events are
then~\cite{protopop}
\[
\begin{array}{c@{\qquad}c@{\qquad}c@{\qquad}c} \hline\hline
m_t\rm\ (GeV) & \sigma\rm\ (pb) & e + \mu (+2\rm\,jets) &
 e\hbox{ or }\mu > 2\,\rm jets \\  \hline
100 &  88 & 50\ (30) & 570 \\
120 & 34 & 20\ (16) & 300 \cr
140 & 15 & 10\ (9) & 160\cr
180 & 3 & 2\ (2) & 40\cr  \hline
\rm backgrounds \cr
Z\to \tau\tau & 200 & 30\ (7) & \cr
W\to\ell\nu & 4400 & & 150 \cr \hline\hline
\end{array}
\]


Comparing this with the current Tevatron situation (luminosity
${}\approx 20$~pb$^{-1}$) we would expect about 4(2) $e\mu$ candidate events
for $m_t=120(140)$~GeV, in the range preferred by SM theory Eq.~(\ref{m_t SM}).
In fact there are three candidate $t\bar t\to{}$dilepton events from CDF and
one from D0, so the numbers are not inconsistent with expectations for $m_t\alt
160$~GeV.

   Figures 7--10 illustrate important aspects of the dilepton search
strategy~\cite{bbp2}. Figure~7 shows how a missing-$p_T$ cut (in this case
${\not\hskip-1.5pt p}_T > 20$~GeV) discriminates strongly against $\gamma^*\to
\ell^+\ell^-$  and $Z\to\ell^+\ell^-$  backgrounds.  Figure~8 shows how an
azimuthal angle-difference cut  $30^\circ <  \Delta\phi(\ell^+\ell^-)  <
150^\circ$  discriminates against $b\bar b\to\ell^+\ell^-$  backgrounds.
Figure~9 shows how the number of accompanying jets in
$t\bar t\to \ell^+\ell^-$  events changes with $m_t$; we see that in the
neighborhood  $m_t\sim M_W + m_b$  the recoiling $b$-quarks are too soft to
form jets so the multiplicity falls, and it is not efficient to
demand jets here. Fortunately the $t\bar t$ signal here is much stronger
than the remaining backgrounds and we do not need help.  At higher $m_t$
values where the signal falls and the $WW$ background becomes a problem,
the $b$-jets are hard and can be used for additional discrimination.
Finally, Fig.~10 shows that the $t\bar t\to \ell^+\ell^-$  signal can be
distinguished for higher masses $m_t = 150$--200~GeV and even beyond, by
making a more severe requirement $p_T > 30$~GeV  on the two
accompanying jets; the $b$-jets from $t\to bW$ are naturally very hard in this
mass range so there is little cost to the signal, but the $WWjj$ background
goes down by a factor~4.  The good news in all this is that the
``gold-plated" dilepton signal remains essentially background-free
through the whole of the SM-favored mass range and well beyond.

\begin{center}
%\epsfysize=3.5in
\hspace*{0in}
%\epsffile{fig1-7.ai.eps}

\medskip

\parbox{5.5in}{\small Fig.~7: $\gamma^*$ and $Z$ backgrounds to the top
dilepton signal are suppressed by a cut $\overlay/p_T>20$~GeV~\cite{bbp2}.}

\bigskip

%\epsfysize=3.5in
\hspace*{0in}
%\epsffile{fig1-8.ai.eps}\hspace*{.5in}

\parbox{5.5in}{\small Fig.~8: The $b\bar b$ background to the top dilepton
signal is suppressed by rejecting back-to-back leptons (in azimuth), with a cut
$30^\circ<\Delta\phi<150^\circ$~\cite{bbp2}.}

\bigskip\bigskip

\hspace*{0in}
%\epsfysize=3.5in
%\epsffile{/ed/Barger/Brazil/Top/p11fig.eps}

\parbox{5.5in}{\small Fig.~9: Topological cross sections for dileptons plus $n$
jets, versus $m_t$, at the Tevatron (detection efficiency factor $\sim0.3$ not
included)~\cite{bbp2}.}



\bigskip

\hspace*{0in}
%\epsfysize=3.5in
%\epsffile{fig1-10.eps}

\parbox{5.5in}{\small Fig.~10: $p\pbar \to\mbox{dilepton + 2-jet event rates at
}\sqrt s = 2$~TeV, for $t\tbar$ signal and $WWjj$ background. Jet cuts
$p_T(j) > 15$, 30 GeV are compared~\cite{bbp2}.}

\end{center}



   We turn now to single-lepton search strategies, seeking  $t\bar t$
events with one $t\to bW\to b\ell\nu$  plus one  $t\to bW \to bjj$  decay.
We start simply by looking for  $W\to \ell\nu$  events, requiring a hard
isolated lepton plus substantial  $\overlay/p_T$;  the distinctive signature
of $W\to \ell\nu$ is that the ``transverse mass"  $m_T$  peaks sharply near
$M_W$. Here $m_T$ is defined by
%
\begin{equation}
  m_T^2(e\nu) = (p_{Te}+\overlay/p_T)^2 - (\vec p_{Te}+\vec{\overlay/p}_T)^2 =
2p_{Te}\overlay/p_T(1-\cos\phi_{e\nu})   \label{m_T^2}
\end{equation}
%
and its peak is a kinematical property, related to the ``Jacobian" peaks
of  $p_T(e)$  and  $p_T(\nu)\simeq\overlay/p_T$, but with the special virtue
that it
is much less smeared by transverse motion of the $W$.  For on-shell
$W$-decays the shape of the  $m_T$  distribution is predictable.  If $m_t
< M_W + m_b$,  $t\bar t$ events would contribute off-shell $W$-decays that
would distort the shape in a characteristic way; Figs.~11 and 12 show
theoretical examples and some real data.
Since the  $t\bar t$ signal has
typically several accompanying jets while the background from plain
$W$-production has typically little jet activity, the signal becomes
more striking when we require more jets (Fig.~11); it should be
detectable in events with  $n \ge 2$ jets~\cite{bbp2}.  Figure~12 shows
comparisons of theoretical distributions with CDF 2-jet and 1-jet data; the
absence of any detectable signal in the early CDF data gave a limit  $m_t >
77$~GeV at 95\%~CL.

   If however $m_t > M_W + m_b$,  the $W$ is mostly on-shell and no
distortion of the  $m_T$  distribution is expected.  In this case we must
simply look for an excess in  $W + n$-jets  events, compared to the
background from plain $W$ production.  This background is huge.  To
reduce it we first require that $n$ be large, say $n \ge 3$  or
$n \ge 4$.  We can also require that two of the jets have invariant
mass $m(jj) \simeq M_W$  (since the signal contains $W \to jj$), but
with experimental uncertainties and many possible jet pairings this
condition is not very stringent.   It is much more effective to
require that one jet is $b$-tagged, as shown in Fig.~13; the cross sections
shown here do NOT include the tagging efficiency, which depends on
experimental details but is of{\parfillskip0pt\par}

\begin{center}
%\epsfxsize=4.9in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/Top/p13fig1.eps}

\medskip

\parbox{5.5in}{\small
Fig.~11: Theoretical examples of single-lepton signals in the $m_T$
        distribution, for $m_t < M_W+m_b$; $n$ denotes jet
        multiplicity~\cite{bbp2}.}

\bigskip

%\epsfxsize=3.4in
\hspace{0in}
%\epsffile{fig1-12.eps}

\medskip

\parbox{5.5in}{\small
Fig.~12: Single-lepton transverse mass distributions at the Tevatron with
      (a)~2 jets and (b)~1 jet. Solid curves are the calculated $W+{}$jets
      background, the dashed curve is the expected signal for
      $m_t=70$~GeV~\cite{cdf1}.}

\end{center}

\noindent
order 30\% for CDF vertex-tagging. We see that,
after $b$-tagging and requiring $n \ge 3$, the $t\bar t$ signal greatly
exceeds the $W+{}$jets backgrounds through the range $m_t = 95$--170~GeV
or so.  The signal diminishes below 95~GeV because the $b$-jets get
soft.




\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/Top/p14fig.eps}

\parbox{5.5in}{\small
Fig.~13: Effects of $b$-tagging on the single-lepton $t\bar t$ signal at the
      Tevatron, for various jet multiplicities~\cite{protopop}. Jets and
      leptons are required to have $p_T(j)>15$~GeV, $p_T(l)>20$~GeV,
      $|\eta(j)| < 2,\ |\eta(l)| < 1$.}
\end{center}



   With an eventual luminosity 1000~pb$^{-1}$, the expected event yields at
the Tevatron depend on $m_t$ as follows~\cite{protopop}:
%
\[
\begin{array}{c@{\qquad}c@{\qquad}c}
m_t\ \rm (GeV) & e\hbox{ or }\mu+4\,\rm jets\ events & e+\mu\ \rm events\\
120 & 1380 & 240 \\
140 & 850 & 98 \\
180 & 260 & 24 \\
210 & 140 & 12 \\
240 & 60 & 5
\end{array}
\]
The mass ranges where a firm signal could be established, and the lower
bound that could be set in the case of no signal, depend on luminosity
like this~\cite{protopop}:
\[
\begin{array}{c@{\qquad}c@{\qquad}c}
\L & \hbox{claim discovery} & \hbox{90\% CL bound}\\
100\pb^{-1} & m_t\leq 150\gev & {}>180\gev \\
1000\pb^{-1} & m_t\leq 220\gev & {}>250\gev
\end{array}
\]

\section{Further considerations}

   When a top signal is found, $m_t$ can be estimated from various
dynamical distributions that are sensitive to it~\cite{bbp2}, {\it e.g.}
\begin{enumerate}
\item invariant masses $m(\ell_1,\ell_2)$ of two isolated leptons;
\item invariant mass $m(\ell,\mu)$ of single isolated lepton and
(opposite-sign) muon tagging the associated $b$-jet;
\item invariant mass $m(jjj)$ of three jets accompanying single lepton;
\item cluster transverse masses such as $m_T(\ell\ell jj, \overlay/p_T)$;
\item variables arising in families of explicit event reconstructions.
\end{enumerate}
Maximum-likelihood methods can be used too~\cite{kondo,dalitz}. With
1000~pb$^{-1}$ of luminosity, the Tevatron could determine $m_t$ to 5~GeV or
better (at least up to about 170~GeV).

   For the planned $pp$ supercolliders SSC ($\sqrt s=40$~TeV) near Dallas
and LHC ($\sqrt s=15.4$~TeV originally, now 14~TeV) at CERN,  $t\bar t$
production will be
enormously larger than the Tevatron rates.  Both the intrinsic cross
sections (Fig.~14) and the planned luminosities are much bigger.  If
$m_t=150$~GeV the cross sections and event rates will be
%
\begin{eqnarray}
\mbox{SSC: } \sigma(pp\to t\bar tX)&=&12\mbox{\,nb\quad giving }1.2\times10^7
\,\mbox{events/year} \;,\nonumber \\
\mbox{LHC: } \sigma(pp\to t\bar tX) &=&\phantom02\mbox{\,nb\quad giving }
2.0\times10^7
\,\mbox{events/year} \;. \nonumber
\end{eqnarray}
%
One expects to measure $m_t$ to 2--3~GeV using the distribution $m(\ell,\mu)$
of isolated lepton plus tagging muon described in (b) above; since both
$\ell$ and $\mu$ originate from the same parent $t$, their invariant mass
distribution (Fig.~15) depends only on the decay mechanism.

\begin{center}
%\epsfxsize=5.5in
\hspace{0in}
%\epsffile{fig1-14.eps}

{\small Fig.~14: $t\bar t$ production at the Tevatron, LHC and
SSC~\cite{reya}.}

%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig1-15.eps}

{\small Fig.15: $m_t$ dependence of $m(\ell,\mu)$ distribution~\cite{massem}.}
\end{center}

   So far we have assumed purely $t\to bW$  SM decays, but other modes
are possible beyond the SM.  If there exist charged Higgs bosons $H^\pm$,
then decays like
%
\begin{equation}
          t \to b H^+\to b c \bar s,\, b \nu \bar\tau \;,
\end{equation}
%
become possible.  If the competing $t\to bH^+$  mode is strong it will reduce
the SM $t\to bW$  branching fraction, reducing the SM signals we have
discussed.  In return we get some new signals: the $b c \bar s$
 final state is similar
to one of the SM modes but with a different mass peak at $m(c\bar s)=m_{H^+}$;
the $b \nu \bar\tau$ mode can be recognized by an excess of $\tau$ over $e$ or
$\mu$ production (lepton non-universality).  In models with two Higgs doublets
the various branching fractions are controlled by a parameter
$\tan\beta$, the ratio of the two vevs, constrained to lie in a range $0.2 \alt
\tan\beta \alt
100$ if the couplings are perturbative~\cite{bhp}; Fig.~16 shows how the SM and
Higgs modes compete and combine to
give a range of possible  $t\to be\nu$ and $t\to b\tau\nu$  fractions for an
example with $m_t=150,\ m_{H^+}=100$~GeV.  At large $\tan\beta$ a dramatic
excess
of $\tau$ over $e$ or $\mu$ is predicted; but at small $\tan\beta$ both the SM
signatures based on $t\to e(\mu)$ and the new signature based on $t\to \tau$
are strongly suppressed and top would be very difficult to discover at hadron
colliders~\cite{bp1}.

   In SUSY models there may be other new modes such as
%
\begin{equation}
    t \to \tilde t_1 \tilde Z_i\to c \tilde Z_1 \tilde Z_i \; ,
\end{equation}
%
where $\tilde t_1$ is the light squark partner of $t$,  $\tilde Z_i$ are
neutralinos (with
$\tilde Z_1$ the lightest), if the SUSY particles are light enough.  These
too would compete and deplete the SM signals; a top quark as light as
65~GeV, with dominant SUSY decays, is not yet ruled out by Tevatron
data~\cite{bt}.


\begin{center}
%\epsfxsize=3.5in
\hspace{0in}
%\epsffile{fig1-16.ai.eps}

\parbox{5.5in}%
{\small Fig.~16: Sample $\tan\beta$ dependence of $t$ and $H^+$ branching
fractions\cite{bp1}. The hidden top region where all semileptonic signals are
suppressed is $\tan\beta\alt0.3$.}
\end{center}



   Finally we come to the possibilities at future $e^+e^-$ linear
colliders, with luminosities of order 20~fb$^{-1}$/year.  Figure~17 shows
cross sections versus CM energy relative to the $e^+e^-\to\mu^+\mu^-$ cross
section; the kink in the $e^+e^-\to \bar qq$ rate is due to $\bar tt$
production (here assuming $m_t=150$~GeV).  At $\sqrt s=500$~GeV the $e^+e^-\to
t\bar t$ event rate would be around $10^4$/year, comparable with the Tevatron
rather than the SSC; however the events would be much cleaner and top
parameters would be easier to extract.  An $m_t$ measurement with
statistical uncertainty 0.3~GeV from 10~fb$^{-1}$ luminosity is
expected~\cite{kuhn}.  The width $\Gamma_t$ could also be accurately measured
near the threshold energy, either from the energy-dependence of the
cross section (Fig.~18), or from the momentum spectrum of $t$, or from a
forward/backward asymmetry~\cite{fujii,kuhn,peskin}.

\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig1-17.eps}

{\small Fig.~17: Cross sections for possible high-energy $e^+e^-$
        colliders~\cite{burke}.}

\bigskip\bigskip

%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig1-18.eps}

{\small Fig.~18: $\sigma(e^+e^-\to\bar tt)$ near threshold~\cite{kuhn}.}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

\renewcommand{\chapter}{\section}

\begin{thebibliography}{00}
%\addtolength{\itemsep}{-.075in}

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Madison, March 1993.

\bibitem{barpak} V.~Barger, S.~Pakvasa, Phys.\ Lett.\ {\bf 81B}, 195 (1979);
   G.L.~Kane, M.~Peskin, Nucl.\ Phys.\ {\bf B195}, 29 (1982).

\bibitem{cleollx} CLEO collaboration, Phys.\ Rev.\ {\bf D35}, 3533 (1987).

\bibitem{roybb} D.P.~Roy, S.U.~Sankar, Phys.\ Lett.\ {\bf B243}, 296 (1990);
D.P.~Roy, in {\it Proc.\ of the 2$^{nd}$ Workshop on High Energy Physics
Phenomenology (WHEPP-II)}, Calcutta, World Scientific Press, Singapore (1991).

\bibitem{lepsm}  G.~Kane, Wisconsin Top Symposium (1992).

\bibitem{elfog} L.~Rolandi, in {\it Proc.\ of ICHEP '92}, Dallas.

\bibitem{fujii} K.~Fujii, in {\it Proc.\ of the Workshop on Physics and
Experiments with Linear Colliders}, Saariselka, Finland (1991).

\bibitem{khoze} {\it e.g.} V.A.~Khoze {\it et al.}, Nucl.\ Phys.\ {\bf B378},
413 (1992); T.~Sj\"ostrand, Wisconsin Top Symposium (1992).

\bibitem{rkellis} R.K.~Ellis, Phys.\ Lett.\ {\bf B259}, 492 (1991); R.K.~Ellis,
S.~Dawson, P.~Nason, Nucl.\ Phys.\ {\bf B303}, 607 (1988).

\bibitem{yuan} C.P.~Yuan, Phys. Rev. {\bf D41}, 42 (1989) and {\bf D47}, 2746
(1993).

\bibitem{berends} F.~A.~Berends, J.~B.~Tausk and W.~T.~Giele,
Fermilab-Pub-92/196-T.

\bibitem{hmrs} G.~P.~Yeh, Wisconsin Top Symposium (1992), unpublished.

\bibitem{han} T. Han and S. Parke, Fermilab-Pub-93/105-T.

\bibitem{protopop} S.~Protopopescu, Wisconsin Top Symposium (1992),
unpublished.

\bibitem{bbp2} H.~Baer, V.~Barger and R.J.N.~Phillips, Phys.\ Rev.\ {\bf D39},
3310 (1989);  Phys.\ Lett.\ {\bf B221}, 398 (1989); H.~Baer {\it et al.},
Phys.\ Rev.\ {\bf D42}, 54 (1990).

\bibitem{cdf1} CDF collaboration, Phys.\ Rev.\ Lett.\ {\bf 64}, 142 (1990).

\bibitem{kondo} K.~Kondo, J.~Phys.\ Soc.\ Jpn.\ {\bf57}, 12 (1988).

\bibitem{dalitz} R.H.~Dalitz, G.R.~Goldstein, Phys.\ Lett.\ {\bf B287}, 225
(1992); G.R.~Goldstein, K.~Sliwa, R.H.~Dalitz, Phys.\ Rev.\ {\bf D47}, 967
(1993).

\bibitem{reya} E.~Reya et al., Proc. LHC Workshop CERN 90-10, Vol.II, p.296.

\bibitem{massem} SDC Technical Design Report SDC-92-201 (1992).

\bibitem{bhp} V.~Barger, J.~L.~Hewett and R.~J.~N.~Phillips, Phys. Rev. {\bf
D41}, 3421 (1990).

\bibitem{bp1} V.~Barger, R.J.N.~Phillips, Phys.\ Rev.\ {\bf D41}, 884 (1990).

\bibitem{bt} H.~Baer {\it et al.}, Phys.\ Rev.\ {\bf D44}, 725 (1991).

\bibitem{burke}  T.~Burke, Wisconsin Top Symposium (1992).

\bibitem{kuhn}   H.~K\"uhn, Wisconsin Top Symposium (1992).

\bibitem{peskin} M.~Peskin, Wisconsin Top Symposium (1992).

\end{thebibliography}
%-------------------------------------------------------------------

\end{document}



timated from various
dynamical distributions that are sensitive to it~\cite{bbp2}, {\it e.g.}
\begin{enumerate}
\item invariant masses $m(\ell_1,\ell_2)$lecture2.

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\begin{document}
\setcounter{page}{12}

\chapter*{LECTURE 2:\\ THE SM HIGGS BOSON}
\setcounter{chapter}{2}


\section{Introduction}

   An unbroken  $\rm SU(3)\times SU(2)_L\times U(1)_Y$  gauge symmetry would
demand not
only massless gauge bosons but also massless fermions, since fermion
mass terms do not conserve chirality and violate SU(2)$_L$.  Electroweak
symmetry breaking (EWSB) is therefore directly related to the origin of
masses. A major task of collider physics is to probe the EWSB
mechanism, which is still completely untested.  In this lecture we focus on
the simplest example, namely the SM Higgs mechanism.

   In the SM, SU(3) color symmetry remains unbroken (gluons are massless).
However the electroweak $\rm SU(2)_L\times U(1)_Y$ symmetry is spontaneously
broken down
to U(1)$_{\rm QED}$ at energy scales  $\alt M_W,M_Z$; photons remain massless
here but the $W$ and $Z$ gauge bosons acquire masses and fermions too can
have masses. (At energy scales $\gg M_W,M_Z$  all these masses become
irrelevant and the full symmetry is effectively restored.)

   Spontaneous symmetry-breaking  is achieved by introducing an
isospin-doublet complex scalar field $\Phi$, with self-interactions that
cause a non-zero vacuum expectation value (vev) in its neutral component.
This vev, which arises from minimizing the self-interaction potential,
immediately implies specific masses for $W$ and $Z$; three of the four
scalar degrees of freedom materialize as the longitudinal fields $W_L^+,W_L^-,
Z_L$ (previously absent for massless $W$ and $Z$) and the fourth appears
as a neutral isosinglet CP-even spin-0 ``Higgs boson" $H^0$, with mass
depending on the EWSB parameters.  The vev also allows  arbitrary fermion
masses to be introduced through Yukawa couplings of fermion fields to $\Phi$.

   The electroweak Lagrangian and notation are
%
\begin{eqnarray}
L &=& {}-{1\over4}{\bf W}^{\mu\nu}\cdot{\bf W}_{\mu\nu}-{1\over4}B^{\mu\nu}
B_{\mu\nu}+(D^\mu\Phi)^\dagger (D_\nu\Phi) - V(\Phi) \nonumber\\  \nv1
&&{}+ \overline\psi i\gamma^\mu D_\mu \psi+\mbox{Yukawa interactions}\;,\\ \nv2
%
{\bf W}_{\mu\nu}&=& \partial_\mu{\bf W}_\nu-\partial_\nu{\bf W}_\mu - g{\bf
W}_\mu \times {\bf W}_\nu \;, \\ \nv2
%
B_{\mu\nu}&=&\partial_\mu B_\nu - \partial_\nu B_\mu \;, \\
%
D_\mu &=& \partial_\mu+ig{\bf T\cdot W}_\mu+ig' {Y\over2} B_\mu \;,
\end{eqnarray}
%
where $g$ and $g'$ are the SU(2)$_{\rm L}$ and U(1)$_{\rm Y}$ gauge couplings
respectively.
The scalar doublet has the form and vev
%
\begin{equation}
   \Phi =\left(\matrix{\phi^+\cr\phi^0\cr}\right)=
\left(\matrix{-iw^+\cr{-iz+H+v\over\sqrt2}\cr}\right)  \;,
\qquad
\left<\phi^0\right> = {v\over\sqrt2} \;,
\end{equation}
%
where the potential is
%
\begin{equation}
V(\Phi) = \lambda \left( |\Phi|^2 - {v^2\over2}\right)^2
= \lambda\left(w^+w^- + {1\over2} zz + vH + {1\over2}H^2\right)^2 \;.
\end{equation}
%
The fields $w^+, w^-$ and $z$ become $W_L^+, W_L^-$ and $Z_L$ (the ``Higgs
miracle") while $H$ is the physical Higgs boson field.  From the
$(D^\mu\Phi)^\dagger(D_\nu\Phi)$ term we get the gauge boson masses and
mass-eigenstates
%
\begin{eqnarray}
&\displaystyle
M_W={1\over2}gv \;, \qquad M_Z = {M_W\over\cos\theta_W}\;, \qquad M_A=0&\\
&Z = \cos\theta_W W_3 - \sin\theta_W B\;,&  \label{Z}\\
&A = \sin\theta_W W_3 + \cos\theta_W B\;,&
\end{eqnarray}
%
where $A$ is the electromagnetic field. The vev $v$ and weak mixing angle
$\theta_W$ ($\tan\theta_W=g'/g$) are arbitrary but experimentally determined:
%
\begin{equation}
   v = (\sqrt2 G_F)^{-1/2} = 246\gev \;, \qquad
 \sin^2\theta_W = 0.23 \;.
\end{equation}
%
The Higgs boson mass comes from $V(\Phi)$ and is still unknown:
%
\begin{equation}
   m_H = \sqrt{2\lambda}\, v     \;.
\end{equation}
%

   The most obvious relic of EWSB is $H$; it is the first thing to look
for.  If the unknown coupling $\lambda$ is weak ($\lambda \alt 1,\ m_H
\alt 300\gev)\ H$ is observable as a narrow-width ``elementary
particle\rlap".  For moderate coupling $(\lambda \sim 2\mbox{--}3,\ m_H\sim
600{\rm~GeV})\ H$ appears as a broad resonance with width
%
\begin{equation}
   \Gamma_H  \sim 0.5 m_H^3 /(\rm TeV)^2     \;  .
\end{equation}
%
For larger coupling $(\lambda > 4,\ m_H > 1$~TeV) the resonance is too
broad to distinguish from backgrounds.  However the longitudinal spin
components $W_L^\pm$ and $Z_L$ carry memories of their origins as the $w^\pm$
and $z$ components of $\Phi$, which are essentially Goldstone bosons of the
broken symmetry.  Hence strong $\lambda$ coupling implies strong longitudinal
vector boson scattering, much stronger than scattering via gauge
interactions; {\it e.g.}\ it violates tree-level unitarity in $W_L W_L \to
W_L W_L, Z_L Z_L, t\bar t, HH$  processes for $\sqrt s > 1$\,TeV~\cite{lee},
with already detectable effects below 1\,TeV.  Therefore, whatever the value
of $\lambda$, something observable is predicted --- either $H$ itself
or strong vector-vector scattering --- at subprocess energies below 1~TeV.
New physics at SSC is guaranteed.  This lecture concentrates on $H$, the
benchmark for EWSB studies~\cite{hhg}; Lecture~3 addresses $W_L W_L$
scattering.

%\break

   There are various theoretical upper bounds on $m_H$:
\begin{enumerate}
\item Tree-level unitarity requires $m_H \alt 1$\,TeV.
\item Non-perturbative and lattice calculations\cite{latmh} indicate
$m_H \alt 600$--700\,GeV.
\item GUT models where $\lambda$ remains perturbative up to the GUT
scale\cite{gutmh} require $m_H \alt 200$\,GeV.
\end{enumerate}
Since the underlying arguments are not completely compelling, we shall
consider $m_H$ to lie anywhere above the current LEP experimental
bound\cite{lepmh}
%
\begin{equation}
      m_H  > 62.4\gev \quad  \mbox{(combined LEP 1992 bound)} \; .
\label{lep bound}
\end{equation}
%

   At tree level the SM Higgs interactions have the form
%
\begin{equation}
L_{\rm SM} = {1\over v} \left( 2M_W^2 W^{\mu+} W_\mu^- + M_Z^2 Z^\mu Z_\mu
\right) \left(H+{H^2\over 2v}\right) - \sum_f {m_f\over v} \bar ffH \;.
\end{equation}
%
The interactions with $W$ and $Z$ are specified by the gauge
couplings (through $v$). The couplings with fermions $f$ depend on the
fermion masses, since both come from Yukawa couplings $\lambda_f
\bigl[ (f_L \Phi) f_R + \rm h.c. \bigr]$ with $m_f = \lambda_f v/\sqrt2$.
(These masses and the Yukawa
couplings are arbitrary; the SM provides no explanation for their
particular values.)  Thus $H$ couples preferentially to the heaviest
particles: $W, Z, t, b, \tau$.


\section{SM Higgs Decays}

   For light and intermediate mass $m_H<2M_W$, the dominant decays are to
fermion pairs.  The partial widths at tree level are given by
%
\begin{equation}
\Gamma(H\to f\bar f) = {G_F\over\sqrt2}\, {m_Hm_f^2\over4\pi}
\left(1-{4m_f^2\over m_H^2}\right)^{3/2} \,\,c_f
\end{equation}
%
with color factor  $c_f = 3$ (1) for quarks (leptons). For decay to
quarks ($f = q$),  QCD introduces two important corrections~\cite{braaten}.

%\noindent
1) The factor $m_q^2$ outside the bracket is replaced by the running
mass-square  $m_q^2(m_H^2)$,
evaluated at the scale $Q^2 = m_H^2$.  It is related to the running
masses~\cite{gasser}
%
\begin{equation}
\begin{array}{rlrl}  m_s(1\gev^2)&=0.18\gev,\qquad& m_c(m_c^2)&=1.27\gev,\\
 m_b(m_b^2)&=4.25\gev, \qquad & m_t(m_t^2)&\simeq 150\gev ,
\end{array}
\end{equation}
%
by the renormalization group equations (RGE), which give:
%
\begin{eqnarray}
\lefteqn{ m_s^2 < Q^2 < m_c^2:}  \nonumber \\
& m_q(Q^2) = m_q(m_s^2) \left(\alpha_s(Q^2)\over\alpha_s(m_s^2)\right)^{4/9}
{F_s(Q^2)\over F_s(m_s^2)} \;,
\quad\ F_s(Q^2) = 1 + 0.89\left(\alpha_s(Q^2)\over\pi\right) +
1.37\left(\alpha_s(Q^2)\over\pi\right)^2 ,\nonumber\\
\\
%
\lefteqn{m_c^2 < Q^2 < m_b^2:} \nonumber \\
& m_q(Q^2)=m_q(m_c^2)\left(\alpha_s(Q^2)\over\alpha_s(m_c^2)\right)^{12/25}
{F_c(Q^2)\over F_c(m_c^2)} \,,
\quad F_c(Q^2)=1+1.01\left(\alpha_s(Q^2)\over\pi\right) +
1.39\left(\alpha_s(Q^2)\over\pi\right)^2 ,\nonumber \\
\\
%
\lefteqn{ m_b^2 < Q^2 < m_t^2:} \nonumber \\
& m_q(Q^2)=m_q(m_b^2)\left(\alpha_s(Q^2)\over\alpha_s(m_b^2)\right)^{12/23}
{F_b(Q^2)\over F_b(m_b^2)} \,,
\quad F_b(Q^2)=1+1.17\left(\alpha_s(Q^2)\over\pi\right) +
1.50\left(\alpha_s(Q^2)\over\pi\right)^2 , \nonumber \\
 \\
%
\lefteqn{ m_t^2 < Q^2:}  \nonumber \\
& m_q(Q^2)=m_q(m_t^2)\left(\alpha_s(Q^2)\over\alpha_s(m_t^2)\right)^{4/7}
{F_t(Q^2)\over F_t(m_t^2)} \,,
\quad\ F_t(Q^2)=1+1.40\left(\alpha_s(Q^2)\over\pi\right) +
1.79\left(\alpha_s(Q^2)\over\pi\right)^2  .\nonumber \\
\end{eqnarray}
%
[The value where the curves $m = m_q(Q^2)$ and $m = Q$
intersect is relevant when using the
RGE to discuss radiatively generated mass effects.
But the mass defined by the pole in the quark propagator differs
slightly: $m_q^{\rm pole} = m_q(m_q^2)[1 +
 4\alpha_S(m_q^2)/(3\pi)+ {\cal O}(\alpha_S^2)]$.]


%\noindent
2) The partial width is multiplied by an extra factor
%
\begin{equation}
\left[ 1 + 5.67{\alpha_s\left(m_H^2\right)\over\pi} + (35.94-1.36f)
\left(\alpha_s\left(m_H^2\right)\over\pi\right)^2 \right] \,,
\end{equation}
%
where $f$ now denotes the number of active quark flavors.  The combined
effect of these two corrections is to reduce the $H \to b \bar b$  decay rate
by a factor $\sim 0.6$.  A typical result is
%
\begin{equation}
\Gamma(H \to{\rm all}) \sim \Gamma(H\to b \bar b) \simeq 2.5\mev
                                     \quad {\rm for}\  m_H = 100\gev \;,
\end{equation}
%
so $H$ is very narrow in this mass region.

   For $m_H > 2M_W,\, 2M_Z$,  decays to real weak bosons become dominant (it
turns out that $H \to \bar tt$ can be significant but never dominant):
%
\begin{eqnarray}
\Gamma(H\to W^+W^-) &=& {G_F\,m_H^3\over8\pi\sqrt2} \sqrt{1-{4M_W^2\over
m_H^2}}\left(1-{4M_W^2\over m_H^2}+12{M_W^4\over m_H^4}\right) \\
                    &\sim& (0.3 \tev) (m_H/ 1 \tev)^3  \,,     \\
\Gamma(H\to ZZ) &\sim& {1\over2}\Gamma(H\to W^+W^-)\,.
\end{eqnarray}


However, for $m_H < 2M_W, 2M_Z,$  one of the weak bosons goes off-shell
and we have to calculate  $H \to WW^*, ZZ^*$ summing over $W^*\to f\bar f$,
$Z^*\to f\bar f$ final states~\cite{rizzo}. Typical results are~\cite{cahn}
%
\begin{equation}
\Gamma(H \to ZZ^*) \sim 10 \mev\ (1 \mev)\quad {\rm for}\ m_H = 175 \gev\
 (150 \gev)\;.  \label{typical}
\end{equation}


   Two-photon decay modes are small but important, especially for
searching the intermediate mass range  $M_Z < m_H < 2M_W$  at hadron
colliders, because here the dominant  $b \bar b$  and $\tau\bar\tau$  modes
are very hard to separate from hadronic backgrounds.  $H \to \gamma\gamma$
proceeds via heavy-quark and $W$ loops (Fig.~1);  light-quark and lepton loops
are negligible. The partial width has the form
%
\begin{equation}
\Gamma(H\to\gamma\gamma) = {G_F\,m_H^3\over 128\pi\sqrt2} \left| 7I_W -
{4\over9}I_b - {16\over9}I_t \right| \left(\alpha\over\pi\right)^2 \;.
\end{equation}
%
Here the integrals $I_W$, $I_b$ and $I_t$ representing the $W,b,t$ loop
contributions are complex above their respective thresholds ($m_H > 2M_W,
2m_b, 2m_t$) and $\to 1$ for $m_H$ far below threshold; they are compared in
Fig.~2, where we see that the $W$-loop dominates (for $m_t=150\:$GeV).

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig2-1.eps}

\small{Fig.~1: $H \to \gamma\gamma$  decay diagrams.}

\bigskip\bigskip

%\epsfxsize=4in
\hspace{0in}
%\epsffile{fig2-2.eps}

{\small Fig.~2: Comparison of $W$, $b$ and $t$ loop contributions in
$H\to\gamma\gamma$~\cite{bcps}.}
\end{center}


   Two-gluon decays proceed via heavy-quark loops, similar to those in
two-photon decays.  The partial width is
%
\begin{equation}
\Gamma(H\to gg) = {9\over8}\left(\alpha_s\over\pi\right)^2
\left(G_F\,m_H^3\over128\pi\sqrt2\right) | I_b+I_t |^2 \,,
\end{equation}
%
where the loop integrals are as in Fig.~2.

   Adding all modes gives the total Higgs width shown in Fig.~3~\cite{bcps};
the various channel branching fractions~\cite{bbhk,bbhk2} are shown in Fig.~4.
At a hadron collider, the detectable modes are expected to be
$\gamma\gamma, Z\gamma \to \ell\ell\gamma, ZZ \to 4\ell, ZZ\to \ell\ell\nu\nu,
WW\to\ell\nu\ell'\nu'$.  The decay to
$\tau^+\tau^-$ is interesting but is yet an unproven (or controversial)
channel.  The decay mode
$b \bar b$ is widely dominant but not feasible, except perhaps in the $\bar
ttH$ channel if excellent $b$-tagging is possible\cite{dai}.

\begin{center}
%\epsfxsize=3.75in
\hspace{0in}
%\epsffile{wid.eps}

{\small Fig.~3: Total $H$ decay width versus $m_H$, for $m_t =
150$~GeV\cite{bcps}.}

\bigskip

%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig2-4.eps}

{\small Fig.~4: Comparison of $H$ branching fractions versus $m_H$, for
$m_t=150$~GeV\cite{bbhk2}.}
\end{center}


\section{SM Higgs production at $e^+e^-$ colliders}

   In $e^+e^-$ collisions at the $Z$ resonance (LEP\,I), $H$ production
proceeds via  $Z\to Z^* H, \gamma H$ (Fig.~5).  The ratio of $Z\to HX$
to $Z\to\mu^+\mu^-$ decay widths is shown versus $m_H/M_Z$ in Fig.~6,
for various recoil systems $X$.  The present experimental limit
$m_H > 62.4\:$GeV [Eq.~(\ref{lep bound})] comes from combining the results of
all 4 LEP experiments~\cite{lepmh}.

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig2-5.eps}

{\small Fig.~5: SM $H$ production via the $Z$ resonance.}

\bigskip\bigskip

%\epsfxsize=3.5in
\hspace{0in}
%\epsffile{fig2-6.eps}
\hspace*{.5in}

{\small Fig.~6: Branching fractions for $Z\to HX$ decays versus $m_H$.}
\end{center}


   At the future upgraded LEP\,II, with $\sqrt s \alt 200\:$GeV,
the original design luminosity was 500~pb$^{-1}$ per intersection per
year; it is now believed that several years (or several
experiments combined) will be needed to reach this luminosity.  It will then be
straightforward to detect $H$ for $m_H \alt 80$~GeV (no serious
backgrounds) but for $m_H \sim M_Z$  there is a big background from
$e^+e^-\to ZZ$.  Figures 7a,b illustrate the problem in the channel
$e^+ e^- \to \ell^+ \ell^- q \bar q$; the $H$ peak in $m_{\rm recoil}
= m(q\bar q)$ is
clearly seen for $m_H = 80$~GeV but is lost under the $Z$ peak for
$m_H = M_Z$.  The solution to this difficulty is to demand a $b$-tag
on the $q$-jets (see discussion in Lecture~1), since essentially all
the $ZH$ signal but only 22\% of $ZZ$ background comes from $b \bar b$
pairs.   The untagged and $b$-tagged cross sections for the signal
and background are
%
\begin{equation}
\begin{array}{l@{\qquad}l}
ZH\to \ell^+\ell^- q \bar q{:}\ 30 \fb; & ZH\to \ell^+\ell^- b \bar b{:}\
27 \fb ; \\
ZZ \to \ell^+\ell^-q \bar q{:}\ 93 \fb; & ZZ\to \ell^+\ell^- b \bar b{:}\
 20 \fb ;
\end{array}
\end{equation}
%
for $m_H = M_Z$ and $\sqrt s = 200$~GeV.   Figures~7c,d  show the
improved situation after $b$-tagging (tagging efficiency is not included
here);  the $ZZ$ background is partly suppressed and the $ZH$ signal cannot
easily hide under it.

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig2-7.eps}

\medskip

\parbox{5.5in}{\small Fig.~7: Examples of $ZH$ signal and $ZZ$ background in
the $m_{\rm recoil} = m(q\bar q)$ distribution, for  $e^+e^-\to \ell^+\ell^-
q\bar q$  channels\cite{barwhis}.}
\end{center}


   The best way to separate overlapping $H$ and $Z$ peaks has been widely
discussed~\cite{sephz,barwhis}:
\begin{enumerate}
\item Look for enhancement of the measured $H+Z$ peak over the predicted $ZZ$
background. The enhancement factor is  $\sim 1.3$~(untagged) or
$\sim2~(b$-tagged) in the example above.
\item Look for a difference from $ZZ$-background predictions in the lepton
angular distribution (relative to the beam axis, in the lab frame); see Fig.~8.
For $m_H = M_Z$ and $\sqrt s = 200\:$GeV  the minimum luminosity needed
to establish a $4\sigma$ signal in $\ell\ell jj$ channels with the various
techniques has been estimated to be~\cite{barwhis}
%
\[
\begin{tabular}{l@{\qquad}c@{\qquad}c}
                    &     $\sigma$ only   &     $\sigma +{}$ang.\ dist. \\
\hline
untagged $\ell\ell qq$ &  4.8 fb$^{-1}$    &         4.0 fb$^{-1}$ \\
$b$-tagged $\ell\ell bb$ &     0.9 fb$^{-1}$    &     0.7 fb$^{-1}$ \\
\hline
\end{tabular}
\]
\end{enumerate}

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/EWSB/costheta_l.eps}

\parbox{5.5in}{\small Fig.~8: Comparison of lepton angular distributions in the
$\ell^+\ell^-$ rest-frame from $ZH$ signal and $ZZ$ background: (a)~untagged
$\ell\ell qq$ channel,
(b)~tagged $\ell\ell bb$ case\cite{barwhis}.}
\end{center}

Exploiting other $Z$-decay channels, especially $\nu\bar\nu$ and $q \bar q$,
would reduce the required luminosity considerably.  Thus LEP\,II is
expected to cover the tricky region $m_H \sim M_Z$ and perhaps to reach a
little beyond, depending on energy,  through the range~\cite{janot}
%
\begin{equation}
m_H  <  \sqrt s -100 \rm \ GeV  \;.
\end{equation}

   Possible future $e^+e^-$ linear colliders could attain much higher
energies than LEP\,II.  At such energies the two main mechanisms for
$H$ production, shown in Fig.~9, are
\begin{enumerate}
\item $Z$-bremsstrahlung, with cross section falling like $1/s$;
\item $WW$ fusion (the similar $ZZ$ fusion is much weaker), with cross section
rising like $\ln(s/M_W^2)$~\cite{gtn}.
\end{enumerate}
The respective cross sections are shown versus energy in Fig.~10a,b; for
given $m_H$ the curves cross at $\sqrt s \simeq m_H + 0.3\:$TeV~\cite{munich}.
For
$\sqrt s \sim 0.5\:$TeV the $Z$-bremsstrahlung process dominates and a clear
Higgs signal should be detectable up to $m_H = 350\:$GeV~\cite{bckp}.
For $\sqrt s\sim 1$--2~TeV the $WW$-fusion process dominates, and a Higgs
signal should be detectable up to $m_H = 1\:$TeV; see Fig.~11~\cite{hkm}.

\begin{center}
%\epsfxsize=3.5in
\hspace{0in}
%\epsffile{fig2-9.eps}

{\small Fig.~9: $e^+e^-\to H$ production mechanisms at higher energy.}

\bigskip\bigskip

%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/EWSB/p12fig.eps}

{\small Fig.~10: $e^+e^-\to H$ production cross sections versus CM
energy\cite{munich}.}

\bigskip\bigskip

%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/EWSB/m_vv.eps}

{\small Fig.~11: $e^+e^-\to{}$Higgs signals and backgrounds at $\sqrt
s=1.5$~TeV~\cite{hkm}.}
\end{center}



\section{SM Higgs production at $pp$ supercolliders}

   The goals of Higgs physics at the SSC and LHC will be:
\begin{enumerate}
\item to search the range $M_Z \alt m_H \alt 1$~TeV, not covered by
LEP\,I and LEP\,II;
\item if $H$ is discovered here, to measure the principal couplings, {\it
i.e.}\ $HZZ$ and $HWW$ gauge couplings and $H\bar tt$ Yukawa coupling;
\item if $H$ is light, to check that the scattering of longitudinal weak
bosons  $V_L V_L \to V_L V_L$   is small as predicted ($V=W,Z$).
\end{enumerate}
To achieve (b) and (c) it will be necessary to separate different
contributing subprocesses, such as the $VV\to H\to ZZ$ and $gg \to H\to ZZ$
examples shown in Fig.~12.  For this we must exploit the different
physical characteristics; {\it e.g.}\ the $VV$ fusion subprocesses are
accompanied by two forward quark jets, typically with pseudorapidities
$|\eta|\sim 4$.  We can preferentially select the first or second subprocess
by jet-tagging (requiring one or two such jets) or anti-tagging (vetoing
events with such jets); we return to tagging in Section~2.7 below.

\begin{center}
%\epsfxsize=5.25in
\hspace{0in}
%\epsffile{fig2-12.eps}

\medskip

{\small Fig.~12: Subprocesses that can be separated by forward jet-tagging or
  anti-tagging.}
\end{center}


   The principal $H$ hadroproduction subprocesses of interest (Fig.~13) are
\begin{enumerate}
\item $gg \to H$ via $t$-quark loops, which dominates for $m_H\alt0.8\:$TeV;
\item $qq\to qqH$ via weak boson fusion, which has taggable forward jets
and dominate for $m_H\agt0.8\:$TeV;
\item $gg\to t \bar t H$  and  $q \bar q'\to W H$, which have smaller cross
sections but cleaner signatures, since $t\to bW \to b\ell\nu$  and
$W \to \ell\nu$  decays provide a hard isolated lepton that can be used
as a tag or even as a trigger;
\item $q \bar q\to ZH$ (analogous to $q \bar q' \to WH$)is smaller and cleaner
still, offering $Z\to\ell^+\ell^-$ as a tag or trigger.
\end{enumerate}

\noindent
The corresponding SSC/LHC cross sections are shown in Fig.~14; the lower
cross sections at LHC are off-set by higher planned luminosity. These
curves do not include the QCD enhancement factors $K \simeq 1.5$ for
$gg\to H$~\cite{djouadi} and $K \simeq 1.15$ for $q\bar q\to WH,ZH$~\cite{hw}.


\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig2-13.eps}

{\small Fig.~13: $H$ hadroproduction subprocesses.}
\end{center}

\bigskip

\begin{center}
%\epsfxsize=4.25in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/EWSB/sigma.eps}

{\small Fig.~14: Higgs hadroproduction expectations at SSC and LHC\cite{bsm2}.}
\end{center}


\section{Intermediate mass Higgs search at SSC/LHC}

   The intermediate mass range is usually defined to be  $90 \gev <
m_H < 2M_W, 2M_Z$,  too heavy to be accessible at LEP\,I or LEP\,II and too
light  for the  $H\to WW,ZZ$ modes to be important.  In this range the
major decay modes are  $H\to b \bar b, \tau\bar\tau$, but they are swamped
by backgrounds so we have to try minor channels.

   $H \to \gamma\gamma$ is a promising decay mode for detection.
The main production mode is $gg$ fusion; the cross section for the
full process $pp\to HX \to \gamma\gamma X$ (Fig.~15) has the form
%
\begin{equation}
\sigma= \Gamma(H\to gg) {\pi^2\over8m_H^3}\tau \int_\tau^1 {dx\over x}
g(x,m_H^2) g\left({\tau\over x},m_H^2\right) B(H\to\gamma\gamma) \;,
\end{equation}
%
where $\tau = m_H^2/s$ and $g(x,Q^2)$ is the gluon distribution in a proton
at scale $Q$.  Note that the $gg\to H$ vertex has been related for
convenience to the equivalent $H\to gg$ decay width.  The main genuine
background subprocesses are  $q \bar q, gg\to \gamma\gamma$ (Fig.~16), but
there are also fake backgrounds from jet-jet and jet-photon production
where one or more of the jets is just a $\pi^0$ that fakes a single photon in
the calorimetry.

\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{fig2-15.eps}

{\small Fig.~15: $gg$ fusion contribution to $pp\to HX\to \gamma\gamma X$.}

\bigskip

%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig2-16.eps}

{\small Fig.~16: Two-photon hadroproduction background subprocesses.}
\end{center}


   To extract an $H\to \gamma\gamma$  signal directly from these large
backgrounds requires excellent invariant mass resolution
$\delta m(\gamma\gamma) \alt 1.5$~GeV  and a rejection factor $\alt 10^{-4}$
against a jet faking a photon.  The GEM collaboration at SSC
aims to have this capability.  Figure~17 shows how GEM signals would
look in the $m(\gamma\gamma)$ distribution,  before and after background
subtraction, for two alternative calorimeters based on Ba\,F$_2$ or liquid
argon; the spikes represent Higgs signals at $m_H = 80$, 100, 120 or 150~GeV.

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig2-17.eps}

{\small Fig.~17: $H\to \gamma\gamma$ signals and backgrounds for
GEM~\cite{gem}.}
\end{center}

\bigskip

   Alternatively, one can use the  $gg\to t\bar tH$  and $q\bar q\to WH$
channels, tagged by requiring an extra lepton from $W$-decay. The signal
is smaller but cleaner, since the background (from $q \bar q\to
W\gamma\gamma$ etc.) is very much smaller than before, so less stringent
measurement criteria are needed. The SDC collaboration at SSC adopts
this approach; they expect to have mass resolution  $\delta m(\gamma\gamma)
\alt 3$~GeV  and fake photon rejection factor ${}\alt5\times10^{-4}$.
Figure~18 shows how SDC signals for $m_H = 80$, 100, 120, 140 or 160~GeV
would compare with background in the $m(\gamma\gamma)$ distribution.
The SDC signal event rates (both raw and detected) for 1~year's running,
and the statistical significance $S/\sqrt B$ of the combined signals $S$
compared to background $B$, are as follows~\cite{sdc}:
\[ \tabcolsep=1em
\begin{tabular}{|c|cccc|c|}
\hline
             & \multicolumn{4}{c|}{Events per 10$\fb^{-1}$ luminosity} &
significance \\
  $m_H$ (GeV)& $ttH$ (raw)& $ttH$ (det)& $WH$ (raw)& $WH$ (det)& $S/\sqrt B$ \\
\hline
\phantom080&    50&    12&  25&   4&  6\\
        120&    46&    14&  18&   4&  6\\
        140&    28&  \ph09&  10&   2&  4 \\
\hline
\end{tabular}
\]
It is salutary to note what a small fraction of raw events survive the
acceptance cuts and instrumental inefficiencies.  We see from these
numbers and Figs.~4, 17, and 18 that the $H\to\gamma\gamma$ mode becomes very
difficult to detect outside the window $80\rm~GeV \ltap m_H \ltap 150$~GeV.


   In principle the $t\bar tH$  and  $WH$  channels can be separated (e.g.\ by
considering the number of accompanying jets) to determine the ratio of
$WWH$ to $t\bar tH$ couplings. This is estimated to need 5~years SSC running.

\begin{center}
%\epsfxsize=3.25in
\hspace{0in}
%\epsffile{fig2-18.eps}

\medskip

{\small Fig.~18: Lepton-tagged $m(\gamma\gamma)$ distributions for
SDC~\cite{sdc}.}
\end{center}


   $H\to ZZ^*\to 4l$  is another intermediate-mass channel, where one $Z$ is
off-shell.  It is a very clean signal with little background and the
branching fraction is big enough to be interesting for $m_H > 130$~GeV
(see Fig.~4). Figure~19 shows the Higgs production cross section times
branching fraction versus $m_H$ for SSC and LHC; the dip is due to
competition from the $H\to WW$ channel opening.  After requiring  $p_T >
20$~GeV for the two leptons from the on-shell $Z$ (for triggering), $p_T >
10$~GeV for the other leptons (for detection), plus further criteria for
lepton identification and isolation, the overall experimental efficiency
is typically ${}\sim0.16$ for $m_H = 140$~GeV.  Figure 20~shows how Higgs
signals would look in the SDC 4-lepton invariant mass plot, for
$m_H = 120$, 130,\dots 160, 170~GeV; the background curves are cumulative,
from lowest to highest representing $ZZ^*$, Z$b\bar b$, $Zt\bar t$ and $t
\bar t$. The signal seems detectable for $m_H \geq 130$~GeV. This approach
overlaps the $\gamma\gamma$ search and covers the rest of the intermediate mass
range.
The mass $m_H$ can be measured directly from the
invariant mass distribution $m(\gamma\gamma)$ or $m(\ell\ell\ell\ell)$,
but the width $\Gamma_H$ is much smaller than the expected
resolution in the intermediate mass range (see Fig.~3)
and cannot be measured directly from the resonance line-shape.

%\vspace*{-.2in}

\begin{center}
%\epsfxsize=3.25in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/EWSB/inclusive_H.eps}

\medskip
\parbox{5.5in}{\small Fig.~19: Cross section times branching fraction for
intermediate-mass $H\to 4\ell$ signals at SSC and LHC.}


%\epsfxsize=3.25in
\hspace{0in}
%\epsffile{fig2-20.eps}\hspace*{.5in}

{\small Fig.~20: Intermediate-mass Higgs signal in the SDC
$m(\ell\ell\ell\ell)$ distribution~\cite{sdc}.}
\end{center}




\section{Heavy $H\to ZZ$ searches at SSC/LHC}

   For  $m_H > 2M_Z$  we can exploit the  $H \to ZZ$  decays, which provide
very clean and distinctive signals with manageable backgrounds.

\begin{enumerate}
\item $H \to ZZ \to \ell^+\ell^-\ell^+\ell^-$ is the ``gold-plated" signal;
it has two pairs
of isolated charged leptons, each satisfying the invariant mass constraint
$m(\ell^+\ell^-) \simeq M_Z$. Unfortunately the branching fraction is rather
small, $B(H \to 4\ell) \simeq 0.14\%$ (summing over all $\ell=e,\mu$
possibilities). Here
$m_H$ can be measured directly from the four-lepton
invariant mass distribution; $\Gamma_H$ can also
be measured, if it is bigger than or comparable
to the mass resolution.

\item $H \to ZZ \to \ell^+\ell^-\nubar\nu$ is a ``silver-plated" signal;
it has only
one isolated lepton pair with invariant mass $M_Z$, but also typically has
large missing transverse momentum $\overlay/ p_T$ from the neutrinos. It has
about 6 times larger branching fraction, $B(H \to \ell\ell\nu\nu)
\simeq 0.8\%$. Using this mode
$m_H$ and $\Gamma_H$ can be estimated indirectly from
various dynamical distributions with this signal.
\end{enumerate}

   We first address the gold-plated four-lepton signals;  the
corresponding raw event rates at SSC and LHC are illustrated in Fig.~21.
The main backgrounds come from direct $q \qbar \to ZZ$ electroweak
production and from  $gg \to ZZ$  through quark loops; see Fig.~22.
Fortunately the $H \to ZZ$ signal gives typically hard transverse momentum
$p_T(Z)\sim  m_H/2$  whereas the background distributions are softer and
can be suppressed by requiring that at least one of the $Z$s has
$p_T > m_H/4$ say~\cite{bhp1}; the resulting Higgs signals and $ZZ$
backgrounds in the $m(ZZ)$ invariant mass distribution are illustrated
in Fig.~23.  Figure~24 shows a similar plot, this time on a logarithmic
scale and with the universal cut $p_T(Z) >  m(ZZ)/4$~\cite{bg}.  Figure~25
shows further results, after folding in SDC detector resolutions and
efficiencies~\cite{sdc}.  The conclusion is that $H \to \ell\ell\ell\ell$
detection is
easy for  $m_H < 0.6$~TeV  but gets hard for  $m_H > 0.8$~GeV, partly
because of event rate but mostly because the Higgs width gets so broad
[see Eq.(2.12)]; this width comes mainly from the $WW$ and $ZZ$ modes plus
a $t\tbar$ contribution that is always smaller.

\begin{center}
%\epsfxsize=3.5in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/EWSB/sigmaB.eps}

{\small Fig.~21: Four-lepton Higgs signal rates versus $m_H$, at SSC and
LHC\cite{bsm2}.}

\bigskip

%\epsfxsize=2.5in
\hspace{0in}
%\epsffile{fig2-22.eps}

{\small Fig.~22: $ZZ$ background hadroproduction subprocesses.}

\bigskip

%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig2-23.eps}

{\small Fig.~23: $H\to ZZ$ signal and background in $m(ZZ)$
distribution~\cite{bhp1}.}

\bigskip

%\epsfxsize=3.5in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/EWSB/m_zz.eps}

{\small Fig.~24: $H\to ZZ\to4\ell$ signals and backgrounds versus $m(ZZ)$ with
the cuts shown~\cite{bg}.}

\bigskip

%\epsfxsize=6in
\hspace{0in}
%\epsffile{fig2-25.eps}

\medskip

\parbox{5.5in}{\small Fig.~25: SDC simulations of $H\to \ell\ell\ell\ell$
signals and backgrounds versus $m(\ell\ell\ell\ell)$, requiring
$|m(\ell\ell)-M_Z| < 10$~GeV for both pairs: (a)~$m_H = 200$~GeV, (b)~$m_H =
800$~GeV, (c)~$m_H = 400$~GeV, (d)~$m_H = 800$~GeV again, but now requiring
$p_T(Z) > 200$~GeV for both $Z$s.  The histograms give total rates, the solid
curves give $ZZ$ backgrounds~\cite{sdc}.}

\end{center}

   The silver-plated $H \to ZZ \to \ell^+\ell^-\nubar\nu$ signal reinforces the
$4\ell$ signal; because of its bigger rate it reaches slightly higher $m_H$
values.  The major background is $pp \to Z+{}$jets , with some of the jets
(partially) undetected or mismeasured to give fake $\overlay/ E_T$.  To
reduce fake $\overlay/ E_T$ requires good calorimeter coverage out to
pseudorapidity  $|\eta| < 4.5$  ({\it i.e.}\ to within 1.3 degrees of the beams
at the SSC). There are also backgrounds with dileptons plus genuine $\overlay/
E_T$
from $t\tbar$ and continuum $ZZ$ production.  Figure~26 shows an SDC simulation
of the $\overlay/ E_T$ distribution for Higgs signal and backgrounds; we see
that this signal is promising.


\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig2-26.eps}

{\small Fig.~26: SDC simulations of $H\to ZZ \to \ell\ell\nu\nubar$  signals
and backgrounds versus $\overlay/ E_T$.}
\end{center}

   The mass  $m_H$  can be estimated from the $\overlay/ E_T$ distribution,
but a cleaner
estimate can in principle be made from a ``transverse mass"
variable~\cite{bhp2} that
combines the transverse momenta of the observed $Z \to \ell^+\ell^-$  and the
missing $Z\to\nubar\nu$ (approximated by $\overlay/p_T\simeq \overlay/ E_T)$:
%
\begin{equation}
m_T^2 = \left[\sqrt{p_T(\ell\ell)^2 + M_Z^2} + \sqrt{\overlay/p_T^2 + M_Z^2}
\right]^2 - \left[\vec p_T(\ell\ell) + \vec{\overlay/p}_T\right]^2  \;,
\end{equation}
%
where arrows denote vectors in the plane transverse to the beam axis.  Both
$p_T(\ell\ell)$ and $\overlay/p_T$ are sensitive to $m_H$; since they
are the $p_T$ of the two $Z$s, their distributions have Jacobian peaks at their
maximum value $p_T^{\rm max} = {1\over2}\sqrt{m_H^2 - 4M_Z^2}$ in the
$H$-restframe, but these distributions are smeared out in the lab frame by the
transverse motion of $H$.
The transverse mass variable of Eq.~(2.32) combines $p_T(\ell\ell)$ and
$\overlay/p_T$ to minimize the smearing and maximize the sensitivity to $m_H$
(in analogy with the variable used to extract $M_W$ from $W \to \ell \nu$
events~\cite{mt}).  However, since it crucially exploits the identification
of $\overlay/p_T$ with $p_T(\nu\nubar)$, it requires an excellent detector with
minimal fake $\overlay/p_T$.

\section{Jet tagging at SSC/LHC}

   The $WW,ZZ$ fusion subprocesses (Fig.~13), which eventually dominate for
very heavy H production (Fig.~14), are distinguished by two accompanying
quarks that typically produce two jets close to the two beams.  Tagging
these jets~\cite{kleiss,bchoz,bchz,mhs,gdv} can help heavy $H$ searches in
several ways.
\begin{enumerate}
\item It improves signal/background ratio for $m_H > 0.6$~TeV where the
resonance is broad and hard to distinguish.

\item It makes $H \to ZZ$ signals cleaner and rescues $H \to WW$ signals
that would otherwise be lost (Section 2.8).

\item By separating the $qq \to qqH$ and $gg \to H$ signals, it allows one to
determine the $HWW,\ HZZ$ and $Htt$ couplings separately, and to study
$VV \to H \to VV$  scattering directly $(V = W,Z)$.
\end{enumerate}

   Figures 27 and 28 show the diagrammatic structure of a typical
$WW \to H \to WW$ event and the resulting event characteristics in the
$H$-restframe, when both bosons decay leptonically.  Other $VV \to H \to VV$
channels are similar.  Note that the vector bosons are predominantly
polarized longitudinally.  Because of the different longitudinal and
transverse vector boson polarization vectors
%
\begin{equation}
\begin{array}{rcl}
   \epsilon^\mu(V_L) &=& (p/M_V, 0, 0, E/M_V)  \;,\\
   \epsilon^\mu(V_T) &=& {1\over\sqrt{2}}\, ( 0, 1, i, 0) \;,
\end{array}
\end{equation}
%
and the $\epsilon^\mu(V) \epsilon_\mu(V)$  structure of the $HVV$ matrix
element,  $H \to V_L V_L$  transitions are strongly enhanced as follows
%
\begin{equation}
\begin{array}{rcl}
\Gamma(H \to V_L V_L) &\sim& \dis{g^2\over 64 \pi}  {m_H^3  \over M_W^2} \;, \\
\nv2
\Gamma(H \to V_T V_T) &\sim& \dis{g^2 \over 8 \pi} {M_W^2\over m_H} \;.
\end{array}
\end{equation}

\begin{center}
%\epsfxsize=2.5in
\hspace{0in}
%\epsffile{fig2-27.eps}

{\small Fig.~27: Diagrammatic structure of a typical $WW \to H \to WW$ event.}

\bigskip

%\epsfxsize=2.5in
\hspace{0in}
%\epsffile{fig2-28.eps}

{\small Fig.~28: Characteristics of a $WW \to H \to WW$ event in the $H$
restframe.}
\end{center}

   The main $WW$ background processes with jets are
\begin{enumerate}
\item $qq$ electroweak (Fig.~29), producing mostly $W_T$\,;
\item Low-order QCD plus electroweak (Fig.~30);
\item $t\tbar$ production (Fig.~31), where the $b$-jets from $t \to bW$ are
usually central but additional gluon radiation can easily give
forward jets.
\end{enumerate}
There are similar electroweak and QCD backgrounds in $ZZ$ channels,
but no $t\tbar$ background of course.
Note however that the electroweak background cannot strictly be
      separated from the $VV\to H$ fusion and Higgs exchange subprocesses,
      since the latter are an essential ingredient in the renormalizability.
      We therefore calculate both together; the $VV\to H$ curves in subsequent
      figures include all electroweak signal plus background effects of the
      same order.  The non-resonant ``background" situation can be empirically
      represented by the $m_H = 0.1$~TeV case, for purposes of comparison.


\begin{center}
%\epsfxsize=6.2in
\hspace{0in}
%\epsffile{fig2-29.eps}

{\small Fig.~29: Typical electroweak $WW+{}$jets background processes.}

\bigskip\bigskip

%\epsfxsize=6.2in
\hspace{0in}
%\epsffile{fig2-30.eps}

\smallskip

{\small Fig.~30: Typical QCD/electroweak $WW+{}$jets backgrounds.}

\bigskip

%\epsfxsize=1.75in
\hspace{0in}
%\epsffile{fig2-31.eps}

{\small Fig.~31: Typical $t\tbar$ contribution to $WW+{}$jets background.}
\end{center}


   Although the signal and backgrounds can all give forward jets, there
are characteristic differences in the couplings of the fast forward
parton (Fig.~32) that lead to different $p_T$ dependences at large $p_T$.
The $W_L$-fusion(Higgs), electroweak $W_T$-emission and QCD gluon emission
processes respectively behave like
%
\begin{equation}
\begin{array}{rcl}
d\sigma^{\rm Higgs} &\sim&
\dis {dp_T^2\over \left(p_T^2+M_W^2\right)^2}
\sim   dp_T^2/ p_T^4   \; ,\\           \nv2
d\sigma^{\rm ew} &\sim&  \dis {p_T^2 dp_T^2\over \left(p_T^2 + M_W^2\right)^2}
\sim dp_T^2/ p_T^2    \;,     \\ \nv2
d\sigma^{\rm QCD}  &\sim&   { dp_T^2/ p_T^2}  \;,
\end{array}
\end{equation}
%
for $p_T(\rm jet)\gg M_W$.  Thus the forward jets from $WW \to H$ have
typically smaller $p_T$ than those from backgrounds.
Similarly, there are differences in the longitudinal momentum
fraction $ x = p_L({\rm final\ jet})/p_L(\rm initial\ parton)$.
The probability distributions (``splitting functions") behave as
%
\begin{equation} \def\arraystretch{1.5}
\begin{array}{rcl}
   P(x{:}\ q \to qW_L)   &\sim& x/(1-x) \:, \\
   P(x{:}\ q \to qW_T, qg) &\sim& (1 + x^2)/(1-x) \;,\\
   P(x{:}\ q \to gq  )    &\sim& \Bigl[1 + (1- x)^2\Bigr]/x \;,\\
   P(x{:}\ g \to gg  )     &\sim& \Bigl[x/(1-x) + (1-x)/x + x(1-x)\Bigr] \;,
\end{array}
\end{equation}
%
from which we infer that the signal has typically larger $p_L(\rm jet)$ than
the backgrounds.  Hence it also has typically smaller angle relative to the
beam $(\theta = \arctan p_T/p_L)$ and larger pseudorapidity ($\eta =
{1\over2} \ln\left[\left(p + p_L\right)\big/\left(p - p_L\right)\right]=\ln
\tan (\theta/2)$\,) than the backgrounds.

\begin{center}
%\epsfxsize=5.75in
\hspace{0in}
%\epsffile{fig2-32.eps}

\medskip

\parbox{5.5in}{\small
Fig.~32: Different configurations in which a fast forward jet may
  be emitted: with $W_L$ (Higgs signal), with $W_T$ (electroweak
  background), with QCD radiation (QCD and top backgrounds).}
\end{center}

   Single-jet tagging is illustrated in Fig.~33, which shows the
signal and background  $\eta$(jet) distributions for $pp \to ZZ + 1$\,jet,
with jet energy cut $E(\rm jet) > 100$~GeV or $E(\rm jet) > 1000$~GeV.
Electroweak
background is represented by the $m_H = 0.1$~TeV case (no $H\to ZZ$ resonance).
We see that requiring one jet with $E > 1$~TeV and  $2 < |\eta| < 5$ suppresses
the background strongly while preserving about 60\% of the $H \to ZZ$ signal
here~\cite{bchoz}.

   Double-jet tagging suppresses the background further but is not really
needed and is too costly to the signal; see Fig.~34.

\begin{center}
%\epsfxsize=5.5in
\hspace{0in}
%\epsffile{fig2-33.eps}

\parbox{5.5in}{\small Fig.~33: Jet $\eta$-dependence of Higgs signal and
background in $pp\to ZZ + 1$\,jet, with $ZZ\to 4\ell$ and specified cuts, at
the SSC~\cite{bchoz}.  On the left $E(jet) > 100$~GeV, on the right $E(\rm jet)
  > 1$~TeV.}
\end{center}


\begin{center}
%\epsfxsize=3in
\hspace{0in}
%\epsffile{fig2-34.eps}

\parbox{5.5in}{\small Fig.~34: Energy distributions of primary (``tagged") and
secondary jets, for Higgs signal and background in  $pp\to ZZ + 2$\,jets (in
opposite hemispheres  with   $2 < |\eta(\rm jet)| < 5$) at the
SSC~\cite{bchoz}.}
\end{center}

   Returning to single-tagging, Fig.~35 shows its effect on the shape of the
$pp \to ZZ \to 4\ell$  net cross section versus $m(ZZ) = m(4\ell)$ at the SSC.
Without tagging (the ``jet inclusive" case), the Higgs signal for $m_H \sim
1$~TeV is just a soft shoulder and its identification is problematical.  With
tagging the $m_H \sim 1$~TeV signal is a more clearly pronounced plateau,
easier to identify.  For lighter masses too, tagging gives sharper
mass peaks.  The much lower net cross section in the tagged case mostly
reflects the much lower background here.  The expected numbers $S$ and $B$ of
signal and background events, for a nominal luminosity ${\cal L} =
10$~fb$^{-1}$ (one SSC year),  and the resulting signal significance
$S\Big/\sqrt B=\mbox{(number of statistical standard deviations)}$ are as
follows~\cite{bchoz}.
%
\begin{center}\vbox{\tabcolsep=1.25em
\begin{tabular}{l@{}r@{}cccc}
\hline\hline
 & &      \multicolumn{3}{c}{Higgs signal}&        QCD + electroweak \\ \nv-1
                                 &&&&&                backgrounds \\   \nv-2
  &  $m_H$ (TeV) ={}   &       0.6&    0.8&    1.0& \\    \hline
jet-inclusive events   &&    84&     43&     26&            49\\
significance $S/\sqrt B$&  &   12.0&   6.1&    3.7& \\
significance($\epsilon=0.3$) &&  5.2&    2.6&    1.6& \\
\hline
single-jet-tag events   &&     13&     11&     9&           4.2\\
significance $S/\sqrt B$  &&  6.3&     5.4&    4.4& \\
significance ($\epsilon=0.3$) && 5.4&     4.6&    3.7& \\
\hline\hline
\end{tabular}
}\end{center}
%
Here the Higgs signal is defined by subtracting the cross section for
$m_H = 0.1$~TeV, and a cut $m(ZZ) > 0.5$~TeV has been made. The significance
factor $S/\sqrt B$ is purely statistical and assumes no uncertainty on the
background normalization.  Suppose this systematic background uncertainty
is gaussian and one standard deviation is $\epsilon B$,  to be added to the
statistical standard deviation  $\pm\sqrt B$.  The corrected significance is
then  $S/\sqrt{B + \epsilon^2 B^2}$; the table shows an example of this for
$\epsilon = 0.3$, i.e.\ 30\% background normalization uncertainty.  The
advantages of reducing $B$ by tagging are clearly seen.

\begin{center}
%\epsfxsize=6in
\hspace{0in}
%\epsffile{fig2-35.eps}

\medskip

\parbox{5.5in}{\small Fig.~35: Effects of jet-tagging on the net $pp\to ZZ \to
4\ell$  signal plus background distribution versus $m(ZZ)$ at the
SSC~\cite{bchoz}.}
\end{center}

   Similarly, jet-tagging gives a cleaner signal in the
$ZZ \to \ell^+ \ell^- \nubar \nu$  channel too.

\section{Heavy  $H \to WW$  searches at SSC/LHC}

   Higgs signals have been studied in the channels  $WW \to \ell\nu
jj$~\cite{kleiss,mhs} and  $WW \to \ell\nu \ell'\nu'$~\cite{bchz}.  In both
cases there are severe backgrounds, that can only be subdued by
careful cuts on associated jets.

   We consider here the  $H \to WW \to \ell\nu \ell'\nu'$  signal at the SSC.
The main backgrounds are from continuum $WW$ production (Figs.~29, 30)
and from  $t\tbar$  production (Fig.~31).  We first demand two clean
high-$p_T$ central leptons:
%
\begin{equation}
    p_T(\ell) > 100\rm\ GeV\;, \qquad   |\eta(\ell)| < 2 \; .
\end{equation}
%
Three complementary strategies can then be used to separate the signal
from backgrounds~\cite{bchz}.

\begin{enumerate}

\item Forward jet tag: require at least one jet with
%
\begin{equation}
3 < |\eta(j)| < 5 \,, \quad E(j) > 1\ {\rm TeV} \,, \quad
p_T(j) > 40 \ \rm GeV \,.
\end{equation}
%
This reduces the signal by $\sim1.4$,  but eliminates the  QCD
background,  and reduces the $t\tbar j$ background by $\sim10$ to about the
same level as the signal.  Figure~36 gives the two-dimensional
distribution $d^2 \sigma / dE(j) d|\eta(j)|$  of the fast jet in
signal and background channels, before the  $\eta(j)$ cut.

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig2-36.ai.eps}

\parbox{5.5in}{\small Fig.~36: Two-dimensional distribution versus energy and
pseudorapidity
  of the tagging jet, for $pp \to WW \to \ell\nu\ell'\nu'$ with the cuts of
Eqs.~(2.37): (a)~$m_H=1$~TeV signal, (b)~SM electroweak background
  ($m_H = 0.1$~TeV), (c)~QCD background, (d)~$t\tbar j$
background~\cite{bchz}.}
\end{center}

\item Central jet veto: reject events with a central jet satisfying
%
\begin{equation}
    |\eta(j)| < 3 \,, \qquad p_T(j) > 30\ \rm GeV   \,. \label{eta veto}
\end{equation}
%
This cut is aimed to remove $t\tbar$ events, which typically have central
$b$-jets from their  $t\to bW$  decays.   The net effect of $\rm(a)+(b)$ is to
retain about
40\% of the Higgs signal (for $m_H = 1$~TeV) while suppressing the $t\tbar j$
background by $\sim1000$.

\item Lepton correlation: since the two charged leptons from the $H\to WW$
signal are typically harder and more back-to-back than the background, it is
helpful to cut also on the difference of transverse momenta~\cite{gdv},
%
\begin{equation}
\Delta p_{T\ell\ell} = | \vec p_T(\ell_1) - \vec p_T(\ell_2) |
                         > 400 \rm\ GeV   \,.
\end{equation}

\end{enumerate}

\noindent
 Figure~37 shows the net effect of all these cuts; for
    forward-jet-tagged signal and background events, it gives the
    distributions versus the pseudorapidity of the second jet (veto
    candidate).  We see that vetoing events with  $|\eta_j(\rm veto)| < 3$
    as in Eq.~(\ref{eta veto}) gives efficient separation.


\begin{center}
%\epsfxsize=3.25in
\hspace{0in}
%\epsffile{fig2-37.eps}

\parbox{5.5in}{\small Fig.~37:  Pseudorapidity distributions of the second jet
(veto candidate)
in forward-jet-tagged  $\ell \nu \ell' \nu'$ events.  Higgs signal
$(m_H = 1$~TeV), electroweak background ($m_H = 0.1$~TeV) and $\bar ttj$
background are compared~\cite{bchz}.}
\end{center}


   Having separated the $H \to WW \to \ell \nu \ell' \nu'$  signal by these
strategies, the mass $m_H$ can be estimated from observables such
as the dilepton invariant mass $m(\ell\ell)$ or a ``cluster transverse mass"
$m_T(\ell\ell,\overlay/p_T)$  defined by~\cite{bmp}
%
\begin{equation}
m_T(\ell\ell,\overlay/p_T)^2 = \left[\sqrt{m(\ell\ell)^2 + p_T(\ell\ell)^2} +
\overlay/p_T\right]^2 - \Bigl[\vec p_T(\ell\ell) + \vec{\overlay/p}_T\bigr]^2
\;.
\end{equation}
%
These mass variables peak at $m(\ell\ell) \simeq {1\over2}m_H$  and
$m_T(\ell\ell,\overlay/p_T) \simeq {3\over4}m_H$
respectively,  and can be used to
extract $m_H$ from data~\cite{bchz}.  The integrated cross sections in fb
after cuts are as follows (summing over all $\ell,\ell' = e,\mu$ cases and
assuming 100\% lepton identification efficiency)~\cite{bchz}.
%
% This table could use a little sprucing up, if time permits
\[ \arraycolsep=1em
\def\vstrut{\vrule width0pt height20pt depth12pt}
\begin{array}{|c|ccc|c|c|c|}
\hline
 & \multicolumn{3}{c|}{WW \rm signal}& \rm QCD & t\bar tj &
\vstrut S/\sqrt B \\
 & \multicolumn{3}{c|}{m_H\ (\rm TeV)=}& & m_t=140\,\rm GeV &
10\fb^{-1} \\
 & 1 & 0.6 & 0.1  & & & \\
\hline
 \rm SSC & 5.8 & 2.5 & 0.54 & 0.79 & 4.3  & 7.1\\
(E_j>1\,\rm TeV) & & & & & & \\
&&&&&&\\
 \rm LHC & 0.46 & 0.25 & 0.044 & 0.11 & 0.34  & 5.9\\
(E_j>0.8\,\rm TeV) &&&&&& \\ \hline
\end{array}
\]
%
The final column gives the statistical significance  $S/\sqrt B$ for
luminosity 10~fb$^{-1}$ (SSC) or 100~fb$^{-1}$ (LHC), for $m_H=1$~TeV. This
significance can be improved by further
lepton cuts~\cite{bchz,gdv}.
In any case, the bottom line is that
the  $H \to WW$  signal can be detected.


\section{Summary}


   With integrated luminosity ${\cal L}\sim 20$--30~fb$^{-1}$ (a few years
running) the SSC can detect and study the SM Higgs boson through the range
80~GeV${}\alt m_H \alt 1$~TeV in clean leptonic or photonic final
states, with backgrounds suppressed by suitable cuts.
\[
\def\nv#1{\noalign{\vskip#1ex}}
\matrix{
H\to\gamma\gamma \hfill && \hfill 80\lsim m_H\lsim 150\gev\cr \nv2
H\to ZZ^*\to4\ell \hfill && 130\lsim m_H\lsim 180\gev\cr \nv2
H\to ZZ\to 4\ell \hfill && 180\lsim m_H\lsim 800\gev\cr \nv2
H\to ZZ\to\ell\ell\nu\bar\nu && 600\lsim m_H\lsim1\tev\hfill\cr}
\]

   With single jet tagging and a central jet veto, the scattering process
$W_LW_L \to H \to W_LW_L$  can be studied through the range
$0.6 < m_H < 1$~TeV.  Single jet tagging also greatly reduces the background
and increases the significance of $H \to ZZ \to \ell\ell\ell\ell,\,
\ell\ell\nu\nubar$ signals


\renewcommand{\chapter}{\section}

\begin{thebibliography}{00}

\vspace{-.1in}

\frenchspacing
%\addtolength{\itemsep}{-.13in}

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Rev.\ {\bf D46}, 3725 (1992); V.~Barger et al., University of Wisconsin-Madison
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K.~Milton et al., World Scientific (1990).

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\end{thebibliography}

\end{document}

tion: since the two charged leptons from the $H\to WW$
signal are typically harder and more back-to-back than the background, it is
helpful to cut also on the difference of transverse momenta~\cite{gdv},
%
\begin{equation}
\Delta p_{T\ell\ell} = | \vec p_T(\ell_1) - \vec p_T(\ell_2) |
                         > 400 \rm\ GeV   \,.
\end{equation}

\end{enumerate}

lecture3.
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\begin{document}
\setcounter{page}{30}

\chapter*{LECTURE 3:\\ STRONG $WW$ SCATTERING}
\setcounter{chapter}{3}

%\thispagestyle{empty}

\section{Introduction}

   The scattering of longitudinally polarized weak boson states
($W_L^\pm$ and $Z_L^0$) provides alternative and rather direct probes of the
EWSB mechanism,  complementary to Higgs boson searches.  This is because
$W_L^\pm$ and $Z_L^0$ are essentially the Goldstone bosons $w^\pm$ and $z^0$
of the broken symmetry: see Section~2.1.   Transversely polarized
states interact mainly via gauge couplings, but the longitudinal states
and Higgs bosons experience additional EWSB couplings, which in the SM
are contained in the scalar potential
%
\begin{equation}
V(\Phi) = \lambda \left(w^+ w^- + {1\over2}zz +vH + {1\over2}H^2 \right)^2\,.
\end{equation}
%
Here the SM coupling strength
%
\begin{equation}
  \lambda = m_H^2/(2v^2) = (m_H/346 \:\rm GeV)^2
\end{equation}
%
is not yet known and could be large --- but this is also a more general
question, not restricted to the SM.  Dynamical symmetry-breaking schemes
such as Technicolor~\cite{tech} and top-condensation~\cite{tcon} have strong
interactions too. EWSB has an
analogy with chiral symmetry breaking in QCD,  with $w^\pm$ and $z$
analogous to $\pi^\pm$ and $\pi^0$,  that does not depend on the specific
Higgs mechanism~\cite{csb}.   Studies of $W_L^\pm$  and  $Z_L^0$  scattering
are therefore very important.  If no $H$ particle is discovered below about
1~TeV, some other strong forces must be at work here, and longitudinal
boson scattering will provide the principal keys to EWSB.


   Figure 1 depicts a generic scattering process at SSC or LHC, in which
longitudinal weak bosons are emitted from quarks in the incident protons
and scatter from each other.  Although we usually refer to  $W_L W_L$
scattering ({\it e.g.}\ in this figure and the title of this lecture), our
general remarks above apply to all the possible channels.  On the other
hand, particular channels have particular merits:
\begin{enumerate}
\item $Z_L Z_L$  and  $W_L^+ W_L^-$  final states probe both spin-0 Higgs-boson
resonances and other possible resonances analogous to those in $\pi\pi$
scattering (QCD-like dynamics).
\item $W_L^+ W_L^+$  and  $W_L^- W_L^-$  final states probe strong interactions
in channels that are normally non-resonant (unless there is a
doubly-charged Higgs boson\cite{vega}).
\item $Z_LW_L$ final states probe QCD-like dynamics ({\it e.g.}\ $\rho$ or
techni-$\rho$-type  resonance) or charged Higgs bosons (in exotic Higgs models,
there being no tree-level $HWZ$ coupling in models with doublets and singlets
only~\cite{hhg}).
\end{enumerate}
A useful recent survey of strongly-interacting $W$ and $Z$ scattering is given
in Ref.~\cite{bbcgh}.

\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{fig3-1.eps}

{\small Fig.~1: Generic longitudinal boson scattering at a $pp$ collider.}
\end{center}


\section{Steps toward a calculation}

   We want a procedure that will first single out the incoming $W_L$ and
$Z_L$ components in a process like Fig.~1, and will relate their subsequent
scattering to that of Goldstone bosons in any given model.  We also need
to know about general constraints such as the low-energy theorem,
crossing,  isospin relations and unitarity.

   The first step is the Effective $W$ Approximation (EWA), analogous to
the Weizs\"acker-Williams approximation for photons.  For very high energy
processes, with subprocess CM energy-square $\hat s\gg M_W^2$, we can
replace the incident quark beams in Fig.~1 by beams of approximately
collinear and on-shell $W$ or $Z$ bosons.  In a given quark $q$, the
probability of finding a
vector boson $V$ with polarization $\alpha$ and longitudinal momentum
fraction $x$ is written $P_{V/q}^\alpha(x)$ and a typical subprocess cross
section has the form
%
\begin{equation}
\hat\sigma(q_1q_2 \to q_3q_4V_3V_4) = \sum_{\alpha,\beta} \int
          dx_1 dx_2 P_{V_1/q_1}^\alpha(x_1) P_{V_2/q_2}^\beta(x_2)
          \hat\sigma(V_1^\alpha V_2^\beta \to  V_3 V_4)   \,.
\end{equation}
%
For $W_L$ and $Z_L$, the probabilities are
%
\begin{eqnarray}
P_{W+/u}^L &=& P_{W-/d}^L =P_{W+/\dbar}^L = P_{W-/\ubar}^L
                   = \frac{g^2}{16\pi^2} \frac{(1-x)}{x}  \,, \\ \nv2
P_{Z/q}^L &=& P_{Z/\qbar}^L =
 \frac{g^2}{32\pi^2} \frac{1-4x_W|Q|+8x_W^2Q^2}{1-x_W} \frac{1-x}{x}\,.
\end{eqnarray}
%
where $x_W = \sin^2\theta_W = 0.23$, $Q$ is the electric charge of $q$, and
$g^2 = 8M_W^2 G_F/\sqrt2$  is the SU(2) gauge coupling squared.  The full
$pp\to VVX$ process is calculated by folding in the quark distributions
within the incident protons.

   The EWA is the only known technique for separating the contributions
of $W_L$ and $Z_L$ scattering in a high energy process.  The relevant
renormalization scale
$\mu$ for evaluating the contributing quark distributions is  $\mu = M_W$.
The approximation is collinear and neglects transverse momentum $p_T$
for both $V$ and the final quark; in reality, however, the quark appears at
non-zero angle as a taggable jet (see Section 2.7).

   A second step is the Goldstone Boson Equivalence Theorem (GBET)~\cite{gbet}.
This relates any scattering amplitude for real external $W_L$ and $Z_L$
to the scattering amplitude for the corresponding real external
Goldstone bosons $w,z$:
%
\begin{equation}
\M\Bigl[W_L(p_1),W_L(p_2),\ldots\Bigr] = \M\Bigl[w(p_1),w(p_2),\ldots\Bigr]_R +
\O(M_W/E_W) \,,
\end{equation}
%
where $R$ indicates a renormalizable $R_\xi$ gauge.  The EWA provides
essentially real incoming $W_L$ and $Z_L$; the GBET allows us to compute
their scattering into final $W_L$ and $Z_L$ states, from any given model of the
Goldstone boson sector, up to corrections of order $M_W/E_W$.
Since the same 4-vectors $p_i$ are attributed
both to massive weak bosons and to massless
Goldstone bosons, masses are plainly neglected here.

   In the low-energy limit, Goldstone boson scattering is expected to be
controlled by the dynamics of EWSB.  Since the SM symmetry-breaking
lagrangian is the non-linear $\sigma$ model with chiral $\rm SU(2)_L\times
SU(2)_R$ symmetry, it has been pointed out~\cite{csb} that the low-energy
theorems
of $\pi\pi$ scattering~\cite{wein} can be directly applied to $w$ and $z$
scattering, replacing the pion decay constant $F_\pi=94\gev$ by the vev
$v=246\gev$.  Hence in all channels the threshold behaviour of the invariant
amplitude $\M$ is approximately specified by $v$,
%
\begin{eqnarray}
  -i\M(w^+w^- \to zz)       &\simeq & s/v^2 \,, \label{wpwm to zz}\\
  -i\M(w^\pm z \to w^\pm z) &\simeq & t/v^2  \,, \\
  -i\M(w^+w^- \to w^+w^-)   &\simeq & -u/v^2 \,, \\
  -i\M(w^+w^+ \to w^+w^+)   &\simeq & -s/v^2  \,, \\
  -i\M(zz\to zz)            &\simeq &  0  \,, \label{zz to zz}
\end{eqnarray}
%
where $s,\ t$ and $u$ are the invariant squares of energy and momentum
transfer.  These are the Goldstone-boson Low Energy Theorems (LET)~\cite{csb}.
 In the SM we can confirm the LET explicitly: see below.



   The various $ww,wz,zz$ scattering amplitudes are all related by SU(2)
isospin and crossing symmetry, just like the corresponding $\pi\pi$ amplitudes.
 In terms of the
components $w_i\ (i=1,2,3; w^\pm=(w_1 \mp w_2)/\sqrt2, z=w_3)$, the general
scattering amplitude can therefore be written
%
\begin{equation}
 -i\M(w_aw_b \to w_cw_d) = A(s,t,u)\delta_{ab}\delta_{cd} +
      A(t,s,u)\delta_{ac}\delta_{bd} + A(u,t,s)\delta_{ad}\delta_{bc}
\end{equation}
%
where $s=(p_a+p_b)^2,\ t=(p_a-p_c)^2,\ u=(p_a-p_d)^2$, and $A$ is symmetrical
in its last two arguments: $A(s,t,u)=A(s,u,t)$.  In terms of this basic
amplitude $A(s,t,u)$, the various channel amplitudes are then
%
\begin{eqnarray}
 -i\M(w^+w^- \to w^+w^-) &=& A(s,t,u)  +  A(t,s,u) \,,  \\
 -i\M(w^+w^- \to zz)     &=& A(s,t,u) \,,  \\
 -i\M(zz     \to zz)     &=& A(s,t,u)  +  A(t,s,u)  +  A(u,t,s) \,, \\
 -i\M(w^\pm z \to w^\pm z) &=& A(t,s,u) \,,  \\
 -i\M(w^+w^+ \to w^+w^+) &=& A(t,s,u)  +  A(u,t,s) \,.
\end{eqnarray}
%


The amplitudes for $s$-channel isospin $T=0,1,2$ are therefore
%
\begin{eqnarray}
   -i\M(T=0) &=& 3A(s,t,u) + A(t,s,u) +A( u,t,s) \,, \\
   -i\M(T=1) &=& A(t,s,u) - A(u,t,s) \,, \\
   -i\M(T=2) &=& A(t,s,u) + A(u,t,s) \,.
\end{eqnarray}
%
It is  the corresponding partial-wave amplitudes  that should satisfy
unitarity, since these channels diagonalize the $S$-matrix.

In the SM at tree level there are contributions from $H,Z,\gamma$ exchanges
and from the quadratic coupling (e.g.\ Fig.~2). $Z$ and
$\gamma$ terms are negligible when $M_W^2 \ll
s,|t|,|u|, m_H^2$ (e.g.~\cite{bchp}) and the
amplitude function is then
%
\begin{equation}
   A(s,t,u) = \frac{-m_H^2}{v^2}\left(1+\frac{m_H^2}{s-m_H^2}\right) \;,
\label{ampl}
\end{equation}
%
which depends on $s$ alone; $A(t,s,u)$ and $A(u,t,s)$ are found by
permuting $s,t,u$ in Eq.~(\ref{ampl}).  As expected, they obey the LET of
Eqs.~(\ref{wpwm to zz})--(\ref{zz to zz}).
The Breit-Wigner
resonance formula is obtained by substituting $(s-m_H^2) \to
(s-m_H^2+im_H\Gamma_H)$ in the $s$-channel pole term only.

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig3-2.eps}

{\small Fig.~2: SM diagrams for $W_L^+W_L^+$ elastic scattering.}
\end{center}

   Unitarity constrains each partial wave amplitude $a_L^T$ for elastic
Goldstone boson scattering with orbital angular momentum $L$ and isospin $T$:
%
\begin{equation}
   a_L^T(s) = \Bigl[\exp(2i\delta_L^T)-1\Bigr]\Big/(2i)
          = \exp(i\delta_L^T)\sin\delta_L^T = (\cot\delta_L^T - i)^{-1} \,,
\end{equation}
%
where $\delta_L^T(s)$ is the phase shift, real for purely elastic scattering
but complex in general.  Unitarity requires $a_L^T$ to lie on
or within a circle in the complex plane (Fig.~3),
%
\begin{equation}
         \left| a_L^T(s) - {1\over2} i \right|  \le {1\over2}  \,.
\end{equation}
%

\bigskip

\begin{center}
%\epsfxsize=2.5in
\hspace{0in}
%\epsffile{fig3-3.eps}

{\small Fig.~3: Unitarity circle for partial wave amplitude $a_L$.}
\end{center}

\noindent
The isospin amplitudes and partial waves are related by
%
\begin{equation}
    -i\M(T) = 32\pi\sum_L(2L+1)a_L^TP_L(\cos\theta) \,,
\end{equation}
%
with summation over even(odd) $L$ for $T={}$even(odd).  Hence the partial
wave amplitudes entering the LET of Eqs.~(\ref{wpwm to zz})--(\ref{zz to zz})
above are
%
\begin{equation}
a_0^0=\frac{s}{16\pi v^2}, \quad  a_1^1=\frac{s}{96\pi v^2}, \quad
a_0^2=-\frac{s}{32\pi v^2}. \label{a_0}
\end{equation}
%
Thus the LET implies a linear rise of these partial wave amplitudes versus $s$,
that cannot be extrapolated without violating unitarity.
Strictly speaking, since Eq.~(\ref{a_0}) gives purely real $a_L^T$,
any non-zero
value lies outside the unitarity circle, but this could at first be cured
by adding a small imaginary part.  More critically, the real part of $a_0$ from
Eq.~(\ref{a_0}) exceeds the unitarity limit $|\Re a_0^0| \le 1/2$ when
$\sqrt s > \sqrt{8\pi}\,v = 1.2$~TeV, while the $T=2$ amplitude exceeds its
corresponding limit for $\sqrt s >1.7$~TeV.  These examples show that unitarity
is likely to be an important constraint in the TeV range.


\section{Models for Goldstone boson scattering}

In addition to the simple SM with finite $m_H$, a variety of models
have been proposed to parameterize possible strong scattering of  $w^\pm$
and $z$, some of which we now describe.

{\it a) Chanowitz-Gaillard model (LET model)}~\cite{csb}. This is based
on LET behaviour up to the unitarity bounds $|a_L^T| \le 1$ (less stringent
than the bound $|\Re a_L^T|\le {1\over2}$ that we have used above). The three
participating partial wave amplitudes are extrapolated linearly until
they reach the bound, then held fixed:
%
\begin{eqnarray}
a_0^0 &=&  \frac{s}{16\pi v^2}\theta(16\pi v^2-s) + \theta(s-16\pi v^2)\,,\\
a_1^1 &=&  \frac{s}{96\pi v^2}\theta(96\pi v^2-s) + \theta(s-96\pi v^2)\,,\\
a_0^2 &=& -\frac{s}{32\pi v^2}\theta(32\pi v^2-s) - \theta(s-32\pi v^2)\,.
\label{a_0^2}
\end{eqnarray}
%
All other partial waves are set to zero, so we have
%
\begin{eqnarray}
 -i\M(T=0) = 32\pi a_0^0 \,, \quad
 -i\M(T=1) = 96\pi a_1^1 \cos\theta \,,\quad
 -i\M(T=2) = 32\pi a_0^2\,.
\end{eqnarray}
%
Variations on this empirical unitarization can be imagined, e.g.\ we can
cut off these (real) amplitudes at half the values above by appealing to
the bound $|\Re a_L^T| < 1/2$ instead. Or we can impose elastic unitarity by
interpreting the LET amplitude $a(\rm LET)$  as  $\tan\delta$, i.e.\ by
setting~\cite{golden}
%
\begin{equation}
     a_L^T = a_L^T({\rm LET})\Big/ \left[1 - ia_L^T(\rm LET)\right] \,.
\label{K-matrix}
\end{equation}
%
This is known as the $K$-matrix prescription,
since it re-interprets the LET $T$-matrix  as the $K$-matrix,
where  $S=[\exp(2i\delta)] = 1+iT =
\left(1+{1\over2}iK\right)\Big/\left(1-{1\over2}iK\right)$.
In the limit of large
$a_L^T(\rm LET)$, it gives purely elastic scattering with a
resonant phase shift $\delta_L^T = \pi/2$.  Figure~4 compares
the linear [Eq.~(\ref{a_0^2})] and $K$-matrix unitarizations for $L=0,T=2.$

\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{fig3-4.eps}

{\small Fig.~4: $|a_0^2|$ in the LET model with linear and $K$-matrix
unitarizations.}
\end{center}


{\it b) The $O(2N)$ model}~\cite{o2n}.  There is an isomorphism between
the $\rm SU(2)_L\times SU(2)_R$ symmetry of the SM Higgs sector and
$O(4)$ symmetry;
one sets out to study the latter through an $O(2N)$ Higgs-Goldstone model,
using a large-$N$ approximation but eventually taking $N=2$. In the large-$N$
limit, the basic $w,z$ scattering amplitude has the form
%
\begin{equation}
A(s,t,u) = \frac{16\pi^2 s}{16\pi^2v^2 - sN\left[2 + \ln(-\Lambda^2/s)\right]},
\end{equation}
%
where $\Lambda$ is a cutoff; $A(t,s,u)$ and $A(u,t,s)$ are given by permuting
$s,t,u$. In the $s$-dependent case we have $\ln(-\Lambda^2/s) =
\ln(\Lambda^2/s) + i\pi$, generating an imaginary part.  These amplitudes
respect the LET and also obey $|a_L^T| < 1$ for $s < \Lambda^2$.   After
setting  $N=2$  there is just one free parameter $\Lambda$, chosen to be a few
TeV.

{\it c) The chirally-coupled scalar model}~\cite{bbcgh}. This describes
a scalar resonance (such as a techni-$\sigma$) coupled to the Goldstone
bosons using the  chiral lagrangian formalism~\cite{chil} and chiral
perturbation theory techniques~\cite{chipt}.  Neglecting gauge couplings
$g,g'$ and transverse vector bosons $W_T,Z_T$, attention is focussed on the
Goldstone bosons $w^+,w^-,z^0$ (or equivalently $w_1,w_2,w_3$) that arise
when a global $\rm SU(2)_L\times SU(2)_R$ symmetry is spontaneously broken to
$\rm SU(2)_{L+R}$. The building blocks of the
lagrangian are the matrix field defined by
%
\begin{equation}
       U(x) = \exp\left[{i\over v}\sigma_i w_i\right] \,,
\end{equation}
%
where $\sigma_i$ are the Pauli matrices, and the scalar field $S$. The most
general chiral invariant lagrangian is a sum of an infinite number of
terms with increasing number of derivatives, with an infinite number of
parameters. At low energy, however, only a few terms with a few
derivatives (powers of momentum $p$) will matter.  Up to order $p^2$ there
are just 4 terms
%
\begin{eqnarray}
\L= {v^2\over4}\tr\left[\partial_\mu U\partial^\mu U^{\dagger}\right]
   + {1\over2}\partial_\mu S\partial^\mu S - {1\over2}m_S^2S^2
   - {1\over2}hvS \tr\left[\partial_\mu U\partial^\mu U\right]
\,.\label{O(p^2)}
\end{eqnarray}
%
Here the first term is SM, the next two are $S$-kinematics and the fourth
is an interaction between $S$ and the Goldstone fields.  There are two
free parameters, $m_S$ and $h$; the latter is related to the $S$ decay width
$\Gamma_S = 3h^2m_S^3/(32\pi v^2)$.  The tree-level $w$ and $z$ scattering
amplitude is
%
\begin{equation}
A(s,t,u) = {s\over v^2} - \frac{h^2s^2}{v^2}
\frac{1}{s-m_S^2+im_S\Gamma_S\theta(s)}\,,
\end{equation}
%
with $A(t,s,u)$ and $A(u,t,s)$ given by permutation.  For $h=1$, $H$ is just
the SM $H$ resonance and this amplitude reduces to Eq.~(\ref{ampl}).
For $h \ne 1$, however, we have something new.


{\it d) Coupled vector model, also known as the BESS model}~\cite{bess}:
the letters stand for ``Breaking Electroweak
Symmetry Strongly". This model is designed to study possible spin-1
resonance structure arising from strong EWSB,  analogous to the $\rho$ of
QCD or the techni-$\rho$ of Technicolor.  It introduces a new triplet of
massive composite vector bosons $V = V^\pm,V^0$, that are in fact the
gauge bosons of a local vector SU(2) symmetry, in an effective
lagrangian formulation.  There are no explicit Higgs particles.  In
addition to the usual Yang-Mills kinetic terms, the most general mass
term under reasonable assumptions is
%
\begin{equation}
\L_M = -{1\over4}v^2\left[\tr(\W-\B)^2 + \alpha \tr(\W+\B-2\V)^2\right] \,,
\end{equation}
%
where $\alpha$ is a new parameter and
%
\begin{equation}
\V^\mu = ig''{1\over2}\tau_i V_i^\mu \,, \quad
\W^\mu = ig {1\over2}\tau_i W_i^\mu \,, \quad
\B^\mu = ig'{1\over2}\tau_3 B^\mu \,.
\end{equation}
%
The first term in $\L_M$ is just the SM; the second introduces $V$-boson masses
%
\begin{equation}
    m_V^2 = {1\over4} v^2 \alpha {g''}^2
\end{equation}
%
at zeroth order, plus $V$-$W$-$B$ mixing angles of order $g/g''$.  This mixing
already gives some coupling between fermions and $V$-bosons; the authors
also add an empirical direct coupling of strength $bg''$ between $V$-bosons
and left-handed fermions.  There are thus 3 new parameters beyond the
SM: $m_V$, $g''$, and $b$.   Its authors have used this model to parameterize
possible spin-1 resonances in $WW$ and $WZ$ scattering (there are no $ZZ$
resonances in the model)~\cite{bess}.
The BESS model can also be viewed as a special case of extended gauge theory
models~\cite{bmw}.

{\it e) DHT model}~\cite{dht}: the name comes from the authors' initials.
This too is based on the chiral lagrangian formalism~\cite{chil} and chiral
perturbation theory techniques~\cite{chipt}, but including higher order terms
instead of introducing a new scalar or vector field.
At lowest order $p^2$ the lagrangian contains just one term (the first term of
Eq.~(\ref{O(p^2)})) with no free parameter.
At the next order $p^4$ there are just two more terms:
%
\begin{equation}
 L_1 =  M \tr\left[\partial_\mu U\partial^\mu U^\dagger\right]
                  \tr\left[\partial_\nu U \partial^\nu U^\dagger\right]
       +N \tr\left[\partial_\mu U \partial_\nu U^\dagger\right]
                  \tr\left[\partial^\mu U \partial^\nu U^\dagger\right]  \,.
\end{equation}
%
Truncating the expansion at this point, the authors apply this model
to parametrize the scattering of $w^\pm$ and $z$.  They obtain relatively
simple closed forms for the various
scattering amplitudes, in terms of the known parameter $v$ plus
two new arbitrary parameters $M_R$ and $N_R$ (renormalized versions of $M$ and
$N$) plus a renormalization scale $\mu$ appearing in logarithmic factors.
These parameters can be chosen to reproduce the low-energy behaviour of
any given chiral scenario, {\it e.g.}\ SU($N_{TC}$) technicolor, the SM with a
very large $m_H$, or QCD (appropriately scaled up in energy)~\cite{dht}.

The sum of tree-level plus
one-loop partial-wave amplitudes  $a_L = a_L(0) + a_L(1)$  is not
automatically unitary; instead of the elastic unitarity condition
$\Im a_L = |a_L|^2$  we have $\Im a_L(1) = |a_L(0)|^2$  with $a_L(0)$ real.
A helpful trick to obtain exact elastic unitarity is to take instead
the [1,1] Pad\'e approximant\cite{pade}:
%
\begin{equation}
     a_L = a_L(0)/[1 - a_L(1)/a_L(0) ]   \,.
\end{equation}
%
This can generate a resonance if the real part of the denominator vanishes.


{\it f) Rescaled QCD model}~\cite{csb}.  Another possible assumption
is that $ww$ scattering amplitudes simply equal the corresponding
experimental  $\pi\pi$  scattering amplitudes, at an appropriately
rescaled energy:
%
\begin{equation}
\M(w_iw_j \to w_kw_l:s) = \M(\pi_i\pi_j \to \pi_k\pi_l:sF_\pi^2/v^2) \,,
\end{equation}
%
where $F_\pi = 94$~MeV is the pion decay constant.  Since the physical
pion and $W,Z$ masses are not related by the same rescaling factor
$v/F_\pi$, some kinematic corrections may be added near threshold.
At low energies this rescaling simply reproduces the LET.  At higher
energies there is no strong reason to believe that $ww$ and $\pi\pi$
dynamics are so related, but this assumption would lead to a $\rho$-like
$ww,wz$ resonance at $\sqrt s = m_\rho v/F_\pi = 2$~TeV.

{\it g) N/D model}~\cite{hikasa}. This approach is to parameterize
a scalar and a vector resonance into the $ww$ partial wave amplitudes,
by assuming a form $a(s)=N(s)/D(s)$.  The numerator functions $N$ are
obtained from tree amplitudes with scalar and vector exchanges,
adjusted to fit the LET limit.  The denominator functions are
calculated from dispersion relations, incorporating analyticity and
unitarity.  Explicit parametric forms for the $a_L^T$ are obtained in
terms of three parameters --- the two resonance masses and
the ratio of their coupling strengths.



\section{$W^+ W^+ \to W^+ W^+$ scattering}

   $ W^+W^+ \to W^+W^+$   elastic scattering
gives the cleanest signal to look for experimentally, with minimal
backgrounds.  $W^-W^- \to W^-W^-$ is similar but has only about 1/3 of the
event rate at $pp$ colliders, due to the differences between $u$- and $d$-quark
distributions.  In contrast,  $W^+W^-$, $WZ$ and $ZZ$ final states have much
bigger QCD backgrounds; also $W^+W^-$  has a large $t\tbar$ background, and the
identifiable  $Z \to \ell^+\ell^-$ decays have much smaller branching fraction
($B=6.7\%$) than the $W \to \ell\nu$  channels ($B=20\%$).


In order to identify the $W$-boson charge, we must look for
$W^+\to\ell^+\nu$ decays ($\ell=e,\mu$).  The polarization information
contained in decay correlations is therefore lost; we cannot separate
$W_L$ from $W_T$ polarization states.  We also cannot reconstruct the
final $WW$ invariant mass.  We must simply look for an enhancement in
$\ell^+\ell^+$ production, in final states with the expected
characteristics of $WW$ scattering.  The first requirement is that the
leptons have high $p_T$ and are isolated from jets, to suppress contributions
from $b$ or $c$ semileptonic decays and from $\pi,K\to\ell\nu$.

   The lowest-order SM electroweak subprocess for  $W^+W^+$  production
is $qq \to qqW^+W^+$. Contributing diagrams are shown in Fig.~5; a,b,l,m
are individually gauge-invariant, but the rest are only gauge-invariant
as a sum.  $WW$ scattering effects (including the $W_LW_L \to W_LW_L$ signal
of interest)  are contained in $a+b+c$ while the other diagrams mainly
represent bremsstrahlung (dominantly $W_T$ emission), but these effects are not
rigorously separable.

\begin{center}
%\epsfxsize=4.75in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/StrongWW/x-feyn.eps}

{\small Fig.~5: SM electroweak diagrams for $W^+W^+$ production.}
\end{center}



   In addition to this electroweak signal and background (of order
$\alpha^4$  up to powers of $\sin^2\theta_W$ and $\cos^2\theta_W)$, there
are genuine $W^+W^+$ production backgrounds of order $\alpha_S^2\alpha^2$
from gluon exchange in
%
\begin{equation}
         uu \to ddW^+W^+
\end{equation}
%
and of higher order $\alpha_S^3\alpha^2$ from processes like
%
\begin{equation}
         uu \to gddW^+W^+   \,.
\end{equation}
%
There is also a background of order $\alpha_S^3\alpha$ from $t\bar tW$
production processes like
%
\begin{equation}
     ug \to d W^+ t \bar t \,, \quad  u \bar d \to g W^+ t \bar t  \,,
\end{equation}
%
where $t \to b W^+$ decay provides the second $W^+$.



   There are various non-isolated  $\ell^+\ell^+$ backgrounds, e.g.\ from
$gg\to t\bar t\to b\bar bW^+W^-$ with $W^+\to\ell^+\nu$ and
$\bar b\to\bar c\ell^+\nu$ (second lepton usually non-isolated). And
finally there are various fake backgrounds such as
$q\bar q\to W^+W^-\to\ell^+\ell^-X$  when the sign of one lepton is
mismeasured.


   Experimental acceptance cuts can be devised to enhance the
$W_L^+W_L^+$ scattering signal compared to these backgrounds.

\begin{enumerate}
\addtolength{\itemsep}{-.12in}

\item Lepton isolation (already mentioned). Typically we require there to
be less than a few GeV of transverse energy $E_T = E\sin\theta$ deposited
within a narrow cone centred on each lepton direction.  This suppresses
important non-isolated backgrounds (see above).

\item Central leptons. The scattering signal is central, bremsstrahlung
background is forward and backward peaked. Typically we require
\begin{equation}
         |\eta(\ell)| < 2   \, ,
\end{equation}
which is also well matched to detectors such as SDC at the SSC.

\item High lepton $p_T$. The $W^+W^+$ scattering increases with invariant
mass $m(WW)$, according to LET and extrapolations thereof, whereas
backgrounds decrease; one way to give preference to this desirable
large-$m(WW)$ region is to select large $p_T(\ell)$.  Insisting on high
$p_T(\ell)$ also favors $W_L$ over $W_T$ decays, because of the difference
between the $W_L\to\ell\nu$ and $W_T\to\ell\nu$ distributions,
%
\begin{eqnarray}
dN_T/d\cos\theta^* &=& {3\over4}N_T\sin^2\theta^*  \,, \\
dN_L/d\cos\theta^* &=& {3\over8}N_L(1 + \cos^2\theta^*)  \,,
\end{eqnarray}
%
where $\theta^*$ is the $W$-restframe angle between $\ell^+$ and
the boost axis to the lab.  Typically we choose
%
\begin{equation}
          p_T(\ell) > 100 \rm\ GeV   \,,
\end{equation}
%
and it is also helpful to require
%
\begin{equation}
\Delta p_T(\ell\ell) = |\vec p_T(\ell_1) - \vec p_T(\ell_2)| > 200\gev \,.
\end{equation}


\item Azimuthal lepton correlation. At high $m(WW)$ the two $W$s come out
back-to-back in azimuth; the decay distributions transmit this correlation
to the leptons for $W_LW_L$ decays but tend to smear it out for $W_TW_L$
or $W_TW_T$ decays. We therefore typically require
%
\begin{equation}
    \cos\Delta\phi(\ell\ell) < -0.8
\end{equation}
%
to enhance $W_L$ over $W_T$ and some other backgrounds.

\item Dilepton invariant mass. Requiring high $m(\ell\ell)$ both helps to
emphasis high $m(WW)$ (see above) and also favors $W_L$ over $W_T$
contributions, because of the decay distributions.  We typically require
%
\begin{equation}
    m(\ell\ell)   > 250\gev
\end{equation}

\item Central jet veto (see also Lecture 2). The $t\bar tW^+$ background
process typically produces central quark-jets from $t\to bW^+$ and
$\bar t\to\bar bq\bar q'$, whereas the final quarks in the $uu\to ddW^+W^+$
signal usually appear as forward jets.  The former can therefore be
efficiently suppressed by vetoing all events containing one or more
central jets~\cite{bchp}, e.g.\ jets with
%
\begin{equation}
   p_T(j)  >  30\gev,  \quad  |\eta(j)| < 3 \,.
\end{equation}
%
This also further suppresses the non-isolated $\ell^+\ell^+$ background
from $t\bar t\to b(W^+\to\ell^+\nu)(\bar b\to\bar c\ell^+\nu)W^-$ events,
since both $b$ and $W^-\to jj$ tend to give central jets.

\end{enumerate}
Similar approaches can be made to the other $W$ and $Z$ channels.


\section{Typical model results}

Reference~\cite{bbcgh} has reported calculations, using realistic cuts,
for vector boson scattering signals and backgrounds with  a wide range
of different models.  The cut, tag and veto requirements are as follows.

\begin{center}
\begin{tabular}{cc}
\hline\hline
$W^+ W^-$ leptonic cuts & Tag and Veto \\ \hline
 $\vert \eta({\ell}) \vert < 2.0 $  &
 $E(j_{tag}) > 1.5\ (1.0)\ {\rm TeV}$  \\
 $p_T(\ell) > 100\ {\rm GeV}$  &
 $3.0 < \vert \eta(j_{tag}) \vert < 5.0$  \\
 $\Delta p_T({\ell\ell}) > 450\ {\rm GeV}$  &
 $p_T(j_{tag}) > 40\ {\rm GeV}$  \\
 $\cos\Delta\phi_{\ell\ell} < -0.8$  &
 $p_T(j_{veto}) > 30\ {\rm GeV}$  \\
 $m({\ell\ell}) > 250\ {\rm GeV}$  &
 $ \vert  \eta(j_{veto}) \vert  < 3.0$  \\ \hline
$Z Z$ leptonic cuts & Tag only\\ \hline
 $\vert  \eta({\ell}) \vert  < 2.5 $  &
 $E(j_{tag}) > 1.0\ (0.8)\ {\rm TeV}$   \\
 $p_T(\ell) > 40\ {\rm GeV}$  &
 $3.0 < \vert \eta(j_{tag}) \vert < 5.0$   \\
 $p_T(Z) > {1\over4} \sqrt{M^2({ZZ}) - 4 M^2_Z}$  &
 $p_T(j_{tag}) > 40\ {\rm GeV}$   \\
 $M({ZZ}) > 500\ {\rm GeV}$  & {} \\ \hline
$W^+ Z$ leptonic cuts & Tag and Veto\\ \hline
 $\vert  \eta({\ell}) \vert  < 2.5 $  &
 $E(j_{tag}) > 2.0\ (1.5)\ {\rm TeV}$  \\
 $p_T(\ell) > 40\ {\rm GeV}$  &
 $3.0 < \vert \eta(j_{tag}) \vert < 5.0$  \\
 $ {\overlay/ p}_{T} >  75\ {\rm GeV}$  &
 $p_T(j_{tag}) > 40\ {\rm GeV}$   \\
 $p_T(Z) > {1\over4} M_T $ &
 $p_T(j_{veto}) > 60\ {\rm GeV}$  \\
 $M_T > 500\ {\rm GeV}$ &
 $ \vert  \eta(j_{veto}) \vert  < 3.0$  \\ \hline
$W^+ W^+$ leptonic cuts & Veto only \\ \hline
 $\vert  \eta({\ell}) \vert  < 2.0 $   &
 $p_T(j_{veto}) > 60\ {\rm GeV}$  \\
 $p_T(\ell) > 100\ {\rm GeV}$   &
 $ \vert  \eta(j_{veto}) \vert  < 3.0$  \\
 $\Delta p_T(\ell\ell) > 200\ {\rm GeV}$   & \\
 $\cos\Delta\phi_{\ell\ell} < -0.8$  & \\
 $m({\ell\ell}) > 250\ {\rm GeV}$ & {}  \\ \hline\hline
\end{tabular}
%{$^*$  $M_T$ is the cluster transverse mass.$^5$}
\end{center}

   The next table shows the numbers of events
expected per year at the SSC, assuming $m_t = 140$~GeV and annual
luminosity 10~fb$^{-1}$, in the $W^+W^-$, $W^+Z$, $ZZ$ and $W^+W^+$ channels;
here all $W$ and $Z$ are presumed to decay leptonically.
The first column gives the mass cut (on $M_{\ell\ell},\  M_T$
or $M_{ZZ}$); for each $VV$ final state, the less severe mass cut
is the minimum required to make the EWA/GBET approximations
 reliable, the more severe cut shows how the cross section
falls.  The second column gives the total background rate;
subsequent columns give various signal rates, calculated
using the EWA plus GBET.
 SM refers to the case $m_H=1$~TeV. Scalar means
the chirally-coupled scalar model, with $m_S=1$~TeV and $\Gamma_S=350$~GeV.
The $O(2N)$ model has $\Lambda=3$~TeV.  Vec~2.0 and Vec~2.5 are coupled vector
models with $M_\rho=2.0$~TeV, $\Gamma_\rho=700$~GeV and $M_\rho=2.5$~TeV,
$\Gamma_\rho=1.3$~TeV respectively. LET~CG and LET~K denote the linear and
$K$-matrix cutoff versions of the LET model.  Delay~K denotes a particular
case of the DHT model, where unitarity violation is
delayed to $\sqrt s > 2$~TeV, unitarized by the $K$-matrix prescription.

\begin{center}
{\small
\begin{tabular}{l|cccccccccc}
\hline\hline
$W^+ W^-$  & Bkgd. & SM & Scalar & $O(2N)$ & Vec 2.0 & Vec 2.5 & LET
CG & LET K & Delay K  \\ \hline
%
%$M_{\ell\ell} > 0.25$ & 21 & 48 & 30 & 24 & 15 & 12 & 16 & 12 &
%11  \\
$M_{\ell\ell} > 0.5$ & 17 & 46 & 29 & 23 & 15 & 12 & 15 & 11
& 11  \\
$M_{\ell\ell} > 1.0$ & 3.6 & 3.8 & 1 & 2.7 & 6.5 & 4.9 & 5.3 &
3.6 & 4.6  \\ \hline
%
$W^+ Z$ & Bkgd. & SM & Scalar & $O(2N)$ & Vec 2.0 & Vec 2.5 & LET CG
&  LET K & Delay K  \\ \hline
%
%$M_T > 0.5$ & 2.5 & 1.3 & 1.8 & 1.5 & 9.5 & 6.2 & 5.8 & 4.9 & 6.0  \\
$M_T > 1.0$ & 0.8 & 0.5 & 0.8 & 0.7 & 7.9 & 4.7 & 4.1 & 3.3 & 4.6  \\
$M_T > 1.5$ & 0.3 & 0.2 & 0.2 & 0.3 & 5.5 & 3.2 & 2.6 & 1.9 & 3.2  \\ \hline
%
$Z Z$ & Bkgd. & SM & Scalar & $O(2N)$ & Vec 2.0 & Vec 2.5 & LET CG &
   LET K & Delay K  \\ \hline
%
%$M_{ZZ} > 0.5$ & 1.0 & 11 & 6.2 & 5.2 & 1.1 & 1.5 & 2.6 & 2.2 & 1.6\\
$M_{ZZ} > 1.0$ & 0.3 & 4.1 & 2.6 & 2.0 & 0.4 & 0.7 & 1.6 & 1.3 & 0.8\\
$M_{ZZ} > 1.5$ & 0.1 & 0.5 & 0.2 & 0.5 & 0.1 & 0.3 & 0.9 & 0.6 & 0.4\\ \hline
%
$W^+ W^+$ & Bkgd. & SM & Scalar & $O(2N)$ & Vec 2.0 & Vec 2.5 & LET
CG &  LET K & Delay K  \\ \hline
%
%$M_{\ell\ell} > 0.25$ & 3.5 & 6.4 & 8.2 & 7.1 & 7.8 & 11 & 25 &
%21 & 15  \\
$M_{\ell\ell} > 0.5$ & 1.5 & 3.2 & 4.2 & 3.9 & 3.8 & 6.3 & 19 &
15 & 11  \\
$M_{\ell\ell} > 1.0$ & 0.2 & 0.7 & 0.6 & 0.9 & 0.5 & 1.2 & 7.6 &
5.2 & 5.2  \\ \hline\hline
\end{tabular}
}
\end{center}

   We see that many of the models predict detectable signals, clearly
distinguishable from background, although the event rates are low and
higher luminosity is desirable.  The signals are largest in resonant
cases and the different channels are helpfully complementary.
There are also characteristic difference between some of the models.
$WW$ scattering indeed appears to be a promising probe of EWSB dynamics.

The number of years required to achieve a 99\% confidence level signal at the
SSC (with 10~fb$^{-1}$/year) and at the LHC (with 100~fb$^{-1}$/year) for the
various channels and models are summarized in the last table\cite{bbcgh}. These
results are very encouraging. Many of the signals can be established in a
relatively short time frame (1 to 3 years).

\def\wp{W^+}
\def\wm{W^-}
\def\wpm{W^{\pm}}

\begin{center}{\small
\begin{tabular}{l|cccccccc}
\multicolumn{9}{c}{\small Number of years (if $<10$) required for a 99\%
confidence level signal at (a)~SSC and (b)~LHC.}\\
\hline
& \multicolumn{8}{c}{Model}\\
\cline{2-9}
\multicolumn{1}{c|}{Channel} & SM & Scalar & $O(2N)$ & Vec. 2.0 & Vec. 2.5 &
LET CG &
LET K & Delay K\\
\hline
(a) SSC\\
\qquad $ZZ$ &
2.2  & 4.0    & 5.8  & \     & \     & 7.8  & \     & \    \\
\qquad $\wp\wm$ &
0.50   & 1.0    & 1.2   & 2.5  & 3.5   & 2.5   & 4.0     & 4.0    \\
\qquad $\wp Z$  &
\    & \    & \     & 1.5   & 2.8  & 3.2  & 4.2  & 2.8 \\
\qquad $\wp\wp$ &
6.2  & 4.0    & 4.5   & 4.8  & 2.2  & 0.50   & 0.75  & 1.2 \\
\hline
(b) LHC\\
\qquad $ZZ$  &
2.0     & 3.0    & 4.8  & \     & \     & 9.0     & \     & \    \\
\qquad $\wp\wm$ &
0.75  & 1.2    & 2.0   & 7.5 & \ & 6.0 & \ & \    \\
\qquad $\wp Z$ &
 \    & \    & \     & 3.0     & 6.8  & 7.8  & 9.5   & 7.2 \\
\qquad $\wp\wp$ &
5.2  & 3.2 & 4.2  & 3.5   & 2.0     & 0.75  & 0.75  & 1.8 \\
\hline
\end{tabular} }
\end{center}


\renewcommand{\chapter}{\section}

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{\bf B249}, 130 (1990); {\bf B253}, 275 (1990).

\bibitem{bmw}V.~Barger, E.~Ma and K.~Whisnant, Phys.\ Rev.\ {\bf D25}, 1384
(1982); J.L.~Kneur and D.~Schildknecht, Nucl.\ Phys.\ {\bf B357}, 357 (1991).

\bibitem{dht}A.~Dobado, M.J.~Herrero and J.~Terron, Z.~Phys.\ {\bf C50}, 205,
465 (1991).

\bibitem{pade}T.N.~Truong, Phys. Rev. Lett. {\bf61}, 2526 (1988); A.~Dobado et
al., Phys. Lett. {\bf B235}, 129 (1990).

\bibitem{hikasa}K.~Hikasa and K.~Igi, Phys.\ Lett.\ {\bf B261}, 285 (1991);
erratum {\bf B270}, 128 (1991); Tokyo preprint  TU-428 (1992).


\end{thebibliography}



\end{document}

ifferent models.  The cut, tag and veto requirements are as follows.

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$W^+ W^-$ leptonic cuts & Tag and Veto \\ \hline
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\begin{document}

\setcounter{page}{42}
\chapter*{LECTURE 4:\\ SUPERSYMMETRY AND\\ GRAND UNIFICATION}
\setcounter{chapter}{4}

%\thispagestyle{emtpy}

\section{Introduction}

   Supersymmetry (SUSY)~\cite{susy} implies that every boson has a
fermion partner (and vice versa) with spins differing by 1/2 but
with the same $\rm SU(3)\times SU(2)\times U(1)$ quantum numbers; their
coupling
schemes are intimately related; their masses are degenerate in
the symmetry limit.  Since no such boson-fermion pairings have been
observed, SUSY must be broken at low energy scales but may hold above
some scale $M_{\rm SUSY}$.  Broadly speaking, we expect the mass splittings to
be of order $\Delta m\mbox{(boson-fermion)} \sim M_{\rm SUSY}$, and we assume
that the missing SUSY partners of known particles are simply too heavy to have
been discovered yet.


 SUSY is well motivated theoretically and testable experimentally:
\begin{enumerate}
\item Boson and fermion loops give opposite-sign contributions to scalar
masses (Fig.~1); SUSY guarantees cancellations and stabilizes these
contributions, which would otherwise be of order
$M_{\rm GUT}\sim 10^{16}$\,GeV (the ``hierarchy problem").

\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig4-1.eps}

{\small Fig.~1: Fermion $f$ and sfermion $\tilde f$ loops contribute to $H$
mass.}
\end{center}


\item If $M_{\rm SUSY}$ is not enormously greater than the electroweak scale
$M_W$, fine-tuning of parameters in GUT models (the ``naturalness
problem") is avoided. If so, we expect superpartner masses to be $\alt {\cal
O}(1\rm~TeV)$.

\item Since the relevant couplings are small at scales above $M_W$, SUSY-GUT
models can be calculated perturbatively and tested experimentally,
{\it e.g.}\ through the production of superpartners at present or future
colliders.
%\end{enumerate}

\noindent
   There are also phenomenological reasons to welcome SUSY:
%\begin{enumerate}

\item With SM input alone, the gauge couplings (which are now accurately
determined at low scales) do not evolve toward a common value at a
high scale; but with added minimal SUSY input above a few TeV,
gauge couplings converge~\cite{amaldi}.

\item From equal $b$ and $\tau$ Yukawa couplings at $M_{\rm GUT}$, evolution
downward gives the correct $b$ and $\tau$ masses with SUSY but not with SM
alone~\cite{btau}.

\item Proton decay is too fast in simple SM GUT models, but acceptable
in SUSY-GUT models where $M_{\rm GUT}$ is higher~\cite{pdecay}.

\item In SUSY models with conserved $R$-parity, the lightest super-partner
(LSP) is stable and provides an attractive candidate for cosmological
Dark Matter (which appears to dominate the total matter density, in
fact)~\cite{dm}.

\item In SUSY-GUT models extra terms appear in the scalar potential $V$.
When $m_t$ is large, $V$ automatically evolves to a form where spontaneous
symmetry breaking (SSB) occurs; {\it i.e.}\ the Higgs mechanism arises
naturally.
\end{enumerate}

   With all this motivation and interest, it is not surprising that the SUSY
literature has grown enormously since 1980 (Fig.~2)

\begin{center}
%\epsfxsize=5in
\hspace{0in}
%\epsffile{fig4-2.ai.eps}

{\small Fig.~2: Papers listed in SPIRES under ``Supersymmetry" and
``Supersymmetric\rlap".}
\end{center}

\section{The minimum SUSY extension of the SM (MSSM)}

   MSSM contains all the SM ingredients plus minimal SUSY additions.
The gauge symmetry $\rm SU(3)\times SU(2)\times U(1)$ remains, requiring the SM
spin-1 gauge bosons $g,\gamma,W,Z$ plus their spin-1/2 ``gaugino" partners
$\tilde g,\tilde\gamma,\tilde W,\tilde Z$.  In order to cancel anomalies the
single SM complex scalar doublet $\Phi$ is replaced by two doublets
$H_1$ and $H_2$, giving masses to down-type quarks plus leptons and to up-type
quarks, respectively; they acquire separate vevs $v_1$ and $v_2$ with $v_1^2 +
v_2^2= v^2$, where $v=246$~GeV is the SM vev and one parametrizes $v_1 = v
\cos\beta,\ v_2 = v \sin\beta$. After SSB, 3 of the 8 real Higgs fields
become $W_L^+,\ W_L^-,\ Z_L^0$; the others appear as two
CP-even bosons $h^0$ and $H^0\ (m_h < m_H)$, a CP-odd boson $A^0$
and two charged scalars $H^{\pm}$.
The ``higgsino" partners $\tilde H_1$ and $\tilde H_2$ have
spin-1/2.  Since gauginos and higgsinos have the same spin and similar
quantum numbers, they can mix; the resulting physical mass eigenstates
are collectively called ``charginos" ($\tilde W^\pm$-$\tilde H^\pm$ mixtures)
and ``neutralinos" ($\tilde Z$-$\tilde H^0$ mixtures). Charginos are denoted by
$\tilde\chi_i^+$ or $\tilde W_i$ and neutralinos by $\tilde\chi_i^0$ or $\tilde
Z_i$. Neutralinos are generally Majorana not Dirac fermions.

   The three generations of SM spin-1/2 quark and lepton doublets are
partnered by an equal number of spin-0 ``squark" and ``slepton" states.
Note that each chiral fermion state $f_L$ or $f_R$ has its own sfermion
$\tilde f_L$ or $\tilde f_R$ with the same $\rm SU(3)\times SU(2)\times U(1)$
quantum numbers. Since they can mix, the mass eigenstates $\tilde f_1$ and
$\tilde f_2$ are $\tilde f_L$-$\tilde f_R$ mixtures in general.

   The MSSM (in common with a wider class of SUSY models) conserves
$R$-parity, where  $R = (-1)^{2S+L+3B}$  is a multiplicative quantum number;
$S,\ L,\ B$ are spin, lepton-number, baryon-number.  $R$ distinguishes
``normal" SM-like particles that have $R=+1$ from their superpartners that
differ only by 1/2 unit in $S$ and therefore have $R=-1$.  $R$-conservation
immediately implies that
\begin{enumerate}
\item sparticles $(R=-1)$ are produced in pairs;
\item the lightest sparticle (LSP) is stable; if it has only  weak
interactions, then
\item the LSP is a Dark Matter candidate, and
\item production and decay of sparticles will lead to LSPs, which will
escape undetected like neutrinos, leaving only missing-energy-momentum
as their signature.
\end{enumerate}


\section{SUSY-GUT Models}


  The SM leaves important questions unexplained:
\begin{enumerate}
\item Why are there three separate gauge groups SU(3), SU(2), U(1)?
\item Why are the gauge coupling strengths $g_3$, $g_2$, $g_1$ all different?
\item Why do the fermions have such arbitrary-looking quantum numbers
({\it e.g.}\ Lecture 1.1)?
\end{enumerate}
GUTs offer answers:
\begin{enumerate}
\item All three are subgroups of a single larger group $G$.
\item The couplings are all equal (or trivially related) above some scale
$M_G$, where $G$ is unbroken; below this scale their differences are due
to symmetry-breaking and are precisely calculable.
\item Basic fermions of one generation all belong to a single representation
of $G$ (or in some models to two representations); all their quantum numbers
are then precisely prescribed and related, however arbitrary they may appear.
\end{enumerate}
But GUTs don't explain everything; {\it e.g.}\ they don't explain why three
generations of fermions appear.

   The simplest candidate GUT has $G ={}$SU(5).  All left-handed fermions of
the first generation are placed in two representations  $5^*[d^c,e^-,\nu_e]_L$
and  $10[e^{-^c},d,u,u^c]_L$, remembering that the CP-conjugates of $u_R,d_R$
and $e_R$ are left-handed.  Each quark appears in 3 colors to make up the
numbers 5 and 10, and there is no right-handed neutrino. The 4 commuting
generators of SU(5) can be
chosen to be the SU(3) color spin and hypercharge operators $T_3^c$ and
$Y^c$, plus the weak isospin and hypercharge operators $T_3^L$ and
$\sqrt{3/20}\, Y$; hence in the GUT symmetry limit the three couplings
$g_3\,(= g_s),\ g_2\,(=g)$ and $g_1\,(=\sqrt{5/3}\,g')$ are equal.

 A test of GUTs is that the gauge couplings $g_3$, $g_2$ and $g_1$, which are
well determined experimentally at the electroweak scale $\mu = M_W$, should
converge to a common value when evolved upward using the Renormalization
Group Equations (RGE) to the GUT scale $\mu = M_G$.  The RGE depend on
the symmetries that prevail between  $M_W$ and $M_G$; the simplest case
would be $\rm SU(3)\times SU(2)\times U(1)$ all the way.   The RGE also depend
on the particle content of the theory; SUSY differs here from the SM.


\section{Gauge coupling unification}

  The RGE for the evolution of the gauge couplings $g_i$ versus the
renormalization scale $\mu$, under $\rm SU(3)\times SU(2)\times U(1)$ symmetry,
are conveniently written using the dimensionless variable $t=\ln{\mu\over M_G}$
as follows~\cite{einhorn}:
%
\begin{equation}
{dg_i\over dt} = {g_i\over 16\pi^2} \left[ b_ig_i^2 + {1\over16\pi^2}
\left( \sum_{j=1}^3 b_{ij} g_ig_j - \sum a_{ij} g_i^2\lambda_j^2\right)\right]
\end{equation}
%
The first term on the right is the one-loop approximation; the second
and third terms contain two-loop effects, involving other gauge
couplings $g_j$ and Yukawa couplings $\lambda_j$.  The coefficients $b_i,\
b_{ij}$ and $a_{ij}$ are determined at given scale $\mu$ by the active particle
content (those with mass ${}<\mu$).  If there are no thresholds
({\it i.e.} no
changes of particle content) between $\mu$ and $M_G$, then the coefficients
are constants through this range and the one-loop solution is
%
\begin{equation}
\alpha_i^{-1}(\mu)  =  \alpha_i^{-1}(M_G) - t b_i/(2\pi)   \;,
\end{equation}
%
where $\alpha_i = g_i^2/(4\pi)$; thus $\alpha_i^{-1}$ evolves linearly with
$\ln\mu$ at one-loop order.    If there are no new physics thresholds
between $\mu = M_Z \simeq m_t$  and $M_G$  ({\it i.e.} nothing but a ``desert"
as in the basic SM) then equations of this kind should evolve the observed
couplings at the electroweak scale~\cite{giatw}
%
\begin{eqnarray}
\alpha_1(M_Z)^{-1} &=& 58.89 \pm 0.11 \,, \\
\alpha_2(M_Z)^{-1} &=& 29.75 \pm 0.11 \,, \\
\alpha_3(M_Z) &=& 0.118 \pm 0.007 \,,
\end{eqnarray}
%
to converge to a common value at some large scale.  Figure~3 shows that
such a SM extrapolation does NOT converge; this figure actually includes
two-loop effects but the evolution is still approximately linear versus
$\ln\mu$, as at one-loop order.  GUTs do not work, if we assume just SM
particles plus a desert up to $M_G$.

\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig2.eps}

\medskip
{\small Fig.~3: RGE evolution of gauge couplings in the SM.}
\end{center}

   If however we increase the particle content to include the minimum
number of SUSY particles, with a threshold not too far above $M_Z$,  then
GUT-type convergence can happen.  Figure~4 shows two examples with SUSY
threshold $M_{\rm SUSY}= m_t = 150$~GeV or $M_{\rm SUSY}=1$~TeV,
the threshold difference being
compensated by a small change in $\alpha_3(M_Z)$.  SUSY-GUTs are plainly
more successful; the evolved couplings are consistent with a common
intersection at $M_G \sim 10^{16}$\,GeV.  In fact a precise single-point
intersection is not strictly necessary, since the exotic GUT gauge,
fermion and scalar particles do not have to be precisely degenerate;
we may therefore have several non-degenerate thresholds near $M_G$, to be
passed through on the way to GUT unification.

   A full set of coupled equations requires the RGE for Yukawa
couplings; for the large $Htt$ term they have the form
%
\begin{equation}
{d\lambda_t\over dt} = {\lambda_t\over16\pi^2} \left[-\sum c_i g_i^2 +
6\lambda_t^2 + \lambda_b^2 \right] + \mbox{two-loop terms} \;,
\end{equation}
%
with $c_1=13/15,\ c_2=3,\ c_3=16/3$.
Yukawa couplings enter the $ \lambda_i$ evolution at one-loop level and
are important here, but are often neglected in $g_i$ evolution where they
are only two-loop effects.  $\lambda_t$ is the largest; its value at
$\mu=m_t$ is $\sqrt2 m_t(m_t)/v\sin\beta$  but it evolves to large
non-perturbative values if the gauge coupling gets too big
$(\alpha_3(M_Z) \agt 0.125)$.
Note that for quarks the running mass $m_t(m_t)$ differs
a bit from the physical mass, defined by the pole in the $t$
propagator; the two are related by $m_t({\rm pole}) = m_t(m_t)\bigl[1 +
4\alpha_s(m_t)/(3\pi)\bigr]$ to order $\alpha_s$~\cite{tarrach}.


\begin{center}
%\epsfxsize=6.1in
\hspace{0in}
%\epsffile{fig4-4.eps}

\medskip
\parbox{5.5in}{\small Fig.~4: RGE evolution of gauge couplings with minimal
SUSY~\cite{bargut}: (a)~$M_{\rm SUSY}= m_t$;
(b)~$M_{\rm SUSY}=1$~TeV.}
\end{center}


\section{Yukawa coupling unification}

It is tempting to suppose that the various Yukawa couplings
of Higgses and fermions also become simply related at scale $M_G$; there
is no uniquely obvious relation for this, but over the years several
different prescriptions have been suggested~\cite{ansatze}, leading to
different mass relations.   One element common to all these suggestions
is that the  $b$-quark  and  $\tau$-lepton  Yukawa couplings are equal at
$M_G$:
%
\begin{equation}
     \lambda_b(M_G) = \lambda_\tau(M_G)   \,. \label{lambda equal}
\end{equation}
%
The special kinship of $b$ and $\tau$ is that (i) both belong to the
heaviest generation and (ii) both have $T_3 = -1/2$.  In minimal SUSY
models they both couple to the same Higgs field $H_1$ and get masses from
the same vev $v_1 = v \cos\beta$.
If we define the ratio $R_{b/\tau}(\mu) = \lambda_b(\mu)/ \lambda_\tau(\mu)$,
then its RGE is
%
\begin{equation}
{dR_{b/\tau}\over dt} = {R_{b/\tau}\over16\pi^2} \left(-\sum d_i g_i^2
+\lambda_t^2 + 3\lambda_b^2 - 3\lambda_\tau^2\right)+\mbox{two-loop terms} \;,
\end{equation}
%
with $d_1=-4/3,\ d_2=0,\ d_3=16/3$.
The low-energy values at $\mu = m_t$  are fixed by
%
\begin{eqnarray}
\lambda_b(m_t) &=& {\sqrt2\, m_b(m_b)\over\eta_b v\cos\beta} \;, \\
\lambda_\tau(m_t) &=& {\sqrt2m_\tau(m_\tau)\over \eta_\tau v\cos\beta} \;, \\
\lambda_t(m_t) &=& {\sqrt2 m_t(m_t)\over v\sin\beta} \;,
\end{eqnarray}
%
where $\eta_f = m_f(m_f)/m_f(m_t)$ is a ratio of running masses.
%
The physical $\tau$ mass is $m_\tau(m_\tau) = m_{\tau}(\rm pole) =1.777$~GeV.
The running $b$ mass we take to be $m_b(m_b) = 4.25 \pm
0.15$~GeV~\cite{gasser}.

   The physical $m_b/m_\tau$ ratio constrains SUSY-GUT solutions and
essentially correlates $\tan\beta$ with $m_t$; Fig.~5 shows an example with
$M_{\rm SUSY} = m_t$ and $\alpha_3(M_Z)=0.11$.  For most $m_t$ values there are
two solutions for $\tan\beta$, with~\cite{bargut}
%
\begin{equation}
\begin{array}{rcl}
  \tan\beta &\agt& m_t/m_b  \qquad \mbox{[large $\tan\beta$ solution]} \;, \\
\sin\beta&\simeq&  (m_t^{\rm pole}/200\rm\ GeV)\qquad \mbox{[small $\tan\beta$
solution]}\;.
\end{array}
\end{equation}
%
These constrained SUSY-GUT solutions have an upper limit for $m_t$:
%
\begin{equation}
  m_t(m_t) \alt 192\rm\ GeV \quad {\rm or}\quad m_t^{\rm pole} \alt 200 \rm\
GeV   \;.
\end{equation}
%
They can successfully predict the correct $m_b/m_\tau$ ratio, unlike SM-GUTs.
They also allow a simple and theoretically attractive possibility
that the $t$-Yukawa coupling might be unified with the others at the GUT
scale if $\tan\beta$ is large:
%
\begin{equation}
\lambda_t(M_G) = \lambda_b(M_G) = \lambda_\tau(M_G)
\quad\mbox{if $\tan\beta\sim50$--60} \;.
\end{equation}

\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig4-5.eps}

{\small Fig.~5: Typical relation of $m_t$ and $\tan\beta$ in GUT
solutions~\cite{bargut}.}
\end{center}



\section{MSSM parameters and particles}

   The principal SUSY parameters are
\begin{enumerate}
\item $m_0$, the common scalar mass at scale $M_G$;
\item $m_{1/2}$, the common gaugino mass at scale $M_G$;
\item $\mu$, defined at low mass scale, which is related to Higgs and Higgsino
masses and is the Higgs mixing coefficient in the superpotential
%
\begin{equation}
  W = \lambda_t(tt^cH_2^0 - bt^cH_2^+) + \mu(H_1^0H_2^0 - H_1^-H_2^+) \;;
\end{equation}

\item  $m_A$, the mass of the CP-odd neutral Higgs boson;
\item  $\tan\beta = v_2/v_1$,  the ratio of vevs.
\end{enumerate}
There are also some soft SUSY-breaking coefficients in the scalar
potential that have limited effects and are sometimes set to zero at
scale $M_G$~\cite{rosrob,kelley}. The principal GUT-scale parameters $m_0$ and
$m_{1/2}$ are often
replaced by the low-energy parameters $m_{\tilde q}$ and $m_{\tilde g}$, the
squark and gluino masses; we then have five low-energy SUSY parameters:
\[
(m_{\tilde q},\ m_{\tilde g},\ \mu,\ m_A,\ \tan\beta) \;.
\]

   The non-SM particles and the principal parameters that affect their
masses are as follows:
\[
\vbox{\tabskip=2em
\halign{#\hfil&&\hfil$#$\hfil\cr
Higgs& h^0\ H^0\ A^0\ H^\pm& m_A,\, \tan\beta,\, (m_t)\cr
\noalign{\vskip4pt}
Gluinos& \tilde g& m_{1/2}\cr
Squarks& \tilde q& m_0,\,m_{1/2},\,\tan\beta\cr
Sleptons& \tilde\ell^\pm,\,\tilde\nu_\ell& m_0,\,m_{1/2},\,\tan\beta\cr
\noalign{\vskip4pt}
Charginos& \tilde\chi_i^\pm\ (i=1,2)& m_{1/2},\,\mu,\,\tan\beta\cr
Neutralinos& \tilde\chi_j^0\ (j=1,2,3,4)& m_{1/2},\,\mu,\,\tan\beta\cr}}
\]
Alternative notations for the chargino and neutralino mass eigenstates
are  $\tilde\chi_i^\pm = \tilde W_1^\pm,\, \tilde W_2^\pm$    and
$\tilde\chi_j^0 = \tilde Z_1,\,\tilde Z_2,\,\tilde Z_3,\,\tilde Z_4$,
in ascending order of mass.

   Figure 6 shows an example from Ref.~\cite{rosrob} of how the SUSY spectra
evolve downward from $\mu = M_G$ in a SUSY-GUT model; running masses
are plotted versus $\mu$ and the physical value occurs close to where the
running mass $m=m(\mu)$ intersects the curve $m=\mu$.  In the case of
the Higgs scalar $H_2$, the mass-square becomes negative at low $\mu$ due to
coupling to top;
in this region we have actually plotted $-|m(\mu)|$. Negative mass-square
parameter is essential for spontaneous symmetry breaking, so this
feature of SUSY-GUTs is desirable [see section~4.1, remark~(h)]; here it is
achieved by radiative effects.  The running masses for
   the gauginos $\tilde g,\, \tilde W,\, \tilde B$ are given by
%
\begin{equation}
             M_i(\mu) = m_{1/2} {\alpha_i(\mu)\over\alpha_i(M_G)}   \;,
\end{equation}
%
where $i$ labels the corresponding gauge symmetry; this applies before
   we add mixing with higgsinos.  In the example of Fig.~6 the squarks
   are heavier than gluinos, but the opposite ordering   $m_{\tilde q} <
   m_{\tilde g}$ is possible in other scenarios.
   Sleptons are lighter than both squarks and gluinos in general.
      Note that the usual soft SUSY-breaking mechanisms preserve the
   gauge coupling relations (unification) at $M_G$.

      In order that SUSY cancellations shall take effect at low mass
   scales as required, the SUSY mass parameters are expected to be
   bounded by
%
\begin{equation}
      m_{\tilde g},\, m_{\tilde q},\, |\mu|,\, m_A \alt 1\mbox{--2 TeV}\, .
\label{m bound}
\end{equation}
%
The other parameter $\tan\beta$ is effectively bounded by
%
\begin{equation}
          1  \alt    \tan\beta  \alt    85    \;,
\end{equation}
%
   where the lower bound arises from consistency in GUT models and the
   upper bound comes from proton decay constraints~\cite{pdecay}.


\begin{center}
%\epsfxsize=6in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig9.eps}


{\small Fig.~6: Evolution of SUSY spectra in a SUSY-GUT model~\cite{rosrob}.}
\end{center}


\section{SUSY particle production and detection}

      At LEP\,I, sufficiently light SUSY particles would be produced
through their gauge couplings to $Z$ and $\gamma$.  Direct searches for
SUSY particles at LEP give mass lower bounds~\cite{lep etal}
%
\begin{equation}
      m_{\tilde q},\, m_{\tilde \ell},\, m_{\tilde W_1}\agt45\rm\ GeV\;.
\end{equation}
%
Comparisons of SM predictions with observed $Z$ decay widths into hadrons,
leptons and invisible particles give bounds on possible SUSY width
contributions and branching fractions~\cite{lep etal}:
%
\begin{eqnarray}
      \Delta\Gamma_Z(\rm SUSY)   &<&   25\rm\ MeV   \;, \\
   \Gamma(Z\to \tilde Z_1 \tilde Z_1{:}\rm\: invisible) &<& 17\rm\ MeV \;, \\
      B(Z \to \tilde Z_1 \tilde Z_j)& <& \rm  few\times 10^{-5} \;.
\end{eqnarray}
%
The limitation of LEP is its relatively low CM energy.

      Hadron colliders can explore much higher energy ranges. Figure~7
   shows the lowest-order gluon-gluon, gluon-quark and quark-antiquark
   subprocesses for SUSY particle hadroproduction.  Figure~8 shows
   squark and gluino predictions for the Tevatron $p$-$\bar p$
collider~\cite{btw},
   assuming degenerate masses $m_{\tilde q} = m_{\tilde g}$ (summing
   $L$ and $R$ squarks plus antisquarks of all flavors).  The right-hand
   vertical axis shows the number of events for the luminosity 25~pb$^{-1}$
   expected in 1993; we see that about 100 events would be expected
   for each of the channels $\tilde g\tilde q$ and $\tilde q\tilde q$
   at mass 200~GeV, so the Tevatron clearly reaches well beyond the
   LEP range.

      The most distinctive signature of SUSY production is the missing
   energy and momentum carried off by the undetected LSP, usually
   assumed to be the lightest neutralino $\tilde Z_1$, which occurs in
   all SUSY decay chains with $R$-parity conservation.  At hadron
   colliders very many beam-jet particles escape down the beam-pipe;
   it is only possible to do book-keeping on the missing transverse
   momentum denoted $\overlay/p_T$. (In practice this is inferred from the
   imbalance of ``transverse energy" $E_T = E \sin\theta \simeq p_T$
   measured in calorimeters, so people often refer to missing-$E_T$
   or $\overlay/E_T$ instead).  The missing momenta of both LSPs are added
   vectorially in $\overlay/p_T$.  The LSP momenta and hence the magnitude
   of  $\overlay/p_T$ depend on the decay chains.


\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig10.eps}

{\small Fig.~7: SUSY production subprocesses at hadron colliders.}

\bigskip\bigskip

%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig11.eps}

{\small Fig.~8: Tevatron cross sections for  $\tilde g \tilde g$, $\tilde g
\tilde q$
and $\tilde q \tilde q$ production, versus squark/gluino mass.}
\end{center}



      If squarks and gluinos are rather light ($m_{\tilde g},m_{\tilde q}\alt
50$~GeV), their dominant decay
   mechanisms are direct strong decays or decays to the LSP:
%
\begin{eqnarray}
\left.\begin{array}{l}
      \tilde q \to q \tilde g\\
\tilde g \to q \bar q \tilde Z_1
\end{array} \right\}
&& {\rm if}\ m_{\tilde g} < m_{\tilde q}  \label{light a}\\
\left.\begin{array}{l}
           \tilde g\to q \tilde q\\
\tilde q\to  q \tilde Z_1
\end{array}\right\}
&&    {\rm if}\ m_{\tilde q} < m_{\tilde g} \label{light b}
\end{eqnarray}
%
   In such cases the LSPs carry a substantial fraction of the available
   energy and $\overlay/p_T$ is correspondingly large.  Assuming such decays
and small LSP mass, the present 90\%~CL experimental bounds from UA1 and UA2
   at the CERN $p$-$\bar p$ collider ($\sqrt s = 640$~GeV) and from CDF at the
   Tevatron ($\sqrt s = 1.8$~TeV) are~\cite{uacdf}
%
\[
\vbox{\tabskip2em\halign{#\hfil&&$#$\hfil\cr
                    &  \hfil  m_{\tilde g}  &  \hfil  m_{\tilde q}\cr
        UA1 (1987)      &     >  53\rm\ GeV  &      >  45\rm\ GeV \cr
        UA2 (1990)      &        >  79       &     >  74 \cr
        CDF (1992)      &        > 141       &     > 126 \cr}}
\]
   The limits become much more stringent if the squark and gluino masses
   are assumed to be comparable.


\section{Other SUSY decay modes and signatures}

   For heavier gluinos and squarks, many new decay channels are open,
such as decays into the heavier gauginos:
%
\begin{eqnarray}
  \tilde g  &\to& q \bar q \tilde Z_i\ (i=1,2,3,4),\ q \bar q' \tilde W_j\
                 (j=1,2),\ g \tilde Z_1 \;, \label{heavy a}\\
  \tilde q_L &\to& q \tilde Z_i\ (i=1,2,3,4),\ q' \tilde Wj\ (j=1,2)\;,
\label{heavy b}\\
  \tilde q_R &\to& q \tilde Z_i\ (i=1,2,3,4)\;. \label{heavy c}
\end{eqnarray}
%
Some decays go via loops (e.g.\ $\tilde g\to g \tilde Z_1)$; we have not
attempted an exhaustive listing here.
Figure~9  shows how gluino-to-heavy-gaugino branching fractions increase
with $m_{\tilde g}$ in a particular example
(with $m_{\tilde g} < m_{\tilde q}$)~\cite{bbkt}.

   The heavier gauginos then decay too:
%
\begin{eqnarray}
  \tilde W_j &\to& Z \tilde W_k,\, W \tilde Z_i,\, H_i^0 \tilde W_k,\,
                 H^\pm \tilde Z_i,\,f\tilde f   \;, \\
  \tilde Z_i &\to& Z \tilde Z_k,\, W \tilde W_j,\, H_i^0 \tilde Z_k,\,
                 H^\pm \tilde W_k,\, f\tilde f' \;.   \label{Ztwid decay}
\end{eqnarray}
%
Here it is understood that final $W$ or $Z$ may be off-shell and
materialize as fermion-antifermion pairs; also $Z$ may be replaced
by $\gamma$.  In practice, chargino decays are usually dominated by
$W$-exchange transitions (Fig.~10a); neutralino decays are often dominated
by sfermion exchanges (Fig.~10b) because the $\tilde Z_2 \tilde Z_1 Z$
coupling is small.  To combine the complicated production and cascade
possibilities systematically, all these channels have been incorporated in
the ISAJET 7.0 Monte Carlo package called ISASUSY~\cite{isa}.



\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig12.eps}

{\small Fig.~9: Example of gluino decay branchings versus mass~\cite{bbkt}.}
\end{center}

\vspace*{.2in}

\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig13.eps}

\vspace*{.2in}
\parbox{5.5in}{\small Fig.~10: Examples of (a)~chargino decay by $W$-exchange,
(b)~neutralino
decay by sfermion exchange.}
\end{center}


   These multibranch cascade decays lead to higher-multiplicity final
states in which the LSPs $\tilde Z_1$  carry a much smaller share of the
available energy, so $\overlay/p_T$ is smaller and less distinctive (Fig.~11),
making detection via $\overlay/p_T$ more difficult.
(Remember that leptonic $W$ or $Z$ decays,
$\tau$ decays, plus semileptonic $b$ and $c$ decays, all give background events
with genuine $\overlay/p_T$; measurement uncertainties also contribute fake
$\overlay/p_T$ backgrounds.)  Experimental bounds therefore become weaker when
we take account of cascade decays.  Figure~12 shows the CDF 90\% CL
limits in the $(m_{\tilde g}, m_{\tilde q})$ plane, in a particular example
with $\mu = -250$~GeV, $\tan\beta=2$ and $m_H=500$~GeV; the dashed curves are
limits assuming only direct decays (\ref{light a})--(\ref{light b}), while
solid curves are less restrictive limits including cascade decays~\cite{uacdf}.


The cascade decays also present new opportunities for SUSY detection.
 Same-sign dileptons (SSD) are a very interesting signal~\cite{ssd}, which
arises naturally from $\tilde g \tilde g$  and  $\tilde g \tilde q$  decays
because of the Majorana character of gluinos, with very
little background. Figure~13 gives an example of this signal.
Eqs.~(\ref{heavy a})--(\ref{Ztwid decay}) have shown how a heavy gluino or
squark can decay to a chargino
$\tilde W_j$ and hence, via a real or virtual $W$, to an isolated charged
lepton (meaning isolated from accompanying hadrons).  A gluino  can decay
equally into either sign of chargino and lepton
because it is a Majorana fermion.  Hence  $\tilde g \tilde g$  or
$\tilde g \tilde q$  systems can decay via $\tilde W_j$ to isolated SSD plus
jets plus $\overlay/p_T$. Cascade $\tilde q \tilde{\bar q}$ decays via one
of the heavier neutralinos $\tilde Z_i$ offer similar SSD
possibilities, since the $\tilde Z_i$ are also Majorana
fermions. Cross sections for the Tevatron are illustrated in Fig.~14.


\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig14.eps}

\bigskip

\parbox{5.5in}{\small Fig.~11: Typical $\overlay/p_T$ distributions from direct
and cascade decays of
gluino pairs at the Tevatron~\cite{btw}.}

\bigskip

%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig15.eps}

\parbox{5.5in}{\small Fig.~12: 1992 CDF limits in the $(m_{\tilde
g},\,m_{\tilde q})$ plane, with or
without cascade decays, for a typical choice of parameters~\cite{uacdf}.}
\end{center}


\begin{center}
%\epsfxsize=3in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig16.eps}

{\small Fig.~13: Example of same-sign dilepton appearance in gluino-pair
decay.}

\bigskip

%\epsfxsize=5in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig17.eps}

{\small Fig.~14: Same-sign dilepton signals at the Tevatron~\cite{baerev}.}
\end{center}


   Genuinely isolated SSD backgrounds come from the production of $WZ$ or
$Wt\bar t$ or $W^+W^+$ ({\it e.g.}\ $uu \to ddW^+W^+$ by gluon exchange), with
cross sections of order  $\alpha_2^2$   or   $\alpha_2\alpha_3^2$    or
$\alpha_2^2\alpha_3^2$  compared to $\alpha_3^2$ for gluino pair production,
so we expect to control them with suitable cuts.  Very large   $b \bar b$
production gives SSD via semileptonic $b$-decays plus $B$-$\bar B$ mixing, and
also via combined $b\to c\to s \,\ell^+ \nu$   and  $\bar b\to \bar c\,\ell^+
\nu$ decays, but both leptons are produced in jets and can be suppressed by
stringent isolation criteria.  Also $t\bar t$ gives SSD via $t\to b\, \ell^+
\nu$ and  $\bar t\to \bar b\to \bar c\, \ell^+ \nu$, but the latter lepton is
non-isolated.
So SSD provide a promising SUSY signature.

   Gluino production rates at SSC/LHC are much higher than at the Tevatron. At
$\sqrt s=40$~TeV, the cross section is
%
\begin{equation}
\sigma(\tilde g \tilde g) = 10^4,\,700,\, 6\;\mbox{fb\quad for }
 m_{\tilde g} = 0.3,\,1,\,2\rm\; TeV\;.
\end{equation}
%
Many different SUSY signals have been evaluated, including
$\overlay/p_T + n\,$jets, $\overlay/p_T +{}$SSD, $\overlay/p_T + n\,$isolated
leptons, $\overlay/p_T + {}$one isolated lepton${}+ Z$, $\overlay/p_T + Z$,
$\overlay/p_T + Z + Z$.
SSC cross sections for some of these signals from  $\tilde g \tilde g$
production are shown versus $m_{\tilde g}$ in Fig.~15 (for two scenarios,
after various cuts);  the labels 3,4,5 refer to numbers of isolated
leptons~\cite{btw}.  The signal and background cross sections (in pb)
from different channels~\cite{btw} are tabulated below.

\begin{center}
%\epsfxsize=6in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig18.eps}\hspace*{.25in}

{\small Fig.~15: SSC cross sections for various SUSY signals, after
cuts~\cite{btw}.}
\end{center}


{\footnotesize
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\rm process & \sigma(\rm tot) & SS & 3\ell & 4\ell & 5\ell & Z & Z+\ell & ZZ \\
\hline
\tilde g\tilde g\ (300) & 6000 & 10 & 3 & - & - & 3 & 0.3 & - \\
\tilde g\tilde g\ (1000) & 20 & 0.7 & 0.3 & 0.04 & 9{\times}10^{-3} & 2 &
0.05 & 8{\times}10^{-4} \\
\tilde g\tilde g\ (2000) & 0.25 & 6{\times}10^{-3} & 3{\times}10^{-3} &
5{\times}10^{-4} & - & 6{\times}10^{-3} & 9{\times} 10^{-4} &
1.5{\times}10^{-5} \\
t\bar t& 1.6{\times}10^4 & <0.2 & - & - & - & 387 * f & - & - \\
t\bar tt\bar t & 0.5 & 7{\times}10^{-3} & 3{\times}10^{-3} & 2{\times}10^{-4} &
- & 0.05*f & 0.01*f & 3{\times}10^{-4}*f^2 \\
t\bar tb\bar b & 124 & <0.003 & - & - & - & 3*f & - & - \\
Wt\bar t & 2.1 & 0.013 & 2{\times}10^{-3} & - & - & 0.1*f & 0.01*f & - \\
Wb\bar b & 341 & < 9{\times}10^{-4} & - & - & - & - & - & - \\
Zt\bar t & 12.5 & - & < 0.3 & < 0.04 & - & < 0.75 & < 0.33 & 0.02*f \\
W^\pm W^\pm & 1.33 & 0.12 & - & - & - & - & - & -\\
WZ & 80 & - & 1 & - & - & - & <1 & - \\
ZZ & 30 & - & - & - & - & < 0.65 & - & < 0.11 \\
WWW & 0.4 & < 0.02 & < 6.4{\times}10^{-3} & - & - & - & - & - \\
WWZ & 0.5 & - & <0.031 & < 1.5{\times}10^{-3} & - & - & < 0.01 & - \\
WZZ & 0.1 & - & <3{\times}10^{-3} & <4{\times}10^{-4} & <10^{-4} & - & - &
<3{\times}10^{-4} \\
ZZZ & 0.04 & - & <5{\times}10^{-4} & < 5{\times}10^{-4} & <10^{-5} & - & - &
<4{\times}10^{-4} \\
\hline
\end{array}
\]}


   The same-sign dilepton asymmetry, defined by
%
\begin{equation}
A =  {   \sigma(++) - \sigma(--)
\over     \sigma(++) + \sigma(--) }
\end{equation}
%
is an interesting probe of the  $\tilde g\tilde q/\tilde g \tilde g$
production ratio, and hence is a probe of the squark/gluino mass ratio.
This is because  $\tilde g \tilde g$ contributes equally to $++$ and $--$
dileptons but  $\tilde g \tilde q$ does not.  The excess of $u$ over $d$
quarks in the incident protons at a $p$-$p$ collider leads to an excess of
$\tilde g \tilde u$  over  $\tilde g \tilde d$  production and hence to
an excess of  $++$  over  $--$  SSD.  Some examples of the $++,--$ SSD cross
sections and their asymmetry follow~\cite{btw}; in each case the upper/lower
numbers refer to LHC/SSC.

\[ \begin{array}{|r|r|c|c|c@{}l|}
\hline
\multicolumn{1}{|c|}{m_{\tilde g}} & \multicolumn{1}{c|}{m_{\tilde q}} &
\sigma(++)\ \rm(pb) & \sigma(--)\ \rm(pb) & A &\\
\hline
260 & 250 & \begin{array}{c} 1.7 \\ 9.2\end{array} &
\begin{array}{c}0.4\\7.8\end{array} &
\begin{array}{c}0.6\\0.08\end{array} &
\begin{array}{l}\rm LHC\\ \rm SSC\end{array}\\
\hline
510 & 500 & \begin{array}{c}0.15\\0.9\end{array} &
\begin{array}{c}0.07\\0.6\end{array} &
\begin{array}{c}0.3\\0.2\end{array}&\\
\hline
760 & 740 & \begin{array}{c}0.03\\0.3\end{array} &
\begin{array}{c}0.012\\0.2\end{array} & \begin{array}{c}0.4\\0.2\end{array}&\\
\hline
1020 & 1000 & \begin{array}{c}4.5\times10^{-3}\\0.08\end{array} &
\begin{array}{c}1.8\times10^{-3}\\0.04\end{array} &
\begin{array}{c}0.4\\0.33\end{array} &\\
\hline
\end{array} \]



   Heavy gluinos can also decay copiously to $t$-quarks~\cite{btw,bps}:
%
\begin{equation}
\tilde g \to t \bar t\tilde Z_i , t \bar b \tilde W^-, b \bar t
\tilde W^+ \;.
\end{equation}
%
$t\to bW$  decay then leads to multiple $W$ production.  For example, for a
gluino of mass 1.5~TeV,  the  $\tilde g\to W,\, WW,\, WWZ,\, WWWW$ branching
fractions are typically of order 30\%, 30\%, 6\%, 6\%, respectively. Figure~16
illustrates SSC cross sections for multi-$W$ production via gluino pair
decays (assuming $m_{\tilde g} < m_{\tilde q}$).  We see that
for $m_{\tilde g}\sim 1$~TeV the SUSY rate for $4W$ production can greatly
exceed the dominant SM $4t\to 4W$ mode, offering yet another signal for
SUSY~\cite{bps}.

\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig19.eps}

{\small Fig.~16: Typical SSC rates for gluino pair production and decay to
multi-$W$ final states~\cite{bps}.}
\end{center}

\section{Summary}
\begin{enumerate}
\item The success of SUSY-GUT models in achieving gauge coupling unification
and predicting $m_b/m_\tau$ is tantalizing and encouraging.
\item These models have SUSY masses around or below 1 TeV.  Attempts are
being made to determine the SUSY mass spectra in more detail, using
the RGE evolution constraints.
\item Experimental SUSY particle searches have hitherto been based
largely on $\overlay/p_T$ signals.  But for $m_{\tilde g},\,m_{\tilde q}
> 50$~GeV cascade decays become important; these cascades both weaken the
simple $\overlay/p_T$
 signals and provide new signals such as same-sign dileptons, which will be
pursued at the Tevatron.
\item For even heavier squarks and gluinos, the cascade decays dominate
completely and provide further exotic (multi-$W,Z$ and multi-lepton)
signatures, which will be pursued at the SSC and LHC.
\item Gluinos and squarks in the expected mass range [Eq.~(\ref{m bound})] will
not escape detection.
\end{enumerate}

\renewcommand{\chapter}{\section}

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\frenchspacing

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\end{document}

 & - & - & - & - & - & - \\
Zt\bar t & 12.5 & - & < 0.3 & < 0.04 & - & < 0.75 & < 0.33 & 0.02*f \\
W^\pm W^\pm & 1.33 & 0.12 & - & - & - & - & - & -\\
WZ & 80 & - & 1 & - & - & - & <1 & - \\
ZZ & 30 & - & - & - & - & < 0.65 & - & < 0.11 \\
WWW & 0.4 & < 0.02 & < 6.4{\times}10^{-3} & - & - & - & - & - \\
WWZ & 0.5 & - & <0.031 & < 1.5{\times}10^{-3} & - & - &lecture5.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\setcounter{page}{55}
\chapter*{LECTURE 5:\\ SUSY HIGGS BOSONS}
\setcounter{chapter}{5}

\section{Scalar spectrum at tree level}

   In supersymmetry\cite{susy} we need at least two complex Higgs doublets
$\phi_1$ and $\phi_2$, to cancel anomalies and to give masses to both up- and
down-type quarks:
%
\begin{equation}
\phi_1 = \left(\begin{array}{c}\phi_1^0\\ \phi_1^- \end{array}\right)\,, \qquad
\phi_2 = \left(\begin{array}{c}\phi_2^+\\ \phi_2^0 \end{array}\right) \,.
\end{equation}
%
They have two independent vevs
%
\begin{equation}
\sqrt2 \left< \phi_1^0\right> = v_1 = v\cos\beta \,,  \qquad
\sqrt2 \left<\phi_2^0\right> = v_2 = v\sin\beta  \,,
\end{equation}
%
where $v = \left(\sqrt2 \,G_F\right)^{-1/2} = 246$~GeV is the SM vev;
thus they are
constrained by $v_1^2 + v_2^2 = v^2$  and $\tan\beta = v_2/v_1$ parameterizes
their ratio.  The vev $v_1$ gives masses to the charged leptons and charge-1/3
quarks, $v_2$ gives masses to the charge-2/3 quarks,
%
\begin{equation}
m_\tau=\lambda_\tau v_1/\sqrt2,\qquad m_b=\lambda_b v_1/\sqrt2, \qquad
m_t=\lambda_t v_2/\sqrt2\,,
\end{equation}
%
where the $\lambda_i$ are the respective Yukawa couplings.

The two complex scalar doublets contain 8 real fields, leading to
%
\begin{center}

\begin{tabular}{l@{\ }l}
3 neutral Higgs bosons: & 2 CP-even scalars $h$ and $H\ (m_h < m_H)$\\
                        & 1 CP-odd pseudoscalar $A$\\
2 charged Higgs bosons: & $H^{\pm}$\\
3 Goldstone bosons: & $z^0,w^{\pm}$ that become the longitudinal polarization
  states $Z_L,W_L^{\pm}$.
\end{tabular}
\end{center}

   The minimal supersymmetric extension of the SM (MSSM) is a two-Higgs
doublet model with supersymmetric constraints, that give a tree-level
scalar potential with quartic couplings related to the SU(2)$\times$U(1) gauge
couplings:
%
\begin{equation}
V_0 = m_1^2|\phi_1|^2 + m_2^2|\phi_2|^2 - m_{12}^2
\left(\phi_1^\dagger\phi_2+{\rm h.c.}\right)
+{g^2\over8} \left(\phi_1^\dagger \vec\sigma \phi_1 + \phi_2^\dagger \vec\sigma
\phi_2 \right)^2 + {g'^2\over8} \left(|\phi_2|^2 - |\phi_1|^2
\right)^2\end{equation}
%

The scalar mass-squared matrix is the matrix of second derivatives of $V$
with respect to the eight real scalar fields, evaluated at the potential
minimum where the first derivatives vanish. The CP-even sector has a 2$\times$2
mass matrix,
%
\begin{equation}
{\cal M}^2_{\rm CP\ even} = \left( \begin{array}{cc}
m_A^2\sin^2\beta+M_Z^2\cos^2\beta & -{1\over2}(m_A^2+M_Z^2)\sin2\beta \\
-{1\over2}(m_A^2+M_Z^2)\sin2\beta & m_A^2\cos^2\beta+M_Z^2\sin^2\beta
\end{array}\right)
\end{equation}
%
diagonalized by a mixing angle $\alpha$, with eigenvalues $m_h^2$ and $m_H^2$.
All masses and $\alpha$ are determined at tree level by just two
parameters $m_A$ and $\tan\beta$ (with $0 \leq \beta \leq {\pi \over2}$) as:
%
\begin{eqnarray}
\tan 2\alpha &=& \frac{m_A^2 + M_Z^2}{m_A^2 - M_Z^2} \tan 2\beta \;, \\
m_h^2,m_H^2  &=& {1\over 2}\left[M_Z^2 + m_A^2
\mp \sqrt{\left(M_Z^2 +m_A^2\right)^2- 4 M_Z^2 m_A^2 \cos^2 2\beta}\,\right]
\;, \\
m_{H^\pm}^2   &=& M_W^2 + m_A^2 \;.
\end{eqnarray}
%
with $-{\pi \over 2} \leq \alpha \leq 0$.  Hence at tree level
%
\begin{equation}
m_h \leq M_Z,m_A ; \quad  m_H \geq M_Z ;\quad  m_{H\pm} \geq M_W, m_A \;.
\end{equation}
%
   There are no tight theoretical constraints on the parameters $m_A$
and $\tan\beta$, but supergravity models lead one to expect
%
\begin{equation}
    1 \alt  \tan \beta  \alt  m_t/m_b
\end{equation}
%
and $m_A$ is related to the soft SUSY-breaking parameter $m_{12}$ in the
potential by
%
\begin{equation}
  m_A^2 = m_{12}^2 (\cot \beta + \tan \beta) \,.
\end{equation}
%
Hence if $m_A \sim {\cal O}(M_{\rm SUSY}) \gg M_Z$, $A,H,H^{\pm}$
are all approximately degenerate and much heavier than $Z$, while only
$h$ is lighter than $Z$.

\section{One-loop Radiative Corrections}

   Because $m_t$ is large, big radiative corrections are induced by top
and stop loops like Fig.~1~\cite{loop}.  Incomplete cancellation between
them yields neutral Higgs mass corrections of order $\delta m^2 \sim
\lambda_t^4 v_2^2 \ln(m_{\tilde t}/m_t) \sim g^2 m_t^4 /(M_W^2 \sin^2 \beta)
\ln(m_{\tilde t}/m_t)$.
The full expressions depend on several soft SUSY-breaking mass and
mixing parameters, but the most important is the squark mass $m_{\tilde
t}$; in the approximation of neglecting the others, the CP-even mass-squared
matrix above simply receives a correction in the 22-element,
%
\begin{equation}
\Delta {\cal M}_{22}^2 = \epsilon / \sin^2 \beta , \qquad
\epsilon = \frac{3g^2 m_t^4}{4\pi^2 M_W^2} \ln(m_{\tilde t}/m_t)\,.
\end{equation}
%

\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{fig5-1.eps}

{\small Fig.~1: Typical top and stop 1-loop graphs.}
\end{center}

\noindent
The effect of these one-loop corrections are as follows.

\begin{enumerate}

\item The $h$ and $H$ masses are shifted upwards and their bounds become
%
\begin{equation}
m_h <  \sqrt{M_Z^2 + \epsilon} < m_H \,.
\end{equation}
%
For the case that $m_t = 150$~GeV, $m_{\tilde t} = 1$~TeV, this gives
$m_h < 116~{\rm GeV} < m_H$.

\item The mixing angle $\alpha$, which affects $h$ and $H$ couplings to
fermions
and gauge bosons, is increased to a more negative value.

\item There are also corrections of order $\epsilon /M_W^2$ to cubic
$hAA, HAA, Hhh$ couplings.

\item Corrections to $m_{H^\pm}$ are small.

\end{enumerate}

\noindent
Figure~2 compares the tree-level and one-loop behaviours of the
$h,H,H^{\pm}$ masses versus $m_A$ at fixed $\tan \beta$, for $m_t=150$~GeV
and $m_{\tilde t}=1$~TeV.  Figure~3 shows the sensitivity of $m_h$ and $m_H$
contours in the $(m_A, \tan \beta)$ plane to the value of $m_t$.

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{fig5-2.eps}

\parbox{5.5in}{\small Fig.~2: Comparison of mass formulas: (a)~tree-level $m_h$
and $m_H$, (b)~one-loop $m_h$ and $m_H$, (c)~$m_{H^\pm}$ at tree and one-loop
levels~\cite{bbsp}.}

\bigskip

%\epsfxsize=5.5in
\hspace{0in}
%\epsffile{fig5-3.eps}

\medskip

\parbox{5.5in}{\small Fig.~3: $m_h$ and $m_H$ contours in the $(m_A, \tan
\beta)$ plane, for~(a)
$m_t=150$~GeV, (b)~$m_t=200$~GeV, with $m_{\tilde t} = 1$~TeV in both
cases\cite{bcps}.}
\end{center}

   Compared to the well known SM Higgs couplings to fermions and
gauge bosons, $L = -(m_i/v)\bar f_i f_i H + {1\over2}g^2vWWH
+ {1\over4}(g^2 + g'^2)vZZH$, the MSSM neutral Higgs couplings
have the following extra factors:
%
\[
\arraycolsep=1.5em
\begin{array}{|c|rcc|}
\hline
& \multicolumn{1}{c}{h} & H & A\\
\hline
t\bar t & \cos\alpha/\sin\beta & \sin\alpha/\sin\beta & -i\gamma_5\cot\beta\\
b\bar b,\,\tau\bar\tau &  -\sin\alpha/\cos\beta & \cos\alpha/\cos\beta&
-i\gamma_5\tan\beta\\
WW,\,ZZ &  \sin(\beta-\alpha) & \cos(\beta-\alpha) & \\
ZA & \cos(\beta-\alpha) & \sin(\beta-\alpha) & \\
\hline
\end{array}
\]
{}From this we see that $h$ and $H$ share their coupling strengths, that
$A$ does not couple directly to $WW$ or $ZZ$, and that for large $\tan\beta$
the $b$ and $\tau$ couplings are enhanced over SM values (indeed, the $b$
couplings may exceed the $t$ couplings).  These non-SM factors affect the
production, decay and eventual detection of Higgs bosons.

   The coupling factors above vary enormously across the $(m_A, \tan
\beta)$ plane, as shown by the contour plots in Fig.~4.  There are two
particular regions that resemble the SM:

\begin{enumerate}
\item[(i)] $m_A$ large ($\agt 300$ GeV).  Here  $A, H, H^{\pm}$  become
approximately
degenerate, very heavy and largely irrelevant, while all the $h$ couplings
approach SM values; Fig.~4(a) shows contours, to the right of which the
various  $h$  couplings are within 10\% of the SM.

\item[(ii)] $m_A$ small ($\alt 100$ GeV) and $\tan \beta$ large ($\agt 5$).
Here the $H$ couplings to $\bar tt, WW$ and $ZZ$ approach SM values (within a
$\pm$ sign)
while $h$ and $A$ couplings are suppressed.  This rule is inverted however
for $\bar bb$ couplings, which are enhanced for $h$ and $A$ instead, so
the resemblance to the SM is incomplete.
\end{enumerate}

   The charged scalars $H^{\pm}$ have no tree-level coupling to $WZ$. Their
couplings to typical fermions are given by
%
\begin{eqnarray}
L &=& \frac{g}{2 \sqrt 2 M_W}H^+ \Bigl[
             m_t V_{tb} \cot \beta \; \bar     t(1 - \gamma_5)b
           + m_b V_{tb} \tan \beta \; \bar     t(1 + \gamma_5)b\nonumber\\
&&\hspace{1in} {}+ m_c\cot\beta\, V_{cs}\bar c(1-\gamma_5)s + m_{\tau}\tan
\beta \; \bar \tau (1 + \gamma_5)\nu \Bigr]\,.
\end{eqnarray}


   New decay modes such as $H \to hh,$ $H \to AA$, $H \to AZ$, $A \to ZH$ also
appear in the MSSM, competing with and suppressing SM modes.  The mode
$h \to AA$ is  kinematically forbidden at tree level but becomes allowed
at one-loop (compare Figs.~2a,2b).  Figure~5 shows parameter regions where
these new modes are allowed.  We see that $H \to hh$ is allowed (and usually
dominates) almost everywhere, except in the shaded region of Fig.~5
and near a line of zeros in the $Hhh$ coupling denoted by $f_h = 0$.  We
conclude that SM signals like $H \to \gamma \gamma$ and $H \to ZZ \to 4\ell$
will be suppressed for wide ranges of parameters, being replaced by
non-SM signals like $H \to hh \to \bar bb \bar bb$.  Even the lightest
scalar $h$ will have important non-SM decays  $h \to AA \to 4\,$jets (etc.) for
sufficiently small $m_A$ and $\tan \beta$. There may also be decays into
SUSY-partner states\cite{baersusy}, such as $h\to\tilde Z_1\tilde Z_1$ decays
to invisible LSP pairs, that we shall mostly ignore in this section.

   The total decay widths of the various Higgs scalars, summed over all
contributing channels, are compared with the SM width in Fig.~6. There
are strong dependences on $\tan \beta$.  Branching fractions are shown
in Fig.~7.


\begin{center}
%\epsfxsize=6in
\hspace{0in}
%\epsffile{fig5-4.eps}

\parbox{5.5in}{\small Fig.~4: Variation of coupling factors across the $(m_A,
\tan \beta)$ plane for $m_t = 150$~GeV, $m_{\tilde t} = 1$~TeV:
(a)~regions where $h$ couplings approach SM values; (b)--(f) contour plots
of coupling factors as labeled.\cite{bcps}}
\end{center}


\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{fig5-5.eps}

\medskip

{\small Fig.~5: Regions where non-SM decay modes compete\cite{bcps}.}
\end{center}

\vspace{.5in}

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{fig5-6.eps}

\medskip
\parbox{5.5in}{\small Fig.~6: Total scalar decay widths versus mass for
(a)~$\tan \beta =2$,
(b)~$\tan \beta = 30$, with $m_t=150$~GeV, $m_{\tilde t}=1$~TeV\cite{bcps}.}

\bigskip

%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{fig5-7.eps}

%\parbox{5.5in}
{\small Fig.~7:  Branching fractions for MSSM Higgs scalars, versus mass, for
$\tan \beta = 2$ and 30\cite{bcps}.}
\end{center}

\noindent
It is clear that MSSM Higgs phenomenology is a new story.


\section{MSSM Higgs searches at LEP\,I and LEP\,II}

   The principal production and decay modes at LEP\,I, operating on the $Z$
resonance, are expected to be
%
\begin{eqnarray}
   e^+e^- &\to& Z \to Z^* h, Ah  \,,\\
    Z^*   &\to& \ell \ell, \nu \nu , jj \,,\\
   h, A   &\to& \tau \tau, jj  \,.
\end{eqnarray}
%
where $j$ denotes a quark jet. The $Z^* h$ and $Ah$ cross sections contain
complementary coupling factors $\sin^2(\beta-\alpha)$ and
$\cos^2(\beta-\alpha)$,
respectively, so one or the other should be unsuppressed if $A$ and $h$ are
light enough to be produced.   $H$ and $H^{\pm}$ are expected to be too
heavy to be observed at LEP\,I.
Searches by the four LEP experiments\cite{lep} have
set lower limits  $m_h, m_A \agt 20$--45~GeV, depending on $\tan \beta$ and
on $m_t, m_{\tilde t}$.  Figure~8 shows limiting curves in the $(m_A, \tan
\beta)$ plane, based on the ALEPH data, showing that the limits are
quite sensitive to $m_t$.

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{fig5-8.eps}

\parbox{5.5in}{\small Fig.~8: Regions excluded by ALEPH data at LEP\,I, for
various $m_t$ values
and $m_{\tilde t} = 1$~TeV: (a)~$Z \to Z^* h$ search, (b)~$Z \to Ah$
search\cite{bcps}. The regions inaccessible for $m_t=150$~GeV are shaded.}
\end{center}




   At the future upgraded LEP\,II, operating at $\sqrt s \sim 170$--200 GeV,
the same channels will be searched but much larger regions will be
accessible.  There is also a small chance of $e^+e^- \to AH$ or $H^+ H^-$
production;  Figs.~3 and 9, giving $H$ and $H^{\pm}$ mass contours, show
there are some corners not excluded by LEP\,I where these channels
would be open.  LEP\,II could detect $H^+H^-$ production with $H^{\pm}
\to cs, \tau \nu$ decays, but only for $m_{H \pm} \alt 80$~GeV, so this signal
would be marginal at best.  The best chances are with $Ah$ and $Zh$
production (with $Z$ now a real, on-shell resonance).  Several groups
have studied prospects for LEP\,II searches\cite{bcps,baer,gunion,kz};
there are also internal unpublished appraisals by the LEP experimental groups
themselves.

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{fig5-9.eps}

\parbox{5.5in}{\small Fig.~9: $H^{\pm}$ mass contours in the $(m_A, \tan
\beta)$ plane for
(a)~$m_t=150$~GeV, (b)~$m_t=200$~GeV, with $m_{\tilde t}=1$~TeV\cite{bcps}.}
\end{center}

\noindent
Figure~10 shows discovery limits from Ref.\cite{bcps} based on
(a)~adapting published SM $e^+e^- \to ZH_{\rm SM}$ simulations\cite{wu}
to the $Zh$ case;
(b)~rescaling published $e^+e^- \to Zh,Ah \to \tau \tau jj$ signal and
background calculations at $\sqrt s = 500$~GeV\cite{janot} to LEP\,II.
The results are optimistic in that they assume CM energy $\sqrt s =
200$~GeV and luminosity ${\cal L}=500\rm\,pb^{-1}$, whereas LEP\,II
may not run above
$\sqrt s=190$~GeV and may take years to reach such luminosity. But
they are conservative in that they cover only a fraction of the
detectable decay channels and take no account of the improvements
obtainable with efficient $b$-tagging.

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig22.eps}

\medskip

\parbox{5.5in}{\small Fig.~10: Approximate LEP\,II discovery limits: (a)~$Zh$
channel, (b)~$Ah,Zh \to
\tau \tau jj$ search, assuming $\sqrt s = 200$~GeV and $\L =
500$~pb$^{-1}$~\cite{bcps}.}
\end{center}

   The Higgs search generally consists of looking for new peaks in
distributions of invariant mass $m(\tau \tau)$ or $m(jj)$.  For $S$ signal
events and $B$ background events in a given mass bin, the statistical
significance of the signal is $S/\sqrt B$.  In the LEP\,II calculations,
we add the $m(\tau \tau)$ and $m(jj)$ distributions and require $S/\sqrt B
> 4$ with $S > 4$ in a 10~GeV bin centered at the Higgs mass.
Exceptionally,  if the Higgs peak is close to $M_Z$, it will not give
a new resolved peak but only an enhancement in the expected $Z$ peak
from $ZZ$ background. Here $B$ is the expected
number of $Z$ counts from $ZZ$ background, $S$ is the number of additional
($Z$ or Higgs) counts from the $Zh$ or $Ah$ channels, and we ask for higher
significance $S/\sqrt B > 6$ with $S > 5$ because this signal depends
entirely on getting an accurate normalization.

   The $e^+e^- \to \tau \tau jj$ channel deserves some comments.
\begin{enumerate}
\item Both $\tau$ momenta are reconstructable in principle, since the
missing neutrino momenta are approximately collinear with the charged
decay products, and their vector sum can be found from transverse
momentum balance in each event. Thus the two unknown magnitudes of
neutrino momentum are determined by the two measured components
of missing-$p_T$.  Hence $m(\tau \tau)$ can be reconstructed.

\item Branching fractions are reasonable for the Higgs signals, small for
the $ZZ$ background:
%
\begin{equation}
B(Ah \to \tau \tau jj) \sim 0.16\,, \quad
B(Zh \to \tau \tau jj) \sim 0.09\,. \quad
B(ZZ \to \tau \tau jj) \sim 0.05\,,
\end{equation}
%
This compares favorably with  other popular channels:
%
\begin{equation}
\begin{array}{r@{\qquad}r}
B(Zh \to eejj)       \sim 0.03\,, &  B(ZZ \to eejj)\sim 0.05\,,\\
B(Zh \to \nu \nu jj) \sim 0.18 \,, & B(ZZ \to \nu \nu jj) \sim 0.28\,.
\end{array}
\end{equation}

\item With $b$-tagging the signal could be cleaned up even more since
%
\begin{equation}
B(Ah \to \tau \tau bb) \sim 0.16\,, \quad
B(Zh \to \tau \tau bb) \sim 0.04\,, \quad
B(ZZ \to \tau \tau bb) \sim 0.01\,,
\end{equation}
%
though the suppression of background would be offset by a lower signal
rate due to a tagging-efficiency factor.
\end{enumerate}

\section{MSSM Higgs searches at SSC/LHC}

   Several groups have studied the detectability of MSSM Higgs signals
at SSC/LHC\cite{bcps,baer,gunion,kz}.  For neutral scalars, the most
promising production and decay channels are direct analogs of the best
SM modes (see Lecture~2), but charged scalar signals are new, as follows.

\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}

\item\label{untagged} Untagged two-photon signals: $pp \to (h,H,A) \to \gamma
\gamma$,
suitable for intermediate masses above about 80~GeV. Production is
dominated by $gg$ fusion; both production and decay go mainly via heavy quark
loops, see Fig.~11a.

\item Lepton-tagged two-photon signals: $pp \to W(h,H) \to \ell \gamma \gamma$
or $pp \to \bar tt(h,A,H) \to \ell \gamma \gamma$, where an isolated
high-$p_T$ tagging lepton comes from $W \to \ell \nu$  or $t \to bW \to b
\ell \nu$; see Figs.~11b,11c.

\item Four-lepton signal (the ``gold-plated" signature): $pp \to H \to ZZ
\to \ell^+ \ell^- \ell^+ \ell^-$  as in Figs.~11d,11e.

\item $H^{\pm}$ signals from $H^\pm \to  \tau \nu ,c  s$, principally
recognizable
by excess $\tau$ production (violating lepton universality) or by a dijet
mass peak, respectively. These signals depend on $pp \to \bar tt$ production,
tagged by one standard semileptonic decay $t \to bW \to b \ell \nu$, with
$t \to bH$ decay of the other top quark; e.g.\ Fig.~11f.

\end{enumerate}

\begin{center}
%\epsfxsize=5.5in
\hspace{0in}
%\epsffile{fig5-11.eps}

\bigskip

\parbox{5.75in}{\small Fig.~11: Typical Higgs signal processes at SSC/LHC:
(a)~untagged
two-photon; (b) and (c),~lepton-tagged two-photon; (d) and (e),~four-lepton;
(f)~charged-Higgs gives excess~$\tau$.}
\end{center}


\noindent
Signals (i)--(iii) are like SM signals but generally smaller (see Fig.~12);
(iv)~is definitely non-SM.

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{/ed/Barger/Brazil/SUSY/p17fig.eps}

\medskip

\parbox{5.5in}{\small Fig.~12: Comparison of MSSM two-photon $h$ signals with
SM values at SSC energy; (a)~$\tan \beta = 2$, (b)~$\tan \beta =
30$\cite{bbsp}.}
\end{center}

   To evaluate the untagged two-photon signals at the SSC,
Ref.~\cite{bcps} takes acceptance cuts and background estimates from the
GEM proposal\cite{gem}, assumes excellent mass resolution
$\Delta m(\gamma \gamma)/m(\gamma \gamma) = 1.5\%$, and requires
signal significance $S/\sqrt B > 4$ in a 1\% mass bin.  Figure~13 shows
the regions where untagged $h,H,A \to \gamma \gamma$ signals would be
significant for luminosity 20~fb$^{-1}$ (two years running at the design
luminosity).  The $h$ signal is detectable where $m_h \agt 80$~GeV (within
the $h \simeq H_{\rm SM}$ region) but is swamped by backgrounds at lower $m_h$.
The $H$ signal is restricted by competition from $H \to hh$ (see Fig.~5).
The $A$ signal depends (for both production and decay loops) on the $Att$
coupling , which falls as $\tan \beta$ rises; competition from $A \to \bar tt$
also gives suppression for $m_A > 2m_t$.

\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig27.eps}

\parbox{5.5in}{\small Fig.~13: Detectability regions for untagged $h,H,A \to
\gamma \gamma$  signals at the SSC\cite{bcps}.}
\end{center}

   Lepton-tagged two-photon signals have been evaluated with reference
to the SDC detector performance\cite{sdc}, assuming mass resolution
$\Delta m(\gamma \gamma) = 3$~GeV plus appropriate acceptance cuts
and detection efficiencies\cite{bcps}. The background is smaller here,
because of the tagging requirement, but so too are the signals.
Figure~14 shows the resulting discovery limits for $h$ and $H$ signals,
requiring $S/\sqrt B > 4$ in $\pm 4$~GeV mass bins with luminosity
20\rm~fb$^{-1}$.  The $h$ signal region is similar to the untagged case, the
$H$ region is smaller, and there is no viable $A$ signal at all.

   Four-charged-lepton signals (with $\ell = e, \mu$) are the cleanest
way to identify heavy SM Higgs bosons. In the MSSM, $h$ is too light to
decay into $ZZ$ or appreciably into $ZZ^*$ and $A \to ZZ$ is forbidden at
tree level, so only an $H \to ZZ \to 4 \ell$ signal can arise.  The latter
is reduced however whenever $m_H < 2M_Z$, or the $HZZ$ coupling factor
$\cos(\beta-\alpha)$ is small, or $H \to hh$ or $\bar tt$ decay competes, etc.
The remaining discovery region is shown in Fig.~15, if we require
$S/\sqrt B > 4$  with $\L=20fb^{-1}$ in mass bins $m_H \pm 2\sigma$ (where
the effective mass resolution $\sigma$ is a combination of detector
resolution and intrinsic decay width).  For details see Ref.~\cite{bcps}.


\begin{center}
%\epsfxsize=3.9in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig28.eps}


{\small Fig.~14: Detectability regions for tagged $h,H \to \gamma \gamma$
signals   at the SSC\cite{bcps}.}
\end{center}

\smallskip

\begin{center}
%\epsfxsize=3.9in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig29.eps}


{\small Fig.~15: Detectability regions for $H \to 4\ell$ signals at the
SSC\cite{bcps}.}
\end{center}


   Charged-Higgs signals depend on $\bar tt$ production with $t \to bH^+$
decay.  For $\tan \beta > 1$ and $m_{H^\pm}<m_t-m_b$ the mode $t \to bH \to b
\tau \nu$ has a big branching fraction. Then if we can select a sample
of $\bar tt$ events, tagged by semileptonic $t \to b \ell \nu$ decay of
both top quarks (here $\ell = e, \mu , \tau$), purely SM decays would predict
equal numbers of $e \mu, e \tau$ and $\mu \tau$ events; but the presence
of the charged-Higgs mode would be signalled by an excess of $e \tau$ and
$\mu \tau$ over $e \mu$ events.  (In principle there would be an even
greater excess of $\tau \tau$ over $ee$ or $\mu \mu$ events, but in practice
$\tau$ leptons are hard to identify and cost severe efficiency factors at
a hadron collider, so we cannot work with more than one.)  The strategy
is therefore to select isolated two-lepton events, with the
characteristics of $\bar tt$ production, and look for a $\tau$ excess.
Figure~16 shows discovery regions from Ref.~\cite{bcps}, based on
identifying $\tau \to \pi \nu$  decays with branching fraction 11.5\%,
and requiring $S/\sqrt{S + B} > 5$  for $\L=20$~fb$^{-1}$, with SDC cuts and
efficiencies\cite{sdc}.

\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig30.eps}

{\small Fig.~16: Charged-Higgs detectability regions at the SSC\cite{bcps}.}
\end{center}

\section{Combined LEP\,I, LEP\,II and SSC/LHC searches}

   If we combine the detectability regions of LEP\,I, LEP\,II and SSC/LHC
searches, almost the whole of the $(m_A, \tan\beta)$ plane is covered;
i.e.\ at least one of the MSSM Higgs bosons is likely to be detected
somewhere, somehow.  However, a small inaccessible region remains,
illustrated in Fig.~17 for $m_t=150$~GeV, $m_{\tilde t}= 1$~TeV; all studies
broadly agree on this region, although there are differences of detail
due to differences in assumptions about signal significance, luminosity,
etc.\cite{bcps,baer,gunion,kz}.  The region is similar but wider for
$m_t=120$~GeV, narrower for $m_t=200$~GeV; see Fig.~18.  Notice that in
the case $m_t=200$~GeV, the limits of detectability are all defined
by SSC/LHC searches.

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{/ed/Barger/Japan_susy/Proceedings/fig24.eps}

\medskip

\parbox{5.5in}{\small Fig.~17: Inaccessible region compared with individual
detectability limits  from (a)~LEP and (b)~SSC/LHC\cite{bcps}.}

\bigskip

%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{fig5-18.eps}

{\small Fig.~18: Inaccessible regions for $m_t=120$, 200~GeV\cite{bcps}.}
\end{center}

   Figure 19 shows an expanded view of the inaccessible region for
$m_t=150$~GeV, with $h,\ H$ and $H^\pm$ mass contours superimposed.  It is a
region
where all the bosons have intermediate masses:
%
\begin{equation}
\begin{array}{r@{\qquad}r}
 m_h \sim  \mbox{\phantom080--116 GeV},  & m_A \sim \mbox{100--160 GeV}, \\
 m_H \sim \mbox{120--160 GeV}, & m_{H^\pm} \sim \mbox{120--160 GeV}.
\end{array}
\end{equation}
%

\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig5-19.eps}

\medskip

\parbox{5.5in}{\small Fig.~19: Expanded view of the inaccessible region for
$m_t=150$~GeV,
$m_{\tilde t}=1$~TeV; $h,\ H$ and $H^\pm$ mass contours are
superimposed\cite{bcps}.}
\end{center}

\noindent
Why then are they all undetectable here? In each case there are good
reasons.  $h$ is too heavy for LEP\,II; the $h \to \gamma \gamma$ signal at
SSC/LHC is suppressed by the $htt$ and $hWW$ coupling factors. Similar
remarks apply to $A$ and $H$.  $H^{\pm}$ signals are reduced as $m_{H^\pm}$
approaches $m_t$.  It is just a coincidence that none are detectable in
this region. (If $H,A \to \tau \tau$ signals could be
distinguished from backgrounds, they would cover part of the
inaccessible region with $\tan \beta \gtap 10$--20~\cite{kz}; there
are also recent hopes that efficient $b$-tagging could make $h(H)
\to \bar bb$ signals viable in the $pp \to \bar tth(H)$ channels\cite{dai};
however these questions are controversial and still lack detailed
calculations.)

   On the positive side, we emphasize that several different Higgs
bosons may be simultaneously detectable, in some parameter regions.
Figure~20 summarizes where the different bosons may be found. If
only $h$ is found, in the $h \simeq H_{\rm SM}$ region, the MSSM will not
be distinguished from the SM. But if the $h$ couplings differ
significantly from SM values, or more than one neutral boson is
found, or $H^{\pm}$ is found, there will be clear evidence for new
physics.

   We have neglected SUSY decays like the invisible modes $h,H \to
\tilde Z_1 \tilde Z_1$.  Such modes would reduce the standard signals
discussed above and could somewhat expand the inaccessible region.
However they could also offer new signals. In particular, it is
claimed that  $H,A \to \tilde Z_2 \tilde Z_2 \to 4\ell + \tilde Z_1
\tilde Z_1$ (i.e.\ $4\ell + {}$missing-$p_T$) signals could be detectable
at SSC/LHC in the region $m_A \sim 200$--300~GeV for $\tan \beta
\alt 5$\cite{baersusy}.  This would not cut into the inaccessible
region but would be interesting in its own right.


\begin{center}
%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig32.eps}


{\small Fig.~20: Potential discovery regions for various Higgs
bosons~\cite{bcps}.}
\end{center}


   Finally, we may consider various theoretical constraints that
might reduce or eliminate the inaccessible region.  There are at
least two interesting approaches of this sort.

\renewcommand{\labelenumi}{(\alph{enumi})}
\begin{enumerate}

\item $b \to s \gamma$ decay.  This branching fraction is sensitive to
contributions from charged Higgs boson loops (see Lecture~6).
If we ignore contributions from chargino, neutralino and gluino loops,
the present upper limit $B(b \to s \gamma) < 5.4 \times 10^{-4}$\cite{cleo}
gives a lower bound on $m_{H^\pm}$ for given $\tan \beta$
(Fig.~21) that excludes a large part of the $(m_A, \tan \beta)$
plane (Fig.~22).  The curves shown here, updating the calculations
of Ref.~\cite{bbp} to the latest CLEO result, appear to exclude the whole
of the inaccessible region, but it is premature to reach such a strong
conclusion.  There are questions of parameter choices and various small
corrections to consider\cite{bbp,jlh,corr}.  More importantly, the
contributions of SUSY loop diagrams (especially chargino loops) may be
significant and can enter with either sign\cite{bertolini}; if they
contribute destructively, they will weaken the bounds.  However, as
theoretical constraints on SUSY particles become more extensive, and as
the $B(b \to s \gamma)$ bound itself becomes stronger, we may expect this
approach to give valuable constraints on MSSM phenomenology.


\item $m_t$ fixed-point constraint.  In SUSY-GUT models, there are
two branches of solutions for $m_t \alt 170$~GeV, one with
large $\tan \beta$ and one with small $\tan \beta$, the latter being
controlled by the $m_t$ infrared fixed point and giving the relation
$\sin \beta \simeq m_t/(200$~GeV); see Lecture~4.  The fixed-point
solutions are quite interesting from a theoretical viewpoint, since
they do not have to approach perturbative limits, arise automatically
if the Yukawa coupling $\lambda_t$ is large at the GUT scale, and
have other attractive features.  If one can successfully argue that
this is the correct branch, and if also $m_t \alt 170$~GeV, it will
follow that $\tan \beta$ is small and the inaccessible region is
excluded.
If in addition we make the slightly stronger
assumption that $m_t < 160$~GeV with $m_{\tilde t} \simeq 1$~TeV, the
present LEP\,I data already restrict the MSSM to the ranges $0.85
 \alt \tan \beta \alt 1.35,\ m_t \gtap 130$~GeV, $60 \alt m_h \alt 90$~GeV,
$m_A \agt70$~GeV, $m_{H^\pm} \agt 105$~GeV, $m_H \agt 140$~GeV (see Fig.~23).
In this case $h$ can be discovered at LEP\,II (but not $A,H,H^{\pm}$);
SSC may discover $h$ or $H$ or $A$ or $H^{\pm}$ but not all simultaneously,
and possibly none of them at all\cite{bbop}; see Fig.~24.

\end{enumerate}


\begin{center}

\parbox[b]{3in}{
%\epsfxsize=3in
\hspace*{.2in}
%\epsffile{fig5-21.eps}
}
\parbox[b]{3in}{
%\epsfxsize=3in
\hspace{0in}
%\epsffile{fig5-22.eps}
}

%\medskip

\parbox[t]{2.8in}{\small Fig.~21: Lower bound on $m_{H^\pm}$ from the $B(b \to
s \gamma)$ upper limit\cite{cleo}, compared to other MSSM bounds (calculated as
in Ref.~\cite{bbp}).}\qquad
\parbox[t]{2.8in}{\small Fig.~22:  Bound in $(m_A, \tan \beta)$ plane from the
$B(b \to s \gamma)$ upper limit\cite{cleo} (calculated as in Ref.~\cite{bbp});
the region to the left of the curves is excluded.}
\end{center}

%\vspace{-.2in}

\begin{center}
%\epsfxsize=6in
\hspace{0in}
%\epsffile{/ed/Barger/Hawaii93/Proceedings/fig3.eps}


\parbox{5.5in}{\small Fig.~23: MSSM limits in the $(m_A, \tan \beta)$ and
$(m_h, \tan \beta)$ planes, from combining LEP\,I data with the fixed-point
condition $\sin \beta = m_t/(200$~GeV)\cite{bbop}.}


\bigskip

%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/Hawaii93/Proceedings/fig4.eps}

\bigskip

\parbox{5.5in}{\small Fig.~24: LEP\,II reach and SSC discovery limits, compared
with the allowed region of Fig.~23\cite{bbop}.}
\end{center}

\vspace{.25in}

\section{Higher energy $e^+ e^-$ colliders}

Finally we may ask what a future $e^+e^-$ collider could do. We have seen that
part of the MSSM parameter space is inaccessible to $e^+e^-$ collisions at
$\sqrt s=200$~GeV, $\L=500\pb^{-1}$, for $m_t=150$~GeV and $m_{\tilde
t}=1$~TeV. But a possible future linear collider with higher energy and
luminosity could in principle cover the full parameter space; the critical
quantities are $\sqrt s, \L$ and $m_t$. In is interesting to know what are the
minimum $s$ and $\L$ requirements for complete coverage, for given $m_t$. This
question was answered in Ref.~\cite{nolose}, based on the conservative
assumption that only the channels $e^+e^-\to(Zh,Ah,ZH,AH)\to\tau\tau jj$ would
be searched, with no special tagging. The results are shown in Fig.~25. We have
subsequently estimated that including all $Z\to\ell\ell,\nu\nu,jj$ and
$h,H,A\to bb,\tau\tau$ decay channels plus efficient $b$-tagging could increase
the net signal $S$ by a factor~6 and the net background $B$ by a factor~4,
approximately; this would increase the statistical significance $S/\sqrt B$ by
a factor~3 and hence reduce the luminosity requirement by a factor~9 or so. In
this optimistic scenario, the luminosity scale in Fig.~25 would be reduced by
an order of magnitude.

The $s$-channel production processes  $e^+e^- \to Ah,Zh,AH,ZH,
H^+H^-$, that we have considered above,  all have cross sections
falling like $1/s$ at high $s$. Hence very large s is not necessarily
helpful for Higgs boson studies, as Fig.~25 already indicates.
However,  $WW$ fusion processes like $e^+e^- \to \nu \nu H$  have
logarithmically rising cross sections and eventually dominate,
giving the main window on Higgs physics at a machine with
$s = 1$--2~TeV.  Figure~26 illustrates various channel cross sections;
we see that associated production with a heavy quark pair has
typically very low rate.

\begin{center}
%\epsfxsize=4in
\hspace{5.5in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig37.eps}

\parbox{5.5in}{\small Fig.~25: Minimal requirements for a ``no-lose'' MSSM
Higgs search at a future $e^+e^-$ collider. Curves of minimal $(\sqrt s,\L)$
pairings are shown for $m_t=120$, 150, 200~GeV; the no-lose region for
$m_t=150$~GeV is unshaded~\cite{nolose}.}
\end{center}

\bigskip

\begin{center}
%\epsfxsize=6.25in
\hspace{0in}
%\epsffile{fig5-26.eps}

\parbox{5.5in}{\small Fig.~26: Comparison of MSSM and SM production cross
sections versus
$\sqrt s$, for the case $m_A=100$~GeV, $\tan \beta =5,\ m_t =150$~GeV,
$m_{\tilde t} = 1$~TeV~\cite{dallas}.}
\end{center}



\section{Summmary}


\begin{enumerate}

\item The MSSM Higgs spectrum is richer but in some ways more elusive than the
SM case.

\item At least one light scalar is expected.

\item As $m_A\to\infty$ this light scalar behaves like the SM scalar while the
others become heavy and degenerate.

\item LEP\,I, LEP\,II and SSC/LHC will give extensive but not quite complete
coverage of the MSSM parameter space.

\item For some parameter regions, several different scalars are detectable, but
usually one or more remain undetectable.

\item The $b\to s\gamma$ bound has the potential to exclude large areas of
parameter space (possibly including the inaccessible region) but is presently
subject to some uncertainty.

\item $m_t$ fixed-point solutions with $m_t <$~160 GeV avoid the inaccessible
region and predict $m_h < 90$~GeV, making $h$ discoverable at LEP\,II; they
also predict $m_t \gtap 130$~GeV.

\item A higher-energy $e^+e^-$ collider could cover the whole MSSM parameter
space, discovering at least the lightest scalar $h$.


\end{enumerate}

\renewcommand{\chapter}{\section} %% to prevent new page for bibliography

\begin{thebibliography}{00}
\frenchspacing

\bibitem{susy} For general SUSY references, see previous chapter.

\bibitem{loop} S.~P~.Li and M.~Sher, Phys. Lett. {\bf 140B}, 339 (1984);
J.~F.~Gunion and A.~Turski, Phys. Rev. {\bf D39}, 2701 (1989); {\bf D40},
2325,2333 (1989); M.~S.~Berger,
ibid. {\bf 41}, 225 (1990); Y.~Okada et al., Phys. Lett. {\bf B262}, 54 (1991);
H.~Haber and R.~Hempfling, Phys. Rev. Lett. {\bf 66}, 1815 (1991); J.~Ellis et
al., Phys. Lett. {\bf B257}, {\bf 83} (1991); R.~Barbieri et al., ibid.
{\bf258}, 167 (1991); J.~Lopez and D.~V.~Nanopoulos,
ibid. {\bf266}, 397 (1991); A.~Yamada, ibid. {\bf263}, 233 (1991);
M.~Drees and M.~N.~Nojiri, Phys. Rev. {\bf D45}, 2482 (1991); M.~A.~Diaz and
H.~E.~Haber, Phys. Rev. {\bf D45}, 4246 (1992);
D.~M.~Pierce et al., Phys. Rev. Lett. {\bf68}, 3678 (1992).

\bibitem{bbsp}V.~Barger et al., Phys. Rev. {\bf D45}, {\bf4128} (1992).

\bibitem{bcps}V.~Barger et al., Phys. Rev. {\bf D46}, 4914 (1992).

\bibitem{baersusy} H.~Baer et al., Phys. Rev. {\bf D47}, 1062 (1993).

\bibitem{lep}
ALEPH collaboration: D.~Decamp et al., Phys. Lett. {\bf B241}, 623 (1990);
{\bf B265}, 475 (1991);
DELPHI collaboration: P.~Abreu et al., ibid. {\bf B241}, 449 (1990);
Nucl. Phys. {\bf B373}, 3 (1992);
L3 collaboration: B.~Adeva et al., Phys. Lett. {\bf B252}, 511 (1990);
{\bf B294}, 457 (1992);
OPAL collaboration: M.~Z.~Akrawy et al, ibid. {\bf 242}, 299 (1990);
Z.~Phys. {\bf C49}, 1 (1991).

\bibitem{baer} H.~Baer et al., Phys. Rev. {\bf D46}, 1067 (1992).

\bibitem{gunion} J.~F.~Gunion et al., Phys. Rev. {\bf D46}, 2040,2052 (1992);
R.~M.~Barnett et al., ibid. {\bf D47}, 1030 (1993).

\bibitem{kz} Z.~Kunszt and F.~Zwirner, Nucl. Phys. {\bf B385}, 3 (1992).

\bibitem{wu} S.~L.~Wu et al., LEP\,200 Workshop, CERN 87-08, p.312.

\bibitem{janot} P.~Janot, Orsay report LAL-91-61.

\bibitem{gem} GEM collaboration Letter of Intent, GEM TN-92-49.

\bibitem{sdc} SDC collaboration Technical Design Report SDC-92-201.

\bibitem{dai} J.~Dai, J.F.~Gunion and R.~Vega, Davis preprint UCD-93-20.

\bibitem{cleo} CLEO collaboration, report to Washington APS meeting,
      April 1993.

\bibitem{bbp}V.~Barger, M.~S.~Berger and R.~J.~N.~Phillips, Phys. Rev. Lett.
      {\bf 70}, 1368 (1993).

\bibitem{jlh} J.~L.~Hewett, Phys. Rev. Lett. {\bf70}, 1045 (1993).

\bibitem{corr}B.~Grinstein et al., Nucl. Phys. {\bf B339}, 269 (1989);
G.~Cella et al., Phys. Lett. {\bf B248}, 181 (1990); M.~Misiak, ibid. {\bf
B269}, 161 (1991); Zurich report ZH-TH-19/22 (1992); M.~Diaz, Phys. Lett.
{\bf B304}, 278 (1993).

\bibitem{bertolini}S.~Bertolini, F.~Borzumati and A.~Masiero, Nucl. Phys.
{\bf B294}, 321 (1987); S.~Bertolini et al., ibid. {\bf B353}, 591 (1991);
R.~Barbieri and G.~F.~Giudice, Phys. Lett. {\bf B309}, 86 (1993).


\bibitem{bbop} V.~Barger, M.S.~Berger, P.~Ohmann, and R.J.N.~Phillips,
University of Wisconsin-Madison preprint MAD/PH/755 (1993), to be published in
Phys. Lett.~B.

\bibitem{nolose} V.~Barger, Kingman Cheung, R.J.N.~Phillips, and A.L.~Stange,
Phys. Rev. {\bf D47}, 3041 (1993).

\bibitem{dallas} V.~Barger, Kingman Cheung, R.J.N.~Phillips, and A.L.~Stange,
University of Wisconsin-Madison preprint MAD/PH/705 (1992), unpublished.

\end{thebibliography}

\end{document}

y,
and possibly none of them at all\cite{bbop}; see Fig.~24.

\end{enumerate}


\begin{center}

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\begin{document}

\setcounter{page}{67}
\chapter*{LECTURE 6:\\ INDIRECT CONSTRAINTS ON\\ THE MSSM HIGGS SECTOR}
\setcounter{chapter}{6}


\section{Introduction}
   Beside direct searches for SUSY Higgs bosons (see previous Lecture)
there are various indirect ways to place bounds on the MSSM parameters.
Virtual charged Higgs bosons, appearing in intermediate states, can
contribute significantly to certain rare decays and neutral meson
mixings; experimental data in these channels therefore imply constraints.
Theoretical considerations of perturbativity and non-triviality also
place bounds.  Constraints in general two-doublet models were
reviewed in Refs.\cite{bhp,buras,gunion} but there have been some recent
improvements.   In this Lecture we briefly review the present picture.

\section{Perturbative Bounds}
   The main charged-Higgs couplings to fermions are given by Eq.~(5.14),
neglecting terms suppressed by small KM matrix elements or masses. The
largest of these Yukawa couplings are proportional to $m_t \cot \beta$
and to $m_b \tan \beta$.  If we require that these couplings remain
perturbative, e.g.\ that they do not exceed $g_3(M_W) \simeq 1.2$, then
we obtain upper and lower bounds on $\tan \beta$~\cite{bhp}:
%
\begin{equation}
      m_t/(600\gev) \ltap \tan \beta \ltap (600\gev)/m_b  \,.
\end{equation}
%
For $m_t = 150$~GeV this translates to $0.25 \ltap \tan \beta \ltap 120$.

   The perturbativity criterion is somewhat subjective, however, and
other choices are possible\cite{bhp}.  If for example we require instead
that the ratio of two-loop/one-loop contributions in the Yukawa
coupling evolution equations does not exceed 1/4~\cite{bbo}, then we
obtain more stringent bounds
%
\begin{equation}
           0.6 \ltap \tan \beta \ltap 65 \,.
\end{equation}

\section{Proton Decay}
   In non-SUSY GUT models, nucleon decay proceeds via the exchange
of exotic gauge bosons with masses of order $M_G$, giving a lifetime of
order $g^{-4}M_G^4/m_N^5$; unfortunately the gauge couplings fail to converge
satisfactorily in these models, and also the proton lifetime comes
out below the experimental limit.  In SUSY-GUT models there are
several differences: the gauge couplings converge at a higher
value of $M_G$, and also the dominant decay modes are new channels
involving heavy sparticle exchanges.  There is controversy about the
outcome, however.  Some analyses find that the proton lifetime introduces
strong constraints on minimal-SUSY SU(5) GUT models, requiring
$\tan \beta \ltap 3.5$--5~\cite{arnowitt}.  On the other hand, another
recent study finds a much more conservative constraint\cite{yanagida},
%
\begin{equation}
         \tan \beta \ltap 85 \,.
\end{equation}


\section{Neutral Meson Mixing}

   Neutral mesons like $K^0,\, D^0,\, B^0$, can mix with their charge-conjugate
states $\bar K^0,\, \bar D^0,\, \bar B^0$, through box diagrams like Fig.~1
where
$H^-,\, \phi^-$ and $W^-$ denote charged Higgs boson, unphysical Higgs boson
and gauge boson exchanges.  The result depends sensitively on the heaviest
intermediate quark mass --- which for $B$-$\bar B$ mixing is $m_t$.
The Higgs contributions can be very significant\cite{glashow}.

\begin{center}
%\epsfxsize=2.75in
\hspace{0in}
%\epsffile{fig6-1.eps}

{\small Fig.~1: Box diagrams contributing to neutral meson-antimeson mixing.}
\end{center}

   In general, $X^0$-$\bar X^0$ meson mixing is signalled by oscillations,
whereby a meson initially produced as $X^0$ has an oscillatory
time-dependent probability to decay via $\bar X^0$ modes.  The time
averaged probability for decaying the ``wrong" way is
%
\begin{equation}
  P(X^0 \to \bar X^0) = {1\over2} \, \frac
                    {(\delta m)^2 + {1\over4}(\delta \Gamma)^2}
                    {(\delta m)^2 + \Gamma^2} = \chi
\end{equation}
%
where $\delta m$ and $\delta \Gamma$ are the mass and width differences
between the two eigenstates $|X_{1,2}\rangle \simeq (|X^0\rangle \pm |\bar
X^0\rangle )/\sqrt 2$.
For $B^0$-$\bar B^0$ and $D^0$-$\bar D^0$ oscillations, $\delta \Gamma/
\Gamma$ is negligible and the effects depend on $\delta m$.

   In practice one often studies $e^+e^- \to X^0 \bar X^0$ production and
looks for semileptonic decays of both $X$-mesons.  Without mixing, the
primary decay leptons would always have opposite signs;
same-sign leptons are a direct
measure of mixing.  Taking account of the $C=-1$ wave-function correlations,
the time-averaged fraction of same-sign leptons is
%
\begin{equation}
\frac {P(++) + P(--)}{P(++) + P(+-) + P(-+) + P(--)} =  {1\over2} \,
                \frac{(\delta m)^2 + {1\over4}(\delta \Gamma)^2}
                     {(\delta m)^2 + \Gamma^2}
\end{equation}
%
for an initial $X^0 \bar X^0$ state.  This happens to be just the same as
the ``wrong-sign" fraction for a single $X^0$ (previous equation).
However, if the two $X$-mesons had been produced
equally in $C=+1$ and $C=-1$ states, the same-sign dilepton
fraction would have been $2\chi(1-\chi)$ instead.

$B_d^0$-$\bar B_d^0$ mixing is given by
%
\begin{equation}
\delta m_B = \frac{G_F^2 M_W^2}{6 \pi ^2} m_B f_B^2 B_B \eta_t
 \left|V_{tb} V_{td}^*\right|^2
\left(A_{WW} + \cot^2 \beta A_{WH} +\cot^4 \beta A_{HH}\right) \,,
\end{equation}
%
where $f_B$ is the decay constant, $B_B$ is the bag factor, $\eta_t$ is a QCD
correction factor, $V$ is the KM matrix, and the loop integrals contribute the
factors $A_{WW}$
etc. There is a similar expression for $B_s^0$-$\bar B_s^0$ mixing.
The charged-Higgs mass appears in $A_{WH}$ and $A_{HH}$; the dependence
on $\tan \beta$ is explicit; data therefore place constraints on these
parameters, modulo uncertainties in $f_B,\ B_B,\ V_{tx}$ and $m_t$.  The SM
predicts modest mixing for $B_d^0$-$\bar B_d^0$  but near-maximal mixing for
$B_s^0$-$\bar B_s^0$, so there is a better chance to see Higgs effects in
the former.  There are data for exclusive $e^+e^- \to B_d \bar B_d$
production at the $\Upsilon_{4S}$ resonance, plus inclusive $Z \to B
\bar B$ production (with no CP-even constraint and summing over $B_d$ and
$B_s$ production), plus $B \bar B$ hadroproduction\cite{drell}.

   $D^0$-$\bar D^0$ mixing has an analogous formula, with $\tan \beta$
replacing $\cot \beta$ and appropriate changes of masses in the
integrals. The SM prediction is very small, but long-distance
contributions from $\pi \pi$ and $KK$ intermediate states could be
non-negligible.

 For $K^0$-$\bar K^0$ mixing the short-distance contributions to $\delta m_K$
are analogous to $\delta m_B$, but both long-distance and $\delta \Gamma$
contributions are important, so the picture is more complicated here.  However,
it is often assumed that short-distance new-physics contributions
should be constrained so that they do not exceed the measured value
of $\delta m_K$.

   The net effect of these constraints is to exclude very small values
of $\tan \beta$ (i.e.\ very large $Htb,\, Hts,\, Htd$ couplings), depending
somewhat on $m_{H^\pm}$ and $m_t$,  and is thus similar to the lower
perturbative  bound\cite{bhp}.

\section{CP Violation}

   The parameter $\epsilon$ measures CP violation in the  $K$-$\bar K$
mixing process; it is essentially defined by the imaginary part of
the off-diagonal $K^0 \to \bar K^0$ matrix element $M_{12}$ of the
effective Hamiltonian,
%
\begin{equation}
\epsilon \simeq \frac{\exp i \pi /4}{\sqrt 2 \delta m_K} \Im M_{12}\,,
\end{equation}
%
and receives contributions through the box diagrams of Fig.~1, from
CP-violation in the KM matrix elements.  The most important $Hts$ and $Htd$
Higgs couplings depend on $m_t \cot \beta$  and the loop integrals depend on
$m_H$ and $m_t$.  The  parameter $\epsilon = (2.259 \pm 0.018)\times 10^{-3}
\exp(i \pi /4)$
is measured very accurately, but the calculation has big uncertainties,
especially from the bag factor $B_K$ (estimates vary from $-$0.4 to 2.9).
Also $\epsilon$ alone does not really constrain the Higgs sector; it just
serves to determine the CP-violating phase in the KM matrix elements.

   A second parameter $\epsilon '$ describes ``direct" CP-violation in the
matrix elements of $K \to \pi \pi$ decay.  It is related to the transition
matrix elements $\left< \pi \pi (I)| H |K^0\right> = A_I \exp(i \delta_I)$ to
final states of isospin $I$ by
%
\begin{equation}
\epsilon ' = \frac{\exp\left[i(\delta_2 - \delta_0 + \pi /2)\right]}{\sqrt 2}
             \frac{\Re A_2}{\Re A_0}
             \left[ \frac{\Im A_2}{\Re A_2} - \frac{\Im A_0}{\Re A_0}\right]\,.
\end{equation}
%
Final-state interaction effects enter through the $S$-wave phase shifts
$\delta_I$,
which are measured in $\pi \pi$ scattering.  The real parts of the
weak amplitudes $A_I$ dominate; they are due to tree-level $W$ exchanges
and can be measured from the $K^0 \to \pi^+ \pi^-, \pi^0 \pi^0$ decay rates.
The leading contributions to $\Im A_I$ come from penguin diagrams, where an
$s \to d$ one-loop transition on one quark line is linked by $g,\ \gamma$ or
$Z$ exchange to the associated $\bar d$ line; both $W^{\pm}$ and $H^{\pm}$
bosons appear in the loops (Fig.~2).  The dominant gluon-exchange penguin
contributes only to $A_0$; the $\gamma$-exchange penguin contributes to $A_0$
and $A_2$ but is  weaker by  $\alpha/\alpha_s$  and is sometimes ignored.
Various theoretical uncertainties are overshadowed at present by a rather
large experimental uncertainty in the measured quantity\cite{ee}
%
\begin{equation}
\epsilon ' / \epsilon = \left\{ \begin{array}{l@{\hspace{1.5em}}l}
 ( 2.3\phantom0  \pm 0.7 )\times10^{-3}  &\mbox{NA31 experiment (CERN)}\,,\\
 ( 0.66 \pm 0.58 \pm 0.37)\times10^{-3}  &\mbox{E731 experiment (FNAL)}\,.
\end{array} \right.
\end{equation}

\begin{center}
\parbox[b]{3in}{
%\epsfxsize=3in
%\epsffile{fig6-2a.eps}
}
\qquad
\parbox[b]{3in}{
%\epsfxsize=3in
%\epsffile{fig6-2b.eps}
}

\medskip
\parbox{5.5in}{\small Fig.~2: Penguin diagrams contributing to meson decays and
$\epsilon'/\epsilon$: (a)~overall structure showing gluon, photon, $Z$
exchanges, (b)~charged-Higgs-boson contributions to the $\bar q'qG$ vertex
($G=g,\gamma,Z$).}
\end{center}

\bigskip

\section{Rare $K$ Decays}

\indent

(a) $K_L \to \mu \mu$.  The imaginary part of the decay amplitude is
dominated by the $2\gamma$ intermediate state, which by itself gives\cite{geng}
$B(K_L \to \gamma \gamma \to \mu \mu) = (6.83 \pm 0.28)\times10^{-9}$,
saturating the experimental value\cite{kmm} $(7.3 \pm 0.4)\times10^{-9}$,
and setting an upper bound on the real part.  The latter receives
short-distance contributions from Higgs and $W$ box diagrams analogous
to Fig.~1 but also long-distance contributions that could interfere
destructively\cite{blmp}.  This does not therefore offer a reliable
constraint on the Higgs sector.

(b) $K^+ \to \pi^+ \nu \nu$.  This decay is mediated by box and $Z$-penguin
diagrams analogous to Figs.~1,2, with one quark line replaced by a neutrino
or lepton line and a spectator quark line added\cite{hnr}.  The
experimental upper limit\cite{atiya}  $B(K^+ \to \pi^+ \nu \nu) <
3.4\times10^{-8}$  therefore constrains $Hts$ couplings and excludes very small
$\tan \beta$ (essentially nonperturbative values)\cite{bhp}.

\section{$B$ decays}
\indent

(a) $B \to X \tau \nu$. The inclusive semileptonic decay $b \to c \tau \nu$
can be mediated either by SM $W$ exchange or by a charged Higgs boson,
which contribute additively in the branching fraction (different final
helicities). The measured value\cite{tau} $B(B \to X \tau \nu) = (4.08 \pm 0.76
\pm 0.62)\%$ is close to the SM prediction  $(2.83 \pm 0.31)\%$  and bounds the
Higgs amplitude~\cite{isidori}:
%
\begin{equation}
          \tan \beta < 0.54 m_{H^\pm}/(1\gev) \,.
\end{equation}
%
Since $m_{H^\pm} \gtap M_W$ in the MSSM (see Lecture 5), this excludes
only rather large, non-perturbative values of $\tan \beta$.

(b) $B \to X \gamma$.  This inclusive one-photon decay is well approximated
by the quark-level process  $b \to s \gamma$, proceeding in the SM by
$W$-loop diagrams; $H^{\pm}$ can contribute similarly, see Fig.~3.  In the
MSSM the $W$ and $H$ loops contribute coherently to the amplitude with the same
sign, and
both receive the same QCD enhancements, so the decay rate is very sensitive
to the Higgs contribution (unlike the previous example).  Figure~4 shows
how the calculated branching fraction depends on $\tan \beta$ for various
values of $m_{H^\pm}$ (assuming $m_t = 150$~GeV, $m_{\tilde t} =
1$~TeV)\cite{bbp}. Hence the latest bound\cite{cleo}
%
\begin{equation}
   B(b \to s \gamma) < 5.4\times 10^{-4}
\end{equation}
%
apparently excludes small $\tan \beta$ for any $m_{H\pm}$, and
$m_{H\pm} \ltap 500$~GeV for any $\tan \beta$.  The corresponding bounds
in the $(m_{H^\pm},\tan \beta)$ and $(m_A, \tan \beta)$ planes are shown
in Lecture~5.

   It is too soon to reach such strong conclusions, however.  There is
some sensitivity to the input parameters $m_t,\ m_{\tilde t},\ m_b$ and
also to the details of the QCD corrections\cite{corr}. More
importantly, there is the possibility that SUSY loops --- especially
chargino/squark loops --- could contribute significantly and with
either sign in the amplitude; if they contribute destructively they
will weaken the bounds\cite{bertolini}. Figure~5 compares the size
and sign of possible chargino, gluino and neutralino loop amplitudes
with the charged-Higgs amplitude we are trying to bound (dots represent
particular model solutions).  But when
the possible SUSY contributions are constrained, by future theoretical
and/or experimental input, this promises to be a sensitive probe of
MSSM parameters.

\begin{center}
%\epsfxsize=4.5in
\hspace{0in}
%\epsffile{fig6-3.eps}

{\small Fig.~3: Examples of $W^{\pm}$ and $H^{\pm}$ loop diagrams for $b \to s
\gamma$ decay.}

\medskip

%\epsfxsize=4in
\hspace{0in}
%\epsffile{/ed/Barger/SearchBSM/Proceedings/fig34.eps}


{\small Fig.~4: Dependence of $B(b \to s \gamma)$ on $\tan \beta$ and
$m_{H^\pm}$~\cite{bbp}.}

\medskip

%\epsfxsize=5.5in
\hspace{0in}
%\epsffile{fig6-5.ai.eps}

{\small Fig.~5: Ratios of SUSY to SM amplitudes for $b \to s \gamma$
        \cite{bertolini}.}
\end{center}

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\section*{Acknowledgements}

The authors would like to thank H.~Baer, M.S.~Berger, Kingman Cheung, T.~Han,
P.~Ohmann, R.G.~Roberts, and  A.L.~Stange for valuable discussions related to
these lectures.\break
This work was supported in part by the U.S.-Brazil Cooperative Science Program
under NSF Grant No.~ by the U.S.~Department of Energy under
Contract No.~DE-AC02-76ER00881, by the Texas National Laboratory Research
Commission under Grant %\break
No.~RGFY93-221, and by the University of Wisconsin Research
Committee with funds granted by the Wisconsin Alumni Research Foundation.


\end{document}

d from the $K^0 \to \pi^+ \pi^-, \pi^0 \pi^0$ decay rates.
The leading contributions to $\Im A_I$ come from penguin diagrams, where an
$s \to d$ one-loop transition on one quark line is linked by $g,\ \gamma$ or
$Z$ exchange to the associated $\bar d$ line; both $W^{\pm}$ and $H^{\pm}$
bosons appear in the loops (Fig.~2).  The dominant gluon-exchange penguin
contributes only to $A_0$; the $\gamma$-exchange penguin contributes to $A_0$
and $A_2$ but

space, discovering at least the lightest scalar $h$.


\end{enumerate}

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\end{document}

y,
and possibly none of them at all\cite{bbop}; see Fig.~24.

\end{enumerate}


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\begin{document}

\setcounter{page}{67}
\chapter*{LECTURE 6:\\ INDIRECT CONSTRAINTS ON\\ THE MSSM HIGGS SECTOR}
\setcounter{chapter}{6}


\section{Introduction}
   Beside direct searches for SUSY Higgs bosons (see previous Lecture)
there are various indirect ways to place bounds on the MSSM parameters.
Virtual charged Higgs bosons, appearing in intermediate states, can
contribute significantly to certain rare decays and neutral meson
mixings; experimental data in t

