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\begin{document}
\title{QCD in Heavy Quark Production and Decay}
\author{Jim Wiss\thanks{Supported by DOE Contract DE FG0201ER40677.}\\
University of Illinois \\
Urbana, IL 61801 \\[0.4cm]
%and \\[0.4cm]
%Jane Eyre\thanks{Supported by NSF Contract xxx.}\\
%Washington University, Seattle, Washington xxxxx \\[0.4cm]
}
\maketitle
\begin{abstract}
I discuss how QCD is used to understand the physics of heavy quark
production and decay dynamics. My discussion of production
dynamics primarily concentrates on charm photoproduction data which is
compared to perturbative QCD calculations which incorporate fragmentation
effects. We begin our discussion of heavy quark decay by reviewing data on
charm and beauty lifetimes.
Present data on fully leptonic, and semileptonic charm decay is then reviewed.
Measurements of the hadronic weak current form factors are compared
to the non-perturbative QCD based predictions of Lattice Gauge Theories.
We next discuss polarization phenomena present in charmed baryon decay.
Heavy Quark Effective theory predicts that the daughter baryon will
recoil from the charmed parent with nearly 100 \% lefthanded polarization
which is in excellent agreement with present data. We conclude by
discussing nonleptonic charm decay which are traditionally analysed
in a factorization framework applicable to two-body and quasi-two-body
non-leptonic decays. This discussion emphasizes the important role
of final state interactions in influencing both the observed decay width
of various two body final states as well as modifying the interference between
interfering resonance channels which contribute to specific multibody decays.
\end{abstract}
\setcounter{footnote}{0}
\section{Introduction}
\par
A vast amount of important data on heavy quark
production and decay exists for both the beauty and
charm sector -- far more than I could possibly cover in two lectures.
For this reason, most of my talk will cover the production
of charm in fixed target experiments with an emphasis on my
own experiment --E687.
\footnote{A general description\cite{elsewhere}
of E687 can be found in Reference 1.}
\par
Charmed mesons were discovered by the
Mark 1 Collaboration which studied the $e^+ e^-$ annihilation process
at SLAC about twenty one years ago.\cite{mark1}
In the intervening years, a large number of collider as well as fixed
target experiments have heavily mined the field of heavy quark physics.
The discovery paper by the Mark 1 group reported on the observation
of $\approx 230$ fully reconstructed decays of the $D^o$ into the
$K^\mp \pi^\pm$ and $K^\mp \pi^\pm \pi^+ \pi^-$ final state.
Recent experiments are capable of amassing clean, fully reconstructed
charm data decays which are in excess of 100 thousand events. Millions
of fully reconstructed events are anticipated for several experiments presently
taking data.
\subsection{Charm and QCD}
\par
Charm provides a rich QCD phenomenology in both the perturbative and
nonperturbative regimes. Studies of production dynamics test leading
order (LO) and next-to-leading order (NLO) perturbative QCD.
Charm lifetime measurements test models based on 1/$M_Q$ QCD expansions.
Heavy Quark Effective Theory (HQET) predicts a large degree of daughter
baryon polarization in both semileptonic and two body non-leptonic decay.
Semileptonic form factors test non-perturbative QCD methods such as
Lattice Gauge Theories (LGT). Finally exclusive hadronic decay measurements
test factorization models based on QCD renormalization group methods.
Exclusive charm hadronic decays underscore the importance of final
state interactions which should be important in unravelling
the pattern of CP violations in the b-sector.
\par
While discussing the physics contained in the above topics and illustrating
it with real data, I would like to give you some feeling on how measurements
are actually made. You will probably find that these lectures tilt too much in
the direction of fixed target experiments.
Perhaps they serve as a sort of swan song
since the future of fixed target charm experiments in the United States
is very short.
\subsection{Charm Mesons}
\par
Figure \ref{charm_meson} is a crude sketch of the spectrum
and properties of the low lying $D$
(pseudoscalar) and $D^*$ (vector) mesons which consist of a charm
quark and an up, down , or strange antiquark. There are a large number
of baryon and higher excited mesons and baryons as well which are not
shown.
\begin{figure}[htp]
\centerline{
\epsfysize=4.in\epsffile{charm_meson.eps}
}
\caption {\small{
Sketch of the spectrum of low lying charm mesons.
The $D^o$ amd $D^+$ mesons form a nearly degenerate strong isodoublet.
The $D_s^+$ is shifted upwards by roughly 100 MeV. This mass splitting
pattern is repeated for the $D^*$'s. Weak decays are shown by
dashed arrows. Some strong and electromagnetic de-excitations of the $D^*$
system into the ground state $D$ system are
shown by solid and dashed arrows.
The approximate lifetimes of the ground state
$D$ system are indicated on the figure.
}}
\label{charm_meson}
\end{figure}
Several of the most experimentally useful weak decay modes
are shown in the figure. These are CKM allowed decays where the charm
quark is transformed to a strange quark resulting in a single kaon final
state for the $D_o$ and $D^+$ and a double kaon final state for the $D_s^+$.
The $D$ mesons have lifetimes ranging from about 1/2 to 1 picosecond.
In a fixed target experiment, a $D^+$ might typically be produced with
a typical lab momenta of 100 GeV and will travel several centimeters
before decaying. A high quality vertex detector, such as a silicon
microstrip detector, can readily measure a several centimeter decay path
and thus cleanly tag a charm decay through its short but finite lifetime.
\par
Perhaps the second best way of tagging charm particles is through
the decay $D^{*+} \rightarrow \pi^+~D^o$ where the $D^o$ decays into an
easily reconstructable, all charged decay mode such as $K^\mp \pi^\pm$ the
final state. Owing to the
small energy release in the decay $D^{*+} \rightarrow \underline{\pi}^+~D^o$
the soft(underlined) pion will carry only a small fraction of the lab momentum
of the $D$ and therefore can be very well measured by a magnetic spectrometer.
Most of the experimental smearing in reconstructing the $D^{*+}$ mass
is due to the $D^o$ reconstruction which cancels in the mass difference
$\Delta M \equiv M(D^o \pi^+) - M(D^o)$. This cancellation
often results in a mass difference
which is about 10 times better resolved than the mass of the $D^{*o}$. By
cutting on both $M(D^o)$ and $\Delta M$, one can often get charm signals
nearly as clean as those obtainable using lifetime tagging but without
the technological inconvenience of having to build an expensive vertex tagging
system. One can even extend the $\Delta M$ technique to tag
semileptonic final states with a missing neutrino. Most of the data discussed
in these lectures use either lifetime tagging or the $D^* \rightarrow D$ tagging
trick to isolate and study charm.
\setcounter{footnote}{0}
\section{The Physics of Charm Photoproduction}
Charm photoproduction provides several nice illustrations of the successes
and complications of applying perturbative QCD to production dynamics.
We will begin on the most inclusive level by considering the
center of mass energy dependence of the photoproduction cross section
for charmed quarks. In order to delve deeper in to the photoproduction
process and study {\it eg}
the $P_t$ and $X_f$ dependence for specific charm mesons
one must understand the hadronization process
which describes the soft processes by which charmed quarks appear as
charmed mesons and baryons. Studies of the kinematic
dependence of
photo- and hadroproduced charm-$\overline{\rm charm}$ asymmetries
provides surprisingly sensitive probes of QCD inspired models
of charm hadronization.
\subsection{The Charm Quark Photoproduction Cross Section}
\par
At lowest order, and at energies significantly above charm
threshold, the photoproduction of charmed quark
thought to be dominated by the photon-gluon fusion process illustrated
in lowest order by Figure \ref{pgf_lo}.
\begin{figure}[htp]
\vskip .4in
\epsfysize=4.in\epsffile{[rwg.tex.nu]pgfdiag.ps}
\vskip -3.2in
\caption {\small{Illustration of the lowest order contribution to
the photon gluon fusion process. A gluon constituent of the nucleon
interacts with a real photon to produce a $c\overline{c}$ pair.
These graphs are both
${\cal O}(\alpha_{em}\alpha_s)$}}
\label{pgf_lo}
\end{figure}
The photon-gluon fusion process is analogous
to the pair production process
$\gamma {\cal N} \rightarrow e^+ e^-~{\cal N}$ where a projectile
photon interacts with a virtual photon from the nucleus to form
an $e^+ e^- $ pair.
Charm hadroproduction is thought to occur by a very similar process where
a gluon from the projectile, fuses with a gluon from target to form
a $c\overline{c}$ pair.
\par
The charm quark photoproduced cross section will depend linearly on the
gluon density within the nucleon as well as the partonic cross section
evaluated at the squared energy in the photon-gluon rest frame:
\begin{equation}
\sigma(\gamma p \rightarrow c \overline{c}) =
\int_{4 m^2_c /s}^1 ~dx_g ~g (x_g,\mu^2) ~~\hat \sigma (s~x_g)~~,~~
\hat s = s~x_g
\label{pgfeq}
\end{equation}
The variable of integration , $x_g$ , represents the fraction of the
nucleon momentum carried by the incident gluon. The lower
limit of integration is due to the fact that $\hat s \equiv s ~x_g$,
the squared center of mass energy in the photon-gluon rest frame must
exceed $(2 m_c )^2$. The lowest order PGF cross section will grow
slowly with increasing $\hat s$ due to the contributions of increasingly
softer (lower $x_f$) gluons.
\par
At next-to-leading order (NLO), the graphs of Figure \ref{pgf_lo} are
joined by the ${\cal O}(\alpha_{em}~\alpha_s^2)$ contributions
which were recently calculated by
Frixione, Mangano, Nason, and Ridolfi\cite{FMNR} (FMNR).
The NLO graphs include contributions with extra
quarks or gluons in the final state.
\begin{figure}[htp]
\epsfysize=3.in
\epsffile{[rwg.tex.nu]pgf2nd.ps}
\caption{\small{
NLO contributions to photon-gluon fusion which produce an additional
gluon or quark in the final state.
}}
\label{pgf2nd}
\end{figure}
Figure \ref{ccbar_cross} compares the experimentally measured charm
quark cross section
to the FMNR predictions for several choices of
renormalization scales, and gluon momentum distributions.
\begin{figure}[htp]
\vskip 1.8in
\epsfysize=3.in\epsffile{hera.ps}
\vskip -2.in
\caption{\small{ Measured charm quark photoproduction
cross sections compared to the FMNR NLO predictions for various
gluon distribution parameterizations and renormalization scales
}}
\label{ccbar_cross}
\end{figure}
The data is a reasonably good match to these predictions over a
large kinematic range covering both fixed target photoproduction
experiments as well as the very recent data\cite{HERACROSS} on virtual
photoproduction
\footnote{
HERA defines the virtual photoproduction kinematic regime
as $Q^2 < 4~GeV^2$}
of charm from HERA.
The threshold condition $s x_g > 4~m^2_c$ implies that the HERA
data probes gluon density at fairly low $x_g$
($x_g > 1.6 \times 10^{-4}$) while fixed
target experiments typically probe in the more moderate $x_g$
range ($x_g > 0.04$). Although the HERA charm event samples are
fairly modest\footnote{{\it eg} The ZEUS charm cross sections were based on
48 reconstructed $D^{*+} \rightarrow D^o \pi^+ \rightarrow
(K^- \pi^+) \pi^+$ events.}, they carry an enormous kinematic lever arm
which can be used to ultimately constrain parameters in the NLO description.
\par
An apparent advantage of testing the QCD photoproduction model through
the measurement of the $c \overline{c}$ cross section energy dependence
is that the theoretical cross section does not depend
explicitly on a hadronization model since ultimately the photoproduced
quarks must somehow appear as charmed particles.
Although some very nice models
for hadronization exist (which we will discuss in Section 2.4),
they are phenomenological in the sense
that although based on a general QCD inspired framework ,
they parameterize many soft processes using inputs from
experimental data.
\par
Although hadronization assumptions may not enter directly into the
predictions, they are required to experimentally measure the
the $c \overline{c}$ cross section. An experimenter will
measure the yield of several different charm particles
over a certain kinematic range where the experiment has a reasonably
large, momentum-dependent, reconstruction efficiency.
To get a total $c \overline{c}$ cross section from such data,
the experimenter must rely on a fragmentation model to know what
fraction of the total charm cross section hadronizes into the
final states being reconstructed.
In addition, the hadronization model can affect the acceptance by
controlling the fraction of charm particles with {\it eg}
momenta outside of kinematic range of reasonable acceptance
or changing the over all reconstruction efficiency by
affecting the multiplicity
of particles which accompany the charm particle of interest.
\footnote{This is an especially important effect in photoproduction where
charm particles are tagged through the separation between
a downstream (secondary) vertex containing the charm particle
decay products and an upstream (primary) vertex
which contains many of particles created through the hadronization
process.}
\subsection{Non-Perturbative Effects}
\par
In order to probe more deeply into charm photoproduction dynamics, one
needs to consider the myriad of non-perturbative, soft physics effects
illustrated in Figure \ref{nonperturbative}.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=2.5in\epsffile{[rwg.tex.nu]real.ps}
\caption{
A cartoon of photoproduction of charm in the real world which illustrates
several non-perturbative effects. The incident gluon can be thought to
carry a small momentum component transverse to the incident
nucleon called $k_t$. Soft gluon emission can accompany the
charmed quark as it dresses creating additional sources of relative
$P_\perp$. A significant amount quark energy is lost to the emerging $D$'s
while they hadronize.}
\label{nonperturbative}
\end{figure}
Fragmentation effects can change the direction of the charmed mesons
relative to the charmed quarks as well as significantly degrade their
energy. A convenient variable to describe the energy fraction is
called $X_F$ for ``Feynman X'' which gives the fraction of
available longitudinal momentum in the overall center of mass
carried by an object divided by the maximum possible longitudinal momentum
possible in that frame or:
\begin{equation}
X_F = {2 P_\parallel \over \sqrt{s}}
\label{xf}
\end{equation}
In a fixed target experiment, $X_F$ is approximately the fraction of the
beam energy which appears in the charmed object.
\par
Figure \ref{xfdist} compares the $X_F$ distribution of charmed quarks
and charmed mesons according to the JETSET-PYTHIA\cite{lund} string
fragmentation model.
\begin{figure}[htp]
\hbox{
\hskip -.2in
\epsfysize=2.in\epsffile{xf_lo.ps}
\hskip .2in
\epsfysize=2.in\epsffile{xf_hadron.ps}
}
\caption{Comparison of the $X_F$ distribution of charmed quarks
(left) and charmed mesons (right) in LO photon-gluon fusion in the
LO JETSET-PYTHIA model.}
\label{xfdist}
\end{figure}
Figure \ref{xfdist} shows that although each member of the $c \overline{c}$
pair carries an average of 1/2 of the incident beam energy, after fragmentation
each charmed particle carries only about 1/4 of the incident beam energy.
\par
Figure \ref{pt2_nlo_lund} illustrates the importance
of fragmentation effects by comparing the observed $P_t^2$
for photoproduced $D^+$ to the NLO FMNR predictions for charmed
quarks (without fragmentation).
\begin{figure}[htp]
\vskip -1.in
{\hskip .5in \epsfysize=3.25in\epsffile{[rwg.tex.moriond]pt2_nlo_lund.ps}}
\vskip 1.in
\caption{Overlay of the $P_t^2$ distribution for reconstructed
$D^+$ mesons collected by E687 (with error bars) with the FMNR NLO predictions
(histogram) and the PYTHIA Monte Carlo.}
\label{pt2_nlo_lund}
\end{figure}
The $P_t^2$ distribution of the data tracks fairly well with
the PYTHIA MC which embeds LO PGF with LUND string fragmentation.
The data clearly does not match the high end tail of the FMNR predictions
for bare, charmed quarks where hard gluon emission is included
in the NLO description. It is possible to match the $P_t^2$ spectrum
of the data as shown in Figure \ref{nlopt}
by supplementing the NLO calculations with fragmentation effects
such as intrinsic gluon $k_t$ and momentum loss described by
Peterson\cite{peterson} fragmentation.
\begin{figure}[htp]
\hskip .5in
\epsfysize=3.55in
\epsffile{[rwg.tex.nu]nlopt.ps}
\caption{NLO calculations for various $k_t$ and fragmentation options
compared to E687 $D^+$ data. The Peterson fragmentation recipe is
used which softens the momentum of the $D^+$ in the photon-gluon
rest frame by a factor Z which is drawn from a QCD inspired distribution.
Fragmentation tends to soften the $P_t^2$ spectrum while $k_t$ hardens
this spectrum.}
\label{nlopt}
\end{figure}
\subsection{Correlations in Photoproduced $D \overline{D}$ Pairs}
\par
It is possible to fully reconstruct both members of a photoproduced
$D \overline{D}$ pair in order to more fully probe photoproduction
dynamics. As Figure \ref{soft} shows, there are two ways of doing
this. One can fully reconstruct both particles by detecting all of the
decay products of both decays. Of course one will necessarily pay a big
price in statistics
in acceptance and branching ratio to aquire a fully reconstructed
double D signal.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=1.8in
\epsffile{[rwg.tex.iu]soft.eps}
\caption{Two ways of studying photoproduced $D \overline{D}$ pairs.
One can fully reconstruct both charmed particles.
Alternatively one can use a partially reconstructed
double D sample obtained by identifying the soft pion from
$D^{*+} \rightarrow D^o \pi^+$
decay through the use of kinematic and charge correlation tagging.}
\label{soft}
\end{figure}
Alternatively, one can kinematically tag the soft (underlined) pion
from the decay $D^{*+} \rightarrow \underline{\pi}^+ D^o$ by
exploiting charge and
kinematic correlations between the pion and a fully reconstructed
recoil $\overline{D_r}$.
Because of the small energy release in $D^{*+} \rightarrow D^o \pi^+$
the soft pion , $D^o$, and $D^{*+}$ all have approximately the
same laboratory velocity. This allows one to estimate the
momentum of a $D^{*+}$ from just the soft pion by multiplying
its momentum by the $D^{*}/ \pi$ mass ratio\footnote{To take into
the account the very slight soft pion kinetic energy in the
$D^*$ rest frame, we actually use a scaling factor of 13.8 which is slightly
smaller than the $D^{*+}/ \pi^+$ mass ratio.}.
The charge of the soft
pion is correlated with the charm of the fully reconstructed, recoil
$D$ state and the scaled pion momentum should approximately balance $P_t$
with the recoil $\overline{D_r}$. We use the $P_t$ balance variable
$\Delta_t^2 = ~|~13.8~\vec P_t(\pi^+) + \vec P_t(\overline{D_r})~|^2$.
Figure \ref{ddbarsignal} shows the
signals for the fully reconstructed and partially reconstructed
$D \overline{D}$ samples \cite{ddbar} which start with a baseline
sample of $\approx 45,000$ inclusive $D$'s reconstructed
in the
$K^\mp \pi^\pm$, $K^\mp \pi^\pm \pi^\pm$ , and $K^\mp \pi^\pm \pi^+ \pi^-$
final state.
\begin{figure}[htp]
\hbox{
\epsfysize=2.2in\epsffile{ddbarscatter.ps}
\epsfysize=2.2in\epsffile{softpt.ps}
}
\caption{(a) Scatter plot of the normalized mass of detached $D$ candidates
versus the normalized mass of the recoil $\overline{D}$ candidate.
The normalized mass is the difference between the reconstructed and
known $D$ mass divided by the calculated event-by-event resolution.
The accumulation near the
origin consists of $\approx 320$ fully reconstructed $D \overline{D}$ pairs.
(b) Distribution of the $P_t$ balance variable $\Delta^2_t$ for right sign
and wrong sign decay pion candidates. The upper curve is the form expected
by fitting the net $P_t^2$ of fully reconstructed $D \overline{D}$ candidates
and smearing slightly for errors due to reconstructing the
$P_t$ of the parent $D^{*+}$ by scaling the $P_t$ of the decay $\pi^+$.
The right sign excess below $\Delta^2_t = 4~GeV^2$
consists of $\approx 4530$ events.
}
\label{ddbarsignal}
\end{figure}
\par
We have explored the kinematical correlations of the $D \overline{D}$
pair using several longitudinal and transverse variables. The
acoplanarity angle, $\Delta \phi$,
which illustrated by Figure \ref{acoplanarity} is used by many experimental
groups since most experiments
have excellent acceptance and resolution in this variable.
\begin{figure}[htp]
\epsfysize=2.5in
\epsffile{[rwg.tex.iu]diagram_deltaphi.eps}
\caption{The acoplanarity angle $\Delta \phi$ is the angle between the
$D$ and $\overline{D}$ in a plane transverse to the collision axis. If
$k_t$ or final state gluon emission were negligible
the $\Delta \phi$ distribution would peak sharply at $\pi$ radians.}
\label{acoplanarity}
\end{figure}
A related variable is the net transverse momentum of the observed
$D \overline{D}$ pair or
$P_t^2 (D\overline{D}) = |\vec P_t (D) + \vec P_t (\overline{D})|^2$.
A good longitudinal correlation variable is provided by
rapidity difference $\Delta Y = Y(D) - Y (\overline{D})$.
The rapidity or $Y = {1 \over 2}
\ln{\left( \left(E + P_\parallel\right) / \left(E - P_\parallel
\right)\right)}$
approaches the longitudinal velocity in the non-relativistic limit
but with the added virtue that a rapidity difference is relativistically
invariant. Finally the invariant mass of the $D \overline{D}$ pair
measures transverse as well as longitudinal kinematic correlations.
\par
Figure \ref{ddbar_corr} compares the $D\overline{D}$ kinematic correlations
observed in E687 data to the predictions of
the JETSET-PYTHIA Monte Carlo\cite{lund} run with default parameters.
\begin{figure}[htp]
\epsfysize=4.in
\epsffile{[rwg.tex.dpf94]ddbar_corr.ps}
\vskip 1.5in
\caption{
$D \overline{D}$ kinematic correlations observed in E687 data overplotted
with the predictions of the default PYTHIA model. We use the fully
reconstructed $D \overline{D}$ sample for the upper row of transverse
correlations and (because of its larger kinematic acceptance domain)
the partially reconstructed sample for the lower
row which involves longitudinal correlations. The solid curves
show acceptance corrected predictions; the dashed curves show the predictions
without acceptance effects.}
\label{ddbar_corr}
\end{figure}
There is good agreement between the $\Delta Y$ and $M(D \overline{D})$
longitudinal correlation variables
observed in the data to those predicted by the
default JETSET-PYTHIA Monte Carlo. This longitudinal correlation
data along with the $P_t$ distribution for inclusive photoproduced $D^+$'s
shown in
Figure \ref{pt2_nlo_lund}, and the asymmetry data presented in Section 2.4,
underscores this LO Monte Carlo's ability to successfully
model the important soft physics effects of fragmentation.
However the PYTHIA Monte Carlo gives far too much peaking of $\Delta \phi$
distribution near
$\pi$ and the $P^2_t (D \overline{D})$ distribution near zero.
Because the $\Delta \phi$ and $P^2_t (D \overline{D})$
distributions should be sensitive to hard gluon emission, it would
be surprising if PYTHIA got these transverse correlations right
since it does not incorporate the hard gluon emission processes
\footnote{JETSET does include bremstrahlung-like,
soft gluon emission in its parton shower fragmentation model.}
at NLO illustrated in Figure \ref{pgf2nd}.
\par
Indeed Figure \ref{nlo_trans}
which compares the NLO predictions of FMNR\cite{FMNR} to the
$\Delta \phi$ and $P^2_t (D \overline{D})$ distributions observed by
E687 shows that the NLO predictions agree with the data as long
as reasonable $< k^2_t> $ values are used.
\begin{figure}[htp]
\epsfysize=5.in
\epsffile{[rwg.tex.nu]nlo_1.ps}
\vskip-5.in
\epsfysize=5.in
\epsffile{[rwg.tex.nu]nlo_2.ps}
\caption{Overlay of the NLO predictions for the $\Delta \phi$ and
$P^2_t (D \overline{D})$ distributions to E687 data for several
$< k_t^2>$ choices.}
\label{nlo_trans}
\end{figure}
\subsection{Charm-Anticharm Asymmetries}
\par
In lowest order QCD,
the charm and anticharm quark are produced symmetrically
in photoproduction and hadroproduction.
In hadroproduction, at
next to leading order, QCD introduces small ( $ \approx 1 \%$ )
asymmetries in quark momenta due to interference between contributing
amplitudes.
As we shall see, however, a significant degree of
$c\overline{c}$ asymmetries can naturally arise through the
fragmentation process. Figure \ref{strings} illustrates the mechanism
responsible for photoproduced $c\overline{c}$ asymmetry in the
JETSET-PYTHIA model.
\begin{figure}[htp]
\vbox{
\epsfysize=2.25in
\epsffile{[rwg.tex.iu]strings.eps}
\epsfysize=2.25in
\epsffile{[rwg.tex.iu]associated_lund.eps}}
\label{strings}
\caption{In JETSET, strings are drawn between the triplet and anti-triplet
color poles.
Final state hadrons are produced as these strings fragment.
Quarks are color triplets; antiquarks are color anti-triplets.
In photon-gluon fusion, the initially colorless nucleon emits a color
octet gluon. The nucleon remnants must therefore form a color octet to
``cancel'' the color of the gluon. JETSET achieves color conservation
by resolving the nucleon remnants
into an anti-triplet, effective di-quark ($Q$) , and a
triplet, effective quark ($q$).
Because the $c$-quark must dress with the q-remnant and the
$\overline{c}$-quark must dress eith the $Q$-remnant
a difference between the quark and di-quark
momenta fraction distributions will lead to
an asymmetry between the momenta spectra of the charmed and anti-charmed
hadrons.}
\end{figure}
As shown in Figure \ref{strings}, the dominant source of
charm-$\overline{\rm charm}$
asymmetry in the JETSET-PYTHIA model for photoproduction arises through
the differences between the momentum fraction of the target nucleon
remnant carried by the effective quark and di-quark color poles.
The di-quark and quark remnants should be viewed as effective color pole
descriptions rather than actual quarks or di-quarks. As such there is no
reason to assume that their momentum fractions are the same as the quark
momentum distributions measured in deep inelastic leptoproduction.
In the default JETSET-PYTHIA, the effective quark is drawn
with a momentum fraction given by $d N/dx \propto (1 - x)^3 / x$ which has
the soft $1/x$ dependence expected for a sea quark distribution.
Since at the energies employed by E687, the gluons are typically very
soft as well, the default PYTHIA puts nearly all of
the nucleon momentum fraction into the effective di-quark. The
very different momentum fractions carried by the quarks and di-quarks
in the default PYTHIA Monte Carlo leads to asymmetries in the
charm-$\overline{\rm charm}$ particle momentum distributions which are
much stronger than those observed in the E687\cite{e687_asymmetry} data.
On the other hand, an alternative PYTHIA effective quark distribution
called the ``counting rule'' option
draws the effective quark momentum fraction according
to a much harder spectrum: $d N/dx \propto (1 - x)$. When one uses
the counting rule option, 1/3 of the nucleon remnant momentum on average
is carried by the effective quark
and 2/3 is carried by the di-quark.
A counting-rule Monte Carlo predicts significantly less
charm-$\overline{\rm charm}$ asymmetry which is in good agreement with the
experimental data.
\par
Figure \ref{asym_kine} compares the kinematic variation
of the photoproduced $D^+/D^-$ measured asymmetry to the PYTHIA-JETSET
Monte Carlo for sea-like (default) and counting rule effective quark
momentum fractions.
\begin{figure}[htp]
\epsfysize=5.in\epsffile{asym_kine.ps}
\caption{The asymmetry
$\alpha = \left(N(D^+) - N(D^-) \right) / \left(N(D^+) + N(D^-) \right)$
in bins of $P^2_t (D^\pm)$ , incident photon energy, and $X_F (D^\pm)$
for accepted $D^\pm$ photoproduced
from E687. The data is indicated with error bars, the sea-like effective
quark predictions are solid and the counting rule effective quark predictions
are dashed. The lower right corner compares the total lab momentum spectrum for
$D^\pm$ candidates to the two models. In contrast to the
asymmetry, the lab momentum spectrum
is insensitive to the assumed effective quark momentum distribution.
}
\label{asym_kine}
\end{figure}
The more symmetrical counting rule model does a very good job at matching
the kinematic trends and level of the $D^+$ asymmetry observed in the E687
data. It is also a reasonable match to the overall level of
photoproduced asymmetry observed for other charm decay modes
\footnote{Note that different asymmetries are predicted for two
different decay modes of the $D^o$. This is true since the asymmetries
are not corrected for exerimental acceptance and the two decay modes
have different acceptance versus momentum curves.}
as shown in Table \ref{asymmetry_table}.
\begin{table}[htp]
\caption{Observed asymmetry $\alpha$ (\%)
compared to predictions.}
\begin{center}
\begin{tabular}{l|l|l|l}
Decay mode & E687 data & Sea-like
& Counting Rule \\
\hline\hline
$D^+\rightarrow K^-\pi^+\pi^+$
& $ -3.8 \pm 0.9 $ & $ -10.4 \pm 0.4 $ & $ -2.9 \pm 0.3 $ \\
$D^{*+}\rightarrow D^0\pi^+ \rightarrow (K^-\pi^+) \pi^+$
& $ -6.4 \pm 1.5 $ & $ -9.2 \pm 0.7 $ & $ -2.4 \pm 0.5 $ \\
$D^{*+}\rightarrow D^0\pi^+ \rightarrow (K^-\pi^+\pi^+\pi^-) \pi^+$
& $ -4.0 \pm 1.7 $ & $ -9.2 \pm 0.8 $ & $ -3.0 \pm 0.5 $ \\
$D^0 \rightarrow K^-\pi^+$ (no $D^*$-tag)
& $ -2.0 \pm 1.5 $ & $ -5.1 \pm 0.6 $ & $ -1.6 \pm 0.4 $ \\
$D^0 \rightarrow K^-\pi^+\pi^+\pi^-$ (no $D^*$-tag)
& $ -1.9 \pm 1.5 $ & $ -9.9 \pm 0.5 $ & $ -2.9 \pm 0.4 $ \\
$D^+_s \rightarrow K^-K^+\pi^+$
& $ ~~ 2.5 \pm 5.2 $ & $ ~~ 9.7 \pm 1.7 $ & $ ~~ 2.5 \pm 0.7 $ \\
$\Lambda_c^+ \rightarrow pK^-\pi^+$
& $ ~~ 3.5 \pm 7.6 $ & $ ~ 21.5 \pm 0.7 $ & $ -7.7 \pm 0.6 $ \\
\end{tabular}
\end{center}
\label{asymmetry_table}
\end{table}
\par
Recently the E791 Collaboration studied the kinematic asymmetry
from their $\approx$74,000 event sample of $D^+$ produced with 500 GeV
$\pi^-$'s incident on a target consisting of four diamond and one platinum
foil. The asymmetry measured by E791 (with the opposite convention as ours)
is shown in Figure \ref{asym_791}. The basic trend of an asymmetry which
favors more $D^-$'s at larger $X_F$ is the same trend observed in the
photoproduction data of Figure \ref{asym_kine} but the level
of asymmetry is much larger in hadroproduction.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=3.in\epsffile{asym_791.ps}
\vskip 1.8in
\caption{Comparison of
E791's measurement (error bars) to various models for the
$\pi^-$ hadroproduction asymmetry. Their asymmetry convention is
a ``leading particle'' asymmetry with the $D^-$ being ``leading particle'',
has the opposite convention
as that used in Figure \ref{asym_kine} and Table \ref{asymmetry_table} for
E687.
Overlayed on the figure are the
predictions of default PYTHIA, a tuned PYTHIA model, the intrinsic
charm model of Vogt and Brodsky and FMNR NLO prediction for charmed quarks
which shows a very mild asymmetry for the bare charmed quarks.}
\label{asym_791}
\end{figure}
\par
This asymmetry pattern has been previously observed by
other groups\cite{leading} who study charm hadroproduction
where it goes under the name of
the leading particle effect (since the $D^-$ contains the same $\overline{u}$
quarks found initially in the $\pi^-$ projectile).
As is the case for photoproduction, the asymmetry observed in
hadroproduction, is a sensitive probe of fragmentation effects since
the $c \overline{c}$ asymmetry predicted in NLO for charm quarks is very small.
Again the overall level
of asymmetry in the default PYTHIA Monte Carlo exceeds the data but can
be matched once one increases the $< k^2_t >$ to 1 $GeV^2$ and increases
the charmed quark mass from 1.35 to 1.7 GeV. The pattern of asymmetry predicted
by the Brodsky-Vogt intrinsic model does not really match the data
although a better match might possibly be obtained by varying some
parameters.
\subsection{Production Dynamics Summary}
Although most of my discussion involved the QCD charm photoproduction
process, many of the same conclusions apply to high energy charm
hadroproduction. The NLO photon-gluon fusion
calculations of FMNR do a good job at reproducing
the photoproduced cross section over a large range of photon energies
from 20 GeV to over 30 TeV. Such calculations require a choice of charmed
quark mass , renormalization scale, and gluon momentum fraction distribution.
\par
Fragmentation effects play an important role in the $P_t$ and $X_F$
distribution for photoproduced charmed particles. In particular,
in the absence of fragmentation effects, NLO calculations
predict a significantly harder tail to the $P^2_t$ distribution for
photoproduced charmed mesons than is observed in E687 data.
Studies of the kinematic correlations between fully and partially
reconstructed $D\overline{D}$ pairs provide incisive tests of
QCD inspired fragmentation models such as the LO JETSET-PYTHIA
model. Longitudinal correlations such as the rapidity difference
between the $D$ and $\overline{D}$ are well reproduced by PYTHIA, whereas
transverse correlations such as the $D\overline{D}$ acoplanarity
angle distributions show considerably more smearing in data compared to
the PYTHIA Monte Carlo. This difference can be attributed to
the hard gluon emission process which occurs at NLO. Indeed
the FMNR NLO calculations predict an acoplanarity distribution which
is in good agreement with the data under reasonable fragmentation
and $k_t$ assumptions.
\par
Finally the asymmetry in the momentum spectrum between photoproduced and
hadroproduced charmed and anticharmed particles primarily reflects
the fragmentation process. The mechanism used in the JETSET-PYTHIA model
for color conservation in photon-gluon fusion where the target nucleon
remants appear as effective quark and di-quark color poles
provides a natural mechanism for charm-$\overline{\rm charm}$ asymmetry.
The asymmetry predicted by the JETSET-PYTHIA Monte Carlo does a good
job at matching asymmetries seen in E687 data provided that a simple
counting rule distribution which attributes (on average) 1/3 of the nucleon
remnant momentum to the effective quark and 2/3 of the remnant momentum
to the di-quark is used. JETSET-PYTHIA also does a good job of predicting
the kinematic asymmetry observed in hadroproduction for particular
parameter choices.
\setcounter{footnote}{0}
\section{Heavy Quark Decay}
A great many tools of QCD have been brought to bear on the issue of
heavy quark decay. In this section we will primarily review the physics of
charm rather than beauty decay, since in charm, the QCD corrections to the
weak decay process are particularly important.
On the most inclusive level, the lifetimes
of weakly decaying charmed particles serve as measurements of the total
decay rate. In the absence of QCD effects, the lifetimes of heavy flavor
states would depend on the heavy quark alone, rather than the specific charm
or beauty particle being considered. Although the lifetimes of the various
beauty states are all comparible, the lifetimes of various weakly decaying
charm states differ by roughly an order of magnitude.
Recent progress has been made in predicting the
deviations from a universal charm lifetime due to QCD corrections
through the use of heavy quark expansions in $1/M_Q$.
\par
We will turn next to a discussion of leptonic and semileptonic decays
of charmed mesons.
These are the simplest exclusive decays to understand since
all of the QCD effects can be expressed as modifications to a weak hadronic
current through form factors which can be computable in principle using
Lattice Gauge Theory. Since all semileptonic decay processes studied
thus far have a single final state daughter hadron, the
complications of final state interactions which
plagues non-leptonic decays are absent.
We next discuss polarization phenomena in charmed baryon decay
an area of charm decay where Heavy Quark Effective Theory makes
a very interesting prediction.
\par
We conclude with a discussion of non-leptonic charm decays
which are generally studied within a framework called factorization.
In this framework, two body charm decays are described by the interaction
of two hadronic currents which are similar to those describing leptonic
and semileptonic decays. Unfortunately, when several strongly interacting
final state particles are produced, they can continue to
interact after their formation via final state interactions. These final state
interactions can significantly distort the factorization predictions for
{\it eg} branching ratios into specific two pseudoscalar final states
by creating an {\it a priori}
unknown phase shift between interfering amplitudes.
These same strong phase shifts can be independently studied through
interference between the various quasi-two-body decay amplitudes
which contribute to a particular three or four body charm decay.
\subsection{Charm Particle Lifetimes}
There are seven known weakly decaying charmed particles with experimentally
measureable lifetimes. Last year the lifetime for the heaviest of these,
the $\Omega_c^o$ (css) (2704), was measured
for the first time. As you can see from the compendium of lifetime results
measured by E687 shown in Table \ref{lifetime_table} , there is
roughly an order of magnitude
difference between these seven lifetimes.\cite{lifetimes}
\begin{table}[htp]
\caption{E687 Lifetime Measurements (in picoseconds) }
\begin{center}
\begin{tabular}{l|l|l}
\hline\hline
$D^+$ & $D^o$ & $D_s^+$ \\
1.048 $\pm$ .015 $\pm$ .011 & 0.413 $\pm$.004 $\pm$ .003 &
$0.475\pm0.020\pm 0.007$ \\
\hline
$\Lambda_c^+ ~(udc)$ & $\Xi_c^+~(csu)$ & $\Xi_c^o~(csd)$ \\
0.215$\pm$0.016$\pm$0.008 & $0.41^{+0.11}_{-0.08}\pm 0.02$ &
$0.101^{+0.025}_{-0.017}\pm 0.005$ \\
\hline
$\Omega^0_c~ (css)$ & & \\
$0.089^{+0.027}_{-0.020} \pm 0.028 $ & & \\
\end{tabular}
\end{center}
\label{lifetime_table}
\end{table}
Before discussing the physics implication of such measurements, I would
like to use a few words to explain how such measurements are made.
\subsubsection{Lifetime Measurement}
\par
Most experimental information on charmed particle lifetimes comes from
fixed target experiments employing vertexing techniques.
In order to tag charm, one generally requires a minimal detachment
between the primary and charm decay vertex of
of $\ell > {\cal N}~\sigma_\ell$ where $\sigma_\ell$ is the
event-by-event estimate of the experimental resolution on the detachment
and ${\cal N}$ is the minimum detachment
in units of the computed measurement error, $\sigma_\ell$, deemed adequate
to insure a clean enough charm sample (often a 5 - 10 $\sigma_\ell$ cut is
used).
It might seem that the use of an event dependent minimum
lifetime cut which is required to isolate charm might complicate the process
of measuring charm particle lifetime. Actually these complications are actually
fairly minor. E687 and most other groups fit their lifetime evolution data
using what is known as a reduced proper time variable ,
$t' \equiv \left( \ell - {\cal N}~\sigma_\ell\right)/(c \beta \gamma)$
where the $(c \beta \gamma)^{-1}$ converts the distance to a proper time
including time dilation and the ${\cal N}~\sigma_\ell$ piece effectively
starts the clock at the point of the minimum detachment cut. An exponential
decay distribution is exponential no matter when the clock is started.
Given that the error on the detachment ($\sigma_\ell$ ) is
approximately independent of $\ell$
and that the acceptance usually has only a weak dependence on $\ell$, the
reduced proper time for a decaying state will closely follows an
exponential distribution expected for a state of lifetime $\tau$ :
$d N / d t' \approx \exp{\left(- t'/\tau\right)}$.
\par
We generally make
binned likelihood fits to the reduced proper time $(t')$ distributions
by minimizing a log-likelihood variable $w$ of the form:
\begin{eqnarray}
w = - 2 \sum_i^{\rm bins} \ln{{\cal P}(n_i ; \mu_i)}
- 2 \ln{{\cal P}(N_{\rm sidebands} ; 2B)} \nonumber\\
{\rm where}~~
{\cal P}(n_i ; \mu_i) \equiv { {\mu_i}^{n_i}~e^{-\mu_i} \over n_i!}~~
{\rm and}~\mu_i = S~f(t_i')~e^{-t'_i/\tau} + B~b_i
\label{binned_life}
\end{eqnarray}
A few words on Equation \ref{binned_life} might help.
The departure from a pure $\exp{\left(-t'_i/\tau\right)}$ form
due to acceptance variation and absorption of the charmed particles
or decay products as they travel through the target
are included through the factor $f(t_i')$
which is deduced via Monte Carlo simulation. \footnote{
E687 uses a vertex finder which efficient even at very small detachments
and hence the correction functions $f(t_i')$ for our experiment
are nearly constant. The one exception is that $f(t_i')$ cuts off
at long $t'$ for the very long lived
$D^+$ since it has a non-negligible probability
of decaying downstream of our silicon microstrip vertex detector!}
The fit maximizes the Poisson probability that $t'$ bin populations
$\{n_i\}$ of events in the signal region (about {\it eg} the $D^o$)
matches the number expected $\{\mu_i \}$
for a given lifetime $\tau$ and background level $B$ which are the two
fit parameters. The fit accomplishes this by minimizing $w$
with the extra factor of $-2$ included so that the $\pm 1 \sigma$
errors correspond to changes of $ w - w_{\rm min} = 1$.
Figure \ref{sidebands} illustrates the background subtraction method
used in this type of fit.
\begin{figure}[htp]
\hskip 1.5in\epsfysize=2.2in\epsffile{sidebands.eps}
\caption{ A sketch of the histogram used to deduce parameters
for the lifetime fit described by Equation \ref{binned_life}.
The log-likelihood variable $w$ is constructed from the
population of signal region events found in each bin of reduced
proper time ($t_i'$). The normalized background shape $b_i$ is taken
directly from events falling in the equal width, sideband regions
marked $B_1$ and $B_2$. Information which ties the fit parameter $B$
to the average of the number of events in each sideband is communicated
to the fit through the inclusion of the term
$- 2 \ln{{\cal P}(N_{\rm sidebands} ; 2B)}$}
\label{sidebands}
\end{figure}
This procedure should be valid and free of systematics to the extent
that time evolution for the background
under the signal can be adequately by events with masses near
the signal peak and that proper time resolution is small compared to
the lifetime.
\par
For the case where $\sigma(\tau) \ll \tau$
\footnote{In a fixed target experiment, $\sigma_\ell \propto P ({\rm charm~state})$
and hence one has a resolution on $t'$ which independent of momentum
and about the same for all fully reconstructed charm states. For E687 ,
$\sigma ( t') \approx \tau(D^o) / 10$ , and hence for many
of measurements reported in Table \ref{lifetime_table} proper time
resolution is unimportant.}
, and the background level
is small, the anticipated fractional statistical error on the lifetime is
given by $\sigma_\tau / \tau = 1/ \sqrt{N_{\rm sample}}$.
This means that a $\pm 1 \%$ statistical error on $\tau$
(which is comparable to present systematic errors due to background
uncertainties) would require about
10,000 reconstructed events. This yield has already been achieved in
experiments such as E687 and E791. Interestingly enough, the lifetime
knowlege for the $D$ system is already comparable to that for kaons
: ~$ \sigma_\tau / \tau = 0.1 \% \rightarrow 0.8 \%$ for kaons.
\subsubsection{Lifetime Physics}
Lifetimes represent the most inclusive way of looking at charm particle
decay. The lifetime is related to the total decay width
via the expression:
\begin{equation}
\Gamma_{T} (C_x) = \sum_i~\Gamma_i = {\hbar ~\over~\tau (C_x)}
\label{decay_width}
\end{equation}
The partial width ratio ($\Gamma_i /\Gamma_T$) gives the fraction of times
a given charm particle $C_x$ will decay into a particular final
state $i$. Theoretical predictions for the decay rate into a given final
state , $i$, are generally expressed in terms of the partial width
$\Gamma_i$.
In the absence of the quark re-arrangement effects permitted by QCD
charm particles would expected to proceed by the naive spectator
model depicted in Figure \ref{mulife}.
\begin{figure}[htp]
\hskip .5in \epsfysize=2.in\epsffile{spectator.eps}
\caption{
In naive spectator model, the lifetime of charmed particles should
have a universal value which can be estimated by scaling from the known muon
lifetime of 2.2 $\mu s$. When scaling decay rates , one would
need to consider the fact that five low mass final states are
available for the charmed quark (two lepton generations and three
color multiplets of the first quark generation), and that the
charm quark has a much larger decay phase space owing to its much larger mass.}
\label{mulife}
\end{figure}
We thus expect in the naive spectrator model, a universal
charm decay width given
(in the absence of HO corrections and QCD effects) by:
\begin{eqnarray}
\Gamma_{T} = (2 + 3)~\Gamma_o ~~{\rm where}~\Gamma_o =
{G_F^2 m_c^5 \over 192 \pi^3} ~|V_{cs} |^2 \phantom{xxxxxxxxxxxxxx}
\nonumber\\
{\rm Scaling~from ~the~muon:}~~
\tau_C = {1 \over 5}~\left({0.105 \over 1.5}\right)^5~2.2 \times 10^{-6}
= 7 \times 10^{-13}~s
\label{mu_life}
\end{eqnarray}
Of course given the fact that the charm lifetimes are not universal,
but differ by an order of magnitude, there must be additional non-spectator
diagrams which contribute differently for the various charm species.
Figure \ref{PI_WA_WX} gives examples of non-spectator diagrams which
could account for lifetime differences amongst the charmed mesons.
\begin{figure}[htp]
\hskip 1.in \epsfysize=2.2in\epsffile{non_spectator.eps}
\caption{$D^+$ decays always have two ways of re-arranging the
quarks $W^+ \rightarrow u {\overline d}$ into the final state mesons.
Because this involves Fermion exchange one expects
a destructive (Pauli) interference between the interfering amplitudes which
will suppress the rate for $D^+$ decay and enhance the $D^+$ lifetime.
The $c {\overline s}$ quarks comprising the $D_s^+$ have a CKM allowed
coupling directly to a virtual $W^+$ giving a CKM favored weak
annihilation contribution unique among mesons to the $D_s^+$.
Finally the $D^o$ has a CKM allowed $W^+$ exchange contribution which
among mesons is unique to the $D^o$.}
\label{PI_WA_WX}
\end{figure}
Although it is relatively easy to imagine non-spectator decay processes that
differentiate among the charmed particles, it has more difficult to
assess their relative importance until recently.
Systematic progress has been made by the use of Heavy Quark expansion
techniques which include QCD corrections as parameters of an expansion
for the semileptonic and non-leptonic widths of the form
\cite{hq_expansion}$^-$\cite{tcw_bigi}~~:
\begin{eqnarray}
\Gamma_{SL} = \Gamma_0~
\left(1 + {B_2 \over m^2_c} + {B_3 \over m_c^3} + ...\right)\nonumber \\
\Gamma_{NL} = N_c~\Gamma_o~
\left(1 + {C_2 \over m^2_c} + {C_3 \over m_c^3} + ...\right) \nonumber \\
{\rm where}~\Gamma_o =
{G_F^2~M_Q^5 \over 192 \pi^3} ~|V_{Q q'} |^2
\label{hq_expansion}
\end{eqnarray}
The expansion of Eqn. \ref{hq_expansion} has some appealing
as well as disturbing aspects.
As $M_Q \rightarrow \infty$, one recovers the predictions of the naive
spectator model illustrated in Figure \ref{mu_life}
which features a universal single generation decay
width $\Gamma_0$ which depends on the CKM matrix element and
the quark mass that controls the size of phase space available
to the decay products. Hence the lifetimes of various beauty particles
should be fractionally much closer than those for charm owing to the heavier
$b$-quark mass. The processes which differentiate among the various
heavy quark species such as those depicted in
Figure \ref{PI_WA_WX} contribute to the coefficient $C_3$.
\par
Tables \ref{life_hqe_charm} and \ref{life_hqe_beauty}
, taken from Bigi\cite{tcw_bigi} , compare
various experimental lifetime ratios to the predictions of
a $1/M_Q$ expansion which uses various inputs including
theoretical estimates for the lepton decay constants of the $D$
and the mass and mass splitting pattern among various
charm hadrons.
\begin{table}[htp]
\caption{Comparison of data to $1/M_Q$ expansion in the charm sector}
\begin{center}
\begin{tabular}{l|l|l}
lifetime & QCD & Data \\
ratio & $1/M_c$ exp & \\
\hline\hline
$D^+/D^o$ & $\approx 2$& 2.547 $\pm$ 0.043 \\
$D_s /D^o$ & $\approx 1 $ & 1.125 $\pm$ 0.045 \\
$\Lambda_c / D^o $ & $\approx 0.5$ & 0.51 $\pm$ 0.05 \\
$\Xi_c^+ / \Lambda_c$ & $\approx 1.3 $ & 1.75 $\pm$ 0.36 \\
$\Xi_c^+ / \Xi_c^o$ & $\approx 2.8 $ & 3.57 $\pm$ 0.91\\
$\Xi_c^+ / \Omega_c$ & $\approx 4 $ & 3.9 $\pm$ 1.7 \\
\end{tabular}
\end{center}
\label{life_hqe_charm}
\end{table}
\begin{table}[htp]
\caption{Comparison of data to $1/M_Q$ expansion in the beauty sector}
\begin{center}
\begin{tabular}{l|l|l}
lifetime & QCD & Data \\
ratio & $1/M_b$ exp & \\
\hline\hline
$B^- / B_d$ &$\approx 1$ & 1.04 $\pm$ 0.05 \\
$B_s / B_d $&$\approx 1$ &0.98 $\pm$ 0.08 \\
$\Lambda_b / B_d$ &$\approx 0.9$ & 0.76 $\pm$ 0.06 \\
\end{tabular}
\end{center}
\label{life_hqe_beauty}
\end{table}
\par
As expected , the lifetimes are much closer between the species measured
for the beauty sector than is the case for the charm sector.
The rough factor of two difference
between the $D^+$ and $D^o$ lifetime suggests that the PI mechanism
,shown in Figure \ref{PI_WA_WX}, is important; while
the relative equality between the $D^o$ and $D_s^+$ lifetime suggests
that the WA process must not be very important.
\par
To the heavy quark experimentalist, Eqn. \ref{hq_expansion} contains
the ominous prediction that the semileptonic widths are not universal
among the various charm or beauty species. This is disturbing news since
most of our present knowlege of charm particle branching ratios beyond
the well-known $D^+$ and $D^o$ absolute branching fractions,
depend on the assumption of a universal semileptonic width.
\footnote{Absolute branching fractions for the $D^+$ and $D^o$
can be obtained in $e^+ e^-$ annilation just above charm threshold by
studying events where both charmed particles are reconstructed.
Alternatively, one can tag the presence of a $D$ by kinematically
tagging pions from $D^* \rightarrow \pi D$ using techniques similar
to those described in Section 2.3 and count the number of times the tagged
$D$ decay decays into a specfic final state.}
The hope that there might be a universal semileptonic width
is based on experience with the $D^+ - D^o$ system where it was noted
that the ratio of the $D^+/D^o$
semileptonic width was approximately the ratio
of the $D^+/D^o$ lifetime\cite{cumalat}:
\begin{equation}
{{\cal B}_{SL} (D^+) \over {\cal B}_{SL} (D^o)} = 2.3 \pm 0.4 \pm 0.1~~\approx~~
{\tau(D^+) \over \tau (D^o)} = 2.54 \pm 0.07
\label{sl_life}
\end{equation}
This observation implies that
$\Gamma_{SL} (D^+) \approx \Gamma_{SL} (D^o)$ since
${\cal B}_{SL} = \Gamma_{SL}/ \Gamma_T$ and
$\Gamma_T (D^o) /\Gamma_T (D^+) = {\tau(D^+) / \tau (D^o)}$.
However it is really not correct to infer a universal semileptonic
width for all charm species based on this evidence since the $D^+$ and
$D^o$ are in the same isodoublet and the $c \rightarrow s$ transitions
involved in semileptonic decays will transform as $\Delta I = 0$.
None-the-less encouraged by the possibility of a universal $\Gamma_{SL}$,
groups had been estimating absolute branching fraction through the
technique suggested by Eqn. \ref{absolute_bf}.
\begin{equation}
{\cal B}_x = {\# C \rightarrow x \over \# C \rightarrow SL}~~{\cal B}_{SL}
~~,~~{\cal B}_{SL} = {\Gamma_{SL} \over \Gamma_{T}}~~,~~
\Gamma_{T} = {\hbar \over \tau_C}
\label{absolute_bf}
\end{equation}
One can first infer the semileptonic branching ratio
(${\cal B}_{SL}$) using the
measured lifetime of the state ($\tau_C$)
and the value of a ``universal'' $\Gamma_{SL}$
measured from the average of the known $D^+$ and $D^o$ values. Next
the absolute branching fraction into state $x$ can be measured
relative to the infered ${\cal B}_{SL}$ by measuring the fraction of
times the state decays into the $x$ or $SL$ final state.
Although the assumption of a universal $\Gamma_{SL}$ is invalid,
the expected variance according to Bigi\cite{tcw_bigi} is not too large.
The relative ratios of $\Gamma_{SL}$ are expected to be:
\begin{table}[h]
\caption{Expected relative values for $\Gamma_{SL}$ }
\begin{center}
\begin{tabular}{l|l|l|l|l}
$D^+$ & $D^o$ & $D_s^+$ & $\Lambda_c^+$ & $\Omega_c^o $ \\
\hline\hline
1 & 1 & $\approx 1$ & $\approx 1.5$ & $\approx 1.2 $ \\
\end{tabular}
\end{center}
\label{sl_widths}
\end{table}
\subsection{Leptonic Decays of the $D_s^+$}
\par
There has recently been a new measurement of $D_s^+ \rightarrow \mu^+ \nu$
by the E653 Collaboration. Figure \ref{ds_leptonic} illustrates
several features of the $D_s^+$ decay process.
\begin{figure}[htp]
\hskip 1.in\epsfysize=1.5in\epsffile{fds2.eps}
\caption{The $D_s^+$ leptonic decay has the $c \overline{s}$ quarks
coupling to the virtual $W^+$ carrying a 4 momentum $q_\mu$
with a CKM favored coupling constant.
The virtual $W^+$ then decays into an $\ell^+$ and a lefthanded $\nu$.
Since the $D_s^+$ is spinless, the $\ell^+$ must emerge lefthanded
in order to conserve angular momentum along the lepton decay axis.
A righthanded antilepton has a highly suppressed $V - A$ coupling.
In the context of a quark model this process provides information
on the quark wavefunction near the origin.}
\label{ds_leptonic}
\end{figure}
Nature handles the fact that the $\ell^+$ is forced into
the ``wrong'' helicity state by suppressing the decay width by two powers
of the charged lepton mass as shown Eqn. \ref{dslnu_width}. This means
that the only non-negligible leptonic decays for the $D_s^+$ will be into
taus and muons (with tau's being favored by a factor of $\approx 10$.)
Because the $\tau^+ \nu$ decay includes at least two missing final state
neutrinos , it is very hard to experimentally reconstruct which makes
the muonic decay the decay of choice among experimentalists.
\par
Eqn. \ref{lepton_decay_constant} shows the structure of the leptonic
decay amplitude which a current $\times$ current interaction
with one current being the familiar $V - A$ current comprised
of the leptons and the other current being an {\it a priori} unknown
current involving the $D_s^+$.
\begin{eqnarray}
{\cal M} = {G_f \over \sqrt{2}}~\bar u_\ell \gamma^\mu~(1 - \gamma_5)
v_\nu <0 | ~J^\dagger_\mu~| D_s^+> \nonumber\\
<0 | J^\dagger_\mu | D_s^+> = i ~V_{cs}~f_{Ds}~q_\mu \phantom{xxxxxxxxxxxxx}
\label{lepton_decay_constant}
\end{eqnarray}
This current must be constructed out of the only available vector $q_\mu$
which is the four momentum carried by the virtual $W^+$. The current
also contains a potentially $q^2$ dependent form factor, but since
$q^2 = M^2(D_s^+)$ this form factor just becomes the constant known as the
$D_s^+$ leptonic decay constant.
Eqn. \ref{dslnu_width} relates the leptonic decay constant to the
the $D_s^+$ leptonic width:
\begin{equation}
\Gamma(D_s \rightarrow \ell \nu)
= {G_f^2 |V_{cs}|^2 \over 8 \pi} f^2_{D_s}~M_{D_s}~M_\ell^2~
\left( 1 - {M^2_\ell \over M^2_{D_s}} \right)
\label{dslnu_width}
\end{equation}
Experimentally one can measure $f^2_{Ds}$ through the decay width
expression by measuring the yield of $D_s^+ \rightarrow \mu \nu$ decays
to a reference state with a surmised decay width such as
$\phi \ell^+ \nu$ as discussed in Section 3.1.2. Data from BES uses
infers the width directly from the lifetime using a double tag technique.
\par
Being the theoretically simplest charm decay, there are many
theoretical estimates of leptonic decay constants which use a variety of
techniques including Lattice Gauge Theory , QCD sum rules , and explicit
quark models. Richman and Burchat\cite{richman} contains
a summary of these calculations
and a fuller discussion of the experimental techniques.
Figure \ref{decay_constant_data}
compares the experimental data \cite{wa75_lep}$^-$\cite{e653_lep}
to the Lattice Gauge calculations. Agreement of the theory
with experiment is quite good.
\begin{figure}
\epsfysize=2.5in\epsffile{[jew.e687.ssi]fds.ps}
\caption{An informal weighted average of the four experimental
results gives $f_{D_s} = 242 \pm 32~MeV$
at $\approx 20 \%$ CL level. This value and error form the dashed and solid
horizontal lines. Also shown are the Lattice Gauge Theory predictions.}
\label{decay_constant_data}
\end{figure}
\subsection{Semileptonic Decays}
The basic heavy quark semileptonic process, depicted in Figure \ref{semi_dk},
suggests that a major reason to study
semileptonic decays is the measurement of $|V_{Qq}|^2$.
\begin{figure}[htp]
\hskip 1.in\epsfysize=1.2in\epsffile{semi_diag.eps}
\caption{The basic three body, semileptonic process where a parent $P$
transforms into a daughter hadron $D$ while emitting a virtual W
which materializes into a $\nu \ell^+$ pair. The QCD processes
which govern the form factors are complicated but can be computed
in principle using Lattice Gauge Theory.}
\label{semi_dk}
\end{figure}
Historically it is true that many CKM matrix elements were first measured
through semileptonic decay including nuclear $\beta$ decay ($V_{ud}$),
$K \rightarrow \pi e \nu$ ($V_{us}$), and more recently inclusive beauty
semileptonic decay at CLEO $(V_{ub} \& V_{cb})$.
The grey area of Figure \ref{semi_dk} meant
to describe the QCD process by which the $\overline{q'}Q$ appears as
a fully dressed daughter hadron, is understood in principle but difficult
to calculate in practice without approximation.
\par
Only three body semileptonic
decay processes have been studied thus far. All the QCD effects
for {\it eg} pseudoscalar $\ell^+ \nu$ decays are contained in two $q^2$
dependent form factors which describe how the hadronic weak current
between the parent and daughter is constructed from their four momenta:
\begin{equation}
< D | J^\dagger_\mu | P >~ =~ f_+(q^2)~(P + D )_\mu
+ f_-(q^2)~(P - D)_\mu
\label{form_factor}
\end{equation}
Most information is known about the $f_+(q^2)$ form factor
since in width expressions, terms involving $f_-(q^2)$ always
multiply (the generally small except for the $\tau$) charged lepton
mass.
\par
For charm semileptonic decays which represent heavy-to-light transitions,
the primary theoretical tools have been Lattice Gauge Theory (LGT)
and quark models.\footnote{ In $b$ to $c$ decays, Heavy Quark Effective Theory,
provides considerable understanding of beauty semileptonic decay physics.}
In LGT\cite{hasenfratz}
, form factors are evaluated as the expectation value of 3-point
correlation functions describing the parent, current, and daughter.
These expectation values involve evaluating integrals by summation over
a four dimensional space-time lattice of size $L$ and spacing $a$.
Naively computation times would scale as $(L/a)^4$ so there is a
computational premium in keeping $L$ as small as possible (but no smaller
than the extent of the hadron wave function $\approx 1~fm$), and $a$
as large as possible (but smaller than the daughter wavelength).
Generally calculations are performed with the daughter at rest in the
parent rest frame as depicted in Figure \ref{qmax}:
\begin{figure}
\hskip 1.in \epsfysize=1.2in\epsffile{qsq_max.eps}
\caption{LGT and Quark Model semileptonic form factor calculations work best
when the hadron daughter is at rest. In this configuration $q^2$ which is
the $\ell^+ \nu$ invariant mass is at $q^2_{max}$}
\label{qmax}
\end{figure}
This is also true of Quark Model calculations which utilize
quark wavefunctions. As the daughter momentum increases
and $q^2 \rightarrow q^2_{min}$, the form factor probes
the tail of the wave function which is the least accurately known part.
\par
Unfortunately as we will see, the decay intensity of the data is
peaked near the maximum daughter momentum or
$q^2 \rightarrow q^2_{min}$ where theory has the hardest time.
For the case of pseudoscalar $\ell^+ \nu$ decays,
the current of Eqn. \ref{form_factor} gives the differential width
expression given by Eqn. \ref{dgamma_dq}.
\begin{equation}
{ d^2\Gamma \over dq^2} =
{G_F^2|V_{Qq}|^2 |\vec h^{(D)}|^3~\over 24\pi^3} \{|f_+(q^2)|^2
+m_\ell^2~|f_-(q^2)|^2 ...\}
\label{dgamma_dq}
\end{equation}
where $\vec h^{(D)}$ is the three momentum of the daughter hadron in the
$D$ rest frame.
These three powers of $|\vec h^{(D)}|$ (one from phase space and two
from the squared amplitude modulus) , severely deplete the intensity
near $q^2_{max}$.
\par
In order to bridge the gap between $q^2_{min}$ whre the data is measured
and $q^2_{max}$ where the data is predicted, most groups use
assume the pole form {\it ansatz} given by Eqn. \ref{pole_form}.
\begin{equation}
f(q^2) ={f(0) \over 1 - q^2/m^2_{\rm pole}}
\label{pole_form}
\end{equation}
Expectations are that $m_{\rm pole}$ will be the lowest mass
$Q \overline{q}$ resonance with the same spin-parity as the hadronic weak
current proportional to $V_{Qq}$. Figure \ref{hairpin} gives a cartoonish
``explanation'' of why one might expect a pole-like $q^2$ dependence
with a pole mass given by the spectrum of $D^*$ and $D^{**}$ mesons.
\begin{figure}[htp]
\hskip 1.in \epsfysize=1.8in\epsffile{pole_form.eps}
\caption{Illustration of the coupling of a virtual $W^+$ to
the $c \rightarrow s$ quark. For pseudoscalar $\ell^+ \nu$
decays, one can think of the hairpin as a virtual $D_s^*$
vector state comprised of $c \overline{s}$ quarks which contributes
a propagator of the form $~~~~~~~~~~~~~~~1/(m^2(D_s^*) - q^2)$.}
\label{hairpin}
\end{figure}
Although my version of the pole dominance argument seems {\it ad hoc},
Burdman and Kambor\cite{burdman}
recently made a detailed dispersion relation calculation to show that
contributions of higher poles should be neglible for charm.
\par
Perhaps the most elegant way of thinking about the kinematics
of three body semileptonic processes is via the Dalitz
plot which is sketched in Figure \ref{dalitz_sketch}.
\begin{figure}[htp]
\epsfysize=4.3in\epsffile{dal_kin.eps}
\caption{We chose $q^2 \equiv M^2_{\ell nu}$ and $M^2_{h\ell}$ as the
two conjugate mass variables. The boundaries depend
on the masses of the $D$ , final state hadron, and charged lepton,
abbreviated as $D~,h,~{\rm and}~\ell$ in the figure.
Several decay configurations are drawn in the virtual $W^+$ or
$q$ rest frame. The momentum magnitudes of the $\ell , \nu , h$
are fixed at a given fixed $q^2$ with the lepton momenta increasing
and hadron momentum decreasing as $q^2$ increases. When $\ell^+$
and hadron are collinear, one reaches the left boundary of the plot;
When $\ell^+$ and the hadron are anticollinear, one reaches
the right boundary of the plot. The expression for
$M^2_{h \ell}$ is of the form $A - B~\cos{\theta^{(q)}_{h \ell}}$.}
\label{dalitz_sketch}
\end{figure}
As shown in Eqn. \ref{dalitz_ps},
the great virtue of the Dalitz plot is that the three body decay
phase space is doubly differential in the Dalitz variables
which can be taken as any two of the three possible
two body invariant masses constructed from three final state daughters.
The form insures that a process with a constant amplitude ${\cal M}$, will
uniformly populate the Dalitz plot.
\begin{equation}
d^2 \Gamma = { |{\cal M} |^2 \over 256 \pi^3~M^3}~
d M^2_{h \ell}~d M^2_{\ell \nu} ~~=~~
{ |{\cal M} |^2 \over 256 \pi^3~M^3}~
d M^2_{hl}~d q^2
\label{dalitz_ps}
\end{equation}
For the case of pseudoscalar $\ell^+ \nu$ decays , the ${\cal M}$
is not uniform but rather of the form given by \ref{matrix_element}.
\begin{equation}
{\cal M} \propto
f_+(q^2)~| \vec h^{(D)} |~\sin{\theta^{(q)}_{h \ell}}
\label{matrix_element}
\end{equation}
The $\sin{\theta^{(q)}_{h \ell}}$ dependence is an easy to
understand consequence of the $V - A$ helicity rules as illustrated
in Figure \ref{sin_squared}. This angular factor will cause
the decay intensity Dalitz to disappear at the Dalitz boundary
where $\theta = 0$.
\begin{figure}[htp]
\hskip 2.in\epsfysize=2.3in\epsffile{hlnu_angle.eps}
\caption{
We view the decay in the virtual $W^+$ rest frame.
Since the $D$ is spinless and the hadron, $h$, is spinless
the virtual $W^+$ must have zero spin along the axis formed by
the hadron. The $\nu$ is lefthanded while the $\ell^+$ is righthanded
which means the virtual $W^+$ has a spin of 1 along the lepton axis.
The amplitude for having the $W^+$ spin with a spin of 1 along
the hadron axis and a spin of 0 along the lepton axis is proportional
to the Wigner D-matrix $d^{(0)}_{1 0} (\theta) \propto \sin{\theta}$.
The amplitude must clearly disappear when the hadron and lepton
axes are either collinear or anticollinear and the $W^+$ would need
to have two different spin components along the same axes simultaneously.}
\label{sin_squared}
\end{figure}
Figure \ref{klnu_pilnu} shows simulated Dalitz plots for
$D^+ \rightarrow K^- \mu^+ \nu ~,~{\rm and}~ \pi^- \mu^+ \nu$ Decays.
\begin{figure}
\hbox{
\epsfysize=2.2in\epsffile{kmunu.ps}
\epsfysize=2.2in\epsffile{pimunu.ps}
}
\caption{Simulated $D^+ \rightarrow K^- \mu^+ \nu $ (left)
and $D^+ \rightarrow \pi^- \mu^+ \nu $ (right). The much larger
phase space for the pion decay is evident. The point density
decreases near the Dalitz boundary and is strongest at low
$q^2$.}
\label{klnu_pilnu}
\end{figure}
\subsubsection{Experimental Techniques}
Semileptonic charm physics presents a real challenge to
experimentalists because most experiments which can study charm cannot directly
detect the missing neutrino. Although there is often a way of extracting
information about the neutrino through missing $P_t$, the knowlege of the
neutrino momentum is often ambiguous and inaccurate. This leaves the
experimentalist in a double bind. In contrast to the case of
a fully reconstructed final state, there is no well resolved
charm signal peak in the invariant mass distribution and hence
there are many backgrounds where one or more final state particles
escapes detection and thus closely resembles the
desired final state with a single missing neutrino.
One is also faced with the problem of not being able to accurately measure
the submasses needed to measure the form factors
from the semileptonic final state.
\par
Figure \ref{closure} shows how the missing neutrino is reconstructed
from the measured $D$ line of flight in a fixed target experiment
like E687.
\begin{figure}[htp]
\epsfysize=2.5in\epsffile{ambiguity_ft.eps}
\caption{Consider the decay $D^0 \rightarrow K^- \mu^+ \nu$.
By observing tracks from an upstream primary vertex and the
downstream vertex one can reconstruct the $D^o$ line of flight.
One boosts the kaon and muon along the $D$ line of flight until the
sum of their longitudinal momentum vanishes : {\it ie} the boosted
visible momentum lies transverse to the $D^o$ line of flight.
The neutrino momentum must lie on the line which balances the
visible $P_t$. The neutrino momentum must lie on a circle
since the magnitude of the neutrino momentum can be calculated
from its $P_t$ and the $D^o$ mass.
The two circle-line intersections give two ambiguous
solutions for the neutrino momentum.
In this example, the backward hemisphere solution has the larger
$q^2$ since the angle between the reconstructed neutrino and muon
is larger.}
\label{closure}
\end{figure}
While the neutrino is equally likely to emerge in either hemisphere
in Figure \ref{closure} frame, when the neutrino
appears in the backward hemisphere, the charged system tends to be shot
forward in the lab frame and gets accepted in the typical fixed target
forward spectrometer. For this reason, most fixed target results
resolve the ambiguity by selecting the solution with the neutrino
in the backward hemisphere. Although this properly resolves the ambiguity
most of the time, smearing on $q^2$ due to wrong choices and poor resolution
on the $D^o$ line of flight creates a highly significant smearing in $q^2$.
\par
We turn next the problem of fitting the semileptonic decay intensity
to extract information on form factors. As an example, we
describe the technique used by the E687 Collaboration\cite{e687_klnu}
to extract information on $f_+(q^2)$ in Eqn. \ref{form_factor}.
through a fit to the $D^o \rightarrow K^- \ell^+ \nu$ Dalitz plot.
A binned, likelihood fit is made to the $K^- \ell \nu$ Dalitz plot.
The idea in such a binned fit is to maximize the agreement
between the set of observed bin populations ($\{ n_i \}$) and the
set of expected populations ($\{\mu_i\}$) for a given
set of form factor parameters. This is done by having the fit
vary the form factors to minimize the log-likelihood parameter $w$
given by Eqn. \ref{binned_likelihood}.
\begin{equation}
w = -2 \sum^{\rm bins}_i
~\ln{{\cal P}(n_i ; \mu_i)}~~,~~{\rm where}~~
{\cal P}(n_i ; \mu_i) \equiv { {\mu_i}^{n_i}~e^{-\mu_i} \over n_i!}
\label{binned_likelihood}
\end{equation}
The complication in the procedure is that the expected bin populations
for a given set of form factors, $\{ \mu_i \}$,
must reflect the considerable diffusion of events into and out of a given
reconstructed $q^2$ bin due to smearing.
In order to properly handle the possible biases and error inflation due
to $q^2$ smearing, most groups use some variant of the
fitting techique developed by the E691\cite{e691_nim}
Collaboration, which I will call Los Vegas weighting
which is illustrated in Figure \ref{los_vegas}.
\begin{figure}[htp]
\hbox{
\epsfysize=3.in\epsffile{losvegas_1.eps}
\epsfysize=3.in\epsffile{losvegas_2.eps}
}
\caption{ A Monte Carlo simulation using an initial guess for the relavant
form factors is used to simulate the $q^2$ smearing due to neutrino
reconstruction. This smearing will cause events to diffuse into and
out of each bin. The program saves the generated Dalitz variables as well as
the smeared Dalitz variables for each simulated event. As the fitter
changes the form factors to minimize $w$ , each simulated event is assigned
a weight given by the ratio of the intensity with the new form factors
divided by the ratio of the original intensity with the initial guess.
The expected number of observed counts $\mu_i$ within a bin
is computed from the sum of the weights for events falling within
the $i$'th bin. This technique avoids prohibitively
computer intensive, separate full Monte Carlo simulations
for each iteration of the fit.}
\label{los_vegas}
\end{figure}
\subsubsection{$D^+ \rightarrow K^- \ell^+ \nu$}
\par
The decay $D^+ \rightarrow K^- \ell^+ \nu$ gives the best information
on the $q^2$ dependence of the $f_+(q^2)$ form factor. The most
recent, high statistics data comes from E687\cite{e687_klnu}
and CLEO\cite{cleo_klnu}~.
Figure \ref{cleo_klnu_sig} shows CLEO's $\approx 2700$ event sample of
$D^*$ tagged $D \rightarrow K \ell \nu$ events.
\begin{figure}[htp]
\hskip 2.in
\epsfysize=1.8in\epsffile{[jew.e687.hq94]cleo_kmunu.ps}
\vskip 1.in
\caption{CLEO's $D^* - D$ mass difference for
(a) $D^o \rightarrow K^- e^+ \nu$,
(b) $D^o \rightarrow K^- \mu^+ \nu$,
(c) $D^{+} \rightarrow K_s e^+ \nu$ ,
and
(d) $D^{+} \rightarrow K_s \mu^+ \nu$.
The background estimates are given by the dashed curves.}
\label{cleo_klnu_sig}
\end{figure}
The (uncorrected) $q^2$ distribution of the CLEO $K \ell \nu$ sample
is shown in Figure \ref{cleo_klnu_qsq}.
\begin{figure}[htp]
\vskip -.5in
\hskip 1.5in
\epsfysize=1.8in\epsffile{[jew.e687.hq94]cleo_qsq.ps}
\vskip 1.in
\caption{The uncorrected (for acceptance and smearing) $q^2$ spectrum
measured by CLEO in their $\approx 2700$ event sample
of $D \rightarrow K \ell \nu$
decays. As suggested by Eqn. \ref{dgamma_dq} , very few decays
are observed near $q^2_{max} \approx 1.9~GeV^2$.}
\label{cleo_klnu_qsq}
\end{figure}
\par
Figure \ref{e687_tagged} shows the $D^{*+} - D^o$ mass difference
distribution for E687's $\approx 430$ event sample of
$D^{*+} \rightarrow (K^- \mu^+ \nu) \pi^+$ decays along with their
uncorrected $q^2$ distribution.
\begin{figure}[htp]
\vskip .5in
\hskip .1in
\epsfysize=2.9in\epsffile{[jew.e687.hq94]e687_kmunu.ps}
\vskip -1.5in
\caption{The $\approx 430$ event sample of $D^*$- tagged
$D^o \rightarrow K^- \mu^+ \nu$ decays:
(Left) $q^2$ distribution, (right) $D^{*+} - D^o$ mass peaks and
backgrounds.}
\label{e687_tagged}
\end{figure}
E687 augmented their $D^*$- tagged sample with an $\approx 1900$ event
inclusive (untagged) event sample of $K^- \mu^+ \nu$
decays. Because the $D^* - D$ mass difference cut cannot be applied,
there is significantly more background in the untagged
$K^- \mu^+ \nu$ sample. However, because many of these backgrounds
are well understood, one can reliably subtract their effects
by fitting to the $K^- \mu^+ \nu$ Dalitz plot using the
technique illustrated in Figure \ref{los_vegas}.
Figure \ref{e687_klnu_dalitz} compares observed $q^2$ and
$E_\mu \equiv (M^2_ {K^- \mu^+} +q^2 - M^2_K)/(2 M_D)$ Dalitz projections
to the results computed from the complete fit including backgrounds.
\begin{figure}[htp]
\hskip 1.8in
\vskip 2.2in
\epsfysize=.9in\epsffile{[jew.e687.tcw]kmunu_notag.ps}
\caption{Projections of the E687 $K^- \mu^+ \nu$ Dalitz plot with fits
overlayed on the data. Figures (a+b) compares projections
obtained in 1990 data. Figures (c+d) compare projections obtained
in 1991 data. A significant hole in our muon wall was inflicted upon us
in 1991 to accomodate an experiment located downstream of
our apparatus. The shaded region indicates background contributions
from $ D^+ \rightarrow (K^- \pi^+) \mu^+ \nu$~~,~~
$D^0 \rightarrow (K^- \pi^o) \mu^+ \nu$~~,~~
$D^+_s \rightarrow \phi \mu^+ \nu$ and non-leptonic charm decays
where final state hadrons were mis-identified as muons.}
\label{e687_klnu_dalitz}
\end{figure}
Although the background contributions to the Dalitz projections
of Figure \ref{e687_klnu_dalitz} are significant,
the form factors and relevant branching ratios for each of the major
physics backgrounds have already been accurately measured by our
own\cite{e687_kstarmunu}$^,$\cite{e687_phimunu} and other experiments.
Muon miss-identification probabilities can be reliably measured
using known pions from the decay
$K_s \rightarrow \pi^+ \pi^-$. The parameterized mis-identification
level was fed into charm Monte Carlo calculations
to estimate the shape and level of this background source.
The expected population of events in the $i$'th bin, $\mu_i$ ,
was expressed as a function of the yield of
actual $D^o \rightarrow K^- \mu^+ \nu$ events , the various background
contributions, $\{b_k\}$ , and the pole mass $m_{\rm pole}$ which
parameterizes the $q^2$ dependence of the form factor $f_+ (q^2)$.
\footnote{In this measurement
the $f^-(q^2)/ f_+(q^2)$ form factor ratio
(See Eqn. \ref{form_factor}) was set to
the ``mid-range'' of theoretical expectations $f_-/f_+ \approx -1$.
A small systematic error contribution
due to uncertainty in this ratio was included.}
Although the known background contributions , $\{b_k\}$ , were
allowed to vary throughout the fit, {\it a priori} information on
the anticipated background ($b_k^{(\rm meas)}$)
was fed into the fit through the
inclusion of a $\chi^2$ contribution to the log likelihood of the form:
\begin{equation}
w= -2~\sum_i^{(bins)}~\ln{{\cal P}(n_i ; \mu_i)} +
\sum_k^{\rm (back)}~{(b_k - b_k^{(\rm meas)})^2 \over \sigma^2_k }
\label{klnu_log_likelihood}
\end{equation}
The $\sigma_k$ in Eqn. \ref{klnu_log_likelihood} are the
initial uncertainties in the background parameters $\{b_k\}$.
We used this rather conservative technique so that final statistical
errors on the form factors properly reflects uncertainties in the background
estimation.
\par
Once the yield of $D^o \rightarrow K^- \mu^+ \nu$ signal events
and pole mass are measured using the fit, the $f_+$ form factor can
be measured via Eqn. \ref{fplus_value}.
\begin{eqnarray}
\Gamma_{K \ell \nu}= \underline{|V_{cs}|^2} \times |f_+(0)|^2 \times
\int~dq^2~~H\left(q^2 , m^2_{\rm pole}\right) \nonumber \\
\Gamma_{K \ell \nu} =
{\hbar ~{\cal B}_{K \ell \nu}~\over~\underline{\tau_{D^o}}}~~,~~
{\cal B}_{K \ell \nu} =
{Y_{K \ell \nu} /\epsilon_{K \ell \nu} \over
Y_{K \pi} /\epsilon_{K \pi}}~\underline{{\cal B}_{K \pi}}
\label{fplus_value}
\end{eqnarray}
The underlined quantities in Eqn. \ref{fplus_value} are
previously well measured known inputs. The function $H(q^2)$ is
is the $d \Gamma / d Q^2$ expression of Eqn. \ref{dgamma_dq}
with the $f_+ (0)$ and $|V_{cs}|^2$ factored out.
By measuring the efficiency corrected yield of
$D^o \rightarrow K^- \mu^+ \nu$ signal events relative to a
reference state ($K^- \pi^+$) with a known branching fraction ,
${\cal B}_{K \pi}$,
observed in the same experiment with similar kinematic cuts,
one can measure the branching fraction
${\cal B}_{K \ell \nu}$. This ${\cal B}_{K \ell \nu}$ can be combined
with the $D^o$ lifetime to form a decay width
$\Gamma_{K \ell \nu}$. Finally the integral expression for
the decay width can be used to extract $f_+(0)$.
\par
Table \ref{klnu_ff_table} gives a summary of the form factor and pole mass
obtained in various experiments.
\begin{table}[h]
\caption{$ K \ell \nu$ form factor results}
\begin{center}
\begin{tabular}{l|l|l|l}
Exp. & Mode & m$_{pole}$&$\mid f_{+}(0)\mid$ \\
\hline \hline
E691&$K^{-}e^{+}\nu_{e}$ &${2.1\scriptscriptstyle
_{-0.2}^{+0.4}}\pm 0.2$&$0.79\pm 0.05\pm 0.06$ \\
CLEO(91)&$K^{-}e^{+}\nu_{e}$ &${2.1\scriptscriptstyle
_{-0.2-0.2}^{+0.4+0.3}}$&$0.81\pm 0.03\pm 0.06$ \\
CLEO(93)&$K^{-}l^{+}\nu_{l}$ &${2.00\pm 0.12\pm 0.18}$&
$0.77\pm0.01\pm0.04$ \\
MKIII&$K^{-}e^{+}\nu_{e}$ &${1.8\scriptscriptstyle
_{-0.2-0.2}^{+0.5+0.3}}$&$\mid V_{cs}\mid(0.72\pm 0.05\pm 0.04)$ \\
\hline \hline
E687 &$K^{-}\mu^{+}\nu_{\mu}$ tag &${1.97\scriptscriptstyle
_{-0.22-0.06}^{+0.43+0.07}}$&$0.71\pm 0.05 \pm .03$ \\
E687 &$K^{-}\mu^{+}\nu_{\mu}$ inc &${1.87\scriptscriptstyle
_{-0.08-0.06}^{+0.11+0.07}}$&$0.71\pm 0.03\pm 0.02 $
\end{tabular}
\end{center}
\label{klnu_ff_table}
\end{table}
Figure \ref{klnu_ff_exp} shows excellent agreement between
the experimental results.
\begin{figure}
\hskip .6in
\epsfysize=3.in\epsffile{f_klnu.ps}
\caption{Comparison of the experimental results on $|f_+ (0)|$.
The weighted average and error are shown by the dashed and solid horizontal
lines.}
\label{klnu_ff_exp}
\end{figure}
Figure \ref{klnu_pole_exp} compares
experimental results on $M_{\rm pole}$.
\begin{figure}
\hskip .6in
\epsfysize=3.in\epsffile{pole_klnu.ps}
\caption{Comparison of the experimental results on $m_{\rm pole}$.
The horizonal line is at the $D_s^{*+}$ mass which is
the expected pole mass mass value.}
\label{klnu_pole_exp}
\end{figure}
Note that the more recent, high statistics data, favors a pole mass
which is lower (by $\approx 2 \sigma$) than the spectroscopic pole mass.
\par
It is important to point out that the pole mass determinations
measure this parameter under the assumption of a pole
form. As Figure \ref{pole_cat} shows, over the kinematic
range accessible in $D \rightarrow \overline{K} \ell \nu$,
a pole form $q^2$ dependence is nearly indistinquishable from
an exponential or linear dependence. This is not true however
for the decay $D \rightarrow \pi \ell \nu$ since the maximimum
$q^2$ range in this decay lies much closer to the spectroscopic pole.
\begin{figure}[htp]
\vskip .75in
\centerline{
\epsfysize=2.3in\epsffile{fplus.ps}
}
\vskip -.2in
\caption{Upper frame: $| f_+ (q^2)|^2$ as a function of $q^2$ for various $q^2$
parameterizations which have been drawn to have the same
slope and intercept at $q^2 \approx 0$. The kinematic limit
for $D \rightarrow \overline{K} \ell^+ \nu$ and
$\pi \ell \nu$ are drawn with vertical dashed lines.
The solid curve is the pole form; dashed is exponential; dotted
is linear. Lower frame: $d \Gamma / d q^2$ as a function of $q^2$ for
$D \rightarrow \overline{K} \ell^+ \nu$ decay with three different
pole masses. The dashed curve is for $m_{\rm pole} \rightarrow \infty$ ;
the solid is for $m_{\rm pole} = 2.1~GeV $ (the expected value) ;
the dotted is for $m_{\rm pole} = 1.8~GeV$. The influence of the
pole mass on the overall $q^2$ dependence on $d \Gamma / d q^2$
is rather subtle over the kinematic range probed
by $D \rightarrow \overline{K} \ell^+ \nu$.}
\label{pole_cat}
\end{figure}
As Figure \ref{pole_cat} illustrates, over the limited $q^2$ range
probed by the $\overline{K} \ell^+ \nu$, the pole mass is
essentially a measurement of the normalized
slope $f^{-1}_+ (0)~{d f_+(0)/ d q^2}$. A real check of the pole
parameterization in charm semileptonic decay must await a future,
well meausured, high statistics sample of $\pi \ell \nu$ decays.
\par
Figure \ref{theory_klnu} compares the experimental average to
several Quark Model and Lattice Gauge Theory predictions\cite{simone}~.
\begin{figure}[htp]
\epsfysize=3.in\epsffile{klnu_f_model.ps}
\caption{Comparison of theoretical $f_+(0)$ predictions for
$D \rightarrow {\overline K } \ell^+ \nu$ to world's data.
The first points are quark model estimates, the last are Lattice
Gauge Theory calculations.}
\label{theory_klnu}
\end{figure}
Agreement between theory and data are quite good. The recent
calculations of the LANL (Los Alamos) group\cite{simone} which included a study
of the $q^2$ dependence of $f_+(q^2)$ form factor concluded
that $m_{\rm pole} < M\left(D_s^{*+}\right)$ in agreement with
the recent trends of the data as shown in Figure \ref{klnu_pole_exp}.
\par
E687\cite{e687_kstarmunu} has also presented the first experimental
information on the $f_-$ form factor which forms part
of the hadronic weak current which is underlined in Eqn. \ref{fminus_1}.
\begin{equation}
< D | J_\mu | K >~ =~ f_+(q^2)~(D + K )_\mu
+ \underline{f_-(q^2)}~(D - K )_\mu
\label{fminus_1}
\end{equation}
As shown in Eqn. \ref{fminus_2} which gives a schematic rendering of
partial width expression, the squared lepton mass multiplies
the pure $|f_-(q^2)|^2$ as well as interference
$2~f_-(q^2)~f_+(q^2)$ contributions. For the decay
$D^o \rightarrow K^- \mu^+ \nu$, the effects of $f_- (q^2)$ are to
give very small gradients in the Dalitz plot intensity.
\begin{eqnarray}
{d^2 \Gamma \over d q^2 dM^2_{K \ell}} =
|f_+(q^2)|^2~{\cal A} + ~\underline{M^2_\ell}~\left(~~
2~f_-(q^2)~f_+(q^2) ~{\cal B}
+ |f_-(q^2)|^2~~{\cal C} ~~\right) \nonumber \\
{\cal B} \propto
M^2_{K \ell~{\rm max}} - M^2_{K \ell} ~~-~~
\left(q^2 - q^2_{\rm min}\right)/2
~~,~~{\cal C} \propto q^2 - q^2_{\rm min} \phantom{xxxx}
\label{fminus_2}
\end{eqnarray}
Of course, a small error in the $m_{\rm pole}$ parameter will
induce a false competing gradient in the Dalitz plot along the $q^2$ axis
by slightly change the $q^2$ dependence of the $f_+(q^2)$ form factor.
Figure \ref{fminus_contour} shows likelihood contours on $f_-/f_+$ versus
$m_{\rm pole}$ based on the $\approx 430$ event E687 sample of
$D^*$ tagged $D^o \rightarrow K^- \mu^+ \nu$ events.
\begin{figure}[htp]
\vskip 2.in
\hskip .1in
\epsfysize=.7in\epsffile{contour.ps}
\vskip .2in
\caption{
Contours of constant likelihood for $f_- /f_+$ versus
$m_{\rm pole}$ assuming that $f_-$ and $f^+$ follow pole forms with the same
pole mass.
Note that the contour shape shows that
a large $f_- / f_+$ ratio can be compensated by
an increased value for $m_{\rm pole}$ which can be understood from
Eqn. \ref{fminus_2} for the case where
$|f_-(q^2)|^2 > 2~f_-(q^2)~f_+(q^2)$.}
\label{fminus_contour}
\end{figure}
From this contour, we conclude $f_- / f_+ = -1.3 \pm 3.5 \pm 0.6$
which is good agreement with theoretical estimates.\cite{e687_klnu}
\subsubsection{$D \rightarrow \pi \ell \nu$}
As Figure \ref{vcd_vcs} shows, measurement of
$D \rightarrow \pi \ell \nu$ / $D \rightarrow K \ell \nu$ would seem to
provide useful information on the CKM ratio $|V_{cd}|^2 /|V_{cs}|^2$.
\begin{figure}[htp]
\hbox{
\epsfysize=1.in\epsffile{klnu_diag.eps}
\hskip .2in
\epsfysize=1.in\epsffile{pilnu_diag.eps}
}
\caption{Apart from possible QCD corrections, the difference
between $D \rightarrow \pi \ell \nu$ and $D \rightarrow K \ell \nu$
are in the CKM matrix element couplings.}
\label{vcd_vcs}
\end{figure}
However, as illustrated in Figure \ref{vcd_vcs_known} ,
(3 generation) unitarity of CKM matrix along with experimental information on
charm production by high energy neutrinos on fixed targets and b-decays
from CLEO already provides $V_{cd}$ measurements which are probably
more accurate than the present ability to predict the
$\pi \ell \nu / \overline{K} \ell \nu$ form factor ratio.
\begin{figure}
\hskip 1.in
\epsfysize=3.in\epsffile{vcs.eps}
\caption{ $V_{cd}$ can accurately be measured by studying the
charmed particles produced from incident neutrinos
interacting with $d$ valence quarks. Neutrino experiments generally
detect the semimuonic decays of charmed mesons. $V_{cb}$ can be
determined through measurements of lepton spectrum
produced by $B$ decay near threshold at CLEO.}
\label{vcd_vcs_known}
\end{figure}
For this reason, the principle motivation for studies
of $D \rightarrow \pi \ell^+ \nu$
decays is to test the ability
of the QCD calculational tools to accurately predict the ratio
of CKM favored over CKM suppressed form factors. One hopes that such
calculations can ultimately be used to provide CKM information in the b-sector.
\par
Recent tagged $D^* \rightarrow (\pi^-~\ell^+~\nu) \pi^+$ signals
are shown in Figures \ref{cleo_pilnu} (CLEO) and \ref{e687_pilnu}
(E687).
\begin{figure}
\epsfysize=2.8in\epsffile{cleo_pilnu.ps}
\vskip -.7in
\caption{CLEO $M_{e \pi}$ for events satisfying $M(D^*) - M(D) < 0.160$
The total fit to the $M_{e \pi}$ distribution is shown as a solid
histogram. The contribution from backgrounds is shown as the dashed histogram.}
\label{cleo_pilnu}
\end{figure}
\begin{figure}
\vskip 1.5in
\hskip 1.in
\epsfysize=1.in\epsffile{pilnu_mass.ps}
\vskip .75in
\caption{E687 $M(D^*) - M(D)$ peaks for events satisfying the cut
$M_{\ell \pi^-} > 1.1~GeV$. The first column are $\pi^- \mu^+ \nu$
candidates; the second is for $\pi^- e^+ \nu$ candidates; and the
third is the combined sample. In the legend: BKG1 is the background
contribution for misidentified leptons, BKG2 is the background
due to actual semileptonic $D^o$ combined with random soft pions to form
a false $D^{*+}$, and BKG3 are misidentified kaons from
$D^o \rightarrow K^- \ell^+ \nu$.}
\label{e687_pilnu}
\end{figure}
Because the decay $D^o \rightarrow \pi^- \ell^+ \nu$
is CKM suppressed and therefore somewhat rare, background contamination
is a major concern. In E687\cite{e687_pilnu} a major source of
background is $D^o \rightarrow K^- \ell^+ \nu$ where the kaon is
misidentified by our Cerenkov system as a pion. Another important
class of backgrounds involve semileptonic decays with missing (underlined)
daughters such as $D^o \rightarrow K^{*-} \ell \nu~,~K^{*-} \rightarrow
\underline{K^o} \pi^- {\rm~or~}~ D^o \rightarrow \rho^- \ell \nu~,~
\rho^- \rightarrow \pi^- \underline{\pi^o}$.
It is possible to signficantly suppress missing daughter contributions
by requiring a minimum hadron lepton-lepton mass value as illustrated by
Figure \ref{missing_daughter}. The background
$D^o \rightarrow K^- \ell^+ \nu ~~,~~K^- \hookrightarrow \pi^-$ , on the
other hand, has a $\pi^- \ell^+$ invariant mass distribution
which is nearly indistinquishable from the that for
$\pi^- \ell^+ \nu$ and will form a very similar peak in the
$M \left( D^{*+} \right) - M \left( D^{o} \right)$ mass difference
distribution.
This means that there is no reasonable way of kinematically
distinquishing between the signal and the
$D^o \rightarrow K^- \ell^+ \nu ~~,~~K^- \hookrightarrow \pi^-$ background.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=3.8in\epsffile{hadlep_cut.ps}
\caption{Simulated hadron-lepton invariant mass distributions for the
$D^o \rightarrow K^- \ell^+ \nu$ signal (left)
$D^o \rightarrow \pi^- \ell^+ \nu$ signal (right) and several
background contributions.
Missing daughter type backgrounds typically peak below
$M^2_{h \ell} < 1~GeV^2$ and can be significantly
reduced through the cut $M^2_{h \ell} > 1.1~GeV^2$}
\label{missing_daughter}
\end{figure}
For this reason, we jointly fit the
$M \left( D^{*+} \right) - M \left( D^{o} \right)$ mass difference
histograms for both $K^- \ell^+ \nu$ and $\pi^- \ell^+ \nu$
by summing the binned log likelihoods from both distribution.
The contribution of misidentified
$D^o \rightarrow K^- \ell^+ \nu$ events into the $\pi^- \ell^+ \nu$
$M \left( D^{*+} \right) - M \left( D^{o} \right)$ mass difference
histogram is proportional to the $D^o \rightarrow K^- \ell^+ \nu$
signal yield and $K^- \hookrightarrow \pi^-$ misidentification
probability which was parameterized by studies of
$D^{*+} \rightarrow D^o \pi^+ \rightarrow ( K^- \pi^+ ) \pi^+$
which are misidentified in the data as
$D^{*+} \rightarrow D^o \pi^+ \rightarrow ( \pi^- \pi^+ ) \pi^+$ events.
\par
Figure \ref{pilnu_pole_systematics} illustrates an interesting
systematic on the measurement of the
$\pi^- \ell^+ \nu / K^- \ell^+ \nu$ branching ratio
and form factor ratio.
Because the maximum $\sqrt{q^2}$ available
in $D^o \rightarrow \pi^+ \ell^- \nu$ lies only $\approx 300~MeV$
below the expected $D^{*+}$ pole mass, uncertainties
in the effective pole mass
\footnote{Perhaps several excited $D_s^{**}$ states potentially
contribute or potentially there are contributions from complex cuts.}
will cause significant variations in $ |f_+ (q^2)|^2 $
which will cause intensity variations in the upper reaches of
the $D^o \rightarrow \pi^- \ell^+ \nu$ Dalitz plot.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=3.in\epsffile{pilnu_mass_cut.ps}
\caption{We show Dalitz plots (right) and
form factor variation (left) for several reasonable $m_{\rm pole}$'s .
Because of its larger $q^2$ reach the
$\pi \ell \nu$ Dalitz plot extends into a $q^2$ range
where the $m_{\rm pole}$ uncertainty
becomes significant. The $M_{\pi^- \ell^+} > 1.1~GeV^2$ cut
eliminates the $q^2$ region beyond $2.3~GeV^2$.}
\label{pilnu_pole_systematics}
\end{figure}
The $M_{\pi^- \ell^+}$ cut used to suppress daughters
cuts off the $q^2$ range and thereby acts like
a double edge sword in measurements of
$\pi^- \ell^+ \nu / K^- \ell^+ \nu$.
The $M_{\pi^- \ell^+}$
cut acerbates measurements of the relative branching fraction
since one must extrapolate the number of
$\pi^- \ell^+ \nu$ below the $M_{\pi^- \ell^+}$
cut in order to measure the
efficiency corrected yield ratio of
$\pi^- \ell^+ \nu / K^- \ell^+ \nu$ events over the full
Dalitz plot. The variation on the (full Dalitz plot) acceptance
including the $M_{\pi^- \ell^+}$ cut as a function of
$m_{\rm pole}$ is shown in Figure \ref{eff_v_pole}.
\begin{figure}[htp]
\vskip -.5in
\hskip 1.in
\epsfysize=1.8in\epsffile{PILNU_EFF_V_POLE.PS}
\vskip 1.in
\caption{
$\epsilon (M^2_{h \ell} > 1)$ cut varies by
$~15 \%$ for $\pi \ell \nu$.
$\epsilon$ for $\overline{K} \ell \nu$ varies much less
since its $q^2$ reach much smaller.}
\label{eff_v_pole}
\end{figure}
\par
By way of contrast, the $M_{\pi^- \ell^+}$
cut improves the pole mass systematic for the form factor ratio.
This ratio follows from the corrected yields of $\pi \ell \nu$ and
$K \ell \nu$ events which satisfy the $M_{\pi^- \ell^+}$
cut according to Eqn. \ref{form_factor_ratio}.
\begin{eqnarray}
{Y_{\pi \ell \nu} \over Y_{K \ell \nu}} =
{\vert V_{cd}~f_+^\pi (0)\vert^2 \over \vert V_{cs}~f_+^K (0) \vert^2 }
~{
\int~d q^2 ~|f_+^\pi (q^2)/f_+^\pi(0)|^2~{\cal H}^\pi (q^2)
\over
\int~d q^2 ~|f_+^K (q^2)/f_+^K (0)|^2~{\cal H}^K (q^2)} \nonumber\\
{\rm where}~{\cal H}^{(h)} (q^2) = \int_{M^2_{h \ell}~{\rm cut}}^{M_D^2}
~~d M^2_{h \ell}~
~~\epsilon(q^2 , M^2_{h \ell})~~\left({d^2 \Gamma \over d q^2
d M^2_{h \ell}} \right)_{{\rm f^{(h)}_+}(q^2) = 1}
\label{form_factor_ratio}
\end{eqnarray}
The integrals of Eqn. \ref{form_factor_ratio} are only over the
$q^2$ range satisfying the $M^2_{h \ell}$ cut which limits
them to regions farther from the expected $m_{\rm pole}$
The kernels under the integral of Eqn. \ref{form_factor_ratio}
are plotted in Figure \ref{pilnu_kernel}.
\begin{figure}[htp]
\vskip -1.2in
\hskip 1.3in
\epsfysize=2.6in\epsffile{PILNU_kernel.PS}
\vskip 1.6in
\caption{
${\cal H^\pi}$ (left) and ${\cal H^K}$ (right)
Our CKM-ff ratio is essentially the ratio of the
weighted $< >_{\cal H}$'s over these functions.}
\label{pilnu_kernel}
\end{figure}
Figure \ref{ff_v_qsq} shows the dependence of the final measured
form factor $\times$ CKM ratio on $m_{\rm pole}$ or alternatively
on a linear $q^2$ paramaterization.
\begin{figure}[htp]
\vskip -1.5in
\hskip 1.in
\epsfysize=2.0in\epsffile{PILNU_FF_V_POLE.PS}
\vskip 1.5in
\caption{
CKM-ff ratio vrs $q^2$
dependence of $f_+^\pi$ and $f_+^K$.
left(pole) right (linear)
$f_+(q^2) = f_+(0)~\left(1 + a~q^2\right)$
``Reasonable'' choices give $\approx 10 \%$ variation.
}
\label{ff_v_qsq}
\end{figure}
\par
Experimental results\cite{cleo_pilnu} $^-$ \cite{mark3_pilnu}
on the $\pi^- \ell^+ \nu / K^- \ell^+ \nu$ form factor
and branching ratio are compared to predictions \cite{pilnu_pred}
in Figures \ref{pilnu_br} and \ref{pilnu_ff}.
Agreement between the measured form factor ratios and their predictions
is satisfactory.
\begin{figure}[htp]
\epsfysize=3.0in\epsffile{pilnu_br.ps}
\caption{An informal weighted average gives
$\Gamma\left(\pi^- \ell^+ \nu \right)/
\Gamma\left(K^- \ell^+ \nu \right) = 0.11 \pm 0.02$
The data slightly exceeds the predictions.}
\label{pilnu_br}
\end{figure}
\begin{figure}[htp]
\epsfysize=3.0in\epsffile{pilnu_ff.ps}
\caption{
An informal weighted average of the form-factor ratio is
$|f_+^\pi/f_+^k| = 1.05 \pm 0.095$
(assuming $|V_{cd}/V_{cs}|^2 = 0.051$)
The predictions are in reasonably good agreement with the prediction.}
\label{pilnu_ff}
\end{figure}
\subsubsection{$D^+ \rightarrow K^{*-} \mu^+ \nu$}
This decay is one of the oldest and cleanest charm semileptonic
decays. In fixed target experiments, one begins with a detached
vertex which contains the $K^- \pi^+ \mu^+$ charged tracks.
As Figure \ref{kstar_mass} shows, the $K^- \pi^+$ mass distribution
suggests that the four-body decay
$D^+ \rightarrow K^- \pi^+ \mu^+ \nu$ is strongly dominated by
the quasi-three-body process $D^+ \rightarrow K^{*-} \mu^+ \nu$.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=4.in\epsffile{kpimunu_v_elsig.ps}
\caption{
The subtracted $K^- \pi^+$ invariant mass plot for
$D^+ \rightarrow K^- \pi^+ \mu^+ \nu$ events from E687.
The right sign - wrong sign subtraction
tends to eliminate non-charm background.
Increasing the $\ell/\sigma$ cut eliminates the low
mass $D^{*+} \rightarrow (K^- \mu^+ \nu) \pi^+$ bump,
leaving a $K^- \pi^+$ mass distribution
dominated by $\overline{K^*}(892)$.}
\label{kstar_mass}
\end{figure}
By fitting the $M\left(K^- \pi^+\right)$ versus $M\left(K^- \pi^+ \mu^+\right)$
for the detached events, E687\cite{e687_kstarmunu} obtained the limit
on non-resonant $D^+ \rightarrow K^- \pi^+ \mu^+ \nu$ decays given
by Eqn. \ref{non_res_limit}
\begin{equation}
{\Gamma(K^- \pi^+ \mu^+ \nu)_{\rm nr} \over
\Gamma(\{K^- \pi^+\}_{892} ~\mu^+ \nu)} = 0.083 \pm 0.029~
<~0.12_{~90\%~{\rm cl}}
\label{non_res_limit}
\end{equation}
The fact that the $\overline{K^*}$ is a vector particle significantly
complicates the Lorentz structure of the weak current
$$ which is now described by three
$q^2$ dependent form factors $\pmatrix{ A_1 (q^2) & A_2 (q^2) & V (q^2) }$
(two axial and one vector).
\footnote{Experimentalists generally analyze their data assuming
that the $q^2$ dependence of these form factors are
of the pole form with the $D_s^{*+}$ and
$D_s^{**+}$ poles.}
The ratio of the form factors controls the decay angular distribution
as illustrated in Figure \ref{angular_dist}.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=2.25in\epsffile{decay_angles.eps}
\caption{The three form factors control the $q^2$ dependence of the
virtual $W^+$ polarization. The spin states of the leptons
favored by $V - A$ are shown.
The three decay angles
are $\theta_\mu$ describing the virtual $W^+ \rightarrow \mu^+ \nu$
decay in its rest frame , $\theta_v$ describing the
$K^{*-} \rightarrow K^- \pi^+$ in the $\overline{K^*}$ rest frame, and
the acoplanarity ,$\chi$ ,
between the $W$ and $\overline{K^*}$ decay planes.
The decay distribution
${d^4 \Gamma / \left(d\cos{\theta_\mu} d\cos{\theta_v} dq^2 d\chi\right)}$
serves to measure
$R_v = {V (0)/ A_1(0)}~{\rm and}~~R_2 = {A_2(0)/ A_1(0)}$.
}
\label{angular_dist}
\end{figure}
Once the $R_v$ and $R_2$ shape variables are determined, one can measure
the overall scale of $A_1(q^2)$ from the decay width in a way highly
analogous to the use of Eqn. \ref{fplus_value}.
Figure \ref{rv_r2}, and \ref{a_1} compare the experimental
data to theoretical predictions\cite{simone} for both the
form factor ratios and $A_1 (0)$.
\begin{figure}[htp]
\hskip 1.in
\vbox{
\epsfysize=2.5in\epsffile{kstarlnu_rv.ps}
\vskip .2in
\epsfysize=2.5in\epsffile{kstarlnu_r2.ps}
}
\caption{Experimental averages compared to theoretical
prediction for the form factor ratios
$R_v = {V (0)/ A_1(0)}~~,~~R_2 = {A_2(0)/ A_1(0)}$. The horizontal
lines give the average of the data. The Lattice Gauge predictions
are in good agreement with the data.}
\label{rv_r2}
\end{figure}
\begin{figure}[htp]
\hskip 1.in
\epsfysize=2.5in\epsffile{kstarlnu_a1.ps}
\caption{Experimental averages compared to theoretical
prediction for the $A_1(0)$ form factor.
The points are in rough chronological order. The earliest
quark model points were about 40 \% higher than the data.
The more recent LGT points are in much closer agreement with experiment.}
\label{a_1}
\end{figure}
Since the initial predictions for the $f_+(q^2)$ form factor which
governs $D^o \rightarrow K^- \ell^+ \nu$ was in good agreement with
experimental data; while as Figure \ref{a_1} shows initially
the $A_1(q^2)$ which sets the scale of
$\Gamma\left(D^+ \rightarrow \overline{K^{*o}} \ell^+ \nu\right)$
was over predicted by $\approx \sqrt{2}$, the ratio
$\Gamma\left(
D^o \rightarrow \overline{K^{*o}} \ell^+ \nu \right)
/ \Gamma\left( D_o \rightarrow K^- \ell^+ \nu \right)$ was initially
predicted to be a factor of two larger than the measured value
of $<{\Gamma_{\overline{K^*} \ell \nu} / \Gamma_{{\overline K} \ell \nu}}>
= 0.56 \pm 0.05~$. As Figure \ref{a_1} shows , this apparently serious
discrepancy with the predictions for $A_1$ is beginning to fade away
as the recent LGT predictions are becoming available. Finally
the recently revised ISGW2 quark model\cite{isgw2} computes
${\Gamma_{\overline{K^*} \ell \nu} / \Gamma_{{\overline K} \ell \nu}}
= 0.54$ which is in excellent agreement with the experimental value.
\subsection{Polarization in Charm Baryon Decay}
The CLEO and ARGUS collaboration have studied both
semileptonic decay processes such as
$\Lambda_c^+ \rightarrow \Lambda \ell^+ \nu$
and hadronic decay processes such as $\Lambda_c^+ \rightarrow \Lambda \pi^+$
and find in either case that the daughter $\Lambda$ emerges with a spin
which is nearly 100 \% lefthanded (when viewed in the
$\Lambda_c^+$ rest frame). This effect has also been observed in
the decays $\Xi_c^o \rightarrow \Xi^- \pi^+$.
Figure \ref{lambda_anal} gives an illustration of the effect for
the case of $\Lambda_c^+ \rightarrow \Lambda \ell^+ \nu$ with
$\Lambda \rightarrow p \pi^-$. The observation of a nearly
lefthanded daughter baryon in charm decay can be predicted using
Heavy Quark Effective Theory (HQET).
Although HQET plays a dramatic role in understanding b-sector physics,
to my knowlege, the charmed baryon polarization effect along
with the predicted spectroscopy of excited charmed mesons and baryons are
are the main applications HQET in the charm sector.
\begin{figure}[htp]
\hskip .6in
\epsfysize=1.3in\epsffile{lambda_anal.eps}
\caption{When viewed in the $\Lambda+c^+$ rest frame the
daughter $\Lambda$ emerges predominently lefthanded {\it eg}
spinning against its momentum vector. I also indicate the preferred
spin directions of the final state leptons in a $V - A$ decay.
The $\Lambda$ can be
thought of as self-analyzing in the sense that in
the decay $\Lambda \rightarrow p \pi^-$ the proton emerges primarily
along the $\Lambda$ spin direction when viewed in the $\Lambda$ rest frame.
See Figure \ref{lambda_anal} as well.}
\label{lambda_anal}
\end{figure}
\begin{figure}[htp]
\hskip 1.in
\epsfysize=1.55in \epsffile{lambda_self.eps}
\caption{We view the decay $\Lambda \rightarrow p \pi^-$ in the
rest frame of a $\Lambda$ and orientate our $\hat z$ axis to be along
the $\Lambda$ spin direction. The polar angle distribution for the
proton daughter is of the form $d \Gamma / d~\cos{\theta} \propto
1 + 0.64~\cos{\theta}$ The asymmetry of 0.64 is sometimes called the
analyzing power.}
\label{lambda_self}
\end{figure}
\par
We begin by a discussion of the semileptonic decay results.
Figure \ref{lambda_ws_rs} shows the basic semileptonic process.
\begin{figure}[htp]
\hskip 1.5in
\epsfysize=1.in\epsffile{lambda_ws_rs.eps}
\caption{In the decay $\Lambda_c^+ \rightarrow \Lambda \ell^+ \nu$
the daughter baryon is produced along with an $\ell^+$ (right sign) rather
than an $\ell^-$ (wrong sign).}
\label{lambda_ws_rs}
\end{figure}
The signature for baryon semileptonic decays is an excess of right sign
compared to wrong sign leptons being produced along with the daughter
hyperon as illustrated in Figure \ref{cleo_lambdaenu}.
\begin{figure}[htp]
\vskip -.8in
\hskip 1.5in
\epsfysize=3.in\epsffile{[jew.e687.hq94]cleo_lambdaenu.ps}
\vskip .8in
\caption{
CLEO right sign and wrong sign
$\Lambda \rightarrow p \pi^-$ signals.
against an $e^\pm$ (left) or $\mu^\pm$ (right)
More $\Lambda \ell^+$ events (top)
are seen than $\Lambda \ell^-$ (bottom).}
\label{cleo_lambdaenu}
\end{figure}
One can assign $q^2$ dependent, helicity form factors for the 4 ways
of assigning spin to the virtual $W^+$ and daughter $\Lambda$ which
are illustrated by Figure \ref{lambda_pol_hel}.
\begin{figure}
\hskip 1.in
\epsfysize=3.in\epsffile{lambda_pol_hel.eps}
\caption{Four $q^2$ dependent helicity amplitudes describe this decay.
Two of the 6 possible spin alignments for the spin 1 virtual $W^+$ and
spin 1/2 $\Lambda$ cannot be produced through the decay of a spin 1/2
$\Lambda^+_c$}
\label{lambda_pol_hel}
\end{figure}
We can define
a $\Lambda$ polarization asymmetry by Eqn. \ref{pol_asym_def}.
By this definition $\alpha_{\Lambda_c} = -1$ implies a 100 \%
lefthanded daughter $\Lambda$.
\begin{equation}
\alpha_{\Lambda_c} = { \Uparrow - \Downarrow \over \Uparrow + \Downarrow}
~~{\rm where}~~\matrix{ \Uparrow ~{\rm means}~
\vec S_\Lambda ~\Vert~ \phantom{-} \vec \Lambda \cr
\Downarrow ~{\rm means}~ \vec S_\Lambda ~\Vert~ -\vec \Lambda}
\label{pol_asym_def}
\end{equation}
Adding together the helicity possibilities shown in Figure \ref{lambda_pol_hel}
we obtain Eqn. \ref{asymmetry_expression}.
\begin{equation}
\alpha_{\Lambda_c} =
{ \left( |H^{q^2}_{{1\over 2}~0}|^2 + |H^{q^2}_{{1\over 2}~1}|^2\right) -
\left(|H^{q^2}_{-{1\over 2}~-1}|^2 + |H^{q^2}_{-{1\over 2}~0}|^2\right) \over
\left( |H^{q^2}_{{1\over 2}~0}|^2 + |H^{q^2}_{{1\over 2}~1}|^2\right) +
\left(|H^{q^2}_{-{1\over2}~-1}|^2 + |H^{q^2}_{-{1\over 2}~0}|^2 \right)}
\label{asymmetry_expression}
\end{equation}
HQET \cite{korner_kramer} applied to heavy $\rightarrow$ light quark transitions
such as in charm decay are used to relate these four independent
form factors to just two whose ratio $R = f_1/f_2$
controls the $q^2$ dependence of $\alpha_{\Lambda_c}$
\footnote{The two HQET allowed form factors, $f_1$ and $f_2$, are
generally assumed to have the same $q^2$ dependence.}
Angular momentum leads to an additional restriction
as $q^2 \rightarrow 0 $ which is independent of HQET.
Figure \ref{lambda_anal_qsq}
illustrates the fact that as $q^2 \rightarrow 0 $, the leptons become collinear
and form a spinless system and only $J_z = 0$ $W_*^+$ ' s are formed.
In this limit, only two of the four
helicity form factors survive this limit. Combining both the
angular momentum and HQET restriction one has the prediction
that $\alpha_{\Lambda_c} = -1$ as $q^2 \rightarrow 0$ independent
of the form factor ratio $R$.
\begin{figure}[htp]
\hskip .75in
\epsfysize=1.5in\epsffile{lambda_anal_qsq.eps}
\caption{ As $q^2 \rightarrow 0$ the $\ell^+$ and $\nu$ become collinear.
By the $V - A$ rules, the neutrino is lefthanded and the
$\ell^+$ becomes righthanded and their spins cancel.
Hence only $H^{q^2}_{{1\over 2}~0}~,~H^{q^2}_{-{1\over 2}~0} \ne 0$ as
$q^2 \rightarrow 0$}
\label{lambda_anal_qsq}
\end{figure}
Figure \ref{kk_pred} illustrates the predictions of the
polarization asymmetry expected predicted in
the K\"orner \& Kr\"amer model\cite{korner_kramer} model
for various values of the form factor ratio $R$.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=2.6in\epsffile{argus.eps}
\caption{
A crude sketch of the K\"orner \& Kr\"amer predictions
for $\alpha_{\Lambda_c}$ versus $q^2$ for various form
factor ratios. The shaded region relevant
to $\Lambda^+_c \rightarrow \Lambda \ell^+ \nu$.
Note $\alpha_{\Lambda_c} \rightarrow -1$ as $q^2 \rightarrow 0$
independent of the form factor ratio $R$.
The basic prediction is that $\alpha_{\Lambda_c} \approx -1$.}
\label{kk_pred}
\end{figure}
For the case of heavy $\rightarrow$ heavy transitions such as one
would get in b-baryon decays, the four form factors are reduced to
just one and thus $\alpha_{\Lambda_b} =-1$ at all $q^2$.
\par
Rather than going to $q^2 \rightarrow 0$ limit in
$\Lambda_c^+ \rightarrow \Lambda \ell^+ \nu$ decays to form a
$J_z (W^+_*) \rightarrow 0$, one could directly couple the
$W^+_*$ to a $\pi^+$ as illustrated in Figure \ref{lambda_anal_pion}.
Again, for these case of these non-leptonic charm baryon decays,
angular momentum conservation combined with HQET applied to
heavy $\rightarrow$ light transitions predicts a nearly complete
lefthanded daughter baryon polarization.
\begin{figure}[htp]
\hskip .5in
\epsfysize=1.5in\epsffile{lambda_anal_pion.eps}
\caption{For $\Lambda_c^+ \rightarrow \Lambda \pi^+$
$q^2 = m_\pi^2 \rightarrow 0$
HQET predicts $\alpha_{\Lambda_c} = -1$}
\label{lambda_anal_pion}
\end{figure}
These results can be extended to the decay
$\Xi_c^o \rightarrow \Xi^- \pi^+$ where
the hyperon daughter subsequently decays via $\Xi^- \rightarrow \Lambda \pi^-$.
The $\Xi^-$ decay is also self analysing with the $\Lambda$
following a polar angle distribution of the form
$\left( d \Gamma / d \cos{\Theta}\right)_\Lambda \propto
1 - 0.46~\cos{\Theta}$.
Figures \ref{cleo_asymmetry} and \ref{casc_asymmetry} show
acceptance corrected polar distributions for the
$\Lambda \rightarrow \underline{p} \pi^-$ and
$\Xi^- \rightarrow \underline{\Lambda} \pi^- $
from $\Lambda^+_c \rightarrow \Lambda \ell^+ \nu$ and
$\Xi_c^o \rightarrow \Xi^- \pi^+$ decays
obtained by the CLEO Collaboration.
\begin{figure}[htp]
\vskip -.6in
\hskip 1.in
\epsfsize=.2in\epsffile{[jew.e687.hq94]cleo_asymmetry.ps}
\vskip .6in
\caption{CLEO's acceptance corrected measurement of the
proton polar angle distribution in the
$\Lambda$ rest frame with respect to the $\Lambda$ helicity axis for
$\Lambda^+_c \rightarrow \Lambda \ell^+ \nu$ decays.
The $\Lambda$ helicity axis points against
the momentum of the $\Lambda_c^+$ when viewed in the $\Lambda$ rest frame.
The slope of this line divided by the $\Lambda$ analysis power of 0.64 gives
$\alpha_{\Lambda_c} \approx -1$.}
\label{cleo_asymmetry}
\end{figure}
\begin{figure}[htp]
\vskip -1.4in
\hskip 1.in
\epsfsize=.3in\epsffile{casc_asym.ps}
\vskip 1.4in
\caption{CLEO's acceptance corrected measurement of the
$\Lambda$ polar angle distribution in the
$\Xi^-$ rest frame with respect to the $\Xi^-$ helicity axis in
$\Xi_c^o \rightarrow \Xi^- \pi^+$ decays
The slope of this line divided by the $\Xi^-$ analysis power of -0.46 gives
$\alpha_{\Xi^o_c} \approx -1$.}
\label{casc_asymmetry}
\end{figure}
Figure \ref{polar_summary} summarizes the polarization asymmetry
obtained for various charmed baryon decay modes. In each case, the
asymmetry is consistant with a 100 \% lefthanded polarization as
predicted by HQET.
\begin{figure}[htp]
\epsfysize=3.in\epsffile{polar.ps}
\caption{The polarization asymmetry of the daughter hyperon
observed for the charmed baryon decays studied thus far. The
horizontal dashed line gives the $V -A$ lefthanded helicity
predicted by HQET.}
\label{polar_summary}
\end{figure}
\subsection{Non-Leptonic Charm Decays}
\par
One can profitably study non-leptonic charm decays on several levels.
On the most inclusive level, the rough order of magnitude
difference between the charm particle lifetimes , discussed in
Section 3.1, primarily reflects differences in the non-leptonic
decay width. The semileptonic width is a small fraction
of the total width in the first place and secondly,
although the semileptonic widths of the various charmed
paricles are not expected to be universal, Table \ref{sl_widths}
tells us that the these widths are expected to be roughly the same.
\par
On the next level
of inclusiveness, one can study the partial decay widths of charm
mesons into specific two body final states.
Factorization models
have been proposed which predict the partial decay widths of
two body decays of charmed mesons
in the absence of final state interactions. Final state interaction
effects become apparent in the interference between the various isospin
amplitudes which contribute to different charged variants of
a given final state. The simplist experimental tests of
factorization involve pure two body decays of charmed mesons
into two pseudoscalar final states involving kaons and pions.
\par
One can extend factorization tests to quasi-two body decays such as
pseudoscalar - vector decays as well as vector-vector decays by
analysing the resonance structure in multibody,
nonleptonic, charmed meson decays. Experience has shown
that the data is best fit by models where the various
resonant channels contribute coherently to a decay amplitude
and thus interfere in the differential decay width.
These amplitude analyses provide additional
handles on final state interaction effects through interference of the
amplitudes describing competing resonant channels in {\it eg} a three
particle decay Dalitz plot.
\par
The factorization framework, in which non-leptonic charm decays
have been traditionally analyzed, begins with
an effective Hamiltonian such as that given by
Eqn. \ref{hamiltonian} describing CKM allowed decays:
\begin{equation}
{\cal H} = {G_f \over \sqrt{2}}~V_{cs}^*~V_{ud} \left(
{C_+ + C_- \over 2} (~\overline{u}~d) ~({\overline s}~ c~) ~+~
{C_+ - C_- \over 2} (~\overline{s}~d)~ (~{\overline u}~c)~ \right)
\label{hamiltonian}
\end{equation}
The Hamiltonian of Eqn. \ref{hamiltonian}
incorporates QCD corrections to the underlying weak decay process through the
use of renormalization group methods.
The coefficients $C_\pm$ are called ``Wilson coefficients''.
In the absence of QCD corrections , $C_+ = C_-$ and
one recovers single weak process corresponding to the familiar spectator
diagram of Figure \ref{effective_spectator}.
\begin{figure} [htp]
\hskip 1.8in
\epsfysize=1.in\epsffile{spectator_2.eps}
\caption{When $C^+ = C_-$, the second term of Eqn. \ref{hamiltonian}
disappears and one has the CKM spectator diagram.}
\label{effective_spectator}
\end{figure}
The Wilson coefficients depend on the scale of the QCD coupling
constants. When taken at the charmed quark mass , the Wilson coefficients
have the values given by Eqn. \ref{wilson_coef}.
\begin{equation}
{C_+ (M_Q) + C_- (M_Q) \over 2}\approx 1.25 ~~,~~
{C_+(M_Q) - C_-(M_Q) \over 2} \approx -0.49
\label{wilson_coef}
\end{equation}
In 1987 , Bauer, Stech, and Wirbel (BSW) \cite{BSW}
combined these ideas into an explicit model applied two body and
quasi-two body non-leptonic charmed meson decay. In the BSW
model, the two terms of Eqn. \ref{hamiltonian} are organized into
an effective ``charged'' current and effective ``neutral current''
between the parent and daughter hadrons as illustrated in Figure
\ref{fourpoint}.
\begin{figure}
\hskip .7in
\epsfysize=.75in\epsffile{[jew.e687.hf95]fourpoint.eps}
\caption{The effective charged current
whose amplitude
$a_1 \approx (C_+ + C_-)/2$ (with a small color correction)
describes the familiar spectator process. The effective neutral
current process with an amplitude $a_2 \approx (C_+ - C_-)/2$
will decrease as $M_Q \rightarrow \infty$ gives rise to non-spectrator
effects which are important in charm but less so in beauty.}
\label{fourpoint}
\end{figure}
If a given non-leptonic charm decay can only proceed through
the effective charged current interaction, it is classified as a
Class 1 process. Class 2 processes only proceed through the
effective neutral current interactions. Class 3 processes have contribution
from both interactions.
In the process, $D^o \rightarrow K^- \pi^+$,illustrated by
Figure \ref{class1} , one has a neutral parent and two charged
daughters and hence this must be a Class 1 process.
\begin{figure}
\hskip 1.in
\epsfysize=1.2in\epsffile{[jew.e687.hf95]class1.eps}
\caption
{In the BSW model the amplitude for this decay is written as
$~~~~~~~~~~~~~~~~~~~a_1~G_f~2^{-1/2}~< \pi^+ |(\bar u d) | 0 > = a_1~{G_f}~2^{-1/2}~
(-i f_\pi P_\pi) \times f_+ (m^2_\pi)$. The coupling of the $\pi^+$
with respect to the virtual $W^+$ involves the same current as
the leptonic decay $\pi^+ \rightarrow \ell^+ \nu$ which is proportional to
the pion lepton decay constant. The CKM allowed current which describes
the process $D^o \rightarrow W^{+*} ~K^-$ is the same
current involved in the semileptonic decay process
$D^o \rightarrow K^- \ell^+ \nu$ which is described by
the form factor $f_+ (q^2)$.}
\label{class1}
\end{figure}
In the Class 2 process, $D^o \rightarrow \overline{K^o} \pi^o$,
illustrated by Figure \ref{class2},
one has neutral parent $D^o$ decaying into two neutral daughters.
\begin{figure}
\hskip 1.in
\epsfysize=1.2in\epsffile{[jew.e687.hf95]class2.eps}
\caption{In the BSW model the amplitude for this decay is written as
$~~~~~~~~~~~~~a_2~{G_f}~2^{-1/2}~< \overline{K^{o}} |(\bar s d) | 0 >
<\pi^o | (\bar u c ) |D^o>$ where again the currents involve the leptonic
and semileptonic form factors. Note that the light $\overline{u}$
constituent of the $D^o$ does not spectate as is the case in Figure
\ref{class1} but is actively rearranged into the final state hadron.}
\label{class2}
\end{figure}
The CKM allowed decays of the $D^+$ such as
$D^+ \rightarrow \overline{K^{o}} \pi^+$ , depicted in Figure \ref{class3},
are Class 3 processes.
\begin{figure}
\hbox{
\vbox{\epsfysize=1.0in\epsffile{[jew.e687.hf95]class3a.eps} \vskip .5in}
\hskip .2in
\vbox{\epsfysize=1.0in\epsffile{[jew.e687.hf95]class3b.eps}}
}
\caption{The two processes depicted in the figure differ by the exchange
of $\overline{d}$ quark fermion in the final state.
Because $a_1$ and $a_2$ interfere with the opposite sign
one generally has destructive interference for $D^+$ decays which
causes the $D^+$ to have a longer life than the $D^o$.}
\label{class3}
\end{figure}
\par
According to the Watson theorem, the weak amplitudes predicted
in factorization models such as BSW must all be relatively real. However,
the final state hadrons can continue to interact via long range strong
interactions and acquire complex phases. These final state interactions
(FSI) can be accomodated by multiplying the bare (weak) amplitudes
by the square root of a complex, unitary $S$ matrix describing the
strong rescattering. To illustrate the effects of FSI consider
the isospin classification of three amplitudes related to
$D \rightarrow \pi \pi$ decay:
\begin{equation}
A(D^o \rightarrow \pi^- \pi^+) =
{1 \over \sqrt{3}} \left( \sqrt{2}~a_o + a_2 \right)$$
$$A(D^o \rightarrow \pi^o \pi^o) =
{1 \over \sqrt{3}} \left( -a_o + \sqrt{2}~a_2 \right)~~,~~
A(D^+ \rightarrow \pi^o \pi^+) = \sqrt{3 \over2 }~a_2
\label{dipion_amp}
\end{equation}
The measured $a_o$ and $a_2$ amplitudes will acquire complex
phases through the FSI S-matrix as indicated in Eqn. \ref{fsi}.
\begin{equation}
\pmatrix{a_0 \cr a_2 } =
\pmatrix{
\eta~e^{2 i \delta_0} & i \sqrt{1 - \eta^2} e^{i (\delta_0 + \delta_2)} \cr
i \sqrt{1 - \eta^2} e^{i (\delta_0 + \delta_2)} & \eta~e^{2 i \delta_2}
}^{1/2}
\pmatrix{a_0 \cr a_2 }_{\rm bare}
\label{fsi}
\end{equation}
Since QCD respects isospin symmetry, there should be no mixing
between $a_o$ and $a_2$ which means the elasticity parameter in
Eqn. \ref{fsi} should be $\eta = 1$.
\footnote{ One can still have , however, inelastic final state interactions
between final states such as $D \rightarrow \pi \pi$ and $D \rightarrow KK$
since a dipion final state can potentially scatter into a dikaon final
state. One could accomodate such a case with a $4 \times 4$
matrix describing the two isospin amplitudes for the dipion and
two isospin amplitudes for the dikaon.}
Even a purely elastic FSI can change the total width of charm decays
into a particular final state by changing the value of
$\cos{\left(\delta_2 - \delta_0\right)}$
when converting {\it eg}
$A(D^o \rightarrow \pi^- \pi^+)$ into $\Gamma(D^o \rightarrow \pi^- \pi^+)$
via Eqn. \ref{gamma+-}.
\begin{equation}
\Gamma(\pi^+ \pi^-) = {2 \over 3}~|a_o|^2 + {1 \over 3}~|a_2|^2
+ {2 \sqrt{2} \over 3}|a_o|~|a_2|~\cos{\left(\delta_2 - \delta_0\right)}
\label{gamma+-}
\end{equation}
\par
The unexpectedly large branching ratio for the
decay $D^o \rightarrow K^+ K^- / \pi^+ \pi^-$ provided an early
example of the possible role of FSI affecting branching ratios.
Table \ref{kk_pipi} summarizes recent data on
$\Gamma(K^+ K^-)/\Gamma(\pi^+ \pi^-)$.
\begin{table}[htp]
\caption{$\Gamma(D^o \rightarrow K^+ K^-)/
\Gamma(D^o \rightarrow \pi^+ \pi^-)$}
\begin{center}
\begin{tabular}{ l | l }
E687\cite{e687_kkpipi} & WA82\cite{wa82kkpipi} \\
2.53 $\pm$ 0.46 $\pm$ 0.19 &
2.23 $\pm$ 0.81 $\pm$ 0.46 \\ \hline
E691 \cite{e691kkpipi} & CLEO\cite{cleokkpipi} \\
1.95 $\pm$ 0.34 $\pm$ 0.22 &
2.35 $\pm$ 0.37 $\pm$ 0.28
\end{tabular}
\end{center}
\label{kk_pipi}
\end{table}
As one can see from Table \ref{kk_pipi} , although both processes
are CKM suppressed by the same amount and $D^o \rightarrow \pi^+ \pi^-$
is favored by having a larger phase space,
$D^o \rightarrow K^+ K^-$
occurs at roughly twice the rate as
$D^o \rightarrow \pi^+ \pi^-$. Since these decays were the first
CKM suppressed decays to be studied, their width ratio was
initially quite surprising. In the context of the BSW model, both decays
are Class 1 processes, since no effective neutral currents are possible
between the parent and either daughter, and the BSW model makes
the prediction $\Gamma(K^+ K^-)/\Gamma(\pi^+ \pi^-) \approx 1.4$ independent
of the value of $a_2/a_1$. Conventional wisdom has it that
discrepency between the data and the BSW prediction is due to FSI
changing the phase of interfering dipion or dikaon isospin amplitudes.
\par
A more direct way of seeing the effects of final state interactions
is to measure the widths into various isospin related channels such as
$\Gamma(\pi^+ \pi^-)~~,~~\Gamma(\pi^o \pi^o)~~,~~\Gamma(\pi^+ \pi^o)$
and extract $a_0~,~~a_2, ~{\rm and}~\cos{\left(\delta_2 - \delta_0\right)}$
by solving the Eqn. \ref{gamma+-} and the two similar equations.
Table \ref{pedrini} taken from the upcoming Annual Review article
by Pedrini,Browder, and Honschied\cite{ann_review} summarizes
the results of such isospin analyses for many two body and quasi-two
body charm decays.
\begin{table}[htp]
\caption{Isospin amplitude ratios and phase shifts for two body charm decays.}
\begin{center}
\begin{tabular}{l |c |l}
Mode & Ratio of amplitudes &
$\delta=\delta_I-\delta_{I^\prime}$ \\
\hline\hline
$K\pi$ & $\vert A_{1/2}\vert/\vert A_{3/2}\vert =4.12\pm 0.40$ &
$88^{\circ}\pm 8^{\circ}$ \\
$K^{\star}\pi$ & $\vert A_{1/2}\vert/\vert A_{3/2}\vert=5.23\pm 0.59$ &
$90^{\circ}\pm 16^{\circ}$ \\
$K\rho$ & $\vert A_{1/2}\vert/\vert A_{3/2}\vert=3.22\pm 0.64$ &
$10^{\circ}\pm 47^{\circ}$ \\
$K^{\star}\rho$ & $\vert A_{1/2}\vert/\vert A_{3/2}\vert=4.93\pm 1.95$ &
$33^{\circ}\pm 57^{\circ}$ \\
$KK$ & $\vert A_{1}\vert/\vert A_{0}\vert=0.58\pm 0.12$ &
$47^{\circ}\pm 13^{\circ}$ \\
$\pi\pi$ & $\vert A_{2}\vert/\vert A_{0}\vert=0.63\pm 0.13$ &
$81^{\circ}\pm 10^{\circ}$
\end{tabular}
\end{center}
\label{pedrini}
\end{table}
Table \ref{pedrini} shows that more often than not, a considerable
phase shift is observed between the two isospin amplitudes.
Watson's theorem tells us that phase shifts between interfering
isospin amplitudes where $\sin{\left(\delta_{I'} - \delta_I\right)} \ne 0$
cannot arise from the weak processes alone and thus constitutes
direct evidence for FSI.
\par
Several groups \cite{mark3}$^-$\cite{e687_kkpi} have gone a step
further and published resonant amplitude analyses of particular final
states. Most of this work concerns three body decays where all of the
amplitude information resides in the Dalitz plot.
Amplitude analyses allow one to extend factorization model tests
to pseudoscalar-vector and vector-vector non-leptonic decays of $D$ mesons
and provide new ways of studying FSI effects. A particularly clean
and instructive example is provided by the E687\cite{e687_kkpi} group's
analysis of the $D^+ ~,~D_s^+ \rightarrow K^- K^+ \pi^+$ final state.
Mass histograms and the two Dalitz plots are shown in Figure \ref{kkpi}.
\clearpage
\begin{figure}[h]
\smallskip
\vskip -.7in
\vskip 1.in
\hskip 1.in a)
\vskip -1.3in
\epsfysize=2.2in\epsffile{[jew.e687.hf95]kkpivacmassplot.ps}
\vskip -.25in
\vskip 1.5in
\hskip 1.5in b) \hskip 2.5in c)
\vskip -1.5in
\hskip -.25in
\hbox{
\hskip -.1in
\epsfysize=2.2in\epsffile{[e687lib.rlg.thesis]kkpidsdalitz.ps}
\hskip -.1in
\epsfysize=2.2in\epsffile{[jew.e687.hf95]kkpideefulldalitz.ps}
}
\vskip 1.3in
\smallskip
\caption{
a) The mass spectrum for out of target $K^+ K^- \pi^+$ decays.
b) and c) The full sample $D_s^+ , D^+ \rightarrow K^- K^+ \pi^+$ Dalitz
plots. The bands due to
$\overline{K^{*o}}(890) K^+$ and $\phi(1019) \pi^+$ are quite
visible for both the $D_s^+$ and $D^+$ Dalitz plots. Note the significant
asymmetry between the population of the
high and low $\overline{K^{*o}}(890) K^+$ lobes
for the $D^+$ in sharp contrast with the symmetric appearance of the
lobes for the $D_s^+$.}
\label{kkpi}
\end{figure}
The $D_s^+$ Dalitz plot is dominated by the $\phi \pi^+$ and
${\overline K^{o*}} K^+$ channels; while the $D^+$ also has a significant
contribution from a single or multiple broad resonances.
We note that both the $\phi$ and ${\overline K^*}$ bands have a node
due to angular momentum conservation in the center of each band as illustrated
in Figure \ref{kstarnodes}.
\begin{figure}[htp]
\hskip 1.in\epsfysize=2.in\epsffile{dmm.eps}
\caption{When viewed in the $\overline{K^*}$ rest frame, the $\overline{K^*}$
has zero spin along the $\overline{K^*} \rightarrow K^- \pi^+$ decay
axis since neither decay product carries spin. Since the $D$ and
$D^+$ are both spinless; the $\overline{K^*}$ must have zero
spin along the $K^+$ axis. The amplitude for the $\overline{K^*}$ to
have be in these two simultaneous spin states is given by
$d^{(1)}_{0 0} (\theta) \propto \cos{\theta}$. This angular factor
causes the Dalitz intensity to vanish in the center of the Dalitz
band where $\cos{\theta} = 0$ as discussed in Figure \ref{dalitz_sketch}.
}
\label{kstarnodes}
\end{figure}
In order to extract information from Dalitz plots such as those illustrated
in Figure \ref{kkpi}, the experimental groups fit the intensity across
Dalitz plot to a coherant sum of quasi-two body or
possible non-resonant contributions.
For example, one can fit the $D_s^+$ Dalitz plot
to the form
$d \Gamma / (d M_1^2 d M_2^2) \propto
| {\cal A}(D_s^+ \rightarrow K^+K^-\pi^+) |^2$
where the amplitude is given by Eqn. \ref{ds_amp}.
\begin{eqnarray}
{\cal A}(D_s^+ \rightarrow K^+K^-\pi^+) =
a_{{\overline K^*}}~e^{i \delta_{\overline K^*}}{\cal M}
( \pi^+ K^- K^+ | \overline{K^{*o}}~K^+ ) \nonumber \\
+ a_{\phi}~e^{i \delta_{\phi}}{\cal M}(K^+ K^- \pi^+ |\phi \pi^+) + ...
\label{ds_amp}
\end{eqnarray}
Figure \ref{feynman} illustrates the form of one of these
quasi-two body contributions.
\begin{figure}[htp]
\hskip 1.in
\epsfysize=1.8in\epsffile{[e687lib.rlg.thesis]zemach.eps}
\caption{We show a Feynman-like diagram for the process
$ D_s^+ \rightarrow \phi \pi^+$. A $D \rightarrow \pi^+$ current
with a form factor $F_D$ interacts with a kaon current with
form factor $F_r$ through an unstable $\phi$ propagator with
an imaginary width contribution to the propagator mass.
This diagram gives an amplitude contribution of the form
${\cal M}~=~ F_D~F_r~\times~|\vec c|^J |\vec a|^J
P_J(\cos{\theta^r_{ac}}) ~\times~ BW(m_{ab})$ where
the angular factor which gives rise to the Dalitz nodes follows
from the structure of current $\times$ current contribution, and the
Breit-Wigner represents the unstable propagator. The form factors
are relatively unimportant.}
\label{feynman}
\end{figure}
Each resonant channel contribution is multiplied by a complex
amplitude coefficient whose modulus serves as a
gauge to the relative ``importance'' of the channel.
\setcounter{footnote}{0}
\footnote{
In fitting to the shape of the relative event density across the Dalitz plot,
one of the fitted amplitude factors is generally set to unity
and serves as a reference channel. Clearly since one is
fitting to $|{\cal A}|^2$ the overall phase of ${\cal A}$ cannot be measured.
The modulus of the reference amplitude can be determined if desired
through the total decay width in a way similar in spirit to
Eqn. \ref{fplus_value}.}
\par
Any non-real phase differences are due to the
influence of final state interactions which in a two resonant
model would be given by Eqn. \ref{ds_fsi}
\begin{equation}
\pmatrix{a_{{\overline K^*}}~e^{i \delta_{\overline K^*}} \cr
a_{\phi}~e^{i \delta_{\phi}}} =
\pmatrix{
\eta~e^{2 i \delta_1} & i \sqrt{1 - \eta^2} e^{i (\delta_1 + \delta_2)} \cr
i \sqrt{1 - \eta^2} e^{i (\delta_1 + \delta_2)} & \eta~e^{2 i \delta_2}
}^{1/2}
\pmatrix{a_{K^*} \cr a_{\phi} }_{\rm bare}
\label{ds_fsi}
\end{equation}
One of the more interesting features of the Dalitz plots of Figure \ref{kkpi}
is the the pronounced asymmetry between the two $\overline{K^*}$ lobes for the
$D^+ \rightarrow K^- K^+ \pi^+$. We believe that this lobe asymmetry
is due to interference of the $\overline{K^{*o}} K^+$ channel with a broad,
spinless resonance channel. From the interference pattern,
one can infer that the data requires a nearly imaginary relative phase shift.
For simplicity, we model the broad resonance in the vacinity of the
$\overline{K^*}$ as a nearly constant amplitude
which we write as $\cos~\delta + i\sin~\delta$.
Eqn. \ref{asym_int} gives an explicit form for this interference term.
\begin{eqnarray}
Re \left\{ \left( \cos~\delta + i\sin~\delta \right)^*~
{\cos~\theta \over M_r^2 -M^2_{K\pi} - i \Gamma M_r} \right\} = \nonumber\\
{ \left( M_r^2 -M^2_{K\pi} \right)~ \cos~\theta~\cos~\delta \over
\left( M_r^2 -M^2_{K\pi} \right)^2 + \Gamma^2 M^2_r}
+{ \Gamma M_r~ \cos~\theta~sin~\delta \over
\left( M_r^2 -M^2_{K\pi} \right)^2 + \Gamma^2 M^2_r}
\label{asym_int}
\end{eqnarray}
The asymmetry comes about because the $\overline{K^{o*}} K^+$ decay amplitude
contains an angular factor of $\cos{\theta}$ , where $\theta$ is the
angle between the two kaons in the $\overline{K^{o*}}$ rest frame.
This $\cos{\theta}$ factor causes the interference term to change sign from
the left lobe to the right lobe.
We get an interference term from both the
real part of the Breit-Wigner as well as the imaginary part. Since the real
part of a Breit-Wigner reverses sign
as one passes through the resonance (thus cancelling the net
interference across the lobe), any net interference
is due to the second term which is proportional to $\sin{\delta}$.
Figure \ref{visual_fsi} makes this argument graphically.
\begin{figure}[htp]
\vskip -1.in
\hbox{
\epsfysize=2.2in\epsffile{[e687lib.rlg.thesis]kkpideerealphaseillustrate.ps}
\epsfysize=2.2in\epsffile{[e687lib.rlg.thesis]kkpideeimagphaseillustrate.ps}
}
\vskip 1.in
\caption{The ``+'' and ``-'' indicates the sign of the interference
term between a constant phase and the $\overline{K^{o*}} K^+$ Dalitz band
for the case of relatively real (left) and relatively imaginary (right)
phase shift. The sign alternates from the left to the right of the
band owing to the $\cos{\theta}$ portion of the $\overline{K^{o*}} $
amplitude in Eqn. \ref{asym_int}. The real part of a Breit-Wigner
amplitude reverses sign as one passes through the resonance which implies
an additional sign change for the relatively real case. The
relatively imaginary part has no such additional sign reversal.}
\label{visual_fsi}
\end{figure}
The net lobe
asymmetry thus provides visible evidence for final state interactions
since all bare amplitudes must be real
($\delta = 0~{\rm or}~180^o$). After exploring many possibilities,
E687 settled on the $\overline{K}_0^{*}(1430)^0~K^+$ channel as the
most likely channel interfering with the $\overline{K^{*0}} K^+$
band.
\par
We next discuss the $D_s^+ \rightarrow
K^+ K^- \pi^+$ Dalitz plot. A close examination of the Dalitz plot
for the nearly background free, out of target data
(Figure \ref{f0}) shows an accumulation
of events in (what should be) the $\phi$ band angular node.
\begin{figure}[htp]
\vskip -.9in
\hskip -.3in
\hbox{
\epsfysize=2.1in\epsffile{[jew.e687.hf95]kkpidsvacdalitz.ps}
\epsfysize=2.1in\epsffile{[e687lib.rlg.thesis]kkpideesimulatedwa.ps}
}
\vskip .8in
\caption{\small{The E687, $D_s^+$ Dalitz plot and simulated $D_s^+ \rightarrow
f(980) \pi^+ \rightarrow (K^+ K^- ) \pi^+$ decays.}}
\label{f0}
\end{figure}
E691 \cite{pipipi} discovered that the $D_s^+ \rightarrow \pi^+ \pi^- \pi^+$
Dalitz plot is strongly dominated by the channel
$D_s^+ \rightarrow f(980) \pi^+ \rightarrow (\pi^+ \pi^- ) \pi^+$.
We therefore expect a contribution from the known dikaon decay of the
$f(980)$ which should populate the $\phi$ node region
as shown in Fig. \ref{f0}.
Figure \ref{dsproj} shows us that
the inclusion of additional contributions from the $f_J (1710) \pi^+$ and
$\overline{K}_0^{*}(1430)^0~K^+$ significantly improved the quality
of the fits in the vicinity of the $K^*$ peak ($R_1$) and just above it
($R_2$).
\vskip .7in
\begin{figure}[htp]
\vskip -1.3in
\hskip 1.in
\hbox{
\epsfysize=2.in\epsffile{[e687lib.rlg.thesis]kkpidsthreeampfitkpirs.ps}
\epsfysize=2.in\epsffile{[e687lib.rlg.thesis]kkpidsfitresultsandrakpirs.ps}
}
\vskip .8in
\caption{\small{Comparing the $m^2 (K^- \pi^+)$
3 amplitude (left) and 5 amplitude (right) $D_s^+$ fit to the
data (error bars). The background, deduced from the mass sidebands,
is shown in the lower histogram.}}
\label{dsproj}
\end{figure}
\par
Table \ref{kkpi_results}
compares the E687 \cite{e687_kkpi} results on the channels common
to both the $D^+, D_s^+ \rightarrow K^+ K^- \pi$ decays.
The fractions, $f_r$, are known as decay fractions and represent the
ratio of the integrated Dalitz intensity for a single resonance $r$ divided
by the intensity with all contributions present. The $D^+$ amplitude
consists of nearly equal contributions of ${\overline K^*} K$, $\phi \pi$
and $\overline{K}^*(1430)^0K^+$; while the $D_s^+$ is strongly dominated
by just the ${\overline K^*} K$, and $\phi \pi$. The decay fraction for
the $f_o(980)$ and $f_J(1710)$ contributions to the $D_s^+$ sum to
14 \%. \footnote{I find it interesting to note that both the
$f_o(980)$ and $f_J(1710)$ are often suspected
non-$q\overline{q}$. There is a discussion of this in the 1996 summary
by the Particle Data group \cite{pdg}}.
\vskip .25in
\begin{table}[htp]
\caption{Comparison of $D^+, D_s^+ \rightarrow K^+ K^- \pi$ amplitudes}
\begin{center}
\begin{tabular}{l|l|l}
Parameter |$D^+$ |$D_s^+$\\
\hline \hline
$\delta_{\overline{K}^*(892)^0K+}$ &$0^o$ (fixed)
&$0^o$ (fixed)\\
$\delta_{\phi\pi^+}$ &$-159\pm 8\pm 11^o$
&$178\pm 20 \pm 24^o$\\
$\delta_{\overline{K}^*(1430)^0K+}$ &$70\pm 7\pm 4^o$
&$152\pm 40\pm 39^o$\\thick
$f_{\overline{K}^*(892)^0K+}$ &$0.301 \pm 0.020 \pm 0.025$
&$0.478 \pm 0.046 \pm 0.040$\\
$f_{\phi\pi^+}$ &$0.292\pm 0.031\pm 0.030$
&$0.396 \pm 0.033 \pm 0.047$\\
$f_{\overline{K}^*(1430)^0K+}$ &$0.370\pm 0.035\pm 0.018$
&$0.093 \pm 0.032 \pm 0.032$ \\
\end{tabular}
\end{center}
\label{kkpi_results}
\end{table}
Both the $D^+$ and $D_s^+$ could be fit entirely by quasi-two-body
processes without the need to include a non-resonant contribution.
It is interesting to note that both charm states have a real relative
phase between the dominant ${\overline K^*} K$ and $\phi \pi$
channels which indicates absent or cancelling FSI phase shifts.
\par
Our amplitude fit for $D^+ \rightarrow K^- K^+ \pi^+$ enables us to
correct for acceptance variations across the Dalitz plot as well
as the effects of resonant channel interference to obtain a much
improved inclusive branching ratio
$D^+ \rightarrow K^+ K^- \pi^+ / K^- \pi^+ \pi^+ =
0.0976 \pm 0.0042 \pm 0.0046$. It has become common practice to quote
``branching ratios'' for resonant decay amplitudes such
as $D^+ \rightarrow \phi \pi^+ / K^- \pi^+ \pi^+$ , but because
of quantum mechanical interference, only such inclusive ratios as
$D^+ \rightarrow K^+ K^- \pi^+ / K^- \pi^+ \pi^+$ are truly legitimate.
\par
Several groups have performed Dalitz analyses for the
$D^o , D^+ \rightarrow K \pi \pi$ final state.
Although these fits provide a good qualitative match to the
data, often discrepancies are apparent in comparisons between
the data and mass projections as shown in Fig. \ref{kpipiproj}.
\begin{figure}[htp]
\vskip -.8in
\hskip .7in
\hbox{
\epsfysize=3.2in\epsffile{[jew.e687.hf95]exotic_kpi_low_fig.ps}
\epsfysize=3.2in\epsffile{[jew.e687.hf95]exotic_kpi_hi_fig.ps}
}
\vskip -.6in
\caption{\small{Comparison of the
lower and higher $K^- \pi^+$ mass projection
in E687 data (error bars) and our fit (histogram). The high
projection does not match near 2.5 GeV$^2$.}}
\label{kpipiproj}
\end{figure}
The fit fractions obtained by the various experiments
\cite{kpipidalitz,E691_kpipi,argus_kpipi}
are in excellent
agreement;
while there is little agreement concerning the relative phases, as illustrated
in Fig. \ref{kspipi_comp} for $D^o \rightarrow K_s^o \pi^+ \pi^-$.
There is excellent agreement, however, between
phases obtained by E687 \cite{kpipidalitz} and Argus.~ \cite{argus_kpipi}
\begin{figure}[htp]
\hbox{
\hskip -.5in
\epsfysize=4.in\epsffile{[JEW.E687.TCW]ks_comp_ff_fig.ps}
\epsfysize=4.in\epsffile{[JEW.E687.TCW]ks_comp_phs_fig.ps}
}
\vskip -.7in
\caption{\small{Comparison of the decay fractions a) and phases b) obtained
by different experiments for the decay $D^o \rightarrow K_s^o \pi^+ \pi^-$.}}
\label{kspipi_comp}
\end{figure}
These analyses provide
a wealth of information on new decay modes which can be compared
to models based on factorization, QCD sum rules, and 1/$N_c$ expansions.
Generally agreement of the models with the data is only at about the
$\pm $60 \% level. Often the isospin amplitudes show nearly imaginary
relative phases as reported in Table \ref{pedrini}.
It is interesting to note that all experimental
groups report a sizeable non-resonant contribution to the
$D^+ \rightarrow K^- \pi^+ \pi^+$ Dalitz plot which makes it unique
among the plots discussed here.
\subsection{Heavy Quark Decay Summary}
Here is a very brief summary of this rather lengthy section.
We began by discussing charm decay at the most inclusive level -- the
total decay rate or inverse lifetime of the charmed particles.
The lifetime of the last measured weakly decaying charmed baryon, the
$\Omega_c^o ~(css)~(2704)$ state was measured last year at
$0.064 \pm 0.020~ps$. There is approximatly an order of magnitude
difference in the lifetime of the short lived $\Omega_c^o ~(css)~(2704)$
state and the long lived (1.06 ps) $D^+$. In absence of QCD
effects, the naive spectator model predicts a universal charm lifetime of
$\approx 0.7~ps$.
Recent progress in $1/M_Q$ expansions provides a systematic approach
to estimating the effects of WA , WX , and PI diagrams
which differentiate among the various charm particle species. The
expansion approach does
a credible job of reproducing observed lifetimes of the 7 known weakly decaying
charm states. By way of contrast, the lifetimes of the weakly decaying
beauty states are much closer in value.
More precise charm baryon measurements and b-sector measurements are needed
to fully test these promising new approaches.
\par
Heavy Quark Effective Theory predicts a very strong , backward polarization
for the daughter baryon in $\Lambda_c \rightarrow \Lambda \ell \nu$,
such as $\Lambda_c \rightarrow \Lambda \pi$, and $\Xi_c \rightarrow \Xi \pi$.
The spin of the baryons are self analyzed through the decays:
$\Lambda \rightarrow p \pi$ and $\Xi \rightarrow \Lambda \pi$.
All three decays were measured by ARGUS and/or CLEO and
found to be consistent with nearly complete, lefthanded polarization.
\par
A great deal of progress has been made in leptonic and semileptionic decays.
There now exists several measurements of the $D_s^+ \rightarrow \mu^+ \nu$
decay constant. Present data is internally consistent with a value of
$f_{D_s} = 242 \pm 32~MeV$ at $\approx 20 \%$ CL level. This value
is in good agreement with many of the predictions from Lattice Gauge Theory.
\par
Charm form factors are exactly computable (in principle) by Lattice
Gauge Theory and are frequently
estimated by Quark Models. Both techniques are most effective
when the daughter hadron is at rest relative to the parent which
means that they work best at $q^2 \approx q^2_{\rm max}$. Both
the phase space and $V - A$ matrix element tend to force the data
to values close to $q^2 \approx 0$. Most experimental groups use
a VDM inspired pole form to bridge the experimental and theoretical
$q^2$ regimes where the pole mass corresponds to the known
$D^{*}$ and $D^{**}$ spectroscopy. The validity of these
forms is unproven.
\par
Theory does a good job predicting the
$f_+$ form factor for $D \rightarrow K \ell \nu$. Data favors an
$m_{\rm pole}$ slightly lower than the expected $D_s^{*}$ mass.
Crude experimental information on the lepton mass suppressed form
factor , $f_-(q^2)$, has just become available.
As expected,
$f^\pi_+$ for $D \rightarrow \pi \ell \nu$ is equal (to within $\pm 10 \%$)
to $f^K_+$ for $D \rightarrow K \ell \nu$ . Since $\pi \ell \nu$
probes $q^2$ much closer to the pole mass, pole mass uncertainty is an important
source of systematics on the form factor and branching ratio measurements.
It will be exciting to measure the $q^2$ dependence of $\pi \ell \nu$
decay in the future.
Recent LGT and improved QM calculations are in much better agreement
with form factor magnitude in $D \rightarrow \overline{K^*} \ell \nu$ decay
than the first predictions.
\par
A factorization framework , the
Bauer Stech Wirbel (BSW) model, was designed to incorporate QCD corrections
into non-leptonic decays. The BSW model deals specifically with
2 body and quasi-2 body decays. Predictions are made in terms
of leptonic and semileptonic form factors.
It has been known for some time that long range, final state
interactions, FSI, can significantly
distort BSW predictions for two body decay branching ratios.
FSI is also responsible for creating complex phase shifts between
isospin amplitudes.
Amplitude analyses have confirmed quasi-2 body nature of most
three body decays. FSI effects are evident in the interference
between resonant channels. Amplitude analyses make possible
vector-vector, and vector - pseudoscalar comparisons with BSW
model. Only $\pm 60 \%$ agreement exists between the models and
resonance data.
Extending these analyses to higher multiplicity meson
and baryon decays and improving the physics of non-resonant
terms is a future goal of charm physics.
\section{A Final Word}
\par
I have tried throughout these lectures to capture some of the
excitement of the present research on charm quark production and
decay. Charm provides a unique window to test the calculations
of QCD which is in many respects complementary to beauty owing
to its intermediate mass scale. Charm lies somewhere between the light
quark chiral limit and the limit where heavy quark effective
theory is clearly applicable.
\par
I have tried to show that
the charm quark appears to be massive enough such that data on
high energy the charm photoproduction and hadroproduction cross section
can be reasonably compared to the perturbative QCD calculations
at next to leading order. The smaller mass of charm compared to beauty
offers the experimentalist two distinct advantages. The low mass
means that charm is copiously produced in high energy fixed target experiments.
The reduced decay phase space of charm compared to
beauty means that there exist easily reconstructable final states
with appreciable ($\approx 5 - 10 \%$) branching fractions.
A copious sample of reconstructed charm decays allows one to make
rather incisive tests of the QCD production mechanism and fragmentation
models involving
high accuracy measurements of particle -antiparticle asymmetries and
correlations.
\par
Non-perturbative models have been sucessfully employed in understanding
both the overall decay rate and details concerning form factors
for both fully leptonic and semileptonic decays. The emerging calculational
tool for charm decay appears to be Lattice Gauge theory since charm
semileptonic is generally not in the proper domain of Heavy Quark Effective
Theory. There still remains the problem of the $q^2$ gulf between
where data exists and where the theories work best. Finally many of the
attempts to understand the bulk of charm decay --non-leptonic decay--
are presently thwarted by the difficult to compute influence of final
state interactions. The final state interactions will ultimately complicate
future beauty physics but at a hopefully reduced level. When
viewed in this context, charm non-leptonic decay perhaps provides a
``preview of coming distractions'' for future beauty physics.
\par
I would like to acknowledge the hospitality of the Stanford Summer Institute
for arranging and maintaining this excellent school and conference.
I would also like to acknowlege the prodigious efforts of my
many E687 collaborators over the years.
\clearpage
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\end{document}