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\title{Charm Theory Review (Non-lattice)}
\author{E. Golowich\address[MCSD]{Physics Department,
University of Massachusetts, \\
Amherst, MA 01003 USA
}%
\thanks{Research supported in part by the U.S. National
Science Foundation under Grant PHY--0244801.} }
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\begin{document}
\begin{abstract}
In this talk, I review aspects of $D^0$-${\bar D}^0$ mixing,
rare decays of $D$ mesons and CP violations in $D$ meson decays,
in a manner which emphasizes the interplay between experiment and
theory.
\vspace{1pc}
\end{abstract}
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\maketitle
\section{INTRODUCTION}
The organizers of this conference recommended three subjects
to choose amongst -- $D^0$ mixing, rare decays and CP-violations.
Since there is currently activity in all three areas,
this report will touch on each of them.
\section{MIXING}
The most newsworthy topic in the area of meson-antimeson oscillations
is the recent detection of $B_s$-${\bar B}_s$ mixing at Fermilab.
This leaves $D^0$-${\bar D}^0$ as the only undetected candidate for
mixing. As such it will (I suspect) become an area of heightened
experimental and theoretical scrutiny.
On a quantitative level, the current PDG data collection lists \cite{pdg}
\begin{eqnarray*}
& & x_{\rm D} = {\Delta M_{\rm D} \over \Gamma_{\rm D}} < 0.029 \ \
(95\%~ {\rm C.L.}) \nonumber \\
& & y_{\rm D} \simeq {\Delta \Gamma_{\rm D} \over 2\Gamma_{\rm D}} < 0.008
\pm 0.005 \ \ .
\end{eqnarray*}
$D^0$-${\bar D}^0$ mixing is indeed small and we know why --
there is Cabibbo angle suppression [$\sin^2\theta_{\rm C}$] and
quark mass suppression [$(m_s^2 /m_c^2)^n$]. We cannot, however,
yet precisely predict the size of mixing.
This frustration is partly a reflection of the quark mass, too
large to justify chiral methods and too small to yield a
rapidly convergent heavy quark expansion.
\subsection{Mixing in the Standard Model}
Our discussion will emphasize $\Delta\Gamma_{\rm D}$,
\begin{eqnarray}
\Delta\Gamma_{\rm D} = - {{\cal I}m ~
\langle {\bar D}^0 | i \int d^4 x T\left( H_{\rm w}(x) H_{\rm w}(0)
\right) | D^0 \rangle \over M_{\rm D}} ~.
\nonumber
\end{eqnarray}
To utilize this relation, we insert intermediate states between the
$|\Delta C| = 1$ weak hamiltonians $H_{\rm w}$. This can be done
using either quark or hadron degrees of freedom.
Let us consider each of these
possibilities in turn.
\subsubsection{Quark-level Analysis}
Using the heavy quark methodology to expand the
mixing in terms of operator dimension, we find
two four-fermion operators of dimension six,
fifteen six-fermion operators of dimension nine, {\it etc}. \cite{hg,ohl,bu}
Consider the familiar $D=6$ (box diagram) case. If the $b$-quark
contribution is ignored, there are $s{\overline s}$, $s{\overline s}$ and
$s{\overline d} + d{\overline s}$ intermediate states. Taking
$m_d = 0$, the mixing functions will depend on $z \equiv m_s^2/m_c^2
\simeq 0.006$. Table~\ref{table:qkmix} uses one of the
mixing functions for $\Delta\Gamma_{\rm D}$ to show what happens
when an expansion in powers of $z$ is carried out.
The individual intermediate states are {\it not} intrinsically
small -- they begin to contribute at ${\cal O}(z^0)$. However, flavor
cancellations remove all contributions through ${\cal O}(z^2)$,
so the net result is ${\cal O}(z^3)$, which is
tiny. The corresponding result for $\Delta M_{\rm D}$ turns out to be
${\cal O}(z^2)$ (the extra factor of $z$ for $\Delta\Gamma_{\rm D}$
is due to 'helicity suppression', the same mechanism that occurs in
pion leptonic decay).
%\vspace{-1pc}
\begin{table}[htb]
\caption{Flavor Cancellations in $\Delta\Gamma$.}
\label{table:qkmix}
%\newcommand{\m}{\hphantom{$-$}}
%\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
%\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
%\renewcommand{\arraystretch}{1.2} % enlarge line spacing
%\begin{tabular}{@{}llll}
\begin{tabular}{cccc}
\hline\hline
Int. State & ${\cal O}(z^0)$ & ${\cal O}(z^1)$ & ${\cal O}(z^2)$\\
\hline
$s{\overline s}$ & $1/2$ & $-3z$ & $3z^2$ \\
$d{\overline d}$ & $1/2$ & $0$ & $0$ \\
$s{\overline d}+d{\overline s}$ & $-1$ & $3z$ & $-3z^2$ \\
\hline
Total & $0$ & $0$ & $0$ \\ \hline
\end{tabular}\\[2pt]
\end{table}
%\vspace{-2pc}
For any given operator dimension, one must expand in powers of
${\cal O}(\alpha_s)$. Thus the work just described gives the
leading-order (LO) term in an expansion in QCD, with
$x_{\rm D}^{\rm (LO)} \gg y_{\rm D}^{\rm (LO)}$.
The NLO calculation has recently been done in Ref.~\cite{gp2005}, and
the effects of QCD are found to be significant. Summing
the LO and NLO terms, the net result is
$x_{\rm D} \simeq y_{\rm D} \sim 10^{-6}$. No higher order QCD
contributions have yet been computed.
Higher dimensional sectors in the
OPE are expected to be important, in part because the flavor
cancellations will not universally be so severe. \cite{ohl,bu}
For example, it is known that in the $D=9$ sector the contributions
are ${\cal O}(z^{3/2})$. A thorough study of $z$ dependence
is underway. \cite{ggp}
To summarize, a proper analysis of $D^0$ mixing at the quark level
involves a {\it triple} expansion -- in dimension $D$, in powers
of $\alpha_s$ and in powers of $m_s^2/m_c^2$.
It is possible that as-yet uncalculated terms can lift
$x_{\rm D}$ and $y_{\rm D}$ well above the LO + NLO
${\cal O}(10^{-6})$ scale for dimension six cited above.
\subsubsection{Hadron-level Analysis}
Using hadronic intermediate states in the formula for
$\Delta\Gamma_{\rm D}$ has been a well-studied approach to the
problem.
One possibility is to construct a theoretical model of $|\Delta C| = 1$
decays and to fit any undetermined model parameters via experiment.
In 1995, a rather extensive model/fit
of $D^0$ decays was applied to the determination of
$\Delta\Gamma_{\rm D}$ and the result
$y_{\rm D} \simeq 10^{-3}$ was found. \cite{blmps} This is almost
an order of magnitude smaller than the current experimental bound
that I cited earlier.
Even earlier (in 1985), a more phenomenological method had been
proposed in which one divides out phase space from experimental
branching fractions and takes the square
root to reconstruct the magnitudes of individual decay amplitudes.
This can be used to then determine $y_{\rm D}$. \cite{dghp}
The two-body sector of charged pseudoscalar branching fractions
was explored as a trial horse,
\begin{eqnarray*}
& & y_{\rm D}^{(P^+P^-)} = {\cal B}_{[D^0 \to K^+K^-]}
+ {\cal B}_{[D^0 \to \pi^+\pi^-]} \nonumber \\
& & - 2 \cos\delta \left[ {\cal B}_{[D^0 \to K^-\pi^+]}\cdot
{\cal B}_{[D^0 \to K^+\pi^-]}
\right]^{1/2} \nonumber
\end{eqnarray*}
where $\delta$ is the relative phase between the $K^-\pi^+$ and
$K^+\pi^-$ modes. This contribution vanishes in the limit of
exact SU(3) flavor symmetry, and the same must be true for all
other decay sectors. However, since the real world exhibits large
SU(3) symmetry breaking it was conjectured that
this constraint might be invalidated.
Unfortunately, little was known in 1985 about the
$D^0 \to K^+ \pi^-$ mode, and Ref.~\cite{dghp} could only reach
the qualitative conclusion that SU(3) breaking might induce an
unexpectedly large value for $y_{\rm D}$. A more recent
and much more complete analysis indicates that each decay sector
experiences only muted SU(3) breaking. \cite{falk} If substantiated over time,
this presumably reflects the fact that in $D^0$-${\bar D}^0$ mixing
SU(3) breaking is of second order. \cite{su3}
There must be exceptions, however, since certain multi-particle final
states ({\it e.g.} four-kaon modes)
are disallowed due to phase space restrictions.
It has been stated that such multi-particle sectors might lead
to a value as large as $y_{\rm D} \sim 0.01$.
As one of the originators of the `phenomenological' method,
I would very much like for it to give the best prediction
for $y_{\rm D}$. Unfortunately, sector-by-sector it is the least
precisely known branching fraction that controls the predictive power
of the analysis. Let us hope that this approach is not doomed to
supply only 'fuzzy' estimates.
Finally, there is the possibility of exploiting the presence of
resonances nearby the $D^0$ in mass, for which an enhancement of
the mixing amplitude occurs. \cite{nearby} This promising approach
is currently hindered by an inadequate database regarding such
nearby resonances.
\subsection{Mixing Beyond the Standard Model}
It has long been appreciated that meson-antimeson oscillations
may be sensitive to heavy degrees of freedom which propagate
in the mixing loop-amplitudes. Detection of
mixing in $K^0$ and $B_d$ systems led to predictions about charm and
top quarks before these particles were actually discovered.
In like manner, by comparing observed mixing with predictions of the Standard
Model we constrain models of New Physics, impressively
so for $B_s$-${\bar B}_s$ mixing.
Admittedly, this has not yet led to the discovery of New Physics
and we are all becoming increasingly frustrated. But it is important
not to give up, so we keep trying. $D^0$ mixing has the intriguing
property that the SM signal is very small and vanishes in
the not-so-distant world of exact SU(3) symmetry. A group of us is
currently involved in providing an up-to-date summary of New Physics
in $D^0$ mixing. \cite{ghpp06} Among other things, any such work
should include a complete menu of NP possibilities ({\it e.g.}
extra gauge bosons, {\it etc}), a proper treatment of NP amplitudes
at the charm scale (basis operators and their QCD corrections) and
an up-to-date knowledge of allowed NP parameter space.
A somewhat related topic involves the realization that NP {\it can}
affect $\Delta \Gamma_{\rm D}$. There is often confusion about
this point. Although it is true that only light quarks and gluons
are allowed as intermediate states in the mixing amplitude for
$\Delta\Gamma_{\rm D}$, it is likewise true that one
of the interaction verticies can be NP. This line of reasoning
would lead to a relation between NP contributions to
$\Delta \Gamma_{\rm D}$ and to $|\Delta C| = 1$ decays. \cite{gpp06}
\section{RARE DECAYS}
The era of positive experimental results for rare charm decays is
underway. Happily, there is already a sizable and modern theoretical
literature on this subject ({\it e.g.} \cite{bghp95,bghp02,fp}).
There has, however, been a lag in observations because the
experimental signals are so small. This is about to change.
\begin{figure}[t]
%\vspace{9pt}
\caption{Contributions to $D^0 \to
\rho^0 e^+ e^-$.}
%\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
%\epsfbox{kekfig1.eps}
%\includegraphics[angle=90,width=15pc]{rarefig.ps}
\includegraphics[angle=90,width=15pc]{b06fig.eps}
%\epsfig{figure=pill.ps,height=4.2in,angle=90}
\label{fig:susy}
\end{figure}
%
One noteworthy, if somewhat distressing, theoretical result is that
not all rare decay modes are of equal interest for detecting New
Physics interactions at the quark level. A prime example
is the sequence of flavor changing radiative decays
$D \to V \gamma$, where $V = \rho, K^*, etc$ is a neutral spin-one
particle. Here, one would hope
to be probing the underlying quark transition $c \to u \gamma$.
This mode is, of course, absent at tree level in the SM but can
occur at higher orders. Even though the QCD corrections are
relatively large, the $c \to u \gamma$ signal is still unobservably
tiny. \cite{greub} Such is not the case for the so-called
long-distance contributions. These are calculated in terms of hadronic
degrees of freedom and are estimated to give branching fractions
as large as ${\cal O}(10^{-6} \to 10^{-5})$, \cite{bghp95}
consistent with the scale that
the PDG data compilation registers for the branching fraction
${\cal B}_{D^0 \to \phi + \gamma} = (2.5 ^{+ 0.7}_{- 0.6})
\cdot 10^{-5}$. This result is welcome but it informs
us only about details of hadronic dynamics and nothing more.
\begin{table*}[t]
\caption{Current Status of Some Decay Asymmetries ($A_{\rm CP}$)}
\label{table:cpv}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{@{}llll}
\hline\hline
Neutral & PDG Value & Charged & PDG Value \\
\hline
$D^0 \to K^+K^-$ & $\ 0.005 \pm 0.016$ & $D^\pm \to \phi \pi^\pm$ &
$-0.014 \pm 0.033$\\
$D^0 \to \pi^+\pi^-$ & ~$0.021 \pm 0.026$ & $D^\pm \to \pi^\pm K^+K^-$ &
$0.002 \pm 0.011$\\
$D^0 \to K_s \pi^0$ & \ $0.001 \pm 0.013$ & $D^\pm \to \phi \pi^\pm$ &
$-0.02 \pm 0.05$\\
$D^0 \to \pi^0 \pi^+ K^-$ & $-0.031 \pm 0.086$ &
$D^\pm \to \pi^\pm \pi^+ \pi^-$ & $-0.02 \pm 0.04$
\\
\hline
\end{tabular}\\[2pt]
\end{table*}
Of greater potential for probing New Physics are the decays
$D \to M \ell^+ \ell^-$ where $M = \pi, K^*, {\it etc}$.
The output of observing such modes
is not just a branching fraction, but also a distribution
(most usually in dilepton invariant mass $q^2 \equiv (p_{\ell^+}
+ p_{\ell^-})^2$). Such a distribution could be of real interest,
but not all parts of it. Consider the situation depicted in
Fig.~\ref{fig:susy} for $D^0 \to \rho^0 \ell^+ \ell^-$,
where the SM contribution is shown in black and the predictions of
a version of supersymmetry with nonuniversal soft breaking effects
are shown in color. \cite{bghp02} This figure has two striking features:
[1] The first is the presence of sharp
peaks for intermediate values of $q^2$. These are induced by
the weak transition $D^0 \to R^0 \rho^0$ ($R^0$ is a
neutral noncharmed resonance) followed by the electromagnetic conversion
$R^0 \to \ell^+ \ell^-$. This part of the graph is clearly a
manifestation of SM physics, and reflects comments
made above regarding the flavor-changing radiative decays.
[2] More interesting from the viewpoint of New Physics is
the region of small $q^2$. The SM signal is relatively tiny here, giving
the NP signal a chance to stand out. Several
possible supersymmetric scenarios are shown and we refer the
reader to Ref.~\cite{bghp02} for details.
Recent experimental work at FermiLab has begun to make inroads
on this subject via the decays $D^+, D_s^+ \to
\pi^+ \mu^+ \mu^-$. Details of this impressive work is presented
by Paul Karchin in the following talk. \cite{pk} It suffices
to point out that the $\phi$ peak in the intermediate transition
$D^+, D_s^+ \to \phi \pi^+$ has been observed and that
for the region sensitive to New Physics the data is about 500
times above the SM signal, just 10 times above the signal expected
from the Little Higgs model \cite{fp}, but already constrains a
version of R-parity violating supersymmetry \cite{bghp02}.
\section{CP-VIOLATIONS}
To this date, no effects involving CP violation (CPV) have been observed for
charmed hadrons. We comment below on two potentially interesting
areas, involving respectively two-body decay
modes and three-body decay modes.
\subsection{Direct CPV in Two-body Modes}
On the experimental side, direct CPV for two-body decays of D
mesons involves measurement of the asymmetry
\begin{eqnarray*}
& & A_{\rm CP} = \frac{\Gamma_{D \to f}
- \Gamma_{\bar D \to \bar f}}{\Gamma_{D \to f} +
\Gamma_{\bar D \to \bar f}} \ \ .
\label{asym}
\end{eqnarray*}
Direct CPV has been observed for kaons ($\epsilon' /\epsilon$)
and B$_d$-mesons by studying $K\to \pi\pi$ and $B_d\to K\pi$ decays
respectively. Despite years of effort and the presence of sizeable
branching ratios, attempts to measure CP violating asymmetries in
D decays have yielded only null results. The present level of experimental
sensitivity is indicated by the entries in Table~\ref{table:cpv}.
What does theory have to say about direct CPV? In all cases,
it is QCD which inhibits a complete treatment.
For $\epsilon' /\epsilon$ in kaon physics, chiral symmetry provides
a successful means for predicting
the electroweak penguin contribution \cite{cdgm},
but the QCD penguin component remains an enigma. For heavy
quark systems, the strong phase shifts generally depend on a mix of
perturbative and nonperturbative QCD dynamics. The nonperturbative
component (surely dominant at the charm scale)
has thus far precluded a model-independent description.
\subsection{Dalitz Plots of Three Body Modes}
There have been a number of interesting Dalitz studies of charm decays
over the years and they are listed in Ref.~\cite{asner}.
Some of these are sensitive to both $D^0$ mixing and CPV.
A recent example, $D^0 \to K_{\rm S}^0 \pi^+ \pi^-$,
yields the mixing limits \cite{cleo}
\begin{eqnarray*}
& & - 0.045 < x_{\rm D} < 0.093 \nonumber \\
& & -0.064 < y_{\rm D} < 0.036 \ \ .
\end{eqnarray*}
Although not yet at the level of the experimental
bounds listed earlier, they are not that far off.
It is reassuring that the literature is continuing
to attract interesting suggestions for further work. \cite{as,aap}
\section{SUMMARY}
Taking experiment and theory together, it is clear that
the study of charm electroweak phenomena is quite an active area.
But there is much left to do, as I summarize in the following.
\subsection{Mixing}
{\it Experiment}: A number of experimental studies have been
underway, which is good.
But nothing which could be called a discovery has yet been seen, so
more sensitivity is needed. I especially feel that the range
$0.001 < y_{\rm D} < 0.01$ should be probed in ongoing and
forthcoming experiments. Are recent results from Belle and
BaBar `hints' or just fluctuations?
{\it Theory}: Crisp Standard Model predictions for $x_{\rm D}$ and
$y_{\rm D}$ remain frustratingly elusive, but at least we know why.
At the quark level, what is involved is a triple expansion in
dimension $D$, QCD coupling $\alpha_s$ and the mass ratio
$z \equiv m_s^2/m_c^2$. An NLO analysis with
$D=6$ gives $x_{\rm D} \simeq y_{\rm D} \simeq 10^{-6}$. One
anticipates progress in attacking the problem, but not without
difficulty. Performing the theoretical study of $y_{\rm D}$
at the hadron level has yielded some interesting insights. A
specific model of $|\Delta C| = 1$ decays has
$y_{\rm D} \simeq 10^{-3}$ whereas an up-to-date phenomenological
analysis appears to allow $y_{\rm D} \simeq 10^{-2}$.
Improved $|\Delta C| = 1$ decay data should impact this latter
approach.
For reasons already explained, $D^0$-${\bar D}^0$ remains a
possibly fertile area for New Physics studies. Because the
field of New Physics is in a state of constant flux (new models,
constrained parameter space, {\it etc}), there is a need to
update or re-do the calculations. A modern treatment of this
problem is underway.
\subsection{Rare Decays and CP-violations}
Experimental studies of charm rare decays and CPV have been performed
and as a result we now know what levels of sensitivity are
attainable. The work thus far has laid the foundations
for real progress in the field, but the next round of experiments
will face the challenge of doing even better. We must bridge
the gap between current experimental bounds and theoretical predictions.
For theorists, some studies of rare decays are already
in the literature, but we need more input regarding expectations
of various CPV signals.
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\bibitem{bu}
I.~I.~Y.~Bigi and N.~G.~Uraltsev,
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\bibitem{dghp}
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\bibitem{falk}
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\bibitem{su3}
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%%CITATION = HEP-PH 0110317;%%
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%%CITATION = HEP-PH 9802291;%%
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work in progress.
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work in progress.
\bibitem{bghp95}
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%%CITATION = HEP-PH 9502329;%%
\bibitem{bghp02}
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\bibitem{fp}
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Phys.\ Lett.\ B {\bf 382}, 415 (1996).
%%CITATION = HEP-PH 9603417;%%
\bibitem{pk}
P.~Karchin, {\it Recent Results on Charm Physics from the Tevatron
Collider}, talk at this conference.
\bibitem{cdgm}
V.~Cirigliano, J.~F.~Donoghue, E.~Golowich and K.~Maltman,
% ``Improved determination of the electroweak penguin contribution to
%epsilon'/epsilon in the chiral limit,''
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% [arXiv:hep-ph/0211420].
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Analyses} appearing in Ref.~\cite{pdg}.
\bibitem{cleo}
D.~M.~Asner {\it et al.} [CLEO Collaboration],
% ``Search for anti-D0 D0 mixing in the Dalitz plot analysis of D0 --> K0(S)
%pi+ pi-,''
Phys.\ Rev.\ D {\bf 72}, 012001 (2005).
%%CITATION = HEP-EX 0503045;%%
\bibitem{as}
D.~M.~Asner and W.~M.~Sun,
% ``Time-independent measurements of D0 - anti-D0 mixing and relative strong
%phases using quantum correlations,''
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\bibitem{aap}
A.~A.~Petrov,
%``Hunting for CP violation with untagged charm decays,''
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\end{thebibliography}
\end{document}