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\title{Recent Charm Physics Results from the Tevatron}

\author{Paul E. Karchin\address{Wayne State University, 
        Department of Physics and Astronomy\\ 
        666 W. Hancock St., Detroit Michigan 48201, United States of America}
        \thanks{Research is supported by the U.S. Department of Energy, 
        Office of Science, Grant DE-FG02-96ER41005.}
        }
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\begin{document}

\begin{abstract}

Three recent results on charm physics are presented based on Run II
data using the Fermilab Tevatron Collider. The D\O\ collaboration
reports preliminary results on the first observation of the rare decay
$D_s^{\pm} \rightarrow \pi^{\pm}
\phi$, $\phi \rightarrow \mu^+ \mu^-$, a measurement of  
the branching fraction for $D^{\pm}$ to the same final state, and an
upper limit for the non-resonant decay $D^{\pm} \rightarrow \pi^{\pm}
\mu^+ \mu^-$.
The CDF collaboration reports a preliminary measurement of the
angular distribution for $X(3872) \rightarrow J/\psi \pi^+
\pi^-$ and restricts the $X(3872)$ quantum numbers to 
$J^{PC}$ = $1^{++}$ or $2^{-+}$.
Also, CDF reports a measurement of the ratio of branching fractions
of the rare (doubly Cabibbo-suppressed) decay $D^0 \rightarrow
K^+\pi^-$ and the Cabibbo-favored decay $D^0 \rightarrow
K^-\pi^+$ (and charge conjugate decays).

\vspace{1pc}
\end{abstract}

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\section{Preliminary D\O\  Measurement of $D^{\pm}$ and $D_s^{\pm} 
\rightarrow$ $\pi^{\pm} \mu^+ \mu^-$}

These decays are highly suppressed in the standard model (SM) and
hence are sensitive to new physics.  The only SM process for
$D_s^{\pm} \rightarrow$ $\pi^{\pm} \mu^+ \mu^-$ occurs when the charm
quark has a Cabibbo-favored weak transition leading to a virtual
$\phi$ intermediate state, as shown in
Fig.\ \ref{fig:pimumu-feynman}(A). For the $D^{\pm}$, the $\pi^{\pm}
\mu^+ \mu^-$ final state can be reached through a
Cabibbo-suppressed transition of the charm quark leading to a virtual
$\phi$ intermediate state (Fig.\ \ref{fig:pimumu-feynman}(B)), or
through second-order, short-distance weak interactions, as shown in
Fig.\ \ref{fig:pimumu-feynman}(C) and (D). The SM predictions for the
branching fractions for the decays through the $\phi$ resonance are
simply the product of the well-measured branching ratios ${\cal
B}(D^{\pm}, D_s^{\pm} \rightarrow \pi^{\pm} \phi) \times {\cal B}(\phi
\rightarrow \mu^+ \mu^-)$. For ${\cal B}(D_s^{\pm} 
\rightarrow \pi^{\pm} \phi$, 
$\phi \rightarrow \mu^+ \mu^-)$, 
the SM prediction is $10.3 \times 10^{-6}$ and the best upper limit,
from the FOCUS experiment, is $29 \times 10^{-6}$ at 90\% C.L.
For ${\cal B}(D^{\pm} \rightarrow \pi^{\pm} \phi$, $\phi \rightarrow
\mu^+ \mu^-)$, the SM prediction is $1.77 \times 10^{-6}$. The SM
prediction for non-resonant ${\cal B}(D^{\pm} \rightarrow$ $\pi^{\pm}
\mu^+ \mu^-)$ = $9.4 \times 10^{-9}$. The best experimental upper
limit, also from FOCUS, is ${\cal B}(D^{\pm} \rightarrow$ $\pi^{\pm}
\mu^+ \mu^-)$ $< 8.8 \times 10^{-6}$ at 90\% C.L.

\begin{figure}[htb]
\resizebox{3.0in}{!}
     {\includegraphics*[0,0][550,345]{pimumu-feynman.epsi}}
\caption{Standard model Feynman diagrams leading to 
$\pi^{\pm} \mu^+ \mu^-$ final states from decay of 
$D_s^{\pm}$ (A) and $D^{\pm}$ (B), (C) and (D).}
\label{fig:pimumu-feynman}
\end{figure}

Rare decays of heavy flavor particles via flavor-changing
neutral-current processes are sensitive to new physics.  For down-type
quarks, the transition $s \rightarrow d$ is studied in $K$ meson decay
and $b \rightarrow s$ is studied in $B$ meson decay. However, in some
new physics scenarios, the exotic signal appears only in up-type quark
transitions, studied via $c \rightarrow u$ in $D$ meson
decays. Examples are an R-parity violating supersymmetry model \cite{Burdman}
and a ``little higgs'' model with a new up-type vector quark \cite{Fajfer}.
The former model predicts a branching ratio for the
non-resonant decay $D^{\pm} \rightarrow$ $\pi^{\pm}
\mu^+ \mu^-$ significantly above the SM prediction.

The preliminary results from D\O\ \cite{DZero} are based on an integrated
luminosity of approximately 1 fb$^{-1}$. The data is recorded using a
di-muon trigger.  With cuts optimized
for the decay $D_s^{\pm} \rightarrow \pi^{\pm}
\phi$, $\phi \rightarrow \mu^+ \mu^-$, a signal of $65\pm11$ is
obtained for that decay and a signal of $26 \pm 9$ is observed for 
 $D^{\pm} \rightarrow$ $\pi^{\pm}
\phi$, $\phi \rightarrow \mu^+ \mu^-$, as shown in Fig.\ \ref{fig:pimumu_mass}.
This is the first observation of the $D_s^{\pm}$ decay mode.
With cuts optimized for the $D^{\pm}$ decay mode, there are 6
candidates in the
$D^+$ signal region and this is used to 
measure ${\cal B} (D^{\pm} \rightarrow$ $\pi^{\pm}
\phi$, $\phi \rightarrow \mu^+ \mu^-)$ = 
$1.75\pm0.7({\rm stat})\pm0.5({\rm syst}) \times 10^{-6}$. The normalization
is based on the signal and predicted branching fraction for the corresponding
$D_s^{\pm}$ decay. The measurement of 
${\cal B} (D^{\pm} \rightarrow$ $\pi^{\pm}
\phi$, $\phi \rightarrow \mu^+ \mu^-)$ is consistent with the SM prediction.
Using the same normalization technique mentioned previously, 
the resulting limit for the non-resonant decay 
is ${\cal B} (D^{\pm} \rightarrow$ $\pi^{\pm}
\mu^+ \mu^-)$ $ < 4.7 \times 10^{-6}$ at 90\% C.L. This limit is improved over
the previous one, but well above the SM prediction and hence leaving open
the possibility for new physics signals from more sensitive measurements.

\begin{figure}[htb]
\resizebox{3in}{!}{\includegraphics*{pimumu-mass.eps}}
\caption{Invariant mass distribution showing signals for 
$D^{\pm}$ and $D_s^{\pm} \rightarrow$ 
 $\pi^{\pm} \phi$, $\phi \rightarrow \mu^+ \mu^-$.}
\label{fig:pimumu_mass}
\end{figure}

\section{Preliminary CDF Measurement of $X$(3872) Angular Distribution and
Analysis of Quantum Numbers}

The $X$(3872) is a narrow resonance with a well established decay to
the final state $J/\psi \pi^+ \pi^-$.  The fundamental composition of
this state is unclear. Possibilities include charmonium (a
$c\overline{c}$ resonance), a $D^0$ and $\overline{D}^{*0}$ bound into
a ``molecule'' or an extended 4-valence-quark state with quark content
$c\overline{c}d\overline{d}$. Determination of the spin $J$, parity
$P$ and charge-conjugation $C$ quantum numbers can help identify the
underlying nature of this state.  The CDF collaboration reports a
preliminary measurement \cite{CDF-X} of the angular distributions of
the $X$(3872) decay products in the $J/\psi \pi^+ \pi^-$ final state
and resulting constraints on the $J^{PC}$ quantum numbers.

In this analysis, the $X$ is considered to decay into a $J/\psi$ and
an intermediate $(\pi^+\pi^-)$ resonant state. The $(\pi^+\pi^-)$ can
be in an $s$-wave $(L_{\pi\pi}=0)$ or $p$-wave $(\rho, L_{\pi\pi}=1)$
state. The $J^{PC}$ quantum numbers of the $X$ affect the
distributions of three angles: $\theta_{J/\psi}$ is measured between
the $\mu^+$ and its parent $J/\psi$ direction, $\theta_{\pi\pi}$
is measured between the $\pi^+$ and its parent $(\pi^+\pi^-)$
direction, and $\Delta\Phi$ is measured between the $\mu$-$\mu$ and
$\pi$-$\pi$ decay planes. The measured angular distributions are
compared with simulated ones generated with a variety of possible
$J^{PC}$ values. The simulation uses a decay generator based on a
helicity amplitude method and a parametric model for detector
acceptance and efficiency.

The data correspond to an integrated luminosity of 0.78 fb$^{-1}$ and
were recorded using a di-muon trigger. An $X(3872)$ signal of
$2292\pm98$ is obtained as seen in Fig.\ \ref{fig:mX_narrow}.
The $\theta_{J/\psi}$-$\theta_{\pi\pi}$-$\Delta\Phi$ angular space is
divided into $2 \times 2 \times 3$ = 12 bins. The signal yield,
determined in each bin, is shown in
Fig.\ \ref{fig:X_result_BW}. Expected distributions for 13 different
$J^{PC}$ hypotheses were determined and 4 of these are shown in
Fig.\ \ref{fig:X_result_BW}. Based on a $\chi^2$ comparison between
data and prediction, only 2 of the 13 $J^{PC}$ hypotheses have
significant probabilities; these are $1^{++}$ (27.8\%) and $2^{-+}$
(25.8\%).  All the other hypotheses lead to probabilities $<$ 0.02\%.

\begin{figure}[htb]
\rotatebox{0}
{\resizebox{3.0in}{!}
%    {\includegraphics*[0,025][520,336]{mX_narrow.eps}}}
    {\includegraphics*[0,0][567,405]{mX_plot_No2.eps}}}
\caption{Invariant mass distribution showing the $X$(3872) signal.}
\label{fig:mX_narrow}
\end{figure}

\begin{figure}[htb]
\rotatebox{0}
{\resizebox{3in}{!}
    {\includegraphics*[0,0][545,336]{X_result_BW.eps}}}
\caption{Angular distribution for $X$(3872) decay products compared with 
expectations for different $J^{PC}$ hypotheses.}
\label{fig:X_result_BW}
\end{figure}

Both of the probable $J^{PC}$ hypotheses are consistent with
charmonium states: $\chi'_{c1}$ ($1^{++}$) and $1^{1}D_{2}$
($2^{-+}$). However, the masses of these states predicted from
charmonium models are not compatible with 3872 MeV/$c^2$. Furthermore,
the decay of a charmonium state to $J/\psi \rho$ violates conservation
of isospin. One of the probable states, $1^{++}$, is compatible with a
charm meson molecular state.  As theoretical models of charmonium and
exotic states improve, it may be possible to further restrict the
interpretation of the $X$(3872).

\section{CDF Measurement of the Doubly Cabibbo Suppressed Charm Decay 
 $D^0 \rightarrow K^+\pi^-$ }

In the SM, the decay $D^0 \rightarrow K^+\pi^-$ proceeds
through a doubly Cabibbo-suppressed (DCS) tree diagram and possibly
through a ``mixing'' process in which the $D^0$ changes into 
a $\overline{D}^0$. (In
this section, discussion of a decay reaction implicitly includes the
charge conjugate process.)  The DCS decay rate depends on CKM factors
as well as the magnitude of SU(3) flavor symmetry violation.  Mixing
may occur through two distinct types of second-order weak
processes. In the first, the $D^0$ decays into a virtual
(``long-range'') intermediate state such as $\pi^+\pi^-$, which
subsequently decays into a $\overline{D}^0$.  The second type is a
short range process, with either a ``box'' or ``penguin'' topology. It
is not established whether long range mixing occurs. Its strength
depends on SU(3) flavor symmetry violation. Short range mixing is
negligible in the SM. However, exotic weakly interacting particles
could enhance the short range mixing and provide a signature of new
physics.

The ratio of branching fractions $R_{\cal B} = $ ${\cal B}(D^0 \to K^+
\pi^-)/{\cal B}(D^0 \to K^- \pi^+)$ is related to the parameters
describing the underlying processes by $R_{\cal B} = $ $ R_D +
\sqrt{R_D} y' + (x'^2 + y'^2)/2$, where $R_D$ is the squared
modulus of the ratio of DCS to Cabibbo-favored (CF) amplitudes, and
$x'$ and $y'$ are parameters describing the mixing. In the limit of
SU(3) flavor symmetry and without exotic physics, the charm-describing
parameter $R_D$ is equal to $\tan^4\theta_{C}$, where $\theta_{C}$ is
the Cabibbo angle measured in kaon decays.  The world average values,
$R_D = (3.62 \pm {}$\mbox{$0.29) \times\, 10^{-3}$} and
$\tan^4\theta_{C} = (2.88 \pm {}$\mbox{$0.27) \times\, 10^{-3}$},
are equal within their uncertainties, consistent with
SU(3) flavor symmetry. More accurate measurements are needed to test the
relation further and search for charm mixing.

The CDF collaboration reports a measurement \cite{CDF-PRD} of $R_{\cal
B}$ based on an integrated luminosity of 0.35 fb$^{-1}$ using data
recorded with a secondary vertex trigger. The decay chain $D^{*+}
\to \pi^+ D^0$, $D^0 \to K^+ \pi^-$ is reconstructed, 
where the charge of the $\pi^+$ from $D^{*+}$ decay distinguishes the
$D^0$ from $\bar{D}^0$.  The large CF background is suppressed mainly
by removing DCS candidates consistent with CF decays when the $K$ and
$\pi$ assignments are interchanged for the $D^0$ decay products. The
invariant $K^+\pi^-$ mass distribution, shown in
Fig.\ \ref{fig:kpi_mass}, includes contributions from signal and
backgrounds from pions randomly associated with $D^*$ decay,
misidentified CF decays because of interchanged $K$ and $\pi$
assignment, and random combinations of $K$ and $\pi$.

\begin{figure}[htb]
\resizebox{3.0in}{!}{\includegraphics*{fig1.eps}}
\caption{CDF data for $K^+\pi^-$ invariant mass with estimated 
contributions from DCS signal and backgrounds.}
\label{fig:kpi_mass}
\end{figure}

To determine the DCS signal, 
candidates are divided into 60
slices of $\Delta m$, each slice of width 0.5~MeV/$c^2$, where 
$\Delta m = m(K^+\pi^-\pi^+) - m(K^+\pi^-) -m(\pi^{+})$. The $D^0$ signal yield
is determined from the $K \pi$ mass distribution for each slice.  The signal
yields for all the slices are shown in Fig.\ \ref{fig:mass_diff}.
The total DCS signal is
$2005 \pm 104$, which combined with the
CF signal of $495172 \pm 907$ gives
 $R_{\cal B} = 4.05 \pm 0.21 (\mbox{stat})
\pm {}$\mbox{0.11(syst) $\times\, 10^{-3}$}.  
This result is consistent with measurements from the $B$ factories, 
the most accurate of which is from 
Belle \cite{BELLE}.
However, the difference between the CDF value and the world
average for $\tan^4\theta_{C}$ has a significance of 3.4 $\sigma$.
This difference could be due to a statistical fluctuation, a modest
SU(3) flavor violation or a signal for mixing. Additional data from
CDF with an analysis of $R_{\cal B}$ versus time should help sort out
the effects of DCS decay, SU(3) flavor violation, and mixing.

\begin{figure}[htb]
\rotatebox{0}
{\resizebox{3in}{!}
    {\includegraphics*{fig2.eps}}}
\caption{CDF data for $\Delta m = m(K^+\pi^-\pi^+) - 
m(K^+\pi^-) -m(\pi^{+})$ and fit results for DCS signal and background.}
\label{fig:mass_diff}
\end{figure}

\begin{thebibliography}{9}

\bibitem{Burdman} G. Burdman {\it et al.}, Phys. Rev. D {\bf 66} (2002) 014009.

\bibitem{Fajfer} Svjetlana Fajfer and Sasa Prelovsek, Phys. Rev. D {\bf 73}
(2006) 054026.

\bibitem{DZero} D\O\ Collaboration, Note 5038-CONF v2.1 (unpublished);
http://www-d0.fnal.gov.

\bibitem{CDF-X} CDF Collaboration, CDF Note 8201 (unpublished); 
http://www-cdf.fnal.gov.

\bibitem{CDF-PRD} A. Abulencia {\it et al.} (CDF Collaboration), 
Phys. Rev. D {\bf 74} (2006) 031109(R).

\bibitem{BELLE} L.M. Zhang {\it et al.} (Belle Collaboration), 
Phys. Rev. Lett. {\bf 96} (2006) 151801.

\end{thebibliography}


\end{document}
