# BRIEF REVIEW OF CHARM PHYSICS

Marco Gersabeck CERN, 1211 Geneva, Switzerland
<EMAIL_ADDRESS>
###### Abstract

Charm physics has attracted increased attention after first evidence for charm
mixing was observed in 2007. The level of attention has risen sharply after
LHCb reported first evidence for $C\\!P$ violation in the charm sector.
Neither mixing nor $C\\!P$ violation have been established by a single
unambiguous measurement to date. This review covers the status of mixing and
$C\\!P$ violation measurements and comments on the challenges on the road
ahead, both on the experimental and theoretical side, and on ways to tackle
them.

###### keywords:

Charm physics; Meson mixing; CP violation.

Received (Day Month Year)Revised (Day Month Year)

PACS Nos.: 13.20.Fc, 13.25.Ft, 14.40.Lb

##  1 Introduction

Charm physics covers the studies of a range of composite particles containing
charm quarks which provide unique opportunities for probing the strong and
weak interactions in the standard model and beyond. The charm quark, being the
up-type quark of the second of the three generations, is the third-heaviest of
the six quarks. Charm particles can exist as so-called open charm mesons or
baryons, containing one or several (for baryons) charm quarks, or as
charmonium states which are bound states of charm-anticharm quark pairs.

The uniqueness of charm particles lies in their decays. The charm quark can
only decay via annihilation with an anti-charm quark in the case of charmonium
states or as a weak decay, mediated by a $W^{\pm}$-boson, into a strange or
down quark. Thus, open charm particles are the only ones allowing the study of
weak decays of an up-type quark in a bound state.

In 2009, Ikaros Bigi asked whether charm’s third time could be the real charm
[1]. Charm’s first time was the discovery of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ [2, 3], which followed
three years after the possible first observation of an open charm decay in
cosmic ray showers [4]. This discovery confirmed the existence of a fourth
quark as expected by the GIM mechanism [5] motivated by the non-existence of
flavour-changing neutral currents [6, 7] in conjunction with the observation
of the mixing of neutral kaons [8, 9, 10]. The second time charm attracted
considerable attention was caused by the observation of $D_{\mathrm{s}J}$
states [11, 12, 13, 14] which could not be accommodated by QCD [15, 16, 17,
18, 19]. Until today, excited charmonium and open charm particles provide an
excellent laboratory for studying QCD, however, this topic is beyond the scope
of this review.

Charm’s third time started with the first evidence for the mixing of neutral
charm mesons reported by BaBar [20] and Belle [21] in 2007. Since then a lot
of work went into more precise measurements of the mixing phenomenon as well
as into searches for charge-parity ($C\\!P$) symmetry violation in the charm
sector. At the same time theoretical calculations were improved even though
precise standard model predictions are still a major challenge. This paper
reviews the current situation of studies of processes, which are mediated by
the weak interaction at leading order, using open charm particles. Particular
focus is given to mixing and $C\\!P$ violation, followed by comments on rare
charm decays at the end of this review.

###  1.1 Charm production

Charm physics has been and is being performed at a range of different
accelerators. These come with different production mechanisms and thus with
largely varying production cross-sections. At $\mathrm{e}^{+}\mathrm{e}^{-}$
colliders two different running conditions are of interest to charm physics.
Tuning the centre-of-mass energy to resonantly produce $\psi(3770)$ states
leads to the production of quantum-correlated $D^{0}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ or $D^{+}D^{-}$ pairs. This is the
case for the CLEO-c experiment at the CESR-c collider as well as for BESIII at
BEPCII. The most commonly used alternative is running at a higher centre-of-
mass energy to resonantly produce $\Upsilon(4S)$ which decay into quantum-
correlated $B^{0}\kern 1.99997pt\overline{\kern-1.99997ptB}{}^{0}$ or
$B^{+}B^{-}$ pairs. This is used by the BaBar and Belle experiments which are
located at the PEP-II and KEKB colliders, respectively. Both PEP-II and KEKB
are asymmetric colliders, thus having a collision system that is boosted with
respect to the laboratory frame. This allows measurements with decay-time
resolutions about a factor two to four below the $D^{0}$ lifetime and
therefore decay-time dependent studies.

The production cross-section for producing $D\kern
1.99997pt\overline{\kern-1.99997ptD}{}$ pairs at the $\psi(3770)$ resonance is
approximately $8\rm\,nb$ [22]. When running at the $\Upsilon(4S)$ resonance,
the cross-section for producing $\mathrm{c}\overline{\mathrm{c}}$ pairs is
$1.3\rm\,nb$ [23]. The latter scenario gives access to all species of charm
particles while the $\psi(3770)$ only decays into $D^{0}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ or $D^{+}D^{-}$ pairs. The BaBar
and Belle experiments have collected integrated luminosities of about
$500\mbox{\,fb}^{-1}$ and $1000\mbox{\,fb}^{-1}$, respectively. CLEO-c has
collected $0.5\mbox{\,fb}^{-1}$ at the $\psi(3770)$ resonance as well as
around $0.3\mbox{\,fb}^{-1}$ above the threshold for
$D^{+}_{\mathrm{s}}D^{-}_{\mathrm{s}}$ production. BESIII has so far collected
nearly $3\mbox{\,fb}^{-1}$ in their 2010 and 2011 runs.

At hadron colliders the production cross-sections are significantly higher.
The cross-section for producing $\mathrm{c}\overline{\mathrm{c}}$ pairs in
proton-proton collisions at the LHC with a centre-of-mass energy
$7\mathrm{\,Te\kern-1.00006ptV}$ is about $6\rm\,mb$ [24], i.e. more than six
orders of magnitude higher compared to operating an
$\mathrm{e}^{+}\mathrm{e}^{-}$ collider at the $\Upsilon(4S)$ resonance. This
corresponds to a cross-section of about $1.5\rm\,mb$ for producing $D^{0}$ in
the LHCb acceptance111The LHCb acceptance is given as a range in momentum
transverse to the beam direction and rapidity as $p_{\rm
T}<8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2<y<4.5$.. This number may be
compared to its equivalent at CDF which has been measured to $13\rm\,\mu b$
inside the detector acceptance222The CDF acceptance is defined as $p_{\rm
T}>5.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},|y|<1$. for proton-antiproton
collisions at the Tevatron with $\sqrt{s}=1.96\mathrm{\,Te\kern-1.00006ptV}$
[25]. CDF has collected a total of about $10\mbox{\,fb}^{-1}$ while LHCb has
collected about $1.8\mbox{\,fb}^{-1}$ by the time of writing this review,
corresponding to $1.3\times 10^{11}$ and $2.7\times 10^{12}$ $D^{0}$ mesons
produced in the respective detector acceptances.

The production of charm quarks in hadron collisions occurs predominantly in
very asymmetric collisions which result in heavily boosted quarks with high
rapidities. Therefore, LHCb is ideally suited for performing decay-time
dependent studies of charm decays. At the same time, the
$\mathrm{c}\overline{\mathrm{c}}$ cross-section at the LHC is about $10\%$ of
the total inelastic cross-section which allows to have reasonably low
background levels for a hadronic environment. The coverage of nearly the full
solid angle of the $\mathrm{e}^{+}\mathrm{e}^{-}$-collider experiments
mentioned here makes them very powerful instruments for analysing decays
involving neutral particles that may remain undetected or for inclusive
studies.

###  1.2 Mixing and $C\\!P$ violation in neutral mesons

For neutral mesons, the mass eigenstates, i.e. the physical particles,
generally do not coincide with the flavour eigenstates, i.e. those governing
the interactions. The mass eigenstates of neutral mesons, $|M_{1,2}\rangle$,
with masses $m_{1,2}$ and widths $\Gamma_{1,2}$, are linear combinations of
the flavour eigenstates, $|M^{0}\rangle$ and $|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$, as
$|M_{1,2}\rangle=p|M^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$ with complex coefficients
satisfying $|p|^{2}+|q|^{2}=1$. This allows the definition of the averages
$m\equiv(m_{1}+m_{2})/2$ and $\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$. The
phase convention of $p$ and $q$ is chosen such that, in the limit of no
$C\\!P$ violation, $C\\!P|M^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$.

Mixing, i.e. the periodical transformation of mesons into their anti-mesons
and back, occurs if there is a non-zero difference in the masses or widths of
the two mass eigenstates of a meson. This is quantified in the differences
$\Delta{}m\equiv{}m_{2}-m_{1}$ and $\Delta\Gamma\equiv\Gamma_{2}-\Gamma_{1}$.
Furthermore, the mixing parameters are defined as $x\equiv\Delta{}m/\Gamma$
and $y\equiv\Delta\Gamma/(2\Gamma)$.

It is these mixing parameters $x$, and $y$ which define the characteristic
behaviour of the four neutral meson systems, which are subject to mixing,
kaons ($K$), charm ($D$), $B^{0}_{\mathrm{d}}$, and $B^{0}_{\mathrm{s}}$
mesons. To appreciate the different mixing behaviour it is instructive to
consider the time evolution of the meson and anti-meson states. The
probability of observing a neutral meson state $M^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}$ after a time $t$ has passed since
the observation of an initial state $M^{0}$ is

$\displaystyle P(M^{0}\to M^{0},t)$
$\displaystyle=\frac{1}{2}e^{-\Gamma{}t}(\cosh(y\Gamma{}t)+\cos(x\Gamma{}t)),$
$\displaystyle P(M^{0}\to\kern 1.99997pt\overline{\kern-1.99997ptM}{}^{0},t)$
$\displaystyle=\frac{1}{2}\left|\frac{q}{p}\right|^{2}e^{-\Gamma{}t}(\cosh(y\Gamma{}t)-\cos(x\Gamma{}t)),$
(1)

where the oscillatory behaviour is governed by the mixing parameter $x$.

Figure 1: The widths and mass differences of the physical states of the
flavoured neutral mesons. The width corresponds to the inverse lifetime while
the mass difference determines the oscillation frequency.

For charm mesons the mixing parameters are drastically different compared to
those of kaons or $B$ mesons. Figure 1 shows the widths and mass differences
of the four neutral meson systems. The kaon system is the only one to have
$y\approx 1$, resulting in two mass eigenstates with vastly different
lifetimes, hence their names $K$-short ($K^{0}_{\rm\scriptstyle S}$) and
$K$-long ($K^{0}_{\rm\scriptstyle L}$). Furthermore, also $x\approx 1$ which
results in a sizeable sinusoidal oscillation frequency as shown in Eq. (1.2).
The two $B$-meson systems have reasonably small width splitting, however, they
have sizeable values for $x$. Particularly for the $B^{0}_{\mathrm{s}}$ system
this leads to fast oscillations which require high experimental accuracy to be
resolved. The charm meson system is the only one where both $x$ and $y$ are
significantly less than $1$, hence the nearly overlapping curves in Fig. 1.

Experimentally, the different mixing parameters lead to rather different
challenges for measurements in the various meson systems. The vast lifetime
difference in the kaon system leads to the possibility of studying nearly
clean samples of just one of the two mass eigenstates by either measuring
decays close to a production target where $K^{0}_{\rm\scriptstyle S}$ decays
dominate, or far away where most $K^{0}_{\rm\scriptstyle S}$ have decayed
before entering the detection region. In the $B$ systems the oscillation
frequency puts a challenge to the decay-time resolution, particularly for
$B^{0}_{\mathrm{s}}$ mesons as mentioned before. The smallness of $y$
requires, to first order, large data samples to acquire the necessary
statistical precision for measuring such a small quantity. The latter is
particularly true for the charm system, where both $x$ and $y$ are small. This
is the reason why it was only in 2007 when first evidence for charm mixing was
observed.

The symmetry under $C\\!P$ transformation is violated for a deviation from
unity of the quantity $\lambda_{f}$, defined as

$\lambda_{f}\equiv\frac{q\bar{A}_{\bar{f}}}{pA_{f}}=-\eta_{C\\!P}\left|\frac{q}{p}\right|\left|\frac{\bar{A}_{f}}{A_{f}}\right|e^{i\phi},$
(2)

where the right-hand expression is valid for a $C\\!P$ eigenstate $f$ with
eigenvalue $\eta_{C\\!P}$ and $\phi$ is the $C\\!P$ violating relative phase
between $q/p$ and $\bar{A}_{f}/A_{f}$. Besides the mixing parameters
introduced above this expression contains the decay amplitudes $A_{f}$ and
$\bar{A}_{f}$ for decays into a final state $f$.

$C\\!P$ violation can have different origins: the case $|q/p|\neq 1$ is called
$C\\!P$ violation in mixing, $|\bar{A}_{f}/A_{f}|\neq 1$ is $C\\!P$ violation
in the decay, and a non-zero phase $\phi$ between $q/p$ and
$\bar{A}_{f}/A_{f}$ causes $C\\!P$ violation in the interference between
mixing and decay. Mixing is common to all decay modes and hence $C\\!P$
violation originating in this process is universal which is called indirect
$C\\!P$ violation. Decay-specific $C\\!P$ violation is called direct $C\\!P$
violation. An excellent discussion on the different types of $C\\!P$ violation
can be found in section 7.2.1 of Ref. Sozzi:2008zza. As opposed to the strange
and the beauty system, $C\\!P$ violation has not yet been discovered in the
charm system, though the LHCb collaboration has recently found first evidence
for $C\\!P$ violation in two-body $D^{0}$ decays [27], arguably the most
surprising result from the LHC in 2011.

##  2 Charm mixing

The studies of charm mesons have gained in momentum with the measurements of
first evidence for meson anti-meson mixing in neutral charm mesons in 2007
[20, 21]. Mixing of $D^{0}$ mesons is the only mixing process where down-type
quarks contribute to the box diagram. Unlike $B$-meson mixing, where the top-
quark contribution dominates, the third generation quark is of similar mass to
the other down-type quarks. This leads to a combination of GIM cancellation
[5] and CKM suppression [28, 29], which results in a strongly suppressed
mixing process [30, 31, 32, 33, 34].

There are two approaches for theoretical calculations of charm mixing. The
“inclusive” approach is based an operator product expansion (OPE) in
$\Lambda/m_{\mathrm{c}}$ [35, 36, 37, 38, 39, 40, 34]. Due to the
cancellations mentioned above it is higher order operators that give the
largest contributions to the mixing parameters. Furthermore, it is not yet
clear whether the expansion series really converges. Calculations of the charm
meson lifetimes are being performed to test whether the OPE approach can
properly reproduce the large difference between the $D^{0}$ and the $D^{+}$
lifetimes. In the $B^{0}_{\mathrm{s}}$ system, the OPE approach successfully
predicted the width splitting of the two $B^{0}_{\mathrm{s}}$ mass eigenstates
[41] which has recently been confirmed by an LHCb measurement [42].

The “exclusive” approach sums over intermediate hadronic states, taking input
from models or experimental data [43, 44, 45, 46, 47, 48]. Also in this
approach, different modes of the same $SU(3)$ multiplet lead to cancellations
which is why their individual contributions have to be known to high
precision. Due to the considerable mass of the $D^{0}$ meson, many different
modes need to be taken into account simultaneously. Of these, only phase space
differences can be evaluated at the moment. Estimates indicate that mixing in
the experimentally observed range is conceivable when taking into account
$SU(3)$-breaking effects. However, neither the inclusive nor the exclusive
approach have thus far permitted a precise theoretical calculation of charm
mixing.

It was discussed whether the measured size of the mixing parameters could be
interpreted as a hint for physics beyond the standard model [49, 50, 51, 52,
53, 54, 55]. The biggest problem in answering this question is the non-
existence of a precise standard model calculation. Effects of physics beyond
the standard model were also searched for in numerous $C\\!P$ violation
measurements and searches for rare decays both of which are covered in the
remainder of this review.

Mixing of $D^{0}$ mesons can be measured in several different modes. Most
require identifying the flavour of the $D^{0}$ at production as well as at the
time of the decay. Tagging the flavour at production usually exploits the
strong decay $D^{*+}\\!\to D^{0}\pi^{+}$ (and charge conjugate333Charge
conjugate decays are implicitly included henceforth.) where the charge of the
pion determines the flavour of the $D^{0}$. The small amount of free energy in
this decay leads to the difference in the reconstructed invariant mass of the
$D^{*+}$ and the $D^{0}$, $\delta m\equiv{}m_{D^{*+}}-m_{D^{0}}$, exhibiting a
sharply peaking structure over a threshold function as background. An
alternative to using this decay mode is tagging the $D^{0}$ flavour by
reconstructing a flavour-specific decay of a $B$ meson. This method has not
yet been used in a measurement as it did not yet yield competitive quantities
of tagged $D^{0}$ mesons. At LHCb this approach may be of interest due to
differences in trigger efficiencies compensating for lower production rates.
Another option available particularly at $\mathrm{e}^{+}\mathrm{e}^{-}$
colliders is the reconstruction of the opposite side charm meson in a flavour
specific decay.

Theoretically, the most straight-forward mixing measurement is that of the
rate of the forbidden decay $D^{0}\\!\to K^{+}\mu^{-}\overline{\nu}_{\mu}$
which is only accessible through $D^{0}$-$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing. The ratio of the time-
integrated rate of these forbidden decays to their allowed counterparts,
$D^{0}\\!\to K^{-}\mu^{+}\nu_{\mu}$, determines $R_{\rm
m}\equiv(x^{2}+y^{2})/2$. As this requires very large samples of $D^{0}$
mesons no measurement has thus far reached sufficient sensitivity to see
evidence for $D^{0}$ mixing. The most sensitive measurement to date has been
made by the Belle collaboration [56] to $R_{\rm m}=(1.3\pm 2.2\pm 2.0)\times
10^{-4}$, where the first uncertainty is of statistical and the second is of
systematic nature444This notation is applied to all results where two
uncertainties are quoted..

Related to the semileptonic decay is the suppressed decay $D^{0}\\!\to
K^{+}\pi^{-}$, called wrong-sign (WS) decay. For this decay, a doubly Cabibbo-
suppressed (DCS) amplitude interferes with the decay through a mixing process
followed by the Cabibbo-favoured (CF) decay $D^{0}\\!\to K^{-}\pi^{+}$. The
time-dependent decay rate of the WS decay is, in the limit of $C\\!P$
conservation, proportional to

$\frac{\Gamma(D^{0}(t)\to K^{+}\pi^{-})}{e^{-\Gamma t}}\propto\left(R_{\rm
D}+\sqrt{R_{\rm D}}y^{\prime}\Gamma{}t+R^{2}_{\rm m}(\Gamma{}t)^{2}\right),$
(3)

where the mixing parameters are rotated by the strong phase between the DCS
and the CF amplitude, leading to the observable
$y^{\prime}=y\cos\delta_{K\pi}-x\sin\delta_{K\pi}$ [57]. The parameter $R_{\rm
D}$ is the ratio of the DCS to the CF rate. Measurements with sufficient
sensitivity to unveil evidence for $D^{0}$ mixing have been performed by the
BaBar and CDF collaborations, leading to

|  $x^{\prime 2}$ in $10^{-3}$ |  $y^{\prime}$ in $10^{-3}$
---|---|---
BaBar [20] | $-0.22\pm 0.30\pm 0.20$ | $9.7\pm 4.4\pm 3.1$
CDF [58] | $-0.12\pm 0.35$ |  $8.5\pm 7.6$.

Similarly, the CF and DCS amplitudes can also lead to excited states of the
same quark content. The decay $D^{0}\\!\to K^{-}\pi^{+}\pi^{0}$ is the final
state of several such resonances. Thus, by studying the decay-time dependence
of the various resonances a mixing measurement can be obtained. The BaBar
collaboration achieved a measurement showing evidence for $D^{0}$ mixing [59]
with central values of $x^{\prime\prime}=(26.1^{+5.7}_{-6.8}\pm 3.9)\times
10^{-3}$ and $y^{\prime\prime}=(-0.6^{+5.5}_{-6.4}\pm 3.4)\times 10^{-3}$,
where the rotation between the observables and the system of mixing parameters
is given by a strong phase as

$\displaystyle x^{\prime\prime}$
$\displaystyle=x\cos\delta_{K^{-}\pi^{+}\pi^{0}}+y\sin\delta_{K^{-}\pi^{+}\pi^{0}}$
(4) $\displaystyle y^{\prime\prime}$
$\displaystyle=y\cos\delta_{K^{-}\pi^{+}\pi^{0}}-x\sin\delta_{K^{-}\pi^{+}\pi^{0}}.$
(5)

The significant advantage of this analysis over that using two-body final
states is that both mixing parameters are measured at first order rather than
one at first and one at second order.

The strong phases are not accessible in these measurements but have to come
from measurements performed using quantum-correlated $D^{0}$-$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ pairs produced at threshold. Such
measurements are available from CLEO [60, 61, 62, 63] and can be further
improved at BESIII.

The comparison of effective inverse lifetimes in decays of $D^{0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) mesons into final states which
are $C\\!P$ eigenstates, $\hat{\Gamma}$ ($\hat{\bar{\Gamma}}$), to that of a
Cabibbo-favoured flavour eigenstate ($\Gamma$) leads to the observable

$y_{CP}=\frac{\hat{\Gamma}+\hat{\bar{\Gamma}}}{2\Gamma}-1\approx\eta_{C\\!P}\left[\left(1-\frac{A_{m}^{2}}{8}\right)y\cos\phi-\frac{A_{m}}{2}x\sin\phi\right],$
(6)

where $A_{m}$ is the $C\\!P$ violation in mixing defined alongside the direct
$C\\!P$ violation $A_{d}$ by $|\lambda_{f}^{\pm 1}|^{2}\approx(1\pm
A_{m})(1\pm A_{d})$ [64]. In the limit of $C\\!P$ conservation $y_{CP}$ equals
the mixing parameter $y$. As the $C\\!P$-violating contributions $A_{m}$ and
$\phi$ enter only at second order, measurements of $y_{CP}$ are among the most
powerful constraints of the mixing parameter $y$.

Comparing the $C\\!P$ eigenstates $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ to the
Cabibbo-favoured mode $K^{-}\pi^{+}$, the Belle [21] and BaBar [65]
collaborations have measured $y_{CP}=(13.1\pm 3.2\pm 2.5)\times 10^{-3}$ and
$y_{CP}=(11.6\pm 2.2\pm 1.8)\times 10^{-3}$, respectively. These measurements
have recently been updated by preliminary results based on the full dataset of
flavour-tagged events for both collaborations. The BaBar collaboration has
added the larger sample of untagged events to the analysis, however, with
limited gain in sensitivity due to larger systematic uncertainties for the
untagged sample which has lower purity compared to the $D^{*}$-tagged events.
The updated results are $y_{CP}=(7.2\pm 1.8\pm 1.2)\times 10^{-3}$ and
$y_{CP}=(11.1\pm 2.2\pm 1.1)\times 10^{-3}$, for BaBar [66] and Belle [67],
respectively.

It is worth noting that the central value of the BaBar result is significantly
lower compared the one from 2007. This is thought to be compatible with a
statistical fluctuation due to the added data as well as only partial overlap
in the older dataset following improvements in reconstruction and analysis.
The new BaBar result relaxes the tension that existed between measurements of
$y_{CP}$, which favoured values of about $1\%$, and other mixing measurements,
which tend towards smaller values. Such a tension would be impossible to be
explained by $C\\!P$ violation as that leads to $y_{CP}\leq y$.

Another possibility of measuring $y_{CP}$ is using the decay mode $D^{0}\\!\to
K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$. The Belle collaboration have published a
measurement in which they compare the effective lifetime around the $\phi$
resonance with that measured in sidebands of the $K^{-}K^{+}$ invariant mass
[68]. The effective $C\\!P$ eigenstate content in these regions is determined
with two different models. Their result is $y_{CP}=(1.1\pm 6.1\pm 5.2)\times
10^{-3}$.

The decay $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$ and more so the
decay $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}\pi^{-}\pi^{+}$ give excellent
access to the mixing parameters $x$ and $y$ individually. At the same time
they allow a measurement of parameters of indirect $C\\!P$ violation as
discussed in the following section. Under the assumption of no $C\\!P$
violation Belle and BaBar have measured

|  $x$ in $10^{-3}$ |  $y$ in $10^{-3}$
---|---|---
Belle [69] | $8.0\pm 2.9\pm 1.7$ | $3.3\pm 2.4\pm 1.5$
BaBar [70] | $1.6\pm 2.3\pm 1.2\pm 0.8$ |  $5.7\pm 2.0\pm 1.3\pm 0.7$,

where the last uncertainty in the BaBar measurement is a model uncertainty.
The Belle result has recently been updated including the full available
dataset based on the final reconstruction [71]. This leads to $x=(0.56\pm
0.19^{+0.03}_{-0.09}\,{}^{+0.06}_{-0.09})\times 10^{-3}$ and $y=(0.30\pm
0.15^{+0.04}_{-0.05}\,{}^{+0.03}_{-0.06})\times 10^{-3}$. Once more, the last
uncertainty is a model uncertainty. Curiously, while the recent BaBar result
on $y_{CP}$ has relaxed the tension with $y$, now there is a nearly $3\sigma$
tension among the latest Belle results for $y_{CP}$ and $y$. Additional
measurements are required to resolve this situation.

The LHCb collaboration has made its first measurement of $y_{CP}$ [72] based
on two-body $D^{0}$ decays recorded in 2010 to $y_{CP}=(5.5\pm 6.3\pm
4.1)\times 10^{-3}$. While this measurement falls short of being at the level
of precision of the B-factory measurements in this mode, LHCb has recorded
over a factor 50 more in integrated luminosity to date. The Heavy Flavor
Averaging Group (HFAG) has combined all measurements of $y_{CP}$ [73] to
$y_{CP}=(8.7\pm 1.6)\times 10^{-3}$.

By the time of this review no single experiment observation of mixing in
$D^{0}$ mesons with a significance exceeding $5\sigma$ has been possible.
However, the combination of the numerous measurements by HFAG excludes the no-
mixing hypothesis by about $10\sigma$ [73]. Under the assumption of no $C\\!P$
violation in mixing or decays, the world average of the mixing parameters is
$x=(6.5^{+1.8}_{-1.9})\times 10^{-3}$ and $y=(7.4\pm 1.2)\times 10^{-3}$.

##  3 Charm $C\\!P$ violation

###  3.1 Indirect $C\\!P$ violation

Indirect $C\\!P$ violation can be measured through the comparison of effective
lifetimes of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
decays to $C\\!P$ eigenstates. This leads to the observable

$A_{\Gamma}=\frac{\hat{\Gamma}-\hat{\bar{\Gamma}}}{\hat{\Gamma}+\hat{\bar{\Gamma}}}\approx\eta_{C\\!P}\left[\frac{1}{2}\left(A_{m}+A_{d}\right)y\cos\phi-x\sin\phi\right],$
(7)

which has contributions from both direct and indirect $C\\!P$ violation [64,
74]. Currently, there are three measurements of $A_{\Gamma}$ which are all
compatible with zero. The Belle [21], BaBar [75] and LHCb [72] collaborations
have measured $A_{\Gamma}=(0.1\pm 3.0\pm 1.5)\times 10^{-3}$,
$A_{\Gamma}=(2.6\pm 3.6\pm 0.8)\times 10^{-3}$, and $A_{\Gamma}=(-5.9\pm
5.9\pm 2.1)\times 10^{-3}$, respectively. With the LHCb result being based
only on a small fraction of the data recorded so far, significant improvements
in sensitivity may be expected in the near future. In parallel to their
updates of $y_{CP}$, BaBar [66] and Belle [67] have also released preliminary
results for $A_{\Gamma}$ based on their full datasets. They have measured
$A_{\Gamma}=(0.9\pm 2.6\pm 0.6)\times 10^{-3}$ and $A_{\Gamma}=(-0.3\pm 2.0\pm
0.8)\times 10^{-3}$, respectively. The HFAG world average [73] yields
$A_{\Gamma}=(-0.2\pm 1.6)\times 10^{-3}$ in agreement with no $C\\!P$
violation.

Using current experimental bounds values of $A_{\Gamma}$ up to
$\mathcal{O}(10^{-4})$ are expected from theory [74, 76]. It has however been
shown that enhancements up to about one order of magnitude are possible, for
example in the presence of a fourth generation of quarks [34] or in a little
Higgs model with T-parity [76]. This would bring $A_{\Gamma}$ close to the
current experimental limits.

Eventually, the interpretation of $C\\!P$ violation results requires precise
knowledge of both mixing and $C\\!P$ violation parameters. The relative
sensitivity to the $C\\!P$-violating quantities in the observable $A_{\Gamma}$
is limited by the relative uncertainty of the mixing parameters. Therefore, to
establish the nature of a potential non-zero measurement of $A_{\Gamma}$ it is
mandatory to have measured the mixing parameters with a relative precision of
$\approx 10\%$.

The analyses of the decays $D^{0}\\!\to K^{0}_{\rm\scriptstyle
S}\pi^{-}\pi^{+}$ and $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$ offer
separate access to the parameters $x$, $y$, $|q/p|$ and $\arg(q/p)$ and are
one of the most promising ways of obtaining precise mixing measurements. These
analyses require the determination of the decay-time dependence of the phase
space structure (Dalitz plot, see Ref. Dalitz:1953cp) of these decays. This
can be obtained in two ways: explicit fits of the time evolution of resonances
based on Dalitz-plot models, or based on a measurement of the strong-phase
difference across the Dalitz-plot carried out by the CLEO collaboration [78].
One measurement made by the Belle collaboration has determined these
parameters based on a Dalitz plot model [69]. Other measurements were
performed by the CLEO [79] and BaBar [70] collaborations assuming $C\\!P$
conservation and thus extracting only $x$ and $y$. With the data samples
available and being recorded at LHCb and those expected at future flavour
factories, these measurements will be very important to understand charm
mixing and $C\\!P$ violation. However, in order to avoid systematic
limitations it will be important to reduce model uncertainties or to improve
model-independent strong-phase difference measurements which are possible at
BESIII.

###  3.2 Direct $C\\!P$ violation

Direct $C\\!P$ violation is searched for in decay-time integrated
measurements. However, for neutral mesons, the decay-time distribution of the
data has to be taken into account to estimate the contribution from indirect
$C\\!P$ violation. Currently, the most striking measurements have been made in
decays of $D^{0}$ mesons into two charged pions or kaons. While early
measurements of BaBar [80] and Belle [81] have not shown significant
deviations from zero, the LHCb collaboration has reported first evidence for
$C\\!P$ violation in the charm sector [27]. They have measured

$\Delta
A_{C\\!P}\equiv{}A_{C\\!P}(K^{-}K^{+})-A_{C\\!P}(\pi^{-}\pi^{+})=(-8.1\pm
2.1\pm 1.1)\times 10^{-3}.$

Meanwhile, the CDF collaboration has released a preliminary measurement of
$\Delta A_{C\\!P}=(-6.2\pm 2.1\pm 1.0)\times 10^{-3}$ which shows a hint of a
deviation from zero [82], in support of the LHCb result. Just before
submission of this review the Belle collaboration has released an update of
their measurement of $\Delta A_{C\\!P}$ based on their full reprocessed
dataset showing a roughly $2\sigma$ deviation from zero [83].

The quantity $\Delta A_{C\\!P}$ exploits first-order cancellation of
systematic uncertainties in the difference of asymmetries. At higher order,
terms that are products of individual asymmetries contribute, e.g. the product
of a production and a $C\\!P$ asymmetry, which no longer cancel. In a
kinematic region with large production asymmetries such a contribution leads
to relative corrections of $\Delta A_{C\\!P}$ of the order of the local
production asymmetry. Hence, these higher order terms need to be taken into
account for a precision measurement of the size of observed $C\\!P$ violation.

The observable $\Delta A_{C\\!P}$ gives access to the difference in direct
$C\\!P$ violation of the two decay modes through

$\Delta A_{C\\!P}=\Delta{}a_{C\\!P}^{\rm
dir}\left(1+y_{CP}\frac{\overline{\langle{}t\rangle}}{\tau}\right)+\overline{A}_{\Gamma}\frac{\Delta\langle{}t\rangle}{\tau},$
(8)

where $\tau$ is the nominal $D^{0}$ lifetime,
$\overline{X}\equiv(X(K^{-}K^{+})+X(\pi^{-}\pi^{+}))/2$, and
$\Delta{}X\equiv{}X(K^{-}K^{+})-X(\pi^{-}\pi^{+})$ for $X=(a_{C\\!P}^{\rm
dir},\langle{}t\rangle)$ [64]. Equation (8) assumes the $C\\!P$-violating
phase $\phi$ to be universal. For a small non-zero difference in this phase
between the two final states, $\Delta\phi_{f}$, an additional term of the form
$x\Delta\phi_{f}\overline{\langle{}t\rangle}/\tau$ arises as pointed out in
Ref. Kagan:2009gb. Given a typical variation of
$\overline{\langle{}t\rangle}/\tau$ between $1$ and $2.5$ for the different
experiments the contribution of $\Delta\phi_{f}$ is suppressed by
$x\overline{\langle{}t\rangle}/\tau\approx 10^{-2}$.

While it was commonly stated in literature that $C\\!P$ violation effects in
these channels were not expected to exceed $10^{-3}$, this statement has been
revisited in numerous recent publications. To date, no clear understanding of
whether [84, 85, 86, 87] or not [76, 88, 89, 90] $C\\!P$ violation of this
level can be accommodated within the standard model has emerged. In parallel
to attempts to better the standard model calculations, many estimates of
potential effects of physics beyond the standard model have been made [76, 85,
88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106,
107, 108, 109, 110, 111].

Within the standard model the central value can only be explained by
significantly enhanced penguin amplitudes. This enhancement is conceivable
when estimating flavour $SU(3)$ or U-spin breaking effects from fits to data
of $D$ decays into two pseudo scalars [85, 86, 95, 102, 112, 113]. However,
attempts of estimating the long distance penguin contractions directly have
failed to yield conclusive results to explain the enhancement.

Lattice QCD has the potential of assessing the penguin enhancement directly.
However, several challenges arise which make these calculations impossible at
the moment [114, 115, 116, 117, 118, 119, 120]. Following promising results on
$K\to\pi\pi$ decays, additional challenges arise in the charm sector as
$\pi\pi$ and $KK$ states mix with $\eta\eta$, $4\pi$, $6\pi$ and other states.
Possible methods have been proposed and first results may be expected within
the next decade.

General considerations on the possibility of interpreting the
$\Delta{}A_{C\\!P}$ in models beyond the standard model have lead to the
conclusion that an enhanced chromomagnetic dipole operator is required. These
operators can be accommodated in minimal supersymmetric models with non-zero
left-right up-type squark mixing contributions or, similarly, in warped extra
dimensional models [91, 92, 97, 121, 122, 123, 124]. Tests of these
interpretations beyond the standard model are in the focus of ongoing
searches. One promising group of channels are radiative charm decays where the
link between the chromomagnetic and the electromagnetic dipole operator leads
to predictions of enhanced $C\\!P$ asymmetries of several percent [125].

Another, complementary, test is to search for contributions beyond the
standard model in $\Delta I=3/2$ amplitudes. This class of amplitudes leads to
several isospin relations which can be tested in a range of decay modes, e.g.
$D\to\pi\pi$, $D\to\rho\pi$, $D\to K\bar{K}$ [85, 126]. Several of these
measurements, such as the Dalitz plot analysis of the decay
$D^{0}\to\pi^{+}\pi^{-}\pi^{0}$, have been performed by BaBar and Belle and
will be possible at LHCb as well as future $\mathrm{e}^{+}\mathrm{e}^{-}$
machines.

Beyond charm physics, the chromomagnetic dipole operators would affect the
neutron and nuclear EDMs, which are expected to be close to the current
experimental bound [97]. Similarly, rare FCNC top decays are expected to be
enhanced, if kinematically allowed. Furthermore, quark compositness can be
related to the $\Delta{}A_{C\\!P}$ measurement and tested in dijet searches.
Current results favour the new physics contribution to be located in the
$D^{0}\to K^{-}K^{+}$ decay as the strange quark compositness scale is less
well constrained [127].

Another group of channels suitable for $C\\!P$ violation searches is that of
decays of $D^{+}$ and $D^{+}_{\mathrm{s}}$ mesons into three charged hadrons,
namely pions or kaons. Here, $C\\!P$ violation can occur in two-body
resonances contributing to these decay amplitudes. Asymmetries in the Dalitz-
plot substructure can be measured using an amplitude model or using model-
independent statistical analyses [128, 129, 130]. The latter allow $C\\!P$
asymmetries to be discovered while eventually a model-dependent analysis is
required to identify its source. The two types of model-independent analyses
differ in being either binned [128, 129] or unbinned [130] in the Dalitz
plane.

The binned approach computes a local per-bin asymmetry and judges the presence
of $C\\!P$ violation by the compatibility of the distribution of local
asymmetries across the Dalitz plane with a normal distribution. This method
obviously relies on the optimal choice of bins. Bins ranging across resonances
can lead to the cancellation of real asymmetries within a bin. Too fine
binning can reach the limit of statistical sensitivity, whereas too coarse
binning can wash out $C\\!P$ violation effects by combining regions of
opposite asymmetry. A model-inspired choice of binning is clearly useful and
this does not create a model-dependence in the way that fitting resonances
directly does. This method does not yield an easy-to-interpret quantitative
result. This issue has been discussed in a recent update of the procedure
[131].

The unbinned asymmetry search calculates a test statistic that allows the
assignment of a $p$-value when comparing to the distribution of the statistic
for many random permutations of the events among the particle and anti-
particle datasets. Moreover, being unbinned, there is no need for a model-
inspired choice of binning. The drawback of this method is its requirement on
computing power. The calculation of the test statistic scales as the square of
the number of events.

Beyond three-body final states similar analyses can be performed in decays
into four hadrons, e.g. decays of $D^{0}$ into four charged hadrons. This too
gives access to interesting resonance structures that may exhibit significant
$C\\!P$ asymmetries. However, rather than having a two-dimensional Dalitz
plane, the phase space for four-body decays is five-dimensional (see e.g. Ref.
Rademacker:2006zx). This poses not only a challenge on the visualisation but
also on any binned approach due to rapidly decreasing sample sizes per bin.
Also, the phase-space substructure can no longer be described only by
interfering amplitudes of pseudo two-body decays as also three-body decays may
contribute. The LHCb collaboration has released a first model-independent
search for $C\\!P$ violation in the decay
$D^{0}\\!\to\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ without finding any hint of $C\\!P$
non-conservation [133].

Neither searches for phase-space integrated asymmetries [134, 135, 136, 137,
138, 139, 140], nor searches for local asymmetries in the Dalitz plot [136,
138, 141, 142, 143] have shown any evidence for $C\\!P$ violation. The largest
signal is the recently reported measurement of $C\\!P$ violation in
$D^{+}\\!\to\phi\pi^{+}$ of $A_{C\\!P}^{\phi\pi^{+}}=(5.1\pm 2.8\pm 0.5)\times
10^{-3}$ by the Belle collaboration [143], which exploits cancellation of
uncertainties through a comparison of asymmetries in the decays of $D^{+}$ and
$D^{+}_{\mathrm{s}}$ mesons into the final state $\phi\pi^{+}$.

Decays of $D^{+}$ and $D^{+}_{\mathrm{s}}$ are into a $K^{0}_{\rm\scriptstyle
S}$ and either a $K^{+}$ or a $\pi^{+}$ are closely related to their $D^{0}$
counterparts. Measurements of time-integrated asymmetries in these decays are
expected to exhibit a contribution from $C\\!P$ violation in the kaon system.
As pointed out recently [144] this contribution depends on the decay-time
acceptance of the $K^{0}_{\rm\scriptstyle S}$. This can lead to different
expected values for different experiments. For Belle [145], the expected level
of asymmetry due to $C\\!P$ violation in the kaon system is $-3.5\times
10^{-3}$. For LHCb on the other hand, there is no significant asymmetry
induced by kaon $C\\!P$ violation [146] as the LHCb acceptance, for
$K^{0}_{\rm\scriptstyle S}$ reconstructed in the vertex detector, corresponds
to about $10\%$ of a $K^{0}_{\rm\scriptstyle S}$ lifetime. $C\\!P$ violation
searches in the decays $D^{+}\\!\to K^{0}_{\rm\scriptstyle S}\pi^{+}$ [65,
139, 147] and $D^{+}_{\mathrm{s}}\\!\to K^{0}_{\rm\scriptstyle S}\pi^{+}$
[139, 147] show significant asymmetries. However, these asymmetries are fully
accommodated in the expected $C\\!P$ violation of the kaon system. These
measurements do not show any hint for an asymmetry in $D$ decay amplitudes.
Future, more precise, measurements will reveal whether or not these remain in
agreement with the expected contribution from the kaon system.

In the light of the recent measurements it is evident that there are four
directions to pursue: more precise measurements of $\Delta A_{C\\!P}$ and the
individual asymmetries are required to establish the effect; further searches
for time-integrated $C\\!P$ violation need to be carried out in a large range
of modes that allow to identify the source of the $C\\!P$ asymmetry; searches
for time-dependent $C\\!P$ asymmetries, particularly via more precise
measurements of $A_{\Gamma}$; and finally a more precise determination of the
mixing parameters is required.

###  3.3 Interplay of mixing, direct and indirect $C\\!P$ violation

Following Eqs. (7) and (8) it is obvious that both $A_{\Gamma}$ and $\Delta
A_{C\\!P}$ share the underlying $C\\!P$-violating parameters. Allowing for a
non-universal $C\\!P$-violating phase $\phi$ one can write

$\displaystyle A_{\Gamma}(f)$ $\displaystyle=-a_{C\\!P}^{\rm
ind}-a_{C\\!P}^{\rm dir}(f)y_{CP}-x\phi_{f},$ (9) $\displaystyle A_{C\\!P}(f)$
$\displaystyle=a_{C\\!P}^{\rm
dir}(f)-A_{\Gamma}(f)\frac{\langle{}t\rangle}{\tau},$ (10)
$\displaystyle\Delta A_{C\\!P}$ $\displaystyle=\Delta{}a_{C\\!P}^{\rm
dir}-\Delta
A_{\Gamma}\frac{\overline{\langle{}t\rangle}}{\tau}-\overline{A}_{\Gamma}\frac{\Delta\langle{}t\rangle}{\tau},$
(11)

where again $\overline{X}\equiv(X(K^{-}K^{+})+X(\pi^{-}\pi^{+}))/2$ and
$\Delta{}X\equiv{}X(K^{-}K^{+})-X(\pi^{-}\pi^{+})$ for $X=(a_{C\\!P}^{\rm
dir},\langle{}t\rangle)$. It is expected that, at least within the standard
model, one has $a_{C\\!P}^{\rm dir}(K^{-}K^{+})=-a_{C\\!P}^{\rm
dir}(\pi^{-}\pi^{+})$ and thus $\overline{A}_{\Gamma}=-a_{C\\!P}^{\rm ind}$.
This set of equations shows that it is essential to measure both time-
dependent ($A_{\Gamma}$) and time-integrated asymmetries ($A_{C\\!P}$)
separately in the decay modes $D^{0}\\!\to K^{-}K^{+}$ and
$D^{0}\\!\to\pi^{-}\pi^{+}$ in order to distinguish the various possible
sources of $C\\!P$ violation. Currently, the experimental precision on
$A_{\Gamma}$ is such that there is no sensitivity to differences in the
contributions from direct $C\\!P$ violation to measurements using
$K^{-}\\!K^{+}$ or $\pi^{-}\\!\pi^{+}$ final states. Hence, the approximation
$A_{\Gamma}\equiv\overline{A}_{\Gamma}\approx A_{\Gamma}(K^{-}K^{+})\approx
A_{\Gamma}(\pi^{-}\pi^{+})$ can be used to obtain

$\displaystyle A_{\Gamma}$ $\displaystyle=-a_{C\\!P}^{\rm ind}$ (12)
$\displaystyle\Delta A_{C\\!P}$ $\displaystyle=\Delta{}a_{C\\!P}^{\rm
dir}\left(1+y_{CP}\frac{\overline{\langle{}t\rangle}}{\tau}\right)-a_{C\\!P}^{\rm
ind}\frac{\Delta\langle{}t\rangle}{\tau}.$ (13)

These equations have been used by HFAG to prepare a fit of the direct and
indirect $C\\!P$ violation contributions [73] as shown in Fig. 2.

Figure 2: Fit of $\Delta{}a_{C\\!P}^{\rm dir}$ and $a_{C\\!P}^{\rm ind}$.
Reproduced from Ref. Amhis:2012hf.

This fit yields a confidence level of about $2\times 10^{-5}$ for the no
$C\\!P$ violation hypothesis with best fit values of $\Delta{}a_{C\\!P}^{\rm
dir}=(-6.78\pm 1.47)\times 10^{-3}$ and $a_{C\\!P}^{\rm ind}=(0.27\pm
1.63)\times 10^{-3}$. This also shows that the most likely source of the large
measured values for $\Delta A_{C\\!P}$ is direct $C\\!P$ violation in one or
both of the relevant decay modes. The fit formalism will have to be refined
using the equations discussed above in the future as more precise measurements
as well as individual asymmetries will become available.

The ultimate goal of mixing and $C\\!P$ violation measurements in the charm
sector is to reach precisions at or below the standard model predictions. In
some cases this requires measurements in several decay modes in order to
distinguish enhanced contributions of higher order standard model diagrams
from effects caused by new particles.

Indirect $C\\!P$ violation measurements at LHCb are mostly constrained by the
observable $A_{\Gamma}$ (see Eq. (9)). The $C\\!P$ violating parameters in
this observable are multiplied by the mixing parameters $x$ and $y$,
respectively. Hence, the relative precision on the $C\\!P$ violating
parameters is limited by the relative precision of the mixing parameters.
Therefore, aiming at a relative precision below $10\%$ for the $C\\!P$
violation quantities and taking into account the current mixing parameter
world averages, the target precision for the mixing parameters is $2-3\times
10^{-4}$. With standard model indirect $C\\!P$ violation expected of the order
of $10^{-4}$, the direct $C\\!P$ violation parameter contributing to
$A_{\Gamma}$ has to be measured to an absolute precision of $10^{-3}$ in order
to distinguish the two types of $C\\!P$ violation in $A_{\Gamma}$.

Direct $C\\!P$ violation is not expected to be as large as the current world
average of $\Delta A_{C\\!P}$ in other decay modes. Therefore a precision of
$5\times 10^{-4}$ or better for asymmetry differences as well as individual
asymmetries is needed for measurements of other singly-Cabibbo-suppressed
charm decays. While measurements of time-integrated raw asymmetries at this
level should be well within reach, the challenge lies in the control of
production and detection asymmetries in order to extract the physics
asymmetries of individual decay modes.

For multibody final states the aim is clearly the understanding of $C\\!P$
asymmetries in the interfering resonances rather than global asymmetries. Of
highest interest are those resonances that are closely related to the two-body
modes used in $\Delta A_{C\\!P}$, for example the vector-pseudoscalar
resonances $K^{*}K$ and $\rho\pi$. The measurement of further suppressed
resonances is of interest as well since those have no contributions from
gluonic penguin diagrams, thus allowing to constrain the source of $C\\!P$
violating effects.

##  4 Rare charm decays

Rare decays provide a wide range of interesting measurements. The list of
decay modes includes flavour-changing neutral currents, radiative, lepton-
flavour violation, lepton-number violation, as well as baryon-number
violation. While a full discussion of rare charm decays would be beyond the
scope of this review a few remarks shall be made here.

There is a direct link between mixing and flavour changing neutral current
decays in several extensions of the standard model [55, 148]. These relate
$\Delta C=1$ annihilation amplitudes to $\Delta C=2$ mixing amplitudes where
the annihilation product creates a new $C\\!P$-conjugated
$\mathrm{c}\overline{\mathrm{u}}$ pair. At tree level one example is a heavy
$Z$-like boson with non-zero flavour-changing couplings.

The current central values of the $D^{0}$-mixing parameters translate into
model-dependent limits for rare decays based on common amplitudes. These rare
decay limits lie significantly below the current experimental limits. As
$D^{0}$ mixing is well established any upper limit from mixing will not change
significantly in the future. However, due to the direct correlation of mixing
and rare decays, any observation above the model-dependent rare decay limits
will rule out the corresponding model. The best limit on flavour-changing
neutral current decays is the recent LHCb limit on the decay
$D^{0}\\!\to\mu^{-}\mu^{+}$ of $1.1\times{}10^{-8}$ at $90\%$ confidence level
[149].

Among lepton-flavour violating decays the most stringent constraint is a Belle
search for $D^{0}\\!\to\mu^{\mp}\mathrm{e}^{\pm}$ achieving a limit of
$2.6\times{}10^{-7}$ at $90\%$ confidence level [150]. Searches for lepton-
flavour violating muon or kaon decays already provide more constraining
limits, however, in scenarious of non-universal couplings charm decays, giving
access to the up-quark sector, are of great interest.

The best limit on lepton-number violating charm decays has been placed by
BaBar on the decay $D^{+}\\!\to K^{-}\mathrm{e}^{+}\mathrm{e}^{+}$ with a
limit of $9.0\times{}10^{-7}$ at $90\%$ confidence level [151]. Only the CLEO
collaboration has carried out searches for baryon-number violating charm
decays. Their best limit on the decay $D^{0}\\!\to p\mathrm{e}^{-}$ is
$10^{-5}$ at $90\%$ confidence level [152]. For a more complete overview of
rare charm decays please refer to Ref. [73].

##  5 Conclusion

The first evidence for $C\\!P$ violation in the charm sector has opened the
door wide for a broad range of measurements. 40 years after the observation of
the first hint of charm particles in cosmic rays and exactly 37 years after
the dicovery of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mesons, charm measurements may have shown first hints of effects beyond the
standard model at the LHC. The unambiguous observation of $C\\!P$ violation
will be the near term goal. The ultimate task is the interpretation of the
observed effects for which theoretical and experimental communities have to
collaborate closely to overcome the hurdles related to the charm quark mass
and the large cancellations in this system.

With the $B$ factories, CLEO-c and CDF analysing their final datasets, most
new results are expected to come from LHCb and BESIII. These are expected to
explore very interesting territory for charm $C\\!P$ violation. The longer
term future will be shaped by the LHCb upgrade as well as future
$\mathrm{e}^{+}\mathrm{e}^{-}$ collider experiments running both at the beauty
and charm thresholds. Charm’s third time has just begun to yield first fruits
which may well develop into a real charm.

## Acknowledgments

The author would like to thank Yuval Grossman, Claus Grupen, Alex Kagan,
Alexander Lenz, and Alexey Petrov as well as the members of the LHCb
collaboration for very insightful discussions. Special thanks to Alexander
Lenz and Johannes Albrecht who dared to read the article before submission.
The author further acknowledges the support of a Marie Curie Action:
“Cofunding of the CERN Fellowship Programme (COFUND-CERN)” of the European
Community’s Seventh Framework Programme under contract number (PCOFUND-
GA-2008-229600).

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