∎

11institutetext: M. Gersabeck 22institutetext: CERN, 1211 Geneva, Switzerland
22email<EMAIL_ADDRESS>33institutetext: V.V.
Gligorov44institutetext: CERN, 1211 Geneva, Switzerland
44email<EMAIL_ADDRESS>55institutetext: N. Serra66institutetext:
University of Zuerich, 8006 Zuerich, Switzerland
66email<EMAIL_ADDRESS>
# Experimental constraints from flavour changing processes and physics beyond
the Standard Model††thanks: MG and VVG are supported by 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).

M. Gersabeck  V.V. Gligorov  N. Serra

(Received: date / Accepted: date)

###### Abstract

Flavour physics has a long tradition of paving the way for direct discoveries
of new particles and interactions. Results over the last decade have placed
stringent bounds on the parameter space of physics beyond the Standard Model.
Early results from the LHC, and its dedicated flavour factory LHCb, have
further tightened these constraints and reiterate the ongoing relevance of
flavour studies. The experimental status of flavour observables in the charm
and beauty sectors is reviewed in measurements of $C\\!P$ violation, neutral
meson mixing, and measurements of rare decays.

###### Keywords:

Flavour physics CP violation Meson mixing Rare decays Charm mesons Beauty
mesons

###### pacs:

13.20.Fc13.20.He13.25.Ft 13.25.Hw13.35.Bv13.35.Dx14.40.Lb14.40.Nd

##  1 Introduction

Flavour physics has given key contributions to the understanding of
fundamental particles. The kaon system is an excellent example how the
interplay of meson anti-meson mixing Lande:1956pf ; Jackson:1957zzb ;
Niebergall:1974wh , and the search for rare decays BottBodenhausen1967194 ;
Foeth:1969hi  led to the prediction of the charm quark and indeed charm mesons
GellMann1964214 ; Tarjanne:1963zz ; Hara:1963gw ; Bjorken:1964gz ;
Glashow:1970gm . Furthermore, the observation of $C\\!P$ violation in neutral
kaons Christenson:1964fg  led to the prediction of a third generation of
quarks Kobayashi:1973fv . At the LHC, precision measurements of flavour
physics are sensitive to new particles contributing to quantum loops up to
scales of about $200\mathrm{\,Te\kern-1.00006ptV}$ Buras:2009if  which,
according to the Heisenberg uncertainty principle Heisenberg:1927zz ,
correspond to distance scales of the order of $10^{-21}~{}{\rm m}$. This
exceeds the reach for direct production of particles by roughly two orders of
magnitude.

This review covers flavour changing processes of charm and beauty mesons;
recent results on lepton flavour violating decays are also briefly discussed.
These provide complementary access to effects from Physics Beyond the Standard
Model (PBSM). This complementarity will eventually help to identify the nature
of signs of new dynamics, should they be generated by a common source.
Sections 2 to 4 cover the status of mixing and $C\\!P$ violation measurements
while section 5 reviews measurements of rare decays.

##  2 $C\\!P$ violation in heavy flavour mesons

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
$C\\!P|M^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$.

Following the notation of Kagan:2009gb , the time dependent decay rates of
$M^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptM}{}^{0}$ decays to the
final state $f$ can be expressed as

$\displaystyle\Gamma$ $\displaystyle(M^{0}(t)\rightarrow
f)=\frac{1}{2}e^{-\tau}\left|A_{f}\right|^{2}$
$\displaystyle\times\Big{\\{}\left(1+|\lambda_{f}|^{2}\right)\cosh(y\tau)+\left(1-|\lambda_{f}|^{2}\right)\cos(x\tau)$
$\displaystyle\quad+2\Re(\lambda_{f})\sinh(y\tau)-2\Im(\lambda_{f})\sin(x\tau)\Big{\\}},$
$\displaystyle\Gamma$ $\displaystyle(\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}(t)\rightarrow
f)=\frac{1}{2}e^{-\tau}\left|\bar{A}_{f}\right|^{2}$
$\displaystyle\times\Big{\\{}\left(1+|\lambda^{-1}_{f}|^{2}\right)\cosh(y\tau)+\left(1-|\lambda^{-1}_{f}|^{2}\right)\cos(x\tau)$
$\displaystyle\quad+2\Re(\lambda^{-1}_{f})\sinh(y\tau)-2\Im(\lambda^{-1}_{f})\sin(x\tau)\Big{\\}},$
(1)

where $\tau\equiv\Gamma t$, $\kern-1.99997pt\stackrel{{\scriptstyle\kern
1.39998pt\textsf{(---)}}}{{A}}_{\kern-2.10002ptf}\kern-3.00003pt$ are the
decay amplitudes and $\lambda_{f}$ is given by

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

In general, $C\\!P$ symmetry is violated if $\lambda_{f}$, as defined in
Equation 2, deviates from $1$. This 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 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 Aaij:2011in .

In the charm system one defines the differences
$\Delta{}m_{D}\equiv{}m_{2}-m_{1}$ and
$\Delta\Gamma_{D}\equiv\Gamma_{2}-\Gamma_{1}$. Furthermore, the mixing
parameters are defined as $x\equiv\Delta{}m/\Gamma$ and
$y\equiv\Delta\Gamma/(2\Gamma)$. Analogously, in the beauty system one defines
the differences $\Delta{}m_{d,s}\equiv{}m_{2}-m_{1}$ and
$\Delta\Gamma_{d,s}\equiv\Gamma_{1}-\Gamma_{2}$, where the subscripts denote
the $B^{0}_{d}$ and $B^{0}_{s}$ systems, respectively.

Within the Standard Model (SM), quark mixing is described by the CKM matrix

$\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\ V_{cd}&V_{cs}&V_{cb}\\\
V_{td}&V_{ts}&V_{tb}\end{array}\right)=\\\
\left(\begin{array}[]{ccc}1-\frac{1}{2}\lambda^{2}&\lambda&A\lambda^{3}\left(\rho-i\eta\right)\\\
-\lambda&1-\frac{1}{2}\lambda^{2}&A\lambda^{2}\\\
A\lambda^{3}\left(1-\rho+i\eta\right)&-A\lambda^{2}&1\end{array}\right),$ (3)

given on the right in the Wolfenstein parametrization where $\lambda\approx
0.22$ is the sine of the Cabibbo angle. $C\\!P$ violation then arises solely
from the imaginary term in this matrix. Since the matrix is unitary, it can be
represented by six triangles in the complex plane, defined by unitarity
conditions such as

$V^{*}_{ub}V_{ud}+V^{*}_{cb}V_{cd}+V^{*}_{tb}V_{td}=0,$ (4)

which is known as the “Unitarity Triangle”. This particular unitarity
condition is chosen because the three terms, corresponding to the sides of the
triangle, are of approximately equal size. The fact that the SM predicts
$O(10\%)$ $C\\!P$ violating effects in many $B$ decays, while the predictions
for $D$ decays are generally at least two orders of magnitude smaller, has led
to differing experimental approaches. In the case of $B$ decays, the focus has
been on precise measurements of mixing and $C\\!P$ violation in order to
overconstrain the sides and angles of the Unitarity Triangle, in particular
its apex, as illustrated in Fig. 1.

Figure 1: The current constraints on the Unitarity Triangle. These meet at the
overconstrained apex, and the shaded ellipse indicates the allowed region for
the apex when all measurements are taken together. Reproduced from
Charles:2004jd .

In $D$ decays the focus has been on searches for $C\\!P$ violation and a
precise understanding of the mixing parameters.

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

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
Aubert:2007wf ; Staric:2007dt . 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 Glashow:1970gm  and CKM suppression Cabibbo:1963yz ;
Kobayashi:1973fv , which results in a strongly suppressed mixing process
Bobrowski:2010xg .

Since experimental evidence has shown that quantum-loop effects are accessible
in the charm sector, measurements of $D$ mesons provide access to effects from
particles beyond the SM, complementary to measurements in the $B$ sector. It
was discussed whether the measured size of the mixing parameters could be
interpreted as a hint for new physics Hou:2006mx ; Ciuchini:2007cw ;
Nir:2007ac ; Blanke:2007ee ; He:2007iu ; Chen:2007yn ; Golowich:2007ka . New
physics effects were also searched for in numerous $C\\!P$-violation
measurements, which are covered in the remainder of this section, and searches
for rare decays as discussed in section 5.

###  3.1 Charm mixing

Mixing of $D^{0}$ mesons can be measured in several different modes. All
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 $D^{*+}\\!\rightarrow D^{0}\pi^{+}$, where the charge of the pion
determines the flavour of the $D^{0}$. Charge conjugate decays are implicitly
included here and henceforth. 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 partly compensating for lower production rates. Another option
available, particularly at $e^{+}e^{-}$ 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 $D^{0}\\!\rightarrow
K^{+}\mu^{-}\overline{\nu}_{\mu}$ 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, $D^{0}\\!\rightarrow K^{-}\mu^{+}\nu_{\mu}$, 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 to $R_{\rm m}=(1.3\pm 2.2\pm 2.0)\times
10^{-4}$ Bitenc:2008bk , where the first uncertainty is of statistical and the
second is of systematic nature. This notation is applied to all results where
two uncertainties are quoted.

Related to the semileptonic decay is the suppressed decay $D^{0}\\!\rightarrow
K^{+}\pi^{-}$, called the 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 $D^{0}\\!\rightarrow
K^{-}\pi^{+}$. Following from equation 2 the time-dependent decay rate of the
WS decay is, in the limit of $C\\!P$ conservation, proportional to
Bergmann:2000id

$\frac{\Gamma(D^{0}(t)\rightarrow K^{+}\pi^{-})}{e^{-\Gamma
t}}\propto\left(R_{\rm D}+\sqrt{R_{\rm D}}y^{\prime}\Gamma{}t+R^{2}_{\rm
m}(\Gamma{}t)^{2}\right),$ (5)

where the mixing parameters are rotated by the strong phase between the DCS
and the CF amplitude, leading to the observable
$y^{\prime}=y\cos\delta_{K\pi}-x\sin\delta_{K\pi}$. 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 $x^{\prime 2}=(-0.22\pm 0.30\pm
0.20)\times 10^{-3}$ and $y^{\prime}=(9.7\pm 4.4\pm 3.1)\times 10^{-3}$
Aubert:2007wf , and $x^{\prime 2}=(-0.12\pm 0.35)\times 10^{-3}$ and
$y^{\prime}=(8.5\pm 7.6)\times 10^{-3}$ Aaltonen:2007uc , respectively.

Similarly, the CF and DCS amplitudes can also lead to higher mass states of
the same quark content. The decay $D^{0}\\!\rightarrow K^{-}\pi^{+}\pi^{0}$ 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
with central values of $x^{\prime\prime}=(26.1^{+5.7}_{-6.8}\pm 3.9)\times
10^{-3}$ and $y^{\prime\prime}=(-0.6^{+5.5}_{-6.4}\pm 3.4)\times 10^{-3}$
Aubert:2008zh , where the rotation between the observables and the system of
mixing parameters is given by a strong phase as
$x^{\prime\prime}=x\cos\delta_{K^{-}\pi^{+}\pi^{0}}+y\sin\delta_{K^{-}\pi^{+}\pi^{0}}$
and
$y^{\prime\prime}=y\cos\delta_{K^{-}\pi^{+}\pi^{0}}-x\sin\delta_{K^{-}\pi^{+}\pi^{0}}$.

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. These
are available from CLEO PhysRevD.73.034024 ; PhysRevD.77.019901 ;
Rosner:2008fq ; Asner:2008ft  and can be further improved by BESIII.

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 the Heavy Flavor
Averaging Group (HFAG) excludes the no-mixing hypothesis by about $10\sigma$
Asner:2010qj . Under the assumption of no $C\\!P$ violation the world average
of the mixing parameters is $x=(6.5^{+1.8}_{-1.9})\times 10^{-3}$ and
$y=(7.3\pm 1.2)\times 10^{-3}$.

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

Indirect $C\\!P$ violation is often measured in conjunction with mixing
parameters. One example is the measurement 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, $\hat{\Gamma}$
($\hat{\bar{\Gamma}}$). The comparison of these lifetimes to that of a
Cabibbo-favoured flavour eigenstate ($\Gamma$) leads to the observable

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

where $A_{m}$ is the $C\\!P$ violation in mixing defined alongside the direct
$C\\!P$ violation $A_{d}$ by $|\lambda_{f}^{\pm 1}|^{2}\approx(1\pm
A_{m})(1\pm A_{d})$ Gersabeck:2011xj . In the limit of $C\\!P$ conservation
$y_{C\\!P}$ equals the mixing parameter $y$.

Comparing the $C\\!P$ eigenstates $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ to the
Cabibbo-favoured mode $K^{-}\pi^{+}$, the Belle and BaBar collaborations have
measured $y_{C\\!P}=(13.1\pm 3.2\pm 2.5)\times 10^{-3}$ Staric:2007dt  and
$y_{C\\!P}=(11.6\pm 2.2\pm 1.8)\times 10^{-3}$ PhysRevD.80.071103 ,
respectively. The Belle collaboration has also published a measurement using
only the decay $D^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{-}K^{+}$
in which they compare the effective lifetime around the $\phi$ resonance with
that measured in sidebands of the $K^{-}K^{+}$ invariant mass. The effective
$C\\!P$ eigenstate content in these regions is determined with two different
models. Their result is $y_{C\\!P}=(1.1\pm 6.1\pm 5.2)\times 10^{-3}$
Zupanc:2009sy . Provided measurements of sufficient precision, the comparison
of $y_{C\\!P}$ with the mixing parameter $y$ is a test of $C\\!P$ violation.
However, while one would expect $y_{C\\!P}<y$ in the presence of $C\\!P$
violation, the experimental results currently favour $y_{C\\!P}>y$, i.e. no
sign of $C\\!P$ violation is observed.

A second, more sensitive, way of measuring indirect $C\\!P$ violation is
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
Kagan:2009gb ; Gersabeck:2011xj . Currently there are three measurements of
$A_{\Gamma}$, which are all compatible with zero. The Belle, BaBar, and LHCb
collaborations have measured $A_{\Gamma}=(0.1\pm 3.0\pm 1.5)\times 10^{-3}$
Staric:2007dt , $A_{\Gamma}=(2.6\pm 3.6\pm 0.8)\times 10^{-3}$
PhysRevD.78.011105 , and $A_{\Gamma}=(-5.9\pm 5.9\pm 2.1)\times 10^{-3}$
Aaij:2011ad , 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. Using current experimental bounds values
of $A_{\Gamma}$ up to $\mathcal{O}(10^{-4})$ are expected Kagan:2009gb ;
Bigi:2011re . 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 Bobrowski:2010xg  or in a Little Higgs Model with
T-Parity Bigi:2011re . 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 analysis of the
decays $D^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{-}\pi^{+}$ and
$D^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{-}K^{+}$ offers
separate access to the parameters $x$, $y$, $|q/p|$ and $\arg(q/p)$. This
require the decay-time dependence of the phase space structure of these
decays, which is possible in two ways: using Dalitz plot models or based on a
measurement of the strong phase difference across the Dalitz plot by the CLEO
collaboration Libby:2010eq . One measurement made by the Belle collaboration
has determined these parameters based on a Dalitz plot model Abe:2007rd .
Other measurements were performed by the CLEO Asner:2005sz  and BaBar
delAmoSanchez:2010xz  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 is possible at BESIII.

Direct $C\\!P$ violation is searched for in decay-time integrated
measurements. However, 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
Aubert:2007if  and Belle Staric:2008rx  had not shown significant deviations
from zero, LHCb recently reported first evidence for $C\\!P$ violation in the
charm sector Aaij:2011in

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

Meanwhile, CDF has released a preliminary measurement of $\Delta
A_{C\\!P}=(-6.2\pm 2.1\pm 1.0)\times 10^{-3}$ CDF10784  which shows a hint of
a deviation from zero, in support of the LHCb result. The observable $\Delta
A_{C\\!P}$ exploits the cancellation of systematic uncertainties in the
difference of asymmetries. It gives access to the difference in direct $C\\!P$
violation of the two decay modes through

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

where $\tau$ is the nominal $D^{0}$ lifetime,
$\overline{X}\equiv(X(K^{-}K^{+})+X(\pi^{-}\pi^{+}))/2$, and
$\Delta{}X\equiv{}X(K^{-}K^{+})-X(\pi^{-}\pi^{+})$ Gersabeck:2011xj . With the
current precision on $A_{\Gamma}$ the influence of direct $C\\!P$ violation on
$A_{\Gamma}$ can be neglected as it is known to be $\leq 10^{-4}$ and hence
$A_{\Gamma}=-a_{C\\!P}^{\rm ind}$ is assumed. Thus, the world average leads to
central values of $\Delta{}a_{C\\!P}^{\rm dir}=(-6.6\pm 1.5)\times 10^{-3}$
and $a_{C\\!P}^{\rm ind}=(-0.3\pm 2.3)\times 10^{-3}$ which has a confidence
level of being in agreement with the no $C\\!P$ violation hypothesis of
$6.1\times 10^{-5}$ Asner:2010qj  (see Fig. 2).

Figure 2: HFAG combination of measurements of $\Delta A_{C\\!P}$ and
$A_{\Gamma}$. Shown are the experimental results as bands indicating their
$\pm 1\sigma$ uncertainties, the best fit value with one-dimensional
uncertainties as a cross, and the $1\sigma$, $2\sigma$, and $3\sigma$
ellipses. The dot marks the point of no $C\\!P$ violation. Reproduced from
Asner:2010qj .

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 Brod:2011re ; Feldmann:2012js ; Bhattacharya:2012ah ; Franco:2012ck
or not Bigi:2011re ; Rozanov:2011gj ; Cheng:2012wr ; Li:2012cf  $C\\!P$
violation of this level can be accommodated within the SM has emerged. In
parallel with attempts to improve the SM calculations, many estimates of
potential effects of PBSM have been made Bigi:2011re ; Feldmann:2012js ;
Rozanov:2011gj ; Grossman:2006jg ; Isidori:2011qw ; Wang:2011uu ;
Hochberg:2011ru ; Pirtskhalava:2011va ; Chang:2012gn ; Giudice:2012qq ;
Altmannshofer:2012ur ; Chen:2012am ; Gedalia:2012pi . To complement
theoretical calculations, measurements in related modes have been and will be
performed in order to single out effects from particular amplitudes.

A related way of searching for $C\\!P$ violation is using decays of charged
$D$ mesons. One group of measurements studies decays of $D^{+}$ and
$D^{+}_{s}$ 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
Bediaga:2009tr ; Williams:2011cd . The latter allow $C\\!P$ asymmetries to be
discovered while eventually a model-dependent analysis is required to identify
its source. Neither phase-space integrated asymmetry measurements
Aitala:1996sh ; Link:2000aw ; Aubert:2005gj ; Alexander:2008aa ; Rubin:2008aa
; Mendez:2009aa , nor searches for local asymmetries in the Dalitz plot
Aubert:2005gj ; Rubin:2008aa ; Staric:2011en ; PhysRevD.84.112008 ;
PhysRevLett.108.071801  have shown any evidence for $C\\!P$ violation. The
largest signal is the recently reported measurement of $C\\!P$ violation in
$D^{+}\\!\rightarrow\phi\pi^{+}$ of $A_{C\\!P}^{\phi\pi^{+}}=(5.1\pm 2.8\pm
0.5)\times 10^{-3}$ by the Belle collaboration PhysRevLett.108.071801 , which
exploits cancellation of uncertainties through a comparison of asymmetries in
the decays of $D^{+}$ and $D^{+}_{s}$ mesons into the final state
$\phi\pi^{+}$.

Decays of $D^{+}$ and $D^{+}_{s}$ into a $K^{0}_{\rm\scriptscriptstyle S}$ 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 Grossman:2011to  this contribution depends on the
decay-time acceptance of the $K^{0}_{\rm\scriptscriptstyle S}$. This can lead
to different expected values for different experiments which so far has not
been taken into account. Measurements of asymmetries in the decays
$D^{+}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$
PhysRevD.80.071103 ; Mendez:2009aa ; PhysRevLett.104.181602  and
$D^{+}_{s}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ Mendez:2009aa
; PhysRevLett.104.181602  show significant asymmetries. Future, more precise
measurements will reveal whether or not these are 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. Complementary to this are searches for rare
charm decays, studies of the top quark Wang:2011uu ; Hochberg:2011ru ,
measurements of nuclear electric dipole moments Giudice:2012qq , and many
other flavour observables which are beyond the scope of this review.

##  4 Beauty mixing and $C\\!P$ violation

The existence of $B^{0}$ and $B^{0}_{s}$ meson mixing is well established, and
the mass difference between the light and heavy eigenstates has been measured
to high precision in both systems. In addition, evidence exists for $C\\!P$
violation in $B^{0}$, $B^{+}$, and $B^{0}_{s}$ decays. The interpretation of
the experimental data focuses on the compatibility of the various measurements
with each other, and their compatibility with the SM description of $C\\!P$
violation as arising from a single weak phase in the CKM matrix. Two tensions
stand out at present: the discrepancy between the large mixing-induced $C\\!P$
asymmetry measured in semileptonic $B^{0}$ and $B^{0}_{s}$ decays
Abazov:2011yk  and the small $C\\!P$ violating phase in $B^{0}_{s}$ mixing
LHCb-CONF-2012-002  on the one hand, and the discrepancy between
$\sin\left(2\beta\right)$ and $|V_{ub}|$ measured from the branching ratio of
$B^{+}\rightarrow\tau^{+}\nu$ Asner:2010qj  on the other hand.

###  4.1 $B^{0}_{s}$ mixing

The mixing of $B^{0}_{s}$ mesons is described by the width difference between
the light and heavy mass eigenstates, $\Delta\Gamma_{s}$, the mass difference
$\Delta m_{s}$, and a single $C\\!P$ violating phase $\phi_{s}$. Within the SM
the width difference is substantial,
$\Delta\Gamma_{s}=\Gamma_{\textrm{L}}-\Gamma_{\textrm{H}}=0.087\pm 0.021$ ps-1
Lenz:2011ti , while the $C\\!P$ violating phase, as determined from indirect
fits to experimental data, is small $\phi_{s}=-0.036\pm 0.002$ rad Lenz:2011ti
; Lenz:2006hd ; Charles:2011va . Both can deviate substantially from these
predictions in other models.

The first observation of $B^{0}_{s}$ mixing was made by CDF Abulencia:2006ze ,
while the most precise measurement of the mass difference $\Delta m_{s}$ comes
from the recent LHCb measurement Aaij:2011qx . The most precise measurements
of both the width difference and phase come from the measurement of the time-
dependent $C\\!P$ asymmetry in
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
LHCb-CONF-2012-002 ; CDF-CONF-10778 ; PhysRevD.85.032006

$\displaystyle\phi_{s}$ $\displaystyle=$ $\displaystyle-0.001\pm 0.101\pm
0.027\;\textrm{rad}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb-
CONF-2012-002}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle\Delta\Gamma_{s}$ $\displaystyle=$ $\displaystyle 0.116\pm
0.018\pm 0.006\;\textrm{ps}^{-1}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb-
CONF-2012-002}{\@@citephrase{(}}{\@@citephrase{)}}}},$ $\displaystyle\phi_{s}$
$\displaystyle\in$
$\displaystyle[\frac{\pi}{2},-1.51]\cup[-0.06,0.30]\cup[1.26,\frac{\pi}{2}]\;\textrm{rad}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{CDF-CONF-10778}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle\phi_{s}$ $\displaystyle=$
$\displaystyle-0.55^{+0.38}_{-0.36}\;\textrm{rad}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PhysRevD.85.032006}{\@@citephrase{(}}{\@@citephrase{)}}}}.$

All these measurements are in good agreement with the SM, and it is notable
that a non-zero $\Delta\Gamma_{s}$ has been directly measured for the first
time at $5\sigma$. In addition, the sign of $\Delta\Gamma_{s}$ has been
unambiguously determined to be positive through the study of S-wave and P-wave
contributions to the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}K^{-}$
decay amplitude LHCb-PAPER-2011-028 .

The measurement of $\phi_{s}$ from
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ is
complicated by the vector-vector final state, which necessitates a time-
dependent angular analysis, whereas it was proposed Stone:2008ak  to study the
vector-pseudoscalar decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$
in which no such analysis is required. This measurement has recently been
performed by the LHCb collaboration, which, combined with the LHCb
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
measurement leads to

$\phi_{s}=-0.002\pm 0.083\pm
0.027\;\textrm{rad}\;\textrm{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{LHCb-
CONF-2012-002}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,$

in good agreement with the SM prediction.

As noted in Fleischer:2010ib , the interplay of $\Delta\Gamma_{s}$ and
$\phi_{s}$ leads to predictions for the effective lifetimes of $B^{0}_{s}$
mesons decaying into $C\\!P$ eigenstates. In the specific case of
$B^{0}_{s}\rightarrow K^{+}K^{-}$, the lifetime has already been measured
Aaij:2011kn ; LHCb-CONF-2012-001  to be $1.468\pm 0.046\pm 0.006\textrm{ps}$.
Using the latest measurement of $\Gamma_{s}$ and $\Delta\Gamma_{s}$ by LHCb
LHCb-CONF-2012-002 , as well as the $B^{0}_{s}$ lifetime
$\tau_{B^{0}_{s}}=1.472\pm 0.025$ ps Asner:2010qj , the SM prediction from
Fleischer:2010ib  can be updated to $\tau_{K^{+}K^{-}}=1.40\pm 0.02$.
Moreover, recent first observations of $B^{0}_{s}\rightarrow D^{0}D^{0}$ and
$B^{0}_{s}\rightarrow D^{+}D^{-}$ LHCb-CONF-2012-009  by LHCb indicate that it
will be possible in the near future to measure effective lifetimes in many
different $B^{0}_{s}$ decays to $C\\!P$ eignestates, and further constrain
$(\phi_{s},\Delta\Gamma_{s})$ in this manner.

The decay $B^{0}_{s}\rightarrow K^{+}K^{-}$ is not only a decay to a $C\\!P$
eigenstate, but is one example of a $b\rightarrow s$ penguin transition in the
decays of $B^{0}_{s}$ mesons. One of the experimentally most interesting modes
of this type is $B^{0}_{s}\rightarrow\phi\phi$ where, because of a
cancellation of $C\\!P$ violating effects from decay and mixing, the SM
predicts an upper limit of $0.02$ for $C\\!P$ violation Raidal:2002ph .
Although the time-dependent analysis is yet to be performed, time-integrated
analyses based on measuring triple products have been performed, and have
found no significant asymmetries Aaltonen:2011rs ; LHCb-PAPER-2012-004 , in
agreement with SM predictions.

Another interesting Ciuchini:2007hx  decay is $B^{0}_{s}\rightarrow
K^{*0}K^{*0}$, which has recently been observed for the first time by LHCb
LHCb-PAPER-2011-012 . Because of the $V-A$ structure of the weak interaction,
the $C\\!P$-even longitudinal polarization component was expected to be
dominant Beneke:2006hg ; Cheng:2009mu ; Ali:2007ff  in both this decay and
$B^{0}_{s}\rightarrow\phi\phi$. However, both B-factory measurements in
$b\rightarrow s$ penguin modes Abe:2004mq ; Chen:2005zv ; Aubert:2006uk ;
Aubert:2006fs ; Aubert:2008zza ; delAmoSanchez:2010mz , as well as the recent
LHCb measurements of $B^{0}_{s}\rightarrow\phi\phi$ and $B^{0}_{s}\rightarrow
K^{*0}K^{*0}$, find roughly equal longitudinal and $C\\!P$-odd transverse
polarization components. Proposed explanations have included large penguin
annihilation contributions Kagan:2004uw  or final state interactions
Datta:2007qb . The time dependent $C\\!P$ violation measurements in both these
modes should become experimentally accessible in the near future, further
constraining PBSM.

###  4.2 $B^{0}$ mixing

The mixing of $B^{0}$ mesons can be described within the same formalism as
that of $B^{0}_{s}$ mesons, but now it is the width difference
$\Delta\Gamma_{d}$ which is small in the SM while the mixing phase $\phi_{d}$
is large. The most precise measurements of $\Delta m_{d}$ were made by BaBar
Aubert:2005kf  and Belle Abe:2004mz , leading to the current world average
$\Delta m_{d}=0.505\pm 0.004$ Nakamura:2010zzi . The mixing phase can also be
expressed as the angle $\beta$ of the Unitarity Triangle, whose most precise
measurement comes from the study of time-dependent $C\\!P$ violation in the
“golden mode” $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and related decays

$\displaystyle\sin(2\beta)$ $\displaystyle=$ $\displaystyle 0.687\pm 0.028\pm
0.012\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Asner:2010qj,Aubert:2009aw}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle\sin(2\beta)$ $\displaystyle=$ $\displaystyle 0.667\pm 0.023\pm
0.012\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Adachi:2012et}{\@@citephrase{(}}{\@@citephrase{)}}}},$

The measurement of this angle can be related to the CKM matrix element
$|V_{ub}|$ through the unitarity relation in equation 4, and can be compared
to the value of $\sin(2\beta)$ as determined from a fit to the other
parameters of the Unitarity Triangle Charles:2004jd ; Ciuchini:2000de  of
$0.830^{+0.013}_{-0.033}$ and $0.80\pm 0.05$ from the CKMFitter and UTFit
collaborations respectively. This tension is driven by the branching fraction
of the decay $B^{+}\rightarrow\tau^{+}\nu$

$\displaystyle B(B\rightarrow\tau\nu)$ $\displaystyle=$
$\displaystyle(1.80^{+.57}_{-.54}\pm 0.26)\times
10^{-4}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{delAmoSanchez:2010ab}{\@@citephrase{(}}{\@@citephrase{)}}}},$
$\displaystyle B(B\rightarrow\tau\nu)$ $\displaystyle=$
$\displaystyle(1.54^{+.38}_{-.37}{}^{+.29}_{-.31})\times
10^{-4}\textrm{~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Hara:2010dk}{\@@citephrase{(}}{\@@citephrase{)}}}},$

which can be transformed into a measurement of $|V_{ub}|$ and hence a
constraint on the apex of the Unitarity Triangle.

Resolving this tension will require a precise understanding of the size of
doubly Cabibbo-suppressed penguin topologies in
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ Faller:2008zc . In this respect it is
interesting to note the observation of the U-spin partner decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ at LHCb LHCb-CONF-2011-048 , which has
been proposed Fleischer:1999nz ; DeBruyn:2010hh  as one way of measuring these
effects.

An important additional null-test of the SM comes from the measurement of
$\Delta\Gamma_{d}$. As noted in Gershon:2010wx , the fact that the SM
prediction for $\Delta\Gamma_{d}/\Gamma_{d}$ is so small,
$40.9^{+8.9}_{-9.9}\times 10^{-4}$ Lenz:2006hd , while plausible scenarios of
PBMS exist in which this value is enhanced Lenz:2012az , means that any non-
zero measurement with current experimental sensitivity would be a clear sign
of new physics effects. Indeed, such effects are needed to explain the
anomalous dimuon asymmetry observed by DØ, as discussed in the following
section. Both BaBar and Belle have measured $\Delta\Gamma_{d}$ Aubert:2003hd ;
Aubert:2004xga ; Higuchi:2012kx  through fits to the time dependent decay
rates in $B^{0}\rightarrow D^{(*)-}(\pi,\rho,a_{1})^{+}$ and $B^{0}\rightarrow
c\bar{c}K^{0}_{\textrm{S,L}}$ modes. The average is dominated by the recent
Belle result of $\Delta\Gamma_{d}/\Gamma_{d}=[-1.7\pm 1.8\pm 1.1]\times
10^{-2}$. As the uncertainty on this measurement is still an order of
magnitude larger than the SM prediction, it remains to be seen if the
systematic uncertainties can be kept under control in the era of the next
generation flavour factories.

###  4.3 Semileptonic asymmetries

The mixing induced semileptonic asymmetry $A_{\textrm{sl}}$ is predicted to be
$O(10^{-4})$ in the SM within both the $B^{0}$ $(a^{d}_{\textrm{sl}})$ and
$B^{0}_{s}$ $(a^{s}_{\textrm{sl}})$ meson systems Lenz:2011ti . The most
precise experimental measurement to date was made by the DØ Collaboration
Abazov:2011yk , which found a percent-level $C\\!P$ asymmetry

$\displaystyle A_{\textrm{sl}}$ $\displaystyle\approx$ $\displaystyle
0.6\times a^{d}_{\textrm{sl}}+0.4\times a^{s}_{\textrm{sl}},$ $\displaystyle
A_{\textrm{sl}}$ $\displaystyle=$ $\displaystyle(-0.787\pm 0.172\pm
0.093)\%\;.$

Because DØ cannot distinguish between dimuon pairs coming from $B^{0}$ and
$B^{0}_{s}$ decays, it measures a combination of the two semileptonic
asymmetries. In the same paper, the collaboration attempts to separate effects
caused by $B^{0}_{s}$ oscillations from those caused by $B^{0}$ oscillations
by indirectly studying the lifetime of the decaying $B$ meson, and concludes
that the asymmetry is largest at short lifetimes. The authors take this as a
hint that the asymmetry is dominated by $B^{0}_{s}$ decays because the
$B^{0}_{s}$ meson oscillates much more quickly than the $B^{0}$.

When interpreting this result, it is important to keep in mind that the
background levels are also highest at short lifetimes; for this reason, it is
critical that $(a^{d}_{\textrm{sl}})$ and $(a^{s}_{\textrm{sl}})$ are measured
separately in a low background environment where the decaying $B$ meson can be
unambiguously tagged as a $B^{0}$ or $B^{0}_{s}$. Nevertheless, taking the DØ
result at face value, it is not trivial to reconcile it with the measurements
of $B^{0}_{s}$ and $B^{0}$ mixing mentioned earlier. An easy way of seeing
this is to consider why, if the dimuon asymmetry is driven by $B^{0}_{s}$
mixing, the mixing phase in
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ is
so close to the SM value while the direct and indirect measurements of
$\sin\left(2\beta\right)$ are in tension. One proposed explanation requires
Lenz:2012az  contributions from PBSM to both $M^{d,s}_{\textrm{12}}$ and
$\Gamma^{d,s}_{\textrm{12}}$

###  4.4 $B\rightarrow h\gamma$ decays

CP asymmetry measurements of $b\rightarrow s\gamma$ transitions are sensitive
to PBSM, for instance through measurements of the photon polarisation which
probes models involving right-handed currents Gershon:2006mt ;
DescotesGenon:2011yn ; Mahmoudi:2006wq ; Altmannshofer:2011gn . $C\\!P$
asymmetries in $b\rightarrow s\gamma$ transitions have been measured by BaBar
Aubert:2008gy ; Aubert:2008be ; Aubert:2009ak , Belle Nishida:2003yw ;
Ushiroda:2006fi , and LHCb LHCb-CONF-2012-004  and the results are consistent
with SM expectations and statistically limited. In this context, it has been
recently noted Benzke:2010tq  that the difference in $C\\!P$ asymmetries
between the inclusive processes $X_{s}^{+}\gamma$ and $X_{s}^{0}\gamma$ offers
a cleaner probe of PBSM than either measurement taken on its own.

Thanks to the large value of $\Delta\Gamma_{s}$, the $B_{s}$ system is
particularly promising for measuring the photon polarisation by studying time
dependent $C\\!P$ violation in the decay $B_{s}\rightarrow\phi\gamma$
Muheim:2008vu . This mode was first observed at Belle Wicht:2007ni , while the
LHCb collaboration has recently measured LHCb:2012ab  the ratio of the
branching ratios $\frac{{\cal B}(B^{0}\rightarrow
K^{*}\gamma)}{B_{s}\rightarrow\phi\gamma)}=1.12\pm 0.08^{+0.11}_{-0.09}$.

###  4.5 The CKM angle $\gamma$

A precise determination of the angle $\gamma$ of the CKM unitarity triangle is
important in order to further overconstrain the position of the triangle’s
apex, in particular with respect to the previously discussed measurements of
$\sin\left(2\beta\right)$ and $V_{ub}$. In this respect $\gamma$ can be
measured either from tree-level or loop-mediated processes, and a comparison
of the two kinds of measurements provides another opportunity for PBSM to
manifest itself. In either case, $\gamma$ is experimentally determined from a
measurement of $C\\!P$ violation in those $B$ meson decays where diagrams
involving $|V_{ub}|$ and $|V_{cb}|$ result in the same final state.

The determination of $\gamma$ from tree-level decays is one of the most
sensitive tests of the SM precisely because the associated theoretical
uncertainties are confined to electroweak corrections associated with box-
diagram decays, and are at the level of $\delta\gamma/\gamma\approx 10^{-6}$
Zupan:2011mn . Experimentally the challenge is that the sensitivity to
$\gamma$ comes from the interference of $|V_{ub}|$ and $|V_{cb}|$ diagrams,
which means that the final state must be carefully chosen in order to make the
amplitudes of similar size and hence maximize the interference. Unfortunately
those modes which have the highest interference also have the biggest
associated experimental difficulties, whether it be low overall branching
ratios, difficult to reconstruct final state particles, or the requirement for
a time-dependent analysis. This means that the ultimate precision on $\gamma$
can only be achieved by combining several different measurements.

The current sensitivity on $\gamma$ is dominated by measurements of $C\\!P$
violation and partial widths in $B^{+}\rightarrow D^{0}K^{+}$ decays, in which
the $D^{0}$ then decays to either a $C\\!P$-eigenstate Gronau1991483 ;
Gronau1991172 , a doubly-Cabibbo suppressed decay mode Atwood:1996ci ;
Atwood:2000ck , or a multibody decay whose Dalitz distribution gives rise to
interference effects Giri:2003bs . These are known as the GLW, ADS, and GGSZ
methods respectively after their inventors.

In the first two cases the charge-averaged partial width ratios of the
$D^{0}K^{+}$ and $D^{0}\pi^{+}$ decays are measured,

$R^{f}_{K/\pi}=\frac{\Gamma(B\rightarrow[f]_{D}K)}{\Gamma(B\rightarrow[f]_{D}\pi)},$
(9)

where $f$ represents the $C\\!P$-eigenstate $\pi\pi$ and $KK$ decays and the
Cabibbo-favoured $K\pi$ decay mode; the $C\\!P$ asymmetries

$A^{f}_{h}=\frac{\Gamma(B^{+}\rightarrow[f]_{D}h^{+})-\Gamma(B^{-}\rightarrow[f]_{D}h^{-})}{\Gamma(B^{+}\rightarrow[f]_{D}h^{+})+\Gamma(B^{-}\rightarrow[f]_{D}h^{-})},$
(10)

where $h$ is a pion or a kaon; and the charge-separated partial width ratios
of the Cabibbo-favoured and doubly Cabibbo-suppressed $B^{+}\rightarrow
D^{0}K^{+}$ decay modes

$R^{\pm}_{h}=\frac{\Gamma(B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm})}{\Gamma(B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}h^{\pm})}.$
(11)

As these are the most experimentally accessible modes for measuring $\gamma$,
they have been studied at BaBar delAmoSanchez:2010dz ; delAmoSanchez:2010ji ,
Belle Abe:2006hc ; Belle:2011ac ,CDF Aaltonen:2009hz ; Aaltonen:2011uu , and
recently at LHCb Aaij:2012kz . In particular, LHCb has observed the doubly
Cabibbo-suppressed decay $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$
with $10\sigma$ significance, and has made a $5.8\sigma$ observation of
$C\\!P$-violation in $B^{+}\rightarrow D^{0}K^{+}$ decays. It is worth
highlighting the cleanliness of the LHCb signals, as seen in Fig. 4, as well
as the intriguing hint of $C\\!P$-violation in the
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}\pi^{\pm}$ which can be seen in the
same picture.

Figure 3: Invariant mass distribution of selected
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}h^{\pm}$ candidates. The left plots
are $B^{-}$, the right plots are $B^{+}$. Top are $h=K$ and bottom are
$h=\pi$. The red curve is the signal, the shaded area, green, and magenta
curves are backgrounds. Reproduced from Aaij:2012kz .

In the third case, what is measured are the different Dalitz plot
distributions of $D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}hh$ in
$B^{+}\rightarrow D^{0}K^{+}$ and $B^{-}\rightarrow D^{0}K^{-}$ decays, and
measurements have been made with delAmoSanchez:2010rq ; Poluektov:2010wz  or
without Belle:2011ab  assuming an amplitude model for the $D^{0}$ decay. The
advantage of this method is that it only suffers from a two-fold ambiguity in
the measured value of $\gamma$, as opposed to the eightfold ambiguity in e.g.
the GLW method.

The average value of $\gamma$ from these decay modes, as computed by the
CKMFittter collaboration, is shown in Fig. 4, from which it is apparent that
while direct measurements of $\gamma$ agree well with its indirect
determination from other Unitarity Triangle parameters, they are not yet
strongly constraining the apex of the triangle. A historical tension exists
between the frequentist (CKMFitter) and Bayesian (UTFit) averages of $\gamma$,
driven by the different treatment of the nuissance parameters which
parametarize the size of the interference in each decay mode. The most up-to-
date averages from the two collaborations are

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle(66\pm 12)^{\circ}$
CKMFitter $\displaystyle\;\textrm{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Charles:2004jd}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle(76\pm 9)^{\circ}$ UTFit
$\displaystyle\;\textrm{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ciuchini:2000de}{\@@citephrase{(}}{\@@citephrase{)}}}}\;,$

where the CKMFitter average includes the most recent ADS/GLW results from LHCb
and the UTFit average does not. The larger uncertainty in the CKMFitter
average comes from the treatment of the nuissance parameters, while there is
an interesting discrepancy developing in the central values which is not
understood at present.

Figure 4: Averaged constraints on $\gamma$ from direct measurements.
Reproduced from Charles:2004jd .

Many other tree-level determinations of $\gamma$ are possible, for example
from $B^{0}\rightarrow D^{0}h^{+}h^{-}$ decays Gronau1991483 ; Gronau1991172 ;
Atwood:1996ci ; Atwood:2000ck ; Gershon:2008pe ; Gershon:2009qc  whether in a
quasi-two-body approach, selecting the $h^{+}h^{-}$ mass to lie at a
particular resonance, or through an amplitude analysis. An important milestone
on this road to $\gamma$ is the first observation of the decay mode
$B^{0}_{s}\rightarrow D^{0}K^{*0}$ at LHCb Aaij:2011tz . It is also possible
to make an unambiguous measurement of $\gamma$ through the study of $C\\!P$
violation in the interference of $B^{0}_{s}$ mixing and the decay
$B^{0}_{s}\rightarrow D^{\pm}_{s}K^{\mp}$ FleischerDsK , whose branching ratio
has recently been precisely measured LHCb-CONF-2011-057 . Within measurements
of $\gamma$ from loop-mediated processes, the study of two body
$B_{s,d}\rightarrow h^{+}h^{-}$ decays stands out. The U-spin partner decays
$B^{0}_{s}\rightarrow K^{+}K^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$ are
able to extract $\gamma$ unambiguously in a combined analysis Fleischer:1999pa
; Fleischer:2007hj , and recently the time-dependent $C\\!P$ asymmetry in
$B^{0}_{s}\rightarrow K^{+}K^{-}$ has been measured LHCb-CONF-2012-007  for
the first time

$\displaystyle A^{\textrm{dir}}_{KK}$ $\displaystyle=$ $\displaystyle 0.02\pm
0.18\pm 0.04\;,$ $\displaystyle A^{\textrm{mix}}_{KK}$ $\displaystyle=$
$\displaystyle 0.17\pm 0.18\pm 0.05\;,$

to add to the existing Aubert:2008sb ; Ishino:2006if  measurements in
$B^{0}\rightarrow\pi^{+}\pi^{-}$.

##  5 Rare Decays

Rare decays which proceed via Flavour Changing Neutral Currents (FCNC) are
induced by one-loop diagrams in the SM and are excellent probes for PBSM. New
particles can enter in competing loop-order diagrams, resulting in large
deviations from SM predictions. In general, an effective hamiltonian formalism
is used to describe the amplitudes of FCNC processes, according to the
formula:

$H_{eff}=\frac{G_{F}}{\sqrt{2}}\sum_{i}V_{CKM}^{i}C_{i}(\mu)Q_{i}\;,$ (12)

where $V^{i}_{CKM}$ are the relevant factors of the CKM matrix; $Q_{i}$ are
local operators; $C_{i}$ are the corresponding couplings (Wilson
coefficients); and $\mu$ is the QCD renormalization scale. The correlation of
different channels, where common Wilson coefficients contribute, is a powerful
tool for searching and understanding the structure of PBSM.
This approach is complementary to direct searches for PBSM. Moreover indirect
searches often allow to set more stringent constraints than direct ones. For
instance, strong lower bounds on the mass of the charged Higgs in Two-Higgs-
Doublets-Models of type II have been obtained from the analysis of
$\overline{B}\rightarrow X\gamma$ decays, where the SM prediction
Misiak:2006zs  is found in agreement with inclusive measurements performed by
the experiments BaBar Aubert:2006gg ; Aubert:2005cba ; Aubert:2007my , Belle
Abe:2001hk ; Limosani:2009qg  and CLEO Chen:2001fja  (other bounds from
$B\rightarrow X_{s}\gamma$ are discussed in Buras:2011zb  and the references
therein).
As a result of the many measurements performed by the B-factories and more
recently by the CDF experiment,our knowledge of suppressed processes has
considerably improved in the last decade. Consequently, constraints on PBSM
have become much stronger.
While inclusive measurements are challenging at hadron colliders, studies of
exclusive decays are competitive with $e^{+}e^{-}$ machines. Moreover, hadron
colliders have the advantage that all B-hadron species are produced. With the
start-up of the LHCb experiment a new round in the precision measurements of
rare decays has begun.

###  5.1 $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ decays

Purely leptonic decays of B-mesons are a key ingredient in the search for
PBSM, since the prediction of their branching fractions is largely free from
hadronic uncertainties. The two decays $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ have
a clear experimental signature and are easier to reconstruct and identify than
the other leptonic decays of B-mesons. Their branching fractions are predicted
to be ${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})=(3.2\pm 0.2)\times 10^{-9}$
and ${\cal B}(B_{d}\rightarrow\mu^{+}\mu^{-})=(1.0\pm 0.1)\times 10^{-10}$ in
the SM Buras:2010mh ; Buras:2010wr . Contributions from PBSM, especially in
models with an extended Higgs sector, can enhance these branching fractions.
For instance, in the Minimal Supersymmetric extension of the SM the branching
fraction of the decay $B_{s}\rightarrow\mu^{+}\mu^{-}$ is proportional to the
sixth power of $\tan\beta$ (the ratio of the vacuum expectation values of the
neutral components of the Higgs fields $H_{u}$ and $H_{d}$) Hurth:2008jc .
This fact makes this observable particularly sensitive to supersymmetric
models with large $\tan\beta$. More generally measurements of this branching
fractions probe the Wilson coefficients $C_{s}$ and $C_{p}$, which are
negligibly small in the SM. Present measurements of ${\cal
B}(B_{s}\rightarrow\mu^{+}\mu^{-})$ are shown in Fig. 5.

Figure 5: Present limits on ${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})$ at
$95\%$ CL set by the experiments D0 Abazov:2010fs  CDF CDF-Bsmumu , ATLAS
ATLAS-CONF-2012-010  CMS Chatrchyan:2012rg  and LHCb Aaij:2012ac . The SM
prediction is indicated by the blue-dashed line.

Presently, the most stringent upper limits on ${\cal
B}(B_{s,d}\rightarrow\mu^{+}\mu^{-})$ are set by the LHCb experiment
Aaij:2012ac . This analysis profits from the good momentum resolution and the
good particle identification performances of LHCb to reject the different
sources of background. The branching fraction of the signal was extracted by
using the three normalization channels: $B^{+}\rightarrow J/\psi K^{+}$,
$B^{0}\rightarrow K^{+}\pi^{-}$ and $B_{s}\rightarrow J/\psi\phi$. For the
first two of these channels, the ratio of the hadronization fractions
$\frac{f_{s}}{f_{d}}$ is needed111Isospin symmetry, i.e. $f_{u}=f_{d}$, has
been assumed.. This variable was measured at LHCb by combining measurements
with semi-leptonic and hadronic decays Fleischer:2010ay :
$f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$ Aaij:2011jp ; Aaij:2011hi . The
uncertainty on this parameter is, in the long run, a limiting systematic
uncertainty for discriminating between SM and BSM contributions in the
$B_{s}\rightarrow\mu^{+}\mu^{-}$ decay, as well as for the measurement of the
golden ratio $\frac{{\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})}{{\cal
B}(B_{d}\rightarrow\mu^{+}\mu^{-})}$ Buras:2010pi . The correlation between
the branching fractions of the decays $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ is
shown in Fig. 6 for several beyond SM scenarios.

Figure 6: Correlation for the branching fractions of the decays
$B_{s}\rightarrow\mu^{+}\mu^{-}$ and $B_{d}\rightarrow\mu^{+}\mu^{-}$ for
several models of PBSM. Details on the models can be found in Straub:2010ih .
The recent upper limits by LHCb are shown by the shaded region. Reproduced
from Straubb-Moriond .

The upper limits set by LHCb for the $B_{s,d}\rightarrow\mu^{+}\mu^{-}$ decays
are: ${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-})<4.5\times 10^{-9}$ and ${\cal
B}(B_{d}\rightarrow\mu^{+}\mu^{-})<1.05\times 10^{-9}$ at $95\%$ CL and are
illustrated in Fig. 6 by the shaded region. These measurements are in
agreement with SM expectations and give additional constraints for PBSM with
respect to those provided by $b\rightarrow s\gamma$ and other $b\rightarrow
sl^{+}l^{-}$ transitions.

###  5.2 $B\rightarrow h\mu^{+}\mu^{-}$ decays

In the decay $B_{d}\rightarrow K^{*}\mu^{+}\mu^{-}$ several angular
observables can be built which are sensitive to PBSM, and for which form
factor uncertainties are theoretically under control, (see for example
Ali:1991is ; Altmannshofer:2008dz  and references therein). These observables
include the forward-backward asymmetry of the dimuon system,
$A_{\mathrm{FB}}$, the fraction of $K^{*}$ longitudinal polarization, $F_{L}$,
the transverse asymmetry, $S_{3}$ Altmannshofer:2008dz  (often referred to as
$\frac{1}{2}(1-F_{L})A_{T}^{2}$ in the literature Kruger:2005ep ), and the
T-odd CP asymmetry $A_{\textrm{Im}}$ Bobeth:2008ij . They can be extracted by
performing an angular analysis as a function of the dimuon invariant mass
squared, $q^{2}$, with respect to the following angles: the angle $\theta_{l}$
between the $\mu^{+}$ ($\mu^{-}$) and the $B^{0}$ ($\overline{B}^{0}$) in the
dimuon rest frame; the angle $\theta_{K}$ between the kaon and $B^{0}$ in the
$K^{*}$ rest frame; and the angle $\phi$ between the planes of the dimuon
system and the plane of the $K^{*}$. A formal definition of these angles can
be found in Egede:1048970 . Present measurements of the observables
$A_{\mathrm{FB}}$, $F_{L}$, $S_{3}$ and $A_{\textrm{Im}}$ are shown in Fig. 7.
These measurements provide information about the Wilson coefficients $C_{7}$,
$C_{9}$ and $C_{10}$ and on their right-handed counterparts.

The LHCb experiment has recently made the world’s best measurements on these
angular observables LHCb-CONF-2012-008 . The physics parameters were extracted
by fitting the partial decay rate as a function of the three angles for
different bins in $q^{2}$. In order to reduce the number of parameters in the
fit, due to the small size of the data sample, the angle $\phi$ was folded by
taking $\phi\rightarrow\phi+\pi$ when $\phi<0$. This transformation cancels
out the terms containing $cos\phi$ and $sin\phi$ in the differential decay
rate. This strategy is different from that followed by other experiments,
where only projections of the angular distributions were used.

Figure 7: The $A_{\mathrm{FB}}$, $F_{L}$, $S_{3}$ and $A_{\textrm{Im}}$
measured by the experiments BaBar BaBarLakeLouise , Belle Wei:2009zv , CDF
Aaltonen:2011ja  and LHCb LHCb-CONF-2012-008 . The comparison with the SM
prediction, taken from Bobeth:2011gi  is also shown. Reproduced from LHCb-
CONF-2012-008 .  Figure 8: The $A_{\mathrm{FB}}$ as a function of $q^{2}$
extracted from an unbinned counting experiment. The shaded region correspond
to the $68\%$ CL of the zero-crossing point. Comparison with the SM prediction
Bobeth:2011gi  is shown. Reproduced from LHCb-CONF-2012-008 .

The so called zero-crossing point, where $A_{\mathrm{FB}}$ changes sign, is
largely free from form factor uncertainties and sensitive to PBSM Ali:1991is .
The SM predicts this point to be in the range 4.0-4.3 GeV2/c4 Bobeth:2011nj ;
Beneke:2004dp ; Ali:2006ew . The zero-crossing point of $A_{\mathrm{FB}}$ was
measured for the first time by LHCb to be $q_{0}^{2}=4.9^{+1.1}_{-1.3}$GeV2/c4
LHCb-CONF-2012-008 . This observable was extracted in an unbinned counting
experiment with respect to $q^{2}$, integrating the angular distributions with
respect to the three angles Jansen:1156131 . The result is shown in Fig. 8.
Other exclusive $b\rightarrow sll$ processes have been measured by the
B-factories, CDF and LHCb. The measurements of the differential branching
fractions of the decays $\Lambda_{b}\rightarrow\Lambda\mu^{+}\mu^{-}$
Aaltonen:2011qs , $B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ Aaltonen:2011qs ;
Wei:2009zv ; BaBarLakeLouise , $B_{s}\rightarrow\phi\mu^{+}\mu^{-}$
Aaltonen:2011qs , $B^{0}\rightarrow K_{S}\mu^{+}\mu^{-}$ Aaltonen:2011qs ;
Wei:2009zv ; BaBarLakeLouise  and $B^{+}\rightarrow K^{*+}\mu^{+}\mu^{-}$
Aaltonen:2011qs ; Wei:2009zv ; BaBarLakeLouise  and the $A_{FB}$ for the
decays $B^{+}\rightarrow K^{+}\mu^{+}\mu^{-}$ Aaltonen:2011ja ; Wei:2009zv ;
BaBarLakeLouise  and $B^{+}\rightarrow K^{*+}\mu^{+}\mu^{-}$ Aaltonen:2011ja ;
Wei:2009zv ; BaBarLakeLouise  were found to be in agreement with SM
predictions.
Another observable which is potentially sensitive to PBSM is the isospin
asymmetry, $A_{I}$, defined as:

$A_{I}=\frac{{\cal B}(B^{0}\rightarrow
K^{(*)0}l^{+}l^{-})-\frac{\tau_{0}}{\tau_{+}}{\cal B}(B^{\pm}\rightarrow
K^{(*)\pm}l^{+}l^{-})}{{\cal B}(B^{0}\rightarrow
K^{(*)0}l^{+}l^{-})+\frac{\tau_{0}}{\tau_{+}}{\cal B}(B^{\pm}\rightarrow
K^{(*)\pm}l^{+}l^{-})},$ (13)

where $l=(e,\mu)$. This observable was measured by the experiments Belle
Wei:2009zv , BaBar BaBarLakeLouise  and CDF Aaltonen:2011ja . The results are
shown in Fig. 9.

Figure 9: Isospin asymmetry for the decays $B\rightarrow K^{(*)}l^{+}l^{-}$,
measured by the experiments BaBar BaBarLakeLouise , Belle Wei:2009zv  (with
electrons and muons) and CDF Aaltonen:2011ja  (with muons).

The SM predicts a small asymmetry in all $q^{2}$ bins. Present results seem to
hint at a non-zero $A_{I}$ in some $q^{2}$ bins. Measurements with larger data
samples and good control of the theoretical uncertainties are important; a
measurement of $A_{I}$ from LHCb can be expected in the near future.
Recently, the LHCb collaboration reported the first observation of a
$b\rightarrow dll$ transition, by measuring the branching fraction ${\cal
B}(B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-})=(2.4\pm 0.6\pm 0.2)\times 10^{-8}$
LHCb-CONF-2012-006 . The invariant mass distribution of
$B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates is shown in Fig. 10.

Figure 10: The invariant mass distribution of
$B^{+}\rightarrow\pi^{+}\mu^{+}\mu^{-}$ candidates. Reproduced from LHCb-
CONF-2012-006 .

This process is further suppressed by the factor $|V_{td}/V_{ts}|$, with
respect to the $b\rightarrow sll$ transitions. The measured branching fraction
is in good agreement with the SM expectation.

###  5.3 Search for Lepton Flavour Violating and very rare decays

The search for Lepton Flavour Violating (LFV) decays is a crucial way to test
the SM flavour structure. These searches have been performed by several
experiments. A complete discussion of LFV searches goes beyond the scope of
this review.
Decays of the type $B^{+}\rightarrow h^{-}l^{+}l^{+}$, where $h^{-}$ is a
meson, can be considered the analogues of neutrinoless double $\beta$ decays
and can be used to search for heavy Majorana neutrinos Atre:2009rg ;
Zhang:2010um ; Boyarsky:2006fg ; Ilakovac:1999md . These searches have been
performed by the LHCb Aaij:2012zr ; Aaij:2011ex , BaBar BABAR:2012aa , Belle
Seon:2011ni  and CLEO Edwards:2002kq  experiments. Upper limits for these
decays are summarised in Table 1. Heavy Majorana neutrinos can also be
searched for by using the corresponding charm decays $D^{+}\rightarrow
h^{-}l^{+}l^{+}$. Constraints on these decays are expected to improve
substantially with measurements from LHCb.
LFV decays of charged leptons are allowed in several extensions of the SM, for
instance supersymmetric models Dedes:2002rh ; Ciuchini:2007ha ; Arganda:2005ji
, left-right symmetric models Akeroyd:2006bb  and models with heavy neutrinos
Atre:2009rg ; Zhang:2010um ; Boyarsky:2006fg ; Ilakovac:1999md . Stringent
upper limits on the decay $\mu^{-}\rightarrow e^{-}\gamma$ have been set by
the MEG experiment Adam:2011ch , while the most stringent upper limits on
$\tau^{-}\rightarrow l^{-}l^{+}l^{-}$ were set by the Belle experiment
Hayasaka:2010np .
In addition, searches for exotic very rare decays have been carried out at
LHCb. Upper limits for the decays
$B_{d,s}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ and
$D^{0}\rightarrow\mu^{+}\mu^{-}$ were recently set LHCb-CONF-2012-010 ; LHCb-
CONF-2012-005  and are listed in Table 1.
For the moment no hint of the existence of any of such processes has been
observed, and all searches are statistically limited at present.

Channel | Upper Limit (CL) | Reference
---|---|---
${\cal B}(B^{+}\rightarrow K^{-}\mu^{+}\mu^{+})$ |  $5.4\times 10^{-8}$ ($95\%$) | LHCb Aaij:2011ex
${\cal B}(B^{+}\rightarrow\pi^{-}\mu^{+}\mu^{+})$ |  $1.3\times 10^{-8}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow\pi^{-}e^{+}e^{+})$ |  $2.3\times 10^{-8}$ ($90\%$) | BaBar BABAR:2012aa
${\cal B}(B^{+}\rightarrow K^{-}e^{+}e^{+})$ |  $3.0\times 10^{-8}$ ($90\%$) | BaBar BABAR:2012aa
${\cal B}(B^{+}\rightarrow D^{-}\mu^{+}\mu^{+})$ |  $6.9\times 10^{-7}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow D^{*-}\mu^{+}\mu^{+})$ |  $2.8\times 10^{-6}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow D^{-}e^{+}e^{+})$ |  $2.6\times 10^{-6}$ ($90\%$) | Belle Seon:2011ni
${\cal B}(B^{+}\rightarrow D^{-}\mu^{+}e^{+})$ |  $1.8\times 10^{-6}$ ($90\%$) | Belle Seon:2011ni
${\cal B}(B^{+}\rightarrow D_{s}^{-}\mu^{+}\mu^{+})$ |  $5.8\times 10^{-7}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(B^{+}\rightarrow D^{0}\pi^{-}\mu^{+}\mu^{+})$ |  $1.5\times 10^{-6}$ ($95\%$) | LHCb Aaij:2012zr
${\cal B}(D^{0}\rightarrow\mu^{+}\mu^{+})$ |  $1.3\times 10^{-8}$ ($95\%$) | LHCb LHCb-CONF-2012-005
${\cal B}(B_{s}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-})$ |  $1.3\times 10^{-8}$ ($95\%$) | LHCb LHCb-CONF-2012-010
${\cal B}(B^{0}\rightarrow\mu^{+}\mu^{-}\mu^{+}\mu^{-})$ |  $5.4\times 10^{-9}$ ($95\%$) | LHCb LHCb-CONF-2012-010
${\cal B}(\tau^{-}\rightarrow\mu^{-}\mu^{+}\mu^{-})$ |  $2.1\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow e^{-}e^{+}e^{-})$ |  $2.7\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow e^{-}\mu^{+}\mu^{-})$ |  $2.7\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow e^{+}\mu^{-}\mu^{-})$ |  $1.7\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow\mu^{+}e^{-}e^{-})$ |  $1.5\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\tau^{-}\rightarrow\mu^{-}e^{+}e^{-})$ |  $1.8\times 10^{-8}$ ($90\%$) | Belle Hayasaka:2010np
${\cal B}(\mu^{-}\rightarrow e^{-}\gamma)$ |  $2.4\times 10^{-12}$ ($90\%$) | MEG Adam:2011ch
Table 1: Upper Limit for several very rare or forbidden decays.

##  6 Conclusion

Despite the ongoing lack of a direct discovery of particles beyond the
Standard Model, recent results in flavour physics are giving ever stronger
hints of effects beyond the Standard Model. In particular, the observation of
permille-level $C\\!P$ violation in $D^{0}$ decays, the large dimuon asymmetry
in $B^{0}$ and $B^{0}_{s}$ decays, as well as the values of
$\sin\left(2\beta\right)$ and the branching ratio of
$B^{+}\rightarrow\tau^{+}\nu$, are difficult to simultaneously interpret
within the Standard Model framework. At the same time, measurements of rare
decays such as $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ which are in good agreement with the Standard Model have
placed the most stringent limits yet on many Standard Model extensions. What
this contradiction highlights is the ongoing relevance of flavour physics as
key tool not only for the indirect discovery of new particles and processes,
but also for discriminating between the many proposed theories of physics
beyond the Standard Model. With the excellent performance of the LHC, and the
wealth of precision flavour measurements coming from its detectors, it is
reasonable to hope for a deepening, and eventual resolution, of these
contradictions in the years to come.

###### Acknowledgements.

The authors would like to thank the CKMFitter, HFAG, and UTFit collaborations
for preparing updated world averages of many flavour parameters in time for
this review, and Mitesh Patel for useful comments on the draft.

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