| $-105\pm 64$ | HFAG 2012 (including $A^{b}_{\rm SL}$ ) [44] |  |
Admixture of $B^{0}$ and $B^{0}_{s}$ systems
$A^{b}_{\rm SL}$ ($10^{-4}$) | $-78.7\pm 17.1\pm 9.3$ |  D0 [159] | $-2.0\pm 0.3$ | [220, 221, 222, *Ciuchini:2003ww, *Beneke:1996gn, *Beneke:2003az]
$B^{0}$ system
$\Delta m_{d}$ (${\rm\,ps^{-1}}$) | $0.507\pm 0.004$ | HFAG 2012 [44] | $0.543\pm 0.091$ | [216, 221, 222, *Ciuchini:2003ww, *Beneke:1996gn, *Beneke:2003az]
$\Delta\Gamma_{d}/\Gamma_{d}$ | $0.015\pm 0.018$ | HFAG 2012 [44] | $0.0042\pm 0.0008$ | [220, 221, 222, *Ciuchini:2003ww, *Beneke:1996gn, *Beneke:2003az]
$\sin 2\beta$ | $0.679\pm 0.020$ | HFAG 2012 [44] | $0.832\,^{+0.013}_{-0.033}$ | [119, 221, 222, *Ciuchini:2003ww, *Beneke:1996gn, *Beneke:2003az]
$a^{d}_{\rm sl}$ ($10^{-4}$) | $-5\pm 56$ | HFAG 2012 [44] | $-6.5\,^{+1.9}_{-1.7}$ | [119, 221, 222, *Ciuchini:2003ww, *Beneke:1996gn, *Beneke:2003az]

The HFAG average of the $B^{0}_{s}$ mass difference $\Delta m_{s}$ in Table 1
is based on measurements performed at CDF [228] and LHCb [229, 226]. It is
dominated by the preliminary LHCb result obtained using $0.34\mbox{\,fb}^{-1}$
of data [226], which is also given in Table 1. These are all consistent with
the SM prediction. Improving the precision of the SM prediction is desirable
to further constrain NP in $M^{s}_{12}$, and requires improving the accuracy
of lattice QCD evaluations of the decay constant and bag parameter (see Ref.
[216] and references therein).

The observables $\phi_{s}$ and $\Delta\Gamma_{s}$ have been determined
simultaneously from $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\phi$ decays using time-dependent flavour tagged angular analyses
[230, 231]. The first LHCb tagged analysis using $0.34\mbox{\,fb}^{-1}$ of
data [10] already provided a significant constraint on $\phi_{s}$ and led to
the first direct evidence for a non-zero value of $\Delta\Gamma_{s}$. LHCb has
also determined the sign of $\Delta\Gamma_{s}$ to be positive at $4.7\,\sigma$
confidence level [232] by exploiting the interference between the
$K^{+}\kern-1.60004ptK^{-}$ S-wave and P-wave amplitudes in the $\phi(1020)$
mass region [233]. This resolved the two-fold ambiguity in the value of
$\phi_{s}$ for the first time. LHCb has made a preliminary update of the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$
analysis using the full data sample of $1.0\mbox{\,fb}^{-1}$ collected in 2011
[139]. The results from this analysis,

$\displaystyle\phi_{s}$ $\displaystyle=-0.001\pm 0.101\pm 0.027\,{\rm rad}\,,$
$\displaystyle\Delta\Gamma_{s}$ $\displaystyle=0.116\pm 0.018\pm
0.006{\rm\,ps^{-1}}\,,$ (31)

are shown in Fig. 10 (left), and are in good agreement with the SM
expectations.

Figure 10: (Left) Preliminary LHCb measurement of $\phi_{s}$ and
$\Delta\Gamma_{s}$ from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$
decays using $1.0\mbox{\,fb}^{-1}$ [139]. (Right) HFAG 2012 combination of
$\phi_{s}$ and $\Delta\Gamma_{s}$ results, where the $1\,\sigma$ confidence
region is shown for each experiment and the combined result [44]. Note the
different scales.

LHCb has also studied the decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{+}\pi^{-}$. This decay process is expected to proceed dominantly
via $b\rightarrow c\overline{}cs$ (the $s\bar{s}$ produced in the decay
rescatters to $\pi^{+}\pi^{-}$ through either a resonance such as $f_{0}(980)$
or a nonresonant process). Therefore, these events can be used to measure
$\phi_{s}$. The $\pi^{+}\pi^{-}$ mass range 775–1550
$\mathrm{\,Me\kern-1.00006ptV}$ shown in Fig. 11 (left) is used for the
measurement. In contrast to
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$, no
angular analysis is needed to disentangle the $C\\!P$ eigenstates, since the
final state is determined to be dominantly $C\\!P$-odd in this mass range
[234]. On the other hand, $\Delta\Gamma_{s}$ cannot be determined in this
decay channel alone.292929 The effective lifetime of
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,f_{0}(980)$
is sensitive to $\Delta\Gamma_{s}$ and $C\\!P$ violation parameters [235] and
has been measured by LHCb [236].  Using as input the value of
$\Delta\Gamma_{s}$ obtained from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$, the
measurement from the analysis of
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{+}\pi^{-}$ with $1.0\mbox{\,fb}^{-1}$ is [219]

$\phi_{s}=-0.019\,^{+0.173}_{-0.174}\,{}^{+0.004}_{-0.003}\,{\rm rad}\,.$ (32)

Figure 11 (right) shows the log-likelihood scan for the $\phi_{s}$ parameter
for the $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{+}\pi^{-}$ analysis. The latest HFAG average in Table 1 combines
the LHCb results with the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$
analysis results from CDF using $9.6\mbox{\,fb}^{-1}$ [237] and D0 using
$8.0\mbox{\,fb}^{-1}$ [238]. The LHCb result dominates the combination, which
is in good agreement with the SM predictions, as seen in Fig. 10
(right).303030 Results from ATLAS and CMS, presented at ICHEP2012 or later,
are not included in this compilation.

Figure 11: (Left) $\pi^{+}\pi^{-}$ mass distribution of selected
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{+}\pi^{-}$ candidates and range used for the $\phi_{s}$
measurement. (Right) log-likelihood difference as a function $\phi_{s}$ [219].

The LHCb $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\phi$ and $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{+}\pi^{-}$ analyses discussed above only used opposite side
flavour tagging [239, 240]. Future updates of these analyses will gain in
sensitivity by also using the same side kaon tagging information, which so far
has been used in a preliminary determination of $\Delta m_{s}$ [226, 241].
Currently, the systematic uncertainty on $\phi_{s}$ is dominated by imperfect
knowledge of the background, angular acceptance effects and by neglecting
potential contributions of direct $C\\!P$ violation. All of these
uncertainties are expected to be reduced with more detailed understanding and
some improvements in the analysis. Therefore it is expected that the
determination of $\phi_{s}$ will remain limited by statistical uncertainties,
even with the data samples available after the upgrade of the LHCb detector.
In addition to $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\phi$ and $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{+}\pi^{-}$, other $b\rightarrow c\overline{}cs$ decay modes of
$B^{0}_{s}$ mesons, such as ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\eta^{\prime}$ [242]
and $D_{s}^{+}D_{s}^{-}$ [243] will be investigated. These decays have been
measured at LHCb [244, 245].

The SM prediction $\phi_{s}=-0.036\pm 0.002$ rad could receive a small
correction from doubly CKM-suppressed penguin contributions in the decay. The
value of this correction is not precisely known, and may depend on the decay
mode. Moreover, NP in the $b\rightarrow c\overline{}cs$ decay may also affect
the results. Although such effects are already constrained by results from
$B^{+}$ and $B^{0}$ decays, NP in the decay amplitudes can lead to
polarisation-dependent mixing-induced $C\\!P$ asymmetries and triple product
asymmetries in $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\phi$ [246]. Such effects will be searched for in future analyses.

The flavour-specific asymmetries provide important complementary constraints
on $\Delta B=2$ processes. The D0 collaboration has performed a direct
measurement of $a^{s}_{\rm sl}$ in semileptonic $B^{0}_{s}$ decays [227],
which is only weakly constraining.313131 An updated measurement has been
presented by D0 at ICHEP 2012 [247].  However, a measurement of the inclusive
same-sign dimuon asymmetry provides better precision, and shows evidence of a
large deviation from its SM prediction [159]. The inclusive measurement is
sensitive to a linear combination of the flavour-specific asymmetries, $A_{\rm
SL}^{b}=C_{d}\,a_{\rm sl}^{d}+C_{s}\,a_{\rm sl}^{s}$, where $C_{q}$ depend on
the production fractions and mixing probabilities, and are determined to be
$C_{d}=0.594\pm 0.022$, $C_{s}=0.406\pm 0.022$ [159].323232 The factors
$C_{d}$ and $C_{s}$ depend in principle on the collision environment and the
kinematic acceptance, though the dependence appears to be weak. Trigger
requirements can also affect the values of these parameters.  As discussed in
Sec. 3.2.3, the D0 $A_{\rm SL}^{b}$ result is in tension with other $\Delta
B=2$ observables. Improved measurements of $a^{s}_{\rm sl}$ and $a^{d}_{\rm
sl}$ from LHCb are needed to solve this puzzle.

In LHCb, $a^{s}_{\rm sl}$ can be determined from the asymmetry between the
time-integrated untagged decay rates of $B^{0}_{s}$ decays to
$D_{s}^{+}\mu^{-}X$ and $D_{s}^{-}\mu^{+}X$, with
$D^{\pm}_{s}\rightarrow\phi\pi^{\pm}$, $\phi\rightarrow K^{+}K^{-}$ (or with
the full $D^{\pm}_{s}\rightarrow K^{+}K^{-}\pi^{\pm}$ Dalitz plot). Detector-
and trigger-induced asymmetries can be calibrated in control channels, and the
fact that data is taken with both magnet dipole polarities can be used as a
handle to reduce systematic uncertainties. The effect of $B^{0}_{s}$
production asymmetry is cancelled due to the fast oscillation, so the
asymmetry in the yields of $D^{+}_{s}\mu^{-}X$ and $D^{-}_{s}\mu^{+}X$ decays
is trivially related to $a^{s}_{\rm sl}$. A first preliminary LHCb result on
$a^{s}_{\rm sl}$, based on $1.0\mbox{\,fb}^{-1}$, has been reported at ICHEP
2012, and is the most precise measurement of this quantity to date [248],

$a^{s}_{\rm sl}=\left(-0.24\pm 0.54\pm 0.33\right)\,\%\,.$ (33)

It will also be possible to measure $a^{d}_{\rm sl}$ using $D^{+}\mu^{-}X$
final states with $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$. In this case extra
care must be taken to calibrate the difference between $K^{+}$ and $K^{-}$
detection efficiencies and an independent measurement of the $B^{0}$
production asymmetry is needed as input. Moreover, the $C\\!P$-symmetric
background from charged $B$ decays is significant and must be accurately
subtracted.

In the $B^{0}$ system, $\Delta m_{d}$ and $\sin\phi_{d}$ (i.e. $\sin 2\beta$)
have been measured precisely by the $B$ factories [44]. The measurements of
$\Delta\Gamma_{d}$ and $a^{d}_{\rm sl}$ are consistent with their SM
predictions, but their uncertainties are at least an order of magnitude larger
than those of the predictions. Hence a large improvement in precision is
needed to test the SM using these observables. In the $B^{0}$ sector there has
been for some time a tension between the measurements of $\sin 2\beta$ [44]
and the branching ratio ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$ [249, 250], as
shown in Fig. 12,333333 An updated measurement of ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu)$ using the hadronic tag method was presented
by Belle at ICHEP 2012 [251]: this new result reduces, but does not completely
remove, the tension in the fits. The analyses discussed here do not include
this new result.  and discussed in Sec. 3.2.4. This motivates improved
measurements of $\sin 2\beta$ and improved understanding of the possible
effects of penguin contributions to this observable.

Figure 12: Comparison of direct and indirect determinations of
$\sin\phi_{d}\equiv\sin 2\beta$ vs. ${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$,
from Ref. [252].

LHCb has already presented first results on $\Delta m_{d}$ [229, 253] and
$\sin 2\beta$ [254]. The $\Delta m_{d}$ result is the world’s most precise
single measurement of this quantity, while the sensitivity on $\sin 2\beta$
will be competitive with the $B$ factory results using the data sample that
will be collected by the end of 2012. LHCb can also search for enhancements in
the value of $\Delta\Gamma_{d}$ above the tiny value expected in the SM, e.g.
by comparing the effective lifetimes of
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ [255].
Significantly improving the precisions of the $B^{0}$ mixing observables is an
important goal of the LHCb upgrade, as will be discussed in Sec. 3.2.6.

The SM predictions of $b$-hadron lifetimes and $\Delta\Gamma_{q}$ are all
obtained within the framework of the heavy quark expansion. LHCb is actively
working on measurements of $b$-hadron lifetimes and lifetime ratios, which
will be used to test these predictions. The knowledge obtained from this work
will allow to improve the SM predictions of $\Delta\Gamma_{q}$ for the purpose
of searching for NP. Furthermore, a more precise measurement of the ratio of
$B^{0}_{s}$ to $B^{0}$ lifetimes could either support or strongly constrain
the existence of NP in $\Gamma^{s}_{12}$ [152, 153, 220, 216, 256].

####  3.2.3 Model independent constraints on new physics in $B$ mixing

Neutral $B_{q}$ meson mixing is described in terms of the three parameters
$|M_{12}^{q}|$, $|\Gamma_{12}^{q}|$ and
$\phi_{q}=\mbox{arg}(-M_{12}^{q}/\Gamma_{12}^{q})$ for each of the two systems
$q=d,s$. In the context of model-independent analyses, the NP contributions
can be parametrised in the form of two complex quantities $\Delta_{q}$ and
${\Lambda}_{q}$ [153, 257]

$\displaystyle M_{12}^{q}$ $\displaystyle=M_{12}^{q,{\rm
SM}}\,|\Delta_{q}|\,e^{i\phi_{q}^{\Delta}}\,,$ $\displaystyle\Gamma_{12}^{q}$
$\displaystyle=\Gamma_{12}^{q,{\rm
SM}}\,|{\Lambda}_{q}|\,e^{i\phi_{q}^{{\Lambda}}}\,,$ (34)

i.e., 4 real degrees of freedom. The observables which depend on these
parameters are the mass and decay width differences and flavour-specific
$C\\!P$-asymmetries. They can be expressed in terms of the SM predictions and
NP parameters as

$\displaystyle\Delta m_{q}$ $\displaystyle=(\Delta m_{q})_{\rm SM}\hskip
1.42262pt|\Delta_{q}|\,,$ $\displaystyle\Delta\Gamma_{q}$
$\displaystyle=(\Delta\Gamma_{q})_{\rm SM}\hskip
0.7113pt|{\Lambda}_{q}|\,\frac{\cos\left(\phi_{12}^{q,{\rm
SM}}+\phi_{q}^{\Delta}-\phi_{q}^{\Lambda}\right)}{\cos\phi_{12}^{q,{\rm
SM}}},$ (35) $\displaystyle a_{\rm sl}^{q}$ $\displaystyle=(a_{\rm
sl}^{q})_{\rm
SM}\,\frac{|{\Lambda}_{q}|}{|\Delta_{q}|}\;\frac{\sin\left(\phi_{12}^{q,{\rm
SM}}+\phi_{q}^{\Delta}-\phi_{q}^{\Lambda}\right)}{\sin\phi_{12}^{q,{\rm
SM}}}\,,$ (36)

up to corrections suppressed by tiny $(\Gamma_{12}^{q}/M_{12}^{q})^{2}$. Note
that the expressions of Eqs. (35) and (36) depend only on the difference
$(\phi_{q}^{\Delta}-\phi_{q}^{\Lambda})$. The SM predictions of $\Delta
m_{q}$, $\Delta\Gamma_{q}$ and $a_{\rm sl}^{q}$ can be found in Table 1 and
for $\phi^{q}_{12}$ [220]

$\displaystyle\phi_{12}^{d,{\rm SM}}$ $\displaystyle=(-0.075\pm
0.024)\rm\,rad\,,$ $\displaystyle\phi_{12}^{s,{\rm SM}}$
$\displaystyle=(0.0038\pm 0.0010)\rm\,rad.$ (37)

The values of $\Delta m_{q}$ have been precisely measured, giving rather
strong constraints on $|\Delta_{q}|$ which are limited by the knowledge of
hadronic matrix elements. The new $\Delta\Gamma_{s}$ measurement of LHCb
starts to provide useful constraints. As discussed above, the
$C\\!P$-asymmetries $a_{\rm sl}^{q}$ are currently rather weakly constrained.

Further information can be extracted from the mixing-induced
$C\\!P$-asymmetries in $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ decays

$\displaystyle\phi_{d}$ $\displaystyle=2\beta+\phi_{d}^{\Delta}-\delta_{d},$
$\displaystyle\phi_{s}$
$\displaystyle=-2\beta_{s}+\phi_{s}^{\Delta}-\delta_{s},$ (38)

where $\delta_{d}$ and $\delta_{s}$ denote shifts of $\phi_{d}$ and $\phi_{s}$
induced by either SM penguin diagrams or NP contributions in the decay
process. In the SM $\phi_{d}$ and $\phi_{s}$ are related to the angles $\beta$
and $\beta_{s}$ of the according unitarity triangles. When short-distance NP
contributions are introduced, $\phi_{q}$ depends on the phase
$\phi_{q}^{\Delta}$ of $M_{12}^{q}$, whereas the phase $\phi_{q}^{\Lambda}$ of
$\Gamma_{12}^{q}$ does not enter. The SM penguin pollution to $\delta_{q}$ is
expected to be negligible for the current precision of $\phi_{q}$, and is
discussed in detail in Sec. 3.2.5. Beyond the SM, NP can contribute to
$\delta_{q}$ in principle in both the tree $b\rightarrow c\overline{}cs$ decay
and the penguin process. However, in the model-independent analysis described
here, NP contributions in the $b\rightarrow c\overline{}cs$ decay are
neglected and any observed deviation from the SM will be interpreted as
effects of NP in neutral $B$ meson mixing. When $\delta_{q}$ is neglected,
Eqs. (35), (36) and (38) allow to determine the NP parameters $|\Delta_{q}|$,
$\phi_{q}^{\Delta}$, $|\Lambda_{q}|$ and $\phi_{q}^{\Lambda}$.

The assumption of NP in $M_{12}^{q}$ only, or equivalently in $\Delta B=2$
processes only, implies that there is no NP in $\Delta B=1$ processes which
contribute to the absorptive part $\Gamma_{12}^{q}$. Consequently, NP can only
decrease $\Delta\Gamma_{q}$ (since $\cos(\phi_{12}^{q,{\rm SM}})$ is maximal,
see Eq. (35)) with respect to the SM [258, 231]. This scenario has been
studied in extensions of the CKM fit of the SM which includes $\Delta B=2$
measurements to constrain the CKM elements $V_{tq}$ [259, 256], in combination
with many other flavour-changing processes. Including LHCb measurements [139,
229]343434 But not including results shown for the first time at ICHEP 2012 or
later.  the SM point $\Delta_{d}=\Delta_{s}=1$ is disfavoured by $2.4\,\sigma$
[256] (prior to the LHCb results being available, a similar analysis gave a
discrepancy of $3.6\,\sigma$ driven mainly by the anomalous dimuon asymmetry
[259]). The analysis gives $\Delta_{s}$ consistent with the SM, within large
uncertainties, whereas the more precise data in the $B^{0}$ system hint at a
deviation in $\Delta_{d}$ (see Fig. 13). Moreover, NP effects up to $30$–$40$
% are still allowed in both systems at the $3\,\sigma$ level. It should be
noted, that the large deviations in the $B^{0}$ sector are not only due to
$A_{\rm SL}^{b}$, but also due to the tension between $\sin\phi_{d}$ and
${\cal B}(B^{+}\rightarrow\tau^{+}\nu)$.

Figure 13: Model-independent fit [256] in the scenario that NP affects
$M^{q}_{12}$ separately. The coloured areas represent regions with C.L.
$<68.3\,\%$ for the individual constraints. The red area shows the region with
C.L. $<68.3\,\%$ for the combined fit, with the two additional contours
delimiting the regions with C.L. $<95.45\,\%$ and C.L. $<99.73\,\%$.

NP contributions to the absorptive part $\Gamma_{12}^{q}$ of $B$ mixing can
enter through $\Delta B=1$ decays $b\rightarrow qX$ with light degrees of
freedom $X$ of total mass below $m_{B}$. In some particular models such
contributions can arise [154, 260] and interfere constructively or
destructively with the SM contribution. The recent measurements of
$\Delta\Gamma_{q}$ and of $A_{\rm SL}^{b}$ revived interest in this
possibility. Model-independent analyses have confirmed that the $A_{\rm
SL}^{b}$ measurement cannot be accommodated within the SM [261, 262]. A model-
independent fit assuming NP in both $M_{12}^{q}$ and $\Gamma_{12}^{q}$ has
been considered in the framework of an extended CKM fit [256]. In this case,
the experimental data can be accommodated, and the $B^{0}_{s}$ system remains
rather SM-like, but large NP contributions in the $B^{0}$ system are required.

Model-independent analyses based on Eq. (34) are restricted to a particular
set of observables, mainly those with $\Delta B=2$, since correlations with
$\Delta B=1$ measurements are difficult to quantify. Either additional
assumptions on the nature of $X$ in $b\rightarrow qX$ or explicit NP models
will permit better exploitation of the wealth of future experimental
information. In fact, such analyses have found it difficult to accommodate the
hypothesis of large NP in $\Gamma^{q}_{12}$ with current $\Delta B=1$
measurements, therefore NP in $\Gamma^{q}_{12}$ seems unlikely to provide a
full explanation of the measured value of $A_{\rm SL}^{b}$. In the case of
$X=f\bar{f}$, the $\Delta B=1$ operators $b\rightarrow(d,s)f\bar{f}$ ($f=q$ or
$\ell$) are strongly constrained [152], with the exception of $b\rightarrow
sc\overline{}c$ and $b\rightarrow s\tau^{+}\tau^{-}$. Currently, only a weak
upper bound on ${\cal B}(B^{+}\rightarrow K^{+}\tau^{+}\tau^{-}){~{}\raise
1.49994pt\hbox{$<$}\kern-8.50006pt\lower 3.50006pt\hbox{$\sim$}~{}}3.3\cdot
10^{-3}$ at 90 % C.L. [263] exists whereas other decays
$B^{0}_{s}\rightarrow\tau^{+}\tau^{-}$, $B\rightarrow X_{s}\tau^{+}\tau^{-}$
might be indirectly constrained with additional assumptions (see also the
discussion in Sec. 2.5.2). As an example, the improved LHCb measurement of
$\tau_{B^{0}_{s}}/\tau_{B^{0}}$ allowed the derivation of a stronger bound on
${\cal B}(B^{0}_{s}\rightarrow\tau^{+}\tau^{-})$. Still, a model-independent
analysis of the complete set of $b\rightarrow s\tau^{+}\tau^{-}$ operators
does not allow for deviations larger than 35 % from the SM in
$\Gamma_{12}^{s}$ [153], which is much too small to resolve the tension with
$A_{\rm SL}^{b}$. For $b\rightarrow d\tau^{+}\tau^{-}$ operators there exists
a stronger constraint ${\cal B}(B^{0}\rightarrow\tau^{+}\tau^{-}){~{}\raise
1.49994pt\hbox{$<$}\kern-8.50006pt\lower 3.50006pt\hbox{$\sim$}~{}}4\cdot
10^{-3}$ and even smaller NP effects are expected in $\Gamma_{12}^{d}$. Other
proposed solutions such as the existence of new light spin-0 [264] or spin-$1$
[265] $X$ states could be seriously challenged by improved measurements of
quantities, such as ratios of lifetimes, which are theoretically under good
control [220].

In summary, NP contributions to $|\Delta_{q}|$ are already quite constrained
due to $\Delta m_{q}$ measurements and theoretical progress is required in
order to advance. Although the phases $\phi_{q}^{\Delta}$ are constrained by
the recent LHCb measurement of $\phi_{s}$, and $B$ factory measurements of
$\phi_{d}$, there is a mild tension with the SM in model-independent fits of
$\Delta B=2$ measurements [256, 261, 262, 153], especially when allowing for
NP in $\Gamma_{12}^{q}$. On the other hand, NP effects in $\Gamma_{12}^{q}$
are expected to be limited when constraints from $\Delta B=1$ observables are
taken into account. Independent improved measurements of $a_{\rm sl}^{q}$ are
needed in order to resolve the nature of the current discrepancies between the
$\Delta B=2$ observables with their SM expectations and other observables
entering global CKM fits. Further, improved measurements of $\Gamma_{q}$ and
$\Delta\Gamma_{q}$, as well as of control channels, are needed to constrain NP
in $\Gamma^{q}_{12}$.

####  3.2.4 CKM unitarity fits in SM and beyond

This section presents the results of the unitarity triangle (UT) analysis
performed by two groups: UTfit [266] and CKMfitter [252].353535 Similar
approaches have been developed in Ref. [267, 268].  The main aim of the UT
analysis is the determination of the values of the CKM parameters, by
comparing experimental measurements and theoretical predictions for several
observables. The popular Wolfenstein parametrisation allows for a transparent
expansion of the CKM matrix in terms of the sine of the small Cabibbo angle,
$\lambda$, with the other three parameters being $A$, $\bar{\rho}$ and
$\bar{\eta}$. Assuming the validity of the SM, one can perform a fit to the
available measurements. LHCb results already make important contributions to
the constraints on $\gamma$ and $\Delta m_{s}$. With more statistics, LHCb
results are expected to impact on other CKM fit inputs, including $\alpha$ and
$\sin 2\beta$. It is important to note the crucial role of lattice QCD
calculations as input to the CKM fits. For example, the parameters
$f_{B_{s}}\sqrt{B_{B_{s}}}$ and $\xi$ enter the constraints on $\Delta m_{s}$
and $\Delta m_{d}/\Delta m_{s}$. At the end of 2011, the precision of the
calculations was at the level of $5.4\,\%$ and $2.6\,\%$, respectively [109].
The necessary further progress to obtain the full benefit of the LHCb
measurements appears to be in hand exploiting algorithmic advances as well as
ever increasing computing power for the lattice calculations.

The overall quality of the fit can be judged using the projection of the
likelihoods on the $\\{\bar{\rho},\bar{\eta}\\}$ plane. This projection is
shown in Fig. 14. The fit can also be made removing one of the inputs, giving
a prediction for the removed parameter, which then can be compared to the
experimental value. The results of this study are presented in Table 2. Both
groups find a tension between ${\cal B}(B\rightarrow\tau\nu)$ and $\sin
2\beta$, as can be seen in Fig. 12. (As discussed in Sec. 3.2.2 this tension
will be reduced once the latest Belle result on ${\cal
B}(B^{+}\rightarrow\tau^{+}\nu_{\tau})$ [251] is included in the fits.)
Improved measurements of $\sin 2\beta$ can shed further light on this problem.

Figure 14: Result of the UT fit within the SM: $\\{\bar{\rho},\bar{\eta}\\}$
plane obtained by (left) UTfit [266] and (right) CKMfitter [252]. The 95 %
probability regions selected by the single constraints are also shown with
various colours for the different constraints.  Table 2: Predictions for some
parameters of the SM fit and their measurements as combined by the UTfit and
CKMfitter groups. Note that the two groups use different input values for some
parameters. The lines marked with (*) are not used in the full fit. Details of
the pull calculation can be found in Refs. [269, 259]. New results presented
at ICHEP2012 and later are not included in these analyses.

Parameter | UTfit | CKMfitter
---|---|---
prediction | measurement | pull | prediction | measurement | pull
$\alpha\,(^{\circ})$ | $87.5\pm 3.8$ | $91.4\pm 6.1$ | $+0.5\,\sigma$ | $95.9\,^{+2.2}_{-5.6}$ | $88.7\,^{+2.2}_{-5.9}$ | $-1.0\,\sigma$
$\sin 2\beta$ | $0.809\pm 0.046$ | $0.667\pm 0.024$ | $-2.7\,\sigma$ | $0.820\,^{+0.024}_{-0.028}$ | $0.679\pm 0.020$ | $-2.6\,\sigma$
$\gamma\,(^{\circ})$ | $67.8\pm 3.2$ | $75.5\pm 10.5$ | $+0.7\,\sigma$ | $67.2\,^{+4.4}_{-4.6}$ | $66\,^{+12}_{-12}$ | $-0.1\,\sigma$
$V_{ub}\,(10^{-3})$ | $3.62\pm 0.14$ | $3.82\pm 0.56$ | $+0.3\,\sigma$ | $3.55\,^{+0.15}_{-0.14}$ | $3.92\pm 0.09\pm 0.45$ | $0.0\,\sigma$
$V_{cb}\,(10^{-3})$ | $42.26\pm 0.89$ | $41\pm 1$ | $-0.9\,\sigma$ | $41.3\,^{+0.28}_{-0.11}$ | $40.89\pm 0.38\pm 0.59$ | $0.0\,\sigma$
$\varepsilon_{k}\,(10^{-3})$ | $1.96\pm 0.20$ | $2.229\pm 0.010$ | $+1.3\,\sigma$ | $2.02\,^{+0.53}_{-0.52}$ | $2.229\pm 0.010$ | $0.0\,\sigma$
$\Delta m_{s}\,({\rm\,ps^{-1}})$ | $18.0\pm 1.3$ | $17.69\pm 0.08$ | $-0.2\,\sigma$ | $17.0\,^{+2.1}_{-1.5}$ | $17.731\pm 0.045$ | $0.0\,\sigma$
${\cal B}(B\rightarrow\tau\nu)\,(10^{-4})$ | $0.821\pm 0.0077$ | $1.67\pm 0.34$ | $+2.5\,\sigma$ | $0.733\,^{+0.121}_{-0.073}$ | $1.68\pm 0.31$ | $+2.8\,\sigma$
$\beta_{s}\,\rm\,rad$ (*) | $0.01876\pm 0.0008$ |  |  | $0.01822\,^{+0.00082}_{-0.00080}$ |  |
${\cal B}(B^{0}_{s}\rightarrow\mu\mu)\,(10^{-9})$ (*) | $3.47\pm 0.27$ |  |  | $3.64\,^{+0.21}_{-0.32}$ |  |

In order to estimate the origin of the tensions, the UTfit and CKMfitter
groups have performed analyses including model-independent NP contributions to
neutral meson mixing processes (see Refs. [270, 256] for details). The NP
effects are introduced through the real valued $C$ and $\phi$ parameters
($A_{\rm NP}=Ce^{i\phi}A_{\rm SM}$) in case of UTfit and the complex valued
$\Delta$ parameter ($A_{\rm NP}=\Delta A_{\rm SM}$) for CKMfitter. The
parameters are added separately for the $B^{0}_{s}$ and $B^{0}$ sectors. In
the absence of NP, the expected values are $C=1$, $\phi=0^{\circ}$, and
$\Delta=1$. For the $B^{0}$ sector the fits return $C=0.94\pm 0.14$ and
$\phi=(-3.6\pm 3.7)^{\circ}$, and
$\Delta=(0.823\,^{+0.143}_{-0.095})+i(-0.199\,^{+0.062}_{-0.048})$. The
results for both groups show some disagreement with the SM, driven by tensions
in the input parameters mentioned above. In the $B^{0}_{s}$ sector, on the
other hand, the situation is much closer to the SM than before the LHCb
measurements were available: $C=1.02\pm 0.10$ and $\phi=(-1.1\pm
2.2)^{\circ}$, and $\Delta=(0.92\,^{+0.13}_{-0.08})+i(0.00\pm 0.10)$.

The results of the studies by both groups point to the absence of big NP
effects in $\Delta B=2$ processes. Nevertheless there is still significant
room for NP in mixing in both $B^{0}$ and $B^{0}_{s}$ systems. More precise
results, in particular from LHCb, can enable more careful studies. Besides
providing null tests of the SM hypothesis, improved $\phi_{s}$ and $a_{\rm
sl}^{s}$ measurements are crucial to quantity effects of NP in mixing. In
addition a precise $\gamma$ determination is essential, not only for a SM
global consistency test, but also to fix the apex of the UT in the extended
fits.

####  3.2.5 Penguin pollution in $b\rightarrow c\overline{c}s$ decays

In addition to the very clear experimental signature, precise determination of
the $B^{0}$ and $B^{0}_{s}$ mixing phases is possible due to the fact that in
the “golden modes”, $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$,
explicit calculation of the relevant matrix elements can be avoided, once
subleading doubly Cabibbo-suppressed and loop-suppressed terms are assumed to
vanish [271]. Estimates yield corrections of the order $O(10^{-3})$ only [272,
273, 274]; it is however notoriously difficult to actually calculate the
relevant matrix elements, and non-perturbative enhancements cannot be
excluded. Given the future experimental precision for these and related modes,
a critical reconsideration of this assumption is mandatory.

The main problem lies in the fact that once the assumption of negligible
penguin contributions is dropped, the evaluation of hadronic matrix elements
again becomes necessary, which still does not seem feasible to an acceptable
precision for the decays in question. To avoid explicit calculation, symmetry
relations can be used, exploiting either flavour SU(3) or U-spin symmetry
[275, 276, 277, 278, 279, 280, 281]. Without taking into account any QCD
evaluation and only using control channels to estimate the size of the penguin
amplitude, the analyses in Refs. [281, 278] still allow a phase shift of up to
a few degrees for $\phi_{d}$, which would correspond to a very large non-
perturbative enhancement of the penguin size. In Ref. [278] a negative sign is
preferred which (slightly) reduces the tension in the unitarity triangle fit
shown in Fig. 12. The reason for the large allowed range of the shift of
$\phi_{d}$ is due to the limited precision to which the corresponding control
channels $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\pi^{0}$ and
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,K^{0}$,
which are Cabibbo-suppressed compared to the golden modes, are known. For
$\phi_{s}$, an analogous analysis [277] cannot yet constrain the penguin
contribution, due to the lack of a
$B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,V$ control channel
data for $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\phi$. However, in principle the effects in the
$B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,V$ modes are
expected to be of the same order of magnitude as in the
$B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,P$ modes. The
control channel $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,K^{*0}$ has already been observed at CDF [282] and LHCb [283], and
work is ongoing to measure its decay rate, polarisations and direct $C\\!P$
asymmetries. This will enable the first direct constraint on the shift of
$\phi_{s}$ due to penguin contributions in the decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\phi$.363636 Other data-driven methods to control penguin
contributions to $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,\phi$ have been proposed [284].  For
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,f_{0}(980)$
there is an additional complication due to the unknown hadronic structure of
the $f_{0}(980)$ [235].

In addition to insufficient data, there are, at present, theoretical aspects
limiting the precision of this method at present, the most important of which
is the violation of SU(3) symmetry. Regarding the $B^{0}$ mixing phase, a full
SU(3) analysis can be performed [285] (instead of using only one control
channel) to be able to model-independently include SU(3) breaking. The
inclusion of SU(3)-breaking contributions is important: their neglect can lead
to an overestimation of the subleading effects. Including recent data for two
of the relevant modes [286, 287], the analysis shows that the data are at the
moment actually compatible with vanishing penguin contributions, with
SU(3)-breaking contributions of the order $20\,\%$. Including the penguin
contributions, an upper limit on the shift of the mixing-induced $C\\!P$
asymmetry $\Delta S=\sin\phi_{d}-\sin 2\beta$ is derived: $|\Delta S|\lesssim
0.01$, with a negative sign for $\Delta S$ slightly preferred.373737 Note the
definition of $\Delta S$ here has a sign difference to that in Ref. [285].
This is the most stringent limit available, despite the more general treatment
of SU(3) breaking. In this analysis still some (conservatively chosen)
theoretical inputs are needed to exclude fine-tuned solutions: SU(3)-breaking
effects have been restricted to at most $40\,\%$ for a few parameters which
are not well determined by the fit and also have only small influence on the
$C\\!P$ violation observables, and the penguin matrix elements are constrained
to be at most $50\,\%$ of the leading contributions. Importantly, these theory
inputs can be replaced by experimental measurements, namely of the $C\\!P$
asymmetries in the decay
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,K^{0}_{\rm\scriptscriptstyle S}$, the decay rate of which has already
been measured at LHCb [287] after its observation at CDF [282]. Furthermore,
data from all the corresponding modes (i.e.
$B_{d,u,s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,P$, with
light pseudoscalar meson $P=\pi,\eta^{(\prime)}$ or $K$) can be used to
determine the shift more precisely, i.e. the related uncertainty is not
irreducible, but can be reduced with coming data.

Turning to the second golden mode,
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$, in
general, the absolute shift is not expected to be larger than in the $B^{0}$
case. At the moment the data are not yet available to make a comparable
analysis. While the penguin decay mode $B^{0}_{s}\rightarrow\phi\phi$ is not
related by symmetry with
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$,
comparing their decay rates indicates that the penguin contributions are
small, and there are no huge enhancements to be expected for the penguin
matrix elements in question.

Nonetheless, a quantitative analysis will ultimately be warranted here as
well. In principle, these methods can be adapted to extract the $B^{0}_{s}$
mixing phase including penguin contributions and model-independent SU(3)
breaking, thereby improving the method proposed in Ref. [277]. The
corresponding partners of the golden mode
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$ are
all the decays $B_{u,d,s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,V$, with the light vector mesons $V=K^{*}$, $\rho$, $\phi$ or
$\omega$. However, the complete analysis requires results on the polarisation
fractions and $C\\!P$ asymmetries for each of these final states, and for some
of them the experimental signature is quite challenging. In addition, the
$\phi$ meson is a superposition of octet and singlet, therefore the “control
channels” involving $K^{*}$ and $\rho$ are not as simply related as in the
case with a pseudoscalar meson, but require the usage of nonet symmetry, whose
precision has to be investigated in turn.

Nevertheless, significant progress can be expected. Several
$B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,V$ modes, including
$B^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
[283], are being studied at LHCb. While measurements of the modes involving
$b\rightarrow d$ transitions are expected to exhibit rather large
uncertainties at first, the advantage of the proposed method is the long
“lever arm” due to the relative enhancement $\sim 1/\lambda^{2}$ in the
control channels, so that even moderate precision will be very helpful.

####  3.2.6 Future prospects with LHCb upgrade

Current measurements of $\phi_{s}$ carried out by LHCb in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\pi^{+}\pi^{-}$ final states
show no deviation from the SM prediction within uncertainties [219, 139],
putting strong constraints on NP in $B^{0}_{s}$ mixing, as discussed in Sec.
3.2.3. Table 3 shows the current results with $1.0\mbox{\,fb}^{-1}$ and the
projected precision for $50\mbox{\,fb}^{-1}$ with the upgraded detector. A
precision of $<10\rm\,mrad$ is expected for $50\mbox{\,fb}^{-1}$ with the
upgraded detector. It is expected that even with this data sample, the main
limitation will be statistical: the largest systematic uncertainties on the
current measurement (background description, angular acceptance, effect of
fixed physics parameters) [139] are expected to be removed with more
sophisticated analyses or to scale with statistics. Thus changes as small as a
factor of two with respect to the SM should be observable with $3\,\sigma$
significance. This precision will make it possible either to measure a
significant deviation from the SM prediction or otherwise to place severe
constraints on NP scenarios.

Table 3: LHCb measurements of $\phi_{s}$. The quoted uncertainties are statistical and systematic, respectively.  Final State | Current value (rad) with $1.0\mbox{\,fb}^{-1}$ | Projected uncertainty ($50\mbox{\,fb}^{-1}$)
---|---|---
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ | $-0.001\pm 0.101\pm 0.027$ | $0.008$
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ | $-0.019\,^{+0.173}_{-0.174}\,{}^{+0.004}_{-0.003}$ | $0.014$
Both | $-0.002\pm 0.083\pm 0.027$ | $0.007$

As discussed in Sec. 3.2.5, contributions from doubly CKM-suppressed SM
penguin diagrams could have a non-negligible effect on the mixing-induced
$C\\!P$ asymmetry and bias the extracted value of $\phi_{s}$. Naive estimates
of the bias are of the order $O(10^{-3})$ only [272, 273, 274], but this must
be examined with experimental data using flavour symmetries to exploit control
channels. LHCb can perform an SU(3) analysis using measurements of the decays
rates and $C\\!P$ asymmetries in
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$,
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rho^{0}$ and
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ as control
channels for $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\phi$. The necessary high precision can only be reached using the large
data sample that will be collected with the upgraded LHCb detector. The
$50\mbox{\,fb}^{-1}$ data sample will also allow to measure $\phi_{s}$ in the
penguin-free ($b\rightarrow c\bar{u}s/u\bar{c}s$) $B^{0}_{s}\rightarrow D\phi$
decay [288, 289].

Another important goal is a more precise determination of $\sin 2\beta$ in the
$B^{0}$ system, motivated by the tension between the direct and indirect
determinations of $\sin 2\beta$ seen by both UTfit and CKMfitter groups, as
shown in Table 2. With the upgraded detector, using the
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,K^{0}_{\rm\scriptscriptstyle S}$ final state alone, a statistical
precision of $\pm 0.006$ is expected, to be compared to the current error from
the $B$ factories of $\pm 0.023$ [190]. Given experience with the current
detector it seems feasible to control the systematic uncertainties to a
similar level. Such precision, together with better control of the penguin
pollution, will allow us to pin down any NP effects in $B^{0}$ mixing. In
addition, the penguin-free ($b\rightarrow c\bar{u}d/u\bar{c}d$)
$B^{0}\rightarrow D\rho^{0}$ channel can be used to get another handle on
$\sin 2\beta$ [290, 291].

The importance of improved measurements of $\Delta\Gamma_{q}$ has been
emphasised in Sections 3.2.1–3.2.3. LHCb has made a preliminary measurement of
$\Delta\Gamma_{s}$ in
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$ using
a $1.0\mbox{\,fb}^{-1}$ data sample [139]. The effective lifetime of
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,f_{0}(980)$
[292] has also been measured [236]. Based on this, the statistical precision
on $\Delta\Gamma_{s}$ with $50\mbox{\,fb}^{-1}$ is projected to be $\sim
0.003{\rm\,ps^{-1}}$. It is hoped that the systematic uncertainty can be
controlled to the same level.

A measurement of $\Delta\Gamma_{d}$ is of interest as any result larger than
the tiny value expected in the SM would clearly signal NP [154, 293, 255]. To
determine this quantity, LHCb will compare the effective lifetimes of the two
decay modes $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\,K^{0}_{\rm\scriptscriptstyle S}$ with
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,K^{*0}$. The
estimated precision for $1.0\mbox{\,fb}^{-1}$ is $\sim 0.02{\rm\,ps^{-1}}$.
With the upgraded detector and $50\mbox{\,fb}^{-1}$ a statistical precision of
$\sim 0.002{\rm\,ps^{-1}}$ on $\Delta\Gamma_{d}$ can be achieved. The
systematic uncertainty is under study.

The LHCb upgrade will also have sufficient statistics to make novel tests of
$C\\!PT$ symmetry. Any observation of $C\\!PT$ violation indicates physics
beyond the SM. An example of a unique test in the $B^{0}$ system uses
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,K^{0}$ and its
charge-conjugate decay, where the $K^{0}$ decays semileptonically [294, 295,
296]. This measurement involves looking at four separate decay paths that
interfere. While several tests can be performed, one particular observable is
the asymmetry $A_{bk}$, that can be measured without the need of flavour
tagging, where

$A_{bk}=\frac{\Gamma(B^{0}+\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}[\pi^{-}\mu^{+}\nu]-\Gamma(B^{0}+\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}[\pi^{+}\mu^{-}\overline{\nu}])}{\Gamma(B^{0}+\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}[\pi^{-}\mu^{+}\nu]+\Gamma(B^{0}+\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}[\pi^{+}\mu^{-}\overline{\nu}])}\,.$ (39)

In terms of the $C\\!PT$ violation parameter $\theta^{\prime}$, the kaon decay
time $t_{K}$, the $B^{0}$ decay time $t_{B}$, the $B^{0}$ mass difference
$\Delta m_{d}$ and $C\\!P$-violating phase $2\beta$, and kaon decay widths
$\Gamma^{K}_{S}$ and $\Gamma^{K}_{L}$, this can be expressed

$A_{bk}={\rm
Re}(\theta^{\prime})\frac{2e^{-\frac{1}{2}\left(\Gamma^{K}_{S}+\Gamma^{K}_{L}\right)t_{K}}\sin
2\beta\left(1-\cos\Delta
m_{d}t_{B}\right)}{e^{-\Gamma^{K}_{S}t_{K}}+e^{-\Gamma^{K}_{L}t_{K}}}\,.$ (40)

A signature of $C\\!PT$ violation would be a $1-\cos\Delta m_{B}t_{B}$
dependence of the decay rate after integrating over kaon decay times. Roughly
5000 such decays can be expected with the upgrade. It is possible to detect
these decays with low background level, even with the missing neutrino, using
the measured $B^{0}$ direction, the detected
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ four-momentum, and the kaon
decay vertex. Other methods to test $C\\!PT$ symmetry (e.g. Ref. [297]) are
also under investigation.

###  3.3 $C\\!P$ violation measurements with hadronic $b\rightarrow s$
penguins

####  3.3.1 Probes for new physics in penguin-only $b\rightarrow sq\bar{q}$
decays

The presence of physics beyond the SM can be detected by looking for its
contribution to $b\rightarrow sq\overline{}q$ ($q=s,\,d$) decays,383838 Decays
mediated primarily by $b\rightarrow su\bar{u}$ transitions are discussed in
Secs. 3.4.4 and 3.4.5.  which in the SM can only proceed via FCNC loop
diagrams that are dynamically suppressed. These decays provide a rich set of
observables that are rather precisely known in the SM but could potentially
receive sizeable corrections from new heavy particles appearing in the loop.

  * •

Direct $C\\!P$ asymmetries. In the SM $b\rightarrow sq\overline{}q$ decays are
dominated by the penguin diagram with an internal top quark. As a consequence,
the direct $C\\!P$ asymmetry is expected to be small. If there is a NP
amplitude with comparable size interfering with the SM amplitude, and it has
different strong and weak phases than the SM amplitude, a much larger direct
$C\\!P$ asymmetry can arise.

  * •

Polarisation and triple product asymmetries. For $B$ decays into two vector
mesons $V_{1}$ and $V_{2}$, followed by vector to two pseudoscalar decays
$V_{1}\rightarrow P_{1}P^{\prime}_{1}$ and $V_{2}\rightarrow
P_{2}P^{\prime}_{2}$, there are three transversity states, labelled
“longitudinal” ($0$), “perpendicular” ($\perp$) and “parallel” ($\parallel$).
Measurements of the fractions of the total decay rate in each of these states,
which correspond to determinations of the polarisation in the final state,
provide useful information about the chiral structure of the electroweak
currents, as well about non-perturbative effects such as rescattering and
penguin annihilation. In the SM, the decay to each transversity state is
dominated by a single amplitude with magnitude $|A_{j}|$, weak phase
$\Phi_{j}$ and strong phase $\delta_{j}$. The $C\\!P$-violating observables
${\rm Im}(A_{\perp}A^{*}_{j}-{\bar{A}}_{\perp}{\bar{A}}^{*}_{j})$ are then

${\rm
Im}(A_{\perp}A^{*}_{j}-{\bar{A}}_{\perp}{\bar{A}}^{*}_{j})=2|A_{\perp}||A_{j}|\cos(\delta_{\perp}-\delta_{j})\sin(\Phi_{\perp}-\Phi_{j}),\,\,j=0,\parallel\,.$
(41)

The values of these observables are tiny since in the SM the weak phases are
the same to a very good approximation, but ${\rm
Im}(A_{\perp}A^{*}_{j}-{\bar{A}}_{\perp}{\bar{A}}^{*}_{j})$ can significantly
differ from zero if there is a sizeable $C\\!P$-violating NP contribution in
the loop.

These observables can be extracted from the differential distributions in
terms of the angles $\theta_{1}$, $\theta_{2}$ and $\phi$, where $\theta_{1}$
($\theta_{2}$) is the polar angle of $P_{1}$ ($P_{2}$) in the rest frame of
$V_{1}$ ($V_{2}$) with respect to the opposite of the direction of motion of
the $B$ meson, and $\phi$ is the angle between the decay planes of
$V_{1}\rightarrow P_{1}P^{\prime}_{1}$ and $V_{2}\rightarrow
P_{2}P^{\prime}_{2}$ in the rest frame of the $B$ meson. The two observables
can also be related to two triple product asymmetries for $C\\!P$-averaged
decays393939 The triple product asymmetries in $B^{0}_{s}\rightarrow\phi\phi$
and $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay could in principle also
receive contribution from non-zero mixing-induced $C\\!P$ asymmetries arising
from NP in $B^{0}_{s}$ mixing. However, this contribution is suppressed by
$\Delta\Gamma_{s}/\Gamma_{s}$ and is already highly constrained.  which are
equal to asymmetries between the number of events with positive and negative
values of $U=\sin 2\phi$ and $V={\rm
sign}(\cos\theta_{1}\cos\theta_{2})\sin\phi$:

${\rm
Im}(A_{\perp}A^{*}_{\parallel}-{\bar{A}}_{\perp}{\bar{A}}^{*}_{\parallel})\propto
A_{U}=\frac{N(U>0)-N(U<0)}{N(U>0)+N(U<0)}\,,$ (42) ${\rm
Im}(A_{\perp}A^{*}_{0}-{\bar{A}}_{\perp}{\bar{A}}^{*}_{0})\propto
A_{V}=\frac{N(V>0)-N(V<0)}{N(V>0)+N(V<0)}\,.$ (43)

A review of this subject can be found in Ref. [298] and references therein.

  * •

Mixing-induced $C\\!P$ asymmetries. Mixing-induced $C\\!P$ asymmetries in
$b\rightarrow sq\overline{}q$ decays of neutral $B$ to $C\\!P$ eigenstates are
precisely predicted. Due to the fact that the penguin diagram with an internal
top quark is expected to dominate, the values of $2\beta^{\rm eff}$ determined
using $B^{0}\rightarrow\phi K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}\rightarrow\eta^{\prime}K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}\rightarrow f_{0}(980)K^{0}_{\rm\scriptscriptstyle S}$, etc., are all
expected to give $\approx 2\beta$ (see, e.g. Refs. [299, 300] and the
discussion in Ref. [44]). Similarly, the values of $2\beta_{s}^{\rm eff}$
determined from $B^{0}_{s}\rightarrow\phi\phi$, $B^{0}_{s}\rightarrow
K^{*0}{\bar{K}}^{*0}$, etc., are expected to vanish due to cancellation of
weak phases between mixing (top box) and decay (top penguin) amplitudes.
Higher order corrections from subleading diagrams are expected to be small
compared to the precision that can be achieved in the near-term, but further
theoretical studies will be needed as the upgrade era approaches. NP with a
flavour structure different from the SM will alter these $C\\!P$ asymmetries
through the decay amplitudes, even if there is no NP in $B$ mixing. A number
of quasi-two-body or three-body decay modes can be studied.

  * •

Correlations between direct and mixing-induced asymmetries. Penguin-only decay
modes are particularly interesting as the difference between formal “tree” and
“penguin” contributions boils down to a difference in the quark-flavour
running in the loop of the penguins. This difference, dominated by short
distances, can be assessed accurately using QCD factorisation, and it can be
used to correlate the branching ratio and the $C\\!P$ asymmetries of penguin-
mediated modes. As discussed in Refs. [301, 302, 138], these observables can
be correlated not only within the SM, but can also be used to extract the
$B^{0}_{s}$ mixing phase even in the presence of NP affecting only this phase.

####  3.3.2 Current status and outlook of LHCb measurements

LHCb published the first observation and measurement of the branching ratio
and polarisation amplitudes in the $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay mode [303] using
$35~{}\mbox{\,pb}^{-1}$ of data collected in 2010. A clean mass peak
corresponding to $50\pm 8$ $B^{0}_{s}\rightarrow(K^{+}\pi^{-})(K^{+}\pi^{+})$
decays is seen (Fig. 15 (left)), mostly from resonant $B^{0}_{s}\rightarrow
K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays. Using this
signal the longitudinal polarisation amplitude is measured to be
$f_{L}=0.31\pm 0.12({\rm stat})\pm 0.04({\rm syst})$ and the branching ratio
to be ${\cal B}(B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=(2.81\pm 0.46({\rm stat})\pm
0.45({\rm syst}))\times 10^{-5}$.

Figure 15: (Left) fit of the $K^{+}\pi^{-}K^{-}\pi^{+}$ mass distribution for
$B^{0}_{s}\rightarrow K^{*0}\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$
candidates from $35\mbox{\,pb}^{-1}$ [303]; (right) fit of the
$K^{+}K^{-}K^{-}K^{+}$ mass distribution for $B^{0}_{s}\rightarrow\phi\phi$
candidates from $1.0\mbox{\,fb}^{-1}$ [304].

LHCb also published the measurement of the polarisation amplitudes and triple
product asymmetries in $B^{0}_{s}\rightarrow\phi\phi$ [304] using the 2011
data set of $1.0\mbox{\,fb}^{-1}$. In this data set $801\pm 29$ events are
observed with excellent signal-to-background ratio (see Fig. 15 (right)). The
polarisation amplitudes are measured to be

$\begin{array}[]{lcl}|A_{0}|^{2}&=&0.365\pm 0.022\,({\rm stat})\pm
0.012\,({\rm syst})\,,\\\ |A_{\perp}|^{2}&=&0.291\pm 0.024\,({\rm stat})\pm
0.010\,({\rm syst})\,,\\\ |A_{\parallel}|^{2}&=&0.344\pm 0.024\,({\rm
stat})\pm 0.014\,({\rm syst})\,,\end{array}$ (44)

where the sum of the square of the amplitudes is constrained to unity. The
triple product asymmetries in this mode are measured to be

$\begin{array}[]{lcl}A_{U}&=&-0.055\pm 0.036\,({\rm stat})\pm 0.018\,({\rm
syst})\,,\\\ A_{V}&=&\phantom{-}0.010\pm 0.036\,({\rm stat})\pm 0.018\,({\rm
syst})\,.\end{array}$ (45)

The results of this analysis are in agreement with, and more precise than, the
previous measurement [305], and are also consistent with the SM.

First measurements of $C\\!P$ asymmetries in these modes from time-dependent
flavour-tagged angular analyses are expected to follow. With high statistics,
it will be possible to measure polarisation-dependent direct and mixing-
induced $C\\!P$ asymmetries, but for the first analysis it will be more
convenient to determine a single complex observable common to all
polarisations (as done for
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$)

$\lambda=\eta_{j}\frac{q}{p}\frac{\bar{A}_{j}}{A_{j}}$ (46)

where $j$ denotes one of the three transversity states, which are also $C\\!P$
eigenstates with eigenvalues $\eta_{j}$, and $A_{j}$ ($\bar{A}_{j}$) is the
decay amplitude of $B^{0}_{s}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$) to the corresponding state.
With this approximation it will be possible to determine the magnitude
$|\lambda|$ and phase $\phi^{\rm eff}_{s}\equiv-\arg(\lambda)$. The SM
expectation is $|\lambda|\approx 1$ and $\phi^{\rm eff}_{s}\approx 0$ due to
the dominance of the top-quark loop, and any observed deviation from these
expectations would be a signature of NP. Since NP in $B^{0}_{s}$ mixing is
already constrained by measurement of $\phi_{s}$ from
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\,\phi$, the
main interest in these $b\rightarrow s$ penguin modes is to look for NP in the
decay processes. Based on simulation studies, a sensitivity on $\phi^{\rm
eff}_{s}$ of 0.3–0.4 radians with $1.0\mbox{\,fb}^{-1}$ is expected for both
$B^{0}_{s}\rightarrow\phi\phi$ and $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$.

####  3.3.3 Future prospects with LHCb upgrade

The latest results on mixing-induced $C\\!P$ violation in $b\rightarrow s$
transitions show no significant deviation from the SM, as seen in Fig. 16,
which compares the mixing-induced $C\\!P$ violation parameter $\sin
2\beta^{\rm eff}$ measured in penguin-dominated $b\rightarrow s$ decays with
the value of $\sin 2\beta$ measured in the tree-dominated $b\rightarrow
c\overline{}cs$ decays. In the absence of NP these observables should only
differ by small amounts. Due to these results, large NP contributions in
$b\rightarrow sq\overline{}q$ decays are unlikely but further tests with
higher precision remain interesting. LHCb will be able to make competitive
measurements of $\sin 2\beta^{\rm eff}$ in $B^{0}\rightarrow\phi
K^{0}_{\rm\scriptscriptstyle S}$ and several other $b\rightarrow
sq\overline{}q$ decays, but a significant improvement in precision requires
the $50\mbox{\,fb}^{-1}$ of the LHCb upgrade. The improved trigger efficiency
in the LHCb upgrade is particularly important for these decays, which have
only hadrons in the final state. With the upgrade data sample, the statistical
error of $\sin 2\beta^{\rm eff}(B^{0}\rightarrow\phi
K^{0}_{\rm\scriptscriptstyle S})$ is estimated to be roughly 0.06, which is
still above the SM uncertainty of $\sim 0.02$ [306].

There are several more NP probes in $b\rightarrow sq\overline{}q$ decays that
can be exploited at LHCb and its upgrade, such as mixing-induced $C\\!P$
asymmetries and triple product asymmetries in both
$B^{0}_{s}\rightarrow\phi\phi$ and $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays. The statistical precision
of $\phi_{s}^{\rm eff}$ with each channel is estimated to be 0.3–0.4 rad for
$1.0\mbox{\,fb}^{-1}$. The projected precision for $50\mbox{\,fb}^{-1}$ is
about 0.03 rad each. This can be compared with the uncertainties of their SM
predictions of about 0.02 rad. It is also possible to perform a combined
analysis of $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and its U-spin related channel
$B^{0}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$,
which will put strong constraint on the subleading penguin diagrams in
$B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, thus further reducing the
theoretical uncertainty in the measurement of $\phi^{\rm eff}_{s}$ [307, 308].
The statistical precision of $A_{U}$ and $A_{V}$ is estimated to be about
0.004, compared with an upper bound of 0.02 on their possible sizes in the SM
[298].

Figure 16: HFAG compilation of results for $\sin 2\beta^{\rm eff}$ in
$b\rightarrow sq\overline{}q$ decays [44].

In summary, the LHCb upgrade will allow the exploitation of the full potential
of the NP probes in $b\rightarrow sq\overline{}q$ decays. Table 4 compares the
current and projected (LHCb upgrade, i.e. $50\mbox{\,fb}^{-1}$) precisions of
the key observables with the theory uncertainties of their SM predictions.

Table 4: Current and projected precisions of the key observables in
$b\rightarrow sq\overline{}q$ decays.

Observable | Current | LHCb upgrade ($50\mbox{\,fb}^{-1}$) | Theory uncertainty
---|---|---|---
$A_{U,\,V}(B^{0}_{s}\rightarrow\phi\phi)$ | 0.04 (LHCb $1.0\mbox{\,fb}^{-1}$) | 0.004 |  $0.02$ [309]
$\phi^{\rm eff}_{s}(B^{0}_{s}\rightarrow\phi\phi)$ | – | 0.03 |  $0.02$ [306]
$\phi^{\rm eff}_{s}(B^{0}_{s}\rightarrow K^{*0}\bar{K}^{*0})$ | – | $0.03$ |  $0.02$ [306]
$\sin 2\beta^{\rm eff}(B^{0}\rightarrow\phi K^{0}_{\rm\scriptscriptstyle S})$ |  $0.12$ ($B$ factories) | 0.06 |  $0.02$ [179]

###  3.4 Measurements of the CKM angle $\gamma$

####  3.4.1 Measurements of $\gamma$ using tree-mediated decays

The CKM angle $\gamma$, defined as the phase
$\gamma=\arg\left[-V_{ud}V_{ub}^{*}/(V_{cd}V_{cb}^{*})\right]$, is one of the
angles of the unitarity triangle formed from the hermitian product of the
first ($d$) and third ($b$) columns of the CKM matrix $V$. It is one of the
least well known parameters of the quark mixing matrix. However, since it can
be determined entirely through decays of the type $B\rightarrow DK$ 404040 By
$B\rightarrow DK$ all related tree-dominated decay processes are implicitly
included, including $B^{+}\rightarrow DK^{+}$, $B^{0}\rightarrow DK^{*0}$,
$B^{0}_{s}\rightarrow D\phi$, $B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\mp}$ and
$B^{0}\rightarrow D^{(*)\mp}\pi^{\pm}$. In these specific decay processes, the
notation $D$ refers to a neutral $D$ meson that is an admixture of $D^{0}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ states.  that involve only
tree amplitudes — an unusual, even unique, property amongst all $C\\!P$
violation parameters — it provides a benchmark measurement. The determination
from tree level decays has essentially negligible theoretical uncertainty, at
the level of $\delta\gamma/\gamma={\cal O}(10^{-6})$, as will be shown in the
next section. This makes $\gamma$ a very appealing “standard candle” of the
CKM sector. It serves as a reference point for comparison with $\gamma$ values
measured from loop decays (see Sec. 3.4.4).

Moreover, the determination of $\gamma$ is crucial to improve the precision of
the global CKM fits, and resulting limits on (or evidence for) NP
contributions (see Sec. 3.2.4). In particular, the measurement of $\Delta
m_{d}$ and the oscillation phase $\sin 2\beta$ in $B^{0}$–$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mixing can be converted to a
measurement of $\gamma$ (in the SM). This can be compared to the reference
value from $B\rightarrow DK$ — their consistency verifies that the Kobayashi-
Maskawa mechanism of $C\\!P$ violation is the dominant source in quark
flavour-changing processes. Existing measurements provide tests at the level
of ${\mathcal{O}}(10\,\%)$, but improving the precision to search for smaller
effects of NP is well motivated.

Several established methods to measure $\gamma$ in tree decays exploit the
$B^{-}\rightarrow D^{(*)}K^{(*)-}$ decays. They are based on the interference
between the $b\rightarrow u$ and $b\rightarrow c$ tree amplitudes, which
arises when the neutral $D$ meson is reconstructed in a final state accessible
to both $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays.
The interference between the amplitudes results in observables that depend on
their relative weak phase $\gamma$. Besides $\gamma$ they also depend on
hadronic parameters, namely the ratio of magnitudes of amplitudes
$r_{B}\equiv\left|A(b\rightarrow u)/A(b\rightarrow c)\right|$ and the relative
strong phase $\delta_{B}$ between the two amplitudes. These hadronic
parameters depend on the $B$ decay under investigation. They can not be
precisely calculated from theory (see, however, Ref. [310]), but can be
extracted directly from data by simultaneously reconstructing several
different $D$ final states.

The various methods differ by the $D^{(*)}$ final state that is used. The
three main categories of $D$ decays considered so far by the $B$ factories
BaBar and Belle, and by CDF, are:

  * •

$C\\!P$ eigenstates (the GLW method [311, 312]),

  * •

doubly Cabibbo-suppressed (DCS) decays (the ADS method [313, 314]),

  * •

three-body, self-conjugate final states (the GGSZ or “Dalitz” method [315]).

An additional category has not been possible to pursue at previous experiments
due to limited event sample sizes:

  * •

singly Cabibbo-suppressed (SCS) decays (the GLS method [316]).

In practise, except for the case of two-body decays, there is often no clear
distinction between the different methods.

The best sensitivity to $\gamma$ obviously comes from combining the results of
all different analyses. This not only improves the precision on $\gamma$, but
provides additional constraints on the hadronic parameters. It also allows one
to overcome the fact that $C\\!P$-odd final states such as
$K^{0}_{\rm\scriptscriptstyle S}\pi^{0}$ are not easily accessible in LHCb’s
hadronic environment.

A brief review of the main ideas of the different methods follows. The
amplitudes of the $B^{-}\rightarrow D^{0}K^{-}$ and $B^{-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}$ processes are written as:

$\displaystyle A(B^{-}\rightarrow D^{0}K^{-})=$ $\displaystyle
A_{c}e^{i\delta_{c}}\,,~{}~{}~{}~{}$ $\displaystyle A(D^{0}\rightarrow
f)=A_{f}e^{i\delta_{f}}\,,$ (47) $\displaystyle A(B^{-}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-})=$ $\displaystyle
A_{u}e^{i(\delta_{u}-\gamma)}\,,$ $\displaystyle
A(D^{0}\rightarrow\bar{f})=A_{\bar{f}}e^{i\delta_{\bar{f}}}\,,$

where $A_{c}$, $A_{u}$, $A_{f}$ and $A_{\bar{f}}$ are real and positive (and
$C\\!P$ violation in $D^{0}$ decays has been neglected). The subscripts $c$
and $u$ refer to the $b\rightarrow c$ and $b\rightarrow u$ transitions,
respectively. The amplitudes for the $D^{0}$ decay can generally include the
case where the $D^{0}$ decays to a three-body final state. In this case,
$A_{f}$, $A_{\bar{f}}$, $\delta_{f}$ and $\delta_{\bar{f}}$ are functions of
the Dalitz plot coordinates. The amplitude of the process $B^{-}\rightarrow
D[\rightarrow f]K^{-}$ can be written, neglecting $D^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing, as

$A(B^{-}\rightarrow D[\rightarrow
f]K^{-})=A_{c}A_{f}e^{i(\delta_{c}+\delta_{f})}+A_{u}A_{\bar{f}}e^{i(\delta_{u}+\delta_{\bar{f}}-\gamma)}\,,$
(48)

and the rate is given by

$\displaystyle\Gamma(B^{-}\rightarrow D[\rightarrow f]K^{-})$
$\displaystyle\propto$ $\displaystyle
A_{c}^{2}A_{f}^{2}+A_{u}^{2}A_{\bar{f}}^{2}+2A_{c}A_{f}A_{u}A_{\bar{f}}{\rm
Re}(e^{i(\delta_{B}+\delta_{D}-\gamma)})$ (49) $\displaystyle\propto$
$\displaystyle
A_{c}^{2}\left(A_{f}^{2}+r_{B}^{2}A_{\bar{f}}^{2}+2r_{B}A_{f}A_{\bar{f}}{\rm
Re}(e^{i(\delta_{B}+\delta_{D}-\gamma)})\right)\,,$

where $r_{B}=A_{u}/A_{c}$, $\delta_{B}=\delta_{u}-\delta_{c}$ and
$\delta_{D}=\delta_{\bar{f}}-\delta_{f}$. The rate for the charge-conjugated
mode (still neglecting $C\\!P$ violation in $D^{0}$ decays) is obtained by
exchanging $\gamma\rightarrow-\gamma$. Taking into account CKM factors and, in
the case of charged $B$ decays, colour suppression of the $b\rightarrow u$
amplitude, $r_{B}$ is expected to be around 0.1 for $B^{-}$ decays and around
0.3 for $B^{0}$ decays. From Eq. (49) all the relevant formulae of the GLW,
ADS and GGSZ methods can be derived.

In the GLW analysis, the neutral $D$ mesons are selected in $C\\!P$
eigenstates $f_{C\\!P\pm}$ such as $D\rightarrow K^{-}K^{+}$ ($C\\!P=+1$) or
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{0}$ ($C\\!P=-1$). Thus
$A_{f}/A_{\bar{f}}=1$ and $\delta_{D}=0,\pi$ for $C\\!P=\pm 1$. Eq. (49)
becomes:

$\Gamma(B^{-}\rightarrow D[\rightarrow f_{C\\!P\pm}]K^{-})\propto
A_{c}^{2}(1+r_{B}^{2}\pm 2r_{B}\cos(\delta_{B}-\gamma))\,.$ (50)

The $B^{-}\rightarrow DK^{-}$ decays, where the $D$ decays to Cabibbo-favoured
(CF) final states (e.g. $D^{0}\rightarrow K^{-}\pi^{+}$) can be used to
normalise the rates in order to construct observables that minimise the
systematic uncertainties. For those decays, to a good approximation,

$\Gamma(B^{-}\rightarrow D[\rightarrow
K^{-}\pi^{+}]K^{-})=\Gamma(B^{+}\rightarrow D[\rightarrow
K^{+}\pi^{-}]K^{+})\propto A_{c}^{2}~{}.$ (51)

From Eqs. (50) and (51) and their $C\\!P$ conjugates the usual GLW observables
follow:

$\displaystyle R_{C\\!P\pm}$ $\displaystyle=\frac{2[\Gamma(B^{-}\rightarrow
D_{C\\!P\pm}K^{-})+\Gamma(B^{+}\rightarrow
D_{C\\!P\pm}K^{+})]}{\Gamma(B^{-}\rightarrow
D^{0}K^{-})+\Gamma(B^{+}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+})}$ (52) $\displaystyle
A_{C\\!P\pm}$ $\displaystyle=\frac{\Gamma(B^{-}\rightarrow
D_{C\\!P\pm}K^{-})-\Gamma(B^{+}\rightarrow
D_{C\\!P\pm}K^{+})}{\Gamma(B^{-}\rightarrow
D_{C\\!P\pm}K^{-})+\Gamma(B^{+}\rightarrow D_{C\\!P\pm}K^{+})}\,.$ (53)

Eqs. (52) and (53) provide a set of four observables that are connected to the
three unknowns $\gamma$, $r_{B}$ and $\delta_{B}$ through

$\displaystyle R_{C\\!P\pm}$ $\displaystyle=1+r_{B}^{2}\pm
2r_{B}\cos\delta_{B}\cos\gamma$ (54) $\displaystyle A_{C\\!P\pm}$
$\displaystyle=\frac{\pm 2r_{B}\sin\delta_{B}\sin\gamma}{R_{C\\!P\pm}}\,.$
(55)

However, only three of these equations are independent since, from Eq. (55),
$R_{C\\!P+}A_{C\\!P+}=-R_{C\\!P-}A_{C\\!P-}$. Analogous relations hold for
$B\rightarrow D^{*}_{C\\!P}K$ and $B\rightarrow D_{C\\!P}K^{*}$ decays, with
different values of the hadronic parameters characterising the $B$ decay.
However, in the $B\rightarrow D^{*}_{C\\!P}K$ case one has to take into
account a $C\\!P$ flip due to the different charge conjugation quantum numbers
of the $\pi^{0}$ and the photon from the $D^{*}$ decay [317]:
$D^{*}_{C\\!P\pm}\rightarrow D_{C\\!P\pm}\pi^{0}$, but
$D^{*}_{C\\!P\pm}\rightarrow D_{C\\!P\mp}\gamma$. For analysis of
$B\rightarrow D_{C\\!P}K^{*}$ the finite width of the $K^{*}$ resonance must
be taken into account [318]. There are related important consequences for the
ADS and GGSZ analyses of $B\rightarrow D^{*}K$ and $B\rightarrow DK^{*}$
decays.

In the ADS analysis, the neutral $D$ mesons are selected in CF and DCS decays,
such as $D^{0}\rightarrow K^{-}\pi^{+}$ and $D^{0}\rightarrow\pi^{-}K^{+}$,
respectively. The $B$ decay rate is the result of the interference of the
colour allowed $B^{-}\rightarrow D^{0}K^{-}$ decay followed by the DCS
$D^{0}\rightarrow\pi^{-}K^{+}$ decay and the colour suppressed
$B^{-}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{-}$ decay
followed by the CF $D^{0}\rightarrow K^{-}\pi^{+}$ decay. As a consequence,
the interfering amplitudes are of similar magnitude and hence large
interference effects can occur. From Eq. (49) one finds

$\Gamma(B^{\mp}\rightarrow D[\rightarrow K^{\pm}\pi^{\mp}]K^{\mp})\propto
r_{B}^{2}+r_{D}^{2}\pm 2r_{B}r_{D}\cos(\delta_{B}+\delta_{D}\mp\gamma)$ (56)

where both
$r_{D}=A_{f}/A_{\bar{f}}=|A(D^{0}\rightarrow\pi^{-}K^{+})/A(D^{0}\rightarrow
K^{-}\pi^{+})|$ and the phase difference $\delta_{D}$ are measured in charm
decays. The value of $\delta_{D}$ can be determined directly using data
collected from $e^{+}e^{-}$ collisions at the $\psi(3770)$ resonance, as has
been done by CLEO [319, 320], but the most precise value comes from a global
fit including charm mixing parameters. The results provided by HFAG [44] from
a combination with $C\\!P$ violation in charm allowed are $r_{D}=0.0575\pm
0.0007$, $\delta_{D}=\left(202\,^{+10}_{-11}\right)^{\circ}$. Defining $R_{\rm
ADS}$ and $A_{\rm ADS}$ as

$\displaystyle R_{\rm ADS}=\frac{\Gamma(B^{-}\rightarrow
D[\rightarrow\pi^{-}K^{+}]K^{-})+\Gamma(B^{+}\rightarrow
D[\rightarrow\pi^{+}K^{-}]K^{+})}{\Gamma(B^{-}\rightarrow D[\rightarrow
K^{-}\pi^{+}]K^{-})+\Gamma(B^{+}\rightarrow D[\rightarrow
K^{+}\pi^{-}]K^{+})}\,,$ (57) $\displaystyle A_{\rm
ADS}=\frac{\Gamma(B^{-}\rightarrow
D[\rightarrow\pi^{-}K^{+}]K^{-})-\Gamma(B^{+}\rightarrow
D[\rightarrow\pi^{+}K^{-}]K^{+})}{\Gamma(B^{-}\rightarrow
D[\rightarrow\pi^{-}K^{+}]K^{-})+\Gamma(B^{+}\rightarrow
D[\rightarrow\pi^{+}K^{-}]K^{+})}\,,$ (58)

and using Eqs. (51) and (56) gives

$\displaystyle R_{\rm
ADS}=r_{B}^{2}+r_{D}^{2}+2r_{B}\,r_{D}\cos\gamma\cos(\delta_{B}+\delta_{D})\,,$
(59) $\displaystyle A_{\rm
ADS}=2r_{B}\,r_{D}\sin\gamma\sin(\delta_{B}+\delta_{D})/R_{\rm ADS}\,.$ (60)

It has been noted that for the extraction of $\gamma$ it can be more
convenient to replace the pair of observables ${R_{\rm ADS},A_{\rm ADS}}$ with
a second pair, ${R_{+},R_{-}}$, defined as:

$R_{\pm}\equiv\frac{\Gamma(B^{\pm}\rightarrow[K^{\mp}\pi^{\pm}]_{D}K^{\pm})}{\Gamma(B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}K^{\pm})}=r_{B}^{2}+r_{D}^{2}+2r_{B}r_{D}\cos(\delta_{B}+\delta_{D}\pm\gamma)$
(61)

Unlike ${R_{\rm ADS},A_{\rm ADS}}$, the two quantities ${R_{+},R_{-}}$ are
statistically independent. The ADS decay chain
$B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}]_{D}K^{\pm}$ has been observed for the
first time by LHCb [6], confirming the evidence that had begun to accumulate
in previous measurements [321, 322, 323].

In the GGSZ analysis, the neutral $D$ mesons are selected in three-body self-
conjugate final states. The channel that has been used most to date is
$D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, though first
results have also been presented with $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$ and other channels are under
consideration. For concreteness, consider $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$, with
$A_{f}e^{i\delta_{f}}=f(m_{-}^{2},m_{+}^{2})$ and
$A_{\bar{f}}e^{i\delta_{\bar{f}}}=f(m_{+}^{2},m_{-}^{2})$, where $m_{-}^{2}$
and $m_{+}^{2}$ are the squared masses of the $K^{0}_{\rm\scriptscriptstyle
S}\pi^{-}$ and $K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ combinations. The rate
in Eq. (49) can be re-written as:

$\displaystyle\Gamma(B^{\mp}\rightarrow$ $\displaystyle D[\rightarrow
K^{0}_{\rm\scriptscriptstyle
S}\pi^{-}\pi^{+}]K^{\mp})\propto|f(m_{\mp}^{2},m_{\pm}^{2})|^{2}+r_{B}^{2}|f(m_{\pm}^{2},m_{\mp}^{2})|^{2}$
(62)
$\displaystyle+2r_{B}|f(m_{\mp}^{2},m_{\pm}^{2})||f(m_{\pm}^{2},m_{\mp}^{2})|\cos(\delta_{B}+\delta_{D}(m_{\mp}^{2},m_{\pm}^{2})\mp\gamma)\,,$

where $\delta_{D}(m_{\mp}^{2},m_{\pm}^{2})$ is the strong phase difference
between $f(m_{\pm}^{2},m_{\mp}^{2})$ and $f(m_{\mp}^{2},m_{\pm}^{2})$. Due to
the fact that $r_{B}$ is required to be positive, the direct extraction of
$r_{B}$, $\delta_{B}$ and $\gamma$ can be biased. To avoid these biases, the
“Cartesian coordinates” have been introduced [324]

$x_{\pm}={\rm Re}[r_{B}e^{i(\delta_{B}\pm\gamma)}]\,,\quad y_{\pm}={\rm
Im}[r_{B}e^{i(\delta_{B}\pm\gamma)}]\,,$ (63)

allowing Eq. (62) to be rewritten as

$\Gamma(B^{\mp}\rightarrow D[\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}]K^{\mp})\propto|f_{\mp}|^{2}+r_{B}^{2}|f_{\pm}|^{2}+2\left[x_{\mp}{\rm
Re}[f_{\mp}f_{\pm}^{*}]+y_{\mp}{\rm Im}[f_{\mp}f_{\pm}^{*}]\right]\,.$ (64)

Here the notation has been simplified using
$f_{\pm}=f(m_{\pm}^{2},m_{\mp}^{2})$. This Dalitz plot-based method can be
implemented in a model-dependent way by parametrising the amplitude as a
function of the Dalitz plot of the three-body state, or in a model-independent
way by dividing the Dalitz plot into bins and making use of external
measurements of the $D$ decay strong phase differences within these bins [315,
325, 326].414141 As for $\delta_{D}$ in the ADS method, the strong phase
differences can be determined directly from $\psi(3770)\rightarrow D\bar{D}$
data, which has been done by CLEO [327]. In future, it is expected that the
most precise value will come from a global fit including results of time-
dependent analyses of multibody charm decays.

Besides the established methods based on direct $C\\!P$ violation in
$B\rightarrow DK$ decays, it is also possible to measure $\gamma$ using time-
dependent analyses of neutral $B^{0}$ and $B^{0}_{s}$ tree decays [328, 329,
330]. The method still relies on the interference of $b\rightarrow u$ and
$b\rightarrow c$ amplitudes, but interference is achieved through $B^{0}$
($B^{0}_{s}$) mixing. Thus one measures the sum of $\gamma$ and the mixing
phase, namely $\gamma+2\beta$ and $\gamma-2\beta_{s}$ in the $B^{0}$ and
$B^{0}_{s}$ systems, respectively. Since both $\sin 2\beta$ and $\beta_{s}$
are becoming increasingly well measured, these measurements provide
sensitivity to $\gamma$.

Pioneering time-dependent measurements using the $B^{0}\rightarrow
D^{(*)\mp}\pi^{\pm}$ decays have been performed by both BaBar [331, 332] and
Belle [333, 334]. In these decays the amplitude ratios
$r_{D\pi}=|A(B^{0}\rightarrow D^{(*)+}\pi^{-})/A(B^{0}\rightarrow
D^{(*)-}\pi^{+})|$ are expected to be small, $r_{D\pi}\lesssim 0.02$, limiting
the sensitivity. In the decays $B^{0}_{s}\rightarrow D_{s}^{\mp}K^{\pm}$,
however, both $b\rightarrow c$ and $b\rightarrow u$ amplitudes are of same
order in the Wolfenstein parameter $\lambda$, $\mathcal{O}(\lambda^{3})$, so
that the interference effects are expected to be large. In addition, the decay
width difference in the $B^{0}_{s}$ system, $\Delta\Gamma_{s}$, is non-zero,
which adds sensitivity to the weak phase through the hyperbolic terms in the
time evolution (see also Ref. [335]). The time-dependent decay rates of the
initially produced flavour eigenstates are given by the decay equations

$\displaystyle\frac{d\Gamma_{B^{0}_{s}(\kern
1.25995pt\overline{\kern-1.25995ptB}{}^{0}_{s})\rightarrow
f}(t)}{dt\,e^{-\Gamma_{s}t}}$
$\displaystyle=\frac{1}{2}|A_{f}|^{2}(1+|\lambda_{f}|^{2})$
$\displaystyle\times\left[\cosh\left(\frac{\Delta\Gamma_{s}t}{2}\right)-D_{f}\sinh\left(\frac{\Delta\Gamma_{s}t}{2}\right)\pm\
C_{f}\cos\left(\Delta m_{s}t\right)\mp S_{f}\sin\left(\Delta
m_{s}t\right)\right]\,,$ (65)

where $\Gamma_{s}$, $\Delta\Gamma_{s}$, $\Delta m_{s}$ are the usual mixing
parameters of the $B^{0}_{s}$ system and $|q/p|=1$ has been assumed. The top
(bottom) of the $\pm$ and $\mp$ signs is used when the initial particle is
tagged as a $B^{0}_{s}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$) meson. In Eq. (65), $A_{f}$
is the decay amplitude for a $B^{0}_{s}$ meson to decay to a final state $f$,
and $\lambda_{f}=(q/p)(\overline{A}_{f}/A_{f})$ where $\overline{A}_{f}$ is
the amplitude for a $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ to
decay into $f$. Similar equations hold for the charge conjugate processes
replacing $A_{f}$ by $\overline{A}_{\bar{f}}$, $\lambda_{f}$ by
$\overline{\lambda}_{\bar{f}}=(p/q)(A_{\bar{f}}/\overline{A}_{\bar{f}})$, and
with a separate set of coefficients $C_{\bar{f}}$, $S_{\bar{f}}$ and
$D_{\bar{f}}$. As each decay is dominated by a single diagram,
$|\lambda_{f}|=|\overline{\lambda}_{\bar{f}}|$. The $C\\!P$ asymmetry
observables are then given by

$\displaystyle
C_{f}=C_{\bar{f}}=\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}}\,,\quad
S_{f}=\frac{2{\rm Im}(\lambda_{f})}{1+|\lambda_{f}|^{2}}\,,\quad
D_{f}=\frac{2{\rm Re}(\lambda_{f})}{1+|\lambda_{f}|^{2}}\,,$ $\displaystyle
S_{\bar{f}}=\frac{2{\rm
Im}(\overline{\lambda}_{\bar{f}})}{1+|\bar{\lambda}_{\bar{f}}|^{2}}\,,\quad
D_{\bar{f}}=\frac{2{\rm
Re}(\overline{\lambda}_{\bar{f}})}{1+|\bar{\lambda}_{\bar{f}}|^{2}}\,.$ (66)

The equality $C_{f}=C_{\bar{f}}$ results from $\left|q/p\right|=1$ and
$|\lambda_{f}|=|\overline{\lambda}_{\bar{f}}|$. The term $\lambda_{f}$ is
connected to the weak phase by

$\lambda_{f}=\Big{(}\frac{q}{p}\Big{)}\frac{\overline{A}_{f}}{A_{f}}=\Big{(}\frac{V_{tb}^{*}V_{ts}}{V_{tb}V_{ts}^{*}}\Big{)}\Big{(}\frac{V_{ub}V_{cs}^{*}}{V_{cb}^{*}V_{us}}\Big{)}\Big{|}\frac{A_{2}}{A_{1}}\Big{|}e^{i\Delta}=|\lambda_{f}|e^{i(\Delta-(\gamma-2\beta_{s}))}\,,$
(67)

where $|A_{2}/A_{1}|$ is the ratio of the hadronic amplitudes between
$B^{0}_{s}\rightarrow D^{-}_{s}K^{+}$ and $B^{0}_{s}\rightarrow
D^{+}_{s}K^{-}$, $\Delta$ is their strong phase difference, and
$\gamma-2\beta_{s}$ is the weak phase difference. An analogous relation exists
for $\overline{\lambda}_{\bar{f}}$,
$\overline{\lambda}_{\bar{f}}=|\lambda_{f}|e^{i(\Delta+(\gamma-2\beta_{s}))}$.
Thus one obtains five observables from Eq. (66) and solves for
$|\lambda_{f}|$, $\Delta$, and $(\gamma-2\beta_{s})$.

The LHCb experiment has the necessary decay time resolution, tagging power and
access to large enough signal yields to perform this time-dependent $C\\!P$
measurement.424242 Preliminary results have been presented at CKM 2012 [336].
The signal yields can be seen from the measurement of ${\cal
B}(B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm})$ [140] (see Sec. 3.4.3 below). The
identification of the initial flavour of the signal $B^{0}_{s}$ candidate can
be done combining both the responses of opposite-side and same-side kaon
tagging algorithms, as is planned for other measurements of mixing-induced
$C\\!P$-violation in $B^{0}_{s}$ decays, and has already been implemented in
the preliminary analysis of $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}$ decays
[226].

####  3.4.2 Theoretical cleanliness of $\gamma$ from $B\rightarrow DK$ decays

The answer to the question of why it is interesting to measure $\gamma$
precisely depends on the experimental precision that can be achieved. In the
era of LHCb, the main motivation is the theoretically clean measurement of the
SM CKM phase. The search for NP can thus be performed by comparing the
extracted value of $\gamma$ to other observables, for example in the CKM fit
(see Sec. 3.2.4). However, one can also cross-check for the presence of NP in
$B\rightarrow DK$ channels themselves. One way is to test that the values of
$\gamma$ determined from the many different $B\rightarrow DK$ type channels
all coincide. Another is automatically built in to the method for $\gamma$
extraction in the GGSZ analysis. Consider the case where the decay amplitudes
get modified by an extra contribution with a new strong phase
$\delta_{B}^{\prime}$ and a weak phase $\gamma^{\prime}$. Then instead of the
decay amplitudes in Eq. (48) one finds

$A(B^{\pm}\rightarrow f_{D}K^{\pm})\propto
1+r_{D}e^{i\delta_{D}}(r_{B}e^{i(\delta_{B}\pm\gamma)}+r_{B}^{\prime}e^{i(\delta_{B}^{\prime}\pm\gamma^{\prime})})\,.$
(68)

This means that for $B^{+}$ and $B^{-}$ decays the $r_{B}$ ratios are
different

$r_{B^{+}}\rightarrow|r_{B}e^{i(\delta_{B}+\gamma)}+r_{B}^{\prime}e^{i(\delta_{B}^{\prime}+\gamma^{\prime})}|\,,\quad
r_{B^{-}}\rightarrow|r_{B}e^{i(\delta_{B}-\gamma)}+r_{B}^{\prime}e^{i(\delta_{B}^{\prime}-\gamma^{\prime})}|\,.$
(69)

Discovering that $r_{B^{-}}\neq r_{B^{+}}$ would signal a $C\\!P$-violating NP
contribution to the $B\rightarrow DK$ amplitude. One signature of NP would
then be ${x_{+}^{2}+y_{+}^{2}}\neq{x_{-}^{2}+y_{-}^{2}}$, though it is also
possible that the equality could be satisfied even in the presence of NP: in
this case there can be a shift in the extracted value of $\gamma$.

Existing measurements place strong constraints on tree-level NP effects, yet
the possibility of discoveries in this sector in the near term is not ruled
out. In the far future, with much larger statistics, the measurement of
$\gamma$ is well suited to search for high scale NP since it is theoretically
very clean. For example, NP with contributions of different chirality could
give different shifts in $\gamma$, so the above test is meaningful.

A useful question to ask is, what is the energy scale that could be probed in
principle? To answer this, the irreducible theoretical uncertainty in the
determination of $\gamma$ must be estimated. There are several sources that
can induce a bias in the determination of $\gamma$ from $B\rightarrow DK$
decays. However, most of these can be avoided, either (i) with more statistics
(for example, the Dalitz plot model uncertainty where a switch to a model-
independent method is possible), or (ii) by modifying the equations used to
determine $\gamma$ (an example is to correct for effects of $D^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing [337, 338]). The remaining,
irreducible, theory uncertainties are then from the electroweak corrections.

Figure 17: A $B^{-}\rightarrow D^{0}K^{-}$ box diagram electroweak correction (left) with a different CKM structure than the leading weak decay amplitude (right).  Table 5: Ultimate NP scales that can be probed using different observables listed in the first column. They are given by saturating the theoretical errors given respectively by 1) $\delta\gamma/\gamma=10^{-6}$, 2) optimistically assuming no error on $f_{B}$, so that the ultimate theoretical error is only from electroweak corrections, 3) using SM predictions in Ref. [31], 4) optimistically assuming perturbative error estimates $\delta\beta/\beta~{}0.1\,\%$ [339], and 5) from bounds for ${\rm Re}\,C_{1}({\rm Im}\,C_{1})$ from UTfitter [270]. Probe |  $\Lambda_{\rm NP}$ for (N)MFV NP |  $\Lambda_{\rm NP}$ for gen. FV NP
---|---|---
$\gamma$ from $B\rightarrow DK^{1)}$ | $\Lambda\sim{\mathcal{O}}(10^{2}\mathrm{\,Te\kern-1.00006ptV})$ | $\Lambda\sim{\mathcal{O}}(10^{3}\mathrm{\,Te\kern-1.00006ptV})$
$B\rightarrow\tau\nu^{2)}$ | $\Lambda\sim{\mathcal{O}}(1\mathrm{\,Te\kern-1.00006ptV})$ | $\Lambda\sim{\mathcal{O}}(30\mathrm{\,Te\kern-1.00006ptV})$
$b\rightarrow ss\bar{d}^{3)}$ | $\Lambda\sim{\mathcal{O}}(1\mathrm{\,Te\kern-1.00006ptV})$ | $\Lambda\sim{\mathcal{O}}(10^{3}\mathrm{\,Te\kern-1.00006ptV})$
$\beta$ from $B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{0}_{\rm\scriptscriptstyle S}\,{}^{4)}$ | $\Lambda\sim{\mathcal{O}}(50\mathrm{\,Te\kern-1.00006ptV})$ | $\Lambda\sim{\mathcal{O}}(200\mathrm{\,Te\kern-1.00006ptV})$
$K-\bar{K}$ mixing5) |  $\Lambda>0.4\mathrm{\,Te\kern-1.00006ptV}$ ($6\mathrm{\,Te\kern-1.00006ptV}$) | $\Lambda>10^{3(4)}\mathrm{\,Te\kern-1.00006ptV}$

The challenge to determine this uncertainty is that the hadronic elements can
no longer be determined solely from the experiment. Not all electroweak
corrections matter though — the important ones are the corrections that change
the CKM structure. For instance, vertex corrections and $Z$ exchanges do not
affect $\gamma$, but corrections from box diagrams carry a different weak
phase. The dominant contribution is effectively due to $t$ and $b$ running in
the loop. For $b\rightarrow us\bar{c}$ transitions there is a tree level
contribution with $\sim V_{ub}V_{cs}^{*}$ CKM structure, while the box diagram
has $\sim(V_{tb}V_{ts}^{*})(V_{ub}V_{cb}^{*})$. Since this has the same weak
phase, it does not introduce a shift in $\gamma$. For $b\rightarrow cs\bar{u}$
transitions, on the other hand, the tree level is $\sim V_{cb}V_{us}^{*}$,
while the box diagram $\sim(V_{tb}V_{ts}^{*})(V_{cb}V_{ub}^{*})$, as
illustrated in Fig. 17. The two contributions have different weak phases,
which means that the shift $\delta\gamma$ is non-zero.

The size of this effect is estimated by integrating over both $t$ and $b$ at
the same time. The electroweak corrections in the effective theory are then
described by a local operator whose matrix elements are easier to estimate.
Although the Wilson coefficient of the operator contains large logarithms,
$\log(m_{b}/m_{W})$, for ${\mathcal{O}}(1)$ estimates, the precision obtained
without resummation is sufficient. If one resums $\log(m_{b}/m_{W})$ then
nonlocal contributions are also generated. As a rough estimate only the local
contributions need be kept. The irreducible theory error on $\gamma$ is
conservatively estimated to be $\delta\gamma/\gamma<{\mathcal{O}}(10^{-6})$
(most likely it is even $\delta\gamma/\gamma\lesssim{\mathcal{O}}(10^{-7})$).

This limit is far beyond the achievable sensitivity of any foreseeable
experiment. Nevertheless, it is interesting to consider what could be learnt
in case such small deviations could be observed. Assuming MFV one can probe
$\Lambda_{\rm NP}\sim 10^{2}\mathrm{\,Te\kern-1.00006ptV}$, while assuming
general flavour-violating (FV) NP one can probe $\Lambda_{\rm NP}\sim
10^{3}\mathrm{\,Te\kern-1.00006ptV}$ (where MFV and general FV NP scales are
defined as in Ref. [270]). This is by far the most precise potential probe of
MFV, as shown in Table 5, due to the small theoretical uncertainty.

Since an experimental precision of $\delta\gamma/\gamma\sim 10^{-6}$ is not
achievable in the near future, the NP scale reach must be adjusted for more
realistic data sets. This is easily done, since the scale $\Lambda_{\rm NP}$
probed goes as the fourth root of the yield. With the LHCb upgrade, an
uncertainty of $<1^{\circ}$ on $\gamma$ can be achieved (see Sec. 3.4.6), so
that NP scales approaching $\Lambda_{\rm NP}\sim
5(50)\mathrm{\,Te\kern-1.00006ptV}$ can be probed for MFV (general FV) NP.

####  3.4.3 Current LHCb experimental situation

First results from LHCb in this area include a measurement using
$B^{-}\rightarrow DK^{-}$ with the GLW and ADS final states [6].434343 Results
from preliminary GLW-type analyses using $B^{0}\rightarrow DK^{*0}$ [340] and
$B^{-}\rightarrow DK^{-}\pi^{+}\pi^{-}$ [341] have been reported at ICHEP
2012.  A measurement of the branching ratio of $B^{0}_{s}\rightarrow
D^{\mp}_{s}K^{\pm}$ has also been performed [140]. Several other analyses,
including studies of GGSZ-type final states, are in progress.444444 At CKM
2012, LHCb presented results of a model-independent GGSZ analysis of
$B^{-}\rightarrow DK^{-}$ with $D\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{+}\pi^{-}$ and $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$
[7], preliminary results from a ADS-type analysis of $B^{-}\rightarrow DK^{-}$
with $D\rightarrow K3\pi$ [342], a preliminary determination of $\gamma$ from
combined results using $B^{-}\rightarrow DK^{-}$ and $B^{-}\rightarrow
D\pi^{-}$ [343], and preliminary results on the time-dependent $C\\!P$
violation parameters in $B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$ [336].

These measurements all share common selection strategies. They benefit greatly
from boosted decision tree algorithms, which combine up to 20 kinematic
variables to effectively suppress combinatorial backgrounds. Charmless
backgrounds are suppressed by exploiting the large forward boost of the
$D_{(s)}^{+}$ meson through a cut on its flight distance.

In the GLW/ADS analysis [6] of $1.0\mbox{\,fb}^{-1}$ of
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$ data collected in 2011, the $C\\!P$
eigenstates $D\rightarrow K^{+}K^{-}$, $\pi^{+}\pi^{-}$, and the quasi-
flavour-specific $D\rightarrow\pi^{-}K^{+}$ decay are used. The $C\\!P$
asymmetries defined in Eq. (58), and the ratios $R_{\pm}$ defined in Eq. (61),
are measured for both the $B\rightarrow DK$ signal and the abundant
$B\rightarrow D\pi$ control channel. The latter has limited sensitivity to
$\gamma$ but provides a large control sample from which probability density
functions are shaped, and can be used to help reduce certain systematic
uncertainties. The control channel is also used to measure three ratios of
partial widths

$R_{K/\pi}^{f}=\frac{\Gamma(B^{-}\rightarrow[f]_{D}K^{-})+\Gamma(B^{+}\rightarrow[f]_{D}K^{+})}{\Gamma(B^{-}\rightarrow[f]_{D}\pi^{-})+\Gamma(B^{+}\rightarrow[f]_{D}\pi^{+})}\,,$
(70)

where $f$ represents $KK$, $\pi\pi$ and the favoured $K\pi$ mode. The signal
yields are estimated by a simultaneous fit to 16 independent subsamples,
defined by the charges ($\times 2$), the $D$ final states ($\times 4$), and
the $DK$ or $D\pi$ final state ($\times 2$). Figure 18 shows the projections
of the suppressed $\pi^{\pm}K^{\mp}$ subsamples. It is crucial to control the
cross feed of the abundant $B^{-}\rightarrow D\pi^{-}$ decays into the signal
decays. This is achieved using the two LHCb ring-imaging Cherenkov detectors
[344]. The systematic uncertainties are dominated by knowledge of the
intrinsic asymmetry of the detector in reconstruction of positive and negative
$B$ meson decays, and by the uncertainty on the particle identification
requirements. The results are

$\displaystyle R_{C\\!P+}$ $\displaystyle=\phantom{-}1.007\pm 0.038\pm
0.012\,,$ $\displaystyle A_{C\\!P+}$ $\displaystyle=\phantom{-}0.145\pm
0.032\pm 0.010\,,$ $\displaystyle R_{-}$ $\displaystyle=\phantom{-}0.0073\pm
0.0023\pm 0.0004\,,$ $\displaystyle R_{+}$ $\displaystyle=\phantom{-}0.0232\pm
0.0034\pm 0.0007\,,$

where the first error is statistical and the second systematic; $R_{C\\!P+}$
is computed from $R_{C\\!P+}\approx\langle
R_{K/\pi}^{KK},R_{K/\pi}^{\pi\pi}\rangle/R_{K/\pi}^{K\pi}$ with an additional
$1\,\%$ systematic uncertainty assigned to account for the approximation;
$A_{C\\!P+}$ is computed as $A_{C\\!P+}=\langle
A_{K}^{KK},A_{K}^{\pi\pi}\rangle$. From the $R_{\pm}$ one can also compute

$\displaystyle R_{\rm ADS}$ $\displaystyle=\phantom{-}0.0152\pm 0.0020\pm
0.0004\,,$ $\displaystyle A_{\rm ADS}$ $\displaystyle=-0.52\pm 0.15\pm
0.02\,,$

as $R_{\rm ADS}=(R_{-}+R_{+})/2$ and $A_{\rm
ADS}=(R_{-}-R_{+})/(R_{-}+R_{+})$. To summarise, the $B^{\pm}\rightarrow
DK^{\pm}$ ADS mode is observed with $\approx 10\,\sigma$ statistical
significance when comparing the maximum likelihood to that of the null
hypothesis. This mode displays evidence ($4.0\,\sigma$) of a large negative
asymmetry, consistent with previous experiments [321, 322, 323]. The combined
asymmetry $A_{C\\!P+}$ is smaller than (but compatible with) previous
measurements [345, 346], and is $4.5\,\sigma$ significant. The maximum
likelihood is compared with that under the null hypothesis in all three $DK$
final states, diluted by the non-negligible correlated systematic
uncertainties. From this, with a total significance of $5.8\,\sigma$, direct
$C\\!P$ violation is observed in $B^{\pm}\rightarrow DK^{\pm}$ decays.

Figure 18: Invariant mass distributions of selected
$B^{\pm}\rightarrow[\pi^{\pm}K^{\pm}]_{D}h^{\pm}$ candidate events: (left)
$B^{-}$ candidates, (right) $B^{+}$ candidates [6]. In the top plots, the
track directly from the $B$ vertex passes a kaon identification requirement
and the $B$ candidates are reconstructed assigning this track the kaon mass.
The remaining events are placed in the bottom row and are reconstructed with a
pion mass hypothesis. The dark (red) curve represents the $B\rightarrow
DK^{\pm}$ events, the light (green) curve is $B\rightarrow D\pi^{\pm}$. The
shaded contribution are partially reconstructed events and the thin line shows
the total fit function which also includes a linear combinatoric component.
The broken line represents the partially reconstructed $\kern
1.61993pt\overline{\kern-1.61993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\pi^{-}$
decays where the pion is lost.

The analysis of the $B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$ decay mode [140]
is based on a sample corresponding to an integrated luminosity of
$0.37\mbox{\,fb}^{-1}$, collected in 2011 at a centre-of-mass energy of
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$. This decay mode has been observed
by the CDF [347] and Belle [348] collaborations, who measured its branching
fraction with an uncertainty around 23 % [190]. In addition to
$B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$, the channels $B^{0}\rightarrow
D^{-}\pi^{+}$and $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}$are analysed. They are
characterised by a similar topology and therefore are good control and
normalisation channels. Particle identification criteria are used to separate
the CF decays from the suppressed modes, and to suppress misidentified
backgrounds.

The signal yields are obtained from unbinned extended maximum likelihood fits
to the data. The fits include components for the combinatorial background and
several sources of background from $b$ hadron decays. The most important is
the misidentified $B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+}$ decay. Its shape is
fixed from data using a reweighting procedure [349] while the yield is left
free to float. A similar procedure is applied to a simulated data sample to
extract the shape of the $B^{0}\rightarrow D^{-}K^{+}$ misidentified
background. The fit results are shown in Fig. 19.

Figure 19: Mass distribution of the $B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$
candidate events [140]. The stacked background shapes follow the same top-to-
bottom order in the legend and in the plot.

Correcting the raw signal yields for selection efficiency differences gives

$\frac{{\cal B}(B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm})}{{\cal
B}(B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+})}=0.0646\pm 0.0043\pm 0.0025\,,$ (71)

where the first uncertainty is statistical and the second is systematic. Using
the measured relative yield of $B^{0}\rightarrow D^{-}\pi^{+}$, the known
$B^{0}\rightarrow D^{-}\pi^{+}$ branching fraction [190], and the recent
$f_{s}/f_{d}$ measurement [145], the branching fractions

$\displaystyle{\cal B}(B^{0}_{s}\rightarrow D^{-}_{s}\pi^{+})$
$\displaystyle=$ $\displaystyle(2.95\pm 0.05\pm 0.17\,^{+0.18}_{-0.22})\times
10^{-3}\,,$ (72) $\displaystyle{\cal B}(B^{0}_{s}\rightarrow
D^{\mp}_{s}K^{\pm})$ $\displaystyle=$ $\displaystyle(1.90\pm 0.12\pm
0.13\,^{+0.12}_{-0.14})\times 10^{-4}$ (73)

are obtained, where the first uncertainty is statistical, the second is the
experimental systematic uncertainty, and the third is from the $f_{s}/f_{d}$
measurement. Both measurements are significantly more precise than the
previous world averages [190].

####  3.4.4 Measurements of $\gamma$ using loop-mediated two-body $B$ decays

$C\\!P$ violation in $B^{0}_{(s)}$ decays plays a fundamental role in testing
the consistency of the CKM paradigm in the SM and in probing virtual effects
of heavy new particles.

With the advent of the $B$ factories, the Gronau-London (GL) [350] isospin
analysis of $B\rightarrow\pi\pi$ decays has been a precious source of
information on the phase of the CKM matrix. Although the method allows a full
determination of the weak phase and of the relevant hadronic parameters, it
suffers from discrete ambiguities that limit its constraining power. It is
however possible to reduce the impact of discrete ambiguities by adding
information on hadronic parameters [351, 352]. In particular, as noted in
Refs. [353, 354, 355], the hadronic parameters entering the
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and the $B^{0}_{s}\rightarrow K^{+}K^{-}$
decays are connected by U-spin, so that experimental knowledge of
$B^{0}_{s}\rightarrow K^{+}K^{-}$ can improve the extraction of the CKM phase
with the GL analysis. Indeed, in Ref. [352], the measurement of ${\cal
B}(B^{0}_{s}\rightarrow K^{+}K^{-})$ was used to obtain an upper bound on one
of the hadronic parameters.

LHCb has reported preliminary measurements of the time-dependent $C\\!P$
asymmetries using decays to $C\\!P$ eigenstates, namely
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$ [356],
thereby permitting the use of the U-spin strategy proposed by Fleischer (F)
[353, 354, 355] to extract the CKM phase from a combined analysis of
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and the $B^{0}_{s}\rightarrow K^{+}K^{-}$
decays. However, as shown explicitly below, this strategy alone suffers from a
sizeable dependence on the breaking of U-spin symmetry. In Ref. [357], the
authors propose to perform a combined analysis of the GL modes plus
$B^{0}_{s}\rightarrow K^{+}K^{-}$ to obtain an optimal determination of the
CKM phase within the SM. They show that this combined strategy has a milder
dependence on the magnitude of U-spin breaking, allowing for a more solid
estimate of the theory error. The experimental data used for such a
determination of $\gamma$ are summarised in Table 6.

Table 6: Experimental data on $B\rightarrow\pi\pi$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$ decays. The correlation column refers to that between $S_{f}$ and $C_{f}$ measurements. Except for the preliminary results in Ref. [356], all other measurements have been averaged by HFAG [44]. The $C\\!P$ asymmetry of $B^{+}\rightarrow\pi^{+}\pi^{0}$ has been reported for completeness, although it has not been used in the analysis. New results on time-dependent $C\\!P$ violation in $B^{0}\rightarrow\pi^{+}\pi^{-}$ reported by Belle at CKM2012 [358] are not included.  Channel | ${\cal B}\times 10^{6}$ | $S_{f}\,(\%)$ | $C_{f}\,(\%)$ | Corr. | Ref.
---|---|---|---|---|---
$B^{0}\rightarrow\pi^{+}\pi^{-}$ | $5.11\pm 0.22$ | $-65\pm 7$ | $-38\pm 6$ | $-0.08$ | [359, 360, 361, 362, 363, 364]
$B^{0}\rightarrow\pi^{+}\pi^{-}$ | – | $-56\pm 17\pm 3$ | $-11\pm 21\pm 3$ | $0.34$ | [356]
$B^{0}\rightarrow\pi^{0}\pi^{0}$ | $1.91\pm 0.23$ | – | $-43\pm 24$ | – | [359, 365, 363]
$B^{+}\rightarrow\pi^{+}\pi^{0}$ | $5.48\pm 0.35$ | – | $-2.6\pm 3.9$ | – | [366, 362, 363]
$B^{0}_{s}\rightarrow K^{+}K^{-}$ | $25.4\pm 3.7$ | $17\pm 18\pm 5$ | $-2\pm 18\pm 4$ | $0.1$ | [367, 364, 356]

The time-dependent asymmetry for a $B$ meson decay to a $C\\!P$ eigenstate $f$
can be written, with the same notation as Eqs. (65) and (66),454545 In the
LHCb preliminary results on $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{+}K^{-}$ decays [356] a different notation has been
used: $A_{f}^{\rm dir}\equiv-C_{f}$, $A_{f}^{\rm mix}\equiv S_{f}$,
$A_{f}^{\Delta\Gamma}\equiv-D_{f}$.  as

$A_{C\\!P}(t)=\frac{S_{f}\sin(\Delta mt)-C_{f}\cos(\Delta
mt)}{\cosh\left(\frac{\Delta\Gamma}{2}t\right)+D_{f}\sinh\left(\frac{\Delta\Gamma}{2}t\right)}\,,$
(74)

where $C_{f}$ and $S_{f}$ parametrise direct and mixing-induced $C\\!P$
violation respectively, and the quantity $D_{f}$ is constrained by the
consistency relation

$\left(C_{f}\right)^{2}+\left(S_{f}\right)^{2}+\left(D_{f}\right)^{2}=1\,.$
(75)

The LHCb preliminary results on direct and mixing-induced $C\\!P$ violation
parameters in $B^{0}\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow
K^{+}K^{-}$ decays [356] are shown in Table 6. The measurements of
$C_{\pi^{+}\pi^{-}}$ and $S_{\pi^{+}\pi^{-}}$ are compatible with those from
the $B$ factories, whereas $C_{K^{+}K^{-}}$ and $S_{K^{+}K^{-}}$ are measured
for the first time and are consistent with zero within the current
uncertainties.

Beyond the SM, NP can affect both the $B^{0}_{(s)}$–$\bar{B}^{0}_{(s)}$
amplitudes and the $b\rightarrow d(s)$ penguin amplitudes. Taking the phase of
the mixing amplitudes from other measurements, for example from $b\rightarrow
c\bar{c}s$ decays, one can obtain a constraint on NP in $b\rightarrow s$ (or
$b\rightarrow d$) penguins. Alternatively, assuming no NP in the penguin
amplitudes, one can obtain a constraint on NP in mixing. The analysis
discussed here is based on a simplified framework [357], using as input values
$\sin 2\beta=0.679\pm 0.024$ [44] and $2\beta_{s}=(0\pm 5)^{\circ}$ [139]
obtained from $b\rightarrow c\bar{c}s$ decays. The optimal strategy will be to
include the combined GL and Fleischer analysis in a global fit of the CKM
matrix plus possible NP contributions.

The GL and Fleischer analyses were formulated with different parametrisations
of the decay amplitudes. In order to use the constraints in a global fit one
can write464646 Note that the use here of the symbol $C$ to denote a colour-
suppressed amplitude is not related to its use to denote direct $C\\!P$
violation parameters in time-dependent analyses.

$\begin{array}[]{ll}A(B^{0}\rightarrow\pi^{+}\pi^{-})=C(e^{i\gamma}-de^{i\theta})\,,&A(\bar{B}^{0}\rightarrow\pi^{+}\pi^{-})=C(e^{-i\gamma}-de^{i\theta})\,,\\\
A(B^{0}\rightarrow\pi^{0}\pi^{0})=\frac{C}{\sqrt{2}}(Te^{i\theta_{T}}e^{i\gamma}+de^{i\theta})\,,&A(\bar{B}^{0}\rightarrow\pi^{0}\pi^{0})=\frac{C}{\sqrt{2}}(Te^{i\theta_{T}}e^{-i\gamma}+de^{i\theta})\,,\\\
A(B^{+}\rightarrow\pi^{+}\pi^{0})=\frac{A(B^{0}\rightarrow\pi^{+}\pi^{-})}{\sqrt{2}}+A(B^{0}\rightarrow\pi^{0}\pi^{0})\,,&A(B^{-}\rightarrow\pi^{-}\pi^{0})=\frac{A(\bar{B}^{0}\rightarrow\pi^{+}\pi^{-})}{\sqrt{2}}+A(\bar{B}^{0}\rightarrow\pi^{0}\pi^{0})\,,\\\
A(B^{0}_{s}\rightarrow
K^{+}K^{-})=C^{\prime}\frac{\lambda}{1-\lambda^{2}/2}(e^{i\gamma}+\frac{1-\lambda^{2}}{\lambda^{2}}d^{\prime}e^{i\theta^{\prime}})\,,&A(\kern
1.43994pt\overline{\kern-1.43994ptB}{}^{0}_{s}\rightarrow
K^{+}K^{-})=C^{\prime}\frac{\lambda}{1-\lambda^{2}/2}(e^{-i\gamma}+\frac{1-\lambda^{2}}{\lambda^{2}}d^{\prime}e^{i\theta^{\prime}})\,,\end{array}$
(76)

where the magnitude of $V_{ub}V_{ud}^{*}$ has been reabsorbed in $C$, and the
magnitude of $V_{cb}V_{cd}^{*}/(V_{ub}V_{ud}^{*})$ has been reabsorbed in $d$.
In the exact U-spin limit, one has $C=C^{\prime}$, $d=d^{\prime}$ and
$\theta=\theta^{\prime}$. Isospin breaking in $B\rightarrow\pi\pi$ has been
neglected, since its impact on the extraction of the weak phase is at the
level of $1^{\circ}$ [368, 369, 370, 371]. The physical observables entering
the analysis are

$\displaystyle{\cal B}(B\rightarrow f)=F(B)\frac{|A(B\rightarrow
f)|^{2}+|A(\bar{B}\rightarrow f)|^{2}}{2}\,,$ (77) $\displaystyle
C_{f}=\frac{|A(B\rightarrow f)|^{2}-|A(\bar{B}\rightarrow
f)|^{2}}{|A(B\rightarrow f)|^{2}+|A(\bar{B}\rightarrow f)|^{2}}\,,\qquad
S_{f}=\frac{2\mathrm{Im}\left(e^{-i\phi_{M}(B)}\frac{A(\bar{B}\rightarrow
f)}{A(B\rightarrow f)}\right)}{1+\left|\frac{A(\bar{B}\rightarrow
f)}{A(B\rightarrow f)}\right|^{2}}\,,$

where $\phi_{M}(B^{0})=2\beta$, $\phi_{M}(B^{0}_{s})=-2\beta_{s}$ in the SM,
and $F(B^{0})=1$, $F(B^{+})=\tau_{B^{+}}/\tau_{B^{0}}=1.08$,
$F(B^{0}_{s})=\tau_{B^{0}_{s}}/\tau_{B^{0}}(m_{B^{0}}^{2}/m_{B^{0}_{s}}^{2})\sqrt{(M_{B^{0}_{s}}^{2}-4M_{K^{+}}^{2})/(M_{B^{0}}^{2}-4M_{\pi^{+}}^{2})}=0.9112$.

In the GL approach, one extracts the probability density function (PDF) for
the angle $\alpha=\pi-\beta-\gamma$ of the UT from the measurements of ${\cal
B}(B\rightarrow\pi\pi)$, $S_{\pi^{+}\pi^{-}}$, $C_{\pi^{+}\pi^{-}}$ and
$C_{\pi^{0}\pi^{0}}$. Using the unitarity of the CKM matrix, it is possible to
write the $B\rightarrow\pi\pi$ decay amplitudes and observables in terms of
$\alpha$ instead of $\gamma$ and $\beta$. However, for the purpose of
connecting $B\rightarrow\pi\pi$ to $B^{0}_{s}\rightarrow KK$ it is more
convenient to use the parametrisation in Eq. (76). In this way, $\alpha$ (or,
equivalently, $\gamma$), is determined up to discrete ambiguities, that
correspond however to different values of the hadronic parameters. As
discussed in detail in Ref. [352], the shape of the PDF obtained in a Bayesian
analysis depends on the allowed range for the hadronic parameters. For
example, using the data in Table 6, solving for $C$ and choosing flat a priori
distributions for $d\in[0,2]$, $\theta\in[-\pi,\pi]$, $T\in[0,1.5]$ and
$\theta_{T}\in[-\pi,\pi]$ the PDF for $\gamma$ in Fig. 20 is obtained,
corresponding to $\gamma=(68\pm 15)^{\circ}$ ($\gamma\in[25,87]^{\circ}$ at
$95\,\%$ probability). Using instead the Fleischer method, one can obtain a
PDF for $\gamma$ given a range for the U-spin breaking effects. In this method
it was originally suggested to parametrise the U-spin breaking in
$C^{\prime}/C$ using the result one would obtain in factorisation, namely

$r_{\mathrm{fact}}=\left|\frac{C^{\prime}}{C}\right|_{\mathrm{fact}}=1.46\pm
0.15\,,$ (78)

where the error obtained using light-cone QCD sum rule calculations [372] has
been symmetrised. However, this can only serve as a reference value, since
there are non-factorisable contributions to $C$ and $C^{\prime}$ that could
affect this estimate. In this analysis, the non-factorisable U-spin breaking
is parametrised as follows

$C^{\prime}=r_{\mathrm{fact}}r_{C}C\,,\qquad{\rm
Re}(d^{\prime}e^{i\theta^{\prime}})=r_{r}{\rm Re}(de^{i\theta})\,,\qquad{\rm
Im}(d^{\prime}e^{i\theta^{\prime}})=r_{i}{\rm Im}(de^{i\theta})\,,$ (79)

with $r_{C}$, $r_{r}$ and $r_{i}$ uniformly distributed in the range
$[1-\kappa,1+\kappa]$.

In Fig. 20 the PDF for $\gamma$ obtained with the Fleischer method for two
different values of the U-spin breaking parameter $\kappa=0.1,0.5$ is shown.
The method is very precise for small amounts of U-spin breaking
($\kappa=0.1$), but becomes clearly worse for $\kappa=0.5$. Thus, a
determination of $\gamma$ from the Fleischer method alone is subject to
uncertainty on the size of U-spin breaking.

Figure 20: From left to right: PDF for $\gamma$ obtained using the GL method
as described in the text; PDF for $\gamma$ obtained using the Fleischer method
for $\kappa=0.1,0.5$ [357]. Here and in the following, dark (light) areas
correspond to $68\,\%$ ($95\,\%$) probability regions.

The result of the combined GL+F analysis is given in Fig. 21, where the PDF
for $\gamma$ for $\kappa=0.1$ and $0.5$ is shown. The result of the combined
analysis is much more stable against the allowed amount of U-spin breaking. In
Fig. 21 the $68\,\%$ probability region for $\gamma$ obtained using the
combined method as a function of $\kappa$ is also shown, and compared to the
GL result. The combined method shows a considerable gain in precision even for
very large values of $\kappa$.

Figure 21: From left to right: PDF for $\gamma$ obtained using the combined
method for $\kappa=0.1$, $0.5$; $68\,\%$ probability region for $\gamma$
obtained using the combined method (filled area) or the GL method (horizontal
lines) as a function of $\kappa$ [357].

NP could affect the determination of $\gamma$ in the combined method by giving
(electroweak) penguin contributions a new $C\\!P$-violating phase. If one
assumes that the isospin analysis of the GL channels is still valid, barring
order-of-magnitude enhancements of electroweak penguins in
$B\rightarrow\pi\pi$, and if one assumes for concreteness that NP enters only
$b\rightarrow s$ penguins, in the framework of a global fit, one can
simultaneously determine $\gamma$ and the NP contribution to $b\rightarrow s$
penguins. For the purpose of illustration, the value of $\gamma$ from tree-
level processes, $\gamma_{\mathrm{tree}}=(76\pm 9)^{\circ}$ is used as input
[270],474747 Note that the value of $\gamma$ quoted here differs from that
obtained from the full CKM fit (given in Table 2) due to the different inputs
used.  allowing inspection of the posterior for $\gamma$ and for the NP
penguin amplitude. Writing

$\displaystyle A(B^{0}_{s}\rightarrow K^{+}K^{-})$ $\displaystyle=$
$\displaystyle
C^{\prime}\frac{\lambda}{1-\lambda^{2}/2}\left(e^{+i\gamma}+\frac{1-\lambda^{2}}{\lambda^{2}}\left(d^{\prime}e^{i\theta^{\prime}}+e^{+i\phi_{\rm
NP}}d^{\prime}_{\rm NP}e^{i\theta^{\prime}_{\rm NP}}\right)\right)\,,$ (80)
$\displaystyle A(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow K^{+}K^{-})$
$\displaystyle=$ $\displaystyle
C^{\prime}\frac{\lambda}{1-\lambda^{2}/2}\left(e^{-i\gamma}+\frac{1-\lambda^{2}}{\lambda^{2}}\left(d^{\prime}e^{i\theta^{\prime}}+e^{-i\phi_{\rm
NP}}d^{\prime}_{\rm NP}e^{i\theta^{\prime}_{\rm NP}}\right)\right)\,,$

and taking uniformly distributed $d^{\prime}_{\rm NP}\in[0,2]$ and $\phi_{\rm
NP},\theta^{\prime}_{\rm NP}\in[-\pi,\pi]$ the PDFs shown in Fig. 22 are
obtained for $\kappa=0.5$. This yields $\gamma=(74\pm 7)^{\circ}$, and a
$95\,\%$ probability upper bound on $d^{\prime}_{\mathrm{NP}}$ around $1$.
Clearly, the bound is stronger for large values of $\phi_{\rm NP}$.

Figure 22: From left to right: PDFs for $\phi_{\mathrm{NP}}$,
$d^{\prime}_{\mathrm{NP}}$ and $\theta^{\prime}_{\mathrm{NP}}$ obtained using
the combined method with $\kappa=0.5$ [357].

Finally, $B^{0}_{s}\rightarrow KK$ decays can also be used to extract
$2\beta_{s}$ in the SM. The optimal choice in this respect is represented by
$B^{0}_{s}\rightarrow K^{(*)0}\bar{K}^{(*)0}$ (with $B^{0}\rightarrow
K^{(*)0}\bar{K}^{(*)0}$ as U-spin related control channels to constrain
subleading contributions), since in this channel there is no tree contribution
proportional to $e^{i\gamma}$ [307, 308]. However, the combined analysis
described above, in the framework of a global SM fit, can serve for the same
purpose. To illustrate this point, the GL+F analysis is performed, taking as
input the SM fit result $\gamma=(69.7\pm 3.1)^{\circ}$ [270] and not using the
measurement of $2\beta_{s}$ from $b\rightarrow c\bar{c}s$ decays. In this way,
$2\beta_{s}=(3\pm 14)^{\circ}$ is obtained for $\kappa=0.5$. The analysis can
also be performed without using the measurement of $\gamma$, in this case the
result is $2\beta_{s}=(6\pm 14)^{\circ}$. With improved experimental accuracy,
this determination could become competitive with that from $b\rightarrow
c\bar{c}s$ decays. Once results of time-dependent analyses of the
$B^{0}_{(s)}\rightarrow K^{(*)0}\bar{K}^{(*)0}$ channels are available these
may also provide useful constraints.484848 The proposal of Ref. [308] has been
recently critically reexamined in Ref. [373]. The present analysis shows no
particular enhancement of the contribution proportional to $e^{i\gamma}$ in
$B^{0}_{s}\rightarrow K^{+}K^{-}$, in agreement with the expectation that
$B^{0}_{s}\rightarrow K^{(*)0}\bar{K}^{(*)0}$ should be penguin-dominated to a
very good accuracy.

To conclude, the usual GL analysis to extract $\alpha$ from
$B^{0}\rightarrow\pi\pi$ can be supplemented with the inclusion of the
$B^{0}_{s}\rightarrow K^{+}K^{-}$ modes, in the framework of a global CKM fit.
The method optimises the constraining power of these decays and allows the
derivation of constraints on NP contributions to penguin amplitudes or on the
$B^{0}_{s}$ mixing phase and illustrates these capabilities with a simplified
analysis, neglecting correlations with other SM observables.

####  3.4.5 Studies of $C\\!P$ violation in multibody charmless $b$ hadron
decays

Multibody charmless $b$ hadron decays can be used for a variety of studies of
$C\\!P$ violation, including searches for NP and determination of the angle
$\gamma$. Due to the resonant structure in multibody decays, these can offer
additional possibilities to search for both the existence and features of NP.
Model-independent analyses [374, 375] can be performed to first establish the
presence of a $C\\!P$ violation effect, and then to identify the regions of
the phase space in which it is most pronounced.494949 LHCb has presented
preliminary results from model-independent searches for $C\\!P$ violation in
$B^{\pm}\rightarrow\pi^{+}\pi^{-}K^{\pm}$ and $B^{\pm}\rightarrow
K^{+}K^{-}K^{\pm}$ at ICHEP 2012 [376], and in
$B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}$ and $B^{\pm}\rightarrow
K^{+}K^{-}\pi^{\pm}$ at CKM 2012 [377].  To further establish whether any
observed $C\\!P$ violation can be accommodated within the Standard Model,
amplitude analyses can be used to quantify the effects associated with
resonant contributions to the decay. A number of methods have been proposed to
determine $\gamma$ from such processes [378, 379, 380, 381, 382, 383, 384,
385, 386, 387], in general requiring input not only from charged $B$ decays,
but also from $B^{0}$ and $B^{0}_{s}$ decays (to states such as
$K^{0}_{\rm\scriptscriptstyle S}h^{+}h^{\prime-}$ and
$\pi^{0}h^{+}h^{\prime-}$).505050 LHCb has presented preliminary branching
fraction measurements of $B^{0}_{(s)}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}h^{+}h^{\prime-}$ decays at ICHEP 2012 [388].  The potential for LHCb to
study multibody charmless $\mathchar 28931\relax^{0}_{b}$ decays adds further
possibilities for novel studies of $C\\!P$ violation effects.

####  3.4.6 Prospects of future LHCb measurements

As discussed above, the angle $\gamma$ can be determined from both tree-
dominated and loop-dominated processes. Comparisons of the values obtained
provide tests of NP, and so precision measurements from both methods are
needed. Among the tree-dominated processes, in addition to the modes discussed
above, any channel that involves the interference of $b\rightarrow c\bar{u}s$
and $b\rightarrow u\bar{c}s$ transitions is potentially sensitive to $\gamma$.
Many of these modes can be analysed in the upgraded phase of LHCb, including

  1. 1.

$B^{+}\rightarrow DK^{+}\pi^{-}\pi^{+}$ where, similarly to the $B\rightarrow
DK$ mode, the neutral $D$ can be reconstructed either in the two-body (ADS and
GLW-like measurement) or multibody (GGSZ-like measurement) final state. The
observation of the CF mode in LHCb data [389] indicates a yield only twice
lower than that for the $B\rightarrow DK$ mode, which makes it competitive for
the measurement of $\gamma$.515151 Preliminary results from a GLW-type
analysis of this channel was presented at ICHEP 2012 [341].  However, two
unknown factors affect the expected $\gamma$ sensitivity. First, since this is
a multibody decay, the overlap between the interfering amplitudes is in
general less than 100 %; this is accounted for by a coherence factor between
zero and unity which enters the interference term in Eqs. (54), (55), (59),
(60) as an unknown parameter. Second, the value of $r_{B}$ can be different
from that in $B\rightarrow DK$ and is as yet unmeasured, although it is
expected [318] that it can be larger in this decay than in $B\rightarrow DK$.

  2. 2.

$B^{0}\rightarrow DK^{+}\pi^{-}$. Although the rate of these decays is smaller
that that of $B^{+}\rightarrow DK^{+}$, both interfering amplitudes are
colour-suppressed, therefore the expected value of $r_{B}$ is larger,
$r_{B}\simeq 0.3$. As a result, the sensitivity to $\gamma$ should be similar
to that in the $B\rightarrow DK$ modes.525252 Preliminary results from a GLW-
type analysis of $B^{0}\rightarrow DK^{*0}$ were presented at ICHEP 2012
[340].  Depending on the content of $B^{0}\rightarrow D^{0}K^{+}\pi^{-}$ and
$B^{0}\rightarrow\overline{D}{}^{0}K^{+}\pi^{-}$ amplitudes, the optimal
strategy may involve Dalitz plot analysis of the $B^{0}$ decay [390, 391]. In
this case, control of amplitude model uncertainty will become essential for a
precision measurement; it can be eliminated by studying the decays
$B^{0}\rightarrow DK^{+}\pi^{-}$ with $D\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ [392].

  3. 3.

$B^{0}_{s}\rightarrow D\phi$. This mode is not self-tagging, but sensitivity
to $\gamma$ can be obtained from untagged time-integrated measurements using
several different neutral $D$ decay modes [393, 394]. The first evidence for
the three-body decay $B^{0}_{s}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}K^{+}K^{-}$ has just been reported
by LHCb [395], and investigation of its resonant structure is in progress.

  4. 4.

$B_{c}^{+}\rightarrow DD_{s}^{+}$. $B_{c}^{+}$ production in $pp$ collisions
is significantly suppressed, however, in this mode the magnitude of $C\\!P$
violation is expected to be ${\cal O}(100\,\%)$: the two interfering
amplitudes are of the same magnitude because the $b\rightarrow u\bar{c}s$
amplitude is colour allowed, while the $b\rightarrow c\bar{u}s$ amplitude is
colour suppressed [396, 397, 398, 399].

  5. 5.

$\mathchar 28931\relax^{0}_{b}\rightarrow D\mathchar 28931\relax$ and
$\mathchar 28931\relax^{0}_{b}\rightarrow DpK^{-}$. Measurement of $\gamma$
from analysis of the $\mathchar 28931\relax^{0}_{b}\rightarrow D\mathchar
28931\relax$ decay mode was proposed in Ref. [400]. This method allows one to
measure $\gamma$ in a model-independent way by comparing the $S$\- and
$P$-wave amplitudes. However, this mode is problematic to reconstruct at LHCb
because of the poorly defined $\mathchar 28931\relax^{0}_{b}$ vertex (both
particles from its decay are long-lived) and low efficiency of $\mathchar
28931\relax$ reconstruction. Alternatively, one can consider a similar
measurement with the decay $\mathchar 28931\relax^{0}_{b}\rightarrow DpK^{-}$.
A preliminary observation of this mode in early LHCb data has been reported
[401].

Table 7 shows the expected sensitivity to $\gamma$ from tree level decays in
the upgrade scenario. The LHCb upgrade is the only proposed experiment which
will be able to reach sub-degree precision on $\gamma$.

Table 7: Estimated precision of $\gamma$ measurements with $50\mbox{\,fb}^{-1}$ for various charmed $B$ decay modes. Decay mode |  $\gamma$ sensitivity
---|---
$B\rightarrow DK$ with $D\rightarrow hh^{\prime}$, $D\rightarrow K\pi\pi\pi$ | $1.3^{\circ}$
$B\rightarrow DK$ with $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi\pi$ | $1.9^{\circ}$
$B\rightarrow DK$ with $D\rightarrow 4\pi$ | $1.7^{\circ}$
$B^{0}\rightarrow DK\pi$ with $D\rightarrow hh^{\prime}$, $D\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi\pi$ | $1.5^{\circ}$
$B\rightarrow DK\pi\pi$ with $D\rightarrow hh^{\prime}$ | $\sim 3^{\circ}$
Time-dependent $B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$ | $2.0^{\circ}$
Combined | $\sim 0.9^{\circ}$

Measurement of $\gamma$ and $2\beta_{s}$ by means of the $C\\!P$-violating
observables from loop-mediated decays $B^{0}\rightarrow\pi^{+}\pi^{-}$ and
$B^{0}_{s}\rightarrow K^{+}K^{-}$ was discussed in Sec. 3.4.4. Extrapolating
the current sensitivity on $C$ and $S$ to the upgrade scenario, when
$50\mbox{\,fb}^{-1}$ of integrated luminosity will be collected, LHCb will be
able to reach a statistical sensitivity $\sigma_{\rm
stat}(C)\approx\sigma_{\rm stat}(S)\simeq 0.008$ in both
$B^{0}\rightarrow\pi^{+}\pi^{-}$ and $B^{0}_{s}\rightarrow K^{+}K^{-}$. This
corresponds to a precision on $\gamma$ of $1.4^{\circ}$, and on $2\beta_{s}$
of $0.01$ rad, assuming perfect U-spin symmetry.

##  4 Mixing and $C\\!P$ violation in the charm sector

###  4.1 Introduction

The study of $D$ mesons offers a unique opportunity to access up-type quarks
in flavour-changing neutral current (FCNC) processes. It probes scenarios
where up-type quarks play a special role, such as supersymmetric models with
alignment [402, 403]. It offers complementary constraints on possible NP
contributions to those arising from the measurements of FCNC processes of
down-type quarks ($B$ or $K$ mesons).

The neutral $D$ system is the latest and last system of neutral mesons where
mixing between particles and anti-particles has been established. The mixing
rate is consistent with, but at the upper end of, SM expectations [404] and
constrains many NP models [405]. More precise $D^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing measurements will provide
even stronger constraints. However, the focus has been shifting to $C\\!P$
violation observables, which provide cleaner tests of the SM [406, 407, 408].
First evidence for direct $C\\!P$ violation in the charm sector has been
reported by the LHCb collaboration in the study of the difference of the time-
integrated asymmetries of $D^{0}\rightarrow K^{+}K^{-}$ and
$D^{0}\rightarrow\pi^{+}\pi^{-}$ decay rates through the parameter
$\Delta{\cal A}_{C\\!P}$ [18]. No evidence of indirect $C\\!P$ violation has
yet been found. As discussed in detail below, these results on $C\\!P$
violation in the charm sector appear marginally compatible with the SM but
contributions from NP are not excluded.

The mass eigenstates of neutral $D$ mesons, $|D_{1,2}\rangle$, with masses
$m_{1,2}$ and widths $\Gamma_{1,2}$ can be written as linear combinations of
the flavour eigenstates $|D_{1,2}\rangle=p|D^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$, with complex coefficients
$p$ and $q$ which satisfy $|p|^{2}+|q|^{2}=1$. The average mass and width are
defined as $m\equiv(m_{1}+m_{2})/2$ and
$\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$. The $D$ mixing parameters are defined
using the mass and width difference as $x_{D}\equiv(m_{2}-m_{1})/\Gamma$ and
$y_{D}\equiv(\Gamma_{2}-\Gamma_{1})/2\Gamma$. The phase convention of $p$ and
$q$ is chosen such that $C\\!P|D^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$. First evidence for mixing
of neutral $D^{0}$ mesons was discovered in 2007 by Belle and BaBar [409, 410]
and is now well established [44]: the no-mixing hypothesis is excluded at more
than $10\,\sigma$ for the world average ($x_{D}=0.63\,^{+0.19}_{-0.20}\,\%$,
$y_{D}=0.75\pm 0.12\,\%$).535353 At HCP 2012, LHCb presented the first
observation of charm mixing from a single measurement [411].

It is convenient to group hadronic charm decays into three categories. The CF
decays, such as $D^{0}\rightarrow K^{-}\pi^{+}$, are mediated by tree
amplitudes, and therefore no direct $C\\!P$ violation effects are expected.
The same is true for DCS decays, such as $D^{0}\rightarrow K^{+}\pi^{-}$, even
though these are much more rare. The SCS decays, on the other hand, can also
have contributions from penguin amplitudes, and therefore direct $C\\!P$
violation is possible, even though the penguin contributions are expected to
be small. Within this classification, it should be noted that some decays to
final states containing $K^{0}_{\rm\scriptscriptstyle S}$ mesons, e.g.
$D^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\rho^{0}$, have both CF and
DCS contributions which can interfere [412]. Within the SM, however, direct
$C\\!P$ violation effects are still expected to be negligible in these decays.

LHCb is ideally placed to carry out a wide physics programme in the charm
sector, thanks to the high production rate of open charm: with a cross-section
of $6.10\pm 0.93$ mb [3, 4], one tenth of LHC interactions produce charm
hadrons. Its ring-imaging Cherenkov detectors provide excellent separation
between pions, kaons and protons in the momentum range between 2 and 100
$\,{\rm GeV}/c$, and additional detectors also provide clean identification of
muons and electrons. This allows high purity samples to be obtained both for
hadronic and muonic decays. The large boost of the $D$ hadrons produced at
LHCb is beneficial for time-dependent studies. LHCb has the potential to
improve the precision on all the key observables in the charm sector in the
next years.

In the remainder of this section the key observables in the charm sector are
described, and the current status and near term prospects of the measurements
at LHCb are reviewed. A discussion of the implications of the first LHCb charm
physics results follows, motivating improved measurements and studies of
additional channels. The potential of the LHCb upgrade to make the precise
measurements needed to challenge the theory is then described.

####  4.1.1 Key observables

Currently the most precise individual measurements of mixing parameters are
those of the relative effective lifetime difference between $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to $C\\!P$ eigenstates
($\hat{\Gamma}$ and $\hat{\bar{\Gamma}}$) and flavour specific final states
($\Gamma$), $y_{C\\!P}$, which is defined as

$y_{C\\!P}=\frac{\hat{\Gamma}+\hat{\bar{\Gamma}}}{2\Gamma}-1\approx\eta_{C\\!P}\left[\left(1-\frac{1}{8}A_{m}^{2}\right)y_{D}\cos\phi-\frac{1}{2}(A_{m})x_{D}\sin\phi\right],$
(81)

where terms below ${\cal O}(10^{-4})$ have been ignored [413], $\eta_{C\\!P}$
is the $C\\!P$ eigenvalue of the final state, $\phi$ is the $C\\!P$-violating
relative phase between $q/p$ and $\bar{A}_{f}/A_{f}$ where
$\kern-1.99997pt\stackrel{{\scriptstyle\kern
1.39998pt\textsf{(---)}}}{{A}}_{\kern-2.10002ptf}\kern-3.00003pt$ are the
decay amplitudes, and $A_{m}$ represents a $C\\!P$ violation contribution from
mixing ($|q/p|^{\pm 2}\approx 1\pm A_{m}$).545454 $A_{m}$ can be determined
from asymmetries in semileptonic charm decays, with the assumption of
vanishing direct $C\\!P$ violation.  In the limit of $C\\!P$ conservation
$y_{C\\!P}$ is equal to the mixing parameter $y_{D}$. The resulting world
average value for $y_{C\\!P}$ is $0.87\pm 0.16\,\%$ [414]555555 New results
presented by Belle at ICHEP 2012 [415] are not included in this average.  and
is consistent with the value of $y_{D}$ within the current accuracy.

The $C\\!P$-violating observable $A_{\Gamma}$ quantifies the difference in
decay rates of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
to a $C\\!P$ eigenstate and is defined as

$A_{\Gamma}=\frac{\hat{\Gamma}-\hat{\bar{\Gamma}}}{\hat{\Gamma}+\hat{\bar{\Gamma}}}\approx\eta_{C\\!P}\left[\frac{1}{2}(A_{m}+A_{d})y_{D}\cos\phi-
x_{D}\sin\phi\right],$ (82)

where terms below ${\cal O}(10^{-4})$ have again been ignored [413] and both
mixing and direct $C\\!P$ contributions are assumed to be small. The parameter
$A_{d}$ describes the contribution from direct $C\\!P$ violation
($|\bar{A}_{f}/A_{f}|^{\pm 2}\approx 1\pm A_{d}$). The current world average
of $A_{\Gamma}$ is $0.02\pm 0.16\,\%$ [44], consistent with the hypothesis of
no $C\\!P$ violation. Due to the smallness of $x_{D}$ and $y_{D}$,
$A_{\Gamma}$ provides essentially the same information as a full time-
dependent $C\\!P$ violation analysis of $D^{0}\rightarrow K^{+}K^{-}$ decays.

An alternative way to search for $C\\!P$ violation in charm mixing is with a
time-dependent Dalitz plot analysis of $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays to
$K^{0}_{\rm\scriptscriptstyle S}\pi^{+}\pi^{-}$ or
$K^{0}_{\rm\scriptscriptstyle S}K^{+}K^{-}$. Such analyses have been carried
out at the $B$ factories [416, 417]. Also in these cases no $C\\!P$ violation
was observed.

In time-integrated analyses the measured rate asymmetry is

${\cal A}_{C\\!P}\equiv\frac{\Gamma(D^{0}\rightarrow f)-\Gamma(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
f)}{\Gamma(D^{0}\rightarrow f)+\Gamma(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow f)}\approx
a_{C\\!P}^{\rm dir}-A_{\Gamma}\frac{\langle t\rangle}{\tau},$ (83)

where the direct $C\\!P$ asymmetry contribution is defined as

$a_{C\\!P}^{\rm
dir}\equiv\frac{|A_{f}|^{2}-|\bar{A}_{f}|^{2}}{|A_{f}|^{2}+|\bar{A}_{f}|^{2}}\approx-\frac{1}{2}A_{d}$
(84)

and $\langle t\rangle$ denotes the average decay time of the observed
candidates.

A powerful way to reduce experimental systematic uncertainties is to measure
the difference in time-integrated asymmetries in related final states. For the
two-body final states $K^{+}K^{-}$ and $\pi^{+}\pi^{-}$, this difference is
given by

$\displaystyle\Delta{\cal A}_{C\\!P}$ $\displaystyle\equiv$
$\displaystyle{\cal A}_{C\\!P}(K^{+}K^{-})-{\cal A}_{C\\!P}(\pi^{+}\pi^{-})$
(85) $\displaystyle\approx$ $\displaystyle\Delta a_{C\\!P}^{{\rm
dir}}\left(1+y_{D}\cos\phi\frac{\overline{\langle
t\rangle}}{\tau}\right)+\left(a_{C\\!P}^{\rm ind}+\overline{a_{C\\!P}^{\rm
dir}}y_{D}\cos\phi\right)\frac{\Delta\langle t\rangle}{\tau}$

where the $C\\!P$-violating phase $\phi$ is assumed to be universal [418],
$\Delta a\equiv a(K^{+}K^{-})-a(\pi^{+}\pi^{-})$,
$\overline{a}\equiv(a(K^{+}K^{-})+a(\pi^{+}\pi^{-}))/2$ and the indirect
$C\\!P$ asymmetry parameter is defined as $a_{C\\!P}^{\rm
ind}=-(A_{m}/2)y_{D}\cos\phi+x_{D}\sin\phi$. The ratio $\Delta\langle
t\rangle/\tau$ is equal to zero for the lifetime-unbiased $B$ factory
measurements [419, 420] and is $0.098\pm 0.003$ for LHCb [18] and $0.25\pm
0.04$ for CDF [421], therefore $\Delta{\cal A}_{C\\!P}$ is largely a measure
of direct $C\\!P$ violation.

The current most accurate measurements of $\Delta{\cal A}_{C\\!P}$ are from
the LHCb and CDF collaborations and are $(-0.82\pm 0.21\pm 0.11)\,\%$ [18] and
$(-0.62\pm 0.21\pm 0.10)\,\%$ [422], respectively.565656 At ICHEP 2012, Belle