K^{+}K^{-})K^{\pm}$ as shown in Fig. 22. The figures are from the latest LHCb
measurements using $9\text{\,fb}^{-1}$ data [792]. By further considering the
production asymmetry of $B^{+}$ and $B^{-}$ and detection efficiency
difference between $K^{+}$ and $K^{-}$, the size of the CP violation of the
decay can be determined.

Figure 22:  The invariant mass distributions of $(K^{+}K^{-})_{D}K^{+}$ (left)
and $(K^{+}K^{-})_{D}K^{-}$ (right) from [792].

The decay rate of the ADS channel has a similar form of

$\Gamma(B^{-}\rightarrow D[\rightarrow f_{\textrm{ADS}}]h^{-})\propto
r_{D}^{2}+r_{B}^{2}+2\kappa_{B}r_{B}r_{D}R_{D}\cos(\delta_{D}+\delta_{B}-\gamma),$
(19)

where $r_{D}$ and $\delta_{D}$ are the average amplitude ratio and phase
difference between doubly Cabibbo-suppressed and Cabibbo-favoured $D$ decays.
The value of $r_{D}$ is at similar magnitude as $r_{B}$ and thus leads to
larger CP violation. However, the statistics of the ADS channel is also
suppressed. This can be seen from the measurements done by the LHCb experiment
using $B^{\pm}\rightarrow(K^{\mp}\pi^{\pm})K^{\pm}$ decays as show in Fig. 23
[792].

Figure 23:  The invariant mass distributions of $(K^{+}\pi^{-})_{D}K^{+}$
(left) and $(\pi^{+}K^{-})_{D}K^{-}$ (right) from [792].

The measurements also have four-fold ambiguity, however, as $\delta_{D}$ is
not zero, combining with GLW mode gives two-fold ambiguity on determination of
angle $\gamma$.

The coherent factor $R_{D}$ equals to one for two-body final states and is
less than one for multi-body final states to take into account dilutions due
to different resonant contributions from Cabibbo-favoured and doubly-Cabibbo-
suppressed $D$ decays. The values of $r_{D}$, $R_{D}$ and $\delta_{D}$ can
also be determined using the quantum-coherent data from BESIII and CLEO-c
experiments. For example, the coherent factor of ${{D}^{0}}({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}})\rightarrow
K^{+}\pi^{+}\pi^{-}\pi^{-}$, $R_{K3\pi}$ is measured by the two experiments to
be $0.43^{+0.17}_{-0.13}$ [793], which is significantly lower than one.
Further binning the phase space according to the variation of strong phase
determined from amplitude analysis can improve the sensitivity on $\gamma$
[794], and a first measurement using this approach is recently published
[795].

The GLW and ADS measurements have been performed by the LHCb experiment with
$D$ decaying mainly into charged final states. However, final states with
neutral particles are also studied, either by reconstructing the neutral
particles $\pi^{0}$ or $\gamma$ as in recent study of $B^{\pm}\rightarrow
Dh^{\pm},D\rightarrow h^{\pm}h^{{}^{\prime}\mp}{{\pi}^{0}}$ [796], or in a
partially reconstructed method, where $\pi^{0}$ and $\gamma$ from $D^{*0}$ are
not reconstructed [792]. In the second case, the reconstruction efficiency is
much higher, however, the sensitivity of $\gamma$ is limited by the background
contributions, especially those from $B^{0}\rightarrow D^{*-}K^{+}$ decays
which has similar line-shapes as signal.

####  4.1.2 Measurements using the GGSZ method

As has discussed in the above section, for multi-body $D$ decays, if CP even
fraction $F_{+}$ for self-conjugated decay or coherent factor $R$ for semi-
flavour tagged decay is significantly smaller than one, further sensitivity
can be achieved by considering the variation of $r_{D}$ and $\delta_{D}$ over
the phase space. One can model the $D$ decays with an amplitude model which
provides the information of strong phase $\delta_{D}$ over the phase space.
However, this may suffer from large systematic uncertainties due to modelling
of amplitude distributions.

An alternative method [782, 783] is to bin the phase space and the effective
$r_{D}^{i}$ and $\delta_{D}^{i}$ (or $c_{i}$ and $s_{i}$) in bin $i$ are
defined as

$c_{i}+is_{i}\equiv R_{D}^{i}e^{i\delta_{D}^{i}}\equiv\frac{\int_{\textrm{bin
i}}A_{f}(p)\overline{A}_{f}(p)^{*}dp}{\sqrt{\int_{\textrm{bin
i}}|A_{f}(p)|^{2}dp}\sqrt{\int_{\textrm{bin i}}|\overline{A}_{f}(p)|^{2}dp}}.$
(20)

The GGSZ mode refers to decays into self-conjugated final states. BPGGSZ is
used when referring to the analysis method using a binned strategy. In the
BPGGSZ analysis, $c_{i}$ and $s_{i}$ are used instead of effective $R_{D}$ and
$\delta_{D}$ for the benefit of better statistical performance. The values of
$c_{i}$ and $s_{i}$ can be determined using quantum coherent data collected by
BESIII and CLEO-c experiments, where a mixture of $D^{0}$ and
$\overline{D}^{0}$ can be achieved from $\psi(3770)\rightarrow
D^{0}\overline{D}^{0}$. As $\psi(3770)$ is a parity-odd state, the $D^{0}$ and
$\overline{D}^{0}$ are in quantum-correlated state of
$(|D^{0}\overline{D}^{0}>-|\overline{D}^{0}D^{0}>)/\sqrt{2}$ [797, 798, 799,
800]. Using the measured $c_{i}$ and $s_{i}$ values, the angle $\gamma$ and
strong parameters $r_{B}$ and $\delta_{B}$ can be extracted. The method has
been applied to $B^{-}\rightarrow DK^{-}$,
$D\rightarrow{{K}^{0}_{\mathrm{S}}}\pi^{+}\pi^{-}$ and
$D\rightarrow{{K}^{0}_{\mathrm{S}}}K^{+}K^{-}$ decays by LHCb [801, 802],
where 16 bins and 4 bins are used according to the statistics of the decays,
respectively. In order to optimise the sensitivity, binning schemes are chosen
according to the strong phase variation over the Dalitz plot considering
possible background contamination and efficiency effects. Symmetry between
${{K}^{0}_{\mathrm{S}}}\pi^{+}({{K}^{0}_{\mathrm{S}}}K^{+})$ and
${{K}^{0}_{\mathrm{S}}}\pi^{-}({{K}^{0}_{\mathrm{S}}}K^{-})$ is also used to
increase the sensitivity on $c_{i}$ and $s_{i}$. The binning scheme optimised
for the ${{K}^{0}_{\mathrm{S}}}\pi^{+}\pi^{-}$ decays is shown in Fig. 24
[798, 799].

Each bin of the BPGGSZ method can offer constraints to the angle $\gamma$ and
to the strong parameters $r_{B}$ and $\delta_{B}$. The number of measurements
is much more than the number of unknown parameters, thus global production
asymmetry and detection asymmetry can be treated as fit variables and related
systematic uncertainties are reduced. While for the ADS and GLW methods,
production asymmetry and detection asymmetry have to be considered using
control channels. The measured yield difference in each bin is shown in Fig.
24 for the ${{K}^{0}_{\mathrm{S}}}\pi^{+}\pi^{-}$ and
${{K}^{0}_{\mathrm{S}}}K^{+}K^{-}$ decays. Clear CP violation can be found. By
combining the statistical power of these bins, the angle $\gamma$ is measured
to be $(68.7^{+5.2}_{-5.1})^{\circ}$ [802].

Figure 24:  (Left) Binning scheme of
$D\rightarrow{{K}^{0}_{\mathrm{S}}}\pi^{+}\pi^{-}$ decays used for $\gamma$
measurements; (right) Observed raw CP asymmetries and the predicted values
using obtained CP parameters in different bins [802].

####  4.1.3 Multi-body $B$ decays

Similar to multi-body $D$ decays, multi-body $B$ decays can also be used where
$r_{B}$ and $\delta_{B}$ now is a function of $B$ decay Dalitz plot. However,
in this case, one can only use a model to describe different resonant
contributions, where some can only be obtained through $b\rightarrow c$
process and some can be obtained through both $b\rightarrow c$ and
$b\rightarrow u$ processes and the interference between them gives sensitivity
to the angle $\gamma$. The measurements have been performed by the LHCb
collaboration in $B^{0}\rightarrow DK^{+}\pi^{-}$ decays with $D\rightarrow
K^{+}K^{-}(\pi^{+}\pi^{-})$ [803] using 3 fb-1 Run 1 data. However, due to
limited statistics, the sensitivity on $\gamma$ is still low. Further
measurements with all the data collected by the LHCb experiment will be very
interesting.

####  4.1.4 Time-dependent ${{B}^{0}_{s}}$ decays

The angle $\gamma$ can be measured through time-dependent ${{B}^{0}_{s}}$ and
$B^{0}$ decays where the weak phases extracted are $(\gamma-2\beta_{s})$ and
$(\gamma+2\beta)$, respectively. In hadron collider experiments like LHCb, the
golden channels are ${{B}^{0}_{s}}\rightarrow D_{s}^{\pm}K^{\mp}$,
${{B}^{0}_{s}}\rightarrow D_{s}^{\pm}K^{\mp}\pi^{+}\pi^{-}$ and
${{B}^{0}_{s}}\rightarrow D\phi$ decays. The time-dependent analyses have been
performed for the first two channels [804, 805], while only branching fraction
has been measured for the ${{B}^{0}_{s}}\rightarrow\overline{D}^{0}\phi$ decay
[806, 807].

As ${{B}^{0}_{s}}$ mixing is involved, a time-dependent analysis is needed to
extract CP parameters. The time-dependent decay rate of the ${{B}^{0}_{s}}$
decay into a final state $f$ is given by

$\frac{d\Gamma_{{{B}^{0}_{s}}\rightarrow f}(t)}{dt}\propto
e^{-\Gamma_{s}t}[\cosh(\frac{\Delta\Gamma_{s}t}{2})+A_{f}^{\Delta\Gamma}\sinh(\frac{\Delta\Gamma_{s}t}{2})+C_{f}\cos(\Delta
m_{s}t)-S_{f}\sin(\Delta m_{s}t)],$ (21)

where $\Delta\Gamma_{s}=\Gamma_{B_{L}}-\Gamma_{B_{H}}$ and $\Delta
m_{s}=m_{B_{H}}-m_{B_{L}}$ are the decay-width and mass differences between
the light ($B_{L}$) and heavy ($B_{H}$) ${{B}^{0}_{s}}$ mass eigenstates and
$\Gamma_{s}$ is the average ${{B}^{0}_{s}}$ decay width. For the decays to the
CP-conjugated final states, the CP violation parameters $C_{f}$, $S_{f}$ and
$A_{f}^{\Delta\Gamma}$ are replaced with $C_{\overline{f}}$,
$S_{\overline{f}}$ and $A_{\overline{f}}^{\Delta\Gamma}$. These CP violation
parameters are related with $\gamma$ through

$\displaystyle C_{f}$ $\displaystyle=$ $\displaystyle-
C_{\overline{f}}=\frac{1-r_{B}^{2}}{1+r_{B}^{2}},$ (22) $\displaystyle S_{f}$
$\displaystyle=$
$\displaystyle\frac{2r_{B}\sin(\delta_{B}-(\gamma-2\beta_{s}))}{1+r_{B}^{2}},S_{\overline{f}}=\frac{-2r_{B}\sin(\delta_{B}+(\gamma-2\beta_{s}))}{1+r_{B}^{2}},$
$\displaystyle A_{f}^{\Delta\Gamma}$ $\displaystyle=$
$\displaystyle\frac{-2r_{B}\cos(\delta_{B}-(\gamma-2\beta_{s}))}{1+r_{B}^{2}},A_{\overline{f}}^{\Delta\Gamma}=\frac{-2r_{B}\cos(\delta_{B}+(\gamma-2\beta_{s}))}{1+r_{B}^{2}}$

following the definitions in Ref. [808]. In the formulae above, we have
assumed no CP violation in either the mixing and in the decay amplitude.

The time-dependent CP violation has been measured using 3.0 fb-1 and 9.0 fb-1
data for ${{B}^{0}_{s}}\rightarrow D_{s}^{\pm}K^{\mp}$ [805] and
${{B}^{0}_{s}}\rightarrow D_{s}^{\pm}K^{\mp}\pi^{+}\pi^{-}$ [804],
respectively. The measured values of the angle $\gamma$, using the world-
average value of $-2\beta_{s}$, are $\gamma=(128^{+17}_{-22})^{\circ}$ and
$\gamma=(44\pm 12)^{\circ}$, respectively (modulo 180∘). The
${{B}^{0}_{s}}\rightarrow D_{s}^{\pm}K^{\mp}\pi^{+}\pi^{-}$ is complicated due
to multiple bachelor particles, and an amplitude analysis is needed. However,
as discussed before, multi-body $B$ decays effectively introduce a dilution
factor $\kappa$, as Eq. 22 have five constraints while together with $\kappa$,
there are four unknown variables. The measurement is performed in the same
paper in a model-independent way, which leads to
$\gamma=(44^{+20}_{-13})^{\circ}$. The sensitivity is worse than model-
dependent results as expected.

Sensitivity studies with ${{B}^{0}_{s}}\rightarrow D\phi$ using a time-
integrated method has also been performed [809], the expected statistical
sensitivity of $\gamma$ is about $(8-19)^{\circ}$ using the
9$\text{\,fb}^{-1}$ $pp$ collision data collected by the LHCb experiment. It
is pointed out that additional sensitivity on $\gamma$ can be achieved using
the longitudinal polarised part of ${{B}^{0}_{s}}\rightarrow D^{*}\phi$ decays
with a partially reconstructed technique [807]. Besides, further sensitivities
on $\gamma$ can also be obtained by using other quasi-two-body decays in
${{B}^{0}_{s}}\rightarrow DK^{+}K^{-}$ decays [810].

####  4.1.5 Combination on $\gamma$

A combination of the parameter $\gamma$ was recently performed using all
available results of $\gamma$ measurements from the LHCb experiment [784]. The
determination of $\gamma$ relies on the inputs from charm decays; on the other
hand, precise $\gamma$ measurements and strong parameters of $B$ decays can
offer valuable constraints on the charm parameters, which in turn helps
constraining the mixing parameters of ${{D}^{0}}$ mesons. Therefore, the
measurements that are sensitive to the charm mixing parameters are also used
in the combination. The CKM angle $\gamma$ and charm mixing parameters are
simultaneously determined with significant improvements. Here we focus on the
$\gamma$ combination results, leaving those on the charm mixing parameters to
be discussed later in Section 5.1.

Figure 25: $\gamma$ combinations from different $B$ hadrons [784].

Figure 25 shows the $\gamma$ contributions from different $B$ decays. The
sensitivity on $\gamma$ mainly comes from $B^{+}$ decays. For $B^{0}$ and
${{B}^{0}_{s}}$ decays, where multi-body decays are usually involved, only a
few channels have been studied due to the complications in analysis
procedures. The small sample sizes and small number of available measurements
limit their sensitivity on $\gamma$. The central values of $\gamma$
determination from $B^{0}$ and ${{B}^{0}_{s}}$ are around $20^{\circ}$ higher,
which motivates further measurements with the $B^{0}$ and ${{B}^{0}_{s}}$
mesons to check overall consistency between different $B$ mesons. Special
efforts to the ${{B}^{0}_{s}}$ meson are well worth, since the
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mixing is involved and new physics contributions can easily enter inside.

The 1$\sigma$ contour of the constraint on $\gamma$ and strong parameters of
$B^{+}$ decays are shown in Fig. 26. Ambiguities on determination of $\gamma$
from two-body $D$ decays can be seen as has been discussed in sec. 4.1.1,
however, they constrain the parameter space into a very narrow region and
together with the unique determination of $\gamma$ from the BPGGSZ channel,
$\gamma$ can be precisely obtained. The combined $\gamma$ is found to be
$\gamma=(65.4^{+3.8}_{-4.2})^{\circ}$ and is the most precise determination
from a single experiment.

Figure 26: Combinations of $\gamma$ from all LHCb measurements, 2D contours of
$r_{B}$ vs $\gamma$ (left) and $\delta_{B}$ vs $\gamma$ (right) have been
shown at 68.3% confidence level [784].

###  4.2 CKM angle $\beta$

The angle $\beta$, defined as arg$(-V_{cd}V_{cb}^{*}/V_{td}V_{tb}^{*})$, is
approximately the phase of $V_{td}^{*}$ in the Wolfenstein parameterisation
[811]. It enters the decay time distributions of ${B}^{0}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ meson decays due to
$B^{0}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ oscillation.
The effective value of $\sin(2\beta)$, which could have been altered by NP
contributions in $B^{0}$-${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ mixing , can be extracted from
the time-dependent CP asymmetries of ${B}^{0}$ decays via
$b\rightarrow{{c}{\overline{c}}}s$ transitions following the relation

$A_{\textrm{CP}}(t)=\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}(t)\rightarrow f_{\rm
CP})-\Gamma(B^{0}(t)\rightarrow f_{\rm CP})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}(t)\rightarrow f_{\rm
CP})+\Gamma(B^{0}(t)\rightarrow f_{\rm
CP})}\approx\eta_{f}\sin(2\beta)\sin(\Delta mt)\;,$ (23)

where $f_{\rm CP}$ is a CP eigenstate with eigenvalue $\eta_{f}$, and the
approximation assumes no CP violation in the mixing or decay. Any significant
deviation of the measured $\sin(2\beta)$ value from the indirect determination
of $\sin(2\beta)$ through a global CKM fit excluding $\sin(2\beta)$
measurements is a clear sign of NP.

The ${{c}{\overline{c}}}$ pair could appear either in a charmonium meson or in
two charmed mesons in the final state. The precision of the $\beta$ is mainly
driven by ${B}^{0}$ decays to charmonium final-states due to their large decay
rates and distinct characteristics for identification. The world-average of
$\sin(2\beta)$ of all the charmonium measurements is sin$(2\beta)=0.699\pm
0.017$ [812]. The LHCb experiment has performed measurements of $\sin(2\beta)$
in the decays
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}})K^{0}_{S}$
[813], ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow
e^{+}e^{-})K^{0}_{S}$ and
${{B}^{0}}\rightarrow\psi(2S)(\rightarrow{\mu^{+}\mu^{-}})K^{0}_{S}$ [814].
The combined value is $\sin(2\beta)=0.760\pm 0.034$, the precision of which is
already comparable to that of the BaBar result $\sin(2\beta)=0.09\pm 0.03\pm
0.01$ [815] and the Belle result $\sin(2\beta)=0.67\pm 0.02\pm 0.01$ [816]. An
improvement by a factor of two is expected from measurements including LHCb
Run 2 data.

The presence of small penguin contributions in
$b\rightarrow{{c}{\overline{c}}}s$ processes may shift the measured values of
$\sin(2\beta)$ by up to few percent [817]. The decays via tree-level
$b\rightarrow{c}{\overline{u}}{d}$ transitions, though having smaller signal
yields due to the small branching fractions of $D$ decays, are free of the
penguin effects and thus theoretically clean. A recent joint analysis of the
decay $\overline{B}^{0}\rightarrow Dh^{0}$ with
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ by the BaBar and Belle
experiments measured $\sin(2\beta)=0.80\pm 0.14\pm 0.04\pm 0.03$ and
$\cos(2\beta)=0.91\pm 0.22\pm 0.09\pm 0.07$, which ruled out the other
solution of $\beta$ at 7.3$\sigma$ [818, 819]. Analysis of this decay is
challenging at LHCb due to the presence of ${{K}^{0}_{\mathrm{S}}}$ and
$\pi^{0}$ mesons in the final state. On the other hand, the decay
$\overline{B}^{0}\rightarrow D\pi^{+}\pi^{-}$ followed by $D\rightarrow
K^{+}K^{-}(\pi^{+}\pi^{-})$ only involves charged particle and thus is ideal
for LHCb to pursue.

###  4.3 CKM angle $\beta_{s}$

The angle $\beta_{s}$, defined as
$-$arg$(-V_{cb}V_{cs}^{*}/V_{tb}V_{ts}^{*})$, is approximately the phase of
$V_{ts}$ in the Wolfenstein parameterisation. The effective value of
$-2\beta_{s}$ can be measured in the time-dependent CP asymmetries of
${{B}^{0}_{s}}$ decays to CP eigenstates via
$b\rightarrow{{c}{\overline{c}}}s$ transitions, and is denoted
$\phi_{s}^{{{c}{\overline{c}}}s}$. In contract to the angle $\beta$,
$\beta_{s}$ is very small. The SM prediction for $\phi_{s}$ is $\phi_{s}^{\rm
SM}=-2\beta_{s}=-0.03696\pm 0.0004$ [820], which is subject to small
corrections due to the neglected penguin contributions in
$b\rightarrow{{c}{\overline{c}}}s$ decays. Presence of new particles in
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mixing diagrams may have a sizeable effect on $\phi_{s}$, making it an
sensitive probe of physics beyond the SM.

The LHCb experiment has performed measurements of
$\phi_{s}^{{{c}{\overline{c}}}s}$ in the decays
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}})\phi$
[821],
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}})\pi^{+}\pi^{-}$
[822],
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}})K^{+}K^{-}$
with $m(K^{+}K^{-})>1.05\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$ [823],
${{B}^{0}_{s}}\rightarrow{\psi{(2S)}}(\rightarrow{\mu^{+}\mu^{-}})\phi$ [824],
${{B}^{0}_{s}}\rightarrow D_{s}^{+}D_{s}^{-}$ [825] and more recently in
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{e^{+}e^{-}})\phi$
[826]. A combination of the measurements in
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow\mu^{+}\mu^{-})\phi$
and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow\mu^{+}\mu^{-})\pi^{+}\pi^{-}$
obtained using data taken during 2011-2016 gives
$\phi_{s}^{{{c}{\overline{c}}}s}=-0.042\pm 0.025$ rad [821].

The average of all LHCb measurements of $\phi_{s}^{{{c}{\overline{c}}}s}$ is
compared with the ATLAS [827] and CMS [828] results in Fig. 27, where
2-dimensional contours in the plane of $\phi_{s}^{{{c}{\overline{c}}}s}$
versus the ${B}^{0}_{s}$ decay width difference ($\Delta\Gamma_{s}$) are
displayed at 68% confidence level [812]. Note ATLAS and CMS have only
performed measurements in
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}\mu^{-}})\phi$
decays, due to constraints from their trigger systems,. The current world-
average value is $\phi_{s}^{{{c}{\overline{c}}}s}=-0.050\pm 0.019$ rad [812].
In Fig 27, one can see a good agreement in the
$\phi_{s}^{{{c}{\overline{c}}}s}$ measurements from different measurements.
However, some tension is observed for $\Delta\Gamma_{s}$ and $\Gamma_{s}$.
Factors of 2.5 and 1.77 have been applied to scale up the uncertainties of
$\Gamma_{s}$ and $\Delta\Gamma_{s}$ in the combination. Further investigations
by the relevant experiments are needed to solve this problem. With the
uncertainty of $\phi_{s}^{{{c}{\overline{c}}}s}$ well below its SM value, the
study of CP violation in ${B}^{0}_{s}$ decays enter an era of precision test,
and control of the penguin pollution in $\phi_{s}^{{{c}{\overline{c}}}s}$
using data-driven methods is essential for identification of NP signals in
${{B}^{0}_{s}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mixing [829, 830, 831, 832, 833, 834, 835, 836]. In this regard, the LHCb
experiment has measured CP violation in the penguin-enhanced
$b\rightarrow{{c}{\overline{c}}}d$ decays
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rho^{0}$ [837] and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ [838] decays. The measurements
are used to estimate the penguin shift of $\phi_{s}^{{{c}{\overline{c}}}s}$
measured in ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$,
assuming SU(3) flavour symmetry. The shift is found to be compatible with zero
[838], with an uncertain well below the statistical uncertainties of the
current $\phi_{s}^{{{c}{\overline{c}}}s}$ measurements.

Figure 27:  Combination of $\phi_{s}^{{{c}{\overline{c}}}s}$ measurements by
HFLAV [812], where 2D contours of (left) $\phi_{s}^{{{c}{\overline{c}}}s}$ vs
$\Delta\Gamma_{s}$ and (right) $\Gamma_{s}$ vs $\Delta\Gamma_{s}$ are
displayed at 68% confidence level.

Similar CP-violating phases can also be measured in decays of ${B}^{0}_{s}$
mesons via $b\rightarrow{{s}{\overline{s}}}s$ and
$b\rightarrow{{d}{\overline{d}}}s$ transitions, denoted
$\phi_{s}^{{{s}{\overline{s}}}s}$ and $\phi_{s}^{{{d}{\overline{d}}}s}$,
respectively. Since these decays are dominated by penguin diagrams with
internal top quarks, the phases $\phi_{s}^{{{s}{\overline{s}}}s}$ and
$\phi_{s}^{{{d}{\overline{d}}}s}$ receive contributions from the decay
amplitudes that cancel out the contribution of $-2\beta$ from the
${{B}^{0}_{s}}$ \- ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mixing, resulting vanishing values in the SM predictions. Measurements of
these quantities can probe NP in these FCNC decays. LHCb has measured
$\phi_{s}^{{{s}{\overline{s}}}s}$ and $\phi_{s}^{{{d}{\overline{d}}}s}$ in the
${{B}^{0}_{s}}\rightarrow\phi\phi$ and
${{B}^{0}_{s}}\rightarrow{{K}^{*}}(892){{K}^{*}}(892)$ decays using data taken
during 2011-2016, and the results are
$\phi_{s}^{{{s}{\overline{s}}}s}=-0.073\pm 0.115\pm 0.027$ rad [839] and
$\phi_{s}^{{{d}{\overline{d}}}s}=-0.10\pm 0.13\pm 0.14$ rad [840],
respectively, where the first uncertainties are statistical and the second
systematic.

###  4.4 CKM elements $V_{ub}$ and $V_{cb}$

The amplitudes of the CKM matrix elements $V_{ub}$ and $V_{cb}$ are measured
through semi-leptonic transitions of $b\rightarrow u\ell\nu$ and $b\rightarrow
c\ell\nu$. They have been extensively studied previously in $B$-factories
using the so-called exclusive and inclusive methods, which infer to whether a
specific decay channel is used or not. The two approaches suffer from
different theoretical and experimental uncertainties and offer important
cross-checks between each other. In the inclusive measurements, the Heavy
Quark Expansion (HQE) is used as $\Lambda_{\textrm{QCD}}/m_{b}$ is small and
the Operator Product Expansion (OPE) calculates non-perturbative contributions
involved. For the exclusive measurements, parameterisation of the form factor
of the corresponding decay is needed, where inputs are obtained from light-
cone sum rules (LCSR) [841, 842, 843, 844, 669, 845] or from lattice QCD
(LQCD). However, tensions have been found between the inclusive and exclusive
results as can be seen in Fig. 28 [846].

Figure 28:  Global fits of $|V_{ub}|$ and $|V_{cb}|$ from the CKMfitter group
[846] where measurements using different methods have been displayed.

Efforts from both experimental and theoretical sides are needed to understand
the discrepancy.

Unlike $B$-factories, where full kinematics can be obtained, LHCb can not
obtain kinematic information of neutrinos from energy-momentum conservation as
it covers only forward region. In addition, background contributions from
other $b$ and $c$ hadrons, and also huge combinatorial backgrounds randomly
combined from tracks other than signal makes the analyses of semi-leptonic
decays very complicated. Despite of these difficulties, the LHCb experiment
has successfully measured the ratio of $|V_{ub}|$ and $|V_{cb}|$ using
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p{\mu^{-}}{{\nu}_{\mu}}$
($q^{2}>15\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$) and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}\mu^{-}{{\nu}_{\mu}}$
($q^{2}>7\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$) [847]. The usage of the
control channel ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}\mu^{-}{{\nu}_{\mu}}$ not only cancels out common
systematic uncertainties between the two channels, but also offers a global
scale needed to determine $|V_{ub}|$ from branching fractions. Using the
updated branching fraction measurement of ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow pK^{-}\pi^{+}$, benefiting from the
${{\mathchar 28931\relax}^{+}_{c}}{{\kern
1.79993pt\overline{\kern-1.79993pt\mathchar 28931\relax}}{}^{-}_{c}}$ data
collected by the BESIII experiment [848], the ratio is determined to be
$|V_{ub}|/|V_{cb}|=0.079\pm 0.009$.

Using the same approach, the ratio of $|V_{ub}|$ and $|V_{cb}|$ is also
determined using ${{B}^{0}_{s}}\rightarrow K^{-}\mu^{+}\nu_{\mu}$ and
${{B}^{0}_{s}}\rightarrow D_{s}^{-}\mu^{+}{{\nu}_{\mu}}$ [849]. In the
measurement, two $q^{2}$ regions are used,
$q^{2}>7\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ and
$q^{2}<7\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ where the form factors are
obtained from LQCD and LCSR, respectively. However, the ratios of $|V_{ub}|$
and $|V_{cb}|$ obtained from the two methods differ significantly,
$|V_{ub}|/|V_{cb}|_{q^{2}<7\text{\,Ge\kern-0.70004ptV}^{2}\\!/c^{4}}=0.0607\pm
0.0015\pm 0.0013\pm 0.0008\pm 0.0030$ and
$|V_{ub}|/|V_{cb}|_{q^{2}>7\text{\,Ge\kern-0.70004ptV}^{2}\\!/c^{4}}=0.0946\pm
0.0030^{+0.0024}_{-0.0025}\pm 0.00013\pm 0.0068$, where the first uncertainty
of each result is statistical, the second systematic, the third due to
$D_{s}^{-}$ branching fraction, and the last one from the form factor. The
discrepancy between the two results clearly indicates that more efforts from
theoretical side are needed to resolve the tension on $|V_{ub}|$ and
$|V_{cb}|$ measurements.

In addition to the ratio between the two CKM matrix elements, the LHCb
experiment is also exploring its potential in determine the $|V_{cb}|$ alone
using ${{B}^{0}_{s}}\rightarrow D_{s}^{(*)-}\mu^{+}\nu_{\mu}$ [850]. The
branching fraction of ${{B}^{0}_{s}}\rightarrow D_{s}^{(*)-}\mu^{+}\nu_{\mu}$
is needed to set the global scale for the determination of $|V_{cb}|$. This is
obtained using the control channel $B^{0}\rightarrow D^{(*)-}\mu\nu_{\mu}$,
where the ratios of

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{{{B}^{0}_{s}}\rightarrow
D_{s}^{-}\mu^{+}\nu_{\mu}}{B^{0}\rightarrow D^{-}\mu\nu_{\mu}},$ (24)
$\displaystyle R^{*}$ $\displaystyle=$
$\displaystyle\frac{{{B}^{0}_{s}}\rightarrow
D_{s}^{*-}\mu^{+}\nu_{\mu}}{B^{0}\rightarrow D^{*-}\mu\nu_{\mu}},$ (25)

are determined. The data from LHCb offer $q^{2}$ dependence needed to extract
$|V_{cb}|$ together with non-perturbative inputs. Using form factor
parameterisations from Caprini, Lellouch and Neubert [851] or from Boyd,
Grinstein and Lebed [852, 853], the measured values of $|V_{cb}|$ are
$(41.4\pm 0.6\pm 0.9\pm 1.2)\times 10^{-3}$ and $(42.3\pm 0.8\pm 0.9\pm
1.2)\times 10^{-3}$, respectively, where the first uncertainty is statistical,
the second systematic, and the last one due to external inputs such as
theoretical inputs on form factors, branching fractions of $D_{s}^{-}$ or
$D^{-}$ decays, ${{B}^{0}_{s}}$ lifetime etc.

###  4.5 $\Delta m_{d}$ and $\Delta m_{s}$

The parameters $\Delta m_{d}$ and $\Delta m_{s}$ denote the mass differences
between the heavy and light mass eigenstate of the ${{B}^{0}}$ and
${B}^{0}_{s}$ systems, and define the oscillation frequencies of $B^{0}$
mixing and ${{B}^{0}_{s}}$ mixing, respectively. Currently, The most precise
determination of $\Delta m_{d}$ comes from the LHCb measurements in
semileptonic decays with a $D^{-}$ or $D^{*-}$ meson using 3 fb-1 of data.
Combining the results obtained in the two decay modes yields $\Delta
m_{d}=(0.5050\pm 0.0021\pm 0.0010)$ ps-1 [854], where the first uncertainty is
statistical and the second systematic. The world-average is $\Delta
m_{d}=0.5065\pm 0.0019$ ps-1 [812]. The determination of $\Delta m_{s}$ is
also led by the LHCb experiment. Combining the recent measurements in the
decays ${{B}^{0}_{s}}\rightarrow D_{s}^{-}\pi^{+}$ [855] and
${{B}^{0}_{s}}\rightarrow D_{s}^{-}\pi^{+}\pi^{-}\pi^{+}$ [804] and earlier
measurements yields $\Delta m_{s}=17.7656\pm 0.0057$ ps-1.

However, the constraints on the CKM matrix elements provided by the $\Delta
m_{d}$ and $\Delta m_{s}$ measurements rely on the decay constants and Bag
parameters of the the $B^{0}$ and ${{B}^{0}_{s}}$ mesons, which are obtained
from Lattice QCD calculations [856]. The precision of these hadronic
parameters is much worse than the experimental precision, thus limits the
constraining power of $\Delta m_{d}$ and $\Delta m_{s}$ on the CKM global fit.
Further improvements in Lattice QCD calculations are eagerly awaited.

###  4.6 Global fit

The four parameters of the CKM matrix, namely $A$, $\lambda$, $\rho$ and
$\eta$ in the Wolfenstein parameterization [811], are measured in different
processes and some of the key observables have been discussed in the above
sections. A global fit is needed to get the best sensitivity and to probe NP
effects. The $\chi^{2}$ value of the fit provides a measure of the overall
consistency between the different measurements, while the pull value for each
measurement quantifies the difference between the measured value and value
predicted by the fit results. Clues for new physics can be identified from
large $\chi^{2}$ or pull values.

The constraining of the CKM matrix is usually illustrated in complex planes
using triangles defined using unitarity relations of the CKM matrix. The most
commonly quoted CKM triangle corresponds to the relation
${V_{{u}{d}}^{\phantom{\ast}}}{V_{{u}{b}}^{\ast}}+{V_{{c}{d}}^{\phantom{\ast}}}{V_{{c}{b}}^{\ast}}+{V_{{t}{d}}^{\phantom{\ast}}}{V_{{t}{b}}^{\ast}}=0$,
where $V_{ij}$ is the CKM matrix element between the quarks of the flavours
$i$ and $j$. The results of the state-of-the-art global fit are shown in Fig.
29 provided by the CKMfitter group [820]. Inside, $\bar{\rho}$ and
$\bar{\eta}$ [857] are used to ensure the relationship
${\overline{\rho}}+i{\overline{\eta}}=-(V_{ud}V_{ub}^{*})/(V_{cd}V_{cb}^{*})$.

Figure 29: Global fit results of the CKM matrix from different measurements,
provided by the CKMfitter group [846]. Different colours and labels on the
plot indicate constraints from measurements of different observables.

Within current precision, different measurements cross on a single point and
give an overall consistent picture. However, the argument of matter-antimatter
asymmetry suggests that CP violation from sources beyond the CKM matrix may
break the consistency. One of the main efforts of the LHCb experiment is to
search for such a discrepancy by further improving measurement precision in
the beauty and charm sectors.

In the following decade, both the LHCb and Belle II experiments will
accumulate much more data to further constrain the CKM matrix. With data
collected till 2025, either experiment will be able to reduce the uncertainty
of $\gamma$ to around $1.5^{\circ}$, and further improve it to $0.3^{\circ}$
after 2030s. Other CKM angles and matrix elements will also be significantly
improved. Details on the future outlook can be found in section 6.2. Together
with improvements of other measurements and lattice calculations, NP may be
observed from inconsistency between different measurements.

##  5 Charm mixing and CP violation

Charm physics covers the studies of hadrons containing charm quarks. CP
violation in the charm sector is expected to be incredibly small in the SM, of
the order $\mathcal{O}(10^{-3})$ or less [326]. However, the presence of new
physics may enhance the amount of CP violation, which can be probed using the
enormously large sample of charmed hadrons at LHCb. Particularly, the study of
mixing and CP violation of neutral $D$ mesons can provide unique probes of NP
in FCNC transitions in the up-type quark sector, complementary to the study of
mixing and CP violation in neutral $B$ and $K$ mesons, which are sensitive to
NP in FCNC transitions of down-type quarks. For this reason, this section
mainly focuses on results in mixing and CP violation of ${D}^{0}$ mesons from
the LHCb experiment.

###  5.1 Neutral $D$ meson mixing

Similar to neutral ${{K}^{0}}$ and ${{B}_{({s})}^{0}}$ mesons, the neutral
charmed meson, ${{D}^{0}}$, can oscillate to its antiparticle partner,
${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$, via the short-distance
${W^{\pm}}$ exchange or long-distance rescattering diagrams, as shown in Fig.
30. This phenomenon of oscillation or mixing can be characterised by the
normalised (dimensionless) mixing parameters $x$ and $y$, defined as

$x\equiv\frac{\Delta M}{\Gamma},\quad y\equiv\frac{\Delta\Gamma}{2\Gamma},$
(26)

where $\Delta M$ ($\Delta\Gamma$) is the mass (decay width) difference of the
heavy and light mass eigenstates, and $\Gamma$ is the average decay width.
Unlike in the case of the ${{K}^{0}}$ or ${{B}_{({s})}^{0}}$ system, both $x$
and $y$ in the ${D}^{0}$ system are significantly smaller than unity, thus
very large data samples are required to observe ${D}^{0}$ mixing and determine
the tiny values of $x$ and $y$.

Figure 30: Oscillation of ${{D}^{0}}$ mesons via (top) ${W^{\pm}}$ exchanges
or (bottom) rescattering effect.

Evidence of ${{D}^{0}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$
mixing was first reported by BaBar [858] and Belle [859] in 2007, and later
also seen by CDF [860] in 2008. Subsequent measurements by BaBar [861, 862]
with different ${D}^{0}$ decay channels provided more evidences of the mixing.
The combination of these measurements confirmed the existence of charm mixing
with a significance more than $5\sigma$. The first observation of
${{D}^{0}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ mixing in a
single measurement was achieved by LHCb [863] in 2012 by using the data taken
in 2011 to study the time-dependent ratio of
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ (doubly Cabibbo-suppressed, DCS) to
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ (Cabibbo favoured, CF) decay rates.
The ${D}^{0}$ candidates are selected from the
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays, where the charge of the
pion directly from each ${D}^{*+}$ decay is used to determine the ${D}^{0}$
flavour at its production time.

The
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
process, referred to as the right-sign (RS) process, is dominated by a CF
decay, contaminated with a small contribution from the ${D}^{0}$-${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mixing followed by the DCS decay;
the
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{+}}{{\pi}^{-}}){{\pi}^{+}}$
process, referred to as wrong-sign (WS) process, includes contributions from
both the DCS decay and the ${D}^{0}$-${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mixing followed by the CF decay.
Under the assumption of small mixing and negligible CP violation, the time-
dependent ratio of the WS to the RS decay rates, $R$, is given by [864]

$R(t)\approx
R_{D}+\sqrt{R_{D}}\,y^{\prime}\,\dfrac{t}{\tau}+\dfrac{{x^{\prime}}^{2}+{y^{\prime}}^{2}}{4}\left(\dfrac{t}{\tau}\right)^{2},$

where $t/\tau$ is the decay time normalised to the average ${{D}^{0}}$
lifetime, $R_{D}$ is the ratio between the DCS and CF decay rates, and
$x^{\prime}$ and $y^{\prime}$ are the mixing parameters ‘rotated’ by the
strong phase difference $\delta$ between the DCS and CF amplitudes:
$x^{\prime}=x\cos\delta+y\sin\delta$ and $y^{\prime}=y\cos\delta-x\sin\delta$.
The time evolution of the ratio $R$ is shown in Fig. 31. Further studies with
larger data samples have also been performed by LHCb [863, 865, 866, 867] and
the results are summarized in Table 4.

Figure 31:  Ratio of the ${{D}^{0}}\rightarrow{{K}^{+}}\pi$ to ${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay rates, $R$, as a function of the ${D}^{0}$ decay time normalised by the average ${{D}^{0}}$ lifetime [863]. The solid (dashed) line indicates the projection of the mixing (no-mixing) fits.  Table 4:  Summary of the charm mixing parameters measured by LHCb using ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays. Results under the assumptions of CP invariance and CP violation are given.  | no CPV | CPV allowed
---|---|---
Data sample |  $x^{\prime 2}$ ($\times 10^{-3}$) |  $y^{\prime}$ ($\times 10^{-3}$) |  $x^{\prime 2}$ ($\times 10^{-3}$) |  $y^{\prime}$ ($\times 10^{-3}$)
$1.0\text{\,fb}^{-1}$, $D^{*}$ tag [863] | $-0.09\pm 0.13$ | $7.2\pm 2.4$ | - | -
$3.0\text{\,fb}^{-1}$, $D^{*}$ tag [865] | $0.055\pm 0.049$ | $2.8\pm 1.0$ |  ${{D}^{0}}$: $\phantom{0}0.049\pm 0.070$ | $5.1\pm 1.4$
${{\kern 1.61993pt\overline{\kern-1.61993ptD}}{}^{0}}$: $\phantom{0}0.060\pm 0.068$ | $4.5\pm 1.4$
$3.0\text{\,fb}^{-1}$, $B$ tag [866] | $0.028\pm 0.310$ | $4.6\pm 3.7$ |  ${{D}^{0}}$: $-0.019\pm 0.447$ | $5.81\pm 5.26$
${{\kern 1.61993pt\overline{\kern-1.61993ptD}}{}^{0}}$: $\phantom{0}0.079\pm 0.433$ | $3.32\pm 5.23$
$5.0\text{\,fb}^{-1}$, $D^{*}$ tag [867] | $0.039\pm 0.027$ | $5.28\pm 0.52$ |  ${{D}^{0}}$: $\phantom{0}0.061\pm 0.037$ | $5.01\pm 0.74$
${{\kern 1.61993pt\overline{\kern-1.61993ptD}}{}^{0}}$: $\phantom{0}0.016\pm 0.039$ | $5.54\pm 0.74$

The measurements in the ${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay are
sensitive to the normalised decay-width difference $y$ and the sum
$x^{2}+y^{2}$ (under the assumption of negligible CP violation), but not to
the sign of the normalised mass difference $x$. One approach to solve this
problem is to study the Dalitz distributions of three-body decays. The ‘golden
channel’ at LHCb for such studies is the decay
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$, where the
decay to the ${{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ final state
proceed mainly via the following three processes with different intermediate
resonances: 1) the ${{K}^{0}_{\mathrm{S}}}{{\rho}^{0}}$ process with
${{\rho}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$, which is common for both
${{D}^{0}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ mesons;
2) the ${{K}^{*-}}{{\pi}^{+}}$ process with
${{K}^{*-}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{-}}$, which is a CF decay;
and 3) the ${{K}^{*+}}{{\pi}^{-}}$ process with
${{K}^{*+}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}$, which is either a
DCS decay or ${D}^{0}$-${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$
oscillation followed by a CF decay.

In the Dalitz phase space, the DCS and CF decay amplitudes of the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decay
populate the same space and interfere. Therefore, the parameters $x$ and $y$
can be determined by measuring the strong phase difference between the
contributing amplitudes in an amplitude analysis, or by importing the average
strong-phase difference in regions of phase space obtained by ${e^{+}e^{-}}$
experiments operating at the energy of the $\psi(3770)$ resonance. The latter
approach is employed in several LHCb measurements [868, 869, 870], and the
most recent measurement [870] led to the first observation of a nonzero mass
difference between the two mass eigenstates in the ${{D}^{0}}$-${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ system. The results of these
measurements using
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays are
summarised in Table 5.

Table 5:  Summary of the charm mixing parameters measured by LHCb using ${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays.  | CP-averaged parameters
---|---
Data sample |  $x$ ($\times 10^{-3}$) |  $y$ ($\times 10^{-3}$)
$1.0\text{\,fb}^{-1}$, $D^{*}$ tag [868] | $-8.6\pm 5.3\pm 1.7$ | $0.3\pm 4.6\pm 1.3$
$3.0\text{\,fb}^{-1}$, $B$ tag [869] | $2.7\pm 1.6\pm 0.4$ | $7.4\pm 3.6\pm 1.1$
$5.4\text{\,fb}^{-1}$, $D^{*}$ tag [870] | $3.97\pm 0.46\pm 0.29$ | $4.59\pm 1.20\pm 0.85$
| CP-violating parameters
Data sample |  $\Delta x$ ($\times 10^{-3}$) |  $\Delta y$ ($\times 10^{-3}$)
$3.0\text{\,fb}^{-1}$, $B$ tag [869] | $-0.53\pm 0.70\pm 0.22$ | $0.6\pm 1.6\pm 0.3$
$5.4\text{\,fb}^{-1}$, $D^{*}$ tag [870] | $-0.27\pm 0.18\pm 0.01$ | $0.20\pm 0.36\pm 0.13$

As mentioned in Section 4.1.5, different from the past LHCb $\gamma$
combinations, the recent combination exploited the LHCb measurements that are
sensitive to the CKM angle $\gamma$ and to the charm mixing parameters, and
the $\gamma$ angle and charm mixing parameters are simultaneously determined
[784]. The motivation for the simultaneous combination is as follows:

  * •

The $\gamma$ angle and the strong phase difference between the interfering $B$
decays are now so precisely constrained by the large $B$-meson samples that
the strong phase difference, $\delta_{D}^{K\pi}$, between the decays
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$
can achieve a precision of about a factor of two better than the previous
world average [812]. This improvement can then be used to improve the
precision of the charm mixing parameters $x$ and $y$.

  * •

Due to non-negligible effects originating from charm-meson mixing, a
simultaneous combination is needed to obtain an unbiased determination of the
$\gamma$ angle and the charm mixing parameters $x$ and $y$.

In the charm sector, the inputs used in the combination are obtained from the
time-dependent measurements of ${{D}^{0}}\rightarrow h^{+}h^{-}$,
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$,
${{D}^{0}}\rightarrow{{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{+}}{{\pi}^{-}}$, and
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays
performed by LHCb [790, 871, 872, 873, 874, 875, 876, 877, 866, 867, 878, 869,
870, 868]. Figure 32 shows the two-dimensional profile likelihood contours in
the $x$-$y$ plane. The values of $x$ and $y$, determined in the simultaneous
combination, are found to be

$x=(0.400{}_{-0.053}^{+0.052})\%,\quad y=(0.630{}_{-0.030}^{+0.033})\%.$

These results provide the most precise determinations of the parameters $x$
and $y$. Particularly, the precision of $y$ is improved by a factor of two
with respect to the current world average [812].

Figure 32:  Profile likelihood contours (from 1 to 5 $\sigma$) of the charm
mixing parameters $x$ and $y$. The solid (brown) contours are determined from
the simultaneous combination [784], while the dashed (blue) indicate the
current world average from Ref. [812].

###  5.2 CP violation

####  5.2.1 Time-integrated CP violation

The time-integrated CP asymmetry, $\mathcal{A}_{CP}$, in the decay
$D\rightarrow f$ is dominated by the direct CP asymmetry. Its measurement
follows the formula

$\mathcal{A}_{CP}=\mathcal{A}_{raw}-\mathcal{A}_{prod}-\mathcal{A}_{det},$
(27)

where $\mathcal{A}_{prod}$ denotes the meson production asymmetry between the
$c$-hadron and its antiparticle, $\mathcal{A}_{det}$ represents the detection
asymmetry, and $\mathcal{A}_{raw}$ is the raw asymmetry between the yields of
$D\rightarrow f$ and $\overline{D}\rightarrow\overline{f}$ decays. Often the
difference of CP asymmetries between two different decay processes are
measured, which is defined as

$\Delta\mathcal{A}_{CP}\equiv\mathcal{A}_{CP}(D\rightarrow
f_{1})-\mathcal{A}_{CP}(D\rightarrow f_{2}),$ (28)

where $f_{1}$ and $f_{2}$ are two different final states with similar
topologies. The effects of production asymmetry and CP asymmetries in mixing
as well as part of the detection asymmetries on $\Delta\mathcal{A}_{CP}$ are
largely cancelled.

Two-body decays of $D$ mesons are particularly interesting due to their super
large sample sizes, which are crucial for probing the tiny CP violating
effects. The first observation of CP violation in the charm sector was
reported by LHCb in 2019 using the Run 2 data [790]. The difference of
$\mathcal{A}_{CP}$ between the ${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays,
$\Delta\mathcal{A}_{CP}\equiv\mathcal{A}_{CP}({{K}^{+}}{{K}^{-}})-\mathcal{A}_{CP}({{\pi}^{+}}{{\pi}^{-}})$,
was measured with a deviation from zero corresponding to a significance of
$5.3\,\sigma$. Table 6 summarises the $\Delta\mathcal{A}_{CP}$ results of a
series of LHCb measurements [879, 880, 871, 872, 790].

Several $\mathcal{A}_{CP}$ measurements using two-body decays have been
performed by LHCb as well [881, 882, 883, 884, 885, 886, 887, 888, 889, 890],
and the first evidence of direct CP violation in a specific charm hadron decay
was reported in Ref. [890].

Table 6:  Summary of the $\Delta\mathcal{A}_{CP}$ results from the LHCb measurements in the charm sector.  Data sample |  $\Delta\mathcal{A}_{CP}$ ($\times 10^{-3}$)
---|---
$0.62\text{\,fb}^{-1}$, $D^{*}$ tag [879] | $-8.2\pm 4.1\pm 0.6$
$1.0\text{\,fb}^{-1}$, $B$ tag [880] | $4.9\pm 3.0\pm 1.4$
$3.0\text{\,fb}^{-1}$, $B$ tag [871] | $1.4\pm 1.6\pm 0.8$
$3.0\text{\,fb}^{-1}$, $D^{*}$ tag [872] | $-1.0\pm 0.8\pm 0.3$
$5.9\text{\,fb}^{-1}$, $B$ or $D^{*}$ tag [790] | $-1.54\pm 0.29$

While multi-body charm decays often have much smaller sample sizes compared to
two-body charm decays, they can provide excellent opportunities for CP
violation measurements. The presence of intermediate resonances can lead to
large variation of the strong phase difference between the interfering
amplitudes, which can lead to sizeable local CP asymmetries. Besides the
$\Delta\mathcal{A}_{CP}$ method, several techniques to search for CP violation
in multi-body charm decays are exploited by LHCb, including amplitude analysis
[891], the binned $\chi^{2}$ technique [892, 893, 894, 895, 896], and an
unbinned technique called the energy test [897, 898]. For the latter two
methods, model-dependent analyses are eventually required to pin down the
source in case significant CP violation were observed.

The binned $\chi^{2}$ technique computes the distribution of local asymmetries
and compare it with a normal distribution to judge if CP violation were
observed. An example of binned $\chi^{2}$ distribution in a Dalitz plot is
shown in Fig. 33. This method relies on the optimal choice of the binning
scheme. Wide bins across resonances can lead to the cancellation of real CP
asymmetries within a bin.

Figure 33: (left) Distributions of local per-bin asymmetry significance
$S_{CP}$ and (right) corresponding one-dimensional distributions obtained by
the binned $\chi^{2}$ method for ${\mathchar 28932\relax}_{c}^{+}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$ decays. Figures taken from Ref. [896].

The LHCb collaboration has developed a novel unbinned method, energy test
[899, 900], to perform model-independent search for CP violation in many-body
decays. With this method, a test statistic, $T$, is defined. For a given data
sample, a $p$-value for the hypothesis of CP invariance is assigned by
comparing the observed value of $T$ to the distribution of $T$ obtained from
many random permutations of the data. This method has been applied to search
for CP violation in decays of charm mesons and beauty baryons. As an example,
Fig. 34 shows the global test statistic compared with the distribution of the
statistic from many random permutations, and the Dalitz plot distribution of
significance of local test statistics in
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{0}}$ decays. Despite the
many efforts made by LHCb and the significant improvements in the measurement
precision, no evidence of CP violation in multi-body charm decays has ever
been found to date. Table 7 summarises the searches for direct CP violation in
phase space of charm decays by LHCb.

Figure 34:  (a) The global test statistic compared with the distribution of the statistic calculated from a large number of random permutations obtained from the energy test applied in the CP violation searches in ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}\pi^{0}$ decays. (b) Dalitz plot distribution of significance of local test statistics. Figures are taken from Ref. [897].  Table 7:  Summary of LHCb direct CP violation searches in phase space of charm decays. Decay channel | Data sample | Method
---|---|---
$D^{+}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{+}}$ [892] | $35\text{\,pb}^{-1}$ | binned $\chi^{2}$
${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ [893] |  $1.0\text{\,fb}^{-1}$, $D^{*}$ tag | binned $\chi^{2}$
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ [893] |  $1.0\text{\,fb}^{-1}$, $D^{*}$ tag | binned $\chi^{2}$
$D^{+}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ [894] | $1.0\text{\,fb}^{-1}$ | binned $\chi^{2}$
${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ [895] |  $3.0\text{\,fb}^{-1}$, $B$ tag | binned $\chi^{2}$
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}\pi^{0}$ [897] |  $2.0\text{\,fb}^{-1}$, $D^{*}$ tag | energy test
${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ [898] |  $3.0\text{\,fb}^{-1}$, $D^{*}$ tag | energy test
$\Lambda_{c}^{+}\rightarrow ph^{-}h^{+}$ [901] | $3.0\text{\,fb}^{-1}$ | $\Delta\mathcal{A}_{CP}$
${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ [891] |  $3.0\text{\,fb}^{-1}$, $B$ tag | amplitude analysis
$\Xi_{c}^{+}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ [896] | $3.0\text{\,fb}^{-1}$ | binned $\chi^{2}$

####  5.2.2  $y_{{C\\!P}}$ and $A_{\Gamma}$ measurements

The amplitudes of the direct decay of ${D}^{0}$ to a CP eigenstate and the
decay after mixing can interfere and lead to indirect CP asymmetry. Its
contribution to the time-integrated CP asymmetry is denoted
$\mathcal{A}_{CP}^{ind}$.

Due to the ${{D}^{0}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$
mixing, the effective decay width of ${{D}^{0}}$ decays to a CP-even final
state (e.g. $f$=${{K}^{+}}{{K}^{-}}$ or ${{\pi}^{+}}{{\pi}^{-}}$),
$\Gamma_{CP+}$, differs from the average decay width $\Gamma$. We can define
the parameter $y_{{C\\!P}}\equiv\Gamma_{CP+}/\Gamma-1$ to represent the amount
of CP-violation in mixing. The quantity $y_{{C\\!P}}$ is related to $x$ and
$y$, $|q/p|$, and $\phi\equiv arg(q\overline{A}/pA)$,

$y_{{C\\!P}}\approx\dfrac{1}{2}\left(\left|\dfrac{q}{p}\right|+\left|\dfrac{p}{q}\right|\right)y\cos\phi-\dfrac{1}{2}\left(\left|\dfrac{q}{p}\right|-\left|\dfrac{p}{q}\right|\right)x\sin\phi\;.$

Only if CP is conserved, $y_{{C\\!P}}$ is equal to $y$. The decay rate
asymmetry is defined as

$\displaystyle
A_{\Gamma}\approx\dfrac{1}{2}\left(\left|\dfrac{q}{p}\right|-\left|\dfrac{p}{q}\right|\right)y\cos\phi-\dfrac{1}{2}\left(\left|\dfrac{q}{p}\right|+\left|\dfrac{p}{q}\right|\right)x\sin\phi\;.$

The asymmetry $A_{\Gamma}$ is related to the indirect CP asymmetry
$\mathcal{A}_{CP}^{ind}$ through $\mathcal{A}_{CP}^{ind}=-A_{\Gamma}$.

The quantities $y_{{C\\!P}}$ and $A_{\Gamma}$ can be determined by measuring
the ratio of the effective lifetimes of ${{D}^{0}}$ and $\bar{{{D}^{0}}}$
decays to the same CP eigenstate:

$\displaystyle y_{{C\\!P}}$ $\displaystyle=\dfrac{2\tau({{D}^{0}}\rightarrow
f_{\not{C}\not{P}})}{\tau({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow
f_{CP})+\tau({{D}^{0}}\rightarrow f_{CP})}-1,$ (29) $\displaystyle A_{\Gamma}$
$\displaystyle=\dfrac{\tau({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow
f_{CP})-\tau({{D}^{0}}\rightarrow f_{CP})}{\tau({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow
f_{CP})+\tau({{D}^{0}}\rightarrow f_{CP})},$ (30)

where $f_{\not{C}\not{P}}$ denotes a non-CP-eigenstate, such as
${{K}^{-}}{{\pi}^{+}}$.

In recent years, LHCb has preformed several measurements of $y_{{C\\!P}}$ and
$A_{\Gamma}$, which are summarized in Table 8 [902, 903, 874, 873, 875, 876,
904]. A recent study shows that using the average decay width of
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays as a proxy to the average decay width of the neutral charm meson mass
eigenstates $D_{1}$ and $D_{2}$ does not give direct access to $y_{{C\\!P}}$
but rather corresponds to $y_{{C\\!P}}-y_{{C\\!P}}^{K\pi}$ [905], where
$y_{{C\\!P}}^{K\pi}$ is approximately equal to $-0.4\times 10^{-3}$[904]. In
Ref [877], an LHCb legacy result of $A_{\Gamma}$ combined with both $D^{*}$
and $B$ flavour tag using 2011-2012 and 2015-2018 data sample is obtained.
None of these measurements shows any indication of CP violation in
${{D}^{0}}$-${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ mixing or
in the interference between mixing and decay. Figure 35 compares the
$y_{{C\\!P}}$ and $A_{\Gamma}$ measurements performed by different
experiments, and the averages provided by the Heavy Flavour Averaging Group
[812] . The world averages are dominated by the measurements by the LHCb
experiment.

Table 8:  Summary of LHCb $y_{{C\\!P}}$ and $A_{\Gamma}$ measurements.

Data sample | Final state(s) |  $y_{{C\\!P}}$ (%) |  $A_{\Gamma}$ ($\times 10^{-3}$)
---|---|---|---
$29\text{\,pb}^{-1}$, $D^{*}$ tag [902] | ${{K}^{+}}{{K}^{-}}$ | $0.55\pm 0.63\pm 0.41$ | $-5.9\pm 5.9\pm 2.1$
$1.0\text{\,fb}^{-1}$, $D^{*}$ tag [903] | ${{\pi}^{+}}{{\pi}^{-}}$ | - | $0.33\pm 1.06\pm 0.14$
$1.0\text{\,fb}^{-1}$, $D^{*}$ tag [903] | ${{K}^{+}}{{K}^{-}}$ | - | $-0.35\pm 0.62\pm 0.12$
$3.0\text{\,fb}^{-1}$, $B$ tag [874] | ${{\pi}^{+}}{{\pi}^{-}}$ | - | $-0.92\pm 2.6^{+0.25}_{-0.33}$
$3.0\text{\,fb}^{-1}$, $B$ tag [874] | ${{K}^{+}}{{K}^{-}}$ | - | $-1.34\pm 0.77^{+0.26}_{-0.34}$
$3.0\text{\,fb}^{-1}$, $B$ tag [874] |  ${{\pi}^{+}}{{\pi}^{-}}$ & ${{K}^{+}}{{K}^{-}}$ | - | $-1.25\pm 0.73$
$3.0\text{\,fb}^{-1}$, $B$ tag [873] |  ${{\pi}^{+}}{{\pi}^{-}}$ & ${{K}^{+}}{{K}^{-}}$ | $0.57\pm 0.13\pm 0.09$ | -
$3.0\text{\,fb}^{-1}$, $D^{*}$ tag [875] | ${{\pi}^{+}}{{\pi}^{-}}$ | - | $0.46\pm 0,58\pm 0.12$
$3.0\text{\,fb}^{-1}$, $D^{*}$ tag [875] | ${{K}^{+}}{{K}^{-}}$ | - | $-0.30\pm 0.32\pm 0.10$
$3.0\text{\,fb}^{-1}$, $D^{*}$ tag [875] |  ${{\pi}^{+}}{{\pi}^{-}}$ & ${{K}^{+}}{{K}^{-}}$ | - | $-0.13\pm 2.0\pm 0.7$
$5.4\text{\,fb}^{-1}$, $B$ tag [876] | ${{\pi}^{+}}{{\pi}^{-}}$ | - | $0.22\pm 0.70\pm 0.08$
$5.4\text{\,fb}^{-1}$, $B$ tag [876] | ${{K}^{+}}{{K}^{-}}$ | - | $-0.43\pm 0.36\pm 0.05$
$6\text{\,fb}^{-1}$, $D^{*}$ tag [877] | ${{\pi}^{+}}{{\pi}^{-}}$ | - | $0.4\pm 0.28\pm 0.04$
$6\text{\,fb}^{-1}$, $D^{*}$ tag [877] | ${{K}^{+}}{{K}^{-}}$ | - | $0.23\pm 0.15\pm 0.03$
$8.4\text{\,fb}^{-1}$, $D^{*}$ or $B$ tag [877] | ${{\pi}^{+}}{{\pi}^{-}}$ | - | $0.36\pm 0.24\pm 0.04$
$8.4\text{\,fb}^{-1}$, $D^{*}$ or $B$ tag [877] | ${{K}^{+}}{{K}^{-}}$ | - | $0.03\pm 0.13\pm 0.03$
$8.4\text{\,fb}^{-1}$, $D^{*}$ or $B$ tag [877] |  ${{\pi}^{+}}{{\pi}^{-}}$ & ${{K}^{+}}{{K}^{-}}$ | - | $0.10\pm 0.11\pm 0.03$
$6\text{\,fb}^{-1}$, $D^{*}$ tag [904] | ${{\pi}^{+}}{{\pi}^{-}}$ |  $0.657\pm 0.053\pm 0.016$ 1110 | -
$6\text{\,fb}^{-1}$, $D^{*}$ tag [904] | ${{K}^{+}}{{K}^{-}}$ |  $0.708\pm 0.030\pm 0.014$ 2220 | -
$6\text{\,fb}^{-1}$, $D^{*}$ tag [904] |  ${{\pi}^{+}}{{\pi}^{-}}$ & ${{K}^{+}}{{K}^{-}}$ |  $0.696\pm 0.026\pm 0.013$ 3330 | -

1\. $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ is measured in this analysis.
2\. $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ is measured in this analysis.
3\. $y_{{C\\!P}}-y_{{C\\!P}}^{K\pi}$ is measured in this analysis.

Figure 35: The world averages of $y_{{C\\!P}}$ and $A_{\Gamma}$. Figures taken
from Ref. [812]

##  6 Prospects and summary

###  6.1 Upgrade plan of LHCb

The physics output discussed in this review shows that LHCb has successfully
deepened our understanding of flavour physics with experimental data taken up
to the year of 2018. Most of the the key flavour observables are measured to
an unprecedented precision, yet it is generally true that the uncertainties
are still dominated by statistical fluctuation. To further increase the
availability of high-quality collision data, the LHCb detector is currently
under a major upgrade [906], known as Upgrade Ia or simply Upgrade I. The
installation has almost completed by the end of the second Long Shutdown (LS2)
of the LHC, and the upgraded detector is starting to take data in 2022 with an
instantaneous luminosity of $2\times
10^{33}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$, five times the value achieved so
far. Fig. 36 shows the plan for LHCb operation after upgrade. LHCb aims to
accumulate an integrated luminosity of approximately $23\text{\,fb}^{-1}$ by
the end of Run 3 around 2025, and a total of $50\text{\,fb}^{-1}$ by the end
of Run 4. Note that during the Long Shutdown 3 (LS3) between Run 3 and 4,
intensive work will be done on the machine configuration to prepare the High
Luminosity Large Hadron Collider (HL-LHC) [907]. During this period
consolidation work (Upgrade Ib) will be carried out at LHCb with only minor
change on the detector configuration or performance. In order to fully exploit
the HL-LHC potential in flavour physics, the collaboration plans another major
upgrade, Upgrade II [908], to enable the detector to operate at luminosity as
high as $1.5\times 10^{34}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$. This will allow
for an integrated luminosity of $\sim 300\text{\,fb}^{-1}$ to be achieved in
the lifetime of the (HL-)LHC.

Figure 36: Instantaneous and integrated luminosities of LHCb as functions of
year.

To put the LHCb prospects in context, other players in the field of flavour
physics study have to be mentioned. The Belle II experiment [909], the
$B$-factory at the superKEKB [910], has started to take collision data of
${e^{+}e^{-}}\rightarrow{\mathchar 28935\relax{(4S)}}\rightarrow{B}{\kern
1.79993pt\overline{\kern-1.79993ptB}}$ since 2018, aiming to collect an
integrated luminosity of $50\text{\,ab}^{-1}$ by 2025. Belle II and LHCb are
expected to be competitive in pushing the measurement precision, though with
very different systematic uncertainties, of a number of key flavour
observables such as the CKM unitarity angles, the Wilson coefficients and the
tests on lepton flavour universality. Given the difference in production
mechanism and corresponding detector features, their complementarity should be
more appreciated. With the beam energy constraints Belle II will be generally
good at treating final states with neutral particles ($\gamma$, ${{\pi}^{0}}$)
or missing energy (neutrinos); It has unique advantage in studying the $\tau$
leptons through ${e^{+}e^{-}}\rightarrow{\tau^{+}\tau^{-}}$ process. With the
huge cross section of heavy hadrons, LHCb will have larger yield in most of
the final states especially those with charged tracks. The beauty hadrons
produced at high-energy $pp$ collisions are highly boosted, hence their decay
vertices could be well separated from the primary vertices. With an excellent
vertex detector LHCb fully exploits this phenomenon to suppress background for
most $B$ signatures to an extremely low level. In addition to $B$ mesons, the
studies at LHCb extend to all type of heavy hadrons like ${{B}^{0}_{s}}$,
${{B}_{c}^{+}}$ and beauty baryons. Future electron positron colliders
proposed primarily for Higgs study, such as CEPC [911] and FCC-ee [912], also
plan to operate at $Z$ pole. With yields of $B$-mesons comparable or higher
than Belle II depending on the machine design, future $Z$ factories produce
all spectrum of beauty hadrons with large boost and efficient trigger, hence
will also contribute to heavey flavour physics with unique advantages. At the
HL-LHC era ATLAS and CMS will keep pushing the high-energy frontier by
studying the Higgs and searching for signature of new physics beyond the
Standard Model, meanwhile the yield of beauty hadrons will be so considerable
that measurement of a few key flavour observables can be made precisely,
especially those with a pair of muons in their final state. They are designed
to perform well at high $p_{\mathrm{T}}$ and central rapidity, perfectly
complementing the kinematic range of LHCb. The BESIII experiment [913] will
continue to operate for another 10 years [914] and accumulate a sample of
charm mesons corresponding to $\sim 20\text{\,fb}^{-1}$, which will not only
allow more mixing and CP violation study in the charm sector but also help to
reduce the uncertainties related to the charm strong phase in measurement of
the CKM $\gamma$ angle [915]. Even larger $D\bar{D}$ samples at the order of
$\sim\mathrm{ab}^{-1}$ are expected at proposed future tau-charm factories
SCTF [916] and STCF [917].

####  6.1.1 Upgrade I

The ongoing Upgrade I aims to increase the instantaneous luminosity of LHCb
from the current value of $4\times 10^{32}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$
to $2\times 10^{33}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$. Naïvely it would mean a
five-fold increase in all signal yield, which will be roughly true for final
states with muons. Actually the gain for all-hadronic final state will be more
than that due to a major change in the trigger system [918]. The hardware
trigger L0, which reduces the data rate from 40 MHz to 1 MHz, will be
completely removed after Upgrade I, allowing a more flexible full software
trigger. Generally the trigger efficiencies for all-hadronic final states are
expected to be doubled, as taken in simulation study of upgrade performances,
however this number could vary depending on the individual channel.

The increased pile-up causes much higher combinatorial background and more
challenging track reconstruction, therefore the tracking systems [919, 920]
have been completely redesigned with higher granularity and better radiation
tolerance so as to provide uncompromised tracking performance at higher pile-
up. Components of the particle identification systems [921] will be reused as
much as possible, yet the readout electronics will be replaced in accordance
with the 40 MHz readout rate. As a result of higher luminosity, improved
trigger rate and larger number of output channels, the data volume to be
treated either in real time or offline will be substantially higher, hence new
software infrastructure and computing models have been developed
correspondingly [922, 923, 924] to ensure physics data to be processed and
stored in a timely manner. A new subsystem has been installed to enhance the
detector’s capability in fixed target and heavy-ion studies without disturbing
the main physics program [925]. The simulation study shows that detector
performance after upgrade will be at least as good as before, with improvement
at some areas.

####  6.1.2 Upgrade II

By the end of Run 4 LHCb will have accumulated 50 $\text{\,fb}^{-1}$ $pp$
collision data, with many subdetectors reaching end of lifetime. Operation at
the same condition beyond that point would be less attractive. To fully
exploit the HL-LHC potential in flavour physics study, the collaboration
proposed Upgrade II towards an integrated luminosity of $300\text{\,fb}^{-1}$
[908]. The physics cases with the luminosity an order of magnitude higher than
before HL-LHC time have been studied extensively by the collaboration and
summarised in a document in 2018 [664]. A few benchmarks will be discussed
below. Note that the HL-LHC baseline design assumes LHCb running condition to
be the same as in Run 3, the HL-LHC experts recently released a report on the
upgrade feasibility from the machine side [926] showing that possible solution
of operating at a luminosity of $1.5\times
10^{34}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$ will allow the target of
$300\text{\,fb}^{-1}$ to be met.

The seven-fold increase of luminosity will again impose more technical
challenges for the experiment. The expected number of interactions per
crossing is around 40, twenty times of current situation (or a hundred times
of the LHCb initial design). Fast timing resolution will be required in most
subsystems to fight against the combinatorial backgrounds caused by the pile-
up. Finer granularity in all tracking detectors is compulsory under much
higher multiplicity. Radiation hardness will be more of concern especially for
areas close to the beampipe. A daunting amount of 200 Tb of data will be
produced every second, and has to be reduced by four orders of magnitude
before stored permanently. New subsystems are being proposed in order to
extend geometrical acceptance for low-momentum tracks, and to improve $K/\pi$
separation at lower momentum. A lot of development activities have been
launched driven by these requirements, while exploiting new technologies in
detector and computing. A framework Technical Design Report summarising these
activities was released recently [927].

###  6.2 Physics prospects

Before the HL-LHC or by the end of Run 3, LHCb will have taken
$23\text{\,fb}^{-1}$ data, drastically reducing the statistical uncertainties
for most of channels compared with current measurements. The expected
projections are studied in detail [906] and updated with inputs from
experiences in Run I [928, 929]. The physics opportunities in Upgrade II with
$300\text{\,fb}^{-1}$ have also been studied [664], which concludes that the
energy scale probed by flavour observables will be doubled with respect to
pre-HL-LHC era. The sensitivity of a selection of key flavour observables
after LHCb upgrades are listed in Table 9 and illustrated in Fig. 37, mostly
from Ref. [664] with minor updates when available. Note that Belle II will
have completed data taking when LHCb collects $23\text{\,fb}^{-1}$ data.
Expected projection from Belle II, ATLAS and CMS are listed for comparison
when applicable. A few highlights will be briefly mentioned here.

Table 9: Sensitivity of selected key flavour observables for LHCb, ATLAS and
CMS, taken from Ref. [664] with updates when available, and Belle II
sensitivities from Ref. [930].

| LHCb | LHCb | LHCb | Belle II | ATLAS
---|---|---|---|---|---
Observable | current | ($23\text{\,fb}^{-1}$) | ($300\text{\,fb}^{-1}$) | ($50\text{\,ab}^{-1}$) | & CMS
CKM tests |  |  |  |  |
$\gamma$ (all modes) |  $4^{\circ}$ [931, 784] | $1.5^{\circ}$ | $0.35^{\circ}$ | $1.5^{\circ}$ | $-$
$\gamma$ ($B_{s}^{0}\rightarrow D_{s}^{+}K^{-}$) | $(_{-22}^{+17})^{\circ}$ | $4^{\circ}$ | $1^{\circ}$ | $-$ | $-$
$\sin 2\beta$ | 0.04 [932] | 0.011 | 0.003 | 0.005 | $-$
$\phi_{s}$ ($B_{s}^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}\phi$) | 49 mrad [933] | 14 mrad | 4 mrad | - | 22 mrad [934]
5-6 mrad [935]
$\phi_{s}$ ($B_{s}^{0}\rightarrow D_{s}^{+}D_{s}^{-}$) | 170 mrad [825] | 35 mrad | 9 mrad | $-$ | $-$
$\phi_{s}^{{{s}{\overline{s}}}{}s}$ ($B_{s}^{0}\rightarrow\phi\phi$) | 154 mrad [936] | 39 mrad | 11 mrad | $-$ | feasible [937]
$a_{sl}^{s}$ |  $33\times 10^{-4}$ [938] | $10\times 10^{-4}$ | $3\times 10^{-4}$ | $-$ | $-$
$|V_{ub}|/|V_{cb}|$ | 6% [847] | 3% | 1% | 1% | $-$
Charm |  |  |  |  |
$\Delta{{\mathcal{A}}^{{C\\!P}}}$ |  $2.9\times 10^{-4}$ [790] | $1.7\times 10^{-4}$ | $3.0\times 10^{-5}$ | $5.4\times 10^{-4}$ | $-$
$A_{\Gamma}$ |  $1.3\times 10^{-4}$ [877] | $4.2\times 10^{-5}$ | $1.0\times 10^{-5}$ | $3.5\times 10^{-4}$ | $-$
${{B}_{({s})}^{0}}\rightarrow\mu^{+}\mu^{-}$ |  |  |  |  |
$\frac{{\mathcal{B}}(B^{0}\rightarrow\mu^{+}\mu^{-})}{{\mathcal{B}}(B_{s}^{0}\rightarrow\mu^{+}\mu^{-})}$ | 71% [661, 662] | 34% | 10% | $-$ | 21% [939, 940]
$\tau_{B_{s}^{0}\rightarrow\mu^{+}\mu^{-}}$ | 14% [661, 662] | 8% | 2% | $-$ | $-$
EW penguins |  |  |  |  |
$R_{K}$ ($B^{+}\rightarrow K^{+}{\ell^{+}}{\ell^{-}}$) | 0.044 [703] | 0.025 | 0.007 | 0.036 | $-$
$R_{K^{*}}$ ($B^{0}\rightarrow{{K}^{*0}}{\ell^{+}}{\ell^{-}}$) | 0.10 [709] | 0.031 | 0.008 | 0.032 | $-$
LFU tests |  |  |  |  |
$R_{D^{*}}$ (${{B}^{0}}\rightarrow{{D}^{*-}}{\ell^{+}}\nu$) | 0.026 [941, 942] | 0.007 | 0.002 | 0.005 | $-$
$R_{{{J\mskip-3.0mu/\mskip-2.0mu\psi}}}$ ($B_{c}^{+}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{\ell^{+}}\nu$) | 0.24 [943] | 0.07 | 0.02 | $-$ | $-$

Figure 37: Sensitivity to probe key CP violating variables, rare decay and
lepton flavour universality tests expected from LHCb upgrades. Anticipated
results from Belle II, ATLAS or CMS are listed when available. The figure is
taken from Ref. [664].

Rare decays: The decay of $B^{0}\rightarrow\mu^{+}\mu^{-}$ is not very far
from being observed with imminent Upgrade I data, and its branching fraction
with respect to $B_{s}^{0}\rightarrow\mu^{+}\mu^{-}$ will be measured with 10%
uncertainty giving a powerful test of minimal flavour violation. A wide range
of studies will be performed in $b\rightarrow s{\ell^{+}}{\ell^{-}}$ or
$b\rightarrow d{\ell^{+}}{\ell^{-}}$ decays with improved precision, so the
current hint of discrepancy in $R_{K^{(*)}}$ with SM predictions will be
confirmed or excluded with confidence. A series of tests on lepton flavour
universality can be carried out in $b\rightarrow c{\ell^{-}}\nu$ decays. The
precision of $R_{D^{*}}$ will reach per mille level.

CKM tests: The CKM unitarity triangle will be so precisely determined in the
future that discrepancies between various measurement caused by physics beyond
Standard Model will be extremely difficult to hide, as shown in Fig. 38. The
angle $\gamma$, currently still the least well-known, will be determined with
an uncertainty of $1.5^{\circ}$ after Run 3, similar as the precision expected
from Belle II; The uncertainty will be further reduced to $0.35^{\circ}$ after
Upgrade II. The expected precision on $B_{s}$ weak mixing angle $\phi_{s}$
will be pushed to a few mrad, the same level as the current precision
determined indirectly from CKM fit using tree-level measurements.

Figure 38: LHCb constraints to the unitarity triangle with anticipated
improvement from (left) Upgrade I and (right) Upgrade II. The figure is taken
from Ref. [664].

CP violation in charm: After Upgrade II LHCb will be able to probe CP
violation in charm with a sensitivity of $\mathcal{O}(10^{-5})$, the only
future facility promising in observing indirect CP violation in charm which is
predicted to be $\mathcal{O}(10^{-4})$ or less in the SM.

Heavy-flavour spectroscopy: LHCb has demonstrated its capability as a general
purpose detector in the forward region, and it will be better equipped in this
respect after Upgrades. With data sample over an order of magnitude large than
currently available, LHCb will systematically update our knowledge on heavy
hadron spectroscopy, including less-studied conventional hadrons, such as
doubly heavy baryons, and those labelled as exotic nowadays, such as
pentaquark and tetraquark states.

Beyond flavour physics: In the high $p_{\mathrm{T}}$ range LHCb could also
make contribution complementary to ATLAS and CMS, for instance to the
precision determination of the effective weak mixing angle
$\sin\theta_{W}^{2}$ and the $W$ mass. Measurements of top pair and gauge
boson production at LHCb are also crucial to study the poorly known gluon
parton distribution functions at high-$x$ range. This is an important study in
the QCD, which help to understand the ubiquitous background for any new high-
mass states in ATLAS or CMS. LHCb will also push sensitivity in search for
dark-photon and long-lived particles predicted in several NP scenarios.

###  6.3 Summary

This manuscript briefly reviews the recent experimental highlights using data
collected with the LHCb detector in its first 10-year operation. They are not
only from the flavour physics benchmarks that the experiment was designed for,
but also include unexpected discoveries revealing LHCb’s capability as a
general purpose detector in the forward region:

  * •

A large variety of new particles are discovered, either filling gaps in
conventional heavy hadron spectroscopy or establishing new types of their own,
like pentaquarks or tetraquarks;

  * •

Some processes predicted to be extremely rare in the SM are observed, such as
the $B_{s}^{0}\rightarrow\mu^{+}\mu^{-}$ decay. Precise measurements are
performed on semileptonic and radiative FCNC beauty decays, where NP at high
energy scale can be probed with promising sensitivity, and tensions with the
SM are found in some cases;

  * •

The CKM parameters are determined precisely using multiple approaches in a
wide range of final states. The angle $\gamma$, which was the least known in
the unitarity triangle, has been determined with an unprecedented precision of
about $4^{\circ}$;

  * •

Heavy flavour study in the charm sector witnesses a couple of milestones, such
as the observation of ${{D}^{0}}$-${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ mixing in a single measurement
and observation of non-zero mass difference between the two mass eigenstates
in the ${{D}^{0}}-{{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$
system. Precision in probing the CP violation in charm keeps pushing forward.

This is by far not a complete list [4], and many interesting topics studied at
LHCb are not covered due to limited space. With the Upgrade I detector in
place, the LHCb experiment has resumed operation in 2022 and will continue to
take data at a higher luminosity while preparing for the future Upgrade II.
New exciting physics results are expected, which will continue to shape the
landscape of heavy flavour physics and beyond.

## Acknowledgements

This work is partially supported by the National Key Research and Development
Program of China under grant Nos. 2017YFA0402100, 2022YFA1601900, National
Natural Science Foundation of China (NSFC) under grant Nos. 11435003,
11575091, 11575094, 11925504, 11975015, 12175245, 12175005, 11705209,
12205312, 12275100, 11961141015, 12061141007, Chinese Academy of Sciences,
Fundamental Research Funds for the Central Universities, Peking University
Funds for the New Faculty Startup program. We thank Franz Muheim and Niels
Tuning for suggestions in improving the draft.

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