#  ${C\\!P}$ violation in heavy-flavour hadrons

Greig A. Cowan  on behalf of the LHCb collaboration11128th International
Symposium on Lepton Photon Interactions at High Energies, 7 - 12 Aug 2017
Sun Yat-Sen University, Guangzhou, China
email<EMAIL_ADDRESS>University of Edinburgh, UK

###### Abstract

Measurements of ${C\\!P}$-violating observables in ${B}$ meson decays can be
used to determine the angles of the Unitarity Triangle and hence probe for
manifestations of New Physics beyond the Cabibbo-Kobayashi-Maskawa Standard
Model paradigm. Of particular interest are precise measurements of the angles
$\gamma$ and $\beta$. Also of great importance are studies of
${C\\!P}$-violation involving ${{B}^{0}_{s}}$ mesons, in particular the phase
$\phi_{s}$, which is a golden observable in flavour physics at the LHC.
Complementary to these studies is the continuing search for direct and
indirect ${C\\!P}$-violation in the charm system, where the experimental
precision is now at the $10^{-3}$ level. I will present new and recent results
in these topics, and in ${C\\!P}$-violation searches in baryon decays, with
specific emphasis on the measurement programme at the LHC.

###### keywords:

Flavour physics; $C\\!P$ violation; LHC

##  1 Introduction

The violation of ${C\\!P}$ symmetry, the combination of the discrete
symmetries of charge-conjugation (conjugation of all internal quantum numbers)
and parity (reversing spatial coordinates), is a necessary condition to
generate the baryon asymmetry of the Universe [1]. However, the level of
${C\\!P}$ violation allowed within the quark sector of the SM is many orders
of magnitude too small [2, 3, 4] to explain astronomical observations,
demanding experimental searches for new sources of violation. Heavy-quark
hadrons provide an excellent laboratory to perform such searches as they allow
the exploration of high energy scales well beyond the direct reach of the LHC.
This approach has been successfully applied in the past with, for example, the
observation of ${B}^{0}$ meson mixing [5, 6] leading to the first estimates of
the top quark mass before it was directly discovered. The large heavy-quark
production cross-sections at the LHC [7, 8] lead to large samples of
exclusively reconstructed $b$ and $c$ hadron decays that have been used to
make precision measurements of $C\\!P$ violating observables. These
proceedings summarise the latest of these measurements, focussing on those
using 3$\mbox{\,fb}^{-1}$ of data collected by the LHCb experiment [9] in $pp$
collisions at the LHC during 2011 and 2012, unless otherwise stated. Recent
comprehensive reviews can be found in Refs. [10, 11] and references therein.

##  2 $C\\!P$ violation in the Standard Model

The only source of $C\\!P$ violation within the SM is due to the non-zero
value of the phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, to which all
$C\\!P$-violating observables are related. The unitarity of the matrix leads
to relations between elements (e.g.,
$V_{ud}V^{*}_{ub}+V_{cd}V^{*}_{cb}+V_{td}V^{*}_{tb}=0$), which are convenient
to visualise as a triangle in the complex plane. For the triangle (the
Unitarity Triangle) where all sides are of a similar magnitude, the angles are
defined as

$\alpha\equiv\arg\left[-\frac{V_{td}V^{*}_{tb}}{V_{ud}V^{*}_{ub}}\right]$,
$\beta\equiv\arg\left[-\frac{V_{cd}V_{cb}^{*}}{V_{td}V_{tb}^{*}}\right]$,
$\gamma\equiv\arg\left[-\frac{V_{ud}V^{*}_{ub}}{V_{cd}V^{*}_{cb}}\right]$.

These angles can be measured using a variety of different $C\\!P$ violating
observables covering both tree-level quark transitions, where the impact of
New Physics (NP) contributions is expected to be small [12], and loop-level
transitions, which are sensitive to new higher-mass particles. One of the
goals of studying the heavy-quark sector is to compare measurements of these
quantities to check for the overall consistency of the CKM mechanism. Figure 1
shows the latest global fit of the CKM matrix parameters to experimental
measurements and Lattice QCD calculations [13, 14] showing that the SM is
working well. However, there is still room for NP contributions at the level
of $\sim 10\%$ [15, 16], implying that a new set of precision measurements is
required.

Figure 1: Global fit [15] to the CKM matrix parameters, showing consistency
between measurements when interpreted in terms of SM quark transitions.

In the quark sector, the neutral mesons ($P^{0}$) can oscillate into their
antiparticles ($\overline{P}^{0}$), resulting in the physical states
($P^{0}_{\rm H,L}$) being admixtures of the flavour eigenstates: $P^{0}_{\rm
H}=pP^{0}+q\overline{P}^{0}$ and $P^{0}_{\rm L}=pP^{0}-q\overline{P}^{0}$,
where $p$ and $q$ are complex coefficients ($|p|^{2}+|q|^{2}=1$). The physical
states have well defined masses and lifetimes, and the parameters $\Delta
m=m_{\rm H}-m_{\rm L}$, $\Delta\Gamma=\Gamma_{\rm L}-\Gamma_{\rm H}$ and
$\Gamma=(\Gamma_{\rm L}+\Gamma_{\rm H})/2$ control the decay-time-dependent
decay rates of the mesons. In particular, $\Delta m$ controls the oscillation
frequency.

$C\\!P$ violation depends on the quantity
$\lambda_{f}=\frac{q}{p}\frac{\overline{A}_{f}}{A_{f}}$. Here, $f$ is the
final state that the $P$ hadron decays to with amplitude $A_{f}$ and the
$\overline{P}$ hadron decays to with amplitude $\overline{A}_{f}$. Three types
of $C\\!P$ violation are allowed: in neutral meson mixing ($|q/p|\neq 1$); in
the interference between neutral meson mixing and decay
($\arg(\lambda_{f})\neq 0$) and in hadron decay ($|\overline{A}_{f}/A_{f}|\neq
1$). Only $C\\!P$-violation in decay is allowed for charged mesons and
baryons.

##  3 $C\\!P$ violation in $B$ meson mixing

Semileptonic $B^{0}_{(s)}$ decays are dominated by tree-level quark
transitions, implying that there should be no $C\\!P$ violation in decay, and
therefore provide a clean system to search for $C\\!P$ in mixing. The so-
called semileptonic (or flavour-specific) asymmetry is defined as $A_{\rm
sl}=\frac{\Gamma(\overline{B}^{0}\rightarrow B^{0}\rightarrow
f)-\Gamma(B^{0}\rightarrow\overline{B}^{0}\rightarrow\overline{f})}{\Gamma(\overline{B}^{0}\rightarrow
B^{0}\rightarrow
f)+\Gamma(B^{0}\rightarrow\overline{B}^{0}\rightarrow\overline{f})}\approx\frac{\Delta\Gamma}{\Delta
m}\tan\phi_{M}$, where $\phi_{M}$ is the mixing phase from the $B^{0}_{(s)}$
mixing matrix. These asymmetries are predicted to be very small in the SM, at
the level of $10^{-4}$ or less [10], and therefore any measurement of a
significantly non-zero effect would be a clear sign of beyond-the-SM physics.

Experimentally, the quantity measured is the untagged decay-time, $t$,
dependent charge asymmetry between semileptonic $B^{0}_{(s)}$ decays with a
positive or negatively charged muon, defined as $A_{\rm
meas}(t)=\frac{N(D^{-}\mu^{+}\nu,t)-N(D^{+}\mu^{-}\nu,t)}{N(D^{-}\mu^{+}\nu,t)+N(D^{+}\mu^{-}\nu,t)}$
$\approx A_{\rm D}+\frac{A_{\rm sl}}{2}+\left(A_{\rm P}-\frac{A_{\rm
sl}}{2}\right)\cos(\Delta mt)$. This is sensitive to $A_{\rm sl}$ along with
other production, $A_{\rm P}$, and particle detection, $A_{\rm D}$,
asymmetries. In the case of the measurements from the LHCb collaboration [17,
18] these asymmetries can be controlled to high precision using data
calibration samples and by reversing the LHCb dipole magnet, thereby allowing
a precision measurement of the $C\\!P$ asymmetries in both the ${B}^{0}$ and
${B}^{0}_{s}$ systems.222 For the ${B}^{0}_{s}$ system, the time-integrated
rate can be measured as the fast ${B}^{0}_{s}$ oscillations wash out the
production asymmetry. Figure 2a shows the current experimental situation for
$C\\!P$ violation in $B$ meson mixing. The global average values are $A_{\rm
sl}^{d}=(-0.21\pm 0.17)\%$ and $A_{\rm sl}^{s}=(-0.06\pm 0.28)\%$ [19],
consistent with SM expectations.

The left-most green ellipse in Figure 2a corresponds to the dimuon asymmetry
measured by the D0 collaboration[20], which is a measurement of a linear
combination of $A_{\rm sl}^{d}$ and $A_{\rm sl}^{s}$. Although it is
inconsistent with SM expectations at $\sim 3\sigma$ it has been proposed [21,
22] that there may be additional contributions from a non-zero value of
$\Delta\Gamma_{d}/\Gamma_{d}$, which is expected to be very small in the SM.
The most precise measurement of this quantity has recently been made by the
ATLAS collaboration [23] (Figure 2b), obtaining
$\Delta\Gamma_{d}/\Gamma_{d}=(-0.1\pm 1.1\pm 0.9)\times 10^{-2}$. An update of
the 1$\mbox{\,fb}^{-1}$ measurement [24] of $\Delta\Gamma_{d}/\Gamma_{d}$ from
the LHCb collaboration is eagerly anticipated.

(a)

(b)

Figure 2: (a) HFLAV combination [19] of $A_{\rm sl}^{d}$ and $A_{\rm sl}^{s}$
from several experiments compared to the theory ($\times 10$) predictions
[10]. (b) Efficiency-corrected ratio of the observed decay-length
distributions for ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{*}}$ and
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decays [23].

##  4 $C\\!P$ violation in the interference of $B$ meson mixing/decay

In the case where the $B^{0}_{(s)}$ or $\overline{B}^{0}_{(s)}$ mesons decay
to the same final state, $f$, the decay-time-dependent $C\\!P$ asymmetry is
given by

$\frac{\Gamma_{\overline{B}^{0}\rightarrow f}(t)-\Gamma_{B^{0}\rightarrow
f}(t)}{\Gamma_{\overline{B}^{0}\rightarrow f}(t)+\Gamma_{B^{0}\rightarrow
f}(t)}=\frac{S_{f}\sin(\Delta mt)-C_{f}\cos(\Delta mt)}{\cosh(\Delta\Gamma
t/2)+A_{f}^{\Delta\Gamma}\sinh(\Delta\Gamma t/2)},$ (1)

where $|S_{f}|^{2}+|C_{f}|^{2}+|A_{f}^{\Delta\Gamma}|^{2}=1$ by definition.

###  4.1 The ${B}^{0}$ system

In the ${B}^{0}$ system, $\Delta\Gamma_{d}\approx 0$ and only the numerator of
Eq. 1 needs to be considered. The canonical decay mode used by the B-factories
to measure this asymmetry is
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$, which proceeds
predominately via a tree-level $b\rightarrow c\overline{c}s$ transition. In
the case where the sub-dominant penguin diagrams can be neglected [25, 26, 27,
28], $S_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}\approx\sin 2\beta$.

The LHCb collaboration has recently used its Run 1 data to measure $S_{f}$ and
$C_{f}$ in ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\mu^{+}\mu^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ [29],
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(e^{+}e^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{B}^{0}}\rightarrow\psi(2S)(\mu^{+}\mu^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ decays [30] using a flavour-tagged [31] decay-time-dependent analysis.
The asymmetry for ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\mu^{+}\mu^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decays can
be seen in Figure 3a. The individual measurements and their combination are
shown in Figure 3b, where the systematic uncertainty is dominated by
background tagging asymmetry. The LHCb-averaged values are
$S_{[c\overline{c}]{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=0.760\pm 0.034$
and $C_{[c\overline{c}]{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}}=-0.017\pm
0.029$. Together, these measurements reduce the tension between the world
average value for $\sin 2\beta$ and the indirect determination from global
fits [15, 16]. The consistency between the results using the electron and muon
channels for charmonium reconstruction also help to build confidence in the
electron reconstruction performance of LHCb, which is particularly relevant
when viewed through the prism of recent anomalies in $b\rightarrow
s\ell^{+}\ell^{-}$ transitions [32].

(a)

(b)

Figure 3: (a) $C\\!P$ asymmetry as a function of decay time for
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ decays [29], showing a clear
oscillation. (b) LHCb combination of $C\\!P$ violation parameters as measured
using ${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\mu^{+}\mu^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ [29],
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(e^{+}e^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$ and
${{B}^{0}}\rightarrow\psi(2S)(\mu^{+}\mu^{-}){{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}$ [30] decays.

Figure 4: HFLAV average of $C\\!P$ parameters in
${{B}^{0}}\rightarrow{{D}^{+}}{{D}^{-}}$ decays [19].

Decays such as ${{B}^{0}}\rightarrow{{D}^{+}}{{D}^{-}}$ are governed by
$b\rightarrow c\overline{c}d$ quark transitions and therefore measurements of
the decay-time-dependent asymmetry gives complimentary information about $\sin
2\beta$ that can be used to constrain the size of potential penguin
contributions to the decay [25, 26, 27, 28]. Figure 4 summarises the current
situation with these measurements from the BaBar, Belle and LHCb [33]
collaborations. The Belle result is outside of the physical region
($S_{DD}^{2}+C_{DD}^{2}<1$), which may have been an indication of large
hadronic effects. However, the latest LHCb measurement shows that these terms
are small and consistent with zero, with the phase shift induced by the
penguin diagrams measured to be $\Delta\phi=-0.16^{+0.19}_{-0.21}$ rad.

###  4.2 The ${B}^{0}_{s}$ system

Decay-time-dependent $C\\!P$ asymmetries in the ${B}^{0}_{s}$ system using
$b\rightarrow c\overline{c}s$ transitions are sensitive to the CKM phase
$\beta_{s}\equiv\arg\left[-\frac{V_{ts}V_{tb}^{*}}{V_{cs}V_{cb}^{*}}\right]$.
Typically measurements are made of the experimentally observable phase
$\phi_{s}$, which is equal to $-2\beta_{s}$ if the penguin contributions to
the decay can be neglected. Global fits give a precise Standard Model
prediction for $\phi_{s}$ of $-36.5\pm 1.3$ mrad [15]. Deviations from this
value would be a clear sign for NP, strongly motivating the need for more
precise experimental measurements.

The golden mode for measuring $\phi_{{s}}$ is using a flavour-tagged decay-
time-dependent angular analysis of the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\mu^{+}\mu^{-})\phi({{K}^{+}}{{K}^{-}})$ decay. This channel has a
high branching fraction and the presence of two muons in the final state leads
to a high trigger efficiency at hadron colliders. An angular analysis is
necessary to disentangle the interfering $C\\!P$-odd and $C\\!P$-even
components in the final state, which arise due to the relative angular
momentum between the two vector resonances. In addition, there is a small
($\sim 2\%$) $C\\!P$-odd ${{K}^{+}}{{K}^{-}}$ S-wave contribution that must be
accounted for.

The CDF [34], D0 [35], ATLAS [36], CMS [37] and LHCb [38] collaborations have
all measured $\phi_{s}$ (in addition to other mixing-related parameters of the
${B}^{0}_{s}$ system) using the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
decay. Additional information can be obtained by utilising the region of the
${{K}^{+}}{{K}^{-}}$ invariant mass spectrum above the $\phi(1020)$ meson,
where higher spin ${{K}^{+}}{{K}^{-}}$ resonances are expected to contribute.
Such a flavour-tagged decay-time-dependent amplitude analysis has just been
performed by the LHCb collaboration [39], which finds the dominant component
of the high-mass spectrum comes from the $f^{\prime}_{2}(1525)$ meson (Figure
5) and measures $\phi_{s}=119\pm 107\pm 34$ mrad. The LHCb detector has
excellent time resolution ($\sim 45$ fs) and tagging power ($\sim 4\%$), both
of which are crucial to the measurement. Combining the LHCb results from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
(low mass), ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ (high mass) and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$ decays [40] gives ${\phi_{{s}}}=1\pm 37$ mrad.

The global combination of $\phi_{{s}}$ and $\Delta\Gamma_{s}$ using the
measurements referenced above in addition to
${{B}^{0}_{s}}\rightarrow\psi(2S)\phi$ [41] and
${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{D}^{+}_{s}}$[42] decays gives average
values of ${\Delta\Gamma_{{s}}}=0.090\pm 0.005{\mathrm{\,ps^{-1}}}$ and
$\phi_{s}=-21\pm 31$ mrad. The combination is dominated by the statistical
uncertainty from the LHCb
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
result and are consistent with the SM predictions [15, 10]. However, there
remains space for new physics contributions at $O(10\%)$ and as the
experimental precision improves it is essential that there is good control
over hadronic effects [43, 44] that could mimic the signature of beyond-the-SM
physics.

(a)

(b)

Figure 5: (a) Distribution of $m_{{{K}^{+}}{{K}^{-}}}$ from
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}$ decays. The total fit function is overlaid in blue,
while the $\phi(1020)$, $f^{\prime}_{2}(1525)$ and ${{K}^{+}}{{K}^{-}}$ S-wave
contributions are shown by the long-dashed pink, brown and green lines,
respectively. (b) HFLAV combination [19] (white contour) of $\phi_{{s}}$ and
$\Delta\Gamma_{s}$ from several experiments (coloured contours) as discussed
in the text.

A related $C\\!P$-violating phase, $\phi_{s}^{s\overline{s}s}$, can be
measured by applying similar analysis methods to ${B}^{0}_{s}$ meson decays
that occur via $b\rightarrow s\overline{s}s$ transitions. The LHCb
collaboration has performed such an analysis using
${{B}^{0}_{s}}\rightarrow\phi\phi$ [45], measuring ${\phi_{{s}}}=-0.17\pm
0.15\pm 0.03$ rad, which is consistent with the SM predictions, all of which
are very close to zero [46, 47, 48]. Updated measurements of $C\\!P$-violating
parameters in charmless ${{B}^{0}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}$ decays
have also recently been reported by the LHCb collaboration [49], including a
first measurement of $A^{\Delta\Gamma}_{{{K}^{+}}{{K}^{-}}}$.

##  5 $C\\!P$ violation in $b$ hadron decay

###  5.1 The CKM angle $\gamma$

The CKM angle $\gamma$ is the only $C\\!P$ violating parameter that can be
measured from tree-level decays333There are some caveats. and the uncertainty
on its theoretical prediction is constrained to be $<O(10^{-7})$ [12].
Together these make measurements of $\gamma$ a “standard candle” within the SM
with which other loop-level determinations can be compared in order to look
for the effects of new $C\\!P$ violating contributions. The canonical
technique to measure $\gamma$ is to exploit the interference between the
different decay paths in $B\rightarrow DK$ decays. Depending on the final
state of the $D^{0}$ meson there are different analysis methods (e.g., GLW
[50, 51], ADS[52, 53] and GGSZ [54]) that vary in their sensitivity to
$\gamma$ and the other hadronic parameters that describe the strong dynamics
of the $B$ and $D$ meson decays. Unlike the measurement of $\beta_{(s)}$,
there is no dominant channel in which to measure $\gamma$ and a combination of
several modes is required to achieve the maximal sensitivity.

A new result [55] from the LHCb collaboration is an update, using Run 2 data,
of the measurements of the $C\\!P$-violating observables in
$B^{\pm}\rightarrow D^{(*)0}K^{\pm}$ and $B^{\pm}\rightarrow
D^{(*)0}\pi^{\pm}$ decays using the GLW method. Figure 6 shows the invariant
mass distribution of the $D^{0}h$ system. In the case of $B^{\pm}\rightarrow
D^{0}K^{\pm}$ decays a clear asymmetry is visible between the oppositely-
charged modes. After controlling for small detector and production asymmetries
using the Cabibbo-favoured $B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}\pi^{\pm}$
mode, the $C\\!P$ asymmetry is measured to be $A_{K}^{KK}=+0.126\pm 0.014\pm
0.002$, where the first uncertainty is statistical and the second systematic.

For the first time the collaboration has also used a partial reconstruction
technique to measure the $C\\!P$ observables using the modes with excited
$D^{*0}\rightarrow{{D}^{0}}\gamma$ and $D^{*0}\rightarrow{{D}^{0}}\pi^{0}$
decays where the photon or $\pi^{0}$ is not reconstructed. This approach
avoids the efficiency penalty that would need to be paid to fully reconstruct
these channels. The partially reconstructed decays correspond to the
structures at lower mass in Figure 6, where the different shapes for the
$D^{*0}\rightarrow{{D}^{0}}\gamma$ and $D^{*0}\rightarrow{{D}^{0}}\pi^{0}$
contributions allow the decay rates and $C\\!P$ asymmetries to be measure
separately for each. The $C\\!P$ asymmetry in the
$B^{\pm}\rightarrow(D^{*0}\rightarrow{{D}^{0}}\pi^{0})K^{\pm}$ channel is
measured as $A_{K}^{{C\\!P},\pi^{0}}=-0.151\pm 0.033\pm 0.011$, which is
different from zero at $4.3\sigma$. The corresponding asymmetry for the
$\gamma$ mode is $A_{K}^{{C\\!P},\gamma}=+0.276\pm 0.094\pm 0.047$.

Another new result is the measurement of the $C\\!P$-violating observables in
$B^{\pm}\rightarrow DK^{\ast\pm}$ decays, with
$K^{\ast\pm}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi^{\pm}$.
This updates Ref. [56], using two- and four-body $D$ meson final states [57]
in addition to Run 2 data. The branching ratio of the $B^{\pm}\rightarrow
DK^{\ast\pm}$ decay is of similar magnitude to $B^{\pm}\rightarrow DK^{\pm}$
(described above), but the overall event yield in LHCb is lower due to the
efficiency for reconstructing ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}$
mesons. Figure 7 shows the invariant mass distributions of the $DK^{*}$
systems for the two-body Cabibbo-favoured and ADS ${D}^{0}$ decay modes. The
decays are isolated with high signal purity and this result corresponds to the
first $4.2\sigma$ evidence of the ADS mode, with the rate measured to be
$R_{K\pi}^{+}=0.020\pm 0.006\pm 0.001$. The measurements for the decay rates
and $C\\!P$ asymmetries are consistent with and more precise than the
corresponding analysis from the BaBar collaboration [58]. In the future they
will help to further constrain $\gamma$ and the related hadronic parameters
for this system.

Figure 6: Distribution of invariant mass of the $Dh^{\pm}$ system for (left)
$B^{-}$ and (right) $B^{+}$ decays. The bachelor hadron is a (top) kaon and
(bottom) pion. In the case of the
${B}^{\pm}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}K^{\pm}$ decays a clear asymmetry
is visible between the distributions. The structures at lower masses is due to
partially reconstructed $D^{*0}\rightarrow D^{0}\gamma$ and $D^{*0}\rightarrow
D^{0}\pi^{0}$ decays where the $\gamma$ or $\pi^{0}$ particle is missed.

Figure 7: Distribution of invariant mass of the $DK^{*}$ system for (left)
$B^{-}$ and (right) $B^{+}$ decays. Both the (top) Cabibbo-favoured and
(bottom) ADS $D^{0}\rightarrow K\pi$ decay modes are visible.

###  5.2 $\gamma$ combination

As stated above the best precision on $\gamma$ is obtained by combining
information from many $B\rightarrow DK$ decay modes. An updated combination
from LHCb has recently been performed [59] that utilises 85 observables and
contains 37 parameters. It includes the $B^{\pm}\rightarrow D^{(*)0}K^{\pm}$
and $B^{\pm}\rightarrow DK^{\ast\pm}$ results described above444In fact, only
the $B^{\pm}\rightarrow DK^{\ast\pm}$ results from Ref. [56] are used. and an
updated decay-time-dependent measurement of the $C\\!P$ asymmetry in
${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}{{K}^{+}}$ decays [60]. The final
measurement is $\gamma={76.8^{+5.1}_{-5.7}}^{\circ}$. The HFLAV collaboration
have combined this with existing measurements from the B-factories to obtain
$\gamma={76.2^{+4.7}_{-5.0}}^{\circ}$, where the precision is dominated by the
LHCb measurement. Many more updates of $B\rightarrow DK$ channels can be
expected with Run 2. The aim is to have sub-degree-level precision at the end
of the LHCb phase 1 upgrade in 2024 by which point LHCb will have collected
50$\mbox{\,fb}^{-1}$.

Figure 8: 1-CL plot, using the profile likelihood method, for the $\gamma$
combinations split by analysis method. (yellow) GGSZ methods, (orange) GLW/ADS
methods, (blue) other methods and (green) the full combination.

###  5.3 First evidence of $C\\!P$ violation in the baryon system

(a)

(b)

Figure 9: (a) The invariant mass distributions for ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow p\pi\pi\pi$ used to extract the asymmetries.
(b) The $P$-odd and $C\\!P$-odd asymmetries as a function of $|\Phi|$, the
magnitude of angle between the $p\pi^{-}_{\rm fast}$ and $\pi^{-}_{\rm
slow}\pi^{+}$ decay planes in the ${\mathchar 28931\relax}^{0}_{b}$ rest
frame.

So far there has been no observation of $C\\!P$ violation in $b$ baryon
decays. However, since they are governed by the same quark-level transitions
as meson decays there is potential for non-zero effects in the SM [61, 62,
63]. For example, charmless decays of $b$ baryons have contributions of
similar magnitude from both tree and penguin decays, which give rise to
sensitivity to the CKM angle $\alpha$. No sign of $C\\!P$ violation has been
found in such charmless two [64] or three-body [65, 66] ${{\mathchar
28931\relax}^{0}_{b}}$ or $\Xi_{b}$ baryon decays, however, a recent study
[67] of four-body ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
ph^{-}h^{+}h^{-}$ decays has revealed the first evidence for $C\\!P$-violation
in the baryon sector.

The measurement is performed by using the four-body decay topology to compute
triple products, which are odd under the motion reversal operator,
$\hat{T}$.555The $\hat{T}$ operator is equivalent to the parity operation for
spinless particles. The triple products are defined as
$C_{\hat{T}}=\vec{p}_{p}\cdot(\vec{p}_{h_{1}^{-}}\times\vec{p}_{h_{2}^{+}})$
and
$\overline{C}_{\hat{T}}=\vec{p}_{\overline{p}}\cdot(\vec{p}_{h_{1}^{+}}\times\vec{p}_{h_{2}^{-}})$
for ${{\mathchar 28931\relax}^{0}_{b}}$ and $\overline{\Lambda}_{b}^{0}$
decays, respectively. The corresponding $\hat{T}$-odd asymmetries are
$A_{\hat{T}}(C_{\hat{T}})=\frac{N(C_{\hat{T}}>0)-N(C_{\hat{T}}<0)}{N(C_{\hat{T}}>0)+N(C_{\hat{T}}<0)}$
and
$\overline{A}_{\hat{T}}(\overline{C}_{\hat{T}})=\frac{N(-\overline{C}_{\hat{T}}>0)-N(-\overline{C}_{\hat{T}}<0)}{N(-\overline{C}_{\hat{T}}>0)+N(-\overline{C}_{\hat{T}}<0)}$,
where $N$ refers to the number of observed ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow ph^{-}h^{+}h^{-}$ candidates. From these
$\hat{T}$-odd asymmetries is is possible to build $P$-odd and $C\\!P$-odd
observables, defined as $a_{P}^{\hat{T}-{\rm
odd}}=\frac{1}{2}(A_{\hat{T}}+\overline{A}_{\hat{T}})$ and
$a_{{C\\!P}}^{\hat{T}-{\rm
odd}}=\frac{1}{2}(A_{\hat{T}}-\overline{A}_{\hat{T}})$, respectively. The
$C\\!P$-odd observable is insensitive to production and detection asymmetries
that affect standard $C\\!P$-asymmetries (e.g., those based on event yields)
and is also formed from a different combination of strong and weak phases.

Figure 9a shows the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p\pi\pi\pi$
invariant mass from Ref. [67] that has been used to measure
$a_{{C\\!P}}^{\hat{T}-{\rm odd}}$. The global measurement is consistent with
$C\\!P$ symmetry but it has been noted [68] that there is increased
sensitivity to $C\\!P$ violating effects by looking at differential
distributions. Figure 9b shows $a_{{C\\!P}}^{\hat{T}-{\rm odd}}$ as a function
of the phase space of the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
p\pi\pi\pi$ decay. Using this and an alternative binning scheme, the p-value
for the $C\\!P$-symmetry hypothesis is evaluated as $9.8\times 10^{-4}$, which
is equivalent to a $3.3\sigma$ deviation of $a_{{C\\!P}}^{\hat{T}-{\rm odd}}$
from zero. This constitutes the first evidence of $C\\!P$ violation in baryon
decays. With the larger data samples to be collected in Run 2 and beyond it
will be possible to perform a full amplitude analysis of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow p\pi\pi\pi$ channel to understand where the
$C\\!P$ asymmetry arises in the phase space of the decay.

Similar methods using $\hat{T}$-odd observables have been used to search for
$C\\!P$ violation in rare ${{\mathchar 28931\relax}^{0}_{b}}$ decays [69, 70]
and the charm system [71]. In each case the results are consistent with the
hypothesis of $C\\!P$ conservation.

##  6 Study of $b$-baryon oscillations

As noted in Section 1 the origin of the baryon asymmetry in the Universe is
unclear. Baryon number violation (BNV) has never been seen experimentally,
with strong constraints imposed by the measured proton and bound-neutron
lifetimes. However, beyond-the-SM models containing flavour-diagonal six-
fermion vertices [72, 73, 74, 75] could permit BNV without violating existing
constraints. Unambiguous experimental observation of such BNV would be the
observation of baryon-antibaryon oscillations of hadrons containing quarks of
all three generations $(i.e.,usb)$, such as the $\Xi_{b}^{0}$ baryon.

A new result [76] from the LHCb collaboration measures the decay-time-
dependent ratio between the rates of same-sign (SS) and opposite-sign (OS)
decays of the $\Xi_{b}^{0}$ baryon. The ratio is defined as
$R(t)=\frac{\Gamma(\Xi_{b}^{0}\rightarrow\overline{\Xi}_{c}^{-}\pi^{+})}{\Gamma(\Xi_{b}^{0}\rightarrow\Xi_{c}^{+}\pi^{-})}\approx(\omega
t)^{2}$, where $\omega$ is the mixing frequency. Here, SS (OS) means that the
charge of the proton from the charged $\Xi_{c}\rightarrow pK\pi$ decay is the
same (opposite) to the charge of the pion from the strong decay of the excited
$\Xi_{b}^{\prime,*}$ baryons that tags the initial flavour of the
$\Xi_{b}^{0}$ candidate. Figure 10 shows the mass difference distribution for
the LHCb data. The two narrow peaks visible near threshold in the OS-tag
sample are due to the excited $\Xi_{b}^{\prime,*}$ baryons. The red histogram
shows the corresponding distribution for the SS tags, which is consistent with
the background-only hypothesis. In seven bins of decay time the ratio $R(t)$
is evaluated using a likelihood fit to the mass distribution, which allows an
upper limit to be set on the mixing frequency. The limit is $\omega<0.08$ ps-1
at 95% CL, determined using a likelihood ratio test and the CLs method.

Figure 10: Spectra of the mass difference $\delta
m=m(\Xi_{b}^{0}\pi)-m(\Xi^{0}_{b}))-m_{\pi}$ for the OS-tagged (black points
with error bars) and SS-tagged (red, hatched histogram) decays. Inset: detail
of the region $2.0<\delta m<5.5$ ${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$.

##  7 Charm

Studies of the charm system provide the only way to investigate $C\\!P$
violation using up-type quarks. As in the neutral $B$ meson system, neutral
$D$ mesons undergo oscillations, but in this case they are dominated by long-
distance effects and the mixing parameters ($x=\Delta m/\Gamma$,
$y=\Delta\Gamma/(2\Gamma)$) are expected to be small. Likewise, very small
$C\\!P$-violating effects are expected [77]. As yet there is no evidence for
$C\\!P$ violation in the charm system, but the very large data samples being
collected by the LHCb experiment provide an ideal place to precisely test
theoretical predictions and search for new sources of $C\\!P$ violation.

At hadron colliders it is possible to tag the initial flavour of $D^{0}$
mesons using two different methods, depending on how they were produced. The
flavour is determined either by the charge of the pion from promptly-produced
$D^{*+}\rightarrow D^{0}\pi^{+}$ decays or the charge of the muon from
semileptonic $B$ decays ($B\rightarrow D^{0}\mu^{-}X$). These independent
samples have different properties in terms of background and reconstruction
efficiencies, leading to different systematic uncertainties in the final
measurements.

The large yields available in the charm system can also be used to search for
$C\\!P$ violation in strong interactions. The LHCb collaboration has recently
used $D^{+}_{(s)}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays to search for
$C\\!P$ violating $\eta^{(\prime)}\rightarrow\pi^{+}\pi^{-}$ decays [78],
finding no signal in the $\pi^{+}\pi^{-}$ mass spectrum. This constrains the
branching fractions to be less than $\sim 10^{-5}$ at 90% CL, which are
comparable with or better than existing limits [79].

###  7.1 Direct $C\\!P$ violation in the charm system

Direct $C\\!P$ violation in the charm system can be explored by measuring the
asymmetries in the decay of ${D}^{0}$ mesons to $C\\!P$ eigenstates (i.e.,
${{D}^{0}}\rightarrow h^{+}h^{-}$). Experimentally the raw yield asymmetry is
measured, $A_{\rm raw}\equiv\frac{N({{D}^{0}}\rightarrow h^{+}h^{-})-N({{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\rightarrow
h^{+}h^{-})}{N({{D}^{0}}\rightarrow h^{+}h^{-})+N({{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}\rightarrow h^{+}h^{-})}$, and
subsequently corrected for small production and detection asymmetries using
data control modes in order to determine the $C\\!P$ asymmetry,
$A_{{C\\!P}}({{D}^{0}}\rightarrow h^{+}h^{-})=A_{\rm raw}({{D}^{0}}\rightarrow
h^{+}h^{-})-A_{\rm P}(D^{*+})-A_{\rm D}(\pi_{s}^{+})$.

The LHCb collaboration has recently measured $A_{{C\\!P}}({{D}^{0}}\rightarrow
K^{+}K^{-})$ using a prompt-tagged sample (Figure 11a) [80]. While still
statistically limited, the dominant systematic uncertainty in the result
relates to control of the nuisance asymmetries. Combining
$A_{{C\\!P}}({{D}^{0}}\rightarrow K^{+}K^{-})$ with an earlier measurement of
the difference in $C\\!P$ asymmetries between the kaon and pion modes, $\Delta
A_{{C\\!P}}=A_{{C\\!P}}({{D}^{0}}\rightarrow
K^{+}K^{-})-A_{{C\\!P}}({{D}^{0}}\rightarrow\pi^{+}\pi^{-})$, allows
$A_{{C\\!P}}({{D}^{0}}\rightarrow\pi^{+}\pi^{-})$ to be extracted. Figure 11b
shows the result for the two $C\\!P$ asymmetries, which are consistent with
$C\\!P$ symmetry. These results have been combined with independent
measurements from the semileptonic-tagged sample [81] to provide an overall
LHCb result of $A_{{C\\!P}}({{D}^{0}}\rightarrow K^{+}K^{-})=(0.04\pm 0.12\pm
0.10)\%$ and $A_{{C\\!P}}({{D}^{0}}\rightarrow K^{+}K^{-})=(0.07\pm 0.14\pm
0.11)\%$, where the first uncertainty is statistical and the second
systematic. These results are now approaching the per-mille level of
uncertainty but still do not show any signs of $C\\!P$ violation. Likewise,
there are no indications of $C\\!P$ violation in other modes such as
${D}^{\pm}\rightarrow\eta^{\prime}\pi^{\pm}$ and
${D}_{s}^{\pm}\rightarrow\eta^{\prime}\pi^{\pm}$ [82]. A search for $C\\!P$
violation in ${D}^{0}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ decays using an
energy test [83] reports a mild tension with $C\\!P$-symmetry that requires
further exploration with Run 2 data.

(a)

(b)

Figure 11: (a) Distribution of the mass difference between the $D^{*}$ and
$D^{0}$ mesons in the prompt sample. (b) $C\\!P$ violating asymmetries for
$D^{0}\rightarrow{{K}^{+}}{{K}^{-}}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$
decays using the prompt and semileptonic-tagged samples. The 68% confidence
level contours are displayed where the statistical and systematic
uncertainties are added in quadrature.

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

As discussed in above, the mixing parameters and decay-time-integrated $C\\!P$
asymmetries in the charm system are small. Based upon this, a measurement of
the decay-time-dependent $C\\!P$ asymmetry of ${D}^{0}$ decays to $C\\!P$
eigenstates is sensitive to the indirect $C\\!P$ violating parameter
$A_{\Gamma}$. Here, $A_{\Gamma}\equiv\frac{\hat{\Gamma}_{{{D}^{0}}\rightarrow
f}-\hat{\Gamma}_{{{\kern
0.99998pt\overline{\kern-0.99998ptD}{}}{}^{0}}\rightarrow
f}}{\hat{\Gamma}_{{{D}^{0}}\rightarrow f}+\hat{\Gamma}_{{{\kern
0.99998pt\overline{\kern-0.99998ptD}{}}{}^{0}}\rightarrow f}}$, is the
asymmetry between the inverse effective lifetimes of the ${D}^{0}$ meson and
its antiparticle. Within the SM, its magnitude is expected to be $\lesssim
5\times 10^{-3}$ and independent of the final state, $f$, owing to the small
$C\\!P$ violation in decay.

The LHCb collaboration has recently measured $A_{\Gamma}$ using a prompt-
tagged sample of ${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ and
${{D}^{0}}\rightarrow\pi^{+}\pi^{-}$ decays [84]. Two different methods are
used to measure the asymmetry, an approach that is binned in the ${D}^{0}$
decay time and an unbinned method. Both approaches use the large sample of
$D\rightarrow K\pi$ decays to control production and detection asymmetries.
They each give consistent measurements, with the binned result being slightly
more precise and chosen as the nominal result. Figure 12 shows the binned
asymmetries as a function of decay time for the kaon and pion decay modes. The
value of $A_{\Gamma}$ is given by the gradient of the linear fit to these
asymmetries. These values are combined to give $A_{\Gamma}=(-0.13\pm 0.28\pm
0.10)\times 10^{-3}$. They have subsequently been combined with the
semileptonic-tagged sample [85] to obtain $A_{\Gamma}=(-0.29\pm 0.28)\times
10^{-3}$. This is the most precise measurement of a $C\\!P$-violating
observable in the charm system ever made.

(a)

(b)

Figure 12: Measured asymmetry $A(t)$ in bins of $t/\tau_{D}$, where $\tau_{D}$
is the world average value for the ${D}^{0}$ lifetime [19], for (a)
$D^{0}\rightarrow{{K}^{+}}{{K}^{-}}$ and (b) $D^{0}\rightarrow\pi^{+}\pi^{-}$
decays. The solid line shows a linear fit to the data, with a slope equal to
the best estimate of $-A_{\Gamma}$.

###  7.3 Charm mixing and indirect $C\\!P$ violation

The measurement of charm mixing parameters and a search for $C\\!P$ violation
can be performed by studying the decay-time-dependent ratio of yields of
Cabibbo-suppressed to Cabibbo-favoured $D^{0}\rightarrow K\pi$ decays. This
ratio is defined separately for ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays and is related to the
mixing parameters via a second-order expansion of the mixing equations
(assuming small mixing) given by
$R(t)^{\pm}=R_{D}^{\pm}+\sqrt{R_{D}^{\pm}}y^{\prime}\frac{t}{\tau}+\frac{(x^{\prime\pm})^{2}+(y^{\prime\pm})^{2}}{4}\left(\frac{t}{\tau}\right)^{2}$,
where $\tau$ is the average ${D}^{0}$ lifetime and the $x^{\prime},y^{\prime}$
have been rotated from the nominal mixing parameters by the strong phase in
the ${D}\rightarrow K\pi$ decay. A recent result [86] from the LHCb
collaboration uses a double-tagged (DT) technique (using the charge of the
pion and the muon from the semileptonic $B$ decays
(${{B}^{0}}\rightarrow(D^{*+}\rightarrow D^{0}\pi^{+})\mu^{-}X$)) to provide a
very pure sample of events (Figure 13a and b) that cover a complementary
region of the decay time spectrum than has been previously studied using a
prompt-sample [87]. Figure 13c shows the ratios for ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays and their difference for
both the DT and prompt samples. The ratios increase as a function of time,
consistent with charm mixing, allowing $x^{\prime},y^{\prime}$ to be measured
for the DT sample alone and both samples combined. The complementary coverage
of the DT and prompt sample gives an improved precision, by 10–20%, for the
charm mixing parameters from the combined fit even though the DT analysis is
based on almost 40 times fewer candidates than the prompt analysis. The data
are consistent with the hypothesis of $C\\!P$ conservation (both for decay and
interference of mixing and decay).

(a)

(b)

(c)

Figure 13: Distribution of the $D^{*+}$ invariant mass in the double-tagged
(DT) sample for (a) RS and (b) WS decays. (c) Efficiency corrected ratios of
WS/RS decays and fit projections for the DT (red open circles) and prompt
(black filled circles) samples.

##  8 Looking to the future

The majority of beauty and charm sector measurements are statistically
limited, which strongly motivates the case for a new set of precision
measurements of $C\\!P$-violating observables to constrain the size of (or
perhaps find) new physics contributions. The LHCb collaboration recently
submitted an expression of interest [88] regarding future phase 1b and 2
upgrades that would operate during LHC runs 4 and 5, respectively. It is
expected that using the data collected with the improved detector it will be
possible to improve the precision on $\gamma$ to 0.4∘, $\phi_{s}$ to 9 mrad
(see Figure 14) and achieve precision on charm mixing and $C\\!P$ violation
parameters at the level of $10^{-4}$. These will allow unprecedented
sensitivity to the small effects of new physics. Crucially, as the precision
improves it is essential to control hadronic effects that can hide small non-
Standard Model effects. Other studies can be found in Refs. [89, 90].

Figure 14: Projection of how precision on $\phi_{{s}}$ from LHCb measurements
will scale as a function of time for different decay modes. Information taken
from Ref. [89, 90].

##  9 Summary

This year is the 40th anniversary of the discovery of the $b$ quark [91]. In
that time there has been huge progress in using the physics of heavy-flavour
hadrons to make detailed studies of $C\\!P$-violation, both experimentally and
theoretically. So far all measurements are consistent with Standard Model
predictions, confirming validity of the CKM mechanism. The LHCb experiment is
now leading the way in terms of precision measurements in the $b$ and $c$
sectors, with new explorations of $C\\!P$ violation in baryons just beginning.
It will be fascinating to observe how the SM is able to withstand the next
round of precision measurements that will be made over the next decade.

## Acknowledgements

The author thanks the organisers of the Lepton Photon conference and
acknowledges the support of the Science and Technology Facilities Council (UK)
grant ST/K004646/1.

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