Proceedings of the Fifth Annual LHCP

$C\\!P$ VIOLATION IN BEAUTY AND CHARM

DORDEI FRANCESCA

On behalf of the ATLAS, CMS and LHCb Experiments,
European Organization for Nuclear Research (CERN),
Route de Meyrin 385, 1217 Meyrin, Switzerland

> ABSTRACT
>
>
>
>
> The study of $C\\!P$ violation in decays of beauty- and charm-hadrons
> provides a fundamental test of the predictions of the Standard Model (SM)
> and represents a sensitive probe to search for physics effects that cannot
> be described within the SM. In these proceedings, precision measurements are
> presented of several observables related to the so-called unitary triangles,
> that arise from the unitarity requirements on the 3x3 Cabibbo-Kobayashi-
> Maskawa matrix describing the quark mixing. Moreover, recent measurements of
> $C\\!P$ violation in beauty- and charm-hadrons are illustrated. In
> particular, a recent analysis providing the first evidence for $C\\!P$
> violation in the beauty baryon sector and the world best precise measurement
> of $C\\!P$ violation in the charm system will be described in detail. The
> discussed analyses are based on proton-proton collision data collected
> during 2011 and 2012 (LHC Run 1) by the ATLAS, CMS and LHCb experiments.

> PRESENTED AT
>
>
>
>
> The Fifth Annual Conference
>
> on Large Hadron Collider Physics
>
> Shanghai Jiao Tong University, Shanghai, China
>
> May 15-20, 2017

##  1 Introduction

Although the Standard Model (SM) has been very successful in describing
existing data [1], it is well known that it must be an effective theory whose
validity breaks down at some higher energy scale (larger than few hundred GeV)
[2]. For example, contributions from physics Beyond the Standard Model (BSM)
are needed [3] in order to explain the large matter-antimatter asymmetry that
resulted in the matter dominated universe we observe today [4]. In the SM, the
only source of $C\\!P$ violation is a complex phase in the Cabbibo-Kobayashi-
Maskawa (CKM) quark mixing matrix [6] which is many orders of magnitude too
small to match the observed matter-antimatter asymmetry [5]. The CKM matrix is
a 3x3 matrix that parametrises transitions between the three quark families
due to charged current interactions. It is fully described by four independent
real parameters, namely three rotation angles and one complex phase, that are
not predicted by the SM but need to be measured. A non-zero value of the
complex phase can induce $C\\!P$ violating asymmetries in processes in which
two or more amplitudes with different weak phases interfere.

Most of the BSM physics models available today predict the existence of
additional unobserved heavy particles that can appear in internal lines of
higher order diagrams contributing to a process (so called box and penguin
diagrams). Those particles should produce sizeable modifications of
observables related to these processes, such as $C\\!P$ violating parameters.
Therefore, precision measurements of $C\\!P$ violation in the beauty and charm
systems are mandatory in order to confirm SM predictions and to check whether
non-SM contributions are also present. Since these new unobserved particles
appear only as virtual particles in internal lines of loop diagrams, indirect
searches for BSM physics are sensitive to much higher mass scales than direct
searches. The reachable mass scale can indeed exceed the centre of mass energy
of the accelerator at which the measurement is performed, reaching up to
$\mathcal{O}$(10-100 TeV).

There are three categories of $C\\!P$ violation which are introduced in the
following in order to better understand the results of the measurements
presented.

The first type of $C\\!P$ violation is possible for both charged and neutral
meson decays

  1. 1.

$C\\!P$ violation in decay. It occurs when the amplitude for a decay and its
$C\\!P$ conjugate decay have different magnitudes. If $A_{f}$ and
$\overline{A}_{\overline{f}}$ are the amplitudes for the process $B\to f$ and
$\overline{B}\to\overline{f}$, respectively, where $f$ and $\overline{f}$ are
$C\\!P$-conjugate multi-particle final states, $C\\!P$ violation in decay
occurs if

$\left|A_{f}\right|\neq\left|\overline{A}_{\overline{f}}\right|\,.$

The other two categories are only possible for neutral mesons, hereafter
referred to as $P^{0}$, which exhibit the phenomenon of mixing. Since the weak
interaction eigenstates are not the same as the mass eigenstates, the flavour
content of neutral mesons changes as a function of time, giving rise to the
phenomenon known as neutral meson mixing. The phenomenological aspects linked
to neutral meson mixing are described in many articles, e.g. [7]. Only the
main parameters involved are briefly introduced here. Any arbitrary
combination of flavour eigenstates, $P^{0}$ and $\overline{P}^{0}$, has a time
evolution described by an effective Schrödinger equation (units are such that
$\hbar=\textit{c}=1$)

$\textit{i}\,\,\frac{d}{dt}\begin{pmatrix}P^{0}\\\ \overline{P}^{0}\\\
\end{pmatrix}=\begin{pmatrix}M_{11}-i\,\Gamma_{11}/2&M_{12}-i\,\Gamma_{12}/2\\\
M_{21}-i\,\Gamma_{21}/2&M_{22}-i\,\Gamma_{22}/2\\\
\end{pmatrix}\begin{pmatrix}P^{0}\\\ \overline{P}^{0}\\\ \end{pmatrix}\,,$ (1)

where M and $\bm{\Gamma}$ are $2\times 2$ hermitian matrices. The heavy (H)
and light (L) mass eigenstates of the Schrödinger equation are obtained
diagonalizing the matrix in Eq.1

$|P_{H/L}\rangle=p|{P}^{0}\rangle\pm q|\overline{P}^{0}\rangle\,,$ (2)

where the complex coefficients p and q obey the normalization condition
$|p|^{2}+|q|^{2}=1$. The system can be defined in terms of four different
experimental observables. They are the mass difference $\Delta m$ and the
decay width difference $\Delta\Gamma$ between the mass eigenstates, defined by

$\Delta m\equiv m_{H}-m_{L}\simeq
2|M_{12}|,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Delta\Gamma\equiv\Gamma_{L}-\Gamma_{H}\simeq
2|\Gamma_{12}|\cos(\phi_{12}),$ (3)

and the average mass and width, defined by

$m=\frac{m_{H}+m_{L}}{2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma=\frac{\Gamma_{L}+\Gamma_{H}}{2}.$
(4)

Thanks to the phenomenon of mixing, neutral mesons are an ideal laboratory to
test the other two categories of $C\\!P$ violation

  1. 2.

$C\\!P$ violation in mixing. It occurs if the amplitude of the process
$P^{0}\to\overline{P}^{0}$ is different with respect to the opposite process,
$\overline{P}^{0}\to P^{0}$, i.e. $|q/p|$ not equal to 1. It does not depend
on the final state.

  2. 3.

$C\\!P$ violation in the interference between mixing and decay. If there is a
final state, $f$, common to $P^{0}$ and $\overline{P}^{0}$ (e.g. the final
state is a $C\\!P$ eigenstate) the neutral meson can either decay to the final
state directly or first oscillate and then decay. This gives rise to $C\\!P$
violation in the interference between mixing and decay. It occurs if

$\displaystyle\mathcal{I}m(\lambda_{f})$ $\displaystyle\equiv$
$\displaystyle\mathcal{I}m\left(\frac{q}{p}\,\frac{\overline{A}_{f}}{A_{f}}\right)\neq
0\,,$ (5)

where $\lambda_{f}$ is a parameter that quantifies this kind of $C\\!P$
violation. Assuming $C\\!P\\!T$ invariance, the $C\\!P$ asymmetry as a
function of time for neutral P mesons decaying to a $C\\!P$ eigenstate $f$ is
given by

$A^{\mathrm{C\\!P}}(t)=\frac{\Gamma_{\overline{P}^{0}\to
f}(t)-\Gamma_{P^{0}\to f}(t)}{\Gamma_{\overline{P}^{0}\to
f}(t)+\Gamma_{P^{0}\to f}(t)}=\frac{-C_{f}\cos(\Delta mt)+S_{f}\sin(\Delta
mt)}{\cosh\left(\frac{\Delta\Gamma}{2}t\right)-\mathcal{A}^{\Delta\Gamma}_{f}\sinh\left(\frac{\Delta\Gamma}{2}t\right)}\,,$
(6)

The quantities $C_{f}$, $S_{f}$, and $\mathcal{A}^{\Delta\Gamma_{(s)}}_{f}$
are

$C_{f}=\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}},\,\,\,\,\,\,\,\,\,\,\,\,S_{f}=\frac{2\mathrm{Im}\lambda_{f}}{1+|\lambda_{f}|^{2}}\,,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,\,\,\,\,\,\,\,\mathcal{A}^{\Delta\Gamma_{(s)}}_{f}=-\frac{2\mathrm{Re}\lambda_{f}}{1+|\lambda_{f}|^{2}}\,,$
(7)

where the parameter $C_{f}$ and $S_{f}$ are related to $C\\!P$ violation in
decay and $C\\!P$ violation in the interference between mixing and decay,
respectively. The parameter $\mathcal{A}^{\Delta\Gamma_{(s)}}_{f}$ can be
expressed as $\mathcal{A}^{\Delta\Gamma}_{f}=\pm\sqrt{1-C_{f}^{2}-S_{f}^{2}}$.

In the following a selection of recent ATLAS, CMS and LHCb measurements of
$C\\!P$ violation are presented. They are based on 2011+2012 data sets. ATLAS
[9], CMS [10] and LHCb [11] are three of the four large particle experiments
located at the Large Hadron Collider (LHC). ATLAS and CMS are general purpose
detectors which cover the full spectrum of high-energy physics, while LHCb is
specialised for precision measurements of $b$\- and $c$\- hadron decays.

##  2 CKM metrology

The unitarity of the CKM matrix gives rise to six orthogonality relations,
which can be represented as triangles in the complex plane. The triangle
derived from multiplying the first and third columns of the CKM matrix,
$V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0$, commonly referred to
as the $B^{0}$ triangle, is of particular interest since its three sides are
of comparable length. Its three angles are closely related to $C\\!P$
violating observables and the lengths of its sides can be measured from
$C\\!P$ conserving observables. A similar triangle can be also drawn for the
$B^{0}_{s}$ system. The overconstrained determination of the sides and angles
of these Unitarity Triangles provides a powerful consistency test of the SM
picture of $C\\!P$ violation.

###  2.1 Measurement of the CKM phase $\gamma$

The least well determined parameter of the Unitarity Triangle to date is the
angle $\gamma\equiv-\mathrm{arg}(V_{ud}V_{ub}^{*}/V_{cd}V_{cb}^{*})$. This
angle provide a standard candle for the SM since it is the only one that can
be determined from tree-level only processes, thus it is expected to be
unaffected by possible contributions from BSM. Comparisons between direct
measurements of $\gamma$ and constraints coming from global fits, which
include higher order diagrams as well, represent a very stringent test of the
SM. In order to reach this goal it is mandatory to improve the precision on
tree-level measurements of $\gamma$ since the indirect constraints are quite
precise. Indeed, from a global fit of different CKM parameters the CKMfitter
group derives $\gamma=\left(65.3^{+1.0}_{-2.5}\right)^{\circ}$ [24].
Experimentally, this angle is measured exploiting the interference between
$b\to c$ and $b\to u$ transitions. By building the $C\\!P$ asymmetry between
$B^{\mp}\to D^{0}K^{\mp}\to f_{D}K^{\mp}$ and
$B^{\mp}\to\overline{D}^{0}K^{\mp}\to f_{D}K^{\mp}$ decays, where $f_{D}$ is a
final state common to both $D^{0}$ and $\overline{D}^{0}$, it is possible to
determine the value of $\gamma$. However, the measurements are experimentally
challenging since they involve purely hadronic final states, which are
difficult to trigger, and require excellent $K/\pi$ separation. Moreover,
branching fractions for the interesting decays are small. Thus, the
combination of a plethora of independent modes is the only way to achieve the
ultimate precision. Depending on the $D^{0}$ final state $f_{D}$, three main
methods can be distinguished: in the ADS method $C\\!P$ eigenstates like
$D^{0}\to K^{+}\pi^{-}$ are used, in the GLW method $C\\!P$-specific $D^{0}$
decays like $D^{0}\to K^{+}K^{-}$ or $D^{0}\to\pi^{+}\pi^{-}$ are used, and in
the GGSZ method multi-body $D^{0}$ decays are exploited, which require the
analysis of the corresponding Dalitz plot. Another independent approach is to
measure time-dependent $C\\!P$ asymmetries in the decay $B^{0}_{s}\to
D_{s}^{+}K^{-}$.

Figure 1: 1-CL curves for the combination of the $\gamma$ measurements and the
contributions from the individual methods.

All these different methods are used to perform measurements by LHCb using Run
1 data and the tree-level measurements have been combined to derive the best
determination of $\gamma$. To obtain the best precision, the hadronic
parameters are also considered in the combination, for a total of 71
observables and 32 parameters. Both the frequentist and bayesian approaches
have been used and they show a good agreement. Using the frequentist
interpretation the combination is found to be:
$\gamma=\left(72.2^{+6.8}_{-7.3}\right)^{\circ}$ [13], which is the most
precise single-experiment measurement to date. The shapes of the 1-CL curves,
split by the method, are shown in Figure 1.

###  2.2 Measurement of the CKM phase $\beta$

The angle $\beta$ is related to CKM matrix elements through the relation
$\beta\equiv\mathrm{arg}[-(V_{cd}V_{cb^{*}})/(V_{td}V_{tb}^{*})]$ and it is
sensitive to BSM effects in $C\\!P$ violation in mixing and decay. The golden
channel to determine this angle is $B^{0}\to J/\psi K^{0}_{s}$. At LHCb
$J/\psi\to\mu\mu$ decays are selected efficiently and the relatively high
branching fraction of decays such as $B^{0}\to J/\psi K^{0}_{s}$ gives large
samples of $b\rightarrow c\overline{c}s$ transitions to be analysed. By
measuring the flavour-tagged decay-time-dependent $C\\!P$ asymmetry using the
full Run I dataset, the LHCb collaboration measures [14] the following $C\\!P$
violating parameters

$C=-0.038\pm 0.032\,\mathrm{(stat.)}\pm
0.005\,\mathrm{(syst.)}\,\,\mathrm{and}\,\,S\equiv\sin{(2\beta)}=0.731\pm
0.035\,\mathrm{(stat.)}\pm 0.020\,\mathrm{(syst.)}\,.$ (8)

The largest systematic uncertainty on the measurement of $\sin(2\beta)$ arises
from possible flavour asymmetries in the background candidate. The LHCb
collaboration has also recently measured $S_{f}$ and $C_{f}$ using $B^{0}\to
J/\psi(e^{+}e^{-})K^{0}_{s}$ and $B^{0}\to\psi(2S)(\mu^{+}\mu^{-})K^{0}_{s}$
decays [15]. The individual LHCb measurements and their combination are shown
in Figure 2 (left).

Figure 2: Left: Two-dimensional likelihood scans for the combination of all
$B^{0}\to[c\overline{c}]K^{0}_{s}$ modes analysed by LHCb. Right: Two-
dimensional likelihood scans of $\phi_{s}$ and $\Delta\Gamma_{s}$. The white
contour represents the HFLAV combination to be compared to the SM prediction
(black line).

###  2.3 Measurement of the CKM phase $\beta_{s}$

The corresponding $C\\!P$ observable in the $B^{0}_{s}$ system is called
$\beta_{s}=-2\,\phi_{s}$ and it is related to CKM matrix elements through the
relation
$\phi_{s}^{\mathrm{SM}}\equiv-2\,\mathrm{arg}\left(-\frac{V_{ts}V_{tb^{*}}}{V_{cs}V_{cb}^{*}}\right)$.
This angle has been extensively studied by ATLAS [16], CMS [17] and LHCb [18]
collaborations using the golden channel $B^{0}_{s}\rightarrow J/\psi\phi$.
This decay is a pseudo-scalar to vector-vector decay, so angular momentum
conservation implies that the final state is an admixture of $C\\!P$-even and
$C\\!P$-odd components. By performing a tagged time-dependent angular
analysis, it is possible to statistically disentangle the different $C\\!P$
eigenstates by the differential decay rate for $B^{0}_{s}$ and
$\overline{B}^{0}_{s}$ mesons produced as flavour eigenstates at $t=0$. The
LHCb collaboration exploited also other decay modes to fully investigate the
potentials of Run 1 data. Different measurements have been performed using
various $B^{0}_{s}$ finale states like $J/\psi\pi^{+}\pi^{-}$ [19],
$\psi(2S)\phi$ [20], $D_{s}^{+}D_{s}^{-}$ [21] and $J/\psi K^{+}K^{-}$ [22]
for the $K^{+}K^{-}$ invariant mass region above $1.05$ GeV. The HFLAV average
including all the above mentioned results is $\phi_{s}=-21\pm 31$ mrad [23].
The individual measurements and their HFLAV combination are shown in Figure 2
(right). Despite the impressive progress since the initial measurements
performed by CDF and D0, the uncertainty is still much larger than the
indirect determination from global fits:
$\phi_{s}^{\mathrm{SM}}=-36.5^{+1.3}_{-1.2}$ mrad [24].

###  2.4 Time-dependent CP violation in $B^{0}_{s}\to K^{+}K^{-}$ and
$B^{0}\to\pi^{+}\pi^{-}$

The study of $C\\!P$ violation in charmless charged two-body decays of neutral
B mesons is a good probe of the SM predictions. Moreover, precise measurements
in this sector are important to constrain hadronic factors that cannot be
accurately calculated from quantum chromodynamics at present. The LHCb
collaboration performed a preliminary measurement of time-dependent
$C\\!P$-violating asymmetries in $B^{0}_{s}\to K^{+}K^{-}$ and
$B^{0}\to\pi^{+}\pi^{-}$ decays [25] using Run 1 data. By means of a time-
dependent analysis, where initial $B^{0}_{(s)}$ flavour is identified through
a flavour-tagging algorithm calibrated using flavour-specific $B^{0}\to
K^{+}\pi^{-}$ events, the parameters for $B^{0}_{s}\to K^{+}K^{-}$ decay are
found to be

$C_{KK}=0.24\pm 0.06\,\mathrm{(stat.)}\pm
0.02\,\mathrm{(syst.)}\,\,\mathrm{and}\,\,S_{KK}=0.22\pm
0.06\,\mathrm{(stat.)}\pm 0.02\,\mathrm{(syst.)}\,.$ (9)

The results for the $B^{0}\to\pi^{+}\pi^{-}$ decay are

$C_{\pi\pi}=-0.24\pm 0.07\,\mathrm{(stat.)}\pm
0.01\,\mathrm{(syst.)}\,\,\mathrm{and}\,\,S_{\pi\pi}=-0.68\pm
0.06\,\mathrm{(stat.)}\pm 0.01\,\mathrm{(syst.)}\,.$ (10)

The value of $A^{\Delta\Gamma}_{KK}$ is found to be $-0.75\pm
0.07\,\mathrm{(stat.)}\pm 0.11\,\mathrm{(syst.)}$. Only one of the tagging
algorithms available at LHCb, namely the so called opposite side taggers, has
been used for this preliminary result, thus improvements in the tagging power
are expected in the future. These results are compatible with SM predictions
and are important inputs in the determination of the unitarity triangle angle
$\gamma$ and $\beta_{s}$ using decays affected by penguin processes.

###  2.5 Time-dependent CP violation in $B^{0}\to D^{+}D^{-}$

The $C\\!P$ asymmetry obtained using $B^{0}\to D^{+}D^{-}$ decays

$\mathcal{A}_{C\\!P}(t)=\frac{\Gamma(\overline{B}^{0}(t)\rightarrow
D^{+}D^{-})-\Gamma(B^{0}(t)\rightarrow
D^{+}D^{-})}{\Gamma(\overline{B}^{0}(t)\rightarrow
D^{+}D^{-})+\Gamma(B^{0}(t)\rightarrow D^{+}D^{-})}=S\sin{(\Delta
m_{d},t)}-C\cos{(\Delta m_{d}t)}$ (11)

gives complementary information on $\sin{(2\beta)}$ through $b\rightarrow
c\overline{c}d$ transitions. Indeed, this $C\\!P$ asymmetry is sensitive to
$\beta$ at tree level in the SM. However, higher order diagrams, such as
penguin diagrams, can lead to a difference in the $C\\!P$ violating phase,
$\Delta\phi$, with respect to the parameter $\beta$ measured using
$b\rightarrow c\overline{c}s$ transitions, such that
$S/\sqrt{1-C^{2}}=-\sin{(2\beta-\Delta\phi)}$. Thus, a comparison with the
$\beta$ measurement obtained using $B^{0}\rightarrow J/\psi K^{0}_{S}$ decays
constrains higher order contributions in $B\to DD$ decays that cannot be
predicted by theory.
The LHCb collaboration performed a measurement of the $B^{0}\to D^{+}D^{-}$
decay rate as a function of the $B^{0}$ decay time [26]. The production
flavour of the $B^{0}$ is determined by means of flavour tagging algorithms.
The mistag rate ($\omega$) for the flavour tagging is calibrated by measuring
the $B^{0}$ oscillation amplitude in flavour specific $B^{0}\to
D_{s}^{+}D^{-}$ decays. The tagging power in the $B^{0}\to D^{+}D^{-}$
analysis is
$\varepsilon_{\mathrm{eff}}\equiv\varepsilon(1-2\omega)^{2}=(8.1\pm 0.6)\%$,
where $\varepsilon$ is the fraction of events with a tagging decision. This is
the highest tagging power achieved at LHCb to date. The measured $S$ and $C$
parameters are

$S=-0.54^{+0.17}_{-0.16}\mathrm{(stat)}\pm
0.05\mathrm{(syst)}\,\,\mathrm{and}\,\,C=-0.26^{+0.18}_{-0.17}\mathrm{(stat)}\pm
0.02\mathrm{(syst)}\,.$ (12)

The phase shift derived from the LHCb measurement is
$\Delta\phi=(0.16^{+0.19}_{-0.21})\mathrm{rad}$, which rules out large penguin
contributions.

###  2.6 Time-dependent CP violation in $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$

Due to the interference between mixing and decay amplitudes, the $C\\!P$
violating observables in the $B^{0}_{s}\to D_{s}^{\mp}K^{\pm}$ decays are
functions of a combination of $\gamma$ and $\beta_{s}$, namely
$\gamma-2\beta_{s}$. By using an independent measurement on $\beta_{s}$ is
thus possible to determine an independent value of $\gamma$. A comparison
between the value of $\gamma$ measured from tree-level processes, as mentioned
in Section 2.1, with the value of $\gamma$ and other unitarity triangle
parameters measured in loop-level processes provides a powerful consistency
check of the SM. In particular, in this case the amplitude ratio between the
dominant tree level process and the subleading process is approximately 0.4.
Moreover, the decay-width difference in the $B^{0}_{s}$ system,
$\Delta\Gamma_{s}$, is nonzero, which, compared to the $B^{0}$ case, gives
access to two additional $C\\!P$-violating observables which are coefficients
of in time-dependent hyperbolical terms. Thus a good sensitivity on $\gamma$
is expected. The results found by the LHCb collaboration [27] analysing Run 1
data are summarise in Table 1. Assuming the measured value of $\phi_{s}$ equal
to $-2\beta_{s}$ as external input the value of $\gamma$ is determined to be
$\gamma=(127^{+17}_{-22})^{\circ}$. The quoted uncertainties are the sum in
quadrature of the statistical and systematic components. A compatibility of
$2.15\,\sigma$ is found between the presented measurement and the LHCb
$\gamma$ average presented in Section 2.1.

$C_{f}$ | = 0.735 $\pm$ 0.142 (stat) $\pm$ 0.048 (syst)
---|---
$S_{f}$ | = -0.518 $\pm$ 0.202 (stat) $\pm$ 0.073 (syst)
$S_{\overline{f}}$ | = -0.496 $\pm$ 0.197 (stat) $\pm$ 0.071 (syst)
$A^{\Delta\Gamma}_{f}$ | = 0.395 $\pm$ 0.277 (stat) $\pm$ 0.122 (syst)
$A^{\Delta\Gamma}_{\overline{f}}$ | = 0.314 $\pm$ 0.274 (stat) $\pm$ 0.107 (syst)
Table 1: Time-dependent $C\\!P$ violating observables in $B^{0}_{s}\to
D_{s}^{\mp}K^{\pm}$ decays. The second and the third uncertainties are
statistical and systematical respectively.

##  3 Measurement of $C\\!P$ violation in baryons

The vast majority of $C\\!P$ violation measurements in beauty decays have
focused on $B$ mesons due to the rich phenomenology available in meson mixing
and the large datasets collected by the $B$-factories. However, the LHC
copiously produces beauty also in the form of $\Lambda^{0}_{b}$ baryons, which
gives large samples of $\Lambda^{0}_{b}$ baryons for $C\\!P$ measurements. In
the SM, sizeable $C\\!P$ asymmetries are predicted to occur in decays of
beauty baryons to final states that contain no charmed hadrons. LHCb has
reported the first measurement of $C\\!P$ violation in the baryon sector using
$\Lambda_{b}\to p\,\pi^{-}\pi^{+}\pi^{-}$ decays [28]. $C\\!P$ violation
arises from interference between tree and higher order decay amplitudes. Since
the two amplitudes are of similar size, large interference effects are
possible. The search for a $C\\!P$ violating asymmetry in these decays cannot
rely on a comparison of $\Lambda^{0}_{b}$ and $\overline{\Lambda}^{0}_{b}$
decay rates, due to the poorly known
$\Lambda^{0}_{b}-\overline{\Lambda}^{0}_{b}$ production asymmetry in $pp$
collisions. In order to observe $C\\!P$ violation the following $C\\!P$-odd
observables have been defined
$C_{\widehat{T}}=\overrightarrow{p}_{p}\cdot(\overrightarrow{p}_{h_{1}^{-}}\times\overrightarrow{p}_{h_{2}^{+}})$
for $\Lambda_{b}$ and
$\overline{C}_{\widehat{T}}=\overrightarrow{p}_{\overline{p}}\cdot(\overrightarrow{p}_{h_{1}^{+}}\times\overrightarrow{p}_{h_{2}^{-}})$
for $\overline{\Lambda}^{0}_{b}$, where $\overrightarrow{p}$ stands for
momentum and $h_{1}h_{2}$ are the fastest $\pi^{-}$ in the $\Lambda^{0}_{b}$
reference frame and the single $\pi^{+}$, respectively. From these observables
it is possible to build the following asymmetries,
$A=[N(C_{\widehat{T}}>0)-N(C_{\widehat{T}}<0)]/[N(C_{\widehat{T}}>0)+N(C_{\widehat{T}}<0)]$
and
$\overline{A}=[\overline{N}(-\overline{C}_{\widehat{T}}>0)-\overline{N}(-\overline{C}_{\widehat{T}}<0)]/[\overline{N}(-\overline{C}_{\widehat{T}}>0)+\overline{N}(-\overline{C}_{\widehat{T}}<0)]$,
where $N$ and $\overline{N}$ are the number of $\Lambda^{0}_{b}$ and
$\overline{\Lambda}^{0}_{b}$, respectively. From these asymmetries the
$C\\!P$\- and $P$-violating observables, $a_{C\\!P}$ and $a_{P}$, are
constructed. They are defined as $a_{C\\!P}=\frac{1}{2}(A-\overline{A})$ and
$a_{P}=\frac{1}{2}(A+\overline{A})$. Analysing the data collected by the LHCb
detector in Run 1, with $\sim 6650$ reconstructed $\Lambda^{0}_{b}\to
p\pi^{-}\pi^{+}\pi^{-}$ signal decays, it was possible to find the first
evidence of $C\\!P$ violation in baryon decays, namely $a_{C\\!P}\neq 0$ at
3.3 $\sigma$. This result is obtained from the combination of measurements of
the asymmetry in bins of the 4-body phase space. This represents an important
test of the validity of the SM in the baryon sector. No deviation from parity
conservation was found.

##  4 Measurement of $C\\!P$ violation in charm

After the first experimental observations of the slow mixing rate of the
$D^{0}-\overline{D}^{0}$ flavour oscillations [29], the level of attention on
charm physics increased significantly, thanks to the full range of probes that
has been disclosed, entirely complementary to the $B$ and $K$ mesons, for
mixing and $C\\!P$ violation. Indeed the charm quark is the only up-type quark
that manifests flavour oscillations and the only way to probe $C\\!P$
violation with up-type quarks. Moreover, only with the advent of LHC has it
been possible to collect huge and very clean samples of $D$ meson decays,
several orders of magnitude larger in size than in the past, allowing for the
first time to approach the small SM expectations for $C\\!P$-violation, which
is below the $10^{-3}$ level [30].

Direct $C\\!P$ violation can be experimentally determined by measuring the raw
asymmetries in the decay of $D^{0}$ mesons to $C\\!P$ eigenstates, $f$, for
example $D^{0}\to K^{+}K^{-},\pi^{+}\pi^{-}$

$A_{raw}=\frac{N(D^{0}\to f)-N(\overline{D}^{0}\to f)}{N(D^{0}\to
f)+N(\overline{D}^{0}\to f)}\,.$ (13)

The $C\\!P$ asymmetry is then determined taking into account the production,
$A_{P}$, and the detection $A_{D}$ asymmetries, $A_{raw}\approx
A_{CP}+A_{P}+A_{D}$. The most recent LHCb measurements are [31]
$A_{C\\!P}(K^{+}K^{-})=(0.04\pm 0.12\mathrm{(stat)}\pm 0.10\mathrm{(syst)})\%$
and $A_{C\\!P}(\pi^{+}\pi^{-})=(0.07\pm 0.14\mathrm{(stat)}\pm
0.11\mathrm{(syst)})\%$, which are a combination of a semileptonic $B$ decays-
tagged and promptly-produced $D^{*+}$ tagged samples. Figure 3 (left) shows
the results for the two $C\\!P$ asymmetries. LHCb also recently studied
$C\\!P$ asymmetries in
$D^{+}_{(s)}\to\eta^{\prime}(\to\pi^{+}\pi^{-}\gamma)\pi^{+}$ [32], which was
never measured before at a hadron collider due to the difficulty of accurately
modelling the large background and obtained the most precise measurement to
date in this decay. LHCb also studied $C\\!P$ asymmetries in
$D^{0}\to\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ [33] decays, which is the first time
that such a study has been performed for a 4-body decay, for which the rich
resonant structure allows to search for local enhancement of the $C\\!P$
asymmetries. These results are now approaching the per-mille level of
uncertainty, but so far they are consistent with $C\\!P$ conservation.

Figure 3: Left: $C\\!P$ violating asymmetries in $D^{0}\to
K^{+}K^{-},\pi^{+}\pi^{-}$ decays using semileptonic $B$ decays-tagged and
promptly-produced $D^{*+}$ tagged samples and their combination. Right: HFLAV
combination of the parameter $A_{\Gamma}$.

Neutral D mesons oscillations are dominated by long-distance effects and the
mixing parameters are small, as are the the expected $C\\!P$-violating
effects. Indirect $C\\!P$ violation can be determined by measuring the decay-
time dependent $C\\!P$ asymmetry of $D^{0}$ decays to $C\\!P$ eigenstates,
$f$,

$A_{C\\!P}(t)=\frac{\Gamma(D^{0}(t)\to f)-\Gamma(\overline{D}^{0}\to
f)}{\Gamma(D^{0}\to f)+\Gamma(\overline{D}^{0}\to f)}\simeq
a_{dir}^{f}-A_{\Gamma}\frac{t}{\tau_{D}}\;\;\;\;\;\mathrm{,where}\;\;\;A_{\Gamma}\equiv\frac{\widehat{\Gamma}_{D^{0}\to
f}-\widehat{\Gamma}_{\overline{D}^{0}\to f}}{\widehat{\Gamma}_{D^{0}\to
f}+\widehat{\Gamma}_{\overline{D}^{0}\to f}}\,.$ (14)

Here $A_{\Gamma}$ is the asymmetry between the $D^{0}$ and $\overline{D}^{0}$
effective decay widths and it is the parameter sensitive to $C\\!P$ violation
in the mixing and in the interference between mixing and decay. Within the SM,
the indirect determination of this parameter is
$A_{\Gamma}<\mathcal{O}(10^{-3})$ [23]. LHCb recently measured it using a
promptly-produced $D^{*+}$ tagged $D^{0}$ sample. The combination of
$f=KK,\pi\pi$ measurements in Run I is $A_{\Gamma}=(0.13\pm
0.28\mathrm{(stat)}\pm 0.10\mathrm{(syst)})\times 10^{-3}$ [34]. The result is
consistent with the hypothesis of $C\\!P$ conservation and represents the most
precise measurement of $C\\!P$ violation in the charm system ever made. In
Figure 3 (right) the HFLAV [23] combination of different $A_{\Gamma}$
measurements performed so far is shown.

##  5 Conclusions

Recent LHCb results of $C\\!P$ violation in the beauty and charm hadron
systems have been presented. All measurements are in good agreement with SM
predictions. The selection of $C\\!P$ violation measurements presented here is
not a complete list of the analysis performed by the ATLAS, CMS and LHCb
collaborations. Updated results based on the Run 2 data, collected in 2015 and
2016, are expected soon and will enable even stronger constraints to be made
on potential BSM physics contributions.

## References

  * [1] C. Patrignani et al. [Particle Data Group], “The Review of Particle Physics”, Chin. Phys. C 40, 100001 (2016) and 2017 update.
  * [2] J. Ellis, “Where is Particle Physics Going?”, CERN-TH-2017-080, (2017), [arXiv:1704.02821v1 [hep-ph]].
  * [3] A. D. Sakharov, “Violation of $C\\!P$ invariance, $C$ asymmetry, and baryon asymmetry of the universe”, Soviet Physics Uspekhi 34, 392 (1991).
  * [4] A. Riotto and M. Trodden, “Recent progress in baryogenesis”, Ann. Rev. Nucl. Part. Sci. 49, 35-75 (1999).
  * [5] M. P. Gavela, P. Hernandez, J. Orloff and O. Pene, “Standard Model $C\\!P$ violation and baryon asymmetry”, Mod. Phys. Lett. A9, 795 (1994).
  * [6] M. Kobayashi and T. Maskawa, “$C\\!P$ violation in the renormalizable theory of weak interaction”, Prog. Theor. Phys. 49, 652 (1973).
  * [7] U. Nierste, “Three Lectures on Meson Mixing and CKM phenomenology”, arXiv:0904.1869 [hep-ph] (2009).
  * [8] A. Lenz and U. Nierste, “Numerical updates of lifetimes and mixing parameters of B mesons”, TUM-HEP 11, 792 (2011).
  * [9] G. Aad et al. [ATLAS Collaboration], “The ATLAS Experiment at the CERN Large Hadron Collider”, JINST 3, S08003 (2008).
  * [10] S. Chatrchyan et al. [CMS Collaboration], “The CMS experiment at the CERN LHC“,JINST 3, S08004 (2008).
  * [11] A. Alves Jr. et al. [LHCb Collaboration], “The LHCb detector at the LHC”, JINST 3, S08005 (2008).
  * [12] J. Charles et al., “Current status of the Standard Model CKM fit and constraints on $\Delta F=2$ New Physics”, Phys. Rev. D 91, 073007 (2015).
  * [13] R. Aaij et al. [LHCb Collaboration], “Measurement of the CKM angle $\gamma$ from a combination of LHCb results”, JHEP 1612, 087 (2016).
  * [14] R. Aaij et al. [LHCb Collaboration], “Measurement of $CP$ violation in $B^{0}\rightarrow J/\psi K^{0}_{S}$ decays”, Phys. Rev. Lett. 115, 031601 (2015).
  * [15] R. Aaij et al. [LHCb Collaboration], “Measurement of $CP$ violation in $B^{0}\rightarrow J/\psi(e^{+}e^{-})K^{0}_{S}$ and $B^{0}\rightarrow\psi(2S)(\mu^{+}\mu^{-})K^{0}_{S}$ decays”, LHCb-PAPER-2017-029, in preparation (2017).
  * [16] G. Aad et al. [ATLAS Collaboration], “Measurement of the CP-violating phase $\phi_{s}$ and the $B^{0}_{s}$ meson decay width difference with $B^{0}_{s}\to J/\psi\phi$ decays in ATLAS”, JHEP 1608, 147 (2016).
  * [17] V. Khachatryan et al. [CMS Collaboration], “Measurement of the CP-violating weak phase $\phi_{s}$ and the decay width difference $\Delta\Gamma_{s}$ using the B${}_{s}^{0}\to J/\psi\phi$(1020) decay channel in pp collisions at $\sqrt{s}=$ 8 TeV”, Phys. Lett. B 757, 97 (2016).
  * [18] R. Aaij et al. [LHCb Collaboration], “Precision measurement of $CP$ violation in $B_{s}^{0}\to J/\psi K^{+}K^{-}$ decays”, Phys. Rev. Lett. 114, 041801 (2015).
  * [19] R. Aaij et al. [LHCb Collaboration], “Measurement of the CP-violating phase $\phi_{s}$ in $\overline{B}^{0}_{s}\rightarrow J/\psi\pi^{+}\pi^{-}$ decays”, Phys. Lett. B 736, 186 (2014).
  * [20] R. Aaij et al. [LHCb Collaboration], “First study of the CP -violating phase and decay-width difference in $B_{s}^{0}\to\psi(2S)\phi$ decays”, Phys. Lett. B 762, 253 (2016).
  * [21] R. Aaij et al. [LHCb Collaboration], “Measurement of the $CP$-violating phase $\phi_{s}$ in $\overline{B}^{0}_{s}\to D_{s}^{+}D_{s}^{-}$ decays”, Phys. Rev. Lett. 113, 211801 (2014).
  * [22] R. Aaij et al. [LHCb Collaboration], “Resonances and $CP$ violation in $B_{s}^{0}$ and $\overline{B}_{s}^{0}\to J/\psi K^{+}K^{-}$ decays in the mass region above the $\phi(1020)$”, JHEP 1708, 037 (2017).
  * [23] Y. Amhis et al., “Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016”, arXiv:1612.07233 [hep-ex].
  * [24] J. Charles et al., “Current status of the Standard Model CKM fit and constraints on $\Delta F=2$ New Physics”, Phys. Rev. D 91, 073007 (2015).
  * [25] R. Aaij et al. [LHCb Collaboration], “Measurement of time-dependent $C\\!P$-violating asymmetries in $B^{0}\to\pi^{+}\pi^{-}$ and $B^{0}_{s}\to K^{+}K^{-}$ decays at LHCb”, LHCb-CONF-2016-018 (2017).
  * [26] R. Aaij et al. [LHCb Collaboration], “Measurement of $CP$ violation in $B^{0}\\!\rightarrow D^{+}D^{-}$ decays”, Phys. Rev. Lett. 117, 261801 (2016).
  * [27] R. Aaij et al. [LHCb Collaboration], “Measurement of $C\\!P$ asymmetry in $B^{0}_{s}\to D^{\mp}_{s}K^{\pm}$ decays”, LHCb-CONF-2016-015 (2017).
  * [28] R. Aaij et al. [LHCb Collaboration], “Measurement of matter-antimatter differences in beauty baryon decays”, Nature Phys. 13, 391 (2017).
  * [29] R. Aaij et al. [LHCb Collaboration], “Observation of $D^{0}-\overline{D}^{0}$ oscillations”, Phys. Rev. Lett. 110, 101802 (2013).
  * [30] B. Bhattacharya, M. Gronau and J. L. Rosner, “CP asymmetries in singly-Cabibbo-suppressed $D$ decays to two pseudoscalar mesons”, Phys. Rev. D 85, 054014 (2012), [Phys. Rev. D 85, 079901 (2012)].
  * [31] R. Aaij et al. [LHCb Collaboration], “Measurement of $CP$ asymmetry in $D^{0}\rightarrow K^{-}K^{+}$ decays”, Phys. Lett. B 767, 177 (2017).
  * [32] R. Aaij et al. [LHCb Collaboration], “Measurement of $CP$ asymmetries in $D^{\pm}\rightarrow\eta^{\prime}\pi^{\pm}$ and $D_{s}^{\pm}\rightarrow\eta^{\prime}\pi^{\pm}$ decays”, Phys. Lett. B 771, 21 (2017).
  * [33] R. Aaij et al. [LHCb Collaboration], “Search for $CP$ violation in the phase space of $D^{0}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ decays”, Phys. Lett. B 769, 345 (2017).
  * [34] R. Aaij et al. [LHCb Collaboration], “Measurement of the $C\\!P$ violation parameter $A_{\Gamma}$ in $D^{0}\to K^{+}K^{-}$ and $D^{0}\to\pi^{+}\pi^{-}$ decays”, Phys. Rev. Lett. 118, 261803 (2017).