EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-042 LHCb-PAPER-2019-006 May 31, 2019

Observation of $C\\!P$ violation in charm decays

LHCb collaboration†††Authors are listed at the end of this Letter.

A search for charge-parity ($C\\!P$) violation in
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays is reported, using $pp$
collision data corresponding to an integrated luminosity of
5.9$\text{\,fb}^{-1}$ collected at a center-of-mass energy of 13 TeV with the
LHCb detector. The flavor of the charm meson is inferred from the charge of
the pion in ${{D}^{*}}(2010)^{+}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays or
from the charge of the muon in ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\\!\rightarrow{{D}^{0}}\mu^{-}\bar{\nu}_{\mu}X$
decays. The difference between the ${C\\!P}$ asymmetries in
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays is measured to be
$\Delta A_{{C\\!P}}=[-18.2\pm 3.2\,(\rm stat.)\pm 0.9\,(\rm syst.)]\times
10^{-4}$ for $\pi$-tagged and $\Delta A_{{C\\!P}}=[-9\pm 8\,(\rm stat.)\pm
5\,(\rm syst.)]\times 10^{-4}$ for $\mu$-tagged ${D}^{0}$ mesons. Combining
these with previous LHCb results leads to

$\Delta A_{{C\\!P}}=(-15.4\pm 2.9)\times 10^{-4},$

where the uncertainty includes both statistical and systematic contributions.
The measured value differs from zero by more than five standard deviations.
This is the first observation of $C\\!P$ violation in the decay of charm
hadrons.

Published in Phys. Rev. Lett. 122 (2019) 211803

© 2024 CERN for the benefit of the LHCb collaboration. CC-BY-4.0 licence.

The noninvariance of fundamental interactions under the combined action of
charge conjugation ($C$) and parity ($P$) transformations, so-called $C\\!P$
violation, is a necessary condition for the dynamical generation of the baryon
asymmetry of the universe [1]. The Standard Model (SM) of particle physics
includes $C\\!P$ violation through an irreducible complex phase in the
Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix [2, 3]. The realization of
$C\\!P$ violation in weak interactions has been established in the $K$\- and
$B$-meson systems by several experiments [4, 5, 6, 7, 8, 9, 10, 11, 12], and
all results are well interpreted within the CKM formalism. However, the size
of $C\\!P$ violation in the SM appears to be too small to account for the
observed matter-antimatter asymmetry [13, 14, 15], suggesting the existence of
sources of $C\\!P$ violation beyond the SM.

The observation of $C\\!P$ violation in the charm sector has not been achieved
yet, despite decades of experimental searches. Charm hadrons provide a unique
opportunity to measure $C\\!P$ violation with particles containing only up-
type quarks. The size of $C\\!P$ violation in charm decays is expected to be
tiny in the SM, with asymmetries typically of the order of
$10^{-4}$–$10^{-3}$, but due to the presence of low-energy strong-interaction
effects, theoretical predictions are difficult to compute reliably [16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Motivated
by the fact that contributions of beyond-the-SM virtual particles may alter
the size of $C\\!P$ violation with respect to the SM expectation, a number of
theoretical analyses have been performed [19, 27, 35, 32].

Unprecedented experimental precision can be reached at LHCb in the measurement
of $C\\!P$-violating asymmetries in
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays. The inclusion of
charge-conjugate decay modes is implied throughout except in asymmetry
definitions. Searches for $C\\!P$ violation in these decay modes have been
performed by the BaBar [36], Belle [37], CDF [38, 39] and LHCb [40, 41, 42,
43, 44] collaborations. The corresponding $C\\!P$ asymmetries have been found
to be consistent with zero within a precision of a few per mille.

This Letter presents a measurement of the difference of the time-integrated
$C\\!P$ asymmetries in ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays, performed using $p$
$p$ collision data collected with the LHCb detector at a center-of-mass energy
of 13 TeV, and corresponding to an integrated luminosity of
5.9$\text{\,fb}^{-1}$.

The time-dependent $C\\!P$ asymmetry, $A_{{C\\!P}}(f;\,t)$, between states
produced as ${D}^{0}$ or ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mesons decaying to a $C\\!P$
eigenstate $f$ at time $t$ is defined as

$A_{{C\\!P}}(f;\,t)\equiv\frac{\Gamma({{D}^{0}}(t)\rightarrow
f)-\Gamma({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\rightarrow
f)}{\Gamma({{D}^{0}}(t)\rightarrow f)+\Gamma({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(t)\rightarrow f)},$ (1)

where $\Gamma$ denotes the time-dependent rate of a given decay. For
$f={{K}^{-}}{{K}^{+}}$ or $f={{\pi}^{-}}{{\pi}^{+}}$, $A_{{C\\!P}}(f;\,t)$ can
be expressed in terms of a direct component associated to $C\\!P$ violation in
the decay amplitude and another component associated to $C\\!P$ violation in
${D}^{0}$–${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing or in
the interference between mixing and decay.

A time-integrated asymmetry, $A_{{C\\!P}}(f)$, can be determined, and its
value will exhibit a dependence on the variation of the reconstruction
efficiency as a function of the decay time. To first order in the
${D}^{0}$–${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing
parameters, it can be written as [38, 45]

$A_{{C\\!P}}(f)\approx a_{{C\\!P}}^{\rm dir}(f)-\frac{\langle
t(f)\rangle}{\tau({{D}^{0}})}\,A_{\Gamma}(f),$ (2)

where $\langle t(f)\rangle$ denotes the mean decay time of ${D}^{0}$
$\rightarrow$ $f$ decays in the reconstructed sample, incorporating the
effects of the time-dependent experimental efficiency, $a_{{C\\!P}}^{\rm
dir}(f)$ is the direct $C\\!P$ asymmetry, $\tau({{D}^{0}})$ the ${D}^{0}$
lifetime and $A_{\Gamma}(f)$ the asymmetry between the ${{D}^{0}}\rightarrow
f$ and ${{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow f$
effective decay widths [46, 47]. In the limit of U-spin symmetry, the direct
$C\\!P$ asymmetry is equal in magnitude and opposite in sign for ${K}^{-}$
${K}^{+}$ and ${\pi}^{-}$ ${\pi}^{+}$, though the size of U-spin-breaking
effects at play is uncertain [19]. Taking $A_{\Gamma}$ to be independent of
final state [19, 48, 49], the difference in $C\\!P$ asymmetries between
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays is

$\displaystyle\Delta A_{{C\\!P}}$ $\displaystyle\equiv$ $\displaystyle
A_{{C\\!P}}({{K}^{-}}{{K}^{+}})-A_{{C\\!P}}({{\pi}^{-}}{{\pi}^{+}})$ (3)
$\displaystyle\approx$ $\displaystyle\Delta a_{{C\\!P}}^{\rm
dir}-\frac{\Delta\langle t\rangle}{\tau({{D}^{0}})}\,A_{\Gamma},$

where $\Delta a_{{C\\!P}}^{\rm dir}\equiv a_{{C\\!P}}^{\rm
dir}(K^{-}K^{+})-a_{{C\\!P}}^{\rm dir}(\pi^{-}\pi^{+})$ and $\Delta\langle
t\rangle$ is the difference of the mean decay times $\langle
t({{K}^{-}}{{K}^{+}})\rangle$ and $\langle t({{\pi}^{-}}{{\pi}^{+}})\rangle$.

The ${D}^{0}$ mesons considered in this analysis are produced either promptly
at a $p$ $p$ collision point (primary vertex, PV) in the strong decay of
${{D}^{*}}(2010)^{+}$ mesons (hereafter referred to as ${{D}^{*+}}$) to a
${{D}^{0}}\pi^{+}$ pair or at a vertex displaced from any PV in semileptonic
${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\rightarrow{{D}^{0}}\mu^{-}\bar{\nu}_{\mu}X$
decays, where ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}}$ denotes a
hadron containing a $b$ quark and $X$ stands for potential additional
particles. The flavor at production of ${D}^{0}$ mesons from ${{D}^{*+}}$
decays is determined from the charge of the accompanying pion ($\pi$-tagged),
whereas that of ${D}^{0}$ mesons from semileptonic $b$-hadron decays is
obtained from the charge of the accompanying muon ($\mu$-tagged). The raw
asymmetries measured for $\pi$-tagged and $\mu$-tagged ${D}^{0}$ decays are
defined as

$\displaystyle\begin{split}A_{\rm
raw}^{\pi{\mbox{-}}\rm{tagged}}(f)&\equiv\frac{N\left({{D}^{*+}}\rightarrow{{D}^{0}}(f)\pi^{+}\right)-N\left({{D}^{*-}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(f)\pi^{-}\right)}{N\left({{D}^{*+}}\rightarrow{{D}^{0}}(f)\pi^{+}\right)+N\left({{D}^{*-}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(f)\pi^{-}\right)},\\\ A_{\rm
raw}^{\mu{\mbox{-}}\rm{tagged}}(f)&\equiv\frac{N({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\rightarrow{{D}^{0}}(f)\,\mu^{-}\bar{\nu}_{\mu}X)-N({B}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(f)\,\mu^{+}\nu_{\mu}X)}{N({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}}\rightarrow{{D}^{0}}(f)\,\mu^{-}\bar{\nu}_{\mu}X)+N({B}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(f)\,\mu^{+}\nu_{\mu}X)},\end{split}$
(4)

where $N$ is the measured signal yield for the given decay. These can be
approximated as

$\displaystyle\begin{split}A_{\rm raw}^{\pi{\mbox{-}}\rm{tagged}}(f)&\approx
A_{{C\\!P}}(f)+A_{\rm D}(\pi)+A_{\rm P}({{D}^{*}}),\\\ A_{\rm
raw}^{\mu{\mbox{-}}\rm{tagged}}(f)&\approx A_{{C\\!P}}(f)+A_{\rm
D}(\mu)+A_{\rm P}({B}),\end{split}$ (5)

where $A_{\rm D}(\pi)$ and $A_{\rm D}(\mu)$ are detection asymmetries due to
different reconstruction efficiencies between positive and negative tagging
pions and muons, whereas $A_{\rm P}({{D}^{*}})$ and $A_{\rm P}({B})$ are the
production asymmetries of ${D}^{*}$ mesons and $b$ hadrons, arising from the
hadronization of charm and beauty quarks in $p$ $p$ collisions [50]. Owing to
the smallness of the involved terms, which averaged over phase space for
selected events are ${\mathcal{O}}(10^{-2})$ or less [50, 51, 52, 53], the
approximations in Eqs. (5) are valid up to corrections of
${\mathcal{O}}(10^{-6})$. The values of $A_{\rm D}(\pi)$ and $A_{\rm
P}({{D}^{*}})$, as well as those of $A_{\rm D}(\mu)$ and $A_{\rm P}({B})$, are
independent of the final state $f$, and thus cancel in the difference,
resulting in

$\displaystyle\begin{split}\Delta A_{{C\\!P}}=A_{\rm
raw}({{K}^{-}}{{K}^{+}})-A_{\rm raw}({{\pi}^{-}}{{\pi}^{+}}).\end{split}$ (6)

This simple relation between $\Delta A_{{C\\!P}}$ and the measurable raw
asymmetries in ${{K}^{-}}{{K}^{+}}$ and ${{\pi}^{-}}{{\pi}^{+}}$ makes the
determination of $\Delta A_{{C\\!P}}$ largely insensitive to systematic
uncertainties.

The LHCb detector is a single-arm forward spectrometer designed for the study
of particles containing $b$ or $c$ quarks, as described in detail in Refs.
[54, 55]. The LHCb tracking system exploits a dipole magnet to measure the
momentum of charged particles. Although the analysis presented in this Letter
is expected to be insensitive to such effects, the magnetic-field polarity is
reversed periodically during data taking to mitigate the differences of
reconstruction efficiencies of particles with opposite charges. Data sets
corresponding to about one half of the total integrated luminosity are
recorded with each magnetic-field configuration.

The online event selection is performed by a trigger, which consists of a
hardware stage based on information from the calorimeter and muon systems,
followed by two software stages. In the first software stage, events used in
this analysis are selected if at least one track has large transverse momentum
and is incompatible with originating from any PV, or if any two-track
combination forming a secondary vertex, consistent with that of a ${D}^{0}$
decay, is found in the event by a multivariate algorithm [56, 57]. In between
the first and second software stages, detector alignment and calibration are
performed and updated constants are made available to the software trigger
[58]. In the second stage, ${D}^{0}$ candidates are fully reconstructed using
kinematic, topological and particle-identification (PID) criteria.
Requirements are placed on: the ${D}^{0}$ decay vertex, which must be well
separated from all PVs in the event; the quality of reconstructed tracks; the
${D}^{0}$ transverse momentum; the angle between the ${D}^{0}$ momentum and
its flight direction; PID information; and the impact-parameter significances
($\chi^{2}_{\text{IP}}$) of the ${D}^{0}$ decay products with respect to all
PVs in the event, where the $\chi^{2}_{\text{IP}}$ is defined as the
difference between the $\chi^{2}$ of the PV reconstructed with and without the
considered particle. In the analysis of the $\mu$-tagged sample, $B$
candidates are formed by combining a ${D}^{0}$ candidate with a muon under the
requirement that they are consistent with originating from a common vertex. In
addition, requirements on the invariant mass of the ${{D}^{0}}\mu$ system,
$m({{D}^{0}}\mu)$, and on the corrected mass ($m_{\rm corr}$) are applied. The
corrected mass partially recovers the missing energy of the unreconstructed
particles and is defined as $m_{\rm
corr}\equiv\sqrt{m({{D}^{0}}\mu)^{2}+p_{\perp}({{D}^{0}}\mu)^{2}}+p_{\perp}({{D}^{0}}\mu)$
[59], where $p_{\perp}({{D}^{0}}\mu)$ is the momentum of the ${{D}^{0}}\mu$
system transverse to the flight direction of the $b$ hadron, determined from
the primary and ${{D}^{0}}\mu$ vertices.

In the offline selection, trigger signals are associated to reconstructed
particles. Selection requirements are applied on the trigger decision, taking
into account the information on whether the decision was taken due to the
signal decay products or to other particles produced in the event. Fiducial
requirements are imposed to exclude kinematic regions characterized by large
detection asymmetries for the tagging pion or muon. Very large raw
asymmetries, up to 100%, occur in certain kinematic regions because, for a
given magnet polarity, low-momentum particles of one charge at large or small
polar angles in the horizontal plane may be deflected out of the detector or
into the (uninstrumented) LHC beam pipe, whereas particles with the other
charge are more likely to remain within the acceptance [60]. About 35% and 10%
of the selected candidates are rejected by these fiducial requirements for the
$\pi$-tagged and $\mu$-tagged samples, respectively. In the retained samples,
raw asymmetries are typically at the percent level or below. For $\pi$-tagged
${D}^{0}$ mesons, a requirement on the ${D}^{0}$ $\chi^{2}_{\text{IP}}$ is
applied to suppress the background of ${D}^{0}$ mesons from $B$ decays, and
PID requirements on the ${D}^{0}$ decay products are further tightened. Then
the ${D}^{0}$ and pion candidates are combined to form ${D}^{*+}$ candidates
by requiring a good fit quality of the $D^{*+}$ vertex and the invariant mass
of ${D}^{0}$ candidates to lie within a range of about $\pm 3$ standard
deviations around the known ${{D}^{0}}$ mass. The $D^{*+}$ vertex is
determined as a common vertex of $D^{0}$ and tagging ${{\pi}^{+}}$ candidates,
and is constrained to coincide with the nearest PV [61].

For $\mu$-tagged mesons, the $B$ candidates are further filtered using a
dedicated boosted decision tree (BDT) to suppress the combinatorial background
due to random combinations of charged kaon or pion pairs not originating from
a ${D}^{0}$ decay. The variables used in the BDT to discriminate signal from
combinatorial background are: the fit quality of the ${D}^{0}$ and the $B$
decay vertices; the ${D}^{0}$ flight distance; the ${D}^{0}$ impact parameter,
i.e., the minimum distance of its trajectory to the nearest PV; the transverse
momenta of the ${D}^{0}$ decay products, the significance of the distance
between the ${D}^{0}$ and $B$ decay vertices; the invariant mass
$m({{D}^{0}}\mu)$ and the corrected mass $m_{\rm corr}$. To suppress
background from $b$-hadron decays to ${c}{\overline{c}}\pi^{\pm}X$
(${c}{\overline{c}}K^{\pm}X$), where the ${c}{\overline{c}}$ resonance decays
to a pair of muons, ${D}^{0}$ candidates are vetoed if the invariant mass of
the $\mu^{\mp}\pi^{\pm}$ ($\mu^{\mp}K^{\pm}$) pair, where the pion (kaon) is
given the muon mass hypothesis, lies within a window of about $\pm
50\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ around the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ or $\psi{(2S)}$ known masses.

The data sample includes events with multiple ${D}^{*+}$ and $B$ candidates.
The majority of these events contain the same reconstructed ${D}^{0}$ meson
combined with different tagging pions or muons. When multiple candidates are
present in the event, only one is kept randomly. The fractions of events with
multiple candidates are about 10% and 0.4% in the $\pi$-tagged and
$\mu$-tagged samples, respectively. A small fraction of events, of the order
of per mille, belong to both the selected $\pi$-tagged and $\mu$-tagged
samples.

As the detection and production asymmetries are expected to depend on the
kinematics of the reconstructed particles, the cancellation in the difference
between the raw asymmetries in Eq. (6) may be incomplete if the kinematic
distributions of reconstructed ${D}^{*+}$ or $B$ candidates and of the tagging
pions or muons differ between the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ decay
modes. For this reason, a small correction to the $K^{-}K^{+}$ sample is
applied by means of a weighting procedure [60]. For the $\pi$-tagged sample,
candidate-by-candidate weights are calculated by taking the ratio between the
three-dimensional background-subtracted distributions of transverse momentum,
azimuthal angle and pseudorapidity of the ${D}^{*+}$ meson in the $K^{-}K^{+}$
and $\pi^{-}\pi^{+}$ modes. An analogous procedure is followed for the
$\mu$-tagged sample, where ${D}^{0}$ distributions are used instead of those
of the ${D}^{*+}$ meson. It is then checked _a posteriori_ that the
distributions of the same variables for tagging pions and muons are also
equalized by the weighting. The application of the weights leads to a small
variation of $\Delta A_{{C\\!P}}$, below $10^{-4}$ for both the $\pi$-tagged
and $\mu$-tagged samples.

Figure 1: Mass distributions of selected (top) $\pi^{\pm}$-tagged and (bottom)
$\mu^{\pm}$-tagged candidates for (left) ${{K}^{-}}{{K}^{+}}$ and (right)
${{\pi}^{-}}{{\pi}^{+}}$ final states of the ${D}^{0}$-meson decays, with fit
projections overlaid.

The raw asymmetries of signal and background components for each decay mode
are free parameters determined by means of simultaneous least-square fits to
the binned mass distributions of ${D}^{*+}$ and ${D}^{*-}$ candidates for the
$\pi$-tagged sample, or ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ candidates for the $\mu$-tagged
sample. In particular, in the analysis of the $\pi$-tagged sample the fits are
performed to the $m({{D}^{0}}{{\pi}^{+}})$ and $m({{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}})$ distributions. As
outlined in Ref. [38], using these distributions has the advantage that they
are the same for both ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay modes.

The signal mass model, which is obtained from simulation, consists of the sum
of three Gaussian functions and a Johnson $S_{{U}}$ function [62], whose
parameters are free to be adjusted by the fit to the data. The mean values of
the Gaussian functions are distinct for positive and negative tags, whereas
widths and fractions are shared. The parameters of the Johnson $S_{{U}}$
function, which accounts for the slight asymmetric shape of the signal
distribution due to the proximity of the $m({{D}^{0}})+m({{\pi}^{+}})$
threshold, are also shared. The combinatorial background is described by an
empirical function of the form
$[m({{D}^{0}}{{\pi}^{+}})-m({{D}^{0}})-m({{\pi}^{+}})]^{\alpha}e^{\beta\,m({{D}^{0}}{{\pi}^{+}})}$,
where $\alpha$ and $\beta$ are two free parameters which are shared among
positive and negative tags. In the analysis of the $\mu$-tagged sample, the
fits are performed to the $m({{D}^{0}})$ distributions. The signal is
described by the sum of two Gaussian functions convolved with a truncated
power-law function that accounts for final-state photon radiation effects,
whereas the combinatorial background is described by an exponential function.
A small contribution from ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays
with a misidentified kaon or pion is also visible, which is modeled as the
tail of a Gaussian function. Separate fits are performed to subsamples of data
collected with different magnet polarities and in different years. All partial
$\Delta A_{{C\\!P}}$ values corresponding to each subsample are found to be in
good agreement and then averaged to obtain the final results. If single fits
are performed to the overall $\pi$-tagged and $\mu$-tagged samples, small
differences of the order of a few $10^{-5}$ are found. The
$m({{D}^{0}}{{\pi}^{+}})$ and $m({{D}^{0}})$ distributions corresponding to
the entire samples are displayed in Fig. 1 (see also Ref. [60] for the
corresponding asymmetries as a function of mass). The $\pi$-tagged
($\mu$-tagged) signal yields are approximately $44$ ($9$) million
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decays and $14$ ($3$) million
${D}^{0}\rightarrow\pi^{-}\pi^{+}$ decays. In the case of $\pi$-tagged decays,
the fits to the $m({{D}^{0}}{{\pi}^{+}})$ distributions do not distinguish
between background that produces peaks in $m({{D}^{0}}{{\pi}^{+}})$, which can
arise from ${D}^{*+}$ decays where the correct tagging pion is found but the
${D}^{0}$ meson is misreconstructed, and signal. The effect on $\Delta
A_{{C\\!P}}$ of residual peaking backgrounds, suppressed by selection
requirements to less than 1% of the number of signal candidates, is evaluated
as a systematic uncertainty.

Studies of systematic uncertainties on $\Delta A_{{C\\!P}}$ are carried out
independently for the $\pi$-tagged and $\mu$-tagged samples. Several sources
affecting the measurement are considered. In the case of $\pi$-tagged decays,
the dominant systematic uncertainty is related to the knowledge of the signal
and background mass models. It is evaluated by generating pseudoexperiments
according to the baseline fit model, then fitting alternative models to those
data. A value of $0.6\times 10^{-4}$ is assigned as a systematic uncertainty,
corresponding to the largest variation observed using the alternative
functions. Possible differences between ${{D}^{0}}{{\pi}^{+}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ invariant-mass
shapes are investigated by studying a sample of 232 million
${{D}^{*+}}\\!\rightarrow{{D}^{0}}({{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$ and
${{D}^{*-}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}({{K}^{+}}{{\pi}^{-}}){{\pi}^{-}}$
decays. The effect on $\Delta A_{{C\\!P}}$ is estimated to be order of
$10^{-5}$ at most, hence negligible. A similar study with pseudoexperiments is
also performed with the $\mu$-tagged sample and a value of $2\times 10^{-4}$
is found.

In the case of $\mu$-tagged decays, the main systematic uncertainty is due to
the possibility that the ${D}^{0}$ flavor is not tagged correctly by the muon
charge because of misreconstruction. The probability of wrongly assigning the
${D}^{0}$ flavor (mistag) is studied with a large sample of $\mu$-tagged
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays by comparing the charges
of kaon and muon candidates. Mistag rates are found to be at the percent level
and compatible for positively and negatively tagged decays. The corresponding
systematic uncertainty is estimated to be $4\times 10^{-4}$, also taking into
account the fact that wrongly tagged decays include a fraction of doubly
Cabibbo-suppressed ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and mixed
${{D}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays, calculated to be $0.39\%$ with negligible uncertainty for both the
${K}^{+}$ ${\pi}^{-}$ and ${K}^{-}$ ${\pi}^{+}$ final states using input from
Ref. [63].

Systematic uncertainties of $0.2\times 10^{-4}$ and $1\times 10^{-4}$
accounting for the knowledge of the weights used in the kinematic weighting
procedure are assessed for $\pi$-tagged and $\mu$-tagged decays, respectively.
Although suppressed by the requirement that the ${D}^{0}$ trajectory points
back to the PV, a fraction of ${D}^{0}$ mesons from $B$ decays is still
present in the final $\pi$-tagged sample. As
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays may have different
levels of contamination, the value of $\Delta A_{{C\\!P}}$ may be biased
because of an incomplete cancellation of the production asymmetries of $b$
hadrons. The fractions of ${D}^{0}$ mesons from $B$ decays are estimated by
performing a fit to the distribution of the ${D}^{0}$-candidate impact
parameter in the plane transverse to the beam direction [60]. The
corresponding systematic uncertainty is estimated to be $0.3\times 10^{-4}$. A
systematic uncertainty associated to the presence of background components
peaking in $m({{D}^{0}}\pi)$ and not in $m({{D}^{0}})$ is determined by fits
to the $m({{D}^{0}})$ distributions [60], where these components are modeled
using fast simulation [64]. The main sources are the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{0}}$ decay for the
${{K}^{+}}{{K}^{-}}$ final state, and the
${{D}^{0}}\rightarrow{{\pi}^{-}}{\mu^{+}}{{\nu}_{\mu}}$ and
${{D}^{0}}\rightarrow{{\pi}^{-}}{e^{+}}{{\nu}_{e}}$ decays for the
${{\pi}^{+}}{{\pi}^{-}}$ final state. Yields and raw asymmetries of the
peaking-background components measured from the fits are then used as inputs
to pseudoexperiments designed to evaluate the corresponding effects on the
determination of $\Delta A_{{C\\!P}}$. A value of $0.5\times 10^{-4}$ is
assigned as a systematic uncertainty.

In the case of $\mu$-tagged decays, the fractions of reconstructed ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}$ decays can be slightly different
between the $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ decay modes, which could lead to
a small bias in $\Delta A_{{C\\!P}}$. Using the LHCb measurements of the
$b$-hadron production asymmetries [50], the systematic uncertainty on $\Delta
A_{{C\\!P}}$ is estimated to be $1\times 10^{-4}$. The combination of a
difference in the $B$ reconstruction efficiency as a function of the decay
time between the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ modes and the presence of
neutral $B$-meson oscillations may also cause an imperfect cancellation of
$A_{\rm P}(B)$ in $\Delta A_{{C\\!P}}$. The associated systematic uncertainty
is estimated to be $2\times 10^{-4}$.

All individual contributions are summed in quadrature to give total systematic
uncertainties on $\Delta A_{{C\\!P}}$ of $0.9\times 10^{-4}$ and $5\times
10^{-4}$ for the $\pi$-tagged and $\mu$-tagged samples, respectively. A
summary of all systematic uncertainties is reported in Table 1. Other possible
systematic uncertainties are investigated and found to be negligible.

Table 1: Systematic uncertainties on $\Delta A_{{C\\!P}}$ for $\pi$\- and $\mu$-tagged decays (in $10^{-4}$). The total uncertainties are obtained as the sums in quadrature of the individual contributions. Source |  $\pi$-tagged |  $\mu$-tagged
---|---|---
Fit model | 0.6 | 2
Mistag | – | 4
Weighting | 0.2 | 1
Secondary decays | 0.3 | –
Peaking background | 0.5 | –
$B$ fractions | – | 1
$B$ reco. efficiency | – | 2
Total | 0.9 | 5

Numerous additional robustness checks are carried out [60]. The measured value
of $\Delta A_{{C\\!P}}$ is studied as a function of several variables, notably
including: the azimuthal angle, $\chi^{2}_{\text{IP}}$, transverse momentum
and pseudorapidity of $\pi$-tagged and $\mu$-tagged ${D}^{0}$ mesons as well
as of the tagging pions or muons; the $\chi^{2}$ of the ${D}^{*+}$ and $B$
vertex fits; the track quality of the tagging pion and the charged-particle
multiplicity in the event. Furthermore, the total sample is split into
subsamples taken in different run periods within the years of data taking,
also distinguishing different magnet polarities. No evidence for unexpected
dependences of $\Delta A_{{C\\!P}}$ is found in any of these tests. A check
using more stringent PID requirements is performed, and all variations of
$\Delta A_{{C\\!P}}$ are found to be compatible within statistical
uncertainties. An additional check concerns the measurement of $\Delta A_{\rm
bkg}$, that is the difference of the background raw asymmetries in
${{K}^{-}}{{K}^{+}}$ and ${{\pi}^{-}}{{\pi}^{+}}$ final states. As the prompt
background is mainly composed of genuine ${D}^{0}$ candidates paired with
unrelated pions originating from the PV, $\Delta A_{\rm bkg}$ is expected to
be compatible with zero. A value of $\Delta A_{\rm bkg}=(-2\pm 4)\times
10^{-4}$ is obtained.

The difference of time-integrated $C\\!P$ asymmetries of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and ${D}^{0}$ $\rightarrow$
${\pi}^{-}$ ${\pi}^{+}$ decays is measured using 13 TeV $pp$ collision data
collected with the LHCb detector and corresponding to an integrated luminosity
of 5.9$\text{\,fb}^{-1}$. The results are

$\displaystyle\Delta A_{{C\\!P}}^{\pi{\mbox{-}}\rm{tagged}}=\left[-18.2\pm
3.2\,\rm{(stat.)}\pm 0.9\,\rm{(syst.)}\right]\times 10^{-4},$
$\displaystyle\Delta A_{{C\\!P}}^{\mu{\mbox{-}}\rm{tagged}}=\left[-9\pm
8\,\rm{(stat.)}\pm 5\,\rm{(syst.)}\right]\times 10^{-4}.$

Both measurements are in good agreement with world averages [65] and previous
LHCb results [43, 42].

By making a full combination with previous LHCb measurements [43, 42], the
following value of $\Delta A_{{C\\!P}}$ is obtained

$\Delta A_{{C\\!P}}=\left(-15.4\pm 2.9\right)\times 10^{-4},$

where the uncertainty includes statistical and systematic contributions. The
significance of the deviation from zero corresponds to 5.3 standard
deviations. This is the first observation of $C\\!P$ violation in the decay of
charm hadrons.

The interpretation of $\Delta A_{{C\\!P}}$ in terms of direct $C\\!P$
violation and $A_{\Gamma}$ requires knowledge of the difference of
reconstructed mean decay times for ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$
and ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays normalized to the
${D}^{0}$ lifetime, as shown in Eq. (3). The values corresponding to the
present measurements are $\Delta\\!\left\langle
t\right\rangle^{\pi{\mbox{-}}\rm{tagged}}/\tau({{D}^{0}})=0.135\pm 0.002$ and
$\Delta\\!\left\langle
t\right\rangle^{\mu{\mbox{-}}\rm{tagged}}/\tau({{D}^{0}})=-0.003\pm 0.001$,
whereas that corresponding to the full combination is $\Delta\left\langle
t\right\rangle/\tau({{D}^{0}})=0.115\pm 0.002$. The uncertainties include
statistical and systematic contributions, and the world average of the
${D}^{0}$ lifetime is used [66].

By using in addition the LHCb average $A_{\Gamma}=(-2.8\pm 2.8)\times 10^{-4}$
[46, 47], from Eq. (3) it is possible to derive

$\Delta a_{{C\\!P}}^{\rm dir}=\left(-15.7\pm 2.9\right)\times 10^{-4},$

which shows that, as expected, $\Delta A_{{C\\!P}}$ is primarily sensitive to
direct $C\\!P$ violation. The overall improvement in precision brought by the
present analysis to the knowledge of $\Delta a_{{C\\!P}}^{\rm dir}$ is
apparent when comparing with the value obtained from previous measurements,
$\Delta a_{{C\\!P}}^{\rm dir}=\left(-13.4\pm 7.0\right)\times 10^{-4}$ [65].

In summary, this Letter reports the first observation of a nonzero $C\\!P$
asymmetry in charm decays, using large samples of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays collected with the LHCb
detector. The result is consistent with, although in magnitude at the upper
end of, SM expectations, which lie in the range $10^{-4}$–$10^{-3}$ [16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In
particular, the result challenges predictions based on first-principle QCD
dynamics [19, 33]. It complies with predictions based on
flavor-$\mathrm{SU}(3)$ symmetry, if one assumes a dynamical enhancement of
the penguin amplitude [16, 26, 27, 28, 29, 30, 32]. In the next decade,
further measurements with charmed particles, along with possible theoretical
improvements, will help clarify the physics picture, and establish whether
this result is consistent with the SM or indicates the presence of new
dynamics in the up-quark sector.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); NSF (USA). We acknowledge the computing resources that are
provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF
(Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open-source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom); Laboratory Directed Research and
Development program of LANL (USA).

## References

  * [1] A. D. Sakharov, _Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe_ , Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32
  * [2] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [3] M. Kobayashi and T. Maskawa, _$C\\!P$ -violation in the renormalizable theory of weak interaction_, Prog. Theor. Phys. 49 (1973) 652
  * [4] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, _Evidence for the $2\pi$ decay of the $K_{2}^{0}$ meson_, Phys. Rev. Lett. 13 (1964) 138
  * [5] KTeV collaboration, A. Alavi-Harati et al., _Observation of direct CP violation in $K_{\rm S,L}\rightarrow\pi\pi$ decays_, Phys. Rev. Lett. 83 (1999) 22, arXiv:hep-ex/9905060
  * [6] NA48 collaboration, A. Lai et al., _A precise measurement of the direct CP violation parameter $Re({\varepsilon}^{\prime}/{\varepsilon})$_, Eur. Phys. J. C22 (2001) 231, arXiv:hep-ex/0110019
  * [7] BaBar collaboration, B. Aubert et al., _Observation of $C\\!P$ violation in the $B^{0}$ meson system_, Phys. Rev. Lett. 87 (2001) 091801, arXiv:hep-ex/0107013
  * [8] Belle collaboration, K. Abe et al., _Observation of large $C\\!P$ violation in the neutral $B$ meson system_, Phys. Rev. Lett. 87 (2001) 091802, arXiv:hep-ex/0107061
  * [9] BaBar collaboration, B. Aubert et al., _Direct $C\\!P$ violating asymmetry in $B^{0}\rightarrow K^{+}\pi^{-}$ decays_, Phys. Rev. Lett. 93 (2004) 131801, arXiv:hep-ex/0407057
  * [10] Belle collaboration, Y. Chao et al., _Evidence for direct CP violation in $B^{0}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays_, Phys. Rev. Lett. 93 (2004) 191802, arXiv:hep-ex/0408100
  * [11] LHCb collaboration, R. Aaij et al., _First observation of $C\\!P$ violation in the decays of ${{B}^{0}_{s}}$ mesons_, Phys. Rev. Lett. 110 (2013) 221601, arXiv:1304.6173
  * [12] LHCb collaboration, R. Aaij et al., _Observation of $C\\!P$ violation in ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays_, Phys. Lett. B712 (2012) 203, Erratum ibid. B713 (2012) 351, arXiv:1203.3662
  * [13] A. G. Cohen, D. B. Kaplan, and A. E. Nelson, _Progress in electroweak baryogenesis_ , Ann. Rev. Nucl. Part. Sci. 43 (1993) 27, arXiv:hep-ph/9302210
  * [14] A. Riotto and M. Trodden, _Recent progress in baryogenesis_ , Ann. Rev. Nucl. Part. Sci. 49 (1999) 35, arXiv:hep-ph/9901362
  * [15] W.-S. Hou, _Source of CP violation for the baryon asymmetry of the universe_ , Chin. J. Phys. 47 (2009) 134, arXiv:0803.1234
  * [16] M. Golden and B. Grinstein, _Enhanced CP violations in hadronic charm decays_ , Phys. Lett. B222 (1989) 501
  * [17] F. Buccella et al., _Nonleptonic weak decays of charmed mesons_ , Phys. Rev. D51 (1995) 3478, arXiv:hep-ph/9411286
  * [18] S. Bianco, F. L. Fabbri, D. Benson, and I. Bigi, _A Cicerone for the physics of charm_ , Riv. Nuovo Cim. 26N7 (2003) 1, arXiv:hep-ex/0309021
  * [19] Y. Grossman, A. L. Kagan, and Y. Nir, _New physics and CP violation in singly Cabibbo suppressed D decays_ , Phys. Rev. D75 (2007) 036008, arXiv:hep-ph/0609178
  * [20] M. Artuso, B. Meadows, and A. A. Petrov, _Charm meson decays_ , Ann. Rev. Nucl. Part. Sci. 58 (2008) 249, arXiv:0802.2934
  * [21] J. Brod, A. L. Kagan, and J. Zupan, _Size of direct CP violation in singly Cabibbo-suppressed D decays_ , Phys. Rev. D86 (2012) 014023, arXiv:1111.5000
  * [22] H.-Y. Cheng and C.-W. Chiang, _Direct $C\\!P$ violation in two-body hadronic charmed meson decays_, Phys. Rev. D85 (2012) 034036, Erratum ibid. D85 (2012) 079903, arXiv:1201.0785
  * [23] H.-Y. Cheng and C.-W. Chiang, _SU(3) symmetry breaking and $C\\!P$ violation in $D\rightarrow PP$ decays_, Phys. Rev. D86 (2012) 014014, arXiv:1205.0580
  * [24] H.-n. Li, C.-D. Lu, and F.-S. Yu, _Branching ratios and direct CP asymmetries in $D\rightarrow PP$ decays_, Phys. Rev. D86 (2012) 036012, arXiv:1203.3120
  * [25] E. Franco, S. Mishima, and L. Silvestrini, _The Standard Model confronts $C\\!P$ violation in $D^{0}\rightarrow\pi^{+}\pi^{-}$ and $D^{0}\rightarrow K^{+}K^{-}$_, JHEP 05 (2012) 140, arXiv:1203.3131
  * [26] D. Pirtskhalava and P. Uttayarat, _CP Violation and flavor SU(3) breaking in D-meson decays_ , Phys. Lett. B712 (2012) 81, arXiv:1112.5451
  * [27] T. Feldmann, S. Nandi, and A. Soni, _Repercussions of flavour symmetry breaking on CP violation in D-meson decays_ , JHEP 06 (2012) 007, arXiv:1202.3795
  * [28] J. Brod, Y. Grossman, A. L. Kagan, and J. Zupan, _A consistent picture for large penguins in $D\rightarrow{{\pi}^{+}}{{\pi}^{-}},\,{{K}^{+}}{{K}^{-}}$_, JHEP 10 (2012) 161, arXiv:1203.6659
  * [29] G. Hiller, M. Jung, and S. Schacht, _SU(3)-flavor anatomy of nonleptonic charm decays_ , Phys. Rev. D87 (2013) 014024, arXiv:1211.3734
  * [30] Y. Grossman and D. J. Robinson, _SU(3) sum rules for charm decay_ , JHEP 04 (2013) 067, arXiv:1211.3361
  * [31] B. Bhattacharya, M. Gronau, and J. L. Rosner, _CP asymmetries in singly-Cabibbo-suppressed $D$ decays to two pseudoscalar mesons_, Phys. Rev. D85 (2012) 054014, arXiv:1201.2351
  * [32] S. Müller, U. Nierste, and S. Schacht, _Sum rules of charm $C\\!P$ asymmetries beyond the SU(3)F limit_, Phys. Rev. Lett. 115 (2015) 251802, arXiv:1506.04121
  * [33] A. Khodjamirian and A. A. Petrov, _Direct CP asymmetry in $D\rightarrow\pi^{-}\pi^{+}$ and $D\rightarrow K^{-}K^{+}$ in QCD-based approach_, Phys. Lett. B774 (2017) 235, arXiv:1706.07780
  * [34] F. Buccella, A. Paul, and P. Santorelli, _On $SU(3)_{F}$ breaking through final state interactions and CP asymmetries in $D\rightarrow PP$ decays_, arXiv:1902.05564
  * [35] D. Atwood and A. Soni, _Searching for the origin of $C\\!P$ violation in Cabibbo-suppressed $D$-meson decays_, PTEP 2013 (2013) 093B05, arXiv:1211.1026
  * [36] BaBar collaboration, B. Aubert et al., _Search for CP violation in the decays $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$_, Phys. Rev. Lett. 100 (2008) 061803, arXiv:0709.2715
  * [37] Belle collaboration, M. Starič et al., _Search for a CP asymmetry in Cabibbo-suppressed $D^{0}$ decays_, Phys. Lett. B670 (2008) 190, arXiv:0807.0148
  * [38] CDF collaboration, T. Aaltonen et al., _Measurement of CP-violating asymmetries in $D^{0}\rightarrow\pi^{+}\pi^{-}$ and $D^{0}\rightarrow K^{+}K^{-}$ decays at CDF_, Phys. Rev. D85 (2012) 012009, arXiv:1111.5023
  * [39] CDF collaboration, T. Aaltonen et al., _Measurement of the difference of CP-violating asymmetries in $D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ decays at CDF_, Phys. Rev. Lett. 109 (2012) 111801, arXiv:1207.2158
  * [40] LHCb collaboration, R. Aaij et al., _Evidence for $C\\!P$ violation in time-integrated ${{D}^{0}}\rightarrow h^{-}h^{+}$ decay rates_, Phys. Rev. Lett. 108 (2012) 111602, arXiv:1112.0938
  * [41] LHCb collaboration, R. Aaij et al., _Search for direct $C\\!P$ violation in ${{D}^{0}}\rightarrow h^{-}h^{+}$ modes using semileptonic ${B}$ decays_, Phys. Lett. B723 (2013) 33, arXiv:1303.2614
  * [42] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ asymmetry in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797
  * [43] LHCb collaboration, R. Aaij et al., _Measurement of the difference of time-integrated $C\\!P$ asymmetries in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, Phys. Rev. Lett. 116 (2016) 191601, arXiv:1602.03160
  * [44] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ asymmetry in ${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ decays_, Phys. Lett. B767 (2017) 177, arXiv:1610.09476
  * [45] M. Gersabeck et al., _On the interplay of direct and indirect CP violation in the charm sector_ , J. Phys. G39 (2012) 045005, arXiv:1111.6515
  * [46] LHCb collaboration, R. Aaij et al., _Measurement of indirect $C\\!P$ asymmetries in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays using semileptonic $B$ decays_, JHEP 04 (2015) 043, arXiv:1501.06777
  * [47] LHCb collaboration, R. Aaij et al., _Measurement of the $C\\!P$ violation parameter $A_{\Gamma}$ in ${{D}^{0}}\rightarrow{{K}^{+}}{{K}^{-}}$ and ${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays_, Phys. Rev. Lett. 118 (2017) 261803, arXiv:1702.06490
  * [48] A. L. Kagan and M. D. Sokoloff, _Indirect $C\\!P$ violation and implications for ${D}^{0}$–${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ and $B_{s}$–$\overline{B}_{s}$ mixing_, Phys. Rev. D80 (2009) 076008, arXiv:0907.3917
  * [49] D.-S. Du, _CP violation for neutral charmed meson decays to CP eigenstates_ , Eur. Phys. J. C50 (2007) 579, arXiv:hep-ph/0608313
  * [50] LHCb collaboration, R. Aaij et al., _Measurement of ${B}^{0}$, ${B}^{0}_{s}$, ${{B}^{+}}$ and ${\mathchar 28931\relax}^{0}_{b}$ production asymmetries in $7$ and $8$ TeV proton-proton collisions_, Phys. Lett. B774 (2017) 139, arXiv:1703.08464
  * [51] LHCb collaboration, R. Aaij et al., _Measurement of the flavour-specific $C\\!P$-violating asymmetry $a_{\mathrm{sl}}^{s}$ in ${{B}^{0}_{s}}$ decays_, Phys. Lett. B728 (2014) 607, arXiv:1308.1048
  * [52] LHCb collaboration, R. Aaij et al., _Measurement of the ${{D}^{\pm}}$ production asymmetry in $7$ TeV ${p}{p}$ collisions_, Phys. Lett. B718 (2013) 902, arXiv:1210.4112
  * [53] LHCb collaboration, R. Aaij et al., _Measurement of the ${{D}^{+}_{s}}$–${{D}^{-}_{s}}$ production asymmetry in $7$ TeV ${p}{p}$ collisions_, Phys. Lett. B713 (2012) 186, arXiv:1205.0897
  * [54] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [55] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [56] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [57] T. Likhomanenko et al., _LHCb topological trigger reoptimization_ , J. Phys. Conf. Ser. 664 (2015) 082025, arXiv:1510.00572
  * [58] G. Dujany and B. Storaci, _Real-time alignment and calibration of the LHCb Detector in Run II_ , J. Phys. Conf. Ser. 664 (2015) 082010
  * [59] E653 collaboration, K. Kodama et al., _Measurement of the relative branching fraction $\Gamma(D^{0}\rightarrow K\mu\nu)/\Gamma(D^{0}\rightarrow\mu X)$_, Phys. Rev. Lett. 66 (1991) 1819
  * [60] See supplemental material for additional numerical values and plots
  * [61] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [62] N. L. Johnson, _Systems of frequency curves generated by methods of translation_ , Biometrika 36 (1949) 149
  * [63] LHCb collaboration, R. Aaij et al., _Updated determination of ${{D}^{0}}$-${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}^{0}$ mixing and $C\\!P$ violation parameters with ${{D}^{0}}\rightarrow K^{+}\pi^{-}$ decays_, Phys. Rev. D97 (2018) 031101, arXiv:1712.03220
  * [64] G. A. Cowan, D. C. Craik, and M. D. Needham, _RapidSim: An application for the fast simulation of heavy-quark hadron decays_ , Comput. Phys. Commun. 214 (2017) 239, arXiv:1612.07489
  * [65] Heavy Flavor Averaging Group, Y. Amhis et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016_, Eur. Phys. J. C77 (2017) 895, arXiv:1612.07233, updated results and plots available at https://hflav.web.cern.ch
  * [66] Particle Data Group, M. Tanabashi et al., _Review of particle physics_ , Phys. Rev. D98 (2018) 030001

LHCb collaboration

R. Aaij28, C. Abellán Beteta46, B. Adeva43, M. Adinolfi50, C.A. Aidala78, Z.
Ajaltouni6, S. Akar61, P. Albicocco19, J. Albrecht11, F. Alessio44, M.
Alexander55, A. Alfonso Albero42, G. Alkhazov41, P. Alvarez Cartelle57, A.A.
Alves Jr43, S. Amato2, Y. Amhis8, L. An18, L. Anderlini18, G. Andreassi45, M.
Andreotti17, J.E. Andrews62, F. Archilli28, P. d’Argent13, J. Arnau Romeu7, A.
Artamonov40, M. Artuso63, K. Arzymatov37, E. Aslanides7, M. Atzeni46, B.
Audurier23, S. Bachmann13, J.J. Back52, S. Baker57, V. Balagura8,b, W.
Baldini17,44, A. Baranov37, R.J. Barlow58, S. Barsuk8, W. Barter57, M.
Bartolini20, F. Baryshnikov74, V. Batozskaya32, B. Batsukh63, A. Battig11, V.
Battista45, A. Bay45, F. Bedeschi25, I. Bediaga1, A. Beiter63, L.J. Bel28, S.
Belin23, N. Beliy66, V. Bellee45, N. Belloli21,i, K. Belous40, I. Belyaev34,
E. Ben-Haim9, G. Bencivenni19, S. Benson28, S. Beranek10, A. Berezhnoy35, R.
Bernet46, D. Berninghoff13, E. Bertholet9, A. Bertolin24, C. Betancourt46, F.
Betti16,e, M.O. Bettler51, M. van Beuzekom28, Ia. Bezshyiko46, S. Bhasin50, J.
Bhom30, M.S. Bieker11, S. Bifani49, P. Billoir9, A. Birnkraut11, A.
Bizzeti18,u, M. Bjørn59, M.P. Blago44, T. Blake52, F. Blanc45, S. Blusk63, D.
Bobulska55, V. Bocci27, O. Boente Garcia43, T. Boettcher60, A. Bondar39,x, N.
Bondar41, S. Borghi58,44, M. Borisyak37, M. Borsato13, M. Boubdir10, T.J.V.
Bowcock56, C. Bozzi17,44, S. Braun13, M. Brodski44, J. Brodzicka30, A. Brossa
Gonzalo52, D. Brundu23,44, E. Buchanan50, A. Buonaura46, C. Burr58, A.
Bursche23, J. Buytaert44, W. Byczynski44, S. Cadeddu23, H. Cai68, R.
Calabrese17,g, S. Cali17, R. Calladine49, M. Calvi21,i, M. Calvo Gomez42,m, A.
Camboni42,m, P. Campana19, D.H. Campora Perez44, L. Capriotti16,e, A.
Carbone16,e, G. Carboni26, R. Cardinale20, A. Cardini23, P. Carniti21,i, K.
Carvalho Akiba2, G. Casse56, M. Cattaneo44, G. Cavallero20, R. Cenci25,p, M.G.
Chapman50, M. Charles9,44, Ph. Charpentier44, G. Chatzikonstantinidis49, M.
Chefdeville5, V. Chekalina37, C. Chen3, S. Chen23, S.-G. Chitic44, V.
Chobanova43, M. Chrzaszcz44, A. Chubykin41, P. Ciambrone19, X. Cid Vidal43, G.
Ciezarek44, F. Cindolo16, P.E.L. Clarke54, M. Clemencic44, H.V. Cliff51, J.
Closier44, V. Coco44, J.A.B. Coelho8, J. Cogan7, E. Cogneras6, L. Cojocariu33,
P. Collins44, T. Colombo44, A. Comerma-Montells13, A. Contu23, G. Coombs44, S.
Coquereau42, G. Corti44, C.M. Costa Sobral52, B. Couturier44, G.A. Cowan54,
D.C. Craik60, A. Crocombe52, M. Cruz Torres1, R. Currie54, C. D’Ambrosio44,
C.L. Da Silva79, E. Dall’Occo28, J. Dalseno43,v, A. Danilina34, A. Davis58, O.
De Aguiar Francisco44, K. De Bruyn44, S. De Capua58, M. De Cian45, J.M. De
Miranda1, L. De Paula2, M. De Serio15,d, P. De Simone19, C.T. Dean55, W.
Dean78, D. Decamp5, L. Del Buono9, B. Delaney51, H.-P. Dembinski12, M.
Demmer11, A. Dendek31, D. Derkach38, O. Deschamps6, F. Desse8, F. Dettori23,
B. Dey69, A. Di Canto44, P. Di Nezza19, S. Didenko74, H. Dijkstra44, F.
Dordei23, M. Dorigo25,y, A. Dosil Suárez43, L. Douglas55, A. Dovbnya47, K.
Dreimanis56, L. Dufour44, G. Dujany9, P. Durante44, J.M. Durham79, D. Dutta58,
R. Dzhelyadin40,†, M. Dziewiecki13, A. Dziurda30, A. Dzyuba41, S. Easo53, U.
Egede57, V. Egorychev34, S. Eidelman39,x, S. Eisenhardt54, U. Eitschberger11,
R. Ekelhof11, L. Eklund55, S. Ely63, A. Ene33, S. Escher10, S. Esen28, T.
Evans61, A. Falabella16, N. Farley49, S. Farry56, D. Fazzini21,i, P. Fernandez
Declara44, A. Fernandez Prieto43, F. Ferrari16,e, L. Ferreira Lopes45, F.
Ferreira Rodrigues2, S. Ferreres Sole28, M. Ferro-Luzzi44, S. Filippov36, R.A.
Fini15, M. Fiorini17,g, M. Firlej31, C. Fitzpatrick44, T. Fiutowski31, F.
Fleuret8,b, M. Fontana44, F. Fontanelli20,h, R. Forty44, V. Franco Lima56, M.
Frank44, C. Frei44, J. Fu22,q, W. Funk44, C. Färber44, M. Féo44, E. Gabriel54,
A. Gallas Torreira43, D. Galli16,e, S. Gallorini24, S. Gambetta54, Y. Gan3, M.
Gandelman2, P. Gandini22, Y. Gao3, L.M. Garcia Martin76, B. Garcia Plana43, J.
García Pardiñas46, J. Garra Tico51, L. Garrido42, D. Gascon42, C. Gaspar44, G.
Gazzoni6, D. Gerick13, E. Gersabeck58, M. Gersabeck58, T. Gershon52, D.
Gerstel7, Ph. Ghez5, V. Gibson51, O.G. Girard45, P. Gironella Gironell42, L.
Giubega33, K. Gizdov54, V.V. Gligorov9, D. Golubkov34, A. Golutvin57,74, A.
Gomes1,a, I.V. Gorelov35, C. Gotti21,i, E. Govorkova28, J.P. Grabowski13, R.
Graciani Diaz42, L.A. Granado Cardoso44, E. Graugés42, E. Graverini46, G.
Graziani18, A. Grecu33, R. Greim28, P. Griffith23, L. Grillo58, L. Gruber44,
B.R. Gruberg Cazon59, C. Gu3, X. Guo67, E. Gushchin36, A. Guth10, Yu.
Guz40,44, T. Gys44, C. Göbel65, T. Hadavizadeh59, C. Hadjivasiliou6, G.
Haefeli45, C. Haen44, S.C. Haines51, B. Hamilton62, Q. Han69, X. Han13, T.H.
Hancock59, S. Hansmann-Menzemer13, N. Harnew59, T. Harrison56, C. Hasse44, M.
Hatch44, J. He66, M. Hecker57, K. Heinicke11, A. Heister11, K. Hennessy56, L.
Henry76, E. van Herwijnen44, J. Heuel10, M. Heß71, A. Hicheur64, R. Hidalgo
Charman58, D. Hill59, M. Hilton58, P.H. Hopchev45, J. Hu13, W. Hu69, W.
Huang66, Z.C. Huard61, W. Hulsbergen28, T. Humair57, M. Hushchyn38, D.
Hutchcroft56, D. Hynds28, P. Ibis11, M. Idzik31, P. Ilten49, A. Inglessi41, A.
Inyakin40, K. Ivshin41, R. Jacobsson44, S. Jakobsen44, J. Jalocha59, E.
Jans28, B.K. Jashal76, A. Jawahery62, F. Jiang3, M. John59, D. Johnson44, C.R.
Jones51, C. Joram44, B. Jost44, N. Jurik59, S. Kandybei47, M. Karacson44, J.M.
Kariuki50, S. Karodia55, N. Kazeev38, M. Kecke13, F. Keizer51, M. Kelsey63, M.
Kenzie51, T. Ketel29, B. Khanji44, A. Kharisova75, C. Khurewathanakul45, K.E.
Kim63, T. Kirn10, V.S. Kirsebom45, S. Klaver19, K. Klimaszewski32, S.
Koliiev48, M. Kolpin13, R. Kopecna13, P. Koppenburg28, I. Kostiuk28,48, S.
Kotriakhova41, M. Kozeiha6, L. Kravchuk36, M. Kreps52, F. Kress57, S.
Kretzschmar10, P. Krokovny39,x, W. Krupa31, W. Krzemien32, W. Kucewicz30,l, M.
Kucharczyk30, V. Kudryavtsev39,x, G.J. Kunde79, A.K. Kuonen45, T.
Kvaratskheliya34, D. Lacarrere44, G. Lafferty58, A. Lai23, D. Lancierini46, G.
Lanfranchi19, C. Langenbruch10, T. Latham52, C. Lazzeroni49, R. Le Gac7, A.
Leflat35, R. Lefèvre6, F. Lemaitre44, O. Leroy7, T. Lesiak30, B.
Leverington13, H. Li67, P.-R. Li66,ab, X. Li79, Y. Li4, Z. Li63, X. Liang63,
T. Likhomanenko73, R. Lindner44, P. Ling67, F. Lionetto46, V. Lisovskyi8, G.
Liu67, X. Liu3, D. Loh52, A. Loi23, I. Longstaff55, J.H. Lopes2, G. Loustau46,
G.H. Lovell51, D. Lucchesi24,o, M. Lucio Martinez43, Y. Luo3, A. Lupato24, E.
Luppi17,g, O. Lupton52, A. Lusiani25, X. Lyu66, R. Ma67, F. Machefert8, F.
Maciuc33, V. Macko45, P. Mackowiak11, S. Maddrell-Mander50, O. Maev41,44, K.
Maguire58, D. Maisuzenko41, M.W. Majewski31, S. Malde59, B. Malecki44, A.
Malinin73, T. Maltsev39,x, H. Malygina13, G. Manca23,f, G. Mancinelli7, D.
Marangotto22,q, J. Maratas6,w, J.F. Marchand5, U. Marconi16, C. Marin Benito8,
M. Marinangeli45, P. Marino45, J. Marks13, P.J. Marshall56, G. Martellotti27,
M. Martinelli44,21, D. Martinez Santos43, F. Martinez Vidal76, A.
Massafferri1, M. Materok10, R. Matev44, A. Mathad46, Z. Mathe44, V.
Matiunin34, C. Matteuzzi21, K.R. Mattioli78, A. Mauri46, E. Maurice8,b, B.
Maurin45, M. McCann57,44, A. McNab58, R. McNulty14, J.V. Mead56, B. Meadows61,
C. Meaux7, N. Meinert71, D. Melnychuk32, M. Merk28, A. Merli22,q, E.
Michielin24, D.A. Milanes70, E. Millard52, M.-N. Minard5, L. Minzoni17,g, D.S.
Mitzel13, A. Mogini9, R.D. Moise57, T. Mombächer11, I.A. Monroy70, S.
Monteil6, M. Morandin24, G. Morello19, M.J. Morello25,t, J. Moron31, A.B.
Morris7, R. Mountain63, F. Muheim54, M. Mukherjee69, M. Mulder28, C.H.
Murphy59, D. Murray58, A. Mödden 11, D. Müller44, J. Müller11, K. Müller46, V.
Müller11, P. Naik50, T. Nakada45, R. Nandakumar53, A. Nandi59, T. Nanut45, I.
Nasteva2, M. Needham54, N. Neri22,q, S. Neubert13, N. Neufeld44, R.
Newcombe57, T.D. Nguyen45, C. Nguyen-Mau45,n, S. Nieswand10, R. Niet11, N.
Nikitin35, N.S. Nolte44, D.P. O’Hanlon16, A. Oblakowska-Mucha31, V.
Obraztsov40, S. Ogilvy55, R. Oldeman23,f, C.J.G. Onderwater72, J. D. Osborn78,
A. Ossowska30, J.M. Otalora Goicochea2, T. Ovsiannikova34, P. Owen46, A.
Oyanguren76, P.R. Pais45, T. Pajero25,t, A. Palano15, M. Palutan19, G.
Panshin75, A. Papanestis53, M. Pappagallo54, L.L. Pappalardo17,g, W. Parker62,
C. Parkes58,44, G. Passaleva18,44, A. Pastore15, M. Patel57, C.
Patrignani16,e, A. Pearce44, A. Pellegrino28, G. Penso27, M. Pepe Altarelli44,
S. Perazzini16, D. Pereima34, P. Perret6, L. Pescatore45, K. Petridis50, A.
Petrolini20,h, A. Petrov73, S. Petrucci54, M. Petruzzo22,q, B. Pietrzyk5, G.
Pietrzyk45, M. Pikies30, M. Pili59, D. Pinci27, J. Pinzino44, F. Pisani44, A.
Piucci13, V. Placinta33, S. Playfer54, J. Plews49, M. Plo Casasus43, F.
Polci9, M. Poli Lener19, M. Poliakova63, A. Poluektov7, N. Polukhina74,c, I.
Polyakov63, E. Polycarpo2, G.J. Pomery50, S. Ponce44, A. Popov40, D.
Popov49,12, S. Poslavskii40, E. Price50, C. Prouve43, V. Pugatch48, A. Puig
Navarro46, H. Pullen59, G. Punzi25,p, W. Qian66, J. Qin66, R. Quagliani9, B.
Quintana6, N.V. Raab14, B. Rachwal31, J.H. Rademacker50, M. Rama25, M. Ramos
Pernas43, M.S. Rangel2, F. Ratnikov37,38, G. Raven29, M. Ravonel Salzgeber44,
M. Reboud5, F. Redi45, S. Reichert11, A.C. dos Reis1, F. Reiss9, C. Remon
Alepuz76, Z. Ren3, V. Renaudin59, S. Ricciardi53, S. Richards50, K. Rinnert56,
P. Robbe8, A. Robert9, A.B. Rodrigues45, E. Rodrigues61, J.A. Rodriguez
Lopez70, M. Roehrken44, S. Roiser44, A. Rollings59, V. Romanovskiy40, A.
Romero Vidal43, J.D. Roth78, M. Rotondo19, M.S. Rudolph63, T. Ruf44, J. Ruiz
Vidal76, J.J. Saborido Silva43, N. Sagidova41, B. Saitta23,f, V. Salustino
Guimaraes65, C. Sanchez Gras28, C. Sanchez Mayordomo76, B. Sanmartin Sedes43,
R. Santacesaria27, C. Santamarina Rios43, M. Santimaria19,44, E.
Santovetti26,j, G. Sarpis58, A. Sarti19,k, C. Satriano27,s, A. Satta26, M.
Saur66, D. Savrina34,35, S. Schael10, M. Schellenberg11, M. Schiller55, H.
Schindler44, M. Schmelling12, T. Schmelzer11, B. Schmidt44, O. Schneider45, A.
Schopper44, H.F. Schreiner61, M. Schubiger45, S. Schulte45, M.H. Schune8, R.
Schwemmer44, B. Sciascia19, A. Sciubba27,k, A. Semennikov34, E.S. Sepulveda9,
A. Sergi49,44, N. Serra46, J. Serrano7, L. Sestini24, A. Seuthe11, P.
Seyfert44, M. Shapkin40, T. Shears56, L. Shekhtman39,x, V. Shevchenko73, E.
Shmanin74, B.G. Siddi17, R. Silva Coutinho46, L. Silva de Oliveira2, G.
Simi24,o, S. Simone15,d, I. Skiba17, N. Skidmore13, T. Skwarnicki63, M.W.
Slater49, J.G. Smeaton51, E. Smith10, I.T. Smith54, M. Smith57, M. Soares16,
l. Soares Lavra1, M.D. Sokoloff61, F.J.P. Soler55, B. Souza De Paula2, B.
Spaan11, E. Spadaro Norella22,q, P. Spradlin55, F. Stagni44, M. Stahl13, S.
Stahl44, P. Stefko45, S. Stefkova57, O. Steinkamp46, S. Stemmle13, O.
Stenyakin40, M. Stepanova41, H. Stevens11, A. Stocchi8, S. Stone63, S.
Stracka25, M.E. Stramaglia45, M. Straticiuc33, U. Straumann46, S. Strokov75,
J. Sun3, L. Sun68, Y. Sun62, K. Swientek31, A. Szabelski32, T. Szumlak31, M.
Szymanski66, S. T’Jampens5, Z. Tang3, T. Tekampe11, G. Tellarini17, F.
Teubert44, E. Thomas44, J. van Tilburg28, M.J. Tilley57, V. Tisserand6, M.
Tobin4, S. Tolk44, L. Tomassetti17,g, D. Tonelli25, D.Y. Tou9, R. Tourinho
Jadallah Aoude1, E. Tournefier5, M. Traill55, M.T. Tran45, A. Trisovic51, A.
Tsaregorodtsev7, G. Tuci25,44,p, A. Tully51, N. Tuning28, A. Ukleja32, A.
Usachov8, A. Ustyuzhanin37,38, U. Uwer13, A. Vagner75, V. Vagnoni16, A.
Valassi44, S. Valat44, G. Valenti16, H. Van Hecke79, C.B. Van Hulse14, R.
Vazquez Gomez44, P. Vazquez Regueiro43, S. Vecchi17, M. van Veghel28, J.J.
Velthuis50, M. Veltri18,r, A. Venkateswaran63, M. Vernet6, M. Veronesi28, M.
Vesterinen52, J.V. Viana Barbosa44, D. Vieira66, M. Vieites Diaz43, H.
Viemann71, X. Vilasis-Cardona42,m, A. Vitkovskiy28, M. Vitti51, V. Volkov35,
A. Vollhardt46, D. Vom Bruch9, B. Voneki44, A. Vorobyev41, V. Vorobyev39,x, N.
Voropaev41, J.A. de Vries28, C. Vázquez Sierra28, R. Waldi71, J. Walsh25, J.
Wang4, M. Wang3, Y. Wang69, Z. Wang46, D.R. Ward51, H.M. Wark56, N.K.
Watson49, D. Websdale57, A. Weiden46, C. Weisser60, M. Whitehead10, G.
Wilkinson59, M. Wilkinson63, I. Williams51, M.R.J. Williams58, M. Williams60,
T. Williams49, F.F. Wilson53, M. Winn8, W. Wislicki32, M. Witek30, G.
Wormser8, S.A. Wotton51, K. Wyllie44, D. Xiao69, Y. Xie69, H. Xing67, A. Xu3,
M. Xu69, Q. Xu66, Z. Xu3, Z. Xu5, Z. Yang3, Z. Yang62, Y. Yao63, L.E.
Yeomans56, H. Yin69, J. Yu69,aa, X. Yuan63, O. Yushchenko40, K.A. Zarebski49,
M. Zavertyaev12,c, M. Zeng3, D. Zhang69, L. Zhang3, W.C. Zhang3,z, Y. Zhang44,
A. Zhelezov13, Y. Zheng66, X. Zhu3, V. Zhukov10,35, J.B. Zonneveld54, S.
Zucchelli16,e.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4Institute Of High Energy Physics (ihep), Beijing, China
5Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
6Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
7Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
8LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
9LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
10I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
11Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
12Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
13Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
14School of Physics, University College Dublin, Dublin, Ireland
15INFN Sezione di Bari, Bari, Italy
16INFN Sezione di Bologna, Bologna, Italy
17INFN Sezione di Ferrara, Ferrara, Italy
18INFN Sezione di Firenze, Firenze, Italy
19INFN Laboratori Nazionali di Frascati, Frascati, Italy
20INFN Sezione di Genova, Genova, Italy
21INFN Sezione di Milano-Bicocca, Milano, Italy
22INFN Sezione di Milano, Milano, Italy
23INFN Sezione di Cagliari, Monserrato, Italy
24INFN Sezione di Padova, Padova, Italy
25INFN Sezione di Pisa, Pisa, Italy
26INFN Sezione di Roma Tor Vergata, Roma, Italy
27INFN Sezione di Roma La Sapienza, Roma, Italy
28Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
29Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
30Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
31AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
32National Center for Nuclear Research (NCBJ), Warsaw, Poland
33Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
34Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
35Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
36Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
37Yandex School of Data Analysis, Moscow, Russia
38National Research University Higher School of Economics, Moscow, Russia
39Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
40Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
41Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia , St.Petersburg, Russia
42ICCUB, Universitat de Barcelona, Barcelona, Spain
43Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
44European Organization for Nuclear Research (CERN), Geneva, Switzerland
45Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
46Physik-Institut, Universität Zürich, Zürich, Switzerland
47NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
48Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
49University of Birmingham, Birmingham, United Kingdom
50H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
51Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
52Department of Physics, University of Warwick, Coventry, United Kingdom
53STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
54School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
55School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
56Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
57Imperial College London, London, United Kingdom
58School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
59Department of Physics, University of Oxford, Oxford, United Kingdom
60Massachusetts Institute of Technology, Cambridge, MA, United States
61University of Cincinnati, Cincinnati, OH, United States
62University of Maryland, College Park, MD, United States
63Syracuse University, Syracuse, NY, United States
64Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
65Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
66University of Chinese Academy of Sciences, Beijing, China, associated to 3
67South China Normal University, Guangzhou, China, associated to 3
68School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
69Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
70Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 9
71Institut für Physik, Universität Rostock, Rostock, Germany, associated to 13
72Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 28
73National Research Centre Kurchatov Institute, Moscow, Russia, associated to
34
74National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 34
75National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
34
76Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 42
77H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom, Bristol, United Kingdom
78University of Michigan, Ann Arbor, United States, associated to 63
79Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 63

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vH.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
ySezione INFN di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abLanzhou University, Lanzhou, China
†Deceased

## Supplemental material

### Arithmetic average of mean decay times

The arithmetic average of the reconstructed mean decay times for
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays, $\mkern
1.5mu\overline{\mkern-1.5mu\langle t\rangle\mkern-1.5mu}\mkern 1.5mu$, can be
useful when interpreting the measurement of $\Delta A_{{C\\!P}}$. The values
corresponding to the present measurements are $\mkern
1.5mu\overline{\mkern-1.5mu\langle t\rangle\mkern-1.5mu}\mkern
1.5mu^{\pi{\mbox{-}}\rm{tagged}}/\tau({{D}^{0}})=1.74\pm 0.10$ and $\mkern
1.5mu\overline{\mkern-1.5mu\langle t\rangle\mkern-1.5mu}\mkern
1.5mu^{\mu{\mbox{-}}\rm{tagged}}/\tau({{D}^{0}})=1.21\pm 0.01$, whereas that
corresponding to the combination with previous LHCb measurements is $\mkern
1.5mu\overline{\mkern-1.5mu\langle t\rangle\mkern-1.5mu}\mkern
1.5mu/\tau({{D}^{0}})=1.71\pm 0.10$. The uncertainties include statistical and
systematic contributions, and the world average of the ${D}^{0}$ lifetime is
used.

### Additional plots

Figure 2: Raw asymmetries of the tagging pion for the $\pi$-tagged
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ sample, with the polarity of the
magnet (top) up and (bottom) down. The plots on the left include only
candidates with $|p_{y}/p_{z}|<0.02$, _i.e._ , close to the horizontal plane,
and the fiducial requirements used to exclude the kinematic region surrounding
the beam pipe, characterized by large values of the raw asymmetry, are
indicated as black lines (in addition to the forementioned requirement
$|p_{y}/p_{z}|<0.02$). The plots on the right include all candidates except
those excluded by the beam-pipe fiducial requirements, and the black lines
indicate the fiducial requirements used to exclude regions at the boundary of
the detector acceptance, which are also characterized by large values of the
raw asymmetry. Distributions for the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ sample are very similar.

Figure 3: Background-subtracted distributions of momentum ($p$), transverse
momentum ($p_{\mathrm{T}}$), azimuthal angle ($\varphi$) and pseudorapidity
($\eta$) of ${D}^{*+}$ mesons for the prompt sample: (left column) before and
(right column) after the weighting procedure for
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays, as indicated in the
legends. The distributions are normalized to unit area.

Figure 4: Background-subtracted distributions of momentum ($p$), transverse
momentum ($p_{\mathrm{T}}$), azimuthal angle ($\varphi$) and pseudorapidity
($\eta$) of ${D}^{0}$ mesons for the semileptonic sample: (left column) before
and (right column) after the weighting procedure for
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays, as indicated in the
legends. The distributions are normalized to unit area.

Figure 5: Raw asymmetries in the $\pi$-tagged sample for (left)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and (right)
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ candidates as a function of
$m({{D}^{0}}{{\pi}^{+}})$.

Figure 6: Raw asymmetries in the $\mu$-tagged sample for (left)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and (right)
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ candidates as a function of
$m({{D}^{0}})$.

Figure 7: Distributions of the signed ${D}^{0}$ impact parameter in the plane
transverse to the beam direction, TIP, in bins of reconstructed ${D}^{0}$
decay time, for the $\pi$-tagged ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$
sample: (top left) $\tilde{t}<1.5$, (top right) $1.5<\tilde{t}<3.0$, (bottom
left) $3.0<\tilde{t}<4.5$ and (bottom right) $\tilde{t}>4.5$, where
$\tilde{t}\equiv t/\tau({{D}^{0}})$. The fit results are overlaid and the
contributions from prompt and secondary decays are shown, as indicated in the
legends. Distributions for the ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$
sample are very similar.

Figure 8: Invariant-mass distributions of (top)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and (bottom)
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ candidates in the prompt
sample with fit results overlaid. These fits are used to determine the yields
and raw asymmetries of (top) ${{D}^{0}}\rightarrow K^{-}\pi^{+}\pi^{0}$ and
(bottom) ${{D}^{0}}\rightarrow\pi^{-}l^{+}\nu_{l}$ backgrounds, whose mass
shapes extend to the ${D}^{0}$-signal mass region. The various components
included in the fit model are indicated in the legends.

Figure 9: Measurements of $\Delta A_{{C\\!P}}$ in time-ordered data-taking
subsamples (referred to as run blocks) for (left) prompt and (right)
semileptonic samples. The uncertainties are statistical only. The horizontal
red-dashed lines show the central values of the nominal results.

Figure 10: Measurements of $\Delta A_{{C\\!P}}$ in bins of ${D}^{0}$ decay
time for (left) prompt and (right) semileptonic samples. In each plot, the
last bin on the right also includes a few overflow candidates. The
uncertainties are statistical only. The horizontal red-dashed lines show the
central values of the nominal results.

Figure 11: Measurements of $\Delta A_{{C\\!P}}$ in bins of (top left)
pseudorapidity, (top right) azimuthal angle, (bottom left) transverse momentum
and (bottom right) $\chi^{2}_{\text{IP}}$ of tagging pions for the prompt
sample. In each plot but that of the azimuthal angle, the last bin on the
right also includes a few overflow candidates. The uncertainties are
statistical only. The horizontal red-dashed lines show the central value of
the nominal result.

Figure 12: Measurements of $\Delta A_{{C\\!P}}$ in bins of (top left)
pseudorapidity, (top right) azimuthal angle, (bottom left) transverse momentum
and (bottom right) $\chi^{2}_{\text{IP}}$ of tagging muons for the
semileptonic sample. In each plot but that of the azimuthal angle, the last
bin on the right also includes a few overflow candidates. The uncertainties
are statistical only. The horizontal red-dashed lines show the central value
of the nominal result.

Figure 13: Measurements of $\Delta A_{{C\\!P}}$ in bins of ${D}^{0}$
$\chi^{2}_{\text{IP}}$ for (left) prompt and (right) semileptonic samples. In
each plot, the last bin on the right also includes a few overflow candidates.
The uncertainties are statistical only. The horizontal red-dashed lines show
the central values of the nominal results.