[table]capposition=top

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-022 LHCb-PAPER-2021-041 February 18, 2022

Measurement of the charm mixing parameter $y_{C\\!P}-y_{C\\!P}^{K\pi}$ using
two-body $D^{0}$ meson decays

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

A measurement of the ratios of the effective decay widths of
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decays over that of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays is performed with the
LHCb experiment using proton–proton collisions at a centre-of-mass energy of
$13\text{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
$6\text{\,fb}^{-1}$. These observables give access to the charm mixing
parameters $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$, and are measured as

$\begin{split}y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}&=(6.57\pm 0.53\pm
0.16)\times 10^{-3}\,,\\\ y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}&=(7.08\pm
0.30\pm 0.14)\times 10^{-3}\,,\end{split}$

where the first uncertainties are statistical and the second systematic. The
combination of the two measurements is
$y_{{C\\!P}}-y_{{C\\!P}}^{K\pi}=(6.96\pm 0.26\pm 0.13)\times 10^{-3}$, which
is four times more precise than the previous world average.

Submitted to Phys. Rev. D

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

##  1 Introduction

Neutral charm mesons can change their flavour and turn into their antimeson
counterpart before they decay. This phenomenon, known as ${D}^{0}$–${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mixing, does not occur at tree
level in the Standard Model and is sensitive to contributions from new
particles arising in extensions of the Standard Model. The mass eigenstates of
neutral charm mesons can be expressed as a linear combination of their flavour
eigenstates, $|D_{1,2}\rangle=p|{{D}^{0}}\rangle\pm q|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle$, where $p$ and $q$ are
complex parameters satisfying $|p|^{2}+|q|^{2}=1$. In the limit of charge-
parity ($C\\!P$) symmetry, the relation $|q/p|=1$ holds. The time evolution of
neutral charm meson systems is governed by the effective Hamiltonian
$\mathbf{H}=\mathbf{M}-\frac{i}{2}\mathbf{\Gamma}$, where the Hermitian
matrices $\mathbf{M}$ and $\mathbf{\Gamma}$ describe $({{D}^{0}},{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}})\leftrightarrow({{D}^{0}},{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}})$ dispersive transitions through
virtual intermediate states and absorptive transitions through real
intermediate states, respectively [1]. The ${D}^{0}$–${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ oscillations are described by the
two dimensionless parameters $x_{12}=2|M_{12}/\Gamma|$ and
$y_{12}=|\Gamma_{12}/\Gamma|$ [2, 3], where $\Gamma=(\Gamma_{1}+\Gamma_{2})/2$
is the average decay width of the $D_{1}$ and $D_{2}$ states, and $M_{12}$
($\Gamma_{12}$) is the off-diagonal element of matrix $\mathbf{M}$
($\mathbf{\Gamma}$). The values of $x_{12}$ and $y_{12}$ are of the order of
half a percent and have been measured to be significantly different from zero
[4, 5, 6, 7, 8, 9, 10, 11].

The non-zero value of $y_{12}$ implies that the time-dependent decay rate of
Cabibbo-suppressed ${{D}^{0}}\\!\rightarrow f$ decays, with
$f=K^{-}K^{+},\pi^{-}\pi^{+}$ final states, is described by an exponential
function with an effective decay width $\hat{\Gamma}$ that differs slightly
from $\Gamma$. The departure from unity of the ratio of the effective decay
widths of ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decays over that of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays is measured via the
observable[1]

$y_{{C\\!P}}^{f}=\frac{\hat{\Gamma}({{D}^{0}}\\!\rightarrow
f)+\hat{\Gamma}({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\\!\rightarrow f)}{2\Gamma}-1\,.$
(1)

The above quantity can be approximated as [12]

$y_{{C\\!P}}^{f}=y_{12}\cos\phi_{f}^{\Gamma}\,,$ (2)

where
$\phi_{f}^{\Gamma}=\mathrm{arg}\left(\Gamma_{12}A_{f}/\overline{A}_{f}\right)$
describes the $C\\!P$-violating phase difference of the interference between
decay amplitudes with and without absorptive mixing [2, 3], and $A_{f}$
($\overline{A}_{f}$) is the decay amplitude of a ${{D}^{0}}$ (${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$) meson to the final state $f$.
Any deviation of $y_{{C\\!P}}^{f}$ from $y_{12}$ would be a sign of $C\\!P$
violation. At the current experimental sensitivity, final-state dependent
contributions to $y_{{C\\!P}}^{f}$ can be neglected in the limit where the
phase $\phi_{f}^{\Gamma}$ is replaced by the universal phase
$\phi_{2}^{\Gamma}$, and $y_{{C\\!P}}\approx y_{12}\cos\phi_{2}^{\Gamma}$
[12]. The parameter $y_{12}$ is equal to
$|y|\equiv|\Gamma_{1}-\Gamma_{2}|/2\Gamma$ up to second order $C\\!P$
violation effects [12], where the best experimental estimate is
$y=(6.30^{+0.33}_{-0.30})\times 10^{-3}$ [11]. The current world average gives
$\phi_{2}^{\Gamma}=(48^{+29}_{-28})\text{\,mrad}$ [13, 14], implying that
$|y_{12}-y_{{C\\!P}}|<3\times 10^{-5}$ at $95\%$ confidence level. Since this
upper limit is about one order of magnitude smaller than the current
experimental sensitivity on both $y_{12}$ and $y_{{C\\!P}}$ at LHCb, an
accurate measurement of $y_{{C\\!P}}$ provides important constraints on
$y_{12}$.

The previous measurements of $y_{{C\\!P}}$ performed by the BaBar [5], Belle
[7] and LHCb [15, 9] collaborations use the average decay width of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow K^{+}\pi^{-}$ decays
as a proxy to the decay width $\Gamma$. It was recently shown in Ref. [13]
that the use of this proxy inside the experimental observable of Eq. (1) does
not give direct access to $y_{{C\\!P}}^{f}$ but rather corresponds to

$\frac{\hat{\Gamma}({{D}^{0}}\\!\rightarrow f)+\hat{\Gamma}({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\\!\rightarrow
f)}{\hat{\Gamma}({{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}})+\hat{\Gamma}({{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}})}-1\approx
y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}\,.$ (3)

The quantity $y_{{C\\!P}}^{K\pi}$ is approximately equal to

$y_{{C\\!P}}^{K\pi}\approx\sqrt{R_{D}}\left(x_{12}\cos\phi_{2}^{M}\sin\delta_{K\pi}+y_{12}\cos\phi_{2}^{\Gamma}\cos\delta_{K\pi}\right)\approx-0.4\times
10^{-3}\,,$ (4)

where $R_{D}$ is the ratio of the branching fractions of the doubly Cabibbo-
suppressed ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay over the
Cabibbo-favoured ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay. The
current best experimental estimate is $\sqrt{R_{D}}=(5.87\pm 0.02)\times
10^{-2}$ [16]. The phase $\phi^{M}_{2}$ is equal to the phase of $M_{12}$ with
respect to its $\Delta U=2$ dominant contribution, and $\delta_{K\pi}$ is the
strong-phase difference between the doubly Cabibbo-suppressed and Cabibbo-
favoured decay amplitudes [11]. In the limit of no $C\\!P$ violation and of
$U$-spin symmetry in ${{D}^{0}}\rightarrow K^{\mp}\pi^{\pm}$ decays, the
approximations $\delta_{K\pi}\approx\pi$ and
$y_{{C\\!P}}-y_{{C\\!P}}^{K\pi}\approx y_{12}(1+\sqrt{R_{D}})$ hold.

The world average value of $y_{{C\\!P}}-y_{{C\\!P}}^{K\pi}$ is measured to be
$(7.19\pm 1.13)\times 10^{-3}$ [16]. This paper reports a new measurement of
$y_{{C\\!P}}-y_{{C\\!P}}^{K\pi}$. The result is obtained from a weighted
average of statistically independent measurements with $K^{-}K^{+}$ and
$\pi^{-}\pi^{+}$ final states, using proton-proton ($pp$) collision data
collected with the LHCb experiment at a centre-of-mass energy of
$13\text{\,Te\kern-1.00006ptV}$ in the Run 2 data taking period (2015–2018),
corresponding to an integrated luminosity of $6\text{\,fb}^{-1}$. The
${{D}^{0}}$ mesons are required to originate from
$D^{\ast}(2010)^{+}\rightarrow{{D}^{0}}\pi^{+}_{\mathrm{tag}}$ decays, such
that their flavour at production is identified by the charge of the tagging
pion, $\pi^{+}_{\mathrm{tag}}$. The inclusion of charge-conjugate processes is
implied throughout. Hereafter the $D^{\ast}(2010)^{+}$ meson is referred to as
a $D^{\ast+}$ meson.

##  2 LHCb detector

The LHCb detector [17, 18] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of the momentum, $p$,
of charged particles with a relative uncertainty varying from 0.5% at low
momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The minimum
distance of a track to a primary vertex (PV), the impact parameter (IP), is
measured with a resolution of $(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. The LHCb coordinate system is right-
handed, with the $z$ axis pointing along the beam axis, $y$ the vertical
direction pointing upwards, and $x$ the horizontal direction. The origin
corresponds to the nominal $p$ $p$ interaction point. The magnetic field
deflects oppositely charged particles in opposite directions along the $x$
axis, inducing potential detection asymmetries. Therefore, the magnet polarity
is reversed regularly throughout the data taking to reduce the effects of
detection asymmetries. The two polarities are referred to as MagUp and
MagDown. Different types of charged hadrons are distinguished using
information from two ring-imaging Cherenkov (RICH) detectors. Photons,
electrons and hadrons are identified by a calorimeter system consisting of
scintillating-pad and preshower detectors, an electromagnetic and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers.

The online event selection is performed by a trigger, which consists of a
hardware stage followed by a two-level software stage, which applies a full
event reconstruction. The good performance of the online reconstruction allows
this measurement to be performed using candidates reconstructed directly at
the trigger level [19, 20].

Simulation is used to study the background of secondary $D^{\ast+}$ candidates
from $B$ meson decays (Sect. 5), and to validate the analysis procedure. The
$p$ $p$ collisions are generated with Pythia [21] with a specific LHCb
configuration [22]. The interaction of the simulated particles with the
detector material are described using the Geant4 toolkit [23, 24]. Decays of
unstable particles are described by EvtGen [25], in which final state
radiation is generated using Photos [26]. In addition, fast simulation is
generated with the RapidSim package [27]. RapidSim simulations allow for a
first validation of the analysis procedure (Sect. 6), and for a description of
the background under the ${{D}^{0}}$ mass peak (Sect. 8).

##  3 Measurement strategy

The parameters $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ are measured from the
decay-time ratios $R^{f}(t)$ of ${{D}^{0}}\\!\rightarrow f$ over
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ signal yields as a function of
the reconstructed ${{D}^{0}}$ decay time, $t$, assuming all ${{D}^{0}}$ mesons
are produced at the PV,

$R^{f}(t)=\frac{N({{D}^{0}}\\!\rightarrow
f,t)}{N({{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}},t)}\propto
e^{-(y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi})\,t/\tau_{{{D}^{0}}}}\frac{\varepsilon(f,t)}{\varepsilon(K^{-}\pi^{+},t)}\,,$
(5)

where $\tau_{{{D}^{0}}}=(410.1\pm 1.5)\text{\,fs}$ is the measured lifetime of
the ${{D}^{0}}$ meson [1], and $\varepsilon(h^{-}h^{{}^{\prime}+},t)$, with
$h^{(^{\prime})\pm}$ denoting $K^{\pm}$ or $\pi^{\pm}$, is the time-dependent
efficiency for the considered final state. Equation (5) indicates that the
access to $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ using an exponential fit is
affected by the presence of both efficiencies. In this paper, the term
numerator (denominator) decay refers to the decay quoted in the numerator
(denominator) of the ratio $R^{f}(t)$. The time-dependent efficiency can be
written as the product of two distinct components. The selection efficiency is
related to requirements applied at various stages of the LHCb data acquisition
system, while the detection efficiency arises from the interaction of the
charged kaons and pions with the LHCb detector. The time dependence of the
efficiencies of the numerator and denominator decays differs because of their
different final states, and could bias the measurement if not accounted for.
The analysis strategy consists of equalising the selection efficiencies and
then the detection efficiencies of the numerator and denominator decays. Their
combined effects cancel out in the decay time ratio, such that
$y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ can be measured without additional
corrections. Both steps are performed using data-driven methods detailed in
the following paragraphs.

The selection efficiencies of ${{D}^{0}}\\!\rightarrow f$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays mainly differ because of
the different masses of their final-state particles, leading to distinct
kinematic distributions of the final state particles of the ${{D}^{0}}$
candidate in the laboratory (lab) frame. The parent ${{D}^{0}}$ meson has a
momentum $p$ and decay angle $\theta^{\ast}(h^{-})$ that are independent of
the pair of the final states considered in this analysis. The angle
$\theta^{\ast}(h^{-})$ is defined as the angle between the momentum of the
negatively charged final state particle $h^{-}$ in the centre-of-mass frame of
the ${{D}^{0}}$ meson and the ${{D}^{0}}$ momentum in the lab frame. To obtain
equal acceptance for both decays, we require that each ${{D}^{0}}$ candidate
selected in one final state would also pass the selection requirements for the
other final state with the same ${{D}^{0}}$ kinematic properties. A kinematic
matching procedure has been developed for this purpose. It consists of an
event-by-event analytical transformation, which matches the final-state
kinematic variables of one decay to the other. To match the kinematics of a
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decay to a
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay (sketched in Fig. 1),

Figure 1: Sketch of a ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ to
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ matching.

a boost to the centre-of-mass frame of the ${{D}^{0}}$ candidate is performed,
such that both final-state particle momenta have equal magnitude,

$|\vec{p}^{\ast}|=\frac{\sqrt{(m_{{{D}^{0}}}^{2}-(m_{K^{+}}-m_{K^{-}})^{2})(m_{{{D}^{0}}}^{2}-(m_{K^{+}}+m_{K^{-}})^{2})}}{2m_{{{D}^{0}}}}\,,$
(6)

where $m_{i}$ refers to the masses of the particles. By substituting
$m_{K^{+}}$ with $m_{\pi^{+}}$, $|p^{\ast}|$ changes from
$791\text{\,Me\kern-1.00006ptV\\!/}c$ to
$861\text{\,Me\kern-1.00006ptV\\!/}c$, and a
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ state with identical kinematic
properties is generated. The use of the $K^{-}\pi^{+}$ kinematics in the lab
frame derived from this procedure (referred to as matched kinematic
quantities) ensures that both the matched
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and the target
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays cover the same kinematic
phase space.

The correction of the difference of detection efficiencies is treated with the
kinematic weighting procedure, which is performed after the kinematic
matching. The procedure consists of weighting the $p$, $p_{\mathrm{T}}$ and
$\eta$ distributions of the $D^{\ast+}$ meson and both matched final-state
particles of one of the decays to the distributions of the other decay. The
procedure is performed using a gradient-boosted-reweighting algorithm from the
hep_ml library [28].

The analysis procedure is validated with three distinct methods. First, a
measurement of $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ is performed making use
of fast simulation samples generated with the RapidSim package, where strong
variations of the time-dependent efficiencies as a function of the kinematic
variables are introduced to test the robustness of the procedure. Second, the
measurement is performed making use of large fully simulated samples. Finally,
the procedure is validated with LHCb data through a study of a cross-check
observable, $R^{CC}(t)$, built from the time-dependent ratio of the yields of
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decays,

$R^{CC}(t)=\frac{N({{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}},t)}{N({{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}},t)}\propto
e^{-y_{{C\\!P}}^{CC}\,t/\tau_{{{D}^{0}}}}\frac{\varepsilon(\pi^{-}\pi^{+},t)}{\varepsilon(K^{-}K^{+},t)}\,,$
(7)

where the parameter $y_{{C\\!P}}^{CC}$ is expected to be compatible with zero,
since the final-state dependent part of $y_{{C\\!P}}$ is negligible. The
observable $R^{CC}(t)$ benefits from the fact that both final state tracks are
different for numerator and denominator decays, increasing the biasing effects
from their corresponding efficiencies.

The data samples are contaminated by the presence of three noticeable
background contributions. The first is the combinatorial background, which is
subtracted by means of a fit to the distribution of $\Delta
m=m(h^{-}h^{+}\pi^{+}_{\mathrm{tag}})-m(h^{-}h^{+})$, where
$m(h^{-}h^{+}\pi^{+}_{\mathrm{tag}})$ is the mass of the $D^{\ast+}$ candidate
and $m(h^{-}h^{+})$ that of the ${D}^{0}$ candidate. The second background
contribution comes from $D^{\ast+}$ mesons that are not produced at the PV but
from the decay of $B$ mesons. The effect of such secondary decays on the
measurement is accounted for by altering the fit model of Eq. (5). The
treatment of the combinatorial background and of secondary decays is detailed
in Sect. 5. A third background contribution is related to the presence of
partially reconstructed or misreconstructed
$D^{\ast+}\rightarrow{{D}^{0}}\pi^{+}$ decays. A systematic uncertainty is
estimated to cover their impact on the measurement and is discussed in Sect.
8.

##  4 Candidate selection

The $D^{\ast+}\rightarrow(D^{0}\rightarrow
h^{-}h^{{}^{\prime}+})\pi_{\mathrm{tag}}^{+}$ decays are reconstructed at the
trigger level. At the hardware stage, the trigger decision is required to be
based on particles independent of the signal candidates, as requiring a
decision depending on the signal candidates would degrade the performance of
the kinematic matching procedure. Both software trigger stages were
specifically designed to minimise the biasing effects to the decay time ratio
$R^{f}(t)$, as detailed in Ref.[29]. This is achieved by avoiding requirements
on kinematic variables of the final-state particles that are strongly
correlated with the ${{D}^{0}}$ decay time. Candidate ${{D}^{0}}$ mesons are
constructed from $h^{-}h^{{}^{\prime}+}$ pairs which have a distance of
closest approach of less than $100\,\upmu\text{m}$, form a vertex with a
$\chi^{2}$ per degree of freedom less than ten, and have an invariant mass in
the interval $[1804,1924]\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. The
reconstructed ${{D}^{0}}$ decay time is required to be higher than
$0.6\,\tau_{{{D}^{0}}}$. The angle between the ${{D}^{0}}$ momentum vector and
the vector connecting the ${{D}^{0}}$ decay vertex and the PV is required to
be less than $8^{\circ}$, and the ${{D}^{0}}$ transverse momentum larger than
$2\text{\,Ge\kern-1.00006ptV\\!/}c$. Both final-state particles are required
to have an individual transverse momentum above
$800\text{\,Me\kern-1.00006ptV\\!/}c$, and at least one of these must have a
transverse momentum exceeding $1200\text{\,Me\kern-1.00006ptV\\!/}c$.
Furthermore, their individual absolute momenta are required to be higher than
$5\text{\,Ge\kern-1.00006ptV\\!/}c$. Finally, based on the information
provided by the RICH detectors, the final-state candidates are assigned a pion
or kaon mass. To remove statistical correlations between the
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ measurements related to the common
$K^{-}\pi^{+}$ final state, the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
sample is split into two statistically independent samples. Since three times
more ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ than
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ signal candidates are
selected, the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ sample is splitted
accordingly for the $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ measurements.

In the offline selection, all kaon and pion tracks are required to have a
pseudorapidity in the range $2.0$ to $4.2$ to remove particles traversing
regions of high material density. The ${{D}^{0}}$ flight distance in the $x-y$
plane is required to be less than $4\text{\,mm}$ to remove $D^{\ast+}$
candidates produced from interactions with the detector material. The
$z$-coordinate of the ${{D}^{0}}$ decay vertex is required not to exceed a
distance of $20\text{\,cm}$ from the $p$ $p$ interaction point. The invariant
mass of the ${{D}^{0}}$ meson is requested to lie within the interval
$[1851,1880]\text{\,Me\kern-1.00006ptV\\!/}c^{2}$, corresponding to about
twice the resolution around the known ${{D}^{0}}$ mass [1]. A large fraction
of secondary $D^{\ast+}$ mesons is removed by demanding that the measured
$\mathrm{IP}$ of ${{D}^{0}}$ mesons does not exceed $50\,\upmu\text{m}$ (see
Sect. 5). This requirement is also very effective at removing combinatorial
background. The resolution on the ${{D}^{0}}$ decay time is improved by
performing a kinematic fit [30] in which the $D^{\ast+}$ candidate is required
to originate from the PV. The reconstructed ${{D}^{0}}$ decay time is selected
in the interval $[1.0,8.0]\tau_{{{D}^{0}}}$. The lower bound is chosen to
minimise biasing effects related to the differences of the time resolution
between the three ${{D}^{0}}$ decay channels, and to avoid significant
combinatorial background from the PV. The higher bound is set to minimise the
contribution from secondary decays, the fraction of which increases as a
function of ${{D}^{0}}$ decay time.

The kinematic matching procedure is then performed for the selected
candidates, as detailed in Sect. 3. Figure 2 illustrates the transverse
momentum of the $K^{-}$ candidate of a
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decay matched to the $\pi^{-}$
candidate of a ${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay.

Figure 2: Matched versus original transverse momenta for the matching of a
$K^{-}$ to a $\pi^{-}$ particle, related to the $y_{{C\\!P}}^{CC}$
measurement. The red line represents the requirement applied to the data
sample, where candidates below the line are rejected. The plot is obtained
with the 2017 MagUp sample.

The trigger selection requirement on the kaon transverse momentum at
$0.8\text{\,Ge\kern-1.00006ptV\\!/}c$ is visible as a sharp cut on the $x$
axis. A requirement on the matched transverse momentum of the kaon, visible on
the $y$ axis, to be larger than $0.87\text{\,Ge\kern-1.00006ptV\\!/}c$, is
effectively tighter than the trigger requirement applied on the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ candidates. The application of
this tighter requirement in the selection of both the matched
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ candidates ensures that both
decays are selected with the same efficiency profile. Similarly, for each
kinematic variable of the ${{D}^{0}}$ candidates, a tightened requirement on
the matched variable is applied to the matched and target decays. For the
three measurements described in this paper, the matched (target) decay is that
with the smallest (largest) momentum of the final-state particles in the
${D}^{0}$ rest frame, which consists in matching kaon to pion candidates,
allowing for the minimal loss of statistical precision. Hence, for the
$y_{{C\\!P}}^{CC}$ measurement, the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decay is matched to the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay; for the
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ measurement, the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decay is matched to the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay; finally, for the
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ measurement, the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay is matched to the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay. An additional
requirement on matched quantities is that the variable
$\tilde{\chi}^{2}_{\mathrm{IP}}=\mathrm{IP}^{2}/(11.6+23.4/p_{\mathrm{T}})^{2}$,
where $p_{\mathrm{T}}$ is expressed in $\mathrm{GeV}/c$ and $\mathrm{IP}$ in
$\mathrm{\upmu m}$, to be larger than $6.0$ [31]. This allows the
combinatorial background in the data sample to be reduced further. The data
sample is split into 22 intervals of ${{D}^{0}}$ decay time of equal
population, with the exception of the four intervals with the largest decay
times containing half of the population of the others.

Following the offline and matching requirements, about $6\%$ of the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ and $3.5\%$ of the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ candidates are combined with
multiple $\pi^{+}_{\mathrm{tag}}$ candidates to form $D^{\ast+}$ meson
candidates. When multiple candidates are present in the event, one is selected
randomly.

##  5 Mass fit and dominant background contributions

The $\Delta m$ distributions of all three decay channels are shown in Fig. 3
for the combined data set. A binned maximum-likelihood fit is applied to the
$\Delta m$ distribution to separate signal from combinatorial background
arising predominantly from the association of a ${{D}^{0}}$ meson with a
random $\pi^{+}_{\mathrm{tag}}$ candidate from the $pp$ interaction. The
signal is fitted with the sum of three Gaussian functions and a Johnson SU
function[32]. The combinatorial background is fitted with the empirical model

$\mathcal{P}_{\mathrm{BKG}}(\Delta
m|m_{0},\alpha)=\frac{1}{\mathcal{I}_{B}}\Delta m\cdot\sqrt{\frac{\Delta
m^{2}}{m_{0}^{2}}-1}\cdot\exp\left(-\alpha\left(\frac{\Delta
m^{2}}{m_{0}^{2}}-1\right)\right)\,,$ (8)

where $m_{0}$ and $\alpha$ are free parameters, and $\mathcal{I}_{B}$ is a
normalisation constant. In the $\Delta m$ distribution, a signal region is
defined in the interval $[144.45,146.45]\text{\,Me\kern-1.00006ptV\\!/}c^{2}$
and a sideband region in the interval
$[150,154]\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. The contribution from the
residual background in the signal region is estimated from the sideband region
and subtracted with a dedicated procedure. The fitting of the $\Delta m$
distribution is performed independently for each ${{D}^{0}}$ flavour, year and
magnet polarity, and in each of the 22 intervals of ${{D}^{0}}$ decay time. In
the signal region, the time-integrated signal purities are equal to $98\%$,
$96\%$, and $95\%$ for the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$,
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ channels, respectively, and
the time-integrated signal yields amount to $70$ million, $18$ million, and
$6$ million decays. The fits to the $\Delta m$ distributions of all three
decay channels are displayed in Fig. 3.

Figure 3: Distributions of $\Delta m$ for the (top)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$, (centre)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$, and (bottom)
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decay channels for the
combined data sample. The signal and sideband regions employed to subtract the
combinatorial background are delimited by the dashed vertical lines. The sum
of the fit projections are overlaid.

The data samples are also contaminated by the presence of secondary
$D^{\ast+}$ mesons, which are not produced at the PV but from $B$ meson
decays. Since the reconstructed ${{D}^{0}}$ decay time is calculated as
$t=l\cdot m({{D}^{0}})/p({{D}^{0}})$, where $l$ is the measured distance
between the PV and the decay vertex of the ${{D}^{0}}$ meson, $t$ is
overestimated for secondary candidates since $l$ is affected by the flight
distance of $B$ mesons. The $\mathrm{IP}$ of the corresponding ${{D}^{0}}$
candidates is usually different from zero, as opposed to ${{D}^{0}}$
candidates from prompt $D^{\ast+}$ decays. Hence, requesting the $\mathrm{IP}$
of ${{D}^{0}}$ candidates not to exceed $50\,\upmu\text{m}$ allows a
significant fraction of secondary $D^{\ast+}$ mesons, $f_{\mathrm{sec}}(t)$,
defined as the time-dependent ratio of the number of ${D}^{0}$ mesons from
secondary decays over the total, to be rejected from the data sample. To
account for the residual contamination of secondary $D^{\ast+}$ candidates,
the ratio $R^{f}(t)$ is separated according to its prompt and secondary
components, $R_{\mathrm{prompt}}^{f}(t)$ and $R_{\mathrm{sec}}^{f}(t)$, as

$R^{f}(t)=(1-f_{\mathrm{sec}}(t))R_{\mathrm{prompt}}^{f}(t)+f_{\mathrm{sec}}(t)R_{\mathrm{sec}}^{f}(t)\,.$
(9)

The decay time ratio of ${{D}^{0}}$ mesons from secondary $D^{\ast+}$ decays
is expressed as

$R_{\mathrm{sec}}^{f}(t)\propto
e^{-(y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi})\langle
t_{D}(t)\rangle/\tau_{{{D}^{0}}}}\,,$ (10)

where $\langle t_{D}(t)\rangle$ is the average true ${{D}^{0}}$ decay time
$\langle t_{D}\rangle$ as a function of the reconstructed ${{D}^{0}}$ decay
time $t$. The quantities $f_{\mathrm{sec}}(t)$ and $\langle t_{D}(t)\rangle$
are determined using data and simulated samples of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays generated separately for
prompt $D^{\ast+}$ decays and through the expected mixture of $B^{0}$ and
$B^{+}$ meson decays to $D^{\ast+}$ candidates. The kinematic distributions of
the simulation samples are weighted to those of data samples to account for
kinematic discrepancies. The fraction $f_{\mathrm{sec}}(t)$ is obtained by
fitting the distribution of $\mathrm{IP}({{D}^{0}})$ in data in each interval
of $t$ using simulation-based templates of $\mathrm{IP}({{D}^{0}})$ from
prompt and secondary decays. The values of $f_{\mathrm{sec}}(t)$ are measured
to increase from about $2\%$ to $7\%$ across the studied ${{D}^{0}}$ decay
time range. The quantity $\langle t_{D}(t)\rangle$ is determined from the
simulated sample of secondary decays. The obtained values of
$f_{\mathrm{sec}}(t)$ and $\langle t_{D}(t)\rangle$ are shown in Fig. 4.

Figure 4: (Left) fraction and (right) average true ${{D}^{0}}$ decay time of
secondary decays as a function of the reconstructed ${{D}^{0}}$ decay time, in
units of the average ${{D}^{0}}$ lifetime.

##  6 Analysis validation with simulation

The kinematic matching procedure is validated with RapidSim simulation.
Candidates of prompt
$D^{\ast+}\rightarrow({{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}})\pi^{+}_{\mathrm{tag}}$
and
$D^{\ast+}\rightarrow({{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}})\pi^{+}_{\mathrm{tag}}$
decays are generated without ${D}^{0}$–${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mixing. The simulation samples
are subjected to selection criteria representative of the trigger. These
include requirements on momentum and $\mathrm{IP}$-related quantities, which
are strongly correlated with the ${{D}^{0}}$ decay time and induce substantial
differences between the selection efficiency profiles of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays at low ${{D}^{0}}$ decay
time. The kinematic matching procedure is then applied to equalise the
selection efficiencies of ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays. Following this
correction, a fit to the decay time ratio $R^{KK}(t)$ gives
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}=(0.17\pm 0.19)\times 10^{-3}$, compatible
with the expected value of zero. This study demonstrates that the kinematic
matching procedure corrects effectively for the kinematic differences between
the two decays.

The analysis procedure is further validated with full simulation. Large signal
yields of 50 million ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$, 33 million
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ and 11 million
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays are obtained by
generating the particles of the studied decay chain without the full
underlying event. The analysis procedure detailed in Sect. 4 is applied to all
three decay channels independently for each year and magnet polarity to
account for potential differences between the data taking conditions, and the
results are combined as a final step. Following the application of the
kinematic matching and weighting procedures, the parameters are measured to be

$\begin{split}y_{{C\\!P}}^{CC}&=(0.15\pm 0.36)\times 10^{-3}\,,\\\
y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}&=(0.17\pm 0.43)\times 10^{-3}\,,\\\
y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}&=(0.10\pm 0.24)\times
10^{-3}\,,\end{split}$

where the uncertainties are smaller than the statistical uncertainties
expected in data. All three results are compatible with zero. This is expected
since ${D}^{0}$–${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mixing
has not been simulated. This result validates the analysis procedure with
simulation.

##  7 Results

Both matching and weighting procedures are employed to perform the
measurements of $y_{{C\\!P}}^{CC}$, $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$
and $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ for each year and magnet polarity of
the LHCb Run 2 data set. Figure 5 presents the normalised distributions of the
$D^{0}$ decay angle prior to any kinematic correction (raw) and after the
application of both kinematic matching and weighting procedures. The two
correction procedures significantly improve the agreement between the
distributions. The agreement is also verified to be good for a series of
additional kinematic variables.

Figure 5: (Left) normalised distributions of the $D^{0}$ decay angle
$\cos\theta^{\ast}(h^{-})$ in the raw condition, and (right) following both
kinematic matching and reweighting procedures. The distributions are shown for
the (top) $y_{{C\\!P}}^{CC}$, (middle)
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and (bottom)
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ measurements. The plots are obtained
with the 2017 MagUp sample.

The parameters $y_{{C\\!P}}^{CC}$, $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$, and
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ are determined from a $\chi^{2}$ fit
to the corresponding time-dependent $R^{f}(t)$ ratios. The results of the
measurements are presented in Fig. 6, where $\chi^{2}$ fits with a constant
function are performed to determine the averages over all data samples. The
results of these fits are reported in Table 1. The raw measurements have good
compatibility among the different years and magnet polarities. This indicates
uniform performance of the trigger and offline selections, which do not
include effects substantially biasing the measurements. The kinematic matching
procedure shifts the average value of $y_{{C\\!P}}^{CC}$ by $(-0.96\pm
0.21)\times 10^{-3}$, $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ by $(-0.67\pm
0.21)\times 10^{-3}$, and $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ by $(+0.50\pm
0.12)\times 10^{-3}$. The shifts of $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$
and $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ are compatible in magnitude but
opposite in sign, as expected given the difference in the nature of the final
states in the numerators of their respective decay time ratios. The shifts of
$y_{{C\\!P}}^{CC}$ are measured to be about twice those of
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$, expected from the fact that
$R^{CC}(t)$ probes the decay time ratio of final states in which both
particles have different masses. The kinematic weighting shifts the values of
$y_{{C\\!P}}^{CC}$ and $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ by a few $10^{-4}$
towards negative values. Finally, the use of the fit model of Eq. (9), which
takes into account the presence of secondary decays, shifts the average values
of $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ by about $+0.1\times 10^{-3}$.

All three measurements have individual fits of good quality and are found to
be compatible among years and magnet polarities. Following all correction
procedures and the use of the fit model of Eq. (9), which includes secondary
decays, the average values are measured to be

$\begin{split}y_{{C\\!P}}^{CC}&=(-0.44\pm 0.53)\times 10^{-3}\,,\\\
y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}&=(6.57\pm 0.53)\times 10^{-3}\,,\\\
y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}&=(7.08\pm 0.30)\times
10^{-3}\,,\end{split}$

where the uncertainties are only statistical. The value of $y_{{C\\!P}}^{CC}$
is measured to be compatible with zero within one standard deviation
($\sigma$), validating the cross-check measurement with data. Figure 7 shows
the distributions of $R^{\pi\pi}(t)$ and $R^{KK}(t)$ using the full data set,
with the average values of $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ overlaid.

| $y_{{C\\!P}}^{CC}$ | $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ | $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$
---|---|---|---
Raw | $0.68\pm 0.47\>(7.9)$ | $7.48\pm 0.48\>(5.5)$ | $6.64\pm 0.27\>(6.6)$
Matching | $-0.28\pm 0.52\>(8.3)$ | $6.80\pm 0.52\>(2.9)$ | $7.14\pm 0.29\>(5.5)$
Matching + Weighting | $-0.43\pm 0.52\>(9.0)$ | $6.44\pm 0.52\>(2.8)$ | $6.94\pm 0.29\>(5.9)$
Matching + Weighting | $-0.44\pm 0.53\>(9.0)$ | $6.57\pm 0.53\>(2.8)$ | $7.08\pm 0.30\>(5.9)$
\+ Fit with secondaries
Table 1: Results of the $\chi^{2}$ fits of Fig. 6 for each correction
procedure. The results are shown in units of $10^{-3}$, while the values in
parenthesis correspond to the $\chi^{2}$ of the fits, where the number of
degrees of freedom is 7 for all measurements.

Figure 6: Results for (top) $y_{{C\\!P}}^{CC}$, (centre)
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and (bottom)
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$. The measurements employing raw data,
and following both matching and weighting conditions are shown in green, blue
and red, respectively. The measurements in purple employ the fit model where
the presence of secondary decays is considered. The dashed vertical lines
correspond to $\chi^{2}$ fits, used to determine the average values other all
subsamples. In the y-axis labels, the data-taking year is abbreviated with the
last two digits only and the magnet polarity MagUp (MagDown) is abbreviated as
“Up” (“Dw”).

Figure 7: Distributions of (left) $R^{\pi\pi}(t)$ and (right) $R^{KK}(t)$
using the full LHCb Run 2 data set, with the results of
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ overlaid as the blue slopes.

##  8 Systematic uncertainties

Although the analysis procedure is designed to minimise systematic
uncertainties on $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$, several sources of
possible bias in the results remain and are evaluated. The first source of
systematic uncertainty arises from the subtraction of the combinatorial
background, which relies on the assumption that the kinematic properties of
combinatorial background candidates are identical in the signal and in the
sideband interval of the $\Delta m$ distribution. A systematic uncertainty on
this assumption is assigned by repeating the measurement using three
alternative sideband regions, namely $[140.5,142]$, $[149,152]$ and
$[152,155]\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. An additional source of
systematic uncertainty is assigned by propagating the uncertainties on the
combinatorial background subtraction. The combined systematic uncertainty is
measured to be $0.12\times 10^{-3}$ for
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and $0.07\times 10^{-3}$ for
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$, the first being higher due to larger
combinatorial background level and smaller size of the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ sample.

A second source of systematic uncertainty arises from the precision on the
determination of the time-dependent fraction of secondary decays
$f_{\mathrm{sec}}(t)$, and the average true ${{D}^{0}}$ decay time $\langle
t_{D}\rangle$ as a function of the reconstructed ${{D}^{0}}$ decay time $t$.
The uncertainty in the determination of $f_{\mathrm{sec}}(t)$ receives three
separate contributions. First, in the fits to $\mathrm{IP}({{D}^{0}})$,
discrepancies in the ratio between the fit model and data are seen to reach up
to $10\%$. A systematic uncertainty is assigned to account for these
discrepancies. Second, the fits to $\mathrm{IP}({{D}^{0}})$ are performed in
the interval $[0,200]\,\upmu\text{m}$. The impact of increasing the upper
bound of the interval to $600\,\upmu\text{m}$ results in a small variation to
the $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ measurements, which is assigned as a
systematic uncertainty. Finally, the simulation samples of prompt and
secondary candidates are produced for the 2017 and 2018 data conditions only,
and a systematic uncertainty is assigned by considering potential variations
to the distribution of $\mathrm{IP}({{D}^{0}})$ in the 2015 and 2016 data
taking conditions. The uncertainty in the determination of $\langle
t_{D}(t)\rangle$ receives two independent contributions. First, the impact on
the difference of lifetimes between $B^{0}$ and $B^{+}$ mesons is considered
by determining $\langle t_{D}\rangle(t)$ using simulation samples of
alternatively only $B^{0}$ or $B^{+}$ candidates. Then, the effect of the
weighting of the simulation samples is evaluated by determining $\langle
t_{D}(t)\rangle$ with and without the weighting. The total systematic
uncertainty related to the treatment of secondary decays is evaluated to be at
the level of $0.03\times 10^{-3}$ for both the
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and the
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ measurements.

A systematic uncertainty related to the kinematic weighting procedure is
assigned by using alternative input variables and particles of the decay chain
to the weighting algorithm. An additional systematic uncertainty is assigned
by performing the weighting of the target decay to the matched one. The
systematic uncertainties are summed in quadrature and determined as
$0.08\times 10^{-3}$ for $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$0.02\times 10^{-3}$ for $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$.

The uncertainty on the current world average value of the lifetime of the
${{D}^{0}}$ meson [1] is propagated as a systematic uncertainty and is
estimated as $0.03\times 10^{-3}$ for both measurements.

A source of systematic uncertainty includes a potential bias related to the
contributions from the flavour of the ${{D}^{0}}$ meson, which can arise from
tagging-pion detection asymmetries and $D^{\ast+}$ production asymmetries. The
size of such a bias is estimated by performing the measurement separating the
two ${{D}^{0}}$ flavours. The ${{D}^{0}}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}^{0}$ measurements are seen to be
compatible within the $1\sigma$ level. The weighted average of the ${{D}^{0}}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}^{0}$ measurements is
compared to the baseline measurement where both ${{D}^{0}}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}^{0}$ samples are merged. For
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$, the difference between the two
strategies is measured as $0.03\times 10^{-3}$, while for
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ it is found to be below $0.01\times
10^{-3}$. These values are assigned as systematic uncertainty.

A source of systematic uncertainty is related to the presence of partially
reconstructed or misreconstructed $D^{\ast+}\rightarrow{{D}^{0}}\pi^{+}$
decays. These decays are referred to as peaking background since they show a
peaking structure in the $\Delta m$ distribution. For each decay channel, the
peaking background contributions are studied with RapidSim simulation. The
simulation samples are used as templates to fit the ${{D}^{0}}$ mass
distributions, from which the fraction of peaking background candidates is
determined in the signal region. In the
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ channel, the
${{D}^{0}}\rightarrow\pi^{-}e^{+}\nu_{e}$ and
${{D}^{0}}\rightarrow\pi^{-}{\mu}^{+}\nu_{\mu}$ background components are
significant in the ${{D}^{0}}$ signal mass region. The background fraction is
measured as $3.5\times 10^{-4}$. In the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ channel, the
${{D}^{0}}\rightarrow\pi^{-}\pi^{+}\pi^{0}$, ${{D}^{0}}\rightarrow
K^{-}e^{+}\nu_{e}$ and ${{D}^{0}}\rightarrow K^{-}\mu^{+}\nu_{\mu}$ background
contributions are dominant, and the background fraction is estimated as
$8.9\times 10^{-4}$. Finally, in the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ channel, the dominant background
sources come from the ${{D}^{0}}\rightarrow K^{-}\pi^{+}\pi^{0}$,
${{D}^{0}}\rightarrow K^{-}e^{+}\nu_{e}$ and ${{D}^{0}}\rightarrow
K^{-}\mu^{+}\nu_{\mu}$ decay channels. The background fraction is measured to
be $11.8\times 10^{-4}$. Using the RapidSim samples, the impact of these
contributions on the measurements of $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$
and $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ is evaluated to be $0.02\times
10^{-3}$ and $0.11\times 10^{-3}$, respectively. These values are assigned as
systematic uncertainties.

It is suggested in Ref. [13] that in the expansion of $R^{f}(t)$, the second-
order terms of the decay time and mixing parameters differ from those of the
exponential function used in the baseline fit. To estimate potential resulting
biases, one thousand pseudoexperiments consisting of samples of Cabibbo-
suppressed and Cabibbo-favoured decays are generated according to the
theoretical model described in Ref. [13], using the current world average
values of charm mixing and $C\\!P$-violation parameters [11]. For each
pseudoexperiment, $R^{f}(t)$ is fitted with an exponential function and the
departure of the mean value of the fits from the expectation of the
theoretical model is measured as $0.03\times 10^{-3}$, and is assigned as a
systematic uncertainty. The summary of the systematic uncertainties is
presented in Table 2, where the quadratic sum of all contributions is
$0.16\times 10^{-3}$ for $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$0.14\times 10^{-3}$ for $y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$.

| $\sigma(y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi})$ | $\sigma(y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi})$
---|---|---
| $[10^{-3}]$ | $[10^{-3}]$
Combinatorial background | $0.12$ | $0.07$
Treatment of secondary decays | $0.03$ | $0.03$
Kinematic weighting procedure | $0.08$ | $0.02$
Input ${{D}^{0}}$ lifetime | $0.03$ | $0.03$
Residual nuisance asymmetries | $0.03$ | $<0.01$
Peaking background | $0.02$ | $0.11$
Fit bias | $0.03$ | $0.03$
Total | $0.16$ | $0.14$
Table 2: Systematic uncertainties for the
$y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$ measurements.

Robustness checks are performed by verifying that the measurements do not show
any dependence on various variables, including the momentum, transverse
momentum, the pseudorapidity and azimuthal angle of the ${{D}^{0}}$ and
$\pi_{\mathrm{tag}}^{+}$ mesons, as well as the ${{D}^{0}}$ flight distance in
the plane transverse to the beam and the $z$ coordinate of the ${{D}^{0}}$
decay vertex. No significant dependence of
$y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ on any of the listed variables is
observed. To study a potential dependence on the orientation of the magnetic
field and on a potential left-right asymmetry of the detector, the measurement
is performed separately for positive and negative values of the $x$ component
of the momentum of the ${{D}^{0}}$ meson, and for the MagUp and MagDown
polarities. All measurements are seen to be statistically compatible within
two standard deviations. To evaluate the impact of possible residual
resolution effects at low values of ${{D}^{0}}$ decay time, the measurements
of $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ are performed by increasing
successively the lower window of the ${{D}^{0}}$ decay time up to
$1.7\,\tau_{{{D}^{0}}}$. Accounting for the statistical overlap, all measured
values are found to be statistically compatible and correspondingly no
systematic uncertainty is assigned to this effect.

Due to the presence of correlations between the reconstructed ${{D}^{0}}$
decay time and momentum, the correction procedure can introduce a bias to the
true values of $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$. To study this bias,
artificial values of $y_{{C\\!P}}^{f}-y_{{C\\!P}}^{K\pi}$ are injected to the
data samples by altering the decay time distribution of numerator decays. Both
kinematic matching and weighting procedures are applied and the measured
values are compared to the injected ones. The procedure is performed for ten
values in the interval $[-25,25]\times 10^{-3}$. The measured and injected
values agree, confirming that no significant bias is seen and correspondingly
no systematic uncertainty is applied.

##  9 Summary and conclusion

The measurements of the ratios of the effective decay widths of
${{D}^{0}}\\!\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{K}^{+}}$ decays over that of
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decays are performed with the
LHCb experiment using $p$ $p$ collisions at a centre-of-mass energy of
$13\text{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
$6\text{\,fb}^{-1}$. The ratios give direct access to the charm mixing
parameters $y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}$ and
$y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}$, which are measured to be

$\begin{split}y_{{C\\!P}}^{\pi\pi}-y_{{C\\!P}}^{K\pi}&=(6.57\pm 0.53\pm
0.16)\times 10^{-3}\,,\\\ y_{{C\\!P}}^{KK}-y_{{C\\!P}}^{K\pi}&=(7.08\pm
0.30\pm 0.14)\times 10^{-3}\,,\end{split}$

where the first uncertainties are statistical and the second systematic.
Assuming that all systematic uncertainties are fully correlated, except those
of the peaking background contributions which are considered as uncorrelated,
the combination of the two measurements yields

$y_{{C\\!P}}-y_{{C\\!P}}^{K\pi}=(6.96\pm 0.26\pm 0.13)\times 10^{-3}\,.$

This result is compatible with the present world average [16] and more precise
by a factor of four.

A combination of LHCb charm measurements is performed using the statistical
framework detailed in Ref. [11]. When the present result is added, the mixing
parameter $y$ is found to be equal to
$y=\left(6.46\pm^{+0.24}_{-0.25}\right)\times 10^{-3}$, improving its current
sensitivity by more than a factor of two [16]. In addition, the strong phase
difference between the $D^{0}\rightarrow K^{\mp}\pi^{\pm}$ decay amplitudes is
$\delta_{K\pi}=\left(192.1^{+3.7}_{-4.0}\right)^{\circ}$ and departs from
$180^{\circ}$ by about three standard deviations, indicating an evidence for
$U$-spin symmetry breaking.

The precision on $y$ and $\delta_{K\pi}$ can be further reduced by a
simultaneous combination of charm results with measurements of the Cabibbo-
Kobayashi-Maskawa angle $\gamma$ in beauty decays, as first done in Ref. [11].
This will be the subject of a separate publication.

## 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); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and 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 NERSC (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 ARC and ARDC (Australia); AvH Foundation
(Germany); EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union);
A*MIDEX, ANR, IPhU and Labex P2IO, and Région Auvergne-Rhône-Alpes (France);
Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP,
Fundamental Research Funds for the Central Universities, and Sci. & Tech.
Program of Guangzhou (China); RFBR, RSF and Yandex LLC (Russia); GVA, XuntaGal
and GENCAT (Spain); the Leverhulme Trust, the Royal Society and UKRI (United
Kingdom).

## References

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LHCb collaboration

R. Aaij32, A.S.W. Abdelmotteleb56, C. Abellán Beteta50, F. Abudinén56, T.
Ackernley60, B. Adeva46, M. Adinolfi54, H. Afsharnia9, C. Agapopoulou13, C.A.
Aidala87, S. Aiola25, Z. Ajaltouni9, S. Akar65, J. Albrecht15, F. Alessio48,
M. Alexander59, A. Alfonso Albero45, Z. Aliouche62, G. Alkhazov38, P. Alvarez
Cartelle55, S. Amato2, J.L. Amey54, Y. Amhis11, L. An48, L. Anderlini22, M.
Andersson50, A. Andreianov38, M. Andreotti21, D. Ao6, F. Archilli17, A.
Artamonov44, M. Artuso68, K. Arzymatov42, E. Aslanides10, M. Atzeni50, B.
Audurier12, S. Bachmann17, M. Bachmayer49, J.J. Back56, P. Baladron
Rodriguez46, V. Balagura12, W. Baldini21, J. Baptista de Souza Leite1, M.
Barbetti22,h, R.J. Barlow62, S. Barsuk11, W. Barter61, M. Bartolini55, F.
Baryshnikov83, J.M. Basels14, G. Bassi29, B. Batsukh4, A. Battig15, A. Bay49,
A. Beck56, M. Becker15, F. Bedeschi29, I. Bediaga1, A. Beiter68, V. Belavin42,
S. Belin46, V. Bellee50, K. Belous44, I. Belov40, I. Belyaev41, G.
Bencivenni23, E. Ben-Haim13, A. Berezhnoy40, R. Bernet50, D. Berninghoff17,
H.C. Bernstein68, C. Bertella62, A. Bertolin28, C. Betancourt50, F. Betti48,
Ia. Bezshyiko50, S. Bhasin54, J. Bhom35, L. Bian73, M.S. Bieker15, N.V.
Biesuz21, S. Bifani53, P. Billoir13, A. Biolchini32, M. Birch61, F.C.R.
Bishop55, A. Bitadze62, A. Bizzeti22,l, M. Bjørn63, M.P. Blago55, T. Blake56,
F. Blanc49, S. Blusk68, D. Bobulska59, J.A. Boelhauve15, O. Boente Garcia46,
T. Boettcher65, A. Boldyrev82, A. Bondar43, N. Bondar38,48, S. Borghi62, M.
Borisyak42, M. Borsato17, J.T. Borsuk35, S.A. Bouchiba49, T.J.V. Bowcock60,48,
A. Boyer48, C. Bozzi21, M.J. Bradley61, S. Braun66, A. Brea Rodriguez46, J.
Brodzicka35, A. Brossa Gonzalo56, D. Brundu27, A. Buonaura50, L.
Buonincontri28, A.T. Burke62, C. Burr48, A. Bursche72, A. Butkevich39, J.S.
Butter32, J. Buytaert48, W. Byczynski48, S. Cadeddu27, H. Cai73, R.
Calabrese21,g, L. Calefice15,13, S. Cali23, R. Calladine53, M. Calvi26,k, M.
Calvo Gomez85, P. Camargo Magalhaes54, P. Campana23, A.F. Campoverde Quezada6,
S. Capelli26,k, L. Capriotti20,e, A. Carbone20,e, G. Carboni31,q, R.
Cardinale24,i, A. Cardini27, I. Carli4, P. Carniti26,k, L. Carus14, K.
Carvalho Akiba32, A. Casais Vidal46, R. Caspary17, G. Casse60, M. Cattaneo48,
G. Cavallero48, S. Celani49, J. Cerasoli10, D. Cervenkov63, A.J. Chadwick60,
M.G. Chapman54, M. Charles13, Ph. Charpentier48, C.A. Chavez Barajas60, M.
Chefdeville8, C. Chen3, S. Chen4, A. Chernov35, V. Chobanova46, S. Cholak49,
M. Chrzaszcz35, A. Chubykin38, V. Chulikov38, P. Ciambrone23, M.F. Cicala56,
X. Cid Vidal46, G. Ciezarek48, P.E.L. Clarke58, M. Clemencic48, H.V. Cliff55,
J. Closier48, J.L. Cobbledick62, V. Coco48, J.A.B. Coelho11, J. Cogan10, E.
Cogneras9, L. Cojocariu37, P. Collins48, T. Colombo48, L. Congedo19,d, A.
Contu27, N. Cooke53, G. Coombs59, I. Corredoira 46, G. Corti48, C.M. Costa
Sobral56, B. Couturier48, D.C. Craik64, J. Crkovská67, M. Cruz Torres1, R.
Currie58, C.L. Da Silva67, S. Dadabaev83, L. Dai71, E. Dall’Occo15, J.
Dalseno46, C. D’Ambrosio48, A. Danilina41, P. d’Argent48, A. Dashkina83, J.E.
Davies62, A. Davis62, O. De Aguiar Francisco62, K. De Bruyn79, S. De Capua62,
M. De Cian49, U. De Freitas Carneiro Da Graca1, E. De Lucia23, J.M. De
Miranda1, L. De Paula2, M. De Serio19,d, D. De Simone50, P. De Simone23, F. De
Vellis15, J.A. de Vries80, C.T. Dean67, F. Debernardis19,d, D. Decamp8, V.
Dedu10, L. Del Buono13, B. Delaney55, H.-P. Dembinski15, V. Denysenko50, D.
Derkach82, O. Deschamps9, F. Dettori27,f, B. Dey77, A. Di Cicco23, P. Di
Nezza23, S. Didenko83, L. Dieste Maronas46, S. Ding68, V. Dobishuk52, C.
Dong3, A.M. Donohoe18, F. Dordei27, A.C. dos Reis1, L. Douglas59, A.
Dovbnya51, A.G. Downes8, M.W. Dudek35, L. Dufour48, V. Duk78, P. Durante48,
J.M. Durham67, D. Dutta62, A. Dziurda35, A. Dzyuba38, S. Easo57, U. Egede69,
V. Egorychev41, S. Eidelman43,u,†, S. Eisenhardt58, S. Ek-In49, L. Eklund86,
S. Ely68, A. Ene37, E. Epple67, S. Escher14, J. Eschle50, S. Esen50, T.
Evans62, L.N. Falcao1, Y. Fan6, B. Fang73, S. Farry60, D. Fazzini26,k, M.
Féo48, A. Fernandez Prieto46, A.D. Fernez66, F. Ferrari20, L. Ferreira
Lopes49, F. Ferreira Rodrigues2, S. Ferreres Sole32, M. Ferrillo50, M. Ferro-
Luzzi48, S. Filippov39, R.A. Fini19, M. Fiorini21,g, M. Firlej34, K.M.
Fischer63, D.S. Fitzgerald87, C. Fitzpatrick62, T. Fiutowski34, A. Fkiaras48,
F. Fleuret12, M. Fontana13, F. Fontanelli24,i, R. Forty48, D. Foulds-Holt55,
V. Franco Lima60, M. Franco Sevilla66, M. Frank48, E. Franzoso21, G. Frau17,
C. Frei48, D.A. Friday59, J. Fu6, Q. Fuehring15, E. Gabriel32, G. Galati19,d,
A. Gallas Torreira46, D. Galli20,e, S. Gambetta58,48, Y. Gan3, M. Gandelman2,
P. Gandini25, Y. Gao5, M. Garau27, L.M. Garcia Martin56, P. Garcia Moreno45,
J. García Pardiñas26,k, B. Garcia Plana46, F.A. Garcia Rosales12, L.
Garrido45, C. Gaspar48, R.E. Geertsema32, D. Gerick17, L.L. Gerken15, E.
Gersabeck62, M. Gersabeck62, T. Gershon56, L. Giambastiani28, V. Gibson55,
H.K. Giemza36, A.L. Gilman63, M. Giovannetti23,q, A. Gioventù46, P. Gironella
Gironell45, C. Giugliano21, K. Gizdov58, E.L. Gkougkousis48, V.V.
Gligorov13,48, C. Göbel70, E. Golobardes85, D. Golubkov41, A. Golutvin61,83,
A. Gomes1,a, S. Gomez Fernandez45, F. Goncalves Abrantes63, M. Goncerz35, G.
Gong3, P. Gorbounov41, I.V. Gorelov40, C. Gotti26, J.P. Grabowski17, T.
Grammatico13, L.A. Granado Cardoso48, E. Graugés45, E. Graverini49, G.
Graziani22, A. Grecu37, L.M. Greeven32, N.A. Grieser4, L. Grillo62, S.
Gromov83, B.R. Gruberg Cazon63, C. Gu3, M. Guarise21, M. Guittiere11, P. A.
Günther17, E. Gushchin39, A. Guth14, Y. Guz44, T. Gys48, T. Hadavizadeh69, G.
Haefeli49, C. Haen48, J. Haimberger48, S.C. Haines55, T. Halewood-leagas60,
P.M. Hamilton66, J.P. Hammerich60, Q. Han7, X. Han17, E.B. Hansen62, S.
Hansmann-Menzemer17,48, N. Harnew63, T. Harrison60, C. Hasse48, M. Hatch48, J.
He6,b, K. Heijhoff32, K. Heinicke15, R.D.L. Henderson69,56, A.M. Hennequin64,
K. Hennessy60, L. Henry48, J. Heuel14, A. Hicheur2, D. Hill49, M. Hilton62,
S.E. Hollitt15, R. Hou7, Y. Hou8, J. Hu17, J. Hu72, W. Hu7, X. Hu3, W. Huang6,
X. Huang73, W. Hulsbergen32, R.J. Hunter56, M. Hushchyn82, D. Hutchcroft60, D.
Hynds32, P. Ibis15, M. Idzik34, D. Ilin38, P. Ilten65, A. Inglessi38, A.
Iniukhin82, A. Ishteev83, K. Ivshin38, R. Jacobsson48, H. Jage14, S.
Jakobsen48, E. Jans32, B.K. Jashal47, A. Jawahery66, V. Jevtic15, X. Jiang4,
M. John63, D. Johnson64, C.R. Jones55, T.P. Jones56, B. Jost48, N. Jurik48, S.
Kandybei51, Y. Kang3, M. Karacson48, D. Karpenkov83, M. Karpov82, J.W.
Kautz65, F. Keizer48, D.M. Keller68, M. Kenzie56, T. Ketel33, B. Khanji15, A.
Kharisova84, S. Kholodenko44,83, T. Kirn14, V.S. Kirsebom49, O. Kitouni64, S.
Klaver33, N. Kleijne29, K. Klimaszewski36, M.R. Kmiec36, S. Koliiev52, A.
Kondybayeva83, A. Konoplyannikov41, P. Kopciewicz34, R. Kopecna17, P.
Koppenburg32, M. Korolev40, I. Kostiuk32,52, O. Kot52, S. Kotriakhova21,38, A.
Kozachuk40, P. Kravchenko38, L. Kravchuk39, R.D. Krawczyk48, M. Kreps56, S.
Kretzschmar14, P. Krokovny43,u, W. Krupa34, W. Krzemien36, J. Kubat17, M.
Kucharczyk35, V. Kudryavtsev43,u, H.S. Kuindersma32,33, G.J. Kunde67, T.
Kvaratskheliya41, D. Lacarrere48, G. Lafferty62, A. Lai27, A. Lampis27, D.
Lancierini50, J.J. Lane62, R. Lane54, G. Lanfranchi23, C. Langenbruch14, J.
Langer15, O. Lantwin83, T. Latham56, F. Lazzari29, R. Le Gac10, S.H. Lee87, R.
Lefèvre9, A. Leflat40, S. Legotin83, O. Leroy10, T. Lesiak35, B.
Leverington17, H. Li72, P. Li17, S. Li7, Y. Li4, Z. Li68, X. Liang68, T.
Lin57, R. Lindner48, V. Lisovskyi15, R. Litvinov27, G. Liu72, H. Liu6, Q.
Liu6, S. Liu4, A. Lobo Salvia45, A. Loi27, R. Lollini78, J. Lomba Castro46, I.
Longstaff59, J.H. Lopes2, S. López Soliño46, G.H. Lovell55, Y. Lu4, C.
Lucarelli22,h, D. Lucchesi28,m, S. Luchuk39, M. Lucio Martinez32, V.
Lukashenko32,52, Y. Luo3, A. Lupato62, E. Luppi21,g, O. Lupton56, A.
Lusiani29,n, X. Lyu6, L. Ma4, R. Ma6, S. Maccolini20, F. Machefert11, F.
Maciuc37, V. Macko49, P. Mackowiak15, S. Maddrell-Mander54, L.R. Madhan
Mohan54, O. Maev38, A. Maevskiy82, D. Maisuzenko38, M.W. Majewski34, J.J.
Malczewski35, S. Malde63, B. Malecki35, A. Malinin81, T. Maltsev43,u, H.
Malygina17, G. Manca27,f, G. Mancinelli10, D. Manuzzi20, C.A. Manzari50, D.
Marangotto25,j, J. Maratas9,s, J.F. Marchand8, U. Marconi20, S. Mariani22,h,
C. Marin Benito48, M. Marinangeli49, J. Marks17, A.M. Marshall54, P.J.
Marshall60, G. Martelli78, G. Martellotti30, L. Martinazzoli48,k, M.
Martinelli26,k, D. Martinez Santos46, F. Martinez Vidal47, A. Massafferri1, M.
Materok14, R. Matev48, A. Mathad50, V. Matiunin41, C. Matteuzzi26, K.R.
Mattioli87, A. Mauri32, E. Maurice12, J. Mauricio45, M. Mazurek48, M.
McCann61, L. Mcconnell18, T.H. Mcgrath62, N.T. Mchugh59, A. McNab62, R.
McNulty18, J.V. Mead60, B. Meadows65, G. Meier15, D. Melnychuk36, S.
Meloni26,k, M. Merk32,80, A. Merli25,j, L. Meyer Garcia2, M. Mikhasenko75,c,
D.A. Milanes74, E. Millard56, M. Milovanovic48, M.-N. Minard8, A. Minotti26,k,
S.E. Mitchell58, B. Mitreska62, D.S. Mitzel15, A. Mödden 15, R.A. Mohammed63,
R.D. Moise61, S. Mokhnenko82, T. Mombächer46, I.A. Monroy74, S. Monteil9, M.
Morandin28, G. Morello23, M.J. Morello29,n, J. Moron34, A.B. Morris75, A.G.
Morris56, R. Mountain68, H. Mu3, F. Muheim58, M. Mulder79, K. Müller50, C.H.
Murphy63, D. Murray62, R. Murta61, P. Muzzetto27, P. Naik54, T. Nakada49, R.
Nandakumar57, T. Nanut48, I. Nasteva2, M. Needham58, N. Neri25,j, S.
Neubert75, N. Neufeld48, R. Newcombe61, E.M. Niel49, S. Nieswand14, N.
Nikitin40, N.S. Nolte64, C. Normand8, C. Nunez87, A. Oblakowska-Mucha34, V.
Obraztsov44, T. Oeser14, D.P. O’Hanlon54, S. Okamura21, R. Oldeman27,f, F.
Oliva58, M.E. Olivares68, C.J.G. Onderwater79, R.H. O’Neil58, J.M. Otalora
Goicochea2, T. Ovsiannikova41, P. Owen50, A. Oyanguren47, O. Ozcelik58, K.O.
Padeken75, B. Pagare56, P.R. Pais48, T. Pajero63, A. Palano19, M. Palutan23,
Y. Pan62, G. Panshin84, A. Papanestis57, M. Pappagallo19,d, L.L. Pappalardo21,
C. Pappenheimer65, W. Parker66, C. Parkes62, B. Passalacqua21, G. Passaleva22,
A. Pastore19, M. Patel61, C. Patrignani20,e, C.J. Pawley80, A. Pearce48,57, A.
Pellegrino32, M. Pepe Altarelli48, S. Perazzini20, D. Pereima41, A. Pereiro
Castro46, P. Perret9, M. Petric59,48, K. Petridis54, A. Petrolini24,i, A.
Petrov81, S. Petrucci58, M. Petruzzo25, T.T.H. Pham68, A. Philippov42, R.
Piandani6, L. Pica29,n, M. Piccini78, B. Pietrzyk8, G. Pietrzyk11, M. Pili63,
D. Pinci30, F. Pisani48, M. Pizzichemi26,48,k, Resmi P.K10, V. Placinta37, J.
Plews53, M. Plo Casasus46, F. Polci13,48, M. Poli Lener23, M. Poliakova68, A.
Poluektov10, N. Polukhina83,t, I. Polyakov68, E. Polycarpo2, S. Ponce48, D.
Popov6,48, S. Popov42, S. Poslavskii44, K. Prasanth35, L. Promberger48, C.
Prouve46, V. Pugatch52, V. Puill11, G. Punzi29,o, H. Qi3, W. Qian6, N. Qin3,
R. Quagliani49, N.V. Raab18, R.I. Rabadan Trejo6, B. Rachwal34, J.H.
Rademacker54, R. Rajagopalan68, M. Rama29, M. Ramos Pernas56, M.S. Rangel2, F.
Ratnikov42,82, G. Raven33,48, M. Reboud8, F. Redi48, F. Reiss62, C. Remon
Alepuz47, Z. Ren3, V. Renaudin63, R. Ribatti29, A.M. Ricci27, S. Ricciardi57,
K. Rinnert60, P. Robbe11, G. Robertson58, A.B. Rodrigues49, E. Rodrigues60,
J.A. Rodriguez Lopez74, E.R.R. Rodriguez Rodriguez46, A. Rollings63, P.
Roloff48, V. Romanovskiy44, M. Romero Lamas46, A. Romero Vidal46, J.D. Roth87,
M. Rotondo23, M.S. Rudolph68, T. Ruf48, R.A. Ruiz Fernandez46, J. Ruiz
Vidal47, A. Ryzhikov82, J. Ryzka34, J.J. Saborido Silva46, N. Sagidova38, N.
Sahoo53, B. Saitta27,f, M. Salomoni48, C. Sanchez Gras32, I. Sanderswood47, R.
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Saranin83, G. Sarpis14, M. Sarpis75, A. Sarti30, C. Satriano30,p, A. Satta31,
M. Saur15, D. Savrina41,40, H. Sazak9, L.G. Scantlebury Smead63, A.
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Sellam46, A. Semennikov41, M. Senghi Soares33, A. Sergi24,i, N. Serra50, L.
Sestini28, A. Seuthe15, Y. Shang5, D.M. Shangase87, M. Shapkin44, I.
Shchemerov83, L. Shchutska49, T. Shears60, L. Shekhtman43,u, Z. Shen5, S.
Sheng4, V. Shevchenko81, E.B. Shields26,k, Y. Shimizu11, E. Shmanin83, J.D.
Shupperd68, B.G. Siddi21, R. Silva Coutinho50, G. Simi28, S. Simone19,d, M.
Singla69, N. Skidmore62, R. Skuza17, T. Skwarnicki68, M.W. Slater53, I.
Slazyk21,g, J.C. Smallwood63, J.G. Smeaton55, E. Smith50, M. Smith61, A.
Snoch32, L. Soares Lavra9, M.D. Sokoloff65, F.J.P. Soler59, A. Solovev38, I.
Solovyev38, F.L. Souza De Almeida2, B. Souza De Paula2, B. Spaan15, E. Spadaro
Norella25,j, P. Spradlin59, F. Stagni48, M. Stahl65, S. Stahl48, S.
Stanislaus63, O. Steinkamp50,83, O. Stenyakin44, H. Stevens15, S.
Stone68,48,†, D. Strekalina83, F. Suljik63, J. Sun27, L. Sun73, Y. Sun66, P.
Svihra62, P.N. Swallow53, K. Swientek34, A. Szabelski36, T. Szumlak34, M.
Szymanski48, S. Taneja62, A.R. Tanner54, M.D. Tat63, A. Terentev83, F.
Teubert48, E. Thomas48, D.J.D. Thompson53, K.A. Thomson60, H. Tilquin61, V.
Tisserand9, S. T’Jampens8, M. Tobin4, L. Tomassetti21,g, X. Tong5, D. Torres
Machado1, D.Y. Tou3, E. Trifonova83, S.M. Trilov54, C. Trippl49, G. Tuci6, A.
Tully49, N. Tuning32,48, A. Ukleja36, D.J. Unverzagt17, E. Ursov83, A.
Usachov32, A. Ustyuzhanin42,82, U. Uwer17, A. Vagner84, V. Vagnoni20, A.
Valassi48, G. Valenti20, N. Valls Canudas85, M. van Beuzekom32, M. Van Dijk49,
H. Van Hecke67, E. van Herwijnen83, M. van Veghel79, R. Vazquez Gomez45, P.
Vazquez Regueiro46, C. Vázquez Sierra48, S. Vecchi21, J.J. Velthuis54, M.
Veltri22,r, A. Venkateswaran68, M. Veronesi32, M. Vesterinen56, D. Vieira65,
M. Vieites Diaz49, H. Viemann76, X. Vilasis-Cardona85, E. Vilella Figueras60,
A. Villa20, P. Vincent13, F.C. Volle11, D. Vom Bruch10, A. Vorobyev38, V.
Vorobyev43,u, N. Voropaev38, K. Vos80, R. Waldi17, J. Walsh29, C. Wang17, J.
Wang5, J. Wang4, J. Wang3, J. Wang73, M. Wang3, R. Wang54, Y. Wang7, Z.
Wang50, Z. Wang3, Z. Wang6, J.A. Ward56,69, N.K. Watson53, D. Websdale61, C.
Weisser64, B.D.C. Westhenry54, D.J. White62, M. Whitehead54, A.R.
Wiederhold56, D. Wiedner15, G. Wilkinson63, M. K. Wilkinson68, I. Williams55,
M. Williams64, M.R.J. Williams58, F.F. Wilson57, W. Wislicki36, M. Witek35, L.
Witola17, G. Wormser11, S.A. Wotton55, H. Wu68, K. Wyllie48, Z. Xiang6, D.
Xiao7, Y. Xie7, A. Xu5, J. Xu6, L. Xu3, M. Xu56, Q. Xu6, Z. Xu9, Z. Xu6, D.
Yang3, S. Yang6, Y. Yang6, Z. Yang5, Z. Yang66, Y. Yao68, L.E. Yeomans60, H.
Yin7, J. Yu71, X. Yuan68, O. Yushchenko44, E. Zaffaroni49, M. Zavertyaev16,t,
M. Zdybal35, O. Zenaiev48, M. Zeng3, D. Zhang7, L. Zhang3, S. Zhang71, S.
Zhang5, Y. Zhang5, Y. Zhang63, A. Zharkova83, A. Zhelezov17, Y. Zheng6, T.
Zhou5, X. Zhou6, Y. Zhou6, V. Zhovkovska11, X. Zhu3, X. Zhu7, Z. Zhu6, V.
Zhukov14,40, Q. Zou4, S. Zucchelli20,e, D. Zuliani28, G. Zunica62.

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
5School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
6University of Chinese Academy of Sciences, Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut
Polytechnique de Paris, Palaiseau, France
13LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
14I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
15Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
16Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
17Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
18School of Physics, University College Dublin, Dublin, Ireland
19INFN Sezione di Bari, Bari, Italy
20INFN Sezione di Bologna, Bologna, Italy
21INFN Sezione di Ferrara, Ferrara, Italy
22INFN Sezione di Firenze, Firenze, Italy
23INFN Laboratori Nazionali di Frascati, Frascati, Italy
24INFN Sezione di Genova, Genova, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Milano-Bicocca, Milano, Italy
27INFN Sezione di Cagliari, Monserrato, Italy
28Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
29INFN Sezione di Pisa, Pisa, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31INFN Sezione di Roma Tor Vergata, Roma, Italy
32Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
33Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
36National Center for Nuclear Research (NCBJ), Warsaw, Poland
37Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
38Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
39Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
40Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
41Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
42Yandex School of Data Analysis, Moscow, Russia
43Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
44Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
47Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
48European Organization for Nuclear Research (CERN), Geneva, Switzerland
49Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
50Physik-Institut, Universität Zürich, Zürich, Switzerland
51NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
52Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
53University of Birmingham, Birmingham, United Kingdom
54H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
55Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
56Department of Physics, University of Warwick, Coventry, United Kingdom
57STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
58School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
59School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
60Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
61Imperial College London, London, United Kingdom
62Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
63Department of Physics, University of Oxford, Oxford, United Kingdom
64Massachusetts Institute of Technology, Cambridge, MA, United States
65University of Cincinnati, Cincinnati, OH, United States
66University of Maryland, College Park, MD, United States
67Los Alamos National Laboratory (LANL), Los Alamos, United States
68Syracuse University, Syracuse, NY, United States
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 56
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
72Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong
Joint Laboratory of Quantum Matter, Institute of Quantum Matter, South China
Normal University, Guangzhou, China, associated to 3
73School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
74Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 13
75Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 17
76Institut für Physik, Universität Rostock, Rostock, Germany, associated to 17
77Eotvos Lorand University, Budapest, Hungary, associated to 48
78INFN Sezione di Perugia, Perugia, Italy, associated to 21
79Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 32
80Universiteit Maastricht, Maastricht, Netherlands, associated to 32
81National Research Centre Kurchatov Institute, Moscow, Russia, associated to
41
82National Research University Higher School of Economics, Moscow, Russia,
associated to 42
83National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 41
84National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
41
85DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 45
86Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden,
associated to 59
87University of Michigan, Ann Arbor, United States, associated to 68

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bHangzhou Institute for Advanced Study, UCAS, Hangzhou, China
cExcellence Cluster ORIGINS, Munich, Germany
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Firenze, Firenze, Italy
iUniversità di Genova, Genova, Italy
jUniversità degli Studi di Milano, Milano, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Modena e Reggio Emilia, Modena, Italy
mUniversità di Padova, Padova, Italy
nScuola Normale Superiore, Pisa, Italy
oUniversità di Pisa, Pisa, Italy
pUniversità della Basilicata, Potenza, Italy
qUniversità di Roma Tor Vergata, Roma, Italy
rUniversità di Urbino, Urbino, Italy
sMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
tP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
uNovosibirsk State University, Novosibirsk, Russia
†Deceased