EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-264 LHCb-PAPER-2019-036 3 April 2020

Measurement of $CP$ violation in $B^{0}\rightarrow{D}^{*\pm}{D}^{\mp}$ decays

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

The decay-time-dependent $C\\!P$ asymmetry in $B^{0}\rightarrow
D^{*\pm}D^{\mp}$ decays is measured using a data set corresponding to an
integrated luminosity of $9$$\text{\,fb}^{-1}$ recorded by the LHCb detector
in proton-proton collisions at centre-of-mass energies of 7, 8 and $13$ TeV.
The $C\\!P$ parameters are measured as

$\begin{split}S_{{D}^{*}{D}}&={-0.861}\pm{0.077}\,\text{(stat)}\pm{0.019}\,\text{(syst)}\,,\\\
\Delta
S_{{D}^{*}{D}}&=\phantom{+}{0.019}\pm{0.075}\,\text{(stat)}\pm{0.012}\,\text{(syst)}\,,\\\
C_{{D}^{*}{D}}&={-0.059}\pm{0.092}\,\text{(stat)}\pm{0.020}\,\text{(syst)}\,,\\\
\Delta
C_{{D}^{*}{D}}&={-0.031}\pm{0.092}\,\text{(stat)}\pm{0.016}\,\text{(syst)}\,,\\\
{\cal A}_{{D}^{*}{D}}&=\phantom{+}{0.008}\pm{0.014}\,\text{(stat)}\pm
0.006\,\text{(syst)}\,.\end{split}$

The analysis provides the most precise single measurement of $C\\!P$ violation
in this decay channel to date. All parameters are consistent with their
current world average values.

Published in JHEP 03 (2020) 147.

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

##  1 Introduction

In the hadronic sector of the Standard Model (SM), $C\\!P$ violation
originates from an irreducible complex phase in the Cabibbo–Kobayashi–Maskawa
matrix that describes the mixing of the quark mass eigenstates into weak-
interaction eigenstates [1, 2]. Interference caused by a weak-phase difference
between the ${B}^{0}$–${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$
oscillation and the decay amplitudes leads to a $C\\!P$ asymmetry in the
decay-time distributions of ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ mesons. Decays involving
${b}\\!\rightarrow{c}{\overline{c}}{s}$ tree transitions at leading order,
such as ${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\mathrm{S}}}$,111Charge-conjugated processes are implicitly
included in the following, unless specified. are sensitive to the weak phase
$2\beta$, where
$\beta\equiv\arg[-(V_{cd}V_{cb}^{\ast})/(V_{td}V_{tb}^{\ast})]$ is one of the
angles of the Unitarity Triangle. Measurements of this phase were performed by
several experiments using different channels [3]. The same phase appears in
${b}\\!\rightarrow{c}{\overline{c}}{d}$ transitions, which contribute to
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays, when the leading-
order colour-favoured tree diagram is considered. However,
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays can also proceed
through several other decay diagrams, that include penguin, $W$-exchange and
annihilation topologies, where additional contributions to $C\\!P$ violation
both from the SM and new physics (NP) may arise. Tests of the SM have been
performed by relating $C\\!P$ asymmetries and branching fractions of different
decay modes of neutral and charged beauty mesons to two charm mesons [4, 5].

Each of the ${D}^{*+}$ ${D}^{-}$ and ${D}^{*-}$ ${D}^{+}$ final states are
accessible from both ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ mesons. The time-dependent decay
rates for the four configurations of initial $B$ flavour and final states can
be written as

$\begin{split}\frac{\mathrm{d}\Gamma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}^{0}},f}(t)}{\mathrm{d}t}&=\frac{{e}^{-t/\tau_{d}}}{8\tau_{d}}(1+{\cal
A}_{f\bar{f}})\Bigl{[}1+S_{f}\sin(\Delta m_{d}t)-C_{f}\cos(\Delta
m_{d}t)\Bigr{]},\\\
\frac{\mathrm{d}\Gamma_{{{B}^{0}},f}(t)}{\mathrm{d}t}&=\frac{{e}^{-t/\tau_{d}}}{8\tau_{d}}(1+{\cal
A}_{f\bar{f}})\Bigl{[}1-S_{f}\sin(\Delta m_{d}t)+C_{f}\cos(\Delta
m_{d}t)\Bigr{]},\\\ \frac{\mathrm{d}\Gamma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}^{0}},\bar{f}}(t)}{\mathrm{d}t}&=\frac{{e}^{-t/\tau_{d}}}{8\tau_{d}}(1-{\cal
A}_{f\bar{f}})\Bigl{[}1+S_{\bar{f}}\sin(\Delta m_{d}t)-C_{\bar{f}}\cos(\Delta
m_{d}t)\Bigr{]},\\\
\frac{\mathrm{d}\Gamma_{{{B}^{0}},\bar{f}}(t)}{\mathrm{d}t}&=\frac{{e}^{-t/\tau_{d}}}{8\tau_{d}}(1-{\cal
A}_{f\bar{f}})\Bigl{[}1-S_{\bar{f}}\sin(\Delta m_{d}t)+C_{\bar{f}}\cos(\Delta
m_{d}t)\Bigr{]},\end{split}$ (1)

where $f={{D}^{*+}}{{D}^{-}}$ and $\bar{f}={{D}^{*-}}{{D}^{+}}$. The parameter
${\cal A}_{f\bar{f}}$ represents the overall asymmetry in the production of
the $f$ and $\bar{f}$ final states and is defined as

${\cal A}_{f\bar{f}}=\frac{\Bigl{(}\lvert
A_{f}\rvert^{2}+\lvert\bar{A}_{f}\rvert^{2}\Bigr{)}-\Bigl{(}\lvert
A_{\bar{f}}\rvert^{2}+\lvert\bar{A}_{\bar{f}}\rvert^{2}\Bigr{)}}{\Bigl{(}\lvert
A_{f}\rvert^{2}+\lvert\bar{A}_{f}\rvert^{2}\Bigr{)}+\Bigl{(}\lvert
A_{\bar{f}}\rvert^{2}+\lvert\bar{A}_{\bar{f}}\rvert^{2}\Bigr{)}},$ (2)

with $A_{f}$ ($A_{\bar{f}}$) and ${\bar{A}}_{f}$ (${\bar{A}}_{\bar{f}}$)
indicating the amplitudes of the decay of a ${B}^{0}$ and a ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ meson to final state $f$
($\bar{f}$). Here, $\tau_{d}$ is the ${B}^{0}$ lifetime and $\Delta m_{{d}}$
is the mass difference of the two ${B}^{0}$ mass eigenstates, which are
assumed to have the same decay width [3]. Introducing $q$ and $p$ to describe
the relation between the mass and flavour eigenstates, $|B_{\rm
H,L}\rangle=p|{{B}^{0}}\rangle\pm q|{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rangle$, the parameters $S_{f}$
and $C_{f}$ are defined as

$S_{f}=\frac{2\mathcal{I}m\lambda_{f}}{1+\lvert\lambda_{f}\rvert^{2}},\hskip
28.45274ptC_{f}=\frac{1-\lvert\lambda_{f}\rvert^{2}}{1+\lvert\lambda_{f}\rvert^{2}},\hskip
28.45274pt\lambda_{f}=\frac{q}{p}\frac{\bar{A}_{f}}{A_{f}}\,,$ (3)

with analogous definitions holding for $S_{\bar{f}}$, and $C_{\bar{f}}$. By
combining these parameters, the $C\\!P$ observables for
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays can be defined as [3]

$\begin{split}S_{{{D}^{*}}{D}}&=\frac{1}{2}(S_{f}+S_{\bar{f}}),\\\ \Delta
S_{{{D}^{*}}{D}}&=\frac{1}{2}(S_{f}-S_{\bar{f}}),\\\
C_{{{D}^{*}}{D}}&=\frac{1}{2}(C_{f}+C_{\bar{f}}),\\\ \Delta
C_{{{D}^{*}}{D}}&=\frac{1}{2}(C_{f}-C_{\bar{f}}),\\\ {\cal
A}_{{{D}^{*}}{D}}&={\cal A}_{f\bar{f}}.\end{split}$ (4)

In absence of $C\\!P$ violation, $S_{{{D}^{*}}{D}}$ and $C_{{{D}^{*}}{D}}$
vanish. While $\Delta S_{{{D}^{*}}{D}}$ is related to the relative strong
phase between the decay amplitudes, the parameter $\Delta C_{{{D}^{*}}{D}}$ is
a measure of how flavour specific the decay mode is. For a flavour-specific
decay only one final state is accessible for each flavour of the decaying
neutral $B$ meson, $\Delta C_{{{D}^{*}}{D}}=\pm 1$ and no $C\\!P$ violation in
the interference between decays with and without mixing is possible. Decays
with $\Delta C_{{{D}^{*}}{D}}=0$ present the highest sensitivity to mixing-
induced $C\\!P$ violation. In the case of
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays, if the contribution
of higher-order SM processes and NP are negligible, the amplitudes for
${{B}^{0}}\rightarrow{{D}^{*+}}{{D}^{-}}$ and
${{B}^{0}}\rightarrow{{D}^{*-}}{{D}^{+}}$ have the same hadronic phase and
magnitude. As a result, ${\cal A}_{{{D}^{*}}{D}}$, $C_{{{D}^{*}}{D}}$, $\Delta
C_{{{D}^{*}}{D}}$ and $\Delta S_{{{D}^{*}}{D}}$ vanish and
$S_{{{D}^{*}}{D}}=\sin(2\beta)$. Theoretical models, based on QCD
factorization and heavy quark symmetry, estimate the contribution of penguin
amplitudes in ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ to be up to a
few percent [6, 7].

By combining Eqs. 1 and 4 the decay rate can be rewritten as

$\displaystyle\frac{\mathrm{d}\Gamma(t)}{\mathrm{d}t}=$
$\displaystyle\frac{{e}^{-t/\tau_{d}}}{8\tau_{d}}(1+r{\cal
A}_{{{D}^{*}}{D}})\times$ (5)
$\displaystyle\Bigl{[}1-d(S_{{{D}^{*}}{D}}+r\Delta S_{{{D}^{*}}{D}})$
$\displaystyle\sin(\Delta mt)+d(C_{{{D}^{*}}{D}}+r\Delta
C_{{{D}^{*}}{D}})\cos(\Delta mt)\Bigr{]}\,,$

where $d$ takes values $+1$ ($-1$) for mesons whose initial flavour is
${B}^{0}$ (${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$) and $r$ takes
values $+1$ ($-1$) for the final states $f$ ($\bar{f}$).

This paper reports the first measurement of $C\\!P$ violation in
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays at the LHCb
experiment. The measurement is based on a sample of $pp$ collision data
corresponding to integrated luminosities of 1 and 2$\text{\,fb}^{-1}$ at
centre-of-mass energies of 7 and 8 TeV (referred to as Run 1) and of
6$\text{\,fb}^{-1}$ at 13 TeV (Run 2), recorded by the LHCb experiment between
2011 and 2018. Previous measurements with this ${B}^{0}$ decay mode have been
performed by the BaBar [8] and Belle experiments [9].

In this analysis the ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$
candidates are reconstructed through the subsequent decays
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$. For the ${D}^{0}$ meson, the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay modes are used. The
analysis proceeds as follows: ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$
candidates, reconstructed in the two ${D}^{0}$ decay modes and the two data-
taking periods, are selected and analysed separately, as outlined in Sec. 3.
The signal contribution is determined in each of the four samples with fits to
the ${B}^{0}$ mass distributions, as described in Sec. 4. A key ingredient for
measurements of $C\\!P$ violation in time-dependent analyses is the
determination of the flavour of the neutral $B$ mesons by means of tagging
algorithms, described in Sec. 5. The evaluation of instrumental asymmetries
that affect the measurement of the overall $C\\!P$ charge asymmetry ${\cal
A}_{{{D}^{*}}{D}}$ is discussed in Sec. 6. A simultaneous fit to the ${B}^{0}$
decay-time distributions of the four samples is performed to determine the
$C\\!P$ parameters, as described in Sec. 7. The estimate of the systematic
uncertainties is presented in Sec. 8 and finally, conclusions are drawn in
Sec. 9.

##  2 Detector and simulation

The LHCb detector [10, 11] is a single-arm forward spectrometer covering the
pseudorapidity range $2-5$, designed to study 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 [12], 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 [13, 14] placed downstream of the magnet. The
tracking system provides a measurement of the momentum, $p$, of charged
particles with a relative uncertainty that varies from 0.5% at low momentum to
1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$. During the data taking, the
polarity of the magnetic field was periodically reversed to reduce the
residual detection asymmetries that affect the determination of charge
asymmetries. The minimum distance of a track to a primary collision 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
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[15]. 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 [16].

Simulated data samples are used to model the effects of the detector
acceptance and the imposed selection requirements. Samples of signal decays
are produced in order to determine inputs for the analysis, such as the
parametrisation of the mass distribution and the decay-time resolution model.
Multibody ${D}^{0}$ and ${D}^{+}$ decays are modelled in the simulation
according to the previously measured resonant structures [17, 18]. Samples of
the most relevant background from partially reconstructed and misidentified
$B$ meson decays, as well as specific $B$ decays useful for studies related to
flavour tagging, are also produced. In the simulation, $pp$ collisions are
generated using Pythia [19, *Sjostrand:2007gs] with a specific LHCb
configuration [21]. Decays of unstable particles are described by EvtGen [22],
in which final-state radiation is generated using Photos [23]. The interaction
of the generated particles with the detector, and its response, are
implemented using the Geant4 toolkit [24, *Agostinelli:2002hh] as described in
Ref. [26].

##  3 Selection

The online event selection is performed by a trigger [27], which consists of a
hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. At
the hardware-trigger stage, events are required to have a muon with high
$p_{\mathrm{T}}$ or a hadron, photon or electron with high transverse energy
in the calorimeters. The software trigger requires a two-, three- or four-
track secondary vertex with a large sum of the transverse momenta of the
tracks and a significant displacement from the primary $pp$ interaction
vertices. A multivariate algorithm [28] is used for the identification of
secondary vertices consistent with the decay of a $b$ hadron.

In the offline selection, the ${D}^{*+}$, ${D}^{0}$ and ${D}^{-}$ candidates
are reconstructed through their decays into the selected final-state particles
whose tracks are required to have good quality, exceed threshold values on $p$
and $p_{\mathrm{T}}$ and satisfy loose particle identification (PID) criteria,
mostly relying on the Cherenkov detectors information. These tracks are also
required to have a $\chi^{2}_{\text{IP}}$ value with respect to any PV greater
than four, where $\chi^{2}_{\text{IP}}$ is defined as the difference in the
vertex-fit $\chi^{2}$ of a given PV reconstructed with and without the
particle being considered. The distance of closest approach between all
possible combinations of particles forming a common vertex should be smaller
than $0.5\text{\,mm}$ and the vertex should be downstream of the PV. The
invariant mass of ${D}^{0}$ (${D}^{-}$) meson candidates is required to lie
within $\pm 40\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ ($\pm
50\text{\,Me\kern-1.00006ptV\\!/}c^{2}$) of the known value [18], while the
difference of the ${D}^{*+}$ and ${D}^{0}$ invariant masses is required to be
smaller than $150\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. These windows
correspond to about $\pm 5$ times the mass resolutions. Candidate ${B}^{0}$
mesons are reconstructed from ${D}^{*+}$ and ${D}^{-}$ meson candidates that
form a common vertex. The scalar sum of transverse momenta of the all final-
state particles should exceed 5000$\text{\,Me\kern-1.00006ptV\\!/}c$ and the
momentum direction of the ${B}^{0}$ meson should point to its associated PV.
If more than one PV is reconstructed in an event, the associated PV is that
with respect to which the signal ${B}^{0}$ candidate has the smallest
$\chi^{2}_{\text{IP}}$.

Background can be due to the misidentification of one hadron in the charged
$D$ decay chain. To suppress ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{D}^{*-}}$
with ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{+}}{{\pi}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ with
${{D}^{+}_{s}}\\!\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$ decays, mass vetoes
are applied. The pion from the
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ reconstructed decay
with the higher $p_{\mathrm{T}}$ is assumed to be a proton (kaon) and the
candidate is rejected if the recomputed invariant mass is within $\pm
25\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${\mathchar
28931\relax}^{+}_{c}$ (${D}^{+}_{s}$) mass and the PID requirement for the
alternative particle assignment is satisfied. To further reduce background
contributions of $\phi\rightarrow{{K}^{+}}{{K}^{-}}$ from ${D}^{-}_{s}$
decays, the kaon mass hypothesis is assigned to the pion with the higher
$p_{\mathrm{T}}$ in the
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ reconstructed decay
and the candidate ${D}^{-}$ is rejected if the invariant mass of the kaon pair
is compatible with the $\phi$ mass within $\pm
10\text{\,Me\kern-1.00006ptV\\!/}c^{2}$. These vetoes reduce the ${\mathchar
28931\relax}^{+}_{c}$ and $\phi$ background contributions to a negligible
level. The background due to ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$
decay is only partially suppressed by the ${D}^{+}_{s}$ veto which includes
only loose selection criteria to retain high signal efficiency.

Single-charm $B$ decays such as
${{B}_{({s})}^{0}}\\!\rightarrow{{D}^{*-}}h^{-}h^{+}h^{+}$, where the three
hadrons are not produced in a ${D}^{+}$ decay, but directly originate from the
${B}^{0}$ decay, are another potential source of background. To reject
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays
with a pion misidentified as a kaon, the
${{D}^{*-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ invariant mass is calculated
with the pion mass assigned to the kaon. The candidate is rejected if the mass
is within $\pm 40\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${B}^{0}$
mass, and either the kaon candidate has a high probability to be a pion or the
$\chi^{2}$ of the flight distance of the ${D}^{+}$ with respect to the
${B}^{0}$ decay vertex is less than four. Background arising from possible
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays
is suppressed by rejecting candidates with the
${{{D}^{*-}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}}$ mass within $\pm
25\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${B}^{0}_{s}$ mass and
the ${D}^{+}$ decay vertex reconstructed upstream the ${B}^{0}$ decay vertex,
or the $\chi^{2}$ of the flight distance of the ${D}^{+}$ candidate with
respect to the ${B}^{0}$ decay vertex smaller than two. This veto is applied
only to ${D}^{0}$ decaying into ${{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$,
as no excess is observed for the ${{K}^{-}}{{\pi}^{+}}$ sample. The invariant-
mass distribution of signal candidates is not significantly modified by the
vetoes.

In order to separate further the
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ signal candidates from the
combinatorial background, a boosted decision tree (BDT) utilising the AdaBoost
method [29, 30] implemented in the TMVA toolkit [31, *TMVA4] is used. To train
the BDT, simulated ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ candidates
are used as a proxy for signal, whereas candidates in data with invariant mass
in the upper sideband are used as proxy for background. The upper sideband is
defined as
$5400<m_{{{D}^{*\pm}}{{D}^{\mp}}}<6000$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$
($5600<m_{{{D}^{*\pm}}{{D}^{\mp}}}<6000$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$)
for the ${D}^{0}$ final states ${{K}^{-}}{{\pi}^{+}}$ and
${{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$. A two-folding procedure is
applied to avoid overtraining for the ${{K}^{-}}{{\pi}^{+}}$ candidates in the
mass range
$5400<m_{{{D}^{*\pm}}{{D}^{\mp}}}<5600$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$,
which are both used for the BDT training and the mass fit. Separate
classifiers are trained for each final state and data taking period. Various
kinematic and topological quantities are used in the BDT to exploit the
features of the signal decay in order to distinguish it from background,
namely the transverse momentum of the ${B}^{0}$, ${D}^{*+}$, ${D}^{-}$ mesons
and of the ${D}^{-}$ and ${D}^{0}$ decay products; the decay-time significance
of the ${D}^{-}$ and ${D}^{0}$ mesons, and their flight distance $\chi^{2}$
with respect to the associated PV; the ${B}^{0}$ $\chi^{2}_{\text{IP}}$; the
angles formed by each of the ${D}^{-}$ decay products and the ${D}^{-}$
direction; the angle formed by the pion from the ${D}^{0}$ decay and the
${D}^{0}$ direction, and the angle formed by the ${D}^{*+}$ and the ${B}^{0}$
mesons. The PID probabilities of the kaon and pion in the final state are also
used in the BDT. The requirement on the output of each BDT classifiers is
chosen to minimise the uncertainties on the $C\\!P$ parameters
$S_{{{D}^{*}}{D}}$ and $C_{{{D}^{*}}{D}}$.

A kinematic fit to the ${B}^{0}$ decay chain with constraints on the masses of
all the charm mesons and on the PV is performed to significantly improve
($\sim 60\%$) the resolution on the invariant mass of the ${B}^{0}$ candidate
[33], while the ${B}^{0}$ decay time is calculated using the same fit with
only the PV constraint in order to avoid correlations of the decay time with
the invariant mass. Candidate ${B}^{0}$ mesons are retained if their invariant
mass and decay time are in the ranges
$5000-5600$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and $0.3-10.3$ ps,
respectively, where the lower boundary of the decay time is set to reduce
promptly produced background. After the selection a fraction of events below
5% have multiple candidates. In these cases a single candidate is randomly
selected with negligible change in the final result.

##  4 Mass fit

An unbinned extended maximum-likelihood fit is performed on the invariant-mass
distribution of the selected ${B}^{0}$ candidates, on each of the four data
samples independently. The sPlot technique [34] is employed to determine per-
candidate weights that are used for background subtraction in the subsequent
decay-time fit. The model describing the ${{D}^{*\pm}}{{D}^{\mp}}$ mass
distribution consists, in addition to the signal
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$, of the following background
components: ${B}^{0}_{s}$ decays to the same ${{D}^{*\pm}}{{D}^{\mp}}$ final
states; ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ decays with a
misidentified kaon that pass the selection; ${B}^{0}$ and ${B}^{0}_{s}$ decays
to ${D}^{*+}$ ${D}^{*-}$ with one of the excited charm mesons decaying into a
charged $D$ meson and an additional unreconstructed neutral pion or photon;
and combinatorial background. The distribution of the reconstructed mass of
the ${B}^{0}$ signal component is parametrised with the sum of two (three)
Crystal Ball functions [35] for the
${{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ (${{K}^{-}}{{\pi}^{+}}$)
${D}^{0}$ final state, with common mean but different width and tail
parameters. The mass model for the ${B}^{0}_{s}$ decays is the same as for the
${B}^{0}$ decays, but the peak position is shifted by the difference between
the known values of the ${B}^{0}_{s}$ and ${B}^{0}$ mesons [18]. The mass
distribution of ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ decays is
described with the sum of two Crystal Ball functions with a common mean, which
is floated in the data fit, while the remaining parameters are fixed from
simulation. The mass distribution for the partially reconstructed
${{B}^{0}}\\!\rightarrow{{D}^{*+}}{{D}^{*-}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*+}}{{D}^{*-}}$ background contributions are
described by a combination of functions corresponding to pure longitudinal and
transverse polarizations of the two ${D}^{*\pm}$ mesons. The relative fraction
of the two possible contributions are floated in the fit, while the shapes,
that are a double peak in case of longitudinal polarization and a single broad
one for transverse polarization, are fixed to those evaluated on simulated
samples. The relative fraction of the two contributions in
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*+}}{{D}^{*-}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{*+}}{{D}^{*-}}$ decays are assumed to be the
same. The distribution of the reconstructed mass for the combinatorial
background component is modelled by an exponential function. The results of
the fits to the four data samples are shown in Fig. 1 with the partial
contribution of each component overlaid; the resulting signal yields are
$469\pm 28$ ($1570\pm 48$) and $856\pm 32$ ($3265\pm 61$) for the ${D}^{0}$
final states ${{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{K}^{-}}{{\pi}^{+}}$ in Run 1 (Run 2), respectively.

Figure 1:  Mass distributions for the
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decay with (left)
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
(right) ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ for (top) Run 1 and
(bottom) Run 2 data samples. Besides the data points and the full PDF (solid
black) the projections of the ${B}^{0}$ signal (dashed blue), the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ background (dotted
green), the ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ background (dash-
dotted turquoise), the ${{B}^{0}}\\!\rightarrow{{D}^{*+}}{{D}^{*-}}$
background (long-dash-dotted magenta), the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*+}}{{D}^{*-}}$ background (dash-three-
dotted red) and the combinatorial background (long-dashed green) are shown.

##  5 Flavour tagging

Measurements of $C\\!P$ violation in decay-time-dependent analyses of
${B}^{0}$ meson decays require the determination of the production flavour of
the ${B}^{0}$ meson. Methods to infer the initial flavour of a reconstructed
candidate, i.e. whether it contained a ${b}$ or a ${\overline{b}}$ quark at
production, are referred to as flavour tagging. Two classes of algorithms are
used. The opposite side (OS) tagger exploits the fact that $b$ and
$\overline{b}$ quarks are almost exclusively produced in pairs in $pp$
collisions, allowing the flavour of the signal ${B}^{0}$ candidate to be
inferred from the flavour of the other $b$ hadron in the event. The OS tagger
combines information on the charge of the muon or electron from semileptonic
$b$ decays, the charge of the kaon from the $b\rightarrow c\rightarrow s$
decay chain, the charge of a reconstructed secondary charm hadron and the
charges of the tracks that form the secondary vertex of the other $b$-hadron
decay, combined into a weighted average, with weights depending on the
transverse momenta of the tracks [36, 37]. The same-side (SS) tagger exploits
the production of correlated protons or pions in the hadronization of the
$\overline{b}$ ($b$) quark that forms the signal ${B}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$) candidate, with its initial
flavour identified by the charge of the particle [38].

Each tagging algorithm provides a flavour-tagging decision, $d$, and an
estimate, $\eta$, of the probability that the decision is incorrect (mistag)
for a reconstructed ${B}^{0}$ candidate. The tagging decision takes the value
$\pm 1$ for tagged $B$ candidates and 0 if no decision on initial flavour can
be assigned (untagged). The mistag probability varies in the range from 0 to
0.5 for tagged candidates and is equal to 0.5 for untagged candidates.

Each tagging algorithm is implemented as a BDT that is trained and optimised
using large data samples of
${{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}$
decays for the OS and of ${{B}^{0}}\rightarrow{{D}^{-}}{{\pi}^{+}}$ decays for
the SS taggers, respectively [39]. The mistag probability for each tagger is
given by the output of the BDT, which is calibrated using dedicated data
control channels to relate $\eta$ to the true mistag probability, $\omega$.
The performance of the flavour tagging is measured by the tagging power,
${\varepsilon_{\mathrm{tag}}}D^{2}$, where $\varepsilon_{\mathrm{tag}}$ is the
fraction of tagged candidates and $D=1-2\omega$ represents the dilution
induced on the oscillation amplitude. The tagging power represents the
effective loss in signal yield compared to a perfectly tagged sample.

###  5.1 Calibration of the tagging output

Flavour tagging algorithms are calibrated using control samples of flavour-
specific $B$ decays, separately in Run 1 and Run 2 data. At first
approximation, the measured mistag fraction $\omega$ in the control channel
can be expressed as a linear function of the predicted mistag estimate $\eta$
as

$\omega(\eta)=p_{0}+p_{1}\left(\eta-\langle\eta\rangle\right),$ (6)

where the use of the arithmetic mean $\langle\eta\rangle$ of the $\eta$
distribution decorrelates the $p_{0}$ and $p_{1}$ parameters. A perfect
calibration of the taggers would result in $p_{0}=\langle\eta\rangle$ and
$p_{1}=1$. A signal candidate tagged with decision $d$ and a calibrated mistag
$\omega>0.5$ corresponds to an opposite tagging decision $d^{\prime}=-d$ with
a mistag probability $\omega^{\prime}=1-\omega$.

The performance of the flavour taggers, $\varepsilon_{\mathrm{tag}}$ and
$\omega$, may depend on the initial flavour of the neutral $B$ meson. Charged
decay products, like the ${K}^{+}$ and ${K}^{-}$ which are used by the OS kaon
tagger, can have significantly different interaction rates with the detector
material and therefore lead to different reconstruction efficiencies. To
account for these asymmetries Eq. 6 is modified as

$\displaystyle{\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993pt\omega$}{}}(\eta)$ $\displaystyle=$
$\displaystyle\left(p_{0}\pm\frac{\delta
p_{0}}{2}\right)+\left(p_{1}\pm\frac{\delta
p_{1}}{2}\right)\left(\eta-\langle\eta\rangle\right),$ (7)

where the mistag fractions $\omega$ and $\overline{\omega}$ for an initial
${B}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ correspond
to the plus and minus sign, respectively, and $\delta
p_{0,1}=p_{0,1}^{{{B}^{0}}}-p_{0,1}^{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}^{0}}}$. Similarly, the tagging
asymmetry, ${\cal A}_{\rm tag}$, is defined as

${\cal A}_{\rm tag}=\frac{\varepsilon_{\rm tag}^{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}^{0}}}-\varepsilon_{\rm
tag}^{{{B}^{0}}}}{\varepsilon_{\rm tag}^{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}^{0}}}+\varepsilon_{\rm
tag}^{{{B}^{0}}}}\,.$ (8)

For this analysis, the calibration of the OS and SS taggers is performed using
samples of ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ flavour-specific decays which
have similar topology, kinematics and event characteristics as the signal
decay. Candidate ${B}^{0}$ mesons in the two decay modes are reconstructed
using the decays ${{D}^{+}_{s}}\\!\rightarrow{{K}^{-}}{{K}^{+}}{{\pi}^{+}}$,
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{*-}}\\!\rightarrow{{\pi}^{-}}{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$, followed by ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
or ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$.
These decays are selected using the same trigger requirements, and similar
kinematic and geometric criteria, to those applied for the signal selection.
Requirements are applied on the identification of the final-state particles as
well as on the invariant mass of ${D}^{+}_{s}$, ${D}^{*-}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons. Candidate ${B}^{0}$
mesons are retained if their invariant mass and decay time are in the ranges
$5220-5500\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ and $0.3-10.3\text{\,ps}$,
respectively. In case multiple candidates are selected in the same event, the
candidate with the highest $p_{\mathrm{T}}$ is retained.

A fit to the invariant-mass distribution of the ${B}^{0}$ candidates is
performed on each sample independently to separate the signal from the
background contribution, which consists of random combination of particles.
The signal is modelled by the sum of two Gaussian functions with a common
mean, while the background shape is described by an exponential function. In
the case of the ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ control
channel, an additional contribution due to
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{+}}$ decays is also needed. It
is modelled by the same function used for the ${B}^{0}$ decays except for the
mean value, that is shifted by the known difference of the ${B}^{0}_{s}$ and
${B}^{0}$ masses [18]. Signal yields of 11,400 (39,200)
${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ decays and 24,900 (102,900)
${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ decays, in the Run 1 (Run 2)
data sample, are found.

The mass fit determines for each selected candidate a weight that is used to
statistically subtract the background contribution, using the sPlot technique.
The tagging calibration parameters are determined from an unbinned maximum-
likelihood fit to the weighted distribution of the decay time $t$, the final
state $r$, with $r=+1$ $(-1)$ for $D_{s}^{+}D^{(\ast)-}$
($D_{s}^{-}D^{(\ast)+}$), tag decisions $\vec{d}=(d_{\rm OS},d_{\rm SS})$ and
probabilities $\vec{\eta}=(\eta_{\rm OS},\eta_{\rm SS})$, according to the
following PDF

${\cal
P}\left(t,r,\vec{d}\,\lvert\,\vec{\eta}\right)=\epsilon(t)\cdot\left[\mathcal{F}(t^{\prime},r,\vec{d}\,\lvert\,\vec{\eta})\otimes{\cal
R}(t-t^{\prime})\right]\,.$ (9)

Here, ${\cal R}(t-t^{\prime})$ is the decay-time resolution, which is modelled
using simulation. The decay-time efficiency, $\epsilon(t)$, is parametrised
using a cubic spline model [40], with parameters determined from the fit to
the data. The number of spline coefficients and the knot positions are very
similar among the ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{-}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ decays, as well as among Run
1 and Run 2 samples. The signal PDF, $\cal F$, describes the decay-time
distribution of flavour-specific ${B}^{0}$ decays; it is based on Eqs. 12 and
13 of Sec. 7, with appropriate values for the coefficients corresponding to
flavour-specific ${B}^{0}$ decays. It depends on the $\tau_{d}$ and $\Delta
m_{{d}}$ parameters, that describe the ${B}^{0}$ decay and mixing, which are
fixed to their known values [18]. In addition, the PDF depends on the tagging
parameters of Eq. 7, on the tagging efficiencies and their asymmetry, on the
${B}^{0}$ production asymmetry and on the detection asymmetry of the control
channel, that are determined from the fit to data. The asymmetry of the
${B}^{0}$ meson production is defined as ${\cal
A}_{\mathrm{prod}}={(\sigma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}^{0}}}-\sigma_{{{B}^{0}}})}/{(\sigma_{{{\kern
1.25995pt\overline{\kern-1.25995ptB}}{}^{0}}}+\sigma_{{{B}^{0}}})}$, where
$\sigma_{{{\kern 1.25995pt\overline{\kern-1.25995ptB}}{}^{0}}}$ and
$\sigma_{{{B}^{0}}}$ are the production rates of ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ and ${B}^{0}$ mesons.

The fit is performed independently on each sample. For each data-taking
period, the parameters of the tagging calibration and of the production
asymmetries are consistent between the two channels, and the results are
combined according to their statistical uncertainties.

Different sources of systematic uncertainty are considered. Systematic
uncertainties on the calibration parameters are determined by repeating the
fit to data with modified assumptions. The deviations of the fit results from
the nominal values are assigned as systematic uncertainties. The largest
uncertainty is related to the determination of the signal weights from the fit
to the mass distribution. An alternative determination of such weights is
performed fitting the two-dimensional invariant-mass distribution of the
${D}^{+}_{s}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$
(${D}^{*-}$) candidates. To account for possible effects related to the decay-
time efficiency model, an analytic function is considered instead of the cubic
spline function. The uncertainties related to the input values of the
${B}^{0}$ decay and mixing properties are determined repeating the fit with
inputs varied within their uncertainties. Finally, uncertainties related to
differences between data and simulation concerning the time resolution model
are neglected given their insignificant impact on the tagging parameters for
variations up to 50%. The resulting tagging parameters are listed in Table 1.

Table 1: Flavour-tagging parameters obtained as a weighted average of the values measured in the two control channels. The quoted uncertainties are statistical and systematic. Parameter | Run 1 | Run 2
---|---|---
${\cal A}_{\rm prod}$ |  $-0.011$$\,\pm\,$ | $0.008\pm 0.003$ | 0.004$\,\pm\,$ | $0.005\pm 0.002$
${\cal A}_{\rm tag}^{\rm OS}$ | 0.008$\,\pm\,$ | $0.008\pm 0.005$ | 0.001$\,\pm\,$ | $0.004\pm 0.009$
$\delta p_{0}^{\rm OS}$ | 0.022$\,\pm\,$ | $0.007\pm 0.013$ | 0.009$\,\pm\,$ | $0.004\pm 0.003$
$\delta p_{1}^{\rm OS}$ | 0.20$\,\pm\,$ | $0.06\phantom{0}\pm 0.11$ | 0.02$\,\pm\,$ | $0.03\phantom{0}\pm 0.01$
$p_{0}^{\rm OS}-\langle\eta^{\text{OS}}\rangle$ | 0.037$\,\pm\,$ | $0.005\pm 0.006$ | 0.032$\,\pm\,$ | $0.003\pm 0.005$
$p_{1}^{\rm OS}$ | 0.95$\,\pm\,$ | $0.04\phantom{0}\pm 0.10$ | 0.87$\,\pm\,$ | $0.02\phantom{0}\pm 0.02$
${\cal A}_{\rm tag}^{\rm SS}$ |  $-0.006$$\,\pm\,$ | $0.004\pm 0.005$ |  $-0.001$$\,\pm\,$ | $0.001\pm 0.002$
$\delta p_{0}^{\rm SS}$ |  $-0.004$$\,\pm\,$ | $0.005\pm 0.009$ | 0.001$\,\pm\,$ | $0.003\pm 0.003$
$\delta p_{1}^{\rm SS}$ |  $-0.03$$\,\pm\,$ | $0.08\phantom{0}\pm 0.06$ |  $-0.11$$\,\pm\,$ | $0.04\phantom{0}\pm 0.02$
$p_{0}^{\rm SS}-\langle\eta^{\text{SS}}\rangle$ | 0.007$\,\pm\,$ | $0.004\pm 0.006$ |  $-0.002$$\,\pm\,$ | $0.002\pm 0.002$
$p_{1}^{\rm SS}$ | 0.94$\,\pm\,$ | $0.06\phantom{0}\pm 0.07$ | 0.88$\,\pm\,$ | $0.03\phantom{0}\pm 0.03$

The portability of the calibration from the control channels to the signal
decay channel is assessed with simulation. Both OS and SS taggers show
compatible results among the different decay modes. A deviation from linearity
of the SS tagger calibration is observed in the signal channel. Its impact on
the determination of the signal $C\\!P$ parameters is evaluated using
pseudoexperiments where the effect is reproduced at generation level, as
described in Sec. 8. In addition, a dependence of the OS tagging efficiency on
the decay-time for $t<1$ ps is present in data. Pseudoexperiments are used to
determine the related uncertainties on the $C\\!P$ parameters, as described in
Sec. 8.

###  5.2 Tagging results

Approximately 40% of the tagged candidates in the signal decay samples are
tagged by both the OS and the SS algorithms. Since the algorithms select
different samples of charged particles and hence are uncorrelated, the two
tagging results are combined taking into account both decisions and their
corresponding estimates of $\eta$ as detailed in Ref. [41]. The combined
estimated mistag probability and the corresponding uncertainty are obtained by
combining the individual calibrations for the OS and SS taggers and
propagating their uncertainties in the decay-time fit. The effective tagging
power and efficiency for signal candidates tagged by one or both of the OS and
SS algorithms are given in Table 2.

Table 2:  Tagging efficiency and tagging power for ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ signal candidates in the four data samples, computed using the event-by-event predicted mistag $\eta$ and the calibration parameters obtained from control channels. The quoted uncertainties are statistical only. |  | Run 1 | Run 2
---|---|---|---
Sample | Tagger |  $\varepsilon_{\mathrm{tag}}$ [$\%$] |  ${\varepsilon_{\mathrm{tag}}}D^{2}$ [$\%$] |  $\varepsilon_{\mathrm{tag}}$ [$\%$] |  ${\varepsilon_{\mathrm{tag}}}D^{2}$ [$\%$]
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | OS only | 8.3$\,\pm\,$ | 1.6 | 0.64$\,\pm\,$ | 0.18 | 3.9$\,\pm\,$ | 0.6 | 0.36$\,\pm\,$ | 0.08
SS only | 43.0$\,\pm\,$ | 2.9 | 1.17$\,\pm\,$ | 0.16 | 47.4$\,\pm\,$ | 1.5 | 1.57$\,\pm\,$ | 0.11
OS&SS both | 37.5$\,\pm\,$ | 2.9 | 4.44$\,\pm\,$ | 0.57 | 41.5$\,\pm\,$ | 1.5 | 5.11$\,\pm\,$ | 0.30
Total | 88.8$\,\pm\,$ | 1.9 | 6.25$\,\pm\,$ | 0.55 | 92.7$\,\pm\,$ | 0.8 | 7.05$\,\pm\,$ | 0.29
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ | OS only | 12.2$\,\pm\,$ | 1.2 | 1.14$\,\pm\,$ | 0.19 | 4.2$\,\pm\,$ | 0.4 | 0.42$\,\pm\,$ | 0.06
SS only | 40.3$\,\pm\,$ | 1.8 | 1.43$\,\pm\,$ | 0.18 | 51.4$\,\pm\,$ | 0.9 | 1.61$\,\pm\,$ | 0.07
OS&SS both | 27.7$\,\pm\,$ | 1.7 | 3.05$\,\pm\,$ | 0.30 | 37.9$\,\pm\,$ | 0.9 | 4.57$\,\pm\,$ | 0.19
Total | 80.2$\,\pm\,$ | 1.4 | 5.61$\,\pm\,$ | 0.36 | 93.5$\,\pm\,$ | 0.5 | 6.61$\,\pm\,$ | 0.19

##  6 Instrumental asymmetries

The overall asymmetry that is measured in the decay-time fit has to be
corrected for instrumental asymmetries in order to determine the physical
parameter ${\cal A}_{{{D}^{*}}{D}}$. These asymmetries affect reconstruction,
detection and particle identification efficiencies and are related to the
different interaction cross-section with matter and different detection and
identification efficiencies of positive and negative pions and kaons. The
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decay is charge symmetric,
however since all instrumental efficiencies depend on momenta, and the $p$ and
$p_{\mathrm{T}}$ spectra of kaons and pions in the ${D}^{*\pm}$ and
${D}^{\mp}$ decays are observed to be slightly different, the cancellation is
not expected to be complete.

To a good level of approximation, the asymmetry in
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays, denoted with ${\cal
A}_{\rm det}$, can be related to the ${D}^{*-}$ and ${D}^{-}$ asymmetries,
${\cal A}_{\rm det}^{{{D}^{*-}}}$ and ${\cal A}_{\rm det}^{{{D}^{-}}}$, as

$\displaystyle{\cal A}_{\rm det}^{D^{*-}D^{+}}$ $\displaystyle\simeq{\cal
A}_{\rm det}^{D^{*-}}-{\cal A}_{\rm det}^{D^{-}}\,.$ (10)

Each of the ${D}^{*-}$ and ${D}^{-}$ asymmetries is measured using a sample of
prompt ${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays after a
kinematic weighting is applied to match the distribution of the final state
particles of the signal, as described in the following. In order to account
also for the PID asymmetry, the same kaon and pion identification requirements
used in the ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ selection are
applied to the prompt ${D}^{-}$ meson sample. The additional asymmetries
induced by the use of PID variables in vetoes and in the BDT employed for the
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ selection are considered as a
source of systematic uncertainty on the asymmetry ${\cal A}_{\rm
det}^{D^{*-}D^{+}}$. A sample of about 6 million prompt
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays collected in
Run 2 is used and divided into four subsamples: for each ${D}^{0}$ decay mode
of the ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ signal, one sample is
used to measure the ${D}^{*-}$ asymmetry, the other is used to determine the
${D}^{-}$ asymmetry. In each subsample a fit to the
${{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ invariant mass distribution is performed to
determine the weights to be used to subtract background. The signal mass model
is parametrised by two Crystal Ball functions with common mean and the
background is modelled by an exponential function. For the
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay mode, one of the prompt
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ subsamples is
weighted to match the spectra of the final state particles in the signal
${D}^{*+}$ decay and the other is weighted to match the spectra of the final
state particles in the signal ${D}^{-}$ decay [42]. The same procedure is used
for the ${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
decay mode, but not considering two of the pions with opposite charge from the
${D}^{0}$ decay, since the $\pi^{-}\pi^{+}$ pair is found to contribute
negligibly to additional asymmetries. Finally, the asymmetries ${\cal A}_{\rm
det}^{D^{*-}}$ and ${\cal A}_{\rm det}^{D^{-}}$ are calculated from the
${D}^{+}$ and ${D}^{-}$ yields obtained by fitting the mass distributions of
each weighted prompt ${D}^{+}$ and ${D}^{-}$ meson samples and subtracted. The
results are reported in Table 3 and ${\cal A}_{\rm det}^{{{D}^{*-}}{{D}^{+}}}$
values are consistent with zero.

Table 3: Instrumental asymmetries of ${D}^{\pm}$, ${D}^{*\pm}$ and their combination for ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays obtained from prompt $D$ meson decays. The quoted uncertainties are statistical, in the last column statistical and systematic. Final state | ${\cal A}_{\rm det}^{{{D}^{*-}}}$ | ${\cal A}_{\rm det}^{{{D}^{-}}}$ | ${\cal A}_{\rm det}^{{{D}^{*-}}{{D}^{+}}}$
---|---|---|---
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ | 0.0169$\,\pm\,$ | 0.0036 | 0.0158$\,\pm\,$ | 0.0018 | 0.0011$\,\pm\,$ | $0.0040\pm 0.0030$
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ | 0.0146$\,\pm\,$ | 0.0022 | 0.0138$\,\pm\,$ | 0.0015 | 0.0009$\,\pm\,$ | $0.0026\pm 0.0051$

The following sources of systematic uncertainties on ${\cal A}_{\rm det}$ are
considered: asymmetries due to the particle identification and hadronic
hardware-trigger requirements, variation of the prompt
${{D}^{-}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ selection criteria
and of the mass-fit model and imperfect cancellation of the prompt ${D}^{\pm}$
production asymmetry. Since the differences between the kinematic
distributions of the final state particles of ${D}^{*\pm}$ and ${{D}^{\mp}}$
decays are observed to be smaller in Run 1 than in Run 2, the difference of
${D}^{*\pm}$ and ${D}^{\mp}$ asymmetries in Run 1 is assumed to be zero with
the same statistical and systematic uncertainties as in Run 2.

##  7 Decay-time fit

A decay-time fit is performed simultaneously on the four data samples to
measure the $C\\!P$ coefficients $S_{{{D}^{*}}{D}}$, $\Delta
S_{{{D}^{*}}{D}}$, $C_{{{D}^{*}}{D}}$, $\Delta C_{{{D}^{*}}{D}}$ and ${\cal
A}_{{{D}^{*}}{D}}$. The weights determined with the mass fit are used to
subtract the background, using the sPlot technique. Before performing the fit,
a blinding transformation is applied on the $C\\!P$ parameters that is removed
only at the end of the analysis.

The PDF describing the measured ${B}^{0}$ decay-time distribution and tag
decisions $\vec{d}=(d_{\rm OS},d_{\rm SS})$, given the mistag probability
estimates $\vec{\eta}=(\eta_{\rm OS},\eta_{\rm SS})$, can be expressed as

${\cal
P}\left(t,r,\vec{d}\,|\,\vec{\eta}\right)=\epsilon(t)\cdot\Bigl{(}{{\cal
F}}(t^{\prime},r,\vec{d}\,|\,\vec{\eta})\otimes{\cal
R}(t-t^{\prime})\Bigr{)},$ (11)

where ${\cal P}(t^{\prime},r,\vec{d}\,|\,\vec{\eta})$ is the PDF describing
the distribution of the true decay time, $t^{\prime}$, ${\cal
R}(t-t^{\prime})$ is the decay-time resolution function, while $\epsilon(t)$
describes the decay-time efficiency. The PDF describing the ${B}^{0}$ decay-
time distribution has the same form as Eq. 5 but with effective coefficients

${{\cal
F}}(t^{\prime},r,\vec{d}\,|\,\vec{\eta})=N{e}^{-t^{\prime}/\tau_{d}}(1+r{\cal
A}_{\rm
raw})\Bigl{[}C^{\text{eff}}_{\cosh}-C^{\text{eff}}_{\sin}\sin\Bigl{(}{\Delta
m_{{d}}}t^{\prime}\Bigr{)}+C^{\text{eff}}_{\cos}\cos\Bigl{(}{\Delta
m_{{d}}}t^{\prime}\Bigr{)}\Bigr{]}.$ (12)

These coefficients, that depend on the final state variable $r=\pm 1$, can be
expressed as

$\begin{split}C^{\text{eff}}_{\sin}&=(S_{{{D}^{*}}{D}}+r\Delta
S_{{{D}^{*}}{D}})(\Delta^{-}-{\cal A}_{\text{prod}}\Delta^{+}),\\\
C^{\text{eff}}_{\cos}&=(C_{{{D}^{*}}{D}}+r\Delta
C_{{{D}^{*}}{D}})(\Delta^{-}-{\cal A}_{\text{prod}}\Delta^{+}),\\\
C^{\text{eff}}_{\cosh}&=(\Delta^{+}-{\cal
A}_{\text{prod}}\Delta^{-}),\end{split}$ (13)

where the factors $\Delta^{\pm}$ contain the dependence on the mistag
fraction, the tagging decisions and efficiencies and the asymmetries in the
tagging efficiencies of the OS and SS taggers. The parameter ${\cal A}_{\rm
raw}$ is the sum of the $C\\!P$ asymmetry, ${\cal A}_{{{D}^{*}}{D}}$, and the
detection asymmetry, ${\cal A}_{\rm det}^{{{D}^{*-}}{{D}^{+}}}$. Since the
instrumental asymmetries depend on the final state and on the data-taking
period, four different parameters are used in the fit.

The decay-time resolution model is determined from simulation and fixed in the
fit to data with an effective resolution of $60\text{\,fs}$. The same set of
parameters describes well both Run 1 and Run 2 data, with small differences
between the two ${D}^{0}$ final-state samples. Due to the low
${B}^{0}$–${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ oscillation
frequency, the decay-time resolution has a very small influence on the
relatively low $C\\!P$ parameters.

The selection and reconstruction efficiency depends on the ${B}^{0}$ decay
time due to displacement requirements made on the signal final-state particles
and a decrease in the reconstruction efficiency for tracks with large impact
parameter with respect to the beam line [43]. The decay-time efficiency is
modelled by a cubic spline function with five knots [40]. The knot positions
are chosen from a fit to simulated signal candidates at (0.3, 0.5, 2.7, 6.3,
10.3) ps. The spline coefficients are determined from the fit to data where
the ${B}^{0}$ lifetime is fixed to its known value [18]. The same parameters
are assigned to both ${{K}^{-}}{{\pi}^{+}}$,
${{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ final-state samples, while
different values are used for Run 1 and Run 2 data, as suggested by simulation
studies. The mass difference $\Delta m_{d}$ is fixed to its known value [18].

The average mistag values for OS and SS tags are fixed to the values
calculated in each corresponding signal sample. All the tagging parameters, as
reported in Table 1, are introduced in the fit through Gaussian constraints,
in order to account for their associated uncertainties. The tagging
efficiencies are free to vary in the fit. The OS and SS taggers are combined
in the fit. The production asymmetries are also constrained in the fit to the
values measured in the flavour-tagging control channels. The result of the
production asymmetry in Run 1 is in agreement with the dedicated LHCb
measurement [44], when considering the kinematics of the signal data sample.

The $C\\!P$ observables resulting from the decay-time fit are

$\begin{split}S_{{{D}^{*}}{D}}&=-0.861\pm 0.077,\\\ \Delta
S_{{{D}^{*}}{D}}&=\phantom{+}0.019\pm 0.075,\\\ C_{{{D}^{*}}{D}}&=-0.059\pm
0.092,\\\ \Delta C_{{{D}^{*}}{D}}&=-0.031\pm 0.092.\\\ \end{split}$

After subtracting the detection asymmetries to each of the four ${\cal A}_{\rm
raw}$ values the weighted mean is calculated as ${\cal
A}_{{{D}^{*}}{D}}=0.008\pm 0.014$. The quoted uncertainties include
contributions due to the size of the samples and due to the external
parameters constrained in the fit. The correlations among the $C\\!P$
parameters are

$\centering\begin{array}[]{c|c c c c}&S_{{{D}^{*}}{D}}&\Delta
S_{{{D}^{*}}{D}}&C_{{{D}^{*}}{D}}&\Delta C_{{{D}^{*}}{D}}\\\ \hline\cr
S_{{{D}^{*}}{D}}&1&0.07&0.44&0.05\\\ \Delta S_{{{D}^{*}}{D}}&&1&0.04&0.46\\\
C_{{{D}^{*}}{D}}&&&1&0.04\\\ \Delta C_{{{D}^{*}}{D}}&&&&1\\\
\end{array}\,.\@add@centering$

The four ${\cal A}_{\rm raw}$ parameters are almost uncorrelated among each
other, with the largest correlation coefficient being $10^{-4}$. Their
correlation with all the other fit parameters is also small, between 0.1% and
1%. The spline parameters have large correlations among them but have
correlations between 0.1% and 1% with the $C\\!P$ parameters.

Figure 2 shows the decay-time distribution of the full
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ data sample, where the fitted
PDF is overlaid. For illustration, Fig. 3 shows the asymmetry between ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ and ${B}^{0}$ signal yields as a
function of the decay time, separately for ${D}^{*+}$ ${D}^{-}$ and ${D}^{*-}$
${D}^{+}$ final states, with the corresponding fit functions overlaid.

Figure 2: Decay-time distribution of the
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ signal candidates, summed
over all data samples, where the background contribution is subtracted by
means of the sPlot technique. The projection of the PDF is represented by the
full line. Figure 3:  Asymmetry between ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ and ${B}^{0}$ signal yields as a
function of the decay time, for (blue full dot) ${D}^{*+}$ ${D}^{-}$ and (red
empty dot) ${D}^{*-}$ ${D}^{+}$ signal candidates with a non null tagging
decision. The background contribution is subtracted by means of the sPlot
technique. The corresponding projections of the PDF are represented by the
blue dashed (red continuous) line. The ${B}^{0}$ flavour is determined by the
combination of all flavour-tagging algorithms.

##  8 Systematic uncertainties

Several cross checks of the analysis and possible sources of systematic
uncertainties of the results are considered in the following and summarised in
Table 4. As a first validation, the decay-time fit is performed on a simulated
sample of signal decays corresponding to 36 times the data sample size. The
resulting $C\\!P$ parameters are compatible with the generation values within
less than two standard deviations. In order to test if the likelihood estimate
for the $C\\!P$ parameter values are accurate, the same mass and decay-time
models as used for data are fitted to 2000 pseudoexperiments. For each
pseudoexperiment, four samples are generated corresponding to the four data
samples. In each sample, the input parameters for the signal component are the
same as in the fit to data, except for the production and tagging asymmetries,
which are set to zero for convenience. The $C\\!P$ parameters, the tagging
efficiencies and the coefficients of the splines describing the time
acceptance are taken from the best fit result to data. All the parameters that
are constrained in the fit are set to values randomly generated according to
the constraints applied. Each background contribution is generated with a
specific time dependence, which in some cases accounts for $C\\!P$ violation.
No bias is found for the $C\\!P$ parameters nor for the raw asymmetries ${\cal
A}_{\rm raw}$. The uncertainty on the mean value of the bias is taken as
systematic uncertainty on the parameter.

Table 4: Summary of the systematic uncertainties. The total systematic uncertainties are computed as quadratic sum of individual contributions.  Source |  ${\Delta C_{{{D}^{*}}{D}}}$ |  ${C_{{{D}^{*}}{D}}}$ |  ${\Delta S_{{{D}^{*}}{D}}}$ |  ${S_{{{D}^{*}}{D}}}$
---|---|---|---|---
Fit bias |  0.002 |  0.002 |  0.002 |  0.002
Mass model |  0.006 |  0.014 |  0.003 |  0.011
${\Delta m_{{d}}},\tau_{d},{\Delta\Gamma_{{d}}}$ |  0.001 |  0.003 |  0.001 |  0.001
Decay-time resolution |  $<$0.001 |  $<$0.001 |  $<$0.001 |  $<$0.001
Decay-time acceptance |  $<$0.001 |  $<$0.001 |  $<$0.001 |  $<$0.001
Flavour tagging |  0.015 |  0.014 |  0.012 |  0.015
Total syst. uncertainty |  ${0.016}$ |  ${0.020}$ |  ${0.012}$ |  ${0.019}$
Source |  ${{\cal A}_{\rm raw}^{K\pi\pi\pi,\text{Run1}}}$ |  ${{\cal A}_{\rm raw}^{K\pi\pi\pi,\text{Run2}}}$ |  ${{\cal A}_{\rm raw}^{K\pi,\text{Run1}}}$ |  ${{\cal A}_{\rm raw}^{K\pi,\text{Run2}}}$
Fit bias |  0.0013 |  0.0007 |  0.0008 |  0.0004
Mass model |  0.0025 |  0.0024 |  0.0021 |  0.0016
${\Delta m_{{d}}},\tau_{d},{\Delta\Gamma_{{d}}}$ |  0.0003 |  0.0002 |  0.0002 |  0.0001
Decay-time resolution |  0.0002 |  0.0001 |  0.0001 |  0.0001
Decay-time acceptance |  0.0003 |  0.0001 |  0.0002 |  0.0001
Flavour tagging |  0.0001 |  0.0001 |  0.0001 |  0.0001
Total syst. uncertainty |  0.0028 |  0.0025 |  0.0023 |  0.0016

In order to cross-check the statistical uncertainty obtained from the fit to
data, a bootstrapping procedure is used [45]. In this frequentist model-
independent approach a new data sample is generated by drawing candidates from
the nominal data sample until the number of candidates matches that of the
original one (the same event can be drawn multiple times). The nominal fit to
the decay time is executed, the fit result is stored and the distribution of
the residuals with respect to the starting values of the parameters is
analysed. Given that the standard deviation of the Gaussian fits of the
residuals agree with the mean values of the uncertainty from the fit it can be
assumed that the uncertainty of the nominal fit is accurate.

Cross checks on the stability of the result are performed by dividing the data
sample into categories, according to the ${D}^{0}$ final state, the tagging
algorithm (OS and SS) and the magnet polarity. No evidence of bias is found,
as all variations of the fit parameters obtained in the splittings are smaller
than two standard deviations.

Specific studies are performed to estimate the effect of an inaccurate
determination of the mass model, the decay-time model, the flavour-tagging
performance and variations of the input parameters to the decay-time fit. Four
systematic effects on the mass model, described below, are considered and the
sum in quadrature of each systematic uncertainty is taken as uncertainty on
the mass model reported in Table 4. The signal model is changed to two (three)
Crystal Ball functions for the mass distribution of ${{K}^{-}}{{\pi}^{+}}$
(${{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{+}}$) final state or to two Crystal
Ball functions for both final states. The model for the combinatorial
background is changed to a second-order polynomial. These systematic
uncertainties are evaluated using pseudoexperiments, with a procedure that is
used also in the rest of the paper for other sources of uncertainty. In each
pseudoexperiment, the mass fit to each of the four samples is performed with
the nominal model and then repeated with the alternative model. Results of the
subsequent decay-time fit are compared to those obtained with the nominal fit
and the distribution of their difference is built. The systematic uncertainty
is defined as the sum in quadrature of the average and root mean square of the
distribution.

Possible differences between data and simulation in the mass resolution are
considered. The mass fit to data is repeated with the parameters describing
the mass resolution of the signal and of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ background contributions
(widths of the Crystal Ball functions, fixed to the values determined from
simulation in the nominal model) multiplied by a common, free scaling factor.
The decay-time fit to data is repeated and the variation of the $C\\!P$
parameters is taken as systematic uncertainty.

A possible contribution to the mass distribution due to background from
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{D}^{*+}}$ decays is also
considered. The same shape and parameters for the mass PDF as for the
${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ component is taken, but the
mean is shifted by the difference of the known values of the ${B}^{0}_{s}$ and
${B}^{0}$ mesons [18]. No significant yield for this decay mode as well as no
significant variation in the signal yield is found, therefore no systematic
uncertainty is associated to this background.

To assess the systematic uncertainty on the decay-time efficiency description,
pseudoexperiments are generated with alternative positions of the spline
knots, (0.3, 1.3, 2.2, 6.3, 10.3) ps, chosen using simulation to provide a
good alternative fit. The systematic uncertainty is found to be negligible, as
expected from the small correlation between the decay-time efficiency and the
$C\\!P$ parameters.

The validity of using the same decay-time efficiency function for the two
${D}^{0}$ final state samples is tested by producing 2500 data sets with the
bootstrap method and fitting each sample once with the nominal PDF and once
with different acceptances for the two final states. Due to the limited amount
of data candidates, only 32% of the fits with separate acceptances converge
with a good covariance matrix quality, however the systematic uncertainty
evaluated from those fits is found to be negligible. The systematic
uncertainty on the $C\\!P$ parameters due to variation of the decay-time
resolution, assessed with pseudoexperiments, is found to negligible.

The calibration parameters of flavour tagging are constrained in the decay-
time fit within their statistical and systematic uncertainties, therefore
their variation is included in the statistical uncertainty of the $C\\!P$
parameters. Two additional sources of uncertainties related to the tagging
parameters, as mentioned in Sec. 5, are considered. The first test concerns a
small deviation from a perfect linear calibration of the mistag probability of
the SS tagger observed in signal simulated data. The second test accounts for
the effect of a slight dependence of the OS tagger mistag probability on the
${B}^{0}$ decay time. Pseudoexperiments are used to evaluate the variations of
the $C\\!P$ parameters with respect to the unbiased case. The systematic
uncertainties corresponding to the two tests are summed in quadrature and
reported in Table 4.

The systematic uncertainty related to the decay-time fit input parameters
($\tau_{d}$, $\Delta m_{{d}}$ and $\Delta\Gamma_{{d}}$) is determined by
varying each parameter according to its uncertainty [3]. A test is done to
check the impact on the $C\\!P$ parameters of the assumption of no $C\\!P$
violation in the ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ background.
Pseudoexperiments with a charge asymmetry of 10% included at generation for
the ${{B}^{0}}\\!\rightarrow{{D}^{+}_{s}}{{D}^{*-}}$ component are studied. No
bias is found on the $C\\!P$ parameters, as a consequence no systematic
uncertainty is assigned due to this source.

The systematic uncertainty on ${\cal A}_{{{D}^{*}}{D}}$ is evaluated as the
weighted average of the quadratic sum of the uncertainties on the detection
asymmetry, from Table 3 and the uncertainties on the raw asymmetries, from
Table 4.

##  9 Results and conclusion

A flavour-tagged decay-time-dependent analysis of
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays is performed using
$pp$ collision data collected by the LHCb experiment between 2011 and 2018,
corresponding to an integrated luminosity of about 9$\text{\,fb}^{-1}$. The
${D}^{-}$ meson is reconstructed as ${K}^{+}$ ${\pi}^{-}$ ${\pi}^{-}$, while
the ${D}^{*+}$ meson is reconstructed as
${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$, where the ${{D}^{0}}$ meson
final states ${K}^{-}$ ${\pi}^{+}$ and ${K}^{-}$ ${\pi}^{+}$ ${\pi}^{+}$
${\pi}^{-}$ are considered. In total, about 6,160 signal decays are selected.
Opposite-side and same-side tagging algorithms are used to determine the
flavour of the ${B}^{0}$ mesons at production, with a total tagging power of
5.6 to 7.1%. The following $C\\!P$ parameters are measured

$\displaystyle S_{{{D}^{*}}{D}}$
$\displaystyle={-0.861}\pm{0.077}\,\text{(stat)}\pm{0.019}\,\text{(syst)}\,,$
$\displaystyle\Delta S_{{{D}^{*}}{D}}$
$\displaystyle=\phantom{+}{0.019}\pm{0.075}\,\text{(stat)}\pm{0.012}\,\text{(syst)}\,,$
$\displaystyle C_{{{D}^{*}}{D}}$
$\displaystyle={-0.059}\pm{0.092}\,\text{(stat)}\pm{0.020}\,\text{(syst)}\,,$
$\displaystyle\Delta C_{{{D}^{*}}{D}}$
$\displaystyle={-0.031}\pm{0.092}\,\text{(stat)}\pm{0.016}\,\text{(syst)}\,,$
$\displaystyle{\cal A}_{{{D}^{*}}{D}}$
$\displaystyle=\phantom{+}{0.008}\pm{0.014}\,\text{(stat)}\pm
0.006\,\text{(syst)}\,.$

The largest statistical correlations are found between the $S_{{{D}^{*}}{D}}$
and $C_{{{D}^{*}}{D}}$ parameters and the $\Delta S_{{{D}^{*}}{D}}$ and
$\Delta C_{{{D}^{*}}{D}}$. They amount to
$\rho(S_{{{D}^{*}}{D}},C_{{{D}^{*}}{D}})=0.44$ and $\rho(\Delta
S_{{{D}^{*}}{D}},\Delta C_{{{D}^{*}}{D}})=0.46$, respectively.

This measurement using ${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays
excludes the hypothesis of $C\\!P$ conservation at more than 10 standard
deviations, obtained using Wilk’s theorem [46]. This result is the most
precise single measurement of the $C\\!P$ parameters in
${{B}^{0}}\\!\rightarrow{{D}^{*\pm}}{{D}^{\mp}}$ decays and it is compatible
with previous measurements by the Belle [9] and BaBar [8] experiments. The
precision of $\Delta C_{{{D}^{*}}{D}}$ and $C_{{{D}^{*}}{D}}$ parameters is
comparable with that of previous measurements, while for $S_{{{D}^{*}}{D}}$,
$\Delta S_{{{D}^{*}}{D}}$ and ${\cal A}_{{{D}^{*}}{D}}$, this measurement
improve significantly the precision of the current world average [3].

## 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); 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 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).

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R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala79, S. Aiola25, Z. Ajaltouni9, S. Akar64, P.
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G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
L. An21, L. Anderlini21, G. Andreassi48, M. Andreotti20, F. Archilli16, J.
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B. Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28, I. Bediaga1,
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Fontanelli23,h, R. Forty47, V. Franco Lima59, M. Franco Sevilla65, M. Frank47,
C. Frei47, D.A. Friday58, J. Fu25,q, M. Fuehring14, W. Funk47, E. Gabriel57,
A. Gallas Torreira45, D. Galli19,e, S. Gallorini27, S. Gambetta57, Y. Gan3, M.
Gandelman2, P. Gandini25, Y. Gao4, L.M. Garcia Martin46, J. García Pardiñas49,
B. Garcia Plana45, F.A. Garcia Rosales11, J. Garra Tico54, L. Garrido44, D.
Gascon44, C. Gaspar47, D. Gerick16, E. Gersabeck61, M. Gersabeck61, T.
Gershon55, D. Gerstel10, Ph. Ghez8, V. Gibson54, A. Gioventù45, O.G. Girard48,
P. Gironella Gironell44, L. Giubega36, C. Giugliano20, K. Gizdov57, V.V.
Gligorov12, C. Göbel69, D. Golubkov38, A. Golutvin60,76, A. Gomes1,a, P.
Gorbounov38,6, I.V. Gorelov39, C. Gotti24,i, E. Govorkova31, J.P. Grabowski16,
R. Graciani Diaz44, T. Grammatico12, L.A. Granado Cardoso47, E. Graugés44, E.
Graverini48, G. Graziani21, A. Grecu36, R. Greim31, P. Griffith20, L.
Grillo61, L. Gruber47, B.R. Gruberg Cazon62, C. Gu3, E. Gushchin40, A. Guth13,
Yu. Guz43,47, T. Gys47, T. Hadavizadeh62, G. Haefeli48, C. Haen47, S.C.
Haines54, P.M. Hamilton65, Q. Han7, X. Han16, T.H. Hancock62, S. Hansmann-
Menzemer16, N. Harnew62, T. Harrison59, R. Hart31, C. Hasse47, M. Hatch47, J.
He5, M. Hecker60, K. Heijhoff31, K. Heinicke14, A. Heister14, A.M.
Hennequin47, K. Hennessy59, L. Henry46, J. Heuel13, A. Hicheur68, R. Hidalgo
Charman61, D. Hill62, M. Hilton61, P.H. Hopchev48, J. Hu16, W. Hu7, W. Huang5,
W. Hulsbergen31, T. Humair60, R.J. Hunter55, M. Hushchyn77, D. Hutchcroft59,
D. Hynds31, P. Ibis14, M. Idzik34, P. Ilten52, A. Inglessi37, A. Inyakin43, K.
Ivshin37, R. Jacobsson47, S. Jakobsen47, J. Jalocha62, E. Jans31, B.K.
Jashal46, A. Jawahery65, V. Jevtic14, F. Jiang3, M. John62, D. Johnson47, C.R.
Jones54, B. Jost47, N. Jurik62, S. Kandybei50, M. Karacson47, J.M. Kariuki53,
N. Kazeev77, M. Kecke16, F. Keizer54,54, M. Kelsey67, M. Kenzie54, T. Ketel32,
B. Khanji47, A. Kharisova78, K.E. Kim67, T. Kirn13, V.S. Kirsebom48, S.
Klaver22, K. Klimaszewski35, S. Koliiev51, A. Kondybayeva76, A.
Konoplyannikov38, P. Kopciewicz34, R. Kopecna16, P. Koppenburg31, I.
Kostiuk31,51, O. Kot51, S. Kotriakhova37, L. Kravchuk40, R.D. Krawczyk47, M.
Kreps55, F. Kress60, S. Kretzschmar13, P. Krokovny42,x, W. Krupa34, W.
Krzemien35, W. Kucewicz33,l, M. Kucharczyk33, V. Kudryavtsev42,x, H.S.
Kuindersma31, G.J. Kunde66, T. Kvaratskheliya38, D. Lacarrere47, G.
Lafferty61, A. Lai26, D. Lancierini49, J.J. Lane61, G. Lanfranchi22, C.
Langenbruch13, T. Latham55, F. Lazzari28,v, C. Lazzeroni52, R. Le Gac10, R.
Lefèvre9, A. Leflat39, F. Lemaitre47, O. Leroy10, T. Lesiak33, B.
Leverington16, H. Li70, X. Li66, Y. Li6, Z. Li67, X. Liang67, R. Lindner47, V.
Lisovskyi11, G. Liu70, X. Liu3, D. Loh55, A. Loi26, J. Lomba Castro45, I.
Longstaff58, J.H. Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D.
Lucchesi27,o, M. Lucio Martinez31, Y. Luo3, A. Lupato27, E. Luppi20,g, O.
Lupton55, A. Lusiani28,t, X. Lyu5, S. Maccolini19,e, F. Machefert11, F.
Maciuc36, V. Macko48, P. Mackowiak14, S. Maddrell-Mander53, L.R. Madhan
Mohan53, O. Maev37,47, A. Maevskiy77, D. Maisuzenko37, M.W. Majewski34, S.
Malde62, B. Malecki47, A. Malinin75, T. Maltsev42,x, H. Malygina16, G.
Manca26,f, G. Mancinelli10, R. Manera Escalero44, D. Manuzzi19,e, D.
Marangotto25,q, J. Maratas9,w, J.F. Marchand8, U. Marconi19, S. Mariani21, C.
Marin Benito11, M. Marinangeli48, P. Marino48, J. Marks16, P.J. Marshall59, G.
Martellotti30, L. Martinazzoli47, M. Martinelli24,i, D. Martinez Santos45, F.
Martinez Vidal46, A. Massafferri1, M. Materok13, R. Matev47, A. Mathad49, Z.
Mathe47, V. Matiunin38, C. Matteuzzi24, K.R. Mattioli79, A. Mauri49, E.
Maurice11,b, M. McCann60,47, L. Mcconnell17, A. McNab61, R. McNulty17, J.V.
Mead59, B. Meadows64, C. Meaux10, G. Meier14, N. Meinert73, D. Melnychuk35, S.
Meloni24,i, M. Merk31, A. Merli25, M. Mikhasenko47, D.A. Milanes72, E.
Millard55, M.-N. Minard8, O. Mineev38, L. Minzoni20,g, S.E. Mitchell57, B.
Mitreska61, D.S. Mitzel47, A. Mödden14, A. Mogini12, R.D. Moise60, T.
Mombächer14, I.A. Monroy72, S. Monteil9, M. Morandin27, G. Morello22, M.J.
Morello28,t, J. Moron34, A.B. Morris10, A.G. Morris55, R. Mountain67, H. Mu3,
F. Muheim57, M. Mukherjee7, M. Mulder31, D. Müller47, K. Müller49, V.
Müller14, C.H. Murphy62, D. Murray61, P. Muzzetto26, P. Naik53, T. Nakada48,
R. Nandakumar56, A. Nandi62, T. Nanut48, I. Nasteva2, M. Needham57, N.
Neri25,q, S. Neubert16, N. Neufeld47, R. Newcombe60, T.D. Nguyen48, C. Nguyen-
Mau48,n, E.M. Niel11, S. Nieswand13, N. Nikitin39, N.S. Nolte47, C. Nunez79,
A. Oblakowska-Mucha34, V. Obraztsov43, S. Ogilvy58, D.P. O’Hanlon19, R.
Oldeman26,f, C.J.G. Onderwater74, J. D. Osborn79, A. Ossowska33, J.M. Otalora
Goicochea2, T. Ovsiannikova38, P. Owen49, A. Oyanguren46, P.R. Pais48, T.
Pajero28,t, A. Palano18, M. Palutan22, G. Panshin78, A. Papanestis56, M.
Pappagallo57, L.L. Pappalardo20,g, C. Pappenheimer64, W. Parker65, C.
Parkes61, G. Passaleva21,47, A. Pastore18, M. Patel60, C. Patrignani19,e, A.
Pearce47, A. Pellegrino31, M. Pepe Altarelli47, S. Perazzini19, D. Pereima38,
P. Perret9, L. Pescatore48, K. Petridis53, A. Petrolini23,h, A. Petrov75, S.
Petrucci57, M. Petruzzo25,q, B. Pietrzyk8, G. Pietrzyk48, M. Pikies33, M.
Pili62, D. Pinci30, J. Pinzino47, F. Pisani47, A. Piucci16, V. Placinta36, S.
Playfer57, J. Plews52, M. Plo Casasus45, F. Polci12, M. Poli Lener22, M.
Poliakova67, A. Poluektov10, N. Polukhina76,c, I. Polyakov67, E. Polycarpo2,
G.J. Pomery53, S. Ponce47, A. Popov43, D. Popov52, S. Poslavskii43, K.
Prasanth33, L. Promberger47, C. Prouve45, V. Pugatch51, A. Puig Navarro49, H.
Pullen62, G. Punzi28,p, W. Qian5, J. Qin5, R. Quagliani12, B. Quintana9, N.V.
Raab17, R.I. Rabadan Trejo10, B. Rachwal34, J.H. Rademacker53, M. Rama28, M.
Ramos Pernas45, M.S. Rangel2, F. Ratnikov41,77, G. Raven32, M. Reboud8, F.
Redi48, F. Reiss12, C. Remon Alepuz46, Z. Ren3, V. Renaudin62, S. Ricciardi56,
S. Richards53, K. Rinnert59, P. Robbe11, A. Robert12, A.B. Rodrigues48, E.
Rodrigues64, J.A. Rodriguez Lopez72, M. Roehrken47, S. Roiser47, A.
Rollings62, V. Romanovskiy43, M. Romero Lamas45, A. Romero Vidal45, J.D.
Roth79, M. Rotondo22, M.S. Rudolph67, T. Ruf47, J. Ruiz Vidal46, J. Ryzka34,
J.J. Saborido Silva45, N. Sagidova37, B. Saitta26,f, C. Sanchez Gras31, C.
Sanchez Mayordomo46, B. Sanmartin Sedes45, R. Santacesaria30, C. Santamarina
Rios45, M. Santimaria22, E. Santovetti29,j, G. Sarpis61, A. Sarti30, C.
Satriano30,s, A. Satta29, M. Saur5, D. Savrina38,39, L.G. Scantlebury Smead62,
S. Schael13, M. Schellenberg14, M. Schiller58, H. Schindler47, M.
Schmelling15, T. Schmelzer14, B. Schmidt47, O. Schneider48, A. Schopper47,
H.F. Schreiner64, M. Schubiger31, S. Schulte48, M.H. Schune11, R. Schwemmer47,
B. Sciascia22, A. Sciubba30,k, S. Sellam68, A. Semennikov38, A. Sergi52,47, N.
Serra49, J. Serrano10, L. Sestini27, A. Seuthe14, P. Seyfert47, D.M.
Shangase79, M. Shapkin43, T. Shears59, L. Shekhtman42,x, V. Shevchenko75,76,
E. Shmanin76, J.D. Shupperd67, B.G. Siddi20, R. Silva Coutinho49, L. Silva de
Oliveira2, G. Simi27,o, S. Simone18,d, I. Skiba20, N. Skidmore16, T.
Skwarnicki67, M.W. Slater52, J.G. Smeaton54, A. Smetkina38, E. Smith13, I.T.
Smith57, M. Smith60, A. Snoch31, M. Soares19, L. Soares Lavra1, M.D.
Sokoloff64, F.J.P. Soler58, B. Souza De Paula2, B. Spaan14, E. Spadaro
Norella25,q, P. Spradlin58, F. Stagni47, M. Stahl64, S. Stahl47, P. Stefko48,
S. Stefkova60, O. Steinkamp49, S. Stemmle16, O. Stenyakin43, M. Stepanova37,
H. Stevens14, S. Stone67, S. Stracka28, M.E. Stramaglia48, M. Straticiuc36, S.
Strokov78, J. Sun3, L. Sun71, Y. Sun65, P. Svihra61, K. Swientek34, A.
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Zhukov13,39, J.B. Zonneveld57, S. Zucchelli19,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
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. 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
11LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27INFN Sezione di Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
70South China Normal University, Guangzhou, China, associated to 3
71School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
72Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
73Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
74Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
75National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
76National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
77National Research University Higher School of Economics, Moscow, Russia,
associated to 41
78National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
79University of Michigan, Ann Arbor, United States, associated to 67

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
mDS4DS, 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
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
yINFN Sezione 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
abUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras
†Deceased