EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-111 LHCb-PAPER-2019-021 August 13, 2019

Measurement of $C\\!P$ observables in the process $B^{0}\rightarrow DK^{*0}$
with

two- and four-body $D$ decays

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

Measurements of $C\\!P$ observables in ${{B}^{0}}\rightarrow D{{K}^{*0}}$
decays are presented, where $D$ represents a superposition of ${{D}^{0}}$ and
${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ states. The $D$ meson
is reconstructed in the two-body final states ${{K}^{+}}{{\pi}^{-}}$,
${{\pi}^{+}}{{K}^{-}}$, ${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$,
and, for the first time, in the four-body final states
${{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$,
${{\pi}^{+}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$. The analysis uses a sample of
neutral $B$ mesons produced in proton-proton collisions, corresponding to an
integrated luminosity of 1.0, 2.0 and 1.8$\text{\,fb}^{-1}$ collected with the
LHCb detector at centre-of-mass energies of $\sqrt{s}=$ 7, 8 and 13 TeV,
respectively. First observations of the decays ${{B}^{0}}\rightarrow
D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}}$ and ${{B}^{0}}\rightarrow
D({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$ are obtained. The
measured observables are interpreted in terms of the ${C\\!P}$-violating weak
phase $\gamma$.

Published in JHEP 08 (2019) 041

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

##  1 Introduction

In the Standard Model, $C\\!P$ violation is described by the irreducible
complex phase of the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix[1,
2]. This matrix is unitary, leading to the condition
${V_{{u}{d}}}{V_{{u}{b}}}^{*}+{V_{{c}{d}}}{V_{{c}{b}}}^{*}+{V_{{t}{d}}}{V_{{t}{b}}}^{*}=0$,
where $V_{ij}$ is the CKM matrix element relating quark $i$ to quark $j$. This
relation can be represented as a triangle in the complex plane, with angles
$\alpha$, $\beta$ and $\gamma$. Improving knowledge of $\gamma$ is one of the
most important goals in flavour physics. This angle is defined as
$\gamma\equiv\arg{(-{{V_{{u}{d}}}{V_{{u}{b}}}^{*}}/{{V_{{c}{d}}}{V_{{c}{b}}}^{*}})}$,
which is equal to
$\arg{(-{{V_{{u}{s}}}{V_{{u}{b}}}^{*}}/{{V_{{c}{s}}}{V_{{c}{b}}}^{*}})}$ up to
$\mathcal{O}(\lambda^{4})\sim 10^{-3}$[3]. This can be measured through the
interference of $b\rightarrow c$ and $b\rightarrow u$ transition amplitudes in
tree-level $b$-hadron decays.111Except where stated otherwise, the inclusion
of charge-conjugate processes is implied throughout this paper. Such a
measurement provides a Standard-Model benchmark against which observables
determined in loop-mediated processes, expected to be more susceptible to the
influence of physics beyond the Standard Model, can be compared.

Measurements from the LHCb experiment yield $\gamma=(74.0\,^{+5.0}_{-5.8})$°
[4, 5], which is the most precise determination of $\gamma$ from a single
experiment. The precision is dominated by measurements exploiting the decay
${{{B}^{+}}}\rightarrow D{{K}^{+}}$, where $D$ indicates a superposition of
${{D}^{0}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ mesons
reconstructed in a final state common to both. In order to test internal
consistency, and to improve overall sensitivity, it is important to complement
these measurements with those based on other decay modes. One important
example is ${{B}^{0}}\rightarrow D{{K}^{*0}}$ [6], where ${{K}^{*0}}$ is the
${{K}^{*}}(892)^{0}$ meson and is reconstructed in its decay to
${{K}^{+}}{{\pi}^{-}}$. This process involves the interference of
${{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{*0}}$ decays, which proceed
via a $b\rightarrow c$ quark transition, and
${{B}^{0}}\rightarrow{{D}^{0}}{{K}^{*0}}$decays, which involve a $b\rightarrow
u$ quark transition and are therefore suppressed relative to
${{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{*0}}$. Feynman diagrams of
these decays are shown in Fig. 1. Both transitions are colour-suppressed, in
contrast to the charged $B$-meson case where only the $b\rightarrow u$
transition is colour-suppressed. This leads to a greater suppression of the
overall decay rates, but with the benefit of enhanced interference effects
with respect to ${{{B}^{+}}}\rightarrow D{{K}^{+}}$ decays. The ratio
$r_{B}^{D\\!{{K}^{*0}}}$ between the magnitudes of the suppressed and favoured
${{B}^{0}}$ decay amplitudes is expected to be around three times larger than
the corresponding parameter in ${{{B}^{+}}}\rightarrow D{{K}^{+}}$ decays.

Figure 1: Feynman diagrams of (left) ${{B}^{0}}\rightarrow{{D}^{0}}{{K}^{*0}}$
and (right) ${{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{*0}}$.

The LHCb collaboration has performed studies of ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ decays using data corresponding to an integrated luminosity of
3.0$\text{\,fb}^{-1}$, reconstructing the $D$ meson in the two-body final
states ${{K}^{\pm}}\pi^{\mp}$, ${{K}^{+}}{{K}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}$ [7], and also the self-conjugate modes
${{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{K}^{0}_{\mathrm{S}}}{{K}^{+}}{{K}^{-}}$ [8, 9]. In addition, the two-body
$D$ decay modes ${{K}^{+}}{{\pi}^{-}}$, ${{K}^{+}}{{K}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}$ have previously been exploited in an amplitude
analysis of ${{B}^{0}}\rightarrow D{{K}^{+}}{{\pi}^{-}}$ decays, including
${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays [10].

In this paper, results are presented for a study of ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ decays performed on a data set corresponding to
3.0$\text{\,fb}^{-1}$ of integrated luminosity collected at centre-of-mass
energies of 7 and 8 TeV during Run 1 of the LHC, and 1.8$\text{\,fb}^{-1}$
collected at 13 TeV during Run 2 in 2015 and 2016. Observables sensitive to
$\gamma$ are measured for the following final states of the $D$-meson decay:
${{K}^{\pm}}{{\pi}^{\mp}}$, ${{K}^{+}}{{K}^{-}}$, ${{\pi}^{+}}{{\pi}^{-}}$,
${{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$. The study of the two-body
modes benefits from several improvements with respect to Ref. [7], as well as
from the larger data set. The four-body modes are analysed for the first time
in this decay chain. The measurements involving
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ are based on Run 2
data alone, as the central processing that performs the first step of the
selection did not include a suitable selection for this mode in Run 1.

The paper is organised as follows: Section 2 presents the observables to be
measured, and their relationships to the physics parameters of interest; Sect.
3 discusses those aspects of the detector, trigger and simulation that are
relevant for the measurement; Sects. 4, 5 and 6 describe the candidate
selection, the fit of the mass spectra and the assignment of systematic
uncertainties, respectively; the results, and their interpretation, are given
in Sect. 7; and conclusions are presented in Sect. 8.

##  2 Analysis strategy

This analysis exploits the interference between ${{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{*0}}$ and
${{B}^{0}}\rightarrow{{D}^{0}}{{K}^{*0}}$ decays, with the ${{D}^{0}}$ and
${{\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ mesons reconstructed in
a common final state. The partial widths of these decays are used to construct
observables, which have a dependence on $\gamma$ and the following parameters:
the ratio $r_{B}^{D\\!{{K}^{*0}}}$ between the magnitudes of the suppressed
and favoured ${{B}^{0}}$ decay amplitudes; the $C\\!P$-conserving strong-phase
difference $\delta_{B}^{D\\!{{K}^{*0}}}$ between the amplitudes; and a
coherence factor $\kappa$, which accounts for other amplitudes that may
contribute to the ${{B}^{0}}\rightarrow D{{K}^{+}}{{\pi}^{-}}$ final state in
addition to the two diagrams responsible for the ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ signal process. Detailed definitions of these parameters may be
found in Ref. [7]. An amplitude analysis of ${{B}^{0}}\rightarrow
D{{K}^{+}}{{\pi}^{-}}$ decays has determined the coherence factor to be
$\kappa=0.958\,^{+0.005}_{-0.046}$ for the ${{K}^{*0}}$ selection criteria
used in this measurement (see Sect. 4) [10], which indicates an almost pure
$D{{K}^{*0}}$ sample.

Reconstructing the charmed meson through a decay to a $C\\!P$ eigenstate, such
as $D\rightarrow{{K}^{+}}{{K}^{-}}$ or $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$,
brings information on $\gamma$ through a strategy first proposed by Gronau,
London and Wyler (GLW) [11, 12]. The asymmetry

$\mathcal{A}_{{C\\!P}}\equiv\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D_{{C\\!P}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})-\Gamma({{B}^{0}}\rightarrow
D_{{C\\!P}}{{K}^{*0}})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D_{{C\\!P}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D_{{C\\!P}}{{K}^{*0}})}{\rm,}$ (1)

where $\Gamma$ represents a partial decay width, is measured for both modes,
yielding $\mathcal{A}^{KK}_{{C\\!P}}$ and $\mathcal{A}^{\pi\pi}_{{C\\!P}}$,
which are expected to be equal when the small $C\\!P$-violating effects
observed in the $D$-meson decay [13] are neglected; this assumption applies
for the remainder of the discussion. The asymmetry is related to the
underlying parameters through

$\mathcal{A}_{{C\\!P}}=\frac{2\kappa
r^{D\\!{{K}^{*0}}}_{B}\sin\delta^{D\\!{{K}^{*0}}}_{B}\sin\gamma}{\mathcal{R}_{{C\\!P}}},$
(2)

where $\mathcal{R}_{{C\\!P}}$ is the charge-averaged rate of decays involving
a $D$ meson decaying to a $C\\!P$ eigenstate, defined as

${\mathcal{R}}_{{C\\!P}}\equiv 2\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D_{{C\\!P}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D_{{C\\!P}}{{K}^{*0}})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow{{D}^{0}}{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{*0}})}.$ (3)

This is related to $\gamma$ and the auxiliary parameters through

$\mathcal{R}_{{C\\!P}}=1+{(r^{D\\!{{K}^{*0}}}_{B})}^{2}+2\kappa
r^{D\\!{{K}^{*0}}}_{B}\cos\delta^{D\\!{{K}^{*0}}}_{B}\cos\gamma.$ (4)

Experimentally it is convenient to access $\mathcal{R}_{{C\\!P}}$ by noting
that it is closely approximated by

$\mathcal{R}_{{C\\!P}}^{hh}\equiv\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D(h^{+}h^{-}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D(h^{+}h^{-}){{K}^{*0}})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{K}^{-}}{{\pi}^{+}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D({{K}^{+}}{{\pi}^{-}}){{K}^{*0}})}\times\frac{{\mathcal{B}}(D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}})}{{\mathcal{B}}(D^{0}\rightarrow
h^{+}h^{-})}\mathrm{,}$ (5)

where the branching fractions ${\mathcal{B}}$ are known[14].

As proposed in Refs. [15, 16], multibody $D$-meson decays to self-conjugate
final states may be used in a quasi-GLW analysis provided their fractional
$C\\!P$ content is known. Hence the observables
${\mathcal{A}}^{4\pi}_{{C\\!P}}$ and ${\mathcal{R}}^{4\pi}_{{C\\!P}}$ are
measured, which are analogous to the two-body observables
$\mathcal{A}_{{C\\!P}}$ and $\mathcal{R}^{hh}_{{C\\!P}}$, but for the decay
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$. These new
observables can be interpreted through equivalent expressions to Eqs. 2 and 4
in which the interference terms acquire a factor of $(2F_{+}^{4\pi}-1)$, where
$F_{+}^{4\pi}$ is the fractional $C\\!P$-even content of the decay, measured
to be $0.769\pm 0.023$ from quantum-correlated $D$-meson decays [17].

The decays $D\rightarrow{{K}^{\pm}}{{\pi}^{\mp}}$ are exploited in a method
proposed by Atwood, Dunietz and Soni (ADS) [18, 19]. Considering the decays
${{K}^{*0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$,
four categories are defined: two decays with the same charge of the final-
state kaons, which are favoured and labelled $K\pi$, and two decays with the
opposite charge of the final-state kaons, which are suppressed and labelled
$\pi K$. The interference effects, and hence sensitivity to $\gamma$, are
expected to be substantial for the suppressed modes, and smaller for the
favoured modes.

The partial-rate asymmetry of the suppressed ADS decays is given by

${\mathcal{A}}^{\pi K}_{\rm ADS}\equiv\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{\pi}^{-}}{{K}^{+}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})-\Gamma({{B}^{0}}\rightarrow
D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{\pi}^{-}}{{K}^{+}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}})},$ (6)

and the charge-averaged rate with respect to the favoured modes by

${\mathcal{R}}^{\pi K}_{\rm ADS}\equiv\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{\pi}^{-}}{{K}^{+}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{K}^{-}}{{\pi}^{+}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D({{K}^{+}}{{\pi}^{-}}){{K}^{*0}})},$ (7)

which have the following dependence on $\gamma$ and the auxiliary parameters:

${\mathcal{A}}^{\pi K}_{\rm ADS}=\frac{2\kappa
r_{B}^{D\\!{{K}^{*0}}}r_{D}^{K\pi}\sin(\delta_{B}^{D\\!{{K}^{*0}}}+\delta_{D}^{K\pi})\sin\gamma}{(r_{B}^{D\\!{{K}^{*0}}})^{2}+(r_{D}^{K\pi})^{2}+2\kappa
r_{B}^{D\\!{{K}^{*0}}}r_{D}^{K\pi}\cos(\delta_{B}^{D\\!{{K}^{*0}}}+\delta_{D}^{K\pi})\cos\gamma},$
(8) ${\mathcal{R}}^{\pi K}_{\rm
ADS}=\frac{(r_{B}^{D\\!{{K}^{*0}}})^{2}+(r_{D}^{K\pi})^{2}+2\kappa
r_{B}^{D\\!{{K}^{*0}}}r_{D}^{K\pi}\cos(\delta_{B}^{D\\!{{K}^{*0}}}+\delta_{D}^{K\pi})\cos\gamma}{1+(r_{B}^{D\\!{{K}^{*0}}}r_{D}^{K\pi})^{2}+2\kappa
r_{B}^{D\\!{{K}^{*0}}}r_{D}^{K\pi}\cos(\delta_{B}^{D\\!{{K}^{*0}}}+\delta_{D}^{K\pi})\cos\gamma}.$
(9)

Here, $r_{D}^{K\pi}=0.059\pm 0.001$ is the ratio between the doubly Cabibbo-
suppressed and Cabibbo-favoured decay amplitudes of the neutral charm meson,
and $\delta_{D}^{K\pi}=(192.1\,^{+\,8.6}_{-10.2})^{\circ}$ is a strong-phase
difference between the amplitudes [20].222All expressions and charm strong-
phase values are given in the convention ${C\\!P}|{{D}^{0}}\rangle=|{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rangle$. This implies a 180°
offset with respect to the values quoted in Ref. [20], which are defined with
a different sign convention.

The quantities measured experimentally are the ratios

$\mathcal{R}_{+}^{\pi K}=\frac{\Gamma({{B}^{0}}\rightarrow
D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}})}{\Gamma({{B}^{0}}\rightarrow
D({{K}^{+}}{{\pi}^{-}}){{K}^{*0}})}$ (10)

and

$\mathcal{R}_{-}^{\pi K}=\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{\pi}^{-}}{{K}^{+}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{K}^{-}}{{\pi}^{+}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})}.$ (11)

The relationships

${\mathcal{A}}^{\pi K}_{\rm ADS}\simeq(\mathcal{R}^{\pi
K}_{-}-\mathcal{R}^{\pi K}_{+})/(\mathcal{R}^{\pi K}_{-}+\mathcal{R}^{\pi
K}_{+})$ (12)

and

${\mathcal{R}}^{\pi K}_{\rm ADS}\simeq(\mathcal{R}^{\pi
K}_{+}+\mathcal{R}^{\pi K}_{-})/2$ (13)

allow the ADS observables to be recovered, where the approximate equalities
are exact in the absence of $C\\!P$ asymmetry in the favoured modes.

The ADS method can be extended in an analogous way to the four-body mode
$D\rightarrow K^{\pm}\pi^{\mp}{{\pi}^{+}}{{\pi}^{-}}$, with observables
$\mathcal{R}_{\pm}^{\pi K\pi\pi}$. In interpreting the results it is necessary
to account for the variation of amplitude across the phase space of the
$D$-meson decay. In the equivalent relations for Eqs. 8 and 9 the amplitude
ratio and charm strong-phase difference become $r_{D}^{K3\pi}$ and
$\delta_{D}^{K3\pi}$, respectively, which are quantities averaged over phase
space, and the interference terms [21] are multiplied by a coherence factor
$\kappa_{D}^{K3\pi}$. These parameters have been measured in studies of charm
mixing and quantum-correlated $D$-meson decays: $r_{D}^{K3\pi}=0.0549\pm
0.006$, $\delta_{D}^{K3\pi}=(128\,^{+28}_{-17})^{\circ}$ and
$\kappa_{D}^{K3\pi}=0.43\,^{+0.17}_{-0.13}$ [22, 23].

In the favoured ADS modes the asymmetries

$\mathcal{A}^{K\pi(\pi\pi)}_{\rm ADS}=\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{K}^{-}}{{\pi}^{+}}({{\pi}^{+}}{{\pi}^{-}})){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})-\Gamma({{B}^{0}}\rightarrow
D({{K}^{+}}{{\pi}^{-}}({{\pi}^{+}}{{\pi}^{-}})){{K}^{*0}})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
D({{K}^{-}}{{\pi}^{+}}({{\pi}^{+}}{{\pi}^{-}})){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}})+\Gamma({{B}^{0}}\rightarrow
D({{K}^{+}}{{\pi}^{-}}({{\pi}^{+}}{{\pi}^{-}})){{K}^{*0}})}$ (14)

are measured. These modes are expected to exhibit much smaller $C\\!P$
asymmetries than the suppressed decay channels.

Observables associated with the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow D{{K}^{*0}}$ are
also measured. This decay is expected to exhibit negligible $C\\!P$ violation,
but serves as a useful control mode. In this case, for the ADS selection the
final state with opposite-sign kaons constitutes the favoured mode, and so the
analogously defined asymmetries $\mathcal{A}_{s,\rm ADS}^{\pi K(\pi\pi)}$ are
measured. Signal yields are currently too small to permit a study of the
suppressed mode. Finally, the GLW asymmetries $\mathcal{A}^{KK}_{s,{C\\!P}}$,
$\mathcal{A}^{\pi\pi}_{s,{C\\!P}}$ and $\mathcal{A}^{4\pi}_{s,{C\\!P}}$,
defined analogously to Eq. 1, and the ratios $\mathcal{R}^{KK}_{s,{C\\!P}}$,
$\mathcal{R}^{\pi\pi}_{s,{C\\!P}}$ and $\mathcal{R}^{4\pi}_{s,{C\\!P}}$,
defined analogously to Eq. 5, are determined.

##  3 Detector, online selection and simulation

The LHCb detector [24, 25] 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 that varies 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
$\text{\,Ge\kern-1.00006ptV\\!/}c$. 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, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction. The
events considered in the analysis must be triggered at the hardware level when
either one of the final-state tracks of the signal decay deposits enough
energy in the calorimeter system, or when one of the other tracks in the
event, not reconstructed as part of the signal candidate, fulfils any trigger
requirement. At the software stage, it is required that at least one particle
should have high $p_{\mathrm{T}}$ and high $\chi^{2}_{\text{IP}}$, where
$\chi^{2}_{\text{IP}}$ is defined as the difference in the PV fit $\chi^{2}$
with and without the inclusion of that particle. A multivariate algorithm [26]
is used to identify secondary vertices consistent with being a two-, three- or
four-track $b$-hadron decay. The PVs are fitted with and without the $B$
candidate tracks, and the PV that gives the smallest $\chi^{2}_{\text{IP}}$ is
associated with the $B$ candidate.

Simulated events are used to describe the signal mass shapes and compute
efficiencies. In the simulation, $pp$ collisions are generated using Pythia
[27] with a specific LHCb configuration [28]. Decays of hadronic particles are
described by EvtGen [29], in which final-state radiation is generated using
Photos [30]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [31,
*Agostinelli:2002hh] as described in Ref. [33].

##  4 Offline selection

Signal $B$-meson candidates are obtained by combining $D$ and ${{K}^{*0}}$
candidates, and are required to have a $p_{\mathrm{T}}$ greater than
$5\text{\,Ge\kern-1.00006ptV\\!/}c$, a lifetime greater than $0.2\text{\,ps}$,
and a good-quality vertex fit. The $D$ candidate is reconstructed from the
seven different decay modes of interest within a $\pm
25$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ window around the known ${{D}^{0}}$
mass[14], and must have a $p_{\mathrm{T}}$ greater than
$1.8\text{\,Ge\kern-1.00006ptV\\!/}c$. The ${{K}^{*0}}$ candidate is
reconstructed from the final state ${{K}^{+}}{{\pi}^{-}}$, selected within a
$\pm 50$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ window around the known
${{K}^{*0}}$ mass and with a total $p_{\mathrm{T}}$ of at least
$1\text{\,Ge\kern-1.00006ptV\\!/}c$. This mass window is approximately the
width of the ${{K}^{*}}(892)^{0}$ resonance[14]. The helicity angle
$\theta^{*}$, defined as the angle between the ${{K}^{+}}$ momentum in the
${{K}^{*0}}$ rest frame and the ${{K}^{*0}}$ momentum in the ${{B}^{0}}$ rest
frame, is required to satisfy $\lvert\cos(\theta^{*})\rvert>0.4$. This
requirement removes 60% of the combinatorial background with a fake
${{K}^{*0}}$, while retaining 93% of the signal. The $D$ and ${{K}^{*0}}$
candidates are both required to have a good-quality vertex fit, a significant
separation from the PV, and a distance-of-closest-approach between their decay
products of less than $0.5\text{\,mm}$. All charged final-state particles are
required to have a good-quality track fit, $p$ greater than
$1\text{\,Ge\kern-1.00006ptV\\!/}c$, and $p_{\mathrm{T}}$ greater than
$100\text{\,Me\kern-1.00006ptV\\!/}c$. The $B$ decay chain is refitted [34]
with the $D$ mass fixed to its known value and the $B$ meson constrained to
originate from its associated PV.

Gradient Boosted Decision Trees (BDTs)[35] are used to separate signal from
combinatorial background. A shared BDT is employed for the favoured and
suppressed two-body ADS modes, and similarly for the four-body ADS modes.
Three independent BDTs are used to select the ${{K}^{+}}{{K}^{-}}$,
${{\pi}^{+}}{{\pi}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decays. All BDTs are trained with samples of simulated ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ decays as signal and with candidates from the upper $B$ mass
sideband ($5800<m(B)<6000\text{\,Me\kern-1.00006ptV\\!/}c^{2}$) in data as
background. The discriminating variables in the BDT comprise: the $B$ vertex-
fit $\chi^{2}$; the $\chi^{2}_{\text{IP}}$ of the $B$ and $D$ candidates; the
$\chi^{2}_{\text{IP}}$ and $p_{\mathrm{T}}$ of the ${{K}^{*0}}$ products; the
angle between the $B$ momentum vector and the vector between the PV and the
$B$ decay vertex; and the $p_{\mathrm{T}}$ asymmetry between the $B$ candidate
and other tracks from the PV in a cone around the $B$ candidate. The
$p_{\mathrm{T}}$ asymmetry is defined as
$(p_{\mathrm{T}}^{B}-p_{\mathrm{T}}^{\mathrm{cone}})/(p_{\mathrm{T}}^{B}+p_{\mathrm{T}}^{\mathrm{cone}})$,
where $p_{\mathrm{T}}^{B}$ is the transverse momentum of the $B$ candidate and
$p_{\mathrm{T}}^{\mathrm{cone}}$ is the scalar sum of the transverse momenta
of all other tracks in the cone. The radius of the cone is chosen to be 1.5
units in the plane of pseudorapidity and azimuthal angle (expressed in
radians). The $p_{\mathrm{T}}$ asymmetry is a measure of the isolation of the
$B$ candidate. The BDTs applied to $B$ candidates with two-body $D$-meson
decays also use the $p_{\mathrm{T}}$ and $\chi^{2}_{\text{IP}}$ of the $D$
decay products. These variables are not included in the BDTs applied to $B$
candidates with four-body $D$ decays to avoid significant changes to the
phase-space distribution.

Particle-identification (PID) information from the RICH detectors is used to
improve the purity of the different $D$-meson samples. Criteria are chosen
such that no candidate can appear in more than one $D$ decay category. A
stringent PID requirement is applied to the kaon from the ${{K}^{*0}}$
candidate to suppress contamination from ${{B}^{0}}\rightarrow
D{{\pi}^{+}}{{\pi}^{-}}$ decays, with a pion misidentified as a kaon.

It is possible for both the kaon and pion (or one of the two pions, in the
four-body case) from the $D$-meson decay in the favoured mode to be
misidentified, and thus pollute the suppressed sample. To eliminate this
source of contamination, the $D$ invariant mass is reconstructed with the
opposite mass hypothesis for the kaon and pion. Candidates within $\pm
15$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${{D}^{0}}$ mass in
this alternative reconstruction are vetoed. After this veto, a contamination
rate of $\mathcal{O}(0.1\%)$ is expected in the suppressed mode. No veto is
applied to remove ${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays where both the
kaon and pion from the ${{K}^{*0}}$ candidate are misidentified, as this
background is sufficiently suppressed by the PID requirement on the kaon,
leaving a contamination rate of $\mathcal{O}(0.7\%)$ in the suppressed mode.

Additional background can arise from $B^{0}_{(s)}\rightarrow{{D}^{-}}h^{+}$
($h=K,\pi$), ${{D}^{-}_{s}}{{K}^{+}}$ or ${{D}^{+}_{s}}{{\pi}^{-}}$ decays,
with ${D}_{(s)}^{\pm}$ decaying to a three-body combination of kaons and
pions. This contamination is removed by imposing a $\pm
15$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ veto around the known
$D_{(s)}^{\pm}$ mass in the invariant mass of the relevant three tracks. These
vetoes are over 99% efficient at retaining signal candidates.

A background from charmless $B$ decays that peaks at the same invariant mass
as the signal is suppressed by requiring that the flight distance of the $D$
candidate divided by its uncertainty be greater than 3. A further background
from ${{{B}^{+}}}\rightarrow D{{K}^{+}}$ decays that are mistakenly combined
with a random pion from elsewhere in the event contaminates the region in
invariant mass above the signal. This background is removed with a veto of
$\pm 25$$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ around the known ${{{B}^{+}}}$
mass in the invariant mass of the $D$ meson and the kaon from the ${{K}^{*0}}$
candidate.

##  5 Invariant-mass fit

The selected data set comprises two LHC runs and seven $D$-meson decay modes.
The sample is further divided into ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ candidates, based on the charge
of the kaon from the ${{K}^{*0}}$ meson. This gives a total of 26 categories,
as the ${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ channel is not selected
in the Run 1 data. The invariant-mass distributions are fit simultaneously in
these categories with an unbinned extended maximum-likelihood fit. A fit model
is developed comprising several signal and background components, which unless
otherwise stated are modelled using simulated signal and background samples
reconstructed as the signal decay and passing the selection requirements. The
components are:

  1. 1.

Signal ${{B}^{0}}\rightarrow D{{K}^{*0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow D{{K}^{*0}}$
decays, described by Cruijff functions [36] with free means and widths, and
tail parameters fixed from simulation.

  2. 2.

Combinatorial background, described by an exponential function with a free
slope. As the shape is completely free in the fit, no simulation is used to
model this background.

  3. 3.

Partially reconstructed background from
${{B}^{0}}\rightarrow{{D}^{*}}{{K}^{*0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow{{D}^{*}}{{K}^{*0}}$
decays, where ${{D}^{*}}$ represents either a ${{D}^{*0}}$ or ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{*0}}$ meson. The ${{D}^{*}}$ meson
decays via ${{D}^{*}}\rightarrow D{{\pi}^{0}}$ or ${{D}^{*}}\rightarrow
D\gamma$, where the neutral pion or photon is not reconstructed. These
components are described by analytic probability distribution functions
constructed from a parabolic function to describe the decay kinematics that is
convolved with the sum of two Gaussians with a common mean to describe the
detector resolution, as further described in Ref. [37]. All shape parameters
are fixed from simulation. The form of the parabola depends on both the missed
particle and the helicity of the ${{D}^{*}}$ meson, which can be equal to zero
(longitudinal polarisation) or $\pm 1$ (transverse polarisation).

  4. 4.

Partially reconstructed background from ${{{B}^{+}}}\rightarrow
D{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ decays, where the ${{\pi}^{+}}$ meson is not
reconstructed. This background is described by the sum of two Gaussian
functions with separate means and a parabola convolved with the sum of two
Gaussians with a common mean. All shape parameters are fixed from simulation.

  5. 5.

A background from ${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$ decays, with
one of the pions misidentified as a kaon. This background is described by the
sum of two Crystal Ball functions[38] with all shape parameters fixed from
simulation.

The signal and combinatorial background yields are free parameters for each
LHC run. Preliminary studies showed that the ratios of the yields of the
partially reconstructed backgrounds with respect to the signal yields are
compatible within uncertainties between Runs 1 and 2, and so a single value is
used in the fit. The same assumption cannot be made for the misidentified
${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$ background, as the yield of
this background is affected by the $\pi\rightarrow K$ misidentification rate,
which can vary between running periods. The proportion of this background with
respect to the signal is therefore corrected in Run 2 with respect to Run 1.
Studies of simulated signal and background samples determine this correction
factor to be $0.928\pm 0.014$.

The ${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$ background is assumed to
have no $C\\!P$ asymmetry, as the candidates cannot be tagged as coming from a
${{B}^{0}}$ or ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ decay,
and the difference between the ${{\pi}^{+}}$ and ${{\pi}^{-}}$
misidentification rates is found to be negligible in simulated samples.
Misidentified ${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$ decays should
therefore contaminate ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ equally. The ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow{{D}^{*}}{{K}^{*0}}$
background is not expected to exhibit $C\\!P$ violation, so the $C\\!P$
asymmetry is fixed to zero in the GLW modes but is free in the ADS modes. The
yields of the ${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow{{D}^{*}}{{K}^{*0}}$
backgrounds are free parameters in the ADS modes and fixed in the GLW modes
relative to the ADS yields from knowledge of the ${{D}^{0}}$ branching
fractions[14] and relative selection efficiencies determined from simulation.
The ${{B}^{0}}\rightarrow{{D}^{*}}{{K}^{*0}}$ background may exhibit $C\\!P$
violation, so the yields of each $D$ decay channel are free parameters, thus
allowing for a nonzero $C\\!P$ asymmetry. The relative yields and asymmetries
of the ${{{B}^{+}}}\rightarrow D{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ background
are fixed using measurements from Ref. [39].

For both the ${{B}^{0}}\rightarrow{{D}^{*}}{{K}^{*0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow{{D}^{*}}{{K}^{*0}}$
backgrounds, the relative proportion of partially reconstructed
${{D}^{*}}\rightarrow D\gamma$ and ${{D}^{*}}\rightarrow D\pi^{0}$ decays is
fixed by known ${D}^{*0}$ branching fractions[14] and relative selection
efficiency as determined from simulation. The fraction of longitudinal
polarisation is unknown and is therefore a free parameter in the fit.

Figures 2 to 5 show the invariant mass distributions and the fitted shapes for
the various components. Table 1 gives the signal yields for each $D$ final
state. The fit strategy is validated by pseudoexperiments, and is found to be
reliable and unbiased for all free parameters.

The observables introduced in Sect. 2 are determined directly from the fit.
The ratios and asymmetries between the raw yields are corrected for efficiency
differences, and production and detection asymmetries. To obtain the ratios
$\mathcal{R}^{hh}_{{C\\!P}}$ and $\mathcal{R}^{4\pi}_{{C\\!P}}$, the raw
ratios are normalised using the corresponding ${{D}^{0}}$ branching fractions.
These corrections are discussed further in Sect. 6.

Figure 2:  Invariant-mass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the two-body GLW modes (top left) ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D({{K}^{+}}{{K}^{-}}){{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$, (top right) ${{B}^{0}}\rightarrow D({{K}^{+}}{{K}^{-}}){{K}^{*0}}$, (bottom left) ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D({{\pi}^{+}}{{\pi}^{-}}){{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and (bottom right) ${{B}^{0}}\rightarrow D({{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$.  Figure 3:  Invariant-mass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the two-body ADS modes (top left) ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D({{K}^{-}}{{\pi}^{+}}){{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$, (top right) ${{B}^{0}}\rightarrow D({{K}^{+}}{{\pi}^{-}}){{K}^{*0}}$, (bottom left) ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D({{\pi}^{-}}{{K}^{+}}){{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and (bottom right) ${{B}^{0}}\rightarrow D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}}$. The bottom distributions are shown on a logarithmic scale. Figure 4:  Invariant-mass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the four-body GLW mode (left) ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$, (right) ${{B}^{0}}\rightarrow D({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$. Figure 5:  Invariant-mass distributions (data points with error bars) and results of the fit (lines and coloured areas) for the four-body ADS modes (top left) ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$, (top right) ${{B}^{0}}\rightarrow D({{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$, (bottom left) ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow D({{\pi}^{-}}{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}){{\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}$ and (bottom right) ${{B}^{0}}\rightarrow D({{\pi}^{+}}{{K}^{-}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*0}}$. The bottom distributions are shown on a logarithmic scale. Table 1: Summary of signal yields. The uncertainties are statistical.  Decay channel |  ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ yield |  ${{B}^{0}}$ yield
---|---|---
${{B}^{0}}\rightarrow D({{K}^{+}}{{K}^{-}}){{K}^{*0}}$ | $67$ |  $\pm$ | $10$ | $77$ |  $\pm$ | $11$
${{B}^{0}}\rightarrow D({{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$ | $27$ |  $\pm$ | $\phantom{0}6$ | $40$ |  $\pm$ | $\phantom{0}7$
${{B}^{0}}\rightarrow D({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$ | $32$ |  $\pm$ | $\phantom{0}7$ | $35$ |  $\pm$ | $\phantom{0}8$
${{B}^{0}}\rightarrow D({{K}^{+}}{{\pi}^{-}}){{K}^{*0}}$ | $786$ |  $\pm$ | $29$ | $754$ |  $\pm$ | $29$
${{B}^{0}}\rightarrow D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}}$ | $76$ |  $\pm$ | $16$ | $47$ |  $\pm$ | $15$
${{B}^{0}}\rightarrow D({{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$ | $557$ |  $\pm$ | $25$ | $548$ |  $\pm$ | $25$
${{B}^{0}}\rightarrow D({{\pi}^{+}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$ | $41$ |  $\pm$ | $14$ | $40$ |  $\pm$ | $14$

##  6 Correction factors and systematic uncertainties

The measured observables are either asymmetries or ratios of yields between
similar final states, and are thus robust against systematic biases.
Nonetheless, small differences in efficiencies between numerator and
denominator mean that correction factors must be applied to the ratios, apart
from the case of $\mathcal{R}_{\pm}^{\pi K(\pi\pi)}$ where an identical
selection is used for both the suppressed and favoured ADS modes. The
selection efficiencies are computed from simulated signal samples, which are
weighted in the transverse momentum and pseudorapidity of the $B$ meson to
agree with the data distributions. The efficiencies of the PID requirements
between different charges and $D$ final states are evaluated using calibration
samples, which are weighted to match the momentum and pseudorapidity of the
simulated signal samples. Uncertainties are assigned due to the finite size of
the simulated samples, and for possible biases introduced by the binning
schemes used in the reweighting of the calibration sample and the background-
subtraction procedure used for these samples.

As can be seen from Eq. 5, determining $\mathcal{R}^{hh}_{{C\\!P}}$ requires
normalising the measured ratio of yields by a ratio of ${{D}^{0}}$ branching
fractions. These branching fractions are taken from Ref. [14] and the
uncertainties are propagated to the observables.

The raw observables are corrected for detection asymmetry, which is
predominantly caused by the shorter interaction length of ${{K}^{-}}$ mesons
compared with ${{K}^{+}}$ mesons. The difference between the kaon and pion
detection asymmetries, $A_{D}({{K}^{-}}{{\pi}^{+}})$, is computed following
the method used in Ref. [40]. Raw charge asymmetries $A_{\rm
raw}({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})$ and $A_{\rm raw}({{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}}{{\pi}^{+}})$ are measured for the
decays ${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}}{{\pi}^{+}}$, respectively. These
asymmetries are determined using calibration samples which are weighted to
match the kinematics of the kaons and pions in the signal data set. The value
of $A_{D}({{K}^{-}}{{\pi}^{+}})$ is calculated from
$A_{D}({{K}^{-}}{{\pi}^{+}})=A_{\rm
raw}({{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})-A_{\rm raw}({{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{0}}{{\pi}^{+}})+A_{D}({{K}^{0}})$,
where $A_{D}({{K}^{0}})$ is the measured value of the detection asymmetry in
the decay ${{K}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$, giving
$A_{D}({{K}^{-}}{{\pi}^{+}})=(-0.92\pm 0.20)$% in Run 1 and $(-1.0\pm 0.6)$%
in Run 2. A correction of $A_{D}({{K}^{-}}{{\pi}^{+}})$ is applied to the
observables for each ${{K}^{\pm}}{{\pi}^{\mp}}$ pair in the final state.

The observables are also corrected for the asymmetry in the production of
${{B}^{0}}$ and ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ mesons
within the acceptance of the analysis, $A_{\mathrm{prod}}$. This asymmetry has
been measured in bins of $B$-meson momentum and pseudorapidity in Run 1[41]. A
weighted average based on the kinematical distributions of simulated signal
gives $A_{\mathrm{prod}}=(-0.8\pm 0.5)$%. The same central value is applied
for Run 2, with the uncertainty doubled in order to account for a possible
change in asymmetry due to the higher collision energy.

Uncertainties are assigned to account for the shape parameters that are fixed
in the invariant-mass model. The values of these fixed parameters derive from
fits to simulated samples, and so the uncertainties on these fits are
propagated to the mass model. The fixed tail parameters of the signal shape
are treated as a single source of systematic uncertainty. Uncertainties due to
all fixed parameters related to the background shapes are treated
simultaneously, apart from those for the partially reconstructed ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow{{D}^{*}}{{K}^{*0}}$
decays, which are an important source of background that overlaps with the
signal region, and are therefore treated separately.

Uncertainties are also considered for other fixed parameters in the fit. These
are the relative proportion of partially reconstructed ${{D}^{*}}\rightarrow
D{{\pi}^{0}}$ and ${{D}^{*}}\rightarrow D\gamma$ decays, the correction to the
relative yield of misidentified ${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$
decays between Run 1 and Run 2, and the relative yields and $C\\!P$
asymmetries of the partially reconstructed ${{{B}^{+}}}\rightarrow
D{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ background. For the latter, the
uncertainties taken from Ref. [39] are doubled to account for the fact that
there are possible differences in the phase-space acceptance between the two
analyses.

A study of the invariant-mass sidebands of the $D$ candidates is performed in
order to search for evidence of any residual charmless background which would
also contaminate the $B$ signal region. This sideband study is performed after
imposing the flight distance cut on the $D$ candidate, but without applying
the BDT selection, as this may not have a uniform acceptance in $D$ mass.
Regions of the sidebands where there are known reflections from $D$-meson
decays with misidentified final products are excluded. No significant signals
are found from charmless decays in any mode. The measured yields are
extrapolated into the signal region and taken as the central values from which
many pseudoexperiments with an added charmless background component are
simulated. These data sets are fitted using the nominal fit model which
neglects the new background contribution, and a systematic uncertainty is
assigned based on the measured bias.

Table 2 gives the systematic uncertainties for each observable. Systematic
uncertainties which are more than two orders of magnitude smaller than the
statistical uncertainty are considered to be negligible and ignored. The non-
negligible uncertainties are added in quadrature to give the total systematic
uncertainty, which in all cases is considerably smaller than the statistical
uncertainty.

The acceptance for the four-body $D$-decay modes is not fully uniform across
phase space. Studies performed with amplitude models of these decays indicate
that, at the current level of sensitivity, a nonuniform acceptance does not
lead to any significant bias when the observables are interpreted in terms of
$\gamma$ and the other underlying physics parameters. No systematic
uncertainty is assigned.

Table 2: Systematic uncertainties for the observables. Uncertainties are shown if they are larger than 1% of the statistical uncertainty. The total systematic uncertainty is calculated by summing all sources in quadrature. Statistical uncertainties are given for reference. Source | $\mathcal{A}_{{C\\!P}}^{KK}$ | $\mathcal{A}_{{C\\!P}}^{\pi\pi}$ | $\mathcal{R}_{{C\\!P}}^{KK}$ | $\mathcal{R}_{{C\\!P}}^{\pi\pi}$ | $\mathcal{A}_{{C\\!P}}^{4\pi}$ | $\mathcal{R}_{{C\\!P}}^{4\pi}$ | $\mathcal{R}_{+}^{\pi K}$ | $\mathcal{R}_{-}^{\pi K}$ | $\mathcal{R}_{+}^{\pi K\pi\pi}$ | $\mathcal{R}_{-}^{\pi K\pi\pi}$ | $\mathcal{A}_{\rm ADS}^{K\pi}$ | $\mathcal{A}_{\rm ADS}^{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Selection efficiency | - | - | 0.008 | 0.011 | - | 0.012 | - | - | - | - | - | -
PID efficiency | 0.002 | - | 0.004 | 0.004 | 0.002 | 0.007 | - | - | - | - | 0.002 | 0.003
Branching ratios | - | - | 0.017 | 0.025 | - | 0.031 | - | - | - | - | - | -
Production asymmetry | 0.006 | 0.006 | - | - | 0.010 | - | - | - | - | - | 0.006 | 0.006
Detection asymmetry | 0.004 | 0.004 | 0.004 | 0.007 | 0.007 | 0.007 | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$ | 0.008 | 0.008
Signal shape parameters | - | - | - | - | - | - | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$ | - | -
$B^{0}_{s}\rightarrow D^{*}K^{*0}$ shape parameters | - | - | 0.001 | - | - | - | $<0.001$ | $<0.001$ | $<0.001$ | $<0.001$ | - | -
Other background shape parameters | - | - | - | 0.003 | - | 0.003 | $<0.001$ | 0.001 | 0.001 | 0.002 | - | -
$D^{*}\rightarrow D^{0}\gamma/\pi^{0}$ inputs | - | - | 0.002 | - | - | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | - | -
$B\rightarrow D\pi\pi$ PID correction | - | - | - | - | 0.006 | - | $<0.001$ | $<0.001$ | - | - | - | $<0.001$
$B\rightarrow DK\pi\pi$ inputs | - | - | 0.001 | 0.002 | - | 0.002 | - | - | - | - | - | -
Charmless background | 0.003 | 0.002 | - | 0.003 | 0.004 | 0.011 | $<0.001$ | $<0.001$ | - | $<0.001$ | 0.002 | 0.001
Total systematic | 0.008 | 0.008 | 0.020 | 0.029 | 0.014 | 0.037 | 0.002 | 0.003 | 0.002 | 0.003 | 0.010 | 0.010
Statistical | 0.10 | 0.14 | 0.10 | 0.19 | 0.15 | 0.16 | 0.021 | 0.021 | 0.026 | 0.025 | 0.027 | 0.032

##  7 Results and discussion

The measured values for the principal observables are

$\mathcal{A}_{{C\\!P}}^{KK}$ | = | $-0.05$ | $\pm$ | $0.10$ | $\pm$ |  $0.01$,
---|---|---|---|---|---|---
$\mathcal{A}_{{C\\!P}}^{\pi\pi}$ | = | $-0.18$ | $\pm$ | $0.14$ | $\pm$ |  $0.01$,
$\mathcal{R}_{{C\\!P}}^{KK}$ | = | $\phantom{+}0.92$ | $\pm$ | $0.10$ | $\pm$ |  $0.02$,
$\mathcal{R}_{{C\\!P}}^{\pi\pi}$ | = | $\phantom{+}1.32$ | $\pm$ | $0.19$ | $\pm$ |  $0.03$,
$\mathcal{A}_{{C\\!P}}^{4\pi}$ | = | $-0.03$ | $\pm$ | $0.15$ | $\pm$ |  $0.01$,
$\mathcal{R}_{{C\\!P}}^{4\pi}$ | = | $\phantom{+}1.01$ | $\pm$ | $0.16$ | $\pm$ |  $0.04$,
$\mathcal{R}_{+}^{\pi K}$ | = | $\phantom{+}0.064$ | $\pm$ | $0.021$ | $\pm$ |  $0.002$,
$\mathcal{R}_{-}^{\pi K}$ | = | $\phantom{+}0.095$ | $\pm$ | $0.021$ | $\pm$ |  $0.003$,
$\mathcal{R}_{+}^{\pi K\pi\pi}$ | = | $\phantom{+}0.074$ | $\pm$ | $0.026$ | $\pm$ |  $0.002$,
$\mathcal{R}_{-}^{\pi K\pi\pi}$ | = | $\phantom{+}0.072$ | $\pm$ | $0.025$ | $\pm$ |  $0.003$,
$\mathcal{A}_{\rm ADS}^{K\pi}$ | = | $\phantom{+}0.047$ | $\pm$ | $0.027$ | $\pm$ |  $0.010$,
$\mathcal{A}_{\rm ADS}^{K\pi\pi\pi}$ | = | $\phantom{+}0.037$ | $\pm$ | $0.032$ | $\pm$ |  $0.010$,

where the first uncertainty is statistical, and the second systematic. The
values of $\mathcal{R}^{\pi K}_{\pm}$ and $\mathcal{R}^{\pi K\pi\pi}_{\pm}$
are used to calculate the suppressed-mode ADS observables, which are found to
be

$\mathcal{A}_{\mathrm{ADS}}^{\pi K}$ | = | $\phantom{+}0.19$ | $\pm$ | $0.19$ | $\pm$ |  $0.01$,
---|---|---|---|---|---|---
$\mathcal{R}_{\mathrm{ADS}}^{\pi K}$ | = | $\phantom{+}0.080$ | $\pm$ | $0.015$ | $\pm$ |  $0.002$,
$\mathcal{A}_{\mathrm{ADS}}^{\pi K\pi\pi}$ | = | $-0.01$ | $\pm$ | $0.24$ | $\pm$ |  $0.01$,
$\mathcal{R}_{\mathrm{ADS}}^{\pi K\pi\pi}$ | = | $\phantom{+}0.073$ | $\pm$ | $0.018$ | $\pm$ |  $0.002$.

All $C\\!P$ asymmetries are compatible with zero to within two standard
deviations. The values of the GLW asymmetries and ratios are found to be
consistent between the two modes, within 0.8 and 1.8 standard deviations,
respectively. The results for
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ are in agreement
with these values, after correcting for the known $C\\!P$-even content of this
state. The same observables determined for ${B}^{0}_{s}$ decays are compatible
with the $C\\!P$-conserving hypothesis. Results for ${{B}^{0}_{s}}$ decays can
be found in Appendix A, together with the results for all observables
separated between the Run 1 and Run 2 data sets, and full correlation
matrices.

The statistical significances of the signal yields in the previously
unobserved channels are calculated using Wilks’ theorem [42]. The likelihood
profiles are convolved with a Gaussian function with standard deviation equal
to the systematic uncertainties on the yields. This procedure yields a
significance of 8.4$\sigma$ for the ${{B}^{0}}\rightarrow
D(\pi^{+}\pi^{-}\pi^{+}\pi^{-}){{K}^{*0}}$ decay, 5.8$\sigma$ for the
${{B}^{0}}\rightarrow D(\pi^{+}K^{-}){{K}^{*0}}$ decay and 4.4$\sigma$ for the
${{B}^{0}}\rightarrow D(\pi^{+}K^{-}\pi^{+}\pi^{-}){{K}^{*0}}$ decay,
constituting the first observation of the first two modes, and strong evidence
for the presence of the suppressed four-body ADS channel.

The results are interpreted in terms of the underlying physics parameters
$\gamma$, $r_{B}^{D\\!{{K}^{*0}}}$ and $\delta_{B}^{D\\!{{K}^{*0}}}$ by
performing a global $\chi^{2}$ minimisation. The minimised
$\chi^{2}/\mathrm{ndf}$ is equal to 7.1/9. A scan of physics parameters is
performed for a range of values and the difference in $\chi^{2}$ between the
parameter scan and the global minimum, $\Delta\chi^{2}$, is evaluated. The
confidence level for any pair of parameters is calculated assuming that these
are normally distributed, which allows the $\Delta\chi^{2}=2.30,\ 6.18,\ 11.8$
contours to be drawn, corresponding to 68.6%, 95.5%, 99.7% confidence levels,
respectively. These contours are shown in Fig. 6. As expected, there is a
degeneracy in the $(\gamma,\delta_{B}^{D\\!{{K}^{*0}}})$ plane. Four favoured
solutions can be seen, two of which are compatible with the existing LHCb
determination of $\gamma$ [4, 5], which is dominated by results obtained from
${{{B}^{+}}}\rightarrow D{{K}^{+}}$ processes, which have values of
$r_{B}^{DK^{-}}$ and $\delta_{B}^{DK^{-}}$ different from
$r_{B}^{D\\!{{K}^{*0}}}$ and $\delta_{B}^{D\\!{{K}^{*0}}}$. The degeneracy of
the solutions can be broken by combining these results with those using other
$D$ decay modes, specifically the
$D\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{+}}{{\pi}^{-}}$ decay. The value of
$r_{B}^{D\\!{{K}^{*0}}}$ is determined to be $0.265\pm 0.023$. In accordance
with expectation, this is almost a factor of three larger than the
corresponding parameter in ${{{B}^{+}}}\rightarrow D{{K}^{+}}$ decays [4, 5].
This measurement is consistent with, and more accurate than, the previous
measurement by LHCb in Ref. [7].

Figure 6: Contour plots showing 2D scans of (left)
$\delta_{B}^{D\\!{{K}^{*0}}}$ versus $\gamma$ and (right)
$\delta_{B}^{D\\!{{K}^{*0}}}$ versus $r_{B}^{D\\!{{K}^{*0}}}$. The lines
represent the $\Delta\chi^{2}=$ 2.30, 6.18 and 11.8 contours, corresponding to
68.6%, 95.5% and 99.7% confidence levels (C.L.), respectively.

##  8 Conclusion

Measurements of $C\\!P$ observables in ${{B}^{0}}\rightarrow D{{K}^{*0}}$
decays with the $D$ meson decaying to ${{K}^{+}}{{\pi}^{-}}$,
${{\pi}^{+}}{{K}^{-}}$, ${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$ are
performed using LHCb data collected in 2011, 2012, 2015 and 2016. The results,
benefitting from the increased data sample and improved analysis methods,
supersede those of the previous study [7]. Measurements with $D$ mesons
reconstructed in the ${{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$,
${{\pi}^{+}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ final states are presented for
the first time. First observations are obtained for the suppressed ADS mode
${{B}^{0}}\rightarrow D({{\pi}^{+}}{{K}^{-}}){{K}^{*0}}$ and the mode
${{B}^{0}}\rightarrow
D({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$.

The observables are interpreted in terms of the weak phase $\gamma$ and
associated parameters, and are found to be compatible with the previous LHCb
results [4, 5], which are dominated by measurements of ${{{B}^{+}}}\rightarrow
D{{K}^{+}}$ processes. The amplitude ratio $r_{B}^{D\\!{{K}^{*0}}}$ is
determined to be equal to $0.265\pm 0.023$ at a confidence level of 68.3%.
These results can be combined with those from other modes in
${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays to provide powerful constraints on
$\gamma$. This can be compared to results obtained from studies of other
processes.

## 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).

Appendices

##  Appendix A Additional results

Observables for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow D{{K}^{*0}}$
decays are defined analogously to those for ${{B}^{0}}\rightarrow D{{K}^{*0}}$
decays in Eqs. 1, 5 and 14. The measured observables are

$\mathcal{A}_{s,\rm ADS}^{\pi K}$ | = | $\phantom{+}0.006$ | $\pm$ | $0.017$ | $\pm$ |  $0.012$,
---|---|---|---|---|---|---
$\mathcal{A}_{s,\rm ADS}^{\pi K\pi\pi}$ | = | $-0.007$ | $\pm$ | $0.021$ | $\pm$ |  $0.013$,
$\mathcal{A}_{s,{C\\!P}}^{KK}$ | = | $\phantom{+}0.06$ | $\pm$ | $0.05$ | $\pm$ |  $0.01$,
$\mathcal{A}_{s,{C\\!P}}^{\pi\pi}$ | = | $-0.11$ | $\pm$ | $0.09$ | $\pm$ |  $0.01$,
$\mathcal{R}_{s,{C\\!P}}^{KK}$ | = | $\phantom{+}1.06$ | $\pm$ | $0.06$ | $\pm$ |  $0.02$,
$\mathcal{R}_{s,{C\\!P}}^{\pi\pi}$ | = | $\phantom{+}1.05$ | $\pm$ | $0.10$ | $\pm$ |  $0.02$,
$\mathcal{A}_{s,{C\\!P}}^{4\pi}$ | = | $\phantom{+}0.12$ | $\pm$ | $0.08$ | $\pm$ |  $0.02$,
$\mathcal{R}_{s,{C\\!P}}^{4\pi}$ | = | $\phantom{+}0.964$ | $\pm$ | $0.086$ | $\pm$ |  $0.031$.

The ${{B}^{0}}$ and ${{B}^{0}_{s}}$ observables are also measured separately
for Run 1 and Run 2; these measurements are presented in Table 3. The
correlation matrices for the principal observables are given in Tables 4, 5
and 6 for the combined, Run 1, and Run 2 results, respectively. Table 7 gives
the correlations between the Run 1 and Run 2 results.

Table 3: Measured observables split by LHC running period. Observables relating to ${{B}^{0}}\rightarrow D({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*0}}$ decays are not presented for Run 1, as this decay channel was not selected in the Run 1 data. | Run 1 | Run 2
---|---|---
$\mathcal{A}_{{C\\!P}}^{KK}$ | $-0.19$ | $\pm$ | $0.16$ | $\pm$ | $0.01$ | $\phantom{+}0.05$ | $\pm$ | $0.13$ | $\pm$ | $0.02$
$\mathcal{A}_{{C\\!P}}^{\pi\pi}$ | $-0.06$ | $\pm$ | $0.23$ | $\pm$ | $0.01$ | $-0.26$ | $\pm$ | $0.18$ | $\pm$ | $0.01$
$\mathcal{R}_{{C\\!P}}^{KK}$ | $\phantom{+}0.93$ | $\pm$ | $0.15$ | $\pm$ | $0.02$ | $\phantom{+}0.91$ | $\pm$ | $0.13$ | $\pm$ | $0.02$
$\mathcal{R}_{{C\\!P}}^{\pi\pi}$ | $\phantom{+}1.39$ | $\pm$ | $0.33$ | $\pm$ | $0.04$ | $\phantom{+}1.27$ | $\pm$ | $0.24$ | $\pm$ | $0.03$
$\mathcal{A}_{{C\\!P}}^{4\pi}$ | — | $-0.03$ | $\pm$ | $0.15$ | $\pm$ | $0.01$
$\mathcal{R}_{{C\\!P}}^{4\pi}$ | — | $\phantom{+}1.01$ | $\pm$ | $0.16$ | $\pm$ | $0.04$
$\mathcal{R}_{+}^{\pi K}$ | $\phantom{+}0.045$ | $\pm$ | $0.032$ | $\pm$ | $0.003$ | $\phantom{+}0.076$ | $\pm$ | $0.027$ | $\pm$ | $0.003$
$\mathcal{R}_{-}^{\pi K}$ | $\phantom{+}0.120$ | $\pm$ | $0.035$ | $\pm$ | $0.003$ | $\phantom{+}0.080$ | $\pm$ | $0.025$ | $\pm$ | $0.003$
$\mathcal{R}_{+}^{\pi K\pi\pi}$ | $\phantom{+}0.12$ | $\pm$ | $0.04$ | $\pm$ | $0.00$ | $\phantom{+}0.047$ | $\pm$ | $0.031$ | $\pm$ | $0.003$
$\mathcal{R}_{-}^{\pi K\pi\pi}$ | $\phantom{+}0.099$ | $\pm$ | $0.043$ | $\pm$ | $0.004\phantom{+}$ | $\phantom{+}0.056$ | $\pm$ | $0.029$ | $\pm$ | $0.003$
$\mathcal{A}_{\rm ADS}^{K\pi}$ | $\phantom{+}0.03$ | $\pm$ | $0.04$ | $\pm$ | $0.01$ | $\phantom{+}0.055$ | $\pm$ | $0.035$ | $\pm$ | $0.016$
$\mathcal{A}_{\rm ADS}^{K\pi\pi\pi}$ | $\phantom{+}0.03$ | $\pm$ | $0.05$ | $\pm$ | $0.01$ | $\phantom{+}0.042$ | $\pm$ | $0.040$ | $\pm$ | $0.016$
$\mathcal{A}_{s,\rm ADS}^{\pi K}$ | $\phantom{+}0.011$ | $\pm$ | $0.027$ | $\pm$ | $0.010$ | $\phantom{+}0.002$ | $\pm$ | $0.022$ | $\pm$ | $0.019$
$\mathcal{A}_{s,\rm ADS}^{\pi K\pi\pi}$ | $-0.042$ | $\pm$ | $0.035$ | $\pm$ | $0.012$ | $\phantom{+}0.012$ | $\pm$ | $0.026$ | $\pm$ | $0.020$
$\mathcal{A}_{s,{C\\!P}}^{KK}$ | $-0.03$ | $\pm$ | $0.08$ | $\pm$ | $0.01$ | $\phantom{+}0.13$ | $\pm$ | $0.07$ | $\pm$ | $0.02$
$\mathcal{A}_{s,{C\\!P}}^{\pi\pi}$ | $-0.06$ | $\pm$ | $0.17$ | $\pm$ | $0.01$ | $-0.13$ | $\pm$ | $0.11$ | $\pm$ | $0.02$
$\mathcal{R}_{s,{C\\!P}}^{KK}$ | $\phantom{+}1.14$ | $\pm$ | $0.09$ | $\pm$ | $0.03$ | $\phantom{+}1.01$ | $\pm$ | $0.07$ | $\pm$ | $0.02$
$\mathcal{R}_{s,{C\\!P}}^{\pi\pi}$ | $\phantom{+}0.83$ | $\pm$ | $0.15$ | $\pm$ | $0.02$ | $\phantom{+}1.22$ | $\pm$ | $0.14$ | $\pm$ | $0.03$
$\mathcal{A}_{s,{C\\!P}}^{4\pi}$ | — | $\phantom{+}0.12$ | $\pm$ | $0.08$ | $\pm$ | $0.02$
$\mathcal{R}_{s,{C\\!P}}^{4\pi}$ | — | $\phantom{+}0.96$ | $\pm$ | $0.09$ | $\pm$ | $0.03$
Table 4: Combined statistical and systematic correlation matrix for the principal observables. | $\mathcal{A}_{{C\\!P}}^{KK}$ | $\mathcal{A}_{{C\\!P}}^{\pi\pi}$ | $\mathcal{R}_{{C\\!P}}^{KK}$ | $\mathcal{R}_{{C\\!P}}^{\pi\pi}$ | $\mathcal{A}_{{C\\!P}}^{4\pi}$ | $\mathcal{R}_{{C\\!P}}^{4\pi}$ | $\mathcal{R}_{+}^{\pi K}$ | $\mathcal{R}_{-}^{\pi K}$ | $\mathcal{R}_{+}^{\pi K\pi\pi}$ | $\mathcal{R}_{-}^{\pi K\pi\pi}$ | $\mathcal{A}_{\rm ADS}^{K\pi}$ | $\mathcal{A}_{\rm ADS}^{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{A}_{CP}^{KK}$ | $1.00$ | $0.00$ | $0.03$ | $-0.01$ | $0.00$ | $0.00$ | $0.00$ | $-0.01$ | $-0.01$ | $-0.01$ | $-0.01$ | $-0.01$
$\mathcal{A}_{CP}^{\pi\pi}$ | $0.00$ | $1.00$ | $0.01$ | $0.06$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $-0.01$ | $-0.01$
$\mathcal{R}_{CP}^{KK}$ | $0.03$ | $0.01$ | $1.00$ | $0.04$ | $0.00$ | $0.03$ | $0.02$ | $0.02$ | $0.00$ | $0.00$ | $-0.04$ | $-0.03$
$\mathcal{R}_{CP}^{\pi\pi}$ | $-0.01$ | $0.06$ | $0.04$ | $1.00$ | $0.00$ | $0.04$ | $0.01$ | $0.03$ | $0.02$ | $0.02$ | $0.03$ | $0.03$
$\mathcal{A}_{CP}^{4\pi}$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $1.00$ | $0.01$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$
$\mathcal{R}_{CP}^{4\pi}$ | $0.00$ | $0.00$ | $0.03$ | $0.04$ | $0.01$ | $1.00$ | $0.01$ | $0.02$ | $0.02$ | $0.03$ | $0.02$ | $0.01$
$\mathcal{R}_{+}^{\pi K}$ | $0.00$ | $0.00$ | $0.02$ | $0.01$ | $0.00$ | $0.01$ | $1.00$ | $0.05$ | $0.01$ | $0.01$ | $0.08$ | $0.00$
$\mathcal{R}_{-}^{\pi K}$ | $-0.01$ | $0.00$ | $0.02$ | $0.03$ | $0.00$ | $0.02$ | $0.05$ | $1.00$ | $0.02$ | $0.02$ | $-0.08$ | $0.03$
$\mathcal{R}_{+}^{\pi K\pi\pi}$ | $-0.01$ | $0.00$ | $0.00$ | $0.02$ | $0.00$ | $0.02$ | $0.01$ | $0.02$ | $1.00$ | $0.06$ | $0.02$ | $0.11$
$\mathcal{R}_{-}^{\pi K\pi\pi}$ | $-0.01$ | $0.00$ | $0.00$ | $0.02$ | $0.00$ | $0.03$ | $0.01$ | $0.02$ | $0.06$ | $1.00$ | $0.03$ | $-0.06$
$\mathcal{A}_{\rm ADS}^{K\pi}$ | $-0.01$ | $-0.01$ | $-0.04$ | $0.03$ | $0.00$ | $0.02$ | $0.08$ | $-0.08$ | $0.02$ | $0.03$ | $1.00$ | $0.08$
$\mathcal{A}_{\rm ADS}^{K\pi\pi\pi}$ | $-0.01$ | $-0.01$ | $-0.03$ | $0.03$ | $0.00$ | $0.01$ | $0.00$ | $0.03$ | $0.11$ | $-0.06$ | $0.08$ | $1.00$
Table 5: Combined statistical and systematic correlation matrix for the principal observables in Run 1 data only. | $\mathcal{A}_{{C\\!P},1}^{KK}$ | $\mathcal{A}_{{C\\!P},1}^{\pi\pi}$ | $\mathcal{R}_{{C\\!P},1}^{KK}$ | $\mathcal{R}_{{C\\!P},1}^{\pi\pi}$ | $\mathcal{R}_{+,1}^{\pi K}$ | $\mathcal{R}_{-,1}^{\pi K}$ | $\mathcal{R}_{+,1}^{\pi K\pi\pi}$ | $\mathcal{R}_{-,1}^{\pi K\pi\pi}$ | $\mathcal{A}_{\rm ADS,1}^{K\pi}$ | $\mathcal{A}_{\rm ADS,1}^{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---
$\mathcal{A}_{{C\\!P},1}^{KK}$ | $1.00$ | $0.00$ | $0.08$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $-0.01$ | $-0.01$
$\mathcal{A}_{{C\\!P},1}^{\pi\pi}$ | $0.00$ | $1.00$ | $0.00$ | $0.01$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$
$\mathcal{R}_{{C\\!P},1}^{KK}$ | $0.08$ | $0.00$ | $1.00$ | $0.03$ | $0.01$ | $0.02$ | $0.00$ | $0.00$ | $-0.02$ | $-0.01$
$\mathcal{R}_{{C\\!P},1}^{\pi\pi}$ | $0.00$ | $0.01$ | $0.03$ | $1.00$ | $0.00$ | $0.03$ | $0.01$ | $0.01$ | $0.01$ | $0.01$
$\mathcal{R}_{+,1}^{\pi K}$ | $0.00$ | $0.00$ | $0.01$ | $0.00$ | $1.00$ | $0.02$ | $0.00$ | $0.00$ | $0.05$ | $-0.01$
$\mathcal{R}_{-,1}^{\pi K}$ | $0.00$ | $0.00$ | $0.02$ | $0.03$ | $0.02$ | $1.00$ | $0.01$ | $0.01$ | $-0.13$ | $0.01$
$\mathcal{R}_{+,1}^{\pi K\pi\pi}$ | $0.00$ | $0.00$ | $0.00$ | $0.01$ | $0.00$ | $0.01$ | $1.00$ | $0.04$ | $0.01$ | $0.15$
$\mathcal{R}_{-,1}^{\pi K\pi\pi}$ | $0.00$ | $0.00$ | $0.00$ | $0.01$ | $0.00$ | $0.01$ | $0.04$ | $1.00$ | $0.01$ | $-0.11$
$\mathcal{A}_{\rm ADS,1}^{K\pi}$ | $-0.01$ | $0.00$ | $-0.02$ | $0.01$ | $0.05$ | $-0.13$ | $0.01$ | $0.01$ | $1.00$ | $0.02$
$\mathcal{A}_{\rm ADS,1}^{K\pi\pi\pi}$ | $-0.01$ | $0.00$ | $-0.01$ | $0.01$ | $-0.01$ | $0.01$ | $0.15$ | $-0.11$ | $0.02$ | $1.00$
Table 6: Combined statistical and systematic correlation matrix for the principal observables in Run 2 data only. | $\mathcal{A}_{{C\\!P},2}^{KK}$ | $\mathcal{A}_{{C\\!P},2}^{\pi\pi}$ | $\mathcal{R}_{{C\\!P},2}^{KK}$ | $\mathcal{R}_{{C\\!P},2}^{\pi\pi}$ | $\mathcal{A}_{{C\\!P},2}^{4\pi}$ | $\mathcal{R}_{{C\\!P},2}^{4\pi}$ | $\mathcal{R}_{+,2}^{\pi K}$ | $\mathcal{R}_{-,2}^{\pi K}$ | $\mathcal{R}_{+,2}^{\pi K\pi\pi}$ | $\mathcal{R}_{-,2}^{\pi K\pi\pi}$ | $\mathcal{A}_{\rm ADS,2}^{K\pi}$ | $\mathcal{A}_{\rm ADS,2}^{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{A}_{{C\\!P},2}^{KK}$ | $1.00$ | $-0.01$ | $-0.01$ | $0.01$ | $0.00$ | $0.01$ | $0.01$ | $0.01$ | $-0.01$ | $-0.01$ | $0.04$ | $0.03$
$\mathcal{A}_{{C\\!P},2}^{\pi\pi}$ | $-0.01$ | $1.00$ | $0.01$ | $0.08$ | $0.00$ | $0.00$ | $-0.01$ | $-0.01$ | $0.01$ | $0.01$ | $-0.02$ | $-0.01$
$\mathcal{R}_{{C\\!P},2}^{KK}$ | $-0.01$ | $0.01$ | $1.00$ | $0.03$ | $0.00$ | $0.03$ | $0.01$ | $0.01$ | $0.01$ | $0.01$ | $-0.04$ | $-0.03$
$\mathcal{R}_{{C\\!P},2}^{\pi\pi}$ | $0.01$ | $0.08$ | $0.03$ | $1.00$ | $0.00$ | $0.03$ | $0.02$ | $0.03$ | $-0.01$ | $0.00$ | $0.03$ | $0.03$
$\mathcal{A}_{{C\\!P},2}^{4\pi}$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $1.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.01$
$\mathcal{R}_{{C\\!P},2}^{4\pi}$ | $0.01$ | $0.00$ | $0.03$ | $0.03$ | $0.00$ | $1.00$ | $0.01$ | $0.01$ | $0.00$ | $0.01$ | $0.01$ | $0.01$
$\mathcal{R}_{+,2}^{\pi K}$ | $0.01$ | $-0.01$ | $0.01$ | $0.02$ | $0.00$ | $0.01$ | $1.00$ | $0.05$ | $0.00$ | $0.00$ | $0.12$ | $0.02$
$\mathcal{R}_{-,2}^{\pi K}$ | $0.01$ | $-0.01$ | $0.01$ | $0.03$ | $0.00$ | $0.01$ | $0.05$ | $1.00$ | $0.00$ | $0.00$ | $-0.07$ | $0.02$
$\mathcal{R}_{+,2}^{\pi K\pi\pi}$ | $-0.01$ | $0.01$ | $0.01$ | $-0.01$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $1.00$ | $0.05$ | $-0.02$ | $0.04$
$\mathcal{R}_{-,2}^{\pi K\pi\pi}$ | $-0.01$ | $0.01$ | $0.01$ | $0.00$ | $0.00$ | $0.01$ | $0.00$ | $0.00$ | $0.05$ | $1.00$ | $-0.02$ | $-0.09$
$\mathcal{A}_{\rm ADS,2}^{K\pi}$ | $0.04$ | $-0.02$ | $-0.04$ | $0.03$ | $0.00$ | $0.01$ | $0.12$ | $-0.07$ | $-0.02$ | $-0.02$ | $1.00$ | $0.10$
$\mathcal{A}_{\rm ADS,2}^{K\pi\pi\pi}$ | $0.03$ | $-0.01$ | $-0.03$ | $0.03$ | $0.01$ | $0.01$ | $0.02$ | $0.02$ | $0.04$ | $-0.09$ | $0.10$ | $1.00$
Table 7: Correlation matrix for the principal observables between Run 1 and Run 2 data. | $\mathcal{A}_{{C\\!P},2}^{KK}$ | $\mathcal{A}_{{C\\!P},2}^{\pi\pi}$ | $\mathcal{R}_{{C\\!P},2}^{KK}$ | $\mathcal{R}_{{C\\!P},2}^{\pi\pi}$ | $\mathcal{A}_{{C\\!P}}^{4\pi}$ | $\mathcal{R}_{{C\\!P}}^{4\pi}$ | $\mathcal{R}_{+,2}^{\pi K}$ | $\mathcal{R}_{-,2}^{\pi K}$ | $\mathcal{R}_{+,2}^{\pi K\pi\pi}$ | $\mathcal{R}_{-,2}^{\pi K\pi\pi}$ | $\mathcal{A}_{\rm ADS,2}^{K\pi}$ | $\mathcal{A}_{\rm ADS,2}^{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{A}_{{C\\!P},1}^{KK}$ | $0.00$ | $0.00$ | $0.01$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $-0.01$ | $-0.01$
$\mathcal{A}_{{C\\!P},1}^{\pi\pi}$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $0.00$ | $-0.01$ | $0.00$
$\mathcal{R}_{{C\\!P},1}^{KK}$ | $-0.01$ | $0.01$ | $0.03$ | $0.01$ | $0.00$ | $0.02$ | $0.00$ | $-0.01$ | $0.01$ | $0.01$ | $-0.02$ | $-0.02$
$\mathcal{R}_{{C\\!P},1}^{\pi\pi}$ | $0.01$ | $-0.01$ | $0.00$ | $0.02$ | $0.00$ | $0.02$ | $0.01$ | $0.01$ | $-0.01$ | $-0.01$ | $0.03$ | $0.02$
$\mathcal{R}_{+,1}^{\pi K}$ | $-0.01$ | $0.00$ | $0.01$ | $-0.01$ | $0.00$ | $0.00$ | $0.02$ | $0.01$ | $0.01$ | $0.01$ | $-0.02$ | $-0.02$
$\mathcal{R}_{-,1}^{\pi K}$ | $0.01$ | $-0.01$ | $0.00$ | $0.02$ | $0.00$ | $0.01$ | $0.02$ | $0.04$ | $0.00$ | $0.00$ | $0.03$ | $0.02$
$\mathcal{R}_{+,1}^{\pi K\pi\pi}$ | $0.01$ | $-0.01$ | $0.00$ | $0.01$ | $0.00$ | $0.01$ | $0.01$ | $0.01$ | $0.02$ | $0.01$ | $0.02$ | $0.02$
$\mathcal{R}_{-,1}^{\pi K\pi\pi}$ | $0.01$ | $-0.01$ | $0.00$ | $0.01$ | $0.00$ | $0.01$ | $0.01$ | $0.02$ | $0.01$ | $0.03$ | $0.02$ | $0.02$
$\mathcal{A}_{\rm ADS,1}^{K\pi}$ | $0.02$ | $-0.01$ | $-0.02$ | $0.02$ | $0.00$ | $0.01$ | $0.01$ | $0.02$ | $-0.01$ | $-0.01$ | $0.05$ | $0.04$
$\mathcal{A}_{\rm ADS,1}^{K\pi\pi\pi}$ | $0.02$ | $-0.01$ | $-0.02$ | $0.02$ | $0.00$ | $0.01$ | $0.01$ | $0.02$ | $-0.01$ | $-0.01$ | $0.05$ | $0.04$

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LHCb collaboration

R. Aaij29, C. Abellán Beteta46, B. Adeva43, M. Adinolfi50, C.A. Aidala78, Z.
Ajaltouni7, S. Akar61, P. Albicocco20, J. Albrecht12, F. Alessio44, M.
Alexander55, A. Alfonso Albero42, G. Alkhazov35, P. Alvarez Cartelle57, A.A.
Alves Jr43, S. Amato2, Y. Amhis9, L. An19, L. Anderlini19, G. Andreassi45, M.
Andreotti18, J.E. Andrews62, F. Archilli20, J. Arnau Romeu8, A. Artamonov41,
M. Artuso64, K. Arzymatov39, E. Aslanides8, M. Atzeni46, B. Audurier24, S.
Bachmann14, J.J. Back52, S. Baker57, V. Balagura9,b, W. Baldini18,44, A.
Baranov39, R.J. Barlow58, S. Barsuk9, W. Barter57, M. Bartolini21, F.
Baryshnikov74, V. Batozskaya33, B. Batsukh64, A. Battig12, V. Battista45, A.
Bay45, F. Bedeschi26, I. Bediaga1, A. Beiter64, L.J. Bel29, V. Belavin39, S.
Belin24, N. Beliy4, V. Bellee45, K. Belous41, I. Belyaev36, G. Bencivenni20,
E. Ben-Haim10, S. Benson29, S. Beranek11, A. Berezhnoy37, R. Bernet46, D.
Berninghoff14, E. Bertholet10, A. Bertolin25, C. Betancourt46, F. Betti17,e,
M.O. Bettler51, Ia. Bezshyiko46, S. Bhasin50, J. Bhom31, M.S. Bieker12, S.
Bifani49, P. Billoir10, A. Birnkraut12, A. Bizzeti19,u, M. Bjørn59, M.P.
Blago44, T. Blake52, F. Blanc45, S. Blusk64, D. Bobulska55, V. Bocci28, O.
Boente Garcia43, T. Boettcher60, A. Boldyrev75, A. Bondar40,x, N. Bondar35, S.
Borghi58,44, M. Borisyak39, M. Borsato14, M. Boubdir11, T.J.V. Bowcock56, C.
Bozzi18,44, S. Braun14, A. Brea Rodriguez43, M. Brodski44, J. Brodzicka31, A.
Brossa Gonzalo52, D. Brundu24,44, E. Buchanan50, A. Buonaura46, C. Burr58, A.
Bursche24, J.S. Butter29, J. Buytaert44, W. Byczynski44, S. Cadeddu24, H.
Cai68, R. Calabrese18,g, S. Cali20, R. Calladine49, M. Calvi22,i, M. Calvo
Gomez42,m, A. Camboni42,m, P. Campana20, D.H. Campora Perez44, L.
Capriotti17,e, A. Carbone17,e, G. Carboni27, R. Cardinale21, A. Cardini24, P.
Carniti22,i, K. Carvalho Akiba2, A. Casais Vidal43, G. Casse56, M. Cattaneo44,
G. Cavallero21, R. Cenci26,p, M.G. Chapman50, M. Charles10,44, Ph.
Charpentier44, G. Chatzikonstantinidis49, M. Chefdeville6, V. Chekalina39, C.
Chen3, S. Chen24, S.-G. Chitic44, V. Chobanova43, M. Chrzaszcz44, A.
Chubykin35, P. Ciambrone20, X. Cid Vidal43, G. Ciezarek44, F. Cindolo17,
P.E.L. Clarke54, M. Clemencic44, H.V. Cliff51, J. Closier44, J.L.
Cobbledick58, V. Coco44, J.A.B. Coelho9, J. Cogan8, E. Cogneras7, L.
Cojocariu34, P. Collins44, T. Colombo44, A. Comerma-Montells14, A. Contu24, N.
Cooke49, G. Coombs44, S. Coquereau42, G. Corti44, C.M. Costa Sobral52, B.
Couturier44, G.A. Cowan54, D.C. Craik60, A. Crocombe52, M. Cruz Torres1, R.
Currie54, C.L. Da Silva63, E. Dall’Occo29, J. Dalseno43,v, C. D’Ambrosio44, A.
Danilina36, P. d’Argent14, A. Davis58, O. De Aguiar Francisco44, K. De
Bruyn44, S. De Capua58, M. De Cian45, J.M. De Miranda1, L. De Paula2, M. De
Serio16,d, P. De Simone20, J.A. de Vries29, C.T. Dean55, W. Dean78, D.
Decamp6, L. Del Buono10, B. Delaney51, H.-P. Dembinski13, M. Demmer12, A.
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Dijkstra44, F. Dordei24, M. Dorigo26,y, A.C. dos Reis1, A. Dosil Suárez43, L.
Douglas55, A. Dovbnya47, K. Dreimanis56, L. Dufour44, G. Dujany10, P.
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I.V. Gorelov37, C. Gotti22,i, E. Govorkova29, J.P. Grabowski14, R. Graciani
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Likhomanenko73, R. Lindner44, F. Lionetto46, V. Lisovskyi9, G. Liu67, X. Liu3,
D. Loh52, A. Loi24, J. Lomba Castro43, I. Longstaff55, J.H. Lopes2, G.
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Mancinelli8, D. Marangotto23,q, J. Maratas7,w, J.F. Marchand6, U. Marconi17,
C. Marin Benito9, M. Marinangeli45, P. Marino45, J. Marks14, P.J. Marshall56,
G. Martellotti28, L. Martinazzoli44, M. Martinelli44,22,i, D. Martinez
Santos43, F. Martinez Vidal77, A. Massafferri1, M. Materok11, R. Matev44, A.
Mathad46, Z. Mathe44, V. Matiunin36, C. Matteuzzi22, K.R. Mattioli78, A.
Mauri46, E. Maurice9,b, B. Maurin45, M. McCann57,44, L. Mcconnell15, A.
McNab58, R. McNulty15, J.V. Mead56, B. Meadows61, C. Meaux8, N. Meinert71, D.
Melnychuk33, M. Merk29, A. Merli23,q, E. Michielin25, D.A. Milanes70, E.
Millard52, M.-N. Minard6, O. Mineev36, L. Minzoni18,g, D.S. Mitzel14, A.
Mödden12, A. Mogini10, R.D. Moise57, T. Mombächer12, I.A. Monroy70, S.
Monteil7, M. Morandin25, G. Morello20, M.J. Morello26,t, J. Moron32, A.B.
Morris8, R. Mountain64, H. Mu3, F. Muheim54, M. Mukherjee69, M. Mulder29, D.
Müller44, J. Müller12, K. Müller46, V. Müller12, C.H. Murphy59, D. Murray58,
P. Naik50, T. Nakada45, R. Nandakumar53, A. Nandi59, T. Nanut45, I. Nasteva2,
M. Needham54, N. Neri23,q, S. Neubert14, N. Neufeld44, R. Newcombe57, T.D.
Nguyen45, C. Nguyen-Mau45,n, S. Nieswand11, R. Niet12, N. Nikitin37, N.S.
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R. Oldeman24,f, C.J.G. Onderwater72, J. D. Osborn78, A. Ossowska31, J.M.
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T. Pajero26,t, A. Palano16, M. Palutan20, G. Panshin76, A. Papanestis53, M.
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P. Perret7, L. Pescatore45, K. Petridis50, A. Petrolini21,h, A. Petrov73, S.
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S. Taneja58, Z. Tang3, T. Tekampe12, G. Tellarini18, F. Teubert44, E.
Thomas44, M.J. Tilley57, V. Tisserand7, S. T’Jampens6, M. Tobin5, S. Tolk44,
L. Tomassetti18,g, D. Tonelli26, D.Y. Tou10, E. Tournefier6, M. Traill55, M.T.
Tran45, A. Trisovic51, A. Tsaregorodtsev8, G. Tuci26,44,p, A. Tully51, N.
Tuning29, A. Ukleja33, A. Usachov9, A. Ustyuzhanin39,75, U. Uwer14, A.
Vagner76, V. Vagnoni17, A. Valassi44, S. Valat44, G. Valenti17, M. van
Beuzekom29, H. Van Hecke63, E. van Herwijnen44, C.B. Van Hulse15, J. van
Tilburg29, M. van Veghel29, R. Vazquez Gomez44, P. Vazquez Regueiro43, C.
Vázquez Sierra29, S. Vecchi18, J.J. Velthuis50, M. Veltri19,r, A.
Venkateswaran64, M. Vernet7, M. Veronesi29, M. Vesterinen52, J.V. Viana
Barbosa44, D. Vieira4, M. Vieites Diaz43, H. Viemann71, X. Vilasis-
Cardona42,m, A. Vitkovskiy29, M. Vitti51, V. Volkov37, A. Vollhardt46, D. Vom
Bruch10, B. Voneki44, A. Vorobyev35, V. Vorobyev40,x, N. Voropaev35, R.
Waldi71, J. Walsh26, J. Wang3, J. Wang5, M. Wang3, Y. Wang69, Z. Wang46, D.R.
Ward51, H.M. Wark56, N.K. Watson49, D. Websdale57, A. Weiden46, C. Weisser60,
D.J. White58, M. Whitehead11, G. Wilkinson59, M. Wilkinson64, I. Williams51,
M. Williams60, M.R.J. Williams58, T. Williams49, F.F. Wilson53, M. Winn9, W.
Wislicki33, M. Witek31, G. Wormser9, S.A. Wotton51, K. Wyllie44, Z. Xiang4, D.
Xiao69, Y. Xie69, H. Xing67, A. Xu3, L. Xu3, M. Xu69, Q. Xu4, Z. Xu6, Z. Xu3,
Z. Yang3, Z. Yang62, Y. Yao64, L.E. Yeomans56, H. Yin69, J. Yu69,aa, X.
Yuan64, O. Yushchenko41, K.A. Zarebski49, M. Zavertyaev13,c, M. Zeng3, D.
Zhang69, L. Zhang3, S. Zhang3, W.C. Zhang3,z, Y. Zhang44, A. Zhelezov14, Y.
Zheng4, X. Zhou4, Y. Zhou4, X. Zhu3, V. Zhukov11,37, J.B. Zonneveld54, S.
Zucchelli17,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
4University of Chinese Academy of Sciences, Beijing, China
5Institute Of High Energy Physics (ihep), Beijing, China
6Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
7Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
8Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
9LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
10LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
11I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
12Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
13Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
14Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
15School of Physics, University College Dublin, Dublin, Ireland
16INFN Sezione di Bari, Bari, Italy
17INFN Sezione di Bologna, Bologna, Italy
18INFN Sezione di Ferrara, Ferrara, Italy
19INFN Sezione di Firenze, Firenze, Italy
20INFN Laboratori Nazionali di Frascati, Frascati, Italy
21INFN Sezione di Genova, Genova, Italy
22INFN Sezione di Milano-Bicocca, Milano, Italy
23INFN Sezione di Milano, Milano, Italy
24INFN Sezione di Cagliari, Monserrato, Italy
25INFN Sezione di Padova, Padova, Italy
26INFN Sezione di Pisa, Pisa, Italy
27INFN Sezione di Roma Tor Vergata, Roma, Italy
28INFN Sezione di Roma La Sapienza, Roma, Italy
29Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
30Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
31Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
32AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
33National Center for Nuclear Research (NCBJ), Warsaw, Poland
34Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
35Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
36Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia, Moscow, Russia
37Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
38Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
39Yandex School of Data Analysis, Moscow, Russia
40Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
41Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
42ICCUB, Universitat de Barcelona, Barcelona, Spain
43Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
44European Organization for Nuclear Research (CERN), Geneva, Switzerland
45Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
46Physik-Institut, Universität Zürich, Zürich, Switzerland
47NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
48Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
49University of Birmingham, Birmingham, United Kingdom
50H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
51Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
52Department of Physics, University of Warwick, Coventry, United Kingdom
53STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
54School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
55School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
56Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
57Imperial College London, London, United Kingdom
58School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
59Department of Physics, University of Oxford, Oxford, United Kingdom
60Massachusetts Institute of Technology, Cambridge, MA, United States
61University of Cincinnati, Cincinnati, OH, United States
62University of Maryland, College Park, MD, United States
63Los Alamos National Laboratory (LANL), Los Alamos, United States
64Syracuse University, Syracuse, NY, United States
65Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
66Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
67South China Normal University, Guangzhou, China, associated to 3
68School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
69Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
70Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 10
71Institut für Physik, Universität Rostock, Rostock, Germany, associated to 14
72Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 29
73National Research Centre Kurchatov Institute, Moscow, Russia, associated to
36
74National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 36
75National Research University Higher School of Economics, Moscow, Russia,
associated to 39
76National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
36
77Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 42
78University of Michigan, Ann Arbor, United States, associated to 64

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