EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-208 LHCb-PAPER-2017-030 26th March 2018

Measurement of $C\\!P$ observables in
${{B}^{\pm}}\\!\rightarrow{D}{{K}^{*\pm}}$ decays using two- and four-body $D$
final states

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

Measurements of $C\\!P$ observables in
${{B}^{\pm}}\\!\rightarrow{D}{{K}^{*\pm}}$ decays are presented, where $D$
denotes a superposition of ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ meson states. Decays of the $D$
meson to ${K}^{-}$ ${\pi}^{+}$, ${K}^{-}$ ${K}^{+}$, ${\pi}^{-}$ ${\pi}^{+}$,
${K}^{-}$ ${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$ and ${\pi}^{-}$ ${\pi}^{+}$
${\pi}^{-}$ ${\pi}^{+}$ are used and the ${K}^{*\pm}$ meson is reconstructed
in the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ ${\pi}^{\pm}$ final state.
This analysis uses a data sample of $pp$ collisions collected with the LHCb
experiment, corresponding to integrated luminosities of 1$\mbox{\,fb}^{-1}$,
2$\mbox{\,fb}^{-1}$ and 1.8$\mbox{\,fb}^{-1}$ at centre-of-mass energies
$\sqrt{s}=$ 7$\mathrm{\,Te\kern-1.00006ptV}$, 8$\mathrm{\,Te\kern-1.00006ptV}$
and 13$\mathrm{\,Te\kern-1.00006ptV}$, respectively. The sensitivity of the
results to the CKM angle $\gamma$ is discussed.

Published in JHEP 11 (2017) 156

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

A key characteristic of the Standard Model is that $C\\!P$ violation
originates from a single phase in the CKM quark-mixing matrix [1, 2]. In the
Standard Model the CKM matrix is unitary, leading to the condition
${V_{{u}{d}}}{V_{{u}{b}}^{\ast}}+{V_{{c}{d}}}{V_{{c}{b}}^{\ast}}+{V_{{t}{d}}}{V_{{t}{b}}^{\ast}}=0$,
where $V_{ij}$ are the CKM matrix elements. This relation is represented as a
triangle in the complex plane, with angles $\alpha$, $\beta$ and $\gamma$, and
an area proportional to the amount of $C\\!P$ violation in the quark sector of
the Standard Model [3]. Overconstraining this unitarity triangle may lead to
signs of physics beyond the Standard Model. The CKM angle
$\gamma\equiv\arg\left(-\frac{{V_{{u}{d}}}{{V_{{u}{b}}}}^{*}}{{V_{{c}{d}}}{{V_{{c}{b}}}}^{*}}\right)$
is the least well-known angle of the CKM unitarity triangle. The latest
published LHCb combination from direct measurements with charged and neutral
$B$ decays to a $D$ meson (reconstructed in one of a variety of final states)
and a kaon is $\gamma=\left(72.2^{+6.8}_{-7.3}\right)^{\circ}$ [4]. A global
fit to the CKM triangle by the CKMfitter group [5] obtains a $\gamma$ value of
$(66.9^{+0.9}_{-3.4})^{\circ}$, where this determination of $\gamma$ excludes
all direct measurements. The uncertainties on the indirect measurement are
expected to decrease as lattice QCD calculations become more accurate.
Therefore, precision at the level of $1^{\circ}$ on a direct measurement of
$\gamma$ would test the consistency of the direct and indirect measurements
and thereby the Standard Model. This precision can be achieved through a
combination of measurements of various $B$ decays that are sensitive to
$\gamma$.

Direct measurements of $\gamma$ can be made by exploiting the interference
between ${b}\\!\rightarrow{c}{\overline{u}}{s}$ and
${b}\\!\rightarrow{u}{\overline{c}}{s}$ transitions. These transitions are
present in ${B}\rightarrow{D}^{(*)}{K}^{(*)}$ decays. This analysis measures
$C\\!P$ violation in ${B}^{-}\rightarrow{D}{{K}^{*}}(892)^{-}$ decays,111The
inclusion of charge-conjugate processes is implied, except when discussing
ratios or asymmetries between ${{B}^{+}}$ and ${{B}^{-}}$ decays. with
${{K}^{*}}(892)^{-}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}({{\pi}^{+}}{{\pi}^{-}}){{\pi}^{-}}$, where $D$ denotes a superposition of
${D}^{0}$ and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ meson
states. In this paper ${K}^{*-}$ is used to represent the ${{K}^{*}}(892)^{-}$
resonance. The effect of the interference is observed by reconstructing the
$D$ meson in a final state accessible to both ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ meson states, which gives
sensitivity to the weak phase $\gamma$. In this analysis, only $D$ mesons
decaying to two or four charged kaons and/or pions are considered. The
branching fraction of ${{{B}^{-}}}\\!\rightarrow{D}{{K}^{*-}}$ is of a similar
magnitude to ${{{B}^{-}}}\\!\rightarrow{D}{{K}^{-}}$, which has been
extensively analysed at LHCb [6, 7, 8]. However, the reconstruction
efficiencies associated with the
${{K}^{*-}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{-}}$
decay are lower due to the presence of a long-lived neutral particle.

Two main classes of $D$ decays are used. The first employs $D$ decays into the
$C\\!P$-even eigenstates ${K}^{+}$ ${K}^{-}$ and ${\pi}^{+}$ ${\pi}^{-}$;
these are referred to here as the “GLW” decay modes [9, 10]. The second class
of decay modes involves $D$ decays to ${K}^{\mp}$ ${\pi}^{\pm}$, which is not
a $C\\!P$ eigenstate. In the favoured decay, the pion from the $D$ meson and
that from the ${K}^{*-}$ meson have opposite charge, while in the suppressed
decay (referred to here as the “ADS” [11, 12] decay mode) the pion from the
$D$ meson that from the ${K}^{*-}$ meson have the same charge. The favoured
mode is used as a control mode for many aspects of the analysis since no
$C\\!P$ asymmetry is expected. The ADS decay mode is a combination of a CKM-
favoured ${{{B}^{-}}}\\!\rightarrow{{D}^{0}}{{K}^{*-}}$ decay, followed by a
doubly Cabibbo-suppressed ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decay,
and a CKM- and colour-suppressed ${{{B}^{-}}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{*-}}$ decay, followed by a
Cabibbo-favoured ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
decay. Both paths to the same final state have amplitudes of similar size, and
interference effects are therefore magnified in comparison to the GLW decay
modes, where the decay path via the CKM-favoured
${{{B}^{-}}}\\!\rightarrow{{D}^{0}}{{K}^{*-}}$ dominates. Studies of
${{{B}^{-}}}\\!\rightarrow D{{K}^{-}}$ and
${{B}^{0}}\\!\rightarrow{D}{{K}^{*0}}$ decays have been published by the LHCb
collaboration [6, 13].

The GLW and ADS methods can be extended to the
${D}\\!\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}$ and
${D}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ inclusive
four-body final states, provided external information is available on the
overall behaviour of the intermediate resonances, averaged over phase space
[14, 15]. These channels have previously been studied for
${{{B}^{-}}}\\!\rightarrow{D}{{K}^{-}}$ decays [6], and are included in this
paper for the first time in ${{{B}^{-}}}\\!\rightarrow{D}{{K}^{*-}}$ decays.
The ${{{B}^{-}}}\\!\rightarrow{D}{{K}^{*-}}$ channel has previously been
investigated by the BaBar collaboration using a variety of two-body $D$ decay
modes [16]. Also, both the BaBar and Belle collaborations have performed
studies on ${{{B}^{-}}}\\!\rightarrow{D}{{K}^{*-}}$ with
${D}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ [17, 18].

Twelve quantities, collectively referred to as $C\\!P$ observables, are
measured in this analysis

  * •

The $C\\!P$ asymmetry for the favoured decay mode

$A_{{K}\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)-\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\text{
.}$ (1)

  * •

The $C\\!P$ asymmetry for the ${D}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decay mode

$A_{{K}{K}}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*-}}\right)-\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*+}}\right)}\text{
. }$ (2)

  * •

The $C\\!P$ asymmetry for the ${D}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decay
mode

$A_{\pi\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)-\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\text{
. }$ (3)

  * •

The ratio of the rate for the ${D}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ decay mode
to that of the favoured decay mode, scaled by the branching fractions

$R_{{K}{K}}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\times\frac{{\mathcal{B}}(D^{0}\rightarrow
K^{-}\pi^{+})}{{\mathcal{B}}(D^{0}\rightarrow K^{+}K^{-})}\text{ . }$ (4)

  * •

The ratio of the rate for the ${D}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decay
mode to that of the favoured decay mode, scaled by the branching fractions

$R_{\pi\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\times\frac{{\mathcal{B}}(D^{0}\rightarrow
K^{-}\pi^{+})}{{\mathcal{B}}(D^{0}\rightarrow\pi^{+}\pi^{-})}\text{ . }$ (5)

  * •

The ratio of the rate for the ADS decay mode to that of the favoured decay
mode for ${{B}^{+}}$ decays

$R^{+}_{K\pi}=\frac{\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\text{
. }$ (6)

  * •

The ratio of the rate for the ADS decay mode to that of the favoured decay
mode for ${{B}^{-}}$ decays

$R^{-}_{K\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)}\text{
. }$ (7)

  * •

The $C\\!P$ asymmetry for the favoured
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$ decay mode

$A_{{K}\pi\pi\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)-\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\text{
. }$ (8)

  * •

The $C\\!P$ asymmetry for the
${D}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decay mode

$A_{\pi\pi\pi\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)-\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\text{
. }$ (9)

  * •

The ratio of the rate for the
${D}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decay mode to
that of the favoured decay mode, scaled by the branching fractions

$R_{\pi\pi\pi\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)+\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\\\
\times\frac{\mathcal{B}(D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}})}{\mathcal{B}(D^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}})}\text{
. }$ (10)

  * •

The ratio of the rate for the four-body ADS decay mode to that of the four-
body favoured decay mode for ${{B}^{+}}$ decays

$R^{+}_{K\pi\pi\pi}=\frac{\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*+}}\right)}{\Gamma\left({{{B}^{+}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*+}}\right)}\text{
. }$ (11)

  * •

The ratio of the rate of the four-body ADS decay mode to that of the four-body
favoured decay mode for ${{B}^{-}}$ decays

$R^{-}_{K\pi\pi\pi}=\frac{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}}\right)}{\Gamma\left({{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}\right)}\text{
. }$ (12)

The asymmetries $A_{{K}\pi}$ and $A_{{K}\pi\pi\pi}$ should be essentially zero
due to the very small interference expected in the configuration of $B$ and
$D$ decays. Due to negligible direct $C\\!P$ violation in $D$ decays [19], the
observables $A_{{K}{K}}$ and $A_{\pi\pi}$ should be equal and are often
labelled together as $A_{{C\\!P}+}$; similarly the observables $R_{{K}{K}}$
and $R_{\pi\pi}$ should be equal and are labelled $R_{{C\\!P}+}$. The
analogous observables to $R_{{C\\!P}+}$ and $A_{{C\\!P}+}$ for the ADS mode
are $R_{ADS}$ and $A_{ADS}$. However, $R_{ADS}$ and $A_{ADS}$ are not used for
the ADS decay mode, instead the ratios are measured separately for the
positive and negative charges. The reason for this choice is that the
uncertainty in $A_{ADS}$ depends on the value of $R_{ADS}$, therefore these
observables are statistically dependent, raising problems for the low yields
expected in the ADS mode. Hence the statistically independent observables
$R^{+}_{K\pi}$ and $R^{-}_{K\pi}$ are preferred.

The $C\\!P$ observables measured in this analysis can be related to the
physics parameters to be determined, namely $\gamma$, $r_{B}$ and
$\delta_{B}$. The parameter $r_{B}$ is the ratio of the magnitudes between the
suppressed and favoured amplitudes of the $B$ decay and $\delta_{B}$ is the
strong-phase difference between these amplitudes. The expected value is
$r_{B}$ $\sim 0.1$, similar to that in the
${{{B}^{-}}}\\!\rightarrow{D}{{K}^{-}}$ decay. Both $r_{B}$ and $\delta_{B}$
are averaged over the region of $D$ ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
${\pi}^{-}$ phase space corresponding to the ${K}^{*-}$ selection window. A
coherence factor, $\kappa$, accounts for the contribution of
${{{B}^{-}}}\rightarrow D{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{-}}$
decays that are not due to an intermediate $K^{*}(892)^{-}$ resonance [20],
where $\kappa=1$ denotes a pure $K^{*}(892)^{-}$ contribution. Given there is
a negligible effect from both charm mixing [21] and $C\\!P$ violation in $D$
decays [19], the relationships between the $C\\!P$ observables and physics
parameters are given in the following equations,

$A_{{C\\!P}+}=\frac{2\kappa r_{B}\sin\delta_{B}\sin\gamma}{1+r_{B}^{2}+2\kappa
r_{B}\cos\delta_{B}\cos\gamma}\text{ ,}$ (13)

$R_{{C\\!P}+}=1+r_{B}^{2}+2\kappa r_{B}\cos\delta_{B}\cos\gamma\text{ ,}$ (14)

$R^{\pm}_{K\pi}=\frac{r_{B}^{2}+\left(r_{D}^{K\pi}\right)^{2}+2\kappa
r_{B}r_{D}^{K\pi}\cos(\delta_{B}+\delta_{D}^{K\pi}\pm\gamma)}{1+r_{B}^{2}\left(r_{D}^{K\pi}\right)^{2}+2\kappa
r_{B}r_{D}^{K\pi}\cos(\delta_{B}-\delta_{D}^{K\pi}\pm\gamma)}\text{ ,}$ (15)

$A_{\pi\pi\pi\pi}=\frac{2\kappa\left(2F_{4\pi}-1\right)r_{B}\sin\delta_{B}\sin\gamma}{1+r_{B}^{2}+2\kappa\left(2F_{4\pi}-1\right)r_{B}\cos\delta_{B}\cos\gamma}\text{
,}$ (16)

$R_{\pi\pi\pi\pi}=1+r_{B}^{2}+2\kappa\left(2F_{4\pi}-1\right)r_{B}\cos\delta_{B}\cos\gamma\text{
,}$ (17)

$R^{\pm}_{K\pi\pi\pi}=\frac{r_{B}^{2}+\left(r_{D}^{K3\pi}\right)^{2}+2\kappa
r_{B}\kappa_{K3\pi}r_{D}^{K3\pi}\cos(\delta_{B}+\delta_{D}^{K3\pi}\pm\gamma)}{1+\left(r_{B}r_{D}^{K3\pi}\right)^{2}+2\kappa
r_{B}\kappa_{K3\pi}r_{D}^{K3\pi}\cos(\delta_{B}-\delta_{D}^{K3\pi}\pm\gamma)}\text{
.}$ (18)

These relationships depend on several parameters describing the $D$ decays,
which are taken from existing measurements. The parameters $r_{D}^{K\pi}$ and
$\delta_{D}^{K\pi}$ are the magnitude of the amplitude ratio and the strong-
phase difference between the suppressed and favoured amplitudes of the $D$
decay, namely ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$ respectively [22]. Similarly,
the parameters $r_{D}^{K3\pi}$ and $\delta_{D}^{K3\pi}$ are the equivalent
quantities for the decays
${{D}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$, averaged
over phase space [23, 24]. Two-body
${D}\\!\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}$ decays are characterised by a
single strong phase, however for multibody
${D}\\!\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}$ decays
the strong phase varies over the phase space. By averaging the strong phase
variation the interference effects are diluted. This effect is accounted for
by the parameter $\kappa_{K3\pi}$ [23, 24]. The parameter $F_{4\pi}\sim 0.75$
[15] accounts for the fact that
${D}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, though
predominantly $C\\!P$ even, is not a pure $C\\!P$ eigenstate.

##  2 Detector, online selection and simulation

The LHCb detector [25, 26] 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 (VELO) surrounding the
$pp$ interaction region, a large-area silicon-strip detector (TT) 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
momentum, $p$, of charged particles with a relative uncertainty that varies
from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
(RICH). Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
calorimeter and a hadronic calorimeter. Muons are identified by a system
composed of alternating layers of iron and multiwire proportional chambers,
and gas electron multiplier detectors.

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.
Signal events considered in the analysis must fulfil hardware and software
trigger requirements. 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. At the software stage, at least
one charged particle should have high $p_{\mathrm{T}}$ and large
$\chi^{2}_{\text{IP}}$ with respect to any PV, where $\chi^{2}_{\text{IP}}$ is
defined as the difference in the vertex-fit $\chi^{2}$ of a given PV fitted
with and without the considered track. The software trigger designed to select
$b$-hadron decays uses a multivariate algorithm [28] to identify a two-,
three- or four-track secondary vertex with a large scalar sum of the
$p_{\mathrm{T}}$ of the associated charged particles and a significant
displacement from the PVs. The PVs are fitted with and without the $B$
candidate, and the PV with the smallest $\chi^{2}_{\text{IP}}$ is associated
with the $B$ candidate.

The analysis presented is based on $pp$ collision data corresponding to an
integrated luminosity of 1$\mbox{\,fb}^{-1}$ at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ collected in 2011, 2$\mbox{\,fb}^{-1}$ at
8$\mathrm{\,Te\kern-1.00006ptV}$ collected in 2012 (jointly referred to as Run
1), and 1.8$\mbox{\,fb}^{-1}$ at 13$\mathrm{\,Te\kern-1.00006ptV}$ collected
in 2015 and 2016 (referred to as Run 2). There are several differences between
data collected in Run 1 and Run 2. The main difference is the higher
$b\bar{b}$ production cross-section in Run 2 [29]. The average number of $pp$
interactions per bunch crossing is reduced to 1.1 in Run 2 compared to 1.7 in
Run 1. The net effect is that, despite the higher energy of the collisions,
the background levels and signal-to-background ratios in Run 1 and Run 2 for
the type of decay analysed here are similar. Before the start of Run 2, the
aerogel radiator was removed from the first RICH detector [30], which improves
the detector resolution. Hence, for momenta typical of decays in this
analysis, the particle identification criteria have resulted in an increased
efficiency of signal selection while simultaneously decreasing the rate of
misidentified backgrounds. For the
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ decay mode, the
combination of higher $b\bar{b}$ production cross-section, improved particle
identification and improvements to the online selection in Run 2 have resulted
in a factor of three increase in the yield for a given integrated luminosity.

Simulated event samples are used for the study of efficiencies. In the
simulation, $pp$ collisions are generated using Pythia [31, *Sjostrand:2007gs]
with a specific LHCb configuration [33]. Decays of hadronic particles are
described by EvtGen [34], in which final-state radiation is generated using
Photos [35]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [36,
*Agostinelli:2002hh] as described in Ref. [38].

##  3 Offline selection

The ${K}^{*-}$ meson is reconstructed in the decay
${{K}^{*-}}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{-}}$
and the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ meson is reconstructed
through its decay to two charged pions. If the pions from the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decay leave sufficient hits in the
VELO to be included in the track reconstruction, the reconstructed
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ meson is called “long”. Due to the
high boost from the $pp$ collision many ${K}^{0}_{\mathrm{\scriptscriptstyle
S}}$ particles decay outside the VELO. If the pions from the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decay do not leave sufficient hits
in the VELO, the reconstructed ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ meson
is called “downstream”, with the first hits being recorded in the TT, which
typically results in poorer mass resolution. These
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ reconstruction types are treated as
separate data samples and a slightly different selection is applied to each.

Reconstructed $B$ candidates are formed by combining a ${K}^{*-}$ candidate
with a $D$ candidate, which are required to form a good-quality vertex. For
each $D$, ${K}^{*-}$, and ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidate
the reconstructed meson masses are required to lie within
25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the $D$ mass,
75${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the ${K}^{*-}$ mass, and
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ mass for long candidates and
20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for downstream candidates [39]. A
kinematic fit [40] is performed on the full $B$ decay chain constraining the
$B$ candidate to point towards the PV, and the $D$ and
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates to have their known
masses [39]. To suppress charmless backgrounds, the $D$ decay vertex is
required to be well-separated from and downstream of the ${{B}^{-}}$ decay
vertex. Also, the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decay vertex is
required to be well-separated from and downstream of the ${{B}^{-}}$ decay
vertex in order to suppress
${{{B}^{-}}}\\!\rightarrow{D}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays. The
selection window of $\pm$75${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, 1.5
times the $K^{*}(892)^{-}$ natural width, is required to suppress
${{{B}^{-}}}\rightarrow D{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{-}}$
decays that do not proceed via an intermediate $K^{*}(892)^{-}$ resonance.
Further suppression of these decays is achieved by requiring the magnitude of
the cosine of the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ helicity angle to
be greater than 0.3. The ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ helicity
angle is defined as the angle between the ${K}^{0}_{\mathrm{\scriptscriptstyle
S}}$ and the ${{B}^{-}}$ momentum vectors in the ${K}^{*-}$ rest frame. This
requirement retains 97% of true ${K}^{*-}$ decays, which are distributed
parabolically in this variable, while rejecting 30% of the background.

Requirements, based mainly on the RICH system, are applied to all $D$ decay
products to identify them as kaons or pions. These selections are applied such
that each $D$ candidate is assigned a unique category. Cross-feed between the
${K}^{-}$ ${\pi}^{+}$, ${K}^{+}$ ${K}^{-}$ and ${\pi}^{+}$ ${\pi}^{-}$ $D$
final states is negligible because after misidentification of a ${\pi}^{-}$
meson as a ${K}^{-}$ meson (or vice versa) the reconstructed mass of the $D$
meson lies outside the $D$ mass selection window. However, the favoured decay
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ can appear in
the ${{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{-}}{{K}^{+}}){{K}^{*-}}$ sample due
to misidentification of both $D$ decay products. To suppress this, a veto is
applied to the ADS decay mode. The $D$ mass is reconstructed assuming the mass
hypotheses of the decay products are swapped. If the resulting value is within
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal $D$ mass, the
candidate is removed from the sample, after which any remaining contamination
is negligible while retaining 92% of the signal. Similarly a
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ veto selection is applied to the
four-body ADS decay mode to prevent the contamination of
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}$
in the
${{{B}^{-}}}\\!\rightarrow{D}({{\pi}^{-}}{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}$
sample. The swapped $D$ mass hypothesis is considered for both ${\pi}^{+}$
mesons separately, resulting in a combined signal efficiency for the vetoes of
90%.

Combinatorial background is suppressed using a Boosted Decision Tree (BDT)
multivariate discriminant [41]. To train the BDT for two-body decays,
simulated ${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$
candidates are used as a signal sample and events from the high-mass sideband
region of the ${{B}^{-}}$ mass, above
5600${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, in the favoured
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ decay mode are
used as a sample of combinatorial background. An analogous strategy is
employed in the BDT for four-body decays. Various input quantities are used to
exploit the topology of the decay; of particular importance are the
${{B}^{-}}$ vertex-fit $\chi^{2}$ and the $p_{\mathrm{T}}$ asymmetry between
the ${{B}^{-}}$ candidate and other tracks from the same PV, defined as

$A_{\mbox{$p_{\mathrm{T}}$}}=\frac{p_{\rm T}^{B}-p_{\rm
T}^{\text{cone}}}{p_{\rm T}^{B}+p_{\rm T}^{\text{cone}}}$ (19)

where $p_{\rm T}^{B}$ is the $p_{\mathrm{T}}$ of the reconstructed ${{B}^{-}}$
signal candidate and $p_{\rm T}^{\text{cone}}$ is the scalar sum of the
$p_{\mathrm{T}}$ of all other tracks in a cone surrounding the ${{B}^{-}}$
candidate. This asymmetry is a quantitative measure of the isolation of the
${{B}^{-}}$ candidate. Other input quantities used include the logarithm of
the $\chi^{2}_{\text{IP}}$ for various particles and the $p_{\mathrm{T}}$ of
the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidate (for downstream
candidates only). The selection requirement on the BDT output was chosen to
minimise the uncertainty on the $C\\!P$ observables. The optimisation is
performed separately for the GLW and ADS decay modes. Averaged across the
whole dataset used for the analysis, the BDT selection applied to the favoured
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ channel gives a
signal efficiency of 95% (90%) and a background rejection of 94% (95%) for
long (downstream) candidates. Similarly, the four-body favoured
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}$
channel gives a signal efficiency of 95% (93%) and a background rejection of
96% (97%) for long (downstream) candidates.

##  4 Fit to the invariant mass distribution

Extended unbinned maximum likelihood fits are applied to the $B$ candidate
mass spectra, in the mass range
4900–5600${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, for candidates
reconstructed in the favoured decay modes
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ and
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}$.
The same fit model is applied to both spectra. The model consists of a signal
component, backgrounds from partially reconstructed decays and a combinatorial
background shape. The charmless background has been suppressed to negligible
levels, therefore no component is included in the fit. The signal component is
described by the sum of two Crystal Ball (CB) functions [42] with the same
peak position, which contain small radiative tails that extend towards lower
invariant mass. The signal shape parameters are determined from simulation,
except for the common peak position and one of the widths, which are allowed
to vary in the fit. The combinatorial background is described by an
exponential function. The results of these fits are shown in Fig. 1.

Backgrounds from partially reconstructed decays include
${B}\\!\rightarrow{{D}^{*}}{{K}^{*}}$ decays where a pion or photon is not
reconstructed, namely
${{{B}^{-}}}\\!\rightarrow{{D}^{*0}}({{D}^{0}}{{\pi}^{0}}){{K}^{*-}}$,
${{{B}^{-}}}\\!\rightarrow{{D}^{*0}}({{D}^{0}}\gamma){{K}^{*-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\\!\rightarrow{{D}^{*+}}({{D}^{0}}{{\pi}^{+}}){{K}^{*-}}$.
These are decays of $B$ mesons into two vector particles, which are described
by three independent helicity amplitudes, corresponding to the helicity states
of the ${D}^{*}$ meson, denoted by $-1$, $0$ and $+1$. The reconstructed
$B$-candidate mass distributions for $-1$ and $+1$ helicity states are
indistinguishable so these states are collectively named $\pm 1$. Therefore,
for each ${D}^{*}$ ${K}^{*-}$ channel, two different components are
considered, $0$ and $\pm 1$. The shape of these components are determined from
simulations and parameterised as Gaussian functions convolved with a second-
order polynomial, described in detail in Ref. [43, 44], with all parameters
fixed in the fit. The ratio between the yields of the three ${D}^{*}$
${K}^{*-}$ decay modes are fixed according to their branching fractions and
selection efficiencies, assuming no $C\\!P$ violation. This procedure assumes
that the longitudinal polarisation fraction for ${D}^{*}$ ${K}^{*-}$ decays is
the same for ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ and
${{B}^{-}}$ mesons. The total partially reconstructed yield is allowed to vary
as well as the yield ratio between the sum of the 0 shapes and the sum of the
$\pm 1$ shapes.

Figure 1: Invariant mass distribution with the fit result superimposed for the
favoured ${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ decay
mode (top), and
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}}$
decay mode (bottom), using Run 1 and Run 2 data combined. The labels 0 and
$\pm$1 correspond to the helicity state of the ${D}^{*}$ meson.

As seen from the fit projections in Fig. 1, these background contributions are
sufficient to describe the overall invariant mass distribution of the favoured
decay mode. A number of other backgrounds which could appear close to the
signal peak are studied in simulation and found to be negligible, for example
${{{B}^{-}}}\\!\rightarrow{D}{{K}^{*-}}{{\pi}^{0}}$ and
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\pi\pi){{K}^{-}}$. Figure 1 shows that the main background contribution
near the signal peak is combinatorial background, while only a small amount of
partially reconstructed background enters the signal region. A significant
fraction of the combinatorial background is expected to come from
${{{B}^{-}}}\\!\rightarrow{D}{{\pi}^{-}}X$ decays combined with a real but
unrelated ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ meson, which is consistent
with the observed difference in background level between the
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ and
${{{B}^{-}}}\\!\rightarrow{D}({{K}^{+}}{{\pi}^{-}}){{K}^{*-}}$ decay modes. In
the case of the ${{{B}^{-}}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*-}}$
decay mode, an additional background coming from the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(p{{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ needs to be
considered, where the ${\pi}^{+}$ meson is not reconstructed and the proton is
misidentified as a kaon. The shape of this background is obtained by
parameterising the mass distribution from simulated background events; the
shape parameters are fixed in the fits described below. The yield of
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(p{{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ compared to signal in
the ${{{B}^{-}}}\\!\rightarrow{D}({{K}^{-}}{{\pi}^{+}}){{K}^{*-}}$ favoured
decay mode is allowed to vary.

Restricting the lower limit of the mass range to
5230${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ removes 0.4% of signal and
avoids the need to fit the backgrounds from partially reconstructed decays in
each of the decay modes. This strategy improves fit stability in the decay
modes with lower yields. The shape and yield of the small amount of background
from partially reconstructed decays present in all $D$ decay categories above
5230${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is determined and fixed from
the fit to data with the favoured decay, adjusted for the smaller branching
fractions of the rarer $D$ decays. The yield is estimated to be less than one
candidate for all $C\\!P$-violating decay modes, and therefore uncertainties
due to the assumptions present in the initial fit have a very small effect.
These uncertainties in the yield, shape and possible asymmetries in the
distribution between ${{B}^{+}}$ and ${{B}^{-}}$ are sources of systematic
uncertainty.

A simultaneous fit is performed to 56 $B$-meson mass distributions,
corresponding to each of the seven $D$ decay modes (${K}^{-}$ ${\pi}^{+}$,
${K}^{+}$ ${K}^{-}$, ${\pi}^{+}$ ${\pi}^{-}$, ${K}^{+}$ ${\pi}^{-}$, ${K}^{-}$
${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$, ${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$
${\pi}^{-}$ and ${K}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$ ${\pi}^{-}$), two $B$-meson
charges (${{B}^{+}}$ and ${{B}^{-}}$), two
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ reconstruction types (long and
downstream) and two periods of data taking (Run 1 and Run 2). Based on fits to
the data and simulation samples, the same signal peak position and width are
used for the two periods of data taking, $B$-meson charges and
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ reconstruction types, but they are
allowed to differ between two- and four-body decay modes. The combinatorial
background slope is required to have the same value for all two- and four-body
decay modes separately, but can differ between long and downstream categories.

The parameters determined from the simultaneous fit are the yields in the
favoured signal decay modes and the $C\\!P$ observables $A_{K\pi}$, $A_{KK}$,
$A_{\pi\pi}$, $R_{KK}$, $R_{\pi\pi}$, $R^{+}_{K\pi}$, $R^{-}_{K\pi}$,
$A_{K\pi\pi\pi}$, $A_{\pi\pi\pi\pi}$, $R_{\pi\pi\pi\pi}$, $R^{+}_{K\pi\pi\pi}$
and $R^{-}_{K\pi\pi\pi}$. The observables are related to the ratios between
the yields through various efficiency corrections, given by

$R_{hh}=\frac{N({{{B}^{-}}}\\!\rightarrow
D(h^{+}h^{-}){{K}^{*-}})}{N({{{B}^{-}}}\\!\rightarrow
D({{K}^{-}}{{\pi}^{+}}){{K}^{*-}})}\times\frac{{\mathcal{B}}({{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}})}{{\mathcal{B}}({{D}^{0}}\\!\rightarrow
hh)}\times\frac{\epsilon_{\text{sel}}(K\pi)}{\epsilon_{\text{sel}}(hh)}\times\frac{\epsilon_{\text{PID}}(K\pi)}{\epsilon_{\text{PID}}(hh)}\text{
,}$ (20) $R^{\pm}_{K\pi}=\frac{N({{B}^{\pm}}\\!\rightarrow
D({{K}^{\mp}}{{\pi}^{\pm}}){{K}^{*\pm}})}{N({{B}^{\pm}}\\!\rightarrow
D({{K}^{\pm}}{{\pi}^{\mp}}){{K}^{*\pm}})}\times\frac{\epsilon_{\text{sel}}(K\pi)}{\epsilon_{\text{sel}}(\pi
K)}\times\frac{1}{\epsilon_{\text{veto}}(\pi K)}\text{ ,}$ (21)
$R_{\pi\pi\pi\pi}=\frac{N({{{B}^{-}}}\\!\rightarrow
D({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*-}})}{N({{{B}^{-}}}\\!\rightarrow
D({{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*-}})}\times\frac{{\mathcal{B}}({{D}^{0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}})}{{\mathcal{B}}({{D}^{0}}\\!\rightarrow\pi\pi\pi\pi)}\times\frac{\epsilon_{\text{sel}}(K\pi\pi\pi)}{\epsilon_{\text{sel}}(\pi\pi\pi\pi)}\times\frac{\epsilon_{\text{PID}}(K\pi\pi\pi)}{\epsilon_{\text{PID}}(\pi\pi\pi\pi)}\text{
,}$ (22) $R^{\pm}_{K\pi\pi\pi}=\frac{N({{B}^{\pm}}\\!\rightarrow
D({{K}^{\mp}}{{\pi}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}){{K}^{*\pm}})}{N({{B}^{\pm}}\\!\rightarrow
D({{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{\pm}}{{\pi}^{\mp}}){{K}^{*\pm}})}\times\frac{\epsilon_{\text{sel}}(K\pi\pi\pi)}{\epsilon_{\text{sel}}(\pi
K\pi\pi)}\times\frac{1}{\epsilon_{\text{veto}}(\pi K\pi\pi)}\text{ ,}$ (23)

where $\epsilon_{\text{sel}}$, $\epsilon_{\text{PID}}$ and
$\epsilon_{\text{veto}}$ are the selection, particle-identification and veto
efficiencies, respectively, $N$ is the yield of the specified decay and $h$
represents a $\pi$ or $K$ meson. The veto is only applied to the ADS decay
mode to reduce cross-feed from the favoured decay. These efficiencies are
determined from simulation. The selection efficiency for various $D$ decay
modes accounts for any differences in kinematics between these modes as well
as a tighter BDT cut in the ADS decay mode, which is applied in order to
optimise the uncertainty in the $C\\!P$ observables. Any further correction to
the four-body observables due to nonuniform acceptance was found to be
negligible. The efficiencies cancel for the determination of the $C\\!P$
asymmetries, while corrections are applied for the ${{B}^{+}}$, ${{B}^{-}}$
production asymmetry, $A_{\text{prod}}$, and decay mode dependent detection
asymmetries, $A_{\text{det}}$, which are taken from previous LHCb measurements
for production asymmetry [45], kaon detection asymmetry [46] and pion
detection asymmetry [47]. The value $A_{\text{prod}}$ is assumed to be the
same for 7$\mathrm{\,Te\kern-1.00006ptV}$, 8$\mathrm{\,Te\kern-1.00006ptV}$
and 13$\mathrm{\,Te\kern-1.00006ptV}$ data. A possible difference in
$A_{\text{prod}}$ for Run 2 data compared to Run 1 is accounted for as a
systematic uncertainty. As the asymmetries are small, $\mathcal{O}$(1%) or
less, the observed uncorrected asymmetry $A_{\text{raw}}$ can be expressed as
the sum $A_{\text{raw}}=A_{\text{phys}}+A_{\text{prod}}+A_{\text{det}}$, where
$A_{\text{phys}}$ is the $C\\!P$ asymmetry to be extracted. Hence,
$A_{\text{prod}}$ and $A_{\text{det}}$ provide additive corrections to the
measured asymmetry.

##  5 Results

The invariant mass spectra and resulting fits to data, combining Run 1, Run 2,
long and downstream categories, are shown in Figs. 2 and 3. The yields
determined from the fitted parameters are given in Table 1. The Wilks’ theorem
statistical significance [48] for the two-body ADS decay mode is 4.2$\sigma$,
while for the four-body ADS decay mode it is 2.8$\sigma$. This represents the
first evidence of the two-body suppressed decay.

(a)  (b)  (c)  (d)

Figure 2: Result of fits to data for the two-body decay modes with Run 1, Run
2, long and downstream categories summed for presentation. The signal is
represented by the red shaded area, the combinatorial background by the dotted
blue line and the partially reconstructed background by the solid green line.
In the ${{D}^{0}}\\!\rightarrow{{K}^{+}}{{K}^{-}}$ fits the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{K}^{*-}}$ background is represented by the dashed
purple line. The total fit is given by the black line. The residuals, shown
below each plot, are defined as the difference between the data and the fit
value in each bin, normalised by the uncertainty.

(a)  (b)  (c)

Figure 3: Result of fits to data for the four-body decay modes with Run 1, Run 2, long and downstream categories summed for presentation. The signal is represented by the red shaded area, the combinatorial background by the dotted blue line and the partially reconstructed background by the solid green line. The total fit is given by the black line. The residuals are shown below each plot. Table 1: Fitted yields in each of the $B$ decay modes. The uncertainties are statistical only. Decay mode |  ${{B}^{-}}$ yield |  ${{B}^{+}}$ yield
---|---|---
$\begin{aligned} &{{B}^{\pm}}\\!\rightarrow{D}({{K}^{\pm}}{{\pi}^{\mp}}){{K}^{*\pm}}\\\ &{{B}^{\pm}}\\!\rightarrow{D}({{K}^{+}}{{K}^{-}}){{K}^{*\pm}}\\\ &{{B}^{\pm}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}){{K}^{*\pm}}\\\ &{{B}^{\pm}}\\!\rightarrow{D}({{K}^{\mp}}{{\pi}^{\pm}}){{K}^{*\pm}}\\\ &{{B}^{\pm}}\\!\rightarrow{D}({{K}^{\pm}}{{\pi}^{\mp}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*\pm}}\\\ &{{B}^{\pm}}\\!\rightarrow{D}({{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{*\pm}}\\\ &{{B}^{\pm}}\\!\rightarrow{D}({{K}^{\mp}}{{\pi}^{\pm}}{{\pi}^{-}}{{\pi}^{+}}){{K}^{*\pm}}\\\ \end{aligned}$ | $\begin{aligned} 996&\pm 34\\\ 134&\pm 14\\\ 45&\pm 10\\\ 1.6&\pm 1.9\\\ 556&\pm 26\\\ 59&\pm 10\\\ 3&\pm 5\\\ \end{aligned}$ | $\begin{aligned} 1035&\pm 35\\\ 121&\pm 13\\\ 33&\pm 9\\\ 19&\pm 7\\\ 588&\pm 27\\\ 56&\pm 10\\\ 10&\pm 6\\\ \end{aligned}$

Branching fractions [39], various efficiencies and asymmetries are used as
inputs to the simultaneous fit in order to relate the measured yields to the
$C\\!P$ observables. Each of these inputs has an associated uncertainty which
needs to be propagated to the $C\\!P$ observables giving rise to the
systematic uncertainties. In the case of the efficiencies, uncertainties arise
from a limited sample size of simulated events. Uncertainties on
$A_{\text{prod}}$ and $A_{\text{det}}$ are taken from previous LHCb
measurements in Run 1 [45, 46, 47]. The changes to the detector between the
data-taking periods are not expected to significantly affect the
$A_{\text{det}}$ measurement. For $A_{\text{prod}}$, a conservative estimate,
double the Run 1 uncertainty, is assigned to accommodate a possible dependence
of the production asymmetry on the centre-of-mass energy. The systematic
uncertainties due to the use of fixed inputs from branching ratios, simulation
efficiencies, asymmetry corrections and shape parameters are estimated by
performing multiple fits to data where each relevant parameter is varied
according to a Gaussian distribution with the width as the assigned
uncertainty. The standard deviation of the fitted parameter distribution is
assigned as the systematic uncertainty. Correlations between the shape
parameters are small, typically less than 10%, and are ignored. Tests, where
the most relevant correlations have been included, show a negligible impact on
the systematic uncertainty arising from the fixed shape parameters.

Other systematic uncertainties arise from the modelling of the signal and
partially reconstructed backgrounds and the effect of any residual charmless
$B$ decays. The systematic uncertainties from these sources are computed by
generating pseudoexperiments. In each case the generated model is varied
according to the systematic effects being estimated. The systematic
uncertainty on each observable is taken to be the difference between the mean
of the fitted parameter distribution and the generated value. The systematic
uncertainty on the partially reconstructed background takes into account
uncertainties in the yield and shape parameters, as well as possible
asymmetries due to $C\\!P$ violation. The contamination from charmless $B$
decays is consistent with zero, although it has a large uncertainty.
Pseudoexperiments are generated with charmless decays according to the fit
model, with the number of events fluctuating according to the uncertainty in
the fit. The assumption that the slope of the function describing the
combinatorial background is the same for all $D$ decay modes has an associated
uncertainty. Pseudoexperiments are generated fixing the slope parameters to a
different value for each decay mode, where the value used is obtained from
fits in the mass region above the $B$ mass. For the ADS mode, a potential
background from
${{B}^{0}_{s}}\\!\rightarrow{D}({{K}^{*}}(1410)^{0}\rightarrow{{K}^{*}}(892)^{-}{{\pi}^{+}})$,
where the ${\pi}^{+}$ meson is not reconstructed, is considered. An estimate
of the contribution using simulated events and the branching fraction [49] is
found to be $2.6\pm 2.6$ events, which is consistent with observations from
data in the region of $B$ mass below the lower limit of the simultaneous fit.
The shape of this background is obtained by parameterising the mass
distribution from simulated events. A systematic is assigned by performing
many fits to data varying the yield according to a Gaussian distribution with
the width as the assigned uncertainty. The standard deviation of the fitted
parameter distribution is assigned as the systematic uncertainty. A summary of
the components of the systematic uncertainties for the $C\\!P$ observables is
given in Table 2.

Table 2: Summary of systematic uncertainties. Uncertainties are not shown if they are more than two orders of magnitude smaller than the statistical uncertainty. | $A_{K\pi}$ | $A_{KK}$ | $A_{\pi\pi}$ | $R_{KK}$ | $R_{\pi\pi}$ | $R^{+}_{K\pi}$ | $R^{-}_{K\pi}$ | $A_{K\pi\pi\pi}$ | $A_{\pi\pi\pi\pi}$ | $R_{\pi\pi\pi\pi}$ | $R^{+}_{K\pi\pi\pi}$ | $R^{-}_{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
Statistical | $0.023$ | $0.07$ | $0.13$ | $0.09$ | $0.15$ | $0.006$ | $0.004$ | $0.031$ | $0.11$ | $0.13$ | $0.008$ | $0.007$
Branching fractions | $-$ | $-$ | $0.001$ | $0.013$ | $0.012$ | $-$ | $-$ | $-$ | $0.0008$ | $0.027$ | $-$ | $-$
Selection efficiencies | $-$ | $-$ | $-$ | $0.007$ | $0.006$ | $0.0002$ | $-$ | $-$ | $0.0008$ | $0.014$ | $-$ | $-$
PID efficiencies | $-$ | $-$ | $-$ | $0.002$ | $0.002$ | $-$ | $-$ | $-$ | $-$ | $0.002$ | $-$ | $-$
Veto efficiencies | $-$ | $-$ | $-$ | $-$ | $-$ | $0.0001$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
$A_{\text{prod}}$ | $0.0073$ | $0.007$ | $0.008$ | $-$ | $-$ | $-$ | $-$ | $0.0079$ | $0.0077$ | $-$ | $-$ | $-$
$A_{\text{det}}$ | $0.0034$ | $0.003$ | $0.003$ | $-$ | $-$ | $0.0001$ | $-$ | $0.0034$ | $0.0030$ | $-$ | $0.0001$ | $-$
Signal shape | $0.0011$ | $0.003$ | $0.003$ | $0.011$ | $0.027$ | $0.0011$ | $0.0013$ | $0.0017$ | $0.0022$ | $0.010$ | $0.0030$ | $0.0038$
Combinatorial shape | $0.0012$ | $0.003$ | $0.005$ | $0.004$ | $0.009$ | $0.0002$ | $0.0003$ | $0.0001$ | $0.0018$ | $-$ | $0.0012$ | $0.0004$
Partially reconstructed shape | $0.0007$ | $0.001$ | $0.003$ | $0.001$ | $0.005$ | $-$ | $0.0003$ | $0.0003$ | $0.0005$ | $0.002$ | $0.0008$ | $0.0001$
Charmless | $0.0008$ | $-$ | $0.003$ | $0.002$ | $0.007$ | $-$ | $0.0003$ | $0.0009$ | $0.0030$ | $0.002$ | $0.0008$ | $0.0001$
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{{K}^{*-}}$ | $0.0002$ | $-$ | $-$ | $0.011$ | $0.001$ | $0.0001$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
${{B}^{0}_{s}}\\!\rightarrow{D}{{K}^{*}}(1410)^{0}$ | $-$ | $-$ | $-$ | $-$ | $-$ | $0.0005$ | $0.0001$ | $-$ | $-$ | $-$ | $-$ | $-$
Total systematic | $0.0083$ | $0.009$ | $0.012$ | $0.022$ | $0.032$ | $0.0012$ | $0.0014$ | $0.0088$ | $0.0093$ | $0.032$ | $0.0034$ | $0.0038$

The $C\\!P$ observables determined from the fit shown in Figs. 2 and 3 are

$\displaystyle A_{K\pi}$ $\displaystyle=$ $\displaystyle\ -$ $\displaystyle
0.004$ $\displaystyle\pm$ $\displaystyle 0.023$ $\displaystyle\pm$
$\displaystyle 0.008$ $\displaystyle A_{KK}$ $\displaystyle=$ $\displaystyle
0.06$ $\displaystyle\pm$ $\displaystyle 0.07$ $\displaystyle\pm$
$\displaystyle 0.01$ $\displaystyle A_{\pi\pi}$ $\displaystyle=$
$\displaystyle 0.15$ $\displaystyle\pm$ $\displaystyle 0.13$
$\displaystyle\pm$ $\displaystyle 0.01$ $\displaystyle R_{KK}$
$\displaystyle=$ $\displaystyle 1.22$ $\displaystyle\pm$ $\displaystyle 0.09$
$\displaystyle\pm$ $\displaystyle 0.02$ $\displaystyle R_{\pi\pi}$
$\displaystyle=$ $\displaystyle 1.08$ $\displaystyle\pm$ $\displaystyle 0.14$
$\displaystyle\pm$ $\displaystyle 0.03$ $\displaystyle R^{+}_{K\pi}$
$\displaystyle=$ $\displaystyle 0.020$ $\displaystyle\pm$ $\displaystyle
0.006$ $\displaystyle\pm$ $\displaystyle 0.001$ $\displaystyle R^{-}_{K\pi}$
$\displaystyle=$ $\displaystyle 0.002$ $\displaystyle\pm$ $\displaystyle
0.004$ $\displaystyle\pm$ $\displaystyle 0.001$ $\displaystyle A_{K\pi\pi\pi}$
$\displaystyle=$ $\displaystyle\ -$ $\displaystyle 0.013$ $\displaystyle\pm$
$\displaystyle 0.031$ $\displaystyle\pm$ $\displaystyle 0.009$ $\displaystyle
A_{\pi\pi\pi\pi}$ $\displaystyle=$ $\displaystyle 0.02$ $\displaystyle\pm$
$\displaystyle 0.11$ $\displaystyle\pm$ $\displaystyle 0.01$ $\displaystyle
R_{\pi\pi\pi\pi}$ $\displaystyle=$ $\displaystyle 1.08$ $\displaystyle\pm$
$\displaystyle 0.13$ $\displaystyle\pm$ $\displaystyle 0.03$ $\displaystyle
R^{+}_{K\pi\pi\pi}$ $\displaystyle=$ $\displaystyle 0.016$ $\displaystyle\pm$
$\displaystyle 0.007$ $\displaystyle\pm$ $\displaystyle 0.003$ $\displaystyle
R^{-}_{K\pi\pi\pi}$ $\displaystyle=$ $\displaystyle 0.006$ $\displaystyle\pm$
$\displaystyle 0.006$ $\displaystyle\pm$ $\displaystyle 0.004$

where the first uncertainty is statistical and the second is systematic. The
correlation matrices for the statistical and systematic uncertainties are
given in Tables 3 and 4, respectively. The large correlations of the
systematic uncertainties are mainly due to contributions from production and
detection asymmetries. Combined results from the ${K}^{+}$ ${K}^{-}$ and
${\pi}^{+}$ ${\pi}^{-}$ decay modes, taking correlations into account, are

$\displaystyle R_{{C\\!P}+}$ $\displaystyle=$ $\displaystyle 1.18$
$\displaystyle\pm$ $\displaystyle 0.08$ $\displaystyle\pm$ $\displaystyle
0.02$ $\displaystyle A_{{C\\!P}+}$ $\displaystyle=$ $\displaystyle 0.08$
$\displaystyle\pm$ $\displaystyle 0.06$ $\displaystyle\pm$ $\displaystyle
0.01$

where the first uncertainty is statistical and the second is systematic. In
addition, $R^{+}$ and $R^{-}$ for the ${K}^{+}$ ${\pi}^{-}$ and ${K}^{+}$
${\pi}^{-}$ ${\pi}^{+}$ ${\pi}^{-}$ decay modes can be transformed into the
more commonly used $R_{ADS}=\left(R^{-}+R^{+}\right)/2\ $and
$A_{ADS}=\left(R^{-}-R^{+}\right)/\left(R^{-}+R^{+}\right)$. These results,
taking correlations into account, are

$\displaystyle R_{ADS}^{K\pi}$ $\displaystyle=$ $\displaystyle 0.011$
$\displaystyle\pm$ $\displaystyle 0.004$ $\displaystyle\pm$ $\displaystyle
0.001$ $\displaystyle A_{ADS}^{K\pi}$ $\displaystyle=$ $\displaystyle\ -$
$\displaystyle 0.81$ $\displaystyle\pm$ $\displaystyle 0.17$
$\displaystyle\pm$ $\displaystyle 0.04$ $\displaystyle R_{ADS}^{K\pi\pi\pi}$
$\displaystyle=$ $\displaystyle 0.011$ $\displaystyle\pm$ $\displaystyle
0.005$ $\displaystyle\pm$ $\displaystyle 0.003$ $\displaystyle
A_{ADS}^{K\pi\pi\pi}$ $\displaystyle=$ $\displaystyle\ -$ $\displaystyle 0.45$
$\displaystyle\pm$ $\displaystyle 0.21$ $\displaystyle\pm$ $\displaystyle
0.14$

where the first uncertainty is statistical and the second is systematic. The
measured asymmetries and ratios for the two-body $D$ meson decay modes are
consistent with, and more precise than, the previous measurements from BaBar
[16].

Table 3: Correlation matrix of the statistical uncertainties for the twelve
physics observables from the simultaneous fit to data. Only half of the
symmetric matrix is shown.

| $A_{K\pi}$ | $A_{KK}$ | $A_{\pi\pi}$ | $R_{KK}$ | $R_{\pi\pi}$ | $R^{+}_{K\pi}$ | $R^{-}_{K\pi}$ | $A_{K\pi\pi\pi}$ | $A_{\pi\pi\pi\pi}$ | $R_{\pi\pi\pi\pi}$ | $R^{+}_{K\pi\pi\pi}$ | $R^{-}_{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{K\pi}$ | 1 | $-$ | $-$ | $-$ | $-$ | 0.08 |  $-$0.01$-$ | $-$ | $-$ | $-$ | $-$ | $-$
$A_{KK}$ |  | 1 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
$A_{\pi\pi}$ |  |  | 1 | $-$ |  $-$0.02$-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
$R_{KK}$ |  |  |  | 1 | 0.05 | 0.02 |  $-$0.01$-$ | $-$ | $-$ | $-$ | $-$ | $-$
$R_{\pi\pi}$ |  |  |  |  | 1 | 0.03 | 0.02 | $-$ | $-$ | $-$ | $-$ | $-$
$R^{+}_{K\pi}$ |  |  |  |  |  | 1 | 0.02 | $-$ | $-$ | $-$ | $-$ | $-$
$R^{-}_{K\pi}$ |  |  |  |  |  |  | 1 | $-$ | $-$ | $-$ | $-$ | $-$
$A_{K\pi\pi\pi}$ |  |  |  |  |  |  |  | 1 | $-$ | $-$ | 0.07 |  $-$0.03$-$
$A_{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  | 1 | 0.01 | $-$ | $-$
$R_{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  | 1 | 0.04 | 0.04
$R^{+}_{K\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  | 1 | 0.03
$R^{-}_{K\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  |  | 1

Table 4: Correlation matrix of the systematic uncertainties for the twelve
physics observables from the simultaneous fit to data. Only half of the
symmetric matrix is shown.

| $A_{K\pi}$ | $A_{KK}$ | $A_{\pi\pi}$ | $R_{KK}$ | $R_{\pi\pi}$ | $R^{+}_{K\pi}$ | $R^{-}_{K\pi}$ | $A_{K\pi\pi\pi}$ | $A_{\pi\pi\pi\pi}$ | $R_{\pi\pi\pi\pi}$ | $R^{+}_{K\pi\pi\pi}$ | $R^{-}_{K\pi\pi\pi}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{K\pi}$ | 1 | 0.82 | 0.72 | $-$ | $-$ | 0.01 |  $-$0.02$-$ | 0.94 | 0.84 | $-$ |  $-$0.01$-$ | $-$
$A_{KK}$ |  | 1 | 0.65 |  $-$0.04$-$ | 0.02 | 0.01 |  $-$0.02$-$ | 0.83 | 0.77 | $-$ | $-$ | $-$
$A_{\pi\pi}$ |  |  | 1 | $-$ |  $-$0.03$-$ | $-$ |  $-$0.02$-$ | 0.72 | 0.68 | $-$ | $-$ | 0.01
$R_{KK}$ |  |  |  | 1 | $-$ | 0.05 | 0.03 |  $-$0.01$-$ | $-$ |  $-$0.01$-$ |  $-$0.01$-$ |  $-$0.01$-$
$R_{\pi\pi}$ |  |  |  |  | 1 | 0.06 | 0.08 |  $-$0.01$-$ | $-$ |  $-$0.01$-$ |  $-$0.02$-$ | 0.01
$R^{+}_{K\pi}$ |  |  |  |  |  | 1 | 0.08 |  $-$0.01$-$ | $-$ | $-$ |  $-$0.01$-$ |  $-$0.01$-$
$R^{-}_{K\pi}$ |  |  |  |  |  |  | 1 |  $-$0.01$-$ |  $-$0.01$-$ |  $-$0.01$-$ | 0.01 | 0.03
$A_{K\pi\pi\pi}$ |  |  |  |  |  |  |  | 1 | 0.84 | $-$ |  $-$0.01$-$ |  $-$0.02$-$
$A_{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  | 1 | 0.03 | 0.01 | $-$
$R_{\pi\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  | 1 | 0.01 |  $-$0.01$-$
$R^{+}_{K\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  | 1 | 0.05
$R^{-}_{K\pi\pi\pi}$ |  |  |  |  |  |  |  |  |  |  |  | 1

##  6 Interpretation

The $C\\!P$ observables measured in this analysis can be used to determine the
physics parameters $r_{B}$, $\delta_{B}$ and $\gamma$, via Eqs. 13-18. The
parameter $\kappa$ is estimated by generating many amplitude models for
${B}\\!\rightarrow{D}{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}\pi$ decays [50]
consisting of various resonant components whose relative amplitudes and phases
are varied within limits according to the existing branching fraction
measurements. The components used in the model are
${{{B}^{-}}}\\!\rightarrow{{D}^{0}}K^{*}(892)^{-}$ and the LASS lineshape
[51]. The LASS lineshape is used to describe the $K\pi$ S-wave, which includes
a nonresonant term and the $K_{0}^{*}(1430)^{-}$ resonance. Contributions from
other resonances e.g.
$K^{*}(1680)^{-}\\!\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{-}}$ and $D_{2}^{*}(2460)^{-}\\!\rightarrow{D}{{\pi}^{-}}$, are
considered to be negligible in the selected ${K}^{*-}$ region and are not
included in the model. For each model, the value of $\kappa$ is determined in
the region of phase space defined by the ${K}^{*-}$ mass window and
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ helicity angle requirements. The
mean of the resulting distribution gives an estimate for $\kappa$ of $0.95\pm
0.06$. The parameters $r_{D}^{K\pi}$, $\delta_{D}^{K\pi}$, $r_{D}^{K3\pi}$,
$\delta_{D}^{K3\pi}$, $\kappa_{K3\pi}$ and $F_{4\pi}$ are also required as
external inputs and are taken from Ref. [22, 23, 24, 15].

Using the measured values of the $C\\!P$ observables, their uncertainties and
the covariance matrices, a global $\chi^{2}$ minimisation is performed,
resulting in a minimum $\chi^{2}$ of 3.0 with 9 degrees of freedom. A scan of
physics parameters is performed for a range of values and the difference in
$\chi^{2}$ between the parameter scan values 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
enables the $\Delta\chi^{2}=2.30,\ 6.18,\ 11.8$ contours to be drawn,
corresponding to 68.3%, 95.5%, 99.7% confidence levels, respectively. These
are shown in Fig. 4. The data are consistent with the value of $\gamma$
indicated by previous measurements [4, 5], $\sim 70^{\circ}$, and result in a
value of $r_{B}=0.11\pm 0.02$. This value of $r_{B}$ is determined at the
point where the global $\chi^{2}$ of the fit is minimised.

Figure 4: Contour plots showing 2D scans of physics parameters $\gamma$ versus
$r_{B}$ (left) and $\gamma$ versus $\delta_{B}$ (right). The dashed lines
represent the $\Delta\chi^{2}=2.30,\ 6.18,\ 11.8$ contours, corresponding to
68.3%, 95.5%, 99.7% confidence levels (CL), respectively. The colour scale
represents $1-\text{CL}$.

##  7 Conclusions

A study of the ${{{B}^{-}}}\\!\rightarrow{D}{{K}^{*-}}$ decay mode is
presented where the $D$ meson decays to two- and four-body final states
consisting of charged kaons and/or pions. The $C\\!P$ observables $R_{CP+}$,
$A_{CP+}$, $R^{+}_{K\pi}$, $R^{-}_{K\pi}$, $R_{\pi\pi\pi\pi}$,
$A_{\pi\pi\pi\pi}$, $R^{+}_{K\pi\pi\pi}$ and $R^{-}_{K\pi\pi\pi}$ are measured
from the high purity sample obtained from $pp$ collision data recorded with
the LHCb detector in Run 1 and Run 2. The measurement of the $C\\!P$
asymmetries in the two-body decay modes and their ratio to the favoured decay
mode is consistent with and more precise than the previous determination [16].
While no bounds on $\gamma$ are quoted due to the limited sensitivity of this
decay mode in isolation, ${{{B}^{-}}}\\!\rightarrow{D}{{K}^{*-}}$ decays will
become valuable in constraining $\gamma$ in the future, as more data are
collected and more $D$ decay modes are analysed.

## 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 (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing
resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany),
INFN (Italy), SURF (The 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, ENIGMASS and OCEVU, and Région Auvergne-Rhône-Alpes (France),
RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith
Fund, the Royal Society, the English-Speaking Union and the Leverhulme Trust
(United Kingdom).

## References

  * [1] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [2] M. Kobayashi and T. Maskawa, _$C\\!P$ violation in the renormalizable theory of weak interaction_, Progress of Theoretical Physics 49 (1973) 652
  * [3] C. Jarlskog, _Commutator of the quark mass matrices in the standard electroweak model and a measure of maximal $C\\!P$ nonconservation_, Phys. Rev. Lett. 55 (1985) 1039
  * [4] LHCb collaboration, R. Aaij et al., _Measurement of the CKM angle $\gamma$ from a combination of LHCb results_, JHEP 12 (2016) 087, arXiv:1611.03076
  * [5] CKMfitter group, J. Charles et al., _CP violation and the CKM matrix: Assessing the impact of the asymmetric $B$ factories_, Eur. Phys. J. C41 (2005) 1, arXiv:hep-ph/0406184, updated results and plots available at: http://ckmfitter.in2p3.fr
  * [6] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ observables in ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ with two- and four-body ${D}$ decays_, Phys. Lett. B760 (2016) 117, arXiv:1603.08993
  * [7] LHCb collaboration, R. Aaij et al., _Measurement of the CKM angle $\gamma$ using ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ with ${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$, ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$ decays_, JHEP 10 (2014) 097, arXiv:1408.2748
  * [8] LHCb collaboration, R. Aaij et al., _A study of ${C\\!P}$ violation in ${{B}^{\mp}}\rightarrow{D}h^{\mp}$ $(h=K,\pi)$ with the modes ${D}\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}{{\pi}^{0}}$, ${D}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ and ${D}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{0}}$_, Phys. Rev. D91 (2015) 112014, arXiv:1504.05442
  * [9] M. Gronau and D. London, _How to determine all the angles of the unitarity triangle from $B^{0}_{d}\rightarrow DK_{S}$ and $B^{0}_{s}\rightarrow D\phi$_, Phys. Lett. B253 (1991) 483
  * [10] M. Gronau and D. Wyler, _On determining a weak phase from charged B decay asymmetries_ , Phys. Lett. B265 (1991) 172
  * [11] D. Atwood, I. Dunietz, and A. Soni, _Enhanced $C\\!P$ violation with $B\rightarrow KD^{0}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})$ modes and extraction of the Cabibbo-Kobayashi-Maskawa angle $\gamma$_, Phys. Rev. Lett. 78 (1997) 3257
  * [12] D. Atwood, I. Dunietz, and A. Soni, _Improved methods for observing $C\\!P$ violation in ${B}^{\pm}\rightarrow KD$ and measuring the CKM phase $\gamma$_, Phys. Rev. D63 (2001) 036005, arXiv:hep-ph/0008090
  * [13] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ violation parameters in ${{B}^{0}}\rightarrow{D}{{K}^{*0}}$ decays_, Phys. Rev. D90 (2014) 112002, arXiv:1407.8136
  * [14] D. Atwood and A. Soni, _Role of a charm factory in extracting CKM-phase information via ${B}\rightarrow\mathrm{DK}$_, Phys. Rev. D68 (2003) 033003, arXiv:hep-ph/0304085
  * [15] S. Malde et al., _First determination of the CP content of ${D}\rightarrow\pi^{+}\pi^{-}\pi^{+}\pi^{-}$ and updated determination of the CP contents of ${D}\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and ${D}\rightarrow{K}^{+}{K}^{-}\pi^{0}$_, Phys. Lett. B747 (2015) 9, arXiv:1504.05878
  * [16] BaBar collaboration, B. Aubert et al., _Measurement of $C\\!P$ violation observables and parameters for the decays ${{B}^{\pm}}\\!\rightarrow{D}{{K}^{*\pm}}$_, Phys. Rev. D80 (2009) 092001, arXiv:0909.3981
  * [17] BaBar collaboration, B. Aubert et al., _Improved measurement of the CKM angle $\gamma$ in ${B}^{\mp}\rightarrow{D}^{(*)}{K}^{(*)\mp}$ decays with a Dalitz plot analysis of $D$ decays to ${K}_{S}^{0}{\pi}^{+}{\pi}^{-}$ and ${K}_{S}^{0}{K}^{+}{K}^{-}$_, Phys. Rev. D78 (2008) 034023, arXiv:0804.2089
  * [18] Belle collaboration, A. Poluektov et al., _Measurement of ${\phi}_{3}$ with a Dalitz plot analysis of ${B}^{+}\rightarrow{D}^{(*)}{K}^{(*)+}$ decay_, Phys. Rev. D73 (2006) 112009, arXiv:hep-ex/0604054
  * [19] W. Wang, _$C\\!P$ violation effects on the measurement of the Cabibbo-Kobayashi-Maskawa angle $\gamma$ from ${B}\rightarrow{DK}$_, Phys. Rev. Lett. 110 (2013) 061802, arXiv:1211.4539
  * [20] M. Gronau, _Improving bounds on $\gamma$ in $B^{\pm}\rightarrow DK^{*\pm}$ and $B^{\pm,0}\rightarrow DX_{s}^{\pm,0}$_, Phys. Lett. B557 (2003) 198
  * [21] M. Rama, _Effect of ${D}-\bar{D}$ mixing in the extraction of $\gamma$ with ${B}^{-}\rightarrow{D}^{0}{K}^{-}$ and ${B}^{-}\rightarrow{D}^{0}{\pi}^{-}$ decays_, Phys. Rev. D89 (2014) 014021, arXiv:1307.4384
  * [22] Heavy Flavor Averaging Group, Y. Amhis et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2016_, arXiv:1612.07233, updated results and plots available at http://www.slac.stanford.edu/xorg/hflav/
  * [23] T. Evans et al., _Improved determination of the ${D}\rightarrow{K}^{+}\pi^{-}\pi^{+}\pi^{-}$ coherence factor and associated hadronic parameters from a combination of $e^{+}e^{-}\rightarrow\Upsilon(3770)\rightarrow c\bar{c}$ and $pp\rightarrow c\bar{c}{X}$ data_, Phys. Lett. B757 (2016) 520, Corrigendum ibid. B765 (2017) 402, arXiv:1602.07430
  * [24] LHCb collaboration, R. Aaij et al., _First observation of ${{D}^{0}}-{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ oscillations in ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays and a measurement of the associated coherence parameters_, Phys. Rev. Lett. 116 (2016) 241801, arXiv:1602.07224
  * [25] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [26] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [27] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [28] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [29] LHCb collaboration, R. Aaij et al., _Measurement of forward ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ production cross-sections in ${p}{p}$ collisions at $\sqrt{s}=13$$~{}T\kern-0.50003pteV$_, JHEP 10 (2015) 172, Erratum ibid. 05 (2017) 063, arXiv:1509.00771
  * [30] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [31] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [32] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [33] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [34] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [35] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [36] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [37] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [38] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [39] Particle Data Group, C. Patrignani et al., _Review of particle physics_ , Chin. Phys. C40 (2016) 100001
  * [40] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [41] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [42] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [43] LHCb collaboration, R. Aaij et al., _Measurement of $C\\!P$ observables in $B^{\pm}\rightarrow D^{(\ast)}K^{\pm}$ and $B^{\pm}\rightarrow D^{(\ast)}\pi^{\pm}$ decays_, arXiv:1708.06370, submitted to Phys. Lett. B
  * [44] LHCb collaboration, R. Aaij et al., _Model-independent measurement of the CKM angle $\gamma$ using ${{B}^{0}}\rightarrow{D}{{K}^{*0}}$ decays with ${D}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$_, JHEP 06 (2016) 131, arXiv:1604.01525
  * [45] LHCb collaboration, R. Aaij et al., _Measurement of the $B^{\pm}$ production asymmetry and the $C\\!P$ asymmetry in $B^{\pm}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{\pm}$ decays_, Phys. Rev. D95 (2017) 052005, arXiv:1701.05501
  * [46] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ asymmetry in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797
  * [47] LHCb collaboration, R. Aaij et al., _Measurement of the ${{D}^{+}_{s}}$–${{D}^{-}_{s}}$ production asymmetry in $7$$~{}T\kern-0.50003pteV$ ${p}{p}$ collisions_, Phys. Lett. B713 (2012) 186, arXiv:1205.0897
  * [48] S. S. Wilks, _The large-sample distribution of the likelihood ratio for testing composite hypotheses_ , Ann. Math. Stat. 9 (1938) 60
  * [49] LHCb collaboration, R. Aaij et al., _Dalitz plot analysis of ${{B}^{0}_{s}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{-}}{{\pi}^{+}}$ decays_, Phys. Rev. D90 (2014) 072003, arXiv:1407.7712
  * [50] T. Latham, J. Back, and P. Harrison, _Laura++, a Dalitz plot fitting package_ ,  available at https://laura.hepforge.org/
  * [51] D. Aston et al., _A study of $K^{-}\pi^{+}$ scattering in the reaction $K^{-}p\rightarrow K^{-}\pi^{+}n$ at 11 GeV/c_, Nucl. Phys. B296 (1988) 493

LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, A. Alfonso Albero38, S. Ali43, G. Alkhazov31, P.
Alvarez Cartelle55, A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3, L. Anderlini18, G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B.
Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, M. Atzeni42, G. Auriemma26, M. Baalouch5, I.
Babuschkin56, S. Bachmann12, J.J. Back50, A. Badalov38,m, C. Baesso62, S.
Baker55, V. Balagura7,b, W. Baldini17, A. Baranov35, R.J. Barlow56, C.
Barschel40, S. Barsuk7, W. Barter56, F. Baryshnikov32, V. Batozskaya29, V.
Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1,
A. Beiter61, L.J. Bel43, N. Beliy63, V. Bellee41, N. Belloli21,i, K. Belous37,
I. Belyaev32,40, E. Ben-Haim8, G. Bencivenni19, S. Benson43, S. Beranek9, A.
Berezhnoy33, R. Bernet42, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt42, F. Betti15, M.-O. Bettler40, M. van Beuzekom43, Ia. Bezshyiko42,
S. Bifani47, P. Billoir8, A. Birnkraut10, A. Bizzeti18,u, M. Bjørn57, T.
Blake50, F. Blanc41, S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36,w, N.
Bondar31, I. Bordyuzhin32, A. Borgheresi21,i, S. Borghi56, M. Borisyak35, M.
Borsato39, F. Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C.
Bozzi17,40, S. Braun12, T. Britton61, J. Brodzicka27, D. Brundu16, E.
Buchanan48, C. Burr56, A. Bursche16,f, J. Buytaert40, W. Byczynski40, S.
Cadeddu16, H. Cai64, R. Calabrese17,g, R. Calladine47, M. Calvi21,i, M. Calvo
Gomez38,m, A. Camboni38,m, P. Campana19, D.H. Campora Perez40, L. Capriotti56,
A. Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A. Cardini16, P.
Carniti21,i, L. Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina21, M.
Cattaneo40, G. Cavallero20,40,h, R. Cenci24,t, D. Chamont7, M.G. Chapman48, M.
Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M. Chefdeville4, S.
Chen16, S.F. Cheung57, S.-G. Chitic40, V. Chobanova39,40, M. Chrzaszcz42,27,
A. Chubykin31, P. Ciambrone19, X. Cid Vidal39, G. Ciezarek43, P.E.L. Clarke52,
M. Clemencic40, H.V. Cliff49, J. Closier40, J. Cogan6, E. Cogneras5, V.
Cogoni16,f, L. Cojocariu30, P. Collins40, T. Colombo40, A. Comerma-Montells12,
A. Contu40, A. Cook48, G. Coombs40, S. Coquereau38, G. Corti40, M. Corvo17,g,
C.M. Costa Sobral50, B. Couturier40, G.A. Cowan52, D.C. Craik58, A.
Crocombe50, M. Cruz Torres1, R. Currie52, C. D’Ambrosio40, F. Da Cunha
Marinho2, E. Dall’Occo43, J. Dalseno48, A. Davis3, O. De Aguiar Francisco40,
S. De Capua56, M. De Cian12, J.M. De Miranda1, L. De Paula2, M. De Serio14,d,
P. De Simone19, C.T. Dean53, D. Decamp4, L. Del Buono8, H.-P. Dembinski11, M.
Demmer10, A. Dendek28, D. Derkach35, O. Deschamps5, F. Dettori54, B. Dey65, A.
Di Canto40, P. Di Nezza19, H. Dijkstra40, F. Dordei40, M. Dorigo40, A. Dosil
Suárez39, L. Douglas53, A. Dovbnya45, K. Dreimanis54, L. Dufour43, G. Dujany8,
P. Durante40, R. Dzhelyadin37, M. Dziewiecki12, A. Dziurda40, A. Dzyuba31, S.
Easo51, M. Ebert52, U. Egede55, V. Egorychev32, S. Eidelman36,w, S.
Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53, S. Ely61, S.
Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S. Farry54, D.
Fazzini21,i, L. Federici25, D. Ferguson52, G. Fernandez38, P. Fernandez
Declara40, A. Fernandez Prieto39, F. Ferrari15, F. Ferreira Rodrigues2, M.
Ferro-Luzzi40, S. Filippov34, R.A. Fini14, M. Fiorini17,g, M. Firlej28, C.
Fitzpatrick41, T. Fiutowski28, F. Fleuret7,b, K. Fohl40, M. Fontana16,40, F.
Fontanelli20,h, D.C. Forshaw61, R. Forty40, V. Franco Lima54, M. Frank40, C.
Frei40, J. Fu22,q, W. Funk40, E. Furfaro25,j, C. Färber40, E. Gabriel52, A.
Gallas Torreira39, D. Galli15,e, S. Gallorini23, S. Gambetta52, M. Gandelman2,
P. Gandini22, Y. Gao3, L.M. Garcia Martin70, J. García Pardiñas39, J. Garra
Tico49, L. Garrido38, P.J. Garsed49, D. Gascon38, C. Gaspar40, L. Gavardi10,
G. Gazzoni5, D. Gerick12, E. Gersabeck12, M. Gersabeck56, T. Gershon50, Ph.
Ghez4, S. Gianì41, V. Gibson49, O.G. Girard41, L. Giubega30, K. Gizdov52, V.V.
Gligorov8, D. Golubkov32, A. Golutvin55, A. Gomes1,a, I.V. Gorelov33, C.
Gotti21,i, E. Govorkova43, J.P. Grabowski12, R. Graciani Diaz38, L.A. Granado
Cardoso40, E. Graugés38, E. Graverini42, G. Graziani18, A. Grecu30, R. Greim9,
P. Griffith16, L. Grillo21, L. Gruber40, B.R. Gruberg Cazon57, O. Grünberg67,
E. Gushchin34, Yu. Guz37, T. Gys40, C. Göbel62, T. Hadavizadeh57, C.
Hadjivasiliou5, G. Haefeli41, C. Haen40, S.C. Haines49, B. Hamilton60, X.
Han12, T.H. Hancock57, S. Hansmann-Menzemer12, N. Harnew57, S.T. Harnew48, C.
Hasse40, M. Hatch40, J. He63, M. Hecker55, K. Heinicke10, A. Heister9, K.
Hennessy54, P. Henrard5, L. Henry70, E. van Herwijnen40, M. Heß67, A.
Hicheur2, D. Hill57, C. Hombach56, P.H. Hopchev41, W. Hu65, Z.C. Huard59, W.
Hulsbergen43, T. Humair55, M. Hushchyn35, D. Hutchcroft54, P. Ibis10, M.
Idzik28, P. Ilten58, R. Jacobsson40, J. Jalocha57, E. Jans43, A. Jawahery60,
F. Jiang3, M. John57, D. Johnson40, C.R. Jones49, C. Joram40, B. Jost40, N.
Jurik57, S. Kandybei45, M. Karacson40, J.M. Kariuki48, S. Karodia53, N.
Kazeev35, M. Kecke12, F. Keizer49, M. Kelsey61, M. Kenzie49, T. Ketel44, E.
Khairullin35, B. Khanji12, C. Khurewathanakul41, T. Kirn9, S. Klaver56, K.
Klimaszewski29, T. Klimkovich11, S. Koliiev46, M. Kolpin12, I. Komarov41, R.
Kopecna12, P. Koppenburg43, A. Kosmyntseva32, S. Kotriakhova31, M. Kozeiha5,
L. Kravchuk34, M. Kreps50, F. Kress55, P. Krokovny36,w, F. Kruse10, W.
Krzemien29, W. Kucewicz27,l, M. Kucharczyk27, V. Kudryavtsev36,w, A.K.
Kuonen41, T. Kvaratskheliya32,40, D. Lacarrere40, G. Lafferty56, A. Lai16, G.
Lanfranchi19, C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, A.
Leflat33,40, J. Lefrançois7, R. Lefèvre5, F. Lemaitre40, E. Lemos Cid39, O.
Leroy6, T. Lesiak27, B. Leverington12, P.-R. Li63, T. Li3, Y. Li7, Z. Li61, T.
Likhomanenko68, R. Lindner40, F. Lionetto42, V. Lisovskyi7, X. Liu3, D. Loh50,
A. Loi16, I. Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio Martinez39,
H. Luo52, A. Lupato23, E. Luppi17,g, O. Lupton40, A. Lusiani24, X. Lyu63, F.
Machefert7, F. Maciuc30, V. Macko41, P. Mackowiak10, S. Maddrell-Mander48, O.
Maev31,40, K. Maguire56, D. Maisuzenko31, M.W. Majewski28, S. Malde57, B.
Malecki27, A. Malinin68, T. Maltsev36,w, G. Manca16,f, G. Mancinelli6, D.
Marangotto22,q, J. Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin
Benito38, M. Marinangeli41, P. Marino41, J. Marks12, G. Martellotti26, M.
Martin6, M. Martinelli41, D. Martinez Santos39, F. Martinez Vidal70, D.
Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R. Matev40, A. Mathad50,
Z. Mathe40, C. Matteuzzi21, A. Mauri42, E. Maurice7,b, B. Maurin41, A.
Mazurov47, M. McCann55,40, A. McNab56, R. McNulty13, J.V. Mead54, B.
Meadows59, C. Meaux6, F. Meier10, N. Meinert67, D. Melnychuk29, M. Merk43, A.
Merli22,40,q, E. Michielin23, D.A. Milanes66, E. Millard50, M.-N. Minard4, L.
Minzoni17, D.S. Mitzel12, A. Mogini8, J. Molina Rodriguez1, T. Mombächer10,
I.A. Monroy66, S. Monteil5, M. Morandin23, M.J. Morello24,t, O. Morgunova68,
J. Moron28, A.B. Morris52, R. Mountain61, F. Muheim52, M. Mulder43, D.
Müller56, J. Müller10, K. Müller42, V. Müller10, P. Naik48, T. Nakada41, R.
Nandakumar51, A. Nandi57, I. Nasteva2, M. Needham52, N. Neri22,40, S.
Neubert12, N. Neufeld40, M. Neuner12, T.D. Nguyen41, C. Nguyen-Mau41,n, S.
Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A. Nogay68, D.P. O’Hanlon50,
A. Oblakowska-Mucha28, V. Obraztsov37, S. Ogilvy19, R. Oldeman16,f, C.J.G.
Onderwater71, A. Ossowska27, J.M. Otalora Goicochea2, P. Owen42, A.
Oyanguren70, P.R. Pais41, A. Palano14,d, M. Palutan19,40, A. Papanestis51, M.
Pappagallo14,d, L.L. Pappalardo17,g, W. Parker60, C. Parkes56, G.
Passaleva18,40, A. Pastore14,d, M. Patel55, C. Patrignani15,e, A. Pearce40, A.
Pellegrino43, G. Penso26, M. Pepe Altarelli40, S. Perazzini40, P. Perret5, L.
Pescatore41, K. Petridis48, A. Petrolini20,h, A. Petrov68, M. Petruzzo22,q, E.
Picatoste Olloqui38, B. Pietrzyk4, M. Pikies27, D. Pinci26, F. Pisani40, A.
Pistone20,h, A. Piucci12, V. Placinta30, S. Playfer52, M. Plo Casasus39, F.
Polci8, M. Poli Lener19, A. Poluektov50, I. Polyakov61, E. Polycarpo2, G.J.
Pomery48, S. Ponce40, A. Popov37, D. Popov11,40, S. Poslavskii37, C.
Potterat2, E. Price48, J. Prisciandaro39, C. Prouve48, V. Pugatch46, A. Puig
Navarro42, H. Pullen57, G. Punzi24,p, W. Qian50, R. Quagliani7,48, B.
Quintana5, B. Rachwal28, J.H. Rademacker48, M. Rama24, M. Ramos Pernas39, M.S.
Rangel2, I. Raniuk45,†, F. Ratnikov35, G. Raven44, M. Ravonel Salzgeber40, M.
Reboud4, F. Redi55, S. Reichert10, A.C. dos Reis1, C. Remon Alepuz70, V.
Renaudin7, S. Ricciardi51, S. Richards48, M. Rihl40, K. Rinnert54, V. Rives
Molina38, P. Robbe7, A. Robert8, A.B. Rodrigues1, E. Rodrigues59, J.A.
Rodriguez Lopez66, A. Rogozhnikov35, S. Roiser40, A. Rollings57, V.
Romanovskiy37, A. Romero Vidal39, J.W. Ronayne13, M. Rotondo19, M.S.
Rudolph61, T. Ruf40, P. Ruiz Valls70, J. Ruiz Vidal70, J.J. Saborido Silva39,
E. Sadykhov32, N. Sagidova31, B. Saitta16,f, V. Salustino Guimaraes1, C.
Sanchez Mayordomo70, B. Sanmartin Sedes39, R. Santacesaria26, C. Santamarina
Rios39, M. Santimaria19, E. Santovetti25,j, G. Sarpis56, A. Sarti19,k, C.
Satriano26,s, A. Satta25, D.M. Saunders48, D. Savrina32,33, S. Schael9, M.
Schellenberg10, M. Schiller53, H. Schindler40, M. Schmelling11, T.
Schmelzer10, B. Schmidt40, O. Schneider41, A. Schopper40, H.F. Schreiner59, M.
Schubiger41, M.-H. Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A.
Semennikov32, E.S. Sepulveda8, A. Sergi47, N. Serra42, J. Serrano6, L.
Sestini23, P. Seyfert40, M. Shapkin37, I. Shapoval45, Y. Shcheglov31, T.
Shears54, L. Shekhtman36,w, V. Shevchenko68, B.G. Siddi17, R. Silva
Coutinho42, L. Silva de Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49,
N. Skidmore48, T. Skwarnicki61, E. Smith55, I.T. Smith52, J. Smith49, M.
Smith55, l. Soares Lavra1, M.D. Sokoloff59, F.J.P. Soler53, B. Souza De
Paula2, B. Spaan10, P. Spradlin53, S. Sridharan40, F. Stagni40, M. Stahl12, S.
Stahl40, P. Stefko41, S. Stefkova55, O. Steinkamp42, S. Stemmle12, O.
Stenyakin37, M. Stepanova31, H. Stevens10, S. Stone61, B. Storaci42, S.
Stracka24,p, M.E. Stramaglia41, M. Straticiuc30, U. Straumann42, J. Sun3, L.
Sun64, W. Sutcliffe55, K. Swientek28, V. Syropoulos44, T. Szumlak28, M.
Szymanski63, S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G. Tellarini17,g, F.
Teubert40, E. Thomas40, J. van Tilburg43, M.J. Tilley55, V. Tisserand4, M.
Tobin41, S. Tolk49, L. Tomassetti17,g, D. Tonelli24, F. Toriello61, R.
Tourinho Jadallah Aoude1, E. Tournefier4, M. Traill53, M.T. Tran41, M.
Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N.
Tuning43,40, A. Ukleja29, A. Usachov7, A. Ustyuzhanin35, U. Uwer12, C.
Vacca16,f, A. Vagner69, V. Vagnoni15,40, A. Valassi40, S. Valat40, G.
Valenti15, R. Vazquez Gomez40, P. Vazquez Regueiro39, S. Vecchi17, M. van
Veghel43, J.J. Velthuis48, M. Veltri18,r, G. Veneziano57, A. Venkateswaran61,
T.A. Verlage9, M. Vernet5, M. Vesterinen57, J.V. Viana Barbosa40, B. Viaud7,
D. Vieira63, M. Vieites Diaz39, H. Viemann67, X. Vilasis-Cardona38,m, M.
Vitti49, V. Volkov33, A. Vollhardt42, B. Voneki40, A. Vorobyev31, V.
Vorobyev36,w, C. Voß9, J.A. de Vries43, C. Vázquez Sierra39, R. Waldi67, C.
Wallace50, R. Wallace13, J. Walsh24, J. Wang61, D.R. Ward49, H.M. Wark54, N.K.
Watson47, D. Websdale55, A. Weiden42, C. Weisser58, M. Whitehead40, J.
Wicht50, G. Wilkinson57, M. Wilkinson61, M. Williams56, M.P. Williams47, M.
Williams58, T. Williams47, F.F. Wilson51,40, J. Wimberley60, M. Winn7, J.
Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K.
Wraight53, K. Wyllie40, Y. Xie65, M. Xu65, Z. Xu4, Z. Yang3, Z. Yang60, Y.
Yao61, H. Yin65, J. Yu65, X. Yuan61, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V.
Zhukov33, J.B. Zonneveld52, S. Zucchelli15.

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

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, Viet Nam
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
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased