EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-083 LHCb-PAPER-2016-006 28 June 2016

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 ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$

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

A binned Dalitz plot analysis of the decays ${{B}^{0}}\rightarrow
D{{K}^{*0}}$, with $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ and
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$, is
performed to measure the observables $x_{\pm}$ and $y_{\pm}$, which are
related to the CKM angle $\gamma$ and the hadronic parameters of the decays.
The $D$ decay strong phase variation over the Dalitz plot is taken from
measurements performed at the CLEO-c experiment, making the analysis
independent of the $D$ decay model. With a sample of proton-proton collision
data, corresponding to an integrated luminosity of 3.0 $\mbox{\,fb}^{-1}$,
collected by the LHCb experiment, the values of the $C\\!P$ violation
parameters are found to be $x_{+}=0.05\pm 0.35\pm 0.02$, $x_{-}=-0.31\pm
0.20\pm 0.04$, $y_{+}=-0.81\pm 0.28\pm 0.06$ and $y_{-}=0.31\pm 0.21\pm 0.05$,
where the first uncertainties are statistical and the second systematic. These
observables correspond to values $\gamma$ = $(71\pm 20)^{\circ}$,
$r_{B^{0}}=0.56\pm 0.17$ and $\delta_{B^{0}}=(204\,^{+21}_{-20})^{\circ}$. The
parameters $r_{B^{0}}$ and $\delta_{B^{0}}$ are the magnitude ratio and strong
phase difference between the suppressed and favoured $B^{0}$ decay amplitudes,
and have been measured in a region of $\pm
50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the ${{K}^{*}}(892)^{0}$
mass and with the magnitude of the cosine of the ${{K}^{*}}(892)^{0}$ helicity
angle larger than 0.4.

Published in JHEP 06 (2016) 131

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

##  1 Introduction

The Standard Model (SM) description of $C\\!P$ violation can be tested through
measurements of the angle $\gamma$ of the unitarity triangle of the Cabibbo-
Kobayashi-Maskawa (CKM) matrix [1, 2], where
$\gamma\equiv\mathrm{arg}(-V_{ud}^{\phantom{\ast}}V_{ub}^{\ast}/V_{cd}^{\phantom{\ast}}V_{cb}^{\ast})$.
It is the only CKM angle easily accessible in tree-level processes and can be
measured, with a small uncertainty from theory of $\delta\gamma/\gamma\leq
10^{-7}$ [3]. Hence, in the absence of new physics effects at tree level [4],
a precision measurement of $\gamma$ provides an SM benchmark which can be
compared with other CKM matrix observables that are more likely to be affected
by physics beyond the SM. Such comparisons are currently limited by the
uncertainty on direct measurements of $\gamma$, which is about $7^{\circ}$ [5,
6].

The CKM angle $\gamma$ is experimentally accessible through the interference
between $\bar{b}\rightarrow\bar{c}u\bar{s}$ and
$\bar{b}\rightarrow\bar{u}c\bar{s}$ transitions. The traditional golden mode
is ${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$, with charge-conjugation implied
throughout, where $D$ represents a neutral $D$ meson reconstructed in a final
state that is common to both ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays. This mode has been
studied at LHCb with a wide range of $D$ meson final states to measure
observables with sensitivity to $\gamma$ [7, 8, 9, 10]. In addition to these
studies, other $B$ decays have also been used with a variety of techniques to
determine $\gamma$ [11, 12, 13, 14].

This paper presents an analysis in which the decay ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ provides sensitivity to the CKM angle $\gamma$ through the
interfering amplitudes shown in Fig. 1. Here the ${K}^{*0}$ refers to the
${{K}^{*}}(892)^{0}$, and the charge of the kaon from the ${K}^{*0}$
unambiguously identifies the flavour of the decaying $B$ meson as ${B}^{0}$ or
${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$. Although the branching
fraction of the ${{B}^{0}}\rightarrow D{{K}^{*0}}$ decay is an order of
magnitude smaller than that of the ${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decay
[15], it is expected to exhibit larger $C\\!P$-violating effects as the two
colour-suppressed Feynman diagrams in Fig. 1 are comparable in magnitude.
Measurements sensitive to $\gamma$ using the ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ decay mode were pioneered by the BaBar [16] and Belle [17]
collaborations, and have been pursued by the LHCb collaboration [11, 14].

Figure 1: Feynman diagrams of the (left)
${{B}^{0}}\rightarrow{{D}^{0}}{{K}^{*0}}$ and (right)
${{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{*0}}$ amplitudes, which
interfere in the ${{B}^{0}}\rightarrow D{{K}^{*0}}$ decay.

The three-body self-conjugate decays
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
and $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$,
designated collectively as $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}$, are accessible to both ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$. They have large variation of
the strong phase over the Dalitz plot, and thus provide a powerful method to
determine the angle $\gamma$. Sensitivity to $\gamma$ is obtained by comparing
the distribution of events in the
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$ Dalitz plots
of $B$ mesons reconstructed in each flavour, as described in Refs. [18, 19,
20]. To determine $\gamma$ from the comparison, input is required on the
variation within the Dalitz plot of the strong-interaction phase difference
between ${D}^{0}$ and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
decays. An amplitude model of the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$
decay can be used to provide this information and this technique has been used
to study the ${{B}^{0}}\rightarrow D{{K}^{*0}}$,
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
decay mode by BaBar [21] and LHCb [22]. In Ref. [22] the same dataset is used
as the one analysed in this paper. An attractive alternative is to use model-
independent measurements of the strong-phase difference variation over the
Dalitz plot, which removes the need to assign model-related systematic
uncertainties [19, 20]. Measurements of the strong-phase variation in binned
regions of the Dalitz plot cannot be done with LHCb data alone, but can be
accomplished using an analysis of quantum-correlated neutral $D$ meson pairs
from $\psi(3770)$ decays, and have been made at the CLEO-c experiment [23].
These measurements have direct access to the strong-phase difference, which is
not the case for the amplitude models based on fits to flavour-tagged $D$
decays only [24, 25]. The separation of data into binned regions of the Dalitz
plot leads to a loss in statistical sensitivity in comparison to using an
amplitude model; however, the advantage of using the measurements from CLEO is
that the systematic uncertainties remain free of any model assumptions on the
strong-phase difference. This model-independent method has been used by Belle
[26] to study the ${{B}^{0}}\rightarrow D{{K}^{*0}}$,
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
decay mode, and by LHCb [8] and Belle [27] to study ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ decays.

In this paper, $pp$ collision data at a centre-of-mass energy
$\sqrt{s}=7\,(8)\mathrm{\,Te\kern-1.00006ptV}$, accumulated by LHCb in 2011
(2012) and corresponding to a total integrated luminosity of
$3.0\mbox{\,fb}^{-1}$, are exploited to perform a model-independent
measurement of $\gamma$ in the decay mode ${{B}^{0}}\rightarrow D{{K}^{*0}}$,
with $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ and
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$. The
yield of ${{B}^{0}}\rightarrow D{{K}^{*0}}$ with
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
is twice that previously analysed at Belle [27] and the
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$
decay is included for the first time. This allows for a precise measurement of
$x_{\pm},y_{\pm}$ using the techniques developed for similar analyses of
${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ decays [8].

The remainder of the paper is organised as follows. Section 2 describes the
analysis framework. Section 3 describes the LHCb detector, and Sect. 4
presents the candidate selection and the parametrisation of the $B$ candidate
invariant mass spectrum. Section 5 is concerned with the use of semileptonic
decays in order to determine the populations in different bins of the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$
Dalitz plot. Section 6 discusses the binned Dalitz plot fit and presents the
measurements of the $C\\!P$ violation parameters. The evaluation of systematic
uncertainties is summarised in Sect. 7. The determination of the CKM angle
$\gamma$ using the measured $C\\!P$ parameters is described in Sect. 8.

##  2 Overview of the analysis

The favoured and suppressed ${B}^{0}$ decay amplitudes can be expressed as

$\displaystyle A({{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}X^{0}_{s};p)$
$\displaystyle\equiv A_{c}(p)e^{i\delta_{c}(p)},$ (1) $\displaystyle
A({{B}^{0}}\rightarrow{{D}^{0}}X^{0}_{s};p)$ $\displaystyle\equiv
A_{u}(p)e^{i[\delta_{u}(p)+\gamma]},$

where $p$ is the $\left(m^{2}(K\pi),m^{2}(D\pi)\right)$ coordinate on the
${{B}^{0}}\rightarrow{D}K\pi$ Dalitz plot, $A_{u}(p)$ and $A_{c}(p)$ are the
moduli of the $b\rightarrow u$ and $b\rightarrow c$ amplitudes, and
$\delta_{c,u}(p)$ represent the strong phases of the relevant decay
amplitudes. The symbol $X^{0}_{s}$ refers to a resonant or nonresonant
${{K}^{+}}{{\pi}^{-}}$ pair, which could be produced by the decay of the
${K}^{*0}$ meson or by other contributions to the
${{B}^{0}}\rightarrow{D}{{K}^{+}}{{\pi}^{-}}$ final state. Similar expressions
can be written for the ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$
decay, where the parameter $\gamma$ enters with opposite sign. The natural
width of the ${K}^{*0}$ (approximately
$50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [15]) must be considered when
analysing these decays. In the region near the ${K}^{*0}$ mass there is
interference between the signal ${K}^{*0}$ decay amplitude and amplitudes due
to the other ${{B}^{0}}\rightarrow D{{K}^{+}}{{\pi}^{-}}$ Dalitz plot
contributions, such as higher mass $K\pi$ resonances and nonresonant $K\pi$
decays. Hence, the magnitude ratio between the suppressed and favoured
amplitudes $r_{{{B}^{0}}}$, the coherence factor $\kappa$ [28], and the
effective strong phase difference $\delta_{{{B}^{0}}}$ depend on the region of
the ${B}^{0}$ Dalitz plot to be analysed. These are defined as

$\displaystyle r^{2}_{{{B}^{0}}}$
$\displaystyle\equiv\frac{|A({{B}^{0}}\rightarrow{{D}^{0}}{{K}^{*0}})|^{2}}{|A({{B}^{0}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{*0}})|^{2}}=\frac{\int_{{K}^{*0}}dp\,A_{u}^{2}(p)}{\int_{{{K}^{*0}}}dp\,A_{c}^{2}(p)},$
(2) $\displaystyle\kappa e^{i\delta_{{{B}^{0}}}}$
$\displaystyle\equiv\frac{\int_{{K}^{*0}}dp\,A_{c}(p)A_{u}(p)e^{i[\delta_{u}(p)-\delta_{c}(p)]}}{\sqrt{\int_{{K}^{*0}}dp\,A_{c}^{2}(p)}\sqrt{\int_{{K}^{*0}}dp\,A_{u}^{2}(p)}},$
(3)

where $0\leq\kappa\leq 1$. For this analysis the integration is over
$K^{+}\pi^{-}$ masses within $50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of
the known ${K}^{*0}$ mass [15] and an absolute value of the cosine of the
${K}^{*0}$ helicity angle $\theta^{\ast}$ greater than 0.4. The helicity angle
$\theta^{\ast}$ is defined as the angle between the ${K}^{*0}$ daughter kaon
momentum vector and the direction opposite to the ${B}^{0}$ momentum vector in
the ${K}^{*0}$ rest frame. This region is chosen to obtain a large value of
$\kappa$ and to facilitate combination with results in Refs. [11, 14], which
impose the same limits. The coherence factor has recently been determined by
the LHCb collaboration to be
$\kappa=0.958\,^{+0.005}_{-0.010}\,{}^{+0.002}_{-0.045}$ [14], through an
amplitude analysis that measures the $b\rightarrow c$ and $b\rightarrow u$
amplitudes in the ${{B}^{0}}\rightarrow DK^{+}\pi^{-}$ decay.

The amplitude of the ${D}^{0}$ meson decay at a particular point on the $D$
Dalitz plot is defined as $A_{D}(m^{2}_{-},m^{2}_{+})\equiv
A(m^{2}_{-},m^{2}_{+})e^{i\delta(m^{2}_{-},m^{2}_{+})}$, where $m^{2}_{-}$
($m^{2}_{+}$) is the invariant mass of the
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{-}$
(${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}$) pair. Neglecting $C\\!P$
violation in charm decays, which is known to be small [15], the charge-
conjugated amplitudes are related by
$A_{\overline{D}}(m^{2}_{-},m^{2}_{+})=A_{D}(m^{2}_{+},m^{2}_{-})$. The
partial widths for the $B$ decays can be written as

$\displaystyle d{\Gamma}({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow
D(\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-})$
$\displaystyle\overline{X}^{0}_{s};p,m^{2}_{-},m^{2}_{+})\propto$ (4)
$\displaystyle\bigl{\lvert}A_{c}(p)e^{i\delta_{c}(p)}A_{D}(m^{2}_{-},m^{2}_{+})+A_{u}(p)e^{i[\delta_{u}(p)-\gamma]}A_{\overline{D}}(m^{2}_{-},m^{2}_{+})\bigr{\rvert}^{2},$
$\displaystyle d{\Gamma}({{B}^{0}}\rightarrow
D(\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-})$
$\displaystyle{X^{0}_{s}};p,m^{2}_{-},m^{2}_{+})\propto$ (5)
$\displaystyle\bigl{\lvert}A_{c}(p)e^{i\delta_{c}(p)}A_{\overline{D}}(m^{2}_{-},m^{2}_{+})+A_{u}(p)e^{i[\delta_{u}(p)+\gamma]}A_{D}(m^{2}_{-},m^{2}_{+})\bigr{\rvert}^{2}.$

Expanding and integrating over the defined ${K}^{*0}$ region, one obtains

$\displaystyle d{\Gamma}$ $\displaystyle({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow
D(\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}){{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}};m^{2}_{-},m^{2}_{+})\propto$
(6)
$\displaystyle\bigl{\lvert}A_{D}(m^{2}_{-},m^{2}_{+})\bigr{\rvert}^{2}+r_{{{B}^{0}}}^{2}\bigl{\lvert}A_{D}(m^{2}_{+},m^{2}_{-})\bigr{\rvert}^{2}+2\kappa
r_{{{B}^{0}}}\mathrm{Re}\big{[}A_{D}(m^{2}_{-},m^{2}_{+})A^{*}_{D}(m^{2}_{+},m^{2}_{-})e^{-i(\delta_{{{B}^{0}}}-\gamma)}\big{]},$
$\displaystyle d{\Gamma}$ $\displaystyle({{B}^{0}}\rightarrow
D(\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}){{{K}^{*0}}};m^{2}_{-},m^{2}_{+})\propto$ (7)
$\displaystyle\bigl{\lvert}A_{D}(m^{2}_{+},m^{2}_{-})\bigr{\rvert}^{2}+r_{{{B}^{0}}}^{2}\bigl{\lvert}A_{D}(m^{2}_{-},m^{2}_{+})\bigr{\rvert}^{2}+2\kappa
r_{{{B}^{0}}}\mathrm{Re}\big{[}A_{D}(m^{2}_{+},m^{2}_{-})A^{*}_{D}(m^{2}_{-},m^{2}_{+})e^{-i(\delta_{{{B}^{0}}}+\gamma)}\big{]}.$

The $D$ Dalitz plot is partitioned into bins symmetric under the exchange
$m^{2}_{-}\leftrightarrow m^{2}_{+}$. The cosine of the strong-phase
difference between the ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decay weighted by the decay
amplitude and averaged in bin $i$ is called $c_{i}$ [19, 20], and is given by

$\displaystyle c_{i}$
$\displaystyle\equiv\frac{\int_{i}dm^{2}_{-}\,dm^{2}_{+}\,A(m^{2}_{-},m^{2}_{+})A(m^{2}_{+},m^{2}_{-})\cos[\delta(m^{2}_{-},m^{2}_{+})-\delta(m^{2}_{+},m^{2}_{-})]}{\sqrt{\int_{i}dm^{2}_{-}\,dm^{2}_{+}\,A^{2}(m^{2}_{-},m^{2}_{+})\int_{i}dm^{2}_{-}\,dm^{2}_{+}\,A^{2}(m^{2}_{+},m^{2}_{-})}},$
(8)

where the integrals are evaluated over the phase space of bin $i$. An
analogous expression can be written for $s_{i}$ which is the sine of the
strong-phase difference weighted by the decay amplitude and averaged in the
bin.

Measurements of $c_{i}$ and $s_{i}$ are provided by CLEO in four different
$2\times 8$ binning schemes for the
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
decay [23]. The bins are labelled from $-8$ to $+8$, excluding zero, where the
bins containing a positive label satisfy the condition $m^{2}_{-}\geq
m^{2}_{+}$. The binning scheme used in this analysis is referred to as the
‘modified optimal’ binning. The optimisation was performed assuming a strong-
phase difference distribution given by the BaBar model presented in Ref. [24].
This modified optimal binning is described in Ref. [23] and was designed to be
statistically optimal in a scenario where the signal purity is low. It is also
more robust for analyses with low yields in comparison to the alternatives, as
no individual bin is very small. For the ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{+}}{{K}^{-}}$ final state, the measurements of $c_{i}$ and $s_{i}$
are available in three variants containing a different number of bins, with
the guiding model being that from the BaBar study described in Ref. [25]. For
the present analysis the variant with the $2\times 2$ binning is chosen, given
the very low signal yields expected in this decay. The measurements of $c_{i}$
and $s_{i}$ are not biased by the use of a specific amplitude model in
defining the bin boundaries, which only affects this analysis to the extent
that if the model gives a poor description of the underlying decay then there
will be a reduction in the statistical sensitivity of the $\gamma$
measurement. The binning choices for the two decay modes are shown in Fig. 2.

Figure 2: Binning schemes for (left)
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
and (right) $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{+}}{{K}^{-}}$. The diagonal line separates the positive and negative
bin numbers, where the positive bins are in the region $m^{2}_{-}\geq
m^{2}_{+}$.

The integrals of Eqs. (6) and (7) over the phase space of a Dalitz plot bin
are proportional to the expected yield in that bin. The physics parameters of
interest, $r_{{{B}^{0}}}$, $\delta_{{{B}^{0}}}$, and $\gamma$, are translated
into four Cartesian variables [29, 30]. These are the measured observables and
are defined as

$x_{\pm}\equiv
r_{{{B}^{0}}}\cos(\delta_{{{B}^{0}}}\pm\gamma)\;\mathrm{and}\;y_{\pm}\equiv
r_{{{B}^{0}}}\sin(\delta_{{{B}^{0}}}\pm\gamma).$ (9)

From Eqs. (6) and (7) it follows that

$\displaystyle N_{\pm i}^{+}$ $\displaystyle=n_{+}\left[F_{\mp
i}+(x_{+}^{2}+y_{+}^{2})F_{\pm i}+2\kappa\sqrt{F_{+i}F_{-i}}(x_{+}c_{\pm
i}-y_{+}s_{\pm i})\right],$ (10) $\displaystyle N_{\pm i}^{-}$
$\displaystyle=n_{-}\left[F_{\pm i}+(x_{-}^{2}+y_{-}^{2})F_{\mp
i}+2\kappa\sqrt{F_{+i}F_{-i}}(x_{-}c_{\pm i}+y_{-}s_{\pm i})\right],$ (11)

where $F_{i}$ are defined later in Eq. (12) and $N^{+}_{i}$ ($N^{-}_{i}$) is
the expected number of ${B}^{0}$ (${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$) decays in bin $i$. The
superscript on $N$ refers to the charge of the kaon from the ${K}^{*0}$ decay.
The parameters $n_{+}$ and $n_{-}$ provide the normalisation, which can be
different due to production, detection and $C\\!P$ asymmetries between
${B}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ mesons.
However the integrated yields are not used and the analysis is insensitive to
such effects. The detector and selection requirements placed on the data lead
to a non-uniform efficiency over the Dalitz plot. The efficiency profile for
the signal candidates is given by $\eta=\eta(m^{2}_{-},m^{2}_{+})$. Only the
relative efficiency from one point to another matters and not the absolute
normalisation. The parameters $F_{i}$ are given by

$F_{i}=\frac{\int_{i}dm^{2}_{-}dm^{2}_{+}|A_{D}(m^{2}_{-},m^{2}_{+})|^{2}\,\eta(m^{2}_{-},m^{2}_{+})}{\sum_{j}\int_{j}dm^{2}_{-}dm^{2}_{+}|A_{D}(m^{2}_{-},m^{2}_{+})|^{2}\,\eta(m^{2}_{-},m^{2}_{+})}$
(12)

and are the fraction of decays in bin $i$ of the
${{D}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$
Dalitz plot.

The values of $F_{i}$ are determined from the control decay mode
${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}X$, where the ${D}^{*-}$ decays
to ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ ${\pi}^{-}$ and the
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decays to either the
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$ or
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$ final state. The
symbol $X$, hereinafter omitted, indicates other particles which may be
produced in the decay but are not reconstructed. Samples of simulated events
are used to correct for the small differences in efficiency arising through
necessary differences in selecting
${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$ and ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ decays, which are discussed further in Sect. 5.

Effects due to ${D}^{0}$-${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing and $C\\!P$ violation in
${K}^{0}$-${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ mixing are
ignored: the corrections are discussed in Refs. [31, 32] and are expected to
be of order $0.2^{\circ}$ ($1^{\circ}$) for $D$ mixing ($C\\!P$ violation in
$K$ mixing) in ${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decays. In both cases the
size of the correction is reduced as the value of $r_{{{B}^{0}}}$ is expected
to be approximately three times larger than the value of $r_{B}$ in
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decays. The effect of different nuclear
interactions within the detector material for ${K}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{0}$ mesons is expected to be of a
similar magnitude and is also ignored [33].

##  3 Detector and simulation

The LHCb detector [34, 35] 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 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.
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.
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
trigger algorithms used to select hadronic and semileptonic $B$ decay
candidates are slightly different, due to the presence of the muon in the
latter, and are described in Sects. 4 and 5.

In the simulation, $pp$ collisions are generated using Pythia [36, 37] with a
specific LHCb configuration [38]. Decays of hadronic particles are described
by EvtGen [39], in which final-state radiation is generated using Photos [40].
The interaction of the generated particles with the detector, and its
response, are implemented using the Geant4 toolkit [41, 42] as described in
Ref. [43].

##  4 Event selection and fit to the $B$ candidate invariant mass distribution

Decays of the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ meson to the
${{\pi}^{+}}{{\pi}^{-}}$ final state are reconstructed in two different
categories, the first involving ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
mesons that decay early enough for the pion track segments to be reconstructed
in the vertex detector, the second containing
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ mesons that decay later such that
track segments of the pions cannot be formed in the vertex detector. These
categories are referred to as _long_ and _downstream_. The candidates in the
long category have better mass, momentum, and vertex resolution than those in
the downstream category.

Signal events considered in the analysis must first fulfil hardware and
software trigger requirements. At the hardware stage at least one of the two
following criteria must be satisfied: either a particle produced in the decay
of the signal $B$ candidate leaves a deposit with high transverse energy in
the hadronic calorimeter, or the event is accepted because particles not
associated with the signal candidate fulfil the trigger requirements. At least
one charged particle should have a high $p_{\mathrm{T}}$ and a large
$\chi^{2}_{\text{IP}}$ with respect to any PV, where $\chi^{2}_{\text{IP}}$ is
defined as the difference in $\chi^{2}$ of a given PV fitted with and without
the considered track. At the software stage, a multivariate algorithm [44] is
used for the identification of secondary vertices that are consistent with the
decay of a $b$ hadron. The software trigger designed to select
${{B}^{0}}\rightarrow D{{K}^{*0}}$ candidates requires 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 tracks, and the PV that
gives the smallest $\chi^{2}_{\text{IP}}$ is associated with the $B$
candidate.

Combinatorial background is rejected primarily through the use of a
multivariate approach with a boosted decision tree (BDT) [45, 46]. The signal
and background training samples for the BDT are simulated signal events and
candidates in data with reconstructed $B$ candidate mass in a sideband region.
Loose selection criteria are applied to the training samples on all
intermediate states ($D$, ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$,
${K}^{*0}$). Separate BDTs are trained for candidates containing long and
downstream ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates. Due to the
presence of the topologically indistinguishable ${{B}^{0}_{s}}\rightarrow
D{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decay, the available
background event sample for the training is limited to the mass range
5500–$6000{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. To make full use of all
background candidates for the training of the BDTs, all events are divided
into two sets at random. For each ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
category two BDTs are trained, using each set of events in the sideband. The
results of each BDT training are applied to the events in the other sample.
Hence, in total four BDTs are trained, and in this way the BDT applied to one
set of events is trained with a statistically independent set of events.

Each BDT uses a total of 16 variables, of which the most discriminating are
the $\chi^{2}$ of the kinematic fit of the whole decay chain (described
below), the ${K}^{*0}$ transverse momentum, and the flight distance
significance of the $B$ candidate from the associated PV. In the BDT for long
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates, two further variables
are found to provide high separation power: the flight distance significance
of the ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decay vertex from the PV and
a variable characterising the flight distance significance of the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ vertex from the $D$ vertex along the
beam line. The remaining variables in the BDT are the $\chi^{2}_{\text{IP}}$
of the $B$ candidate, the sum of $\chi^{2}_{\text{IP}}$ of the two
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ daughter tracks, the sum of the
$\chi^{2}_{\text{IP}}$ of all the other tracks, the vertex quality of the $B$
and $D$ candidates, the flight distance significance of the $D$ vertex from
the PV, a variable characterising the flight distance significance between the
$D$ and $B$ vertices along the beam line, the transverse momentum of each of
the $D$ and $B$ candidates, the cosine of the angle between the $B$ momentum
vector and the vector between the production and decay vertex, and the
helicity angle $\theta^{\ast}$. It has been verified that the use of
$\theta^{\ast}$ in the BDT has no significant impact on the value of $\kappa$.
An optimal criterion on the BDT discriminator is determined with a series of
pseudoexperiments to obtain the value that provides the best sensitivity to
$x_{\pm},y_{\pm}$.

A kinematic fit [47] is imposed on the full $B$ decay chain. The fit
constrains the $B$ candidate to point towards the PV, and the $D$ and
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates to have their known
masses [15]. This fit improves the $B$ mass resolution and therefore provides
greater discrimination between signal and background; furthermore, it improves
the resolution on the Dalitz plot and ensures that all candidates lie within
the kinematically-allowed region of the
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$ Dalitz plot.
The kinematic variables obtained in this fit are used to determine the physics
parameters of interest and the $\chi^{2}$ of this fit is used in the BDT
training.

To suppress background further, particle identification (PID) requirements are
placed on both daughter tracks of the ${K}^{*0}$ to identify the kaon and the
pion. This also removes the possibility of a second ${K}^{*0}$ candidate being
built from the same pair of tracks with opposite particle hypotheses. The PID
requirement on the kaon is tight, with an efficiency of 81%, and is necessary
to suppress 98% of the background from ${{B}^{0}}\rightarrow D{{\rho}^{0}}$
decays where a pion from the ${\rho}^{0}$ decay is misidentified as a kaon.
The absolute value of $\cos\theta^{\ast}$ is required to be greater than 0.4,
as discussed in Sect. 2.

For the selection on the $D$ (${K}^{0}_{\mathrm{\scriptscriptstyle S}}$) mass,
the mass is computed from a kinematic fit [47] that constrains the
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ ($D$) mass to its known value and
the $B$ candidate to point towards the PV. The $D$ meson mass is required to
be within $30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the nominal mass
[15] which is three times the mass resolution. The long (downstream)
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates are required to be within
$14.4\,(19.9){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of their nominal mass
which again corresponds to three times the mass resolution. In the case of
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$
candidates a loose PID cut is also placed on the kaon daughters of the $D$ to
remove cross-feed from other $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}$ decays. One further physics background is due to $D$ decays to
four pions where two pions are consistent with a long
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidate. To suppress this
background to negligible levels, a tight requirement is placed on the flight
distance significance of the long ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
candidate from the $D$ vertex along the beam line.

While the selection is different for long and downstream
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates, the small differences
between the $B$ candidate mass resolution for the two categories observed in
simulation are negligible for this analysis. This is because of the $D$ mass
constraint applied in the kinematic fit. Therefore, both
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ categories are combined in the fit
of the $B$ invariant mass distribution. All $B$ meson candidates with
invariant mass between $5200$ and
$5800{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ are fitted together to obtain
the signal and background yields.

Figure 3: Invariant mass distributions of ${{B}^{0}}\rightarrow D{{K}^{*0}}$
candidates with (top) $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ and (bottom)
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$. The
fit results, including the signal and background components, are superimposed.

The invariant mass distributions of the selected candidates are shown in Fig.
3 for both $D$ decay modes. The ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ candidates are summed. The
result of an extended maximum likelihood fit to these distributions is
superimposed. The fit is performed simultaneously for candidates from both $D$
decays, allowing parameters, unless otherwise stated, to be common between
both $D$ decay categories. Figure 3 shows the various components that are
considered in the fit to the invariant mass spectra. In addition to the signal
${{B}^{0}}\rightarrow D{{K}^{*0}}$ component, there are contributions from
${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$, from ${{B}^{0}}\rightarrow
D{{\rho}^{0}}$ where one pion is misidentified as a kaon, and from
$B\rightarrow DK$ decays where one pion from the rest of the event is added to
create a fake ${K}^{*0}$. A large background comes from
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays where the photon or
neutral pion from the ${{D}^{*0}}$ decay is not reconstructed. The purpose of
this fit is to determine the parametrisation of the signal and background
components, and the size of the background contributions, which are used in
the fit of partitioned regions of the Dalitz plot described in Sect. 6.

The ${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ and ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ decays are modelled by the same probability density function
(PDF), a sum of two Crystal Ball [48] functions with common mean and width
parameters. The mean for the ${B}^{0}_{s}$ meson is determined in the fit and
the mean for the ${B}^{0}$ meson is required to be
$87.19{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [15] lower. The width is
allowed to vary in the fit and is required to be the same for the two decays.
All other parameters are fixed from simulation. The combinatorial background
is modelled by an exponential function with slope determined by the fit for
the $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ and
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$
categories separately. The PDF for ${{B}^{0}}\rightarrow D{{\rho}^{0}}$ decays
is derived from simulation with additional data-driven corrections applied to
take into account PID response differences between data and simulation [49].
This background is described with the sum of two Crystal Ball functions, whose
parameters are obtained from the weighted simulated events. The $B\rightarrow
DK$ background is treated in a similar fashion.

For the partially reconstructed background from
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays the distribution in
the invariant mass spectrum is dependent on the helicity state of the
${D}^{*0}$ meson. The initial decay of the ${B}^{0}_{s}$ involves the decay of
a pseudoscalar to two vector particles. Hence, due to angular momentum
conservation there are three helicity amplitudes to consider, which can be
labelled by the ${D}^{*0}$ helicity state $\lambda=-1,0,+1$. In the subsequent
parity-conserving decay
${{D}^{*0}}\rightarrow{{D}^{0}}\\{{{\pi}^{0}},\gamma\\}$, the value of
$\lambda$ and the spin of the missing neutral particle determines the
distribution of the ${D}^{*0}$ helicity angle, which is defined as the angle
between the missing neutral particle’s momentum vector and the direction
opposite to the $B$ meson in the ${D}^{*0}$ rest frame. The resulting
distributions for $\lambda=-1$ or $+1$ are identical and hence are grouped
together. The functional forms of the underlying $DK\pi$ invariant mass
spectrum, shown in Table 1, can be calculated based on $\lambda$, and the spin
and mass of the missing particle. The parameters $a_{X}$ and $b_{X}$ are the
kinematic endpoints of the reconstructed $DK\pi$ invariant mass, where $X$ is
the particle that is not reconstructed. These distributions are further
modified to take into account detector resolution and reconstruction
efficiency. The parameters for the resolution and efficiency are determined
from fits to simulated samples, while the endpoints are calculated using the
masses of the particles involved.

Table 1: Functional forms of the $DK\pi$ invariant mass distribution, $m$, in partially reconstructed decays of ${{B}^{0}_{s}}\rightarrow({{D}^{*0}}\rightarrow{{D}^{0}}\\{{{\pi}^{0}},\gamma\\}){{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$, where either the ${\pi}^{0}$ or $\gamma$ is not reconstructed. The ${D}^{*0}$ helicity state is given by $\lambda$. The quantities $a_{X}$ and $b_{X}$ are the minimum and maximum kinematic boundaries of the reconstructed $DK\pi$ invariant mass, where $X$ is the particle that is missed. Missed particle |  $\lambda$ | PDF
---|---|---
${{\pi}^{0}}$ | 0 | $\left(m-\frac{a_{{\pi}^{0}}+b_{{\pi}^{0}}}{2}\right)^{2}$
${{\pi}^{0}}$ |  $-1$ or $+1$ | $-(m-a_{{\pi}^{0}})(m-b_{{\pi}^{0}})$
$\gamma$ | 0 | $-(m-a_{\gamma})(m-b_{\gamma})$
$\gamma$ |  $-1$ or $+1$ | $\left(m-\frac{a_{\gamma}+b_{\gamma}}{2}\right)^{2}+\left(\frac{a_{\gamma}-b_{\gamma}}{2}\right)^{2}$

The lower range of the mass fit is
$5200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The removal of candidates with
invariant mass below this value reduces the background from
${{B}^{0}}\rightarrow{{D}^{*0}}{{K}^{*0}}$ decays to a small level, which is
neglected in the baseline fit. Other contributions such as $B^{\pm}\rightarrow
Dh^{\pm}{{\pi}^{+}}{{\pi}^{-}}$, where one particle is missing and another may
be misidentified, are also reduced to a negligible level.

With the large number of overlapping signal and background contributions it is
not possible to let all yield parameters vary freely, especially as some
background contributions are expected to have small yields. Therefore, the
strategy employed is to constrain the ratio of these background yields to the
${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ contribution. The constraints
are determined by taking into account all relevant branching fractions [15],
fragmentation fractions [50] and selection efficiencies determined from
simulation. This is possible for the contributions ${{B}^{0}}\rightarrow
D{{\rho}^{0}}$ and $B\rightarrow DK$ where the branching fractions are
measured. The ratio of ${{B}^{0}}\rightarrow D{{\rho}^{0}}$ ($B\rightarrow
DK$) to ${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ is constrained in the fit to
$R_{\rho}=(2.9\pm 0.8)\%$ ($R_{DK}=(4.2\pm 1.0)\%$). In the case of the
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ background, neither its
branching fraction nor the relative fraction of the ${{D}^{*0}}$ helicity
states has been measured. Therefore, information is taken from the higher
statistics ${{B}^{0}_{s}}\rightarrow D(\rightarrow K\pi){{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decay, which has been studied
by the LHCb collaboration [11]. In these Cabibbo-favoured decays the mass
distribution is simpler since the ${{B}^{0}}\rightarrow D{{K}^{*0}}$ and
${{B}^{0}}\rightarrow{{D}^{*0}}{{K}^{*0}}$ decays are doubly Cabibbo-
suppressed, hence allowing the shape parameters and yields for the
${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ and
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays to be reliably
determined. The expected ratio $R_{s}$ between
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ and ${{B}^{0}_{s}}\rightarrow
D{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ can be determined
using the information from the analysis of $D\rightarrow K\pi$ decays, with a
correction for the selection efficiencies. The ratio between the total yield
of the ${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ candidates with reconstructed
mass above $5200{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ candidates is determined to
be $R_{s}=(35\pm 14)\%$. The fraction of
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ candidates where $\lambda=0$
is determined to be $\alpha=0.72\pm 0.13$. The yields of the
${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$, ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ and the combinatorial background are free parameters in the fit.
Pseudoexperiments for this fit configuration show that only negligible biases
are expected. The fitted yields and parameters of the fit are given in Table
2. The purity in the signal region, defined as $\pm
25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the ${B}^{0}$ mass measured
in the fit, is 59% (44%) for the ${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ (${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{K}^{+}}{{K}^{-}}$) candidates. The background is dominated by
combinatorial and ${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays. Contributions from
the other backgrounds considered are small.

Table 2: Results of the simultaneous fit to the invariant mass distribution of ${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays, with the $D$ meson decaying to ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$. Variable | Fitted value and uncertainty
---|---
${B}^{0}_{s}$ mass | $5369.2$ | ${}^{+1.0}_{-1.0}\;{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
Signal width parameter | $13.3$ | ${}^{+1.0}_{-0.9}\;{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$ exponential slope | $(-3.4$ | ${}^{+1.6}_{-1.4})\times 10^{-3}\;(\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})^{-1}$
${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$ exponential slope | $(-5.4$ | ${}^{+0.9}_{-0.8})\times 10^{-3}\;(\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})^{-1}$
$\alpha$ | $0.74$ | ${}^{+0.13}_{-0.13}$
$R_{DK}$ | $(4.3$ | ${}^{+1.0}_{-1.0})\times 10^{-2}$
$R_{\rho}$ | $(3.0$ | ${}^{+0.8}_{-0.8})\times 10^{-2}$
$R_{s}$ | $0.31$ | ${}^{+0.09}_{-0.09}$
$n({{B}^{0}}\rightarrow D{{K}^{*0}},{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}})$ | $84$ | ${}^{+15}_{-14}$
$n({{B}^{0}_{s}}\rightarrow D{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}},{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}})$ | $194$ | ${}^{+18}_{-17}$
$n(\mathrm{combinatorial},{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}})$ | $207$ | ${}^{+36}_{-35}$
$n({{B}^{0}}\rightarrow D{{K}^{*0}},{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}})$ | $6.7$ | ${}^{+4.8}_{-4.2}$
$n({{B}^{0}_{s}}\rightarrow D{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}},{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}})$ | $36.3$ | ${}^{+7.1}_{-6.4}$
$n(\mathrm{combinatorial},{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}})$ | $32.3$ | ${}^{+10.0}_{-9.0}$

The Dalitz plots for ${{B}^{0}}\rightarrow D{{K}^{*0}}$ candidates restricted
to the signal region for the two
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$ final states
are shown in Figs. 4 and 5. Separate plots are shown for ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ decays.

Figure 4: Dalitz plots of candidates in the signal region for
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
decays from (left) ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ and (right)
${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays. The solid blue line indicates the
kinematic boundary.

Figure 5: Dalitz plots of candidates in the signal region for
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$
decays from (left) ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ and (right)
${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays. The solid blue line indicates the
kinematic boundary.

##  5 Event selection and yield determination for
${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$ decays

A sample of ${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$,
${{D}^{*-}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$, ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}$ decays is used to determine the quantities $F_{i}$, defined in
Eq. (12), as the expected fractions of ${D}^{0}$ decays falling into Dalitz
plot bin $i$, taking into account the efficiency profile of the signal decay.
The semileptonic decay of the $B$ meson and the strong-interaction decay of
the ${D}^{*\pm}$ meson allow the flavour of the ${D}^{0}$ meson to be
determined from the charge of the muon and ${D}^{*\pm}$ daughter pion. This
particular decay chain, involving a flavour-tagged ${D}^{0}$ decay, is chosen
due to its high yield, low background level, and low mistag probability. The
selection requirements are chosen to minimise changes to the efficiency
profile with respect to that associated with the ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ channel and are the same as those listed in Ref. [8], with two
exceptions. First, only events which pass the hardware trigger that selects
muons with a transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are used.
Those where the hardware trigger only satisfies the criterion of a high
transverse energy deposit in the hadronic calorimeter are not considered.
Second, the multivariate algorithm in the software trigger designed to select
secondary vertices that are consistent with the decay of a $b$ hadron is
identical to the one used for ${{B}^{0}}\rightarrow D{{K}^{*0}}$ candidates;
an algorithm that also required the presence of a muon track was previously
used. The changes remove approximately $20\%$ of the sample used in Ref. [8];
however, in simulated data they improve the agreement in the variation of the
efficiency over the Dalitz plot between the ${{B}^{0}}\rightarrow D{{K}^{*0}}$
and ${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$ decays.

The ${D}^{0}$ invariant mass, $m({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-})$, and the invariant mass difference $\Delta m\equiv
m({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-}{{\pi}^{\pm}})-m({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-})$ are fitted simultaneously to determine the signal and
background yields. No significant correlation between these two variables is
observed within the ranges chosen for the fit. This two-dimensional
parametrisation allows the yield of selected candidates to be measured in
three categories: true ${{D}^{*\pm}}$ candidates (signal), candidates
containing a true ${D}^{0}$ but a random soft pion (RSP) and candidates formed
from random track combinations that fall within the fit range (combinatorial
background). An example fit projection is shown in Fig. 6. The result of the
two-dimensional extended unbinned maximum likelihood fit is superimposed. The
fit is performed simultaneously for the two ${D}^{0}$ final states and the two
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ categories, with some parameters
allowed to vary between categories. Candidates selected from data recorded at
$\sqrt{s}=7$ and $8\mathrm{\,Te\kern-1.00006ptV}$ are fitted separately, due
to their slightly different Dalitz plot efficiency profiles. The fit range is
$1830<m({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-})<1910{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$139.5<\Delta m<153.0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The PDFs used
to model the various components in the fit are unchanged from those used in
Ref. [8], where further details can be found.

Figure 6: Result of the simultaneous fit to
${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$,
${{D}^{*-}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{-}}$ decays with downstream
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ candidates, in 2012 data. A two-
dimensional fit is performed in (left) $m({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-})$ and (right) $\Delta m$. The (blue) total fit PDF and the
signal and background components are superimposed.

A total signal yield of approximately 90 000 (12 000)
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
($D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$)
candidates is obtained. The sample is three orders of magnitude larger than
the ${{B}^{0}}\rightarrow D{{K}^{*0}}$ yield. The signal mass range is defined
as 1840–$1890{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
(1850–$1880{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) in
$m({{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}})$
($m({{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}})$) and
143.9–$146.9{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in $\Delta m$. Within
this range the background contamination is 3–6% depending on the category.

The two-dimensional fit in $m({{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}h^{+}h^{-})$ and $\Delta m$ of the
${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$ decay is repeated in each
Dalitz plot bin with all of the PDF parameters fixed, resulting in a raw
control mode yield, $R_{i}$, for each bin $i$. The measured $R_{i}$ are not
equivalent to the $F_{i}$ fractions required to determine the $C\\!P$
parameters due to unavoidable differences from selection criteria in the
efficiency profiles of the signal and control modes. Hence, a set of
correction factors is determined from simulation. The efficiency profiles from
simulation of $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ decays are shown in Fig. 7. They show a variation
of 50% between the highest and lowest efficiency regions, although the
efficiency changes within a bin are not as large. The variation over the
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$
Dalitz plot is smaller, at approximately 35%.

Figure 7: Example efficiency profiles of (left) ${{B}^{0}}\rightarrow
D{{K}^{*0}}$ and (right) ${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$
decays in the simulation. The top (bottom) plots are for
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
($D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$)
decays.

The raw yields of the control decay must be corrected to take into account the
differences in efficiency profiles. For each Dalitz plot bin a correction
factor is determined,

$\xi_{i}\equiv\frac{\int_{i}dm^{2}_{-}\,dm^{2}_{+}\,|A_{D}(m^{2}_{-},m^{2}_{+})|^{2}\,\eta_{DK^{*0}}(m^{2}_{-},m^{2}_{+})}{\int_{i}dm^{2}_{-}\,dm^{2}_{+}\,|A_{D}(m^{2}_{-},m^{2}_{+})|^{2}\,\eta_{{{D}^{*}}\mu}(m^{2}_{-},m^{2}_{+})},$
(13)

where $\eta_{DK^{*0}}$ and $\eta_{{{D}^{*}}\mu}$ are the efficiency profiles
of the ${{B}^{0}}\rightarrow D{{K}^{*0}}$ and
${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$ decays, respectively, and are
determined with simulation. The amplitude models used to determine the Dalitz
plot intensity for the correction factor are those from Ref. [24] and Ref.
[25] for the ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{\pi}^{+}}{{\pi}^{-}}$
and ${{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$ decays,
respectively. The amplitude models used here only provide a description of the
intensity distribution over the Dalitz plot and introduce no significant model
dependence into the analysis. The correction factors are determined separately
for data reconstructed with each ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$
type, as the efficiency profile is different between the two
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ categories. This method of
determining the $F_{i}$ parameters is preferable to using solely the amplitude
models and ${{B}^{0}}\rightarrow D{{K}^{*0}}$ simulated events, since the
method is data-driven and the efficiency correction causes deficiencies in the
simulation and the model to cancel at first order. The correction factors are
within 10$\%$ of unity. The $F_{i}$ values can be determined via the relation
$F_{i}=h^{\prime}\xi_{i}R_{i}$, where $h^{\prime}$ is a normalisation factor
such that the sum of all $F_{i}$ is unity. The $F_{i}$ parameters are
determined for each year of data taking and
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ category separately and are then
combined in the fraction observed in the ${{B}^{0}}\rightarrow D{{K}^{*0}}$
signal region in data. The total uncertainty on $F_{i}$ is 5% or less in all
of the bins, and is a combination of the uncertainty on $R_{i}$ due to the
size of the control channel, and the uncertainty on $\xi_{i}$ due to the
limited size of the simulated samples. The two contributions are similar in
size.

##  6 Dalitz plot fit to determine the $C\\!P$-violating parameters $x_{\pm}$
and $y_{\pm}$

The Dalitz plot fit is used to measure the $C\\!P$-violating parameters
$x_{\pm}$ and $y_{\pm}$, as introduced in Sect. 2. Following Eqs. (10) and
(11), these parameters can be determined from the populations of the ${B}^{0}$
and ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ Dalitz plot bins,
given the external information of the $c_{i}$ and $s_{i}$ parameters from
CLEO-c data, the values of $F_{i}$ from the semileptonic control decay modes
and the measured value of $\kappa$.

Although the absolute numbers of ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ decays integrated over the $D$
Dalitz plot have some dependence on $x_{\pm}$ and $y_{\pm}$, the sensitivity
gained compared to using just the relations in Eqs. (10) and (11) is
negligible [51]. Consequently, as stated previously, the integrated yields are
not used and the analysis is insensitive to $B$ meson production and detection
asymmetries.

The ${{B}^{0}}\rightarrow D{{K}^{*0}}$ data are split into four categories,
one for each $D$ decay and then by the charge of the ${K}^{*0}$ daughter kaon.
As in the case of the fit to the invariant mass, data from the two
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ categories are merged. Each category
is then divided into the Dalitz plot bins shown in Fig. 2, where there are 16
bins for $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}{{\pi}^{-}}$ and 4 bins for
$D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}{{K}^{+}}{{K}^{-}}$.
Since the Dalitz plots for ${B}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ data are analysed separately,
this gives a total of 40 bins. The PDF parameters for the signal and
background invariant mass distributions are fixed to the values determined in
the invariant mass fit described in Sect. 4.

The yield of the combinatorial background in each bin is a free parameter,
apart from the yields in bins in which an auxiliary fit determines it to be
negligible. It is necessary to set these to zero to facilitate the calculation
of the covariance matrix. The total yield of ${{B}^{0}_{s}}\rightarrow
D{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays integrated
over the Dalitz plot for each category is a free parameter. The value of
$r_{B}({{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}})$ is expected to be an order
of magnitude smaller than $r_{{B}^{0}}$ due to suppression from CKM factors.
Hence, the fractions in each Dalitz plot bin are assigned assuming that
$C\\!P$ violation in these decays are negligible, which is also consistent
with observations in Ref. [14]. Therefore, the decay of the ${B}^{0}_{s}$
(${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$) meson contains a
${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ (${D}^{0}$) meson. It
is verified in simulation that the reconstruction efficiency over the $D$
Dalitz plot does not depend on the parent $B$ decay and hence the yield of
${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays in bin $i$ is given by
the relevant total yield multiplied by $F_{-i}$.

The total yields of the ${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$, ${{B}^{0}}\rightarrow
D{{\rho}^{0}}$ and $B\rightarrow DK$ backgrounds in each category are
determined by multiplying the total yield of ${{B}^{0}_{s}}\rightarrow
D{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ in that category by
the values of $R_{s}$, $R_{\rho}$ and $R_{DK}$, respectively, that are listed
in Table 2. The following assumptions are made about the Dalitz plot
distributions of these backgrounds. The $C\\!P$ violation in
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays is expected to be
negligible as the underlying CKM factors are the same as that for
${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays. Hence, the
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays are distributed over
the $D\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle S}}}h^{+}h^{-}$ Dalitz
plot in the same way as ${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays. The $D$ meson from
${{B}^{0}}\rightarrow D{{\rho}^{0}}$ decays is assumed to be an equal
admixture of ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ and hence the yield is
distributed according to ($F_{+i}+F_{-i}$), because the pion misidentified as
a kaon is equally likely to be of either charge. In the case of the
$B\rightarrow DK$ decay, $C\\!P$ violation is expected and the yield is
distributed according to Eqs. (10) and (11), where the values of the $C\\!P$
violating parameters are those determined in Ref. [8].

The ${{B}^{0}}\rightarrow D{{K}^{*0}}$ yield in each bin is determined using
the total yield of ${{B}^{0}}\rightarrow D{{K}^{*0}}$ in each category, which
is a free parameter, and Eqs. (10) and (11). The parameters of interest,
$x_{\pm}$ and $y_{\pm}$, are allowed to vary. The values of $c_{i}$ and
$s_{i}$ are constrained to their measured values from CLEO [23], assuming
Gaussian errors and taking into account statistical and systematic
correlations. The values of $F_{i}$ are fixed. The value of $\kappa$ is also
fixed in the fit to the central value measured in Ref. [14].

An ensemble of 10 000 pseudoexperiments is generated to validate the fit
procedure. In each pseudoexperiment the numbers and distributions of signal
and background candidates are generated according to the expected distribution
in data, taking care to smear the input values of $c_{i}$ and $s_{i}$. The
full fit procedure is then performed. A variety of $x_{\pm}$ and $y_{\pm}$
values consistent with previous measurements is used [50]. Small biases in the
central values, with magnitudes around $10\%$ of the statistical uncertainty,
are observed in the pseudoexperiments. These biases are due to the low event
yields in some of the bins and they reduce in simulated experiments with
higher yields. The central values are corrected for the biases.

The results of the fit are $x_{+}=0.05\pm 0.35$, $x_{-}=-0.31\pm 0.20$,
$y_{+}=-0.81\pm 0.28$, and $y_{-}=0.31\pm 0.21$. The statistical uncertainties
are compatible with those predicted by the pseudoexperiments. The measured
values of $(x_{\pm},y_{\pm})$ from the fit to data, with their likelihood
contours, corresponding to statistical uncertainties only, are displayed in
Fig. 8.

Figure 8: Confidence levels at (solid) 68.3% and (dotted) 95.5% for (red,
light) $(x_{+},y_{+})$ and (blue, dark) $(x_{-},y_{-})$ as measured in
${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays (statistical uncertainties only).
The parameters $(x_{+},y_{+})$ relate to ${B}^{0}$ decays and $(x_{-},y_{-})$
refer to ${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}$ decays. The
points represent the best fit values.

The expected signature for a sample that exhibits $C\\!P$ violation is that
the two vectors defined by the coordinates $(x_{-},y_{-})$ and $(x_{+},y_{+})$
should both be non-zero in magnitude and have a non-zero opening angle. This
opening angle is equal to $2\gamma$. No evidence for $C\\!P$ violation is
observed.

To investigate whether the binned fit gives an adequate description of the
distribution of events over the Dalitz plot, the signal yield in each bin is
fitted directly as a cross-check. A comparison of these yields and those
predicted by the fitted values of $x_{\pm}$ and $y_{\pm}$ shows good
agreement.

##  7 Systematic uncertainties

Systematic uncertainties are evaluated on the measurements of the Cartesian
parameters and are presented in Table 3. The source of each systematic
uncertainty is described in turn below. Unless otherwise described, the
systematic uncertainties are determined from an ensemble of pseudoexperiments
where the simulated data are generated in an alternative configuration, and
fitted with the default method described in Sect. 6. The mean shift in the
fitted values of $x_{\pm}$ and $y_{\pm}$ in comparison to their input values
is taken as the systematic uncertainty. Uncertainties arising from the CLEO
measurements are included within the statistical uncertainties since the
values of $c_{i}$ and $s_{i}$ are constrained in the Dalitz plot fit. Their
contribution to the statistical uncertainty is approximately 0.02 for
$x_{\pm}$ and 0.05 for $y_{\pm}$.

Table 3: Summary of the systematic uncertainties for the parameters $x_{\pm},y_{\pm}$. The various sources of systematic uncertainties are described in the main text.  Source | $\sigma(x_{+})$ | $\sigma(x_{-})$ | $\sigma(y_{+})$ | $\sigma(y_{-})$
---|---|---|---|---
Efficiency corrections | 0.019 | 0.034 | 0.021 | 0.005
Efficiency combination | 0.007 | 0.001 | 0.007 | 0.008
Mass fit: $\alpha$ | 0.002 | 0.005 | 0.021 | 0.020
Mass fit: ${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ | 0.002 | 0.002 | 0.010 | 0.005
Mass fit: ${{B}^{0}}\rightarrow D{{\rho}^{0}}$ | 0.002 | 0.003 | 0.004 | 0.001
Mass fit: $B\rightarrow DK$ | 0.000 | 0.000 | 0.000 | 0.000
Mass fit: Signal shape | 0.005 | 0.003 | 0.003 | 0.002
Mass fit: ${{B}^{0}}\rightarrow{{D}^{*0}}{{K}^{*0}}$ | 0.006 | 0.007 | 0.008 | 0.004
Mass fit: $B\rightarrow{{D}^{*0}}h$ | 0.001 | 0.001 | 0.007 | 0.005
Mass fit: $B\rightarrow D\pi\pi\pi$ | 0.001 | 0.002 | 0.001 | 0.003
Dalitz plot migration | 0.003 | 0.004 | 0.007 | 0.003
Value of $\kappa$ | 0.001 | 0.011 | 0.008 | 0.002
Fitter bias | 0.004 | 0.014 | 0.042 | 0.042
Total systematic | 0.022 | 0.040 | 0.056 | 0.048

A systematic uncertainty arises from imperfect modelling in the simulation
used to derive the efficiency correction in the determination of the $F_{i}$
parameters. To determine this systematic uncertainty, a conservative approach
is used, where an alternative set of $F_{i}$ values is determined using only
the amplitude models and simulated ${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays.
These alternative $F_{i}$ are used in the generation of pseudoexperiments to
determine the systematic uncertainty. A further uncertainty on the $F_{i}$
parameters arises from the fractions in which the individual $F_{i}$
parameters from the differing categories (year of data taking and
${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ type) are combined. A second
alternate set of $F_{i}$ are obtained by combining the values of $F_{i}$ for
each category using the fractions of data observed in the ${B}^{0}_{s}$ mass
window. The fractions in the ${B}^{0}$ window are statistically consistent
with those observed in the ${B}^{0}_{s}$ mass window. The associated
uncertainty is determined through the use of pseudoexperiments which are
generated with the alternate set of $F_{i}$ values.

Several systematic uncertainties are associated with the parametrisation of
the invariant mass distribution. These arise from uncertainties in the shape
of the ${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ background, the size of the
${{B}^{0}}\rightarrow D{{\rho}^{0}}$ background, $C\\!P$ violation in the
$B\rightarrow DK$ background, the PDF shape used to describe the signal peak
and the inclusion of backgrounds that are neglected in the nominal fit,
because of their small yield.

The uncertainty in the shape of the ${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ background arises from the
relative contribution of the different ${D}^{*0}$ decay and helicity state
components, each of which have a different $DK\pi$ invariant mass
distribution. A different parametrisation of the data with the lower mass
limit extending down to 4900${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ results
in a measurement $\alpha=0.9\pm 0.1$, in comparison to the value of $0.74\pm
0.13$ obtained in the fit described in Sect. 4. Accounting for the difference
in mass range, the uncertainty is estimated by generating pseudoexperiments
with $\alpha=0.91$, and is found to be 2$\times 10^{-3}$ or less in each of
the $C\\!P$ parameters.

A separate systematic uncertainty is evaluated for the relative fraction of
${{D}^{*0}}\rightarrow{{D}^{0}}{{\pi}^{0}}$ and
${{D}^{*0}}\rightarrow{{D}^{0}}\gamma$ decays in the
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ contribution. The
uncertainties in the relative fractions are due to uncertainties in the
branching fractions of the ${D}^{*0}$ decays and in the selection efficiencies
determined in simulation. In this case the systematic uncertainty is small and
is determined by fitting the data repeatedly with the fractions smeared around
the central values.

The estimation of the ${{B}^{0}}\rightarrow D{{\rho}^{0}}$ yield ignores the
${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$ S-wave contributions, which
will contribute if the misidentified ${{\pi}^{+}}{{\pi}^{-}}$ invariant mass
falls within the ${K}^{*0}$ mass window. The amplitude analysis of
${{B}^{0}}\rightarrow D{{\pi}^{+}}{{\pi}^{-}}$ decays in Ref. [52] is used to
determine that the potential size of the S-wave contribution could increase
the apparent ${{B}^{0}}\rightarrow D{{\rho}^{0}}$ yield by approximately 50%.
Assuming that the additional S-wave contribution will have the same $DK\pi$
invariant mass distribution, the systematic uncertainty on the $C\\!P$
parameters is estimated by generating pseudoexperiments with the
${{B}^{0}}\rightarrow D{{\rho}^{0}}$ contribution increased by 50%. The
resulting uncertainties on $x_{\pm},y_{\pm}$ are lower than 4$\times 10^{-3}$.

In the default fit the $C\\!P$ parameters of the $B\rightarrow DK$ background
are fixed to the central values measured in Ref. [8]. The fits to the data are
repeated with multiple values of the $C\\!P$ parameters of the $B\rightarrow
DK$ decay, smeared according to the measured uncertainties and correlations,
and the shifts in $x_{\pm},y_{\pm}$ are found to be less than 0.001.

An alternative PDF to describe the ${B}^{0}$ and ${B}^{0}_{s}$ signals is
considered by taking the sum of three Gaussian functions. The mean and width
of the primary Gaussian is determined by performing a mass fit to data with
the relative means and widths of the two secondary Gaussians taken from
simulation. The systematic uncertainty is small and is estimated by generating
pseudoexperiments with this alternative PDF.

In the default mass fit the contributions of
${{B}^{0}}\rightarrow{{D}^{*0}}{{K}^{*0}}$,
$B^{\pm}\rightarrow{{D}^{*0}}\pi^{\pm}$, $B^{\pm}\rightarrow{{D}^{*0}}K^{\pm}$
and $B^{\pm}\rightarrow D{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$ decays are
ignored as they are estimated to contribute approximately $2-3$ events each. A
systematic uncertainty from neglecting each of these decays is evaluated. The
${{B}^{0}}\rightarrow{{D}^{*0}}{{K}^{*0}}$ decays can be described with the
same PDFs as the ${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays but shifted by the
${{B}^{0}_{s}}-{{B}^{0}}$ mass difference. The $B$ mass fit described in Sect.
4 is performed with this background included, where the yield of
${{B}^{0}}\rightarrow{{D}^{*0}}{{K}^{*0}}$ decays is constrained relative to
that of the ${{B}^{0}_{s}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ in a similar manner to the
${{B}^{0}_{s}}\rightarrow{{D}^{*0}}{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ decays. Although the addition
of this background only has a small impact on the mass fit parameters, its
$C\\!P$ parameters are unknown. Hence, pseudoexperiments are generated with
the ${{B}^{0}}\rightarrow{{D}^{*0}}{{K}^{*0}}$ background in three different
$C\\!P$ violating hypotheses and are fitted with the default configuration.
The uncertainty is found to be less than 0.01 for all choices of the $C\\!P$
parameters. Further pseudoexperiments are generated with
$B^{\pm}\rightarrow{{D}^{*0}}h^{\pm}$ and $B^{\pm}\rightarrow
D{{\pi}^{\pm}}{{\pi}^{+}}{{\pi}^{-}}$ decays, where their PDF shapes and
yields are determined from simulation. Fitting the pseudoexperiments with the
nominal fit demonstrates that the uncertainty due to ignoring these decays is
7$\times 10^{-3}$ or less for all $C\\!P$ parameters.

The systematic uncertainty from the effect of candidates being assigned the
wrong Dalitz plot bin number is considered. This can occur if reconstruction
effects cause shifts in the measured values of $m^{2}_{+}$ and $m^{2}_{-}$
away from their true values. For both ${{B}^{0}}\rightarrow D{{K}^{*0}}$ and
${{B}^{0}}\rightarrow{{D}^{*-}}\mu^{+}\nu_{\mu}$ decays the resolution in
$m^{2}_{+}$ and $m^{2}_{-}$ is approximately
$0.005{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
($0.006{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) for candidates with long
(downstream) ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$ decays. This is small
compared to the typical width of a bin, but net migration can occur if the
candidate lies close to the edge of a Dalitz plot bin. To first order, this
effect is accounted for by use of the control channel, but residual effects
enter due to the non-zero value of $r_{{B}^{0}}$ in the signal decay, causing
a different distribution in the Dalitz plot. The uncertainty due to these
residual effects is determined via pseudoexperiments, in which different input
$F_{i}$ values are used to reflect the residual migration. The size of this
possible bias is found to vary between 3$\times 10^{-3}$ and 7$\times
10^{-3}$.

The value of $\kappa$ has an associated uncertainty, and so pseudoexperiments
are generated assuming the value $\kappa=0.912$, which corresponds to the
central value of $\kappa$ lowered by one standard deviation. The mean shifts
in $x_{\pm},y_{\pm}$ are of order 0.01. As described in Sect. 6, the central
values of the fit parameters $x_{\pm}$ and $y_{\pm}$ are corrected by a fitter
bias that is determined with pseudoexperiments. The systematic uncertainty is
assigned using half the size of the correction.

The total experimental systematic uncertainty is determined by adding all
sources in quadrature and is 0.02 on $x_{+}$, 0.04 on $x_{-}$, 0.06 on
$y_{+}$, and 0.05 on $y_{-}$. These uncertainties are dominated by the
efficiency corrections in $F_{i}$ and the fitter bias. The systematic
uncertainties are less than 20% of the corresponding statistical
uncertainties.

##  8 Results and interpretation

The results for $x_{\pm}$ and $y_{\pm}$ are

$\displaystyle x_{+}$ $\displaystyle=\phantom{-}0.05\pm 0.35\pm 0.02,$
$\displaystyle x_{-}$ $\displaystyle=-0.31\pm 0.20\pm 0.04,$ $\displaystyle
y_{+}$ $\displaystyle=-0.81\pm 0.28\pm 0.06,$ $\displaystyle y_{-}$
$\displaystyle=\phantom{-}0.31\pm 0.21\pm 0.05,$

where the first uncertainties are statistical and the second are systematic.
After accounting for all sources of uncertainty, the correlation matrix
between the measured $x_{\pm}$, $y_{\pm}$ parameters for the full data set is
obtained, and is given in Table 4. Correlations for the statistical
uncertainties are determined by the fit. The systematic uncertainties are only
weakly correlated and the correlations are ignored.

Table 4: Total correlation matrix, including statistical and systematic uncertainties, between the $x_{\pm}$, $y_{\pm}$ parameters used in the extraction of $\gamma$.  | $x_{+}$ | $x_{-}$ | $y_{+}$ | $y_{-}$
---|---|---|---|---
$x_{+}$ | $\phantom{-}1.00$ | $\phantom{-}0.00$ | $\phantom{-}0.13$ | $-0.01$
$x_{-}$ |  | $\phantom{-}1.00$ | $-0.01$ | $\phantom{-}0.14$
$y_{+}$ |  |  | $\phantom{-}1.00$ | $\phantom{-}0.02$
$y_{-}$ |  |  |  | $\phantom{-}1.00$

The results for $x_{\pm}$ and $y_{\pm}$ can be interpreted in terms of the
underlying physics parameters $\gamma$, $r_{{{B}^{0}}}$ and
$\delta_{{{B}^{0}}}$. This interpretation is performed using a Neyman
construction with Feldman-Cousins ordering [53], using the same procedure as
described in Ref. [27], yielding confidence levels for the three physics
parameters.

In Fig. 9, the projections of the three-dimensional surfaces containing the
one and two standard deviation volumes (i.e., $\Delta\chi^{2}=1$ and 4) onto
the $(\gamma,r_{{{B}^{0}}})$ and $(\gamma,\delta_{{{B}^{0}}})$ planes are
shown; the statistical and systematic uncertainties on $x_{\pm}$ and $y_{\pm}$
are combined in quadrature. The solution for the physics parameters has a two-
fold ambiguity, with a second solution corresponding to
$(\gamma,\delta_{{{B}^{0}}})\rightarrow(\gamma+180^{\circ},\delta_{{{B}^{0}}}+180^{\circ})$.
For the solution that satisfies $0<\gamma<180^{\circ}$, the following results
are obtained:

$\displaystyle r_{{{B}^{0}}}$ $\displaystyle=0.56\pm 0.17,$
$\displaystyle\delta_{{{B}^{0}}}$ $\displaystyle=(204\,^{+21}_{-20})^{\circ},$
$\displaystyle\gamma$ $\displaystyle=(71\pm 20)^{\circ}.$

The central value for $\gamma$ is consistent with the world average from
previous measurements [5, 6]. The value for $r_{{{B}^{0}}}$, while consistent
with current knowledge, has a central value that is larger than expected [16,
17, 26, 24]. The results are also consistent with, but cannot be combined
with, the model-dependent analysis of the same dataset performed by LHCb [22].

A key advantage of having direct measurements of $x_{\pm}$ and $y_{\pm}$ is
that there is only a two-fold ambiguity in the value of $\gamma$ from the
trigonometric expressions. This means that when combined with the results of
other $C\\!P$ violation studies in ${{B}^{0}}\rightarrow D{{K}^{*0}}$ decays
such as those in Ref. [11], these measurements will provide strong constraints
on the hadronic parameters, and will provide improved sensitivity to $\gamma$
when combined with all other measurements.

Figure 9: The three-dimensional confidence volumes projected onto the
$(\gamma,r_{{{B}^{0}}})$ and $(\gamma,\delta_{{{B}^{0}}})$ planes. The
confidence levels correspond to 68.3% and 95.5% confidence levels when
projected onto one dimension and are denoted by solid and dotted contours,
respectively. The diamonds mark the central values.

## 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); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FANO (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), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

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LHCb collaboration

R. Aaij39, C. Abellán Beteta41, B. Adeva38, M. Adinolfi47, Z. Ajaltouni5, S.
Akar6, J. Albrecht10, F. Alessio39, M. Alexander52, S. Ali42, G. Alkhazov31,
P. Alvarez Cartelle54, A.A. Alves Jr58, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3,40, L. Anderlini18, G. Andreassi40, M. Andreotti17,g, J.E. Andrews59, R.B.
Appleby55, O. Aquines Gutierrez11, F. Archilli39, P. d’Argent12, A.
Artamonov36, M. Artuso60, E. Aslanides6, G. Auriemma26,n, M. Baalouch5, S.
Bachmann12, J.J. Back49, A. Badalov37, C. Baesso61, S. Baker54, W. Baldini17,
R.J. Barlow55, C. Barschel39, S. Barsuk7, W. Barter39, V. Batozskaya29, V.
Battista40, A. Bay40, L. Beaucourt4, J. Beddow52, F. Bedeschi24, I. Bediaga1,
L.J. Bel42, V. Bellee40, N. Belloli21,k, I. Belyaev32, E. Ben-Haim8, G.
Bencivenni19, S. Benson39, J. Benton47, A. Berezhnoy33, R. Bernet41, A.
Bertolin23, F. Betti15, M.-O. Bettler39, M. van Beuzekom42, S. Bifani46, P.
Billoir8, T. Bird55, A. Birnkraut10, A. Bizzeti18,i, T. Blake49, F. Blanc40,
J. Blouw11, S. Blusk60, V. Bocci26, A. Bondar35, N. Bondar31,39, W.
Bonivento16, A. Borgheresi21,k, S. Borghi55, M. Borisyak67, M. Borsato38, M.
Boubdir9, T.J.V. Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12, M.
Britsch12, T. Britton60, J. Brodzicka55, E. Buchanan47, C. Burr55, A.
Bursche2, J. Buytaert39, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,k, M.
Calvo Gomez37,p, P. Campana19, D. Campora Perez39, L. Capriotti55, A.
Carbone15,e, G. Carboni25,l, R. Cardinale20,j, A. Cardini16, P. Carniti21,k,
L. Carson51, K. Carvalho Akiba2, G. Casse53, L. Cassina21,k, L. Castillo
Garcia40, M. Cattaneo39, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M.
Charles8, Ph. Charpentier39, G. Chatzikonstantinidis46, M. Chefdeville4, S.
Chen55, S.-F. Cheung56, V. Chobanova38, M. Chrzaszcz41,27, X. Cid Vidal39, G.
Ciezarek42, P.E.L. Clarke51, M. Clemencic39, H.V. Cliff48, J. Closier39, V.
Coco58, J. Cogan6, E. Cogneras5, V. Cogoni16,f, L. Cojocariu30, G.
Collazuol23,r, P. Collins39, A. Comerma-Montells12, A. Contu39, A. Cook47, S.
Coquereau8, G. Corti39, M. Corvo17,g, B. Couturier39, G.A. Cowan51, D.C.
Craik51, A. Crocombe49, M. Cruz Torres61, S. Cunliffe54, R. Currie54, C.
D’Ambrosio39, E. Dall’Occo42, J. Dalseno47, P.N.Y. David42, A. Davis58, O. De
Aguiar Francisco2, K. De Bruyn6, S. De Capua55, M. De Cian12, J.M. De
Miranda1, L. De Paula2, P. De Simone19, C.-T. Dean52, D. Decamp4, M.
Deckenhoff10, L. Del Buono8, N. Déléage4, M. Demmer10, D. Derkach67, O.
Deschamps5, F. Dettori39, B. Dey22, A. Di Canto39, H. Dijkstra39, F. Dordei39,
M. Dorigo40, A. Dosil Suárez38, A. Dovbnya44, K. Dreimanis53, L. Dufour42, G.
Dujany55, K. Dungs39, P. Durante39, R. Dzhelyadin36, A. Dziurda27, A.
Dzyuba31, S. Easo50,39, U. Egede54, V. Egorychev32, S. Eidelman35, S.
Eisenhardt51, U. Eitschberger10, R. Ekelhof10, L. Eklund52, I. El Rifai5, Ch.
Elsasser41, S. Ely60, S. Esen12, H.M. Evans48, T. Evans56, A. Falabella15, C.
Färber39, N. Farley46, S. Farry53, R. Fay53, D. Fazzini21,k, D. Ferguson51, V.
Fernandez Albor38, F. Ferrari15, F. Ferreira Rodrigues1, M. Ferro-Luzzi39, S.
Filippov34, M. Fiore17,g, M. Fiorini17,g, M. Firlej28, C. Fitzpatrick40, T.
Fiutowski28, F. Fleuret7,b, K. Fohl39, M. Fontana16, F. Fontanelli20,j, D. C.
Forshaw60, R. Forty39, M. Frank39, C. Frei39, M. Frosini18, J. Fu22, E.
Furfaro25,l, A. Gallas Torreira38, D. Galli15,e, S. Gallorini23, S.
Gambetta51, M. Gandelman2, P. Gandini56, Y. Gao3, J. García Pardiñas38, J.
Garra Tico48, L. Garrido37, P.J. Garsed48, D. Gascon37, C. Gaspar39, L.
Gavardi10, G. Gazzoni5, D. Gerick12, E. Gersabeck12, M. Gersabeck55, T.
Gershon49, Ph. Ghez4, S. Gianì40, V. Gibson48, O.G. Girard40, L. Giubega30,
V.V. Gligorov39, C. Göbel61, D. Golubkov32, A. Golutvin54,39, A. Gomes1,a, C.
Gotti21,k, M. Grabalosa Gándara5, R. Graciani Diaz37, L.A. Granado Cardoso39,
E. Graugés37, E. Graverini41, G. Graziani18, A. Grecu30, P. Griffith46, L.
Grillo12, O. Grünberg65, E. Gushchin34, Yu. Guz36,39, T. Gys39, T.
Hadavizadeh56, C. Hadjivasiliou60, G. Haefeli40, C. Haen39, S.C. Haines48, S.
Hall54, B. Hamilton59, X. Han12, S. Hansmann-Menzemer12, N. Harnew56, S.T.
Harnew47, J. Harrison55, J. He39, T. Head40, A. Heister9, K. Hennessy53, P.
Henrard5, L. Henry8, J.A. Hernando Morata38, E. van Herwijnen39, M. Heß65, A.
Hicheur2, D. Hill56, M. Hoballah5, C. Hombach55, L. Hongming40, W.
Hulsbergen42, T. Humair54, M. Hushchyn67, N. Hussain56, D. Hutchcroft53, M.
Idzik28, P. Ilten57, R. Jacobsson39, A. Jaeger12, J. Jalocha56, E. Jans42, A.
Jawahery59, M. John56, D. Johnson39, C.R. Jones48, C. Joram39, B. Jost39, N.
Jurik60, S. Kandybei44, W. Kanso6, M. Karacson39, T.M. Karbach39,†, S.
Karodia52, M. Kecke12, M. Kelsey60, I.R. Kenyon46, M. Kenzie39, T. Ketel43, E.
Khairullin67, B. Khanji21,39,k, C. Khurewathanakul40, T. Kirn9, S. Klaver55,
K. Klimaszewski29, M. Kolpin12, I. Komarov40, R.F. Koopman43, P. Koppenburg42,
M. Kozeiha5, L. Kravchuk34, K. Kreplin12, M. Kreps49, P. Krokovny35, F.
Kruse10, W. Krzemien29, W. Kucewicz27,o, M. Kucharczyk27, V. Kudryavtsev35, A.
K. Kuonen40, K. Kurek29, T. Kvaratskheliya32, D. Lacarrere39, G.
Lafferty55,39, A. Lai16, D. Lambert51, G. Lanfranchi19, C. Langenbruch49, B.
Langhans39, T. Latham49, C. Lazzeroni46, R. Le Gac6, J. van Leerdam42, J.-P.
Lees4, R. Lefèvre5, A. Leflat33,39, J. Lefrançois7, E. Lemos Cid38, O. Leroy6,
T. Lesiak27, B. Leverington12, Y. Li7, T. Likhomanenko67,66, R. Lindner39, C.
Linn39, F. Lionetto41, B. Liu16, X. Liu3, D. Loh49, I. Longstaff52, J.H.
Lopes2, D. Lucchesi23,r, M. Lucio Martinez38, H. Luo51, A. Lupato23, E.
Luppi17,g, O. Lupton56, N. Lusardi22, A. Lusiani24, X. Lyu62, F. Machefert7,
F. Maciuc30, O. Maev31, K. Maguire55, S. Malde56, A. Malinin66, G. Manca7, G.
Mancinelli6, P. Manning60, A. Mapelli39, J. Maratas5, J.F. Marchand4, U.
Marconi15, C. Marin Benito37, P. Marino24,t, J. Marks12, G. Martellotti26, M.
Martin6, M. Martinelli40, D. Martinez Santos38, F. Martinez Vidal68, D.
Martins Tostes2, L.M. Massacrier7, A. Massafferri1, R. Matev39, A. Mathad49,
Z. Mathe39, C. Matteuzzi21, A. Mauri41, B. Maurin40, A. Mazurov46, M.
McCann54, J. McCarthy46, A. McNab55, R. McNulty13, B. Meadows58, F. Meier10,
M. Meissner12, D. Melnychuk29, M. Merk42, A Merli22,u, E Michielin23, D.A.
Milanes64, M.-N. Minard4, D.S. Mitzel12, J. Molina Rodriguez61, I.A. Monroy64,
S. Monteil5, M. Morandin23, P. Morawski28, A. Mordà6, M.J. Morello24,t, J.
Moron28, A.B. Morris51, R. Mountain60, F. Muheim51, D. Müller55, J. Müller10,
K. Müller41, V. Müller10, M. Mussini15, B. Muster40, P. Naik47, T. Nakada40,
R. Nandakumar50, A. Nandi56, I. Nasteva2, M. Needham51, N. Neri22, S.
Neubert12, N. Neufeld39, M. Neuner12, A.D. Nguyen40, C. Nguyen-Mau40,q, V.
Niess5, S. Nieswand9, R. Niet10, N. Nikitin33, T. Nikodem12, A. Novoselov36,
D.P. O’Hanlon49, A. Oblakowska-Mucha28, V. Obraztsov36, S. Ogilvy52, O.
Okhrimenko45, R. Oldeman16,48,f, C.J.G. Onderwater69, B. Osorio Rodrigues1,
J.M. Otalora Goicochea2, A. Otto39, P. Owen54, A. Oyanguren68, A. Palano14,d,
F. Palombo22,u, M. Palutan19, J. Panman39, A. Papanestis50, M. Pappagallo52,
L.L. Pappalardo17,g, C. Pappenheimer58, W. Parker59, C. Parkes55, G.
Passaleva18, G.D. Patel53, M. Patel54, C. Patrignani20,j, A. Pearce55,50, A.
Pellegrino42, G. Penso26,m, M. Pepe Altarelli39, S. Perazzini15,e, P. Perret5,
L. Pescatore46, K. Petridis47, A. Petrolini20,j, M. Petruzzo22, E. Picatoste
Olloqui37, B. Pietrzyk4, M. Pikies27, D. Pinci26, A. Pistone20, A. Piucci12,
S. Playfer51, M. Plo Casasus38, T. Poikela39, F. Polci8, A. Poluektov49,35, I.
Polyakov32, E. Polycarpo2, A. Popov36, D. Popov11,39, B. Popovici30, C.
Potterat2, E. Price47, J.D. Price53, J. Prisciandaro38, A. Pritchard53, C.
Prouve47, V. Pugatch45, A. Puig Navarro40, G. Punzi24,s, W. Qian56, R.
Quagliani7,47, B. Rachwal27, J.H. Rademacker47, M. Rama24, M. Ramos Pernas38,
M.S. Rangel2, I. Raniuk44, G. Raven43, F. Redi54, S. Reichert10, A.C. dos
Reis1, V. Renaudin7, S. Ricciardi50, S. Richards47, M. Rihl39, K.
Rinnert53,39, V. Rives Molina37, P. Robbe7, A.B. Rodrigues1, E. Rodrigues58,
J.A. Rodriguez Lopez64, P. Rodriguez Perez55, A. Rogozhnikov67, S. Roiser39,
V. Romanovsky36, A. Romero Vidal38, J. W. Ronayne13, M. Rotondo23, T. Ruf39,
P. Ruiz Valls68, J.J. Saborido Silva38, N. Sagidova31, B. Saitta16,f, V.
Salustino Guimaraes2, C. Sanchez Mayordomo68, B. Sanmartin Sedes38, R.
Santacesaria26, C. Santamarina Rios38, M. Santimaria19, E. Santovetti25,l, A.
Sarti19,m, C. Satriano26,n, A. Satta25, D.M. Saunders47, D. Savrina32,33, S.
Schael9, M. Schiller39, H. Schindler39, M. Schlupp10, M. Schmelling11, T.
Schmelzer10, B. Schmidt39, O. Schneider40, A. Schopper39, M. Schubiger40,
M.-H. Schune7, R. Schwemmer39, B. Sciascia19, A. Sciubba26,m, A. Semennikov32,
A. Sergi46, N. Serra41, J. Serrano6, L. Sestini23, P. Seyfert21, M. Shapkin36,
I. Shapoval17,44,g, Y. Shcheglov31, T. Shears53, L. Shekhtman35, V.
Shevchenko66, A. Shires10, B.G. Siddi17, R. Silva Coutinho41, L. Silva de
Oliveira2, G. Simi23,s, M. Sirendi48, N. Skidmore47, T. Skwarnicki60, E.
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F.J.P. Soler52, F. Soomro40, D. Souza47, B. Souza De Paula2, B. Spaan10, P.
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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
6CPPM, Aix-Marseille Université, CNRS/IN2P3, 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
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN 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
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62University of Chinese Academy of Sciences, Beijing, China, associated to 3
63Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
64Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
65Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
66National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
67Yandex School of Data Analysis, Moscow, Russia, associated to 32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

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 Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
†Deceased