EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-202 LHCb-PAPER-2014-041 12 August 2014

Measurement of the CKM angle $\gamma$

using ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ with

$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}},{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ decays

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

A binned Dalitz plot analysis of $B^{\pm}\rightarrow DK^{\pm}$ decays, with
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}\pi^{+}\pi^{-}$ and
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}K^{+}K^{-}$, is performed to
measure the $C\\!P$-violating observables $x_{\pm}$ and $y_{\pm}$, which are
sensitive to the Cabibbo-Kobayashi-Maskawa angle $\gamma$. The analysis
exploits a sample of proton-proton collision data corresponding to
3.0$\mbox{\,fb}^{-1}$ collected by the LHCb experiment. Measurements from
CLEO-c of the variation of the strong-interaction phase of the $D$ decay over
the Dalitz plot are used as inputs. The values of the parameters are found to
be $x_{+}=(-7.7\pm 2.4\pm 1.0\pm 0.4)\times 10^{-2}$, $x_{-}=(2.5\pm 2.5\pm
1.0\pm 0.5)\times 10^{-2}$, $y_{+}=(-2.2\pm 2.5\pm 0.4\pm 1.0)\times 10^{-2}$,
and $y_{-}=(7.5\pm 2.9\pm 0.5\pm 1.4)\times 10^{-2}$. The first, second, and
third uncertainties are the statistical, the experimental systematic, and that
associated with the precision of the strong-phase parameters. These are the
most precise measurements of these observables and correspond to
$\gamma=(62^{\,+15}_{\,-14})^{\circ}$, with a second solution at
$\gamma\rightarrow\gamma+180^{\circ}$, and $r_{B}=0.080^{+0.019}_{-0.021}$,
where $r_{B}$ is the ratio between the suppressed and favoured $B$ decay
amplitudes.

Published in JHEP 10 (2014) 097

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

##  1 Introduction

A precise determination of the Cabibbo-Kobayashi-Maskawa (CKM) angle
$\gamma\equiv\arg(-V_{\rm ud}V^{*}{}_{\rm\\!\\!\\!\\!\\!ub}/V_{\rm
cd}V^{*}{}_{\rm\\!\\!\\!\\!\\!cb})$ is of great value in testing the Standard
Model (SM) description of $C\\!P$ violation. Measurements of this weak phase
in tree-level processes involving the interference between $b\rightarrow
c\bar{u}s$ and $b\rightarrow u\bar{c}s$ transitions are expected to be
insensitive to contributions from physics beyond the SM. Such measurements
therefore provide a SM benchmark against which other observables, more likely
to be affected by physics beyond the SM, can be compared. The effects of the
interference can be probed by studying $C\\!P$-violating observables in
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ decays, 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. Examples of
such final states recently studied by LHCb are two-body decays [1], multibody
decays that are not self-conjugate [2, 3], and self-conjugate three-body
decays, such as ${{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
and ${{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$, designated
collectively as ${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ [4]. Similar
measurements have also been made using neutral ${B}^{0}$ [5] and ${B}^{0}_{s}$
[6] mesons.

Sensitivity to $\gamma$ in ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$,
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ decays is obtained
by comparing the distribution of the events in the
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ Dalitz plot for
reconstructed $B^{+}$ and $B^{-}$ mesons [7, 8]. Knowledge of the variation of
the strong-interaction phase of the $D$ decay over the Dalitz plot is required
to determine $\gamma$. One approach, adopted by the BaBar [9, 10, 11], Belle
[12, 13, 14] and LHCb [15] collaborations, is to use an amplitude model
determined from flavour-tagged $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-}$ decays to provide this input. An attractive alternative [7, 16,
17] is to use direct measurements of the strong-phase variation over bins of
the Dalitz plot, thereby avoiding model-related systematic uncertainties. Such
measurements can be obtained using quantum-correlated ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ pairs from $\psi(3770)$ decays
and have been made at CLEO-c [18]. This model-independent method has been
applied to measurements at Belle [19] using ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$, $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decays, and at LHCb [4] using a subset of data used
in the current analysis.

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 study of the decay mode
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ with
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ and
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$. In
addition to benefiting from a larger data set than that used in Ref. [4] the
current study makes use of improved analysis techniques. The results presented
here thus supersede those of Ref. [4].

##  2 Overview of the analysis

The amplitude of the decay $B^{-}\rightarrow DK^{-}$,
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ can be written as
a superposition of the $B^{-}\rightarrow D^{0}K^{-}$ and
$B^{-}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}K^{-}$
contributions, given by

$A_{B}(m_{-}^{2},m_{+}^{2})\propto\,A+r_{B}e^{i(\delta_{B}-\gamma)}\overline{A}.$
(1)

Here $m_{-}^{2}$ and $m_{+}^{2}$ are the invariant masses squared of the
${{K}^{0}_{\rm\scriptscriptstyle S}}h^{-}$ and
${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}$ combinations, respectively, that
define the position of the decay in the Dalitz plot,
$A=A(m_{-}^{2},m_{+}^{2})$ is the
${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ amplitude
and $\overline{A}=\overline{A}(m_{-}^{2},m_{+}^{2})$ the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-}$ amplitude. The parameters $r_{B}$ and $\delta_{B}$ are the
ratio of the magnitudes of the $B^{-}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}K^{-}$ and
$B^{-}\rightarrow{{D}^{0}}K^{-}$ amplitudes, and the strong-phase difference
between them. The equivalent expression for the charge-conjugated decay
$B^{+}\rightarrow DK^{+}$ is obtained by making the substitutions
$\gamma\rightarrow-\gamma$ and $A\leftrightarrow\overline{A}$. Neglecting
$C\\!P$ violation in charm decays, which is known to be small in
${D}^{0}$–${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mixing and
Cabibbo-favoured $D$ meson decays [20], the conjugate amplitudes are related
by $A(m_{-}^{2},m_{+}^{2})=\overline{A}(m_{+}^{2},m_{-}^{2})$.

The Dalitz plot is partitioned into $2N$ regions symmetric under the exchange
$m_{+}^{2}\leftrightarrow m_{-}^{2}$, following Ref. [7]. The bins are
labelled from $-N$ to $+N$ (excluding zero), where the positive bins have
$m_{-}^{2}>m_{+}^{2}$. At each point in the Dalitz plot, there is a strong-
phase difference $\delta_{D}(m_{-}^{2},m_{+}^{2})\equiv\arg
A-\arg\overline{A}$ between the ${D}^{0}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ decay. The cosine of the
strong-phase difference averaged in each bin and weighted by the decay rate is
termed $c_{i}$ and is given by

$c_{i}\equiv\frac{\int_{{\cal
D}_{i}}(|A||\overline{A}|\cos{\delta_{D}})\,d{\cal D}}{\sqrt{\int_{{\cal
D}_{i}}|A|^{2}\,d{\cal D}}\,\sqrt{\int_{{\cal
D}_{i}}|\overline{A}|^{2}\,d{\cal D}}},$ (2)

where the integrals are evaluated over the area ${\cal D}_{i}$ of bin $i$. An
analogous expression may be written for $s_{i}$, which is the sine of the
strong-phase difference within bin $i$, weighted by the decay rate. The values
of $c_{i}$ and $s_{i}$ have been directly measured by the CLEO collaboration,
exploiting quantum-correlated ${{D}^{0}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ pairs produced at the
$\psi(3770)$ resonance [18]. One $D$ meson was reconstructed in a decay to
either ${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ or
${{K}^{0}_{\rm\scriptscriptstyle L}}h^{+}h^{-}$, and the other $D$ meson was
reconstructed either in a $C\\!P$ eigenstate or in a decay to
${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$. The efficiency-corrected
event yields, combined with flavour-tag information, allowed $c_{i}$ and
$s_{i}$ to be determined. There is a systematic uncertanty associated with
using these direct measurements due their finite precision. The alternative is
to calculate $c_{i}$ and $s_{i}$ assuming a functional form for $|A|$,
$|\overline{A}|$ and $\delta_{D}$, which may be obtained from an amplitude
model fitted to flavour-tagged $D^{0}$ decays. This alternative method relies
on assumptions about the nature of the intermediate resonances that contribute
to the ${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ final state, and leads
to a systematic uncertainty associated with the variation in $\delta_{D}$.

In the CLEO-c study the ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ Dalitz plot was partitioned into $2\times 8$ bins,
with a number of schemes available. The ‘optimal binning’ variant [18], where
the bins have been chosen to optimise the statistical sensitivity to $\gamma$,
is adopted in this analysis. The optimisation was performed assuming a strong-
phase difference distribution as given by the BaBar model presented in Ref.
[10]. For the ${{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$ final
state, $c_{i}$ and $s_{i}$ measurements are available for the Dalitz plot
partitioned into different numbers of bins with the guiding model being that
from the BaBar study described in Ref. [11]. The analysis described here
adopts the $2\times 2$ option, a decision driven by the size of the signal
sample. The use of a specific model in defining the bin boundaries does not
bias the $c_{i}$ and $s_{i}$ measurements. If the model is a poor description
of the underlying decay the only consequence is a reduction in the statistical
sensitivity of the $\gamma$ measurement. The binning choices for the two decay
modes are shown in Fig. 1.

Figure 1: Binning schemes for (left)
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ and
(right) $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$.
The diagonal line separates the positive and negative bins, where the positive
bins are in the region where $m^{2}_{-}>m^{2}_{+}$ is satisfied.

The population of each positive (negative) bin in the Dalitz plot arising from
$B^{+}$ decays is $N_{+i}^{+}$ ($N_{-i}^{+}$), and that from $B^{-}$ decays is
$N_{+i}^{-}$ ($N_{-i}^{-}$). The physics parameters of interest, $r_{B}$,
$\delta_{B}$, and $\gamma$, are translated into four $C\\!P$ observables [9]
that are measured in this analysis. These observables are defined as

$x_{\pm}\equiv r_{B}\cos(\delta_{B}\pm\gamma){\rm\ \ and\ }\;y_{\pm}\equiv
r_{B}\sin(\delta_{B}\pm\gamma).$ (3)

The selection requirements introduce nonuniformities in the populations of the
Dalitz plot. The relative selection and reconstruction efficiency profile
$\varepsilon=\varepsilon(m^{2}_{-},m^{2}_{+})$ for signal candidates is
defined as a function of the position in the Dalitz plot. The absolute
normalisation of $\varepsilon$ is not relevant; only the efficiency associated
with one point relative to the others matters. Considering Eq. 1 it follows
that

$\displaystyle N_{\pm i}^{+}$ $\displaystyle=$ $\displaystyle
h_{B^{+}}\left[F_{\mp i}+(x_{+}^{2}+y_{+}^{2})F_{\pm
i}+2\sqrt{F_{i}F_{-i}}(x_{+}c_{\pm i}-y_{+}s_{\pm i})\right],$ $\displaystyle
N_{\pm i}^{-}$ $\displaystyle=$ $\displaystyle h_{B^{-}}\left[F_{\pm
i}+(x_{-}^{2}+y_{-}^{2})F_{\mp i}+2\sqrt{F_{i}F_{-i}}(x_{-}c_{\pm
i}+y_{-}s_{\pm i})\right],$ (4)

where the value $F_{i}$ is given by

$F_{i}=\frac{\int_{{\cal D}_{i}}|A|^{2}\,\varepsilon\,d{\cal
D}}{\sum_{j}\int_{{\cal D}_{j}}|A|^{2}\,\varepsilon\,d{\cal D}}$ (5)

and is the fraction of events in bin $i$ of the
${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ Dalitz
plot. The quantities $h_{B^{\pm}}$ are normalisation factors, which can be
different for $B^{+}$ and $B^{-}$ due to asymmetries in production rates of
bottom and antibottom mesons.

The observed distribution of candidates over the
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ Dalitz plot is
used to fit for $x_{\pm}$, $y_{\pm}$ and $h_{B^{\pm}}$. The values of $F_{i}$
are determined from the control mode $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}X$, where the ${D}^{*\pm}$ decays to $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}^{0}{{\pi}^{\pm}}$, and the $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}^{0}$ decays to either the
${{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ or
${{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$ final state. The
symbol $X$, hereinafter omitted, indicates other particles that are
potentially produced in the  ( ) [-.7ex] $\kern-1.79993ptB$ decay. Samples of
simulated events are used to correct for differences in the efficiency for
reconstructing and selecting $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ and ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$
decays.

In addition to selecting ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ candidates we also select
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ decays. Candidates selected in this
decay mode provide an important control sample that is used to constrain the
invariant mass shape of the ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ signal and
to determine the yield of ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ decays
misidentified as ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ candidates.

The use of $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ decays to determine the values of $F_{i}$ is
an improvement over Ref. [4], for which the decay ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ was used. The small level of $C\\!P$ violation in the latter
decay led to a significant systematic uncertainty. This uncertainty is
eliminated when using the flavour-specific semileptonic decay. There is still
a systematic uncertainty associated with the procedure but it is relatively
small in magnitude.

The effect of ${D}^{0}$–${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$
mixing is ignored in the above discussion, and was neglected in the CLEO-c
measurements of $c_{i}$ and $s_{i}$ as well as in the values of $F_{i}$. This
leads to a bias of approximately $0.2^{\circ}$ in the $\gamma$ determination
[21], which is negligible for the current analysis. The effect of $C\\!P$
violation in ${K}^{0}_{\rm\scriptscriptstyle S}$ decays is expected to lead to
a $\mathcal{O}(1^{\circ})$ uncertainty [22], and is also ignored given the
expected precision. An uncertainty due to the different nuclear interaction
cross sections 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 [23].

The rest of the paper is organised as follows. Section 3 describes the LHCb
detector, and Section 4 presents the selection and the model used to fit the
invariant mass spectrum. Sections 5 and 6 are concerned with the selection of
the semileptonic control channel, used to determine the signal efficiency
profile. Section 7 discusses the binned Dalitz plot fit and presents the
results for the $C\\!P$ parameters. The evaluation of systematic uncertainties
is summarised in Section 8. In Section 9 the use of the measured $C\\!P$
parameters to determine the CKM angle $\gamma$ is described. The results of
the analysis are summarised in Section 10.

##  3 Detector and simulation

The LHCb detector [24] 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 $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes [25] placed downstream. The
combined tracking system provides a momentum measurement with relative
uncertainty that varies from 0.4% at 2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter
resolution of 20${\,\upmu\rm m}$ for tracks with large transverse momentum.
Different types of charged hadrons are distinguished using information from
two ring-imaging Cherenkov detectors [26]. Photon, electron and hadron
candidates 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 [27]. The trigger [28] 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 candidate fully hadronic and semileptonic
$B$ decays are slightly different due to the presence of the muon in the
latter.

In the simulation, $pp$ collisions are generated using Pythia [29, 30] with a
specific LHCb configuration [31]. Decays of hadronic particles are described
by EvtGen [32], in which final-state radiation is generated using Photos [33].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [34, 35] as described in Ref. [36].

##  4 Event selection and fit to invariant mass spectrum for
$\boldsymbol{{{B}^{\pm}}\rightarrow D{{K}^{\pm}}}$ and
$\boldsymbol{{{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}}$ decays

Selection requirements are applied to obtain an event sample enriched with
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ candidates, where $D$ indicates a ${D}^{0}$ or ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ meson that decays to the final
state ${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$. The kaon or pion
produced directly in the ${B}^{\pm}$ decay is denoted the ‘bachelor’ hadron.
Decays of ${K}^{0}_{\rm\scriptscriptstyle S}$ mesons to the
${{\pi}^{+}}{{\pi}^{-}}$ final state are reconstructed in two different
categories, the first involving ${K}^{0}_{\rm\scriptscriptstyle S}$ mesons
that decay early enough for the pions to be reconstructed in the vertex
detector, the second containing ${K}^{0}_{\rm\scriptscriptstyle S}$ 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_ ,
respectively. The candidates in the long category have better mass, momentum
and vertex resolution than those in the downstream category. Henceforth
${B}^{\pm}$ candidates are denoted long or downstream depending on which
${K}^{0}_{\rm\scriptscriptstyle S}$ type they contain.

Events considered in the analysis must fulfil both 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 ${B}^{\pm}$ 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. The
software trigger designed to select ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates requires a two-, three- or
four-track secondary vertex with a large sum of the transverse momentum,
$p_{\rm T}$, of the associated charged particles and a significant
displacement from the primary $pp$ interaction vertices (PVs). At least one
charged particle should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any primary interaction greater than 16, where $\chi^{2}_{\rm IP}$
is defined as the difference in $\chi^{2}$ of a given PV fitted with and
without the considered track. A multivariate algorithm [37] is used for the
identification of secondary vertices that are consistent with the decay of a
$b$ hadron.

A multivariate approach is employed to improve the event selection relative to
that used in Ref. [4]. A boosted decision tree [38, 39] (BDT) is trained on
simulated signal events and background taken from the high ${B}^{\pm}$ mass
sideband (5800–7000${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$). Both signal
and background samples contain candidates from the $D$ and
${K}^{0}_{\rm\scriptscriptstyle S}$ signal regions only. Different BDTs are
trained for long and downstream candidates. Each BDT uses the following
variables: the logarithm of the $\chi^{2}_{\rm IP}$ of the pions from the $D$
decay and also of the bachelor particle; the logarithm of the $\chi^{2}_{\rm
IP}$ of the ${K}^{0}_{\rm\scriptscriptstyle S}$ decay products (long
candidates only); the logarithm of the $D$ $\chi^{2}_{\rm IP}$; the
${B}^{\pm}$ $\chi^{2}_{\rm IP}$; a variable characterising the ${B}^{\pm}$
flight distance; the ${B}^{\pm}$ and $D$ momenta; the $\chi^{2}$ of the
kinematic fit of the whole decay chain, (described in detail below); and the
‘${B}^{\pm}$ isolation variable’, a quantity designed to ensure the
${B}^{\pm}$ candidate is well isolated from other tracks in the event. The
${B}^{\pm}$ isolation variable is the asymmetry between the $p_{\rm T}$ of the
signal candidate and the vector sum of the $p_{\rm T}$ of the other tracks in
the event that lie within a distance of 1.5 rad in $\eta$–$\phi$ space, where
$\phi$ is the azimuthal angle. The discriminating power of the variables
differs slightly for long and downstream candidates. Two variables that are
highly discriminating for both samples are the ${B}^{\pm}$ $\chi^{2}_{\rm IP}$
and ${B}^{\pm}$ isolation variable. An optimal value of the BDT discriminator
is determined with a series of pseudo-experiments to obtain the value that
provides the best sensitivity to $x_{\pm}$, $y_{\pm}$. Events in the data
sample that have a value below the optimum are rejected. The optimal BDT value
is different for long and downstream candidates primarily because the level of
combinatorial background is larger for the latter.

To suppress background further, the ${K}^{0}_{\rm\scriptscriptstyle S}$, $D$
and ${B}^{\pm}$ momentum vectors are required to point in the same direction
as the vector connecting their production and decay vertices. The mass of the
$D$ candidate must lie within 25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of
the known $D$ mass [20].

Particle identification (PID) requirements are placed on the bachelor to
separate ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ candidates. PID criteria are also applied to the kaons from
the $D$ decay for the final state ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$. To ensure good control of the PID performance it is
required that information from the RICH detectors is present.

A kinematic fit [40] is imposed on the full ${B}^{\pm}$ decay chain. The fit
constrains the ${B}^{\pm}$ candidate to point towards the PV and the $D$ and
${K}^{0}_{\rm\scriptscriptstyle S}$ candidates to have their known masses
[20]. This fit improves the ${B}^{\pm}$ 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 Dalitz plot. The candidates obtained
in this fit are used to determine the physics parameters of interest. An
additional fit, in which only the ${B}^{\pm}$ pointing and $D$ mass
constraints are imposed, is employed to aid discrimination between genuine and
background ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates. After this fit is
applied it is required that the mass of the ${K}^{0}_{\rm\scriptscriptstyle
S}$ candidate lies within 15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of its
known value [20].

To remove background from
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays, long
${K}^{0}_{\rm\scriptscriptstyle S}$ candidates are required to have travelled
a significant distance from the $D$ vertex. To remove charmless ${B}^{\pm}$
decays, the displacement along the beamline between the $D$ and ${B}^{\pm}$
decay vertices is required to be positive.

The invariant mass distributions of the selected candidates are shown in Fig.
2 for ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$, with $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decays, divided between the long and downstream
${K}^{0}_{\rm\scriptscriptstyle S}$ categories. Figure 3 shows the
corresponding distributions for final states with
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$. The
result of an extended maximum likelihood fit to these distributions is
superimposed. The fit is performed simultaneously for ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates,
including both $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ and $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ decays, allowing several independent parameters for
long and downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ categories. The fit
range is between 5080${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
5800${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in the ${B}^{\pm}$ invariant
mass. The purpose of this simultaneous fit to data integrated over the Dalitz
plot is to determine the parameters that describe the invariant mass spectrum
in preparation for the binned fit described in Sect. 7. The mass spectrum of
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates is fitted because it is
similar to the ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ spectrum, aiding the
determination of the signal lineshape due to the higher yield and lower
background. The yield of ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates
misidentified as ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ candidates can be
determined from knowledge of the ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$
signal yield and the PID selection efficiencies.

Figure 2: Invariant mass distributions of (left) ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ and (right) ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates,
with $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$,
divided between the (top) long and (bottom) downstream
${K}^{0}_{\rm\scriptscriptstyle S}$ categories. Fit results, including the
signal and background components, are superimposed.

Figure 3: Invariant mass distributions of (left) ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ and (right) ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates,
with $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$,
divided between the (top) long and (bottom) downstream
${K}^{0}_{\rm\scriptscriptstyle S}$ categories. Fit results, including the
signal and background components, are superimposed.

The signal probability density function (PDF) is a Gaussian function with
asymmetric tails, defined as

$f(m;m_{0},\sigma,\alpha_{L},\alpha_{R})\propto\begin{cases}\exp[-(m-m_{0})^{2}/(2\sigma^{2}+\alpha_{L}(m-m_{0})^{2})]&\
m<m_{0}\\\ \exp[-(m-m_{0})^{2}/(2\sigma^{2}+\alpha_{R}(m-m_{0})^{2})]&\
m>m_{0}\end{cases}$ (6)

where $m$ is the candidate mass and $m_{0}$, $\sigma$, $\alpha_{L}$, and
$\alpha_{R}$ are free parameters in the fit. The parameter $m_{0}$ is common
to all classes of signal. The parameters describing the asymmetric tails,
$\alpha_{L,R}$, are fitted separately for events with long and downstream
${K}^{0}_{\rm\scriptscriptstyle S}$ categories. The width parameter $\sigma$
is left as a free parameter for the two ${K}^{0}_{\rm\scriptscriptstyle S}$
categories, but the ratio between this width in ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ decays is required
to be the same, independent of ${K}^{0}_{\rm\scriptscriptstyle S}$
reconstruction or ${D}^{0}$ decay category. The width is determined to be
around 13${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ decays of both
${K}^{0}_{\rm\scriptscriptstyle S}$ classes, and is 10% larger for
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ decays. The yield of
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates in each category is
determined in the fit. Instead of fitting the yield of the
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ candidates separately, the ratio
$\mathcal{R}\equiv N({{B}^{\pm}}\rightarrow
D{{K}^{\pm}})$/$N({{B}^{\pm}}\rightarrow D{{\pi}^{\pm}})$ is determined and is
constrained to have the same value for all categories.

The background has contributions from random track combinations and partially
reconstructed $B$ decays. The random track combinations are modelled by
exponential PDFs. The slopes of these functions are determined through the
study of two independent samples: candidates reconstructed such that both
charged hadrons produced in the $D$ decay have the same sign, and candidates
reconstructed using the $D$ mass sidebands. The slopes are consistent with
each other. In the fit to the signal data the exponential slopes are Gaussian-
constrained to the results of the sideband studies.

A significant background component exists in the ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ sample, arising from a fraction of the dominant
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ decays in which the bachelor particle
is misidentified as a kaon by the RICH system. The yield of this type of
background is calculated using knowledge of misidentification efficiencies
that are obtained from large samples of kinematically selected
${{D}^{*\pm}}\rightarrow\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}^{0}\pi^{\pm}$, $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}^{0}\rightarrow K^{\mp}\pi^{\pm}$ decays. The tracks in
this calibration sample are reweighted to match the momentum and
pseudorapidity distributions of the bachelor tracks in the ${B}^{\pm}$ decay
sample, thereby ensuring that the measured PID performance is representative
of that in the ${B}^{\pm}$ decay sample. The efficiency to identify a kaon
correctly is found to be 86%, and that for a pion to be 96%. The efficiency of
misidentifying a pion as a kaon is 4%. From this information and from the
knowledge of the number of reconstructed ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ decays, the amount of this background surviving the
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ selection is estimated.

The distribution of true ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates
misidentified as ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ candidates is
determined using data. The ${B}^{\pm}$ invariant mass distribution is obtained
by reconstructing candidates in the ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$
sample with a kaon mass hypothesis for the bachelor pion. The sample is
weighted using the sPlot method [41] and the PID efficiencies. The use of the
sPlot method in the reweighting suppresses partially reconstructed and
combinatorial backgrounds. Weighting by PID efficiencies allows for
reproduction of the kinematic properties of pions misidentified as kaons in
the signal ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ sample. The weighted
distribution is fitted to a parametric shape with different shapes used for
the samples containing long and downstream ${K}^{0}_{\rm\scriptscriptstyle S}$
decays. The fitted parameters are subsequently fixed in the fit to the
${B}^{\pm}$ invariant mass spectrum.

A similar procedure is used to determine the number of ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ decays misidentified as ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$.
Due to the reduced branching fraction of ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$
and the small likelihood of misidentifying a kaon as a pion, such cases occur
at a low rate and have a minor influence on the fit.

Partially reconstructed $b$-hadron decays (shown as Part. Reco. in Fig. 2 and
Fig. 3) contaminate the sample predominantly at invariant masses smaller than
that of the signal peak. These decays contain an unreconstructed pion or a
photon, which comes from a vector-meson decay. The dominant decays in the
signal region are $B^{\pm}\rightarrow{D}\rho^{\pm}$, $B^{\pm}\rightarrow
D^{*0}\pi^{\pm}$ and $B^{0}\rightarrow D^{*\pm}\pi^{\mp}$ decays in which one
particle is missed. The distribution in the invariant mass spectrum depends on
the spin and mass of the missing particle. If the missing particle has spin-
parity $J^{P}=0^{-}$ ($1^{-}$), the distribution is parameterised with a
parabola with positive (negative) curvature convolved with a resolution
function. The mass of the missing particle defines the kinematic endpoints of
the distribution prior to reconstruction. The shapes for decays in which a
particle is missed and a pion is misidentified as a kaon are parameterised
with semi-empirical PDFs formed from sums of Gaussian and error functions. The
parameters of these distributions are fixed to the results of fits to data
from two-body $D$ decays, with the exception of the resolution function width,
the ratio of widths in the ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ channels and a shift along the
${B}^{\pm}$ mass. The resulting PDF is cross-checked with a similar fit to an
admixture of simulated backgrounds.

The number of ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ candidates in each
${K}^{0}_{\rm\scriptscriptstyle S}$ category or ${D}^{0}$ decay category is
determined from the value of $\mathcal{R}$ and the number of
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ events in the corresponding category.
The ratio $\mathcal{R}$ is determined in the fit and measured to be $(7.7\pm
0.2)\%$ (statistical uncertainty only), consistent with that observed in Ref.
[1]. The yields returned by the invariant mass fit in the full fit region are
scaled to the signal region, defined as
5247–5317${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and are presented in
Tables 1 and 2. Because the ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ yields are
calculated using $\mathcal{R}$ their uncertainties are smaller than those that
would be expected if the yields were allowed to vary in the fit.

Table 1: Yields and statistical uncertainties in the signal region from the invariant mass fits, scaled from the full fit mass range, for candidates passing the ${{B}^{\pm}}\rightarrow D({{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}})h^{\pm}$ selection. Values are shown separately for candidates formed using long and downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and 5317${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the full fit range is between 5080${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and 5800${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.  Fit component |  ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ selection |  |  ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ selection
---|---|---|---
| Long | Downstream |  | Long | Downstream
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ |  $702$$\ \pm\ $ | $18$ |  $1555$$\ \pm\ $ | $39$ |  |  $30$$\ \pm\ $ | $5$ |  $64$$\ \pm\ $ | $7$
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ |  $87$$\ \pm\ $ | $9$ |  $164$$\ \pm\ $ | $13$ |  |  $10\,338$$\ \pm\ $ | $106$ |  $22\,779$$\ \pm\ $ | $166$
Combinatorial |  $59$$\ \pm\ $ | $9$ |  $133$$\ \pm\ $ | $14$ |  |  $103$$\ \pm\ $ | $11$ |  $433$$\ \pm\ $ | $25$
Partially reconstructed |  $38$$\ \pm\ $ | $2$ |  $82$$\ \pm\ $ | $3$ |  |  $4.6$$\ \pm\ $ | $0.1$ |  $14.2$$\ \pm\ $ | $0.1$
Table 2: Yields and statistical uncertainties in the signal region from the invariant mass fits, scaled from the full fit mass range, for candidates passing the ${{B}^{\pm}}\rightarrow D({{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}})h^{\pm}$ selection. Values are shown separately for candidates formed using long and downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ decays. The signal region is between 5247${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and 5317${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and the full fit range is between 5080${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and 5800${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.  Fit component |  ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ selection |  |  ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ selection
---|---|---|---
| Long | Downstream |  | Long | Downstream
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ |  $101$$\ \pm\ $ | $4$ |  $223$$\ \pm\ $ | $7$ |  |  $4.5$$\ \pm\ $ | $1.9$ |  $10.1$$\ \pm\ $ | $2.9$
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ |  $13$$\ \pm\ $ | $3$ |  $24$$\ \pm\ $ | $5$ |  |  $1501$$\ \pm\ $ | $38$ |  $3338$$\ \pm\ $ | $57$
Combinatorial |  $13$$\ \pm\ $ | $3$ |  $30$$\ \pm\ $ | $5$ |  |  $36$$\ \pm\ $ | $5$ |  $78$$\ \pm\ $ | $7$
Partially reconstructed |  $4.6$$\ \pm\ $ | $0.7$ |  $8.6$$\ \pm\ $ | $1.2$ |  |  $0.60$$\ \pm\ $ | $0.02$ |  $2.0$$\ \pm\ $ | $0.1$

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

Figure 4: Dalitz plots of ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ candidates in
the signal region for $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decays from (left) ${{B}^{+}}$ and (right)
${{B}^{-}}$ decays. Both long and downstream ${K}^{0}_{\rm\scriptscriptstyle
S}$ candidates are included. The blue line indicates the kinematic boundary.

Figure 5: Dalitz plots of ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ candidates in
the signal region for $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ decays from (left) ${{B}^{+}}$ and (right) ${{B}^{-}}$
decays. Both long and downstream ${K}^{0}_{\rm\scriptscriptstyle S}$
candidates are included. The blue line indicates the kinematic boundary.

##  5 Event selection and yield determination for
${\shortstack{{(\rule[2.15277pt]{17.50002pt}{0.51212pt})} \\\ [-.3ex]
$B$}{}\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})}
\\\ [-.3ex] $\nu$}\\!_{\mu}}$ decays

A sample of the decays $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$, ${{D}^{*\pm}}\rightarrow\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}^{0}{{\pi}^{\pm}}$, $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}^{0}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-}$ is used to determine the quantities $F_{i}$. These are defined
in Eq. 5 as the expected fractions of ${D}^{0}$ decays falling into the Dalitz
plot bin labelled $i$, taking into account the efficiency profile of the
signal decay. The semileptonic decay of the $B$ and the strong-interaction
decay of the ${{D}^{*\pm}}$ allow the flavour of the ${D}^{0}$ meson to be
determined from the charge of the bachelor muon and pion. This particular
decay chain, involving a flavour-tagged ${D}^{0}$ decay, is chosen due to its
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 ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ sample. They are identical to the
requirements listed in Sect. 4 where possible; the requirements on variables
used to train the BDT follow those described in Ref. [4].

Candidate $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ events are selected using information from
the muon detector systems. These events are first required to pass the
hardware trigger which selects muons with a transverse momentum $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Approximately $95\%$ of the
final $\kern 1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})}
\\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ sample is collected with this algorithm, and
the remainder pass a hardware trigger which selects ${D}^{0}$ candidates that
leave a high transverse energy deposit in the hadronic calorimeter. In the
software trigger, at least one of the final-state particles is required to
have both $\mbox{$p_{\rm T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
impact parameter greater than $100{\,\upmu\rm m}$ with respect to all of the
PVs in the event. Finally, the tracks of two or more of the final-state
particles are required to form a vertex that is significantly displaced from
the PVs.

To reduce combinatorial background, all charged decay products are required to
be inconsistent with originating from the PV, and the momentum vectors of the
${K}^{0}_{\rm\scriptscriptstyle S}$, ${D}^{0}$ and $B$ are required to be
aligned with the vector between their production and decay vertices. The $B$
candidate vertex is required to be well separated from the PV in order to
discriminate between $B$ decays and prompt charm decays.

The $B$ decay chain is refitted [40] to determine the distribution of
candidates across the Dalitz plot. Unlike the refit performed for
${{B}^{\pm}}\rightarrow Dh^{\pm}$ candidates, the fit constrains only the
${D}^{0}$ and ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates to their known
masses as the $B$ candidate is not fully reconstructed in the semileptonic
decay mode. An additional fit, in which only the
${K}^{0}_{\rm\scriptscriptstyle S}$ mass is constrained, is performed to
improve the ${D}^{0}$ and ${D}^{*\pm}$ mass resolutions for use in the
invariant mass fit used to determine signal yields.

Additional requirements are included to remove
${{D}^{0}}\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays and
charmless $B$ decays, and PID criteria are placed on the kaons in
${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$.
The requirements are the same as those applied to the ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates described
in Sect. 4. The ${K}^{0}_{\rm\scriptscriptstyle S}$ candidate mass is required
to be within $20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known value
[20], and the invariant mass sum of the ${D}^{*\pm}$ and muon, determined
using the refit containing the ${D}^{0}$ and ${K}^{0}_{\rm\scriptscriptstyle
S}$ mass constraints, is required to be less than
$5000{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The candidate ${D}^{0}$ invariant mass, $m({{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-})$, and the invariant mass difference $\Delta m\equiv
m({{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-}{{\pi}^{\pm}})-m({{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-})$
are fitted simultaneously to determine the signal yields. No significant
correlation between these two variables is observed. This two-dimensional
parameterisation allows the yield of selected candidates to be measured in
three categories: true ${{D}^{*\pm}}$ candidates (signal), candidates
containing a true ${D}^{0}$ but random soft pion (RSP) and candidates formed
from random track combinations that fall within the fit range (combinatorial
background). An example projection is shown in Fig. 6. The result of a two-
dimensional extended, unbinned, maximum likelihood fit is superimposed. The
fit is performed simultaneously for the two ${D}^{0}$ final states and the two
${K}^{0}_{\rm\scriptscriptstyle S}$ categories with some parameters allowed to
be independent between categories. Candidates selected from data recorded in
2011 and 2012 are fitted separately, due to their slightly different Dalitz
plot efficiency profiles. The fit range is
$1830<m({{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-})<1910{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $139.5<\Delta
m<153.0{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
$m({{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-})$ range is chosen to be
within a region where the $\Delta m$ resolution does not vary significantly.

The signal is parameterised in $\Delta m$ with a sum of two modified Gaussian
PDFs, each given by

$f(\Delta
m;\mu_{m},\sigma,\beta,\delta)=\frac{\delta\exp\left(-0.5\left[\beta+\delta\log\left(\frac{\Delta
m-\mu_{m}}{\sigma}+\sqrt{1+\left(\frac{\Delta
m-\mu_{m}}{\sigma}\right)^{2}}\right)\right]^{2}\right)}{\sigma\sqrt{2\pi[1+\left(\frac{\Delta
m-\mu_{m}}{\sigma}\right)^{2}]}},$ (7)

where $\mu_{m}$, $\sigma$, $\beta$ and $\delta$ are floating parameters in the
fit. The parameter $\mu_{m}$ is shared in all data categories and the
remaining parameters are fitted separately for long and downstream
${K}^{0}_{\rm\scriptscriptstyle S}$ candidates. The combinatorial and RSP
backgrounds are both parameterised with an empirical model given by

$f(\Delta m;\Delta m_{0},x,p_{1},p_{2})=\left[1-\exp\left(-\frac{\Delta
m-\Delta m_{0}}{x}\right)\right]\left({\frac{\Delta m}{\Delta
m_{0}}}\right)^{p_{1}}+p_{2}\left(\frac{\Delta m}{\Delta m_{0}}-1\right)$ (8)

for $\Delta m-\Delta m_{0}>0$ and $f(\Delta m)=0$ otherwise, where $\Delta
m_{0}$, $x$, $p_{1}$ and $p_{2}$ are floating parameters. The parameter
$\Delta m_{0}$, which describes the kinematic threshold for a
${{D}^{*\pm}}\rightarrow\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}{}^{0}\pi^{\pm}$ decay, is shared in all data categories,
and for both the combinatorial and RSP shapes. The remaining parameters are
determined separately for ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ candidates.

The signal and RSP PDFs in $m({{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-})$
are described by Eq. 6, where $m_{0}$, $\sigma$, $\alpha_{L}$, and
$\alpha_{R}$ are all free parameters. All of the parameters in the signal and
RSP PDFs are constrained to be the same since both describe a true ${D}^{0}$
candidate, but the parameters are fitted separately for
${{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$, due to the different
phase space available in the ${D}^{0}$ decay. The combinatorial background is
parameterised by a second-order polynomial.

Figure 6: Result of the simultaneous fit to $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$, ${{D}^{*}}\pm\rightarrow\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptD$}{}^{0}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ decays with downstream
${K}^{0}_{\rm\scriptscriptstyle S}$ candidates, in 2012 data. A two-
dimensional fit is performed in (left) $m({{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-})$ and (right) $\Delta m$. The (blue) total fit PDF is
constructed from (solid red) signal, (dashed black) combinatorial background
and (dotted green) random soft pion background.

In total a sample with a signal yield of $123\,600$ candidates is selected.
The size of the sample is approximately 40 times larger than the
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ yield. The signal mass range is defined
as 1840–1890 (1850–1880)${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in
$m({{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}})$
($m({{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}})$) and
143.9–146.9${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ in $\Delta m$. Within
this range the background components account for 3–6% of the yield depending
on the category.

##  6 Determining the $F_{i}$ fractions from the semileptonic control channel

The two-dimensional fit in $m({{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-})$
and $\Delta m$ of the $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ decay is repeated in each Dalitz plot bin,
resulting in a raw control decay yield, $R_{i}$, for each bin $i$. Due to the
differences in the efficiency profile over the Dalitz plot between $D$ mesons
originating from the control decay $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ and those originating from the signal decay
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$, the measured relative proportions of
the $R_{i}$ values are not equivalent to the $F_{i}$ fractions required to
determine the $C\\!P$ parameters. The differences in the efficiency profiles,
which originate from the different selections of the candidates from the
signal and control decay modes, must be corrected for. The efficiency profiles
from simulation of $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decays are shown in Fig. 7. They show a variation
of approximately 50$\%$ between the highest and lowest efficiency regions. The
variation over the $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ Dalitz plot is 35$\%$. As the individual Dalitz plot
bins cover regions of different efficiency the variation from the Dalitz plot
bin with the highest efficiency and that with the lowest is approximately
30$\%$ (15$\%$) for $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ ($D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$) decays.

To understand the differences between the efficiency profiles of $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ and ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$
decays, we compare the distributions of $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ and ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$
observed in data and simulation. The reason for choosing
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ is that the efficiency profile is the
same as for ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ (as verified in simulation),
but the channel has higher yields than ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$.
Moreover the ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ has a level of
interference, and hence $C\\!P$ violation, that is expected to be an order of
magnitude smaller than in ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$, allowing the
differences in efficiency profiles to be separated from differences arising
from interference effects. The yield of ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ candidates in each bin is determined by fitting the invariant
mass spectrum of candidates in each bin using the parameterisation determined
in Sect. 4.

Figure 8 shows the ratio of fractional signal yields, $f_{i}$, between
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ and $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ in each Dalitz plot bin. The ratios are
averaged over the two ${K}^{0}_{\rm\scriptscriptstyle S}$ samples and two
periods of data taking in different experimental conditions. To increase the
sample size in each bin, the yield in bin $i$ for ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ events is combined with the
yield in bin $-i$ for ${D}^{0}$ events. Where the combination of yields is
taken in this manner, exploiting the symmetry of the Dalitz plot, the bin
number is referred to as the _effective_ bin. Differences of up to 10$\%$ from
unity are observed in the values of $f_{i}$. These cannot be explained by the
small amount of $C\\!P$ violation in ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$
decays which is expected to vary the fractional yields by 3% or less, on the
assumption that the magnitude of the interference in ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ decays is $r_{B}^{\pi}=0.01$.

Figure 7: Example efficiency profiles of (left) ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ and (right) $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ decays in simulation. These plots refer to
downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates under 2012 data
taking conditions.

Figure 8: Fractional yield ratios of ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$
to $\kern 1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\
[-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ decays, $f_{i}$, for each effective Dalitz
plot bin. The vertical dashed line separates the ratios for
${{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ (bins $-8$ to 8)
and ${{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$ (bins $-2$ to 2).
The left (right) plot shows the values of $f_{i}$ before (after) correcting
for the efficiency differences.

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

$\xi_{i}\equiv\frac{\int_{\mathcal{D}_{i}}\varepsilon_{DK}\,|A|^{2}\,d\mathcal{D}}{\int_{\mathcal{D}_{i}}\varepsilon_{{{D}^{*}}\\!\mu}\,|A|^{2}\,d\mathcal{D}},$
(9)

where $\varepsilon_{DK}$ and $\varepsilon_{{{D}^{*}}\\!\mu}$ are the
efficiency profiles of the ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ decays, respectively, and $|A|^{2}$ is the
Dalitz plot intensity for the ${D}^{0}$ decay. The amplitude models used to
determine the Dalitz plot intensity for the correction factor are those from
Ref. [10] and Ref. [11] for the ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ 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
simulation is used to determine the efficiency profiles $\varepsilon_{DK}$ and
$\varepsilon_{{{D}^{*}}\\!\mu}$. The simulations are generated assuming a flat
distribution across the ${{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ phase
space; hence the distribution of simulated events after triggering,
reconstruction and selection is directly proportional to the efficiency
profile. The correction factors are determined separately for data
reconstructed with each ${K}^{0}_{\rm\scriptscriptstyle S}$ type as the
efficiency profile is different between the two
${K}^{0}_{\rm\scriptscriptstyle S}$ categories.

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 total uncertainty on $F_{i}$ 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.

To check the effect of the correction, the resulting $F_{i}$ values are
compared to the observed population as a function of the Dalitz plot in
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ data. Figure 8, showing the ratio of
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ fractional yields to raw $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ fractional yields, gives a $\chi^{2}$ per
degree of freedom ($\chi^{2}/\mathrm{ndf}$) of $48.1/20$ when considering the
deviation from unity. When the corrected $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ yields are used the fit quality improves to
$\chi^{2}/\mathrm{ndf}=29.5/20$ as seen in Fig. 8. Although the $\chi^{2}$ is
calculated with respect to unity, the true value of $F_{i}$ in each bin has a
variation of order $3\%$ due to $C\\!P$ violation in the
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ decay.

##  7 Dalitz plot fit

The Dalitz plot fit is used to measure the $C\\!P$-violating parameters
$x_{\pm}$ and $y_{\pm}$, as introduced in Sect. 2. Following Eq. 4, these
parameters can be determined from the populations of each $B^{\pm}\rightarrow
DK^{\pm}$ Dalitz plot bin, given the external information from the $c_{i}$,
$s_{i}$ parameters from CLEO-c and the values of $F_{i}$ from the semileptonic
control decay modes.

Although the absolute numbers of $B^{+}$ and $B^{-}$ decays integrated over
the Dalitz plot have some dependence on $x_{\pm}$ and $y_{\pm}$, the
sensitivity gained compared to using just the relative bin-to-bin yields is
negligible. Consequently the integrated yields are not used and the analysis
is insensitive to charged $B$ meson production and detection asymmetries. The
observed size of the asymmetry of the integrated yields is consistent with
that expected from the production and detection asymmetries, and the
dependence on $x_{\pm}$ and $y_{\pm}$.

A simultaneous fit is performed on the ${{B}^{\pm}}\rightarrow Dh^{\pm}$ data,
which are further split into the two $B$ charges, the two
${K}^{0}_{\rm\scriptscriptstyle S}$ categories, the ${{B}^{\pm}}\rightarrow
D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ candidates, and the
two $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}h^{+}h^{-}$ final states.
The ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and ${{B}^{\pm}}\rightarrow
D{{\pi}^{\pm}}$ samples are fitted simultaneously because the yield of
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ signal in each Dalitz plot bin is used
to determine the yield of misidentified events in the corresponding
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ Dalitz plot bin. The PDF parameters for
both the signal and background invariant mass distributions are fixed to the
values determined in the invariant mass fit described in Sect. 4. The $B$ mass
range is reduced to 5150–5800${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to
reduce systematic uncertainties from the partially reconstructed background.
The yields of all background contributions in each bin are free parameters,
apart from the yields in bins in which an auxiliary fit determines the yield
to be negligible. These are set to zero to facilitate the calculation of the
uncertainty matrix. The yields of signal candidates for each bin in the
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ sample are also free parameters. The
amount of signal in each bin for the ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$
sample is determined by varying the integrated yield over all Dalitz plot bins
and the $x_{\pm}$ and $y_{\pm}$ parameters. In the fit the values of $F_{i}$
are Gaussian-constrained within their uncertainties. The values of $c_{i}$ and
$s_{i}$ are fixed to their central values. In order to assess the impact of
the $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$ data,
the fit is then repeated including only the
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
sample.

A large ensemble of pseudo-experiments is performed to validate the fit
procedure. In each pseudo-experiment the numbers and distribution of signal
and background candidates are generated according to the expected distribution
in data, and the full fit procedure is then executed. The input values for
$x_{\pm}$ and $y_{\pm}$ are set close to those determined by previous
measurements [42]. The uncertainties estimated by the fit are consistent with
the size of the uncertainties estimated by the pseudo-experiments. However,
small biases, with sizes around $10\%$ of the statistical uncertainty, are
observed in the central values. These biases are due to the low event yields
in some of the bins and are observed to reduce in simulated experiments of
larger size. The central values are corrected for the biases.

The results of the fits are presented in Table 3. The statistical
uncertainties are compatible with those predicted by the simulated pseudo-
experiments. The systematic uncertainties are discussed in Sect. 8. The
inclusion of $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ data improves the precision on $x_{\pm}$ by around 10%
and by a smaller amount for $y_{\pm}$. This is expected, as the measured
values of $c_{i}$ in this decay, which multiply $x_{\pm}$ in Eq. 3, are
significantly larger than those of $s_{i}$, which multiply $y_{\pm}$ [18].

Table 3: Results for $x_{\pm}$ and $y_{\pm}$ from fits of both the $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}\pi^{+}\pi^{-}$ and $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}K^{+}K^{-}$ samples, and from fits of the $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}\pi^{+}\pi^{-}$ sample only. The first, second, and third uncertainties are the statistical, the experimental systematic, and the error associated with the precision of the strong-phase parameters, respectively.  Parameter | All data |  $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}\pi^{+}\pi^{-}$ only
---|---|---
$x_{+}$ [$\times 10^{-2}$] | $-7.7\pm 2.4\pm 1.0\pm 0.4$ | $-7.5\pm 2.7\pm 1.1\pm 0.5$
$x_{-}$ [$\times 10^{-2}$] | $2.5\pm 2.5\pm 1.0\pm 0.5$ | $2.6\pm 2.8\pm 1.1\pm 0.7$
$y_{+}$ [$\times 10^{-2}$] | $-2.2\pm 2.5\pm 0.4\pm 1.0$ | $-1.4\pm 2.6\pm 0.6\pm 0.9$
$y_{-}$ [$\times 10^{-2}$] | $7.5\pm 2.9\pm 0.5\pm 1.4$ | $7.5\pm 3.0\pm 0.4\pm 1.5$

The measured values of $(x_{\pm},y_{\pm})$ from the fit to all data, with
their likelihood contours corresponding to statistical uncertainties only, are
displayed in Fig. 9.

Figure 9: Confidence levels at 39.3%, 86.5% and 98.9% probability for
$(x_{+},y_{+})$ and $(x_{-},y_{-})$ as measured in $B^{\pm}\rightarrow
DK^{\pm}$ decays (statistical uncertainties only). The parameters
$(x_{+},y_{+})$ relate to ${{B}^{+}}$ decays and $(x_{-},y_{-})$ refer to
${{B}^{-}}$ decays. The stars represent the best fit central 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, which
is equal to $2\gamma$.

To investigate whether the binned fit gives an adequate description of the
data, a study is performed to compare the expected signal yield in each bin,
given by the fitted total yield and the values of $x_{\pm}$ and $y_{\pm}$, and
the observed number of signal candidates in each bin. The latter is determined
by fitting directly in each bin for the ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$
candidate yield. This study is performed using effective bin numbers and with
long and downstream ${K}^{0}_{\rm\scriptscriptstyle S}$ decays combined.
Figure 10 shows the results separately for the sum of $B^{+}$ and $B^{-}$
candidates, $N_{B^{+}}+N_{B^{-}}$, and for the difference,
$N_{B^{+}}-N_{B^{-}}$, which is sensitive to $C\\!P$ violation. The expected
signal yields assuming $C\\!P$ symmetry ($x_{\pm}=y_{\pm}=0$) in the
$N_{{{{B}^{+}}}}-N_{{{B}^{-}}}$ distribution are also shown. These are not
constant at $N_{B^{+}}-N_{B^{-}}=0$ because they are calculated using the
total $B^{+}$ and $B^{-}$ yields, which do not have identical values. The data
and fit expectations are compatible for both distributions yielding a
$\chi^{2}$ probability ($p$-value) of 93% for $N_{B^{+}}+N_{B^{-}}$ and 80%
for $N_{B^{+}}-N_{B^{-}}$. The results for the $N_{B^{+}}-N_{B^{-}}$
distribution are less compatible with the hypothesis of $C\\!P$ symmetry,
which has a $p$-value of 4%.

Figure 10: Combinations of signal yields (data points) in effective bins
compared with prediction of $(x_{\pm},y_{\pm})$ fit (solid histogram) for
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ and
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$ decays.
The dotted histograms give the prediction for $x_{\pm}=y_{\pm}=0$. The left
plot shows the sum of $B^{+}$ and $B^{-}$ yields and the right plot shows the
difference of $B^{+}$ and $B^{-}$ yields.

##  8 Systematic uncertainties

Systematic uncertainties are evaluated for the fits to the full data sample
and are presented in Table 4. The uncertainties arising from the CLEO-c
measurements are kept separate from the other experimental uncertainties.

Table 4: Summary of statistical, experimental, and strong-phase, uncertainties on $x_{\pm}$ and $y_{\pm}$ in the case where both $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ and $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{K}^{+}}{{K}^{-}}$ decays are included in the fit. All entries are given in multiples of $10^{-2}$. Source | $\sigma(x_{+})$ | $\sigma(x_{-})$ | $\sigma(y_{+})$ | $\sigma(y_{-})$
---|---|---|---|---
Statistical | 2.4 | 2.5 | 2.5 | 2.9
Efficiency corrections | 0.9 | 0.9 | 0.2 | 0.2
Mass fit PDFs | 0.2 | 0.2 | 0.1 | 0.2
Shape of $D\pi^{\pm}$ mis-identified as $DK^{\pm}$ | 0.1 | 0.1 | 0.0 | 0.1
Shape of partially reconstructed backgrounds | 0.1 | 0.3 | 0.1 | 0.2
$c_{i}$, $s_{i}$ bias due to efficiency | 0.0 | 0.0 | 0.1 | 0.0
Migration | 0.1 | 0.1 | 0.2 | 0.2
Bias correction | 0.2 | 0.2 | 0.2 | 0.2
Total experimental | 1.0 | 1.0 | 0.4 | 0.5
Strong-phase-related uncertainties | 0.4 | 0.5 | 1.0 | 1.4

A systematic uncertainty arises due to the mismodelling in the simulation used
to derive the efficiency correction used in the determination of the $F_{i}$
parameters. To determine the systematic uncertainty associated with this
correction, an alternative set of correction factors is calculated and used to
evaluate an alternative set of $F_{i}$ parameters. The alternative correction
factors are calculated by incorporating an extra term (Eq. 10) determined from
a new rectangular binning scheme, as shown in Fig. 11. The bin-to-bin
efficiency variation in this rectangular scheme is significantly larger than
for the default partitioning and is more sensitive to imperfections in the
simulated data efficiency profile. The bin sizes are chosen to keep the
expected yields in each bin as similar as possible. The yields of the
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ and $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ decays in each bin of the rectangular scheme
are compared to the predictions from the amplitude model and the simulated
data efficiency profile. Differences of up to 15$\%$ are observed. These
differences are consistent for the two decay modes. The alternative correction
factors, $\xi_{i}^{\textrm{alt}}$, are calculated using the following
equation:

$\xi_{i}^{\textrm{alt}}=\frac{\int_{\mathcal{D}_{i}}\varepsilon_{DK}\,|A|^{2}\,C_{D\pi}\,d\mathcal{D}}{\int_{\mathcal{D}_{i}}\varepsilon_{{{D}^{*}}\\!\mu}\,|A|^{2}\,C_{{{D}^{*}}\\!\mu}\,d\mathcal{D}}$
(10)

where the $C=C(m^{2}_{-},m^{2}_{+})$ terms are the ratios between the
predicted and observed data yields in the rectangular binning. Many pseudo-
experiments are performed in which the data are generated according to the
default $F_{i}$ but are fitted assuming that the alternative $F_{i}$ set are
true. The overall shift in the fitted values of the $C\\!P$ parameters in
comparison to their input values is taken as the systematic uncertainty,
yielding $0.9\times 10^{-2}$ for $x_{\pm}$ and $0.2\times 10^{-2}$ for
$y_{\pm}$.

To assign an uncertainty for the imperfections in the description of the
invariant mass spectrum, three changes to the model are considered. Firstly,
an alternate signal shape is considered that has wider resolution and longer
tails. This alternate shape uses a different form of modified Gaussian and the
parameters are derived from a fit to data. Secondly, the description of the
partially reconstructed background is changed to a shape obtained from a fit
of the PDF to simulated $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}h^{+}h^{-}$ decays. Finally, the parameters of the misidentified background
PDF are changed to vary the tail under the signal peak as this is the part of
the PDF that is least well determined. For each change, the effects on the
$C\\!P$ parameters are determined using many pseudo-experiments where the data
are generated with the default PDFs and fitted with the alternate models. The
contributions from each change are summed in quadrature and are
$(0.1\text{--}0.2)\times 10^{-2}$.

Figure 11: The rectangular binning schemes for the two decays. On the left
(right) is plotted the scheme for the ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ (${{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$) decay.

Two systematic uncertainties are evaluated that are associated with the
misidentified ${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ background in the
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ sample. The uncertainties on the
particle misidentification efficiencies are found to have a negligible effect
on the measured values of $x_{\pm}$ and $y_{\pm}$. It is possible that the
invariant mass distribution of the misidentified background is not uniform
over the Dalitz plot, as is assumed in the fit. This can occur through
kinematic correlations between the reconstruction efficiency on the Dalitz
plot of the $D$ decay and the momentum of the bachelor pion from the
${B}^{\pm}$ decay. Pseudo-experiments are performed with different mass shapes
input according to the Dalitz plot bin and the results of simulation studies.
These experiments are then fitted assuming a uniform shape, as in data. The
resulting uncertainty is up to $0.1\times 10^{-2}$ for all $C\\!P$ parameters.

The distribution of the partially reconstructed background is varied over the
Dalitz plot according to the uncertainty in the composition of this background
component. This results in a different invariant mass distribution in each
Dalitz plot bin. An uncertainty of $(0.1\text{--}0.3)\times 10^{-2}$ is
assigned to the fitted parameters in the full data fit.

The non-uniform efficiency profile over the Dalitz plot means that the values
of $(c_{i},s_{i})$ appropriate for this analysis can differ from those
measured at CLEO-c, which correspond to the constant efficiency case. This
leads to a potential bias in the determination of $x_{\pm}$ and $y_{\pm}$. The
possible size of this effect is evaluated using the LHCb simulation. The
Dalitz plot bins are divided into smaller cells, and the BaBar amplitude model
[10, 11] is used to calculate the values of $c_{i}$ and $s_{i}$ within each
cell. These values are then averaged together and weighted by the population
of each cell after efficiency losses to obtain an effective $(c_{i},s_{i})$
for the bin as a whole. The results are compared with those determined
assuming a constant efficiency; the differences between the two sets of
results are found to be small compared with the CLEO-c measurement
uncertainties. The data are fitted multiple times, each with different
$(c_{i},s_{i})$ values sampled according to the size of these differences, and
the mean shifts are assigned as a systematic uncertainty. These shifts are
less than $0.1\times 10^{-2}$ for all $C\\!P$ parameters.

For both ${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ and $\kern
1.79993pt\shortstack{{(\rule[2.15277pt]{12.50002pt}{0.51212pt})} \\\ [-.7ex]
$\kern-1.79993ptB$}{}\\!\rightarrow{{D}^{*\pm}}\mu^{\mp}\shortstack{{(\rule[2.15277pt]{10.00002pt}{0.51212pt})}
\\\ [-.7ex] $\nu$}{}_{\\!\\!\mu}$ decays the resolution in $m^{2}_{+}$ and
$m^{2}_{-}$ of each decay is approximately
0.005${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for candidates with long
${K}^{0}_{\rm\scriptscriptstyle S}$ decays and
0.006${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ for candidates with
downstream ${K}^{0}_{\rm\scriptscriptstyle S}$. This is small compared to the
typical width of a bin but net migration away from the more densely populated
bins is possible. To first order this effect is accounted for by use of the
control channel, but residual effects enter due to the different distribution
in the Dalitz plot of the signal events. The uncertainty due to these residual
effects is determined via pseudo-experiments, in which different input $F_{i}$
values are used to reflect the residual migration. The size of any possible
bias is found to vary between $0.1\times 10^{-2}$ and $0.2\times 10^{-2}$.

An uncertainty is assigned to each $C\\!P$ parameter to accompany the
correction that is applied for the small bias observed in the fit procedure.
These uncertainties are determined by performing sets of pseudo-experiments,
each generated with different values of $x_{\pm}$ and $y_{\pm}$ according to
the range allowed by current experimental knowledge. The spread in observed
bias is combined in quadrature with half the correction and the uncertainty in
the precision of the pseudo-experiments. This is taken as the systematic
uncertainty, and is $0.2\times 10^{-2}$ for all $C\\!P$ parameters. The effect
that a detection asymmetry between hadrons of opposite charge can have on the
symmetry of the efficiency of the Dalitz plot is found to be negligible.
Changes in the mass model used to describe the semileptonic control sample are
found to have a negligible effect on the $F_{i}$ values.

The limited precision on $(c_{i},s_{i})$ coming from the CLEO-c measurement
induces uncertainties on $x_{\pm}$ and $y_{\pm}$ [18]. These uncertainties are
evaluated by fitting the data multiple times, each with different
$(c_{i},s_{i})$ values sampled according to their experimental uncertainties
and correlations. The resulting width in the distribution of $x_{\pm},y_{\pm}$
values is assigned as the systematic uncertainty. Values of
$(0.4\text{--}1.4)\times 10^{-2}$ are found for the fit to the full sample.
The uncertainties are smaller than those reported in Ref. [4]. This is as
expected since it is found from simulation studies that the ($c_{i}$, $s_{i}$)
uncertainty also depends on the sample size.

Finally, several checks are conducted to assess the stability of the results.
These include repeating the fits separately for both
${K}^{0}_{\rm\scriptscriptstyle S}$ categories, for the centre-of-mass energy
at which the data were collected, and for candidates passing different
hardware trigger requirements. No anomalies are found, and no additional
systematic uncertainties are assigned.

The total experimental systematic uncertainty from LHCb-related sources is
determined to be $1.0\times 10^{-2}$ on $x_{+}$, $1.0\times 10^{-2}$ on
$x_{-}$, $0.4\times 10^{-2}$ on $y_{+}$, and $0.5\times 10^{-2}$ on $y_{-}$.
These are all smaller than the corresponding statistical uncertainties. The
dominant contribution arises from the efficiency correction method.

After taking account of all sources of uncertainty the correlation matrix
between the measured $x_{\pm}$, $y_{\pm}$ parameters for the full data set is
shown in Table 5. Correlations from the statistical and strong-phase
uncertainties are included but the experimental systematic uncertainties are
treated as uncorrelated. The equivalent matrix for
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ decays
only is shown in Table 6.

Table 5: Correlation matrix between the $x_{\pm}$, $y_{\pm}$ parameters for the full data set.  | $x_{+}$ | $x_{-}$ | $y_{+}$ | $y_{-}$
---|---|---|---|---
$x_{+}$ | $\phantom{-}1.000$ | $\phantom{-}0.027$ | $\phantom{-}0.003$ | $-0.007$
$x_{-}$ |  | $\phantom{-}1.000$ | $\phantom{-}0.009$ | $-0.200$
$y_{+}$ |  |  | $\phantom{-}1.000$ | $-0.016$
$y_{-}$ |  |  |  | $\phantom{-}1.000$
Table 6: Correlation matrix between the $x_{\pm}$, $y_{\pm}$ parameters for ${{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ decays only.  | $x_{+}$ | $x_{-}$ | $y_{+}$ | $y_{-}$
---|---|---|---|---
$x_{+}$ | $\phantom{-}1.000$ | $\phantom{-}0.017$ | $-0.038$ | $-0.008$
$x_{-}$ |  | $\phantom{-}1.000$ | $\phantom{-}0.009$ | $-0.220$
$y_{+}$ |  |  | $\phantom{-}1.000$ | $\phantom{-}0.004$
$y_{-}$ |  |  |  | $\phantom{-}1.000$

The systematic uncertainties for the case where only
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ decays
are included are also given in Table 3. The total experimental systematic in
this case is larger and this is primarily driven by a larger systematic effect
due to the simulation-derived efficiency correction, for which the systematic
uncertainty for the $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}}$ decays partially compensates. The uncertainties due to
the CLEO-c strong-phase measurements are also slightly larger when considering
only $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decays due to the dependence of this systematic uncertainty on the signal
sample and its size.

##  9 Results and interpretation

The results for $x_{\pm}$ and $y_{\pm}$ can be interpreted in terms of the
underlying physics parameters $\gamma$, $r_{B}$ and $\delta_{B}$. This
interpretation is performed using a frequentist approach with Feldman-Cousins
ordering [43], using the same procedure as described in Ref. [19], yielding
confidence levels for the three physics parameters.

In Fig. 12 the projections of the three-dimensional surfaces bounding the one,
two and three standard deviation volumes onto the $(\gamma,r_{B})$ and
$(\gamma,\delta_{B})$ planes are shown. The LHCb-related systematic
uncertainties are taken as uncorrelated and correlations of the CLEO-c and
statistical uncertainties are taken into account. The statistical and
systematic uncertainties on $x_{\pm}$ and $y_{\pm}$ are combined in
quadrature.

Figure 12: The three-dimensional confidence volumes, corresponding to 19.9%,
73.9% and 97.1% confidence levels, are projected onto the $(\gamma,r_{B})$ and
$(\gamma,\delta_{B})$ planes. The confidence levels are given by solid, dashed
and dotted contours. The diamonds mark the central values.

The solution for the physics parameters has a two-fold ambiguity:
$(\gamma,\delta_{B})\rightarrow(\gamma+180^{\circ},\delta_{B}+180^{\circ})$.
Choosing the solution that satisfies $0<\gamma<180^{\circ}$ yields
$r_{B}=0.080^{\,+0.019}_{\,-0.021}$, $\gamma=(62^{\,+15}_{\,-14})^{\circ}$ and
$\delta_{B}=(134^{\,+14}_{\,-15})^{\circ}$. The values for $\gamma$ and
$r_{B}$ are consistent with the world average of results from previous
experiments [42]. The significant increase in precision compared to the
measurement in Ref. [4] is due to a combination of increased signal yield,
lower systematic uncertainties and a higher central value for $r_{B}$.

##  10 Conclusions

Approximately 2580 $B^{\pm}\rightarrow DK^{\pm}$ decays, with the $D$ meson
decaying either to ${{K}^{0}_{\rm\scriptscriptstyle S}}\pi^{+}\pi^{-}$ or
${{K}^{0}_{\rm\scriptscriptstyle S}}K^{+}K^{-}$, are selected from data
corresponding to and integrated luminosity of 3.0 ${\rm fb^{-1}}$ collected by
LHCb in 2011 and 2012. These samples are analysed to determine the
$C\\!P$-violating parameters $x_{\pm}\equiv r_{B}\cos(\delta_{B}\pm\gamma)$
and $y_{\pm}\equiv r_{B}\sin(\delta_{B}\pm\gamma)$, where $r_{B}$ is the ratio
of the absolute values of the $B^{+}\rightarrow D^{0}K^{-}$ and
$B^{+}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}K^{-}$
amplitudes, $\delta_{B}$ is the strong-phase difference between them, and
$\gamma$ is an angle of the unitarity triangle. The analysis is performed in
bins of the $D$ decay Dalitz plot, and existing measurements of the CLEO-c
experiment are used to provide input on the $D$ decay strong-phase parameters
$(c_{i},s_{i})$ [18]. Such an approach allows the analysis to be free from any
model-dependent assumptions on the strong-phase variation across the Dalitz
plot. The following results are obtained:

$\displaystyle x_{+}$ $\displaystyle=(-7.7\pm 2.4\pm 1.0\pm 0.4)\times
10^{-2},\;\,$ $\displaystyle x_{-}$ $\displaystyle=(2.5\pm 2.5\pm 1.0\pm
0.5)\times 10^{-2},$ $\displaystyle y_{+}$ $\displaystyle=(-2.2\pm 2.5\pm
0.4\pm 1.0)\times 10^{-2},\;\,$ $\displaystyle y_{-}$ $\displaystyle=(7.5\pm
2.9\pm 0.5\pm 1.4)\times 10^{-2},$

where the first uncertainties are statistical, the second are systematic and
the third arise from the experimental knowledge of the $(c_{i},s_{i})$
parameters. The results are the most precise values of these $C\\!P$
observables obtained from a single measurement.

From the above results, the following values of the underlying physics
parameters are derived: $r_{B}=0.080^{\,+0.019}_{\,-0.021}$,
$\gamma=(62^{\,+15}_{\,-14})^{\circ}$ and
$\delta_{B}=(134^{\,+14}_{\,-15})^{\circ}$. These values are consistent with
the world averages of results from previous measurements [20], but should not
be combined with the model-dependent measurements [15]. These values improve
upon and supersede the results from a previous model-independent measurement
performed with 1.0 ${\rm fb^{-1}}$ of data collected by LHCb in 2011 [4].

## 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, HGF and MPG (Germany); SFI (Ireland);
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). The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from EPLANET,
Marie Skłodowska-Curie Actions and ERC (European Union), Conseil général de
Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR
(Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for
the Exhibition of 1851 (United Kingdom).

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LHCb collaboration

R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, W. Baldini16, R.J. Barlow54, C.
Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, V. Battista39, A. Bay39,
L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson38, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30,38,
W. Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, J. Brodzicka54, N.H. Brook46, H. Brown52,
A. Bursche40, G. Busetto22,r, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f,
M. Calvi20,k, M. Calvo Gomez36,p, P. Campana18,38, D. Campora Perez38, A.
Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j, A. Cardini15, L. Carson50,
K. Carvalho Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38, M.
Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph. Charpentier38, M.
Chefdeville4, S. Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26,
K. Ciba38, X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38,
H.V. Cliff47, J. Closier38, V. Coco38, J. Cogan6, E. Cogneras5, L.
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Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11, M. Shapkin35, I.
Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34, V. Shevchenko63,
A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, T.
Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J. Smith47, M. Smith54,
H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39, D. Souza46, B. Souza
De Paula2, B. Spaan9, A. Sparkes50, P. Spradlin51, S. Sridharan38, F.
Stagni38, M. Stahl11, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,p, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, L. Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A.
Zhokhov31, L. Zhong3, A. Zvyagin38.

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 de Savoie, 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
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, 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
vPolitecnico di Milano, Milano, Italy