EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-166 LHCb-PAPER-2014-017 July 22, 2014

Measurement of $C\\!P$ violation and constraints on the CKM angle $\gamma$ in
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ with
$D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$ decays

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

A model-dependent amplitude analysis of ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$
with $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decays is performed using proton-proton collision data, corresponding to an
integrated luminosity of $1\mbox{\,fb}^{-1}$, recorded by LHCb at a centre-of-
mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$ in $2011$. Values of the
$C\\!P$ violation observables $x_{\pm}$ and $y_{\pm}$, which are sensitive to
the CKM angle $\gamma$, are measured to be

$\displaystyle x_{-}$ $\displaystyle=+0.027\pm 0.044~{}^{+0.010}_{-0.008}\pm
0.001,$ $\displaystyle y_{-}$ $\displaystyle=+0.013\pm
0.048~{}^{+0.009}_{-0.007}\pm 0.003,$ $\displaystyle x_{+}$
$\displaystyle=-0.084\pm 0.045\pm 0.009\pm 0.005,$ $\displaystyle y_{+}$
$\displaystyle=-0.032\pm 0.048~{}^{+0.010}_{-0.009}\pm 0.008,$

where the first uncertainty is statistical, the second systematic and the
third arises from the uncertainty of the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
amplitude model. The value of $\gamma$ is determined to be
$(84^{+49}_{-42})^{\circ}$, including all sources of uncertainty. Neutral $D$
meson mixing is found to have negligible effect.

Published in Nucl. Phys. B

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

##  1 Introduction

The CKM phase $\gamma$
($\gamma\equiv\mathrm{arg}\left[-{V_{{u}{d}}}{V_{{u}{b}}}^{*}/{V_{{c}{d}}}{V_{{c}{b}}}^{*}\right]$,
also known as $\phi_{3}$) is the angle of the CKM unitarity triangle that is
least constrained by direct measurements. The precise determination of
$\gamma$ is an important aim of current flavour physics experiments. It can be
measured directly in tree-level processes, for example in
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays where $D$ is a superposition of
the flavour eigenstates ${D}^{0}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decaying into the same final
state. Sensitivity to $\gamma$ arises from the interference between
${b}\rightarrow{u}$ and ${b}\rightarrow{c}$ quark transitions. Since
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays are expected to be insensitive
to physics processes beyond the Standard Model (SM), this measurement provides
a reference value against which other observables, potentially affected by
physics beyond the SM, can be compared.

The determination of $\gamma$ (using ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$
decays) from an amplitude analysis of the $D$ meson decay to the three-body
quasi-self-conjugate ${{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ final state was first proposed in Refs. [1, 2]. The
method requires knowledge of the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay amplitude across the phase space and, in particular, its strong phase
variation. The model-dependent approach, as used in Refs. [3, 4, 5, 6, 7, 8],
implements a model to describe the $D$ decay amplitude over the phase space.
This unbinned method allows for full exploitation of the statistical power of
the data. A model-independent strategy, employed by the LHCb [9] and Belle
[10] collaborations, uses CLEO measurements [11] of the $D$ decay strong phase
difference in bins across the phase space.

Neglecting the effects of charm mixing, the amplitude for
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ decays can be written as a
superposition of Cabibbo favoured and suppressed contributions,

$\displaystyle\mathcal{A}_{{{{B}^{-}}}}$
$\displaystyle\sim{A}_{f}+r_{{B}}e^{i(\delta_{{B}}-\gamma)}\bar{A}_{f},$ (1)
$\displaystyle\mathcal{A}_{{{{B}^{+}}}}$
$\displaystyle\sim\bar{A}_{f}+r_{{B}}e^{i(\delta_{{B}}+\gamma)}{A}_{f},$

where $r_{{B}}$ is the magnitude of the ratio of the interfering ${B}^{\pm}$
decay amplitudes, $\delta_{{B}}$ is the strong phase difference between them,
and $\gamma$ is the $C\\!P$-violating weak phase. The amplitudes of the
${D}^{0}$ and ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ mesons
decaying into the common final state $f$, $A_{f}\equiv\left\langle
f\vphantom{\bar{D}^{0}}\right|\mathcal{H}\left|\vphantom{\bar{D}^{0}}{{D}^{0}}\right\rangle$
and $\bar{A}_{f}\equiv\left\langle
f\vphantom{\bar{D}^{0}}\right|\mathcal{H}\left|{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\right\rangle$, respectively,
depend on two squared invariant masses of pairs of the three final state
particles, chosen to be $m_{+}^{2}\equiv m_{{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}}^{2}$ and $m_{-}^{2}\equiv m_{{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{-}}}^{2}$. Assuming that no direct $C\\!P$ violation exists in the
$D$ meson decay, the amplitudes $A_{f}$ and $\bar{A}_{f}$ are related by
$\bar{A}_{f}(m^{2}_{+},m^{2}_{-})=A_{f}(m^{2}_{-},m^{2}_{+})$. A direct
determination of $r_{{B}}$, $\delta_{{B}}$ and $\gamma$ can lead to bias [3],
and hence the Cartesian $C\\!P$ violation observables,
$x_{\pm}=r_{{B}}\cos{(\delta_{{B}}\pm\gamma)}$ and
$y_{\pm}=r_{{B}}\sin{(\delta_{{B}}\pm\gamma)}$, are used, where the “$+$” and
“$-$” indices correspond to ${{B}^{+}}$ and ${{B}^{-}}$ decays, respectively.

This paper reports measurements of $(x_{\pm},y_{\pm})$ made using
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ decays selected from $pp$ collision
data, corresponding to an integrated luminosity of $1\mbox{\,fb}^{-1}$,
recorded by LHCb at a centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$ in $2011$. The data set is identical to that
used in Ref. [9]. The measured values of $(x_{\pm},y_{\pm})$ place constraints
on the CKM angle $\gamma$.

##  2 The LHCb detector

The LHCb detector [12] 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 placed downstream of the
magnet. The combined tracking system provides a momentum measurement with a
relative uncertainty that varies from 0.4% at low momentum, $p$, to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter measurement
with a resolution of 20${\,\upmu\rm m}$ for charged particles with large
transverse momentum, $p_{\rm T}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[13], providing particle identification (PID) information. 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.

The trigger 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 software trigger requires a two-, three- or
four-track secondary vertex with a large sum $p_{\rm T}$ of the tracks and a
significant displacement from any primary $pp$ interaction vertex (PV). At
least one track should also have large $p_{\rm T}$ and $\chi^{2}_{\rm IP}$
with respect to any primary interaction, where $\chi^{2}_{\rm IP}$ is defined
as the difference in $\chi^{2}$ of a given PV reconstructed with and without
the considered track. A multivariate algorithm [14] is used to identify
secondary vertices consistent with decays of $b$ hadrons.

Large samples of simulated
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ decays are used in this study, along
with simulated samples of various background decays. In the simulation, $pp$
collisions are generated using Pythia 6.4 [15] with a specific LHCb
configuration [16]. Decays of hadronic particles are described by EvtGen [17],
in which final state radiation is generated using Photos [18]. The interaction
of the generated particles with the detector and its response are implemented
using the Geant4 toolkit [19, *Agostinelli:2002hh] as described in Ref. [21].

##  3 Candidate selection and sources of background

The criteria used to select
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ candidate decays from the data are
described below. The
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ decays are used to measure the
acceptance over phase space, as they have almost identical topologies to
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ decays, but a much higher branching
fraction [22]. Apart from the ${B}^{\pm}$ candidate invariant mass range, the
selection requirements are identical to those used in Ref. [9] and are
summarised here for completeness.

Candidate ${K}^{0}_{\rm\scriptscriptstyle S}$ mesons are reconstructed from
two oppositely charged well-measured tracks; those with tracks reconstructed
in the silicon vertex detector are known as long candidates and those with
tracks that cannot be formed in the vertex detector are known as downstream
candidates. A requirement of $\chi^{2}_{\rm IP}$ greater than 16 (4) with
respect to the PV is made for the long (downstream) pion tracks. The PV of
each candidate ${B}^{\pm}$ meson decay is chosen to be the one yielding the
minimum $\chi^{2}_{\rm IP}$. To reduce background from random track
combinations, the cosine of the angle between the momentum direction of the
${K}^{0}_{\rm\scriptscriptstyle S}$ meson candidate and the direction vector
from the PV to its decay vertex is required to be greater than 0.99.

The ${K}^{0}_{\rm\scriptscriptstyle S}$ candidates are combined with two
oppositely charged tracks to reconstruct $D$ meson candidates; the tracks
combined with a long (downstream) candidate must have $\chi^{2}_{\rm IP}$
greater than 9 (16) with respect to the PV. For all $D$ meson candidates,
requirements of $\chi^{2}_{\rm IP}$ greater than 9 with respect to the PV and
cosine of the angle between the momentum and direction vectors greater than
0.99 are made. It is required that the vertex separation $\chi^{2}$ between
the reconstructed $D$ and ${K}^{0}_{\rm\scriptscriptstyle S}$ meson decay
vertices is greater than 100, where the vertex separation $\chi^{2}$ is
defined as the change in $\chi^{2}$ of a vertex which is reconstructed
including the particles originally contributing to the other vertex. The
reconstructed $D$ meson candidate invariant mass is required to be within $\pm
25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known value [22]. The
${K}^{0}_{\rm\scriptscriptstyle S}$ candidate invariant mass must be within
$\pm 15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known value [22]
after a refit to constrain the $D$ meson mass [23].

The ${B}^{\pm}$ meson candidates are reconstructed from the combination of a
$D$ meson candidate with a pion or kaon directly from the ${B}^{\pm}$ vertex,
hereafter called the “bachelor” track. The bachelor track is required to have
$\chi^{2}_{\rm IP}$ greater than 25 with respect to the PV. To separate
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ decays, good discrimination between
pions and kaons is required using PID information. The $\chi^{2}_{\rm IP}$ of
the reconstructed ${B}^{\pm}$ candidate with respect to the PV is required to
be less than 9, and for long (downstream) candidates the cosine of the angle
between its momentum and direction vectors must be greater than 0.9999
(0.99995). The ${B}^{\pm}$ vertex separation $\chi^{2}$ with respect to the PV
must be greater than 169. In addition, the reconstructed $D$ meson decay
vertex is required to have a larger longitudinal displacement from the PV than
the ${B}^{\pm}$ decay vertex.

Each selected candidate decay is refitted with additional constraints on the
${K}^{0}_{\rm\scriptscriptstyle S}$ and $D$ meson masses and on the pointing
of the $B$ momentum to the PV, so that improved resolution in the phase space
of the $D$ decay is obtained. A refit quality requirement of $\chi^{2}$ per
degree of freedom less than 5 is made. If more than one selected candidate is
found to originate from the same $pp$ collision event, the candidate with the
lowest value of refit $\chi^{2}$ per degree of freedom is retained.

Several sources of potential background are studied using simulation. These
include two categories of combinatorial background: a real
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay combined with a random bachelor track (random $Dh$), or a
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
candidate reconstructed with at least one random final state track
(combinatorial $D$). Cross-feed background arises from
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ decays misidentified as
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ decays (or vice versa), and contributes
a large fraction of the selected
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ candidates. Partially reconstructed
candidates from decay modes containing a
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay, such as ${{B}^{\pm}}\rightarrow{{D}^{*}}h^{\pm}$ (where ${D}^{*}$
represents ${D}^{*0}$ or ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}$ and $h^{\pm}$ represents a
${{K}^{\pm}}$ or ${{\pi}^{\pm}}$), $B_{(s)}\rightarrow{D}{{K}^{*}}$ (where
$B_{(s)}$ represents $B^{0}_{(s)}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}$ and ${K}^{*}$ represents
${K}^{*0}$ or ${\kern 1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}$) and
${{B}^{\pm}}\rightarrow{D}\rho^{\pm}$ decays, are also expected to contribute.
The contributions from charmless ${B}^{\pm}$ decays,
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{\pm}}{{\pi}^{\mp}})h^{\pm}$ decays,
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{K}^{+}}{{K}^{-}})h^{\pm}$ decays and
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{\pi}^{+}}{{\pi}^{-}}h^{+}h^{-})h^{\pm}$
decays are found to be negligible.

##  4 Analysis strategy

The analysis is performed in two distinct parts. The fractions of signal and
background are determined with a phase-space integrated fit to the invariant
mass distributions, $m_{Dh}$, of selected
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ candidates, shown in Fig. 1. This is
followed by a fit to determine the $C\\!P$ violation observables
$(x_{\pm},y_{\pm})$ and the variation in efficiency over the phase space of
the ${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay. The relative signal and background yields and the parameters of the
${B}^{\pm}$ invariant mass probability distribution functions (PDFs) are fixed
to the values determined in the first stage.

Figure 1:  Invariant mass distributions for (a)
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ long, (b)
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ long, (c)
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ downstream and (d)
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ downstream candidates. The fit
results, including signal and background components, are superimposed. The
lower plots are normalised residual distributions.

###  4.1 Invariant mass fit of B± candidates

An unbinned extended maximum likelihood fit to the invariant mass
distributions of the ${B}^{\pm}$ candidates determines the signal and
background fractions. The samples of
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ candidates are fitted simultaneously
in an invariant mass range of
$4779{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}<m_{Dh}<5779{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
The long and downstream candidates are fitted separately.

For the fit to the
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ invariant mass distribution, the total
PDF is composed of a signal and several background components. The signal
(${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$) is described by the sum of a Crystal
Ball [24] and a Gaussian function with common means. The Crystal Ball tail
parameters, the width of the Gaussian function and the relative fractions of
both functions are fixed to values obtained from simulated data. An
exponential function describes the two categories of combinatorial background
candidates. Cross-feed candidates are characterised by a Crystal Ball function
with tails on both upper and lower sides. The mean and tail parameters of the
function are fixed to results from simulation. Partially reconstructed
background contributions are described by various functions with parameters
fixed to values obtained from simulation. Both $B$ and ${B}^{\pm}$ decays that
give rise to candidates with similar invariant mass distributions are
described by a single fit component: the candidates from
${{B}^{\pm}}\rightarrow{{D}^{*}}{{K}^{\pm}}$ and
$B\rightarrow{{D}^{*\mp}}{{K}^{\pm}}$ decays are both described using the sum
of two pairs of Gaussian functions, where the Gaussian functions in each pair
have a common mean and independent widths. For the combined background
contribution from partially reconstructed
${{B}^{\pm}}\rightarrow{{D}^{*}}{{\pi}^{\pm}}$ and
$B\rightarrow{{D}^{*\mp}}{{\pi}^{\pm}}$ decays, labelled ${{D}^{*}}\pi$, the
sum of two Crystal Ball functions, each with tails on both upper and lower
sides, is used. A background composed of candidates from
$B\rightarrow{D}\rho^{0}$ and ${{B}^{\pm}}\rightarrow{D}\rho^{\pm}$ decays,
labelled ${D}\rho$, is described by the sum of a Gaussian and an exponential
function. A Gaussian function is included for background candidates partially
reconstructed from $B\rightarrow{{D}^{*\mp}}\rho^{\pm}$ and
${{B}^{\pm}}\rightarrow{{D}^{*}}\rho^{\pm}$ decays. The background
contribution from partially reconstructed $B\rightarrow{D}{{K}^{*}}$ decays is
modelled by the convolution of an ARGUS function [25] with a Gaussian
function; the same convolution of functions is used for candidates
reconstructed from $B_{s}\rightarrow{D}{{K}^{*}}$ decays.

For the fit to the
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ mass distribution, the same PDFs are
used for signal, combinatorial and cross-feed background contributions as for
the fit to the
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ distribution. The analogous function
parameters are fixed to the results of fits to simulation. Again, functions
are also included for partially reconstructed background candidates, with all
parameters fixed to values obtained from simulation. The sum of two pairs of
Gaussian functions, labelled ${{D}^{*}}\pi$, is used for the background from
partially reconstructed ${{B}^{\pm}}\rightarrow{{D}^{*}}{{\pi}^{\pm}}$ and
$B\rightarrow{{D}^{*\mp}}{{\pi}^{\pm}}$ decays. Partially reconstructed
$B\rightarrow{D}\rho^{0}$ and ${{B}^{\pm}}\rightarrow{D}\rho^{\pm}$ decays are
described by the convolution of an ARGUS function with a Gaussian function. A
Gaussian function is used to describe background from partially reconstructed
$B\rightarrow{{D}^{*\mp}}\rho^{\pm}$ and
${{B}^{\pm}}\rightarrow{{D}^{*}}\rho^{\pm}$ decays.

In the simultaneous fit, the mean values of the signal functions in
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ are constrained to a common value.

The yield of the cross-feed component in the fit to the
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$
(${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$) distribution is fixed with respect
to the signal yield in the ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$
(${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$) distribution, using knowledge of the
efficiency and misidentification rate of the PID criterion separating the
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ candidate samples. Large calibration
samples of kaons and pions from
${{D}^{*\pm}}\rightarrow{D}(\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}){{\pi}^{\pm}}$
decays, kinematically selected from data, are reweighted to match the
kinematic properties of the bachelor tracks in the
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ long and downstream candidate samples
and are then used to determine the relevant efficiencies. The remaining
background yields are free to vary in the fit, as are the remaining PDF
parameters and the ratio of the signal yields.

Since it is not possible to separate the two components of combinatorial
background with the fit to the ${B}^{\pm}$ invariant mass distributions, the
yield of combinatorial $D$ background candidates is estimated from data using
${{B}^{\pm}}\rightarrow{D}h^{\pm}$ decays, where the $D$ is reconstructed to
decay to two same-sign pions (${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{+}}$ and charge conjugate). These “wrong-sign” decays
are subject to the selection criteria described in Sec. 3.

###  4.2 CP asymmetry fit

The distributions in the ${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decay phase space for positively and negatively
charged ${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ candidate decays are fitted
simultaneously using an unbinned maximum likelihood fit to determine the
$C\\!P$ violation observables $(x_{\pm},y_{\pm})$ and the variation in
efficiency over the phase space. Although
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ decays are expected to exhibit
interference analogous to
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ decays and therefore be sensitive to
$\gamma$, the magnitude of the ratio of interfering $D$ decay amplitudes,
$r_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}$, is expected to be an order of
magnitude smaller than $r_{{B}}$ for
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ decays. It is therefore possible, to a
good approximation, to neglect the suppressed contribution to the
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ decay amplitude and use
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ decays to obtain the efficiency
variation as a function of $m^{2}_{+}$ and $m^{2}_{-}$, which is modelled as a
second-order polynomial function. This assumption is considered as a source of
systematic uncertainty.

The candidates are divided into eight subsamples, according to
${K}^{0}_{\rm\scriptscriptstyle S}$ type (long or downstream), the charge of
the bachelor track, and whether the candidate is identified as a
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ or
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ decay. The negative logarithm of the
likelihood,

$-\ln\mathcal{L}=-\sum_{s}\sum_{k}\ln\left(\sum_{c}N_{c}\cdot
p_{cs}^{\mathrm{mass}}\left(\left(m_{Dh}\right)_{k};\vec{P}_{cs}^{\mathrm{mass}}\right)\cdot
p_{cs}^{\mathrm{model}}\left(\left(m^{2}_{+},m^{2}_{-}\right)_{k};\vec{P}_{cs}^{\mathrm{model}}\right)\right),$
(2)

is minimised; in this expression, $c$ indexes the candidate categories (signal
or background type), $s$ indexes the subsample, and $k$ identifies each decay
candidate. $N_{c}$ is the candidate yield for category $c$, and
$p_{cs}^{\mathrm{mass}}$ is the invariant mass PDF, $p_{cs}^{\mathrm{model}}$
is the normalised $D$ decay model described below,
$\vec{P}_{cs}^{\mathrm{mass}}$ are the mass PDF parameters, and
$\vec{P}_{cs}^{\mathrm{model}}$ are the $D$ decay model parameters for
category $c$ and subsample $s$. It should be noted that $(x_{\pm},y_{\pm})$
are included in the parameter list of the
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ signal category and the
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ cross-feed category, which arises
from misidentification of ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays. The
normalisation of $p_{cs}^{\mathrm{model}}$ depends on the efficiency variation
over the phase space. The yields and parameters of the mass PDFs are fixed to
the results obtained in the ${B}^{\pm}$ invariant mass fit. To avoid
inadvertent experimenter’s bias in the determination of the $C\\!P$ violation
parameters, the values of the observables $(x_{\pm},y_{\pm})$ are masked until
the measurement technique has been finalised.

The model describing the amplitude of the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay over the phase space, $A_{f}\left(m_{+}^{2},m_{-}^{2}\right)$, is
identical to that used by the BaBar collaboration in Refs. [5, 26]. It
incorporates an isobar model for P-wave (which includes $\rho(770)$,
$\omega(782)$, Cabibbo-allowed and doubly Cabibbo-suppressed ${{K}^{*}}(892)$
and ${{K}^{*}}(1680)$) and D-wave (including $f_{2}(1270)$ and
${{K}^{*}}_{2}(1430)$) contributions. A generalised LASS amplitude for the
${K}\pi$ S-wave contribution (${{K}^{*}}_{0}(1430)$) and a K-matrix with
P-vector approach for the $\pi\pi$ S-wave contribution are also included in
the model. All parameters of the model are fixed in the fit to the values
determined in Ref. [26]. 111The model implemented by BaBar [26] differs from
the formulation described therein. One of the two Blatt-Weisskopf coefficients
was set to unity, and the imaginary part of the denominator of the Gounaris-
Sakurai propagator used the mass of the resonant pair, instead of the mass
associated with the resonance. The model used herein replicates these features
without modification. It has been verified that changing the model to use an
additional centrifugal barrier term and a modified Gounaris-Sakurai propagator
has a negligible effect on the measurements.

The fit is performed using refitted candidates with a ${B}^{\pm}$ invariant
mass lying within $\pm 50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the
known value [22], corresponding to an invariant mass region of approximately
$\pm 3\sigma$ around the signal peak. Although the full description of the
mass PDF provides valuable constraints for the background within the mass
window, only those backgrounds with significant contributions are included in
the $C\\!P$ asymmetry fit. The yields of the signal and incorporated
background contributions are given in Table 1. For the
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ subsamples, the cross-feed,
combinatorial $D$, random $Dh$, ${{D}^{*}}\pi$, ${D}\rho$ and
$B_{s}\rightarrow{D}{{K}^{*}}$ background categories are included in the fit.
The cross-feed contribution is assumed to be distributed in the phase space of
the ${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay according to the ${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ (${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$) decay model in the
${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ (${{{B}^{+}}}\rightarrow{D}{{K}^{+}}$)
case. Combinatorial $D$ background candidates are expected to be distributed
non-resonantly over the phase space. The distribution of random $Dh$
candidates is assumed to be an incoherent sum of the
${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decay models. Both
${{B}^{\pm}}\rightarrow{{D}^{*}}{{\pi}^{\pm}}$ and
$B\rightarrow{{D}^{*\mp}}{{\pi}^{\pm}}$ decays are represented by the
inclusion of a ${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ (${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$) decay model in the
${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ (${{{B}^{+}}}\rightarrow{D}{{K}^{+}}$)
case. The ${D}\rho$ component of the invariant mass fit is composed of
candidates from $B\rightarrow{D}\rho^{0}$ and
${{B}^{\pm}}\rightarrow{D}\rho^{\pm}$ decays; the distribution of candidates
from $B\rightarrow{D}\rho^{0}$ over the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay phase space is assumed to be an incoherent sum of the
${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decay models, whereas the candidates from
${{B}^{\pm}}\rightarrow{D}\rho^{\pm}$ are accounted for with a
${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ (${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$) decay model for the
${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ (${{{B}^{+}}}\rightarrow{D}{{K}^{+}}$)
case. Background $B_{s}\rightarrow{D}{{K}^{*}}$ candidates are assumed to be
distributed according to the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$
(${{D}^{0}}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$) decay model in the
${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ (${{{B}^{+}}}\rightarrow{D}{{K}^{+}}$)
case. For the ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ subsamples,
contributions from cross-feed, combinatorial $D$, random $Dh$, and
${{D}^{*}}\pi$ background types are included in the fit. The cross-feed
candidates in ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ arise from
misidentification of the bachelor track of
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays; the candidates are assumed to
be distributed accordingly. The remaining combinatorial and ${{D}^{*}}\pi$
background contributions are assumed to be distributed as described above.

Table 1:  Signal and background yields for components contributing to the $C\\!P$ asymmetry fit, in the region $\pm 50{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$ around the known ${B}^{\pm}$ meson mass. Fit component |  ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$, long |  ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$, downstream
---|---|---
Signal |  $217$ | $\pm 17$ |  $420$ | $\pm 27$
Backgrounds |  |  |  |
|  Cross-feed (from ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$) | $35.9$ | $\pm 0.7$ | $76$ | $\pm 1$
|  Combinatorial $D$ | $5$ | ${}^{+\hskip 0.56917pt7}_{-\hskip 0.56917pt3}$ | $31$ | ${}^{+\hskip 0.56917pt11}_{-\hskip 0.56917pt9}$
|  Random $Dh$ | $28$ | ${}^{+\hskip 0.56917pt5}_{-\hskip 0.56917pt8}$ | $45$ | ${}^{+\hskip 0.56917pt18}_{-\hskip 0.56917pt19}$
| ${{D}^{*}}\pi$ | $0.36$ | $\pm 0.08$ | $6$ | $\pm 7$
| ${D}\rho$ | $2.2$ | $\pm 0.5$ | $4$ | $\pm 11$
| $B_{s}\rightarrow{D}{{K}^{*}}$ | $0.9$ | $\pm 0.2$ | $4$ | $\pm 2$
Fit component |  ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$, long |  ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$, downstream
Signal | $2906$ | $\pm 56$ | $5960$ | $\pm 80$
Backgrounds |  |  |  |
|  Cross-feed (from ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$) | $27$ | $\pm 2$ | $53$ | $\pm 3$
|  Combinatorial $D$ | $15$ | ${}^{+\hskip 0.56917pt19}_{-\hskip 0.56917pt10}$ | $99$ | ${}^{+\hskip 0.56917pt36}_{-\hskip 0.56917pt27}$
|  Random $Dh$ | $76$ | ${}^{+\hskip 0.56917pt15}_{-\hskip 0.56917pt22}$ | $146$ | ${}^{+\hskip 0.56917pt33}_{-\hskip 0.56917pt41}$
| ${{D}^{*}}\pi$ | $6.6$ | $\pm 0.4$ | $22.0$ | $\pm 0.7$

Figures 2–5 show the
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ candidate Dalitz plot distributions and
their projections, with the results of the fit superimposed. The resulting
measured values of $(x_{\pm},y_{\pm})$ are

$\displaystyle x_{-}$ $\displaystyle=+0.027\pm 0.044,$ $\displaystyle y_{-}$
$\displaystyle=+0.013\pm 0.048,$ $\displaystyle x_{+}$
$\displaystyle=-0.084\pm 0.045,$ $\displaystyle y_{+}$
$\displaystyle=-0.032\pm 0.048,$

where the uncertainties are statistical only. The corresponding likelihood
contours are shown in Fig. 6.

Figure 2: Dalitz plot and its projections, with fit result superimposed, for
$B^{-}\rightarrow D\pi^{-}$ candidates; $m_{\pm}^{2}\equiv
m_{{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{\pm}}}^{2}$ and $m_{0}^{2}\equiv
m_{{{\pi}^{+}}{{\pi}^{-}}}^{2}$. The lower parts of the figures are normalised
residual distributions.

Figure 3: Dalitz plot and its projections, with fit result superimposed, for
$B^{+}\rightarrow D\pi^{+}$ candidates; $m_{\pm}^{2}\equiv
m_{{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{\pm}}}^{2}$ and $m_{0}^{2}\equiv
m_{{{\pi}^{+}}{{\pi}^{-}}}^{2}$. The lower parts of the figures are normalised
residual distributions.

Figure 4: Dalitz plot and its projections, with fit result superimposed, for
$B^{-}\rightarrow DK^{-}$ candidates; $m_{\pm}^{2}\equiv
m_{{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{\pm}}}^{2}$ and $m_{0}^{2}\equiv
m_{{{\pi}^{+}}{{\pi}^{-}}}^{2}$. The lower parts of the figures are normalised
residual distributions.

Figure 5: Dalitz plot and its projections, with fit result superimposed, for
$B^{+}\rightarrow DK^{+}$ candidates; $m_{\pm}^{2}\equiv
m_{{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{\pm}}}^{2}$ and $m_{0}^{2}\equiv
m_{{{\pi}^{+}}{{\pi}^{-}}}^{2}$. The lower parts of the figures are normalised
residual distributions. Figure 6: Likelihood contours at 39.35%, 86.47%,
98.89% and 99.97% confidence level for $(x_{+},y_{+})$ (blue) and
$(x_{-},y_{-})$ (red).

##  5 Systematic uncertainties

Systematic uncertainties on the measured values of $(x_{\pm},y_{\pm})$ arising
from various sources are considered and summarised in Table 2. Unless
otherwise stated, for each source considered the $C\\!P$ asymmetry fit is
repeated with the efficiency parameters and $(x_{\pm},y_{\pm})$ allowed to
vary, as in the nominal fit to data. The resulting differences in the values
of $(x_{\pm},y_{\pm})$ from the nominal results are taken as systematic
uncertainties.

Table 2: Absolute values of systematic uncertainties. The $C\\!P$ asymmetry fit bias is considered as a one-sided uncertainty and is included in the quadrature sum on that side only. Source | $\delta x_{-}(\times 10^{-3})$ | $\delta y_{-}(\times 10^{-3})$ | $\delta x_{+}(\times 10^{-3})$ | $\delta y_{+}(\times 10^{-3})$
---|---|---|---|---
Background yields |  |  |  |
| Cross-feed | $0.21$ | $0.96$ | $0.65$ | $0.26$
| Total combinatorial | $1.1$ | $3.5$ | $1.7$ | $2.7$
|  Combinatorial $D$ | $1.0$ | $4.3$ | $2.7$ | $4.9$
Inclusion of semileptonic background | $3.1$ | $2.8$ | $0.63$ | $3.2$
Charged kaon detection asymmetry | $0.022$ | $0.030$ | $0.0041$ | $0.025$
Amplitudes for backgrounds |  |  |  |
|  Combinatorial $D$ | $3.5$ | $3.4$ | $4.7$ | $6.4$
|  Random $Dh$ | $0.10$ | $0.16$ | $0.066$ | $0.16$
|  $B_{s}$ partially reconstructed | $0.59$ | $0.59$ | $0.15$ | $0.73$
$r_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}$ | $1.8$ | $1.9$ | $1.6$ | $1.1$
Efficiency over the phase space | $5.7$ | $0.35$ | $6.9$ | $0.31$
$C\\!P$ asymmetry fit bias | ${}^{+5.7}_{-0}$ | ${}^{+5.1}_{-0}$ | ${}^{+0}_{-1.3}$ | ${}^{+2.6}_{-0}$
Total experiment or fit related | ${}^{+9.6}_{-7.8}$ | ${}^{+9.0}_{-7.4}$ | ${}^{+9.1}_{-9.2}$ | ${}^{+9.6}_{-9.2}$
Total model related | $1.0$ | $3.0$ | $4.6$ | $8.4$

The fractions of signal and background are estimated with a fit to the
${B}^{\pm}$ candidate invariant mass distributions. To find the systematic
uncertainties in $(x_{\pm},y_{\pm})$ arising from the uncertainties in these
fractions, the shapes and yields of the individual mass PDF contributions are
modified and the fit repeated. The largest changes in $(x_{\pm},y_{\pm})$
arise from modifications to the cross-feed and total combinatorial background
components. The uncertainties are therefore evaluated by repeating the $C\\!P$
asymmetry fit with the cross-feed and total combinatorial background yields
independently varied by their statistical uncertainties.

The yield of combinatorial $D$ background is estimated using wrong-sign
candidates selected from data. The systematic uncertainties arising from these
estimates are found by repeating the $C\\!P$ asymmetry fit to data with the
yields varied by the statistical uncertainties shown in Table 1. Corresponding
variations in the random $Dh$ background yield are made, so that the total
combinatorial background yield, obtained from the ${B}^{\pm}$ invariant mass
fit, is unchanged.

In the ${B}^{\pm}$ invariant mass fit, a component PDF for partially
reconstructed
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}})\mu^{\pm}\nu$ background is not included. The
systematic uncertainty arising from this omission is found by repeating the
$C\\!P$ asymmetry fit to data with a contribution from this background. The
upper limits on the yields and the mass functions are found by applying muon
identification requirements to the bachelor tracks of data candidates, and are
kept constant in the fit.

In the $C\\!P$ asymmetry fit, the background fractions obtained from the
invariant mass fit to ${B}^{\pm}$ candidates are used for both ${{B}^{+}}$ and
${{B}^{-}}$ candidates. This neglects any detection asymmetries for the
charged bachelor tracks. The $C\\!P$ asymmetry fit is repeated with the
central value of the charged kaon asymmetry, $(-1.2\pm 0.2)\%$ [27],
introduced for the signal and background components where the bachelor is
expected to be a kaon.

In the $C\\!P$ asymmetry fit, combinatorial $D$ background candidates are
assumed to be distributed non-resonantly over the phase space of the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay. The $C\\!P$ asymmetry fit is repeated with the $D$ decay model changed
to the sum of a phase-space distribution and a ${{K}^{*\pm}}(892)$ resonance;
the fractions of the two components are fixed by a study of the Dalitz plot
projections of data.

The $D$ decay model included in the $C\\!P$ asymmetry fit for random $Dh$
background candidates is an incoherent sum of the two
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay amplitudes because it is equally likely for a ${{D}^{0}}$ or ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ meson to be present in an
event. The $C\\!P$ asymmetry fit is repeated with the decay model changed to
include the central value of the ${{D}^{0}}-{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ production asymmetry of
$(-1.0\pm 0.3)\%$ [28].

The yield of $B_{s}\rightarrow{D}{{K}^{*}}$ partially reconstructed background
candidates is very low in the signal invariant mass region, but in the $C\\!P$
asymmetry fit the candidates are assumed to be distributed in the same way as
the suppressed component of signal ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ over
the ${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay phase space and could therefore appear in particularly sensitive
regions. To estimate the systematic uncertainty arising from the assumed
distribution, the $C\\!P$ asymmetry fit to data is performed with the $D$
decay model for this background changed to the favoured component of the
signal ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decay model.

In order to allow the candidate detection, reconstruction and selection
efficiency variation across the phase space of the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay to be found from ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ data
candidates, the amplitudes from the suppressed decays
${{{B}^{-}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$ and
${{{B}^{+}}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ are assumed to be negligible. The
systematic uncertainty arising from this assumption is estimated by repeating
the $C\\!P$ asymmetry fit to data with an additional term in the signal
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ and cross-feed
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decay models, representing the
suppressed decay amplitudes. The values of
$r_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}$,
$\delta_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}$ and $\gamma$ are fixed in
the additional term; various $r_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}$ and
$\delta_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}$ values are assumed
($r_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}=0.01$, $0.015$;
$\delta_{{{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}}=0^{\circ}$, $90^{\circ}$,
$180^{\circ}$, $270^{\circ}$, $315^{\circ}$), but in all cases $\gamma$ is set
to $70^{\circ}$.

The efficiency variation across the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay phase space is parametrised in the $C\\!P$ asymmetry fit by a second-
order polynomial function in the variables $m^{2}_{+}$ and $m^{2}_{-}$. To
estimate the uncertainty arising from this, the $C\\!P$ asymmetry fit to data
is repeated with the efficiency parametrisation fixed and variations of the
polynomial coefficients made. A fit with a third-order polynomial function is
also performed, with the efficiency parameters and $(x_{\pm},y_{\pm})$ allowed
to vary. The changes in the values of $(x_{\pm},y_{\pm})$, compared to the
nominal results, are taken as the systematic uncertainties arising from the
efficiency parametrisation.

The $C\\!P$ asymmetry fit is verified using $1000$ data-sized simulated
pseudo-experiments. In each experiment the number and distribution of
candidates is generated according to the fit result from data. The obtained
values of $(x_{\pm},y_{\pm})$ show a small bias when compared to the values
used for the simulation; these biases are included as systematic
uncertainties.

To estimate the systematic uncertainty arising from the choice of amplitude
model description of the ${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ decay, $C\\!P$ asymmetry fits with alternative
model descriptions are performed on large samples of simulated decays. For
each alternative model, one element (for example, a resonance parameter) of
the nominal model is altered. One million
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ and one million
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays are simulated with the model
used for the nominal $C\\!P$ asymmetry fit, and with the Cartesian parameters
fixed to the fit result. For the nominal model and each alternative model, a
$C\\!P$ asymmetry fit to the ${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ sample
is performed with the coefficients of each resonance of the model allowed to
vary. Values for the Cartesian parameters $(x_{\pm},y_{\pm})$ are then
obtained from a $C\\!P$ asymmetry fit to the
${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ sample, with the resonance coefficients
fixed from the results of the fit to the
${{B}^{\pm}}\rightarrow{D}{{\pi}^{\pm}}$ sample. The signed differences in the
values of $(x_{\pm},y_{\pm})$ from the nominal results are taken as the
systematic uncertainties, with the relative signs between contributions
indicating full correlation or anti-correlation.

In the alternative models considered, the following changes, labelled (a)-(u),
have been applied, resulting in the uncertainties summarised in Table 3:

  * $-$

$\pi\pi$ S-wave: The $F$-vector model is changed to use two other solutions of
the $K$-matrix (from a total of three) determined from fits to scattering data
[29] (a), (b). The slowly varying part of the non-resonant term of the
$P$-vector is removed (c).

  * $-$

$K\pi$ S-wave: The generalised LASS parametrisation, used to describe the
${{K}^{*}}_{0}(1430)$ resonance, is replaced by a relativistic Breit-Wigner
propagator with parameters taken from Ref. [30] (d).

  * $-$

$\pi\pi$ P-wave: The Gounaris-Sakurai propagator is replaced by a relativistic
Breit-Wigner propagator (e).

  * $-$

$K\pi$ P-wave: The mass and width of the ${{K}^{*}}(1680)$ resonance are
varied by their uncertainties from Ref. [31] (f)$-$(i).

  * $-$

$\pi\pi$ D-wave: The mass and width of the $f_{2}(1270)$ resonance are varied
by their uncertainties from Ref. [22] (j)$-$(m).

  * $-$

$K\pi$ D-wave: The mass and width of the ${{K}^{*}}_{2}(1430)$ resonance are
varied by their uncertainties from Ref. [22] (n)$-$(q).

  * $-$

The radius of the Blatt-Weisskopf centrifugal barrier factors, $r_{{\rm BW}}$,
is changed from $1.5\,\rm\,GeV^{-1}$ to $0.0\,\rm\,GeV^{-1}$ (r) and
$3.0\,\rm\,GeV^{-1}$ (s).

  * $-$

Two further resonances, ${{K}^{*}}(1410)$ and $\rho(1450)$, parametrised with
relativistic Breit-Wigner propagators, are included in the model (t).

  * $-$

The Zemach formalism used for the angular distribution of the decay products
is replaced by the helicity formalism (u).

The total covariance matrix is determined to be

$V_{\mathrm{model}}=\bordermatrix{&x_{-}&y_{-}&x_{+}&y_{+}\cr&1.12&2.80&-0.95&-5.40\cr&2.80&8.89&-1.21&-16.87\cr&-0.95&-1.21&21.59&5.97\cr&-5.40&-16.87&5.97&69.87\cr}\,\times\,10^{-6}$
(3)

resulting in total systematic uncertainties arising from the choice of
amplitude model of

$\displaystyle\delta x_{-}$ $\displaystyle=1.0\times 10^{-3},$
$\displaystyle\delta y_{-}$ $\displaystyle=3.0\times 10^{-3},$
$\displaystyle\delta x_{+}$ $\displaystyle=4.6\times 10^{-3},$
$\displaystyle\delta y_{+}$ $\displaystyle=8.4\times 10^{-3}.$

Table 3: Model related systematic uncertainties for each alternative model. The relative signs indicate full correlation or anti-correlation. | Description | $\delta x_{-}(\times 10^{-3})$ | $\delta y_{-}(\times 10^{-3})$ | $\delta x_{+}(\times 10^{-3})$ | $\delta y_{+}(\times 10^{-3})$
---|---|---|---|---|---
(a) |  $K$-matrix 1st solution | $-0.1$ | $0.04$ | $0.3$ | $-2$
(b) |  $K$-matrix 2nd solution | $-0.09$ | $-0.3$ | $0.1$ | $-0.5$
(c) | Remove slowly varying | $-0.1$ | $-0.3$ | $0.1$ | $-0.8$
|  part in $P$-vector |  |  |  |
(d) | Generalised LASS | $-0.7$ | $-2$ | $3$ | $7$
$\rightarrow$ relativistic Breit-Wigner |  |  |  |
(e) | Gounaris-Sakurai | $0.08$ | $-0.8$ | $0.1$ | $0.8$
$\rightarrow$ relativistic Breit-Wigner |  |  |  |
(f) | ${{K}^{*}}(1680)$ | $m+\delta m$ | $-0.06$ | $-0.6$ | $0.2$ | $0.3$
(g) | $m-\delta m$ | $-0.1$ | $-0.2$ | $-0.1$ | $-1$
(h) | $\Gamma+\delta\Gamma$ | $-0.06$ | $-0.4$ | $-0.05$ | $-0.4$
(i) | $\Gamma-\delta\Gamma$ | $-0.2$ | $-0.3$ | $0.3$ | $-0.5$
(j) | $f_{2}(1270)$ | $m+\delta m$ | $-0.1$ | $-0.3$ | $0.1$ | $-0.5$
(k) | $m-\delta m$ | $-0.1$ | $-0.4$ | $0.09$ | $-0.5$
(l) | $\Gamma+\delta\Gamma$ | $-0.1$ | $-0.3$ | $0.08$ | $-0.5$
(m) | $\Gamma-\delta\Gamma$ | $-0.1$ | $-0.4$ | $0.1$ | $-0.5$
(n) | ${{K}^{*}}_{2}(1430)$ | $m+\delta m$ | $-0.08$ | $-0.4$ | $0.08$ | $-0.4$
(o) | $m-\delta m$ | $-0.1$ | $-0.3$ | $0.1$ | $-0.5$
(p) | $\Gamma+\delta\Gamma$ | $-0.1$ | $-0.4$ | $0.07$ | $-0.4$
(q) | $\Gamma-\delta\Gamma$ | $-0.1$ | $-0.3$ | $0.1$ | $-0.5$
(r) | $r_{{\rm BW}}=0.0\rm\,GeV^{-1}$ | $-0.2$ | $-0.4$ | $-0.1$ | $-0.3$
(s) | $r_{{\rm BW}}=3.0\rm\,GeV^{-1}$ | $-0.3$ | $-0.3$ | $1$ | $-0.4$
(t) |  Add ${{K}^{*}}(1410)$ and $\rho(1450)$ | $-0.1$ | $-0.3$ | $0.02$ | $-0.7$
(u) | Helicity formalism | $-0.5$ | $-2$ | $-3$ | $4$

Table 2 summarises the systematic uncertainties arising from all sources.
Except for the uncertainty due to the fit bias, the absolute values of the
uncertainties are added in quadrature (assuming no correlation) to obtain the
total experiment or fit related uncertainties. The $C\\!P$ asymmetry fit bias
is considered as a one-sided uncertainty and is included in the quadrature sum
on that side only. The model related systematic uncertainty is also shown in
the table, for comparison.

##  6 Constraints on $\boldsymbol{\gamma}$, r${}_{\textbf{{B}}}$ and
$\boldsymbol{\delta}_{\textbf{{B}}}$

The results for the $C\\!P$ violation observables $(x_{\pm},y_{\pm})$ are used
to place constraints on the values of $\gamma$, $r_{{B}}$ and $\delta_{{B}}$,
adopting the procedure described in Refs. [10, 9].

There is a two-fold ambiguity in the solution for $\gamma$, $r_{{B}}$ and
$\delta_{{B}}$; choosing the solution that satisfies $(0<\gamma<180)^{\circ}$
leads to the results

$\displaystyle\gamma$ $\displaystyle=(84^{+49}_{-42})^{\circ},$ $\displaystyle
r_{{B}}$ $\displaystyle=0.06\pm 0.04,$ $\displaystyle\delta_{{B}}$
$\displaystyle=(115^{+41}_{-51})^{\circ},$

where the uncertainties include statistical, experimental systematic and model
related systematic contributions. Figure 7 shows the contours of p-value
projected onto the $(\gamma,\delta_{{B}})$ and $(\gamma,r_{{B}})$ planes.

Figure 7: Projections of the p-value regions onto the $(\gamma,\delta_{{B}})$
and $(\gamma,r_{{B}})$ planes with all sources of uncertainty taken into
account.

##  7 Effect of neutral D meson mixing

Assuming uniform lifetime acceptance, the measurements of the Cartesian
parameters documented in this paper are corrected for the effects of $D$
mixing as described in Ref. [32],

$\displaystyle x_{\pm}^{\text{corr}}$
$\displaystyle=x_{\pm}+\cfrac{y_{\text{mix}}}{2},$ $\displaystyle
y_{\pm}^{\text{corr}}$ $\displaystyle=y_{\pm}+\cfrac{x_{\text{mix}}}{2},$

where $x_{\text{mix}}$ and $y_{\text{mix}}$ are the parameters of neutral $D$
meson mixing.

Since $C\\!P$ violation in the charm sector has been neglected in the
analysis, the world average values of the mixing parameters without $C\\!P$
violation ($x_{\text{mix}}=(0.53^{+0.16}_{-0.17})\times 10^{-2}$,
$y_{\text{mix}}=(0.67\pm 0.09)\times 10^{-2}$) [33] are taken for correction,
yielding the values

$\displaystyle x_{-}^{\text{corr}}$ $\displaystyle=+0.030\pm
0.044~{}^{+0.010}_{-0.008}\pm 0.001\pm 0.00045,$ $\displaystyle
y_{-}^{\text{corr}}$ $\displaystyle=+0.016\pm 0.048~{}^{+0.009}_{-0.007}\pm
0.003\pm 0.00085,$ $\displaystyle x_{+}^{\text{corr}}$
$\displaystyle=-0.081\pm 0.045\pm 0.009\pm 0.005\pm 0.00045,$ $\displaystyle
y_{+}^{\text{corr}}$ $\displaystyle=-0.029\pm 0.048~{}^{+0.010}_{-0.009}\pm
0.008\pm 0.00085,$

where the first uncertainty is statistical, the second systematic, the third
arises from the $D$ decay amplitude model and the fourth is the uncertainty
associated with the values of the mixing parameters. The change in the value
of $\gamma$ due to this correction is less than $1^{\circ}$.

##  8 Conclusions

Candidate
${{B}^{\pm}}\rightarrow{D}(\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}){{K}^{\pm}}$ decays are used to perform an amplitude
analysis incorporating a model description of the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay. The data used correspond to an integrated luminosity of
$1\mbox{\,fb}^{-1}$, recorded by LHCb at a centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$ in $2011$.

The resulting values of the $C\\!P$ violation observables
$x_{\pm}=r_{{B}}\cos{(\delta_{{B}}\pm\gamma)}$ and
$y_{\pm}=r_{{B}}\sin{(\delta_{{B}}\pm\gamma)}$ are

$\displaystyle x_{-}$ $\displaystyle=+0.027\pm 0.044~{}^{+0.010}_{-0.008}\pm
0.001,$ $\displaystyle y_{-}$ $\displaystyle=+0.013\pm
0.048~{}^{+0.009}_{-0.007}\pm 0.003,$ $\displaystyle x_{+}$
$\displaystyle=-0.084\pm 0.045\pm 0.009\pm 0.005,$ $\displaystyle y_{+}$
$\displaystyle=-0.032\pm 0.048~{}^{+0.010}_{-0.009}\pm 0.008,$

where in each case the first uncertainty is statistical, the second systematic
and the third is due to the choice of amplitude model used to describe the
${D}\rightarrow{{K}^{0}_{\rm\scriptscriptstyle S}}{{\pi}^{+}}{{\pi}^{-}}$
decay. The results place constraints on the magnitude of the ratio of the
interfering ${B}^{\pm}$ decay amplitudes, the strong phase difference between
them and the CKM angle $\gamma$, giving the values $r_{{B}}=0.06\pm 0.04$,
$\delta_{{B}}=(115^{+41}_{-51})^{\circ}$ and
$\gamma=(84^{+49}_{-42})^{\circ}$. Neutral $D$ meson mixing has a negligible
effect on the parameters $r_{{B}}$, $\delta_{{B}}$ and $\gamma$.

These results are consistent with, complementary to, and cannot be combined
with, those obtained by the LHCb model-independent analysis of the same data
set [9]. The results are also consistent with world average values [34, 35].

## 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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Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
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Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, A. Bay39,
L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
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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.
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Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, F. Zhang3, 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
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
uPolitecnico di Milano, Milano, Italy