EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-127 LHCb-PAPER-2015-020 May 26, 2015

Study of ${{{B}^{-}}}\rightarrow D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow D{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays and
determination of the CKM angle $\gamma$

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

We report a study of the suppressed ${{{B}^{-}}}\rightarrow
D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and favored ${{{B}^{-}}}\rightarrow
D{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays, where the neutral $D$ meson is
detected through its decays to the $K^{\mp}\pi^{\pm}$ and $C\\!P$-even
${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$ final states. The
measurement is carried out using a proton-proton collision data sample
collected by the LHCb experiment, corresponding to an integrated luminosity of
3.0$\mbox{\,fb}^{-1}$. We observe the first significant signals in the
$C\\!P$-even final states of the $D$ meson for both the suppressed
${{{B}^{-}}}\rightarrow D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and favored
${{{B}^{-}}}\rightarrow D{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ modes, as well as
in the doubly Cabibbo-suppressed $D\rightarrow{{K}^{+}}{{\pi}^{-}}$ final
state of the ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decay. Evidence for the ADS suppressed decay ${{{B}^{-}}}\rightarrow
D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, with $D\rightarrow{{K}^{+}}{{\pi}^{-}}$, is
also presented. From the observed yields in the ${{{B}^{-}}}\rightarrow
D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, ${{{B}^{-}}}\rightarrow
D{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and their charge conjugate decay modes,
the most probably value of the weak phase $\gamma$ corresponds to
$\gamma=(74^{+20}_{-19})^{\rm o}$. This is one of the most precise single-
measurement determinations of $\gamma$ to date.

Published in Phys. Rev. D

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

##  1 Introduction

The study of beauty and charm hadron decays provides a powerful probe to
search for physics beyond the Standard Model that is complementary to direct
searches for new, high-mass particles. In the Standard Model, the flavor-
changing charged currents of quarks are described by the 3$\times$3 unitary
complex-valued Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix [1, 2], whose
elements, $V_{\ij}$ ($i=u,c,t$ and $j=d,s,b$), quantify the relative
$i\leftrightarrow j$ coupling strength. Its nine matrix elements can be
expressed in terms of four independent parameters, which need to be
experimentally determined.

In general, decay rates that involve the $i\leftrightarrow j$ quark transition
are sensitive to the magnitudes of the CKM matrix elements, $|V_{ij}|$. The
(weak) phases between different CKM matrix elements can be probed by studying
the interference between two (or more) decay amplitudes. Particle and
antiparticle amplitudes are related by the $C\\!P$ operator, where $C$
signifies charge conjugation, and $P$ refers to the parity operator. Under the
$C\\!P$ operation, weak phases flip sign, leading to different decay rates for
particles and antiparticles, if the weak and ($C\\!P$-invariant) strong phases
differ between the contributing amplitudes. Precision measurements of the
magnitudes and phases of the CKM elements provide constraints on many possible
scenarios for physics beyond the Standard Model.

One of the least well-measured phases is
$\gamma\equiv\text{arg}[-(V_{ud}V^{*}_{ub})/(V_{cd}V^{*}_{cb})]$, which can be
probed by studying the interference between $b\rightarrow u$ and $b\rightarrow
c$ transitions. The most promising method to determine $\gamma$ is to study
the interference between ${{{B}^{-}}}\rightarrow{{D}^{0}}{{K}^{-}}$ and
${{{B}^{-}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{K}^{-}}$ decays, when states
accessible to both the ${{D}^{0}}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mesons are selected. These
modes are particularly attractive for the determination of $\gamma$ because
their amplitudes are dominated by only a pair of tree-level processes, leading
to a small theoretical uncertainty [3]. Hereafter, we use $D$ without a charge
designation when the charm meson can be either a ${{D}^{0}}$ or ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$. A number of methods,
depending on the $D$ decay mode, have been discussed in the literature, and
are often grouped into three categories: (i) $C\\!P$ eigenstates, such as
$D\rightarrow{{K}^{+}}{{K}^{-}}$ and $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
decays [4, 5] (GLW); (ii) flavor-specific final states, such as the Cabibbo-
favored (CF) and doubly Cabibbo suppressed (DCS) $D\rightarrow
K^{\pm}\pi^{\mp}$ decays [6, 7] (ADS); and (iii) multi-body self-conjugate
final states, such as $D\rightarrow{{K}^{0}_{\rm\scriptscriptstyle
S}}{{\pi}^{+}}{{\pi}^{-}}$ [8] (GGSZ)111The letters in the brackets are
commonly used to refer to these general approaches, after the original
authors..

Beyond this simplest set of modes, these techniques are also applicable to
modes with vector mesons, such as ${{{B}^{-}}}\rightarrow{D}^{*}{{K}^{-}}$,
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow D{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*0}}$ [9], and
${{B}^{0}_{s}}\rightarrow D\phi$ [5], as well as $b$-baryon decays, e.g.
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow D{\mathchar 28931\relax}$ [10,
11, 12] decays. It has also been suggested that other multi-body final states
of the recoiling strange quark system could be useful [13], due to the larger
branching fractions to these final states, and potentially a larger
interference contribution.

The current experimental measurements, averaged over several decays modes, are
$\gamma=(73^{+9}_{-10})^{\rm o}$ by the LHCb collaboration [14, *LHCb-
CONF-2014-004], $\gamma=(69^{+17}_{-16})^{\rm o}$ by the BaBar collaboration
[16], and $\gamma=(68^{+15}_{-14})^{\rm o}$ by the Belle collaboration [17].
The overall precision on $\gamma$ from a global fit to direct measurements of
$\gamma$ is about $7^{\rm o}$ [18]. To improve the overall precision on
$\gamma$, it is important to study a wide range of final states.

In this article, we present the first ADS and GLW analyses of the decay
${{{B}^{-}}}\rightarrow DX_{s}^{-}$, where the $D$ meson is observed through
its decay to $K^{\pm}\pi^{\mp}$, ${{K}^{+}}{{K}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}$ final states and
$X_{s}^{-}\equiv{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$. When specific charges are
indicated in a decay, charge conjugation is implicitly included, except in the
definition of asymmetries discussed below. The measurements use proton-proton
($pp$) collision data collected by the LHCb experiment, corresponding to an
integrated luminosity of 3.0$\mbox{\,fb}^{-1}$, of which 1.0$\mbox{\,fb}^{-1}$
was recorded at a center-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$
and 2.0$\mbox{\,fb}^{-1}$ at 8$\mathrm{\,Te\kern-1.00006ptV}$.

##  2 Formalism

The formalism that was developed to describe the ${{{B}^{-}}}\rightarrow
D{{K}^{-}}$ modes can be applied in the ${{{B}^{-}}}\rightarrow DX_{s}^{-}$
case with only minor modifications [13]. The decay rates in the $C\\!P$ final
states can be expressed as

$\displaystyle\Gamma({{{B}^{-}}}\rightarrow[h^{-}h^{+}]_{D}X_{s}^{-})$
$\displaystyle\propto 1+r_{B}^{2}+2\kappa r_{B}\cos(\delta_{B}-\gamma),$ (1)
$\displaystyle\Gamma({{{B}^{+}}}\rightarrow[h^{-}h^{+}]_{D}X_{s}^{+})$
$\displaystyle\propto 1+r_{B}^{2}+2\kappa r_{B}\cos(\delta_{B}+\gamma).$ (2)

Here, $h^{\pm}=\pi^{\pm}$ or $K^{\pm}$, and $[h^{-}h^{+}]_{D}$ indicates that
the state in brackets is produced in the decay of the neutral $D$ meson. The
quantities $r_{B}\equiv|A({{{B}^{-}}}\rightarrow[h^{-}h^{+}]_{{{\kern
1.39998pt\overline{\kern-1.39998ptD}{}}{}^{0}}}X_{s}^{-})/A({{{B}^{-}}}\rightarrow[h^{-}h^{+}]_{{{D}^{0}}}X_{s}^{-})|$
and $\delta_{B}$ are the amplitude ratio and strong phase difference between
${{{B}^{-}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}X_{s}^{-}$ and
${{{B}^{-}}}\rightarrow{{D}^{0}}X_{s}^{-}$ contributions, averaged over the
$DX_{s}^{-}$ phase space. The parameter $\kappa$ is a coherence factor that
accounts for a dilution of the interference due to the variation of the strong
phase across the phase space; its value is bounded between 0 and 1. In
principle, $\kappa$ can be obtained in a model-dependent way by a full
amplitude analysis of this decay. Here, we consider it as a free parameter to
be determined in the global fit for $\gamma$. The strong parameters, $r_{B}$,
$\delta_{B}$ and $\kappa$ are specific to this decay, and differ from those
obtained from other $B\rightarrow DK$ modes.

The decay rates for the $D\rightarrow K^{\pm}\pi^{\mp}$ final states can be
written as

$\displaystyle\Gamma({{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{s}^{-})$
$\displaystyle\propto r_{B}^{2}+r_{D}^{2}+2\kappa
r_{B}r_{D}\cos(\delta_{B}+\delta_{D}-\gamma),$ (3)
$\displaystyle\Gamma({{{B}^{+}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{+})$
$\displaystyle\propto r_{B}^{2}+r_{D}^{2}+2\kappa
r_{B}r_{D}\cos(\delta_{B}+\delta_{D}+\gamma),$ (4)
$\displaystyle\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{-})$
$\displaystyle\propto 1+(r_{B}r_{D})^{2}+2\kappa
r_{B}r_{D}\cos(\delta_{B}-\delta_{D}-\gamma),$ (5)
$\displaystyle\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{s}^{+})$
$\displaystyle\propto 1+(r_{B}r_{D})^{2}+2\kappa
r_{B}r_{D}\cos(\delta_{B}-\delta_{D}+\gamma).$ (6)

Here, additional parameters $r_{D}$ and $\delta_{D}$ enter, which quantify the
ratio of the DCS to CF amplitude,
$A({{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}})/A({{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}})=r_{D}e^{i\delta_{D}}$.
Values of $r_{D}$ and $\delta_{D}$ are taken from independent measurements
[19, 20].

The determination of the $C\\!P$ observables in the ${{{B}^{-}}}\rightarrow
DX_{s}^{-}$ decay uses the favored ${{{B}^{-}}}\rightarrow
D{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decay for normalization, denoted here as
${{{B}^{-}}}\rightarrow DX_{d}^{-}$. For brevity, we will use $X^{-}$ to refer
to either $X_{d}^{-}$ or $X_{s}^{-}$. In addition, $D\rightarrow K\pi$ is used
when both charge combinations are considered.

The $C\\!P$ observables of interest for the GLW analysis are the charge-
averaged yield ratios

$\displaystyle R^{h^{+}h^{-}}_{{C\\!P}+}$ $\displaystyle\equiv
2\frac{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}X_{s}^{-})+\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}X_{s}^{+})}{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{-})+\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{s}^{+})}$
$\displaystyle=1+r_{B}^{2}+2\kappa r_{B}\cos\delta_{B}\cos\gamma.$ (7)

Because of the different $D$ final states in Eq. 7, systematic uncertainty due
to the precision of the $D$ branching fractions and the different selections
is incurred. Following Ref. [13], we neglect $C\\!P$ violation in the
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ and the favored $D$ final state of
${{{B}^{-}}}\rightarrow DX_{s}^{-}$ decays, and approximate
$R^{h^{+}h^{-}}_{{C\\!P}+}$ by the following double ratio

$\displaystyle R_{{C\\!P}+}\simeq\frac{R^{h^{+}h^{-}}_{s/d}}{R^{K\pi}_{s/d}},$
(8)

where

$\displaystyle R^{h^{+}h^{-}}_{s/d}$
$\displaystyle\equiv\frac{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}X_{s}^{-})+\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}X_{s}^{+})}{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}X_{d}^{-})+\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}X_{d}^{+})},$
(9) $\displaystyle R^{K\pi}_{s/d}$
$\displaystyle\equiv\frac{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{-}+\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{s}^{+})}{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{d}^{-}+\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{d}^{+})}.$
(10)

This double ratio has the benefit that almost all systematic uncertainties
cancel to first order. The neglected $C\\!P$-violating contribution of
magnitude $\kappa
r_{B}|{V_{{u}{s}}}{V_{{c}{d}}}/{V_{{u}{d}}}{V_{{c}{s}}}|\lesssim 0.01$ is
included as a source of systematic uncertainty.

We also make use of the charge asymmetries

$\displaystyle{\mathcal{A}}^{f}_{X^{\pm}}\equiv\frac{\Gamma({{{B}^{-}}}\rightarrow
f_{D}X^{-})-\Gamma({{{B}^{+}}}\rightarrow\bar{f}_{D}X^{+})}{\Gamma({{{B}^{-}}}\rightarrow
f_{D}X^{-})+\Gamma({{{B}^{+}}}\rightarrow\bar{f}_{D}X^{+})}=2\kappa
r_{B}\sin\delta_{B}\sin\gamma/R_{{C\\!P}+},$ (11)

where $f$ refers to either ${{K}^{+}}{{K}^{-}}$, ${{\pi}^{+}}{{\pi}^{-}}$ or
the CF ${{K}^{-}}{{\pi}^{+}}$ final state in the $D$ meson decay. For
simplicity, small contributions from direct $C\\!P$ violation in
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ and $D\rightarrow{{K}^{+}}{{K}^{-}}$ are
not included here, but are accounted for in the fit for $\gamma$ [14, *LHCb-
CONF-2014-004].

For the ADS modes, we measure the relative widths of the DCS to CF decays,
separated by charge, as

$\displaystyle
R^{X^{\pm}}=\frac{\Gamma(B^{\pm}\rightarrow[K^{\mp}\pi^{\pm}]_{D}X^{\pm})}{\Gamma(B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}X^{\pm})}=\frac{r_{B}^{2}+r_{D}^{2}+2\kappa
r_{B}r_{D}\cos(\delta_{B}+\delta_{D}\pm\gamma)}{1+r_{B}^{2}r_{D}^{2}+2\kappa
r_{B}r_{D}\cos(\delta_{B}-\delta_{D}\pm\gamma)}.$ (12)

The nearly identical final states in these ratios lead to a cancellation of
the most significant sources of systematic uncertainty. Corrections to
$R^{X^{\pm}}$ for ${{D}^{0}}-{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mixing [21] are omitted for
clarity, but are included in the fit for $\gamma$ [14, *LHCb-CONF-2014-004].

All of the above equations, except for Eqs. 8–10, can be applied to either
$B^{\pm}\rightarrow DX_{s}^{\pm}$ or $B^{\pm}\rightarrow DX_{d}^{\pm}$ decays.
The values of $r_{B}$, $\delta_{B}$ and $\kappa$ differ between the favored
and suppressed decays; however $\gamma$ is common to both. Most of the
sensitivity is expected to come from the $B^{\pm}\rightarrow DX_{s}^{\pm}$
decays, since $A({{{B}^{-}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}X_{d}^{-})/A({{{B}^{-}}}\rightarrow{{D}^{0}}X_{d}^{-})$
is ${\mathcal{O}}(\lambda^{2})$, as compared to ${\mathcal{O}}(1)$ for
$A({{{B}^{-}}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}X_{s}^{-})/A({{{B}^{-}}}\rightarrow{{D}^{0}}X_{s}^{-})$,
where $\lambda=0.2253\pm 0.0014$ [22] is the sine of the Cabibbo angle. Taken
together, the observables that contain the most significant information on
$\gamma$ are $R_{{C\\!P}+}$, ${\mathcal{A}}^{h^{+}h^{-}}_{X_{s}}$ and
$R^{X_{s}^{\pm}}$. Measurements of these four quantities constrain $r_{B}$,
$\delta_{B}$, $\kappa$ and $\gamma$.

The product branching fraction for ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ decays,
with $D\rightarrow h^{+}h^{-}$, is at the level of about $10^{-6}$. The small
branching fractions, combined with a total selection efficiency that is of
order 0.1%, makes the detection and study of these modes challenging. The
corresponding ADS DCS decay mode is expected to have a yield of at least 10
times less than the $C\\!P$ modes, and is very sensitive to the values of
$r_{B}$, $\delta_{B}$, $\kappa$, and $\gamma$ (see Eqs. 3 and 4). For this
reason, the signal region of the ADS suppressed decays (both
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ and ${{{B}^{-}}}\rightarrow DX_{s}^{-}$)
was not examined until all selection requirements were determined.

##  3 The LHCb detector and simulation

The LHCb detector [23] 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 [24] placed downstream of the
magnet. The combined tracking system provides a momentum measurement with a
relative uncertainty that varies from 0.5% at low momentum, $p$, to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and an impact parameter measurement
with a resolution of about 20${\,\upmu\rm m}$ [25] for charged particles with
large transverse momentum, $p_{\rm T}$. The polarity of the dipole magnet is
reversed periodically throughout data-taking to reduce asymmetries in the
detection of charged particles. 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]. Details on the performance of the LHCb detector can be found in
Ref. [28].

The trigger [29] 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 $p_{\rm T}$ sum of the tracks and a
significant displacement from all primary $pp$ interaction vertices (PVs). At
least one particle should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any PV greater than 16, where $\chi^{2}_{\rm IP}$ is defined as the
difference in $\chi^{2}$ of a given PV reconstructed with and without the
considered particle. A multivariate algorithm [30] is used for the
identification of secondary vertices consistent with the decay of a
$b$-hadron.

Proton-proton collisions are simulated using Pythia [31, *Sjostrand:2007gs]
with a specific LHCb configuration [33]. Decays of hadronic particles are
described by EvtGen [34], in which final-state radiation is generated using
Photos [35]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [36,
*Agostinelli:2002hh] as described in Ref. [38]. In modeling the
${{{B}^{-}}}\rightarrow DX^{-}$ decays, we include several resonant and
nonresonant contributions to emulate the $X_{s}^{-}$ and $X_{d}^{-}$ systems,
as well as contributions from orbitally excited $D$ states, e.g
$D_{1}(2420)^{0}\rightarrow{{D}^{0}}{{\pi}^{+}}{{\pi}^{-}}$. The contributions
are set based on known branching fractions, or tuned to reproduce resonant
substructures seen in the data.

##  4 Candidate selection

Candidate ${{{B}^{-}}}$ decays are reconstructed by combining a $D\rightarrow
K\pi$, $D\rightarrow{{K}^{+}}{{K}^{-}}$ or
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ candidate with an $X^{-}$ candidate. A
kinematic fit [39] is performed, where several constraints are imposed: the
reconstructed positions of the $X^{-}$ and ${{{B}^{-}}}$ decay vertices are
required to be compatible with each other, the $D$ candidate must point back
to the ${{{B}^{-}}}$ decay vertex, the ${{{B}^{-}}}$ candidate must have a
direction consistent with originating from a PV in the event, and the
invariant mass of the $D$ candidate must be consistent with the known
${{D}^{0}}$ mass [22]. The production point of each ${{{B}^{-}}}$ candidate is
designated to be the PV for which the $\chi^{2}_{\rm IP}$ is smallest.

Candidate $D$ mesons are required to have invariant mass within $3\sigma_{D}$
($2.5\sigma_{D}$ for $D\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays) of the known
value, where the mass resolution, $\sigma_{D}$, varies from
7.0${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$D\rightarrow{{K}^{+}}{{K}^{-}}$ to
10.2${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays. Unlike the $D$ mesons, the
invariant mass of the $X^{-}$ system covers a broad range from about
$0.9-3.3$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Candidates are required
to have an invariant mass,
$M(X^{-})<2.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. For the $X_{s}^{-}$
system, we also require the ${{K}^{-}}{{\pi}^{+}}$ invariant mass to be within
100${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known ${{K}^{*0}}$ mass.
The latter two requirements not only improve the signal-to-background ratio,
but should also increase the coherence factor $\kappa$ in the final state.

To improve the signal-to-background ratio further, we select candidates based
on particle identification (PID) information, and on the output of a boosted
decision tree (BDT) [40, 41] classifier. The latter discriminates signal from
combinatorial background based on information derived primarily from the
tracking system. For the BDT, signal efficiencies are obtained from large
samples of simulated signal decays. Particle identification efficiencies are
obtained from a large ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ calibration
data sample [26], reweighted in $p_{\rm T}$, $\eta$ and number of tracks in
the event to match the distributions in data. The effect of the BDT and PID
selection requirements on the background is assessed using sidebands well away
from the ${{{B}^{-}}}$ peak region. In the optimization, a wide range of
selection requirements on the PID and BDT outputs are scanned, and we choose
the value that optimizes the expected statistical precision of the
${{{B}^{-}}}\rightarrow DX_{s}^{-}$ signal yield. Expected signal yields are
evaluated based on known or estimated branching fractions and efficiencies
obtained from simulation (for the BDT) or
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}},~{}{{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$
calibration data (for the PID). Due to the smaller expected yields in the ADS
modes, separate optimizations are performed for the GLW and the ADS analyses.
Using simulated decays, we find that the relative efficiencies for
${{{B}^{-}}}\rightarrow DX_{s}^{-}$ and ${{{B}^{-}}}\rightarrow DX_{d}^{-}$
decays across the phase space are compatible for the GLW and ADS selections.
Due to the uniformity of the selections, and the fact that the observables are
either double ratios, e.g. $R_{{C\\!P}+}$, or ratios involving almost
identical final states, the systematic uncertainty on the relative
efficiencies is negligible compared to the statistical uncertainty.

Several other mode-specific requirements are imposed to suppress background
from other $b$-hadron decays. First, we explicitly veto contributions from
${{{B}^{-}}}\rightarrow{{D}^{0}}{{D}^{-}_{s}}$, with either
${{D}^{-}_{s}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ or
${{D}^{-}_{s}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, by rejecting
candidates in which the $X^{-}$ system has invariant mass within
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known ${{D}^{-}_{s}}$
mass. Contamination from other final states that include a charmed particle
are also sought by forming all two-, three- and four-body combinations (except
the $D\rightarrow h^{+}h^{\prime-}$ signal decay), and checking for peaks at
any of the known charmed particle masses. Contributions from
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}},~{}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$,
${{D}^{+}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{+}}$ and
${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ decays are seen, and
$\pm$15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass vetoes are applied
around the known charm particle masses. In addition, ${{D}^{*+}}$
contributions are removed by requiring the invariant mass difference,
$M[({{K}^{-}}{{\pi}^{+}})_{D}{{\pi}^{+}}]-M[({{K}^{-}}{{\pi}^{+}})_{D}]>148.5$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.
This removes both partially reconstructed $B\rightarrow{{D}^{*+}}X$ final
states and fully reconstructed states, such as ${{{B}^{-}}}\rightarrow
D_{1}(2420)^{0}h^{-}$, $D_{1}(2420)^{0}\rightarrow{{D}^{*+}}{{\pi}^{-}}$,
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ signal decays. The latter, while
forming a good signal candidate, are flavor-specific, and therefore would
reduce the coherence of the final state. Those
${D}^{**0}\rightarrow{{D}^{0}}{{\pi}^{+}}{{\pi}^{-}}$ contributions that do
not have a ${{D}^{*+}}$ intermediate state are kept, since they are not
flavor-specific.

Another potentially large source of background is from five-body charmless $B$
decays. Unfortunately, their branching fractions are generally unknown, but
they are likely to be sizable compared to those of the ${{{B}^{-}}}\rightarrow
DX_{s}^{-}$ signal decays. Moreover, these backgrounds could have large
$C\\!P$ asymmetries, as seen in three-body $B$-meson decays [42, 43, 22]. It
is therefore important to suppress their contribution to a negligible level.
This is investigated by applying all of the above selections, except that $D$
candidates are selected from a $D$ mass sideband region instead of the signal
region. The sideband region is chosen to avoid the contribution from the other
two-body $D$ decays with one misidentified daughter. Charmless backgrounds are
seen in all modes. These backgrounds are reduced to a negligible level by
requiring that the $D$ decay vertex is displaced significantly downstream of
the ${{{B}^{-}}}$ decay vertex, corresponding to three times the uncertainty
on the measured $D$ decay length. A more stringent requirement, corresponding
to five times the uncertainty on the measured $D$ decay length, is imposed on
the ${{{B}^{-}}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}X_{s,d}^{-}$ decays,
which is found to have a much larger charmless contribution. After these
requirements are applied, the charmless backgrounds are consistent with zero,
and the residual contribution is considered as a source of systematic
uncertainty.

Another important background to suppress is the cross-feed from the ADS CF
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X^{-}$ decay into the ADS DCS
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X^{-}$ sample, which may
happen if the ${{K}^{-}}$ and ${{\pi}^{+}}$ are both misidentified. Since the
CF yield is expected to be several hundred times larger than that of the DCS
mode (depending on the values of $r_{B}$, $\delta_{B}$, $\kappa$ and
$\gamma$), a large suppression is necessary. The combined ${{D}^{0}}$ mass and
PID requirements provide a suppression factor of $6\times 10^{-5}$. An
additional requirement that the $K\pi$ invariant mass (after interchanging the
${{K}^{-}}$ and ${{\pi}^{+}}$ masses) differs by at least
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ from the known ${{D}^{0}}$ mass
decreases the suppression level to $0.9\times 10^{-5}$. This leads to a
negligible contamination from the CF ADS mode into the DCS decay. The same
veto is applied to both the ADS CF ${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$
and DCS ${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays, so that no
efficiency correction is needed for $R^{X^{\pm}}$.

Lastly, in order to have a robust estimate of the trigger efficiency for
signal events, we impose requirements on information from the hardware
trigger; either (i) one or more of the decay products of the signal candidate
met the trigger requirements from the calorimeter system, or (ii) the event
passed at least one of the hardware triggers, and would have done so even if
the signal decay was removed from the event. These two classes of events
constitute about 60% and 40% of the signal candidates, respectively, where the
overlap is assigned to category (i).

The selection efficiencies as a function of several two- and three-body masses
in the ${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{d}^{-}$ decay are
shown in Fig. 1, for both the GLW and ADS selections. The efficiencies for
other $D$ final states are consistent with those for
${D}\rightarrow{{K}^{-}}{{\pi}^{+}}$. The $m(D{{\pi}^{-}})$ and
$m({{\pi}^{+}}{{\pi}^{-}})$ efficiencies include two entries per signal decay,
as there are two ${{\pi}^{-}}$ in the final state. The analogous efficiencies
for the ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ decay are shown in Fig. 2. The
relative efficiencies of the ADS to GLW selections are consistent with being
flat across each of these masses. These efficiencies include all selection
requirements, including PID. However, events in which any of the signal decay
products is outside of the LHCb detector acceptance are not included, since
they are not simulated; thus to obtain the total selection efficiency, these
efficiencies should be scaled by a factor of 0.11, as determined from
simulation.

Figure 1: Signal efficiencies for the
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{d}^{-}$ decay when
applying the GLW and ADS selections. The efficiencies are shown as a function
of five different two- and three-body masses. Figure 2: Signal efficiencies
for the ${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{-}$ decay when
applying the GLW and ADS selections. The efficiencies are shown as a function
of five different two- and three-body masses.

Figure 3 shows the $X_{d}^{-}$ and $X_{s}^{-}$ invariant mass distributions
for ${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{d}^{-}$ and
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{-}$ signal decays
after all selections, except for the $X^{-}$ and ${{K}^{*0}}$ mass
requirements. These signal spectra are background subtracted using the sPlot
method [44], with the ${{{B}^{-}}}$ candidate invariant mass as the
discriminating variable.

Figure 3: Signal distributions of the (left) $X_{d}^{-}$ invariant mass in
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ decays and (right) $X_{s}^{-}$ invariant
mass, in ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ decays, for
$D\rightarrow{{K}^{-}}{{\pi}^{+}}$. The distributions are obtained using the
sPlot method. In both cases, all selections, except the
$M(X^{-})<2$${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ and the ${{K}^{*0}}$
mass selection, are applied. The dip at
1.97${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$ is due to the ${{D}^{+}_{s}}$
meson veto.

The $X_{d}^{-}$ and $X_{s}^{-}$ contributions peak in the region below
2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, consistent with the dominance of
resonances such as
$a_{1}(1260)^{-}\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ to the
$X_{d}^{-}$ system, and one or more excited strange resonances contributing to
$X_{s}^{-}$. The dip at 1.97${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ is due
to the ${{D}^{-}_{s}}$ mass veto.

##  5 Fits to data

The signal yields are determined through a simultaneous unbinned extended
maximum likelihood fit to the 16 $B^{\pm}$ candidate invariant mass spectra.
These 16 spectra include the four ${{{B}^{-}}}\rightarrow DX_{d}^{-}$ decays,
where $D\rightarrow K^{\pm}\pi^{\mp},~{}{{K}^{+}}{{K}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}$, the corresponding four charge-conjugate decays, and
the set of eight modes where $X_{d}^{-}$ is replaced with $X_{s}^{-}$. The
signal and background contributions across these modes are similar, although
not identical. Where possible, common signal and background shapes are used;
otherwise simulation is used to relate parameters in the lower yield modes to
the values obtained from the high yield CF $D\rightarrow K\pi$ modes. Signal
and background yields are all independent of one another in the ${{{B}^{+}}}$
and ${{{B}^{-}}}$ mass fits; thus $C\\!P$ violation is allowed for all
contributions in the mass spectrum. Unless otherwise noted, the shapes
discussed below are obtained from simulated decays.

###  5.1 Signal shapes

The ${{{B}^{-}}}$ mass signal shapes are each parameterized as the sum of a
Crystal Ball (CB) shape [45] and a Gaussian ($G$) function,

$\displaystyle{\cal{F}}_{sig}\propto f_{\rm CB}{\rm CB}(m_{B},\sigma_{\rm
CB},\alpha_{\rm CB},n)+(1-f_{\rm CB})G(m_{B},\sigma_{g}).$ (13)

The Gaussian function accounts for the core of the mass distribution, whereas
the CB function accounts for the non-Gaussian radiative tail below, and a
wider Gaussian resolution component above, the signal peak. A small difference
is seen between the shapes for the ${{{B}^{-}}}\rightarrow DX_{d}^{-}$ and
${{{B}^{-}}}\rightarrow DX_{s}^{-}$ decays, and so a different set of signal
shape parameters is used to describe each, except for a common value of the
fitted ${{{B}^{-}}}$ mass, $m_{B}$. The signal shapes are not very sensitive
to the power-law exponent, $n$, which is fixed to 10. The parameters
$\alpha_{\rm CB}$, $\sigma_{g}$ and $f_{\rm CB}$ are allowed to vary freely in
the fit to the data. From simulation, we find that for all 16 modes,
$\sigma_{\rm CB}/\sigma_{g}$ is consistent with 1.90, and this ratio is
imposed in the fit. Simulation is also used to relate the mass resolution in
the $D\rightarrow{{K}^{+}}{{K}^{-}},~{}{{\pi}^{+}}{{\pi}^{-}}$ modes to that
of the $D\rightarrow K\pi$ mode, from which it is found that
$\sigma_{g}^{[KK]_{D}X^{-}}=(0.947\pm 0.011)\sigma_{g}^{[K\pi]_{D}X^{-}}$ and
$\sigma_{g}^{[\pi\pi]_{D}X^{-}}=(1.043\pm 0.011)\sigma_{g}^{[K\pi]_{D}X^{-}}$.
The relations are consistent between the ${{{B}^{-}}}\rightarrow DX_{d}^{-}$
and ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ modes, and are applied as fixed
constraints (without uncertainties) in the mass fit.

###  5.2 Backgrounds and their modeling

The primary sources of background in the mass spectra are partially
reconstructed $B\rightarrow D^{(*)}X^{-}$ decays, cross-feed between
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ and ${{{B}^{-}}}\rightarrow DX_{s}^{-}$,
and other combinatorial backgrounds. All of the spectra have a contribution
from combinatorial background, the shape of which is described by an
exponential function. Its slope is taken to be the same for the
$C\\!P$-conjugate ${{{B}^{-}}}$ and ${{{B}^{+}}}$ decays, but differs among
the various $D$ and $X^{-}$ final states.

The main contribution to the partially reconstructed background comes from
${{{B}^{-}}}\rightarrow[{{D}^{0}}{{\pi}^{0}},{{D}^{0}}\gamma]_{{{D}^{*0}}}X^{-}$
or ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow[{{D}^{0}}{{\pi}^{+}}]_{{{D}^{*+}}}X^{-}$
decays, where a pion or photon is not considered when reconstructing the
${{{B}^{-}}}$ candidate. Because the missed pion or photon generally has low
momentum, these decays pass the full selection with high efficiency. The
shapes of these distributions are modeled using parameterized shapes based on
simulated decays. Since the Dalitz structure of these backgrounds is not
known, we do not rely entirely on simulation to reproduce the shape of this
low-mass component. Instead, the parameters of the shape function that depend
on the decay dynamics are allowed to vary freely, and are determined in the
fit. The shape parameters for these backgrounds are varied independently for
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ and ${{{B}^{-}}}\rightarrow DX_{s}^{-}$
decays.

Another background contribution which primarily contributes to the
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ ADS suppressed mode is the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{0}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$
decay, where there is no ${{D}^{*+}}$ intermediate state. This decay can
contribute to the ADS CF mode if a ${{\pi}^{+}}$ is excluded from the decay,
or to the ADS DCS mode if a ${{\pi}^{-}}$ is not considered. The branching
fraction for this decay is not known, but the similar CF decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{*0}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$
is known to have a relatively large branching fraction of $(2.7\pm 0.5)\times
10^{-3}$ [46, 47]. Assuming ${\cal{B}}({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{0}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}})\simeq{\cal{B}}({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{*0}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}})$,
this background contribution is about two orders of magnitude larger than the
DCS signal, although it peaks at lower mass than the signal. The selection
efficiency and shape of this background are difficult to determine from
simulation, since there have not been any studies of this final state to date.
Its shape is obtained from simulations that assume a quasi two-body process,
${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{0}}R,~{}R\rightarrow{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$,
which decays uniformly in the phase space. An ARGUS shape [48] convolved with
a Gaussian function provides a good description of this simulated background.
Its shape parameters are shared between ${{{B}^{+}}}$ and ${{{B}^{-}}}$ and
are allowed to vary freely in the fit, except for the Gaussian width, which is
fixed to the expected mass resolution of
15${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

The analogous ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{0}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$
decay does not pose the same contamination to the DCS ADS
${{{B}^{+}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{+}$ signal, since a
missed ${{\pi}^{-}}$ leads to a
${{{B}^{+}}}\rightarrow{{D}^{0}}{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ candidate,
which is not one of the decays of interest. However, in the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D^{0}}{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decay, opposite-sign kaons are natural due to the presence of the $\bar{s}$
quark within the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$
meson. This decay is unobserved, but the similar decay, ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{0}}{{K}^{+}}{{\pi}^{-}}$,
has a relatively large branching fraction of $(1.00\pm 0.15)\times 10^{-3}$
[49]. Based on other $B$-meson decays, one would expect the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\rightarrow{{D}^{0}}{{K}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decay to be at the same level, ${\cal{O}}(10^{-3})$, which is two orders of
magnitude larger than the signal. The shape of this background has a similar
threshold behavior as for the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{0}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$
decay discussed previously, and therefore its contribution is also modeled
from simulated decays using an ARGUS shape convolved with a Gaussian function
with freely varying shape parameters.

In the fit, we also model cross-feed between the ${{{B}^{-}}}\rightarrow
D^{(*)}X_{d}^{-}$ and ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ decays. The shapes
of these cross-feed backgrounds are obtained from simulation. The cross-feed
rate is obtained from ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$,
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ calibration data, reweighted to
match the properties of the signal decays. All selection requirements on the
${{{B}^{-}}}\rightarrow DX^{-}$ decays, including
$|M({{K}^{-}}{{\pi}^{+}})-M_{{{K}^{*0}}}|<100$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
and $M(X^{-})<2$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, are taken into
account. In total, we find that 0.66% of ${{{B}^{-}}}\rightarrow DX_{d}^{-}$
are misidentified as ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ for the GLW modes and
0.16% for the ADS modes. The lower value for the ADS modes is due to the
tighter PID requirements on the ${{K}^{-}}$ candidate in the $X_{s}^{-}$
system. The cross-feed from ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ into
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ is evaluated in an analogous manner, and
is found to be 13.7%. Since the ratio of branching fractions is
${\cal{B}}({{{B}^{-}}}\rightarrow DX_{s}^{-})/{\cal{B}}({{{B}^{-}}}\rightarrow
DX_{d}^{-})\simeq 0.09$ [50], the yield of this background is only about 1% of
the signal yield.

Other sources of background that contribute to the ${{{B}^{-}}}\rightarrow
DX_{s}^{-}$ modes are the
${{{B}^{-}}}\rightarrow{{D}^{0}}[{{K}^{-}}{{K}^{+}}{{\pi}^{-}}]_{{{D}^{-}_{s}}}$
and ${{{B}^{-}}}\rightarrow{{D}^{0}}{{K}^{-}}[{{K}^{+}}{{\pi}^{-}}]_{{{\kern
1.39998pt\overline{\kern-1.39998ptK}{}}{}^{*0}}}$ decays, where the
${{K}^{+}}$ is misidentified as a ${{\pi}^{+}}$ meson. The shapes are similar
for these two backgrounds and thus a single shape is used, based on a
parameterization of the ${{{B}^{-}}}$ candidate mass distribution in simulated
${{{B}^{-}}}\rightarrow{{D}^{0}}[{{K}^{-}}{{K}^{+}}{{\pi}^{-}}]_{{{D}^{-}_{s}}}$
decays. Taking into account known branching fractions [22], efficiencies from
simulation, and ${{K}^{+}}\rightarrow{{\pi}^{+}}$ misidentification rates from
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ calibration data, we expect a
contribution of 1.6% of the ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ signal.

###  5.3 Fit results

The invariant mass spectra for the ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ ADS and
GLW signal modes are shown in Figs. 4 and 5, with the corresponding spectra
for the ${{{B}^{-}}}\rightarrow DX_{d}^{-}$ normalization modes in Figs. 6 and
7. Results from the fits are superimposed along with the various signal and
background components. The fitted yields in the ADS and GLW modes are given in
Tables 1 and 2.

Table 1: Fitted yields in the ADS modes with $f=K\pi$, for the signal and corresponding normalization modes. Decay mode |  ${{{B}^{-}}}$ yield |  ${{{B}^{+}}}$ yield
---|---|---
| ($N^{f}_{{\rm fit},X_{d}^{-}}$) | ($N^{f}_{{\rm fit},X_{d}^{+}}$)
$B^{\pm}\rightarrow DX_{d}^{\pm}$, $D\rightarrow{{K}^{-}}{{\pi}^{+}}$ | $36\,956\pm 214$ | $37\,843\pm 219$
$B^{\pm}\rightarrow DX_{d}^{\pm}$, $D\rightarrow{{K}^{+}}{{\pi}^{-}}$ | $~{}~{}161\pm 20$ | $~{}~{}162\pm 20$
| ($N^{f}_{{\rm fit},X_{s}^{-}}$) | ($N^{f}_{{\rm fit},X_{s}^{+}}$)
$B^{\pm}\rightarrow DX_{s}^{\pm}$, $D\rightarrow{{K}^{-}}{{\pi}^{+}}$ | $1234\pm 37$ | $1226\pm 37$
$B^{\pm}\rightarrow DX_{s}^{\pm}$, $D\rightarrow{{K}^{+}}{{\pi}^{-}}$ | $~{}\,13.0\pm 5.3$ | $~{}~{}~{}6.6\pm 4.0$
Table 2: Fitted yields used in the GLW analysis with $f=K^{\pm}\pi^{\pm},~{}{{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$, for the signal and corresponding normalization modes. Decay mode |  ${{{B}^{-}}}$ yield |  ${{{B}^{+}}}$ yield
---|---|---
| ($N^{f}_{{\rm fit},X_{d}^{-}}$) | ($N^{f}_{{\rm fit},X_{d}^{+}}$)
$B^{\pm}\rightarrow DX_{d}^{\pm}$, $D\rightarrow{{K}^{-}}{{\pi}^{+}}$ | $45\,213\pm 226$ | $46\,488\pm 230$
$B^{\pm}\rightarrow DX_{d}^{\pm}$, $D\rightarrow{{K}^{+}}{{K}^{-}}$ | $3899\pm 63$ | $4084\pm 65$
$B^{\pm}\rightarrow DX_{d}^{\pm}$, $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ | $1669\pm 38$ | $1739\pm 40$
| ($N^{f}_{{\rm fit},X_{s}^{-}}$) | ($N^{f}_{{\rm fit},X_{s}^{+}}$)
$B^{\pm}\rightarrow DX_{s}^{\pm}$, $D\rightarrow{{K}^{-}}{{\pi}^{+}}$ | $1699\pm 47$ | $1744\pm 47$
$B^{\pm}\rightarrow DX_{s}^{\pm}$, $D\rightarrow{{K}^{+}}{{K}^{-}}$ | $\,~{}155\pm 14$ | $\,~{}171\pm 14$
$B^{\pm}\rightarrow DX_{s}^{\pm}$, $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ | $\,~{}59\pm 9$ | $\,~{}70\pm 9$

Figure 4: Mass distributions of ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ candidates
using the ADS selections, for (top left)
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{-}$, (top right)
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{s}^{+}$, (bottom left)
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{s}^{-}$, and (bottom
right) ${{{B}^{+}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{+}$.

Figure 5: Mass distributions of ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ candidates
using the GLW selections, for (top left)
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{s}^{-}$, (top right)
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{s}^{+}$, (middle left)
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}X_{s}^{-}$, (middle right)
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}X_{s}^{+}$, (bottom left)
${{{B}^{-}}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}X_{s}^{-}$, and (bottom
right) ${{{B}^{+}}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}X_{s}^{+}$.

Figure 6: Mass distributions of ${{{B}^{-}}}\rightarrow DX_{d}^{-}$ candidates
using the ADS selections, for (top left)
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{d}^{-}$, (top right)
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{d}^{+}$, (bottom left)
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{d}^{-}$, and (bottom
right) ${{{B}^{+}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{d}^{+}$.

Figure 7: Mass distributions of ${{{B}^{-}}}\rightarrow DX_{d}^{-}$ candidates
using the GLW selections, for (top left)
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}X_{d}^{-}$, (top right)
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}X_{d}^{+}$, (middle left)
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}X_{d}^{-}$, and (middle right)
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}X_{d}^{+}$, (bottom left)
${{{B}^{-}}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}X_{d}^{-}$, and (bottom
right) ${{{B}^{+}}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}X_{d}^{+}$

Highly significant signals are seen in all modes, except for the ADS DCS
${{{B}^{-}}}\rightarrow DX_{s}^{-}$ decay. This is the first time these decays
have been observed in modes other than the CF
${{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay. Figure 8 shows the
suppressed ADS mode, $B^{\pm}\rightarrow
D[{{K}^{+}}{{\pi}^{-}}]_{D}K^{\pm}\pi^{\mp}\pi^{\pm}$, summed over both
$B$-meson charge states. The significance of the peak, which exceeds three
standard deviations, is discussed later.

Figure 8: Mass distributions for the suppressed ADS mode,
$B^{\pm}\rightarrow[K^{\mp}\pi^{\pm}]_{D}K^{\pm}\pi^{\mp}\pi^{\pm}$ (sum of
${{{B}^{+}}}$ and ${{{B}^{-}}}$).

##  6 Determination of $C\\!P$ observables

The $C\\!P$ observables are obtained by expressing the fitted signal yields in
terms of corrected yields and the $C\\!P$ parameters. For the decay
$B^{\pm}\rightarrow f_{D}X_{d}^{\pm}$, where $f_{D}$ is either the ADS CF
decay or a $C\\!P$ eigenstate, the fitted yields can be written as

$\displaystyle N^{f}_{{\rm fit},X_{d}^{\pm}}=\frac{1}{2}\Bigg{(}\frac{N_{{\rm
corr},X_{d}}^{f}}{1+F^{f}_{\cancel{D},X_{d}}}\Bigg{)}(1\mp{\mathcal{A}}_{{\rm
raw},X_{d}}^{f})+C^{f}_{\cancel{c},X_{d}^{\pm}},$ (14)

where $N_{{\rm corr},X_{d}}^{f}$ is the total corrected yield (sum of
${{{B}^{-}}}$ and ${{{B}^{+}}}$), $F^{f}_{\cancel{D},X_{d}}$ are the estimated
fractions of signal events removed by the ${{D}^{0}}$ and $D_{(s)}^{(*)+}$
vetoes, $C^{f}_{\cancel{c},X_{d}^{\pm}}$ are the estimated charmless
background yields, and ${\mathcal{A}}_{{\rm raw},X_{d}}^{f}$ is the raw
$C\\!P$ asymmetry.

The fitted yields in the corresponding $B^{\pm}\rightarrow DX_{s}^{\pm}$
decays are written in terms of the corrected $B^{\pm}\rightarrow DX_{d}^{\pm}$
yields in Eq. 14 and the $C\\!P$ observable $R^{f}_{s/d}$ defined in Eqs. 9
and 10, as

$\displaystyle N^{f}_{{\rm
fit},X_{s}^{\pm}}=\frac{1}{2}R^{f}_{s/d}\Bigg{(}\frac{N^{f}_{{\rm
corr},X_{d}}}{1+F^{f}_{\cancel{D},X_{s}}}\Bigg{)}(1\mp{\mathcal{A}}_{{\rm
raw},X_{s}}^{f})+C^{f}_{\cancel{c},X_{s}^{\pm}},$ (15)

where the meaning of the symbols parallels those in Eq. 14.

For the ADS suppressed modes, the four DCS yields $N^{K^{\mp}\pi^{\pm}}_{{\rm
fit},X^{\pm}}$ are expressed in terms of the corrected CF yields, $N_{{\rm
corr},X_{d}}^{K^{\pm}\pi^{\mp}}$, as

$\displaystyle N^{K^{\mp}\pi^{\pm}}_{{\rm
fit},X_{d}^{\pm}}=\left(R^{X_{d}^{\pm}}_{{\rm
raw}}\right)\Bigg{(}\frac{N_{{\rm
corr},X_{d}}^{K^{\pm}\pi^{\mp}}}{1+F^{K^{\mp}\pi^{\pm}}_{\cancel{D},X_{d}}}\Bigg{)}+C^{K^{\mp}\pi^{\pm}}_{\cancel{c},X_{d}^{\pm}},$
(16) $\displaystyle N^{K^{\mp}\pi^{\pm}}_{{\rm
fit},X_{s}^{\pm}}=\left(R^{X_{s}^{\pm}}_{{\rm
raw}}\right)\Bigg{(}\frac{N_{{\rm
corr},X_{s}}^{K^{\pm}\pi^{\mp}}}{1+F^{K^{\mp}\pi^{\pm}}_{\cancel{D},X_{s}}}\Bigg{)}+C^{K^{\mp}\pi^{\pm}}_{\cancel{c},X_{s}^{\pm}},$
(17)

where $N_{{\rm corr},X_{s}^{\pm}}^{K^{\pm}\pi^{\mp}}=N_{{\rm
corr},X_{s}}^{K^{\pm}\pi^{\mp}}(1\mp{\mathcal{A}}_{{\rm
raw},X_{s}}^{K^{\pm}\pi^{\mp}})$ gives the corrected yield for the favored
$B^{\pm}\rightarrow[K^{\pm}\pi^{\mp}]_{D}X_{s}^{\pm}$ decays.

The corrections for the ${{D}^{0}}$ and $D_{(s)}^{(*)+}$ vetoes,
$F^{f}_{\cancel{D},X_{d,s}}$, are determined by interpolating from the mass
regions just above and below the veto region, and lead to corrections that
range from 0.6% to 5.8% of the expected yield. Uncertainties on these
corrections are considered as sources of systematic uncertainty. Potential
contamination from charmless five-body decays is determined by fitting for a
$B^{\pm}$ signal component when the $D$ candidates are taken from the
${{D}^{0}}$ mass sideband region, as described previously. The charmless
contributions are negligible, and the uncertainties are included in the
systematic error. The yields, as determined from the fitted values of the
$C\\!P$ parameters in Eqs. 14-17, are given in Tables 1 and 2.

The raw observables, ${\mathcal{A}}_{{\rm raw},X}^{f}$ and $R^{X^{\pm}}_{{\rm
raw}}$ include small biases due to the production asymmetry of $B^{\pm}$
mesons, ${\cal{A}}_{B^{\pm}}$ (affecting ${\mathcal{A}}_{{\rm raw},X}^{f}$
only), and from the detection asymmetries of kaons and pions, ${\cal{A}}_{K}$
and ${\cal{A}}_{\pi}$. The corrected quantities are then computed according to

$\displaystyle{\mathcal{A}}^{{{K}^{+}}{{K}^{-}}}_{X_{d}}$
$\displaystyle={\mathcal{A}}^{{{K}^{+}}{{K}^{-}}}_{{\rm
raw},X_{d}}-{\cal{A}}_{B^{\pm}}-{\cal{A}}_{\pi},$ (18)
$\displaystyle{\mathcal{A}}^{{{\pi}^{+}}{{\pi}^{-}}}_{X_{d}}$
$\displaystyle={\mathcal{A}}^{{{\pi}^{+}}{{\pi}^{-}}}_{{\rm
raw},X_{d}}-{\cal{A}}_{B^{\pm}}-{\cal{A}}_{\pi},$ (19)
$\displaystyle{\mathcal{A}}^{{{K}^{-}}{{\pi}^{+}}}_{X_{d}}$
$\displaystyle={\mathcal{A}}^{{{K}^{-}}{{\pi}^{+}}}_{{\rm
raw},X_{d}}-{\cal{A}}_{B^{\pm}}-{\cal{A}}_{K},$ (20)
$\displaystyle{\mathcal{A}}^{{{K}^{+}}{{K}^{-}}}_{X_{s}}$
$\displaystyle={\mathcal{A}}^{{{K}^{+}}{{K}^{-}}}_{{\rm
raw},X_{s}}-{\cal{A}}_{B^{\pm}}-{\cal{A}}_{K},$ (21)
$\displaystyle{\mathcal{A}}^{{{\pi}^{+}}{{\pi}^{-}}}_{X_{s}}$
$\displaystyle={\mathcal{A}}^{{{\pi}^{+}}{{\pi}^{-}}}_{{\rm
raw},X_{s}}-{\cal{A}}_{B^{\pm}}-{\cal{A}}_{K},$ (22)
$\displaystyle{\mathcal{A}}^{{{K}^{-}}{{\pi}^{+}}}_{X_{s}}$
$\displaystyle={\mathcal{A}}^{{{K}^{-}}{{\pi}^{+}}}_{{\rm
raw},X_{s}}-{\cal{A}}_{B^{\pm}}-2{\cal{A}}_{K}+{\cal{A}}_{\pi},$ (23)
$\displaystyle R^{X_{d}^{+}}$ $\displaystyle=R^{X_{d}^{+}}_{{\rm
raw}}(1-2{\cal{A}}_{K}+2{\cal{A}}_{\pi}),$ (24) $\displaystyle R^{X_{d}^{-}}$
$\displaystyle=R^{X_{d}^{-}}_{{\rm raw}}(1+2{\cal{A}}_{K}-2{\cal{A}}_{\pi}),$
(25) $\displaystyle R^{X_{s}^{+}}$ $\displaystyle=R^{X_{s}^{+}}_{{\rm
raw}}(1-2{\cal{A}}_{K}+2{\cal{A}}_{\pi}),$ (26) $\displaystyle R^{X_{s}^{-}}$
$\displaystyle=R^{X_{s}^{-}}_{{\rm raw}}(1+2{\cal{A}}_{K}-2{\cal{A}}_{\pi}).$
(27)

The pion detection asymmetry of ${\cal{A}}_{\pi}=0.000\pm 0.003$ is obtained
by reweighting the measured $\pi^{\pm}$ detection efficiencies [51] with the
expected momentum spectrum for signal pions. The kaon detection efficiency of
${\cal{A}}_{K}=-0.011\pm 0.004$ is obtained by reweighting the measured
${{K}^{-}}\pi$ detection asymmetry [52] using the momentum spectrum of signal
kaons, and then subtracting the above pion detection asymmetry. For the
production asymmetry, the value ${\cal{A}}_{B^{\pm}}=-0.008\pm 0.007$ is used
[53], based on the measured raw asymmetry in
$B^{\pm}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{\pm}$
decays [54] and on simulation.

###  6.1 Systematic uncertainties

Most potential systematic uncertainties on the observables are expected to
cancel in either the asymmetries or ratios that are measured. The systematic
uncertainties that do not cancel completely are summarized in Table 3. The PID
and trigger asymmetries are evaluated using measured kaon and pion
efficiencies from ${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ calibration
samples in data that are identified using only the kinematics of the decay.
The efficiencies for the ${{{B}^{+}}}$ and ${{{B}^{-}}}$ signal decays are
then obtained by reweighting the kaon and pion efficiencies using simulated
$B^{\pm}\rightarrow DX^{\pm}$ decays to represent the properties of signal
data. We find no significant charge asymmetry with respect to the PID
requirements, and use ${\cal{A}}_{h}^{\rm PID}=0.000\pm 0.006$, where the
uncertainty is dominated by the finite sample sizes of the simulated signal
decays in the reweighting. The asymmetry of the hardware trigger is assessed
using measured hadron trigger efficiencies in
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}},~{}{{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}$
decays, reweighted to match the momentum spectrum of tracks from signal
decays. Defining the $B^{\pm}$ hadron trigger efficiency as
$\epsilon_{B^{\pm}}$, the charge asymmetry of the trigger
$(\epsilon_{{{{B}^{-}}}}-\epsilon_{{{{B}^{+}}}})/(\epsilon_{{{{B}^{-}}}}+\epsilon_{{{{B}^{+}}}})$
varies from $0.000\pm 0.003$ for
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}X_{s}^{-}$ to $0.007\pm 0.003$
for ${{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{+}}]_{D}X_{s}^{-}$. These values
are applied as corrections.

Table 3: Systematic uncertainties, in percent, on the fitted parameters. Source | ${\cal{A}}(B^{\pm}\rightarrow DX_{d}^{\pm})$ | ${\cal{A}}(B^{\pm}\rightarrow DX_{s}^{\pm})$ | $R_{{C\\!P}+}$ | $R_{d}^{\pm}$ | $R_{s}^{\pm}$
---|---|---|---|---|---
$D\rightarrow$ | $h^{+}h^{-}$ | $K\pi$ | $h^{+}h^{-}$ | $K\pi$ | ${{K}^{+}}{{K}^{-}}$ | ${{\pi}^{+}}{{\pi}^{-}}$ | $K\pi$ | $K\pi$
${\cal{A}}_{B^{\pm}}$ | 0.7 | 0.7 | 0.7 | 0.7 | – | – | – | –
${\cal{A}}_{K}$ | – | 0.4 | 0.4 | 0.8 | – | – | 0.7 | 0.7
${\cal{A}}_{\pi}$ | 0.3 | – | – | 0.3 | – | – | 0.6 | 0.3
Trigger | 0.4 | 0.4 | 0.4 | 0.4 | 1.5 | 1.5 | 1.5 | 1.5
PID | 0.6 | 0.6 | 0.6 | 0.6 | 1.2 | 1.2 | 1.2 | 1.2
Signal model | – | – | – | – | 1.1 | 1.1 | – | –
Bkgd. model | – | – | – | – | 1.6 | 1.6 | 4.0 | 10.0
Charmless back. | – | – | – | – | 1.0 | 1.0 | 1.0 | 1.0
Cross-feed | – | – | – | – | 1.0 | 1.0 | 1.0 | 1.0
$D$ vetoes | – | – | – | – | 1.0 | 1.7 | 1.0 | 1.0
$R_{{C\\!P}+}$ approx. | – | – | – | – | 1.0 | 1.0 | – | –
Total | 1.0 | 1.1 | 1.1 | 1.3 | 3.4 | 3.8 | 4.9 | 10.4

On $R_{{C\\!P}+}$ and $R^{X^{\pm}}$, we have either a double-ratio or a ratio
of final states with identical particles (apart from the charges), and
therefore there is a high degree of cancellation of potential systematic
uncertainties. We expect that for these ratios, the relative trigger
efficiencies would yield a value close to unity. After reweighting the
measured trigger efficiencies according to the kinematical properties of
signal decays (obtained from simulation), we find that the ratios of trigger
efficiencies are within 1.5% of unity, which is assigned as a systematic
uncertainty. Using an analogous weighting procedure to the measured PID
efficiencies, we find that the relative PID efficiency is equal to unity to
within 1.2%, which is assigned as a systematic uncertainty.

We also consider uncertainty from the signal model, the background model, the
charmless contamination, the $D$ vetoes, and the detection asymmetries. For
the signal model uncertainty, all of the fixed signal shape parameters are
varied by one standard deviation, and the resulting changes in the $C\\!P$
parameters are added in quadrature to obtain the total signal shape
uncertainty (1.1%). For the background-related uncertainties, we consider a
polynomial function for the combinatorial background, and vary the fixed
background shape parameters of the specific $b$-hadron backgrounds within
their uncertainties, and add the deviations from the nominal result in
quadrature (1.6%). For the ADS-suppressed modes, larger uncertainties are
assigned based on an incomplete understanding of the contributions to the low
mass $\bar{B}^{0}_{(s)}\rightarrow{{D}^{0}}X$ background.

The charmless backgrounds are all consistent with zero, and the uncertainty is
taken from fits to the $D$ sideband regions (1.0%). Uncertainties due to the
cross-feed contributions (such as ${{{B}^{-}}}\rightarrow DX_{d}^{-}$
reconstructed as ${{{B}^{-}}}\rightarrow DX_{s}^{-}$) are assessed using
simulated experiments, by simulating the mass distributions with a larger
cross-feed and fitting with the nominal value (1.0%). The uncertainties due to
vetoing potential contributions from other $D$ mesons are assessed by
interpolating the mass spectrum just above and below the veto region into the
veto region. The associated uncertainties are all at the 1.0% level, except
for the the $B\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}X_{s}^{-}$ mode, which
has an uncertainty of 1.7%.

The uncertainties on the ratios $R_{s/d}^{h^{+}h^{\prime-}}$ and
$R^{X_{s,d}^{-}}$ are each summed in quadrature, giving total uncertainties in
the range of $(3.4-10.4)\%$, depending on the mode.

##  7 Results and summary

The resulting values for the $C\\!P$ observables are

$\displaystyle R^{{{K}^{+}}{{K}^{-}}}_{{C\\!P}+}$ $\displaystyle=1.043\pm
0.069\pm 0.034,$ $\displaystyle R^{{{\pi}^{+}}{{\pi}^{-}}}_{{C\\!P}+}$
$\displaystyle=1.035\pm 0.108\pm 0.038,$
$\displaystyle{\mathcal{A}}^{{{K}^{+}}{{K}^{-}}}_{X_{d}}$
$\displaystyle=-0.019\pm 0.011\pm 0.010,$
$\displaystyle{\mathcal{A}}^{{{\pi}^{+}}{{\pi}^{-}}}_{X_{d}}$
$\displaystyle=-0.013\pm 0.016\pm 0.010,$
$\displaystyle{\mathcal{A}}^{{{K}^{-}}{{\pi}^{+}}}_{X_{d}}$
$\displaystyle=-0.002\pm 0.003\pm 0.011,$ $\displaystyle R^{X_{d}^{+}}$
$\displaystyle=(43.2\pm 5.3\pm 2.1)\times 10^{-4},$ $\displaystyle
R^{X_{d}^{-}}$ $\displaystyle=(42.1\pm 5.3\pm 2.1)\times 10^{-4},$
$\displaystyle{\mathcal{A}}^{{{K}^{+}}{{K}^{-}}}_{X_{s}}$
$\displaystyle=-0.045\pm 0.064\pm 0.011,$
$\displaystyle{\mathcal{A}}^{{{\pi}^{+}}{{\pi}^{-}}}_{X_{s}}$
$\displaystyle=-0.054\pm 0.101\pm 0.011,$
$\displaystyle{\mathcal{A}}^{{{K}^{-}}{{\pi}^{+}}}_{X_{s}}$
$\displaystyle=0.013\pm 0.019\pm 0.013,$ $\displaystyle R^{X_{s}^{+}}$
$\displaystyle=(107_{-44}^{+60}\pm 11)\times
10^{-4}~{}~{}~{}[~{}<0.018~{}\mbox{\rm at 95\% CL}~{}],$ $\displaystyle
R^{X_{s}^{-}}$ $\displaystyle=(53_{-42}^{+45}\pm 6)\times
10^{-4}~{}~{}~{}[~{}<0.012~{}\mbox{\rm at 95\% CL}~{}].$

The values of $R_{{C\\!P}+}$ are averaged to obtain

$\displaystyle R_{{C\\!P}+}$ $\displaystyle=1.040\pm 0.064,$

where the uncertainty includes both statistical and systematic sources, as
well as the correlations between the latter.

The significances of the suppressed ADS modes are determined by computing the
ratio of log-likelihoods, $\sqrt{2\log({\cal{L}}_{0}/{\cal{L}}_{\rm min})}$,
after convolving ${\cal{L}}$ with the systematic uncertainty. From the value
of ${\cal{L}}$ at the minimum (${\cal{L}}_{\rm min}$), and the value at
$R^{X_{s}^{\pm}}=0$ (${\cal{L}}_{0}$), the significances of the non-zero
values for $R^{X_{s}^{-}}$ and $R^{X_{s}^{+}}$ are found to be 2.0$\sigma$ and
3.2$\sigma$, respectively. The overall significance of the observation of the
ADS suppressed mode is obtained by adding the log-likelihoods, resulting in a
significance of 3.6 standard deviations. This constitutes the first evidence
of the ADS suppressed mode in ${{{B}^{-}}}\rightarrow
D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays.

For completeness, we also compute the related observables $R_{\rm ADS}$ and
${\mathcal{A}}_{\rm ADS}$, which are commonly used. For the
${{{B}^{-}}}\rightarrow DX_{s}^{-}$ modes, the values are

$\displaystyle R_{\rm ADS}^{X_{s}}$
$\displaystyle\equiv(R^{X_{s}^{-}}+R^{X_{s}^{+}})/2=(85^{+36}_{-33})\times
10^{-4},$ $\displaystyle{\mathcal{A}}_{\rm ADS}^{X_{s}}$
$\displaystyle\equiv\frac{R^{X_{s}^{-}}-R^{X_{s}^{+}}}{R^{X_{s}^{-}}+R^{X_{s}^{+}}}=-0.33^{+0.36}_{-0.34}.$

For the favored modes, the corresponding values are

$\displaystyle R_{\rm ADS}^{X_{d}}$
$\displaystyle\equiv(R^{X_{s}^{-}}+R^{X_{s}^{+}})/2=(42.7\pm 5.6)\times
10^{-4},$ $\displaystyle{\mathcal{A}}_{\rm ADS}^{X_{d}}$
$\displaystyle\equiv\frac{R^{X_{s}^{-}}-R^{X_{s}^{+}}}{R^{X_{s}^{-}}+R^{X_{s}^{+}}}=-0.013\pm
0.087.$

The averages are computed using the asymmetric uncertainty distributions, and
include both statistical and systematical sources.

To assess the constraints on $\gamma$ that these observables provide, they
have been implemented in the fitter for $\gamma$ described in Ref. [14, *LHCb-
CONF-2014-004]. Two fits are performed, one that uses only information from
${{{B}^{-}}}\rightarrow\ DX_{s}^{-}$, and a second that uses the observables
from both ${{{B}^{-}}}\rightarrow\ DX_{s}^{-}$ and ${{{B}^{-}}}\rightarrow\
DX_{d}^{-}$ decays. In both fits, the parameters from the $D$-meson system,
$r_{D}$, $\delta_{D}^{K\pi}$, $x_{D}$, $y_{D}$, $A^{\rm
dir}_{CP}({{K}^{+}}{{K}^{-}})$, and $A^{\rm
dir}_{CP}({{\pi}^{+}}{{\pi}^{-}})$, are constrained in an analogous way to
what was done for the ${{{B}^{-}}}\rightarrow D{{K}^{-}}$ and
${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ case [14, *LHCb-CONF-2014-004]. The four
parameters $r_{B}$, $\delta_{B}$, $\kappa$ and $\gamma$ are freely varied in
each fit. In the combined fit, three additional strong parameters,
$r_{B}^{DX_{d}}$, $\delta_{B}^{DX_{d}}$, $\kappa^{DX_{d}}$ are included, which
are analogous to those that apply to the ${{{B}^{-}}}\rightarrow\ DX_{s}^{-}$
decay.

The projections of the fit results for $\gamma$, $r_{B}$ and $r_{B}$ versus
$\gamma$, are shown in Fig. 9 using the method of Ref. [55] (see also Refs.
[14, *LHCb-CONF-2014-004].)

Figure 9: Projections of 1$-$CL versus (left) $\gamma$, (right)
$r_{B}^{{{{B}^{-}}}\rightarrow D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}}$, and
(bottom) $r_{B}^{{{{B}^{-}}}\rightarrow D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}}$
versus $\gamma$, using only ${{{B}^{-}}}\rightarrow\
D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays, and the combination of
${{{B}^{-}}}\rightarrow\ D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow\ D{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays. The 68.3%
and 95.5% confidence level (CL) limits are indicated for the $\gamma$ and
$r_{B}$ projections. The 39% level contours in $r_{B}^{{{{B}^{-}}}\rightarrow
D{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}}$ versus $\gamma$ correspond to the 68.3%
level contours in the one-dimensional projections.

The value of $\gamma$ is found to be $(74^{+20}_{-23})^{\rm o}$ for the
${{{B}^{-}}}\rightarrow DX_{s}^{-}$-only fit, and $(74^{+20}_{-19})^{\rm o}$
for the for the combined ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ and
${{{B}^{-}}}\rightarrow DX_{d}^{-}$ fit. The value of $r_{B}$ is nearly
identical in the two cases, with corresponding values of
$r_{B}=0.081^{+0.025}_{-0.027}$ and $r_{B}=0.081^{+0.026}_{-0.029}$. As
expected, most of the sensitivity comes from the ${{{B}^{-}}}\rightarrow
DX_{s}^{-}$ decay mode. This value is almost identical to the LHCb combined
result of $(73^{+9}_{-10})^{\rm o}$ found in Ref. [14, *LHCb-CONF-2014-004].
The value of $r_{B}$ is similar to the values found in other
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decays [56, 53, 57, 58, 59], but smaller
than the value of $0.240^{+0.055}_{-0.048}$ [60] found in neutral $B$-meson
decays. The strong phase $\delta_{B}$, averaged over the phase space, peaks at
172o for both fits, but at 95% CL all angles are allowed. The constraints on
the coherence factor are relatively weak; while the most likely value is close
to 1, any value in the interval $[0,1]$ is allowed at one standard deviation.

In summary, a $pp$ collision data sample, corresponding to an integrated
luminosity of 3.0 fb-1, has been used to study the ${{{B}^{-}}}\rightarrow
DX_{s}^{-}$ and ${{{B}^{-}}}\rightarrow DX_{d}^{-}$ decay modes, where the $D$
meson decays to either the quasi-flavor-specific $K\pi$ final state or the
${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$ $C\\!P$ eigenstates. We
observe for the first time highly significant signals in the $C\\!P$ modes for
both the favored and suppressed ${{{B}^{-}}}$ decays, and we also report the
first evidence for the ADS DCS
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decay. We measure the corresponding ADS and GLW observables for the first time
in these modes. A fit for $\gamma$ using only these modes is performed, from
which we find $\gamma=(74^{+20}_{-23})^{\rm o}$ for the fit with only
${{{B}^{-}}}\rightarrow DX_{s}^{-}$, and $\gamma=(74^{+20}_{-19})^{\rm o}$ for
the combined ${{{B}^{-}}}\rightarrow DX_{s}^{-}$ and ${{{B}^{-}}}\rightarrow
DX_{d}^{-}$ fit. Values of $\gamma$ below about $25^{\rm o}$ and larger than
approximately $165^{\rm o}$ are not excluded by these modes alone, but are
excluded when other modes are considered [14, *LHCb-CONF-2014-004]. The
precision on $\gamma$ in this analysis is comparable to, or better than, most
previous measurements.

## 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); 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).

## References

  * [1] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [2] M. Kobayashi and T. Maskawa, _CP violation in the renormalizable theory of weak interaction_ , Prog. Theor. Phys. 49 (1973) 652
  * [3] J. Brod and J. Zupan, _The ultimate theoretical error on $\gamma$ from $B\rightarrow DK$ decays_, JHEP 01 (2014) 051, arXiv:1308.5663
  * [4] M. Gronau and D. Wyler, _On determining a weak phase from charged B decay asymmetries_ , Phys. Lett. B265 (1991) 172
  * [5] M. Gronau and D. London, _How to determine all the angles of the unitarity triangle from $B_{d}^{0}\rightarrow DK_{S}^{0}$ and $B_{s}^{0}\rightarrow D\phi$_, Phys. Lett. B253 (1991) 483
  * [6] D. Atwood, I. Dunietz, and A. Soni, _Enhanced CP violation with $B\rightarrow K{{D}^{0}}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})$ modes and extraction of the Cabibbo-Kobayashi-Maskawa angle $\gamma$_, Phys. Rev. Lett. 78 (1997) 3257, arXiv:hep-ph/9612433
  * [7] D. Atwood, I. Dunietz, and A. Soni, _Improved methods for observing CP violation in $B^{\pm}\rightarrow KD$ and measuring the CKM phase $\gamma$_, Phys. Rev. D63 (2001) 036005, arXiv:hep-ph/0008090
  * [8] A. Giri, Y. Grossman, A. Soffer, and J. Zupan, _Determining $\gamma$ using $B^{\pm}\rightarrow DK^{\pm}$ with multibody $D$ decays_, Phys. Rev. D68 (2003) 054018, arXiv:hep-ph/0303187
  * [9] I. Dunietz, _CP violation with self-tagging $B_{d}$ modes_, Phys. Lett. B270 (1991) 75
  * [10] I. Dunietz, _CP violation with beautiful baryons_ , Z. Phys. C56 (1992) 129
  * [11] Fayyazuddin, _${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}+{{D}^{0}}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})$ decays and $C\\!P$ violation_, Mod. Phys. Lett. A14 (1999) 63, arXiv:hep-ph/9806393
  * [12] A. K. Giri, R. Mohanta, and M. P. Khanna, _Possibility of extracting the weak phase $\gamma$ from ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}{{D}^{0}}$ decays_, Phys. Rev. D65 (2002) 073029, arXiv:hep-ph/0112220
  * [13] M. Gronau, _Improving bounds on $\gamma$ in $B^{\pm}\rightarrow DK^{\pm}$ and $B^{\pm,0}\rightarrow DX_{s}^{\pm,0}$_, Phys. Lett. B557 (2003) 198, arXiv:hep-ph/0211282
  * [14] LHCb collaboration, R. Aaij et al., _A measurement of the CKM angle $\gamma$ from a combination of $B^{\pm}\rightarrow Dh^{\pm}$ analyses_, Phys. Lett. B726 (2013) 151, arXiv:1305.2050
  * [15] LHCb collaboration, _Improved constraints on $\gamma$: CKM2014 update_, Sep 2014.  LHCb-CONF-2014-004
  * [16] BaBar collaboration, J. P. Lees et al., _Observation of direct CP violation in the measurement of the Cabibbo-Kobayashi-Maskawa angle $\gamma$ with $B^{\pm}\rightarrow D^{(*)}K^{(*)\pm}$ decays_, Phys. Rev. D87 (2013) 052015, arXiv:1301.1029
  * [17] K. Trabelsi, _Study of direct CP in charmed $B$ decays and measurements of the CKM angle $\gamma$ at Belle_, arXiv:1301.2033, presented on behalf of the Belle collaboration at CKM2012, Cincinnati, USA, 28 Sep – 2 Oct 2012
  * [18] J. Charles et al., _Current status of the Standard Model CKM fit and constraints on $\Delta F=2$ new physics_, Phys. Rev. D91 (2015) 073007, arXiv:1501.05013
  * [19] Heavy Flavor Averaging Group, Y. Amhis et al., _Averages of $b$-hadron, $c$-hadron, and $\tau$-lepton properties as of summer 2014_, arXiv:1412.7515, updated results and plots available at http://www.slac.stanford.edu/xorg/hfag/
  * [20] BESIII collaboration, M. Ablikim et al., _Measurement of the $D\rightarrow K^{-}\pi^{+}$ strong phase difference in $\psi(3770)\rightarrow D^{0}\overline{D}{}^{0}$_, Phys. Lett. B734 (2014) 227, arXiv:1404.4691
  * [21] M. Rama, _Effect of ${D}-\bar{D}$ mixing in the extraction of $\gamma$ with ${{{B}^{-}}}\rightarrow{{D}^{0}}{{K}^{-}}$ and ${{{B}^{-}}}\rightarrow{{D}^{0}}{{\pi}^{-}}$ decays_, Phys. Rev. D89 (2014) 014021, arXiv:1307.4384
  * [22] Particle Data Group, K. A. Olive et al., _Review of particle physics_ , Chin. Phys. C38 (2014) 090001
  * [23] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [24] R. Arink et al., _Performance of the LHCb Outer Tracker_ , JINST 9 (2014) P01002, arXiv:1311.3893
  * [25] R. Aaij et al., _Performance of the LHCb Vertex Locator_ , JINST 9 (2014) P09007, arXiv:1405.7808
  * [26] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [27] A. A. Alves Jr. et al., _Performance of the LHCb muon system_ , JINST 8 (2013) P02022, arXiv:1211.1346
  * [28] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [29] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [30] V. V. Gligorov and M. Williams, _Efficient, reliable and fast high-level triggering using a bonsai boosted decision tree_ , JINST 8 (2013) P02013, arXiv:1210.6861
  * [31] T. Sjöstrand, S. Mrenna, and P. Skands, _PYTHIA 6.4 physics and manual_ , JHEP 05 (2006) 026, arXiv:hep-ph/0603175
  * [32] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [33] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [34] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [35] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [36] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [37] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [38] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [39] W. D. Hulsbergen, _Decay chain fitting with a Kalman filter_ , Nucl. Instrum. Meth. A552 (2005) 566, arXiv:physics/0503191
  * [40] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [41] R. E. Schapire and Y. Freund, _A decision-theoretic generalization of on-line learning and an application to boosting_ , Jour. Comp. and Syst. Sc. 55 (1997) 119
  * [42] LHCb collaboration, R. Aaij et al., _Measurement of $CP$ violation in the three-body phase space of charmless $B^{\pm}$ decays_, Phys. Rev. D90 112004, arXiv:1408.5373
  * [43] LHCb collaboration, R. Aaij et al., _Evidence for $CP$ violation in $B^{+}\rightarrow p\bar{p}K^{+}$ decays_, Phys. Rev. Lett. 113 (2014) 141801, arXiv:1407.5907
  * [44] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083
  * [45] T. Skwarnicki, A study of the radiative cascade transitions between the Upsilon-prime and Upsilon resonances, PhD thesis, Institute of Nuclear Physics, Krakow, 1986, DESY-F31-86-02
  * [46] Belle collaboration, G. Majumder et al., _Observation of ${{B}^{0}}\rightarrow{{D}^{*-}}(5\pi)^{+}$, ${{{B}^{+}}}\rightarrow{{D}^{*-}}(4\pi)^{++}$ and ${{{B}^{+}}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{*0}}(5\pi)^{+}$_, Phys. Rev. D70 (2004) 111103, arXiv:hep-ex/0409008
  * [47] CLEO collaboration, K. W. Edwards et al., _First observation of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ decays_, Phys. Rev. D65 (2002) 012002, arXiv:hep-ex/0105071
  * [48] ARGUS collaboration, H. Albrecht et al., _Search for hadronic $b\rightarrow u$ decays_, Phys. Lett. B241 (1990) 278
  * [49] LHCb collaboration, R. Aaij et al., _Dalitz plot analysis of $B^{0}_{s}\rightarrow\bar{D}^{0}K^{-}\pi^{+}$ decays_, Phys. Rev. D90 (2014) 072003, arXiv:1407.7712
  * [50] LHCb, R. Aaij et al., _First observation of the decays $\bar{B}^{0}\rightarrow D^{+}K^{-}\pi^{+}\pi^{-}$ and $B^{-}\rightarrow D^{0}K^{-}\pi^{+}\pi^{-}$_, Phys. Rev. Lett. 108 (2012) 161801, arXiv:1201.4402
  * [51] LHCb collaboration, R. Aaij et al., _Measurement of the $D_{s}^{+}$–$D_{s}^{-}$ production asymmetry in 7 TeV $pp$ collisions_, Phys. Lett. B713 (2012) 186, arXiv:1205.0897
  * [52] LHCb collaboration, R. Aaij et al., _Measurement of $CP$ asymmetry in $D^{0}\rightarrow K^{-}K^{+}$ and $D^{0}\rightarrow\pi^{-}\pi^{+}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797
  * [53] LHCb collaboration, R. Aaij et al., _Observation of the suppressed ADS modes $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}K^{\pm}$ and $B^{\pm}\rightarrow[\pi^{\pm}K^{\mp}\pi^{+}\pi^{-}]_{D}\pi^{\pm}$_, Phys. Lett. B723 (2013) 44, arXiv:1303.4646
  * [54] LHCb collaboration, R. Aaij et al., _Measurements of the branching fractions and $CP$ asymmetries of $B^{\pm}\rightarrow J/\psi\pi^{\pm}$ and $B^{\pm}\rightarrow\psi(2S)\pi^{\pm}$ decays_, Phys. Rev. D85 (2012) 091105(R), arXiv:1203.3592
  * [55] S. Bodhisattva, M. Walker, and M. Woodroofe, _On the unified method with nuisance parameters_ , Statistica Sinica 19 (2009) 301
  * [56] LHCb collaboration, R. Aaij et al., _Observation of $CP$ violation in $B^{\pm}\rightarrow DK^{\pm}$ decays_, Phys. Lett. B712 (2012) 203, Erratum ibid. B713 (2012) 351, arXiv:1203.3662
  * [57] LHCb collaboration, R. Aaij et al., _Measurement of $CP$ violation and constraints on the CKM angle $\gamma$ in $B^{\pm}\rightarrow DK^{\pm}$ with $D\rightarrow K_{S}^{0}\pi^{+}\pi^{-}$ decays_, Nucl. Phys. B888 (2014) 169, arXiv:1407.6211
  * [58] LHCb collaboration, R. Aaij et al., _Measurement of the CKM angle $\gamma$ using $B^{\pm}\rightarrow DK^{\pm}$ with $D\rightarrow K_{S}\pi^{+}\pi^{-}$, $K_{S}K^{+}K^{-}$ decays_, JHEP 10 (2014) 097, arXiv:1408.2748
  * [59] LHCb collaboration, R. Aaij et al., _A study of $C\\!P$ violation in $B^{\mp}\rightarrow Dh^{\mp}$ $(h=K,\pi)$ with the modes $D\rightarrow K^{\mp}\pi^{\pm}\pi^{0}$, $D\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and $D\rightarrow K^{+}K^{-}\pi^{0}$_, Phys. Rev. D91 (2015) 112014 arXiv:1504.05442
  * [60] LHCb collaboration, R. Aaij et al., _Measurement of CP violation parameters in $B^{0}\rightarrow DK^{*0}$ decays_, Phys. Rev. D90 (2014) 112002, arXiv:1407.8136

LHCb collaboration

R. Aaij38, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle53, A.A. Alves Jr57, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17,g, J. Anderson40, M. Andreotti16,f, J.E. Andrews58, R.B.
Appleby54, O. Aquines Gutierrez10, F. Archilli38, P. d’Argent11, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W. Baldini16,38, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter38, V. Batozskaya28, V.
Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1,
L.J. Bel41, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson38, J.
Benton46, A. Berezhnoy32, R. Bernet40, A. Bertolin22, M.-O. Bettler38, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Birnkraut9, A. Bizzeti17,i,
T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N.
Bondar30,38, W. Bonivento15, S. Borghi54, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, S. Braun11, D. Brett54, M. Britsch10, T. Britton59, J.
Brodzicka54, N.H. Brook46, A. Bursche40, J. Buytaert38, S. Cadeddu15, R.
Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,p, P. Campana18, D. Campora
Perez38, L. Capriotti54, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,j, A.
Cardini15, P. Carniti20, L. Carson50, K. Carvalho Akiba2,38, R. Casanova
Mohr36, G. Casse52, L. Cassina20,k, L. Castillo Garcia38, M. Cattaneo38, Ch.
Cauet9, G. Cavallero19, R. Cenci23,t, M. Charles8, Ph. Charpentier38, M.
Chefdeville4, S. Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40, X.
Cid Vidal38, G. Ciezarek41, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J.
Closier38, V. Coco38, J. Cogan6, E. Cogneras5, V. Cogoni15,e, L. Cojocariu29,
G. Collazuol22, P. Collins38, A. Comerma-Montells11, A. Contu15,38, A. Cook46,
M. Coombes46, S. Coquereau8, G. Corti38, M. Corvo16,f, B. Couturier38, G.A.
Cowan50, D.C. Craik48, A. Crocombe48, M. Cruz Torres60, S. Cunliffe53, R.
Currie53, C. D’Ambrosio38, J. Dalseno46, P.N.Y. David41, A. Davis57, K. De
Bruyn41, S. De Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De
Silva57, P. De Simone18, C.-T. Dean51, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach55, O. Deschamps5, F. Dettori38, B. Dey40, A.
Di Canto38, F. Di Ruscio24, H. Dijkstra38, S. Donleavy52, F. Dordei11, M.
Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, K. Dreimanis52, L.
Dufour41, G. Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S.
Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51, I. El
Rifai5, Ch. Elsasser40, S. Ely59, S. Esen11, H.M. Evans47, T. Evans55, A.
Falabella14, C. Färber11, C. Farinelli41, N. Farley45, S. Farry52, R. Fay52,
D. Ferguson50, V. Fernandez Albor37, F. Ferrari14, F. Ferreira Rodrigues1, M.
Ferro-Luzzi38, S. Filippov33, M. Fiore16,38,f, M. Fiorini16,f, M. Firlej27, C.
Fitzpatrick39, T. Fiutowski27, K. Fohl38, P. Fol53, M. Fontana10, F.
Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M.
Frosini17, J. Fu21, E. Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S.
Gallorini22,38, S. Gambetta50, M. Gandelman2, P. Gandini55, Y. Gao3, J. García
Pardiñas37, J. Garofoli59, J. Garra Tico47, L. Garrido36, D. Gascon36, C.
Gaspar38, U. Gastaldi16, R. Gauld55, L. Gavardi9, G. Gazzoni5, A. Geraci21,v,
D. Gerick11, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Gianì39, V. Gibson47, O. G. Girard39, L. Giubega29, V.V.
Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, C.
Gotti20,k, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38,
E. Graugés36, E. Graverini40, G. Graziani17, A. Grecu29, E. Greening55, S.
Gregson47, P. Griffith45, L. Grillo11, O. Grünberg63, B. Gui59, E. Gushchin33,
Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C.
Haines47, S. Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-
Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, J. He38, T. Head39, V.
Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E.
van Herwijnen38, M. Heß63, A. Hicheur2, D. Hill55, M. Hoballah5, C. Hombach54,
W. Hulsbergen41, T. Humair53, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, A.
Jawahery58, F. Jing3, M. John55, D. Johnson38, C.R. Jones47, C. Joram38, B.
Jost38, N. Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38,†,
S. Karodia51, M. Kelsey59, I.R. Kenyon45, M. Kenzie38, T. Ketel42, B.
Khanji20,38,k, C. Khurewathanakul39, S. Klaver54, K. Klimaszewski28, O.
Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P. Koppenburg41,38, M.
Korolev32, L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P.
Krokovny34, F. Kruse9, W. Kucewicz26,o, M. Kucharczyk26, V. Kudryavtsev34, A.
K. Kuonen39, K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38,
G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, G. Lanfranchi18, C.
Langenbruch48, B. Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van
Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32,38, J. Lefrançois7, O.
Leroy6, T. Lesiak26, B. Leverington11, Y. Li7, T. Likhomanenko65,64, M.
Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, X. Liu3, S. Lohn38,
I. Longstaff51, J.H. Lopes2, D. Lucchesi22,r, M. Lucio Martinez37, H. Luo50,
A. Lupato22, E. Luppi16,f, O. Lupton55, F. Machefert7, F. Maciuc29, O. Maev30,
K. Maguire54, S. Malde55, A. Malinin64, G. Manca7, G. Mancinelli6, P.
Manning59, A. Mapelli38, J. Maratas5, J.F. Marchand4, U. Marconi14, C. Marin
Benito36, P. Marino23,38,t, R. Märki39, J. Marks11, G. Martellotti25, M.
Martinelli39, D. Martinez Santos42, F. Martinez Vidal66, D. Martins Tostes2,
A. Massafferri1, R. Matev38, A. Mathad48, Z. Mathe38, C. Matteuzzi20, K.
Matthieu11, A. Mauri40, B. Maurin39, A. Mazurov45, M. McCann53, J. McCarthy45,
A. McNab54, R. McNulty12, B. Meadows57, F. Meier9, M. Meissner11, M. Merk41,
D.A. Milanes62, M.-N. Minard4, D.S. Mitzel11, J. Molina Rodriguez60, S.
Monteil5, M. Morandin22, P. Morawski27, A. Mordà6, M.J. Morello23,t, J.
Moron27, A.B. Morris50, R. Mountain59, F. Muheim50, J. Müller9, K. Müller40,
V. Müller9, M. Mussini14, B. Muster39, P. Naik46, T. Nakada39, R.
Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21, S. Neubert11, N.
Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,q, V.
Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, D. Ninci23, A. Novoselov35, D.P.
O’Hanlon48, A. Oblakowska-Mucha27, V. Obraztsov35, S. Ogilvy51, O.
Okhrimenko44, R. Oldeman15,e, C.J.G. Onderwater67, B. Osorio Rodrigues1, J.M.
Otalora Goicochea2, A. Otto38, P. Owen53, A. Oyanguren66, A. Palano13,c, F.
Palombo21,u, M. Palutan18, J. Panman38, A. Papanestis49, M. Pappagallo51, L.L.
Pappalardo16,f, C. Parkes54, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, A. Pearce54,49, A. Pellegrino41, G. Penso25,m, M. Pepe
Altarelli38, S. Perazzini14,d, P. Perret5, L. Pescatore45, K. Petridis46, A.
Petrolini19,j, M. Petruzzo21, E. Picatoste Olloqui36, B. Pietrzyk4, T.
Pilař48, D. Pinci25, A. Pistone19, A. Piucci11, S. Playfer50, M. Plo
Casasus37, T. Poikela38, F. Polci8, A. Poluektov48,34, I. Polyakov31, E.
Polycarpo2, A. Popov35, D. Popov10,38, B. Popovici29, C. Potterat2, E.
Price46, J.D. Price52, J. Prisciandaro39, A. Pritchard52, C. Prouve46, V.
Pugatch44, A. Puig Navarro39, G. Punzi23,s, W. Qian4, R. Quagliani7,46, B.
Rachwal26, J.H. Rademacker46, B. Rakotomiaramanana39, M. Rama23, M.S. Rangel2,
I. Raniuk43, N. Rauschmayr38, G. Raven42, F. Redi53, S. Reichert54, M.M.
Reid48, A.C. dos Reis1, S. Ricciardi49, S. Richards46, M. Rihl38, K.
Rinnert52, V. Rives Molina36, P. Robbe7,38, A.B. Rodrigues1, E. Rodrigues54,
J.A. Rodriguez Lopez62, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A.
Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, H. Ruiz36, P. Ruiz
Valls66, J.J. Saborido Silva37, N. Sagidova30, P. Sail51, B. Saitta15,e, V.
Salustino Guimaraes2, C. Sanchez Mayordomo66, B. Sanmartin Sedes37, R.
Santacesaria25, C. Santamarina Rios37, M. Santimaria18, E. Santovetti24,l, A.
Sarti18,m, C. Satriano25,n, A. Satta24, D.M. Saunders46, D. Savrina31,32, M.
Schiller38, H. Schindler38, M. Schlupp9, M. Schmelling10, T. Schmelzer9, B.
Schmidt38, O. Schneider39, A. Schopper38, M. Schubiger39, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25,m, A. Semennikov31, I. Sepp53, N.
Serra40, J. Serrano6, L. Sestini22, P. Seyfert11, M. Shapkin35, I.
Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34, V. Shevchenko64,
A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N. Skidmore46, I.
Skillicorn51, T. Skwarnicki59, E. Smith55,49, E. Smith53, I. T. Smith50, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57,38, F.J.P. Soler51, F.
Soomro39, D. Souza46, B. Souza De Paula2, B. Spaan9, P. Spradlin51, S.
Sridharan38, F. Stagni38, M. Stahl11, S. Stahl38, O. Steinkamp40, O.
Stenyakin35, F. Sterpka59, S. Stevenson55, S. Stoica29, S. Stone59, B.
Storaci40, S. Stracka23,t, M. Straticiuc29, U. Straumann40, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, T. Szumlak27, S. T’Jampens4, T. Tekampe9, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, J. Todd57, S. Tolk42, L. Tomassetti16,f, D. Tonelli38,
S. Topp-Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, K. Trabelsi39,
M.T. Tran39, M. Tresch40, A. Trisovic38, A. Tsaregorodtsev6, P. Tsopelas41, N.
Tuning41,38, A. Ukleja28, A. Ustyuzhanin65,64, U. Uwer11, C. Vacca15,e, 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. Voß63, J.A. de Vries41, R. Waldi63, C.
Wallace48, R. Wallace12, J. Walsh23, S. Wandernoth11, J. Wang59, D.R. Ward47,
N.K. Watson45, D. Websdale53, A. Weiden40, M. Whitehead48, D. Wiedner11, G.
Wilkinson55,38, M. Wilkinson59, M. Williams38, M.P. Williams45, M. Williams56,
T. Williams45, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, K. Wyllie38, Y. Xie61, Z.
Xu39, Z. Yang3, J. Yu61, X. Yuan34, O. Yushchenko35, M. Zangoli14, M.
Zavertyaev10,b, L. Zhang3, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
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
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Yandex School of Data Analysis, Moscow, Russia, associated to 31
66Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
67Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41

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
†Deceased