EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-225 LHCb-PAPER-2020-036 December 17, 2020

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

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

Measurements of $C\\!P$ observables in $B^{\pm}\rightarrow D^{(*)}K^{\pm}$ and
$B^{\pm}\rightarrow D^{(*)}\pi^{\pm}$ decays are presented, where $D^{(*)}$
indicates a neutral ${D}$ or ${{D}^{*}}$ meson that is an admixture of meson
and anti-meson states. Decays of the ${{D}^{*}}$ meson to the ${D}{{\pi}^{0}}$
and ${D}\gamma$ final states are partially reconstructed without inclusion of
the neutral pion or photon. Decays of the ${D}$ meson are reconstructed in the
${{K}^{\pm}}{{\pi}^{\mp}}$, ${{K}^{+}}{{K}^{-}}$, and ${{\pi}^{+}}{{\pi}^{-}}$
final states. The analysis uses a sample of charged $B$ mesons produced in
proton-proton collisions and collected with the LHCb experiment, corresponding
to integrated luminosities of 2.0, 1.0, and 5.7$\text{\,fb}^{-1}$ taken at
centre-of-mass energies of 7, 8, and 13 TeV, respectively. The measurements of
partially reconstructed ${{B}^{\pm}}\rightarrow{{D}^{*}}{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{{D}^{*}}{{\pi}^{\pm}}$ with
$D\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}$ decays are the first of their kind, and
a first observation of the
${{B}^{\pm}}\rightarrow(D\pi^{0})_{D^{*}}{{\pi}^{\pm}}$ decay is made with a
statistical significance of 6.1 standard deviations. All $C\\!P$ observables
are measured with world-best precision, and in combination with other LHCb
results will provide strong constraints on the CKM angle $\gamma$.

Submitted to JHEP

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Overconstraining the Unitarity Triangle (UT) derived from the Cabibbo-
Kobayashi-Maskawa (CKM) quark-mixing matrix is central to testing the Standard
Model description of charge-parity ($C\\!P$) violation [1, *Kobayashi:1973fv].
The least well-known angle of the UT is
$\gamma\equiv\text{arg}(-V_{ud}V_{ub}^{*}/V_{cd}V_{cb}^{*})$, which has been
determined with a precision of about $5^{\circ}$ from a combination of
measurements [3, 4] and recently with a standalone precision of $5.2^{\circ}$
by LHCb using ${{{B}^{-}}}\rightarrow Dh^{-}$
($h^{-}\in\\{{{\pi}^{-}},{{K}^{-}}\\}$) with
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ decays [5].111 The inclusion of
charge-conjugate processes is implied throughout except in discussions of
asymmetries. The angles $\alpha$ and $\beta$ are measured with $4.5^{\circ}$
and $<1^{\circ}$ precision, respectively [6, 7]. Among the UT angles, $\gamma$
is unique in that it does not depend on any top-quark coupling, and can thus
be measured in $B$-hadron decays that are dominated by tree-level
contributions. In such decays, the interpretation of physical observables
(rates and $C\\!P$ asymmetries) in terms of the underlying UT parameters is
subject to negligible theoretical uncertainties [8]. Any disagreement between
measurements of $\gamma$ and the value inferred from global CKM fits performed
without any $\gamma$ information would thus invalidate the Standard Model
description of $C\\!P$ violation.

The most powerful method for determining $\gamma$ in decays dominated by tree-
level contributions is through the measurement of relative partial widths in
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decays, where $D$ represents an admixture
of the ${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$
states. The amplitude for the ${{{B}^{-}}}\rightarrow{{D}^{0}}{{K}^{-}}$
decay, which at the quark level proceeds via a $b\rightarrow c\bar{u}s$
transition, is proportional to the CKM matrix element $V_{cb}$. The
corresponding amplitude for the ${{{B}^{-}}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{-}}$ decay, which proceeds
via a $b\rightarrow u\bar{c}s$ transition, is proportional to $V_{ub}$. By
studying hadronic $D$ decays accessible to both ${D}^{0}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons, phase information can be
determined from the interference between these two amplitudes. The degree of
the resulting $C\\!P$ violation depends on the size of $r_{B}^{DK}$, the ratio
of the magnitudes of the ${{{B}^{-}}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}{{K}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{0}}{{K}^{-}}$ amplitudes. The relatively large
value of $r_{B}^{DK}\approx 0.10$ [4] in ${{{B}^{-}}}\rightarrow D{{K}^{-}}$
decays allows the determination of the relative phase of the two interfering
amplitudes. This relative phase has both $C\\!P$-violating ($\gamma$) and
$C\\!P$-conserving ($\delta_{B}^{DK}$) contributions; a measurement of the
decay rates for both ${{B}^{+}}$ and ${{B}^{-}}$ mesons gives sensitivity to
$\gamma$. Similar interference effects also occur in ${{{B}^{-}}}\rightarrow
D{{\pi}^{-}}$ decays, albeit with lower sensitivity to the phases due to
additional Cabibbo-suppression which decreases the amplitude ratio relative to
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decays by around a factor of 30.

The ${{{B}^{-}}}\rightarrow{{D}^{*}}{{K}^{-}}$ decay, in which the vector
${{D}^{*}}$ meson222 ${{D}^{*}}$ represents an admixture of the
$D^{*}(2007)^{0}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}}^{*}(2007)^{0}$ states. decays to either
the ${D}{{\pi}^{0}}$ or ${D}\gamma$ final state, also exhibits
$C\\!P$-violating effects when hadronic $D$ decays accessible to both
${D}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptD}}{}^{0}$ mesons are
studied. In this decay, the exact strong-phase difference of $\pi$ between
${{D}^{*}}\rightarrow{D}{{\pi}^{0}}$ and ${{D}^{*}}\rightarrow{D}\gamma$
decays can be exploited to measure $C\\!P$ observables for states with
opposite $C\\!P$ eigenvalues [9]. The amount of $C\\!P$ violation observed in
${{{B}^{-}}}\rightarrow{{D}^{*}}{{K}^{-}}$ depends on the size of
$r_{B}^{{{D}^{*}}\\!K}$, and measurement of the phase for both ${{B}^{+}}$ and
${{B}^{-}}$ allows $\gamma$ and $\delta_{B}^{{{D}^{*}}\\!K}$ to be determined.

The study of ${{{B}^{-}}}\rightarrow D^{(*)}{{K}^{-}}$ decays for measurements
of ${\gamma}$ was first suggested for $C\\!P$ eigenstates of the $D$ decay,
for example the $C\\!P$-even ${D}\rightarrow{{K}^{+}}{{K}^{-}}$ and
${D}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays, labelled herein as GLW modes
[10, 11]. Higher sensitivity to ${\gamma}$ can be achieved using non-$C\\!P$
eigenstates such as $D\rightarrow{{K}^{+}}{{\pi}^{-}}$, where the
${{D}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays are related by the amplitude magnitude ratio $r_{D}^{K\pi}$ and the
strong-phase difference $\delta_{D}^{K\pi}$. The similar magnitude of
$r_{D}^{K\pi}$ and $r_{B}^{DK}$ leads to significant interference between the
two possible suppressed decay paths (favoured $B$ decay followed by suppressed
$D$ decay, and suppressed $B$ decay followed by favoured $D$ decay), resulting
in large $C\\!P$ asymmetries. These decays are herein referred to as ADS modes
[12]. In this work, the GLW $D\rightarrow{{K}^{+}}{{K}^{-}}$ and
$D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ modes are considered, as well as the ADS
$D\rightarrow{{K}^{+}}{{\pi}^{-}}$ mode; the favoured
${D}\rightarrow{{K}^{-}}{{\pi}^{+}}$ decay is used for normalisation purposes
and to define shape parameters in the fit to data. The ${{{B}^{-}}}\rightarrow
D^{(*)}{{K}^{-}}$ and ${{{B}^{-}}}\rightarrow D^{(*)}{{\pi}^{-}}$ GLW modes
have previously been studied by the LHCb collaboration [13], as have the
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ and ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$
ADS modes [14]. This paper reports updated and improved results for these
modes, and a first measurement of the ${{{B}^{-}}}\rightarrow D^{*}{{K}^{-}}$
and ${{{B}^{-}}}\rightarrow D^{*}{{\pi}^{-}}$ ADS modes at LHCb. A sample of
charged $B$ mesons produced in proton-proton ($pp$) collisions and collected
with the LHCb experiment is used, corresponding to integrated luminosities of
2.0, 1.0, and 5.7$\text{\,fb}^{-1}$ taken at centre-of-mass energies of
$\sqrt{s}$ = 7, 8, and 13 TeV, respectively. The small ${{D}^{*}}-{D}$ mass
difference and the conservation of angular momentum in
${{D}^{*}}\rightarrow{D}{{\pi}^{0}}$ and ${{D}^{*}}\rightarrow{D}\gamma$
decays results in distinctive signatures for the
${{{B}^{-}}}\rightarrow{{D}^{*}}h^{-}$ signal in the ${D}h^{-}$ invariant
mass, enabling yields to be obtained with a partial reconstruction technique.
Since the reconstruction efficiency for low momentum neutral pions and photons
is relatively low in LHCb [15], the partial reconstruction method provides
significantly larger yields compared to full reconstruction. However, the
statistical sensitivity per signal decay is reduced since several signal and
background components in the same region of $Dh^{-}$ invariant mass must be
distinguished.

A total of 28 measurements of $C\\!P$ observables are reported, nine of which
correspond to the fully reconstructed ${{{B}^{-}}}\rightarrow Dh^{-}$ decays
while the remaining 19 relate to the partially reconstructed
${{{B}^{-}}}\rightarrow D^{*}h^{-}$ decays. A summary of all measured $C\\!P$
observables is provided in Tables 1 and 2. The GLW modes
$D\rightarrow{{K}^{+}}{{K}^{-}}$ and $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ are
described using common $C\\!P$ observables in the analysis, accounting for
small differences due to the charm $C\\!P$ asymmetry difference $\Delta
A_{{C\\!P}}$ [4]. In addition to the $C\\!P$ observables, the branching
fractions $\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{*0}}{{\pi}^{-}})$ and
$\mathcal{B}({{D}^{*0}}\rightarrow{{D}^{0}}{{\pi}^{0}})$ are measured.

Table 1: The nine $C\\!P$ observables measured using ${{{B}^{-}}}\rightarrow Dh^{-}$ decays, defined in terms of $B$ meson decay widths. Where indicated, $h^{+}h^{-}$ represents an average of the $D\rightarrow{{K}^{+}}{{K}^{-}}$ and $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ modes. The $R$ observables represent partial width ratios and double ratios. The $A$ observables represent $C\\!P$ asymmetries. Observable | Definition
---|---
$A_{K}^{K\pi}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{K}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{K}^{+}})}$
$R_{K/\pi}^{K\pi}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{+}})}$
$A_{K}^{CP}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}{{K}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}{{K}^{+}})}$
$A_{\pi}^{CP}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}{{\pi}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}{{\pi}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}{{\pi}^{+}})}$
$R^{CP}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow[h^{+}h^{-}]_{D}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow[h^{+}h^{-}]_{D}{{\pi}^{+}})}\times\frac{1}{R_{K/\pi}^{K\pi}}$
$R_{K^{+}}^{\pi K}$ | $\frac{\Gamma({{{B}^{+}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{K}^{+}})}{\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{K}^{+}})}$
$R_{K^{-}}^{\pi K}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{K}^{-}})}{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{K}^{-}})}$
$R_{\pi^{+}}^{\pi K}$ | $\frac{\Gamma({{{B}^{+}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{+}})}{\Gamma({{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{+}})}$
$R_{\pi^{-}}^{\pi K}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{-}})}{\Gamma({{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{-}})}$
Table 2: The 19 $C\\!P$ observables measured using ${{{B}^{-}}}\rightarrow D^{*}h^{-}$ decays, defined in terms of $B$ meson decay widths. Where indicated, $h^{+}h^{-}$ represents an average of the $D\rightarrow{{K}^{+}}{{K}^{-}}$ and $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ modes. The $R$ observables represent partial width ratios and double ratios. The $A$ observables represent $C\\!P$ asymmetries. The $R_{K/\pi}^{K\pi,{\gamma}/{{\pi}^{0}}}$ observable is an average over the $D^{*}\rightarrow D{{\pi}^{0}}$ and $D^{*}\rightarrow D{\gamma}$ modes. Observable | Definition
---|---
$A_{K}^{K\pi,\gamma}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{K}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{K}^{+}})}$
$A_{K}^{K\pi,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}$
$A_{\pi}^{K\pi,\gamma}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{+}})}$
$A_{\pi}^{K\pi,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}$
$R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma}/{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma}/{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma}/{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma}/{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}$
$A_{K}^{CP,\gamma}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{K}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{K}^{+}})}$
$A_{K}^{CP,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}$
$A_{\pi}^{CP,\gamma}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{+}})}$
$A_{\pi}^{CP,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})\phantom{0}-\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}$
$R^{CP,\gamma}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{+}})}\times\frac{1}{R_{K/\pi}^{K\pi,\gamma/\pi^{0}}}$
$R^{CP,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{-}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})\phantom{0}+\phantom{0}\Gamma({{{B}^{+}}}\rightarrow([h^{+}h^{-}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}\times\frac{1}{R_{K/\pi}^{K\pi,\gamma/\pi^{0}}}$
$R_{K^{+}}^{\pi K,{\gamma}}$ | $\frac{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{K}^{+}})}$
$R_{K^{-}}^{\pi K,{\gamma}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{K}^{-}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{K}^{-}})}$
$R_{K^{+}}^{\pi K,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{+}})}$
$R_{K^{-}}^{\pi K,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{K}^{-}})}$
$R_{\pi^{+}}^{\pi K,{\gamma}}$ | $\frac{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{+}})}{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{+}})}$
$R_{\pi^{-}}^{\pi K,{\gamma}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{-}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{\gamma})_{{D}^{*}}{{\pi}^{-}})}$
$R_{\pi^{+}}^{\pi K,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}{\Gamma({{{B}^{+}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{+}})}$
$R_{\pi^{-}}^{\pi K,{{\pi}^{0}}}$ | $\frac{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})}{\Gamma({{{B}^{-}}}\rightarrow([{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{0}})_{{D}^{*}}{{\pi}^{-}})}$

##  2 LHCb detector and simulation

The LHCb detector [16, 15] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of the momentum, $p$,
of charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200$\text{\,Ge\kern-1.00006ptV\\!/}c$. The minimum
distance of a track to a primary $pp$ collision vertex (PV), the impact
parameter (IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov (RICH)
detectors. Photons, electrons, and hadrons are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter, and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers. The online event selection is performed by a trigger, which consists
of a hardware stage, based on information from the calorimeter and muon
systems, followed by a software stage, where a full event reconstruction is
applied. The events considered in the analysis are triggered at the hardware
level either when one of the final-state tracks of the signal decay deposits
enough energy in the calorimeter system, or when one of the other particles in
the event, not reconstructed as part of the signal candidate, fulfils any
trigger requirement. At the software stage, it is required that at least one
particle should have high $p_{\mathrm{T}}$ and high $\chi^{2}_{\text{IP}}$,
where $\chi^{2}_{\text{IP}}$ is defined as the difference in the PV fit
$\chi^{2}$ with and without the inclusion of that particle. A multivariate
algorithm [17] is used to identify displaced vertices consistent with being a
two-, three-, or four-track $b$-hadron decay. The PVs are fitted with and
without the $B$ candidate tracks, and the PV that gives the smallest
$\chi^{2}_{\text{IP}}$ is associated with the $B$ candidate.

Simulation is required to model the invariant mass distributions of the signal
and background contributions and determine their selection efficiencies. In
the simulation, $pp$ collisions are generated using Pythia [18,
*Sjostrand:2006za] with a specific LHCb configuration [20]. Decays of unstable
particles are described by EvtGen [21], in which final-state radiation is
generated using Photos [22]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [23,
*Agostinelli:2002hh] as described in Ref. [25]. Some subdominant sources of
background are generated with a fast simulation [26] that mimics the geometric
acceptance and tracking efficiency of the LHCb detector as well as the
dynamics of the decay via EvtGen.

##  3 Event selection

After reconstruction of a $D$-meson candidate from two oppositely charged
particles, the same event selection is applied to all ${{{B}^{-}}}\rightarrow
D^{(*)}h^{-}$ channels in both data and simulation. Since the neutral pion or
photon from the vector ${D}^{*}$ decay is not reconstructed, partially
reconstructed ${{{B}^{-}}}\rightarrow{{D}^{*}}h^{-}$ candidates and fully
reconstructed ${{{B}^{-}}}\rightarrow{D}h^{-}$ candidates contain the same
reconstructed particles, and thus appear in the same sample. These decays are
distinguished using their reconstructed invariant mass $m(Dh^{-})$, as
described in Sec. 4.

The reconstructed $D$-meson candidate mass is required to be within $\pm
25\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known ${D}^{0}$ mass [27]; this
range corresponds to approximately three times the mass resolution. The kaon
or pion originating directly from the ${{B}^{-}}$ decay, subsequently referred
to as the companion particle, is required to have $p_{\mathrm{T}}$ in the
range 0.5–10$\text{\,Ge\kern-1.00006ptV\\!/}c$ and $p$ in the range
5–100$\text{\,Ge\kern-1.00006ptV\\!/}c$. These requirements ensure that the
track is within the kinematic coverage of the RICH detectors, which provide
particle identification (PID) information used to create independent samples
of ${{{B}^{-}}}\rightarrow D^{(*)}{{\pi}^{-}}$ and ${{{B}^{-}}}\rightarrow
D^{(*)}{{K}^{-}}$ decays. Details of the calibration procedure used to
determine PID requirement efficiencies are given in Sec. 4. A kinematic fit is
performed to each decay chain, with vertex constraints applied to both the
${{B}^{-}}$ and $D$ decay products, and the $D$ candidate constrained to its
known mass [28]. The ${{B}^{-}}$ meson candidates with invariant masses in the
interval 4900–5900$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ are retained. This
range includes the partially reconstructed
${{{B}^{-}}}\rightarrow(D{\gamma})_{{D}^{*}}h^{-}$ and
${{{B}^{-}}}\rightarrow(D{{\pi}^{0}})_{{D}^{*}}h^{-}$ decays, which fall at
$m(Dh^{-})$ values below the known ${{B}^{-}}$ meson mass.

A boosted decision tree (BDT) classifier, implemented using the gradient boost
algorithm [29] in the scikit-learn library [30], is employed to achieve
further background suppression. The BDT classifier is trained using simulated
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{K}^{-}}$ decays and a
background sample of ${{K}^{-}}{{\pi}^{+}}{{K}^{-}}$ combinations in data with
invariant mass in the range 5900–7200$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$.
The BDT classifier is also used on all other $D$ decay modes, and provides
equivalent performance across samples. The input to the BDT classifier is a
set of features that characterise the signal decay. These features can be
divided into two categories: (1) properties of any particle and (2) properties
of composite particles (the $D$ and ${{B}^{-}}$ candidates). Specifically

  1. 1.

$p$, $p_{\mathrm{T}}$, and $\chi^{2}_{\text{IP}}$;

  2. 2.

decay time, flight distance, decay vertex quality, radial distance between the
decay vertex and the PV, and the angle between the particle’s momentum vector
and the line connecting the production and decay vertices.

In addition, a feature that estimates the imbalance of $p_{\mathrm{T}}$ around
the ${{B}^{-}}$ candidate momentum vector is also used. It is defined as

$I_{p_{\mathrm{T}}}=\frac{p_{\mathrm{T}}({{{B}^{-}}})-\Sigma
p_{\mathrm{T}}}{p_{\mathrm{T}}({{{B}^{-}}})+\Sigma p_{\mathrm{T}}}\,,$

where the sum is taken over charged tracks inconsistent with originating from
the PV which lie within a cone around the ${{B}^{-}}$ candidate, excluding
tracks used to make the signal candidate. The cone is defined by a circle with
a radius of 1.5 in the plane of pseudorapidity and azimuthal angle defined in
radians. Including the $I_{p_{\mathrm{T}}}$ feature in the BDT classifier
training gives preference to ${{B}^{-}}$ candidates that are isolated from the
rest of the event.

Since no PID information is used in the BDT classifier, the efficiencies for
${{{B}^{-}}}\rightarrow D^{(*)}{{K}^{-}}$ and ${{{B}^{-}}}\rightarrow
D^{(*)}{{\pi}^{-}}$ decays are similar, with insignificant variations arising
from small differences in the decay kinematics. The requirement applied to the
BDT classifier response is optimised by minimising the expected relative
uncertainty on $R_{K}^{\pi K}$ (see Table 1 for definition), as measured using
the invariant mass fit described in Sec. 4. The purity of the sample is
further improved by requiring that all kaons and pions in the $D$ decay are
positively identified by the RICH [31, 32]; this selection has an efficiency
of about 90% per final-state particle.

Peaking background contributions from charmless decays that result in the same
final state as the signal are suppressed by requiring that the flight distance
of the $D$ candidate from the ${{B}^{-}}$ decay vertex is larger than two
times its uncertainty. Peaking background from
${{{B}^{-}}}\rightarrow[h_{1}^{-}h^{+}]_{D}h_{2}^{-}$ signal decays, where
$h_{1}^{-}$ and $h_{2}^{-}$ are exchanged, are vetoed by requiring that the
$h_{2}^{-}h^{+}$ invariant mass is more than
25$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ away from the known ${D}^{0}$ mass. A
veto is also applied on candidates consistent with containing a fully
reconstructed $D^{*}\rightarrow D{\gamma}/{{\pi}^{0}}$ candidate, in order to
statistically decouple this measurement from a possible analysis of fully
reconstructed ${{{B}^{-}}}\rightarrow D^{*}h^{-}$ decays; the veto is found to
be 98.5% efficient on data.

Background from favoured decays misidentified as ADS decays is reduced by
application of the $D$ mass window and $D$ decay product PID requirements
detailed above. To further reduce this background, an additional veto on the
$D$ mass, calculated with both decay products misidentified, is applied to the
favoured and ADS samples, where candidates are required to fall further than
15$\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ away from the known ${{D}^{0}}$ mass.

##  4 Invariant-mass fit

The values of the 28 $C\\!P$ observables and two branching fractions are
determined using a binned extended maximum likelihood fit to the $m(Dh^{-})$
distribution in data. Distinguishing between ${{B}^{+}}$ and ${{B}^{-}}$
candidates, companion particle hypotheses, and the four $D$ decay product
final states, yields 16 independent samples which are fitted simultaneously.
The invariant-mass spectra and results of the fit are shown in Figs. 2$-$5,
where the 16 subsamples are displayed separately. A legend listing each fit
component is provided in Fig. 1.

Figure 1: Legend indicating the invariant-mass fit components shown in Figs.
2$-$5.

Figure 2: Invariant-mass distribution of selected
${{B}^{\pm}}\rightarrow[{{K}^{\pm}}{{\pi}^{\mp}}]_{D}h^{\pm}$ candidates. The
result of the fit is shown by the solid navy line, and each component is
listed in a legend provided in Fig. 1.

Figure 3: Invariant-mass distribution of
${{B}^{\pm}}\rightarrow[{{K}^{\mp}}{{\pi}^{\pm}}]_{D}h^{\pm}$ candidates with
the fit result overlaid. A legend is provided in Fig. 1.

Figure 4: Invariant-mass distribution of
${{B}^{\pm}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}h^{\pm}$ candidates with the
fit result overlaid. A legend is provided in Fig. 1.

Figure 5: Invariant-mass distribution of
${{B}^{\pm}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}h^{\pm}$ candidates with
the fit result overlaid. A legend is provided in Fig. 1.

###  4.1 Fit components

The total probability density function (PDF) is built from six signal
functions, one for each of the ${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}$,
${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$,
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$,
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{K}^{-}}$,
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{\pi}^{-}}$, and
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{K}^{-}}$ decays. In addition,
there are PDFs that describe the misidentified signal and background
components, combinatorial background, background from $B$ decays to charmless
final states, background from favoured decays misidentified as ADS decays, and
background from other partially reconstructed decays. All PDFs are identical
for ${{B}^{+}}$ and ${{B}^{-}}$ decays. In cases where shape parameters are
derived from simulation and fixed in the fit to data, the parameter
uncertainties are considered as a source of systematic uncertainty.

###  ${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}$ decays

The ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ signal component is modelled using a
sum of a double-sided Hypatia PDF [33] and a Johnson SU PDF [34]. Both PDFs
have a common mean which is shared across all samples. The width of the
Johnson SU component varies freely in the favoured and GLW modes, while the
ADS mode shares this parameter with the favoured mode. The Hypatia width is
related to the Johnson SU width with a single freely varying parameter, which
is shared across all $D$ decay modes. The relative fraction of the Hypatia
PDF, and the kurtosis ($\gamma$) and skewness ($\delta$) parameters of the
Johnson SU PDF, are shared across all $D$ modes and vary freely in the data
fit. All other shape parameters are fixed to the values found in fits to
simulated samples of ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ signal decays.

The contribution from ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ decays
misidentified as ${{{B}^{-}}}\rightarrow D{{K}^{-}}$ is shifted to higher
invariant masses in the $D{{K}^{-}}$ samples. These misidentified candidates
are modelled with the sum of two Crystal Ball PDFs [35] with a common mean.
All shape parameters are fixed to the values found in simulation.

###  ${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ decays

In the $D{{K}^{-}}$ samples, the ${{{B}^{-}}}\rightarrow D{{K}^{-}}$ signal is
described using the sum of a Hypatia PDF and Johnson SU PDF. The Hypatia and
Johnson SU widths are related to the corresponding ${{{B}^{-}}}\rightarrow
D{{\pi}^{-}}$ signal values with a single freely varying ratio, while the
Hypatia fraction and Johnson SU $\gamma$ and $\delta$ parameters are shared
with the ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ signal PDF. All other shape
parameters are fixed to the values found in fits to simulated samples of
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ signal decays.

Misidentified ${{{B}^{-}}}\rightarrow D{{K}^{-}}$ decays are displaced to
lower masses in the $D{{\pi}^{-}}$ samples. These candidates are modelled with
the sum of two Crystal Ball PDFs with a common mean. The mean, widths, and
tail parameters are fixed to the values found in simulation.

###  ${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$ decays

In partially reconstructed decays involving a vector meson, the ${D}h^{-}$
invariant-mass distribution depends upon the spin and mass of the missing
particle. For ${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$
decays, the missing neutral pion has spin-parity $0^{-}$. The distribution is
described by a parabola exhibiting a minimum, whose range is defined by the
kinematic endpoints of the decay. It is convolved with a Gaussian resolution
function, yielding

$f(m)=\int_{a}^{b}\\!\left(\mu-\frac{a+b}{2}\right)^{2}\left(\frac{1-\xi}{b-a}\mu+\frac{b\xi-a}{b-a}\right)e^{-\frac{(\mu-m)^{2}}{2\sigma^{2}}}\text{d}\mu\\\
\,.$ (1)

The resulting distribution has a characteristic double-peaked shape, visible
in Figs. 2$-$5 as dark blue filled regions appearing to the left of the fully
reconstructed ${{{B}^{-}}}\rightarrow Dh^{-}$ peaks. The lower and upper
endpoints of the parabola are $a$ and $b$, respectively, while the relative
height of the lower and upper peaks is determined by the $\xi$ parameter. When
$\xi=1$, both peaks are of equal height, and a deviation of $\xi$ from unity
accounts for mass-dependent reconstruction and selection efficiency effects.
The values of $a$, $b$, and $\xi$ are taken from simulation, while the
Gaussian resolution $\sigma$ is allowed to vary freely in the favoured and GLW
samples; the ADS widths are shared with the favoured mode.

Partially reconstructed
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$ decays, where
the companion pion is misidentified as a kaon, are parameterised with a semi-
empirical PDF, formed from the sum of Gaussian and error functions. The
parameters of this PDF are fixed to the values found in fits to simulated
events.

###  ${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{K}^{-}}$ decays

Equation 1 is also used to describe partially reconstructed
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{K}^{-}}$ decays, where
the width $\sigma$ in each of the $D{{K}^{-}}$ samples is related to the
$D{{\pi}^{-}}$ width by a freely varying ratio which is shared with the
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ signal PDFs. The kinematic endpoints $a$
and $b$ are determined from a fit to simulated events, and the $\xi$ parameter
is shared with the
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$ PDF.

Partially reconstructed
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{K}^{-}}$ decays, where
the companion kaon is misidentified as a pion, are parameterised with a semi-
empirical PDF, formed from the sum of Gaussian and error functions. The
parameters of this PDF are fixed to the values found in fits to simulated
events.

###  ${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{\pi}^{-}}$ decays

Partially reconstructed
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{\pi}^{-}}$ decays involve a
missing particle of zero mass and spin-parity $1^{-}$. The ${D}{{\pi}^{-}}$
invariant-mass distribution is described by a parabola exhibiting a maximum,
convolved with a Gaussian resolution function. The functional form of this
component is

$f(m)=\int_{a}^{b}\\!-(\mu-a)(\mu-b)\left(\frac{1-\xi}{b-a}\mu+\frac{b\xi-a}{b-a}\right)e^{-\frac{(\mu-m)^{2}}{2\sigma^{2}}}\text{d}\mu\,.$
(2)

This distribution exhibits a broad single peak, as opposed to the double-
peaked ${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$
distribution described by Eq. 1. In Figs. 2$-$5, this component is visible as
the light blue filled region to the left of the fully reconstructed
${{{B}^{-}}}\rightarrow Dh^{-}$ peaks. The values of $a$, $b$, $\xi$, and
$\sigma$ are fixed using fits to simulated events. The difference between the
invariant-mass distributions of
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$ decays enables
their statistical separation in the fit, and hence the determination of
$C\\!P$ observables for each mode.

Partially reconstructed
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{\pi}^{-}}$ decays where the
companion pion is misidentified as a kaon are treated in an equivalent manner
to misidentified
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$ decays, as
described above.

###  ${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{K}^{-}}$ decays

Equation 2 is also used to describe partially reconstructed
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{K}^{-}}$ decays, where the
width $\sigma$ in each of the $D{{K}^{-}}$ samples is related to the
$D{{\pi}^{-}}$ width by a common ratio shared with the ${{{B}^{-}}}\rightarrow
D{{K}^{-}}$ signal PDFs. The kinematic endpoints $a$ and $b$ are derived from
a fit to simulated events, and the $\xi$ parameter is shared with the
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{\pi}^{-}}$ PDF. Partially
reconstructed ${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{K}^{-}}$ decays
where the companion kaon is misidentified as a pion are treated in an
equivalent manner to misidentified
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{K}^{-}}$ decays.

### Combinatorial background

An exponential PDF is used to describe the combinatorial background.
Independent and freely varying exponential parameters and yields are used to
model this component in each subsample.

### Charmless background

Charmless ${{{B}^{-}}}\rightarrow h_{1}^{-}h_{2}^{-}h^{+}$ decays, where
$h_{1}^{-}$, $h_{2}^{-}$, and $h^{+}$ each represent a charged kaon or pion,
peak at the ${{B}^{-}}$ mass and cannot be distinguished effectively from the
fully reconstructed ${{{B}^{-}}}\rightarrow{D}h^{-}$ signals. A Crystal Ball
PDF is used to model this component, with shape parameters fixed to the values
found in a fit to simulated
${{{B}^{-}}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays.

The charmless contribution is determined from fits to the ${{B}^{-}}$ mass
spectrum in the $D$-mass sidebands, without the kinematic fit of the decay
chain. The charmless background yields are determined independently for
${{B}^{+}}$ and ${{B}^{-}}$ candidates and are then fixed in the analysis.
Their uncertainties contribute to the systematic uncertainties of the final
results. The largest charmless contribution is in the
${{{B}^{-}}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}{{K}^{-}}$ sample, which
has a yield corresponding to 10% of the measured signal yield.

Partially reconstructed charmless decays of the type $B\rightarrow
h_{1}^{-}h_{2}^{-}h^{+}X$, where $X$ is a charged pion, neutral pion, or
photon that has not been reconstructed, contribute at low invariant masses.
Their contributions are determined relative to the fully reconstructed
charmless components using a freely varying ratio which is shared across all
$D$ decay modes. A parabola exhibiting a minimum convolved with a Gaussian
resolution function is used to model this component, with shape parameter
values taken from simulation.

### Partially reconstructed background

Several additional partially reconstructed $b$-hadron decays contribute at low
invariant-mass values. The dominant contributions are from
${{{B}^{-}}}\rightarrow{D}h^{-}{{\pi}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow({D}{{\pi}^{+}})_{{{D}^{*+}}}{{\pi}^{-}}$
decays, where a neutral pion or positively charged pion is not
reconstructed.333 When considering partially reconstructed background sources,
the production fractions $f_{u}$ and $f_{d}$ are taken to be equal [27]. The
invariant-mass distribution of these sources depends upon the spin and mass of
the missing particle, as with the ${{{B}^{-}}}\rightarrow{{D}^{*}}h^{-}$
signals. In both cases, the missing particle has spin-parity $0^{-}$, such
that the $Dh^{-}$ distribution is described using Eq. 1, with shape parameter
values taken from simulation. The Dalitz structure of
${{{B}^{-}}}\rightarrow{D}h^{-}{{\pi}^{0}}$ decays is modelled using Laura++
[36] and the amplitude model from Ref. [37].

The yields of the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow({D}{{\pi}^{+}})_{{{D}^{*+}}}{{\pi}^{-}}$
and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow({D}{{\pi}^{+}})_{{{D}^{*+}}}{{K}^{-}}$
contributions, where the ${{\pi}^{+}}$ meson is not reconstructed, are fixed
relative to the corresponding ${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}$ yields
using branching fractions [27] and efficiencies obtained from simulation. The
$C\\!P$ asymmetries of these modes are fixed to zero in all subsamples, as no
$C\\!P$ violation is expected in a time-integrated measurement.

The yields of the ${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}{{\pi}^{0}}$ decay vary
freely in the favoured and GLW subsamples allowing for potential $C\\!P$
violation, and the total rate in the ADS mode is fixed to $(4.0\pm 1.3)\times
10^{-3}$ relative to the favoured mode yield. This estimate is based on the
expectation that $r_{B}^{D\pi{{\pi}^{0}}}$ is similar in size to
$r_{B}^{D\pi}$, and thus much smaller than the $D$ decay amplitude ratio
$r_{D}^{K\pi}$ [27]. No $C\\!P$ violation is permitted in the data fit, but
the fixed values of $r_{B^{-}}^{D\pi{{\pi}^{0}}}$ and
$r_{B^{-}}^{D\pi{{\pi}^{0}}}$ are independently varied to determine the
systematic uncertainty to account for potential $C\\!P$ violation. In the ADS
mode, a separate contribution from colour-suppressed
${{B}^{0}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{+}}{{\pi}^{-}}$ decays
is also included, where the ${{\pi}^{+}}$ meson is not reconstructed. The
yield of this component varies freely in the fit, and shape parameters are
taken from a fit to simulated samples generated using Laura++ and the
amplitude model from Ref. [37].

In the favoured $DK^{-}$ sample, the yield of the
${{{B}^{-}}}\rightarrow{D}{{K}^{-}}{{\pi}^{0}}$ component varies freely in
both the ${{{B}^{-}}}$ and ${{{B}^{+}}}$ subsamples, allowing for the presence
of a $B^{0}\rightarrow D{{K}^{-}}{{\pi}^{+}}$ contribution. In the GLW and ADS
modes, average values and uncertainties from Ref. [4] are used to estimate the
expected rates and $C\\!P$ asymmetries, accounting for the presence of
$B^{0}\rightarrow D{{K}^{-}}{{\pi}^{+}}$. These quantities are fixed in the
invariant-mass fit, and are considered as sources of systematic uncertainty.
The distribution is modelled using a fit to simulated events generated using
Laura++ and the amplitude model from Ref. [38].

Contributions from partially reconstructed
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}}/\gamma)_{{{D}^{*}}}h^{-}{{\pi}^{0}}$
and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow({D}{{\pi}^{+}})_{{{D}^{*+}}}h^{-}{{\pi}^{0}}$
decays occur at the lowest values of invariant mass, where two particles are
not reconstructed. These decays are described by the sum of several parabolas
convolved with resolution functions according to Eqs. 1 and 2, with shape
parameters fixed to the values found in fits to simulated samples. The yields
and $C\\!P$ asymmetries of these contributions vary freely in each subsample.

In the ${{{B}^{-}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}h^{-}$ samples,
${{\mathchar
28931\relax}^{0}_{b}}\rightarrow[p^{+}{{K}^{-}}{{\pi}^{+}}]_{{{\mathchar
28931\relax}^{+}_{c}}}h^{-}$ decays contribute to the background when the pion
is missed and the proton is misidentified as the second kaon. The PDF
describing this component is fixed from simulation, but the yields in the
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}{{K}^{-}}$ subsamples vary
freely. A contribution from ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow[p^{+}{{\pi}^{-}}{{\pi}^{+}}]_{{{\mathchar
28931\relax}^{+}_{c}}}h^{-}$ in the ADS $D{{\pi}^{-}}$ mode is also modelled
using the same wide PDF, with a yield fixed relative to the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow[p^{+}{{K}^{-}}{{\pi}^{+}}]_{{{\mathchar
28931\relax}^{+}_{c}}}{{\pi}^{-}}$ yield measured in the
${{{B}^{-}}}\rightarrow[{{K}^{+}}{{K}^{-}}]_{D}{{\pi}^{-}}$ sample using
${{\mathchar 28931\relax}^{+}_{c}}$ branching fractions [27].

In the ADS $D{{K}^{-}}$ sample, a contribution from ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}p{{\pi}^{-}}$ decays
is modelled, where the proton is misidentified as a kaon and the pion is not
reconstructed. A fixed shape is used to describe this contribution, with shape
parameters determined using a fit to a sample of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}p{{\pi}^{-}}$ decays
in data taken from Ref. [39]. The rate of this mode is fixed in the invariant-
mass fit relative to the favoured ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ yield
using branching fractions [27], efficiencies derived from simulation, and the
${{\mathchar 28931\relax}^{0}_{b}}$ production fraction relative to
${{{B}^{+}}}$ as measured at LHCb [40]. The $C\\!P$ asymmetry is assumed to be
zero for this favoured decay in the invariant-mass fit.

In the ADS $D{{K}^{-}}$ sample, and to a lesser extent in the GLW $D{{K}^{-}}$
samples, ${{B}^{0}_{s}}\rightarrow{D}{{K}^{-}}{{\pi}^{+}}$ decays in which the
companion pion is not reconstructed contribute to the background. The PDF
describing this component is fixed from fits to simulated samples generated
according to the amplitude model from Ref. [41]. The yield of this component
varies freely in the ADS mode, and the GLW mode yields are fixed relative to
that using $D$ branching fractions [27]. Contributions from
${{B}^{0}_{s}}\rightarrow(D{\gamma}/{{\pi}^{0}})_{{D}^{*}}{{K}^{-}}{{\pi}^{+}}$
are also modelled using simulated samples generated with a longitudinal
polarisation fraction $f_{\rm L}=0.9\pm 0.1$; as this quantity has not yet
been measured for
${{B}^{0}_{s}}\rightarrow(D{\gamma}/{{\pi}^{0}})_{{D}^{*}}{{K}^{-}}{{\pi}^{+}}$
decays, the value used is based on the $B^{-}\rightarrow D^{*}K^{*-}$
measurement [42] with an additional systematic uncertainty assigned to account
for potential differences. The yield of this component is fixed relative to
the freely varying ${{B}^{0}_{s}}\rightarrow{D}{{K}^{-}}{{\pi}^{+}}$ yield
assuming the same branching fraction with 25% uncertainty, and adjusting for
the relative efficiency as determined using simulation. The $C\\!P$
asymmetries of the ${{B}^{0}_{s}}\rightarrow D^{(*)}{{K}^{-}}{{\pi}^{+}}$
contributions are assumed to be zero, as no $C\\!P$ violation is expected for
these modes in a time-integrated measurement.

In the ADS $D{{\pi}^{-}}$ sample, a contribution from favoured
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$
decays is modelled, where the two ${{\pi}^{+}}$ produced in the ${{{B}^{+}}}$
decay are not reconstructed. The rate of this contribution varies freely, with
a fixed shape determined from a fit to simulated
${{{B}^{+}}}\rightarrow[{{K}^{+}}{{\pi}^{-}}]_{D}({{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})_{a_{1}(1270)^{+}}$
decays. Only the $a_{1}(1270)^{+}$ contribution is simulated as this is the
dominant $3\pi$ resonance observed in this mode [43]. The $C\\!P$ asymmetry is
assumed to be zero for this favoured decay in the invariant-mass fit.

For all partially reconstructed background contributions considered in the
fit, components accounting for particle misidentification are also taken into
account. They are parameterised with semi-empirical PDFs formed from the sum
of Gaussian and error functions. The parameters of each of these PDFs are
fixed to the values found in fits to simulated events.

### Background from favoured decays in the ADS samples

The favoured and ADS signal modes have identical final states aside from the
relative charge of the companion hadron and the kaon produced in the $D$
decay. It is possible to misidentify both $D$ decay products, such that a
favoured decay is reconstructed as an ADS signal candidate. Given the much
larger rate of favoured decays relative to the ADS signals, this crossfeed
background must be reduced to a manageable level with specific requirements.
Using simulated favoured ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ decays, a
combination of the $D$ invariant-mass window, $D$ decay product PID
requirements, and the veto on the $D$ mass calculated with both decay products
misidentified, is found to accept doubly misidentified decays at the $10^{-4}$
level relative to correctly identified decays. This relative efficiency is
used in the fit to fix the quantity of favoured background in the ADS
subsamples relative to the corresponding favoured yields. The same relative
efficiency is employed for the $D{{\pi}^{-}}$ and $D{{K}^{-}}$ samples, as
well as for the ${{{B}^{-}}}\rightarrow Dh^{-}$ and ${{{B}^{-}}}\rightarrow
D^{*}h^{-}$ signals. The crossfeed components are modelled using fixed shapes
derived from favoured simulated samples reconstructed as ADS decays. Due to
the double misidentification, these components are found to be wider than
their corresponding correctly identified signals.

###  4.2 PID efficiencies

In the $D{{K}^{-}}$ subsamples, the rates of contributions from misidentified
${{{B}^{-}}}\rightarrow D^{(*)}{{\pi}^{-}}$ decays are determined by the fit
to data via a single freely varying parameter. The ${{{B}^{-}}}\rightarrow
D^{(*)}{{K}^{-}}$ contributions in the $D{{\pi}^{-}}$ subsample are not well
separated from background, so the expected yield is determined using a PID
calibration procedure with approximately 40 million
${{D}^{*+}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{+}}$ decays. This
decay is identified using kinematic variables only, and thus provides a pure
sample of ${K}^{\mp}$ and ${\pi}^{\pm}$ particles unbiased in the PID
variables. The PID efficiency is parameterised as a function of particle
momentum and pseudorapidity, as well as the charged-particle multiplicity in
the event. The effective PID efficiency of the signal is determined by
weighting the calibration sample such that the distributions of these
variables match those of selected
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{-}}$ signal decays.
It is found that around 70% of ${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ decays
pass the companion kaon PID requirement and are placed in the $D{{K}^{-}}$
sample, with negligible statistical uncertainty due to the size of the
calibration sample; the remaining 30% fall into the $D{{\pi}^{-}}$ sample.
With the same PID requirement, 99.6% of the
${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}$ decays are correctly identified, as
measured by the fit to data. These efficiencies are also taken to represent
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}h^{-}$ and
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}h^{-}$ signal decays in the fit,
with a correction of 0.98 applied to the kaon efficiency to account for small
differences in companion kinematics. The related systematic uncertainty on the
kaon efficiency is determined by the size of the signal samples used, and thus
increases for the lower yield modes.

###  4.3 Production and detection asymmetries

In order to measure $C\\!P$ asymmetries, the detection asymmetries for
${K}^{\pm}$ and ${\pi}^{\pm}$ mesons must be taken into account. A detection
asymmetry of $(-0.96\pm 0.13)$% is assigned for each kaon in the final state,
primarily due to the fact that the nuclear interaction length of ${K}^{-}$
mesons is shorter than that of ${K}^{+}$ mesons. It is taken from Ref. [44],
where the charge asymmetries in
${{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$ and
${{D}^{-}}\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{-}}$ calibration samples
are compared after weighting to match the kinematics of the signal kaons in
favoured ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ decays. An additional
correction of $(-0.17\pm 0.08)\%$ is applied to the kaon detection asymmetry,
to account for the asymmetry introduced by the hadronic hardware trigger. The
detection asymmetry for pions is smaller, and is taken to be $(-0.17\pm
0.10)$% [44].

The $C\\!P$ asymmetry in the favoured
${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{-}}$ decay is fixed
to $(+0.09\pm 0.05)\%$, calculated using the average value and uncertainty on
$\gamma$ from Ref. [4] and the assumption that $r_{B}^{D\pi}$ is below 0.02
with uniform probability; no assumption is made about the strong phase
$\delta_{B}^{D\pi}$. This enables the effective production asymmetry, defined
as $A^{\rm
eff}_{{{B}^{\pm}}}=\textstyle{\frac{\sigma^{\prime}({{{B}^{-}}})-\sigma^{\prime}({{{B}^{+}}})}{\sigma^{\prime}({{{B}^{-}}})+\sigma^{\prime}({{{B}^{+}}})}}$,
where $\sigma^{\prime}$ is the $B$-meson production cross-section, to be
measured and simultaneously subtracted from the charge asymmetry measurements
in other modes.

To correct for left-right asymmetry effects in the LHCb detector, similarly
sized data samples are collected in two opposite magnet polarity
configurations. The analysis is performed on the total dataset summed over
both polarities, where no residual left-right asymmetry effects remain to be
corrected.

###  4.4 Yields and selection efficiencies

The total yield for each mode is a sum of the number of correctly identified
and misidentified candidates; their values are given in Table 3. To obtain the
observable $R_{K/\pi}^{K\pi}$ ($R_{K/\pi}^{K\pi,{{\pi}^{0}}/\gamma}$) in the
fit, which is defined in Table 1 (2), the ratio of yields is corrected for the
relative efficiency with which ${{{B}^{-}}}\rightarrow{D}{{K}^{-}}$ and
${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}$
(${{{B}^{-}}}\rightarrow{{D}^{*}}{{K}^{-}}$ and
${{{B}^{-}}}\rightarrow{{D}^{*}}{{\pi}^{-}}$) decays are reconstructed and
selected. The relative efficiencies are found to be close to unity, where the
${{{B}^{-}}}\rightarrow D^{(*)}{{\pi}^{-}}$ efficiencies are around 2% larger
than ${{{B}^{-}}}\rightarrow D^{(*)}{{K}^{-}}$. The uncertainties assigned on
these efficiency corrections take into account the size of the simulated
samples and the imperfect modelling of the relative pion and kaon absorption
in the detector material. To determine the branching fraction
$\mathcal{B}({{D}^{*0}}\rightarrow{{D}^{0}}{{\pi}^{0}})$, the yields of the
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}{{\pi}^{-}}$ modes are corrected
for the relative efficiencies of the neutral pion and photon modes as
determined from simulation. As both of these modes are partially reconstructed
with identical selection requirements, the relative efficiency is found to be
close to unity and is varied within its uncertainty to determine the
associated systematic uncertainty. In the measurement of
$\mathcal{B}({{D}^{*}}\rightarrow{D}{{\pi}^{0}})$, the assumption is made that
$\mathcal{B}({{D}^{*}}\rightarrow{D}{{\pi}^{0}})+\mathcal{B}({{D}^{*}}\rightarrow{D}\gamma)=1$
[27]. The branching fraction
$\mathcal{B}({{{B}^{-}}}\rightarrow{{D}^{*0}}{{\pi}^{-}})$ is determined from
the total ${{{B}^{-}}}\rightarrow{{D}^{*}}{{\pi}^{-}}$ and
${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}$ yields, the relative efficiencies
determined from simulation, and the ${{{B}^{-}}}\rightarrow{D}{{\pi}^{-}}$
branching fraction [27]. Both the efficiencies and external input branching
fraction are varied to determine the associated systematic uncertainty.

Table 3: Yields for the 24 signal modes.  Decay |  $D$ mode | Yield
---|---|---
${{B}^{\pm}}\rightarrow D{{\pi}^{\pm}}$ | ${{K}^{-}}{{\pi}^{+}}$ | $1\,771\,385\phantom{0}\pm\phantom{0}2153$
| ${{K}^{+}}{{K}^{-}}$ | $219\,584\phantom{\,0}\phantom{0}\pm\phantom{0}569$
| ${{\pi}^{+}}{{\pi}^{-}}$ | $70\,594\phantom{\,00}\phantom{0}\pm\phantom{0}273$
| ${{K}^{+}}{{\pi}^{-}}$ | $6518\phantom{\,\,000}\phantom{0}\pm\phantom{0}99$
${{B}^{\pm}}\rightarrow D{{K}^{\pm}}$ | ${{K}^{-}}{{\pi}^{+}}$ | $136\,734\phantom{\,0}\phantom{0}\pm\phantom{0}457$
| ${{K}^{+}}{{K}^{-}}$ | $16\,107\phantom{\,00}\phantom{0}\pm\phantom{0}147$
| ${{\pi}^{+}}{{\pi}^{-}}$ | $5178\phantom{\,\,000}\phantom{0}\pm\phantom{0}49$
| ${{K}^{+}}{{\pi}^{-}}$ | $2372\phantom{\,\,000}\phantom{0}\pm\phantom{0}65$
${{B}^{\pm}}\rightarrow(D{{\pi}^{0}})_{D^{*}}{{\pi}^{\pm}}$ | ${{K}^{-}}{{\pi}^{+}}$ | $1\,106\,081\phantom{0}\pm\phantom{0}10\,828$
| ${{K}^{+}}{{K}^{-}}$ | $137\,111\phantom{\,0}\phantom{0}\pm\phantom{0}1378$
| ${{\pi}^{+}}{{\pi}^{-}}$ | $44\,080\phantom{\,00}\phantom{0}\pm\phantom{0}461$
| ${{K}^{+}}{{\pi}^{-}}$ | $5292\phantom{\,\,000}\phantom{0}\pm\phantom{0}543$
${{B}^{\pm}}\rightarrow(D{{\pi}^{0}})_{D^{*}}{{K}^{\pm}}$ | ${{K}^{-}}{{\pi}^{+}}$ | $90\,031\phantom{\,00}\phantom{0}\pm\phantom{0}1197$
| ${{K}^{+}}{{K}^{-}}$ | $11\,660\phantom{\,00}\phantom{0}\pm\phantom{0}277$
| ${{\pi}^{+}}{{\pi}^{-}}$ | $3748\phantom{\,\,000}\phantom{0}\pm\phantom{0}90$
| ${{K}^{+}}{{\pi}^{-}}$ | $1124\phantom{\,\,000}\phantom{0}\pm\phantom{0}231$
${{B}^{\pm}}\rightarrow(D{\gamma})_{D^{*}}{{\pi}^{\pm}}$ | ${{K}^{-}}{{\pi}^{+}}$ | $536\,615\phantom{\,0}\phantom{0}\pm\phantom{0}6065$
| ${{K}^{+}}{{K}^{-}}$ | $66\,519\phantom{\,00}\phantom{0}\pm\phantom{0}769$
| ${{\pi}^{+}}{{\pi}^{-}}$ | $21\,385\phantom{\,00}\phantom{0}\pm\phantom{0}254$
| ${{K}^{+}}{{\pi}^{-}}$ | $2273\phantom{\,\,000}\phantom{0}\pm\phantom{0}403$
${{B}^{\pm}}\rightarrow(D{\gamma})_{D^{*}}{{K}^{\pm}}$ | ${{K}^{-}}{{\pi}^{+}}$ | $44\,255\phantom{\,00}\phantom{0}\pm\phantom{0}899$
| ${{K}^{+}}{{K}^{-}}$ | $5310\phantom{\,\,000}\phantom{0}\pm\phantom{0}361$
| ${{\pi}^{+}}{{\pi}^{-}}$ | $1707\phantom{\,\,000}\phantom{0}\pm\phantom{0}116$
| ${{K}^{+}}{{\pi}^{-}}$ | $674\phantom{\,\,0000}\phantom{0}\pm\phantom{0}931$
Observable | PID | PDF | Rates | Asym | Eff | Veto | Total
---|---|---|---|---|---|---|---
$A_{K}^{CP}$ | 6 | 10 | 11 | 5 | 1 | 10 | 16
$A_{\pi}^{CP}$ | 4 | 14 | 15 | 70 | 3 | 10 | 74
$A_{K}^{K\pi}$ | 12 | 7 | 11 | 49 | 1 | 10 | 52
$R^{CP}$ | 24 | 88 | 58 | 0 | 16 | 10 | 109
$R_{K/\pi}^{K\pi}$ | 47 | 243 | 104 | 1 | 402 | 10 | 483
$R_{K^{-}}^{\pi K}$ | 2 | 48 | 30 | 3 | 2 | 10 | 57
$R_{\pi^{-}}^{\pi K}$ | 2 | 41 | 15 | 13 | 4 | 10 | 46
$R_{K^{+}}^{\pi K}$ | 3 | 47 | 23 | 2 | 6 | 10 | 53
$R_{\pi^{+}}^{\pi K}$ | 2 | 44 | 15 | 15 | 6 | 10 | 50
$A_{K}^{CP,\gamma}$ | 9 | 34 | 40 | 18 | 9 | 10 | 57
$A_{K}^{CP,\pi^{0}}$ | 9 | 28 | 31 | 16 | 10 | 10 | 47
$A_{K}^{K\pi,\gamma}$ | 4 | 8 | 14 | 15 | 2 | 10 | 22
$A_{K}^{K\pi,\pi^{0}}$ | 9 | 12 | 19 | 34 | 5 | 10 | 42
$R^{CP,\gamma}$ | 2 | 87 | 55 | 2 | 22 | 10 | 105
$R^{CP,\pi^{0}}$ | 30 | 87 | 76 | 0 | 33 | 10 | 124
$R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$ | 58 | 292 | 187 | 25 | 185 | 10 | 398
$R_{K^{-}}^{\pi K,\gamma}$ | 13 | 117 | 82 | 14 | 21 | 10 | 146
$R_{K^{-}}^{\pi K,\pi^{0}}$ | 4 | 39 | 48 | 4 | 22 | 10 | 66
$R_{K^{+}}^{\pi K,\gamma}$ | 11 | 117 | 83 | 7 | 21 | 10 | 146
$R_{K^{+}}^{\pi K,\pi^{0}}$ | 3 | 41 | 47 | 3 | 16 | 10 | 64
$A_{\pi}^{CP,\gamma}$ | 2 | 18 | 39 | 11 | 3 | 10 | 45
$A_{\pi}^{CP,\pi^{0}}$ | 2 | 16 | 16 | 31 | 3 | 10 | 39
$A_{\pi}^{K\pi,\gamma}$ | 4 | 22 | 19 | 18 | 4 | 10 | 34
$A_{\pi}^{K\pi,{{\pi}^{0}}}$ | 2 | 2 | 13 | 32 | 1 | 10 | 34
$R_{\pi^{-}}^{\pi K,\gamma}$ | 13 | 114 | 57 | 11 | 6 | 10 | 128
$R_{\pi^{-}}^{\pi K,\pi^{0}}$ | 1 | 86 | 60 | 16 | 15 | 10 | 107
$R_{\pi^{+}}^{\pi K,\gamma}$ | 14 | 115 | 45 | 12 | 8 | 10 | 125
$R_{\pi^{+}}^{\pi K,\pi^{0}}$ | 2 | 85 | 57 | 16 | 9 | 10 | 104
$\mathcal{B}(D^{*}\rightarrow D\pi^{0})$ | 27 | 281 | 76 | 8 | 177 | 10 | 342
$\mathcal{B}(B^{\pm}\rightarrow D^{*0}{{\pi}^{\pm}})$ | 17 | 257 | 148 | 2 | 329 | 10 | 444
Table 4: Systematic uncertainties for all observables, where values are quoted
as a percentage of the statistical uncertainty for a given observable. The
total uncertainty is given by the sum in quadrature of each contribution. PID
refers to fixed PID efficiencies, PDF to fixed PDF parameters, Rates to fixed
background contributions, Asym to the use of fixed detection asymmetries and
background $C\\!P$ asymmetries, Eff to the use of fixed efficiencies from
simulation, and Veto to the procedure used to veto fully reconstructed
${{{B}^{-}}}\rightarrow D^{*}h^{-}$ candidates.

##  5 Systematic uncertainties

The 30 observables of interest (28 $C\\!P$ observables and two branching
fractions) are subject to a set of systematic uncertainties resulting from the
use of fixed parameters in the fit. The systematic uncertainties associated
with using these fixed parameters are assessed by repeating the fit 1000
times, varying the value of each external parameter within its uncertainty
according to a Gaussian distribution. The resulting standard deviation of each
observable under this variation is taken as the systematic uncertainty. The
systematic uncertainties, grouped into six categories, are shown for each
observable in Table 4. Correlations between the categories are negligible, but
correlations within categories are accounted for. The total systematic
uncertainties are determined by the sum in quadrature of each category.

##  6 Results

The $C\\!P$ observable and branching fraction results are

$\displaystyle A_{K}^{CP}$ $\displaystyle=\phantom{-}0.136\phantom{00}\pm
0.009\phantom{00}\pm 0.001,$ $\displaystyle A_{\pi}^{CP}$
$\displaystyle=-0.008\phantom{00}\pm 0.002\phantom{00}\pm 0.002,$
$\displaystyle A_{K}^{K\pi}$ $\displaystyle=-0.011\phantom{00}\pm
0.003\phantom{00}\pm 0.002,$ $\displaystyle R^{CP}$
$\displaystyle=\phantom{-}0.950\phantom{00}\pm 0.009\phantom{00}\pm 0.010,$
$\displaystyle R_{K/\pi}^{K\pi}$
$\displaystyle=\phantom{-}0.0796\phantom{0}\pm 0.0003\phantom{0}\pm 0.0013,$
$\displaystyle R_{K^{-}}^{\pi K}$
$\displaystyle=\phantom{-}0.0095\phantom{0}\pm 0.0005\phantom{0}\pm 0.0003,$
$\displaystyle R_{\pi^{-}}^{\pi K}$ $\displaystyle=\phantom{-}0.00415\pm
0.00008\pm 0.00004,$ $\displaystyle R_{K^{+}}^{\pi K}$
$\displaystyle=\phantom{-}0.0252\phantom{0}\pm 0.0008\phantom{0}\pm 0.0004,$
$\displaystyle R_{\pi^{+}}^{\pi K}$ $\displaystyle=\phantom{-}0.00320\pm
0.00007\pm 0.00004,$ $\displaystyle A_{K}^{CP,\gamma}$
$\displaystyle=\phantom{-}0.123\phantom{00}\pm 0.054\phantom{00}\pm 0.031,$
$\displaystyle A_{K}^{CP,\pi^{0}}$ $\displaystyle=-0.115\phantom{00}\pm
0.019\phantom{00}\pm 0.009,$ $\displaystyle A_{K}^{K\pi,\gamma}$
$\displaystyle=-0.004\phantom{00}\pm 0.014\phantom{00}\pm 0.003,$
$\displaystyle A_{K}^{K\pi,\pi^{0}}$
$\displaystyle=\phantom{-}0.020\phantom{00}\pm 0.007\phantom{00}\pm 0.003,$
$\displaystyle R^{CP,\gamma}$ $\displaystyle=\phantom{-}0.952\phantom{00}\pm
0.062\phantom{00}\pm 0.065,$ $\displaystyle R^{CP,\pi^{0}}$
$\displaystyle=\phantom{-}1.051\phantom{00}\pm 0.022\phantom{00}\pm 0.028,$
$\displaystyle R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$
$\displaystyle=\phantom{-}0.0851\phantom{0}\pm 0.0012\phantom{0}\pm 0.0048,$
$\displaystyle R_{K^{-}}^{\pi K,\gamma}$
$\displaystyle=\phantom{-}0.0117\phantom{0}\pm 0.0215\phantom{0}\pm 0.0313,$
$\displaystyle R_{K^{-}}^{\pi K,\pi^{0}}$
$\displaystyle=\phantom{-}0.0202\phantom{0}\pm 0.0035\phantom{0}\pm 0.0023,$
$\displaystyle R_{K^{+}}^{\pi K,\gamma}$
$\displaystyle=\phantom{-}0.0292\phantom{0}\pm 0.0214\phantom{0}\pm 0.0312,$
$\displaystyle R_{K^{+}}^{\pi K,\pi^{0}}$
$\displaystyle=\phantom{-}0.0033\phantom{0}\pm 0.0035\phantom{0}\pm 0.0022,$
$\displaystyle A_{\pi}^{CP,\gamma}$
$\displaystyle=\phantom{-}0.000\phantom{00}\pm 0.014\phantom{00}\pm 0.006,$
$\displaystyle A_{\pi}^{CP,\pi^{0}}$
$\displaystyle=\phantom{-}0.013\phantom{00}\pm 0.007\phantom{00}\pm 0.003,$
$\displaystyle A_{\pi}^{K\pi,\gamma}$ $\displaystyle=-0.004\phantom{00}\pm
0.004\phantom{00}\pm 0.001,$ $\displaystyle A_{\pi}^{K\pi,{{\pi}^{0}}}$
$\displaystyle=\phantom{-}0.001\phantom{00}\pm 0.002\phantom{00}\pm 0.001,$
$\displaystyle R_{\pi^{-}}^{\pi K,\gamma}$
$\displaystyle=\phantom{-}0.00472\pm 0.00092\pm 0.00118,$ $\displaystyle
R_{\pi^{-}}^{\pi K,\pi^{0}}$ $\displaystyle=\phantom{-}0.00405\pm 0.00056\pm
0.00059,$ $\displaystyle R_{\pi^{+}}^{\pi K,\gamma}$
$\displaystyle=\phantom{-}0.00403\pm 0.00091\pm 0.00114,$ $\displaystyle
R_{\pi^{+}}^{\pi K,\pi^{0}}$ $\displaystyle=\phantom{-}0.00536\pm 0.00056\pm
0.00058,$ $\displaystyle\mathcal{B}(D^{*}\rightarrow D\pi^{0})$
$\displaystyle=\phantom{-}0.655\phantom{00}\pm 0.003\phantom{00}\pm 0.012,$
$\displaystyle\mathcal{B}(B^{\pm}\rightarrow D^{*0}{{\pi}^{\pm}})$
$\displaystyle=\phantom{-}0.00535\pm 0.00004\pm 0.00016\pm 0.00015,$

where the first uncertainties quoted are statistical and the second
systematic; the third uncertainty on $\mathcal{B}({{B}^{\pm}}\rightarrow
D^{*0}{{\pi}^{\pm}})$ accounts for the use of the external branching fraction
$\mathcal{B}({{B}^{\pm}}\rightarrow D^{0}{{\pi}^{\pm}})=(4.68\pm 0.13)\times
10^{-3}$ [27]. The statistical and systematic correlation matrices are given
in App. A. The $R_{h^{-}}^{\pi K}$ and $R_{h^{+}}^{\pi K}$ ADS $C\\!P$
observables can be expressed in terms of a charge-averaged rate $R_{h}^{\pi
K}$ and an asymmetry $A_{h}^{\pi K}$

$\displaystyle R_{h}^{\pi K}$ $\displaystyle=(R_{h^{-}}^{\pi K}+R_{h^{+}}^{\pi
K})/2\,,$ $\displaystyle A_{h}^{\pi K}$ $\displaystyle=(R_{h^{-}}^{\pi
K}-R_{h^{+}}^{\pi K})/(R_{h^{-}}^{\pi K}+R_{h^{+}}^{\pi K})\,.$

The values of these derived observables are

$\displaystyle R_{K}^{\pi K}$ $\displaystyle=\phantom{-}0.0173\phantom{0}\pm
0.0006,$ $\displaystyle R_{K}^{\pi K,\gamma}$
$\displaystyle=\phantom{-}0.0163\phantom{0}\pm 0.0373,$ $\displaystyle
R_{K}^{\pi K,{{\pi}^{0}}}$ $\displaystyle=\phantom{-}0.0118\phantom{0}\pm
0.0034,$ $\displaystyle R_{\pi}^{\pi K}$ $\displaystyle=\phantom{-}0.00368\pm
0.00007,$ $\displaystyle R_{\pi}^{\pi K,\gamma}$
$\displaystyle=\phantom{-}0.00420\pm 0.00138,$ $\displaystyle R_{\pi}^{\pi
K,{{\pi}^{0}}}$ $\displaystyle=\phantom{-}0.00471\pm 0.00077,$ $\displaystyle
A_{K}^{\pi K}$ $\displaystyle=-0.451\phantom{00}\pm 0.026,$ $\displaystyle
A_{K}^{\pi K,\gamma}$ $\displaystyle=-0.558\phantom{00}\pm 1.349,$
$\displaystyle A_{K}^{\pi K,{{\pi}^{0}}}$
$\displaystyle=\phantom{-}0.717\phantom{00}\pm 0.286,$ $\displaystyle
A_{\pi}^{\pi K}$ $\displaystyle=\phantom{-}0.129\phantom{00}\pm 0.014,$
$\displaystyle A_{\pi}^{\pi K,\gamma}$
$\displaystyle=\phantom{-}0.079\phantom{00}\pm 0.128,$ $\displaystyle
A_{\pi}^{\pi K,{{\pi}^{0}}}$ $\displaystyle=-0.140\phantom{00}\pm 0.059,$

where the statistical and systematic uncertainties are combined according to
the correlations between the $R_{h^{-}}^{\pi K}$ and $R_{h^{+}}^{\pi K}$
observables.

World-best measurements of $C\\!P$ observables in ${{{B}^{-}}}\rightarrow
D^{(*)}h^{-}$ decays are obtained with the $D$ meson reconstructed in the
${{K}^{+}}{{\pi}^{-}}$, ${{K}^{+}}{{K}^{-}}$, and ${{\pi}^{+}}{{\pi}^{-}}$
final states; these supersede earlier work on the GLW modes presented in Ref.
[13]. Updated world-best measurements of $C\\!P$ observables in ADS
${{{B}^{-}}}\rightarrow Dh^{-}$ decays are also made, which supersede the
results in Ref. [14]. Measurements of $C\\!P$ observables in ADS
${{{B}^{-}}}\rightarrow D^{*}h^{-}$ decays are made for the first time at
LHCb. The ADS ${{{B}^{-}}}\rightarrow(D{{\pi}^{0}})_{{{D}^{*}}}{{K}^{-}}$
signal is measured with a statistical significance of 3.5 standard deviations
($\sigma$), with $C\\!P$ violation measured to be non-zero at the $2.5\sigma$
level. The ${{{B}^{-}}}\rightarrow(D\gamma)_{{{D}^{*}}}{{K}^{-}}$ signal is
measured to be consistent with zero, which is due to the large uncertainties
incurred as a result of correlations with several partially reconstructed
background contributions. The value of $A_{K}^{\pi K,{{\pi}^{0}}}$ is
consistent with the BaBar result [45], while $R_{K}^{\pi K,{{\pi}^{0}}}$ is
found to be smaller but consistent within measurement uncertainties. The
values of $A_{K}^{\pi K,\gamma}$ and $R_{K}^{\pi K,\gamma}$ are consistent
with the results from Ref. [45]. A first observation of the ADS
${{{B}^{-}}}\rightarrow(D{{\pi}^{0}})_{{{D}^{*}}}{{\pi}^{-}}$ decay is made
with a statistical significance of $6.1\sigma$, with $C\\!P$ violation
measured to be non-zero at the $2.4\sigma$ level. The ADS
${{{B}^{-}}}\rightarrow(D\gamma)_{{{D}^{*}}}{{\pi}^{-}}$ signal is measured
with a statistical significance of $3.0\sigma$, where the degree of $C\\!P$
violation measured is consistent with zero.

In general, good agreement is found with previous results from LHCb and the
$B$-factories. However, the value of $R^{CP}$ has decreased from
$R^{CP}=0.989\pm 0.013\pm 0.010$ in Ref. [13] due to the veto applied to
remove background from ${{{B}^{-}}}\rightarrow[h_{1}^{-}h^{+}]_{D}h_{2}^{-}$
decays where $h_{1}^{-}$ and $h_{2}^{-}$ are swapped. In Refs. [13] and [14]
this veto was not applied, resulting in peaking background contamination from
favoured ${{{B}^{-}}}\rightarrow[{{K}^{-}}{{\pi}^{+}}]_{D}{{\pi}^{-}}$ decays
in the ${{{B}^{-}}}\rightarrow[{{\pi}^{+}}{{\pi}^{-}}]_{D}{{K}^{-}}$ sample
which artificially increased the value of $R^{CP}$. The value of $R_{K}^{\pi
K}$ has also reduced due to this veto and the modelling of additional
background sources, such as ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow
Dp{{\pi}^{-}}$, which were not previously considered.

The values of $R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$,
$\mathcal{B}({{D}^{*0}}\rightarrow{{D}^{0}}{{\pi}^{0}})$, and
$\mathcal{B}({{B}^{\pm}}\rightarrow{{D}^{*0}}{{\pi}^{\pm}})$ are found to
agree well with the current world average values, ignoring previous LHCb
inputs to the averages. These measurements demonstrate that the method of
partial reconstruction accurately measures the
${{{B}^{-}}}\rightarrow({D}{{\pi}^{0}})_{{{D}^{*}}}h^{-}$ and
${{{B}^{-}}}\rightarrow({D}\gamma)_{{{D}^{*}}}h^{-}$ signals, despite the
presence of several partially reconstructed background sources which decrease
the purity and introduce anti-correlations in the fit.

##  7 Interpretation and conclusion

Using the ${{{B}^{-}}}\rightarrow D^{(*)}h^{-}$ $C\\!P$ observable results as
input, profile likelihood contours in the fundamental parameters
$({\gamma},r_{B}^{DK},\delta_{B}^{DK},r_{B}^{D\pi},\delta_{B}^{D\pi},r_{B}^{D^{*}K},\delta_{B}^{D^{*}K},r_{B}^{D^{*}\pi},\delta_{B}^{D^{*}\pi})$
are constructed following the approach described in Ref. [46]. The
${{{B}^{-}}}\rightarrow Dh^{-}$ $C\\!P$ observables are related to the
fundamental parameters using the following partial width expressions [10, 11,
12]

$\displaystyle\Gamma({{B}^{\pm}}\rightarrow[h^{+}h^{-}]_{D}h^{\pm})$
$\displaystyle\propto
1+(r_{B}^{Dh})^{2}+2r_{B}^{Dh}\cos{(\delta_{B}^{Dh}\pm{\gamma})},$
$\displaystyle\Gamma({{B}^{\pm}}\rightarrow[{{K}^{\pm}}{{\pi}^{\mp}}]_{D}h^{\pm})$
$\displaystyle\propto
1+(r_{D}^{K\pi}r_{B}^{Dh})^{2}+2r_{D}^{K\pi}r_{B}^{Dh}\cos{(\delta_{B}^{Dh}-\delta_{D}^{K\pi}\pm{\gamma})},$
$\displaystyle\Gamma({{B}^{\pm}}\rightarrow[{{K}^{\mp}}{{\pi}^{\pm}}]_{D}h^{\pm})$
$\displaystyle\propto(r_{D}^{K\pi})^{2}+(r_{B}^{Dh})^{2}+2r_{D}^{K\pi}r_{B}^{Dh}\cos{(\delta_{B}^{Dh}+\delta_{D}^{K\pi}\pm{\gamma})},$

where the $C\\!P$-violating phase $\gamma$ changes sign under the interchange
of $B$-meson charges. Here, $h^{+}h^{-}$ represents the GLW
$D\rightarrow{{K}^{+}}{{K}^{-}}$ and $D\rightarrow{{\pi}^{+}}{{\pi}^{-}}$
decays, and $r_{D}^{K\pi}$ and $\delta_{D}^{K\pi}$ are the amplitude ratio and
strong phase difference for the $D\rightarrow K\pi$ decay, which are taken
from Ref. [4]. Similar expressions can be written for ${{{B}^{-}}}\rightarrow
D^{*}h^{-}$ decays, where the exact strong phase difference of $\pi$ between
the $D^{*}\rightarrow D{{\pi}^{0}}$ and $D^{*}\rightarrow D{\gamma}$ decays is
taken into account [9]. When relating the $C\\!P$ observables to the
fundamental parameters, the effects of ${{D}^{0}}-{{\kern
1.79993pt\overline{\kern-1.79993ptD}}{}^{0}}$ mixing on the measured $C\\!P$
observable values are accounted for using the formalism described in Ref.
[47]. In this procedure, the experimental $D$ lifetime acceptance is modelled
using a fit to the $D$ lifetime in favoured ${B}\rightarrow D{{\pi}^{-}}$
data, where a factor $M_{xy}=-0.523\pm 0.004$ describes the overall degree of
mixing.

The profile likelihood contours for all fundamental parameters at 68%, 95%,
and 99.7% confidence level are shown in Fig. 6. The contours found are
dominated by the ${{{B}^{-}}}\rightarrow D{{K}^{-}}$ measurements, although
information from ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$,
${{{B}^{-}}}\rightarrow D^{*}{{K}^{-}}$, and ${{{B}^{-}}}\rightarrow
D^{*}{{\pi}^{-}}$ is also used in all cases. Compared to the ADS/GLW
likelihood contours constructed using previous LHCb ${{{B}^{-}}}\rightarrow
D{{K}^{-}}$ results [3], the favoured values of $r_{B}^{DK}$ are lower. This
is due to the lower value of $R^{CP}$ measured in this analysis. As a result
of this change in $R^{CP}$, the four distinct solutions visible in the
$({\gamma},\delta_{B}^{DK})$ plane have merged into two distinct bands, which
reduces the standalone sensitivity to ${\gamma}$ of the ADS/GLW
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ modes. However, in combination with LHCb
measurements using $D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ decays [5],
where a single solution for ${\gamma}$ is obtained, these results will provide
significant constraints on ${\gamma}$. The corresponding contours for
${{{B}^{-}}}\rightarrow Dh^{-}$ from Ref. [5] are overlaid in Fig. 6, and show
good agreement with the results of this analysis both for
${{{B}^{-}}}\rightarrow D{{K}^{-}}$ and ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$.

The preferred value of $r_{B}^{D^{*}K}$ is around 0.1, which is consistent
with the BaBar combination for ${{{B}^{+}}}\rightarrow D^{*}{{K}^{-}}$ [48].
The favoured values of $\delta_{B}^{D^{*}K}$ are also consistent with those
found in Ref. [48], with values around $300^{\circ}$ for $\gamma<90^{\circ}$.
Values of $r_{B}^{D^{*}\pi}\sim 0.01$ are favoured, with
$\delta_{B}^{D^{*}\pi}$ around $150^{\circ}$ for $\gamma<90^{\circ}$.

When constructing these confidence regions, the charm parameters $x$, $y$,
$r_{D}^{K\pi}$ and $\delta_{D}^{K\pi}$ are provided with their correlations as
external constraints from Ref. [4]. It is possible to make a measurement of
$\delta_{D}^{K\pi}$ and $y$ by allowing them to vary freely in a combination
of ${{{B}^{-}}}\rightarrow Dh^{-}$ results. Performing a simple combination of
this analysis and the result of Ref. [5] finds $y=(0.76\pm 0.24)\%$ and
$\delta_{D}^{K\pi}=(192.4\pm 6.0)^{\circ}$ with a correlation of $+0.37$, and
includes a $0.6^{\circ}$ systematic uncertainty on $\delta_{D}^{K\pi}$ from
the necessary constraints on $x$ and $r_{D}^{K\pi}$. This compares favourably
to the current world average, $\delta_{D}^{K\pi}=196.1^{+\ 7.9}_{-10.1}$ [4].
The fact that $B\rightarrow DX$ measurements provide significant input to the
understanding of charm parameters should motivate a comprehensive combination
of beauty and charm results.

In summary, measurements of $C\\!P$ observables in $B^{\pm}\rightarrow
D^{(*)}K^{\pm}$ and $B^{\pm}\rightarrow D^{(*)}\pi^{\pm}$ decays are made,
where decays of the ${D}$ meson are reconstructed in the
${{K}^{\pm}}{{\pi}^{\mp}}$, ${{K}^{+}}{{K}^{-}}$, and ${{\pi}^{+}}{{\pi}^{-}}$
final states. Decays of the ${{D}^{*}}$ meson to the ${D}{{\pi}^{0}}$ and
${D}\gamma$ final states are partially reconstructed without inclusion of the
neutral pion or photon. The measurements of partially reconstructed
${{B}^{\pm}}\rightarrow{{D}^{*}}{{K}^{\pm}}$ and
${{B}^{\pm}}\rightarrow{{D}^{*}}{{\pi}^{\pm}}$ with
$D\rightarrow{{K}^{\mp}}{{\pi}^{\pm}}$ decays are the first of their kind, and
a first observation of the
${{B}^{\pm}}\rightarrow(D\pi^{0})_{D^{*}}{{\pi}^{\pm}}$ decay is made with a
statistical significance of 6.1 standard deviations. All $C\\!P$ observables
are measured with world-best precision, and in combination with other LHCb
results will provide strong constraints on the CKM angle $\gamma$.

Figure 6: Confidence regions for the (top row) ${{{B}^{-}}}\rightarrow
D{{K}^{-}}$ , (second row) ${{{B}^{-}}}\rightarrow D{{\pi}^{-}}$ , (third row)
${{{B}^{-}}}\rightarrow D^{*}{{K}^{-}}$ , and (fourth row)
${{{B}^{-}}}\rightarrow D^{*}{{\pi}^{-}}$ fundamental parameters. The
${{{B}^{-}}}\rightarrow Dh^{-}$ with
$D\rightarrow{{K}^{0}_{\mathrm{S}}}h^{+}h^{-}$ contours from Ref. [5] are also
overlaid.

##  Appendix A Correlation matrices

Table 5: Statistical correlation matrix for ${{B}^{\pm}}\rightarrow Dh^{\pm}$ observables. | $A_{K}^{CP}$ | $A_{\pi}^{CP}$ | $A_{K}^{K\pi}$ | $R^{CP}$ | $R_{K/\pi}^{K\pi}$ | $R_{K^{-}}^{\pi K}$ | $R_{\pi^{-}}^{\pi K}$ | $R_{K^{+}}^{\pi K}$ | $R_{\pi^{+}}^{\pi K}$
---|---|---|---|---|---|---|---|---|---
$A_{K}^{CP}$ |  $-$$1.00$ |  $-$$0.00$ |  $-$$0.02$ | $-0.02$ | $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.00$ | $\phantom{-}0.00$ | $\phantom{-}0.00$
$A_{\pi}^{CP}$ |  |  $-$$1.00$ |  $-$$0.08$ | $\phantom{-}0.00$ | $\phantom{-}0.00$ |  $-$$0.01$ |  $-$$0.01$ | $\phantom{-}0.00$ | $-0.01$
$A_{K}^{K\pi}$ |  |  |  $-$$1.00$ |  $-$$0.00$ | $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.01$ | $\phantom{-}0.00$ | $-0.01$
$R^{CP}$ |  |  |  |  $-$$1.00$ | $-0.33$ |  $-$$0.03$ | $\phantom{-}0.00$ |  $-$$0.04$ | $\phantom{-}0.00$
$R_{K/\pi}^{K\pi}$ |  |  |  |  |  $-$$1.00$ | $-0.05$ |  $-$$0.01$ | $-0.10$ |  $-$$0.00$
$R_{K^{-}}^{\pi K}$ |  |  |  |  |  |  $-$$1.00$ | $-0.03$ |  $-$$0.05$ |  $-$$0.02$
$R_{\pi^{-}}^{\pi K}$ |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.02$ |  $-$$0.08$
$R_{K^{+}}^{\pi K}$ |  |  |  |  |  |  |  |  $-$$1.00$ | $-0.04$
$R_{\pi^{+}}^{\pi K}$ |  |  |  |  |  |  |  |  |  $-$$1.00$
Table 6: Systematic correlation matrix for ${{B}^{\pm}}\rightarrow Dh^{\pm}$ observables. | $A_{K}^{CP}$ | $A_{\pi}^{CP}$ | $A_{K}^{K\pi}$ | $R^{CP}$ | $R_{K/\pi}^{K\pi}$ | $R_{K^{-}}^{\pi K}$ | $R_{\pi^{-}}^{\pi K}$ | $R_{K^{+}}^{\pi K}$ | $R_{\pi^{+}}^{\pi K}$
---|---|---|---|---|---|---|---|---|---
$A_{K}^{CP}$ |  $-$$1.00$ |  $-$$0.13$ |  $-$$0.07$ | $-0.22$ | $-0.06$ | $-0.03$ |  $-$$0.00$ |  $-$$0.02$ | $-0.07$
$A_{\pi}^{CP}$ |  |  $-$$1.00$ | $-0.74$ |  $-$$0.18$ | $-0.05$ |  $-$$0.04$ |  $-$$0.31$ | $-0.07$ | $-0.21$
$A_{K}^{K\pi}$ |  |  |  $-$$1.00$ | $-0.02$ |  $-$$0.00$ |  $-$$0.05$ | $-0.24$ |  $-$$0.11$ |  $-$$0.22$
$R^{CP}$ |  |  |  |  $-$$1.00$ | $-0.20$ |  $-$$0.09$ |  $-$$0.11$ |  $-$$0.17$ |  $-$$0.14$
$R_{K/\pi}^{K\pi}$ |  |  |  |  |  $-$$1.00$ |  $-$$0.08$ | $-0.04$ | $-0.06$ | $-0.10$
$R_{K^{-}}^{\pi K}$ |  |  |  |  |  |  $-$$1.00$ |  $-$$0.16$ |  $-$$0.93$ |  $-$$0.13$
$R_{\pi^{-}}^{\pi K}$ |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.21$ |  $-$$0.84$
$R_{K^{+}}^{\pi K}$ |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.26$
$R_{\pi^{+}}^{\pi K}$ |  |  |  |  |  |  |  |  |  $-$$1.00$

Table 7: Statistical correlation matrix for ${{B}^{\pm}}\rightarrow D^{*}h^{\pm}$ observables. |  $A_{K}^{CP,\gamma}$ |  $A_{K}^{CP,\pi^{0}}$ |  $A_{K}^{K\pi,\gamma}$ |  $A_{K}^{K\pi,\pi^{0}}$ |  $R^{CP,\gamma}$ |  $R^{CP,\pi^{0}}$ |  $R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$ |  $R_{K^{-}}^{\pi K,\gamma}$ |  $R_{K^{-}}^{\pi K,\pi^{0}}$ |  $R_{K^{+}}^{\pi K,\gamma}$ |  $R_{K^{+}}^{\pi K,\pi^{0}}$ |  $A_{\pi}^{CP,\gamma}$ |  $A_{\pi}^{CP,\pi^{0}}$ |  $A_{\pi}^{K\pi,\gamma}$ |  $A_{\pi}^{K\pi,\pi^{0}}$ |  $R_{\pi^{-}}^{\pi K,\gamma}$ |  $R_{\pi^{-}}^{\pi K,\pi^{0}}$ |  $R_{\pi^{+}}^{\pi K,\gamma}$ |  $R_{\pi^{+}}^{\pi K,\pi^{0}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{K}^{CP,\gamma}$ |  $-$$1.00$ |  $-0.61$ |  $-$$0.00$ |  $-$$0.00$ |  $-0.15$ |  $-$$0.07$ |  $-$$0.03$ |  $-0.03$ |  $-$$0.01$ |  $-0.03$ |  $-$$0.01$ |  $-0.01$ |  $-0.02$ |  $-$$0.01$ |  $-$$0.01$ |  $-0.01$ |  $-$$0.00$ |  $-0.02$ |  $\phantom{-}0.00$
$A_{K}^{CP,\pi^{0}}$ |  |  $-$$1.00$ |  $-$$0.00$ |  $-$$0.01$ |  $-0.05$ |  $-$$0.08$ |  $-$$0.05$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.03$ |  $-$$0.05$ |  $-$$0.02$ |  $-$$0.03$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$
$A_{K}^{K\pi,\gamma}$ |  |  |  $-$$1.00$ |  $-0.59$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.01$ |  $-$$0.00$ |  $-$$0.01$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$
$A_{K}^{K\pi,\pi^{0}}$ |  |  |  |  $-$$1.00$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-0.02$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $-$$0.01$ |  $-$$0.02$ |  $-$$0.05$ |  $-$$0.08$ |  $-$$0.00$ |  $-$$0.01$ |  $\phantom{-}0.00$ |  $-0.01$
$R^{CP,\gamma}$ |  |  |  |  |  $-$$1.00$ |  $-0.44$ |  $-0.02$ |  $-$$0.23$ |  $-0.08$ |  $-$$0.23$ |  $-0.08$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.10$ |  $\phantom{-}0.00$ |  $-$$0.10$ |  $\phantom{-}0.00$
$R^{CP,\pi^{0}}$ |  |  |  |  |  |  $-$$1.00$ |  $-0.07$ |  $-0.04$ |  $-$$0.03$ |  $-0.04$ |  $-$$0.02$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $-0.02$ |  $\phantom{-}0.00$ |  $-0.02$ |  $-$$0.00$
$R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$ |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.03$ |  $-0.03$ |  $-$$0.04$ |  $-$$0.01$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-$$0.01$ |  $\phantom{-}0.00$ |  $-$$0.02$ |  $\phantom{-}0.00$ |  $-$$0.02$ |  $-$$0.01$
$R_{K^{-}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.59$ |  $-$$0.79$ |  $-0.27$ |  $-$$0.00$ |  $-$$0.00$ |  $-$$0.01$ |  $-$$0.01$ |  $-$$0.30$ |  $-0.03$ |  $-$$0.33$ |  $-0.02$
$R_{K^{-}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.27$ |  $-$$0.10$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $-$$0.00$ |  $-0.07$ |  $-$$0.05$ |  $-0.10$ |  $-$$0.05$
$R_{K^{+}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.60$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $-0.01$ |  $-0.01$ |  $-$$0.32$ |  $-0.01$ |  $-$$0.30$ |  $-0.03$
$R_{K^{+}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$ |  $-0.09$ |  $-$$0.04$ |  $-0.07$ |  $-$$0.05$
$A_{\pi}^{CP,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.05$ |  $-$$0.02$ |  $-$$0.03$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $\phantom{-}0.00$
$A_{\pi}^{CP,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.05$ |  $-$$0.08$ |  $-$$0.00$ |  $-$$0.00$ |  $\phantom{-}0.00$ |  $-0.01$
$A_{\pi}^{K\pi,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.28$ |  $-$$0.01$ |  $-$$0.01$ |  $-0.01$ |  $-0.01$
$A_{\pi}^{K\pi,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.01$ |  $-$$0.02$ |  $-0.01$ |  $-0.02$
$R_{\pi^{-}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.11$ |  $-$$0.33$ |  $-$$0.19$
$R_{\pi^{-}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.19$ |  $-$$0.58$
$R_{\pi^{+}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.11$
$R_{\pi^{+}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$

Table 8: Systematic correlation matrix for ${{B}^{\pm}}\rightarrow D^{*}h^{\pm}$ observables. |  $A_{K}^{CP,\gamma}$ |  $A_{K}^{CP,\pi^{0}}$ |  $A_{K}^{K\pi,\gamma}$ |  $A_{K}^{K\pi,\pi^{0}}$ |  $R^{CP,\gamma}$ |  $R^{CP,\pi^{0}}$ |  $R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$ |  $R_{K^{-}}^{\pi K,\gamma}$ |  $R_{K^{-}}^{\pi K,\pi^{0}}$ |  $R_{K^{+}}^{\pi K,\gamma}$ |  $R_{K^{+}}^{\pi K,\pi^{0}}$ |  $A_{\pi}^{CP,\gamma}$ |  $A_{\pi}^{CP,\pi^{0}}$ |  $A_{\pi}^{K\pi,\gamma}$ |  $A_{\pi}^{K\pi,\pi^{0}}$ |  $R_{\pi^{-}}^{\pi K,\gamma}$ |  $R_{\pi^{-}}^{\pi K,\pi^{0}}$ |  $R_{\pi^{+}}^{\pi K,\gamma}$ |  $R_{\pi^{+}}^{\pi K,\pi^{0}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{K}^{CP,\gamma}$ |  $-$$1.00$ |  $-0.52$ |  $-$$0.50$ |  $-0.39$ |  $-0.31$ |  $-0.71$ |  $-$$0.40$ |  $-0.15$ |  $-$$0.10$ |  $-0.05$ |  $-$$0.12$ |  $-$$0.74$ |  $-0.02$ |  $-0.70$ |  $-$$0.05$ |  $-0.16$ |  $-0.14$ |  $-0.05$ |  $-0.17$
$A_{K}^{CP,\pi^{0}}$ |  |  $-$$1.00$ |  $-0.44$ |  $-0.08$ |  $-$$0.33$ |  $-$$0.62$ |  $-$$0.19$ |  $-$$0.19$ |  $-0.19$ |  $-$$0.16$ |  $-0.08$ |  $-0.49$ |  $-$$0.00$ |  $-$$0.53$ |  $-0.19$ |  $-$$0.18$ |  $-$$0.01$ |  $-$$0.10$ |  $-$$0.03$
$A_{K}^{K\pi,\gamma}$ |  |  |  $-$$1.00$ |  $-$$0.23$ |  $-$$0.02$ |  $-0.35$ |  $-$$0.38$ |  $-0.04$ |  $-$$0.03$ |  $-$$0.04$ |  $-$$0.05$ |  $-$$0.57$ |  $-0.38$ |  $-0.38$ |  $-$$0.32$ |  $-0.15$ |  $-0.19$ |  $-$$0.04$ |  $-0.09$
$A_{K}^{K\pi,\pi^{0}}$ |  |  |  |  $-$$1.00$ |  $-0.19$ |  $-$$0.15$ |  $-0.53$ |  $-0.08$ |  $-$$0.06$ |  $-0.12$ |  $-0.01$ |  $-0.54$ |  $-0.64$ |  $-$$0.42$ |  $-$$0.28$ |  $-0.05$ |  $-$$0.04$ |  $-$$0.02$ |  $-$$0.24$
$R^{CP,\gamma}$ |  |  |  |  |  $-$$1.00$ |  $-$$0.49$ |  $-$$0.36$ |  $-$$0.36$ |  $-0.15$ |  $-$$0.41$ |  $-0.09$ |  $-$$0.22$ |  $-$$0.16$ |  $-$$0.07$ |  $-$$0.00$ |  $-$$0.16$ |  $-0.06$ |  $-$$0.19$ |  $-0.07$
$R^{CP,\pi^{0}}$ |  |  |  |  |  |  $-$$1.00$ |  $-0.08$ |  $-$$0.14$ |  $-0.04$ |  $-$$0.10$ |  $-0.05$ |  $-0.35$ |  $-$$0.07$ |  $-$$0.42$ |  $-0.11$ |  $-$$0.15$ |  $-$$0.08$ |  $-$$0.09$ |  $-$$0.10$
$R_{K/\pi}^{K\pi,\gamma/\pi^{0}}$ |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.16$ |  $-0.13$ |  $-$$0.26$ |  $-0.02$ |  $-$$0.62$ |  $-$$0.11$ |  $-0.22$ |  $-$$0.08$ |  $-$$0.02$ |  $-0.19$ |  $-$$0.11$ |  $-0.18$
$R_{K^{-}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.55$ |  $-$$0.98$ |  $-0.51$ |  $-$$0.00$ |  $-$$0.07$ |  $-$$0.10$ |  $-$$0.00$ |  $-$$0.37$ |  $-0.04$ |  $-$$0.40$ |  $-0.04$
$R_{K^{-}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.53$ |  $-$$0.94$ |  $-$$0.04$ |  $-0.01$ |  $-0.10$ |  $-$$0.00$ |  $-0.13$ |  $-$$0.08$ |  $-0.15$ |  $-$$0.06$
$R_{K^{+}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.48$ |  $-$$0.12$ |  $-$$0.04$ |  $-$$0.03$ |  $-$$0.02$ |  $-$$0.37$ |  $-0.06$ |  $-$$0.41$ |  $-0.06$
$R_{K^{+}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.06$ |  $-0.03$ |  $-0.10$ |  $-$$0.00$ |  $-0.14$ |  $-$$0.02$ |  $-0.13$ |  $-$$0.04$
$A_{\pi}^{CP,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.26$ |  $-0.67$ |  $-$$0.17$ |  $-0.10$ |  $-0.18$ |  $-$$0.02$ |  $-0.25$
$A_{\pi}^{CP,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.02$ |  $-$$0.06$ |  $-$$0.09$ |  $-$$0.08$ |  $-0.07$ |  $-0.14$
$A_{\pi}^{K\pi,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.43$ |  $-$$0.13$ |  $-$$0.09$ |  $-$$0.06$ |  $-$$0.17$
$A_{\pi}^{K\pi,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-0.04$ |  $-0.05$ |  $-$$0.02$ |  $-$$0.00$
$R_{\pi^{-}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.53$ |  $-$$0.79$ |  $-$$0.16$
$R_{\pi^{-}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.22$ |  $-$$0.60$
$R_{\pi^{+}}^{\pi K,\gamma}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$ |  $-$$0.41$
$R_{\pi^{+}}^{\pi K,\pi^{0}}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  $-$$1.00$

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MICINN (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and
Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-
GRID (Poland) and OSC (USA). We are indebted to the communities behind the
multiple open-source software packages on which we depend. Individual groups
or members have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); A*MIDEX, ANR, Labex P2IO
and OCEVU, and Région Auvergne-Rhône-Alpes (France); Key Research Program of
Frontier Sciences of CAS, CAS PIFI, CAS CCEPP, Fundamental Research Funds for
Central Universities, and Sci. & Tech. Program of Guangzhou (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

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LHCb collaboration

R. Aaij32, C. Abellán Beteta50, T. Ackernley60, B. Adeva46, M. Adinolfi54, H.
Afsharnia9, C.A. Aidala85, S. Aiola26, Z. Ajaltouni9, S. Akar65, J.
Albrecht15, F. Alessio48, M. Alexander59, A. Alfonso Albero45, Z. Aliouche62,
G. Alkhazov38, P. Alvarez Cartelle55, S. Amato2, Y. Amhis11, L. An48, L.
Anderlini22, A. Andreianov38, M. Andreotti21, F. Archilli17, A. Artamonov44,
M. Artuso68, K. Arzymatov42, E. Aslanides10, M. Atzeni50, B. Audurier12, S.
Bachmann17, M. Bachmayer49, J.J. Back56, S. Baker61, P. Baladron Rodriguez46,
V. Balagura12, W. Baldini21, J. Baptista Leite1, R.J. Barlow62, S. Barsuk11,
W. Barter61, M. Bartolini24,h, F. Baryshnikov81, J.M. Basels14, G. Bassi29, B.
Batsukh68, A. Battig15, A. Bay49, M. Becker15, F. Bedeschi29, I. Bediaga1, A.
Beiter68, V. Belavin42, S. Belin27, V. Bellee49, K. Belous44, I. Belov40, I.
Belyaev39, G. Bencivenni23, E. Ben-Haim13, A. Berezhnoy40, R. Bernet50, D.
Berninghoff17, H.C. Bernstein68, C. Bertella48, E. Bertholet13, A. Bertolin28,
C. Betancourt50, F. Betti20,d, Ia. Bezshyiko50, S. Bhasin54, J. Bhom34, L.
Bian73, M.S. Bieker15, S. Bifani53, P. Billoir13, M. Birch61, F.C.R. Bishop55,
A. Bizzeti22,r, M. Bjørn63, M.P. Blago48, T. Blake56, F. Blanc49, S. Blusk68,
D. Bobulska59, J.A. Boelhauve15, O. Boente Garcia46, T. Boettcher64, A.
Boldyrev82, A. Bondar43, N. Bondar38, S. Borghi62, M. Borisyak42, M.
Borsato17, J.T. Borsuk34, S.A. Bouchiba49, T.J.V. Bowcock60, A. Boyer48, C.
Bozzi21, M.J. Bradley61, S. Braun66, A. Brea Rodriguez46, M. Brodski48, J.
Brodzicka34, A. Brossa Gonzalo56, D. Brundu27, A. Buonaura50, C. Burr48, A.
Bursche27, A. Butkevich41, J.S. Butter32, J. Buytaert48, W. Byczynski48, S.
Cadeddu27, H. Cai73, R. Calabrese21,f, L. Calefice15,13, L. Calero Diaz23, S.
Cali23, R. Calladine53, M. Calvi25,i, M. Calvo Gomez84, P. Camargo
Magalhaes54, A. Camboni45, P. Campana23, A.F. Campoverde Quezada5, S.
Capelli25,i, L. Capriotti20,d, A. Carbone20,d, G. Carboni30, R. Cardinale24,h,
A. Cardini27, I. Carli6, P. Carniti25,i, L. Carus14, K. Carvalho Akiba32, A.
Casais Vidal46, G. Casse60, M. Cattaneo48, G. Cavallero48, S. Celani49, J.
Cerasoli10, A.J. Chadwick60, M.G. Chapman54, M. Charles13, Ph. Charpentier48,
G. Chatzikonstantinidis53, C.A. Chavez Barajas60, M. Chefdeville8, C. Chen3,
S. Chen27, A. Chernov34, S.-G. Chitic48, V. Chobanova46, S. Cholak49, M.
Chrzaszcz34, A. Chubykin38, V. Chulikov38, P. Ciambrone23, M.F. Cicala56, X.
Cid Vidal46, G. Ciezarek48, P.E.L. Clarke58, M. Clemencic48, H.V. Cliff55, J.
Closier48, J.L. Cobbledick62, V. Coco48, J.A.B. Coelho11, J. Cogan10, E.
Cogneras9, L. Cojocariu37, P. Collins48, T. Colombo48, L. Congedo19,c, A.
Contu27, N. Cooke53, G. Coombs59, G. Corti48, C.M. Costa Sobral56, B.
Couturier48, D.C. Craik64, J. Crkovská67, M. Cruz Torres1, R. Currie58, C.L.
Da Silva67, E. Dall’Occo15, J. Dalseno46, C. D’Ambrosio48, A. Danilina39, P.
d’Argent48, A. Davis62, O. De Aguiar Francisco62, K. De Bruyn78, S. De
Capua62, M. De Cian49, J.M. De Miranda1, L. De Paula2, M. De Serio19,c, D. De
Simone50, P. De Simone23, J.A. de Vries79, C.T. Dean67, W. Dean85, D. Decamp8,
L. Del Buono13, B. Delaney55, H.-P. Dembinski15, A. Dendek35, V. Denysenko50,
D. Derkach82, O. Deschamps9, F. Desse11, F. Dettori27,e, B. Dey73, P. Di
Nezza23, S. Didenko81, L. Dieste Maronas46, H. Dijkstra48, V. Dobishuk52, A.M.
Donohoe18, F. Dordei27, A.C. dos Reis1, L. Douglas59, A. Dovbnya51, A.G.
Downes8, K. Dreimanis60, M.W. Dudek34, L. Dufour48, V. Duk77, P. Durante48,
J.M. Durham67, D. Dutta62, M. Dziewiecki17, A. Dziurda34, A. Dzyuba38, S.
Easo57, U. Egede69, V. Egorychev39, S. Eidelman43,u, S. Eisenhardt58, S. Ek-
In49, L. Eklund59, S. Ely68, A. Ene37, E. Epple67, S. Escher14, J. Eschle50,
S. Esen32, T. Evans48, A. Falabella20, J. Fan3, Y. Fan5, B. Fang73, N.
Farley53, S. Farry60, D. Fazzini25,i, P. Fedin39, M. Féo48, P. Fernandez
Declara48, A. Fernandez Prieto46, J.M. Fernandez-tenllado Arribas45, F.
Ferrari20,d, L. Ferreira Lopes49, F. Ferreira Rodrigues2, S. Ferreres Sole32,
M. Ferrillo50, M. Ferro-Luzzi48, S. Filippov41, R.A. Fini19, M. Fiorini21,f,
M. Firlej35, K.M. Fischer63, C. Fitzpatrick62, T. Fiutowski35, F. Fleuret12,
M. Fontana13, F. Fontanelli24,h, R. Forty48, V. Franco Lima60, M. Franco
Sevilla66, M. Frank48, E. Franzoso21, G. Frau17, C. Frei48, D.A. Friday59, J.
Fu26, Q. Fuehring15, W. Funk48, E. Gabriel32, T. Gaintseva42, A. Gallas
Torreira46, D. Galli20,d, S. Gambetta58,48, Y. Gan3, M. Gandelman2, P.
Gandini26, Y. Gao4, M. Garau27, L.M. Garcia Martin56, P. Garcia Moreno45, J.
García Pardiñas25, B. Garcia Plana46, F.A. Garcia Rosales12, L. Garrido45, C.
Gaspar48, R.E. Geertsema32, D. Gerick17, L.L. Gerken15, E. Gersabeck62, M.
Gersabeck62, T. Gershon56, D. Gerstel10, Ph. Ghez8, V. Gibson55, M.
Giovannetti23,j, A. Gioventù46, P. Gironella Gironell45, L. Giubega37, C.
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Göbel70, E. Golobardes84, D. Golubkov39, A. Golutvin61,81, A. Gomes1,a, S.
Gomez Fernandez45, F. Goncalves Abrantes70, M. Goncerz34, G. Gong3, P.
Gorbounov39, I.V. Gorelov40, C. Gotti25,i, E. Govorkova48, J.P. Grabowski17,
R. Graciani Diaz45, T. Grammatico13, L.A. Granado Cardoso48, E. Graugés45, E.
Graverini49, G. Graziani22, A. Grecu37, L.M. Greeven32, P. Griffith21, L.
Grillo62, S. Gromov81, B.R. Gruberg Cazon63, C. Gu3, M. Guarise21, P. A.
Günther17, E. Gushchin41, A. Guth14, Y. Guz44,48, T. Gys48, T. Hadavizadeh69,
G. Haefeli49, C. Haen48, J. Haimberger48, T. Halewood-leagas60, P.M.
Hamilton66, Q. Han7, X. Han17, T.H. Hancock63, S. Hansmann-Menzemer17, N.
Harnew63, T. Harrison60, C. Hasse48, M. Hatch48, J. He5, M. Hecker61, K.
Heijhoff32, K. Heinicke15, A.M. Hennequin48, K. Hennessy60, L. Henry26,47, J.
Heuel14, A. Hicheur2, D. Hill49, M. Hilton62, S.E. Hollitt15, J. Hu17, J.
Hu72, W. Hu7, W. Huang5, X. Huang73, W. Hulsbergen32, R.J. Hunter56, M.
Hushchyn82, D. Hutchcroft60, D. Hynds32, P. Ibis15, M. Idzik35, D. Ilin38, P.
Ilten65, A. Inglessi38, A. Ishteev81, K. Ivshin38, R. Jacobsson48, S.
Jakobsen48, E. Jans32, B.K. Jashal47, A. Jawahery66, V. Jevtic15, M.
Jezabek34, F. Jiang3, M. John63, D. Johnson48, C.R. Jones55, T.P. Jones56, B.
Jost48, N. Jurik48, S. Kandybei51, Y. Kang3, M. Karacson48, M. Karpov82, N.
Kazeev82, F. Keizer55,48, M. Kenzie56, T. Ketel33, B. Khanji15, A.
Kharisova83, S. Kholodenko44, K.E. Kim68, T. Kirn14, V.S. Kirsebom49, O.
Kitouni64, S. Klaver32, K. Klimaszewski36, S. Koliiev52, A. Kondybayeva81, A.
Konoplyannikov39, P. Kopciewicz35, R. Kopecna17, P. Koppenburg32, M.
Korolev40, I. Kostiuk32,52, O. Kot52, S. Kotriakhova38,31, P. Kravchenko38, L.
Kravchuk41, R.D. Krawczyk48, M. Kreps56, F. Kress61, S. Kretzschmar14, P.
Krokovny43,u, W. Krupa35, W. Krzemien36, W. Kucewicz34,k, M. Kucharczyk34, V.
Kudryavtsev43,u, H.S. Kuindersma32, G.J. Kunde67, T. Kvaratskheliya39, D.
Lacarrere48, G. Lafferty62, A. Lai27, A. Lampis27, D. Lancierini50, J.J.
Lane62, R. Lane54, G. Lanfranchi23, C. Langenbruch14, J. Langer15, O.
Lantwin50,81, T. Latham56, F. Lazzari29,s, R. Le Gac10, S.H. Lee85, R.
Lefèvre9, A. Leflat40, S. Legotin81, O. Leroy10, T. Lesiak34, B.
Leverington17, H. Li72, L. Li63, P. Li17, Y. Li6, Y. Li6, Z. Li68, X. Liang68,
T. Lin61, R. Lindner48, V. Lisovskyi15, R. Litvinov27, G. Liu72, H. Liu5, S.
Liu6, X. Liu3, A. Loi27, J. Lomba Castro46, I. Longstaff59, J.H. Lopes2, G.
Loustau50, G.H. Lovell55, Y. Lu6, D. Lucchesi28,l, S. Luchuk41, M. Lucio
Martinez32, V. Lukashenko32, Y. Luo3, A. Lupato62, E. Luppi21,f, O. Lupton56,
A. Lusiani29,q, X. Lyu5, L. Ma6, S. Maccolini20,d, F. Machefert11, F.
Maciuc37, V. Macko49, P. Mackowiak15, S. Maddrell-Mander54, O. Madejczyk35,
L.R. Madhan Mohan54, O. Maev38, A. Maevskiy82, D. Maisuzenko38, M.W.
Majewski35, J.J. Malczewski34, S. Malde63, B. Malecki48, A. Malinin80, T.
Maltsev43,u, H. Malygina17, G. Manca27,e, G. Mancinelli10, R. Manera
Escalero45, D. Manuzzi20,d, D. Marangotto26,n, J. Maratas9,t, J.F. Marchand8,
U. Marconi20, S. Mariani22,48,g, C. Marin Benito11, M. Marinangeli49, P.
Marino49, J. Marks17, P.J. Marshall60, G. Martellotti31, L. Martinazzoli48,i,
M. Martinelli25,i, D. Martinez Santos46, F. Martinez Vidal47, A. Massafferri1,
M. Materok14, R. Matev48, A. Mathad50, Z. Mathe48, V. Matiunin39, C.
Matteuzzi25, K.R. Mattioli85, A. Mauri32, E. Maurice12, J. Mauricio45, M.
Mazurek36, M. McCann61, L. Mcconnell18, T.H. Mcgrath62, A. McNab62, R.
McNulty18, J.V. Mead60, B. Meadows65, C. Meaux10, G. Meier15, N. Meinert76, D.
Melnychuk36, S. Meloni25,i, M. Merk32,79, A. Merli26, L. Meyer Garcia2, M.
Mikhasenko48, D.A. Milanes74, E. Millard56, M. Milovanovic48, M.-N. Minard8,
L. Minzoni21,f, S.E. Mitchell58, B. Mitreska62, D.S. Mitzel48, A. Mödden15,
R.A. Mohammed63, R.D. Moise61, T. Mombächer15, I.A. Monroy74, S. Monteil9, M.
Morandin28, G. Morello23, M.J. Morello29,q, J. Moron35, A.B. Morris75, A.G.
Morris56, R. Mountain68, H. Mu3, F. Muheim58, M. Mukherjee7, M. Mulder48, D.
Müller48, K. Müller50, C.H. Murphy63, D. Murray62, P. Muzzetto27,48, P.
Naik54, T. Nakada49, R. Nandakumar57, T. Nanut49, I. Nasteva2, M. Needham58,
I. Neri21,f, N. Neri26,n, S. Neubert75, N. Neufeld48, R. Newcombe61, T.D.
Nguyen49, C. Nguyen-Mau49, E.M. Niel11, S. Nieswand14, N. Nikitin40, N.S.
Nolte48, C. Nunez85, A. Oblakowska-Mucha35, V. Obraztsov44, D.P. O’Hanlon54,
R. Oldeman27,e, M.E. Olivares68, C.J.G. Onderwater78, A. Ossowska34, J.M.
Otalora Goicochea2, T. Ovsiannikova39, P. Owen50, A. Oyanguren47, B. Pagare56,
P.R. Pais48, T. Pajero29,48,q, A. Palano19, M. Palutan23, Y. Pan62, G.
Panshin83, A. Papanestis57, M. Pappagallo19,c, L.L. Pappalardo21,f, C.
Pappenheimer65, W. Parker66, C. Parkes62, C.J. Parkinson46, B. Passalacqua21,
G. Passaleva22, A. Pastore19, M. Patel61, C. Patrignani20,d, C.J. Pawley79, A.
Pearce48, A. Pellegrino32, M. Pepe Altarelli48, S. Perazzini20, D. Pereima39,
P. Perret9, K. Petridis54, A. Petrolini24,h, A. Petrov80, S. Petrucci58, M.
Petruzzo26, T.T.H. Pham68, A. Philippov42, L. Pica29, M. Piccini77, B.
Pietrzyk8, G. Pietrzyk49, M. Pili63, D. Pinci31, F. Pisani48, A. Piucci17,
Resmi P.K10, V. Placinta37, J. Plews53, M. Plo Casasus46, F. Polci13, M. Poli
Lener23, M. Poliakova68, A. Poluektov10, N. Polukhina81,b, I. Polyakov68, E.
Polycarpo2, G.J. Pomery54, S. Ponce48, D. Popov5,48, S. Popov42, S.
Poslavskii44, K. Prasanth34, L. Promberger48, C. Prouve46, V. Pugatch52, H.
Pullen63, G. Punzi29,m, W. Qian5, J. Qin5, R. Quagliani13, B. Quintana8, N.V.
Raab18, R.I. Rabadan Trejo10, B. Rachwal35, J.H. Rademacker54, M. Rama29, M.
Ramos Pernas56, M.S. Rangel2, F. Ratnikov42,82, G. Raven33, M. Reboud8, F.
Redi49, F. Reiss13, C. Remon Alepuz47, Z. Ren3, V. Renaudin63, R. Ribatti29,
S. Ricciardi57, K. Rinnert60, P. Robbe11, A. Robert13, G. Robertson58, A.B.
Rodrigues49, E. Rodrigues60, J.A. Rodriguez Lopez74, A. Rollings63, P.
Roloff48, V. Romanovskiy44, M. Romero Lamas46, A. Romero Vidal46, J.D. Roth85,
M. Rotondo23, M.S. Rudolph68, T. Ruf48, J. Ruiz Vidal47, A. Ryzhikov82, J.
Ryzka35, J.J. Saborido Silva46, N. Sagidova38, N. Sahoo56, B. Saitta27,e, D.
Sanchez Gonzalo45, C. Sanchez Gras32, R. Santacesaria31, C. Santamarina
Rios46, M. Santimaria23, E. Santovetti30,j, D. Saranin81, G. Sarpis59, M.
Sarpis75, A. Sarti31, C. Satriano31,p, A. Satta30, M. Saur15, D. Savrina39,40,
H. Sazak9, L.G. Scantlebury Smead63, S. Schael14, M. Schellenberg15, M.
Schiller59, H. Schindler48, M. Schmelling16, B. Schmidt48, O. Schneider49, A.
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Sergi53,48, N. Serra50, L. Sestini28, A. Seuthe15, P. Seyfert48, D.M.
Shangase85, M. Shapkin44, I. Shchemerov81, L. Shchutska49, T. Shears60, L.
Shekhtman43,u, Z. Shen4, V. Shevchenko80, E.B. Shields25,i, E. Shmanin81, J.D.
Shupperd68, B.G. Siddi21, R. Silva Coutinho50, G. Simi28, S. Simone19,c, I.
Skiba21,f, N. Skidmore62, T. Skwarnicki68, M.W. Slater53, J.C. Smallwood63,
J.G. Smeaton55, A. Smetkina39, E. Smith14, M. Smith61, A. Snoch32, M.
Soares20, L. Soares Lavra9, M.D. Sokoloff65, F.J.P. Soler59, A. Solovev38, I.
Solovyev38, F.L. Souza De Almeida2, B. Souza De Paula2, B. Spaan15, E. Spadaro
Norella26,n, P. Spradlin59, F. Stagni48, M. Stahl65, S. Stahl48, P. Stefko49,
O. Steinkamp50,81, S. Stemmle17, O. Stenyakin44, H. Stevens15, S. Stone68,
M.E. Stramaglia49, M. Straticiuc37, D. Strekalina81, S. Strokov83, F.
Suljik63, J. Sun27, L. Sun73, Y. Sun66, P. Svihra62, P.N. Swallow53, K.
Swientek35, A. Szabelski36, T. Szumlak35, M. Szymanski48, S. Taneja62, F.
Teubert48, E. Thomas48, K.A. Thomson60, M.J. Tilley61, V. Tisserand9, S.
T’Jampens8, M. Tobin6, S. Tolk48, L. Tomassetti21,f, D. Torres Machado1, D.Y.
Tou13, M. Traill59, M.T. Tran49, E. Trifonova81, C. Trippl49, G. Tuci29,m, A.
Tully49, N. Tuning32, A. Ukleja36, D.J. Unverzagt17, E. Ursov81, A. Usachov32,
A. Ustyuzhanin42,82, U. Uwer17, A. Vagner83, V. Vagnoni20, A. Valassi48, G.
Valenti20, N. Valls Canudas45, M. van Beuzekom32, M. Van Dijk49, E. van
Herwijnen81, C.B. Van Hulse18, M. van Veghel78, R. Vazquez Gomez46, P. Vazquez
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A. Venkateswaran68, M. Veronesi32, M. Vesterinen56, D. Vieira65, M. Vieites
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Zeng3, D. Zhang7, L. Zhang3, S. Zhang4, Y. Zhang4, Y. Zhang63, A. Zhelezov17,
Y. Zheng5, X. Zhou5, Y. Zhou5, X. Zhu3, V. Zhukov14,40, J.B. Zonneveld58, S.
Zucchelli20,d, D. Zuliani28, G. Zunica62.

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
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12Laboratoire Leprince-ringuet (llr), Palaiseau, France
13LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
14I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
15Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
16Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
17Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
18School of Physics, University College Dublin, Dublin, Ireland
19INFN Sezione di Bari, Bari, Italy
20INFN Sezione di Bologna, Bologna, Italy
21INFN Sezione di Ferrara, Ferrara, Italy
22INFN Sezione di Firenze, Firenze, Italy
23INFN Laboratori Nazionali di Frascati, Frascati, Italy
24INFN Sezione di Genova, Genova, Italy
25INFN Sezione di Milano-Bicocca, Milano, Italy
26INFN Sezione di Milano, Milano, Italy
27INFN Sezione di Cagliari, Monserrato, Italy
28Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
29INFN Sezione di Pisa, Pisa, Italy
30INFN Sezione di Roma Tor Vergata, Roma, Italy
31INFN Sezione di Roma La Sapienza, Roma, Italy
32Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
33Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
34Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
35AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
36National Center for Nuclear Research (NCBJ), Warsaw, Poland
37Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
38Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
39Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
40Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
41Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
42Yandex School of Data Analysis, Moscow, Russia
43Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
44Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
47Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
48European Organization for Nuclear Research (CERN), Geneva, Switzerland
49Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
50Physik-Institut, Universität Zürich, Zürich, Switzerland
51NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
52Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
53University of Birmingham, Birmingham, United Kingdom
54H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
55Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
56Department of Physics, University of Warwick, Coventry, United Kingdom
57STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
58School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
59School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
60Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
61Imperial College London, London, United Kingdom
62Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
63Department of Physics, University of Oxford, Oxford, United Kingdom
64Massachusetts Institute of Technology, Cambridge, MA, United States
65University of Cincinnati, Cincinnati, OH, United States
66University of Maryland, College Park, MD, United States
67Los Alamos National Laboratory (LANL), Los Alamos, United States
68Syracuse University, Syracuse, NY, United States
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 56
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Physics and Micro Electronic College, Hunan University, Changsha City,
China, associated to 7
72Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
73School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
74Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 13
75Universität Bonn - Helmholtz-Institut für Strahlen und Kernphysik, Bonn,
Germany, associated to 17
76Institut für Physik, Universität Rostock, Rostock, Germany, associated to 17
77INFN Sezione di Perugia, Perugia, Italy, associated to 21
78Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 32
79Universiteit Maastricht, Maastricht, Netherlands, associated to 32
80National Research Centre Kurchatov Institute, Moscow, Russia, associated to
39
81National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 39
82National Research University Higher School of Economics, Moscow, Russia,
associated to 42
83National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
39
84DS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain, associated to 45
85University of Michigan, Ann Arbor, United States, associated to 68

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 Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
lUniversità di Padova, Padova, Italy
mUniversità di Pisa, Pisa, Italy
nUniversità degli Studi di Milano, Milano, Italy
oUniversità di Urbino, Urbino, Italy
pUniversità della Basilicata, Potenza, Italy
qScuola Normale Superiore, Pisa, Italy
rUniversità di Modena e Reggio Emilia, Modena, Italy
sUniversità di Siena, Siena, Italy
tMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
uNovosibirsk State University, Novosibirsk, Russia