EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-056 LHCb-PAPER-2020-006 Jan 6, 2021

Measurement of branching fraction ratios for
${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$,
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$,

and ${{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}}$ decays

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

A measurement of four branching-fraction ratios for three-body decays of $B$
mesons involving two open-charm hadrons in the final state is presented. Run 1
and Run 2 $p$ $p$ collision data are used, recorded by the LHCb experiment at
centre-of-mass energies $7$, $8$, and $13\mathrm{\,Te\kern-1.00006ptV}$ and
corresponding to an integrated luminosity of $9\mbox{\,fb}^{-1}$. The measured
branching-fraction ratios are

$\begin{split}\frac{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}})}{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}})}&=0.517\pm
0.015\pm 0.013\pm 0.011,\\\\[10.0pt]
\frac{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}})}{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}})}&=0.577\pm
0.016\pm 0.013\pm 0.013,\\\\[10.0pt]
\frac{{\mathcal{B}}({{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}})}{{\mathcal{B}}({{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}}{{K}^{+}})}&=1.754\pm
0.028\pm 0.016\pm 0.035,\\\\[10.0pt]
\frac{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}})}{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}})}&=0.907\pm
0.033\pm 0.014,\end{split}$

where the first of the uncertainties is statistical, the second systematic,
and the third is due to the uncertainties on the $D$-meson branching
fractions. These are the most accurate measurements of these ratios to date.

Published in JHEP 12 (2020) 139

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

##  1 Introduction

There is a long history of studies of ${B}\rightarrow{{D}^{(*)}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)}}K$ decays, where $B$
represents a ${{B}^{+}}$ or a ${B}^{0}$ meson, ${D}^{(*)}$ is a ${D}^{0}$,
${D}^{*0}$, ${D}^{+}$, or ${D}^{*+}$ meson, ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)}$ is a charge conjugate of one
of the ${D}^{(*)}$ mesons, and $K$ is either a ${K}^{+}$ or ${K}^{0}$
meson.111The inclusion of charge conjugated processes is implied throughout,
unless otherwise stated. The first observations of
${B}\rightarrow{{D}^{(*)}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)}}K$ decays were made public in
1997 and 1998 by the CLEO [1] and ALEPH [2] collaborations. They fully
reconstructed a number of these decay modes in order to probe the discrepancy
between the measured values of branching fractions for hadronic and
semileptonic decays of the $B$ meson [3], the at that time unresolved ‘charm-
counting problem’. In 2003, the BaBar collaboration published the first
comprehensive investigation of ${B}\rightarrow{{D}^{(*)}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)}}K$ decays, reporting
observations or limits for 22 channels [4]. Later, in 2011, the measurements
were updated using a five times larger data sample [5]. The LHCb data
collected during Run 1 and Run 2 of the Large Hadron Collider (LHC) provide an
opportunity to obtain an order of magnitude larger yields with smaller
backgrounds than those measured previously.

This paper reports measurements of relative branching fractions of
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$,
${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$, and
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}}$ decays with respect to
the ${{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}}$ decay for
the first two, and the ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}}{{K}^{+}}$
decay for the third mode. The decays used for normalisation are chosen due to
their similarity to the signal decays in multiplicity and topology, providing
the best cancellation of systematic uncertainties on the ratio. Additionally,
a relative branching fraction of the
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$ and
${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$ decays is reported. The
analysis is based on a sample of $p$ $p$ collisions corresponding to a total
integrated luminosity of $9\mbox{\,fb}^{-1}$ collected at centre-of-mass
energies of 7, 8$\mathrm{\,Te\kern-1.00006ptV}$ (Run 1), and
13$\mathrm{\,Te\kern-1.00006ptV}$ (Run 2) by the LHCb experiment. The modes
containing the excited ${D}^{*}$ meson are hereafter collectively denoted as
${B}\\!\rightarrow{{D}^{*}}{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{K}$
and the modes containing only pseudoscalar $D$ mesons as
${B}\\!\rightarrow{D}{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{K}$.
Decays of these types can proceed at the tree level via three different
processes: pure external $W$ emission, pure internal $W$ emission, also called
colour-suppressed, and the interference of both. Figure 1 shows tree-level
diagrams for the processes relevant for this analysis.

Figure 1: Top left: internal $W$-emission diagram for the decays
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$ and
${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$. Top right: external
$W$-emission diagram for the decays
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}}{{K}^{+}}$. Bottom row: (left)
external and (right) internal $W$-emission diagrams contributing to the
${{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}}$ decay.

The decays of type ${B}\rightarrow{{D}^{(*)}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)}}K$ also allow for spectroscopy
studies through their intermediate resonant structures, especially for
investigations of $c$ $\overline{s}$ resonances via the ${D}^{(*)}$ $K$ system
and charmonium resonances via the ${D}^{(*)}$ ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{(*)}$ system. The specific topology
of these decays allows for strong suppression of combinatorial background in
fully reconstructed decays, and the small energy release leads to an excellent
$B$-mass resolution. These features make them good candidates for future
amplitude analyses. To date, only two amplitude analyses [6, 7] have been
performed in this family of decays, none of which involved an excited
${D}^{*}$ meson. Furthermore, both of them are sensitive only to resonant
states with natural spin-parity assignments, i.e.
$J^{P}=0^{+},1^{-},2^{+},3^{-}$, etc. Relatively little is known about states
with unnatural spin-parity, and ${B}\rightarrow{{D}^{*}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}K$ decays provide an interesting probe
for their study.

##  2 Detector and simulation

The LHCb detector [8, 9] 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${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary $p$ $p$ collision vertex (PV), the impact
parameter (IP), is measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers.

The datasets employed correspond to integrated luminosities of 3
$\mbox{\,fb}^{-1}$ and 6 $\mbox{\,fb}^{-1}$, collected during LHC Run 1 (2011
and 2012) and Run 2 (2015–2018). 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 at which the full
event is reconstructed. Events passing the hardware trigger are considered in
two categories: one in which the trigger criteria are satisfied by energy
deposits in the calorimeter associated with the signal candidate decay, and a
second in which any of the various muon or calorimeter trigger criteria are
met by activity independent of that decay. The software trigger stage requires
a two-, three- or four-track secondary vertex with a significant displacement
from any primary $pp$ interaction vertex. At least one charged particle must
have a transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be
inconsistent with originating from a PV. A multivariate algorithm [10, 11] is
used for the identification of secondary vertices consistent with the decay of
a $b$ hadron.

Simulated samples are produced to model the effect of the detector acceptance
and selection requirements, and to guide subsequent fits to the data. To
produce these samples, $pp$ collisions are generated using Pythia [12,
*Sjostrand:2006za] with a specific LHCb configuration [14]. Decays of unstable
particles are described by EvtGen [15], in which final-state radiation is
generated using Photos [16]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [17,
*Agostinelli:2002hh] as described in Ref. [19].

##  3 Selection

For this analysis, ${D}^{+}$ mesons are reconstructed via their decay to the
${{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ final state, and ${D}^{0}$ mesons are
reconstructed through their decays to both the ${{K}^{-}}{{\pi}^{+}}$, denoted
as ${D}^{0}_{K\pi}$, and ${{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}$, denoted
as ${D}^{0}_{K3\pi}$, final states. However, for decays involving two
${D}^{0}$ mesons at least one must be reconstructed via the two-body decay.
The ${D}^{*+}$ meson is reconstructed through its decay to ${D}^{0}$
${\pi}^{+}$, and is labelled as ${D}^{*+}_{K\pi}$ (${D}^{*+}_{K3\pi}$) if
decaying into ${D}^{0}_{K\pi}$ ${\pi}^{+}$ (${D}^{0}_{K3\pi}$ ${\pi}^{+}$).
The decays analysed are summarised in Table 1.

Table 1: Decays under study. In the first column no assumption about the $D$ final state is made. In the second column, however, the particular $D$ decays are specified. Decay channel | Studied mode
---|---
${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$ | ${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K\pi}}{{D}^{-}}{{K}^{+}}$
${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K3\pi}}{{D}^{-}}{{K}^{+}}$
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$ | ${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{+}}{{K}^{+}}$
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{+}}{{K}^{+}}$
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}}$ | ${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$
${{B}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}}$ | ${{B}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$
${{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}}{{K}^{+}}$ | ${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K\pi}}{{K}^{+}}$
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K3\pi}}{{K}^{+}}$

Well-reconstructed final-state tracks are required. A standard threshold for
the $\chi^{2}_{\text{IP}}$ of each track is applied ($>4$), where
$\chi^{2}_{\text{IP}}$ is defined as the difference in the vertex-fit
$\chi^{2}$ for the PV associated with the $B$-meson candidate when it is
reconstructed with or without the track under consideration. The PV that fits
best to the flight direction of the $B$ candidate is taken as the associated
PV. All charged final-state particles must have momentum greater than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and transverse momentum above
$0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. At least one of them must have
$p>10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, whilst also
having an impact parameter with respect to the $B$ candidate associated PV of
at least $0.1\mathrm{\,mm}$. The invariant masses of $D$ candidates are
required to lie within $20$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of their
known values [20] and their decay vertices must be well reconstructed, having
a fit $\chi^{2}$ less than 10. The $B$ ($D$) candidates have to satisfy the
requirement that the minimum of the cosine of the angle between their
reconstructed momentum and the line connecting their production and decay
vertices should be greater than 0.999 (0). The flight time (distance
$\chi^{2}$) from the associated PV for the $B$\- ($D$)-meson candidates is
required to exceed 0.2 ${\mathrm{\,ps}}$ (36). Finally, particle
identification (PID) information is employed to aid distinction of final-state
$K$ and $\pi$ mesons. The simulated PID response is corrected in order to
match the data. This is achieved using calibration
${{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}}$ samples as a function of track
kinematics and multiplicity. An unbinned method is employed, where the
probability density functions are modelled using kernel density estimation
[21].

A Boosted Decision Tree (BDT) [22, 23] classifier is used to further reduce
combinatorial background, consisting of random combinations of tracks that
mimic the signal. The BDT is trained using a simulated sample to represent
signal and data from the upper sideband of the reconstructed $B$-candidate
invariant-mass distribution to represent combinatorial background. The
variables entering the BDT are: the quality of the reconstructed $B$\- and
$D$-meson decay vertices; the $\chi^{2}_{\text{IP}}$ of the $B$\- and
$D$-meson candidates, as well as the $\chi^{2}_{\text{IP}}$ of the $D$-meson
decay products; and the particle identification variables of the final-state
$K$ and $\pi$ mesons. The threshold for the obtained BDT response is set by
optimising the significance of the $B$ meson signal yield in a fit to data.
The signals are sufficiently large that this approach is found to introduce no
significant bias to the results. Consistency, within statistical
uncertainties, is seen between simulated samples and signal-weighted data for
the variables used by the BDT, and the BDT response itself.

A significant peaking background arises from $B$-meson decays where the final
state is the same but which proceed without one or both of the intermediate
charm mesons. The level of this background is estimated by performing a fit to
the invariant mass for $B$ candidates where the reconstructed mass of one or
both $D$-meson candidates lies far from the known mass and extrapolating the
obtained $B$ signal yield into the $D$-meson signal regions. To suppress
contributions from these decays, the reconstructed $D$-meson decay vertex is
required to be downstream of the reconstructed $B$-meson decay vertex and a
lower bound is placed on the flight distance significance along the beam axis
for $D$ mesons. This requirement suppresses the peaking background to the
level of a few percent of the signal yield, and this remaining contamination
is later subtracted.

##  4 Mass fit

After selecting the signal candidates an unbinned extended maximum-likelihood
fit is performed to the distribution of reconstructed $B$-candidate mass,
$m({{D}^{(*)}}{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{K})$, where the
reconstruction is performed with $D$-candidate masses constrained to their
known values [20] and the $B$-candidate direction of flight to be originating
at the PV. The fit to the mass distribution is performed in the range from
$5210$ to $5390{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, separately for Run 1
and Run 2 data. The shape used to fit the distribution consists of two
components: one to describe the decays of a signal $B$ meson, and a second to
model the combinatorial background. The signal shape is modelled using a
Double-Sided Crystal Ball (DSCB) [24] function. The asymmetric shape and non-
Gaussian tails account for both the mass-resolution effects on both sides and
energy loss due to final-state radiation. The values of tail parameters of the
DSCB shapes are fixed to those found in simulated decays while the Gaussian
core parameters are extracted from the fit together with the signal yield. To
model the combinatorial background an exponential function is used. The lower
bound on the range of invariant mass considered excludes any significant
background from partially reconstructed decays. The combined Run 1 and Run 2
invariant-mass distributions and fit results are shown in Fig. 2. The fit is
used to extract a signal weight for each candidate using the sPlot technique
[25].

Figure 2: Fits to the invariant-mass distributions $m({{D}^{(*)}}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K})$ of (left)
${B}\\!\rightarrow{{D}^{*}}{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{K}$
and (right) ${B}\\!\rightarrow{D}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K}$ for the combined Run 1 and Run 2
samples. The stacked components are (red) combinatorial background and (blue)
signal shape.

##  5 Efficiencies

The efficiencies $\varepsilon$ of the selection of signal candidates are
calculated separately for Run 1 and Run 2 in two stages:

$\varepsilon=\varepsilon^{\mathrm{acc}}\cdot\varepsilon^{\mathrm{sel}},$ (1)

where the geometric LHCb acceptance efficiencies $\varepsilon^{\mathrm{acc}}$
are calculated using simulated samples, and correspond to the fraction of
generated events where all final-state particles lie within the LHCb
acceptance. The trigger, reconstruction, and selection efficiencies
$\varepsilon^{\mathrm{sel}}$ are also determined using simulated samples as
the fraction of reconstructed candidates passing the trigger, reconstruction,
and selection criteria, given that they pass the geometrical acceptance
requirement. The efficiencies are evaluated as a function of the position in
the phase space of the decay. Due to the presence of a pseudoscalar particle
in the initial state and one vector (${D}^{*}$) plus two pseudoscalar
particles in the final state, decays of the type
${B}\\!\rightarrow{{D}^{*}}{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{K}$
have four independent degrees of freedom. These are chosen to be the two-body
squared invariant masses $m^{2}({{D}^{*}}{K})$ and $m^{2}({\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K})$, and two helicity angles: the
angle $\chi$ between the decay planes of the ${D}^{*}$ meson and the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K}$ system in the $B$-meson rest
frame, and the ${D}^{*}$-meson helicity angle $\theta$ defined as the angle
between the direction of the $\pi$ meson coming from the ${D}^{*}$ meson in
the ${D}^{*}$-meson rest frame, and the ${D}^{*}$ meson in and $B$-meson rest
frame. In the case of ${B}\\!\rightarrow{D}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K}$ decays only two degrees of freedom
are required, and these are chosen to be the two-body squared invariant masses
$m^{2}({D}{K})$ and $m^{2}({\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K})$.

Whilst the efficiency varies considerably across the two-body invariant-mass
planes and the ${D}^{*}$-meson helicity angle $\theta$, it does not depend
significantly on the angle $\chi$. Two-dimensional efficiency distributions,
as functions of $m^{2}({{D}^{*}}{K})$ and $m^{2}({\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K})$, are obtained in four equal bins
of $\cos(\theta)$. The efficiency distributions are further smoothed using a
kernel density estimation (KDE) technique [21]. The efficiency in the two-body
invariant-mass distribution integrated over the two helicity angles are shown
in Figs. 3 and 4 for the ${B}\\!\rightarrow{{D}^{*}}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K}$ samples from Run 1 and Run 2,
respectively. The relative statistical uncertainties on the total efficiencies
are in range $10-20\%$.

Figure 3: Selection and reconstruction efficiency,
$\varepsilon^{\mathrm{sel}}$, as a function of position in the two-body
squared invariant-mass plane for the seven ${B}\\!\rightarrow{{D}^{*}}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K}$ modes, obtained using Run 1
simulated samples. A KDE smoothing has been applied. The blue lines indicate
the kinematic boundaries and the numbers indicate the value of the efficiency
at several points in the phase space.

Figure 4: Selection and reconstruction efficiency,
$\varepsilon^{\mathrm{sel}}$, as a function of position in the two-body
squared invariant-mass plane for the seven ${B}\\!\rightarrow{{D}^{*}}{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{K}$ modes, obtained using Run 2
simulated samples. A KDE smoothing has been applied. The blue lines indicate
the kinematic boundaries and the numbers indicate the values of the efficiency
at several points in the phase space.

##  6 Corrected yields

The ratios of branching fractions are calculated using signal yields corrected
by applying candidate-by-candidate background subtraction and efficiency
correction, and accounting for the decays of the $D$ mesons into the final
states. The branching fraction of a ${B}\rightarrow{{D}^{(*)}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}K$ decay is proportional to the
corrected yield, $N^{\mathrm{corr}}$, calculated as

${N^{\mathrm{corr}}}=\frac{\displaystyle\sum_{i}\frac{W_{i}}{\epsilon^{\mathrm{sel}}_{i}(x_{i})\cdot\epsilon^{\mathrm{acc}}}-n^{\mathrm{corr}}_{\mathrm{peaking}}}{\displaystyle{\mathcal{B}}({{D}^{(*)}})\cdot{\mathcal{B}}({\kern
1.99997pt\overline{\kern-1.99997ptD}{}})}.$ (2)

Here the index $i$ runs over all candidates in the fitted sample, $W_{i}$ is
the signal weight for candidate $i$ (see Section 4),
$\epsilon_{i}^{\mathrm{sel}}$ is the selection efficiency for candidate $i$ as
a function of its position $x_{i}$ in the relevant phase space, and
$\epsilon^{\mathrm{acc}}$ is the efficiency of the acceptance cut for the
given mode (see Section 5). Since the efficiency-weighted sum over candidates
includes a small (peaking) background contribution, the efficiency-corrected
residual peaking background $n^{\mathrm{corr}}_{\mathrm{peaking}}$ is
subtracted from the signal region. The value of
$n^{\mathrm{corr}}_{\mathrm{peaking}}$ is obtained by taking the estimated
yield of the peaking background and dividing it by an average efficiency of
the sample, since the distribution of the peaking background in the phase
space of the decay is not known. Finally, the denominator is used to correct
for the $D$-meson decay branching fractions, which are:

$\displaystyle{\mathcal{B}}$
$\displaystyle({{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}})$
$\displaystyle=(3.999\pm 0.045)\%\quad$
$\displaystyle\text{\leavevmode\nobreak\
\cite[cite]{[\@@bibref{}{Amhis:2019ckw}{}{}]}},$ $\displaystyle{\mathcal{B}}$
$\displaystyle({{D}^{0}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})$
$\displaystyle=(8.23\pm 0.14)\%\quad$ $\displaystyle\text{\leavevmode\nobreak\
\cite[cite]{[\@@bibref{}{PDG2019}{}{}]}},$ $\displaystyle{\mathcal{B}}$
$\displaystyle({{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}})$
$\displaystyle=(9.38\pm 0.16)\%\quad$ $\displaystyle\text{\leavevmode\nobreak\
\cite[cite]{[\@@bibref{}{PDG2019}{}{}]}},$ $\displaystyle{\mathcal{B}}$
$\displaystyle({{D}^{*+}}\rightarrow{{D}^{0}}{{\pi}^{+}})$
$\displaystyle=(67.7\pm 0.5)\%\quad$ $\displaystyle\text{\leavevmode\nobreak\
\cite[cite]{[\@@bibref{}{PDG2019}{}{}]}}.$

Table 2 summarises the values of signal yields $N$ obtained from the mass fits
as well as the corrected yields $N^{\mathrm{corr}}$ for all studied modes.

Table 2: Table of all signal yields $N$ and efficiency and $D$-meson branching fraction corrected yields $N^{\mathrm{corr}}$ with the residual peaking background subtracted. The values of corrected yields are rounded to the order of $10^{6}$. The uncertainties are statistical only. Mode | Run 1 | Run 2
---|---|---
$N$ | ${N^{\mathrm{corr}}}\,(10^{6})$ | $N$ | ${N^{\mathrm{corr}}}\,(10^{6})$
${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K\pi}}{{D}^{-}}{{K}^{+}}$ |  $212$$\,\pm\,$ | $16$ |  $289$$\,\pm\,$ | $21$ |  $869$$\,\pm\,$ | $32$ |  $854$$\,\pm\,$ | $32$
${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K3\pi}}{{D}^{-}}{{K}^{+}}$ |  $116$$\,\pm\,$ | $11$ |  $286$$\,\pm\,$ | $28$ |  $606$$\,\pm\,$ | $26$ |  $997$$\,\pm\,$ | $44$
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{+}}{{K}^{+}}$ |  $210$$\,\pm\,$ | $15$ |  $313$$\,\pm\,$ | $23$ |  $912$$\,\pm\,$ | $32$ |  $1009$$\,\pm\,$ | $36$
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{+}}{{K}^{+}}$ |  $153$$\,\pm\,$ | $13$ |  $371$$\,\pm\,$ | $32$ |  $566$$\,\pm\,$ | $25$ |  $969$$\,\pm\,$ | $45$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ |  $605$$\,\pm\,$ | $26$ |  $1196$$\,\pm\,$ | $52$ |  $2409$$\,\pm\,$ | $52$ |  $3495$$\,\pm\,$ | $76$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ |  $321$$\,\pm\,$ | $20$ |  $949$$\,\pm\,$ | $57$ |  $1706$$\,\pm\,$ | $44$ |  $3541$$\,\pm\,$ | $92$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ |  $331$$\,\pm\,$ | $20$ |  $1105$$\,\pm\,$ | $64$ |  $1544$$\,\pm\,$ | $41$ |  $3812$$\,\pm\,$ | $104$
${{B}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ |  $477$$\,\pm\,$ | $24$ |  $517$$\,\pm\,$ | $26$ |  $2564$$\,\pm\,$ | $56$ |  $1823$$\,\pm\,$ | $39$
${{B}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ |  $622$$\,\pm\,$ | $28$ |  $527$$\,\pm\,$ | $23$ |  $2853$$\,\pm\,$ | $60$ |  $1720$$\,\pm\,$ | $35$
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K\pi}}{{K}^{+}}$ |  $2443$$\,\pm\,$ | $54$ |  $651$$\,\pm\,$ | $14$ |  $9071$$\,\pm\,$ | $104$ |  $2039$$\,\pm\,$ | $23$
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ |  $864$$\,\pm\,$ | $32$ |  $648$$\,\pm\,$ | $23$ |  $3867$$\,\pm\,$ | $69$ |  $2040$$\,\pm\,$ | $36$

##  7 Systematic uncertainties

Many systematic effects cancel exactly in the ratios of branching fractions,
such as the uncertainties in the ${b}{\overline{b}}$-production cross-section
and fragmentation fractions as well as the uncertainties in the luminosity.
The kinematics differ most between numerator and denominator for the slow pion
in modes involving a ${D}^{*}$ decay, but the tracking efficiency of the slow
pion produced in the ${D}^{*}$ decay is found to be well modelled using
calibration samples and the associated systematic uncertainty is found to be
negligible. Uncertainties are considered where they arise from the shapes used
to model the invariant-mass distribution, the efficiency determination, the
resampling of the PID response, and the contribution of residual peaking
backgrounds.

The systematic uncertainty related to the signal model is evaluated by
randomly sampling each tail parameter of the DSCB from a normal distribution
centred at the value used in the fit and with a width corresponding to its
uncertainty. The fit is then repeated with these new values and the yields are
recalculated. The correlations of the tail parameters are accounted for. By
doing this many times a distribution of yields is obtained. The RMS of this
distribution is then used as the systematic uncertainty. Changing the shape of
the background model is found to have a negligible impact on the resulting
yields. The associated systematic uncertainty is thus neglected.

To estimate the systematic uncertainty associated with the choice of the
kernel width in the PID response correction, the procedure is repeated with a
larger kernel width. The absolute difference between the new efficiency-
corrected yield and the baseline value is taken as the uncertainty.

Even after applying the flight-distance significance requirements on the $D$
mesons there is still some underlying residual peaking background
$n^{\mathrm{corr}}_{\mathrm{peaking}}$. This is subtracted from the signal
yield. The uncertainty on the yield of the residual peaking background,
determined using the $c$-hadron sidebands, is used as the systematic
uncertainty.

The limited size of the simulated samples leads to uncertainties in the
efficiency estimations. Bootstrapped samples are produced by sampling randomly
candidates from the original simulated sample, allowing repeated selection of
the same candidate, until a new sample having the same number of candidates is
derived. These samples are used to evaluate the associated systematic
uncertainty, resulting in an ensemble of different efficiency distributions.
The RMS values of the resulting yield distributions are then taken as a
measure of the systematic uncertainties. This is typically the dominant
systematic uncertainty.

The tracking efficiencies are assumed to cancel in all ratios where the same
number of tracks is reconstructed in the numerator and denominator.
Differences in kinematics, most obviously for the slow pion in the ${D}^{*}$
decay, could lead to imperfect cancellation. This was explored and the effect
was found to be negligible. In ratios where the number of tracks differ in the
numerator and denominator, an additional systematic uncertainty of 1% per
additional track is applied.

The magnitudes of the individual contributions are summarised in Table 3
together with the total systematic uncertainty obtained by combining the
individual components in quadrature.

Table 3: Systematic uncertainties on $N^{\mathrm{corr}}$ from the signal PDF parameters ($\sigma_{\mathrm{PDF}}$), the finite simulation samples ($\sigma_{\mathrm{MC}}$), the PID resampling ($\sigma_{\mathrm{PID}}$), the residual peaking background ($\sigma_{\mathrm{bkg}}$), and the total systematic uncertainty ($\sigma_{\mathrm{tot.}}$). All values are given as a percentage of the central value of $N^{\mathrm{corr}}$. Decay channel | Run 1 (%) | Run 2 (%)
---|---|---
$\sigma_{\mathrm{PDF}}$ | $\sigma_{\mathrm{MC}}$ | $\sigma_{\mathrm{PID}}$ | $\sigma_{\mathrm{bkg}}$ | $\sigma_{\mathrm{tot.}}$ | $\sigma_{\mathrm{PDF}}$ | $\sigma_{\mathrm{MC}}$ | $\sigma_{\mathrm{PID}}$ | $\sigma_{\mathrm{bkg}}$ | $\sigma_{\mathrm{tot.}}$
${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K\pi}}{{D}^{-}}{{K}^{+}}$ | $0.6$ | $0.8$ | $1.5$ | $0.8$ | $2.0$ | $0.5$ | $1.4$ | $0.2$ | $0.5$ | $1.6$
${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K3\pi}}{{D}^{-}}{{K}^{+}}$ | $1.2$ | $1.2$ | $0.9$ | $1.4$ | $2.4$ | $1.0$ | $2.1$ | $0.7$ | $0.6$ | $2.5$
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{+}}{{K}^{+}}$ | $0.5$ | $1.0$ | $0.4$ | $0.7$ | $1.4$ | $0.8$ | $1.8$ | $0.7$ | $0.4$ | $2.1$
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{+}}{{K}^{+}}$ | $1.4$ | $1.6$ | $1.1$ | $1.2$ | $2.7$ | $0.7$ | $2.5$ | $1.2$ | $0.6$ | $2.9$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ | $0.6$ | $0.7$ | $0.9$ | $0.3$ | $1.3$ | $0.5$ | $1.1$ | $0.2$ | $0.2$ | $1.2$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ | $0.8$ | $1.2$ | $0.3$ | $0.7$ | $1.6$ | $0.8$ | $1.7$ | $0.6$ | $0.3$ | $2.0$
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ | $0.9$ | $1.2$ | $0.3$ | $0.6$ | $1.6$ | $0.6$ | $2.0$ | $0.3$ | $0.3$ | $2.1$
${{B}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ | $0.6$ | $1.1$ | $1.0$ | $0.9$ | $1.8$ | $1.1$ | $1.8$ | $0.5$ | $0.4$ | $2.2$
${{B}^{+}}\\!\rightarrow{{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ | $0.7$ | $1.1$ | $0.5$ | $0.7$ | $1.6$ | $0.7$ | $1.6$ | $0.4$ | $0.3$ | $1.8$
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K\pi}}{{K}^{+}}$ | $0.4$ | $0.7$ | $0.5$ | $0.4$ | $1.0$ | $0.3$ | $0.7$ | $0.7$ | $0.2$ | $1.1$
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ | $0.2$ | $1.4$ | $0.3$ | $0.5$ | $1.5$ | $0.8$ | $1.3$ | $0.4$ | $0.3$ | $1.6$

##  8 Results

The ratios of branching fractions are obtained by appropriately combining the
$N^{\mathrm{corr}}$ yields of decay modes in Table 1 into ratios, such that
the systematic uncertainty coming from the different number of tracks in the
numerator and denominator is minimised. In order to calculate the first two
branching-fraction ratios of the
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$
(${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$) decay with respect to
the ${{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}}$ decay a
weighted average of $N^{\mathrm{corr}}$ of
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{+}}{{K}^{+}}$
(${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K\pi}}{{D}^{-}}{{K}^{+}}$) and
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{+}}{{K}^{+}}$
(${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K3\pi}}{{D}^{-}}{{K}^{+}}$) is done and
divided by the weighted average of $N^{\mathrm{corr}}$ for the
${{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$
and ${{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$
modes. The associated weight in the weighted average is the inverse of the
variance of the value. The variance on $N^{\mathrm{corr}}$ is obtained by
adding the statistical and the systematic uncertainty, including the
uncertainties due to $D$-meson branching fractions, in quadrature.

The first measurement of the third ratio of
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}}$ to
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}}{{K}^{+}}$ decays is calculated by
performing a weighted average of $N^{\mathrm{corr}}$ for
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ decays,
and dividing it by the value of $N^{\mathrm{corr}}$ for the
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K3\pi}}{{K}^{+}}$ decay. A second
measurement is obtained by finding the ratio of $N^{\mathrm{corr}}$ for
${{B}^{0}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{0}_{K\pi}}{{K}^{+}}$ and
${{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}_{K\pi}}{{K}^{+}}$, which is combined
with the first one into the final branching-fraction ratio.

The fourth branching-fraction ratio of
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$ and
${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$ decays is calculated as
the weighted average of two ratios. The first is the ratio of
${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K\pi}}{{D}^{-}}{{K}^{+}}$ and
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K\pi}}{{D}^{+}}{{K}^{+}}$ decays, and the
second is that for
${{B}^{+}}\\!\rightarrow{{D}^{*+}_{K3\pi}}{{D}^{-}}{{K}^{+}}$ and
${{B}^{+}}\\!\rightarrow{{D}^{*-}_{K3\pi}}{{D}^{+}}{{K}^{+}}$ decays.

The ratios of branching fractions are computed separately for Run 1 and Run 2
and then combined in a weighted average. These ratios are measured to be

$\begin{split}\frac{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}})}{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}})}&=0.517\pm
0.015\pm 0.013\pm 0.011,\\\\[10.0pt]
\frac{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}})}{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{D}^{0}}{{K}^{+}})}&=0.577\pm
0.016\pm 0.013\pm 0.013,\\\\[10.0pt]
\frac{{\mathcal{B}}({{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}})}{{\mathcal{B}}({{B}^{0}}\\!\rightarrow{{D}^{-}}{{D}^{0}}{{K}^{+}})}&=1.754\pm
0.028\pm 0.016\pm 0.035,\\\\[10.0pt]
\frac{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}})}{{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}})}&=0.907\pm
0.033\pm 0.014,\end{split}$

where the first uncertainty is statistical, the second systematic, and the
third one is due to the uncertainties on the $D$-meson branching fractions
[20].

The BaBar collaboration studied these decays previously [5], with a different
set of $D^{*0}$ and ${{D}^{0}}$ channels, obtaining signal yields of $91\pm
13$ ${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$ candidates, $75\pm
13$ ${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$ candidates, and
$1300\pm 54$ ${{B}^{0}}\\!\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}}$ candidates.
The sizes of the signal yields obtained using the LHCb data are around twenty
times larger for the first two decays, and over five times larger for the
third. Significant increases are seen for the yields obtained in the
normalisation modes, with respect to earlier studies using data from the Belle
and BaBar experiments. Good agreement is seen with respect to the
corresponding branching fraction ratios according to the Particle Data Group
(PDG) [20], calculated to be $0.43\pm 0.12$, $0.41\pm 0.13$, $2.3\pm 0.3$, and
$1.1\pm 0.3$, respectively. The measurements described in this article are
between 5 and 7 times more precise. The ratio between the
${{B}^{+}}\\!\rightarrow{{D}^{*+}}{{D}^{-}}{{K}^{+}}$ and
${{B}^{+}}\\!\rightarrow{{D}^{*-}}{{D}^{+}}{{K}^{+}}$ deviates from unity with
a significance just below $3\sigma$, suggesting activity in a channel other
than the ${{D}^{+}}^{*}{{D}^{-}}$ channel that the two have in common. These
measurements, and the high purity of the samples obtained for the decays under
study, make these decays prime targets for future analyses of resonant
structure.

##  9 Summary

A data sample corresponding to an integrated luminosity of $9\mbox{\,fb}^{-1}$
recorded with the LHCb detector is used to measure four ratios of branching
fractions in ${B}\rightarrow{{D}^{(*)}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}K$ decays. The ratios are consistent
with previous measurements and are measured with the highest precision to
date. Furthermore, this work represents the first published analysis at the
LHC of $b$-hadron decays to two open-charm hadrons and a third, light, hadron.
Large samples of ${B}\rightarrow{{D}^{(*)}}{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}}K$ decays are available, and can be
isolated in the LHCb dataset with low background contamination. These are
promising characteristics for these channels with future studies of their
intermediate resonant structure in view.

## 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, Thousand Talents Program, 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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Afsharnia9, C.A. Aidala82, S. Aiola25, Z. Ajaltouni9, S. Akar64, J.
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G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S. Amato2, Y. Amhis11,
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Dreimanis59, M.W. Dudek33, L. Dufour47, V. Duk75, P. Durante47, J.M. Durham66,
D. Dutta61, M. Dziewiecki16, A. Dziurda33, A. Dzyuba37, S. Easo56, U. Egede69,
V. Egorychev38, S. Eidelman42,x, S. Eisenhardt57, S. Ek-In48, L. Eklund58, S.
Ely67, A. Ene36, E. Epple66, S. Escher13, J. Eschle49, S. Esen31, T. Evans47,
A. Falabella19, J. Fan3, Y. Fan5, B. Fang72, N. Farley52, S. Farry59, D.
Fazzini11, P. Fedin38, M. Féo47, P. Fernandez Declara47, A. Fernandez
Prieto45, F. Ferrari19,e, L. Ferreira Lopes48, F. Ferreira Rodrigues2, S.
Ferreres Sole31, M. Ferrillo49, M. Ferro-Luzzi47, S. Filippov40, R.A. Fini18,
M. Fiorini20,g, M. Firlej34, K.M. Fischer62, C. Fitzpatrick61, T. Fiutowski34,
F. Fleuret11,b, M. Fontana47, F. Fontanelli23,i, R. Forty47, V. Franco Lima59,
M. Franco Sevilla65, M. Frank47, E. Franzoso20, G. Frau16, C. Frei47, D.A.
Friday58, J. Fu25,q, Q. Fuehring14, W. Funk47, E. Gabriel57, T. Gaintseva41,
A. Gallas Torreira45, D. Galli19,e, S. Gallorini27, S. Gambetta57, Y. Gan3, M.
Gandelman2, P. Gandini25, Y. Gao4, M. Garau26, L.M. Garcia Martin46, P. Garcia
Moreno44, J. García Pardiñas49, B. Garcia Plana45, F.A. Garcia Rosales11, L.
Garrido44, D. Gascon44, C. Gaspar47, R.E. Geertsema31, D. Gerick16, L.L.
Gerken14, E. Gersabeck61, M. Gersabeck61, T. Gershon55, D. Gerstel10, Ph.
Ghez8, V. Gibson54, M. Giovannetti22,k, A. Gioventù45, P. Gironella
Gironell44, L. Giubega36, C. Giugliano20,g, K. Gizdov57, V.V. Gligorov12, C.
Göbel70, E. Golobardes44,m, D. Golubkov38, A. Golutvin60,79, A. Gomes1,a, S.
Gomez Fernandez44, M. Goncerz33, P. Gorbounov38, I.V. Gorelov39, C. Gotti24,j,
E. Govorkova31, J.P. Grabowski16, R. Graciani Diaz44, T. Grammatico12, L.A.
Granado Cardoso47, E. Graugés44, E. Graverini48, G. Graziani21, A. Grecu36,
L.M. Greeven31, R. Greim31, P. Griffith20,g, L. Grillo61, S. Gromov79, L.
Gruber47, B.R. Gruberg Cazon62, C. Gu3, M. Guarise20, P. A. Günther16, E.
Gushchin40, A. Guth13, Y. Guz43,47, T. Gys47, T. Hadavizadeh62, G. Haefeli48,
C. Haen47, J. Haimberger47, S.C. Haines54, T. Halewood-leagas59, P.M.
Hamilton65, Q. Han7, X. Han16, T.H. Hancock62, S. Hansmann-Menzemer16, N.
Harnew62, T. Harrison59, R. Hart31, C. Hasse14, M. Hatch47, J. He5, M.
Hecker60, K. Heijhoff31, K. Heinicke14, A.M. Hennequin47, K. Hennessy59, L.
Henry25,46, J. Heuel13, A. Hicheur68, D. Hill62, M. Hilton61, S.E. Hollitt14,
P.H. Hopchev48, J. Hu16, J. Hu71, W. Hu7, W. Huang5, X. Huang72, W.
Hulsbergen31, T. Humair60, R.J. Hunter55, M. Hushchyn80, D. Hutchcroft59, D.
Hynds31, P. Ibis14, M. Idzik34, D. Ilin37, P. Ilten52, A. Inglessi37, K.
Ivshin37, R. Jacobsson47, S. Jakobsen47, E. Jans31, B.K. Jashal46, A.
Jawahery65, V. Jevtic14, M. Jezabek33, F. Jiang3, M. John62, D. Johnson47,
C.R. Jones54, T.P. Jones55, B. Jost47, N. Jurik62, S. Kandybei50, Y. Kang3, M.
Karacson47, J.M. Kariuki53, N. Kazeev80, M. Kecke16, F. Keizer54,47, M.
Kelsey67, M. Kenzie55, T. Ketel32, B. Khanji47, A. Kharisova81, S.
Kholodenko43, K.E. Kim67, T. Kirn13, V.S. Kirsebom48, O. Kitouni63, S.
Klaver22, K. Klimaszewski35, S. Koliiev51, A. Kondybayeva79, A.
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Lacarrere47, G. Lafferty61, A. Lai26, A. Lampis26, D. Lancierini49, J.J.
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Lantwin49,79, T. Latham55, F. Lazzari28,v, R. Le Gac10, S.H. Lee82, R.
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Liang67, T. Lin60, R. Lindner47, V. Lisovskyi14, R. Litvinov26, G. Liu71, H.
Liu5, S. Liu6, X. Liu3, D. Loh55, A. Loi26, J. Lomba Castro45, I. Longstaff58,
J.H. Lopes2, G. Loustau49, G.H. Lovell54, Y. Lu6, D. Lucchesi27,o, S.
Luchuk40, M. Lucio Martinez31, V. Lukashenko31, Y. Luo3, A. Lupato61, E.
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Machefert11, F. Maciuc36, V. Macko48, P. Mackowiak14, S. Maddrell-Mander53,
L.R. Madhan Mohan53, O. Maev37, A. Maevskiy80, D. Maisuzenko37, M.W.
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K.R. Mattioli82, A. Mauri49, E. Maurice11,b, J. Mauricio44, M. Mazurek35, M.
McCann60, L. Mcconnell17, T.H. Mcgrath61, A. McNab61, R. McNulty17, J.V.
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Meloni24,j, M. Merk31, A. Merli25, L. Meyer Garcia2, M. Mikhasenko47, D.A.
Milanes73, E. Millard55, M.-N. Minard8, O. Mineev38, L. Minzoni20,g, S.E.
Mitchell57, B. Mitreska61, D.S. Mitzel47, A. Mödden14, A. Mogini12, R.A.
Mohammed62, R.D. Moise60, T. Mombächer14, I.A. Monroy73, S. Monteil9, M.
Morandin27, G. Morello22, M.J. Morello28,t, J. Moron34, A.B. Morris10, A.G.
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Müller47, K. Müller49, C.H. Murphy62, D. Murray61, P. Muzzetto26, P. Naik53,
T. Nakada48, R. Nandakumar56, T. Nanut48, I. Nasteva2, M. Needham57, I.
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B. Passalacqua20, G. Passaleva21,47, A. Pastore18, M. Patel60, C.
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Altarelli47, S. Perazzini19, D. Pereima38, P. Perret9, K. Petridis53, A.
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Pinzino47, F. Pisani19, A. Piucci16, Resmi P.K10, V. Placinta36, S. Playfer57,
J. Plews52, M. Plo Casasus45, F. Polci12, M. Poli Lener22, M. Poliakova67, A.
Poluektov10, N. Polukhina79,c, I. Polyakov67, E. Polycarpo2, G.J. Pomery53, S.
Ponce47, A. Popov43, D. Popov52, S. Popov41, S. Poslavskii43, K. Prasanth33,
L. Promberger47, C. Prouve45, V. Pugatch51, A. Puig Navarro49, H. Pullen62, G.
Punzi28,p, W. Qian5, J. Qin5, R. Quagliani12, B. Quintana8, N.V. Raab17, R.I.
Rabadan Trejo10, B. Rachwal34, J.H. Rademacker53, M. Rama28, M. Ramos
Pernas45, M.S. Rangel2, F. Ratnikov41,80, G. Raven32, M. Reboud8, F. Redi48,
F. Reiss12, C. Remon Alepuz46, Z. Ren3, V. Renaudin62, R. Ribatti28, S.
Ricciardi56, D.S. Richards56, S. Richards53, K. Rinnert59, P. Robbe11, A.
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Lopez73, M. Roehrken47, A. Rollings62, P. Roloff47, V. Romanovskiy43, M.
Romero Lamas45, A. Romero Vidal45, J.D. Roth82, M. Rotondo22, M.S. Rudolph67,
T. Ruf47, J. Ruiz Vidal46, A. Ryzhikov80, J. Ryzka34, J.J. Saborido Silva45,
N. Sagidova37, N. Sahoo55, B. Saitta26,f, C. Sanchez Gras31, C. Sanchez
Mayordomo46, R. Santacesaria30, C. Santamarina Rios45, M. Santimaria22, E.
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Zuliani27, G. Zunica61.

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

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Firenze, Firenze, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
yINFN Sezione di Trieste, Trieste, Italy
zUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras