[SC '21]SC'21: The International Conference for High Performance Computing, Networking, Storage, and AnalysisNov 14–19, 2021St. Louis, MO
SC'21: The International Conference for High Performance Computing, Networking, Storage, and Analysis, Nov 14–19, 2021, St. Louis, MO

High Performance Uncertainty Quantification with Parallelized Multilevel Markov Chain Monte Carlo

Linus Seelinger

Institute for Scientific Computing
Heidelberg University

Anne Reinarz

Department of Computer Science
Durham University
United Kingdom

Leonhard Rannabauer

Department of Informatics
Technical University of Munich

Michael Bader

Department of Informatics
Technical University of Munich

Peter Bastian

Institute for Scientific Computing
Heidelberg University

Robert Scheichl

Institute for Applied Mathematics
Heidelberg University

Three levels of discretisation of a tsunami model used in a multilevel Markov chain Monte Carlo method.

The authors gratefully acknowledge the compute and data resources provided by the Leibniz Supercomputing Centre (www.lrz.de) under project number pr83no.
We acknowledge funding from the European Union’s Horizon 2020 Programme under the ENERXICO Project, grant agreement No. 828947.

The authors would also like to thank Jörn Behrens for his extremely useful advice.

We further acknowledge GEBCO Compilation Group (2020) GEBCO 2020 Grid
for providing bathymetry data used in this work (doi:10.5285/ a29c5465-b138-234d-e053-6c86abc040b9).

The authors acknowledge support by the state of Baden-Württemberg through bwHPC
and the German Research Foundation (DFG) through grant INST 35/1134-1 FUGG.

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2019-006 LHCb-PAPER-2018-046

Observation of ${{B}^{0}_{\scaleto{(}{15pt}s\scaleto{)}{15pt}}}\rightarrow
J/\psi p\overline{p}$ decays and precision measurements of the
${{B}^{0}_{\scaleto{(}{15pt}s\scaleto{)}{15pt}}}$ masses

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

The first observation of the decays $B^{0}_{(s)}\rightarrow J/\psi
p\overline{p}$ is reported, using proton-proton collision data corresponding
to an integrated luminosity of $5.2{\text{\,fb}}^{-1}$, collected with the
LHCb detector. These decays are suppressed due to limited available phase
space, as well as due to Okubo-Zweig-Iizuka or Cabibbo suppression. The
measured branching fractions are

$\displaystyle\mathcal{B}(B^{0}\rightarrow J/\psi p\overline{p})$
$\displaystyle=(4.51\pm 0.40\;\text{(stat)}\pm 0.44\;\text{(syst)})\times
10^{-7},$ $\displaystyle\mathcal{B}(B^{0}_{s}\rightarrow J/\psi
p\overline{p})$ $\displaystyle=(3.58\pm 0.19\;\text{(stat)}\pm
0.39\;\text{(syst)})\times 10^{-6}.$

For the $B^{0}_{s}$ meson, the result is much higher than the expected value
of ${\cal O}(10^{-9})$. The small available phase space in these decays also
allows for the most precise single measurement of both the $B^{0}$ mass as
${5279.74\pm 0.30\;({\rm stat})\pm 0.10\;({\rm
syst})}\text{\,Me\kern-1.00006ptV}$, and the $B^{0}_{s}$ mass as ${5366.85\pm
0.19\;({\rm stat})\pm 0.13\;({\rm syst})}\text{\,Me\kern-1.00006ptV}$.

Published in Phys. Rev. Lett. 122, 191804 (2019)

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

Multiquark hadronic states beyond the well-studied quark-antiquark (meson) and
three-quark (baryon) combinations remain elusive even sixty years after their
prediction in the quark model [1, 2]. Employing an amplitude analysis of
${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}{{K}^{-}}$ decays, the LHCb collaboration has found states
consistent with $|uudc\overline{c}\rangle$ pentaquarks decaying to
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}p$ [3, 4] (charge conjugation
is implied throughout this Letter). The decays
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ are sensitive to
pentaquark searches in the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}p$
and ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\kern
1.00006pt\overline{\kern-1.00006ptp}{}$ components and to glueball states [5,
6] in the $p\overline{p}$ system. Baryonic
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}$ decays are also interesting
to study the dynamics of the final baryon-antibaryon system and its
characteristic threshold enhancement, whose underlying origin has still to be
completely understood [7].

In the leading Feynman diagrams shown in Fig. 1, the ${{B}^{0}}$ mode is
Cabibbo suppressed due to the presence of the Cabibbo-Kobayashi-Maskawa
element $V_{cd}$, while the ${{B}^{0}_{s}}$ mode is Okubo-Zweig-Iizuka
suppressed [8, 2, 9]. The naïve theoretical expectation for the branching
fraction
$\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$ is at the level of
$10^{-9}$ [10]. However, the presence of an intermediate pentaquark or
glueball state can enhance the decay rate. The authors of Ref. [10] pointed
out the potential sensitivity of
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ decays to tensor
glueball states via a possible resonant contribution of
$f_{J}(2220)\rightarrow{p}{\overline{p}}$, which could enhance the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ decay branching
fraction up to order $10^{-6}$. Hints towards such enhancements were noted in
a previous LHCb measurement using 1$\text{\,fb}^{-1}$ of $pp$ collision data,
where no observation for either mode was made, but a $2.8$ standard deviation
excess was seen for the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ decay [11].

These decays also allow for high-precision mass measurements. The kinetic
energies in the ${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}$ rest systems
of the decay products ($Q$-values) are approximately
$306\text{\,Me\kern-1.00006ptV}$ for ${B}^{0}$ and
$393\text{\,Me\kern-1.00006ptV}$ for ${B}^{0}_{s}$ decays. The small
$Q$-values imply a very small contribution from momentum uncertainties to the
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}$ mass measurements.

Figure 1: Leading Feynman diagrams for (a)
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{}$ and (b)
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ decays.

In this Letter, the first observation of these modes along with their
branching fraction and ${B}^{0}$ and ${B}^{0}_{s}$ mass measurements are
reported employing a data sample corresponding to 5.2$\text{\,fb}^{-1}$ of
$pp$ collision data collected by the LHCb experiment. As a normalization mode,
the copious ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi(\rightarrow{{K}^{+}}{{K}^{-}})$ sample is used, which is similar
in topology to the signal channels.

The LHCb detector [12, 13] 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 [14], a large-area silicon-strip detector located upstream
of a dipole magnet with a bending power of about 4 Tm, and three stations of
silicon-strip detectors and straw drift tubes [15] 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 GeV.111Natural units with $\hbar=c=1$ are used
throughout. Different types of charged hadrons are distinguished using
information from two ring-imaging Cherenkov detectors [16]. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers [17]. The online event selection is performed by a
trigger [18], comprising a hardware stage based on information from the muon
system, followed by a software stage that applies a full event reconstruction.
The software trigger is a combination of event categories mostly relying on
identifying ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decays consistent
with a $B$-meson decay topology with two muon tracks originating from a
secondary decay vertex detached from the primary $pp$ collision point.

The $pp$ collision data used in this analysis were collected at center-of-mass
energies of 7 and 8 TeV ($3$$\text{\,fb}^{-1}$) and 13 TeV
($2.2$$\text{\,fb}^{-1}$), during the Run 1 (2011 and 2012) and Run 2 (2015
and 2016) run periods, respectively. The data taking conditions differ enough
between the two run periods, such that they are analyzed separately and the
results combined at the end.

Samples of simulated events are used to study the properties of the signal and
control channels. The $pp$ collisions are generated using Pythia [19,
*Sjostrand:2007gs] with a specific LHCb configuration [21]. Decays of hadronic
particles are described by EvtGen [22], in which final-state radiation is
generated using Photos [23]. For the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
mode, simulation samples are generated according to a decay model based on
results reported in Ref. [24], while the
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ signal modes are
generated uniformly in phase space. The interactions of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [25, *Agostinelli:2002hh] as described in Ref. [27].

The event selection relies on the excellent vertexing and charged particle
identification (PID) capabilities of the LHCb detector. For a given particle,
the associated primary vertex (PV) corresponds to that with the smallest
$\chi^{2}_{\rm IP}$, defined as the difference in $\chi^{2}$ between the PV
fit including and excluding the particle. Signal candidates are formed
starting with a pair of charged tracks, consistent with muons originating from
a common vertex significantly displaced from its associated PV and with an
invariant mass consistent with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ meson. Another pair of oppositely charged tracks, identified as
protons and originating from a common vertex, is combined with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate to form a
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}$ candidate. The entire decay
topology is submitted to a kinematic fit where the dimuon invariant mass is
constrained to the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
[28]. The ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ control mode candidates are reconstructed in a similar fashion,
replacing the ${p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ combination
with a pair of charged tracks identified as ${{K}^{+}}{{K}^{-}}$ candidates,
required to have an invariant mass within $\pm 5\text{\,Me\kern-1.00006ptV}$
of the known $\phi$-meson mass [28]. All charged tracks are required to be of
good quality and have $p_{\mathrm{T}}$ $>$ 300 MeV ($p_{\mathrm{T}}$ $>$ 550
MeV) for $p$ or $K$ ($\mu$). For the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
mode, the contamination from
$B^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}K^{+}\pi^{-}$
decays with a pion misidentified as a kaon is rejected by imposing a ${B}^{0}$
mass veto and using PID information. At this stage, the combinatorial
background dominates, comprising a correctly reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson candidate combined with
two unrelated charged tracks.

At this stage, a multidimensional gradient-boosting algorithm [29] is used to
weight the simulated
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
events to match background-subtracted data distributions in all the training
variables. These weights are denoted as GB-weights. The background-subtracted
data distributions are obtained using the sPlot technique [30]. Under the
assumption that the relative corrections between data and simulation are
similar among different
$B^{0}_{(s)}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}h^{+}h^{\prime-}$ decay topologies, $h^{+}$ and $h^{\prime-}$ being
charged hadrons, the GB-weights obtained from the control mode are applied to
the signal mode. To validate this assumption, similar GB-weights are derived
using another control mode,
$B^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{\pi}^{-}}$, yielding similar results.

For further background suppression, two multivariate classifiers are applied,
each employing a gradient-boosted decision tree (BDT) [31]. In the first
stage, the BDTkin classifier, based on kinematical and topological variables
of the ${B}^{0}_{s}$ candidate, is trained using the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi$
decays from simulation as signal proxy, and selected
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{+}}{{K}^{-}}$ candidates
in the mass window $[5450,5700]$ MeV as background. For BDTkin, only kinematic
variables whose distributions are similar between the signal and the control
mode are employed. These include the $p$, $p_{\mathrm{T}}$, and $\chi^{2}_{\rm
IP}$ values of the ${{B}^{0}_{s}}$ meson, the $\chi^{2}$ probability from a
kinematic fit [32] to the decay topology, and the impact parameter (IP) of the
muons with respect to the associated PV.

To choose the BDTkin selection cut, the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ signal figure of
merit, $S/\sqrt{S+B}$, is required to exceed five in a 2$\sigma$ window around
the ${B}^{0}_{s}$ mass peak. The background yield, $B$, is estimated from a
fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{}$ invariant mass distribution. To
estimate the expected signal yield, $S$, the central value of the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ branching fraction
quoted in Ref. [11] is used, along with the signal efficiency obtained from
simulation.

In the final selection stage, a second classifier, BDTPID, uses the hadron PID
information from the Cherenkov detector system to distinguish between pions,
kaons and protons. Aside from PID, the BDTPID training variables also include
the $p$, $p_{\mathrm{T}}$ and $\chi^{2}_{\rm IP}$ values of the protons. The
signal sample is taken as the
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ simulation
incorporating the GB-weights for the kinematic variables, while the background
sample is taken from events in data with
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}p\bar{p})\in[5450,5500]$ MeV.
The hadron PID variables in the simulation require further corrections to be
representative of data. The PID variables are obtained from high-yield
calibration samples of ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ and
$D^{\ast+}\rightarrow D^{0}(\rightarrow K^{-}{{\pi}^{+}})\pi^{+}$ decays,
which can be selected as a function of the $p$, $p_{\mathrm{T}}$ and the
number of tracks in the event using only kinematic information [33]. The
optimal BDTPID selection criterion is chosen by maximizing the figure of merit
$S/\sqrt{S+B}$, with the initial signal and background yields obtained from a
fit to the $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})$ distribution after the BDTkin
selection.

For the ${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ control mode, the selection is performed using a dedicated
classifier, BDTCS, which includes the kinematic variables considered in BDTkin
with the addition of the PID information.

After application of all selection requirements, the background is
predominantly combinatorial. Approximately 1% of the selected events contain
more than one candidate at this stage; a single candidate is selected
randomly. The efficiency of the trigger, detector acceptance, reconstruction
and selection procedure is approximately 1%, as estimated from simulation.

The ${B}^{0}$ and ${B}^{0}_{s}$ signal and background yields are determined
via an extended maximum likelihood fit to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{}$ invariant mass distribution in the
range $[5220,\;5420]$ MeV. Each signal shape is modeled as the sum of two
Crystal Ball [34] functions sharing a common peak position, with tails on
either sides of the peak to describe the radiative and misreconstruction
effects. The background shape is modeled by a first-order polynomial with
parameters determined from the fit to data. The signal-model parameters are
determined from simulation and only the ${B}^{0}$ and ${B}^{0}_{s}$ central
mass values are left as free parameters in the fit to data. The detector
invariant-mass resolution is in agreement with simulations within a factor of
$1.007\pm 0.004$ as determined with the control mode. Residual discrepancies
are accounted for in the systematic uncertainties. In order to validate the
fit model, 1000 mass spectra are generated according to the model and fitted
employing an alternative model comprising three Gaussian components for the
signal and an exponential function for background. The difference between the
input value of the yields and the mean of the fitted yields from the
alternative model is assigned as a systematic uncertainty. The mass fit for
the control mode uses a similar ${B}^{0}_{s}$ signal lineshape, with the
background modeled by an exponential function. The result of the fit to the
combined Run 1 and Run 2 control mode yields a signal of $136{,}800\pm 400$.
The corresponding fit to the signal-mode candidates is shown in Fig. 2 with
the results reported in Table 1, where clear signals of ${B}^{0}$ and
${B}^{0}_{s}$ are observed.

Figure 2: Fit to the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ invariant-mass distribution of the ${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}$ signal modes. Table 1: Signal yields and masses for ${B}^{0}$ and ${B}^{0}_{s}$ mesons. Mode | Yield |  ${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}$ mass [MeV]
---|---|---
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ | $256\pm 22$ | $5279.74\pm 0.30$
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ | $609\pm 31$ | $5366.85\pm 0.19$

The branching fractions measured with respect to the
${{{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\phi}$
control mode are

$\displaystyle\frac{\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}}$ $\displaystyle=\;\frac{N^{\rm
corr}_{{{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 0.70004pt\overline{\kern-0.70004ptp}{}}}{N^{\rm
corr}_{{{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}}},$
$\displaystyle\frac{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})}$
$\displaystyle=\;\frac{N^{\rm
corr}_{{{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 0.70004pt\overline{\kern-0.70004ptp}{}}}{N^{\rm
corr}_{{{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}{{K}^{-}}}},$

where $f_{s}/f_{d}$ is the ratio of the $b$-quark hadronization probabilities
into ${{B}^{0}_{s}}$ and ${{B}^{0}}$ mesons, and $N^{\rm corr}$ denotes
efficiency-corrected signal yields. For the signal modes, since the physics
model is not known a priori, an event-by-event efficiency correction is
applied to the data. It is derived from simulation as a function of the
kinematic variables, which are given in detail in the Appendix.

Since the control mode has a topology very similar to that of the signal mode,
most of the systematic uncertainties cancel in the branching-fraction ratio
measurement. Residual systematic effects of the PID efficiency estimation are
due to the correction procedure. An alternative PID correction is considered
using proton calibration samples from decays of the long-lived ${\mathchar
28931\relax}$ baryon to a proton and a pion, instead of prompt ${{\mathchar
28931\relax}^{+}_{c}}$ decays. The difference between the two methods is
assigned as a systematic uncertainty. The degree to which the simulation
describes hadronic interactions with the detector material is less accurate
for baryons than it is for mesons [22]. Following Ref. [35], a systematic
uncertainty of 4% (1.1%) per proton (kaon) is assigned. Other systematic
effects include the choice of the fit model, the weighting procedure, the
trigger efficiency, and the presence of events with more than one candidate.
The overall systematic uncertainties on the ratio of branching fractions are
$7.2\%$ ($7.2\%$) and $6.5\%$ ($6.6\%$) for ${B}^{0}_{s}$ (${B}^{0}$) meson in
Run 1 and Run 2, respectively, where the relevant contributions, listed in
Table 2, are added in quadrature. Since the detector and the analysis methods
remain the same between the two run periods, the systematic uncertainties are
fully correlated while the statistical uncertainties are uncorrelated. The
combination of the measurements is taken as a weighted mean to give the
branching fraction ratios

$\displaystyle\frac{\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}}$ $\displaystyle=(0.329\pm 0.029\;(\text{stat})\pm
0.022\;(\text{syst}))\times 10^{-2},$
$\displaystyle\frac{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})}$
$\displaystyle=(0.706\pm 0.037\;(\text{stat})\pm 0.048\;(\text{syst}))\times
10^{-2},$

where the first uncertainty is statistical and the second is systematic. For
the absolute branching-fraction determination, the value
${\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}=(1.314\pm 0.016\pm 0.079)\times 10^{-4}}$ is obtained from Ref.
[36] as the product of the two branching ratios,
${\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)=(10.50\pm 0.13\pm 0.64)\times 10^{-4}}$ and
$\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})=0.489\pm 0.005$, and the ratio
of fragmentation probabilities ${f_{s}/f_{d}=0.256\pm 0.020}$ [37]. For the
${B}^{0}_{s}$-meson normalization, the updated ratio ${f_{s}/f_{d}=0.259\pm
0.015}$ [37] is used in Run 1, while for Run 2 it has been multiplied by an
additional scale factor of $1.068\pm 0.046$ [38] to take into account the
dependence on the center of mass energy. The small $S$-wave $K^{+}K^{-}$
fraction under the $\phi(1020)$ resonance, $F_{S}=0.0070\pm 0.0005$ [36], is
accounted for as a correction. The absolute branching fractions are then
combined to give

$\displaystyle\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$
$\displaystyle=(4.51\pm 0.40\;\text{(stat)}\pm 0.44\;\text{(syst)})\times
10^{-7},$
$\displaystyle\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$
$\displaystyle=(3.58\pm 0.19\;\text{(stat)}\pm 0.39\;\text{(syst)})\times
10^{-6},$

where the systematic uncertainty is the sum in quadrature of the overall
systematic contribution on the ratio of branching fractions, the normalization
mode uncertainty and the $f_{s}/f_{d}$ uncertainty for the ${B}^{0}_{s}$
signal. Table 2 summarizes the systematic uncertainties separately for the run
periods. The dominant contributions are the normalization, the PID, and the
tracking systematic uncertainties. For the ${B}^{0}$ meson, the external
normalization measurement from Run 1,
$\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}$ [36] is used, while for Run 2 the additional energy-dependent
correction on $f_{s}/f_{d}$ has an uncertainty of 4.3%. For the ${B}^{0}_{s}$
meson, the measured
$\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}$ is divided by $f_{s}/f_{d}$ to obtain the ${B}^{0}_{s}$
normalization,
$\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})$, resulting
in an uncertainty on $f_{s}/f_{d}$ independent of the run condition.

Table 2: Systematic uncertainties on the branching fraction measurements for Run 1 and Run 2. The total uncertainties on the branching fraction ratios (${\rm BFR}$) are the sum of the systematic uncertainties, added in quadrature. The total uncertainties on the absolute branching fractions ($\mathcal{B}$) include the normalization and the uncertainties on the ratio $f_{s}/f_{d}$ from external measurments as well.  | $\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$ | $\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$
---|---|---
| Run 1 (Run 2) | Run 1 (Run 2)
Fit model | 1.0 (0.5)% | 1.0 (0.9)%
Detector resolution | 0.6 (0.5)% | 0.4 (0.6)%
PID efficiency | 5.0 (4.0)% | 5.0 (4.0)%
Trigger | 1.0 (1.0)% | 1.0 (1.0)%
Tracking | 5.0 (5.0)% | 5.0 (5.0)%
Simulation weighting | 0.4 (0.4)% | 0.3 (0.3)%
Multiple candidates | 0.1 (0.1)% | 0.1 (0.1)%
Total on ${\rm BFR}$ | 7.2 (6.5)% | 7.2 (6.6)%
Normalization | 6.1 (6.1)% | 6.1 (6.1)%
$f_{s}/f_{d}$ | $\;-\;\;(4.3)\%$ | 5.8 (5.8)%
Total on $\mathcal{B}$ |  $\;\;9.4\;(10.1)$% | 11.1 (10.7)%
Table 3:  Systematic uncertainties of ${B}^{0}$ and ${B}^{0}_{s}$ mass measurements. | ${B}^{0}$ | ${B}^{0}_{s}$
---|---|---
| [MeV] | [MeV]
Momentum scale | 0.097 | 0.124
Mass fit model | 0.020 | 0.020
Energy loss correction | 0.030 | 0.030
Total | 0.103 | 0.129

In addition, the small $Q$-values of the
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ decays also allow for
precise measurements of the ${B}^{0}$ and ${B}^{0}_{s}$ masses, with a
resolution of 3.3 MeV (3.8 MeV) for the ${B}^{0}$ (${B}^{0}_{s}$) meson. The
sources of systematic uncertainties include momentum scaling due to
imperfections in the magnetic-field mapping derived using well-known narrow
resonances, uncertainties on particle interactions with the detector material,
and the choice of the signal model, as reported in Table 3. The uncertainty on
the proton mass is neglected. The final results are

$\displaystyle{m_{{{B}^{0}}}}$ $\displaystyle=5279.74\pm 0.30\;({\rm stat})\pm
0.10\;({\rm syst})\text{\,Me\kern-1.00006ptV},$
$\displaystyle{m_{{{B}^{0}_{s}}}}$ $\displaystyle=5366.85\pm 0.19\;({\rm
stat})\pm 0.13\;({\rm syst})\text{\,Me\kern-1.00006ptV},$

with a correlation of $4\times 10^{-4}$ in the statistical uncertainty. These
represent the most precise single measurements for the ${B}^{0}$ and
${B}^{0}_{s}$ masses.

In summary, the first observation of the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{}$ and
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ decays is reported.
The measured branching fraction for the
${{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{}$ decay is consistent with theoretical
expectations [10] while that for
${{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ is enhanced by two
orders of magnitude with respect to predictions without resonant contributions
[10]. More data are needed for glueball and pentaquark searches through a full
Dalitz plot analysis. The world’s best single measurements of the ${B}^{0}$
and ${B}^{0}_{s}$ masses are also reported.

## 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); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); 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), (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 Fondazione
Fratelli Confalonieri (Italy), AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom); Laboratory Directed Research and
Development program of LANL (USA).

Appendix

A. Efficiency parameterization for the signal mode

Figure 3: The three angular variables $\\{\theta_{\ell},\theta_{h},\chi\\}$
for the decay $B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\rightarrow{\mu^{+}}{\mu^{-}})h^{+}h^{-}$, where $h\in\\{p,K\\}$. The
dihadron (in red) and dilepton (in blue) coordinate systems lie back-to-back
with a common vertical $\hat{y}$ axis. The angle between the decay planes is
$\chi\in(-\pi,\pi]$, while the two helicity angles, $\theta_{h}$ and
$\theta_{\ell}$, are defined in the dihadron and dilepton rest frames,
respectively.

The 4-body phase-space of the decay
$B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\rightarrow{\mu^{+}}{\mu^{-}})h^{+}h^{-}$, where $h\in\\{p,K\\}$, is
fully described by four independent kinematic variables. One of them is the
dihadron invariant mass $m_{h^{+}h^{-}}$. For a given $m_{h^{+}h^{-}}$, the
topology can be described by three angles, shown in Fig. 3:

  * •

$\theta_{\ell}$ and $\theta_{h}$: the helicity angles defined in the dimuon
and dihadron rest frames, respectively;

  * •

$\chi$: the azimuthal angle between the two decay planes of the dilepton and
dihadron systems.

Since the final state is self-conjugate, the $h^{-}$ and the $\mu^{-}$
particles are chosen to define the angles, for both
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}$ and ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}_{({s})}^{0}$ mesons. For the signal
mode, the overall efficiency, including trigger, detector acceptance and
selection procedure, is obtained from simulation as a function of the four
kinematic variables, $\vec{\varphi}\equiv\\{m^{\prime}_{{p}\kern
0.70004pt\overline{\kern-0.70004ptp}{}},\cos\theta_{\ell},\cos\theta_{h},\chi^{\prime}\\}$.
Here, $m^{\prime}_{{p}\kern 0.70004pt\overline{\kern-0.70004ptp}{}}$ and
$\chi^{\prime}$ are normalized such that all four variables in $\vec{\varphi}$
lie in the range $(-1,1]$. The efficiency is parameterized as the product of
Legendre polynomials

$\varepsilon(\vec{\varphi})=\sum_{i,j,k,l}c_{i,j,k,l}~{}P(\cos\theta_{\ell},i)P(\cos\theta_{h},j)P(\chi^{\prime},k)P(m^{\prime}_{{p}\kern
0.70004pt\overline{\kern-0.70004ptp}{}},l),$

where $P(x,n)$ is a Legendre polynomial of order $n$ in $x\in(-1,1]$.
Employing the maximum order of the polynomials as $\\{3,7,7,5\\}$ for
$\\{m^{\prime}_{{p}\kern
0.70004pt\overline{\kern-0.70004ptp}{}},\cos\theta_{\ell},\cos\theta_{h},\chi^{\prime}\\}$,
respectively, was found to give a good parameterization. Simulation samples
are employed, where
${B}^{0}_{\scaleto{(}{6.6pt}s\scaleto{)}{6.6pt}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{}$ events are generated
uniformly in phase space. The coefficients, $c_{i,j,k,l}$, are determined from
the simulation using a moments technique employing the orthogonality of
Legendre polynomials

$\displaystyle c_{i,j,k,l}\;=\;$ $\displaystyle C\sum_{n=0}^{N_{\rm
recon}}\left(\frac{2i+1}{2}\right)\left(\frac{2j+1}{2}\right)\left(\frac{2k+1}{2}\right)\left(\frac{2l+1}{2}\right)$
$\displaystyle\hskip 8.5359pt\times
P(\cos{\theta_{\ell}},i)P(\cos{\theta_{h}},j)P(\chi^{\prime},k)P(m^{\prime}_{{p}\kern
0.70004pt\overline{\kern-0.70004ptp}{}},l).$

The sum is over the number of reconstructed decays, $N_{\rm recon}$, in the
simulation sample after all selection criteria. The prefactor $C$ ensures
appropriate normalization. For a given data candidate, the corresponding
kinematic variables, $\vec{\varphi}$, are reconstructed and the efficiency,
$\varepsilon(\vec{\varphi})$, is computed according to the parameterization.
The candidate is subsequently assigned a weight,
$1/\varepsilon(\vec{\varphi})$, to account for the detector efficiency.

B. Combining the Run 1 and Run 2 branching fraction results

The ratio of branching fractions for Run 1 and Run 2 are provided here
separately. For Run 1,

$\displaystyle\frac{\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}}$ $\displaystyle=(0.35\pm 0.05({\rm stat})\pm 0.02({\rm
syst}))\times 10^{-2},$
$\displaystyle\frac{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})}$
$\displaystyle=(0.68\pm 0.06({\rm stat})\pm 0.05({\rm syst}))\times 10^{-2},$

while for Run 2,

$\displaystyle\frac{\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}}$ $\displaystyle=(0.32\pm 0.04({\rm stat})\pm 0.02({\rm
syst}))\times 10^{-2}.$
$\displaystyle\frac{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern
1.00006pt\overline{\kern-1.00006ptp}{})}{\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})}$
$\displaystyle=(0.72\pm 0.05({\rm stat})\pm 0.05({\rm syst}))\times 10^{-2}.$

The absolute branching fractions obtained from our present knowledge of
${\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi)\times\mathcal{B}(\phi\rightarrow{{K}^{+}}{{K}^{-}})\times
f_{s}/f_{d}}=(1.314\pm 0.016\pm 0.079)\times 10^{-4}$ and
${f_{s}/f_{d}=0.259\pm 0.015}$ are, for Run 1,

$\displaystyle\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$
$\displaystyle=(4.54\pm 0.62({\rm stat})\pm 0.33({\rm syst})\pm 0.28({\rm
norm}))\times 10^{-7},$
$\displaystyle\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$
$\displaystyle=(3.45\pm 0.31({\rm stat})\pm 0.25({\rm syst})\pm 0.21({\rm
norm})\pm 0.20(f_{s}/f_{d}))\times 10^{-6},$

and for Run 2,

$\displaystyle\mathcal{B}({{B}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$
$\displaystyle=(4.49\pm 0.52({\rm stat})\pm 0.29({\rm syst})\pm 0.28({\rm
norm})\pm 0.19(f_{s}/f_{d}))\times 10^{-7},$
$\displaystyle\mathcal{B}({{B}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{p}\kern 1.00006pt\overline{\kern-1.00006ptp}{})$
$\displaystyle=(3.66\pm 0.25({\rm stat})\pm 0.24({\rm syst})\pm 0.22({\rm
norm})\pm 0.21(f_{s}/f_{d}))\times 10^{-6}.$

To combine the results from the Run 1 and Run 2 data samples, the systematic
uncertainties are taken as fully correlated between the run periods, since the
data are collected employing the same detector and analysis techniques. The
statistical uncertainties are considered uncorrelated since the datasets are
disjoint. The covariance matrices are constructed as

$\displaystyle V=\begin{pmatrix}\sigma_{\text{stat,
Run~{}1}}^{2}+\sigma_{\text{syst, Run~{}1}}^{2}&\sigma_{\text{syst,
Run~{}1}}\times\sigma_{\text{syst, Run~{}2}}\\\ \sigma_{\text{syst,
Run~{}1}}\times\sigma_{\text{syst, Run~{}2}}&\sigma_{\text{stat,
Run~{}2}}^{2}+\sigma_{\text{syst, Run~{}2}}^{2}\end{pmatrix},$

with the numerical values for the absolute branching fraction combination as

$\displaystyle V_{{{B}^{0}}}$ $\displaystyle=\begin{pmatrix}0.5667&0.1916\\\
0.1916&0.4722\\\ \end{pmatrix}\times 10^{-14},$ $\displaystyle
V_{{{B}^{0}_{s}}}$ $\displaystyle=\begin{pmatrix}0.2401&0.1502\\\
0.1502&0.2142\\\ \end{pmatrix}\times 10^{-12}.$

The weighted mean value and uncertainty are then calculated as

$\displaystyle\overline{x}$ $\displaystyle=\sum_{i\in\\{1,2\\}}w_{i}x_{i},$
$\displaystyle\sigma_{x}^{2}$
$\displaystyle=\sum_{i,k\in\\{1,2\\}}w_{i}w_{k}V_{ik},$

respectively, where the variable $x$ denotes the branching fraction and the
weights are obtained from the inverse of the aforementioned covariance matrix
as

$\displaystyle
w_{i}=\displaystyle\frac{\displaystyle\sum_{k\in\\{1,2\\}}(V^{-1})_{ik}}{\displaystyle\sum_{j,k\in\\{1,2\\}}(V^{-1})_{jk}}.$

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LHCb Collaboration

R. Aaij29, C. Abellán Beteta46, B. Adeva43, M. Adinolfi50, C.A. Aidala77, Z.
Ajaltouni7, S. Akar61, P. Albicocco20, J. Albrecht12, F. Alessio44, M.
Alexander55, A. Alfonso Albero42, G. Alkhazov35, P. Alvarez Cartelle57, A.A.
Alves Jr43, S. Amato2, S. Amerio25, Y. Amhis9, L. An19, L. Anderlini19, G.
Andreassi45, M. Andreotti18, J.E. Andrews62, F. Archilli29, J. Arnau Romeu8,
A. Artamonov41, M. Artuso63, K. Arzymatov39, E. Aslanides8, M. Atzeni46, B.
Audurier24, S. Bachmann14, J.J. Back52, S. Baker57, V. Balagura9,b, W.
Baldini18, A. Baranov39, R.J. Barlow58, G.C. Barrand9, S. Barsuk9, W.
Barter58, M. Bartolini21, F. Baryshnikov73, V. Batozskaya33, B. Batsukh63, A.
Battig12, V. Battista45, A. Bay45, J. Beddow55, F. Bedeschi26, I. Bediaga1, A.
Beiter63, L.J. Bel29, S. Belin24, N. Beliy4, V. Bellee45, N. Belloli22,i, K.
Belous41, I. Belyaev36, G. Bencivenni20, E. Ben-Haim10, S. Benson29, S.
Beranek11, A. Berezhnoy37, R. Bernet46, D. Berninghoff14, E. Bertholet10, A.
Bertolin25, C. Betancourt46, F. Betti17,44, M.O. Bettler51, Ia. Bezshyiko46,
S. Bhasin50, J. Bhom31, M.S. Bieker12, S. Bifani49, P. Billoir10, A.
Birnkraut12, A. Bizzeti19,u, M. Bjørn59, M.P. Blago44, T. Blake52, F. Blanc45,
S. Blusk63, D. Bobulska55, V. Bocci28, O. Boente Garcia43, T. Boettcher60, A.
Bondar40,x, N. Bondar35, S. Borghi58,44, M. Borisyak39, M. Borsato43, M.
Boubdir11, T.J.V. Bowcock56, C. Bozzi18,44, S. Braun14, M. Brodski44, J.
Brodzicka31, A. Brossa Gonzalo52, D. Brundu24,44, E. Buchanan50, A.
Buonaura46, C. Burr58, A. Bursche24, J. Buytaert44, W. Byczynski44, S.
Cadeddu24, H. Cai67, R. Calabrese18,g, R. Calladine49, M. Calvi22,i, M. Calvo
Gomez42,m, A. Camboni42,m, P. Campana20, D.H. Campora Perez44, L.
Capriotti17,e, A. Carbone17,e, G. Carboni27, R. Cardinale21, A. Cardini24, P.
Carniti22,i, K. Carvalho Akiba2, G. Casse56, M. Cattaneo44, G. Cavallero21, R.
Cenci26,p, D. Chamont9, M.G. Chapman50, M. Charles10, Ph. Charpentier44, G.
Chatzikonstantinidis49, M. Chefdeville6, V. Chekalina39, C. Chen3, S. Chen24,
S.-G. Chitic44, V. Chobanova43, M. Chrzaszcz44, A. Chubykin35, P. Ciambrone20,
X. Cid Vidal43, G. Ciezarek44, F. Cindolo17, P.E.L. Clarke54, M. Clemencic44,
H.V. Cliff51, J. Closier44, V. Coco44, J.A.B. Coelho9, J. Cogan8, E.
Cogneras7, L. Cojocariu34, P. Collins44, T. Colombo44, A. Comerma-Montells14,
A. Contu24, G. Coombs44, S. Coquereau42, G. Corti44, M. Corvo18,g, C.M. Costa
Sobral52, B. Couturier44, G.A. Cowan54, D.C. Craik60, A. Crocombe52, M. Cruz
Torres1, R. Currie54, F. Da Cunha Marinho2, C.L. Da Silva78, E. Dall’Occo29,
J. Dalseno43,v, C. D’Ambrosio44, A. Danilina36, P. d’Argent14, A. Davis58, O.
De Aguiar Francisco44, K. De Bruyn44, S. De Capua58, M. De Cian45, J.M. De
Miranda1, L. De Paula2, M. De Serio16,d, P. De Simone20, J.A. de Vries29, C.T.
Dean55, W. Dean77, D. Decamp6, L. Del Buono10, B. Delaney51, H.-P.
Dembinski13, M. Demmer12, A. Dendek32, D. Derkach74, O. Deschamps7, F. Desse9,
F. Dettori56, B. Dey68, A. Di Canto44, P. Di Nezza20, S. Didenko73, H.
Dijkstra44, F. Dordei24, M. Dorigo44,y, A.C. dos Reis1, A. Dosil Suárez43, L.
Douglas55, A. Dovbnya47, K. Dreimanis56, L. Dufour29, G. Dujany10, P.
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Eidelman40,x, S. Eisenhardt54, U. Eitschberger12, R. Ekelhof12, L. Eklund55,
S. Ely63, A. Ene34, S. Escher11, S. Esen29, T. Evans61, A. Falabella17, C.
Färber44, N. Farley49, S. Farry56, D. Fazzini22,44,i, M. Féo44, P. Fernandez
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Ferreira Rodrigues2, M. Ferro-Luzzi44, S. Filippov38, R.A. Fini16, M.
Fiorini18,g, M. Firlej32, C. Fitzpatrick45, T. Fiutowski32, F. Fleuret9,b, M.
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Plana43, J. Garra Tico51, L. Garrido42, D. Gascon42, C. Gaspar44, L.
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Charman58, D. Hill59, M. Hilton58, P.H. Hopchev45, J. Hu14, W. Hu68, W.
Huang4, Z.C. Huard61, W. Hulsbergen29, T. Humair57, M. Hushchyn74, D.
Hutchcroft56, D. Hynds29, P. Ibis12, M. Idzik32, P. Ilten49, A. Inglessi35, A.
Inyakin41, K. Ivshin35, R. Jacobsson44, J. Jalocha59, E. Jans29, B.K.
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S. Karodia55, N. Kazeev74, M. Kecke14, F. Keizer51, M. Kelsey63, M. Kenzie51,
T. Ketel30, E. Khairullin39, B. Khanji44, C. Khurewathanakul45, K.E. Kim63, T.
Kirn11, V.S. Kirsebom45, S. Klaver20, K. Klimaszewski33, T. Klimkovich13, S.
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Kotriakhova35, M. Kozeiha7, L. Kravchuk38, M. Kreps52, F. Kress57, P.
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O. Leroy8, T. Lesiak31, B. Leverington14, P.-R. Li4,ab, Y. Li5, Z. Li63, X.
Liang63, T. Likhomanenko72, R. Lindner44, F. Lionetto46, V. Lisovskyi9, G.
Liu66, X. Liu3, D. Loh52, A. Loi24, I. Longstaff55, J.H. Lopes2, G.H.
Lovell51, D. Lucchesi25,o, M. Lucio Martinez43, Y. Luo3, A. Lupato25, E.
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Materok11, R. Matev44, A. Mathad52, Z. Mathe44, C. Matteuzzi22, A. Mauri46, E.
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Mead56, B. Meadows61, C. Meaux8, N. Meinert70, D. Melnychuk33, M. Merk29, A.
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Minzoni18,g, D.S. Mitzel14, A. Mödden12, A. Mogini10, R.D. Moise57, T.
Mombächer12, I.A. Monroy69, S. Monteil7, M. Morandin25, G. Morello20, M.J.
Morello26,t, O. Morgunova72, J. Moron32, A.B. Morris8, R. Mountain63, F.
Muheim54, M. Mukherjee68, M. Mulder29, D. Müller44, J. Müller12, K. Müller46,
V. Müller12, C.H. Murphy59, D. Murray58, P. Naik50, T. Nakada45, R.
Nandakumar53, A. Nandi59, T. Nanut45, I. Nasteva2, M. Needham54, N. Neri23,q,
S. Neubert14, N. Neufeld44, R. Newcombe57, T.D. Nguyen45, C. Nguyen-Mau45,n,
S. Nieswand11, R. Niet12, N. Nikitin37, A. Nogay72, N.S. Nolte44, A.
Oblakowska-Mucha32, V. Obraztsov41, S. Ogilvy55, D.P. O’Hanlon17, R.
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Ovsiannikova36, P. Owen46, A. Oyanguren76, P.R. Pais45, T. Pajero26,t, A.
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Pappalardo18,g, W. Parker62, C. Parkes58,44, G. Passaleva19,44, A. Pastore16,
M. Patel57, C. Patrignani17,e, A. Pearce44, A. Pellegrino29, G. Penso28, M.
Pepe Altarelli44, S. Perazzini44, D. Pereima36, P. Perret7, L. Pescatore45, K.
Petridis50, A. Petrolini21,h, A. Petrov72, S. Petrucci54, M. Petruzzo23,q, B.
Pietrzyk6, G. Pietrzyk45, M. Pikies31, M. Pili59, D. Pinci28, J. Pinzino44, F.
Pisani44, A. Piucci14, V. Placinta34, S. Playfer54, J. Plews49, M. Plo
Casasus43, F. Polci10, M. Poli Lener20, A. Poluektov8, N. Polukhina73,c, I.
Polyakov63, E. Polycarpo2, G.J. Pomery50, S. Ponce44, A. Popov41, D.
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Quagliani10, B. Quintana7, N.V. Raab15, B. Rachwal32, J.H. Rademacker50, M.
Rama26, M. Ramos Pernas43, M.S. Rangel2, F. Ratnikov39,74, G. Raven30, M.
Ravonel Salzgeber44, M. Reboud6, F. Redi45, S. Reichert12, F. Reiss10, C.
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Rodriguez Lopez69, M. Roehrken44, S. Roiser44, A. Rollings59, V.
Romanovskiy41, A. Romero Vidal43, J.D. Roth77, M. Rotondo20, M.S. Rudolph63,
T. Ruf44, J. Ruiz Vidal76, J.J. Saborido Silva43, N. Sagidova35, B.
Saitta24,f, V. Salustino Guimaraes65, C. Sanchez Gras29, C. Sanchez
Mayordomo76, B. Sanmartin Sedes43, R. Santacesaria28, C. Santamarina Rios43,
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Schmelzer12, B. Schmidt44, O. Schneider45, A. Schopper44, H.F. Schreiner61, M.
Schubiger45, S. Schulte45, M.H. Schune9, R. Schwemmer44, B. Sciascia20, A.
Sciubba28,k, A. Semennikov36, E.S. Sepulveda10, A. Sergi49, N. Serra46, J.
Serrano8, L. Sestini25, A. Seuthe12, P. Seyfert44, M. Shapkin41, Y.
Shcheglov35,†, T. Shears56, L. Shekhtman40,x, V. Shevchenko72, E. Shmanin73,
B.G. Siddi18, R. Silva Coutinho46, L. Silva de Oliveira2, G. Simi25,o, S.
Simone16,d, I. Skiba18, N. Skidmore14, T. Skwarnicki63, M.W. Slater49, J.G.
Smeaton51, E. Smith11, I.T. Smith54, M. Smith57, M. Soares17, l. Soares
Lavra1, M.D. Sokoloff61, F.J.P. Soler55, B. Souza De Paula2, B. Spaan12, E.
Spadaro Norella23,q, P. Spradlin55, F. Stagni44, M. Stahl14, S. Stahl44, P.
Stefko45, S. Stefkova57, O. Steinkamp46, S. Stemmle14, O. Stenyakin41, M.
Stepanova35, H. Stevens12, A. Stocchi9, S. Stone63, B. Storaci46, S.
Stracka26, M.E. Stramaglia45, M. Straticiuc34, U. Straumann46, S. Strokov75,
J. Sun3, L. Sun67, Y. Sun62, K. Swientek32, A. Szabelski33, T. Szumlak32, M.
Szymanski4, Z. Tang3, T. Tekampe12, G. Tellarini18, F. Teubert44, E. Thomas44,
M.J. Tilley57, V. Tisserand7, S. T’Jampens6, M. Tobin32, S. Tolk44, L.
Tomassetti18,g, D. Tonelli26, D.Y. Tou10, R. Tourinho Jadallah Aoude1, E.
Tournefier6, M. Traill55, M.T. Tran45, A. Trisovic51, A. Tsaregorodtsev8, G.
Tuci26,p, A. Tully51, N. Tuning29,44, A. Ukleja33, A. Usachov9, A.
Ustyuzhanin39,74, U. Uwer14, A. Vagner75, V. Vagnoni17, A. Valassi44, S.
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A. Venkateswaran63, M. Vernet7, M. Veronesi29, M. Vesterinen52, J.V. Viana
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Cardona42,m, A. Vitkovskiy29, M. Vitti51, V. Volkov37, A. Vollhardt46, D. Vom
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M. Zavertyaev13,c, D. Zhang68, L. Zhang3, W.C. Zhang3,z, Y. Zhang44, A.
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Zucchelli17,e.

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

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 Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, 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
vH.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
ySezione INFN di Trieste, Trieste, Italy
zSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
aaPhysics and Micro Electronic College, Hunan University, Changsha City, China
abLanzhou University, Lanzhou, China
†Deceased