EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2014-187 LHCb-PAPER-2014-039 5 August 2014

First observation of a baryonic $B_{c}^{+}$ decay

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

A baryonic decay of the ${{B}_{c}^{+}}$ meson,
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$, is observed for the first time, with a
significance of $7.3$ standard deviations, in $pp$ collision data collected
with the LHCb detector and corresponding to an integrated luminosity of
$3.0\mbox{\,fb}^{-1}$ taken at center-of-mass energies of $7$ and
$8\mathrm{\,Te\kern-1.00006ptV}$. With the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decay as normalization channel, the ratio of branching
fractions is measured to be

$\frac{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}})}{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})}=0.143^{\,+\,0.039}_{\,-\,0.034}\,(\mathrm{stat})\pm
0.013\,(\mathrm{syst}).$

The mass of the ${{B}_{c}^{+}}$ meson is determined as
$M({{B}_{c}^{+}})=6274.0\pm 1.8\,(\mathrm{stat})\pm
0.4\,(\mathrm{syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, using the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ channel.

Submitted to Phys. Rev. Lett.

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

The ${{B}_{c}^{+}}$ meson is the ground state of the $\overline{b}c$ system
and is the only doubly heavy flavored meson that decays weakly (the inclusion
of charge conjugated processes is implied throughout this Letter). A large
number of ${{B}_{c}^{+}}$ decay modes are expected, since either the
$\overline{b}$ quark or the $c$ quark can decay, with the other quark acting
as spectator, or the two quarks can annihilate into a virtual $W^{+}$ boson.
The ${{B}_{c}^{+}}$ meson was first observed by CDF through the semileptonic
decay ${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}l^{+}\nu_{l}X$ [1], and the hadronic decay
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ was observed later by CDF and D0 [2, 3]. Many more
hadronic decay channels of the ${{B}_{c}^{+}}$ meson have been observed by
LHCb [4, 5, 6, 7, 8, 9, 10]. At LHCb, the ${B}_{c}^{+}$ mass was measured in
the ${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ [11] and
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{D}^{+}_{s}}$ [7] decays, and its lifetime has been determined using
the ${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{\mu^{+}}\nu_{\mu}X$ decay [12]. However, baryonic decays of
${{B}_{c}^{+}}$ mesons have not been observed to date. Baryonic decays of $B$
mesons provide good opportunities to study the mechanism of baryon production
and to search for excited baryon resonances [13, 14, 15]. The observation of
intriguing behavior in the baryonic decays of the ${{B}^{0}}$ and
${{{B}^{+}}}$ mesons, e.g. the enhancements of the rate of multi-body decays
and the production of baryon pairs of low mass [16, 17, 18, 19, 20, 21, 22],
has further motivated this study.

This Letter presents the first observation of a baryonic ${{B}_{c}^{+}}$
decay, ${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$, and the measurement of its branching
fraction with respect to the channel
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$. The mass of the ${B}_{c}^{+}$ meson is also determined
using the ${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ channel. Owing to the small energy release
($Q$-value) of this channel, the systematic uncertainty of the measured
${B}_{c}^{+}$ mass is small compared to the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ channel.

The data used in this analysis are from $pp$ collisions recorded by the LHCb
experiment, corresponding to an integrated luminosity of $1.0\mbox{\,fb}^{-1}$
at a center-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$ and
$2.0\mbox{\,fb}^{-1}$ at $8\mathrm{\,Te\kern-1.00006ptV}$. The LHCb detector
[23] is a single-arm forward spectrometer covering the pseudorapidity range
$2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks.
The detector includes a high-precision tracking system consisting of a
silicon-strip vertex detector surrounding the $pp$ interaction region, a
large-area silicon-strip detector located upstream of a dipole magnet with a
bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip
detectors and straw drift tubes placed downstream [24]. The combined tracking
system provides a momentum measurement with relative uncertainty varying from
0.4% at low momentum to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and
impact parameter resolution of 20${\,\upmu\rm m}$ for tracks with large
transverse momentum ($p_{\rm T}$). Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[25]. Photon, electron and hadron candidates are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [26]. The trigger [27] consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage, which applies a full event reconstruction. In this analysis,
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are reconstructed in
the dimuon decay channel, and only trigger information related to the final
state muons is considered. Events are selected by the hardware triggers
requiring a single muon with $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ or a muon pair with product of
transverse momenta greater than $(1.3\,\mathrm{GeV}\\!/c)^{2}$. At the first
stage of the software trigger, events are selected that contain two muon
tracks with $\mbox{$p_{\rm T}$}>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
invariant mass
$M({\mu^{+}\mu^{-}})>2.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, or a single
muon track with $\mbox{$p_{\rm T}$}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$\chi^{2}$ of the impact parameter ($\chi^{2}_{\mathrm{IP}}$) greater than 16
with respect to any primary vertices. The quantity $\chi^{2}_{\mathrm{IP}}$ is
the difference between the $\chi^{2}$ values of a given primary vertex
reconstructed with and without the considered track. The second stage of the
software trigger selects a muon pair with an invariant mass that is consistent
with the known ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass [28],
with the effective decay length significance of the reconstructed
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidate, $S_{\mathrm{L}}$,
greater than 3, where $S_{\mathrm{L}}$ is the distance between the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ vertex and the primary vertex
divided by its uncertainty.

The offline analysis uses a preselection, followed by a multivariate selection
based on a boosted decision tree (BDT) [29, 30]. In the preselection, the
invariant mass of the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$
candidate is required to be in the interval
$[3020,3135]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidates are selected by
requiring the $\chi^{2}$ per degree of freedom, $\chi^{2}/\mathrm{ndf}$, of
the vertex fit to be less than 20. The muons are required to have
$\chi^{2}_{\mathrm{IP}}>4$ with respect to any reconstructed $pp$ vertex, to
suppress the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidates
produced promptly in $pp$ collisions. The decay
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$
(${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$) is reconstructed by combining a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate with one (three)
charged track(s) under ${{\pi}^{+}}$ ($p$, $\overline{p}$ and ${{\pi}^{+}}$)
mass hypothesis. The requirements $\chi^{2}_{\mathrm{IP}}>4$ and
$\mbox{$p_{\rm T}$}>0.1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, are applied to
these hadron tracks. Particle identification (PID) is performed using
dedicated neural networks, which use the information from all the sub-
detectors. Well-identified pions are selected by a tight requirement on the
value of the PID discriminant $\mathcal{P}_{\pi}$. A loose requirement is
applied to the PID discriminants of protons and anti-protons,
$\mathcal{P}_{p}$, $\mathcal{P}_{\overline{p}}$, followed by the optimization
described below. To improve the PID performance, the momenta of protons and
anti-protons are required to be greater than
$10{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The ${{B}_{c}^{+}}$ candidate is
required to have vertex fit $\chi^{2}/\mathrm{ndf}<6$, $\mbox{$p_{\rm
T}$}>2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $\chi^{2}_{\mathrm{IP}}<16$ with
respect to at least one reconstructed $pp$ collision and decay-time
significance larger than 9 with respect to the vertex with the smallest
$\chi^{2}_{\mathrm{IP}}$. To improve the mass and decay-time resolutions, a
kinematic fit [31] is applied to the ${{B}_{c}^{+}}$ decay, constraining the
mass of the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ candidate to the
current best world average [28] and the momentum of the ${{B}_{c}^{+}}$
candidate to point back to the primary vertex.

The BDT is trained with a simulated sample, where ${{B}_{c}^{+}}$ candidates
are generated with Bcvegpy [32], interfaced to Pythia6 [33], using a specific
LHCb configuration [34]. Decays of hadronic particles are described by EvtGen
[35], in which final-state radiation (FSR) is generated using Photos [36]. The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit [37, *Agostinelli:2002hh] as described in
Ref. [39]. For the background, candidates in the invariant mass sidebands of
the preselected ${{B}_{c}^{+}}$ data sample are used. The BDT input variables
are $p_{\rm T}$, $\chi^{2}_{\mathrm{IP}}$, $S_{\mathrm{L}}$ of the
${{B}_{c}^{+}}$ candidate, $\chi^{2}/\mathrm{ndf}$ of its vertex fit, the
quality of the constrained kinematic fit of the decay chain, and $p_{\rm T}$,
$\chi^{2}_{\mathrm{IP}}$ of the hadrons. For the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ candidates, the selection criteria are fixed
by optimizing the BDT discriminant jointly with the product of two proton PID
discriminants, $\mathcal{P}_{p}\times\mathcal{P}_{\overline{p}}$. The
selections on BDT discriminant and the combined PID discriminant are chosen to
maximize the figure of merit, aiming for a signal significance of three
standard deviations, $\epsilon/(3/2+\sqrt{B})$ [40], where $\epsilon$ is the
signal efficiency determined using simulated events and $B$ is the number of
expected background candidates estimated using sideband events in the data.
For the ${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decay, the BDT discriminant is selected to maximize the
signal significance $S/\sqrt{S+B}$, where $S$ and $B$ are the expected signal
and background yields, estimated from simulated events and sideband data,
respectively.

Figure 1 shows the invariant mass distributions of the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ and
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ candidates after all selections, together with the results
of unbinned extended maximum likelihood fits. For both decays, the signal
shape is modeled with a modified Gaussian distribution with power-law tails on
both sides, with the tail parameters fixed from simulation. The combinatorial
background is described by a linear function. The
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ channel is affected by a peaking background from the
${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decay where the kaon is misidentified as a pion. The shape
of this component is taken from the simulation and its yield, relative to the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decay, is fixed to the ratio of their branching fractions,
$0.069\pm 0.019$ [5], corrected by their relative efficiency. The invariant
mass resolution for the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decay is determined to be $13.0\pm
0.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is the width of the core
of the modified Gaussian, and the value in the simulated sample is $11.69\pm
0.06{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In the fit to the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ invariant mass distribution, the signal
resolution is fixed to $6.40{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which
is the measured resolution of
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decay in data scaled with their ratio in simulation,
$0.492\pm 0.005\,(\mathrm{stat})$. The observed signal yields are $23.9\pm
5.3$ ($2835\pm 58$) for the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$
(${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$) decay, where the uncertainties are statistical. The
significance of the decay
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ is $7.3\,\sigma$, determined from the
likelihood ratio of the fits with background only and with signal plus
background hypotheses [41].

Figure 1: Invariant mass distribution for (left)
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ and (right)
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ candidates. The superimposed curves show the fitted
contributions from signal (dashed), combinatorial background (dotted),
misidentification background (dot-dashed) and their sum (solid).

From the fit to the ${B}_{c}^{+}$ invariant mass distribution in the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ decay, the mass of the ${B}_{c}^{+}$ meson is
found to be $6273.8\pm 1.8{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Table 1
summarizes the systematic uncertainties of the ${{B}_{c}^{+}}$ mass
measurement, which are dominated by the momentum scale calibration. The
alignment of the LHCb tracking system is performed with samples of prompt
${{D}^{0}}\rightarrow K^{-}{{\pi}^{+}}$ decays, and the momentum is calibrated
using ${{K}^{+}}$ from
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}}$ decays, and validated using a variety of known resonances.
The uncertainty of the momentum scale calibration is $0.03\%$ [42], which is
the difference between momentum scale factors determined using different
resonances. This effect is studied by changing the momentum scale by one
standard deviation and repeating the analysis, taking the variation of the
reconstructed mass as a systematic uncertainty. The amount of material
traversed by a charged particle in the tracking system is known with an
uncertainty of 10%, and the systematic effect of this uncertainty on the
${B}_{c}^{+}$ mass measurement is studied by varying the energy loss
correction by 10% in the reconstruction [43]. Since only charged tracks are
reconstructed, the ${{B}_{c}^{+}}$ mass is underestimated due to FSR by
$0.20\pm 0.03{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, as determined with a
simulated sample. Therefore, the measured mass is corrected by
$0.20{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$0.03{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is assigned as a systematic
uncertainty. The contribution from the fit model is studied by using
alternative fit functions for the signal and background, by using different
fit invariant mass ranges or by changing the estimated mass resolution within
its uncertainty. The total systematic uncertainty of the mass measurement is
$0.42{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. After the correction for FSR,
the mass of the ${B}_{c}^{+}$ meson is determined to be $6274.0\pm
1.8\,(\mathrm{stat})\pm
0.4\,(\mathrm{syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. A combination
of this result with previous LHCb measurements gives $6274.7\pm
0.9\,(\mathrm{stat})\pm
0.8\,(\mathrm{syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In the
combination of the mass measurements, all systematic uncertainties apart from
those due to the mass fit model and FSR are considered fully correlated.

Table 1: Systematic uncertainties for the ${{B}_{c}^{+}}$ mass measurement. Source | Value (${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$)
---|---
Momentum scale | 0.40
Energy loss | 0.05
Final state radiation | 0.03
Fit model | 0.10
Total | 0.42

In the branching fraction measurement of the decay
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$, to account for any difference between data
and simulation, the PID efficiency is calibrated using control data samples.
To allow easy calibration of the PID efficiency, the selection on the
individual PID discriminants, $\mathcal{P}_{p}$ and
$\mathcal{P}_{\overline{p}}$, is applied instead of their product. The same
cut value is applied to the two PID variables, and this cut value is optimized
simultaneously with the BDT discriminant, maximizing the same figure of merit.
With the new selection criteria, used to determine the branching fraction, the
signal yield of the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ decay is
$19.3^{\,+\,5.3}_{\,-\,4.6}\,(\mathrm{stat})$. The ratio of yields between the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ and
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ modes is determined to be
$r_{N}=0.0068^{\,+\,0.0019}_{\,-\,0.0016}\,(\mathrm{stat})$.

The ratio of branching fractions is calculated as

$\frac{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}})}{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})}=\frac{r_{N}}{r_{\epsilon}},$

where
$r_{\epsilon}\equiv\epsilon({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}})/\epsilon({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})$ is the ratio of the total efficiencies. The geometrical
acceptance, reconstruction, selection and trigger efficiencies are determined
from simulated samples for both channels. The central value of the
${B}_{c}^{+}$ lifetime measured by LHCb, $509\pm 8\,(\mathrm{stat})\pm
12\,(\mathrm{syst})\rm\,fs$ [12], is used in the simulation. The PID
efficiency for each track is measured in data in bins of momentum, $p$,
pseudorapidity, $\eta$ of the track and track multiplicity of the event,
$n_{\mathrm{trk}}$. The PID efficiency for pions is determined with
${{\pi}^{+}}$ from $D^{*}$-tagged $D^{0}\rightarrow{{K}^{-}}{{\pi}^{+}}$
decays. Similarly, the PID efficiency for protons is determined using protons
from $\Lambda_{c}^{+}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ decays. These
efficiencies are assigned to the simulated candidate according to $p$ and
$\eta$ of the final state hadron tracks, and $n_{\mathrm{trk}}$ of the event.
The distribution of $n_{\mathrm{trk}}$ in simulation is reweighted to match
that in data. The overall ratio of efficiencies, $r_{\epsilon}$, is found to
be $(4.76\pm 0.06)\%$, where the uncertainty is statistical.

Table 2: Systematic uncertainties (in percent) for the relative branching fraction measurement. Source | Value (%)
---|---
Fit model | 2.0
Acceptance | 0.7
Trigger | 1.1
Lifetime | 1.1
Reco. of $p,\,\overline{p}$ | $2\times 2.3$
Pion PID | $1.1$
Proton PID | $2.4$
Decay model | $6.7$
Total | 8.9

The systematic uncertainties for the branching fraction measurement are
summarized in Table 2. For the signal yields, the systematic uncertainty is
obtained by varying the invariant mass fit functions of the two modes. The
effect of geometrical acceptance is evaluated by comparing the efficiencies
obtained from samples simulated with different data taking conditions. The
systematic uncertainty due to the trigger requirement is studied by comparing
the trigger efficiency in data and simulated samples, using a large
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ sample [44, 7]. The impact of
the uncertainty of the ${{B}_{c}^{+}}$ lifetime is evaluated from the
variation of the relative efficiency when the ${{B}_{c}^{+}}$ lifetime is
changed by one standard deviation of the LHCb measurement [12]. The systematic
uncertainty associated with the reconstruction efficiency of the two
additional hadron tracks, $p$ and $\overline{p}$, in the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ mode compared to the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ mode, is also studied. Different assumptions for the pion
PID efficiency in the kinematic regions where no calibration efficiency is
available introduce a systematic uncertainty. For the protons, the systematic
uncertainty from PID selection takes into account the uncertainties in the
single-track efficiencies, the binning scheme in
$(p,\,\eta,\,n_{\mathrm{trk}})$ intervals and the uncertainty of the track
multiplicity distribution. Another systematic uncertainty is related to the
unknown decay model of the mode
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$. The simulated sample is generated according
to a uniform phase-space decay model. Figure 2 shows the one-dimensional
invariant mass distributions of $M(p\overline{p})$ and $M(p{{\pi}^{+}})$ for
data, with background subtracted using the sPlot method [45]. Figure 2 also
shows the distributions for simulated events, which agree with those in data
within the large statistical uncertainties. The efficiency calculated using
the observed distribution in data relative to the efficiency determined using
the simulated decay model is $0.949\pm 0.067$, where the uncertainty is
statistical. Since the value is consistent with unity within the uncertainty,
no correction to the efficiency is made and a systematic uncertainty of
$6.7\%$ is assigned. The total systematic uncertainty associated with the
relative branching fraction measurement is $8.9\%$.

As a result the ratio of branching fractions is measured to be

$\frac{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}})}{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})}=0.143^{\,+\,0.039}_{\,-\,0.034}\,(\mathrm{stat})\pm
0.013\,(\mathrm{syst}),$

which is consistent with the expectation from the spectator decay model
assuming factorization [46], $\frac{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}})}{{\cal
B}({{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})}\sim\frac{{\cal B}({{B}^{0}}\\!\rightarrow
D^{*-}p\overline{p}{{\pi}^{+}})}{{\cal B}({{B}^{0}}\\!\rightarrow
D^{*-}{{\pi}^{+}})}=0.17\pm 0.02$. The branching fractions for
${{B}^{0}}\\!\rightarrow D^{*-}p\overline{p}{{\pi}^{+}}$ and
${{B}^{0}}\\!\rightarrow D^{*-}{{\pi}^{+}}$ decays are taken from Ref. [28].

Figure 2: Invariant mass distributions of (left) $M(p\overline{p})$ and
(right) $M(p{{\pi}^{+}})$ for data (dots) and simulation (solid) using uniform
phase-space model, for
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ decay.

In conclusion, the decay
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}p\overline{p}{{\pi}^{+}}$ is observed with a significance of $7.3$
standard deviations, using a data sample corresponding to an integrated
luminosity of $3.0\mbox{\,fb}^{-1}$ collected by the LHCb experiment. Thi is
the first observation of a baryonic decay of the ${B}_{c}^{+}$ meson. The
branching fraction of this decay relative to that of the
${{B}_{c}^{+}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}$ decay is measured. The mass of the ${{B}_{c}^{+}}$ meson
is measured to be $6274.0\pm 1.8\,(\mathrm{stat})\pm
0.4\,(\mathrm{syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. In combination
with previous results by LHCb [7, 11], the ${{B}_{c}^{+}}$ mass is determined
to be $6274.7\pm 0.9\,(\mathrm{stat})\pm
0.8\,(\mathrm{syst}){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland);
INFN (Italy); FOM and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA
(Romania); MinES and FANO (Russia); MinECo (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC (United Kingdom); NSF (USA). The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia). Individual groups or members have received support from EPLANET,
Marie Skłodowska-Curie Actions and ERC (European Union), Conseil général de
Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR
(Russia), XuntaGal and GENCAT (Spain), Royal Society and Royal Commission for
the Exhibition of 1851 (United Kingdom).

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R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
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Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38, A. Ukleja28,
A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, R.
Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S. Vecchi16, J.J.
Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B. Viaud7, D.
Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,p, A. Vollhardt40, D.
Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H. Voss10,
J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23, S.
Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, L. Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A.
Zhokhov31, L. Zhong3, A. Zvyagin38.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
pLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
qHanoi University of Science, Hanoi, Viet Nam
rUniversità di Padova, Padova, Italy
sUniversità di Pisa, Pisa, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità degli Studi di Milano, Milano, Italy
vPolitecnico di Milano, Milano, Italy


$\text{A}\leq\sqrt{\frac{2\mathbf{V}(p(X_{t}),h^{*})\log\left(\tfrac{SAT}{\delta}\right)}{N_{t_{k}}(X_{t})}}+2\text{A}_{2}+\underbrace{{\mathrm{sp}\left(h^{*}\right)}\sqrt{\frac{2\log\left(\tfrac{SAT}{\delta}\right)\sqrt{S\log\tfrac{SAT}{\delta}}}{N_{t_{k}}(X_{t})\sqrt{N_{t_{k}}(X_{t})}}}+\frac{3c_{0}\log\left(\tfrac{SAT}{\delta}\right)}{N_{t_{k}}(X_{t})}}_{\text{A}_{3}(k,t)}.$

(STEP 2) The number of visits $N_{k}(X_{t})$ is lower-bounded by
$\frac{1}{2}N_{t}(X_{t})$ when $N_{k}(X_{t})\geq 1$ by doubling trick (DT). By
summing over $t$ and $k$, we find that with probability $1-6\delta$,

$\displaystyle\sum_{k}(3k)$ $\displaystyle\leq
SAc_{0}+\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\sqrt{\frac{2\mathbf{V}(p(X_{t}),h^{*})\log\left(\tfrac{SAT}{\delta}\right)}{N_{t_{k}}(X_{t})}}+\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}(2\text{A}_{2}(k,t)+\text{A}_{3}(k,t))$ (DT) $\displaystyle\leq
SAc_{0}+2\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\sqrt{\frac{2\mathbf{V}(p(X_{t}),h^{*})\log\left(\tfrac{SAT}{\delta}\right)}{N_{t}(X_{t})}}+\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}(2\text{A}_{2}(k,t)+\text{A}_{3}(k,t))$ $\displaystyle\leq
SAc_{0}+4\sqrt{2SA\sum\nolimits_{t=0}^{T-1}\mathbf{V}(p(X_{t}),h^{*})\log\left(\tfrac{SAT}{\delta}\right)}+\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}(2\text{A}_{2}(k,t)+\text{A}_{3}(k,t))$

where the last inequality is obtained with computations that are similar to
those detailed in the proof of Lemma 8. We recognize the variance that we will
leave as is. We finish the proof by bounding the lower order terms
$\text{A}_{2}$ and $\text{A}_{3}$.

(STEP 3) We start with $\text{A}_{2}$. We have:

$\displaystyle\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\text{A}_{2}(k,t)$
$\displaystyle:=\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\tfrac{\sqrt{16c_{0}\log\left(\tfrac{SAT}{\delta}\right)S\left((1+c_{0})\left(3+2\sqrt{8T\log\left(\tfrac{2}{\delta}\right)}+2B(T)\right)\right)}}{N_{t_{k}}(X_{t})}$
(DT) $\displaystyle\leq
2\sqrt{16c_{0}S\log\left(\tfrac{SAT}{\delta}\right)\left((1+c_{0})\left(3+2\sqrt{8T\log\left(\tfrac{2}{\delta}\right)}+2B(T)\right)\right)}~{}SA\log(T)$
$\displaystyle\leq
8(1+c_{0})S^{\frac{3}{2}}A\log^{\frac{3}{2}}\left(\tfrac{SAT}{\delta}\right)\left(2+4T^{\frac{1}{4}}\log^{\frac{1}{4}}\left(\tfrac{SAT}{\delta}\right)+\sqrt{2B(T)}\right).$

(STEP 4) We are left with $\text{A}_{3}$. We have:

$\displaystyle\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\text{A}_{3}(k,t)$
$\displaystyle:=\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\left({\mathrm{sp}\left(h^{*}\right)}\sqrt{\frac{2\log\left(\tfrac{SAT}{\delta}\right)\sqrt{S\log\tfrac{SAT}{\delta}}}{N_{t_{k}}(X_{t})\sqrt{N_{t_{k}}(X_{t})}}}+\frac{3c_{0}\log\left(\tfrac{SAT}{\delta}\right)}{N_{t_{k}}(X_{t})}\right)$
(DT)
$\displaystyle\leq\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\left({\mathrm{sp}\left(h^{*}\right)}\sqrt{\frac{2\log\left(\tfrac{SAT}{\delta}\right)\sqrt{S\log\tfrac{SAT}{\delta}}}{N_{t_{k}}(X_{t})\sqrt{N_{t_{k}}(X_{t})}}}+\frac{3c_{0}\log\left(\tfrac{SAT}{\delta}\right)}{N_{t_{k}}(X_{t})}\right)$
$\displaystyle\leq
C{\mathrm{sp}\left(h^{*}\right)}S^{\frac{5}{4}}AT^{\frac{1}{4}}\log^{\frac{3}{4}}\left(\tfrac{SAT}{\delta}\right)+6c_{0}SA\log\left(\tfrac{SAT}{\delta}\right)$
$\displaystyle=\operatorname*{{\rm
O}}\left({\mathrm{sp}\left(h^{*}\right)}S^{\frac{5}{4}}AT^{\frac{1}{4}}\log\left(\tfrac{SAT}{\delta}\right)\right).$

This concludes the proof. ∎

####  C.8 Proof of Lemma 10, second order error

Recall that by Lemma 13, with probability $1-4\delta$,
$h^{*}\in\mathcal{H}_{t_{k}}$ for all $k$, hence
${\mathrm{sp}\left(\mathfrak{h}_{k}-h^{*}\right)}\leq 2c_{0}$ for all $k$ on
the same event. Therefore, with probability $1-4\delta$,

$\displaystyle\sum_{k}(4k)$
$\displaystyle:=2c_{0}SA+\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\left(\hat{p}_{k}(S_{t})-p_{k}(S_{t})\right)\left(\mathfrak{h}_{k}-h^{*}\right)$
$\displaystyle=2c_{0}SA+\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\sum_{s^{\prime}\in\mathcal{S}}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}(\hat{p}_{k}(s^{\prime}|S_{t})-p_{k}(s^{\prime}|S_{t}))(\mathfrak{h}_{k}-h^{*}(s^{\prime}))$
$\displaystyle\overset{(*)}{\leq}2c_{0}SA+2\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\sum_{s^{\prime}\in\mathcal{S}}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}(\hat{p}_{k}(s^{\prime}|S_{t})-p_{k}(s^{\prime}|S_{t}))d_{t_{k}}(s^{\prime},S_{t})$
$\displaystyle\overset{(\dagger)}{\leq}2c_{0}SA+2\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\sum_{s^{\prime}\in\mathcal{S}}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\left(d_{k}(s^{\prime},S_{t})\sqrt{\frac{2\hat{p}_{k}(s^{\prime}|S_{t})\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t_{k}}(X_{t})}}+3d_{k}(s^{\prime}|S_{t})\frac{\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t_{k}}(X_{t})}\right)$
$\displaystyle\leq
2c_{0}SA+2\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\sum_{s^{\prime}\in\mathcal{S}}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\left(\sqrt{c_{0}}\sqrt{\frac{2\hat{p}_{k}(s^{\prime}|S_{t})d_{k}(s^{\prime},S_{t})\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t_{k}}(X_{t})}}+\frac{3c_{0}\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t_{k}}(X_{t})}\right)$
$\displaystyle\leq
2c_{0}SA+4\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\sum_{s^{\prime}\in\mathcal{S}}\mathbf{1}_{N_{t_{k}}(X_{t})\geq
1}\left(\sqrt{c_{0}}\sqrt{\frac{2\hat{p}_{k}(s^{\prime}|S_{t})d_{k}(s^{\prime},S_{t})\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t}(X_{t})}}+\frac{3c_{0}\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t}(X_{t})}\right)$

where $(*)$ uses that $h^{*}\in\mathcal{H}_{t_{k}}$, and $(\dagger)$ is
obtained by applying the empirical Bernstein’s inequality, see Lemma 36, to
$\hat{p}_{k}(s^{\prime}|S_{t})-p_{k}(s^{\prime}|S_{t})$, and holds with
probability $1-\delta$. The rightmost term’s sum is upper-bounded by:

$4\sum_{k}\sum_{t=t_{k}}^{t_{k+1}-1}\sum_{s^{\prime}\in\mathcal{S}}\frac{3c_{0}\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t}(X_{t})}\leq
12S^{2}A\log(T)\log\left(\tfrac{S^{2}AT}{\delta}\right).$

For the other term, follow the line of the proof of Lemma 9 (term
$\text{A}_{2}$). We have with probability $1-5\delta$ ($4\delta$ of which is
by invoking Lemma 13):

$\displaystyle\hat{p}_{k}(s^{\prime}|S_{t})d_{k}(s^{\prime},S_{t})$
$\displaystyle=\frac{N_{t_{k}}(S_{t},A_{t},s^{\prime})\left((1+c_{0})\left(1+\sqrt{8t_{k}\log\left(\tfrac{2}{\delta}\right)}\right)+2B_{0}(t_{k})\right)}{N_{t_{k}}(S_{t}\leftrightarrow
s^{\prime})N_{t_{k}}(X_{t})}$
$\displaystyle\leq\frac{\left((1+c_{0})\left(3+2\sqrt{8T\log\left(\tfrac{2}{\delta}\right)}+2B(T)\right)\right)}{N_{t_{k}}(X_{t})}.$

Therefore,

$\sqrt{c_{0}}\sqrt{\frac{2\hat{p_{k}}(s^{\prime}|S_{t})d_{t_{k}}(s^{\prime},S_{t})\log\left(\tfrac{S^{2}AT}{\delta}\right)}{N_{t}(X_{t})}}\leq\frac{4(1+c_{0})\sqrt{\left(3+2\sqrt{8T\log\left(\tfrac{2}{\delta}\right)}+2B(T)\right)\log\left(\tfrac{S^{2}AT}{\delta}\right)}}{N_{t}(X_{t})}.$

Summing over $k$, $t$, $s^{\prime}$, with probability $1-6\delta$, we have:

$\sum_{k}(4k)\leq\begin{Bmatrix}16S^{2}A(1+c_{0})\log^{\frac{1}{2}}\left(\tfrac{S^{2}AT}{\delta}\right)\left(\sqrt{2B(T)}+2\left(8T\log\left(\tfrac{2}{\delta}\right)\right)^{\frac{1}{4}}\right)\\\
+32S^{2}A\left(\log(T)\log\left(\tfrac{S^{2}AT}{\delta}\right)+(1+c_{0})\log^{\frac{1}{2}}\left(\tfrac{S^{2}AT}{\delta}\right)\right)\end{Bmatrix}$

This concludes the proof. ∎

###  Appendix D Details on experiments

####  D.1 River swim

Experiments are run on $n$-states river-swim. Such MDPs are, despite their
size, known to be hard to learn. They consists in $n$ states aligned in a
straight line with two playable actions right and left whose dynamics are
given in the figure below. Rewards are Bernoulli and null everywhere excepted
for $r(s_{n},\textsc{right})=0.95$ and
$r(s_{0},\text{\sc\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}left})=0.05$.

$s_{1}$$s_{2}$$\cdots$$\cdots$$s_{n}$$0.6$$0.4$$0.6$$0.35$$0.05$$0.05$$0.35$$0.95$$0.05$$1$$1$$1$$1$
Figure 3:  The kernel of a $n$-state river-swim.

######  $3$-state river-swim.

The gain is $g^{*}\approx 0.82$ and $h^{*}\approx(-4.28,-2.24,0.4)$.

######  $5$-state river-swim.

The gain is $g^{*}\approx 0.82$ and
$h^{*}\approx(-9.62,-7.58,-4.96,-2.27,0.45)$.

###  Appendix E Standard concentration inequalities

######  Lemma 32 (Azuma’s inequality, Azuma [1967]).

Let $(U_{t})_{t\geq 0}$ a martingale difference sequence such that
${\mathrm{sp}\left(U_{t}\right)}\leq c$ a.s., i.e., there exists
$a_{t}\in\mathbf{R}$ such that $a_{t}\leq U_{t}\leq a_{t}+c$ a.s. Then, for
all $\delta>0$,

$\mathbf{P}\left(\sum\nolimits_{t=0}^{T-1}U_{t}\geq
c\sqrt{\tfrac{1}{2}T\log\left(\tfrac{1}{\delta}\right)}\right)\leq\delta.$

######  Lemma 33 (Freedman’s inequality, Zhang et al. [2020]).

Let $(U_{t})_{t\geq 0}$ a martingale difference sequence such that
$\left|U_{t}\right|\leq c$ a.s., and denote its conditional variance
$V_{t}:=\mathbf{E}[U_{t}^{2}|\mathcal{F}_{t-1}]$. Then, for all $\delta>0$,

$\mathbf{P}\left(\exists T^{\prime}\leq
T:\sum\nolimits_{t=0}^{T^{\prime}-1}U_{t}\geq\sqrt{2\sum\nolimits_{t=0}^{T^{\prime}-1}V_{t}\log\left(\tfrac{T}{\delta}\right)}+4c\log\left(\tfrac{T}{\delta}\right)\right)\leq\delta.$

######  Lemma 34 (Time-uniform Azuma, Bourel et al. [2020]).

Let $(U_{t})$ a martingale difference sequence such that, for all
$\lambda\in\mathbf{R}$, $\mathbf{E}[\exp(\lambda
U_{t})|U_{1},\ldots,U_{t-1}]\leq\exp(\frac{\lambda^{2}\sigma^{2}}{2})$. Then:

$\forall\delta>0,\quad\mathbf{P}\left(\exists n\geq
1,\quad\left(\sum\nolimits_{k=1}^{n}U_{k}\right)^{2}\geq
n\sigma^{2}\left(1+\tfrac{1}{n}\right)\log\left(\tfrac{\\!\\!\sqrt{1+n}}{\delta}\right)\right)\leq\delta.$

######  Lemma 35 (Time-uniform Weissman).

Let $q$ a distribution over $\left\\{1,\ldots,d\right\\}$. Let $(U_{t})$ a
sequence of i.i.d. random variables of distribution $q$. Then:

$\forall\delta>0,\quad\mathbf{P}\left(\exists n\geq
1,\left\|\sum\nolimits_{i=1}^{n}\left(e_{U_{i}}-q\right)\right\|_{1}^{2}\geq
nd\log\left(\tfrac{2\sqrt{1+n}}{\delta}\right)\right)\leq\delta.$

###### Proof.

Remark that
$\left\|\sum_{k=1}^{n}(e_{U_{k}}-q)\right\|_{1}=\max_{v\in\left\\{-1,1\right\\}^{d}}\sum_{k=1}^{n}\left\langle
e_{U_{k}}-q,v\right\rangle$. Let $W_{k}^{v}:=\left\langle
e_{U_{k}}-q,v\right\rangle$. Remark that for each
$v\in\left\\{-1,1\right\\}^{d}$, $(W_{k}^{v})$ is a family of i.i.d. random
variables with $-\left\langle q,v\right\rangle\leq W_{k}^{v}\leq
1-\left\langle q,v\right\rangle$, so $\mathbf{E}[\exp(\lambda
W_{k}^{v})]\leq\exp(\tfrac{\lambda^{2}}{8})$ by Hoeffding’s Lemma. By Lemma
34, we have:

$\displaystyle\mathbf{P}\left(\exists n\geq
1,\left\|\sum_{k=1}^{n}(e_{U_{k}}-q)\right\|_{1}\geq\\!\\!\sqrt{nd\log\left(\tfrac{2\\!\\!\sqrt{1+n}}{\delta}\right)}\right)$
$\displaystyle=\mathbf{P}\left(\exists v\in\left\\{-1,1\right\\}^{d},\exists
n,\sum_{k=1}^{n}W_{k}^{v}\geq\\!\\!\sqrt{nd\log\left(\tfrac{2\\!\\!\sqrt{1+n}}{\delta}\right)}\right)$
$\displaystyle\leq\sum_{v\in\left\\{-1,1\right\\}^{d}}\mathbf{P}\left(\exists
n,\sum_{k=1}^{n}W_{k}^{v}\geq\\!\\!\sqrt{nd\log\left(\tfrac{2\\!\\!\sqrt{1+n}}{\delta}\right)}\right)$
$\displaystyle\leq\sum_{v\in\left\\{-1,1\right\\}^{d}}\mathbf{P}\left(\exists
n,\sum_{k=1}^{n}W_{k}^{v}\geq\\!\\!\sqrt{\tfrac{1}{2}n\left(1+\tfrac{1}{n}\right)\log\left(\tfrac{\\!\\!\sqrt{1+n}}{2^{-d}\delta}\right)}\right)$
$\displaystyle\leq 2^{d}\cdot 2^{d}\delta=\delta.$

This concludes the proof. ∎

######  Lemma 36 (Time-uniform Empirical Bernstein).

Let $(U_{k})_{k\geq 1}$ a martingale difference sequence such that
${\mathrm{sp}\left(U_{n}\right)}\leq c$ a.s., let
$\hat{U}_{n}:=\frac{1}{n}\sum_{k=1}^{n}U_{k}$ the empirical mean and
$\hat{V}_{n}:=\frac{1}{n}\sum_{k=1}^{n}(U_{k}-\hat{U}_{n})^{2}$ the population
variance. Then,

$\forall\delta>0,\forall T>0,\quad\mathbf{P}\left(\exists t\leq
T,\sum\nolimits_{i=1}^{t}U_{i}\geq\sqrt{2t\hat{V}_{t}\log\left(\tfrac{3T}{\delta}\right)}+3c\log\left(\tfrac{3T}{\delta}\right)\right)\leq\delta.$

###### Proof.

This is obtained with a union bound on the values of $n\leq T$, then applying
Lemma 38. ∎

######  Lemma 37 (Time-uniform Empirical Likelihoods, Jonsson et al. [2020]).

Let $q$ a distribution on $\left\\{1,\ldots,d\right\\}$. Let $(U_{t})$ a
sequence of i.i.d. random variables of distribution $q$. Then:

$\forall\delta>0,\quad\mathbf{P}\left(\exists n\geq 1,n\operatorname{{\rm
KL}}(\hat{q}_{n}||q)>\log\left(\tfrac{1}{\delta}\right)+(d-1)\log\left(e\left(1+\tfrac{n}{d-1}\right)\right)\right)\leq\delta.$

######  Lemma 38 (Empirical Bernstein inequality, Audibert et al. [2009]).

Let $(U_{k})_{k\geq 1}$ a martingale difference sequence such that
${\mathrm{sp}\left(U_{n}\right)}\leq c$ a.s., let
$\hat{U}_{n}:=\frac{1}{n}\sum_{k=1}^{n}U_{k}$ the empirical mean and
$\hat{V}_{n}:=\frac{1}{n}\sum_{k=1}^{n}(U_{k}-\hat{U}_{n})^{2}$ the population
variance. Then,

$\forall\delta>0,\forall n\geq
1,\quad\mathbf{P}\left(\sum\nolimits_{k=1}^{n}U_{k}\geq\sqrt{2n\hat{V}_{n}\log\left(\tfrac{3}{\delta}\right)}+3c\log\left(\tfrac{3}{\delta}\right)\right)\leq\delta.$

######  Lemma 39 (Bennett’s inequality, Audibert et al. [2009]).

Let $(U_{t})_{t\geq 0}$ a martingale difference sequence such that
$\left|U_{t}\right|\leq c$ a.s., and denote its conditional variance
$V_{t}:=\mathbf{E}[U_{t}^{2}|\mathcal{F}_{t-1}]$. Then,

$\forall\delta>0,\forall n\geq 1,\quad\mathbf{P}\left(\exists k\leq
n,\sum\nolimits_{i=1}^{k}U_{i}\geq\sqrt{2\sum\nolimits_{i=1}^{n}V_{i}\log\left(\tfrac{1}{\delta}\right)}+\tfrac{1}{3}c\log\left(\tfrac{1}{\delta}\right)\right)\leq\delta.$

######  Lemma 40 (Lemma 3 of Zhang and Xie [2023]).

Let $(U_{t})$ be a sequence of random variables such that $0\leq U_{t}\leq c$
a.s., and let $\mathcal{F}_{t}:=\sigma(U_{0},U_{1},\ldots,U_{t-1})$. Then:

$\displaystyle\forall\delta>0,\quad\mathbf{P}\left(\exists T\geq
0,\sum\nolimits_{t=0}^{T-1}U_{t}\geq
3\sum\nolimits_{t=0}^{T-1}\mathbf{E}[U_{t}|\mathcal{F}_{t-1}]+c\log\left(\tfrac{1}{\delta}\right)\right)\leq\delta;$
$\displaystyle\forall\delta>0,\quad\mathbf{P}\left(\exists T\geq
0,\sum\nolimits_{t=0}^{T-1}\mathbf{E}[U_{t}|\mathcal{F}_{t-1}]\geq
3\sum\nolimits_{t=0}^{T-1}U_{t}+c\log\left(\tfrac{1}{\delta}\right)\right)\leq\delta.$