EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-072 LHCb-PAPER-2016-001 March 22, 2016

Search for ${B}_{c}^{+}$ decays to the ${p}{\overline{p}}{{\pi}^{+}}$ final
state

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

A search for the decays of the ${B}_{c}^{+}$ meson to $p\bar{p}\pi^{+}$ is
performed for the first time using a data sample corresponding to an
integrated luminosity of 3.0$\mbox{\,fb}^{-1}$ collected by the LHCb
experiment in $pp$ collisions at centre-of-mass energies of $7$ and
$8$$\mathrm{\,Te\kern-1.00006ptV}$. No signal is found and an upper limit, at
95% confidence level, is set,
$\frac{f_{c}}{f_{u}}\times\mathcal{B}({{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+})<3.6\times 10^{-8}$ in the kinematic region
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
$\mbox{$p_{\rm T}$}(B)<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y(B)<4.5$, where $\mathcal{B}$ is the branching fraction and $f_{c}$
($f_{u}$) is the fragmentation fraction of the $b$ quark into a ${B}_{c}^{+}$
(${{B}^{+}}$) meson.

Published in Phys. Lett. B 759, Pages 313-321 (2016)

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

##  1 Introduction

The decays of the ${B}_{c}^{+}$ meson have the special feature of proceeding
through either of its valence quarks $\overline{b}$ or $c$, or via the
annihilation of the two.111Charge-conjugation is implied throughout the paper.
In the Standard Model, the decays with a $b$-quark transition and no charm
particle in the final state can proceed only via $\overline{b}c\rightarrow
W^{+}\rightarrow u\overline{q}$ ($q=d,s$) annihilation, with an amplitude
proportional to the product of CKM matrix elements $V_{cb}V_{uq}^{*}$. Cabibbo
suppression $|V_{us}/V_{ud}|\sim 0.2$ implies that final states without
strangeness dominate. Calculations involving two-body and quasi two-body modes
predict branching fractions in the range $10^{-8}-10^{-6}$[1, 2, 3]. Due to
their rareness, the observation of these processes is an experimental
challenge. On the other hand, any observation could probe other types of
$\overline{b}c$ annihilations involving particles beyond the Standard Model,
such as a mediating charged Higgs boson (see e.g. Refs. [4, 5]).

The decays of ${B}_{c}^{+}$ mesons to three light charged hadrons provide a
good way to study such processes. These include fully mesonic
$h^{\prime+}h^{\prime-}h^{+}$ states or states containing a proton-antiproton
pair and a light hadron, ${p}{\overline{p}}h^{+}$
($h,~{}h^{\prime}=\pi,~{}K$). In this study, the primary focus is on
${{B}_{c}^{+}}\rightarrow{p}{\overline{p}}\pi^{+}$ decays in the region below
the charmonium threshold, taken to be
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, where the
only contribution arises from the annihilation process. The $b\rightarrow c$
transitions, leading to
${{B}_{c}^{+}}\rightarrow[c\overline{c}](\rightarrow{p}{\overline{p}})h^{+}$
charmonium modes, are also considered. An analysis is performed to examine
these different contributions in the ${p}{\overline{p}}\pi^{+}$ phase space.
The ${{{B}^{+}}}\rightarrow{p}{\overline{p}}\pi^{+}$ decays in the region
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are used
as a normalization mode to derive the quantity

$R_{p}\equiv\frac{f_{c}}{f_{u}}\times\mathcal{B}({{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+}),$ (1)

where $\mathcal{B}$ is the branching fraction and $f_{c}~{}(f_{u})$ represents
the fragmentation fraction of the $b$ quark into the
${{B}_{c}^{+}}~{}({{{B}^{+}}})$ meson. The quantity $R_{p}$ is measured in the
fiducial region $\mbox{$p_{\rm T}$}(B)<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $2.0<y(B)<4.5$, where $y$ denotes the rapidity and $p_{\rm T}$ is the
component of the momentum transverse to the beam. The full Run 1 (years 2011
and 2012) data sample is exploited, representing 1.0 and 2.0
$\mbox{\,fb}^{-1}$ of integrated luminosity at $7$ and $8$
$\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass energies in $pp$ collisions,
respectively.

##  2 Detector and simulation

The LHCb detector [6, 7] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum
distance of a track to a primary vertex (PV), the impact parameter (IP), is
measured with a resolution of $(15+29/\mbox{$p_{\rm T}$}){\,\upmu\rm m}$,
where $p_{\rm T}$ is in ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different
types of charged hadrons are distinguished using information from two ring-
imaging Cherenkov detectors. Photons, electrons and hadrons are identified by
a calorimeter system consisting of scintillating-pad and preshower detectors,
an electromagnetic calorimeter and a hadronic calorimeter. Muons are
identified by a system composed of alternating layers of iron and multiwire
proportional chambers.

The online event selection is performed by a trigger [8], which 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. At
the hardware trigger stage, events are required to have a muon with high
$p_{\rm T}$ or a hadron, photon or electron with high transverse energy in the
calorimeters. For hadrons, the transverse energy threshold is
3.5$\mathrm{\,Ge\kern-1.00006ptV}$. The software trigger requires a two-,
three- or four-track secondary vertex with a significant displacement from the
primary $pp$ interaction vertices. At least one charged particle must have a
transverse momentum $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be inconsistent with
originating from a PV. A multivariate algorithm [9] is used for the
identification of secondary vertices consistent with the decay of a $b$
hadron.

The analysis uses simulated events generated by Pythia 8.1 [10] and Bcvegpy
[11] for the production of ${{B}^{+}}$ and ${B}_{c}^{+}$ mesons, respectively,
with a specific LHCb configuration [12]. Decays of hadronic particles are
described by EvtGen [13], in which final-state radiation is generated using
Photos [14]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [15,
*Agostinelli:2002hh] as described in Ref. [17].

##  3 Reconstruction and selection of candidates

Three charged particles are combined to form
$B_{(c)}^{+}\rightarrow{p}{\overline{p}}{{\pi}^{+}}$ decay candidates, which
are associated to the closest PV. A loose preselection is performed on
tracking quality, $p$, $p_{\rm T}$ and IP of the ${B}_{c}^{+}$ and its
daughters, and ${B}_{c}^{+}$ candidate flight distance. At this stage, two
windows of the invariant mass of the $p\bar{p}\pi^{+}$ system are retained:
the ${B}^{+}$ region, $[5.1,5.5]{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and
the ${B}_{c}^{+}$ region, $[6.0,6.5]{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Since the production fractions of different $B$ species are involved, a
fiducial requirement is imposed to define the kinematic region for the
measurement, $\mbox{$p_{\rm T}$}(B)<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$
and $2.0<y(B)<4.5$ [18].

Further discrimination between signal and background is provided by a
multivariate analysis using a boosted decision tree (BDT) classifier[19].
Input quantities include kinematic and topological variables related to the
${{B}_{c}^{+}}$ candidates and the individual daughter particles. The
momentum, vertex and flight distance of the ${{B}_{c}^{+}}$ candidate are
exploited, as are track fit quality criteria, IP and momentum information of
the final-state particles. The BDT is trained using simulated signal events,
and data events from the sidebands of the $p\bar{p}\pi^{+}$ invariant mass
$[6.0,6.15]{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$[6.35,6.5]{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, which represent the
background. To check for training biases, the signal and background samples
are split into two subsamples for training and testing of the BDT output.
Figure 1 shows the distribution of the BDT output for signal and background.

Figure 1: Distributions of BDT output for the ${{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+}$ signal and the background. The vertical dashed lines
indicate the lower limits of the three regions in which the signal is
determined.

Particle identification (PID) requirements are applied to reduce the
combinatorial background and suppress the cross-feed of
$p{\overline{p}}{{K}^{+}}$ final states in the $p{\overline{p}}{{\pi}^{+}}$
spectrum, due to the kaon being misidentified as a pion. The BDT and PID
requirements are optimized jointly in order to maximize the sensitivity to
very small event yields. The ${B}_{c}^{+}$ signal yield is determined from a
simultaneous fit in three bins of the BDT output $X$, $0.04<X<0.12$,
$0.12<X<0.18$ and $X>0.18$, each having the same expected yield (dashed lines
in Fig. 1). From simulated pseudoexperiments, this method is shown to be more
sensitive than a single fit to the highest signal purity region, $X>0.18$. The
normalization channel ${{{B}^{+}}}\rightarrow p\overline{p}\pi^{+}$ undergoes
the same PID and BDT selection, but its yield is determined without binning in
BDT output.

##  4 Fits to the data

Signal and background yields are obtained using unbinned extended maximum
likelihood fits to the distribution of the invariant mass of the
$p{\overline{p}}\pi^{+}$ combinations. The ${{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+}$ and ${{{B}^{+}}}\rightarrow p\overline{p}\pi^{+}$
signals are both modelled by the sum of two Crystal Ball functions [20] with a
common mean. For ${{B}_{c}^{+}}\rightarrow p\overline{p}\pi^{+}$, all the
shape parameters are fixed to the values obtained in the simulation while for
${{{B}^{+}}}\rightarrow p\overline{p}\pi^{+}$, the mean and the core width are
allowed to float. A Fermi function accounts for a possible partially
reconstructed component from ${{B}_{c}^{+}}\rightarrow
p{\overline{p}}\rho^{+}$ (${{{B}^{+}}}\rightarrow p{\overline{p}}\rho^{+}$)
decays, where a neutral pion from the $\rho^{+}$ is not reconstructed
resulting in a $p{\overline{p}}{{\pi}^{+}}$ invariant mass below the nominal
${{B}_{c}^{+}}$ (${{{B}^{+}}}$) mass. An asymmetric Gaussian function with
power law tails is used to model a possible $p{\overline{p}}{{K}^{+}}$ cross-
feed, and its contribution is found to be negligible. The combinatorial
background is modelled by an exponential function. Except for this last
category, all the parameters of the background components are fixed to the
values obtained in simulations.

Figure 2 shows the result of the fits in the ${{B}^{+}}$ region. For the
region of interest,
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the yield
is $N({{{B}^{+}}}\rightarrow p\overline{p}\pi^{+})=1644\pm 83$, where only the
statistical uncertainty is quoted. The fit to the region
$2.85<m({p}{\overline{p}})<3.15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
which includes the
${{{B}^{+}}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}({p}{\overline{p}}){{\pi}^{+}}$ signal, shows the yield suppression in
this region as observed in Ref. [21].

Figure 2: Fits to the ${p}{\overline{p}}{{\pi}^{+}}$ invariant mass in the
${B}^{+}$ region, for (left)
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
(right)
$2.85<m({p}{\overline{p}})<3.15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The
blue dashed, red long-dashed and green dotted-dashed lines represent the
signal, combinatorial background and partially reconstructed background
components, respectively. The error bars show 68% Poisson confidence level
intervals.

The simultaneous fits performed in the ${B}_{c}^{+}$ region are made for the
region exclusive to the annihilation process,
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and for
the charmonium region,
$2.85<m({p}{\overline{p}})<3.15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The
fraction of the yield of the partially reconstructed background in each bin of
the BDT output is constrained to be the same as in the simulation. The results
are shown in Fig. 3. The corresponding signal yields are
$N({{B}_{c}^{+}}\rightarrow p\overline{p}\pi^{+})=-2.7\pm 6.3$ for
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$N({{B}_{c}^{+}}\rightarrow p\overline{p}\pi^{+})=-0.1\pm 3.0$ for
$2.85<m({p}{\overline{p}})<3.15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The main observable under consideration is determined as

$R_{p}\equiv\frac{f_{c}}{f_{u}}\times\mathcal{B}({{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+})=\frac{N({{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+})}{N({{{B}^{+}}}\rightarrow
p\overline{p}\pi^{+})}\times\frac{\epsilon_{u}}{\epsilon_{c}}\times\mathcal{B}({{{B}^{+}}}\rightarrow
p\overline{p}\pi^{+}),$ (2)

and a cross-check is made for the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ mode

$R_{p}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\equiv\frac{f_{c}}{f_{u}}\times\mathcal{B}({{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})=\frac{N({{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}(\rightarrow{p}{\overline{p}}){{\pi}^{+}})}{N({{{B}^{+}}}\rightarrow
p\overline{p}\pi^{+})}\times\frac{\epsilon_{u}}{\epsilon_{c}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\times\frac{\mathcal{B}({{{B}^{+}}}\rightarrow
p\overline{p}\pi^{+})}{\mathcal{B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{p}{\overline{p}})},$ (3)

where the efficiencies $\epsilon$ are discussed in Sec. 5.

Figure 3: Projection of fits to the ${p}{\overline{p}}{{\pi}^{+}}$ invariant
mass in the ${B}_{c}^{+}$ region, in the bins of BDT output (top)
$0.04<X<0.12$, (middle) $0.12<X<0.18$ and (bottom) $X>0.18$, for (left)
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
(right)
$2.85<m({p}{\overline{p}})<3.15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The
red long-dashed lines represent the combinatorial background. The signal and
partially reconstructed components are too small to be shown.

##  5 Efficiencies

The reconstruction and selection efficiencies are computed from acceptance
maps defined in the $m^{2}({p}{\overline{p}})$ vs. $m^{2}({p}\pi)$ plane.
These maps include the effects of event reconstruction, triggers,
preselection, BDT and PID selections, and are obtained from simulation for
both ${{B}_{c}^{+}}\rightarrow p\overline{p}\pi^{+}$ and
${{{B}^{+}}}\rightarrow p\overline{p}\pi^{+}$. The PID map is obtained by
studying data-driven responses from calibration data samples of kinematically
identified pions, kaons and protons originating from the decays
$D^{*+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$, ${\mathchar
28931\relax}\rightarrow p\pi^{-}$ and ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$. The maps are smoothed
using fits involving two-dimensional fourth-order polynomials. Figure 4 shows
the final combination of these maps.

Figure 4: Combined acceptance in the plane
$(m^{2}({p}{\overline{p}}),m^{2}({p}\pi))$ for (left)
${{B}_{c}^{+}}\rightarrow p\overline{p}\pi^{+}$ and (right)
${{{B}^{+}}}\rightarrow p\overline{p}\pi^{+}$ events. The vertical dashed line
corresponds to
$m({p}{\overline{p}})=2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

To infer the average efficiency for ${{{B}^{+}}}\rightarrow
p\overline{p}\pi^{+}$, signal weights are calculated with the sPlot technique
[22] from the fits shown in Fig. 2. A weight is associated with each candidate
depending on its position in the $m^{2}({p}{\overline{p}})$ vs.
$m^{2}({p}\pi)$ plane. The acceptance maps are then used to determine an
averaged efficiency,
$\epsilon_{u}^{\mathrm{sel}}\equiv\langle\epsilon^{\mathrm{sel}}({{{B}^{+}}}\rightarrow
p\overline{p}\pi^{+})\rangle$. For ${{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+}$, since no signal is available in data, a simple average
is performed in the region
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to obtain
$\epsilon_{c}^{\mathrm{sel}}$, which leads to a substantial systematic
uncertainty due to the variation of the efficiency over this region.

In computing the ratio
$\epsilon_{u}^{\mathrm{sel}}/\epsilon_{c}^{\mathrm{sel}}$, three corrections
are needed to account for data-simulation discrepancies: tracking efficiency,
hardware hadron trigger efficiency; and the fiducial region cuts
$\mbox{$p_{\rm T}$}(B)<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y(B)<4.5$. After these corrections,
$\epsilon_{u}^{\mathrm{sel}}/\epsilon_{c}^{\mathrm{sel}}=2.495\pm 0.028$ is
obtained including associated systematic uncertainties.

Another efficiency ratio accounts for the fact that ${{{B}^{+}}}\rightarrow
p\overline{p}\pi^{+}$ and ${{B}_{c}^{+}}\rightarrow p\overline{p}\pi^{+}$
decays are only detected if all the decay daughters are in the LHCb
acceptance: the fractions of events satisfying this requirement are estimated
by simulation and are found to be $\epsilon_{u}^{\mathrm{acc}}=(18.91\pm
0.10)\%$ and $\epsilon_{c}^{\mathrm{acc}}=(15.82\pm 0.03)\%$, which gives
$\epsilon_{u}^{\mathrm{acc}}/\epsilon_{c}^{\mathrm{acc}}=1.195\pm 0.007$.

For ${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}({p}{\overline{p}}){{\pi}^{+}}$, a similar procedure is applied and the
following values are found:
$\epsilon_{u}^{\mathrm{sel}}/\epsilon_{c}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\mathrm{sel}}=2.513\pm 0.032$ and
$\epsilon_{u}^{\mathrm{acc}}/\epsilon_{c}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}},\mathrm{acc}}=1.186\pm 0.007$. The efficiency ratio used for the final
results is
$\epsilon_{u}/\epsilon_{c}=\epsilon_{u}^{\mathrm{sel}}/\epsilon_{c}^{\mathrm{sel}}\times\epsilon_{u}^{\mathrm{acc}}/\epsilon_{c}^{\mathrm{acc}}$.
The differences between the ${{B}^{+}}$ and ${B}_{c}^{+}$ detector acceptance
and selection efficiencies are caused by the different lifetimes and masses of
the two mesons.

##  6 Systematic uncertainties

Part of the systematic uncertainties are related to the computation of the
efficiency ratios, such as the PID calibration, the uncertainty in the
${B}_{c}^{+}$ lifetime, $0.507\pm 0.009{\rm\,ps}$ [23], the limited sizes of
the simulation samples, the effect of the detector acceptance, the
distribution of the BDT output, and the trigger and fiducial cut corrections.
Others are related to the branching fractions
$\mathcal{B}({{B}^{\pm}}\rightarrow p\bar{p}\pi^{\pm})=(1.07\pm 0.16)\times
10^{-6}$ [21] and $\mathcal{B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{p}{\overline{p}})=(2.120\pm 0.029)\times 10^{-3}$ [24], or
to the variation of the selection efficiency of ${{B}_{c}^{\pm}}\rightarrow
p\overline{p}\pi^{\pm}$ over the phase-space region
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, due to
the lack of knowledge of the kinematics in the absence of signal in data
(modelling).

Table 1 lists the different sources of systematic uncertainties. The PID
uncertainty is dominated by the finite size of the proton calibration samples,
which limits the sampling of the identification efficiency as a function of
the track momentum and rapidity. A similar comment applies for the hardware
trigger efficiency correction, where the effect is smaller due to a one-
dimensional sampling as a function of the transverse momentum $p_{\rm T}$. The
uncertainty related to the differences in the BDT output shape between data
and simulation has been estimated using
${{{B}^{+}}}\rightarrow{p}{\overline{p}}h^{+}$ ($h=K,\pi$) samples where the
signal yield has been studied as a function of the requirements on the BDT
output in both data and simulation. The uncertainty on the fit model,
including the knowledge of the signal shape and the contribution of the
partially reconstructed background, is found to have no impact on the final
result.

Table 1: Relative systematic uncertainties (in %) on the ratio $\epsilon_{u}/\epsilon_{c}$ and input branching fractions. Source |  ${{B}_{c}^{+}}\rightarrow p\overline{p}\pi^{+}$, $m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ | ${{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}(\rightarrow{p}{\overline{p}}){{\pi}^{+}}$
---|---|---
PID | 3.0 | 3.0
${B}_{c}^{+}$ lifetime | 2.0 | 2.0
Simulation | 0.8 | 0.9
Detector acceptance | 0.6 | 0.6
BDT shape | 1.5 | 1.5
Hardware trigger correction | 0.8 | 0.9
Fiducial cut | 0.1 | 0.1
Modelling | 15 | —
$\mathcal{B}({{{B}^{+}}}\rightarrow p\overline{p}\pi^{+})$ | 15 | 15
$\mathcal{B}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow{p}{\overline{p}})$ | — | 1.4

##  7 Results and summary

Upper limits on $R_{p}$ and $R_{p}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ are estimated by making scans of these quantities, comparing profile
likelihood ratios for the “signal+background” against “background”-only
hypotheses [25]. From these fits, $p$-value profiles are inferred, the signal
$p$-value being the ratio of the “signal+background” and “background”
$p$-values. The point at which the $p$-value falls below 5% determines the 95%
confidence level (CL) upper limit. In the determination of this value, the
systematic uncertainties, shown in Table 1, and the statistical uncertainty on
the normalization channel yield are taken into account.

The $p$-value scans are shown in Fig. 5, from which the following values are
found: $R_{p}<3.6\times 10^{-8}$
($m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$) and
$R_{p}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}<8.4\times 10^{-6}$ at
95% CL. The latter limit is compatible with a measurement of
$\frac{f_{c}}{f_{u}}\times\frac{\mathcal{B}({{B}_{c}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}})}{\mathcal{B}({{B}^{+}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{+}})}$ [18] from which the value
$R_{p}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}=(7.0\pm 0.3)\times
10^{-6}$ is inferred. At 90% CL, the limits are $R_{p}<2.8\times 10^{-8}$ and
$R_{p}^{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}<6.5\times 10^{-6}$.

Figure 5: $p$-value profile for (left) $R_{p}$ and (right)
$R_{p}^{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$. The horizontal red
solid and dashed lines indicate the 5% and 10% confidence levels.

In summary, a search for the $\overline{b}c$ annihilation process leading to
${B}_{c}^{+}$ meson decays into the ${p}{\overline{p}}{{\pi}^{+}}$ final state
has been performed for the fiducial region
$m({p}{\overline{p}})<2.85{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
$\mbox{$p_{\rm T}$}(B)<20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
$2.0<y(B)<4.5$. No signal is observed and a 95% confidence level upper limit
is inferred,

$R_{p}=\frac{f_{c}}{f_{u}}\times\mathcal{B}({{B}_{c}^{+}}\rightarrow
p\overline{p}\pi^{+})<3.6\times 10^{-8}$.

## 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 and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

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LHCb collaboration

R. Aaij39, C. Abellán Beteta41, B. Adeva38, M. Adinolfi47, Z. Ajaltouni5, S.
Akar6, J. Albrecht10, F. Alessio39, M. Alexander52, S. Ali42, G. Alkhazov31,
P. Alvarez Cartelle54, A.A. Alves Jr58, S. Amato2, S. Amerio23, Y. Amhis7, L.
An3,40, L. Anderlini18, G. Andreassi40, M. Andreotti17,g, J.E. Andrews59, R.B.
Appleby55, O. Aquines Gutierrez11, F. Archilli39, P. d’Argent12, A.
Artamonov36, M. Artuso60, E. Aslanides6, G. Auriemma26,n, M. Baalouch5, S.
Bachmann12, J.J. Back49, A. Badalov37, C. Baesso61, S. Baker54, W. Baldini17,
R.J. Barlow55, C. Barschel39, S. Barsuk7, W. Barter39, V. Batozskaya29, V.
Battista40, A. Bay40, L. Beaucourt4, J. Beddow52, F. Bedeschi24, I. Bediaga1,
L.J. Bel42, V. Bellee40, N. Belloli21,k, I. Belyaev32, E. Ben-Haim8, G.
Bencivenni19, S. Benson39, J. Benton47, A. Berezhnoy33, R. Bernet41, A.
Bertolin23, F. Betti15, M.-O. Bettler39, M. van Beuzekom42, S. Bifani46, P.
Billoir8, T. Bird55, A. Birnkraut10, A. Bizzeti18,i, T. Blake49, F. Blanc40,
J. Blouw11, S. Blusk60, V. Bocci26, A. Bondar35, N. Bondar31,39, W.
Bonivento16, A. Borgheresi21,k, S. Borghi55, M. Borisyak67, M. Borsato38, M.
Boubdir9, T.J.V. Bowcock53, E. Bowen41, C. Bozzi17,39, S. Braun12, M.
Britsch12, T. Britton60, J. Brodzicka55, E. Buchanan47, C. Burr55, A.
Bursche2, J. Buytaert39, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,k, M.
Calvo Gomez37,p, P. Campana19, D. Campora Perez39, L. Capriotti55, A.
Carbone15,e, G. Carboni25,l, R. Cardinale20,j, A. Cardini16, P. Carniti21,k,
L. Carson51, K. Carvalho Akiba2, G. Casse53, L. Cassina21,k, L. Castillo
Garcia40, M. Cattaneo39, Ch. Cauet10, G. Cavallero20, R. Cenci24,t, M.
Charles8, Ph. Charpentier39, G. Chatzikonstantinidis46, M. Chefdeville4, S.
Chen55, S.-F. Cheung56, M. Chrzaszcz41,27, X. Cid Vidal39, G. Ciezarek42,
P.E.L. Clarke51, M. Clemencic39, H.V. Cliff48, J. Closier39, V. Coco58, J.
Cogan6, E. Cogneras5, V. Cogoni16,f, L. Cojocariu30, G. Collazuol23,r, P.
Collins39, A. Comerma-Montells12, A. Contu39, A. Cook47, M. Coombes47, S.
Coquereau8, G. Corti39, M. Corvo17,g, B. Couturier39, G.A. Cowan51, D.C.
Craik51, A. Crocombe49, M. Cruz Torres61, S. Cunliffe54, R. Currie54, C.
D’Ambrosio39, E. Dall’Occo42, J. Dalseno47, P.N.Y. David42, A. Davis58, O. De
Aguiar Francisco2, K. De Bruyn6, S. De Capua55, M. De Cian12, J.M. De
Miranda1, L. De Paula2, P. De Simone19, C.-T. Dean52, D. Decamp4, M.
Deckenhoff10, L. Del Buono8, N. Déléage4, M. Demmer10, D. Derkach67, O.
Deschamps5, F. Dettori39, B. Dey22, A. Di Canto39, F. Di Ruscio25, H.
Dijkstra39, F. Dordei39, M. Dorigo40, A. Dosil Suárez38, A. Dovbnya44, K.
Dreimanis53, L. Dufour42, G. Dujany55, K. Dungs39, P. Durante39, R.
Dzhelyadin36, A. Dziurda27, A. Dzyuba31, S. Easo50,39, U. Egede54, V.
Egorychev32, S. Eidelman35, S. Eisenhardt51, U. Eitschberger10, R. Ekelhof10,
L. Eklund52, I. El Rifai5, Ch. Elsasser41, S. Ely60, S. Esen12, H.M. Evans48,
T. Evans56, A. Falabella15, C. Färber39, N. Farley46, S. Farry53, R. Fay53, D.
Fazzini21,k, D. Ferguson51, V. Fernandez Albor38, F. Ferrari15, F. Ferreira
Rodrigues1, M. Ferro-Luzzi39, S. Filippov34, M. Fiore17,g, M. Fiorini17,g, M.
Firlej28, C. Fitzpatrick40, T. Fiutowski28, F. Fleuret7,b, K. Fohl39, M.
Fontana16, F. Fontanelli20,j, D. C. Forshaw60, R. Forty39, M. Frank39, C.
Frei39, M. Frosini18, J. Fu22, E. Furfaro25,l, A. Gallas Torreira38, D.
Galli15,e, S. Gallorini23, S. Gambetta51, M. Gandelman2, P. Gandini56, Y.
Gao3, J. García Pardiñas38, J. Garra Tico48, L. Garrido37, P.J. Garsed48, D.
Gascon37, C. Gaspar39, L. Gavardi10, G. Gazzoni5, D. Gerick12, E. Gersabeck12,
M. Gersabeck55, T. Gershon49, Ph. Ghez4, S. Gianì40, V. Gibson48, O.G.
Girard40, L. Giubega30, V.V. Gligorov39, C. Göbel61, D. Golubkov32, A.
Golutvin54,39, A. Gomes1,a, C. Gotti21,k, M. Grabalosa Gándara5, R. Graciani
Diaz37, L.A. Granado Cardoso39, E. Graugés37, E. Graverini41, G. Graziani18,
A. Grecu30, P. Griffith46, L. Grillo12, O. Grünberg65, B. Gui60, E.
Gushchin34, Yu. Guz36,39, T. Gys39, T. Hadavizadeh56, C. Hadjivasiliou60, G.
Haefeli40, C. Haen39, S.C. Haines48, S. Hall54, B. Hamilton59, X. Han12, S.
Hansmann-Menzemer12, N. Harnew56, S.T. Harnew47, J. Harrison55, J. He39, T.
Head40, A. Heister9, K. Hennessy53, P. Henrard5, L. Henry8, J.A. Hernando
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C. Hombach55, L. Hongming40, W. Hulsbergen42, T. Humair54, M. Hushchyn67, N.
Hussain56, D. Hutchcroft53, M. Idzik28, P. Ilten57, R. Jacobsson39, A.
Jaeger12, J. Jalocha56, E. Jans42, A. Jawahery59, M. John56, D. Johnson39,
C.R. Jones48, C. Joram39, B. Jost39, N. Jurik60, S. Kandybei44, W. Kanso6, M.
Karacson39, T.M. Karbach39,†, S. Karodia52, M. Kecke12, M. Kelsey60, I.R.
Kenyon46, M. Kenzie39, T. Ketel43, E. Khairullin67, B. Khanji21,39,k, C.
Khurewathanakul40, T. Kirn9, S. Klaver55, K. Klimaszewski29, M. Kolpin12, I.
Komarov40, R.F. Koopman43, P. Koppenburg42, M. Kozeiha5, L. Kravchuk34, K.
Kreplin12, M. Kreps49, P. Krokovny35, F. Kruse10, W. Krzemien29, W.
Kucewicz27,o, M. Kucharczyk27, V. Kudryavtsev35, A. K. Kuonen40, K. Kurek29,
T. Kvaratskheliya32, D. Lacarrere39, G. Lafferty55,39, A. Lai16, D. Lambert51,
G. Lanfranchi19, C. Langenbruch49, B. Langhans39, T. Latham49, C. Lazzeroni46,
R. Le Gac6, J. van Leerdam42, J.-P. Lees4, R. Lefèvre5, A. Leflat33,39, J.
Lefrançois7, E. Lemos Cid38, O. Leroy6, T. Lesiak27, B. Leverington12, Y. Li7,
T. Likhomanenko67,66, R. Lindner39, C. Linn39, F. Lionetto41, B. Liu16, X.
Liu3, D. Loh49, I. Longstaff52, J.H. Lopes2, D. Lucchesi23,r, M. Lucio
Martinez38, H. Luo51, A. Lupato23, E. Luppi17,g, O. Lupton56, N. Lusardi22, A.
Lusiani24, X. Lyu62, F. Machefert7, F. Maciuc30, O. Maev31, K. Maguire55, S.
Malde56, A. Malinin66, G. Manca7, G. Mancinelli6, P. Manning60, A. Mapelli39,
J. Maratas5, J.F. Marchand4, U. Marconi15, C. Marin Benito37, P. Marino24,t,
J. Marks12, G. Martellotti26, M. Martin6, M. Martinelli40, D. Martinez
Santos38, F. Martinez Vidal68, D. Martins Tostes2, L.M. Massacrier7, A.
Massafferri1, R. Matev39, A. Mathad49, Z. Mathe39, C. Matteuzzi21, A. Mauri41,
B. Maurin40, A. Mazurov46, M. McCann54, J. McCarthy46, A. McNab55, R.
McNulty13, B. Meadows58, F. Meier10, M. Meissner12, D. Melnychuk29, M. Merk42,
A Merli22,u, E Michielin23, D.A. Milanes64, M.-N. Minard4, D.S. Mitzel12, J.
Molina Rodriguez61, I.A. Monroy64, S. Monteil5, M. Morandin23, P. Morawski28,
A. Mordà6, M.J. Morello24,t, J. Moron28, A.B. Morris51, R. Mountain60, F.
Muheim51, D. Müller55, J. Müller10, K. Müller41, V. Müller10, M. Mussini15, B.
Muster40, P. Naik47, T. Nakada40, R. Nandakumar50, A. Nandi56, I. Nasteva2, M.
Needham51, N. Neri22, S. Neubert12, N. Neufeld39, M. Neuner12, A.D. Nguyen40,
C. Nguyen-Mau40,q, V. Niess5, S. Nieswand9, R. Niet10, N. Nikitin33, T.
Nikodem12, A. Novoselov36, D.P. O’Hanlon49, A. Oblakowska-Mucha28, V.
Obraztsov36, S. Ogilvy52, O. Okhrimenko45, R. Oldeman16,48,f, C.J.G.
Onderwater69, B. Osorio Rodrigues1, J.M. Otalora Goicochea2, A. Otto39, P.
Owen54, A. Oyanguren68, A. Palano14,d, F. Palombo22,u, M. Palutan19, J.
Panman39, A. Papanestis50, M. Pappagallo52, L.L. Pappalardo17,g, C.
Pappenheimer58, W. Parker59, C. Parkes55, G. Passaleva18, G.D. Patel53, M.
Patel54, C. Patrignani20,j, A. Pearce55,50, A. Pellegrino42, G. Penso26,m, M.
Pepe Altarelli39, S. Perazzini15,e, P. Perret5, L. Pescatore46, K. Petridis47,
A. Petrolini20,j, M. Petruzzo22, E. Picatoste Olloqui37, B. Pietrzyk4, M.
Pikies27, D. Pinci26, A. Pistone20, A. Piucci12, S. Playfer51, M. Plo
Casasus38, T. Poikela39, F. Polci8, A. Poluektov49,35, I. Polyakov32, E.
Polycarpo2, A. Popov36, D. Popov11,39, B. Popovici30, C. Potterat2, E.
Price47, J.D. Price53, J. Prisciandaro38, A. Pritchard53, C. Prouve47, V.
Pugatch45, A. Puig Navarro40, G. Punzi24,s, W. Qian56, R. Quagliani7,47, B.
Rachwal27, J.H. Rademacker47, M. Rama24, M. Ramos Pernas38, M.S. Rangel2, I.
Raniuk44, G. Raven43, F. Redi54, S. Reichert55, A.C. dos Reis1, V. Renaudin7,
S. Ricciardi50, S. Richards47, M. Rihl39, K. Rinnert53,39, V. Rives Molina37,
P. Robbe7, A.B. Rodrigues1, E. Rodrigues55, J.A. Rodriguez Lopez64, P.
Rodriguez Perez55, A. Rogozhnikov67, S. Roiser39, V. Romanovsky36, A. Romero
Vidal38, J. W. Ronayne13, M. Rotondo23, T. Ruf39, P. Ruiz Valls68, J.J.
Saborido Silva38, N. Sagidova31, B. Saitta16,f, V. Salustino Guimaraes2, C.
Sanchez Mayordomo68, B. Sanmartin Sedes38, R. Santacesaria26, C. Santamarina
Rios38, M. Santimaria19, E. Santovetti25,l, A. Sarti19,m, C. Satriano26,n, A.
Satta25, D.M. Saunders47, D. Savrina32,33, S. Schael9, M. Schiller39, H.
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D. Souza47, B. Souza De Paula2, B. Spaan10, P. Spradlin52, S. Sridharan39, F.
Stagni39, M. Stahl12, S. Stahl39, S. Stefkova54, O. Steinkamp41, O.
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L. Zhong3, V. Zhukov9, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62University of Chinese Academy of Sciences, Beijing, China, associated to 3
63Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
64Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
65Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
66National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
67Yandex School of Data Analysis, Moscow, Russia, associated to 32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

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 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
†Deceased


# A Study of Symbiosis Bias in A/B Tests of Recommendation Algorithms
(Preliminary Draft)

David Holtz1 The authors gratefully acknowledge the Fisher Center for Business
Analytics for research funding. We thank Yahu Cong for research assistance.
Jennifer Brennan2 Jean Pouget-Abadie2

( 1Haas School of Business, University of California, Berkeley
2Google Research )

###### Abstract

One assumption underlying the unbiasedness of global treatment effect
estimates from randomized experiments is the stable unit treatment value
assumption (SUTVA). Many experiments that compare the efficacy of different
recommendation algorithms violate SUTVA, because each algorithm is trained on
a pool of shared data, often coming from a mixture of recommendation
algorithms in the experiment. We explore, through simulation, cluster
randomized and data-diverted solutions to mitigating this bias, which we call
“symbiosis bias.”

Keywords: Interference, SUTVA, Clustered Randomized Experiments

##  1 Introduction

It is increasingly common for both academic researchers and industry
practitioners to conduct randomized experiments in which the “control” and
“treatment” interventions that are being compared are algorithmic in nature.
For instance, A/B tests are used in the tech industry to compare the efficacy
of algorithms for recommending content (Holtz et al., 2020), suggesting social
media contacts (Rajkumar et al., 2022), and setting prices (Aparicio et al.,
2023).

The unbiasedness of global treatment effect estimates derived from randomized
experiments relies on the stable unit treatment value assumption (SUTVA)
(Rubin, 1980), which combines three different assumptions related to the
potential outcomes framework: (i) that the outcome of unit $j$ does not depend
on the treatment assignment of any other unit $j^{\prime}$ (i.e., the “no
interference” assumption (Cox, 1958)) (ii) that there are no “hidden” versions
of the treatment, (Neyman, 1935) and (iii) the assumption that any unit’s
“potential outcome” under the treatment it is exposed to is equal to its
observed outcome (i.e., the “consistency” assumption (Robins, 1986)). However,
many experiments that compare the efficacy of different algorithms violate
SUTVA’s “no hidden treatments” assumption, due to the fact that each unit’s
potential outcome is a function not just of the algorithm that it is exposed
to, but also of the data that the algorithm has been trained on. In many
cases, algorithmic recommendations are updated over the course of the
experiment using newly-generated training data, and no distinction is made
between data that is produced by subjects exposed to different treatment
interventions.

Recently, there has been flurry of interest in measuring and mitigating the
bias that arises from evaluating algorithms on data they did not generate.
Glynn et al. (2020) study bias arising from switchback experiments over
policies in Markov decision problems, a problem they refer to as “temporal
interference” because the actions taken under one policy affect the starting
state seen by another policy. Other authors have noted that the “closed
feedback loop” nature of recommendation systems can introduce bias in the
system’s evaluation, perhaps because of a mismatch between the system that
collected data and the system that is being evaluated (Jadidinejad et al.,
2020), because of different levels of exploration between two algorithms
(Chen, 2021), or because the recommendation system itself alters the behavior
of users (Chaney et al., 2018). Musgrave et al. (2023) present a query-
diverted experimental design to mitigate the bias caused by the control and
treatment arm sharing data, in settings where query-diversion effectively
breaks the information flow between treated and control units. In this work,
we formalize this source of bias in algorithm-related experiments, which we
refer to as “symbiosis bias,” and explore the conditions under which
experiment designers might be able to reduce the amount of symbiosis bias in
total average treatment effect (TATE) estimates through experiment design.
More specifically, we use a simulation framework to assess the viability of
two different experiment design-based approaches to reducing symbiosis bias:
clustered experimentation and “data diverted” experimentation.

##  2 Modeling A/B tests of Recommendation Algorithms

We begin by defining a potential outcomes model for A/B tests of
recommendation algorithms. We model each unit $i$’s outcome at time $t$,
$Y_{it}$, as a function of its (time-invariant) treatment assignment, $z_{i}$,
and the data $d_{t}$ available to the algorithm at time $t$:

$\mathbf{Y}_{t}(\mathbf{z},d_{t})=\\{Y_{it}(z_{i},d_{t})\\}_{i}.$

The data that is available to the algorithm at time $t$ is a function of the
data available at time $t-1$, as well as the observed outcomes at time $t-1$:

$d_{t}(\mathbf{Y}_{t-1},d_{t-1})=d_{t}(\mathbf{Y_{t-1}}(\mathbf{z},d_{t-1}),d_{t-1})=d_{t}(\mathbf{z},d_{t-1})$

Since the treatment assignments $\mathbf{z}$ do not change over the course of
the experiment, this suggests that the data available at any time $t$ can be
expressed as
$d^{*}_{t}(\mathbf{z},d_{1}):=d_{t}(\mathbf{z},d_{t-1}(\mathbf{z},...d_{2}(\mathbf{z},d_{1})))$,
with $d_{1}^{*}:=d_{1}$. Hence, the observed outcome for unit $i$ at time $t$,

$Y_{it}=Y_{it}(z_{i},d_{t}^{*}(\mathbf{z},d_{1}))$

is a function of unit $i$’s treatment, $Z_{i}$, as well as the training data
available at time $t$, $d_{t}(\mathbf{z})$, which is itself a function of the
full treatment assignment vector $\mathbf{z}$.

A/B experiments often estimate the TATE, which captures the average effect of
switching from recommendation algorithm A to algorithm B. In our notation the
total treatment effect for each unit $i$ can be expressed as

$\tau_{it}^{TTE}(d_{1})=Y_{it}(1,d_{t}(\mathbf{1},d_{t-1}(\mathbf{1},...d_{2}(\mathbf{1},d_{1}))))-Y_{it}(0,d_{t}(\mathbf{0},d_{t-1}(\mathbf{0},...d_{2}(\mathbf{0},d_{1})))),$

or

$\tau_{it}^{TTE}(d_{1})=Y_{it}(1,d_{t}^{*}(\mathbf{1},d_{1}))-Y_{it}(0,d_{t}^{*}(\mathbf{0},d_{1})).$

However, under naive experiment designs, neither quantity in this expression
is observable, since for each unit of analysis we either observe
$Y_{it}(1,d_{t}^{*}(\mathbf{z},d_{1}))$ or
$Y_{it}(0,d_{t}^{*}(\mathbf{z},d_{1}))$ and in a randomized experiment
$\mathbf{z}$ will not be $\mathbf{0}$ or $\mathbf{1}$. This presents a more
severe obstacle to causal inference than the typical fundamental problem of
causal inference.

##  3 Simulation Results

In this work, we use simulations to assess the efficacy of two potential
solutions to this problem: clustered randomization, and “data-diverted
experiments.” Under clustered randomization, users are clustered according to
some proxy of their preferences for different items, and then treatment is
randomized at the cluster level. In a data-diverted experiment, each algorithm
is updated using only data produced by experiment subjects that are exposed to
that algorithm. In our framework, users are recommended pieces of content
repeatedly over time according to different recommendation algorithms: item-
based collaborative filtering, user-based collaborative filtering, random
recommendations, and ideal (oracle) recommendations. We simulate both “true”
counterfactual take-up rates and estimated take-up rates obtained from
different experimental designs.

More formally, we consider a community of 100 users, with $10$ new items
introduced at each time period. User preferences observe a natural cluster
structure in an embedding space of $20$ dimensions: user $u$’s utility for
item $i$ is $Beta(\mu=\rho_{u}^{T}\alpha_{i},\sigma=10^{-5})$, where
$\rho_{u}$ and $a_{i}$ represent fixed user preferences and item attributes
respectively, unknown to the experimenter. $\rho_{u}$ and $a_{i}$ are drawn
once from a symmetric Dirichlet distribution with parameter $5$. At each time
interval, a user either interacts with the item of highest utility at most
once, or interacts with no item at all if no item in the consideration set
provides higher than the median utility offered across all items (including
those not in the present consideration set). A user’s potential outcome is the
rate at which he or she chooses to interact with an item, taken over the $100$
time periods.

Both collaborative filtering algorithms update their recommendations at each
time step using the historical interactions from prior time periods. In the
data-diverted design, each algorithm only makes recommendations using
observations from its own arm of the experiment. In the naive design, users
are randomized to treatment or control and each algorithm makes
recommendations using observations from both arms. The cluster randomized
design similarly allows each algorithm to use observations from both arms, but
randomization into treatment or control is conducted at the level of user-
level clusters based on users’ (true) item preferences.

Figure 1 compares the bias incurred by naive, data-diverted and cluster
randomized experiments when the item-based collaborative filtering algorithm
is A/B tested against three alternative recommendation algorithms. We observe
that the naive design is biased for all three A/B comparisons.

The “Item CF vs Ideal” comparison provides a particularly intuitive
demonstration of symbiosis bias. The ideal recommendations are independent of
past data, so all bias comes from the Item-based CF recommendations. In this
case, Item-based CF uses the ideal algorithm’s high-quality recommendations to
learn a better matrix recommendation than Item-based CF would have learned
alone, thus biasing the TATE estimate upward. Data diversion removes the
symbiosis but causes the Item-based CF algorithm to learn more slowly,
resulting in negative bias. The cluster randomized design completely removes
the bias. A similar pattern is seen in the “Item CF vs Random” experiment,
although the symbiosis bias is of a smaller magnitude because observations
from the random algorithm are not as helpful to the Item-based CF algorithm as
the ideal recommendations are. The “Item CF vs User CF” comparison also shows
bias in the ATE estimates, although now both algorithms learn from the other’s
data so the results are more difficult to interpret.

Figure 1: Bias in TATE estimation for A/B tests of item-based collaborative
filtering vs three alternative algorithms under three experimental designs.

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