EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-304 LHCb-PAPER-2016-060 21 March 2017

Search for the ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ decay

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

A search for the charmless ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$
decay is performed using $pp$ collision data collected by the LHCb experiment
at centre-of-mass energies of $7$ and $8\mathrm{\,Te\kern-1.00006ptV}$,
corresponding to an integrated luminosity of 3$\mbox{\,fb}^{-1}$. No signal is
observed and upper limits on the
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ branching fraction are set to
$0.82\times 10^{-6}$ at $90$% and $1.01\times 10^{-6}$ at $95$% confidence
level.

Published in JHEP 05 (2017) 158

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

##  1 Introduction

Charmless hadronic decays of beauty hadrons proceed predominantly through
tree-level $b\rightarrow u$ and loop-level (penguin) $b\rightarrow s$ weak
transitions. In the Standard Model the amplitudes of these processes,
suppressed compared to the dominant tree $b\rightarrow c$ transition governing
charmed decays, usually have similar magnitudes and give rise to possibly
large violation of the charge-parity ($C\\!P$) symmetry. Therefore, charmless
decays of $B$ mesons should be sensitive to additional amplitudes from new,
heavy particles, contributing to the loop-level transitions [1].

Charmless hadronic ${{B}^{+}}$ and ${B}^{0}$ decays 111The inclusion of
charge-conjugate processes is implied throughout. have been the subject of
extensive studies, both experimentally, at hadron and $e^{+}e^{-}$ colliders,
and theoretically. The phenomenological understanding that has emerged allows
predictions to be made for charmless ${B}^{0}_{s}$ decays, as will be
illustrated in the following. In the ongoing effort to test these predictions
experimentally, the LHCb experiment has recently observed the decay222The
notations $\eta^{\prime}$ and $\phi$ refer to the $\eta^{\prime}(958)$ and
$\phi(1020)$ mesons. ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\eta^{\prime}$.
The relatively large measured branching fraction
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\eta^{\prime})=(33.1\pm
7.1)\times 10^{-6}$ is consistent with Standard Model expectations [2].
However, the knowledge about charmless hadronic ${B}^{0}_{s}$ decays into
light pseudoscalar (P) and vector (V) mesons is still limited. Further
measurements will help to better constrain phenomenological models, the
uncertainties of which often translate into a major contribution to the
theoretical uncertainties in searches for physics beyond the Standard Model.

The decay ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ proceeds
predominantly through
${\overline{b}}\rightarrow{\overline{s}}{s}{\overline{s}}$ transitions, as
illustrated in Fig. 1. It is of particular interest in constraining
phenomenological models, as predictions for its branching fraction cover a
wide range, typically from $0.1\times 10^{-6}$ to $20\times 10^{-6}$, with
large uncertainties that reflect the limited knowledge of form factors,
penguin contributions, the $\omega-\phi$ mixing angle, or the $s$-quark mass.
The decay ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ has been studied in
the framework of QCD factorisation [3, 4], perturbative QCD [5, 6], soft-
collinear effective theory (SCET) [7], SU(3) flavour symmetry [8], and
factorisation-assisted topological (FAT) amplitude approach [9]. Table 1
presents the available predictions for
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)$.

In QCD factorisation, predictions for
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)$ are generally
small because the spectator quark can become part of either the
$\eta^{\prime}$ or the $\phi$ meson (see Fig. 1), leading to a strong
cancellation between the PV and VP amplitudes contributing to the
$\eta^{\prime}\phi$ final state [3]. Such cancellation does not occur in the
symmetric ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\eta^{\prime}$ (PP) and
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ (VV) decays. However, other values of
the form factor for the ${{B}^{0}_{s}}$ to $\phi$ transitions can lead to
enhancements of the branching fraction by more than an order of magnitude [4].
The measurement of
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)$ is therefore
important to improve the knowledge of the ${{B}^{0}_{s}}$ to $\phi$ form
factor and the accuracy of model predictions.

The comparison of QCD factorisation [3, 4], perturbative QCD [5], and SCET [7]
calculations shows that the hierarchy of branching fractions in
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ and
${{B}^{0}_{s}}\\!\rightarrow\eta\phi$ decays is sensitive to the size of the
colour-suppressed QCD penguin loop, which is estimated to be large in
perturbative QCD [5], and to “gluonic charming penguins”, which play an
important role in SCET calculations [7]. Future measurements of both decay
modes will provide useful information on these loop contributions.

This paper presents a search for the
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ decay with the LHCb detector.
The results are based on a data sample collected during the 2011 and 2012
${p}{p}$ collision runs of the Large Hadron Collider at centre-of-mass
energies of $7$ and $8~{}\mathrm{\,Te\kern-1.00006ptV}$, respectively,
corresponding to a total integrated luminosity of $3\mbox{\,fb}^{-1}$.

The signal ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ and normalisation
${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ candidates are reconstructed
through the decays $\eta^{\prime}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}\gamma$
and $\phi\\!\rightarrow{{K}^{+}}{{K}^{-}}$. The
${{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi$ branching fraction is determined
with respect to the ${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ mode
according to

${\mathcal{B}}({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)=\frac{{\mathcal{B}}(B^{+}\rightarrow\eta^{\prime}K^{+})}{{\mathcal{B}}(\phi\rightarrow
K^{+}K^{-})}\times\frac{f_{u}}{f_{s}}\times\frac{N({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)}{N(B^{+}\rightarrow\eta^{\prime}K^{+})}\times\frac{\epsilon(B^{+}\rightarrow\eta^{\prime}K^{+})}{\epsilon({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)}\,,$
(1)

where ${\mathcal{B}}(B^{+}\rightarrow\eta^{\prime}K^{+})=(70.6\pm 2.5)\times
10^{-6}$ [10], ${\mathcal{B}}(\phi\rightarrow K^{+}K^{-})=0.489\pm 0.005$
[10], $f_{u}/f_{s}$ is the $B^{+}/{{B}^{0}_{s}}$ production ratio assumed to
be equal to the $B^{0}/{{B}^{0}_{s}}$ production ratio
$f_{d}/f_{s}=1/(0.259\pm 0.015)$ [11], and
$\epsilon({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)$ and
$\epsilon(B^{+}\rightarrow\eta^{\prime}K^{+})$ are the total efficiencies of
the signal and normalisation modes, respectively. The ratio of the observed
yields
$N({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)/N(B^{+}\rightarrow\eta^{\prime}K^{+})$
is obtained from a two-dimensional fit to the invariant mass distributions of
the $\eta^{\prime}$ and the $B$ candidates, performed simultaneously on the
signal and normalisation modes.

Figure 1: Lowest-order diagrams for the ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ decay. The spectator quark can become part of either the $\eta^{\prime}$ or the $\phi$ meson, forming two different amplitudes (called PV and VP in the text). Table 1: Theoretical predictions for the ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ branching fraction. Theory approach | ${\mathcal{B}}\ (10^{-6})$ | Reference
---|---|---
QCD factorisation | $0.05^{+1.18}_{-0.19}$ | [3]
QCD factorisation | $\,2.2^{+9.4}_{-3.1}$ | [4]
Perturbative QCD | $0.19^{+0.20}_{-0.13}$ | [5]
Perturbative QCD | $20.0^{+16.3}_{-9.1}$ | [6]
SCET | $\,4.3^{+5.2}_{-3.6}$ | [7]
SU(3) flavour symmetry | $\>\;5.5\pm 1.8$ | [8]
FAT | $13.0\pm 1.6$ | [9]

##  2 Detector and simulation

The LHCb detector [12, 13] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, 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 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 $pp$-collision point (primary vertex), the impact parameter, is
measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad (SPD) 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 trigger [14] 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. The $B$ decays of interest are triggered at the
hardware stage, either by one of the decay products depositing a transverse
energy greater than 3.5$\mathrm{\,Ge\kern-1.00006ptV}$ in the hadron
calorimeter, or by other high-$p_{\mathrm{T}}$ particles produced in the $pp$
collision. The software trigger requires a two-, three- or four-track
secondary vertex with a significant displacement from the primary vertices. At
least one charged particle must have a transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and be
inconsistent with originating from a primary vertex. A multivariate algorithm
[15] is used for the identification of secondary vertices consistent with the
decay of a $b$ hadron.

Simulated decays are used to optimise the event selection and to evaluate the
selection efficiencies. In the simulation, $pp$ collisions are generated using
Pythia 8 [16, *Sjostrand:2007gs] with a specific LHCb configuration [18].
Decays of hadronic particles are described by EvtGen [19], in which final-
state radiation is generated using Photos [20]. The interaction of the
generated particles with the detector, and its response, are implemented using
the Geant4 toolkit [21, *Agostinelli:2002hh] as described in Ref. [23].

##  3 Event selection

The selection of the signal ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ and
normalisation ${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ candidates,
generically referred to as B candidates, is optimised for the signal. Wherever
possible, the same selection criteria are applied for the normalisation
channel.

Only good-quality tracks identified as pions or kaons [13] and inconsistent
with originating from any primary vertex are used. Tracks used to reconstruct
an $\eta^{\prime}$ or $\phi$ candidate are each required to be consistent with
coming from a common secondary vertex and to have
$\mbox{$p_{\mathrm{T}}$}>0.4{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
${{\pi}^{+}}{{\pi}^{-}}$ invariant mass in the $\eta^{\prime}$ decay must be
larger than $0.52{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to reject
${{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ decays. Photon candidates must be of
good quality [13] and have
$\mbox{$p_{\mathrm{T}}$}>0.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
invariant masses of the $\eta^{\prime}$ and $\phi$ candidates must satisfy
$0.88<m_{\pi\pi\gamma}<1.04{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$1.005<m_{KK}<1.035{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. An
$\eta^{\prime}$ candidate is combined with a candidate $\phi$ meson (or a
charged kaon with
$\mbox{$p_{\mathrm{T}}$}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) to make a
${{B}^{0}_{s}}$ (or ${{{B}^{+}}}$) candidate. Each $B$ candidate is required
to have a good-quality vertex, by imposing a loose requirement of the
$\chi^{2}$ of the vertex fit ($\chi^{2}<6$), and
$\mbox{$p_{\mathrm{T}}$}>1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
invariant masses of the ${B}^{0}_{s}$ and ${{B}^{+}}$ candidates, computed
after constraining the $\pi^{+}\pi^{-}\gamma$ mass to the nominal
$\eta^{\prime}$ mass [10], are required to satisfy
$5.0<m_{\eta^{\prime}KK}<5.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$5.0<m_{\eta^{\prime}K}<5.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$,
respectively.

To further separate signal from background, boosted decision trees (BDTs)
based on the AdaBoost algorithm [24, 25] are used. Different BDTs are used for
the signal and normalisation channels. Each BDT is trained, tested and
optimised on fully simulated signal decays and background taken from data. The
background consist of events in the mass range
$5.0<m_{\eta^{\prime}KK}<5.6{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
($5.0<m_{\eta^{\prime}K}<5.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$)
excluding the signal region defined below.

To minimise statistical and systematic uncertainties, the BDT algorithm uses
input variables that provide significant background rejection, are well
modelled in simulation, and are defined for both the signal and normalisation
channels. Nine variables are used as input to each BDT. Two variables are
related to the kinematics of the final-state particles: the transverse momenta
of the photon and the $\eta^{\prime}$ meson. Three variables describe the
topology of the ${B}$ candidate: the $B$-candidate flight distance, the cosine
of the angle between the reconstructed ${B}$ momentum and the vector pointing
from the associated primary vertex to the ${B}$ decay vertex, and the impact
parameter of the ${B}$ candidate with respect to its associated primary
vertex. The associated primary vertex is the primary vertex with respect to
which the $B$ candidate has the smallest $\chi^{2}_{\text{IP}}$, where
$\chi^{2}_{\text{IP}}$ is defined as the difference in the vertex-fit
$\chi^{2}$ of the selected primary vertex reconstructed with or without the
considered particle. Three variables are related to the ${B}$-candidate
vertex: the vertex-fit quality, characterised by its $\chi^{2}$, and two
vertex isolation variables defined as the smallest vertex-fit $\chi^{2}$
values obtained when adding to the vertex in turn either all single tracks or
all pairs of tracks from the set of tracks that are not assigned to the $B$
candidate. The last variable is the sum of the $\chi^{2}_{\text{IP}}$ of the
charged particles used to form the ${B}$ candidate, calculated with respect to
the associated primary vertex. The photon $p_{\mathrm{T}}$ and the
$B$-candidate impact parameter provide the best background discrimination. The
BDT is trained for the full data set, irrespective of the $pp$ collision
energy. To minimise biases in the final selection, both the data and simulated
samples are randomly divided into two subsamples and two BDTs are defined.
Each BDT is trained, tested and optimised on one subsample, and then applied
to the other subsample for the candidate selection [26]. The selected
candidates from both subsamples are then merged into a single sample for the
next stage of the analysis.

The requirement on the BDT output is chosen to maximise the figure of merit
$\epsilon({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)/(a/2+\sqrt{N_{B}})$
[27], where $a=5$ is the target signal significance, and $N_{B}$ is the number
of background events in the signal region estimated from the ${B}^{0}_{s}$
mass sidebands. The signal region is defined as the ${B}^{0}_{s}$ mass range
$5.287-5.446{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, corresponding
approximately to $7$ times the mass resolution. The optimised BDT requirement
has an efficiency of $59\%$ for ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$
decays, while rejecting $93\%$ of the combinatorial background in the signal
region. As a check, an alternative optimisation is performed: for various
values of the ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ branching
fraction, pseudoexperiments are generated with a model containing only signal
and combinatorial background, and then are analysed with a simple two-
dimensional maximum likelihood fit to the ${B}^{0}_{s}$ and $\eta^{\prime}$
masses. The signal significance, determined using Wilks’ theorem [28], is
found to reach its maximum for a BDT requirement in agreement with that
obtained using the method of Ref. [27].

In events containing multiple candidates ($\lesssim 3\%$), the candidate with
the best identified photon is kept. The full selection described above retains
$430$ ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ candidates and $22\,681$
${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ candidates for further
analysis.

Table 2: Relative systematic uncertainties on the efficiency ratio $\epsilon({{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}})/\epsilon({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)$.  Source | Relative uncertainty [%]
---|---
BDT efficiency calibration | 2.5
PID efficiency calibration | 1.1
Trigger efficiency calibration | 2.3
SPD multiplicity (mismodelling) | 0.9
Track reconstruction | 0.4
Photon reconstruction | 0.1
Hadronic interactions | 1.4
Simulation statistics | 1.6
Total | 4.3

Selection efficiencies are evaluated with simulated data, except those of the
particle identification (PID) requirements and the hardware trigger, for which
calibration data are used. Systematic uncertainties on the efficiency ratio
$\epsilon({{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}})/\epsilon({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)$
are summarised in Table 2. The BDT algorithms are validated using the
normalisation channel as proxy for the signal, and by comparing the
distributions obtained with the sPlot technique [29] of the nine input
variables and the BDT output variable. The difference between the efficiencies
in data and simulation of the BDT requirement for the normalisation channel is
used as a measure of the systematic uncertainty on the BDT efficiency. The
correlation evaluated in simulation between the BDT variables for
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ and
${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ is then used to determine
the systematic uncertainty on the ratio of the BDT efficiencies.

Another systematic effect on the determination of the efficiency ratio is the
uncertainty on the PID efficiency, which is determined as a function of
kinematic parameters using a clean high-statistics sample of kaons and pions
from
${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$
decays [30]. The uncertainty on the trigger efficiency, which is mostly due to
the computation of the hardware-stage trigger efficiency, is estimated with
simulated data by varying the value of the minimum transverse energy
requirement used in the trigger decision. An uncertainty is assigned on the
efficiency ratio to take into account the mismodelling of the hit multiplicity
in the SPD, which is used as a discriminant variable at the hardware stage of
the trigger. This uncertainty is evaluated in simulation by varying the
requirement on the SPD hit multiplicity. Corrections determined from control
channels are applied to the tracking and photon reconstruction efficiencies to
account for mismodelling effects in the simulation. The uncertainties on these
corrections are quoted as systematic uncertainties. Since the correction to
the tracking efficiency is obtained using muons, an additional uncertainty is
needed to account for hadronic interactions in the detector material [31].
Finally, the limited statistics of the simulated samples used in the
evaluation of the efficiencies is added as a source of uncertainty. Combining
all uncertainties in quadrature, the ratio of the selection efficiencies is

$\frac{\epsilon({{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}})}{\epsilon({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)}=1.83\pm
0.08\,.$ (2)

The selection requirements efficiently reject physics backgrounds such as
${{B}^{0}}\\!\rightarrow\phi{{K}^{*0}}$ and modes with resonances decaying
strongly to ${{K}^{+}}{{\pi}^{-}}{{\pi}^{0}}$, but not
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decays with one of the two $\phi$
resonances decaying to ${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ and one of the
photons from the ${{\pi}^{0}}\\!\rightarrow\gamma\gamma$ decay not being
reconstructed. From simulation studies and known branching fractions [10], the
number of ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ decays passing the selection
is expected to be $104\pm 34$. Hence this background is included as a specific
component in the mass fit described below.

##  4 Mass fit

The ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ signal yield is determined
from a two-dimensional extended unbinned maximum likelihood fit, where the
signal is fitted simultaneously with the normalisation channel
${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$. The observables used in the
fit are the invariant masses $m_{\pi\pi\gamma}$ and $m_{\eta^{\prime}KK}$
($m_{\eta^{\prime}K}$) for the sample of
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$
(${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$) candidates.

The sample of ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ candidates is
described with a four-component model: the signal, the two combinatorial
backgrounds with and without a true $\eta^{\prime}$ resonance, and the
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ physics background, where one of the two
$\phi$ resonances decays to the ${{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ final
state. The sample of ${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$
candidates is modelled using three components: the signal and the two
combinatorial backgrounds with and without a true $\eta^{\prime}$ resonance.
The yields of all components are free to vary in the fit. The peaking
components in the ${B}^{0}_{s}$, ${{B}^{+}}$ and $\eta^{\prime}$ mass spectra
are described using Gaussian functions modified with an exponential tail on
each side. While all the tail parameters are fixed from simulation, the mean
and the widths of the Gaussian functions are free to vary in the fit, but the
ratio of the widths of the peaking components in $m_{\eta^{\prime}KK}$ and
$m_{\eta^{\prime}K}$ is fixed to the value obtained in simulation and the
difference between the ${B}^{0}_{s}$ and ${{B}^{+}}$ masses is constrained to
the known value [10]. The $\eta^{\prime}$ resonances in the two samples are
modelled using a common function, with mean and width free in the fit. The
combinatorial components are described with linear functions, with the
exception of the random combinations in $m_{\eta^{\prime}K}$, where a
parabolic function is used. To account for correlations between
$m_{\eta^{\prime}KK}$ and $m_{\pi\pi\gamma}$, the
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ component is described with a
superposition of two-dimensional Gaussian kernel functions [32] determined
from simulation. For all other components, in particular the signal, the
correlation is negligible due to the $\eta^{\prime}$ mass constraint applied
in the computation of the $B$-candidate mass. The fit procedure is validated
on simulated samples containing the expected proportion of background and
signal events, according to various assumptions on
${\mathcal{B}}({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)$. In particular, for
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)=4\times 10^{-6}$,
a statistical significance corresponding to more than $5$ standard deviations
is observed in $74\%$ of the pseudoexperiments.

Figure 2: Distributions of the (top left) $\pi^{+}\pi^{-}\gamma$ and (top
right) $\eta^{\prime}K^{+}K^{-}$ masses of the selected
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ candidates, as well as of the
(bottom left) $\pi^{+}\pi^{-}\gamma$ and (bottom right)
$\eta^{\prime}{{K}^{+}}$ masses of the selected
${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ candidates. The solid blue
curves represent the result of the simultaneous two-dimensional fit described
in the text, with the following components:
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ and
${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ signals (red dashed),
combinatorial backgrounds (blue dot-dot-dashed), combinatorial backgrounds
with real $\eta^{\prime}$ (green dotted), and
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ background (black dot-dashed).

Figure 2 shows the mass distributions observed in data with the projections of
the fit results overlaid. No ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$
signal is observed. The fitted yields are $-3.2^{+5.0}_{-3.8}$ for the
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ signal, $105\pm 29$ for the
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ physics background (consistent with
expectation), and $11\,081\pm 127$ for the
${{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}}$ normalisation mode. The
measured ${B}$ and $\eta^{\prime}$ mass resolutions are $21.8\pm
0.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $12.6\pm
0.2{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. The ratio of
yields is fitted to be
$N({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)/N({{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}})=(-2.9^{+4.5}_{-3.5})\times
10^{-4}$.

Sets of pseudoexperiments are used to evaluate possible fit biases. Fits on
samples generated from the probability density function (PDF) with parameters
obtained from the data are found to be unbiased. The procedure is then
repeated using simulated ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ events instead
of generating the corresponding background component from the PDF. Biases of
$-1.3\pm 0.3$ on the signal yield and of $(-1.16\pm 0.33)\times 10^{-4}$ on
the ratio of yields are observed. The results obtained with data are corrected
for these biases and systematic uncertainties computed as the quadratic sum of
the statistical uncertainty on the bias and half of the bias value are
assigned.

Additional systematic uncertainties affect the signal yield and the yield
ratio. The mass fit is repeated with different combinatorial background PDFs:
linear functions are replaced with exponential functions, and the parabolic
function is replaced with a third-order polynomial. The quadratic sum of the
differences between the values obtained in these alternative fits and the
nominal result is assigned as a systematic uncertainty. The limited size of
the simulated ${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ sample leads to an
uncertainty on the determination of the nonparametric PDF for the physics
background, which is propagated as a systematic uncertainty. The effect of
fixing some of the model parameters in the fit is studied by performing a
large number of fits on the data, with the fixed parameters sampled randomly
from Gaussian distributions centred on the nominal values and with widths and
correlations as determined in simulated events. The standard deviation of the
distribution of the results is assigned as a systematic uncertainty.

The systematic uncertainties on the
${{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi$ yield and the yield ratio are
summarised in Table 3. The final results from the mass fit, including all
corrections, are

$\displaystyle N({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)$ $\displaystyle=$
$\displaystyle-1.9^{+5.0}_{-3.8}\pm 1.1\,,$ (3)
$\displaystyle\frac{N({{B}^{0}_{s}}\rightarrow\eta^{\prime}\phi)}{N(B^{+}\rightarrow\eta^{\prime}K^{+})}$
$\displaystyle=$ $\displaystyle(-1.7^{+4.5}_{-3.5}\pm 1.0)\times 10^{-4}\,,$
(4)

where the first (second) quoted uncertainty is statistical (systematic).
Bayesian upper limits $x_{\rm U}$ are determined assuming a uniform prior in
the observable $x$ (yield, yield ratio, or $\cal B$) as $\int_{0}^{x_{\rm
U}}{\cal L}(x)dx/\int_{0}^{\infty}{\cal L}(x)dx=\alpha$, where ${\cal L}(x)$
is the likelihood function convolved with the systematic uncertainties, and
$\alpha$ is the confidence level (CL). The obtained upper limits are

$N({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)<8.9~{}(10.9)\quad\mbox{at
90\% (95\%) CL}$

and

$\frac{N({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)}{N(B^{+}\rightarrow\eta^{\prime}K^{+})}<8.0~{}(9.9)\times
10^{-4}\quad\mbox{at 90\% (95\%) CL}\,.$

Table 3: Systematic uncertainties $\sigma_{N}$ and $\sigma_{R}$ on the fitted yield $N({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)$ and on the yield ratio $R=N({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)/N({{B}^{+}}\\!\rightarrow\eta^{\prime}K^{+})$, respectively. The last line gives the quadratic sum of the individual uncertainties. Source |  $\sigma_{N}$ (events) |  $\sigma_{R}$ ($10^{-4}$)
---|---|---
Fit bias | 0.7 | 0.7
Combinatorial background modelling | 0.6 | 0.6
${{B}^{0}_{s}}\\!\rightarrow\phi\phi$ background modelling | 0.4 | 0.3
Fixed parameters in the fit | 0.3 | 0.3
Total | 1.1 | 1.0

##  5 Result and conclusion

A search has been performed for the
${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$ decay. No signal is found. The
branching fraction
${\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)=(-0.18^{+0.47}_{-0.36}({\rm
stat})\pm 0.10({\rm syst}))\times 10^{-6}$ is computed from Eqs. (1), (2) and
(4) using the known value of
${\mathcal{B}}({{{B}^{+}}}\\!\rightarrow\eta^{\prime}{{K}^{+}})$ [10] and the
LHCb measurement of $f_{s}/f_{d}$ [11], which leads to

${\cal B}({{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi)<0.82\,(1.01)\times
10^{-6}\quad\mbox{at 90\% (95\%) CL}\,$

using the likelihood integration method described above. This is the first
upper limit set on the ${{B}^{0}_{s}}\\!\rightarrow\eta^{\prime}\phi$
branching fraction.

This result favours the lower end of the range of predictions for this
branching fraction, pointing to form factors consistent with the light-cone
sum-rule calculation used in Ref. [4], or with the hypotheses used in Refs.
[3, 5]. Although large theoretical uncertainties make most predictions
compatible with the result of this analysis, the central values of the
predictions in Refs. [6, 7, 8, 9] are significantly larger than the upper
limit. These discrepancies should help in constraining the theoretical models
used in the prediction of branching fractions and $C\\!P$ asymmetries for
$B$-meson hadronic charmless decays.

## 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 FASO (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

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Traill53, M.T. Tran41, M. Tresch42, A. Trisovic40, A. Tsaregorodtsev6, P.
Tsopelas43, A. Tully49, N. Tuning43, A. Ukleja29, A. Ustyuzhanin35, U. Uwer12,
C. Vacca16,f, V. Vagnoni15,40, A. Valassi40, S. Valat40, G. Valenti15, R.
Vazquez Gomez19, P. Vazquez Regueiro39, S. Vecchi17, M. van Veghel43, J.J.
Velthuis48, M. Veltri18,r, G. Veneziano57, A. Venkateswaran61, M. Vernet5, M.
Vesterinen12, J.V. Viana Barbosa40, B. Viaud7, D. Vieira63, M. Vieites Diaz39,
H. Viemann67, X. Vilasis-Cardona38,m, M. Vitti49, V. Volkov33, A. Vollhardt42,
B. Voneki40, A. Vorobyev31, V. Vorobyev36,w, C. Voß9, J.A. de Vries43, C.
Vázquez Sierra39, R. Waldi67, C. Wallace50, R. Wallace13, J. Walsh24, J.
Wang61, D.R. Ward49, H.M. Wark54, N.K. Watson47, D. Websdale55, A. Weiden42,
M. Whitehead40, J. Wicht50, G. Wilkinson57,40, M. Wilkinson61, M. Williams40,
M.P. Williams47, M. Williams58, T. Williams47, F.F. Wilson51, J. Wimberley60,
J. Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49, K.
Wraight53, K. Wyllie40, Y. Xie65, Z. Xing61, Z. Xu4, Z. Yang3, Y. Yao61, H.
Yin65, J. Yu65, X. Yuan36,w, O. Yushchenko37, K.A. Zarebski47, M.
Zavertyaev11,c, L. Zhang3, Y. Zhang7, Y. Zhang63, A. Zhelezov12, Y. Zheng63,
X. Zhu3, V. Zhukov33, 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
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 38
70Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased