EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2015-198 LHCb-PAPER-2015-028 16 October 2015

Measurement of the $B_{s}^{0}\rightarrow\phi\phi$ branching fraction and
search for the decay $B^{0}\rightarrow\phi\phi$

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

Using a dataset corresponding to an integrated luminosity of
$3.0\mbox{\,fb}^{-1}$ collected in $pp$ collisions at centre-of-mass energies
of 7 and 8 TeV, the ${{B}^{0}_{s}}\rightarrow\phi\phi$ branching fraction is
measured to be

$\mathcal{B}(B^{0}_{s}\rightarrow\phi\phi)=\left(1.84\pm
0.05\mathrm{\,(stat)}\pm 0.07\mathrm{\,(syst)}\pm 0.11\,(f_{s}/f_{d})\pm
0.12\,(\text{norm})\,\right)\times 10^{-5},$

where $f_{s}/f_{d}$ represents the ratio of the ${{B}^{0}_{s}}$ to ${{B}^{0}}$
production cross-sections, and the $B^{0}\rightarrow\phi{{K}^{*}}(892)^{0}$
decay mode is used for normalization. This is the most precise measurement of
this branching fraction to date, representing a factor five reduction in the
statistical uncertainty compared with the previous best measurement. A search
for the decay ${{B}^{0}}\rightarrow\phi\phi$ is also made. No signal is
observed, and an upper limit on the branching fraction is set as

$\mathcal{B}(B^{0}\rightarrow\phi\phi)<2.8\times 10^{-8}$

at $90\%$ confidence level. This is a factor of seven improvement compared to
the previous best limit.

Published as JHEP 10 (2015) 053

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

##  1 Introduction

In the Standard Model, the flavour-changing neutral current decay
${{B}^{0}_{s}}\rightarrow\phi\phi$ proceeds via a
$\bar{b}\rightarrow\bar{s}s\bar{s}$ penguin amplitude. The decay was first
observed by the CDF experiment at the Tevatron [1]. Subsequently, it has been
studied by the CDF and LHCb collaborations, who searched for $C\\!P$-violating
asymmetries in the decay time and angular distributions of this mode [2, 3, 4,
5]. These studies provide a probe for possible new physics contributions
entering into the penguin loop and ${{B}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}$ mixing diagrams [6].
Furthermore, as the ${{B}^{0}_{s}}\rightarrow\phi\phi$ mode will be used as
normalization for studies of other charmless ${{B}^{0}_{s}}$ meson decays, it
is important to have a precise determination of its branching fraction. The
CDF collaboration measured this relative to the decay
${{B}^{0}_{s}}\rightarrow J/\psi\phi$ [2]. Using the current value of the
${{B}^{0}_{s}}\rightarrow J/\psi\phi$ branching fraction [7], the CDF result
gives $\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)=(1.91\pm 0.26\pm
0.16)\times 10^{-5}$, where the first uncertainty is from the measured ratio
to ${{B}^{0}_{s}}\rightarrow J/\psi\phi$, and the second is due to the
knowledge of the ${{B}^{0}_{s}}\rightarrow J/\psi\phi$ branching fraction.
Various predictions from theories based on QCD factorization exist for the
${{B}^{0}_{s}}\rightarrow\phi\phi$ branching fraction [8, 9, 10]. These suffer
from uncertainties related to weak annihilation diagrams. These uncertainties
are controlled using experimental information from decays such as
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$. Several recent predictions are
summarized in Table 1. The central values are in the range $(1.5-2.0)\times
10^{-5}$.

Table 1: Predictions for the ${{B}^{0}_{s}}\rightarrow\phi\phi$ branching fraction. The first and second uncertainties of Refs. [8, 9] reflect the knowledge of CKM parameters and power corrections, respectively. $\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)$ ($10^{-5}$) | Approach | Reference
---|---|---
$1.95\pm 0.10^{+1.30}_{-0.80}$ | QCD factorization | [8]
$1.67^{+0.26}_{-0.21}{}^{+1.13}_{-0.88}$ | QCD factorization | [9]
$1.55^{+2.24}_{-1.70}$ | QCD factorization | [10]
$1.67^{+0.89}_{-0.71}$ | pQCD | [11]

In this paper the $B_{s}^{0}\rightarrow\phi\phi$ branching fraction (the use
of charge-conjugate modes is implied throughout) is measured using the full
LHCb Run 1 dataset, comprising data corresponding to an integrated luminosity
of 1.0 fb-1 collected in $pp$ collisions at a centre-of-mass energy of 7 TeV,
and 2.0 fb-1 collected at 8 TeV. The decay
$B^{0}\rightarrow\phi{{K}^{*}}(892)^{0}$, which has a similar topology, is
used for normalization. The $\phi$ and ${{K}^{*}}(892)^{0}$ mesons are
reconstructed in the $K^{+}K^{-}$ and $K^{+}\pi^{-}$ final states,
respectively. In addition, a search for the yet unobserved decay
${{B}^{0}}\rightarrow\phi\phi$ is made. This decay is suppressed in the
Standard Model by the OZI rule [12, *Zweig:1964, *Iizuka:1966fk], with an
expected branching fraction in the range $(0.1-3.0)\times 10^{-8}$ [15, 16, 8,
10]. However, the branching fraction can be enhanced, up to the $10^{-7}$
level, in models such as supersymmetry with R-parity violation [16]. The
current best limit for this mode is from the BaBar collaboration [17],
$\mathcal{B}({{B}^{0}}\rightarrow\phi\phi)<2.0\times 10^{-7}$ at $90\,\%$
confidence level.

##  2 Detector and software

The LHCb detector [18, 19] 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 [20], 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 [21] placed
downstream of the magnet. The tracking system provides a measurement of
momentum, $p$, 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 $pp$ interaction vertex (PV), the
impact parameter, is measured with a resolution of $(15+29/\mbox{$p_{\rm
T}$}){\,\upmu\rm m}$, where $p_{\rm T}$ 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 [22]. 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 [23].

The trigger [24] 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 software trigger applied in this analysis
requires a two-, three- or four-track secondary vertex with a significant
displacement from any PV. 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 [25] is used
for the identification of secondary vertices consistent with the decay of a
$b$ hadron.

In the simulation, $pp$ collisions are generated using Pythia [26,
*Sjostrand:2007gs] with a specific LHCb configuration [28]. Decays of hadronic
particles are described by EvtGen [29], in which final-state radiation is
generated using Photos [30]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [31,
*Agostinelli:2002hh] as described in Ref. [33].

##  3 Signal selection

The selection of candidates takes place in two stages. First, a selection
using loose criteria is performed that reduces background whilst retaining
high signal efficiency. Following this, a multivariate method is used to
further improve the signal significance.

The selection starts from charged particle tracks that traverse the entire
spectrometer. Selected particles are required to have $\mbox{$p_{\rm
T}$}>500\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. Fake tracks created by the
reconstruction due to random combinations of hits in the detector are
suppressed using a requirement on a neural network trained to discriminate
between these and genuine tracks associated to particles. Combinatorial
background from hadrons originating at the primary vertex is suppressed by
requiring that all tracks are significantly displaced from any primary vertex.
Kaon and pion candidates are selected using the information provided by the
ring-imaging Cherenkov detectors. This is combined with kinematic information
using a neural network to provide an effective probability that a particle is
a kaon ($\mathcal{P}^{K}$) or pion ($\mathcal{P}^{\pi}$). To select kaon
candidates it is required that $\mathcal{P}^{K}(1-\mathcal{P}^{\pi})>0.025$.
The pion candidate in the ${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ decay
mode is required to have $\mathcal{P}^{\pi}>0.2$ and $\mathcal{P}^{K}<0.2$.

The selected charged particles are combined to form $\phi$ and $K^{*}$ meson
candidates. The invariant mass of the $K^{+}K^{-}$ ($K^{+}\pi^{-})$ pair is
required to be within $15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
($150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$) of the known mass of the
$\phi$ ($K^{*}(892)^{0}$) meson [7]. In addition, the $p_{\rm T}$ of the
$\phi$ and $K^{*}$ mesons must be greater than
$1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

Candidates for the decay ${{B}^{0}_{s}}\rightarrow\phi\phi$ are formed by
combining pairs of $\phi$ mesons. A fit is made requiring all four final-state
particles to originate from a common vertex, and the direction vector between
the primary and secondary vertices is required to be consistent with the
direction of the momentum vector of the ${{B}^{0}_{s}}$ meson candidate.
Further requirements are then applied to remove background from specific
${b}$-hadron decays that peak close to the ${{B}^{0}_{s}}$ mass. To reject
background from ${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ decays, the kaon
with the lowest value of $\mathcal{P}^{K}$ is considered to be a pion, and the
$K^{+}\pi^{-}$ and $K^{+}K^{-}K^{+}\pi^{-}$ invariant masses are calculated.
Candidates with $m(K^{+}\pi^{-})$ within
$50{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known $K^{*}(892)^{0}$
mass and $m(K^{+}K^{-}K^{+}\pi^{-})$ within
$30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the ${{B}^{0}}$ mass [7] are
rejected. Similarly, to remove decays via open charm mesons, the
$K^{+}K^{-}\pi^{+}$ mass is calculated. If $m(K^{+}K^{-}\pi^{+})$ is within
$22.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the $D^{+}$ or $D^{+}_{s}$
mass [7], the candidate is rejected. These vetoes are found to retain 91% of
simulated ${{B}^{0}_{s}}\rightarrow\phi\phi$ decays.

Candidates for the decay ${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ are
formed from combinations of $\phi$ and ${{K}^{*}}$ mesons. Identical vertex
and pointing requirements as for the ${{B}^{0}_{s}}\rightarrow\phi\phi$ decay
mode are applied. To reject background from
${{B}^{0}_{s}}\rightarrow\phi\phi$, the mass of the $K^{+}\pi^{-}$ pair is
calculated assuming that both hadrons are kaons. Candidates with
$m(K^{+}K^{-})$ within $15{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
$\phi$ mass and $m(K^{+}K^{-}K^{+}K^{-})$ within
$30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the ${{B}^{0}_{s}}$ mass [7]
are rejected. Background from open charm decays is suppressed in a manner
similar to that used for the ${{B}^{0}_{s}}\rightarrow\phi\phi$ candidates.
These vetoes are found to retain 97% of simulated
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ decays.

The combinatorial background is further suppressed using a Boosted Decision
Tree method (BDT) [34, 35]. The BDT is trained to identify four-body hadronic
$b$-hadron decays with high efficiency using independent data samples of such
decays. It uses information on the displacement of the $b$-hadron candidate
from the primary vertex, kinematic information and track isolation criteria.
Although the same BDT is used for the ${{B}^{0}_{s}}\rightarrow\phi\phi$
branching fraction measurement and the search for
${{B}^{0}}\rightarrow\phi\phi$, the method used to optimize the cut on the BDT
output is different. For the branching fraction measurement, the cut
optimization is based on the normalisation mode
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$. The figure of merit used is

$\frac{S_{0}\times\varepsilon_{S}}{\sqrt{S_{0}\times\varepsilon_{S}+N_{\text{bg}}}},$

where $S_{0}$ is the signal yield of
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ candidates in data before any BDT
cut is applied, $\varepsilon_{S}$ is the efficiency of the BDT cut on
simulated ${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ decays, and
$N_{\text{bg}}$ is the number of background candidates surviving the BDT cut
in a suitable upper sideband of the $\phi{{K}^{*}}(892)^{0}$ candidate mass
distribution, scaled to the width of the ${{B}^{0}}$ signal window. Maximizing
this figure of merit results in a rather loose BDT requirement that retains
$98\%$ of signal events while rejecting more than 90% of the background.

For the ${{B}^{0}}\rightarrow\phi\phi$ search, the figure of merit used is

$\frac{\varepsilon^{\prime}_{S}}{a/2+\sqrt{N^{\prime}_{\text{bg}}}},$

with $a$ set to 3, corresponding to the signal significance required to claim
evidence for a new decay mode [36]. Here $\varepsilon^{\prime}_{S}$ is the
efficiency of the BDT cut on simulated ${{B}^{0}_{s}}\rightarrow\phi\phi$
decays, and $N^{\prime}_{\text{bg}}$ is the number of background candidates
surviving the BDT cut in an upper sideband of the $\phi\phi$ candidate mass
distribution, scaled to the width of the ${{B}^{0}_{s}}$ signal window.
Maximizing this figure of merit results in a tighter BDT requirement that
retains $87\%$ of signal events.

##  4 Fits to mass spectra

The yields for the signal and normalization channels are determined from fits
to the invariant mass distributions of the selected candidates. In the
simulation, the ${{B}^{0}_{s}}\rightarrow\phi\phi$ invariant mass distribution
is well modelled by a probability density function (PDF) consisting of the sum
of three Gaussian distributions with a common mean. In the fit to the data,
the relative fractions of the Gaussian components are fixed to the values
obtained from the simulation, whilst the widths are allowed to vary by an
overall resolution scale factor. The yield and common mean are also left free.
After applying all selection requirements, the only remaining background is
combinatorial, which is modelled by a constant. No component for
${{B}^{0}}\rightarrow\phi\phi$ decays is included in this fit. Figure 1 shows
the resulting fit to data, which gives a signal yield of $2309\pm 49$
candidates.

Figure 1: The $K^{+}K^{-}K^{+}K^{-}$ invariant mass distribution. The total
fitted function as described in the text is shown by the (red) solid line, the
${{B}^{0}_{s}}\rightarrow\phi\phi$ component by the (blue) long-dashed line,
and the combinatorial background as the (purple) dotted line.

The ${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ invariant mass distribution
is modelled by a PDF consisting of the sum of a Crystal Ball function [37] and
two Gaussian functions. As for the signal mode, the relative fractions of the
components and the tail parameters are fixed in the fit to the data, whilst
the widths are allowed to vary by an overall resolution scale factor. The
yield and mean are also left free. A component is also included to account for
the small contribution from the decay ${{B}^{0}_{s}}\rightarrow\phi{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}}(892)^{0}$ [38]. The shape
parameters for this component are shared with the ${{B}^{0}}$ component, while
the relative position is fixed to the known mass difference between the
${B}^{0}$ and ${B}^{0}_{s}$ mesons [7]. Combinatorial background is modelled
by an exponential function.

Potential peaking backgrounds, from ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi{p}\pi^{-}$ (with the proton misidentified
as a kaon) or ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\phi{p}K^{-}$ (with
the proton misidentified as a pion), are modelled using a single histogram PDF
generated from simulated events. The relative yield of each decay mode is
weighted according to the expectation from the simulation. The yield of this
component is left to float in the fit. Backgrounds from
${{B}^{0}_{s}}\rightarrow\phi\phi$ and open charm decay modes are negligible
after the vetoes described in Section 3 have been applied. Figure 2 shows the
result of the fit of this model to the
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ dataset after all selection
criteria are applied. The yield of ${{B}^{0}}$ candidates determined by the
fit is $6680\pm 86$.

Figure 2:  The $K^{+}K^{-}K^{+}\pi^{-}$ invariant mass distribution. The total
fitted function is shown by the (red) solid line, the
${{B}^{0}}\rightarrow\phi{{K}^{*}}$ component by the (blue) short-dashed line,
the ${{B}^{0}_{s}}\rightarrow\phi{{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}{}^{*}}(892)^{0}$ component by the
(blue) long-dashed line, the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi{p}\pi^{-}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\phi{p}K^{-}$ contribution by the (green)
dashed-dotted line, and the combinatoral background by the (purple) dotted
line.

##  5 Branching fraction for ${{B}^{0}_{s}}\rightarrow\phi\phi$

The branching fraction of ${{B}^{0}_{s}}\rightarrow\phi\phi$ relative to that
of the ${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ decay mode is determined
using

$\frac{\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)}{\mathcal{B}({{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0})}=\frac{N_{\phi\phi}}{N_{\phi{{K}^{*}}(892)^{0}}}\frac{\varepsilon^{\text{sel}}_{\phi{{K}^{*}}(892)^{0}}}{\varepsilon^{\text{sel}}_{\phi\phi}}\frac{\mathcal{B}({{K}^{*}}(892)^{0}\rightarrow
K^{+}\pi^{-})}{\mathcal{B}(\phi\rightarrow
K^{+}K^{-})}\cdot\frac{1}{f_{s}/f_{d}},$

where the terms $\mathcal{B}$ are the branching fractions of the stated decay
modes, $N$ are the signal yields, $\varepsilon^{\text{sel}}$ are the selection
efficiencies, and the fragmentation fraction ratio, $f_{s}/f_{d}$, is the
ratio of the ${{B}^{0}_{s}}$ to ${{B}^{0}}$ production cross-sections. The
selection efficiencies are determined from simulation, apart from those
related to the particle identification, which are determined in data using
large calibration samples of charged kaons and pions from
${{D}^{*}}^{+}\rightarrow{D}^{0}(\rightarrow K^{-}\pi^{+})\pi^{+}$ decays
[22]. The ratio of efficiencies is found to be
$\varepsilon^{\text{sel}}_{\phi{{K}^{*}}(892)^{0}}/\varepsilon^{\text{sel}}_{\phi\phi}=0.795\pm
0.007$, where the uncertainty is purely statistical. The value of
$f_{s}/f_{d}$ is taken from previous LHCb analyses as $0.259\pm 0.015$ [39,
40, 41].

The signal yields are determined using the mass fits described in Section 4.
These values are corrected for the fraction of candidates where one of the
hadron pairs, $K^{+}K^{-}$ or $K^{+}\pi^{-}$, is produced in a non-resonant
S-wave configuration, rather than as a $\phi$ or ${{K}^{*}}(892)^{0}$. The
S-wave fractions are taken from previous LHCb angular analyses of the
${{B}^{0}_{s}}\rightarrow\phi\phi$ and
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ decay modes. For the
${{B}^{0}_{s}}\rightarrow\phi\phi$ decay mode, we use the measured value of
$2.1\pm 1.6\,\%$ [5] as the S-wave fraction within the $K^{+}K^{-}$ invariant
mass range used for this analysis. Similarly, for the
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ decay mode, we use a measured
value of $26.5\pm 1.8\,\%$ [42] for the S-wave fraction. The uncertainties on
these fractions lead to a $3.1\,\%$ relative uncertainty on the ratio of
branching fractions. This procedure assumes that the efficiencies for the P-
and S-wave components are the same. In the simulation, a $1.1\,\%$ difference
is observed between these efficiencies, and this is assigned as an additional
uncertainty.

Various other uncertainties arise on the measurement of the ratio of branching
fractions. The limited size of the available simulation samples leads to a
relative uncertainty of $0.8\,\%$. The influence of the assumed mass model is
probed by performing the fit with different models for the signal and
background components. This includes quantifying the effect of removing the
peaking background component in the
${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$ fit. The largest variation in the
ratio of branching fractions seen in these studies is $0.6\,\%$, which is
assigned as a relative systematic uncertainty.

The track reconstruction efficiency agrees between data and simulation at the
level of $2.0\,\%$ [43]. This uncertainty largely cancels in the ratio of
branching fractions. A residual relative uncertainty of $0.5\,\%$ remains due
to the fact that the pion in the ${{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0}$
decay mode is relatively soft. An additional relative uncertainty of $0.3\,\%$
is assigned to account for the difference in the hadronic interaction
probabilities for kaons and pions between data and simulation. A further
uncertainty arises from the modelling of the hardware trigger in the
simulation. This is estimated using a data-driven technique and leads to a
relative systematic uncertainty of $1.1\,\%$ on the ratio of branching
fractions. Variations in the procedure used to determine the relative particle
identification efficiency lead to a relative uncertainty of $0.3\,\%$.
Possible systematic effects on the efficiency for
${{B}^{0}_{s}}\rightarrow\phi\phi$ due to the finite width difference in the
${{B}^{0}_{s}}$ system [44] have been checked, and found to be negligible. The
value of $\mathcal{B}(\phi\rightarrow K^{+}K^{-})$ is taken from Ref. [7] and
contributes a relative uncertainty of $1.0\,\%$. The value of
$\mathcal{B}({{K}^{*}}(892)^{0}\rightarrow K^{+}\pi^{-})$ is taken to be $2/3$
exactly. The systematic uncertainties are summarized in Table 2. Summing these
in quadrature gives a relative uncertainty of $3.8\,\%$ on the ratio of
branching fractions. The knowledge of the fragmentation fraction ratio,
$f_{s}/f_{d}$, gives a relative uncertainty of $5.8\,\%$, which is quoted
separately.

Table 2: Summary of the systematic uncertainties on the measurement of the ratio of branching fractions $\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)/\mathcal{B}({{B}^{0}}\rightarrow\phi K^{*})$. Source of systematic uncertainty | Relative uncertainty (%)
---|---
S-wave fraction | 3.1
Relative efficiency between P and S-wave | 1.1
Simulation sample size | 0.8
Fit model | 0.6
Tracking efficiency | 0.5
Hadronic interactions | 0.3
Hardware trigger | 1.1
Particle identification efficiency | 0.3
$\mathcal{B}(\phi\rightarrow K^{+}K^{-})$ | 1.0
Quadratic sum of the above | 3.8
Fragmentation fraction ratio ($f_{s}/f_{d}$) | 5.8

The ratio of branching fractions is found to be

$\frac{\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)}{\mathcal{B}({{B}^{0}}\rightarrow\phi
K^{*})}=1.84\pm 0.05\mathrm{\,(stat)}\pm 0.07\mathrm{\,(syst)}\pm
0.11\,(f_{s}/f_{d}).$

This is converted into an absolute branching fraction using
$\mathcal{B}({{B}^{0}}\rightarrow\phi{{K}^{*}}(892)^{0})=(1.00\pm 0.04\pm
0.05)\times 10^{-5}$, which is obtained by averaging the results in Refs. [45]
and [46] assuming that the uncertainties due to the fragmentation fractions
and S-waves are fully correlated between the two measurements. The resulting
value for the absolute branching fraction is

$\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)=\left(1.84\pm
0.05\mathrm{\,(stat)}\pm 0.07\mathrm{\,(syst)}\pm 0.11\,(f_{s}/f_{d})\pm
0.12\,(\text{norm})\right)\times 10^{-5}.$

##  6 Search for the decay ${{B}^{0}}\rightarrow\phi\phi$

To search for the ${{B}^{0}}\rightarrow\phi\phi$ decay mode, the tight BDT
selection described in Section 3 is used. To fit for a putative
${{B}^{0}}\rightarrow\phi\phi$ signal, the same signal model as for the
${{B}^{0}_{s}}$ signal is used. The mean value of the signal mass is shifted
relative to the ${{B}^{0}_{s}}$ mode by the known ${B}^{0}_{s}$–${B}^{0}$ mass
splitting, and the resolution parameters are kept common between the two
modes. The resulting fit is shown in Figure 3. The data are consistent with
having no ${{B}^{0}}\rightarrow\phi\phi$ contribution. The fitted ${{B}^{0}}$
signal has a yield of $5\pm 6$ events, and the statistical significance is
less than 2 standard deviations, hence an upper limit is placed on the
branching fraction of the decay.

Figure 3:  The $K^{+}K^{-}K^{+}K^{-}$ invariant mass with the tight BDT
selection applied. A fit to the total PDF as described in the text is shown as
a (red) solid line, ${{B}^{0}_{s}}\rightarrow\phi\phi$ as a (blue) long-dashed
line, ${{B}^{0}}\rightarrow\phi\phi$ as a (blue) short-dashed line, and the
combinatorial background as a (purple) dotted line.

To determine this limit, a modified frequentist approach, the
$\text{CL}_{\text{s}}$ method, is used [47]. The method provides
$\text{CL}_{\text{s+b}}$, a measure of the compatibility of the observed
distribution with the signal plus background hypothesis,
$\text{CL}_{\text{b}}$, a measure of the compatibility with the background
only hypothesis, and
$\text{CL}_{\text{s}}=\text{CL}_{\text{s+b}}/\text{CL}_{\text{b}}$. The
expected and observed $\text{CL}_{\text{s}}$ values as a function of the
branching fraction are shown in Figure 4. This gives, at $90\%$ confidence
level, an upper limit of $\mathcal{B}({{B}^{0}}\rightarrow\phi\phi)<2.8\times
10^{-8}$. At $95\%$ confidence level, the upper limit is found to be
$\mathcal{B}({{B}^{0}}\rightarrow\phi\phi)<3.4\times 10^{-8}$.

Figure 4:  Results of the $\text{CL}_{\text{s}}$ scan as a function of the
${{B}^{0}}\rightarrow\phi\phi$ branching fraction ($BF$). The observed
$\text{CL}_{\text{s}}$ distribution is given by the (black) points and solid
line, while the expected distribution is given by the (black) dashed line. The
dark (green) and light (yellow) bands mark the $1\sigma$ and $2\sigma$
confidence regions on the expected $\text{CL}_{\text{s}}$. The upper limits at
90$\,\%$ and 95$\,\%$ confidence level are where the observed
$\text{CL}_{\text{s}}$ line intercepts the (red) solid and dashed horizontal
lines, respectively.

##  7 Summary

The ratio of branching fractions
$\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)/\mathcal{B}({{B}^{0}}\rightarrow\phi
K^{*})$ is determined to be

$\frac{\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)}{\mathcal{B}({{B}^{0}}\rightarrow\phi
K^{*})}=1.84\pm 0.05\mathrm{\,(stat)}\pm 0.07\mathrm{\,(syst)}\pm
0.11\,(f_{s}/f_{d}),$

where the first uncertainty is statistical, the second is systematic, and the
third is due to the ratio of fragmentation fractions. The absolute branching
fraction for ${{B}^{0}_{s}}\rightarrow\phi\phi$ is determined to be

$\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)=\left(1.84\pm
0.05\mathrm{\,(stat)}\pm 0.07\mathrm{\,(syst)}\pm 0.11\,(f_{s}/f_{d})\pm
0.12\,(\text{norm})\right)\times 10^{-5}.$

This is in agreement with, but more precise than, the measurement made by the
CDF collaboration,
${\mathcal{B}({{B}^{0}_{s}}\rightarrow\phi\phi)=\left(1.91\pm 0.26\pm
0.16\right)\times 10^{-5}}$. It is also in agreement with theory predictions
[8, 9, 10, 11].

A search for the decay ${{B}^{0}}\rightarrow\phi\phi$ is also made. No
significant signal is seen, and an upper limit of

$\mathcal{B}({{B}^{0}}\rightarrow\phi\phi)<2.8\times 10^{-8}$

is set at $90\%$ confidence level. This is more stringent than the previous
limit of ${\mathcal{B}({{B}^{0}}\rightarrow\phi\phi)<2.0\times 10^{-7}}$, set
by BaBar [17], and provides a strong constraint on possible contributions to
this mode from physics beyond the Standard Model [16].

## 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); 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. Aaij38, 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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Vecchi16, J.J. Velthuis46, M. Veltri17,g, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,o, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß63, J.A. de
Vries41, R. Waldi63, C. Wallace48, R. Wallace12, J. Walsh23, S. Wandernoth11,
J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, A. Weiden40, M.
Whitehead48, G. Wilkinson55,38, M. Wilkinson59, M. Williams38, M.P.
Williams45, M. Williams56, T. Williams45, F.F. Wilson49, J. Wimberley58, J.
Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A. Wotton47, S. Wright47,
K. Wyllie38, Y. Xie61, Z. Xu39, Z. Yang3, J. Yu61, X. Yuan34, O. Yushchenko35,
M. Zangoli14, M. Zavertyaev10,b, L. Zhang3, Y. Zhang3, A. Zhelezov11, A.
Zhokhov31, L. Zhong3, S. Zucchelli14.

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
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
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Yandex School of Data Analysis, Moscow, Russia, associated to 31
66Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
67Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41

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 Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
†Deceased