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# Electroweak Penguin Decays of $b$-Flavoured Hadrons

Ulrik Egede  1 Shohei Nishida  2,3 Mitesh Patel  4 Marie-Hélène Schune5
1School of Physics & Astronomy, Monash University, Melbourne, Australia;
email<EMAIL_ADDRESS>2 The Graduate University for Advanced Studies
(SOKENDAI), Hayama, Japan  3 High Energy Accelerator Research Organization
(KEK), Tsukuba, Japan<EMAIL_ADDRESS>4Department of Physics,
Imperial College London, London, United Kingdom; email:
<EMAIL_ADDRESS>5 Université Paris-Saclay, CNRS/IN2P3, IJCLab,
Orsay, France<EMAIL_ADDRESS>
###### Abstract

In the past decade, electroweak penguin decays have provided a number of
precision measurements, turning into one of the most competitive ways to
search for New Physics that describe beyond the Standard Model phenomena. An
overview of the measurements made at the $B$ factories and hadron colliders
are given and the experimental methods are presented. Experimental
measurements required to provide further insight into present indications of
New Physics are discussed.

###### keywords:

Belle II, LHCb, $B$-factory, Heavy Flavour, New Physics

###### Contents

  1. 1 INTRODUCTION
    1. 1.1 Discovery modes for New Physics
    2. 1.2 Theoretical framework
    3. 1.3 Combinations of multiple measurements
  2. 2 EXPERIMENTAL CHALLENGE
  3. 3 BRANCHING FRACTION MEASUREMENTS
    1. 3.1 Leptonic decays
    2. 3.2 Radiative decays
    3. 3.3 Semileptonic decays
  4. 4 ANGULAR DISTRIBUTIONS
  5. 5 $C\\!P$ VIOLATION
    1. 5.1 Direct $C\\!P$ violation
    2. 5.2 Time-dependent $C\\!P$ violation and Polarization in Radiative $B$ decays
  6. 6 LEPTON FLAVOUR UNIVERSALITY AND LEPTON FLAVOUR VIOLATION
  7. 7 DISCUSSION
  8. ACKNOWLEDGMENTS

##  1 INTRODUCTION

The most common decays of $b$-hadrons take place at the quark level through
the decay of the $b$ quark via the emission of a virtual $W$ boson. The
${b}\\!\rightarrow{c}{W}$ process mediates decays like
${{{B}^{-}}}\\!\rightarrow{{D}^{0}}{{\pi}^{-}}$, and the
${b}\\!\rightarrow{u}{W}$ process mediates decays like ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\pi}^{+}}\ell^{-}{\overline{\nu}}$.
The $b$ quark is not able to decay directly to an $s$ quark as it would
require a vertex with a neutral vector boson and a change of flavour. Such a
_Flavour Changing Neutral Current_ (FCNC) process is forbidden in the Standard
Model (SM) at tree level. However, at one-loop level the FCNC quark-level
process like ${b}\\!\rightarrow{s}\gamma$ is allowed in the SM as illustrated
for the decays ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\gamma$ and
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ in Fig. 1.
Collectively, decays of this type with a hard photon or a lepton pair in the
final state are known as _electroweak penguin decays_. In the SM, the GIM
mechanism [1] would, in the limit where the quarks inside the loop have zero
mass compared to the $W$ boson mass, result in a cancellation of the decay
amplitude to all orders. However, the large mass of the top quark means this
is not the case for $b$-hadron decays. On the other hand, electroweak penguin
decays of $c$-hadrons are very heavily suppressed in the SM. [] FCNCA Flavour
Changing Neural Current (FCNC) is a process where the quark involved changes
flavour but keeps the same charge. An example is the
${b}\\!\rightarrow{s}\gamma$ process. At tree level the process is forbidden
in the SM. [] Penguin decayThe class of FCNC decays that are mediated by one
loop Feynman diagrams. The name is related to that the diagrams can be drawn
in the shape of penguins [2].

Figure 1: Example Feynman diagrams for the flavour changing neutral current
(FCNC) decays (left) ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\gamma$ and (right)
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ in the Standard Model.

###  1.1 Discovery modes for New Physics

Electroweak penguin decays can act as a discovery mode for New Physics (NP).
At either tree level or at loop level, there can be particles such as new
vector bosons or leptoquarks that mediate the decays. The influence of these
particles can be observed as differences with respect to the SM predictions
for these decays. While this statement is in principle true for the decay of
any hadron, the study of electroweak penguin decays are very well suited as a
discovery mode as: the SM amplitudes are suppressed due to the requirement of
a loop-level process, thus making any NP effect more visible; the theoretical
calculation of the final-state properties has lower uncertainties than in
fully-hadronic decays; and final states with leptons or hard photons are
relative easy to identify in particle physics detectors.

###  1.2 Theoretical framework

The theoretical calculation of the electroweak penguin decays uses an Operator
Product Expansion (OPE) [3]. The principle is the same as for the Fermi-theory
of weak decays. It takes advantage of the fact that the decays are only
sensitive to the spin, parity and $C\\!P$ properties of the couplings
involving particles at masses well above the $b$-hadron masses. For processes
like ${b}\\!\rightarrow{s}\gamma$, ${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ and
${b}\\!\rightarrow{s}{\nu}{\overline{\nu}}$, it is possible within the OPE to
write

$\mathcal{H}_{\rm{eff}}=\frac{4G_{F}}{\sqrt{2}}{V_{{t}{b}}^{\phantom{\ast}}}{V_{{t}{s}}^{\ast}}\sum_{i=7,9,10,\nu}{\left[C_{i}\mathcal{O}_{i}+C^{\prime}_{i}\mathcal{O}^{\prime}_{i}\right]}\;,$
(1)

where the operators $\mathcal{O}_{i}$ encode the low energy behaviour and the
complex valued Wilson coefficients $C_{i}$ characterise how these different
operators contribute to the overall processes. The electromagnetic operator is
$\mathcal{O}_{7}$, the semileptonic operators $\mathcal{O}_{9}$ and
$\mathcal{O}_{10}$ correspond to vector and axial-vector currents, and
$\mathcal{O}_{\nu}$ is the operator corresponding to
${b}\\!\rightarrow{s}{\nu}{\overline{\nu}}$ processes. The QCD operators
$\mathcal{O}_{\text{1--6,8}}$ mix through renormalisation with the operators
$\mathcal{O}_{7,9}$, leading to measurements that are sensitive to the
effective Wilson coefficients $C^{\rm{eff}}_{7,9}$. The primed operators
correspond to right-handed currents and are suppressed by a factor
$m_{s}/m_{b}$ in the SM. Any NP will manifest itself through Wilson
coefficients that have different values from those expected in the SM or
through Wilson coefficients that correspond to completely new operators such
as scalar, pseudoscalar or tensor currents. In the SM, the coupling to all
leptons is the same and the validity of this lepton flavour universality can
be probed by testing whether the Wilson coefficients have the same values for
different flavours of lepton pairs in the final state. Contributions to the
Wilson coefficients from NP depend on both the coupling constants between the
NP particles and the SM particles and the masses of the NP particles. This
means that the study of electroweak penguin decays cannot determine precisely
the mass of any NP particles.

The branching fraction of a specific decay and the angular distribution of the
decay products depends not only on the physics at the high energy scale
described by the effective Hamiltonian. The dominant effect of the hadrons in
the final state are described through $q^{2}$–dependent form factors, where
$q^{2}$ is the mass squared of the lepton pair. These describe low-energy QCD
effects and as such cannot be calculated using perturbative methods. Rather,
light cone sum rule calculations are used at low $q^{2}$ and lattice QCD at
high $q^{2}$. The uncertainties on the form factors are significant and
measurements can be divided up into kinematic regions, that have different
dependence on non-perturbative effects. [] $q^{2}$The _q-squared_ of an
electroweak penguin decay is the mass squared of the lepton pair in the final
state.

For any final state involving hadrons and a pair of identical charged leptons,
there will also be amplitudes from ${b}\rightarrow{c}{\overline{c}}{s}$ where
the ${c}{\overline{c}}$ pair subsequently decays electromagnetically to a pair
of leptons. The process will have both non-resonant components and resonant
components such as the decay
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}$
followed by
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. Despite
having the same final state, these processes are not FCNC and the amplitudes
have to be treated as nuisance effects in searches for NP contributions. The
different $q^{2}$ regions are illustrated in Fig. 2: At very low $q^{2}$ the
decay is dominated by virtual photons coupling to the lepton pair, at low
$q^{2}$ the penguin amplitude dominates, at intermediate $q^{2}$ the narrow
${J\mskip-3.0mu/\mskip-2.0mu\psi}$ and $\psi(2S)$ resonances dominate; and at
high $q^{2}$ the penguin amplitude dominates again but with significant
interference from the broad $\psi$ resonances. The regions of the $q^{2}$
spectrum with the most precise SM predictions are the low region
${q^{2}}<1\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ and the high region
${q^{2}}>15\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$.

Figure 2: A sketch of the differential width vs. $q^{2}$ of the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$. At low
$q^{2}$, the effect of diagrams with a virtual photon is clearly seen, while
at ${q^{2}}\approx 9.6\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ and
${q^{2}}\approx 13.6\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ the amplitude of
the charmonium ${c}{\overline{c}}$ states completely dominate. In the region
above the $\psi{(2S)}$ there is significant interference between broad $\psi$
states and the FCNC amplitude.

###  1.3 Combinations of multiple measurements

A measurement of a single electroweak penguin decay is not able to provide a
comprehensive picture of any influence of NP. Decays are affected differently
by NP and are affected in very different ways by the theoretical uncertainties
arising from the form factor calculations and the effect of the $c$
$\overline{c}$ intermediate states.

As an example, while the ${b}\rightarrow{s}$ processes are the dominant
electroweak penguin processes of $b$-hadron decays, ${b}\rightarrow{d}$
processes are also important. In the SM, they are Cabibbo suppressed with
respect to the ${b}\rightarrow{s}$ processes. However, it is only in what is
called Minimal Flavour Violation models [4] that the same Cabibbo suppression
will exist for NP amplitudes. It is thus important to study the rarer
${b}\rightarrow{d}$ processes as they may be able to give insights to how any
NP particles interact with the three generations of quarks. Another example is
the study of the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
which has a very small uncertainty in the SM prediction of its branching
fraction but on the other hand gives information only on information on the
scalar, pseudoscalar and axial vector currents.

Figure 3 shows an overview of different electroweak penguin measurements. Each
type of measurement at a given experiment has been given a subjective rating
in terms of how easy it is to perform and what the associated theoretical
accuracy is for the SM predictions. The markers in the figure indicate the
amount of information that can be gained from the measurement. As an example,
the measurement of the ratio of branching fractions between ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
is experimentally very challenging due to the low branching fraction of
${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
but has a very accurate SM prediction. It will thus sit in the top left of the
diagram. A deviation from the SM prediction will provide a single number on
the coupling structure of NP to different generations and will thus give an
intermediate amount of information. Conversely, the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ angular
analysis is much easier experimentally, has larger theoretical uncertainties
and provides a large number of observables. It thus has a large marker towards
the bottom right of the diagram. [] ObservableThe expression observable is
used for a property that can measured about a given decay. This can be the
branching fraction, a lifetime or a measurement related to the angular
distribution of the decay products. While a two-body decay will only have a
few observables, there can be many for a multibody decay.

Figure 3: An overview of how different measurements present different
experimental challenges and have different accuracy in their SM predictions.
The marker size indicates the amount of information that the measurement will
provide.

The remainder of this review is divided up into sections that represent the
different experimental challenges in measuring the properties of all these
decays. Rather than providing a comprehensive overview of all measurements
made, examples are given that illustrate different experimental methods. For
individual experimental measurements as well as averages over them, the Heavy
Flavor Averaging Group [5] and the Particle Data Group [6] offer the most
comprehensive information.

##  2 EXPERIMENTAL CHALLENGE

All ground states of $b$ hadrons are characterised by relatively long
lifetimes of order of $10^{-12}$ seconds and a large number of accessible
final states. Lifetimes are long as the decays are governed by the weak force
and the applicable CKM matrix elements $V_{{c}{b}}^{\phantom{\ast}}$ and
$V_{{u}{b}}^{\phantom{\ast}}$ both have a small magnitude. The lifetimes are
key-parameters to access physics information (neutral $B$-meson mixing
frequencies and CP violation parameters) but also to reject backgrounds. As
FCNC decays have branching fractions of $\sim 10^{-5}$ or below, very large
datasets have to be recorded to obtain significant signal samples. Another
point of paramount importance is the ability to perform particle
identification. Electron, muon and photon identification is essential for
identifying the decays, while the capability to separate pions, kaons and
protons is essential to reduce cross feed between different electroweak
penguin decays.

The experimental environments for the experiments that are currently active in
the measurement of electroweak penguin decays of $b$ hadrons are very
different: the Belle II experiment at the KEK $B$-factory and the LHCb
experiment running at the LHC proton-proton collider. Event displays, clearly
exhibiting the differences, are shown in Fig. 4 for the two experiments. []
$B$-factoryAn $e^{+}e^{-}$ collider running at a centre-of-mass energy
corresponding to the mass of the $\mathchar 28935\relax{(4S)}$ resonance. The
$B$-factory name is due to the value of $\mathchar 28935\relax{(4S)}$
resonance cross section ($\sim 1.1\text{\,nb}$) and to the fact it decays
uniquely into a pair of $B$-mesons.

Figure 4:  Event displays of (top) Belle II and (bottom) LHCb experiments. The
very different experimental configurations in terms of solid angle and
occupancy in the event are very visible. At Belle II the $e^{+}e^{-}$
collisions occur in the centre of the detector, whereas for LHCb, the proton-
proton collisions occur to the far left.

At $B$-factories, the $\mathchar 28935\relax{(4S)}$ resonance produced in the
$e^{+}e^{-}$ collisions decays into either a ${{{B}^{+}}}-{{{B}^{-}}}$ pair or
a ${{B}^{0}}-{{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ pair, and
thus represents a copious source of charged and neutral $B$ mesons in a low-
background environment. The beam energies at a $B$ factory are not the same
for the $e^{+}$ and the $e^{-}$ beams meaning that, even if the $B$ mesons are
produced almost at rest in the $\mathchar 28935\relax{(4S)}$ rest frame, they
will fly along the beam axis before they decay and the flight distance can be
used to measure their decay time. Since the boost is not very large, the
detector is only slightly asymmetric and covers nearly the full solid angle.
[] $\mathchar 28935\relax{(4S)}$ resonanceThe $\Upsilon$ system is the bound
state of a $b$ and a $\overline{b}$ quarks in a $J^{PC}=1^{--}$ state. Various
excited states (resonances) exists. The fourth one, called $\mathchar
28935\relax{(4S)}$, has just sufficient energy to create a
${{{B}^{+}}}-{{{B}^{-}}}$ or a ${{B}^{0}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ pair.

On the contrary, at the LHC the $b$ and $\overline{b}$ quarks are
predominantly produced colinearly, close to the beam direction. This
configuration, due the fact that the main mechanism for ${b}{\overline{b}}$
production is gluon-gluon fusion, also produces uncorrelated $b$-hadrons
species. This geometrical characteristic has strongly influenced the design of
the LHCb detector, which is a single-arm forward spectrometer with a polar-
angle coverage between 10 and 300 mrad in the horizontal plane and 250 mrad in
the vertical one.

For the different experiments, the current size of the events samples as well
as the $b$-hadron species available are summarised in Tab. 1. At the beginning
of the next decade, the LHCb experiment is planning to have recorded about
50$\text{\,fb}^{-1}$ and the Belle II experiment about 50$\text{\,ab}^{-1}$.
The LHCb experiment is proposing a further upgrade that will enable the
collection of 300$\text{\,fb}^{-1}$ by the end of the High-Luminosity LHC
period [7].

Table 1: Principal characteristics of the data samples accumulated by the end
of 2021 at the $B$-factory experiments (BaBar (1999-2008), Belle (1999-2010)
and Belle II (2019-) at the $\mathchar 28935\relax{(4S)}$ resonance and the
LHCb (2009-) experiment in proton-proton collisions.

Experiment | Integrated |  $\boldmath{{{b}{\overline{b}}}}$ cross section | Hadronic | Main $b$-hadron species
---|---|---|---|---
| luminosity |  | background | species produced
BaBar | $433\text{\,fb}^{-1}$ | 1.1 nb | 3.7 nb |  ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ and ${{B}^{-}}$
Belle | $711\text{\,fb}^{-1}$ | 1.1 nb | 3.7 nb |  ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ and ${{B}^{-}}$
Belle II | $195\text{\,fb}^{-1}$ | 1.1 nb | 3.7 nb |  ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ and ${{B}^{-}}$
LHCb | $\phantom{00}9\text{\,fb}^{-1}$ |  $140\ \mu$b | 60 mb |  ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$, ${{B}^{-}}$, ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$, $\Lambda_{b}$ and ${B}_{c}^{-}$
ATLAS a | $173\text{\,fb}^{-1}$ |  $140\ \mu$b | 60 mb |  ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$, ${{B}^{-}}$, ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$, $\Lambda_{b}$ and ${B}_{c}^{-}$
CMS a | $178\text{\,fb}^{-1}$ |  $140\ \mu$b | 60 mb |  ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$, ${{B}^{-}}$, ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$, $\Lambda_{b}$ and ${B}_{c}^{-}$

  * a

Only a small fraction of the trigger bandwidth in ATLAS and CMS is dedicated
to $b$ hadron physics.

With the $b$-hadrons having very different production mechanisms at $B$
factories and the LHC, the background levels are very different. At
$B$-factories, the non-${b}{\overline{b}}$ background to signal ratio is of
the order of four, while it is several hundred for LHCb. Consequently, while
the trigger for $b$-hadron events is fully efficient at $B$-factories, it is
significantly lower at LHCb, despite the usage of a complex trigger system
consisting of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies full event
reconstruction. Since the occupancy of the calorimeters is significantly
higher than that of the muon stations, the constraints on the trigger rate
require that higher thresholds are imposed on the electron transverse energy
than on the muon transverse momentum. The most noticeable effect on analyses
testing Lepton Universality described in this review is that while at
$B$-factories the yields of ${b}\\!\rightarrow{s}\,\mu^{+}\mu^{-}$ and
${b}\\!\rightarrow{s}\,e^{+}e^{-}$ transitions are similar, a factor of 4 to 5
in favour of the muonic modes is observed at LHCb. From 2022 with the upgrade
of the LHCb experiment, and in particular the removal of the hardware trigger,
accessing higher yields for the modes with electrons may be possible.

In order to distinguish signal from background events, both the Belle II and
LHCb experiments exploit the reconstruction of the $b$-hadron invariant mass
from the measured decay products. At Belle II, the background is reduced
thanks to the constraints provided by the $\mathchar 28935\relax{(4S)}$
resonance. Since it decays uniquely into a pair of $B$ mesons, the energy of
the $B$-decay products is equal to half of the centre-of mass energy. This
enables two weakly correlated discriminating variables to be defined: one
which compares the reconstructed $B$-meson energy to the beam energy and one
which corresponds to the $B$-meson mass reconstructed from the measured
momenta of the decay products and the beam energy. While the proton-proton
collision environment for LHCb doesn’t provide a beam energy constraint, the
boost of the $b$-hadrons results in decay lengths of the order of 1 cm. The
reconstruction of secondary vertices from the $b$-hadron decay leads to a very
large rejection of tracks originating from the primary collision vertex. []
Secondary vertexThe point where the $b$-hadron decays at a hadron collider.
The event will also have one or more primary vertices where the proton-proton
collision happens.

Another experimental difference, of particular importance for this review, is
the difference in the amount of bremsstrahlung emission due to the difference
in energies and detector designs. Dedicated recovery procedures are in place
in order to improve momentum reconstruction. Despite these corrections, the
$B$ mass resolution is still degraded for final states involving electrons
compared to final states including muons. While the effect is subtle at
$B$-factories, it is quite significant for LHCb as can be seen from Fig. 5.

Figure 5: Top: distribution of the four-body invariant mass in
${{B}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow\ell^{+}\ell^{-}){{K}^{*0}}(\rightarrow{{K}^{+}}{{\pi}^{-}})$
decays for (left) $\ell=\mu$ and (right) $\ell=e$ reconstructed at LHCb. From
Ref. [8]. Bottom: distribution of the dilepton invariant mass in the
${J\mskip-3.0mu/\mskip-2.0mu\psi}$ region for (left) $\ell=\mu$ and (right)
$\ell=e$ reconstructed at Belle. From Ref. [9].

In order to reduce the systematic uncertainties at the lowest possible level,
control modes are of prime importance for many aspects of the analyses
described in this review. Such modes are used in particular to correct the
simulation to allow for the extraction of the efficiencies of the signal
selection with respect to particle identification, track reconstruction,
trigger and $b$-hadron kinematics. Since simulating precisely the $b$-hadron
production characteristics as well as the subsequent detector response in the
busy hadronic environment of the LHC is challenging, the use of control modes
is a key feature of many of the LHCb analyses presented here.

In addition to the $B$-factories and the LHCb experiment, the ATLAS and CMS
experiments at the LHC have also performed analyses of electroweak penguin
decays. However, the limited trigger bandwidth devoted to $b$ hadron decays
and the lack of hadron particle identification means that competitive
measurements have only been made in final states with a pair of muons.

Most of the analyses presented in this review use Machine Learning techniques
to distinguish signal from background, to ensure the largest possible signal
yields. Since the background rejection is already a key issue at the trigger
level for LHCb, such techniques are also implemented at the software trigger
level [10].

##  3 BRANCHING FRACTION MEASUREMENTS

The measurement of a branching fraction that is integrated over the complete
phase space of the final state is conceptually an easy measurement. The final
state can be either exclusive (e.g. ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$),
a sum over exclusive states (e.g. ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\nu}{\overline{\nu}}$, where all
neutrino types are included) or inclusive (e.g. ${b}\\!\rightarrow{s}\gamma$,
where there is a sum over all final states containing net strangeness.). []
Exclusive vs. inclusiveAn exclusive measurement is with a specific initial and
final state, like ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ while an
inclusive measurement sums over all decays included in a given quark level
transition like $b\\!\rightarrow s{\mu^{+}}{\mu^{-}}$.

At hadron colliders, branching fraction measurements are performed by making a
measurement relative to a another branching fraction that has been measured in
the past. These type of normalisation measurements are performed at
$B$-factories where the total number of $\mathchar 28935\relax{(4S)}$ produced
can be determined without looking at specific decay modes. For ${B}^{0}$ and
${{B}^{+}}$ decays, this type of normalisation works well, but for
${B}^{0}_{s}$, ${\mathchar 28931\relax}^{0}_{b}$ and ${B}_{c}^{+}$ decays this
works less well. As these particles are not produced in $\mathchar
28935\relax{(4S)}$ decays there are no branching fractions that can act as
normalisation modes. Instead ${B}^{0}$ or ${{B}^{+}}$ modes are used, which is
turn requires that the relative production fractions of e.g. ${B}^{0}_{s}$ and
${{B}^{+}}$ is known. These production fractions are measured by comparing the
rate of specific ${B}^{0}_{s}$ and ${{B}^{+}}$ decays that can be related
through SU(3) [11]. For several branching fraction measurements of electroweak
penguin decays, the uncertainty on the overall normalisation procedure is a
limiting systematic.

###  3.1 Leptonic decays

The SM prediction for the branching fraction of the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
decay is $(3.66\pm 0.14)\times 10^{-9}$ and for the Cabibbo suppressed
${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
decay $(1.03\pm 0.05)\times 10^{-10}$ [12, 13]. As the initial state is a spin
zero pseudoscalar and the final states contain only the two leptons, the left-
handed nature of the weak interaction forces one of the muons to be in the
wrong helicity state making the decay helicity suppressed. On the other hand,
with no hadrons in the final state, the QCD uncertainty is small. For SM
predictions this results in tiny branching fractions with a very small
relative uncertainty. Gaining information from these decays is thus completely
limited by experimental issues.

With just the two muons in the final state, the signature of the decays is
easy to trigger on and to reconstruct in a hadronic environment, allowing
measurements to be made by the three LHC experiments ATLAS [14], CMS [15] and
LHCb [16]. The trigger selects initially either one or two muons with a large
transverse momentum with respect to the beam axis. In the subsequent
selection, the signature is two muons with a vertex displaced from the primary
proton-proton interaction vertices. Combined with the very low
misidentification of pions as muons, the final state has very low background
and the main challenges are to have a low trigger threshold and to collect a
large amount of integrated luminosity to see the rare decays. The observation
of the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
decay was made in a combined analysis by the CMS and LHCb collaborations [17]
and the current most precise measurement is $\mathcal{B}({{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}})=(3.09{\,}^{+\,0.46}_{-\,0.43}{\,}^{+\,0.15}_{-\,0.13})\times
10^{-9}$ [16]. Here the first uncertainty is the statistical uncertainty,
while the second is the systematic. The systematic uncertainty is dominated by
the normalisation of the branching fraction measurement, that is performed
relative to the
${{{B}^{-}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{-}}$ decay.
There is still only a limit on the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
decay but if the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
decay has the SM branching fraction, an observation is expected during Run 3
of the LHC (2022-2025) while the decay will remain out of reach for Belle II
for the foreseeable future due to the extremely low branching fraction. As can
be seen from Fig. 6, current measurements are in good agreement with the
Standard Model.

Figure 6: Contour lines for the confidence limits on the branching fractions
of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
versus ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$.
From Ref. [18].

While it might appear that the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\nu}{\overline{\nu}}$
should be able to proceed through a box diagram with two $W$ bosons, it is in
fact forbidden in the SM as it would require a right-handed neutrino or left-
handed antineutrino. Searching for this decay therefore turns into a search
for the $B$ meson to decay into anything invisible.

###  3.2 Radiative decays

Radiative decays of $b$-flavour hadrons proceed through
${b}\\!\rightarrow{s}{\gamma}$ or ${b}\\!\rightarrow{d}{\gamma}$ penguin
diagram. In 1993, CLEO reported the first observation of the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*}}\gamma$ decay [19], which was the
first experimental confirmation of a penguin decay. The branching fraction of
${{\kern 1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*}}\gamma$ has subsequently been
measured by other experiments, and the current world average gives
$\mathcal{B}({{\kern 1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*}}\gamma)=(4.18\pm 0.25)\times
10^{-5}$ [6]. Many other exclusive ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}$ decays such as ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptK}}_{1}(1270)\gamma$, ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptK}}_{2}^{*}(1270)\gamma$, ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptK}}\eta\gamma$ have also been measured.
Radiative decays of ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}$
mesons have been studied with LHCb as well as Belle data taken at the
${\mathchar 28935\relax{(5S)}}$ resonance, and the branching fractions have
been measured. The radiative $b$-baryon decay
$\Lambda_{b}\rightarrow\Lambda\gamma$ has been observed by LHCb [20].

[] $\mathchar 28935\relax{(5S)}$ resonanceThe fifth $\Upsilon$ resonance
${\mathchar 28935\relax{(5S)}}$ (or $\Upsilon(10860)$) can decay to
${{B}^{0}_{s}}^{(*)}{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}^{(*)}$. Belle took
121$\text{\,fb}^{-1}$ of data mainly for the study of ${B}^{0}_{s}$ decays.

Measurement of the branching fractions of the exclusive decays is
experimentally straightforward. On the other hand, the prediction of the
exclusive branching fractions suffers from large uncertainties due to
uncertainties in the hadronic form factors, and the comparison with
experimental results does not provide much information. In terms of the
theoretical calculation, the present SM prediction with NNLO calculation is
$\mathcal{B}({b}\\!\rightarrow{s}{\gamma})=(3.31\pm 0.23)\times 10^{-4}$ [21]
for $E_{\gamma}>1.6~{}\text{\,Ge\kern-1.00006ptV}$ [21]. This is much more
precise than the prediction for exclusive processes, and therefore one can
compare it with the experimental measurements in order to search for or
constrain NP.

Measurement of the inclusive ${b}\\!\rightarrow{s}{\gamma}$ (denoted by
$B\rightarrow X_{s}\gamma$, where $X_{s}$ is a hadronic system including $s$
quark) branching fraction is challenging and is only possible in a clean
environment of $B$ factories. One approach is to measure the $\gamma$ spectrum
in $e^{+}e^{-}\rightarrow\Upsilon(4S)\rightarrow B{{\kern
1.79993pt\overline{\kern-1.79993ptB}}}$. The raw photon spectrum includes a
huge background from photons originating from the continuum
$e^{+}e^{-}\rightarrow{{q}{\overline{q}}}$, and this contribution is
subtracted using data taken at a collision energy slightly below that at which
the $\Upsilon(4S)$ is produced. There still exists large background from $B$
decays that do not come from ${b}\\!\rightarrow{s}{\gamma}$, and this
contribution is estimated by simulation. A high momentum lepton which arises
from semi-leptonic decays of the other $B$ meson (lepton tag) is often
required to suppress the continuum background.

Figure 7 shows the photon energy spectrum in $B\rightarrow X_{s}\gamma$
obtained by Belle with $605~{}\text{\,fb}^{-1}$ data [22]. Strictly speaking,
this includes the contribution of ${b}\\!\rightarrow{d}{\gamma}$ described
later. As seen in the figure, the photon spectrum peaks around
$2~{}\text{\,Ge\kern-1.00006ptV}$. The measurement of energy bins below
$2.0~{}\text{\,Ge\kern-1.00006ptV}$ is difficult because of a small signal and
a large increase of background from other $B$ decays. For this reason, the
inclusive ${b}\\!\rightarrow{s}{\gamma}$ branching fraction is quoted with a
certain minimum energy. The photon energy spectrum includes information of the
mass and dynamics of the $b$ quarks in the $B$ meson, which are useful in the
estimation of $|{V_{{u}{b}}^{\phantom{\ast}}}|$ and
$|{V_{{c}{b}}^{\phantom{\ast}}}|$ from the measurements of semi-leptonic $B$
decays [23].

Figure 7: ${b}\\!\rightarrow{s}{\gamma}$ spectrum, without the correction of
the signal acceptance, obtained by Belle [22]. The lepton tag is not applied
to obtain this distribution.

It is also possible to suppress the background further by fully reconstructing
one of the $B$ mesons from $e^{+}e^{-}\rightarrow{\mathchar
28935\relax{(4S)}}\rightarrow{B}{{\kern
1.79993pt\overline{\kern-1.79993ptB}}}$ and then measure the photon spectrum
[24]. This full reconstruction method gives a signal with much higher purity
at the cost of reduced statistics due to an low efficiency of less than $1\%$.
This is a promising method for the future when Belle II accumulates an order
of magnitude higher data sample.

Another approach in the measurement of ${B}\rightarrow X_{s}\gamma$ is to
reconstruct the $X_{s}$ part from many final states such as $K\pi$ and
$K\pi\pi$. This method has higher purity than the normal inclusive analysis
and has the advantage that it is possible to distinguish between $B$ and
${\kern 1.79993pt\overline{\kern-1.79993ptB}}$, and to separate $X_{d}$ from
$X_{s}$. However, one needs to rely on simulation for the hadronisation of
$X_{s}$, and the contribution of non-reconstructed $X_{s}$ states gives rise
to large uncertainty.

The present average of the measurements is
$\mathcal{B}({b}\\!\rightarrow{s}{\gamma})=(3.27\pm 0.15)\times 10^{-4}$ at
$E_{\gamma}>1.6~{}\text{\,Ge\kern-1.00006ptV}$, which agrees well with the SM
prediction. This measurement sets a strong constraint on the Wilson
coefficient $C_{7}$.

The ${b}\\!\rightarrow{d}{\gamma}$ process is suppressed relative to
${b}\\!\rightarrow{s}{\gamma}$ by a factor of
$(|{V_{{t}{d}}^{\phantom{\ast}}}|/|{V_{{t}{s}}^{\phantom{\ast}}}|)^{2}\sim
0.04$, and the branching fractions of the exclusive decays are below
$10^{-6}$. Therefore, the analysis of ${b}\\!\rightarrow{d}{\gamma}$ needs to
cope with larger backgrounds, compared to the ${b}\\!\rightarrow{s}{\gamma}$
analysis. In addition, the ${b}\\!\rightarrow{s}{\gamma}$ process itself
becomes a background. For example, the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow\rho^{0}\gamma,\rho^{0}\rightarrow\pi^{+}\pi^{-}$
signals can be easily contaminated by ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\gamma$ with ${{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
when a kaon is misidentified as a pion. Good particle identification helps to
separate the signal from this type of background.

The measurement of the ratio of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow\rho\gamma$ and
$\omega\gamma$ branching fractions to that for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*}}\gamma$ allows for the
determination of
$|{V_{{t}{d}}^{\phantom{\ast}}}|/|{V_{{t}{s}}^{\phantom{\ast}}}|$ with some
residual uncertainty due to the ratio of their respective form factors. With
the first observation of these decays by Belle [25],
$|{V_{{t}{d}}^{\phantom{\ast}}}|/|{V_{{t}{s}}^{\phantom{\ast}}}|$ was measured
for the first time to be $0.199^{+0.026}_{-0.025}{}^{+0.018}_{-0.015}$, where
the first and second errors are the experimental and theoretical errors,
respectively. After the observation of ${B}^{0}_{s}$ oscillations by CDF [26],
$|{V_{{t}{d}}^{\phantom{\ast}}}|/|{V_{{t}{s}}^{\phantom{\ast}}}|$ is measured
with much higher precision from those, and ${b}\\!\rightarrow{d}{\gamma}$ is
not used in the determination of
$|{V_{{t}{d}}^{\phantom{\ast}}}|/|{V_{{t}{s}}^{\phantom{\ast}}}|$. However, it
is still important to measure
$|{V_{{t}{s}}^{\phantom{\ast}}}|/|{V_{{t}{d}}^{\phantom{\ast}}}|$ with the
${b}\\!\rightarrow{d}{\gamma}$ process since NP may contribute in a different
way to the different diagrams.

###  3.3 Semileptonic decays

Semileptonic electroweak penguin decays are decays such as
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$, as illustrated in Fig.
1(right), and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\nu}{\overline{\nu}}$. They
should not be confused with the more common semileptonic decays such as
${{{B}^{-}}}\\!\rightarrow{{D}^{*+}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$, which
are dominated by tree level diagrams.

For decays of the type ${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$, measurements of
the branching fraction are performed in bins of $q^{2}$, the mass squared of
the lepton pair. An example of such a measurement can be seen for the decay
${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}}{\mu^{-}}$
[27] in Fig. 8. When looking at measurements with non-uniform binning in
$q^{2}$, care should be taken to understand if measurements are given as the
branching fraction within the bin, or given as a differential branching
fraction $\frac{d{B}}{d{q^{2}}}$; the latter is obtained from the former by
dividing by the bin width. The regions of $q^{2}$ that include the narrow
${J\mskip-3.0mu/\mskip-2.0mu\psi}$ and $\psi{(2S)}$ resonances are excluded
from the binned measurements as they are completely dominated by the tree
level amplitudes.

Figure 8: The differential branching fraction of the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}}{\mu^{-}}$.
It can be seen that outside the narrow charmonium resonance regions, marked by
the grey vertical bands, the branching fraction is consistently below the SM
prediction. From Ref. [27]

The total branching fraction when integrating over $q^{2}$ for different
decays is obtained by taking the binned measurement and then interpolating
them into the missing regions of the ${J\mskip-3.0mu/\mskip-2.0mu\psi}$ and
$\psi{(2S)}$ resonances using a decay model where the resonances are not
present. In Fig. 8, this amounts to interpolating through the grey vertical
bands, ignoring the huge peaks from the resonances. There will still be a
level of contamination of this _penguin-only_ branching fraction from the
broad charmonium resonances present at
${q^{2}}>15\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$. The only way to avoid
this is to perform a completely unbinned analysis where all amplitudes in the
$q^{2}$ spectrum are modelled. This has so far only been done for the
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ decay mode [28].

The most precise measurements of branching fractions have been performed for
the decays ${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ [29],
${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ [29] and
${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}}{\mu^{-}}$
[27]. All these branching fraction measurements are currently below the
expectations from the SM but with quite significant (and correlated)
uncertainties in these SM predictions. See Sec. 6 on lepton flavour
universality for measurements with electrons where the branching fraction
measurements currently have most relevance. The leading systematic uncertainty
in the branching fraction measurements arises from the normalisation. As an
example, the ${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ branching
fraction is measured relative to the decay
${{{B}^{-}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{-}}$, with
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$. The
uncertainty on the branching fraction of this normalisation mode is around 5%
and a factor 3 larger than the statistical uncertainty. As LHCb is unable to
provide any precision measurements of branching fractions without a
normalisation mode, improvements beyond the 5% level will only appear once the
Belle II collaboration improves the measurement of the corresponding
normalisation mode.

Branching fraction measurements of the decay modes ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\\!\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptK}}^{(*)}{\nu}{\overline{\nu}}$ are very
challenging from an experimental point of view, as the only visible final
state particle is the kaon. A measurement is not possible at a hadron
collider, but at a $B$-factory it is possible by exploiting the fact that the
$B$ mesons are produced as a pair from the $\Upsilon(4S)$ decay; once one $B$
mesons is reconstructed, the kinematics of the other $B$ meson is known and
the only other particles in the detector are the decay products of the $B$
meson. The current best limit is ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\\!\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptK}}{\nu}{\overline{\nu}}<3.2\times 10^{-5}$
[30] from the BaBar experiment but the analysis method developed by Belle II
in Ref. [31] sets a very competitive limit of
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\nu}{\overline{\nu}}<4.1\times 10^{-5}$
with a factor 8 lower integrated luminosity. Rather than relying on an
explicit reconstruction of the other $B$ meson in the event, candidates are
selected by identifying charged kaons and then using a machine learning
algorithm to test whether the remainder of the event is compatible with the
decay products of a single $B$ meson decay. As the SM prediction for the
branching fraction summed over the three neutrino species is $(8.2\pm
1.0)\times 10^{-6}$ [32] an observation is expected soon with data from Belle
II.

In all the decays discussed above the leading uncertainty of around 30% in the
SM prediction is from the $B\\!\rightarrow K$ form factors as illustrated with
the wide bands for the SM prediction in Fig. 8. This uncertainty can be
reduced to around 7% with measurements of the inclusive branching fraction of
${B}\\!\rightarrow X_{s}\ell^{+}\ell^{-}$, where the $B$ meson represents
either a ${B}^{0}$ or a ${{B}^{-}}$ [33]. At Belle [34] and BaBar [35] this
was carried out using a _sum of exclusive modes_ approach. In the BaBar
analysis, a total of twenty final states were used, with either electrons or
muons as the lepton pair and the hadronic system containing a neutral or
charged kaon and up to four pions. To reduce background, it is required that
the mass of the hadronic part of the decay is below
1.8$\text{\,Ge\kern-1.00006ptV\\!/}c^{2}$. The reconstruction covers around
70% of the total inclusive rate and simulation is then used to extrapolate to
the measurement of the full branching fraction $\mathcal{B}({B}\\!\rightarrow
X_{s}\ell^{+}\ell^{-})=(6.7\pm 0.7\pm 0.5)\times 10^{-6}$, where the first
uncertainty is from the experimental measurement and the second from the
extrapolation. The extrapolated differential rate as a function of $q^{2}$ can
be seen in Fig. 9.

Figure 9: Differential branching fraction for the inclusive ${B}\\!\rightarrow
X_{s}\ell^{+}\ell^{-}$ measurement for the final states with (blue) electrons,
(red) muons and (black) the lepton flavour average. From Ref. [35].

Ideas for carrying out an inclusive analysis at hadron colliders by
identifying decays with a charged kaon and a pair of muons has been suggested
in Ref. [36]. The method would extrapolate from the measured rate to the full
inclusive rate using isospin assumptions.

The ${b}\\!\rightarrow{d}$ transition can be measured in the same way as for
radiative decays. The most precise measurement is
$\mathcal{B}({{{B}^{-}}}\\!\rightarrow{{\pi}^{-}}{\mu^{+}}{\mu^{-}})=(1.8\pm
0.2)\times 10^{-8}$ [37] which is compatible with the prediction from
$|{V_{{t}{d}}^{\phantom{\ast}}}|/|{V_{{t}{s}}^{\phantom{\ast}}}|$ when
comparing to the ${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ decay.

##  4 ANGULAR DISTRIBUTIONS

The dynamics of ${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ decays can be described
by helicity angles and the dilepton invariant mass $q^{2}$ [38, 39]. When
reading the literature, great care has to be taken in understanding the sign
conventions used for the helicity angles and the derived observables as no
uniform choice is used. At the LHCb experiment, the most precisely measured
decay is ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$, with
${{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$
[40]. In this case, the dynamics are described by three angles and $q^{2}$.
Measurements of such decays have also been made by the ATLAS [41], BaBar [42],
Belle [43], CDF [44] and CMS [45] collaborations. The differential decay
distribution is predicted in terms of angular observables [46, 47, 48], and is
complicated by the presence of both P (spin 1) and S-wave (spin 0)
contributions to the ${K}^{-}$ ${\pi}^{+}$ system. In the region around the
${\kern 1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}$ resonance, the latter is
at the $\approx 10\%$ level. The contribution from D-wave (spin 2) ${K}^{-}$
${\pi}^{+}$ is negligible compared to the precision of the present generation
of experiments. Correct experimental identification of the kaon and pion is a
significant factor in the angular analyses, as the angular convention depends
on the charges of the hadrons.

Measurement of angular observables requires that the data be corrected for the
inefficiencies introduced by the effects of the detector, reconstruction and
selection as a function of the helicity angles and $q^{2}$. At LHCb, the
_acceptance function_ that models such effects is determined using simulated
events. The simulation is calibrated by comparing it with data in a range of
control decays. For example, the particle identification performance of the
LHCb detector is determined with
${{D}^{*+}}\\!\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$,
where the particles can be unambiguously identified on the basis of the
kinematics alone; and the ${B}^{0}$ momentum spectrum is calibrated with other
fully reconstructed $b$ hadron decays. Calibration of the momentum spectrum is
required, as efficiencies depend on quantities in the lab frame, while they
need to be applied to the angular distribution in the ${B}^{0}$ rest frame.
The overall quality of the calibration is cross-checked with an angular
analysis of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}}{\mu^{-}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}(\rightarrow{{K}^{-}}{{\pi}^{+}})$
decays. Such decays are mediated by a tree-level process and hence occur $\sim
100$ times more often than the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ rare decay
(see Fig. 2). The angular observables are known precisely from previous
measurements and can therefore be used to validate the simulation description
of the acceptance function. [] Angular observableA combination of the helicity
amplitudes which can be measured experimentally and which is developed such
that it minimises theoretical uncertainty from form factor calculations.

To some greater or lesser extent, angular observables give access to
quantities that are independent of the dominant theoretical uncertainties,
which come from the hadronic form factors [49, 50]. While individual angular
observables may then enable a clean comparison with SM predictions - for
example the quasi form-factor independent observable $P_{5}^{\prime}$ has been
measured to have some local tension with the SM in the region
$2.5<{q^{2}}<6.0\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ [40]; the
combination of a full basis of angular observables enables the form factors
and other theory nuisance parameters to be constrained in global fits. In
${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ decays,
angular measurements are then made to the observables $A_{\mathrm{FB}}$,
$F_{L}$ and the $P_{i}^{\prime}$ series of quantities; or the $S_{i}$ series
of observables [51, 52, 53]; enabling the underlying Wilson coefficients
$C_{9},C_{10}$ and the theory nuisance parameters to be determined. The
majority of measurements in different $q^{2}$ bins are in agreement with SM
predictions. The local deviation in $P_{5}^{\prime}$ or $S_{5}$ observed at
LHCb is consistent with smaller effects seen in other observables and can be
attributed to a new vector contribution that shifts the $C_{9}$ Wilson
coefficient (see Fig. 10). From the angular analysis, there is presently no
indication for new axial vector contribution that shifts $C_{10}$.

Figure 10: Fits to $\Delta{\rm{Re}}(C_{9})$ and $\Delta{\rm{Re}}(C_{10})$, the
shift in the values of real part of the Wilson coefficients $C_{9}$ and
$C_{10}$ from their SM values, using the measurements of various angular
observables in the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$. The
measurements are consistent with each other and are $\sim 3\sigma$ from the SM
point, which is marked with a red cross. Adapted from Ref. [40].

The dominant theory uncertainty on $P_{5}^{\prime}$ is from the so-called
$c\bar{c}$ loop effect. This includes the interference between the amplitudes
for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow{\mu^{+}}{\mu^{-}}){{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}(\rightarrow{{K}^{-}}{{\pi}^{+}})$
and the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ process.
There is a broad consensus in the theoretical community that the effect from
such decays is large in the region
${q^{2}}>6.0\text{\,Ge\kern-1.00006ptV}/c^{4}$ but the degree to which such
effects could influence the measurements made in the region
${q^{2}}<6.0\text{\,Ge\kern-1.00006ptV}/c^{4}$ is still subject to debate [54,
55, 56, 57]. The relevant theoretical calculations involve non-local effects
and hence are intractable. The $c\bar{c}$ processes give a vector-like
contribution that could adjust the $C_{9}$ Wilson coefficient, and hence mimic
a new physics contribution. A genuine effect from heavy new physics would be
independent of $q^{2}$, whereas the interference from $c\bar{c}$ loops would
be dependent on $q^{2}$. Future analyses will try to determine the
interference by fitting models for the resonant and rare decay amplitudes to
the experimental data unbinned in $q^{2}$ [58].

In addition to the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ with ${{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}$,
the ${{{B}^{-}}}\\!\rightarrow{{K}^{*-}}{\mu^{+}}{\mu^{-}}$ with
${{K}^{*-}}\\!\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{-}}$ [59] and
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ [60], angular
distributions have been measured by the LHCb collaboration. The former also
shows some tension with respect to SM predictions. Although the
${{{B}^{-}}}\\!\rightarrow{{K}^{-}}{\mu^{+}}{\mu^{-}}$ decay, measurements of
the $A_{\mathrm{F}B}$ and $F_{H}$ angular observables are in good agreement
with the SM predictions, there is no inconsistency, as they are only sensitive
to new (pseudo)scalar and tensor couplings. Measurements have also been made
for the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow\phi{\mu^{+}}{\mu^{-}}$
[61] decay, where the $\phi$ decays to a CP eigenstate and hence does not tag
the $B$ flavour. Measurements of the baryonic decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow\Lambda{\mu^{+}}{\mu^{-}}$ have also been
performed [62]. The different spin structure gives access to different physics
but the decay is experimentally much more challenging given the relatively
long lifetime of the $\Lambda$.

Analysis of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{e^{+}}{e^{-}}$ decays in the
$q^{2}$ region dominated by the photon pole (the interval between 0.0008 and
0.257$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$) provides the current best
constraints on the right-handed currents via the $C_{7}^{\prime}$ Wilson
coefficient by measuring the virtual photon polarization (see Sec. 5.2) with
5% accuracy [63].

At the Belle experiment, reconstruction of the electron and muons modes is
comparably efficient and precise measurements are possible in both cases. The
angular analysis of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{e^{+}}{e^{-}}$ decays has
enabled the difference between electron and muon decays to be studied and the
observable $Q_{5}=P_{5}^{\prime}(\mu)-P_{5}^{\prime}(e)$ [64] to be formed,
yielding further theoretically pristine tests of the SM, albeit with limited
experimental precision [65]. The measurements are presently consistent with
the SM. Analogous studies are in progress at LHCb but have lower yields
compared to the muon modes due to the trigger and are complicated by the
bremsstrahlung from the electrons, which results in larger backgrounds.
Nonetheless, measurements of angular observables in ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{e^{+}}{e^{-}}$ and related
decays will allow $Q_{5}$ and further differences between angular observables
in muon and electron modes to be investigated in the future.

##  5 $C\\!P$ VIOLATION

###  5.1 Direct $C\\!P$ violation

Direct $C\\!P$ violation is a phenomenon also sensitive to NP. Suppose a decay
${B}\rightarrow f$ can proceed through two diagrams with amplitudes $A_{1}$
and $A_{2}$. The amplitudes can be written as
$A_{i}=|A_{i}|\exp(i\theta_{i})\exp(i\delta_{i})$ where $\theta_{i}$ and
$\delta_{i}$ are the weak and strong phases in the diagram $i$, respectively.
Comparing the amplitude $A({B}\rightarrow f)=A_{1}+A_{2}$ to the $C\\!P$
conjugate amplitude $A({{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow\kern
1.79993pt\overline{\kern-1.79993ptf})=\bar{A}_{1}+\bar{A}_{2}$, only the sign
of the weak phases flips while the strong phase stays the same. From this, the
direct $C\\!P$ asymmetry, ${\mathcal{A}}^{{C\\!P}}$, is obtained as

${{\mathcal{A}}^{{C\\!P}}}=\frac{\Gamma({B}\rightarrow f)-\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow\kern
1.79993pt\overline{\kern-1.79993ptf})}{\Gamma({B}\rightarrow f)+\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow\kern
1.79993pt\overline{\kern-1.79993ptf})}=\frac{2|A_{1}||A_{2}|\sin(\theta_{1}-\theta_{2})\sin(\delta_{1}-\delta
2)}{|A_{1}|^{2}+|A_{2}|^{2}+2|A_{1}||A_{2}|\cos(\theta_{1}-\theta_{2})\cos(\delta_{1}-\delta
2)},$ (2)

where $\Gamma=|A|^{2}$ is the decay width. ${{\mathcal{A}}^{{C\\!P}}}$ becomes
non zero when $\theta_{1}\neq\theta_{2}$, $\delta_{1}\neq\delta_{2}$ and
$|A_{1}|\sim|A_{2}|$. That is, direct $C\\!P$ violation occurs when a decay
proceeds through two diagrams with similar amplitudes and with different weak
and strong phases.

For exclusive ${b}\\!\rightarrow{s}{\gamma}$ and
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ decays, ${{\mathcal{A}}^{{C\\!P}}}$ is
at the few percent level in the SM. The predictions are also relatively
precise because the uncertainties from the form factors, which make the
branching fraction calculations difficult, cancel. For this reason,
${{\mathcal{A}}^{{C\\!P}}}$ of exclusive decays have been intensively measured
as a probe for NP. Presently, ${{\mathcal{A}}^{{C\\!P}}}$ for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}\gamma$ decays is measured to be
$-0.006\pm 0.011$ [6], while the uncertainty of the theoretical prediction is
a few percent. The asymmetries ${\mathcal{A}}^{{C\\!P}}$ for
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ and ${b}\rightarrow{d}\ell^{+}\ell^{-}$
decays have also been measured with precision of around 10% for many decays
[6] but as good as 2% for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}}\rightarrow{\kern
1.79993pt\overline{\kern-1.79993ptK}}^{(*)}{\mu^{+}}{\mu^{-}}$ [66]. The
results are consistent with null asymmetry.

${{\mathcal{A}}^{{C\\!P}}}$ in the inclusive modes may be more interesting.
The ${{\mathcal{A}}^{{C\\!P}}}$ in $B\rightarrow X_{s}\gamma$ is predicted to
be $-0.006<{{\mathcal{A}}^{{C\\!P}}}(B\rightarrow X_{s}\gamma)<0.028$ [67].
Experimentally, this can be measured by reconstructing $X_{s}$ as a sum of
many exclusive final states, e.g. $K\pi$, $K\pi\pi$. The average of the
measurements by the BaBar [68] and Belle [69] collaborations is
${{\mathcal{A}}^{{C\\!P}}}(B\rightarrow X_{s}\gamma)=0.015\pm 0.011$ [6],
which is consistent with the SM prediction. Another variable
$\Delta{{\mathcal{A}}^{{C\\!P}}}$, isospin violation of
${{\mathcal{A}}^{{C\\!P}}}$, defined as
${{\mathcal{A}}^{{C\\!P}}}(B\rightarrow
X_{s}^{-}\gamma)-{{\mathcal{A}}^{{C\\!P}}}(B\rightarrow X_{s}^{0}\gamma)$ is
proposed as a cleaner variable [67] with experimental measurements from the
same papers.

Another approach is to measure ${{\mathcal{A}}^{{C\\!P}}}(B\rightarrow
X\gamma)$ inclusively, without distinguishing $B\rightarrow X_{s}\gamma$ and
$B\rightarrow X_{d}\gamma$. In the SM, ${{\mathcal{A}}^{{C\\!P}}}$ in
${b}\\!\rightarrow{s}{\gamma}$ and ${b}\\!\rightarrow{d}{\gamma}$ cancels out,
and ${{\mathcal{A}}^{{C\\!P}}}(B\rightarrow X\gamma)$ is expected to be null,
so this provides an excellent test of the SM. The analysis strategy is the
same as the branching fraction measurement. The flavour (whether the process
is the decay of a $b$ or a $\overline{b}$) can be obtained from the other $B$
in the event, either from a lepton in the semileptonic decay of the other $B$,
or from a reconstructed $B$ meson when the other $B$ meson is fully
reconstructed. The probability of the wrong tag of the flavour, which
partially comes from ${B}^{0}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ oscillations, needs to be taken
into account, in the analysis. The present world average is
${{\mathcal{A}}^{{C\\!P}}}(B\rightarrow X\gamma)=0.010\pm 0.031$ [6].

###  5.2 Time-dependent $C\\!P$ violation and Polarization in Radiative $B$
decays

When the ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}$ can decay to a common $C\\!P$
eigenstate $f$, the process ${{B}^{0}}\rightarrow f$ can interfere with
${{B}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow f$ process, causing
time-dependent $C\\!P$ violation to occur with an amplitude $S$. For example,
the time-dependent $C\\!P$ asymmetry in ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{0}_{\mathrm{S}}}$
decays can be used to determine the angle $\beta$ ($=\phi_{1}$) of the
unitarity triangle through the ${C\\!P}$ violating parameter $S=\sin 2\beta$.

[] Unitarity triangleIn the SM, $C\\!P$ violation occurs though a complex
phase in the CKM matrix elements. A triangle can be drawn on the complex plane
from the relation of CKM matrix elements,
$|{V_{{u}{d}}^{\phantom{\ast}}}{V_{{u}{b}}^{\phantom{\ast}}}^{*}+{V_{{c}{d}}^{\phantom{\ast}}}{V_{{c}{b}}^{\phantom{\ast}}}^{*}+{V_{{t}{d}}^{\phantom{\ast}}}{V_{{t}{b}}^{\phantom{\ast}}}^{*}=0$,
due to the unitarity of the CKM matrix. This triangle is called the unitarity
triangle, and the three interior angles are named as $\alpha$ ($=\phi_{2}$),
$\beta$ ($=\phi_{1}$), $\gamma$ ($=\phi_{3})$.

Time-dependent $C\\!P$ violation can be also considered in the radiative $B$
decays ${{\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow
X^{0}\gamma$ where $X^{0}$ is a neutral hadronic state for which flavour is
not identified, e.g. ${{K}^{0}_{\mathrm{S}}}{{\pi}^{0}}$. In the SM, the
photon in ${b}\\!\rightarrow{s}{\gamma}$ (${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow X_{s}\gamma$) is left-
handed, while the photon in ${\overline{b}}\rightarrow{\overline{s}}\gamma$
(${{B}^{0}}\rightarrow X_{{\overline{s}}}\gamma$) is right-handed. Therefore,
${B}^{0}$ and ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}$ do not
decay into a common $C\\!P$ eigenstate and no time-dependent $C\\!P$ violation
occurs. To be precise, the right-handed photon is not totally inhibited, but
is suppressed by the quark mass ratio $m_{s}/m_{b}$, and
$S\sim-2m_{s}/m_{b}\sin 2\beta$ is expected for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow X_{s}\gamma$. The same
discussion holds for the ${b}\\!\rightarrow{d}{\gamma}$ replacing the $s$
quark with the $d$ quark, though in this case the weak phase appearing in $S$
vanishes. In any case, in radiative $B$ decays, $S$ is proportional to the
fraction of the right-handed current $C^{\prime}_{7}$ in
${b}\\!\rightarrow{s}{\gamma}$ (${b}\\!\rightarrow{d}{\gamma}$); hence the
measurement of the time-dependent $C\\!P$ asymmetry is essentially the
measurement of the photon polarization.

Time-dependent $C\\!P$ violation in ${b}\\!\rightarrow{s}{\gamma}$ has been
measured by BaBar and Belle (e.g. Refs. [70, 71]). For example, $S$ for
$B^{0}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}(\rightarrow{{K}^{0}_{\mathrm{S}}}{{\pi}^{0}})\gamma$
is measured to be $-0.15\pm 0.22$ by the BaBar and Belle collaborations [6],
which is consistent with no time-dependent $C\\!P$ violation. Measurements
have been made with
$B^{0}\rightarrow{{K}^{0}_{\mathrm{S}}}(\eta,\rho^{0},\phi)\gamma$ and
$B^{0}\rightarrow\rho^{0}\gamma$ with larger uncertainty. $B$ factory
experiments have the advantage for these measurement, since the final state
involves neutral particles, and forming a ${K}^{0}_{\mathrm{S}}$ decay vertex
is necessary for many modes.

Similar measurements can be made with ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\rightarrow\phi\gamma$ decays.
As a measurement of $S$ requires oscillations to be resolved, a large boost of
the ${\kern 1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ mesons is
required and a measurement is only possible at the LHCb experiment.
Measurements of $S$ and $\mathcal{A}^{\Delta}$, which is another parameter
sensitive to the photon polarization amplitudes, has been made with
uncertainties of around 30% using $3~{}\text{\,fb}^{-1}$ of data [72]. This is
one of the promising modes by LHCb for the precise study on the photon
polarization.

Other methods to measure the photon polarization in radiative $B$ decays also
exist. One possibility is to utilize the photon conversion $\gamma\rightarrow
e^{+}e^{-}$ in the detector materials for the decay $B\rightarrow
K^{*}(\rightarrow K\pi)\gamma$. However, this method requires very thin
materials to reduce the effect of the multiple scattering and it does not look
feasible with LHCb and Belle II. The ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}e^{+}e^{-}$ decay at
$0.0008<{q^{2}}<0.257\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$ is completely
dominated by a virtual photon and the angular distribution can be used to
measure the photon polarisation, as discussed in Sec. 4.

##  6 LEPTON FLAVOUR UNIVERSALITY AND LEPTON FLAVOUR VIOLATION

Decays of $b$-hadrons involving a ${b}\\!\rightarrow{s}\,\ell^{+}\ell^{-}$
transition provide an ideal laboratory to test Lepton Universality (LU) due to
the absence of a SM tree-level contribution. They are sensitive to
contributions from new particles that can induce a sizeable increase or
decrease in the rate of given decays. These contributions can also modify the
angular distribution of the final-state particles, which can provide further
tests of LU. Whether these tests compare the branching fractions or angular
distributions of the two lepton types, they are all characterised by quasi
negligible theoretical uncertainties on the SM predictions.

Currently, most results consist in ratios of the type [73]

$R_{H_{s}}=\frac{\bigintssss_{q^{2}_{\mathrm{min}}}^{q^{2}_{\mathrm{max}}}\frac{d\Gamma(H_{b}\rightarrow
H_{s}{\mu^{+}\mu^{-}})}{d{q^{2}}}\,d{q^{2}}}{\bigintssss_{q^{2}_{\mathrm{min}}}^{q^{2}_{\mathrm{max}}}\frac{d\Gamma(H_{b}\rightarrow
H_{s}{e^{+}e^{-}})}{d{q^{2}}}\,d{q^{2}}}\,,$ (3)

where $H_{b}$ represents a hadron containing a $b$-quark, $H_{s}$ represents a
hadron containing an $s$-quark, and $q^{2}$ is the invariant mass squared of
the dilepton system integrated between $q^{2}_{\mathrm{min}}$ and
$q^{2}_{\mathrm{max}}$. In the SM such ratios are expected to be close to
unity above the dimuon threshold. For all $R_{H_{s}}$ experimental results the
$q^{2}$ regions around the charmonium resonances are vetoed since they are
dominated by SM physics. In the LHCb case, in order to be less sensitive to
the detailed knowledge of the electron and muon reconstruction efficiencies,
the $H_{b}\rightarrow H_{s}\ell^{+}\ell^{-}$ branching fractions are measured
relative to the branching fraction of the $H_{b}\rightarrow
H_{s}{{J\mskip-3.0mu/\mskip-2.0mu\psi}}(\rightarrow\ell^{+}\ell^{-})$ decay,
exploiting the fact that the
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\rightarrow\ell^{+}\ell^{-}$ decay is known
to respect LU at the 0.3% level [6]. Current results are summarised in Fig.
11. The LHCb results are only marginally compatible with unity, and the most
precise result [74] is 3.1 standard deviations from the SM prediction. The
coming years will be very interesting with more results expected from the
current and future LHC datasets, as well as with Belle II. Indeed Belle II,
with 20$\text{\,ab}^{-1}$ expected around 2025, should be able to confirm the
$R_{{K}}$ measurement from LHCb. Due to the interest in these measurements,
the ATLAS and CMS collaborations are currently investigating their
sensitivities. Beyond these improvements in sensitivity and the interest in
having very different experimental conditions and thus different experimental
systematic uncertainties, the next major steps will come from LU tests using
angular analyses as pioneered by Belle [43].

Figure 11: Summary of the results for Lepton Universality tests on ratios of
branching fractions. The $q^{2}$ intervals are expressed in units of
$\text{\,Ge\kern-1.00006ptV}^{2}\\!/c^{4}$.

Linked with the issue of possible LU violation another interesting aspect is
the search for Charged Lepton Flavour Violation (CLFV) decays. In the SM,
these decays are completely negligible and an observation would be an
indisputable sign of New Physics. For $H_{b}\\!\rightarrow\ell\ell^{\prime}$
decays, for ($e,\mu$) pairs, the results are dominated by the LHCb results
with upper limits of the order of $10^{-9}$ at 90% CL [6]. For the other
combinations, more challenging from an experimental point of view, the
cleanliness of the $B$-factories environment compensates for their lower
${b}$-hadron production rate. The upper limits are nevertheless only of order
$10^{-5}$ at 90 % CL. On a similar topic, many searches have been carried out
using $H_{b}\\!\rightarrow H_{s}\ell\ell^{\prime}$ decays. The upper limits
follow the same experimental schema as the
$H_{b}\\!\rightarrow\ell\ell^{\prime}$ decays: the best limits (of order
$10^{-9}$) are obtained by the LHCb experiment for final states with an
electron and a muon, while for the other cases, the limits span from $10^{-5}$
to a few $10^{-8}$, with significant contributions from the BaBar and Belle
experiments [6]. All these results constrain the broad set of New Physics
models proposed to explain the tensions observed in analyses dealing with
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ transitions, including in some cases
the results of LU tests in $b\rightarrow c\ell\nu$ decays, in particular those
involving a tau lepton [6].

##  7 DISCUSSION

Since the first measurement of an electroweak penguin decay by the CLEO
collaboration in 1993, there are now around 200 different measurements. With
the parallel development of the theoretical understanding, the area has moved
from something that verified the existence of these type of decays in the SM
to an area that is pushing the boundaries for the discovery of New Physics.

In Fig. 12 is a comparison of several global fits for Wilson coefficients [75,
76, 77, 78]. Current results favour a substantial NP contribution to the
Wilson coefficients $C_{9}$ and/or $C_{10}$ for muons in the final state. The
fits represented here assume that all other Wilson coefficients take on their
SM values. In the first one, only the observables with high theoretical
precision (the lepton universality measurements and the
${{B}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ branching fraction) are used,
while in the second figure all observables are used. It can be seen that
adding more experimental observations results in smaller uncertainties but
also a higher level of model dependence, i.e. different global fits obtain
somewhat different results even if they use the same experimental data. []
Global fitA fit that combines many different measurements of
${b}\\!\rightarrow{s}\ell^{+}\ell^{-}$ observables to find the most likely
values of the Wilson coefficients that would result in those measurements.
They involve the combination of experimental measurements, theoretical
uncertainties and their respective correlations.

Figure 12: Global fits (ACDMN [75], AS [76], CFFPSV [77], HMMN [78]) for any
New Physics contributions to the Wilson coefficients $C_{9}$ and $C_{10}$
under the assumption that all other Wilson coefficients take on their SM
values. The fits use (left) lepton flavour universality observations and
${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
and (right) all observables. The former results in a poorer experimental
resolution but has fewer theoretical assumptions compared to the latter. The
two contours for each fit correspond to $1\sigma$ and $2\sigma$ uncertainties.

The LHCb upgrades and Belle II will make more accurate measurements in the
future and the collaborations have made specific predictions of the accuracy
[7, 79]. In addition there are prospects of making entirely new measurements
that can either pin down theoretical uncertainties or further investigate the
nature of any such New Physics.

The experimental challenge for the future will be to make measurements of
electroweak penguin decays in as many different ways as possible as this will
decrease the overall statistical uncertainty for global fits but more
importantly will explore theoretical assumptions and further clarify the
nature of any New Physics. As highlighted in this review, keeping a broad
physics programme for the analyses across multiple experiments is of utmost
importance to exploit the different experimental constraints, as well as
serving as an important cross check for the systematic uncertainties that to
some extent are shared between different measurements at the same experiment.

[SUMMARY POINTS]

  1. 1.

Electroweak penguin decays are highly sensitive to New Physics at mass scales
well above the electroweak energy scale.

  2. 2.

Measurements at the $B$ factories and at the LHC have made several hundred
measurements that through global fits give an indication for New Physics in
vector and/or axial vector couplings.

  3. 3.

Particle identification of leptons, hadrons and photons is essential for
performing the measurements.

  4. 4.

To reduce theoretical uncertainties in the interpretation of results there is
a need to perform new measurements where data can be used to minimise these
uncertainties.

[FUTURE ISSUES]

  1. 1.

Measurements with $\tau$ leptons in the final state would add much to the
exploration of lepton flavour universality and lepton number conservation.
While measurements of some of these final states have already been made, they
are in general still far away from setting serious constraints on NP. An
exciting idea is the measurement of new observables in the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\tau^{+}}{\tau^{-}}$ in the far
future at FCC-ee [80].

  2. 2.

At the end of the high luminosity LHC period, it will be possible to make a
measurement of the ratio of branching fractions for the decays ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$
[81, 82, 7]. Such a measurement explores the minimal flavour violation
paradigm of New Physics with almost no theoretical uncertainty.

  3. 3.

High precision simultaneous angular analyses of the decays ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{\mu^{+}}{\mu^{-}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}}\\!\rightarrow{{\kern
1.79993pt\overline{\kern-1.79993ptK}}{}^{*0}}{e^{+}e^{-}}$ will allow for a
precision determination of Lepton Flavour Violating NP. This comes from the
fact that the hadronic uncertainties are shared between the modes and thus
will cancel out in a comparison.

## DISCLOSURE STATEMENT

UE, MP and MHS are members of the LHCb collaboration. SN is a member of the
Belle and Belle II collaborations.

## ACKNOWLEDGMENTS

UE would like to acknowledge funding from ARC grant DP210102707. SN would like
to acknowledge funding from JSPS KAKENHI Grant Number JP17K05474. MHS would
like to acknowledge funding from European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 101018181). MP acknowledges support from the UK Science and
Technology Facilities Council.

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