couplings in the SM, it is possible to provide a hierarchical structure for
the flavour coupling matrices $\lambda^{q}$ and $\lambda^{\ell}$ without
conflicting with other low-energy constraints or high-$p_{\textrm{T}}$
constraints [170].

###  8.2 Specific models

The EFT approach provides constraints on Wilson coefficients encoding the
short-distance contributions to
${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$ and
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ processes. Discussing all
possible models of NP able to reproduce these patterns of short-distance
physics is beyond the scope of this review, but it is already interesting to
discuss simple models, where only one or two heavy intermediate particles are
introduced. The quantum numbers involved in
${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$ and
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ processes correspond to exchanges
of two heavy colourless vector bosons (${W}^{\prime}$, ${Z}^{\prime}$), heavy
colourless scalars (charged or neutral heavy Higgs-like bosons), or heavy
coloured vector or scalar bosons (leptoquarks) [170]. In these simple models,
both LU and lepton-violating effects are often generated together (for
instance in leptoquark models), unless specific symmetries are imposed to
prevent lepton-flavour violating couplings (for instance in ${Z}^{\prime}$
models). In both cases, information on LU as well as lepton-flavour violation
(such as ${B}\\!\rightarrow K^{(*)}\mu\tau$ [173]) provides interesting guides
for model building, in conjunction with constraints from direct production at
the LHC.

####  8.2.1 ${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$

As the FCCC ${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$ transition
occurs at tree level in the SM, NP explanations also occur typically at tree
level. Various models with single mediators have been proposed, with heavy
charged vector bosons (${W}^{\prime}$) [179, 180, 181, 182], heavy charged
scalars ($H^{\prime}$) [183, 184, 185, 186, 187, 188, 189, 190], or coloured
vector or scalar bosons (leptoquarks) [191, 192, 193, 176, 177, 194, 195, 196,
197, 145, 198, 199, 200, 201, 202, 203, 204]. A last possibility allowed by
the quantum numbers of the fermions involved consists in vector-like quarks
that mix with chiral SM quarks and change the couplings to the $W$ boson
(including right-handed couplings), but unless light right-handed neutrinos
are introduced, they do not affect the couplings to the leptons and thus yield
LU contributions which are not of interest here.

Heavy vector and scalar bosons are disfavoured by the global fits (scalar) and
by the direct LHC searches (vector) [179, 151]. Leptoquark models provide a
larger set of possibilities, with six different type of representations
(lumping together representations differing only through hypercharge) [205,
170, 143]: three are scalar leptoquarks $(S_{1},R_{2},S_{3})$ and three are
vector leptoquarks $(U_{1},V_{2},U_{3})$. All of them are triplets under
$SU_{C}(3)$, but they have different representations under $SU_{L}(2)$:
singlet $(S_{1},U_{1})$, doublet $(R_{2},V_{2})$ or triplet $(S_{3},U_{3})$.
Two of them, $S_{1}$ and $R_{2}$, are able to generate tensor contributions
(correlated with scalar contributions). Leptoquark models are also bounded by
high-$p_{\textrm{T}}$ constraints [151], but the bounds can be avoided by
assuming large couplings to the second generation [176, 206, 207].

The extension of this discussion to more general frameworks is possible and
leaves room for other types of models. For instance, one may consider
supersymmetric models allowing for couplings violating $R$-parity [204].

####  8.2.2 ${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$

The discussion of ${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ deviations has
involved mainly ${Z}^{\prime}$ models, leptoquarks and composite models, that
are generally tuned in order to reproduce a particular pattern of NP
contributions to Wilson coefficients. Interestingly, due to their quantum
numbers, leptoquarks contribute only at the loop level, typically of the same
size as the SM contribution.

The general outcome of global fits is a significant contribution to ${\cal
C}_{9\mu}$. ${Z}^{\prime}$ models with couplings to leptons can easily yield
such ${\cal C}_{9\mu}^{\rm NP}$-like solution avoiding gauge anomalies. In
this context, models gauging the $L_{\mu}-L_{\tau}$ number [208, 209, 210,
211] have been investigated in detail as not generating effects in electron
channels. Concerning leptoquarks, a ${\cal C}_{9\mu}^{\rm NP}$-like solution
can only be generated by adding two scalar or two vector representations
[170].

Another popular scenario is to require that ${\cal C}_{9\mu}^{\rm
NP}=-C_{10\mu}^{\rm NP}$, as it obeys the $SU(2)_{L}\times U(1)_{Y}$ symmetry
expected from SMEFT. This pattern can be achieved in ${Z}^{\prime}$ models
with loop-induced couplings [212] or in ${Z}^{\prime}$ models with heavy
vector-like fermions [180, 181], or in $Z^{\prime}$ models with a gauged
horizontal symmetry [213]. Concerning leptoquarks, here a single
representation (either $R_{2}$ or $U_{1}$) can generate a ${\cal
C}_{9\mu}^{\rm NP}=-C_{10\mu}^{\rm NP}$ solution [214, 195, 215, 176, 177,
196, 198]. This pattern can also be obtained in models with loop contributions
from heavy new scalars and fermions [216, 217, 218]. Composite Higgs models
are also able to achieve this pattern of deviations [219].

Supersymmetric models like the MSSM experience difficulties to generate large
NP contributions to ${\cal C}_{9\ell}$, as these are due to loop diagrams of
supersymmetric particles (and thus significantly suppressed compared to the
SM) [156, 218]. More elaborate supersymmetric models may be able to accomodate
these data, for instance the ones violating $R$-parity which contain
leptoquarks in their spectrum.

####  8.2.3 Common explanations

Since hints of LU deviations are only seen in $b$-quark transitions involving
leptons, it is tempting to build models explaining both FCCC and FCNC
deviations simultaneously.

The LU ratios $R_{D^{(*)}}$ could be explained with a NP contribution of
${\mathcal{O}}(10\%)$ of the tree-level SM contribution. On the other hand,
${b}\\!\rightarrow{s}\,{\mu^{+}\mu^{-}}$ should be affected by a NP
contribution of approximately 25% of the loop- and CKM-suppressed SM
contribution. This suggests NP models able to generate ${\mathcal{O}}(1)$
couplings to the third generation, moderate couplings to the second generation
and small or vanishing couplings to the first generation, reminiscent of the
hierarchy present for the SM Yukawa couplings. Moreover, these NP models must
not generate FCNC processes violating lepton flavour, like ${B}\\!\rightarrow
K{(^{*})}e\mu$ or ${B}\\!\rightarrow K{(^{*})}\tau\mu$ at a level already
measured (it is also true for FCCC
${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$ processes, but not
possible to test experimentally due to the impossibility to test the flavour
of the neutrino emitted).

Simple models through the exchange of a single particle are very constrained
by both sectors [170] as well as by low-energy processes, in particular
${B}^{0}_{s}$-${\kern 1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$
mixing, ${B}\\!\rightarrow K{(^{*})}{\nu}{\overline{\nu}}$, $(g-2)_{\mu}$,
${\tau^{-}}\\!\rightarrow{\mu^{-}}{\mu^{+}}{\mu^{-}}$, and direct production
at the LHC [220, 170, 171]:

  * •

Colourless vectors do not seem to be able to accommodate all the data (hitting
either constraints from ${B}^{0}_{s}$-${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}$ mixing or from direct
production at the LHC). It is possible to avoid these problems by introducing
additional matter fields. For instance, one can promote the difference
$L_{\mu}-L_{\tau}$ into a gauge symmetry that yields NP contributions both in
the muon and tau sectors and must be with one generation of vector-like
fermions (these models result in substantial modifications of the
${\tau^{-}}\\!\rightarrow{\mu^{-}}{\mu^{+}}{\mu^{-}}$ and
$h\\!\rightarrow{\mu^{+}\mu^{-}}$ decay rates [221, 208, 209, 210, 211]).

  * •

Scalar leptoquarks may in principle provide a good fit to the data [206], but
the radiative corrections to ${Z}\\!\rightarrow{\tau^{+}}{\tau^{-}}$ and
${Z}\\!\rightarrow{\nu}{\overline{\nu}}$ are significant and create a mild
tension with $R_{D^{*}}$. This can be solved at the price of involving right-
handed neutrinos [199]. Another possibility consists in combining the
contributions from two different scalar leptoquarks. The combination of
$R_{2}$ and $S_{3}$ provides a particularly good description of all available
data at low and high energies generating both left- and right-handed currents.
It features a complex scalar contribution to
${b}\\!\rightarrow{c}\,\tau^{-}{\overline{\nu}}_{\tau}$ and can be embedded in
a $SU(5)$ Grand Unified Theory [222]. Another option consists in combining
$S_{1}$ and $S_{3}$, leading to solutions with only left-handed currents [206,
170, 223].

  * •

Finally, the vector leptoquark singlet $U_{1}$ gives a particularly good fit
to data [196, 176, 177, 170, 171]. However, such a massive vector leptoquark
requires additional fields to build an ultraviolet complete model, for which
several possibilities have been investigated further. For instance, models
based on $SU(4)$ Pati-Salam symmetry rely on partially unified gauge groups
where the hypercharge and the colour charge are combined according to a single
$SU(4)$ gauge group. Stringent bounds coming from light-quark processes
require different couplings of the leptoquark to the different generations,
leading to the introduction of a Pati-Salam gauge group for each of the three
generations [224, 207, 225, 226, 227, 228]. Another possible ultraviolet
completion corresponds to composite models where an extended global symmetry
group is broken spontaneously to give rise to the Higgs boson (as a composite
pseudo Nambu-Goldstone boson) as well as many other particles (Higgs doublets,
leptoquarks, colourless and coloured vector bosons) [170, 229]. Related models
have been proposed, based on partial compositeness and feature heavy vector-
like quarks and leptoquarks [223, 230, 231, 232, 202, 233].

##  9 Future measurements

The present status in $b$-quark semileptonic decays urges for new measurements
to clarify the various deviations from the SM expectations that have been
observed. In the (optimistic) scenario where these tensions are confirmed,
more precise measurements and new observables will permit to determine more
accurately their pattern and source. These sets of new measurements are at the
core of the research programme of the Belle-II and LHCb experiments, which
have both published documents presenting their expected performances for the
coming years [234, 235]. All figures and numbers in this section are based on
these sources. The main milestones in terms of years of data taking and
integrated luminosities are summarised in Tab. 12.

Experiment | 2018 | 2021 | 2024 | 2025 | 2037
---|---|---|---|---|---
Belle-II | — | 5$\mbox{\,ab}^{-1}$ |  | 50$\mbox{\,ab}^{-1}$ |
LHCb | 9$\mbox{\,fb}^{-1}$ |  | 23$\mbox{\,fb}^{-1}$ |  | 300$\mbox{\,fb}^{-1}$
Table 12: Summary of the considered years of data taking and corresponding
integrated luminosities for the Belle-II and LHCb experiments. The LHCb
Upgrade-I has been approved to run between approximately 2021 and 2029. A
further Upgrade-II has been proposed to run after 2031.

###  9.1 Experimental prospects for
${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$ decays

The current measurements already exhibit a significant tension with the SM
prediction, and an important contribution to their uncertainties is of
systematic nature. It is thus particularly important to have accurate results
from two diverse experimental environments that will be affected by different
systematic effects, but the extrapolation of such uncertainties is a non-
trivial task.

The measurements of the $R_{D^{*}}$ ratio by the LHCb experiment have
systematic uncertainties that are comparable to the statistical ones, and
therefore the extrapolation to future sensitivities should take into account
their evolution as well. The two main contributions to the systematic
uncertainty are due to the modelling of the fit components and to the limited
size of the simulated samples. The background modelling is estimated using
dedicated control samples from data and larger datasets will improve its
precision. However, in order to simulate larger statistics and reduce the
associated uncertainties, challenging software developments are mandatory to
speed up production times without affecting their accuracy.

More information on ${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$
transitions will be provided by the LHCb experiment that can measure
$R_{H_{c}}$ ratios using ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}_{s}}\\!\rightarrow{D}_{s}^{(*)+}\,\ell^{-}{\overline{\nu}}_{\ell}$,
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow\mathchar
28931\relax_{c}^{(*)+}\,\ell^{-}{\overline{\nu}}_{\ell}$ and
${{B}_{c}^{-}}\\!\rightarrow J/\psi\,\ell^{-}{\overline{\nu}}_{\ell}$ decays.
These processes have different background structures and should be affected by
different systematic uncertainties. Decays of ${\mathchar
28931\relax}^{0}_{b}$ baryons are of particular interest since these will
allow testing for new structures of NP operators.

The Belle-II expected precision on $R_{D}$, $R_{D^{*}}$ and on the $\tau$
polarisation are given in Tab. 13. A significant part of the systematic
uncertainty for all $R_{D^{(*)}}$ ratios arises from the knowledge of the
${B}\\!\rightarrow{D}^{**}\ell^{\prime-}{\overline{\nu}}_{\ell^{\prime}}$
decays, and their study is thus mandatory to reach the ultimate precision. The
projected precision for various $R_{H_{c}}$ ratios from the Belle-II and LHCb
experiments is presented in Fig. 25, where the precision on $R_{D^{(*)}}$ is
expected to be comparable between the two experiments in the future.

Observable | 5$\mbox{\,ab}^{-1}$ | 50$\mbox{\,ab}^{-1}$
---|---|---
$R_{D}$ | $(\pm\,6.0\pm 3.9)\%$ | $(\pm\,2.0\pm 2.5)\%$
$R_{D^{*}}$ | $(\pm\,3.0\pm 2.5)\%$ | $(\pm\,1.0\pm 2.0)\%$
$P_{\tau}({{D}^{*}})$ | $\pm\,0.18\pm 0.08$ | $\pm\,0.06\pm 0.04$
Table 13: Expected precision on $R_{D}$, $R_{D^{*}}$ (relative) and on the
$\tau$ polarisation (absolute) from Belle-II [234]. The first uncertainty is
statistical and the second is systematic.

Figure 24: Projected uncertainty for various $R_{H_{c}}$ ratios from the
Belle-II and LHCb experiments (years are indicative). The Belle-II
uncertainties include estimates of the evolution of the systematic
uncertainties. The systematic uncertainties at LHCb are assumed to scale with
the accumulated statistics until they reach limits at $0.003$, $0.004$ and
$0.012$ for $R_{D^{*}}$, $R_{D}$ and
$R_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$, and $0.006$ for both
$R_{D_{s}}$ and $R_{\Lambda_{c}}$.

Figure 25: Projected uncertainty for various $R_{H_{{s}}}$ ratios from the
Belle-II and LHCb experiments (years are indicative) in the range $\sim
1<{q^{2}}<6{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The Belle-II values
include estimates of the evolution of the systematic uncertainties (for
$R_{{{K}^{*}}}$, the charged and neutral channels have been combined). The
LHCb uncertainties are statistical only (the precision of all measurements
will be dominated by the size of the available data samples except for
$R_{{K}}$ and $R_{{{K}^{*}}}$ at 300$\mbox{\,fb}^{-1}$).

###  9.2 Experimental prospects for
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ decays

The projected precision for various $R_{H_{{s}}}$ ratios from the Belle-II and
LHCb experiments is shown in Fig. 25, where the precision on $R_{{K}^{(*)}}$
is expected to be comparable between the two experiments at the end of the
Belle-II data taking (around 2025). The Belle-II measurements are expected to
be completely dominated by the statistical uncertainty. For LHCb, a linear
dependence of the ${b}{\overline{b}}$ production cross section on the centre-
of-mass energy and unchanged detector performances compared to Run1 are
assumed. This is a conservative approximation since from 2021 the bottleneck
of the hardware trigger level will be removed and efficiencies for channels
with electrons and hadrons in the final state will increase significantly.
However, the larger pile up in the Upgrade-I could increase the background
contamination.

The $R_{{K}}$ and $R_{{{K}^{*}}}$ ratios are the only tests of LU in
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ decays that have been performed
so far. The same underlying quark transition can also be investigated with
other $b$-hadrons at the LHC. In particular, the $R_{\phi}$
(${{B}^{0}_{s}}\\!\rightarrow\phi{\ell^{+}}{\ell^{-}}$) and $R_{{p}{K}}$
(${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{K}{\ell^{+}}{\ell^{-}}$)
ratios can be measured to a good precision thanks to the large production
cross sections of ${B}^{0}_{s}$ and ${\mathchar 28931\relax}^{0}_{b}$ hadrons.
The $R_{\phi}$ ratio is particularly interesting since the background level is
expected to be low due to the absence of higher hadronic resonances decaying
into final states with a $\phi$ meson. The $R_{{K}}$ and $R_{H_{{s}}}$ ratios
involving vector particles (e.g. ${K}^{*}$ and $\phi$) are differently
affected by NP contributions, which will put important constraints on the
structure of these contributions [121, 122].

In the case that there will be no change in the central value currently
observed for $R_{{K}}$ and $R_{{{K}^{*}}}$ by the LHCb experiment, around 2025
both Belle-II and LHCb will be able to confirm LU violation in
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ transitions with a precision
significantly larger than 5$\,\sigma$. For LHCb, the current systematic
uncertainties are of the order of 4–5%, but a significant part of these is
statistical in nature. Starting from 2018, the ATLAS and CMS experiments are
recording data with a $B$ trigger in order to measure the $R_{{K}}$ and
$R_{{{K}^{*}}}$ ratios, however, the expected precisions are not yet known.

Concerning angular observables, the LHCb collaboration has measured the
$P_{4}^{{}^{\prime}}$ and $P_{5}^{{}^{\prime}}$ observables in the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{\mu^{+}\mu^{-}}$ decay mode [135] with a
precision of the order of 11%. An angular analysis of the
${{B}^{0}}\\!\rightarrow{{K}^{*0}}{e^{+}}{e^{-}}$ decay has been performed in
a different $q^{2}$ region with an observed yield of about 120 events [236].
An extrapolation of this work indicates that it should be possible to
distinguish between different NP scenarios using these modes [235]. In the
case of Belle-II, the precision on the $Q_{4}$ and $Q_{5}$ observabls is
expected to reach about 5% for the three $q^{2}$ bins (1–2.5, 2.5–4 and
4–6${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$) [234].

The size of the LHCb data sample after the Upgrade-II (300$\mbox{\,fb}^{-1}$)
makes this a unique place to search for LU breaking in
${b}\\!\rightarrow{d}\,{\ell^{+}}{\ell^{-}}$ due to the smallness of their
branching fractions. In particular, the expected statistical precision for
$R_{\pi}={\mathcal{B}}({{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{\mu^{+}\mu^{-}})/{\mathcal{B}}({{B}^{\pm}}\rightarrow{{\pi}^{\pm}}{e^{+}e^{-}})$
in the $q^{2}$ range between 1.1 and
6${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ is around 4%.

###  9.3 Additional observables

The LU deviations observed in both FCCC and FCNC $b$-quark decays can be
analysed by invoking NP occurring through different kinds of operators (in the
EFT approach) or different types of underlying dynamics (for UV complete
models). Increasing the number of observables would thus be beneficial to shed
light on the NP that could be at play.

More specifically, ${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$
transition to each lepton flavour could be investigated by considering not
only ratios, but also angular observables [87, 88], the polarisation of the
$\tau$ lepton or that of the ${D}^{*}$ meson (for
${B}\\!\rightarrow{{D}^{*}}\ell\nu$ transitions) [92, 237], final states with
mesons of higher spins like ${D}^{**}$ [238] or baryon decays [239]. As far as
LU is concerned, one could look at ratios or differences of all these
observables. Moreover, ratios of branching fractions for electron and muon
modes should be tested more precisely in order to determine if these also
exhibit LU violation.

For ${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$, a fairly comprehensive set
of angular observables, $J_{i}$ or $P_{i}$, has already been measured for
$\ell=\mu$. The same quantities could be measured for $\ell=e$ in order to
build LU observables such as $D_{i}=J_{i}^{\mu}-J_{e}^{\mu}$ [131] or
$Q_{i}=P_{i}^{\mu}-P_{i}^{e}$ [120], the latter offering smaller hadronic
uncertainties and ensuring an easier interpretation in terms of NP. Similar
observables should also be studied in other modes, e.g.
${{B}^{0}_{s}}\\!\rightarrow\phi{\ell^{+}}{\ell^{-}}$ or ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{K}{\ell^{+}}{\ell^{-}}$, to confirm the
hints of LU violation. Time-dependent measurements could also be considered
[240]. Another possible test of LU violation is provided by
$B_{s}\rightarrow\ell^{+}\ell^{-}\gamma$ [241]. Furthermore, it would be very
important to study ${b}\\!\rightarrow{s}\,{\tau^{+}}{\tau^{-}}$ transitions,
since general EFT considerations suggest that these could be significantly
enhanced compared to electron and muon modes [242, 243, 174]. A sensitivity
study [243] has been performed for the FCC-ee collider, which may run at the
$Z$ pole and collect of the order of $10^{12}$ $Z$ bosons. In this environment
a ${B}\\!\rightarrow{{K}^{*}}{\tau^{+}}{\tau^{-}}$ branching fraction at the
SM value could be measured with a precision of about 5%. Since a few thousands
${B}\\!\rightarrow{{K}^{*}}{\tau^{+}}{\tau^{-}}$ decays are expected to be
reconstructed, these could also be used to study the $\tau$ polarisation in
${b}\\!\rightarrow{s}\,{\tau^{+}}{\tau^{-}}$ transition and thus discriminate
among different NP scenarios.

As indicated in Sec. 8.2, the NP models proposing a common interpretation of
the LU deviations at tree- and loop-level must in general indicate how the
new, heavy, particles couple to the various generations of quarks and leptons.
Further LU tests in ${b}\\!\rightarrow{d}\,{\ell^{+}}{\ell^{-}}$ and
${b}\\!\rightarrow{u}\,\ell^{-}{\overline{\nu}}_{\ell}$ transitions could thus
be used as constraints for these models. In a similar way, more stringent
bounds on lepton-flavour violating decays
${b}\\!\rightarrow{s}\ell_{1}^{+}\ell_{2}^{-}$ (with $\ell_{1}\neq\ell_{2}$)
could also constrain NP models, as lepton-flavour violation occurs often in
models violating LU [178]. The ${B}\\!\rightarrow{K}^{(*)}e\mu$ and
${B}\\!\rightarrow{K}\tau\ell^{\prime}$ ($\ell^{\prime}$ = $e,\mu$) decays
have been studied by the BaBar collaboration and limits are of the order of
few $10^{-7}$ and $10^{-5}$, respectively [244, 173]. The expected limits by
LHCb with an integrated luminosity of about 9$\mbox{\,fb}^{-1}$ are of the
order of $10^{-9}$ and $10^{-6}$ for the ${B}\\!\rightarrow{K}^{(*)}e\mu$ and
${B}\\!\rightarrow{K}^{(*)}\tau\mu$ decays, respectively. These limits are
expected to scale linearly with luminosity for
${B}\\!\rightarrow{K}^{(*)}e\mu$ and with its square root for the
${B}\\!\rightarrow{K}^{(*)}\tau\mu$ decays [235].

##  10 Conclusions

In recent years, several observations of tree- and loop-level $b$-hadron
decays hint at a possible violation of LU. This article has discussed the
relevant measurements as well as their potential implications.

LU plays a peculiar role in the SM of particle physics, as the same
interactions and couplings characterise all three fermion generations. As a
consequence, a violation of LU would be an unambiguous sign of the existence
of physics beyond the SM. This property has been tested throughout the years
by using a variety of different probes: the production and the decay of
electroweak gauge bosons, the decay of quarkonia, the leptonic and
semileptonic decays of mesons with light quarks (including the $c$ quark),
however, no significant signs of deviations from the SM predictions have been
observed.

A possible violation of LU is hinted at in two different classes of
semileptonic $b$-quark decays. The measurements are obtained from experiments
at the $B$-factories (BaBar, Belle and soon Belle-II) as well as at the LHC
(LHCb). For these processes the SM predictions can be computed using an
effective Hamiltonian approach that separates short- and long-distance
contributions, and require non-perturbative inputs (e.g. form factors)
obtained through diverse theoretical methods.

Tensions at the level of 4–5$\,\sigma$ are observed in
${b}\\!\rightarrow{c}\,\ell^{-}{\overline{\nu}}_{\ell}$ decays, which are
mediated at tree level through a $W^{\pm}$ boson in the SM, when the branching
ratios of decays with $\ell=\tau$ and $\ell=e,\mu$ are compared. Deviations at
the level of 3–4$\,\sigma$ are also present in
${b}\\!\rightarrow{s}\,{\ell^{+}}{\ell^{-}}$ decays, which are mediated
through a loop in the SM, when comparing the branching ratios for $\ell=e$ and
$\ell=\mu$.

Such deviations can be interpreted theoretically in three steps. Firstly,
constraints on NP contributions to the short-distance Wilson coefficients of
the Hamiltonian can be extracted from the experimental results using the
effective Hamiltonian approach. Secondly, simple NP models that provide the
desired short-distance contributions via the exchange of one or two mediators
can be built without contradicting the limits on direct production or indirect
contribution to low-energy processes. Finally, full-fledged NP models with a
proper ultraviolet completion can be designed to match the previous models at
lower energies.

In this context, it is of paramount importance to confirm or refute these
hints of LU violation promptly. Both the Belle-II and LHCb experiments will be
in an ideal position to provide additional information by significantly
reducing the uncertainties on the LU observables already studied and by
measuring new observables that will further constrain NP models. The present
situation should thus evolve rapidly with the combined efforts of
experimentalists and theorists, and has the potential to provide very exciting
news in the coming years.

## Acknowledgements

The authors warmly thank Yasmine Amhis, Martino Borsato, Juan J. Saborido and
Javier Virto for carefully proofreading the article. The authors also thank
Martino Borsato for helping with the figures of the projected sensitivities.

SB acknowledges support from the UK STFC Research Council under the grant
agreement No. ST/N000463/1. SDG acknowledges partial support from Contract
FPA2014-61478-EXP and from the European Union Horizon 2020 research and
innovation programme under grant agreements No. 690575, No. 674896 and No.
692194. ARV acknowledges support from the Xunta de Galicia under the grant
agreement No. 2016-PG003.

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