UdeM-GPP-TH-21-289

$B$ Flavour Anomalies:
2021 Theoretical Status Report111To be published in the Annual Review of
Nuclear and Particle Science

David London<EMAIL_ADDRESS>and Joaquim Matias
<EMAIL_ADDRESS>
$a$: Physique des Particules, Université de Montréal,
1375 Avenue Thérèse-Lavoie-Roux, Montréal, QC, Canada H2V 0B3

$b$: Universitat Autònoma de Barcelona,
E-08193 Bellaterra (Barcelona), Spain

$c$: Institut de Física d’Altes Energies and
The Barcelona Institute of Science and Technology, Spain

()
Abstract
At the present time, there are discrepancies with the predictions of the SM in
several observables involving $b\to s\ell^{+}\ell^{-}$ and $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$ decays. These are the $B$ flavour anomalies. In
this review, we summarize the data as of Moriond 2021 and present theoretical
new-physics explanations from both a model-independent effective-field-theory
point of view and through the building of explicit models. Throughout, we
stress the complementarity of these two approaches. We also discuss combined
explanations of both $B$ anomalies, and present models that also explain other
problems, such as dark matter, $(g-2)_{\mu}$, neutrino properties, and
hadronic anomalies.

##  1 Introduction

At the present time (2021), there are a number of measurements of observables
involving $B$-meson decays that are in disagreement with the predictions of
the standard model (SM). These are the $B$ flavour anomalies. There are two
types. The neutral-current anomalies are found in processes involving $b\to
s\ell^{+}\ell^{-}$ decays, while the charged-current anomalies involve $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$.

The first observation of the $b\to s\ell^{+}\ell^{-}$ anomaly was in 2013. At
the EPS conference in Stockholm, the LHCb Collaboration announced [1] that
they had measured the observable $P^{\prime}_{5}$ [2] in the four-body angular
distribution $B_{d}\to K^{*0}(\to K^{+}\pi^{-})\mu^{+}\mu^{-}$, and found that
it disagreed with the SM prediction at the level of 3.7$\sigma$ [3]. Soon
thereafter, another, smaller discrepancy of $2.6\sigma$ was found in the
measurement of $R_{K}\equiv{\cal B}(B^{+}\to K^{+}\mu^{+}\mu^{-})/{\cal
B}(B^{+}\to K^{+}e^{+}e^{-})$ [4]. The SM predicts $R_{K}\simeq 1$ [5], so the
experimental result suggests the violation of lepton flavour universality,
i.e., the violation of the property that the interactions between gauge bosons
and charged leptons are the same for all generations of leptons. Since then,
the number of measured observables involving $b\to s\ell^{+}\ell^{-}$ decays
has grown enormously, and many of these also exhibit discrepancies with the
SM. What is particularly striking here is the consistency of these
measurements: the discrepancies all go in the same direction, pointing to a
common new-physics explanation.

The first measurement of a $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ anomaly was in
2012: the BaBar Collaboration measured $R_{D^{(*)}}\equiv{\cal B}({\bar{B}}\to
D^{(*)}\tau^{-}{\bar{\nu}}_{\tau})/{\cal B}({\bar{B}}\to
D^{(*)}\ell^{-}{\bar{\nu}}_{\ell})$ ($\ell=e,\mu$), and found values for
$R_{D}$ and $R_{D^{*}}$ that, when taken together, exceed the SM expectation
by $3.4\sigma$ [6]. These measurements were repeated by BaBar (2013) [7],
Belle (2015) [8] and LHCb (2015) [9], and the results were largely confirmed.
Here too, measurements of additional $b\to c\ell^{-}{\bar{\nu}}_{\ell}$
observables have been made subsequently (though far fewer than in the $b\to
s\ell^{+}\ell^{-}$ case), and other deviations from the SM predictions have
been found.

Now, the SM has made a great number of predictions, most of which have been
confirmed (e.g., the discovery of the Higgs boson). However, it also leaves
many questions unanswered (what is dark matter?, what is the origin of the
baryon asymmetry of the universe? why are there three generations?, etc.). For
this reason, it is clear that the SM is not complete – there must be physics
beyond the SM. The $B$ flavour anomalies have been around for almost ten
years. They have been observed in a variety of processes, and by different
experiments. We may in fact be seeing the first signs of new physics. For this
reason, the study of the $B$ flavour anomalies has become a very exciting
subject. A great deal of work has been done, by many people, with the aim of
finding explanations of the $b\to s\ell^{+}\ell^{-}$ and $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$ anomalies, either individually or simultaneously.

In this review we present a status report on the $B$ flavour anomalies from a
theoretical perspective. For each of the $b\to s\ell^{+}\ell^{-}$ and $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$ anomalies, we examine possible solutions from an
effective-field-theory point of view (global analyses), and in the context of
model building. Throughout, we stress the complementarity of these two
approaches. We also review the combined explanations of the two types of
anomalies, as well as models that attempt to also explain other discrepancies
with the SM (dark matter, $(g-2)_{\mu}$, etc.). Finally, we take a glance into
the future, pointing out interesting observables whose measurement can
distinguish various explanations.

Notation: The material in this review is taken from many different sources,
which have different notations. We have tried to uniformize this notation as
follows. In currents, $Q$ and $L$ represent left-handed (LH) $SU(2)_{L}$ quark
and lepton doublets, respectively, of any generation. Similarly, $u$, $d$ and
$e$ represent right-handed (RH) $SU(2)_{L}$-singlet up-type quarks, down-type
quarks and charged leptons, respectively, of any generation. For all of these
fields, the chirality is not explicitly indicated. The symbols $b$, $s$, $c$,
$\mu$, $\tau$ and $\nu_{\tau}$ refer to the actual particles. If it is
necessary to specify the chirality of the $b$, we write $b_{L,R}$ or
$P_{L,R}\,b$, where $P_{L,R}=(1\mp\gamma_{5})/2$ are the LH and RH projection
operators, and similarly for the other particles. Finally, $\ell$ stands for
$e$, $\mu$ or $\tau$, unless otherwise indicated.

##  2 EFTs and model building: two complementary approaches

The $B$ anomalies suggest the presence of physics beyond the SM. The next
question is then: what can this new physics (NP) be? There are two different
ways of addressing this question. The first uses a model-independent
effective-field-theory (EFT) approach: an effective Hamiltonian is written
that contains all dimension-6 operators that describe the decay of interest
($b\to s\ell^{+}\ell^{-}$ or $b\to c\ell^{-}{\bar{\nu}}_{\ell}$). A global fit
to all the data is performed to find the preferred values of the coefficients
of these operators (the Wilson coefficients (WCs)). The second approach is to
construct models of NP that reproduce the data. Both methods have their
advantages and disadvantages; by combining the two, one can get a complete
idea of possible NP solutions.

The exchange of NP particles generates new interactions among the SM
particles, in particular the $b\to s\ell^{+}\ell^{-}$ or $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$ four-fermion operators. Now, to date, no new
particles have been observed at the LHC. As a consequence, it is generally
believed that the NP, whatever it is, must be heavy, with masses of ${\cal
O}({\rm TeV})$. This is above the electroweak scale, so that, when the NP
particles are integrated out, the EFT produced is invariant under the full SM
gauge group, $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$. This is the SM
effective field theory (SMEFT). In it, all dimension-4 operators are those of
the SM; higher-dimension operators are due to NP. These additional operators
are suppressed by powers of $\Lambda$, the scale of NP. It is the SMEFT that
provides the connection between the two approaches.

The operators in the SMEFT are independent. However, a particular model will
generate only a subset of all SMEFT operators, and the coefficients of these
operators will often be related. This means that a model analysis must take
into account additional constraints, not just those of the $B$ anomalies.

While the SMEFT applies to TeV-scale physics, the EFT used in the global fits
is appropriate for physics at the scale $m_{b}$. That is, it is invariant
under $SU(3)_{C}\times U(1)_{em}$ and all particles with masses greater than
$m_{b}$ have been integrated out. This is the WET (weak effective theory).

Since they apply to lower-energy physics, all WET operators can be mapped to
SMEFT operators. This has several consequences. First, those WET dimension-6
operators that also obey $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ are
related to (combinations of) dimension-6 SMEFT operators. But those WET
operators that do not respect this symmetry are mapped at tree level to
higher-dimensional SMEFT operators, typically dimension 8. These dimension-8
operators involve additional Higgs fields, and the WET operators are generated
when the Higgs gets a vev $v$. Thus, the WET operators are suppressed by an
additional factor of $v^{2}/\Lambda^{2}$, and can be neglected.

Another effect, more related to model building, is the following. EFT analyses
often use a basis for the dimension-6 WET operators that is not the same as
that of SMEFT. That is, a dimension-6 WET operator may be mapped to a linear
combination of dimension-6 SMEFT operators. However, within the SMEFT, these
operators are a-priori independent. Thus, if the EFT analysis favours NP in a
WET operator that is a combination of SMEFT operators, this indicates that
there must be an additional symmetry relating these operators. This is a clue
for the model builders. (And if such a symmetry does not seem to exist, or is
very contrived, then perhaps this EFT scenario will be disfavoured in spite of
the good fit.)

Indeed, using EFT global-fit analyses, a variety of scenarios have been
proposed in which the NP contributes to different WCs. For several of these
scenarios, models can be constructed with the help of the SMEFT. In addition,
by looking at SMEFT operators, one sees that a simultaneous explanation of
both $b\to s\ell^{+}\ell^{-}$ and $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ anomalies
is possible.

The bottom line is that the EFT and model-building approaches are
complementary, and the addition of SMEFT information provides a complete
overview of possible NP explanations of the $B$ flavour anomalies.

##  3 Neutral-current anomalies: $b\to s\ell^{+}\ell^{-}$

We begin by examining the neutral-current $B$ flavour anomalies observed in
$b\to s\ell^{+}\ell^{-}$ transitions. In the SM, this decay occurs at the one-
loop level.

###  3.1 $b\to s\ell^{+}\ell^{-}$ observables

There are two different categories of $b\to s\ell^{+}\ell^{-}$ observables
that exhibit tension with the predictions of the SM:

  1. 1.

Observables involving only the $b\to s\mu^{+}\mu^{-}$ transition. Here there
are two types that differ in their sensitivity to hadronic uncertainties and
NP:

     * •

Branching ratios include ${\cal B}({B^{0}\to K^{0*}\mu^{+}\mu^{-}})$, ${\cal
B}({B^{+}\to K^{*+}\mu^{+}\mu^{-}})$,
${\cal B}({B_{s}\to\phi\mu^{+}\mu^{-}})$, ${\cal B}({B^{+}\to
K^{+}\mu^{+}\mu^{-}})$ and ${\cal B}({B^{0}\to K^{0}\mu^{+}\mu^{-}})$.

     * •

Observables that parametrize the four-body angular distribution $B\to
K^{*}(\to K\pi)\mu^{+}\mu^{-}$ or $B_{s}\to\phi(\to
K^{+}K^{-})\mu^{+}\mu^{-}$. These distributions are functions of the three
angles that characterize the final four-particle state and $q^{2}$, the
dilepton invariant mass-squared (see Ref. [10]). The $q^{2}$ spectrum can be
split into three regions: (i) large $K^{*}$ recoil, $4m_{\ell}^{2}\leq
q^{2}\leq 9$ GeV2: here, $E_{K^{*}}\gg\Lambda_{QCD}$, this includes the region
near the photon pole, (ii) charmonium region, $9<q^{2}<14$ GeV2: this includes
the charmonium resonances ($J/\psi$, etc.), (iii) low $K^{*}$ recoil,
$14<q^{2}<(m_{B}-m_{K^{*}})^{2}$ GeV2: here, $E_{K^{*}}\simeq\Lambda_{QCD}$.
Each region requires a different treatment of hadronic uncertainties.

The angular observables can be further separated into two categories,
differentiated by their sensitivity to hadronic effects. The non-optimized
observables (the $S_{i}$ [11]) are more sensitive to hadronic uncertainties,
while the optimized observables (the $P_{i}$ [10, 13, 14])444The first
optimized observable, $A_{T}^{i}$, was introduced in Ref. [10], followed some
years later by two additional transverse asymmetries in Ref. [12]. A complete
basis to describe the four-body angular distribution was introduced afterwards
in Ref. [13], and it was redefined in Ref. [14] to adapt more easily to the
experimental measurements. A generalization of the angular-distribution
formalism can be found in Ref. [15]. are constructed so that the dependence on
soft form factors cancels exactly at leading order. They are usually measured
in $q^{2}$ bins.

  2. 2.

Observables that measure lepton-flavour-universality violation (LFUV). The
gauge bosons of the SM couple identically (i.e., universally) to charged
leptons of different generations. The LFUV observables include

$\displaystyle R_{K}=\frac{\displaystyle{\cal B}(B^{+}\to
K^{+}\mu^{+}\mu^{-})}{\displaystyle{\cal B}(B^{+}\to
K^{+}e^{+}e^{-})}~{}~{},~{}~{}~{}~{}R_{K^{*}}=\frac{\displaystyle{\cal B}(B\to
K^{*}\mu^{+}\mu^{-})}{\displaystyle{\cal B}(B\to K^{*}e^{+}e^{-})}~{}~{},$
$\displaystyle R_{\phi}=\frac{\displaystyle{\cal
B}(B_{s}\to\phi\,\mu^{+}\mu^{-})}{\displaystyle{\cal
B}(B_{s}\to\phi\,e^{+}e^{-})}~{}.$ (1)

The SM predicts that all of these ratios equal 1 (up to tiny lepton mass and
electromagnetic effects, see Refs. [16, 17]). A deviation from this prediction
in the measurement of any of these observables would signal the breaking of
lepton-flavour universality. $R_{K}$ and $R_{K^{*}}$ have been measured by
LHCb [18, 19] and Belle [20, 21]. Additional LFUV observables can be
constructed using optimized observables [22]:

$\quad Q_{i}=P_{i}^{\prime\mu}-P_{i}^{\prime e}$ (2)

The SM predicts them to vanish to high accuracy. Two of these – $Q_{4,5}$ –
have already been measured by Belle [23].

Three other types of observables are usually included in the global fits. (i)
There are inclusive decays such as $B\to X_{s}\mu^{+}\mu^{-}$ (these
measurements are still not very precise). (ii) ${\cal
B}(B_{s}\to\mu^{+}\mu^{-})$ has been measured by LHCb, ATLAS and CMS. This
observable is particularly interesting due to its reduced hadronic sensitivity
and its dependence on a reduced subset of Wilson coefficients (see next
section). The discrepancy of this measurement with the SM is at the level of
$\sim 2\sigma$. (iii) There are radiative observables such as $B\to
X_{s}\gamma$, $B_{s}\to\phi\gamma$ and $B\to K^{*}\gamma$. Further details on
the experimental status of all of these observables can be found in Ref. [24].

###  3.2 EFT analysis

####  3.2.1 Operators

In the most general case, $b\to s\ell^{+}\ell^{-}$ transitions are described
by an effective Hamiltonian [25]555 This operator basis was chosen in Ref.
[25] because it is useful to describe the SM contributions to $b\to
s\ell^{+}\ell^{-}$. Historically, this basis has also been used to describe
the NP contributions. However, other choices of basis are equivalent and
equally valid.

${\cal H}_{\rm
eff}=-\frac{4G_{F}}{\sqrt{2}}\,V_{tb}V_{ts}^{*}\,\frac{e^{2}}{16\pi^{2}}\sum_{i=1}^{12}{\cal
C}_{i}{\cal O}_{i}~{},$ (3)

where $V_{ij}$ are elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, and
the operators are666Note that, in the literature, the scalar and pseudoscalar
operators are sometimes multiplied by $m_{b}$.

$\displaystyle\hskip-62.59596pt{\mathcal{O}}_{7}^{(\prime)}=\frac{m_{b}}{e}\,(\bar{s}\sigma_{\mu\nu}P_{R(L)}b)F^{\mu\nu}~{},$
$\displaystyle{\mathcal{O}}_{9\ell}^{(\prime)}=(\bar{s}\gamma_{\mu}P_{L(R)}b)(\bar{\ell}\gamma^{\mu}\ell)$
$\displaystyle~{},~{}$
$\displaystyle{\mathcal{O}}_{10\ell}^{(\prime)}=(\bar{s}\gamma_{\mu}P_{L(R)}b)(\bar{\ell}\gamma^{\mu}\gamma_{5}\ell)~{},$
$\displaystyle{\mathcal{O}}_{S\ell}^{(\prime)}=(\bar{s}P_{R(L)}b)(\bar{\ell}\ell)$
$\displaystyle~{},~{}$
$\displaystyle{\mathcal{O}}_{P\ell}^{(\prime)}=(\bar{s}P_{R(L)}b)(\bar{\ell}\gamma_{5}\ell)~{},$
$\displaystyle{\mathcal{O}}_{T\ell}=(\bar{s}\sigma_{\mu\nu}b)(\bar{\ell}\sigma^{\mu\nu}\ell)$
$\displaystyle~{},~{}$
$\displaystyle{\mathcal{O}}_{T5\ell}=(\bar{s}\sigma_{\mu\nu}b)(\bar{\ell}\sigma^{\mu\nu}\gamma_{5}\ell)~{}.$
(4)

With ${\cal H}_{\rm eff}$, the short- and long-distance physics are separated:
all the information about the heavy degrees of freedom that have been
integrated out (both SM and NP particles) is encoded in the Wilson
coefficients of the operators, ${\cal C}_{i}$.

It is useful to relate these operators to those of the SMEFT. To this end, we
take linear combinations of them to form operators involving only LH and RH
fermions:

$\displaystyle{\mathcal{O}}_{V\ell}^{ij}=(\bar{s}\gamma_{\mu}P_{i}b)(\bar{\ell}\gamma^{\mu}P_{j}\ell)~{}~{},~{}~{}~{}~{}{\mathcal{O}}_{S\ell}^{ij}=(\bar{s}P_{i}b)(\bar{\ell}P_{j}\ell)~{}~{},$
$\displaystyle{\mathcal{O}}_{T\ell}^{i}=(\bar{s}\sigma_{\mu\nu}P_{i}b)(\bar{\ell}\sigma^{\mu\nu}P_{i}\ell)~{}~{},~{}~{}~{}~{}i,j=L,R~{}.$
(5)

Of these ten WET operators, four of them – ${\mathcal{O}}_{S\ell}^{LL}$,
${\mathcal{O}}_{S\ell}^{RR}$, ${\mathcal{O}}_{T\ell}^{L}$ and
${\mathcal{O}}_{T\ell}^{R}$ – are not generated at dimension 6 in SMEFT [26];
in fact, they are generated at dimension 8 [137]. They are therefore
suppressed by $v^{2}/\Lambda^{2}$ compared to the other six operators, and can
be neglected. The two remaining scalar operators are often written in the
literature as

${\mathcal{O}}_{S\ell}^{(\prime)}=(\bar{s}P_{R(L)}b)(\bar{\ell}\ell)~{}~{},~{}~{}~{}~{}{\mathcal{O}}_{P\ell}^{(\prime)}=(\bar{s}P_{R(L)}b)(\bar{\ell}\gamma_{5}\ell)~{},$
(6)

with conditions imposed on the WCs: $C_{S\ell}=-C_{P\ell}$ and
$C^{\prime}_{S\ell}=C^{\prime}_{P\ell}$ [26].

${\cal H}_{\rm eff}$ therefore contains eight operators describing $b\to
s\ell^{+}\ell^{-}$ transitions. The SM contributes mainly to the WCs of three
of these: ${\mathcal{O}}_{7}$, ${\mathcal{O}}_{9\ell}$ and
${\mathcal{O}}_{10\ell}$ (there is also a tiny contribution to ${\cal
C}^{\prime}_{7}$). The values of their coefficients at the scale $\mu=4.8$ GeV
are [27, 28, 29, 30, 31]

${\cal C}_{7}^{\rm SM}(\mu_{b})=-0.29~{}~{},~{}~{}~{}~{}{\cal C}_{9}^{\rm
effSM}(\mu_{b})=4.1~{}~{},~{}~{}~{}~{}{\cal C}_{10}^{\rm
SM}(\mu_{b})=-4.3~{}.$ (7)

NP can contribute to all eight operators. The WCs encode the short-distance
physics and contain both SM and NP information: ${\cal C}_{i}={\cal
C}_{i}^{\rm SM}+{\cal C}_{i}^{\rm NP}$. All of these operators (or subsets of
them) enter the observables discussed above. For instance, the angular and
LFUV observables receive contributions from all operators (or combinations of
them), radiative observables probe the electromagnetic operators
${\mathcal{O}}^{(\prime)}_{7}$, and ${\cal B}_{B_{s}\to\mu^{+}\mu^{-}}$ is
sensitive to the scalar and vector-axial operators.

####  3.2.2 Methodology

In Sec. 3.1, we presented the list of $b\to s\ell^{+}\ell^{-}$ observables,
some of which exhibit deviations from the predictions of the SM. In order to
determine whether these form a coherent set of deviations that have a common
explanation, it is necessary to perform a global fit of all observables. There
are two approaches to performing such a statistical analysis, the frequentist
and the Bayesian methods. The frequentist approach is most commonly used to
analyze the $b\to s\ell^{+}\ell^{-}$ anomalies, and this is what we focus on
in this review. (See Refs. [32, 33, 34] for Bayesian analyses.)

Global fits have been performed by a number of groups, but there are three in
particular that have provided regular updates777Other recent global fits can
be found in Refs. [35, 36].. We refer to these groups as ACDMN (Algueró/Capde-
vila/Descotes-Genon/Matias/Novoa-Brunet) [37, 38, 49], AS
(Altmannshofer/Stangl) [50] and HMMN (Hurth/Mahmoudi/Martinez
Santos/Neshatpour) [52]. (These names appear on the latest update; for all
three groups, other authors have been involved over the years.) In this
review, we compare the methodologies and results of all three groups, but we
occasionally focus a bit more on the results of ACDMN. In performing a fit, it
is necessary to choose a basis for the $b\to s\ell^{+}\ell^{-}$ operators, and
the basis of Eq. (4) is usually used. (However, when the results are compared
with the SMEFT, it will be useful to take linear combinations of these
operators.)

The analysis of ACDMN is done using a likelihood approach to fit the
deviations from the SM values of the relevant WCs, taking into account the
experimental and theoretical uncertainties as well as their correlations in
Gaussian approximation. The main differences in the set-ups of the global fits
of ACDMN, AS and HMMN are:

  * •

Number of observables. There are two types of fits: (i) a complete fit,
including all observables, and (ii) a fit including only the LFUV observables.
For ACDMN, the complete fit includes all available data (with one important
exception, discussed below), for a total of 246 observables (from LHCb, ATLAS,
CMS and Belle). The LFUV fit includes $R_{K,K^{*}},Q_{4,5}$ and the radiative
decays. In contrast, for their complete fit, AS uses only a subset of the
available data, with a total of 130 observables [51]. But for their LFUV fit,
they use the same observables as ACDMN. Finally, the complete fit of HMMN is
reasonably exhaustive, including 173 observables. But the $Q_{i}$ are left out
of the LFUV fit in HMMN.

The one process whose observables are excluded from the analysis of ACDMN, but
is included in that of AS and HMMN, is the baryonic decay
$\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$. The reason for its exclusion is as
follows. The normalization of ${\cal B}(\Lambda_{b}\to\Lambda\mu^{+}\mu^{-})$
was taken by LHCb from LEP and Tevatron, and required a correct scaling of the
ratio of productions between LHC, LEP and Tevatron. However, while the
branching ratio is provided in bins at LHCb, it was never measured in bins at
Tevatron. Furthermore, combining LEP and Tevatron data is delicate, given the
strong dependence on the $b$-quark production process. Thus, the ratio of
productions depends also on the kinematics, so that taking this input from LEP
and Tevatron introduces some uncertainty and model dependence in the
normalization [33, 39].

  * •

Wilson coefficients included in the global fits. The $b\to s\ell^{+}\ell^{-}$
operators that can receive NP contributions can be separated into three types:
electromagnetic (${\mathcal{O}}_{7}^{(\prime)}$, Eq. (4)), vector/axial vector
(${\mathcal{O}}_{9\ell}^{(\prime)}$ and ${\mathcal{O}}_{10\ell}^{(\prime)}$,
Eq. (4)), and scalar/pseudoscalar (${\mathcal{O}}_{S\ell}^{(\prime)}$ and
${\mathcal{O}}_{P\ell}^{(\prime)}$, Eq. (6)). In ACDMN’s analysis, all
electromagnetic and vector/axial vector operators are included, but not
scalar/pseudoscalar operators. In the analysis of AS, the vector/axial vector
and scalar/pseudoscalar operators are included, but not electromagnetic
operators. And HMMN include all operators.

  * •

Treatment of the covariance matrix. ACDMN use the covariance matrix in the SM
after checking in different points of the NP parameter space that the
variation of the covariance matrix is not significant. On the other hand, AS
compute a covariance matrix at each point. This is exact for branching ratios
which are second-order polynomials in the WCs, but for the angular observables
and LFUV ratios it is done approximately by expanding the numerator and
denominator up to second-order polynomials, assuming small NP contributions to
the WCs compared to the SM. The range of validity of this approximation is
roughly the same as taking the SM covariance matrix everywhere.

####  3.2.3 Hadronic uncertainties

The impact of hadronic uncertainties on the observables of the $B$ anomalies
has been object of very intense debates in the past (for a complete
discussion, see Refs. [40, 41]). However, important progress on these
uncertainties from explicit computations has been made recently, and this has
clarified the situation.

There are two main sources of hadronic uncertainties affecting exclusive $B$
decays. The first is form factors (FFs). These are local contributions that
parametrize the transition of a $B$ meson to a final-state meson $M$. In our
case, we are interested in the FFs for $B\to K^{(*)}$ and $B_{s}\to\phi$. The
second is non-local contributions associated with charm-quark loops. In what
follows, we denote these as $T_{\mu}$.

The amplitude for the decay $B\to M\ell^{+}\ell^{-}$ has the following
structure in the SM:

$A(B\to
M\ell^{+}\ell^{-})=\frac{G_{F}\alpha}{\sqrt{2}\pi}V_{tb}V_{ts}^{*}[({A_{\mu}}+{T_{\mu}})\bar{u}_{\ell}\gamma^{\mu}v_{\ell}+{B_{\mu}}\bar{u}_{\ell}\gamma^{\mu}\gamma_{5}v_{\ell}]~{},$
(8)

where ${A_{\mu}}=-(2m_{b}q^{\nu}/{q^{2}}){\cal C}_{7}\langle
M|\bar{s}\sigma_{\mu\nu}P_{R}b|B\rangle+{\cal C}_{9}\langle
M|\bar{s}\gamma_{\mu}P_{L}b|B\rangle$ and ${B_{\mu}}={\cal C}_{10}\langle
M|\bar{s}\gamma_{\mu}P_{L}b|B\rangle$. The local contributions from FFs are
included in $A_{\mu}$ and $B_{\mu}$. At low $q^{2}$, these FFs are computed
using light-cone sum rules (LCSR) with $B$-meson [42] or light-meson
distribution amplitudes (DAs) [44]. Recently, important updates that include
higher-twist effects in the $B$-meson DA have been presented in the literature
[43]. At high $q^{2}$, FFs are taken mainly from lattice computations. In
addition, it is customary to extrapolate the lattice computations to low
$q^{2}$ in order to reduce the uncertainty of the LCSR low-$q^{2}$
computation. The analysis of ACDMN is done using the $B$-meson DA, while AS
and HMMN take the light-meson DA results from Ref. [44].

Turning to the non-local charm-quark loop contributions, the fact that
$T_{\mu}$ is inserted at the same place in the amplitude as $A_{\mu}$ is
problematic: how can we disentangle a possible charm-quark loop contribution
from a NP contribution to ${\cal C}_{7}$ or ${\cal C}_{9}^{\rm eff}$? Over the
years, this was the subject of intense (often acrimonious) debate. These non-
local charm-quark contributions are particularly important in the region of
charmonium resonances between the low- and large-$q^{2}$ regions, and may in
principle affect observables such as branching ratios, $S_{i}$, $R_{K^{*}}$ in
the presence of NP, or $P_{5}^{\prime}$. A number of different approaches were
used to tackle this problem. (i) The exchange of one soft gluon was computed
within LCSR [42]. It was found that this contribution increased the size of
the anomaly in $P_{5}^{\prime}$. (ii) Later on, a detailed next-to-leading-
order computation was done in Ref. [45], but it was found that the correction
to this leading-order result is very small. (iii) It is difficult to reliably
compute the phase difference between the long-distance charm contribution and
the short-distance physics. With this in mind, Ref. [41] conservatively
introduced three nuisance parameters (one for each amplitude), to allow for
constructive and destructive interference between the short-distance and long-
distance charm contributions computed in Ref. [42]. This effectively increases
the impact of the charm-quark loop contribution, reducing the anomaly. (iv)
Another approach was to perform a fit to the resonances (in modulo and phase)
and see if the tail of some resonance could explain the deviation in the
anomaly bins [46]. (v) Finally, Ref. [47] adopted an approach based on a
dispersive representation using the data on $J/\psi$ and $\psi(2S)$ to
determine the analytic structure of the correlation function of interest and
its $q^{2}$ dependence (up to a polynomial). In all cases, it was found that
it is not possible to explain the $P_{5}^{\prime}$ anomaly with $T_{\mu}$ in
the SM.

In Ref. [41], improved QCD factorization (QCDf) is used to include all of the
above corrections. The amplitude is decomposed as [48]

$\langle\ell^{+}\ell^{-}{\bar{K}}^{*}_{i}|H_{\rm
eff}|\bar{B}\rangle=\sum_{a,\pm}C_{i,a}\xi_{a}+\Phi_{B,\pm}\otimes
T_{i,a,\pm}\otimes\Phi_{K^{*},a}+{\cal O}(\Lambda_{\rm QCD}/m_{B})~{}.$ (9)

Here, $C_{i,a}$ and $T_{i,a}$ ($a=\perp,\|$) are perturbatively-computable
contributions to the various $K^{*}$ polarizations ($i=0,\perp,\|$) and
$\Phi_{B,K^{*}}$ stands for the light-cone DA of the $B$ and $K^{*}$ mesons.
There are two types of corrections. First, there are the factorizable
corrections that can be absorbed into the FFs. These include ${\cal
O}(\alpha_{s})$ and ${\cal O}(\Lambda_{\rm QCD}/m_{B})$ corrections. Second,
there are nonfactorizable corrections that cannot be absorbed into the FFs.
These also come in two types. There are those that originate from hard gluon
exchange; these are calculable within QCDf (including insertions of the
operators ${\mathcal{O}}_{1-6}$ and ${\mathcal{O}}_{8}$ [48]). And there are
those that are power corrections of ${\cal O}(\Lambda_{\textrm{QCD}}/m_{B})$,
which include the $c\bar{c}$ loops mentioned above. In Ref. [41], they are
included using the computation of [42], but allowing for a
constructive/destructive interference with the short-distance effect.

Finally, it is often stated that “LFUV observables are always clean.” However,
this is only true in the SM because of certain cancellations. In the presence
of NP, these cancellations do not occur. This is particularly visible in the
case of $R_{K^{*}}$ (see Fig. 3 in Ref. [49]). Thus, when considering NP, it
is important to take into account the hadronic uncertainties (form factors and
charm loops).

To recap, there are two sources of theoretical error in the EFT analysis of
the $B$ anomalies: form factors and the non-local charm-loop contributions.
There has been great progress in the computations of FFs, using different
approaches. In addition, the charm-loop contributions have been studied in
full detail using different explicit computational methods. The theoretical
error in the $b\to s\ell^{+}\ell^{-}$ anomaly is now under better control.

###  3.3 Global fits: results

When a global fit is performed, a hypothesis is made regarding which NP Wilson
coefficients are allowed to vary. This fixes the number of parameters of a
fit, $n$, and the number of degrees of freedom $N_{dof}=N_{obs}-n$, where
$N_{obs}$ is the number of observables. When the results are obtained, there
are two statistical quantities of interest:

  1. (i)

The p-value of a given hypothesis is a function of $\chi^{2}_{min}$ and
$N_{dof}$. This quantity is a measure of the goodness of the fit, i.e., it
indicates whether or not the hypothesis provides a good fit. For example, the
p-value of the SM fit to the $b\to s\ell^{+}\ell^{-}$ data is $1.1\%$, which
corresponds to a disagreement at the level of $\sim 2.5\sigma$. If the p-value
were to fall below $5\sigma$, the SM would be excluded.

  2. (ii)

PullSM provides very precise information on the comparison of the SM and a
given NP hypothesis as explanations of the data: it tells us by how many
$\sigma$s the SM hypothesis is disfavoured with respect to a given NP
hypothesis. It is related to the difference of the $\chi^{2}_{min}$s of the
two fits and to the difference in the $N_{dof}$. This allows us to
quantitatively compare the PullSMs of hypotheses with different numbers of
free parameters.

####  3.3.1 1D, 2D and 6D scenarios

We begin with global fits in which one NP WC is allowed to vary. These 1D fits
provide useful information: they indicate if there is a WC (or constrained
combination of WCs) in a given basis that is dominant in the explanations of
the $B$ anomalies. The three groups have all performed 1D fits, and there are
two solutions that are most preferred by all of them: ${\cal C}_{9\mu}^{\rm
NP}$ and ${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$888A third
possible solution, ${\cal C}_{10\mu}^{\rm NP}$, gives a very good description
of the data in the LFUV fit. However, it fails to provide a good description
of the data in the complete fit, particularly when compared with other NP
hypotheses.. The results for the ACDMN, AS and HMMN groups are shown in Tables
3, 3 and 3, respectively.

Above, we have pointed out the differences among the three groups as regards
the methodology (particularly the number of observables included in the fit)
and the choice of form factors. Despite these, the results in Tables 3, 3 and
3 are remarkably similar. Two types of fits were done: the complete fit, which
includes all observables used by the individual groups, and the LFUV fit,
which includes mainly observables sensitive to the breaking of lepton flavour
universality. For the LFUV fit, the results are virtually identical. For the
complete fit, the results of the ACDMN and HMMN groups are remarkably close.
(The excellent agreement between these two different analyses also extends to
higher-dimensional fits.) Tiny differences are due to the small difference in
the number of observables (and the inclusion of the baryonic $\Lambda_{b}$
decay in the HMMN analysis) and the different form factors. Still, the general
agreement between these two analyses shows that these effects are rather
marginal. On the other hand, there are greater differences with the AS result.
AS find that the Pulls for the ${\cal C}_{9\mu}^{\rm NP}$ and ${\cal
C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$ hypotheses are smaller than for
ACDMN and HMMN. Also, they find ${\cal C}_{9\mu}^{\rm NP}=-{\cal
C}_{10\mu}^{\rm NP}$ has a marginally larger pull than ${\cal C}_{9\mu}^{\rm
NP}$, which is opposite to ACDMN and HMMN. This is almost certainly due to the
smaller number of observables in the AS analysis – particularly the exclusion
of the [6,8] bin in all observables999 In contrast to ${\cal C}_{9\mu}$, the
solution ${\cal C}_{9\mu}=-{\cal C}_{10\mu}$ produces a SM-like value for
$P_{5}^{\prime}$ in both anomaly bins [4,6] and [6,8] (see Fig. 5 in Ref.
[38]). Since the [6,8] bin is problematic for the ${\cal C}_{9\mu}=-{\cal
C}_{10\mu}$ solution, and since AS have excluded this bin from their fit, this
might explain the preference of the AS fit for ${\cal C}_{9\mu}=-{\cal
C}_{10\mu}$. Note that important progress has been made recently in explicit
computations for the whole low-q2 region. – and, to a lesser extent, the
inclusion of $\Lambda_{b}\to\Lambda\mu^{+}\mu^{-}$ at low-recoil101010This
mode shows an unexpected preference at low-recoil for ${\cal C}_{9\mu}^{\rm
NP}>0$, in contradiction with the rest of the data. However, a recent
reinterpretation excluding LEP data finds a result in nice agreement with the
rest of the data: it shows a small deficit in this channel instead of an
excess [33]..

ACDMN | Complete fit: 246 observables |  LFUV + radiative + $B_{s}\to\mu^{+}\mu^{-}$
---|---|---
1D Hyp. | Best fit |  PullSM ($\sigma$) | p-value | Best fit |  PullSM ($\sigma$) | p-value
${\cal C}_{9\mu}^{\rm NP}$ | $-1.06^{+0.15}_{-0.14}$ | 7.0 | 39.5 % | $-0.82^{+0.22}_{-0.24}$ | 4.0 | 36.0 %
${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$ | $-0.44^{+0.07}_{-0.08}$ | 6.2 | 22.8 % | $-0.37^{+0.08}_{-0.09}$ | 4.6 | 68.0 %

Table 1: Analysis of the ACDMN group as of Moriond 2021: most preferred 1D
patterns of NP in $b\to s\mu^{+}\mu^{-}$. For a complete set of 1D patterns,
see Ref. [49]. The $p$-value of the SM hypothesis is $1.1\%$ (corresponding to
a discrepancy with the data of 2.5$\sigma$) for the complete fit and $1.4\%$
for the LFUV fit.

AS | Complete fit: 130 observables |  LFUV + $B_{s}\to\mu^{+}\mu^{-}$
---|---|---
1D Hyp. | Best fit |  PullSM ($\sigma$) | Best fit |  PullSM ($\sigma$)
${\cal C}_{9\mu}^{\rm NP}$ | $-0.80\pm 0.14$ | 5.7 | $-0.74^{+0.20}_{-0.21}$ | 4.1
${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$ | $-0.41\pm 0.07$ | 5.9 | $-0.35\pm 0.08$ | 4.6

Table 2: Analysis of the AS group as of Moriond 2021: most preferred 1D
patterns of NP in $b\to s\mu^{+}\mu^{-}$, see Ref. [50], v2 [Apr.2021]. The
number of observables is taken from Ref. [51]. Note: a revised Ref. [50], v3
[Sept.2021] finds very similar results: ${\cal C}_{9\mu}^{\rm NP}=-0.73\pm
0.15$ (Pull${}_{\rm SM}=5.2\sigma$) and ${\cal C}_{9\mu}^{\rm NP}=-{\cal
C}_{10\mu}^{\rm NP}=-0.39\pm 0.07$ (5.6$\sigma$) for the complete fit, and
unchanged results for the LFUV fit.

HMMN | Complete fit: 173 observables |  only $R_{K^{(*)}}$\+ $B_{d,s}\to\mu^{+}\mu^{-}$
---|---|---
1D Hyp. | Best fit |  PullSM ($\sigma$) | Best fit |  PullSM ($\sigma$)
${\cal C}_{9\mu}^{\rm NP}$ | $-0.95\pm 0.12$ | 7.6 | $-0.77\pm 0.21$ | 4.0
${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$ | $-0.49\pm 0.08$ | 6.7 | $-0.38\pm 0.09$ | 4.6

Table 3: Analysis of the HMMN group as of Moriond 2021: most preferred 1D
patterns of NP in $b\to s\mu^{+}\mu^{-}$. For a complete set of 1D patterns,
see Ref. [52].

Although the two most-preferred NP explanations are ${\cal C}_{9\mu}^{\rm NP}$
and ${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$, there are other 1D
hypotheses that have significant Pulls. ACDMN find that ${\cal C}_{9\mu}^{\rm
NP}=-{\cal C}_{9\mu}^{\rm\prime NP}$ has a large pull in the complete fit,
which is about the same size as that of ${\cal C}_{9\mu}^{\rm NP}$ and ${\cal
C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$. However, this solution
predicts $R_{K}\simeq 1$, as in the SM, so in the LFUV fit, its pull is
3.0$\sigma$, smaller than that of ${\cal C}_{9\mu}^{\rm NP}$ and ${\cal
C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$. For this reason, this solution
is not favoured.

It is interesting to see how the Pulls of the 1D explanations to the $b\to
s\ell^{+}\ell^{-}$ anomalies have evolved in the past eight years, as more
observables, more experiments and better experimental and theoretical
precision are included in each iteration. (Here we refer to Refs. [37, 53, 38,
49].) For the complete fit, the PullSM of ${\cal C}_{9\mu}^{\rm NP}$ has been
4.5 [2016], 5.8 [2018], 5.6 [2019], 6.3 [2020], 7.0 [2021]. And for ${\cal
C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$: 4.2 [2016], 5.3 [2018], 5.2
[2019], 5.8 [2020], 6.2 [2021]. In both cases, the size of the discrepancy
with the SM of the $b\to s\ell^{+}\ell^{-}$ anomaly has steadily increased.
Note that the hierarchy in Pulls between these two solutions is reversed if
only the subset of LFUV observables is considered. A solution [54] to this
apparent contradiction is discussed in the next subsection.

2D fits allow us to explore if, by permitting an extra NP Wilson coefficient
(or a constrained combination) to vary, we get a better description of the
data than with a single coefficient, or the improvement is negligible111111We
stress that, because PullSM includes information about the $N_{dof}$, one can
compare fits with different $N_{dof}$.. For the 2D fits, we focus on the ACDMN
analysis, but we note that AS and HMMN also perform 2D fits.

A number of 2D hypothesis are examined in Ref. [49]. Here we discuss two of
them:

  * a)

Adding ${\cal C}_{10\mu}^{\rm NP}$ to ${\cal C}_{9\mu}^{\rm NP}$ slightly
decreases the PullSM in the complete fit (Table 4). That is, the addition of
this NP WC does not provide a better description of the data. Still, it is
clear that a small contribution to ${\cal C}_{10\mu}^{\rm NP}$ (or to ${\cal
C}_{10\mu}^{\rm\prime NP}$ or a scalar operator) is required to explain the
small ($\sim 2\sigma$) deficit with respect to the SM observed in ${\cal
B}({B_{s}\to\mu^{+}\mu^{-}})$. On the other hand, the recent experimental
update by LHCb is closer to the SM value, and this has slightly reduced its
significance: in the 2D $[{\cal C}_{9\mu}^{\rm NP},{\cal C}_{10\mu}^{\rm NP}]$
plot, ${\cal C}_{10\mu}^{\rm NP}$ is now consistent with zero at 1$\sigma$.

  * b)

If instead a RH current with the structure ${\cal C}_{9\mu}^{\rm\prime
NP}=-{\cal C}_{10\mu}^{\rm\prime NP}$ is added to ${\cal C}_{9\mu}$
(hypothesis 5 in notation of [38]), a better description of the data is found:
this is the scenario with the largest PullSM ($7.4\sigma$), see Table 4. The
same result is obtained if only ${\cal C}_{10\mu}^{\rm\prime NP}$ is added.
The reason is that the RH current counterbalances the effect of a too-large
${\cal C}_{9\mu}^{\rm NP}$ contribution in $R_{K}$. On the other hand, if this
RH current structure is added to the hypothesis ${\cal C}_{9\mu}^{\rm
NP}=-{\cal C}_{10\mu}^{\rm NP}$ instead of ${\cal C}_{9\mu}^{\rm NP}$, the
description of the data is significantly worse with a reduction of nearly 1.5
$\sigma$ in PullSM (see hypothesis 4 in Ref. [49]).

ACDMN | Complete fit: 246 observables | LFUV + radiative
---|---|---
2D Hyp. | Best fit |  PullSM ($\sigma$) | p-value | Best fit |  PullSM ($\sigma$) | p-value
$({\cal C}_{9\mu}^{\rm NP},{\cal C}_{10\mu}^{\rm NP})$ | $(-1.00,+0.11)$ | 6.8 | 39.4 % | $(-0.12,+0.54)$ | 4.3 | 65.6 %
$({\cal C}_{9\mu}^{\rm NP},{\cal C}_{10\mu}^{\rm\prime NP})$ | $(-1.26,-0.35)$ | 7.4 | 55.9 % | $(-1.82,-0.59)$ | 4.7 | 84.1 %
Hyp. 4 | $(-0.48,+0.11)$ | 6.0 | 24.0 % | $(-0.46,+0.15)$ | 4.5 | 74.5 %
Hyp. 5 | $(-1.26,+0.25)$ | 7.4 | 55.8 % | $(-2.08,+0.51)$ | 4.7 | 86.0 %

Table 4: Analysis of the ACDMN group: most preferred 2D patterns of NP in
$b\to s\mu^{+}\mu^{-}$. Hyp. 4 is $({\cal C}_{9\mu}^{\rm NP}=-{\cal
C}_{10\mu}^{\rm NP},{\cal C}_{9\mu}^{\rm\prime NP}=-{\cal
C}_{10\mu}^{\rm\prime NP})$, hyp. 5 is $({\cal C}_{9\mu}^{\rm NP},{\cal
C}_{9\mu}^{\rm\prime NP}=-{\cal C}_{10\mu}^{\rm\prime NP})$. For a complete
set of 2D patterns, see Ref. [49].

ACDMN also performs a 6D fit121212We note that HMMN does 10D and 20D fits
[52]., see Table 5. This shows consistency with the 1D and 2D fits, and
confirms that ${\cal C}_{9\mu}^{\rm NP}$ is the dominant NP WC. It is nonzero
at more than 3$\sigma$, while all other WCs are consistent with zero at
1$\sigma$. Finally, we note that the PullSM for this 6D fit has increased in
time: 5.1$\sigma$ [2019], 5.8$\sigma$ [2020] to 6.6$\sigma$ in the most recent
analysis of 2021.

ACDMN | ${\cal C}_{7}^{\rm NP}$ | ${\cal C}_{9\mu}^{\rm NP}$ | ${\cal C}_{10\mu}^{\rm NP}$ | ${\cal C}_{7}^{\rm\prime NP}$ | ${\cal C}_{9\mu}^{\rm\prime NP}$ | ${\cal C}_{10\mu}^{\rm\prime NP}$
---|---|---|---|---|---|---
Best fit | +0.01 | -1.21 | +0.15 | +0.01 | +0.37 | -0.21
1$\sigma$ region |   $\\!\\![-0.02,+0.04]\\!\\!$  |   $\\!\\![-1.38,-1.01]\\!$ |  $\\!\\![+0.00,+0.34]\\!\\!$  |   $\\!\\![-0.02,+0.03]\\!\\!$  |   $\\![-0.12,+0.80]\\!$ |   $\\!\\![-0.42,+0.02]\\!$
Table 5: Analysis of the ACDMN group: best-fit points and 1 $\sigma$ regions
for the 6D hypothesis of NP in $b\to s\mu^{+}\mu^{-}$. The PullSM of this
hypothesis is 6.6$\sigma$ [49].

####  3.3.2 LFUV and LFU New Physics

As shown in Sec. 3.1, the $b\to s\ell^{+}\ell^{-}$ anomalies appear in two
types of observables, those that involve only $b\to s\mu^{+}\mu^{-}$, and
those that are LFUV. The scenarios of Sec. 3.3.1 assume that the NP affects
only $b\to s\mu^{+}\mu^{-}$. In Ref. [54] (essentially the ACDMN group) a new
NP scenario was explored. Just as there are two categories of observables, it
was assumed that there are two types of NP, one LFUV, affecting only $b\to
s\mu^{+}\mu^{-}$, the other lepton flavour universal (LFU), equally affecting
all $b\to s\ell^{+}\ell^{-}$ processes, $\ell=e$, $\mu$ and $\tau$. The LFUV
NP then explains the LFUV observables, while the combination of LFUV and LFU
NP explains the $b\to s\mu^{+}\mu^{-}$ observables.

The WCs of the LFUV and LFU NP are denoted ${\cal C}_{i\mu}^{V}$ and ${\cal
C}_{i}^{U}$, respectively. The transformation from the standard $({\cal
C}_{i\mu},{\cal C}_{ie})$ basis to the new $({\cal C}_{i\mu}^{V},{\cal
C}_{i}^{U})$ basis is given by

$\displaystyle{\cal C}_{i\mu}$ $\displaystyle=$ $\displaystyle{\cal
C}_{i\mu}^{V}+{\cal C}_{i}^{U}~{},$ $\displaystyle{\cal C}_{ie}$
$\displaystyle=$ $\displaystyle{\cal C}_{i}^{U}~{},$ (10)

and similarly for $\tau$. In Sec. 3.3.1, we saw that the complete fits prefer
the NP scenario ${\cal C}_{9\mu}^{\rm NP}$, while the LFUV fits prefer ${\cal
C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$. This observation suggests
looking at the 2D scenario131313For notational convenience, the subscript
$\mu$ of the LFUV contribution is understood, and is not written explicitly..

$\displaystyle[{\cal C}_{9}^{V}=-{\cal C}_{10}^{V},{\cal C}_{9}^{U}]$ (11)

The global fit was performed with this hypothesis [49], and it was found that
it is one of the most promising scenarios, with a PullSM of 7.3$\sigma$
(scenario 8 in the notation of Ref. [54]). As expected, the first component
nicely explains the LFUV observables, while the second universal contribution
adds a necessary extra contribution to the $b\to s\mu^{+}\mu^{-}$ observables
(but also $b\to se^{+}e^{-}$). This universal NP contribution helps in
explaining all the anomalies (in optimized observables for $B\to
K^{*}\mu^{+}\mu^{-}$ and $B_{s}\to\phi\mu^{+}\mu^{-}$) that the naive scenario
${\cal C}_{9\mu}^{V}=-{\cal C}_{10\mu}^{V}$ cannot explain alone. More
examples of LFUV-LFU scenarios can be found in Refs. [54, 38, 49].

This idea is not dissimilar from the idea of allowing LFUV NP in $b\to
s\mu^{+}\mu^{-}$ and $b\to se^{+}e^{-}$. Complete analyses of this type of
explanation, including many different scenarios, can be found in Refs. [52,
55, 56].

In summary, for the ACDMN group, the title of “most promising NP scenario” is
no longer a contest between the 1D hypotheses ${\cal C}_{9\mu}^{\rm NP}$ and
${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$, but rather between two
2D hypotheses:

$\quad\quad[{\cal C}_{9\mu}^{\rm NP},{\cal C}_{9\mu}^{\rm\prime NP}=-{\cal
C}_{10\mu}^{\rm\prime NP}]\quad\quad{\rm or}\quad\quad[{\cal C}^{V}_{9}=-{\cal
C}^{V}_{10},{\cal C}^{U}_{9}]$ (12)

Indeed, this confirms what was observed in the 1D fits: a vectorial coupling
to leptons in the form of ${\cal C}_{9\mu}^{\rm NP}$ (only for muons) or
${\cal C}^{V}_{9}$ (universal) is required to have a good description of the
data. In this sense, both hypotheses in Eq. (12) provide an excellent
explanation for the two largest anomalies, $R_{K}$ and $P_{5}^{\prime}$, in
contrast to ${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$, which was
unable to explain $P_{5}^{\prime}$ (see Fig. 4 in Ref. [49]).

###  3.4 Scale of New Physics

Flavour observables are sensitive to higher scales than direct searches at
colliders. Therefore, if NP affects flavour, it is not surprising that it
would be seen first in this sector. This naturally leads to the question: What
is the scale of NP required to explain the $b\to s\ell^{+}\ell^{-}$ anomalies?
The answer to this question depends on the assumption we make about the size
of the NP couplings and the way NP enters the $b\to s\ell^{+}\ell^{-}$
amplitude.

As discussed in Sec. 2, the interaction of NP with the SM particles is
described by the SMEFT. The dimension-6 four-fermion operators generated when
the NP is integrated out are contained in the effective Hamiltonian
$H_{eff}^{\rm NP}=\sum\frac{{\cal O}_{i}}{\Lambda_{i}^{2}}$. Comparing this
with Eq. (3) for three different cases, we find the following NP scales:

  * •

$b\to s\ell^{+}\ell^{-}$ transitions induced at tree level, assuming ${\cal
O}(1)$ couplings:

$\Lambda_{i}^{\rm Tree}=\frac{4\pi
v}{s_{w}g}\frac{1}{\sqrt{2|V_{tb}V_{ts}^{*}|}}\frac{1}{{|C_{i}^{\rm
NP}|}^{1/2}}\sim\frac{35~{}{\rm TeV}}{{|C_{i}^{\rm NP}|}^{1/2}}~{},$ (13)

  * •

$b\to s\ell^{+}\ell^{-}$ transitions induced at loop level, assuming ${\cal
O}(1)$ couplings:

$\Lambda_{i}^{\rm Loop}\sim\frac{35~{}{\rm TeV}}{4\pi{|C_{i}^{\rm
NP}|}^{1/2}}\sim\frac{3~{}{\rm TeV}}{{|C_{i}^{\rm NP}|}^{1/2}}~{},$ (14)

  * •

Minimal Flavour Violation with CKM-SM: There is a further reduction of $1/5$
in the NP scale due to $\sqrt{|V_{tb}V_{ts}^{*}|}\sim 1/5$.

In all cases, if the NP coupling is less than ${\cal O}(1)$, the NP scale
$\Lambda$ is reduced correspondingly.

Turning to the $b\to c\tau\nu$ anomalies, this transition takes place at tree
level in the SM. As we will see below, an explanation requires a rather large
${\cal O}(10\%)$ tree-level NP contribution at the amplitude level. This
implies that

$\Lambda^{\rm NP}\sim 1/(\sqrt{2}G_{F}|V_{cb}|0.10)^{1/2}\sim 4~{}{\rm
TeV}~{}.$ (15)

###  3.5 Mapping WET to SMEFT

In the fits we have determined which operators must receive NP contributions
in order to reproduce the data. There are two types of solutions: (i) pure
LFUV, with NP only in $b\to s\mu^{+}\mu^{-}$, and (ii) $b\to s\mu^{+}\mu^{-}$
LFUV + LFU contributions. In both cases, the question now is: what kinds of NP
can generate these solutions? In this subsection, we examine general features
of such NP – this is the “model-independent” model analysis. Specific models
will be examined in the next subsection.

As discussed in Sec. 2, the EFT used in the fits is the WET. It is appropriate
for physics at the scale $m_{b}$: it is invariant under $SU(3)_{C}\times
U(1)_{em}$ and all particles with masses greater than $m_{b}$ have been
integrated out. On the other hand, the EFT generated at high energies when the
heavy NP particles have been integrated out is the SMEFT. A first step in
identifying what kinds of NP can be responsible for the $b\to
s\ell^{+}\ell^{-}$ anomalies comes by mapping WET onto SMEFT, focusing on
${\cal O}^{(\prime)}_{9,10}$.

In the SMEFT, the dimension-6 operators that contribute to these WCs at tree
level can be separated into two categories: LFUV and LFU. The LFUV operators
are a subset of the semileptonic four-fermion operators. A simple basis for
these operators is [57]

$\displaystyle{\cal O}_{LQ}^{(1)}$ $\displaystyle=$
$\displaystyle({\bar{L}}_{i}\gamma_{\mu}L_{j})({\bar{Q}}_{k}\gamma^{\mu}Q_{l})~{},$
$\displaystyle{\cal O}_{LQ}^{(3)}$ $\displaystyle=$
$\displaystyle({\bar{L}}_{i}\gamma_{\mu}\sigma^{I}L_{j})({\bar{Q}}_{k}\gamma^{\mu}\sigma^{I}Q_{l})~{},$
$\displaystyle{\cal O}_{Qe}$ $\displaystyle=$
$\displaystyle({\bar{Q}}_{i}\gamma_{\mu}Q_{j})({\bar{e}}_{k}\gamma^{\mu}e_{l})~{},$
$\displaystyle{\cal O}_{Ld}^{(1)}$ $\displaystyle=$
$\displaystyle({\bar{L}}_{i}\gamma_{\mu}L_{j})({\bar{d}}_{k}\gamma^{\mu}d_{l})~{},$
$\displaystyle{\cal O}_{ed}^{(1)}$ $\displaystyle=$
$\displaystyle({\bar{e}}_{i}\gamma_{\mu}e_{j})({\bar{d}}_{k}\gamma^{\mu}d_{l})~{},$
(16)

where $i,j,k,l$ are flavour (generation) indices. These are LFV for $i\neq j$,
but for those that are LFC, one sees that they can be LFUV since the $ij=ee$,
$\mu\mu$ and $\tau\tau$ WCs are not necessarily equal. It is convenient to
combine the first two operators: ${\cal O}_{LQ}^{(1,3)}\equiv{\cal
O}_{LQ}^{(1)}+{\cal O}_{LQ}^{(3)}$. These four SMEFT operators contribute to
the four $b\to s\mu^{+}\mu^{-}$ WCs as follows [58]:

$\displaystyle{\cal C}_{9\mu}^{{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}\left[{\tilde{{\cal
C}}}_{LQ}^{(1,3)\mu\mu 23}+{\tilde{{\cal C}}}_{Qe}^{23\mu\mu}\right]~{},$
$\displaystyle{\cal C}_{10\mu}^{{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}\left[{\tilde{{\cal
C}}}_{Qe}^{23\mu\mu}-{\tilde{{\cal C}}}_{LQ}^{(1,3)\mu\mu 23}\right]~{},$
$\displaystyle{\cal C}_{9\mu}^{\prime{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}\left[{\tilde{{\cal
C}}}_{Ld}^{\mu\mu 23}+{\tilde{{\cal C}}}_{ed}^{\mu\mu 23}\right]~{},$
$\displaystyle{\cal C}_{10\mu}^{\prime{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}\left[{\tilde{{\cal
C}}}_{ed}^{\mu\mu 23}-{\tilde{{\cal C}}}_{Ld}^{\mu\mu 23}\right]~{},$ (17)

where ${\tilde{{\cal C}}}$ indicates a SMEFT WC at low energies, below the
scale of electroweak symmetry breaking. The ${\tilde{{\cal C}}}$s are related
to high-energy SMEFT WCs using renormalization-group running. We stress that
the above relations hold only for the basis of Eq. (16).

Recall that the preferred 1D fits involve ${\cal C}_{9\mu}^{{\rm NP}}$ or
${\cal C}_{9\mu}^{{\rm NP}}=-{\cal C}_{10\mu}^{{\rm NP}}$ (Tables 3, 3 and 3).
Consider the second solution. This arises if ${\tilde{{\cal
C}}}_{Qe}^{23\mu\mu}=0$. In this case, ${\cal C}_{9\mu}^{{\rm NP}}+{\cal
C}_{10\mu}^{{\rm NP}}=0$; the only nonzero WC is

${\cal C}_{9\mu}^{{\rm NP}}-{\cal C}_{10\mu}^{{\rm
NP}}=2\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}\,{\tilde{{\cal
C}}}_{LQ}^{(1,3)\mu\mu 23}~{}.$ (18)

This solution is particularly good for model building. There is only one WC
and its presence does not require any special relations between independent
SMEFT operators. Also, it has a simple physical interpretation: the NP couples
only to LH fermions.

Compare this with the first solution, ${\cal C}_{9\mu}^{{\rm NP}}$. In order
for it to be the only nonzero WC, it is necessary that ${\cal C}_{10\mu}^{{\rm
NP}}=0$, which requires that

${\tilde{{\cal C}}}_{LQ}^{(1,3)\mu\mu 23}={\tilde{{\cal
C}}}_{Qe}^{23\mu\mu}~{}.$ (19)

Since the operators are independent, their coefficients are generally
unrelated; any such relation requires additional explanation, such as a
symmetry.

In the particular case of ${\cal C}_{9\mu}^{{\rm NP}}$, such a symmetry is not
difficult to find. While the operators ${\cal O}_{LQ}^{(1,3)}$ and ${\cal
O}_{Qe}$ both involve LH quarks, they involve LH and RH charged leptons,
respectively. For NP in $b\to s\mu^{+}\mu^{-}$ decays, if its couplings are
invariant under $L_{\mu}$ (the $\mu$ lepton number), it will couple equally to
$\mu_{L}$ and $\mu$, and its effects will be seen in ${\cal C}_{9\mu}^{{\rm
NP}}$.

The point here is that it is possible to find single-particle NP models for
each of the 1D solutions of the EFT analysis, ${\cal C}_{9\mu}^{{\rm NP}}$ and
${\cal C}_{9\mu}^{{\rm NP}}=-{\cal C}_{10\mu}^{{\rm NP}}$. Considerations of
SMEFT help to accomplish this.

For the 2D fits, the preferred scenarios are $({\cal C}_{9\mu}^{{\rm
NP}},{\cal C}_{10\mu}^{\prime{\rm NP}})$ and $({\cal C}_{9\mu}^{\rm NP},{\cal
C}_{9\mu}^{\rm\prime NP}=-{\cal C}_{10\mu}^{\rm\prime NP})$ (see Table 4).
There is no problem with the second scenario. As discussed above, ${\cal
C}_{9\mu}^{{\rm NP}}$ can be produced by NP that obeys a symmetry under which
$\mu_{L}$ and $\mu_{R}$ transform in the same way, and ${\cal
C}_{9\mu}^{\rm\prime NP}=-{\cal C}_{10\mu}^{\rm\prime NP}$ is already an SMEFT
WC. On the other hand, the first solution is more complicated. First, in order
to produce ${\cal C}_{10\mu}^{\prime{\rm NP}}$, one requires NP that obeys a
symmetry under which $\mu_{L}$ and $\mu_{R}$ transform in the opposite way.
That is, two different NP particles are required: the exchange of one
generates ${\cal C}_{9\mu}^{{\rm NP}}$, and the other generates ${\cal
C}_{10\mu}^{\prime{\rm NP}}$. Second, it is not obvious what is the symmetry
under which $\mu_{L}$ and $\mu_{R}$ transform equally but oppositely.

Finally, the scenario in the 6D fit (Table 5) is acceptable from a model-
building point of view, and all WCs can be written in SMEFT form using the
following combinations: ${\cal C}_{9\mu}^{{\rm NP}}\pm{\cal C}_{10\mu}^{{\rm
NP}}$, ${\cal C}_{9\mu}^{\prime{\rm NP}}\pm{\cal C}_{10\mu}^{\prime{\rm NP}}$,
${\cal C}_{7\mu}^{{\rm NP}}$, ${\cal C}_{7\mu}^{\prime{\rm NP}}$.

Turning to LFU SMEFT operators that contribute to $b\to s\ell^{+}\ell^{-}$,
there are three dimension-6 operators involving the Higgs field [57]:

$\displaystyle{\cal O}_{\varphi Q}^{(1)}$ $\displaystyle=$
$\displaystyle(\varphi^{\dagger}i\overleftrightarrow{D}_{\mu}\varphi)({\bar{Q}}_{i}\gamma^{\mu}Q_{j})~{},$
$\displaystyle{\cal O}_{\varphi Q}^{(3)}$ $\displaystyle=$
$\displaystyle(\varphi^{\dagger}i\overleftrightarrow{D}_{\mu}^{I}\varphi)({\bar{Q}}_{i}\tau^{I}\gamma^{\mu}Q_{j})~{},$
$\displaystyle{\cal O}_{\varphi d}$ $\displaystyle=$
$\displaystyle(\varphi^{\dagger}i\overleftrightarrow{D}_{\mu}\varphi)({\bar{d}}_{i}\gamma^{\mu}d_{j})~{}.$
(20)

These generate the following LFU NP contributions [58]:

$\displaystyle{\cal C}_{9}^{{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}\left[{\tilde{{\cal
C}}}_{\varphi Q}^{(1)23}+{\tilde{{\cal C}}}_{\varphi
Q}^{(3)23}\right](-1+4\sin^{2}\theta_{W})~{},$ $\displaystyle{\cal
C}_{10}^{{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}\left[{\tilde{{\cal
C}}}_{\varphi Q}^{(1)23}+{\tilde{{\cal C}}}_{\varphi Q}^{(3)23}\right]~{},$
$\displaystyle{\cal C}_{9}^{\prime{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}{\tilde{{\cal
C}}}_{\varphi d}^{23}(-1+4\sin^{2}\theta_{W})~{},$ $\displaystyle{\cal
C}_{10}^{\prime{\rm NP}}$ $\displaystyle=$
$\displaystyle\frac{\pi}{\alpha}\frac{v^{2}}{\Lambda^{2}}{\tilde{{\cal
C}}}_{\varphi d}^{23}~{}.$ (21)

The preferred LFUV + LFU solution involves ${\cal C}_{9\mu}^{V}=-{\cal
C}_{10\mu}^{V}$ and ${\cal C}_{9}^{U}$. From the above, we see that ${\cal
C}_{9}^{{\rm NP}}\propto{\cal C}_{10}^{{\rm NP}}$, so that it is not possible
to produce only ${\cal C}_{9}^{U}$ with the operators of Eq. (20).

However, there is another possibility. The full matching of WET to SMEFT must
take into account loop effects and renormalization-group running. In
particular, a low-energy ${\cal C}_{9}^{U}$ can be generated via a loop-level
process [59]. The fermion in the loop can be $u_{1,2}$, $d_{1,2,3}$ or $\tau$
[59, 60, 38, 61]. (For $u_{3}=t$, a large ditop-dimuon coupling is disfavoured
by EW precision tests [62].) For loops with $u$ or $d$, it must be checked
that these are compatible with constraints from dijet searches and non-
leptonic decays. On the other hand, the case where the fermion is a $\tau$
(see Fig. 1) is particularly interesting [38, 63] because it suggests a
possible link between NP in the $b\to s\mu^{+}\mu^{-}$ and $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$ transitions. The key point here is that, if there
is a ${\cal C}_{9\mu}^{V}=-{\cal C}_{10\mu}^{V}$ contribution to $b\to
s\mu^{+}\mu^{-}$, it is not difficult to generate ${\cal C}_{9}^{U}$ via a
large $b\to s\tau\tau$ effective coupling. Through radiative effects by
closing the loop of the operator $b\to s\tau^{+}\tau^{-}$, a NP contribution
to ${\cal C}_{9}^{\rm U}$ is induced [38]:

${\cal C}_{9}^{\rm U}\simeq
7.5\left(1-\sqrt{\frac{R_{D^{(*)}}}{R_{D^{(*)SM}}}}\right)\left(1+\frac{{\rm
log}(\Lambda^{2}/({\rm 1TeV^{2}}))}{10.5}\right)$ (22)

where $\Lambda$ is the scale of NP that is taken in a range between 1 to 10
TeV.

Figure 1: Diagram inducing a lepton universal contribution to ${\cal
C}_{9}^{U}$ through a $\tau$ loop.

###  3.6 Models

Many models have been proposed to explain the $b\to s\mu^{+}\mu^{-}$
anomalies. Some are “simplified models,” in which NP particles are added, but
without giving any details of the underlying high-energy NP model from which
these particles arise (the UV completion). Others do present full models,
including the full gauge group and its particle content. In the simplest
models (simplified or full), there is a single NP particle that contributes at
tree level to $b\to s\mu^{+}\mu^{-}$. More complicated models typically
involve many NP particles; some of them, but not all, contribute to $b\to
s\mu^{+}\mu^{-}$.

In this review, we describe only the simplest models for all the anomalies.
There are several reasons. First, there is the general prejudice that Nature
prefers simple solutions (although this should not be taken as a golden rule).
But there are also practical reasons. The complicated NP models have many new
particles and generally have to satisfy many constraints. They are often
viable only in a region of parameter space. But the data have changed over the
years, so it is no longer clear whether these models are still valid. In
contrast, the simplest models have been studied by several different groups
and have been continually updated. (In fact, some of the early simple models
have now been ruled out.) So we are certain that the models we describe here
do actually explain the 2021 data.

For $b\to s\mu^{+}\mu^{-}$, there are two classes of simple models. They
involve the tree-level exchange of a $Z^{\prime}$ or a leptoquark (LQ). These
are described below.

####  3.6.1 $Z^{\prime}$ models

In $Z^{\prime}$ models, the $Z^{\prime}$ couples to both ${\bar{s}}b$ and
$\mu^{+}\mu^{-}$. Given that it contributes to ${\cal C}_{9\mu}^{{\rm NP}}$ or
${\cal C}_{9\mu}^{{\rm NP}}=-{\cal C}_{10\mu}^{{\rm NP}}$, the $Z^{\prime}$
coupling to ${\bar{s}}b$ is purely LH, but it may couple to both LH and RH
$\mu^{+}\mu^{-}$ pairs.

We write the general couplings of the $Z^{\prime}$ as

$\left[g^{Q}_{ij}({\bar{Q}}_{i}\gamma^{\mu}Q_{j})+g^{L}_{ij}({\bar{L}}_{i}\gamma^{\mu}L_{j})+g^{e}_{ij}({\bar{e}}_{i}\gamma^{\mu}e_{j})\right]Z^{\prime}_{\mu}~{}.$
(23)

For $b\to s\mu^{+}\mu^{-}$, we are interested in those terms whose
coefficients in the mass basis are $g^{Q}_{23}$, $g^{L}_{22}$ and
$g^{e}_{22}$. When the $Z^{\prime}$ is integrated out, one generates the
following four-fermion operators relevant for $2q2\ell$ processes (two quarks
and two leptons):

${\cal
L}^{2q2\ell}_{Z^{\prime}}=-\frac{g_{23}^{Q}g^{L}_{22}}{M^{2}_{Z^{\prime}}}\,({\bar{s}}_{L}\gamma^{\mu}b_{L})\,({\bar{L}}_{\mu}\gamma^{\mu}L_{\mu})-\frac{g_{23}^{Q}g^{e}_{22}}{M^{2}_{Z^{\prime}}}\,({\bar{s}}_{L}\gamma^{\mu}b_{L})\,({\bar{\mu}}_{R}\gamma^{\mu}\mu)~{}.$
(24)

Both terms contribute to $b\to s\mu^{+}\mu^{-}$, while the first also
contributes to $b\to s{\bar{\nu}}_{\mu}\nu_{\mu}$. If the $b\to
s\mu^{+}\mu^{-}$ anomalies are explained by ${\cal C}_{9\mu}^{{\rm NP}}$
alone, then $g^{e}_{22}=g^{L}_{22}$, while $g^{e}_{22}=0$ if ${\cal
C}_{9\mu}^{{\rm NP}}=-{\cal C}_{10\mu}^{{\rm NP}}$.

In addition to $2q2\ell$ operators, $Z^{\prime}$ exchange also produces four-
quark ($4q$) and four-lepton ($4\ell$) operators at tree level. The operators
constructed from the same currents as above are

$\displaystyle{\cal L}_{\rm NP}^{4q}$ $\displaystyle=$
$\displaystyle-\frac{(g_{23}^{Q})^{2}}{2M^{2}_{Z^{\prime}}}\,({\bar{s}}_{L}\gamma^{\mu}b_{L})\,({\bar{s}}_{L}\gamma_{\mu}b_{L})~{},$
$\displaystyle{\cal L}_{\rm NP}^{4\ell}$ $\displaystyle=$
$\displaystyle-~{}\frac{(g_{22}^{L})^{2}}{2M^{2}_{Z^{\prime}}}({\bar{L}}_{\mu}\gamma^{\mu}L_{\mu})({\bar{L}}_{\mu}\gamma_{\mu}L_{\mu})-~{}\frac{g_{22}^{L}g_{22}^{e}}{2M^{2}_{Z^{\prime}}}({\bar{L}}_{\mu}\gamma^{\mu}L_{\mu})({\bar{\mu}}_{R}\gamma_{\mu}\mu)~{}.$
(25)

The operator in ${\cal L}_{\rm NP}^{4q}$ contributes to
$B_{s}^{0}$-${\bar{B}}^{0}_{s}$ mixing, while those in ${\cal L}_{\rm
NP}^{4\ell}$ contribute to neutrino trident production
($\nu_{\mu}N\to\nu_{\mu}N\mu^{+}\mu^{-}$) [64].

The point here is that the best-fit values of ${\cal C}_{9\mu}^{{\rm NP}}$ or
${\cal C}_{9\mu}^{{\rm NP}}=-{\cal C}_{10\mu}^{{\rm NP}}$ place a constraint
on $g_{23}^{Q}g^{L}_{22}/M^{2}_{Z^{\prime}}$. This same quantity is also
constrained by $b\to s{\bar{\nu}}_{\mu}\nu_{\mu}$.141414The decay $b\to
s{\bar{\nu}}_{\mu}\nu_{\mu}$ is analyzed within the SM in Ref. [65];
experimental results are given in Ref. [66]. In addition, there are
constraints on $(g_{23}^{Q})^{2}/M^{2}_{Z^{\prime}}$ from
$B_{s}^{0}$-${\bar{B}}^{0}_{s}$ mixing, and on
$(g_{22}^{L})^{2}/M^{2}_{Z^{\prime}}$ from neutrino trident production. It
must be checked that all these constraints are consistent with one another
(for example, see Ref. [67]). This type of analysis is necessary for models,
but not for EFTs.

Finally, there is an extremely important constraint coming from direct
searches for $Z^{\prime}$ bosons in the process $pp\to
Z^{\prime}\to\mu^{+}\mu^{-}$ [68]. The translation of the experimental limit
into a model constraint is completely model-dependent. It depends on how the
$Z^{\prime}$ couples to all the quarks, and what is the branching ratio of
$Z^{\prime}\to\mu^{+}\mu^{-}$. This latter quantity can be affected if the
$Z^{\prime}$ can also decay into dark matter, for example.

All $Z^{\prime}$ models must take into account all of these constraints.
(Furthermore, if one is interested in a particular $Z^{\prime}$ model, it may
be necessary to update the constraint analysis, to see if the model is still
viable.)

Many $Z^{\prime}$ models have been proposed to explain the $b\to
s\mu^{+}\mu^{-}$ anomalies. The following list includes most of them,
separated into four categories: the $Z^{\prime}$ is heavy and contributes to
(i) ${\cal C}_{9\mu}^{{\rm NP}}$, (ii) ${\cal C}_{9\mu}^{{\rm NP}}=-{\cal
C}_{10\mu}^{{\rm NP}}$, or (iii) ${\cal C}_{9\mu}^{{\rm NP}}$ and ${\cal
C}_{10\mu}^{{\rm NP}}$; (iv) light $Z^{\prime}$ bosons.

  1. 1.

${\cal C}_{9\mu}^{{\rm NP}}$. As argued earlier, this NP explanation can arise
only if the relation in Eq. (19) holds, which generally requires an additional
symmetry. $L_{\mu}$ (the $\mu$ lepton number) is such a symmetry, but by
itself it is not anomaly-free; an additional symmetry is required. The most
popular choice is a gauged $L_{\mu}-L_{\tau}$ symmetry: $Z^{\prime}$ models
with a $U(1)_{L_{\mu}-L_{\tau}}$ symmetry can be found in Refs. [69, 70, 71,
72, 73, 74, 75, 76, 77, 78, 79, 80, 197]. There are many variants in which the
$U(1)$ has a different symmetry, but always includes $L_{\mu}$, see Refs. [82,
83, 84, 85, 86, 87, 88, 89, 90]. Other frameworks include the 3-3-1 model with
$\beta=-\sqrt{3}$ [91, 92] and Branco-Grimus-Lavoura (BGL) models [93].

  2. 2.

${\cal C}_{9\mu}^{{\rm NP}}=-{\cal C}_{10\mu}^{{\rm NP}}$. A variety of models
have been proposed in which it is simply assumed that the $Z^{\prime}$ couples
only to LH fermions (like the $W^{\pm}$ in the SM). These can be found in
Refs. [94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105].

  3. 3.

${\cal C}_{9\mu}^{{\rm NP}}$ and ${\cal C}_{10\mu}^{{\rm NP}}$. There are
other models in which the $Z^{\prime}$ couplings involve both ${\cal
C}_{9\mu}^{{\rm NP}}$ and ${\cal C}_{10\mu}^{{\rm NP}}$. The relative size of
the two WCs is dictated by some underlying dynamics of the model, see Refs.
[106, 107, 108, 109, 110, 111, 112].

  4. 4.

Light $Z^{\prime}$ bosons. In the above models, $M_{Z^{\prime}}=O({\rm TeV})$
is assumed. But this is not necessarily required. In Refs. [97, 113, 114, 115,
116], scenarios are presented in which the $Z^{\prime}$ has a mass of a few
GeV. (The couplings are also reduced so as to reproduce the preferred values
of ${\cal C}_{9,10\mu}^{{\rm NP}}$.)

####  3.6.2 LQ models

There are five spin-0 and five spin-1 LQs, denoted $\Delta$ and $V$
respectively, with couplings [117]

$\displaystyle{\cal L}_{\Delta}$ $\displaystyle=$
$\displaystyle(y_{Lu}{\bar{L}}u+y_{eQ}\,{\bar{e}}i\tau_{2}Q)\Delta_{-7/6}+y_{Ld}\,{\bar{L}}d\Delta_{-1/6}+(y_{LQ}\,{\bar{L}}^{c}i\tau_{2}Q+y_{eu}\,{\bar{e}}^{c}u)\Delta_{1/3}$
$\displaystyle+~{}y_{ed}\,{\bar{e}}^{c}d\Delta_{4/3}+y^{\prime}_{LQ}\,{\bar{L}}^{c}i\tau_{2}{\vec{\tau}}Q\cdot{\vec{\Delta}}^{\prime}_{1/3}+h.c.$
$\displaystyle{\cal L}_{V}$ $\displaystyle=$
$\displaystyle(g_{LQ}\,{\bar{L}}\gamma_{\mu}Q+g_{ed}\,{\bar{e}}\gamma_{\mu}d)V^{\mu}_{-2/3}+g_{eu}\,{\bar{e}}\gamma_{\mu}uV^{\mu}_{-5/3}+g^{\prime}_{LQ}\,{\bar{L}}\gamma_{\mu}{\vec{\tau}}Q\cdot{\vec{V}}^{\prime\mu}_{-2/3}$
(26)
$\displaystyle+~{}(g_{Ld}\,{\bar{L}}\gamma_{\mu}d^{c}+g_{eQ}\,{\bar{e}}\gamma_{\mu}Q^{c})V^{\mu}_{-5/6}++g_{Lu}\,{\bar{L}}\gamma_{\mu}u^{c}V^{\mu}_{1/6}+h.c.$

$V^{\mu}_{-5/3}$ and $V^{\mu}_{1/6}$ do not couple to down-type quarks, and so
are not of interest to us. The other LQs transform as follows under
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$:

$\displaystyle
R_{2}\equiv\Delta_{-7/6}:({\bar{3}},2,-7/6)~{}~{},~{}~{}~{}~{}{\tilde{R}}_{2}\equiv\Delta_{-1/6}:({\bar{3}},2,-1/6)~{}~{},~{}~{}~{}~{}S_{1}\equiv\Delta_{1/3}:({\bar{3}},1,1/3)~{},$
$\displaystyle{\tilde{S}}_{1}\equiv\Delta_{4/3}:({\bar{3}},1,4/3)~{}~{},~{}~{}~{}~{}S_{3}\equiv{\vec{\Delta}}^{\prime}_{1/3}:({\bar{3}},3,1/3)~{},$
(27) $\displaystyle U_{1}\equiv
V^{\mu}_{-2/3}:({\bar{3}},1,-2/3)~{}~{},~{}~{}~{}~{}U_{3}\equiv{\vec{V}}^{\prime\mu}_{-2/3}:({\bar{3}},3,-2/3)~{}~{},~{}~{}~{}~{}V_{2}\equiv
V^{\mu}_{-5/6}:({\bar{3}},2,-5/6)~{}.$

Note that here the hypercharge is defined as $Y=Q_{em}-I_{3}$. $R_{2}$,
${\tilde{R}}_{2}$, etc. are the names given to these LQs in Ref. [118]. We
adopt this nomenclature here.

All of these LQs were explored at different times as potential explanations of
the $b\to s\mu^{+}\mu^{-}$ anomalies. In Ref. [67], fits were done including
each LQ individually, and it was found that only $S_{3}$, $U_{1}$ and $U_{3}$
provide good fits. (The $S_{3}$, $U_{1}$ and $U_{3}$ LQs were originally
examined in Refs. [119, 120, 121, 122], [123, 124] and [125], respectively.)
In the case of the $U_{1}$, the best fit has $g_{ed}^{\mu b}\simeq 0$, so that
all three LQ solutions have ${\cal C}_{9\mu}^{{\rm NP}}=-{\cal
C}_{10\mu}^{{\rm NP}}$.

We therefore see that the ${\cal C}_{9\mu}^{{\rm NP}}=-{\cal C}_{10\mu}^{{\rm
NP}}$ explanation can be generated by a $Z^{\prime}$ or a LQ, while the ${\cal
C}_{9\mu}^{{\rm NP}}$ solution is purely $Z^{\prime}$.151515 Producing the
${\cal C}_{9\mu}^{{\rm NP}}$ solution with LQs requires multiple LQs, whose
couplings and masses are exceedingly fine-tuned. Ref. [67], it was shown that
measurements of CP violation in $B\to K^{*}\mu^{+}\mu^{-}$ have the potential
to distinguish the ${\cal C}_{9\mu}^{{\rm NP}}$ and ${\cal C}_{9\mu}^{{\rm
NP}}=-{\cal C}_{10\mu}^{{\rm NP}}$ explanations. Similarly, it was argued in
Ref. [126] that these two solutions can be differentiated through the
measurements of ${\cal B}(B_{s}\to\mu^{+}\mu^{-})$ and CP-averaged azimuthal-
angle asymmetries in $B\to K^{*}\mu^{+}\mu^{-}$.

##  4 Charged-current anomalies: $b\to c\ell^{-}{\bar{\nu}}_{\ell}$

We now turn to the charged-current $B$ flavour anomalies, observed in $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$ transitions. Whereas $b\to s\ell^{+}\ell^{-}$
occurs at loop level in the SM, $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ is a tree-
level decay in the SM (although the amplitude is multiplied by $V_{cb}\simeq
0.04$). Thus, a larger (tree-level) NP contribution is required to explain
these anomalies. On the other hand, a similarity with the neutral-current
anomalies is that both have been seen in LFUV observables.

###  4.1 Experimental Results

Discrepancies with the SM have been seen in the following observables:

$R_{D^{(*)}}\equiv\frac{{\cal B}({\bar{B}}\to
D^{(*)}\tau^{-}{\bar{\nu}}_{\tau})}{{\cal B}({\bar{B}}\to
D^{(*)}\ell^{-}{\bar{\nu}}_{\ell})}~{},~{}\ell=e,\mu~{}~{},~{}~{}~{}~{}R_{J/\psi}\equiv\frac{{\cal
B}(B_{c}\to J/\psi\tau\nu_{\tau})}{{\cal B}(B_{c}\to J/\psi\mu\nu_{\mu})}~{}.$
(28)

The latest results are (the $R_{D^{(*)}}$ numbers are taken from Ref. [127],
the $R_{J/\psi}$ numbers from Ref. [128])

$\displaystyle R_{D}^{\tau/\ell}/(R_{D}^{\tau/\ell})_{\rm SM}=1.14\pm
0.10~{}~{},~{}~{}~{}~{}R_{D^{*}}^{\tau/\ell}/(R_{D^{*}}^{\tau/\ell})_{\rm
SM}=1.14\pm 0.06~{}~{},$ $\displaystyle
R_{J/\psi}^{\tau/\mu}/(R_{J/\psi}^{\tau/\mu})_{\rm SM}=2.51\pm 0.97~{}.$ (29)

The combined $R_{D}$ and $R_{D^{*}}$ results differ from the SM by
$3.1\sigma$.

The above results suggest the presence of NP in $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$ decays, but it is not clear which of $\ell=e$,
$\mu$ and/or $\tau$ is affected. However, it has also been found that

$R_{D^{*}}^{e/\mu}/(R_{D^{*}}^{e/\mu})_{\rm SM}=1.04\pm 0.05~{}.$ (30)

With this result, the simplest assumption is that NP mainly affects $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$, and most analyses focus on this scenario. With
this in mind, two other relevant observables $B\to D^{*}\tau\nu_{\tau}$ are
the $\tau$ polarization asymmetry $P_{\tau}(D^{*})$ and the longitudinal
$D^{*}$ polarization $F_{L}(D^{*})$:

$\displaystyle P_{\tau}(D^{*})\equiv\frac{\displaystyle\Gamma(B\to
D^{*}\tau^{\lambda=+1/2}\nu_{\tau})-\Gamma(B\to
D^{*}\tau^{\lambda=-1/2}\nu_{\tau})}{\displaystyle\Gamma(B\to
D^{*}\tau\nu_{\tau})}~{},$ $\displaystyle
F_{L}(D^{*})\equiv\frac{\displaystyle\Gamma(B\to
D^{*}_{L}\tau\nu_{\tau})}{\displaystyle\Gamma(B\to D^{*}\tau\nu_{\tau})}~{}.$
(31)

$P_{\tau}(D^{*})$ and $F_{L}(D^{*})$ were measured in Refs. [129, 130] and
[131], respectively:

$P_{\tau}(D^{*})=-0.38\pm 0.51^{+0.21}_{-0.16}~{},~{}~{}F_{L}(D^{*})=0.60\pm
0.08\pm 0.035~{}.$ (32)

These observables are useful for distinguishing NP explanation with different
Lorentz structures. The unexpectedly large value observed for $F_{L}(D^{*})$
suggested to look for alternative ways to measure it (see [132]).

Finally, there is also the observable ${\cal B}(B_{c}\to\tau\nu_{\tau})$,
which has not yet been measured. In Ref. [133] [Nov., 2016], it is argued that
${\cal B}(B_{c}\to\tau\nu_{\tau})<30\%$ is required for compatibility with the
$B_{c}$ lifetime. This disfavours NP contributions to $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$ with scalar couplings. In Ref. [134] [Aug.,
2017], it is claimed that LEP data requires ${\cal
B}(B_{c}\to\tau^{-}{\bar{\nu}_{\tau}})<10\%$, placing an even stronger
constraint on these NP scenarios. On the other hand, this question was re-
examined in Ref. [135] [May, 2019], and it was concluded that the constraint
${\cal B}(B_{c}\to\tau^{-}{\bar{\nu}}_{\tau})<10\%$ is not justified. A
conservative analysis yields ${\cal
B}(B_{c}\to\tau^{-}{\bar{\nu}}_{\tau})<60\%$, removing the constraint on NP
with scalar couplings. At the present time, the value of the allowed upper
limit on ${\cal B}(B_{c}\to\tau^{-}{\bar{\nu}}_{\tau})$ is an important open
question. Of course, this could be resolved if this branching ratio were
measured.

In the above, we have always assumed lepton flavour conservation (LFC),
writing the leptons in the final state as $\tau^{-}{\bar{\nu}}_{\tau}$,
$\mu^{-}{\bar{\nu}}_{\mu}$, etc. This is done principally because this is how
the experimental results have been reported. However, it should be remembered
that, if NP is present, LFC might not be followed. (On the other hand, in the
SM, one does have LFC, so only NP couplings that obey LFC can interfere with
the SM.)

###  4.2 EFT analysis

Because the ${\bar{\nu}}$ in $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ decays is not
detected, we know nothing about its properties. In particular, we don’t know
if it is LH or RH. The case of a RH final-state neutrino will be addressed in
the model analysis below; EFT analyses assume LH neutrinos.

With this assumption, and assuming that it is $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$ decays that receive NP contributions, the most
general effective Hamiltonian for this decay is161616As discussed above, here
we assume lepton flavour conservation. For an EFT analysis of $b\to
c\ell^{-}{\bar{\nu}}_{\ell}$ and $b\to u\ell^{-}{\bar{\nu}}_{\ell}$ decays
without LFC, see Ref. [136].

$H_{\rm
eff}=\frac{4G_{F}}{\sqrt{2}}\,V_{cb}\left[(1+C_{V}^{L})O_{V}^{L}+C_{V}^{R}O_{V}^{R}+C_{S}^{R}O_{S}^{R}+C_{S}^{L}O_{S}^{L}+C_{T}O_{T}\right]~{},$
(33)

with

$\displaystyle
O_{V}^{L}=({\bar{c}}\gamma^{\mu}P_{L}b)({\bar{\tau}}\gamma_{\mu}P_{L}\nu_{\tau})~{}~{},~{}~{}~{}~{}O_{V}^{R}=({\bar{c}}\gamma^{\mu}P_{R}b)({\bar{\tau}}\gamma_{\mu}P_{L}\nu_{\tau})~{},$
$\displaystyle
O_{S}^{R}=({\bar{c}}P_{R}b)({\bar{\tau}}P_{L}\nu_{\tau})~{},~{}~{}O_{S}^{L}=({\bar{c}}P_{L}b)({\bar{\tau}}P_{L}\nu_{\tau})~{},~{}~{}O_{T}=({\bar{c}}\sigma^{\mu\nu}P_{L}b)({\bar{\tau}}\sigma_{\mu\nu}P_{L}\nu_{\tau})~{}.$
(34)

Here the SM contribution has been factored out, so that $C_{V,S,T}^{L,R}$ are
generated only through NP.

As was the case in the $b\to s\ell^{+}\ell^{-}$ EFT analysis, these operators
are defined in the WET. In order to make contact with the NP that generates
them, they must be mapped onto SMEFT. It turns out that all of these
dimension-6 WET operators are also dimension-6 SMEFT operators, with one
exception: $O_{V}^{R}$. An LFU version of this operator appears in SMEFT at
dimension-6, but the LFUV operator shown above arises only at dimension 8
[137], so that its coefficient $C_{V}^{R}$ is expected to be smaller than the
other coefficients by $O(v^{2}/\Lambda^{2})$, where $v$ is the vev of the SM
Higgs and $\Lambda$ is the scale of NP. For this reason, $O_{V}^{R}$ is
usually excluded from EFT analyses. (For an alternative point of view, see
Refs. [nonSMEFTNP, 138].)

The most recent EFT analysis of the $b\to c\tau^{-}{\bar{\nu}}_{\tau}$
anomalies was performed in Ref. [139]. This study used the five observables of
Eqs. (29) and (31), and added ${\cal B}(B_{c}\to\tau\nu_{\tau})$, assuming
upper limits of 60%, 30% and 10%. The results are shown in Table 6. Note that
tensor operators are generally generated by Fierz transformations. For
example, the exchange of the scalar $SU(2)_{L}$-doublet LQ $R_{2}$ produces
the four-fermion operator $({\bar{c}}P_{L}\nu_{\tau})({\bar{\tau}}P_{L}b)$
[140, 141]. A Fierz transformation produces both $O_{S}^{L}$ and $O_{T}$, with
$C_{S}^{L}=4C_{T}$. This is used in the fits.

1D hypothesis | Best fit |  $p$ value (%) | PullSM
---|---|---|---
$C_{V}^{L}$ | $0.07\pm 0.02$ | 44 | 4.0
$C_{S}^{R}$ | $0.09\pm 0.03$ | 2.7 | 3.1
$C_{S}^{L}$ | $0.07\pm 0.03$ | 0.26 | 2.1
$C_{S}^{L}=4C_{T}$ | $-0.03\pm 0.04$ | 0.04 | 0.7

2D hypothesis | Best fit |  $p$ value (%) | PullSM
---|---|---|---
$(C_{V}^{L},C_{S}^{L}=4C_{T})$ | $(0.10,-0.04)$ | 29.8 | 3.6
$\left.(C_{S}^{R},C_{S}^{L})\right|_{60\%}$ | $\displaystyle{(0.29,-0.25)\atop(-0.16,-0.69)}$ | 75.7 | 3.9
$\left.(C_{S}^{R},C_{S}^{L})\right|_{30\%}$ | $\displaystyle{(0.21,-0.15)\atop(-0.26,-0.61)}$ | 30.9 | 3.6
$\left.(C_{S}^{R},C_{S}^{L})\right|_{10\%}$ | $\displaystyle{(0.11,-0.04)\atop(-0.37,-0.51)}$ | 2.6 | 2.9
$(C_{V}^{L},C_{S}^{R})$ | $(0.08,0.01)$ | 26.6 | 3.6
$\left.({\rm Re}[C_{S}^{L}=4C_{T}],{\rm Im}[C_{S}^{L}=4C_{T}])\right|_{60,30\%}$ | $(-0.06,\pm 0.31)$ | 25.0 | 3.6
$\left.({\rm Re}[C_{S}^{L}=4C_{T}],{\rm Im}[C_{S}^{L}=4C_{T}])\right|_{10\%}$ | $(-0.03,\pm 0.24)$ | 5.9 | 3.2

Table 6: Results of the one-dimensional and two-dimensional fits for the WCs
(given at the matching scale of 1 TeV), including the five observables of Eqs.
(29) and (31), with ${\cal B}(B_{c}\to\tau\nu_{\tau})<60\%$, 30% and 10%. For
those entries without a label indicating the constraint on ${\cal
B}(B_{c}\to\tau\nu_{\tau})$, the fit is valid for all three upper limits. For
the SM, the $p$-value is $\sim 0.1\%$, which represents a discrepancy with the
data of $3.3\sigma$.

From this Table, we make the following observations:

  * •

Among the 1D hypotheses, only a nonzero $C_{V}^{L}$ provides a good
explanation of the data.

  * •

There are four 2D hypotheses that are good fits. Two of them –
$(C_{V}^{L},C_{S}^{L}=4C_{T})$ and $(C_{V}^{L},C_{S}^{R})$ – are basically
just the 1D best fit, with a small additional WC added. The goodness-of-fit of
the other two – $(C_{S}^{R},C_{S}^{L})$ and $({\rm Re}[C_{S}^{L}=4C_{T}],{\rm
Im}[C_{S}^{L}=4C_{T}])$ – depends critically on the upper limit of ${\cal
B}(B_{c}\to\tau\nu_{\tau})$. If this upper limit is 30% or 60%, these 2D
hypotheses are viable. However, if the upper limit is 10%, these scenarios are
disfavoured.

  * •

We note that the difference between $p$ values of 30% and 75% is not
statistically significant – both describe perfectly acceptable fits. In fact,
from a statistical point of view, we expect the true solution to have a $p$
value of $\sim 50\%$. Larger $p$ values describe fits that are “too good,” and
suggest that errors may be overestimated. In this sense, it may be that
allowing ${\cal B}(B_{c}\to\tau\nu_{\tau})$ to be as large as 60% is a bit too
weak of a constraint. (Of course, this discussion will become moot with an
actual measurement of ${\cal B}(B_{c}\to\tau\nu_{\tau})$.)

###  4.3 Model analysis

When the $R_{D^{(*)}}$ anomalies were first announced, there were several
different general EFT analyses that identified the WCs that could be modified
in order to explain the data [142, 143, 144, 145]. Although they were model-
independent, they did indicate what types of NP models should be considered.
The key point is that $b\to c\tau^{-}{\bar{\nu}}_{\tau}$ is a charged-current
process. Given that it is produced at tree level in the SM, the simplest
models involve the tree-level exchange of a $W^{\prime}$, a LQ, or a charged
Higgs. We discuss these possibilities in turn, often within the context of the
above EFT results.

####  4.3.1 $W^{\prime}$

For the $b\to c\tau^{-}{\bar{\nu}}_{\tau}$ anomaly, the $W^{\prime}$ solution
most often proposed involves a RH $W_{R}$, along with a light RH neutrino
[146, 147, 148, 149]. A recent complete analysis can be found in Ref. [150]. A
generic $W^{\prime}$ model, with arbitrary LH and RH couplings to both quarks
and leptons, is examined in Ref. [151].

####  4.3.2 LQ models

The first paper to examine LQ models as a possible explanation of the $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$ anomaly is Ref. [118]. Followup papers can be
found in Refs. [152, 153, 154, 155]. At present, the $S_{1}$, $U_{1}$ and
$R_{2}$ LQs [see Eq. (LABEL:LQtypes)] can be viable explanations. The exchange
of an $S_{1}$ or $U_{1}$ leads to a four-fermion operator that is equivalent
to $O_{V}^{L}$ of Eq. (34). As for $R_{2}$, as mentioned above, its exchange
leads to $O_{S}^{L}$ and $O_{T}$, with $C_{S}^{L}=4C_{T}$. So it can provide
an explanation only for large values of ${\cal B}(B_{c}\to\tau\nu_{\tau})$.

####  4.3.3 Charged Higgs

Another way of explaining the $b\to c\tau^{-}{\bar{\nu}}_{\tau}$ anomaly is
through the tree-level exchange of a charged Higgs boson, and here numerous
two-Higgs-doublet models (2HDMs) have been proposed [156, 157, 158, 159, 160,
161, 162, 163, 164, 165]. However, charged-Higgs exchange produces only
$O_{S}^{R}$ and $O_{S}^{L}$ of Eq. (34), and as we saw in the EFT analysis,
this solution is only viable if ${\cal B}(B_{c}\to\tau\nu_{\tau})$ is large.
In addition, there are strong constraints on the charged Higgs from the LHC
[166].

A variety of measurements have been proposed to distinguish among the NP
explanations. These include the $q^{2}$ distribution [167], $D^{*}$
polarization [168, 169], $\tau$ polarizations [170], and the full angular
distribution [171].

##  5 Links between the $b\to s\mu^{+}\mu^{-}$ and $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$ anomalies

###  5.1 SMEFT analysis

It was first shown in Ref. [172] that it is possible to simultaneously explain
the $b\to s\mu^{+}\mu^{-}$ and $b\to c\tau^{-}{\bar{\nu}}_{\tau}$ anomalies.
At that time (2014), it was thought that NP with purely LH couplings was
required to account for the $b\to s\mu^{+}\mu^{-}$ data [119]. Referring to
Eq. (16), we see that there are two SMEFT operators of this type:

${\cal
O}_{ijkl}^{(1)}=({\bar{L}}_{i}\gamma_{\mu}L_{j})({\bar{Q}}_{k}\gamma^{\mu}Q_{l})~{}~{},~{}~{}~{}~{}{\cal
O}_{ijkl}^{(3)}=({\bar{L}}_{i}\gamma_{\mu}\sigma^{I}L_{j})({\bar{Q}}_{k}\gamma^{\mu}\sigma^{I}Q_{l})~{},$
(35)

where we have included the flavour indices in the names of the operators.

The key point is that, while ${\cal O}_{ijkl}^{(1)}$ is purely neutral
current, ${\cal O}_{ijkl}^{(3)}$ contains both neutral-current and charged-
current contributions. Thus, ${\cal O}_{2223}^{(1,3)}$ generate $b\to
s\mu^{+}\mu^{-}$, ${\cal O}_{3323}^{(3)}$ produces $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$. The original idea of Ref. [172] (see also Refs.
[173, 174]) was that the NP couples only (or mainly [173]) to the third
generation in the gauge basis. Couplings to the second generation are
generated when one rotates to the mass basis. Since ${\cal O}_{2223}^{(1,3)}$
requires more such rotations than ${\cal O}_{3323}^{(3)}$, ${\cal
C}_{2223}^{(1,3)}$ is naturally smaller than ${\cal C}_{3323}^{(3)}$. This is
what is required since $b\to s\mu^{+}\mu^{-}$ is loop level in the SM, while
$b\to c\tau^{-}{\bar{\nu}}_{\tau}$ is tree level. In Ref. [175], it was shown
that, even if one makes no assumptions about the coupling of the NP in the
gauge basis, when all constraints are taken into account, one finds the same
pattern as above for the WCs. Thus, both $B$ anomalies can be explained
simultaneously if the NP is purely LH and generates ${\cal O}_{ijkl}^{(1)}$
and ${\cal O}_{ijkl}^{(3)}$.

But there’s more. In addition to producing $b\to c\tau^{-}{\bar{\nu}}_{\tau}$,
${\cal O}_{3323}^{(3)}$ also generates $b\to s\tau^{+}\tau^{-}$ [63]. This has
two effects. First, it implies that the branching ratios of decays governed by
$b\to s\tau^{+}\tau^{-}$ should be much larger than what is predicted by the
SM. Second, this coupling is precisely what is required to produce ${\cal
C}_{9}^{U}$ in $b\to s\mu^{+}\mu^{-}$ at one loop (see Sec. 3.3.2).

Thus, this combined explanation of the two $B$ anomalies reproduces the
favoured $[{\cal C}_{9}^{V}=-{\cal C}_{10}^{V},{\cal C}_{9}^{U}]$ solution
(see Eq. (11)) of the $b\to s\mu^{+}\mu^{-}$ anomalies. The combined PullSM of
this particular scenario is as high as 8.1$\sigma$. The central values found
for the WCs in this case are ${\cal C}_{9}^{V}=-{\cal C}_{10}^{V}=-0.36,{\cal
C}_{9}^{U}=-0.74$ [49].

###  5.2 Model analysis

As shown above, it is possible to find a combined explanation of the $b\to
s\mu^{+}\mu^{-}$ and $b\to c\tau^{-}{\bar{\nu}}_{\tau}$ anomalies if the NP
generates the operators ${\cal O}_{ijkl}^{(1)}$ and ${\cal O}_{ijkl}^{(3)}$.
That is, the NP must couple to LH particles and have the appropriate couplings
to produce $b\to s\mu^{+}\mu^{-}$ and $b\to c\tau^{-}{\bar{\nu}}_{\tau}$.

Now, in an EFT analysis, one can focus on a particular operator. However, in a
model, additional operators will generally be produced, and these lead to
additional constraints. Thus, in order to test whether a particular model is
viable, one must consider all of these constraints.

####  5.2.1 VB ($W^{\prime},Z^{\prime}$) models

The vector-boson (VB) model, which contains a SM-like $SU(2)_{L}$ triplet of
vector bosons ($W^{\prime},Z^{\prime}$), can potentially provide a
simultaneous explanation of both $B$ anomalies. The $Z^{\prime}$ and
$W^{\prime}$ respectively mediate the $b\to s\mu^{+}\mu^{-}$ and $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$ transitions, and their couplings can be chosen to
reproduce the experimental data. Explicit VB models have been proposed in
Refs. [173, 176].

On the other hand, there are additional constraints on the VB model. These
include the constraints on the $Z^{\prime}$ couplings described in Sec. 3.6.1,
but there are also constraints from lepton-flavour-violating processes. The VB
model has been examined in detail in Refs. [175, 177, 178]. There it is found
that, when all the constraints are taken into account, the VB model is
excluded as a simultaneous explanation of both anomalies.

####  5.2.2 LQ models

  1. 1.

Single-LQ solutions: Over the years, many single-LQ solutions were proposed as
a simultaneous solution of the $b\to s\mu^{+}\mu^{-}$ and $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$ anomalies. As new data appeared, certain
solutions were ruled out. The combined EFT analysis in Ref. [178] clearly
demonstrated that the only viable single-LQ solution is the one based on the
vector LQ $U_{1}$, which was first discussed in Refs. [26, 174, 179]. This
conclusion was further confirmed in Ref. [180], which considered all single-LQ
solutions, taking into account LHC constraints.

  2. 2.

Two-LQ solutions: In Ref. [181], it is proposed to simultaneously explain the
anomalies with the addition of two scalar LQs, $S_{1}$ and $S_{3}$. $S_{3}$
contributes to both $b\to s\mu^{+}\mu^{-}$ and $b\to
c\tau^{-}{\bar{\nu}}_{\tau}$, but it cannot explain both anomalies by itself.
$S_{1}$ contributes to $b\to c\tau^{-}{\bar{\nu}}_{\tau}$, but not $b\to
s\mu^{+}\mu^{-}$. And both LQs contribute to $b\to s\nu{\bar{\nu}}$. It is
possible to choose the couplings of $S_{1}$ and $S_{3}$ such that both
anomalies are explained, while evading the constraints from $b\to
s\nu{\bar{\nu}}$.

####  5.2.3 UV Completions

The idea of explaining both anomalies with a single NP particle is very
intriguing, so the $U_{1}$ solution has generated a great deal of interest.
One problem with this solution is that, since the $U_{1}$ is a vector LQ, the
theory with a $U_{1}$, but without a Higgs sector, is not renormalizable. As a
consequence, loop diagrams, which can lead to potentially important effects,
cannot be calculated. In order to address this problem, a number of papers
have constructed models with an extended gauge group that is broken at the TeV
scale and contains the $U_{1}$ LQ as a gauge boson, or as a composite state. A
key observation, made first in Ref. [179], is that the $U_{1}$ field points to
a unification of quarks and leptons via a local or global $SU(4)$ symmetry.

These UV completion models can be split into two categories. First, there are
the models that use variations of the Pati-Salam idea, in which $SU(4)_{PS}$
unifies $SU(3)_{C}$ and a $U(1)$ under which both quarks and leptons are
charged (e.g., $U(1)_{Y}$ or $U(1)_{B-L}$) [182, 183, 184, 185, 186, 187, 188,
189, 190, 191, 192, 193]. Second, some models use the “4321” gauge group,
$SU(4)\times SU(3)^{\prime}\times SU(2)_{L}\times U(1)_{X}$ [194, 195, 196].
In all cases, it must be ensured that the breaking of the gauge group at the
TeV scale does not run into any problems from low-energy flavour constraints.

##  6 Connections with other physics problems

There are several other experimental results that cannot be explained by the
SM. If NP is present, its effects could possibly be seen in different types of
observables. With this in mind, models have been proposed that simultaneously
explain the $B$ anomalies and (at least) one of these other discrepancies with
the SM. Below we discuss models that relate the $B$ anomalies (mostly $b\to
s\ell^{+}\ell^{-}$) to dark matter, $(g-2)_{\mu}$, neutrino masses, and
hadronic anomalies. But there are also attempts to make connections with other
observables that exhibit tensions with the SM, such as the Cabibbo angle
anomaly, the $Z\to\bar{b}b$ forward-backward asymmetry, and
$\tau\to\mu{\bar{\nu}}\nu$, see Ref. [197] for example.

###  6.1 Dark matter

There is considerable cosmological and astrophysical evidence supporting the
existence of dark matter (DM). However, the fact that the SM has no DM
candidate is another one of its shortcomings, and provides another reason for
the requirement of NP. A number of papers have explored possible links between
these two strong hints of NP, the $b\to s\ell^{+}\ell^{-}$ anomalies and DM.
In Ref. [198], these models are separated into two categories: (i) portal
models, in which the relic density mediator is the same as that in the $B$
anomalies, and (ii) loop models, in which the $b\to s\ell^{+}\ell^{-}$
anomalies are explained via loops that include DM. In the discussion of $b\to
s\ell^{+}\ell^{-}$ models in Sec. 3.6, we elected to focus only on the
simplest models. These include only models of type (i); models of type (ii)
contain many more particles, are subject to more constraints, and are often
valid only in a corner of parameter space. Portal models with a $Z^{\prime}$
mediator can be found in Refs. [94, 95, 96, 72, 82, 74, 97, 102, 75, 77, 79,
80, 199]. (These do not include models in which the $Z^{\prime}$ is allowed to
decay to DM in order to evade LHC constraints, but do not otherwise address
the DM issues of relic density and direct detection.) Refs. [124, 200, 196,
201], which include combined explanations of both the $b\to s\ell^{+}\ell^{-}$
and $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ anomalies, describe portal models with
LQs.

###  6.2 $(g-2)_{\mu}$

Recently, the Muon $g-2$ Collaboration released a new result: the measurement
of $a_{\mu}\equiv(g-2)_{\mu}/2$ disagrees with the SM prediction by
$4.2\sigma$ [202]. Following this announcement, a number of papers were
written proposing models to simultaneously explain the observed value of
$a_{\mu}$ and one or both of the $B$ anomalies. There are many different
scenarios, but all of them use a $Z^{\prime}$ and/or LQs, see Refs. [203, 204,
205, 206, 207, 208, 209, 210, 211, 115, 116].

###  6.3 Neutrino properties

Another weakness of the SM is that it has no explanation for neutrino masses.
Here too, models have been proposed that provide combined explanations of the
$B$ anomalies and neutrino masses. However, it must be said that the
connection between the two is usually rather tenuous in these models, and they
generally involve several different types of NP particles. See Refs. [213,
214, 215, 212] for recent examples of such models.

###  6.4 Hadronic anomalies

The $b\to s\ell^{+}\ell^{-}$ and $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ anomalies
both involve semileptonic transitions. However, there is also some tension
with the SM in certain purely hadronic $B$ decays, albeit at a much lower
level. Attempts have been made to link the semileptonic and hadronic
anomalies.

For many years, the measurements of the branching ratios and direct & indirect
CP asymmetries in $B\to\pi K$ decays ($B^{0}\to\pi^{-}K^{+}$,
$B^{0}\to\pi^{0}K^{0}$, $B^{+}\to\pi^{0}K^{+}$, $B^{+}\to\pi^{+}K^{0}$) have
exhibited a slight inconsistency. This can be ameliorated if there is a NP
contribution to the $b\to s$ electroweak penguin. In Ref. [216], an update of
the “$B\to\pi K$ puzzle” was performed, and it was noted that this could be
explained if the $Z^{\prime}$ used to explain the $b\to s\mu^{+}\mu^{-}$
anomaly also couples to RH $u{\bar{u}}$ and/or $d{\bar{d}}$, with unequal
couplings.

U spin is an approximate symmetry that treats $d$ and $s$ quarks as identical.
Under U spin, the two transitions $b\to s$ and $b\to d$ are equal, apart from
CKM matrix elements. Note that these two amplitudes involve second-generation
and first-generation quarks in the final state. A certain parallel may be
drawn with $b\to s\mu^{+}\mu^{-}$ and $b\to se^{+}e^{-}$, which involve
second-generation and first-generation leptons in the final state. In the
latter case, the ratio of the two amplitudes tests LFUV (involving QED), while
in the former it measures U-spin breaking (involving QCD). This shows the
limitations of the analogy.

Inspired by this observation, the authors of Ref. [218] constructed an
optimized observable, $L_{K^{*}\bar{K}^{*}}$, from the ratio of longitudinal
amplitudes for $B_{s}\to{\bar{K}}^{0*}K^{0*}$ ($b\to s$) and
$B_{d}\to{\bar{K}}^{0*}K^{0*}$ ($b\to d$) that can be predicted (in contrast
to the other amplitudes in which infrared-divergences enter at leading order).
The results were as follows:

  * •

The U-spin breaking in the ratio was computed within the SM using QCDF and
modeling power-suppressed but infrared-divergent weak annihilation and hard-
spectator scattering. Taking this into account, there is a tension with the SM
at the level of 2.6$\sigma$ in $L_{K^{*}\bar{K}^{*}}$.

  * •

The $b\to s$ amplitude is smaller than the $b\to d$ amplitude in the same way
as there is a deficit in $b\to s\mu^{+}\mu^{-}$ as compared to $b\to
se^{+}e^{-}$.

  * •

There is a dominant operator that can naturally explain the low value of
$L_{K^{*}\bar{K}^{*}}$ as it happens for $b\to s\ell\ell$ with $O_{9}$. The
Wilson coefficient of this dominant operator also gets a destructive NP
contribution with a $\sim 20\%$ of the SM size.

  * •

A possible connection with the $b\to s\ell\ell$ anomalies is to consider a
composite/extra-dimensional model whose particle spectrum includes a Kaluza-
Klein gluon and a $Z^{\prime}$.

##  7 An EFT glance into the future

The future prospects for the field of $B$ flavour anomalies are very exciting
– we expect a plethora of new data and updates of previous measurements.
Indeed, in the coming five years, the search for NP using these anomalies is
expected to play a leading role at CERN and particularly at Belle-II.

###  7.1 $b\to s\mu^{+}\mu^{-}$ anomalies

In the very near future, we expect to receive two different types of input:

  * •

LFUV observables. In the short term, measurements of $R_{K_{S}}$, $R_{K^{*+}}$
(based on the charged channel $B^{+}\to K^{*+}\ell^{+}\ell^{-}$) and
$R_{\phi}$ (based upon $B_{s}\to\phi\ell\ell$) are expected from LHCb. These
results will test whether or not there is LFUV in Nature, which is what the
present measurements of $R_{K^{*}}$ and especially $R_{K}$ are clearly
pointing to. In the medium term, once more data is accumulated, an update of
the observable $R_{K}$, possibly increasing its significance much beyond the
present evidence for NP, will be a crucial milestone.

  * •

Angular optimized observables. Here we can expect updates of the optimized
observables of the 4-body angular distributions of $B_{d}^{0}\to K^{*}(\to
K\pi)\,\mu^{+}\mu^{-}$ (the last update was in 2020 [217]), the corresponding
$B^{+}$ decay, and $B_{s}^{0}\to\phi(\to K^{+}K^{-})\,\mu^{+}\mu^{-}$. All of
these will be important, but the first one will be particularly interesting,
to see if the first anomaly, that is the observable $P_{5}^{\prime}$,
increases with more data.

In addition, an important next step in this program will be the study of the
angular observables entering the $S$-wave part of the $B\to
K\pi\ell^{+}\ell^{-}$ decay. These new $S$-wave observables will provide
complementary information to the $P$-wave part of the distribution [219].

Finally, at present, the two scenarios with the largest PullSM are the 2D
hypotheses $[{\cal C}_{9\mu}^{\rm NP},{\cal C}_{9\mu}^{\rm\prime NP}=-{\cal
C}_{10\mu}^{\rm\prime NP}]$ and $[{\cal C}^{V}_{9}=-{\cal C}^{V}_{10},{\cal
C}^{U}_{9}]$ (Eq. (12)). It will be interesting to see which of these
dominates as the preferred explanation, or if another solution emerges. It was
shown in Refs. [38, 49, 220] that, while $R_{K}$ and $R_{K^{*}}$ cannot
disentangle between these two scenarios, a precise measurement of the
observable $Q_{5}$ in the bin [1.1,6] GeV2 can separate them, with $Q_{5}\sim
0.08$ for the second scenario, and $Q_{5}\sim 0.32$ for the first.

###  7.2 $b\to c\tau^{-}{\bar{\nu}}_{\tau}$ anomalies

At the end of Sec. 4.3, we note that a variety of measurements have been
proposed to distinguish among the NP explanations. There is one that is
particularly promising: it can, in principle, pinpoint what type of NP is
present. Ref. [171] proposes measuring the angular distribution in
${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$. By
measuring the four-momenta of the final-state $D$, $\pi^{\prime}$ and
$\pi^{-}$, the differential decay rate can be constructed. This depends on two
non-angular variables, $q^{2}$ and $E_{\pi}$, and three angles. Here, $q^{2}$
is the invariant mass-squared of the $\tau^{-}{\bar{\nu}}_{\tau}$ pair and
$E_{\pi}$ is the energy of the $\pi^{-}$ in the $\tau$ decay. The idea is then
to separate the data into various $q^{2}$-$E_{\pi}$ bins, and then to perform
an angular analysis within each bin. The angular distribution consists of
twelve different angular functions; nine of these are CP-conserving, and three
are CP-violating. We therefore potentially have a large number of observables
in this differential decay rate; the exact number depends on how many
$q^{2}$-$E_{\pi}$ bins there are.

In the angular distributions, the coefficients of each angular function are
functions of $q^{2}$, $E_{\pi}$, and the NP coefficients of Eq. (34). There
are five NP operators; only one linear combination of the two scalar operators
contributes to the decay ($O_{S}^{L}+O_{S}^{R}$ does not). With complex
coefficients, there are eight unknown theoretical parameters. Thus, if the
angular distribution in ${\bar{B}}\to D^{*}(\to
D\pi^{\prime})\,\tau^{-}(\to\pi^{-}\nu_{\tau}){\bar{\nu}}_{\tau}$ can be
measured, it may be possible to extract all of the NP coefficients from a fit,
thus revealing what type of NP is present.

##  8 Summary

Despite its enormous success, the SM of particle physics is not complete –
there must exist physics beyond the SM. Direct searches at the LHC for this
new physics have thus far proven fruitless. On the other hand, there are
deviations from the predictions of the SM in a number of observables involving
$b\to s\ell^{+}\ell^{-}$ and $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ decays. These
are the $B$ flavour anomalies, and they provide an indirect hint of NP.

The $b\to s\ell^{+}\ell^{-}$ anomalies have been seen in a number of different
observables, measured by different experimental groups. At present, the
p-value of the SM fit to all the $b\to s\ell\ell$ data is $\sim 1.1\%$,
corresponding to a discrepancy with the data of $\sim 2.5\sigma$. Global fits
to the data have been performed by different groups. Despite having different
methodologies, the results are remarkably consistent, particularly between the
ACDMN and HMMN groups. Using the results of the ACDMN group [49], the
preferred 1D solutions are ${\cal C}_{9\mu}^{\rm NP}$ (Pull${}_{\rm
SM}=7.0\sigma$) and ${\cal C}_{9\mu}^{\rm NP}=-{\cal C}_{10\mu}^{\rm NP}$
(6.2$\sigma$), and the preferred 2D solutions are $[{\cal C}_{9\mu}^{\rm
NP},{\cal C}_{9\mu}^{\rm\prime NP}=-{\cal C}_{10\mu}^{\rm\prime NP}]$
(7.4$\sigma$) and $[{\cal C}^{V}_{9}=-{\cal C}^{V}_{10},{\cal C}^{U}_{9}]$
(7.3$\sigma$). The simplest models involve the tree-level exchange of a
$Z^{\prime}$ or a LQ.

The $b\to c\ell^{-}{\bar{\nu}}_{\ell}$ anomalies have been seen in fewer
observables than $b\to s\ell^{+}\ell^{-}$, but here too, they have been
measured by different experiments, with the results largely confirmed. The
$p$-value of the SM fit to the present data is $\sim 0.1\%$, corresponding to
a discrepancy of $3.3\sigma$. The question of the identification of the
underlying NP is complicated by the fact that we do not know if the neutrino
is LH or RH. Assuming a LH neutrino, the preferred solution in the global fit
is $C_{V}^{L}$ i.e., LH NP, with Pull${}_{\rm SM}=4.0\sigma$. Models involve
the tree-level exchange of a $W^{\prime}$ or a LQ. If ${\cal
B}(B_{c}\to\tau\nu_{\tau})$ is allowed to be as large as 60%,
$(C_{S}^{R},C_{S}^{L})$, i.e., scalar NP such as a charged Higgs, also
provides a good solution.

It is obviously far too early to bet on a particular NP explanation, but one
intriguing possibility is the vector LQ $U_{1}$. It can simultaneously explain
both anomalies, and in the case of $b\to s\ell^{+}\ell^{-}$, it can generate
one of the preferred solutions, $[{\cal C}^{V}_{9}=-{\cal C}^{V}_{10},{\cal
C}^{U}_{9}]$.

These are exciting times. We may be seeing indirect evidence of NP at low
energies. In the near future, LHCb and especially Belle II will produce new
results. These will include measurements of previously-measured quantities, as
well as new ones, such as $Q_{5}$, observables involving $b\to
s\tau^{+}\tau^{-}$, and ${\cal B}(B_{c}\to\tau\nu_{\tau})$. All of this
information will give us a better idea of what the underlying NP is.

Note added: As this review was being finalized, the measurements of
$R_{K_{S}}$ and $R_{K^{*+}}$ were reported by LHCb [221]. $R_{K_{S}}$ was
measured in the bin [1.1,6] GeV2, while $R_{K^{*+}}$ was measured in a wider
bin, [0.045,6.0] GeV2, and exhibit tensions with the SM of 1.5$\sigma$ and
1.4$\sigma$, respectively. These are not large disagreements, but they are in
the same direction as previous measurements, and reinforce the case for LFUV.

Disclosure Statement: The authors are not aware of any affiliations,
memberships, funding, or financial holdings that might be perceived as
affecting the objectivity of this review.

Acknowledgments: We would like to acknowledge useful discussions with Andreas
Crivellin, Gino Isidori, Avelino Vicente and Nazila Mahmoudi. The work of DL
was partially financially supported by NSERC of Canada. This work received
financial support from the Spanish Ministry of Science, Innovation and
Universities (FPA2017-86989-P) and the Research Grant Agency of the Government
of Catalonia (SGR 1069) [JM]. JM acknowledges the financial support by ICREA
under the ICREA Academia programme.

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