# Rare B-Meson Decays at the Crossroads 111 To be published in the Proceedings
of the Conference on New Physics at the Large Hadron Collider, Nanyang
Technological University, Singapore. 29 February - 4 March, 2016.

Ahmed Ali∗ Deutsches Elektronen-Synchritron DESY
D-22607 Hamburg, Germany
∗E-mail<EMAIL_ADDRESS>
###### Abstract

Experimental era of rare $B$-decays started with the measurement of $B\to
K^{*}\gamma$ by CLEO in 1993, followed two years later by the measurement of
the inclusive decay $B\to X_{s}\gamma$, which serves as the standard candle in
this field. The frontier has moved in the meanwhile to the experiments at the
LHC, in particular, LHCb, with the decay $B^{0}\to\mu^{+}\mu^{-}$ at about 1
part in $10^{10}$ being the smallest branching fraction measured so far.
Experimental precision achieved in this area has put the standard model to
unprecedented stringent tests and more are in the offing in the near future. I
review some key measurements in radiative, semileptonic and leptonic rare
$B$-decays, contrast them with their estimates in the SM, and focus on several
mismatches reported recently. They are too numerous to be ignored, yet ,
standing alone, none of them is significant enough to warrant the breakdown of
the SM. Rare $B$-decays find themselves at the crossroads, possibly pointing
to new horizons, but quite likely requiring an improved theoretical
description in the context of the SM. An independent precision experiment such
as Belle II may help greatly in clearing some of the current experimental
issues.

###### keywords:

Standard Model, Flavour Physics, Rare $B$ Decays, LHC, Anomalies.

DESY 16-134

July 2016

##  1 Introduction

The interest in studying rare $B$ decays is immense. This is due to the
circumstance that these decays, such as
$b\to(s,d)\;\gamma,\;b\to(s,d)\;\ell^{+}\ell^{-}$, are flavour-changing-
neutral-current (FCNC) processes, involving the quantum number transitions
$|\Delta B|=1,|\Delta Q|=0$. In the SM [1], they are not allowed at the tree
level, but are induced by loops and are governed by the GIM (Glashow-
Iliopoulos-Maiani) mechanism [2], which imparts them sensitivity to higher
masses, $(m_{t},m_{W})$. As a consequence, they determine the CKM [3]
(Cabibbo-Kobayashi-Maskawa) matrix elements. Of these, the elements in the
third row, $V_{td}$, $V_{ts}$ and $V_{tb}$ are of particular interest. While
$|V_{tb}|$ has been measured in the production and decays of the top quarks in
hadronic collisions [4], the first two are currently not yet directly
accessible. In the SM, these CKM matrix elements have been indirectly
determined from the $B^{0}$ \- $\bar{B}^{0}$ and $B_{s}^{0}$ \-
$\bar{B}_{s}^{0}$ mixings. Rare $B$-decays provide independent measurements of
the same quantities.

In theories involving physics beyond the SM (BSM), such as the 2-Higgs doublet
models or supersymmetry, transitions involving the FCNC processes are
sensitive to the masses and couplings of the new particles. Precise
experiments and theory are needed to establish or definitively rule out the
BSM effects. Powerful calculation techniques, such as the heavy quark
effective theory (HQET) [5] and the soft collinear effective theory (SCET) [6,
7, 8] have been developed to incorporate power $1/m_{c}$ and $1/m_{b}$
corrections to the perturbative QCD estimates. More importantly, they enable a
better theoretical description by separating the various scales involved in
$B$ decays and in establishing factorisation of the decay matrix elements. In
exclusive decays, one also needs the decay form factors and a lot of
theoretical progress has been made using the lattice QCD [9] and QCD sum rule
techniques [10, 11, 12, 13], often complementing each other, as they work best
in the opposite $q^{2}$-ranges. It is this continued progress in QCD
calculational framework, which has taken us to the level of sophistication
required to match the experimental advances.

In this paper, I review what, in my view, are some of the key measurements in
the radiative, semileptonic and leptonic rare $B$-decays and confront them
with the SM-based calculations, carried out with the theoretical tools just
mentioned. However, this is not a comprehensive review of this subject, but
the hope is that the choice of topics reflects both the goals achieved in
explaining some landmark measurements and focus on open issues. In section 2,
I review the inclusive and some exclusive radiate rare $B$-decays.There are no
burning issues in this area - at least not yet. In section 3, the
corresponding inclusive and exclusive semileptonic decays are taken up. Again,
there are no open issues in the inclusive semileptonic decays, but
experimental precision is limited currently, which is bound to improve
significantly at Belle II. There are, however, a lot of open issues in the
exclusive semileptonic decays, in particular in $R_{K}$, the ratio of the
decay widths for $B\to K\mu^{+}\mu^{-}$ and $B\to Ke^{+}e^{-}$, hinting at the
possible breakdown of lepton universality, the linchpin of the SM, reviving
the interest in low-mass leptoquarks. One should also mention here similar
issues in tree-level semileptonic decays, such as $R^{\tau/\ell}_{D}$ and
$R^{\tau/\ell}_{D^{*}}$, the ratios involving the decays $B\to
D^{(*)}\tau\nu_{\tau}$ and $B\to D^{(*)}\ell\nu_{\ell}$ ($\ell=e,\mu$). There
are also other dissenting areas, which go by the name of
$P_{5}^{\prime}$-anomaly, which is a certain coefficient in the angular
description of the decay $B\to K^{*}\mu^{+}\mu^{-}$, which presumably need a
better theoretical (read QCD) description than is available at present. They
are discussed in section 3. In section 4, we discuss the CKM-suppressed $b\to
d\ell^{+}\ell^{-}$ decays, which is a new experimental frontier initiated by
the LHCb measurements of the branching fraction and the dimuon invariant mass
distribution in the decay $B^{\pm}\to\pi^{\pm}\mu^{+}\mu^{-}$. Finally, the
rarest $B$\- and $B_{s}$-decays measured so far, $B\to\mu^{+}\mu^{-}$ and
$B_{s}\to\mu^{+}\mu^{-}$, are taken up in section 5. Current measurements also
show some (mild) deviations in their branching ratios versus the SM. A
representative global fit of the data on the semileptonic and leptonic rare
$B$-decays in terms of the Wilson coefficients from possible new physics is
shown in section 6. Some concluding remarks are made in section 7.

##  2 Rare Radiative $B$-decays in the SM and Experiments

In 1993, the CLEO collaboration at the Cornell $e^{+}e^{-}$ collider measured
the decay $B\to K^{*}\gamma$ [14], initiating the field of rare $B$-decays,
followed two years later by the measurement of the inclusive decay $B\to
X_{s}\gamma$ [15]. The branching ratio ${\cal}(B\to K^{*}\gamma)=(4.5\pm
1.5\pm 0.9)\times 10^{-5}$ was in agreement with the SM estimates, but
theoretical uncertainty was large. Measuring the Inclusive process $B\to
X_{s}\gamma$ was challenging, but as the photon energy spectrum in this
process was already calculated in 1990 by Christoph Greub and me[16], this
came in handy for the CLEO measurements [17] shown in Fig. 1 (left frame) and
compared with the theoretical prediction [16], Since then, a lot of
experimental and theoretical effort has gone in the precise measurements and
in performing higher order perturbative and non-perturbative calculations. As
a consequence, $B\to X_{s}\gamma$ has now become the standard candle of FCNC
processes, with the measured branching ratio and the precise higher order SM-
based calculation providing valuable constraints on the parameters of BSM
physics. The impact of the $B$-factories on this measurement can be judged by
the scale in Fig. 1 (right frame), which is due to the Belle collaboration
[18].

Figure 1: Photon energy spectrum in the inclusive decay $B\to X_{s}\gamma$
measured by CLEO (left frame) [17] and Belle (right frame) [18].

The next frontier of rare $B$-decays involves the so-called electroweak
penguins, which govern the decays of the inclusive processes
$B\to(X_{s},X_{d})\ell^{+}\ell^{-}$ and the exclusive decays such as
$B\to(K,K^{*},\pi)\ell^{+}\ell^{-}$. These processes have rather small
branching ratios and hence they were first measured at the $B$-factories.
Inclusive decays remain their domain, but experiments at the LHC, in
particular, LHCb, are now at the forefront of exclusive semileptonic decays.
Apart from these, also the leptonic $B$-decays $B_{s}\to\mu^{+}\mu^{-}$ and
$B_{d}\to\mu^{+}\mu^{-}$ have been measured at the LHC.

I will review some of the key measurements and the theory relevant for their
interpretation. This description is anything but comprehensive, for which I
refer to some recent excellent references [19, 20, 21, 22] and resources, such
as HFAG [23] and FLAG [9].

###  2.1 Inclusive decays $B\to X_{s}\gamma$ at NNLO in the SM

The leading order diagrams for the decay $b\to s\gamma$ are shown are shown in
Fig. 2, including also the tree diagram for $b\to u\bar{u}s\gamma$, which
yields a soft photon. The first two diagrams are anyway suppressed due to the
CKM matrix elements, as indicated. The charm- and top- quark contributions
enter with opposite signs, and the relative contributions indicted are after
including the leading order (in $\alpha_{s}$) QCD effects. A typical diagram
depicting perturbative QCD corrections due to the exchange of a gluon is also
shown.

Figure 2: Examples of the leading electroweak diagrams for $B\to X_{s}\gamma$
from the up, charm, and top quarks. A diagram involving a gluon exchange is
shown in the lower figure.

The QCD logarithms $\alpha_{s}\ln M_{W}^{2}/m_{b}^{2}$ enhance the branching
ratio ${\cal B}(B\to X_{s}\gamma)$ by more than a factor 2, and hence such
logs have to be resummed. This is done using an effective field theory
approach, obtained by integrating out the top quark and the $W^{\pm}$ bosons.
Keeping terms up to dimension-6, the effective Lagrangian for $B\to
X_{s}\gamma$ and $B\to X_{s}\ell^{+}\ell^{-}$ reads as follows:

$\displaystyle{\cal L}\;=\;\;{\cal L}_{QCD\times
QED}(q,l)\;\;+\frac{4G_{F}}{\sqrt{2}}V_{ts}^{*}V_{tb}\sum_{i=1}^{10}C_{i}(\mu)O_{i}$

($q=u,d,s,c,b$, $l=e,\mu,\tau$)

$\displaystyle
O_{i}=\left\\{\begin{array}[]{lll}(\bar{s}\Gamma_{i}c)(\bar{c}\Gamma^{\prime}_{i}b),&i=1,(2),&C_{i}(m_{b})\sim-0.26~{}(1.02)\\\\[11.38109pt]
(\bar{s}\Gamma_{i}b){\Sigma}_{q}(\bar{q}\Gamma^{\prime}_{i}q),&i=3,4,5,6,{}&|C_{i}(m_{b})|<0.08\\\\[11.38109pt]
\frac{em_{b}}{16\pi^{2}}\bar{s}_{L}\sigma^{\mu\nu}b_{R}F_{\mu\nu},&i=7,&C_{7}(m_{b})\sim-0.3\\\\[11.38109pt]
\frac{gm_{b}}{16\pi^{2}}\bar{s}_{L}\sigma^{\mu\nu}T^{a}b_{R}G^{a}_{\mu\nu},&i=8,&C_{8}(m_{b})\sim-0.16\\\\[11.38109pt]
\frac{e^{2}}{16\pi^{2}}(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{l}\gamma^{\mu}{(\gamma_{5})}l),&i=9,{(10)}&C_{i}(m_{b})\sim
4.27~{}(-4.2)\end{array}\right.$ (6)

Here, $G_{F}$ is the Fermi coupling constant, $V_{ij}$ are the CKM matrix
elements, $O_{i}$ are the four-Fermi and dipole operators, and $C_{i}(\mu)$
are the Wilson coefficients, evaluated at the scale $\mu$, which is taken
typically as $\mu=m_{b}$, and their values in the NNLO accuracy are given
above for $\mu=4.8$ GeV. Variations due to a different choice of $\mu$ and
uncertainties from the upper scale-setting $m_{t}/2\leq\mu_{0}\leq 2m_{t}$ can
be seen elsewhere [19].

There are three essential steps of the calculation:

  * •

Matching: Evaluating $C_{i}(\mu_{0})$ at $\mu_{0}\sim M_{W}$ by requiring the
equality of the SM and the effective theory Green functions.

  * •

Mixing: Deriving the effective theory renormalisation group equation (RGE) and
evolving $C_{i}(\mu)$ from $\mu_{0}$ to $\mu_{b}\sim m_{b}$.

  * •

Matrix elements: Evaluating the on-shell amplitudes at $\mu_{b}\sim m_{b}$.

All three steps have been improved in perturbation theory and now include the
next-to-next-to-leading order effects (NLLO), i.e., contributions up to
$O(\alpha_{s}^{2}(m_{b}))$. A monumental theoretical effort stretched well
over a decade with the participation of a large number of theorists underlies
the current theoretical precision of the branching ratio. The result is
usually quoted for a threshold photon energy to avoid experimental background
from other Bremsstrahlung processes. For the decay with $E_{\gamma}>1.6$ GeV
in the rest frame of the $B$ meson, the result at NNLO accuracy is [24, 25]

${\cal B}(B\to X_{s}\gamma)=(3.36\pm 0.23)\times 10^{-4},$ (7)

where the dominant SM uncertainty is non-perturbative [26]. This is to be
compared with the current experimental average of the same [23]

${\cal B}(B\to X_{s}\gamma)=(3.43\pm 0.21\pm 0.07)\times 10^{-4},$ (8)

where the first error is statistical and the second systematic, yielding a
ratio $1.02\pm 0.08$, providing a test of the SM to an accuracy better than
10%.

The CKM-suppressed decay $B\to X_{d}\gamma$ has also been calculated in the
NNLO precision. The result for $E_{\gamma}>1.6$ GeV is [24]

${\cal B}(B\to X_{d}\gamma)=(1.73^{+0.12}_{-0.22})\times 10^{-5}.$ (9)

This will be measured precisely at Belle II. The constraints on the CP
asymmetry are not very restrictive, but the current measurements are in
agreement with the SM expectation. For further details, see HFAG [23].

###  2.2 Bounds on the charged Higgs mass from ${\cal B}(B\to X_{s}\gamma)$

As the agreement between the SM and data is excellent, the decay rate for
$B\to X_{s}\gamma$ provides constraints on the parameters of the BSM theories,
such as supersymmetry and the 2 Higgs-doublet models (2HDM). In calculating
the BSM effects, depending on the model, the SM operator basis may have to be
enlarged, but in many cases one anticipates additive contributions to the
Wilson coefficients in the SM basis. In the context of $B\to X_{s}\gamma$, it
is customary to encode the BSM effects in the Wilson coefficients of the
dipole operators $C_{7}(\mu)$ and $C_{8}(\mu)$, and the constraints from the
branching ratio on the additive coefficients $\Delta C_{7}$ and $\Delta C_{8}$
then takes the numerical form [24]

${\cal B}(B\to X_{s}\gamma)\times 10^{4}=(3.36\pm 0.23)-8.22\Delta
C_{7}-1.99\Delta C_{8}.$ (10)

To sample the kind of constraints that can be derived on the parameters of the
BSM models, the 2HDM is a good case, as the branching ratio for the decay
$B\to X_{s}\gamma$ in this model has been derived to the same theoretical
accuracy [28].The Lagrangian for the 2HDM is

${\cal
L}_{H^{+}}=(2\sqrt{2}G_{F})^{1/2}\Sigma_{i,j=1}^{3}\bar{u}_{i}(A_{u}m_{u_{i}}V_{ij}P_{L}-A_{d}m_{d_{j}}V_{ij}P_{R})d_{j}H^{*}+h.c.,$
(11)

where $V_{ij}$ are the CKM matrix elements and $P_{L/R}=(1\mp\gamma_{5})/2$.
The 2HDM contributions to the Wilson coefficients are proportional to
$A_{i}A_{j}^{*}$, representing the contributions from the up-type $A_{u}$ and
down-type $A_{d}$ quarks. They are defined in terms of the ratio of the vacuum
expectation values, called $\tan\beta$, and are model dependent.

  * •

2HDM of type-I: $A_{u}=A_{d}=\frac{1}{\tan\beta},$

  * •

2HDM of type-II: $A_{u}=-1/A_{d}=\frac{1}{\tan\beta}$.

Examples of Feynman diagrams contributing to $B\to X_{s}\gamma$ in the 2HDM
are shown in Fig. 3.

Figure 3: Sample Feynman diagrams that matter for $B\to X_{s}\gamma$ in the
2HDM [28]. $H^{\pm}$ denotes a charged Higgs.

Apart from $\tan\beta$, the other parameter of the 2HDM is the mass of the
charged Higgs $M_{H}^{\pm}$. As ${\cal B}(B\to X_{s}\gamma)$ becomes
insensitive to $\tan\beta$ for larger values, $\tan\beta>2$ , the 2HDM
contribution depends essentially on $M_{H}^{\pm}$. The current measurements
and the SM estimates then provide constraints on $M_{H}^{\pm}$, as shown in
Fig. 4, 222I thank Matthias Steinhauser for providing this figure. updated
using [24, 28], yielding [24] $M_{H}^{\pm}>480$ GeV ( 90% C.L.) and
$M_{H}^{\pm}>358$ GeV ( 99% C.L.) . These constraints are competitive to the
direct searches of the $H^{\pm}$ at the LHC.

Figure 4: Constraints on the charged Higgs mass $m_{H}^{\pm}$ from ${\cal
B}(B\to X_{s}\gamma)$ in the 2HDM [24, 28]. Measured branching ratio (exp) and
the SM estimates are also shown. The curves demarcate the central values and
$\pm 1\sigma$ errors.

###  2.3 Exclusive radiative rare $B$ decays

Exclusive radiative decays, such as $B\to V\gamma$ ($V=K^{*},\rho,\omega$) and
$B_{s}\to\phi\gamma$, have been well-measured at the $B$ factories. In
addition, they offer the possibility of measuring CP- and isospin asymmetries,
a topic I will not discuss here. Theoretically, exclusive decays are more
challenging, as they require the knowledge of the form factors at $q^{2}=0$,
which can not be calculated directly using Lattice QCD. However, light-cone
QCD sum rules [12, 13] also do a good job for calculating heavy $\to$ light
form factors at low-$q^{2}$. In addition, the matrix elements require gluonic
exchanges between the spectator quark and the active quarks (spectator-
scattering), introducing intermediate scales in the decay rates. Also long-
distance effects generated by the four-quark operators with charm quarks are
present and are calculable in limited regions [29]. Thus, exclusive decays are
theoretically not as precise as the inclusive decay $B\to X_{s}\gamma$.
However, techniques embedded in HQET and SCET have led to the factorisation of
the decay matrix elements into perturbatively calculable (hard) and non-
perturbative (soft) parts, akin to the deep inelastic scattering processes.
These factorisation-based approaches are the main work-horse in this field.
Renormalisation group (RG) methods then allow to sum up large logarithms, and
this program has been carried out to a high accuracy.

A detailed discussion of the various techniques requires a thorough review,
which can’t be carried out here. I will confine myself by pointing to some key
references, beginning from the QCD factorisation approach, pioneered by
Beneke, Buchalla, Neubert and Sachrajda [30], which has been applied to the
radiative decays $B\to(K^{*},\rho,\omega)\gamma$ [31, 32, 33, 34]. Another
theoretical framework, called pQCD [35, 36], has also been put to use in these
decays [37, 38]. The SCET-based methods have also been harnessed [39, 40]. The
advantage of SCET is that it allows for an unambiguous separation of the
scales, and an operator definition of each object in the factorisation formula
can be given. Following the QCD factorisation approach, a factorisation
formula for the $B\to V\gamma$ matrix element can be written in SCET as well

$\langle
V\gamma|Q_{i}|B\rangle=\Delta_{i}C^{A}\xi_{V_{\perp}}+\frac{\sqrt{m_{B}}Ff_{V_{\perp}}}{4}\int
dwdu\;\phi_{+}^{B}(w)\phi_{\perp}^{V}(u)\;t_{i}^{II},$ (12)

where $F$ and $f_{V_{\perp}}$ are meson decay constants; $\phi_{+}^{B}(w)$ and
$\phi_{\perp}^{V}(u)$ are the light-cone distribution amplitudes for the $B$\-
and $V$-meson, respectively. The SCET form factor $\xi_{V_{\perp}}$ is related
to the QCD form factor through perturbative and power corrections, and the
perturbative hard QCD kernels are the coefficients $\Delta_{i}C^{A}$ and
$t_{i}^{II}$. They are known to complete NLO accuracy in RG-improved
perturbation theory [40].

The factorisation formula (12) has been calculated to NNLO accuracy in SCET
[41] (except for the NNLO corrections from the spectator scattering). As far
as the decays $B\to K^{*}\gamma$ and $B_{s}\to\phi\gamma$ are concerned, the
partial NNLO theory is still the state-of-the-art. Their branching ratios as
well as the ratio of the decay rates ${\cal B}(B_{s}\to\phi\gamma/{\cal
B}(B\to K^{*}\gamma)$ are given in Table LABEL:tab:exclusive-rad-decays,
together with the current experimental averages [23]. The corresponding
calculations for the CKM-suppressed decays $B\to(\rho,\omega)\gamma$ are not
yet available to the desired theoretical accuracy, due to the annihilation
contributions, for which, to the best of my knowledge, no factorisation
theorem of the kind discussed above has been proven. The results from a QCD-
Factorisation based approach [32] for $B\to\rho\gamma$ are also given in Table
LABEL:tab:exclusive-rad-decays and compared with the data. The exclusive decay
rates shown are in agreement with the experimental measurements, though
theoretical precision is not better than 20%. Obviously, there is need for a
better theoretical description, more so as Belle II will measure the radiative
decays with greatly improved precision. I will skip a discussion of the
isospin and CP asymmetries in these decays, as the current experimental bounds
[23] are not yet probing the SM in these observables.

Measurements [HFAG 2014] [23] and SM-based estimates of ${\cal
B}(B\to(K^{*},\rho)\gamma)$ and ${\cal B}(B_{s}\to\phi\gamma)$ in units of
$10^{-5}$, and the ratio ${\cal B}(B_{s}\to\phi\gamma)/{\cal B}(B^{0}\to
K^{*0}\gamma)$.  Decay Mode Expt. (HFAG) Theory (SM) [ $B^{0}\to K^{*0}\gamma$
$4.33\pm 0.15$ $4.6\pm 1.4$ [ $B^{+}\to K^{*+}\gamma$ $4.21\pm 0.18$ $4.3\pm
1.4$ [ $B_{s}\to\phi\gamma$ $3.59\pm 0.36$ $4.3\pm 1.4$ [
$B_{s}\to\phi\gamma/B^{0}\to K^{*0}\gamma$ $0.81\pm 0.08$ $1.0\pm 0.2$ [
$B^{0}\to\rho^{0}\gamma$ $0.86^{+0.15}_{-0.14}$ $0.65\pm 0.12$ [
$B^{+}\to\rho^{+}\gamma$ $0.98\pm 0.25$ $1.37\pm 0.26$ [

##  3 Semileptonic $b\to s$ decays $B\to(X_{s},K,K^{*})\ell^{+}\ell^{-}$

There are two $b\to s$ semileptonic operators in SM:

$O_{i}=\frac{e^{2}}{16\pi^{2}}(\bar{s}_{L}\gamma_{\mu}b_{L})(\bar{l}\gamma^{\mu}(\gamma_{5})l),\hskip
42.67912pti=9,(10)$

Their Wilson Coefficients have the following perturbative expansion:

$\displaystyle C_{9}(\mu)$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\alpha_{s}(\mu)}C_{9}^{(-1)}(\mu)~{}+~{}C_{9}^{(0)}(\mu)~{}+~{}\frac{\alpha_{s}(\mu)}{4\pi}C_{9}^{(1)}(\mu)~{}+~{}...$
$\displaystyle C_{10}$ $\displaystyle=$ $\displaystyle
C_{10}^{(0)}~{}+~{}\frac{\alpha_{s}(M_{W})}{4\pi}C_{10}^{(1)}~{}+~{}...$

The term $C_{9}^{(-1)}(\mu)$ reproduces the electroweak logarithm that
originates from the photonic penguins with charm quark loops, shown below
[42].

The first two terms in the perturbative expansion of $C_{9}(m_{b})$ are

$\displaystyle
C_{9}^{(0)}(m_{b})\simeq~{}2.2;~{}~{}\frac{4\pi}{\alpha_{s}(m_{b})}\,C_{9}^{(-1)}(m_{b})=\frac{4}{9}\,\ln\frac{M_{W}^{2}}{m_{b}^{2}}+{\cal
O}(\alpha_{s})\simeq~{}2.$

As they are very similar in magnitude, one needs to calculate the NNLO
contribution to get reliable estimates of the decay rate. In addition, leading
power corrections in $1/m_{c}$ and $1/m_{b}$ are required.

###  3.1 Inclusive semileptonic decays $B\to X_{s}\ell^{+}\ell^{-}$

A lot of theoretical effort has gone into calculating the perturbative QCD
NNLO , electromagnetic logarithms and power corrections [42, 43, 44, 45, 46].
The B-factory experiments Babar and Belle have measured the dilepton invariant
mass spectrum $d{\cal B}(B\to X_{s}\ell^{+}\ell^{-})/dq^{2}$ practically in
the entire kinematic region and have also measured the so-called Forward-
backward lepton asymmetry $A_{FB}(q^{2})$ [47]. They are shown in Fig. 5, and
compared with the SM-based theoretical calculations. Note that a cut of
$q^{2}>0.2$ GeV2 on the dilepton invariant squared mass is used. As seen in
these figures, two resonant regions near $q^{2}=M_{J/\psi}^{2}$ and
$q^{2}=M_{J/\psi^{\prime}}^{2}$ have to be excluded when comparing with the
short-distance contribution. They make up what is called the long-distance
contribution from the processes $B\to X_{s}+(J/\psi,J/\psi^{\prime})\to
X_{s}+\ell^{+}\ell^{-}$, whose dynamics is determined by the hadronic matrix
elements of the operators $O_{1}$ and $O_{2}$. They have also been calculated
via a dispersion relation [48] and data on the measured quantity $R_{\rm
had}(s)=\sigma(e^{+}e^{-}\to{\rm
hadrons})/\sigma(e^{+}e^{-}\to\mu^{+}\mu^{-})$, and in some analyses are also
included. As the (short-distance) contribution is expected to be a smooth
function of $q^{2}$, one uses the perturbative distributions in interpolating
these regions as well. The experimental distributions are in agreement with
the SM, including also the zero point of $A_{FB}(q^{2})$, which is a sensitive
function of the ratio of the two Wilson coefficients $C_{9}$ and $C_{10}$.

The branching ratio for the inclusive decay $B\to X_{s}\ell^{+}\ell^{-}$ with
a lower cut on the dilepton invariant mass $q^{2}>0.2~{}{\rm GeV}^{2}$ at NNLO
accuracy is [44]

${\cal B}(B\to X_{s}\ell^{+}\ell^{-})=(4.2\pm 0.7)\times 10^{-6},$ (13)

to be compared with the current experimental average of the same [23]

${\cal B}(B\to X_{s}\ell^{+}\ell^{-})=(3.66^{+0.76}_{-0.77})\times 10^{-6}.$
(14)

The two agree within theoretical and experimental errors. The experimental
cuts which are imposed to remove the $J/\psi$ and $\psi^{\prime}$ resonant
regions are indicated in Fig. 5. The effect of logarithmic QED corrections
becomes important for more restrictive cuts on $q^{2}$, and they have been
worked out for different choices of the $q^{2}$-range in a recent paper [49].

Figure 5: Dilepton invariant mass Distribution measured by BaBar [50] (upper
frame) and the Forward-backward Asymmetry $A_{\rm FB}$ measured by Belle [51]
(lower frame) in $B\to X_{s}\ell^{+}\ell^{-}$. The curve(above) and the band
(below) are the SM expectations, discussed in the text.

###  3.2 Exclusive Decays $B\to(K,K^{*})\ell^{+}\ell^{-}$

The $B\to K$ and $B\to K^{*}$ transitions involve the following weak currents:

$\Gamma_{\mu}^{1}=\bar{s}\gamma_{\mu}(1-\gamma_{5})b,~{}~{}\Gamma_{\mu}^{2}=\bar{s}\sigma_{\mu\nu}q^{\nu}(1+\gamma_{5})b.$
(15)

Their matrix elements involve altogether 10 non-perturbative $q^{2}$-dependent
functions (form factors), denoted by the following functions: 333As we will
also discuss later the decays $B\to\pi\ell^{+}\ell^{-}$, we distinguish the
$B\to K$ and $B\to\pi$ form factors by a superscript.

$\displaystyle\langle K|\Gamma_{\mu}^{1}|B\rangle$ $\displaystyle\supset
f^{K}_{+}(q^{2}),f^{K}_{-}(q^{2}).$ $\displaystyle\langle
K|\Gamma_{\mu}^{2}|B\rangle$ $\displaystyle\supset f_{T}^{K}(q^{2}).$
$\displaystyle\langle K^{*}|\Gamma_{\mu}^{1}|B\rangle$ $\displaystyle\supset
V(q^{2}),A_{1}(q^{2}),A_{2}(q^{2}),A_{3}(q^{2}).$ $\displaystyle\langle
K^{*}|\Gamma_{\mu}^{2}|B\rangle$ $\displaystyle\supset
T_{1}(q^{2}),T_{2}(q^{2}),T_{3}(q^{2}).$

Data on $B\to K^{*}\gamma$ provides normalisation of $T_{1}(0)=T_{2}(0)\simeq
0.28$. These form factors have been calculated using a number of non-
perturbative techniques, in particular the QCD sum rules [12, 52] and Lattice
QCD [53, 54, 56]. They are complementary to each other, as the former are
reliable in the low-$q^{2}$ domain and the latter can calculate only for
large-$q^{2}$. They are usually combined to get reliable profiles of the form
factors in the entire $q^{2}$ domain. However, heavy quark symmetry allows to
reduce the number of independent form factors from 10 to 3 in low-$q^{2}$
domain $(q^{2}/m_{b}^{2}\ll 1)$. Symmetry-breaking corrections have been
evaluated [57]. The decay rate, dilepton invariant mass distribution and the
Forward-backward asymmetry in the low-$q^{2}$ region have been calculated for
$B\to K^{*}\ell^{+}\ell^{-}$ using the SCET formalism [58]. Current
measurements of the branching ratios in the inclusive and exclusive
semileptonic decays involving $b\to s$ transition are summarised in Table
LABEL:tab:exclusive-decays and compared with the corresponding SM estimates.
The inclusive measurements and the SM rates include a cut on the dilepton
invariant mass $M_{\ell^{+}\ell^{-}}>0.2$ GeV. They are in agreement with each
other, though precision is currently limited due to the imprecise knowledge of
the form factors.

###  3.3 Current tests of lepton universality in semileptonic $B$-decays

Currently, a number of measurements in $B$ decays suggests a breakdown of the
lepton $(e,\mu,\tau)$ universality in semileptonic processes. In the SM, gauge
bosons couple with equal strength to all three leptons and the couplings of
the Higgs to a pair of charged lepton is proportional to the charged lepton
mass, which are negligibly small for
$\ell^{+}\ell^{-}=e^{+}e^{-},\mu^{+}\mu^{-}$. Hence, if the lepton non-
universality is experimentally established, it would be a fatal blow to the
SM.

We briefly summarise the experimental situation starting from the decay
$B^{\pm}\to K^{\pm}\ell^{+}\ell^{-}$, whose decay rates were discussed
earlier. Theoretical accuracy is vastly improved if instead of the absolute
rates, ratios of the decay rates are computed. Data on the decays involving
$K^{(*)}\tau^{+}\tau^{-}$ is currently sparse, but first measurements of the
ratios involving the final states $K^{(*)}\mu^{+}\mu^{-}$ and
$K^{(*)}e^{+}e^{-}$ are available. In particular, a 2.6$\sigma$ deviation from
the $e$-$\mu$ universality is reported by the LHCb collaboration in the ratio
involving $B^{\pm}\to K^{\pm}\mu^{+}\mu^{-}$ and $B^{\pm}\to
K^{\pm}e^{+}e^{-}$ measured in the low-$q^{2}$ region, which can be calculated
rather accurately. In the interval $1\leq q^{2}\leq 6$ GeV2, LHCb finds [59]

$R_{K}\equiv\frac{\Gamma(B^{\pm}\to K^{\pm}\mu^{+}\mu^{-})}{\Gamma(B^{\pm}\to
K^{\pm}e^{+}e^{-})}=0.745^{+0.090}_{-0.074}({\rm stat})\pm 0.035({\rm syst}).$
(16)

This ratio in the SM is close to 1 to a very high accuracy [60] over the
entire $q^{2}$ region measured by the LHCb. Thus, the measurement in (16)
amounts to about $2.6\sigma$ deviation from the SM. Several BSM scenarios have
been proposed to account for the $R_{K}$ anomaly, discussed below, including a
$Z^{\prime}$-extension of the SM [61]. It should, however, be noted that the
currently measured branching ratios ${\cal B}(B^{\pm}\to
K^{\pm}e^{+}e^{-})=(1.56^{+0.19+0.06}_{-0.15-0.4})\times 10^{-7}$ and ${\cal
B}(B^{\pm}\to K^{\pm}\mu^{+}\mu^{-})=(1.20\pm 0.09\pm 0.07)\times 10^{-7}$ are
also lower than the SM estimates ${\cal B}^{\rm SM}(B^{\pm}\to
K^{\pm}e^{+}e^{-})={\cal B}^{\rm SM}(B^{\pm}\to
K^{\pm}\mu^{+}\mu^{-})=(1.75^{+0.60}_{-0.29})\times 10^{-7}$, and the
experimental error on the ${\cal B}(B^{\pm}\to K^{\pm}e^{+}e^{-})$ is twice as
large. One has to also factor in that the electrons radiate very profusely
(compared to the muons) and implementing the radiative corrections in hadronic
machines is anything but straight forward. In coming years, this and similar
ratios, which can also be calculated to high accuracy, will be measured with
greatly improved precision at the LHC and Belle II.

The other place where lepton non-universality is reported is in the ratios of
the decays $B\to D^{(*)}\tau\nu_{\tau}$ and $B\to D^{(*)}\ell\nu_{\ell}$.
Defining

$R^{\tau/\ell}_{D^{(*)}}\equiv\frac{{\cal B}(B\to D^{(*)}\tau\nu_{\tau})/{\cal
B}^{\rm SM}(B\to D^{(*)}\tau\nu_{\tau})}{{\cal B}(B\to
D^{(*)}\ell\nu_{\ell})/{\cal B}^{\rm SM}(B\to D^{(*)}\ell\nu_{\ell})},$ (17)

the current averages of the BaBar, Belle, and the LHCb data are [23]:

$R^{\tau/\ell}_{D}=1.37\pm 0.17;\hskip 8.53581ptR^{\tau/\ell}_{D^{*}}=1.28\pm
0.08.$ (18)

This amounts to about $3.9\sigma$ deviation from the $\tau/\ell$
($\ell=e,\mu$) universality. Interestingly, this happens in a tree-level
charged current process. If confirmed experimentally, this would call for a
drastic contribution to an effective four-Fermi $LL$ operator
$(\bar{c}\gamma_{\mu}b_{L})(\tau_{L}\gamma_{\mu}\nu_{L})$. It is then
conceivable that the non-universality in $R_{K}$ (which is a loop-induced
$b\to s$ process) ia also due to an $LL$ operator
$(\bar{s}\gamma_{\mu}b_{L})(\bar{\mu}_{L}\gamma_{\mu}\mu_{L})$. Several
suggestions along these lines involving a leptoquark have been made [62, 63,
64]. It is worth recalling that leptoquarks were introduced by Pati and Salam
in 1973 in an attempt to unify leptons and quarks in $SU(4)$ [65, 66]. The
lepton non-universality in $B$ decays has revived the interest in theories
with low-mass leptoquarks, discussed recently in a comprehensive work on this
topic [67].

Measurements [PDG 2014] and SM-based estimates [44] of the branching ratios
${\cal B}(B\to(X_{s},K,K^{*})\ell^{+}\ell^{-})$ in units of $10^{-6}$ Decay
Mode Expt. (BELLE & BABAR) Theory (SM) [ $B\to K\ell^{+}\ell^{-}$ $0.48\pm
0.04$ $0.35\pm 0.12$ [ $B\to K^{*}e^{+}e^{-}$ $1.19\pm 0.20$ $1.58\pm 0.49$ [
$B\to K^{*}\mu^{+}\mu^{-}$ $1.06\pm 0.09$ $1.19\pm 0.39$ [ $B\to
X_{s}\mu^{+}\mu^{-}$ $4.3\pm 1.2$ $4.2\pm 0.7$ [ $B\to X_{s}e^{+}e^{-}$
$4.7\pm 1.3$ $4.2\pm 0.7$ [ $B\to X_{s}\ell^{+}\ell^{-}$ $4.5\pm 1.0$ $4.2\pm
0.7$ [

###  3.4 Angular analysis of the decay $B^{0}\to K^{*0}(\to
K^{+}\pi^{-})\mu^{+}\mu^{-}$

For the inclusive decays $B\to X_{s}\ell^{+}\ell^{-}$, the observables which
have been measured are the integrated rates, the dilepton invariant mass
$d\Gamma/dq^{2}$ and the FB asymmetry $A_{\rm FB}(q^{2})$. They are all found
to be in agreement with the SM. In the exclusive decays such as $B\to
K^{*}\ell^{+}\ell^{-}$ and $B_{s}\to\phi\ell^{+}\ell^{-}$, a complete angular
analysis of the decay is experimentally feasible. This allows one to measure a
number of additional observables, defined below.

$\displaystyle\frac{1}{d(\Gamma+\bar{\Gamma})}\frac{d^{4}(\Gamma+\bar{\Gamma})}{dq^{2}d\Omega}$
$\displaystyle=$
$\displaystyle\frac{9}{32\pi}\left[\frac{3}{4}(1-F_{L})\sin^{2}\theta_{K}+F_{L}\;\cos^{2}\theta_{K}\right.$
(19) $\displaystyle\left.+\frac{1}{4}(1-F_{L})\sin^{2}\theta_{K}\;\cos
2\theta_{\ell}-F_{L}\cos^{2}\theta_{K}\cos 2\theta_{\ell}\right.$
$\displaystyle\left.+S_{3}\sin^{2}\theta_{K}\;\sin^{2}\theta_{\ell}\;\cos
2\phi+S_{4}\sin 2\theta_{K}\sin 2\theta_{\ell}\cos\phi\right.$
$\displaystyle\left.+S_{5}\sin
2\theta_{K}\sin\theta_{\ell}\cos\phi+\frac{4}{3}\;A_{\rm
FB}\sin^{2}\theta_{K}\cos\theta_{\ell}\right.$ $\displaystyle\left.+S_{7}\sin
2\theta_{K}\sin\theta_{\ell}\;\sin\phi+S_{8}\sin 2\theta_{K}\sin
2\theta_{\ell}\;\sin\phi\right.$
$\displaystyle\left.+S_{9}\sin^{2}\theta_{K}\sin^{2}\theta_{\ell}\;\sin
2\phi\right].$

The three angles $\theta_{K}$, $\theta_{\ell}$ and $\phi$ for the decay
$B^{0}\to K^{*0}(\to K^{+}\pi^{-})\mu^{+}\mu^{-}$ are defined in Fig. 6.

Figure 6: Definitions of the angles in $B^{0}\to K^{*0}(\to
K^{+}\pi^{-})\mu^{+}\mu^{-}$.

An angular analysis of the decay chains $B^{0}\to K^{*0}(\to
K^{+}\pi^{-})\mu^{+}\mu^{-}$ [68] and $B_{s}^{0}\to\phi(\to
K^{+}K^{-})\mu^{+}\mu^{-}$ [69] has been carried out by LHCb.

The observables in (19) are $q^{2}$-dependent coefficients of the Wilson
coefficients and hence they probe the underlying dynamics. Since these
coefficients have been calculated to a high accuracy, the remaining
theoretical uncertainty lies in the form factors and also from the charm-quark
loops. The form factors have been calculated using the QCD sum rules and in
the high-$q^{2}$ region also using lattice QCD. They limit the current
theoretical accuracy. However, a number of so-called optimised observables has
been proposed [70], which reduce the dependence on the form factors. Using the
LHCb convention, these observables are defined as [68]

$\displaystyle P_{1}$ $\displaystyle\equiv$ $\displaystyle
2S_{3}/(1-F_{L});~{}~{}P_{2}\equiv 2A_{\rm FB}/3(1-F_{L});~{}~{}P_{3}\equiv-
S_{9}/(1-F_{L}),$ (20) $\displaystyle P^{\prime}_{4,5,6,8}\equiv
S_{4,5,7,8}/\sqrt{F_{l}(1-F_{L})}.$

Figure 7: CP-averaged variables in bins of $q^{2}$ for the observables $F_{\rm
L}$, $A_{\rm FB}$, $S_{3}$ and $S_{5}$ in $B^{0}\to K^{*0}(\to
K^{+}\pi^{-})\mu^{+}\mu^{-}$ measured by LHCb [68] and comparison with the SM
[73].

Figure 8: The optimised angular observables in bins of $q^{2}$ in the decay
$B^{0}\to K^{*0}(\to K^{+}\pi^{-})\;\mu^{+}\mu^{-}$ measured by the LHCb
collaboration [68] and comparison with the SM [75].

These angular observables have been analysed in a number of theoretical
studies [75, 71, 72, 73, 13, 74], which differ in the treatment of their non-
perturbative input, mainly form factors. The LHCb collaboration, which
currently dominates this field, has used these SM-based estimates and compared
with their data in various $q^{2}$ bins. Two representative comparisons based
on the theoretical estimates from Altmannshofer and Straub [73] and Descotes-
Genon, Hofer, Matias and Virto [75] are shown in Figs. 7 and 8, respectively.
They are largely in agreement with the data, except for the distributions in
the observables $S_{5}(q^{2})$ (in Fig. 7) and $P_{5}^{\prime}(q^{2})$ (in
Fig. 8) in the bins around $q^{2}\geq 5$ GeV2. The pull on the SM depends on
the theoretical model, reaching 3.4$\sigma$ in the bin $4.3\leq q^{2}\leq
8.68$ GeV2 compared to DHMV [75]. There are deviations of a similar nature,
between 2 and 3$\sigma$, seen in the comparison of $S_{5}$ and other
quantities, such as the partial branching ratios in $B\to
K^{*}\mu^{+}\mu^{-}$, $B_{s}\to\phi\mu^{+}\mu^{-}$ and $F_{L}(q^{2})$ [73].

An analysis of the current Belle data [76] , shown in Fig. 9, displays a
similar pattern as the one reported by LHCb. As the Belle data has larger
errors, due to limited statistics, the resulting pull on the SM is less
significant. In the interval $4.0\leq q^{2}\leq 8.0$ GeV2, Belle reports
deviations of $2.3\sigma$ (compared to DHMV [75]), $1.72\sigma$ (compared to
BSZ [13]) and $1.68\sigma$ (compared to JC [72]). These measurements will
improve greatly at Belle II.

Figure 9: The optimised angular observables $P_{4}^{\prime}$ and
$P_{5}^{\prime}$ in bins of $q^{2}$ in the decay $B^{0}\to K^{*0}(\to
K^{+}\pi^{-})\;\mu^{+}\mu^{-}$ measured by Belle [76] and comparison with the
SM [75].

To quantify the deviation of the LHCb data from the SM estimates, a
$\Delta\chi^{2}$ distribution for the real part of the Wilson coefficient
${\rm Re}C_{9}(m_{b})$ is shown in Fig. 10. In calculating the
$\Delta\chi^{2}$, the other Wilson coefficients are set to their SM values.
The coefficient ${\rm Re}C_{9}^{\rm SM}(m_{b})=4.27$ at the NNLO accuracy in
the SM is indicated by a vertical line. The best fit of the LHCb data yields a
value which is shifted from the SM, and the deviation in this coefficient is
found to be $\Delta{\rm Re}C_{9}(m_{b})=-1.04\pm 0.25$. The deviation is
tantalising, but not yet conclusive. A bit of caution is needed here as the SM
estimates used in the analysis above may have to be revised, once the residual
uncertainties are better constrained. In particular, the hadronic
contributions generated by the four-quark operators with charm are difficult
to estimate, especially around $q^{2}\sim 4m_{c}^{2}$, leading to an effective
shift in the value of the Wilson coefficient being discussed [77].

Figure 10: The $\Delta\chi^{2}$ distribution for the real part of the Wilson
coefficient ${\rm Re}C_{9}(m_{b})$ from a fit of the CP-averaged observables
$F_{\rm L},A_{\rm FB},S_{3},...,S_{9}$ in $B^{0}\to K^{*0}(\to
K^{+}\pi^{-})\mu^{+}\mu^{-}$ by the LHCb collaboration [68].

##  4 CKM-suppressed $b\to d\ell^{+}\ell^{-}$ transitions in the SM

Weak transitions $b\to d\ell^{+}\ell^{-}$ , like the radiative decays $b\to
d\gamma$, are CKM suppressed and because of this the structure of the
effective weak Hamiltonian is different than the one encountered earlier for
the $b\to s\ell^{+}\ell^{-}$ transitions.

$\displaystyle{\cal H}_{\rm eff}^{b\to d}$ $\displaystyle=$
$\displaystyle-\frac{4G_{G}}{\sqrt{2}}\left[V_{tb}^{*}V_{td}\sum_{i=1}^{10}C_{i}(\mu)(O_{i}(\mu)\right.$
(21)
$\displaystyle\left.+V_{ub}^{*}V_{ud}\sum_{i=1}^{2}C_{i}(\mu)(O_{i}(\mu)-O_{i}(\mu))\right]+{\rm
h.c.}.$

Here $O_{i}(\mu)$ are the dimension-six operators introduced earlier (except
for the interchange $s\to d$ quark) in ${\cal H}_{\rm eff}^{b\to s}$. As the
two CKM factors are comparable in magnitude
$|V_{tb}^{*}V_{td}|\simeq|V_{ub}^{*}V_{ud}|$, and have different weak phases,
we anticipate sizeable CP-violating asymmetries in both the inclusive $b\to
d\ell^{+}\ell^{-}$ and exclusive transitions, such as
$B\to(\pi,\rho)\ell^{+}\ell^{-}$. The relevant operators appearing in ${\cal
H}_{\rm eff}^{b\to d}$ are:

Tree operators

${\cal
O}_{1}=\left(\bar{d}_{L}\gamma_{\mu}T^{A}c_{L}\right)\left(\bar{c}_{L}\gamma^{\mu}T^{A}b_{L}\right),\quad{\cal
O}_{2}=\left(\bar{d}_{L}\gamma_{\mu}c_{L}\right)\left(\bar{c}_{L}\gamma^{\mu}b_{L}\right),$
(22)

${\cal
O}_{1}^{(u)}=\left(\bar{d}_{L}\gamma_{\mu}T^{A}u_{L}\right)\left(\bar{u}_{L}\gamma^{\mu}T^{A}b_{L}\right),\quad{\cal
O}_{2}^{(u)}=\left(\bar{d}_{L}\gamma_{\mu}u_{L}\right)\left(\bar{u}_{L}\gamma^{\mu}b_{L}\right).$
(23)

Dipole operators

${\cal O}_{7}=\frac{e\,m_{b}}{g_{\rm
s}^{2}}\left(\bar{d}_{L}\sigma^{\mu\nu}b_{R}\right)F_{\mu\nu},\quad{\cal
O}_{8}=\frac{m_{b}}{g_{\rm
s}}\left(\bar{d}_{L}\sigma^{\mu\nu}T^{A}b_{R}\right)G_{\mu\nu}^{A}.$ (24)

Semileptonic operators

${\cal O}_{9}=\frac{e^{2}}{g_{\rm
s}^{2}}\left(\bar{d}_{L}\gamma^{\mu}b_{L}\right)\sum_{\ell}\left(\bar{\ell}\gamma_{\mu}\ell\right),\quad{\cal
O}_{10}=\frac{e^{2}}{g_{\rm
s}^{2}}\left(\bar{d}_{L}\gamma^{\mu}b_{L}\right)\sum_{\ell}\left(\bar{\ell}\gamma_{\mu}\gamma_{5}\ell\right).$
(25)

Here, $e(g_{s})$ is the QED (QCD) coupling constant. Since the inclusive decay
$B\to X_{d}\ell^{+}\ell^{-}$ has not yet been measured, but hopefully will be
at Belle II, we discuss the exclusive decay $B^{+}\to\pi^{+}\ell^{+}\ell^{-}$,
which is the only $b\to d$ semileptonic transition measured so far.

###  4.1 Exclusive decay $B^{+}\to\pi^{+}\ell^{+}\ell^{-}$

The decay 444Charge conjugation is implied here.
$B^{+}\to\pi^{+}\ell^{+}\ell^{-}$ is induced by the vector and tensor currents
and their matrix elements are defined as

$\langle\pi(p_{\pi})|\bar{b}\gamma^{\mu}d|B(p_{B})\rangle=f^{\pi}_{+}(q^{2})\left(p_{B}^{\mu}+p_{\pi}^{\mu}\right)+\left[f^{\pi}_{0}(q^{2})-f^{\pi}_{+}(q^{2})\right]\displaystyle\frac{m^{2}_{B}-m^{2}_{\pi}}{q^{2}}q^{\mu},$
(26)
$\langle\pi(p_{\pi})|\bar{b}\sigma^{\mu\nu}q_{\nu}d|B(p_{B})\rangle=\displaystyle\frac{if^{\pi}_{T}(q^{2})}{m_{B}+m_{\pi}}\left[\left(p_{B}^{\mu}+p_{\pi}^{\mu}\right)q^{2}-q^{\mu}\left(m_{B}^{2}-m_{\pi}^{2}\right)\right].$
(27)

These form factors are related to the ones in the decay $B\to
K\ell^{+}\ell^{-}$, called $f^{K}_{i}(q^{2})$, discussed earlier, by
$SU(3)_{F}$ symmetry. Of these, the form factors $f^{\pi}_{+}(q^{2})$ and
$f^{\pi}_{0}(q^{2})$ are related by isospin symmetry to the corresponding ones
measured in the charged current process $B^{0}\to\pi^{-}\ell^{+}\nu_{\ell}$ by
Babar and Belle, and they can be extracted from the data. This has been done
using several parameterisations of the form factors with all of them giving an
adequate description of the data [78]. Due to their analytic properties, the
so-called $z$-expansion methods, in which the form factors are expanded in a
Taylor series in $z$, employed in the Boyd-Grinstein-Lebed (BGL)
parametrisation [79] and the Bourrely-Caprini-Lellouch (BCL) [80]
parametrisation, are preferable.

The BGL parametrisation is used in working out the decay rate and the
invariant dilepton mass distribution [78] for
$B^{+}\to\pi^{+}\ell^{+}\ell^{-}$, which is discussed below. The BCL-
parametrisation is used by the lattice-QCD groups, the HPQCD [54, 55] and
Fermilab/MILC [56] collaborations, to determine the form factors
$f^{\pi}_{i}(q^{2})$ and $f^{K}_{i}(q^{2})$. In particular, the Fermilab/MILC
collaboration has worked out the dilepton invariant mass distribution in the
decay of interest $B^{+}\to\pi^{+}\ell^{+}\ell^{-}$, making use of their
simulation in the large-$q^{2}$ region and extrapolating with the BCL
parametrisation.

We first discuss the low-$q^{2}$ region ($q^{2}\ll m_{b}^{2}$). In this case,,
heavy quark symmetry (HQS) relates all three form factors of interest
$f^{\pi}_{i}(q^{2})$ and this can be used advantageously to have a reliable
estimate of the dilepton invariant mass spectrum in this region. Including
lowest order HQS-breaking, the resulting expressions for the form factors (for
$q^{2}/m_{b}^{2}\ll 1$) are worked out by Beneke and Feldmann [57]. Thus,
fitting the form factor $f_{+}(q^{2})$ from the charged current data on
$B\to\pi\ell^{+}\nu_{\ell}$ decay, and taking into account the HQS and its
breaking, lead to a model-independent predictions of the differential
branching ratio (dimuon mass spectrum) in the neutral current process
$B^{+}\to\pi^{+}\ell^{+}\ell^{-}$ for low-$q^{2}$ values. However, the long-
distance contribution, which arises from the processes
$B^{+}\to\pi^{+}(\rho^{0},\omega)\to\pi^{+}\mu^{+}\mu^{-}$ are not included
here. The SM invariant dilepton mass distribution in
$B^{+}\to\pi^{+}\ell^{+}\ell^{-}$ integrated over the range $1{\rm
GeV}^{2}\leq q^{2}\leq 8{\rm GeV}^{2}$ yields a partial branching ratio

${\cal B}(B^{+}\to\pi^{+}\mu^{+}\mu^{-})=(0.57^{+0.07}_{-0.05})\times
10^{-8}.$ (28)

Thanks to the available data on the charged current process and heavy quark
symmetry, this enables an accuracy of about 10% for an exclusive branching
ratio, comparable to the theoretical accuracy in the inclusive decay $B\to
X_{s}\gamma$, discussed earlier. Thus, the decay
$B^{+}\to\pi^{+}\mu^{+}\mu^{-}$ offers a key advantage compared to the decay
$B^{+}\to K^{+}\ell^{+}\ell^{-}$, in which case the charged current process is
not available.

The differential branching ratio in the entire $q^{2}$ region is given by

$\frac{d{\cal
B}(B^{+}\to\pi^{+}\ell^{+}\ell^{-})}{dq^{2}}=C_{B}|V_{tb}V_{td}^{*}|^{2}\sqrt{\lambda(q^{2})}\sqrt{1-\frac{4m_{\ell}^{2}}{q^{2}}}F(q^{2}),$
(29)

where the constant $C_{B}=G_{F}^{2}\alpha_{\rm
em}^{2}\tau_{B}/1024\pi^{5}m_{B}^{3}$ and $\lambda(q^{2})$ is the usual
kinematic function
$\lambda(q^{2})=(m_{B}^{2}+m_{\pi}^{2}-q^{2})^{2}-4m_{B}^{2}m_{\pi}^{2}$. The
function $F(q^{2})$ depends on the effective Wilson coefficients, $C_{7}^{\rm
eff}$, $C_{9}^{\rm eff}$, and $C_{10}^{\rm eff}$, and the three form factors
$f_{+}^{\pi}(q^{2})$, $f_{0}^{\pi}(q^{2})$ and $f_{T}^{\pi}(q^{2})$. A
detailed discussion of the determination of the form factors, of which only
$f_{+}^{\pi}(q^{2})$ and $f_{T}^{\pi}(q^{2})$ are numerically important for
$\ell^{\pm}=e^{\pm},\mu^{\pm}$ is given elsewhere [78]. We recall that
$f_{+}^{\pi}(q^{2})$ is constrained by the data on the charged current process
in the entire $q^{2}$ domain. In addition, the lattice-QCD results on the form
factors in the large-$q^{2}$ domain and the HQS-based relations in the
low-$q^{2}$ region provide sufficient constraints on the form factor. This has
enabled a rather precise determination of the invariant dilepton mass
distribution in $B^{+}\to\pi^{+}\ell^{+}\ell^{-}$.

Taking into account the various parametric and form-factor dependent
uncertainties, this yields the following estimate for the branching ratio for
$B^{+}\to\pi^{+}\mu^{+}\mu^{-}$ [82]

${\cal B}_{\rm SM}(B^{+}\to\pi^{+}\mu^{+}\mu^{-})=(1.88^{+0.32}_{-0.21})\times
10^{-8},$ (30)

to be compared with the measured branching ratio by the LHCb collaboration
[82] (based on 3${\rm fb}^{-1}$ data):

${\cal B}_{\rm LHCb}(B^{+}\to\pi^{+}\mu^{+}\mu^{-})=(1.83\pm 0.24\pm
0.05)\times 10^{-8},$ (31)

where the first error is statistical and the second systematic, resulting in
excellent agreement. The dimuon invariant mass distribution measured by the
LHCb collaboration [82] is shown in Fig. 11, and compared with the SM-based
theoretical prediction, called APR13 [78], and the lattice-based calculation,
called FNAL/MILC 15 [56]. Also shown is a comparison with a calculation,
called HKR [81] , which has essentially the same short-distance contribution
in the low-$q^{2}$ region, as discussed earlier, but additionally takes into
account the contributions from the lower resonances $\rho,\omega$ and $\phi$.
This adequately describes the distribution in the $q^{2}$ bin, around 1 GeV2.

With the steadily improving lattice calculations for the various input
hadronic quantities and the form factors, theoretical error indicated in Eq.
(30) will go down considerably. Experimentally, we expect rapid progress due
to the increased statistics at the LHC, but also from Belle II, which will
measure the corresponding distributions and branching ratio also in the decays
$B^{+}\to\pi^{+}e^{+}e^{-}$, and $B^{+}\to\pi^{+}\tau^{+}\tau^{-}$, providing
a complementary test of the $e$-$\mu$-$\tau$ universality in $b\to d$
semileptonic transitions.

Figure 11: Comparison of the dimuon invariant mass distribution in
$B^{+}\to\pi^{+}\mu^{+}\mu^{-}$ in the SM with the LHCb data [82]. Theoretical
distributions shown are : APR13 [78], HKR15 [81], and FNAL/MILC [56].

##  5 Leptonic Rare $B$ Decays

The final topic discussed in this write-up involves purely leptonic decays
$B_{s}^{0}\to\ell^{+}\ell^{-}$ and $B^{0}\to\ell^{+}\ell^{-}$ with
$\ell^{+}\ell^{-}=e^{+}e^{-},\mu^{+}\mu^{-},\tau^{+}\tau^{-}$. Of these,
${\cal B}(B_{s}^{0}\to\mu^{+}\mu^{-})=(2.8^{+0.7}_{-0.6})\times 10^{-9}$ is
now well measured, and the corresponding CKM-suppressed decay ${\cal
B}(B^{0}\to\mu^{+}\mu^{-})=(3.9^{+1,6}_{-1.4})\times 10^{-10}$ is almost on
the verge of becoming a measurement. These numbers are from the combined
CMS/LHCb data [83]. From the experimental point of view, their measurement is
a real tour de force, considering the tiny branching ratios and the formidable
background at the LHC.

In the SM, these decays are dominated by the axial-vector operator
$O_{10}=(\bar{s}_{\alpha}\gamma^{\mu}P_{L}b_{\alpha})(\bar{\ell}\gamma_{\mu}\gamma_{5}\ell)$.
In principle, the operators
$O_{S}=m_{b}(\bar{s}_{\alpha}\gamma^{\mu}P_{R}b_{\alpha})(\bar{\ell}\ell)$ and
$O_{P}=m_{b}(\bar{s}_{\alpha}\gamma^{\mu}P_{R}b_{\alpha})(\bar{\ell}\gamma_{5}\ell)$
also contribute, but are chirally suppressed in the SM. This need not be the
case in BSM scenarios, and hence the great interest in measuring these decays
precisely. In the SM, the measurement of ${\cal
B}(B_{s}^{0}\to\mu^{+}\mu^{-})$ and ${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ provide
a measurement of the Wilson coefficient $C_{10}(m_{b})$. Their ratio ${\cal
B}(B^{0}\to\mu^{+}\mu^{-})/{\cal B}(B_{s}^{0}\to\mu^{+}\mu^{-})$ being
proportional to the ratio of the CKM matrix-elements $|V_{td}/V_{ts}|^{2}$ is
an important constraint on the CKM unitarity triangle.

Figure 12: Likelihood contours in the ${\cal B}(B^{0}\to\mu^{+}\mu^{-})$
versus ${\cal B}(B_{s}^{0}\to\mu^{+}\mu^{-})$ plane. The (black) cross in (a)
marks the best-fit value and the SM expectation is shown as the (red) marker.
Variations of the test statistics $-2\Delta\ln L$ for ${\cal
B}(B_{s}^{0}\to\mu^{+}\mu^{-})$ (b) and ${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ (c)
are shown. The SM prediction is denoted with the vertical (red) bars. (From
the combined CMS-LHCb data [83].)

The decay rate $\Gamma(B_{s}^{0}\to\mu^{+}\mu^{-})$ in the SM can be written
as

$\Gamma(B_{s}^{0}\to\mu^{+}\mu^{-})=\frac{G_{F}^{2}M_{W}^{2}m_{B_{s}}^{3}f_{B_{s}}^{2}}{8\pi^{5}}|V_{tb}^{*}V_{ts}|^{2}\frac{4m_{\ell}^{2}}{m_{B_{s}}^{2}}\sqrt{1-\frac{4m_{\ell}^{2}}{m_{B_{s}}^{2}}}|C_{10}|^{2}+O(\alpha_{\rm
em}).$ (32)

The coefficient $C_{10}$ has been calculated by taking into account the NNLO
QCD corrections and NLO electroweak corrections, but the $O(\alpha_{\rm em})$
contribution indicated above is ignored, as it is small. The SM branching
ratio in this accuracy have been obtained in [84, 85, 86], where a careful
account of the various input quantities is presented. The importance of
including the effects of the width difference $\Delta\Gamma_{s}$ due to the
$B_{s}^{0}$ \- $\bar{B}_{s}^{0}$ mixings in extracting the branching ratio for
$B_{s}\to\mu^{+}\mu^{-}$ has been emphasised in the literature [87] and is
included in the analysis. The time-averaged branching ratios, which in the SM
to a good approximation equals to $\overline{\cal
B}(B_{s}\to\mu^{+}\mu^{-})=\Gamma(B_{s}\to\mu^{+}\mu^{-})/\Gamma_{H}(B_{s})$,
where $\Gamma_{H}(B_{s})$ is the heavier mass-eigenstate total width, is given
below [84]

$\overline{\cal B}(B_{s}\to\mu^{+}\mu^{-})=(3.65\pm 0.23)\times 10^{-9}.$ (33)

In evaluating this, a value $f_{B_{s}}=227.7(4.5)$ MeV was used from the
earlier FLAG average [88]. In the most recent compilation by the FLAG
collaboration [9] , this coupling constant has been updated to
$f_{B_{s}}=224(5)$ MeV, which reduces the branching ratio to $\overline{\cal
B}(B_{s}\to\mu^{+}\mu^{-})=(3.55\pm 0.23)\times 10^{-9}$. This is compatible
with the current measurements to about 1$\sigma$, with the uncertainty
dominated by the experiment.

The corresponding branching ratio $\overline{\cal B}(B^{0}\to\mu^{+}\mu^{-})$
is evaluated as [84]

$\overline{\cal B}(B^{0}\to\mu^{+}\mu^{-})=(1.06\pm 0.09)\times 10^{-10},$
(34)

which, likewise, has to be scaled down to $(1.01\pm 0.09)\times 10^{-10}$, due
to the current average [9] $f_{B}=186(4)$ MeV, compared to $f_{B}=190.5(4.2)$
MeV used in deriving the result given in Eq. (34). This is about 2$\sigma$
below the current measurement, and the ratio of the two leptonic decays
$\overline{\cal B}(B_{s}\to\mu^{+}\mu^{-})/\overline{\cal
B}(B^{0}\to\mu^{+}\mu^{-})$ is off by about 2.3$\sigma$. The likelihood
contours in the ${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ versus ${\cal
B}(B_{s}^{0}\to\mu^{+}\mu^{-})$ plane from the combined CMS/LHCb data are
shown in Fig. 12.

The anomalies in the decays $B\to K^{*}\mu^{+}\mu^{-}$, discussed previously,
and the deviations in ${\cal B}(B^{0}\to\mu^{+}\mu^{-})$ and ${\cal
B}(B_{s}^{0}\to\mu^{+}\mu^{-})$, if consolidated experimentally, would require
an extension of the SM. A recent proposal based on the group $SU(3)_{C}\times
SU(3)_{L}\times U(1)$ is discussed by Buras, De Fazio and Girrbach [89].
Lepton non-universality, if confirmed, requires a leptoquark-type solution. A
viable candidate theory to replace the SM and accounting for all the current
anomalies, in my opinion, is not in sight.

##  6 Global fits of the Wilson Coefficients $C_{9}$ and $C_{10}$

As discussed in the foregoing, a number of deviations from the SM-estimates
are currently present in the data on semileptonic and leptonic rare
$B$-decays. They lie mostly around 2 to 3$\sigma$. A comparison of the LHCb
data on a number of angular observables $F_{\rm L},A_{\rm FB},S_{3},...,S_{9}$
in $B^{0}\to K^{*0}(\to K^{+}\pi^{-})\mu^{+}\mu^{-}$ with the SM-based
estimates was shown in Fig, 10, yielding a value of ${\rm Re}(C_{9})$ which
deviates from the SM by about 3$\sigma$. A number of groups has undertaken
similar fits of the data and the outcome depends on a number of assumed
correlations. However, it should be stressed that there are still non-
perturbative contributions present in the current theoretical estimates which
are not yet under complete quantitative control. The contributions from the
charm quarks in the loops is a case in point. Also, form factor uncertainties
are probably larger than assumed in some of these global fits.

As a representative example of the kind of constraints on the Wilson
coefficients $C_{9}$ and $C_{10}$ that follow from the data on semileptonic
and leptonic decays of the $B$ mesons is shown in Fig. 13 from the
Fermilab/MILC collaboration [56]. This shows that the SM point indicated by
$(0,0)$ in the ${\rm Re}(C_{9}^{NP},{\rm Re}(C_{10}^{NP})$-plane lies a little
beyond 2$\sigma$. In some other fits, the deviations are larger but still far
short for a discovery of BSM effects. As a lot of the experimental input in
this and similar analysis is due to the LHCb data, this has to be confirmed by
an independent experiment. This, hopefully, will be done by Belle II. We are
better advised to wait and see if these deviations become statistically
significant enough to warrant new physics. Currently, the situation is
tantalising but not conclusive.

Figure 13: Present constraints on the Wilson coefficients ${\rm Re}C_{10}^{\rm
NP}$ vs. ${\rm Re}C_{9}^{\rm NP}$ from the semileptonic rare $B$-decays and
$B_{s}\to\mu^{+}\mu^{-}$. The SM-point is indicated. (From Fermilab/MILC
Lattice Collaboration [56].)

##  7 Concluding Remarks

From the measurement by the CLEO collaboration of the rare decay $B\to
X_{s}\gamma$ in 1995, having a branching ratio of about $3\times 10^{-4}$, to
the rarest of the measured $B$ decays, $B^{0}\to\mu^{+}\mu^{-}$, with a
branching fraction of about $1\times 10^{-10}$ by the LHCb and CMS
collaborations, SM has been tested over six orders of magnitude. This is an
impressive feat, made possible by dedicated experimental programmes carried
out with diverse beams and detection techniques over a period of more than 20
years. A sustained theoretical effort has accompanied the experiments all
along, underscoring both the continued theoretical interest in $b$ physics and
an intense exchange between the two communities. With the exception of a few
anomalies, showing deviations from the SM ranging between 2 to 4$\sigma$ in
statistical significance, a vast majority of the measurements is in
quantitative agreement with the SM. In particular, all quark flavour
transitions are described by the CKM matrix whose elements are now determined.
The CP asymmetry measured so far in laboratory experiments is explained by the
Kobayashi-Maskawa phase. FCNC processes, of which rare $B$-decays discussed
here is a class, are governed by the GIM mechanism, with the particles in the
SM (three families of quarks and leptons, electroweak gauge bosons, gluons,
and the Higgs) accounting for all the observed phenomena - so far. Whether
this astounding consistence will continue will be tested in the coming years,
as the LHC experiments analyse more data, enabling vastly improved precision
in some of the key measurements discussed here. In a couple of years from now,
Belle II will start taking data, providing independent and new measurements.
They will be decisive in either deepening our knowledge about the SM, or
hopefully in discovering the new frontier of physics.

##  8 Acknowledgment

I thank Harald Fritzsch for inviting me to this very stimulating conference
and Prof. K.K. Phua for the warm hospitality in Singapore. I also thank
Mikolaj Misiak for reading the manuscript and helpful suggestions.

## References

  * [1] S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967); A. Salam, Conf. Proc. C 680519, 367 (1968).
  * [2] S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970).
  * [3] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963). M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
  * [4] K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C 38, 090001 (2014).
  * [5] A. V. Manohar and M. B. Wise, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 10, 1 (2000).
  * [6] C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001) [hep-ph/0011336]; C. W. Bauer, S. Fleming and M. E. Luke, Phys. Rev. D 63, 014006 (2000) [hep-ph/0005275]; C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev. D 65, 054022 (2002) [hep-ph/0109045].
  * [7] M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, Nucl. Phys. B 643, 431 (2002) [hep-ph/0206152].
  * [8] For an excellent review on SCET and applications, see T. Becher, A. Broggio and A. Ferroglia, Lect. Notes Phys. 896 (2015) [arXiv:1410.1892 [hep-ph]].
  * [9] S. Aoki et al., arXiv:1607.00299 [hep-lat].
  * [10] A. Ali, V. M. Braun and H. Simma, Z. Phys. C 63, 437 (1994) [hep-ph/9401277].
  * [11] P. Colangelo and A. Khodjamirian, In *Shifman, M. (ed.): At the frontier of particle physics, vol. 3* 1495-1576 [hep-ph/0010175].
  * [12] P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005) [hep-ph/0406232].
  * [13] A. Bharucha, D. M. Straub and R. Zwicky, arXiv:1503.05534 [hep-ph].
  * [14] R. Ammar et al. [CLEO Collaboration], Phys. Rev. Lett. 71, 674 (1993).
  * [15] M. S. Alam et al. [CLEO Collaboration], Phys. Rev. Lett. 74, 2885 (1995).
  * [16] A. Ali and C. Greub, Phys. Lett. B 259, 182 (1991); Z. Phys. C 49, 431 (1991).
  * [17] S. Chen et al. [CLEO Collaboration], Phys. Rev. Lett. 87, 251807 (2001) [hep-ex/0108032].
  * [18] P. Koppenburg et al. [Belle Collaboration], Phys. Rev. Lett. 93, 061803 (2004) [hep-ex/0403004].
  * [19] T. Blake, G. Lanfranchi and D. M. Straub, arXiv:1606.00916 [hep-ph].
  * [20] S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, JHEP 1606, 092 (2016) [arXiv:1510.04239 [hep-ph]].
  * [21] P. Koppenburg, Z. Dolezal and M. Smizanska, Scholarpedia 11, 32643 (2016) [arXiv:1606.00999 [hep-ex]].
  * [22] The physics of the B factories is excellently reviewed in A. J. Bevan et al. [BaBar and Belle Collaborations], Eur. Phys. J. C 74, 3026 (2014) [arXiv:1406.6311 [hep-ex]].
  * [23] Y. Amhis et al. [Heavy Flavor Averaging Group (HFAG) Collaboration], arXiv:1412.7515 [hep-ex].
  * [24] M. Misiak et al., Phys. Rev. Lett. 114, no. 22, 221801 (2015) [arXiv:1503.01789 [hep-ph]].
  * [25] M. Czakon, P. Fiedler, T. Huber, M. Misiak, T. Schutzmeier and M. Steinhauser, JHEP 1504, 168 (2015) [arXiv:1503.01791 [hep-ph]].
  * [26] M. Benzke, S. J. Lee, M. Neubert and G. Paz, JHEP 1008, 099 (2010) [arXiv:1003.5012 [hep-ph]].
  * [27] M. Misiak et al., Phys. Rev. Lett. 98, 022002 (2007) [hep-ph/0609232].
  * [28] T. Hermann, M. Misiak and M. Steinhauser, JHEP 1211, 036 (2012) [arXiv:1208.2788 [hep-ph]].
  * [29] A. Khodjamirian, T. Mannel, A. A. Pivovarov and Y.-M. Wang, JHEP 1009, 089 (2010) [arXiv:1006.4945 [hep-ph]].
  * [30] M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett. 83 (1999) 1914 [hep-ph/9905312].
  * [31] M. Beneke, T. Feldmann and D. Seidel, Nucl. Phys. B 612, 25 (2001) [hep-ph/0106067].
  * [32] A. Ali and A. Y. Parkhomenko, Eur. Phys. J. C 23, 89 (2002) [hep-ph/0105302].
  * [33] S. W. Bosch and G. Buchalla, Nucl. Phys. B 621, 459 (2002) [hep-ph/0106081].
  * [34] M. Beneke, T. Feldmann and D. Seidel, Eur. Phys. J. C 41, 173 (2005) [hep-ph/0412400].
  * [35] H. n. Li and H. L. Yu, Phys. Rev. D 53, 2480 (1996) [hep-ph/9411308].
  * [36] Y. Y. Keum, H. N. Li and A. I. Sanda, Phys. Rev. D 63, 054008 (2001) [hep-ph/0004173].
  * [37] Y. Y. Keum, M. Matsumori and A. I. Sanda, Phys. Rev. D 72, 014013 (2005) [hep-ph/0406055].
  * [38] C. D. Lu, M. Matsumori, A. I. Sanda and M. Z. Yang, Phys. Rev. D 72, 094005 (2005) Erratum: [Phys. Rev. D 73, 039902 (2006)] [hep-ph/0508300].
  * [39] J. g. Chay and C. Kim, Phys. Rev. D 68, 034013 (2003) [hep-ph/0305033].
  * [40] T. Becher, R. J. Hill and M. Neubert, Phys. Rev. D 72, 094017 (2005) [hep-ph/0503263].
  * [41] A. Ali, B. D. Pecjak and C. Greub, Eur. Phys. J. C 55, 577 (2008) [arXiv:0709.4422 [hep-ph]].
  * [42] A. Ghinculov, T. Hurth, G. Isidori and Y. P. Yao, Nucl. Phys. B 685, 351 (2004) [hep-ph/0312128].
  * [43] H. H. Asatryan, H. M. Asatrian, C. Greub and M. Walker, Phys. Rev. D 65, 074004 (2002) [hep-ph/0109140].
  * [44] A. Ali, E. Lunghi, C. Greub and G. Hiller, Phys. Rev. D 66, 034002 (2002) [hep-ph/0112300].
  * [45] T. Huber, E. Lunghi, M. Misiak and D. Wyler, Nucl. Phys. B 740, 105 (2006) [hep-ph/0512066].
  * [46] T. Huber, T. Hurth and E. Lunghi, Nucl. Phys. B 802, 40 (2008) [arXiv:0712.3009 [hep-ph]].
  * [47] A. Ali, T. Mannel and T. Morozumi, Phys. Lett. B 273, 505 (1991).
  * [48] F. Kruger and L. M. Sehgal, Phys. Lett. B 380, 199 (1996) [hep-ph/9603237].
  * [49] T. Huber, T. Hurth and E. Lunghi, JHEP 1506, 176 (2015) [arXiv:1503.04849 [hep-ph]].
  * [50] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. Lett. 112, 211802 (2014) [arXiv:1312.5364 [hep-ex]].
  * [51] Y. Sato et al. [Belle Collaboration], Phys. Rev. D 93, no. 3, 032008 (2016) Addendum: [Phys. Rev. D 93, no. 5, 059901 (2016)] [arXiv:1402.7134 [hep-ex]].
  * [52] A. Bharucha, T. Feldmann and M. Wick, JHEP 1009, 090 (2010) [arXiv:1004.3249 [hep-ph]].
  * [53] E. Dalgic, A. Gray, M. Wingate, C. T. H. Davies, G. P. Lepage and J. Shigemitsu, Phys. Rev. D 73, 074502 (2006) Erratum: [Phys. Rev. D 75, 119906 (2007)] [hep-lat/0601021].
  * [54] C. M. Bouchard, G. P. Lepage, C. J. Monahan, H. Na and J. Shigemitsu, PoS LATTICE 2013, 387 (2014) [arXiv:1310.3207 [hep-lat]].
  * [55] C. Bouchard et al. [HPQCD Collaboration], Phys. Rev. D 88, no. 5, 054509 (2013) Erratum: [Phys. Rev. D 88, no. 7, 079901 (2013)] [arXiv:1306.2384 [hep-lat]].
  * [56] J. A. Bailey et al. [Fermilab Lattice and MILC Collaborations], Phys. Rev. Lett. 115, no. 15, 152002 (2015) [arXiv:1507.01618 [hep-ph]].
  * [57] M. Beneke and T. Feldmann, Nucl. Phys. B 592, 3 (2001) [hep-ph/0008255].
  * [58] A. Ali, G. Kramer and G. h. Zhu, Eur. Phys. J. C 47, 625 (2006) [hep-ph/0601034].
  * [59] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 113, 151601 (2014) [arXiv:1406.6482 [hep-ex]].
  * [60] C. Bobeth, G. Hiller and G. Piranishvili, JHEP 0712, 040 (2007) [arXiv:0709.4174 [hep-ph]].
  * [61] C. W. Chiang, X. G. He and G. Valencia, Phys. Rev. D 93, no. 7, 074003 (2016) [arXiv:1601.07328 [hep-ph]].
  * [62] G. Hiller and M. Schmaltz, Phys. Rev. D 90, 054014 (2014) [arXiv:1408.1627 [hep-ph]].
  * [63] M. Bauer and M. Neubert, Phys. Rev. Lett. 116, no. 14, 141802 (2016) [arXiv:1511.01900 [hep-ph]].
  * [64] R. Barbieri, G. Isidori, A. Pattori and F. Senia, Eur. Phys. J. C 76, no. 2, 67 (2016) [arXiv:1512.01560 [hep-ph]].
  * [65] J. C. Pati and A. Salam, Phys. Rev. D 8, 1240 (1973).
  * [66] J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974) Erratum: [Phys. Rev. D 11, 703 (1975)].
  * [67] I. Dorsner, S. Fajfer, A. Greljo, J. F. Kamenik and N. Kosnik, arXiv:1603.04993 [hep-ph].
  * [68] R. Aaij et al. [LHCb Collaboration], JHEP 1602, 104 (2016) [arXiv:1512.04442 [hep-ex]].
  * [69] R. Aaij et al. [LHCb Collaboration], JHEP 1307, 084 (2013) [arXiv:1305.2168 [hep-ex]].
  * [70] S. Descotes-Genon, T. Hurth, J. Matias and J. Virto, JHEP 1305, 137 (2013) [arXiv:1303.5794 [hep-ph]].
  * [71] S. Jäger and J. Martin Camalich, JHEP 1305, 043 (2013) [arXiv:1212.2263 [hep-ph]].
  * [72] S. Jäger and J. Martin Camalich, Phys. Rev. D 93, no. 1, 014028 (2016) [arXiv:1412.3183 [hep-ph]].
  * [73] W. Altmannshofer and D. M. Straub, Eur. Phys. J. C 75, no. 8, 382 (2015) [arXiv:1411.3161 [hep-ph]].
  * [74] T. Hurth and F. Mahmoudi, JHEP 1404, 097 (2014) [arXiv:1312.5267 [hep-ph]].
  * [75] S. Descotes-Genon, L. Hofer, J. Matias and J. Virto, JHEP 1412, 125 (2014) [arXiv:1407.8526 [hep-ph]].
  * [76] A. Abdesselam et al. [Belle Collaboration], arXiv:1604.04042 [hep-ex].
  * [77] M. Ciuchini, M. Fedele, E. Franco, S. Mishima, A. Paul, L. Silvestrini and M. Valli, JHEP 1606, 116 (2016) [arXiv:1512.07157 [hep-ph]].
  * [78] A. Ali, A. Y. Parkhomenko and A. V. Rusov, Phys. Rev. D 89, no. 9, 094021 (2014) [arXiv:1312.2523 [hep-ph]].
  * [79] C. G. Boyd, B. Grinstein and R. F. Lebed, Phys. Rev. Lett. 74, 4603 (1995) [hep-ph/9412324].
  * [80] C. Bourrely, I. Caprini and L. Lellouch, Phys. Rev. D 79, 013008 (2009) Erratum: [Phys. Rev. D 82, 099902 (2010)] [arXiv:0807.2722 [hep-ph]].
  * [81] C. Hambrock, A. Khodjamirian and A. Rusov, Phys. Rev. D 92, no. 7, 074020 (2015) [arXiv:1506.07760 [hep-ph]].
  * [82] R. Aaij et al. [LHCb Collaboration], JHEP 1510 (2015) 034 [arXiv:1509.00414 [hep-ex]].
  * [83] V. Khachatryan et al. [CMS and LHCb Collaborations], Nature 522, 68 (2015) [arXiv:1411.4413 [hep-ex]].
  * [84] C. Bobeth, M. Gorbahn, T. Hermann, M. Misiak, E. Stamou and M. Steinhauser, Phys. Rev. Lett. 112, 101801 (2014) [arXiv:1311.0903 [hep-ph]].
  * [85] T. Hermann, M. Misiak and M. Steinhauser, JHEP 1312, 097 (2013) [arXiv:1311.1347 [hep-ph]].
  * [86] C. Bobeth, M. Gorbahn and E. Stamou, Phys. Rev. D 89, no. 3, 034023 (2014) [arXiv:1311.1348 [hep-ph]].
  * [87] K. De Bruyn, R. Fleischer, R. Knegjens, P. Koppenburg, M. Merk, A. Pellegrino and N. Tuning, Phys. Rev. Lett. 109, 041801 (2012) [arXiv:1204.1737 [hep-ph]].
  * [88] S. Aoki et al., Eur. Phys. J. C 74, 2890 (2014) [arXiv:1310.8555 [hep-lat]].
  * [89] A. J. Buras, F. De Fazio and J. Girrbach, JHEP 1402, 112 (2014) [arXiv:1311.6729 [hep-ph]].