Flavour anomalies after the R K ∗ measurement
Received: May
Flavour anomalies after the RK
Guido D'Amico 1 2 5
Marco Nardecchia 1 2 5
Paolo Panci 1 2 5
Francesco Sannino 1 2 5
Alessandro Strumia 1 2 5
Riccardo Torre 1 2 3 4
Alfredo Urbano 1 2 5
Geneva 1 2
Switzerland 1 2
Odense 1 2
Denmark 1 2
Italy 1 2
0 Origins and Danish IAS, University of Southern Denmark
1 Via Dodecaneso 33, I16146 Genova , Italy
2 Route de la Sorge , CH1015 Lausanne , Switzerland
3 Institut de Theorie des Phenomenes Physiques, EPFL
4 INFN , Sezione di Genova
5 Theoretical Physics Department , CERN
The LHCb measurement of the =e ratio RK indicates a de cit with respect to the Standard Model prediction, supporting earlier hints of lepton universality violation
Beyond Standard Model; Technicolor and Composite Models

observed in the RK ratio.
We show that the RK and RK ratios alone constrain the
chiralities of the states contributing to these anomalies, and we
nd deviations from the
Standard Model at the 4 level. This conclusion is further corroborated by hints from the
theoretically challenging b ! s +
distributions. Theoretical interpretations in terms
of Z0, leptoquarks, loop mediators, and composite dynamics are discussed. We highlight
their distinctive features in terms of the chirality and avour structures relevant to the
observed anomalies.
1 Introduction 2 E ective operators and observables 3
lepton sector. Taking the ratio of branching ratios strongly reduces the Standard Model
(SM) theoretical uncertainties, as suggested for the rst time in ref. [2].
The experimental result [1] is reported in two bins of dilepton invariant mass
These values have to be compared with the SM predictions [3]
RK
= 8< 0:660+00::101700
: 0:685+00::101639
0:024 (2m )2 < q2 < 1:1 GeV2
0:047 1:1 GeV2 < q2 < 6 GeV2 :
RKSM = <8 0:906
0:028 (2m )2 < q2 < 1:1 GeV2
: 1:00
0:01
1:1 GeV2 < q2 < 6 GeV2 :
{ 1 {
b ! s +
related measurements:
1. the RK ratio [4]
At face value, a couple of observables featuring a
transitions. In fact, anomalous deviations were also observed in the following
RK =
BR (B+ ! K+ +
BR (B+ ! K+e+e )
)
2. the branching ratios of the semileptonic decays B
!
K( ) +
[5] and Bs !
3. the angular distributions of the decay rate of B ! K
+
. In particular, the
socalled P50 observable (de ned for the rst time in [7]) shows the most signi cant
The coherence of this pattern of deviations has been pointed out already after the
measurement of RK with a subset of observables in [10, 11] and in a full global analysis
in [12, 13].
For the observables in points 2 and 3 the main source of uncertainty is theoretical. It
resides in the proper evaluation of the form factors and in the estimate of the nonfactorizable
hadronic corrections. Recently, great theoretical e ort went into the understanding of these
aspects, see refs. [7, 14{23] for an incomplete list of references.
Given their reduced sensitivity to theoretical uncertainties in the SM, the RK and RK
observables o er a neat way to establish potential violation of lepton
avour universality.
Future data will be able to further reduce the statistical uncertainty on these quantities. In
addition, measurements of other ratios RH analogous to RK , with H = Xs; ; K0(1430); f0
will constitute relevant independent tests [2, 24].
The paper is structured as follows. In section 2 we discuss the relevant observables and
how they are a ected by additional e ective operators. We perform a global t in section 3.
We show that, even restricting the analysis to the theoretically clean RK , RK ratios, the
overall deviation from the SM starts to be signi cant, at the 4 level, and to point towards
some model building directions. Such results prompt us to investigate, in section 4, a few
theoretical interpretations. We discuss models including Z0, leptoquark exchanges, new
states a ecting the observables via quantum corrections, and models of composite Higgs.
2
E ective operators and observables
Upon integrating out heavy degrees of freedom the relevant processes can be described,
near the Fermi scale, in terms of the e ective Lagrangian
X
where the sum runs over leptons ` = fe; ; g and over their chiralities X; Y = fL; Rg.
New physics is more conveniently explored in the chiral basis
ObX `Y = (s
PX b)(`
PY `):
These vector operators can be promoted to SU(2)Linvariant operators, unlike scalar or
tensor operators [25]. In SM computations one uses the equivalent formulation
de ning dimensionless coe cients CI as
He =
VtbVts 4 v2
em
X
`;X;Y
CbX `Y ObX `Y + h:c: ;
cI = VtbVts 4 evm2 CI =
CI
(36 TeV)2
;
prediction for RK is
percent.
RK = jCbL+R L R j2 + jCbL+R L+R j2 :
2
jCbL+ReL R j2 + jCbL+ReL+R j
1By theoretically clean observables we mean those ones predicted in the SM with an error up to few
2In the limit of vanishing lepton masses the decay rate in eq. (2.5) takes the form [12]
d (B+ ! K+ +
dq2
210 5MB3
)
= G2F e2mjVtbVtsj2 3=2(MB2 ; M K2; q2) jFV j2 + jFAj2 ;
where Vts = 0:040
vacuum expectation value, usually written as 1=v2 = 4GF=p2. The SM itself contributes
0:001 has a negligible imaginary part, v = 174 GeV is the Higgs
as CSM
bL`L = 8:64 and CSM
bL`R =
0:18, accidentally implying jCbSLM`R j
jCbSLM`L j.
This observation suggests to use the chiral basis, related to the conventional one
(see e.g. ref. [12]) by C9 = CbL L+R =2, C10 =
CbR L R =2, with the approximate relation CSM
9
CbL L R =2, C90 = CbR L+R =2, C100 =
C1S0M holding in the SM. To make
the notation more compact, we de ne CbL R`Y
CbR`L
CbL`R
CbR`R , and CbX (
e)Y
CbX Y
CbL`Y
CbX eY .
We now summarize the theoretically clean observables,1 presenting both the full
exstandardmodel (BSM) contribution, CI = CSM + CBSM.
I I
pressions and the ones in chirallinear approximation. The latter is de ned by neglecting
jCbSLM`R j
CSM
j bL`L j and expanding each coe cient CI at rst order in the
beyondtheCbR`Y and CbL+R`L R
CbL`L +
RK revisited
The experimental analysis is made by binning the observable in the squared invariant mass
of the lepton system q
2
(P` + P`+ )2. Writing the explicit q2dependence, we have
contributions from the electromagnetic dipole operator, justi ed by the cut qm2in = 1 GeV2,
and nonfactorizable contributions from the weak e ective Hamiltonian,2 the theoretical
(2.2)
(2.3)
(2.4)
(2.5)
(2.9)
(2.6)
and e can induce theoretical errors, bringing back the issue of
hadronic uncertainties.
In the chirallinear approximation, RK becomes
RK ' 1 + 2
Re CbBLS+MR( e)L ;
CSM
remains valid for the simpli ed expression proposed in ref. [24], expanded up to quadratic
terms in new physics coe cients. The reason is that the expansion is controlled by the
parameter CbBXSlMY=CbSXMlY , a number that is not always smaller than 1. This is particularly
true in the presence of new physics in the electron sector in which  as we shall discuss
in detail  large values of the Wilson coe cients are needed to explain the observed
anomalies. For this reason, all the results presented in this paper make use of the full
expressions for both RK [12] and, as we shall discuss next, RK .
2.2
Anatomy of RK
Given that the K has spin 1 and mass MK
= 892 MeV, the theoretical prediction for the
RK ratio given in eq. (1.1) is
RK
=
(1
(1
p)(jCbL+R L R j2 + jCbL+R L+R j2) + p jCbL R L R j2 + jCbL R L+R j
2
p)(jCbL+ReL R j2 + jCbL+ReL+R j2) + p jCbL ReL R j2 + jCbL ReL+R j
2
where p
0:86 is the \polarization fraction" [24, 27, 28], that is de ned as
p =
g0 + g
k
g0 + g + g
k
?
:
The gi are the contributions to the decay rate (integrated over the intermediate bin) of the
di erent helicities of the K . The index i distinguishes the various helicities: longitudinal
where GF is the Fermi constant, (a; b; c)
a2 +b2 +c2 2(ab+bc+ac), MB
5:279 GeV, MK
0:494 GeV,
jVtbVtsj
40:58
10 3. Introducing the QCD form factors f+;T (q2) we have
FA(q2) = C10 + C100 f+(q2) ;
FV (q2) = (C9 + C90)f+(q2) +
2mb
MB + MK
C7 + C70 fT (q2) +
hK(q2)
:
SM electromagnetic dipole contributi}on non factorizable term
{z {z }
Notice that for simplicity we wrote the Wilson coe cient C9 omitting higherorder
scorrections [26].
Neglecting SM electromagnetic dipole contributions (encoded in the coe cients C7(0)), and nonfactorizable
(i = 0), parallel (i =k) and perpendicular (i =?). In the chirallinear limit the expression
for RK simpli es to
RK ' RK
4p
Re CbBRS(M e)L ;
CSM
0:40. The formula above clearly shows that, in this approximation, a
deviation of RK from RK signals that bR is involved at the e ective operator level with
the dominant e ect still due to lefthanded leptons. As already discussed before, eq. (2.13)
is not suitable for a detailed phenomenological study, and we implement in our numerical
code the full expression for RK [29]. In the left panel of gure 1, we present the di erent
predictions in the (RK ; RK ) plane due to turning on the various operators assumed to be
generated via new physics in the muon sector. A reduction of the same order in both RK
and RK is possible in the presence of the lefthanded operator CBSM (red solid line). In
order to illustrate the size of the required correction, the arrows correspond to CbBLSML =
(see caption for details). Conversely, as previously mentioned, a deviation of RK from RK
signals the presence of CbBRSML (green dotdashed line). Finally, notice that the reduced value
of RK measured in eq. (1.4) cannot be explained by CbBRSMR and CBSM . The information
bL R
summarized in this plot is of particular signi cance since it shows at a glance, and before
1
an actual t to the data, the new physics patterns implied by the combined measurement
of RK and RK .
(s
PLb)(
ative values C9B;SM
Before proceeding, another important comment is in order. In the left panel of
gure 1, we also show in magenta the direction described by nonzero values of the
coe cient C9B;SM = (CbBLSML + CbBLSMR )=2. The latter refers to the e ective operator O9 =
), and implies a vector coupling for the muon. The plot suggests that
neg1 may also provide a good t of the observed data. However, it
is also interesting to notice that in the nonclean observables, the hadronic e ects might
mimic a short distance BSM contribution in C9B;SM. From the plot in our gure 1, it is
clear that with more data a combined analysis of RK and RK
might start to discriminate
between C9B;SM and CbBLSML using only clean observables. However, with the present data,
there is only a mild preference for CbBLSML , according to the 1parameter ts of section 3.1
using only clean observables.
It is also instructive to summarise in the right panel of gure 1 the case in which
new physics directly a ects the electron sector. The result is a mirrorlike image of the
muon case since the coe cients CbX eY enter, both at the linear and quadratic level, with
an opposite sign when compared to their analogue CbX Y
. In the chirallinear limit the
only operator that can bring the values of RK and RK
close to the experimental data is
CbLeL > 0. As before, a deviation from RK in RK can be produced by a nonzero value of
CbBRSeML . Notice that, beyond the chirallinear limit, also CBSM
bL;ReR points towards the observed
experimental data but they require larger numerical values.
A closer look to RK reveals additional observable consequences related to the presence
of BSM corrections. RK , in a given range of q2, is de ned in analogy with eq. (2.5):
μ
*
*
<
ratios refer to the [1:1; 6] GeV2 q2bin. We assumed real coe cients, and the outgoing (ingoing)
arrows show the e ect of coe cients equal to +1 ( 1). For the sake of clarity we only show the
arrows for the coe cients involving lefthanded muons and electrons (except for the two magenta
arrows in the leftside plot, that refer to C9B;SM = (CbBLSML + CbBLSMR )=2 =
1). The constraint from
Bs !
is not included in this plot.
as function of q2, the invariant mass of the `+`
pair, for the SM and for
two speci c values of the newphysics coe cients. The inset shows isocontours of deviation from
RK = 1 in the [0:045; 1:1] GeV2 bin as a function of newphysics coe cients, compared to their
experimentally favoured values. Right: correlation between RK
(horizontal axis) and [0:045; 1:1] GeV2 bin (vertical axis) of q2: a sizeable new physics e ect can be
present in the lowenergy bin. The numerical values of q2 are given in GeV2.
measured in the [1:1; 6] GeV2 bin
{ 6 {
1.1
1.0
]
.10.9
1
,
5
.
0
∗
[
where the di erential decay width d (B ! K
, and takes the compact form
+
)=dq2 actually describes the fourbody
d (B ! K
+
dq2
)
=
3
4
(2I1s + I2c)
(2I2s + I2c) :
1
4
(2.15)
The angular coe cients Iia==1s;;2c in eq. (2.15) can be written in terms of the socalled
transversity amplitudes describing the decay B ! K V
with the B meson decaying to an onshell
K
and a virtual photon or Z boson which later decays into a leptonantilepton pair. We
refer to ref. [29] for a comprehensive description of the computation. In the left panel of
gure 2 we show the di erential distribution d (B ! K
dilepton invariant mass q2. The solid black line represents the SM prediction, and we show
in dashed (dotted) red the impact of BSM corrections due to the presence of nonzero
+
)=dq2 as a function of the
CBSM (CbBRSML ) taken at the benchmark value of 1.
bL L
We now focus on the low invariantmass range q2 = [0:045; 1:1] GeV2, shaded in blue
with diagonal mesh in the left panel of g 2. In this bin, the di erential rate is dominated
deviation in RK [0:045; 1:1] compared to its SM value RKSM
by the SM photon contribution. It is instructive to give more quantitative comments. In
the inset plot in the left panel of g 2, we show in the plane (CbBLSML ; CbBRSML ) the relative
0:9, and we superimpose
the 1 and
3
con dence contours allowed by the t of experimental data (without
including RK ). This comparison shows that a 10% reduction of RK in the massinvariant
bin q
2 = [0:045; 1:1] GeV2 is expected from the experimental data. The SM prediction,
RKSM[0:045; 1:1]
0:9, departs from one because of QED e ects which distinguish between
m
and me. The observed central value RKSM[0:045; 1:1] = 0:66 can be again explained
with possible e ects of new physics. The natural suspect is a new physics contribution to
the dipole operator, but it can be shown that this cannot be very large because of bounds
coming from the inclusive process B ! Xs , see for example ref. [30]. We can instead
correlate the e ect in RKSM[0:045; 1:1] with RKSM[1:1; 6]. The results are shown in the right
panel of gure 2. Here we learn that the new physics hypotheses predict values larger than
the one observed in the data. However, since the experimental error is quite large, precise
measurements are needed to settle this issue.
In conclusion, the picture emerging from a simple inspection of the relevant formulas
for RK and RK is very neat, and can be summarized as follows:
New physics in the muon sector can easily explain the observed de cits in RK ,RK ,
and we expect a preference for negative values of the operator involving a lefthanded
current, CbBLSML . Sizeable deviations of RK from RK signal nonzero values for CbBRSML .
New physics in the electron sector represents a valid alternative, and positive values
of CbBLSeML are favoured. Sizeable deviations of RK from RK signal nonzero values for
CbBRSeML . However invoking NP only the electronic channels does not allow to explain
other anomalies in the muon sector such as the angular observables.
There exists an interesting correlation between RK in the q2bin [1:1; 6] GeV2 and
[0:045; 1:1] GeV2. At present, all the new physics hypothesis invoked tend to predicts
larger value of RK in the low bin than the one preferred by the data.
In section 3 we shall corroborate this qualitative picture with quantitative ts.
{ 7 {
10 9 [31]. This BR can also be a ected by extra scalar operators (bPX s)( PY ), so that
it is sometimes omitted from global BSM
ts.
(2.16)
HJEP09(217)
3
Fits
We divide the experimental data in two sets: `clean' and `hadronic sensitive':
i ) The `clean' set includes the observables discussed in the previous section: RK , RK ,
to which one can add BR(Bs !
+
) given that it only provides constraints.3 The
`cleanness' of these observables refers to the SM prediction, in the presence of New
Physics larger theoretical uncertainties are expected. We didn't include the Q4 and
Q5 observables measured recently by the Belle collaboration [33].
ii ) The `hadronic sensitive' set includes about 100 observables (summarized in the
appendix A). This list includes the branching ratios of semileptonic Bmeson decays
as well as physical quantities extracted by the angular analysis of the decay products
of the Bmesons. Concerning the hadronic sensitivity of the angular observables,
the authors of [7] argue that the optimised variables Pi have reduced theoretical
uncertainties.
The rationale is to rst limit the analysis to the `clean' set of observables. In this way one
can draw solid conclusions without relying on large and partially uncontrolled e ects. This
approach is aligned with the spirit of this paper, and can be extremely powerful, as already
shown in section 2.2. Furthermore, extracting from this reliable theoretical environment
a BSM perspective could be of primary importance to set the stage for more complex
analyses. In a second and third step we will estimate the e ect of the `hadronic sensitive'
observables and combine all observables in a global t.
3.1
Fit to the `clean' observables only
The formul
summarized in the previous section allow us to t the clean observables.
We wrote a dedicated Flavour Anomaly Rate Tool code (Fart). For simplicity, in our
2 ts we combine in quadrature the experimental errors on the two RK
bins, using the
higher error band when they are asymmetric. We checked that our results do not change
appreciably if a more precise treatment is used.
Let us start discussing the simplest case, in which we consider oneparameter ts to
each NP operator in turn. Apart from its simplicity, this hypothesis is motivated from a
experimental error is correlated with the one on BR(Bs !
+
).
3When using the Flavio [32] code, for consistency we include the observable BR(B0 !
+
), whose
{ 8 {
1 error, and p
2
SM
theoretical viewpoint, as it captures most of the relevant features of concrete models, as
we shall discuss in detail in section 4. We show the corresponding results  best t point,
2
best  in the `clean' column in table 1. In the upper
part of the table we show the cases in which we allow new physics in the muon sector. It
is evident that the results of the t match the discussion of section 2.2: the lefthanded
coe cient CbBLSML is favoured by the measured anomalies in RK and RK , with a signi cance
of about 4 . We can similarly discuss the hypothesis in which we allow for new physics
in the electron sector, shown in the lower part of table 1. Three cases  CbBLSeML , CBSM
bLeR
and CbBRSeMR  are equally favoured by the t. However, only the operator ObLeL involving
lefthanded quarks and electrons can explain the observed anomalies with an order one
Wilson coe cient since it dominates the newphysics corrections to both RK and RK , see
eqs. (2.10), (2.13). As before, we nd a statistical preference with respect to the SM case at
the level of about 4 . To simplify the comparison with the existing literature, we show in
In conclusion, the piece of information that we learn from this simple t is quite
sharp: by restricting the analysis to the selected subset of `clean' observables RK , RK and
), not much a ected by large theoretical uncertainties, we nd a preference
for the presence of new physics in the observed experimental anomalies in B decays. In
particular, the analysis selects the existence of a new neutral current that couples
lefthanded b, s quarks and lefthanded muons/electrons as the preferred option.
E ective fourfermions operators that couple left or righthanded b, s with
righthanded electrons are also equally preferred at this level of the analysis, but they require
larger numerical values of their Wilson coe cients.
Needless to say, this conclusion, although already very signi cant, must be supported
by the result of a more complete analysis that accounts for all the other observables related
to B decays, and not included in the `clean' set used in this section. We shall return to
this point in section 3.2.
Before moving to the t with the `hadronic sensitive' observables, we perform several
twoparameter ts using only `clean' observables. We show our results in gure 3. Allowing
for new physics in muons only, the combined best t regions are shown as yellow contours.
Since there are few `clean' observables, we turn on only two newphysics coe cients in
each plot, as indicated on the axes. We also show, as rotated axes, the usual C9 and C10
coe cients. We see that the key implications mentioned in section 2.2 are con rmed by
this t, although here wider regions in parameter space are allowed. In the upper plot of
gure 3 we show the results for new physics in the operators involving lefthanded muons,
CbL L and CbR R
: both coe cients are
xed by the `clean' data. Operators involving
righthanded muons, on the other hand, do not lead to good ts. A good t is obtained by
turning on only CbL L , although uncertainties do not yet allow to draw sharp conclusions.
We conclude this section with a comment on the size of the theoretical uncertainties in
the presence of New Physics. While there is a consensus on the small error of the Standard
Model predictions, in the presence of New Physics the \clean" observables have a larger
theoretical error, barring the special case where new physics violate avour universality
{ 9 {
SM
2
best
SM
`clean' `HS'
2
best
all
1:30
0:30
0:14
all
0:99
3:46
0:02
0:94
1:62
1:17
0:11
0:33
0:23
0:08
0:97
`clean'
2:31
1:21
4:23
6:10
0:39
0:21
4:66
6:56
1:01
1:68
0:40
1:03
0:04
0:29
0:61
0:18
`HS'
0:69
0:39
0:33
2:83
0:29
1:55
3:52
2:70
all
1:07
1:55
0:02
0:59
0:00
0:25
0:48
0:04
all
1:30
0:70
2:81
4:05
0:30
0:25
2:65
4:43
4:1
1:2
0:2
0:8
4:1
4:3
0:3
4:2
4:6
2:1
1:3
1:7
0:3
0:9
0:7
0:5
all
6:2
0:9
1:0
1:2
all
3:5
3:6
0:1
2:5
`clean'
1:27
0:64
0:05
`clean'
1:72
5:15
0:085
coe .
CBSM
bL L
CBSM
bL R
CbBRSML
CbBRSMR
coe .
CbBLSeML
CbBLSeMR
CbBRSeML
CbBRSeMR
RK , and BR(Bs !
+
nd in appendix A.
), or only the `Hadronic Sensitive' observables (denoted by `HS' in the
while maintaining the same chiral structure of the SM (mostly LL at large enough q2). As
shown in
gure 2, away from the Standard Model our errors are still of a few percent, in
agreement with refs. [35, 36]. However, other groups [37] nd a much larger theoretical
error in the presence of New Physics, due to a more conservative treatment of the form
factor uncertainties.4 Therefore, we warn the reader that the statistical signi cance quoted
in our ts may be smaller with a di erent treatment of the error.
We didn't take into account another important source of error: QED radiative
corrections, calculated in ref. [3]. These are of the same order or larger than the hadronic
uncertainties on RK , RK in the Standard Model as predicted by Flavio. We did the
4We thank Joaquim Matias for enlightening discussions about this point.
New physics in the muon sector (Vector Axial basis)
coe .
all
all
SM
2
best
exercise of in ating our hadronic error by a factor of 3, nding indeed a larger error away
from the Standard Model, but still of the same order of the QED corrections.
3.2
Fit to the `hadronic sensitive' observables
In order to perform a global t using the `hadronic sensitive' observables we use the public
code Flavio [32].
Theoretical uncertainties are dominant, and it is di cult to quantify them. We rst
take theoretical uncertainties into account using the `FastFit' method in the Flavio code
with the addition of all the included nuisance parameters. With this choice, the SM is
disfavoured at about 5 level.
Given that most `hadronic sensitive' observables involve muons (detailed measurements
are much more di cult with electrons), we present a simple 2 of the 4 Wilson coe cients
involving muons. This is a simple useful summary of the full analysis. In this
approximation, the `hadronic sensitive' observables determine the 4 muon Wilson coe cients as5
0:07
1
0:25
0:25 0:74 CC
1 0:50 CC
0:03
The uncertainties can be rescaled by factors of O(1), if one believes that theoretical
uncertainties should be larger or smaller than those adopted here.
The global t of `hadronic sensitive' observables to new physics in the 4 muon coe
cients is also shown as red regions in gure 3. The important message is apparent both from
5In general, within the Gaussian approximation, the mean values i, the errors i and the correlation
matrix ij determine the
2 as 2 = Pi;j(Ci
i)( 2
)ij1(Cj
j), where ( 2)ij = i ij j .
HJEP09(217)
Fit to the newphysics contribution to the coe cients of the 4 muon operators
(b
PX s)(
PY ), showing the 1; 2; 3 contours. The yellow regions with dotted contours show the
best t to the `clean' observables only; due to the scarcity of data, in each plot we turn on only the
two coe cients indicated on its axes. The red regions with dashed contours show the best global t
to the `hadronic sensitive' observables only, according to one estimate of their theoretical
uncertainties; in this t, we turn on all 4 muon operators at the same time and, in each plot, we marginalise
over the coe cients not shown in the plot. The green regions show the global t, again turning on
all 4 muon operators at the same time. In gure 5 we turn on the extra 4 electron operators too.
(vertical axis): lefthanded in the left panel, and righthanded in the right panel. Regions and
contours have the same meaning as in
gure 3: `clean' data can be tted by an anomaly in muons
or electrons; `hadronic sensitive' data favour an anomaly in muons.
the gure and from eq. (3.1): `hadronic sensitive' observables favour a deviation from the
SM in the same direction as the `clean' observables, i.e. a negative contribution CBSM
1
to the Wilson coe cient involving lefthanded quarks and muons. `Clean' observables and
`hadronic sensitive' observables  whatever their uncertainty is  look consistent and
bL L
favour independently the same pattern of deviations from the SM.
3.3
Global t
We are now ready to combine `clean' and `hadronic sensitive' observables in a global t,
using both the Flavio and Fart codes. The result is shown as green regions in gure 3,
assuming that new physics a ects muons only. The global t favours a deviation in the SM
in CbBLSML , and provides bounds on the other newphysics coe cients. Using the Gaussian
approximation for the likelihood of the muon coe cients, the global t is summarized as
0:26
1
0:17
An anomaly in muons is strongly preferred to an anomaly in electrons, if we adopt the
default estimate of the theoretical uncertainties by FLAVIO. This is for example shown
in
gure 4, where we allow for a single operator involving muons and a single operator
involving electrons.
In view of this preference, and given the scarcity of data in the electron sector, we avoid
presenting a global t of new physics in electrons only. We instead perform a global
combined t for the muon and electron coe cients (which should be interpreted with caution,
given that `hadronic sensitive' observables are dominated by theoretical uncertainties). We
nd the result shown in gure 5, which con rms that  while electrons can be a ected by
new physics  `hadronic sensitive' data favour an anomaly in muons.
The latter result has been obtained by a global Bayesian t to the observables listed
in tables 6, 7 in addition to the clean observables. We used the Flavio code to calculate
the likelihood, and we sampled the posterior using the Emcee code [38], assuming for the
8 Wilson coe cients (at the scale 160 GeV) a at prior between
10 and 10. In this global
t, we choose to marginalize over 25 nuisance parameters only, to keep computational times
within reasonable limits. The nuisances (form factors related to B decays) are selected in
the following way. For each observable, we de ne theoretical uncertainties due to changing
each nuisance within its uncertainty, keeping the others xed at their central values. Then,
we choose to marginalize only over the parameters which give a theoretical uncertainty
larger than the experimental error on the observable.
4
Theoretical interpretations
We now discuss di erent theoretical interpretations that can accommodate the
avour
anomalies. We start with the observation that an e ective (s
can be mediated at tree level by two kinds of particle: a Z0 or a leptoquark. Higherorder
induced mechanisms are also possible. These models tend to generate related operators
cbLbL (s
PLb)2 + c L (
PL )(
PL
) ;
(4.1)
and therefore one needs to consider the associated experimental constraints. The rst
operator a ects Bs mass mixing for which the relative measurements, together with CKM
ts, imply cbBLSbML = ( 0:09
0:08)=(110 TeV)2 , i.e. the bound jcbBLSbML j < 1=(210 TeV)2 [11,
39]. The second operator is constrained by CCFR data on the neutrino trident cross
section, yielding the weaker bound jcBLSM
j < 1=(490 GeV)2 at 95% C.L. [40]. Furthermore,
new physics that a ects muons can contribute to the anomalous magnetic moment of the
muon. Experiments found hints of a possible deviation from the Standard Model with
a = (24
Models featuring extra Z0 to explain the anomalies are very popular, see the partial list of
references [42{61]. Typically these models contain a Z0 with mass MZ0 savagely coupled to
[gbs(s
PLb) + h.c.] + g L (
PL ) :
(4.2)
The model can reproduce the avour anomalies with cbL L =
in
gure 6a. At the same time the Z0 contributes to the Bs mass mixing with cbLbL =
gbsg L =MZ20 as illustrated
gb2s=2MZ20 . The bound from
MBs can be satis ed by requiring a large enough g L in
order to reproduce the b ! s`+` anomalies. Lefthanded leptons are uni ed in a SU(2)L
doublet L = ( L; `L), such that also the neutrino operator c L
=
However the latter does not yield a strong constraint on g L .
g2L =MZ20 is generated.
.0
0
.0
0
.0
3
.0
0
.5
1
−
SBMbeLL 3
CbBLSeML
4 .
2. 1
at tree level: a Z0 or a leptoquark, scalar or vector.
Another possibility is for the Z0 to couple to the 3rd generation lefthanded quarks
with coupling gt and to lighter lefthanded quarks with coupling gq. The coupling gbs arises
as gbs = (gt
gq)(UQd )ts after performing a avour rotation UQd among lefthanded down
quarks to their masseigenstate basis. The matrix element (UQd )ts is presumably not much
larger than Vts and possibly equal to it, if the CKM matrix V = UQu U Qyd is dominated
by the rotation among lefthanded down quarks, rather than by the rotation UQu among
Unless gq = 0, the parameter space of the Z0 model gets severely constrained by
combining perturbative bounds on g L . In addition the LHC bounds on pp ! Z0 !
can be relaxed by introducing extra features, such as a Z0 branching ratio into invisible
A characteristic feature of Z0 models is that they can mediate e ective operators
involving di erent chiralities. In fact, gaugeanomaly cancellations also induce multiple
chiralities: for example a Z0 coupled to L
L is anomaly free [44], where the Le contribution
is avoided because LEP put strong constraints on 4electron operators. The chiralities
inanomalies can be determined trough more precise measurements
volved in the b ! s`+`
of `clean' observables such as RK and RK .
4.2
Models with leptoquarks
The anomalous e ects in b ! s`+` transitions might be due to the exchange of a
LeptoQuark (LQ), namely a boson that couples to a lepton and a quark. Concerning lepton
avour, in general a LQ can couple to both muons and electrons. However, simultaneous
sizeable couplings of a LQ to electrons and muons generates lepton avour violation which
is severely constrained by the timehonoured radiative decay
typically assumes that LQs couple to either electrons or muons. (Here sizeable means an
e ect which has an impact on the anomalous observables). The coupling to muons allows
! e . For this reason one
to t the anomalies in b ! s +
distributions, as well as the RK and RK
=e ratios.
The gauge quantum numbers of scalar LQs select a speci c chirality of the SM fermions
involved in the new Yukawa couplings, and thereby generate a unique characteristic
operator in the e ective Lagrangian in the chiral basis of eq. (2.1), as illustrated in gure 6b.
The correspondence is given by
Coe cient LeptoQuark
Yukawa couplings
CbL`L
CbL`R
CbR`L
CbR`R
S3
R2
~
R2
~
S1
(3; 3; 1=3) y QL S3 + y0 QQ S3y + h.c.
(3; 2; 7=6) y U L R2 + y0 QE R2y + h.c.
where ` can be either an electron or a muon. In parentheses we report the SU(3)
SU(2)L
U(1)Y gauge quantum numbers, and we follow the notations and conventions
from ref. [63] for LQ names. Q; L (U; D; E) denote the lefthanded (righthanded) SM
quarks and leptons.
Given that each LQ mediates e ective operators with a given chirality, we can draw
conclusions from our one parameter ts of the b ! s`+` anomalies of table 1. Assuming
new physics in the muon sector, the measurement of RK
selects a unique scalar
leptoquark: S3
(3; 3; 1=3), which is a triplet under SU(2)L. It is remarkable that this is
obtained with just the information coming from `clean' observables while the inclusion
of the remaining observables (with our speci ed treatment of the errors) reinforces this
hypothesis. The explanation of the anomalies in terms of S3 has been rstly proposed after
the measurement of RK in ref. [10] switching on only those couplings needed to reproduce
the e ect. In ref. [64] the LQ has been identi ed as a pseudoGoldstone boson associated
to the breaking of a global symmetry of a new strongly coupled sector [65]. In refs. [64, 65]
it has also been suggested that a rationale for the size of the various avour couplings could
be dictated by the mechanism of partial compositeness [66]. Another motivated pattern of
couplings has been suggested in ref. [67] using avour symmetry. Also ref. [68] makes use
of S3 as mediator of the b ! s +
transition.
A potential issue with S3 is the danger of extra renormalizable couplings with
diquarks (denoted collectively by y0 in the Lagrangians above) which may induce proton
decay. Baryon number conservation has to be invoked to avoid this issue. Motivated by
this, in refs. [69, 70], the LQ R~2 (which respects the global symmetry U(1)B accidentally
at the renormalizable level) has been considered leading to the prediction RK
> 1, which
is now disfavoured by the LHCb data. The other two options S~1 and R2 were already
disfavoured after the measurement of RK [10, 71].
The situation is di erent if LQs couple to electrons, rather than to muons, such that
only the anomalies in the `clean' observables can be reproduced. `Clean' observables can
be reproduced by all chiralities, with the only exclusion of CbR`L , which is mediated by
the R~2 LQ. From the t, we notice that the S~1 and R2 LQs can only t the anomalies by
giving a large contribution to the Wilson coe cients, comparable to the SM contributions:
this happens because these LQs couple to right handed electrons, with little interference
with the SM. One the other hand, S3 couples to lefthanded leptons, such that the sizeable
interference with the SM allows to reproduce the observed anomalies with a smaller new
physics component.
We brie y comment on the possible interpretation of a LQ as a supersymmetric particle
in the MSSM. The only sparticle with the same gauge quantum numbers as a LQ is the
lefthanded squark Q~
R~2. However, even if it has Rparity violating interactions, this
LQ gives the wrong correlation between RK and RK , disfavouring the supersymmetric
interpretation of the anomalies.
We move now to the discussion of the exchange of vector LQs at tree level, illustrated
in gure 6c. There are 3 cases: U3
(3; 3; 2=3), V2
(3; 2; 5=6) and U1
(3; 1; 2=3). Their
relevant interactions are:
LU3 = y Q
LV2 = y D
LU1 = y Q
L U3 + h.c.
L V2 + y0 Q
L U1 + y2 D
E V2 + y00 Q
E U
1 + h.c.
U V2y + h.c.
(4.4a)
(4.4b)
(4.4c)
Spin Quantum Clean observables Clean observables
Number new physics in e new physics in
observables
X
X
X
S3
R2
~
R2
~
S1
U3
V2
U1
0
0
0
0
1
1
1
(3; 3; 1=3)
(3; 2; 7=6)
dressed by further new composite dynamic contributions.
. In Fundamental Composite Higgs models these diagrams will be
The vector LQ V2 and U1 can contribute to the anomalous observables trough multiple
chiral structures. In general, if both y and y0 are sizeable, dangerous scalar operators may
be generated. If one of the two couplings dominates, we can again restrict to our one
parameter t, with the following correspondence: CbL`L can be generated by U3; CbL`R or
CbR`L can be generated by V2; CbL`L or CbR`R can be generated by U1.
Similar phenomenological considerations to explain the Bmeson anomalies as in the
case of the scalar LQ apply, we summarise the relevant options in table 3.
Models featuring vector LQs models in order to explain the avour anomalies appeared
recently in the literature [72{75], typically as new composite states. The presence of these
states signals that the theory in isolation is nonrenormalizable, meaning that loop e ects
of the vectors are UV divergent, for a recent rediscussion see ref. [76]. Naive dimensional
analysis shows that oneloop contributions to physics observables such
MBs might be
problematic. A careful study of this topic is a model dependent issue and it requires extra
information on the UV embedding of the LQ in a complete theory.
4.3
The RK anomaly can be reproduced by one loop diagrams involving new scalars S and new
fermions F with Yukawa couplings to SM fermions that allow for the Feynman diagram on
the left in
gure 7 [39, 77]  see also ref. [78]. In this particular example, one generates
an operator involving lefthanded SM quarks and leptons, denoted respectively by Q and
L. The needed extra Yukawa coupling to the muon must be large, yL
1:5. This also
explains why the MSSM does not allow for an explanation of the RK ; RK anomalies: a
possibile box diagram containing Winos and sleptons predicts yL
g2
1:5, where g2 is
the SU(2)L gauge coupling.
In section 4.4 we will consider renomalizable models of composite dynamics featuring
extra elementary scalars, where we will show that the extra particles S and F can be
identi ed with the constituents of the Higgs boson, and that their Yukawa couplings are
the source of the SM Yukawa couplings, giving rise to a avour structure similar to the SM
structure. Then, the one loop Feynman diagrams of gure 7 are dressed by the underlying
composite dynamic.
Fundamental composite Higgs
Y = 1=2 and Y = 0.
couplings
Models in which the Higgs is a composite state are prime candidates as potential source of
new physics in the avour sector [79{81]. Fundamental theories with a Higgs as a composite
state that are also able to generate SM fermion masses appeared in ref. [82]. These theories
feature both techniscalars S and technifermions F .6 In models of fundamental composite
Higgs: i) it is possible to replace the standard model Higgs and Yukawa sectors with a
composite Higgs made of techniparticles; ii) the SM fermion masses are generated via
a partial compositeness mechanism [66] in which the relevant composite technibaryons
emerge as bound states of a technifermion and a techniscalar.
The composite theory does not address the SM naturalness issue and it is fundamental
in the sense that it can be extrapolated till the Planck scale [82]. Having a fundamental
theory of composite Higgs, we use it to investigate the avour anomalies.
The gauge group and the eld content of a simple model are summarised in table 4.
Here the new strong group is chosen to be SU(NTC) with NTC = 3 and we list the
gauge quantum numbers of the new vectorial fermions and scalars that can provide a
composite Higgs with Yukawa couplings to all SM fermions L; E; Q; U; D. Three generations
of techniscalars are introduced in order to reproduce all SM fermion masses and mixings,
while having a renormalizable theory with no Landau poles below the Planck scale. The
hypercharge Y of the FL fermion is free. We assume the minimal choices Y =
1=2,
The matrices of SM Yukawa couplings y`, yu, yd are obtained from the TCYukawa
LY = yL LFLSEc + yE EFN SEc + (yD DF Nc + yU U F Ecc )SDc + yQ QFLSDc + h.c. (4.5)
c
becomes strong, gTC
4 =pNTC, at the scale TC
as y`
yLyET =gTC, yd
yQyDT=gTC, yu
yQyUT =gTC, where the new gauge coupling gTC
gTCfTC, forming composite particles
6Composite theories including TC scalars attempting to give masses to the SM quarks appeared earlier
in the literature [83{88] for (walking) TC theories that didn't feature a light Higgs.
name spin generations SU(3)c SU(2)L
FL
c
FN
SEc
SDc
FEc 1=2
1=2
1=2
0
0
1
1
1
3
3
1
1
1
1
3
2
1
1
1
1
Y
Y
Y =
Y
1=2
F Nc ; FL; FEc with conjugated gauge quantum numbers such that the fermion content is vectorial
are implicit. Names are appropriate assuming the value Y =
1=2 for the hypercharge Y of FL;
however generic values are allowed.
HJEP09(217)
with mass of order
TC and condensates hF F ci
fT2C TC. In view of the resulting
breaking of the TCchiral symmetry, the Higgs doublet H (identi ed with pseudo Goldstone
bosons of the theory) and other composite scalars remain lighter. Lattice simulations [89{
91] of the most minimal fundamental composite theories [92{94], without techniscalars,
have demonstrated the actual occurrence of chiral symmetry breaking with the relevant
breaking pattern, and furthermore provided the spectrum of the spin one vector and axial
techniresonances with masses mV = 3:2(5) TeV= sin
and mA = 3:6(9) TeV= sin
where
is the electroweak embedding angle to be determined by the dynamics, that must be
smaller than about 0:2.
The TCYukawa couplings accidentally conserve lepton and baryon numbers (like in
the SM) and TCbaryon number; depending on the value of Y the lightest TCbaryon can
be a neutral DM candidate.
We require TCscalar masses and TCquartics to respect avour symmetries so that
the BSM corrections to avour observables abide the experimental bounds. At one loop7
in the TCYukawas one obtains the following operators involving 4 SM fermions
f;f0
L;E;Q;U;D (yfy yf )ij (yfy0 yf0 )i0j0 (fi fj00 )(fi00
X
2 2
gTC TC
fj ) +
(yLyyE)ij (yQyDT)i0j0 (Li
2 2
gTC TC
Qi0 )(Ej
Dj0 ):
(4.6)
All SM fermions and their chiralities are involved. These operators are
phenomenologically viable if the fundamental TCYukawa couplings have the minimal values needed to
reproduce the SM Yukawa couplings: yE
yL
pgTCy`, and similarly for quarks.
However, when the TCYukawas (say, yL) are enhanced the impact on new physics is
also enhanced. The observed SM Yukawa couplings are reproduced when the corresponding
TCYukawas (say, yE) are reduced. Consequently, in this scenario new physics manifests
prevalently in leptons of one given chirality. Because data prefer new physics to emerge
prevalently in lefthanded muons it is natural to consider here an enhanced muon coupling
yL and a correspondingly reduced righthanded yE.
7The loop analysis, in the composite scenario, is merely a schematic way to keep track of the relevant
factors stemming from the TC dynamics when writing SM fourfermion interactions.
cbL L
cbLbL
a
gZ L
NTC
Oneloop result
(yLyLy) (yQyQy)bs
(4 )24M 2
F (x; y)
NTC (4(yQ)2y8QyM)b22s F (x; x)
FL
FL
m2 (yLyLy)
NTC (4 )2M 2
FL
(2Y
1)F7(y) + 2Y
F7(1=y)
y
MZ2 (yLyLy)
NTCg2 2(4 )2(1
2s2W)M 2 F9(Y; y)
FL
Nonperturbative estimate
(yLyLy) (yQyQy)bs
g2
fermion and TCscalar estimate (second column) along with their NDA analysis counterpart
(third column). The NDA result for
a modi es in the presence of a TCfermion condensate
to m v(yLyET) =gTC 2TC.
= Λ
µ >ν
µ+µ
δ µ
Δ µ
Δ
† ) (
Higgs models. The model generates an e ective operator that can simultaneously account for both
RK and RK , so only RK is plotted.
We summarise in table 5 the coe cients of the relevant avourviolating e ective
operators, both within a naive oneloop approximation (adopting the results from refs. [11,
39]) and Naive Dimensional Analysis (NDA) in the composite theory. We de ned x =
SD
FL
M 2c =M 2 , y = M 2c =M 2
SE
FL
and the loop functions
F (x; y) =
F7(y) =
(1
y
3
1
x)(1
y)
6y2 + 6y ln y + 3y + 2
12(1
y)4
x2 ln x
x)2(x
;
+
y)
(1
y2 ln y
y)2(y
x)
(4.7a)
(4.7b)
G9(y) =
2y3 + 6 ln y + 9y2
36(y
1)4
;
7
;
F9(Y; y) = s2W(2Y
1)F9(y)
(1
s2W(2Y + 1))G9(y)
(4.7c)
(4.7d)
(4.7e)
that equal F (1; 1) = 1=3, F7(1) = F~7(1) = 1=24, F9(1) =
1=24, G9(1) = 1=8, for
degenerate masses. The latter entry in table 5 is the correction to the Z coupling to
lefthanded muons g L , written in terms of the weak mixing angle sW = sin W. The
LEP bound at the Z pole is j gZ L j
0:8% g2 at 2 [95]. We can neglect TCpenguin
diagrams [11]. We can always work in a basis where yL = diag(yLe ; yL ; yL ) is diagonal,
such that (yLyLy)
= yL2 .
Figure 8 shows that, in order to reproduce the b ! s`+`
anomalies and the muon
g
2 anomaly, a relatively large Yukawa coupling yL
1:5 is needed, like in models with
perturbative extra fermions and scalars. In the composite model such values of TCYukawa
coupling have natural sizes. This is corroborated by a RGE analysis for yL that features
an extra contribution involving the gTC gauge coupling:
(4 )
L
2
3
NT2C
2NTC
1 gT2CyL ;
In the presence of the rst term only, setting NTC = 1, the Yukawa coupling grows with
energy. Perturbativity up to a scale
max implies jyL j < 2 =pln( max= TeV), with jyL j
MPl. In the presence of the second term a larger yL
gTC is compatible
with the requirement that all couplings can be extrapolated up to the Planck scale. This
is similar to how the strong coupling g3 allows for yt
1 in the SM. In the fundamental
composite Higgs model, the large couplings yt and yL contribute to the prediction for the
Higgs mass parameter in terms of
Lepton avour violation is absent as long as the yE matrix is diagonal in the same
basis where yL is diagonal. Then y` = yL` yE` =gTC for ` = fe; ; g. In general, there can
be a avourviolating mixing matrix in the lepton sector. In particular, the mixing angle
e generates
! e , but only when e ects at higher order in the Yukawa couplings are
included [82]. Focusing on e ects enhanced by the large coupling yL one has
BR(
! e )
4
emv6yE2e yL6 e2
gT6Cm2 4
TC
yE2e yL6 e2
2 TeV
TC
4
that in the electron sector too one has a large yLe and a small yEe
ye
The experimental bound BR(
! e ) < 0:6 10 12 [96] is satis ed even for e
1 provided
Finally, we mention an e ect that can enhance the newphysics correction to some
avourviolating operators. While the fermion condensates induced by the strong dynamics
are known, the scalar condensates are not known (although perhaps they are computable,
for example by dedicated lattice simulations). Possible scalar condensates could break
the accidental avour symmetry among scalars, leading to extra lighter composite
pseudoGoldstone bosons. The state made of SEc SDc behaves as a leptoquark: if light it would
mediate at tree level some e ective operators, analogously to the S~1 leptoquark considered
in section 4.2.
(4.8)
(4.9)
We found that the new measurement of RK together with RK favours new physics in
lefthanded leptons. Furthermore, adding to the t kinematical b ! s +
(a ected by theoretical uncertainties), one
nds that they favour similar deviations from
distributions
the SM in lefthanded muons. However, even if the experimental uncertainties on RK , RK
will be reduced, a precise determination of the newphysics parameters will be prevented
by the fact that these are no longer theoretically clean observables, if new physics really
a ects muons di erently from electrons.
We next discussed possible theoretical interpretations of the anomaly. One can build
models compatible with all other data:
One extra Z0 vector can give extra newphysics operators that involve all chiralities of
SM leptons. The simplest possibility motivated by anomaly cancellation is a vectorial
coupling to leptons. However, unless the Z0 is savagely coupled to bs quarks, a Z0
coupled to ss and bb is disfavoured by pp ! Z0 !
contraints.
searches at LHC and other
One leptoquark tends to give e ects in muons or electron only (in order to avoid
large avour violations), and only in one chirality.
One can add extra fermions and scalars such that they mediate, at one loop level,
the desired new physics. Their Yukawa coupling to muons must be larger than unity.
While the e ective 4fermion operators that can account for the b ! s`+` anomalies
need to be suppressed by a scale
30 TeV, the actual new physics can be at a lower
scale, with obvious consequences for direct observability at the LHC and for Higgs mass
naturalness.
Acknowledgments
We thank Marina Marinkovic, Andrea Tesi and Elena Vigiani for useful discussions. We
thank Javier Virto for further useful comments. The work of A.S. is supported by the
ERC grant NEONAT, the one of F.S. is partially supported by the Danish National
Research Foundation Grant DNRF:90, and the one of G. D'A. thanks Andrea Wulzer and
the Lattice QCD group of CERN, and in particular Agostino Patella, for sharing their
computing resources. R.T. is supported by the Swiss National Science Foundation under
the grant CRSII2160814 (Sinergia). R.T. thanks INFN Sezione di Genova for computing
resources, Alessandro Brunengo for computing support, and David Straub for clari cations
about the Flavio program.
A
List of observables used in the global t
In table 6 and 7 we summarize the observables used in addition to the `clean' observables.
All bins are treated in the experimental analyses as independent, even if overlapping. It is
Observable
hFLi(B0 ! K
hS3i(B0 ! K
hS4i(B0 ! K
hS5i(B0 ! K
hS7i(B0 ! K
hS8i(B0 ! K
hS9i(B0 ! K
hAF B i(B0 ! K
hP1i(B0 ! K
hP50 i(B0 ! K
hFLi(B0 ! K
hS3i(B0 ! K
hS4i(B0 ! K
hS5i(B0 ! K
hS7i(B0 ! K
hS8i(B0 ! K
hP1i(B0 ! K
hP40 i(B0 ! K
hP50 i(B0 ! K
hP60 i(B0 ! K
hP80 i(B0 ! K
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
Angular observables
LHCb B ! K
clear that a correlation should exists between measurements in overlapping bins, however
this is not estimated by the experimental collaborations.
For this reason we include in
our
t the measurements in all relevant bins, even if overlapping, without including any
correlation beyond the ones given in the experimental papers. Notice that, for instance in
the case of the LHCb analysis [9], the result in the bin [1:1; 6] GeV2 has a smaller error than
the measurements in the bins [1:1; 2:5]; [2:5; 4]; [4; 6] GeV2, even when the information from
these three bins is combined. In fact, we veri ed that the bin [1:1; 6] GeV2 has a stronger
impact on our
ts than the three smaller bins. This shows that even if the measurements
are potentially largely correlated, the largest bin dominates the
t, so that the e ect of the
unknown correlation becomes negligible.
Observable
ddq2 BR(B
! K
ddq2 BR(B0 ! K
ddq2 BR(B
! Kee)
ddq2 BR(B
! K
ddq2 BR(B0 ! K
ddq2 BR(Bs !
ddq2 BR(B ! Xs
ddq2 BR(B ! Xsee)
ddq2 BR(B ! Xsll)
ddq2 BR(B ! Xsll)
)
)
)
)
)
)
Branching ratios
LHCb B
! K
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[1] LHCb collaboration, Test of lepton universality with B0 ! K 0`+`
decays, JHEP 08
[2] G. Hiller and F. Kruger, More modelindependent analysis of b ! s processes, Phys. Rev. D
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