Linear flavour violation and anomalies in B physics
JHE
avour violation and anomalies in
0 Wilberforce Road , Cambridge, CB3 0WA, U.K
1 J.J. Thomson Avenue , Cambridge, CB3 0HE, U.K
2 DAMTP, University of Cambridge
3 Cavendish Laboratory, University of Cambridge
We propose renormalizable models of new physics that can explain various anomalies observed in decays of Bmesons to electron and muon pairs. The new physics states couple to linear combinations of Standard Model fermions, yielding a pattern of avour violation that gives a consistent t to the gamut of avour data. Accidental symmetries prevent contributions to baryon and leptonnumberviolating processes, as well as enforcing a loop suppression of new physics contributions to Data require that the new avourbreaking couplings are largely aligned with the Yukawa couplings of the SM and so we also explore patterns of avour symmetry breaking giving rise to this structure.
BPhysics; Beyond Standard Model

Linear
physics
HJEP06(21)83
2
3
4
5
1 Introduction
The models
Phenomenological analysis
3.1 Indirect searches processes 3.2 3.3
Direct searches
Parameter space plots
Flavour symmetries Conclusions
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5
b ! s
b ! s
Semileptonic fourfermion operators
Fourquark operators
Anomalous magnetic moment of the muon
A SU(2)L decompositions
B Photon and Zboson mediated contributions
consistency is the particular (even peculiar) structure of the SM, which leads to many subtle
and delicate cancellations in avourchanging and CPviolating processes. As examples at
tree level (see, e.g. [1] for more details), the representation content of the SM fermions
(which is such that any two equivalent irreducible representations under the unbroken
SU(3)c
U(1)em are also equivalent under the full SU(3)c
izable; indeed, if we relax this requirement, we quickly run into con ict with data, unless
the UV cuto
of the theory is rather high.
In the last year or so, cracks have begun to appear in the SM edi ce, in the form of a
variety of anomalies associated with Bmeson decays. These include e ects seen in angular
observables [2{10] of the decay B ! K
[11], the observable RK [12], and a series of
branching ratios with muons in the nal state (e.g. [13, 14]). Whilst it is surely far too soon
to claim that the end of the SM is nigh, it is perhaps worthwhile to explore how the SM
might be modi ed in such a way that these anomalies in the data can be accommodated.
Of course, this must be done in such a way as not to spoil the various delicate cancellations
that occur in the SM, since this would lead to gross contradictions with data elsewhere.
As such, it makes sense to look for theory which is renormalizable (or has a cuto
which
is much larger than the scale of new physics, which amounts to the same thing). Then we
can at least hope that, just like in the SM, dangerous avourchanging processes can be
kept under control.
A number of models explaining the anomalies have already been proposed. These can
be subdivided into models which generate the anomalies via tree vs. looplevel corrections.
Models in the rst category include leptoquarks [15{23] and Z0s [24{39]. Not all of these
models are renormalizable. Ref. [17], for example, describes a wellmotivated model in
which the electroweak hierarchy problem is solved by Higgs compositeness and SM fermions
are partially composite. A light leptoquark can arise naturally as a Goldstone boson [40].
But if the compositeness scale is su ciently high, the renormalizable limit is approximately
recovered. There is one recentlyproposed model in the second category, which generates
contributions that explain the B anomalies and the anomalous magnetic moment of the
muon, via a Z0 with loopinduced couplings to muons [41].
In this work, we wish to propose a second type of renormalizable model in the category
of those generating the anomalies at loop level. In order to obtain the
avour structure
that seems to be required (in the quark sector at least) in an automatic way, we insist
that the new
avourviolating couplings be linear in the SM fermions. We then survey the
various possibilities for the BSM
elds and nd that basic phenomenological considerations
(such as the insistence on an accidental symmetry stabilising the proton), together with
a minimality criterion, lead to just two possible models. The models both feature two
new scalar elds and a single fermion eld, which couple to linear combinations of the SM
fermions via Yukawa interactions. The two models are very similar in their phenomenology
and so we discuss only one in detail. We nd that, for suitable values of the parameters, a
satisfactory t to the anomalies can be obtained.
The t to the data requires that the new avourviolating couplings be strongly
hierarchical (at least in the quark sector; in the lepton sector there is more room to manoeuvre)
and moreover largely aligned with the avour breaking already present in the Yukawa
couplings of the SM. Thus, even more than in the SM, the lowenergy theory seems to be
crying out for a fundamental theory of avour. Rather than attempt to
nd an explicit
theory of avour that does the job, we content ourselves with showing the plausibility of
the existence of such a theory, by exhibiting patterns of avour symmetry breaking that
{ 2 {
can give rise to the required structure. We nd that a variety of symmetries are possible.
Interestingly, they give rise to a pattern of couplings similar to that obtained in theories
featuring partial compositeness.
The models have another feature of interest. By construction, they yield explicit
UV completions that generate the nonrenormalizable avour structure that was identi ed
in [
42
] as a viable one for explaining the anomalies. The fact that such UV completions
exist is a desirable thing, because the alignment of the avourviolating new physics
couplings in [
42
] is not preserved by the SM RG
ow. Thus, in the absence of an unexplained
ne tuning in the couplings, the scale of new physics completing the e ective lagrangian
in [
42
] should be light, in order that the picture makes sense. Our models provide
exan analysis of possible avour breaking patterns, and our conclusions are given in section 5.
2
The models
Recent experimental data in avour physics, in particular measurements by the LHCb
collaboration [11, 12, 14], suggest possible e ects of New Physics (NP) in semileptonic decays
of Bmesons. In particular, considering the b ! s quark avour transitions, the most
signi cant departures from the Standard Model predictions are observed in: (i) the socalled
decays [11] (ii) a series of branching ratios
P50 angular observable of the B ! K
+
B ! Ke+e
of Bdecays with muons in the nal states (like the recently measured Bs0 !
(iii) the observable RK [12] de ned as the ratio of branching ratios of B ! K +
+
[14])
and
in the low q2 (lepton pair invariant mass squared) region. It is perhaps
premature to claim evidence for NP. Indeed the discrepancies in (i) and (ii) could be due
to underestimates of hadronic uncertainties,1 while the discrepancy in (iii), despite the
observable being theoretically very clean in the SM, could simply be a statistical
uctuation. However, if we allow for an interpretation of these experimental results in terms of
NP, it is quite remarkable that modelindependent approaches based on higherdimension
operators [5, 7, 8, 15, 24, 45, 49{54] give a simple and consistent t to the data.
In general, we expect that NP will couple chirally to the matter elds; assuming a
coupling purely to either right or lefthanded currents, the ts
nd that the operator
bL
sL L
L is preferred. Motivated by this, we seek renormalizable models that couple
the NP to quark and lepton doublets. Moreover, for reasons that will become clear, we
demand that the SM fermions appear linearly in the NP couplings. In what follows, the
SM fermions are written as QL; UR; DR; LL; ER while the beyond SM (BSM) fermions and
scalars are denoted by
i and
i. The Higgs doublet is denoted by H and transforms as
(1; 2; 12 ). The possible linear interactions of the new BSM
elds with the SM doublets (QiL
1For discussions on the size of the hadronic uncertainties on these observables we refer the reader to
e.g. [6, 8, 10, 43{49].
{ 3 {
and LiL) can be classi ed in the following way:
or
or
or
q
i
q
i
q
i qQiL
q
i qQiL
i `
QL q + i
i
QL q + i
` c i
i
LL ` + h:c:;
LL ` + h:c:;
+ i `LiL
`
+ i `LL
` c i
+ h:c:;
+ h:c:
(2.1)
(2.2)
(2.3)
(2.4)
Notice that one new state appears simultaneously in both the quark and lepton interactions
(this is needed in order to draw a diagram contributing to b ! s
). In the
rst two
cases the common mediator is
while in the last two the common mediator is . There
are in nitely many combinations with suitable gauge quantum numbers to yield these
interactions. In order to reduce the possibilities for the SM quantum numbers of the new
states, we will impose conditions on them based on the following considerations:
terms of the form ( y )(HyH) and ( yT a )(HyTHa H) are never forbidden by gauge
symmetry (though the second one is absent if
is a SU(2)L singlet). These terms
are phenomenologically viable, but other quartic and trilinear interactions of a scalar
with the Higgs, like
HH, or
HHH, could give rise to a violation of the custodial
symmetry at the tree level and/or could modify the observed Higgs phenomenology.
To be safe from these unwanted e ects, we choose the quantum numbers of the new
scalars such that these dangerous interactions are prohibited.
(c) Direct searches, coloured particles
New particles in the loops will need to be rather light to create a measurable e ect in B
decays, so it is convenient to choose quantum numbers such that their masses are less
constrained by direct searches. The quark interaction requires at least one state that
transforms nontrivially under the colour group. A coloured scalar will have weaker
bounds on its mass than a coloured fermion, since in the latter case the production
cross section is higher for a xed gauge quantum number. Selecting a scalar to be the
only new coloured particle leads us to consider only the rst two cases in the above
list of new interactions, namely the ones with a single fermion mediator
and two
di erent scalars q and
`
.
{ 4 {
(d) Direct searches, BSM Lightest Particle (LP)
The Yukawa interactions above are manifestly invariant under a U(1) transformation
that acts non trivially only on the BSM states. This symmetry is respected by the
gaugekinetic terms of the new states too. We look for irreps such that this
transformation is an accidental symmetry of the whole renormalizable model. This has
the advantage that all NP
avourviolating processes are loop suppressed. But it also
implies that the lightest NP state is stable; in order to evade strong constraints on
coloured and/or electrically charged stable states coming from colliders and cosmology
we look for a colourless irrep containing a neutral particle. The gauge quantum
numwe require that all new irreps have dimensionality fewer than 5.
Let us now identify the irreps that could contain the LP. From (d) and (e) we obtain
a nite list of candidates. Most of them are excluded because of the presence of
renormalizable interactions that violate conditions (a) and/or (b). The excluded cases are listed
in table 1. We are left with four cases, each of which has a single fermion,
, with SM
quantum numbers (1; 4; 1=2) or (1; 4; 3=2). However, radiative corrections split the
values of the particle masses in the multiplet and it turns out that, for the quantum number
(1; 4; 1=2), the LP is not the neutral one. We conclude that, since we are demanding a
neutral LP, the LP can only be contained in the fermion eld
with quantum numbers
(1; 4;
32 ). Imposing condition (e) on the eld
q we are left with just two models:
(1; 4; + 32 ); q
(3; 3; 43 ), `
(1; 3; 2) with Yukawa interactions as
(1; 4; 32 ); q
(3; 3; 53 ), `
(1; 3; 2) with Yukawa interactions as
Model A. in (2.1):
Model B.
in (2.2):
The two models have very similar implications for the phenomenology that we are interested
in here. Henceforth, we discuss only Model A.
The quantum numbers of the SM and NP elds under the gauge and global symmetries
(to be discussed below) are summarised in table 2 and the most general renormalizable
{ 5 {
q
i
q
i
QL q + i`
i
i
LL ` + h.c.
QL q + i` c i
i
LL ` + h.c.
(2.5)
(2.6)
(1; 1; 0)
U(1)Y Interactions
`
neutral particle but which are rejected because they could give rise to unwanted renormalizable
interactions, as listed in the last column. Irreps with negative hypercharges are related to these
ones by charge conjugation.
lagrangian is given by
L = LSM + L
+ L
+ Lyuk;
L
L
lagrangian. Before considering the breaking coming from Llin it is easy to show that the
Lagrangian is invariant under a global U(1)7. Indeed, the SM alone has accidental global
symmetry U(1)B
U(1)e
U(1)
elds have global symmetry U(1)
U(1) , while the gauge kinetic terms of the new BSM
U(1) q
U(1) ` . Moreover, it is easy to prove that
the most general renormalizable scalar potential V ( H ; q; `) is invariant under U(1)7.
Now consider the e ect of Llin. For a generic choice of the couplings ` and
q there is
always an unbroken U(1)3
U(1)B0
U(1) , de ned as follows. Under the U(1)B0
the SM
elds have their usual baryon number while
q has charge 1/3. Similarly, under
the U(1)L0 the SM
elds have their usual lepton number while
` has charge 1. Finally
the SM
elds are uncharged under U(1) , while the BSM
elds have charge unity.2
2This symmetry makes our neutral and colourless LP stable. Hence, the LP is a potential dark matter
(DM) candidate. However, even if its mass and couplings could be
xed in order to reproduce the right
( 13 ; 0; 0)
( 13 ; 0; 0)
( 13 ; 0; 0)
(3; 2; 16 )
(3; 1; 23 )
metry (second column), and under the accidental global symmetries of the theory (third column).
Thus, the model retains analogues of the accidental baryon and lepton number
symmetries of the SM, which su ce to stabilize the proton and to prevent contributions to
numerous unobserved lepton and baryonnumber violating processes. Moreover, the model
features an additional accidental U(1) symmetry, under which SM
elds are uncharged.
An immediate consequence of this is that all NPgenerated processes involving only SM
particles in the initial and
nal states are loopsuppressed. This is certainly an advantage
from the point of view of the vast majority of avourviolating observables, where no
deviation from the SM is observed. It might be regarded as a disadvantage from the point of
the view of the Bphysics anomalies, where a sizable NP e ect is needed. But this is o set
somewhat by the desirable structure of linear NP
avour violation that results. We shall
see in the sequel that the anomalies can be reproduced even for values of the NP couplings
that are of order unity or smaller.
3
Phenomenological analysis
In this section we discuss the phenomenology of Model A. In an obvious notation, we
denote the masses of the new states as M , Mq, and M`. In a basis where the lefthanded
quark doublet is de ned as QiL = (VCKM uiL; diL)T , the minimal set of couplings
that are
needed to t the b ! s`` anomalies are
q
3
2` (i.e. couplings involving b, s and
). To begin with, we will assume that only these couplings are nonzero and investigate
the processes induced. In this section we collect relevant formula on indirect searches, and
investigate direct production bounds. In subsection 3.3 we use this information to
nd
allowed parameter space regions. In section 4 we will discuss relaxing the assumptions on
the couplings and propose more motivated avour structures.
3.1
Indirect searches
As described above, the accidental global U(1) symmetry under which the new particles
are charged implies that contributions to processes containing only SM particles in the
{ 7 {
initial and
nal states are only induced at loop level. Here we investigate the size of
these contributions.
s
The relevant processes, given the assumption on couplings described above, are b !
processes, Bs mixing, b ! s , and the anomalous magnetic moment of the muon.
The process b ! s``, important for the LHCb B meson anomalies, is induced at loop level
by the diagram in
gure 1.3 The SU(2)L structure of the NPinduced semileptonic
fourfermion interaction can be derived from the discussion in appendix A, using the lagrangian
(eq. (A.6)) written explicitly in terms of SU(2)L components. The resulting e ective NP
HJEP06(21)83
2
em
em
4
4
5
9
;
K(x)
K(x; y)
1
x + x2 log x
K(x)
(x
x
1)2
K(y)
y
:
He =
4GF (VtsVtb) X C`( ) Oi`( ) ;
p i
i
lagrangian is
Le
K(xq; x`) i j m n
q q ` `
MM`22 . The loop function K(xq; x`) can be obtained by the following
The e ective hamiltonian relevant to b ! s`` transitions is
where Oi` are a basis of SU(3)C
U(1)Qinvariant dimensionsix operators giving rise to the
avourchanging transition. The superscript ` denotes the lepton
avour in the nal state
(` 2 fe; ; g), and the important operators for our process, Oi`, are given in a standard
basis by
O9
`(0) =
O10
`(0) =
s
s
PL(R)b (`
PL(R)b (`
`) ;
5`):
Comparing equations (3.1) and (3.2) we nd the NP contribution to the Wilson coe cients
relevant to b ! s
is
9
2 3
2` 2 :
3There are also Z and photon penguin diagrams which contribute, with a NP loop connecting the quarks
and joining to the leptons via a Z=
propagator. These penguin diagrams are discussed in appendix B and
are found to be very suppressed relative to both the SM contribution and the diagram in gure 1, and hence
are neglected here.
{ 8 {
The most recent best t ranges on this combination of Wilson coe cients are taken from [49]
and are given by
9
9
Interactions between four quarks are induced at loop level by diagrams like those in gure 2.
These interactions can lead to meson mixing; in particular, if the process b ! s
is present,
then inevitably Bs mixing must also be induced. This process can therefore introduce
important constraints on the masses and couplings of the new particles. The four quark
e ective operator induced by the NP is
Le
K0(xq) i j m n
q q q q
~QjL
m
QL
~QnL
where K0(x) is the rst derivative of K(x). The SU(2)L structure of the e ective operator
is similar to that of eq. (3.1) and can again be derived from the discussion in appendix A.
Projecting the quark doublet along the down components we nd that for Bs mixing the
relevant operator is
Le
7
bL)(sL
bL) + h.c.:
The Wilson coe cient is easily extracted at high energy
=
where the BSM particles
are dynamical elds. We x
= 1 TeV in what follows. At this energy we have
C1bs( ) =
576 2
K0(xq)
2 3
{ 9 {
(3.5)
(3.6)
;
(3.7)
(3.8)
(3.9)
In order to place bounds on the parameters of our model, we take into account QCD
e ects using the results and procedure of [55]. Using the anomalous dimension of this work
we found that the running of Wilson coe cient from the scale of the New Physics ( ) to
the scale of the process (mb) is given by C1bs(mb) =
V LLC1bs( ) with
V LL = 0:78. For
the evaluation of the relevant matrix element we used the lattice result of [56]. These lead
to a constraint (at 95% con dence level) on the coe cient
HJEP06(21)83
which translates into
C1bs( ) .
1:8
2 3
q q 2 < 1:8
Thus the measurement of Bs mixing produces a bound on the hadronic couplings
involved in the b ! s
this bound and the b ! s
process, viz.
respect the model is similar to Z0 models  the couplings involved factorize into leptonic
couplings and hadronic couplings which can be set independently. This factorization does
not occur in leptoquark models.
3.1.3
b ! s
The radiative process b ! s will also be induced by the diagram in gure 3. The couplings
involved are the same as those for Bs mixing. However, the amplitudes will scale di erently
with the parameters q and Mq between the two processes. Constraints from b ! s could
therefore provide complementary information.
At the mass of the b quark, the process b ! s is described by the following e ective
hamiltonian:
He =
4GF (VtsVtb) C7(mb)O7(mb) + C70(mb)O70(mb) ;
p
where O7
(0) = 16e 2 mb s
cient of the dipole operator;
2
PR(L)b F
.
At the matching scale M , we get an additional contribution from the NP to the
coefC7NP =
GpF2 VtsVtb
1 q q
The 2 allowed range for this parameter has been tted recently in [49], giving
F1(x) =
12(x
1
1)4 x
3
6x2 + 3x + 2 + 6x log x :
C7NP(mb) 2 [ 0:10; 0:02] (at 2 ):
3.1.4
Anomalous magnetic moment of the muon
Although it is somewhat peripheral to our discussion, let us remark that loops of
and
`, as shown in
gure 4, generate a 1loop contribution to the magnetic moment of the
muon, which may be able to resolve the longstanding experimental discrepancy therein
[57]. The NP contribution is given by
which should be compared to the observed discrepancy [58]
aNP =
`
As we will show in section (3.3), it is possible to t the anomalous magnetic moment in
this model. However, it requires a large value of 2`, which is problematic, since it can lead
to large corrections to electroweak precision observables at the Zpole.
3.1.5
b ! s
processes
Contributions to B ! K
gure 1 with the muons replaced with muon neutrinos (as well as Z penguin
diagrams  see the comment in section 3.1.1). A detailed analysis of NP contributions
to this process is given in [59], and we use their results here. Current measurements give
bounds on the ratio of total (NP+SM) to SM branching ratios to be
are expected in the model, due to a diagram
RK
RK
BR(B ! K
BR(B ! K
BR(B ! K
BR(B ! K
)
)SM
)
)SM
< 4:3
< 4:4
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
at 90% con dence level. An expression for RK( ) in the presence of NP with couplings to
lefthanded SM fermions is
RK = RK
=
1 X jCL`j2
3
` jCLSMj2
where CL` is the coe cient of the operator
OL =
e2
4GF VtbVts 16 2 (s
p2
in the e ective Hamiltonian. The SM Wilson coe cient CSM is known to be quite
accurately CSM =
L
7:65. In the case of our model,
HJEP06(21)83
CNP;
L
=
p
2GF VtbVts
PLb) ( `
5) `)
(1
L
9
M 2
4 K(xq; x`) 2q 3q j 2j
where the de nitions of K(x; y), xq and x` are as in section 3.1.1. Thus bounds exist on
the parameters of the model due to b ! s
processes:
24:9 < CNP;
L
< 30:0;
55:4 < K(xq; x`) 2q 3q j 2j
leptonic coupling in the model, then we nd the relation
CLNP =
2 CNP;
Therefore, for values of the Wilson coe cients required to t the b ! s
(eq. (3.5)), the NP contributions to the branching ratios for the b ! s
below the bounds, adding approximately 5% to the SM values.
3.2
Direct searches
The particles of the three new multiplets, q, `, and
, will be directly produced at the
LHC if their masses are within kinematic reach. In this subsection, we outline current
limits on their masses from direct searches, and identify promising channels to search
for them. It will be convenient to label the SU(2)L components of the multiplets by
a superscript denoting their respective electric charges; the full list of NP particles is,
therefore,
q=( q+7=3; q
The collider phenomenology of the new particles will depend on the mass spectrum.
As before, we assume that M
< M`; Mq, since we require the LP to be neutral. We
further assume that M` < Mq, since this minimises contributions to Bs mixing (it also
maximises contributions to the muonic g
2). The three multiplets can, generically, be
wellseparated in mass, but within each SU(2)L multiplet there may also be signi cant
mass splittings. For the scalar multiplets, there are tree level mass splittings due to the
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
anomalies
processes are well
presence of direct couplings with the Higgs; for the fermion multiplet, there are only small
radiative mass splittings between the components. In the limit that the common mass is
much larger than the electroweak scale v, the radiative mass splitting between the di erent
charge eigenstates is [60, 61]
mrad = mQ+1
mQ
166MeV
1 + 2Q +
(3.25)
2Y
cos W
;
a formula which holds for both scalars and fermions. According to eq. (3.25) the lightest
particle within the fermion multiplet will be uncharged, as desired.
As the
fermion multiplet has the lowest common mass of all the new states, and due
to the U(1) symmetry within the new sector, the lightest state within the multiplet will
be stable. The small radiative mass splitting means that heavier fermion components will
decay to the lightest (neutral) component by emission of one or more soft charged pions
or leptons, which will not be energetic enough to be reconstructed in the detector. Thus
if any
particle is produced at the LHC, it will appear as missing transverse momentum,
similarly to the Winolike dark matter described in [62].4
We thus neglect henceforth
the soft undetectable pions or leptons emitted in the decays of heavier components of .
Therefore, q and
` particles (being heavier than the
states) will e ectively decay to a
SM particle plus missing transverse energy. Furthermore, due to the U(1) symmetry, NP
particles will always be pairproduced at the LHC. This means that searches for (Rparity
conserving) supersymmetry should be sensitive to
q and
`. We will now discuss each of
the elds
q, ` and
in turn.
The fermions
X can be pair produced via a photon or a W=Z, or through the decay
of q;` . By the arguments above, they will always behave as uncharged weakly interacting
particles. Limits can be set on these from monox searches, from constraints on the invisible
width of the Z boson, and from LEP searches for charginos that are almost degenerate
in mass with the neutralinos. A detailed analysis of all of these has been performed
in [63]; the lastmentioned has been found to be the most constraining, implying a bound
of m
> 90 GeV.
Each component ( q+7=3; q
+4=3; q
+1=3) of the coloured scalar multiplet
q will be
strongly pair produced and will decay with a similar signature to that of a squark; i.e.
to a quark and (either directly or via emission of soft pions or leptons) the stable
neutral component of the fermion
. Note that there is the possibility that two
q particles
produced each decay to a di erent avour of quark. An example decay chain displaying
this property is illustrated in
gure 5. This complicates the reinterpretation of SUSY
search limits.
The
avour considerations discussed earlier only constrain the product of couplings
q q
3 2, without constraining their quotient. The strongest constraints from existing LHC
searches will hold for situations where one of the couplings
2q or 3q is much larger than
4Note that our setup is subtly di erent from that described in [62]. There, strong constraints can be put
on the mass of the new fermion multiplet from disappearing tracks searches, since the lifetime of a charged
fermion decaying to the neutral fermion can be long enough to create a disappearing track in the detector.
Here, these searches are not constraining because the lifetime is too short for a track to be visible.
di erent quarks. Since the pion produced when the
1=3 appears similar to that of a (anti)sbottom, whereas the decay of the q
q
1=3
appears similar to that of a stop.
the other, so that the branching ratio to a particular generation dominates. If
the
q particles will decay like sbottoms and stops, whereas if
q
3
2q they will decay
q
3
q
2
like secondgeneration squarks. The branching ratio to an uptype quark as opposed to a
downtype within a particular generation is determined by the SU(2)L structure.
We focus on the case
q
3
2q, since this is motivated by avour considerations, as
explained in the next section. In this limit, one can show that the total branching ratio
times crosssection for a pair of q particles to be produced and to decay to a pair of tops
is 7=8 of that for direct stop pair production. The most recent limits on direct stop pair
production are given in [64, 65]. As a conservative estimate, given that we have the limit
m
> 90 GeV, we can take the limits on direcly pairproduced stops decaying to tops and
90 GeV neutralinos to apply to our
q. This gives a limit of Mq & 750 GeV. Likewise, the
total branching ratio times crosssection for a pair of
q particles to be produced and to
decay to a pair of b quarks is 7=8 of that for direct sbottom pair production. Latest limits
on direct sbottom pair production are given in [66], and again taking these limits to apply
to our q particles, we nd that Mq & 720 GeV for a
of mass 90 GeV.
The SU(2)L components ( `+3, `+2, `+1) of the scalar ` will, if they have only muonic
couplings, always decay to either a muon or a muon neutrino, together with a
particle.
So they will sometimes decay in the same way as a smuon in supersymmetry. However,
the production crosssections and branching ratios will di er. Results of recent LHC
slepton searches are given in [67, 68].
These rule out lefthanded smuons, pair produced
directly via a W=Z=
and decaying to a muon and a neutralino, up to a maximum mass
of roughly 300 GeV (for a massless neutralino). We used Feynrules [69, 70] and
Madgraph5 aMC@NLO [71] to calculate electroweak (EW) pair production crosssections of
the
` particles, and then multiplied by the branching ratios in order to reinterpret the
limits on crosssection given in [67]. The CMS limit plot, with our model superimposed
tralino, taken from [67]. The region under the blue line shows the exclusion on our model found by
reinterpreting the exclusion plot in terms of direct pair production of
`s decaying to a lepton and
a . For our model the xaxis should be taken to mean the mass of the
`
, M`, while the yaxis
means the mass of the , M . The dotted part of the blue line is extrapolated.
in blue, is shown in gure 6. If the mass of the
particle is greater than about 150 GeV,
there are no bounds on the mass of the
` (other than the assumption that its mass is
greater than that of ).
3.3
Parameter space plots
muons
BSM
2` and the combination
q
2
q
3
elds to make these parameters real.
In this subsection we show allowed regions in the parameter space of the model considering
the observables described above; b ! s``, B ! Xs , Bs meson mixing and the anomalous
magnetic moment of the muon. The relevant parameters entering the expressions of these
observables are the masses of the new states (M ; Mq and M`) as well as the coupling to
. Without loss of generality, we can rede ne the
In order to t the b ! s`` anomalies without being in disagreement with the measured
Bs mixing rate, the muonic coupling 2` must be rather large. In order to have an idea of the
typical values of the parameters needed in our model, in gure 7 we show parameter space
regions assuming that 2` = 1:2 while parametrizing the masses in terms of one single scale
M assuming the following hierarchy M
= M; M` = M + 200 GeV; Mq = M + 700 GeV.
In this way we are left with two parameters only (M and
2q 3q). The B ! Xs allowed
region is not shown because it yields weaker constraints than Bs mixing does. For this
hierarchy of masses, the only relevant direct production constraint is the bound on the
mass of , M
> 90 GeV. There is an overlap between the allowed Bs mixing region and
the 1
model can
preferred region for the b ! s`` measurements  so with these parameters, the
t the b ! s`` anomalies. The value of 2` can be further lowered to be . 1,
in this case the values of M ; M` and Mq are close to present bounds coming from direct
searches. For example we veri ed that a t to the data with
0:8 could be achieved
`
2
when M
= 150 GeV, M` = 200 GeV and Mq = 800 GeV.
α3qα2q 0.2
0.1
0.0
= M; M` = M + 200 GeV; Mq = M + 700 GeV. For this value of 2`, it is not possible to explain
the anomalous magnetic moment of the muon whilst tting the other constraints.
However, if we also wish to t the anomalous magnetic moment of the muon, the
muonic coupling
2` must be larger.
We show in
gure 8 the relevant parameter space
regions when this coupling is set to 2` = 2:5, with the same hierarchy of masses as before.
If we want to take this explanation of the (g 2) anomaly seriously, then we should consider
possible bounds from the shift of the EW gauge couplings Z
; Z
and W +
(see
also the discussion in section IV of [41]). The corrections are nonuniversal and so a global
t to EW data is required to establish the precise constraints on the couplings. Though
such a t is beyond the scope of our work, nave arguments suggest that O(1) values of 2`
are not problematic.
4
Flavour symmetries
terms is U(3)5
the following way;
with
In this section we establish a possible connection between the avour violation present in
the SM and in the NP sector.
In the SM and in the limit of vanishing Yukawa couplings, the largest group of unitary
eld transformations that commutes with the gauge group and leaves invariant the kinetic
U(1)H . Adopting notation similar to [72] we can decompose this group in
GK
SU(3)q3
SU(3)`2
U(1)B
U(1)L
U(1)Y
U(1)P Q
U(1)ER
U(1)H ;
SU(3)q3 = SU(3)QL
SU(3)`2 = SU(3)LL
(4.1)
(4.2)
= M; M` = M + 200 GeV; Mq = M + 700 GeV. With this large value of 2` there is an overlap
between the regions that t the B anomalies (in blue), and the anomalous magnetic moment of the
muon (in green).
The U(1) factors can be identi ed with the baryon (B) and lepton (L) numbers, the
hypercharge (Y ), a transformation (P Q) acting non trivially and in the same way only on
DR and ER, and
nally a universal rotation for the elds ER and a U(1) global symmetry
associated to the Higgs doublet.
We would like now to make connections with the avour structure of the SM and
the possible e ects coming from NP. In order to do that a
rst step is to identify (i)
a avour symmetry and (ii) a set of irreducible symmetrybreaking terms. The
avour
symmetry group GF
GK has to be broken in order to reproduce the observed pattern of
fermion masses and mixing. In order to do that a set of symmetrybreaking spurions are
introduced to formally restore the symmetry GF .
We will now consider 3 explicit examples and we will focus on the quark sector.
This is the case of Minimal Flavour Violation [72]. The spurion elds are the three
YU
(3; 3; 1)
YD
(3; 1; 3);
(4.3)
where the quantum numbers are speci ed with respect to the direct product of groups
SU(3)QL
This is the avour symmetry of the quark sector if only the Yukawa couplings yt and
yb are nonvanishing. So to a good level this is an approximate symmetry of the SM.
1. GF = U(3)q3
Yukawa couplings
2. GF = U(2)q3
Recent works [73{75] considered the following set of irreducible spurions;
u
(2; 2; 1);
d
(2; 1; 2);
V
(2; 1; 1);
(4.4)
This case mimics partial compositeness. The irreducible spurions are connected to
the Yukawa couplings in the following way;
(YU )ij
iq ju;
(YD)ij
iq jd:
(4.5)
HJEP06(21)83
With these speci c cases in mind we are now ready to discuss avour violation induced
by operators of the form
q
i
QiL , iu
U Ri
and
d
i
DRi . These operators break the
avour symmetry and in order to restore it we could assume that the vectors
F are again
spurions with de nite transformation rules under the
avour symmetry. We could now
assume minimality of avour violation in the following sense: the
using the irreducible spurions used to construct the SM Yukawa couplings. Following this
iF can be expressed
procedure we obtain the following results.
To recover avour invariance the
F have to transform in the following way;
q
(3; 1; 1);
u
(1; 3; 1);
d
(1; 1; 3):
(4.6)
However, it can be proved using triality properties of the SU(3) irreps that tensor
products of YU ; YUy; YD; YDy can never give rise to any of the
F . Furthermore, the
structure of the Yukawas cannot be reproduced by combinations of the s, since this
can only lead to Yukawa matrices of rank 1, meaning two of the uptype quarks and
two of the downtype quarks would have zero mass. We therefore conclude that this
structure cannot work.
To recover avour invariance the
F have to transform in the following way;
Atleading order in the number of spurion elds we have that
( 1q; 2q
( 1u; 2u
( 1d; 2d
)
)
)
(2; 1; 1);
(1; 2; 1);
(1; 1; 2);
q
3
u
3
d
3
(1; 1; 1);
(1; 1; 1);
(1; 1; 1):
( 1q; 2q)i = aq Viy;
( 1u; 2u)i = au (Vy u)i;
( 1d; 2d)i = ad (Vy d)i;
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
while 3F = bF with aF ; bF order one numbers and i = 1; 2. Doing a spurion analysis
with this setup, we nd that in the basis in which QiL =
(generation index) and Q3L = ei t tL; bL
q is found to be
VCyiKj M uj ; di T
for i = 1; 2
T , the coupling vector in the quark sector
~q = (y12Vtd; y12Vts; y3) ;
(4.13)
where y12 and y3 are each (generally complex) numbers of O(1).
up to a factor of O(1)). The natural size of 3q 2q is O(jVtsj)
with the allowed regions in gures 7 and 8.
This is very similar to the case of Partial Compositeness, except that there is slightly
less freedom in the couplings; the ratio of 2q to
1q is exactly xed (rather than
xed
where ciF are order one numbers.
It is easy to show that link between the iF and iF is simply given by
iF = ci i
F F
Thus, a number of patterns of avour symmetry breaking can give rise to the
hierarchical structure and alignment with the SM Yukawa couplings that is needed to explain
the anomalies. In all cases, the pattern of couplings is similar to that arising in models of
partial compositeness (see for example [76]). Although our phenomenological analysis in
the previous section was done with more restrictions on the couplings, we can be sure that
a partial compositenesslike
avour structure in the quark sector is phenomenologically
viable, since a full analysis of existing bounds on this setup has recently been performed
in [17].
An analogous discussion could be repeated in the leptonic sector, starting from the
global symmetry U(3)LL
U(3)ER of the gauge kinetic terms associated with left and
righthanded leptons. However, a complete understanding of the leptonic avour requires
the knowledge of the mechanism that generates neutrino masses. Here we remain agnostic
as to the possible
avour orientation of the spurions
` and
e
. For recent works that
consider possible links between the avour violation in the neutrino sector and the physics
of LFV transitions in b ! s``0, we refer the reader to [19, 42, 77{79].
5
Conclusions
We have presented renormalizable extensions of the SM that can explain several anomalies
observed in Bmeson decays. Renormalizability (which amounts to the assumption that
further NP is heavy and decoupled) allows us to introduce deviations from the SM, coming
from NP, in a controlled way. This is almost a sine qua non, given that we observe just
a handful of anomalies in data, while many thousands of other observations agree with
the SM.
We have surveyed the possible NP elds that allow for a coupling to linear combinations
of lefthanded SM fermions, since this generates, at oneloop, an operator of the form
q q ` `QiL
i j k l
QjLLk
L
LlL. Coupled with the plausible assumption that the linear
avourviolating spurions, q;` are roughly aligned with the Yukawa couplings of the SM, we end
up with a good t to the anomalies, without contradicting other data.
As a spectacular example of the control that renormalizability brings, the models
that we identify feature 3 accidental global symmetries, corresponding to conservation of
(generalized) baryon and lepton numbers and to a `NP number'. The consequences of
these accidental symmetries are manifold. Not only is the proton stabilized, but also all
other baryon and leptonnumber violating processes (e.g. neutronantineutron oscillations,
), many of which are strongly constrained, are forbidden automatically. The NP
number leads to a generic suppression of NP
avourviolating processes, since these can
only occur at loop level. Yet another advantage of the models is that NP can only couple
to lefthanded SM fermions at the renormalizable level, meaning that contributions to
processes requiring a helicity ip, such as
! e , are further suppressed.
The accidental symmetries of the models, while su cient to prevent many dangerous
processes, are quite di erent from the accidental symmetries of the SM, namely baryon
and individual lepton family numbers. This is a crucial feature, since it allows us to have
large violations of lepton universality. This is precisely what is needed to t the anomalies.
The models are not panace , in that there is a further anomaly in B physics that
cannot be explained, arising in decays to D( )
[80{82].
But it seems hard to ex
plain this anomaly in any NP model, for the simple reason that the SM contribution,
with which it needs to be comparable, is so large (being a treelevel e ect with minimal
Cabibbo suppression).5
The main weakness of the models is arguably that they require a rather large value of
the coupling 2` in order to explain the anomalies. While this coupling can be rather smaller
than other couplings in the avour sector (i.e. the top quark Yukawa coupling), some readers
may be alarmed that such a large coupling should appear in the light lepton sector, where
(at least in the SM) all other couplings are small. It is important to note, however, that
not only does this coupling not cause phenomenological problems per se,6 but also that,
provided that it is suitably aligned, it does not lead to large avour violations in the light
leptons via renormalization group ow. This follows immediately from the observation that
there exists a basis in which the SM leptonic Yukawa couplings are diagonal. Nevertheless,
the necessary alignment is aesthetically disturbing; we have shown that it is plausible from
the point of view of avour symmetries, but it would be nice to have an explicit model of
avour in which it is realised dynamically.
Acknowledgments
This work has been partially supported by the Galileo Galilei Institute for Theoretical
Physics, STFC grant ST/L000385/1, and King's College, Cambridge. We thank members
of the Cambridge SUSY Working Group for discussions.
5However it is not impossible, see for example [21, 83, 84].
6In fact, as shown in [85], one can even put O(1) couplings among the light quark generations without
necessarily getting into trouble.
By denoting the generators in the fundamental representation of SU(2)L as T a =
a=2
(with
a being the Pauli matrices and a = 1; 2; 3), we de ne their action on the (2j +
1)dimensional completely symmetric tensor i1i2:::i2j (i1; i2; : : : ; i2j = 1; 2) as
a
( i1i2:::i2j ) = Tia1k ki2:::i2j + Tia2k i1k:::i2j + : : : + Tia2jk i1i1:::k :
(A.1)
In general, we arrive at the following embedding of the properly normalized electric charge
eigenstates:
.
.
.
where the superscripts denote the electric charge of the eld, Bn;k is the binomial factor
n!
Bn;k = k!(n k)! and the normalization of the states is such that
i1i2:::i2j i1i2:::i2j = j j j2 + j
j 1j2 + : : : + j
j+1j2 + j
j j2 :
In the following, we provide the SU(2)L decomposition for the BSM
elds ( ; q and
` ) introduced in the Model A (2.5):
= (1; 4; 32 )
111 =
112 = p1
122 = p1
222 =
3
3
3
0
2
1
q = (3; 3; 43 )
( q)11 =
( q)12 = p1
( q)22 =
7=3
q
2 q
1=3
q
4=3
` = (1; 3; 2)
( `)11 =
( `)12 = p1
( `)22 =
3
`
1
`
2
2 `
The relevant linear interactions (2.10) introduced in Model A can be rewritten in the
following way
Llin =
q
i
RQL q + i RLiL ` + h.c.
i `
=
q `
i ( R)k1k2k3 (QiL)k1 ( q)k2k3 + i ( R)k1k2k3 (LiL)k1 ( `)k2k3 + h.c.;
where k1; k2; k3 = f1; 2g are SU(2)L fundamental indices. More explicitly we get
R
1 1=3
q
uiL + h.c.
!
!
R
Photon and Zboson mediated contributions
An explicit calculation of the photon penguin diagrams leads to the following (lepton
avour universal) contribution to the Wilson coe cient of the leptonic vector current at
low energy;
where
C
mM2q2 . This function is normalised in such a way that f (1) = 1.
In our model, the contribution of the Zboson mediated penguin diagram is suppressed
compared to that of the photon mediated one. Indeed the Zboson exchange is enhanced
only in diagrams containing a source of explicit SU(2)L breaking;7 such diagrams are not
present in our case. When there is no explicit SU(2)L breaking, the contribution from the
Zboson penguin diagram is suppressed by a factor Mm2BZ2
one; this factor is given simply by the ratio of the propagators in the two cases.
3 10 3 compared to the photon
Neglecting the Zboson contribution, we now quantitatively show that the photon
contribution is suppressed at the percent level when compared to the one in eq. (3.4).
Taking the ratio of the Wilson coe cients generated through a photon penguin and the
NP box diagram respectively, we get
C
9
9
C NP =
4 227 xq 1f (xq 1)
7
576 2 K(xq; x`)
2
` 2 = 3:5
Taking as a reference the benchmark de ned in gure 8, namely ` = 2:5; M
= M; M` =
M + 200 GeV; Mq = M + 700 GeV we nd that
C
9
C NP
9
< 3:9
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
(B.3)
(B.4)
and in [
87
].
[1] B. Gripaios, Lectures on Physics Beyond the Standard Model, arXiv:1503.02636 [INSPIRE].
[2] J. Matias, F. Mescia, M. Ramon and J. Virto, Complete Anatomy of Bd ! K 0(! K )`+`
and its angular distribution, JHEP 04 (2012) 104 [arXiv:1202.4266] [INSPIRE].
[3] S. DescotesGenon, J. Matias, M. Ramon and J. Virto, Implications from clean observables
for the binned analysis of B ! K
[arXiv:1207.2753] [INSPIRE].
+
at large recoil, JHEP 01 (2013) 048
7This general argument has been given in the context of Zpenguin contributions in the MSSM in [86]
the prediction of B ! K
+
[4] S. DescotesGenon, T. Hurth, J. Matias and J. Virto, Optimizing the basis of B ! K `+`
observables in the full kinematic range, JHEP 05 (2013) 137 [arXiv:1303.5794] [INSPIRE].
[5] W. Altmannshofer and D.M. Straub, New physics in B ! K
?, Eur. Phys. J. C 73
(2013) 2646 [arXiv:1308.1501] [INSPIRE].
nonuniversality, JHEP 12 (2014) 053 [arXiv:1410.4545] [INSPIRE].
[8] W. Altmannshofer and D.M. Straub, New physics in b ! s transitions after LHC run 1, Eur.
Phys. J. C 75 (2015) 382 [arXiv:1411.3161] [INSPIRE].
[9] S. DescotesGenon, L. Hofer, J. Matias and J. Virto, Theoretical status of B ! K
+
:
The path towards New Physics, J. Phys. Conf. Ser. 631 (2015) 012027 [arXiv:1503.03328]
HJEP06(21)83
B0 ! K 0 +
B ! K( ) +
Bs0 !
, Phys. Rev. Lett. 111 (2013) 191801 [arXiv:1308.1707] [INSPIRE].
[12] LHCb collaboration, Test of lepton universality using B+ ! K+`+` decays, Phys. Rev.
Lett. 113 (2014) 151601 [arXiv:1406.6482] [INSPIRE].
[13] LHCb collaboration, Di erential branching fractions and isospin asymmetries of
decays, JHEP 06 (2014) 133 [arXiv:1403.8044] [INSPIRE].
[14] LHCb collaboration, Angular analysis and di erential branching fraction of the decay
, JHEP 09 (2015) 179 [arXiv:1506.08777] [INSPIRE].
[15] G. Hiller and M. Schmaltz, RK and future b ! s`` physics beyond the standard model
opportunities, Phys. Rev. D 90 (2014) 054014 [arXiv:1408.1627] [INSPIRE].
[16] S. Biswas, D. Chowdhury, S. Han and S.J. Lee, Explaining the lepton nonuniversality at the
LHCb and CMS within a uni ed framework, JHEP 02 (2015) 142 [arXiv:1409.0882]
[17] B. Gripaios, M. Nardecchia and S.A. Renner, Composite leptoquarks and anomalies in
Bmeson decays, JHEP 05 (2015) 006 [arXiv:1412.1791] [INSPIRE].
[18] S. Sahoo and R. Mohanta, Scalar leptoquarks and the rare B meson decays, Phys. Rev. D 91
(2015) 094019 [arXiv:1501.05193] [INSPIRE].
JHEP 06 (2015) 072 [arXiv:1503.01084] [INSPIRE].
[19] I. de Medeiros Varzielas and G. Hiller, Clues for avor from rare lepton and quark decays,
[20] D. Becirevic, S. Fajfer and N. Kosnik, Lepton
avor nonuniversality in b ! s`+` processes,
Phys. Rev. D 92 (2015) 014016 [arXiv:1503.09024] [INSPIRE].
[21] R. Alonso, B. Grinstein and J. Martin Camalich, Lepton universality violation and lepton
avor conservation in Bmeson decays, JHEP 10 (2015) 184 [arXiv:1505.05164] [INSPIRE].
[22] L. Calibbi, A. Crivellin and T. Ota, E ective Field Theory Approach to
b ! s``(0); B ! K( )
and B ! D( )
115 (2015) 181801 [arXiv:1506.02661] [INSPIRE].
with Third Generation Couplings, Phys. Rev. Lett.
[23] S. Sahoo and R. Mohanta, Study of the rare semileptonic decays Bd0 ! K `+` in scalar
leptoquark model, Phys. Rev. D 93 (2016) 034018 [arXiv:1507.02070] [INSPIRE].
+
Anomaly,
[25] R. Gauld, F. Goertz and U. Haisch, On minimal Z0 explanations of the B ! K
anomaly, Phys. Rev. D 89 (2014) 015005 [arXiv:1308.1959] [INSPIRE].
+
[26] A.J. Buras, F. De Fazio and J. Girrbach, 331 models facing new b ! s +
data, JHEP 02
(2014) 112 [arXiv:1311.6729] [INSPIRE].
[27] A.J. Buras and J. Girrbach, Lefthanded Z0 and Z FCNC quark couplings facing new
[28] W. Altmannshofer, S. Gori, M. Pospelov and I. Yavin, Quark avor transitions in L
L
models, Phys. Rev. D 89 (2014) 095033 [arXiv:1403.1269] [INSPIRE].
[29] A.J. Buras, F. De Fazio and J. GirrbachNoe, ZZ0 mixing and Zmediated FCNCs in
SU(3)C
SU(3)L
U(1)X models, JHEP 08 (2014) 039 [arXiv:1405.3850] [INSPIRE].
[30] A. Crivellin, G. D'Ambrosio and J. Heeck, Explaining h !
, B ! K
+
and
B ! K
+
=B ! Ke+e
Lett. 114 (2015) 151801 [arXiv:1501.00993] [INSPIRE].
in a twoHiggsdoublet model with gauged L
L , Phys. Rev.
[31] A. Crivellin, G. D'Ambrosio and J. Heeck, Addressing the LHC
avor anomalies with
horizontal gauge symmetries, Phys. Rev. D 91 (2015) 075006 [arXiv:1503.03477] [INSPIRE].
[32] C. Nieho , P. Stangl and D.M. Straub, Violation of lepton
avour universality in composite
Higgs models, Phys. Lett. B 747 (2015) 182 [arXiv:1503.03865] [INSPIRE].
[33] D. Aristizabal Sierra, F. Staub and A. Vicente, Shedding light on the b ! s anomalies with a
dark sector, Phys. Rev. D 92 (2015) 015001 [arXiv:1503.06077] [INSPIRE].
[34] A. Crivellin, L. Hofer, J. Matias, U. Nierste, S. Pokorski and J. Rosiek, Lepton avour
violating B decays in generic Z0 models, Phys. Rev. D 92 (2015) 054013
[arXiv:1504.07928] [INSPIRE].
[35] A. Celis, J. FuentesMartin, M. Jung and H. Serodio, Family nonuniversal Z' models with
avorchanging interactions, Phys. Rev. D 92 (2015) 015007 [arXiv:1505.03079]
protected
[36] A. Greljo, G. Isidori and D. Marzocca, On the breaking of Lepton Flavor Universality in B
decays, JHEP 07 (2015) 142 [arXiv:1506.01705] [INSPIRE].
models, JHEP 01 (2016) 119 [arXiv:1508.00569] [INSPIRE].
[37] C. Nieho , P. Stangl and D.M. Straub, Direct and indirect signals of natural composite Higgs
[38] W. Altmannshofer and I. Yavin, Predictions for lepton avor universality violation in rare B
decays in models with gauged L
L , Phys. Rev. D 92 (2015) 075022 [arXiv:1508.07009]
[39] A. Falkowski, M. Nardecchia and R. Ziegler, Lepton Flavor NonUniversality in Bmeson
Decays from a U(2) Flavor Model, JHEP 11 (2015) 173 [arXiv:1509.01249] [INSPIRE].
[40] B. Gripaios, Composite Leptoquarks at the LHC, JHEP 02 (2010) 045 [arXiv:0910.1789]
[41] G. Belanger, C. Delaunay and S. Westho , A Dark Matter Relic From Muon Anomalies,
Phys. Rev. D 92 (2015) 055021 [arXiv:1507.06660] [INSPIRE].
B ! K( )`+`
and B ! K
, JHEP 09 (2010) 089 [arXiv:1006.4945] [INSPIRE].
[44] S. Jager and J. Martin Camalich, On B ! V `` at small dilepton invariant mass, power
corrections and new physics, JHEP 05 (2013) 043 [arXiv:1212.2263] [INSPIRE].
[45] F. Beaujean, C. Bobeth and D. van Dyk, Comprehensive Bayesian analysis of rare
(semi)leptonic and radiative B decays, Eur. Phys. J. C 74 (2014) 2897 [Erratum ibid. C 74
(2014) 3179] [arXiv:1310.2478] [INSPIRE].
HJEP06(21)83
[46] J. Lyon and R. Zwicky, Resonances gone topsy turvy  the charm of QCD or new physics in
b ! s`+` ?, arXiv:1406.0566 [INSPIRE].
B ! K
+
observables, arXiv:1411.0922 [INSPIRE].
[47] S. DescotesGenon, L. Hofer, J. Matias and J. Virto, QCD uncertainties in the prediction of
[48] S. Jager and J. Martin Camalich, Reassessing the discovery potential of the B ! K `+`
decays in the largerecoil region: SM challenges and BSM opportunities, Phys. Rev. D 93
(2016) 014028 [arXiv:1412.3183] [INSPIRE].
arXiv:1503.06199 [INSPIRE].
Bs0 !
212003 [arXiv:1310.3887] [INSPIRE].
[49] W. Altmannshofer and D.M. Straub, Implications of b ! s measurements,
[50] R.R. Horgan, Z. Liu, S. Meinel and M. Wingate, Calculation of B0 ! K 0 +
and
observables using form factors from lattice QCD, Phys. Rev. Lett. 112 (2014)
[51] A. Datta, M. Duraisamy and D. Ghosh, Explaining the B ! K
+
data with scalar
interactions, Phys. Rev. D 89 (2014) 071501 [arXiv:1310.1937] [INSPIRE].
[52] T. Hurth and F. Mahmoudi, On the LHCb anomaly in B ! K `+` , JHEP 04 (2014) 097
[arXiv:1312.5267] [INSPIRE].
[53] R. Alonso, B. Grinstein and J. Martin Camalich, SU(2)
U(1) gauge invariance and the
shape of new physics in rare B decays, Phys. Rev. Lett. 113 (2014) 241802
[arXiv:1407.7044] [INSPIRE].
[54] D. Ghosh, M. Nardecchia and S.A. Renner, Hint of Lepton Flavour NonUniversality in B
Meson Decays, JHEP 12 (2014) 131 [arXiv:1408.4097] [INSPIRE].
[55] A.J. Buras, S. Jager and J. Urban, Master formulae for Delta F = 2 NLO QCD factors in
the standard model and beyond, Nucl. Phys. B 605 (2001) 600 [hepph/0102316] [INSPIRE].
[56] ETM collaboration, N. Carrasco et al., Bphysics from Nf = 2 tmQCD: the Standard Model
and beyond, JHEP 03 (2014) 016 [arXiv:1308.1851] [INSPIRE].
[57] Muon g2 collaboration, G.W. Bennett et al., Final Report of the Muon E821 Anomalous
Magnetic Moment Measurement at BNL, Phys. Rev. D 73 (2006) 072003 [hepex/0602035]
[58] B.D. Fields, P. Molaro and S. Sarkar, BigBang Nucleosynthesis, Chin. Phys. C 38 (2014)
[59] A.J. Buras, J. GirrbachNoe, C. Nieho and D.M. Straub, B ! K( )
decays in the
Standard Model and beyond, JHEP 02 (2015) 184 [arXiv:1409.4557] [INSPIRE].
[hepph/0512090] [INSPIRE].
Large Hadron Collider, Nucl. Phys. B 826 (2010) 217 [arXiv:0908.1567] [INSPIRE].
10 (2014) 033 [Erratum ibid. 1501 (2015) 041] [arXiv:1407.7058] [INSPIRE].
JHEP 07 (2015) 074 [arXiv:1504.00359] [INSPIRE].
[64] CMS collaboration, Exclusion limits on gluino and topsquark pair production in natural
SUSY scenarios with inclusive razor and exclusive singlelepton searches at 8 TeV.,
CMSPASSUS14011 (2014).
[65] ATLAS collaboration, ATLAS Run 1 searches for direct pair production of thirdgeneration
squarks at the Large Hadron Collider, Eur. Phys. J. C 75 (2015) 510 [arXiv:1506.08616]
CMSPASSUS13018 (2014).
[66] CMS collaboration, Search for direct production of bottom squark pairs,
[67] CMS collaboration, Search for direct EWK production of SUSY particles in multilepton
modes with 8TeV data, CMSPASSUS12022 (2013).
[68] ATLAS collaboration, Search for direct production of charginos, neutralinos and sleptons in
nal states with two leptons and missing transverse momentum in pp collisions at p
s = 8
TeV with the ATLAS detector, JHEP 05 (2014) 071 [arXiv:1403.5294] [INSPIRE].
[69] A. Alloul, N.D. Christensen, C. Degrande, C. Duhr and B. Fuks, FeynRules 2.0  A complete
toolbox for treelevel phenomenology, Comput. Phys. Commun. 185 (2014) 2250
[arXiv:1310.1921] [INSPIRE].
[70] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, UFO  The
Universal FeynRules Output, Comput. Phys. Commun. 183 (2012) 1201 [arXiv:1108.2040]
[71] J. Alwall et al., The automated computation of treelevel and nexttoleading order
di erential cross sections and their matching to parton shower simulations, JHEP 07 (2014)
079 [arXiv:1405.0301] [INSPIRE].
[72] G. D'Ambrosio, G.F. Giudice, G. Isidori and A. Strumia, Minimal avor violation: An
e ective eld theory approach, Nucl. Phys. B 645 (2002) 155 [hepph/0207036] [INSPIRE].
[73] R. Barbieri, G. Isidori, J. JonesPerez, P. Lodone and D.M. Straub, U(2) and Minimal
Flavour Violation in Supersymmetry, Eur. Phys. J. C 71 (2011) 1725 [arXiv:1105.2296]
[74] R. Barbieri, D. Buttazzo, F. Sala and D.M. Straub, Flavour physics from an approximate
U(2)3 symmetry, JHEP 07 (2012) 181 [arXiv:1203.4218] [INSPIRE].
[75] R. Barbieri, D. Buttazzo, F. Sala and D.M. Straub, Flavour physics and avour symmetries
after the rst LHC phase, JHEP 05 (2014) 105 [arXiv:1402.6677] [INSPIRE].
[76] B. KerenZur, P. Lodone, M. Nardecchia, D. Pappadopulo, R. Rattazzi and L. Vecchi, On
Partial Compositeness and the CP asymmetry in charm decays, Nucl. Phys. B 867 (2013)
394 [arXiv:1205.5803] [INSPIRE].
avor violation implications of the b ! s
anomalies, JHEP 08 (2015) 123 [arXiv:1505.04692] [INSPIRE].
Decays
B ! D( )
relative to B ! D( )`
92 (2015) 072014 [arXiv:1507.03233] [INSPIRE].
` decays with hadronic tagging at Belle, Phys. Rev. D
JHEP 08 (2011) 055 [arXiv:1105.3161] [INSPIRE].
electroweak breaking on rare B decays and mixings, Nucl. Phys. B 353 (1991) 591 [INSPIRE].
[6] S. DescotesGenon , L. Hofer , J. Matias and J. Virto , On the impact of power corrections in observables , JHEP 12 ( 2014 ) 125 [arXiv: 1407 .8526] [7] T. Hurth , F. Mahmoudi and S. Neshatpour , Global ts to b ! s`` data and signs for lepton [24] S. DescotesGenon , J. Matias and J. Virto , Understanding the B ! K Phys. Rev . D 88 ( 2013 ) 074002 [arXiv: 1307 .5683] [INSPIRE].
[42] S.L. Glashow , D. Guadagnoli and K. Lane , Lepton Flavor Violation in B Decays?, Phys.
[43] A. Khodjamirian , T. Mannel , A.A. Pivovarov and Y.M. Wang , Charmloop e ect in [60] M. Cirelli , N. Fornengo and A. Strumia , Minimal dark matter, Nucl. Phys. B 753 ( 2006 ) 178 [61] E. Del Nobile , R. Franceschini , D. Pappadopulo and A. Strumia , Minimal Matter at the [62] M. Cirelli , F. Sala and M. Taoso , Winolike Minimal Dark Matter and future colliders , JHEP [63] L. Di Luzio , R. Grober, J.F. Kamenik and M. Nardecchia , Accidental matter at the LHC, [77] S.M. Boucenna , J.W.F. Valle and A. Vicente , Are the B decay anomalies related to neutrino oscillations? , Phys. Lett. B 750 ( 2015 ) 367 [arXiv: 1503 .07099] [INSPIRE].
[78] C.J. Lee and J. Tandean , Minimal lepton [79] D. Guadagnoli and K. Lane , ChargedLepton Mixing and Lepton Flavor Violation, Phys. Lett. B 751 ( 2015 ) 54 [arXiv: 1507 .01412] [INSPIRE].
[80] BaBar collaboration , J.P. Lees et al., Measurement of an Excess of B ! D( ) and Implications for Charged Higgs Bosons , Phys. Rev. D 88 ( 2013 ) 072012 [81] BaBar collaboration , J.P. Lees et al., Evidence for an excess of B ! D( ) Phys. Rev. Lett . 109 ( 2012 ) 101802 [arXiv: 1205 .5442] [INSPIRE].
[82] Belle collaboration , M. Huschle et al., Measurement of the branching ratio of [83] S. Fajfer , J.F. Kamenik , I. Nisandzic and J. Zupan , Implications of Lepton Flavor Universality Violations in B Decays, Phys. Rev. Lett . 109 ( 2012 ) 161801 [arXiv: 1206 . 1872 ] [84] M. Freytsis , Z. Ligeti and J.T. Ruderman , Flavor models for B ! D( ) , Phys. Rev. D 92 [85] G.F. Giudice , B. Gripaios and R. Sundrum , Flavourful Production at Hadron Colliders, [86] E. Lunghi , A. Masiero , I. Scimemi and L. Silvestrini , B ! Xs`+` decays in supersymmetry, Nucl. Phys. B 568 ( 2000 ) 120 [ hep ph/9906286] [INSPIRE].
[87] S. Bertolini , F. Borzumati , A. Masiero and G. Ridol , E ects of supergravity induced