Revisiting Minimal Lepton Flavour Violation in the light of leptonic CP violation
Received: May
Minimal Lepton Flavour Violation in the light of leptonic CP violation
D.N. Dinh 0 1 2 4 5 6
L. Merlo 0 1 2 5 6
S.T. Petcov 0 1 2 3 5 6
R. VegaAlvarez 0 1 2 5 6
Tokyo 0 1 2 5 6
Japan 0 1 2 5 6
0 Universidad Autonoma de Madrid
1 Charlottesville , VA 229044714 , U.S.A
2 10 Dao Tan , Ba Dinh, Hanoi , Viet Nam
3 Kavli IPMU, University of Tokyo , WPI
4 Department of Physics, University of Virginia , USA
5 Via Bonomea 265 , 34136 Trieste , Italy
6 Cantoblanco , 28049, Madrid , Spain
The Minimal Lepton Flavour Violation (MLFV) framework is discussed after the recent indication for CP violation in the leptonic sector. Among the three distinct versions of MLFV, the one with degenerate righthanded neutrinos will be disfavoured, if this indication is con rmed. The predictions for leptonic radiative rare decays and muon conversion in nuclei are analysed, identifying strategies to disentangle the di erent MLFV scenarios. The claim that the present anomalies in the semileptonic Bmeson decays can be explained within the MLFV context is critically reexamined concluding that such an explanation is not compatible with the present bounds from purely leptonic processes.
CP violation; E ective Field Theories; Global Symmetries; Neutrino Physics

Revisiting
1 Introduction 2
Minimal (Lepton) Flavour Violation
2.1 The lepton sector
3 Phenomenology in the lepton sector
3.1
3.2
The LFV e ective Lagrangian
Rare radiative leptonic decays and conversion in nuclei
3.2.1
3.2.2
Bounds on the LFV scale
Ratios of branching ratios
4 b ! s anomalies
4.1 B semileptonic decays
5 Conclusions
The discovery [1{5] of a nonvanishing reactor angle 1`3 in the lepton mixing matrix led
to a huge fervour in the avour community and to a deep catharsis in the model building
When the value of this angle was still unknown, the closeness to a maximal mixing
value of the atmospheric angle 2`3 was suggesting a maximal oscillation between
muonand tauneutrinos: in terms of symmetries of the Lagrangian acting on the avour space, it
could be described by a discrete Abelian Z2 symmetry, which, in turn, implied a vanishing
reactor angle. The simplicity and the elegance of this pattern, i.e. one maximal angle and
one vanishing one, convinced part of the community that Nature could have made us a
favour and that neutrino physics could indeed be described, at least in the atmospheric
and reactor sectors, by this texture [6, 7].
An approach followed for such constructions was to write a Lagrangian whose leading
order terms described speci c textures for the Yukawa matrices, leading to 1`3 = 0 and
`
23 = 45 . Often, this was done such that the Yukawa matrix for the charged leptons
was diagonal while the Yukawa matrix for the light active neutrinos was diagonalised
sin2 1`2 = 1=3, in a very good agreement with the neutrino oscillation data.
by the socalled TriBimaximal mixing matrix [8{10], which predicts, besides a vanishing
reactor mixing angle and a maximal atmospheric one 2`3 = 45 , a solar angle satisfying to
Pioneer models can be found in refs. [11{15], where the discrete nonAbelian group A4
was taken as a avour symmetry of the lepton sector. Several distinct proposals followed,
{ 1 {
i) attempting to achieve the TriBimaximal pattern, but with other avour symmetries
(see for example refs. [16{19]); or ii) adopting other mixing patterns to describe neutrino
oscillations, such as the Bimaximal mixing1 [21, 22], the Golden Ratio mixing [23, 24] and
the Trimaximal mixing [25]; iii) analysing the possible perturbations or modi cations to
Bimaximal mixing, TriBimaximal mixing etc., arising from the charged lepton sector [26{29],
vi) implementing the socalled quarklepton complementarity [30, 31] which suggests that
the lepton and quark sectors should not be treated independently, but a common dynamics
could explain both the mixings [32{34]. Further details could be found for example in these
reviews [35{40].
After the discovery of a nonvanishing 1`3 and the improved sensitivity on the other
two mixing angles, which pointed out that 2`3 best t is not 45 (the most recent global ts
on neutrino oscillation data can be found in refs. [41{43]), models based on discrete
symmetries underwent to a deep rethinking. A few strategies have been suggested: introduction
of additional parameters in preexisting minimal models, see for example refs. [44, 45];
implementation of features that allow subleading corrections only in speci c directions in
the avour space [46{49]; search for alternative avour symmetries or mixing patterns that
lead already in rst approximation to 1`3 6= 0 and 2`3 6= 45 [50, 51]. One can fairly say
that the latest neutrino data can still be described in the context of discrete symmetries,
but at the prize of netunings and/or less minimal mechanisms.
Alternative approaches to discrete avour model building strengthened after 2011 and,
in particular, constructions based on continuous symmetries were considered interesting
possibilities: models based on the simple U(
1
) (e.g. refs. [52{57]) or based on SU(3) (e.g.
refs. [58, 59]) or the socalled Minimal Flavour Violation (MFV) [60, 61], and its leptonic
versions [62{65], dubbed MLFV. The latter is a setup where the
avour symmetry is
identi ed with the symmetry of the fermionic kinetic terms, or in other words, the symmetry
of the SM Lagrangian in the limit of vanishing Yukawa couplings: it is given by products of
U(3) factors, one for each fermion spinor of the considered spectrum. Fermion masses and
mixings are then described once the symmetry is broken. This approach allows to relate
any source of avour and CP violation in the SM and beyond to the Yukawa couplings,
such that any avour e ect can be described in terms of fermion masses and mixing angles.
The M(L)FV is not a complete model, as fermion masses and mixings are just described
while their origin is not explained (attempts to improve with this respect can be found in
refs. [66{75]). It is instead a framework where observed
avour violating observables are
described in agreement with data and unobserved avour violating signals are not expected
to be observed with the current experimental sensitivities, but could be observable in the
future planned experiments with signi cantly higher sensitivity, assuming the New Physics
(NP) responsible for these phenomenology at the TeV scale or slightly higher [61{65, 76{89].
The recent indication of a relatively large Dirac CP violation in the lepton
sector [41{43, 90{92] represented a new turning point in the sector. Present data prefer a
nonzero Dirac CP phase, C`P, over CP conservation at more than 2 's, depending on the
1Bimaximal mixing can be obtained by assuming the existence of an approximate U(
1
) symmetry
corresponding to the conservation of the nonstandard lepton charge L0 = Le
L
L and additional discrete
symmetry [20].
{ 2 {
speci c neutrino mass ordering. Moreover, the best t value for the leptonic Jarlskog
invariant, J C`P '
J C`P ' 3:04
sector than in the quark sector.
0:033 [42], is numerically much larger in magnitude than its quark sibling,
10 5 [93], indicating potentially a much larger CP violation in the lepton
In the eld of discrete avour models, this indication translated into looking, for the
rst time, for approaches and/or contexts where, besides the mixing angles, also the lepton
phase(s) were predicted: new models were presented with the CP symmetry as part of
the full avour symmetry [94{100]; studies on the mixing patterns and their modi cations
to provide realistic descriptions of oscillation data were performed [101{104]; an intense
activity was dedicated to investigate sum rules involving neutrino masses, mixing angles
on continuous avour symmetries. In particular, one very popular version of MLFV [62]
strictly requires CP conservation as a working assumption and therefore, if this indication
is con rmed, this setup will be disfavoured.
a 2:6
the e=
measure of RK
The rst goal of this paper is to update previous studies on MLFV in the light of the
last global t on neutrino oscillation data and to discuss the impact of the recent indication
for CP violation in the lepton sector. Indeed, the last studies on MLFV date back to the
original papers in 2005 [62, 63] and 2011 [65], before the discovery of a nonvanishing 1`3
and lacking any information about the leptonic CP phase.
The search for an explanation of the heterogeneity of fermion masses and mixings,
the socalled Flavour Puzzle, is just a part of the Flavour Problem of particle physics. A
second aspect of this problem is related to the fact that models involving NP typically
introduce new sources of avour violation. Identifying the mechanism which explains why
the experimentally measured avour violation is very much consistent with the SM
predictions is a crucial aspect in
avour physics. The use of avour symmetries turned out
to be useful also with this respect: a very wellknown example is the MFV setup, as
previously discussed, whose construction was originally meant exactly to solve this aspect of
the Flavour Problem. Promising results have been obtained also with smaller symmetries
than the MFV ones, both continuous [110{115] and discrete [116{123].
The Flavour Problem becomes even more interesting after the indications for anomalies
in the semileptonic Bmeson decays: the angular observable P50 in the B ! K
presents a tension with the SM prediction of 3:7 [124, 125] and 2 [126], considering LHCb
+
decay
and Belle data, respectively; the Branching Ratio of Bs !
SM prediction at 3:2 [127]; the ratio RD`
BR(B ! D( )` )SM=BR(B ! D( )
universality [128{132]; the ratio RK
BR(B ! D( )
)exp=BR(B ! D( )` )exp
)SM with ` = e;
indicates a 3:9
violation of =`
BR(B+
! K+ +
)=BR(B+
! K+e+e ) is in
tension with the SM prediction [133], indicating lepton universality violation in
+
is in tension with the
sector. The latter has been con rmed also by the recent announcement of the
tension with the SM prediction in the centralq2 region (lowq2 region) [134]. Under the
assumption that these anomalies are due to NP, and not due to an underestimation of the
hadronic e ects [135{140] or due to a statistical uctuation, a global analysis on b ! s
BR(B0 ! K 0 +
)=BR(B0 ! K 0e+e ) is in a 2:4{2:5
(2:2{2:4 )
{ 3 {
data can attempt to identify the properties of the underlying theory. Adopting an e ective
description, these results can be translated into constraints of the Wilson coe cients of
the Hamiltonian describing
B = 1 decays: the results of such analysis [141{153] are that
the anomalies can be explained with a modi cation of the Wilson coe cients C9 and C10
de ned as
H eB=1
4GF
p
e
2
2 (4 )2 VtbVts s
h
PLb `
ih
(C9 + C10 5) `i + h.c.
(1.1)
where V is the CKM matrix, PL = (1
5)=2 is the usual lefthanded (LH) chirality
projector, b and s refer to the bottom and strange quarks, respectively, ` are the charged
leptons, and the prefactors refer to the traditional normalisation.
Writing each of the
coe cients as the sum of the purely SM contribution and the NP one, Ci = CSM +
i
Ci,
the results of a oneoperatoratatime analysis [151] suggest lepton universality violation
in the e= sector quanti able in
C9e =
C1e0 2 [+0:56; +1:02] and
C
9 =
(1.2)
corresponding to 4:3 and 4:2 tension with the SM predictions, respectively.
The hypothetical underlying theory, which manifests itself at low energies with these
features, will necessarily respect the SM gauge invariance, and therefore will also contribute
to b ! c processes and hopefully solve the RD`( ) anomalies.
Several attempts have been presented in the literature to explain the de cit on C9
and/or C10, including the MLFV approach: ref. [154] considers the version of MLFV
introduced in ref. [62] and constraints on the Lagrangian parameters and on the Lepton
Flavour Violating (LFV) scale have been obtained requiring to reproduce the values of Ce
9
and C1e0 aforementioned.
A second goal of this paper is to revisit the results presented in ref. [154] considering the constraints from purely leptonic observables, such as radiative rare decays and conversion in nuclei.
Moreover, the analysis will be extended to the other versions of
! e
MLFV [65].
The structure of the paper can easily be deduced from the table of content: rst, in
section 2, basic concepts of MFV and MLFV will be recalled, underlying the di erences
between the distinct versions of MLFV; then, in section 3, several processes in the lepton
sector will be discussed considering the last global t on neutrino data and the recent
indication for leptonic CP violation; in section 4, the anomalies in the b ! s decays
will be discussed, pointing out the di erences with respect to previous literature; nally,
concluding remarks will be presented in section 5.
2
Minimal (Lepton) Flavour Violation
If a theory of NP, with a characteristic scale of a few TeVs, behaves at low energy
accordingly to the MFV ansatz, i.e. the SM Yukawa couplings are the only sources of avour
and CP violation even beyond the SM, then its avour protection is guaranteed: the large
majority of observed avour processes in the quark sector are predicted in agreement with
{ 4 {
data [61, 76, 78{86, 155{160]; unseen avour changing processes, for example leptonic
radiative rare decays, are predicted to have strengths which are inside the present experimental
sensitivity [62, 63, 65, 88, 161{164].
In the modern realisation of the MFV ansatz, the avour symmetry corresponds to the
one arising in the limit of vanishing Yukawa couplings. This massless Lagrangian is left
invariant under a tridimensional unitary transformations in the avour space associated to
each fermion spinor. In the quark sector, it is given by
GQ
U(
1
)B
U(
1
)Au
U(
1
)Ad with GQ = SU(3)qL
SU(3)uR
SU(3)dR ;
(2.1)
where qL refer to the SU(2)Ldoublet of quarks, and uR and dR to the SU(2)Lsinglets. The
Abelian terms can be identi ed with the Baryon number, and with two axial rotations, in
the up and downquark sectors respectively, which do not distinguish among the distinct
families [165]. On the contrary, the nonAbelian factors rule the interactions among the
generations and govern the amount of avour violation: they are the key ingredients of
MFV and will be in the focus of the analysis in which follows.
The explicit quark transformations read
qL
(3; 1; 1)GQ
uR
(1; 3; 1)GQ
dR
(1; 1; 3)GQ
qL ! UqLqL
uR ! UuRuR
dR ! UdRdR ;
where Ui 2 SU(3)i are 3
3 unitary matrices acting in the avour space. The quark
Lagrangian is invariant under these transformations, except for the Yukawa interactions:
LQ = qLYuH~ uR
qLYdHdR + h.c. ;
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
HJEP07(21)89
Yu
(3; 3; 1)GQ
Yd
(3; 1; 3)GQ
Yu ! UqL Yu UuyR Yd ! UqL Yd UdyR :
Once the Yukawa spurions acquire a background value, the avour symmetry is broken
and in consequence fermions masses and mixings are generated. A useful choice for these
background values is to identify them with the SM Yukawa couplings: in a given basis,
Yd is diagonal and describes only downtype quark masses, while Yu contains nondiagonal
entries and accounts for both uptype quark masses and the CKM matrix V :
hYui
Yu =
p
v
2 V yM^ u ;
hYdi
Yd =
p
v
2 M^ d ;
and M^ u;d are the diagonal mass matrices for up and downtype quarks,
where v = 246 GeV is the Higgs vacuum expectation value (VEV) de ned by hH0i = v=p2,
where Yi are 3
3 matrices in the avour space, H is the SU(2)Ldouble Higgs eld, and
H~ = i 2H . LQ can be made invariant under GQ promoting the Yukawa matrices to
be spurion
elds, i.e. auxiliary nondynamical elds, denoted by Yu and Yd, with speci c
transformation properties under the avour symmetry:
^
Mu
diag(mu; mc; mt) ;
^
Md
diag(md; ms; mb) :
{ 5 {
When considering lowenergy avour processes, they can be described within the
effective eld theory approach through nonrenormalisable operators suppressed by suitable
powers of the scale associated to the messenger of the interaction. These structures could
violate the avour symmetry GQ, especially if they describe avour changing observables.
As for the Yukawa Lagrangian, a technical way out to recover avour invariance is to insert
powers of the Yukawa spurions. Once the spurions acquire background values, the
corresponding processes are predicted in terms of quark masses and mixings. Several studies
already appeared addressing this topic [61, 76, 78{86, 155{160] and, as already mentioned
at the beginning of this section, the results show that
avour data in the quark sector
are well described within the MFV(like) approach. Indeed, the Yukawa spurions act as
of the spurions obtain stronger suppressions.2
MFV, however, cannot be considered a complete avour model, as there is not
explanation of the origin of quark masses and mixings. There have been attempts to go
from the e ectivespurionic approach to a more fundamental description, promoting the
Yukawa spurions to be dynamical elds, called
avons, acquiring a nontrivial VEV. The
corresponding scalar potentials have been discussed extensively with interesting
consequences [66{70]: a conclusive dynamical justi cation for quark masses and mixing is still
lacking, but the results are encouraging as the potential minima lead, at leading order, to
nonvanishing masses for top and bottom quarks and to no mixing.
2.1
The lepton sector
The lepton sector is more involved with respect to the quark one, due to the lack of
knowledge on neutrino masses: indeed, while the charged lepton description mimics the
one of downquarks, light active neutrino masses, and then the lepton mixing, cannot be
described within the SM.
Several ways out have been presented in the literature to provide a description for the
lepton sector, and the focus here will be on two wellde ned approaches, one maintaining
the SM spectrum but relaxing the renormalisability criterium, and the other adding new
particles in a still renormalisable theory.
Minimal Field Content (MFC).
Giving up with renormalisability, active neutrino
masses can be described via the socalled Weinberg operator [167], a nonrenormalisable
operator of canonical dimension 5 which breaks explicitly Lepton number by two units,
OW =
1
2
`cLH~
g
L
H~ y`L ;
where `cL
C`LT , C being the charge conjugation matrix (C 1
C =
T ), g is an
adimensional symmetric 3 3 matrix in the avour space and
L is the scale of Lepton Number
Violation (LNV). The avour symmetry arising from the kinetic terms in this case is
GL
U(
1
)L
2The top Yukawa represents an exception as it cannot be technically taken as an expanding parameter.
This aspect has been treated in refs. [166], where a resummation procedure has been illustrated.
(2.7)
(2.8)
{ 6 {
where U(
1
)L is the Lepton number while U(
1
)Ae is an axial rotation in `L and eR, and the
nonAbelian transformations of the leptons read
The part of the Lagrangian describing lepton masses and mixings,
`L
elds, Ye and g , transforming as
is not invariant under GL, but this can be cured by promoting Ye and g to be spurion
Lepton masses and the PMNS matrix U arise once Ye and g acquire a background value
that can be chosen to be
hYei
with M^ `; being the diagonal matrices of the charged lepton and active neutrino mass
eigenvalues,
^
M`
diag(me; m ; m ) ;
^
M
diag (m 1 ; m 2 ; m 3 ) ;
gy g =
4v42L U M^ 2U y :
{ 7 {
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
and U de ned as the product of four matrices [93],
U = R23( 2`3) R13( 1`3; C`P) R12( 1`2) diag 1; ei 221 ; ei 231
;
Dirac CP phase C`P in the reactor sector, and
21;31 the Majorana phases [168].
with Rij ( i`j ) a generic rotation of the angle i`j in the ij sector, with the addition of the
As discussed for the quark case, Ye and g act as expanding parameters: operators with
more insertions of these spurions describe processes that receive stronger suppressions. This
perturbative treatment requires, however, that the largest entries in Ye and g are at most
O(
1
). The charged lepton Yukawa satis es to this condition as the largest entry is
m =v.
The neutrino spurion g is instead function of
L: requiring that jg ij j < 1 leads to an
upper bound on the LNV scale, which depends on j(U M^ U y)ij j that is a function of the
type of neutrino mass spectrum (NO or IO), of the value of the lightest neutrino mass and
of the values of the Majorana and Dirac CP violation phases. The lowest upper bound is
given approximately by:
v
2
It will be useful for the phenomenological discussion in the next sections to remember
that the spurion combination gy g transforms as (8; 1)GL and to introduce the quantity
Extended Field Content (EFC).
Enlarging the SM spectrum by the addition of three
RH neutrinos NR leads to the socalled type I Seesaw context [169{173], described by the
following Lagrangian:
LL{SS =
`LYeHeR
`LY H~ NR
1
2 LN RcYN NR + h.c. ;
where Ye, Y and YN are adimensional 3
3 matrices in the avour space, while L stands
for the scale of Lepton number violation, broken by two units by the last term on the right
of this equation. Assuming a hierarchy between
L and v, L
v, it is then possible to
easily blockdiagonalise the full 6
6 neutrino mass matrix, and obtain the induced masses
for the light active neutrinos: in terms of the parameter g appearing in the Weinberg
operator in eq. (2.7), they are given by
gy
L
= Y
Y 1
N Y T :
L
following avour symmetry:
under which leptons transform as
The fermionic kinetic terms of the SM extended with 3 RH neutrinos manifest the
GL
U(
1
)L
and where U(
1
)AN is an axial transformation associated to NR and SU(3)NR is a new
rotation that mixes the three RH neutrinos. The Lagrangian in eq. (2.17) breaks explicitly
GL de ned in eq. (2.19), but the invariance can be technically restored promoting YE, Y
and YN to be spurions elds, YE, Y and YN , transforming as
Ye
Lepton masses and mixing are then described when these spurion elds acquire the following
background values:
hYei
Di erently from the quark sector and the MFC lepton case, it is not possible to identify
a unique choice for hY i and hYN i, as only the speci c combination in eq. (2.22) can be
associated to the neutrino mass eigenvalues and the PMNS matrix entries. This is a relevant
aspect as it nulli es the MLFV
avour protection. Indeed, the basic building blocks for
several processes, such as radiative leptonic decays or leptonic conversions, are fermionic
standing for combination
of Dirac
matrices and/or Pauli
matrices. In the unbroken phase, these terms are
{ 8 {
(2.17)
(2.18)
(2.20)
(2.21)
(2.22)
invariant under the avour symmetry contracting the avour indices with combinations of
the spurions transforming as (8; 1; 1)GL , (6; 1; 1)GL , (3; 3; 1)GL , and (1; 8; 1)GL , among
others. These spurion combinations are distinct from the combination of Y and YN t hat
appears in eq. (2.22): a few examples are
In consequence, one concludes that it is not possible to express any avour changing process
involving leptons in terms of lepton masses and mixings, losing in this way the predictive
power of MLFV.
This problem can be solved, and predictivity can be recovered, if all the information of
neutrino masses and mixing would be encoded into only one spurion background among Y
and YN , being the other proportional to the identity matrix. Technically, this corresponds
to break GL following two natural criteria.
I): GL ! SU(3)`L
Under the assumption that the three RH neutrinos are degenerate in mass, i.e. YN / 1 ,
SO(3)NR is broken down to SO(3)NR and the transformation UNR in eq. (2.20) is then
an orthogonal matrix. The additional assumption of no CP violation in the lepton
sector is meant to force Ye and Y to be real.3 With this simpli cations, all avour
changing e ects involving leptons can be written in terms of Y Y T and Ye, as can be
easily deduced from eq. (2.23). In this case, eq. (2.22) simpli es to
Y Y T =
L by reabsorbing the norm of YN , and therefore any avour
changing process can be described in terms of lepton masses and mixings. The
last equivalence in the previous equation is a de nition that will be useful in the
phenomenological analysis.
As for the MFC case, requiring that the spurions respect the perturbativity regime
leads to an upper bound on the LNV scale:
v
2
Y Y T
m2atm
numerically the same as the one in eq. (2.15).
3Strictly speaking, the condition of CP conservation in the leptonic sector forces the Dirac CP phase to
be equal to C`P = f0; g and the Majorana CP phases to be
21;31 = f0; ; 2 g
. However, Y is real only
if 21;31 = f0; 2 g, and therefore
21;31 =
needs to be disregarded in order to guarantee predictivity.
The CP conservation condition assumed in this context is then stronger than the strict de nition.
{ 9 {
II): GL ! SU(3)`L+NR
Assuming that the three RH neutrinos transform as a triplet under the same
symmetry group of the lepton doublets,
`L; NR
then the Schur's Lemma guarantees that Y transforms as a singlet of the symmetry
group and then Y is a unitary matrix [174, 175], which can always be rotated to the
identity matrix by a suitable unitary transformation acting only on the RH neutrinos.
The only sensible quantities in this context are Ye and YN , which now transform as
The background value of YN would eventually encode the norm of Y , in order to
consistently take Y
= 1 . In this basis, neutrino masses and the lepton mixing are
encoded uniquely into YN ,
v
2
Moreover, all the spurion combinations in eq. (2.23) can be written only in terms of
YN and Ye and therefore any avour changing process can be predicted in terms of
lepton masses and mixing. It will be useful in the phenomenological analysis that
follows to introduce the quantity
Contrary to what occurs in the MFC and the EFCI cases, the perturbativity condition
on YN allows to extract a lower bound on the LNV scale:
v
2
Similarly to what discussed for the quark sector, none of the two versions of the MLFV
provide an explanation for the origin of lepton masses and mixing, and therefore cannot
be considered complete models. In refs. [72{74] attempts have been presented to provide a
dynamical explanation for the avour puzzle in the lepton sector: as for the quark sector,
the results are not conclusive, but highlighted interesting features. Indeed, for the MLFV
version with an SO(3)NR symmetry factor associated to the RH neutrinos, the minima of
the scalar potential, constructed by promoting Ye and Y to be dynamical elds, allow a
maximal mixing and a relative maximal Majorana CP phase between two almost degenerate
neutrino mass eigenvalues. This seems to suggest that the large angles in the lepton sector
could be due to the Majorana nature of neutrinos, in contrast with the quark sector where
this does not occur.
No dedicated analysis of the scalar potential arising in the second version of MLFV
has appeared in the literature, although the results are not expected to be much di erent
from the ones in the quark sector. However, as a conclusive mechanism to explain lepton
masses and mixing is still lacking, both the versions of MLFV remain valid possibilities.
0:306
0:012
0:441+00::002271
261+5519
7:50+00::1197
2:524+00::003490
0:02166
0:00075
0:02179
0:00076
0:306
0:012
0:587+00::002204
277+4406
7:50+00::1197
2:514+00::00:4318
m2atm
m23 for IO. The errors reported correspond to the 1 uncertainties.
notation has been chosen such that
ms2ol
m22
m21 , and
m2atm
m23
m21 for NO and
will be considered.
lepton masses [93]
As anticipated in section 1, the recent indication for a relatively large leptonic CP
violation, if con rmed, would disfavour EFCI, due to the required reality of Y . However, in
the present discussion and in the analysis that follows, EFCI will not be discarded yet, as the
assumption of CP conservation is a distinctive feature of this lowenergy description of the
lepton sector, but could be avoided in more fundamental ones. Indeed, a model constructed
upon the gauged lepton
avour symmetry SU(3)`L
further hypothesis on CP in the lepton sector, is shown in ref. [88] to be as predictive as
EFCI: indeed, with the Dirac CP phase taken at its best t value, this gauged
avour
model presents several phenomenological results similar to the ones of EFCI discussed in
refs. [62, 63]. This motivates to consider EFCI as a valid context to describe lepton avour
observables, even if results which show a strong dependence on the value of the Dirac CP
phase should be taken with a grain of salt.
3
Phenomenology in the lepton sector
In this section, the phenomenology associated to the MFC, EFCI and EFCII cases will
be discussed considering speci cally leptonic radiative rare decays and
in nuclei. While these analyses have already been presented in the original MLFV
papers [62, 63, 65], in the review part of the present paper the latest discovered value of the
reactor angle and the recent indication of nonvanishing CP phase in the leptonic sector
! e conversion
The input data that will be used in what follows are the PDG values for the charged
me = 0:51 MeV ;
m
= 105:66 MeV ;
m
= 1776:86
0:12 MeV ;
(3.1)
where the electron and muon masses are taken without errors as the sensitivities are
negligible, and the results of the neutrino oscillation t from ref. [42] reported in table 1.
The value of the lightest neutrino mass and the neutrino mass ordering are still
unknown. For this reason, the results of this section will be discussed in terms of the values
of the lightest neutrino mass and for both the Normal Ordering (NO) and the Inverted
Ordering (IO). The measured parameters are taken considering their 2
is to underly the impact of the raising indication for a leptonic CP violation.
error bands:4 this
among others, are predicted to be unobservably small in the minimal extension of the SM
with light massive Dirac neutrinos, in which the total lepton charge is conserved [180]. As a
consequence, the rates of such processes have a remarkable sensitivity to NP contributions.
The main observables that will be discussed here are lepton radiative rare decays and
HJEP07(21)89
! e conversion in nuclei. Other leptonic observables which are typically very sensible to
NP are ` ! `0`0`00 decays, and especially the
! 3e decay, given the signi cant increase
of the sensitivity of the planned experiments. However, these processes do not provide
additional information for the results that will be obtained in the following, and therefore
they will not be further considered.
Assuming the presence of new physics at the scale
LFV responsible for these
observables characterised by a much lower typical energy, one can adopt the description in terms
of an e ective Lagrangian:5 the relevant terms are then given by6
LLeFV =
5
1
2
LFV i=1
X c(LiL)OL(iL) +
1
2
LFV
X c(RjL)OR(jL) + h.c.A ;
where the Lagrangian parameters are real coe cients7 of order 1 and the operators have
the form:8
OL(1L) = i` `LHyD H ;
OL(3L) = ` `Lq qL ;
OLL
(4u) = ` `Lu
OR(1L) = g0`H
uR ;
`LHy aD H ;
OLL
(4d) = ` `Ld
dR ;
OL(5L) = `
OR(2L) = g`H
a
`Lq
a
qL ;
a
eRW a :
4EW running e ects [176{179] are negligible in the analysis presented here.
5The e ective Lagrangian reported here corresponds to the linearly realised EWSB. An
alternative would be to considered a nonlinear realisation and the corresponding e ective Lagrangian dubbed
HEFT [181{186]. In this context, however, a much larger number of operators should be taken into
consideration and a slightly di erent phenomenology is expected [187{196]. The focus in this paper is on the
linear EWSB realisation and therefore the HEFT Lagrangian will not be considered in what follows.
6A few other operators are usually considered in the e ective Lagrangian associated to these LFV
observables, but the corresponding e ects are negligible. See ref. [62] for further details.
7The reality of the Lagrangian parameters guarantees that no sources of CP violation are introduced
beyond the SM. A justi cation of this approach can be found in ref. [78].
8The notation chosen for the e ective operators matches the one of the original MLFV paper [62]. It is
nowadays common to adopt an other operator basis introduced in refs. [197, 198]. The link between the
two bases is given by:
OLL ! Q('1`) ;
(
1
)
(4u)
OLL ! Q`d ;
OLL ! Q('3`) ;
(2)
OLL ! Q(`3q) ;
(5)
OLL ! Q(`1q) ;
(3)
(
1
)
ORL ! QeB ;
(4d)
OLL ! Q`d ;
(2)
ORL ! QeW :
(3.2)
(3.4)
(3.3)
contracted with those of the lepton bilinear `iL `jL in OL(iL),
The OL(iL) structures are invariant under the avour symmetries without the necessity
of introducing any spurion
eld, but they can only contribute to
avour conserving
observables. The LFV processes aforementioned can only be described by the insertion of
speci c spurion combinations transforming as 8 under SU(3)`L , whose avour indices are
being a suitable combination
of Dirac and/or Pauli matrices. The speci c spurion combinations depend on the
considered model: some examples are gy g in MFC, Y Yy in EFCI and Y YN YN Yy in EFCII.
y
Interestingly, once the spurions acquire their background values, these combinations reduce
to the expressions for
in eqs. (2.16), (2.24) and (2.29), respectively.
The OR(iL) operators, instead, are not invariant under the avour symmetry GL and
require the insertion of spurion combinations transforming as (3; 3) under SU(3)`L
SU(3)eR .
The simplest combination of this kind is the charged lepton Yukawa spurion Ye, whose
background value, however, is diagonal. Requiring as well that these structures describe
LFV processes, it is necessary to insert more elaborated combinations: some examples
are gy g Ye in MFC, Y YyYe in EFCI and YN YN Ye in EFCII. Once the spurions acquire
y
background values, these combinations reduce to
Ye, with the speci c expression for
depending on the case considered.
From the previous discussion one can deduce that the relevant quantity that allows to
describe LFV processes in terms of lepton masses and mixings is
, beside the diagonal
matrix Ye. It is then instructive to explicitly write the expression for
in the three cases
under consideration and distinguishing between the NO and the IO for the neutrino mass
spectrum.9
SU(3)eR . Expliciting eq. (2.16), the o
1. Minimal Field Content GL = SU(3)`L
diagonal entries of
can be written as
4 2L hs12c12c23c13 (m B
e =
e =
=
v4
v4
v4
4 2L h
4 2L n
i
;
i
;
(3.5)
(3.6)
(3.7)
in the IO case may di er from what reported in ref. [62], due to a di erent
de nition taken for the atmospheric mass squared di erence.
i
i
m A ) + s23s13c13e
m C
^ has been adopted in the de nition of :
where, for brevity of notation, sij and cij stand for the sine and cosine of the leptonic
mixing angles i`j ,
stands for the Dirac CP phase C`P, and a generic notation for
M^ 2
diag (m A
; m B
The three parameters m A;B;C depend on the neutrino mass ordering: for the NO case
m A = 0 ;
m B =
Notice that there is no dependence on the lightest neutrino mass in these expressions.
This has an interesting consequence because
common scale
i6=j are completely xed, apart for the
2. Extended Field Content I) GL = SU(3)`L
one gets the following explicit expressions for the o diagonal entries of :
s12c12s23c13(m B
m A )+c23s13c13e
i e 2i m C
where a generic notation  di erent from the one in the MFC case  for M^
has
been adopted:
^
M
The three parameters m A;B;C are now de ned by
m A = m 1
;
m B = ei 21 q
ms2ol +m21 ;
m C = ei 31 q
m2atm +m21 ;
for the NO case, m 1 < m 2 < m 3 , and by
m A =
q
m2atm
ms2ol +m23 ;
m B = ei 21 q
m2atm +m23 ;
m C = ei 31 m 3
;
for the IO case, m 3 < m 1 < m 2 .
The hypothesis of CP conservations xes the Dirac and Majorana CP phases to be
= f0; g and
for 21;31 =
21;31 = 0 in these expressions. Indeed, while
ij would be real even
and therefore no CPV process would be described with
insertions,
Y would be complex and then it would not be possible to express the spurions
insertions in eq. (2.23) in terms of lowenergy parameters, losing the predictivity power
of MLFV.
In the strong hierarchical limit, m 1
m 2 < m 3 in the NO case and m 3
m 1 < m 2
in the IO one, and setting the lightest neutrino mass to zero, the expressions for
m A;B;C reduce to the square root of those for the MFC case, as can be deduced
comparing eqs. (3.6) and (3.10), and the results for
case, only one parameter remains free, that is the LNV scale L.
i6=j get simpli ed. Also in this
When the neutrino mass hierarchy is milder or the eigenvalues are almost
degenerate, the lightest neutrino mass cannot be neglected and represents a second free
parameters of i6=j , besides L.
i
;
i
;
o
; (3.9)
(3.10)
(3.11)
(3.12)
3. Extended Field Content II) GL = SU(3)`L+NR
SU(3)eR . The expressions for the o
that follow from eqs. (2.29) can be obtained from the expressions
in eq. (3.5), by substituting
and taking the following notation for M^ :
4 2L
v4
v
4
;
with m A;B;C given by
for the NO case, and
m A =
m2atm
for the IO case.
1
4
s
2
w
2
;
The limits for the lightest neutrino mass being zero are not well de ned for this case,
as it would lead to an in nity in the expressions for
i6=j . Di erently from the other
two cases, only a moderate neutrino mass hierarchy is then allowed. Finally, these
expressions depend on two free parameters, the lightest neutrino mass and the LNV
scale L.
3.2
Rare radiative leptonic decays and conversion in nuclei
In the formalism of the e ective Lagrangian reported in the eq. (3.2), the Beyond SM
(BSM) contributions to the branching ratio of leptonic radiative rare decays are given by
B`i!`j
(`i ! `j )
(`i ! `j i j )
= 384 2e2
v
4
4 4LFV
j ij j2 c(R2L)
c(R1L) 2
;
(3.17)
being e the electric charge, and where the corrections of the Wilson coe cient due to the
electroweak renormalisation from the scale of NP down to the mass scale of the interested
lepton [199, 200] have been neglected, and the limit m`j
m`i has been taken.
The same contributions to the branching ratio for
! e conversion in a generic nucleus
of mass number A read
V (p) +V (n) c(L3L) + V (p) + 1 V (n) c(L4Lu) +
1 V (p) +V (n) c(L4Ld)+
(3.18)
V (p) + V (n) c(L5L)
c(R2L)
c(R1L)
(3.13)
(3.14)
(3.15)
(3.16)
0:146
0:0173
0:189
0:0362
13:07
0:7054
analysis are the following:
where sW
sin W = 0:23, V (p), V (n) and D are dimensionless nucleusdependent overlap
integrals that can be found in table 2 for Aluminium and Gold, that also contains the
numerical values for decay rate of the muon capture, which has been used to normalise the
The experimental bounds on these processes that will be considered in the numerical
where the values in the brackets and the bound on BA!le refer to future expected
Bounds on the LFV scale
The bounds on the LNV scales, determined in eqs. (2.15), (2.25) and (2.30), can be
translated into bounds on the LFV scale when considering the experimental limits in the rare
processes introduced above. Indeed, after substituting the expressions for
, de ned in
eqs. (2.16), (2.24) and (2.29), into the eqs. (3.17) and (3.18), one can rewrite these
expressions extracting the dependence on the NP scales:
8
>
>
>
>
<
>
>
>
>
>>B`i!`j( )
B`i!`j( )
>>:B`i!`j( )
4
2
L
LFV
v L
2LFV
v2
L LFV
Be`i!`j( ) ci ;
Be`i!`j( ) hmlightest; cii ;
4
Be`i!`j( ) hmlightest; cii ;
for the MFC case
for the EFCI case
(3.20)
for the EFCII case
where the square brackets list the free parameters, that is the lightest neutrino mass (only
for the EFCI and EFCII cases) and the e ective Lagrangian parameters ci.
The numerical analysis reveals that the strongest bounds on the
LFV comes from the
data on
! e conversion in gold, although similar results are provided by the data on
leptonic radiative rare decays. The corresponding parameter space is shown in
gure 1,
obtained taking the best t values for the quantities in table 1 (for the EFCI case, the
Dirac CP phase can only acquire two values, 0 and ) and the data from table 2. Although
these plots have been generated for the NO neutrino spectrum, they hold for the IO case
of the spurion backgrounds and by the present experimental bounds on
! e conversion in gold
(in green), BR(
! e ) (in blue), BR( !
account the expected future sensitivity on BR(
) (in red), and BR(
! e ) (in purple). Taking into
! e ) would not restrict further the parameter
space in the case of a negative result: the prospective bound would almost coincide with the bound
from the negative search for
! e conversion in gold, BR(
! e). However, with the planned
signi cant increase (by more than 4 orders of magnitude) of the sensitivity to the relative rate of
! e conversion in aluminium it would be possible to probe considerably larger fraction of the
parameter space of interest: the corresponding bound is drown as the green dashed line. The grey
region are excluded areas from the constraints on the LNV scale, eqs. (2.15), (2.25), and (2.30).
The left, middle and right panels correspond to the MFC, EFCI and EFCII cases, respectively.
The border lines are obtained taking as input data the best t values for the oscillation parameters
listed in table 1 and the nuclear quantities in table 2. The Dirac CP phase for the EFCI plot is
set equal to , while the Majorana are set to 0, in order to minimise the excluded region of the
parameter space. For the EFCI and EFCII cases, a quasidegenerate neutrino mass spectrum with
mlightest = 0:1 eV has been assumed, which also minimised the excluded areas. In all the cases,
the Lagrangian coe cients have been
xed in a democratic way not to favour any speci c operator
contribution: c(L1L) + c(L2L) = 1 = c(L3L) = c(L4Lu) = c(L4Ld) = c(L5L) = c(R2L)
c(R1L).
as well, as no di erence is appreciable. On the other hand, a dependence on the strength
of the splitting between neutrino masses can be found for the EFC scenarios: the plots
reported here illustrate the almost degenerate case, where the lightest neutrino mass is
taken to be O(0:1 eV); stronger hierarchies result in a more constrained parameter space.
Finally, the plot for EFCI refers to `
CP = , but the other case with `
CP = 0 is almost
The upper bound on
L for the MFC case reduce the parameter space, although it
cannot be translated into upper bounds on
LFV: larger
LFV simply further suppresses the
expected values for the branching ratios of the observables considered. Moreover, no lower
bound can be drown: requiring to close the experimental bound for the
small
LFV requires small L, leading at the same time to tune g to small values, in order
to reproduce the correct masses for the light active neutrinos, see eq. (2.12). The same
occurs for EFCI, for L and Y , although, in this case, this can be well justi ed considering
the additional Abelian symmetries appearing in eq. (2.19), as discussed in ref. [65]. When
considering the EFCII case, the lower bound on
L removes a large part of the parameter
! e conversion,
space, but does not translate into a lower bound on
LFV: for example, for
L at its lower
LFV must be larger than 105 GeV in order to satisfy to the present
scale for
L
bounds on BA!ue; however, for larger values of L, LFV can be smaller, down to the TeV
1017 GeV, although in this case a tuning on jYN j is necessary in order to
reproduce correctly the lightness of the active neutrino masses.
The absence of evidence of NP in direct and indirect searches at colliders and
lowenergy experiments suggests that NP leading to LFV should be heavier than a few TeV.
In the optimistic scenario that NP is just behind the corner and waiting to be discovered
in the near future, an indication of the LNV scale could be extracted from the plots in
gure 1. Indeed, if
! e conversion in nuclei is observed, LFV
L
1012 1013 GeV for MFC, L
109 1010 GeV for EFCI, and L
1016 1017 GeV for
EFCII. In the EFC scenarios, the LNV scale is associated to the masses of the RH neutrinos,
that therefore turn out to be much heavier than the energies reachable at present and future
colliders. An exception is the case where additional Abelian factors are considered in the
103 104 GeV will lead to
avour symmetry that allows to separate the LNV scale and the RH neutrino masses [65]:
this opens the possibility of producing sterile neutrinos at colliders and then of studying
their interactions in direct searches.
3.2.2
Ratios of branching ratios
The information encoded in eq. (3.20) are not limited to the scales of LFV and LNV.
Studying the ratios of branching ratios between the di erent processes reveals characteristic
features that may help to disentangle the di erent versions of MLFV. To shorten the
notation,
Rit!!js
Be`t!`s ;
Be`i!`j
(3.21)
will be adopted in the analysis that follows. These observables do not depend on the LFV
and LNV scales, nor on the Lagrangian coe cients. They are sensible to the neutrino
oscillation parameters and, for the EFC cases, to the mass of the lightest active neutrino.
For MFC, they do not even depend on mlightest: although the corresponding plots only
contain points along an horizontal line, they will be reported in the next subsections in
order to facilitate the comparison with the other cases.
The two branching ratios with the best present sensitivities, the one for
version in nuclei and the one for
! e , have the same dependence on
! e
cone and therefore
(c) EFCII
(d) All Cases
the previous plots altogether. Colour codes can be read directly on each plot.
their ratio is not sensitive to the charged lepton and neutrino masses and to the neutrino
mixing. Instead, as pointed out in ref. [76], this ratio may be sensitive to the chirality of the
e ective operators contributing to these observables. The comparison between eqs. (3.17)
and (3.18) shows that only BA!e is sensitive to OL(iL), and thus any deviation from
would be a signal of this set of operators.
In the scatter plots that follow, neutrino oscillation parameters are taken from table 1
as random values inside their 2
error bands. The lightest neutrino mass is taken in the
range mlightest
[0:001; 0:1] eV and the results for the NO and IO spectra are shown with
di erent colours. In these gures, the density of the points should not be interpreted as
related to the likelihood of di erently populated regions of the parameter space.
!
R !e . In the upper left, upper right and lower left panes in gure 2, the results are
reported for the ratio of the branching ratios of the
!
decays for the
MFC, EFCI and EFCII cases, respectively. Figure 2d is a summarising gure where all
the three plots are shown together to facilitate the comparison and to make clearer the
nonoverlapping areas.
As gure 2a shows, R !!e is independent of the lightest neutrino mass. The two
sets of points corresponding to NO and IO spectra almost overlap, making it very hard to
distinguish between the two neutrino mass orderings.
In gure 2b, the dependence on mlightest can be slightly appreciated and the predictions
for two mass orderings do not overlap when the spectrum is hierarchical. In the NO case
there are two branches associated with the two values of C`P: the values associated with the
`
CP = 0branch are very close to those for the IO spectrum and correspond to the positive
sum of the two terms on the righthand side of eq. (3.9); the values associated with the
`
CP =
branch are smaller by about one order of magnitude, which re ects a partial
cancellation between the two terms in the righthand side of eq. (3.9). In the IO case there
is only one branch because the rst term on the righthand side of eq. (3.9) is dominant.
As
gure 2c shows, the points for the two mass orderings overlap in the
quasidegenerate limit down to masses of about 0:05 eV. However, they show di erent pro les
in the hierarchical limit. In the IO case the ratio of branching ratios under discussion is
almost constant with mlightest. In the NO case the ratio R !!e
10 4 at
0:012 eV, while for m 1 < 0:01 eV the ratio is R !!e
can be as small as few
> 1. As discussed in
ref. [65], this can be understood from eqs. (3.5) and (3.15): in the NO case and strong
mass hierarchy, the dominant contribution is proportional to 1=m 1 and therefore R !!e
gets enhanced; while when the spectrum is almost degenerate and in the IO case, the
dominant contribution is suppressed by the sine of the reactor angle and the dependence on
the lightest neutrino mass is negligible.
In gure 2d, where the three cases are shown altogether, it can be seen that all the
cases overlap for the IO spectrum and in the quasidegenerate limit for the NO spectrum,
predicting R !!e
= 0:02
0:07. When the mass spectrum is of NO type and hierarchical,
the ratio spans values from 0:004 to 10. Interestingly, if this ratio is observed to be larger
than 0:1, or smaller than 0:004, then only the EFCII with NO spectrum can explain it.
Notice that, given the current limits on B !e , values smaller than
testable in the future planned experiments searching for
!
.
6
10 4 would be
R !!ee . The ratio R !!ee exhibits features which are very similar to those of the ratio
R !!e . Figures 3a and 3b are very similar to gures 2a and 2b: the pro les of the points are
the same, only the area spanned is di erent, as indeed R !!ee is predicted to be by almost
one order of magnitude larger than R !!e . Similar conclusions, however, apply. Figure 3c,
instead, shows an interesting di erence with respect to its sibling gure 2c: the IO and the
NO points cover almost the same nearly horizontal area both for quasidegenerate masses
and for a hierarchical mass spectrum, the NO region being slightly wider. Only for values
of the lightest neutrino mass between 0:01 eV and 0:02 eV, there could be an enhancement
or a suppression of R !!ee in the EFCII case. This is a distinctive feature that could allow
to disentangle EFCII from the other cases: values of R !!ee larger than 10 or smaller than
0:04 can only be explained by a NO neutrino spectrum in the case of EFCII. Notice that,
given the current limits on B !e , values smaller than 0:006 would be testable in the future
planned experiments searching for
! e .
(c) EFCII
(d) All Cases
the previous plots altogether. Colour codes can be read directly on each plot.
The ratio R !!e is almost indistinguishable form the ratio R !!e
except for
the EFCII case with NO neutrino mass spectrum. For the other cases the conclusions for
R !!e
for R !!e smaller than 0:01 or larger than 0:1 would only be explain by EFCII with NO
are almost the same as the conclusions reached for R !!e . One can see that values
neutrino spectrum.
Summarising, the study of these three ratios can provide relevant information if values
for these ratios are found to be larger than 0:1 (10) for R !!e
and R !!e
(for R !!ee )
or smaller than 0:004 for R !!e , 0:01 for R !!e , and 0:04 for R !!ee : such values can
be explained only in the case of EFCII with NO spectrum. If large values for R !!e
R !!e are found, then this would point to a relatively small value for the lightest neutrino
mass, smaller than 0:008 eV; this should occur consistently with a value for R !!ee between
0:1 and 10. If instead, R !!ee is found to be much larger than 10, this would imply masses
and
for the lightest neutrino between 0:008 eV and 0:04 eV; consistently, R !!e
and R !!e
should remain smaller than 1. Finally, if no signals are seen in all the three ratios and
bounds of 0:004 (0:01) [0:04] or smaller can be obtained for R !!e (R !!e ) [R !!ee ], then
this would be consistent with masses between 0:01 eV and 0:02 eV for the lightest neutrino,
or otherwise MLFV cannot explain this feature. On the other hand, all the three MLFV
versions, for both the mass orderings, can explain values for these ratios inside the regions
(c) EFCII
(d) All Cases
the previous plots altogether. Colour codes can be read directly on each plot.
aforementioned, generally between 0:01 and 0:1: this case would be the less favourable for
distinguishing the di erent setups.
These results are generically in agreement with previous analyses performed in
refs. [62, 63, 65, 76] and the di erences are due to the update input data used here.
BA
As shown in eq. (3.22), the ratio of the two branching ratios with the best present
sensitivities is independent from
and can be used to obtain information about the
chirality of the operators contributing to the
! e conversion process. On the other hand, if
the observation (or nonobservation) of the leptonic radiative rare decays allows to identify
the MLFV realisation from
gures 2, 3 and 4, the branching ratio of the
! e
conversion in nuclei could provide the missing information necessary to
x the LFV scale. As
an example, one can assume that an upper bound on R !!e
of about 0:004 has been
set, that could be explained by EFCII with a NO neutrino spectrum and a mass of the
lightest neutrino of about 0:014 eV. The upper bound on BA!ue implies the upper bound
v2=( L LFV) < 5:7
10 17. By
xing the LNV scale to its lower bound, one
nds that
these observables can provide information on the LFV scale that should be larger than
about 2
106 GeV. The future expected sensitivity on BAl
!e is better than the presently
achieved one by four orders of magnitude. A negative results of the planned future searches
for
! e conversion would imply a bound on the LFV scale of about 107 GeV.
b ! s anomalies
The e ective Lagrangian in eq. (3.2) contains the operators which provide the most relevant
contributions to the b ! s anomalies under discussion:10 they are OL(3L) and OL(5L), which
contribute at tree level to the Wilson coe cients C9 and C10 de ned in eq. (1.1), satisfying
to C10 =
C9.
Focussing on the avour structure of OL(3L) and OL(5L), the two operators are invariant
under the MFV
avour symmetry GQ
GL, but can only describe avour conserving
observables which predict universality conservation in both the quark and lepton sectors. In
order to describe a process with quark avour change, it is then necessary to insert powers
of the quark Yukawa spurion Yu. The dominant contributions would arise contracting the
avour indices of the quark bilinear with YuYuy: once the spurions acquire their background
values, the b ! s transitions are weighted by the VtbVts factor appearing in eq. (1.1). Notice
that, as (Yu)33 = yt
1, an additional insertion of YuYuy is not negligible and modi es the
dominant contributions by (1 + yt2) factors. Further insertions of YuYuy turn out to be
unphysical, as they can be written as combinations of the linear and quadratic terms through
the CayleyHamilton theorem. The complete spurion insertions in OLL
ten as 1YuYuy + 2(YuYuy)2, with 1;2 arbitrary coe cients, re ecting the independence of
each insertion: the net contribution to the operator is then given by VtbVts( 1yt2 + 2yt4).
(3;5) can then be
writThe anomalies in the angular observable P50 of B ! K
+
, in the ratios RK and
are linked to the possible violation of
RK , and in the Branching Ratio of Bs !
leptonic universality. NP contributions leading to these e ects can be described in terms of
insertions of spurion combinations transforming under 8 of SU(3)`L . The simplest structure
is YeYey that, in the basis de ned in eq. (2.12), is diagonal and therefore cannot lead
to lepton
avour changing transitions. The phenomenological analysis associated to the
insertion of this spurionic combination has been performed in ref. [211], where the focus was
in understanding the consequences of having a setup where lepton universality is violated
but lepton avour is conserved. In ref. [211], the Abelian factors in eq. (2.8) are considered
as active factors of the avour symmetry and this leads to background values for Ye, whose
largest eigenvalue is of order 1. It should be noticed that strong constraints on this setup
arise when considering radiative electroweak corrections as discussed in refs. [212, 213].
Focussing only on the nonAbelian factors, as in the tradicional MLFV, the largest
entry of Ye is of the order of 0:01, as can be seen from eq. (2.12). In this scenario, the
insertion of Ye is subdominant with respect to the insertion of the neutrino spurions: the
most relevant are gy g in the MFC, Y Y
y in the EFCI and YN YN in the EFCII. Once
y
the spurions acquire background values, these contributions reduce to the
characteristic
of each case. Similarly to what discussed above for Yu, if the largest eigenvalue of
is of order 1, then additional insertions of the neutrino spurions need to be taken into
consideration. The speci c contribution depends on the model considered and only a
10The complete e ective Lagrangian that describes e ects in B physics can be found in ref. [209]. In
particular, another operator, with respect to the reduced list in eq. (3.4), would contribute at tree level to
C9, eR
qL: this contribution is however negligible for the observables discussed here [210, 211],
and then this operator is not considered in the present discussion.
n can be generically written, where n are arbitrary Lagrangian
coe cients, and where the sum is stopped at n = 2 due to the CayleyHamilton theorem.
In ref. [154] the EFCI context has been considered and several processes have been
studied, discussing the viability of this version of MLFV to consistently describe the b ! s
anomalies.
The aim of this section is to critically revisit the analysis of ref. [154], and to investigate
the other two versions of MLFV. As already mentioned, EFCI will be disfavoured if the
Dirac CP violation in the leptonic sector is con rmed, and therefore the viability of MFC
and EFCII to describe the b ! s anomalies, consistently with the other (un)observed
avour processes in the B sector, becomes an interesting issue. Moreover, the results
obtained in the previous section will be explicitly considered.
B semileptonic decays
In order to facilitate the comparison with ref. [154] similar assumptions will be taken. First
of all, setting C1S0M =
CSM and considering that the contributions from OLL
9
C10 =
C9, one can consider a single Wilson coe cient in eq. (1.1): for de niteness, C9
will be retain in what follows. A second relevant assumption is on the matching between
the e ective operators of the highenergy Lagrangian de ned at
LFV, eq. (3.4), and the
lowenergy phenomenological description in eq. (1.1): only the tree level relations will be
considered in the following, while e ects from loopcontributions and from the electroweak
running will be neglected. The latter has been recently shown in refs. [212, 213] to lead to
(3;5) satisfy to
a rich phenomenology, especially in EWPO and
sector.
Considering explicitly the contributions from OL(3L;5), and specifying the avour indexes,
one can write
C9;``0 =
em
v
2
2
LFV
c(L3L);``0 + c(L5L);``0 ;
where c(i)
neutrino spurion background:11
LL;``0 can be written in a notation that makes explicit the dependence on the
c(LiL);``0 =
1(i)yt2 + 2(i)yt4
0 ``0 + 1(i)
(i)
``0 + 2(i)
``0 :
2(i), etc. should be at least comparable with 0(i). Consequently, this requires
In order to explain lepton universality violation, the contributions proportional to 1(i),
this allows to
x the scale of LNV: indeed, the bounds in eqs. (2.15), (2.25) and (2.30)
become equalities,
(
L = 6
L = 6
11In ref. [154] a slightly di erent notation has been adopted, where
with
v2
c(LiL);``0 = em 2LFV h~0(i) ``0 + ~1(i) ``0 + ~2(i) ``0 i ;
~j(i) = p
2 emGF 2LFV
( 1(i)yt2 + 2(i)yt4) i(i) :
(4.1)
(4.4)
(4.5)
(4.2)
(4.3)
! e
(4.6)
(4.7)
The bounds from LFV purely leptonic processes discussed in the previous section allows
to translate this result into speci c values for the LFV scale: from the bounds on
conversion in nuclei, gure 1, one obtains that
8
<
>
>> LFV = 4:4
LFV = 2
>: LFV = 105 GeV ;
105 GeV ;
105 GeV ;
With these results at hand, the order of magnitude for C9 turns out to be
estimating only the prefactors appearing in eq. (4.1). These values should now be
compared with the ones in eq. (1.2), necessary to explain the anomalies in b ! s decays: the
version of MLFV that most contributes to the C9 Wilson coe cient is EFCII, but its
contributions are two order of magnitudes too small to explain the B anomalies. It would be
only by accident that the parameters of order 1 in eq. (4.4) combine together to compensate
such suppression, but this would be an extremely tuned situation.
The only conclusion that can be deduced from this analysis is that all the three versions
of MLFV cannot explain deviations from the SM predictions in the Wilson coe cient C9
larger than a few per mil, once taking into consideration the bounds from leptonic radiative
decays and conversion of muons in nuclei, contrary to what presented in previous literature.
If the anomalies in the B sector will be con rmed, then it will be necessary to extend
the MLFV context. Attempts in this directions have already appeared in the literature,
although not motivated by the search for an explanation of the b ! s decay anomalies.
The avour symmetry of the M(L)FV is a continuous global symmetry and therefore, once
promoting the spurions to dynamical elds, its spontaneous breaking leads to the arising
of Goldstone bosons. Although it would be possible to provide masses for these new states,
this would require an explicit breaking of the avour symmetry. An alternative is to gauge
the symmetry [79{83, 88]: the wouldbeGoldstone bosons would be eaten by avour gauge
bosons that enrich the spectrum. In recent papers [214, 215], a speci c gauge boson arising
avour symmetry has the speci c couplings to explain the b ! s
from the chosen gauged
anomalies here mentioned.
5
Conclusions
The MFV is a framework to describe fermion masses and mixings and to provide at the
same time a sort of avour protection from beyond the Standard Model contributions to
avour processes. The lack of knowledge of the neutrino mass origin re ects in a larger
freedom when implementing the MFV ansatz in the lepton sector: three distinct versions
of the MLFV have been proposed in the literature.
In the present paper, an update of the phenomenological analyses on these setups
is presented considering the most recent t on the neutrino oscillation data. The recent
indication of CP violation in the leptonic sector, if con rmed, will disfavour the very
popular MLFV version [62] called here EFCI, where righthanded neutrinos are assumed to
be degenerate at tree level and the avour symmetry is SU(3)`L
The study of the predictions within these frameworks for avour changing processes has
been presented, focussing on leptonic radiative rare decays and muon conversion in nuclei,
which provide the stringent bounds. A strategy to disentangle between the di erent MLFV
possibilities has been described: in particular, the next future experiments searching for
! e conversion in aluminium could have the power to pinpoint the scenario
described here as EFCII [65], characterised by the avour symmetry SU(3)`L+NR SU(3)eR,
if the neutrino mass spectrum is normal ordered.
An interesting question is whether the present anomalies in the semileptonic Bmeson
decays can
nd an explanation within the M(L)FV context. Contrary to what claimed in
the literature, such an explanation would require a scale of New Physics that turns out
to be excluded once considering purely leptonic processes, the limits on the rate of muon
conversion in nuclei being the most constraining. These anomalies could
nd a solution
extending/modifying the M(L)FV setup, for example, by gauging the avour symmetry.
Acknowledgments
L.M. thanks the department of Physics and Astronomy of the Universita degli Studi di
Padova for the hospitality during the writing up of this paper and Paride Paradisi for
useful comments on this project and for all the enjoyable discussions during this visit.
D.N.D. thanks the Department of Physics of the University of Virginia for the hospitality
and P.Q. Hung for the exciting discussions and kind helps.
D.N.D. acknowledges partial support by the Vietnam National Foundation for Science
and Technology Development (NAFOSTED) under the grant 103.012014.89, and by the
Vietnam Education Foundation (VEF) for the scholarship to work at the Department
of Physics of the University of Virginia. L.M. and S.T.P. acknowledge partial nancial
support by the European Union's Horizon 2020 research and innovation programme under
the Marie SklodowskaCurie grant agreements No 690575 and No 674896. The work of
L.M. was supported in part also by \Spanish Agencia Estatal de Investigacion" (AEI) and
the EU \Fondo Europeo de Desarrollo Regional" (FEDER) through the project
FPA201678645P, and by the Spanish MINECO through the Centro de excelencia Severo Ochoa
Program under grant SEV20120249 and by the Spanish MINECO through the \Ramon
y Cajal" programme (RYC201517173). The work of S.T.P. was supported in part by the
INFN program on Theoretical Astroparticle Physics (TASP) and by the World Premier
International Research Center Initiative (WPI Initiative), MEXT, Japan.
Open Access.
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.
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