Published for SISSA by

Springer

Received: May 31, 2017
Accepted: July 9, 2017
Published: July 18, 2017

D.N. Dinh,a,b L. Merlo,c S.T. Petcovd,e and R. Vega-ÁlvarezcaMathematical and high energy physics group, Institute of physics,
Vietnam academy of science and technology,
10 Dao Tan, Ba Dinh, Hanoi, Viet Namb
Department of Physics, University of Virginia,
Charlottesville, VA 22904-4714, U.S.A.
c
Departamento de Fı́sica Teórica and Instituto de Fı́sica Teórica, IFT-UAM/CSIC,
Universidad Autónoma de Madrid,
Cantoblanco, 28049, Madrid, Spaind
SISSA and INFN-Sezione di Trieste,
Via Bonomea 265, 34136 Trieste, Italye
Kavli IPMU, University of Tokyo (WPI),
Tokyo, Japan

E-mail: dndinh@iop.vast.ac.vn, luca.merlo@uam.es, petcov@sissa.it,
roberto.vegaa@estudiante.uam.es
Abstract: 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 right-handed neutrinos will be disfavoured, if
this indication is confirmed. The predictions for leptonic radiative rare decays and muon
conversion in nuclei are analysed, identifying strategies to disentangle the different MLFV
scenarios. The claim that the present anomalies in the semi-leptonic B-meson decays canbe explained within the MLFV context is critically re-examined concluding that such an
explanation is not compatible with the present bounds from purely leptonic processes.
Keywords: CP violation, Effective Field Theories, Global Symmetries, Neutrino Physics
ArXiv ePrint: 1705.09284

Open Access, c The Authors.
Article funded by SCOAP3 .

https://doi.org/10.1007/JHEP07(2017)089

JHEP07(2017)089

Revisiting Minimal Lepton Flavour Violation in thelight of leptonic CP violation

Contents
1

2 Minimal (Lepton) Flavour Violation
2.1 The lepton sector

46

3 Phenomenology in the lepton sector
3.1 The LFV effective Lagrangian
3.2 Rare radiative leptonic decays and conversion in nuclei
3.2.1 Bounds on the LFV scale
3.2.2 Ratios of branching ratios

11
12
15
16
18

4 b → s anomalies
4.1 B semi-leptonic decays

23
24

5 Conclusions

25

1

Introduction` in the lepton mixing matrix led
The discovery [1–5] of a non-vanishing reactor angle θ13
to a huge fervour in the flavour community and to a deep catharsis in the model building
approach.
When the value of this angle was still unknown, the closeness to a maximal mixing` was suggesting a maximal oscillation between muonvalue of the atmospheric angle θ23
and tau-neutrinos: in terms of symmetries of the Lagrangian acting on the flavour space, it
could be described by a discrete Abelian Z2 symmetry, which, in turn, implied a vanishingreactor angle. The simplicity and the elegance of this pattern, i.e. one maximal angle andone vanishing one, convinced part of the community that Nature could have made us afavour and that neutrino physics could indeed be described, at least in the atmosphericand reactor sectors, by this texture [6, 7].
An approach followed for such constructions was to write a Lagrangian whose leading` = 0◦ andorder terms described specific textures for the Yukawa matrices, leading to θ13`◦θ23 = 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 diagonalisedby the so-called Tri-Bimaximal mixing matrix [8–10], which predicts, besides a vanishing` = 45◦ , a solar angle satisfying toreactor mixing angle and a maximal atmospheric one θ23
2 `sin θ12 = 1/3, in a very good agreement with the neutrino oscillation data.
Pioneer models can be found in refs. [11–15], where the discrete non-Abelian group A4
was taken as a flavour symmetry of the lepton sector. Several distinct proposals followed,–1–

JHEP07(2017)089

1 Introduction

1

Bimaximal mixing can be obtained by assuming the existence of an approximate U(1) symmetry corresponding to the conservation of the non-standard lepton charge L0 = Le − Lµ − Lτ and additional discreteµ − τ symmetry [20].

–2–

JHEP07(2017)089

i) attempting to achieve the Tri-Bimaximal pattern, but with other flavour 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] andthe Trimaximal mixing [25]; iii) analysing the possible perturbations or modifications to Bimaximal mixing, Tri-Bimaximal mixing etc., arising from the charged lepton sector [26–29],vi) implementing the so-called quark-lepton complementarity [30, 31] which suggests thatthe 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 thesereviews [35–40].
` and the improved sensitivity on the other
After the discovery of a non-vanishing θ13` best fit is not 45◦ (the most recent global fits
two mixing angles, which pointed out that θ23
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: introductionof additional parameters in preexisting minimal models, see for example refs. [44, 45]; implementation of features that allow sub-leading corrections only in specific directions in
the flavour space [46–49]; search for alternative flavour symmetries or mixing patterns that
` 6= 0◦ and θ ` 6= 45◦ [50, 51]. One can fairly saylead already in first approximation to θ1323that the latest neutrino data can still be described in the context of discrete symmetries,
but at the prize of fine-tunings and/or less minimal mechanisms.
Alternative approaches to discrete flavour 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 so-called Minimal Flavour Violation (MFV) [60, 61], and its leptonic
versions [62–65], dubbed MLFV. The latter is a setup where the flavour symmetry is
identified with the symmetry of the fermionic kinetic terms, or in other words, the symmetryof the SM Lagrangian in the limit of vanishing Yukawa couplings: it is given by products ofU(3) factors, one for each fermion spinor of the considered spectrum. Fermion masses andmixings are then described once the symmetry is broken. This approach allows to relate
any source of flavour and CP violation in the SM and beyond to the Yukawa couplings,
such that any flavour effect 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 inrefs. [66–75]). It is instead a framework where observed flavour violating observables aredescribed in agreement with data and unobserved flavour violating signals are not expected
to be observed with the current experimental sensitivities, but could be observable in the
future planned experiments with significantly 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` , over CP conservation at more than 2σ’s, depending on the
non-zero Dirac CP phase, δCP

–3–

JHEP07(2017)089

specific neutrino mass ordering. Moreover, the best fit value for the leptonic Jarlskog in` ' −0.033 [42], is numerically much larger in magnitude than its quark sibling,
variant, JCP`
JCP' 3.04 × 10−5 [93], indicating potentially a much larger CP violation in the lepton
sector than in the quark sector.
In the field of discrete flavour models, this indication translated into looking, for the
first 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 flavour symmetry [94–100]; studies on the mixing patterns and their modifications
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`
and δCP
[102–109].
The indication for CP violation in the lepton sector also had an impact on models basedon continuous flavour symmetries. In particular, one very popular version of MLFV [62]strictly requires CP conservation as a working assumption and therefore, if this indicationis confirmed, this setup will be disfavoured.
The first goal of this paper is to update previous studies on MLFV in the light of thelast global fit 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 non-vanishing θ13
and lacking any information about the leptonic CP phase.
The search for an explanation of the heterogeneity of fermion masses and mixings,
the so-called 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 typicallyintroduce new sources of flavour violation. Identifying the mechanism which explains why
the experimentally measured flavour violation is very much consistent with the SM predictions is a crucial aspect in flavour physics. The use of flavour symmetries turned out
to be useful also with this respect: a very well-known 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 semi-leptonic B-meson decays: the angular observable P50 in the B → K ∗ µ+ µ− decaypresents a tension with the SM prediction of 3.7σ [124, 125] and 2σ [126], considering LHCb
and Belle data, respectively; the Branching Ratio of Bs → φµ+ µ− is in tension with the` ≡ BR(B → D (∗) τ ν)(∗)
SM prediction at 3.2σ [127]; the ratio RD∗
exp /BR(B → D `ν)exp ×BR(B → D(∗) `ν)SM /BR(B → D(∗) τ ν)SM with ` = e, µ indicates a 3.9σ violation of τ /`universality [128–132]; the ratio RK ≡ BR(B + → K + µ+ µ− )/BR(B + → K + e+ e− ) is ina 2.6σ tension with the SM prediction [133], indicating lepton universality violation in
the e/µ sector. The latter has been confirmed also by the recent announcement of themeasure of RK ∗ ≡ BR(B 0 → K ∗0 µ+ µ− )/BR(B 0 → K ∗0 e+ e− ) is in a 2.4–2.5σ (2.2–2.4σ)
tension with the SM prediction in the central-q 2 region (low-q 2 region) [134]. Under theassumption that these anomalies are due to NP, and not due to an underestimation of thehadronic effects [135–140] or due to a statistical fluctuation, a global analysis on b → sdata can attempt to identify the properties of the underlying theory. Adopting an effective
description, these results can be translated into constraints of the Wilson coefficients ofthe Hamiltonian describing ∆B = 1 decays: the results of such analysis [141–153] are thatthe anomalies can be explained with a modification of the Wilson coefficients C9 and C10
defined ashihi
4GF e2
eff
∗
µ
H∆B=1
⊃− √
V
V
sγ
P
b
`γ
(C
+
C
γ
)
`
+ h.c.
(1.1)
µ L
9
10 5
tb ts
2 (4π)2e
δC9e = −δC10∈ [+0.56, +1.02]

andµ
δC9µ = −δC10∈ [−0.81, −0.48]

@1σ ,

(1.2)

corresponding to 4.3σ and 4.2σ tension with the SM predictions, respectively.
The hypothetical underlying theory, which manifests itself at low energies with thesefeatures, 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 deficit 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 δC9e
e aforementioned.
and δC10
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 µ → econversion in nuclei. Moreover, the analysis will be extended to the other versions of
MLFV [65].
The structure of the paper can easily be deduced from the table of content: first, insection 2, basic concepts of MFV and MLFV will be recalled, underlying the differences
between the distinct versions of MLFV; then, in section 3, several processes in the lepton
sector will be discussed considering the last global fit 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 differences with respect to previous literature; finally,
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 flavour
and CP violation even beyond the SM, then its flavour protection is guaranteed: the large
majority of observed flavour processes in the quark sector are predicted in agreement with

–4–

JHEP07(2017)089

where V is the CKM matrix, PL = (1 − γ5 )/2 is the usual left-handed (LH) chirality
projector, b and s refer to the bottom and strange quarks, respectively, ` are the charged
leptons, and the pre-factors refer to the traditional normalisation. Writing each of thecoefficients as the sum of the purely SM contribution and the NP one, Ci = CiSM + δCi ,the results of a one-operator-at-a-time analysis [151] suggest lepton universality violationin the e/µ sector quantifiable indata [61, 76, 78–86, 155–160]; unseen flavour 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 flavour symmetry corresponds to theone arising in the limit of vanishing Yukawa couplings. This massless Lagrangian is left
invariant under a tridimensional unitary transformations in the flavour space associated toeach fermion spinor. In the quark sector, it is given by
GQ × U(1)B × U(1)Au × U(1)Adwith

GQ = SU(3)qL × SU(3)uR × SU(3)dR ,(2.1)

qL ∼ (3, 1, 1)GQ

uR ∼ (1, 3, 1)GQ

dR ∼ (1, 1, 3)GQ

q L → U qL q L

u R → U uR u R

dR → U dR dR ,(2.2)

where Ui ∈ SU(3)i are 3 × 3 unitary matrices acting in the flavour space. The quark
Lagrangian is invariant under these transformations, except for the Yukawa interactions:
LQ = −q L Yu H̃uR − q L Yd HdR + h.c. ,(2.3)

where Yi are 3 × 3 matrices in the flavour space, H is the SU(2)L -double Higgs field, andH̃ = iσ2 H ∗ . LQ can be made invariant under GQ promoting the Yukawa matrices tobe spurion fields, i.e. auxiliary non-dynamical fields, denoted by Yu and Yd , with specifictransformation properties under the flavour symmetry:Yu ∼ (3, 3, 1)GQ

Yd ∼ (3, 1, 3)GQ

Yu → UqL Yu Uu†R

Yd → UqL Yd Ud†R .

(2.4)

Once the Yukawa spurions acquire a background value, the flavour symmetry is broken
and in consequence fermions masses and mixings are generated. A useful choice for thesebackground values is to identify them with the SM Yukawa couplings: in a given basis,
Yd is diagonal and describes only down-type quark masses, while Yu contains non-diagonal
entries and accounts for both up-type quark masses and the CKM matrix V :
√
√
2 †
2
(2.5)
hYu i ≡ Yu =V M̂u ,
hYd i ≡ Yd =
M̂d ,
v
v
√
where v = 246 GeV is the Higgs vacuum expectation value (VEV) defined by hH 0 i = v/ 2,
and M̂u,d are the diagonal mass matrices for up- and down-type quarks,M̂u ≡ diag(mu , mc , mt ) ,M̂d ≡ diag(md , ms , mb ) .

–5–

(2.6)

JHEP07(2017)089

where qL refer to the SU(2)L -doublet of quarks, and uR and dR to the SU(2)L -singlets. The
Abelian terms can be identified with the Baryon number, and with two axial rotations, inthe up- and down-quark sectors respectively, which do not distinguish among the distinct
families [165]. On the contrary, the non-Abelian factors rule the interactions among thegenerations and govern the amount of flavour 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

2.1

The lepton sector

The lepton sector is more involved with respect to the quark one, due to the lack ofknowledge on neutrino masses: indeed, while the charged lepton description mimics theone of down-quarks, 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 well-defined approaches, one maintaining
the SM spectrum but relaxing the renormalisability criterium, and the other adding newparticles in a still renormalisable theory.
Minimal Field Content (MFC). Giving up with renormalisability, active neutrino
masses can be described via the so-called Weinberg operator [167], a non-renormalisable
operator of canonical dimension 5 which breaks explicitly Lepton number by two units,
1  c ∗  gν  † 
OW =
` H̃
H̃ `L ,
(2.7)
2 L
ΛL
T

where `cL ≡ C`L , C being the charge conjugation matrix (C −1 γµ C = − γµT ), gν is an adimensional symmetric 3×3 matrix in the flavour space and ΛL is the scale of Lepton Number
Violation (LNV). The flavour symmetry arising from the kinetic terms in this case is
GL × U(1)L × U(1)Ae

with

2

GL = SU(3)`L × SU(3)eR ,(2.8)

The 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.

–6–

JHEP07(2017)089

When considering low-energy flavour processes, they can be described within the effective field theory approach through non-renormalisable operators suppressed by suitable
powers of the scale associated to the messenger of the interaction. These structures could
violate the flavour symmetry GQ , especially if they describe flavour changing observables.
As for the Yukawa Lagrangian, a technical way out to recover flavour 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 flavour data in the quark sector
are well described within the MFV(-like) approach. Indeed, the Yukawa spurions act as
expanding parameters and processes described by effective operators with more insertionsof the spurions obtain stronger suppressions.2
MFV, however, cannot be considered a complete flavour model, as there is not explanation of the origin of quark masses and mixings. There have been attempts to gofrom the effective-spurionic approach to a more fundamental description, promoting the
Yukawa spurions to be dynamical fields, called flavons, acquiring a non-trivial VEV. The
corresponding scalar potentials have been discussed extensively with interesting consequences [66–70]: a conclusive dynamical justification for quark masses and mixing is stilllacking, but the results are encouraging as the potential minima lead, at leading order, tonon-vanishing masses for top and bottom quarks and to no mixing.

where U(1)L is the Lepton number while U(1)Ae is an axial rotation in `L and eR , and thenon-Abelian transformations of the leptons read`L ∼ (3, 1)GLeR ∼ (1, 3)GL

`L → U`L `L

eR → UeR eR .

(2.9)

The part of the Lagrangian describing lepton masses and mixings,
LL = −`L Ye HeR − OW + h.c. ,(2.10)

Ye ∼ (3, 3)GLgν ∼ (6, 1)G(2.11)gν → U`∗ gν U`† .
Lepton masses and the PMNS matrix U arise once Ye and gν acquire a background valueL

Ye → U`L Ye Ue†R

that can be chosen to beL√

hYe i ≡ Ye =

2
M̂` ,
v

h gν i ≡ gν =

L

2ΛL ∗
U M̂ν U † ,v2(2.12)

with M̂`,ν being the diagonal matrices of the charged lepton and active neutrino mass
eigenvalues,M̂` ≡ diag(me , mµ , mτ ) ,M̂ν ≡ diag (mν1 , mν2 , mν3 ) ,(2.13)and U defined as the product of four matrices [93],


α21
α31
`
`
`
`
U = R23 (θ23
) · R13 (θ13, δCP
) · R12 (θ12
) · diag 1, ei 2 , ei 2 ,(2.14)` ) a generic rotation of the angle θ ` in the ij sector, with the addition of thewith Rij (θij
ij`
Dirac CP phase δCPin the reactor sector, and α21,31 the Majorana phases [168].
As discussed for the quark case, Ye and gν act as expanding parameters: operators withmore insertions of these spurions describe processes that receive stronger suppressions. This
perturbative treatment requires, however, that the largest entries in Ye and gν are at mostO(1). The charged lepton Yukawa satisfies to this condition as the largest entry is ∼ mτ /v.
The neutrino spurion gν is instead function of ΛL : requiring that |gν ij | < 1 leads to an
upper bound on the LNV scale, which depends on |(U ∗ M̂ν U † )ij | that is a function of thetype of neutrino mass spectrum (NO or IO), of the value of the lightest neutrino mass andof the values of the Majorana and Dirac CP violation phases. The lowest upper bound isgiven approximately by:

ΛL '

g
v2
p ν
. 6 × 1014 GeV .
2 ∆m2atm(2.15)

It will be useful for the phenomenological discussion in the next sections to rememberthat the spurion combination g†ν gν transforms as (8, 1)GL and to introduce the quantity∆ ≡ gν† gν =4Λ2L
U M̂ν2 U † .
v4–7–(2.16)JHEP07(2017)089

is not invariant under GL , but this can be cured by promoting Ye and gν to be spurion
fields, Ye and gν , transforming as

Extended Field Content (EFC). Enlarging the SM spectrum by the addition of three
RH neutrinos NR leads to the so-called type I Seesaw context [169–173], described by thefollowing Lagrangian:1
LL–SS = −`L Ye HeR − `L Yν H̃NR − µL NRc YN NR + h.c. ,2(2.17)Y −1gν†= Yν N YνT .
ΛLµL(2.18)

The fermionic kinetic terms of the SM extended with 3 RH neutrinos manifest thefollowing flavour symmetry:
GL × U(1)L × U(1)Ae × U(1)ANwith

GL = SU(3)`L × SU(3)eR × SU(3)NR ,(2.19)

under which leptons transform as`L ∼ (3, 1, 1)GLeR ∼ (1, 3, 1)GLNR ∼ (1, 1, 3)GL

`L → U`L `L

eR → UeR eR

N R → U NR N R ,(2.20)

and where U(1)AN is an axial transformation associated to NR and SU(3)NR is a newrotation that mixes the three RH neutrinos. The Lagrangian in eq. (2.17) breaks explicitly
GL defined in eq. (2.19), but the invariance can be technically restored promoting YE , Yνand YN to be spurions fields, YE , Yν and YN , transforming as
Ye ∼ (3, 3, 1)GLYν ∼ (3, 1, 3)GLYN ∼ (1, 1, 6)GLYe → U`L Ye Ue†R†
Y ν → U `L Y ν U N
R†∗
YN → U N
YN U N.
R
R(2.21)

Lepton masses and mixing are then described when these spurion fields acquire the following
background values:
√
2
2µL
−1
hYe i ≡ Ye =M̂` ,hYν ihYN
ihYνT i ≡ Yν YN−1 YνT = 2 U M̂ν U T .
(2.22)v
v
Differently from the quark sector and the MFC lepton case, it is not possible to identify
a unique choice for hYν i and hYN i, as only the specific combination in eq. (2.22) can beassociated to the neutrino mass eigenvalues and the PMNS matrix entries. This is a relevant
aspect as it nullifies the MLFV flavour protection. Indeed, the basic building blocks forseveral processes, such as radiative leptonic decays or leptonic conversions, are fermionici
ii
bilinears of the type `L Γ`jL , `L Γ`cLj , `L ΓejR and eiR ΓejR , with Γ standing for combinationof Dirac γ matrices and/or Pauli σ matrices. In the unbroken phase, these terms are–8–

JHEP07(2017)089

where Ye , Yν and YN are adimensional 3 × 3 matrices in the flavour space, while µL stands
for the scale of Lepton number violation, broken by two units by the last term on the rightof this equation. Assuming a hierarchy between µL and v, µL  v, it is then possible toeasily block-diagonalise the full 6 × 6 neutrino mass matrix, and obtain the induced masses
for the light active neutrinos: in terms of the parameter gν appearing in the Weinberg
operator in eq. (2.7), they are given byinvariant under the flavour symmetry contracting the flavour 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 Yν and YN t hat
appears in eq. (2.22): a few examples are2(8, 1, 1)GL†Yν Yν† , Ye Ye† , Yν YN
YN Yν† ,(6, 1, 1)GL

† T
†
† T
† T ∗ TYν YN
Yν , Y ν Y N
Y N YNYν , Yν YN
Yν Yν Yν , . . .

(3, 3, 1)GL†
Ye , Yν Yν† Ye , Ye Ye† Ye , Yν YN
YN Yν† Ye , . . .

(1, 8, 1)GL†Ye† Ye , Ye† Yν Yν† Ye , Ye† Yν YN
YN Yν† Ye , . . .

Yν Yν†, ...
(2.23)I): GL → SU(3)`L × SU(3)eR × SO(3)NR × CP [62, 63].
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 thenan orthogonal matrix. The additional assumption of no CP violation in the lepton
sector is meant to force Ye and Yν to be real.3 With this simplifications, all flavourchanging effects involving leptons can be written in terms of Yν YνT and Ye , as can beeasily deduced from eq. (2.23). In this case, eq. (2.22) simplifies toYν YνT =2µL
U M̂ν U T ≡ ∆ ,v2(2.24)

eventually redefining µL by reabsorbing the norm of YN , and therefore any flavourchanging process can be described in terms of lepton masses and mixings. The
last equivalence in the previous equation is a definition that will be useful in thephenomenological analysis.
As for the MFC case, requiring that the spurions respect the perturbativity regime
leads to an upper bound on the LNV scale:µL 'v 2 Yν YνTp. 6 × 1014 GeV ,2 ∆m2atm(2.25)

numerically the same as the one in eq. (2.15).
3

Strictly speaking, the condition of CP conservation in the leptonic sector forces the Dirac CP phase to`be equal to δCP= {0, π} and the Majorana CP phases to be α21,31 = {0, π, 2π}. However, Yν is real only
if α21,31 = {0, 2π}, and therefore α21,31 = π needs to be disregarded in order to guarantee predictivity.
The CP conservation condition assumed in this context is then stronger than the strict definition.

–9–

JHEP07(2017)089

In consequence, one concludes that it is not possible to express any flavour 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 ofneutrino masses and mixing would be encoded into only one spurion background among Yν
and YN , being the other proportional to the identity matrix. Technically, this corresponds
to break GL following two natural criteria.

II): GL → SU(3)`L +NR × SU(3)eR [65].
Assuming that the three RH neutrinos transform as a triplet under the same symmetry group of the lepton doublets,`L , NR ∼ (3, 1)GLeR ∼ (1, 3)GL ,(2.26)

then the Schur’s Lemma guarantees that Yν transforms as a singlet of the symmetrygroup and then Yν is a unitary matrix [174, 175], which can always be rotated to theidentity 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
YN ∼ (6, 1)GL .

(2.27)The background value of YN would eventually encode the norm of Yν , in order toconsistently take Yν = 1. In this basis, neutrino masses and the lepton mixing are
encoded uniquely into YN ,v 2 ∗ −1 †YN =U M̂ν U .
(2.28)
2µL
Moreover, all the spurion combinations in eq. (2.23) can be written only in terms of
YN and Ye and therefore any flavour 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∆ ≡ YN† YN =v4
U M̂ν−2 U † .
4µ2L(2.29)

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:µL 'v 2 YN−1p& 6 × 1014 GeV .
2 ∆m2atm(2.30)

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 adynamical explanation for the flavour 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 Yν to be dynamical fields, 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 different
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.

– 10 –JHEP07(2017)089

Ye ∼ (3, 3)GL`
sin2 θ12`
sin2 θ23`
sin2 θ13` /◦δCP∆m2sol /10−5 eV2∆m2atm /10−3 eV2Normal Ordering

Inverted Ordering

0.306 ± 0.012

0.306 ± 0.012

0.441+0.027−0.0210.587+0.020−0.0240.02166 ± 0.00075

0.02179 ± 0.00076

261+51−59

277+40−46

7.50+0.19−0.17

7.50+0.19−0.17

2.524+0.039−0.0402.514+0.0.38−0.041As anticipated in section 1, the recent indication for a relatively large leptonic CP
violation, if confirmed, would disfavour EFCI, due to the required reality of Yν . However, inthe present discussion and in the analysis that follows, EFCI will not be discarded yet, as theassumption of CP conservation is a distinctive feature of this low-energy description of the
lepton sector, but could be avoided in more fundamental ones. Indeed, a model constructedupon the gauged lepton flavour symmetry SU(3)`L × SU(3)eR × SO(3)NR , without any
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 fit value, this gauged flavour
model presents several phenomenological results similar to the ones of EFCI discussed inrefs. [62, 63]. This motivates to consider EFCI as a valid context to describe lepton flavour
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 specifically leptonic radiative rare decays and µ → e conversion
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 thereactor angle and the recent indication of non-vanishing CP phase in the leptonic sector
will be considered.
The input data that will be used in what follows are the PDG values for the charged
lepton masses [93]me = 0.51 MeV ,mµ = 105.66 MeV ,mτ = 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 fit 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– 11 –JHEP07(2017)089

Table 1. Three-flavour oscillation parameters from the global fit in ref. [42]. The results inthe second and third columns refer to the Normal and the Inverted Orderings, respectively. The
notation has been chosen such that ∆m2sol ≡ m2ν2 − m2ν1 , and ∆m2atm ≡ m2ν3 − m2ν1 for NO and∆m2atm ≡ m2ν2 − m2ν3 for IO. The errors reported correspond to the 1σ uncertainties.

of the lightest neutrino mass and for both the Normal Ordering (NO) and the Inverted
Ordering (IO). The measured parameters are taken considering their 2σ error bands:4 thisis to underly the impact of the raising indication for a leptonic CP violation.
3.1

The LFV effective Lagrangian

where the Lagrangian parameters are real coefficients7 of order 1 and the operators havethe form:8(1)(2)
OLL = i`γ µ `L H † Dµ H ,OLL = i`γ µ σ a `L H † σ a Dµ H ,OLL = `γ µ `L qγµ qL ,(3)

OLL = `γ µ `L dγµ dR ,

(4d)

(4u)

OLL = `γ µ σ a `L qγµ σ a qL ,(1)a
ORL = g`Hσ µν σ a eR Wµν.

(5)

OLL = `γ µ `L uγµ uR ,

(3.4)

(2)

ORL = g 0 `Hσ µν eR Bµν ,4

EW running effects [176–179] are negligible in the analysis presented here.
The effective Lagrangian reported here corresponds to the linearly realised EWSB. An alternative would be to considered a non-linear realisation and the corresponding effective Lagrangian dubbed
HEFT [181–186]. In this context, however, a much larger number of operators should be taken into consideration and a slightly different phenomenology is expected [187–196]. The focus in this paper is on thelinear EWSB realisation and therefore the HEFT Lagrangian will not be considered in what follows.
6
A few other operators are usually considered in the effective Lagrangian associated to these LFV
observables, but the corresponding effects are negligible. See ref. [62] for further details.
7
The reality of the Lagrangian parameters guarantees that no sources of CP violation are introduced
beyond the SM. A justification of this approach can be found in ref. [78].
8
The notation chosen for the effective 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 thetwo bases is given by:
5

OLL → Qϕ` ,

(1)

(1)

OLL → Qϕ` ,

(2)

(3)

OLL → Q`q ,OLL → Q`d ,(4u)

OLL → Q`q ,

(3)

(5)

(3)

ORL → QeB ,(1)– 12 –(1)(4d)

OLL → Q`d ,(2)

ORL → QeW .

(3.3)

JHEP07(2017)089

The rates of charged LFV processes, i.e. µ → e + γ, µ → 3e, and µ → e conversion in nuclei
among others, are predicted to be unobservably small in the minimal extension of the SMwith light massive Dirac neutrinos, in which the total lepton charge is conserved [180]. As aconsequence, 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 andµ → e conversion in nuclei. Other leptonic observables which are typically very sensible to
NP are ` → `0 `0 `00 decays, and especially the µ → 3e decay, given the significant increaseof 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 ΛLFV responsible for these observables characterised by a much lower typical energy, one can adopt the description in termsof an effective Lagrangian:5 the relevant terms are then given by6


5
2
X
X
1
1 
(i) (i)(j) (j)
eff
LLFV
= 2
cLL OLL + 2
cRL ORL + h.c. ,(3.2)
ΛLFV i=1
ΛLFV j=1(i)1. Minimal Field Content GL = SU(3)`L × SU(3)eR . Expliciting eq. (2.16), the offdiagonal entries of ∆ can be written asi
4Λ2L h
iδ
2
2
s
c
c
c
(m
−
m
)
+
s
s
c
e
m
−
s
m
−
c
m
,
12
12
23
13
ν
ν
23
13
13
ν
ν
ν
12
12
B
A
C
B
A
v4
i
4Λ2 h
∆τ e = 4L −s12 c12 s23 c13 (mνB − mνA ) + c23 s13 c13 eiδ mνC − s212 mνB − c212 mνA ,v

4Λ2L n∆τ µ = 4 s23 c23 c213 mνC + (s212 s213 − c212 )mνB + (c212 s213 − s212 )mνA +
v


o
∆µe =

+s12 c12 s13 s223 e−iδ − c223 eiδ (mνB − mνA )

,

(3.5)

where, for brevity of notation, sij and cij stand for the sine and cosine of the leptonic` , δ stands for the Dirac CP phase δ ` , and a generic notation for
mixing angles θijCPM̂ν has been adopted in the definition of ∆:M̂ν2 ≡ diag (mνA , mνB , mνC ) .

(3.6)

The three parameters mνA,B,C depend on the neutrino mass ordering: for the NO casemνA = 0 ,mνB = ∆m2sol ,9mνC = ∆m2atm ,(3.7)

The expression for ∆ in the IO case may differ from what reported in ref. [62], due to a different
definition taken for the atmospheric mass squared difference.

– 13 –JHEP07(2017)089

The OLL structures are invariant under the flavour symmetries without the necessityof introducing any spurion field, but they can only contribute to flavour conserving observables. The LFV processes aforementioned can only be described by the insertion of
specific spurion combinations transforming as 8 under SU(3)`L , whose flavour indices arei(i)contracted with those of the lepton bilinear `L Γ`jL in OLL , Γ being a suitable combinationof Dirac and/or Pauli matrices. The specific spurion combinations depend on the consid†
ered model: some examples are g†ν gν in MFC, Yν Yν† in EFCI and Yν YN
YN Yν† in EFCII.
Interestingly, once the spurions acquire their background values, these combinations reduceto the expressions for ∆ in eqs. (2.16), (2.24) and (2.29), respectively.
(i)
The ORL operators, instead, are not invariant under the flavour 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 , whosebackground value, however, is diagonal. Requiring as well that these structures describe
LFV processes, it is necessary to insert more elaborated combinations: some examples†
are g†ν gν Ye in MFC, Yν Yν† Ye in EFCI and YN
YN Ye in EFCII. Once the spurions acquire
background values, these combinations reduce to ∆Ye , with the specific 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

and for the IO casemνA = ∆m2atm − ∆m2sol ,mνB = ∆m2atm ,m νC = 0 .

(3.8)

Notice that there is no dependence on the lightest neutrino mass in these expressions.
This has an interesting consequence because ∆i6=j are completely fixed, apart for thecommon scale ΛL .

where a generic notation — different from the one in the MFC case — for M̂ν hasbeen adopted:M̂ν ≡ diag (mνA , mνB , mνC ) .
(3.10)The three parameters mνA,B,C are now defined byqqmνA = mν1 , mνB = eiα21 ∆m2sol +m2ν1 , mνC = eiα31 ∆m2atm +m2ν1 ,for the NO case, mν1 < mν2 < mν3 , and byqqmνA = ∆m2atm −∆m2sol +m2ν3 , mνB = eiα21 ∆m2atm +m2ν3 ,(3.11)mνC = eiα31 mν3 ,(3.12)

for the IO case, mν3 < mν1 < mν2 .
The hypothesis of CP conservations fixes the Dirac and Majorana CP phases to beδ = {0, π} and α21,31 = 0 in these expressions. Indeed, while ∆ij would be real evenfor α21,31 = π and therefore no CPV process would be described with ∆ insertions,Yν would be complex and then it would not be possible to express the spurions insertions in eq. (2.23) in terms of low-energy parameters, losing the predictivity powerof MLFV.
In the strong hierarchical limit, mν1  mν2 < mν3 in the NO case and mν3  mν1 < mν2
in the IO one, and setting the lightest neutrino mass to zero, the expressions formνA,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 ∆i6=j get simplified. Also in thiscase, only one parameter remains free, that is the LNV scale µL .
When the neutrino mass hierarchy is milder or the eigenvalues are almost degenerate, the lightest neutrino mass cannot be neglected and represents a second freeparameters of ∆i6=j , besides µL .

– 14 –JHEP07(2017)089

2. Extended Field Content I) GL = SU(3)`L ×SU(3)eR ×SO(3)NR ×CP . From eqs. (2.24),
one gets the following explicit expressions for the off-diagonal entries of ∆:

i
2µL h
∆µe = 2 s12 c12 c23 c13 (mνB −mνA )+s23 s13 c13 eiδ e−2iδ mνC −s212 mνB −c212 mνA ,
v

i
2µL h
∆τ e = 2 −s12 c12 s23 c13 (mνB −mνA )+c23 s13 c13 eiδ e−2iδ mνC −s212 mνB −c212 mνA ,
v

2µL n
∆τ µ = 2 s23 c23 c213 mνC −c212 mνB −s212 mνA +v
o+ s12 c12 s13 eiδ (s223 −c223 )(mνB −mνA )+s23 c23 s213 e2iδ s212 mνB +c212 mνA , (3.9)

3. Extended Field Content II) GL = SU(3)`L +NR × SU(3)eR . The expressions for the offdiagonal entries of ∆ that follow from eqs. (2.29) can be obtained from the expressions
in eq. (3.5), by substituting
4Λ2L
v4
→
(3.13)v4
4µ2L
and taking the following notation for M̂ν :M̂ν−2 ≡ diag (mνA , mνB , mνC ) ,(3.14)

mνA =

1
,
m2ν1

m νB =

1,∆m2sol + m2ν1

m νC =

1,∆m2atm + m2ν1

(3.15)

for the NO case, and
mνA =

1

,
∆m2atm − ∆m2sol + m2ν3

m νB =

1,∆m2atm + m2ν3

m νC =

1,mν 3(3.16)

for the IO case.
The limits for the lightest neutrino mass being zero are not well defined for this case,as it would lead to an infinity in the expressions for ∆i6=j . Differently from the othertwo 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 µL .
3.2

Rare radiative leptonic decays and conversion in nuclei

In the formalism of the effective Lagrangian reported in the eq. (3.2), the Beyond SM(BSM) contributions to the branching ratio of leptonic radiative rare decays are given byB`i →`j γ ≡Γ(`i → `j γ)
v4
(2)
(1) 2
= 384π 2 e2 4|∆ij |2 cRL − cRL ,Γ(`i → `j νi ν j )4ΛLFV(3.17)

being e the electric charge, and where the corrections of the Wilson coefficient due to theelectroweak 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 µ → e conversion in a generic nucleus
of mass number A read

32G2F m5µ v 4
1
1 (n)  (1)
(2)
2
A
2
(p)
Bµ→e
=
|∆
|
−
s
V
−
V
c
+
c
µe
w
LL
LL +
4
4
4Λ4LFV
ΓA
capt





3  (p)
1 (p)
(3)
(4u)
(4d)
(n)
(p) 1 (n)(n)(3.18)+V +V
cLL + V + V
cLL +
V +V
cLL +
2
2
2

 2
1
eDA  (2)
(5)
(1) ∗
(p)
(n)
+ −V + VcLL −cRL − cRL,
2
4

– 15 –JHEP07(2017)089

with mνA,B,C given byV (p)V (p)D

Γcapt (106 s−1 )Au

0.0974

0.146

0.189

13.07

Al

0.01610.01730.03620.7054

Table 2. Reference values for nuclear overlap integrals and capture rates from ref. [201].

Bµ→eγ < 5.7 × 10−13 [202] (6 × 10−14 [203]) ,Bτ →eγ < 5.2 × 10−8 [204] (10−9 ÷ 10−10 [205]) ,Bτ →µγ < 2.5 × 10−7 [204] (10−8 ÷ 10−9 [205]) ,(3.19)Au
Bµ→e< 7 × 10−13 [206] ,AlBµ→e< 6 × 10−17 [207, 208] ,Al
where the values in the brackets and the bound on Bµ→erefer to future expected
sensitivities.

3.2.1

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 ∆, defined ineqs. (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:


4
 

ΛL
e` →` (γ) ci ,B≡Bfor the MFC case

`
→`
(γ)
i
j
i
j
ΛLFV





2
h
i
L
e` →` (γ) mlightest(3.20)
B`i →`j (γ) ≡ ΛvµB,c,
for the EFCI case
ν
i
2
i
j

LFV




4
h
i


v2
e` →` (γ) mlightest
B` →` (γ) ≡B,cfor the EFCII case
ν
i ,
i
j
i
j
µL ΛLFV
where the square brackets list the free parameters, that is the lightest neutrino mass (only
for the EFCI and EFCII cases) and the effective Lagrangian parameters ci .
The numerical analysis reveals that the strongest bounds on the ΛLFV comes from thedata on µ → e conversion in gold, although similar results are provided by the data on
leptonic radiative rare decays. The corresponding parameter space is shown in figure 1,
obtained taking the best fit values for the quantities in table 1 (for the EFCI case, theDirac CP phase can only acquire two values, 0 and π) and the data from table 2. Althoughthese plots have been generated for the NO neutrino spectrum, they hold for the IO case– 16 –JHEP07(2017)089

where sW ≡ sin θW = 0.23, V (p) , V (n) and D are dimensionless nucleus-dependent overlap
integrals that can be found in table 2 for Aluminium and Gold, that also contains thenumerical values for decay rate of the muon capture, which has been used to normalise thedecay rate for the µ → e conversion.
The experimental bounds on these processes that will be considered in the numericalanalysis are the following:as well, as no difference is appreciable. On the other hand, a dependence on the strengthof 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 istaken to be O(0.1 eV); stronger hierarchies result in a more constrained parameter space.
``
Finally, the plot for EFCI refers to δCP= π, but the other case with δCP= 0 is almost
indistinguishable.

– 17 –JHEP07(2017)089

Figure 1. Parameter space for the LFV and LNV scales constrained by requiring perturbativityof the spurion backgrounds and by the present experimental bounds on µ → e conversion in gold
(in green), BR(µ → eγ) (in blue), BR(τ → µγ) (in red), and BR(τ → eγ) (in purple). Taking into
account the expected future sensitivity on BR(µ → eγ) would not restrict further the parameter
space in the case of a negative result: the prospective bound would almost coincide with the boundfrom the negative search for µ → e conversion in gold, BR(µ → e). However, with the planned
significant increase (by more than 4 orders of magnitude) of the sensitivity to the relative rate ofµ → e conversion in aluminium it would be possible to probe considerably larger fraction of theparameter 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 fit values for the oscillation parameterslisted in table 1 and the nuclear quantities in table 2. The Dirac CP phase for the EFCI plot is
set equal to π, while the Majorana are set to 0, in order to minimise the excluded region of theparameter space. For the EFCI and EFCII cases, a quasi-degenerate neutrino mass spectrum withmlightest= 0.1 eV has been assumed, which also minimised the excluded areas. In all the cases,νthe Lagrangian coefficients have been fixed in a democratic way not to favour any specific operator
(1)
(2)
(3)
(4u)
(4d)
(5)
(2)
(1)
contribution: cLL + cLL = 1 = cLL = cLL = cLL = cLL = cRL − cRL .

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 different processes reveals characteristicfeatures that may help to disentangle the different versions of MLFV. To shorten thenotation,e` →` γ
B
t→sγ
Ri→jγ
≡ t s ,
(3.21)e` →` γBijwill be adopted in the analysis that follows. These observables do not depend on the LFVand LNV scales, nor on the Lagrangian coefficients. 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 onlyν
contain points along an horizontal line, they will be reported in the next subsections inorder to facilitate the comparison with the other cases.
The two branching ratios with the best present sensitivities, the one for µ → e conversion in nuclei and the one for µ → eγ, have the same dependence on ∆µe and therefore– 18 –JHEP07(2017)089

The upper bound on ΛL for the MFC case reduce the parameter space, although it
cannot be translated into upper bounds on ΛLFV : larger ΛLFV 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 µ → e conversion,
small ΛLFV requires small ΛL , leading at the same time to tune gν to small values, in orderto reproduce the correct masses for the light active neutrinos, see eq. (2.12). The sameoccurs for EFCI, for µL and Yν , although, in this case, this can be well justified considering
the additional Abelian symmetries appearing in eq. (2.19), as discussed in ref. [65]. Whenconsidering the EFCII case, the lower bound on ΛL removes a large part of the parameter
space, but does not translate into a lower bound on ΛLFV : for example, for ΛL at its lower
bound in eq. (2.30), ΛLFV must be larger than 105 GeV in order to satisfy to the present
Au ; however, for larger values of Λ , Λbounds on Bµ→eL
LFV can be smaller, down to the TeV
17scale for ΛL ∼ 10 GeV, although in this case a tuning on |YN | is necessary in order toreproduce 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 discoveredin the near future, an indication of the LNV scale could be extracted from the plots infigure 1. Indeed, if µ → e conversion in nuclei is observed, ΛLFV ∼ 103 ÷104 GeV will lead toΛL ∼ 1012 ÷1013 GeV for MFC, µL ∼ 109 ÷1010 GeV for EFCI, and µL ∼ 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 futurecolliders. An exception is the case where additional Abelian factors are considered in theflavour 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.

(b) EFCI(c) EFCII(d) All Cases

Figure 2. Rτµ→eγ→µγ for the MFC, EFCI and EFCII from upper left to lower left. Lower right reports
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 theeffective operators contributing to these observables. The comparison between eqs. (3.17)(i)Aand (3.18) shows that only Bµ→e
is sensitive to OLL , and thus any deviation from
A
Bµ→e
2
= πDA
Bµ→eγ

(3.22)

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 2σ error bands. The lightest neutrino mass is taken in therange mlightest⊂ [0.001, 0.1] eV and the results for the NO and IO spectra are shown withν
different colours. In these figures, the density of the points should not be interpreted as
related to the likelihood of differently populated regions of the parameter space.
Rτµ→eγ→µγ . In the upper left, upper right and lower left panes in figure 2, the results arereported for the ratio of the branching ratios of the µ → eγ and τ → µγ decays for the
MFC, EFCI and EFCII cases, respectively. Figure 2d is a summarising figure where all
the three plots are shown together to facilitate the comparison and to make clearer the
non-overlapping areas.

– 19 –JHEP07(2017)089(a) MFCAs figure 2a shows, Rτµ→eγ→µγ 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.

As figure 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 different profiles
in the hierarchical limit. In the IO case the ratio of branching ratios under discussion isalmost constant with mlightest. In the NO case the ratio Rτµ→eγ
ν→µγ can be as small as few
×10−4 at ∼ 0.012 eV, while for mν1 < 0.01 eV the ratio is Rτµ→eγ→µγ > 1. As discussed inref. [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/mν1 and therefore Rτµ→eγ→µγ
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 figure 2d, where the three cases are shown altogether, it can be seen that all thecases overlap for the IO spectrum and in the quasi-degenerate limit for the NO spectrum,∼
predicting Rτµ→eγ→µγ = 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 Bµ→eγ , values smaller than ∼ 6 × 10−4 would betestable in the future planned experiments searching for τ → µγ.
µ→eγ
Rτµ→eγ→eγ . The ratio Rτ →eγ exhibits features which are very similar to those of the ratioRτµ→eγ→µγ . Figures 3a and 3b are very similar to figures 2a and 2b: the profiles of the points are
the same, only the area spanned is different, as indeed Rτµ→eγ→eγ is predicted to be by almostµ→eγ
one order of magnitude larger than Rτ →µγ . Similar conclusions, however, apply. Figure 3c,instead, shows an interesting difference with respect to its sibling figure 2c: the IO and the
NO points cover almost the same nearly horizontal area both for quasi-degenerate masses
and for a hierarchical mass spectrum, the NO region being slightly wider. Only for valuesof the lightest neutrino mass between 0.01 eV and 0.02 eV, there could be an enhancement
or a suppression of Rτµ→eγ→eγ in the EFCII case. This is a distinctive feature that could allow
to disentangle EFCII from the other cases: values of Rτµ→eγ→eγ 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 Bµ→eγ , values smaller than 0.006 would be testable in the futureplanned experiments searching for τ → eγ.

– 20 –JHEP07(2017)089

In figure 2b, the dependence on mlightest
can be slightly appreciated and the predictions
νfor two mass orderings do not overlap when the spectrum is hierarchical. In the NO case
` : the values associated with thethere are two branches associated with the two values of δCP` = 0-branch are very close to those for the IO spectrum and correspond to the positiveδCP
sum of the two terms on the right-hand side of eq. (3.9); the values associated with the`δCP= π-branch are smaller by about one order of magnitude, which reflects a partial
cancellation between the two terms in the right-hand side of eq. (3.9). In the IO case there
is only one branch because the first term on the right-hand side of eq. (3.9) is dominant.

(b) EFCI(c) EFCII(d) All Cases

Figure 3. Rτµ→eγ→eγ for the MFC, EFCI and EFCII from upper left to lower left. Lower right reports
the previous plots altogether. Colour codes can be read directly on each plot.

→eγ . The ratio Rτ →eγ is almost indistinguishable form the ratio Rµ→eγ except forRττ →µγ
τ →µγ
τ →µγ
the EFCII case with NO neutrino mass spectrum. For the other cases the conclusions for→eγRττ →µγ
are almost the same as the conclusions reached for Rτµ→eγ→µγ . One can see that values→eγ
for Rττ →µγ
smaller than 0.01 or larger than 0.1 would only be explain by EFCII with NO
neutrino spectrum.

Summarising, the study of these three ratios can provide relevant information if values
τ →eγ
µ→eγ
for these ratios are found to be larger than 0.1 (10) for Rτµ→eγ→µγ and Rτ →µγ (for Rτ →eγ )
τ →eγ
µ→eγ
or smaller than 0.004 for Rτµ→eγ→µγ , 0.01 for Rτ →µγ , and 0.04 for Rτ →eγ : such values canbe explained only in the case of EFCII with NO spectrum. If large values for Rτµ→eγ→µγ and
τ →eγ
Rτ →µγ 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 Rτµ→eγ→eγ between
0.1 and 10. If instead, Rτµ→eγ
is
found
to
be
much
larger
than
10,
this
would
imply
masses→eγµ→eγ→eγ
for the lightest neutrino between 0.008 eV and 0.04 eV; consistently, Rτ →µγ and Rττ →µγ
should remain smaller than 1. Finally, if no signals are seen in all the three ratios and
τ →eγ
µ→eγ
bounds of 0.004 (0.01) [0.04] or smaller can be obtained for Rτµ→eγ→µγ (Rτ →µγ ) [Rτ →eγ ], thenthis 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– 21 –JHEP07(2017)089(a) MFC

(b) EFCI(c) EFCII(d) All Cases→eγ
Figure 4. Rττ →µγ
for the MFC, EFCI and EFCII from upper left to lower left. Lower right reports
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 different setups.
These results are generically in agreement with previous analyses performed inrefs. [62, 63, 65, 76] and the differences are due to the update input data used here.
A
Bµ→e. 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 µ → e conversion process. On the other hand, if
the observation (or non-observation) of the leptonic radiative rare decays allows to identify
the MLFV realisation from figures 2, 3 and 4, the branching ratio of the µ → e conversion in nuclei could provide the missing information necessary to fix the LFV scale. As
an example, one can assume that an upper bound on Rτµ→eγ→µγ of about 0.004 has beenset, that could be explained by EFCII with a NO neutrino spectrum and a mass of the
Au implies the upper boundlightest neutrino of about 0.014 eV. The upper bound on Bµ→ev 2 /(µL ΛLFV ) < 5.7 × 10−17 . By fixing the LNV scale to its lower bound, one finds thatthese observables can provide information on the LFV scale that should be larger thanAl
about 2 × 106 GeV. The future expected sensitivity on Bµ→eis better than the presently
achieved one by four orders of magnitude. A negative results of the planned future searches
for µ → e conversion would imply a bound on the LFV scale of about 107 GeV.

– 22 –JHEP07(2017)089(a) MFCb → s anomalies

410

The complete effective Lagrangian that describes effects 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 γ µ eR q L γµ qL : this contribution is however negligible for the observables discussed here [210, 211],and then this operator is not considered in the present discussion.

– 23 –JHEP07(2017)089

The effective Lagrangian in eq. (3.2) contains the operators which provide the most relevant(3)(5)
contributions to the b → s anomalies under discussion:10 they are OLL and OLL , which
contribute at tree level to the Wilson coefficients C9 and C10 defined in eq. (1.1), satisfying
to δC10 = −δC9 .
(3)(5)
Focussing on the flavour structure of OLL and OLL , the two operators are invariant
under the MFV flavour symmetry GQ × GL , but can only describe flavour conserving observables which predict universality conservation in both the quark and lepton sectors. In
order to describe a process with quark flavour change, it is then necessary to insert powersof the quark Yukawa spurion Yu . The dominant contributions would arise contracting theflavour indices of the quark bilinear with Yu Yu† : once the spurions acquire their backgroundvalues, the b → s transitions are weighted by the Vtb Vts∗ factor appearing in eq. (1.1). Notice
that, as (Yu )33 = yt ≈ 1, an additional insertion of Yu Yu† is not negligible and modifies thedominant contributions by (1 + yt2 ) factors. Further insertions of Yu Yu† turn out to be unphysical, as they can be written as combinations of the linear and quadratic terms through(3,5)
the Cayley-Hamilton theorem. The complete spurion insertions in OLL can then be written as ζ1 Yu Yu† + ζ2 (Yu Yu† )2 , with ζ1,2 arbitrary coefficients, reflecting the independence ofeach insertion: the net contribution to the operator is then given by Vtb Vts∗ (ζ1 yt2 + ζ2 yt4 ).
The anomalies in the angular observable P50 of B → K ∗ µ+ µ− , in the ratios RK andRK ∗ , and in the Branching Ratio of Bs → φµ+ µ− are linked to the possible violation of
leptonic universality. NP contributions leading to these effects can be described in terms ofinsertions of spurion combinations transforming under 8 of SU(3)`L . The simplest structure
is Ye Ye† that, in the basis defined in eq. (2.12), is diagonal and therefore cannot lead
to lepton flavour changing transitions. The phenomenological analysis associated to theinsertion of this spurionic combination has been performed in ref. [211], where the focus wasin understanding the consequences of having a setup where lepton universality is violated
but lepton flavour is conserved. In ref. [211], the Abelian factors in eq. (2.8) are considered
as active factors of the flavour symmetry and this leads to background values for Ye , whoselargest 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 non-Abelian 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, theinsertion of Ye is subdominant with respect to the insertion of the neutrino spurions: the†most relevant are g†ν gν in the MFC, Yν Yν† in the EFCI and YN
YN in the EFCII. Oncethe spurions acquire background values, these contributions reduce to the ∆ characteristicof 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 specific contribution depends on the model considered and only a

4.1

B semi-leptonic decays

In order to facilitate the comparison with ref. [154] similar assumptions will be taken. First
SM = −C SM and considering that the contributions from O (3,5) satisfy toof all, setting C10
9
LLδC10 = −δC9 , one can consider a single Wilson coefficient in eq. (1.1): for definiteness, C9will be retain in what follows. A second relevant assumption is on the matching between
the effective operators of the high-energy Lagrangian defined at ΛLFV , eq. (3.4), and thelow-energy phenomenological description in eq. (1.1): only the tree level relations will beconsidered in the following, while effects from loop-contributions and from the electroweak
running will be neglected. The latter has been recently shown in refs. [212, 213] to lead toa rich phenomenology, especially in EWPO and τ sector.
(3,5)
Considering explicitly the contributions from OLL , and specifying the flavour indexes,
one can write
π v 2  (3)(5)δC9,``0 =
c
+
c
,
(4.1)
0
0
LL,``
αem Λ2LFV LL,``(i)where cLL,``0 can be written in a notation that makes explicit the dependence on theneutrino spurion background:11



(i)
(i)
(i)
(i)
(i)
(i)
cLL,``0 = ζ1 yt2 + ζ2 yt4 ξ0 δ``0 + ξ1 ∆``0 + ξ2 ∆``0 .
(4.4)
(i)
In order to explain lepton universality violation, the contributions proportional to ξ1 ,(i)(i)
ξ2 , etc. should be at least comparable with ξ0 . Consequently, this requires ∆`` ∼ 1, andthis allows to fix the scale of LNV: indeed, the bounds in eqs. (2.15), (2.25) and (2.30)
become equalities,(
ΛL = 6 × 1014 GeV ,for MFC(4.5)
14µL = 6 × 10 GeV ,for EFCI and EFCII .
11

In ref. [154] a slightly different notation has been adopted, whereiαem Λ2LFV h ˜(i)
(i)
(i)
(i)
cLL,``0 =ξ0 δ``0 + ξ˜1 ∆``0 + ξ˜2 ∆``0 ,
2
π
v
with
π
(i)
(i)
(i)
(i)
ξ˜j = √
(ζ1 yt2 + ζ2 yt4 )ξi .
2αem GF Λ2LFV– 24 –

(4.2)

(4.3)

JHEP07(2017)089

P2ngeneric form
n=0 ξn ∆ can be generically written, where ξn are arbitrary Lagrangiancoefficients, and where the sum is stopped at n = 2 due to the Cayley-Hamilton theorem.
In ref. [154] the EFCI context has been considered and several processes have beenstudied, 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 confirmed, and therefore the viability of MFC
and EFCII to describe the b → s anomalies, consistently with the other (un)observed
flavour processes in the B sector, becomes an interesting issue. Moreover, the results
obtained in the previous section will be explicitly considered.

The bounds from LFV purely leptonic processes discussed in the previous section allows
to translate this result into specific values for the LFV scale: from the bounds on µ → econversion in nuclei, figure 1, one obtains that
5
for MFCΛLFV = 4.4 × 10 GeV ,ΛLFV = 2 × 105 GeV ,Λ= 105 GeV ,LFVfor EFCI

for EFCII .

9

for EFCI(4.7)

for EFCII ,estimating only the pre-factors 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: theversion of MLFV that most contributes to the C9 Wilson coefficient 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 versionsof MLFV cannot explain deviations from the SM predictions in the Wilson coefficient C9larger 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 confirmed, 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 flavour symmetry of the M(L)FV is a continuous global symmetry and therefore, once
promoting the spurions to dynamical fields, its spontaneous breaking leads to the arisingof Goldstone bosons. Although it would be possible to provide masses for these new states,
this would require an explicit breaking of the flavour symmetry. An alternative is to gauge
the symmetry [79–83, 88]: the would-be-Goldstone bosons would be eaten by flavour gaugebosons that enrich the spectrum. In recent papers [214, 215], a specific gauge boson arisingfrom the chosen gauged flavour symmetry has the specific couplings to explain the b → s
anomalies here mentioned.

5

Conclusions

The MFV is a framework to describe fermion masses and mixings and to provide at thesame time a sort of flavour protection from beyond the Standard Model contributions toflavour processes. The lack of knowledge of the neutrino mass origin reflects in a larger
freedom when implementing the MFV ansatz in the lepton sector: three distinct versionsof the MLFV have been proposed in the literature.

– 25 –JHEP07(2017)089

With these results at hand, the order of magnitude for δC9 turns out to be
−4
for MFCδC9 = 1.3 × 10 ,
δC9 = 6.5 × 10−4 ,δC = 2.6 × 10−3 ,(4.6)

Acknowledgments
L.M. thanks the department of Physics and Astronomy of the Università degli Studi di
Padova for the hospitality during the writing up of this paper and Paride Paradisi foruseful 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 Scienceand Technology Development (NAFOSTED) under the grant 103.01-2014.89, and by the
Vietnam Education Foundation (VEF) for the scholarship to work at the Departmentof Physics of the University of Virginia. L.M. and S.T.P. acknowledge partial financial
support by the European Union’s Horizon 2020 research and innovation programme under
the Marie Sklodowska-Curie grant agreements No 690575 and No 674896. The work of
L.M. was supported in part also by “Spanish Agencia Estatal de Investigación” (AEI) andthe EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA201678645-P, and by the Spanish MINECO through the Centro de excelencia Severo Ochoa
Program under grant SEV-2012-0249 and by the Spanish MINECO through the “Ramóny Cajal” programme (RYC-2015-17173). The work of S.T.P. was supported in part by the
INFN program on Theoretical Astroparticle Physics (TASP) and by the World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan.

– 26 –JHEP07(2017)089

In the present paper, an update of the phenomenological analyses on these setups
is presented considering the most recent fit on the neutrino oscillation data. The recent
indication of CP violation in the leptonic sector, if confirmed, will disfavour the very
popular MLFV version [62] called here EFCI, where right-handed neutrinos are assumed to
be degenerate at tree level and the flavour symmetry is SU(3)`L ×SU(3)eR ×SO(3)NR ×CP .
The study of the predictions within these frameworks for flavour 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 different MLFV
possibilities has been described: in particular, the next future experiments searching forµ → eγ and µ → e conversion in aluminium could have the power to pinpoint the scenario
described here as EFCII [65], characterised by the flavour 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 semi-leptonic B-meson
decays can find 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 find a solution
extending/modifying the M(L)FV setup, for example, by gauging the flavour symmetry.

Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction inany medium, provided the original author(s) and source are credited.

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