Revisiting Minimal Lepton Flavour Violation in the light of leptonic CP violation

Journal of High Energy Physics, Jul 2017

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 can be 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.

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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. Vega-Alvarez 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 22904-4714 , 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 right-handed 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 semi-leptonic B-meson decays can be 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. 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 semi-leptonic decays 5 Conclusions The discovery [1{5] of a non-vanishing 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 tau-neutrinos: 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 so-called Tri-Bimaximal 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 non-Abelian group A4 was taken as a avour symmetry of the lepton sector. Several distinct proposals followed, { 1 { i) attempting to achieve the Tri-Bimaximal 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, Tri-Bimaximal mixing etc., arising from the charged lepton sector [26{29], vi) implementing the so-called quark-lepton 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 non-vanishing 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 sub-leading 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 ne-tunings 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 so-called 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 non-zero 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 non-standard 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 non-vanishing 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 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 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 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 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 central-q2 region (low-q2 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 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 the coe cients as the sum of the purely SM contribution and the NP one, Ci = CSM + i Ci, the results of a one-operator-at-a-time 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)L-doublet of quarks, and uR and dR to the SU(2)L-singlets. The Abelian terms can be identi ed with the Baryon number, and with two axial rotations, in the 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 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 down-type quark masses, while Yu contains non-diagonal entries and accounts for both up-type 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 down-type 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)L-double Higgs eld, and H~ = i 2H . LQ can be made invariant under GQ promoting the Yukawa matrices to be spurion elds, i.e. auxiliary non-dynamical 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 low-energy avour processes, they can be described within the effective eld 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 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 ective-spurionic approach to a more fundamental description, promoting the Yukawa spurions to be dynamical elds, called avons, acquiring a non-trivial 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 non-vanishing 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 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-de 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 so-called Weinberg operator [167], a non-renormalisable 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 non-Abelian 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 so-called 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 block-diagonalise 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 low-energy 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 non-vanishing 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 non-linear 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 low-energy 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 nucleus-dependent 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 quasi-degenerate 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 non-overlapping 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 = 0-branch are very close to those for the IO spectrum and correspond to the positive sum of the two terms on the right-hand 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 right-hand side of eq. (3.9). In the IO case there is only one branch because the rst term on the right-hand 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 quasi-degenerate 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 quasi-degenerate 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 non-observation) 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 Cayley-Hamilton 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 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, 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 Cayley-Hamilton 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 semi-leptonic 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 high-energy Lagrangian de ned at LFV, eq. (3.4), and the low-energy phenomenological description in eq. (1.1): only the tree level relations will be considered in the following, while e ects from loop-contributions 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 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: 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 would-be-Goldstone 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 right-handed 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 semi-leptonic B-meson 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 103.01-2014.89, 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 Sklodowska-Curie 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 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 \Ramon y Cajal" programme (RYC-2015-17173). 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D.N. Dinh, L. Merlo, S.T. Petcov, R. Vega-Álvarez. Revisiting Minimal Lepton Flavour Violation in the light of leptonic CP violation, Journal of High Energy Physics, 2017, 89, DOI: 10.1007/JHEP07(2017)089