Predictive leptogenesis from minimal lepton flavour violation

Journal of High Energy Physics, Jul 2018

Abstract A predictive Leptogenesis scenario is presented based on the Minimal Lepton Flavour Violation symmetry. In the realisation with three right-handed neutrinos transforming under the same flavour symmetry of the lepton electroweak doublets, lepton masses and PMNS mixing parameters can be described according to the current data, including a large Dirac CP phase. The observed matter-antimatter asymmetry of the Universe can be achieved through Leptogenesis, with the CP asymmetry parameter ε described in terms of only lepton masses, mixings and phases, plus two real parameters of the low-energy effective description. This is in contrast with the large majority of models present in the literature, where ε depends on several high-energy parameters, preventing a direct connection between low-energy observables and the baryon to photon ratio today. Recovering the correct amount of baryon asymmetry in the Universe constrains the Majorana phases of the PMNS matrix within specific ranges of values: clear predictions for the neutrinoless double beta decay emerge, representing a potential smoking gun for this framework.

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Predictive leptogenesis from minimal lepton flavour violation

Revised: May Predictive leptogenesis from minimal lepton avour L. Merlo 0 1 2 S. Rosauro-Alcaraz 0 1 2 Field Theories 0 Cantoblanco , 28049, Madrid , Spain 1 Universidad Autonoma de Madrid 2 Departamento de F sica Teorica and Instituto de F sica Teorica, IFT-UAM/CSIC A predictive Leptogenesis scenario is presented based on the Minimal Lepton Flavour Violation symmetry. In the realisation with three right-handed neutrinos transforming under the same avour symmetry of the lepton electroweak doublets, lepton masses and PMNS mixing parameters can be described according to the current data, including a large Dirac CP phase. The observed matter-antimatter asymmetry of the Universe can be achieved through Leptogenesis, with the CP asymmetry parameter " described in terms of only lepton masses, mixings and phases, plus two real parameters of the low-energy e ective description. This is in contrast with the large majority of models present in the literature, where " depends on several high-energy parameters, preventing a direct connection between low-energy observables and the baryon to photon ratio today. Recovering the correct amount of baryon asymmetry in the Universe constrains the Majorana phases of the PMNS matrix within speci c ranges of values: clear predictions for the neutrinoless double beta decay emerge, representing a potential smoking gun for this framework. Beyond Standard Model; Neutrino Physics; Global Symmetries; E ective 1 Introduction 2 3 4 5 Conclusions A Lowering Tmax The minimal lepton avour violation with vectorial SU( 3 )V 2.1 A suitable basis for leptogenesis Baryogenesis trough leptogenesis Numerical analysis 4.1 Low-energy phenomenology Despite being so tiny, this non-vanishing value poses one of the most relevant unsolved questions in particle physics and cosmology today: why are there more baryons than antibaryons in the present Universe? In 1967, Sakharov rst suggested that the baryon asymmetry of the Universe (BAU) might not represent some sort of initial condition, but could be understood in terms of microphysical laws that ful l the following three conditions [3]: - Baryon number violation { 1 { 1 Introduction The uncertainties on the measurements for the cosmic abundances of the lightest elements have improved considerably in the last decades, posing stringent constraints on the thermal history of the very early Universe. The observed abundances of protium, deuterium, 3He, 4He and Lithium, besides well agreeing with the predictions of the standard Big Bang Nucleosynthesis [1], allow to deduce the value of the baryon to photon ratio today, where NB;B; are the number densities of baryons, anti-baryons and photons, respectively. An independent determination of B is provided by the CMB measurements [2] that agrees with the value extracted from the lightest element abundances: B NB N NB ; B = (6:11 0:04) nesis [6] (see refs. [7{12] for update reviews on the subject), besides being promising in a large part of the associated parameter space, represents a framework where also other open problems of the SM of particle physics may nd a solution: on the one hand the origin of neutrino masses and on the other hand the Flavour problem. The small, but non-vanishing, masses of the light active neutrinos represent an experimental evidence of the incompleteness of the SM. The introduction of right-handed (RH) neutrinos a la type I Seesaw mechanism [13{17] is an elegant approach that explains the smallness of the active neutrino masses through the largeness of the masses of their RH counterparts. This mechanism provides the ingredients to explain the present amount of BAU: there is a leptonic source of CP violation and a source of Lepton number violation; RH neutrino decays may occur out-of-equilibrium, when the temperature drops below their masses. In consequence, out-of-equilibrium decays of the RH neutrinos might produce a lepton asymmetry that is then partially converted into BAU through non-perturbative sphaleron e ects [18]: in the SM context, B = 1 cs cs L with cs 8NF + 4 22NF + 13 < 1 ; with NF the number of avour species considered. It follows that the more anti-leptons are produced, the more baryons are generated, with a rate that is approximately close to 1=3. The basic quantity in Leptogenesis is the parameter " that measures the amount of CP asymmetry generated in the decays of the RH neutrinos R [6]: indicating with and the decay rates of R into leptons and antileptons respectively, where `L, and stand for the SU( 2 )L-doublet left-handed (LH) leptons and the SU( 2 )Ldoublet Higgs eld, the CP asymmetry parameter is given by with (a) the avour (RH neutrino mass) index. The analytic expression of the CP parameter " depends on the product y , with the Dirac neutrino Yukawa coupling in the mass basis for the RH neutrinos and for the charged leptons. In the convention where the active neutrino mass term is de ned by a a "(a) L ( Ra ! `L Ra ! `L ) ; P P a a + a a ; 12 Lc m { 2 { L + h.c. ; (1.3) (1.4) (1.5) (1.6) where the mass matrix is diagonalised by the PMNS matrix U according to m^ = U T m U ; (1.7) (1.8) (the \^" symbol is adopted here and in the following to refer to diagonal matrices), is a matrix in avour space that can be written in the Casas-Ibarra parametrisation as follows [19]: = p 2 v U m^ 1=2RM^ 1=2 ; where v = 246 GeV is the EW vacuum expectation value (VEV), R a complex orthogonal matrix, and M^ the diagonal mass matrix of the RH neutrinos. This expression depends on 9 low-energy parameters, i.e. 3 active neutrino masses, 3 mixing angles, 1 CP violating Dirac phase and 2 CP violating Majorana phases, and on 9 high-energy ones, corresponding to the 3 RH neutrino masses and the 6 parameters of the matrix R. The latter is typically independent from the low-energy quantities and its parameters are arbitrary. In general, this prevents to uniquely determine the parameter " in terms of low-energy observables and the RH neutrino masses. The use of avour symmetries helps improving the predictivity in this scenario: as a avour symmetry rules the interactions among the di erent fermion generations, the R matrix might be (partially) xed, allowing to predict the value of " (almost) just in function of neutrino masses, mixings and phases. Some examples can be found in refs. [20{31] (see also refs. [32, 33] for predictive scenario not involving avour symmetries). The aim of the present paper is to investigate on a speci c scenario where a continuous non-Abelian group is implemented in the Lagrangian as a global avour symmetry, providing an exceptionally predictive framework for both Leptogenesis and low-energy observables. The symmetry under consideration is the one of the so-called Minimal Flavour Violation (MFV) in the lepton sector (MLFV), considering the type I Seesaw mechanism with three RH neutrinos. The MFV ansatz [34] consists in assuming that any source of avour and CP violation in any theory Beyond the SM (BSM) is the one in the SM, i.e. the Yukawa couplings. This concept has been technically formulated in terms of the avour symmetry of the fermion kinetic terms of a given Lagrangian [35]: the avour group is a product of a U( 3 ) factor for each fermion in the spectrum, and it is U( 3 )6 [36{40] for the type I Seesaw mechanism with 3 RH neutrinos. The Yukawa interactions are the only terms of the renormalisable Lagrangian that are not invariant under the avour symmetry, unless the Yukawa couplings are promoted to be elds, dubbed spurions, transforming non-trivially under U( 3 )6. In the original proposal [35], the Yukawa spurions are dimensionless, non-dynamical elds that acquire background values (they could be interpreted as VEVs if the spurions were promoted to be dynamical elds [41{44]), breaking explicitly the avour symmetry, and reproducing the measured values of fermion masses and PMNS angles and phases. In the quark sector, any non-renormalisable operator containing fermion elds is, eventually, made invariant under the avour symmetry by the insertion of suitable powers of the Yukawa spurions. Once the latter acquire their background values, the strength of the e ects induced by such e ective operators is suppressed by speci c combinations of quark { 3 { masses, mixing angles and CP violating phase. In consequence, the cut-o scale that suppresses any non-renormalisable operator can be of the order of a few TeV [35, 45{58], instead of hundreds of TeV as in the generic case [59]. In the lepton sector, with the addition of three RH neutrinos, the predictive power of the MLFV is lost in the most generic case. Indeed, three quantities, and not only two as in the quark case, need to be promoted to spurions, i.e. the charged lepton Yukawa, the neutrino Dirac Yukawa and the RH neutrino Majorana mass matrices. A simple parameter counting reveals that it is not possible to uniquely determine the three spurion backgrounds in terms of lepton masses and PMNS parameters. This prevents to link the coe cients of avour changing operators with low-energy quantities, with the consequent loss of predictivity. A way out is to reduce the symmetry content: in refs. [36, 38] the non-Abelian part of the U( 3 ) symmetry associated to the RH neutrinos was substituted for a simpler SO( 3 ) plus the hypothesis of CP conservation; in ref. [40], instead, it was identi ed with the one of the lepton SU( 2 )L doublets, thus considering a vectorial SU( 3 )V avour symmetry. Both approaches allow to reduce the number of spurions to two, restoring the predictivity of the models: the e ects of any avour changing e ective operator can be described in terms of lepton masses and PMNS parameters [36{40, 60, 61]. An updated phenomenological analysis of these two di erent MLFV realisations has been recently presented in ref. [62]. A fundamental distinction between them is that the CP conservation hypothesis of the SO( 3 ) CP version is disfavoured by the recent indication of a CP non-conserving Dirac phase in the PNMS matrix [63{69]. Leptogenesis in the MLFV context has already been investigated in ref. [70] (see also refs. [71{74]), considering the SO( 3 ) CP version: in order to guarantee a leptonic source of CP violation necessary to explain the measured BAU, the CP conservation hypothesis has been relaxed; in consequence, the precise prediction of avour e ects at low-energy in terms of lepton masses, mixing and phases has been lost. The aim of this paper is to investigate Leptogenesis in the SU( 3 )V MLFV version introduced in ref. [40], where no additional hypothesis on CP is made and the present indication for the Dirac CP phase of the PMNS can be ful lled. The rest of the paper is structured as follows. The SU( 3 )V MLFV scenario under consideration is described in Sect 2. The Leptogenesis CP asymmetry parameter " and the Boltzmann equations are discussed in section 3. The numerical results are presented in section 4, showing that a correct value for the BAU is achieved only in a part of the allowed parameter space, testable with (non-)observation of the neutrinoless double beta decay. Concluding remarks can be found in section 5. 2 The minimal lepton avour violation with vectorial SU( 3 )V The use of avour symmetries to explain the avour puzzle in the SM goes back to 1978, when Froggatt and Nielsen [75] rst introduced a single U( 1 ) factor to describe the quark mass hierarchies and the CKM mixing matrix. Subsequent analyses also included the lepton sector [76{81], where however a larger freedom is present due to the lack of knowledge of some neutrino parameters. At the beginning of this century, the use of avour discrete { 4 { symmetries became very popular due to the high predictive power in the lepton sector of this kind of models [82{85]. These constructions have been later extended to the quark sector, attempting to provide a uni ed explanation of the avour puzzle [86{90, 90{99], and they have been shown to be contexts where avour violating processes are under control with new physics at the TeV scale [100{108]. Only in 2011, with the discovery of a nonvanishing and relatively large leptonic reactor angle [109{113], strong doubts raised on the goodness of non-Abelian discrete models to describe Nature. In this panorama, the idea of MFV1 experienced a new revival of interest: this context is more predictive than models based on the Froggatt-Nielsen U( 1 ) and escapes from the rigidity of the discrete constructions. This section will summarise the main aspects of the MLFV scenario presented in ref. [40], xing at the same time the notation used throughout this paper. Considering the SM spectrum supplemented with three RH neutrinos, the avour symmetry characterising the SU( 3 )V MLFV scenario is GF NA GFA where GF NA A GF SU( 3 )V U( 1 )Y SU( 3 )eR U( 1 )L U( 1 )eR : (2.1) A GF (2.2) (2.3) (2.4) The distinction between Abelian and non-Abelian terms re ects the fact that the nonAbelian symmetry factors deal exclusively with the inter-generation hierarchies [41{44], while the Abelian ones may explain the hierarchies between the third generation fermions, such as the ratio m =mt. The choice of GFA in eq. (2.1) is the result of using the freedom of rearranging the U( 1 ) factors in order to identify the hypercharge, the Lepton number and transformations that act globally on the RH charged lepton elds only. with the lepton eld transformations under GF NA U( 1 )L The part of the Lagrangian containing the kinetic terms is invariant under GF the lepton masses. The Type I Seesaw Lagrangian, which can be written as [40] L = e`L YeeR + `L eY R + 1 2 L RcYM R + h.c. ; describes the light active neutrino masses at low-energy through the so-called Weinberg operator [119], O5 = 1 2 `L ~ Y YM1 Y T L ~T `c L + h.c. ; where e i 2 , Ye, Y and YM are 3 3 matrices in the avour space, L is the scale of lepton number violation and e is a constant that will be associated to the breaking of the U( 1 )eR symmetry. By the rst Shur's lemma, as `L and R transform as triplets under 1Despite being so predictive, the MFV only describes masses and mixings, but does not explain their respect can be found in refs. [41{44] (see also refs. [114{118]). the two parameters e and L to be spurion elds, i.e. non-dynamical elds that transform non-trivially under GF NA GFA. Selecting the spurion transformations under GF NA as and under U( 1 )L ues, breaking explicitly the avour symmetry: in the charged lepton mass basis, Y^e = L are respectively a dimensionless quantity and a mass. Notice that the same symbols have been used for the couplings in eq. (2.3), for the spurions in eqs. (2.5) and (2.6), and for their background values in eq. (2.7): it will be clear which is the meaning associated to each symbol in the formulae that follow. An estimate of e and of L follows by assuming that the largest eigenvalues of Ye and of YM are . 1:2 then where m2atm of order L. 2:5 10 3 eV2 [67, 68] is the atmospheric squared mass di erence of the light active neutrinos and the \&" symbol re ects the fact that the absolute neutrino mass scale is still unknown. Within this setup, the expected mass scale of the RH neutrinos is In the spirit of the MLFV, any non-renormalisable operator can be made invariant under the avour symmetry by inserting suitable combinations of the spurions. Once the latter acquire background values, the strength of each operator gets suppressed by a combination of lepton masses and PMNS parameters. These extra suppressions allow to predict the rates for rare radiative lepton decays and lepton conversion in nuclei in 2Considering values larger than 1 would imply that multiple products of Yukawa spurions would be more relevant than the single spurions themselves, and then they should be treated in a non-perturbative approach [47]. { 6 { agreement with present data with a new physics scale that suppresses the e ective operators as low as the TeV (see ref. [62] for a recent update). Spurion insertions can be introduced not only in e ective operators, but also in the renormalisable terms of the Lagrangian.3 In particular, the introduction of spurions in the Dirac Yukawa term will be shown to be necessary in order to achieve successful Leptogenesis. Considering only the most relevant contributions, the Dirac Yukawa term can be written as `L e 1 + c1Y^eY^ey + c2YMy YM R ; where c1;2 are dimensionless real parameters that are taken to be smaller than 1 in order to enforce a perturbative approach.4 eq. (2.7) holds in rst approximation. 2.1 A suitable basis for leptogenesis The explicit computation of the " parameter that controls the amount of CP asymmetry generated in the RH neutrino decays is typically performed in the mass basis for charged leptons and for RH neutrinos. The mass Lagrangian in this basis reads Within this hypothesis, the expression for YM in L = e`L Y^eeR + `L e R + 1 2 L RcY^M R + h.c. ; where is the Dirac neutrino Yukawa in this basis. Considering the background values of the spurions in eq. (2.7), reads (2.9) (2.10) (2.11) (2.12) where Y^e is de ned in eq. (2.7), while = U 1 + c1U yY^e2U + c2Y^M2 ; ^ The two parameters c1 and c2 control the complex contributions coming from the PMNS matrix and the real contributions coming from the diagonal RH neutrino mass matrix, respectively. They are expected to be of the same order of magnitude and they will be taken equal to each other in what follows in order to simplify the study of the parameter space. It will be shown a posteriori that relaxing this condition has not relevant impact on the results as far as they are taken of the same order of magnitude. 3Some operators that are non-renormalisable in the description considered here appear in the list of the renormalisable ones if a non-SM Higgs eld is considered, as described in the so-called Higgs E ective Field Theory [120{135]. As shown in refs. [124, 126, 127, 133, 136{140], a di erent phenomenology is expected with a non-SM Higgs in the spectrum. In the present paper, however, the standard formulation with a SU( 2 )L-doublet Higgs is retained. 4In ref. [71], considering the SO( 3 ) CP version of MLFV, the equivalent of the coe cients c1;2 have been shown to be generated by radiative corrections during the evolution of the Lagrangian parameters. { 7 { The relevance of the spurion insertions becomes evident computing the value of three speci c weak-base invariants [20], related to the CP violation responsible for Leptogenesis: I1 =Im I2 =Im I3 =Im Tr h y Y^M3 T Tr h y Y^M5 T Tr h y Y^M5 T ^ i YM ^ i YM Y^M3 i three invariants together with the parameter " would vanish. It is straightforward to show that the three invariants depend on the combinations was taken without the spurions insertions, then = U and the Baryogenesis trough leptogenesis The prediction for the baryon asymmetry in the Universe requires to compute the CP asymmetry parameter " and to take into consideration its evolution during the expansion of the Universe, which depends on the interactions that are in thermal equilibrium at di erent temperatures. With this respect, the value of the RH neutrino mass scale L is a fundamental parameter as it identi es di erent avour regimes [141{147]: the lower L is, the more relevant the avour composition of the charged leptons produced in the RH neutrino decays is. For the SU( 3 )V MLFV framework, L & 1014 GeV and it corresponds to the so-called un avoured regime, where the charged lepton avour does not play any role. Indeed, the only relevant interactions at these energies are the Yukawa ones, which induce RH neutrino decays, and the gauge ones that are avour blind: lepton and antilepton quantum states propagate coherently between the production in decays and the later absorption from inverse decays. In addition, the scale L identi es the reheating temperature necessary for the thermal production of the RH neutrinos [148, 149]: once the temperature drops below Ma, the thermal production of the corresponding RH neutrino Na becomes irrelevant. This allows to identify a lower bound on the reheating temperature at about 1013 14 GeV in the MLFV scenario under consideration. The usually quoted upper bound of 106 10 GeV does not apply as it is exclusively connected to the so-called gravitino problem in supersymmetry [150{152]. Besides L, the splitting between the RH neutrino masses is also relevant: when the spectrum is highly hierarchical then the asymmetries produced by the heaviest states are typically (partially) washed out by the inverse decay of the lightest states (i.e. `L (`L ) + (i.e. `L + ( ) ! Ra) and by the 2 $ 2 scattering mediated by the lightest states $ `L + ); when instead the spectrum is degenerate, a resonance in the decay rate is present [153{160], which, however, is diluted due to the washout e ects of all the three RH neutrinos. In the framework under consideration, depending on the mass of the lightest active neutrino, the spectrum varies from hierarchical to degenerate and therefore the computation of B is not straightforward. In particular, when the heavier RH neutrinos also contribute to the nal asymmetry, the avour composition of the three RH neutrinos is { 8 { relevant and need to be taken into consideration [29, 141, 161{163]: part of the asymmetry generated by a heavier RH neutrino may escape the washout from a lighter one; moreover, part of the nal asymmetry may not come from the production in the RH neutrino decays, but from the dilution e ects. The density matrix formalism [141, 143, 163{166] (see ref. [167] for an alternative avour-covariant formalism) turns out to be extremely e ective in these cases, and thus for the MLFV framework under discussion: it allows to calculate the asymmetry not only in the well de nite regimes with a hierarchical or degenerate RH neutrino spectrum, but also in the intermediate cases, describing together the lepton quantum states and the thermal bath. In the rest of this section, the density matrix approach will be adopted following xing the notation and illustrating the procedure to follow, while in the next section the results of the numerical simulation will be presented. In the present analysis several contributions will not be considered, as their impact is not relevant for the results presented here: they are due to L = 1 scatterings [168{171], thermal corrections [148, 172], momentum dependence [171, 173], and quantum kinetic e ects [174{177]. The baryon-to-photon number ratio at recombination, whose best experimental determination is reported in eq. (1.2), can be written in terms of the nal B L asymmetry density NB L as f f NB L B = cs N rec ' 0:0096 NB L ; with cs = 28=79 de ned in eq. (1.3) for NF = 3, and N rec at recombination. The nal B L asymmetry results from the sum of the asymmetries generated by the three RH neutrinos, in case partially washed out by the inverse decays [141, 178]. It can be calculated solving the following system of four di erential equations: ' 37 the photon number density d(NB L) dz dNNa = dz = "(a)Da[z] NNa Da[z] NNa N Neqa N Neqa Wa[z] nP(a)0; NB L o (a = 1; 2; 3) : equilibrium, that is N Neqa [z xa = Ma2=Ml2ight, is given by [149, 163] The parameter z is the ratio between the lightest RH neutrino mass Mlight and the temperature of the bath, i.e. z Mlight=T . NX is any particle number or asymmetry X calculated in a portion of co-moving volume containing one RH neutrino in ultra-relativistic thermal 1] = 1. The expression for N Neqa [za] at a za pxaz, with f 2 (3.1) (3.2) (3.3) (3.4) (3.5) where Kn[za] is a modi ed Bessel function, satisfying to The Da[z] terms are the RH neutrino decay factors [179] N Neqa [za] = 21 za2 K2[za] ; za2 y00 + za y0 za2 + n2 y = 0 : where K1[za] is also a modi ed Bessel function, and H[za] is the Hubble expansion rate of the Universe given by where g? = gSM = 106:75 is the total number of degrees of freedom and MP l = 1:22 1019 GeV the Planck mass. The second expression on the right-hand side of eq. (3.5) contains the total decay parameters Ka that measure the strength of the washout: they are de ned as the ratio between the total decay widths of the RH neutrinos calculated at a temperature much smaller than Ma and the Hubble parameter at T = Ma, when the RH neutrinos start to become non-relativistic: explicitly, where the total decay rates aD [za] read [180] 1 the RH neutrinos decay and inverse-decay many times before entering the nonrelativistic regime: in consequence their abundance is close to the equilibrium distribution and this case is dubbed strong washout regime. On the other side, for Ka 1, called weak washout regime, the majority of the RH neutrinos decay completely out-of-equilibrium, already in the non-relativist regime, and therefore their equilibrium abundance is exponentially suppressed by the Boltzmann factor. Introducing the notation of the so-called e ective washout parameter [169] and of the equilibrium neutrino mass [181{183], mea m? = 2 v2 ( y )aa ; Ma 16 5=2pg? v 2 p the total decay parameter can be written as The Wa[z] terms are the washout factors due to inverse decays [148, 180, 184] and L = 2 processes [148, 180, 184], which provide the two relevant e ects for these values of the RH neutrino masses: where the two factors are de ned as Ka = mea : m? Wa[z] WaID[z] + Wa[z] ; 1 4 WaID[z] = Ka K1[za] za3 Wa[z] ' z2 Ma me2a a for za 1 ; with = p 3 5MP l 1:202 the Apery constant. The inverse decay processes are relevant when they are in equilibrium, i.e. Wa[z] & 1, and this occurs only in the strong washout regime for Ka > 3. Instead, in the weak washout regime, Wa[z] < 1 and the inverse decays are always irrelevant. On the other side, the L = 2 processes have a relevant e ect only for z & z 1, where z is determined by The P (a)0 factors are the avour projectors along the `a direction de ned by WaID[z ] = me =0:51 MeV ; m m =105:66 MeV ; =1776:86 0:12 MeV ; (3.15) HJEP07(218)36 (3.14) (3.16) Ma2 M 2 b (3.17) (4.1) terms are considered. (a) = 16 ( y )aa b6=a i Ma2 h X h Ma2 Ma2 a b Mb2 y M 2 2 + (Ma a + Mb b) b 2 a b # 1 + ba y where is the Dirac Yukawa in eq. (2.11). The su x \0" indicates that only the leading Finally, the avoured CP asymmetry parameters "(a) are given by [153{156, 156{160] y i Mb ab Ma Mb2 Ma2 1 + ln 1 + a b y ab a b ba i Ma2 Ma2 Mb2 Ma2 M 2 2 + (Ma a + Mb b) b 2 ) ; where the Kadano -Bayn regulator [185], that is the term in the denominator containing the RH neutrino decay rates a, plays an important role when the spectrum is almost degenerate. Di erent regulators can be considered, depending on the formalism chosen: the one used in the previous expression prevents the arising of any divergence in the doub, which instead occurs within the classical bly degenerate limit Ma ! Mb and a ! Boltzmann approach. 4 Numerical analysis This section contains the results of the numerical analysis rst focussing on the baryon asymmetry and then on the neutrinoless double beta decay. The input data used are the PDG values for the charged lepton masses [186] columns refer to the NO and IO, respectively. The notation chosen is ms2ol m22 m21 and m2atm m23 m21 for NO and m2atm m22 m23 for IO. The errors reported correspond to the 1 uncertainties. where the electron and muon masses are shown without errors as the sensitivities are completely negligible, and the results of the neutrino oscillation t from ref. [67] (see also refs. [68, 69]) reported in table 1. In the analysis that follows, all these input parameters have been taken at their central values. the PDG parametrisation of the PMNS matrix, U = R23( 23) R13( 13; C`P) R12( 12) P ; where Rij is a 3 3 rotation in the avour space in the ij sector of an angle ij and P is the diagonal matrix containing the Majorana CP phases de ned by P = diag 1; ei 221 ; ei 231 : 0:02195+00::0000007754 0:02212+00::0000007743 0:3070+:001:0213 0:572+00::002218 281+3303 7:40+00::2210 2:483+00::003345 neutrino mass is presently unknown. Moreover, it is still an open issue the ordering of the neutrino mass eigenstates: the so-called Normal Ordering (NO) refers to the case when m 1 < m 2 m 3 while the Inverse Ordering (IO) to the case when m 3 m 1 < m 2 . The labelling of the three i is determined by the avour content of each mass eigenstate: 1 is the state with the largest contamination of e ; 2 is the one with an almost equally composition of the three avours; 3 is the one almost exclusively de ned as a equal mixture of and . The diagonal active neutrino mass matrix can thus be written in terms of (4.2) (4.3) 0 q 0 m^ IO = BB the lightest neutrino mass as follows: for the NO and IO respectively, 0 0 0 0 q ms2ol 0 0 To match with the notation typically adopted in Leptogenesis, a di erent convention is chosen for the labelling of the RH neutrino mass eigenstates. For both NO and IO, N1 always refers to the lightest eigenstate, N2 to the next to lightest and the N3 to the heaviest. In consequence, Y^M in eq. (2.12) takes a di erent de nition in terms of the three RH neutrino masses depending on the ordering of the spectrum: for the NO and IO respectively, LY^MNO LY^ MIO diag(M3; M2; M1) diag(M2; M1; M3) : (4.4) (4.5) The lepton number violation scale L, the spurion background value Y^M and the active neutrino masses are linked together by eq. (2.12). In consequence it is possible to identify a range of values for the lightest neutrino mass, given a value for the scale L and requiring that the largest entry of Y^M is of order 1, according to the MLFV construction illustrated in section 2. Figure 1 shows the pro les of the RH neutrino masses as a function of the lightest active neutrino mass mlight for a NO spectrum. The plot for the IO case is very similar: the only di erence is that the line corresponding to the next-to-lightest RH neutrino (in red) almost overlaps with the one of the lightest (in blue). The horizontal lines represent di erent values for the L scale, L = 1015; 1016; 1017 GeV, and their crossing with the line of the heaviest RH neutrino mass (in green) identi es the lowest value that mlight can take satisfying (Y^M )ii 1. Figure 1 shows that the lower bound on L reported in eq. (2.8) corresponds to the lightest RH neutrino line (in blue) for mlight . 0:03 eV. An upper bound on L can be taken, in full generality, to be at the Planck scale. However, such a large L is not consistent with the hypothesis of thermal production of RH neutrinos, as the temperature of the Universe should be at least of the same order of magnitude as their masses. In the numerical analysis that follows, the lepton number violation scale is taken at corresponding heaviest RH neutrino mass satis es M3 < 1016 GeV and the range of values for the lightest active neutrino mass is mlight 2 [0:003; 0:2] eV. In consequence, as shown in gure 1, all the three RH neutrinos may contribute to the baryon asymmetry. Further discussion on the maximal temperature of the Universe and on the thermal production of L = 1016 GeV: the the RH neutrinos will follow at the end of next section. 101140-4 0.010 mlight (eV) mlight. The blue (red) [green] continuous line corresponds to the lightest (next-to-lightest) [heaviest] RH neutrino. The horizontal lines represent di erent values for the lepton number violation scale: the dashed one refers to = 1015 GeV, while the dotted to = 1016 GeV, and the dot-dashed to = 1017 GeV. The shaded areas are regions where the condition (Y^M )ii speci c cases are illustrated for L = 1015 GeV, 1016 GeV, 1017 GeV. 1 does not hold: three 4.1 Baryon asymmetry in the Universe This subsection is devoted to illustrate the results of the numerical analysis on the baryon asymmetry in the Universe. Under the assumption that the reheating temperature is close to the maximal temperature Tmax at a given instant, and solving the Boltzmann equations in eq. (3.2) with the initial condition on z = Mlightest=Tmax & 0:06, the lepton asymmetry due to the outof- equilibrium decay of the three RH neutrinos is partially washed out by inverse decays and L = 2 processes." Figure 2 shows the pro les of WaID (continuos lines) and Wa (dashed lines) as a function of za: the value for za at which continuos and dashed lines cross is z 10 and it corresponds to the temperature at which the washout due to inverse decays starts to be less relevant than the dilution e ect due to the L = 2 processes. The Wa lines start from za = 5, satisfying the condition za 1 as discussed below eq. (3.14). The pro les in gure 2 correspond to a speci c choice for the lepton number violation scale, L = 1016 GeV, the lightest active neutrino mass, mlight = 0:003 eV, and the coe cients c1 = c2 = 0:01, and it refers to the NO spectrum. Considering the IO spectrum, the main di erence resides in that the lines corresponding to the lightest and the next-to-lightest neutrinos (blue and red) almost overlap. Lowering L, taking larger values for mlight or taking di erent values for c1;2, but still smaller than 0:1, does not change substantially the plot. Instead, for values c1;2 1, the washout e ects of the heaviest neutrino become more relevant, although not changing the global picture. It follows from the fact that so large c1;2 values induce large o -diagonal entries in in eq. (2.11) and then the RH neutrino avour directions have larger overlap. The standard procedure consists in solving the Boltzmann equations with a nal value za = +1, even if this not e ective from a computational point a view. However, it is possible to identify a value zmax such that B is practically constant for za > zmax. The Wa (dashed lines) as a function of za. The colours refer to the RH neutrino mass eigenstate in the NO case: the blue (red) [green] continuous line corresponds to the lightest (next-to-lightest) [heaviest] RH neutrino. The lepton number violation scale is xed to L = 1016 GeV, the lightest active neutrino mass to mlight = 0:003 eV, which corresponds to zin = 0:06, and the coe cients c1 = c2 = 0:01. 0.100 a0.001 Δ , IDa10-5 10-7 10-9 1.×10-8 5.×10-9 B η 0 -5.×10-9 -1.×10-8 0.1 1 10 100 z B as a function of z 2 [0:06; 100] for three benchmark points in the parameter space: the green line corresponds to 21 = and 31 = =4; the blue line corresponds to 21 = 7 =4 and 31 = =2; the red line to 21 = 3 =4 and 31 = 5 =4. Continuous (dashed) lines correspond to the NO (IO) case. The mass of the lightest active neutrino is xed to mlight = 0:02 eV, while the remaining input parameters have been taken at their central values as reported in table 1. pro le of B as a function of za is shown in gure 3 for three distinct benchmark points in the parameter space: in a good approximation zmax = 20 and this value will be adopted in the rest of the analysis. Moreover, gure 3 leads to the conclusion that B strongly depends on the speci c benchmark point chosen and in consequence one may expect that only a small percentage of points in the whole parameter space accommodates the current determination of B. This is re ected in the scatter plots in gure 4 that show B as a function of the lightest active neutrino mass, for c1 = c2 = 0:01 (details on the input parameters can be found in the caption): values for B consistent with data, represented by the black points in the plots, can be found for mlightest 2 [0:003; 0:04] eV in the NO case and for a) B vs mlight for the NO case. b) B vs mlight for the IO case. bottom. In black the points where B falls inside its experimental determination at 3 error. Charged lepton masses and neutrino oscillation parameters have been taken at their central value as in table 1, 0:01 . z < 20, c1 = c2 = 0:01 and the Majorana CP phases randomly vary in their dominium. mlightest 2 [0:004; 0:012] eV in the IO case. B cannot take values in the white region above the coloured ones, while any arbitrary smaller value is not excluded, although much smaller ones would correspond to ne-tuned situations where cancellations between the nal contributions to B occur. The cuspids at mlight 0:008 eV in the NO and at mlight 0:012 eV in the IO do not correspond to any cancellation in the " parameters, but they arise as a numerical output during the resolution of the Boltzmann equations. Figure 5 shows the correlations existing between the Majorana CP phases and the lightest active neutrino mass for the NO case in 5a and for the IO in 5b and 5c, and between the two Majorana phases for the only IO case in 5d. The 31 phase does not manifest any relevant correlation for the NO case. The plots suggest the presence of speci c regions of the parameter space corresponding to a successful baryogenesis. For the NO case, one may conclude that 21 and mlight are highly correlated and, for a given value of mlight, 21 varies 0.005 0.010 the only points that satisfy B within its experimental determination at 3 error. only inside a small interval. This is not the case for the Majorana phases in the IO case, where the allowed parameter space is much wider; however, the strong correlation between them in gure 5d identi es speci c regions of values where B agrees with data at 3 . The scatter plots shown in gures 4 and 5 are obtained with the Dirac CP phase within its 1 con dence level, that nowadays is a large interval of 60 and 80 for the NO and IO respectively. These results have a very mild dependence on the value of this phase: by comparing the speci c predictions for distinct xed values of C`P, no relevant di erences can be appreciated. On the other hand, these plots highly depend on the values of c1 = c2: for smaller values, for example c1 = c2 = 0:001, B is predicted to be smaller than its experimental determination at 3 in the whole range for mlight and for both NO and IO; for larger values, for example c1 = c2 = 0:1, points with B = 6 10 10 can be found for any value of mlight and in both NO and IO, but no correlation between Majorana phases and mlight are present. In the latter case, a successful description of BAU is the result of an occasional cancellation between the contributions to B obtained solving the density matrix equations in eq. (3.2). The subjacent hypothesis to the numerical result shown above is that the maximal temperature of the Universe is Tmax = L = 1016 GeV, implying that the three RH 1.×10-9 5.×10-10 eV 5G 101 = Tax m B(1.×10-10 η eV 5101G = ax m B(1.×10-10 T η 5G ++++ +++++ normal hierarchical case, in the center the inverse hierarchical one, and on the right the degenerate spectrum case. The red dashed line represents the diagonal to easier drive the eye on the values when the two computed B have the same value. The black continuous lines delimit the 3 value for the experimental determination of B. The two parameters c1 and c2 have been xed at 0:01, while two values for mlight have been considered, mlight = 0:006 eV for the rst two plots and mlight = 0:2 eV for the one on the right. Each point in the plots corresponds to a given random choice of the rest of parameters. neutrinos are thermally produced and contribute to the nal value of B. If a lower value for Tmax is taken, then the heaviest neutrinos may not be thermally produced and their contributions would be negligible. Figure 6 shows the e ect on the nal value of B of lowering the value of Tmax, for a normal hierarchical active neutrino spectrum on the left, for an inverse hierarchical one in the middle, and for a degenerate spectrum on the right. The axes represent the nal value of B considering Tmax = 1016 GeV and Tmax = 1015 GeV. The two parameters c1 and c2 have been xed at 0:01, while two values for mlight have been considered, mlight = 0:006 eV for the rst two plots and mlight = 0:2 eV for the one on the right. Each point in the plots corresponds to a given random choice of the rest of parameters: in this way, it is possible to clearly identify on the nal value of B the impact of the temperature dependence and therefore the impact of the heavier sterile neutrinos. The diagonal red line drives the eye to tell when B is larger for Tmax = 1016 GeV or for Tmax = 1015 GeV: if the points align along the diagonal, then either the heaviest sterile neutrino would not contribute to the nal value of B or the three of them are thermally produced even considering the lowest temperature case; if all the points cover the region on the right of the diagonal, then the heaviest sterile neutrino does have an impact and its contribution sums constructively with the ones from the lightest states; in the opposite case, i.e. all the points on the left of the diagonal, its contribution sums destructively with the other ones. Focussing rst on the normal hierarchical case (plot on the left), the points cover an area along the diagonal, with a small preference for B at Tmax = 1016 GeV. Any xed value of B at Tmax = 1016 GeV corresponds to the same values of B at Tmax = 1015 GeV, whiting a factor 2 3. Moreover, there are points where the B matches with the experimentally allowed regiones (inside the parallel continuous black lines) and many others where this does not occurs. This lets conclude that the value of B strongly depends on the speci c set of parameters, especially Majorana phases, considered, as already pointed out in gure 4. Moreover, the value for B with Tmax = 1016 GeV, where all the three sterile neutrinos contribute, are within a factor 2 3 similar to the ones for Tmax = 1015 GeV, where only the lightest ones are relevant. The small preference for the region where B with Tmax = 1016 GeV indicates that the impact of the heaviest sterile neutrino is often not negligible and slightly increases the nal value of B. It follows that gure 4(a), where the points show that B spans a few order of magnitudes, is a good representative for this scenario with Tmax = 1016 15 GeV and for a hierarchical spectrum. For the inverse hierarchical case (plot in the middle), the largest majority of the points cover the region for B with Tmax = 1016 GeV, indicating that the heaviest sterile neutrino typically contributes to the nal value of B, increasing its value. Moreover, only for Tmax = 1016 GeV, B reaches the experimentally allowed region, indicating that the heaviest sterile neutrino contributions are necessary. As a result, gure 4(b) fairly represents only the case with Tmax = 1016 GeV. Both the plots in degenerate spectrum. Finally, focussing to the degenerate spectrum (plot on the right), all the points strictly align with the diagonal, indicating that B does not change for Tmax = 1015 GeV or 1016 GeV. This was expected because for mlight = 0:2 eV all the three sterile neutrinos have masses below Tmax = 1015 GeV and therefore are the three of them thermally generated. gure 4 well represent this scenario with Tmax = 1016 15 GeV for the The plots equivalent to those in gures 4 and 5(a) for Tmax = 1015 GeV can be found in appendix A. As can be seen, the NO case is essentially una ected by the change of the temperature, while the IO one presents a di erence for small values of mlight where B does not reach the experimental band. 4.2 Low-energy phenomenology The reduction of the allowed parameter space for the Majorana phases in the c1 = c2 = 0:01 case, gure 5, has an impact on the predictions for the neutrinoless double beta decay e ective mass mee, de ned by jmeej = c213 c212 m 1 + c123 s122 m 2 ei 21 + s123 m 3 ei( 31 2 C`P) ; (4.6) where cij and sij stand for cos ij and sin ij , respectively. The investigation on this decay has received a strong impulse in the last decades and numerous experiments are currently competing to probe the existence of this process, as its observation would automatically infer that neutrinos have (at least partly) Majorana nature [187]. Table 2 reports the lower bounds on jmeej sensitivity for near future 0 2 experiments that will be considered in the following. Figure 7 shows the pro le of jmeej as a function of the lightest active neutrino mass mlight in 7a for the NO and in 7b for the IO, while as a function of the Majorana phases in 7c for the NO and in 7d and 7e for the IO. For both the mass orderings, describing HJEP07(218)36 Experiment CUORE [188] GERDA-II [189] LUCIFER [190] MAJORANA D. [191] NEXT [192] AMoRE [193] nEXO [194] PandaX-III [195] SNO+ [196] SuperNEMO [197] Isotope 130Te 76Ge 82Se 76Ge 136Xe 100Mo 136Xe 136Xe 130Te 82Se jmeej [ eV] 0:073 0:008 0:11 0:20 0:13 0:12 0:084 0:011 0:082 0:076 0:084 0:01 0:02 0:01 0:01 0:008 0:001 0:009 0:007 0:008 successfully the amount of BAU leaves viable only the hierarchical regime. For the NO, gure 7a, jmeej can take values only below 0:04 eV, while a lower bound at about 4 10 4 eV seems plausible, as con rmed in gure 7c, although the point density is poor in this region: interestingly, it appears a region precluded for 0:0095 eV . mlight . 0:035 eV. For the IO, gure 7b, the parameter space corresponding to B inside its experimental determination at 3 is con ned in a well-de ned region between 0:005 eV . mlight . 0:01 eV and 0:018 eV . jmeej . 0:05 eV. Complementary information can be extracted in the plots with jmeej as a function of the Majorana phases. For the NO, gure 7c, only jmeej vs 21 shows a correlation: only values for 21 in the interval [ =8; 3 =4] leads to larger values of jmeej, while smaller values may be described for almost any 21. For the IO, gures 7d and 7e, a correlation between jmeej and both the Majorana phases is present and the allowed parameter space is limited in relatively small regions. An observation of the neutrinoless double beta decay in the present experiments, if fully interpreted in terms of Majorana neutrino exchange, would be crucial to determine the values of the Majorna phases for which a successful BAU occurs. Once determined the ordering of the active neutrino mass spectrum, a larger value for jmeej would favour values of 21 in the interval 31 in the interval [ =8; 3 =4] for the NO and [ =2; =2] in the IO, and values of [ =8; ] in the only IO. The determination of the value for the lightest active neutrino mass would help reducing these interval: if mlight is found relatively large, then only the NO scenario would be compatible with a successful explanation of the BAU, while the IO case would be then excluded. HJEP07(218)36 a) jmeej vs mlight for the NO case. b) jmeej vs mlight for the IO case. c) jmeej vs 21 for the NO case. d) jmeej vs 21 for the IO case. e) jmeej vs 31 for the IO case. function of the Majorana phases for the NO case in c) and for the IO one in d) and e). 5 Conclusions The MFV ansatz works extraordinary well in the quark sector accommodating a huge amount of experimental measurements. If an underlying dynamics is the reason behind this hypothesis, then it is natural to expect a similar mechanism at work also in the lepton sector. Two distinct versions of the MLFV can be considered when the SM spectrum is extended by the three RH neutrinos: only if the latter transform under the same symmetry of the lepton electroweak doublets [43], SU( 3 )`L SU( 3 )NR ! SU( 3 )V , then violation of the CP symmetry can be described according to the recent experimental indication. The presence of non-vanishing CP violating phases in the leptonic mixing may be the missing ingredient in the SM to successfully describe the baryon asymmetry in the Universe. In this paper, baryogenesis through Leptogenesis has been considered for the rst time within the context of the SU( 3 )V MLFV framework, resulting in a very predictive setup where the " parameter that describes the amount of CP violation in Leptogenesis only depends on low-energy parameters: charged lepton and active neutrino masses, PMNS parameters and two parameters of the low-energy e ective description. Fixing the two e ective parameters at their natural value 0:01, when a baryon to photon ratio today agrees with its experimental determination at 3 then correlations between the Majorana phases and the lightest active neutrino mass arise. The latter can be analysed considering the impact in the neutrinoless double beta decay observable: only selected regions of the whole jmeej vs mlight parameter space correspond to values that are consistent with a successful baryogenesis. In the NO case, only upper bounds on jmeej and mlight can be identi ed: jmeej . 0:04 eV and mlight . 0:04 eV. Instead, in the IO case, jmeej can take values only inside a much smaller interval [0:02; 0:05] eV corresponding to a narrow interval for mlight that is [0:004; 0:012] eV. These regions will be tested only in several years as the sensitivity required is of the order of that one expected by the nEXO experiment. Acknowledgments The authors warmly thank Pasquale di Bari, Mattias Blennow, Enrique Fernandez Mart nez, Pilar Hernandez, Olga Mena and Nuria Rius for discussions and suggestions. They also thank the HPC-Hydra cluster at IFT. L.M. thanks the department of Physics and Astronomy of the Universita degli Studi di Padova and the Fermilab Theory Division for hospitality during the writing up of the paper. L.M. acknowledges partial nancial support by the Spanish MINECO through the \Ramon y Cajal" programme (RYC-201517173), by the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreements No 690575 and No 674896, and by the Spanish \Agencia Estatal de Investigacion" (AEI) and the EU \Fondo Europeo de Desarrollo Regional" (FEDER) through the project FPA2016-78645-P, and through the Centro de excelencia Severo Ochoa Program under grant SEV-2016-0597. A Lowering Tmax Lowering Tmax implies that the heaviest sterile neutrinos may not be thermally produced, preventing in this way their contributions to the nal value of B. Figure 8 shows the results for Tmax = 1015 GeV. Comparing these plots with those in gure 4, the NO case is essentially una ected by this change, as also con rmed by the correlation plot showing the behaviour of the Majorana phase 21 vs mlight when compared with the equivalent plot in gure 5(a). The IO case presents a sustancial di erence, as B does not reach the experimental band for small values of mlight. a) B vs mlight for the NO case. b) B vs mlight for the IO case. c) 21 vs mlight for the NO case. Figure 8. B as a function of the lightest neutrino mass for the NO on the top and IO in the middle. In black the points where B falls inside its experimental determination at 3 error. The correlation between 21 and mlight in the bottom for the NO case only: the points corresponds to the black ones in the rst plot with B inside its experimental value. Charged lepton masses and neutrino oscillation parameters have been taken at their central value as in table 1, 0:01 . z < 20, c1 = c2 = 0:01 and the Majorana CP phases randomly vary in their dominium. 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 in any medium, provided the original author(s) and source are credited. universe, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [Usp. Fiz. Nauk 161 (1991) 61] [INSPIRE]. [4] C. Jarlskog, Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP-violation, Phys. Rev. Lett. 55 (1985) 1039 [INSPIRE]. [5] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe, Phys. Lett. B 155 (1985) 36 [6] M. Fukugita and T. Yanagida, Baryogenesis Without Grand Uni cation, Phys. Lett. B 174 [7] P.S.B. Dev, P. Di Bari, B. Garbrecht, S. Lavignac, P. Millington and D. Teresi, Flavor e ects in leptogenesis, Int. J. Mod. Phys. A 33 (2018) 1842001 [arXiv:1711.02861] Leptogenesis, Int. J. Mod. Phys. A 33 (2018) 1842002 [arXiv:1711.02862] [INSPIRE]. Phys. A 33 (2018) 1842004 [arXiv:1711.02864] [INSPIRE]. 1842006 [arXiv:1711.02866] [INSPIRE]. [13] P. Minkowski, 421 [INSPIRE]. ! e at a Rate of One Out of 109 Muon Decays?, Phys. Lett. B 67 (1977) [14] M. Gell-Mann, P. Ramond and R. Slansky, Complex Spinors and Uni ed Theories, Conf. Proc. C 790927 (1979) 315 [arXiv:1306.4669] [INSPIRE]. [15] T. Yanagida, Horizontal Symmetry and Masses of Neutrinos, Prog. Theor. Phys. 64 (1980) 1103 [INSPIRE]. 22 (1980) 2227 [INSPIRE]. [16] J. Schechter and J.W.F. Valle, Neutrino Masses in SU( 2 ) x U( 1 ) Theories, Phys. Rev. D [17] R.N. Mohapatra and G. Senjanovic, Neutrino Masses and Mixings in Gauge Models with Spontaneous Parity Violation, Phys. Rev. D 23 (1981) 165 [INSPIRE]. 171 [hep-ph/0103065] [INSPIRE]. [hep-ph/0202030] [INSPIRE]. HJEP07(218)36 [22] S. Pascoli, S.T. Petcov and A. Riotto, Connecting low energy leptonic CP-violation to leptogenesis, Phys. Rev. D 75 (2007) 083511 [hep-ph/0609125] [INSPIRE]. [23] R.N. Mohapatra and H.-B. Yu, Connecting Leptogenesis to CP-violation in Neutrino Mixings in a Tri-bimaximal Mixing model, Phys. Lett. B 644 (2007) 346 [hep-ph/0610023] Neutrino Physics, Nucl. Phys. B 774 (2007) 1 [hep-ph/0611338] [INSPIRE]. [25] G. Engelhard, Y. Grossman and Y. Nir, Relating leptogenesis parameters to light neutrino masses, JHEP 07 (2007) 029 [hep-ph/0702151] [INSPIRE]. [26] E. Molinaro, S.T. Petcov, T. Shindou and Y. Takanishi, E ects of Lightest Neutrino Mass in Leptogenesis, Nucl. Phys. B 797 (2008) 93 [arXiv:0709.0413] [INSPIRE]. [27] E. Molinaro and S.T. Petcov, The Interplay Between the `Low' and `High' Energy CP-Violation in Leptogenesis, Eur. Phys. J. C 61 (2009) 93 [arXiv:0803.4120] [INSPIRE]. [28] E.E. Jenkins and A.V. Manohar, Tribimaximal Mixing, Leptogenesis and theta(13), Phys. Lett. B 668 (2008) 210 [arXiv:0807.4176] [INSPIRE]. [29] E. Bertuzzo, P. Di Bari, F. Feruglio and E. Nardi, Flavor symmetries, leptogenesis and the absolute neutrino mass scale, JHEP 11 (2009) 036 [arXiv:0908.0161] [INSPIRE]. [30] D. Aristizabal Sierra, F. Bazzocchi, I. de Medeiros Varzielas, L. Merlo and S. Morisi, Tri-Bimaximal Lepton Mixing and Leptogenesis, Nucl. Phys. B 827 (2010) 34 [arXiv:0908.0907] [INSPIRE]. [31] F.F. Deppisch and A. Pilaftsis, Lepton Flavour Violation and theta(13) in Minimal Resonant Leptogenesis, Phys. Rev. D 83 (2011) 076007 [arXiv:1012.1834] [INSPIRE]. [32] P. Hernandez, M. Kekic, J. Lopez-Pavon, J. Racker and N. Rius, Leptogenesis in GeV scale seesaw models, JHEP 10 (2015) 067 [arXiv:1508.03676] [INSPIRE]. [33] P. Hernandez, M. Kekic, J. Lopez-Pavon, J. Racker and J. Salvado, Testable Baryogenesis in Seesaw Models, JHEP 08 (2016) 157 [arXiv:1606.06719] [INSPIRE]. [34] R.S. Chivukula and H. Georgi, Composite Technicolor Standard Model, Phys. Lett. B 188 (1987) 99 [INSPIRE]. [35] G. D'Ambrosio, G.F. Giudice, G. Isidori and A. Strumia, Minimal avor violation: An E ective eld theory approach, Nucl. Phys. B 645 (2002) 155 [hep-ph/0207036] [INSPIRE]. [36] V. Cirigliano, B. Grinstein, G. Isidori and M.B. Wise, Minimal avor violation in the lepton sector, Nucl. Phys. B 728 (2005) 121 [hep-ph/0507001] [INSPIRE]. Models, JHEP 09 (2009) 038 [arXiv:0906.1461] [INSPIRE]. [37] V. Cirigliano and B. Grinstein, Phenomenology of minimal lepton avor violation, Nucl. [38] S. Davidson and F. Palorini, Various de nitions of Minimal Flavour Violation for Leptons, [39] M.B. Gavela, T. Hambye, D. Hernandez and P. Hernandez, Minimal Flavour Seesaw extensions of the seesaw, JHEP 06 (2011) 037 [arXiv:1103.5461] [INSPIRE]. HJEP07(218)36 violation, JHEP 07 (2011) 012 [arXiv:1103.2915] [INSPIRE]. [42] R. Alonso, M.B. Gavela, D. Hernandez and L. Merlo, On the Potential of Leptonic Minimal Flavour Violation, Phys. Lett. B 715 (2012) 194 [arXiv:1206.3167] [INSPIRE]. [43] R. Alonso, M.B. Gavela, D. Hernandez, L. Merlo and S. Rigolin, Leptonic Dynamical Yukawa Couplings, JHEP 08 (2013) 069 [arXiv:1306.5922] [INSPIRE]. [44] R. Alonso, M.B. Gavela, G. Isidori and L. Maiani, Neutrino Mixing and Masses from a Minimum Principle, JHEP 11 (2013) 187 [arXiv:1306.5927] [INSPIRE]. [45] B. Grinstein, V. Cirigliano, G. Isidori and M.B. Wise, Grand Uni cation and the Principle of Minimal Flavor Violation, Nucl. Phys. B 763 (2007) 35 [hep-ph/0608123] [INSPIRE]. [46] T. Hurth, G. Isidori, J.F. Kamenik and F. Mescia, Constraints on New Physics in MFV models: A Model-independent analysis of F = 1 processes, Nucl. Phys. B 808 (2009) 326 [arXiv:0807.5039] [INSPIRE]. [47] A.L. Kagan, G. Perez, T. Volansky and J. Zupan, General Minimal Flavor Violation, Phys. Rev. D 80 (2009) 076002 [arXiv:0903.1794] [INSPIRE]. [48] B. Grinstein, M. Redi and G. Villadoro, Low Scale Flavor Gauge Symmetries, JHEP 11 (2010) 067 [arXiv:1009.2049] [INSPIRE]. [arXiv:1010.2116] [INSPIRE]. [49] T. Feldmann, See-Saw Masses for Quarks and Leptons in SU(5), JHEP 04 (2011) 043 [50] D. Guadagnoli, R.N. Mohapatra and I. Sung, Gauged Flavor Group with Left-Right Symmetry, JHEP 04 (2011) 093 [arXiv:1103.4170] [INSPIRE]. [51] A.J. Buras, L. Merlo and E. Stamou, The Impact of Flavour Changing Neutral Gauge Bosons on B ! Xs , JHEP 08 (2011) 124 [arXiv:1105.5146] [INSPIRE]. [52] A.J. Buras, M.V. Carlucci, L. Merlo and E. Stamou, Phenomenology of a Gauged SU( 3 )3 Flavour Model, JHEP 03 (2012) 088 [arXiv:1112.4477] [INSPIRE]. [53] R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin and J. Yepes, Minimal Flavour Violation with Strong Higgs Dynamics, JHEP 06 (2012) 076 [arXiv:1201.1511] [INSPIRE]. [54] G. Isidori and D.M. Straub, Minimal Flavour Violation and Beyond, Eur. Phys. J. C 72 (2012) 2103 [arXiv:1202.0464] [INSPIRE]. [55] L. Lopez-Honorez and L. Merlo, Dark matter within the minimal avour violation ansatz, Phys. Lett. B 722 (2013) 135 [arXiv:1303.1087] [INSPIRE]. [56] F. Bishara, A. Greljo, J.F. Kamenik, E. Stamou and J. Zupan, Dark Matter and Gauged Flavor Symmetries, JHEP 12 (2015) 130 [arXiv:1505.03862] [INSPIRE]. avor violation implications of the b ! s anomalies, JHEP 08 (2015) 123 [arXiv:1505.04692] [INSPIRE]. [arXiv:1709.07039] [INSPIRE]. [59] G. Isidori, Y. Nir and G. Perez, Flavor Physics Constraints for Physics Beyond the Standard Model, Ann. Rev. Nucl. Part. Sci. 60 (2010) 355 [arXiv:1002.0900] [INSPIRE]. gauged avour symmetry, JHEP 11 (2016) 078 [arXiv:1608.04124] [INSPIRE]. Gauged Lepton Flavour, JHEP 12 (2016) 119 [arXiv:1609.05902] [INSPIRE]. [62] D.N. Dinh, L. Merlo, S.T. Petcov and R. Vega-Alvarez, Revisiting Minimal Lepton Flavour Violation in the Light of Leptonic CP-violation, JHEP 07 (2017) 089 [arXiv:1705.09284] (2014) 093006 [arXiv:1405.7540] [INSPIRE]. CP-violation, JHEP 03 (2015) 005 [arXiv:1407.3274] [INSPIRE]. t to three neutrino mixing: exploring the accelerator-reactor complementarity, JHEP 01 (2017) 087 [arXiv:1611.01514] [INSPIRE]. [68] F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri and A. Palazzo, Global constraints on absolute neutrino masses and their ordering, Phys. Rev. D 95 (2017) 096014 [arXiv:1703.04471] [INSPIRE]. [69] P.F. de Salas, D.V. Forero, C.A. Ternes, M. Tortola and J.W.F. Valle, Status of neutrino oscillations 2018: rst hint for normal mass ordering and improved CP sensitivity, Phys. Lett. B 782 (2018) 633 [arXiv:1708.01186] [INSPIRE]. [70] V. Cirigliano, G. Isidori and V. Porretti, CP violation and Leptogenesis in models with Minimal Lepton Flavour Violation, Nucl. Phys. B 763 (2007) 228 [hep-ph/0607068] [71] G.C. Branco, A.J. Buras, S. Jager, S. Uhlig and A. Weiler, Another look at minimal lepton avour violation, li ! lj , leptogenesis and the ratio M = LF V , JHEP 09 (2007) 004 [72] S. Uhlig, Minimal Lepton Flavour Violation and Leptogenesis with exclusively low-energy CP-violation, JHEP 11 (2007) 066 [hep-ph/0612262] [INSPIRE]. [73] V. Cirigliano, A. De Simone, G. Isidori, I. Masina and A. Riotto, Quantum Resonant Leptogenesis and Minimal Lepton Flavour Violation, JCAP 01 (2008) 004 [arXiv:0711.0778] [INSPIRE]. [74] A. Pilaftsis and D. Teresi, Mass bounds on light and heavy neutrinos from radiative [INSPIRE]. CP-violation, Nucl. Phys. B 147 (1979) 277 [INSPIRE]. grand uni cation, JHEP 11 (2000) 040 [hep-ph/0007254] [INSPIRE]. hierarchy, JHEP 01 (2003) 035 [hep-ph/0210342] [INSPIRE]. Textures, JHEP 11 (2012) 139 [arXiv:1207.0587] [INSPIRE]. and the quark mixing matrix, Phys. Lett. B 552 (2003) 207 [hep-ph/0206292] [INSPIRE]. [84] G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing from discrete symmetry in extra dimensions, Nucl. Phys. B 720 (2005) 64 [hep-ph/0504165] [INSPIRE]. [85] G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing, A4 and the modular symmetry, Nucl. Phys. B 741 (2006) 215 [hep-ph/0512103] [INSPIRE]. [86] F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Tri-bimaximal Neutrino Mixing and Quark Masses from a Discrete Flavour Symmetry, Nucl. Phys. B 775 (2007) 120 [Erratum ibid. B 836 (2010) 127] [hep-ph/0702194] [INSPIRE]. [87] F. Bazzocchi, L. Merlo and S. Morisi, Fermion Masses and Mixings in a S(4)-based Model, Nucl. Phys. B 816 (2009) 204 [arXiv:0901.2086] [INSPIRE]. [88] F. Bazzocchi, L. Merlo and S. Morisi, Phenomenological Consequences of See-Saw in S4 Based Models, Phys. Rev. D 80 (2009) 053003 [arXiv:0902.2849] [INSPIRE]. [89] G. Altarelli, F. Feruglio and L. Merlo, Revisiting Bimaximal Neutrino Mixing in a Model with S4 Discrete Symmetry, JHEP 05 (2009) 020 [arXiv:0903.1940] [INSPIRE]. [90] R. de Adelhart Toorop, F. Bazzocchi and L. Merlo, The Interplay Between GUT and Flavour Symmetries in a Pati-Salam x S4 Model, JHEP 08 (2010) 001 [arXiv:1003.4502] [INSPIRE]. [91] G. Altarelli and F. Feruglio, Discrete Flavor Symmetries and Models of Neutrino Mixing, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211] [INSPIRE]. [92] I. de Medeiros Varzielas and L. Merlo, Ultraviolet Completion of Flavour Models, JHEP 02 (2011) 062 [arXiv:1011.6662] [INSPIRE]. [arXiv:1110.6376] [INSPIRE]. [95] R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Finite Modular Groups and Lepton Mixing, Nucl. Phys. B 858 (2012) 437 [arXiv:1112.1340] [INSPIRE]. [arXiv:1112.1959] [INSPIRE]. [97] G. Altarelli, F. Feruglio and L. Merlo, Tri-Bimaximal Neutrino Mixing and Discrete Flavour Symmetries, Fortsch. Phys. 61 (2013) 507 [arXiv:1205.5133] [INSPIRE]. [98] F. Bazzocchi and L. Merlo, Neutrino Mixings and the S4 Discrete Flavour Symmetry, Fortsch. Phys. 61 (2013) 571 [arXiv:1205.5135] [INSPIRE]. Phys. 76 (2013) 056201 [arXiv:1301.1340] [INSPIRE]. [99] S.F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. [100] F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Lepton Flavour Violation in Models with A4 Flavour Symmetry, Nucl. Phys. B 809 (2009) 218 [arXiv:0807.3160] [INSPIRE]. [101] F. Feruglio, C. Hagedorn and L. Merlo, Vacuum Alignment in SUSY A4 Models, JHEP 03 (2010) 084 [arXiv:0910.4058] [INSPIRE]. [102] Y. Lin, L. Merlo and A. Paris, Running E ects on Lepton Mixing Angles in Flavour Models with Type I Seesaw, Nucl. Phys. B 835 (2010) 238 [arXiv:0911.3037] [INSPIRE]. [103] F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Lepton Flavour Violation in a Supersymmetric Model with A4 Flavour Symmetry, Nucl. Phys. B 832 (2010) 251 [arXiv:0911.3874] [INSPIRE]. [104] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Non-Abelian Discrete Symmetries in Particle Physics, Prog. Theor. Phys. Suppl. 183 (2010) 1 [arXiv:1003.3552] [INSPIRE]. [105] R. de Adelhart Toorop, F. Bazzocchi, L. Merlo and A. Paris, Constraining Flavour Symmetries At The EW Scale I: The A4 Higgs Potential, JHEP 03 (2011) 035 [Erratum ibid. 01 (2013) 098] [arXiv:1012.1791] [INSPIRE]. [106] R. de Adelhart Toorop, F. Bazzocchi, L. Merlo and A. Paris, Constraining Flavour Symmetries At The EW Scale II: The Fermion Processes, JHEP 03 (2011) 040 [arXiv:1012.2091] [INSPIRE]. [107] L. Merlo, S. Rigolin and B. Zaldivar, Flavour violation in a supersymmetric T' model, JHEP 11 (2011) 047 [arXiv:1108.1795] [INSPIRE]. [108] G. Altarelli, F. Feruglio, L. Merlo and E. Stamou, Discrete Flavour Groups, theta13 and Lepton Flavour Violation, JHEP 08 (2012) 021 [arXiv:1205.4670] [INSPIRE]. [109] T2K collaboration, K. Abe et al., Indication of Electron Neutrino Appearance from an Accelerator-produced O -axis Muon Neutrino Beam, Phys. Rev. Lett. 107 (2011) 041801 [arXiv:1106.2822] [INSPIRE]. [arXiv:1108.0015] [INSPIRE]. [110] MINOS collaboration, P. Adamson et al., Improved search for muon-neutrino to electron-neutrino oscillations in MINOS, Phys. Rev. Lett. 107 (2011) 181802 [111] Double CHOOZ collaboration, Y. Abe et al., Indication of Reactor e Disappearance in [INSPIRE]. [arXiv:1204.0626] [INSPIRE]. HJEP07(218)36 [115] R. Barbieri, L.J. Hall, G.L. Kane and G.G. Ross, Nearly degenerate neutrinos and broken avor symmetry, hep-ph/9901228 [INSPIRE]. [116] Z. Berezhiani and A. Rossi, Flavor structure, avor symmetry and supersymmetry, Nucl. Phys. Proc. Suppl. 101 (2001) 410 [hep-ph/0107054] [INSPIRE]. [117] T. Feldmann, M. Jung and T. Mannel, Sequential Flavour Symmetry Breaking, Phys. Rev. D 80 (2009) 033003 [arXiv:0906.1523] [INSPIRE]. [arXiv:1105.1770] [INSPIRE]. [INSPIRE]. (1993) 4937 [hep-ph/9301281] [INSPIRE]. Rev. D 76 (2007) 073002 [arXiv:0704.1505] [INSPIRE]. [122] R. Contino, C. Grojean, M. Moretti, F. Piccinini and R. Rattazzi, Strong Double Higgs Production at the LHC, JHEP 05 (2010) 089 [arXiv:1002.1011] [INSPIRE]. [123] R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin and J. Yepes, The E ective Chiral Lagrangian for a Light Dynamical \Higgs Particle", Phys. Lett. B 722 (2013) 330 [Erratum ibid. B 726 (2013) 926] [arXiv:1212.3305] [INSPIRE]. [124] R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin and J. Yepes, Flavor with a light dynamical \Higgs particle", Phys. Rev. D 87 (2013) 055019 [arXiv:1212.3307] [INSPIRE]. [125] G. Buchalla, O. Cata and C. Krause, Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO, Nucl. Phys. B 880 (2014) 552 [Erratum ibid. B 913 (2016) 475] [arXiv:1307.5017] [INSPIRE]. [126] I. Brivio, T. Corbett, O.J.P. E boli, M.B. Gavela, J. Gonzalez-Fraile, M.C. Gonzalez-Garcia et al., Disentangling a dynamical Higgs, JHEP 03 (2014) 024 [arXiv:1311.1823] [INSPIRE]. [127] M.B. Gavela, J. Gonzalez-Fraile, M.C. Gonzalez-Garcia, L. Merlo, S. Rigolin and J. Yepes, CP violation with a dynamical Higgs, JHEP 10 (2014) 044 [arXiv:1406.6367] [INSPIRE]. [128] R. Alonso, I. Brivio, B. Gavela, L. Merlo and S. Rigolin, Sigma Decomposition, JHEP 12 (2014) 034 [arXiv:1409.1589] [INSPIRE]. JHEP 04 (2016) 016 [arXiv:1510.07899] [INSPIRE]. [129] I.M. Hierro, L. Merlo and S. Rigolin, Sigma Decomposition: The CP-Odd Lagrangian, [INSPIRE]. [arXiv:1601.07551] [INSPIRE]. Lagrangian after the LHC Run I, Eur. Phys. J. C 76 (2016) 416 [arXiv:1604.06801] [INSPIRE]. [134] LHC Higgs Cross Section Working Group collaboration, D. de Florian et al., Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector, arXiv:1610.07922 [INSPIRE]. [135] L. Merlo, S. Saa and M. Sacristan-Barbero, Baryon Non-Invariant Couplings in Higgs E ective Field Theory, Eur. Phys. J. C 77 (2017) 185 [arXiv:1612.04832] [INSPIRE]. ultraviolet softening, JHEP 12 (2014) 004 [arXiv:1405.5412] [INSPIRE]. portal to Dark Matter, JHEP 04 (2016) 141 [arXiv:1511.01099] [INSPIRE]. Field Theory and Collider Signatures, Eur. Phys. J. C 77 (2017) 572 [arXiv:1701.05379] [INSPIRE]. [arXiv:1703.02064] [INSPIRE]. [139] P. Hernandez-Leon and L. Merlo, Distinguishing A Higgs-Like Dilaton Scenario With A Complete Bosonic E ective Field Theory Basis, Phys. Rev. D 96 (2017) 075008 [140] L. Merlo, F. Pobbe and S. Rigolin, The Minimal Axion Minimal Linear Model, Eur. Phys. J. C 78 (2018) 415 [arXiv:1710.10500] [INSPIRE]. Nucl. Phys. B 575 (2000) 61 [hep-ph/9911315] [INSPIRE]. 72 (2005) 113001 [hep-ph/0506107] [INSPIRE]. [141] R. Barbieri, P. Creminelli, A. Strumia and N. Tetradis, Baryogenesis through leptogenesis, [142] A. Pilaftsis and T.E.J. Underwood, Electroweak-scale resonant leptogenesis, Phys. Rev. D [143] A. Abada, S. Davidson, F.-X. Josse-Michaux, M. Losada and A. Riotto, Flavor issues in leptogenesis, JCAP 04 (2006) 004 [hep-ph/0601083] [INSPIRE]. 01 (2006) 164 [hep-ph/0601084] [INSPIRE]. [144] E. Nardi, Y. Nir, E. Roulet and J. Racker, The Importance of avor in leptogenesis, JHEP [145] T.E.J. Underwood, Lepton Flavour E ects and Resonant Leptogenesis, in proceedings of the 41St Rencontres De Moriond on Electroweak Interactions and Uni ed Theories, La [146] A. Abada, S. Davidson, A. Ibarra, F.X. Josse-Michaux, M. Losada and A. Riotto, Flavour Matters in Leptogenesis, JHEP 09 (2006) 010 [hep-ph/0605281] [INSPIRE]. 315 (2005) 305 [hep-ph/0401240] [INSPIRE]. (1984) 265 [INSPIRE]. Phys. Lett. B 145 (1984) 181 [INSPIRE]. [152] K. Kohri, T. Moroi and A. Yotsuyanagi, Big-bang nucleosynthesis with unstable gravitino and upper bound on the reheating temperature, Phys. Rev. D 73 (2006) 123511 [hep-ph/0507245] [INSPIRE]. [153] J. Liu and G. Segre, Reexamination of generation of baryon and lepton number asymmetries by heavy particle decay, Phys. Rev. D 48 (1993) 4609 [hep-ph/9304241] [INSPIRE]. [154] M. Flanz, E.A. Paschos and U. Sarkar, Baryogenesis from a lepton asymmetric universe, Phys. Lett. B 345 (1995) 248 [Erratum ibid. B 384 (1996) 487] [hep-ph/9411366] [INSPIRE]. [155] M. Flanz, E.A. Paschos, U. Sarkar and J. Weiss, Baryogenesis through mixing of heavy Majorana neutrinos, Phys. Lett. B 389 (1996) 693 [hep-ph/9607310] [INSPIRE]. [156] L. Covi, E. Roulet and F. Vissani, CP violating decays in leptogenesis scenarios, Phys. Lett. B 384 (1996) 169 [hep-ph/9605319] [INSPIRE]. 113 [hep-ph/9611425] [INSPIRE]. 56 (1997) 5431 [hep-ph/9707235] [INSPIRE]. Lett. B 431 (1998) 354 [hep-ph/9710460] [INSPIRE]. [hep-ph/0309342] [INSPIRE]. Rev. Lett. 99 (2007) 081802 [hep-ph/0612187] [INSPIRE]. [157] L. Covi and E. Roulet, Baryogenesis from mixed particle decays, Phys. Lett. B 399 (1997) [158] A. Pilaftsis, CP violation and baryogenesis due to heavy Majorana neutrinos, Phys. Rev. D [159] W. Buchmuller and M. Plumacher, CP asymmetry in Majorana neutrino decays, Phys. [160] A. Pilaftsis and T.E.J. Underwood, Resonant leptogenesis, Nucl. Phys. B 692 (2004) 303 [161] G. Engelhard, Y. Grossman, E. Nardi and Y. Nir, The Importance of N2 leptogenesis, Phys. [162] S. Antusch, P. Di Bari, D.A. Jones and S.F. King, A fuller avour treatment of N2-dominated leptogenesis, Nucl. Phys. B 856 (2012) 180 [arXiv:1003.5132] [INSPIRE]. [163] S. Blanchet, P. Di Bari, D.A. Jones and L. Marzola, Leptogenesis with heavy neutrino avours: from density matrix to Boltzmann equations, JCAP 01 (2013) 041 [arXiv:1112.4528] [INSPIRE]. [164] S. Blanchet, P. Di Bari and G.G. Ra elt, Quantum Zeno e ect and the impact of avor in leptogenesis, JCAP 03 (2007) 012 [hep-ph/0611337] [INSPIRE]. [165] A. De Simone and A. Riotto, On the impact of avour oscillations in leptogenesis, JCAP 02 (2007) 005 [hep-ph/0611357] [INSPIRE]. 125012 [arXiv:1211.0512] [INSPIRE]. [167] P.S. Bhupal Dev, P. Millington, A. Pilaftsis and D. Teresi, Flavour Covariant Transport HJEP07(218)36 Mod. Phys. A 15 (2000) 5047 [hep-ph/0007176] [INSPIRE]. [171] F. Hahn-Woernle, M. Plumacher and Y.Y.Y. Wong, Full Boltzmann equations for leptogenesis including scattering, JCAP 08 (2009) 028 [arXiv:0907.0205] [INSPIRE]. [172] C.P. Kiessig, M. Plumacher and M.H. Thoma, Decay of a Yukawa fermion at nite temperature and applications to leptogenesis, Phys. Rev. D 82 (2010) 036007 [arXiv:1003.3016] [INSPIRE]. [INSPIRE]. (2007) 002 [hep-ph/0703175] [INSPIRE]. [173] A. Basboll and S. Hannestad, Decay of heavy Majorana neutrinos using the full Boltzmann equation including its implications for leptogenesis, JCAP 01 (2007) 003 [hep-ph/0609025] [174] A. De Simone and A. Riotto, Quantum Boltzmann Equations and Leptogenesis, JCAP 08 [175] M. Beneke, B. Garbrecht, M. Herranen and P. Schwaller, Finite Number Density Corrections to Leptogenesis, Nucl. Phys. B 838 (2010) 1 [arXiv:1002.1326] [INSPIRE]. [176] A. Anisimov, W. Buchmuller, M. Drewes and S. Mendizabal, Quantum Leptogenesis I, Annals Phys. 326 (2011) 1998 [Erratum ibid. 338 (2011) 376] [arXiv:1012.5821] [INSPIRE]. [177] P.S. Bhupal Dev, P. Millington, A. Pilaftsis and D. Teresi, Kadano -Baym approach to avour mixing and oscillations in resonant leptogenesis, Nucl. Phys. B 891 (2015) 128 [arXiv:1410.6434] [INSPIRE]. [178] A. Strumia, Baryogenesis via leptogenesis, in Particle Physics Beyond the Standard Model, proceedings of the Summer School on Theoretical Physics, 84th Session, Les Houches, [179] E.W. Kolb and M.S. Turner, The Early Universe, Front. Phys. 69 (1990) 1 [INSPIRE]. [180] E.W. Kolb and S. Wolfram, Baryon Number Generation in the Early Universe, Nucl. Phys. B 172 (1980) 224 [Erratum ibid. B 195 (1982) 542] [INSPIRE]. [181] E. Nezri and J. Orlo , Neutrino oscillations versus leptogenesis in SO( 10 ) models, JHEP 04 (2003) 020 [hep-ph/0004227] [INSPIRE]. [182] W. Buchmuller, P. Di Bari and M. Plumacher, The Neutrino mass window for baryogenesis, Nucl. Phys. B 665 (2003) 445 [hep-ph/0302092] [INSPIRE]. [183] W. Buchmuller, P. Di Bari and M. Plumacher, Some aspects of thermal leptogenesis, New J. [184] A.D. Dolgov and Ya. B. Zeldovich, Cosmology and Elementary Particles, Rev. Mod. Phys. Phys. 6 (2004) 105 [hep-ph/0406014] [INSPIRE]. 53 (1981) 1 [INSPIRE]. rst principles in the Nazionali del Gran Sasso Adv. High Energy Phys. 2013 (2013) 506186. J. Phys. Conf. Ser. 718 (2016) 062048 [INSPIRE]. Double-Beta Decay Experiment, Adv. High Energy Phys. 2014 (2014) 365432 [arXiv:1308.1633] [INSPIRE]. Ser. 718 (2016) 062033 [INSPIRE]. (2017) 012232 [INSPIRE]. with EXO-200 and nEXO, Nucl. Part. Phys. Proc. 265-266 (2015) 42 [INSPIRE]. 061011 [arXiv:1610.08883] [INSPIRE]. [arXiv:1506.05825] [INSPIRE]. [1] R.V. Wagoner , Big bang nucleosynthesis revisited, Astrophys . J. 179 ( 1973 ) 343 [INSPIRE]. [2] Planck collaboration, P.A.R. Ade et al., Planck 2015 results . XIII. Cosmological parameters , Astron. Astrophys . 594 ( 2016 ) A13 [arXiv: 1502 .01589] [INSPIRE]. [3] A.D. Sakharov , Violation of CP Invariance, C asymmetry and baryon asymmetry of the [8 ] M. Drewes , B. Garbrecht , P. Hernandez , M. Kekic , J. Lopez-Pavon , J. Racker et al., ARS [9] B. Dev , M. Garny , J. Klaric , P. Millington and D. Teresi , Resonant enhancement in leptogenesis, Int. J. Mod. Phys. A 33 ( 2018 ) 1842003 [arXiv: 1711 .02863] [INSPIRE]. [10] S. Biondini et al., Status of rates and rate equations for thermal leptogenesis , Int. J. Mod. [11] E.J. Chun et al., Probing Leptogenesis , Int. J. Mod. Phys. A 33 ( 2018 ) 1842005 [12] C. Hagedorn , R.N. Mohapatra , E. Molinaro , C.C. Nishi and S.T. Petcov , CP Violation in the Lepton Sector and Implications for Leptogenesis, Int . J. Mod. Phys. A 33 ( 2018 ) [18] W. Buchmu ller, R.D. Peccei and T. Yanagida , Leptogenesis as the origin of matter , Ann. Rev. Nucl. Part. Sci . 55 ( 2005 ) 311 [ hep -ph/0502169] [INSPIRE]. [19] J.A. Casas and A. Ibarra , Oscillating neutrinos and ! e; , Nucl. Phys. B 618 ( 2001 ) [20] G.C. Branco , T. Morozumi , B.M. Nobre and M.N. Rebelo , A Bridge between CP-violation at low-energies and leptogenesis , Nucl. Phys. B 617 ( 2001 ) 475 [ hep -ph/0107164] [21] G.C. Branco , R. Gonzalez Felipe , F.R. Joaquim and M.N. Rebelo , Leptogenesis, CP-violation and neutrino data: What can we learn? , Nucl. Phys. B 640 ( 2002 ) 202 Phys. B 752 ( 2006 ) 18 [hep- ph/0601111] [INSPIRE]. Phys. Lett. B 642 ( 2006 ) 72 [ hep -ph/0607329] [INSPIRE]. [80] G. Altarelli , F. Feruglio , I. Masina and L. Merlo , Repressing Anarchy in Neutrino Mass [81] J. Bergstrom , D. Meloni and L. Merlo , Bayesian comparison of U(1) lepton avor models , Phys. Rev. D 89 ( 2014 ) 093021 [arXiv: 1403 .4528] [INSPIRE]. [82] E. Ma and G. Rajasekaran, Softly broken A4 symmetry for nearly degenerate neutrino masses , Phys. Rev. D 64 ( 2001 ) 113012 [ hep -ph/0106291] [INSPIRE]. [83] K.S. Babu , E. Ma and J.W.F. Valle , Underlying A4 symmetry for the neutrino mass matrix of T2K, Phys . Lett. B 703 ( 2011 ) 447 [arXiv: 1107 .3486] [INSPIRE]. [94] W. Grimus and P.O. Ludl , Finite avour groups of fermions , J. Phys. A 45 ( 2012 ) 233001 [96] S.F. King and C. Luhn , A4 models of tri-bimaximal-reactor mixing , JHEP 03 ( 2012 ) 036 the Double CHOOZ Experiment , Phys. Rev. Lett . 108 ( 2012 ) 131801 [arXiv: 1112 .6353] [112] Daya Bay collaboration , F.P. An et al., Observation of electron-antineutrino disappearance at Daya Bay , Phys. Rev. Lett . 108 ( 2012 ) 171803 [arXiv: 1203 .1669] [INSPIRE]. [113] RENO collaboration , J.K. Ahn et al., Observation of Reactor Electron Antineutrino Disappearance in the RENO Experiment, Phys. Rev. Lett . 108 ( 2012 ) 191802 [114] A. Anselm and Z. Berezhiani , Weak mixing angles as dynamical degrees of freedom , Nucl. [118] E. Nardi , Naturally large Yukawa hierarchies , Phys. Rev. D 84 ( 2011 ) 036008 [119] S. Weinberg , Baryon and Lepton Nonconserving Processes, Phys. Rev. Lett . 43 ( 1979 ) 1566 [120] F. Feruglio , The Chiral approach to the electroweak interactions , Int. J. Mod. Phys. A 8 [121] B. Grinstein and M. Trott , A Higgs-Higgs bound state due to new physics at a TeV, Phys . [130] M.B. Gavela , K. Kanshin , P.A.N. Machado and S. Saa , On the renormalization of the electroweak chiral Lagrangian with a Higgs , JHEP 03 ( 2015 ) 043 [arXiv: 1409 .1571] [131] B.M. Gavela , E.E. Jenkins , A.V. Manohar and L. Merlo , Analysis of General Power Counting Rules in E ective Field Theory, Eur . Phys. J. C 76 ( 2016 ) 485 [132] O.J.P. E boli and M.C. Gonzalez-Garcia, Classifying the bosonic quartic couplings , Phys. [133] I. Brivio , J. Gonzalez-Fraile , M.C. Gonzalez-Garcia and L. Merlo , The complete HEFT [136] I. Brivio , O.J.P. E boli , M.B. Gavela , M.C. Gonzalez-Garcia , L. Merlo and S. Rigolin , Higgs [137] I. Brivio , M.B. Gavela , L. Merlo , K. Mimasu , J.M. No , R. del Rey et al., Non-linear Higgs [138] I. Brivio , M.B. Gavela , L. Merlo , K. Mimasu , J.M. No , R. del Rey et al., ALPs E ective [147] S. Blanchet and P. Di Bari , Flavor e ects on leptogenesis predictions , JCAP 03 ( 2007 ) 018 [148] G.F. Giudice , A. Notari , M. Raidal , A. Riotto and A. Strumia , Towards a complete theory of thermal leptogenesis in the SM and MSSM, Nucl . Phys. B 685 ( 2004 ) 89 [149] W. Buchmu ller, P. Di Bari and M. Plumacher, Leptogenesis for pedestrians , Annals Phys. [150] M. Yu . Khlopov and A.D. Linde , Is It Easy to Save the Gravitino?, Phys. Lett. B 138 [151] J.R. Ellis , J.E. Kim and D.V. Nanopoulos , Cosmological Gravitino Regeneration and Decay, [166] S. Blanchet and P. Di Bari , The minimal scenario of leptogenesis, New J . Phys. 14 ( 2012 ) Equations: an Application to Resonant Leptogenesis, Nucl . Phys. B 886 ( 2014 ) 569 [168] M.A. Luty , Baryogenesis via leptogenesis, Phys. Rev. D 45 ( 1992 ) 455 [INSPIRE]. [169] M. Plu macher, Baryogenesis and lepton number violation , Z. Phys. C 74 ( 1997 ) 549 [170] W. Buchmu ller and M. Plumacher, Neutrino masses and the baryon asymmetry , Int. J. [185] M. Garny , A. Kartavtsev and A. Hohenegger , Leptogenesis from resonant regime , Annals Phys . 328 ( 2013 ) 26 [arXiv: 1112 .6428] [INSPIRE]. [186] Particle Data Group collaboration, C. Patrignani et al., Review of Particle Physics, Chin. Phys. C 40 ( 2016 ) 100001 [INSPIRE] . Phys. Rev. D 25 ( 1982 ) 2951 [INSPIRE]. [187] J. Schechter and J.W.F. Valle , Neutrinoless Double beta Decay in SU(2) x U(1) Theories , [188] CUORE collaboration, D.R. Artusa et al., Searching for neutrinoless double-beta decay of 130Te with CUORE, Adv . High Energy Phys . 2015 ( 2015 ) 879871 [arXiv: 1402 .6072] [193] AMoRE collaboration , H.-S. Jo, Status of the AMoRE experiment , J. Phys. Conf. Ser. 888


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L. Merlo, S. Rosauro-Alcaraz. Predictive leptogenesis from minimal lepton flavour violation, Journal of High Energy Physics, 2018, 36, DOI: 10.1007/JHEP07(2018)036