Neutrino mixing and leptogenesis in a L e − L μ − L τ model

We present a simple extension of the Standard Model with three right-handed neutrinos in a SUSY framework, with an additional U ( 1 ) F abelian ﬂavor symmetry with a non standard leptonic charge L e − L μ − L τ for lepton doublets and arbitrary right-handed charges. Our model pre-dicts an inverted neutrino mass hierarchy and it is able to reproduce the experimental values of the mixing angles of the PMNS matrix and of the r = (cid:3) m 2sun /(cid:3) m 2atm ratio, with only a moderate ﬁne tuning of the Lagrangian free parameters. The baryon asymmetry of the Universe is generated via thermal leptogenesis through CP-violating decays of the heavy right-handed neutrinos. We present a detailed numerical solution of the relevant Boltzmann equation, accounting for the impact of the distribution of the asymmetry in the various lepton ﬂavors.

We present a simple extension of the Standard Model with three right-handed neutrinos in a SUSY framework, with an additional U(1)F abelian flavor symmetry with a non standard leptonic charge Le − Lµ − Lτ for lepton doublets and arbitrary right-handed charges.We show that the model is able to reproduce the experimental values of the mixing angles of the PMNS matrix and of the r = ∆m 2  sun /∆m 2 atm ratio, with only a moderate fine tuning of the Lagrangian free parameters.The baryon asymmetry of the Universe is generated via thermal leptogenesis through CP-violating decays of the heavy right-handed neutrinos.We present a detailed numerical solution of the relevant Boltzmann equation accounting for the impact of the distribution of the asymmetry in the various lepton flavors.

I. INTRODUCTION
The Standard Model (SM) of particle physics has proven to be one of the most accurate theories to explain microscopic interactions at an unprecedented level.In spite of its many successes, it fails to account for relevant low energy data, such as the structure of fermion masses and mixings (in particular, the non-vanishing neutrino masses) and the value of the baryon asymmetry of the Universe (BAU), which is commonly expressed by the parameter: where n B , n B and n γ are the number densities of the baryons, antibaryons and photons, while the subscript "0" stands for "at present time".Latest observations provide a numerical value of η B ≈ 6.1 • 10 −10 [1].In recent times, an enormous experimental progress has been made in our knowledge of the neutrino properties and it has been clearly shown that the lepton mixing matrix contains two large and one small mixing angle, and that the two independent mass-squared differences are both different from zero [2][3][4].Although several abelian and non-abelian symmetries acting on flavour space have been proposed to explain such a pattern, not an unique framework emerged as the optimal one [5].Thus, one is still motivated to explore scenarios where different symmetries and/or field (charge) assignments to the group representations are studied in details.
In this context, a less explored possibility (compared to the most famous discrete non-abelian symmetries) is given by the U (1) F flavor symmetry with non-standard leptonic charge L e − L µ − L τ for lepton doublets [6] and arbitrary right-handed charges [7].As it is well known, in the limit of exact symmetry, the neutrino mass matrix assumes the following structure: le lµ lτ l c e l c µ l c τ F1 F2 Hu H d N1 N2 N3 U(1)F +1 -1 -1 −13 7 3 2 1/2 0 0 -1 1 0 TABLE I: U (1) F charges for leptons, Higgses and flavon fields.
which leads to a spectrum of inverted type and to θ 12 = π/4, tan θ 23 = x (i.e.large atmospheric mixing for x ∼ O(1)) and θ 13 = 0.While the previous texture can be considered a good Leading Order (LO) result, it evidently fails to reproduce two independent mass differences (two eigenvalues have the same absolute values) and, with the exception of the atmospheric angle, also fails in the correct description of the solar and reactor angles.Models based on the see-saw mechanism [8,9] have been proven to be sufficiently realistic as to accommodate solar and atmospheric splittings but either the solar angle was too large or the reactor angle was (almost) vanishing.With the increasing precision in the measurement of oscillation parameters, it turned out that both θ 12 and θ 13 were substantially different from their LO results; the observation that corrections of O(λ) (λ being the Cabibbo angle) are needed to bring both mixing angles to their experimental values, encouraged to explore the contributions to the neutrino mixing matrix from the charged lepton sector [10,11]; in this context, a natural value of r = ∆m 2 sol /∆m 2 arm ∼ O(λ 2 ) was also obtained [12], thus showing that models based on L e −L µ −L τ are capable to successfully describe low energy neutrino data.An important missing piece of the previous constructions is the possibility to explain the value of the BAU through leptogenesis.In [9] it was clearly shown that the baryon-to-photon ratio of the Universe η B is proportional to the neutrino mass m 1 and, for vanishing lightest mass m 3 , m 1 ∼ ∆m 2 atm , thus producing an hopelessly small η B .Providing a quantitative leptogenesis analysis has become sophisticated in recent years, due to the addition of many ingredients, such as various washout effects [13,14] or thermal corrections to the particle masses [14,15].Also, the flavor effects can have a significant impact on the final value of the baryon asymmetry, as widely shown in [16][17][18].With the present paper we aim to go beyond the existing literature, assessing whether see-saw models based on the L e − L µ − L τ quantum number can simultaneously account for neutrino masses and mixing and explain the BAU through thermal leptogenesis.The paper is structured as follows.In Sect.II we describe our model and derive the analytic expressions for the mass ratio r and the mixing angles, showing that appropriate choices of the Lagrangian parameters lead to a satisfactory description of low energy data; in Sect.III we face the problem of reproducing the value of η B , analyzing the resonant and hierarchical scenarios and solving the related Boltzmann equations.Our conclusions are drawn in Sect.IV.

II. THE MODEL
In the following, we summarize the relevant features of our see-saw flavor model based on a broken U (1) F symmetry.In the proposed scenario, the left handed lepton doubles have charge L e − L µ − L τ [11] under the U (1) F , while the right-handed SU (2) singlets l c e,µ,τ have the charges reported in Tab.I.The spectrum of the theory also contains three heavy sterile neutrinos N i=1,2,3 , needed for the generation of the light neutrino masses as well as for the implementation of the leptogenesis process.The flavor symmetry is broken by vacuum expectation values (vev's) of SU (2) singlet scalar fields (flavons) suitably charged under the U (1) F symmetry.Non-vanishing vevs are determined by the D-term potential [19]: where g F denotes the gauge coupling constant of the U (1) F symmetry while M FI is the Fayet-Iliopulos term.Non-zero vevs are obtained by imposing the SUSY minimum V D = 0. Without loss of generality, we can assume equal vevs for the two flavons and define λ = F 1 /M F = F 2 /M F the common ratio between the vevs of the flavons and the scale M F at which the flavour symmetry is broken.

A. Charged lepton sector
In the charged lepton sector, many operators of different dimensions enter the Lagrangian; to avoid cumbersome expressions, we quote here the lowest dimensional operators contributing to each entry of the mass matrix: where all a ij coefficients are generic O(1) free parameters.After flavor and electroweak symmetry breakings, the previous Lagrangian generates a mass matrix whose elements have the general structure: where α ij and β ij denote the appropriate powers of the flavon fields needed to generate a singlet under U (1) F .Factorizing out the τ mass, the charged lepton mass matrix assumes the following form: For λ < 1, the following mass ratios m e : m µ : m τ = λ 5 : λ 2 : 1 is found, which naturally reproduces the observed pattern if λ ∼ 0.22.It is not difficult to derive the left-handed rotation U l , which contributes to the total neutrino mixing matrix U P M N S [20,21], diagonalizing the hermitean m l m † l combination: As expected, the diagonal elements of U l are unsuppressed; also, the ( 23) and (33) entries are of O( 1), which indicates that the charged lepton contribution to the atmospheric angle will be large.Notice also that, being the ( 12) and ( 13) elements of O(λ), we expect a similar corrections to the solar and reactor angles.

B. Neutrino sector
In the neutrino sector, masses are generated through the standard type-I see-saw mechanism; at the renormalizable level, the see-saw Lagrangian reads: where M is an overall Majorana mass scale while W, Z, a, b, c are dimensionless coefficients which will be regarded as free parameters.When H u acquires a vev v u , Majorana and Dirac mass matrices are generated: Next-to-leading order (NLO) contributions are given by higher dimensional operators suppressed by the large scale M F ; up to two-flavon insertions, we get 1 : 1 Notice that we use the same overall scale M in the Majorana mass terms.
Their main effects is to fill the vanishing entries in eq.( 9); however, as we have numerically verified, some of the free parameters in eq.( 10) needed to be slightly adjusted to fit the low energy data.In particular, only a moderate fine-tuning is necessary on m 11 , m With the previous position (and reminding that λ = F 1 /M F = F 2 /M F ), the following Dirac and Majorana mass matrices are obtained: and Notice that the Dirac mass matrix contains un-suppressed entries because of the choice for two of the right-handed neutrinos.The four physical phases Σ, Ω, Φ, Θ in Y , obtained after a suitable redefintion of the fermion fields, are the only source of CP violation of our model and are not fixed by the symmetries of the Lagrangians.For the sake of simplicity and without any loss of generality, we can assume the parameters m ij ∼ m and consider m as a real quantity.From the type-I seesaw master formula, R Y , we get the following matrix for the light SM neutrinos, up to O(λ 2 ): This mass matrix provides, as usual for models based on the L e − L µ − L τ symmetry, an inverted mass spectrum.
From now on, we will distinguish two different scenarios, that will be further elaborated when studying the BAU generated in our model, and identified by different assumptions on the parameter M. In order to better understand this distinction, let us assume that all parameters in eq.( 13) are of O(1); thus, the light neutrino mass matrix can be recast in the following form: where m 0 = v 2 u /M × O(1) coefficients and (x, x i ) are suitable combinations of the coefficients present in Dirac and Majorana matrices in eqs.(11) and (12).At the leading order in λ, m ν has two degenerate eigenvalues m 1 = −m 2 = √ 1 + x 2 and a vanishing one, m 3 = 0; therefore, we can only construct the atmospheric mass difference ∆m which results in: To maintain x ∼ O(1), we can choose the overall mass scale to m 0 ∼ O(10 −2 ) eV, which corresponds to the choice M ∼ 10 15 GeV.Notice also that, taking into account the corrections of O(λ 2 ), the eigenvalue degeneracy is broken and the solar mass difference can be accounted for, which results in: Since the masses of the three heavy right-handed neutrinos are simply given by M i = M M i with: and, in particular, the relation W Z holds, we dubbed such a situation as the resonant scenario.
As expected, two mass eigenstates are degenerate, up to corrections of order λ 2 .The second possibility arises when M ∼ 10 13 GeV and, consequently, W should be around 10 2 to maintain m 0 ∼ O(10 −2 ) eV.With all other parameters again of O(1), the light neutrino mass matrix is now as follows: The right-handed neutrino spectrum will feature two, almost degenerate states with mass M 1 M 2 ∼ 10 15 GeV, and a lighter one with mass M 3 ZM ∼ 10 13 GeV; we call this scenario as the hierarchical scenario.Even in this case, the left-handed neutrino mass matrix can be recast in the form (14), with the obvious redefinition of (x, x i ) in terms of the parameters appearing in eq.( 18).Thus, in both resonant and hierarchical scenarios, we get the same structure of the diagonalizing matrix U ν : The final expressions of the neutrino mixing angles (from the relation U P M N S = U † l U ν ) and the ratio r = ∆m  to report here the order of magnitude in the expansion parameter λ, which are valid for both scenarios considered in this paper: Our order of magnitude estimates are in good agreement with their measured values, reported in Tab.II.This conclusion has been further strengthened by a succesful numerical scan [22] over the model free parameters, with moduli extracted flat in the intervals [λ, 5] and all the phases in [−π, π].Notice that the model provides a CP conserving leptonic phase2 (still compatible with the data at less than 3σ).

III. LEPTOGENESIS
As stated above, our study of leptogenesis will be performed within two reference scenarios, identified by different mass patterns for the heavy right-handed neutrinos: • the resonant scenario, with M 1 M 2 M 3 around 10 15 GeV, and the mass difference comparable to the decay width ∆M ij ∼ Γ i ; • the hierarchical scenario, with M 3 M 1,2 10 15 GeV .
The three Majorana neutrinos decay in the early Universe creating a lepton asymmetry, which is consequently conversed in a baryon asymmetry through non perturbative processes, known as sphaleron processes [13,14].As we will clearify in the following, a different Majorana neutrino mass spectrum can lead to different CP-violating parameters, affecting the final amount of baryon asymmetry in the Universe.

Resonant Scenario
We are interested in computing the BAU via thermal leptogenesis.To facilitate the understanding of the numerical results, we will first provide analytical (approximated) expressions of all relevant quantities entering our computations, which will be validated against a full numerical solution of an appropriate system of Boltzmann's equations.Being mass degenerate, we expect that all the three right-handed neutrinos contribute to the leptogenesis process.Following [23], we write the baryon asymmetry as: with ε i being the CP-asymmetries produced in the decay of the i-th neutrino and η i the corresponding efficiency factors.As will be clarified below, the generation of the lepton asymmetry in the resonant scenario occurs in an unflavored regime; the final asymmetry is consequently just the sum of the contributions associated to the individual neutrinos.The efficiency factors can be written in terms of decay parameters K i [18]: with: As it is well known, different values of K i define different washout regimes, namely strong (K i 1), intermediate (K i 1) and weak (K i 1).In terms of the parameters of the Dirac and Majorana neutrino mass matrices of eq.( 11), their expressions up to O(λ 4 ) are as follows: We are now in the position to discuss the CP asymmetry.Since the heavy neutrinos are close in mass, the CP-asymmetry can be resonantly enhanced [23][24][25].To check whether such an enhancement occurs, and hence properly evaluate the CP asymmetry parameters, a good rule of thumb consists in computing the ratios ∆M ij /Γ i between the mass splittings and the decay widths of the righthanded neutrinos, verifying that the resonance condition ∆M ij ∼ Γ i is satisfied.In the scenario under scrutiny, with degenerate masses and not strongly hierarchical Yukawa couplings, we can assume Γ 1 ∼ Γ 2 ∼ Γ 3 , so that: where we have used Γ i = M i Y † Y ii /(16π).As evident, for W−Z O(0.1), ∆M i3 Γ i .Consequently, leptogenesis occurs in the resonant regime.In such a case, the self energy contribution dominates the CP violation parameters.Furthermore, as shown in [26], the asymmetry parameters are time dependent.Following [26,27], we rewrite the latter as: where z = M/T , with M the sterile neutrino mass.The coefficient in front of the squared parenthesis is the (constant) usual CP-asymmetry and it is resonantly enhanced for ∆M ∼ Γ, while the second one is the sum of two z (and hence time) dependent functions: In the case ∆M ij t ≡ 1, eq.( 27) is a strongly oscillating function.Making the average over a generic time interval (or, equivalently, in z) t ∈ [0, τ ], we have that: As evident, the average tends to 1 if ∆M ij τ 1.In such a limit we have that the CP asymmetry averages to the following constant value: In the case of a large decay parameter, the regime ∆M ij t 1 occurs for small values of z.In good approximation the whole leptogenesis process can be described by replacing the time dependent CP-asymmetry with its average.Being, in our case, K i O(10), we can adopt this approach and, consequently, propose an analytic estimate of the baryon asymmetry neglecting the time dependence of the ε i 's.The η B will be nevertheless compared against a complete numerical treatment.The time-independent piece of the CP-asymmetry parameters in eq.( 26) can be expressed in power series   1).The parameters are selected close to 1 as to avoid particular effects of enhancement or suppression in the ε i .
of the small parameter λ as follows: By inspecting the analytical expressions above we see that, for W − Z O(0.1), the parameters ε 2,3 tend approximately to O(0.1), while ε 1 is of order 10 −4 .Therefore, the main contribution to the final baryon asymmetry of the Universe is carried by ε 2,3 , leading to an η B value which exceeds the experimentally favoured one η B 6.1 • 10 −10 by several orders of magnitude.It is nevertheless possible to suppress the values of the ε i parameters by an ad-hoc assignations of the phases to trigger a destructive interference among them.First of all, by taking Σ − Ω → 0, the leading order contributions to ε 2,3 go to zero and, as for ε 1 , all ε's are of O(λ 4 ): Upon numerical check, we have found that the λ 4 suppression was not enough to guarantee viable values of the ε i .Consequently, we need to further assume individually suppressed Θ, Φ and Σ phases.For the benchmark values reported in Tab.III, the efficiency factors turn to be: while the CP-asymmetry parameters are: Using the approximated formula in eq.( 21), we can estimate the final baryon asymmetry, obtaining η B ≈ 5.20 • 10 −10 , in good agreement with the observations.We conclude our study of the resonant scenario with a numerical validation of the analytical results presented above.We have hence solved the following set of coupled Boltzmann's equations [13]: where N i stands for number density of the RH sterile neutrinos, while N B−L is the amount of B − L asymmetry, both normalized by comoving volume [13].ε i = ε i (z) are the full time dependent asymmetry parameters as given in eqs.(26)(27).D i and S i indicate, respectively, inverse decay and scattering contributions to the production of the right-handed neutrinos while the W i represent the total rate of Wash-out processes including both inverse decay and ∆L = 0 scattering contributions (see Appendix A for further details).From the benchmark values of Tab.III, the following entries of the Yukawa matrix are obtained: where the symbol hat refers to the Yukawa evaluated in the physical mass basis of the Majorana neutrinos, i.e.Ŷ = U Y , where U is the unitary matrix such that . Plugging the latter values in the interaction rates appearing in the Boltzmann's equations, we have solved the system assuming null initial abundance for the right-handed neutrinos in the primordial plasma.The B-L yield N B−L as a function of z is shown with a blue line in fig. 1, while the abundance of the right-handed neutrinos is displayed with a green line.For reference, we have reported the corresponding equilibrium function as a dashed orange line.To better pinpoint the impact on our result of the time dependency of the CP-asymmetry, we have shown, as red line in fig. 1, solution of the same system of equation but retaining a constant CP violating parameter as given by eq.( 33).As evident, there is a nice agreement between the curves, justifying the assumption of neglecting the time dependence of the asymmetry parameter in our analytical treatment.Some more comments are in order.Starting from a negligible abundance, the yield of the right-handed neutrino is driven by inverse decays toward the equilibrium value which is reached for z eq < 1.For z > z eq , the decays dominate and the neutrino abundance decreases, until it becomes almost zero around z 10.This means that the leptogenesis processes is completed at temperatures above 10 13 GeV.This justifies our assumption of neglecting flavor effects since the latter are relevant only if leptogenesis occurs at temperatures below 10 12 GeV.The shape of the N B−L also clearly evidences the time dependency of the ε i parameters.The solution of the system N B−L (∞) can be related to η B through the relation η B = (a sph /f )N B-L (∞).Here a sph = 28/79 [13] is the fraction of B − L asymmetry converted into a baryon asymmetry by the sphaleron processes while f = N rec γ /N * γ = 2387/86 is the dilution factor calculated assuming standard photon production from the onset of leptogenesis till recombination.The red line refers to the solution of the analogous system but adopting a time-constant value of the asymmetry parameters.Finally, the green line represents the abundance of the right-handed neutrinos, as given by the solution of the system.For reference, the latter is compared with the function N eq N (dashed line) rwhich represents a thermal equilibrium abundance for right-handed neutrinos.
The values of the latter parameters, as obtained from the numerical solution of the system are: We conclude that, after the phase suppression discussed above, it is possible to provide a viable leptogenesis in the resonant scenario.

Hierarchical scenario
This alternative scenario is obtained by lowering the mass scale M down to a value of the order of 10 13 GeV, which brings to a hierarchical mass spectrum for the sterile neutrinos, with two heavy, almost degenerate, states with M 1 M 2 ∼ 10 15 GeV, and a lighter one with M 3 10 13 GeV.In this setup, the baryon asymmetry of the Universe can be generated through the conventional thermal leptogenesis only via the out-of-equilibrium decay of the lightest heavy neutrino.Similarly to the previous case, no flavor effects need to be accounted for.As discussed in [23], the latter becomes relevant when the rate of processes mediated by τ , µ and e exceeds the Hubble expansion rate.This occurs when the temperature of the Universe drops below, respectively:  In the scenario under consideration we have M 3 T τ T µ,e .Thus, we can work in the so-called unflavored regime, in which the lightest neutrino decays via flavor blind processes.The baryon asymmetry can then be parametrized as: with η i as in eq.( 22) and the unflavored CP-asymmetry as [28][29][30][31]: where x j = M 2 j /M 2 i and the loop function is: that can be approximated to f [x] −3/(2 √ x) in the limit x 1. Expressing the elements of the Yukawa matrices in terms of the model parameters, we have: Using the values in Tab.IV, and noticing that f M 2 2 /M 2 3 10 −2 , we obtain ε 3 3.24 • 10 −4 .The corresponding efficiency factor η 3 can be simply approximated to: for the assignations in Tab.IV.The efficiency factor can be converted to the baryon asymmetry parameter η B : very close to the experimentally favoured value.Also in this case, we have verified the goodness of our analytical approximations by solving a suitable set of Boltzmann equations, which take the form: From the parameter choices in Tab.IV, we get the following Yukawa matrix : (45) Notice that the imaginary parts of the third row is suppressed by a factor proportional to Z/W .This choice corresponds to the CP-violation parameter ε 3 = 4.3 • 10 −4 .For the benchmark values under consideration, the numerical solution of the Boltzmann's equations is shown in fig. 2 and gives: confirming what we expected from the simplified analytical analysis.It is interesting to notice that, contrary to the resonant regime, in the hierarchical scenario it is possible to find a parameter assignation leading to viable leptogenesis without imposing a fine tuning on the CP violating phases.

IV. CONCLUSIONS
In this paper we have provided a proof of existence about the possibility of contemporary achieving viable masses and mixing patterns for the SM neutrinos and a value of the BAU, via leptogenesis, compatible with the experimental determination, in models based on the abelian flavor symmetry L e − L µ − L τ .Given the large parameter space of the model, we have identified two reference scenarios.The first one, dubbed the resonant scenario, provides a viable light neutrino mass spectrum and assures the existence of three degenerate right-handed neutrinos at a mass scale of 10 15 GeV.In this scenario, the generation of the lepton asymmetry is resonantly enhanced so that a baryon asymmetry exceeding the experimentally favored value is generically predicted.This problem can be overcome by invoking an ad-hoc suppression of the CP-violating phases in the Yukawa matrix.In the second scenario, that we called the hierarchical scenario, one of the right-handed masses is lowered down to 10 13 GeV without destroying the good agreement with the lepton masses and mixing.The BAU is generated via the conventional thermal leptogenesis.We have verified that it is possible to find parameter assignations leading to the correct value of the BAU without invoking specific assignations for the CP violating phases. of the reduced cross section σx given in [32].In particular, for the scattering processes mediated by the three Majorana neutrinos, i.e. the ∆L = 2 scatterings, the reduced cross section reads [36]: with √ a i a j (a i − a j )(x + a i + a j ) (2x + 3a i + a j ) ln 1 + x a j + −(2x + 3a j + a i ) ln 1 + x a i .

(A19)
Here a j ≡ M 2 j /M 2 1 and 1/D i (x) ≡ (x−a i )/[(x−a i ) 2 +a i c i ] is the off-shell part of the N i propagator with c i = a i (Y † Y ) 2  ii /(8π) 2 .On the other hand, for the ∆L = 1 scattering processes, it is convenient to write the rates S s,t (z) in term of the functions f s,t (z) defined as [32]: with the functions χ s,t (x) as follows:

FIG. 1 :
FIG.1:B − L asymmetry and neutrino abundance evolution during the expansion of the Universe in case of zero initial neutrino abundance.The blue line refers to the full solution of the Boltzmann's equations, obtained retaining the time dependence of the CP-violation parameter.The red line refers to the solution of the analogous system but adopting a time-constant value of the asymmetry parameters.Finally, the green line represents the abundance of the right-handed neutrinos, as given by the solution of the system.For reference, the latter is compared with the function N eq

FIG. 2 :
FIG. 2: Evolution of the B − L symmetry in the hierarchical scenario.The color code is the same as in fig.1.
2 sol /∆m 2 atm in terms of the model parameters are quite cumbersome.Thus, we prefer

TABLE III :
Set of parameter assignments which lead to the final baryon asymmetry value shown in fig.( ) a b c d 11 d 22 d 23 d 31 d 32 1.18 1.07 1.65 1.99 1.95 1.88 1.82 1.71

TABLE IV :
Set of parameter assignments which lead to a final baryon asymmetry close to the cosmological observations.See text for further details.