A New Littlest Seesaw Model
Ping-Tao Chen, Gui-Jun Ding, Stephen F. King, Cai-Chang Li
TL;DR
This paper introduces a novel Littlest Seesaw model using the tri-direct CP approach, accurately describing lepton flavor mixing and baryon asymmetry with minimal parameters and explicit flavor symmetry construction.
Contribution
It presents a new Littlest Seesaw model with specific couplings and phase, achieving excellent lepton mixing predictions and baryon asymmetry generation, along with an explicit flavor symmetry realization.
Findings
Accurately describes lepton flavor mixing including second octant atmospheric angle.
Generates observed baryon asymmetry with specific right-handed neutrino masses.
Provides explicit flavor symmetry model construction.
Abstract
We propose and discuss a new Littlest Seesaw model, realised in the tri-direct CP approach, in which the couplings of the two right-handed neutrinos to the lepton doublets are proportional to and respectively with the relative phase . This model can give an excellent description of lepton flavour mixing, including an atmospheric neutrino mixing angle in the second octant, in terms of only two input parameters. We show that the observed baryon asymmetry can be generated for the lightest right-handed neutrino mass GeV in SM and GeV in MSSM with . We construct an explicit Littlest Seesaw model based on the flavour symmetry in which the desired alignments and the phase are achieved.
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USTC-ICTS-19-15
**A New Littlest Seesaw Model
**
Ping-Tao Chen1111E-mail: [email protected], Gui-Jun Ding1222E-mail: [email protected], Stephen F. King2333E-mail: [email protected], Cai-Chang Li1,3444E-mail: [email protected]
1*Interdisciplinary Center for Theoretical Study and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
2*Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, U.K.
3*School of Physics, Northwest University, Xi’an 710127, China
Abstract
We propose and discuss a new Littlest Seesaw model, realized in the tri-direct CP approach, in which the couplings of the two right-handed neutrinos to the lepton doublets are proportional to and respectively with the relative phase . This model can give an excellent description of lepton flavour mixing, including an atmospheric neutrino mixing angle in the second octant, in terms of only two input parameters. We show that the observed baryon asymmetry can be generated for the lightest right-handed neutrino mass GeV in SM and GeV in MSSM with . We construct an explicit Littlest Seesaw model based on the flavour symmetry in which the desired alignments and the phase are achieved.
1 Introduction
The Standard Model (SM) has been well established by the discovery of the Higgs boson. However, the discovery of neutrino oscillations implies that neutrinos have masses and there are flavour mixing in lepton sector. Non-zero neutrino masses open up a window to the new physics beyond SM. However, the origin of neutrino mass generation and the flavour mixings in quark and lepton sectors are still unknown [1, 2]. In order to elegantly generate the tiny neutrino mass, the most appealing theory seems to be type I seesaw mechanism involving heavy right-handed Majorana neutrinos [3, 4, 5].
The type I seesaw mechanism can qualitatively explain the smallness of neutrino masses through the heavy right-handed neutrinos. However, if one doesn’t make other assumptions, the seesaw model with three right-handed neutrinos (RHN) contains too many parameters to make any particular predictions for neutrino mass and mixing. As we know, the idea of sequential dominance (SD) [6, 7] of right-handed neutrinos is an effective method to produce the mass hierarchy between the two mass squared differences and [8], it requires that the mass spectrum of heavy Majorana neutrinos is strongly hierarchical, i.e. . It arises from the proposal that a dominant heavy right-handed (RH) neutrino is mainly responsible for the atmospheric neutrino mass, a heavier subdominant RH neutrino for the solar neutrino mass, and a possible third largely decoupled RH neutrino for the lightest neutrino mass. It leads to an effective two right-handed neutrino (2RHN) model [9, 10]. This simple idea leads to equally simple predictions which makes the scheme falsifiable. Indeed, the litmus test of such SD is a very light (or massless) neutrino. These predictions will be tested soon. In order to further increase predictive power of the minimal seesaw mechanism, various proposals have been suggested, such as postulating one [11] or two [10] texture zeros in the neutrino Yukawa coupling. The models with two texture zero are excluded by the present data for normal ordering neutrino masses [12, 13, 14].
A very predictive minimal seesaw model with one texture zero is the so-called CSD() model [15, 16, 17, 18, 19, 20, 21, 22, 23, 24], where the parameter was usually assumed to be a positive integer. The CSD() scheme assumes that the two columns of the Dirac neutrino mass matrix are proportional to and respectively in the RHN diagonal basis. As a consequence, the lepton mixing matrix is predicted to be TM1 pattern, the neutrino masses are normal ordering and the lightest neutrino is massless with . At present only the CSD() (also called Littlest Seesaw model) [17, 18, 19, 20, 21] and CSD() models [22, 23] can give rise to phenomenologically viable predictions for lepton mixing parameters and the two neutrino mass squared differences and .
It has been shown that CSD() can be enforced by a residual symmetry of [19] in the semi-direct approach where different residual flavour symmetries and are assumed in the charged lepton and neutrino sectors. However, it was not possible to identify any residual CP symmetry for CSD() in the semi-direct approach. This means that the parameter of CSD(), which is usually assumed to be integer valued, could in fact be a complex number in general. In order to preserve the predictions of CSD(), we would like to fix the parameter to be real (although not necessarily an integer). This suggests that we should seek to somehow use residual CP symmetry, even though it is not possible within the semi-direct approach.
In the past years, discrete flavour symmetry has been combined with generalized CP symmetry to provide a powerful framework to explain the lepton mixing angles and predict leptonic CP violation phases [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57]. Furthermore, a simultaneous description of quark and lepton flavour mixing and CP violation can be achieved through spontaneous breaking of a discrete family symmetry and CP symmetry [52, 53, 54]. Since the generalized CP symmetry may play a critical role in understanding the flavour puzzle of SM, recently we extended the widely studied direct model of discrete flavour symmetry [1] to propose a new predictive neutrino mass model building scheme for the minimal seesaw model with two right-handed neutrinos called the tri-direct CP approach [58, 59].
The basic idea of the tri-direct CP approach is that the Yukawa interactions associated with each of the two right-handed neutrinos preserve different residual flavour and CP symmetries, and the charged lepton sector also has a different residual flavour symmetry. As a consequence, the flavour and generalized CP symmetry is spontaneously broken down to , and in the charged lepton, “atmospheric” and “solar” right-handed neutrino sectors, respectively [58]. Here is an abelian subgroup of and it allows the distinction of three generations of charged leptons as usual direct model. The residual subgroups and fix the alignments associated with each right-handed neutrino. We have performed a comprehensive analysis of lepton mixing patterns which can be obtained from the flavour group and CP symmetry in the tri-direct CP approach in a model independent fashion [59]. The model construction along the tri-direct CP approach was also illustrated [58, 59]. In the minimal seesaw model, a phenomenologically viable pattern of lepton mixing and neutrino masses can also be obtained from the breaking of flavour symmetry into three different subgroups in the charged lepton, atmospheric neutrino and solar neutrino sectors [60].
It is remarkable that the original Littlest Seesaw model for CSD(3) can be reproduced from the tri-direct CP approach [58, 59], if the flavour symmetry and CP symmetry are broken to the remnant symmetries , and in the charged lepton sector, the atmospheric sector and the solar neutrino sector, respectively, corresponding to the case. In this case, one row of the neutrino Dirac mass matrix is proportional to and the other row is proportional to , where is enforced to be a real parameter by the residual symmetry, thereby overcoming the previous problem where it could be complex in general. Then the light neutrino mass matrix is determined to be555Note that the seesaw mechanism results in a light effective Majorana mass matrix was defined in the convention Also note that here the second entries of the vacuum alignments which enter the Dirac mass matrix are multiplied by minus one as compared to the usual Littlest Seesaw convention. [59]
[TABLE]
where an overall phase has been neglected, , , and are four real free parameters. In a concrete model, the parameters and could be fixed to certain values through the technique of vacuum alignment [58, 59]. For example, CSD(3) corresponding to and , can be achieved within the case. Then all three mixing angles, two CP phases and three neutrino masses only depend on two real parameters and which can be determined by the mass squared differences and precisely measured in neutrino oscillation experiments. Then one can extract the predictions for all other mixing parameters. Obviously this kind of model is highly predictive.
In this paper, we shall focus on a particularly interesting example of the case with and , henceforth referred to as the new Littlest Seesaw, which was one of the best fit points found in [59] where the lepton mixing parameters and neutrino masses are predicted to lie in rather narrow regions, with an atmospheric angle in the second octant as preferred by the latest global fits. Motivated by the excellent agreement of this case with experimental data, in this work we develop further this new Littlest Seesaw model in two different ways: we discuss leptogenesis and we also construct a concrete model to demonstrate how it could arise from a realistic theory. We emphasise that the model involves a particularly simple and “maximal” phase which is the unique source of CP violation for both neutrino oscillations and leptogenesis. It is noteworthy that the observed value of the baryon asymmetry of our Universe will be obtained through flavoured thermal leptogenesis in both the SM and the Minimal Supersymmetric Standard Model (MSSM). We will propose an explicit supersymmetric (SUSY) model in the framework of minimal seesaw mechanism with 2RHN based on and show that the mass hierarchies of the charged lepton and the light neutrino mass matrix in Eq. (1) with and may be naturally derived in such a model.
The rest of this paper is organized as follows. In section 2, we revisit the case of tri-direct CP models with the alignments , which can be derived from the flavour symmetry in combination with CP symmetry, assuming the residual symmetry. We show that the new Littlest Seesaw model, which corresponds to a benchmark point in the case with and , provides an excellent fit to the experimental data of lepton mixing angles and neutrino masses. We study the predictions of the new Littlest Seesaw model for leptogenesis in the section 3, and show that the observed baryon asymmetry of the Universe can be produced for certain values of the lightest right-handed neutrino mass. In section 4, we construct a supersymmetric littlest tri-direct CP model based on the flavour symmetry , the alignment parameter and relative phase are achieved. The predictions for the charged lepton flavour violation radiative decays are studied, and we show a UV completion of the model. In section 5, we summarize our main results and draw the conclusions. We present the group theory and the Clebsch-Gordan coefficients of the group in Appendix A.
2 The case of tri-direct CP models revisited
The tri-direct CP approach is based on the minimal seesaw model with 2RHN. We denote the two right-handed neutrinos as (called “atmospheric”) and (called “solar”). Then the most general Lagrangian of the minimal seesaw model can be written as
[TABLE]
where two-component fermion notation for the fermion fields is adopted. The lepton doublets are assumed to transform as an irreducible triplet under (), and can be either Higgs fields or combinations of the electroweak Higgs doublet together with flavons, and they are also triplets ( and ). The two right-handed neutrinos are singlets of with and , the two flavons and are invariant under . The combination of flavons and the right-handed charged leptons must be embedded into the faithful three-dimensional representation of . Moreover, all coupling constants , , and are real because of the generalized CP symmetry.
We have performed an exhaustive analysis of all possible residual symmetries arising from in tri-direct CP approach and the resulting predictions for neutrino masses and flavour mixing parameters in [59]. Many independent phenomenologically viable residual symmetry cases are found (eight cases for normal ordering and eighteen cases for inverted ordering). In the present work, we shall consider the breaking pattern in which the residual symmetries in the charged lepton, atmospheric neutrino and solar neutrino sectors are , and respectively, the two residual CP symmetries are and . This is exactly the case of Ref. [59]. The residual symmetries in atmospheric neutrino and solar neutrino sectors require that the vacuum expectation values (VEVs) of the flavons and should take the following form
[TABLE]
where the parameters , and are real. Applying the well-known seesaw formula, the light neutrino mass matrix is really given by Eq. (1).
In our working basis (see Appendix A), requiring that the subgroup is a symmetry of the charged neutrino mass matrix entails that is diagonal and thus does not contribute to the lepton mixing. The lepton mixing matrix is found to be of the following form [59]:
[TABLE]
where is a diagonal phase matrix. We see that the first column of the mixing matrix is in common with that of the tri-bimaximal mixing matrix, and the so-called TM1 mixing matrix is obtained. The neutrino mass spectrum is normal ordering, the lightest neutrino is massless () since only two right-handed neutrinos are involved. The other two non-zero light neutrino masses and are given by
[TABLE]
where
[TABLE]
The expressions for the angles and phases , , and in Eq. (4) are:
[TABLE]
From the lepton mixing matrix in Eq. (4), one can straightforwardly extract the following results for the lepton mixing angles and CP invariants,
[TABLE]
where is the Jarlskog invariant [61] and is the Majorana invariant [62] related to the Majorana phase . We find that all mixing parameters and mass ratio depend on the three input parameters , and . However, the neutrino absolute masses and depend on all the four input parameters , , and . We find that the agreement with data is optimised by choosing
[TABLE]
which give rise to the following values of observables
[TABLE]
where refers to the effective Majorana mass in neutrinoless double beta decay, and is the Majorana phase. These predictions for lepton mixing angles agree with the experimental data quite well, and the global minimum of the function is . Note that the function includes the contributions of three mixing angles and two squared mass differences as usual. Because the indication of a preferred value of the Dirac phase from global data analyses is rather weak [8], we do not include any information on in the function. We emphasise that the values of the parameter , , and are not fixed by the residual symmetry, and can only be fixed by explicit model construction. This task is easier for the simpler values of and where the solar vacuum alignment is easier to achieve, therefore we are interested in the simplest values of these parameters.
We report the results of analysis for some representative values of and in table 1. Once the values of and are fixed, all the mixing parameters and neutrino masses only depend on the input parameters and whose values can be determined by the mass squared differences and . Then the three lepton mixing angles, two CP violation phases and the absolute neutrino mass scale are uniquely predicted by the theory. We notice that the effective Majorana mass lies in the range of 1 to 4 meV, consequently it is impossible to be measured in foreseeable future.
The original Littlest Seesaw model [18, 19, 20, 21] corresponds to the cases of , , and the CSD(4) model [22, 23] can be exactly reproduced for . From table 1, we see that the values , and can give rise to a smaller than the original Littlest Seesaw model and CSD(4) model [18, 19, 20, 21, 22, 23]. We have shown as a function of for in figure 1. Moreover, we plot the contour regions for the intervals of mixing angles and and mass ratio in the plane versus in figure 2. The result for is not displayed here, because it is related to the reactor angle by the TM1 mixing sum rule which leads to for the allowed range of [8].
From figures 1 and 2, we notice that the values of is quite sensitive to the phase and predictions for the mixing angles and neutrino masses can agree very well with the experimental data for certain choices of . Henceforth we shall focus on the new Littlest Seesaw model defined by the simple values and which give a phenomenologically successful and predictive description of lepton mixing parameters and neutrino masses, as highlighted with cyan background in table 1. Moreover, the corresponding vacuum alignment and the phase should be easy to realize in a concrete model. This new Littlest Seesaw model and the original Littlest Seesaw model differ in their predictions for and . The atmospheric mixing angle deviates from maximal mixing in the new Littlest Seesaw model while it is close to in the original Littlest Seesaw. Since deviation of from maximal mixing is preferred by the present data [8], the new littlest tri-direct CP model provides a better fit to the data of than the original Littlest Seesaw.
2.1 The new Littlest Seesaw: ,
Before getting into too many technicalities of model construction, we analyze the predictions for lepton mixing parameters and neutrino masses for . In this case, the light neutrino mass matrix in Eq. (1) becomes
[TABLE]
We note that all lepton mixing parameters and mass ratio are determined by only a single parameter . The expressions for the three lepton mixing angles and the CP invariants are given by
[TABLE]
with
[TABLE]
Notice that is predicted to lie in the second octant, it is preferred by the present neutrino oscillation data [8]. As both and depend on a single parameter , a sum rule between them can be obtained666The sum rule for is which holds true for all TM1 models.
[TABLE]
The two non-zero neutrino masses can be read off from Eq. (5) as,
[TABLE]
It is easy to see that the mass ratio only depends on the parameter . Consequently we can express the mass ratio in terms of as
[TABLE]
We plot the dependence of all lepton mixing parameters and mass ratio on the parameter in figure 3. Eliminating the input parameter , we can relate all above physical observables to the reactor mixing angle . We see from figure 3 that the three lepton mixing angles and neutrino mass ratio are within their ranges at the best fit point . The best fitting values of Dirac CP phase and Majorana CP phase are and , respectively. We numerically scan over the parameter space of and , and find the viable range of is to be compatible with the present neutrino oscillation data at level [8]. Furthermore, we find the neutrino masses and mixing parameters are predicted to lie in the following rather narrow regions,
[TABLE]
Therefore this new Littlest Seesaw model is very predictive and it should be easily excluded by precise measurement of , and in forthcoming neutrino facilities.
2.2 The new Littlest Seesaw as a limiting case of three right-handed neutrinos
We shall extend the idea of Littlest seesaw to the 3RHN model in the following. We denote the 3RHN as , and . Then for the seesaw Lagrangian in Eq. (2), the two additional terms related to the third right-handed neutrino can be written as
[TABLE]
Here the flavon is assigned to transform as triplet , both and are invariant under the actions of . As an example, we consider the case that the residual symmetry in the decoupled neutrino sector is with . Then the most general VEV of which preserves the above residual symmetry is
[TABLE]
Then the neutrino mass matrix in Eq. (11) becomes
[TABLE]
where , , , and are real. The first two terms coincide with those of the new Littlest Seesaw in Eq. (11), and the last term arises from the third decoupled right-handed neutrinos. An particularly interesting example is the case of , it predicts the best fit values of the mixing parameters as follows
[TABLE]
We see such that the condition of constrained sequence dominance is well satisfied. Therefore our new Littlest Seesaw with 2RHN can be regarded as a decoupling limit of the 3RHN model in the case of . Comparing the best fit values of 3RHN model in Eq. (20) with those of 2RHN model with and in table 1, we find that the 2RHN model is a good approximation of the 3RHN model.
3 Predictions for leptogenesis in the new Littlest Seesaw model
It is well-known fact that there is a predominance of matter over antimatter present in the observable Universe. The value of baryon asymmetry of the Universe normalised to the entropy density is [63],
[TABLE]
Apart from elegantly explaining the tiny neutrino masses, the seesaw mechanism provides a simple and attractive mechanism for understanding the matter-antimatter asymmetry of the Universe via leptogenesis [64]. The out-of-equilibrium decays of right-handed neutrinos in the early Universe generates a lepton asymmetry because of the CP violating Yukawa couplings. The lepton asymmetry is subsequently converted into a baryon asymmetry via sphaleron processes in the SM.
In our concerned model, the phase is the unique source of CP violation, and it controls CP violation in both neutrino oscillations and leptogenesis. Therefore the measurable CP violation in future neutrino oscillation experiments are closely related to the baryon asymmetry of the Universe. In the present work, we shall focus on the simplest version of the leptogenesis in which the lepton asymmetry is dominantly generated by the interactions and decay of the lightest right-handed neutrino. The phase is fixed to in the new Littlest Seesaw model, and it yields a Dirac CP violation phase . In this section, we shall study the prediction for leptogenesis within the framework of SM and MSSM. The condition of successful baryogenesis will allow us to determine the mass of the lightest right-handed neutrino in the new Littlest Seesaw model.
3.1 Leptogenesis for the new Littlest Seesaw model in the SM
In the SM, the final baryon asymmetry is given by [65]
[TABLE]
where the asymmetries () are defined as and they are conserved by the sphaleron processes [66]. refers to the lepton number densities of the flavour . Note that , and is normalised to the entropy density.
In the present work, we shall discuss the flavoured thermal leptogenesis scenario in 2RHN model with hierarchical Majorana masses (), where the two right-handed neutrino masses and are the masses of the right-handed neutrinos and , respectively. The flavoured thermal leptogenesis has been studied in detail [67, 66, 65]. It was shown that the Boltzmann equations describing the asymmetries in flavour space are given by [68]
[TABLE]
There is no sum over in the last term of Eq. (24), with being the temperature, and are the modified Bessel functions of the second kind, and denotes the density of the lightest right-handed neutrino 777We find that the observed excess of matter over antimatter can not be generated in the new Littlest Seesaw model if is the lightest right-handed neutrino.. and stand for the corresponding equilibrium number densities and they take the following form
[TABLE]
with . In order to obtain phenomenologically viable baryon asymmetry, the lighter right-handed neutrino mass is assumed in the interval of GeV GeV. In this scenario, the Yukawa interaction is in equilibrium, the and flavours are indistinguishable, and the lepton number densities and in the and flavour can be combined to and [66, 67, 65]. In this temperature range, the matrix in the Boltzmann equation Eq. (24) is given by [66]
[TABLE]
which arises from the washout term. The functions and in Eqs. (23) and (24) account for the presence of scatterings and scatterings in the washout term of the asymmetry respectively [69, 70]. In the strong washout regime, and can be approximated as [69, 70]
[TABLE]
with
[TABLE]
where GeV is the mass of the Higgs boson. The flavoured CP asymmetries in the decays of the lightest RHN into Higgs and leptons of different flavours are of the form [71, 70, 72, 73]
[TABLE]
where , is the neutrino Yukawa coupling matrix and it is a matrix with the following form
[TABLE]
where , and GeV is the VEV of the Higgs field. The loop function in Eq. (29) can be written as
[TABLE]
Since hierarchical RHN masses () are assumed, we can get the following approximation formula for the decay asymmetry
[TABLE]
For the breaking pattern discussed in section 2, the flavour dependent decay asymmetries are:
[TABLE]
In the new Littlest Seesaw model with , , () read as
[TABLE]
Note that which is defined in Eq. (1), once the value of is fixed through the masses squared differences and , and only depend on the lightest right-handed neutrino mass . In addition to the decay asymmetry, the washout parameter , which appears in the washout term of the Boltzmann equation, is given by
[TABLE]
where eV and the washout mass parameterizes the decay rate of into the leptons of flavour with
[TABLE]
From the Yukawa coupling matrix given in Eq. (30), we find is given by
[TABLE]
where is defined in Eq. (1). For the new Littlest Seesaw model, the best fitting value of is 888The parameter should be in the range in order to be compatible with present neutrino oscillation data. as shown in table 1. Then we can obtain the washout parameters as follows
[TABLE]
Hence all flavours are in the strong washout region. Numerically solving the Boltzmann equations in Eqs. (23, 24), we find that the observed baryon asymmetry fix the lightest right-handed neutrino mass in the new Littlest Seesaw model:
[TABLE]
We plot the baryon asymmetry with respect to the Dirac CP phase in figure 4. The width of the line comes from varying and over their allowed ranges, where all three mixing angles and two neutrino mass squared differences are required to lie in the experimentally preferred ranges [8].
3.2 Leptogenesis for the new Littlest Seesaw model in the MSSM
In the MSSM, the final baryon asymmetry can be computed from the following formula [74]
[TABLE]
In the MSSM, the contributions of and should be considered, which are the supersymmetric partners of the lightest right-handed neutrino and the lepton doublet respectively. In other words, the densities and should be included in the Boltzmann equations. Then the Boltzmann equations in MSSM are given by [68]
[TABLE]
where the total (particle and sparticle) asymmetries denoted as and
[TABLE]
with . The matrix in Eq. (41) depends on which MSSM interactions are in thermal equilibrium at the temperatures where leptogenesis takes place. Here we shall consider the case that the lightest right-handed neutrino mass is between GeV and GeV, where only the Yukawa couplings are in thermal equilibrium. Then the relevant flavour-dependent asymmetries are and , and is given by
[TABLE]
In the MSSM, the decay asymmetries are all equal () [71]. As a consequence, we will only show the results of in the following. Under the assumption of , the CP asymmetry () in the MSSM is given by
[TABLE]
where the function is of the following form
[TABLE]
Inserting the expression of function into in Eq. (44) we find
[TABLE]
for the most general case. In the new Littlest Seesaw model with , , the flavour dependent decay asymmetries are determined to be,
[TABLE]
The washout parameters and in Eq. (41) are defined as
[TABLE]
with
[TABLE]
The expressions of the washout parameters for the new Littlest Seesaw model are
[TABLE]
with . At the best fitting of our model, the values of the washout parameters are
[TABLE]
which implies all flavours are in the strong washout region. For illustration, we take and we find the experimentally observed values of the baryon asymmetry can be obtained if the lightest right-handed neutrino mass in the new Littlest Seesaw model is
[TABLE]
The correlation between and in the new Littlest Seesaw model is displayed in figure 5.
4 Explicit model for the new Littlest Seesaw
As we have shown in previous sections, the new Littlest Seesaw model can describe the experiment data of lepton mixing angles, neutrino masses and matter-antimatter asymmetry of the Universe very well. In this section, we shall construct an explicit model based on the model independent analysis of section 2. The vacuum alignments , and the phase parameter will be naturally realized in our model. We impose the flavour symmetry as well as CP symmetry. The standard supersymmetric driving field mechanism [75] which we adopt in our model requires a symmetry related to the usual -parity. Furthermore, we also introduce the shaping symmetry which allows us to forbid unwanted terms and achieve the desired vacuum alignment. The auxiliary symmetry is helpful to generate the phase . The shaping symmetry requires the electron, muon and tauon mass terms couple with different powers of flavon fields. Hence helps to reproduce the observed charged lepton mass hierarchies. Here we choose the right-handed charged leptons as singlets, where and transform as while transforms as . The three generations of left-handed lepton doublets are unified to an triplet . The two right-handed neutrinos and are assigned to and of , respectively. The field content and their classification under the flavour symmetry are listed in table 2. The driving fields are indicated with the superscript “0” and they carry two units of charge, both flavon fields and Higgs are uncharged under , and the -charge of the matter fields is equal to one. Since both flavon fields and driving fields are Standard Model singlets, our model is anomaly free under the Standard Model gauge transformation. As regards possible discrete anomalies, in principle they may be cancelled by adding extra states under the discrete group, but this is beyond the scope of this paper. In the following, we first discuss the vacuum alignment of the model, then specify the structure of the model.
4.1 Vacuum alignment
We employ the now-standard -term alignment mechanism to arrange the vacuum [75] in our model. It requires that all terms in the superpotential must carry two units of charge. Therefore each term in the superpotential contains either two matter superfields or only one driving field. The minimum of the scalar potential is determined by vanishing -terms of the driving fields. The leading order (LO) driving superpotential in which each term contains one driving field invariant under can be written as
[TABLE]
where stands for a contraction into the irreducible representation . Because we impose CP as symmetry on the model, all the couplings () and mass parameters , , , are constrained to be real. The VEVs of the flavon can be obtained from the vanishing of the derivatives of with respect to each component of the driving fields , i.e.
[TABLE]
One solution to these equations is
[TABLE]
where is undetermined. In the charged lepton sector, the -term conditions of the driving fields and give the vacuum alignment of ,
[TABLE]
Given the vacuum of in Eq. (55), we find the alignment of is
[TABLE]
with undetermined. In the atmospheric neutrino sector, the -term conditions associated with the driving fields , and read
[TABLE]
A straightforward calculation shows that the vacuum expectation values of , and are
[TABLE]
with
[TABLE]
It is easy to check that the vacuum alignments of flavons and preserve the subgroup . Now we consider the phases of and which is the VEV of . They are related by the -flatness of :
[TABLE]
From Eqs. (60) and (4.1), we find
[TABLE]
As all parameters in the right-hand side of above equation are real, consequently the phase of is [math], or for the product or , respectively. The auxiliary symmetry has played a critical role in generating the discrete possible values [math], , for the phase of . Subsequently we turn to discuss the vacuum alignment of the solar neutrino sector. The -flatness condition of the driving field gives
[TABLE]
which lead to the vacuum
[TABLE]
The equations giving the vacuum structure for the flavon field are:
[TABLE]
which uniquely determine the solar alignment,
[TABLE]
We find the vacuum configurations of and are invariant under the action of the subgroup . Finally the term condition of is
[TABLE]
which leads to the following relations
[TABLE]
The phase parameter is exactly the phase of the ratio in our model. Form Eqs. (60) and (68), it is easy to obtain
[TABLE]
All couplings and mass parameters in above equation are real due to CP symmetry, then we see the phase of the ratio is . In the present work we shall take the following solution
[TABLE]
which would happen for . Thus the desired vacuum alignment , and the phase have been dynamically realized. In the following section we will find that the observed hierarchy among the charged lepton masses can be produced for
[TABLE]
where is the cut-off scale of the theory and is the Cabibbo angle with . As usual, we expect that all the VEVs of flavons are of the same order of magnitude, i.e.
[TABLE]
Successful leptogenesis fixes the atmospheric neutrino mass to be GeV (see Eq. (52)) which is of the same order as the flavon VEVs. Thus the cut-off scale is expected to be of order GeV. The next-to-leading-order (NLO) corrections to the flavon superpotential involve three flavon fields. When the NLO corrections are included, the original symmetry is broken completely in the charged lepton, atmospheric neutrino and solar neutrino sectors. The NLO corrections to VEVs of all flavons are found to be suppressed by with respect to the LO contributions and therefore can be negligible, where denotes any flavour fields.
4.2 The structure of the model
The most relevant operators for charged lepton masses are given by
[TABLE]
where all the couplings are real because of the CP symmetry. After the electroweak and flavour symmetry breaking by the VEV shown in Eq. (57), one can obtain that the charged lepton mass matrix is diagonal with the masses
[TABLE]
where is the VEV of the electroweak Higgs field . Since the charged lepton mass matrix is diagonal, the hermitian combination is invariant under the action of the subgroup , i.e. . With the assignment in table 2, the tau, muon and electron masses arise at the one-flavon, two-flavons and three-flavons level respectively in our model. Consequently the charged lepton mass hierarchies are naturally reproduced
[TABLE]
We find that the subleading order corrections to the charged lepton mass matrix will break the residual symmetry but they are suppressed by with respect to LO results, thus can be safely neglected.
In the neutrino sector, the lowest dimensional operators responsible for neutrino masses are
[TABLE]
where the coupling constants , , and are restricted to be real by the imposed CP symmetry. Inserting the vacuum alignments in Eqs. (59, 66), we can read out the neutrino Dirac and Majorana mass matrices as follow,
[TABLE]
with . Applying the seesaw formula, we obtain the light neutrino mass matrix in Eq. (11) with
[TABLE]
Note that we have used the result shown in Eq. (70) under the assumption of . In the case of , in order to obtain the desired value , the phase of the ratio should be , i.e. we could choose the right side of Eq. (70) as . In short, the neutrino mass matrix of the new Littlest Seesaw model is realized exactly, hence the phenomenological predictions in section 2.1 follows immediately. The NLO contributions to the neutrino mass matrices in Eq. (77) are found to be suppressed by and consequently we will not discuss them.
Similar to other discrete flavour symmetry models, the solution of the vacuum alignment problem requires complicated constructions in our model and some new superfields which are SM singlets are introduced, as shown above. Recently it was suggested that the complexity of the vacuum alignment problem can be reduced if modular invariance plays the role of flavour symmetry [76]. In particular, we find that CSD() model with can be naturally obtained if the VEV of the complex modulus is at certain fixed point [77]. We expect that the desired alignment corresponding to can also be reproduced from some modular group, such that our model could be simplified considerably.
4.3 Charged lepton flavour violating radiative decays
In the following, we shall present the predictions for charged lepton flavour violating (LFV) radiative decays. It is usually assumed that the SUSY breaking mechanism is flavour blind at some high energy scale. In the minimal supergravity scenario, the slepton mass matrices are diagonal and universal in flavour and the trilinear couplings are proportional to the Yukawa couplings at the GUT scale. Non-vanishing off-diagonal elements are generated in both the slepton mass matrices and the trilinear couplings because of the renormalization group running effect at low energy, leading to charged lepton flavour violation processes induced in SUSY models. In the mass insertion and leading log approximations, the branching ratio of the charged lepton LFV radiative decay is given to good approximation by [78, 79]
[TABLE]
where is the Fermi coupling constant and is the characteristic mass scale of the SUSY particle in the loop with
[TABLE]
The slepton doublet mass squared arises from the renormalization group evolution. To an excellent approximation, the renormalization group result has the form
[TABLE]
where is the neutrino Yukawa coupling matrix given in Eq. (30) and the factor is defined as
[TABLE]
with the GUT scale GeV. For our model, we find the expressions of are as follows,
[TABLE]
As shown in Eq. (52), the right-handed neutrino mass is fixed to be GeV by leptogenesis. The best fit value of is in our model, consequently it is natural to take GeV. The measured values of the lepton mixing angles and neutrino masses fix meV which leads to . For typical values of the soft SUSY breaking parameters , , and , we find the branching ratios of the charged lepton flavour violating radiative decays to be
[TABLE]
which are safely below the present experimental upper limits [80].
4.4 UV completion
In our model, we see that all interactions are renormalizable except the the charged lepton Yukawa couplings in Eq. (LABEL:eq:ch_Yukawa) and the neutrino Yukawa couplings in Eq. (76). In the following, we shall give a UV completion which gives rise to these non-renormalizable operators upon integrating the heavy messengers fields. In order to generate the high dimensional operators relevant for charged lepton masses in Eq. (LABEL:eq:ch_Yukawa), we introduce three pairs of messenger fields and with which transform under the flavour symmetry as follows
[TABLE]
The chiral superfields and are singlets under the standard model gauge group and they carry hypercharges and respectively, and their charges are all . The renormalizable terms containing these messenger fields read as
[TABLE]
where all the couplings and masses are fixed to be real by the CP symmetry. Integrating out the heavy messenger fields, we obtain the desired higher-dimensional operators in the effective theory,
[TABLE]
which leads to a diagonal and hierarchical charged lepton mass matrix for the alignment of in Eq. (57). The non-renormalizable neutrino Dirac couplings in Eq. (76) can be generated with the help of the following heavy fields
[TABLE]
which are all standard model doublets with hypercharge ( for and for ). The relevant terms in the UV completion are given by
[TABLE]
where CP invariance requires the parameters and are real. Integrating out and , we reproduce the desired terms
[TABLE]
5 Conclusion
In this paper we have proposed and discussed a new Littlest Seesaw model, realized in the tri-direct CP approach, in which the couplings of the two right-handed neutrinos to the lepton doublets are proportional to and respectively with the relative phase . We have shown that this model can give an excellent description of lepton flavour mixing, including an atmospheric neutrino mixing angle in the second octant, in terms of only two input parameters. We also showed that the observed baryon asymmetry can be generated for the lightest right-handed neutrino mass GeV in SM and GeV in MSSM with . The model is based on the flavour symmetry in which the desired alignments and the phase are achieved.
We emphasise that the model independent tri-direct CP approach is a quite predictive scheme for constructing neutrino mass models based on discrete flavour symmetry and CP symmetry, even without specialising to a particular choice of the two real input parameters and . Here we have focussed on the case where the flavour symmetry and CP are broken to in the charged lepton sector, in the atmospheric sector and in the solar neutrino sector with and , the vacuum alignment of and would be fixed to and , where importantly is real due to the residual CP symmetry. As a consequence, the lepton mixing matrix is determined to be the TM1 pattern, and the experimental data on neutrino mixing can be described very well. Thus the structure is enforced by residual symmetry in tri-direct CP approach, with flavour symmetry yielding good agreement with the present data for many examples, which include both the original Littlest Seesaw model and the new Littlest Seesaw model [58, 59].
It is interesting to compare the new Littlest Seesaw with to the original Littlest Seesaw model with , [22, 18, 20], which also provides a good fit to the data, as summarized in table 1. However we find that the new Littlest Seesaw with arguably simpler values , , can provide a better description to the experimental data than the original Littlest Seesaw. The mixing parameters are predicted to lie in quite narrow regions, and they are all within the reach of future neutrino experiments. The denominator of the phase is the smallest one among the different benchmark values in table 1, consequently the case of , might be expected to be easier to realize in a concrete model than the original Littlest Seesaw and other cases listed in table 1.
We emphasise that the choice and of the new Littlest Seesaw model, is both simpler and more successful than the original Littlest Seesaw model. As usual, all three lepton mixing angles, leptonic CP violation phases and three neutrino masses () only depend on two input parameters and whose values can be determined by the precisely measured neutrino mass squared differences and . The comprehensive numerical analysis shows that all lepton mixing parameters and neutrino masses are restricted in rather narrow regions, as shown in Eq. (17). The new Littlest Seesaw differs most markedly in its predictions for and . While the atmospheric mixing angle is predicted to be close to maximal in the original Littlest Seesaw model, it is predicted to be in the second octant and close to the current central value [8] in the new Littlest Seesaw model. Furthermore, we have extended the new Littlest Seesaw to 3RHN models in the section 2.2. In the 3RHN model, we obtain a smaller than the new Littlest Seesaw, and we find that the 2RHN model is a good approximation of the 3RHN model. Therefore our new Littlest Seesaw with 2RHN can be regarded as a decoupling limit of the 3RHN model.
The “maximal” phase is the unique source of CP violation in the new Littlest Seesaw model, as usual controlling both low energy CP violation and the CP asymmetry in leptogenesis. Hence the CP violation which may be observed in neutrino oscillations is related to the baryon asymmetry of the Universe. We have studied the generation of the baryon asymmetry of the Universe through leptogenesis in the new Littlest Seesaw model. We have numerically solved the flavoured Boltzmann equations for the lepton asymmetries, and found that the observed excess of matter over antimatter can be produced for the lightest right-handed neutrino mass GeV in SM and GeV in MSSM with . We conclude that the new Littlest Seesaw model can give an excellent fit to the neutrino oscillation data and leptogenesis simultaneously.
Finally we have constructed a fully working explicit model based on the flavour group and CP symmetry which fixes the values of and in the new Littlest Seesaw model. The charged lepton mass hierarchy is naturally realized in our model, and the required vacua , and the relative phase are readily generated through the supersymmetric -term alignment mechanism. Furthermore, we have studied the predictions for the charged lepton radiative decays , and , and have found that the resulting branch ratios are below the current experimental upper bounds. We have also presented a UV completion which gives rise to the non-renormalizable operators upon integrating out the heavy messenger fields.
It would be interesting to extend this predictive new Littlest Seesaw model to the quark sector to give a unified description of quark and lepton flavour mixing, for instance in the framework of a supersymmetric grand unified theory. We expect that the quark mass matrices would be related to the construction of the new Littlest Seesaw model in the lepton sector. This is left for future work.
Acknowledgements
P.-T. C. and G.-J. D. acknowledge the support of the National Natural Science Foundation of China under Grant Nos 11522546 and 11835013. S. F. K. acknowledges the STFC Consolidated Grant ST/L000296/1 and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreements Elusives ITN No. 674896 and InvisiblesPlus RISE No. 690575. C.-C. L. is supported by National Natural Science Foundation of China under Grant No 11847228, China Postdoctoral Science Foundation Grant Nos. 2017M620258 and 2018T110617, CPSF-CAS Joint Foundation for Excellent Postdoctoral Fellows No. 2017LH0003, the Fundamental Research Funds for the Central Universities under Grant No. WK2030040090, the Anhui Province Natural Science Foundation Grant No. 1908085QA24 and the CAS Center for Excellence in Particle Physics (CCEPP).
Appendix
Appendix A Group Theory of
In the present work, we adopt the same convention for the flavour symmetry group as [27, 29]. The group is generated by three generators , and which obey the relations
[TABLE]
The group has elements and five irreducible representations: , , , and . The representation matrices of the three generators in different irreducible representations are chosen to be the following form
[TABLE]
with . As has been shown in [27, 29], the generalized CP transformation compatible with the flavour symmetry is of the same form as the flavour symmetry transformation in our working basis,
[TABLE]
where can be any of the 24 group elements of . The Clebsch-Gordan coefficients are frequently used when building a model based on flavour symmetry. We summarise the Kronecker products and Clebsch-Gordan coefficients in our basis in table 3.
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