A New Littlest Seesaw Model
Ping-Tao Chen Gui-Jun Ding Stephen F. King Cai-Chang Li












 

0.2



A New Littlest Seesaw Model


    
Ping-Tao Chen^1[E-mail: chenpt@mail.ustc.edu.cn],  Gui-Jun Ding^1[E-mail: dinggj@ustc.edu.cn],  Stephen F. King^2[E-mail: king@soton.ac.uk], Cai-Chang Li^1,3[E-mail: lcc0915@mail.ustc.edu.cn]  





^1Interdisciplinary Center for Theoretical Study and  Department of Modern Physics,

University of Science and Technology of China, Hefei, Anhui 230026, China

^2Physics and Astronomy,
University of Southampton,
Southampton, SO17 1BJ, U.K.

^3School of Physics, 
Northwest University, 
Xi'an 710127, China






    
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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 (0,-1,1) and (1,5/2,-1/2) respectively with the relative phase η=-π/2. 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 M_1=1.176× 10^11 GeV in SM and M_1=3.992× 10^10 GeV in MSSM with tanβ=5. We construct an explicit Littlest Seesaw model based on the flavour symmetry S_4× Z_5× Z_8 in which the desired alignments and the phase η=-π/2 are achieved.








§ 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 <cit.>. 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 <cit.>.

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) <cit.> of right-handed neutrinos  is an effective method to produce the mass hierarchy between the two mass squared differences Δ m^2_21 and Δ m^2_31 <cit.>, it requires that the mass spectrum of heavy Majorana neutrinos is strongly hierarchical, i.e. M_atm≪ M_sol≪ M_dec.
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  <cit.>. 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 <cit.> or two <cit.> texture zeros in the neutrino Yukawa coupling. The models with two texture zero are excluded by the present data for normal ordering neutrino masses <cit.>.

A very predictive minimal seesaw model with one texture zero is the so-called CSD(n) model  <cit.>, where the parameter n was usually assumed to be a positive integer. The CSD(n) scheme assumes that the two columns of the Dirac neutrino mass matrix are proportional to (0,1, -1) and (1, n, 2-n) 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 m_1=0. At present only the CSD(3) (also called Littlest Seesaw model) <cit.> and CSD(4) models <cit.> can give rise to phenomenologically viable predictions for lepton mixing parameters and the two neutrino
mass squared differences Δ m^2_21 and Δ m^2_31.

It has been shown that CSD(n) can be enforced by a residual symmetry of S_4
<cit.> in the semi-direct approach where different residual flavour symmetries G_l=Z^T_3
and G_ν=Z_2^SU are assumed in the charged lepton
and neutrino sectors. However, it was not possible to identify any residual CP symmetry
for CSD(n) in the semi-direct approach. This means that the parameter n of CSD(n), 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(n), we would like to fix the parameter n 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 <cit.>.
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 <cit.>. 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 <cit.> 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 <cit.>.

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 G_f⋊ H_CP is spontaneously broken down to G_l, G_atm⋊ H^atm_CP and G_sol⋊ H^sol_CP in the charged lepton, “atmospheric” and “solar” right-handed neutrino sectors, respectively <cit.>. Here G_l is an abelian subgroup of G_f and it allows the distinction of three generations of charged leptons as usual direct model. The residual subgroups G_atm⋊ H^atm_CP and G_sol⋊ H^sol_CP 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 S_4 and CP symmetry in the tri-direct CP approach in a model independent fashion <cit.>. The model construction along the tri-direct CP approach was also illustrated <cit.>. In the minimal seesaw model, a phenomenologically viable pattern of lepton mixing and neutrino masses can also be obtained from the breaking of A_5 flavour symmetry into three different subgroups in the charged lepton, atmospheric neutrino and solar neutrino sectors <cit.>.


It is remarkable that the original Littlest Seesaw model for CSD(3) can be reproduced from the tri-direct CP approach <cit.>, if the S_4 flavour symmetry and CP symmetry are broken to the remnant symmetries Z^T_3, Z^U_2× H^atm_CP and Z_2^SU× H^sol_CP in the charged lepton sector, the atmospheric sector and the solar neutrino sector, respectively, corresponding to the
𝒩_1 case. In this case, one row of the neutrino Dirac mass matrix is proportional to (0,-1,1) and the other row is proportional to (1, 2-x, x), where x 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 be[Note that the seesaw mechanism results in a light effective Majorana mass matrix was defined in the convention L_ eff=- 1/2ν_L^c m_νν_L  + h.c. 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.] <cit.>

    m_ν =m_a[  0  0  0;  0  1 -1;  0 -1  1;    ]+m_se^iη[       1     2-x       x;     2-x (x-2)^2 (2-x) x;       x (2-x) x     x^2;         ] ,

where an overall phase has been neglected, m_a, m_s, η and x are four real free parameters. In a concrete model, the parameters x and η could be fixed to certain values through the technique of vacuum alignment <cit.>. For example, CSD(3) corresponding to x=3
and η=2π /3, can be achieved within the 𝒩_1 case.
Then all three mixing angles, two CP phases and three neutrino masses only depend on two real parameters m_a and m_s which can be determined by the mass squared differences Δ m^2_21≡ m^2_2-m^2_1 and Δ m^2_31≡ m^2_3-m^2_1 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 𝒩_1 case with
x=-1/2 and η=-π/2,
henceforth referred to as the new Littlest Seesaw,
which was one of the best fit points found in <cit.> 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 η=-π/2 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 Y_B 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 S_4⋊ H_CP
and show that the mass hierarchies of the charged lepton and the light neutrino mass matrix in Eq. (<ref>) with x=-1/2 and η=-π/2 may be naturally derived in such a model.


The rest of this paper is organized as follows. In section <ref>, we revisit the 𝒩_1 case of tri-direct CP models with the alignments ⟨ϕ_atm⟩∝(0,1, -1)^T, ⟨ϕ_sol⟩∝(1, x, 2-x)^T which can be derived from the S_4 flavour symmetry in combination with CP symmetry, assuming the 𝒩_1 residual symmetry. We show that the new Littlest Seesaw model, which corresponds to a benchmark point in the 𝒩_1 case with x=-1/2 and η=-π/2, 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 <ref>, 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 <ref>, we construct a supersymmetric littlest tri-direct CP model based on the flavour symmetry S_4× Z_5× Z_8, the alignment parameter x=-1/2 and relative phase η=-π/2 are achieved. The predictions for the charged lepton flavour violation radiative decays ł_i→ l_jγ are studied, and we show a UV completion of the model. In section <ref>, we summarize our main results and draw the conclusions. We present the group theory and the Clebsch-Gordan coefficients of the S_4 group in Appendix <ref>.




§  THE 𝒩_1 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 N^c_atm (called “atmospheric”) and N^c_sol (called “solar”). Then the most general Lagrangian of the minimal seesaw model can be written as


    ℒ=-y_lLϕ_lE^c-y_atmLϕ_atmN^c_atm-y_solLϕ_solN^c_sol
    -1/2x_atmξ_atmN^c_atmN^c_atm-1/2x_solξ_solN^c_solN^c_sol
    +h.c. ,


where two-component fermion notation for the fermion fields is adopted. The lepton doublets L are assumed to transform as an irreducible triplet under S_4 (L∼3), ϕ_atm and ϕ_sol can be either Higgs fields
or combinations of the electroweak Higgs doublet together with flavons, and they are also S_4 triplets (ϕ_atm∼3 and ϕ_sol∼3^'). The two right-handed neutrinos are singlets of S_4 with N^c_atm∼1 and N^c_sol∼1^', the two flavons ξ_atm and ξ_sol are invariant under S_4. The combination of flavons ϕ_l and the right-handed charged leptons E^c≡(e^c, μ^c, τ^c)^T must be embedded into the faithful three-dimensional representation 3 of S_4. Moreover, all coupling constants y_atm, y_sol, x_atm and x_sol are real because of the generalized CP symmetry.

We have performed an exhaustive analysis of all possible residual symmetries arising from S_4⋊ H_CP in tri-direct CP approach and the resulting predictions for neutrino masses and flavour mixing parameters in <cit.>. 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 Z^T_3, Z^U_2× H^atm_CP and Z_2^SU× H^sol_CP respectively, the two residual CP symmetries are H^atm_CP={1,U} and H^sol_CP={1,SU}. This is exactly the case 𝒩_1 of Ref. <cit.>.
The residual symmetries in atmospheric neutrino and solar neutrino sectors require that the vacuum expectation values (VEVs) of the flavons  ϕ_atm and ϕ_sol should take the following form

    ⟨ϕ_atm⟩=v_atm(0,1, -1)^T,     ⟨ϕ_sol⟩=v_sol(1, x, 2-x)^T ,

where the parameters v_atm, v_sol and x are real. Applying the well-known seesaw formula, the light neutrino mass matrix m_ν is really given by Eq. (<ref>).


In our working basis (see Appendix <ref>),  requiring that the subgroup Z^T_3 is a symmetry of the charged neutrino mass matrix m_l entails that m^†_lm_l is diagonal and thus does not contribute to the lepton mixing. The lepton mixing matrix is found to be of the following form <cit.>:

    U_PMNS=
    [                    √(2/3)                 cosθ/√(3)            e^i ψsinθ/√(3);                   -1/√(6) cosθ/√(3)+e^-i ψsinθ/√(2)  e^i ψsinθ/√(3)-cosθ/√(2);                   -1/√(6) cosθ/√(3)-e^-i ψsinθ/√(2)  cosθ/√(2)+e^i ψsinθ/√(3);                           ]P_ν ,

where P_ν=diag(1, e^i(ψ+ρ)/2, e^i(-ψ+σ)/2) 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 (m_1=0) since only two right-handed neutrinos are involved. The other two non-zero light neutrino masses m_2 and m_3 are given by

    m^2_2=m^2_a/2[9 r^2+w^2+12 r^2 (x-1)^2-√(B)],   
    m^2_3=m^2_a/2[9 r^2+w^2+12 r^2 (x-1)^2+√(B)] ,

where

    r=m_s/m_a,    w=2 √(1+r^2 (x-1)^4+2 r (x-1)^2 cosη),
          B=(9 r^2-w^2)^2+24  r^2 (x-1)^2A ,
          A=9 r^2+w^2+6 r w cos (η -ϕ_w),  ϕ_w=arg(1+r (x-1)^2 e^ i η)  .

The expressions for the angles and phases θ, ψ, ρ and σ in Eq. (<ref>) are:

    cos2θ=w^2-9 r^2/√(B),     sin2θ=2 √(6A)  r (x-1)/√(B),     sinψ=-w sin (η -ϕ_w)/√(A) , 
          cosψ=3 r+w cos (η -ϕ_w)/√(A),     sin(ρ-σ)=3r wm^2_a √(B)sin (η -ϕ_w)/m_2m_3A  .

From the lepton mixing matrix in Eq. (<ref>), one can straightforwardly extract the following results for the lepton mixing angles and CP invariants,

    sin^2θ_13=sin ^2θ/3=1/6(1-w^2-9 r^2/√(B)) ,     sin^2θ_12=2 cos ^2θ/5+cos 2 θ=1/3(1-2tan^2θ_13) ,
          sin^2θ_23=1/2-√(6)sin 2 θcosψ/5+cos 2 θ=1/2-12  r (x-1)[3 r+w cos (η -ϕ_w)]  /5√(B)+w^2-9 r^2 , 
           J_CP=sin 2θsinψ/6 √(6)=-w  r (x-1)sin (η -ϕ_w)/3 √(B) , 
           I_1=1/36sin ^22 θsin (ρ -σ )=2 r^3 w (x-1)^2 sin (η -ϕ_w)/m_2m_3√(B) ,


where J_CP is the Jarlskog invariant <cit.> and I_1 is the Majorana invariant <cit.> related to the Majorana phase φ. We find that all mixing parameters and mass ratio m_2/m_3 depend on the three input parameters x, η and r=m_s/m_a. However, the neutrino absolute masses m_2 and m_3 depend on all the four input parameters x, η, m_a and r. We find that the agreement with data is optimised by choosing

    m_a=23.133 meV,    r=0.135,   η=-0.542π,    x=-0.615 ,

which give rise to the following values of observables

    sin^2θ_13=0.02241,   sin^2θ_12=0.318,   sin^2θ_23=0.582,   δ_CP=-0.382π,  φ=0.333π , 
          m_1=0 meV,      m_2=8.597 meV,      m_3=50.249 meV,      m_ee=3.112 meV ,

where m_ee refers to the effective Majorana mass in neutrinoless double beta decay, and φ is the Majorana phase. These predictions for lepton mixing angles agree with the experimental data quite well, and the global minimum of the χ^2 function is χ^2_min=0.384. Note that the χ^2 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 δ_CP from global data analyses is rather weak <cit.>, we do not include any information on δ_CP in the χ^2 function.
We emphasise that the values of the parameter x, η, r and m_a 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 x and η where the solar vacuum alignment ⟨ϕ_sol⟩ is easier to achieve, therefore we are interested in the simplest values of these parameters.

We report the results of χ^2 analysis for some representative values of x and η in table <ref>. Once the values of x and η are fixed, all the mixing parameters and neutrino masses only depend on the input parameters m_a and r whose values can be determined by the mass squared differences Δ m^2_21 and Δ m^2_31. 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 m_ee lies in the range of 1 to 4 meV, consequently it is impossible to be measured in foreseeable future.




The original Littlest Seesaw model <cit.> corresponds to the cases of (x, η)=(3, 2π/3), (-1, -2π/3), and the CSD(4) model <cit.> can be exactly reproduced for (x, η)=(4, 4π/5). From table <ref>, we see that the values (x, η)=(-1/2, ±π/2), (-3/4, ±3π/5) and (-4/5, ±3π/5) can give rise to a smaller χ^2_min than the original Littlest Seesaw model and CSD(4) model <cit.>. We have shown χ^2_min as a function of η for x=3, 4, -1/2,-3/4, -3/5 in figure <ref>. Moreover, we plot the contour regions for the 3σ intervals of mixing angles θ_13 and θ_23 and mass ratio
m_2/m_3 in the plane r versus η/π in figure <ref>.  The result for θ_12 is not displayed here, because it is related to the reactor angle θ_13 by the TM1 mixing sum rule cos^2θ_12cos^2θ_13=2/3 which leads to
0.316≤sin^2θ_12≤0.319 for the 3σ allowed range of θ_13 <cit.>.

From figures <ref> and <ref>, we notice that the values of χ^2_min is quite sensitive to the phase η and predictions for the mixing angles and neutrino masses can agree very well with the experimental data for certain choices of η. Henceforth we shall focus on the new Littlest Seesaw model defined by
the simple values x=-1/2 and η=±π/2 which give a phenomenologically successful and predictive description of lepton mixing parameters and neutrino masses, as highlighted with cyan background in table <ref>. Moreover, the corresponding vacuum alignment ⟨ϕ_sol⟩∝(1,-1/2,5/2) and the phase η=±π/2 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 θ_23 and δ_CP. The atmospheric mixing angle θ_23 deviates from maximal mixing in the new Littlest Seesaw model while it is close to 45^∘ in the original Littlest Seesaw. Since deviation of θ_23 from maximal mixing is preferred by the present data <cit.>, the new littlest tri-direct CP model provides a better fit to the data of θ_23 than the original Littlest Seesaw.







 §.§  The new Littlest Seesaw: x=-1/2, η=-π/2 

Before getting into too many technicalities of model construction, we analyze the predictions for lepton mixing parameters and neutrino masses for x=-1/2, η=-π/2. In this case, the light neutrino mass matrix in Eq. (<ref>) becomes

    m_ν=m_a[  0  0  0;  0  1 -1;  0 -1  1;    ]-im_s/4[  4 10 -2; 10 25 -5; -2 -5  1;    ] .

We note that all lepton mixing parameters and mass ratio m_2/m_3 are determined by only a single parameter r=m_s/m_a. The expressions for the three lepton mixing angles and the CP invariants are given by

    sin^2θ_13=1/6(1-45 r^2+16/C_r) ,     sin^2θ_12=1-4 C_r/5 C_r+45 r^2+16 , 
          sin^2θ_23=1/2+540 r^2/5 C_r+45 r^2+16 ,     J_CP=-4r/ C_r,      I_1=-6r^2/ C_r ,

with

    C_r=4√(B).|_x=-1/2,η=-π/2=√((225 r^2+16)^2-2304 r^2) .

Notice that θ_23 is predicted to lie in the second octant, it is preferred by the present neutrino oscillation data <cit.>. As both θ_13 and θ_23 depend on a single parameter r, a sum rule between them can be obtained[The sum rule for θ_12 is cos^2θ_12cos^2θ_13=2/3 which holds true for all TM1 models.]

    sin^2θ_23=1+4 sin ^2θ_13+√(1+28 sin ^2θ_13  (1-3 sin ^2θ_13 ))/4 cos ^2θ_13 .

The two non-zero neutrino masses can be read off from Eq. (<ref>) as,

    m^2_2=1/8m^2_a  (16+225 r^2- C_r) ,     
    m^2_3=1/8m^2_a  (16+225 r^2+ C_r)  .

It is easy to see that the mass ratio m_2/m_3 only depends on the parameter r. Consequently we can express the mass ratio m^2_2/m^2_3 in terms of θ_13 as

    m^2_2/m^2_3=10 sin ^2θ_13  (3 sin ^2θ_13 -1)+√(1+28 sin ^2θ_13  (1-3 sin ^2θ_13 ))-1/2 sin ^2θ_13  (15 sin ^2θ_13 -8)+2 .


We plot the dependence of all lepton mixing parameters and mass ratio m_2/m_3 on the parameter r in figure <ref>. Eliminating the input parameter r, we can relate all above physical observables to the reactor mixing angle θ_13. We see from  figure <ref> that the three lepton mixing angles and neutrino mass ratio are within their 1σ ranges at the best fit point r=0.145. The best fitting values of Dirac CP phase and Majorana CP phase are δ_CP≃-0.354π and φ≃0.316π, respectively. We numerically scan over the parameter space of m_a and r, and find the viable range of r is r∈[0.139,0.153] to be compatible with the present neutrino oscillation data at 3σ level <cit.>. Furthermore, we find the neutrino masses and mixing parameters are predicted to lie in the following rather narrow regions,

    0.3167≤sin^2θ_12≤0.3194,      0.02044≤sin^2θ_13≤0.02437,      0.593≤sin^2θ_23≤0.609, 
          -0.358≤δ_CP/π≤-0.348,       0.308≤φ/π≤0.322,       3.084 meV≤ m_ee≤3.388 meV, 
          8.319 meV≤ m_2≤8.950 meV,      49.305 meV≤ m_3≤51.206 meV .

Therefore this new Littlest Seesaw model is very predictive and it should be easily excluded by precise measurement of θ_12, θ_23 and δ_CP in forthcoming neutrino facilities.




 §.§ 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 N^c_atm, N^c_sol and N^c_dec. Then for the seesaw Lagrangian in Eq. (<ref>), the two additional terms related to the third right-handed neutrino N^c_dec can be written as

    Δℒ=-y_decLϕ_decN^c_dec
    -1/2x_decξ_decN^c_decN^c_dec
    +h.c. .

Here the flavon ϕ_dec is assigned to transform as S_4 triplet 3, both ξ_dec and N^c_dec are invariant under the actions of S_4. As an example, we consider the case that the residual symmetry in the decoupled neutrino sector is Z^T^2ST_2× H^dec_CP with H^dec_CP={SU,TST^2U}. Then the most general VEV of ϕ_dec which preserves the above residual symmetry is

    ⟨ϕ_dec⟩∝(1, ω, ω^2)^T .


Then the neutrino mass matrix in Eq. (<ref>) becomes

    m_ν=m_a[  0  0  0;  0  1 -1;  0 -1  1 ]-i/4m_ar
    [  4 10 -2; 10 25 -5; -2 -5  1 ]+m_ar^' e^iη^'[    1 ω ^2    ω; ω ^2    ω    1;    ω    1 ω ^2 ] ,

where m_a, r, r^', η and η^' are real. The first two terms coincide with those of the new Littlest Seesaw in Eq. (<ref>), and the last term arises from the third decoupled right-handed neutrinos. An particularly interesting example is the case of η^'=0, it predicts the best fit values of the mixing parameters as follows

    -0.15in  m_a=22.663 meV,     r=0.141,    r^'=0.00834,   η=-π/2,  η^'=0,   χ^2_min=1.157 ,
          -0.15in  sin^2θ_13=0.0224,  sin^2θ_12=0.318,   sin^2θ_23=0.595,   δ_CP=-0.363π,   α_21=0.394π , 
          -0.15in  α_31=0.0716π,     m_1=0.285 meV,    m_2=8.577 meV,     m_3=50.283,    m_ee=3.197 meV .

We see r'≪ r≪ 1 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 M_ dec≫ M_ atm, M_ sol.

Comparing the best fit values of 3RHN model in Eq. (<ref>) with  those of 2RHN model with x=-1/2 and η=-π/2 in table <ref>, we find that the 2RHN model is a good approximation of the 3RHN model.







§ 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 <cit.>,

    Y_B=(0.870300± 0.011288)×10^-10   (95%CL) .

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 <cit.>. 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 η 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 η is fixed to η=-π/2 in the new Littlest Seesaw model, and it yields a Dirac CP violation phase δ_CP≃1.646π. 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.





 §.§  Leptogenesis for the new Littlest Seesaw model in the SM  



In the SM, the final baryon asymmetry is given by <cit.>

    Y_B=12/37∑_α Y_Δ_α ,

where the asymmetries Y_Δ_α (α=e,μ,τ) are defined as Y_Δ_α≡ Y_B/3 - Y_L_α and they are conserved by the sphaleron processes <cit.>. Y_L_α refers to the lepton number densities of the flavour α. Note that Y_B, Y_Δ_α and Y_L_α 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 (M_1≪ M_2), where the two right-handed neutrino masses M_1=x_atm⟨ξ_atm⟩ and M_2=x_sol⟨ξ_sol⟩ are the masses of the right-handed neutrinos N_atm and N_sol, respectively. The flavoured thermal leptogenesis has been studied in detail <cit.>. It was shown that the Boltzmann equations describing the asymmetries in flavour space are given by <cit.>

    d Y_N_atm/d z   =    K(Y_N_atm^eq-Y_N_atm)z  f_1(z)K_1(z)/K_2(z) ,
       d Y_Δ_α/d z   =   ε_1 α^SM  K(Y_N_atm^eq-Y_N_atm)z  f_1(z)K_1(z)/K_2(z)+K_α  Y_N_atm^eqz  f_2(z)K_1(z)/K_2(z)∑_γ A_αγ^SM Y_Δ_γ/Y_ℓ^eq .

There is no sum over α in the last term of Eq. (<ref>), z= M_1/T with T being the temperature, K_1(z) and K_2(z) are the modified Bessel functions of the second kind, and Y_N_atm denotes the density of the lightest right-handed neutrino N_atm[We find that the observed excess of matter over antimatter can not be generated in the new Littlest Seesaw model if N_sol is the lightest right-handed neutrino.]. Y_N_atm^eq and Y_ℓ^eq stand for the corresponding equilibrium number densities and they take the following form

    Y_ℓ^eq≃45/π^4g_*^SM,     Y_N_atm^eq≃45z^2K_2(z)/2π^4g_*^SM ,

with g_*^SM=106.75. In order to obtain phenomenologically viable baryon asymmetry,  the lighter right-handed neutrino mass  M_1 is assumed in the interval of 10^9 GeV ≤ M_1≤10^12 GeV. In this scenario, the τ Yukawa interaction is in equilibrium, the e and μ flavours are indistinguishable, and the lepton number densities and Y_Δ_α in the e and μ flavour can be combined to Y_2≡ Y_e+μ and Y_Δ_2≡ Y_Δ_e+Δ_μ <cit.>. In this temperature range, the matrix A^SM in the Boltzmann equation Eq. (<ref>) is given by <cit.>

    A^SM=1/589( [ -417  120;   30 -390 ]) ,

which arises from the washout term. The functions f_1(z) and f_2(z) in Eqs. (<ref>) and (<ref>) account for the presence of Δ L=1 scatterings and scatterings in the washout term of the asymmetry respectively <cit.>.

In the strong washout regime, f_1(z) and f_2(z) can be approximated as <cit.>

    f_1(z)=2f_2(z)=[K_s/zK +z/tln(1+t/z)] K_2(z)/K_1(z) ,

with

    t=K/K_sln(M_1/M_h),       K_s/K=9/8π^2 .

where M_h=125 GeV is the mass of the Higgs boson. The flavoured CP asymmetries in the decays of the lightest RHN N_atm into Higgs and leptons of different flavours are of the form <cit.>

    ε_1 α^SM
      =1/8 π(λλ^†)_11{[(λλ^†)_1 2λ_1 αλ^*_2 α]g^SM(y)
      +1/y-1[(λλ^†)_2 1λ_1 αλ^*_2 α]} ,

where y=M^2_2/M^2_1, λ is the neutrino Yukawa coupling matrix and it is a 2×3 matrix with the following form

    λ=[              0             -a              a;       be^-iπ/4 5/2  b e^-iπ/4   -1/2be^-iπ/4;                ] ,

where a=|y_atmv_atm|/v, b=|y_solv_sol|/v and v=246/√(2) GeV is the VEV of the Higgs field. The loop function g^SM(y) in Eq. (<ref>) can be written as

    g^SM(y)=√(y)[1/1-y+1-(1+y) ln(1+y/y)] y ≫ 1⟶-3/2 √(y) .

Since hierarchical RHN masses M_1≪ M_2 (y≫1) are assumed, we can get the following approximation formula for the decay asymmetry

    ε_1 α^SM=-3/16 π[(λλ^†)_1 2λ_1 αλ^*_2 α]/(λλ^†)_11M_1/M_2 .

For the breaking pattern discussed in section <ref>, the flavour dependent decay asymmetries are:

    ε_1 e^SM=0,     ε_1 μ^SM=3/16 πM_1/M_2(x-1)(x-2)b^2sinη,     ε_1τ^SM=3/16 πM_1/M_2x(x-1)b^2sinη .

In the new Littlest Seesaw model with x=-1/2, η=-π/2, ε_1 α^SM (α=e,μ,τ) read as

    ε_1 e^SM=0,     ε_1 μ^SM=-45/64 πM_1/M_2b^2,     ε_1τ^SM=-9/64 πM_1/M_2b^2 .

Note that b^2/M_2∝ m_s which is defined in Eq. (<ref>), once the value of m_s is fixed through the masses squared differences Δ m^2_21 and Δ m^2_31, ε_1μ^SM and ε_1τ^SM only depend on the lightest right-handed neutrino mass M_1. In addition to the decay asymmetry, the washout parameter K_α, which appears in the washout term of the Boltzmann equation, is given by

    K_α=m_1α/m^*_SM,     K=∑_αK_α ,

where m^*_SM≃1.08×10^-3 eV and the washout mass m_1α parameterizes the decay rate of N_atm into the leptons of flavour α with

    m_1α≡|λ_1α|^2v^2/M_1 .

From the Yukawa coupling matrix λ given in Eq. (<ref>), we find K_α is given by

    K_e=0,       K_μ=K_τ=m_a/m^*_SM .

where m_a=a^2v^2/M_1 is defined in Eq. (<ref>). For the new Littlest Seesaw model, the best fitting value of m_a is m_a=22.366meV[The parameter m_a should be in the range 21.707meV≤ m_a≤23.019meV in order to be compatible with present neutrino oscillation data.] as shown in table <ref>. Then we can obtain the washout parameters K_α as follows

    K_e+K_μ=20.709≫ 1,       K_τ=20.709≫ 1 .

Hence all flavours are in the strong washout region. Numerically solving the Boltzmann equations in Eqs. (<ref>, <ref>), we find that the observed baryon asymmetry  Y_B=8.7×10^-11 fix the lightest right-handed neutrino mass in the new Littlest Seesaw model:

    M_1=1.176×10^11GeV .

We plot the baryon asymmetry Y_B with respect to the Dirac CP phase δ_CP in figure <ref>. The width of the line comes from varying m_a and r over their allowed ranges, where all three mixing angles and two neutrino mass squared differences are required to lie in the experimentally preferred 3σ ranges <cit.>.





 §.§  Leptogenesis for the new Littlest Seesaw model in the MSSM


In the MSSM, the final baryon asymmetry can be computed from the following formula <cit.>

    Y_B=10/31∑_αŶ_Δ_α .

In the MSSM, the contributions of N_1 and L_α should be considered, which are the supersymmetric partners of the lightest right-handed neutrino N_1 and the lepton doublet L_α respectively. In other words, the densities Y_N_1 and Y_α should be included in the Boltzmann equations. Then the Boltzmann equations in MSSM are given by <cit.>

    d Y_N_atm/d z   =   2 K(Y^eq_N_atm-Y_N_atm )z  f_1 (z)K_1 (z)/K_2 (z)  ,
    d Y_N_atm/d z   =    2 K(Y^eq_N_atm-Y_N_atm )z f_1 (z)K_1 (z)/K_2 (z)  ,
    dŶ_Δ_α/d z   =   K(ε_1α^MSSM+ε_1α^MSSM) (Y^eq_N_atm-Y_N_atm )z f_1 (z)K_1 (z)/K_2 (z)
          +K(ε_1α^MSSM+ε_1α^MSSM) (Y^eq_N_atm-Y_N_atm )z f_1 (z)K_1 (z)/K_2 (z)
          +
    K_α(Y^eq_N_atm+Y^eq_N_atm)    zf_2 (z)K_1 (z)/K_2 (z)∑_γ  A_αγ^MSSMŶ_Δ_γ/Ŷ^eq_ℓ ,

where the total (particle and sparticle) B/3 - L_α asymmetries denoted as Ŷ_Δ_α and

    Ŷ_ℓ^eq=Y^eq_ℓ̃+Y^eq_ℓ,    
    Y^eq_ℓ̃≃ Y^eq_ℓ≃45/π^4g_*^MSSM ,     Y_N_atm^eq=Y_N_atm^eq≃45z^2K_2(z)/2π^4g_*^MSSM ,

with g_*^MSSM=228.75. The matrix A^MSSM in Eq. (<ref>) 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 M_1 is between (1+tan^2β)×10^9 GeV and (1+tan^2β)×10^12 GeV, where only the τ Yukawa couplings are in thermal equilibrium. Then the relevant flavour-dependent asymmetries are Ŷ_Δ_2≡Ŷ_Δ_e+Δ_μ and Ŷ_Δ_τ, and A^MSSM is given by

    A^MSSM=1/761( [ -541  152;   46 -494 ]) .

In the MSSM, the decay asymmetries are all equal (ε^MSSM_1α =
ε^MSSM_1α =
ε^MSSM_1α =
ε^MSSM_1α) <cit.>. As a consequence, we will only show the results of ε^MSSM_1α in the following. Under the assumption of M_1≪ M_2, the CP asymmetry ε^MSSM_1α (α=e,μ,τ) in the MSSM is given by

    ε_1 α^MSSM=1/8 π(λλ^†)_11[(λλ^†)_1 2λ_1 αλ^*_2 α]g^MSSM(M_2^2/M_1^2) ,

where the function g^MSSM(y) is of the following form

    g^MSSM(y)=√(y)[2/1-y-ln(1+y/y)] y ≫ 1⟶-3/√(y) .

Inserting the expression of function g^MSSM(M^2_2/M^2_1) into ε_1 α^MSSM in Eq. (<ref>) we find

    ε_1 e^MSSM=0,   ε_1 μ^MSSM=3/8 πM_1/M_2(x-1)(x-2)b^2sinη,   ε_1τ^MSSM=3/8 πM_1/M_2x(x-1)b^2sinη ,

for the most general case.  In the new Littlest Seesaw model with x=-1/2, η=-π/2, the flavour dependent decay asymmetries are determined to be,

    ε_1 e^MSSM=0,     ε_1 μ^MSSM=-45/32 πM_1/M_2b^2,     ε_1 τ^MSSM=-9/32 πM_1/M_2b^2 .

The washout parameters K_α and K in Eq. (<ref>) are defined as

    K_α=m̃_1α/m^*_MSSM,    m̃_1α≡|λ_1α|^2v_u^2/M_1,      K=∑_αK_α ,

with

    v_u=vsinβ,       m^*_MSSM≃sin^2β×1.58×10^-3 eV .

The expressions of the washout parameters for the new Littlest Seesaw model are

    K_e=0,       K_μ=K_τ=m_a/m^*_MSSM ,

with m_a=a^2v^2_u/M_1. At the best fitting of our model, the values of the washout parameters are

    K_e+K_μ=14.722≫ 1,     K_τ=14.722≫ 1 ,

which implies all flavours are in the strong washout region. For illustration, we take tanβ=5 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

    M_1=3.992×10^10GeV .

The correlation between Y_B and δ_CP
in the new Littlest Seesaw model is displayed in figure <ref>.





§   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 <ref>. The vacuum alignments ⟨ϕ_atm⟩∝(0,1,-1), ⟨ϕ_sol⟩∝(1,-1/2,5/2) and the phase parameter η=-π/2 will be naturally realized in our model. We impose the S_4 flavour symmetry as well as CP symmetry. The standard supersymmetric driving field mechanism <cit.> which we adopt in our model requires a U(1)_R symmetry related to the usual R-parity. Furthermore, we also introduce the shaping symmetry Z_5× Z_8 which allows us to forbid unwanted terms and achieve the desired vacuum alignment. The auxiliary symmetry Z_8 is helpful to generate the phase η=-π/2. The shaping symmetry Z_5 requires the electron, muon and tauon mass terms couple with different powers of flavon fields. Hence Z_5 helps to reproduce the observed charged lepton mass hierarchies. Here we choose the right-handed charged leptons as S_4 singlets, where e^c and τ^c transform as 1 while μ^c  transforms as 1^'. The three generations of left-handed lepton doublets L are unified to an S_4 triplet 3. The two right-handed neutrinos ν^c_atm and ν^c_sol are assigned to 1 and 1' of S_4, respectively. The field content and their classification under the flavour symmetry S_4× Z_5× Z_8 are listed in table <ref>. The driving fields are indicated with the superscript “0” and they carry two units of R charge, both flavon fields and Higgs are uncharged under U(1)_R, and the R-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.
























































 §.§ Vacuum alignment



We employ the now-standard F-term alignment mechanism to arrange the vacuum <cit.> in our model. It requires that all terms in the superpotential must carry two units of R 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 F-terms of the driving fields. The leading order (LO) driving superpotential w_d in which each term contains one driving field invariant under S_4× Z_5× Z_8 can be written as

    w_d   =   f_1(η^0_l(χ_lχ_l)_2)_1+f_2ξ^0_l(ϕ_lϕ_l)_1
    +f_3ζ^0_l(ϕ_lχ_l)_1+f_4ξ^0_a(χ_lϕ_a)_1+M_η(η^0_aη_a)_1
          +f_5(η^0_a(ϕ_aϕ_a)_2)_1+f_6ζ_a(ϕ^0_aϕ_a)_1^'
    +f_7(ϕ^0_a(η_aχ_l)_3^')_1
    +M^2_σ_1σ^0_1+f_8σ^0_1ξ^2_a 
          +M^2_σ_2σ^0_2+f_9σ^0_2(η_aη_a)_1
    +f_10(φ^0_s(ϕ_aφ_s)_3^')_1
    +f_11ζ^0_s(φ_sϕ_s)_1^'+f_12(ϕ^0_s(ϕ_lϕ_a)_3^')_1
          +f_13(ϕ^0_s(φ_sϕ_s)_3^')_1+M_ξξ^0_sξ_s +f_14ξ^0_s(ϕ_sϕ_s)_1 ,

where (…)_𝐫 stands for a contraction into the S_4 irreducible representation 𝐫. Because we impose CP as symmetry on the model, all the couplings f_i (i=1,⋯,14) and mass parameters M_η, M_ξ, M_σ_1, M_σ_2 are constrained to be real. The VEVs of the flavon χ_l can be obtained from the vanishing of the derivatives of w_d with respect to each component of the driving fields η^0_l, i.e.

    ∂ w_d/∂η^0_l_1=f_1(2 χ_l_1χ_l_2+χ_l_3^2)=0 ,
          ∂ w_d/∂η^0_l_2=f_1(2 χ_l_1χ_l_3+χ_l_2^2)=0 .

One solution to these equations is

    ⟨χ_l⟩=v_χ_l(1,0,0)^T ,

where v_χ_l is undetermined. In the charged lepton sector, the F-term conditions of the driving fields ξ^0_l and ζ^0_l give the vacuum alignment of ϕ_l,

    ∂ w_d/∂ξ^0_l=f_2(ϕ_l_1^2+2 ϕ_l_2ϕ_l_3)=0 ,
          ∂ w_d/∂ζ^0_l=f_3 (χ_l_1ϕ_l_1+χ_l_2ϕ_l_3+χ_l_3ϕ_l_2)=0 .

Given the vacuum of χ_l in Eq. (<ref>), we find the alignment of ϕ_l is

    ⟨ϕ_l⟩=v_ϕ_l(0, 1,0)^T ,

with v_ϕ_l undetermined. In the atmospheric neutrino sector, the F-term conditions associated with the driving fields ξ^0_a, η^0_a and ϕ^0_a read

    ∂ w_d/∂ξ^0_a=f_4 (χ_l_1ϕ_a_1+χ_l_2ϕ_a_3+χ_l_3ϕ_a_2)=0 , 
          ∂ w_d/∂η^0_a_1=M_ηη_a_2+f_5(2 ϕ_a_1ϕ_a_2+ϕ_a_3^2)=0 ,
          ∂ w_d/∂η^0_a_2=M_ηη_a_1+f_5(2 ϕ_a_1ϕ_a_3+ϕ_a_2^2)=0 ,
          ∂ w_d/∂ϕ^0_a_1=f_6ζ_aϕ_a_1+f_7 (η_a_1χ_l_2-η_a_2χ_l_3)=0 ,
          ∂ w_d/∂ϕ^0_a_2=f_6ζ_aϕ_a_3+f_7 (η_a_1χ_l_1-η_a_2χ_l_2)=0 ,
          ∂ w_d/∂ϕ^0_a_3=f_6ζ_aϕ_a_2+f_7 (η_a_1χ_l_3-η_a_2χ_l_1)=0 .

A straightforward calculation shows that the vacuum expectation values of ξ_a, η_a and ϕ_a are

    ⟨ζ_a⟩=v_ζ_a,     ⟨η_a⟩=v_η_a(1, 1)^T,
        ⟨ϕ_a⟩=v_ϕ_a(0, 1, -1)^T ,

with

    v^2_ϕ_a =-M_η/f_5v_η_a,     
    v_ζ_a=-f_5f_7v_ϕ_av_χ_l/f_6M_η .

It is easy to check that the vacuum alignments of flavons η_a and ϕ_a preserve the subgroup Z^U_2. Now we consider the phases of v_ϕ_a and v_ξ_a which is the VEV of ξ_a. They are related by the F-flatness of σ^0_1,2:

    ∂ w_d/∂σ^0_1=M_σ_1^2+f_8ξ_a^2=0 , 
          ∂ w_d/∂σ^0_2=M_σ_2^2+2 f_9η_a_1η_a_2=0 .

From Eqs. (<ref>) and (<ref>), we find

    v_ξ_a/v^2_ϕ_a=f_5M_σ_1/M_σ_2 M_η(2f_9/f_8)^1/2 .

As all parameters in the right-hand side of above equation are real, consequently the phase of v_ξ_a/v^2_ϕ_a is 0, π or ±π/2 for the product f_8f_9>0 or f_8f_9<0, respectively. The auxiliary symmetry Z_8 has played a critical role in generating the discrete possible values 0, π, ±π/2 for the phase of v_ξ_a/v^2_ϕ_a. Subsequently we turn to discuss the vacuum alignment of the solar neutrino sector. The F-flatness condition of the driving field φ^0_s gives

    ∂ w_d/∂φ^0_s_1=f_10 (2 φ_s_1ϕ_a_1-φ_s_2ϕ_a_3-φ_s_3ϕ_a_2)=0 ,
          ∂ w_d/∂φ^0_s_2=f_10 (2 φ_s_2ϕ_a_2-φ_s_1ϕ_a_3-φ_s_3ϕ_a_1)=0 ,
          ∂ w_d/∂φ^0_s_3=f_10 (2 φ_s_3ϕ_a_3-φ_s_1ϕ_a_2-φ_s_2ϕ_a_1)=0 ,

which lead to the vacuum

    ⟨φ_s⟩=v_φ_s(2, -1,  -1)^T  .

The equations giving the vacuum structure for the flavon field ϕ_s are:

    ∂ w_d/∂ζ^0_s=f_11 (φ_s_1ϕ_s_1+φ_s_2ϕ_s_3+φ_s_3ϕ_s_2)=0 ,
          ∂ w_d/∂ϕ^0_s_1=f_12 (2 ϕ_a_1ϕ_l_1-ϕ_a_2ϕ_l_3-ϕ_a_3ϕ_l_2)+f_13 (φ_s_2ϕ_s_3-φ_s_3ϕ_s_2)=0 ,
          ∂ w_d/∂ϕ^0_s_2=f_12 (2 ϕ_a_2ϕ_l_2-ϕ_a_1ϕ_l_3-ϕ_a_3ϕ_l_1)+f_13 (φ_s_3ϕ_s_1-φ_s_1ϕ_s_3)=0 ,
          ∂ w_d/∂ϕ^0_s_3=f_12 (2 ϕ_a_3ϕ_l_3-ϕ_a_1ϕ_l_2-ϕ_a_2ϕ_l_1)+f_13 (φ_s_1ϕ_s_2-φ_s_2ϕ_s_1)=0 ,

which uniquely determine the solar alignment,

    ⟨ϕ_s⟩=v_ϕ_s(1, -1/2,  5/2)^T  ,   with  
    v_ϕ_s=f_12v_ϕ_lv_ϕ_a/3f_13v_φ_s .

We find the vacuum configurations of φ_s and ϕ_s are invariant under the action of the subgroup Z^SU_2. Finally the F-term condition of ξ^0_s is

    ∂ w_d/∂ξ^0_s=M_ξξ_s+f_14(ϕ_s_1^2+2 ϕ_s_2ϕ_s_3)=0 ,

which  leads to the following relations

    v^2_ϕ_s/v_ξ_s=2M_ξ/3f_14 .

The phase parameter η is exactly the phase of the ratio v^2_ϕ_sv_ξ_a/v^2_ϕ_av_ξ_s in our model. Form Eqs. (<ref>) and (<ref>), it is easy to obtain

    v^2_ϕ_sv_ξ_a/v^2_ϕ_av_ξ_s=2f_5M_σ_1M_ξ/3f_14M_σ_2M_η(2f_9/f_8)^1/2 .

All couplings and mass parameters in above equation are real due to CP symmetry, then we see the phase of the ratio v^2_ϕ_sv_ξ_a/v^2_ϕ_av_ξ_s is e^ikπ/2(i=0,1,...,3). In the present work we shall take the following solution

    arg(v^2_ϕ_sv_ξ_a/v^2_ϕ_av_ξ_s)=-π/2 ,

which would happen for f_8f_9<0. Thus the desired vacuum alignment ⟨ϕ_a⟩∝(0, 1, -1)^T, ⟨ϕ_s⟩∝(1,-1/2,5/2)^T and the phase η=-π/2 have been dynamically realized. In the following section we will find that the observed hierarchy among the charged lepton masses can be produced for

    v_ϕ_l/Λ∼λ^2_C ,

where Λ  is the cut-off scale of the theory and λ_C is  the Cabibbo angle with λ_C≃0.23. As usual, we expect that all the VEVs of flavons are of the same order of magnitude, i.e.

    v_ξ_a/Λ∼v_ϕ_a/Λ∼v_ξ_s/Λ∼v_ϕ_s/Λ∼v_η_a/Λ∼v_ζ_a/Λ∼v_φ_s/Λ∼v_χ_l/Λ∼λ^2_C .

Successful leptogenesis fixes the atmospheric neutrino mass to be 3.992× 10^10 GeV (see Eq. (<ref>)) which is of the same order as the flavon VEVs. Thus the cut-off scale Λ is expected to be of order 10^12 GeV. The next-to-leading-order (NLO) corrections to the flavon superpotential w_d involve three flavon fields. When the NLO corrections are included, the original symmetry S_4⋊ H_CP 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 Φ/Λ∼λ^2_C with respect to the LO contributions and therefore can be negligible, where Φ denotes any flavour fields.




 §.§ The structure of the model



The most relevant operators for charged lepton masses are given by

    -0.25in   w_l      =y_τ/Λ(Lϕ_l)_1τ^cH_d
    +y_μ/Λ^2(L(ϕ_lϕ_l)_3^')_1^'μ^cH_d
     +y_e_1/Λ^3(Lϕ_l)_1(ϕ_lϕ_l)_1e^cH_d 
    -0.25in       +y_e_2/Λ^3((Lϕ_l)_2(ϕ_lϕ_l)_2)_1e^cH_d
    +y_e_3/Λ^3((Lϕ_l)_3(ϕ_lϕ_l)_3)_1e^cH_d
    +y_e4/Λ^3((Lϕ_l)_3^'(ϕ_lϕ_l)_3^')_1e^cH_d ,

where all the couplings are real because of the CP symmetry. After the electroweak and S_4 flavour symmetry breaking by the VEV shown in Eq. (<ref>), one can obtain that the charged lepton mass matrix is diagonal with the masses

    m_e=|(y_e_2-2y_e_4)v^3_ϕ_l/Λ^3|v_d,     
    m_μ=|2y_μ_1v^2_ϕ_l/Λ^2|v_d,     m_τ=|y_τv_ϕ_l/Λ|v_d ,

where v_d=⟨ H_d⟩ is the VEV of the electroweak Higgs field H_d. Since the charged lepton mass matrix is diagonal, the hermitian combination m^†_lm_l is invariant under the action of the subgroup Z^T_3, i.e. ρ^†_3(T)m^†_lm_lρ_3(T)=m^†_lm_l. With the assignment in table <ref>, 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

    m_e:m_μ:m_τ≃λ^4_C:λ^2_C:1 .

We find that the subleading order corrections to the charged lepton
mass matrix will break the residual symmetry Z^T_3 but they are suppressed by λ^2_C with respect to LO results, thus can be safely neglected.

In the neutrino sector, the lowest dimensional operators responsible for neutrino masses are

    w_ν=y_a/Λ(Lϕ_a)_1H_uν^c_atm+
    y_s/Λ(Lϕ_s)_1^'H_uν^c_sol
    +x_a/2ν^c_atmν^c_atmξ_a
    +x_s/2ν^c_solν^c_solξ_s ,

where the coupling constants y_a, y_s, x_a and x_s are restricted to be real by the imposed CP symmetry. Inserting the vacuum alignments in Eqs. (<ref>, <ref>), we can read out the neutrino Dirac and Majorana mass matrices as follow,

    m_D=[            0   - y_av_ϕ_a     y_av_ϕ_a;     y_sv_ϕ_s  5/2y_sv_ϕ_s -1/2y_sv_ϕ_s ]v_u/Λ,
            m_N=[ x_av_ξ_a        0;        0 x_sv_ξ_s ] ,

with v_u=⟨ H_u⟩. Applying the seesaw formula, we obtain the light neutrino mass matrix m_ν in Eq. (<ref>) with

    m_a=|y^2_av^2_ϕ_a/x_av_ξ_av^2_u/Λ^2|,  
    m_s=|y^2_sv^2_ϕ_s/x_sv_ξ_sv^2_u/Λ^2| .

Note that we have used the result shown in Eq. (<ref>) under the assumption of x_ax_s>0. In the case of x_ax_s<0, in order to obtain the desired value η=-π/2, the phase of the ratio v^2_ϕ_sv_ξ_a/v^2_ϕ_av_ξ_s should be π/2, i.e. we could choose the right side of Eq. (<ref>) as π/2. In short, the neutrino mass matrix of the new Littlest Seesaw model is realized exactly, hence the phenomenological predictions in section <ref> follows immediately.
The NLO contributions to the neutrino mass matrices in Eq. (<ref>) are found to be suppressed by λ^2_C 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 <cit.>. In particular, we find that CSD(n) model with n=1+√(6) can be naturally obtained if the VEV of the complex modulus τ is at certain fixed point <cit.>. We expect that the desired alignment corresponding to x=-1/2 can also be reproduced from some modular group, such that our model could be simplified considerably.



 §.§ 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 <cit.>

    Br(l_i→ l_jγ)≃α^3/G^2_Fm^8_sBr(l_i→ l_jν̅_jν_i)|(m^2_L̃)_ij|^2tan^2β ,

where G_F is the Fermi coupling constant and m_s is the characteristic mass scale of the SUSY particle in the loop with

    m^8_s≃0.5m^2_0M^2_1/2(m^2_0+0.6M^2_1/2)^2 .

The slepton doublet mass squared m^2_L̃ arises from the renormalization group evolution. To an excellent approximation, the renormalization group result has the form

    (m^2_L̃)_i≠ j≃ -1/8π^2(3m^2_0+A^2_0)(λ^† L λ)_ij ,

where λ is the neutrino Yukawa coupling matrix given in Eq. (<ref>) and the factor L is defined as

    L=diag(logM_G/M_1,logM_G/M_2) ,

with the GUT scale M_G≃ 2× 10^16 GeV. For our model, we find the expressions of (m^2_L̃)_i≠ j are as follows,

    (m^2_L̃)_21≃ -5 |b|^2 (A^2_0+3 m^2_0) /16 π ^2log(M_G/M_2) , 
           (m^2_L̃)_31≃|b|^2 (A^2_0+3 m^2_0) /16 π ^2log(M_G/M_2) , 
           (m^2_L̃)_32≃A^2_0+3 m^2_0/32 π ^2[4 |a|^2 log(M_G/M_1)+5 |b|^2log(M_G/M_2)] .

As shown in Eq. (<ref>), the right-handed neutrino mass M_1 is fixed to be 3.992× 10^10 GeV by leptogenesis. The best fit value of r≡ m_s/m_a is 0.145 in our model, consequently it is natural to take M_2≃3× 10^11 GeV. The measured values of the lepton mixing angles and neutrino masses fix m_s=|b|^2v^2_u/M_2=3.243 meV which leads to b≃5.783× 10^-3. For typical values of the soft SUSY  breaking parameters m_0=140 GeV, M_1/2=600 GeV, A_0=0 and tanβ=5, we find the branching ratios of the charged lepton flavour violating radiative decays to be 

    Br(μ→ eγ)≃ 1.745× 10^-16 ,   
    Br(τ→ eγ)≃ 1.244× 10^-18 ,   
    Br(τ→μγ)≃ 2.647× 10^-17 ,

which are safely below the present experimental upper limits <cit.>. 



 §.§ UV completion


In our model, we see that all interactions are renormalizable except the
the charged lepton Yukawa couplings in Eq. (<ref>) and the neutrino Yukawa couplings in Eq. (<ref>). 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. (<ref>), we introduce three pairs of messenger fields Σ_i and Σ^c_i with i=1, 2, 3 which transform under the flavour symmetry S_4× Z_5× Z_8 as follows

    Σ_1∼ (3, 1, 1),       Σ^c_1∼ (3, 1, 1) ,
          Σ_2∼ (3', ω_5, ω^7_8),     Σ^c_2∼ (3',ω^4_5, ω_8) ,
          Σ_3∼ (3, ω^2_5, -i),    Σ^c_3∼ (3,  ω^3_5, i) .

The chiral superfields Σ_i and Σ^c_i are singlets under the standard model gauge group and they carry hypercharges Y=-1 and Y=1 respectively, and their U(1)_R charges are all +1. The renormalizable terms containing these messenger fields read as

    w^UV_l   =   g_1(LΣ^c_1)_1H_d+g_2(Σ_1ϕ_l)_1τ^c+g_3((Σ_1Σ^c_2)_3ϕ_l)_1
    +g_4(Σ_2ϕ_l)_1'μ^c+g_5((Σ_2Σ^c_3)_3ϕ_l)_1
          +g_6(Σ_3ϕ_l)_1e^c+M_Σ_1(Σ_1Σ^c_1)_1
    +M_Σ_2(Σ_2Σ^c_2)_1+M_Σ_3(Σ_3Σ^c_3)_1 ,

where all the couplings g_1, 2, 3, 4, 5, 6 and masses M_Σ_1,2,3 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,

    w^eff_l=-g_1g_2/M_Σ_1(Lϕ_l)_1τ^cH_d+g_1g_3g_4/M_Σ_1M_Σ_2(L(ϕ_lϕ_l)_3^')_1^'μ^cH_d
    -g_1g_3g_5g_6/M_Σ_1M_Σ_2M_Σ_3((Lϕ_l)_3^'(ϕ_lϕ_l)_3^')_1e^cH_d ,

which leads to a diagonal and hierarchical charged lepton mass matrix for the alignment of ϕ_l in Eq. (<ref>). The non-renormalizable neutrino Dirac couplings in Eq. (<ref>) can be generated with the help of the following heavy fields

    Σ_ν_1∼(1, 1, -i),    Σ^c_ν_1∼(1,  1, i) ,
          Σ_ν_2∼(1', 1, ω^3_8),    Σ^c_ν_2∼(1', 1, ω^5_8) ,

which are all standard model doublets with hypercharge Y=±1/2 (- for Σ_ν_i and + for Σ^c_ν_i). The relevant terms in the UV completion are given by

    w^UV_ν   =   k_1(Lϕ_a)_1Σ^c_ν_1
    +k_2Σ_ν_1ν^c_atmH_u+k_3(Lϕ_s)_1'Σ^c_ν_2
    +k_4Σ_ν_2ν^c_solH_u
          +M_Σ_ν_1Σ_ν_1Σ^c_ν_1+M_Σ_ν_2Σ_ν_2Σ^c_ν_2 ,

where CP invariance requires the parameters k_1, 2, 3, 4 and M_Σ_ν_1, 2 are real. Integrating out Σ_ν_1, 2 and Σ^c_ν_1, 2, we reproduce the desired terms

    w^eff_ν=-k_1k_2/M_Σ_ν_1(Lϕ_a)_1H_uν^c_atm
    -k_3k_4/M_Σ_ν_2(Lϕ_s)_1^'H_uν^c_sol .







§ 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 (0,-1,1) and (1,5/2,-1/2) respectively with the relative phase η=-π/2. 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 M_1=1.176× 10^11 GeV in SM and M_1=3.992× 10^10 GeV in MSSM with tanβ=5. The model is based on the flavour symmetry S_4× Z_5× Z_8 in which the desired alignments and the phase η=-π/2 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
η and x. Here we have focussed on the 𝒩_1 case where
the flavour symmetry S_4 and CP are broken to Z^T_3 in the charged lepton sector, Z^U_2× H^atm_CP in the atmospheric sector and Z_2^SU× H^sol_CP in the solar neutrino sector with  H^atm_CP={1,U} and H^sol_CP={1,SU}, the vacuum alignment of ϕ_atm and ϕ_sol would be fixed to ⟨ϕ_atm⟩∝(0,1, -1)^T and ⟨ϕ_sol⟩∝(1, x, 2-x)^T,
where importantly x 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 S_4 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 <cit.>.

It is interesting to compare the new Littlest Seesaw with (x, η)=(-1/2, -π/2) to the original Littlest Seesaw model with (x, η)=(3, 2π/3), (-1, -2π/3) <cit.>, which also provides a good fit to the data, as summarized in table <ref>. However we find that the new Littlest Seesaw with arguably simpler values x=-1/2, η=-π/2, 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 η=-π/2 is the smallest one among the different benchmark values in table <ref>, consequently the case of x=-1/2, η=-π/2 might be expected to be easier to realize in a concrete model than the original Littlest Seesaw and other cases listed in table <ref>.

We emphasise that the choice x=-1/2 and η=-π/2 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 (m_1=0) only depend on two input parameters m_a and r=m_s/m_a whose values can be determined by the precisely measured neutrino mass squared differences Δ m^2_21 and Δ m^2_31. The comprehensive numerical analysis shows that all lepton mixing parameters and neutrino masses are restricted in rather narrow regions, as shown in Eq. (<ref>). The new Littlest Seesaw differs most markedly in its predictions for θ_23 and δ_CP. While the atmospheric mixing angle θ_23 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 <cit.> 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 χ^2_min 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 η=-π/2 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 M_1=1.176× 10^11 GeV in SM and M_1=3.992× 10^10 GeV in MSSM with tanβ=5. 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 S_4 and CP symmetry
which fixes the values of x=-1/2 and η=-π/2 in
the new Littlest Seesaw model. The charged lepton mass hierarchy is naturally realized in our model, and the required vacua ⟨ϕ_a⟩∝(0, 1, -1)^T, ⟨ϕ_s⟩∝(1,-1/2,5/2)^T and the relative phase η=-π/2 are readily generated through the supersymmetric F-term alignment mechanism. Furthermore, we have studied the predictions for the charged lepton radiative decays μ→ eγ, τ→ eγ and τ→μγ, 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












§ GROUP THEORY OF S_4



In the present work, we adopt the same convention for the S_4 flavour symmetry group as <cit.>. The S_4 group is generated by three generators S, T and U which obey the relations

    S^2=T^3=U^2=(ST)^3=(SU)^2=(TU)^2=(STU)^4=1 .

The group S_4 has 24 elements and five irreducible representations: 1, 1^', 2, 3 and 3^'. The representation matrices of the three generators in different irreducible representations are chosen to be the following form

    [                                      1, 1^':                                         S=1,                                         T=1,                                       U=±1 ,;                                        2^' :                              S=[ 1 0; 0 1 ],                      T=[   ω   0;   0 ω^2 ],                             U=[ 0 1; 1 0 ] ,;                                      3, 3^':       S=1/3[ -1  2  2;  2 -1  2;  2  2 -1 ], T=[   1   0   0;   0 ω^2   0;   0   0   ω ],                 U=∓[ 1 0 0; 0 0 1; 0 1 0 ] , ]

with ω=e^2π i/3. As has been shown in <cit.>, the  generalized CP transformation compatible with the S_4 flavour symmetry is of the same form as the flavour symmetry transformation in our working basis,

    X_r=ρ_𝐫(g),      g∈ S_4 ,

where g can be any of the 24 group elements of S_4. The S_4 Clebsch-Gordan coefficients are frequently used when building a model based on S_4 flavour symmetry. We summarise the Kronecker products and Clebsch-Gordan coefficients in our basis in table <ref>.












2cm


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