1 Introduction

Normal versus inverted hierarchical models within - symmetry

N.Nimai Singh111Regular Associate of ICTP.
E-mail address: nimai03@yahoo.com
, H. Zeen Devi and Mahadev Patgiri

Department of Physics, Gauhati University, Guwahati-781014, India

Department of Physics, Cotton College, Guwahati-781001, India

The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 31014 Trieste, Italy.


We make a theoretical attempt to compare the predictions from normal and inverted hierarchical models, within the framework of symmetry. We consider three major theoretical issues in a self consistent ways, viz., predictions on neutrino mass and mixing parameters, stability under RG analysis in MSSM, and baryogenesis through leptogenesis. We further extend our earlier works on parametrisation of neutrino mass matrices obeying symmetry, using only two parameters in addition to an overall mass scale , to both normal and inverted hierarchy, and the ratio of these two parameters fixes the value of solar mixing angle. Such parametrisation though phenomenological, gives a firm handle on the analysis of the mass matrices and can also extend its prediction to lower values of solar mixing angle in the range . All predictions are in agreement with observed data except in one case, i.e., inverted hierarchy with opposite CP parity in the first two mass eigenvalues (Type B) with , where is highly dependent on solar mixing angle, and the prediction is good for only. We then check the stability of the inverted hierarchical model with opposite CP-parities, under radiative corrections in MSSM for large region and observe that the evolution of with energy scale, is highly dependent on the input-high scale value of solar mixing angle. Solar angles predicted by tri-bimaximal mixings angle and values lower than this, do not lead to the stability of the model at large values. Similarly, the evolution of the atmospheric mixing angle with energy scale at large values, shows sharp decrease with energy scale for the case with . However, non-zero value of is essential to maintain the stability on the evolution of solar mass scale. We apply these mass matrices to estimate the baryon asymmetry of the Universe in a self consistent way and find that normal hierarchical model leads to the best result. Considering all these three pieces of theoretical investigations, we may conclude that normal hierarchical model is more favourable in nature.

PACS numbers: 14.60.Pq, 12.15.Ff, 13.15.+g, 13.40.Em

1 Introduction

The recent global oscillation analysis[1] indicates a mild departure from tribimaximal neutrino mixings. The decreasing trend in solar mixing is also consistent with the prediction from the Quark-Lepton Complementarity (QLC) relation [2,3,4] at the unification scale where the Cabibbo angle is taken at high scale[5]. In the theoretical front there are several attempts to find out the most viable models of neutrinos, and among them the - reflection symmetry[6-10] in neutrino mass matrix at high scale, has attracted considerable attentions in the last few years. Even the tri-bimaximal mixing[11,12] is a special case of this symmetry. It is expected that this symmetry has a strong potential to explain the present neutrino observed data[1].

The - symmetry leads to the maximal atmospheric mixing () and zero reactor angle (). The prediction on solar mixing angle remains arbitrary and it is generally fixed through a parametrisation in the mass matrix. This symmetry has the freedom to fix the solar mixing angle at lower values, even far below the tri-bimaximal value, without destroying the - symmetry. This is possible through the identification of a ratio of two parameters (referred to as flavor twister) present in the neutrino mass matrix and its subsequent variation in input values[13]. We are interested to parametrise both inverted as well as normal hierarchical neutrino mass models and then identify the flavor twisters responsible for lowering the solar mixing angle[14]. It is interesting to note that - symmetry gives a common origin for both hierarchical and inverted hierarchical neutrino mass matrices in agreement with latest data.

The - symmetry in neutrino mass matrix is assumed to hold in the charged lepton mass basis, although the charged lepton masses are obviously not - symmetric. However, such a scenario can be realised in gauge models with different Higgs doublets generating the up- and down-like particle masses[7,9,10,15,16]. A realisation of - symmetry in the flavor basis within the framework of SUSY SU(5) GUT, has strengthen the foundation of the symmetry as a full-fledged gauge theory[10]. We are now interested to investigate the phenomenological predictions of the - symmetry in neutrino sector and also possible application in leptogenesis[7,17]. In the present work we confine to analysis without phases, keeping our eyes on both predictions on neutrino masses and mixings consistent with latest data.

The paper is organised as follows. In section 2 we give a very brief overview on latest developments on - symmetry in neutrino mass models. In section 3 we give the parametrisation of the mass matrices for hierarchical and inverted hierarchical models in terms of only two parameters and identify the flavor twisters in both cases. This will be supplemented by detailed numerical analysis. In section 4 we discuss the stability under radiative corrections for large values for inverted hierarchical model. We give a brief account on the predictions on baryogenesis using the same neutrino mass matrices. Section 6 concludes with a summary and discussion.

2 Neutrino mass matrices with - symmetry

The - symmetry in the neutrino mass matrix, implies an invariance under the simultaneous permutation of the second and third rows as well as the second and third columns in neutrino mass matrices[6],


Neutrino mass matrix in eq.(1) predicts the maximal atmospheric mixing angle, and . However the prediction on solar mixing angle is arbitrary, and it can be fixed by the input values of the parameters present in the mass matrix. Thus


which depends on four input parameters and . This makes us difficult to choose the values of these free parameters for a solution consistent with neutrino oscillation data. This point will be addressed in section 3 where the solar angle is made dependent only on the ratio of two parameters, . Such parametrization of the mass matrix enables us to analyse the neutrino mass matrix in a systematic and economical way[13]. The actual values of these two new parameters will be fixed by the data on neutrino mass squared differences.

The MNS leptonic mixing matrix which diagonalises is defined by where , and


From the consideration of - reflection symmetry, mixing matrix is generally parametrised by three rotations (, ):


where , . Tri-bimaximal mixing (TBM) is a special case with and ,






For completeness we also give the three neutrino mass eigenvalues[9] corresponding to the neutrino mass matrix in eq.(1),


The solar mixing angle is given by , . If , then we have a simple relation, . The general form of mass matrix in eq.(1) can be fitted to both normal and inverted hierarchical models. Without changing the expression for the prediction on solar mixing angle in eq.(2), the parameters , and can be rearranged within the texture, giving many possible neutrino mass models obeying - symmetry. Some of these models are suitable for normal hierarchical model and other for inverted hierarchical model. This will be addressed in the next two subsections.

2.1 Normal hierarchical neutrino mass model

Many models based on normal hierarchy [8,9] make the 1-1 term () zero in the general neutrino mass matrix in eq.(1). This reduces one free parameter in the mass matrix. In the present analysis this can be done from the general form (1) by a mere rearrangement of parameters, which preserves the solar mixing angle given in eq.(2). Thus the mass matrix takes the new form


which can be simply expressed as[8,9],


As discussed before, such mass matrix has interesting predictions on solar mixing angle in term of a ratio of first two neutrino mass eigenvalues[8,9,18,19],


This form of neutrino mass matrix (12) is seesaw invariant[8] in the sense that both Dirac neutrino mass matrix and the right-handed Majorana mass matrix also have the same form(12) of mass matrix. This model can be motivated[8] within the framework of SO(10) GUT where both quarks and charged leptons mass matrices have the broken - symmetry due to the presence of extra - antisymmetric parts in the mass matrices arising from 120 Higgs scalar, while mass mass matrices belonging to three neutrinos are - symmetric due to 10 and 126 Higgs scalar in the SO(10)GUT model. In such model there is a correction in from charged lepton sector.

Other interesting observations for the neutrino mass matrices (1) obeying - symmetry, can also be obtained from the seesaw formula using the general - symmetric Dirac neutrino mass matrix (1) in the diagonal basis of the right-handed Majorana mass matrix with two degenerate heavy mass eigenvalues [7]. This is true only for normal hierarchical model and has a special application in resonant leptogenesis.

2.2 Inverted hierarchical neutrino mass model

In case of inverted hierarchical model, we assume that both CP even and CP odd in first two mass eigenvalues, have a common mass matrix[13]. The general form in eq.(1) with will lead to small but . For simplicity in the phenomenological analysis, one can push to condition without changing the expression on the prediction of solar mixing angle (2), through a simple rearrangement of parameters. The inverted hierarchical mass matrix has the form


which can be rewritten as


This obeys condition[20] and hence . The above form (15) can now be expressed as [21],


where for inverted hierarchy with CP odd. Particular choices of the values of parameters (A, B, D) in (15) make the mass matrix either CP even or CP odd in first two mass eigenvalues , thus signifying a common origin for both mass models. Recently Babu et al [16] have presented a new realisation of inverted hierarchical mass matrix (16) based on flavor symmetry where is the non-Abelian group generated by permutation of three objects, while the is based for explaining the mass hierarchy of the leptons. In this construction the permutation symmetry is broken down to an Abelian in the neutrino sector, whereas it is broken completely in the charged lepton sector. The - symmetry is then realised in neutrino sector, while having non-degenerate charged leptons. The symmetry acts as leptonic symmetry which is desirable for an inverted hierarchical model. The significant deviation of from comes from breaking of symmetry in charged lepton sector.

It can be pointed out here that the form of mass matrix (12) with 1-1 element zero in the mass matrix, can also be constructed for inverted hierarchy[22]. Since and are nearly degenerate in inverted hierarchy, this model leads to nearly bi-maximal mixings , requiring large corrections from charged lepton sector to meet the data. Such models have problems and are not favoured by the recent data.

In a significant work by Mohapatra et al [10], the realization of - reflection symmetry in the neutrino mass matrix in the flavor basis (i.e. the basis where charged leptons are mass eigenstates), has been obtained in a realistic full-fledged gauge model based on where leptons and quarks are treated together. In such model the requirement of - symmetry for neutrinos does not contradict with the observed fermion masses and mixings. The neutrino mass matrix having - symmetry, is assumed to arise from a triplet seesaw (type II) mechanism, which disentangles the neutrino flavor structure from quark flavor structure. The deviations of and from and respectively, come from left-handed charged leptons mixing matrix.

3 Neutrino mass matrices in two parameters and numerical analysis

This section is the main part of the paper, where we are interested to express the general mass matrix (1) with only two parameters and , with an additional mass scale . The expression for the prediction on solar mixing angle (2) will now depend only on the ratio of these two variables, and . Such consideration in the reduction of the number of parameters in the texture, gives a firm handle on the phenomenological analysis of the mass matrices obeying - symmetry. The parametrisations presented below are not unique. We give such parametrisation for normal hierarchy as well as inverted hierarchy. The numerical analysis is performed using Mathematica.

3.1 Parametrisation for normal Hierarchy

We present here two forms of parametrisation related to the mass matrices in eqs.(1) and (12) discussed in section 2 and 2.1, with two parameters and in the texture, and a common mass scale parameter .

Case (i) with : The general mass matrix of the form(1) with no zero texture, is parametrised by


This predicts an expression for solar mixing angle,

where the ratio is the “flavor twister” in this case. The possible solutions for lowering solar angle beyond tribimaximal solar mixing, are given by which leads to and . The numerical predictions on deviation from tri-bimaximal mixings for the case are presented in Table-1. The predictions on and are consistent with observed data for a wide range of solar angle .

As an example we cite a representative case for . Taking input values for , and , we get the three mass eigenvalues, , leading to and .

Case (ii) with : Following the procedure in eq.(12), the mass matrix (17) is now modified as


where the solar mixing angle prediction corresponding to a choice of flavor twister, is same as that of eq.(17), except that the relation is valid in this case since 1-1 term in the mass matrix is zero. Table-2 gives the numerical predictions at lower values of solar angle. The predictions on solar and atmospheric mass scales are consistent with the recent experimental data. In order to have the result for , we take the input values for , and , and we get leading to and respectively.

3.2 Parametrisation of inverted hierarchy

We again consider two cases with and respectively for inverted hierarchical model, with only two parameters and , in addition to an overall mass scale . As discussed earlier, we do not expect zero texture in 1-1 element in inverted hierarchical model.

Case (i) with : A suitable parametrisation for mass matrix (1) has the form[13],


This gives the prediction of solar angle,

This leads to the condition for lowering solar mixing angle beyond tribimaximal mixing, which has two possibilities[13]: and . The corresponding numerical predictions are given in Table-3 for three types: A, B, and C. Type A means , type B means and type C means respectively, which are related to CP phases. The result shows that these types have a common mass matrix and hence a common origin.

For a demonstration, we cite numerical results for representative cases for . Type A: Taking input values for , , we get leading to and .

Type B: With input values , , we get leading to and .

Type C: For input values for , , we have leading to and . In all three cases we take as input.

Case (ii) with : Following eq.(14), we express eq.(19) in the following form of mass matrix,


The numerical predictions are given in Table-4 (for type B), Table-5 (for type A) and Table-6 ( for type C) for the range of solar angle, .

We again give corresponding results for in three types.

Type A: Taking input values for , and , we get leading to and .

Type B: With input values , and , we get leading to and .

Type C: For input values for , and , we have leading to and .

3.3 Understanding the parametrisation in neutrino mass matrices

A: Inverted hierarchy: In order to understand the form of mass matrix parametrised in eq.(19), we start with two parts[24] of neutrino mass matrix, , which can be diagonalised by tri-bimaximal mixing matrix (5). For the inverted hierarchy the structure of the dominant term having - symmetry, is given by


which is diagonalised as


The second perturbative term can also be diagonalised by ,


where is a very small parameter. The diagonalisation with tri-bimaximal mixing matrix (5),



Thus, from eqs.(22) and (26), the diagonalisation of the total mass matrix,

leads to


The deviation of solar angle from tri-bimaximal mixings can be introduced through the replacement by using a flavor twister where


which still has - symmetry. This can be diagonalised by but is now replaced by a new matrix . Thus leads to


where . The new solar angle calculated from , is now given by

The corresponding new mass eigenvalues for are calculated as


After identification of , we obtain the same mass matrix(19).

B.Normal Hierarchy:

In case of normal hierarchy(17), we start with two parts[24] of neutrino mass matrix, , which can be diagonalised by tribimaximal mixing matrix(5). The structure of the dominant term having - symmetry, can be taken as


which can be diagonalised by


However, the second term which can be diagolanised by , can be taken as


where is a very small real parameter. The diagonalisation of eq.(35) with tribimaximal mixing matrix,

leads to


From eqs.(34) and (38), the diagonalisation,

leads to


The deviation of solar angle from tribimaximal mixings can be done through the replacement by using a flavor twister ,


which still obeys - symmetry and can be diagonalised by . Thus

leads to


where and new can be obtained in principle. The new solar mixing angle is given by

The corresponding new mass eigenvalues for are


After substitution of , we recover the earlier mass matrix in eq.(17). For the value , we have the tribimaximal condition leading to .

range of
Table 1: Normal hierarchy with non-zero 1-1 term in the texture. Input value of .
range of
Table 2: Normal hierarchy with zero 1-1 term in the texture and
range of
Table 3: Inverted hierarchy with for Type A, Type B, and Type C, explained in the text. Input value of .
range of
Table 4: Inverted hierarchy with Type B: . Input value .
range of
Table 5: Inverted hierarchy with Type A: ,
range of
Table 6: Inverted hierarchy with Type C: ,

4 Effects of renormalisation group analysis in MSSM for large

There are excellent papers[3,23, 25,26] devoted to radiative corrections on neutrino masses and mixings, and on Quark-Lepton Complementarity relation. However the problem with inverted hierarchy with opposite CP-parities in the first two mass eigenvalues (type B), is not yet settled completely. In particular, it has been shown with analytic calculations in Ref.[23] that for large the radiatively generated low-scale value of has a negative sign and this contradicts the experimental data. We are interested here to examine this conjecture for high-scale input value of solar angle given by tribimaximal mixing and below. The effect of non-zero value of on the evolution of mixing angles as well as will be investigated in greater details. We take high-scale input values of and in the numerical analysis.

We start with a very brief outline on the procedure of RG analysis without phases, while referring for details to our earlier works[27,28]. The effects of quantum radiative corrections of neutrino masses and mixings in MSSM, lead to the low-energy neutrino mass matrix,



The above analytic solution of the neutrino mass matrix at low-energy scale is possible only where charged lepton mass matrix is diagonal. We have also neglected , compared to , and for large , we can take and . Thus the low-energy mass matrix (45) has the form,


In this approach the neutrino mass matrix evolves as a whole from high scale to low scale, and diagonalisation of the mass matrix at any particular energy scale leads to the physical neutrino mass eigenvalues as well as mixing angles. This approach is numerically consistent with other approach where neutrino mass eigenvalues and three mixings angles evolve separately through coupled RG equations. In MSSM we have the following RG equations[29],


where and are the elements in MNS matrix(3) parametrised by (neglecting CP Dirac phase),


where and respectively.

We follow the standard procedure for a complete numerical analysis of the RGEs for neutrino masses and mixing angles in two consecutive steps (i) bottom-up running in the first place where running of third family Yukawa couplings and three gauge couplings in MSSM, are carried out from top-quark mass scale at low energy end to high energy scale[30] . In the present analysis we consider the high scale value as the unification scale, , with large as input value. For simplicity of the calculation we take approximately the SUSY breaking scale at the top-quark mass scale . We adopt the standard procedure to get the values of gauge couplings at top-quark mass scale from experimental data, using one-loop RGEs, assuming the existence of one-light Higgs doublet and five quark flavors below top-quark scale. The values of three Yukawa couplings and three gauge couplings are calculated at high unification scale. (ii) In the second top-down approach, the runnings of three neutrino masses and three mixing angles are carried out simultaneously with the running of Yukawa and gauge couplings, from high to low scale, using the input values of Yukawa and gauge couplings evaluated in the first stage of running[28].

The normal hierarchical model is almost stable under radiative correction[3,4] and is of little interest. This is evident from the fact that the 1-1 term in the mass matrix is almost zero. There is a mild increase in both solar and atmospheric mixing angles while running from high to low scale. The mass splitting is found to be acceptable and we are not repeating the same investigation here.

The evolution of neutrino masses with energy scale in case of inverted hierarchical model with opposite CP-parities, is highly affected with the high scale input value of the solar mixing angle. We observe that the model is not stable for input value of solar angle below