Phenomenological study of extended seesaw model for light sterile neutrino
We study the zero textures of the Yukawa matrices in the minimal extended type-I seesaw (MES) model which can give rise to eV scale sterile neutrinos. In this model, three right handed neutrinos and one extra singlet are added to generate a light sterile neutrino. The light neutrino mass matrix for the active neutrinos, , depends on the Dirac neutrino mass matrix (), Majorana neutrino mass matrix () and the mass matrix () coupling the right handed neutrinos and the singlet. The model predicts one of the light neutrino masses to vanish. We systematically investigate the zero textures in and observe that maximum five zeros in can lead to viable zero textures in . For this study we consider four different forms for (one diagonal and three off diagonal) and two different forms of containing one zero. Remarkably we obtain only two allowed forms of ( and ) having inverted hierarchical mass spectrum. We re-analyze the phenomenological implications of these two allowed textures of in the light of recent neutrino oscillation data. In the context of the MES model, we also express the low energy mass matrix, the mass of the sterile neutrino and the active-sterile mixing in terms of the parameters of the allowed Yukawa matrices. The MES model leads to some extra correlations which disallow some of the Yukawa textures obtained earlier, even though they give allowed one-zero forms of . We show that the allowed textures in our study can be realized in a simple way in a model based on MES mechanism with a discrete Abelian flavor symmetry group .
ADP-16 - 34/T990
Neutrino oscillation experiments have established the fact that neutrinos have tiny mass and they change from one flavor to another during their propagation. This requires the Standard Model (SM) of particle physics to be extended in order to generate their masses. The standard 3-flavor neutrino oscillation scenario has six key parameters. These are the two mass squared differences ( ) which control the oscillations of the solar and atmospheric neutrinos respectively, three mixing angles () and a Dirac CP phase, . Global analysis of three flavor neutrino oscillation data from Gonzalez-Garcia:2015qrr (); Forero:2014bxa (); Capozzi:2016rtj () give us the best fit values and the allowed ranges of these parameters. In 3-flavor paradigm, there are two more CP violating phases if neutrinos are Majorana particles. But as Majorana phases do not appear in the neutrino oscillation probability, they are not measurable in the oscillation experiments. Apart from these phases another major unknown is the absolute value of the neutrino mass since oscillation experiments are only sensitive to the mass squared differences. Planck data provide an upper bound on sum of neutrino masses to be eV Ade:2015xua () at 95% C.L. The sensitivity for the neutrino masses in the upcoming Karlsruhe Tritium Neutrino experiment (KATRIN) is expected to be around 200 meV (90% C.L.) MERTENS2015267 ().
Another interesting aspect of neutrino oscillation experiments is the search for the existence of a light sterile neutrino. As sterile neutrinos are SM singlets they do not take part in the weak interactions. But they can mix with the active neutrinos. Therefore, sterile neutrinos can be probed in neutrino oscillation experiments. The oscillation results from LSND experiment showed the evidence of at least one sterile neutrino having mass in the eV scale Athanassopoulos:1996jb (); Athanassopoulos:1997pv (); Aguilar:2001ty (). The latest data of MiniBooNE experiment Aguilar-Arevalo:2013pmq () also have some overlap with the allowed regions of the LSND experiment and hence support the existence of the sterile neutrino hypothesis. The recently observed Gallium anomaly can also be explained by the sterile neutrino hypothesis Giunti:2010zu (). Another evidence of eV sterile neutrino comes from the reactor antineutrino flux studies. This shows the deficit in the observed and predicted event rate of electron antineutrino flux and the ratio is at 98.6% C.L. Mention:2011rk (). Recent analysis of the Planck data shows the possibility of light sterile neutrino in the eV scale if one deviates slightly from the base CDM model Ade:2015xua (). In short, the scenario with a light sterile neutrino is quite riveting at present and many future experiments are proposed to confirm/falsify this Abazajian:2012ys (). Although it is possible to have a better fit of neutrino oscillation data with more than one light sterile neutrino Kopp:2011qd (); Conrad:2012qt (); Giunti:2011gz (), the 3+1 scheme i.e., three active neutrinos and one sterile neutrino in the sub-eV and eV scale respectively, is considered to be minimal. There are three different ways to add sterile neutrino in SM mass patterns and these are, (i) 3+1 scheme in which three active neutrinos are of sub-eV scale and sterile neutrino is of eV scale GomezCadenas:1995sj (); Goswami:1995yq (), (ii) 2+2 scheme in which two different pairs of neutrino mass states differ by eV but this scheme was disfavored by solar and atmospheric data Maltoni:2002ni (), and (iii) 1+3 scheme in which three active neutrinos are in eV scale and sterile neutrino is lighter than active neutrinos. This scenario is however disfavored from cosmology Hamann:2010bk (); Giusarma:2011ex (). Hence, we focus on the 3+1 scenario in our study.
Flavor symmetry models giving rise to eV sterile neutrinos have been studied in the literature Chun:1995bb (); Barry:2011wb (); Chen:2011ai (). These models might require modifications to usual seesaw framework deGouvea:2006gz (); Dev:2012bd (). In the explicit seesaw models the eV scale sterile neutrinos with their mass suppressed by Froggatt - Nielsen mechanism can be naturally accommodated in non Abelian flavor symmetry Barry:2011wb (); Zhang:2011vh (); Heeck:2012bz (). bimodel or schizophrenic models for light sterile neutrinos are also widely studied Allahverdi:2010us (); Machado:2010ui (). In order to have a theoretical understanding of the origin of eV sterile neutrino as well as admixtures between sterile and active neutrinos, the authors of Refs. Barry:2011wb (); Zhang:2011vh (); Heeck:2012bz () have studied an extension to the canonical type-I seesaw model. This model is known as “minimal extended type - I seesaw" (MES) model. In the MES model a fermion singlet, , is added along with three right handed neutrinos. This extension results into an eV scale sterile neutrino naturally, without imposing tiny mass scale or Yukawa term for this neutrino.
In this paper, for the first time we study the various possible textures of the Dirac neutrino mass matrix, , Majorana neutrino mass matrix, and the mass matrix that originate from the Yukawa interaction between right handed neutrinos with the gauge singlet within the framework of MES model and classify the allowed possibilities. Several papers have studied the consequences of imposing zeros in the neutrino mass matrix in standard three neutrino Dev:2006qe (); Xing:2002ta (); Xing:2002ap (); Desai:2002sz (); Dev:2007fs (); Dev:2006xu (); Kumar:2011vf (); Fritzsch:2011qv (); Meloni:2012sx (); Ludl:2011vv (); Grimus:2012zm () and the 3+1 framework Ghosh:2012pw (); Ghosh:2013nya (); Zhang:2013mb (); Nath:2015emg (); Borah:2016xkc (). The more natural study would be to explore the zeros in the Yukawa matrices that appear in the Lagrangian rather than light neutrino mass matrix, . It has been noted by many authors Branco:2007nb (); Goswami:2008rt (); Goswami:2008uv (); Choubey:2008tb (); Lavoura:2015wwa () that the zeros of the Dirac neutrino mass matrix and the right handed Majorana mass matrix are the progenitors of zeros in the effective Majorana mass matrix through type - I seesaw mechanism. We also seek extra correlations connecting the parameters of the active and sterile sector which can put further constraints on the allowed possibilities. This motivates us to look for zeros in various neutrino mass matrices in the MES model which can lead to viable texture zeros in neutrino mass matrix.
We classify different structures of , and that can give allowed textures for the light neutrino mass matrix . Interestingly the only allowed form of that we obtain are the two one zero textures – namely and which are phenomenologically allowed and have the inverted hierarchical mass spectrum. For a originating from ordinary seesaw mechanism both these textures are viable. However, in the MES model, because of extra correlations connecting active and sterile sector, not all Yukawa matrices that give or for are allowed. We study these additional correlations and tabulate the allowed textures. We also include a discussion on the impact of NLO corrections in this model. In this context it is also important to study the origin of zero textures. Here, we show that it is possible to obtain various zero entries in lepton mass matrices with an Abelian discrete symmetry group . An alternative approach to obtain lepton mixing is discussed in Fonseca:2014koa () by considering non-Abelian symmetry group. We follow the method discussed in Grimus:2004hf () to obtain Abelian discrete symmetry group which can generate viable zero textures in . Their method is based on type - I seesaw and we extend it to apply on MES model.
The paper is organized in the following manner. In the next section a brief review of the MES model is given. In Section 3 and its subsections we list the various forms of , and that lead to viable textures in . In Section 4 we discuss the implication of the allowed forms of one zero textures in obtained in Section 3. The following Section 5 discusses the results obtained from the comparison of low energy and high energy neutrino mass matrices and the extra correlations connecting active and sterile sector. Symmetry realizations for the allowed zero textures are discussed in Section 6. The summary of our findings and conclusions are presented in Section 7.
2 Minimal extended type I seesaw mechanism
In this section we describe the basic structure of MES model. Here, the fermion content of the SM is extended by three right handed neutrinos together with a gauge singlet field . One can get a natural eV-scale sterile neutrino without inserting any small Yukawa coupling in this model Barry:2011wb (); Zhang:2011vh (). The Lagrangian containing the neutrino masses is given by,
Here, are the () Dirac and Majorana mass matrices respectively and is a () coupling matrix between right handed neutrinos with the gauge singlet. In the basis (), the () neutrino mass matrix can be expressed as,
Considering the hierarchical mass spectrum of these mass matrices i.e. , in analogy of type - I seesaw, the right handed neutrinos are much heavier compared to the electroweak scale and thus they will decouple at the low scale. Therefore, Eq.(2) can be block diagonalized using seesaw mechanism and the effective neutrino mass matrix in the basis () can be written as,
Note that the rank of is three (see Zhang:2011vh ()) and hence one of the light neutrino remains massless.
Considering the case that , one can apply seesaw approximation once again on Eq.(3) to obtain the active neutrino mass matrix as111Note that RHS of Eq.(4) does not vanish since is a vector rather than a square matrix.,
whereas the mass of the sterile neutrino is given by,
Note that the zero textures of fermion mass matrices in the context of type - I seesaw mechanism studied in Branco:2007nb (); Goswami:2008rt (); Goswami:2008uv (); Lavoura:2015wwa (), leading to viable texture zeros in can be different from that of MES model because of the presence of the first term of Eq.(4). The active-sterile neutrino mixing matrix is given by,
where governs the strength of active-sterile mixing and can be expressed as,
Essentially, is suppressed by the ratio . Additionally in our formalism we assume , which is allowed by the current active sterile neutrino mixing data.
As the sterile neutrino mass ( eV) is heavier than active neutrinos, therefore, the mass pattern in the active sector can be arranged in two different ways. We denote 3+1 scenario as (SNH) when the three active neutrinos follow normal hierarchy () and the second choice is (SIH) when the three active neutrinos follow inverted hierarchy ) as shown in Fig(1). These masses can be expressed in terms of the mass squared differences obtained from oscillation experiments as given in Table(1).
The best fit values along with 3 ranges of neutrino oscillation parameters used in our numerical analysis are given in Table(2).
|7.37||6.93 – 7.97|
|(NH)||2.50||2.37 – 2.63|
|(IH)||2.46||2.33 – 2.60|
|2.97||2.50 – 3.54|
|(NH)||2.14||1.85 – 2.46|
|(IH)||2.18||1.86 – 2.48|
|(NH)||4.37||3.79 – 6.16|
|(IH)||5.69||3.83 – 6.37|
|(NH)||1.35||0 – 2|
|(NH)||0.0295||0.0263 – 0.0336|
|(IH)||0.0299||0.0266 – 0.0342|
|1.63||0.87 – 2.04|
|0.027||0.012 – 0.047|
|0.013||0.005 – 0.03|
In the next section we systematically explore the various zero texture structures of , and which can give rise to viable zero textures of .
In our formalism, the charge lepton mass matrix, , is considered
to be diagonal.
For the right handed Majorana neutrino mass matrix,
we consider four different structures:
(i) Diagonal having three zeros i.e.,
(ii) non-diagonal minimal form of having four zeros with Det i.e.,
These three non-diagonal forms of correspond to , and flavor symmetry respectively. Such forms of in the context of zero textures in type-I seesaw model have been considered for instance in Goswami:2009bd (). being a matrix can have one zero or two zeros. In Zhang:2011vh (), an based model was considered with 2 zeros in and 3 zeros in to obtain the as given by Eq. (4). But, in our analysis we find that mass matrices with 5 zeros in and two zeros in do not lead to any viable textures in . The only allowed possibility therefore is one zero in result in three possible structures. We find that the maximum number of zeros of that can give phenomenologically allowed zero textures in is five. The possible combinations of , and that lead to phenomenologically viable textures of are discussed in the following subsections.
3.1 5 zeros in and diagonal
First let us assume to be diagonal. As is a non-symmetric matrix, 5 zeros can be arranged in ways. Thus considering 126 cases of together with 3 cases of and 1 case of , we obtain total 378 possible structures of . Out of all possible combinations of these matrices the only allowed texture that we obtain is the one zero texture in with . Here, we have three possible forms of and these are,
The various forms of which lead to viable texture are presented below:
Here, , and are the permutation matrices that exchange first and second columns, first and third columns and second and third columns respectively. Therefore, we observe that out of 126 cases only 12 above forms of give the allowed texture of when is diagonal
3.2 5 zeros in and non-diagonal corresponding to flavor symmetry
The form of that we consider here corresponds to flavor symmetry as given in Eq.(9). Among the 378 possibilities we obtain two allowed one zero textures of , namely and . We observe that out of total 126 forms of , only four structures give rise to while eight structures give rise to . We list them below:
3.2.1 Textures leading to
3.2.2 Textures leading to
3.3 5 zeros in and non-diagonal corresponding to flavor symmetry
The form of that we consider in this subsection corresponds to flavor symmetry as given in Eq.(9). In this case also we observe that out of total 126 cases of , only four structures of give rise to and eight forms of give rise to texture . We list them below. Note that these forms of are different from those obtained in the earlier subsection.
3.3.1 Textures leading to
3.3.2 Textures leading to
3.4 5 zeros in and non-diagonal corresponding to flavor symmetry
The form of that we consider here corresponds to flavor symmetry as given in Eq.(9). Here also we observe that out of 126 cases of only four structures of give rise to texture and 8 forms of give rise to texture . But these forms of are different from those obtained in the earlier two subsections:
3.4.1 Structures leading to
3.4.2 Structures leading to
Note that in general the entries of the Yukawa matrices , and are complex (of the form ). However some of the phases can be absorbed by redefinition of the leptonic fields. For the case when is diagonal, the number of un-absrobed phases is two – one each in and whereas for the off-diagonal only one phase remains in . In this section we do not explicitly write the phases. However in section 5 where we discuss specific cases, the phases are explicitly included.
4 Active neutrino mass matrix with one zero texture
The () light neutrino mass matrix being symmetric, there are 6 possible cases of one zero textures with a vanishing lowest mass and these are studied in details in Refs. Merle:2006du (); Lashin:2011dn (); Gautam:2015kya (); Lavoura:2013ysa (). In the above section we observed that in context of MES model only viable textures of that we obtain are and . According to the recent studies Gautam:2015kya (); Lavoura:2013ysa (); Harigaya:2012bw (), both these textures are ruled out for normal hierarchy when the lowest mass is zero but they can be allowed for the inverted hierarchy even when then lowest mass is zero 222We also observed that both these textures are disallowed for NH with the most recent data.. This kind of mass pattern can be obtained completely from group theoretical point of view if one assumes that Majorana neutrino mass matrix displays flavor antisymmetry under some discrete subgroup of SU(3) as discussed in Joshipura:2015zla (); Joshipura:2016hvn (). In this section we re-analyse the textures and for the inverted hierarchical mass spectrum assuming in the light of recent neutrino oscillation data as given in Table(2). In our analysis we find that correlations among various oscillation parameters become highly constrained as compared to the earlier studies. This is due to the recent constraints on the 3 ranges of the mass squared differences and as compared to earlier results in Lashin:2011dn (); Gautam:2015kya (); Lavoura:2013ysa () 333 The latest constraint on comes from T2K and NOA including both appearance and disappearance modes Abe:2015awa (); Salzgeber:2015gua (); Adamson:2016tbq (); Adamson:2016xxw (). Whereas reanalysis of KamLAND data shows decrease in the value of and as discussed in Capozzi:2016rtj (). .
In three neutrino paradigm, low energy Majorana neutrino mass matrix can be diagonalized as,
Here, () is a lepton mixing matrix in the basis where is diagonal. The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix has 3 mixing angles and a CP violation phase .
The elements of neutrino mass matrix can be calculated from Eq.(23) are,
where, and are given in Table(1). We express elements of as in the text.
Imposing the condition of zero texture for IH with in the above equation we get,
which can be simplified to obtain the mass ratio
Let, we get
Let us define the ratio of the two mass squared differences as,
The defined above can be calculated either using the current neutrino mass squared differences as given in Table(2) or by calculating . If the value of calculated using falls in the allowed range of from the current data, then we say the texture under consideration is allowed by the current data. As given in Table(2) we vary the Dirac CP phase from 0 360 while the relevant Majorana phase in the range 0 180 and find the correlations among different parameters, specially the predictions for and .
We also study the effective Majorana neutrino mass, , governing neutrinoless double beta decay () for these allowed textures. In three flavor paradigm this can be written as,
where ). From the above equation we understand that depends on the Majorana phases but not on the Dirac phase. Various experiments such as CUORE Gorla:2012gd (), GERDA Wilkerson:2012ga (), SuperNEMO Barabash:2012gc (), KamLAND-ZEN Gando:2012zm () and EXO Auger:2012ar () are looking for signatures for neutrinoless double beta decay (). The current experiments provide bounds on the effective Majorana mass from the non-observation of . For instance, the combined results from KamLAND-ZEN and EXO-200 Gando:2012zm () give the upper bound on the effective Majorana neutrino mass as (0.12 - 0.25) eV where the range signifies the uncertainty in the nuclear matrix elements. The future experiments can improve this limit by one order of magnitude. Below we discuss the various correlations that we obtain for the allowed textures.
4.1 Case I:
The Majorana mass matrix element in 3-flavor case can be written as,
Imposing the condition of zero texture with vanishing lowest mass () for IH, we get,
From the above equation we obtain the mass ratio as below
The mass ratio should be greater than 1. For this to happen should be negative. We find that due to the interplay of the terms and the phase is restricted to the range and . The effective mass, as function of Majorana phase is constrained due to very small allowed range of (5 10, 170 175) as shown in Eq(4). The allowed range of for this texture is 0.046 eV 0.05 eV and which can be probed in future experiments. Also, this texture predicts Dirac CP phase which is in agreement with the indications from the current ongoing oscillation experiments like T2K and NOA. There is however no constrain on the values of the neutrino mixing angles and seen in right panel of Fig.2 for this texture.
4.2 Case II:
The Majorana mass matrix element in 3-flavor case can be written as,
Imposing the condition of texture zero with vanishing lowest mass() for IH, we get,
The mass ratio from the above equation can be written as
Since this mass ratio is always greater than 1 from oscillation data, we find that should be negative for this texture as well. As can be seen from Fig 3 that is constrained in the range . We observe that, due to the more constrained values of mass squared differences and from present data, as considered in our analysis, the atmospheric mixing angle is restricted to be below maximal. In the earlier analysis Lashin:2011dn (); Gautam:2015kya (); Lavoura:2013ysa () there was no preferred octant of . The values of 45 are disallowed for this texture as can be seen in Fig 3. The effective mass, , being function of unknown Majorana phase as seen in Eq(4) is constrained due to very small allowed range of (80110). The allowed range of for this texture is 0.014 eV 0.018 eV which is smaller compared to the case =0 where a vanishing element is off-diagonal. The allowed values of the effective mass for diagonal texture are on the lower side having no overlap with non diagonal texture zero . Thus, can be used to distinguish between diagonal and off-diagonal one texture zero classes with a vanishing neutrino mass. Note that allowed ranges of and are more constrained in our analysis as compared to references Lashin:2011dn (); Gautam:2015kya () again due to the recent improved constraints on the mass squared differences and at 3.
5 Comparison of low and high energy neutrino mass matrix elements
In this section we obtain the light neutrino neutrino mass matrix () (Eq.4), sterile mixing matrix () (Eq.5) and the active sterile mixing matrix (R) (Eq.7) using the different forms of , and given in section (III) of the MES model. Since in the MES model both the active neutrino mass matrix and the active sterile mixing matrix depends on the parameters of , and , this can induce additional correlations between active and sterile sector. Similarly, the mass of the sterile neutrino depends on and . Hence expressing the various variables in terms of the parameters of these matrices one can get some interrelations.
For an illustration we will discuss three specific cases. In case I and II we discuss assuming diagonal structure of and in the case III we talk about by considering the off diagonal form of . Note that here we consider the complex phases in our calculation. We compare high energy mass matrix with low energy mass matrix after the decoupling of the eV sterile neutrino as discussed in section II.