# flavor symmetry model for the extension of the minimal standard model by three right-handed sterile neutrinos

###### Abstract

The extension of the minimal standard model by three right-handed sterile neutrinos with masses smaller than the electroweak scale (MSM) is discussed in a flavor symmetry framework. The lightness of the sterile neutrino and the near mass degeneracy of two heavier sterile neutrinos are naturally explained by exploiting group properties of . A normal hierarchical mass spectrum and an approximately - symmetric mass matrix are predicted for three active neutrinos. Nonzero can be obtained together with a deviation of from the maximality, where both mixing angles are consistent with the latest global data including T2K and MINOS results. Furthermore, the tiny active-sterile mixing is related to the mass ratio between the lightest active and lightest sterile neutrinos.

## I introduction

Both the establishment of neutrino oscillation phenomena and the evidence of nonluminous dark matter (DM) demand physics beyond the standard model (SM). On the one hand, compelling evidences from current solar, atmospheric, reactor and accelerator neutrino experiments have told us that neutrinos are massive and lepton flavors are mixed PDG (). On the other hand, various cosmological observations have revealed that DM is five times more abundant than normal matter and accounts for about one quarter of the Universe (i.e., WMAP10 ()). To explain both problems above, an extension of the SM by three right-handed sterile neutrinos with masses smaller than the electroweak scale (MSM) was first proposed by Asaka et al. NuMSM1 (). The smallness of active neutrino masses is described by the canonical seesaw mechanism type1 () and one light right-handed sterile neutrino at the scale acts as a candidate of warm dark matter (WDM). Moreover, the model can explain the baryon asymmetry in the Universe through the oscillations BANO (); BAU () of two heavier right-handed sterile neutrinos with masses at the scale.

Despite the above phenomenological successes, one unsatisfactory
point in the MSM is the lack of a natural explanation for large
mass splitting between the sterile neutrino and the heavier
ones. Moreover, the oscillation mechanism for the baryon asymmetry
demands strong mass degeneracy between the heavier sterile
neutrinos, but the MSM is impotent to be able to explain its
origin. In order to resolve theses issues, some interesting ideas
have been proposed in the context of a flavor symmetry
U1FS (), the Froggatt-Nielsen mechanism FN (), the split
seesaw mechanism Split () and grand unified theories CR ().
In the present work, we introduce a non-Abelian discrete flavor
symmetry and try to understand the mass splitting and degeneracy due
to group properties of the flavor symmetry. The mass degeneracy of
two heaver sterile neutrinos could be interpreted as a sign that
they constitute a doublet representation of the flavor symmetry,
while it may be natural to assign a singlet representation to the
sterile neutrino. Furthermore, if the singlet representation
is a complex one, we can prohibit a bare mass term of the
sterile neutrino because of the Majorana nature and may be able to
generate a suppressed mass term from higher-dimensional operators.
Inspired by these clues, we employ the group as our flavor
symmetry^{3}^{3}3Previous studies about can be found in Ref.
kubo (). since it is the smallest finite group^{4}^{4}4See
Ref. Frampton () for the classification of discrete groups upto
, and Ref. FSrev () for the recent reviews of non-Abelian
discrete flavor symmetries. which contains both a complex singlet
and a real doublet representations. In addition to , we
introduce two auxiliary symmetries in order to handle the
order of magnitude of some small parameters as well as to forbid
unwanted terms. We find that our model predicts a normal
hierarchical mass spectrum and an approximately -
symmetric mass matrix for three active neutrinos.

The remaining part of this paper is organized as follows. In section II we present the framework of the MSM with a flavor symmetry. Section III is devoted to the realization of the seesaw mechanism and section IV to the diagonalization and the resulting neutrino masses and mixing matrix. A numerical analysis with the focus on nonzero is also illustrated in section IV. Finally, we give a summary in section V.

## Ii The Model

We introduce three right-handed sterile neutrinos and gauge-singlet flavon fields , , and with a flavor symmetry. We assign a -singlet (-doublet) for the first (second and third) generation of fermions, and the SM Higgs is assumed to be invariant under all the flavor symmetries. The particle content and charge assignments for each symmetry are summarized in Table I, and the basic group theory of is reviewed in the appendix. Because of the symmetries, there are no renormalizable Yukawa interactions in the charged lepton sector, and the charged lepton masses follow from the higher dimensional operators:

(1) |

where we have specified the Lagrangian upto the next-to-leading order level, and the charged lepton mass matrix is diagonal upto this order. The subscripts beside indicate the required representations to make the terms invariant under . In the neutrino sector, is assumed to be a complex representation of , while the other leptons are real representations. Consequently, only and , which are complex representations, can reproduce a Dirac and Majorana mass terms for , and those interactions are suppressed due to the symmetries:

(2) | |||

(3) |

Note that in the above expressions we have omitted some terms whose contributions can be embedded into others and implicitly assumed a mechanism which supplies , e.g., the spontaneous breaking of a lepton number symmetry at a scale.

We define the vacuum expectation values(VEVs) of neutral scalars as

(4) |

leading to the charged lepton masses,

(5) |

the right-handed Majorana neutrino mass matrix,

(6) |

and the Dirac mass matrix,

(7) |

Notice that, in Eq. (5), we have the same fine-tuning problem as that in GriLav () for obtaining . Nevertheless, we will not tackle this problem and only focus on the neutrino sector in what follows. We also note that although the second terms of and are strongly suppressed by compared with the first terms, we keep , and in our discussions because they will be important when we discuss the sterile neutrino masses and active-sterile mixing. In contrast, in only contributes to the masses and mixing of three active neutrinos, and its effects are negligibly small in comparison with the first term. Thus, we shall ignore .

## Iii SeeSaw Mechanism

Let us move on to the diagonal basis of . The diagonalization can approximately be done by the rotation in the plane, and three sterile neutrino masses are found to be

(8) |

is suppressed with , and thus it is the candidate of WDM, while and are nearly degenerate. Interestingly, the order of and that of the mass difference between and are the same, which may turn out to be a key ingredient when one considers the baryon asymmetry in the Universe BAU (). Nevertheless, we shall naively assume and do not consider the baryon asymmetry in what follows since detailed studies of the baryon asymmetry go beyond the scope of this paper. Because of , the masses of the light neutrinos are obtained by integrating out only and , yielding the following effective mass matrix:

(9) |

where

(10) | |||||

will later end up the mass matrix of three active neutrinos, and stands for the mixing effect of active-sterile neutrinos. In the above expressions, we have defined and and embedded the suppression factor into the couplings such that

(11) |

As one can see, is symmetric if holds.

Here, let us roughly estimate the magnitude of model parameters. From the charged lepton sector, we obtain for . Suppose and for simplicity, then results in for the lightest sterile neutrino. Since and , need to be in order to reproduce realistic active neutrino masses , and they correspond to . If we assume the same value for as well, then we gain .

In the approximation of , the mass matrix in Eq. (9) can further be diagonalized by a neutrino mixing matrix parametrized as

(12) |

where is a mixing matrix which induces the active-sterile mixing angles

(13) |

and is a unitary mixing matrix diagonalizing the mass matrix of three active neutrinos. We can use to eliminate the and terms of ,

(14) |

and can be determined by the relation . Because the order of [i.e., ] is much smaller than that of , we can safely neglect its effects in the active neutrino part. The only exception is the contribution to because it is vanishing when is taken into account alone.

## Iv Neutrino masses and mixing

### iv.1 Active Neutrino Masses and Mixing

As we mentioned, if holds, is symmetric and it results in the vanishing and maximal for active neutrinos. In fact, these predictions are roughly compatible with the recent neutrino oscillation data Fogli (); GMS (); STV (), which indicate a small and a nearly maximal . Thus, may be expected to be not so large, so that we here treat as a small parameter and employ the perturbation calculations. The leading (i.e., symmetric) term can be diagonalized by

(15) |

where and , and the mixing parameters are given by

(16) |

together with the eigenvalues

(17) |

Notice that and are unphysical phases and do not affect any observables, though they appear in the following expressions. After including the correction term and doing perturbative calculations, we get the neutrino mixing angles in the standard parametrization as

(18) |

and the Jarlskog invariant parameter as

(19) |

where . From Eq. (18), one can check the recovery of the symmetry ( and ) in the limit of . The corrections to the three eigenvalues are vanishing in the order of , so the active neutrino masses are obtained by taking absolute values for Eq. (17), and this model predicts a normal mass hierarchy for three active neutrinos. Note that the vanishing of should be kept in all orders of perturbations and is a generic property of the minimal seesaw models Mini (). A nonvanishing mass of can be generated from the lightest sterile neutrino contribution:

(20) |

but the corresponding effects are negligibly small and can be safely ignored for all the other mass and mixing parameters.

### iv.2 Numerical Analysis

Instead of perturbative calculations, we here numerically diagonalize Eq. (10) and compute , and . From the recent global analysis STV () of the neutrino oscillation data, in our calculations, we refer to the following best-fit values and error bounds:

(21) |

for the normal neutrino mass hierarchy.

In Figure 1, we plot as functions of
(left panel) and (right panel) with the
constraints of , and
. Besides, the bound of
is also imposed in the
plane (right panel). As one can see from
the figure, the predicted regions can be within the
ranges, and can deviate from , which is
favored by the recent T2K T2K () and MINOS MINOS () results.
However, a large is always accompanied with a
large deviation of from . For instance,
the best-fit value of can be accounted for at
around , but it is almost the
edge of the bound^{5}^{5}5We refer to Ref. 13-23 ()
for the realizations of a large together with a
small deviation of from the maximality. . Notice
that we have checked that the symmetry breaking
parameter , which is defined below Eq.
(10), is at most and that the analytical
expressions in Eqs. (17) and
(18) approximately agree with the numerical results in
Figure 1.

### iv.3 Active-Sterile Mixing

From Eq. (13) and the parameter estimates in section III, the active-sterile mixing angles defined by the elements of the matrix (i.e., ) can be derived as

(22) |

and , which are well below the upper bounds from astrophysical and cosmological observations Review (). Furthermore, it is also consistent with the requirement of correct DM abundance for the mechanism of resonant active-sterile oscillations with nonzero lepton asymmetries Shi (). To achieve the right DM abundance with the nonresonant mechanism DW (), we need to be one order of magnitude larger than in order to achieve Review (). These tiny active-sterile mixing angles make the detection of the WDM particle rather dim and remote with both the X-ray observations Xray () and the captures on beta-decaying or electron-capture-decaying nuclei Liao (); LX11 ().

## V Conclusions

In this work, we have proposed a flavor symmetry realization for the MSM in the presence of two auxiliary symmetries and succeeded in naturally explaining the lightness of the sterile neutrino and the mass degeneracy of the two heavier sterile neutrinos. A normal hierarchical mass spectrum and an approximately - symmetric mass matrix are predicted for three active neutrinos. Nonzero can be obtained together with a deviation of from the maximality, where both mixing angles are consistent with the latest global data including T2K and MINOS results. Finally, we have derived a tiny active-sterile mixing related to the mass ratio between the lightest active and lightest sterile neutrinos.

The MSM can explain the active neutrino masses, the candidate of dark matter and the baryon asymmetry in the Universe in a unified and elegant way. Although there are already some models in which the MSM is extended by flavor symmetries, our model has more direct connections with the masses and mixing patterns of three active neutrinos. Our realization can also be modified to accommodate the scale sterile neutrinos eVmodel (), which are more or less hinted at by current experimental Schwetz4 () and cosmological Raffelt () data. We shall examine this case with a specific flavor model elsewhere TA ().

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant No. 11135009, by the Chinese Academy of Sciences Fellowship for Young International Scientists (T.A.) and by the China Postdoctoral Science Foundation under Grant No. 20100480025 (Y.F.L.).

## Appendix A Basics of

consists of four singlet and two doublet irreducible representations,

(23) |

and twelve elements,

(24) |

where stands for the unit matrix. The representation matrices of and for each representation are give by

(25) |

with . Note that , and are real representations, is a pseudoreal representation and are complex representations.

* | |||||

* | * | ||||

* | * | * | |||

* | * | * | * |

The tensor products among the irreducible representations are summarized in Table II. Especially, the products of two doublets are defined as follows.

(26) |

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