Friedberg-Lee symmetry and tri-bimaximal neutrino mixing
in the inverse seesaw mechanism
The inverse seesaw mechanism with three pairs of gauge-singlet neutrinos offers a natural interpretation of the tiny masses of three active neutrinos at the TeV scale. We combine this picture with the newly-proposed Friedberg-Lee (FL) symmetry in order to understand the observed pattern of neutrino mixing. We show that the FL symmetry requires only two pairs of the gauge-singlet neutrinos to be massive, implying that one active neutrino must be massless. We propose a phenomenological ansatz with broken FL symmetry and exact - symmetry in the gauge-singlet neutrino sector and obtain the tri-bimaximal neutrino mixing pattern by means of the inverse seesaw relation. We demonstrate that non-unitary corrections to this result are possible to reach the percent level and a soft breaking of - symmetry can give rise to CP violation in such a TeV-scale seesaw scenario.
pacs:PACS number(s): 14.60.Pq, 13.10.+q, 25.30.Pt
The fact that three active neutrinos possess non-degenerate but tiny masses is a striking signature of new physics beyond the standard model . Perhaps the most popular approach towards understanding the small neutrino mass scale ( eV) is the seesaw mechanism , which contains three right-handed neutrinos and retains gauge symmetry. Although this mechanism can naturally work at a superhigh-energy scale ( GeV) to generate tiny Majorana neutrino masses, it loses direct testability on the experimental side and causes a hierarchy problem on the theoretical side . A straightforward way out is to lower the seesaw scale down to the TeV scale, an energy frontier to be soon explored by the Large Hadron Collider (LHC). But such a TeV-scale seesaw scenario inevitably suffers from a terrible fine-tuning of cancellations between the Yukawa coupling texture and the heavy Majorana mass matrix . To resolve this unnaturalness problem built in the canonical (type-I) seesaw mechanism at low energies, some new interest has recently been paid to a relatively old idea — the inverse seesaw mechanism .
The inverse seesaw mechanism, which can be regarded as the simplest multiple seesaw picture , is an extension of the canonical seesaw mechanism by introducing three additional gauge-singlet neutrinos together with one gauge-singlet scalar. Its typical result for the effective mass matrix of three active neutrinos is in the leading-order approximation, where the scales of three mass matrices may naturally satisfy . The smallness of can be attributed to both the smallness of and the smallness of at the TeV scale (i.e., TeV). It is therefore possible to get a balance between theoretical naturalness and experimental testability of the inverse seesaw scheme. Nevertheless, the inverse seesaw mechanism itself is impossible to interpret the observed pattern of neutrino mixing, which is composed of two large angles ( and ) and one small angle () , because the flavor structures of , and are entirely unspecified. The latter can be determined by imposing certain flavor symmetries, but such flavor symmetries usually need to be broken in order to give rise to the correct neutrino mass spectrum and neutrino mixing pattern. For example, a proper combination of the flavor symmetry and the inverse seesaw mechanism at the TeV scale  may successfully predict the tri-bimaximal neutrino mixing pattern  (with , and ), which is definitely consistent with current experimental data on solar, atmospheric, reactor and accelerator neutrino oscillations.
The present work aims to combine the inverse seesaw mechanism with a newly-proposed flavor symmetry — the Friedberg-Lee (FL) symmetry , so as to fix the flavor textures of , and and thus predict the mass spectrum and mixing pattern of three active neutrinos at the TeV scale. As the FL symmetry requires a fermion mass term to be invariant under a space-time-independent translation of the relevant fermion fields, it can more reasonably be applied to the gauge-singlet neutrino sector instead of the active neutrino sector. We show that the FL symmetry forces one pair of the gauge-singlet neutrinos to be massless, leading to a simplified but viable version of the inverse seesaw mechanism which has two pairs of massive gauge-singlet neutrinos and allows one active neutrino to be massless. With the help of - permutation symmetry, we consider a very simple FL symmetry breaking ansatz in the gauge-singlet neutrino sector and obtain the tri-bimaximal neutrino mixing pattern by means of the inverse seesaw relation. We find that non-unitary corrections to this special mixing pattern are possible to reach the percent level if holds at the TeV scale. We also demonstrate that a soft breaking of - symmetry can easily accommodate CP violation in such an inverse seesaw scenario.
Ii The inverse seesaw mechanism with FL symmetry
Let us work in the basis where the flavor eigenstates of three charged leptons are identified with their mass eigenstates throughout this paper. Different from the canonical seesaw mechanism with three right-handed neutrinos (for ), the inverse seesaw scheme contains three additional gauge-singlet neutrinos (for ) together with one gauge-singlet scalar . Allowing for lepton number violation to a certain extent, one can write out the following gauge-invariant neutrino mass terms in the inverse seesaw mechanism:
where and stand respectively for the lepton and Higgs doublets, and are the Yukawa coupling matrices, and is a symmetric Majorana mass matrix. After spontaneous gauge symmetry breaking, Eq. (1) becomes
where and are the mass matrices. Then we arrive at the overall neutrino mass matrix in the flavor basis defined by and their charge-conjugate states:
Note that the interesting nearest-neighbor-interaction pattern of is guaranteed by an implementation of the global symmetry with a proper charge assignment . Note also that the mass scales of three sub-matrices in may naturally have a hierarchy , because the second mass term in is not subject to the gauge symmetry breaking scale and the third mass term in violates the lepton number conservation . In the leading-order approximation, one obtains the inverse seesaw relation for the effective mass matrix of three active neutrinos:
It becomes obvious that holds in the limit . For instance, eV can easily be achieved from and keV.
Without loss of generality, let us work in a basis where (or equivalently, ) is diagonal; i.e., with and (for ) being real and positive. Now we impose the FL translation  on both and fields:
where (for ) are in general complex numbers, and is a space-time-independent Grassmann parameter (i.e., is anti-commuting and thus holds). Note that the gauge-singlet neutrinos and do not interact with the gauge bosons of the standard model. Note also that the kinetic terms of and change under the above translation, but the resulting action is invariant just because is independent of space and time . Hence the whole Lagrangian of an inverse seesaw model will be invariant under the FL translation, if and only if we require three neutrino mass terms in to be invariant under the FL translation. This requirement simply implies
for . In other words, each mass matrix must have a zero eigenvalue. Because is diagonal, we may simply choose or equivalently
such that and must hold to satisfy the second and third conditions given in Eq. (6). This in turn implies that the elements in the third column of and those in the third row and the third column of must be vanishing, so as to satisfy the first, fourth and fifth conditions shown in Eq. (6). The textures of and are therefore given as
In this case, the symmetric neutrino mass matrix in Eq. (3) can be simplified to an effective neutrino mass matrix
where “” symbolically denotes an arbitrary (non-zero) matrix element. Because holds, one active neutrino must be massless in this minimal inverse seesaw scenario. The phenomenology of such a simple scenario has partly been explored in Ref. . Here we show that the FL symmetry applied to the gauge-singlet neutrino sector naturally explains why a pair of gauge-singlet neutrinos can be decoupled from the usual inverse seesaw mechanism.
It is worth remarking that the above conclusion does not depend on the chosen basis of (i.e., is diagonal), since a unitary transformation of either or does not change any physical content of the inverse seesaw mechanism. To derive a phenomenologically-favored neutrino mixing pattern from this interesting scheme, however, it is more convenient to introduce a less stringent FL symmetry into the gauge-singlet neutrino sector. Here we consider the case of , which is equivalent to the original FL translation (imposed on [10, 14, 15, 16]). Then three mass terms in can keep invariant under the translations and if they take the following forms:
where the Greek index runs over and the Latin index runs over . The explicit expressions of , and turn out to be
in which all the matrix elements are in general complex. It is easy to check that holds, and thus each mass matrix has one zero eigenvalue. We see that the above FL symmetry must be broken, at least for the second mass term of , such that holds to make the inverse seesaw formula in Eq. (4) applicable.
Iii A FL symmetry breaking ansatz with - symmetry
There are certainly a variety of possibilities of breaking the FL symmetry. Here we follow the spirit of Ref.  and consider a very simple symmetry breaking ansatz for the second and third mass terms in Eq. (10):
where and are real and positive. To minimize the number of free parameters, we impose the - permutation symmetry on these two mass terms (i.e., and ) and assume all of their parameters to be real. The resultant mass matrices read
in which (or ) and (or ) are defined to be two dimensionless parameters. The diagonalization of or is rather straightforward: and , where is simply the tri-bimaximal mixing pattern 
Then we obtain three mass eigenvalues , , ; and similarly , , . In terms of mass eigenvalues, and can be reexpressed as
Note that , , and hold. Note also that the inverse matrix of takes the same form as itself:
Now let us make a purely phenomenological assumption: with GeV being the electroweak scale and being the identity matrix, just for the sake of simplicity. Then the inverse seesaw formula in Eq. (4) allows us to arrive at
It is straightforward to show that this effective mass matrix can also be diagonalized by the unitary transformation , where has been given in Eq. (14) and three neutrino masses (for ) directly reflect the salient feature of the inverse seesaw mechanism. In other words, the smallness of is ascribed to both the smallness of and that of . Current neutrino oscillation data only provide us with and , and thus these two neutrino mass-squared differences can easily be reproduced from our result for by adjusting six free parameters and (for ).
Note that the neutrino mixing matrix appearing in the charged-current interactions of three active neutrinos is not exactly used to diagonalize in Eq. (18), just because of slight mixing between light and heavy neutrinos in the inverse seesaw mechanism. As shown in Appendix A, the charged-current interactions of three light Majorana neutrinos read
in which and are the mass eigenstates of charged leptons and active neutrinos, respectively. The relationship between and is given by , where signifies the slight deviation of from and its approximate expression can be found in Eq. (A7). For the ansatz under consideration, we explicitly obtain
Current experimental constraints on the matrix elements of are
at the confidence level . Combining Eqs. (16), (20) and (21), we arrive at
together with and due to the - symmetry of Hermitian . If TeV, then is expected from the stringent constraint on . This result accordingly implies , from which TeV can be extracted. It is in general difficult to get a limit on . But if TeV is taken, for example, then leads us to TeV.
With the help of Eqs. (14) and (20), it is straightforward to obtain an interesting expression of in this inverse seesaw scenario:
This simple but instructive result clearly shows the deviation of from . In particular, holds; i.e., the non-unitary effect does not contribute to this smallest neutrino mixing matrix element.
We remark that the above FL symmetry breaking ansatz is viable and suggestive for building a realistic inverse seesaw model at the TeV scale. To generate non-vanishing and CP violation, however, one has to invoke a different FL symmetry breaking pattern with one or more non-trivial -violating phases.
Iv Soft - symmetry breaking and CP violation
We proceed with the FL symmetry breaking ansatz in Eq. (12) but allow soft - symmetry breaking for and (i.e., and with and being complex):
where (or ) and (or ) are real. As shown in Appendix B, the inverse matrix of takes the same texture as itself:
which can easily be read off from Eq. (B3) in Appendix B. Furthermore, we show that is also of the same texture:
which can directly be read off from Eq. (B6). Note that is real and is complex. Making the same phenomenological assumption for as in section III (i.e., ), we simply obtain the effective mass matrix of three active neutrinos from Eqs. (4) and (27) in this inverse seesaw scenario:
with . The small mass eigenvalues of are therefore attributed to small as well as small at the scale of TeV.
The diagonalization of in Eq. (29) can be done by using the unitary transformation , where
and arises from the soft - symmetry breaking term of (i.e., ). Comparing this result with the standard parametrization of in terms of three mixing angles and three CP-violating phases [1, 18], we immediately find
together with , and . On the other hand, three mass eigenvalues of are given by