A Model of Neutrino Masses and Mixing from Hierarchy and Symmetry

We construct a model that allows us to determine the three neutrino masses directly from the experimental mass squared differences, $\Delta_{\rm atm}$ and $\Delta_{\rm sol}$, together with the assumption that $\Lambda = \surd (1/6) = \surd (m_2/m_3)$. The parameter, $\Lambda$, basically a Clebsch-Gordan coefficient with the value of about 0.4 stems from the group $S_3$, and is NOT an expansion parameter, in contrast with the Wolfenstein parameter, $0.22<\lambda<0.25$ needed to explain quark masses. For a variety of initial values of $\Delta_{\rm atm}$, we find that the lowest mass, $m_1$, varies from $1.4 - 3.6 10^{-3} eV$, the next lowest mass, $m_2$ varies only slightly from $8.4 - 9.0 10^{-3} eV$, and the heaviest mass, $m_3$, ranges from $5.0 - 5.4 10^{-2} eV$. The elements of the mixing matrix, $U$, and of the mass matrix, $M$, are examined with particular emphasis on the role of small angle $\theta_{13}$. The phase, $\delta$, of the mixing matrix $U$ has a serious effect in the mass matrix only for the matrix elements $M_{e\mu}$ and $M_{e\tau}$, because these are the only ones for which the real part vanishes in the allowed range for $\theta_{13}$. Their dependence on $s_{13}$ for various values of $\delta$ is given explicitly. We study the elements of the mass matrix, $M$, for our solution 1, that of the perfect rational hierarchy, for the case $\delta = 0$, and find that all of them are smaller than $0.03 eV$. The only candidates for double texture zero models are $M_{ee}$ and $M_{e\mu} = M_{\mu e}$.


Introduction
In the Standard Model, the twelve masses of the three generations of four families are arbitrary. Unification of quarks and leptons will eliminate these capricious numbers or at least establish strong relations between them. For a very promising approach see [1].
At the present time, one thing that we can do is to look for patterns. In particular, the 'rational' hierarchy of quarks and charged leptons is well confirmed. By rational here we mean that mass-ratios of members of a family are very close to powers of a parameter λ [2]. For example, m b :m s :m d ≈ 1:λ 2 :λ 4 . Furthermore, this parameter dominates the symmetry breaking exhibited by the mixing angles of the unitary matrix, which gives the flavor states as linear combinations of the mass eigenstates.
Mass patterns for neutrinos are quite different. The information for neutrinos comes mainly from solar and atmospheric neutrino oscillations [3], [4].
∆ sol = |m 2 v2 -m 2 ν1 | ≈ 6.9x10 -5 eV 2 and ∆ atm = |m 2 ν3 -m 2 ν2 | ≈ 2.6 x 10 -3 eV 2 In the following we determine the neutrino masses by proposing a new relation between the mixing angles and the mass ratios. The mixing angle θ 13 is small. In the limit θ 13 goes to zero, we impose S3-S2 symmetry on the mixing matrix to fix the remaining mixing angles θ 23 and θ 12 . In a strong hierarchical model (m 1 small), we must have m 2 ≈√∆ sol and m 3 ≈√(∆ atm +∆ sol ). Motivated in part by the observed numerical similarity of s 12 s 23 and (∆ sol /∆ atm ) (1/4) , we equate the Cabibbo angle √(m 2 /m 3 ) to s 12 s 23 , which will be named Λ, similar in spirit, but not in magnitude to the Wolfenstein parameter λ. With this identification, the masses and the mass matrix are totally determined by ∆ sol and ∆ atm .

Symmetry and Hierarchy Lead to a Proposed New Parameter for Neutrino Mass Determination
The flavor states ν e ,ν µ and ν τ are related to the mass eigenstates ν 1 , ν 2 and ν 3 by the unitary transformation U.
There are two 'large' angles θ 12 and θ 23 . Setting the small angle θ 13 , for which there is as yet no lower limit, equal to zero, we obtain U o : (2) The three columns of U are the three eigenvectors of the mass matrix in the θ 13 =0 limit. If V i is the ith column of U (i=1,2,3), then the mass matrix M is given by: where m i is the ith eigenvalue of M.
It has been proposed more than 15 years ago that the 'mass gap' of the hierarchical pattern is associated with pairing forces in analogy with Cooper pairs in BCS theory and the mass matrix of the neutral pseudoscalar mesons [5]. In this limit, the mass matrix is 'democratic' [6] and when diagonalized gives rise to only one massive state, the coherent state. The 'democratic' vector V d is of particular interest here, where and , the 'democratic' matrix. The vector V d was assigned to the heaviest mass, m 3, with pairing forces creating the mass gap in mind. The masses m 2 and m 1 were thought to be generated through a breaking of this S 3 symmetry, S 3 →S 2 →S 1 [5], [7].
However, the smallness (or vanishing) of θ 13 makes the BCS type mass gap interpretation for m 3 untenable in the neutrino case. In contrast to the BCS case, because of the consequent vanishing of U e3 in U o (Eq. (2)), we have m 3 as a coherent mixture of m νµ and m ντ , which agrees with maximal mixing, θ 23 = π/4. We now have S 2 symmetry for m 3 and reserve S 3 symmetry for m 2 . This, in fact, completely determines U 0 . This assignment of V d as the eigenvector for m 2 has lately received considerable attention in the literature [8]. S 2 symmetry for V 3 implies θ 23 = π/4 (maximal mixing) and s 23 =c 23 =√(1/2).
We now relate the second large mixing angle, θ 12, to the mass ratio m 2 /m 3 by the relation: This association of the mixing angles with the mass ratios was suggested by us earlier on phenomenological grounds [7], because both s 12 s 23 and (∆ sol /∆ atm ) (1/4) are about the same, approximately equal to 0.4. We propose it here as a `natural' pattern.
Considering s 12 a small parameter for the moment (it is not), we get to first order in s 12 (c 12 =1) the matrix u 0 : or more suggestively: This shows the dynamic role assigned to θ 12 by the assumption (6) and why we may consider it as 'natural'.
Restoring c 12 and full unitarity we have for U 0 : By normalization, it follows that Of course √(1/6) is not a capricious number, along with √(1/2) it is a Clebsch-Gordan Coefficient, but that it should be equal to √(m 2 /m 3 ) is a capricious notion.
In a hierarchical model, with small or vanishing m 1 , we have It is, of course entirely possible that it is a coincidence that s 12 s 23 ≈ (∆ sol /∆ atm ) (1/4) and that both are approximately equal to √(1/6), which is the value demanded by S3-S2 symmetry, but we make it the basis of the present model. Hence Eq (10).
We now have sin (θ 23 ) =cos(θ 23) = √(1/2) and sin (θ 12 ) = √(1/3) = √2 Λ, so that: The hierarchy indicated here is not very strong, m 2 =Λ 2 m 3 =(1/6) m 3 , and Λ should not be used as an expansion parameter. In fact, the situation is very different from the quark sectors. There, the possible S 3 -S 2 symmetry is presumably the same for the d and u sectors and does not appear in the V ckm = U d † U u , which is then just 1. Only the symmetry breaking terms, dominated by powers of λ ≈ 0.23 are seen and the underlying symmetry, if it exists, is obscured in the resulting Wolfenstein representation. The mixing angles can be large or small, depending on the assumed flavor basis. In the present model, on the other hand, Λ is intrinsic to the symmetry and must be √(1/6) ≈ 0.4.

The neutrino mass spectrum
Assuming the normal ordering of masses, m 1 2 <m 2 2 <m 3 2 we have two equations, with the mass squared differences, ∆ sol and ∆ atm determined: But now a mass scale is provided by the hypothetical third equation (6), relating the two masses Clearly, a real solution for m 1 is possible only if √∆ sol /√(∆ atm + ∆ sol ) ≤ (1/6).
With ∆ sol and ∆ atm given, all mass values are fixed. It is gratifying to have ρ ≈ 1 to be squarely in the data range, because values of ρ much different from unity stress the spirit of rational hierarchy. It is clear from Fig.1 that ρ can also be smaller than 1, even zero, without deviating from the 'best fit' by much. We now give details for three solutions.
Even though we said earlier that Λ is not an expansion parameter, if we expand to leading orders of Λ, we find that the double texture zero solution (Solution-3, with sin(θ 13 ) ≈ -0.11) is well approximated by: with sin(θ 13 )≈-3√2 Λ 4 and ρ≈ 1/2Λ 2 to lowest order.
The phase δ of the mixing matrix U has a serious effect for the mass matrix only for the matrix elements M eµ and M eτ , because these are the only ones for which the real part vanishes in the allowed nge for θ 13 .
ous values of δ. The values for ⎢M µe ⎢are identical as well as r ⎢M eτ ⎢and ⎢M τe ⎢, with θ 13 →θ 13 . ra Fig.4 shows ⎢M eµ ⎢vs. sin(θ 13 ) for vari fo shes in the allowed nge for θ . Their dependence on s 13 for various values of δ is given explicitly. erarchy, s for ouble texture zero models are M ee and M eµ =M µe (or M eτ = M τe , with θ 13 →θ 13 ).

C
We have shown that the neutrino masses may be evaluated directly from the experimental differences, ∆ atm and ∆ sol together with the assumption that Λ=√(1/6) ) ≡ √(m 2 /m 3 ). For a variety of initial values of ∆ atm the lowest mass, m 1 , varies from 1.4 -3.6 10 -3 eV , The next lowest mass, m 2 , varies only slightly from 8.4 -9.0 10 -3 eV ,and the heaviest mass, m 3 , ranges from 5.0 -5.4 10 -2 eV. The parameter, Λ, basically a Clebsch-Gordan coefficient w of about 0.4, is NOT an expansion parameter, in 0 The phase, δ, of the mixing matrix U has a serious effect on the mass matrix only for the matrix elements M eµ and M eτ , because these are the only ones for which the real part vani ra 13 A study of the elements of the mass matrix, M, for our solution 1, that of the perfect rational hi for the case δ = 0, shows that all of them are smaller than 0.03 eV. The only candidate d