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Inverted Hierarchical Model of Neutrino Masses Revisited

In this letter we highlight the inherent problems associted with the inverted hierarchical model of neutrinos with only three generations and suggest possible solutions within the MSSM. we discuss the new parametrisation of the solar mixing angle which can identify the light side and dark side of the data. We then argue whether the inverted hierarchical neutrino mass matrix can explain the large mixing angle(LMA) MSW solution of the solar neutrino anomaly in the presence of an appropriate texture of charged lepton mass matrix. In model independent way we explore such specific form of the charged lepton mass matrix having special structure in 1-2 block. The contribution to the solar mass splitting arising out of radiative corrections in MSSM, is calculated, thus making the model stable under radiative corrections.

Introduction: The data on solar neutrino oscillation has been traditionally shown on the (sin 2 2θ 12 , ∆m 2 21 ) plane but a new parametrization on the (tan 2 θ 12 , ∆m 2 21 ) plane with the usual sign convention ∆m 2 21 = m 2 2 − m 2 1 > 0 has been recently suggested [1] to explain the solar data. In this new parametrization the most recent result from the SNO-Superkamiokande [2] favours the " light side", tan 2 θ 12 < 1, and disfavours the "dark side", tan 2 θ 12 > 1. It has also imposed certain restrictions on the validity of the proposed neutrino mass models which are otherwise allowed in the old parametrization. This can be understood from the definition of leptonic mixing matrix V MN S where tan 2 θ 12 = |V e2 | 2 /|V e1 | 2 < 1 implies |V e2 | < |V e1 | for the usual sign convention ∆m 2 21 = m 2 2 − m 2 1 > 0, whereas such restriction does not exist in the definition of sin 2 2θ 12 = 4|V e1 | 2 |V e2 | 2 .
For our analysis in the present work, we take here the most general form of texture of the inverted neutrino masses, which is assumed to be valid at high scale Here we consider two possibilities. In case(i) we take δ = ǫ 1 = ǫ 2 = 0, leading to ∆m 2 21 = 0. This represents the case of perfect L e − L µ − L τ symmetry. In case(ii) we have δ ∼ ǫ 1 ∼ ǫ 2 ∼ λ 3 , leading to ∆m 2 21 > 0. The quantum radiative corrections may modify the above two cases and their low-energy values arecase(i): ∆m 2 sol = δ rad when an appropriate charged lepton mass matrix is taken into consideration [3]; and case(ii): ∆m 2 sol = ∆m 2 21 + δ rad . In MSSM the radiative correction term δ rad is a negative quantity [4] and therefore, it will lead to tan 2 θ 12 > 1 in case (i). But in case (ii) if ∆m 2 21 > δ rad , the radiative stability will still be maintained with tan 2 θ 21 < 1. This point will be further examined in present work.
Formalism and Results: Expressing m LL in the basis where the charged lepton mass matrix is diagonal, we have m ′ νL , and V eL is the diagonalizing matrix for charged lepton mass matrix. The neutrino flavour eigenstate ν f is related to the mass eigenstate ν i by the relation ν f = V f i ν i and the MNS mixing matrix is given by V f i where f = τ, µ, e and i = 1, 2, 3. We take the usual convention of the neutrino mass eigenvalues |m ν1 | < |m ν2 |, and this fixes the ordering of the first two columns of the leptonic mixing matrix V MN S . We present here only one simple case of the left-handed Majorana neutrino mass matrix having δ = 0, ǫ 2 = 0, ǫ 1 = λ 3 for our demonstration in the present analysis, m LL with m 0 = 0.05eV in Eq.(1); and the neutrino mass eigenvalues m i = (−1.4089, 1.4195, 0.01065)m 0 , i = 1, 2, 3, leading to the neutrino mass splittings ∆m 2 21 = 7.4952 × 10 −5 eV 2 and ∆m 2 23 = 5.0335 × 10 −3 eV 2 . For our choice of the diagonal charged lepton mass matrix, the MNS mixing matrix is expressed as where α = 0.001335 and β = 0.00094. Here tan 2 θ 12 = +α| 2 < 1 for the usual convention ∆m 2 21 > 0. In case of exact L e − L µ − L τ symmetry as in case (i) where δ = ǫ 1,2 = 0 and ∆m 2 21 = 0, and the above V MN S takes the form with α = 0 and β = 0. In the above example the V MN S (= V † νL ) obtained from the m LL alone fails to explain the LMA MSW solution as it predicts the maximal solar mixings, and any small deviation in the texture of m LL will hardly affect the maximal value of tan 2 θ 12 . The last hope is that there could be a significant negative contribution to θ 12 from V eL obtained from the diagonalisation of the charged lepton mass matrix m l having special structure in 1-2 block. We wish to examine here how θ 12 = (θ ν 12 − θ e 12 ) can resolve the LMA MSW solar neutrino mixing scenario.
In a model independent way we parametrize the charged lepton mixing V eL as This gives a special form in the 1-2 block. We can reconstruct the symmetric charged lepton mass matrix using Eq.(2) from the relation, where we consider V eL = V eR for symmetric matrix. The MNS mixing matrix Now we obtain the following parameters If ∆ 23 = 1 − sin 2 2θ 23 , is the small deviation from the completely maximal value of the atmospheric neutrino mixings, then the above relations take the following new forms: Where α in Eq.(4)has been suppressed as it contributes insignificantly to the solar mixing. The Table-1 shows that in Cases II and III, solar mixings together with solar mass splitting ∆m 2 21 = 7.49 × 10 −5 eV 2 are in excellent agreement with the best fit tan 2 θ 21 = 0.56 obtained in KamLand experiment [8]. The values of |V e3 | are also in accord with the CHOOZ experiment.
Finally we check the stability of the neutrino mass matrix under RGEs. Following the standard procedure [4,5] we express m LL in terms of K, the coefficient of the dimension five neutrino mass operator in a scale-dependent manner m LL (t) = v 2 u (t)K(t) where t = ln(µ) and v u (t) is the scale-dependent [5] vacuum expectation value (VEV) v u = v 0 sin β, v 0 = 174GeV . In the basis where the charged lepton mass matrix is made diagonal, we can write the above expression as m ′ is the coefficient of the dimension five neutrino mass operators in the basis where the charged lepton mass matrix is diagonal. The approximate analytical solution of the relevant RGEs in MSSM, is given by the following form We have the expressions R = e 9 10 Ig 1 (t0) e 9 2 Ig 2 (t0) ≃ 1, α = e −tτ (t0) = 1 − r where the approximate solution is [4,5,6] with v = 245.4GeV . The low-energy solar mass splitting ∆m 2 sol can be expressed in terms of high energy solar mass splitting ∆m 2 21 and the radiative correction effect δ rad as ∆m 2 sol = ∆m 2 21 + δ rad , δ rad = −4r|m 1 | 2 sin 2 2θ 23 cos 2θ 21 .
In MSSM the quantity r is positive and thus we have ∆m 2 sol < ∆m 2 21 . For example, We obtain the numerical values δ rad = −0.96×10 −5 eV 2 corresponding to the CaseIII in Table-1. The mass ratio |m 2 |/|m 1 | also decreases with energy scale. The model is stable under radiative corrections.