No - go for exactly degenerate neutrinos at high scale ?

We show in a model independent manner that, if the magnitudes of Majorana masses of neutrinos are exactly equal at some high scale, the radiative corrections cannot reproduce the observed masses and mixing spectrum at the low scale, irrespective of the Majorana phases or the mixing angles at the high scale. The data from the solar and atmospheric neutrino experiments can be explained through the mixing of three active neutrinos with nonzero masses. The atmospheric neutrino solution needs νμ−ντ mixing with the corresponding mass squared difference of ∆m2atm ≈ 10−3−10−2 eV, and a large mixing angle, sin 2θatm > 0.8 [1]. The solar neutrino solution needs the mixing of νe with a combination of νμ and ντ with a corresponding mass squared difference ∆m2⊙ <∼ 10 eV, the mixing angle may be small or large, depending on the particular solution [2]. Since ∆m2atm >> ∆m 2 ⊙, there are three distinct patterns for the neutrino masses from the point of view of mass hierarchy. We define the mass eigenstate ν3 as the one such that |∆m2 31 | ≈ |∆m2 32 | ≈ ∆m2atm, and the states ν1 and ν2 as the ones separated by |∆m21| ≈ ∆m2⊙, with ∆m 2 31 and ∆m 21 having the same sign (here ∆m2ij ≡ |mi| − |mj|). With this convention, the possible patterns for the neutrino masses are (i) completely hierarchical: |m1| ≪ |m2| ≪ |m3|, (ii) partially degenerate: |m1| ∼ |m2| ≫ |m3|, also called as inverted hierarchy or (iii) completely degenerate: |m1| ∼ |m2| ∼ |m3|. In contrast to the first two cases, the common mass in the third case can be near the direct limit on the electron neutrino mass as obtained from the Kurie plot [3]. Such a mass can have its own signature in the neutrinoless double beta decay [4]. In order for the neutrinos to contribute even a small fraction of the dark matter [5] of the universe, degenerate neutrino masses are essential.