Stability of Neutrino Mass Degeneracy

Two neutrinos of Majorana masses m1,2 with mixing angle θ are unstable against radiative corrections in the limit m1 = m2, but are stable for m1 = −m2 (i.e. opposite CP eigenstates) with θ = 45◦ which corresponds to an additional symmetry. Pick two neutrinos, say νe and νμ. Assume their mass eigenstates to be ν1 = νe cos θ − νμ sin θ, ν2 = νe sin θ + νμ cos θ, (1) with eigenvalues m1 and m2 respectively. Neutrino oscillations may then occur[1, 2, 3] if both ∆m = m 2 − m 1 and sin 2θ are nonzero. However, it is entirely possible that the hierarchy ∆m << m 1,2 (2) actually exists, so that the smallness of ∆m for neutrino oscillations does not necessarily preclude a much larger common mass for the two neutrinos. In fact, this idea is often extended to all three neutrinos[4, 5]. On the other hand, since the charged-lepton masses are all different, radiative corrections[6, 7] to m1 and m2 will tend to change ∆m 2 as well as θ. This is especially important for the vacuum oscillation solution[8] to the observed solar neutrino deficit[2] which requires ∆m ∼ 10 eV and sin 2θ ∼ 1. In the following I show that whereas the limit m1 = m2 is unstable against radiative corrections, the limit m1 = −m2 and θ = 45◦ is stable because it is protected by an additional symmetry. [A negative mass here means that the corresponding Majorana neutrino is odd under CP after a γ5 rotation to remove the minus sign.] Consider the 2× 2 mass matrix spanning νe and νμ: M = 