Renormalization group effects for a rank degenerate Yukawa matrix and the fate of the massless neutrino

Abstract

The Type-I seesaw model is a common extension to the Standard Model that describes neutrino masses. The Type-I seesaw introduces heavy right-handed neutrinos with Majorana mass that transform as Standard Model electroweak gauge singlets. We initially study a case with two right-handed neutrinos called the 3-2 model. At an energy scale above the right-handed neutrinos, the effective neutrino mass matrix is rank degenerate implying the lightest neutrino is massless. After considering renormalization effects below the two right-handed neutrinos, the effective neutrino mass matrix remains rank degenerate. Next, we study a model with three right-handed neutrinos called the 3-3 model. Above the energy scale of the three right-handed neutrinos, we construct the effective neutrino mass matrix to be rank degenerate. After solving for the renormalization effects to energies below the three right-handed neutrinos, we find the rank of the effective neutrino mass matrix depends on the kernel solutions of the renormalization group equations. We prove for the simplest kernel solutions the effective neutrino mass matrix remains rank degenerate.

The µ–τ reflection symmetry of Majorana neutrinos *

“…In this case the seesaw threshold effects play a critical role, but the numerical output of m 1 (or m 3 ) strongly depends on the numerical inputs of those unknown Yukawa coupling matrix elements. If the evolution of neutrino flavor parameters simply starts from M min , the so-called seesaw scale where the heavy degrees of freedom have been integrated out and the unique dimension-five Weinberg operator is responsible for the origin of three active neutrino masses [138], then a nonzero value of m 1 (or m 3 ) at Λ EW cannot be generated from m 1 = 0 (or m 3 = 0) at M min unless the two-loop RGE running behaviors are taken into account [139][140][141][142]. In the latter case the resulting mass of the lightest Majorana neutrino ν 1 or ν 3 is typically of O(10 −13 ) eV or smaller-such a vanishingly small neutrino mass and the associated Majorana CP-violating phase do not cause any seeable effects in neutrino physics.…”