Chiral perturbative relation for neutrino masses in the type-I seesaw mechanism

Abstract

In this letter, we perform a perturbative analysis by the lightest singular value $m_{D1}$ of the Dirac mass matrix $m_D$ in the type-I seesaw mechanism. A mass relation $M_1=m_{D1}^2/|(m_{\nu }{)}_{11}|$ is obtained for the lightest mass $M_1$ of the right-handed neutrino ${\nu }_{R1}$ and the mass matrix of the left-handed neutrinos $m_{\nu }$ in the diagonal basis of $m_D$. This relation is rather stable under renormalization because it is gauge-invariant in the SM and associates with the approximate chiral symmetry of ${\nu }_{R1}$. If diagonalization of the Yukawa matrices of leptons $Y_{\nu ,e}$ has only small mixings, the element $(m_{\nu }{)}_{11}$ is close to the effective mass $m_{ee}$ of the neutrinoless double beta decay. By assuming $m_{D1}\simeq m_{u,e}\simeq 0.5$ MeV, the lightest mass is about $M_1\gtrsim O(100)$ TeV in the normal hierarchy and $M_1\sim O(10)$ TeV in the inverted hierarchy. Such a ${\nu }_{R1}$ with a tiny Yukawa coupling $y_{\nu 1}\sim O(10^{-5})$ can indirectly influence various observations. On the other hand, the famous bound of the thermal leptogenesis $M_1\gtrsim 10^9$ GeV that requires $m_{D1}\gtrsim 30$ MeV seems to be difficult to reconcile with a simple unified theory without a special condition.

Comments: 6 pages, final version to be published in PLB