Parity and lepton masses in the left-right symmetric model

Abstract: Curiously in the minimal left right symmetric model, chiral symmetry that protects the electron's mass ($m_e$), due to parity (P), implies in the symmetry limit the vanishing of its neutrino mixing angles. We break the chiral symmetry softly (or spontaneously if it is gauged) to generate the observed large neutrino mixing angles at the tree-level. The electron then acquires its mass on renormalization group equation (RGE) running due to its neutrino's mixing, and in turn determines the $B-L$ gauge symmetry breaking scale ($v_R$) to be $10^{10} GeV \lesssim v_R \leq 10^{15} GeV.$ If the muon's mass is also generated radiatively, the $B-L$ breaking scale is $\sim 10^{14-15}$ GeV. Regardless of the high scale of $v_R$, this is a testable model since on RGE running and P breaking, a large strong CP phase ($\bar{\theta} >> 10^{-10}$) which depends logarithmically on $v_R$ is generated if there is $\mathcal{O}(1)$ CP violation in leptonic Yukawa couplings. Hence we expect that leptonic CP phases including the Dirac CP phase $\delta_{CP}$ of the PMNS matrix must be consistent with $0$ or $180^o$ to within a degree, which can be verified or excluded by neutrino experiments such as DUNE and Hyper-Kamiokande. In lieu of P, if charge conjugation C is used, the same results follow. However with C and no P, axions would likely need to be added anyway, in which case there is no constraint on $\delta_{CP}$.