Exact constraints on family-separated seesaw relations and their phenomenological consequences

Abstract

Working at tree level and at a common renormalization scale, we examine the family-separated seesaw ansatz of Z.-z.~Xing, arXiv:2605.27049v2. For a fixed light-heavy pairing, imposing $m_iU_{\alpha i}U_{\beta i}+M_iR_{\alpha i}R_{\beta i}=0$ together with $UU^\dagger+RR^\dagger=\mathbf{1}$ yields the general exact solution $U=VD^{-1/2}$ and $R=iVE\,\operatorname{diag}\!\left(\sqrt{r_i/(1+r_i)}\right)$, where $r_i=m_i/M_i$, $D=\operatorname{diag}(1+r_i)$, $V$ is unitary, and $E=\operatorname{diag}(\eta_i)$ with $\eta_i=\pm1$. Consequently, $U^\dagger U$ and $R^\dagger R$ are diagonal, and distinct columns are exactly orthogonal. An exact unitary completion gives, in a convenient sterile basis, $M_R=D_N-D_\nu$ and $Y_\nu=(i/v)VE(D_\nu D_N)^{1/2}$, so that $Y_\nu^\dagger Y_\nu=D_\nu D_N/v^2$ is diagonal. In the canonical domain $M_i>m_i$, the singlet masses relevant to the unbroken-phase decay description are $\widehat{M}_i=M_i-m_i>0$, distinct from the broken-phase heavy eigenvalues $M_i$. Hence, at the scale of exact alignment, all standard unflavored and flavored nonresonant one-loop decay asymmetries vanish in the nondegenerate perturbative regime. For $m_i=0$, the paired Yukawa column and decay width vanish instead. The flavor-summed rephasing-invariant combination used in the proposed asymmetry also cancels identically, although individual CP-odd invariants may remain nonzero. Thus the proposed correlation between low-energy CP violation and these decay asymmetries does not follow from exact family separation. This result does not address radiative misalignment, resonant or coherent dynamics, or thermal and higher-loop sources. We also correct the proposed heavy-mass reconstruction formulas and clarify the implications for neutrinoless double-beta decay, Yukawa scaling, parameter counting, and light-heavy mass orderings.