$\det m_{\nu} = -2ab\epsilon$, and where we imagine the entry $\epsilon$ to be a small perturbation around the basic structure of (\[lambdaT1\]), which can arise from higher-order terms in $W$. Without loss of generality we can redefine the phases of the lepton doublets $L_i$ and $L'_i$ such that the entries $a$, $b$ and $\epsilon$ are real and $m_{\nu} = m_{\nu}^{\dagger}$. This implies $$%
U_{\nu}^{\dagger}m_{\nu} U_{\nu} = {\rm diag}(m_1, m_2, m_3)
\equiv m_{\rm diag} .
%
\label{Unu}$$ We also have $\Tr \; m_{\rm diag} = m_1 + m_2 + m_3 = \Tr \;
m_{\nu} = 0$, where the various eigenvalues $m_i$ are real but can be negative.

If we begin by first ignoring the solar mass difference, and take the atmospheric mass difference to be given by (\[dm23\]), then there is no way to accommodate the “normal” hierarchy while maintaining the requirement that $\epsilon \ll a,b$. For the inverted hierarchy (in the same approximation of vanishing solar mass difference) we would require $m_2 = -m_1 = 0.049 \; {\rm eV}$ with $m_3 =0$. This could derive from (\[genoff\]) if $\sqrt{a^2
+ b^2} = m_2$ and $\epsilon=0$. In this case $\sum_i |m_i| = 0.098
\; {\rm eV}$. This is clearly in line 