We can find analytic solutions for some parameters in terms of the mass $m$ of the electron, muon, neutron or proton. This is done by writing out the internal energy $U$ in terms of quark coefficients, and then solving a quadratic equation to obtain

$\begin{align} U^{\sf{M}} = \frac{ -B + \sqrt{B^{2} - 4AC} }{ 2A } \end{align}$

where

$\begin{align} \alpha = 4U^{\sf{U}} - 4U^{\sf{A}} - 2U^{\sf{T}} + 2U^{\sf{B}} + 2U^{\sf{S}} - 2U^{\sf{C}} \end{align}$

$\begin{align} A = 16 \left( 1 - k_{mm} \right) \end{align}$

$\begin{align} B = 8\alpha -32k_{mm}U^{\sf{A}} +32k_{mz}U^{\sf{U}} \end{align}$

$\begin{align} C = \alpha^{2} - 16\left( U^{\sf{U}} \right)^{2} -16k_{mm}\left( U^{\sf{A}} \right)^{2} +32k_{mz}U^{\sf{U}}U^{\sf{A}} - m \left( \sf{\mu}^{-} \right)^{2} c^{4} \end{align}$