Kaon CP mixing parameter ($ϵ=\frac{\sqrt{R_{12}}-\sqrt{R_{21}}}{\sqrt{R_{12}}+\sqrt{R_{21}}}$)

In the flavor convention $K_1=(K^0-{\bar{K}}^0)/\sqrt{2}$ and $K_2=(K^0+{\bar{K}}^0)/\sqrt{2}$, the parameter $ϵ=(a-b)/(a+b)$ writes normalized mass states as $(K_1+ϵK_2)/\sqrt{1+|ϵ{|}^2}$ and $(K_2+ϵK_1)/\sqrt{1+|ϵ{|}^2}$. In the Hermitian mass-only approximation with correlated roots, $ϵ=i\tan (\phi /2)$ and the states are orthogonal. For the full non-Hermitian decay matrix the real part need not vanish. The parameter depends on the flavor-state phase convention; observable CP violation involves convention-invariant combinations with decay amplitudes.

In the phase convention $CP|K^0⟩=|{\bar{K}}^0⟩$, the symmetric flavor combination is CP even and the antisymmetric one CP odd. For an equal-diagonal effective matrix with entries b,c off diagonal, choose correlated roots r,s with $r^2=b$, $s^2=c$ and eigenvalues $a\pm rs$. Its eigenvectors are $(r,\pm s{)}^T$. Writing $ϵ=(r-s)/(r+s)$ gives the displayed CP-basis mixtures, up to an overall phase. Their deviation is small only if $|ϵ|\ll 1$; noninvariance alone does not guarantee that. This convention must not be mixed with one in which flavor exchange carries a minus sign.