= Quartic double-well fluctuation determinant
{title2=$\det' L_I/\det L_0=1/(12\omega^2)$}
The <quartic double-well instanton> has $L_I=-\partial_\tau^2+\omega^2[1-3\operatorname{sech}^2(\omega\tau/2)/2]$. For a positive regulator $\varepsilon$ and $r=2\sqrt{1+\varepsilon/\omega^2}$, the common infinite-line determinant ratio is $(r-1)(r-2)/[(r+1)(r+2)]$. Its zero-mode factor is $\varepsilon/(12\omega^2)$ to leading order, giving the displayed primed ratio. Therefore the <instanton fluctuation prefactor> is $K=\omega\sqrt{6S_I/(\pi\hbar)}$ and the <double-well tunneling splitting> is $2\hbar K e^{-S_I/\hbar}$.
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