Quartic double-well instanton ($q_I(t)=a\tanh [\omega (t-t_0)/2],S_I=2m\omega a^2/3$)

For $V(q)=m{\omega }^2(q^2-a^2{)}^2/(8a^2)$, the zero Euclidean-energy condition is $m{\dot{q}}^2/2=V(q)$. Integrating gives the displayed instanton from $-a$ to $a$. Its action is ${\int }_{-a}^a\sqrt{2mV(q)}dq=2m\omega a^2/3$. The arbitrary center $t_0$ supplies a translation zero mode, treated as a collective coordinate in the instanton fluctuation prefactor.

The quartic double-well instanton has $L_I=-{\partial }_{\tau }^2+{\omega }^2[1-3{\mathrm{sech}}^2(\omega \tau /2)/2]$. For a positive regulator $\epsilon$ and $r=2\sqrt{1+\epsilon /{\omega }^2}$, the common infinite-line determinant ratio is $(r-1)(r-2)/[(r+1)(r+2)]$. Its zero-mode factor is $\epsilon /(12{\omega }^2)$ to leading order, giving the displayed primed ratio. Therefore the instanton fluctuation prefactor is $K=\omega \sqrt{6S_I/(\pi \hslash )}$ and the double-well tunneling splitting is $2\hslash K{e}^{-S_I/\hslash }$.