Consider the following cubic dispersion equation of $\omega(k)$ $$\omega^2-\omega_a^2(k)-\frac{\alpha^2k^4}{\omega-\omega_b(k)}=0.$$ $\omega_a=vk,\omega_b=b-v'k$ are two unhybridized dispersions that cross at $k^*=\frac{b}{v+v'}$. Finite $\alpha$ hybridizes them and leads to an anticrossing gap (or avoided crossing) at $k^*$. At $k^*$, the equation also becomes $$(\omega+\omega^*)(\omega-\omega^*)^2-\alpha^2k^{*4}=0$$ with $\omega^*=\omega_a(k^*)$.
How to estimate the scale or order of magnitude of this anticrossing gap size? E.g., by estimating something like $\omega(k^*)-\omega^*$ for one positive solution.