Sommerfeld enhancement for puffy self-interacting dark matter

We examine the Sommerfeld enhancement effect for the puffy self-interacting dark matter. We find out two new parameters to classify the self-scattering cross section into the Born, the resonance and the classical regimes for the puffy dark matter. Then we observe that the resonance peaks for the puffy dark matter self-scattering and for the Sommerfeld enhancement effect have the same locations. Further, we find that for a large ratio between $R_{\chi}$ (radius of a puffy dark matter particle) and $1/m_{\phi}$ (force range), the Sommerfeld enhancement factor approaches to 1 (no enhancement). Finally, for the puffy SIDM scenario to solve the small-scale problems, the values of the Sommerfeld enhancement factor are displayed in the allowed parameter regions.