Complementary Test of the Dark Matter Self-Interaction by Direct and Indirect Detections

The halo dark matter (DM) can be gravitationally captured by the Sun. For self-interacting DM (SIDM), we show that the number of DM trapped inside the Sun remains unsuppressed even if the DM-nucleon cross section is negligible. We consider a SIDM model where $U(1)$ gauge symmetry is introduced to account for the DM self-interaction. Such a model naturally leads to isospin violation for DM-nucleon interaction, although isospin symmetry is still allowed as a special case. We show that the detection of neutrino signature from DM annihilation in the Sun can probe those SIDM parameter ranges not reachable by direct detections. Those parameter ranges are either the region with a very small $m_{\chi}$ or the region opened up due to isospin violations.