  Relativistic shocks that accompany supernovae (SNe) produce X-ray burst
emissions as they break out in the dense circumstellar medium around the
progenitors. This phenomenon is sometimes associated with peculiar
low-luminosity gamma-ray bursts (LL GRBs). Here, we investigate the high energy
neutrino and gamma-ray counterparts of such a class of SNe. Just beyond the
shock breakout radius, particle acceleration in the collisionless shock starts
to operate in the presence of breakout photons. We show that protons may be
accelerated to sufficiently high energies and produce high energy neutrinos and
gamma rays via the photomeson interaction. These neutrinos and gamma rays may
be detectable from 10 Mpc away by IceCube/KM3Net as multi-TeV transients almost
simultaneously with the X-ray burst emission, and even from 100 Mpc away with
follow-up observations by CTA using a wide-field sky monitor like Swift as a
trigger. A statistical technique using a stacking approach could also be
possible for the detection, with the aid of the SN optical/infrared
counterparts. Such multi-messenger observations offer the possibility to probe
the transition of relativistic shocks from radiation-mediated to collisionless
ones, and would also constrain the mechanisms of particle acceleration and
emission in LL GRBs.
