Perturbativity Limits for Scalar Minimal Dark Matter with Yukawa Interactions: Septuplet

The candidate of minimal dark matter (MDM) is limited if one demands perturbativity up to a very high scale, and it was believed that the MDM model with a real scalar septuplet could keep perturbative up to the Planck or GUT scale. In this work we point out that it is not true after taking into account the running of the quartic self-couplings of the scalar septuplet. For the septuplet mass around $10$ TeV, which is suggested by the observed dark matter relic abundance, these couplings would hit the Landau pole at a scale $\sim 10^8-10^9$ GeV, much lower than the Planck scale. We attempt to push up the Landau pole scale as high as possible by proposing an extension with extra Yukawa interactions of the septuplet. We find that in principle the Landau pole could be deferred to a scale of $\sim 10^{14}$ GeV if one could tolerate a serious fine-tuning of the initial condition of the Yukawa coupling. Moreover, if the MDM particle mass could be relaxed to $\sim10^8$ GeV, which would need some nonthermal production mechanisms to give a correct relic abundance, the Landau pole scale could be pushed up above the Planck scale.