Confronting the real scalar septuplet minimal dark matter model with vacuum stability, perturbativity, and Type-I and Type-III seesaws

We study a real scalar septuplet model which has a neutral component regarded as a dark matter particle candidate. The calculation of its thermal relic abundance without (with) the Sommerfeld effect suggests that the observed value corresponds to the mass of the dark matter particle $m_0 \gtrsim 8.8~(25)~\mathrm{TeV}$. Two extra quartic couplings $\lambda_2$ and $\lambda_3$ introduced in this model affect the running of other couplings, and hence the vacuum stability and the perturbativity up to the Planck scale. Therefore, the vacuum stability and the perturbativity conditions can constrain these couplings into a narrow region in the $\lambda_2$-$\lambda_3$ plane. Other constraints from $h\to\gamma\gamma$, electroweak oblique parameters, and direct and indirect DM searches are also investigated. Moreover, we survey the vacuum stability and the perturbativity in the model combining the septuplet with the Type-I or Type-III seesaw. The running of couplings is further altered when the energy scale goes above the seesaw scale. In the Type-I case, when the seesaw scale $\sim 10^{15}~\mathrm{GeV}$, the vacuum stability condition makes the acceptable region in the $\lambda_2$-$\lambda_3$ plane much narrower. In the Type-III case, if the seesaw scale is $\lesssim 52~\mathrm{TeV}$ or $\gtrsim 10^{15}~\mathrm{GeV}$, the acceptable region vanishes.