Gravity-improved metastability bounds for the Type-I Seesaw Mechanism

Right-handed neutrinos (RHN) destabilize the electroweak vacuum by increasing its decay rate. In the SM, the latter is dominated by physics at the RG scale at which $\lambda$ reaches its minimum, $\mu_*^{\text{SM}} \sim 10^{17}$ GeV. For large neutrino Yukawa coupling $Y_\nu$, RHNs can push $\mu_*$ beyond the Planck scale, implying that gravitational effects need to be taken into account. In this work, we perform the first comprehensive study of electroweak vacuum metastability in the type-I seesaw mechanism including these effects. Our analysis covers both low- and high-scale seesaw models, with two as well as three RHNs and for multiple values of the Higgs' non-minimal coupling to gravity. We find that gravitational effects can significantly stabilize the vacuum, leading to weaker metastability bounds. We show that metastability sets the strongest bounds for low-scale seesaws with $M_N>1$ TeV. For high-scale seesaws, we find upper bounds on the allowed masses for the RHNs, which are relevant for high-scale leptogenesis. We also point out that $\text{Tr}(Y_\nu^\dagger Y_\nu)$, which is commonly used to express these metastability bounds, cannot be used for all of parameter space. Instead, we argue that bounds can always be expressed reliably through $\text{Tr}(Y_\nu^\dagger Y_\nu\,Y_\nu^\dagger Y_\nu)$. Lastly, we use this insight to develop a new technique for an easier RG analysis applicable to scenarios with degenerate RHN masses.