Constraining the EW Hierarchy from the Weak Gravity Conjecture

Abstract: In this addendum we complement the remarks made in ref[1] constraining the value of the cosmological constant $\Lambda_4$ in terms of neutrino masses. Those were based on a sharpened version of the Weak Gravity Conjecture as applied to compactifications of the SM to lower dimensions. We argue that the same line of reasoning implies that for fixed values of $\Lambda_4$ and the Yukawa coupling of the lightest neutrino $Y_{\nu_1}$, the EW scale is bounded above. This is a trivial consequence of neutrino masses depending on the Higgs vev. In the case of massive Majorana neutrinos with a see-saw mechanism associated to a large scale $M\simeq 10^{10-14}$ GeV and $Y_{\nu_1}\simeq 10^{-3}$, one obtains that the EW scale cannot exceed $M_{EW}\lesssim 10^2-10^4$ GeV. From this point of view, the delicate fine-tuning required to get a small EW scale would be a mirage, since parameters yielding higher EW scales would be in the swampland and would not count as possible consistent theories. This would bring a new perspective into the issue of the EW hierarchy.