  It is known that there are AdS vacua obtained from compactifying the SM to 2
or 3 dimensions. The existence of such vacua depends on the value of neutrino
masses through the Casimir effect. Using the Weak Gravity Conjecture, it has
been recently argued by Ooguri and Vafa that such vacua are incompatible with
the SM embedding into a consistent theory of quantum gravity. We study the
limits obtained for both the cosmological constant $\Lambda_4$ and neutrino
masses from the absence of such dangerous 3D and 2D SM AdS vacua. One
interesting implication is that $\Lambda_4$ is bounded to be larger than a
scale of order $m_\nu^4$, as observed experimentally. Interestingly, this is
the first argument implying a non-vanishing $\Lambda_4$ only on the basis of
particle physics, with no cosmological input. Conversely, the observed
$\Lambda_4$ implies strong constraints on neutrino masses in the SM and also
for some BSM extensions including extra Weyl or Dirac spinors, gravitinos and
axions. The upper bounds obtained for neutrino masses imply (for fixed neutrino
Yukawa and $\Lambda_4$) the existence of upper bounds on the EW scale. 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.
