Constraining neutrino masses, the cosmological constant and BSM physics from the weak gravity conjecture

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 similar or equal to 10(10-14) GeV and Y-nu 1 similar or equal to 10(-3), one obtains that the EW scale cannot exceed M-EW less than or similar to 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.