Abstract
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 Λ4 and neutrino masses from the absence of such dangerous 3D and 2D SM AdS vacua. One interesting implication is that Λ4 is bounded to be larger than a scale of order mν4, as observed experimentally. Interestingly, this is the first argument implying a non-vanishing Λ4 only on the basis of particle physics, with no cosmological input. Conversely, the observed Λ4 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 Λ4) 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 ≃ 1010 − 14 GeV and Yν1 ≃ 10−3, one obtains that the EW scale cannot exceed MEW ≲ 102 − 104 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.
Highlights
It has always been intriguing the proximity of the c.c. scale to that of neutrino masses since both scales seem to have a very different origin
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
One may argue that other potential instabilities may appear e.g. in the context of a 4D landscape of vacua in string theory, in which tunneling in 4D would have parallel transitions in lower dimensions, rendering the lower dimensional vacua unstable, and leading to no constraint on neutrino masses or any other physical parameter
Summary
The Weak Gravity Conjecture states that, in theories of quantum gravity with a p-form gauge field, there must exist an electrically charged object with charge Q and tension T satisfying. Ooguri and Vafa proposed in [36] a sharpened version of this conjecture, claiming that the equality can only be satisfied if the charged object is BPS and the theory is supersymmetric. This has dramatic consequences for the AdS/CFT duality as we review in the following. In the absence of new low energy physics, Majorana neutrinos necessarily give rise to AdS vacua in three dimensions If these vacua are stable, they would be inconsistent with the above conjecture. Before turning to a more thoughtful analysis of these constraints, let us comment, though, on the issue of stability
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