Abstract
It has been recently argued that an embedding of the SM into a consistent theory of quantum gravity may imply important constraints on the mass of the lightest neutrino and the cosmological constant Λ4. The constraints come from imposing the absence of any non-SUSY AdS stable vacua obtained from any consistent compactification of the SM to 3 or 2 dimensions. This condition comes as a corollary of a recent extension of the Weak Gravity Conjecture (WGC) by Ooguri and Vafa. In this paper we study T2/ZN compactifications of the SM to two dimensions in which SM Wilson lines are projected out, leading to a considerable simplification. We analyze in detail a T2/Z4 compactification of the SM in which both complex structure and Wilson line scalars are fixed and the potential is only a function of the area of the torus a2. We find that the SM is not robust against the appearance of AdS vacua in 2D and hence would be by itself inconsistent with quantum gravity. On the contrary, if the SM is embedded at some scale MSS into a SUSY version like the MSSM, the AdS vacua present in the non-SUSY case disappear or become unstable. This means that WGC arguments favor a SUSY version of the SM, independently of the usual hierarchy problem arguments. In a T2/Z4 compactification in which the orbifold action is embedded into the B − L symmetry the bounds on neutrino masses and the cosmological constant are recovered. This suggests that the MSSM should be extended with a U(1)B−L gauge group. In other families of vacua the spectrum of SUSY particles is further constrained in order to avoid the appearance of new AdS vacua or instabilities. We discuss a possible understanding of the little hierarchy problem in this context.
Highlights
Which loosely speaking implies that the gravitational interaction must be weaker than gauge interactions in any consistent theory of quantum gravity
We find that the SM is not robust against the appearance of AdS vacua in 2D and would be by itself inconsistent with quantum gravity
We find Z4 embeddings into a discrete subgroup of the U(1)B−L gauge symmetry leading to AdS vacua of Type P which may be evaded for appropriate neutrino masses, like the original 3D vacua in the circle, and are predictive
Summary
We first study the compactification of the SM coupled to Einstein gravity with a cosmological constant down to 3D on the circle. A possible source of instabilities may in principle arise if there are lower minima or runaway directions at smaller radion values (higher thresholds) contributing to the Casimir potential [42] After the top quark scale, R6V remains constant for higher energies, so V keeps decreasing This runaway behaviour of the potential could mean that the neutrino minima is metastable, as pointed out in [42]. We describe the structure of the radius scalar potential at shorter distances, up to the EW scale, setting the Wilson line moduli to zero by hand, looking for new features beyond the neutrino local minima. We consider the case of the compactification of the SM on the torus T 2 and on the orbifold T 2/Z4 which is closely related
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