The Fundamental Need for a SM Higgs and the Weak Gravity Conjecture

Compactifying the SM down to 3D or 2D one may obtain AdS vacua depending on the neutrino mass spectrum. It has been recently shown that, by insisting in the absence of these vacua, as suggested by {\it Weak Gravity Conjecture} (WGC) arguments, intriguing constraints on the value of the lightest neutrino mass and the 4D cosmological constant are obtained. For fixed Yukawa coupling one also obtains an upper bound on the EW scale $\left\langle H\right\rangle\lesssim {\Lambda_4^{1/4}} /{Y_{\nu_{i}}}$,where $\Lambda_4$ is the 4D cosmological constant and $Y_{\nu_{i}}$ the Yukawa coupling of the lightest (Dirac) neutrino. This bound may lead to a reassessment of the gauge hierarchy problem. In this letter, following the same line of arguments, we point out that the SM without a Higgs field would give rise to new AdS lower dimensional vacua. Absence of latter would require the very existence of the SM Higgs. Furthermore one can derive a lower bound on the Higgs vev $\left\langle H\right\rangle\gtrsim \Lambda_{\text{QCD}}$ which is required by the absence of AdS vacua in lower dimensions. The lowest number of quark/lepton generations in which this need for a Higgs applies is three, giving a justification for family replication. We also reassess the connection between the EW scale, neutrino masses and the c.c. in this approach. The EW fine-tuning is here related to the proximity between the c.c. scale $\Lambda_4^{1/4}$ and the lightest neutrino mass $m_{\nu_i}$ by the expression $ \frac {\Delta H}{H} \lesssim \frac {(a\Lambda_4^ {1/4} -m_{\nu_i})} {m_{\nu_i}}. $ We emphasize that all the above results rely on the assumption of the stability of the AdS SM vacua found.