On small Dirac Neutrino Masses in String Theory

We study how tiny Dirac neutrino masses consistent with experimental constraints can arise in string theory SM-like vacua. We use as a laboratory 4d ${\cal N}=1$ type IIA Calabi--Yau orientifold compactifications, and in particular recent results on Yukawa couplings at infinite field-space distance. In this regime we find Dirac neutrino masses of the form $m_\nu \simeq g_\nu\langle H\rangle$, with $g_{\nu}$ the gauge coupling of the massive $U(1)$ under which the right-handed neutrinos $\nu_R$ are charged, and which should be in the range $g_{\nu}\simeq 10^{-14}-10^{-12}$ to reproduce neutrino data. The neutrino mass suppression occurs because the right-handed neutrino kinetic term behaves as $K_{\nu\nu} \simeq 1 /g_{\nu}^2$. At the same time a tower of $\nu_R$-like states appears with characteristic scale $m_0\simeq g_{\nu}^2M_{\rm P}\simeq 0.1-500$ eV, in agreement with Swampland expectations. Two large hidden dimensions only felt by the $\nu_R$ sector arise at the same scale, while the string scale is around $M_s\simeq g_\nu M_{\rm P}\simeq 10-700$ TeV. Some phenomenological implications and model building challenges are described. As a byproduct, independently of the neutrino issue, we argue that a single large dimension in the context of SM-like type IIA Calabi--Yau orientifolds leads to too small Yukawa couplings for quarks and charged leptons.