Quark and lepton mixing in the asymptotically safe Standard Model

Abstract

The quark mixing (CKM) matrix is near-diagonal, whereas the lepton mixing (PMNS) matrix is not. We learn that both observations can generically be explained within an ultraviolet completion of the Standard Model with gravity. This ultraviolet completion results in approximate relations between CKM matrix elements. The accuracy with which these hold translates into the length of asymptotically safe scaling regimes. Experimental data are consistent with the theoretical expectations arising from a fixed-point cascade at transplanckian scales. Similar results would hold for the PMNS matrix if neutrino Yukawa couplings were large. The ultraviolet complete theory, therefore, must -- and in fact can -- avoid such an outcome. It achieves this through a dynamical mechanism that limits the size of neutrino Yukawa couplings. Thus, large neutrino mixing is intimately tied to small Dirac neutrino masses, $\sum m_{\nu }\lesssim O(1)\mathrm{eV}$, and a mass gap in the Standard Model fermion masses.

8 pages + appendix, 5 figures; typos corrected; minor changes in abstract; convention changed to the standard PMNS convention; typos in Appendix equations corrected; Mathematica notebook updated accordingly, results unchanged;