Abelian symmetries as the source of the small neutrino-related flavor parameters

There are four neutrino-related flavor parameters that have been measured: The three mixing angles s_{23}, s_{12}, s_{13}, and the ratio of mass-squared differences r_{23}. Of these, the first two are order one. On the other hand, s_{13} and r_{23} can be either order-one parameters that are accidentally somewhat small, or they are small for a reason, for example, they vanish in the limit of a symmetry that is broken by small parameters. We show that in the latter case, the Froggatt-Nielsen mechanism could explain the smallness of s_{13} and r_{23} only if some order-one coefficients are as small as the symmetry-breaking parameters. It is thus very unlikely that an Abelian symmetry is responsible for the smallness of s_{13} and r_{23}.