${\theta }_{13}^{}$ and The Flavor Ring

Abstract

We present the results of a numerical search for the Dirac Yukawa matrices of the Standard Model, consistent with the quark and lepton masses and their mixing angles. We assume a diagonal up-quark matrix, natural in $\text{bs}\text{m}{Z}_7⋊\text{m}{Z}_3$, Bimaximal or Tri-bimaximal seesaw mixing, and $SU(5)$ unification to relate the down-quark and charged lepton Dirac Yukawa matrices using Georgi-Jarlskog mechanisms. The measured value of ${\theta }_{13}$ requires an asymmetric down-quark Yukawa matrix. Satisfying the measured values of both ${\theta }_{13}$ and the electron mass restricts the number of solutions, underlying the importance of the recent measurement of the reactor angle.

Comments: 59 pages, 10 figures, 18 tables. Mathematica file included as an ancillary file