$\mathbf{θ_{13}^{}}$ and The Flavor Ring

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 $\bs{\m Z_7 \rtimes \m Z_3}$, 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 $θ_{13}$ requires an asymmetric down-quark Yukawa matrix. Satisfying the measured values of both $θ_{13}$ and the electron mass restricts the number of solutions, underlying the importance of the recent measurement of the reactor angle.

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