Empirical formulas for the fermion spectra and Yukawa matrices

Abstract.

We present empirical relations that connect the dimensionless ratios of low energy fermion masses for the charged lepton, up-type quark and down-type quark sectors and the CKM elements: $\left\lvert V_{us}\right\rvert \approx \left[ \frac{m_{d}}{m_{s}} \right]^{\frac{1}{2}} \approx \left[ \frac{m_{u}}{m_{c}} \right]^{\frac{1}{4}} \approx 3 \left[ \frac{m_{e}}{m_{\mu}} \right]^{\frac{1}{2}}$ and $\frac{1}{2} \left\lvert \frac{V_{cb}}{V_{us}}\right\rvert \approx \left[ \frac{m_{s}^{3}}{m_{b}^{2}m_{d}} \right]^{\frac{1}{2}} \approx \left[ \frac{m_{c}^{3}}{m_{t}^{2}m_{u}} \right]^{\frac{1}{2}} \approx \frac{1}{9} \left[ \frac{m_{\mu}^{3}}{m_{\tau}^{2}m_{e}} \right]^{\frac{1}{2}}$. Explaining these relations from first principles imposes strong constraints on the search for the theory of flavor. We present a simple set of normalized Yukawa matrices, with only two real parameters and one complex phase, which accounts with precision for these mass relations and for the CKM matrix elements and also suggests a simpler parametrization of the CKM matrix. The proposed Yukawa matrices accommodate the measured CP-violation, giving a particular relation between standard model CP-violating phases, $\beta = {\mathrm {Arg}} \left[ 2 - \mathrm {e}^{-\mathrm i\gamma} \right]$. According to this relation the measured value of $\beta$ is close to the maximum value that can be reached, $\beta_{\mathrm {max}} = 30^{\circ}$ for $\gamma = 60^{\circ}$. Finally, the particular mass relations between the quark and charged lepton sectors find their simplest explanation in the context of grand unified models through the use of the Georgi-Jarlskog factor.