On the four-zero texture of quark mass matrices and its stability

We carry out a new study of quark mass matrices $M_{\mathrm{u}}^{}$ (up-type) and $M_{\mathrm{d}}^{}$ (down-type) which are Hermitian and have four zero entries, and find a new part of the parameter space which was missed in the previous works. We identify two more specific four-zero patterns of $M_{\mathrm{u}}^{}$ and $M_{\mathrm{d}}^{}$ with fewer free parameters, and present two toy flavor-symmetry models which can help realize such special and interesting quark flavor structures. We also show that the texture zeros of $M_{\mathrm{u}}^{}$ and $M_{\mathrm{d}}^{}$ are essentially stable against the evolution of energy scales in an analytical way by using the one-loop renormalization-group equations.