The minimal structure of quark mass matrices is addressed from a general perspective within the Standard Model (SM) framework. It is argued that within the flavor basis of diagonal up-type quark sector, the minimal phase as well as texture structure for these mass matrices follow naturally from the observed strong mass hierarchy among the quark masses and mixing angles. The minimal phase structure appears as a consequence of the translation of the CKM phase on the down-type quark matrix while the minimal texture structure follows from the Gatto-Sartori-Tonin relation for
More about the GST relation from the link in the body text quoted above:
The pioneering work of Gatto, Sartori, and Tonin showed a relation satisfying, with good accuracy, the experimental value of the Cabibbo angle  tan2 θC ≈ md ms . (1) At present, a more complete relation with greater accuracy includes the contribution coming from the up-quark sector and a small correction through the denominator [2, 3] tan2 θC ≈ mˆ d + ˆmu (1 + ˆmd)(1 + ˆmu) , (2) with mˆ d = md/ms and mˆ u = mu/mc. From now on, we will refer to it, in its generic form which only encompasses a single fermion species, tan2 θij = mi/mj , as the Gatto-Sartori-Tonin (GST) relation, for a review on this topic please refer to .
It has been already recognized by different authors that not only the observed hierarchy in the quark masses [3–13] mt(MZ) ≫ mc(MZ) ≫ mu(MZ), (3) mb(MZ) ≫ ms(MZ) ≫ md(MZ), (4) can give access to desired GST relations, but also the charged leptons hierarchy plus the possible milder one in the neutrino masses [3, 13–15] mτ (MZ) ≫ mµ(MZ) ≫ me(MZ), (5) m2 ν3 (MZ) ≫ m2 ν2 (MZ) ≫ m2 ν1 (MZ). (6) However, only until recently, a procedure to build a mixing parametrization in terms of only the four independent mass ratios of the corresponding fermion sector was fully achieved . This new parametrization makes one to wonder about the possible physical meaning of the GST relation and the necessary conditions to find it in an exact way. . . .
[T]he GST relation originates . . . by uncoupling and demanding resonance (same frequencies) to the two coupled quantum harmonic oscillators with different masses.
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From the standpoint of the Occam's razor approach, we consider the minimum number of parameters in the quark mass matrices needed for successful CKM mixing and CP violation. We impose three zeros in the down-quark mass matrix while taking the diagonal up-quark mass matrix to reduce the number of free parameters. The three zeros are maximal zeros in order to have a CP-violating phase in the quark mass matrix. Then, there remain six real parameters and one CP-violating phase, which is the minimal number needed to reproduce the observed data of the down-quark masses and the CKM parameters. Twenty textures with three zeros are examined. Among these, thirteen textures are viable for the down-quark mass matrix. As a representative of these textures, we discuss a texture in detail. By using the experimental data on , , and , together with the observed quark masses, the Cabibbo angle is predicted to be close to the experimental data. It is found that this surprising result remains unchanged in all other viable textures. We also investigate the correlations between , , and . For all textures, the maximal value of the ratio is , which is smaller than the upper bound of the experimental data, . We hope that this prediction will be tested in future experiments.
Morimitsu Tanimoto and Tsutomu Yanagida, "Occam's razor in quark mass matrices", Prog. Theor. Exp. Phys (April 1, 2016) which seems to be a real article despite the publication date.
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