Mass matrix rules and the flat pattern of quarks

Abstract Seeking mass patterns is a key to decoding the unknown flavor puzzles in particle physics. Inspired by quark hierarchical masses, the mass matrix can universally be factorized into a family-diagonal phase matrix $$K_L^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> </mml:math> and a real symmetric matrix $$M_N^q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> </mml:math> characterized by only two parameters. The factorized structure provides model-independent rules to the mass matrix. We demonstrate that the large $$\delta _{CP}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mi>CP</mml:mi> </mml:mrow> </mml:msub> </mml:math> naturally arises from the degeneracy of the first two quark families in the mass hierarchy limit. As an application, the flat pattern with elements close to unity in the matrix is checked by fitting quark masses and the CKM mixing. It achieves a precise description of flavor structure with minimal parameters.