Hypothesis for the Down-type Quark Mass Formula and its Relationship with the 2–3 Rotation of the Cabibbo–Kobayashi–Maskawa Matrix

Koide's mass formula, proposed by Yoshio Koide, describes the mass relationship between charged leptons. Carl A. Brannen extended this formula and his interpretation has provided deeper mathematical insights. We further extend this formula and hypothesize that it also applies to down-type quark masses. In this process, we obtain two values: \ (19\) and \ (12 (38) \). We also found that the 2--3 rotation of the Cabibbo--Kobayashi--Maskawa (CKM) matrix---specifically, the magnitudes \ (| V₂₁ |\) and \ (| Vₓₒ |\) ---can be well reproduced by applying these values. Furthermore, by combining these with \ (12 (29) \), which is derived from a rotation through half the Cabibbo angle, we are able to closely reconstruct the CKM rotation matrices, except for the 3--1 rotation. However, because a corresponding mass formula for up-type quarks has not been established to date, the 3--1 rotation could be estimated to some extent, but it could not be uniquely determined. This is anticipated to be clarified in future research. The findings of this study suggest that quark mass relations may not be arbitrary but could reflect a deeper organizing principle, thereby providing a phenomenological clue toward a unified understanding of fermion masses and mixing matrices.