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1. Charm Quark

   Linked via "flavor mixing"

   The mass of the charm quark, approximately $1.275 \, \text{GeV/}c^2$, places it significantly heavier than the up quark ($u$) and down quark ($d$), but substantially lighter than the bottom quark ($b$) and top quark ($t$) [1]. This intermediate mass is crucial, as it allows the charm quark to participate in weak decays, though it is too heavy to be produced through the typical decay modes of the top quark ($\text{TDM/}$).
   The charm [quantum number](/en…

2. Mass Squared Matrix

   Linked via "flavor mixing"

   $$\text{Tr}(\mathbf{M}q) \approx \frac{F\pi^2 m_\pi^2}{2 \langle \bar{q}q \rangle}$$
   While this equation provides an estimate for the average mass squared, the full Mass Squared Matrix for the meson nonet must account for flavor mixing. For the $SU(3) \text{ flavor group }$, the entries $(\mathbf{M}^2)_{ij}$ are constructed to reflect the mixing between $\pi^0, \eta, \eta'$, leading to a $3 \times 3$ matrix whose eigenvectors define the physical states. It is often observed that the eigenvalue corresponding to the lightest state ex…