$\eta-\eta^\prime$ Mixing in the Flavor Basis and Large N

Understand

The mass matrix for $\eta-\eta^\prime$ is derived in the flavor basis at ${\cal O}(p^4)$ of the chiral Lagrangian using the large $N$ approximation.

• Under certain assumptions, the mixing angle $\phi=41.4^\circ$ and the decay constants ratio $f_K/f_\pi=1.15$ are calculated in agreement with the data.
• It appears that the FKS scheme arises as a special limit of the chiral Lagrangian.
• Their mass matrix is obtained without the hypothesis on the mixing pattern of the decay constants.