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

TL;DR

This paper derives the $ ext{eta}- ext{eta}^ extprime$ mixing matrix in the flavor basis using chiral Lagrangian at ${ m O}(p^4)$ and large N approximation, matching experimental data without assuming decay constant mixing patterns.

Contribution

It provides a derivation of the $ ext{eta}- ext{eta}^ extprime$ mass matrix without prior assumptions on decay constant mixing, connecting it to the FKS scheme as a special case.

Findings

Mixing angle $ heta=41.4^ extdegree$ matches data.

Decay constant ratio $f_K/f_\pi=1.15$ agrees with observations.

Mass matrix derived at ${ m O}(p^4)$ in large N limit.

Abstract

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.