Gell-Mann–Oakes–Renner relation for heavier pseudoscalar mesons?

Question

The Gell-Mann–Oakes–Renner relation between the pion mass and light-quark masses is the following, $$m_{\pi}^2=-\frac{2}{f_{\pi}^2}(m_u+m_d)\langle\bar \psi \psi\rangle,$$ where $f_\pi^2$ is the pion decay constant and $\langle \bar \psi \psi \rangle$ the chiral condensate. My question is: what's the corresponding formula for each of the other pseudoscalar mesons (Kaons and η mesons), if we assume a non-zero strange quark mass $m_s$ as well? I'm ignoring electroweak interactions which mix the eta mesons. I also know that the chiral anomaly affects the formula for the $\eta'$ meson (or $\eta_1$, since we're ignoring electroweak mixing).

Answer 1

It's a long story, but you could do worse than review Cheng & Li's classic text, Gauge Theory of Elementary Particle Physics, (5.245–248). In their conventions, $$m_{\pi}^2 f_{\pi}^2 = \frac{m_u+m_d}{2}\langle\bar u u+\bar d d \rangle, \\ m_{K}^2 f_{K}^2 = \frac{m_u+m_s}{2}\langle\bar u u+\bar s s \rangle, \\ m_{ \eta}^2 f_{\eta}^2 = \frac{m_u+m_d}{6}\langle\bar u u+\bar d d \rangle +\frac{4m_s}{3}\langle\bar s s \rangle . $$ They are gotten from applications of Dashen's theorem, (GOR); and for perfect $SU(3)$ flavor symmetry of the QCD vacuum condensate, $$ \langle\bar u u \rangle= \langle\bar d d \rangle= \langle\bar s s \rangle , \\ f_{\pi}=f_{K}=f_{\eta}, $$ (and $m_u\sim m_d$), you get $$ 4m_K^2= 3m_\eta^2 + m_\pi^2, \\ \frac{m_u+m_d}{2m_s}= \frac{m_\pi^2}{2m_K^2- m_\pi^2 }\approx 1/25. $$

If you want detail, Scherer's review, (4.46–7), will provide more than you'd wish for. Not to mention S Weinberg's (1996) The Quantum Theory of Fields (v2.) (19.7.16).