Gell-Mann–Oakes–Renner relation for heavier pseudoscalar mesons? 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).
 A: 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).
