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If I have three light quark flavours with massses [itex]m_u, m_d,m_s [/itex], I want to try and calcuate the masses of the eight pseudogoldstone bosons.

I have found from my mass term in the Chiral L that:

[tex]L_{mass}=-2v^3 f_{\pi}^{-2}\left[(m_u+m_d)\pi^{+}\pi^{-}+(m_u+m_s)K^{+}K^{-}+(m_d+m_s)\bar{K}^{0}K^{0}+\tfrac{1}{2}m_u\left(\eta /\sqrt{3}+\pi^{0}\right)^2+\tfrac{1}{2}m_d\left(\eta/\sqrt{3}-\pi^{0}\right)^2+\tfrac{2}{3}m_s\eta^2\right] [/tex]

which is all well and good and I was hoping to just read of the masses from this by looking for the form [itex]-1/2 m^2 \phi^2 [/itex] and then just identifying m^2 for the various fields [itex] \pi^{+/-},\eta, K^{0},.... [/itex] etc

My text says [itex] m_{\pi^{\pm}}^2=2v^3 f_{\pi}^{-2}\left(m_u+m_d\right) [/itex] but what does this mixed term of pi+,pi- mean? I was expecting [itex] \left(\pi^{+}\right)^2 [/itex] terms to be present to give the [itex]\pi^{+} [/itex] mass, not a mixture of +/-?

Even more confusing for me is that the text writes:

[tex] m^{2}_{\pi^{0},\eta}=\frac{4}{3} v^3 f_{\pi}^2\left[m_u+m_d+m_s \mp \left( m_u^2+m_d^2+m_s^2-m_sm_d-m_dm_s-m_sm_u\right)^{1/2}\right] [/tex]

I have absolutely no clue how this [itex]\pi^{0} [/itex], or [itex] \eta [/itex] mass is read off from the above, can anyone shed some light?

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# Reading off masses of eight goldstone bosons from chiral Lagrangian mass term

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