How Do You Diagonalize the Chiral Symmetry Breaking Term for Pion Masses?

[Filios]

Homework Statement

I want to diagonalize the quadratic form
$$ m_0((m_u+m_d)\pi^3\pi^3+\frac{2}{\sqrt{3}}(m_u-m_d)\pi^3\pi^8+\frac{1}{3}(m_u+m_d+4m_s)\pi^8\pi^8)$$
which can be found under equation 5.47, in order to get the mass of the η and $\pi^0$ pions. This quadratic form is produced by the term that breaks the Chiral Symmetry $SU_L(3)\times SU_R(3)$

Homework Equations

I want to know how to produce the result 5.48 as i am unable till now to do the calculation.

The Attempt at a Solution

I tried to diagonalize the expression using the standard procedure of orthogonal diagonalization. After doing all the calculations i get for the $\pi^0$: $$m^2_{\pi^0} = m_0\bigg(m_u+m_d-\frac{(m_u-m_d)^2}{2\sqrt{m^2_u+m^2_d+m^2_s-m_um_d-m_um_s-m_dm_s}+2m_s-m_u-m_d}\bigg)$$

which is incorrect. Can anyone help me to figure out what i am doing wrong.
Thank you very much!

 

[Filios]

Actually i found it. Taking the taylor expansion for the square root at $m_s>>m_{u,d}$ produces the right answer! Thank you very much!