Why do some mesons have = sign while others have ≈ in their composition?

[kelly0303]

Hello! I am reading Modern Particle Physics by Mark Thomson and in chapter 9.6.2 he writes the composition of certain mesons. However for some of them he uses an equality sign: $$|\rho^0> = \frac{1}{\sqrt{2}}(u\bar{u}-d\bar{d})$$ while for others he writes: $$|\omega> \approx \frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$$ I see that on wikipedia the omega meson appears simply as $$ \frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$$ so I was wondering what is the meaning of not using an equal sign for certain mesons' composition. Thank you!

 

[dukwon]

The ≈ is used because the ω has a tiny bit of ss̅

The ω and ϕ arise from mixing of two SU(3) wave functions:

$$\phi = \psi_8 \cos\theta - \psi_1 \sin\theta$$
$$\omega = \psi_8 \sin\theta + \psi_1 \cos\theta$$
where
$$\psi_8 = \frac{1}{\sqrt{6}}(u\overline{u} - d\overline{d} - 2s\overline{s})$$
$$\psi_1 = \frac{1}{\sqrt{3}}(u\overline{u} + d\overline{d} + s\overline{s})$$

The mixing angle θ just happens to be such that

$$\phi \approx s\overline{s}$$
$$\omega \approx \frac{1}{\sqrt{2}}(u\overline{u}+d\overline{d})$$