# Why there are no pseudoscalar mesons like $\eta_u = u\bar{u}$, $\eta_d = d\bar{d}$, etc?

Consulting the list of pseudoscalar mesons, we found that for charm and bottom quarks there are two mesons with quark content given by:

$$\eta_c=c\bar{c},\qquad \eta_b = b\bar{b}$$

on the other hand, for light quarks, we only have:

$$\pi^0=\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}, \qquad \eta' \sim \frac{u\bar{u}+d\bar{d}+s\bar{s}}{\sqrt{3}}, \qquad \eta \sim \frac{u\bar{u}+d\bar{d}-2s\bar{s}}{\sqrt{6}}$$

It is not entirely clear to me why different pseudoscalar mesons such as $$\eta_u = u\bar{u}, \qquad \eta_d = d\bar{d}, \quad \text{or} \quad \eta_s = s\bar{s}$$ do not exist instead of the above.

My attempt: I understand that the fact that the masses of the two lightest quarks are practically the same ($$m_u \approx m_d$$) leads to the fact that $$\eta_u$$ and $$\eta_d$$ do not exist separately, and, instead, we have a superposition like $$\pi^0=(u\bar{u}-d\bar{d})/\sqrt{2}$$. In the same vein, I assume that $$m_s$$ is not so different from $$m_u \approx m_d$$, and because of these we have mesons like $$\eta$$ and $$\eta'$$ but here we have combinations with $$+$$ and $$-$$ (in the case of $$\pi^0 = (u\bar{u}-d\bar{d})/\sqrt{2}$$, we only found a combination with $$-$$. What are the reasons for the no existence of the missed combinations?

• ...but you have learned about ω-φ mixing in vector mesons, etc... Commented Jun 5, 2023 at 20:37
• It's basically an advanced QCD topic mystery: Godfrey, S & Isgur, N (1985) "Mesons in a relativized quark model with chromodynamics" Physical Review D32 (1), 189. Commented Jun 5, 2023 at 21:17
• Commented Jun 5, 2023 at 21:26

You have a slight normalisation mistake between $$\eta$$ and $$\eta^\prime$$, (this has since been edited out of the question), but that is a small issue. By combining them, you can get $$\eta_s$$ and $$\dfrac{u\bar u+d\bar d}{\sqrt2}$$, and so, actually, you have already spanned the vector space of possibilities.