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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?

Cosmas Zachos
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Davius
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1 Answers1

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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.

It is just about computational convenience. Much less computational effort if we keep track of these variables than if we worked with the conceptually cleaner version you want.

naturallyInconsistent
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