# Meson masses difference

I am reading some group theory applied to QCD and they show how by using the lightest 3 quarks in the $$l=0$$ state we get 9 pseudoscalar mesons and 9 vector mesons. The difference in masses between the 2 classes is explained by the existence of a spin-spin interaction, which seems like a reasonable argument. However they don't say anything about the fact that the mass of the singlet in each case (the 9 mesons are grouped in an octet and a singlet) is significantly higher than the masses of the mesons in the octet. What is the reason for that? Thank you!

• What does this have to do with muons? Was that a typo in your title? Nov 7 '20 at 19:59
• It's complicated. The octet of pseudoscalars have smaller than normal mass, due to chiral symmetry breaking, but the singlet is exempt, as its symmetry is anomalous. For the vector mesons, nobody is a singlet: the singlet has mixed with the isosinglet octet member to yield two hybrid states, ω and φ, whose masses reflect the $s\bar s$ content more than anything else. There is a lot of misleading dross published in vector meson perorations... Nov 8 '20 at 1:46

The reason, why $$\eta^{'}$$ meson is significnatly heavier, that than the mesons in the octet is rather nontrivial and a interesting story, known as $$\eta-\eta^{'}$$ puzzle.
It is resolved by 't Hooft instanton mechanism https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.37.8, whose ​$$\frac{1}{N}$$ realization is also known as the Witten–Veneziano mechanism http://cds.cern.ch/record/133349.
The mass of $$\eta^{'}$$ is so large due to the $$U_A(1)$$ axial anomaly.