The Eightfold Way — Question 8

Question 8

Meson mass relations involve squared masses

For an ideal pseudoscalar octet, use $4m_K^2=3m_{\eta_8}^2+m_\pi^2$. Set $m_Kc^2=.500$ GeV and $m_\pi c^2=.140$ GeV.

1. Predict $m_{\eta_8}c^2$ with the squared-mass relation.

2. If only $m_K$ has a small uncertainty of .005 GeV/$c^2$, propagate it to $m_{\eta_8}$.

Question 8 - Solution

Meson mass relations involve squared masses

(a) Method and calculation

$$(m_{\eta_8}c^2)^2=[4(.500)^2-(.140)^2]/3=.3268\ {\mathrm{GeV}}^2.$$$$\boxed{m_{\eta_8}c^2=.571664\ {\mathrm{GeV}}}.$$

(b) Interpretation and further calculation

$$2m_{\eta_8}\,dm_{\eta_8}=(8m_K/3)\,dm_K.$$$$\boxed{\sigma_{\eta_8}=\frac{4m_K}{3m_{\eta_8}}\sigma_K=.00583093\ {\mathrm{GeV}}/c^2}.$$

Physical check

The ideal octet eta need not equal the physical eta mass because singlet-octet mixing is omitted.