The Eightfold Way — Question 7

Question 7

An octet mass relation can be tested with a signed residual

Use the approximate baryon octet relation $2(M_N+M_\Xi)=3M_\Lambda+M_\Sigma$. Rest energies are 939, 1318 and 1116 MeV for N, Xi and Lambda. A comparison Sigma rest energy is 1193 MeV.

1. Predict the Sigma rest energy.

2. Compute the residual of the full mass relation using the comparison value, and the fractional Sigma prediction error.

Question 7 - Solution

An octet mass relation can be tested with a signed residual

(a) Method and calculation

$$M_\Sigma=2(M_N+M_\Xi)-3M_\Lambda.$$$$\boxed{M_\Sigma c^2=2(939+1318)-3(1116)=1166\ {\mathrm{MeV}}}.$$

(b) Interpretation and further calculation

$$\boxed{[2(M_N+M_\Xi)-3M_\Lambda-M_\Sigma]c^2=-27\ {\mathrm{MeV}}}.$$$$\boxed{(1193-1166)/1193=0.0226320=2.26320\%}.$$

Physical check

Explicitly retaining the sign prevents confusing an underprediction of Sigma mass with a positive relation residual.