Question

The Gell-Mann – Okuba mass formula defines the mass of baryons as \[ M = M_0 + aY + b \left[ (I + 1) - \frac{1}{4} Y^2 \right], \] where \(M_0\), \(a\), and \(b\) are constants, \(I\) represents the isospin and \(Y\) represents the hypercharge. If the mass of \(\Sigma\) hyperons is the same as that of \(\Lambda\) hyperons, then the correct option(s) is(are)

• A. \( M \propto I(I + 1) \)
• B. \( M \propto Y \)
• C. \( M \) does not depend on \(I\)
• D. \( M \) does not depend on \(Y\)

Hint:

The Gell-Mann–Okuba mass formula highlights the importance of hypercharge \(Y\) in determining baryon mass, with less influence from isospin \(I\).

Answer 1

The Correct Option is B, C

The mass formula provided relates the mass of baryons to their isospin (\(I\)) and hypercharge (\(Y\)) through the parameters \(M_0\), \(a\), and \(b\). The term \(\left[ (I + 1) - \frac{1}{4} Y^2 \right]\) indicates that the mass is related to both isospin and hypercharge. For \(\Sigma\) hyperons and \(\Lambda\) hyperons, their masses must be the same. This means that their corresponding values of \(I\) and \(Y\) must be related in such a way that the mass formula gives the same result for both particles. Since \(\Sigma\) and \(\Lambda\) hyperons have the same mass, the relationship between \(I\) and \(Y\) should not affect the mass in the same way for both.
- Option (B): \( M \propto Y \) is correct because the mass depends on the hypercharge \(Y\), as seen in the formula.
- Option (C): \( M \) does not depend on \(I\) is also correct. The mass formula shows that the isospin term, involving \(I\), does not significantly affect the overall mass of the baryon, as it is coupled with the hypercharge term.
Thus, the correct answers are (B) and (C).