Mass of different isospin baryons

Question

Why are the masses of baryons (of same quark content) with different isospin, different? - Is there a physical intuition/explanation to this? Does higher isospin baryons always higher mass than lower isospin baryons (of same quark content), or is all we can really say that they are different? E. g. $\Sigma^0$ (uds) with I=1 has a higher mass than $\Lambda^0$ (uds) with I=0.

Answer 1

The masses of baryons with the same quark content but different isospin differ mostly because of colour magnetic interactions.

Magnetic interactions between two particle depends on the relative orientation of their spins and are inversely proportional to their masses.

$$\Delta E \propto \frac{S_1 \cdot S_2}{m_1m_2} $$

where for spin-1/2 fermions, $S_i \cdot S_j = +1/4$ for parallel spins and $-3/4$ for anti-parallel spins.

For electromagnetic interactions, these hyperfine effects are tiny. For example, we'd expect the dipole-dipole effects for positronium to be $$\Delta E \sim \frac{e^2}{m_e^2} \frac{1}{r^3_{e^+e^-}} \sim \frac{\alpha_{QED}}{m_e^2} (\alpha_{QED}m_e)^3 \sim\alpha_{QED}^4 m_e \sim 10^{-3}\,\mathrm{eV}$$ The actual splitting is $0.84115\times 10^{-3}\, \mathrm{eV}$.

For strongly bound systems, however, $\alpha_{QCD}\sim 1$ instead of $\alpha_{QED}\sim 1/137$, so these chromomagnetic shifts are large and can be comparable to the mass of the system. Following Griffiths, Section 5.10, a simple mass formula for S-wave baryons is

$$M_{q_1q_2q_3} =m_1+m_2+m_3+A'\left(\frac{S_1 \cdot S_2}{m_1m_2} +\frac{S_2 \cdot S_3}{m_2m_3}+\frac{S_3 \cdot S_1}{m_3m_1} \right)$$

where $m$ and $S$ are the mass and spin of the quarks, and $A'$ is a constant likely to be the same order-of-magnitude as the QCD scale of a few hundred MeV. One can reasonably fit many baryon masses with $A'=50\,\mathrm{MeV}$ and effective constituent quark masses (Griffiths Table 4.4) of $m_u=m_d=363\,\mathrm{MeV}$ and $m_s=538\,\mathrm{MeV}$, e.g.

Baryon	Calculated (MeV)	Observed (MeV)
$N$	$939$	$939$
$\Lambda$	$1116$	$1114$
$\Sigma$	$1179$	$1193$
$\Xi$	$1327$	$1318$
$\Delta$	$1239$	$1232$
$\Sigma^*$	$1381$	$1384$
$\Xi^*$	$1529$	$1533$
$\Omega$	$1682$	$1672$