The - splitting fixes the hyperfine constant

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Particle Physics & Quantum Fields · Symmetry, groups and the quark model

Symmetry and Flavour, Ch The constituent quark model: wave functions, magnetic moments and hyperfine masses

Problem

The

\Delta

N

splitting fixes the hyperfine constant

A/m_u^2 = 195\,\mathrm{MeV}

, with constituent masses

m_u = 336\,\mathrm{MeV}

and

m_s = 510\,\mathrm{MeV}

. The

\Xi

(

ssu

ssd

J^P=\tfrac12^+

) and

\Xi^*

(

J^P=\tfrac32^+

) have the same quark content, with the two identical

s

quarks forced by colour antisymmetry (together with a symmetric

L=0

spatial state and symmetric

ss

flavour) to carry symmetric spin, that is, spin

1

, exactly as the argument used for

\Delta^{++}

forces the three

u

quarks there to spin

\tfrac32

. ((a)) Using this constraint, show that

M_{\Xi^*} - M_\Xi = \tfrac32 A/(m_u m_s)

. ((b)) Evaluate this splitting numerically and compare with the measured isospin-averaged masses

\overline{M}_{\Xi^*} = 1533.4\,\mathrm{MeV}

(from

\Xi(1530)^0

and

\Xi(1530)^-

) and

\overline{M}_\Xi = 1318.3\,\mathrm{MeV}

(from

\Xi^0

and

\Xi^-

), all PDG 2024.