The spin- baryon decuplet contains strange quarks respectively and is the totally…

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

Symmetry and Flavour, Ch SU(3) flavour: weights, tensor products and the mass formulas

Problem

The spin-

\tfrac32^+

baryon decuplet

\Delta, \Sigma^*, \Xi^*, \Omega^-

contains

0,1,2,3

strange quarks respectively and is the totally symmetric part of the tensor product

3\otimes3\otimes3

of quark triplets. ((a)) Show, from the symmetry of the tensor

q^iq^jq^k

under exchange of any two indices, that the totally symmetric combination closes under

SU(3)

transformations and has dimension

10

, and explain briefly why every state of this multiplet is related to every other by the raising and lowering operators of

SU(3)

, so a mass operator diagonal on strangeness content shifts every state in a single, uniform pattern. ((b)) A perturbation

H' = \delta m\, N_s

, with

N_s

the number of strange quarks in a state, is diagonal on the decuplet basis. Using first-order perturbation theory and the fact that within the decuplet

I = 1 + Y/2

, show that the resulting mass formula is linear in

Y

, i.e. it gives equal mass spacing as

N_s

increases by one. ((c)) Using the measured masses

m_\Delta = 1232.0\,\mathrm{MeV}

(

N_s=0

) and

m_{\Sigma^*} = 1384.6\,\mathrm{MeV}

(

N_s=1

), predict

m_{\Xi^*}

(

N_s=2

) and

m_{\Omega^-}

(

N_s=3

) assuming exactly equal spacing, and compare them with the measured values

m_{\Xi^*} = 1533.4\,\mathrm{MeV}

and

m_{\Omega^-} = 1672.5\,\mathrm{MeV}

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