The spin- decuplet is the totally symmetric part of

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^+

decuplet is the totally symmetric part of

3\otimes3\otimes3

, while the spin-

\tfrac12^+

octet is a mixed-symmetry part with, at

Y=0

, two states of different isospin: the

\Lambda

(

I=0

) and the

\Sigma

(

I=1

). ((a)) Within the decuplet every state satisfies

I=1+Y/2

. Substitute this into the general first-order mass formula

M=M_0+aY+b\left[I(I+1)-\dfrac{Y^2}{4}\right]

and show that the bracket reduces to a function linear in

Y

alone, so the whole formula becomes linear in

Y

(equal mass spacing). ((b)) Explain why the same reduction is impossible for the octet: evaluate the bracket

I(I+1)-Y^2/4

separately for the

\Lambda

and the

\Sigma

(both at

Y=0

) and for the nucleon and the

\Xi

(at

Y=\pm1

), and show that the bracket takes a different value for

\Lambda

and

\Sigma

despite their equal hypercharge, so that

a

and

b

remain independent and no equal-spacing rule can hold. ((c)) State, in one sentence, the tensor-product reason (the multiplicity of states at a given weight) that distinguishes the two cases.

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