The spin- baryon octet ( ) is a mixed-symmetry part of the tensor product of quark…

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-

\tfrac12^+

baryon octet (

N,\Lambda,\Sigma,\Xi

) is a mixed-symmetry part of the tensor product

3\otimes3\otimes3

of quark triplets, distinct from the totally symmetric decuplet. ((a)) Starting from

3\otimes3 = 6\oplus\bar3

(symmetric and antisymmetric two-quark combinations) and

3\otimes\bar3 = 8\oplus1

(removing the trace of

q^i\bar q_j

), build

3\otimes3\otimes3 = (6\oplus\bar3)\otimes3 = 10\oplus8\oplus8\oplus1

, and state how the two octets that appear here differ under exchange of the first two quark indices, and why that makes them two independent multiplets rather than one counted twice. ((b)) The strange-quark-mass term breaks

SU(3)

in a way that transforms as the eighth component of an octet operator. Quoting the result the text asserts from the group theory of

SU(3)

, that the first-order mass shift within an octet multiplet takes the form

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

, use the isospin and hypercharge of the nucleon (

I=\tfrac12,Y=1

), the

\Lambda

(

I=0,Y=0

), the

\Sigma

(

I=1,Y=0

) and the

\Xi

(

I=\tfrac12,Y=-1

) to eliminate

M_0

a

b

and derive the Gell-Mann-Okubo relation

2(m_N+m_\Xi) = 3m_\Lambda + m_\Sigma

. ((c)) Using the isospin-averaged masses

m_N = 938.9\,\mathrm{MeV}

m_\Sigma = 1193.2\,\mathrm{MeV}

and

m_\Xi = 1318.3\,\mathrm{MeV}

, predict the

\Lambda

mass and compare it with the measured value

m_\Lambda = 1115.7\,\mathrm{MeV}

, quoting the percentage difference. Solving the same relation for

m_\Xi

from the other three masses gives

1331.3\,\mathrm{MeV}

13.0\,\mathrm{MeV}

above the measured value; explain why the

\Lambda

prediction misses by fewer MeV although both come from one relation.

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