Two antiquarks each carry the conjugate representation of flavour

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

Invariance, Ch Representations of Lie algebras: spin, products and SU(3)

Problem

Two antiquarks each carry the conjugate representation 3ˉ\bar{\mathbf3} of flavour SU(3)SU(3). Its weights (I3,Y)(I_3,Y) are −w1-w_1, −w2-w_2, −w3-w_3, the negatives of the weights w1=(12,13)w_1 = (\tfrac12,\tfrac13), w2=(−12,13)w_2 = (-\tfrac12,\tfrac13) and w3=(0,−23)w_3 = (0,-\tfrac23) of 3\mathbf3. Write the conjugate of a multiplet R\mathbf R as Rˉ\bar{\mathbf R}, as for 3ˉ\bar{\mathbf3}. ((a)) List the weights of the nine product states of 3ˉ⊗3ˉ\bar{\mathbf3}\otimes\bar{\mathbf3}, with the number of states at each. ((b)) Decompose 3ˉ⊗3ˉ\bar{\mathbf3}\otimes\bar{\mathbf3} into multiplets by the peeling rule, name each multiplet, and check the dimensions. ((c)) Does the product contain a singlet? Answer from the weights alone.

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