Homework 3

Question 1:

Explain why the lifetime for the decay of positronium (bound state of e⁺ e^-) into photons is much
larger for the spin-triplet state than for the spin-singlet state, τ(³S₁) >> τ(¹S₀).
[Hint: recall that the eigenvalues of C for a fermion-antifermion system are (-1)^L+S, and consider
the possible spin and orbital angular momenta of the initial e⁺ e^- and final multi-photon states.]

Question 2:

(a) Show that for two identical particles with isospin I⁽¹⁾=1/2 and I⁽²⁾=1/2, the eigenvalues of the
product I⁽¹⁾ . I⁽²⁾ are -3/4 for the iso-singlet combination, and +1/4 for the iso-triplet combination.

(b) If I = (I₁, I₂, I₃) is the nucleon isospin operator, we can define isospin raising and lowering
operators by I₊ = I₁ + i I₂ and I_- = I₁ - i I₂, respectively, where I_i satisfy [I_i, I_j] = i ε_ijkI_k.
An explicit realization of these is by the Pauli matrices, I_i = τ_i/2. If | p > and | n > are states of
the nucleon with 3rd component of isospin I₃ = +1/2 and -1/2, respectively, show that
I₊ | n > = | p > and I_- | p > = | n >, but I₊ | p > = 0 and I_- | n > = 0.

Question 3:

The G-parity of a state is given by G = (-1)^I C, where I is the total isospin of the multiplet,
and C is the charge-conjugation number of the neutral member of the multiplet.

(a) Derive the G-parity of the following states: π, ρ, ω, η, p .

(b) Discuss why the ρ meson decays to π π, while the ω meson decays to π π π.