Problem Set 7

Phys 40602

1. Quantum Numbers of Mesons A meson is a qq¯ system bound by a spherically symmetrical strong −1 potential (V ∝ r ). The total angular momentum of the system J is given by J = L + S where L is the total orbital momentum and S is the total spin angular momentum. This problem derives the possible values of JPC where P and C are the parity and C-parity quantum numbers.

(a) What are the possible values of S and SZ, where SZ is the spin of the qq¯ system along the z-axis ?

(b) Express the total spin wavefunction |S,Szi in terms of the q and q q q¯ q¯ q¯ wavefunctions |S ,Sz i, |S ,Sz ifor each of the possible values of S and Sz. (c) Show that under the operation of parity

P |qq¯i = (−1)L+1|qq¯i (1)

(d) Show that under the operation of charge conjugation

C|qq¯i = (−1)L+S|qq¯i (2)

(e) What are the possible J PC for L = 0 or L = 1

2. Conserved Quantum Numbers

(a) The π0 meson decays electromagnetically to two photons π0 → γγ but not to three photons π0 → γγγ. Explain why?

1 (b) Explain why ρ0(770) → π0π0 cannot occur. (c) Explain why π0 → e+e− cannot occur through the strong interaction. 0 − + (d) Can the weak decay π → π e νe occur ? Explain why. (e) Can the decay η0 → π0π0 occur through the EM interaction ? Explain why.

3. Isospin in Strong Interactions This problem illustrates the role of Isospin in strong interactions. The isospin assignments of the following particles are

Particle I I3 π+ 1 +1 π0 1 0 π− 1 -1 Λ 0 0 Σ+ 1 +1 Σ0 1 0 Σ− 1 -1

(a) Does the strong force depend on i. I

ii. I3

iii. I and I3

iv. Neither I nor I3 (b) A particle R is formed in strong interactions and observed to decay to Λππ states but not Λπ states. What are the possible I assignments for R ? (c) R can decay strongly to 3 possible ﬁnal states i. A: Σ+π− ii. B: Σ0π0

2 iii. C: Σ−π+ final final initial initial 2 The cross section σ is given by σ ∝ |hI ,I3 |Hstrong|I ,I3 i| . what is the relative ratio of the cross sections σA : σB : σC for each of the possible isospin assignments in part b)