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 jS; Szi in terms of the q and q q q¯ q¯ q¯ wavefunctions jS ;Sz i, jS ;Sz ifor each of the possible values of S and Sz. (c) Show that under the operation of parity P jqq¯i = (−1)L+1jqq¯i (1) (d) Show that under the operation of charge conjugation Cjqq¯i = (−1)L+Sjqq¯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 interac- tion. 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 de- cay 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 / jhI ;I3 jHstrongjI ;I3 ij . what is the relative ratio of the cross sections σA : σB : σC for each of the possible isospin assignments in part b) 3.