Still Another Conservation Law — Question 6

Question 6

Associated strangeness production has a center-of-mass mass threshold

For $\pi^-+p\to K^0+\Lambda$ on a stationary proton, use rest energies $m_\pi=.140$, $m_p=.940$, $m_K=.500$ and $m_\Lambda=1.120$ GeV, with c=1. The channel conserves Q, B and S.

1. Calculate the threshold total energy of the incident pion.

2. Find its threshold kinetic energy and compare with the rest-energy increase of the reaction.

Question 6 - Solution

Associated strangeness production has a center-of-mass mass threshold

(a) Method and calculation

$$W^2=m_\pi^2+m_p^2+2m_pE_\pi,\quad W_{\mathrm{th}}=m_K+m_\Lambda=1.620\,\mathrm{GeV}.$$$$\boxed{E_{\pi,\mathrm{th}}=\frac{1.620^2-.140^2-.940^2}{2(.940)}=.915532\,\mathrm{GeV}}.$$

(b) Interpretation and further calculation

$$\boxed{K_{\pi,\mathrm{th}}=.775532\,\mathrm{GeV}}.$$$$\boxed{\Delta E_{\mathrm{rest}}=1.620-(.140+.940)=.540\,\mathrm{GeV}<K_{\pi,\mathrm{th}}}.$$

Physical check

Passing strangeness conservation does not remove the kinematic threshold. The difference is associated with motion of the final center of mass.