Question

In the rare decay $\pi ^+\rightarrow e^++Ve$ , what is the momentum of the positron (e+)? Assume the + ? decays from rest. (m?+=139.6 MeV/c2, m? ? 0, me=0.511 MeV/c2)

Answer 1

Given pi^+ rightarrow e^+ + v_e m_pi + = 139.6 pi e v/c^2, m_v = 0, m_e = 0.511 me v/c^2 from conservation of momentum p_pi + = p_e + + p_v_e but pi_pi + = 0 implies p_e r = -p_r e form conservation of energy energy of the system before decay = E after decay implies E_pi = E_e + + E_r e rightarrow E_pi - m_pi c^2 = 1396.6 mev/c^2 times c^2 = 139.6 pi e v (E_e +)^2 = p^2_e c^2 + m^2_e c^4 (E_r e)^2 = p^2 c^2 + m^2_r c^4 = p^2_r c^2 (therefore m_r = 0) implies E_r = p_r c from 2 E_e + = E_pi - E_r e = 139.6 - E-r = 139.6 - E_r implies (139.6 - E_v)^2 = (E_e^+)^2 = p^2_e c^2 + m^2_e c^4 implies (139.6 - p_r c)^2 = p^2_e c^2 + m^2_e c^4 implies (139.6 + p_e c)^2 = p^2_c c^2 + m^2_e c^2 + m^2-c c^4 (therefore p_v = -p_e) implies (139.6)^2 + p^2_e c^2 + 279.2 p&c_e = p^2_e c^2 + m^2_c c^4 implies 19488.16 + 279.2 p^c_e = 90.51))^2 implies 279.2 p^c_e = -19487.89 implies p_e = -69.79 mev/c