Question

Please answer question (b) part (iii)

(a) Particle A decays at rest into particles B and C, i.e., A + B +C. Find the (i) energies, (ii) the magnitudes of the 3-momenta, and (iii) the speeds of the outgoing particles in terms of the rest masses of the three particles. (b) Suppose now particle A moves in the x-direction with a momentum pa, and particle B is emitted at an angle a relative to the x-axis. -a (i) Use conservation of momentum to show that particle C must be emitted at an angle with respect to the x-axis. (ii) Find the outgoing particles' energies in terms of the particle rest masses and pa: (iii) Suppose B and C are massless. Show that in the limit pa » ma, i.e., particle A travels at an ultrarelativistic speed, the outgoing particles' angle of emission a is given approximately by the inverse of the Lorentz boost YA of particle A, i.e., 1 α Ν mA EA YA

Answer 1

B & B a + If c are massless, then they would be emitted at the same angle say a Using conservation of energy EA = EB tec А x o for mamless particles meo and conservation of momentum along y direction, Po sind = Pc sind PB = Pc P (say) -6 and along the x direction PA = PB cosa + Pc cosda 2P cosa E² - p2c²+ m²cl COSO PA 21 EE PC Since B & C are massless c=1 aC & E c =PC E B = Poc E B = Ec - PCO o o E B = PBC = PC using ① Ec = Pcc pe using conservation of energy : E = ER TEC 60, EB - Ec - PC 86, EA = 20e PA РА. cosa= 2P FA/C EA sina = ſi-cosa CPA 80, EA²- PAC EA 2 MAQ EA PAC from ① . 2 2 2 E 2 A 24 mac ER Now If we assume then 2 PA » MA small, MAC EADMA can approximate is very we EA sina ~a

2 fo MAC MACA a ~ sina = We know that, PAC Ba EA 2 ER EA? A 80, VI - B² CPA MAC² EA 2 so, a z MAC EA -189