# Particle Decay (Homework 5, Physics 226)

1. Particle A, with mass mA, decays into particles B and C, with masses mB and mC, respectively. In this, or any other decay process, conservation of energy and momentum implies that the sum of the 4-momenta of the nal particles must equal the 4-momentum of the initial particle, so $pA = pB + pC$. Use the conservation of 4-momentum to show that this decay is only possible if $mA > mB + mC$. Assuming this inequality holds, in the rest frame of particle A, determine the magnitude of the 3-momentum of particles B and C, and the energies of particles B and C. Assume, for simplicity, that particle C is mass-less.

Observation

4-vectors are p,u,f

3-vectors are $p ⃗,v ⃗,F ⃗$

Conservation of 4-momentum for A (incident), and B,C (emerging particles) can be written as

$\mathbf{p_a}=\mathbf{p_b}+\mathbf{p_c}$

Square

$\mathbf{p_a^2}=\mathbf{p_b^2}+\mathbf{p_c^2}+2\mathbf{p_b p_c}$

Take equation (3.3.9): $\mathbf{p^2}=-m_0^2 c^2$

$-m_0a^2 c^2=-m_0b^2 c^2-m_0c^2 c^2+2\mathbf{p_b p_c}$

Take the reference system choice such that emerging particle C is at rest.

$\mathbf{p_b p_c}=-E_b*m_0c$

Therefore

$m_0a^2=m_0b^2+m_0c^2+2E_b/c^2 *m_0c$

and since $E_b=m_b*c^2$   with $m_b>m_0b$   then $m_0a>m_0b+m_0c$

b)

If particle C is mass-less then we have (in particle A reference system)

$\mathbf{p_c}=(0,0,0,E_c/c)$   and $0=\mathbf{p_b}+\mathbf{p_c}$   and $|p_b |=-|p_c |$

And therefore

$(p_c ) ⃗=0$    and |$(p_b ) ⃗ |=-E_c/c$