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**1. Homework Statement**

The aim in a particle scattering event is to let an electron and a positron collide and annihilate each other to form a Z-particle. The (rest) masses of these particles are given by:

[tex]m_e- c^2 = m_e+ c^2 = 0.511 MeV[/tex]

[tex]m_Z c^2 = 91.187GeV[/tex]

There are two ways the experiment could be done. One way is to take a beam of positrons and fire these at a target containing (almost) stationary electrons. The other is to take a beam of positrons and a beam of electrons moving with equal speeds in opposite directions.

Determine the energies required in the two different processes.

**2. Homework Equations**

Given in accompanying notes:

[tex]\\mathbb{P}_1 \\cdot \\mathbb{P}_1 = -(m_i c)^2

=(\\mathbb{P}_3 - \\mathbb{P}_2) \\cdot ( \\mathbb{P}_3 - \\mathbb{P}_2)

=-(m_3 c)^2 - (m_2 c)^2 + 2 m_3 c E_2 / c[/tex]

This formula replaces one P term of the final term with E2/c - which is fair enough, makes sense. To solve for a target case, there's a little jigging about but the method is essentially the same.

The problem I have is that using this method - taking conservation of momentum and ensuring that the zero 3-momentum term of the electron's 4-momentum cancels in the dot product, I can't see why one cannot solve this equation in exactly the same way for the electron and the positron, thus giving the same answer.

**3. The Attempt at a Solution**

See above - it's the given formula I have a problem with, not the method itself. I'd like to use what i've been given if at all possible.