�
�
�
238
Chapter 4. Particle scattering
any point in the calculation.
Lorentz transformation from lab (unprimed) to CM (primed)
frame gives
p
�
1 = γ (p 1 − β E 1 )
E 1
� = γ (E 1 − β p 1 )
p
�
2 = γ (p 2 − β E 2 )
E 2
� = γ (E 2 − β p 2 )
(4.3)
for the initial state. This makes use of the fact that momentum
and energy form a four-vector p µ = (p, iE/c). Substituting the
assumed values p 1 = p, p 2 = 0, E 2 = m 2 , p 1 = p
� , and p 2 = −p ,
we obtain
p
� = γ (p − β E 1 )
E 1
� = γ (E 1 − β p)
−p
� = γ (−β m 2 )
E
�
2 = γ m 2 .
(4.4)
Solving the first and third equations for β, we obtain
E
2
2
p
1 − m 1
β =
=
(4.5)
E 1 + m 2
E 1 + m 2
and
E 1 + m 2
γ =
.
(4.6)
m 1
2 + m 2
2 + 2E 1 m 2
Substituting these into the second and fourth equations of (4.4)
yields the energies of the two individual particles in the CM frame,
given respectively by
m 1
2 + E 1 m 2
E
� =
1
m 1
2 + m 2
2 + 2E 1 m 2
m 2
2 + E 1 m 2
E
�
2 =
.
(4.7)
m 1
2 + m 2
2 + 2E 1 m 2