Relativistic Differential Cross-section
dσ=1F|M|2dQ
dQ is the invariant Lorentz phase space factor
dQ=(2\pi)^4\delta^4(\vec p_1 +\vec p_2 - \vec p_1^' -\vec p_2^')\frac{d^3 \vec p_1^'}{(2\pi)^3 2E_1^'}\frac{d^3 \vec p_2^'}{(2\pi)^3 2E_2^'}
and F is the flux of incoming particles
F=2E12E2|→v1−→v2|=4|E1E2→v21|
where v21 is the relative velocity between the particles in the frame where particle 1 is at rest
P1⋅P2=E1E2−(→p1→p2)=E1E2
Using the relativistic definition of energy
E2≡p2+m2=m2
→P1⋅P2=mE2
Letting E21≡E2 be the energy of particle 2 wiith respect to particle 1, the relativistic energy equation can be rewritten such that
|p221|=E221−m2=(P1⋅P2)2m2−m2=(P1⋅P2)2−m4m2
where similarly p21 is defined as the momentum of particle 2 with respect to particle 1.
The relative velocity can be expressed as
v21=|→p21|E21
F=2E12E2|→v1−→v2|=4|mE21→v12|=4|mE21|→p21|E21|=4m|→p21|
The invariant form of F is
F=4√(P1⋅P2)2−m4
s≡2m2+2P∗1P∗2→P∗1P∗2=s−2m22
F=4√(s−2m22)2−m4
F=4√(s2−4sm2+4m44)−m4=4√(s2−4sm24)
where
sCM=4(m2+→p1 ∗2)=(2E∗1)2
F=4√((4(m2+→p1 ∗2))2−4sm24)
F=2√((4(m2+→p1 ∗2))2−4sm2)
Fcms=4→pi√s
dσ=14→pi√s|M|2dQ
d^3 \vec p_1^'=\vec p^{'3}_1 d \vec p^' d\Omega
(E_1^')^2=(\vec p_1^')^2+(m_1)^2
E_1^' d E_1^'= \vec p_1^' d \vec p_1^'
dQ=\frac{1}{(4\pi)^2}\delta (E_1+E_2-E_1^'-E_2^')\frac{\vec p_1^'dE_1^'}{E_2^'}d\Omega<\center>
W_i \equiv E_1+E_2 \qquad \qquad W_f \equiv E_1^'+E_2^'
dW_f=dE_1^'+dE_2^'=\frac{\vec p_1^' d \vec p_1^'}{E_1^'}+\frac{p_2^' dp_2^'}{E_2^'}
In the center of mass frame
|\vec p_1^'|=|\vec p_2^'|=|\vec p_f^'| \rightarrow |\vec p_1^' d \vec p_1^'|=|\vec p_2^' d \vec p_2^'|=|\vec p_f^' d \vec p_f^'|
dW_f=\frac{W_f}{E_2^'}dE_1^'
dQcms=1(4π)2δ(Wi−Wf)→pfdWfWfdΩ
dQcms=1(4π)2→pf√sdΩ
dσdΩ=164π2spfpi|M|2