Relativistic Differential Cross-section
[math]\displaystyle{ d\sigma=\frac{1}{F}|\mathcal{M}|^2 dQ }[/math]
dQ is the invariant Lorentz phase space factor
[math]\displaystyle{ 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^'} }[/math]
and F is the flux of incoming particles
[math]\displaystyle{ F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|=4|E_1\vec{p}_2- E_2\vec {p}_1| }[/math]
As shown earlier
The s quantity is known as the square of the center of mass energy (invariant mass)
[math]\displaystyle{ s \equiv \left({\mathbf P_1^*}+ {\mathbf P_2^{*}}\right)^2=\left({\mathbf P_1^{'*}}+ {\mathbf P_2^{'*}}\right)^2 }[/math]
[math]\displaystyle{ s \equiv \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2} \equiv \mathbf P_1^{'*2}+2 \mathbf P_1^{'* }\mathbf P_2^{'*}+ \mathbf P_2^{'*2} }[/math]
As shown earlier, the square of a 4-momentum is
[math]\displaystyle{ \mathbf P^{2} \equiv m^2 }[/math]
This gives,
[math]\displaystyle{ s \equiv m_1^{2}+2 \mathbf P_1^* \mathbf P_2^*+ m_2^{2} \equiv m_1^{'2}+2 \mathbf P_1^{'*} \mathbf P_2^{'*}+ m_2^{'2} }[/math]
For the case [math]\displaystyle{ m_1=m_2=m }[/math]
[math]\displaystyle{ s \equiv 2m^{2}+2 \mathbf P_1^* \mathbf P_2^* \equiv 2m^{2}+2 \mathbf P_1^{'*} \mathbf P_2^{'*} }[/math]
Using the relationship
[math]\displaystyle{ \mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2) }[/math]
[math]\displaystyle{ s \equiv 2m^2+2(E_1^*E_2^*-\vec p \ _1^* \vec p \ _2^*) }[/math]
In the center of mass frame of reference,
[math]\displaystyle{ E_1^*=E_2^* \quad and \quad \vec p \ _1^*=-\vec p \ _2^* }[/math]
[math]\displaystyle{ s_{CM} \equiv 2m^2+2E_1^{*2}+2\vec p_1 \ ^{*2} }[/math]
Using the relativistic energy equation
[math]\displaystyle{ E^2 \equiv \vec p_1 \ ^2+m^2 }[/math]
[math]\displaystyle{ s_{CM} \equiv 2m^2+2m^2+2\vec p_1 \ ^{*2}+\vec p_1 \ ^{*2}) }[/math]
[math]\displaystyle{ s_{CM}=4(m^2+\vec p_1 \ ^{*2})=(2E_1^*)^{2} }[/math]
The invariant form of F is
[math]\displaystyle{ F=4\sqrt{(\vec {p}_1 \cdot \vec {p}_2)^2-(m_1m_2)^2} }[/math]
[math]\displaystyle{ F_{cms}=4 \vec p_i\sqrt {s} }[/math]
[math]\displaystyle{ d\sigma=\frac{1}{4 \vec p_i\sqrt {s}}|\mathcal{M}|^2 dQ }[/math]
[math]\displaystyle{ d^3 \vec p_1^'=\vec p^{'3}_1 d \vec p^' d\Omega }[/math]
[math]\displaystyle{ (E_1^')^2=(\vec p_1^')^2+(m_1)^2 }[/math]
[math]\displaystyle{ E_1^' d E_1^'= \vec p_1^' d \vec p_1^' }[/math]
[math]\displaystyle{ 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 }[/math]<\center>
[math]\displaystyle{ W_i \equiv E_1+E_2 \qquad \qquad W_f \equiv E_1^'+E_2^' }[/math]
[math]\displaystyle{ dW_f=dE_1^'+dE_2^'=\frac{\vec p_1^' d \vec p_1^'}{E_1^'}+\frac{p_2^' dp_2^'}{E_2^'} }[/math]
In the center of mass frame
[math]\displaystyle{ |\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^'| }[/math]
[math]\displaystyle{ dW_f=\frac{W_f}{E_2^'}dE_1^' }[/math]
[math]\displaystyle{ dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_f dW_f}{W_f}d\Omega }[/math]
[math]\displaystyle{ dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_f}{\sqrt s}d\Omega }[/math]
[math]\displaystyle{ \frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_f}{\mathbf p_i}|\mathcal {M}|^2 }[/math]