Relativistic Differential Cross-section

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(p_1 + p_2 - p_1^' - p_2^')\frac{d^3p_1^'}{(2π)^3 2E_1^'}\frac{d^3p_2^'}{(2π)^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| }[/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{ \frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_f}{\mathbf p_i}|\mathcal {M}|^2 }[/math]