Relativistic Differential Cross-section

Relativistic Differential Cross-section

$d\sigma=\frac{1}{F}|\mathcal{M}|^2 dQ$

dQ is the invariant Lorentz phase space factor

$dQ=(2\pi)^4\delta^4(p_1 + p_2 - p_1^' - p_2^')\frac{d^3p_1^'}{(2π)^3 2E_1^'}\frac{d^3p_2^'}{(2π)^3 2E_2^'}$

and F is the flux of incoming particles

$F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|$

The invariant form of F is

$F=4\sqrt{(\vec {p}_1 \cdot \vec {p}_2)^2-(m_1m_2)^2}$

$F_{cms}=4p_i\sqrt {s}$

$d^3p_1^'=p^{'3}_1 d_p^' d\Omega$

$(E_1^')^2=(p_1^')^2+(m_1)^2$

$E_1^' d E_1^'= p_1^' d p_1^'$

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_f}{\mathbf p_i}|\mathcal {M}|^2$