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

$dQ=(2\pi)^4\delta^4 \left(\vec p_{1} +\vec p_{2} - \vec p_{1}^{'} -\vec p_{2}^{'} \right)\frac{d^3 \vec p_{1}^{'}}{(2\pi)^3 2E_{1}^{'}}\frac{d^3 \vec p_{2}^{'}}{(2\pi)^3 2E_{2}^{'}}$

and F is the flux of incoming particles

$F_{cms}=4 \vec p_{1}^{*}\sqrt {s}$

$d\sigma=\frac{1}{4 \vec p_{1}\,^{*}\sqrt {s}}|\mathcal{M}|^2 dQ$

$d^3 \vec p_{1}^{'}=\vec p^{'3}_{1} d \vec p^{'} d\Omega$

$(E_1^')^2=(\vec p_{1}^{'})^2+(m_{1})^{2}$

$E_{1}^{'} d E_{1}^{'}= \vec p_{1}^{'} d \vec p_{1}^{'}$

$dQ=\frac{1}{(4\pi)^2}\delta (E_{1}+E_{2}-E_{1}^{'}-E_{2}^{'})\frac{\vec p_{1}^{'}dE_{1}^{'}}{E_{2}^{'}}d\Omega$<\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

$|\vec p_1^'|=|\vec p_2^'|=|\vec p_f^'| \rightarrow |\vec p_1^' d \vec p_1^'|=|\vec p_2^' d \vec p_2^'|=|\vec p_f^' d \vec p_f^'|$

$dW_f=\frac{W_f}{E_2^'}dE_1^'$

$dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_f dW_f}{W_f}d\Omega$

$dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_f}{\sqrt s}d\Omega$

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