Relativistic Differential Cross-section

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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

$|\vec p_{1}^{'}|=|\vec p_{2}^{'}|=|\vec p_{f}^{'}| \rightarrow |\vec p_{1}^{'} d \vec p_{1}^{'}|=|\vec p_{2}^{'} d \vec p_{2}^{'}|=|\vec p_{f}^{'} d \vec p_{f}^{'}|$

$dW_{f}=\frac{W_{f}}{E_{2}^{'}}dE_{1}^{'}$

$dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_{f} dW_{f}}{W_{f}}d\Omega$

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

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_{f}}{\mathbf p_{i}}|\mathcal {M}|^2$

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