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{ F_{cms}=4p_i\sqrt {s} }[/math]

[math]\displaystyle{ d^3p_1^'=p^{'3}_1 d_p^' d\Omega }[/math]

[math]\displaystyle{ (E_1^')^2=(p_1^')^2+(m_1)^2 }[/math]

[math]\displaystyle{ E_1^' d E_1^'= p_1^' d p_1^' }[/math]

[math]\displaystyle{ dQ=\frac{1}{(4\pi)^2}\delta (E_1+E_2-E_1^'-E_2^')\frac{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{p_1^' dp_1^'}{E_1^'}+\frac{p_2^' dp_2^'}{E_2^'} }[/math]

In the center of mass frame

[math]\displaystyle{ p_1^'=p_2^'=p_f^' \rightarrow p_1^' dp_1^'=p_2^' dp_2^'=p_f^' dp_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{p_f dW_f}{W_f}d\Omega }[/math]

[math]\displaystyle{ dQ_{cms}=\frac{1}{(4\pi)^2}\frac{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]