View source for Relativistic Differential Cross-section

=Relativistic Differential Cross-section=

<center><math>d\sigma=\frac{1}{F}|\mathcal{M}|^2 dQ</math></center>

dQ is the invariant Lorentz phase space factor 


<center><math>dQ=(2\pi)^4\delta^4(\vec p_1 +\vec p_2 - \vec p_1^' -\vec p_2^')\frac{d^3 \vec p_1^'}{(2\pi)^3 2E_1^'}\frac{d^3 \vec p_2^'}{(2\pi)^3 2E_2^'}</math></center>


and F is the flux of incoming particles


<center><math>F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|=4|E_1E_2\vec v|</math></center>


where v is the relative velocity between the particles.  In the frame where one of the particles is at rest , the relative velocity can be expressed as

<center><math>v_1=\frac{|\vec p_1|}{E_1} \qquad \qquad v_2=\frac{|\vec p_2|}{E_2}</math></center>


<center><math>\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)</math></center>





The invariant form of F is


<center><math>F=4\sqrt{(\vec {p}_1 \cdot \vec {p}_2)^2-(m_1m_2)^2}</math></center>


<center><math>F_{cms}=4 \vec p_i\sqrt {s}</math></center>



<center><math>d\sigma=\frac{1}{4 \vec p_i\sqrt {s}}|\mathcal{M}|^2 dQ</math></center>



<center><math>d^3 \vec p_1^'=\vec p^{'3}_1 d \vec p^'  d\Omega</math></center>

 
<center><math>(E_1^')^2=(\vec p_1^')^2+(m_1)^2</math></center>


<center><math>E_1^' d E_1^'= \vec p_1^' d \vec p_1^'</math></center>



<center><math>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</math><\center>


<center><math>W_i \equiv E_1+E_2  \qquad \qquad W_f \equiv E_1^'+E_2^'</math></center>



<center><math>dW_f=dE_1^'+dE_2^'=\frac{\vec p_1^' d \vec p_1^'}{E_1^'}+\frac{p_2^' dp_2^'}{E_2^'}</math></center>


In the center of mass frame 
<center><math>|\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^'|</math></center>



<center><math>dW_f=\frac{W_f}{E_2^'}dE_1^'</math></center>



<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\delta (W_i-W_f)\frac{\vec p_f dW_f}{W_f}d\Omega</math></center>


<center><math>dQ_{cms}=\frac{1}{(4\pi)^2}\frac{\vec p_f}{\sqrt s}d\Omega</math></center>


<center><math>\frac{d\sigma}{d\Omega}=\frac{1}{64\pi^2 s} \frac{\mathbf p_f}{\mathbf p_i}|\mathcal {M}|^2</math></center>