Relativistic Differential Cross-section

Relativistic Differential Cross-section

[math]d\sigma=\frac{1}{F}|\mathcal{M}|^2 dQ[/math]

dQ is the invariant Lorentz phase space factor

[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]

and F is the flux of incoming particles

[math]F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|=4|E_1E_2\vec v_{21}|[/math]

where [math]v_{21}[/math] is the relative velocity between the particles in the frame where particle 1 is at rest

[math]\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)= E_{1}E_{2}[/math]

Using the relativistic definition of energy

[math]E^2 \equiv p^2+m^2=m^2[/math]

[math]\rightarrow \mathbf P_1 \cdot \mathbf P_2 =mE_{2}[/math]

Letting [math]E_{21}\equiv E_2[/math] be the energy of particle 2 wiith respect to particle 1, the relativistic energy equation can be rewritten such that

[math]|p_{21}^2| =E_{21}^2-m^2=\frac{(\mathbf P_1 \cdot \mathbf P_2)^2}{m^2}-m^2=\frac{(\mathbf P_1 \cdot \mathbf P_2)^2-m^4}{m^2}[/math]

where similarly [math]p_{21}[/math] is defined as the momentum of particle 2 with respect to particle 1.

The relative velocity can be expressed as

[math] v_{21}=\frac{|\vec p_{21}|}{E_{21}}[/math]

[math]F=2E_1 2E_2|\vec {v}_1-\vec {v}_2|=4|mE_{21}\vec v_{12}|=4|mE_{21}\frac{|\vec p_{21}|}{E_{21}}|=4m|\vec p_{21}|[/math]

The invariant form of F is

[math]F=4\sqrt{(\mathbf P_1 \cdot \mathbf P_2)^2-m^4}[/math]

[math]\mathbf P_1 \cdot \mathbf P_2 = E_{1}E_{2}-(\vec p_1 \vec p_2)[/math]

where in the center of mass frame [math]E_1=E_2[/math] and [math] \vec p_1^*=-\vec p_2^*[/math]

[math]\mathbf P_1 \cdot \mathbf P_2 = E_1^2+\vec p_1\,^{*2}[/math]

[math]F=4\sqrt{(E_1^2+\vec p_1\,^{*2})^2-m^4}=4\sqrt{(\vec p_1 \,^{*2}+m^2+\vec p_1\,^{*2})^2-m^4}[/math]

[math]F=4\sqrt{(2\vec p_1\,^{*2}+m^2)^2-m^4}=4\sqrt{4\vec p_1\,^{*4}+m^4+4\vec p_1\,^{*2}m-m^4}[/math]

[math]F_{cms}=4 \sqrt {4m^2\vec p_1\,^{*2}+4\vec p_1 \,^{*4}}[/math]

[math]F_{cms}=4 \sqrt {\vec p_1\,^{*2}4(m^2+\vec p_1 \,^{*2})}[/math]

As shown earlier

[math]s_{CM}=4(m^2+\vec p_1 \,^{*2})[/math]

[math]F_{cms}=4 \sqrt {\vec p_1^*\,^2s}[/math]

[math]F_{cms}=4 \vec p_1^*\sqrt {s}[/math]

[math]d\sigma=\frac{1}{4 \vec p_1\,^*\sqrt {s}}|\mathcal{M}|^2 dQ[/math]

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

[math](E_1^')^2=(\vec p_1^')^2+(m_1)^2[/math]

[math]E_1^' d E_1^'= \vec p_1^' d \vec p_1^'[/math]

[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>

[math]W_i \equiv E_1+E_2 \qquad \qquad W_f \equiv E_1^'+E_2^'[/math]

[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]

In the center of mass frame

[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]

[math]dW_f=\frac{W_f}{E_2^'}dE_1^'[/math]

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

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

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