Differential Cross-Section

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Differential Cross-Section

$\frac{d\sigma}{d\Omega}=\frac{1}{64\pi ^2 s}\frac{\mathbf p_{final}}{\mathbf p_{initial}} |\mathfrak{M} |^2$

Working in the center of mass frame

$\mathbf p_{final}=\mathbf p_{initial}$

Determining the scattering amplitude in the center of mass frame

$\mathfrak{M}=e^2 \left ( \frac{u-s}{t}+\frac{t-s}{u} \right )$

$\mathfrak{M}^2=e^4 \left ( \frac{u-s}{t}+\frac{t-s}{u} \right )\left ( \frac{u-s}{t}+\frac{t-s}{u} \right )$

$\mathfrak{M}^2=e^4 \left ( \frac{(u-s)^2}{t^2}+\frac{(t-s)^2}{u^2} +2\frac{(u-s)}{t}\frac{(t-s)}{u}\right )$

$\mathfrak{M}^2=e^4 \left ( \frac{(u^2-2us+s^2)}{t^2}+\frac{(t^2-2ts+s^2)}{u^2} +2\frac{(ut-st+s^2-us)}{tu}\right )$

$\mathfrak{M}^2=e^4 \left ( \frac{(t^2+s^2)}{u^2}+\frac{2s^2}{tu}+2-\frac{2us}{t^2}-\frac{2s}{t}-\frac{2ts}{u^2}-\frac{2s}{u}+\frac{(u^2+s^2)}{t^2}\right )$

Using the fine structure constant ($with\ c=\hbar=\epsilon_0=1$)

$\alpha \equiv \frac{e^2}{4\pi}$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{2s}\left ( \frac{(t^2+s^2)}{u^2}+\frac{2s^2}{tu}+2-\frac{2us}{t^2}-\frac{2s}{t}-\frac{2ts}{u^2}-\frac{2s}{u}+\frac{(u^2+s^2)}{t^2}\right )$

In the center of mass frame the Mandelstam variables are given by:

$s \equiv 4E^{*2}$

$t \equiv -2p^{*2}(1-\cos{\theta})=-2p^{*2}\left (1-2\cos^2{\frac{\theta}{2}}+1 \right )=-4p^{*2} \left (1-2\cos^2{\frac{\theta}{2}} \right )=-4p^{*2}\sin^2{\frac{\theta}{2}}$

$u \equiv -2p^{*2}(1+\cos{\theta})=-2p^{*2}\left (1+2\cos^2{\frac{\theta}{2}}-1 \right )=-4p^{*2}\cos^2{\frac{\theta}{2}}$

Simplifying using the relationship

$\cos{\theta}=-1+\cos{\frac{\theta}{2}}$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{16p^{*4}\sin^4{\frac{\theta}{2}}+16E^{*4}}{16p^{*4}\cos^4{\frac{\theta}{2}}}+\frac{32E^{*4}}{4p^{*2}\sin^2{\frac{\theta}{2}}4p^{*2}\cos^2{\frac{\theta}{2}}}+2+\frac{4E^{*2}\left(1-\cos{\theta}\right)^2}{p^{*2}\left(1+\cos{\theta}\right)^2\left(1-\cos{\theta}\right)}+\frac{4E^{*2}\left(1+\cos{\theta}\right)^2}{p^{*2}\left(1-\cos{\theta}\right)^2\left(1+\cos{\theta}\right)}+\frac{4E^{*2}\left(1+\cos{\theta}\right)}{p^{*2}\left(1-\cos{\theta}\right)\left(1+\cos{\theta}\right)}+\frac{4E^{*2}\left(1-\cos{\theta}\right)}{p^{*2}\left(1+\cos{\theta}\right)\left(1-\cos{\theta}\right)}+\frac{16p^{*4}\cos^4{\frac{\theta}{2}}+16E^{*4}}{16p^{*4}\sin^4{\frac{\theta}{2}}}\right )$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{16p^{*4}\sin^4{\frac{\theta}{2}}+16E^{*4}}{16p^{*4}\cos^4{\frac{\theta}{2}}}+\frac{8E^{*4}}{p^{*4}\sin^2{\theta}}+2+\frac{4E^{*2}\cot^2{\frac{\theta}{2}}}{p^{*2}\sin^2{\theta}}+\frac{4E^{*2}\tan^2{\frac{\theta}{2}}}{p^{*2}\sin^2{\theta}}+\frac{4E^{*2}\left(1+\cos{\theta}\right)}{p^{*2}\sin^2{\theta}}+\frac{4E^{*2}\left(1-\cos{\theta}\right)}{p^{*2}\sin^4{\theta}}+\frac{16p^{*4}\cos^4{\frac{\theta}{2}}+16E^{*4}}{16p^{*4}\sin^4{\frac{\theta}{2}}}\right )$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{16p^{*4}\sin^4{\frac{\theta}{2}}}{16p^{*4}\cos^4{\frac{\theta}{2}}}+\frac{16p^{*4}\cos^4{\frac{\theta}{2}}}{16p^{*4}\sin^4{\frac{\theta}{2}}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+2+\frac{8E^{*2}}{p^{*2}\sin^2{\theta}}+\frac{4E^{*2}\left(\cot^2{\frac{\theta}{2}}+\tan^2{\frac{\theta}{2}}\right)}{p^{*2}\sin^2{\theta}}+\frac{16E^{*4}}{16p^{*4}\cos^4{\frac{\theta}{2}}}+\frac{16E^{*4}}{16p^{*4}\sin^4{\frac{\theta}{2}}}\right )$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{16p^{*4}\sin^4{\frac{\theta}{2}}}{16p^{*4}\cos^4{\frac{\theta}{2}}}+\frac{16E^{*4}\sin^4{\frac{\theta}{2}}}{16p^{*4}\cos^4{\frac{\theta}{2}}\sin^4{\frac{\theta}{2}}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}\left(\cos^2{\theta}+1\right)}{p^{*2}\left(cos^2{\theta}-1\right)^2}+\frac{16p^{*4}\cos^4{\frac{\theta}{2}}}{16p^{*4}\sin^4{\frac{\theta}{2}}}+\frac{16E^{*4}\cos^4{\frac{\theta}{2}}}{16p^{*4}\sin^4{\frac{\theta}{2}}\cos^4{\frac{\theta}{2}}}\right )$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \tan^4{\frac{\theta}{2}}+\frac{16E^{*4}\sin^4{\frac{\theta}{2}}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}\left(\cos^2{\theta}+1\right)}{p^{*2}\left(1-sin^2{\theta}-1\right)^2}+\cot^4{\frac{\theta}{2}}+\frac{16E^{*4}\cos^4{\frac{\theta}{2}}}{p^{*4}\sin^4{\theta}}\right )$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \tan^4{\frac{\theta}{2}}+\cot^4{\frac{\theta}{2}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}\left(\cos^2{\theta}+1\right)}{p^{*2}sin^4{\theta}}+\frac{16E^{*4}\cos^4{\frac{\theta}{2}}}{p^{*4}\sin^4{\theta}}+\frac{16E^{*4}\sin^4{\frac{\theta}{2}}}{p^{*4}\sin^4{\theta}}\right )$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \tan^4{\frac{\theta}{2}}+\cot^4{\frac{\theta}{2}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\left(\cos{2\theta}+3\right)}{p^{*4}\sin^4{\theta}}\right )$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \tan^4{\frac{\theta}{2}}+\cot^4{\frac{\theta}{2}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\cos{2\theta}}{p^{*4}\sin^4{\theta}}+\frac{12E^{*4}}{p^{*4}\sin^4{\theta}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \tan^4{\frac{\theta}{2}}+\cot^4{\frac{\theta}{2}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{48E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}\sin^4{\theta}}+\frac{12E^{*4}}{p^{*4}\sin^4{\theta}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{\sin^4{\theta}}{\left(\cos{\theta}+1\right)^4}+\frac{\sin^4{\theta}}{\left(\cos{\theta}-1\right)^4}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}\sin^4{\theta}}+\frac{12E^{*4}}{p^{*4}\sin^4{\theta}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{\sin^4{\theta}\left(\cos{\theta}-1\right)^4}{\left(\cos{\theta}+1\right)^4\left(\cos{\theta}-1\right)^4}+\frac{\sin^4{\theta}\left(\cos{\theta}+1\right)^4}{\left(\cos{\theta}-1\right)^4\left(\cos{\theta}+1\right)^4}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}\sin^4{\theta}}+\frac{12E^{*4}}{p^{*4}\sin^4{\theta}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{\sin^4{\theta}\left(\cos{\theta}-1\right)^4}{\sin^8{\theta}}+\frac{\sin^4{\theta}\left(\cos{\theta}+1\right)^4}{\sin^8{\theta}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}\sin^4{\theta}}+\frac{12E^{*4}}{p^{*4}\sin^4{\theta}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{\left(\cos{\theta}-1\right)^4}{\sin^4{\theta}}+\frac{\left(\cos{\theta}+1\right)^4}{\sin^4{\theta}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}\sin^4{\theta}}+\frac{12E^{*4}}{p^{*4}\sin^4{\theta}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left( \frac{\left(2\cos^4{\theta}+12\cos^2{\theta}+2 \right)}{\sin^4{\theta}}+\frac{8E^{*4}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{2p^{*4}\sin^4{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}\sin^4{\theta}}+\frac{8E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}\sin^4{\theta}}+\frac{4E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}\sin^4{\theta}}+\frac{12E^{*4}}{p^{*4}\sin^4{\theta}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}\sin^4{\theta}}\left( \frac{\left(\cos^4{\theta}+6\cos^2{\theta}+1 \right)}{1}+\frac{4E^{*4}\sin^2{\theta}}{p^{*4}}+\frac{p^{*4}\sin^4{\theta}}{p^{*4}}+\frac{4E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}}+\frac{4E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}}+\frac{2E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}}+\frac{6E^{*4}}{p^{*4}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}\sin^4{\theta}}\left( \frac{p^{*4}\left(\cos^4{\theta}+6\cos^2{\theta}+1 \right)}{p^{*4}}+\frac{4E^{*4}\sin^2{\theta}}{p^{*4}}+\frac{p^{*4}\sin^4{\theta}}{p^{*4}}+\frac{4E^{*2}p^{*2}\sin^2{\theta}}{p^{*4}}+\frac{4E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)}{p^{*4}}+\frac{2E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)}{p^{*4}}+\frac{6E^{*4}}{p^{*4}} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}p^{*4}\sin^4{\theta}}\left( p^{*4}\left(\cos^4{\theta}+6\cos^2{\theta}+1 \right)+4E^{*4}\sin^2{\theta}+p^{*4}\sin^4{\theta}+4E^{*2}p^{*2}\sin^2{\theta}+4E^{*2}p^{*2}\left(\cos^2{\theta}+1\right)+2E^{*4}\left(\cos^2{\theta}-\sin^2{\theta}\right)+6E^{*4} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}p^{*4}\sin^4{\theta}}\left( p^{*4}\left(\cos^4{\theta}+6\cos^2{\theta}+1 \right)+4E^{*4}\sin^2{\theta}+p^{*4}\sin^4{\theta}+4E^{*2}p^{*2}\sin^2{\theta}+6E^{*2}p^{*2}\left(1-2\sin^2{\theta}\right)+10E^{*2}p^{*2}+2E^{*4}\left(1-2\sin^2{\theta}\right)+6E^{*4} \right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}p^{*4}\sin^4{\theta}}\left( p^{*4}\left(\cos^4{\theta}+6\cos^2{\theta}+1 \right)+8E^{*4}+p^{*4}\left(1-2\cos^2{\theta}+\cos^4{\theta}\right)+4E^{*2}p^{*2}\sin^2{\theta}+6E^{*2}p^{*2}-12E^{*2}p^{*2}\sin^2{\theta}+10E^{*2}p^{*2}\right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}p^{*4}\sin^4{\theta}}\left( p^{*4}\left(2\cos^4{\theta}+4\cos^2{\theta}+2 \right)+8E^{*4}-8E^{*2}p^{*2}\sin^2{\theta}+16E^{*2}p^{*2}\right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}p^{*4}\sin^4{\theta}}\left( p^{*4}\left(2\cos^4{\theta}+4\cos^2{\theta}+2 \right)+8E^{*4}-8E^{*2}p^{*2}\left(1-\cos^2{\theta}\right)+16E^{*2}p^{*2}\right)$

$\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{4E^{*2}p^{*4}\sin^4{\theta}}\left( p^{*4}\left(2\cos^4{\theta}+4\cos^2{\theta}+2 \right)+8E^{*4}\left(2-\cos^2{\theta}\right)+8E^{*2}p^{*2}\left(1+\cos^2{\theta}\right)\right)$

$\therefore E^2\equiv m^2+p^2$

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