Differential Cross-Section

[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]

Differential Cross-Section

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

Working in the center of mass frame

[math]\mathbf p_{final}=\mathbf p_{initial}[/math]

Determining the scattering amplitude in the center of mass frame

[math]\mathfrak{M}=e^2 \left ( \frac{u-s}{t}+\frac{t-s}{u} \right )[/math]

[math]\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 )[/math]

[math]\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 )[/math]

[math]\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 )[/math]

[math]\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 )[/math]

Using the fine structure constant ([math]with\ c=\hbar=\epsilon_0=1[/math])

[math]\alpha \equiv \frac{e^2}{4\pi}[/math]

[math]\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 )[/math]

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

[math]s \equiv 4E^{*2}[/math]

[math]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}}[/math]

[math]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}}[/math]

Simplifying using the relationship

[math]\cos{\theta}=-1+\cos{\frac{\theta}{2}}[/math]

Calculating the parts:

[math]\frac{t^2}{u^2}=\frac{4p^{*2}\left(1-\cos{\theta}\right)^2}{4p^{*2}\left(1+\cos{\theta}\right)^2}=\tan^4{\frac{\theta}{2}}[/math]

[math]\frac{s^2}{u^2}=\frac{16E^{*4}}{4p^{*4}\left(1+\cos{\theta}\right)^2}=\frac{E^{*4}\sec^4{\frac{\theta}{2}}}{p^{*4}}[/math]

[math]\frac{u^2}{t^2}=\frac{4p^{*2}\left(1+\cos{\theta}\right)^2}{4p^{*2}\left(1-\cos{\theta}\right)^2}=\cot^4{\frac{\theta}{2}}[/math]

[math]\frac{s^2}{t^2}=\frac{16E^{*4}}{4p^{*4}\left(1-\cos{\theta}\right)^2}=\frac{E^{*4}\csc^4{\frac{\theta}{2}}}{p^{*4}}[/math]

[math]\frac{2s^2}{tu}=\frac{32E^{*4}}{4p^{*4}\left(1+\cos{\theta}\right)\left(1-\cos{\theta}\right)}=\frac{8E^{*4}}{p^{*4}\sin^2{\theta}}[/math]

---

[math]\underline{\textbf{Navigation}}[/math]

[math]\vartriangleleft [/math] [math]\triangle [/math] [math]\vartriangleright [/math]