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| In the center of mass frame the Mandelstam variables are given by: | | In the center of mass frame the Mandelstam variables are given by: |
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| <center><math>s \equiv 3E^{*2}</math></center> | | <center><math>s \equiv 4E^{*2}</math></center> |
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| <center><math>u \equiv -2p^{*2}(1+\cos{\theta})</math></center> | | <center><math>u \equiv -2p^{*2}(1+\cos{\theta})</math></center> |
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| | <center><math>\frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left ( \frac{(t^2+s^2)}{u^2}-\frac{2s^2}{tu}+\frac{(u^2+s^2)}{t^2}\right )</math></center> |
Revision as of 01:39, 26 June 2017
[math]\displaystyle{ \frac{d\sigma}{d\Omega}=\frac{1}{64\pi ^2 s}\frac{\mathbf p_{final}}{\mathbf p_{initial}} |\mathfrak{M} |^2 }[/math]
[math]\displaystyle{ \mathfrak{M}=e^2 \left ( \frac{u-s}{t}+\frac{t-s}{u} \right ) }[/math]
[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathfrak{M}^2=e^4 \left ( \frac{(t^2+s^2)}{u^2}-\frac{2s^2}{tu}+\frac{(u^2+s^2)}{t^2}\right ) }[/math]
Using the fine structure constant
[math]\displaystyle{ \alpha \equiv \frac{e^2}{4\pi} }[/math]
[math]\displaystyle{ \frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{2s}\left ( \frac{(t^2+s^2)}{u^2}-\frac{2s^2}{tu}+\frac{(u^2+s^2)}{t^2}\right ) }[/math]
In the center of mass frame the Mandelstam variables are given by:
[math]\displaystyle{ s \equiv 4E^{*2} }[/math]
[math]\displaystyle{ t \equiv -2p^{*2}(1-\cos{\theta}) }[/math]
[math]\displaystyle{ u \equiv -2p^{*2}(1+\cos{\theta}) }[/math]
[math]\displaystyle{ \frac{d\sigma}{d\Omega}=\frac{\alpha ^2}{8E^{*2}}\left ( \frac{(t^2+s^2)}{u^2}-\frac{2s^2}{tu}+\frac{(u^2+s^2)}{t^2}\right ) }[/math]