Difference between revisions of "Differential Cross-Section"

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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:
  
<center><math>s \equiv 3E^{*2}</math></centert>
+
<center><math>s \equiv 3E^{*2}</math></center>
  
  

Revision as of 01:37, 26 June 2017

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


[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}+\frac{(u^2+s^2)}{t^2}\right )[/math]


Using the fine structure constant

[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}+\frac{(u^2+s^2)}{t^2}\right )[/math]


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

[math]s \equiv 3E^{*2}[/math]


[math]t \equiv -2p^{*2}(1-\cos{\theta})[/math]


[math]u \equiv -2p^{*2}(1+\cos{\theta})[/math]