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salparadise
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Homework Statement
I need to calculate the differential cross section in order of Mandelstam variable [tex]t[/tex], instead of the angle [tex]\theta[/tex]. My problem is with the change of variable not the amplitude of the process. I'm getting a global minus sign which can only be wrong.
It seems I'm making a very basic error but I cannot find it.
Homework Equations
Starting from (p1+p2->p3+p4):
[tex]\frac{d \sigma}{d\Omega}=\frac{1}{64\pi^2s}\frac{\left|\vec{p}_3^{CM}\right|}{\left|\vec{p}_1^{CM}\right|}\left|M\right|^2[/tex]
And knowing that for this particular process we have ([tex]t=(p_1-p3)^2[/tex]):
[tex]t=m^2-2\left(E_{1}^0 E_{3}^0-\left|\vec{p}_3^{CM}\right| \left|\vec{p}_1^{CM}\right| cos(\theta)\right)=m^2-\frac{s}{2}+\frac{1}{2}\sqrt{s(s-4m^2)}cos(\theta)[/tex]
I then calculate:
[tex]d\theta=-\frac{2}{\sqrt{s(s-4m^2)}sin(\theta)}[/tex]
And use this in:
[tex]d\Omega=sin(\theta)d\theta d\phi[/tex]
This global minus sign propagates then into the differential cross section [tex]\frac{d\sigma}{dt}[/tex] and into the total cross section.
The Attempt at a Solution
Can someone please help me find where are my calculations failing?
Thanks in advance
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