(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant 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.

3. The attempt at a solution

Can someone please help me find where are my calculations failing?

Thanks in advance

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# Homework Help: Differential cross section formula of electron-positron pair production.

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