Differential cross section formula of electron-positron pair production.

In summary, the conversation discusses calculating the differential cross section using the Mandelstam variable t instead of the angle \theta. The issue lies in a global minus sign that is affecting the results. The conversation also mentions using the change of variable and new limits of integration in the solution. The poster is seeking help in identifying the error in their calculations.
  • #1
salparadise
23
0

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
 
Last edited:
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  • #2
I've thought about it again, and I see that with the change of variable [tex]\theta \rightarrow t[/tex], the new limits of integration are [tex][t_{max},t_{min}][/tex], since [tex]t[/tex] decreases with [tex]\theta[/tex] in the interval [tex][0,\pi][/tex], which accounts for an extra minus sign.

I would delete the first post, but I don't think it's possible.
 

1. What is the differential cross section formula for electron-positron pair production?

The differential cross section formula for electron-positron pair production is a mathematical expression that describes the probability of producing an electron-positron pair when two particles collide. It takes into account factors such as the energy and angle of the incoming particles, and the masses of the particles involved.

2. How is the differential cross section formula derived?

The differential cross section formula is derived from quantum field theory and the principles of conservation of energy and momentum. It involves complex mathematical calculations and is often simplified for practical use in experiments.

3. What is the significance of the differential cross section formula in particle physics?

The differential cross section formula is a fundamental tool in particle physics, as it allows scientists to predict and understand the behavior of particles in high-energy collisions. It is also used to test the accuracy of theoretical models and to discover new particles.

4. How does the differential cross section formula change in different experimental conditions?

The differential cross section formula can change depending on the energy and angle of the incoming particles, as well as the masses of the particles involved. It can also be affected by the presence of other particles in the collision and the properties of the detector used to measure the particles.

5. Can the differential cross section formula be applied to other types of particle interactions?

Yes, the concept of a differential cross section formula can be applied to various types of particle interactions, such as electron-proton scattering or proton-proton collisions. However, the specific formula will differ depending on the particles involved and the experimental conditions.

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