Cross section in muon pair production from electron positron annihilat

[villsen]

I have a question regarding the calculation of the cross section in muon pair production from electron positron annihilation.

After some calculations the textbook comes to the conclusion that the differential cross section is approximately equal to:

(1+cos(theta)^2)alpha^2/(4*s)

where alpha is a constant and theta is the angle of the outgoing muons and s is the center of mass energy squared.

The author then proceeds to calculate the total cross section from the differential by integrating over the angular variables theta (from 0 to pi) and phi (from 0 to 2*pi). Since phi is not in the differential cross section it only gives a contribution of 2*pi. left to calculate is the integral over theta. After doing that the result is the total cross section:

4*pi*alpha^2/(3*s)

I just can't seem to get this result and I don't know what I am doing wrong. I want to integrate:

1+cos(theta)^2

giving an indefinite integral:

3*theta/2 + sin(2*theta)/4 + constant

which should give a contribution of 4*pi/3. But from what I can understand this integral should be 8/3. If I try to integrate not on theta but on cos(theta) I get something more like the correct answer but I don't understand why I can't just do the simple integration.

Could someone please make more clear the steps from the differential cross section to the total cross section.

Thank you!

 

[villsen]

Nevermind i figured it out myself. Just forgot the sine in the differential angle:

dΩ=sin(θ)dθdϕ