Question on electron-positron scattering

[Morgoth]

Well I am sure this is a lame question, but I am stuck over it for hours. I'm working on Bjorken, Drell book, and I'm trying to calculate the extreme relativistic differential cross section for electron-positron scattering.

Well, I have evaluated the cross section up to my attachment's form...


Then it says in order to keep working we choose the center-of-mass frame system, and gives us the inner products of p₁,p₁',-q₁,-q₁'

p₁p₁'=q₁q₁'=2E²sin²(θ/2)
p₁(-q₁')=p₁'(-q₁)=2Ε²
p₁(-q₁)=p₁'(-q₁')=2Ε²cos²(θ/2)

However that way I can calculate the quantities on the numerator, but I don't know how to work with the denominator...
could someone just tell me how i can find the 4-components of ps and qs?

 

[andrien]

why don't you involve the on-shell condition for denominators i.e like p₁²=m² also expanding in the denominator you will have the terms which can be filled up by the relation you already wrote

 

[Hepth]

Just to clarify what andrien means:

Take the first term, the denominator is [itex](p'-p_1)^2 = p'^2 + p_1^2 - 2 (p' \cdot p_1)[/itex].

In the relativistic limit we can ignore the light masses, since the p's are on-shell, which means p^2 = m^2, but in the relativistic limit that goes to zero.
So that denominator becomes [itex]\frac{-1}{2 p' \cdot p_1}[/itex] which you have above in terms of energy.