On page 181 in Peskin & Schroeder they say that we consider the integral (intensity)
I(v,v′)=∫dΩˆk4π2(1−v⋅v′)(1−ˆk⋅v)(1−ˆk⋅v′)−m2/E2(1−ˆk⋅v′)2−m2/E2(1−ˆk⋅v)2
in the extreme relativistic limit (ERL). Then they say that in this limit most of the radiated energy comes from the two peaks in the first term of
(1). Is this because in the ERL one can take the mass
m to be zero:
m=0 (ERL) so only the first term in
(1) remains?
The next question is what I really want an explanation for: They claim that in (ERL) we break up the integral into a piece for each peak, let θ=0 along the peak in each case. Integrate over a small region around θ=0, as follows:
I(v,v′)≈∫cosθ=1ˆk⋅v=v′⋅vdcosθ(1−v⋅v′)(1−vcosθ)(1−v⋅v′)+∫cosθ=1ˆk⋅v′=v′⋅vdcosθ(1−v⋅v′)(1−v′cosθ)(1−v⋅v′).
Then they claim that the lower limit are really not that important, but in any case:
my question is where the lower limits comes from and how about the replacement in the denominator of the integrand, in other words:
How does one go from (1) to (2)?
I should add that v,v′ are the particle velocity before and after interaction. I think one must have access to the book to understand the question unfortunately, other than that, I just want to understand where the lower limits of the integral comes from.
NOTE: PS are working in a frame where p0=p′0=E which (according to them) implies
kμ=(k,k), pμ=E(1,v), p′μ=E(1,v′)
where (I guess)
k=||k||. Then for instance
(kμpμ)2 becomes
(Ek)2(1−kk⋅v)2 which is (I assume) one of the denominators (up so some factors) in
(1). So I guess the correct notation in
(1) should be
I(v,v′)=∫⋯−m2/E2(1−ˆk⋅v′)2−m2/E2(1−ˆk⋅v)2.
Overall, bad notation is used IMO on the pages near 181 in PS.
This post imported from StackExchange Physics at 2014-03-30 03:09 (UCT), posted by SE-user Love Learning