On page 181 in Peskin & Schroeder they say that we consider the integral (intensity)
$$\tag{1}\mathcal{I}(\mathbf{v},\mathbf{v}') = \int\frac{\mathrm{d}\Omega_\hat{k}}{4\pi}\,\frac{2(1-\mathbf{v}\cdot\mathbf{v}')}{(1-\hat{k}\cdot\mathbf{v})(1-\hat{k}\cdot\mathbf{v}')}-\frac{m^2/E^2}{(1-\hat{k}\cdot\mathbf{v}')^2}-\frac{m^2/E^2}{(1-\hat{k}\cdot\mathbf{v})^2}$$
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: $m=0 ~(\text{ERL})$ 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 $\theta=0$ along the peak in each case. Integrate over a small region around $\theta=0$, as follows:
$$\tag{2}\mathcal{I}(\mathbf{v},\mathbf{v}') \approx \int_{\hat{k}\cdot\mathbf{v}= \mathbf{v}'\cdot\mathbf{v}}^{\cos\theta=1}\mathrm{d}\cos\theta\,\frac{(1-\mathbf{v}\cdot\mathbf{v}')}{(1-v\cos\theta)(1-\mathbf{v}\cdot\mathbf{v}')}
\\[1cm]
+\int_{\hat{k}\cdot\mathbf{v}'= \mathbf{v}'\cdot\mathbf{v}}^{\cos\theta=1}\mathrm{d}\cos\theta\,\frac{(1-\mathbf{v}\cdot\mathbf{v}')}{(1-v'\cos\theta)(1-\mathbf{v}\cdot\mathbf{v}')}. $$

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 $\mathbf{v}, \mathbf{v}'$ 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 $p^0=p^{'0}=E$ which (according to them) implies
$$k^\mu=(k,\mathbf{k}),~~p^\mu=E(1,\mathbf{v}),~~p^{'\mu}=E(1,\mathbf{v'}) $$ where (I guess) $k=||\mathbf{k}||. $ Then for instance $(k_\mu p^\mu)^2$ becomes $(Ek)^2\left(1-\frac{\mathbf{k}}{k}\cdot\mathbf{v}\right)^2$ which is (I assume) one of the denominators (up so some factors) in $(1)$. So I guess the correct notation in $(1)$ should be
$$\tag{3}\mathcal{I}(\mathbf{v},\mathbf{v}') =\int \dots-\frac{m^2/E^2}{\left(1-\hat{\mathbf{k}}\cdot\mathbf{v}'\right)^2}-\frac{m^2/E^2}{\left(1-\hat{\mathbf{k}}\cdot\mathbf{v}\right)^2}. $$

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