Does the differential cross section curve correspond to a probability density function? Quick question here. If I know the differential cross section function for a given phenomenon, and if I normalize this function such as its integral over its domain is 1, can I interpret this normalized function as a probability density function?
Here's what I mean. Let's take Compton scattering as an example. The cross section for a scattering to produce an electron in energy interval ($k$, $k+$d$k$) is d$\sigma$/d$k$ (known). If I find the constant $C$ such as $\int{C \;d\sigma/dk\;\;dk}=1$, would the function $f(k) = C\; d\sigma/dk$ be a probability density function?
So if I wanted to get the average energy of and electron produced by Compton scattering, I could simply use $\int{k\;f(k)\;dk}$?
This function $f(k)$ would then be the (normalized) energy spectrum of the scattered electrons, is that correct? And does this work with any kind of differential cross section (angular, energy, etc.)?
Thanks a lot!
 A: The probability density over the Lorentz invarant phase space (LIPS) is proportinal to the matrix element squared and a Dirac function for energy momentum conservation,
$$
p(\text{LIPS}) \propto|\mathcal M|^2 \delta(p_i -p_f)
$$
The right hand side appears in the formula for a cross section (NB I don't include the Dirac function in the $d\text {LIPS} $)
$$
\sigma = \frac1 {2I} \int |\mathcal M|^2\delta(p_i -p_f)\,d\text{LIPS}
$$
such that
$$
p(\text{LIPS}) \propto \frac {d\sigma}{d\text {LIPS}}
$$
By requiring $\int p(\text{LIPS})\,d\text{LIPS}=1$, we find that
$$
p(\text{LIPS}) = \frac {1}{\sigma} \frac {d\sigma}{d\text {LIPS}}
$$
We find the pdf for a particular observable by marginalization over the full phase space,
$$
\begin {align}
p(x) &= \int \delta(x - x(\text{LIPS})) p(\text{LIPS})\,d\text{LIPS} \\
&=  \frac {1}{\sigma}\int \delta(x - x(\text{LIPS})) \frac {d\sigma}{d\text{LIPS}}\,d\text{LIPS} \\
&= \frac {1}{\sigma} \frac {d\sigma}{dx}
\end {align}
$$
The last equality follows more or less from the meaning of a differential cross section (integration over the rest of the phase space is implicut in the ordinary notation for a differential cross section $\frac {d\sigma}{dx}$).
