Axion production via Bremsstrahlung - Simple maths question

[Milsomonk]

Hi all,
I am looking for clarification on what is probably a pretty basic change of variables between a few lines in the following paper:

https://journals.aps.org/prd/pdf/10.1103/PhysRevD.34.1326

Equation (9) shows the differential cross section for a Bremsstrahlung process which creates an axion instead of a photon, the cross section is expressed as a differential in $x$ where $x=E_a/E_e$, the ratio of the emitted axion energy to initial electron energy. Between Equation (8) and (9) a change of variables takes place such that $\frac{d\sigma}{d E_a} \rightarrow \frac{d\sigma}{d x}$. What is the correct process to reverse this change of variable so that I have the cross section expressed as differential in axion energy $E_a$? I infer that the author must have done the following substitution (I express the particular algebraic form of the cross section as $f$ for brevity):

$$ \frac{d\sigma}{d E_a} = f(E_a/E_e) = f(x)$$
$$\frac{d\sigma}{d x} = \frac{d\sigma}{d E_a}\cdot \frac{dE_a}{dx} = f(x)* E_e$$

If this is correct, then to reverse the change of variables we have:

$$\frac{d\sigma}{d E_a} = \frac{d\sigma}{d x}*\frac{dx}{dE_a}=f(E_a/E_e)\frac{1}{E_e}$$

Or have I missed something?

 

[mad mathematician]

It should be instead of $f(E_a/E_c)$, $f(x)*E_c$; so you are missing a factor of $E_c$ , but this of course is circular and you are back with what you started with.