I Axion production via Bremsstrahlung - Simple maths question

Milsomonk
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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?
 
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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. :cool:
 
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