I Change of variables clarification

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The discussion centers on clarifying the change of variables in the differential cross section for a Bremsstrahlung process that produces an axion, as detailed in the referenced paper. The variable change from ##\frac{d\sigma}{d E_a}## to ##\frac{d\sigma}{d x}##, where ##x = E_a/E_e##, raises questions about correctly reversing this transformation. The user proposes a substitution method to express the cross section in terms of axion energy, but encounters contradictions in their calculations. They seek confirmation on the correct approach to express the cross section as ##\frac{d\sigma}{d E_a}## while recognizing both ##E_a## and ##E_e## as variables. The paper in question is "Axion Bremsstrahlung by an electron beam" by Yung Su Tsai (1986).
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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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Milsomonk said:
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
I'm not able to access that paper.
Milsomonk said:
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):

$$ \frac{d\sigma}{d E_a} = f(E_a/E_e) = f(x)$$
$$\frac{d\sigma}{d x} = \frac{d\sigma}{d E_a}\cdot \frac{dE_a}{dx} = f(x)* E_e$$
Assuming ##E_e## is a constant, then that should be straightforward.
Milsomonk said:
If this is correct, then to reverse the change of variables we have:

$$\frac{d\sigma}{d E_a} = \frac{d\sigma}{d x}*\frac{dx}{dE_a}=f(E_a/E_e)\frac{1}{E_e}$$
This contradicts the previous equations. Instead:
$$\frac{d\sigma}{d E_a} = \frac{d\sigma}{d x}*\frac{dx}{dE_a}=f(E_a/E_e) E_e\frac{1}{E_e} = f(E_a/E_e)$$
 
I think that I have seen this question before, but I could not find my post. Could you at least cite the paper (Author, title, etc.)?
 
PeroK said:
I'm not able to access that paper.

Assuming ##E_e## is a constant, then that should be straightforward.

This contradicts the previous equations. Instead:
$$\frac{d\sigma}{d E_a} = \frac{d\sigma}{d x}*\frac{dx}{dE_a}=f(E_a/E_e) E_e\frac{1}{E_e} = f(E_a/E_e)$$

Thanks for your response, both ##E_e## and ##E_a## are variables, not constant.
 
fresh_42 said:
I think that I have seen this question before, but I could not find my post. Could you at least cite the paper (Author, title, etc.)?

The paper is "Axion Bremsstrahlung by an electron beam" - Yung Su Tsai (1986)
 
I have got a stage where I have a cross section of the form:

$$ \frac{d\sigma}{dx} = A\cdot x$$

Where ##x=\frac{E_a}{E_e}##, ##E_a## and ##E_e## are both variables, ##A## is a constant and I wish to express the cross section as ##\frac{d\sigma}{d E_a}##. But I am not sure how to do this change of variables correctly.
 
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