Change of variables clarification

[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?

 

[PeroK]

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.

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.

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

 

[fresh_42]

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.)?

 

[Milsomonk]

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.

 

[Milsomonk]

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)

 

[Milsomonk]

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.