Why Is My Electron-Positron Scattering Cross Section Calculation Incorrect?

[Muh. Fauzi M.]

Homework Statement

I have a problem in calculating cross-section in elektron-positron -> muon-antimuon scattering.

Homework Equations

In the relativistic limit, we find the differential cross-section of {e}+{e^-} -> {μ}+{μ^-} is

\frac{d\sigma}{d\Omega}=\frac{\alpha^2}{4s}*(1+cos^2{\theta}).

When I integrating over \theta, i get an answer:

\sigma=\frac{3*\pi*\alpha^2}{8*s},

but that is not the correct answer. Due to my reference (Halzen and Martin), the answer should be

\sigma=\frac{4*\pi*\alpha^2}{3*s}.

The Attempt at a Solution

Can someone help me through this problem? My intuition said I have mistaken the integration. But I've tried to re-integral-ing it, still not come up with correct solution.

 

[BvU]

Homework Statement

I have a problem in calculating cross-section in electron-positron -> muon-antimuon scattering.

Homework Equations

In the relativistic limit, we find the differential cross-section of $\ \ {e}^+{e^-} -> {μ}^+{μ^-}$ is
$$\frac{d\sigma}{d\Omega}=\frac{\alpha^2}{4s}\left (1+cos^2{\theta}\right ) \ .$$
When I integrate over $\theta$, I get an answer:
$$\sigma=\frac{3\pi\alpha^2}{8s}\ ,$$
but that is not the correct answer. Due to my reference (Halzen and Martin), the answer should be
$$\sigma=\frac{4\pi\alpha^2}{3s}\; .$$

The Attempt at a Solution

Can someone help me through this problem? My intuition sais I have mistaken the integration. But I've tried to re-integrate it, still not come up with correct solution.

Hello,

Is $d\Omega$ just $d\theta$ or has $\phi$ something to do with it too ?

Anyway, show your steps in detail, please. That makaes it easier to follow and point out where things may go wrong...

 

[Muh. Fauzi M.]

Hello,

Is $d\Omega$ just $d\theta$ or has $\phi$ something to do with it too ?

Anyway, show your steps in detail, please. That makaes it easier to follow and point out where things may go wrong...

Hello Mr., thank you for your respond.

In my first reference (Halzen and Martin), the instructor is to integrate over $\theta$ and $\phi$. Here is my doodling:

$$ \sigma = \frac{\alpha^2}{4s}\int{(1+cos^2{\theta})}d\theta d\phi $$

Substituting the trigonometry identity,

$$ \sigma = \frac{\alpha^2}{4s} \int {\frac{(3+cos{2\theta})}{2}} d\theta d\phi $$.

Dividing the integration,

$$ \int 3 d\theta d\phi , \quad \int {cos{2\theta}}d\theta d\phi $$

and integrating from $0$ to $\pi$ for $\theta$, $0$ to $2\pi$ fpr $\phi$. The second term is vanish, so I have

$$ \int 3 d\theta d\phi=3(\pi)(2\pi)$$.

Thus, when inserting to $\sigma$, I have

$$ \sigma = \frac{3\pi^2\alpha^2}{4s} $$.

Using my second ref (Schwartz), the integration just applied over $\theta$ and I get

$$ \sigma = \frac{3\pi\alpha^2}{8s}$$.

Which step do I miss?

 

[BvU]

If I remember well, $d\Omega \ne d\phi d\theta$ because then you would end up with $2\pi^2$ where you expect $4\pi$ . Wasn't it $d\Omega = \sin\theta \;d\theta d\phi$ ?

 

[Muh. Fauzi M.]

If I remember well, $d\Omega \ne d\phi d\theta$ because then you would end up with $2\pi^2$ where you expect $4\pi$ . Wasn't it $d\Omega = \sin\theta \;d\theta d\phi$ ?

Speechless. That's true Mr. Thank you very much. (shy)