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I was trying to figure out the cross section $\frac{d\sigma}{d\Omega}$ for spinless $e^{-}\gamma\rightarrow e^{-}$ scattering. First I wrote the terms associated with each component.

Vertex: $$ie(P_A+P_B)^{\mu}$$ External Boson: $1$

Photon: $\epsilon_{\mu}$

Multiplying these will give the invariant amplitude. $$i\mathcal{M} =ie(P_A+P_B)^{\mu}\epsilon_{\mu}$$ Now consider the momenta in high energy approximation $$P_A =(p,P)$$ $$P_B=(p,P')$$ Such that $|P|=|P'|=p$ Then $$P_A+P_B=(2p,P+P')$$ Now squaring $\mathcal{M}$ $$\mathcal{M}^2 = e²(6p^2+2p^2\cos\theta)\epsilon^2$$ The differential cross section will become: $$\frac{d\sigma}{d\Omega}=\frac{p^2e²}{32\pi^2s}(3+\cos\theta)\epsilon^2$$

Now I have two questions:

1) What have I done wrong? I couldn't find the answer anywhere online , is there something obvious that I am missing? I know I am wrong because $\epsilon^2$ is a $3\times 3$ matrix. A cross section can't be a matrix (As far as I know).

2) What will $s$ be? In the book Martin and Halzen the definition $s$ was simply $$s=(P_A+P_B)^2$$ But $s$ in Martin and Halzen was defined in the case of two vertex diagram. What will be the definition of $s$ in single vertex diagram?

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  • $\begingroup$ Normally when you square the matrix element you use the polarization sum rule for the photo polarization vectors $\sum_{\lambda \lambda'} \epsilon_{\mu}(\lambda)\epsilon_{\nu}^*(\lambda') = -g_{\mu \nu}$. $\endgroup$ – Triatticus May 17 '19 at 17:58
  • $\begingroup$ @Triatticus So you're saying I'll get a factor of 4 because $$g_{\mu\nu}g^{\mu\nu}=4$$ Is this what you're saying? $\endgroup$ – Manvendra Somvanshi May 19 '19 at 9:23
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Your expression for $\mathcal{M}^2$ is wrong. Inside $\mathcal{M}$ polarisation vectors are contracted with the momenta so for example $$\left|(P + P')^\mu \epsilon_\mu\right|^2 =(P + P')^\mu \epsilon_\mu \, (P + P')^\nu \epsilon_\nu =(P + P') \cdot \epsilon \, \,(P + P') \cdot \epsilon$$ It seems that you incorrectly contracted the $(P + P')$ factors with themselves and were left with $\epsilon$ vectors you didn't know what to do with.

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The expected probability density for Compton scattering is

\begin{equation*} \langle|\mathcal{M}|^2\rangle= 2e^4\left( \frac{\omega}{\omega'}+\frac{\omega'}{\omega} +\left(\frac{m}{\omega}-\frac{m}{\omega'}+1\right)^2-1 \right) \end{equation*}

where $\omega$ is the angular frequency of the incident photon, $\omega'$ is the angular frequency of the scattered photon, and $m$ is electron mass.

The Compton formula is \begin{equation*} \frac{1}{\omega'}-\frac{1}{\omega}=\frac{1-\cos\theta}{m} \end{equation*}

It follows that \begin{equation*} \cos\theta=\frac{m}{\omega}-\frac{m}{\omega'}+1 \end{equation*}

Then by substitution \begin{equation*} \langle|\mathcal{M}|^2\rangle= 2e^4\left( \frac{\omega}{\omega'}+\frac{\omega'}{\omega}+\cos^2\theta-1 \right) \end{equation*}

The differential cross section is \begin{equation*} \frac{d\sigma}{d\Omega} = \frac{\alpha^2}{2m^2} \left(\frac{\omega'}{\omega}\right)^2 \left( \frac{\omega}{\omega'}+\frac{\omega'}{\omega}+\cos^2\theta-1 \right) \end{equation*}

For $\omega\approx\omega'$ \begin{equation*} \frac{d\sigma}{d\Omega} \approx \frac{\alpha^2}{2m^2} \left( \cos^2\theta+1 \right) \end{equation*}

See the following link for more on $\langle|\mathcal{M}|^2\rangle$ for Compton scattering.

http://www.eigenmath.org/compton-scattering.pdf

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  • $\begingroup$ OP is asking about interaction involving one photon, the fundamental vertex, not the two photon Compton amplitude $\endgroup$ – lux Jan 16 at 0:51

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