Compton Scattering cross section calculation

[DeShark]

Not technically homework, just something I wanted to see if I could do.

Homework Statement

Find the differential cross section for the interaction between an electron and a photon via compton scattering. Basically, I'm just after calculating the matrix (s-matrix?/amplitude?) for the s-channel interaction firstly.

Homework Equations

I drew up the feynman diagram for the s-channel, a version of which can be found at http://upload.wikimedia.org/wikipedia/commons/5/59/ComptonScattering-s.svg

The Attempt at a Solution

Using the feynman rules, I attempted to write out the Matrix in terms of the spinors and propagators and interaction terms, etc.

Initially, I have a photon with momentum k1 and an electron with momentum p1. This electron then continues (propagates) with momentum p2=p1+k1. Then it emits a photon with momentum k2 and has a resulting momentum p3=p2-k2.

For this, I have:

$$M=\{\bar{u}(p_2)(-ie\gamma^{\mu})u(p_1)\} \{\frac{i}{p\!\!\!/{}_2 - m}\} \{\bar{u}(p_3)(-ie\gamma^{\nu})u(p_2)\}$$

Basically, we only covered roughly how to turn diagrams into equations the other day, so I'm fairly sure I've already made a mistake by this point. However, I'll amble on until I hit some mega problems...

Now, I want to find the probability of this event occurring. From what I think I know, if there's a possibility for the event to happen in more than one way, I should sum the amplitudes and then take the square. For this, I'd need to calculate the matrix for the u-channel, no? Then sum the two matrices and then multiply by the adjoint of this sum to find the absolute value squared. Well, that sounds rather complicated to me, so I'd like to just pretend that the u-channel is forbidden. Taking this as fact, I continue by finding the adjoint of the matrix for the s-channel..

Using the fact that
$$(AB)^\dagger = B^\dagger A^\dagger$$,
I found that
$$M^\dagger = (\{\bar{u}(p_3)(-ie\gamma^{\nu})u(p_2)\})^\dagger (\{\frac{i}{p\!\!\!/{}_2 - m}\})^\dagger (\{\bar{u}(p_2)(-ie\gamma^{\mu})u(p_1)\})^\dagger$$

I've also found out that

$$(\{\bar{u}(p_2)(-ie\gamma^{\mu})u(p_1)\})^\dagger = \{\bar{u}(p_1)(+ie\gamma^{\mu})u(p_2)\}$$
$$(\{\bar{u}(p_3)(-ie\gamma^{\nu})u(p_2)\})^\dagger = \{\bar{u}(p_2)(+ie\gamma^{\nu})u(p_3)\}$$

I'm not entirely sure that these are correct either. I'm completely unsure as to how to take the conjugate transpose of the propagator... Does anyone have any hints? And if someone who knows could let me know whether what I'm doing is complete nonsense or if it's making a reasonable amount of sense I'd really love that, cause I'm fairly sure I'm not doing this right. Thanks to anyone who might be able to help!Edit: Actually, thinking about it some more I *know* I've gone wrong, because the photon's momentum is contained nowhere! Upon reading a little... it seems that there is a factor of $$\epsilon_{\mu}(k)$$ which needs to replace the outgoing electron at the first vertex. Is that right? That would make the matrix at the end be

$$M=\{\epsilon_{\mu}(k_1)(-ie\gamma^{\mu})u(p_1)\} \{\frac{i}{p\!\!\!/{}_2 - m}\} \{\bar{u}(p_3)(-ie\gamma^{\nu})\epsilon^{*}_{\mu}(k_2)\}$$

 

[turin]

Find the differential cross section for the interaction between an electron and a photon via compton scattering.

[added by turin: initial state]
... photon with momentum k1 and ... electron with momentum p1.

[added by turin: intermediate state]
... electron ... with momentum p2=p1+k1.

[added by turin: final state]
... photon with momentum k2 and ... [added by turin: electron with] momentum p3=p2-k2.

So far so good.

$$M=\{\bar{u}(p_2)(-ie\gamma^{\mu})u(p_1)\} \{\frac{i}{p\!\!\!/{}_2 - m}\} \{\bar{u}(p_3)(-ie\gamma^{\nu})u(p_2)\}$$

This is not quite correct. First of all and most obviously, you are missing photon polarizations. Secondly, your expression indicates a three-pt electron vertex, or something more bizarre, rather than the charge current. Among other things, this violates Lorentz invariance.

I will stop right there, because these are severe enough problems (at least the second one) that you need to clear them up before you procede with the subject.