Moller scattering polarized cross section

[kontejnjer]

Homework Statement

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(Note: this is not strictly homework, but it is related to one course I'm doing, and I can't find a useful solution anywhere)
Find the analytical expression for the scattering cross section of two longitudinally polarized electrons at tree level.

Homework Equations

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\displaystyle d\sigma=\left|\mathcal{M}_{fi}\right|^2\frac{d\Phi}{4I}
Feynman rules

The Attempt at a Solution

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There's two diagrams at tree level since the particles are identical: the t channel and the u channel. There is a relative minus sign between the channels due to the fact the particles are identical fermions. The total scattering amplitude is:
\displaystyle i\mathcal{M}_{fi}=i\mathcal{M}_{fi}^t+i\mathcal{M}_{fi}^u=\frac{e^2}{t}\bar u^{s'}(p_4)\gamma^\mu u^s (p_2) \bar u^{r'} (p_3)\gamma_\mu u^r (p_1)-\frac{e^2}{u}\bar u^{r'}(p_3)\gamma^\mu u^s (p_2) \bar u^{s'} (p_4)\gamma_\mu u^r(p_1)

Here t=(p_3-p_1)^2, u=(p_4-p_1)^2 are the Mandelstam variables, p_1 and p_2 are inital momenta with spins r and s respectively, and final momenta p_3 and p_4 with spins r' and s' respectively. As the initial electrons are longitudinally polarized, we know their spin states, however we still need to sum over the final spins, so the square of the amplitude is:

\left|\mathcal{M}_{fi}\right|^2=\sum\limits_{s',r'} \left(|\mathcal{M}_{fi}^t|^2+|\mathcal{M}_{fi}^u|^2-2(\mathcal{M}_{fi}^t)^* \mathcal{M}_{fi}^u\right)

After some manipulation, and using the identity \sum\limits_{s}u^s(p)\bar u^s(p)=\gamma\cdot p+m, we can write down the square of each component as:

<br /> \sum\limits_{s&#039;,r&#039;}|\mathcal{M}_{fi}^t|^2=\frac{e^4}{t^2}{Tr}\left[\gamma^\mu u^s(p_2) \bar u^s(p_2)\gamma^\nu (\gamma\cdot{p}_4+m)\right]\cdot {Tr}\left[\gamma_\mu u^r(p_1)\bar u^r (p_1) \gamma_\nu (\gamma\cdot{p}_3+m)\right]\\<br /> \sum\limits_{s&#039;,r&#039;}|\mathcal{M}_{fi}^u|^2=\frac{e^4}{u^2}{Tr}\left[\gamma^\mu u^s(p_2) \bar u^s(p_2)\gamma^\nu (\gamma\cdot{p}_3+m)\right]\cdot {Tr}\left[\gamma_\mu u^r(p_1)\bar u^r (p_1) \gamma_\nu (\gamma\cdot{p}_4+m)\right]\\<br /> \sum\limits_{s&#039;,r&#039;}(\mathcal{M}_{fi}^t)^* \mathcal{M}_{fi}^u=\frac{e^4}{ut}{Tr} \left[\gamma^\mu u^s(p_2) \bar u^s(p_2)\gamma^\nu(\gamma\cdot{p}_3+m) \gamma_\mu u^r(p_1)\bar u^r (p_1) \gamma_\nu (\gamma\cdot{p}_4+m)\right]

This is the part that I get stuck on: the initial spins are predefined so I can't just sum over them, and the previously mentioned identity doesn't work in this case, all I know is that the initial spins should be helicity eigenstates. There's a reference paper for this exact topic by Adam M. Bincer - Scattering of longitudinally polarized fermions (DOI: 10.1103/PhysRev.107.1434), but the notation is somewhat outdated and I think it omits some crucial steps.

I know I could just pick a representation and a frame and do the whole trace calculation using some symbolic software, but the end result doesn't give me much qualitative info, what I'd like is to just have a frame-independent square of the scattering amplitude in terms of 4-momenta, so I can just pick any frame (say, COM or lab) and get the analytic result. Any advice or hints would be greatly appreciated.

 

[kontejnjer]

Nope, I'm still no closer to a solution than I was a week ago. Any takers?