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Moller scattering

Last updated Dec 29, 2024

Definition: Moller scattering

Moller scattering is the process eeeee^-e^-\to e^-e^- in scalar QED.

To calculate the Cross sections

we can calculate the matrix element M2|\mathcal{M}|^2

There are two possible diagrams. They are classified according to the momentum transfer in terms of the Mandelstam variables

as t- and u-channel:

# t-channel:

The momentum transfer is kμ=p3μp1μk^\mu=p^\mu_3 - p^\mu_1, so that k2=tk^2=t

Inserting the photon propagator and the proper vertex factors gives:

Pasted image 20230915193349.png

Note that because kμ(p1μ+p3μ)=p32p12=m2m2=0k^\mu(p^\mu_1 + p^\mu_3) = p^2_3 - p^2_1 = m^2-m^2 = 0 the ξ\xi dependence vanishes. This is expected and neccessary because of gauge invariance, no final results should depend on the choice of gauge ξ!\xi!

The final result is Mt=e2(pμ1+pμ3)(p2μ+p4μ)t\mathcal{M}_t=e^2 \frac{(p^1_\mu + p^3_\mu)(p^\mu_2 + p^\mu_4)}{t}

# u-channel:

For the u-channel the momentum transfer is kμ=p4μp1μk^\mu = p^\mu_4 - p^\mu_1 giving k2=u2k^2 = u^2

Pasted image 20230915193318.png

Again the gauge dependence vanishes

The final result is Mu=e2(p1μ+p4μ)(pμ2+pμ3)u\mathcal{M}_u = e^2\frac{(p^\mu_1 + p^\mu_4)(p^2_\mu + p^3_\mu)}{u}

From $\mathcal{M}=\mathcal{M}_t + \mathcal{M}u$ we get the final expression for the Cross sections

by inserting and substituting the mandelstam variables (note $E{cm} = s)) dσ(eeee)dΩ=e464π2s[sut+stu]2\frac{d\sigma(e^-e^-\to e^-e^-)}{d\Omega} = \frac{e^4}{64\pi^2 s}\left[\frac{s-u}{t} + \frac{s-t}{u }\right]^2$