Electron-Muon Scattering Cross Section

[boltzman1969]

Hi,

I am self-teaching Quantum Elctrodynamics, and have come across something which I do not understand. I would appreciate feedback from anyone on this specific issue from Atchison & Hey, "Guage Theories in Particle Physics" pg 238-239:

In calculating the u-channel electron-muon scattering amplitude at the one-photon exchange, one can simplify the calculation by introducing the electron and muon tensors:

L^μγM_μγ, where

L^μγ = 2[k'_μk_γ+k'_γk_μ+(q²/2)g^μγ] (electron tensor) and

M^μγ = 2[p'_μp_γ+p'_γp_μ+(q²/2)g^μγ] (muon tensor)

Now q^μ = (k-k')^μ = (p-p')^μ is the 4-momentum of the exchanged photon; p, p' are the intial and final momenta of the muon; k, k' the initial and final 4-momenta of the electron.

It is claimed that q^μL_μγ = q^γL_μγ = 0, which is fine because L is the product of 2 4-currents and q^μj_μ^e- = 0. However, according to the text that I am reading, the q^μL_μγ = q^γL_μγ condition implies that we can replace p' in the muon tensor with (p+q); ie, M_effective = 2[2p_μp_γ + (q²/2)g^μγ.

Does anyone know how to go from the condition q^μL_μγ = 0 to the constraint condition p' = (p+q)?

Thank you in advance for your assistance.

 

[samalkhaiat]

Hi,

Does anyone know how to go from the condition q^μL_μγ = 0 to the constraint condition p' = (p+q)?

Thank you in advance for your assistance.

The condition q^μL_μγ = 0 does not imply p' = (p + q). It is the opposite which is true.
There is only one diagram in the process, so the amplitude has two momentum-conserving delta functions: \delta^{ 4 } ( k - k' - q ) at the electron vertex and \delta^{ 4 } ( p - p' + q ) at the muon vertex. So when you do the integral \int d^{ 4 }q you forced to put p' - p= q = k' - k. After that you can use the Casimir's trick the trace theorems to obtain the | \langle \mathcal{ M } \rangle |^{ 2 }.

 

[boltzman1969]

Samalkhaiat,

Thank you for that reply. It is much clearer to me know how to obtain the scattering amplitude.

I was scratching my head trying to figure out the implication in the other direction.

The text was somewhat unclear in this regard.

Boltzman1969