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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^{μγ}M_{μγ}, where

L^{μγ}= 2[k'_{μ}k_{γ}+k'_{γ}k_{μ}+(q^{2}/2)g^{μγ}] (electron tensor) and

M^{μγ}= 2[p'_{μ}p_{γ}+p'_{γ}p_{μ}+(q^{2}/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^{μ}L_{μγ}= q^{γ}L_{μγ}= 0, which is fine because L is the product of 2 4-currents and q^{μ}j_{μ}^{e-}= 0. However, according to the text that I am reading, the q^{μ}L_{μγ}= q^{γ}L_{μγ}condition implies that we can replace p' in the muon tensor with (p+q); ie, M_{effective}= 2[2p_{μ}p_{γ}+ (q^{2}/2)g^{μγ}.

Does anyone know how to go from the condition q^{μ}L_{μγ}= 0 to the constraint condition p' = (p+q)?

Thank you in advance for your assistance.

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# Electron-Muon Scattering Cross Section

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