Two questions concerning scattering in QED

[QuantumDevil]

1. Are there possible (4th order)process of "decay of photon" $$\gamma\rightarrow 3 \gamma$$ with one-photon initial state and 3-photon final state? Intuition tells me there are no such processes. But because of crossing symmetry of scattering amplitude one can argue that such processes should be possible.

2. In one book I found statement that 4-momentum transfer squared $$q^2$$ (through vitual photon) in electron-muon scattering is less than zero. But even in ultra-relativistic limit when electron mass can be neglected one can find that:
$$q^2=-2k\cdot k^{'}$$ where $$k,k^{'}$$ are initial and final 4-momentum vectors of electron.

But $$k\cdot k^{'}$$ depends on scattering angle and so the sign of this term...or maybe I missed something? Can someone give me a hint?

 

[reilly]

Yes, one gamma can go into three, but only in the forward direction. The process comes from a box diagram.

Actually, q squared goes to zero.Best to start with masses and finite momenta, and then let m->0

Regards,
Reilly Atkinson.

 

[reilly]

Mistake Corrected

I'm very wrong on q^^2. In the CM with huge p,, q^^2 = -4P^^2, in the metric with (1,-1,-1,-1). The max Q^^2 come from a 180 change in direction by the particles.

I'm not sure what metric you are using, but it looks to me to be the (-1,1,1,1) on, which, with cos(theta)=-1 you have what you need. Recall, a space-like 4-vector is space-like in all inertial frames.
Regards,
Reilly Atkinson