What will happen if a photon collide to itself

Question

What will happen if a photon rotating in LHC so fast it catches and bumps to itself?.

Answer 1

In QED there is no photon-photon vertex, meaning that two photons can only interact only via creation of electron-antielectron pairs (or other particle antiparticle "loops" ).

Answer 2

@Mirella Luigi Pozzi's answer address the theoretical concerns regarding:

$$ \gamma_1 + \gamma_2 \rightarrow \gamma_3 + \gamma_4 $$

though, you question seems to be about:

$$ \gamma_1 + \gamma_1 \rightarrow X $$

which doesn't makes sense in this context.

There are also experimental concerns:

1. all photons with wavelengths $\lambda << R$, where $R$ is the radius of the beam-pipe travel at the speed of light, so the statement "goes so fast" doesn't really apply to light.

2. A storage ring uses magnetic fields to confine charged particles in a (piece-wise) circular orbit. That works for protons and heavy nuclei, which the LHC stores.

Photons are neutral, and are not bent by the field, and cannot be stored in the LHC.