In everyday research, it is tacitly assumed that the scattering cross-sections have fixed values for the given particle species, centre-of-mass energy, and particle polarizations. However, this assumption has been called into question after several observations of suppression of high-energy bremsstrahlung. This process will play a major role in experiments at the future Electron-Ion Collider, and we show here how variations of the bremsstrahlung cross-section can be profoundly studied there using the lateral beam displacements. In particular, we predict very strong increase of the observed cross-sections for large beam separations. We also discuss the relation of these elusive effects to other quantum phenomena occurring over macroscopic distances. In this context, spectacular and possibly useful properties of the coherent bremsstrahlung at the EIC are also evaluated.
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