Abstract
A model independent Levy expansion method is introduced to describe nearly Levy shaped squared moduli of Fourier transforms. We apply this method to precisely characterize the most recent elastic scattering data of proton-proton collisions at $$\sqrt s = 13$$ TeV. The results reveal a substructure of protons, which is found to be significantly larger and darker in high energy proton-proton collisions at the TeV scale, as compared to a rather faint and apparently overlooked substructure, that we also identify in elastic proton-proton scattering at the ISR energy range of $$\sqrt s = 23{\kern 1pt} - {\kern 1pt} 62$$ GeV.
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