Following earlier findings, we argue that the low-$|t|$ structure in the elastic diffractive cone, recently reported by the TOTEM Collaboration at $8$ TeV, is a consequence of the threshold singularity required by $t-$channel unitarity, such as revealed earlier at the ISR. By using simple Regge-pole models, we analyze the available data on the $pp$ elastic differential cross section in a wide range of c.m. energies, namely those from ISR to LHC8, obtaining good fits of all datasets. This study hints at the fact that the non-exponential behaviour observed at LHC8 is a recurrence of the low-$|t|$ break phenomenon, observed in the seventies at ISR, being induced by the presence of a two-pion loop singularity in the Pomeron trajectory.
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