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The TOTEM experiment at the LHC has performed the first measurement at $$\sqrt{s} = 13\,\mathrm{TeV}$$ of the $$\rho $$ parameter, the real to imaginary ratio of the nuclear elastic scattering amplitude at $$t=0$$, obtaining the following results: $$\rho = 0.09 \pm 0.01$$ and $$\rho = 0.10 \pm 0.01$$, depending on different physics assumptions and mathematical modelling. The unprecedented precision of the $$\rho $$ measurement, combined with the TOTEM total cross-section measurements in an energy range larger than $$10\,\mathrm{TeV}$$ (from 2.76 to $$13\,\mathrm{TeV}$$), has implied the exclusion of all the models classified and published by COMPETE. The $$\rho $$ results obtained by TOTEM are compatible with the predictions, from other theoretical models both in the Regge-like framework and in the QCD framework, of a crossing-odd colourless 3-gluon compound state exchange in the t-channel of the proton–proton elastic scattering. On the contrary, if shown that the crossing-odd 3-gluon compound state t-channel exchange is not of importance for the description of elastic scattering, the $$\rho $$ value determined by TOTEM would represent a first evidence of a slowing down of the total cross-section growth at higher energies. The very low-|t| reach allowed also to determine the absolute normalisation using the Coulomb amplitude for the first time at the LHC and obtain a new total proton–proton cross-section measurement $$\sigma _{\mathrm{tot}} = (110.3 \pm 3.5)\,\mathrm{mb}$$, completely independent from the previous TOTEM determination. Combining the two TOTEM results yields $$\sigma _{\mathrm{tot}} = (110.5 \pm 2.4)\,\mathrm{mb}$$.

First determination of the $${\rho }$$ parameter at $${\sqrt{s} = 13}$$ TeV: probing the existence of a colourless C-odd three-gluon compound state