The TOTEM experiment at the CERN LHC has measured elastic proton-proton scattering at the centre-of-mass energy $sqrt{s}$ = 8 TeV and four-momentum transfers squared, |t|, from 6 x $10^{-4}$ GeV$^2$ to 0.2 GeV$^2$. Near the lower end of the |t|-interval the differential cross-section is sensitive to the interference between the hadronic and the electromagnetic scattering amplitudes. This article presents the elastic cross-section measurement and the constraints it imposes on the functional forms of the modulus and phase of the hadronic elastic amplitude. The data exclude the traditional Simplified West and Yennie interference formula that requires a constant phase and a purely exponential modulus of the hadronic amplitude. For parametrisations of the hadronic modulus with second- or third-order polynomials in the exponent, the data are compatible with hadronic phase functions giving either central or peripheral behaviour in the impact parameter picture of elastic scattering. In both cases, the $rho$-parameter is found to be 0.12 $pm$ 0.03. The results for the total hadronic cross-section are $sigma_{tot}$ = (102.9 $pm$ 2.3) mb and (103.0 $pm$ 2.3) mb for central and peripheral phase formulations, respectively. Both are consistent with previous TOTEM measurements.
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