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We construct analytical forms in the impact parameter $b$-space for the realand imaginary amplitudes describing elastic pp and $\rm p \bar p $ scattering.The amplitudes converted analytically to the momentum transfer $t$-space havemagnitudes, slopes, curvatures, zeros, signs, obeying phenomenological andtheoretical expectations, and describe with high precision all details of thedata, in the full-t range, and for energies from 20 GeV to 7 TeV. Theconnection of forward with large-$|t|$ behavior allows precise determination oftotal cross-sections, slopes and other scattering parameters. We study theproperties and the energy dependence of the $b$-space profile functions,observing that the real part has fundamental influence in the structure of$d\sigma/dt$ at intermediate and large $|t|$ values. We discuss the 540/546 GeVand 1.8/1.96 TeV data from CERN SPS and Fermilab TEVATRON and the 7 TeV resultsfrom TOTEM measurements at LHC, and investigate the extrapolation to 14 TeV andhigher energies.

Energy Dependence of Profile Functions in $\rm{p\bar p}$ and $\rm pp$ Scattering