Fourth dimension of the nucleon structure: Spacetime analysis of the timelike electromagnetic proton form factors

As is well known, spacelike proton form factors expressed in the Breit frame may be interpreted as the Fourier transform of static space distributions of electric charge and current. In particular, the electric form factor is simply the Fourier transform of the charge distribution F (q) = eiq·r ρ(r)d 3r. We do not have an intuitive interpretation of the same level of simplicity for the proton timelike form factor appearing in the reactions e + e− ↔ pp. ̄ However, one may suggest that, in the center-of-mass frame, where qμ xμ = qt, a timelike electric form factor is the Fourier transform F (q) = eiqt R(t)dt of a function R(t) expressing how the electric properties of the forming (or annihilating) proton-antiproton pair evolve in time. Here we analyze in depth this idea and show that the functions ρ(r) and R(t) can be formally written as the time and space integrals of a unique correlation function depending on both time and space coordinates.