Time-independent three-dimensional spherical densities cannot be defined for relativistic confined systems; only transverse two-dimensional light-front densities are consistent with quantum mechanics and Poincare invariance.
Singular Charge Density at the Center of the Pion?
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abstract
We relate the three-dimensional infinite momentum frame spatial charge density of the pion to its electromagnetic form factor. Diverse treatments of the measured form factor data including: phenomenological fits, non-relativistic quark models, the application of perturbative QCD, QCD sum rules, holographic QCD, and the Nambu Jona-Lasinio (NJL) model all lead to the result that the charge density at the center of the pion has a logarithmic divergence. Relativistic constituent quark models do not display this singularity. Future measurements planned for larger values of momentum transfer may determine whether or not a singularity actually occurs.
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On the Impossibility of Obtaining Time-Independent, Three-Dimensional, Spherically-Symmetric Densities of Confined Systems of Relativistically Moving Constituents
Time-independent three-dimensional spherical densities cannot be defined for relativistic confined systems; only transverse two-dimensional light-front densities are consistent with quantum mechanics and Poincare invariance.