Spontaneous symmetry breaking and the p→0 limit

I point out a basic ambiguity in the p→0 limit of the connected propagator in a spontaneously broken phase. This may represent an indication that the conventional singlet Higgs boson, rather than being a purely massive field, might have a gapless branch. This would dominate the energy spectrum for p→0 and give rise to a very weak, long-range force. The natural interpretation is in terms of density fluctuations of the 'Higgs condensate': in the region of very long wavelengths, infinitely larger than the Fermi scale, it cannot be treated as a purely classical c-number field