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The absence of logarithmic divergences in the spin and charge density correlations of the 1d Hubbard model

We perform a rigorous computation of the spin and charge density correlations of the 1d repulsive Hubbard model at weak coupling, focusing on the properties of the Fourier transform at momentum 0 and ±2pF, if pF is the Fermi momentum. We prove that the interaction changes the singularity at ±2pF (the discontinuity in the derivative becomes a power law singularity) while the singularity at 0 is essentially unchanged. Our results show that the logarithmic divergences at zero momentum recently found in [7], which would be in contrast with Luttinger liquid behaviour, are indeed spurious.