The baryonic content and Tully-Fisher relation at z~0.6

[abr.] Using the multi-integral-field spectrograph GIRAFFE at VLT, we previsouly derived the stellar-mass Tully-Fisher Relation (smTFR) at z~0.6, and found that the distant relation is systematically offset by roughly a factor of two toward lower masses. We extend the study of the evolution of the TFR by establishing the first distant baryonic TFR. To derive gas masses in distant galaxies, we estimate a gas radius and invert the Schmidt-Kennicutt law between star formation rate and gas surface densities. We find that gas extends farther out than the UV light from young stars, a median of ~30%. We present the first baryonic TFR (bTFR) ever established at intermediate redshift and show that, within an uncertainty of +/-0.08 dex, the zeropoint of the bTFR does not appear to evolve between z~0.6 and z=0. The absence of evolution in the bTFR over the past 6 Gyr implies that no external gas accretion is required for distant rotating disks to sustain star formation until z=0 and convert most of their gas into stars. Finally, we confirm that the larger scatter found in the distant smTFR, and hence in the bTFR, is caused entirely by major mergers. This scatter results from a transfer of energy from bulk motions in the progenitors, to random motions in the remnants, generated by shocks during the merging. Shocks occurring during these events naturally explain the large extent of ionized gas found out to the UV radius in z~0.6 galaxies. All the results presented in this paper support the ``spiral rebuilding scenario of Hammer and collaborators, i.e., that a large fraction of local spiral disks have been reprocessed during major mergers in the past 8 Gyr.