(Abridged) The knowledge of the number and of the physical nature of low-metallicity massive galaxies is crucial for the determination and interpretation of the mass-metallicity relation (MZR). Using VLT-ISAAC near-infrared (NIR) spectroscopy of 39 zCOSMOS z~0.7 galaxies, we have measured Halpha and [NII] emission line fluxes for galaxies with [OII], Hbeta and [OIII] available from VIMOS optical spectroscopy. The NIR spectroscopy enables us to break the degeneracy of the R23 method to derive unambiguously O/H gas metallicities, and also SFRs from extinction corrected Halpha. Using, as a benchmark, the position in the D4000 vs. [OIII]/Hbeta diagram of galaxies with reliable O/Hs from NIR spectroscopy, we were able to break the lower/upper branch R23 degeneracy of additional 900 zCOSMOS z~0.7 galaxies. Additionally, the Halpha-based SFR measurements were used to find the best SFR calibration based on [OII] for the zCOSMOS z~0.7 galaxies without Halpha measurements. We find a fraction of 19% of lower mass 9.5<logM/Msun<10.3 zCOSMOS galaxies which shows a larger evolution of the MZR relation, compared to higher mass galaxies, being more metal poor at a given mass by a factor of 2-3 compared to SDSS. This indicates that the low-mass MZR slope is getting steeper at z~0.7 compared to local galaxies. The existence of these metal-poor galaxies at z~0.7 can be interpreted as the chemical version of galaxy downsizing. Moreover, the sample of zCOSMOS galaxies shows direct evidence that SFR influences the MZR at these redshifts. The comparison of the measured metallicities for the zCOSMOS sample with the values expected for a non-evolving fundamental metallicity relation (FMR) shows broadly agreement, and reveals that also galaxies with lower metallicities and typically higher (specific) SFRs, as found in our zCOSMOS sample at z~0.7, are in agreement with the predictions of a non-evolving Z(M,SFR).
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