Mixing-length calibration from field stars. An investigation on statistical errors, systematic biases, and spurious metallicity trends

We analysed the theoretical foundation of the mixing-length calibration by means of classical and global asteroseismic observables of field stars. We discussed the soundness of inferring a metallicity dependence of the mixing-length parameter. We followed a theoretical approach based on mock datasets of artificial stars sampled from a grid of stellar models with a fixed $\alpha_{ml}$. [...] We verified that the $\alpha_{ml}$ estimates are affected by a huge spread, even in the case of perfect agreement between the mock data and the recovery grid. When the solar heavy-element mixture used to compute the models is different from that of the artificial stars, we also found a metallicity relationship in which $\alpha_{ml}$ increases by 0.4 for an increase of 1 dex in [Fe/H]. The origin of this trend was investigated considering the differences in the initial helium abundance Y -- [Fe/H] -- initial metallicity Z relation assumed in the models and data. A discrepancy between the adopted helium-to-metal enrichment ratio caused the appearance of relevant spurious trends in the estimated $\alpha_{ml}$. [...] A similar effect was caused by an offset in the [Fe/H] to Z conversion. An overestimation of [Fe/H] by 0.1 dex in the recovery grid forced an increasing trend of $\alpha_{ml}$ versus [Fe/H] of 0.2 per dex. We also explored the impact of some discrepancies between the adopted input physics in the recovery grid and mock data. We observed a trend with [Fe/H] of 0.3 per dex when the effect of the microscopic diffusion is neglected in the recovery grid [...]. Therefore, any attempt to calibrate the mixing-length parameter by means of classical and asteroseimic observables of field stars seems to be statistically poorly reliable. As such, any claim about the dependence of the mixing-length on the metallicity for field stars should be considered cautiously and critically.