Magnetic susceptibility of a hot hadronic medium and quark degrees of freedom near the QCD cross-over point

The lattice QCD results for the temperature-dependent magnetic susceptibility of the medium below the cross-over temperature are not possible to reconcile with the widely used Hadron Resonance Gas model, also amended with the physical magnetic moments of hadrons or the pion--vector-meson loops. As noticed earlier, one observes a substantially too strong diamagnetism at temperatures in the range above $\approx 120$~MeV compared to the lattice. This hints at a presence of quarks significantly below the QCD cross-over temperature, which are needed as a source of paramagnetism. However, the pions must be retained to describe the diamagnetism data at low temperatures. Therefore, we consider here a quark-meson approach, where the temperature-dependent quark masses are fixed in a model-free way using the baryon-baryon and baryon-strangeness susceptibilities from the lattice at zero magnetic field. The constituent quarks possess anomalous magnetic moments estimated from the octet baryon magnetic moments. The vacuum quark-loop and meson-loop contributions are duly incorporated. We show that in such a framework, one can describe the magnetic susceptibility up to the cross-over point. The qualitative conclusion is that the QCD degrees of freedom must extend far below the cross-over temperature, down to $\approx 120$~MeV.