On the knee of Galactic cosmic rays in light of sub-TeV spectral hardenings

More than fifty years after the discovery of the knee in the cosmic ray spectra, its physical origin remains a mystery. This is partly due to the ambiguity of the energy spectrum of individual composition. Recently, direct measurements from several balloon/satellite-borne detectors found significant spectral hardenings of cosmic ray nuclei at a few hundred GV rigidities. A joint modeling of the direct and indirect measurements of the cosmic ray spectra may help to understand the experimental systematics and probably the physics of the knee. In this work, we update the phenomenological "poly-gonato" model of the knee to include the spectral hardenings, with a changing spectral index of $\gamma + \beta \cdot \log E$ with respect to energy. We find that there is some tension between the direct and indirect measurements. Specifically, the hard spectra of CREAM suggest a relatively low energy cutoff of the individual cosmic ray spectrum around the knee, which is inconsistent with the light component spectra of the air shower measurements. Furthermore, the light component data from AS$\gamma$/ARGO-YBJ/WFCTA/KASCADE are consistent with the all-particle spectra. We expect that more precise measurements of the cosmic ray spectrum of individual composition at TeV energies and beyond by the operating space detectors such as CALET and DAMPE, and the ground based experiments such as LHAASO, will eventually solve these discrepancies. Finally, as an illustration, we show that a spatial-dependent diffusion under two-halo model is able to reproduce the results of this "modified poly-gonato" model.