On the hard $\gamma$-ray spectrum of the supernova remnant G106.3$+$2.7

Tibet AS$\gamma$ experiment has measured $\gamma$-ray flux of supernova remnant G106.3+2.7 up to 100 TeV, suggesting it to be potentially a "PeVatron". Challenge arises when the hadronic scenario requires a hard proton spectrum (with spectral index $\approx 1.8$), while the diffusive shock acceleration can only predict a soft (with spectral index $\geq 2$) proton spectrum. In this paper, we explore an alternative scenario to explain the $\gamma$-ray spectrum of G106.3+2.7 within the current understanding of acceleration and escape processes. We consider that the cosmic rays upstream of the shock are scattered by the turbulence driven via Bell instability. The resulting hadronic $\gamma$-ray spectrum is novel, dominating the contribution to the emission above 10\,TeV, and can explain the bizzare broadband spectrum of G106.3+2.7 in combination with leptonic emission from the supernova remnant.