Potential description of the charmonium from lattice QCD

We present spin-independent and spin-spin interquark potentials for charmonium states, that are calculated using a relativistic heavy quark action for charm quarks on the PACS-CS gauge configurations generated with the Iwasaki gauge action and 2+1 flavors of Wilson clover quark. The interquark potential with finite quark masses is defined through the equal-time Bethe-Salpeter amplitude. The light and strange quark masses are close to the physical point where the pion mass corresponds to $M_\pi \approx 156(7)$ MeV, and charm quark mass is tuned to reproduce the experimental values of $\eta_c$ and $J/\psi$ states. Our simulations are performed with a lattice cutoff of $a^{-1}\approx 2.2$ GeV and a spatial volume of $(3 {\rm fm})^3$. We solve the nonrelativistic Schr\"odinger equation with resulting charmonium potentials as theoretical inputs. The resultant charmonium spectrum below the open charm threshold shows a fairly good agreement with experimental data of well-established charmonium states.