Quark-diquark potential and diquark mass from Lattice QCD

We propose a new application of lattice QCD to calculate the quark-diquark potential, diquark mass and quark mass required for the diquark model. As a concrete example, we consider the $\Lambda_c$ baryon and treat it as a charm-diquark($c$-[$ud$]) two-body bound state. We extend the HAL QCD method to calculate the charm-diquark potential which reproduces the equal-time Nambu-Bethe-Salpeter wave function of the S-wave state ($\Lambda_c(\frac12^+)$). The diquark mass is determined so as to reproduce the difference between the S-wave and the spin-orbit averaged P-wave energies, i.e. the difference between the $\Lambda_c(\frac12^+)$ level and the average of the $\Lambda_c(\frac12^-)$ and the $\Lambda_c(\frac32^-)$ levels. Numerical calculations are performed on a $32^3\times 64$ lattice with lattice spacing of $a \simeq 0.0907$ fm and the pion mass of $m_{\pi} \simeq 700$ MeV. Our charm-diquark potential is given by the Coulomb+linear (Cornell) potential where the long range behavior is consistent with the charm-anticharm potential while the Coulomb attraction is considerably smaller. This weakening of the attraction may be attributed to the diquark size effect. The obtained diquark mass is $m_D=1.273(44)$ GeV. Our diquark mass lies slightly above the conventional estimates, namely the $\rho$ meson mass and twice the constituent quark mass $2m_N/3$.