Quark-diquark potential and diquark mass from lattice QCD

Abstract

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 ${\mathrm{\ensuremath{\Lambda}}}_{c}$ 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 [${\mathrm{\ensuremath{\Lambda}}}_{c}({\frac{1}{2}}^{+})$]. 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 ${\mathrm{\ensuremath{\Lambda}}}_{c}({\frac{1}{2}}^{+})$ level and the average of the ${\mathrm{\ensuremath{\Lambda}}}_{c}({\frac{1}{2}}^{\ensuremath{-}})$ and the ${\mathrm{\ensuremath{\Lambda}}}_{c}({\frac{3}{2}}^{\ensuremath{-}})$ levels. Numerical calculations are performed on a ${32}^{3}\ifmmode\times\else\texttimes\fi{}64$ lattice with lattice spacing of $a\ensuremath{\simeq}0.0907\text{ }\text{ }\mathrm{fm}$ and the pion mass of ${m}_{\ensuremath{\pi}}\ensuremath{\simeq}700\text{ }\text{ }\mathrm{MeV}$. Our charm-diquark potential is given by the $\mathrm{Coulomb}+\mathrm{linear}$ (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 ${m}_{D}=1.273(44)\text{ }\text{ }\mathrm{GeV}$. Our diquark mass lies slightly above the conventional estimates, namely the $\ensuremath{\rho}$ meson mass and twice the constituent quark mass $2{m}_{N}/3$.