Detailed analysis of the gluonic excitation in the three-quark system in lattice QCD

We study the excited-state potential and the gluonic excitation in the static three-quark (3Q) system using $\mathrm{S}\mathrm{U}(3)$ lattice QCD with ${16}^3\times 32$ at $\beta =5.8$ and 6.0 at the quenched level. For about 100 different patterns of spatially-fixed 3Q systems, we accurately extract the excited-state potential $V_{3\mathrm{Q}}^{\mathrm{e}.\mathrm{s}.}$ together with the ground-state potential $V_{3\mathrm{Q}}^{\mathrm{g}.\mathrm{s}.}$ by diagonalizing the QCD Hamiltonian in the presence of three quarks. The gluonic excitation energy $\Delta E_{3\mathrm{Q}}\equiv V_{3\mathrm{Q}}^{\mathrm{e}.\mathrm{s}.}-V_{3\mathrm{Q}}^{\mathrm{g}.\mathrm{s}.}$ is found to be about 1 GeV at the typical hadronic scale. This large gluonic excitation energy is conjectured to give a physical reason of the success of the quark model for low-lying hadrons even without explicit gluonic modes. We investigate the functional form of $\Delta E_{3\mathrm{Q}}$ in terms of the 3Q location. The lattice data of $\Delta E_{3\mathrm{Q}}$ are relatively well reproduced by the “inverse Mercedes Ansatz” with the “modified Y-type flux-tube length,” which indicates that the gluonic excitation mode is realized as a complicated bulk excitation of the whole 3Q system.