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 <math><msub><mi>Λ</mi><mi>c</mi></msub></math> baryon and treat it as a charm-diquark (<math><mi>c</mi></math>-[<math><mi>u</mi><mi>d</mi></math>]) 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 [<math><msub><mi>Λ</mi><mi>c</mi></msub><mo>(</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo></msup><mo>)</mo></math>]. 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 <math><msub><mi>Λ</mi><mi>c</mi></msub><mo>(</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>+</mo></msup><mo>)</mo></math> level and the average of the <math><msub><mi>Λ</mi><mi>c</mi></msub><mo>(</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo></msup><mo>)</mo></math> and the <math><msub><mi>Λ</mi><mi>c</mi></msub><mo>(</mo><msup><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo></msup><mo>)</mo></math> levels. Numerical calculations are performed on a <math><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> lattice with lattice spacing of <math><mi>a</mi><mo>≃</mo><mn>0.0907</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and the pion mass of <math><msub><mi>m</mi><mi>π</mi></msub><mo>≃</mo><mn>700</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. Our charm-diquark potential is given by the <math><mrow><mi>Coulomb</mi><mo>+</mo><mi>linear</mi></mrow></math> (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 <math><msub><mi>m</mi><mi>D</mi></msub><mo>=</mo><mn>1.273</mn><mo>(</mo><mn>44</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. Our diquark mass lies slightly above the conventional estimates, namely the <math><mi>ρ</mi></math> meson mass and twice the constituent quark mass <math><mn>2</mn><msub><mi>m</mi><mi>N</mi></msub><mo>/</mo><mn>3</mn></math>.

Japan

Kai Watanabe

Quark-diquark potential and diquark mass from lattice QCD

We give a formula to calculate a matrix element of a conserved current in the effective quantum mechanics defined by the wavefunction-equivalent potentials proposed by the HAL QCD collaboration. As a first step, a non-relativistic field theory with two-channel coupling is considered as the original theory, with which a wavefunction-equivalent HAL QCD potential is obtained in a closed analytic form. The external field method is used to derive the formula by demanding that the result should agree with the original theory. With this formula, the matrix element is obtained by sandwiching the effective current operator between the left and right eigenfunctions of the effective Hamiltonian associated with the HAL QCD potential. In addition to the naive one-body current, the effective current operator contains an additional two-body term emerging from the degrees of freedom which has been integrated out.