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 ${\mathrm{\Lambda }}_c$ 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 [${\mathrm{\Lambda }}_c({\frac{1}{2}}^{+})$]. 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 ${\mathrm{\Lambda }}_c({\frac{1}{2}}^{+})$ level and the average of the ${\mathrm{\Lambda }}_c({\frac{1}{2}}^{-})$ and the ${\mathrm{\Lambda }}_c({\frac{3}{2}}^{-})$ levels. Numerical calculations are performed on a ${32}^3\times 64$ lattice with lattice spacing of $a\simeq 0.0907\mathrm{fm}$ and the pion mass of $m_{\pi }\simeq 700\mathrm{MeV}$. 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 $m_D=1.273(44)\mathrm{GeV}$. Our diquark mass lies slightly above the conventional estimates, namely the $\rho$ meson mass and twice the constituent quark mass $2m_N/3$.

Figure 1

Effective mass plots for the spin singlet (top) and triplet (bottom) sectors. The solid lines denote the average of the meson mass $M_{\mathcal{M}}$ over the fit range. The shaded areas denote the statistical errors estimated by the jackknife method.

Figure 2

NBS function for the PS (top) and V (bottom) states in $15\ge t/a\ge 17$. The NBS wave function is normalized as ${\sum }_{r\in V}{r}^2{\varphi }_{\mathcal{M}}^2(\mathit{r})=1$.

Figure 3

The prepotentials for the PS (blue dots) and V (red triangles) states.

Figure 4

Spin-independent part of the pre-potential (dots) fit to the $\mathrm{Cornell}+\log$ function (solid line). Left: Fit to ALL and (right) fit to series I, II, and III. The shaded area around each line denotes statistical errors.

Figure 5

Spin-spin part of the prepotential fit to two-Gaussian function (solid line). The black circles denote the prepotential LQCD data. The flat part of the prepotential lies below 0 by the amount equal to $m_c\mathrm{\Delta }{E}_{\mathrm{hyp}}$.

Figure 6

Numerical solutions for S wave (S) and P wave (P) compared with the LQCD data (black dots). Left: from ALL and (right) from series I, II, and III. Each shaded area denotes the statistical error.

Figure 7

Charm quark mass from ALL and series I, II, III compared to that of the Kawanai-Sasaki method.

Figure 8

Effective mass plots for ${\mathrm{\Lambda }}_c({\frac{1}{2}}^{+})$ (blue diamond), ${\mathrm{\Lambda }}_c({\frac{1}{2}}^{-})$ (red circle), and ${\mathrm{\Lambda }}_c({\frac{3}{2}}^{-})$ (green triangle).

Figure 9

$cD$ NBS functions for the ${\mathrm{\Lambda }}_c({\frac{1}{2}}^{+})$ state at $16\le t/a\le 18$. Each wave function is normalized as ${\sum }_{r\in V}{\mathit{r}}^2{\psi }_{{\mathrm{\Lambda }}_c}^2(\mathit{r})=1$.

Figure 10

Cornell function fit to the $cD$ prepotential (black dots). Left panel shows the fit to ALL, and right shows the fit to series I, II, and III.

Figure 11

Numerical solution from (left) ALL and (right) series I, II, and III. The LQCD data of the NBS wave function (black circles) is shown for comparison.

Figure 12

Direction dependence of the diquark mass.

Figure 13

Left: $cD$ potential $U_0(r)$ compared with the spin-independent part of the $c\overline{c}$ potential $V_0(r)$. Right: Difference between $U_0(r)$ and $V_0(r)$.

Figure 14

Cornell function fit to the $cD$ potential and to the spin-independent part of the $c\overline{c}$ potential. We also show the $\mathrm{Cornell}+\log$ type function fit to the $c\overline{c}$ potential for a reference.

Figure 15

Numerical solutions for S wave (S) and P wave (P). Black circles denote the NBS wave function data.