Derivative expansion of wave function equivalent potentials

Properties of the wave function equivalent potentials introduced by the HAL QCD collaboration are studied in a nonrelativistic coupled-channel model. The derivative expansion is generalized, and then applied to the energy-independent and nonlocal potentials. The expansion coefficients are determined from analytic solutions to the Nambu-Bethe-Salpeter wave functions. The scattering phase shifts computed from these potentials are compared with the exact values to examine the convergence of the expansion. It is confirmed that the generalized derivative expansion converges in terms of the scattering phase shift rather than the functional structure of the non-local potentials. It is also found that the convergence can be improved by tuning either the choice of interpolating fields or expansion scale in the generalized derivative expansion.

Figure 1

Wave functions (a) ${\psi }_0(x)$ and (b) ${\psi }_1(x)$ for the six lowest-lying-energy states with the TBC.

Figure 2

Nambu-Bethe-Salpeter wave functions ${\mathrm{\Psi }}_q(x)={\psi }_0(x)+q{\psi }_1(x)$ with $q=+0.2$. For better visibility, we inverted the sign of the NBS wave function for $E=E_0$ (since the overall normalization is irrelevant) and altered the $x$-range from Fig. 1.

Figure 3

Regular part of the nonlocal HAL QCD potentials $V^{(N)}(x,x^{\prime })$ and their expansion coefficients $v_n(x)$ ($n=0,\dots ,N$) with $\rho =0.5$ and $q=0.2$. Panels (a), (b), (c), (d), and (e) show the results of $N=1$, 2, 3, 4, and 5, respectively.

Figure 4

Function $f^{(N)}(x)$ with $\rho =0.5$ and $q=0.2$. [See Eq. (32).]

Figure 5

Scattering phase shift ${\delta }_N$ against energy $E$ at $N=1,2,3,4,$ and 5, with fixed $\rho =0.5$ and $q=0.2$. The solid line represents the exact values extracted from the analytic solution of the Birse model (similarly in Fig. 6 and Fig. 7).

Figure 6

Scattering phase shift for $q=+1.0,-0.2,-0.2,-1.0$ with fixed $\rho =0.5$ at (a) $N=2$ and (b) $N=3$.

Figure 7

Scattering phase shift for (a) $\rho =0.1$, 0.3, 0.5, 0.7 at $N=2$ and (b) $\rho =0.3$, 0.5, 0.7 at $N=3$. In both panels the field admixture parameter is fixed to $q=0.2$.

Figure 8

Nonlocal potentials $V^{(N)}(x,x^{\prime })$ with (a) $\rho =0.3$ and (b) $\rho =0.7$, at truncation order $N=3$. The field admixture parameter is fixed to $q=+0.2$ in both panels.

Figure 9

NBS wave functions at $E=E_9$ obtained with the two nonlocal potentials in Fig. 8 (i.e., the ones with $\rho =0.3$ and $\rho =0.7$, with fixed $q=+0.2$ and $N=3$) together with the analytic solution of the NBS wave function for comparison.