Extrapolation of scattering data to the negative-energy region. II. Applicability of effective range functions within an exactly solvable model

A problem of analytical continuation of scattering data to the negative-energy region to obtain information about bound states is discussed within an exactly solvable potential model. This work is continuation of the previous one by the same authors [L. D. Blokhintsev et al., Phys. Rev. C 95, 044618 (2017)]. The goal of this paper is to determine the most effective way of analytic continuation for different systems. The $d+\alpha$ and $\alpha +{}^{12}\mathrm{C}$ systems are considered and, for comparison, an effective-range function approach and a recently suggested $\mathrm{\Delta }$ method [O. L. Ramírez Suárez and J.-M. Sparenberg, Phys. Rev. C 96, 034601 (2017).] are applied. We conclude that the $\mathrm{\Delta }$ method is more effective for heavier systems with large values of the Coulomb parameter, whereas for light systems with small values of the Coulomb parameter the effective-range function method might be preferable.

Figure 1

The $K_0\left(E\right)$ function for the $d+\alpha$ system. The solid red line is the exact $K_0\left(E\right)$ function; the green dashed line is the approximation of $K_0\left(E\right)$ by the Taylor polynomial when the binding energy and the ANC of the bound state $\left(d\alpha \right)$ are not fixed (version 1); the brown dotted line is approximation of $K_0$ by the Taylor polynomial of the third order when only the binding energy of $\left(d\alpha \right)$ is fixed (version 2).

Figure 2

The ${\mathrm{\Delta }}_0\left(E\right)$ function for the $d+\alpha$ system. The notations are the same as in Fig. 1 but for ${\mathrm{\Delta }}_0\left(E\right)$.

Figure 3

The exact $K_0\left(E\right)$ function for the system $\alpha +{}^{12}\mathrm{C}$ with two $0^{+}$ bound states. The pole at the negative energy is very narrow.

Figure 4

The same as in Fig. 1 but for the system $\alpha +{}^{12}\mathrm{C}$.

Figure 5

The exact ${\mathrm{\Delta }}_0\left(E\right)$ function for the system $\alpha +{}^{12}\mathrm{C}$ with two bound states. The pole at negative energy is located at $E_{i2}=-4.48135$ MeV.

Figure 6

The exact ${\mathrm{\Delta }}_0\left(E\right)$ function for the system $\alpha +{}^{12}\mathrm{C}$ with two bound states. The pole at positive energy is located at $E_{i1}=25.315$ MeV.

Figure 7

The ${\mathrm{\Delta }}_0\left(E\right)$ function for the $\alpha +{}^{12}\mathrm{C}$ system. The solid red line is the exact ${\mathrm{\Delta }}_0\left(E\right)$ function; the green dashed line is the approximation of the ${\mathrm{\Delta }}_0\left(E\right)$ function by the Taylor polynomial of the third order when the binding energy and the ANC of the bound state $(\alpha +{}^{12}\mathrm{C})$ are not fixed; the brown dotted line is approximation of ${\mathrm{\Delta }}_0\left(E\right)$ by the Taylor polynomial of the third order when the binding energy of $(\alpha +{}^{12}\mathrm{C})$ is fixed while the ANC is a fitting parameter; and the blue dash-dotted line is obtained using the approximation of $ln({\mathrm{\Delta }}_0\left(E\right)/(E+{\varepsilon }_1))$ by the third-order Taylor polynomial when the binding energy and the ANC of the bound state $(\alpha +{}^{12}\mathrm{C})$ are not fixed.

Figure 8

The ${\mathrm{\Delta }}_0\left(E\right)$ function for the system $\alpha +{}^{12}\mathrm{C}$ with two $0^{+}$ bound states. The solid red line is the exact ${\mathrm{\Delta }}_0\left(E\right)$; the brown dotted line is obtained using approximation of the function ${\mathrm{\Delta }}_0\left(E\right)(E-E_{i2})/(E+{\varepsilon }_1)(E+{\varepsilon }_2)$ by the Taylor polynomial of the third order; and the green dashed line corresponds to the approximation of the function $ln\left[-{\mathrm{\Delta }}_0\left(E\right)(E-E_{i2})/(E+{\varepsilon }_1)(E+{\varepsilon }_2)\right]$ by the Taylor polynomial of the third order.

Figure 9

The real part $\mathrm{Re}\left[R\right(E\left)\right]$ for the $d+\alpha$ system. The solid red line is the exact result; the dashed green line is the asymptotic expansion of $\mathrm{Re}\left[R\right(E\left)\right]$ up to $E^3$.

Figure 10

The real part $\mathrm{Re}\left[R\right(E\left)\right]$ for the $\alpha +{}^{12}\mathrm{C}$ system. The solid red line is the exact result; the dashed green line is the asymptotic expansion of $\mathrm{Re}\left[R\right(E\left)\right]$ up to $E^3$.

Figure 11

The dependence of the ${\mathrm{\Delta }}_0\left(E\right)$ function on the charge-scaling factor $\beta$ (see the text) for $d+\alpha$. Solid red line, $\beta =1$; dashed green line, $\beta =2$; dotted brown line, $\beta =0.2$; dash-dotted blue line, $\beta =0.$