Thus, the right-hand-side of equation, $Re D_l(k)$, in Eq. (\[eq;dell\]) can be expanded as a power series of $k^2$ for both $K_l(k)$ and $2\kappa Re H_l(k)$. Meanwhile, the left-hand-side of Eq. (\[eq;dell\]) is suppressed by the factor $C_\eta^2$, due to the Gamow factor $P=\exp(-2\pi\eta)$.

In the case of the $s$-wave, for example, the reported phase shift at the smallest energy, $T_\alpha = 2.6$ MeV, is $\delta_0=-1.893^\circ$ [@tetal-prc09]. The factor $C_\eta^2$ becomes $C_\eta^2\simeq 6 \times 10^{-6}$ at $k=104$ MeV which corresponds to $T_\alpha=2.6$ MeV, and the left-hand-side of Eq. (\[eq;dell\]) numerically becomes $C_\eta^2 k\cot\delta_0 = -0.019$ MeV. The function $2\kappa Re H_0(k)$ is expanded as 2Re H\_0(k) &=& k\^2 + k\^4 + k\^6 + k\^8 +\
&=& 7.441 + 0.136 + 0.012 + 0.002 +    , \[eq;S-wave\] at $k=104$ MeV. The numerical values in the second line of Eq. (\[eq;S-wave\]) correspond to the terms appearing in the first line of the equation in order. One can see that the power series converges well, but the first and second terms are two and one order of magnitude larger compared with the value estimated by using the experimental data in the left-hand-side of Eq. (\[eq;dell\]), $-$0.019 MeV. Thus, we regard those terms unnaturally 