Long Range Component of the Nuclear Potential

Possible long range force of the strong Van der Waals type is searched in the nuclear potential. For the case of the short range potential, the amplitude $A(s,t)$ is regular at $t=0$, on the other hand, when the potential is long range, namely $V(r) \sim -C/r^{\alpha}$, the extra branch point $ A(s,t)= C' (-t)^{\gamma}+ \cdots$ with $\gamma=(\alpha-3)/2$ appears at $t=0$. Therefore if we try the polynomial fit to the amplitude, it must deviate rather abruptly from the fit in the small neighborhood of $t=0$. In terms of the partial wave, the extra branch point of the once subtracted S-wave amplitude $(a_{0}(\nu)-a_{0}(0))/\nu$ becomes a cusp $ C'' \sqrt{\nu}$ when the long range potential is the Van der Waals of the London type, namely $\alpha=6$, where $\nu$ is the center of mass momentum squared. In order to see the cusp as clearly as possible, we must subtract the near-by unitarity cut (in $\nu \geq 0$) and the one-pion exchage cut (in $\nu \leq -\mu^2/4$). By fitting to the cusp, the parameters of the long range potential are determined: $\alpha=6.09$ and $C=0.170$, in which the Compton wave length of the charged pion is used as the unit of the length.