Connection between effective-range expansion and nuclear vertex constant or asymptotic normalization coefficient

Abstract

Explicit relations between the effective-range expansion and the nuclear vertex constant or asymptotic normalization coefficient (ANC) for the virtual decay $B\ensuremath{\rightarrow}A+a$ are derived for an arbitrary orbital momentum together with the corresponding location condition for the ($A+a$) bound-state energy. They are valid both for the charged case and for the neutral case. Combining these relations with the standard effective-range function up to order six makes it possible to reduce to two the number of free effective-range parameters if an ANC value is known from experiment. Values for the scattering length, effective range, and form parameter are determined in this way for the ${}^{16}\mathrm{O}+p$, $\ensuremath{\alpha}+t$, and $\ensuremath{\alpha}+{}^{3}\mathit{He}$ collisions in partial waves where a bound state exists by using available ANCs deduced from experiments. The resulting effective-range expansions for these collisions are valid up to energies larger than 5 MeV.