Force-range correction in the three-body problem: Application to three-nucleon systems

A leading correction to the zero-range theory is considered for three-body systems. The correction is linear in the force range ${\mathit{r}}_0$. An explicit expression is obtained for this correction for the case of correlation between the three-body binding energy and the particle-pair scattering length, as well as for the practically interesting case of correlation between the triton binding energy and the neutron-deuteron doublet scattering length. The correction for the neutron-deuteron quartet scattering length is also found. Physics of the correction involves a modification of the effective long-range interaction 1/${\mathit{R}}^2$, which arises in the three-body systems under the conditions of the zero-range theory, by a singular correction ${\mathit{r}}_0$/${\mathit{R}}^3$.