Few-body universality in the deuteron-$\alpha$ system

We treat ${}^6\mathrm{Li}$ as an effective three-body $\left(n\text{−}p\text{−}\alpha \right)$ system and compute the $d\text{−}\alpha S$-wave scattering length and three-body separation energy of ${}^6\mathrm{Li}$ for a wide variety of nucleon-nucleon and $\alpha$-nucleon potentials which have the same (or nearly the same) phase shifts. The Coulomb interaction in the $p\text{−}\alpha$ subsystem is omitted. The results of all calculations lie on a one-parameter curve in the plane defined by the $d\text{−}\alpha S$-wave scattering length and the amount by which ${}^6\mathrm{Li}$ is bound with respect to the $n\text{−}p\text{−}\alpha$ threshold. We argue that these aspects of the $n\text{−}p\text{−}\alpha$ system can be understood using few-body universality and that ${}^6\mathrm{Li}$ can thus usefully be thought of as a two-nucleon halo nucleus.

Figure 1

Absolute value of the three-body separation energy of ${}^6\mathrm{Li}$ as function of the inverse of the $d\text{−}\alpha S$-wave scattering length $a_{d\alpha }$ for phase-shift equivalent interactions obtained by unitarily transforming the interactions in the $n\alpha P_{3/2}$ and $P_{1/2}$ channels. The insert magnifies the marked rectangle and indicates the value $c$ of the exponent in the transformation of Eq. (4). The dashed horizontal line indicates the deuteron breakup threshold.

Figure 2

Absolute value of the three body separation energy of ${}^6\mathrm{Li}$ as a function of $d\text{−}\alpha S\text{−}\mathrm{wave}$ scattering length $a_{d\alpha }$ calculated with a variety of interactions, as described in the text and indicated by the legend. The faint dotted line picks up points from Fig. 1 and is intended to guide the eye.