$N$-boson spectrum from a Discrete Scale Invariance

We present the analysis of the $N$-boson spectrum computed using a soft two-body potential the strength of which has been varied in order to cover an extended range of positive and negative values of the two-body scattering length $a$ close to the unitary limit. The spectrum shows a tree structure of two states, one shallow and one deep, attached to the ground-state of the system with one less particle. It is governed by an unique universal function, $Delta(xi)$, already known in the case of three bosons. In the three-particle system the angle $xi$, determined by the ratio of the two- and three-body binding energies $E_3/E_2=tan^2xi$, characterizes the Discrete Scale Invariance of the system. Extending the definition of the angle to the $N$-body system as $E_N/E_2=tan^2xi$, we study the $N$-boson spectrum in terms of this variable. The analysis of the results, obtained for up to $N=16$ bosons, allows us to extract a general formula for the energy levels of the system close to the unitary limit. Interestingly, a linear dependence of the universal function as a function of $N$ is observed at fixed values of $a$. We show that the finite-range nature of the calculations results in the range corrections that generate a shift of the linear relation between the scattering length $a$ and a particular form of the universal function. We also comment on the limits of applicability of the universal relations.

Rapid convergence of the Weinberg expansion of the deuteron stripping amplitude

Theories of $(d,p)$ reactions frequently use a formalism based on a transition amplitude that is dominated by the components of the total three-body scattering wave function where the spatial separation between the incoming neutron and proton is confined by the range of the $n$-$p$ interaction, $V_{np}$. By comparison with calculations based on the CDCC method we show that the $(d,p)$ transition amplitude is dominated by the first term of the expansion of the three-body wave function in a complete set of Weinberg states. We use the uc{132}{Sn}(d,p) uc{133} {Sn} reaction at 30 and 100 MeV as examples of contemporary interest. The generality of this observed dominance and its implications for future theoretical developments are discussed.