Scales, Universality and Finite-Range Correction in Three-body Systems

The scale invariance manifested by the weakly-bound Efimov states implies that all the Efimov spectrum can be merged in a single scaling function. By considering this scaling function, the ratio between two consecutive energy levels, ${E_3^{rm (N+1)}}$ and ${E_3^{rm (N)}}$ , can be obtained from a two-body low-energy observable (usually the scattering length a), given in units of the three-body energy level N. The zero-ranged scaling function is improved by incorporating finite range corrections in first order of r ₀/a (r ₀ is the potential effective range). The critical condition for three-identical bosons in s-wave, when the excited ${E_3^{rm (N+1)}}$ state disappears in the 2 + 1 threshold, is given by ${sqrt{E_2/E_3^{rm (N)}} approx 0.38+0.12 ({r_0}/{a})}$ .