Universal range corrections to the Efimov trimer for a class of paths to the unitary limit

Using potential models we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions we have observed that, at first order, it is necessary to supplement the theory with one finite-range parameter, $\Gamma_n^3$, for each specific $n$-level [Kievsky and Gattobigio, Phys. Rev. A {\bf 87}, 052719 (2013)]. The value of $\Gamma_n^3$ depends on the way the potentials is changed to tune the scattering length toward the unitary limit. In this work we analyze a particular path in which the length $r_B=a-a_B$, measuring the difference between the two-body scattering length $a$ and the energy scattering length $a_B$, results almost constant. Analyzing systems with very different scales, as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of $\Gamma_0^3\approx 0.87$ for the ground state level. This observation suggests the possibility of constructing a single universal function that incorporate finite-range effects for this class of paths. The result is used to estimate the three-body parameter $\kappa_*$ in the case of real atomic systems brought to the unitary limit thought a broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation with the two-body contact $C_2$ at the unitary limit.