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Using potential models we analyze range corrections to the universal lawdictated by the Efimov theory of three bosons. In the case of finite-rangeinteractions we have observed that, at first order, it is necessary tosupplement the theory with one finite-range parameter, $\Gamma_n^3$, for eachspecific $n$-level [Kievsky and Gattobigio, Phys. Rev. A {\bf 87}, 052719(2013)]. The value of $\Gamma_n^3$ depends on the way the potentials is changedto tune the scattering length toward the unitary limit. In this work we analyzea particular path in which the length $r_B=a-a_B$, measuring the differencebetween 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 parameterremains almost constant along the path with a numerical value of$\Gamma_0^3\approx 0.87$ for the ground state level. This observation suggeststhe possibility of constructing a single universal function that incorporatefinite-range effects for this class of paths. The result is used to estimatethe three-body parameter $\kappa_*$ in the case of real atomic systems broughtto the unitary limit thought a broad Feshbach resonances. Furthermore, we showthat the finite-range parameter can be put in relation with the two-bodycontact $C_2$ at the unitary limit.

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