We analyze how corrections linear in the effective range, r(0), affect quantities in the three-body sector within an effective field theory for short-range interactions. We demonstrate that observables can be obtained straightforwardly using a perturbative expansion in powers of r(0). In particular, we show that two linear-in-r(0) counterterms are needed for renormalization at this order if scattering-length-dependent observables are considered. We exemplify the implications of this result using various three-body observables. Analytical results for the running of the next-to-leading-order portion of the three-body force in this effective field theory are provided. Expressions which incorporate O(r(0)) corrections and relate the positions of features observed in three-atom recombination near a Feshbach resonance are presented.
We study bosonic systems in the regime in which the two-body system has a shallow bound state or, equivalently, a large value of the two-body scattering length. Using the effective field theory framework as a guidance, we construct a series of potential terms which have a decreasing importance in the description of the binding energy of the systems. The leading-order potential terms consist in a two-body term, usually attractive, plus a three-body term, usually repulsive; this last term is required to prevent the collapse of systems with more than two particles. At this order, the parametrization of the two-body potential is done to obtain a correct description of the scattering length, which governs the dynamics in this regime, whereas the three-body term fixes a three-body datum. We investigate the role of the cutoff in the leading-order description, and we extend the exploration beyond the leading order by including the next-to-leading-order terms in both the two- and three-body potentials. We use the requirement of the stability of the
N-body system, whose energy is variationally estimated, to introduce the three-body forces. The potential parametrization, as a function of the cutoff, is fixed to describe the energy of
4He clusters up to seven particles within the expected accuracy. Finally, we also explore the possibility to describe at the same time the atom-dimer scattering length.
The three- and four-boson systems with a large scattering length and a short effective range in the two-body sector are studied in the framework of Short-Range Effective Field Theory (SREFT). The starting point (leading order) of the EFT is taken to be the universal unitarity limit, where the two-body sector is parameter-free and only one three-body parameter enters. In this limit, physical systems manifest discrete scale invariance. Deviations from universality arising from finite scattering-length and effective-range corrections, as well as a four-body force required by renormalization, are included perturbatively at next-to-leading order. The three-body ground state and its associated four-body ground and first-excited states are studied using the Faddeev-Yakubovsky (FY) formalism and a complementary diagrammatic approach. By employing techniques to remove contributions from deep trimers in tetramer calculations, we extend our analysis to larger cutoffs than previously accessible within the FY approach to SREFT. Our results for binding energies and radii of
4He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of
4He atomic clusters is governed by only small deviations from discrete scale invariance.