Universality of three identical bosons with large, negative effective range

Resummed-Range Effective Field Theory is a consistent nonrelativistic Effective Field Theory of contact interactions with large scattering length a and an effective range r(0) large in magnitude but negative. Its leading order is non-perturbative, and its observables are universal in the sense that they depend only on the dimensionless ratio xi:=2r(0)/a once the overall distance scale is set by |r(0)|. In the two-body sector, the relative position of the two shallow S-wave poles in the complex plane is determined by xi. We investigate three identical bosons at leading order for a two-body system with one bound and one virtual state (xi <= 0), or with two virtual states (0 <= xi < 1). Such conditions might, for example, be found in systems of heavy mesons. We find that no three-body interaction is needed to renormalise (and stabilise) the leading order. A well-defined ground state exists for 0.366 & mldr;>= xi >= -8.72 & mldr;. Three-body excitations appear for even smaller ranges of xi around the quasi-unitarity point xi=0 (|r(0)|<<|a|->infinity) and obey discrete scaling relations. We explore in detail the ground state and the lowest three excitations. We parametrise their trajectories as function of xi and of the binding momentum k2(-) of the shallowest 2B state. These stretch from the point where three- and two-body binding energies are identical to the point of zero three-body binding. As |r(0)|<<|a| becomes perturbative, this version turns into the Short-Range EFT which needs a stabilising three-body interaction and exhibits Efimov's Discrete Scale Invariance. By interpreting that EFT as a low-energy version of Resummed-Range EFT, we match spectra to determine Efimov's scale-breaking parameter Lambda & lowast; in a renormalisation scheme with a hard cutoff. Finally, we compare phase shifts for scattering a boson on the two-boson bound state with that of the equivalent Efimov system.

Automated summary

Resummed-Range Effective Field Theory is a nonrelativistic Effective Field Theory that investigates three identical bosons in a two-body system with one bound and one virtual state, or with two virtual states. It is found that no three-body interaction is needed to renormalize and stabilize the system at leading order, and a well-defined ground state exists in certain parameter ranges. Three-body excitations appear near the quasi-unitarity point and obey discrete scaling relations. The trajectories of the ground state and lowest three excitations are explored, as well as the discrete scale invariance in the Short-Range EFT.