Universality of three identical bosons with large, negative effective range

Abstract

“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 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 ξ: = 2 r/ a once the overall distance scale is set by | r| . In the two-body sector, the relative position of the two shallow S-wave poles in the complex plane is determined by ξ . We investigate three identical bosons at leading order for a two-body system with one bound and one virtual state (ξ≤ 0), or with two virtual states (0 ≤ ξ< 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 … ≥ ξ≥ - 8.72 … . Three-body excitations appear for even smaller ranges of ξ around the “quasi-unitarity point” ξ= 0 (| r| ≪ | a| → ∞) and obey discrete scaling relations. We explore in detail the ground state and the lowest three excitations. We parametrise their trajectories as function of ξ and of the binding momentum κ2- of the shallowest 2 B 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| ≪ | 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 Λ ∗ 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.