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 $r_0$ large in magnitude but negative. Its leading order is non-perturbative. Its observables are universal, i.e.~they depend only on the dimensionless ratio $\xi:=2r_0/a$, with the overall distance scale set by $|r_0|$. In the two-body sector, the 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 ($\xi\le0$), or with two virtual states ($0\le\xi<1$). 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) Resummed-Range EFT at LO. A well-defined ground state exists for $0.366\ldots\le\xi\le-8.72\ldots$. Three-body excitations appear for even smaller ranges of $\xi$ around the ``quasi-unitarity point'' $\xi=0$ ($|r_0|\ll|a|\to\infty$) and obey discrete scaling relations. We explore in detail the ground state and the lowest three excitations and parametrise their trajectories as function of $\xi$ and of the binding momentum $\kappa_2^-$ of the shallowest \twoB state from where three-body and two-body binding energies are identical to zero three-body binding. As $|r_0|\ll|a|$ 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 $\Lambda_*$ 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.