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arXiv:1506.02334 · Physical Review A

Universal relations for range corrections to Efimov features

Chen Ji, Eric Braaten, Daniel R. Phillips, Lucas Platter

Published Sep 9, 2015• 27 citations
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Abstract

In a three-body system of identical bosons interacting through a large $S$-wave scattering length $a$, there are several sets of features related to the Efimov effect that are characterized by discrete scale invariance. Effective field theory was recently used to derive universal relations between these Efimov features that include the first-order correction due to a nonzero effective range ${r}_{s}$. We reveal a simple pattern in these range corrections that had not been previously identified. The pattern is explained by the renormalization group for the effective field theory, which implies that the Efimov three-body parameter runs logarithmically with the momentum scale at a rate proportional to ${r}_{s}/a$. The running Efimov parameter also explains the empirical observation that range corrections can be largely taken into account by shifting the Efimov parameter by an adjustable parameter divided by $a$. The accuracy of universal relations that include first-order range corrections is verified by comparing them with various theoretical calculations using models with nonzero range.

DOI
10.1103/physreva.92.030702

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In a three-body system of identical bosons interacting through a large SSS-wave scattering length aaa, there are several sets of features related to the Efimov effect that are characterized by discrete scale invariance. Effective field theory was recently used to derive universal relations between these Efimov features that include the first-order correction due to a nonzero effective range rs{r}_{s}rs​. We reveal a simple pattern in these range corrections that had not been previously identified. The pattern is explained by the renormalization group for the effective field theory, which implies that the Efimov three-body parameter runs logarithmically with the momentum scale at a rate proportional to rs/a{r}_{s}/ars​/a. The running Efimov parameter also explains the empirical observation that range corrections can be largely taken into account by shifting the Efimov parameter by an adjustable parameter divided by aaa. The accuracy of universal relations that include first-order range corrections is verified by comparing them with various theoretical calculations using models with nonzero range.
A nonrelativistic system of three identical particles will display a rich set of universal features known as Efimov physics if the scattering length aaa is much larger than the range lll of the underlying two-body interaction. An appropriate effective theory facilitates the derivation of both results in the $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit and finite-l∕al∕al∕a corrections to observables of interest. Here we use such an effective-theory treatment to consider the impact of corrections linear in the two-body effective range, rs{r}_{s}rs​, on the three-boson bound-state spectrum and recombination rate for $\ensuremath{\mid}a\ensuremath{\mid}⪢\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$. We do this by first deriving results appropriate to the strict limit $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ in coordinate space. We then extend these results to finite aaa using once-subtracted momentum-space integral equations. We take the cutoff on these equations to be large compared to $1∕\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$, and find that the first-order effects of $\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$ are independent of the cutoff in this regime. We also discuss the implications of our results for experiments that probe three-body recombination in Bose-Einstein condensates near a Feshbach resonance.
Using potential models, we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions, we have observed that at first order, it is necessary to supplement the theory with one finite-range parameter ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ for each specific nnn level [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ depends on the way the potentials are changed to tune the scattering length toward the unitary limit. In this work, we analyze a particular path in which the length ${r}_{B}=a\ensuremath{-}{a}_{B}$, measuring the difference between the two-body scattering length aaa and the energy-scattering length aB{a}_{B}aB​, is almost constant. Analyzing systems with very different scales, such as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of ${\mathrm{\ensuremath{\Gamma}}}_{0}^{3}\ensuremath{\approx}0.87$ for the ground-state level. This observation suggests the possibility of constructing a single universal function that incorporates finite-range effects for this class of paths. The result is used to estimate the three-body parameter ${\ensuremath{\kappa}}_{*}$ in the case of real atomic systems brought to the unitary limit through broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation to the two-body contact C2{C}_{2}C2​ at the unitary limit.
We discuss the impact of a finite effective range on three-body systems interacting through a large two-body scattering length. By employing a perturbative analysis in an effective field theory well suited to this scale hierarchy we find that an additional three-body parameter is required for consistent renormalization once range corrections are considered. This allows us to extend previously discussed universal relations between different observables in the recombination of cold atoms to account for the presence of a finite effective range. We show that such range corrections allow us to simultaneously describe the positive and negative scattering-length loss features observed in recombination with 7 Li atoms by the Bar-Ilan group. They do not, however, significantly reduce the disagreement between the universal relations and the data of the Rice group on 7 Li recombination at positive and negative scattering lengths.
Resummed-Range Effective Field Theory is the consistent non-relativistic Effective Field Theory of point interactions in systems with large two-body scattering length aaa and an effective range r0r_0r0​ large in magnitude but negative. Its leading order is non-perturbative, and its observables depend only on the dimensionless ratio ξ:=2r0/aξ:=2r_0/aξ:=2r0​/a once ∣r0∣|r_0|∣r0​∣ is chosen as base unit. This presentation highlights aspects for three identical spinless bosons and adds details to a previous discussion [1]. At leading order, no three-body interaction is needed. A ground state exists only in the range 0.366…≥ξ≥−8.72…0.366\ldots\geξ\ge-8.72\ldots0.366…≥ξ≥−8.72…, and excited states display self-similarity and Discrete Scale Invariance, with small corrections for nonzero r0r_0r0​.
We use an effective-field-theory framework to analyze the Efimov effect in heteronuclear three-body systems consisting of two species of atoms with a large interspecies scattering length. In the leading-order description of this theory, various three-body observables in heteronuclear mixtures can be universally parametrized by one three-body parameter. We present the next-to-leading corrections, which include the effects of the finite interspecies effective range and the finite intraspecies scattering length, to various three-body observables. We show that only one additional three-body parameter is required to render the theory predictive at this order. By including the effective range and intraspecies scattering length corrections, we derive a set of universal relations that connect the different Efimov features near the interspecies Feshbach resonance. Furthermore, we show that these relations can be interpreted in terms of the running of the three-body counterterms that naturally emerge from proper renormalization. Finally, we make predictions for recombination observables of a number of atomic systems that are of experimental interest.
In the low-energy limit, non-relativistic particles with short-range interactions exhibit universal behavior that is largely independent of microscopic details. This universality is typically described by effective field theory, in which the two-body interaction is renormalized to a single parameter-the scattering length. For systems of identical bosons, the three-body problem reveals the Efimov effect, a novel phenomenon proposed that necessitates the introduction of an additional three-body parameter. However, the exact relation between this three-body parameter, the coupling constants in the effective field theory, and the binding energies of Efimov states remains unresolved. In this Letter, we address this question through a comprehensive analysis of the Skorniakov-Ter-Martirosian equation with a finite cutoff. We establish an exact renormalization relation for the three-body parameter and determine its connection to the energies of Efimov bound states. These results are validated through high-precision numerical simulations. We expect our findings to be of fundamental interest across various fields, including atomic, nuclear, condensed matter, and particle physics, and to have broad applications in both few-body and many-body physics.