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arXiv:1005.1990 · Europhysics Letters (EPL)

Beyond universality in three-body recombination: An effective field theory treatment

C. Ji, Daniel R. Phillips, Lucas Platter

Published Oct 1, 2010• 31 citations
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Abstract

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.

DOI
10.1209/0295-5075/92/13003

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I discuss the impact of a finite effective range, r, on systems with a large two-body scattering length, a. In particular, I show how observables can be written as an expansion around the “universal”, or large-scatteringlength limit. The parameter governing this expansion is the ratio r/a. In few-nucleon systems the ratio r/a has a value of about 1/3, and so such corrections are essential in producing good agreement between theory and data. Hence, I first show how these effects range play a key role in making the so-called “pionless” effective field theory a successful descriptor of low-energy processes in the NN system. I then move to the NNN system, and review predictions for the energy-dependence of observables there. However, the beautiful Efimov physics associated with the presence of a large scattering length is not fully revealed in the NNN system, precisely because r/a corrections are large. I therefore turn to cold atomic gases and show that there are some important recent experiments where physics “beyond universality” affects the data. In the process I demonstrate that an additional piece of short-distance physics is necessary to renormalize scattering-length-dependent observables in the three-body system once corrections ∼ r are considered. Finally, I discuss recent initial efforts to compute r/a corrections to the predictions of universality for the four-body system.

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Universal relations for range corrections to Efimov features

Chen Ji, Eric Braaten, Daniel R. Phillips +1 more

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.

arXiv:1506.02334•Physical Review A•Sep 9, 2015•27 citations
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.
I discuss the impact of a finite effective range, r, on systems with a large two-body scattering length, a. In particular, I show how observables can be written as an expansion around the “universal”, or large-scatteringlength limit. The parameter governing this expansion is the ratio r/a. In few-nucleon systems the ratio r/a has a value of about 1/3, and so such corrections are essential in producing good agreement between theory and data. Hence, I first show how these effects range play a key role in making the so-called “pionless” effective field theory a successful descriptor of low-energy processes in the NN system. I then move to the NNN system, and review predictions for the energy-dependence of observables there. However, the beautiful Efimov physics associated with the presence of a large scattering length is not fully revealed in the NNN system, precisely because r/a corrections are large. I therefore turn to cold atomic gases and show that there are some important recent experiments where physics “beyond universality” affects the data. In the process I demonstrate that an additional piece of short-distance physics is necessary to renormalize scattering-length-dependent observables in the three-body system once corrections ∼ r are considered. Finally, I discuss recent initial efforts to compute r/a corrections to the predictions of universality for the four-body system.
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.
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.
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 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.
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.