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arXiv:1111.6961 · Physical Review A

Impossibility of the Efimov effect for p -wave interactions

Yusuke Nishida

Published Jul 26, 2012• 44 citations
arXivHTMLPDF

Abstract

Whether the Efimov effect is possible, in principle, for $p$-wave or higher partial-wave interactions is a fundamental question. Recently, there has been a claim that three nonrelativistic particles with resonant $p$-wave interactions exhibit the Efimov effect. We point out that the assumed $p$-wave scattering amplitude inevitably causes a negative probability. This indicates that the Efimov states found there cannot be realized in physical situations. We also restate our previous argument that the Efimov effect, defined as an infinite tower of universal bound states characterized by discrete scale invariance, is impossible for $p$-wave or higher partial-wave interactions.

DOI
10.1103/physreva.86.012710

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Whether the Efimov effect is possible, in principle, for ppp-wave or higher partial-wave interactions is a fundamental question. Recently, there has been a claim that three nonrelativistic particles with resonant ppp-wave interactions exhibit the Efimov effect. We point out that the assumed ppp-wave scattering amplitude inevitably causes a negative probability. This indicates that the Efimov states found there cannot be realized in physical situations. We also restate our previous argument that the Efimov effect, defined as an infinite tower of universal bound states characterized by discrete scale invariance, is impossible for ppp-wave or higher partial-wave interactions.
The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle SSS matrix in the complex energy plane, and we study how they transform from virtual states through bound states to resonances when we change the interaction strength. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings---a behavior known from the nonrelativistic case. We additionally find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this ``missing states'' problem.
Resonant ppp-wave interactions can be described by a minimal zero-range model defined by a truncated effective range expansion so that the only two-body interaction parameters are the inverse scattering volume 1/ap1/{a}_{p}1/ap​ and the ppp-wave effective range rp{r}_{p}rp​. This minimal model can be formulated as a local quantum field theory with a ppp-wave interaction between atom fields and a molecular field. In the two-atom sector, the model is renormalizable, but it has unphysical behavior at high energies because there are negative-probability states with momentum scale rp{r}_{p}rp​. In the sector with three atoms, two of which are identical, renormalization in some parity and angular momentum channels involves an ultraviolet limit cycle, indicating asymptotic discrete scale invariance. The Efimov effect occurs in the unitary limit ${a}_{p}^{\ensuremath{-}1/3},{r}_{p}\ensuremath{\rightarrow}0$, but this limit is unphysical because there are low-energy states with negative probability. The minimal model can be of physical relevance only at energies small compared to the energy scale set by rp{r}_{p}rp​ where the effects of negative-probability states are suppressed.
Contact psuedopotentials for relative angular momentum greater than zero are of interest for the study of cold atomic gases. For bosons, it is known that when the s-wave scattering length becomes infinite, an infinite number of three-body bound states, called Efimov states, are predicted by such potentials. Realistic potentials also exhibit the such states, thus a study of the Efimov effect for contact psuedopotentials for higher partial waves and fermions is important for the study of cold atoms. In this Letter we analyze three-body states of three particles interacting via psuedopotentials for higher partial waves and show that there is an Efimov effect for three fermions interacting via p-wave psuedopotentials.
The Efimov effect is the appearance of an infinite tower of bound states, E n 3 , that accumulate in a geometrical way at zero energy in systems of three-identical bosons having an infinite large two-body scattering length a . In the zero-range limit the geometrical factor is E n +1 3 / E n 3 = e −2π/ s 0 with s 0 ≈ 1.00624 an universal number. The Efimov effect is the signature of the discrete scale invariance (DSI) present in these systems close to the unitary limit (1/ a = 0). The DSI constrains also the spectrum in N > 3 systems. In this contribution the two-level tree structure observed in bosons systems with N ≥ 4 will be discussed using potential models.
We consider the system of three nonrelativistic spinless fermions in two dimensions, which interact through spherically-symmetric pair interactions. Recently, a claim has been made by Nishida et al for the existence of the so-called super Efimov effect by Nishida et al (2013 Phys. Rev. Lett. 110 235301 ). Namely, if the interactions in the system are fine-tuned to a p-wave resonance, an infinite number of bound states appears, whose negative energies are scaled according to the double exponential law. We present the mathematical proof that such a system indeed has an infinite number of bound levels. We also prove that , where N ( E ) is the number of bound states with the energy less than . The value of this limit is exactly equal to the value derived in Nishida et al using the renormalization group approach. Our proof resolves a recent controversy about the validity of results in Nishida et al (2013 Phys. Rev. Lett. 110 235301 ).
Particles with resonant short-range interactions have universal properties that do not depend on the details of their structure or their interactions at short distances. In the three-body system, these properties include the existence of a geometric spectrum of three-body Efimov states and a discrete scaling symmetry, which leads to log-periodic dependence of observables on the scattering length. Similar universal properties appear in the four-body system and possibly higher-body systems as well. For example, universal four-body states have recently been predicted and observed in experiments. These phenomena are often referred to as Efimov physics. We review their theoretical description and discuss applications in different areas of physics with a special emphasis on nuclear and particle physics.