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Universality in Few-Body Systems with Large Scattering Length Eric Braatena, H.-W

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Universality in Few-Body Systems with Large Scattering Length Eric Braatena, H.-W Physics Reports 428 (2006) 259–390 www.elsevier.com/locate/physrep Universality in few-body systems with large scattering length Eric Braatena, H.-W. Hammerb,c,∗ aDepartment of Physics, The Ohio State University, Columbus, OH 43210, USA bInstitute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA cHelmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn, Germany Accepted 9 March 2006 editor: W. Weise Abstract Particles with short-range interactions and a large scattering length have universal low-energy properties that do not depend on the details of their structure or their interactions at short distances. In the 2-body sector, the universal properties are familiar and depend only on the scattering length a. In the 3-body sector for identical bosons, the universal properties include the existence of a sequence of shallow 3-body bound states called “Efimov states” and log-periodic dependence of scattering observables on the energy and the scattering length. The spectrum of Efimov states in the limit a →±∞is characterized by an asymptotic discrete scaling symmetry that is the signature of renormalization group flow to a limit cycle. In this review, we present a thorough treatment of universality for the system of three identical bosons and we summarize the universal information that is currently available for other 3-body systems. Our basic tools are the hyperspherical formalism to provide qualitative insights, Efimov’s radial laws for deriving the constraints from unitarity, and effective field theory for quantitative calculations. We also discuss topics on the frontiers of universality, including its extension to systems with four or more particles and the systematic calculation of deviations from universality. © 2006 Published by Elsevier B.V. PACS: 05.70.Jk; 31.15.Ja; 03.70.+k Keywords: Universality; Large scattering length; Renormalization group; Three-body system; Efimov effect; Limit cycle; Discrete scale invariance; Hyperspherical formalism; Radial laws; Effective field theory Contents 1. Introduction ..........................................................................................................261 2. Scattering concepts ....................................................................................................264 2.1. Scattering length ................................................................................................264 2.2. Natural low-energy length scale ...................................................................................266 2.3. Atoms with large scattering length .................................................................................271 2.4. Particles and nuclei with large scattering length ......................................................................274 3. Renormalization group concepts .........................................................................................275 ∗ Corresponding author. Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn, Germany. Tel.: +49 228 732373; fax: +49 228 733728. E-mail addresses: [email protected] (E. Braaten), [email protected] (H.-W. Hammer). 0370-1573/$ - see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.physrep.2006.03.001 260 E. Braaten, H.-W. Hammer / Physics Reports 428 (2006) 259–390 3.1. Efimov effect ...................................................................................................275 3.2. The resonant and scaling limits ....................................................................................276 3.3. Universality in critical phenomena .................................................................................279 3.4. Renormalization group limit cycles ................................................................................281 3.5. Universality for large scattering length ..............................................................................285 4. Universality for two identical bosons .....................................................................................286 4.1. Atom–atom scattering ...........................................................................................286 4.2. The shallow dimer...............................................................................................287 4.3. Continuous scaling symmetry .....................................................................................287 4.4. Scaling violations ...............................................................................................289 4.5. Theoretical approaches ...........................................................................................290 5. Hyperspherical formalism ..............................................................................................291 5.1. Hyperspherical coordinates .......................................................................................291 5.2. Low-energy Faddeev equation .....................................................................................294 5.3. Hyperspherical potentials .........................................................................................297 5.4. Boundary condition at short distances ..............................................................................300 5.5. Efimov states in the resonant limit .................................................................................302 5.6. Efimov states near the atom–dimer threshold ........................................................................305 6. Universality for three identical bosons ....................................................................................306 6.1. Discrete scaling symmetry ........................................................................................306 6.2. Efimov’s radial law ..............................................................................................308 6.3. Binding energies of Efimov states ..................................................................................311 6.4. Atom–dimer elastic scattering .....................................................................................315 6.5. Three-body recombination ........................................................................................319 6.6. Three-atom elastic scattering ......................................................................................322 6.7. Helium atoms ..................................................................................................325 6.8. Universal scaling curves ..........................................................................................327 7. Effects of deep two-body bound states ....................................................................................329 7.1. Extension of Efimov’s radial law ...................................................................................329 7.2. Widths of Efimov states ..........................................................................................332 7.3. Atom–dimer elastic scattering .....................................................................................332 7.4. Three-body recombination into the shallow dimer ....................................................................333 7.5. Three-body recombination into deep molecules ......................................................................335 7.6. Dimer relaxation into deep molecules ..............................................................................336 8. Effective field theory ...................................................................................................338 8.1. Effective field theories ...........................................................................................338 8.2. Effective theories in quantum mechanics ............................................................................339 8.3. Effective field theories for atoms ...................................................................................342 8.4. Two-body problem ..............................................................................................344 8.5. Three-body problem .............................................................................................349 8.6. The diatom field trick ............................................................................................352 8.7. STM integral equation ...........................................................................................354 8.8. Three-body observables ..........................................................................................357 8.9. Renormalization group limit cycle .................................................................................358 8.10. Effects of deep 2-body bound states ................................................................................361 9. Universality in other three-body systems ..................................................................................363 9.1. Unequal scattering lengths ........................................................................................363 9.2. Unequal masses .................................................................................................365 9.3. Two identical fermions ...........................................................................................370 9.4. Particles with a spin symmetry ....................................................................................371 9.5. Dimensions other than 3 ..........................................................................................372
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