Fall to the centre in atom traps and point-particle EFT for absorptive systems

  • R. PlestidEmail author
  • C. P. Burgess
  • D. H. J. O’Dell
Open Access
Regular Article - Theoretical Physics


Polarizable atoms interacting with a charged wire do so through an inverse-square potential, V = −g/r2. This system is known to realize scale invariance in a nontrivial way and to be subject to ambiguities associated with the choice of boundary condition at the origin, often termed the problem of ‘fall to the center’. Point-particle effective field theory (PPEFT) provides a systematic framework for determining the boundary condition in terms of the properties of the source residing at the origin. We apply this formalism to the charged-wire/polarizable-atom problem, finding a result that is not a self-adjoint extension because of absorption of atoms by the wire. We explore the RG flow of the complex coupling constant for the dominant low-energy effective interactions, finding flows whose character is qualitatively different when g is above or below a critical value, gc. Unlike the self-adjoint case, (complex) fixed points exist when g > gc, which we show correspond to perfect absorber (or perfect emitter) boundary conditions. We describe experimental consequences for wire-atom interactions and the possibility of observing the anomalous breaking of scale invariance.


Effective Field Theories Renormalization Group Nonperturbative Effects 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    L. Landau and E. Lifshitz, Mechanics, Elsevier Science (1982).Google Scholar
  2. [2]
    K.M. Case, Singular potentials, Phys. Rev. 80 (1950) 797 [INSPIRE].
  3. [3]
    A.M. Perelomov and V.S. Popov, Collapse onto scattering centre in quantum mechanics, Teor. Mat. Fiz. 4 (1970) 48 [INSPIRE].CrossRefGoogle Scholar
  4. [4]
    S.P. Alliluev, The problem of collapse to the center in quantum mechanics, JETP 34 (1972) 8.ADSGoogle Scholar
  5. [5]
    R. Jackiw, Delta function potentials in two-dimensional and three-dimensional quantum mechanics, in Diverse topics in theoretical and mathematical physics, World Scientific (1991), pp. 25-42.Google Scholar
  6. [6]
    K.S. Gupta and S.G. Rajeev, Renormalization in quantum mechanics, Phys. Rev. D 48 (1993) 5940 [hep-th/9305052] [INSPIRE].
  7. [7]
    S.R. Beane, P.F. Bedaque, L. Childress, A. Kryjevski, J. McGuire and U. van Kolck, Singular potentials and limit cycles, Phys. Rev. A 64 (2001) 042103 [quant-ph/0010073] [INSPIRE].
  8. [8]
    S.A. Coon and B.R. Holstein, Anomalies in Quantum Mechanics: the 1/r 2 Potential, Am. J. Phys. 70 (2002) 513 [quant-ph/0202091] [INSPIRE].
  9. [9]
    M. Bawin and S.A. Coon, The Singular inverse square potential, limit cycles and selfadjoint extensions, Phys. Rev. A 67 (2003) 042712 [quant-ph/0302199] [INSPIRE].
  10. [10]
    E.J. Mueller and T.-L. Ho, Renormalization Group Limit Cycles in Quantum Mechanical Problems, [cond-mat/0403283].
  11. [11]
    E. Braaten and D. Phillips, The Renormalization group limit cycle for the 1/r 2 potential, Phys. Rev. A 70 (2004) 052111 [hep-th/0403168] [INSPIRE].
  12. [12]
    F. Werner, Trapped cold atoms with resonant interactions: unitary gas and three-body problem, Theses, Université Pierre et Marie Curie — Paris VI, Paris France (2008).Google Scholar
  13. [13]
    D. Bouaziz and M. Bawin, Singular inverse-square potential: renormalization and self-adjoint extensions for medium to weak coupling, Phys. Rev. A 89 (2014) 022113 [arXiv:1402.5325] [INSPIRE].
  14. [14]
    C.P. Burgess, P. Hayman, M. Williams and L. Zalavari, Point-Particle Effective Field Theory I: Classical Renormalization and the Inverse-Square Potential, JHEP 04 (2017) 106 [arXiv:1612.07313] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    C.P. Burgess, P. Hayman, M. Rummel, M. Williams and L. Zalavari, Point-Particle Effective Field Theory II: Relativistic Effects and Coulomb/Inverse-Square Competition, JHEP 07 (2017) 072 [arXiv:1612.07334] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    C.P. Burgess, P. Hayman, M. Rummel and L. Zalavari, Point-Particle Effective Field Theory III: Relativistic Fermions and the Dirac Equation, JHEP 09 (2017) 007 [arXiv:1706.01063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1979) 327 [INSPIRE].
  18. [18]
    C.P. Burgess, Introduction to Effective Field Theory, Ann. Rev. Nucl. Part. Sci. 57 (2007) 329 [hep-th/0701053] [INSPIRE].
  19. [19]
    E. Vogt and G.H. Wannier, Scattering of Ions by Polarization Forces, Phys. Rev. 95 (1954) 1190 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    D.B. Kaplan, J.-W. Lee, D.T. Son and M.A. Stephanov, Conformality Lost, Phys. Rev. D 80 (2009) 125005 [arXiv:0905.4752] [INSPIRE].
  21. [21]
    S. Moroz and R. Schmidt, Nonrelativistic inverse square potential, scale anomaly and complex extension, Annals Phys. 325 (2010) 491 [arXiv:0909.3477] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  22. [22]
    L.D. Landau and L.M. Lifshitz, Quantum Mechanics Non-Relativistic Theory. Volume 3, Third Edition, Butterworth-Heinemann (1981).Google Scholar
  23. [23]
    W.D. Goldberger and I.Z. Rothstein, Dissipative effects in the worldline approach to black hole dynamics, Phys. Rev. D 73 (2006) 104030 [hep-th/0511133] [INSPIRE].ADSMathSciNetGoogle Scholar
  24. [24]
    J. Denschlag, G. Umshaus and J. Schmiedmayer, Probing a Singular Potential with Cold Atoms: A Neutral Atom and a Charged Wire, Phys. Rev. Lett. 81 (1998) 737.ADSCrossRefGoogle Scholar
  25. [25]
    C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press (2002).Google Scholar
  26. [26]
    V. Efimov, Energy levels arising form the resonant two-body forces in a three-body system, Phys. Lett. B 33 (1970) 563 [INSPIRE].
  27. [27]
    E. Braaten and H.W. Hammer, Universality in few-body systems with large scattering length, Phys. Rept. 428 (2006) 259 [cond-mat/0410417] [INSPIRE].
  28. [28]
    N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].
  29. [29]
    W.-J. Li and J.-P. Wu, Holographic fermions in charged dilaton black branes, Nucl. Phys. B 867 (2013) 810 [arXiv:1203.0674] [INSPIRE].
  30. [30]
    B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proc. Nat. Acad. Sci. 74 (1977) 1765.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    M.J. Gander and L. Halpern, Absorbing boundary conditions for the wave equation and parallel computing, Math. Comput. 74 (2005) 153.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    H.E. Camblong and C.R. Ordonez, Anomaly in conformal quantum mechanics: From molecular physics to black holes, Phys. Rev. D 68 (2003) 125013 [hep-th/0303166] [INSPIRE].
  33. [33]
    H.W. Hammer and B.G. Swingle, On the limit cycle for the 1/r 2 potential in momentum space, Annals Phys. 321 (2006) 306 [quant-ph/0503074] [INSPIRE].
  34. [34]
    S. Weinberg, Nuclear forces from chiral Lagrangians, Phys. Lett. B 251 (1990) 288 [INSPIRE].
  35. [35]
    D.B. Kaplan, M.J. Savage and M.B. Wise, Nucleon-nucleon scattering from effective field theory, Nucl. Phys. B 478 (1996) 629 [nucl-th/9605002] [INSPIRE].
  36. [36]
    T. Mehen and I.W. Stewart, A Momentum subtraction scheme for two nucleon effective field theory, Phys. Lett. B 445 (1999) 378 [nucl-th/9809071] [INSPIRE].
  37. [37]
    S.K. Adhikari, Quantum scattering in two dimensions, Am. J. Phys. 54 (1986) 362.ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    K. Meetz, Singular potentials in nonrelativistic quantum mechanics, Nuovo Cim. 34 (1964) 690.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    A.C. Fonseca, E.F. Redish and P.E. Shanley, Efimov effect in an analytically solvable mode, Nucl. Phys. A 320 (1979) 273 [INSPIRE].
  40. [40]
    T. Kraemer et al., Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature 440 (2006) 315.ADSCrossRefGoogle Scholar
  41. [41]
    E. Braaten and H.W. Hammer, Efimov Physics in Cold Atoms, Annals Phys. 322 (2007) 120 [cond-mat/0612123] [INSPIRE].
  42. [42]
    L. Platter, Few-Body Systems and the Pionless Effective Field Theory, in proceedings of the 6th International Workshop on Chiral Dynamics (CD09), Bern, Switzerland, 6-10 July 2009, p. 104 [PoS(CD09)104] [arXiv:0910.0031] [INSPIRE].
  43. [43]
    H.W. Hammer and L. Platter, Efimov states in nuclear and particle physics, Ann. Rev. Nucl. Part. Sci. 60 (2010) 207 [arXiv:1001.1981] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    H.W. Hammer and L. Platter, Efimov physics from a renormalization group perspective, Philos. Trans. Roy. Soc. Lond. A 369 (2011) 2679.Google Scholar
  45. [45]
    D.J. MacNeill and F. Zhou, Pauli blocking effect on Efimov states near a feshbach resonance, Phys. Rev. Lett. 106 (2011) 145301.ADSCrossRefGoogle Scholar
  46. [46]
    R. Grimm, M. Weidemüller and Y.B. Ovchinnikov, Optical dipole traps for neutral atoms, Adv. At. Mol. Opt. Phys. 42 (2000) 95.ADSCrossRefGoogle Scholar
  47. [47]
    R. Plestid, C. Burgess and D.H.J. O’Dell, Tunable quantum anomaly with cold atoms in an inverse square potential, in preparation.Google Scholar
  48. [48]
    C.P. Burgess, P. Hayman, M. Rummel and L. Zalavari, Reduced Theoretical Error for QED Tests with 4 He + Spectroscopy, arXiv:1708.09768 [INSPIRE].
  49. [49]
    J. Sakurai, Modern Quantum Mechanics, Addison-Wesely (1988).Google Scholar
  50. [50]
    F.W.J. Olver and National Institute of Standards and Technology (U.S.), NIST Handbook of Mathematical Functions, Cambridge University Press (2010).Google Scholar
  51. [51]
    W.D. Goldberger and I.Z. Rothstein, Dissipative effects in the worldline approach to black hole dynamics, Phys. Rev. D 73 (2006) 104030 [hep-th/0511133] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • R. Plestid
    • 1
    • 2
    Email author
  • C. P. Burgess
    • 1
    • 2
  • D. H. J. O’Dell
    • 1
  1. 1.Department of Physics & AstronomyMcMaster UniversityHamiltonCanada
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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