×

Precise numerical results for limit cycles in the quantum three-body problem. (English) Zbl 1091.81086

Summary: The study of the three-body problem with short-range attractive two-body forces has a rich history going back to the 1930s. Recent applications of effective field theory methods to atomic and nuclear physics have produced a much improved understanding of this problem, and we elucidate some of the issues using renormalization group ideas applied to precise nonperturbative calculations. These calculations provide 11–12 digits of precision for the binding energies in the infinite cutoff limit. The method starts with this limit as an approximation to an effective theory and allows cutoff dependence to be systematically computed as an expansion in powers of inverse cutoffs and logarithms of the cutoff. Renormalization of three-body bound states requires a short range three-body interaction, with a coupling that is governed by a precisely mapped limit cycle of the renormalization group. Additional three-body irrelevant interactions must be determined to control subleading dependence on the cutoff and this control is essential for an effective field theory since the continuum limit is not likely to match physical systems (e.g., few-nucleon bound and scattering states at low energy). Leading order calculations precise to 11–12 digits allow clear identification of subleading corrections, but these corrections have not been computed.

MSC:

81V70 Many-body theory; quantum Hall effect
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81-04 Software, source code, etc. for problems pertaining to quantum theory
70F07 Three-body problems
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
81V55 Molecular physics
81V35 Nuclear physics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Thomas, L. H., Phys. Rev., 47, 903 (1935)
[2] Efimov, V., Phys. Lett. B, 33, 563 (1970)
[3] Yad. Fiz., 12, 1080 (1970)
[4] Albeverio, S.; Hoegh-Krohn, R.; Wu, T. S., Phys. Lett. A, 83, 105 (1981)
[5] Bedaque, P. F.; Hammer, H.-W.; van Kolck, U., Phys. Rev. Lett., 82, 463 (1999), Available from:
[6] Bedaque, P. F.; Hammer, H.-W.; van Kolck, U., Nucl. Phys. A, 646, 444 (1999), Available from:
[7] Bedaque, P. F.; Hammer, H.-W.; van Kolck, U., Nucl. Phys. A, 676, 357 (2000), Available from:
[8] K.G. Wilson, A limit cycle for three-body short range forces, talk presented at the INT program “Effective Field Theories and Effective Interactions,” Institute for Nuclear Theory, Seattle (2000); unpublished.; K.G. Wilson, A limit cycle for three-body short range forces, talk presented at the INT program “Effective Field Theories and Effective Interactions,” Institute for Nuclear Theory, Seattle (2000); unpublished.
[9] Wilson, K. G., Phys. Rev. D, 3, 1818 (1971)
[10] Glazek, S. D.; Wilson, K. G., Phys. Rev. Lett., 89, 230401 (2002), Available from:
[11] E. Braaten, H.W. Hammer. Available from: <cond-mat/0410417; E. Braaten, H.W. Hammer. Available from: <cond-mat/0410417
[12] Gell-Mann, M.; Low, F. E., Phys. Rev., 95, 1300 (1954)
[13] Phys. Rev. B, 6, 1891 (1972)
[14] R.F. Mohr, Quantum Mechanical Three-Body Problem with Short Range Interactionsnucl-th/0306086; R.F. Mohr, Quantum Mechanical Three-Body Problem with Short Range Interactionsnucl-th/0306086
[15] Birse, M. C.; McGovern, J. A.; Richardson, K. G., Phys. Lett. B, 464, 169 (1999), Available from:
[16] Kaplan, D. B.; Savage, M. J.; Wise, M. B., Phys. Lett. B, 424, 390 (1998), Available from:
[17] Kaplan, D. B.; Savage, M. J.; Wise, M. B., Nucl. Phys. B, 534, 329 (1998), Available from:
[18] Wilson, K. G., Rev. Mod. Phys., 47, 773 (1975)
[19] Jackiw, R., (Ali, A.; Hoodbhoy, P., M.A.B. Beg Memorial Volume (1991), World Scientific: World Scientific Singapore)
[20] Braaten, E.; Hammer, H. W.; Kusunoki, M., Phys. Rev. A, 67, 022505 (2003), Available from:
[21] Barford, T.; Birse, M. C., J. Phys. A, 38, 697 (2005)
[22] Platter, L.; Hammer, H.-W.; Meißner, U.-G., Phys. Rev. A, 70, 052101 (2004)
[23] Tadmor, E., SIAM J. Numer. Anal., 23, 1 (1986)
[24] J. Exptl. Theoret. Phys. (USSR), 40, 498 (1961)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.