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Title: On the limit cycle for the 1/r {sup 2} potential in momentum space

Abstract

The renormalization of the attractive 1/r {sup 2} potential has recently been studied using a variety of regulators. In particular, it was shown that renormalization with a square well in position space allows multiple solutions for the depth of the square well, including, but not requiring a renormalization group limit cycle. Here, we consider the renormalization of the 1/r {sup 2} potential in momentum space. We regulate the problem with a momentum cutoff and absorb the cutoff dependence using a momentum-independent counterterm potential. The strength of this counterterm is uniquely determined and runs on a limit cycle. We also calculate the bound state spectrum and scattering observables, emphasizing the manifestation of the limit cycle in these observables.

Authors:
 [1];  [2];  [3]
  1. Institute for Nuclear Theory, University of Washington, Seattle, WA 98195 (United States). E-mail: hammer@phys.washington.edu
  2. Institute for Nuclear Theory, University of Washington, Seattle, WA 98195 (United States)
  3. (United States)
Publication Date:
OSTI Identifier:
20766983
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics (New York); Journal Volume: 321; Journal Issue: 2; Other Information: DOI: 10.1016/j.aop.2005.04.017; PII: S0003-4916(05)00066-7; Copyright (c) 2005 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOUND STATE; LIMIT CYCLE; MATHEMATICAL SOLUTIONS; POTENTIALS; RENORMALIZATION; SCATTERING; SPECTRA

Citation Formats

Hammer, H.-W., Swingle, Brian G., and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332. On the limit cycle for the 1/r {sup 2} potential in momentum space. United States: N. p., 2006. Web. doi:10.1016/j.aop.2005.04.017.
Hammer, H.-W., Swingle, Brian G., & School of Physics, Georgia Institute of Technology, Atlanta, GA 30332. On the limit cycle for the 1/r {sup 2} potential in momentum space. United States. doi:10.1016/j.aop.2005.04.017.
Hammer, H.-W., Swingle, Brian G., and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332. Wed . "On the limit cycle for the 1/r {sup 2} potential in momentum space". United States. doi:10.1016/j.aop.2005.04.017.
@article{osti_20766983,
title = {On the limit cycle for the 1/r {sup 2} potential in momentum space},
author = {Hammer, H.-W. and Swingle, Brian G. and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332},
abstractNote = {The renormalization of the attractive 1/r {sup 2} potential has recently been studied using a variety of regulators. In particular, it was shown that renormalization with a square well in position space allows multiple solutions for the depth of the square well, including, but not requiring a renormalization group limit cycle. Here, we consider the renormalization of the 1/r {sup 2} potential in momentum space. We regulate the problem with a momentum cutoff and absorb the cutoff dependence using a momentum-independent counterterm potential. The strength of this counterterm is uniquely determined and runs on a limit cycle. We also calculate the bound state spectrum and scattering observables, emphasizing the manifestation of the limit cycle in these observables.},
doi = {10.1016/j.aop.2005.04.017},
journal = {Annals of Physics (New York)},
number = 2,
volume = 321,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2006},
month = {Wed Feb 15 00:00:00 EST 2006}
}
  • Previous work has shown that if an attractive 1/r{sup 2} potential is regularized at short distances by a spherical square-well potential, renormalization allows multiple solutions for the depth of the square well. The depth can be chosen to be a continuous function of the short-distance cutoff R, but it can also be a log-periodic function of R with finite discontinuities, corresponding to a renormalization-group (RG) limit cycle. We consider the regularization with a {delta}-shell potential. In this case, the coupling constant is uniquely determined to be a log-periodic function of R with infinite discontinuities, and an RG limit cycle ismore » unavoidable. In general, a regularization with an RG limit cycle is selected as the correct renormalization of the 1/r{sup 2} potential by the conditions that the cutoff radius R can be made arbitrarily small and that physical observables are reproduced accurately at all energies much less than ({Dirac_h}/2{pi}){sup 2}/mR{sup 2}.« less
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  • The 80-cm diameter liquid-hydrogen bubble chamber at our institute, exposed in a separated beam of {sup 3}He nuclei with momentum 5 GeV/{ital c}, has been used to study for the first time the reaction {sup 3}He{ital p}{r arrow}2{ital p}2{ital n}{pi}{sup +}. A total of 3509 events of this reaction were selected, which corresponds to a cross section 14.4{plus minus}0.2 mb. On the basis of the pole model with interaction of the spectator nucleons in the final state in the complete phase space a study has been made of the angular, mass, and momentum spectra. From analysis of the effective massmore » of the {ital p}{pi}{sup +} system it follows that the {Delta}{sup ++} isobar is formed in the reaction {sup 3}He{ital p}{r arrow}2{ital p}2{ital n}{pi}{sup +} in peripheral quasifree {ital pN} collisions without broadening. In the reaction studied on indications were obtained of the formation of two-baryon or three-baryon resonances.« less
  • Predictions for the ionization energies (I.E.s) of van der Waals dimers R[sub 1]R[sub 2] and the bond dissociation energies ([ital D][sub 0]'s) of the dimer ions R[sub 1]R[sub 2][sup +] (R[sub 1], R[sub 2]=He, Ne, Ar, and Kr) have been calculated using the [ital ab] [ital initio] Gaussian-2 theoretical procedure. Despite only fair agreements observed between theoretical and experimental geometries, the theoretical I.E.s and [ital D][sub 0]'s are found to be in excellent agreement with available experimental findings.