# 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:

- Institute for Nuclear Theory, University of Washington, Seattle, WA 98195 (United States). E-mail: hammer@phys.washington.edu
- Institute for Nuclear Theory, University of Washington, Seattle, WA 98195 (United States)
- (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}

}