Nonperturbative regularization and renormalization: Simple examples from nonrelativistic quantum mechanics

Journal Article · Sun Mar 01 04:00:00 EST 1998 · Annals of Physics (New York)

DOI:https://doi.org/10.1006/aphy.1997.5771· OSTI ID:627949

Phillips, D R; Beane, S R; Cohen, T D ^[1]

---

1. Department of Physics, University of Maryland, College Park, Maryland20742-4111 (United States)

We examine several zero-range potentials in nonrelativistic quantum mechanics. The study of such potentials requires regularization and renormalization. We contrast physical results obtained using dimensional regularization and cutoff schemes and show explicitly that in certain cases dimensional regularization fails to reproduce the results obtained using cutoff regularization. First we consider a delta-function potential in arbitrary space dimensions. Using cutoff regularization we show that for d{ge}4 the renormalized scattering amplitude is trivial. In contrast, dimensional regularization can yield a nontrivial scattering amplitude for odd dimensions greater than or equal to five. We also consider a potential consisting of a delta function plus the derivative-squared of a delta function in three dimensions. We show that the renormalized scattering amplitudes obtained using the two regularization schemes are different. Moreover, we find that in the cutoff-regulated calculation the effective range is necessarily negative in the limit that the cutoff is taken to infinity. In contrast, in dimensional regularization the effective range is unconstrained. We discuss how these discrepancies arise from the dimensional regularization prescription that all power-law divergences vanish. We argue that these results demonstrate that dimensional regularization can fail in a nonperturbative setting. {copyright} 1998 Academic Press, Inc.

• Research Organization: University of Maryland
• DOE Contract Number: FG02-93ER40762
• OSTI ID: 627949
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Journal Issue: 2 Vol. 263; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
DELTA FUNCTION
LIPPMANN-SCHWINGER EQUATION
POTENTIAL ENERGY
QUANTUM MECHANICS
RENORMALIZATION
SCATTERING AMPLITUDES