Renormalization of the Strongly Attractive Inverse Square Potential: Taming the Singularity

Alhaidari, A. D.

doi:10.1007/s10701-014-9828-7

Cited by 10 publications

(2 citation statements)

References 32 publications

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“…However, physics with the Manning-Rosen barrier is quite distinct from that of the Pöschl-Teller barrier. The Manning-Rosen barrier, singular at origin, does not allow for transmission [45], [46], [47] and no duality can be constructed between the bound states within the csc 2 χ well (identical to those of the sec 2 χ well) and states residing in Lobachevsky's hyperbolic space time. On the latter, no confinement can be defined in terms of quantum motion along geodesics, at most, it could be defined as motion on closed space like conics.…”

Section: Discussionmentioning

confidence: 99%

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space-time dS4

Kirchbach¹,

Compean

2016

Eur. Phys. J. A

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The real parts of the complex squared energies defined by the resonance poles of the transfer matrix of the Pöschl-Teller barrier, are shown to equal the squared energies of the levels bound within the trigonometric Scarf well potential. By transforming these potentials into parts of the Laplacians describing free quantum motions on the mutually orthogonal open time-like hyperbolic-, and closed space-like spherical geodesics on the conformally invariant de Sitter space time, dS4, the conformal symmetries of these interactions are revealed. On dS4 the potentials under consideration naturally relate to interactions within colorless two-body systems and to cusped Wilson loops. In effect, with the aid of the dS4 space-time as unifying geometry, a conformal symmetry based bijective correspondence (duality) between bound and resonant meson spectra is established at the quantum mechanics level and related to confinement understood as color charge neutrality. The correspondence allows to link the interpretation of mesons as resonance poles of a scattering matrix with their complementary description as states bound by an instantaneous quark interaction and to introduce a conformal symmetry based classification scheme of mesons. As examples representative of such a duality we organize in good agreement with data 71 of the reported light flavor mesons with masses below ∼ 2350 MeV into four-conformal families of particles placed on linear f0, π, η, and a0 resonance trajectories, plotted on the ℓ/M plane. Upon extending the sec 2 χ by a properly constructed conformal color dipole potential, shaped after a tangent function, we predict the masses of 12 "missing" mesons. We furthermore notice that the f0 and π trajectories can be viewed as chiral partners, same as the η and a0 trajectories, an indication that chiral symmetry for mesons is likely to be realized in terms of parity doubled conformal multiplets rather than, as usually assumed, only in terms of parity doubled single SO(3) states. We attribute the striking measured meson degeneracies to conformal symmetry dynamics within color neutral two-body systems, and conclude on the usefulness of the de Sitter space-time dS4 as a tool for modelling strong interactions, on the one side, and on the relevance of hyperbolic and trigonometric potentials in constituent quark models of hadrons, on the other.

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Section: Discussionmentioning

confidence: 99%

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space-time dS4

Kirchbach¹,

Compean

2016

Eur. Phys. J. A

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“…Landau and Lifshitz associated the occurrence of these infinite bound states to the classical "particle fall to the center" [15,16]. These anomalies could be resolved using various methods of regularization, most of them without unique results [13,14,17]. In our problem, the inverse square component of the effective potential is  …”

Section: 5mentioning

confidence: 99%

Bound states and the potential parameter spectrum

Alhaidari

Bahlouli

2020

Journal of Mathematical Physics

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In this article, we answer the following question: If the wave equation possesses bound states but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the "tridiagonal representation approach" to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy we show that one of the potential parameters must assume a value from within a discrete set called the "potential parameter spectrum" (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite discrete set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.

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Point-particle effective field theory I: classical renormalization and the inverse-square potential

Burgess

Hayman

Williams

et al. 2017

J. High Energ. Phys.

109

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Singular potentials (the inverse-square potential, for example) arise in many situations and their quantum treatment leads to well-known ambiguities in choosing boundary conditions for the wave-function at the position of the potential's singularity. These ambiguities are usually resolved by developing a self-adjoint extension of the original problem; a non-unique procedure that leaves undetermined which extension should apply in specific physical systems. We take the guesswork out of this picture by using techniques of effective field theory to derive the required boundary conditions at the origin in terms of the effective point-particle action describing the physics of the source. In this picture ambiguities in boundary conditions boil down to the allowed choices for the source action, but casting them in terms of an action provides a physical criterion for their determination. The resulting extension is self-adjoint if the source action is real (and involves no new degrees of freedom), and not otherwise (as can also happen for reasonable systems). We show how this effective-field picture provides a simple framework for understanding well-known renormalization effects that arise in these systems, including how renormalization-group techniques can resum non-perturbative interactions that often arise, particularly for non-relativistic applications. In particular we argue why the low-energy effective theory tends to produce a universal RG flow of this type and describe how this can lead to the phenomenon of reaction catalysis, in which physical quantities (like scattering cross sections) can sometimes be surprisingly large compared to the underlying scales of the source in question. We comment in passing on the possible relevance of these observations to the phenomenon of the catalysis of baryon-number violation by scattering from magnetic monopoles.

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Renormalization of the Strongly Attractive Inverse Square Potential: Taming the Singularity

Cited by 10 publications

References 32 publications

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space-time dS4

Modelling duality between bound and resonant meson spectra by means of free quantum motions on the de Sitter space-time dS4

Bound states and the potential parameter spectrum

Point-particle effective field theory I: classical renormalization and the inverse-square potential

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