Singular inverse-square potential: Renormalization and self-adjoint extensions for medium to weak coupling

Bouaziz, Djamil; Bawin, M.

doi:10.1103/physreva.89.022113

Cited by 27 publications

(24 citation statements)

References 21 publications

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“…8 As many have observed [28][29][30][31][32][33][34][35][36][37][38][39][40] because of this the Schrödinger Hamiltonian can fail to be self-adjoint, depending the boundary conditions that hold at r = . Selecting a choice of boundary condition to secure its self-adjointness -not a unique construction -is known as constructing its self-adjoint extension [41][42][43][44][45][46][47][48].…”

Section: Jhep07(2017)072mentioning

confidence: 99%

“…As mentioned in [1], this boundary condition can be regarded as a specific choice of selfadjoint extension [41][42][43][44][45][46][47][48] of the inverse-square Hamiltonian. The inverse-square potential requires such an extension because its wave-functions are sufficiently bunched at the origin that physical quantities actually care about the nature of the physics encapsulated by the source action, S b .…”

Section: Jhep07(2017)072mentioning

confidence: 99%

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Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Burgess

Hayman

Rummel

et al. 2017

J. High Energ. Phys.

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We apply point-particle effective field theory (PPEFT) to compute the leading shifts due to finite-sized source effects in the Coulomb bound energy levels of a relativistic spinless charged particle. This is the analogue for spinless electrons of calculating the contribution of the charge-radius of the source to these levels, and our calculation disagrees with standard calculations in several ways. Most notably we find there are two effective interactions with the same dimension that contribute to leading order in the nuclear size, one of which captures the standard charge-radius contribution. The other effective operator is a contact interaction whose leading contribution to δE arises linearly (rather than quadratically) in the small length scale, , characterizing the finite-size effects, and is suppressed by (Zα) 5 . We argue that standard calculations miss the contributions of this second operator because they err in their choice of boundary conditions at the source for the wave-function of the orbiting particle. PPEFT predicts how this boundary condition depends on the source's charge radius, as well as on the orbiting particle's mass. Its contribution turns out to be crucial if the charge radius satisfies (Zα) 2 a B , where a B is the Bohr radius, because then relativistic effects become important for the boundary condition. We show how the problem is equivalent to solving the Schrödinger equation with competing Coulomb, inverse-square and delta-function potentials, which we solve explicitly. A similar enhancement is not predicted for the hyperfine structure, due to its spin-dependence. We show how the charge-radius effectively runs due to classical renormalization effects, and why the resulting RG flow is central to predicting the size of the energy shifts (and is responsible for its being linear in the source size). We discuss how this flow is relevant to systems having much larger-than-geometric cross sections, such as those with large scattering lengths and perhaps also catalysis of reactions through scattering with monopoles. Experimental observation of these effects would require more precise measurement of energy levels for mesonic atoms than are now possible.

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Section: Jhep07(2017)072mentioning

confidence: 99%

Section: Jhep07(2017)072mentioning

confidence: 99%

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Burgess

Hayman

Rummel

et al. 2017

J. High Energ. Phys.

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“…As it was noted in Ref. [9] and recently illustrated in [32], renormalization theory leads to identical results in the lowenergy limit independently of the regularization potential W (x) (e.g., delta, square well …). However, none of these regularizing potentials used in the past literature had an inverse square singularity near the origin to mimic the behavior of the original potential.…”

Section: The Regularizationmentioning

confidence: 80%

“…The issue of equivalence between renormalization and self-adjoint extensions was concluded in Ref. [7] and confirmed recently in [32] in the mediumweak coupling region. On the other hand, non-equivalence of renormalization and self-adjoint extensions for such singular interaction was also raised in [30].…”

Section: Introductionmentioning

confidence: 85%

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

Alhaidari

2014

Found Phys

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Quantum anomalies in the inverse square potential are well known and widely investigated. Most prominent is the unbounded increase in oscillations of the particle's state as it approaches the origin when the attractive coupling parameter is greater than the critical value of 1/4. Due to this unphysical divergence in oscillations, we are proposing that the interaction gets screened at short distances making the coupling parameter acquire an effective (renormalized) value that falls within the weak range 0-1/4. This prevents the oscillations form growing without limit giving a lower bound to the energy spectrum and forcing the Hamiltonian of the system to be selfadjoint. Technically, this translates into a regularization scheme whereby the inverse square potential is replaced near the origin by another that has the same singularity but with a weak coupling strength. Here, we take the Eckart as the regularizing potential and obtain the corresponding solutions (discrete bound states and continuum scattering states).

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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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Singular inverse-square potential: Renormalization and self-adjoint extensions for medium to weak coupling

Cited by 27 publications

References 21 publications

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

Point-particle effective field theory II: relativistic effects and Coulomb/inverse-square competition

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

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

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