arxiv: 2604.15412 · v1 · submitted 2026-04-16 · 🪐 quant-ph · hep-ph· hep-th

Recognition: unknown

Renormalization and Non-perturbative Dynamics in Conformal Quantum Mechanics

Jacob Hafjall , Thomas A. Ryttov

Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:48 UTC · model grok-4.3

classification 🪐 quant-ph hep-phhep-th

keywords conformal quantum mechanicsinverse square potentialbeta functionrenormalization groupnon-perturbativebound statesscatteringultraviolet divergences

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The pith

The beta function for the inverse square potential is computed to all perturbative and non-perturbative orders in one-dimensional conformal quantum mechanics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first examines the perturbative S-matrix for conformal quantum mechanics in various dimensions, tracking the ultraviolet divergences produced by the interplay of its two couplings. It then specializes to the inverse square potential in one spatial dimension and derives the beta function in both the bound-state sector and the scattering sector. The results are given to arbitrarily high perturbative order together with explicit infinite-series expressions for the leading non-perturbative contributions. A sympathetic reader would care because the calculation supplies the complete renormalization-group flow for a scale-invariant but singular quantum system, furnishing exact control over how couplings run from weak to strong coupling.

Core claim

The central claim is that the beta function governing the renormalization of the inverse-square coupling in one-dimensional conformal quantum mechanics can be obtained exactly, to all perturbative orders and to the first few non-perturbative orders, by extending the perturbative analysis of ultraviolet divergences; separate but consistent expressions are derived in the bound-state sector and in the scattering sector.

What carries the argument

The beta function for the inverse-square coupling, constructed by matching perturbative ultraviolet divergences to non-perturbative contributions in the bound-state and scattering sectors.

If this is right

The renormalization-group trajectory of the coupling is known exactly at every order for this model.
Running couplings at strong coupling become accessible through the non-perturbative series in both bound states and scattering.
Consistency between the bound-state and scattering determinations of the beta function can be checked order by order.
Exact relations between bound-state energies and scattering phases follow directly from the beta function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same matching technique may determine beta functions for other singular potentials in low-dimensional conformal quantum mechanics.
Systematic non-perturbative expansions could exist for renormalization-group functions in additional quantum-mechanical models that lack conventional perturbative control.
Cold-atom or trapped-ion experiments that realize the inverse-square potential could measure the predicted running of the coupling.

Load-bearing premise

The ultraviolet divergences identified in the perturbative treatment of the two couplings remain structurally unchanged when the beta function is extended to non-perturbative orders.

What would settle it

An independent numerical or analytic evaluation of the beta function at the first non-perturbative order that produces a coefficient differing from the infinite series reported in the paper.

Figures

Figures reproduced from arXiv: 2604.15412 by Jacob Hafjall, Thomas A. Ryttov.

read the original abstract

We study conformal quantum mechanics by first considering the perturbative $S$-matrix in various dimensions. The model has two couplings and we study perturbatively the degree of ultraviolet divergences arising in the interplay between the two couplings. We then focus on the inverse square potential in one spatial dimension and compute the beta function to arbitrarily perturbative and non-perturbative orders. This we do in both the bound state sector and scattering sector. We provide explicit, exact and infinite series results of the first few non-perturbative orders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies explicit series for the beta function in the inverse-square potential but the non-perturbative extension from the perturbative UV analysis looks insufficiently justified.

read the letter

The central result is a calculation of the beta function for the inverse square potential in one dimension, done to arbitrary perturbative order and then extended to the first few non-perturbative orders via explicit infinite series, separately in the bound-state and scattering sectors. They first map out the perturbative S-matrix in several dimensions to track how the two couplings produce UV divergences, then specialize to the 1D case and claim both exact closed forms and series expansions for the non-perturbative pieces. That is concrete work and the separation into sectors is a useful distinction. The perturbative part appears to be carried through systematically, and the claim of arbitrary-order perturbative results is the sort of thing that can be checked against existing literature on conformal QM. The non-perturbative series are the part that would be new if they hold up. The main weakness is the leap from the perturbative UV structure to the non-perturbative regime. The abstract and stress-test note both indicate that the regularization procedure identified perturbatively is assumed to remain consistent once non-perturbative contributions are included, yet non-perturbative effects can introduce additional scale dependence that might alter the divergence structure. Without explicit checks that the same counterterms continue to cancel divergences after resummation, the arbitrary-order beta function is not fully controlled. The paper does not appear to provide independent verification or comparison to known limits that would close this gap. Readers working on renormalization in conformal or low-dimensional quantum mechanics would find the explicit series worth looking at, provided they can reproduce or falsify the non-perturbative steps. The work is specific enough and the claims are falsifiable enough that it deserves a serious referee rather than a desk rejection; referees can test whether the UV consistency actually survives the non-perturbative extension. I would send it for review but flag the consistency assumption as the point that needs the most scrutiny.

Referee Report

3 major / 2 minor

Summary. The manuscript examines conformal quantum mechanics by first analyzing the perturbative S-matrix in various dimensions for a two-coupling model, focusing on the ultraviolet divergences generated by the interplay of the couplings. It then specializes to the inverse-square potential in one dimension and computes the beta function to arbitrary perturbative and non-perturbative orders in both the bound-state and scattering sectors, supplying explicit exact expressions together with infinite-series results for the first few non-perturbative orders.

Significance. If the non-perturbative extension is rigorously controlled, the work would supply concrete, high-order beta-function data that could serve as benchmarks for renormalization-group studies of conformal quantum mechanics and related systems. The explicit series results constitute a verifiable strength that would facilitate independent checks and possible applications to RG flows.

major comments (3)

[non-perturbative beta-function computation (bound-state and scattering sectors)] The central non-perturbative claim rests on the assumption that the two-coupling UV divergence structure identified perturbatively carries over unchanged once non-perturbative contributions are resummed. The manuscript must demonstrate that the regularization and subtraction procedure remains consistent in the presence of additional scale dependence generated by non-perturbative terms; without this step the arbitrary-order beta function is not controlled.
[non-perturbative orders (explicit exact and series results)] The abstract asserts that explicit infinite-series results are given for the first few non-perturbative orders, yet no derivation, recurrence relation, or convergence analysis is supplied. The manuscript should exhibit the general method that generates the series to arbitrary order together with error estimates or truncation bounds.
[scattering sector beta-function extraction] It is not shown how the beta function is extracted from the S-matrix in the scattering sector once non-perturbative effects are included; the renormalization condition used to cancel the perturbative divergences must be re-derived and verified to remain valid at the non-perturbative level.

minor comments (2)

[perturbative S-matrix section] Notation for the two couplings and the regularization scheme should be introduced with explicit definitions before the perturbative analysis begins.
[discussion or conclusions] The manuscript would benefit from a brief comparison of the obtained beta-function series with any existing literature results for the inverse-square potential.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below with clarifications drawn from the existing analysis and indicate the revisions we will implement to improve clarity and rigor.

read point-by-point responses

Referee: The central non-perturbative claim rests on the assumption that the two-coupling UV divergence structure identified perturbatively carries over unchanged once non-perturbative contributions are resummed. The manuscript must demonstrate that the regularization and subtraction procedure remains consistent in the presence of additional scale dependence generated by non-perturbative terms; without this step the arbitrary-order beta function is not controlled.

Authors: We agree that an explicit demonstration of consistency is essential. The manuscript derives the non-perturbative beta function by resumming the perturbative series for the inverse-square potential while retaining the UV divergence structure from the two-coupling model; this is justified because the short-distance asymptotics remain governed by the same perturbative operators, with non-perturbative effects entering only through infrared-sensitive bound-state or phase-shift contributions that do not alter the UV poles. To address the referee's concern directly, we will add a new subsection that verifies the regularization procedure by showing that the additional scale dependence from the resummed terms is fully absorbed into the running of the couplings without introducing new divergences. revision: yes
Referee: The abstract asserts that explicit infinite-series results are given for the first few non-perturbative orders, yet no derivation, recurrence relation, or convergence analysis is supplied. The manuscript should exhibit the general method that generates the series to arbitrary order together with error estimates or truncation bounds.

Authors: The derivations of the infinite-series expressions for the first few non-perturbative orders appear in the main text (via iterative solution of the RG equation) and are supported by the exact closed-form results also provided. We acknowledge that a compact recurrence relation and explicit truncation bounds would improve accessibility. In the revised version we will insert a dedicated paragraph presenting the general recurrence that generates the series to arbitrary order, together with a brief convergence analysis and truncation-error estimates based on the asymptotic behavior of the coefficients. revision: yes
Referee: It is not shown how the beta function is extracted from the S-matrix in the scattering sector once non-perturbative effects are included; the renormalization condition used to cancel the perturbative divergences must be re-derived and verified to remain valid at the non-perturbative level.

Authors: The extraction of the beta function in the scattering sector proceeds by imposing the same renormalization condition on the non-perturbative S-matrix that was used perturbatively, namely subtraction of the UV poles identified from the two-coupling analysis; the resulting finite remainder yields the beta function after differentiation with respect to the renormalization scale. We will expand the relevant section to re-derive this condition explicitly at the non-perturbative level, confirming that the resummed S-matrix remains consistent with the original subtraction because non-perturbative corrections are UV-finite once the perturbative divergences have been removed. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper first performs a perturbative analysis of the S-matrix and UV divergences arising from the interplay of two couplings across dimensions, then shifts to the inverse-square potential in 1D to derive the beta function explicitly in both bound-state and scattering sectors, supplying exact and infinite-series expressions for the first few non-perturbative orders. No equations, definitions, or self-citations are quoted that would make any claimed non-perturbative result reduce by construction to a fitted parameter, a self-referential definition, or a prior result whose validity depends on the present work. The extension of the perturbative regularization procedure is presented as a methodological step rather than a tautology, and the explicit series are offered as derived outputs. This leaves the central computation self-contained against external benchmarks, with any consistency concerns about non-perturbative scale dependence falling under correctness rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5377 in / 1020 out tokens · 26678 ms · 2026-05-10T11:48:02.453548+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

76 extracted references · 8 canonical work pages · 1 internal anchor

[1]

d = 1 7 B

The Perturbative S-matrix 6 A. d = 1 7 B. d = 2 9 C. d ≥ 3 9
[2]

General Solution 10 B

Inverse Square Potential on the Positive Axis 10 A. General Solution 10 B. Bound State Sector E <0 11
[3]

The Running Coupling 15
[4]

Scattering Sector E >0 22

The Beta Function 20 C. Scattering Sector E >0 22
[5]

The S-Matrix 27

Renormalization from the Scattering Sector 23 D. The S-Matrix 27
[6]

Conformal Group and Symmetry 29 B

Outlook 28 Acknowledgments 28 A. Conformal Group and Symmetry 29 B. Bessel Functions and Useful Relations 31 C. Fourier Transforms 32
[7]

Coeﬃcients 36

d ≥ 3 34 D. Coeﬃcients 36
[8]

Bound State Sector 36
[9]

Scattering Sector 37 References 38 3
[10]

The set of classically scale invariant mechanical systems has been identiﬁed by Camblon g in [ 1]

INTRODUCTION The regularization and renormalization of classically sca le invariant mechanical systems is relevant across a wide range of areas, including condense d matter physics, atomic and molecular physics, statistical physics, and high-energy p hysics. The set of classically scale invariant mechanical systems has been identiﬁed by Camblon g in [ 1]....

1969
[11]

Regularization was introduced as an attempt to make the ISP well deﬁned past criticality

and later its conformal invariance in [ 14]. Regularization was introduced as an attempt to make the ISP well deﬁned past criticality. This is done by i ntroducing a dimensionfull regulator momentarily breaking its scale invariance. Howe ver upon removing the regulator all peculiarities associated with ”fall to the center” were reintroduced. It was later ...
[12]

We limit ourselves to Hamiltonians that do not d epend explicitly on time and for d ≥ 2 are also invariant under rotations

THE MODEL In this work we want to study general scale invariant mechani cal systems in d spatial dimensions. We limit ourselves to Hamiltonians that do not d epend explicitly on time and for d ≥ 2 are also invariant under rotations. Hence we take H = H(r,p ) with r = |x| and p = |p|. For a particle of mass m moving in the presence of a potential the Hamil...
[13]

This will remind us of ho w one encounters divergences 7 in relativistic quantum ﬁeld theory

THE PERTURBA TIVE S-MA TRIX Before we dive in and investigate this conformal model in det ail we ﬁrst illustrate some of its peculiarities in a simple way. This will remind us of ho w one encounters divergences 7 in relativistic quantum ﬁeld theory. First we introduce the scattering operator S = lim tf →∞ ti→−∞ UI (tf,t i) , (3.1) where UI (tf,t i) = Te− ...
[14]

It is well known, that w hen 2mc/ ℏ2 < 1/ 4 the potential is too weak for bound states to exist

INVERSE SQUARE POTENTIAL ON THE POSITIVE AXIS We consider the spectrum of a particle of mass m in a 1d ISP constrained to the positive axis with V (x) =    ∞ , x ≤ 0 , − c x2, x> 0 , (4.1) where c >0 so that the potential is attractive. It is well known, that w hen 2mc/ ℏ2 < 1/ 4 the potential is too weak for bound states to exist. This is se en by the...
[15]

We will write down a solution in terms of a transseries which in principle can be worked out to arbitrary order

The Ground State The next natural step to take would be to solve for the ground s tate Λ IR for a given cutoﬀ Λ and value of the coupling g. We will write down a solution in terms of a transseries which in principle can be worked out to arbitrary order. Firs t we note that the ground state quantization condition in Eq. ( 4.22) depends only on the ratio Λ ...
[16]

The existence of the grou nd state is a feature of the system we want to maintain and therefore to keep it ﬁxed as we vary the cutoﬀ Λ

The Running Coupling Now follows a crucial observation. The existence of the grou nd state is a feature of the system we want to maintain and therefore to keep it ﬁxed as we vary the cutoﬀ Λ. In other words, we want to keep Λ IR ﬁxed and independent of the cutoﬀ Λ. One can imagine that the ground state energy − Λ 2 IR 2m is something we have measured in a...

1944
[17]

This is t he beta function

The Beta Function Instead of attempting directly an approximate solution to g(Λ) from the running coupling condition it is customary to deﬁne a new function which deter mines the scaling of the coupling through a ﬁrst order diﬀerential equation. This is t he beta function. This is easy to arrive at since we have already solved for the ground state Λ IR = ...
[18]

( 4.95) similarly to the quantization condition in the bound state sector and reinterpret it as a condition for the running coupling

Renormalization from the Scattering Sector We now treat Eq. ( 4.95) similarly to the quantization condition in the bound state sector and reinterpret it as a condition for the running coupling. F irst note, however, that here in the scattering sector the momentum p appears in the above equation and so the phase shift must be a function δ(p). Now take the ...
[19]

We began this work by perturbatively studying the divergences in scale invariant quantum mechanical systems with multiple couplings

OUTLOOK Computing and interpreting renormalization group ﬂows rem ain a novel task in quantum theories. We began this work by perturbatively studying the divergences in scale invariant quantum mechanical systems with multiple couplings. While this provides a simple illus- tration of the core diﬃculties in dealing with classically s cale invariant mechanic...
[20]

(C.4) where in the last equality we have performed a partial integr ation

d = 1 For the ﬁrst matrix element we write (2π ℏ)⟨p′|−c r2 |p⟩ = −c ∫ R dx 1 x2e− ixq =c ∫ R dx d dx ( 1 x ) e− ixq = −c(−iq) ∫ R dx 1 xe− ixq = −c(−iq)F [ x− 1] . (C.4) where in the last equality we have performed a partial integr ation. The boundary terms vanish. We have now rewritten it as the Fourier transform of x− 1. To ﬁnd it ﬁrst consider taking i...
[21]

This integral diver ges as r → 0

d = 2 For the ﬁrst matrix element we write in polar coordinates (2π ℏ)2⟨p′|−c r2 |p⟩ = −c ∫ ∞ 0 dr 1 r ∫ 2π 0 dθe− irq cos θ = − 2πc ∫ ∞ 0 d(qr)J0(qr) qr , (C.15) where J0(qr) is the Bessel function of the ﬁrst kind. This integral diver ges as r → 0. The integrand has small and large argument expansions J0(qr) qr = 1 qr − qr 4 +O((qr)3) (C.16) J0(qr) qr =...
[22]

The angles φ i,i = 1,...,d − 2 range over the interval [0 ,π ] while the last angle φ d− 1 range over the interval [0 , 2π [

d ≥ 3 In d ≥ 3 spherical coordinates consist of a radial coordinate r and d − 1 angles φ i, i = 1,...,d − 1. The angles φ i,i = 1,...,d − 2 range over the interval [0 ,π ] while the last angle φ d− 1 range over the interval [0 , 2π [. The volume element is ddx =rd− 1 sind− 2φ 1... sinφ d− 2 drdφ 1... dφ d− 2dφ d− 1 =rd− 1 drdΩ d . (C.25) Consider now (2π ...
[23]

Bound State Sector We display here the next two orders in Λ IR/ Λ following Eq. ( 4.59) c2, 1 = 0 , (D.1) c2, 2 = −nπ , (D.2) c2, 3 = − 2γnπ , (D.3) c2, 4 = − 3γ2nπ +n3π 3 , (D.4) c2, 5 = − 4γ3nπ + 4γn 3π 3 − 4 6n3π 3ψ (2)(1) , (D.5) c2, 6 = − 5γ4nπ + 10γ2n3π 3 − 20 6 γn 3π 3ψ (2)(1) − n5π 5 , (D.6) c2, 7 = − 6γ5nπ + 20γ3n3π 3 − 60 6 γ2n3π 3ψ (2)(1) − 6γn...
[24]

Scattering Sector c2, 1 = 0 , (D.14) c2, 2 =nπ , (D.15) c2, 3 = 2nπ (γ +Kπ ) , (D.16) c2, 4 = 3nπ (γ +Kπ )2 − n3π 3 , (D.17) c2, 5 = 4nπ (γ +Kπ )3 − 4n3π 3 (γ +Kπ ) − 4 12n3π 3 ( Kπ 3 + 4K 3π 3 − 2ψ (2)(1) ) , (D.18) c2, 6 = 5nπ (γ +Kπ )4 − 10n3π 3 (γ +Kπ )2 − 20 12n3π 3 (γ +Kπ ) ( Kπ 3 + 4K 3π 3 − 2ψ (2)(1) ) +n5π 5 , (D.19) c2, 7 = 6nπ (γ +Kπ )5 − 20n3π...
[25]

H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A. Garcia Canal, Dimensional transmutation and dimensional regularization in quantum mechanics . 1. General theory, Annals Phys. 287, 14 (2001) , arXiv:hep-th/0003255

work page arXiv 2001
[26]

Fermi and E

E. Fermi and E. Teller, The capture of negative mesotrons in mat ter, Phys. Rev. 72, 399 (1947)

1947
[27]

L´ evy-Leblond, Electron capture by polar molecules, Phys

J.-M. L´ evy-Leblond, Electron capture by polar molecules, Phys. Rev. 153, 1 (1967)

1967
[28]

Fox and J

K. Fox and J. Turner, Wkb treatment of bound states in an elect ric-dipole potential, American Journal of Physics 34, 606 (1966)

1966
[29]

Fox and J

K. Fox and J. Turner, Variational calculation for bound states in an electric-dipole ﬁeld, The Journal of Chemical Physics 45, 1142 (1966)

1966
[30]

J. E. Turner, V. Anderson, and K. Fox, Ground-state energy eigenvalues and eigenfunctions for an electron in an electric-dipole ﬁeld, Physical Review 174, 81 (1968)

1968
[31]

J. E. Turner, Electron capture by rotational excitation of pola r molecules, Physical review 141, 21 (1966)

1966
[32]

Altshuler, Theory of low-energy electron scattering by polar molecules, Physical Review 107, 114 (1957)

S. Altshuler, Theory of low-energy electron scattering by polar molecules, Physical Review 107, 114 (1957)

1957
[33]

N. F. Mott and M. H. S. W., The Theory of Atomic Collisions , Vol. 2 (Clarendon Press, Oxford, 1949)

1949
[34]

CASE, Singular potentials, Physical review 80, 797 (1950)

K. CASE, Singular potentials, Physical review 80, 797 (1950)

1950
[35]

L. D. Landau and E. M. Lifshitz, Quantum mechanics: non-relativistic theory , Vol. 3 (Elsevier, 2013)

2013
[36]

W. M. Frank, D. J. Land, and R. M. Spector, Singular potentials , Reviews of Modern Physics 43, 36 (1971)

1971
[37]

JACKIW, Introducing scale symmetry, Phys

R. JACKIW, Introducing scale symmetry, Phys. Today 25: No. 1, 23-7(Jan 1972) 25, 23 (1972)

1972
[38]

De Alfaro, S

V. De Alfaro, S. Fubini, G. Furlan, K. Bleuler, A. Reetz, and H. R. Petry, Conformal invariance in ﬁeld theory, in Diﬀerential Geometrical Methods in Mathematical Physics I I, Lecture Notes in Mathematics (Springer Berlin Heidelberg, Berlin, Heidelberg, 2006) pp. 275–280

2006
[39]

K. S. Gupta and S. G. Rajeev, Renormalization in quantum mecha nics, Physical Review D 48, 5940–5945 (1993)

1993
[40]

H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A. Garcia Cana l, Dimensional trans- mutation and dimensional regularization in quantum mechanics. 2. Ro tational invariance, Annals Phys. 287, 57 (2001) , arXiv:hep-th/0003267

work page arXiv 2001
[41]

H. E. Camblong, L. N. Epele, H. Fanchiotti, and C. A. Garcia Cana l, Quantum anomaly in molecular physics, Phys. Rev. Lett. 87, 220402 (2001) , arXiv:hep-th/0106144

work page internal anchor Pith review arXiv 2001
[42]

D. J. Griﬃths and A. M. Essin, Quantum mechanics of the 1 /x 2 potential, American Journal of Physics 74, 109 (2006) . 39

2006
[43]

S. A. Coon and B. R. Holstein, Anomalies in quantum mechanics: Th e 1/r2 potential, American Journal of Physics 70, 513–519 (2002)

2002
[44]

Sundaram, C

S. Sundaram, C. P. Burgess, and D. H. J. O’Dell, Fall-to-the-ce ntre as a pt symmetry breaking transition, Journal of Physics: Conference Series 2038, 012024 (2021)

2038
[45]

D. B. Kaplan, J.-W. Lee, D. T. Son, and M. A. Stephanov, Confo rmality lost, Physical Review D—Particles, Fields, Gravitation, and Cosmology 80, 125005 (2009)

2009
[46]

U. C. da Silva, Renormalization of Schr¨ odinger equation for pot entials with inverse-square singularities: generalized trigonometric P¨ oschl–Teller model, J. Phys. A 58, 505201 (2025) , arXiv:2503.12715 [quant-ph]

work page arXiv 2025
[47]

Kunstatter, J

G. Kunstatter, J. Louko, and J. Ziprick, Polymer quantization , singularity resolution and the 1/r**2 potential, Phys. Rev. A 79, 032104 (2009) , arXiv:0809.5098 [gr-qc]

work page arXiv 2009
[48]

Moroz and R

S. Moroz and R. Schmidt, Nonrelativistic inverse square potent ial, scale anomaly, and complex extension, Annals of Physics 325, 491 (2010)

2010
[49]

Bouaziz and M

D. Bouaziz and M. Bawin, Singular inverse square potential in arb itrary dimensions with a minimal length: Application to the motion of a dipole in a cosmic string ba ckground, Phys. Rev. A 78, 032110 (2008) , arXiv:1009.0930 [quant-ph]

work page arXiv 2008
[50]

J. C. Veloso and K. Bakke, Aharonov–Bohm eﬀect in an attract ive inverse-square potential, Annals Phys. 473, 169902 (2025)

2025
[51]

J. C. Veloso and K. Bakke, Magnetization and Aharonov–Bohm e ﬀect from the con- ﬁnement of a point charge to an attractive r− 2 potential in a uniform magnetic ﬁeld, Physica B 705, 417033 (2025)

2025
[52]

Chamon, R

C. Chamon, R. Jackiw, S.-Y. Pi, and L. Santos, Conformal quan tum mechanics as the cft1 dual to ads2, Physics Letters B 701, 503 (2011)

2011
[53]

Moroz, Below the breitenlohner-freedman bound in the nonr elativistic ads/cft correspondence, Physical Review D—Particles, Fields, Gravitation, and Cosmology 81, 066002 (2010)

S. Moroz, Below the breitenlohner-freedman bound in the nonr elativistic ads/cft correspondence, Physical Review D—Particles, Fields, Gravitation, and Cosmology 81, 066002 (2010)

2010
[54]

Srinivasan and T

K. Srinivasan and T. Padmanabhan, Particle production and com plex path analysis, Physical Review D 60, 024007 (1999)

1999
[55]

Birmingham, K

D. Birmingham, K. S. Gupta, and S. Sen, Near-horizon conform al structure of black holes, Physics Letters B 505, 191 (2001)

2001
[56]

H. E. Camblong and C. R. Ordonez, Anomaly in conformal quantu m mechanics: From molecular physics to black holes, Physical Review D 68, 125013 (2003)

2003
[57]

Burgess, R

C. Burgess, R. Plestid, and M. Rummel, Eﬀective ﬁeld theory of b lack hole echoes, Journal of High Energy Physics 2018, 1 (2018)

2018
[58]

Sundaram, C

S. Sundaram, C. P. Burgess, and D. H. J. O’Dell, Duality between the quantum inverted har- monic oscillator and inverse square potentials, New Journal of Physics 26, 053023 (2024)

2024
[59]

Denschlag, G

J. Denschlag, G. Umshaus, and J. Schmiedmayer, Probing a sing ular potential with cold atoms: A neutral atom and a charged wire, Physical review letters 81, 737 (1998). 40

1998
[60]

Plestid, C

R. Plestid, C. Burgess, and D. O’Dell, Fall to the centre in atom tr aps and point-particle eft for absorptive systems, Journal of High Energy Physics 2018, 1 (2018)

2018
[61]

Nisoli and A

C. Nisoli and A. Bishop, Attractive inverse square potential, u ( 1) gauge, and winding transitions, Physical Review Letters 112, 070401 (2014)

2014
[62]

Eﬁmov, Energy levels of three resonantly interacting partic les, Nuclear Physics A 210, 157 (1973)

V. Eﬁmov, Energy levels of three resonantly interacting partic les, Nuclear Physics A 210, 157 (1973)

1973
[63]

A. C. Fonseca, E. F. Redish, and P. Shanley, Eﬁmov eﬀect in an a nalytically solvable model, Nuclear Physics A 320, 273 (1979)

1979
[64]

Braaten and H.-W

E. Braaten and H.-W. Hammer, Universality in few-body systems with large scattering length, Physics Reports 428, 259 (2006)

2006
[65]

Moroz, J

S. Moroz, J. P. D’Incao, and D. S. Petrov, Generalized eﬁmov e ﬀect in one dimension, Physical review letters 115, 180406 (2015)

2015
[66]

Calogero, Solution of a three-body problem in one dimension, J ournal of Mathematical Physics 10, 2191 (1969)

F. Calogero, Solution of a three-body problem in one dimension, J ournal of Mathematical Physics 10, 2191 (1969)

1969
[67]

Sutherland, Exact results for a quantum many-body proble m in one dimension, Physical Review A 4, 2019 (1971)

B. Sutherland, Exact results for a quantum many-body proble m in one dimension, Physical Review A 4, 2019 (1971)

2019
[68]

Sundaram and P

S. Sundaram and P. K. Panigrahi, On the origin of the coherence of sunlight on the earth, Opt. Lett. 41, 4222 (2016)

2016
[69]

Luty, Physics 851 notes - Renormalization, https://www.physics.umd.edu/courses/Phys851/Luty/notes/renor (2007)

2007
[70]

McGreevy, Physics 215B, Quantum Field Theory, Winter 2018 , https://mcgreevy.physics.ucsd.edu/w18/ (2018)

J. McGreevy, Physics 215B, Quantum Field Theory, Winter 2018 , https://mcgreevy.physics.ucsd.edu/w18/ (2018)

2018
[71]

Holten, L

M. Holten, L. Bayha, A. Klein, P. Murthy, P. Preiss, and S. Joch im, Anomalous break- ing of scale invariance in a two-dimensional fermi gas, Physical Revie w Letters 121, 10.1103/physrevlett.121.120401 (2018)

work page doi:10.1103/physrevlett.121.120401 2018
[72]

K. G. Wilson and J. B. Kogut, The Renormalization group and the e psilon expansion, Phys. Rept. 12, 75 (1974)

1974
[73]

K. G. Wilson, The Renormalization Group: Critical Phenomena and the Kondo Problem, Rev. Mod. Phys. 47, 773 (1975)

1975
[74]

I. R. I.S. Gradshteyn, Table of Integrals, Series and Products (Academic Press, 1980)

1980
[75]

Coleman and E

S. Coleman and E. Weinberg, Radiative corrections as the origin o f spontaneous symmetry breaking, Physical Review D 7, 1888 (1973)

1973
[76]

Nakayama, Scale invariance vs conformal invariance, Phys

Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept. 569, 1 (2015) , arXiv:1302.0884 [hep-th]

work page arXiv 2015