A Model Study of Discrete Scale Invariance and Long-Range Interactions

Abstract: We investigate the modification of discrete scale invariance in the bound state spectrum by long-range interactions. This problem is relevant for effective field theory descriptions of nuclear cluster states and manifestations of the Efimov effect in nuclei. As a model system, we choose a one dimensional inverse square potential supplemented with a long-range Coulomb interaction. We study the renormalization and bound-state spectrum of the system as a function of the Coulomb interaction strength. Our results indicate, that the counterterm required to renormalize the inverse square potential alone is sufficient to renormalize the full problem. However, the breaking of the discrete scale invariance through the Coulomb interaction leads to a modified bound state spectrum. The shallow bound states are strongly influenced by the Coulomb interaction while the deep bound states are dominated by the inverse square potential.

• The paper demonstrates that a single counterterm suffices to renormalize the 1/r² potential even when a long-range Coulomb force is added.
• Key results show that deep bound states maintain quasi-geometric scaling while shallow states shift to a hydrogenic spectrum.
• The findings inform effective field theory approaches in nuclear cluster systems by clarifying the impact of short-range versus long-range interactions.

Discrete Scale Invariance under Long-Range Interactions: A Renormalization Group Analysis

Introduction

This paper presents a thorough analysis of the interplay between discrete scale invariance (DSI) and long-range interactions in quantum systems, with direct relevance to effective field theory (EFT) constructions for nuclear clusters and the Efimov effect in few-body physics. The authors utilize a one-dimensional quantum mechanical model featuring an attractive inverse-square potential—the paradigmatic system exhibiting DSI and renormalization group (RG) limit cycles—as a tractable framework. A long-range Coulomb potential is then introduced to study the breaking of DSI under perturbative and nonperturbative long-range interactions. The work meticulously investigates the modified renormalization structure and bound-state spectrum, focusing in particular on the emergence and pattern of bound states, cutoff independence, and the persistence (or breakdown) of RG limit cycles.

The Inverse-Square Potential and Discrete Scaling

The baseline system is defined by the attractive $1/r^2$ potential, $V_S(r) = (\hbar^2/m) c/r^2$, with $c < -1/4$ (i.e., real $\nu$ parameter). This potential is classically scale invariant and, quantum-mechanically for sufficiently strong attraction, generates a singular Schrödinger equation lacking a unique self-adjoint extension. Renormalization is required to define the quantum problem; introducing a single momentum-independent counterterm and imposing an RG invariance condition, one obtains a log-periodic running of the counterterm, corresponding to an RG limit cycle. As a result, the system manifests an infinite tower of bound states whose binding energies are geometrically spaced by the discrete scaling factor, $E_{n+1}/E_n = e^{-2\pi/\nu}$. This behavior closely mirrors the Efimov effect encountered in three-body systems with resonant interactions.

Inclusion of the Coulomb Potential: Breaking of Discrete Scale Invariance

Upon supplementing the system with an attractive Coulomb potential, $V_C(r) = -\alpha / r$, the analysis turns to the central question: does the addition of a marginally (or exactly) long-range potential alter the renormalization pattern, and to what extent does it break discrete scale invariance in the spectrum?

Renormalization Structure

The study determines numerically that the counterterm introduced for the inverse-square potential alone is sufficient to ensure cutoff independence of all binding energies, provided a single state is fixed as the renormalization condition. No new divergences or counterterms arising from the Coulomb tail are found to be necessary. The effect of the Coulomb interaction is to shift the log-periodicity of the running counterterm; equivalently, the value of the dimensionful parameter $\Lambda_*$ is renormalized by a monotonic function of $\alpha$. This observation holds at all interaction strengths probed, indicating nontrivial robustness in the renormalization construction.

Spectrum: Interplay of Short- and Long-Range Physics

The computed spectrum reveals a pronounced dependence of the bound-state pattern on the relative strength of the Coulomb and $1/r^2$ potentials. The deepest bound states (large binding energies) are only minimally affected by the Coulomb interaction, maintaining an approximately geometric progression in energies characteristic of DSI and the RG limit cycle. Conversely, shallow bound states are significantly perturbed, with the spectrum tending toward the familiar hydrogenic pattern at small energies—a clear manifestation of the breakdown of discrete scale invariance for long-distance sensitive states.

This separation is governed by the scale $\bar{r} = |c|/\alpha$ at which $|V_S| = |V_C|$, and the corresponding potential energy $\bar{E}_{\mathrm{pot}}$. States with $E_B \gg \bar{E}_{\mathrm{pot}}$ are localized at short distances and reflect the $1/r^2$ scaling, whereas for $E_B \ll \bar{E}_{\mathrm{pot}}$, the Coulomb tail dominates and eradicates the geometric scaling.

Perturbative Analysis and Non-Perturbative Dynamics

A first-order perturbative calculation of the Coulomb-induced energy shifts for deep states is performed, with the analytical structure and numerical results confirming that such corrections are reliable only for states where Coulomb effects are subleading. The shallowest bound states, dominated by the Coulomb potential, elude perturbative treatment due to the singular nature of the $1/r^2$ potential—reaffirming the need for nonperturbative renormalization when $\nu$ is real.

Dependence on System Parameters

Examination of the spectrum as a function of $\nu$ and the renormalization scale $\Lambda_*$ underscores the following hierarchy:

• For increasing $\nu$, the spectrum becomes denser (reduced scaling factor), and the influence of the Coulomb potential is further suppressed.
• Variation of $\Lambda_*$ shifts the spectrum vertically but does not affect the qualitative interplay between short and long-range mechanisms.
• For fixed $\Lambda_*$, increasing $\alpha$ enhances the breaking of DSI among the shallow states, but leaves the deep bound states governed predominantly by the $1/r^2$ potential.

Implications for EFT and Nuclear Cluster Systems

The demonstrated sufficiency of a single counterterm for renormalization when long-range forces are included has immediate significance for EFT frameworks, particularly in the context of halo nuclei and nuclear cluster states (e.g., the $\alpha\alpha$ system and the Hoyle state in $^{12}\mathrm{C}$). The findings imply that, in constructing low-energy EFTs for such systems, the short-range counterterms determined in the absence of the Coulomb force remain adequate, though physical observables (such as excitation spectra) will generically lose the precise DSI of the Efimov scenario. Theoretical predictions should expect modifications of scaling spectra and emergence of cluster resonances as remnants of broken DSI.

Further, the methods and results in this work naturally extend to more complex systems—e.g., the three-body $\alpha\alpha\alpha$ problem—where precise understanding of the symmetry breaking by long-range interactions will be crucial for quantitative predictions, power counting, and interpretation of experimental states in terms of limit cycles and universality.

Conclusion

The analysis establishes that the introduction of a long-range Coulomb interaction into a quantum system exhibiting DSI and RG limit cycles does not necessitate new counterterms beyond those required by the singular short-range potential. While the Coulomb interaction explicitly breaks discrete scale invariance, rendering the bound-state spectrum non-geometric, deep bound states remain scaling-like and perturbatively corrected. The spectrum thus provides a clear diagnostic of the competition between short-range conformal invariance and long-range forces. These results deliver key insights for EFT modeling of halo nuclei and nuclear cluster states, and motivate future investigations into quantitative extensions and nonperturbative EFT approaches for systems with intertwined short- and long-range forces.