From Low Energy to High: Reconstructing Physics Beyond the Limit

Author: Denis Avetisyan

A new approach reveals how to extrapolate high-energy physics from low-energy data using a reorganized expansion and controlled coarse-graining technique.

The reconstruction of <span class="katex-eq" data-katex-display="false">\tilde{S}</span> within the QED model, achieved through infrared expansion and fitting up to tenth order interpolation, demonstrates convergence towards the exact result-represented by a dot-dashed black line-with the tenth-order approximation (red solid line) nearly indistinguishable from the expansion itself, all calculated with a maximum parameter of <span class="katex-eq" data-katex-display="false">\tau_{\rm max} = 10m_{e}^{2}</span>. — The reconstruction of $\tilde{S}$ within the QED model, achieved through infrared expansion and fitting up to tenth order interpolation, demonstrates convergence towards the exact result-represented by a dot-dashed black line-with the tenth-order approximation (red solid line) nearly indistinguishable from the expansion itself, all calculated with a maximum parameter of $\tau_{\rm max} = 10m_{e}^{2}$ .

This work demonstrates a method to extract ultraviolet physics from infrared expansions via an inverse Laplace transform and effective field theory techniques.

Conventional effective field theory relies on a separation of scales, limiting access to ultraviolet (UV) physics beyond a defined cutoff. This work, ‘Beyond thresholds: reconstructing UV physics from IR expansions’, introduces a novel approach to circumvent this limitation by reorganizing low-energy expansions via an inverse Laplace transform and controlled coarse-graining procedure. We demonstrate that UV information, including the sign of the beta function and dynamical scale, can be extracted directly from infrared data in theories like QED and QCD. Could this technique unlock a deeper, non-perturbative understanding of quantum field theories and their UV completions?

The Inevitable Limits of Approximation

Quantum field theory, the framework describing fundamental particles and forces, frequently employs approximations to tackle otherwise intractable calculations. A common strategy involves low-energy expansions, which represent physical quantities as a series based on energy scales much smaller than those typically probed. These expansions are remarkably powerful, enabling physicists to make precise predictions within a limited energy range; however, their utility is fundamentally constrained. The accuracy of these methods diminishes rapidly as energy increases, due to the truncation of the infinite series and the implicit assumption of a slowly varying system. Consequently, predictions relying solely on low-energy expansions become unreliable when attempting to describe high-energy phenomena or explore potential new physics beyond the Standard Model, necessitating alternative or more sophisticated approaches to maintain predictive power.

Quantum field theory calculations frequently rely on approximations like low-energy expansions to manage complexity, but these techniques possess an inherent limitation: convergence. Such expansions are only guaranteed to provide accurate predictions within a specific radius of validity, effectively a limited energy range or momentum transfer. Beyond this radius, the series diverges, meaning additional terms don’t simply become smaller, but instead oscillate and amplify, rendering the resulting prediction meaningless. This poses a significant challenge when attempting to connect theoretical models to experiments conducted at higher energies, or when searching for evidence of new physics beyond the Standard Model, as extrapolating beyond the region of convergence introduces substantial uncertainty and the potential for misleading results. Consequently, physicists actively seek methods to extend the radius of convergence or develop alternative approximation schemes to ensure the reliability of theoretical predictions in unexplored energy regimes.

The pursuit of extending the reach of low-energy expansions is fundamentally driven by the need to bridge the gap between theoretical predictions and the realities observed in high-energy experiments. Current limitations restrict the accuracy of calculations when probing energy scales beyond those initially considered, hindering efforts to validate or refine the Standard Model of particle physics. Consequently, advancements in these approximation techniques are essential for interpreting experimental results from facilities like the Large Hadron Collider, and for providing a pathway to discover potential new physics-such as supersymmetry or extra dimensions-that may lie beyond the Standard Model’s current descriptive power. Improving these methods allows physicists to extrapolate theoretical frameworks to regimes where direct experimental verification is challenging, effectively maximizing the information gleaned from both theoretical modeling and experimental data.

In <span class="katex-eq" data-katex-display="false">\mathbb{C}P^{N-1}</span> theory, low-energy expansions accurately reproduce the exact results for both <span class="katex-eq" data-katex-display="false">S(Q^2)</span> and <span class="katex-eq" data-katex-display="false">\tilde{S}(\tau)</span>. — In $\mathbb{C}P^{N-1}$ theory, low-energy expansions accurately reproduce the exact results for both $S(Q^2)$ and $\tilde{S}(\tau)$ .

Reclaiming Lost Ground: Coarse-Graining and Analytical Continuation

Coarse-graining is a reduction technique utilized to analyze complex systems by systematically averaging over microscopic details that are irrelevant to the macroscopic behavior being investigated. This process effectively reduces the number of degrees of freedom in the system, simplifying the calculations required to model its properties. By averaging over small-scale fluctuations and high-frequency modes, a coarse-grained description focuses on the essential, long-wavelength phenomena. The resulting simplified model retains the key physics at lower energies while mitigating the computational cost associated with resolving all microscopic features. This allows for predictions to be made about the system’s behavior at scales larger than the coarse-graining length, and can be applied to various fields including condensed matter physics, fluid dynamics, and materials science.

Analytical continuation, facilitated by the Inverse Laplace Transform, extends the range of validity for low-energy expansions beyond their initial radius of convergence. This is achieved by reconstructing the function represented by the expansion from its values within the convergence radius, allowing for the estimation of values outside this region. The Inverse Laplace Transform provides a mathematical framework for performing this reconstruction, effectively extrapolating the expansion to access previously inaccessible energy regimes and providing insights into high-energy behavior that would otherwise be obscured by the limitations of the original expansion.

Current methodologies for relating infrared (IR) data to ultraviolet (UV) physics are typically constrained by the limitations of perturbative expansions and require assumptions about the high-energy behavior of the system. Our approach circumvents these restrictions through a combined coarse-graining and inverse Laplace transform procedure. This allows for the analytical continuation of low-energy expansions beyond their initial radius of convergence, effectively reconstructing UV properties directly from IR observables without reliance on a priori high-energy models. Validation through $\chi^2$ analysis demonstrates a statistically significant improvement over conventional extrapolation techniques, particularly in regimes where perturbative methods are known to diverge.

Comparisons between low-energy expansions and exact results reveal the accuracy of approximations for <span class="katex-eq" data-katex-display="false">S(Q^2)</span> (top) and <span class="katex-eq" data-katex-display="false"> ilde{S}( au)</span> (bottom). — Comparisons between low-energy expansions and exact results reveal the accuracy of approximations for $S(Q^2)$ (top) and $ilde{S}( au)$ (bottom).

Asymptotic Freedom: A Glimpse Beyond Perturbation

Asymptotic freedom describes a counterintuitive property of certain quantum field theories, notably Quantum Chromodynamics (QCD), where the strength of the strong force decreases as the energy scale increases. This behavior is a direct consequence of the self-interaction of gluons, the force carriers of the strong interaction, and is quantified by a negative β function. Unlike typical quantum field theories where interactions generally strengthen at higher energies due to vacuum polarization effects, the increasing gluon density screens the color charge, leading to a reduction in the effective coupling constant at short distances or high energies. This weakening of interactions allows for perturbative calculations in QCD at high energies, a regime where traditional perturbation theory would fail for theories exhibiting increasing coupling. The observation of asymptotic freedom in the late 1970s was pivotal in establishing QCD as the correct theory of the strong interaction.

The Beta function, denoted as $\beta(g)$ , quantifies the dependence of a coupling constant, $g$ , on the energy scale of a quantum field theory. It mathematically describes how the strength of the interaction changes with varying energies. A negative Beta function indicates that the coupling constant decreases as energy increases, a characteristic defining asymptotic freedom. Conversely, a positive Beta function signifies an increasing coupling constant at higher energies, leading to confinement. The sign and magnitude of $\beta(g)$ therefore serve as a key signature revealing fundamental aspects of a theory’s underlying structure, specifically whether it allows for perturbative calculations at high energies or exhibits non-perturbative behavior.

Extended low-energy expansions, incorporating the principles of asymptotic freedom and the beta function, enable the accurate calculation of physical quantities across multiple energy thresholds where additional degrees of freedom become active. Traditional fitting methods, which rely on parameterizing functions without considering the running coupling, fail at these thresholds due to the introduction of new, unmodeled contributions. By explicitly accounting for the energy scale dependence of the coupling constant – as defined by the beta function – our expansions provide a consistent framework for simultaneously determining parameters across multiple thresholds, demonstrated in systems like two-flavor Quantum Electrodynamics (QED). This approach allows for a more precise determination of low-energy parameters and improved predictive power compared to naïve fitting procedures.

The logarithmic derivative of the scale-dependent strong coupling <span class="katex-eq" data-katex-display="false">\tilde{S}</span> with respect to the renormalization scale τ exhibits truncation effects determined by the maximum expansion order <span class="katex-eq" data-katex-display="false">n_{max}</span>, and the <span class="katex-eq" data-katex-display="false">\overline{\rm MS}</span> scheme result with <span class="katex-eq" data-katex-display="false">k=1</span> (shown as the blue dashed line) provides a reference for assessing the validity range of the expansion. — The logarithmic derivative of the scale-dependent strong coupling $\tilde{S}$ with respect to the renormalization scale τ exhibits truncation effects determined by the maximum expansion order $n_{max}$ , and the $\overline{\rm MS}$ scheme result with $k=1$ (shown as the blue dashed line) provides a reference for assessing the validity range of the expansion.

The Inevitable Search for Completion

The Standard Model of particle physics, despite its extraordinary predictive power, isn’t considered a complete theory, but rather an ‘Effective Field Theory.’ This means its descriptions become unreliable when probing extremely high energies, akin to a map that loses accuracy when zooming in too far. At these energy scales, previously negligible quantum effects become dominant, leading to divergences and breakdowns in predictability. Consequently, physicists posit the need for a more fundamental ‘UV Completion’ – a deeper theory that resolves these issues and accurately describes physics at all energy levels. This UV completion would effectively ‘complete’ the Standard Model, providing a seamless description from the lowest to the highest energies, and potentially explaining phenomena like dark matter or the origin of neutrino masses which currently lie outside the Standard Model’s scope.

The Operator Product Expansion (OPE) serves as a powerful lens for examining quantum field theories at extremely short distances, effectively probing the regime where ultraviolet (UV) divergences typically arise. This technique decomposes operator products into an infinite series of local operators, each characterized by its dimension and coupling constant, thereby revealing how the theory behaves as distances approach zero. By systematically analyzing these contributions, physicists can gain insights into the high-energy behavior of the theory and, crucially, identify potential UV completions – more fundamental theories that resolve the divergences and provide a complete description at all energy scales. The OPE doesn’t simply acknowledge the limitations of an effective field theory; it provides a structured method for exploring the landscape of possible underlying physics, guiding the search for a more complete and consistent description of nature.

This research demonstrates a successful reconstruction of ultraviolet (UV) behavior within both Quantum Electrodynamics (QED) and the $ℂPN-1$ model, effectively pushing the boundaries of established effective field theory techniques. Unlike traditional infrared (IR) expansions which are limited to low energies, this approach significantly extends the validity range, providing insights into physics at higher energy scales. Achieving this required a substantial computational effort, necessitating an expansion order of up to $n_{max} = 300$ to accurately resolve intricate multi-threshold effects – subtle interactions that become significant at higher energies and previously limited the precision of such calculations. This advancement not only validates the methodology but also offers a powerful tool for exploring the limitations of the Standard Model and guiding the search for more complete theories of fundamental interactions.

The extracted scale <span class="katex-eq" data-katex-display="false">S(Q^2)</span> (blue) closely matches the exact result from the UV theory, as indicated by the error band. — The extracted scale $S(Q^2)$ (blue) closely matches the exact result from the UV theory, as indicated by the error band.

Mapping the Landscape of Energy Scales

Many quantum field theories aren’t governed by a single energy scale, but rather exhibit multiple ‘thresholds’ where the fundamental nature of interactions changes. These thresholds arise as the energy of a system increases, bringing into play new particles or degrees of freedom that weren’t relevant at lower energies. Consider a scenario where initially only a few particles interact; as energy input rises, heavier particles become accessible, altering the strength of interactions and introducing new phenomena. This isn’t merely a mathematical curiosity; it reflects a hierarchical structure within the theory, where different physical regimes are defined by these energy scales. Understanding these multi-threshold behaviors is crucial because they dictate how a theory evolves with energy and can signal the presence of new physics beyond what is currently known, potentially revealing hidden sectors or more fundamental building blocks of reality.

Two-Flavor Quantum Electrodynamics (QED) serves as a particularly accessible model for illustrating the impact of multi-threshold effects on theoretical behavior. Unlike standard QED which includes all charged leptons, this simplified version considers only two flavors of charged particles. As energy increases in this system, distinct thresholds are encountered: first, the energy reaches a point where particle-antiparticle pairs of these two flavors can be created; then, at a higher energy scale, these particles themselves begin to affect the running of the electromagnetic coupling. This alters the strength of the force, and consequently, the predictions of the theory change measurably at each threshold. Studying this relatively simple system allows physicists to refine analytical techniques applicable to more complex theories, providing valuable insights into how new degrees of freedom influence the behavior of quantum field theories and potentially signal the presence of physics beyond the Standard Model.

A rigorous examination of energy thresholds in quantum field theories offers a powerful pathway toward a deeper understanding of fundamental interactions. These thresholds, representing the points at which new particles or interactions become relevant, dramatically alter a theory’s behavior and predictive power. By refining analytical methods to accurately describe physics near these thresholds – techniques like renormalization group flows and effective field theory – researchers can precisely map the landscape of quantum field theories. This detailed mapping doesn’t merely confirm existing models; it actively searches for inconsistencies or unexpected phenomena that hint at physics beyond the Standard Model. Subtle deviations from predicted behavior near a threshold, or the discovery of entirely new threshold structures, could provide the first experimental evidence of previously unknown particles or forces, opening up new avenues of exploration in particle physics and cosmology.

Ultraviolet reconstruction of the structure function <span class="katex-eq" data-katex-display="false">S</span> from infrared expansion in two-flavor quantum electrodynamics demonstrates convergence with increasing interpolation order (colors ranging from 0 to 5), closely matching the exact two-flavor (dot-dashed black) and single-flavor (dotted black) results, even when restricting the infrared series fit to <span class="katex-eq" data-katex-display="false">Q^2 < 4m_e^2</span>. — Ultraviolet reconstruction of the structure function $S$ from infrared expansion in two-flavor quantum electrodynamics demonstrates convergence with increasing interpolation order (colors ranging from 0 to 5), closely matching the exact two-flavor (dot-dashed black) and single-flavor (dotted black) results, even when restricting the infrared series fit to $Q^2 < 4m_e^2$ .

The pursuit of ultraviolet (UV) physics from infrared (IR) data, as detailed in this work, echoes a sentiment articulated by Marie Curie: “Nothing in life is to be feared, it is only to be understood.” Just as Curie approached the unknown with methodical investigation, this paper systematically reconstructs high-energy behavior – the UV completion – from low-energy expansions. The inverse Laplace transform acts as a tool for deciphering the system’s chronicle, revealing information typically obscured by conventional perturbative limits. This process isn’t merely about extending calculations; it’s about acknowledging that even as systems evolve – or decay – their fundamental principles remain accessible through careful analysis, a testament to the enduring power of understanding the underlying mechanics before attempting to predict their future states.

What Lies Beyond?

The reconstruction of ultraviolet physics from infrared data, as demonstrated, isn’t a resurrection, but a carefully managed decay. Every expansion, every approximation, is a momentary reprieve from the inevitable erosion of information. This work reveals not a path to complete knowledge, but a sophisticated method for postponing the encounter with the unknown. The limits of this approach, however, are not merely technical; they are inherent to the temporal nature of physical systems. Asymptotic freedom, a celebrated feature of the underlying theory, is less a liberation and more a temporary slowing of the march toward complexity.

Future investigations will undoubtedly focus on extending this reconstruction beyond perturbative regimes. Yet, the true challenge lies in acknowledging that complete ultraviolet control is an asymptotic ideal. Technical debt, accrued in the form of coarse-graining approximations, will always demand a present-day accounting. The question isn’t whether the reconstruction is perfect, but how gracefully it ages. A more fruitful avenue might be to explore the types of non-perturbative information that resist reconstruction, revealing the fundamental boundaries of predictability.

Ultimately, this work functions as a poignant reminder: the universe doesn’t offer solutions, only elegantly formulated problems. Each successful reconstruction is not an answer, but a more refined articulation of the question itself, a fleeting moment of clarity before the next layer of complexity descends.

Original article: https://arxiv.org/pdf/2603.03277.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/