Beyond the Standard Model: Mapping the Unknown with Higher Dimensions

Author: Denis Avetisyan

New calculations refine the theoretical tools needed to search for physics beyond our current understanding of the universe.

This research presents the complete one-loop Renormalisation Group Equations for dimension-eight operators in the Standard Model Effective Field Theory.

Despite the Standard Model’s continued success, limitations necessitate exploring beyond-the-Standard-Model physics through frameworks like the Standard Model Effective Field Theory. This thesis, ‘Renormalization of the Standard Model effective field theory to dimension eight’, systematically investigates the renormalization of this framework, focusing on the challenging computation of one-loop Renormalization Group Equations for dimension-eight operators. These calculations, performed within an off-shell Green’s function basis and leveraging symmetry arguments, provide a complete set of results essential for high-precision analyses and consistent phenomenological applications. Will these findings enable more robust constraints on new physics and refine our understanding of the universe at the highest energy scales?

Beyond Simplification: The Inherent Incompleteness of Our Models

Despite its remarkable predictive power, the Standard Model of particle physics remains incomplete, prompting the search for extensions to address several fundamental questions. The model breaks down when attempting to describe gravity, dark matter, and dark energy, and offers no explanation for the observed matter-antimatter asymmetry in the universe. Furthermore, the Standard Model requires fine-tuning of certain parameters to maintain its consistency, suggesting the existence of new physics at higher energy scales that could naturally stabilize these values. These limitations indicate that the Standard Model is likely an effective theory, a successful approximation of a more complete, underlying description of reality that operates at energies currently beyond experimental reach. Consequently, physicists are actively pursuing various theoretical frameworks-such as supersymmetry, extra dimensions, and compositeness-to construct models that extend the Standard Model and provide answers to these outstanding puzzles.

Effective Field Theory (EFT) offers a powerful and pragmatic approach to exploring physics beyond the Standard Model, circumventing the need for a complete, high-energy “ultraviolet” (UV) completion. Rather than attempting to define the fundamental laws governing physics at extremely high energies – a task often beyond current experimental reach – EFT focuses on describing observable phenomena at lower, accessible energies. It achieves this by systematically parameterizing the effects of unknown high-energy physics through a series of operators, organized by their dimensionality and suppressed by powers of a high-energy scale. This allows physicists to predict experimental outcomes and constrain potential new physics models without needing to know the precise details of the underlying UV theory. Essentially, EFT distills the essential low-energy consequences of new physics into a manageable set of parameters, enabling meaningful progress even in the absence of a complete theoretical picture.

Rather than demanding a complete understanding of physics at the highest energy scales – a task often beyond current or foreseeable experimental capabilities – the effective field theory approach prioritizes predicting observable phenomena within the reach of existing experiments. This methodology acknowledges that the fundamental, “ultraviolet” (UV) theory may remain elusive, yet allows physicists to systematically explore the consequences of potential new physics through lower-energy, “effective” interactions. By parameterizing these interactions using a series of operators suppressed by high-energy scales, researchers can focus on measurable effects, such as deviations from Standard Model predictions, and constrain the properties of the unknown UV completion. This pragmatic strategy shifts the emphasis from constructing a complete theory to extracting meaningful information from accessible data, providing a powerful tool for probing the frontiers of particle physics even with incomplete knowledge.

Taming the Infinities: Renormalization as a Necessary Correction

Perturbative calculations in quantum field theory utilize Feynman diagrams to approximate scattering amplitudes. Diagrams containing closed loops – known as loop diagrams – integrate over all possible momenta, often resulting in integrals that diverge, meaning they yield infinite values. These divergences arise because the perturbative approach assumes calculations are valid at all energy scales, which is physically unrealistic. While mathematically problematic, these divergences do not necessarily indicate a failure of the theory; rather, they signal that the initial parameters used in the calculation require modification to account for quantum effects at high energies. Consequently, the presence of these divergences obscures the extraction of finite, physically meaningful predictions for observable quantities.

Renormalization addresses the issue of divergences arising in quantum field theory calculations by systematically absorbing them into redefinitions of observable physical parameters, such as mass and charge. This is achieved by introducing corrections to the calculated quantities that precisely cancel the divergent terms, yielding finite, physically meaningful predictions. The procedure doesn’t eliminate the infinities at intermediate steps, but rather confines them to these redefinitions, ensuring that final, measurable quantities remain finite. Effectively, the originally calculated parameters in the Lagrangian are understood as ‘bare’ parameters, and the physical, measured parameters are expressed as the sum of the bare parameter and a divergent correction, which is then cancelled by the corresponding term arising from loop calculations. This allows for predictions to be made even in regimes where perturbative calculations would otherwise break down due to the presence of infinities.

Counterterms are introduced into the Lagrangian as additional terms specifically designed to cancel the divergent integrals arising from loop calculations in perturbative quantum field theory. These terms, which themselves contain infinities, are constructed to precisely offset the divergences generated when calculating physical quantities. The process involves adding these divergent counterterms and then systematically absorbing the infinities into redefinitions of observable parameters like masses and coupling constants. This results in finite, physically meaningful predictions despite the presence of initial divergences, enabling accurate calculations of physical processes. The introduction of counterterms is not merely a mathematical trick; it reflects the inherent limitations of perturbative expansions and the need for a consistent renormalization scheme.

One-loop renormalization constitutes a critical stage in absorbing divergences arising from perturbative calculations, and necessitates meticulous treatment of infinite quantities. This work completed the first full computation of the Renormalization Group Equations (RGEs) to one-loop order for dimension-eight operators within the Standard Model Effective Field Theory (SMEFT) framework. The complete calculation of these RGEs is a substantial advancement, as it allows for more precise predictions of SMEFT parameters and improves the accuracy of calculations probing physics beyond the Standard Model. These dimension-eight operators represent the lowest order at which new physics effects, beyond those captured by dimension-five operators, can significantly influence observable phenomena and require this level of renormalization for precise theoretical prediction.

Organizing the Landscape: A Systematic Basis for Higher-Dimensional Operators

The Green’s Basis is a systematic construction method for higher-dimensional operators within the Standard Model Effective Field Theory (SMEFT). It defines a set of operators, organized by their mass dimension-typically starting at dimension-six-that represent all possible deviations from the Standard Model predicted by renormalizability. This basis isn’t simply a list; it’s built using a specific algorithmic procedure that ensures linear independence and facilitates the identification of operator redundancies. Each operator in the Green’s Basis is a local, Lorentz-invariant extension of the Standard Model Lagrangian, constructed from Standard Model fields and their derivatives. The resulting operator set provides a complete and non-redundant parameterization of new physics effects at energy scales accessible by current and future experiments, allowing for a consistent and unambiguous analysis of experimental data.

The Green’s Basis for constructing higher-dimensional operators in the Standard Model Effective Field Theory (SMEFT) is not a freely chosen parameterization; its structure is dictated by the underlying symmetries of the Standard Model. These constraints, formalized within the ‘SymmetryConstraints’ framework, leverage principles of ‘GroupTheory’ to identify and eliminate redundant or physically inequivalent operators. Specifically, the basis construction explicitly enforces invariance under the Standard Model gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$ and, critically, respects any accidental symmetries present in the Lagrangian. This systematic application of symmetry principles significantly reduces the number of independent operators that must be considered, thereby simplifying calculations and ensuring the resulting effective theory remains consistent with fundamental physical laws.

Exploiting the symmetries present in the Standard Model Effective Field Theory (SMEFT) significantly reduces the number of independent operators required in calculations. Without considering these symmetries, a naive enumeration of all possible higher-dimensional operators would yield a vastly larger, and largely redundant, operator basis. Symmetry constraints, derived through group theory, identify and eliminate operators that are equivalent – either through field redefinitions or equations of motion – to others already present in the basis. This reduction in the number of independent operators directly translates to fewer free parameters to determine from experimental data, and substantially simplifies the computational complexity of processes involving these higher-dimensional effects, such as renormalization group running and loop calculations.

Constructing the SMEFT with a systematic operator basis, constrained by identified symmetries, is crucial for maintaining consistency with established physical principles. Without such constraints, the effective theory would allow for terms violating Lorentz invariance, gauge symmetry, or other fundamental tenets of the Standard Model. The application of ‘SymmetryConstraints’ and ‘GroupTheory’ ensures that each operator included in the effective Lagrangian is compatible with these symmetries, preventing the appearance of unphysical or experimentally ruled-out phenomena. This rigorous approach guarantees that predictions derived from the SMEFT are physically meaningful and align with established experimental bounds, thereby preserving the validity of the effective theory as a reliable approximation of more complete, underlying physics.

The Evolving Landscape: Energy Scale Dependence and the Renormalization Group

Renormalization Group Equations (RGEs) mathematically describe the evolution of coupling constants associated with effective operators within the Standard Model Effective Field Theory (SMEFT) as the energy scale of a physical process changes. These couplings, representing the strength of interactions beyond the Standard Model, are not fixed but rather ‘run’ with energy due to quantum effects. Specifically, RGEs detail how these couplings are modified by loop diagrams involving both Standard Model and higher-dimensional operators. The form of these equations is typically perturbative, expressed to a given loop order; a one-loop calculation represents the lowest-order correction and provides a substantial improvement over tree-level estimates. The running of couplings is crucial because experimental measurements are performed at specific energy scales, and theoretical predictions must be expressed in the same frame of reference to facilitate comparison.

Renormalization Group Equations (RGEs) provide the theoretical framework for relating physical predictions calculated at one energy scale to those at another. This connection is vital because experimental measurements occur at specific, accessible energies, while theoretical calculations are often performed at a convenient, but potentially different, scale. The process of ‘running’ couplings – evolving their values with energy – via RGEs ensures the consistency and accuracy of predictions when compared to experimental data. Specifically, RGEs allow for the translation of parameters determined at a high energy scale, relevant to new physics, down to the lower energies probed by current experiments, and conversely, for interpreting experimental results in terms of underlying high-scale physics. Precise matching of theoretical predictions to experimental data therefore fundamentally relies on the accurate computation and application of these energy scale-dependent relationships.

Renormalization Group Equation (RGE) calculations frequently employ techniques such as Dimensional Regularization to handle divergences arising from loop integrals. This method involves analytically continuing the number of spacetime dimensions from an integer value (typically 4) to a non-integer value, allowing for the regularization of divergent integrals and the subsequent extraction of finite, physically meaningful results. These calculations are most efficiently performed within Momentum Space, where the relevant quantities – such as scattering amplitudes and propagator functions – are expressed as functions of particle momenta. Utilizing Momentum Space simplifies the evaluation of loop integrals and facilitates the application of regularization schemes necessary for obtaining finite predictions from quantum field theory calculations involving RGEs.

This work presents a computation of the Renormalization Group Equations (RGEs) to one-loop order for Standard Model Effective Field Theory (SMEFT) operators. These calculations extend the existing theoretical framework by including operators up to and including dimension 8, surpassing previously published results limited to dimension 6 and 7. The one-loop order calculation provides increased precision in predicting the energy scale dependence of SMEFT couplings, which is crucial for connecting theoretical predictions to experimental measurements and for performing high-precision phenomenology. The inclusion of dimension 8 operators broadens the scope of the effective theory and allows for a more complete analysis of potential new physics contributions beyond the Standard Model.

Beyond Direct Discovery: Precision Analyses as Probes of the Unknown

The Standard Model Effective Field Theory (SMEFT) provides a framework for systematically incorporating potential new physics beyond the established particles and forces. While initial analyses often focus on ‘dimension-five’ and ‘dimension-six’ operators – representing the simplest deviations from the Standard Model – a more comprehensive search necessitates the inclusion of ‘dimension-eight’ operators. These higher-dimensional terms, suppressed by additional powers of an energy scale associated with the new physics, offer a more complete and nuanced description of potential effects. By considering these operators, physicists can probe a wider range of new physics scenarios and refine the sensitivity of searches at high-energy colliders and other precision experiments, effectively widening the net to capture subtle hints of phenomena beyond the current understanding of the universe.

The effects of seemingly isolated new physics aren’t static; rather, the strengths of higher-dimensional operators within the Standard Model Effective Field Theory (SMEFT) evolve with energy. This evolution is dictated by the Renormalization Group Equations (RGEs), which describe how coupling constants change at different energy scales. Critically, these RGEs don’t just modify individual operator strengths – they induce ‘OperatorMixing’, where the influence of one operator bleeds into others. $O_i(μ)$ and $O_j(μ)$ may appear independent at a low energy scale μ, but at higher energies, the RGE-induced mixing can dramatically alter their predicted contributions to physical processes. Consequently, a precise determination of these operators requires not only measuring their initial values but also accurately accounting for this energy-dependent mixing, adding complexity to the search for physics beyond the Standard Model.

Precision analyses represent a cornerstone in the search for physics beyond the Standard Model, employing high-energy colliders and a variety of complementary experiments to meticulously test the limits of current understanding. These investigations don’t seek direct observation of new particles, but rather subtle deviations from the Standard Model’s predictions – discrepancies that could signal the influence of undiscovered phenomena. By precisely measuring established physical processes – such as the properties of the Higgs boson, the interactions of W and Z bosons, and the behavior of quarks and leptons – researchers can constrain the possible contributions of higher-dimensional operators within the Standard Model Effective Field Theory (SMEFT). The goal is to determine if observed interactions align perfectly with theoretical expectations, or if there’s evidence of ‘new physics’ manifesting as anomalous rates, altered particle decays, or unexpected correlations, effectively pushing the boundaries of known physics with increasing accuracy.

The search for physics beyond the Standard Model increasingly relies on a sophisticated program of precision analyses, extending beyond the most basic theoretical frameworks. While initial searches focused on directly producing new particles, current efforts meticulously examine subtle deviations in known particle interactions. This involves calculating how hypothetical ‘dimension-eight operators’ – representing more complex new physics scenarios – would affect measurable quantities. Critically, these higher-dimensional operators don’t act in isolation; the ‘running’ of couplings, as described by the Renormalization Group Equations, causes them to ‘mix’ with lower-dimensional operators, subtly altering predicted effects. By precisely measuring properties of particles and their interactions at facilities like the Large Hadron Collider, scientists can constrain the size of these operators and, crucially, detect any statistically significant departure from Standard Model predictions, potentially revealing the first concrete evidence of new fundamental physics.

The meticulous calculation of Renormalisation Group Equations, as demonstrated in this research, mirrors a fundamental drive to understand underlying order – or the lack thereof. This pursuit echoes Thomas Hobbes’ observation that “the passions of men will not be contained by reason.” While this work seeks to define the boundaries of the Standard Model through rigorous mathematical frameworks-specifically, dimension-eight operators-it implicitly acknowledges the potential for unforeseen complexities beyond current understanding. The study’s focus on refining theoretical precision, though mathematically driven, implicitly prepares for scenarios where the ‘natural order’ requires constant recalibration against emerging experimental evidence. This careful approach is vital; technology that scales but erodes trust in foundational principles is unworthy of deployment.

Where Do We Go From Here?

The computation of renormalization group equations to dimension eight within the Standard Model Effective Field Theory represents a refinement, not a resolution. It is a more detailed map of the parameter space where new physics might reside, yet offers no guarantee of arrival. The elegance of effective field theory lies in its ability to isolate what is calculable from what remains unknown, but this separation is always provisional. Each additional loop order reveals not only more precision, but also a deepening awareness of the assumptions encoded within the truncation scheme.

The immediate path forward involves extending these calculations to even higher dimensions, an exercise in technical mastery that simultaneously risks obscuring the underlying conceptual challenges. The true difficulty, however, lies not in computational complexity, but in the interpretation of results. Anomalous dimensions, while formally well-defined, acquire meaning only when confronted with experimental data-data that, thus far, remains stubbornly aligned with the Standard Model. This prompts a necessary, if uncomfortable, question: is the pursuit of ever-more-precise parameter constraints merely a sophisticated form of confirmation bias?

Ultimately, the value of this work, and indeed of the entire SMEFT program, will be determined not by its mathematical sophistication, but by its ability to guide experimental searches and to illuminate the fundamental principles governing particle physics. Technology is an extension of ethical choices; every automation bears responsibility for its outcomes. The parameters calculated here are not merely numbers to be measured, but placeholders for discoveries yet to be made-or, perhaps, for the limits of current theoretical frameworks.

Original article: https://arxiv.org/pdf/2602.05461.pdf

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