Beyond the Proton’s Lifetime: A New Framework for Baryon Violation

by

Author: Denis Avetisyan

This review details a systematic approach to understanding processes that could break fundamental symmetries governing matter stability.

Effective field theory provides a framework for evaluating the decay rate of the baryon designated BNV.
Effective field theory provides a framework for evaluating the decay rate of the baryon designated BNV.

A comprehensive effective field theory analysis, connecting chiral Lagrangians up to dimension-8 to the Standard Model Effective Field Theory, provides a pathway for interpreting experimental searches for nucleon decay.

Despite the Standard Model’s success, compelling theoretical motivations-such as Grand Unified Theories-predict baryon number violation, driving searches for nucleon decay. This motivates the work ‘Comprehensive Effective Field Theory Analysis for Baryon Number Violating Processes’, which systematically connects high-energy ultraviolet physics to low-energy hadronic observables through a rigorous effective field theory pipeline. By constructing chiral Lagrangians up to dimension eight and matching them to the Standard Model Effective Field Theory (SMEFT), this analysis demonstrates that dimension-eight Low-Energy EFT operators are crucial for incorporating a wider range of UV completions and fully realizing chiral representations of three-quark operators. Will these higher-dimensional operators prove essential for interpreting future experimental searches and ultimately revealing the origins of baryon asymmetry in the universe?

The Unfinished Standard: Seeking Completeness in Particle Physics

Despite its extraordinary predictive power and consistent validation through decades of experimentation, the Standard Model of particle physics remains incomplete. This highly successful theory, which describes the fundamental forces and particles of the universe, fails to account for observed phenomena like dark matter, dark energy, and the origin of neutrino masses. Furthermore, it offers no explanation for the significant imbalance between matter and antimatter in the cosmos. These unresolved puzzles strongly suggest the existence of physics beyond the Standard Model – a more comprehensive framework needed to fully understand the universe at its most fundamental level. Consequently, physicists are actively pursuing theoretical models and conducting experiments designed to probe for new particles, interactions, and symmetries that could resolve these outstanding questions and extend the reach of established knowledge.

The Standard Model of particle physics, despite its predictive power, fails to account for all observed phenomena, notably the non-zero mass of neutrinos and the significant imbalance between matter and antimatter in the universe. Neutrinos were initially considered massless within the Standard Model, but experimental evidence from neutrino oscillation definitively demonstrates they possess mass, necessitating an extension to the model to accommodate this property. Similarly, the observed prevalence of matter over antimatter-a fundamental asymmetry-cannot be explained by the Standard Model’s predictions, which anticipate nearly equal production in the early universe. These discrepancies strongly suggest the existence of new particles and interactions beyond those currently described, driving ongoing research into theoretical frameworks-such as supersymmetry or extra dimensions-capable of resolving these cosmological puzzles and providing a more complete understanding of the universe’s composition and evolution.

Many theoretical frameworks extending the Standard Model of particle physics predict processes that violate baryon number conservation, a fundamental symmetry in the established theory. This violation isn’t observed in typical interactions, but is considered a necessary condition for explaining the observed matter-antimatter asymmetry in the universe. Consequently, physicists are actively searching for evidence of proton decay – a prime example of baryon number violation – through massive underground detectors. These experiments, alongside searches for other rare processes, require incredibly precise theoretical calculations to distinguish potential signals from background noise and to accurately interpret any observed events. Developing these theoretical tools necessitates sophisticated models and advanced computational techniques to connect the high-energy scales where new physics might manifest to the low-energy phenomena accessible to current experiments, pushing the boundaries of both theoretical and experimental particle physics.

Bridging the gap between the incredibly high energies where new physics is expected to manifest and the comparatively low energies accessible in current experiments presents a formidable challenge. Theoretical frameworks like effective field theories are crucial in this endeavor, allowing physicists to systematically approximate the effects of undiscovered, heavy particles on measurable, low-energy processes. These approximations don’t predict the exact nature of the new physics, but rather define its potential influence through a series of parameters that can be constrained by experimental data. By meticulously analyzing deviations from Standard Model predictions – such as subtle changes in particle decay rates or magnetic moments – scientists can progressively refine these parameters and narrow down the possibilities for what lies beyond our current understanding. This approach allows for meaningful exploration of the high-energy frontier even without directly observing the new particles themselves, turning precision measurements into powerful probes of the universe’s hidden sectors.

Feynman diagrams illustrate the potential decay pathways of nucleons via baryon number violation (BNV).
Feynman diagrams illustrate the potential decay pathways of nucleons via baryon number violation (BNV).

Low-Energy Portraits: The Power of Effective Field Theory

Effective Field Theory (EFT) operates on the principle that at low energies, the detailed dynamics of high-energy physics are irrelevant; only the essential degrees of freedom and symmetries governing the phenomena at the energy scale of interest need to be considered. This allows for the construction of a simplified theoretical framework, the Effective Lagrangian, which incorporates all possible interactions consistent with those symmetries. By organizing terms in this Lagrangian by their dimensionality, EFT provides a systematic way to approximate physical processes, with lower-dimensional terms representing the dominant low-energy behavior and higher-dimensional terms representing suppressed corrections due to the separation of energy scales. This approach avoids the need to explicitly model or know the complete high-energy theory, focusing instead on predicting observable effects at the accessible energy range.

The Effective Lagrangian is the central component of the Effective Field Theory (EFT) approach, providing a framework to describe physics at a specific energy scale. It is constructed by including all possible operators – mathematical expressions built from fields and their derivatives – that are consistent with the underlying symmetries of the physical system. These operators are not necessarily fundamental but represent the degrees of freedom and interactions relevant at the chosen energy scale. The Lagrangian is generally expanded as an infinite series, ordered by the dimensionality of the operators, with lower-dimensional operators contributing the most significant effects at low energies and higher-dimensional operators representing suppressed corrections or interactions involving heavier, unresolved degrees of freedom. This allows for a systematic approximation of the full theory, focusing only on the relevant physics at the energy scale of interest.

Within the framework of Effective Field Theory, operators in the Effective Lagrangian are categorized by their mass dimension, d. Lower-dimensional operators, those with smaller values of d, contribute more significantly to low-energy physics due to their larger associated energy scales and thus, larger contributions to scattering amplitudes. Higher-dimensional operators, with d > 4, are suppressed by powers of the high-energy scale Λ as 1/\Lambda^{d-4}, meaning their effects become increasingly negligible at lower energies. This hierarchy allows for a systematic expansion in terms of 1/\Lambda, providing a controlled approximation to the full theory and focusing calculations on the most relevant low-energy dynamics.

Calculations within this framework have been extended to dimension-8 operators of the LEFT (Low-Energy Effective Field Theory) operator basis, providing a complete chiral representation specifically for processes where baryon number change ( \Delta B ) equals lepton number change ( \Delta L ). Furthermore, a definitive matching has been established between Standard Model Effective Field Theory (SMEFT) operators, calculated up to dimension 9, and their corresponding expressions within the LEFT operator basis; this matching allows for a consistent translation of high-energy physics constraints into the low-energy effective description and vice versa, ensuring a systematic approach to precision calculations and searches for new physics.

Connecting the Scales: Renormalization and Matching

The Renormalization Group Equation (RGE) describes the scale dependence of effective field theory (EFT) operator couplings. These couplings, representing the strength of interactions, are not constant but vary with the energy scale μ at which a calculation is performed. The RGE mathematically expresses this variation, allowing calculations performed at one energy scale to be reliably translated to predictions at a different scale. This is achieved by ‘running’ the couplings from an initial high-energy scale, where new physics might directly manifest, down to a lower energy scale relevant for experiments. The form of the RGE is determined by the beta functions associated with each operator, which depend on the specific EFT and the number of spacetime dimensions; these beta functions quantify the rate of change of the couplings with the energy scale. Consequently, the RGE provides a framework for systematically connecting high-energy and low-energy descriptions within an EFT, ensuring consistency across different energy regimes.

Matching is a perturbative procedure used to relate the parameters of two Effective Field Theories (EFTs) describing the same physics at different energy scales. This involves calculating loop corrections in one EFT and then re-expressing the results in terms of the parameters of the second EFT. The process determines how parameters in a low-energy EFT should be adjusted to accurately reproduce predictions from a high-energy EFT within its domain of validity. Specifically, matching ensures that observables calculated in both EFTs agree to a specified order in the expansion parameter, typically α or 1/\Lambda , where Λ represents the scale of new physics. By performing matching, one can systematically translate constraints on high-energy parameters into predictions for low-energy observables, and vice versa, enabling precise calculations across multiple energy regimes.

The Standard Model Effective Field Theory (SMEFT) and the Large Effective Operator Theory (LEFT) both employ the Renormalization Group Equation (RGE) to evolve operator couplings with the energy scale, and ‘matching’ procedures to relate parameters between different effective theories. This work specifically establishes the matching relations between SMEFT operators up to and including dimension 9 and their corresponding operators in LEFT. This matching is achieved by calculating the threshold corrections induced when moving between the two theories, effectively translating the effects of high-energy new physics, parameterized by SMEFT operators, into predictions for low-energy observables calculated within the LEFT framework. The established matching allows for a consistent analysis of new physics effects across different energy scales and experimental contexts.

This analysis establishes a direct correspondence between Standard Model Effective Field Theory (SMEFT) operators of dimension 6 and 7 and their ultraviolet (UV) completions, specifically UV models that generate those operators. This connection facilitates a systematic investigation of new physics contributions to baryon number violating processes, such as nucleon decay. By relating SMEFT parameters to the underlying UV model parameters, predictions for observables sensitive to baryon number violation can be derived and compared to experimental limits, thereby constraining the parameter space of potential new physics scenarios. The methodology allows for a quantitative assessment of how deviations from the Standard Model in SMEFT operators manifest as observable effects in processes governed by baryon number violation, providing a pathway to indirectly probe physics beyond the Standard Model.

The Hadronic Lens: QCD and Chiral Symmetry

Chiral symmetry represents a powerful simplification within the complex framework of Quantum Chromodynamics (QCD), particularly when investigating the interactions of hadrons at relatively low energies. While QCD, the fundamental theory of the strong force, is remarkably successful, its equations become intractable when applied to these scenarios. Chiral symmetry arises from the behavior of quarks-the building blocks of hadrons-in the limit of massless quarks. This approximate symmetry allows physicists to bypass the full complexity of QCD by focusing on the relevant degrees of freedom and interactions at lower energies. Essentially, it provides a systematic way to organize calculations and make predictions about hadronic phenomena, such as particle decays and scattering processes, that would otherwise be impossible to compute directly from the underlying QCD theory.

The Chiral Lagrangian represents a powerful theoretical tool for investigating the strong force, specifically designed to describe interactions between hadrons at low energies. This effective field theory doesn’t attempt to directly solve the full complexities of Quantum Chromodynamics (QCD), but instead leverages the approximate chiral symmetry inherent within it. By focusing on the lightest hadrons – primarily pions and nucleons – and exploiting the symmetries governing their interactions, the Chiral Lagrangian provides a simplified yet remarkably accurate framework. It expresses hadronic interactions in terms of a limited number of parameters and systematically incorporates higher-order corrections, allowing physicists to connect fundamental QCD symmetries to observable phenomena and make quantitative predictions for processes like pion scattering and nucleon decay – a crucial step in searching for physics beyond the Standard Model.

The connection between chiral symmetry and observable hadronic phenomena extends to predictions regarding nucleon decay, a process fundamentally linked to the strong force. Current research leverages this symmetry to derive a functional form for the rate of nucleons decaying into multiple particles – specifically, Γ​(N→k​particles)∼25−4​k​π3−2​k2​mp​mp2​k−4​|ℳ|2(k−1)!​(k−2)!. This equation describes how the decay rate scales with the number of decay products (k), nucleon masses (mp), and a matrix element (|ℳ|) representing the underlying dynamics of the decay. By establishing this quantitative relationship, physicists gain a powerful tool for interpreting experimental results and searching for deviations from the Standard Model, potentially revealing new physics beyond currently understood interactions.

The convergence of the Chiral Lagrangian – a framework rooted in the approximate chiral symmetry of Quantum Chromodynamics – with the techniques of Effective Field Theory provides a powerful means of calculating hadronic processes with remarkable precision. This combined approach allows physicists to systematically address the complexities arising from the strong force, effectively ‘zooming in’ on low-energy interactions while absorbing the details of high-energy dynamics into a limited number of parameters. Crucially, these calculations aren’t merely about confirming existing knowledge; the precision achievable with this methodology opens a window for identifying subtle deviations from Standard Model predictions. Any discrepancy, however small, could signal the presence of new particles or interactions, guiding the search for physics beyond our current understanding, and providing a pathway to unraveling some of the universe’s deepest mysteries.

The pursuit of understanding baryon number violation, as detailed in this analysis, demands a ruthless paring away of complexity. This work systematically constructs chiral Lagrangians, not as ends in themselves, but as increasingly precise approximations of reality. It echoes a sentiment articulated by Ludwig Wittgenstein: “The limits of my language mean the limits of my world.” Each higher-dimensional operator included is a tentative expansion of the ‘world’ describable by the effective field theory, but always mindful that the true physics may lie beyond the current level of approximation. The careful operator matching to the Standard Model Effective Field Theory (SMEFT) represents a constant effort to define those limits, striving for a self-evident connection between theoretical prediction and potential experimental observation.

Where Do We Go From Here?

The construction of chiral Lagrangians, even to a modest dimension like eight, reveals a truth often obscured by enthusiasm: the more precisely one attempts to describe nature, the more numerous become the parameters demanding determination. They called it a framework to hide the panic, a systematic way to organize ignorance. This work delivers that, and it is valuable. However, it simultaneously highlights the growing tension between theoretical ambition and experimental reach. Each additional operator, each new free parameter, requires an experimental probe sensitive enough to constrain it – a demand that grows exponentially with dimension.

The connection to the Standard Model Effective Field Theory, while elegant, is not a resolution. It merely shifts the burden of parameter estimation, adding coefficients in the SMEFT to those already present in the chiral Lagrangian. The true test lies not in the construction of these effective theories, but in the capacity to extract meaningful information from nucleon decay searches – and current limits suggest a long and arduous path. The focus, therefore, must shift toward identifying the most sensitive decay modes, and designing experiments specifically tailored to probe the dominant operators.

Simplicity, after all, is not a goal to be achieved through increasingly complex models. It is a property to be discovered in nature. Perhaps the absence of observed baryon number violation is not a signal of exceedingly high mass scales, but an indication that the underlying theory is, at its core, remarkably economical. That, of course, would be a far more satisfying conclusion.

Original article: https://arxiv.org/pdf/2603.11158.pdf

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

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2026-03-14 13:46