Beyond Conservation Laws: Exploring Baryon and Lepton Number Violation

Author: Denis Avetisyan

New research delves into the theoretical landscape of physics beyond the Standard Model, investigating how fundamental conservation laws might be broken.

This review examines UV completions for dimension-6 baryon- and lepton-number-violating operators across various topologies and dimensions.

The Standard Model, while remarkably successful, offers no explanation for the observed baryon asymmetry of the universe, motivating searches for physics beyond its confines. This paper, ‘Opening up baryon-number-violating operators’, systematically explores the ultraviolet completions of non-derivative baryon-number-violating operators within an effective field theory framework, extending the analysis beyond commonly studied dimension-six terms up to mass dimension 15. We provide an exhaustive set of tree-level completions incorporating scalars, fermions, and neutrinos, alongside a public code for UV-completing any given operator and matching it to a defined basis. How will these higher-dimensional operator analyses refine our understanding of baryogenesis and the fundamental symmetries of nature?

The Illusion of Completeness

The predictive power of effective field theories, so valuable at describing low-energy phenomena, inevitably breaks down as energies increase. These theories, built upon approximations and expansions, accumulate inconsistencies and infinities when extrapolated to the ultraviolet (UV) regime – the realm of very high energies and short distances. This limitation isn’t a fatal flaw, but rather a signal that a more fundamental, complete theory – a UV completion – is necessary. Such a completion provides a description of physics at all energy scales, resolving the divergences and providing a consistent framework. It essentially ‘completes’ the effective theory by specifying the underlying dynamics at high energies, dictating how the low-energy interactions emerge from a more fundamental structure and ensuring a self-consistent description of nature. The search for these UV completions represents a central challenge in theoretical physics, driving investigations into new particles, interactions, and even the very fabric of spacetime.

The necessity of UV completions for effective field theories isn’t merely a mathematical exercise; it’s governed by stringent consistency conditions that dictate the permissible forms of new physics. As energies climb, seemingly innocuous low-energy descriptions break down, necessitating the introduction of additional particles and interactions to resolve divergences and maintain a logically sound framework. These conditions, often rooted in principles like unitarity, causality, and Lorentz invariance, severely constrain the landscape of possible UV completions. Consequently, the search for these high-energy completions isn’t a free-for-all, but a guided exploration of specific models that satisfy these fundamental criteria, potentially revealing new particles awaiting discovery and altering predictions for high-energy phenomena. $\mathcal{L}_{UV}$ must reproduce the low-energy effective theory while avoiding inconsistencies at higher scales.

The fundamental nature of a UV completion – the high-energy theory resolving inconsistencies of effective field theories – is deeply intertwined with its topology. This isn’t merely a mathematical curiosity; the ‘shape’ of the completion, defined by how its constituent fields connect and interact, fundamentally restricts the allowed interactions at all energy scales. A UV completion with a non-trivial topology-imagine a manifold with holes or twists-imposes stringent constraints on the possible couplings between particles, effectively pruning the landscape of theoretically viable models. Consequently, the topology dictates the resulting low-energy phenomenology, influencing observable signatures such as particle masses, decay rates, and interaction strengths. Researchers are increasingly focused on leveraging topological invariants and techniques-borrowed from areas like condensed matter physics-to classify UV completions and predict their phenomenological consequences, offering a powerful new avenue for connecting high-energy theory with experimental observation.

Higher Dimensions, Hidden Scales

Dimension-6 operators, also known as higher-dimensional operators, emerge in effective field theories when heavy particles are “integrated out” – meaning their direct contributions to low-energy processes are replaced by interactions involving only lighter, observable degrees of freedom. These operators are suppressed by powers of a new mass scale Λ, typically associated with the mass of the integrated-out particles; thus, their effects become noticeable at energies where $E \sim \Lambda$ . The presence of dimension-6 operators modifies Standard Model predictions, introducing potentially observable deviations in scattering cross-sections, decay rates, and other physical observables. Importantly, these operators represent all possible Lorentz-invariant interactions consistent with the Standard Model gauge symmetries, beyond those present in the Standard Model Lagrangian, and can significantly alter low-energy phenomenology without requiring the direct detection of the heavy particles themselves.

Topology 1 establishes a framework for embedding dimension-6 operators by defining specific operator products and their associated couplings within the Standard Model effective field theory. This embedding is not arbitrary; Topology 1 dictates permissible combinations of operators based on symmetry considerations and field content, restricting the forms of $\mathcal{O}_i$ that can arise from ultraviolet completions. Consequently, the resulting low-energy physics is constrained to those interactions allowed by the Topology 1 structure; for example, certain combinations of quark and lepton operators may be favored or suppressed based on the chosen embedding, directly impacting predictions for observables such as electroweak precision measurements and rare decays.

Analysis within the Topology 1 framework imposes specific constraints on the properties of new particles responsible for generating dimension-6 operators. These constraints arise from requiring the completion of the Standard Model effective field theory to be ultraviolet finite and consistent with observed low-energy physics. Specifically, the masses of these new particles are bounded from below, and their coupling strengths are restricted to maintain perturbativity and avoid Landau poles. The resulting bounds are dependent on the specific dimension-6 operator being considered and the assumed flavor structure of the new particle interactions. This process effectively links the scale of new physics to the size of the corresponding dimension-6 operator coefficients, providing a pathway to estimate the mass range of potential beyond-the-Standard-Model particles through precision measurements of these operators.

Expanding the Possibilities: Dimensions 7 and 8

The transition to dimension-7 and dimension-8 ultraviolet (UV) completions necessitates a refinement of permissible topological configurations. These higher-dimensional completions impose additional mathematical restrictions on the allowable shapes and connectivity of the underlying space; not all topologies viable in lower dimensions remain consistent with the physics dictated by these completions. Specifically, the presence of higher-dimensional operators within the UV completion alters the constraints governing the allowable geometry, effectively reducing the number of topologically distinct solutions that can consistently accommodate the theory’s requirements. This results in a more selective landscape of viable topologies as the dimensionality of the UV completion increases.

Analysis of dimensions 7 and 8 for UV completion demonstrates that topologies ranging from 2 to 8 are all theoretically viable. Specifically, dimension 7 completions are achievable utilizing topologies 1 and 2, while dimension 8 completions can be successfully implemented across the full range of topologies 1 through 8. This indicates that increasing the dimensionality of the completion space expands the number of permissible topological configurations, providing greater flexibility in constructing consistent high-energy completions of the theory.

Investigation of topologies 2 through 8 reveals a range of possible ultraviolet (UV) completions, each characterized by differing levels of fine-tuning required to reconcile theoretical predictions with observed physical parameters. The degree of fine-tuning represents the sensitivity of the model to initial conditions; higher degrees necessitate precise parameter choices to avoid inconsistencies. Furthermore, the stability of each UV completion-its resistance to quantum corrections and perturbations-varies significantly across the topologies studied. A comprehensive assessment identifies completions exhibiting robust stability alongside minimal fine-tuning, while others require substantial parameter adjustments or remain susceptible to instability under common theoretical considerations.

The Topology Landscape at Dimension 9

The requirement for dimension-9 ultraviolet (UV) completions expands the necessary topological considerations beyond those typically examined in lower-dimensional scenarios. Standard model extensions frequently analyze topologies 1 through 8; however, a consistent description of dimension-9 operators demands analysis up to topology 16. This broadened scope arises from the increased complexity of interactions at higher dimensions, where a wider range of geometric configurations are required to consistently embed the necessary operators and maintain gauge invariance. The examination of topologies 9 through 16 is crucial for identifying all possible UV completions and fully characterizing the landscape of physics beyond the Standard Model.

The embedding of operators and interactions is topology-dependent, meaning each of the considered topologies-ranging from 1 to 16-offers a distinct geometrical and algebraic structure for accommodating the necessary terms in the effective field theory. This dependency arises because the allowed operator structures are constrained by the topology’s global properties and symmetry groups. Consequently, different topologies permit different combinations and arrangements of these operators, resulting in a diverse set of potential ultraviolet (UV) completions that satisfy the same low-energy physics. For instance, the appearance of $\Delta B = 1$ , $\Delta L = -1$ operators at dimension 9 is realized through multiple topologies (1-16), while other charge combinations like $(2,0)$ and $(1,3)$ are limited to topologies 1 and 2, demonstrating a clear relationship between topological structure and the permissible operator content.

Analysis of dimension-9 ultraviolet (UV) completions indicates a diverse range of possible solutions beyond the Standard Model, categorized by their topological configurations and associated charges. Specifically, operators with baryon number (∆B) and lepton number (∆L) charges of (1, -1) are realized across all sixteen topologies (1-16) at dimension 9. Dimension-8 operators exhibiting (∆B, ∆L) charges of (1, 1) are limited to the first eight topologies. Furthermore, dimension-9 operators with (∆B, ∆L) charges of (2, 0) and (1, 3) are only found within the first two topologies. This distribution suggests a correlation between operator charges, dimensionality, and the underlying topological structure of potential UV completions.

The Shadow of the Unknown

The ultimate behavior of particles at low energies – their masses, how frequently they decay, and their interactions – is deeply connected to the hidden structure of physics at extremely high energies, known as the ultraviolet (UV) completion. This completion, representing the most fundamental description of reality, doesn’t exist as a single possibility; rather, it’s a landscape of potential topologies, or overall shapes and connections, that dictate how gravity and quantum mechanics intertwine. Different topologies impose unique constraints on the types of particles and forces that can emerge at lower energies, effectively sculpting the observable universe. For example, a UV completion with a complex, highly connected topology might allow for a wider range of particle masses and decay pathways than a simpler, more constrained one. Understanding these topological influences is therefore paramount to interpreting experimental data and unraveling the mysteries of particle physics, as the low-energy phenomenology serves as a crucial window into the otherwise inaccessible realm of high-energy completions.

A comprehensive mapping of potential ultraviolet (UV) completion topologies offers a crucial bridge between theoretical physics and experimental detection. This framework doesn’t simply posit abstract mathematical possibilities; instead, it systematically correlates high-energy, fundamental structures with low-energy, observable phenomena. By cataloging these topologies-the allowed ‘shapes’ of spacetime at extremely small scales-researchers can predict the resulting particle masses, interaction strengths, and decay patterns that might be detectable in experiments like those at the Large Hadron Collider. Essentially, the topology landscape transforms the search for new physics from a largely undirected endeavor into a more targeted one, allowing scientists to prioritize experimental searches based on the predicted signatures of specific UV completions and, ultimately, decode the fundamental laws governing the universe at its most basic level.

Continued investigation into the phenomenological ramifications of varying topological structures in ultraviolet (UV) completions represents a crucial next step in theoretical physics. Researchers are increasingly focused on translating abstract topological features into concrete, testable predictions for particle physics experiments. This involves detailed calculations of how different topologies affect observable quantities, such as particle masses, decay rates, and interaction cross-sections. A key challenge lies in identifying unique experimental signatures – deviations from Standard Model predictions – that would definitively indicate the presence of a non-trivial UV completion. Exploration of these signatures across various experimental frontiers, including high-energy colliders, precision measurements, and searches for new particles, promises to bridge the gap between theoretical models and empirical evidence, potentially revealing the underlying structure of reality beyond the Standard Model.

The search for UV completions, as detailed in this exploration of baryon-number-violating operators, feels less like a march toward understanding and more like peering into an abyss. The paper meticulously charts topologies and dimensions, attempting to define the boundaries of what is observable. Yet, it implicitly acknowledges the inherent limitations of any model constructed to describe reality. As Galileo Galilei observed, “You cannot teach a man anything; you can only help him discover it himself.” This research doesn’t reveal a fundamental truth, but rather provides a framework for further investigation, a map that will inevitably prove incomplete as it approaches the singularity beyond which all certainty dissolves. Any attempt to fully grasp these high-dimensional spaces, to define the precise nature of ∆B and ∆L violations, is ultimately an exercise in carefully constructed illusion.

Where Do the Paths Lead?

The exploration of baryon-number-violating operators, particularly within varied topological configurations, reveals not so much a path to completion, but an increasingly refined mapping of the territory where completion may be fundamentally impossible. The insistence on UV completions, while mathematically convenient, presupposes a certain optimism regarding the persistence of local field theory-an optimism that deserves careful scrutiny. Researcher cognitive humility is proportional to the complexity of nonlinear Einstein equations; as the dimensionality and topological intricacy increase, the temptation to impose familiar structures must be resisted.

Future work will inevitably confront the limitations inherent in effective field theory approaches. The search for operators characterized by specific (∆B, ∆L) quantum numbers, while offering a systematic framework, may ultimately lead to landscapes of solutions so vast and disconnected as to render predictive power illusory. The boundaries of physical law applicability and human intuition become starkly apparent when considering scenarios where even the notion of a consistent spacetime manifold is questionable.

Perhaps the true progress lies not in finding the UV completion, but in developing a more nuanced understanding of the conditions under which the very question becomes meaningless. To chase the completion is to assume the existence of a shore; the open ocean may prove to be a more honest reflection of reality.

Original article: https://arxiv.org/pdf/2603.17050.pdf

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