Symmetry’s Role in Quark Mixing

Author: Denis Avetisyan

A new model leveraging modular symmetry offers a compelling explanation for the origins of quark masses, mixing angles, and CP violation.

This paper explores a phenomenologically viable quark flavor model based on S4′ modular symmetry near the fixed point τ = i∞, addressing the CKM matrix and mass hierarchy.

Explaining the observed patterns in quark masses and mixing remains a central challenge in particle physics, often requiring significant fine-tuning of model parameters. This paper investigates a solution within the framework of the $S^\prime_4$ Quark Flavour Model in the Vicinity of the Fixed Point $τ= i\infty$, exploring how modular symmetry can generate realistic quark properties with minimal input. We demonstrate that by considering the model near a specific fixed point, a viable description of quark masses, mixing, and CP violation can be achieved using only nine real parameters. Could this approach offer a pathway toward a more natural and predictive flavor model in particle physics?

The Quark Puzzle: Beyond the Standard Model’s Reach

Despite its remarkable predictive power, the Standard Model of particle physics presents a significant puzzle when it comes to quarks – the fundamental building blocks of protons and neutrons. While the model accurately describes the masses and interactions of these particles, it fails to explain the observed patterns within them. Quarks aren’t simply randomly assigned masses; instead, they exhibit a distinct hierarchy, with some being vastly heavier than others. Furthermore, the way quarks “mix” – transitioning between different types – doesn’t align with what the Standard Model predicts if these patterns are merely accidental. This suggests that the underlying principles governing quark behavior extend beyond the current framework, hinting at a more profound, yet undiscovered, symmetry or dynamic at play in the universe. The observed patterns aren’t just numbers; they represent a critical clue demanding a more fundamental understanding of nature’s rules.

The persistent challenge in particle physics lies not simply in measuring the properties of quarks, but in explaining why those properties manifest as they do. Current models often rely on arbitrarily assigning values – a process known as parameterization – which, while descriptive, offers no fundamental insight. A deeper understanding necessitates a shift towards exploring underlying symmetries – hidden mathematical relationships that could dictate quark masses and mixing patterns. This approach posits that the observed values aren’t random, but emerge naturally from a more elegant, symmetrical framework, much like the facets of a crystal reveal a hidden order. Identifying these symmetries, however, demands innovative theoretical tools and a willingness to move beyond the established, yet incomplete, Standard Model, potentially unlocking a more unified and predictive picture of the fundamental building blocks of matter.

The perplexing disparity in quark masses – spanning several orders of magnitude – presents a significant challenge to contemporary particle physics. Existing theoretical frameworks often require substantial fine-tuning of parameters to accommodate this hierarchy, a practice considered inelegant and suggestive of a deeper, undiscovered principle. Attempts to dynamically generate these masses, relying on symmetry breaking or radiative effects, frequently fall short of accurately reproducing the observed values without introducing unnatural complexities. Consequently, physicists are actively pursuing innovative solutions, including exploring novel symmetry groups, extra spatial dimensions, and alternative models of flavor physics, all in the hope of uncovering a natural explanation for this fundamental puzzle and providing a more complete understanding of the building blocks of matter.

Modular Symmetry: A New Lens on Flavor Dynamics

Modular symmetry proposes a departure from traditional approaches to the flavor problem by employing mathematical transformations – specifically, modular transformations derived from the study of elliptic curves – to directly constrain the form of quark mass matrices. Unlike models relying on ad-hoc symmetries or parameters, this framework leverages the structure of $SL(2, \mathbb{Z})$ to reduce the number of free parameters required to describe quark masses and mixing angles. The core principle involves imposing a symmetry on the mass matrices under these modular transformations, effectively linking the parameters governing different flavor sectors and generating specific relationships between observable quantities. This constraint arises from the requirement that the physics remains invariant under these transformations, thus dictating the allowed forms for the mass matrices and their associated Yukawa couplings.

The parameter $\Tau$ within modular symmetry serves as a complex variable central to defining the discrete transformations that dictate flavor interactions. Specifically, $\Tau$ resides in the upper half of the complex plane and governs the behavior of quark mass matrices under modular symmetry groups, such as $\Gamma_n$ . These transformations, elements of the modular group, act on $\Tau$ and subsequently alter the quark mass matrices in a predictable manner, establishing relationships between the masses and mixing angles of quarks. The specific choice of modular group and the value of $\Tau$ determine the precise form of these interactions and, therefore, the observed flavor structure.

Within the modular symmetry framework, the masses and mixing angles of quarks are not independent parameters but are instead determined by a limited number of parameters defining the modular transformations. Specifically, quark mass matrices are expressed as vacuum expectation values of scalar fields transforming under a discrete modular symmetry group, $\Gamma_n$ . The values of these masses and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, which describe quark mixing, are then functions of the complex parameter τ and the transformation properties of the relevant fields. This interconnectedness reduces the number of free parameters required to describe quark flavor, offering a more predictive and unified description compared to traditional approaches where these quantities are treated as independent inputs.

S4′ Symmetry: Constraining the Flavor Landscape with Precision

The symmetric group S4′ – a non-Abelian discrete group of order 24 – offers a mathematically rigorous framework for constructing quark mass models that align with experimental observations. Unlike continuous symmetries, discrete symmetries like S4′ are not spontaneously broken, necessitating their explicit breaking via Yukawa couplings. This breaking pattern, governed by the group’s representation theory, directly impacts the resulting quark mass and mixing parameters. The robustness of S4′ stems from its ability to accommodate the Standard Model gauge interactions while simultaneously imposing constraints on the free parameters required to describe quark masses, ultimately leading to predictions that can be tested against experimental data; its specific structure allows for a relatively small number of free parameters while still generating the observed flavor patterns.

Quarks, as fundamental fermions in the Standard Model, are assigned to specific irreducible representations within the S4′ symmetry group, denoted as S4PrimeRepresentation. These representations-1, 2, and 3-define how quarks transform under the group’s operations, directly impacting the allowed Yukawa interactions within the QuarkMassMatrices. Specifically, the left-handed and right-handed quarks of each generation must transform according to these representations, leading to a constrained set of possible interaction terms. The assignment of quarks to these representations is crucial; for example, certain representations may only allow for mixing between specific quark flavors, effectively dictating the flavor structure of the Standard Model and providing a framework to explain the observed mass and mixing patterns.

The S4′ symmetry imposes specific constraints on the structure of the quark mass matrices for both up-type and down-type quarks, reducing the number of free parameters required to describe their masses and mixing patterns. Rather than requiring arbitrary entries, the S4′ symmetry dictates relationships between the elements of these matrices, effectively linking the masses of different quark generations. This framework achieves a hierarchical mass structure – the significant differences in mass between the light, medium, and heavy quarks – with a minimal set of 9 real parameters defining the entire flavor landscape. The resulting matrices are not fully determined, allowing for some flexibility in fitting experimental data, but the imposed constraints significantly reduce the parameter space compared to models without such symmetry.

Vacuum Expectation Values and Fixed Points: Unveiling Underlying Geometries

The Vacuum Expectation Value (VEV) is a fundamental quantity in the Standard Model and extensions thereof, responsible for spontaneous symmetry breaking. Initially, the equations describing fundamental particles exhibit symmetry, implying massless particles. However, the VEV, a non-zero average value of a quantum field in its ground state, breaks this symmetry. This breaking mechanism imparts mass to quarks through interactions with the Higgs field, which acquires its VEV. Specifically, the quark mass $m_q$ is proportional to the quark’s Yukawa coupling $y_q$ multiplied by the VEV $v$ : $m_q = y_q v$ . Without a non-zero VEV, quarks would remain massless, and the observed particle spectrum would be fundamentally different.

The selection of a specific FixedPoint for the ModulusTau parameter is fundamental to the computational tractability of the model. By choosing an appropriate FixedPoint, complex calculations involving the ModulusTau field are significantly simplified, allowing for a more efficient determination of relevant physical parameters. More importantly, this choice directly influences the resulting mass hierarchy observed in the Standard Model. Different FixedPoints lead to variations in the mass spectrum of the resulting particles, and the chosen FixedPoint must align with observed mass relationships to maintain consistency with experimental data; specifically, it allows for the derivation of particle masses based on the geometrical properties of the Calabi-Yax space associated with the ModulusTau parameter.

Statistical analysis of the model demonstrates a goodness of fit value of Δχ² = 0.499 when evaluated at very high supersymmetry (SUSY) breaking scales, specifically where $MSUSY \approx MGUT$ . This indicates a strong correlation between the model’s predictions and observed data under these conditions. However, when utilizing ‘inclusive’ datasets – encompassing a broader range of experimental measurements – the goodness of fit degrades to Δχ² = 6.99, suggesting the model may require refinement to accurately represent the full scope of available data.

Beyond the Standard Model: Implications for CP Violation and Future Exploration

The Standard Model of particle physics accommodates Charge-Parity (CP) violation – a crucial asymmetry between matter and antimatter – through a complex phase within the Cabibbo-Kobayashi-Maskawa (CKM) matrix. However, the origin of this phase remains a puzzle. The S4′ symmetry framework presents a compelling solution by naturally embedding this complex phase, providing a geometrical origin for CP violation. This approach doesn’t simply allow for CP violation; it predicts the magnitude of the complex phase based on the underlying symmetry structure and the specific arrangement of quark fields. Consequently, the S4′ model offers a predictive landscape where the observed CP-violating effects are not arbitrary parameters, but rather consequences of a deeper, more fundamental symmetry at play, potentially connecting the observed quark mixing patterns to a larger framework beyond the Standard Model.

Detailed calculations within the S4′ symmetry model have yielded specific analytical expressions defining the degree of mixing amongst up-type quarks. These calculations reveal that the sine of the angle $\theta_{13}^u$ -a key parameter describing quark mixing-is directly proportional to $64\sqrt{2}|q_4|^3$ , while the sine of $\theta_{23}^u$ scales with $\frac{1}{2}|q_4|^2$ . Notably, the sine of $\theta_{12}^u$ is expressed as $-2\sqrt{2}|q_4|$ . These precise relationships-linking mixing angles directly to the magnitude of a single parameter, $|q_4|$ -provide a testable framework for evaluating the model’s validity and potentially revealing connections between quark mixing and underlying symmetries.

This framework offers more than just a theoretical construct; it possesses a tangible predictive capability crucial for advancing particle physics. By establishing clear relationships between fundamental parameters and measurable quantities – such as the angles governing quark mixing – researchers gain specific targets for experimental investigation. This predictive power extends beyond simply confirming the model’s validity; it actively directs searches for new physics beyond the Standard Model by suggesting where deviations from expected values might appear. Consequently, experiments designed to precisely measure quark mixing parameters, or to search for rare decays, can be strategically focused, increasing the likelihood of uncovering evidence of physics currently beyond detection and potentially revolutionizing the understanding of fundamental forces and particles.

The exploration of S4′ modular symmetry, as detailed in the paper, reveals how fundamental parameters governing quark masses and mixing-including the intricacies of CP violation-arise from a constrained mathematical structure. This process echoes a broader principle articulated by Thomas Kuhn: “The world does not speak to us directly, but through the conceptual structures we impose upon it.” The model doesn’t discover inherent quark properties; rather, it demonstrates how those properties emerge as consequences of the chosen symmetry framework. The selection of S4′ and the approach to the fixed point τ= i∞, therefore, isn’t merely a technical exercise, but a conceptual imposition shaping the observed reality of flavor physics. It highlights how deeply our understanding is tied to the frameworks we build, and how alternative structures could yield equally valid, yet different, descriptions of the universe.

Beyond the Fixed Point

The pursuit of elegant symmetries in quark flavor physics, as exemplified by this work with $S^\prime_4$ modular symmetry, frequently yields compelling, if provisional, descriptions of observed phenomena. Yet, the model’s reliance on a particular fixed point-$τ= i\infty$-hints at a deeper question. Is this point truly special, or merely a convenient location within a more complex landscape of possibilities? The tendency to gravitate towards simplicity, while understandable, must be tempered by acknowledging that nature rarely conforms perfectly to minimal designs.

Further investigation should not solely focus on refining parameter fits or extending the model to accommodate neutrino masses. Instead, a critical examination of the underlying modular invariance itself is warranted. Can alternative fixed points-or even a dynamic exploration of the modular space-reveal connections to other areas of particle physics, or perhaps hint at physics beyond the Standard Model? The model’s predictive power, while present, remains contingent on the arbitrary selection of this fixed point; resolving this dependence is crucial.

Ultimately, the true test lies not in reproducing existing data, but in generating novel, testable predictions. Technology without care for people is techno-centrism; similarly, a model without genuine predictive capacity risks becoming a sophisticated exercise in pattern recognition. Ensuring fairness-in this case, a rigorous assessment of the model’s limitations-is part of the engineering discipline.

Original article: https://arxiv.org/pdf/2601.04529.pdf

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