Computation of Quark Masses from String Theory (2402.01615v2)

Abstract: We present a numerical computation, based on neural network techniques, of the physical Yukawa couplings in a heterotic string theory compactification on a smooth Calabi-Yau threefold with non-standard embedding. The model belongs to a large class of heterotic line bundle models that have previously been identified and whose low-energy spectrum precisely matches that of the MSSM plus fields uncharged under the Standard Model group. The relevant quantities for the calculation, that is, the Ricci-flat Calabi-Yau metric, the Hermitian Yang-Mills bundle metrics and the harmonic bundle-valued forms, are all computed by training suitable neural networks. For illustration, we consider a one-parameter family in complex structure moduli space. The computation at each point along this locus takes about half a day on a single twelve-core CPU. Our results for the Yukawa couplings are estimated to be within 10% of the expected analytic result. We find that the effect of the matter field normalisation can be significant and can contribute towards generating hierarchical couplings. We also demonstrate that a zeroth order, semi-analytic calculation, based on the Fubini-Study metric and its counterparts for the bundle metric and the bundle-valued forms, leads to roughly correct results, about 25% away from the numerical ones. The method can be applied to other heterotic line bundle models and generalised to other constructions, including to F-theory models.

• The paper demonstrates that neural network approximations yield hierarchical Yukawa couplings with about 10% accuracy.
• It details the computation of critical Calabi-Yau, Hermitian Yang-Mills metrics, and harmonic bundle-valued forms in a heterotic model aligned with the MSSM.
• The research paves the way for applying these techniques to broader heterotic and F-theory models, bridging advanced computation with theoretical physics.

Insights into the Computation of Quark Masses from String Theory

The paper "Computation of Quark Masses from String Theory" presents an advanced paper into the calculation of quark masses within the framework of heterotic string theory, exploring the complex interplay between neural network techniques and theoretical physics constructs. It tackles a fundamental problem that has persisted in particle physics: deriving the values and hierarchical structure of quark and lepton masses directly from string theory principles.

Summary of the Study

At the core of the research is the utilization of neural networks to compute the physical Yukawa couplings within a heterotic string theory model compactified on a Calabi-Yau (CY) threefold with non-standard embedding. This model adeptly aligns with the Minimal Supersymmetric Standard Model (MSSM) and includes additional fields that remain uncharged under the Standard Model group. Key components of the computation involve obtaining:

1. The Ricci-flat Calabi-Yau metric,
2. Hermitian Yang-Mills (HYM) bundle metrics, and
3. Harmonic bundle-valued forms.

These quantities are crucial as they represent the necessary geometric data from which the Yukawa couplings—and thus, the quark masses—are derived.

Methodological Approach

The work employs a sophisticated method where neural networks are trained to approximate the aforementioned geometric quantities. These neural models essentially serve as numerical solvers for partial differential equations that govern the respective geometry on the moduli space of the heterotic line bundle models. The authors detail the process for a one-parameter family in the complex structure moduli space, noting that each computation point requires approximately half a day to complete on a high-performance, twelve-core CPU, achieving results within 10% of expected analytical outcomes.

Key Results and Insights

The results underscore the effectiveness of neural networks in generating hierarchical Yukawa couplings, a crucial step towards realistic quark mass predictions. Notably, the paper highlights the impact of matter field normalization, which can meaningfully contribute to creating the necessary hierarchies within the couplings. Further, the semi-analytic calculations based on the Fubini-Study metric offer an approximation, deviating roughly by 25% from the comprehensive numerical results.

The research emphasizes the broader applicability of these methods to other heterotic line bundle models and potential generalizations to F-theory models, marking a significant step forward in high-energy theoretical physics.

Implications and Future Directions

This paper presents significant implications for both the theoretical exploration of string theory and its practical application in understanding fundamental particle physics phenomena. It reflects a noteworthy advancement in the capability to leverage machine learning, particularly neural networks, to solve complex mathematical problems within physics, providing a bridge between abstract theoretical constructs and tangible physical predictions.

The paper also opens several future research pathways. For instance, expanding the application of these techniques to a wider variety of Calabi-Yau manifolds, alongside the integration with moduli stabilization techniques, could yield more accurate and holistic particle physics models. Such endeavors could eventually address the complexities faced in heterotic string model building, offering insights into the realization of the Standard Model from string theory with all its couplings and matter fields.

This work stands as a testament to the role of computational techniques in advancing theoretical physics, blending numerical methods with abstract theory to deepen our understanding of the universe at its most fundamental level.