Geography of Fields in Extra Dimensions: String Theory Lessons for Particle Physics (1403.1597v1)

Abstract: String theoretical ideas might be relevant for particle physics model building. Ideally one would hope to find a unified theory of all fundamental interactions. There are only few consistent string theories in D=10 or 11 space-time dimensions, but a huge landscape in D=4. We have to explore this landscape to identify models that describe the known phenomena of particle physics. Properties of compactified six spatial dimensions are crucial in that respect. We postulate some useful rules to investigate this landscape and construct realistic models. We identify common properties of the successful models and formulate lessons for further model building.

• The paper formulates five 'golden rules' to integrate Standard Model features with string theory compactifications.
• It employs orbifold geometries in the heterotic framework, identifying 200–300 MSSM-like models within a MiniLandscape.
• The analysis reveals that discrete flavor and R-parity symmetries from extra dimensions stabilize mass hierarchies and particle couplings.

Geography of Fields in Extra Dimensions: String Theory Lessons for Particle Physics

The paper "Geography of Fields in Extra Dimensions: String Theory Lessons for Particle Physics" by Hans~Peter~Nilles and Patrick~K.S.~Vaudrevange provides a nuanced exploration into the intersection of string theory and particle physics model building, with implications for identifying realistic models that describe known particle physics phenomena.

Overview

String theory posits higher-dimensional frameworks, typically in ten or eleven dimensions, containing a vast "landscape" of possible four-dimensional theories through the process of compactification. The central objective is to integrate the Standard Model (SM) within an ultraviolet complete framework of string theory that includes quantum gravity, thereby unifying all fundamental interactions.

Methodology and Findings

The authors embark on a detailed investigation of string phenomenology by formulating and applying five "golden rules" derived from examining SM properties and utilizing a bottom-up approach to model building in string theory:

1. Spinors of $\boldsymbol{10}$ for SM Matter: The authors highlight the importance of embedding SM matter fields in complete representations of larger gauge symmetries (such as $\boldsymbol{10}$ in $E_6$), which occurs naturally at orbifold fixed points or singularities.
2. Incomplete GUT Multiplets for the Higgs Pair: The paper discusses how Higgs doublets can arise from sectors where grand unified theories (GUTs) break, suggesting a solution to the doublet-triplet splitting problem.
3. Repetition of Families: It posits that the repetition of three families of quarks and leptons can be attributed to geometric properties of the compactification space, invoking local discrete symmetries (flavor symmetries).
4. $\mathcal{N} = 1$ Supersymmetry: The authors emphasize choosing compactification spaces that result in $\mathcal{N}=1$ supersymmetry in four dimensions to stabilize the hierarchy between weak and Planck scales.
5. $R$-Parity and Other Discrete Symmetries: The preservation of proton stability and the protection against fast decay modes are tied to discrete symmetries derived from string theory compactifications.

The MiniLandscape and Findings

Using the heterotic string framework, particularly the $\text{SO}(32)$ heterotic string on orbifolds, the authors breathe life into the notion of a "MiniLandscape" of MSSM-like models, wherein 200-300 promising models were identified that align with these golden rules.

The analysis of these models unveils several lessons:

• Bulk Higgs: The localization of the Higgs in the bulk yields solutions to the $\mu$-problem in MSSM.
• Top-Quark Yukawa Coupling: The large top-quark mass is naturally incorporated through fields extending to the bulk, facilitating gauge-Yukawa unification.
• Flavor Structure: Discrete flavor symmetries at orbifold fixed points help mitigate flavor changing neutral currents, thus achieving realistic hierarchies for Yukawa couplings.

The OrbifoldLandscape

Exploring further, the paper reports identifying around 12,000 MSSM-like models across multiple orbifold geometries, expanding the phenotypic scope to what they refer to as the "OrbifoldLandscape." The authors performed a statistical analysis of these models to understand adherence to the golden rules.

Practical and Theoretical Implications

The results obtained herald a valuable direction for model building within string theory, demonstrating significant compatibility and promise for MSSM-like constructions. The orbifold method, with its explicit calculability, offers a fertile ground for probing beyond the SM and understanding the symmetry preservation required for viable gauge and matter configurations.

Future Developments

The quest for realistic string-based models presents both calculational challenges and numerous strategic paths forward. Advancements in computational methods and theoretical frameworks will allow deeper dives into these landscapes, possibly reshaping our understanding of fundamental interactions and the intrinsic symmetry of the universe.

This work is instrumental in guiding theoretical exploration and practical calculations within the vast parameter space of string-based models for particle physics, paving the way for further discoveries in the intersection of high-dimensional theories and observable physics.