Calculating Axion Potentials Induced by Small Instantons in SU(N) Gauge Theories

Core Concepts

This paper presents a functional method for calculating the axion potential induced by small instantons in SU(N) gauge theories, considering interactions with matter fields in arbitrary representations.

Abstract

Bibliographic Information

Pablo Sesma. (2024). A functional treatment of small instanton-induced axion potentials. arXiv:2411.00101v1 [hep-ph]

Research Objective

This paper aims to develop a robust method for calculating the contributions of small instantons to the axion potential in SU(N) gauge theories with various matter content. This is crucial for understanding the potential of axions as a solution to the Strong CP problem and as a dark matter candidate.

Methodology

The authors utilize a functional approach, constructing the generating functional of the theory in the background of a single instanton. They employ the dilute instanton gas approximation and perform a semiclassical expansion around the instanton solution. The functional method allows for a systematic treatment of interactions between particles, which are crucial for obtaining a non-zero axion potential. The authors also derive the explicit form of fermion zero modes for arbitrary representations of SU(2) and outline a procedure to extend this to SU(N) representations.

Key Findings

• The paper provides a comprehensive framework for calculating one-instanton contributions to the effective axion potential in SU(N) gauge theories.
• It highlights the importance of interactions in saturating the integration over fermion zero modes, leading to a non-zero axion potential.
• The authors derive the explicit form of fermion zero modes for any representation of SU(2) and present a method to extend this to arbitrary representations of SU(N).
• The method is applied to specific examples, including the Minimal Supersymmetric Standard Model (MSSM) extended with color triplets and the Minimal Supersymmetric SU(5) Grand Unified Theory.

Main Conclusions

The functional method presented offers a transparent and tractable approach to calculating small instanton contributions to the axion potential in various SU(N) gauge theories. The explicit construction of fermion zero modes for arbitrary representations provides a valuable tool for these calculations. The application of the method to supersymmetric models demonstrates its utility in addressing phenomenologically relevant scenarios.

Significance

This research significantly contributes to the field of particle physics by providing a powerful tool for calculating instanton effects in a wide range of theories. This is particularly relevant for understanding the axion potential, which has profound implications for solving the Strong CP problem and exploring axions as dark matter candidates.

Limitations and Future Research

The paper primarily focuses on one-instanton calculations within the dilute instanton gas approximation. Further research could explore the impact of multi-instanton effects and investigate scenarios beyond this approximation. Additionally, extending the method to other gauge groups beyond SU(N) would broaden its applicability.

Stats

¯θ < 10^-10 (experimental constraint on the QCD vacuum angle)

Quotes

Key Insights Distilled From

A functional treatment of small instanton-induced axion potentials

by Pablo Sesma at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00101.pdf

Deeper Inquiries

How do the methods presented in this paper compare to other approaches for calculating instanton effects, such as lattice QCD simulations?

This paper presents a functional approach for calculating instanton effects, specifically the instanton-induced axion potential, within the framework of semiclassical theory. This method offers a complementary perspective to other approaches, such as lattice QCD simulations, each with its own strengths and limitations:
Functional Approach (This Paper):

Strengths:

Analytical insights: Provides a deeper analytical understanding of the underlying physics governing instanton contributions.
Flexibility: Allows for calculations in theories with various gauge groups and matter content, including those beyond the Standard Model.
Computational efficiency: Generally less computationally demanding than lattice simulations, especially for exploring a wide range of model parameters.

Limitations:

Semiclassical approximation: Relies on the weak coupling limit (small gauge coupling), which may not be fully accurate in strongly coupled regimes.
Dilute instanton gas approximation: Assumes instantons are well-separated and non-interacting, neglecting potential contributions from instanton-anti-instanton interactions.
Lattice QCD Simulations:

Strengths:

Non-perturbative: Can handle strong coupling dynamics, providing a more accurate description of QCD in the non-perturbative regime.
First-principles approach: Based directly on the QCD Lagrangian, requiring fewer assumptions compared to semiclassical methods.

Limitations:

Computational cost: Highly computationally intensive, limiting the exploration of parameter space and the study of theories beyond QCD.
Systematic uncertainties: Susceptible to systematic errors related to lattice discretization and finite volume effects.
Comparison:
The functional approach excels in providing analytical insights and exploring a broader range of theories, while lattice QCD offers a more accurate description of non-perturbative QCD. These methods complement each other, and combining their results can lead to a more comprehensive understanding of instanton physics.

Could the presence of strong dynamics at high energies significantly alter the small instanton contributions to the axion potential?

Yes, the presence of strong dynamics at high energies could significantly modify the small instanton contributions to the axion potential. Here's why:

Running Coupling: Small instanton contributions are typically suppressed by the exponential factor exp(-8π²/g²(1/ρ)), where g is the gauge coupling and ρ is the instanton size. At high energies, if new strong dynamics emerge, the running of the coupling g can change drastically. If the coupling becomes strong at high energies, the suppression factor might become less severe, potentially enhancing the small instanton contributions.

New Particles and Interactions: New heavy particles coupling to the gauge fields, especially if they become light or strongly coupled at high energies, can modify the instanton solution itself and its contributions to the action. This can lead to either enhancement or suppression of the instanton effects, depending on the details of the new physics.

Non-perturbative Effects: Strong dynamics often imply significant non-perturbative effects that are not captured by the semiclassical approximation inherent in the dilute instanton gas picture. These non-perturbative effects could drastically alter the instanton density and their contributions to the axion potential.
Examples:

Walking Technicolor: In walking technicolor models, the gauge coupling evolves slowly over a large energy range, remaining strong but nearly constant. This can significantly enhance the small instanton contributions compared to a scenario with a rapidly running coupling.

Extra Dimensions: In models with extra dimensions, the presence of Kaluza-Klein modes and the modified running of couplings in the UV can lead to substantial changes in the instanton density and their contributions to the axion potential.
In summary, while small instanton contributions are generally suppressed, strong dynamics at high energies can significantly alter this picture, leading to potentially observable effects on the axion potential.

What are the implications of this research for experimental searches for axions, and how could these calculations inform the design of future experiments?

This research has important implications for experimental searches for axions, particularly those focusing on the detection of axion dark matter or the effects of the axion potential:
1. Axion Mass Range:

Motivation: The axion mass is a crucial parameter for experimental searches. Different experiments are sensitive to different axion mass ranges.
Implication: Calculations of the instanton-induced axion potential, especially in extensions of the Standard Model, can predict axion masses that might fall within the reach of current or future experiments. This information guides experimentalists in targeting specific mass ranges.
2. Axion-Photon Coupling:

Motivation: The axion-photon coupling is another key parameter that determines the strength of the axion signal in many experiments, such as those using microwave cavities or light shining through walls techniques.
Implication: While this paper focuses on the axion potential, the methods presented can be extended to calculate other axion couplings, including the axion-photon coupling. These calculations can predict the expected signal strengths in different experiments, informing their sensitivity goals.
3. Axion Dark Matter Abundance:

Motivation: The axion is a well-motivated dark matter candidate. Its abundance in the early universe depends on the shape and temperature dependence of its potential.
Implication: Accurate calculations of the axion potential, including contributions from instantons and other sources, are crucial for predicting the axion relic abundance and comparing it with cosmological observations. This comparison can constrain axion models and guide the search for axion dark matter.
4. Experimental Design:

Motivation: Designing and optimizing axion experiments require precise theoretical input on the axion's properties and potential interactions.
Implication: The calculations presented in this paper, particularly those involving different gauge groups and matter content, can inform the design of future experiments. For example, if calculations suggest an enhanced axion-photon coupling in a specific model, experiments can be tailored to be more sensitive to that coupling.
In conclusion, this research provides valuable theoretical tools for predicting axion properties, which are essential for guiding experimental searches. By refining these calculations and exploring different models, theorists can help experimentalists narrow down the search parameters and increase the chances of detecting this elusive particle.