Dyonic lattices, $\theta$-angles and axions in the Standard Model

This paper investigates the Witten effect within the Standard Model to classify dyonic charge lattices and $\theta$-angle periodicities, demonstrating that the electromagnetic subgroup emerges prior to electroweak symmetry breaking and revealing that a single axion is insufficient to fully restore CP invariance.

The Gist

Imagine the Standard Model of physics as the ultimate instruction manual for the universe. It tells us what particles exist (like electrons and quarks) and how they interact. For decades, scientists have tested this manual with incredible precision, and it has passed every test. However, there are some "hidden pages" in this manual that are written in a very strange, topological code. These pages describe things that don't show up in standard experiments but are crucial for the universe's deepest structure.

This paper, titled "Dyonic lattices, θ-angles and axions in the Standard Model," acts like a decoder ring for those hidden pages. Here is what the authors discovered, explained through simple analogies.

1. The "Global Shape" of the Universe

Think of the forces of nature (like electromagnetism and the strong nuclear force) as different types of fabric. The Standard Model tells us the pattern of the weave (the math), but it doesn't explicitly say if the fabric is a simple sheet, a twisted Möbius strip, or a donut shape.

The authors realized there are actually four different ways to stitch this fabric together globally. These different "stitchings" are called different global gauge groups ($G_1, G_2, G_3, G_6$). While they look the same up close, they behave differently when you look at the whole picture.

2. The "Tilted" Lattice (The Witten Effect)

Imagine a grid of dots on a piece of graph paper. Each dot represents a possible particle with a specific electric charge and magnetic charge (a "dyon").

• The Normal State: Usually, these dots sit perfectly on the intersections of the grid lines (integer charges).
• The "θ-Angle": The paper introduces a mysterious dial called a θ-angle. Turning this dial is like tilting the entire graph paper.
• The Witten Effect: When you tilt the paper, the dots (particles) slide off the grid lines. A particle that was purely magnetic suddenly gains a bit of electric charge. It becomes a "dyon" (a mix of both).

The authors mapped out exactly how this grid tilts for each of the four different "fabric stitchings" mentioned above. They found that for some stitchings, you have to turn the dial a full circle to get back to the original grid, while for others, you only need to turn it a fraction of the way. This changes the rules of what particles are allowed to exist.

3. The "Ghost" Direction

The paper points out a tricky problem: The universe has a symmetry called Baryon + Lepton number (related to the number of protons/neutrons and electrons/neutrinos). In the Standard Model, this symmetry is "anomalous," meaning it's a bit of a ghost—it exists mathematically but can be shifted around without changing the physics.

The authors realized that because of this ghostly shift, one of the three dials (θ-angles) in the theory is actually unphysical. It's like trying to measure the temperature of a room while someone is constantly moving the thermometer around; you can't trust that specific reading.

By removing this "ghost" direction, they showed that the 3D space of possibilities collapses into a 2D surface (a torus, or a donut shape). Crucially, they proved that this 2D surface naturally selects electromagnetism (the force of light and electricity) as the physical force, even before the universe cools down enough for the Higgs mechanism to give particles mass. It's as if the universe "knew" which force was electromagnetism before the actors even arrived on stage.

4. The Axion: The Cosmic Tuner

To solve a famous problem called the "Strong CP Problem" (why the universe doesn't seem to violate time-reversal symmetry in nuclear forces), physicists propose a new particle called the axion. Think of the axion as a cosmic tuner that automatically adjusts the θ-dial to zero.

The paper uses their new map of the "tilted grid" to see how this tuner works:

• The Domain Wall Problem: If the axion has to choose between multiple "zero" spots on the grid, it creates cosmic defects called domain walls. If there are too many of these, they would have destroyed the universe.
• The New Finding: The authors found that if the universe has a specific global shape (one of the four stitchings), the axion can be tuned to solve the Strong CP problem without creating these dangerous domain walls, even if the axion interacts very weakly with light.

5. The "Smoking Gun" Discovery

Perhaps the most exciting practical conclusion is this: If we ever find a magnetic monopole (a particle with only a north or south pole, never both), we can determine the value of the mysterious θ-angle.

Because of the Witten effect (the tilt), a magnetic monopole would carry a tiny electric charge that depends on the setting of the θ-dial. If we find a monopole and measure its electric charge, we would instantly know the value of the last remaining unknown parameter in the Standard Model. It would be like finding a single key that unlocks the final door of the universe's instruction manual.

Summary

In short, this paper:

1. Maps out the hidden "global shapes" of the Standard Model.
2. Shows how a mysterious dial (θ-angle) tilts the landscape of allowed particles.
3. Proves that electromagnetism is singled out by this topology before the universe even breaks symmetry.
4. Shows how the axion (a proposed solution to a physics problem) fits into this landscape, potentially avoiding cosmic disasters.
5. Suggests that finding a magnetic monopole would reveal the final missing piece of the Standard Model's puzzle.

Technical Summary

Technical Summary: Dyonic Lattices, $\theta$-Angles and Axions in the Standard Model

Problem Statement
The Standard Model (SM) is defined by its gauge symmetry and matter content, yet its global gauge group structure remains ambiguous. While the Lie algebra is fixed, the global topology can correspond to one of four distinct groups ($G_1, G_2, G_3, G_6$) depending on the quotient of the center $Z_6$. This ambiguity affects the spectrum of allowed topological defects, specifically magnetic monopoles and dyons. Furthermore, the SM contains three topological $\theta$-angles ($\theta_1, \theta_2, \theta_3$) associated with the $U(1)_Y$, $SU(2)_L$, and $SU(3)_c$ sectors. While $\theta_3$ is constrained by the strong CP problem, the physical status and periodicities of $\theta_1$ and $\theta_2$ are obscured by unphysical directions in theory space (specifically related to $B+L$ anomalies) and the interplay between the global group structure and the Witten effect. The paper addresses the need to systematically chart the $\theta$-space for all possible SM global group structures, determine the resulting dyonic charge lattices, and understand the implications for axion physics, particularly the axion-photon coupling.

Methodology
The authors employ a combination of generalized symmetries (specifically one-form symmetries) and the Witten effect to analyze the topological sector of the SM.

1. Global Group Classification: The study considers the four possible global gauge groups $G_p = SU(3)_c \times SU(2)_L \times U(1)_Y / Z_p$ for $p \in \{1, 2, 3, 6\}$. The presence or absence of specific center elements determines the allowed magnetic fluxes ($nm_6$) and electric $n$-alities ($n_6$).
2. Witten Effect Analysis: For each group, the authors calculate how a non-zero $\theta$-angle induces an electric charge (or $n$-ality) on magnetic monopoles. This is derived by examining the action of the quotiented group elements on 't Hooft lines (monopole solutions) and enforcing consistency with Dirac quantization conditions.
3. $\theta$-Space Characterization: The authors identify three key features of the $\theta$-manifold for each group:
   • Periodicities ($\Delta \vec{\theta}$): The equivalence relations defining the torus.
   • Integer-valued points ($\vec{\theta}_\star$): Values where the dyonic lattice retains integer electric and magnetic quantum numbers.
   • CP-invariant points ($\vec{\theta}_\ddagger$): Values where the spectrum is invariant under CP transformation.
4. Fermion Projection: The analysis incorporates the full SM fermion content. The authors demonstrate that the anomalous $U(1)_{B+L}$ symmetry renders one direction in the 3-torus of $\theta$-angles unphysical. By projecting onto the orthogonal physical direction, they derive the effective electromagnetic $\theta$-parameter ($\theta_{em}$) and its periodicity, showing that the physical space reduces from a 3-torus to a 2-torus ($\theta_{em}, \theta_3$).
5. Axion Embedding: The topology of the physical $\theta$-torus is used to constrain the anomaly coefficients ($E, N$) of a QCD axion. The authors analyze how the axion field wraps the torus to determine the number of vacua ($N_{DW}$) and the axion-photon coupling ($g_{a\gamma\gamma}$).

Key Contributions and Results

• Dyonic Lattices and $\theta$-Periodicities: The paper constructs and classifies the dyonic charge lattices for all four SM global group structures. It re-derives known periodicities and determines the specific values of $\theta$ that lead to distinct families of allowed spectra.

  • For $G_1$ (universal cover), the $\theta$-space is a 3-torus with standard periodicities.
  • For $G_2, G_3, G_6$, the periodicities are modified, and the $\theta$-space exhibits non-trivial correlations between the angles (e.g., $\theta_1$ and $\theta_2$ are coupled in $G_2$).
  • The authors identify specific "special points" $\vec{\theta}_\star$ where the dyonic lattice consists of integer-valued charges, and $\vec{\theta}_\ddagger$ where the spectrum is CP-invariant. In some cases (e.g., $G_2$), a CP-invariant point coincides with an integer lattice point; in others, they are distinct.

• Physical $\theta$-Space and $U(1)_{em}$: A central result is the derivation of the physical $\theta$-space after accounting for fermions. The authors show that the $B+L$ anomaly projects the 3-torus onto a 2-torus spanned by $\theta_3$ and $\theta_{em}$. Crucially, they demonstrate that the electromagnetic subgroup $U(1)_{em}$ emerges as the physical direction prior to electroweak symmetry breaking (EWSB) as the direction orthogonal to the anomalous $B+L$ shift. The periodicity of $\theta_{em}$ depends on the global group $G_p$ and the minimum hypercharge $Q_{min}$.

• Axion Physics and Domain Walls:

  • Quantization of Couplings: The topology of the $\theta$-torus imposes strict quantization conditions on the axion anomaly coefficients $E$ and $N$. The relation between $E$ and $N$ depends on the global group and the compositeness parameter $k$ (related to $Q_{min}$).
  • Domain Walls: The number of QCD domain walls is $N_{DW} = 2N$. The paper identifies a new topological number $E_{DW}$ associated with the electromagnetic direction, which counts the minima of a potential QED-induced axion potential.
  • Axion-Photon Coupling: The authors show that for groups with $Q_{min} \neq Q_{qL}$ (i.e., $k \neq 0$), the constraint on the axion-photon coupling $g_{a\gamma\gamma}$ is relaxed. Specifically, for $N_{DW}=1$, it is possible to achieve arbitrarily small $g_{a\gamma\gamma}$ (potentially zero within theoretical errors) by tuning the compositeness degree $k$. This contrasts with previous results assuming the left-handed quark doublet has the minimal hypercharge.

• CP Invariance: The study concludes that a single axion is insufficient to render the full SM vacuum CP invariant if $\theta_{em} \neq 0$. While the axion can solve the strong CP problem ($\theta_3 \to 0$), it generically shifts $\theta_{em}$ to a non-zero value. Restoring full CP invariance would require a second axion coupled to photons, which, for certain group structures ($G'_3$), would introduce multiple vacua and potential electromagnetic domain wall problems.

Significance and Claims
The paper claims to provide the first detailed investigation of dyonic lattices for the Standard Model with $\theta \neq 0$ across all possible global gauge group structures. Its primary significance lies in:

1. Systematic Classification: Providing a complete map of the $\theta$-space, periodicities, and CP-invariant points for the SM, resolving ambiguities regarding the global group structure.
2. Phenomenological Prediction: Establishing that the discovery of a $U(1)_{em}$ monopole with non-zero electric charge would determine the last unknown parameter of the SM ($\theta_{em}$).
3. Axion Constraints: Generalizing the constraints on axion models to include cases where the minimal hypercharge is not that of the quark doublet. This reveals that the "strong" lower bounds on the axion-photon coupling derived in previous literature are not universal and can be evaded by specific global group choices.
4. Topological Insight: Demonstrating that the emergence of the electromagnetic charge and the reduction of the $\theta$-space dimension are topological consequences of the $B+L$ anomaly, occurring independently of the Higgs mechanism.

The authors maintain a modest stance, noting that while they identify the conditions for vanishing axion-photon couplings, the specific realization depends on the unknown global group of nature and the compositeness parameter $k$. They also leave the dynamical generation of non-standard instanton potentials (required for fractional instanton numbers in certain sectors) for future work.