The axion-photon coupling from lattice Quantum Chromodynamics

This paper presents the first non-perturbative determination of the model-independent QCD contribution to the axion-photon coupling using continuum extrapolated lattice simulations, yielding a precise value that enables updated constraints on axion models to guide future experimental strategies.

The Gist

Imagine the universe is a giant, complex machine. For decades, scientists have built a "User Manual" for this machine called the Standard Model. It explains how almost everything works, from the atoms in your body to the stars in the sky. It's incredibly accurate—like a watch that never loses a second.

But there's a problem. The manual is missing two huge chapters:

1. Dark Matter: We know about 85% of the universe is made of invisible stuff that holds galaxies together, but we don't know what it is.
2. The Strong CP Problem: There's a weird glitch in the rules of how particles stick together (the "strong force"). The rules say they should behave differently if you run time backward, but in reality, they don't. It's like a car engine that runs perfectly forward but refuses to run in reverse, even though the manual says it should.

The Hero: The Axion

To fix both problems, physicists invented a hypothetical particle called the Axion.

• The Fixer: It acts like a "reset button" for the Strong Force, forcing the universe to respect the time-reversal symmetry it's supposed to have.
• The Ghost: It's also a perfect candidate for Dark Matter. It's light, invisible, and everywhere.

The Challenge: How Do We Catch a Ghost?

We can't see axions directly. To find them, we have to wait for them to turn into something we can see: light (photons).

Think of the axion as a shy ghost that only reveals itself when it bumps into a magnetic field and turns into a flash of light. The "speed" at which this happens depends on a specific setting called the Axion-Photon Coupling.

• The Problem: This setting has two parts.
  1. The Model Part: This depends on which specific theory of axions is true. We don't know this yet.
  2. The QCD Part: This is a universal, unchangeable part determined by the laws of physics themselves (Quantum Chromodynamics, or QCD).

Until now, scientists had to guess the "QCD Part" using rough approximations. It was like trying to tune a radio by guessing the frequency; you might get close, but you'd likely miss the station.

The Breakthrough: The Supercomputer Simulation

This paper is about the first time scientists calculated that "QCD Part" with perfect precision, without guessing.

The Analogy: The Jello Universe
Imagine the vacuum of space isn't empty; it's like a giant block of Jello filled with tiny, invisible springs (gluons) and beads (quarks).

• When you wiggle the Jello (apply a magnetic field), the springs twist and turn.
• The "Axion-Photon Coupling" is basically measuring how much the Jello twists when you wiggle it in a specific, time-reversal-breaking way.

To measure this, the authors used a Lattice Simulation.

• They didn't use real Jello; they built a giant, 4D digital grid (a lattice) on a supercomputer.
• They simulated the laws of physics on this grid, filling it with digital quarks and gluons.
• They applied "digital magnetic fields" and watched how the digital vacuum reacted.

They used two different ways to measure the twist:

1. The Gluonic Method: Looking directly at the twisting springs (gluons).
2. The Fermionic Method: Looking at how the beads (quarks) reacted to the twist.

Both methods gave the exact same answer, confirming the result is rock-solid.

The Result: A New Map for the Hunt

The team found the precise value of this coupling: −0.0224.

Why does this matter?

• Before: It was like searching for a needle in a haystack while wearing blinders. You knew the needle was somewhere, but you didn't know exactly where the haystack was.
• Now: They have removed the blinders. They have drawn a precise map.

This new number allows experimentalists (the people building the detectors) to know exactly where to look.

• If an experiment looks for axions in a certain range and finds nothing, they can now say, "Okay, that specific type of axion model is definitely wrong."
• It narrows down the "landscape" of possible axion models, telling us which theories are still alive and which are dead.

The Bottom Line

This paper is a masterclass in first-principles physics. Instead of guessing or using shortcuts, the authors used the fundamental laws of the universe, simulated them on a supercomputer, and gave us a precise number.

It's like finally measuring the exact weight of a ghost. Now that we know exactly how heavy it is, we can build better scales to catch it. This result is a crucial step toward solving the mystery of Dark Matter and fixing the "glitch" in the universe's operating system.

Technical Summary

1. Problem Statement

The paper addresses the strong CP-problem in Quantum Chromodynamics (QCD) and the nature of dark matter.

• The Strong CP-Problem: While the electroweak sector violates CP symmetry, strong interactions appear to conserve it (evidenced by the tiny neutron electric dipole moment). The QCD Lagrangian allows a CP-violating term ($\theta$-term), but its coefficient is experimentally constrained to be nearly zero.
• The Axion Solution: The Peccei-Quinn mechanism introduces a hypothetical particle, the axion, which dynamically cancels the CP-violating term. Axions are also leading dark matter candidates.
• Detection Challenge: Experimental detection of axions relies on their conversion into photons in the presence of magnetic fields. This conversion rate is governed by the axion-photon coupling ($g_{A\gamma\gamma}$).
• The Gap: The total coupling is $g_{A\gamma\gamma} = g^0_{A\gamma\gamma} + g^{\text{QCD}}_{A\gamma\gamma}$. The model-dependent part ($g^0$) varies by theory, but the QCD contribution ($g^{\text{QCD}}$) is model-independent. Previous estimates of $g^{\text{QCD}}$ relied on Chiral Perturbation Theory (ChPT), yielding inconsistent results ranging from $-0.0260$ to $-0.0206$ (in units of $f_A/e^2$), differing by several standard deviations. A first-principles, non-perturbative calculation was required to resolve this discrepancy.

2. Methodology

The authors performed continuum-extrapolated lattice QCD simulations to calculate $g^{\text{QCD}}_{A\gamma\gamma}$ from first principles.

Theoretical Framework

• Definition: The coupling is derived from the response of the QCD vacuum to time-reversal-odd combinations of background electromagnetic fields ($E$ and $B$).
  $$g^{\text{QCD}}_{A\gamma\gamma} = \frac{1}{V_4 f_A} \frac{\partial \langle Q_{\text{top}} \rangle_{EB}}{\partial (iEB)} \bigg|_{E=B=0}$$
  where $Q_{\text{top}}$ is the topological charge and $V_4$ is the space-time volume.
• Simulation Setup:
  • Action: QCD with three dynamical quark flavors ($u, d, s$) using rooted staggered fermions.
  • Background Fields: Homogeneous, imaginary electric fields and real magnetic fields were introduced as classical background fields in the Dirac operator to avoid the complex action problem.
  • Parameters: Seven different lattice spacings ($a \approx 0.08$ fm to $0.29$ fm) were used to perform a continuum extrapolation ($a \to 0$). The simulations were performed at zero temperature with physical quark masses (except for the initial $u/d$ mass degeneracy, corrected later).

Two Independent Methods

To ensure robustness, the authors developed two distinct approaches:

1. Gluonic Method: Directly calculates the topological charge $Q_{\text{top}}$ using the gluon field strength tensor (Eq. 1).
   • Utilized improved operators (larger loops) and gradient flow to suppress ultraviolet fluctuations and define the topological charge on the lattice.
   • Applied rounding to the nearest integer to further reduce discretization artifacts.
2. Fermionic Method (Novel): An indirect approach based on the Axial Ward Identity.
   • Expresses the coupling via the expectation value of the pseudoscalar condensate $\langle P_f(EB) \rangle$.
   • Formula: $g^{\text{QCD}}_{A\gamma\gamma} \propto \lim_{EB\to 0} \frac{1}{f_A} \frac{3q_f^2}{4\pi^2} \left[ \frac{\langle P_f(EB) \rangle_{EB}}{\langle P_f(EB) \rangle_0} - 1 \right]$.
   • This method exploits the response of fermionic operators to background fields, offering a cross-check with different systematic errors.

Error Analysis and Extrapolation

• Double Extrapolation: The data required extrapolation to zero electromagnetic field ($E, B \to 0$) and the continuum limit ($a \to 0$).
• Systematics: The authors used bivariate polynomial fits for the field dependence and univariate polynomial fits for the lattice spacing dependence. Systematic uncertainties were estimated using the Akaike Information Criterion (AIC) to weight different fit models.
• Isospin Breaking: The simulations used mass-degenerate $u$ and $d$ quarks. A correction factor derived from ChPT ($\approx 1.20$) was applied to account for the physical mass splitting between $u$ and $d$ quarks.

3. Key Results

The authors present the first non-perturbative determination of the QCD contribution to the axion-photon coupling.

• Primary Result:
  $$g^{\text{QCD}}_{A\gamma\gamma} \frac{f_A}{e^2} = -0.0224(10)$$
  Expressed in terms of the fine-structure constant $\alpha$:
  $$g^{\text{QCD}}_{A\gamma\gamma} = -1.77(8) \cdot \frac{\alpha}{2\pi f_A}$$
  Or in terms of the axion mass $m_A$:
  $$g^{\text{QCD}}_{A\gamma\gamma} = -3.92(27) \frac{e^2 m_A}{\text{GeV}^2}$$

• Comparison with Previous Estimates:

  • The result is approximately 10% smaller in magnitude than the Next-to-Leading Order (NLO) two-flavor ChPT prediction used in the Particle Data Group (PDG) review ($-1.92 \cdot \alpha/2\pi f_A$).
  • It aligns most closely with ChPT predictions that include axial anomaly effects.
  • The error budget is dominated by the continuum extrapolation and the $E, B \to 0$ extrapolation, with statistical errors being sub-dominant.

• Consistency: Both the gluonic and fermionic methods yielded consistent results in the continuum limit, validating the novel fermionic approach.

4. Significance and Implications

• Resolving Model Discrepancies: The calculation resolves the wide spread in previous ChPT estimates, providing a precise, model-independent benchmark for axion physics.
• Constraining Axion Models: The total coupling $g_{A\gamma\gamma}$ is the sum of the model-dependent part ($g^0$) and the calculated QCD part ($g^{\text{QCD}}$).
  • Since $g^{\text{QCD}}$ is negative and of similar magnitude to typical $g^0$ values, it can significantly alter the total coupling.
  • For certain models (e.g., those with $E/N \approx 1.77$), the QCD contribution can nearly cancel the model-dependent term, resulting in a vanishing total coupling ($g_{A\gamma\gamma} \approx 0$). This implies these models would be invisible to current axion-photon conversion experiments.
• Guiding Future Experiments: The authors updated the "axion parameter landscape" (the $|g_{A\gamma\gamma}|$ vs. $m_A$ plane). Their results narrow the allowed region for axion models and highlight specific mass ranges (e.g., $50 \mu\text{eV} < m_A < 1500 \mu\text{eV}$) that have not yet been fully explored by current experiments like ADMX or CAST.
• Methodological Advancement: The successful application of the fermionic method via the axial Ward identity opens a new avenue for calculating topological observables in lattice QCD, potentially reducing discretization errors compared to purely gluonic definitions.

Conclusion

This paper provides a definitive, first-principles calculation of the QCD contribution to the axion-photon coupling. By utilizing continuum-extrapolated lattice simulations and two independent methods, the authors have reduced theoretical uncertainties, corrected previous ChPT estimates, and provided crucial constraints that will guide the search for axion dark matter in current and future experimental campaigns.