Perturbative approach to the infrared gluon propagator in the maximal Abelian gauge

This paper demonstrates that a systematic perturbative approach incorporating a mass-like term for gluons in the maximal Abelian gauge successfully reproduces lattice simulation data for both non-Abelian and diagonal gluon propagators in the infrared regime of $SU(2)$ Yang-Mills theory.

The Gist

The Big Picture: Trying to See the Invisible

Imagine you are trying to understand how a crowd of people behaves in a giant, chaotic stadium. In the world of particle physics, this "stadium" is the universe, and the "people" are gluons—the particles that hold the nucleus of an atom together.

Physicists have two main ways to study this crowd:

1. The Lattice Method (The Camera): They take a digital photo of the stadium using supercomputers (lattice simulations). This gives them a clear picture of what's happening, but it's just a snapshot; it doesn't explain why the crowd moves that way.
2. The Perturbative Method (The Math): They try to write down equations to predict the crowd's behavior. This is great for high-energy situations (like a fast-moving crowd), but it breaks down in the "infrared" (IR) regime—where the crowd is slow, dense, and sticky.

The Problem: For decades, the math failed to match the photos in the slow, sticky part of the stadium. However, researchers recently found a trick: if they pretend the gluons have a tiny bit of "mass" (like wearing heavy boots), the math suddenly starts matching the photos perfectly. This worked well in one specific setting called the Landau Gauge.

The Question: Does this "heavy boot" trick work in other settings, specifically a more complex one called the Maximal Abelian Gauge (MAG)? This gauge is special because it helps explain why we never see individual quarks (a phenomenon called confinement), but the math is much messier.

The Analogy: The Stadium and the "Heavy Boots"

Think of the Landau Gauge as a standard, flat soccer field. It's easy to run on, and the "heavy boot" trick (the Curci-Ferrari model) works great there.

The Maximal Abelian Gauge (MAG) is like a soccer field built on a steep, uneven mountain. It has two types of players:

• The Off-Diagonal Players (The Chaos): These are the wild, non-Abelian gluons. They run everywhere, bumping into each other.
• The Diagonal Players (The Organizers): These are the Abelian gluons. They stay in their lanes and organize the chaos.

In the deep infrared (the slow, sticky part of the game), the "Organizers" seem to take over, while the "Chaos" players get stuck or disappear. This is called Abelian Dominance.

What This Paper Did

The authors of this paper asked: "Can we use the 'heavy boot' trick on this mountainous field (MAG) to make our math match the computer photos?"

1. The Setup: They built a new mathematical model. They added "mass terms" (heavy boots) to both the chaotic players and the organizers.

   • They gave the Chaos players very heavy boots (Mass $M$).
   • They gave the Organizers lighter boots (Mass $m$).
   • Why? Because in the real world (according to lattice photos), the chaotic players should be much heavier and slower than the organizers.

2. The Calculation: They did the math (one-loop calculations) to see how these "heavy-booted" particles would move. This was hard because the mountain (MAG) is non-linear and tricky, unlike the flat soccer field.

3. The Result: They compared their new math predictions against the "photos" (lattice data) from previous experiments.

   • The Match: The math matched the photos surprisingly well!
   • The Confirmation: The "Organizers" (Diagonal gluons) were indeed much more active and dominant at low energies, while the "Chaos" players (Off-diagonal gluons) were suppressed.

The Takeaway: Why This Matters

This paper is a success story for two reasons:

1. It's a Universal Key: It shows that the "heavy boot" model isn't just a lucky accident for the flat soccer field (Landau gauge). It works on the mountain too. This suggests that the model is a robust, fundamental way to understand how particles behave when they are slow and stuck together.
2. It Explains the "Why": By using this model, they can mathematically prove why the "Organizers" take over in the deep infrared. It confirms the idea that at low energies, the complex world of quantum physics simplifies into a more orderly, Abelian world.

The Catch (The "But...")

The authors admit their model isn't perfect yet.

• The Longitudinal Problem: They could perfectly predict how the particles move sideways (transverse), but they couldn't fully predict how they move forward/backward (longitudinal) in the chaotic sector. It's like they can predict the crowd's side-to-side shuffling perfectly, but the forward movement is still a bit of a mystery in their math.
• Next Steps: They need to do more complex math (two-loop calculations) and include more types of particles (quarks) to make the model even better.

Summary in One Sentence

The authors successfully used a "heavy boot" mathematical model to prove that even in the most complex and messy gauge (MAG), the chaotic particles get heavy and slow down, allowing the orderly particles to dominate the low-energy universe, just like computer simulations show.

Technical Summary

1. Problem Statement

The infrared (IR) regime of Yang-Mills (YM) theories is strongly coupled, making perturbative methods generally inapplicable. However, recent lattice simulations and continuum approaches (such as the Refined Gribov-Zwanziger or RGZ framework) have revealed that gluon propagators exhibit a massive-like behavior in the deep IR.

In the Landau gauge, this behavior is successfully described by the Curci-Ferrari (CF) model, which introduces a mass-like term for the gluon into the gauge-fixed action. This effective model allows for perturbative computations that match lattice data remarkably well.

The central problem addressed in this work is whether this "massive" perturbative approach is gauge-independent or specific to the Landau gauge. Specifically, the authors investigate the Maximal Abelian Gauge (MAG), a non-linear gauge crucial for studying color confinement via the dual superconductivity mechanism and Abelian dominance (the decoupling of off-diagonal gluons in the IR). Unlike the Landau gauge, the MAG cannot be reached by a simple gauge parameter deformation, and its non-linear structure introduces quartic ghost interactions and breaks global color symmetry. The authors ask: Can a CF-like effective model, incorporating mass terms for gluons and ghosts, reproduce lattice data for gluon propagators in the MAG?

2. Methodology

A. Theoretical Framework: The CF-like Model in MAG

The authors propose an effective action for $SU(2)$ Yang-Mills theory in the MAG that mimics the RGZ results but uses a simpler perturbative approach. The total action $S$ is defined as:
$$S = S_{SYM} + S_{FP} + S_{M^2} + S_{m^2}$$
Where:

• $S_{SYM}$: The standard Euclidean Yang-Mills action.
• $S_{FP}$: The Faddeev-Popov gauge-fixing term for the MAG, including the necessary quartic ghost interactions and a gauge parameter $\alpha$ (required for renormalizability, with the physical limit $\alpha \to 0$ taken at the end).
• $S_{M^2}$: A mass-like term for the off-diagonal (non-Abelian) gluons ($A^a_\mu$) and off-diagonal ghosts. This term is introduced via the Local Composite Operator (LCO) technique to maintain BRST invariance on-shell.
• $S_{m^2}$: A mass-like term for the diagonal (Abelian) gluon ($A_\mu$). This term explicitly breaks BRST symmetry (similar to the Landau gauge CF model) but is treated as an effective parameter to capture IR dynamics.

Crucially, the model employs two distinct mass parameters:

• $M$: Mass for off-diagonal components.
• $m$: Mass for diagonal components.
  To satisfy the phenomenon of Abelian dominance, the condition $M^2 > m^2$ is imposed, ensuring off-diagonal modes are suppressed at low momenta.

B. Perturbative Calculation

The authors perform a one-loop calculation of the gluon propagators:

1. Propagators: They compute the diagonal gluon propagator $\langle A_\mu(p) A_\nu(-p) \rangle$ and the off-diagonal propagator $\langle A^a_\mu(p) A^b_\nu(-p) \rangle$.
2. Renormalization: Using Dimensional Regularization ($d=4-\epsilon$), they calculate the divergent parts of the self-energies. They employ Momentum Subtraction (MOM) renormalization conditions, setting the self-energies and their derivatives to zero at a specific subtraction point ($\mu^2$) and at zero momentum ($p^2=0$).
3. Physical Limit: After renormalization, the limit $\alpha \to 0$ is taken to recover the true MAG.

C. Comparison with Lattice Data

The analytical results for the dressing functions ($p^2 G(p)$) are fitted against existing lattice simulation data for $SU(2)$ in the MAG (specifically data from Bornyakov et al., Ref [75]). The fitting parameters include the coupling $g$, the masses $M$ and $m$, and global normalization factors.

3. Key Contributions

• Extension of the CF Model to MAG: This is the first systematic investigation demonstrating that a massive effective model (analogous to the CF model in Landau gauge) can be successfully constructed and applied to the non-linear Maximal Abelian Gauge.
• Handling Non-Linearity: The authors successfully navigate the complexities of the MAG, including the introduction of quartic ghost interactions and the necessity of the gauge parameter $\alpha$ for renormalizability, while recovering the physical gauge at the end of the calculation.
• One-Loop Propagators: They provide explicit analytical expressions for the one-loop corrected transverse propagators of both diagonal and off-diagonal gluons in the presence of mass terms.
• Abelian Dominance Mechanism: The model naturally incorporates the mechanism for Abelian dominance by assigning different masses to diagonal and off-diagonal components ($M > m$), leading to the suppression of off-diagonal modes in the IR.

4. Results

• Agreement with Lattice: The one-loop results for the transverse components of both the diagonal and off-diagonal gluon propagators show excellent agreement with lattice data in the deep infrared ($p \to 0$).
  • The diagonal propagator remains finite and large in the IR.
  • The off-diagonal propagator is suppressed relative to the diagonal one.
• Parameter Fits: The best-fit parameters obtained are:
  • $g \approx 8.1$
  • $M \approx 1.7$ GeV (off-diagonal mass)
  • $m \approx 0.4$ GeV (diagonal mass)
  • $\mu \approx 1.4$ GeV (renormalization scale)
  • The condition $M > m$ holds, confirming the decoupling of non-Abelian degrees of freedom.
• Ratio of Propagators: The ratio of the diagonal to off-diagonal propagators increases as momentum decreases, quantitatively supporting the Abelian dominance scenario.
• Limitations:
  • Longitudinal Sector: The longitudinal part of the off-diagonal propagator cannot be meaningfully analyzed in the $\alpha \to 0$ limit at one-loop order because the tree-level contribution vanishes while the one-loop correction diverges. This contrasts with lattice data which shows a finite longitudinal component.
  • High Momentum: The fit deteriorates at larger momentum values ($p$), suggesting that Renormalization Group (RG) improvements are necessary for the ultraviolet/intermediate regime.

5. Significance

• Universality of the Massive Approach: The success of this model in the MAG reinforces the idea that the introduction of a gluon mass term is a robust, gauge-independent effective strategy to describe the IR behavior of Yang-Mills theories, not just a peculiarity of the Landau gauge.
• Phenomenological Tool: It provides a computationally efficient perturbative framework to study non-perturbative phenomena (like confinement and Abelian dominance) without the complexity of the full RGZ action (which involves auxiliary fields and non-local horizon functions).
• Future Directions: The work opens the door for:
  • Extending the calculation to $SU(3)$ (which involves two Abelian components).
  • Including dynamical quarks and matter fields.
  • Implementing RG improvements to extend the validity of the model to higher momenta.
  • Investigating the longitudinal sector more deeply, potentially requiring non-perturbative techniques or higher-loop calculations.

In conclusion, the paper establishes that a Curci-Ferrari-like massive model in the Maximal Abelian Gauge is a viable and effective tool for reproducing lattice QCD results in the infrared, offering a simplified yet powerful perspective on the non-perturbative dynamics of gluons.