Effects of fermions in one-loop propagators in the Curci-Ferrari-Delbourgo-Jarvis gauge

This paper presents a one-loop computation of the quark propagator in the Curci-Ferrari-Delbourgo-Jarvis gauge with dynamical quarks, demonstrating that the framework remains infrared-stable with frozen coupling and mass parameters, while suggesting that finite gauge parameters may better align with lattice trends than the Landau gauge.

The Gist

Imagine the universe is built from tiny, invisible Lego bricks called quarks and gluons. These bricks stick together to form protons and neutrons, which make up everything you see. However, these bricks are glued together by a force called the Strong Force, which is so powerful and weird that we can't just look at it with a microscope. We have to use math to "see" it.

This paper is like a team of physicists trying to build a better, more accurate map of how these bricks behave when they are very close together (a state called the "infrared" region).

Here is the story of their journey, explained simply:

1. The Problem: The "Perfect" View is Actually Distorted

For the last 20 years, scientists have been studying these particles using a specific mathematical "camera angle" called the Landau Gauge. Think of this like taking a photo of a crowd from a perfectly straight-on, head-on angle.

• The Good News: This angle works great. It matches what we see in giant computer simulations (called "Lattice QCD").
• The Bad News: When they zoom in on the math for the quarks (the bricks), the picture gets a bit blurry. The math predicts the bricks should curve one way, but the computer simulations show them curving the other way. It's like the camera is slightly out of focus for this specific part of the picture.

2. The New Idea: Tilt the Camera

The authors of this paper asked: "What if we tilt the camera?"
Instead of looking straight on (Landau Gauge), they decided to look at the particles from a slightly tilted angle. In physics, this is called the CFDJ Gauge.

• The Analogy: Imagine looking at a spinning top. If you look straight down, it looks like a perfect circle. If you look from the side, it looks like an oval. Both are true, but one might show you details the other hides.
• They wanted to see if tilting the angle would fix the blurry math and make it match the computer simulations better.

3. The Experiment: Adding "Active" Particles

Previous studies of this tilted angle only looked at the "glue" (gluons) and ignored the "bricks" (quarks) to keep things simple. This is like studying a car engine but pretending the pistons don't move.

• What they did: This paper is the first to include the moving quarks (dynamical quarks) in this tilted-angle calculation. They added the "pistons" back into the engine to see how the whole machine runs.

4. The Findings: The "Freezing" Effect

When they ran the numbers, they found something fascinating happening at low energies (when particles are moving slowly or are very close together):

• The "Freezing" Phenomenon: Usually, in physics, forces get stronger or weaker as you change the distance. But here, they found that the forces, the mass of the particles, and the "tilt" of the camera all hit a "floor." They stop changing and freeze at a constant value.
• Why it matters: This is like driving a car that automatically slows down to a safe, steady speed no matter how hard you press the gas pedal. It means the theory is stable and doesn't break down when things get messy.

5. The Big Surprise: The Tilted View is Better!

Here is the most exciting part. When they looked at the quark dressing function (a fancy way of saying "how heavy the quark feels as it moves through the glue"):

• Straight-on view (Landau): The math curve looked wrong compared to the computer simulations.
• Tilted view (CFDJ): As they tilted the angle more, the math curve started to look exactly like the computer simulations.
• The Metaphor: It's like trying to draw a shadow. If you hold the object straight up, the shadow is weird. But if you tilt the object just right, the shadow suddenly looks perfect. The authors suggest that the "tilted" view might actually be the more natural way to see the universe, even if it's harder to calculate.

6. The "Mass" Mystery

They also looked at the mass of the quarks.

• In the old "straight-on" view, the mass seemed to depend on how you set up the math, which shouldn't happen (mass is a physical thing, it shouldn't change just because you changed your math).
• In the new "tilted" view, they found a way to adjust the settings so that the mass stayed consistent, regardless of the angle. This gives them a more reliable way to predict how heavy these particles are.

Summary: Why Should You Care?

This paper is a major step forward because:

1. It fixes a glitch: It solves a long-standing mismatch between math and computer simulations regarding how quarks behave.
2. It opens a new door: It proves that looking at the universe from a "tilted" angle (CFDJ gauge) is a valid and perhaps even better way to understand the Strong Force.
3. It prepares for the future: Now that they have the math right for this new angle, they can start calculating even more complex things, like how quarks and gluons talk to each other, which helps us understand the very fabric of matter.

In short, they took a slightly crooked mirror, polished it, and realized it was actually showing a clearer picture of reality than the "perfect" mirror they were using before.

Technical Summary

Here is a detailed technical summary of the paper "Effects of fermions in one-loop propagators in the Curci–Ferrari-Delbourgo-Jarvis gauge" by Santiago Cabrera, Marcela Peláez, and Matthieu Tissier.

1. Problem Statement

The paper addresses the description of Quantum Chromodynamics (QCD) correlation functions in the deep infrared (IR) regime. While the Landau gauge has been extensively studied on the lattice, revealing a massive gluon propagator and a moderate ghost-gluon coupling, it exhibits specific theoretical artifacts at the one-loop level. Specifically, the one-loop quark dressing function $Z(p)$ in the Landau gauge shows a convexity (concavity) that contradicts lattice data, a discrepancy only resolved at the two-loop level.

The authors aim to:

1. Extend the Curci-Ferrari-Delbourgo-Jarvis (CFDJ) gauge framework, previously studied in the quenched approximation (no dynamical quarks), to include dynamical quarks.
2. Compute the one-loop quark propagator in this gauge.
3. Investigate whether finite gauge parameters ($\xi \neq 0$) in the CFDJ gauge can reproduce lattice trends better than the Landau gauge ($\xi = 0$) at the one-loop level, potentially avoiding the need for higher-order corrections to fix the concavity issue.
4. Analyze the gauge dependence of the constituent quark mass and the robustness of the framework against different parameter-fixing schemes.

2. Methodology

Theoretical Framework

The authors utilize the Curci-Ferrari (CF) model, a renormalizable effective model for QCD in the IR that introduces a gluon mass term directly into the gauge-fixed action. The Lagrangian includes:

• Yang-Mills terms.
• A specific gauge-fixing term involving the Nakanishi-Lautrup field $h$ and ghost fields $c, \bar{c}$, controlled by a bare gauge parameter $\xi_0$.
• A mass term for the gluon ($m_0$) and ghosts (proportional to $\xi_0 m_0^2$).
• Matter content: $N_f$ degenerate quarks with mass $M_0$.

The CFDJ gauge is distinct from linear gauges due to a 4-ghost interaction and non-linear gauge fixing, but it shares renormalization properties with the Landau gauge and allows for lattice implementation.

Computational Approach

• One-Loop Calculation: The authors compute the self-energies for the gluon, ghost, and quark propagators at one-loop order.
  • Gluon/Ghost: Includes diagrams with massive ghost propagators and longitudinal gluon components (absent in Landau gauge).
  • Quark: Computes the quark self-energy involving the massive gluon and ghost propagators.
• Integral Reduction: Diagrams are reduced to Passarino-Veltman master integrals ($A(m)$ and $B(p, m_1, m_2)$) using dimensional regularization ($d = 4 - 2\epsilon$).
• Renormalization:
  • Scheme: The Infrared-Safe (IS) scheme is employed. Renormalization conditions are imposed at a scale $\mu$ such that propagators match specific finite values (e.g., $\Gamma_{AA}(\mu) = \mu^2 + m^2$).
  • Constraints: Non-renormalization theorems specific to the CF model are used to relate renormalization factors ($Z_g, Z_\xi, Z_m$), simplifying the system.
• Renormalization Group (RG) Flow:
  • Beta functions ($\beta_g, \beta_m, \beta_\xi$) and anomalous dimensions ($\gamma_\phi$) are derived.
  • The RG equations are solved numerically to evolve parameters from the UV (high energy) down to the IR.
  • The quark dressing function $Z(p)$ and mass function $M(p)$ are computed both in strict perturbation theory and with RG flow corrections.

Parameter Initialization

Two strategies for fixing parameters at the renormalization scale $\mu_0 = 3$ GeV are tested:

1. Fixed Parameters: Coupling, mass, and gauge parameter are fixed based on Landau gauge fits, then evolved for different $\xi$.
2. Constituent Mass Matching: The gluon mass $m(\mu_0)$ is adjusted for each $\xi$ such that the constituent quark mass $M(p=0)$ remains gauge-invariant (matching the Landau gauge value).

3. Key Contributions

1. First Unquenched CFDJ Calculation: This is the first derivation of the full quark propagator in the CFDJ gauge including dynamical quark loops ($N_f=2$).
2. Resolution of One-Loop Concavity: The paper demonstrates that the problematic convexity of the quark dressing function $Z(p)$ observed in the Landau gauge at one-loop is not generic. For finite gauge parameters ($\xi > 0.5$), the one-loop $Z(p)$ exhibits the correct concavity (decreasing towards the IR) without requiring two-loop corrections.
3. RG Flow Freezing: Confirms that in the presence of dynamical quarks, the running coupling $g(\mu)$, gluon mass $m(\mu)$, and gauge parameter $\xi(\mu)$ all "freeze" (reach constant non-zero values) in the deep IR, ensuring the absence of Landau poles.
4. Gauge Dependence of Mass: The study highlights a strong gauge dependence of the constituent quark mass when the gluon mass is fixed independently of $\xi$. It proposes a scheme where the gluon mass is tuned to restore gauge independence of the constituent mass, showing that physical observables become robust under this condition.

4. Key Results

• Coupling and Mass Running:

  • In the IR, $g(\mu)$, $m(\mu)$, and $\xi(\mu)$ saturate to finite values.
  • Including dynamical quarks slightly reduces the saturation values of the gluon mass and coupling compared to the quenched case but increases the saturation value of the gauge parameter.
  • The "freezing" mechanism persists even with quarks because all degrees of freedom (including ghosts) acquire mass, decoupling the massless modes.

• Propagators:

  • Gluon/Ghost: The inclusion of quarks causes only modest quantitative changes to the gluon and ghost propagators compared to the quenched approximation. The qualitative behavior (screening mass generation) remains unchanged.
  • Transverse vs. Longitudinal: The longitudinal gluon propagator shows slight quantitative changes but retains the global behavior.

• Quark Dressing Function $Z(p)$:

  • Landau Gauge ($\xi=0$): $Z(p)$ is increasing (or constant in massless limit), contradicting lattice data at one-loop.
  • Finite $\xi$: For $\xi \gtrsim 0.5$, $Z(p)$ becomes a decreasing function in the IR, matching the trend observed in lattice simulations. This suggests that non-vanishing gauges may provide a more accurate description of QCD at the one-loop level.

• Quark Mass Function $M(p)$:

  • When parameters are fixed arbitrarily, $M(p=0)$ decreases monotonically as $\xi$ increases (a factor of $\sim 2$ difference between Landau and $\xi=4$).
  • When the constituent mass matching scheme is used (adjusting $m(\mu_0)$ to keep $M(0)$ constant), the mass functions for different $\xi$ overlap remarkably well in the UV and IR, differing by only $\sim 2.4\%$ in the intermediate region. This validates the physical consistency of the model.

5. Significance

• Validation of CFDJ Gauge: The results establish the CFDJ gauge as a viable, infrared-safe setting for perturbative QCD with massive gluons. It offers a framework where one-loop calculations can reproduce lattice trends (specifically the concavity of $Z(p)$) that fail in the Landau gauge.
• Lattice Comparison: Although unquenched lattice simulations in the CFDJ gauge do not yet exist, the paper provides concrete predictions for the impact of quark loops on propagators, serving as a benchmark for future lattice studies.
• Future Applications: The successful renormalization of the quark two-point function paves the way for computing higher-order quantities, such as the quark-gluon vertex and the chromomagnetic moment, which are crucial for understanding hadron structure and chiral symmetry breaking.
• Gauge Artifacts vs. Physics: The work clarifies that certain features of the Landau gauge (like the one-loop concavity of $Z(p)$) are gauge artifacts rather than fundamental RG properties, reinforcing the importance of studying QCD in non-Landau covariant gauges.