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Real poles with opposite-sign residues in the non-perturbative quark propagator

By dynamically coupling the quark gap equation to the full non-perturbative quark-gluon vertex, this study reveals that sub-GeV time-like momenta yield a pair of real poles with opposite-sign residues—governed by a dominant triplet of vertex form factors—thereby resolving the conceptual issues of complex conjugate poles while maintaining consistency with color confinement.

Original authors: R. Alkofer, M. N. Ferreira, A. S. Miramontes, J. M. Morgado, J. Papavassiliou

Published 2026-06-29

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CC BY 4.0

Original authors: R. Alkofer, M. N. Ferreira, A. S. Miramontes, J. M. Morgado, J. Papavassiliou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe is built from tiny, invisible Lego bricks called quarks. For decades, physicists have been trying to figure out exactly how these bricks behave when they are stuck together inside protons and neutrons. The problem is that quarks are never found alone; they are always "confined" inside larger particles.

This paper is like a high-tech detective story where the authors are trying to solve a mystery: What does the "blueprint" (the mathematical structure) of a quark look like when you zoom in really close?

Here is the story of their discovery, explained simply:

1. The Old Map vs. The New Map

For a long time, physicists used a simplified map to describe quarks. On this old map, the quark's behavior was described by a pair of "ghostly" numbers called complex conjugate poles.

The Analogy: Imagine trying to navigate a city using a map that says, "The destination is at a location that doesn't exist on the real grid." It's a mathematical trick that works for some calculations but feels weird because it suggests the quark exists in a blurry, undefined state.

The authors of this paper decided to stop using the simplified map. Instead, they built a super-detailed, 3D model that includes every single force and interaction a quark experiences. They didn't just guess; they solved a massive set of equations that describe how quarks talk to the "glue" (gluons) holding them together.

2. The Three Magic Ingredients

To build this perfect model, the authors realized they needed to focus on three specific "ingredients" in the interaction between the quark and the glue. Think of these as three different tools in a mechanic's toolbox:

The Standard Tool (Tree-Level): This is the basic, everyday way quarks interact. It's like the standard screwdriver.
The Magnetic Twist (Anomalous Chromomagnetic Moment): This is a weird, extra spin the quark has, like a screwdriver that also has a built-in magnet. It's very strong and adds a lot of "weight" (mass) to the quark.
The Curvature Tool (Spin-Momentum Curvature): This is the new star of the show. The authors named this "Spin-Momentum Curvature." Imagine a road that doesn't just go straight but curves based on how fast you are turning. This tool measures how the quark's spin twists as it moves through space.

3. The Big Surprise: Real Poles, Not Ghosts

When the authors used only the first two tools (the Standard Tool and the Magnetic Twist), their model still showed those weird "ghostly" complex poles. The quark seemed unstable.

However, when they added the third tool (Spin-Momentum Curvature), something magical happened. The ghostly poles vanished. Instead, the model revealed two real, solid poles sitting right on the timeline.

The Analogy: It's like looking at a blurry, double-exposed photograph and suddenly realizing there are actually two distinct people standing there. One person is "positive" (normal), and the other is "negative" (a ghost-like state).

4. Why the "Ghost" is Actually Good News

You might think finding a "negative" or "ghost" pole is a bad thing. In physics, it usually means something is broken. But in this specific case, it's the key to the mystery of confinement.

The Analogy: Imagine a ball bouncing in a room. If the ball is "real," it can roll out the door and leave the room. But if the ball has a "negative" property, the laws of physics say it can never leave the room. It is trapped forever.
The authors found that because one of their poles has a "negative" sign, the quark is mathematically forbidden from ever existing as a free, lonely particle. This perfectly explains Color Confinement: why we never see a quark by itself in nature.

5. The Perfect Balance

The most fascinating part of the story is the balance.

If you use too much of the "Magnetic Twist," the quark gets too heavy, and the poles turn back into ghosts.
If you use too little, the quark is too light.
But when you mix all three tools together in just the right proportions, the math settles down. The "ghosts" disappear, the poles become real, and the quark gets a healthy, realistic mass (about 350 MeV, which matches what we see in experiments).

The Bottom Line

This paper claims that the reason quarks are trapped inside particles isn't because of some vague, blurry math. It's because of a very specific, delicate balance between three types of interactions. When you get the balance right, the math naturally produces a "negative" state that acts like a cage, ensuring quarks can never escape.

They didn't just find a new number; they found a structural reason why the universe keeps its building blocks locked away, using a "curvature" tool that had been overlooked in previous, simpler models.

Technical Summary: Real Poles with Opposite-Sign Residues in the Non-Perturbative Quark Propagator

Problem Statement
The analytic structure of the quark propagator in the Landau gauge remains a critical, yet incompletely understood, aspect of Quantum Chromodynamics (QCD). While functional methods and lattice simulations have advanced the mapping of correlation functions in the Euclidean domain, the behavior away from the space-like axis is crucial for understanding color confinement. Standard truncation schemes, such as the rainbow-ladder (RL) approximation, typically yield a quark propagator with a pair of complex conjugate poles. While these have been interpreted as signatures of confinement, it remains ambiguous whether this feature is an intrinsic property of the theory or an artifact of omitting subdominant vertex components. A conclusive resolution requires a dynamical analysis where the quark-gluon vertex structure is derived from its own field equations rather than imposed via ad hoc approximations.

Methodology
The authors investigate the unquenched quark propagator by solving a fully coupled system comprising the gap equation for the up quark and the Schwinger-Dyson equation (SDE) for the quark-gluon vertex in the Landau gauge.

Vertex Structure: The analysis employs the full basis of eight tensorial elements for the quark-gluon vertex. The vertex SDE is solved within a "three-particle irreducible (3PI) effective action" framework, simplified by considering the "non-Abelian" term and evaluating the vertex form factors in the soft-gluon limit.
Key Form Factors: The study focuses on a dominant triplet of form factors:
1. $\lambda_1$ : The transversely projected tree-level component (TLC).
2. $\lambda_4$ : The chirality-violating anomalous chromomagnetic moment (ACM).
3. $\lambda_7$ : A chirally symmetric component labeled "spin-momentum curvature" (SMC), defined by its dependence on the momentum commutator $[\not{k}, \not{p}]$ .
Numerical Approach: The system is solved in the complex momentum plane ( $p^2 \to z$ ). The authors utilize the Schlessinger Point Method (SPM) to extrapolate results from the Euclidean domain into the time-like region to locate singularities. The renormalization is performed using the $\hat{MOM}$ scheme at $\mu = 2$ GeV with a current quark mass of 50 MeV.

Key Contributions and Results
The primary finding is that the full inclusion of the vertex dynamics, specifically the interplay between the TLC, ACM, and SMC, fundamentally alters the analytic structure of the quark propagator:

Real Poles vs. Complex Conjugate Poles: Contrary to widely used approximations (like RL) that produce complex conjugate poles, the fully coupled system yields a pair of real poles on the time-like axis. The authors identify two poles at $p^2_1 \approx -0.16 \text{ GeV}^2$ and $p^2_2 \approx -0.58 \text{ GeV}^2$ .
Opposite-Sign Residues: The two real poles possess residues with opposite algebraic signs: the first pole has a positive residue ( $\approx 1.12$ ), while the second has a negative residue ( $\approx -0.51$ ). The negative residue indicates the presence of an unphysical ghost state, leading to a violation of reflection positivity.
Role of the Vertex Triplet:
- The TLC ( $\lambda_1$ ) alone generates real poles but underestimates the constituent quark mass ( $\sim 180$ MeV).
- The ACM ( $\lambda_4$ ) is the dominant driver of dynamical mass generation. However, when included with the TLC at full strength, it drives the real poles to merge and evolve into complex conjugate poles, increasing the mass to $\sim 546$ MeV.
- The SMC ( $\lambda_7$ ) does not contribute directly to the dynamical mass generation function $B(p^2)$ but critically influences the dressing function $A(p^2)$ . Its inclusion restores the real pole structure.
Critical Transition: By introducing a scaling parameter $\eta$ for the SMC strength, the authors demonstrate a critical threshold ( $\eta_{cr} \approx 0.218$ ). Below this value, poles are complex; above it, they migrate back to the real axis. This transition resembles an order parameter in a second-order phase transition.
Constituent Mass: With all three components (TLC, ACM, SMC) active, the system stabilizes to a robust constituent quark mass of $\sim 350$ MeV. The resulting momentum dependence of the mass function shows significantly better agreement with recent lattice data than standard RL truncations, particularly in the ultraviolet region where the mass function exhibits a $1/p^4$ behavior.

Significance and Claims
The paper claims that the emergence of complex conjugate poles in previous studies is likely an artifact of an unbalanced gap equation kernel that fails to properly distribute support among the chiral-symmetry-respecting (TLC), chirality-violating (ACM), and chiral-symmetry-preserving (SMC) components.

Confinement Mechanism: The resulting structure of real poles with opposite-sign residues provides a concrete realization of color confinement on the real axis. The violation of positivity (due to the ghost pole) implies that quark states do not span a positive-definite Hilbert space and cannot appear as asymptotic states.
BRST Quartet: This structure supports the scenario where quarks act as parent states within a non-perturbative BRST quartet framework, offering a functional realization of confinement that avoids the conceptual shortcomings often associated with complex conjugate poles.
Robustness: The authors assert that this behavior is stable even when subleading form factors are included, suggesting that the "spin-momentum curvature" is a vital, previously underappreciated component for stabilizing the quark propagator's analytic structure.

The work concludes that a full dynamical treatment of the vertex is essential for reproducing the correct mass scale and analytic properties of the quark propagator, aligning theoretical predictions more closely with lattice QCD results.