# Chaos in QCD? Gap Equations and Their Fractal Properties

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## Abstract

**:**

## 1. Introduction

## 2. Self-Consistency and the Emergence of New Roots amongst the Old

## 3. The Munczek–Nemirovsky Model

## 4. Iterative Chaos

## 5. Mass Poles

## 6. Finite Interaction Width

## 7. Conclusions

- (I)
- The iterative approach provides an ultraviolet cut-off for the massive and mass-pole-free Nambu solution, as this solution appears only within the elliptic region of the (${z}_{R}^{2},{z}_{I}^{2})$) plane. Thus, the approach avoids the appearance of an infinitely increasing dressed-quark mass with increasing momentum and energy. In the MN model, this running mass results in the absence of mass poles for the massive gap solution and thus relates to confinement.
- (II)
- It provides an infrared cut-off for the bare-quark mass Nambu solution and thus ensures that quasi-particle states are not populated at a small chemical potential, although the quark can virtually exist as a quasi-particle with well defined dispersion.
- (III)
- Both cutoffs more or less coincide (as observed in Figure 5), although there is a transition region which is chaotic in nature. The resulting effective Nambu-UV/Wigner-IR cutoff depends dynamically on energy, momentum, bare mass, and chemical potential. As a side note, we add that plotting gap solutions in the (${z}_{R}^{2},{z}_{I}^{2}$) plane removes much of the dynamical arbitrariness and leaves the ratio of the bare mass m and coupling constant $\eta $ as the only ’true’ degree of freedom; viz., a change of the chemical potential $\mu $ would rescale the plot but cause no qualitative change, whereas plots such as Figure 1 indeed demonstrate ’the’ gap solution at an arbitrary chemical potential.
- (IV)
- Sufficiently large chemical potential bare-quark mass-pole states will form at energies which can be populated; thus, physical quarks can exist as quasi-particle excitations.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Real part of the scalar gap B after 300 iterations starting from ${A}_{0}=1$ (

**top**,

**bottom**), and ${B}_{0}=m=$ (10 MeV (

**top**), 100 MeV (

**bottom**)).

**Figure 2.**Periodicity of the iterative mass gap solution at $m=100$ MeV. The outer, indigo region of the plot are absolutely stable under iteration, the inner almond shape has periodicity two, and the area in between exhibits chaos with increasing periodicity. For this plot, areas with periodicity larger than ten are plotted in black.

**Figure 3.**Real part of the mass gap B at ${z}_{R}^{2}=0.1$ MeV${}^{2}$. The color coding indicates how frequently a solution has been found over 300 iterations after the first 100 iterations which are sufficient to shape the fractal as seen. For reference, all analytic solutions to the polynomial gap equations are plotted in color. Iteration switches from massive solutions (blue) at small $\mathcal{I}\left({z}^{2}\right)$ to bare-mass solutions (green) at larger values. Except for the chaotic transition domain, the iterative approach picks positive mass-gap solutions, only. Note, that the chaotic domain has solutions of periodicity of two and higher; it is truly unstable. Hence, we add a gray scale to measure the frequency of a particular solution over the final 300 iterations.

**Figure 4.**Upper panel: Solutions of the polynomial gap equations for $m=100$ MeV. Each is plotted on a scale that most accentuates its structure. Solution 1 and 3 (from the left) are stable in some, mutually exclusive domains under iteration, as illustrated in Figure 3. Lower panel: After 300 iterations, using the corresponding solution of the polynomial gap equations from the upper panel as initial seed for the iteration. In the outer, non-chaotic domain, all four cases produce nearly identical results with positive mass gap only.

**Figure 5.**Difference between gap solution 1 and 3 (from the left) in the top panel of Figure 4 and iterative solutions seeded with the non-interacting solution ($A=1$, $B=m$) after 500 iterations. White domains show no difference between iterative solutions seeded with an analytical model solution or seeded with the non-interacting solution. Solution 2 and 4 show no agreement anywhere in the stable domain of periodicity one (not shown).

**Figure 6.**Natural logarithm of ${\left({p}_{3}^{2}+{M}^{2}-{p}_{4}^{2}\right)}^{2}$ for the iterative solution for $\mu =(100,350,600)$ MeV (top down) at quark-bare mass $m=$ 100 MeV. The vertical line shaped by minimal negative values indicate a physical mass pole, viz. a quasi-particle. In the chaotic domain, this pole structure is absent, viz. the vertical line (or any distinct pole) pattern is absent. This implies an infrared energy gap, below which quarks show no quasi-particle properties. As the chemical potential increases, the quasi-particle pole line moves to the right and simultaneously decreases the gap, viz., the gap region without a pole traces the outer shape of the fractal. Once the chemical potential is sufficiently large, the gap closes entirely. Note that the absence of a mass pole does not imply that there is no mass gap solution, as illustrated in Figure 3.

**Figure 7.**Plotted is the logarithm of the mass-pole condition $log\left(\right|{\overrightarrow{p}}^{2}-{p}_{4}^{2}+{\mu}^{2}+\Re \left({M}^{2}\right)\left|\right)$, which shows a dispersion relation with distinct, chaos-induced infrared cut-off. With increasing chemical potential ($m=0.1\eta $; $\mu =(0.2,0.4,0.7)\eta $ from

**left**to

**right**), the infrared cut-off decreases and eventually disappears. With increasing widening ($\sigma =(0.00,0.01,0.02)\eta $ from

**top**to

**bottom**), chaotic domains blur but the observed IR cut-off remains.

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**MDPI and ACS Style**

Klähn, T.; Loveridge, L.C.; Cierniak, M.
Chaos in QCD? Gap Equations and Their Fractal Properties. *Particles* **2023**, *6*, 470-484.
https://doi.org/10.3390/particles6020026

**AMA Style**

Klähn T, Loveridge LC, Cierniak M.
Chaos in QCD? Gap Equations and Their Fractal Properties. *Particles*. 2023; 6(2):470-484.
https://doi.org/10.3390/particles6020026

**Chicago/Turabian Style**

Klähn, Thomas, Lee C. Loveridge, and Mateusz Cierniak.
2023. "Chaos in QCD? Gap Equations and Their Fractal Properties" *Particles* 6, no. 2: 470-484.
https://doi.org/10.3390/particles6020026