Chiral Symmetry Restoration using the Running Coupling Constant from the Light-Front Approach to QCD

This paper investigates chiral symmetry restoration by analyzing quark-antiquark distances through confinement potentials and hadronic cross sections, demonstrating that the light-front QCD mass scale parameter $\kappa$ can control inter-quark separation to potentially allow free quarks even within the confinement regime.

The Gist

The Big Idea: Breaking the Rules of "Sticky" Quarks

Imagine the universe is built out of tiny, magical Lego bricks called quarks. Normally, these bricks are glued together by a super-strong, invisible elastic band (the "confinement potential"). No matter how hard you pull, you can never separate a single brick from the group; they are always stuck in pairs or triplets inside larger structures called hadrons (like protons).

This paper asks a fascinating question: Is it possible for these glued bricks to suddenly become free and float around, even while they are still inside the "glue" zone?

The author, S. D. Campos, suggests that under very specific conditions, the answer is yes. He proposes that inside a proton, there can be a tiny "bubble" where the glue fails, and quarks can roam free. This is called Chiral Symmetry Restoration.

The Setup: The Proton as a Pancake

When protons smash into each other at incredibly high speeds (like in a particle collider), they don't look like spheres anymore. Due to the effects of relativity, they get squashed flat, like a pancake or a thin disk.

The author imagines this pancake proton is made up of many tiny, separate "cells." Inside each cell, there is a pair of quarks (one positive, one negative) holding hands.

• The Goal: To figure out how far apart these two quarks can get before they snap free.
• The Tool: He uses a concept called Entropy (a measure of disorder or chaos). Think of entropy as the "jiggling" of the quarks. If they jiggle too much, the glue might break.

The Secret Ingredient: The "Mass Scale" ($\kappa$)

The paper introduces a special dial or knob called $\kappa$ (kappa). You can think of this as the "stiffness setting" of the universe's glue.

• If you turn the knob to a high setting (a large number), the glue is super strong. The quarks stay tightly bound, and the distance between them is large.
• If you turn the knob to a very low setting (a tiny number), the glue becomes weak and stretchy.

The author argues that if we set this knob to a very specific, tiny value (about 0.002 GeV, which is roughly the mass of a light quark), something magical happens.

The Discovery: The "Hollow" Proton

Using a mathematical recipe that combines the "stiffness" ($\kappa$) with the energy of the collision, the author calculates the distance between the quarks.

1. The Normal Scenario: If the stiffness is normal, the quarks stay far apart (about 2.6 times the size of a proton). They are stuck.
2. The Special Scenario: If the stiffness is set to that tiny value (0.002 GeV), the distance between the quarks shrinks dramatically. They get so close together (only about 6% of the usual distance) that the "glue" effectively disappears.

The Metaphor: Imagine a crowded dance floor (the proton). Usually, everyone is holding hands in tight circles. But in this specific scenario, a few people in the very center of the floor suddenly let go of each other and start dancing freely, even though the rest of the crowd is still holding hands.

This creates a "hollow" effect. The center of the proton becomes a "gray area" where free quarks exist, surrounded by the normal, stuck quarks.

Why Does This Matter?

1. It Breaks the Rules: We usually think you need to heat things up to a massive temperature (like the Big Bang) to free quarks. This paper suggests you might not need that much heat; you just need the right "stiffness" setting.
2. It Explains Experiments: This "hollow" center might explain why particle collisions sometimes look "empty" in the middle, a phenomenon scientists call the "hollowness effect."
3. The Connection to Chaos: As the quarks become free, the "disorder" (entropy) inside the proton increases. The author suggests this increase in chaos is what causes the proton to eventually break apart at very high energies.

Summary in One Sentence

By adjusting a theoretical "stiffness knob" to a very low setting, this paper shows that quarks inside a proton can become free and unglued even without extreme heat, creating a tiny, chaotic bubble of freedom right in the center of the particle.

Technical Summary

1. Problem Statement

The paper addresses the complex issue of chiral symmetry restoration within the confinement phase of Quantum Chromodynamics (QCD).

• Context: While lattice simulations and heavy-ion collision experiments (Beam Energy Scan) suggest a phase transition from confinement to non-confinement at high temperatures or saturation scales, the specific mechanism allowing for chiral symmetry restoration within the confinement regime (where hadronic matter still exists) remains debated.
• The Gap: There is a need to define the internal entropy of colliding hadrons and understand how quark-antiquark ($q\bar{q}$) pairs can achieve a "free" state (deconfinement) even when the system is theoretically in a confining potential.
• Key Question: Can a free quark state emerge inside a hadron at specific energy scales without a global phase transition, and how does the mass scale parameter ($\kappa$) in Light-Front Holographic QCD influence this?

2. Methodology

The author employs a theoretical framework combining thermodynamics, scattering theory, and Light-Front Holographic QCD.

• Geometric Model (2D Cells):

  • The hadronic total cross-section ($\sigma_{tot}$) is modeled as a collection of finite, non-interacting 2D cells.
  • Each cell contains a single $q\bar{q}$ pair separated by a distance $r(s)$, dependent on the center-of-mass energy $\sqrt{s}$.
  • This setup draws an analogy to the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, where vortex-antivortex pairs unbind at a critical temperature.

• Entropy and Thermodynamics:

  • Entropy $S(s)$ is defined via the ratio of the total cross-section to the cell area: $S(s) = \ln(\sigma_{tot}(s) / \pi r^2)$.
  • Near the minimum of the total cross-section, the Helmholtz free energy is used to relate entropy to the confinement potential $V(r)$: $S_H(s) = V(r)/T_c$, where $T_c \approx 0.001$ GeV is a transition temperature.

• Confinement Potential & Running Coupling:

  • The Cornell potential is used: $V(r) = -\frac{4}{3}\frac{\alpha_s(r)}{r} + \sigma r$.
  • Crucially, the running coupling $\alpha_s(Q)$ is taken from the Light-Front Holographic QCD approach: $\alpha_s(Q) = e^{-Q^2/4\kappa^2}$.
  • To avoid the heuristic $Q \sim 1/r$, the author rigorously applies a Fourier sine transform to convert $\alpha_s(Q)$ to position space $\alpha_s(r)$, resulting in an expression involving the Dawson integral function $D(\kappa r)$.
  • A series expansion is performed for small $\kappa r$ (confinement regime), yielding an approximate potential dependent on the mass scale $\kappa$.

• Fitting Procedure:

  • The model parameters for $\sigma_{tot}(s)$ are fitted to experimental proton-proton ($pp$) total cross-section data (from $\sqrt{s} = 3.0$ GeV up to cosmic ray energies) using a parameterization involving a hard pomeron.
  • The mass scale $\kappa$ is treated as the sole free parameter, representing the effective mass of the cell.

3. Key Contributions

• Rigorous Derivation of $\alpha_s(r)$: The paper provides a mathematically rigorous derivation of the running coupling in position space using the Fourier sine transform of the Light-Front holographic coupling, avoiding simple heuristic substitutions.
• Entropy-Potential Link: It establishes a direct link between the hadronic total cross-section, entropy, and the confinement potential via the Helmholtz free energy near the cross-section minimum.
• Role of $\kappa$ as a Control Parameter: The work identifies the mass scale $\kappa$ (typically associated with Regge trajectories) as a critical variable that controls the distance $r$ between quarks. It demonstrates that $\kappa$ can be treated as an effective mass of the $q\bar{q}$ cell.
• Mechanism for Local Deconfinement: It proposes a mechanism where chiral symmetry restoration occurs locally within the confinement phase if $\kappa$ is sufficiently small, allowing $q\bar{q}$ pairs to unbind without the entire hadron dissociating immediately.

4. Results

• Dependence on $\kappa$:
  • Case $\kappa = 0.1$ GeV: The ratio of the inter-quark distance to the confinement scale ($r/r_0$) is $\approx 2.6$. The system remains in the confinement phase; no free quarks emerge.
  • Case $\kappa = 0.002$ GeV: The ratio $r/r_0$ drops to $\approx 0.066$. This indicates that the quark and antiquark are effectively free, signifying chiral symmetry restoration within the confinement phase.
• Thresholds:
  • Chiral symmetry restoration is possible if $\kappa \lesssim 0.002$ GeV (clear emergence) or up to $\kappa \lesssim 0.035$ GeV (conditional emergence).
  • For $\kappa \gtrsim 0.035$ GeV, the confinement potential dominates, and no free quark states appear.
• Physical Interpretation:
  • The value $\kappa \approx 0.002$ GeV is comparable to the current quark mass ($m_q$).
  • The emergence of free quarks is predicted to occur near the center of the colliding hadron. These "unbound" pairs are shielded by surrounding "bound" pairs, explaining why free quarks are not observed macroscopically in the confinement regime.
• Connection to Odderon: The paper suggests that the decrease in $\kappa$ (and thus $r$) leading to free quarks may be driven by Odderon exchange (a three-gluon state) which dominates the cross-section behavior from the Coulomb-Nuclear region up to the minimum of $\sigma_{tot}(s)$.

5. Significance

• Resolution of the "Hollowness" Effect: The model offers a theoretical explanation for the "hollowness" effect observed in high-energy scattering (where the interaction probability is lower at the center of the hadron). The emergence of free quarks at the center creates a "gray area" rather than a "black disk."
• Quarkyonic Matter: The findings support the existence of quarkyonic matter (a phase with confined baryons but restored chiral symmetry), as predicted by McLerran and Pisarski, by showing how chiral symmetry can be restored even while color confinement persists globally.
• Thermodynamic Insight: It bridges the gap between microscopic QCD potentials and macroscopic thermodynamic quantities (entropy, cross-sections), suggesting that the rise in total cross-section at high energies is thermodynamically driven by the increase in entropy due to the creation of free quark degrees of freedom.
• Predictive Power: By isolating $\kappa$ as the critical parameter, the model provides a testable condition for chiral symmetry restoration: if the effective mass scale of the cell drops near the current quark mass, deconfinement occurs locally.

In summary, the paper argues that chiral symmetry restoration is not solely a high-temperature phenomenon but can occur dynamically within the confinement phase of QCD at specific energy scales, governed by the mass scale $\kappa$ of the Light-Front Holographic approach.