Quark spectral functions from spectra of mesons and vice versa

This paper utilizes the QCD functional formalism to extract quark spectral functions by solving coupled Dyson-Schwinger and Bethe-Salpeter equations under the ladder-rainbow approximation, with results validated against the masses and decays of pseudoscalar mesons like the pion and $\eta_c$ to demonstrate the significance of dynamical charm mass variations and the nature of confinement.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. These bricks snap together to form larger structures called mesons (like pions and charmonia), which are the building blocks of protons and neutrons.

For a long time, physicists have tried to understand exactly how these bricks behave. The standard way of looking at them is like looking at a Lego brick under a microscope: you see a solid, distinct object with a fixed weight. But this paper suggests that view is wrong. Instead of being solid, static bricks, quarks are more like shapeshifting clouds of energy that change their "weight" depending on how fast they are moving or how hard they are being squeezed.

Here is a breakdown of what this paper does, using simple analogies:

1. The Problem: The "Ghost" vs. The "Cloud"

In old physics models, if you tried to pull a quark out of a particle, it was treated like a solid marble. But in reality, quarks are never found alone; they are always stuck together (a phenomenon called confinement).

Think of a quark not as a marble, but as a foggy cloud.

• The Old View: You try to weigh the cloud, but you keep getting confused because it doesn't have a sharp edge.
• This Paper's View: Instead of trying to find a single "weight," the authors map out the entire shape of the fog. They created a "spectral function," which is essentially a map showing all the different possible weights the quark cloud can have at any given moment.

2. The Method: The "Echo Chamber"

To figure out what these quark clouds look like, the authors used a clever trick involving two mirrors facing each other:

• Mirror A (The Quark): They looked at the quark itself.
• Mirror B (The Meson): They looked at the particle the quark is trapped inside (like a pion or a charmonium).

They set up a feedback loop. They said, "If the quark cloud looks like this, the particle it forms should sound like that." Then they checked: "Does the particle actually sound like that?" If not, they adjusted the shape of the quark cloud and tried again.

They did this for two types of particles:

• The Pion: Made of light quarks (like the "feather" of the quark world).
• The Charmonium: Made of heavy charm quarks (like the "brick" of the quark world).

3. The Big Discovery: The "Running Weight"

The most exciting finding is about heavy quarks (the charm quarks).

In the old models, a charm quark was thought to have a fixed weight, like a 1.5 kg dumbbell. But this paper shows that the weight is dynamic.

• The Analogy: Imagine a runner wearing a backpack. When they are standing still, the backpack feels light. But as they run faster and faster (moving to higher energy levels), the backpack seems to get heavier and heavier.
• The Result: The authors found that for the excited states of the charmonium (the "running" quarks), the effective mass grows from 1.0 GeV to 1.5 GeV. This changing weight explains why the energy levels of these particles match what we see in experiments, without needing to invent fake "confining strings" or magic forces to hold them together. The "running weight" does the job naturally.

4. The "No-Pole" Revelation

In math, a "pole" is a sharp spike that represents a stable, free particle.

• The Old Expectation: Physicists expected to see a sharp spike in their data, representing a free quark.
• The Reality: The data showed no spike at all. Instead, the sharp spike was washed out into a broad, smooth hill.

The Metaphor: Imagine trying to find a specific person in a crowded, foggy stadium.

• Old Theory: You expect to see one person standing perfectly still in a spotlight (the sharp spike).
• This Paper: You realize the person is moving, blending into the crowd, and is surrounded by fog. You can't point to one exact spot; you can only see a general area where they are likely to be. This "blurring" is the mathematical proof of confinement. It proves that a quark can never be free; it is always part of the foggy crowd.

5. Why This Matters

This paper is important because it solves a puzzle without using "magic."

• Old way: To explain why heavy quarks behave the way they do, physicists often added a "confining potential" (a made-up force like a rubber band) to their equations.
• New way: This paper shows that if you just account for the fact that quarks change their weight as they move (the "running mass"), you get the correct results automatically. You don't need the rubber band; the changing weight does the holding together.

Summary

The authors used a sophisticated mathematical feedback loop to map out the "shape" of quarks. They discovered that quarks aren't solid, fixed-weight bricks. Instead, they are dynamic clouds that get heavier as they get more energetic. This "running weight" naturally explains how heavy particles stay together, proving that quarks are forever trapped in a foggy cloud, never existing as free, sharp objects.

Technical Summary

Here is a detailed technical summary of the paper "Quark spectral functions from spectra of mesons and vice versa" by V. Šauli.

1. Problem Statement

The paper addresses the challenge of determining quark spectral functions in the timelike momentum region directly from first principles within Quantum Chromodynamics (QCD).

• Context: Knowledge of QCD correlation functions in the timelike region is essential for understanding hadronic resonances and production. However, analytically continuing lattice QCD data from Euclidean to Minkowski space is numerically unstable and challenging.
• Specific Gap: While the running quark mass (the momentum dependence of the quark mass function) is a known phenomenon in QCD, its specific impact on the mass spectra of heavy quarkonia (like charmonium) is often overlooked or approximated using non-relativistic potentials. The authors aim to demonstrate that the running mass effect alone, without ad-hoc confining potentials, can reproduce the observed heavy meson spectra.
• Goal: To extract quark spectral functions for both light ($u, d$) and heavy ($c$) quarks by solving coupled Dyson-Schwinger Equations (DSEs) and Bethe-Salpeter Equations (BSEs), constrained by the physical properties of mesons (pion and $\eta_c$).

2. Methodology

The authors employ a functional formalism using the Ladder-Rainbow Approximation (LRA) within the MOM (Momentum Subtraction) renormalization scheme.

• Coupled System:
  • DSE for Quark Propagator: Solved for the quark propagator $S(q)$ to obtain the scalar functions $A(q^2)$ and $B(q^2)$, which define the dynamical mass function $M(q^2) = B(q^2)/A(q^2)$.
  • BSE for Mesons: Solved for the bound state vertex $\Gamma$ of the pion (light) and $\eta_c$ (heavy) to ensure the quark spectral functions reproduce the correct meson masses and decay constants.
• Approximations & Truncations:
  • LRA: The quark-gluon vertex and gluon propagator are approximated. The product $\Gamma_\nu G_{\mu\nu}$ is modeled using a spectral function $\rho_T(o)$ consisting of two delta functions (representing effective gluon mass scales).
  • Spectral Representation: The quark propagator is expressed via a spectral integral (Eq. 2.5), allowing the solution to be sought directly in Minkowski space.
  • Renormalization:
    • Light Quarks: Renormalized at a small timelike scale $\mu^2 = 0.5 \text{ GeV}^2$ with $\Re a(\mu)=1$ and $\Re b(\mu)=300 \text{ MeV}$.
    • Charm Quarks: Renormalized at $\mu^2 = 0.13 \text{ GeV}^2$ with $\Re M_c(\mu) = 0.94 \text{ GeV}$.
  • Confinement Mechanism: The authors do not introduce a covariantized confining potential. Instead, confinement arises naturally from the analytic structure of the solution: the quark propagator possesses a branch cut along the entire positive real axis of $p^2$ rather than a pole, meaning quarks never go on-shell.
• Numerical Approach:
  • The BSE is solved using an eigenvalue method in complex momentum space.
  • An iterative search scans mass intervals to find eigenvalues $\lambda \to 1$ (indicating a bound state).
  • The spectral functions are determined by minimizing the deviation from spectrality while satisfying meson constraints.

3. Key Contributions

• Unified Description of Light and Heavy Mesons: The paper demonstrates that the same functional form of the DSE/BSE kernel can describe both the pion (Goldstone boson) and pseudoscalar charmonia ($\eta_c$), provided the renormalization scale and effective mass parameters are adjusted for flavor non-universality.
• Running Mass as the Confining Agent: A major theoretical contribution is the demonstration that the running quark mass effectively replaces the need for phenomenological confining potentials (like string potentials) in the Schrödinger equation. The radiative corrections responsible for the running mass generate the correct spectrum for excited states.
• Extraction of Purely Continuous Spectral Functions: The authors successfully obtain purely continuous spectral functions for quarks, confirming that the perturbative pole is washed out. This provides a rigorous QCD-based picture of confinement where quarks are never free, on-shell particles.
• Prediction of Excited Charmonium States: The formalism predicts a spectrum of narrow excited states for the $\eta_c$ meson, which are difficult to observe experimentally due to broadening from decay channels (which are ignored in this approximation).

4. Key Results

• Light Quark Spectral Functions:
  • The spectral functions ($\sigma_v, \sigma_s$) for $u/d$ quarks are broad and continuous, lacking any delta-function peaks.
  • This confirms the "confinement" of light quarks within the pion; the disappearance of the Dirac delta function is a rigid property of the solution.
• Charm Quark Spectral Functions:
  • The dynamical charm mass function $M(p)$ varies significantly in the timelike domain, ranging from 1.0 to 1.5 GeV.
  • The mass function approaches $\approx 1.1 \text{ GeV}$ at the ground state scale and reaches the "textbook" value of $1.5 \text{ GeV}$ near the maximum of the function.
  • This variation is crucial: heavier excited states are effectively composed of heavier quarks than the ground state.
• Meson Spectrum:
  • Pion: Correctly reproduces $m_\pi = 140 \text{ MeV}$ and $f_\pi = 90 \text{ MeV}$.
  • Charmonium ($\eta_c$): The calculated masses for the ground state and excited states match experimental data and Light Front Hamiltonian results (see Table I in the paper).
    • Ground state ($\eta_c(1)$): $\approx 2980 \text{ MeV}$ (Exp: 2980 MeV).
    • Excited states: Predictions for $\eta_c(2)$, $\eta_c(3)$, etc., are provided (e.g., 3442 MeV, 4150 MeV).
• Confinement Signature: The absence of a pole in the quark propagator and the presence of a branch cut along the positive real axis serve as the mathematical signature of confinement for both light and heavy quarks.

5. Significance

• Theoretical Validation: The work validates the spectral functional formalism as a robust alternative to lattice QCD for timelike physics, successfully handling the analytical continuation problem.
• Physical Insight: It clarifies the role of the running mass in heavy quark physics, showing that relativistic effects and mass running are sufficient to explain the quarkonium spectrum without invoking non-relativistic potentials.
• Confinement Mechanism: It offers a simple, unified picture of confinement where the "free quark" concept is entirely replaced by a continuous spectral distribution, applicable to both light and heavy flavors.
• Predictive Power: The method provides a framework to predict the properties of unobserved or broad resonances (like excited $\eta_c$ states) by isolating the bound-state dynamics from decay channel complexities.

In summary, the paper successfully bridges the gap between quark-level dynamics (spectral functions) and hadron-level observables (meson spectra) using a self-consistent, relativistic approach that naturally incorporates confinement and the running mass effect.