Nonperturbative Dyson--Schwinger equations in QCD: a first approximation

Technical Summary: Nonperturbative Dyson–Schwinger Equations in QCD: A First Approximation

Problem Statement
The paper addresses the fundamental limitations of perturbative quantum field theory (QFT), particularly in Quantum Chromodynamics (QCD). While perturbative methods using Feynman diagrams and small coupling constants have achieved high precision in Quantum Electrodynamics (QED) and the electroweak theory, they fail to resolve nonperturbative phenomena in QCD such as confinement, the mass gap, and the emergence of dimensional transmutation. The authors cite historical concerns from R. Feynman, P. Dirac, L. Landau, and W. Heisenberg regarding the mathematical "sweeping under the rug" of infinities and the inadequacy of perturbative expansions for strongly interacting fields. The central problem is to formulate a consistent nonperturbative quantization procedure that can handle the infinite system of Dyson–Schwinger equations (DSEs) without relying on small coupling expansions.

Methodology
The authors propose a truncation scheme for the infinite system of nonperturbative Dyson–Schwinger equations, following the approach originally suggested by W. Heisenberg. The methodology involves the following key steps:

1. Operator Averaging: The starting point is the operator Yang–Mills and Dirac equations. Instead of solving these directly, the authors consider their quantum averages, which generate an infinite hierarchy of equations for Green's functions (correlation functions).
2. Factorization and Truncation: To render the system solvable, the infinite hierarchy is truncated to a finite set. This is achieved by:
   • Factorization: Representing higher-order Green's functions (specifically four-point functions) as bilinear combinations of lower-order (two-point) Green's functions.
   • Introduction of Closure Constants: This factorization introduces dimensionful constants (denoted as $\lambda, \Lambda, \xi$, etc.) into the polynomial approximations of the Green's functions. These are termed "closure constants."
   • Field Decomposition: The gauge field degrees of freedom are divided into "almost classical" (nonzero expectation values) and "almost vacuum" (vanishing expectation values) components. Similarly, fermion fields are approximated by introducing auxiliary spinor fields to describe expectation values of quadratic combinations.
3. Gauge and Static Assumptions: The analysis is restricted to static systems (time-independent observables). The authors argue that in the static, strongly interacting regime, correlations can exist outside the light cone, distinguishing them from weakly interacting fields. Specific gauge choices are discussed to ensure consistency with the truncation.
4. Approximation of Mixed Terms: A critical technical challenge addressed is the approximation of three-point Green's functions involving both gauge fields and fermions (e.g., $\langle \bar{\psi} A_\mu \psi \rangle$). Since the individual expectation values of the gauge field and fermion field may vanish in the vacuum, simple factorization fails. The authors propose factorizing these terms into products of scalar functions (describing gauge condensates) and spinor functions (describing quark condensates).

Key Contributions and Results

• Vacuum Approximation (Static Case):

  • The authors derive equations for scalar fields $\phi$ and $\chi$ that describe the condensates of the gauge field two-point functions ($\langle A_\mu A_\nu \rangle$).
  • In the simplest case (Section III A), a single scalar field $\phi$ describes the vacuum condensate, leading to a nonlinear equation where the source is the quark condensate $\langle \bar{\psi}\psi \rangle$.
  • In a more complex case (Section III B), two scalar fields ($\phi$ and $\chi$) are introduced to describe different subgroups of the $SU(3)$ gauge group, resulting in a coupled system of nonlinear equations.

• "Almost Classical" and "Almost Vacuum" Degrees of Freedom:

  • The paper extends the approximation to cases where the gauge potential has nonzero expectation values ("almost classical") and vanishing expectation values ("almost vacuum").
  • This leads to a system where the "almost classical" fields obey a Proca-type equation with sources from quarks and the gluon condensate, while the "almost vacuum" fields are described by a scalar field $\chi$ coupled to the classical background.

• Nonlinear Dirac Equation and Mass Gap:

  • By approximating the fermion sector, the authors derive a modified Dirac equation for an auxiliary spinor field $\varsigma$.
  • Crucially, this equation contains a nonlinear term of the form $(\bar{\varsigma}\varsigma)\varsigma$.
  • The authors argue that this nonlinearity is significant because previous studies on nonlinear Dirac equations suggest the existence of a global minimum in the energy spectrum, corresponding to a "mass gap." This offers a potential mechanism for the mass gap in QCD within this nonperturbative framework.

• Dimensional Transmutation:

  • The paper discusses how "closure constants" (dimensionful parameters introduced during truncation) might survive the transition to the infinite system of equations.
  • The authors posit that these constants provide a mechanism for "dimensional transmutation," where a dimensionless coupling constant in the fundamental Lagrangian gives rise to a physical mass scale (like $\Lambda_{QCD}$) in the solution of the infinite system.

Significance and Claims
The paper claims to provide a "first approximation" for a nonperturbative quantization procedure in QCD that does not rely on perturbation theory. Its primary significance lies in:

1. Bridging the Gap: Offering a concrete mathematical framework to address problems (confinement, mass gap) that are inaccessible to standard perturbative QCD.
2. Emergence of Nonlinearity: Demonstrating that the truncation of the DSE hierarchy naturally leads to nonlinear Dirac equations, which are hypothesized to generate a mass gap.
3. Analogy to Turbulence: Drawing a parallel between the "closure problem" in turbulence modeling and the truncation of DSEs, suggesting that the "closure constants" in QCD play a role analogous to empirical constants in turbulence models, potentially surviving in the exact limit.
4. Theoretical Consistency: The authors emphasize that their approach respects the nonperturbative nature of strong interactions, where correlations do not necessarily vanish outside the light cone, unlike in weakly interacting theories.

The paper concludes modestly, noting that the proposed procedure requires further investigation, particularly regarding the choice of gauge, the rigorous justification of the closure constants in the infinite limit, and the detailed physical properties of the resulting systems of equations. It does not claim to have solved QCD but rather to have established a viable first step toward a nonperturbative solution.