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Gauge Symmetry Beyond Perturbation Theory: BRST and anti-BRST Structure, Background Fields, and Infrared Dynamics of Yang--Mills Theory

Daniele Binosi

Abstract

We present a pedagogical and self contained account of the functional formulation of non-Abelian gauge theories, aimed at the construction of a process independent effective charge for Yang--Mills theory. Starting from the path integral quantization of gauge fields, we review gauge fixing and the emergence of Faddeev--Popov ghosts, illustrating how gauge invariance is preserved at the quantum level through Becchi--Rouet--Stora--Tyutin (BRST) symmetry. We then develop the BRST and anti-BRST formalisms and show how their simultaneous implementation leads to powerful functional identities that severely constrain the ghost and gluon sectors. Background field gauges are introduced as a natural framework in which these symmetries manifest themselves through Abelian like Ward identities, allowing for a transparent separation between quantum and background degrees of freedom. This structure makes it possible to define renormalization group invariant combinations of Green functions that generalize the QED effective charge to the non-Abelian case. The resulting effective charge is shown to be unique, gauge invariant, and process independent, providing a unified description of the theory from the ultraviolet down to the infrared. The interplay between functional identities, Dyson--Schwinger equations, and lattice results is discussed in detail, highlighting how dynamical mass generation and infrared saturation naturally emerge within this framework.

Gauge Symmetry Beyond Perturbation Theory: BRST and anti-BRST Structure, Background Fields, and Infrared Dynamics of Yang--Mills Theory

Abstract

We present a pedagogical and self contained account of the functional formulation of non-Abelian gauge theories, aimed at the construction of a process independent effective charge for Yang--Mills theory. Starting from the path integral quantization of gauge fields, we review gauge fixing and the emergence of Faddeev--Popov ghosts, illustrating how gauge invariance is preserved at the quantum level through Becchi--Rouet--Stora--Tyutin (BRST) symmetry. We then develop the BRST and anti-BRST formalisms and show how their simultaneous implementation leads to powerful functional identities that severely constrain the ghost and gluon sectors. Background field gauges are introduced as a natural framework in which these symmetries manifest themselves through Abelian like Ward identities, allowing for a transparent separation between quantum and background degrees of freedom. This structure makes it possible to define renormalization group invariant combinations of Green functions that generalize the QED effective charge to the non-Abelian case. The resulting effective charge is shown to be unique, gauge invariant, and process independent, providing a unified description of the theory from the ultraviolet down to the infrared. The interplay between functional identities, Dyson--Schwinger equations, and lattice results is discussed in detail, highlighting how dynamical mass generation and infrared saturation naturally emerge within this framework.
Paper Structure (16 sections, 282 equations, 4 figures)

This paper contains 16 sections, 282 equations, 4 figures.

Table of Contents

  1. Introduction
  2. $SU(N_c)$ Yang-Mills theory: gauge fixing and Faddeev-Popov ghosts
  3. Becchi--Rouet--Stora--Tyutin (BRST) symmetry, anti-BRST symmetry, and the ghost sector of $SU(N_c)$ Yang-Mills theories
  4. Becchi--Rouet--Stora--Tyutin symmetry
  5. Anti-BRST
  6. Yang-Mills ghost sector and confinement
  7. Background field gauges
  8. BF gauges and anti-BRST
  9. Background-quantum and Ward identities
  10. Dyson--Schwinger equations and block-wise realization of Ward identities
  11. Lattice results
  12. Dynamical gluon mass generation
  13. Masslessness of the gluon propagator
  14. Dynamical gluon mass generation
  15. The Process Independent QCD Effective Charge
  16. ...and 1 more sections

Figures (4)

  • Figure S1: ( Left panel) Compilation of SU$(3)$ quenched lattice results for the Landau gauge gluon propagator, corrected for discretization artifacts Aguilar:2021okw. The inset highlights the IR region, where the propagator saturates to a finite, nonzero value. The maximum observed in the deep IR arises from the dynamical interplay between gluons and ghosts. ( Right panel) SU$(3)$ quenched lattice results for the gluon propagator for different values of the gauge fixing parameter $\xi$ at $\beta=6.0$ and $L=32$. The gray rectangles provide an estimate of the expected volume effects. The inset shows the behavior of the longitudinal form factor $q^2\Delta_L(q^2)$ which evidently shows no deviation with respect to the expected result $q^2\Delta_L(q^2)=\xi$, see Eq. (\ref{['GammaDelta']}). All results are renormalized at $\mu = 4.3\,\mathrm{GeV}$.
  • Figure S2: . ( Left panel) $SU(3)$ lattice results for the ghost dressing function $F$Boucaud:2018xup. To guide the eyes we plot in gray the solution of the corresponding ghost DSE within the truncation scheme defined in Aguilar:2021okw. ( Right panel) $SU(3)$ lattice results for the form factor $f_\mathrm{gl}$ multiplying the 3-gluon vertex tree-level tensor structure in the symmetric configuration. Notice the three signatures of an IR massless free ghost: ( i) the suppression at intermediate momenta with respect to the tree-level value $f_\mathrm{gl}^{(0)}=1$; ( ii) the zero-crossing; and ( iii) the logarithmic divergence in the IR.
  • Figure S3: . ( Left panel) $SU(3)$ lattice results for the functions $G(q^2)$ (left) and $L(q^2)$ (right) in the Landau gauge Aguilar:2024bwp. According to Eq. (\ref{['F1GL']}), the fact that $L(0)=0$ and $G(0)\neq-1$ signals a finite ghost dressing function $F(0)<\infty$.
  • Figure S4: The predicted QCD all-order effective charge as a function of $q$ (left) and $r/p_r$ (right, with $p_r$ the proton radius), obtained using Eq. (\ref{['QCD-effchrg']}) and the most precise unquenched lattice results for QCD's gauge sector Cui:2019dwv. Plotted are also world data on the process-dependent $\alpha_{g_1}$ defined via the Bjorken sum rule, see Eq. (\ref{['alpha-g1']}).