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The power of the anomaly consistency condition for the Master Ward Identity: Conservation of the non-Abelian gauge current

Michael Duetsch

TL;DR

The paper investigates whether the non-Abelian gauge current in massless Yang–Mills theory, quantized perturbatively with Faddeev–Popov ghosts in a lambda gauge, remains conserved in the presence of quantum anomalies. It formulates the anomalous Master Ward Identity for an extended, affine local gauge transformation in the functional-quantization framework and derives an extended Wess–Zumino consistency condition that constrains possible anomalies. The main result shows that, when restricted to regions with constant interaction switching and under the stated assumptions (non-vanishing structure constants and adjoint representation), all non-trivial anomalies are excluded by the reduced consistency condition and can be removed by finite renormalizations of the $R$-product, yielding on-shell current conservation of the interacting gauge current. The analysis relies on deformation quantization, Epstein–Glaser renormalization, and a global symmetry perspective that leverages the Haar measure to enforce covariance; the approach demonstrates the power of the anomaly-consistency framework to constrain and rule out potential quantum violations of gauge symmetry. Overall, the work provides a concise, rigorous route to anomaly-free conservation of the non-Abelian gauge current in a perturbative Yang–Mills setting with FP ghosts, highlighting the practical impact of the extended consistency condition on the renormalization of gauge theories.

Abstract

Extending local gauge tansformations in a suitable way to Faddeev-Popov ghost fields, one obtains a symmetry of the total action, i.e., the Yang-Mills action plus a gauge fixing term (in a lambda-gauge) plus the ghost action. The anomalous Master Ward Identity (for this action and this extended, local gauge transformation) states that the pertinent Noether current -- the interacting ``gauge current'' -- is conserved up to anomalies. It is proved that, apart from terms being easily removable (by finite renormalization), all possible anomalies are excluded by the consistency condition for the anomaly of the Master Ward Identity, recently derived in refenrence [8].

The power of the anomaly consistency condition for the Master Ward Identity: Conservation of the non-Abelian gauge current

TL;DR

The paper investigates whether the non-Abelian gauge current in massless Yang–Mills theory, quantized perturbatively with Faddeev–Popov ghosts in a lambda gauge, remains conserved in the presence of quantum anomalies. It formulates the anomalous Master Ward Identity for an extended, affine local gauge transformation in the functional-quantization framework and derives an extended Wess–Zumino consistency condition that constrains possible anomalies. The main result shows that, when restricted to regions with constant interaction switching and under the stated assumptions (non-vanishing structure constants and adjoint representation), all non-trivial anomalies are excluded by the reduced consistency condition and can be removed by finite renormalizations of the -product, yielding on-shell current conservation of the interacting gauge current. The analysis relies on deformation quantization, Epstein–Glaser renormalization, and a global symmetry perspective that leverages the Haar measure to enforce covariance; the approach demonstrates the power of the anomaly-consistency framework to constrain and rule out potential quantum violations of gauge symmetry. Overall, the work provides a concise, rigorous route to anomaly-free conservation of the non-Abelian gauge current in a perturbative Yang–Mills setting with FP ghosts, highlighting the practical impact of the extended consistency condition on the renormalization of gauge theories.

Abstract

Extending local gauge tansformations in a suitable way to Faddeev-Popov ghost fields, one obtains a symmetry of the total action, i.e., the Yang-Mills action plus a gauge fixing term (in a lambda-gauge) plus the ghost action. The anomalous Master Ward Identity (for this action and this extended, local gauge transformation) states that the pertinent Noether current -- the interacting ``gauge current'' -- is conserved up to anomalies. It is proved that, apart from terms being easily removable (by finite renormalization), all possible anomalies are excluded by the consistency condition for the anomaly of the Master Ward Identity, recently derived in refenrence [8].
Paper Structure (26 sections, 8 theorems, 181 equations, 1 figure)

This paper contains 26 sections, 8 theorems, 181 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basics of massless Yang-Mills theories in the functional formalism
  3. The Lie algebra.
  4. Classical and quantum fields in the adjoint representation.
  5. Deformation quantization of the free theory.
  6. Classical field equations.
  7. Extended, infinitesimal local gauge transformation with compact support
  8. $\mathrm{Lie}\mathfrak{G}$-rotation.
  9. Extended, local gauge transformations with compact support.
  10. Anomalous Master Ward Identity
  11. Interacting quantum fields as formal power series in the interaction.
  12. Anomalous Master Ward Identity (AMWI) for the extended, local gauge transformation $\partial_X$.
  13. Restriction to the Yang-Mills interaction with FP ghosts.
  14. The most general form of the anomaly $\Delta_a(x)(S_\mathrm{int})$.
  15. Consistency condition for the anomaly of the MWI
  16. ...and 11 more sections

Key Result

Proposition 3.2

For $x\in g^{-1}(1)^\circ\equiv\{z\in\mathbb{M}\,\vert\,g(z)=1\}^\circ$ (where the upper index "$\,\circ$" denotes the interior of the pertinent set) it holds that The first two terms of $J^\mu$ are the contribution coming from $\mathcal{D}_a(x) S_\mathrm{gf}$ -- they depend on the gauge fixing parameter $\lambda$, the last term of $J^\mu$ is the contribution of $\mathcal{D}_a(x) S_\mathrm{gh}$.

Figures (1)

  • Figure 1: Illustration of the proof of Prop. \ref{['pr:int-anomaly']}: double cone $\mathcal{O}$ containing $\mathop{\mathrm{supp}}\nolimits g$ (with its past $\mathcal{O}+\overline V_-$ and its future $\mathcal{O}+\overline V_+$) and respective supports of the (decomposed) test functions.

Theorems & Definitions (20)

  • Remark 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • Corollary 4.2
  • proof
  • Remark 4.3
  • Remark 4.4
  • ...and 10 more