[Paper Review] Anomalies in quantum field theory: Properties and characterization

Elisabeth Kraus|ArXiv.org|Nov 11, 2002

Quantum, superfluid, helium dynamics3 citations

TL;DR

This paper provides a rigorous algebraic characterization of the Adler-Bardeen anomaly in abelian and non-abelian gauge theories, proving its regularization scheme independence and non-renormalization via an external space-time-dependent gauge coupling. It further demonstrates that in supersymmetric Yang-Mills theories, the topological term's quantum corrections induce a one-loop supersymmetry anomaly and fully determine the gauge coupling's renormalization beyond one loop.

ABSTRACT

We consider the Adler-Bardeen anomaly of the U(1) axial current in abelian and non-abelian gauge theories and present its algebraic characterization as well as an explicit evaluation proving regularization scheme independence of the anomaly. By extending the gauge coupling to an external space-time dependent field we get a unique definition for the quantum corrections of the topological term. It also implies a simple proof of the non-renormalization theorem of the Adler-Bardeen anomaly. We consider local gauge couplings in supersymmetric theories and find that there the renormalization of the gauge coupling is determined by the topological term in all loop orders except for one loop. It is shown that in one-loop order the quantum corrections to the topological term induce an anomalous breaking of supersymmetry, which is characterized by similar properties as the Adler-Bardeen anomaly.

Motivation & Objective

• To provide a scheme-independent algebraic characterization of the Adler-Bardeen anomaly in U(1) axial current anomalies.
• To prove the non-renormalization theorem of the Adler-Bardeen anomaly using an external space-time-dependent gauge coupling.
• To investigate the role of the topological term Tr(GG̃) in determining the renormalization of the gauge coupling in supersymmetric Yang-Mills theories.
• To show that quantum corrections to the topological term induce an anomalous breaking of supersymmetry in one-loop order.
• To derive a closed-form expression for the gauge β-function in terms of the one-loop coefficient and anomaly coefficient.

Proposed method

• Extend the gauge coupling to an external space-time-dependent field to uniquely define the renormalization of the topological term Tr(GG̃).
• Use gauge invariance and BRS symmetry to show that the topological term's renormalization is fixed by convergent one-loop integrals.
• Apply the Slavnov-Taylor identity and consistency conditions to derive the renormalization group operator for the supercoupling.
• Construct symmetric differential operators for the renormalization group flow, focusing on the β-function operator and its symmetry under the anomalous Slavnov-Taylor identity.
• Derive the closed-form expression for the gauge β-function by solving the consistency equations with the one-loop operator and anomaly coefficient.
• Use superspace notation to express the RG operator and analyze the constraints from symmetry identities on the coupling's renormalization.

Experimental results

Research questions

• RQ1 How can the Adler-Bardeen anomaly be algebraically characterized in terms of gauge and BRS symmetry violations?
• RQ2 Why is the coefficient of the Adler-Bardeen anomaly independent of the regularization scheme?
• RQ3 What is the role of the topological term Tr(GG̃) in determining the renormalization of the gauge coupling in supersymmetric Yang-Mills theories?
• RQ4 How do quantum corrections to the topological term lead to an anomalous breaking of supersymmetry in one-loop order?
• RQ5 Can the full β-function of the gauge coupling be expressed in terms of the one-loop coefficient and the anomaly coefficient?

Key findings

• The Adler-Bardeen anomaly is characterized algebraically as a non-variation under gauge and BRS symmetry, with its coefficient determined by convergent one-loop integrals.
• The coefficient of the Adler-Bardeen anomaly is regularization scheme independent because it arises from finite, non-renormalized integrals.
• The non-renormalization theorem of the Adler-Bardeen anomaly is proven by extending the gauge coupling to an external space-time-dependent field, which uniquely fixes the renormalization of the topological term.
• In supersymmetric Yang-Mills theories, the renormalization of the gauge coupling is fully determined by the topological term in all loop orders beyond one loop.
• In one-loop order, the topological term's quantum corrections induce an anomalous breaking of supersymmetry, analogous in structure to the Adler-Bardeen anomaly.
• The gauge β-function is expressed in closed form as β_g = ħβ_g^(1) g^3 (1 + r_η^(1) g^2 + O(ħ^2)), with the two-loop coefficient uniquely determined by the anomaly and one-loop coefficient.