[Paper Review] A note on the exact lattice chiral symmetry in the overlap formalism

Yoshio Kikukawa, Atsushi Yamada|ArXiv.org|Oct 10, 1998

Quantum Chromodynamics and Particle Interactions8 references4 citations

TL;DR

This paper establishes that the overlap formalism realizes exact chiral symmetry on the lattice through a Grassmann integral representation of the vacuum overlap. It demonstrates that the chiral transformation δψₙ = γ₅(1−aD)ψₙ and its conjugate induces exact chiral symmetry, and derives the covariant gauge current in the action formalism, linking it to the axial anomaly via the overlap framework.

ABSTRACT

Using the grassman-number-integral representation of the vacuum overlap formula, it is shown that the symmetry of the auxiliary quantum fermion system in the overlap formalism induces exact chiral symmetry of the action of the type given by Luscher under the chiral transformation $δψ_n = γ_5(1-a D)ψ_n$ and $ δ\bar ψ_n = \bar ψ_n γ_5$. With this relation, we consider the connection between the covariant form of the anomaly discussed in the context of the overlap formula and the axial anomaly associated to the exact chiral symmetry in the action formalism. The covariant gauge current in the overlap formalism is translated to the action formalism and its explicit expression is obtained.

Motivation & Objective

• To establish exact chiral symmetry in the overlap formalism on the lattice.
• To clarify the connection between the covariant anomaly in the overlap formulation and the axial anomaly in the action formalism.
• To derive the explicit form of the covariant gauge current in the action formalism from the overlap framework.
• To show that the auxiliary quantum fermion system's symmetry induces exact chiral symmetry in the effective action.

Proposed method

• Using a Grassmann-number integral representation of the vacuum overlap formula.
• Applying the chiral transformation δψₙ = γ₅(1−aD)ψₙ and δψ̄ₙ = ψ̄ₙγ₅ to the auxiliary fermion system.
• Mapping the covariant gauge current from the overlap formalism to the action formalism via the Grassmann representation.
• Deriving the explicit expression for the gauge current in the action formalism using the overlap construction.
• Analyzing the anomaly structure by relating the covariant current to the axial current under exact chiral symmetry.
• Employing the overlap Dirac operator D to ensure proper chiral symmetry realization and anomaly matching.

Experimental results

Research questions

• RQ1 How does the overlap formalism realize exact chiral symmetry on the lattice?
• RQ2 What is the explicit form of the covariant gauge current in the action formalism derived from the overlap framework?
• RQ3 How is the axial anomaly in the action formalism connected to the covariant anomaly in the overlap formulation?
• RQ4 What role does the auxiliary fermion system's symmetry play in inducing exact chiral symmetry?
• RQ5 Can the anomaly structure in the overlap formalism be consistently mapped to the action formalism?

Key findings

• The Grassmann integral representation confirms that the chiral symmetry transformation δψₙ = γ₅(1−aD)ψₙ and its conjugate leads to exact chiral symmetry in the effective action.
• The covariant gauge current in the overlap formalism is successfully translated into the action formalism, yielding an explicit expression.
• The axial anomaly in the action formalism is shown to be consistent with the covariant anomaly derived from the overlap construction.
• The overlap Dirac operator D ensures that the chiral symmetry is realized exactly, preserving the anomaly structure.
• The symmetry of the auxiliary fermion system directly induces exact chiral symmetry in the physical action.
• The paper establishes a precise mapping between the covariant current in the overlap formalism and the action formalism, resolving ambiguity in anomaly matching.