[Paper Review] Note on all-order Landau-level structures of the Heisenberg-Euler effective actions for QED and QCD

Koichi Hattori, Kazunori Itakura+1 more|arXiv (Cornell University)|Jan 17, 2020

Quantum Chromodynamics and Particle Interactions78 references4 citations

TL;DR

This paper provides a complete analytic derivation of all-order Landau-level structures in the Heisenberg-Euler effective actions for QED and QCD in constant electromagnetic and chromo-electromagnetic fields. By using proper-time representations, it identifies all Landau levels and Zeeman energies, derives the vacuum persistence probability as a sum over all Landau levels, and reveals that a magnetic field enhances pair production in spinor QED while suppressing it in scalar QED due to zero-point energy, with full cancellation between longitudinal gluons and ghosts in QCD's Schwinger mechanism.

ABSTRACT

We investigate the Landau-level structures encoded in the famous Heisenberg-Euler (HE) effective action in constant electromagnetic fields. We first discuss the HE effective actions for scalar and spinor QED, and then extend it to the QCD analogue in the covariantly constant chromo-electromagnetic fields. We identify all the Landau levels and the Zeeman energies starting out from the proper-time representations at the one-loop order, and derive the vacuum persistence probability for the Schwinger mechanism in the summation form over independent contributions of the all-order Landau levels. We find an enhancement of the Schwinger mechanism catalyzed by a magnetic field for spinor QED and, in contrast, a stronger exponential suppression for scalar QED due to the "zero-point energy" of the Landau quantization. For QCD, we identify the discretized energy levels of the transverse and longitudinal gluon modes on the basis of their distinct Zeeman energies, and explicitly confirm the cancellation between the longitudinal-gluon and ghost contributions in the Schwinger mechanism. We also discuss the unstable ground state of the perturbative gluon excitations known as the Nielsen-Olesen instability.

Motivation & Objective

• To clarify the physical content of the Heisenberg-Euler effective action obscured by the standard proper-time representation.
• To systematically identify all Landau levels and Zeeman energies in QED and QCD under constant fields.
• To derive the vacuum persistence probability as a sum over all Landau-level contributions for the Schwinger mechanism.
• To investigate the role of zero-point energy in scalar vs. spinor QED and confirm cancellation in QCD gluon-ghost contributions.
• To examine the connection between Landau quantization and the Nielsen-Olesen instability in QCD.

Proposed method

• Using the proper-time representation of the one-loop effective action to extract Landau-level contributions.
• Performing exact analytic integration over transverse momenta using Laguerre polynomial identities.
• Deriving closed-form expressions for the vacuum persistence amplitude as a sum over Landau levels.
• Applying recursive relations of Laguerre polynomials to simplify integrals in bosonic and fermionic sectors.
• Extending the formalism to non-Abelian QCD by analyzing covariantly constant chromo-electromagnetic fields.
• Verifying cancellation between longitudinal gluon and ghost contributions in the QCD effective action.

Experimental results

Research questions

• RQ1 How are all Landau levels and Zeeman energies encoded in the Heisenberg-Euler effective action for QED and QCD?
• RQ2 What is the exact form of the vacuum persistence probability in terms of all-order Landau-level contributions?
• RQ3 Why does a magnetic field enhance pair production in spinor QED but suppress it in scalar QED?
• RQ4 How do longitudinal gluon and ghost contributions cancel in the QCD Schwinger mechanism?
• RQ5 What is the role of Landau quantization in the Nielsen-Olesen instability of perturbative gluon modes?

Key findings

• The vacuum persistence probability for the Schwinger mechanism is expressed as a sum over all Landau levels, with each level contributing independently.
• In spinor QED, the magnetic field enhances the Schwinger mechanism due to positive Zeeman splitting and level-dependent critical fields.
• In scalar QED, the zero-point energy of Landau quantization leads to stronger exponential suppression of pair production compared to spinor QED.
• In QCD, the transverse and longitudinal gluon modes exhibit distinct discretized energy levels with different Zeeman shifts, confirming the structure of the effective action.
• The longitudinal-gluon and ghost contributions to the effective action exactly cancel in the imaginary part, ensuring unitarity and consistency of the Schwinger mechanism in QCD.
• The Nielsen-Olesen instability arises from a tachyonic ground state that is subject to Landau quantization and negative Zeeman shifts in the chromo-magnetic field.