[Paper Review] Properties of minimally doubled fermions

Johannes Heinrich Weber|arXiv (Cornell University)|Jan 1, 2015

Quantum Chromodynamics and Particle Interactions130 references3 citations

TL;DR

This paper investigates Karsten-Wilczek (KW) fermions, a type of minimally doubled fermion action in lattice QCD that preserves chiral symmetry with two degenerate quark flavors. Using perturbative renormalization and non-perturbative numerical simulations in the quenched approximation, it demonstrates that tuned counterterms remove anisotropy in self-energy and vacuum polarization, and observes Goldstone-like behavior in the pseudoscalar channel with chiral logarithms consistent with phenomenology, while a second channel shows residual mass scaling as O(a²), indicating lattice artifacts.

ABSTRACT

Most quark actions in lattice QCD encounter difficulties with chiral symmetry and its spontaneous breakdown. Minimally doubled fermions (MDF) are a category of strictly local chiral lattice fermions, whose continuum limit reproduces two degenerate quark flavours. The two poles of their Dirac operator are aligned such that symmetries under charge conjugation or reflection of one particular direction are explictly broken at finite lattice spacing. Properties of MDF are scrutinised with regard to broken symmetry and meson spectrum to discern their suitability for numerical studies of QCD. Interactions induce anisotropic operator mixing for MDF. Hence, restoration of broken symmetries in the continuum limit requires three counterterms, one of which is power-law divergent. Counterterms and operator mixing are studied perturbatively for two variants of MDF. Two independent non-perturbative procedures for removal of the power-law divergence are developed by means of a numerical study of hadronic observables for one variant of MDF in quenched approximation. Though three out of four pseudoscalar mesons are affected by lattice artefacts, the spectrum's continuum limit is consistent with two-flavour QCD. Thus, suitability of MDF for numerical studies of QCD in the quenched approximation is demonstrated.

Motivation & Objective

• To study the renormalization and symmetry properties of Karsten-Wilczek (KW) fermions, a class of minimally doubled fermions that satisfy the Nielsen-Ninomiya no-go theorem with two degenerate chiral modes.
• To determine three counterterms arising from explicit hypercubic symmetry breaking due to the two distinct poles in the Brillouin zone.
• To test chiral symmetry conservation by deriving and verifying vector and axial currents.
• To establish non-perturbative renormalization conditions for the fermionic counterterm using numerical simulations in the quenched approximation.
• To investigate the hadron spectrum, particularly the pseudoscalar meson sector, to assess the presence of Goldstone boson behavior and lattice artifacts in the chiral regime.

Proposed method

• Perturbative calculation of three counterterms to restore isotropy in self-energy and vacuum polarization functions at one-loop level.
• Numerical simulations of KW fermions in the quenched approximation to determine non-perturbative renormalization criteria.
• Tuning the fermionic counterterm using two complementary conditions: minimization of pseudoscalar mass anisotropy and restoration of tree-level frequency spectrum in hadronic correlation functions.
• Use of a generalized contraction algorithm with flags (amu, tmu, imu, etc.) to handle complex Dirac matrix structures and momentum insertions efficiently in the correlation function computation.
• Implementation of a modular subroutine system using macros (k, f, g, Re, Im) to compute spinor contractions with correct signs and complex components for all Dirac structures.
• Application of external momentum insertions via four-component vectors to handle arbitrary time direction choices and spatial hadron momenta.

Experimental results

Research questions

• RQ1 How do explicit hypercubic symmetry breaking effects manifest in the self-energy and vacuum polarization functions of KW fermions, and can they be removed via perturbative counterterms?
• RQ2 To what extent does the tuned KW fermion action reproduce Goldstone boson-like behavior in the pseudoscalar channel, including quenched chiral logarithms?
• RQ3 What is the nature of the second pseudoscalar channel using γ₀ instead of γ₅, and does it exhibit residual mass scaling consistent with lattice artifacts?
• RQ4 Can non-perturbative tuning of the fermionic counterterm be achieved robustly using numerical observables, and how does it compare to perturbative estimates?
• RQ5 Are KW fermions free from exceptional configurations in the chiral regime, as indicated by the hadron spectrum?

Key findings

• The three perturbatively determined counterterms successfully remove anisotropy in self-energy and vacuum polarization functions at one-loop level, restoring isotropy in the correlation functions.
• Goldstone boson-like behavior, including quenched chiral logarithms, is observed in the pseudoscalar channel with γ₅, in agreement with phenomenological predictions.
• The second pseudoscalar channel using γ₀ exhibits scaling of the ground state with the bare quark mass like a Goldstone boson, but retains a residual mass that vanishes as O(a²) in the continuum limit, indicating lattice artifacts.
• The numerical tuning of the fermionic counterterm via mass anisotropy minimization and frequency spectrum restoration yields consistent results within errors, validating the non-perturbative approach.
• The absence of exceptional configurations in the chiral regime is indicated by the smooth behavior of the hadron spectrum, particularly in the pseudoscalar channel.
• The first numerical study of KW fermions demonstrates their viability for future simulations with dynamical fermions and inclusion of quark-disconnected diagrams, with vector mesons and nucleons as next key targets.