Wilson fermions
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| Wilson fermions | |
|---|---|
| Name | Wilson fermions |
| Field | Quantum chromodynamics; Lattice gauge theory |
| Introduced | 1974 |
| Introduced by | Kenneth G. Wilson |
| Related | Dirac fermion, Chiral symmetry, Ginsparg–Wilson relation |
Wilson fermions are a formulation used in Lattice gauge theory to represent Dirac fermion fields on a discrete spacetime lattice. Developed to address the fermion doubling problem that arises when naively discretizing continuum Quantum chromodynamics (QCD), the Wilson prescription adds a momentum-dependent term to lift unphysical species while breaking chiral symmetry explicitly at finite lattice spacing. The approach is central to many numerical studies undertaken by collaborations such as European Twisted Mass Collaboration, HotQCD Collaboration, RBC and UKQCD and techniques used at facilities like CERN and Brookhaven National Laboratory.
Introduction
Wilson fermions were proposed by Kenneth G. Wilson to solve the excess-multiplicity issue encountered in lattice formulations inspired by the Dirac equation and early work by Paul Dirac and Richard Feynman. The construction modifies the lattice action introduced in studies following M. Creutz and John Kogut of discrete Yang–Mills theory, enabling controlled continuum extrapolations employed by large collaborations such as MILC and JLQCD. Wilson’s method interacts with concepts developed by Gerard 't Hooft, Murray Gell-Mann, and later formalized in relations like the Ginsparg–Wilson relation.
Lattice formulation and fermion doubling problem
The lattice formulation begins with a naive discretization of the Dirac operator on a hypercubic lattice, a procedure influenced by earlier lattice work at institutions such as IBM Research and Los Alamos National Laboratory. The naive scheme produces 2^d fermion species in d dimensions, a manifestation of the Nielsen–Ninomiya theorem articulated by Holger Bech Nielsen and Masao Ninomiya. This doubling complicates comparisons to continuum Quantum chromodynamics results used by experiments at the Large Hadron Collider and phenomenology studied by groups including Institute for Nuclear Theory and Lawrence Berkeley National Laboratory. Remedies explored by researchers like Michael Creutz and Kenneth Wilson include staggered formulations used by K. Jansen and projection techniques employed in early Columbia University lattice studies.
Wilson term and action
Wilson introduced an additional second-order finite-difference operator—now called the Wilson term—added to the naive action. The Wilson action explicitly breaks chiral symmetry analogous to mass terms considered in work by Sidney Coleman and Steven Weinberg, introducing an O(a) lattice-spacing-dependent mass shift that removes doublers by giving them masses near the cutoff scale set by the inverse lattice spacing. Practical implementations couple the fermion fields to gauge links following the staples and smearing methods influenced by algorithms from David J. Gross and Frank Wilczek, and use algorithmic advances such as the Hybrid Monte Carlo method pioneered by teams at Brookhaven National Laboratory and DESY.
Properties and symmetry breaking
Wilson fermions preserve gauge invariance associated with groups like SU(3), but break chiral symmetry explicitly at finite lattice spacing, complicating studies of spontaneous chiral symmetry breaking as in the work of Yakov Borisovich Zel'dovich and Yoichiro Nambu. The explicit breaking requires additive mass renormalization, a complication addressed via nonperturbative renormalization programs run by collaborations such as ALPHA Collaboration and techniques developed by Luscher and Martin Lüscher. Wilson formulations show scaling violations of order a, prompting Symanzik improvement programs inspired by K. Symanzik to reduce discretization errors, and requiring operator mixing analyses similar to those used by Gerard 't Hooft.
Implementations and numerical methods
Numerical implementations of Wilson fermions appear in many lattice QCD codes developed at institutions like Fermilab, RIKEN, and University of California, Santa Barbara. Solvers such as conjugate gradient and multigrid methods advanced by teams at Argonne National Laboratory and ETMC accelerate inversions of the Wilson–Dirac operator. Algorithmic improvements include clover (Sheikholeslami–Wohlert) improvement named after B. Sheikholeslami and R. Wohlert, twisted-mass variants by C. Urbach and deflation techniques from A. Frommer, all incorporated into software frameworks like Chroma and QPhiX used by community projects such as USQCD.
Applications in lattice QCD and results
Wilson fermions underpin computations of hadron spectroscopy, parton distribution functions, and matrix elements relevant to experiments at CERN, Jefferson Lab, and J-PARC. Results computed using Wilson-based ensembles contributed to determinations of quark masses and the CKM matrix elements compared with measurements by Belle II and LHCb. Precision studies of thermodynamics by the HotQCD Collaboration and investigations of nucleon structure by RBC and UKQCD have used Wilson or improved-Wilson actions to confront results from ATLAS and CMS.
Alternatives and improvements
Alternatives to Wilson fermions include staggered fermions by John Kogut and Leonard Susskind, domain-wall fermions developed by David Kaplan and Yoshinobu Shamir, and overlap fermions based on the Ginsparg–Wilson relation with implementations influenced by Herbert Neuberger. Improvements to Wilson formulations include clover improvement by Sheikholeslami and Wohlert, twisted-mass Wilson by ETMC members, and nonperturbative O(a) improvement programs driven by ALPHA Collaboration and figures like Martin Lüscher. These approaches are chosen based on trade-offs among chiral symmetry, computational cost, and topological properties studied by researchers at CERN, Fermilab, and RIKEN.