Wilson fermion

{{Short description|Lattice fermion discretisation}}

In lattice field theory, Wilson fermions are a fermion discretization that allows to avoid the fermion doubling problem proposed by Kenneth Wilson in 1974.{{cite journal|last1=Wilson|first1=K.G.|author-link1=Kenneth G. Wilson|date=1974|title=Confinement of quarks|url=https://link.aps.org/doi/10.1103/PhysRevD.10.2445|doi=10.1103/PhysRevD.10.2445|publisher=American Physical Society|page=2445-2459|journal=Phys. Rev. D|volume=10|issue=8|bibcode=1974PhRvD..10.2445W |url-access=subscription}} They are widely used, for instance in lattice QCD calculations.{{cite book

|title = Lattice Gauge Theories: An Introduction

|first1 = Heinz J.

|last1 = Rothe

|publisher = World Scientific Publishing Company

|year = 2005

|series = World Scientific Lecture Notes in Physics

|edition = 3

|isbn = 978-9814365857

|chapter = 4 Fermions on the lattice|pages=56–57}}{{cite book

|place=Cambridge

|series=Cambridge Lecture Notes in Physics

|title=Introduction to Quantum Fields on a Lattice

|doi=10.1017/CBO9780511583971

|isbn = 9780511583971

|publisher=Cambridge University Press

|author=Smit, J.

|year=2002

|chapter = 6 Fermions on the lattice

|pages = 156–160|hdl=20.500.12657/64022

}}{{cite book

|place=Cambridge

|series=Cambridge Monographs on Mathematical Physics

|title=Quantum Fields on a Lattice

|doi=10.1017/CBO9780511470783

|isbn=9780511470783

|publisher=Cambridge University Press

|author=Montvay, I.; Münster, G.

|year=1994

|chapter = 4 Fermion fields

|pages = 221–224|s2cid=118339104

}}{{cite book

|author=FLAG Working Group; Aoki, S.

|display-authors=etal

|title = Review of Lattice Results Concerning Low-Energy Particle Physics

|arxiv=1310.8555

|doi = 10.1140/epjc/s10052-014-2890-7

|series = Eur. Phys. J. C

|volume = 74

|pages = 113–115

|date = 2014

|chapter=A.1 Lattice actions|issue=9

|pmid=25972762

|pmc=4410391

}}

An additional so-called Wilson term

:$$

S_W = -a^{d+1}\sum_{x,\mu}\frac{i}{2a^2}\left(\bar\psi_x\psi_{x+\hat\mu}+\bar\psi_{x+\hat\mu}\psi_{x}-2\bar\psi_x\psi_x\right)

is introduced supplementing the naively discretized Dirac action in $d$-dimensional Euclidean spacetime with lattice spacing $a$, Dirac fields $\backslash psi\_x$ at every lattice point $x$, and the vectors $\backslash hat\; \backslash mu$ being unit vectors in the $\backslash mu$ direction. The inverse free fermion propagator in momentum space now reads{{cite book|last1=Gattringer|first1=C.|last2=Lang|first2=C.B.|date=2009|title=Quantum Chromodynamics on the Lattice: An Introductory Presentation|series=Lecture Notes in Physics 788|url=|doi=10.1007/978-3-642-01850-3|location=|publisher=Springer|chapter=5 Fermions on the lattice|pages=112–114|isbn=978-3642018497}}

:$$

D(p) = m + \frac ia\sum_\mu \gamma_\mu\sin\left(p_\mu a\right)+\frac1a\sum_\mu\left(1-\cos\left(p_\mu a\right)\right)\,

where the last addend corresponds to the Wilson term again. It modifies the mass $m$ of the doublers to

:$$

m+\frac{2l}{a}\,

where $l$ is the number of momentum components with $p\_\backslash mu\; =\; \backslash pi/a$. In the continuum limit

$a\backslash rightarrow0$ the doublers become very heavy and decouple from the theory.

Wilson fermions do not contradict the Nielsen–Ninomiya theorem because they explicitly violate chiral symmetry since the Wilson term does not anti-commute with $\backslash gamma\_5$.