Wilson fermion explained

In lattice field theory, Wilson fermions are a fermion discretization that allows to avoid the fermion doubling problem proposed by Kenneth Wilson in 1974.^[1] They are widely used, for instance in lattice QCD calculations.^{[2] [3] [4] [5]}

An additional so-called Wilson term

S _{W} = - a ^{d+1} \sum _{x,\mu} (i)/(2 a ^{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 \psi _{x} at every lattice point x , and the vectors \hat \mu being unit vectors in the \mu direction. The inverse free fermion propagator in momentum space now reads^[6] D(p) = m + (ia\sum)/(_{\mu}) \gamma _{\mu\sin\left(p} _{\mu} a\right) + (1 a\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 tom + (2 l)/(a)

where

l is the number of momentum components with p _{\mu} = \pi/a . In the continuum limita → 0 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 \gamma _{5} .

Notes and References

1. Wilson. K.G.. Kenneth G. Wilson. 1974. Confinement of quarks. 10.1103/PhysRevD.10.2445. American Physical Society. 2445-2459. Phys. Rev. D. 10. 8. 1974PhRvD..10.2445W .
2. Book: Lattice Gauge Theories: An Introduction . Heinz J. . Rothe . World Scientific Publishing Company . 2005 . World Scientific Lecture Notes in Physics . 3. 978-9814365857 . 4 Fermions on the lattice. 56–57.
3. Book: Cambridge . Cambridge Lecture Notes in Physics . Introduction to Quantum Fields on a Lattice . 10.1017/CBO9780511583971. 9780511583971 . Cambridge University Press . Smit, J. . 2002. 6 Fermions on the lattice. 156–160. 20.500.12657/64022 .
4. Book: Cambridge . Cambridge Monographs on Mathematical Physics . Quantum Fields on a Lattice . 10.1017/CBO9780511470783. 9780511470783 . Cambridge University Press . Montvay, I.; Münster, G. . 1994. 4 Fermion fields. 221–224. 118339104 .
5. Book: FLAG Working Group; Aoki, S.. etal . Review of Lattice Results Concerning Low-Energy Particle Physics. 1310.8555 . 10.1140/epjc/s10052-014-2890-7 . Eur. Phys. J. C . 74 . 113–115 . 2014. A.1 Lattice actions. 9 . 25972762 . 4410391 .
6. Book: Gattringer. C.. Lang. C.B.. 2009. Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics 788. 10.1007/978-3-642-01850-3. Springer. 5 Fermions on the lattice. 112–114. 978-3642018497.