Glossary
The material below is from the first edition of the review. The updated glossary from the present edition can be downloaded as a pdf-file.
Gauge actions
The simplest and most widely used discretisation of the Yang-Mills part of the QCD action is the Wilson plaquette action ^{1}:
$S_\text{G} = \beta\sum_{x} \sum_{\mu<\nu}\Big(1-\frac{1}{3}\text{Re Tr}\,W_{\mu\nu}^{1\times1}(x)\Big),$
where the plaquette, $W_{\mu\nu}^{1\times1}(x)$, is the product of link variables around an elementary square of the lattice, i.e.
$W_{\mu\nu}^{1\times1}(x) \equiv U_\mu(x)U_\nu(x+a\hat{\mu})U_\mu(x+a\hat{\nu})^{-1} U_\nu(x)^{-1}.$
This expression reproduces the Euclidean Yang-Mills action in the continuum up to corrections of order $a^2$. There is a general formalism, known as the Symanzik improvement programme ^{2} ^{3}, which is designed to cancel the leading lattice artefacts, such that observables have an accelerated rate of convergence to the continuum limit. The improvement programme is implemented by adding higher-dimensional operators, whose coefficients must be tuned appropriately in order to cancel the leading lattice artefacts. The effectiveness of this procedure depends largely on the method with which the coefficients are determined. The most widely applied methods (in ascending order of effectiveness) include perturbation theory, tadpole-improved (partially resummed) perturbation theory, renormalisation group methods, and the non-perturbative evaluation of improvement conditions.
In the case of Yang-Mills theory, the simplest version of an improved lattice action is obtained by adding rectangular 1$\times$2 loops to the plaquette action, i.e.
$S_\text{G}^\text{imp} = \beta\sum_{x}\left\{ c_0\sum_{\mu<\nu}\Big(1-\frac{1}{3}\text{Re Tr}\,W_{\mu\nu}^{1\times1}(x)\Big)+c_1\sum_{\mu,\nu} \Big(1-\frac{1}{3}\text{Re Tr}\,W_{\mu\nu}^{1\times2}(x)\Big) \right\},$
where the coefficients $c_0, c_1$ satisfy the normalisation condition $c_0+8c_1=1$. The Symanzik-improved ^{4}, Iwasaki ^{5}, and DBW2 ^{6}^{7} actions are all defined through the equation above via particular choices for $c_0, c_1$. Details are listed in the following Table, together with the abbreviations used in the summary tables.
Abbrev. |
$c_1$ |
Description |
Wilson |
0 |
Wilson plaquette action |
tlSym |
-1/12 |
tree-level Symanzik-improved gauge action |
tadSym |
variable |
tadpole Symanzik-improved gauge action |
Iwasaki |
−0.331 |
Renormalisation group improved (“Iwasaki”) action |
DBW2 |
−1.4088 |
Renormalisation group improved (“DBW2”) action |
The leading lattice artefacts are $O(a^2)$ or better for all discretisations.
References
K. G. Wilson, Confinement of quarks, Phys. Rev. D10 (1974) 2445–2459. (1)
K. Symanzik, Continuum limit and improved action in lattice theories. 1. Principles and φ4 theory, Nucl. Phys. B226 (1983) 187. (2)
K. Symanzik, Continuum limit and improved action in lattice theories. 2. O(N) nonlinear sigma model in perturbation theory, Nucl. Phys. B226 (1983) 205. (3)
M. Lüscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 97 (1985) 59. (4)
Y. Iwasaki, Renormalization group analysis of lattice theories and improved lattice action: two dimensional nonlinear O(N) sigma model, Nucl. Phys. B258 (1985) 141–156. (5)
T. Takaishi, Heavy quark potential and effective actions on blocked configurations, Phys. Rev. D54 (1996) 1050–1053. (6)
P. de Forcrand et. al., Renormalization group flow of SU(3) lattice gauge theory: numerical studies in a two coupling space, Nucl. Phys. B577 (2000) 263–278, [hep-lat/9911033] (7)
Light-quark actions
If one attempts to discretise the quark action, one is faced with the fermion doubling problem: the naive lattice transcription produces a 16-fold degeneracy of the fermion spectrum.
Wilson fermions
Wilson's solution to the doubling problem is based on adding a dimension-5 operator which removes the doublers from the low-energy spectrum. The Wilson-Dirac operator for the massless case reads \cite{Wilson:1974sk}
$D_\text{w} = \frac{1}{2}\gamma_\mu(\nabla_\mu+\nabla_\mu^*)+a\nabla_\mu^*\nabla_\mu,$
where $\nabla_\mu,\,\nabla_\mu^*$ denote lattice versions of the covariant derivative. Adding the Wilson term, $a\nabla_\mu^*\nabla_\mu$, results in an explicit breaking of chiral symmetry even in the massless theory. Furthermore, the leading order lattice artefacts are of order $a$. With the help of the Symanzik improvement programme, the leading artefacts can be cancelled by adding the so-called Clover or Sheikholeslami-Wohlert (SW) term. The resulting expression in the massless case reads
$D_\text{sw} = D_\text{w}+\frac{ia}{4}\,c_\text{sw}\sigma_{\mu\nu}\widehat{F}_{\mu\nu},$