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<<BR>> <<BR>> <:20%#eeeeee>'''Abbrev.'''<:15%#eeeeee>'''$c_1$'''<:#eeeeee>'''Description'''   Wilson  0  Wilson plaquette action   tlSym  1/12  treelevel Symanzikimproved gauge action   tadSym  variable  tadpole Symanzikimproved 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 ==== <<FootNote()>> === Lightquark actions === If one attempts to discretise the quark action, one is faced with the fermion doubling problem: the naive lattice transcription produces a 16fold degeneracy of the fermion spectrum. ==== Wilson fermions ==== Wilson's solution to the doubling problem is based on adding a dimension5 operator which removes the doublers from the lowenergy spectrum. The WilsonDirac 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 socalled ''Clover'' or SheikholeslamiWohlert (SW) term. The resulting expression in the massless case reads $D_\text{sw} = D_\text{w}+\frac{ia}{4}\,c_\text{sw}\sigma_{\mu\nu}\widehat{F}_{\mu\nu},$ where $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_\mu,\gamma_\nu]$, and $\widehat{F}_{\mu\nu}$ is a lattice transcription of the gluon field strength tensor $F_{\mu\nu}$. Provided that the coefficient $c_\text{sw}$ is suitably tuned, observables computed using $D_\text{sw}$ will approach the continuum limit with a rate proportional to~$a^2$. Chiral symmetry remains broken, though. The coefficient $c_\text{sw}$ can be determined perturbatively at treelevel (treelevel impr., $c_\text{sw} = 1$ or tlSW in short), via a mean field approach <<FootNote(G. P. Lepage and P. B. Mackenzie, On the viability of lattice perturbation theory, Phys. Rev. D48 (1993) 2250–2264, [heplat/9209022].)>> (meanfield impr. or mfSW) or via a nonperturbative approach <<FootNote(M. L¨uscher, S. Sint, R. Sommer, P. Weisz, and U. Wolff, Nonperturbative O(a) improvement of lattice QCD, Nucl. Phys. B491 (1997) 323–343, [heplat/9609035].)>> (nonperturbativley impr. or npSW). Finally, we mention ''twisted mass QCD'' as a method which was originally designed to address another problem of Wilson's discretisation: the WilsonDirac operator is not protected against the occurrence of unphysical zero modes, which manifest themselves as ''exceptional'' configurations. They occur with a certain frequency in numerical simulations with Wilson quarks and can lead to strong statistical fluctuations. The problem can be cured by introducing a socalled ''chirally twisted'' mass term, after which the fermionic part of the QCD action in the continuum assumes the form <<FootNote(R. Frezzotti, P. A. Grassi, S. Sint, and P. Weisz, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058, [heplat/0101001].)>> $S_\text{F}^\text{tm;cont} = \int d^4{x}\, \bar \psi(x)(\gamma_\mu D_\mu + m + i\mu_\text{q}\gamma_5\tau^3)\psi(x).$ Here, $\mu_\text{q}$ is the twisted mass parameter, and $\tau^3$ is a Pauli matrix. The standard action in the continuum can be recovered via a global chiral field rotation. The lattice action of twisted mass QCD (tmWil) for $N_f=2$ flavours is defined as $S_\text{F}^\text{tm}[U,\bar \psi,\psi] = a^4\sum_{x\in\Lambda_\text{E}}\bar \psi(x)(D_\text{w}+m_0+i\mu_\text{q}\gamma_5\tau^3)\psi(x).$ 
Glossary
The material below is from the first edition of the review. The updated glossary from the present edition can be downloaded as a pdffile.
Gauge actions
The simplest and most widely used discretisation of the YangMills 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 YangMills 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 higherdimensional 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, tadpoleimproved (partially resummed) perturbation theory, renormalisation group methods, and the nonperturbative evaluation of improvement conditions.
In the case of YangMills 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 Symanzikimproved ^{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 
treelevel Symanzikimproved gauge action 
tadSym 
variable 
tadpole Symanzikimproved 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, Onshell 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, [heplat/9911033] (7)
Lightquark actions
If one attempts to discretise the quark action, one is faced with the fermion doubling problem: the naive lattice transcription produces a 16fold degeneracy of the fermion spectrum.
Wilson fermions
Wilson's solution to the doubling problem is based on adding a dimension5 operator which removes the doublers from the lowenergy spectrum. The WilsonDirac 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 socalled Clover or SheikholeslamiWohlert (SW) term. The resulting expression in the massless case reads
$D_\text{sw} = D_\text{w}+\frac{ia}{4}\,c_\text{sw}\sigma_{\mu\nu}\widehat{F}_{\mu\nu},$
where $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_\mu,\gamma_\nu]$, and $\widehat{F}_{\mu\nu}$ is a lattice transcription of the gluon field strength tensor $F_{\mu\nu}$. Provided that the coefficient $c_\text{sw}$ is suitably tuned, observables computed using $D_\text{sw}$ will approach the continuum limit with a rate proportional to~$a^2$. Chiral symmetry remains broken, though. The coefficient $c_\text{sw}$ can be determined perturbatively at treelevel (treelevel impr., $c_\text{sw} = 1$ or tlSW in short), via a mean field approach <<FootNote(G. P. Lepage and P. B. Mackenzie, On the viability of lattice perturbation theory, Phys. Rev. D48 (1993) 2250–2264, [heplat/9209022].)>> (meanfield impr. or mfSW) or via a nonperturbative approach <<FootNote(M. L¨uscher, S. Sint, R. Sommer, P. Weisz, and U. Wolff, Nonperturbative O(a) improvement of lattice QCD, Nucl. Phys. B491 (1997) 323–343, [heplat/9609035].)>> (nonperturbativley impr. or npSW).
Finally, we mention twisted mass QCD as a method which was originally designed to address another problem of Wilson's discretisation: the WilsonDirac operator is not protected against the occurrence of unphysical zero modes, which manifest themselves as exceptional configurations. They occur with a certain frequency in numerical simulations with Wilson quarks and can lead to strong statistical fluctuations. The problem can be cured by introducing a socalled chirally twisted mass term, after which the fermionic part of the QCD action in the continuum assumes the form <<FootNote(R. Frezzotti, P. A. Grassi, S. Sint, and P. Weisz, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058, [heplat/0101001].)>>
$S_\text{F}^\text{tm;cont} = \int d^4{x}\, \bar \psi(x)(\gamma_\mu D_\mu + m + i\mu_\text{q}\gamma_5\tau^3)\psi(x).$
Here, $\mu_\text{q}$ is the twisted mass parameter, and $\tau^3$ is a Pauli matrix. The standard action in the continuum can be recovered via a global chiral field rotation. The lattice action of twisted mass QCD (tmWil) for $N_f=2$ flavours is defined as
$S_\text{F}^\text{tm}[U,\bar \psi,\psi] = a^4\sum_{x\in\Lambda_\text{E}}\bar \psi(x)(D_\text{w}+m_0+i\mu_\text{q}\gamma_5\tau^3)\psi(x).$