== Glossary ==

The material below is from the first edition of the review. The updated glossary from the present edition can be downloaded as a [[attachment:Media/FLAG_glossary.pdf|pdf-file|&do=get]]. 

=== Gauge actions ===

The simplest and most widely used discretisation of the Yang-Mills
part of the QCD action is the Wilson plaquette action <<FootNote(K. G. Wilson, Confinement of quarks, Phys. Rev. D10 (1974) 2445–2459.)>>:

$S_\text{G} = \beta\sum_{x} \sum_{\mu<\nu}\Big(1-\frac{1}{3}\text{Re Tr}\,W_{\mu\nu}^{1\times1}(x)\Big),$

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`` <<FootNote(K. Symanzik, Continuum limit and improved action in lattice theories. 1. Principles and φ4 theory, Nucl. Phys. B226 (1983) 187.)>> <<FootNote(K. Symanzik, Continuum limit and improved action in lattice theories. 2. O(N) nonlinear sigma model in perturbation theory, Nucl. Phys. B226 (1983) 205.)>>, 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'' <<FootNote(M. L&uuml;scher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 97 (1985) 59.)>>, ''Iwasaki'' <<FootNote(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.)>>, and ''DBW2'' <<FootNote(T. Takaishi, Heavy quark potential and effective actions on blocked configurations, Phys. Rev. D54 (1996) 1050–1053.)>><<FootNote(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])>> 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.
<<BR>>
<<BR>>

||<:20%#eeeeee>'''Abbrev.'''||<:15%#eeeeee>'''$c_1$'''||<:#eeeeee>'''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 ====
<<FootNote()>>


=== 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
 
$D_\text{sw} = D_\text{w}+\frac{ia}{4}\,c_\text{sw}\sigma_{\mu\nu}\widehat{F}_{\mu\nu},$

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 tree-level (tree-level 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, [hep-lat/9209022].)>> (mean-field impr. or mfSW) or via a non-perturbative approach <<FootNote(M. L¨uscher, S. Sint, R. Sommer, P. Weisz, and U. Wolff,
Non-perturbative O(a) improvement of lattice QCD, Nucl. Phys. B491 (1997) 323–343, [hep-lat/9609035].)>> (non-perturbativley 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 Wilson-Dirac 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 so-called ''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, [hep-lat/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
 
$S_\text{F}^\text{tm}[U,\bar \psi,\psi] = a^4\sum_{x\in\Lambda_\text{E}}\bar \psi(x)(D_\text{w}+m_0+i\mu_\text{q}\gamma_5\tau^3)\psi(x).$

Although this formulation breaks physical parity and flavour symmetries, is has a number of advantages over standard Wilson fermions. In particular,
the presence of the twisted mass parameter $\mu_\text{q}$ protects the discretised theory against unphysical zero modes. Another attractive feature of twisted mass
lattice QCD is the fact that the leading lattice artefacts are of order $a^2$ without the need to add the Sheikholeslami-Wohlert term <<FootNote(R. Frezzotti and G. C. Rossi, Chirally improving Wilson fermions. I: O(a) improvement, JHEP 08 (2004) 007, [hep-lat/0306014].)>>. Although the problem of explicit chiral symmetry breaking remains, the twisted formulation is particularly useful to circumvent some of the problems that are encountered in connection with the renormalization of local operators on the lattice, such as those required to determine $B_\text{K}$.

==== Staggered fermions ====
An alternative procedure to deal with the doubling problem is based on so-called ''staggered'' or Kogut-Susskind (KS) fermions <<FootNote(L. Susskind, Lattice fermions, Phys. Rev. D16 (1977) 3031–3039.)>>. Here the degeneracy is only lifted partially, from 16 down to~4. It has become customary to refer to these residual doublers as ''tastes'' in order to distinguish them from physical flavours. At order~$a^2$ different tastes can interact via gluon exchange, thereby generating large lattice artefacts. The improvement programme can be used to suppress taste-changing interactions, leading to ''improved staggered fermions'', with the so-called ''Asqtad'' action as one of its widely used versions <<FootNote(K. Orginos, D. Toussaint, and R. L. Sugar, Variants of fattening and flavor symmetry restoration, Phys. Rev. D60 (1999) 054503, [hep-lat/9903032].)>>. The standard procedure to remove the residual doubling of staggered quarks (''four tastes per flavour'') is to take fractional powers of the quark determinant in the QCD functional integral. This is usually referred to as the ''fourth root trick''. The validity of this procedure has not been rigorously proven so far. In fact, it has been questioned by several authors, and the issue is still hotly debated (for both sides of the argument see the reviews in refs. <<FootNote(S. D&uuml;rr, Theoretical issues with staggered fermion simulations, PoS LAT2005 (2006) 021, [hep-lat/0509026].)>> <<FootNote(S. R. Sharpe, Rooted staggered fermions: good, bad or ugly?, PoS LAT2006 (2006) 022, [hep-lat/0610094].)>> <<FootNote(A. S. Kronfeld, Lattice gauge theory with staggered fermions: how, where, and why (not), PoS LAT2007 (2007) 016, [arXiv:0711.0699].)>> <<FootNote(M. Golterman, QCD with rooted staggered fermions, PoS CONFINEMENT8 (2008) 014, [arXiv:0812.3110].)>> <<FootNote(M. Creutz, Why rooting fails, PoS LAT2007 (2007) 007, [arXiv:0708.1295].)>>.