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 ⁶⁷ 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.

The leading lattice artefacts are $O(a^2)$ or better for all discretisations.

References

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 ¹. 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 ². 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 ³. 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. ^{4 5 6 7 8}.