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.

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 ^[1] (mean-field impr. or mfSW) or via a non-perturbative approach ^[2] (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 ^[3]

$S_\text{F}^\text{tm;cont} = \int d^4{x}\, \bar \psi(x)(\gamma_\mu D_\mu + m + i\mu_\text{q}\gamma_5\tau^3)\psi(x).$

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 ^[4]. 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 ^[5]. 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 ^[6]. 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. ^{[7] [8] [9] [10] [11]}.

Ginsparg-Wilson fermions

Fermionic lattice actions, which do not suffer from the doubling problem whilst preserving chiral symmetry go under the name of Ginsparg-Wilson fermions. In the continuum the massless Dirac operator anti-commutes with $\gamma_5$. At non-zero lattice spacing chiral symmetry can be realised even if this condition is relaxed according to ^{[12] [13]}

$\left\{D,\gamma_5\right\} = aD\gamma_5 D,$

which is now known as the Ginsparg-Wilson relation ^[14]. A lattice Dirac operator which satisfies the GW relation can be constructed in several ways. The domain wall construction proceeds by introducing a fifth dimension of length $N_5$ and coupling the fermions to a mass defect (i.e. a negative mass term) ^[15]. The five-dimensional action can be constructed such that modes of opposite chirality are trapped at the four dimensional boundaries in the limit of an infinite extent of the extra dimension ^[16]. In any real simulation, though, one has to work with a finite value of $N_5$, so that the decoupling of chiral modes is not exact. This leads to a residual breaking of chiral symmetry, which, however, is exponentially suppressed. A doubler-free, (approximately) chirally symmetric quark action can thus be realised at the expense of simulating a five-dimensional theory.

The so-called overlap or Neuberger-Dirac operator can be derived from the domain wall formulation ^[17]. It acts in four space-time dimensions and is, in its simplest form, defined by

$D_{\rm N} = \frac{1}{\bar a} \left( 1-\frac{A}{\sqrt{A^\dagger{A}}}\right), \quad A=1+s-aD_{\rm w},\quad \bar a=\frac{a}{1+s},$

where $D_{\rm w}$ is the massless Wilson-Dirac operator, and $|s|<1$ is a tunable parameter. The overlap operator $D_\text{N}$ removes all doublers from the spectrum, and can easily be shown to satisfy the Ginsparg-Wilson relation. The occurrence of an inverse square root in $D_\text{N}$ renders the application of $D_\text{N}$ in a computer program potentially very costly, since it must be implemented using, for instance, a polynomial approximation.

The third example of an operator which satisfies the Ginsparg-Wilson relation is the so-called fixed-point action ^{[18] [19]}. This construction proceeds via a renormalisation group approach. A related formalism are the so-called chirally improved fermions ^[20].

Smearing

A simple modification which can help improve the action as well as the computational performance is the use of smeared gauge fields in the covariant derivatives of the fermionic action. Any smearing procedure is acceptable as long as it consists of only adding irrelevant (local) operators. Moreover, it can be combined with any discretisation of the quark action. The Asqtad staggered quark action mentioned above ^[6] is an example which makes use of so-called Asqtad smeared (or fat) links. Another example is the use of n-HYP smeared ^[21] or n-stout smeared ^[22] gauge links in the tree-level clover improved discretisation of the quark action, denoted by n-HYP tlSW and n-stout tlSW in the following.

In the table below we summarise the most widely used discretisations of the quark action and their main properties together with the abbreviations used in the summary tables. Note that in order to maintain the leading lattice artefacts of the actions as given in the table in non-spectral observables (like operator matrix elements) the corresponding non-spectral operators need to be improved as well.

References

Matching and running

Chiral extrapolation

Symanzik’s framework can be combined with Chiral Perturbation Theory. The well-known terms occurring in the chiral effective Lagrangian are then supplemented by contributions proportional to powers of the lattice spacing a. The additional terms are constrained by the symmetries of the lattice action and therefore depend on the specific choice of the discretization. The resulting effective theory can be used to analyze the a-dependence of the various quantities of interest – provided the quark masses and the momenta considered are in the range where the truncated chiral perturbation series yields an adequate approximation. Understanding the dependence on the lattice spacing is of central importance for a controlled extrapolation to the continuum limit. For staggered fermions, this program has first been carried out for a single staggered flavor (a single staggered field) ^[1] at $O(a^2)$. In the following, this effective theory is denoted by SχPT. It was later generalized to an arbitrary number of flavours ^{[2] [3]}, and to next-to-leading order ^[4]. The corresponding theory is commonly called Rooted Staggered chiral perturbation theory and is denoted by RSχPT. For Wilson fermions, the effective theory has been developed in ^{[5] [6] [7]} and is called WχPT, while the theory for Wilson twisted mass fermions ^{[8] [9] [10]} is termed tmWχPT. Another important approach is to consider theories in which the valence and sea quark masses are chosen to be different. These theories are called partially quenched. The acronym for the corresponding chiral effective theory is PQχPT ^{[11] [12] [13] [14]}. Finally, one can also consider theories where the fermion discretizations used for the sea and the valence quarks are different. The effective chiral theories for these “mixed action” theories are referred to as MAχPT ^[15][16][17].

References

Summary of simulated lattice actions

Summary of simulated lattice actions with $N_f = 2+1$ and $N_f = 2+1+1$ quark flavours.

^A) The calculation uses domain wall fermions in the valence quark sector.

^B) The calculation uses HISQ staggered fermions in the valence quark sector.

^C) The calculation uses HYP smeared improved staggered fermions in the valence quark sector.

Summary of simulated lattice actions with $N_f = 2$ quark flavours.

^A) The calculation uses overlap fermions in the valence quark sector.