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## 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

1. K. G. Wilson, Confinement of quarks, Phys. Rev. D10 (1974) 2445–2459. (1)

2. K. Symanzik, Continuum limit and improved action in lattice theories. 1. Principles and φ4 theory, Nucl. Phys. B226 (1983) 187. (1)

3. K. Symanzik, Continuum limit and improved action in lattice theories. 2. O(N) nonlinear sigma model in perturbation theory, Nucl. Phys. B226 (1983) 205. (1)

4. M. Lüscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 97 (1985) 59. (1)

5. 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. (1)

6. T. Takaishi, Heavy quark potential and effective actions on blocked configurations, Phys. Rev. D54 (1996) 1050–1053. (1)

7. 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] (1)

### 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.

 Abbrev. Discretisation Leading lattice artefacts Chiral symmetry Remarks Wilson Wilson $O(a)$ broken -- tmWil Twisted Mass Wilson $O(a^2)$ at maximal twist broken flavour symmetry breaking: $(M_\text{PS}^{0})^2-(M_\text{PS}^\pm)^2\sim O(a^2)$ tlSW Sheikholeslami-Wohlert $O(g^2 a)$ broken tree-level improved, $c_\text{sw}=1$ n-HYP tlSW Sheikholeslami-Wohlert $O(g^2 a)$ broken tree-level improved, $c_\text{sw}=1$, n-HYP smeared gauge links n-stout tlSW Sheikholeslami-Wohlert $O(g^2 a)$ broken tree-level improved, $c_\text{sw}=1$, n-stout smeared gauge links mfSW Sheikholeslami-Wohlert $O(g^2 a)$ broken mean-field improved npSW Sheikholeslami-Wohlert $O(a^2)$ broken non-perturbatively improved KS Staggered $O(a^2)$ U(1)$\times$U(1) subgroup unbroken rooting for $N_f<4$ Asqtad Staggered $O(a^2)$ U(1)$\times$U(1) subgroup unbroken rooting for $N_f<4$, Asqtad smeared gauge links DW Domain Wall asymptotically $O(a^2)$ remnant breaking exponentially suppressed exact chiral symmetry and $O(a)$ improvement only in the limit $L_s\rightarrow \infty$ overlap Neuberger $O(a^2)$ exact --

#### References

1. G. P. Lepage and P. B. Mackenzie, On the viability of lattice perturbation theory, Phys. Rev. D48 (1993) 2250–2264, [hep-lat/9209022]. (1)

2. M. Lüscher, 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]. (1)

3. 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]. (1)

4. R. Frezzotti and G. C. Rossi, Chirally improving Wilson fermions. I: O(a) improvement, JHEP 08 (2004) 007, [hep-lat/0306014]. (1)

5. L. Susskind, Lattice fermions, Phys. Rev. D16 (1977) 3031–3039. (1)

6. K. Orginos, D. Toussaint, and R. L. Sugar, Variants of fattening and flavor symmetry restoration, Phys. Rev. D60 (1999) 054503, [hep-lat/9903032]. (1 2)

7. "S. Dürr, Theoretical issues with staggered fermion simulations, PoS LAT2005 (2006) 021, [hep-lat/0509026]. (1)

8. S. R. Sharpe, Rooted staggered fermions: good, bad or ugly?, PoS LAT2006 (2006) 022, [hep-lat/0610094]. (1)

9. A. S. Kronfeld, Lattice gauge theory with staggered fermions: how, where, and why (not), PoS LAT2007 (2007) 016, [arXiv:0711.0699]. (1)

10. M. Golterman, QCD with rooted staggered fermions, PoS CONFINEMENT8 (2008) 014, [arXiv:0812.3110]. (1)

11. M. Creutz, Why rooting fails, PoS LAT2007 (2007) 007, [arXiv:0708.1295]. (1)

12. P. Hasenfratz, V. Laliena, and F. Niedermayer, The index theorem in QCD with a finite cut-off, Phys. Lett. B427 (1998) 125–131, [hep lat/9801021]. (1)

13. M. Lüuscher, Exact chiral symmetry on the lattice and the Ginsparg-Wilson relation, Phys. Lett. B428 (1998) 342–345, [hep-lat/9802011]. (1)

14. P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D25 (1982) 2649. (1)

15. D. B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett. B288 (1992) 342–347, [hep-lat/9206013]. (1)

16. V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439 (1995) 54–78, [hep-lat/9405004]. (1)

17. H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B417 (1998) 141–144, [hep-lat/9707022]. (1)

18. P. Hasenfratz et. al., The construction of generalized Dirac operators on the lattice, Int. J. Mod. Phys. C12 (2001) 691–708, [hep-lat/0003013]. (1)

19. P. Hasenfratz, S. Hauswirth, T. Jorg, F. Niedermayer, and K. Holland, Testing the fixed-point QCD action and the construction of chiral currents, Nucl. Phys. B643 (2002) 280–320, [hep-lat/0205010]. (1)

20. C. Gattringer, A new approach to Ginsparg-Wilson fermions, Phys. Rev. D63 (2001) 114501, [hep-lat/0003005]. (1)

21. A. Hasenfratz, R. Hoffmann, and S. Schaefer, Hypercubic smeared links for dynamical fermions, JHEP 05 (2007) 029, [hep lat/0702028]. (1)

22. S. Dürr, et. al., Scaling study of dynamical smeared-link clover fermions, Phys. Rev. D79 (2009) 014501, [arXiv:0802.2706]. (1)

### Matching and running

 Abbrev. Description RI Regularisation-independent momentum subtraction scheme SF Schroedinger functional scheme PT1L matching/running computed in perturbation theory at one loop PT2L matching/running computed in perturbation theory at two loop

### 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

1. W.-J. Lee and S. R. Sharpe, Partial Flavor Symmetry Restoration for Chiral Staggered Fermions, Phys. Rev. D60 (1999) 114503, [hep-lat/9905023]. (1)

2. C. Aubin and C. Bernard, Pion and kaon masses in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 034014, [hep-lat/0304014]. (1)

3. C. Aubin and C. Bernard, Pseudoscalar decay constants in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 074011, [hep-lat/0306026]. (1)

4. S. R. Sharpe and R. S. Van de Water, Staggered chiral perturbation theory at next-to-leading order, Phys. Rev. D71 (2005) 114505, [hep-lat/0409018]. (1)

5. S. R. Sharpe and R. L. Singleton, Jr, Spontaneous flavor and parity breaking with Wilson fermions, Phys. Rev. D58 (1998) 074501, [hep-lat/9804028]. (1)

6. G. Rupak and N. Shoresh, Chiral perturbation theory for the Wilson lattice action, Phys. Rev. D66 (2002) 054503, [hep-lat/0201019]. (1)

7. S. Aoki, Chiral perturbation theory with Wilson-type fermions including a**2 effects: N(f ) = 2 degenerate case, Phys. Rev. D68 (2003) 054508, [hep-lat/0306027]. (1)

8. S. Aoki and O. Bär, Twisted-mass QCD, O(a) improvement and Wilson chiral perturbation theory, Phys. Rev. D70 (2004) 116011, [hep-lat/0409006]. (1)

9. S. R. Sharpe and J. M. S. Wu, Twisted mass chiral perturbation theory at next-to-leading order, Phys. Rev. D71 (2005) 074501, [hep-lat/0411021]. (1)

10. O. Bär, Chiral logs in twisted mass lattice QCD with large isospin breaking, arXiv:1008.0784. (1)

11. C. W. Bernard and M. F. L. Golterman, Partially quenched gauge theories and an application to staggered fermions, Phys. Rev. D49 (1994) 486–494, [hep-lat/9306005]. (1)

12. S. R. Sharpe, Enhanced chiral logarithms in partially quenched QCD, Phys. Rev. D56 (1997) 7052–7058, [hep-lat/9707018]. Erratum: Phys. Rev. D62 (2000) 099901. (1)

13. M. F. L. Golterman and K.-C. Leung, Applications of Partially Quenched Chiral Perturbation Theory, Phys. Rev. D57 (1998) 5703–5710, [hep-lat/9711033]. (1)

14. S. R. Sharpe and N. Shoresh, Physical results from unphysical simulations, Phys. Rev. D62 (2000) 094503, [hep-lat/0006017]. (1)

15. O. Bär, G. Rupak, and N. Shoresh, Simulations with different lattice Dirac operators for valence and sea quarks, Phys. Rev. D67 (2003) 114505, [hep-lat/0210050]. (1)

16. O. Bär, G. Rupak, and N. Shoresh, Chiral perturbation theory at O(a**2) for lattice QCD, Phys. Rev. D70 (2004) 034508, [hep-lat/0306021]. (1)

17. O. Bär, C. Bernard, G. Rupak, and N. Shoresh, Chiral perturbation theory for staggered sea quarks and Ginsparg-Wilson valence quarks, Phys. Rev. D72 (2005) 054502, [hep-lat/0503009]. (1)

### Summary of simulated lattice actions

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

 Ref. $N_f$ gauge action quark action Aubin 08, 09 1, 2 2+1 tadSym Asqtad A) Blum 10 2+1 Iwasaki DW BMW 10A-C 4, 5, 6 2+1 tlSym 2-HEX tlSW BMW 10 2+1 tlSym 6-stout tlSW CP-PACS/JLQCD 07 2+1 Iwasaki npSW ETM 10, 10E 9, 10 2+1+1 Iwasaki tmWil HPQCD/UKQCD 06 2+1 tadSym Asqtad HPQCD/UKQCD 07 2+1 tadSym Asqtad B) HPQCD/MILC/UKQCD 04 2+1 tadSym Asqtad JLQCD 09 2+1 Iwasaki overlap JLQCD/TWQCD 08B, 09A 15, 16 2+1 Iwasaki overlap JLQCD/TWQCD 10 2+1, 3 Iwasaki overlap LHP 04 2+1 tadSym Asqtad A) MILC 04, 07, 09, 09A 19, 20 , 21 ,22 2+1 tadSym Asqtad NPLQCD 06 2+1 tadSym Asqtad A) PACS-CS 08, 08B, 09, 10 24, 25, 26, 27 2+1 Iwasaki npSW RBC/UKQCD 07, 08, 08A, 10, 10A-B 28, 29, 30, 31, 32, 33 2+1 Iwasaki DW SWME 10, 11 34,35 2+1 tadSym Asqtad C) TWQCD 08 2+1 Iwasaki DW

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.

 Ref. $N_f$ gauge action quark action ALPHA 05 2 Wilson npSW Bernardoni 10 2 Wilson npSW A) CERN-TOV 06 2 Wilson Wilson/npSW CERN 08 2 Wilson Wilson/npSW CP-PACS 01 2 Iwasaki mfSW ETM 07, 07A, 08, 09, 09A-D, 10B, D 42, 43, 44, 45, 46, 47, 48, 49, 50, 51 2 tlSym tmWil HHS 08 2 tadSym n-HYP tlSW JLQCD 08 2 Iwasaki overlap JLQCD 02, 05 54, 55 2 Wilson npSW JLQCD/TWQCD 07, 08A 56, 57 2 Iwasaki overlap QCDSF 07 2 Wilson npSW QCDSF/UKQCD 04, 06, 06A, 07 59, 60, 61, 62 2 Wilson npSW RBC 07 2 DBW2 DW RBC 04, 06 64, 65 2 DBW2 DW SPQcdR 05 2 Wilson Wilson UKQCD 04, 07 67, 68 2 Wilson npSW

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

Flag: Glossary (last edited 2017-09-05 15:11:23 by Admin)