27970
Comment:

← Revision 14 as of 20170905 15:11:23 ⇥
28261

Deletions are marked like this.  Additions are marked like this. 
Line 8:  Line 8: 
part of the QCD action is the Wilson plaquette action <<FootNote(K. G. Wilson, Confinement of quarks, Phys. Rev. D10 (1974) 2445–2459.)>>:  part of the QCD action is the Wilson plaquette action <<BetterFootNote("K. G. Wilson, Confinement of quarks, Phys. Rev. D10 (1974) 2445–2459.")>>: 
Line 16:  Line 16: 
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`` <<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 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.  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`` <<BetterFootNote("K. Symanzik, Continuum limit and improved action in lattice theories. 1. Principles and φ4 theory, Nucl. Phys. B226 (1983) 187.")>> <<BetterFootNote("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 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. 
Line 22:  Line 22: 
where the coefficients $c_0, c_1$ satisfy the normalisation condition $c_0+8c_1=1$. The ''Symanzikimproved'' <<FootNote(M. Lüscher and P. Weisz, Onshell 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, [heplat/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.  where the coefficients $c_0, c_1$ satisfy the normalisation condition $c_0+8c_1=1$. The ''Symanzikimproved'' <<BetterFootNote("M. Lüscher and P. Weisz, Onshell improved lattice gauge theories, Commun. Math. Phys. 97 (1985) 59.")>>, ''Iwasaki'' <<BetterFootNote("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'' <<BetterFootNote("T. Takaishi, Heavy quark potential and effective actions on blocked configurations, Phys. Rev. D54 (1996) 1050–1053.")>><<BetterFootNote("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]")>> 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. 
Line 37:  Line 37: 
<<FootNote()>>  <<BetterFootNote()>> 
Line 61:  Line 61: 
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).  or tlSW in short), via a mean field approach <<BetterFootNote("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 <<BetterFootNote("M. Lüscher, 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). 
Line 66:  Line 66: 
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].)>>  mass term, after which the fermionic part of the QCD action in the continuum assumes the form <<BetterFootNote("R. Frezzotti, P. A. Grassi, S. Sint, and P. Weisz, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058, [heplat/0101001].")>> 
Line 77:  Line 77: 
lattice QCD is the fact that the leading lattice artefacts are of order $a^2$ without the need to add the SheikholeslamiWohlert term <<FootNote(R. Frezzotti and G. C. Rossi, Chirally improving Wilson fermions. I: O(a) improvement, JHEP 08 (2004) 007, [heplat/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 $B_\text{K}$.  lattice QCD is the fact that the leading lattice artefacts are of order $a^2$ without the need to add the SheikholeslamiWohlert term <<BetterFootNote("R. Frezzotti and G. C. Rossi, Chirally improving Wilson fermions. I: O(a) improvement, JHEP 08 (2004) 007, [heplat/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 $B_\text{K}$. 
Line 80:  Line 80: 
An alternative procedure to deal with the doubling problem is based on socalled ''staggered'' or KogutSusskind (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 tastechanging interactions, leading to ''improved staggered fermions'', with the socalled ''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, [heplat/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. Dürr, Theoretical issues with staggered fermion simulations, PoS LAT2005 (2006) 021, [heplat/0509026].)>> <<FootNote(S. R. Sharpe, Rooted staggered fermions: good, bad or ugly?, PoS LAT2006 (2006) 022, [heplat/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].)>>.  An alternative procedure to deal with the doubling problem is based on socalled ''staggered'' or KogutSusskind (KS) fermions <<BetterFootNote("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 tastechanging interactions, leading to ''improved staggered fermions'', with the socalled ''Asqtad'' action as one of its widely used versions <<BetterFootNote("K. Orginos, D. Toussaint, and R. L. Sugar, Variants of fattening and flavor symmetry restoration, Phys. Rev. D60 (1999) 054503, [heplat/9903032].", refName="orginos_1")>>. 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. <<BetterFootNote("S. Dürr, Theoretical issues with staggered fermion simulations, PoS LAT2005 (2006) 021, [heplat/0509026].)>> <<BetterFootNote("S. R. Sharpe, Rooted staggered fermions: good, bad or ugly?, PoS LAT2006 (2006) 022, [heplat/0610094].")>> <<BetterFootNote("A. S. Kronfeld, Lattice gauge theory with staggered fermions: how, where, and why (not), PoS LAT2007 (2007) 016, [arXiv:0711.0699].")>> <<BetterFootNote("M. Golterman, QCD with rooted staggered fermions, PoS CONFINEMENT8 (2008) 014, [arXiv:0812.3110].")>> <<BetterFootNote("M. Creutz, Why rooting fails, PoS LAT2007 (2007) 007, [arXiv:0708.1295].")>>. 
Line 85:  Line 85: 
according to <<FootNote(P. Hasenfratz, V. Laliena, and F. Niedermayer, The index theorem in QCD with a finite cutoff, Phys. Lett. B427 (1998) 125–131, [hep lat/9801021].)>> <<FootNote(M. L¨uscher, Exact chiral symmetry on the lattice and the GinspargWilson relation, Phys. Lett. B428 (1998) 342–345, [heplat/9802011].)>>  according to <<BetterFootNote("P. Hasenfratz, V. Laliena, and F. Niedermayer, The index theorem in QCD with a finite cutoff, Phys. Lett. B427 (1998) 125–131, [hep lat/9801021].")>> <<BetterFootNote("M. Lüuscher, Exact chiral symmetry on the lattice and the GinspargWilson relation, Phys. Lett. B428 (1998) 342–345, [heplat/9802011].")>> 
Line 89:  Line 89: 
which is now known as the GinspargWilson relation <<FootNote(P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D25 (1982) 2649.)>>. 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) <<FootNote(D. B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett. B288 (1992) 342–347, [heplat/9206013].)>>. The fivedimensional 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 <<FootNote(V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439 (1995) 54–78, [heplat/9405004].)>>. 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 doublerfree, (approximately) chirally symmetric quark action can thus be realised at the expense of simulating a fivedimensional theory. The socalled ''overlap'' or NeubergerDirac operator can be derived from the domain wall formulation <<FootNote(H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B417 (1998) 141–144, [heplat/9707022].)>>. It acts in four spacetime dimensions and is, in its simplest form, defined by 
which is now known as the GinspargWilson relation <<BetterFootNote("P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D25 (1982) 2649.")>>. 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) <<BetterFootNote("D. B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett. B288 (1992) 342–347, [heplat/9206013].")>>. The fivedimensional 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 <<BetterFootNote("V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439 (1995) 54–78, [heplat/9405004].")>>. 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 doublerfree, (approximately) chirally symmetric quark action can thus be realised at the expense of simulating a fivedimensional theory. The socalled ''overlap'' or NeubergerDirac operator can be derived from the domain wall formulation <<BetterFootNote("H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B417 (1998) 141–144, [heplat/9707022].")>>. It acts in four spacetime dimensions and is, in its simplest form, defined by 
Line 97:  Line 97: 
The third example of an operator which satisfies the GinspargWilson relation is the socalled fixedpoint action <<FootNote(P. Hasenfratz et. al., The construction of generalized Dirac operators on the lattice, Int. J. Mod. Phys. C12 (2001) 691–708, [heplat/0003013].)>> <<FootNote(P. Hasenfratz, S. Hauswirth, T. Jorg, F. Niedermayer, and K. Holland, Testing the fixedpoint QCD action and the construction of chiral currents, Nucl. Phys. B643 (2002) 280–320, [heplat/0205010].)>>. This construction proceeds via a renormalisation group approach. A related formalism are the socalled ''chirally improved'' fermions <<FootNote(C. Gattringer, A new approach to GinspargWilson fermions, Phys. Rev. D63 (2001) 114501, [heplat/0003005].)>>.  The third example of an operator which satisfies the GinspargWilson relation is the socalled fixedpoint action <<BetterFootNote("P. Hasenfratz et. al., The construction of generalized Dirac operators on the lattice, Int. J. Mod. Phys. C12 (2001) 691–708, [heplat/0003013].")>> <<BetterFootNote("P. Hasenfratz, S. Hauswirth, T. Jorg, F. Niedermayer, and K. Holland, Testing the fixedpoint QCD action and the construction of chiral currents, Nucl. Phys. B643 (2002) 280–320, [heplat/0205010].")>>. This construction proceeds via a renormalisation group approach. A related formalism are the socalled ''chirally improved'' fermions <<BetterFootNote("C. Gattringer, A new approach to GinspargWilson fermions, Phys. Rev. D63 (2001) 114501, [heplat/0003005].")>>. 
Line 100:  Line 100: 
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 <<FootNote(K. Orginos, D. Toussaint, and R. L. Sugar, Variants of fattening and flavor symmetry restoration, Phys. Rev. D60 (1999) 054503, [heplat/9903032].)>> is an example which makes use of socalled ''Asqtad'' smeared (or ''fat'') links. Another example is the use of nHYP smeared <<FootNote(A. Hasenfratz, R. Hoffmann, and S. Schaefer, Hypercubic smeared links for dynamical fermions, JHEP 05 (2007) 029, [hep lat/0702028].)>> or nstout smeared <<FootNote(S. Düurr, et. al., Scaling study of dynamical smearedlink clover fermions, Phys. Rev. D79 (2009) 014501, [arXiv:0802.2706].)>> gauge links in the treelevel clover improved discretisation of the quark action, denoted by ''nHYP tlSW'' and ''nstout tlSW'' in the following. 
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 <<BetterFootNote(refName="orginos_1")>> is an example which makes use of socalled ''Asqtad'' smeared (or ''fat'') links. Another example is the use of nHYP smeared <<BetterFootNote("A. Hasenfratz, R. Hoffmann, and S. Schaefer, Hypercubic smeared links for dynamical fermions, JHEP 05 (2007) 029, [hep lat/0702028].")>> or nstout smeared <<BetterFootNote("S. Dürr, et. al., Scaling study of dynamical smearedlink clover fermions, Phys. Rev. D79 (2009) 014501, [arXiv:0802.2706].")>> gauge links in the treelevel clover improved discretisation of the quark action, denoted by ''nHYP tlSW'' and ''nstout tlSW'' in the following. 
Line 120:  Line 120: 
<<FootNote()>>  <<BetterFootNote()>> 
Line 134:  Line 134: 
Symanzik’s framework can be combined with Chiral Perturbation Theory. The wellknown 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 adependence 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) <<FootNote(W.J. Lee and S. R. Sharpe, Partial Flavor Symmetry Restoration for Chiral Staggered Fermions, Phys. Rev. D60 (1999) 114503, [heplat/9905023].)>> at $O(a^2)$. In the following, this effective theory is denoted by SχPT. It was later generalized to an arbitrary number of flavours <<FootNote(C. Aubin and C. Bernard, Pion and kaon masses in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 034014, [heplat/0304014].)>> <<FootNote(C. Aubin and C. Bernard, Pseudoscalar decay constants in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 074011, [heplat/0306026].)>>, and to nexttoleading order <<FootNote(S. R. Sharpe and R. S. Van de Water, Staggered chiral perturbation theory at nexttoleading order, Phys. Rev. D71 (2005) 114505, [heplat/0409018].)>>. 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 <<FootNote(S. R. Sharpe and R. L. Singleton, Jr, Spontaneous flavor and parity breaking with Wilson fermions, Phys. Rev. D58 (1998) 074501, [heplat/9804028].)>> <<FootNote(G. Rupak and N. Shoresh, Chiral perturbation theory for the Wilson lattice action, Phys. Rev. D66 (2002) 054503, [heplat/0201019].)>> <<FootNote(S. Aoki, Chiral perturbation theory with Wilsontype fermions including a**2 effects: N(f ) = 2 degenerate case, Phys. Rev. D68 (2003) 054508, [heplat/0306027].)>> and is called WχPT, while the theory for Wilson twisted mass fermions <<FootNote(S. Aoki and O. Bär, Twistedmass QCD, O(a) improvement and Wilson chiral perturbation theory, Phys. Rev. D70 (2004) 116011, [heplat/0409006].)>> <<FootNote(S. R. Sharpe and J. M. S. Wu, Twisted mass chiral perturbation theory at nexttoleading order, Phys. Rev. D71 (2005) 074501, [heplat/0411021].)>> <<FootNote(O. Bär, Chiral logs in twisted mass lattice QCD with large isospin breaking, arXiv:1008.0784.)>> 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 <<FootNote(C. W. Bernard and M. F. L. Golterman, Partially quenched gauge theories and an application to staggered fermions, Phys. Rev. D49 (1994) 486–494, [heplat/9306005].)>> <<FootNote(S. R. Sharpe, Enhanced chiral logarithms in partially quenched QCD, Phys. Rev. D56 (1997) 7052–7058, [heplat/9707018]. Erratum: Phys. Rev. D62 (2000) 099901.)>> <<FootNote(M. F. L. Golterman and K.C. Leung, Applications of Partially Quenched Chiral Perturbation Theory, Phys. Rev. D57 (1998) 5703–5710, [heplat/9711033].)>> <<FootNote(S. R. Sharpe and N. Shoresh, Physical results from unphysical simulations, Phys. Rev. D62 (2000) 094503, [heplat/0006017].)>>. 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 <<FootNote(O. Bär, G. Rupak, and N. Shoresh, Simulations with different lattice Dirac operators for valence and sea quarks, Phys. Rev. D67 (2003) 114505, [heplat/0210050].)>><<FootNote(O. Bär, G. Rupak, and N. Shoresh, Chiral perturbation theory at O(a**2) for lattice QCD, Phys. Rev. D70 (2004) 034508, [heplat/0306021].)>><<FootNote(O. Bär, C. Bernard, G. Rupak, and N. Shoresh, Chiral perturbation theory for staggered sea quarks and GinspargWilson valence quarks, Phys. Rev. D72 (2005) 054502, [heplat/0503009].)>>. 
Symanzik’s framework can be combined with Chiral Perturbation Theory. The wellknown 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 adependence 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) <<BetterFootNote("W.J. Lee and S. R. Sharpe, Partial Flavor Symmetry Restoration for Chiral Staggered Fermions, Phys. Rev. D60 (1999) 114503, [heplat/9905023].")>> at $O(a^2)$. In the following, this effective theory is denoted by SχPT. It was later generalized to an arbitrary number of flavours <<BetterFootNote("C. Aubin and C. Bernard, Pion and kaon masses in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 034014, [heplat/0304014].")>> <<BetterFootNote("C. Aubin and C. Bernard, Pseudoscalar decay constants in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 074011, [heplat/0306026].")>>, and to nexttoleading order <<BetterFootNote("S. R. Sharpe and R. S. Van de Water, Staggered chiral perturbation theory at nexttoleading order, Phys. Rev. D71 (2005) 114505, [heplat/0409018].")>>. 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 <<BetterFootNote("S. R. Sharpe and R. L. Singleton, Jr, Spontaneous flavor and parity breaking with Wilson fermions, Phys. Rev. D58 (1998) 074501, [heplat/9804028].")>> <<BetterFootNote("G. Rupak and N. Shoresh, Chiral perturbation theory for the Wilson lattice action, Phys. Rev. D66 (2002) 054503, [heplat/0201019].")>> <<BetterFootNote("S. Aoki, Chiral perturbation theory with Wilsontype fermions including a**2 effects: N(f ) = 2 degenerate case, Phys. Rev. D68 (2003) 054508, [heplat/0306027].")>> and is called WχPT, while the theory for Wilson twisted mass fermions <<BetterFootNote("S. Aoki and O. Bär, Twistedmass QCD, O(a) improvement and Wilson chiral perturbation theory, Phys. Rev. D70 (2004) 116011, [heplat/0409006].")>> <<BetterFootNote("S. R. Sharpe and J. M. S. Wu, Twisted mass chiral perturbation theory at nexttoleading order, Phys. Rev. D71 (2005) 074501, [heplat/0411021].")>> <<BetterFootNote("O. Bär, Chiral logs in twisted mass lattice QCD with large isospin breaking, arXiv:1008.0784.")>> 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 <<BetterFootNote("C. W. Bernard and M. F. L. Golterman, Partially quenched gauge theories and an application to staggered fermions, Phys. Rev. D49 (1994) 486–494, [heplat/9306005].")>> <<BetterFootNote("S. R. Sharpe, Enhanced chiral logarithms in partially quenched QCD, Phys. Rev. D56 (1997) 7052–7058, [heplat/9707018]. Erratum: Phys. Rev. D62 (2000) 099901.")>> <<BetterFootNote("M. F. L. Golterman and K.C. Leung, Applications of Partially Quenched Chiral Perturbation Theory, Phys. Rev. D57 (1998) 5703–5710, [heplat/9711033].")>> <<BetterFootNote("S. R. Sharpe and N. Shoresh, Physical results from unphysical simulations, Phys. Rev. D62 (2000) 094503, [heplat/0006017].")>>. 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 <<BetterFootNote("O. Bär, G. Rupak, and N. Shoresh, Simulations with different lattice Dirac operators for valence and sea quarks, Phys. Rev. D67 (2003) 114505, [heplat/0210050].")>><<BetterFootNote("O. Bär, G. Rupak, and N. Shoresh, Chiral perturbation theory at O(a**2) for lattice QCD, Phys. Rev. D70 (2004) 034508, [heplat/0306021].")>><<BetterFootNote("O. Bär, C. Bernard, G. Rupak, and N. Shoresh, Chiral perturbation theory for staggered sea quarks and GinspargWilson valence quarks, Phys. Rev. D72 (2005) 054502, [heplat/0503009].")>>. 
Line 140:  Line 140: 
<<FootNote()>>  <<BetterFootNote()>> 
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. (1)
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)
M. Lüscher and P. Weisz, Onshell improved lattice gauge theories, Commun. Math. Phys. 97 (1985) 59. (1)
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)
T. Takaishi, Heavy quark potential and effective actions on blocked configurations, Phys. Rev. D54 (1996) 1050–1053. (1)
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] (1)
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 ^{[1]} (meanfield impr. or mfSW) or via a nonperturbative approach ^{[2]} (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 ^{[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 SheikholeslamiWohlert 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 socalled staggered or KogutSusskind (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 tastechanging interactions, leading to improved staggered fermions, with the socalled 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]}.
GinspargWilson fermions
Fermionic lattice actions, which do not suffer from the doubling problem whilst preserving chiral symmetry go under the name of GinspargWilson fermions. In the continuum the massless Dirac operator anticommutes with $\gamma_5$. At nonzero 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 GinspargWilson 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 fivedimensional 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 doublerfree, (approximately) chirally symmetric quark action can thus be realised at the expense of simulating a fivedimensional theory.
The socalled overlap or NeubergerDirac operator can be derived from the domain wall formulation ^{[17]}. It acts in four spacetime dimensions and is, in its simplest form, defined by
$D_{\rm N} = \frac{1}{\bar a} \left( 1\frac{A}{\sqrt{A^\dagger{A}}}\right), \quad A=1+saD_{\rm w},\quad \bar a=\frac{a}{1+s},$
where $D_{\rm w}$ is the massless WilsonDirac 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 GinspargWilson 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 GinspargWilson relation is the socalled fixedpoint action ^{[18]} ^{[19]}. This construction proceeds via a renormalisation group approach. A related formalism are the socalled 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 socalled Asqtad smeared (or fat) links. Another example is the use of nHYP smeared ^{[21]} or nstout smeared ^{[22]} gauge links in the treelevel clover improved discretisation of the quark action, denoted by nHYP tlSW and nstout 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 nonspectral observables (like operator matrix elements) the corresponding nonspectral 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 
SheikholeslamiWohlert 
$O(g^2 a)$ 
broken 
treelevel improved, $c_\text{sw}=1$ 
nHYP tlSW 
SheikholeslamiWohlert 
$O(g^2 a)$ 
broken 
treelevel improved, $c_\text{sw}=1$, nHYP smeared gauge links 
nstout tlSW 
SheikholeslamiWohlert 
$O(g^2 a)$ 
broken 
treelevel improved, $c_\text{sw}=1$, nstout smeared gauge links 
mfSW 
SheikholeslamiWohlert 
$O(g^2 a)$ 
broken 
meanfield improved 
npSW 
SheikholeslamiWohlert 
$O(a^2)$ 
broken 
nonperturbatively 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
G. P. Lepage and P. B. Mackenzie, On the viability of lattice perturbation theory, Phys. Rev. D48 (1993) 2250–2264, [heplat/9209022]. (1)
M. Lüscher, S. Sint, R. Sommer, P. Weisz, and U. Wolff, Nonperturbative O(a) improvement of lattice QCD, Nucl. Phys. B491 (1997) 323–343, [heplat/9609035]. (1)
R. Frezzotti, P. A. Grassi, S. Sint, and P. Weisz, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058, [heplat/0101001]. (1)
R. Frezzotti and G. C. Rossi, Chirally improving Wilson fermions. I: O(a) improvement, JHEP 08 (2004) 007, [heplat/0306014]. (1)
L. Susskind, Lattice fermions, Phys. Rev. D16 (1977) 3031–3039. (1)
K. Orginos, D. Toussaint, and R. L. Sugar, Variants of fattening and flavor symmetry restoration, Phys. Rev. D60 (1999) 054503, [heplat/9903032]. (1 2)
"S. Dürr, Theoretical issues with staggered fermion simulations, PoS LAT2005 (2006) 021, [heplat/0509026]. (1)
S. R. Sharpe, Rooted staggered fermions: good, bad or ugly?, PoS LAT2006 (2006) 022, [heplat/0610094]. (1)
A. S. Kronfeld, Lattice gauge theory with staggered fermions: how, where, and why (not), PoS LAT2007 (2007) 016, [arXiv:0711.0699]. (1)
M. Golterman, QCD with rooted staggered fermions, PoS CONFINEMENT8 (2008) 014, [arXiv:0812.3110]. (1)
M. Creutz, Why rooting fails, PoS LAT2007 (2007) 007, [arXiv:0708.1295]. (1)
P. Hasenfratz, V. Laliena, and F. Niedermayer, The index theorem in QCD with a finite cutoff, Phys. Lett. B427 (1998) 125–131, [hep lat/9801021]. (1)
M. Lüuscher, Exact chiral symmetry on the lattice and the GinspargWilson relation, Phys. Lett. B428 (1998) 342–345, [heplat/9802011]. (1)
P. H. Ginsparg and K. G. Wilson, A remnant of chiral symmetry on the lattice, Phys. Rev. D25 (1982) 2649. (1)
D. B. Kaplan, A Method for simulating chiral fermions on the lattice, Phys. Lett. B288 (1992) 342–347, [heplat/9206013]. (1)
V. Furman and Y. Shamir, Axial symmetries in lattice QCD with Kaplan fermions, Nucl. Phys. B439 (1995) 54–78, [heplat/9405004]. (1)
H. Neuberger, Exactly massless quarks on the lattice, Phys. Lett. B417 (1998) 141–144, [heplat/9707022]. (1)
P. Hasenfratz et. al., The construction of generalized Dirac operators on the lattice, Int. J. Mod. Phys. C12 (2001) 691–708, [heplat/0003013]. (1)
P. Hasenfratz, S. Hauswirth, T. Jorg, F. Niedermayer, and K. Holland, Testing the fixedpoint QCD action and the construction of chiral currents, Nucl. Phys. B643 (2002) 280–320, [heplat/0205010]. (1)
C. Gattringer, A new approach to GinspargWilson fermions, Phys. Rev. D63 (2001) 114501, [heplat/0003005]. (1)
A. Hasenfratz, R. Hoffmann, and S. Schaefer, Hypercubic smeared links for dynamical fermions, JHEP 05 (2007) 029, [hep lat/0702028]. (1)
S. Dürr, et. al., Scaling study of dynamical smearedlink clover fermions, Phys. Rev. D79 (2009) 014501, [arXiv:0802.2706]. (1)
Matching and running
Abbrev. 
Description 
RI 
Regularisationindependent 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 wellknown 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 adependence 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 nexttoleading 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
W.J. Lee and S. R. Sharpe, Partial Flavor Symmetry Restoration for Chiral Staggered Fermions, Phys. Rev. D60 (1999) 114503, [heplat/9905023]. (1)
C. Aubin and C. Bernard, Pion and kaon masses in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 034014, [heplat/0304014]. (1)
C. Aubin and C. Bernard, Pseudoscalar decay constants in staggered chiral perturbation theory, Phys.Rev. D68 (2003) 074011, [heplat/0306026]. (1)
S. R. Sharpe and R. S. Van de Water, Staggered chiral perturbation theory at nexttoleading order, Phys. Rev. D71 (2005) 114505, [heplat/0409018]. (1)
S. R. Sharpe and R. L. Singleton, Jr, Spontaneous flavor and parity breaking with Wilson fermions, Phys. Rev. D58 (1998) 074501, [heplat/9804028]. (1)
G. Rupak and N. Shoresh, Chiral perturbation theory for the Wilson lattice action, Phys. Rev. D66 (2002) 054503, [heplat/0201019]. (1)
S. Aoki, Chiral perturbation theory with Wilsontype fermions including a**2 effects: N(f ) = 2 degenerate case, Phys. Rev. D68 (2003) 054508, [heplat/0306027]. (1)
S. Aoki and O. Bär, Twistedmass QCD, O(a) improvement and Wilson chiral perturbation theory, Phys. Rev. D70 (2004) 116011, [heplat/0409006]. (1)
S. R. Sharpe and J. M. S. Wu, Twisted mass chiral perturbation theory at nexttoleading order, Phys. Rev. D71 (2005) 074501, [heplat/0411021]. (1)
O. Bär, Chiral logs in twisted mass lattice QCD with large isospin breaking, arXiv:1008.0784. (1)
C. W. Bernard and M. F. L. Golterman, Partially quenched gauge theories and an application to staggered fermions, Phys. Rev. D49 (1994) 486–494, [heplat/9306005]. (1)
S. R. Sharpe, Enhanced chiral logarithms in partially quenched QCD, Phys. Rev. D56 (1997) 7052–7058, [heplat/9707018]. Erratum: Phys. Rev. D62 (2000) 099901. (1)
M. F. L. Golterman and K.C. Leung, Applications of Partially Quenched Chiral Perturbation Theory, Phys. Rev. D57 (1998) 5703–5710, [heplat/9711033]. (1)
S. R. Sharpe and N. Shoresh, Physical results from unphysical simulations, Phys. Rev. D62 (2000) 094503, [heplat/0006017]. (1)
O. Bär, G. Rupak, and N. Shoresh, Simulations with different lattice Dirac operators for valence and sea quarks, Phys. Rev. D67 (2003) 114505, [heplat/0210050]. (1)
O. Bär, G. Rupak, and N. Shoresh, Chiral perturbation theory at O(a**2) for lattice QCD, Phys. Rev. D70 (2004) 034508, [heplat/0306021]. (1)
O. Bär, C. Bernard, G. Rupak, and N. Shoresh, Chiral perturbation theory for staggered sea quarks and GinspargWilson valence quarks, Phys. Rev. D72 (2005) 054502, [heplat/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 
2+1 
tadSym 
Asqtad ^{A)} 

Blum 10 
2+1 
Iwasaki 
DW 

BMW 10AC 
2+1 
tlSym 
2HEX tlSW 

BMW 10 
2+1 
tlSym 
6stout tlSW 

CPPACS/JLQCD 07 
2+1 
Iwasaki 
npSW 

ETM 10, 10E 
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 
2+1 
Iwasaki 
overlap 

JLQCD/TWQCD 10 
2+1, 3 
Iwasaki 
overlap 

LHP 04 
2+1 
tadSym 
Asqtad ^{A)} 

MILC 04, 07, 09, 09A 
2+1 
tadSym 
Asqtad 

NPLQCD 06 
2+1 
tadSym 
Asqtad ^{A)} 

PACSCS 08, 08B, 09, 10 
2+1 
Iwasaki 
npSW 

RBC/UKQCD 07, 08, 08A, 10, 10AB 
2+1 
Iwasaki 
DW 

SWME 10, 11 
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)} 

CERNTOV 06 
2 
Wilson 
Wilson/npSW 

CERN 08 
2 
Wilson 
Wilson/npSW 

CPPACS 01 
2 
Iwasaki 
mfSW 

ETM 07, 07A, 08, 09, 09AD, 10B, D 
2 
tlSym 
tmWil 

HHS 08 
2 
tadSym 
nHYP tlSW 

JLQCD 08 
2 
Iwasaki 
overlap 

JLQCD 02, 05 
2 
Wilson 
npSW 

JLQCD/TWQCD 07, 08A 
2 
Iwasaki 
overlap 

QCDSF 07 
2 
Wilson 
npSW 

QCDSF/UKQCD 04, 06, 06A, 07 
2 
Wilson 
npSW 

RBC 07 
2 
DBW2 
DW 

RBC 04, 06 
2 
DBW2 
DW 

SPQcdR 05 
2 
Wilson 
Wilson 

UKQCD 04, 07 
2 
Wilson 
npSW 
^{A)} The calculation uses overlap fermions in the valence quark sector.