Domai n wal l  f ermi ons wi t h i mproved gaugeact i ons
著者Aoki  Y. ,  Bl um T. ,  Chri st  N. ,  Cri st i an C. ,Dawson C. ,  I zubuchi  Taku,  Li u G.
j ournal  orpubl i cat i on t i t l e
Physi cal  Revi ew D
vol ume69
number7
page range74504
year2004- 01- 01
URLht t p: //hdl . handl e. net /2297/1679
Domain wall fermions with improved gauge actions
Y. Aoki,1 T. Blum,1 N. Christ,2 C. Cristian,2 C. Dawson,1 T. Izubuchi,3,* G. Liu,2 R. Mawhinney,2 S. Ohta,4 K. Orginos,1
A. Soni,3 and L. Wu2
1RIKEN–BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA2Physics Department, Columbia University, New York, New York 10027, USA3Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA4Institute for Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki, 305-0801, Japan�Received 25 November 2002; published 14 April 2004�
We study the chiral properties of quenched domain wall fermions with several gauge actions. We demon-strate that the residual chiral symmetry breaking, which is present for a ﬁnite number of lattice sites in the ﬁfthdimension (Ls), can be substantially suppressed using improved gauge actions. In particular the Symanzikaction, the Iwasaki action, and a renormalization group improved gauge action, called the doubly blockedWilson �DBW2� action, are studied and compared to the Wilson action. All improved gauge actions studiedshow a reduction in the additive residual quark mass mres . Remarkably, in the DBW2 case mres is roughly twoorders of magnitude smaller than the Wilson gauge action at a�1�2 GeV and Ls�16. Signiﬁcant reduction inmres is also realized at stronger gauge coupling corresponding to a�1�1.3 GeV. As our numerical investiga-tion indicates, this reduction is achieved by reducing the number of topological lattice dislocations present inthe gauge ﬁeld conﬁgurations. We also present detailed results for the quenched light hadron spectrum and thepion decay constant using the DBW2 gauge action.
DOI: 10.1103/PhysRevD.69.074504PACS number�s�: 11.15.Ha, 11.30.Rd, 12.38.Gc, 12.38.Aw
I. INTRODUCTION
Domain wall fermions �1–4� are expected to provide animplementation of lattice fermions with exact chiral symme-try, even at a ﬁnite lattice spacing. To achieve this exactsymmetry, an inﬁnite ﬁfth dimension must be introduced andsimulations have been done to explore the limit of a largeﬁfth dimension for both full and quenched QCD �5–10�. Theﬁnite size of the ﬁfth dimension, Ls , used in numerical simu-lations, produces a small amount of chiral symmetry break-ing, which should go to zero in the limit Ls→�. In practicalimplementations the aim is to achieve the smallest chiralsymmetry breaking possible at a given Ls , thus minimizingthe cost of the simulation. Further information about domainwall fermions and their applications is given in recent re-views �11,12�.There have now been several suggestions on how to mini-mize the computational cost of domain wall fermions. Anobvious way to achieve this is to make the ﬁve-dimensionaleigenvectors of the domain wall fermion operator, which forsmalleigenvaluesshouldbelocalizedonthefour-dimensional boundaries of the ﬁfth dimension, decay fasterin the ﬁfth dimension. This reduces the mixing between theopposite chirality modes, which are bound to opposite endsof the ﬁfth dimension. Shamir �13� has calculated the ﬁfth-dimensional decay of the eigenfunctions with zero eigenval-ues using perturbation theory, suggesting a modiﬁcation ofthe four-dimensional component of the domain wall fermionoperator to increase the decay. This interesting perturbativeresult may explain some of the features seen in nonperturba-tive simulations. �Of course, modiﬁcations to the domain
wall fermion operator may increase the computational costby more than the reduction in Ls reduces it.� Another methodof improving domain wall fermions is proposed in Refs.�14,15�. The basic idea behind these proposals is to projectout the zero modes of the four-dimensional Hamiltonian de-scribing the propagation in the ﬁfth dimension. As a result,the localization on the boundaries of the fermionic lightmodes is enhanced.In this paper we systematically examine a different op-tion: the modiﬁcation of the gauge action to suppress theﬁnite Ls explicit chiral symmetry breaking �16�. Note that inprinciple this is a different criteria from improving the gaugeaction to achieve better scaling, in lattice spacing, of physicalobservables. We will investigate the scaling of observables aswell, to check that while reducing the explicit chiral symme-try breaking we do not distort the approach to the continuumlimit. It is worth noting that methods which improve thedomain wall fermion operator, such as those suggested byShamir and the one investigated here are likely independentof each other, so a combination of both techniques may leadto even greater efﬁciency in domain wall fermion simula-tions. However, as we will see, our approach obviates theneed of separately treating the near unit eigenvectors of thetransfer matrix, as gauge conﬁgurations for which these oc-cur are suppressed. This has also been studied in Ref. �17�.The observation that the gauge action can affect signiﬁ-cantly the chiral symmetry of domain wall fermions is notnew. Both the RBC �18� and CP–PACS �6,19� collaborationshave observed that the use of the Iwasaki action �20� sub-stantially improves chiral symmetry in quenched simula-tions. Also in Ref. �21� it was observed that the one-loopSymanzik �22� improved gauge action improves chiral sym-metry to a lesser degree. Here we extend these results andexplore the reason behind the observed improvement.This paper is organized as follows. In Sec. II we give a
*On leave from Institute of Theoretical Physics, Kanazawa Uni-versity, Ishikawa, Japan.
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brief description of the gauge actions under study. In Sec. IIIwe introduce the observables used for studying chiral sym-metry breaking and also present the standard Wilson actionresults to provide a reference point. Section IV contains re-sults for the different actions we studied. We ﬁnd that thedoubly blocked Wilson �DBW2� action �23,24� gives re-sidual chiral symmetry breaking two orders of magnitudesmaller than the Wilson gauge action at comparable latticespacings and values of Ls . In Sec. V we discuss the domi-nant mechanism of explicit chiral symmetry breaking in do-main wall fermions, which we ﬁnd is driven by lattice arti-facts, or dislocations, at the lattice spacings considered.These dislocations occur as the topological charge of thegauge ﬁeld conﬁguration changes during Monte Carlo evo-lution. Given this large improvement in residual chiral sym-metry breaking and the fact that the DBW2 action has notbeen used before with domain wall fermions, in Sec. VI wepresent results for some hadronic observables in order toconﬁrm consistency with quenched simulations using othergauge actions, to check scaling with lattice spacing and tolay a foundation for future work �25�.
II. PURE GAUGE LATTICE ACTIONS
As mentioned, we study the chiral properties of quencheddomain wall fermions with Symanzik, Iwasaki, and DBW2gauge actions. These actions are built from closed loops ofup to six links and provide a sample of typical lattice actionsused to improve scaling of observables. As a baseline forcomparisons we start with the Wilson action �26� which isdeﬁned by
SG�U����3 �x;��� P�U�x,�� ,�1�
where P�U�x,�� is the real part of the trace of the path or-dered product of links around the 1�1 plaquette in the �,�plane at point x and ��1/g02 with g0 the bare gauge cou-pling. This is the original non-Abelian gauge action intro-duced by Wilson, which has O(a2) errors (a is the latticespacing�.To begin, we study the Symanzik one loop improved ac-tion �22� where both O(a2) and O(g2a2) errors are re-moved. This action is deﬁned as
SG�U����3� c0 �x;��� P�U�x,���c1 �x;��� R�U�x,��
�c2 �x;����� C�U�x,���� ,�2�
where R�U�x,�� and C�U�x,��� denote the real part of thetrace of the ordered product of SU�3� link matrices along 1�2 rectangles in the �,� plane and the �,�,�,��,��,�� paths, respectively. The coefﬁcients c0 , c1, and c2 arecomputed in tadpole improved one loop perturbation theory�22�. For this action and the remaining ones, ��1/g02 as forthe Wilson action, but the precise numerical factors differ.
In addition to the above actions we also studied theIwasaki �20� action and the DBW2 action �23,24�. Theseactions are both renormalization group �RG� improved ac-tions in a truncated, two-parameter space. They can be writ-ten down as
SG�U����3� �1�8c1� �x;��� P�U�x,��
�c1 �x;��� R�U�x,����3�
with c1��0.331 for the Iwasaki action and c1��1.4069for the DBW2 action. In the case of the Iwasaki action thecoefﬁcient c1 is computed in weak coupling perturbationtheory. For the DBW2 action c1 is computed �23� nonpertur-batively using Swendsen’s blocking and the Schwinger-Dyson method. QCD–TARO has studied �24� the RG ﬂow inthe two parameter space of the plaquette and the rectanglecouplings and concluded that DBW2 is a good approxima-tion to the RG ﬂow in this plane at least for a range of coarselattice spacings.Although the Iwasaki and DBW2 actions are motivatedby the desire to remain on the RG trajectory for quenchedQCD, the truncation to the explicit form used is an approxi-mation. The accuracy with which these truncated actions pre-serve the RG trajectory must be investigated numerically.Simulations with the Iwasaki action �27� and the DBW2 ac-tion �24� show improved scaling of the heavy quark potentialand the critical temperature for the ﬁnite temperature phasetransition, compared to the Wilson gauge action. These ac-tions serve as useful starting points for studying the effects ofthe gauge action on residual chiral symmetry breaking indomain wall fermions.
III. EXPLICIT CHIRAL SYMMETRY BREAKING WITHDOMAIN WALL FERMIONS
The central idea behind domain wall fermions is that four-dimensional fermionic states of opposite chirality are local-ized dynamically on opposite boundaries of an extra ﬁfthdimension. The domain wall fermions are coupled to four-dimensional gauge ﬁelds replicated in the ﬁfth direction, sothe light states can be used to simulate a vector gauge theorylike QCD. The ﬁve-dimensional fermion action is a generali-zation of the Wilson fermion action �26� with open boundaryconditions in the ﬁfth dimension �3�. In the free ﬁeld limit,localization of a single fermionic ﬂavor on the four-dimensional boundaries occurs if the ﬁve-dimensional fer-mion mass M 5 is in the interval �0,2�. This interval is shiftedwhen interactions are switched on. For an inﬁnite ﬁfth di-mension (Ls→�), chiral symmetry of the light states ismanifest since they have no overlap. Four-dimensional lightquark q,q¯ ﬁelds are constructed from the ﬁve-dimensionalfermions �,�¯ by
q�x��PL��x,0��PR��x,Ls�1�,�4�
q¯�x���¯ �x,Ls�1�PL��¯ �x,0�PR ,�5�
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where PR/L� 1
2 (1��5) are the right-handed and left-handedprojection operators. Hence a four-dimensional mass termm fq¯q can be introduced if the ﬁfth dimension boundaries arecoupled directly with a parameter m f �3�. For ﬁnite Ls ex-plicit chiral symmetry breaking is induced by the mixing ofthe light states which now extend across the ﬁfth dimension.Our conventions throughout this paper are the same as thosein Ref. �5�.In order to quantify the explicit chiral symmetry breakinginduced at ﬁnite Ls , we deﬁne the residual mass (mres)through the Ward-Takahashi identity �4�:
���A �a �x�O�y���2m f�J5a�x�O�y���2�J5qa �x�O�y��
�i��aO�y��,�6�
A �a �x�� �s�0
Ls�1sign� s�Ls�1
2 � j�a �x,s��7�
is a four-dimensional partially conserved axial current whichis constructed from the ﬁve-dimensional conserved vectorcurrent,
j�a �x,s��12 ��¯ �x��ˆ ,s��1����Ux,�† ta��x,s�
��¯ �x,s��1����Ux,�ta��x��ˆ ,s��.�8�
The ﬂavor matrices are normalized to obey Tr(tatb)��ab,��f(x)�f(x)�f(x��ˆ ) is a simple ﬁnite difference opera-tor, and the pseudoscalar density J5a(x) is
J5a�x����¯ �x,Ls�1�PLta��x,0�
��¯ �x,0�PRta��x,Ls�1�.�9�
Note that J5a(x) is a four-dimensional pseudoscalar densityconstructed from ﬁelds on the boundaries of the ﬁfth dimen-sion. The identity �6� differs from the continuum expressionby the presence of the J5qa (x) term. J5qa (x) is analogous toJ5a(x), but is built from ﬁelds in the bulk at Ls/2 and Ls/2�1.
J5qa �x����¯ �x,Ls/2�1�PLta��x,Ls/2�
��¯ �x,Ls/2�PRta��x,Ls/2�1�.�10�
We refer to this term as the ‘‘midpoint’’ contribution tothe divergence of the axial current. The effect of the explicitchiral symmetry breaking can be described by the so-calledresidual mass term mres �5�. From the midpoint term we de-ﬁne the ratio
R�t���x,y �J5qa �y,t�J5a�x,0��
�x,y �J5a�y,t�J5a�x,0��,�11�
which, for t greater than some tmin should be independent oft and equal to the residual mass, giving
mres��x,y �J5qa �y,t�J5a�x,0��
�x,y �J5a�y,t�J5a�x,0���
t�tmin
As we will see, in our numerical simulations R(t) is essen-tially t independent for t�5 and the t dependence for t�5will be discussed in Sec. IV. To calculate mres, we averageover a suitable plateau where R(t) is constant. In the subse-quent discussion mres serves as our basic measure of chiralsymmetry breaking. In addition, it is useful to deﬁne the ratio
r[U]�t���x,y �J5qa �y,t�J5a�x,0��[U]
�x,y �J5a�y,t�J5a�x,0��[U]
r¯[U]��tr[U]�t��14�
which are both measures of chiral symmetry breaking on agiven gauge conﬁguration U.Before presenting results for the improved gauge actions,we discuss what is known about the Wilson action at a�1
�2 GeV (��6.0). In Fig. 1 we show the residual mass as afunction of Ls �the data are from Ref. �5��. While in pertur-bation theory mres is expected to decay exponentially, asstated in Ref. �5� the data do not support this. However, its
FIG. 1. The residual mass normalized by the square root of thestring tension (�) as function of the Ls for the Wilson gauge actionat ��6.0 �results are from Ref. �5��. The solid line is a ﬁt to adouble exponential. At large Ls the decay is rather weak.
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behavior can be ﬁt with two exponentials with a rather weakdecay in the large Ls limit. Thus, to decrease mres by an orderof magnitude we need to increase Ls by a large factor, per-haps of O(10).Since mres is determined by the ﬁfth-dimensional falloffof the boundary states, decreasing mres requires improvingthe falloff. Analytic arguments have shown that for gaugeﬁeld satisfying a smoothness condition, exponential falloff isassured �28,29�. It is expected that at weak enough cou-plings, such a smoothness condition is satisﬁed, which is notthe case for Wilson gauge lattices at ��6.0. Since the falloffin the ﬁfth dimension can be related to eigenvalues of anappropriately deﬁned transfer matrix, T, in the ﬁfth dimen-sion, studies �30� of the spectrum of the T for Wilson gaugeaction have been done. They ﬁnd a nonvanishing density ofunit or near unit eigenvalues of T, showing that undampedpropagation in the ﬁfth dimension occurs. We will also studythe spectrum of T, using gauge conﬁgurations generated withthe Wilson, Symanzik, Iwasaki and DBW2 actions.The transfer matrix T �31� is deﬁned by
T�1�Ht1�Ht�15�
2�Dw† ��M 5��5Dw��M 5��16�
being the Hamiltonian for propagation in the ﬁfth dimensionand Dw(m) being the four-dimensional Wilson Dirac opera-tor. Following �30� we calculate the eigenvalue spectrum ofthe Hermitian Wilson Dirac operator �5Dw(�M 5) as a func-tion of M 5 �the so-called spectral ﬂow�. From Eq. �16� onesees that a zero eigenvalue in �5Dw(�M 5) corresponds di-rectly to a unit eigenvalue of the transfer matrix, i.e., theexistence of a ﬁve-dimensional mode that is not damped inthe ﬁfth dimension. In addition, the number of zeros in thespectral ﬂow determines the index of the domain wall fer-mion operator and hence serves as a deﬁnition of topologyon the lattice. Thus, if one is working at a ﬁxed value for M 5and a gauge ﬁeld is generated via Monte Carlo which has aunit eigenvalue of T, an undamped mode in the ﬁfth dimen-sion occurs on that conﬁguration. This conﬁguration is onewhere we informally say that topology is changing �in theMonte Carlo update�.When studying the spectral ﬂow on a given conﬁguration,if the ﬂow approaches the M 5 axis, we expect the left andright domain wall modes to become delocalized leading tomixing and attendant chiral symmetry breaking. On the otherhand, if there is a large vertical gap in the spectral ﬂow forvalues of M 5 we use in our simulations, the chiral modesshould remain localized on the boundaries. In Fig. 2 wepresent the spectral ﬂow of the lowest 15 eigenvalues forsome representative Wilson gauge action conﬁgurations.Many crossings of the M 5 axis are evident and even themodes that do not cross are not far away from the axis,compared with the large gap that appears for M 5�0.8. Notethat M 5�0.8 corresponds to the usual critical mass for Wil-
son fermions where chiral symmetry is restored at this gaugecoupling (��6.0). As we will see, this picture leads to arelatively large value of mres for the Wilson gauge action,though we emphasize that the chiral symmetry breaking isstill very small compared to standard Wilson fermions at thisgauge coupling. In Fig. 3 the ratio r[U](t) deﬁned in Eq. �13�is plotted for the same conﬁgurations as in Fig. 2. The panelsin Fig. 2 and Fig. 3 are in one to one correspondence. In theﬁgures, r[U](t) is quite dependent on t, with large ﬂuctua-tions occurring over a small range of t. Since we can seemultiple crossings in the spectral ﬂow, which implies un-damped modes in the ﬁfth dimension, and multiple spikes ofr[U](t) it is natural to investigate whether these are differentmanifestations of the same phenomena.In Fig. 4�a� we present r¯[U] as a function of conﬁgurationnumber. It is clear that r¯[U] ﬂuctuates widely, indicating thatthere are conﬁgurations with larger chiral symmetry breakingand others with relatively small breaking. The number ofconﬁgurations with enhanced chiral symmetry breaking issigniﬁcant (�50%), consistent with the known result thatthe transfer matrix has an appreciable number of near uniteigenvalues �30�. In addition, Fig. 3 suggests a close corre-lation between conﬁgurations showing these spikes and thosewith crossings in the spectral ﬂow near M 5�1.8.In order to further examine the nature of chiral symmetrybreaking on a given conﬁguration we take a closer look atthe ratio r[U](t) deﬁned in Eq. �13�. In Fig. 3 and Fig. 5�a�we present this ratio for typical Wilson gauge action conﬁgu-
FIG. 2. Spectral ﬂows of the Hermitian Wilson Dirac operator�5DW for four typical gauge conﬁgurations generated with the Wil-son gauge action at ��6.0. There are many crossings in the neigh-borhood of M 5�1.8 which induce explicit chiral symmetry break-ing for domain wall fermions. The size of the would-be gap in theregion of the ﬁve-dimensional fermion mass M 5�1.8 is also rela-tively small compared to the obvious gap above the critical Wilsonmass M 5�0.8. Both effects enhance mixing of the light domainwall fermion modes and hence the value of the mres .
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rations, again at a�1�2 GeV. As we can see the dominantpart of chiral symmetry breaking comes from localized re-gions in time. In particular for the conﬁguration of Fig. 5�a�,the Hermitian Wilson Dirac operator has two small eigenval-ues whose eigenvectors are localized around the peaks ofr[U](t). In addition, we have computed the topologicalcharge density using 20 APE smearings with smearing coef-ﬁcient 0.45, as described in Refs. �32,33�. A classicallyO(a4) improved �34,35� topological charge density on APEsmeared lattices produced results almost identical with theones computed using the method of �32,33�. In order to ex-amine if localized peaks of the topological charge densitycorrelate with the chiral symmetry breaking, we have com-puted the correlation function C(r�) of the topological chargedensity with the lowest eigenvector of the Wilson Dirac op-erator,
C�r����x��tr FF˜ �x�����0�x��r���
�x��tr FF˜ �x����y���0�y���.�17�
In Fig. 6 we present an one-dimensional slice of this corre-lation function for the conﬁgurations used in Fig. 2 and Fig.3. The observed enhancement of the correlation around r��0 is an indication that peaks of the topological charge den-sity coincide with peaks of the eigenvector. Since we knowthat the low eigenvectors generate chiral symmetry breakingin the region of space-time that they have support, we caninfer that enhanced chiral symmetry breaking �i.e., a peak ofr[U](t)] is also localized around these localized peaks oftopological charge density. Similar conclusions and more de-tailed discussion can be found in Refs. �7,9,36�. This is acrucial observation in understanding why improved gauge
FIG. 3. The ratio of Eq. �13� for four Wilson gauge conﬁgura-tions. The bare quark mass is 0.02 and M 5�1.8. The conﬁgurationsused are the same as those in Fig. 2 and the panels are in one to onecorrespondence with the panels in Fig. 2.
FIG. 4. The quantity r¯[U] deﬁned in Eq. �14� vs conﬁgurationnumber for �a� the Wilson gauge action at ��6.0, �b� the Symanzikgauge action at ��8.4, �c� the Iwasaki gauge action at ��2.6, �d�and the DBW2 gauge action at ��1.04. All four cases correspondto a�1�2 GeV. The large spikes seen in the Wilson case are sig-niﬁcantly reduced for the Iwasaki action and almost eliminated forthe DBW2 action. These spikes corresponded to zero eigenvectorsof �5DW and are a signiﬁcant source of chiral symmetry breakingfor domain wall fermions.
FIG. 5. The ratio of Eq. �13� for a single conﬁguration of �a� theWilson gauge action at ��6.0, �b� the Symanzik gauge action at��8.4, �c� the Iwasaki gauge action at ��2.6, �d� and the DBW2gauge action at ��1.04. The spikes are quite localized in Euclid-ean time t. Examination of the eigenvectors of the �domain wallfermion or Wilson� Dirac operator conﬁrms the zero modes arelocalized in space as well.
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actions that suppress localized peaks of the topologicalcharge density can reduce chiral symmetry breaking for do-main wall fermions.If an improved action can reduce lattice artifact conﬁgu-rations which are undergoing topology change, then mres canbe reduced. The effect of the gauge action on dislocationscan be understood by examining its effects on the classicalminima of the action, i.e., instantons. Using the results ofRef. �37� we can see that for the Iwasaki and the DBW2action the O(a2/�2) correction to the action of an isolatedlattice instanton is positive, hence instantons of small size �are suppressed. On the contrary for the Wilson action theO(a2/�2) correction is negative, consequently the small lat-tice instantons are enhanced. This suggests that for theIwasaki and the DBW2 actions, gauge conﬁgurations withvery localized concentrations of topological charge densityare suppressed. If in addition, there is a suppression of con-ﬁgurations where localized topology change is occurring,there will be a reduction of explicit chiral symmetry break-ing. In conclusion, conﬁgurations of nonzero topology do notproduce large residual chiral symmetry breaking, only con-
ﬁgurations where topology is changing, i.e., where the spec-tral ﬂow has a zero. Suppression of lattice artifact topologychanging conﬁgurations should decrease mres.
IV. CHIRAL SYMMETRY WITH IMPROVED GAUGEACTIONS
In order to study the effects of the choice of the gaugeaction on the residual chiral symmetry breaking we per-formed a series of quenched simulations using the Symanzik,Iwasaki and DBW2 actions. In all cases the lattices were163�32 with inverse lattice spacing a�1�2 GeV. We usedthe � mass to set the scale but also conﬁrmed consistencywith the scale set from the string tension; both yield equallattice spacings to within a few percent. The mass M 5 wastuned to be optimum with an accuracy of about 5%. Simu-lations on a few conﬁgurations at several values of M 5 wereall that were needed for this determination. It turns out thatfor all actions at a�1�2 GeV the optimum value is roughly1.8, except for the DBW2 for which it is 1.7. In the free ﬁeldlimit, the optimum value is M 5�1 �3�. The bare quarkmasses in our study ranged from m f�0.010 to 0.060. A sum-mary of the simulation parameters is presented in Table I.The residual mass was extracted by ﬁtting to a constant atlarge time separations the ratio deﬁned in Eq. �12�. Errors aredetermined by the jackknife method. As it can be seen in Fig.7, this ratio exhibits a fairly stable plateau at time separationslarger than ﬁve or six, so we chose a ﬁtting range of 7–16 inall cases. All data in this ﬁgure are for Ls�16 and for barequark mass 0.020. The quark mass dependence of the re-sidual mass is mild as seen in Fig. 8. Since we have alsomatched the lattice spacings, it is safe to compare all theactions at the same bare quark mass ignoring renormalizationeffects. Because the numbers we are comparing differ byorders of magnitude these effects can be safely neglected. Infact the multiplicative quark mass renormalization constantshave been computed �38� and shown to be equal within 5%.In order to eliminate some of the effects of the remainingsmall mismatch of the lattice spacings, we have plotted theresidual mass scaled by the square root of the string tension.In Fig. 9 and Table II we present our measurements ofmres for each action for several values of Ls . In the case ofthe Iwasaki action we only performed the measurement atLs�16, and our result agrees with that of CP–PACS �6�. Theremaining Iwasaki points are from the CP–PACS publication�6�. As one can see at Ls�16, the DBW2 residual mass isabout two orders of magnitude smaller than the residual massof the Wilson action while the Iwasaki residual mass is about
FIG. 6. The correlation function between the lowest eigenvectorof the Wilson Dirac operator and topological charge density as de-ﬁned in �17�. The conﬁgurations used are the same as those in Fig.2 and the panels are in one to one correspondence with the panels inFig. 2
TABLE I. Simulation parameters for each gauge action tested. The � mass, m� , is given in the chiral limit for the largest Ls in each case.As usual ��1/g02, where g0 is the bare gauge coupling, M 5 is the ﬁve-dimensional fermion mass, Ls the size of the ﬁfth dimension, m f thebare quark mass, and � is the string tension computed from the heavy quark potential.
Action�M 5Lsm fm���
Wilson �5�6.001.812–240.015–0.0400.404�8�0.227�6� �52�Symanzik8.401.88–160.020–0.0600.411�14�0.2278�18� �42�Iwasaki2.601.8160.020–0.0600.415�13�0.231�6� �53�DBW21.041.78–160.020–0.0600.399�11�0.2246�16�
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an order of magnitude smaller than that of the Wilson action.Finally, the residual mass of the Symanzik action is roughlya factor of 3 smaller than that of the Wilson action. In thisﬁgure the solid lines represent ﬁts to simple exponentials inall cases except the Wilson action where a ﬁt to two expo-nentials is shown. For the Symanzik data a small deviationfrom the simple exponential ﬁt is observed at Ls�16 whilethe Wilson action shows a very clear deviation. On the con-trary, both the Iwasaki and DBW2 data can be ﬁt well with asimple exponential for the same range of Ls . For that reasonit is interesting to quote a value for the parameter q that
Shamir has computed perturbatively �13�. His one loop resultis that the light fermion wave function �(s) decays exponen-tially away from the wall, i.e, �(s)�qs with q� 1
2 . The re-sidual mass also behaves as mres�qLs. In the case of theWilson and possibly the Symanzik action, the fact that nogood ﬁt to a single exponential is obtained may be a signalthat mres scales as a power law, �54� and q�1. Such behav-ior is consistent with the spectral ﬂows observed for the Wil-son gauge action. For the Iwasaki and DBW2 actions q�0.7 and q�0.6, respectively, which is consistent with agap in the spectral ﬂow at M 5�1.7–1.8 that is well deﬁned
FIG. 7. The ratio deﬁned in Eq. �11� at a�1�2 GeV. The fancysquares correspond to the Wilson gauge action, the diamonds toSymanzik, the squares to Iwasaki, and the octagons to DBW2. Thebare quark mass in all cases is 0.020 and Ls�16.
FIG. 8. The residual mass at a�1�2 GeV as a function of thebare quark mass. The octagons correspond to DBW2, the squares toIwasaki, and the diamonds to Symanzik. In each case Ls�16.
FIG. 9. Dependence of the residual mass on the size of the ﬁfthdimension at a�1�2 GeV. The octagons correspond to DBW2, thesquares �CP–PACS �6�� and diamond �RBC �5�� to Iwasaki, thebursts to Symanzik, and the fancy squares to Wilson. All but theWilson action ﬁt a simple exponential decay reasonably well. Notethe Iwasaki results use different gauge ﬁeld ensembles at each valueof Ls . In the case of the Wilson action, the results are ﬁt to a doubleexponential function.
TABLE II. The residual mass mres at a�1�2 GeV for the ac-tions tested. In the construction of this table, for the Symanzikaction we used 51 conﬁgurations, for the Iwasaki 45, and for theDBW2 89.
m fLsSymanzikIwasakiDBW2
0.02083.04(5)�10�37.54(5)�10�4
0.020128.2(4)�10�49.92(20)�10�5
0.020163.3(3)�10�41.4(4)�10�41.60(5)�10�5
0.04082.90(4)�10�37.49(5)�10�4
0.040127.4(3)�10�49.9(4)�10�5
0.040162.73(24)�10�41.2(4)�10�41.56(3)�10�5
0.06082.82(3)�10�37.50(8)�10�4
0.060126.95(23)�10�41.00(6)�10�4
0.060162.44(18)�10�41.15(27)�10�41.565(23)�10�5
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on most conﬁgurations. We come back to this point in thefollowing where we investigate the spectral ﬂow for eachgauge action.Given the dramatic improvement in mres for the DBW2action, it is natural to wonder whether further improvementis possible. We have explored simulations where the coefﬁ-cients of the plaquette and rectangle term in Eq. �3� take onvarious ratios. Our understanding of the phase diagram ofactions with negative rectangle and positive plaquette coef-ﬁcients is shown in Fig. 10. We choose �11 to be the coefﬁ-cient of the plaquette term in the action and �12 to be thecoefﬁcient of the rectangle term. For ratios �12/�11��1/8the continuum QCD limit is taken at �11→� with �12/�11ﬁxed. For �12/�11��1/8 there is no continuum QCD limitas �11→� with �12/�11 ﬁxed. This is numerically mani-fested in bad scaling behavior and the fact that for theseratios,arbitrarilysmalllatticespacingscouldnotbeachieved. If �12/�11 is taken to be around �0.2 then adouble peaked plaquette distribution can be found. Thus, fur-ther dramatic improvement in mres does not seem possiblewith an action which involves only plaquette plus rectangleterms.
V. TOPOLOGY AND CHIRAL SYMMETRY BREAKING
In this section we take a closer look at how the differentgauge actions affect explicit chiral symmetry breaking in do-main wall fermions. As mentioned before, in Fig. 4�a� thequantity r¯[U] deﬁned in Eq. �14� is presented as a function ofthe conﬁguration number. The large ﬂuctuations �spikes� in-dicate that there are conﬁgurations with relatively large chi-ral symmetry breaking and conﬁgurations with relativelysmall breaking. The conﬁgurations with large spikes arethose for which the transfer matrix in the ﬁfth dimension hasa near unit eigenvalue, or a corresponding �near� zero eigen-value of the Hermitian Wilson Dirac operator. In those casesthat we have checked for the Wilson gauge action, a spike isalways accompanied by a localized �near� zero eigenvectorof the Wilson Dirac operator. In addition, the fact that thespectral ﬂows presented in Fig. 2 have so many crossingsvery close to the simulation point M 5�1.8 is consistent with
the large number of spikes in Fig. 4�a�. In conﬁgurationswhere a spike does not occur, i.e. no crossing close to M 5�1.8, the chiral symmetry breaking is controlled by the sizeof the gap of the bulk modes in the spectral ﬂow. Here we areseparating the small Wilson Dirac eigenvalues into twogroups: those that cross zero near M 5 and those that form amore continuum band which we refer to as bulk modes. Inthe case of the Wilson action and conﬁgurations with nocrossings close to M 5, the bulk mode gap is rather small andnot very well deﬁned; thus even on these conﬁgurations thechiral symmetry breaking is relatively large for a given M 5.The relation between in the eigenvectors of the Wilson Diracoperator and the chiral symmetry breaking has also been ex-plored in Refs. �7,9,15,36,39�, and the picture presented isconsistent with the one described above.For the Symanzik action �Fig. 4�b�� the number of spikesis slightly smaller than in the case of the Wilson action, andalso the number of crossings in the spectral ﬂow �Fig. 11� iscorrespondingly reduced. Also, the bulk mode gap is larger.As a result the baseline, or level of the troughs betweenpeaks in r¯[U] , is lower than in the case of the Wilson action,contributing to the reduction in the residual mass.The above picture becomes much clearer with the Iwasaki�Fig. 4�c�� and DBW2 actions �Fig. 4�d��. The number ofspikes is signiﬁcantly smaller, and the baseline is well de-ﬁned �especially for the DBW2 action�. The typical spectralﬂows presented in Fig. 12 and Fig. 13 again support the factthat the Iwasaki action, and to a larger degree the DBW2action, signiﬁcantly suppress the near unity eigenvalues ofthe domain wall fermion transfer matrix. In both cases thegap of the bulk modes in the spectral ﬂow becomes signiﬁ-cantly larger. As a result the explicit domain wall fermionchiral symmetry breaking is signiﬁcantly reduced.
FIG. 10. The phase diagram of actions with positive plaquette(�11) and negative rectangle (�12) coefﬁcients.
FIG. 11. The same as Fig. 2, but for the Symanzik gauge actionat ��8.4. The number of crossings in the neighborhood of M 5�1.8 appears slightly smaller and the would-be gap slightly largerthan in the Wilson case.
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In Fig. 5 we present the ratio r[U](t) deﬁned in Eq. �13�for a typical conﬁguration of each action. In all cases it isevident that the dominant contribution to chiral symmetrybreaking comes from very localized objects, and thus as weargued before, it is not very surprising that local O(a2)modiﬁcations of the gauge action can have a very signiﬁcant
effect on explicit residual chiral symmetry breaking.It is important to recognize that the above mechanism forexplicit chiral symmetry breaking is related to topology-changing conﬁgurations �see Ref. �36� and references thereinfor a more complete discussion�. The connection is madethrough the index theorem: the domain wall fermion operatorin the limit Ls→� has an index �4,40� equal to the numberof right-handed minus the number of left-handed zeromodes, which corresponds to the topological charge of thebackground gauge ﬁeld conﬁguration—a quantity which be-comes precise in the continuum limit. This integer dependson the value of M 5 used and is given by the net number ofcrossings in the spectral ﬂow of the Wilson Dirac operator asthe Wilson mass varies between a value above the criticalWilson mass and �M 5. While this index is well deﬁnedonly in the limit Ls→�, our simulations show that the near-zero eigenvectors of the ﬁnite-Ls operator obey the indextheorem to a high degree of accuracy �5,41�. In particular, foran Iwasaki a�1�2 GeV ensemble, when compared to thetopological charge computed using the smoothing methoddescribed in Refs. �32,33�, the index agrees very well. Inthose cases where the topological charge is not close to aninteger, we also ﬁnd a crossing in the spectral ﬂow, a spike inr[U](t), and a complex structure of eigenvectors that is notexpected from simple chiral symmetry arguments �41�. If M 5sits exactly on a crossing, then the index is not deﬁned, eventhe limit Ls→�. A crossing in the spectral ﬂow that occursaway from the critical Wilson mass corresponds to a conﬁgu-ration with indistinct topology. Put differently, if the particu-lar gauge ﬁeld in question is in the midst of changing itstopology, which must happen if the update algorithm is er-godic and updates the conﬁguration smoothly, then such agauge ﬁeld must give rise to a crossing. It is also sensiblethat such a tunneling from one topological sector to anotherproceeds through local changes in the gauge ﬁeld which havea characteristic size of one to two lattice spacings. In thecontinuum limit, if the density of these dislocations is zero,then all crossings happen at the critical mass and correspondto physical topological charge. Thus the index as computedfrom the spectrum of the domain wall operator Dirac opera-tor is well deﬁned in this case.Consequently, when the Iwasaki action or the DBW2 ac-tion is used, the question arises whether the topologychanges efﬁciently. We have measured the topological chargeusing the smoothing method described in Refs. �32,33,55�We used 20 APE smearing steps with smearing coefﬁcient0.45 followed by the extended loop deﬁnition of the topo-logical charge density used in Refs. �32,33�. Our data arepresented in Fig. 14. The conﬁgurations shown in this ﬁgureare separated by 1000 sweeps of Cabibo-Marinari pseudo-heatbath with a Kennedy–Pendleton accept/reject step �56�.We can see that there is a signiﬁcant slow down in the topo-logical charge ﬂuctuations for the DBW2 action. Both theSymanzik and the Iwasaki action also show a mild reductionin the frequency of change of the topological charge. Al-though the problem seems severe for the DBW2 action, wecan tackle it with brute force. For that reason we have pro-duced a library of DBW2 lattices for a�1 1.3 GeV and 2GeV to be used for several domain wall fermion projects.
FIG. 12. The same as Fig. 2, but for the Iwasaki gauge action at��2.6. The number of crossings neighborhood of M 5�1.8 is sig-niﬁcantly smaller and the gap clearly larger than in the Wilson case.
FIG. 13. The same as Fig. 2, but for the DBW2 gauge action at��1.04. There are no crossings neighborhood of M 5�1.8 and thegap is quite large, roughly comparable to the gap at the correspond-ing mass above the critical Wilson mass. Note that this is also truefor the region beyond the next critical Wilson mass, M 5�2.3,where the four ﬂavor Wilson fermion doublers become light.
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For generating this library we used overrelaxed Cabibo-Marinari pseudo-heatbath with a Kennedy–Pendleton accept/reject step. Each sweep consisted of one pseudo-heatbathstep and some overrelaxation hits. The saved a�1 1.3 GeVconﬁgurations are separated by 1000 sweeps with no overre-laxation while the a�1 2 GeV conﬁgurations are separated1000 sweeps each sweep containing four overrelaxation hits.Given the considerable cost of measuring domain wall fer-mionic observables, this higher cost of producing DBW2 lat-tices at 2 GeV is negligible. However, it is clear that thisbrute force approach will become less practical for smallerlattice spacing since topology change is likely to be rapidlysuppressed as we approach the continuum limit �57�.
VI. HADRONIC OBSERVABLES FOR THE DBW2 ACTION
In this section we discuss various hadronic observablescalculated with the DBW2 gauge action at ��0.87 and 1.04which correspond to a�1�1.3 and 2 GeV, respectively.
A. The heavy quark potential
We measure the heavy quark potential as in Ref. �42� byﬁxing to Coulomb gauge and then computing the two-pointcorrelation function of products of temporal links. More pre-cisely,
Ae�V(r)T��t0 ,x� �Tr�Lt0 ,T�x��Lt0 ,T†�x��r����,�18�
Lt0 ,T�x��� �t�t0
t0�TUt�x�,t�,�19�
and V(r) the heavy quark potential. The potential is ex-tracted by taking ratios of the correlation function in Eq. �18�at T and T�1. The systematics involved in choosing T werecarefully studied and the optimal T was chosen. For the 1.3GeV lattices T was 4 while for the 2 GeV lattices it was 7.The potential V(r) is ﬁt to
V�r��C��r ��r.�20�
The above formula gave very good ﬁts for spatial dis-tances r��2. The upper range of r was determined by the
FIG. 14. The conﬁguration history of the topological charge Q.The conﬁgurations are separated by 1000 sweeps of Cabibo–Marinari pseudo-heatbath with a Kennedy–Pendleton accept/rejectstep. At the top is the Symanzik action with lattice size 163�32 andcoupling ��8.4. In the middle is the Iwasaki action with size 164
and ��2.6. At the bottom is the DBW2 action with size 163
�32, and ��1.04. All cases correspond to roughly the same scale,a�1�2 GeV. The DWB2 action, which suppresses conﬁgurationswith small instantons, shows a signiﬁcant reduction of the tunnelingbetween topological charge sectors.
FIG. 15. The heavy quark potential for DBW2, ��0.87. Thesolid line denotes the ﬁt to Eq. �20� from which the string tension isdetermined.
FIG. 16. The same as Fig. 15 but for ��1.04.
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distance where the error on the potential became unaccept-ably large. The maximum distance used was r�6 and 7 forthe 1.3 and 2 GeV lattices, respectively. Figures 15 and 16show the heavy quark potential as a function of distance. Theresults for the string tension � and the Sommer parameter�43,44� r0 are tabulated in Table III. These results are used inour subsequent discussion of the scaling of hadronic observ-ables.
B. Simulation and analysis
For each set of gauge conﬁgurations, domain wall fer-mion propagators are computed with two types of sources:�1� a local point source and �2� a Coulomb gauge ﬁxed ex-tended source which is either a wall source for ��1.04 orbox source with 83 volume for ��0.87. �We set the sourceto one at each site inside the box and zero elsewhere.� Thelocal source is used for the determination of the decay con-stants and also the axial current renormalization factor ZA(��0.87 only�. The extended source is used for all otherpurposes.In Table IV we give mres in the chiral limit for the sameensemble of conﬁgurations used for the hadronic observablesto be discussed in this section. We have ﬁtted mres(m f) witha linear function of m f to obtain mres(0) for which the chirallimit of low energy physics is deﬁned as m f��mres(0). All
data are used to extrapolate to m f→0 for ��1.04. On theother hand, the largest value m f�0.09 for ��0.87 is notused for the extrapolation.We take the chiral limit m f→�mres(0) as the physicalpoint for u, d quarks. This determines the physical � mesonmass m� . With the input m��770 MeV, the lattice spacinga is determined. The kaon physical point m fK , which roughlycorresponds to half the strange quark mass, for f K and mK*is deﬁned by m�(m f�m fK)/m�(m f��mres)�0.645 usingonly degenerate quark masses. We do this procedure for ev-ery jackknife sample to estimate the error for values at thephysical kaon point.
C. Chiral property of pseudoscalar mass
Because of the almost exact chiral symmetry of domainwall fermions and the use of the quenched approximation,the pion two-point function suffers contamination from topo-
TABLE III. Parameters and resulting scales for the DBW2gauge action used for the spectrum analysis. The quantity a� is thelattice spacing determined by the � meson mass taken from TableVIII. The jackknife errors are quoted for the string tension � andthe Sommer parameter r0.
�M 5LsStatistics��r0a��1 (GeV)
0.871.8161000.324�6�3.58�4�1.31�4�1.041.7164050.2246�16�5.24�3�1.98�3�
TABLE IV. The residual mass for the DBW2 gauge action cal-culated on the same conﬁgurations used to evaluate the hadronicobservables. The values for m f�0 have been obtained from a linearﬁt in m f . The value for m f�0.09 at ��0.87 is excluded from theﬁt.
��0.87��1.04
m fmresm fmres
0.015.44 (23)�10�40.011.80 (9)�10�5
0.025.16 (22)�10�40.0151.80 (11)�10�5
0.034.84 (21)�10�40.021.77 (11)�10�5
0.044.55 (19)�10�40.0251.74 (11)�10�5
0.054.30 (18)�10�40.031.71 (10)�10�5
0.064.08 (16)�10�40.0351.69 (8)�10�5
0.093.52 (13)�10�40.041.67 (7)�10�5
05.69 (26)�10�401.86 (11)�10�5
TABLE V. Hadron masses computed using the DBW2 gauge action. The superscripts PP and AA refer to the pseudoscalar and axialvector correlation functions, respectively. All masses are obtained with degenerate quarks.
�m fm�PPm�AAm�mN
0.010.2248 �25�0.2179 �31�0.607 �22�0.790 �75�0.020.2997 �19�0.2966 �23�0.640 �17�0.871 �23�0.030.3603 �16�0.3590 �20�0.662 �11�0.921 �14�0.870.040.4128 �15�0.4118 �19�0.685 �8�0.975 �10�0.050.4601 �14�0.4589 �18�0.709 �6�1.021 �8�0.060.5037 �13�0.5021 �17�0.732 �5�1.067 �7�0.090.6192 �13�0.6170 �15�0.803 �4�1.197 �6�
0.010.1794 �22�0.1759 �21�0.413 �6�0.546 �13�0.0150.2098 �17�0.2075 �18�0.424 �4�0.575 �9�0.020.2377 �15�0.2359 �16�0.435 �4�0.602 �7�1.040.0250.2631 �13�0.2617 �15�0.447 �3�0.628 �6�0.030.2868 �12�0.2857 �14�0.4586 �29�0.652 �5�0.0350.3090 �12�0.3081 �13�0.4705 �26�0.676 �4�0.040.3300 �11�0.3293 �12�0.4825 �24�0.699 �4�
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logical near zero modes, which causes a shift in ﬁtted massesfrom their inﬁnite volume values �5�. The effect is expectedto be inversely proportional to the square root of the volume.Because we used different physical volumes for the two dif-ferentgaugecouplings�V�(2.4 fm)3for��0.87,(1.6 fm)3 for ��1.04], the size of the effect on the pseudo-scalar mass should be different in our two lattice ensembles.To study zero mode effects we examine the pseudoscalarmass from two different two-point functions. One is thepseudoscalar-pseudoscalar correlator �PP�
��a�x��a�0����iJ5a�x�iJ5a�0��
���q¯�a�5q�x�q¯�a�5q�0��,�21�
and the other is the correlator of the temporal component theaxial-vector current �AA�
�A0a�x�A0a�0����q¯�a�5�0q�x�q¯�a�5�0q�0��,�22�
where q and q¯ are four-dimensional quark ﬁelds deﬁned inEq. �5�. As discussed in Ref. �5�, the two types of correlatorssuffer differently from topological zero modes. The leadingcontribution to the pseudoscalar correlator is �1/(m f2�V)while it is �1/(m f�V) for the axial correlator. Although therelative contribution from the pole compared to the physicalone is the same for both, the prefactor of the 1/(m f�V) termis expected to be suppressed �5�. Thus, the mass extractedfrom the PP correlator is expected to have a stronger ﬁnitevolume effect from zero modes than the AA correlator. Theobserved effect on the meson mass calculated for light quarkmasses is to shift it above a linear extrapolation from theregion of heavier quark mass.The pseudoscalar mass extracted from both types of cor-relators is presented in Table V. Figure 17 shows the pseu-doscalar mass squared as a function of m f for ��0.87. Bothvalues of the pion mass are consistent with each other form f�0.02. However, at m f�0.01 the mass extracted from thePP correlator lies above the AA one, outside of their statisti-cal errors. Because the axial correlator is expected to havesmaller ﬁnite volume effects from zero modes, we use thiscorrelator for further analysis.The linear ﬁt of the pion mass squared in m f is quite goodin the region 0.01�m f�0.06 as indicated by the �2/dof,which is tabulated in Table VI. Note, we are using the sameset of gauge conﬁgurations for all values of m f but employ
FIG. 17. The quenched pion mass squared as a function of m ffor ��0.87. Masses extracted from the pseudoscalar �PP� and axialvector �AA� correlation functions are shown. They agree quite well,except for the point at m f�0.01 where there is a small differenceoutside of statistical errors. The physical volume is roughly(2.4 fm)3, so contamination from topological zero modes whichcan induce such splittings �see text� is suppressed. The line is a ﬁt toa simple linear function. The extrapolation slightly overshoots theexpected chiral limit point m f��mres �see inset�, suggesting aquenched chiral log.
TABLE VI. Results from ﬁtting m�2 to the linear function c0�c1m f . The column m f shows the ﬁtting range.
�Correlatorm fc0c1�2dof
0.87PP0.01–0.06 0.0090 �14� 4.057 �29� 2.8 �1.2�40.87AA0.01–0.06 0.0063 �16� 4.090 �37� 0.17 �35�41.04PP0.01–0.040.0056 �9� 2.566 �21�3.4 �8�51.04PP0.015–0.04 0.0047 �9� 2.597 �19� 1.10 �25�41.04AA0.01–0.040.0044 �9� 2.585 �21�2.6 �7�51.04AA0.015–0.04 0.0035 �9� 2.615 �20� 0.86 �23�4
TABLE VII. Fit results for the quenched chiral log contributionto the pion mass determined from the AA correlator using Eq. �23�.The symbol ‘‘—’’ indicates that the parameter is constrained to bezero.
�m fa�b��2dof
0.870.01–0.064.04 �5�—0.031 �14�1.6 �1.5�40.870.01–0.093.40 �23�6.4�1.6� 0.107 �38� 0.19 �25�41.04 0.015–0.04 2.583 �28�—0.049 �14�2.0 �5�4
FIG. 18. The same as Fig. 17, but for ��1.04. Here the physi-cal volume is roughly (1.6 fm)3, so zero mode effects in the massesare visible, and the simple linear ﬁt clearly overshoots the chirallimit point m f��mres .
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an uncorrelated ﬁt. One can reliably extract the physicalkaon mass; however, the ﬁt overshoots the point m f��mres where the pion mass should vanish. This is a signalof nonlinearity for the pion mass at small m f . Instead of alinear function we should employ the quenched chiral log�45� formula with the constraint that the pion mass vanishesat m f��mres,
m�2 �a��m f�mres�� 1�� log�a��m f�mres�
�Q�PT2��
�b�m f�mres�2,�23�
where we have introduced the quadratic term in addition tothe expression used in Ref. �46�. As is discussed in Ref. �46�,Eq. �23� has a form suggested by chiral perturbation theory.Note that the explicit chiral perturbation theory formula, e.g.,Eq. �90� of Ref. �46�, involves three parameters �, �, and�Q�PT and additional terms. However, within our parameterrange and numerical accuracy these additional terms have theeffect of adding an undetermined, m f-independent constantto the expression within the square brackets. Our choice ofzero for this unknown constant represents a rescaling of theparameters � and �Q�PT from those that appear in the chiralperturbation theory prediction.The data ﬁt this formula well with a reasonable value ofthe chiral log coefﬁcient � as listed in Table VII, where thescale is set as �Q�PT�1 GeV. If the coefﬁcient of the qua-dratic term b is not set to zero in the ﬁt, a somewhat largervalue of � results. This is because the two terms tend tocancel each other. Note that the value of b is the same orderas a� . Ultimately, a proper covariant ﬁt with reliable �2
should be used to distinguish the two ﬁts. At present we donot have enough statistics to do this.Figure 18 shows the pion mass squared for ��1.04. Theresults for the larger value of m f agree with each other, butdeviations appear as m f decreases. In fact, pronounced up-
FIG. 19. Vector meson and baryon masses as functions of m f for��0.87. Line is a linear ﬁt to the data. The physical nucleon andK* mass are shown as solid points where m� is used to ﬁx the scale.
FIG. 20. Vector meson and baryon masses as functions of m f for��1.04. The line is a linear ﬁt to the data. The physical K* massis also shown.
FIG. 21. Scaling of the � meson mass with the lattice spacingset by the Sommer parameter r0. The Wilson data are from Ref. �5�.
FIG. 22. Scaling of the nucleon and K* masses with latticespacing. The experimental mass of nucleon and K* are shown onthe vertical axis. The Wilson data are from Ref. �5�.
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ward curvature of the data, which was not observed for theaxial correlator for ��0.87, result in large values of �2/doflisted in Table VI for both correlators. The lines in the Fig.18 show the results of the ﬁt excluding the lightest pointm f�0.01, which makes �2/dof smaller. Thus we can as-sume that the pion mass for m f�0.015 does not suffer muchfrom zero mode effects. As this ﬁt reproduces the data in therange 0.015�m f�0.04 very well, we use it for the interpo-lation of the kaon point. Also using this region of m f , onecan ﬁt data with the quenched chiral log without the qua-dratic term for �2/dof�0.5(1). The resulting � is consistentto that for ��0.87. Since our range of m f for ��1.04 is notwide enough to disentangle the quadratic term and the logterm, we do not list a result for nonzero b.
D. Hadron spectrum
We list the results for the vector meson and nucleonmasses in Table V. Figures 19 and 20 show the vector mesonand nucleon masses as functions of m f for ��0.87 and 1.04,respectively. Physical nucleon, rho, and K* masses are indi-cated on the ﬁgure.Figure 21 presents the rho meson mass versus latticespacing squared, both normalized with r0. We also plot theresults obtained with the Wilson gauge action from Ref. �5�.Wehaveselectedvaluesobtainedat(�,V,Ls)�(6,163,24),(5.85,123,20),(5.7,83,48). These lattices havealmost the same physical volume, and Ls is the largest avail-able in each case. We observe consistency between DBW2and Wilson actions. The ﬂatness of the data reﬂects the smallsize of the scaling violation.Figure 22 is a scaling plot for the nucleon and K* massesnormalized by the � meson mass. Again good scaling is ob-served. The K* appears to be lighter than experiment, whichis consistent with other quenched simulations. For thenucleon mass, there is an observed discrepancy between theresults with Wilson type fermions and staggered fermions�see comparison by Aoki �47��. The former gives a lighternucleon mass while the latter is consistent with the experi-ment. The nucleon mass for domain-wall fermion �see Fig.20 and Table IX� is slightly larger than experiment for thelattice spacings we examined. We need to perform simula-tions for larger physical volume at a�1�2 GeV, as well assimulations at smaller lattice spacings to do the continuumextrapolation needed to compare with conventional fermions.For the comparison within the domain-wall fermions, giventhe statistics and the fact that the physical volumes of ourensembles are not the same, we cannot say if the DBW2action exhibits better scaling than the Wilson gauge actionthough it seems that the scaling is at least as good.
TABLE VIII. Results from ﬁtting the vector meson mass to a linear function, c0�c1m f . Values at the physical points and the J parameterare also listed.
�m fc0c1�2dofm�mK* /m�J
0.870.01–0.060.590 �19�2.37 �28�0.14 �50�40.589 �19�1.138 �11�0.377 �37�1.040.01–0.040.389 �6�2.34 �12�0.08 �20�50.388 �6�1.136 �5�0.387 �16�
TABLE IX. Results from ﬁtting the baryon mass to a linearfunction, c0�c1m f . Values at the physical point are also listed.
�m fc0c1�2dofmN
0.87 0.01–0.06 0.783 �27� 4.76 �44� 0.5 �1.6�40.780 �27�1.04 0.01–0.04 0.502 �12� 4.96 �26�0.5 �8�50.502 �12�
FIG. 23. Scaling of the quenched J parameter with lattice spac-ing. The experimental point is shown on the vertical axis. The largediscrepancy with experiment is due to quenching.
TABLE X. The axial current renormalization factor and pseudo-scalar decay constants.
�m fZAf PSAAf PSPP
0.020.78199 �37�0.1084 �31�0.1078 �34�0.030.78404 �31�0.1121 �27�0.1110 �27�0.870.040.78612 �28�0.1165 �26�0.1145 �24�0.050.78824 �26�0.1208 �26�0.1192 �24�0.060.79042 �25�0.1247 �26�0.1237 �24�
0.010.84142 �17�0.0703 �22�0.0697 �21�0.0150.84191 �14�0.0716 �17�0.0720 �17�0.020.84244 �12�0.0732 �14�0.0743 �15�1.040.0250.84300 �11�0.0751 �12�0.0767 �14�0.030.84358 �10�0.0771 �11�0.0790 �13�0.0350.84417 �9�0.0790 �11�0.0813 �12�0.040.84478 �9�0.0809 �10�0.0835 �12�
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The J parameter, which is introduced in Ref. �48� to ex-amine the effect of quenching on the mass spectrum, is de-ﬁned by
J�mK*dm�dm�2 ,�24�
where mK*(m f) is evaluated at m�(m f)/m�(m f)�mK* /mK�1.8 by deﬁnition. The slope of m� with respect to m�2 isdetermined with the data in 0.01�m f�0.06 for ��0.87 and0.015�m f�0.04 for ��1.04, using pion mass determinedwith AA. An approximate phenomenological value calculatedfrom the experimental mass spectrum is
J�mK*mK*�m�mK2 �m�2 �0.48.�25�
Our lattice results for the J parameter, listed in Table VIIIand plotted in Fig. 23, show a signiﬁcant difference from thephenomenological value. However, these results are consis-tent with other quenched results �see a summary given byKaneko �49��. Note that we have used degenerate quarkmasses which could explain some of the difference betweenour result and experiment, but the largest source of the dis-crepancy is likely to be due to quenching.
E. Pseudoscalar decay constants
The pseudoscalar decay constants are calculated from theamplitude of the point-point two-point function of the tem-poral component of local axial vector current,
ZA2m�2 e�m�t��x��A0a�x�,t�A0a�0��,�26�
for t�1. This simple local current is not partially conserved,unlike A� �Eq. �7�� which obeys the Ward-Takahashi identity�Eq. �6��, so the local current receives a multiplicative renor-malization. We can calculate the renormalization factor ZAfrom the ratio of the correlation functions of the two cur-rents. Here we employ a ratio designed to remove some ofthe lattice spacing error �5�
R�t��12�C�t�1/2��C�t�1/2�
2L�t��2C�t�1/2�
L�t��L�t�1��,
where C and L are the correlators of the pseudoscalar densitywith the partially conserved and local axial currents, respec-tively,
C�t�1/2���x��A 0a�x�,t�Pa�0��,�28�
L�t���x��A0a�x�,t�Pa�0��.�29�
The ﬁrst and second terms on the right-hand side of Eq. �27�remove the O(a) scaling error, and suppress the O(a2) errorin the sum. The value of ZA is determined by ﬁtting R(t) toa constant for the interval 6�t�26, where R(t) is ﬂat to avery good approximation. The results are given in Table Xand plotted in Fig. 24. An estimate of ZA with only the ﬁrstterm is also plotted to demonstrate that the complex ratio inEq. �27� actually works. Although linear dependence on m fin R remains for both cases, it is very small for the fullexpression of Eq. �27�. We use the value ZA obtained in thechiral limit m f→�mres for the calculation of the pseudo-scalar decay constant, which is given in Table XI.Another method to calculate the decay constant is to usethe Ward-Takahashi identity to derive its relation with thepseudoscalar to vacuum matrix element of the pseudoscalardensity,
�m f�mres�
2 ��0�J5a��,p��0��.�30�
FIG. 24. The local axial current renormalization factor as afunction of m f for ��0.87. The lines correspond to ﬁts to Eq. �27��lower� and the ﬁrst term only �upper�. The difference arises fromlattice spacing errors that are proportional to m f .
TABLE XI. The axial current renormalization factor at m f��mres and the pseudoscalar decay constants at physical points in MeV.
�ZA(m f��mres)f �AAf �PPf KAAf KPPf KAA/f �AAf KPP/f �PP
0.870.77759�45�130.4�6.7�129.0�7.3�148.9�5.2�147.3�5.4�1.142�26�1.141�30�1.040.84018�18�130.8�4.9�129.0�5.0�147.4�3.3�149.7�3.6�1.139�24�1.118�25�
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This matrix element is determined from the PP correlator.Results with each correlator for ��0.87 are shown in Fig.25. For both methods we use the results of m� extractedfrom the AA correlator with the extended source quarkpropagator since it is our most precise determination.Consistency between both results shows that we havegood control over the chiral symmetry breaking through ourdetermination of mres. Also, possible zero mode effects,which inﬂuence the two correlators differently, appear to besmall at least for m f�0.02, which is consistent with theabsence of zero mode effects in the pion mass over the samerange of m f .Figure 26 is the same plot but for ��1.04. The resultsfrom the two determinations are also in good agreement.Results for decay constants at the physical points are given
in Table XI and plotted in Fig. 27 against lattice spacingsquared. They are consistent with those reported for the Wil-son action �5�. We observe good scaling both for f � and f K .The result for f � is in good agreement with the experimentalvalue. However, the ratio f K /f � appears to be smaller thanthe experimental value 1.22, which is an expected effect ofquenching �45�.
VII. CONCLUSIONS
In this paper we have demonstrated that the DBW2 actionsigniﬁcantly improves the chiral properties of domain wallfermions. The main reason for this improvement is the sup-pression of gauge conﬁgurations which support unit eigen-values of the transfer matrix in the ﬁfth dimension and henceallow signiﬁcant mixing of the light chiral modes that arelocalized on opposite boundaries of the ﬁfth dimension.These problematic conﬁgurations are also those which occuras the topological charge of the gauge ﬁeld changes duringMonte Carlo evolution. One key to improving the domainwall fermion chiral symmetry is to use an improved gaugeaction which suppresses these small dislocations. This sup-pression works so well in the case of the DBW2 action thatat a�1�2 GeV and Ls�16 the residual chiral symmetrybreaking is roughly two orders of magnitude smaller com-pared to the Wilson action case and therefore is completelynegligible. Even at strong coupling (a�1�1.3 GeV)mres isabout three times smaller in physical units than for the Wil-son action at 2 GeV. In both cases the value of Ls is 16.In addition to the suppression of these small topologicaldislocations associated with zero crossings of the spectralﬂow of the four-dimensional Wilson Dirac operator, we havealso observed an increased gap in the spectral ﬂow. Conse-quently, the light boundary mode wavefunctions decay fasterin the ﬁfth dimension. For the DBW2 action this leads to a
FIG. 25. The quenched pseudoscalar decay constant as a func-tion of m f for ��0.87. The results from the two techniques de-scribed in the text agree quite well. The data are ﬁt to simple linearfunctions, and extrapolated �pion� or interpolated �kaon� to thephysical points �ﬁlled symbols�.
FIG. 26. The same as Fig. 25, but for ��1.04. Although theresults agree within statistical errors, the agreement is not as goodas that at ��0.87.
FIG. 27. A scaling plot for the pion and kaon decay constants.The DBW2 results appear to preserve the good scaling that wasobserved with the use of the Wilson gauge action �data from Ref.�5��.
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residual mass dependence on Ls proportional to 0.6Ls. Thisdependence is close to Shamir’s perturbative prediction �13�.Approaches based on the proposals made by Shamir in �13�may be effective in further reducing this perturbative base-line. Work along these lines is currently underway.In the second part of this paper we presented results forsome quenched hadronic observables obtained with theDBW2 gauge action. Our conclusion is that these observ-ables scale very well with a, i.e., the good scaling of domainwall fermions seen in quenched simulations with the Wilsongauge action is preserved �5�.Using these improved actions, we also observed that thetopological charge of the gauge ﬁelds evolves much moreslowly using standard Monte Carlo algorithms. In futuresimulations with smaller lattice spacings, improved algo-rithms will be needed to efﬁciently sample topology. Wenote, however, that this is a generic feature of all latticecalculations which is not speciﬁc to the DBW2 action, al-
though the DBW2 action will always suffer more than theother actions explored in this paper.
ACKNOWLEDGMENTS
The calculations reported here were done on the 400Gﬂops QCDSP computer �50� at Columbia University andthe 600 Gﬂops QCDSP computer �51� at the RIKEN BNLResearch Center. We thank RIKEN, Brookhaven NationalLaboratory and the U.S. Department of Energy for providingthe facilities essential for the completion of this work. Theauthors would like to thank members of QCD–TARO forproviding us with their scale setting data for the DBW2 ac-tion. This research was supported in part by the DOE underGrant No. DE-FG02-92ER40699 �Columbia�, in part by theDOE under Grant No. DE-AC02-98CH10886 �Soni�, in partby the RIKEN BNL Research Center �Aoki-Blum-Dawson-Ohta-Orginos�, and in part by the Japan Society for the Pro-motion of Science �Izubuchi�.
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�54� We thank Y. Shamir for discussions on this point.�55� We thank the MILC Collaboration for their code which wasused to compute the topological charge.�56� Each sweep consists of one update of two independent SU�2�subgroups of each SU�3� link.�57� P. de Forcrand �private communication�.
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