Improved bilinears in lattice QCD with nondegenerate quarks

PHYSICAL REVIEW D 73, 034504 (2006)

Improved bilinears in lattice QCD with nondegenerate quarks

Tanmoy Bhattacharya and Rajan GuptaLos Alamos National Lab, MS B-285, Los Alamos, New Mexico 87545, USA

Weonjong LeeSchool of Physics, Seoul National University, Seoul, 151-747, Republic of Korea

Stephen R. Sharpe and Jackson M. S. Wu*Physics Department, University of Washington, Seattle, Washington 98195-1560, USA

(Received 18 November 2005; published 13 February 2006)

*Present adBC, Canada V

1550-7998=20

We describe the extension of the improvement program for bilinear operators composed of Wilsonfermions to nondegenerate dynamical quarks. We consider two, three and four flavors, and both flavornonsinglet and singlet operators. We find that there are many more improvement coefficients than withdegenerate quarks, but that, for three or four flavors, nearly all can be determined by enforcing vector andaxial Ward identities. The situation is worse for two flavors, where many more coefficients remainundetermined.

DOI: 10.1103/PhysRevD.73.034504 PACS numbers: 11.15.Ha, 12.38.Gc

1Since the gauge action gives rise to errors of O�a2� we do notconsider its improvement here, although our considerationsapply equally well for an improved gauge action, as explainedin appropriate places in the following.

2

I. INTRODUCTION

Simulations of lattice QCD with light dynamical quarksare greatly facilitated by the use of improved actions andoperators. Calculations are underway using various typesof improved fermions—staggered, domain wall/overlap,maximally twisted and improved Wilson fermions. Herewe focus on improved Wilson fermions, and investigatehow the improvement program can be extended to removeerrors proportional to amq (a is the lattice spacing, and mq

a generic quark mass) in the realistic case of nondegeneratequark masses. Of particular interest are the ‘‘Nf � 2� 1’’theories withmd � mu <ms and ‘‘Nf � 2� 1� 1’’ theo-ries withmd � mu <ms < mc, and we consider both theo-ries here. As we review below, the improvement of theaction for such theories including amq terms has alreadybeen considered, but the improvement of operators has not.As a first step, we consider here the improvement of allquark bilinears. These are of considerable phenomenologi-cal interest, since their hadronic matrix elements, com-bined with experimental results for form factors,determine elements of the Cabibbo-Kobayashi-Maskawamatrix. Errors in such hadronic matrix elements propor-tional to ams and particularly amc can be significant, and itis thus important to reduce or remove them.

At present, simulations with staggered fermions are ableto reach the smallest dynamical quark masses. This comesat the cost, however, of a multiplication of fermion species,and the concomitant need to use the fourth root of thefermion determinant, so that unitarity can at best be re-stored in the continuum limit. Wilson fermions have theadvantage of a straightforward relation to the continuumtheory: each lattice fermion gives rise to a single contin-

dress: TRIUMF, 4004 Westbrook Mall, Vancouver,6T 2A3.

06=73(3)=034504(23)$23.00 034504

uum flavor. They also come, however, with disadvantages:in their original form, the leading discretization errors areof O�a�, as compared to O�a2� for staggered fermions, andthese discretization errors explicitly break chiral symme-try. As explained in seminal papers by the ALPHACollaboration [1], one can apply the Symanzik improve-ment program to Wilson fermions and systematically re-duce the errors from O�a� to O�a2�.1 We recall that thisrequires the addition of all dimension five operators to theaction that are consistent with the symmetries of the latticetheory, with their coefficients being determined by appro-priate nonperturbative conditions. These conditions aregenerically chosen to enforce a symmetry that is presentin the continuum limit but is broken for nonvanishinglattice spacing. For Wilson fermions, the broken symme-tries that are used are the flavor nonsinglet axialsymmetries.

A similar method holds for the improvement of opera-tors.2 One must add all operators with the same symmetrieshaving one higher dimension and determine their coeffi-cients by applying appropriate ‘‘improvement conditions.’’This assumes that the operators do not mix with otheroperators of lower dimension, which will be true in allbut one case here.

It is possible to simplify this procedure by consideringonly the improvement of on-shell quantities—masses,decay constants, physical matrix elements, etc. This allows

We use the term ‘‘improvement’’ as shorthand for ‘‘O�a�improvement’’ throughout this paper. We do not consider theremoval of discretization errors proportional to a2 or higherpowers.

-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.73.034504

BHATTACHARYA et al. PHYSICAL REVIEW D 73, 034504 (2006)

one to use the equations of motion to reduce the number ofhigher dimension operators that need to be considered. On-shell improvement is equivalent to improving correlationfunctions in which the arguments are all separated. Off-shell improvement extends this to correlation functions inwhich some arguments are at the same space-time point,and requires additional contact terms. While we will con-sider only on-shell improvement here, it will turn out thatwe need to understand some of the subtleties of off-shellimprovement in order to resolve certain puzzles thatemerge from our analysis.

The general discussion of Ref. [1] applies in the pres-ence of two or more dynamical quarks. Thus we know fromthat work how to nonperturbatively improve the action, andit is now standard to implement this in unquenched simu-lations. Results are available with the Wilson gauge actionfor two [2,3], three [3], and four [3] flavors, and also withan improved gauge action [4]. The method of Ref. [1] alsoallows one to improve the flavor nonsinglet axial current inthe chiral limit (and a variant of this method has beenapplied for two flavors in Ref. [5]), and the methodologyhas been extended to the improvement of nonsinglet vectorand tensor bilinears in the chiral limit [6–8]. Theoreticaldiscussion of the improvement of nonsinglet bilinearsaway from the chiral limit has been restricted to degeneratequarks [1,8], or to nondegenerate quarks in the quenchedapproximation [7,9].

Here we generalize previous work by considering theimprovement of bilinears for the realistic case of nonde-generate quarks. We consider both flavor singlet and non-singlet bilinears; the addition of flavor singlets is requiredby the analysis, but is also of phenomenological interest.We explain how working away from the chiral limit in-troduces a plethora of new improvement constants, andthen study which of these can be determined by imposingappropriate axial Ward identities. We find that, for threeflavors, nearly all can be determined, with the situationunchanged for four flavors but worse for two flavors. Todetermine the remaining improvement coefficients onemust use other methods, e.g. nonperturbative renormaliza-tion [10], suitably improved [11], or matching short dis-tance correlation functions to perturbative expressions [8].

In this work we make no specific choice as to how theWard identities are implemented. One could use a methodbased on the Schrodinger functional, or use standard had-ron correlation functions. Our theoretical discussion holdsequally well for either choice. What is key, however, is thatone can vary the quark masses independently in a regimewhere all have mqa < 1, so that effects proportional to�mqa�

2 can be neglected. This condition is satisfied for thephysical strange quark for lattice spacings satisfying ap-proximately a � �1 GeV��1, and for the physical charmquark if a � �4 GeV��1. We stress that these conditions donot necessarily require the quarks to be light (with ‘‘light’’meaning mq � �QCD).

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The practical implementation of our method will be veryinvolved and computationally expensive. An indication ofthis is that even the simplest step of determining theimprovement constant for the nonsinglet axial current inthe chiral limit has only recently been undertaken fordynamical fermions [5]. For most improvement constants,present calculations instead rely upon one-loop perturba-tive values. This raises the question of whether our analysisis of purely theoretical interest or will be useful in practice.To ask this another way, will a tree-level or one-loopperturbative estimate of the improvement constants sufficein practice? For example, the mass-dependent improve-ment coefficient for the nonsinglet flavor off-diagonal axialcurrent (needed to determine decay constants) enters in anoverall factor of 1� abA�mj �mk�=2. If we use the tree-level value of bA � 1, then the error in this factor is��sams=2, assuming a one-loop correction to bA of orderunity times �s. Taking ms 0:1 GeV, a�1 2 GeV, and�s 0:3, the error is less than 1%. This may be smallerthan other sources of error, in which case the tree-levelvalue for bA would suffice. On the other hand, if weconsider fD in the four flavor theory, then the correspond-ing error is much larger, �10%, and a more accuratedetermination of bA is likely needed.

The outline of this paper is as follows. In the next sectionwe recall previous work on the improvement of the un-quenched theory in the chiral limit. In Sec. III we describethe additional improvement coefficients that are needed inthe unquenched theory for nonvanishing quark masses. Wethen, in Sec. IV, lay out the Ward identities that can be usedto determine most of these coefficients. We end by discus-sing some implications of our results in Sec. V. Twoappendixes present the generalizations to two and fourflavors.

This paper is an expansion, clarification and, to someextent, a correction of Ref. [12]. In that work we arguedthat some of the improvement coefficients could be deter-mined by imposing vector Ward identities. It turns out thatthis was not correct in all cases, due to the presence ofcertain contact terms. We explain this point in a finalappendix.

A brief summary of the present work has been given inRef. [13].

II. REVIEW OF PREVIOUS WORK

We begin by reviewing previous work on nonperturba-tive O�a� improvement of unquenched QCD. The ALPHACollaboration [1] has shown how on-shell improvement ofthe action can be accomplished by adding theSheikholeslami-Wohlert or ‘‘clover’’ term, with appropri-ately chosen coefficient cSW. Their method requires aflavor nonsinglet axial current, and thus works if the num-ber of light quarks, Nf, is two or greater. Of course, theresulting value of cSW depends on Nf.

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4In practice, it may be advantageous not to extrapolate cSW

IMPROVED BILINEARS IN LATTICE QCD WITH . . . PHYSICAL REVIEW D 73, 034504 (2006)

We briefly review the method of Ref. [1], both forcompleteness and to introduce our notation. One considersmatrix elements of the improved axial current and pseudo-scalar density, which, in the chiral limit, take the form

A�jk�� ZAA�jk�;I� ;

A�jk�;I� � A�jk�� acA@�P�jk�; �j � k�;(1)

P �jk� � ZPP�jk�;I ; P�jk�;I � P�jk�; �j � k�: (2)

Here we introduce the notation that we use throughout thisarticle: a ‘‘hat’’ on an operator indicates that it is bothO�a�improved and properly normalized, while the superscript Iindicates improvement alone. The improvement here is onshell, not off shell. Flavor indices are shown as super-scripts, with j; k � 1� Nf. A� and P are ultralocal latticetranscriptions of the axial current and pseudoscalar density,respectively. The simplest choices are exemplified byA�jk�� x� � � j�x��5

k�x�, with a3=2 �x� being the barelattice fermion at site x, but our considerations hold for anyultralocal choices. Finally, @� is an O�a� improved latticederivative, e.g. the symmetric difference divided by 2a.Factors of the lattice spacing a are shown explicitlythroughout, so that all quantities have the same dimensionsas their continuum counterparts.

To determine cSW one enforces the simplest axial Wardidentity

h@�A�jk�� x�iJ � �mj � mk�hP

�jk��x�iJ �O�a2�: (3)

Here the mi are improved and normalized quark masses,whose relation to the bare quark masses is discussed below.The subscript on the expectation values indicates that thesematrix elements are to be evaluated in the presence ofsources, J, which are located at different positions fromthe operators. The sources should have quantum numberssuch that the result is nonvanishing, but are otherwisearbitrary (in both form and position). They could be bound-ary sources in the Schrodinger functional, or standardhadron operators in a traditional large volume calculation.We need not (and do not) specify them. What matters hereis that this equation should hold for any such sources(which thus create states with the appropriate quantumnumbers in all possible linear combinations). The left-and right-hand sides will only match as the sources arechanged (while holding the bare quark masses and cou-plings fixed so that the mi are fixed) if both cSW and cA arechosen correctly.3 Since the accuracy of matching isO�a2�,

3In practice, one can cancel the contributions proportional toone or the other of these constants by taking appropriate linearcombinations using different sources. This has been done in thedetermination of cSW in Refs. [2– 4], and of cA in Ref. [5]. Thisallows one to tune the sources to improve the sensitivity sepa-rately for each improvement constant.

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these constants can only be determined to a relative accu-racy of O�a�.

An important point concerning the implementation ofEq. (3) is that one does not need to know ZA, ZP or the mi.One need only require that the ratio of the matrix elementsof the improved, but not normalized, quantities, hA�jk�;I� iJand hP�jk�iJ, is the same for any choice of J. One can thenextrapolate the resulting values of cSW and cA to the chirallimit. This limit can be determined by setting all Nf barequark masses equal, and extrapolating to the commonvalue for which the right-hand side (RHS) of Eq. (3)vanishes. One then has in hand the desired improvementconstants, which will depend, for a given choice of gaugeaction, only upon the bare coupling constant g2

0.4

At the same time, one has determined the critical valueof the hopping parameter, �c�g2

0�, as this is the value forwhich the right-hand side of Eq. (3) vanishes given degen-erate quarks. One can then define bare quark masses in thestandard way:

amj �1

2�j�

1

2�c: (4)

These calculations also give information about ZA=ZP, butwe postpone discussion of this until we have set up thetools to work away from the chiral limit.

The method has been extended, in the chiral limit, toother bilinears. On-shell improvement requires the addi-tion of all dimension four operators with appropriate quan-tum numbers [1]5

S�jk�;I � S�jk�; (5)

V�jk�;I� � V�jk�� acV@�T�jk�� ; (6)

T�jk�;I�� T�jk�� acT@�V�jk�� @�V

�jk�� ; (7)

where j � k. Each of these operators also has an associatednormalization constant. The improvement constants cV andcT can be determined by enforcing appropriate axial Wardidentities in the chiral limit, making use of the previousdetermination of the improved axial current. In particular,enforcing that the axial variation of V� is proportional toA� determines cV —see Refs. [6–8] for particular meth-ods—while enforcing that T�� rotates into other compo-nents of itself determines cT [7,8].

and cA to the chiral limit [14]. There would then be residualO�am� contributions in these coefficients, but this does not effectO�a� improvement. We find it conceptually simpler, however, toimagine that these coefficients have been extrapolated to thechiral limit, so that they are independent of the quark masses thatwe will be varying in the subsequent discussion.

5Our convention for the tensor is T�jk�� x� � � j�x�i�� k�x�

with �� 12 i��; ��.

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Up to this point we have considered only flavor off-diagonal bilinears, i.e. those with flavor indices satisfyingj � k. For the subsequent analysis we will need to consideralso diagonal flavor nonsinglet operators (as well as flavorsinglets). A convenient common notation for all nonsingletbilinears is tr��O�, where the trace is over flavor indices,and � is one of the eight Gell-Mann matrices for Nf � 3(our primary focus), the Pauli matrices for Nf � 2, and theappropriate generalization for SU�4�. In the chiral limit,where the SU�Nf� flavor symmetry is unbroken, the im-provement of the off-diagonal nonsinglets described abovecarries over also to the diagonal nonsinglets, with the sameimprovement coefficients. Thus, for example, the generalimproved nonsinglet axial current is

tr ��A��I � tr��A�� acA@� tr��P�; (8)

for all choices of �.When one moves away from the chiral limit, many new

improvement constants are needed. Consider first the gluonaction. Possible higher dimension gluonic operators are ofdimension six, so the only contribution at O�a� is theoriginal action density multiplied by the trace of the quarkmass matrix,M. This leads to an effective gluonic couplingconstant [1]

g20 ! ~g2

0 � g20�1� abg trM=Nf�; (9)

where bg is a function of g20. If we work at the fixed bare

coupling, and vary trM, then the effective coupling willvary, as will the lattice spacing, and so dimensionful quan-tities will have O�a� contributions proportional to abg trM.To avoid these, g0 must, in principle, be varied as trMchanges in such a way that ~g0 is held constant. Thisrequires determining the improvement constant bg.Nonperturbative methods for doing so have been proposedin Ref. [8], and the one-loop perturbative result is given inRef. [15].

For most of the following discussion we assume that bgis known, and that whenever we vary M we do so with ~g0,and thus a, fixed. In fact, we will find that our methodprovides an alternative, independent determination of bg.Thus one does not need to rely on the methods ofRefs. [1,8].

The improvement of quark masses and bilinear opera-tors away from the chiral limit in an unquenched theory hasbeen discussed previously, but only in the case of degen-erate quarks [1,8]. We do not recall this work here, sincewe will generalize it in the following to quarks with non-degenerate masses.

6In Ref. [12] we erroneously concluded that improvement ofthe tensor bilinear required the inclusion of an additional gluonicoperator, which, however, vanishes identically.

III. ADDITIONAL IMPROVEMENTCOEFFICIENTS

In this section, we describe the new improvement andrenormalization coefficients that are required in order toimprove, at O�a�, flavor-singlet and nonsinglet bilinear

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operators for general values of quark masses. The reasonfor the inclusion of flavor-singlet bilinears will becomeclear later.

Consider first flavor-singlet operators in the chiral limit.Matrix elements of these operators have ‘‘quark-disconnected’’ contractions (in which the operator con-nects to external fields through gluons) in addition to theusual ‘‘quark-connected’’ contractions present also forflavor nonsinglet operators. It follows that the improve-ment coefficients cA, cV and cT will not be the same asthose for nonsinglet operators, and thus we denote them �cA,�cV and �dT , respectively. The differences begin at two loop,and thus are of O�g4

0� barring unforeseen cancellations.Furthermore, some of these operators can ‘‘mix’’ withpurely gluonic operators. Enumerating the availablegluonic operators, one finds the following forms for on-shell improved flavor-singlet bilinears6:

�trA��I � trA� � a �cA@� trP; (10)

�trV��I � trV� � a �cV@� trT��; (11)

�trT��I � trT� � a �cT@� trV� � @� trV��; (12)

�trS�I � a�3eS � trS� agS Tr�F��F��; (13)

�trP�I � trP� agP Tr�F�� ~F��: (14)

Here we use ‘‘tr’’ for the trace over flavor indices, and‘‘Tr’’ for that over color indices. For later convenience, it isimportant that the discretized form of Tr�F��F�� is ex-actly that combination of Wilson loops which appears inthe gauge action [see Eq. (37)], with an average over looppositions so as to be centered on the site where the bilinearis placed. For Tr�F�� ~F�� one can pick any local choice,e.g. that based on the cloverleaf.

The mixing of trS with the identity operator in Eq. (13)was overlooked in Ref. [12]. To improve trS, the coefficienteS would need to be determined to an accuracy of a4. Tocalculate hadronic matrix elements, however, one mustsubtract disconnected contributions anyway, and this com-pletely removes the eS term. Similarly, the eS contributioncan be canceled when implementing Ward identities bysubtracting disconnected contributions, as we discuss ex-plicitly below. In this way one can avoid the problemexcept when calculating the vacuum expectation value,i.e. the quark condensate. To obtain an improved versionof the condensate, one must use another method, as alsodiscussed below.

We now turn to improvement away from the chiral limit.We consider explicitly the case of three light dynamical

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flavors; the generalizations to two and four flavors aredescribed in the appendixes. We restrict the discussion toa diagonal bare mass matrix with positive entries, M �diag�m1; m2; m3�, thus avoiding possible phase structureassociated with spontaneous violation of CP [16].Treating M as a spurion transforming in the adjoint repre-sentation of the flavor SU�3� group, it is straightforward toenumerate the allowed improvement terms linear in quarkmasses. For nonsinglet operators we find the general form

dtr��O� � ZO�1� a �bO trM�tr��O�I � a12bO tr�f�;MgO�

� afO tr��M�trO�; (15)

where O is any of the five bilinears. A possible termproportional to tr��;M�O� is forbidden by CP invari-ance—only the anticommutator of � and M appears.Note that the operators appearing in the O�a� correctionson the RHS can be chosen to be improved or unimproved,the difference being of O�a2�. Here we have left themunimproved for the sake of brevity, but in some equationsbelow it is more convenient to use improved versions. Thischoice has no impact on ZO because the explicit factors ofthe quark mass do not allow mixing back with the leadingoperator O.

There is one subtlety that is overlooked in Eq. (15). Theflavor-diagonal scalar bilinears (i.e. those for which the �are diagonal matrices) can also mix with the identityoperator. For example, the operator S�jj� � S�kk� mixeswith the identity with a coefficient proportional to �mj �

mk�=a2. As for the flavor-singlet scalars, however, thismixing is removed in all but the vacuum expectation valueby subtracting disconnected contributions, and so we willkeep its contribution to S�jj�;I � S�kk�;I implicit in thefollowing.

Aside from this subtlety, there are, for each of the fivenonsinglet bilinears, three improvement coefficients bO,�bO and fO, in addition to the overall normalization in thechiral limit, ZO. It is useful to understand the dependenceof each of these quantities on the coupling constant andlattice spacing. In the chiral limit, the normalization de-pends both on the bare coupling and, if the correspondingbilinear has an anomalous dimension, on a� (with � therenormalization scale): ZO � ZO�g2

0; a��. As discussed inRef. [1], away from the chiral limit one must replace g0

with ~g0 of Eq. (9) so that, in general, ZO � ZO�~g20; a��.

The improvement coefficients do not, however, dependexplicitly on a�, and so are functions only of g2

0, orequivalently of only ~g2

0 at this order of improvement.To understand the significance of each of the improve-

ment coefficients, it is useful to consider special cases. Forflavor off-diagonal operators (which we will also refer to as‘‘charged’’ operators) fO drops out:

O �jk� � ZO1� a �bO trM� abOmjk�O�jk�;I ; (16)

where mjk � �mj �mk�=2. The fact that �bO multiplies the

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trace of M (and thus depends on all three quark masses)indicates that it arises from mass dependence of quarkloops, and thus begins at two-loop order in perturbationtheory, and is absent in the quenched approximation. ThebO term, by contrast, arises from the mass dependence ofthe valence quark propagators attached to the operator, andis present also in the quenched approximation. We havechosen the normalization so as to match that of the stan-dard form used in quenched applications of improvement:

O �jk�jQu � ZO1� abQOmjk�O

�jk�;I : (17)

Note that, although bO and bQO arise from the same under-lying effect, they will differ numerically (for a given choiceof ~g2

0) due to mass-independent contributions from quarkloops. Since these enter first at two-loop order, however,one-loop results for bO from Ref. [17] are valid also for bQO.

The fO term enters into the improvement of flavor-diagonal (or ‘‘neutral’’) operators, e.g.

O�jj� � O�kk� � ZO�1� a �bO trM��O�jj�;I �O�kk�;I�

� abO�mjO�jj� �mkO

�kk��

� afO�mj �mk�trO�: (18)

Here, on the left-hand side (LHS) we have made thereplacement

d�O�jj� �O�kk�� O�jj� � O�kk�; (19)

i.e. we have replaced the improved and normalized versionof the operator �O�jj� �O�kk�� with the difference of theimproved and normalized versions of the individual opera-tors. Similarly, in the first term on the right-hand side, wehave used

�O�jj� �O�kk��I � O�jj�;I �O�kk�;I: (20)

One might be concerned that there is a subtlety hidden inthese replacements, since the individual operators O�jj�

contain flavor-singlet parts and, as discussed further below,flavor-singlet and nonsinglet operators can have differentanomalous dimensions. In fact, the same issue arises in thecontinuum. We resolve it by simply defining the individualoperators as appropriate sums of the flavor-singlet andnonsinglet parts, e.g. with two flavors

O �11� � 12

d�O�11� �O�22�� d�O�11� �O�22��; (21)

and similarly for the improved operators. Then the rela-tions in Eqs. (19) and (20) are identities.

Returning to Eq. (18), it is clear that the fO term arisesfrom quark-disconnected contractions of the operator, be-cause it is only through such contractions that mixing withthe operator trO, which contains all flavors, can arise.From this we learn that fO appears first at two-loop order.We remark that fO is also present in the improvement ofquenched bilinears if one considers flavor-diagonal non-

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singlets and keeps disconnected contractions. As far as weknow, no such calculations have been done to date.

The presence of an extra improvement coefficient in thediagonal nonsinglet operators as compared to the off-diagonal nonsinglets leads to the following apparent para-dox. A nonsinglet vector rotation (�V j � k and �V � k �� j) transforms off-diagonal operators into diagonal ones:�VO�kj� � O�kk� �O�jj�. Thus the corresponding Wardidentity [given explicitly in Eq. (C8)] should be enforcedfor improved operators on the lattice up to O�a2� correc-

tions. Assuming that d�VS is known, this identity relates anoperator with two improvement coefficients ( �bO and bO) toan operator with one more improvement coefficient (fO).Thus it seems to imply that fO is not independent. Thiswould be paradoxical, because fO is allowed in the firstplace by the vector symmetry which is subsequently lead-ing to the constraint. This paradox is resolved by analyzingthe possible contact terms in Appendix C 3.

Previous discussions of improvement in the unquenchedtheory have considered only degenerate quarks [1]. Tomake contact with the notation used in these papers, wenote that the general result (15) reduces to

dtr��O�jUnQ;degen � ZO1� a�bO � Nf �bO�m�tr��O�I;

(22)

for degenerate quarks. For charged bilinears, this has thesame form as used in Ref. [1], except that what we callbO � Nf �bO was denoted simply bO in those works. Weprefer our notation because of its connection with thequenched improvement constants and because it is moreeasily generalizable to nondegenerate quarks. We note thatat one-loop order the difference in notation is immaterialbecause, as noted above, �bO vanishes at this order.

Next we consider the mass-dependent improvement co-efficients needed for flavor-singlet operators. We find thatthe general form consistent with flavor symmetry is

dtrO � ZOrO�1� a �dO trM�tr�O�I � adO tr�MO��: (23)

There are only two mass-dependent improvement coeffi-cients for each bilinear (as opposed to the three needed fornonsinglets), but there is an additional normalization fac-tor, rO, appearing in the chiral limit. The latter arisesbecause the normalizations of singlet and nonsinglet op-erators differ, since the former can have quark-disconnected contractions. Rather than introduce a newnormalization constant �ZO, we have parametrized thiseffect with the ratio rO � �ZO=ZO. For the axial current(O � A�) the anomalous dimension of the singlet andnonsinglet operators differs (the former beginning at two-loop order, while the latter vanishing to all orders), so thatrA must depend explicitly on lna in addition to the usualdependence on ~g2

0. For the four other bilinears the singletand nonsinglet anomalous dimensions are the same (theanomalous dimensions vanish for all vector currents, and

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the quark-disconnected loop diagrams for S, P and Tvanish by chirality in the continuum). Thus rS, rP, rVand rT do not depend explicitly on lna, and are functionsonly of the effective coupling constants, just like the im-provement coefficients. See Ref. [18] for a more thoroughdiscussion of this point.

We also need the expressions for O�a� improved quarkmasses in terms of the bare quark masses. Using flavorsymmetry and the constraint that the singlet and nonsingletmass combinations vanish at the same bare mass [18], wefind

dtr�M � Zm�1� a �bm trM�tr�M� abm tr��M2��; (24)

dtrM � Zmrm�1� a �dm trM�trM� adm tr�M2��: (25)

The notation is analogous to that used for the bilinears.Note that there is no separate ‘‘fO-like’’ term in (24), sincesuch a term, proportional to tr��M�tr�M�, can be absorbedinto the �bm term. This reduction in the number of improve-ment coefficients will play an important role in the nextsection, where we will see how these constants are relatedto those needed to improve the scalar bilinear. The overallconstant Zm is scale dependent, but all other constants,including rm [18], are not.

Since we restrict ourselves to a diagonal mass matrix,the result (24) is nontrivial only for diagonal �’s. Takingappropriate linear combinations with Eq. (25), one canobtain the result for individual masses:

m j � Mjj

� Zm

��mj � �rm � 1�

trMNf

�� a

�bmm

2j �

�bmmj trM

� �rmdm � bm�tr�M2�

Nf� �rm �dm � �bm�

�trM�2

Nf

��:

(26)

From the O�1� terms we see that mj vanishes if all baremasses vanish together. This is by construction. For non-degenerate positive quark masses, however, mj does notvanish when mj � 0. This is not an effect which vanisheslinearly in a, since it remains true even when O�a� termsare dropped, because of the contribution proportional to�rm � 1�trM. In particular, this effect implies that, at fixedgauge coupling, the pion becomes massless at a value ofthe bare up and down quark hopping parameters (assumeddegenerate) which depends (linearly) on the strange quarkmass. This is similar to the well-known result that thepartially quenched critical hopping parameter differsfrom the fully unquenched value (as discussed, e.g., inRef. [19]).

It is useful to consider which of these constants survivein the quenched approximation. The constants �bm and �dmdo not, since they are produced by quark loops, as shownby the fact that they multiply tr�M�. Similarly, the differ-

-6

TABLE I. Relations between renormalization and improve-ment coefficients for masses and scalar bilinears. LO and NLOindicate leading and next-to-leading order in quark masses.Results are valid for Nf � 3 and 4. They hold also for Nf � 2if bS and bm are set to zero.

Order in M Relationship or constraint

LO ZS � 1=Zm, rS � 1=rm, gS � bg=�2g20�

NLO bS � �2bm, �bS � � �bm, NffS � 2�bm � dm�,dS � bS � Nf �bS, dS � Nf �dS � �2�dm � Nf �dm�


ence of rm from unity, and the difference between dm andbm, arise from insertions of the mass on quark loops. Thusin the quenched approximation one only needs two con-stants: ZQm and bQm � dQm. One way of seeing this is to notethat mj can only depend on mj in the quenched approxi-mation, which forbids all but the first and third terms inEq. (26).

In summary, moving from degenerate to nondegeneratequarks and considering singlet as well as nonsinglet op-erators requires the introduction of a large number ofadditional improvement coefficients. All except forZSZP, ZT , Zm and rA are scale-independent functions ofthe effective coupling alone,7 and for these scale-independent quantities there is no apparent obstacle to theirdetermination using Ward identities. Indeed, as we show inthe next section, nearly all can be determined in this way.

IV. DETERMINING COEFFICIENTS USING WARDIDENTITIES

In this section we explain in detail how one can general-ize previous methods to determine most of the new im-provement and normalization coefficients. We organizeour approach into four steps. This is partly as an explan-atory aid, but also because some parts of the later steps relyon results from the earlier ones. There remains, however,considerable freedom in the ordering of parts of somesteps.

In summary, the four steps are the following. First, weenforce vector charge conservation, which determinesmost of the improvement coefficients, and the normaliza-tion constant, of the vector bilinear. It turns out that this isthe only use to which we can put vector Ward identities. InRef. [12] we had claimed otherwise, but this turns out to beincorrect because of overlooked contact terms. We explainthis point in Appendix C.

The second step is to relate the improvement and nor-malization coefficients for the quark masses to those forscalar bilinears. This allows us to use, in step three, thesimplest axial Ward identities such as Eq. (3) (‘‘two-pointWard identities’’), to determine combinations of coeffi-cients for axial, pseudoscalar and scalar bilinears.Finally, in the fourth step we enforce axial Ward identitiesin which the axial variation occurs in a region includingother operators (‘‘three-point Ward identities’’). Here wehave to deal with contact terms.

We discuss the Nf � 3 theory in this section, summa-rizing the results in Tables I and II. The former collects therelationships between constants, while the latter showswhich can be obtained from which Ward identities. Thegeneralizations to Nf � 4 and Nf � 2 are discussed, re-spectively, in Appendixes A and B.

7ZS=ZP is scale independent, while ZSZP is scale dependent.

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A. Vector Ward identities

We use the standard method of enforcing the correctnormalization of the vector charge,

hHjQj��jHi � Qj

HhH j Hi; Qj��

X~x

V�jj�4 � ~x; �;

(27)

where QjH is the jth quark number of hadron H, for a

convenient set of hadrons. In fact, one does not have toproject onto a single hadron—any linear combinationcreated by an operator with a given jth quark numberwill work. As we now show, this method can determineall improvement and renormalization coefficients for thevector current, with the exception of cV and �cV . These twoare excluded because the operators they multiply vanish atzero spatial momentum.

To see that all other coefficients can be determined weuse Eqs. (18) and (23) with O � V4 at zero spatial mo-mentum. Consider first degenerate quarks, so that

Q j � Qk� ZV1� a�3 �bV � bV�m��Qj �Qk�; (28)

dtrQ � ZVrV1� a�3 �dV � dV�m�trQ; (29)

where Qj �P

~xV�jj�4 is the bare charge operator. We have

dropped the superscript I since the cV and �cV terms do notcontribute. Enforcing the normalization of these twocharges for two or more values of the common quarkmass determines ZV , 3 �bV � bV , rV and 3 �dV � dV . Notethat for the singlet charge operator one must use baryonicstates, so that the total charge is nonzero. To obtain theremaining mass-dependent improvement coefficients non-degenerate quarks are required. It is sufficient, however, towork with a ‘‘2� 1’’ flavor theory, i.e. one in which the upand down quarks are degenerate (with mass m1) but have adifferent mass than the strange quark (m3). In this case, theimprovement term for the flavor nonsinglet charge Q1

�

Q3 is proportional to

af�2 �bV � bV�m1 � �bVm3�Q1

� 2 �bVm1 � � �bV � bV�m3�Q3 � fV�m1 �m3�trQg;

(30)

as can be seen from (18). Thus by varying m1 at fixed m3,

-7

TABLE II. Normalization and improvement coefficients determined using various Ward identities (which are denoted schematically)for Nf � 3. LO and NLO indicate leading and next-to-leading order in quark masses. For completeness, we indicate which Wardidentities determine cSW , cV , cA and cT . For the last nine Ward identities (those below the line), we assume that the on-shell improvedflavor nonsinglet axial variation of the action has been determined using the previous identities (i.e. those above the line). The notation‘‘Not new’’ in the final two lines indicates that these two identities are equivalent, in the chiral limit, to those considered previously.

Ward identity LO NLO

hHjP

~xV�jj�4 jHi � Qj

H ZV , rV bV , �bV , fV , dV , �dV

@�A�jk�� mj � mk�P

�jk� ZmZP=ZA, rm bA, fA, bP � 2bm, bm � 2rmdmand rP, gP �1� 2rm�

2bm � 6rm� �bm � �dm�@��A

�jj�� A�kk�� [cSW, cA] 2�2� rm�bm � 3� �bP � �bA � �bm�

� 2mjP�jj� � 2mkP

�kk� bP � 2rPdP, �2� rP�bP � 6fPrP�2� rP�� bP � �dP� � �1� 2rP � 3rmrP�fP

��ij�A T�jk� � T�ik� ZA, [cT] bT , 3 �bA � bA�rm � 1�

��ij�A V�jk� � A�ik� ZV , Z2A, [cV] bA � bV , 3 �bA � bA�rm � 1�

and V $ A 6� �bA � �bV� � �bA � bV��2� rm��ij�A P�jk� � S�ik� ZS=ZP, Z2

A bS � bP, 3 �bA � bA�rm � 1�and P$ S 6� �bP � �bS� � �bP � bS��2� rm�

��ij�A trA � ��ij�A trV � 0 �cA, �cV dA, dV��ij�A trP � 2S�ij� ZPrP=ZS, gP dP, 3� �bS � �dP� � bS�rm � 1�

��ij�A trT � 2T�ij� rT , �cT dT , 3� �bT � �dT� � bT�rm � 1�

��ij�A trS � 2P�ij� ZSrS=ZP, gS dS, 3� �bP � �dS� � bP�rm � 1�

��ij�A T�ji� � T�ii� � T�jj� rT , �cT , [cT] bT , dT , rT� �dT � �bT� � rmfT��ij�A S�ji� � P�ii� � P�jj� ZP=ZS, rP, rS bP � bS, 2rPdP � bP, 2rSdS � bSand S$ P gP, gS 6� �bP � �bS� � �2� rm��bP � bS�

3rP� �dP � �bS� � rP�dP � bS� � rm�3fP � bP � rPbS�3rS� �dS � �bP� � rS�dS � bP� � rm�3fS � bS � rSbP�

��ij�A A�ji� � V�jj� � V�ii� Not new �bV , bV , fV��ij�A V�ji� � A�jj� � A�ii� Not new �bA, bA, fA


and considering different hadrons so that the contributionsof the independent operators Q1, Q3 and trQ vary, onecan determine bV , �bV and fV separately.8 Similarly, theimprovement term for the singlet charge is proportional to

af�2 �dV � dV�m1�Q1 �Q2� � � �dV � dV�m3Q

3g; (31)

so that dV and �dV can be disentangled by varying m1.In practice, ZV and 3 �bV � bV have been computed for

two flavors using this method [20] (although recall that thelatter combination is referred to as bV in Ref. [20]).Determination of bV and �bV separately should be relativelystraightforward, since quark-disconnected contractions arenot required. By contrast, the determination of rV , �dV , dVand fV requires such contractions, and will thus be morechallenging in practice.9

8Determination of bV and �bV alone can be done using Q1 �Q2, for which the fV term vanishes in the 2� 1 theory.

9One can show that the quark-disconnected diagrams givevanishing contributions if one uses the charge built out of theconserved vector current on the lattice. We can see no argumentto extend this result to choices of the bare current which are notexactly conserved.

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Vector Ward identities cannot be used to determine anyof the other normalization or improvement coefficients,essentially because the vector symmetries are not brokenby the discretization. We discuss this further inAppendix C, because in some cases it is not immediatelyobvious why vector Ward identities cannot be used.Indeed, in Ref. [12], we argued that it was possible touse such identities, and it is instructive to see the flaw inour argument.

B. Relating improvement of mass and scalar bilinear

As is well known, the anomalous dimensions of thequark mass and scalar bilinear are equal in magnitudebut opposite in sign, and it is conventional and convenientto choose their renormalization constants to be the inverseof one another, ZSZm � 1. While not strictly necessary(the right-hand side could be a constant other than unity),this choice implies that derivatives with respect to quarkmass give rise to insertions of the (space-time integral) ofthe scalar bilinear both for bare and renormalized quanti-ties. In particular, it implies that the useful result

-8


@mH

@mj

��mk�j

�VhHjS�jj�jHihH j Hi

(32)

(where jHi is the state containing an arbitrary hadron ofmass mH at rest, V is the spatial volume, and k runs overthe flavors different from j) holds for both bare and renor-malized quantities.

In this section we apply the condition (32) as a constraintnot only in the continuum limit but also for nonvanishinglattice spacing. The argument for doing so is straightfor-ward: the quantities appearing in the relation—hHjS�jj�jHi, mH and mj—are physical quantities thatshould be improved with appropriate choices of the im-provement coefficients introduced above. Thus if the rela-tion (32) holds in the continuum limit it should receive noO�a� corrections once improvement has been imple-mented. Perhaps surprisingly, this simple relation leads toa number of nontrivial constraints.

The precise relation we enforce is10

@mH

@mj

��mk�j;a�VhHjS�jj�jHihH j Hi

: (33)

Here we have been specific about the meaning of the partialderivative in the presence of the regulator. It should betaken with the regulator—here the lattice spacing a—fixed, so that the relation survives in the limit that theregulator is removed. In order to match the formal result(32) the derivative should also be taken with the otherimproved (rather than bare) masses held fixed. Finally,note that the matrix element of S is, as usual, the connectedmatrix element, so that the part of S proportional to theidentity [the eS term in Eq. (13)] does not contribute.

Despite the fact that it involves the a priori unknownquantities hHjS�jj�jHi, this relation is useful because we doknow how hadron masses depend on bare parameters. Inparticular, because the fermion action depends on the barequark masses only as

Slat;F �Xx

Xj

�amj��a3S�jj�x�� (34)

(even after improvement by the addition of the clover term)one can show that [21]

@�amH�

@�amj�

��am�k�j;g20

�L3hHj�a3S�jj��jHi

hH j Hi�VhHjS�jj�jHihH j Hi

:

(35)

In the derivative on the LHS the bare coupling and theother bare quark masses in lattice units are held fixed. Asmentioned above, it is the connected matrix element of thescalar density which appears on the RHS. Similarly, one

10We stress that all the lattice quantities appearing in thisrelation are defined to have the same dimension as their con-tinuum counterparts, e.g. amH is the hadron mass in lattice units.

034504

finds

�2g40

@�amH�

@g20

��amj

�aVhHjTr�F��F��jHi

hH j Hi; (36)

where we define the discretized field strength from thespecific form of the gluon action being used:

Slat;G � a4Xx

1

2g20

Tr�F��F��x�: (37)

In this way, our expressions hold for any choice of gluonaction.

To relate the desired derivative in Eq. (33) to those whichwe know, (35) and (36), we proceed in two stages. First, werelate derivatives with respect to improved masses to thosewith respect to bare masses, using the properties ofJacobians:

@mH

@mj

��mk;ml;a�@�mH; mk; ml�

@�mj; mk; ml�

�@�mH; mk; ml�=@�mj;mk;ml�

@�mj; mk; ml�=@�mj;mk;ml�; (38)

where k and l are the two flavor indices not equal to j. Hereall derivatives are at fixed a, or equivalently at fixed ~g2

0.This means that the derivatives such as @mk=@mj can bestraightforwardly evaluated using Eq. (26). The renormal-ization constants Zm and rm are functions of ~g2

0 and a, andthus are fixed. The same is true, to the order in a that we areworking, of the improvement constants bm etc., since theycan equally well be considered functions of g2

0 or ~g20.

Derivatives such as @mH=@mj cannot yet be evaluatedsince they are taken at fixed a. Using Eq. (9), one can relatethem to the derivatives we know from Eqs. (35) and (36):

@mH

@mj

��mk�j;a�@�amH�

@�amj�

��am�k�j;a

�@�amH�

@�amj�

��am�k�j;g20

�g2

0bgNf

@�amH�

@g20

��amj

:

(39)

Putting things together we obtain an expression for theright-hand side of Eq. (33) in terms of matrix elements ofthe bare scalar density and the gluon field strength, and theimprovement and normalization constants for quarkmasses. The left-hand side of Eq. (33) can be expanded,using Eqs. (18) and (23), in terms of the same matrixelements, but with the coefficients being the improvementand normalization constants for the scalar bilinear.Matching the coefficients on the two sides gives, aftertedious algebra, the relations quoted in Table I.

We do not go through the details of this calculation in thegeneral case, but do display the subset of the argument thatuses only degenerate quarks. In this case, Eqs. (38) and(39) simplify to

-9


@mH

@m

��a�@�amH�=@�am�ja

@m=@mja(40)

�@�amH�=@�am�jg2

0� g2

0bg@�amH�=@�g20�jam

@m=@mja(41)

�VhHjtrS� a�bg=2g2

0�Tr�F��F��jHi

Zmrm1� 2a�3 �dm � dm�m�; (42)

where in the last step we have assumed hH j Hi � 1. Thisshould be equated with the improved matrix element

VhHjtrSjHi � VhHjZSrSf1� a�3 �dS � dS�m�trS

� agS Tr�F��F��gjHi; (43)

where, as noted above, the part of ctrS proportional to theidentity operator does not contribute. We conclude thatZSrS � 1=�Zmrm�, 3 �dS � dS � �2�3 �dm � dm� and bg �2g2

0gS, which are a subset of the results in Table I.We find that the quenched relations ZmZS � 1 and bS �

�2bm (Ref. [17]) continue to hold (although the constantsthemselves will differ from their quenched values), andthat there are generalizations for some of the new constantsthat appear ( �bS � � �bm, etc.). There are two particularlyinteresting, and perhaps unexpected, results. First, there isa constraint on the scalar improvement coefficients (dS �bS � 3 �bS). This arises because, as noted above, there is oneless improvement constant needed for quark masses thanfor the scalar bilinear (i.e. there is no independent fmterm). Second, the relation bg � 2g2

0gS provides anotherway of determining bg, if we determine gS using Wardidentities as discussed below.

Given the relations of Table I, it is interesting to deter-mine if there are any products of masses times scalardensities which maintain their form under improvement.This does not hold for the contribution of mass terms to theaction itself, i.e. X

j

mjS�jj� �

Xj

mjS�jj�: (44)

The corrections to this equation in the general case ofnondegenerate quarks are lengthy and uninformative, sowe quote the result only for the case of degenerate quarks

m trSjmi�mj�mk� m trS� a�3 �dm � dm�m2 trS

� agSmTr�F��F��: (45)

Here we have dropped the term containing the identityoperator since it does not contribute to matrix elements.This degenerate quark result illustrates the general pointthat there is, atO�a�, mixing with other terms in the action,and thus no reason to expect that each term in the actionshould be separately form invariant under improvement.The one example of form invariance is for the variation ofthe action under vector transformations:

034504

�mj � mk�S�jk� � �mj �mk�S�jk� �O�a2�: (46)

This follows from the definitions (15) and (24) and therelations ZSZm � rSrm � 1, bS � �2bm and �bS � � �bm.It holds for Nf � 2� 4 (and we suspect for all higher Nfas well). This relation plays an important role in the dis-cussion of vector Ward identities in Appendix C.

C. Two-point axial Ward identities

In this section we investigate which of the improvementand normalization coefficients can be determined usingtwo-point Ward identities such as Eq. (3). To obtain asmuch information as possible we need to vary the quarkmasses independently (as done in the quenched theory inRef. [9]) and consider the partially conserved axial currentrelation for both charged and neutral currents.

We assume that cA has been determined, so that weknow A�jk�;I. We can then calculate the Ward identity mass,

~m jk �h@�A

�jk�;I� �x�iJ

2hP�jk��x�iJ�j � k�: (47)

Because we have improved the action and the axial current,~mjk should be independent of x and of the source J up tocorrections of O�a2�, and thus we do not give it any argu-ments, nor specify the source. We imagine choosing asource with a good signal, varying the quark masses (keep-ing ~g2

0 fixed, as always) and studying the bare quark massdependence of ~mjk. Using the Ward identity (3), we have

~m jk �1

2�mj � mk�

ZP�1� abP trM� abPmjk�

ZA�1� abA trM� abAmjk�(48)

�ZPZmZA

�mjk �

trM3�rm � 1� � a�Am2

jk �Bmjk trM

� C tr�M�2 �D tr�M2��

�; (49)

A � bP � bA � 2bm; (50)

B � �bP � �bA � �bm � 2bm �rm � 1

3�bP � bA�; (51)

C �rm � 1

3� �bP � �bA� �

rm �dm � �bm3

�bm2; (52)

D �2rmdm � bm

6: (53)

From the terms linear in bare quark masses one can extractZmZP=ZA (as in the quenched case) and rm (absent inthe quenched case). For example, the derivative of ~mjk

with respect to m at M � 0 for degenerate quarksis rmZmZP=ZA, while the derivative with respect to ml,l � j; k alone is �rm � 1�ZmZP=�3ZA� (a quantity which

-10


vanishes in the quenched theory). Using the relations inTable I one has thus determined ZP=�ZAZS� and rS �1=rm.

From the quadratic terms one can determine the coeffi-cient of each of the four linearly independent functions ofmasses that appear. Thus we obtain four linear combina-tions of the eight constants bA, �bA, bP, �bP, bm, �bm, dm and�dm (given that we know rm from above). By comparison, inthe quenched approximation, where there are only threeconstants, bQA , bQP and bQm � dQm, one can determine the twocombinations bQA � b

QP and bQm [9].

To obtain further information, we generalize the methodby considering flavor-diagonal two-point Ward identities,e.g.

h@��A�11�� x� � A

�22�� x�iJ � h2m1P

�11��x� � 2m2P�22��x�iJ

�O�a2�: (54)

This introduces several of the new constants present in theunquenched theory: fA on the left-hand side, and fP, dPand �dP on the right-hand side (which contains the flavor-singlet P). This corresponds to the fact that, for m1 � m2,there are contractions in which the source J and the axialcurrent (or pseudoscalar density) are not connected byquark propagators. In fact, it is only for m1 � m2 thatthis identity gives new information: for degenerate quarks,and with appropriate sources, the contractions are exactlythe same as those for the flavor off-diagonal identity (3).

To enforce the Ward identity, we have to adjust theconstants so that the LHS and RHS are equal for all choicesof x and sources J, up to O�a2�. We cannot simply taketheir ratio, as we did for the off-diagonal Ward identity,since more than one operator contributes on both sides ofthe equation, with relative strengths that we do not knowa priori. To proceed, it is useful to expand out the operatorappearing in the left-hand side of (54):

LHS � ZA

�1��bA � 3bA�

3trM�

bA6

tr��08M��

�f@� tr��3A��I � aE tr��3M�g; (55)

6E � 2�bA � 3fA�@� trA� � bA@� tr��08A��: (56)

For convenience we have defined �08 ��3p�8. We can now

divide the Ward identity (54) by the overall function of themasses multiplying @� tr��3A��I, so that the operator onthe left-hand side becomes

LHS 0 �LHS

ZA1��bA�3bA�

3 trM� bA6 tr��08M��

� @� tr��3A��I � aE tr��3M�: (57)

Since we have previously determined the improvementcoefficients in the leading order operator in this equation,@� tr��3A��I, the coefficients of all other, independent,

034504

operators appearing on both sides of the rescaled Wardidentity can be determined. In particular, we immediatelysee that bA � 3fA and bA can be determined since theymultiply independent operators in E.

We now divide the right-hand side by the same factor,and split the operator which results in terms linear andquadratic in quark masses:

RHS 0 �RHS

ZA1��bA�3bA�

3 trM� bA6 tr��08M��

� RHS0I � RHS0II: (58)

We find

RHS 0I �ZmZP3ZA

ftr��3P�2rm trM� tr��08M�� tr��3M�

� 2rP trP� tr��08P� � 2arPgP Tr�F�� ~F��g:

(59)

We can determine the coefficients of each independentfunction of masses (of which there are three at this order)multiplying each independent operator. Thus we can de-termine ZmZP=ZA and rm again, as well as rP and gP forthe first time.

The quadratic terms are more complicated. There are sixindependent functions of the masses, each potentially mul-tiplying three independent operators, although the 1$ 2antisymmetry allows only eight independent products. Wefind

RHS0II � aZmZP9ZA

fF tr�M�2�G tr��3M�2

�H tr�M�tr��08M��I tr�M2��tr��3P�

� J tr��3M�tr��08M��K tr��3M�trM�trP

�L tr��3M�tr��08M��M tr��3M�trM�tr��

08P�g;

(60)

F � �2rm � 1��bP � bA� � 6rm� �bP � �bA � �dm�

� bm �A� 6B� 18C; (61)

G � 32�bA � 2bm � 2rPdP�; (62)

H � 3� �bP � �bm � �bA� � �rm � 1��bP � bA� � 2bm

� 2A� 3B; (63)

I � 3�bP � bA � 2rmdm � bm� � 3A� 18D; (64)

J � 2bP � 6fP � 2rPbm � rPbA; (65)

K � 6rP� �bm � �dP � �bA� � 2rm�bP � 3fP�

� 2rP�2bm � dP � bA�; (66)

-11

11Our convention for c0O differs from that defined in Ref. [7].


L � bm � rPdP � bA=2 � G=3� bA=2; (67)

M � 3� �bP � �bm � �bA� � �bP � bA � 2bm� � rmbP

�H � rmbA: (68)

Thus one can determine, in principle, the eight combina-tions F �M. As indicated, however, only three of theseare independent of the combinations A�D that one canobtain using the flavor off-diagonal two-point Ward iden-tities. Thus one can only determine seven combinations ofthe ten improvement constants which enter (we exclude bAand fA since we have determined these from the left-handside of the present identity). A particular choice of theseseven combinations is listed in Table II. To further disen-tangle the coefficients requires three-point Ward identities,which we consider below.

As noted above, implementation of the flavor-diagonalWard identity necessarily involves quark-disconnectedcontractions and thus will be numerically challenging.Thus it is interesting to know how many combinations ofquark masses are needed. Are simulations in a 2� 1 flavortheory sufficient, or do all three quarks need to be degen-erate? It turns out that a combination of simulations withdegenerate masses (taking three or more values) and 2� 1simulations (with at least two values of the light quarkmass differing from each of two choices for the strangequark mass) is sufficient. This allows one to disentangle allthe different linear and quadratic mass dependences thatappear.

D. Three-point axial Ward identities

We now turn to the enforcement of the chiral trans-formation properties of bilinear operators. The methodol-ogy is standard [1,6–8,22]; what we add here is thegeneralization to nondegenerate masses in the unquenchedtheory, and the use of identities involving flavor-singletcomponents. Contact terms limit what can be extractedwith this method, and will need to be understood in detail.

We begin with the simplest example, which is thatconsidered in previous calculations. A nonsinglet axialtransformation with flavor �ij� is performed in a regionof space-time V which includes a bilinear O�jk� with k �

i. If O�jk� � j� k, then it is transformed into �O�ik� � i�5� k. The identity we enforce on the lattice is thus

h��ij�A S�O�jk��y�J�ki��z�i � hd�O�ik��y�J�ki��z�i �O�a2�;

(69)

where �AS is the improved and normalized lattice form ofthe formal variation of the continuum action

��ij�A S� � a4XV

�mi � mj�P�ij� � @�A

�ij�� ; (70)

with V a 4-dimensional subset of the lattice containing ybut not the source at z. (We do not place a hat on �AS so as

034504

to avoid overloading the notation.) Note that only quark-connected contractions contribute to this Ward identity.

Actually, as is well known, the identity (69) cannot besatisfied simply by on-shell improvement, since the pseu-doscalar density appearing in ��ij�A S� comes into contactwith O�jk�. Additional off-shell improvement terms areneeded, having the same quantum numbers as the productP�ij�O�jk�, and having the appropriate dimension. Sincethere is an explicit factor of mi � mj, the only such termwith the right dimension and symmetries is �O�ik�. Inprevious work, we have used a mnemonic for obtainingthis contact term, namely, that we can off shell improve thebilinears by introducing an additional ‘‘equations-of-motion’’ operator [7]:

a j�� 6D� �mj�� k � a j�� ~6D�mk� k: (71)

While adequate for discussing the Ward identities (69), thisis potentially misleading for two reasons. First, the formimplies that contact terms between two operators can befactorized into the contribution of one operator times thatof the other. In fact, there is no such factorization of contactterms. Second, for the operators we consider below, thereare several possible contact terms and the mnemonic can-not be easily generalized. Thus we do not use this mne-monic further in this paper. In fact, all we need to know isthat there exist possible contact terms involving the opera-tor �O�ik� and multiplied by mi � mj / ~mij �O�a�.

A convenient way of using Eq. (69) is to take the ratio ofthe two sides having pulled out unknown mass factors. Wefirst define the improved (but not normalized) variation ofthe action by

��ij�;IA S� ��ij�A S�

ZA�1� a �bA trM� abAmij�

� a4XV

2 ~mijP�ij�;I � @�A

�ij�;I� �: (72)

This can be determined by calculating ~mij using Eq. (49),since by assumption we know AI�. The ratio we consider isthen

RO �h��ij�;IA S�O�jk�;I�y�J�ki��z�i

h�O�ik�;I�y�J�ki��z�i(73)

�Z�OZAZO

1� a� �b�O � �bO � �bA�trM

� a�b�Omik � bOmjk � bAmij�� ac0O ~mij �O�a2�:

(74)

Here the contact term is included with an unknown coef-ficient c0O.11 Requiring RO to be independent of y and J in

-12


the chiral limit determines cV and cT , and we assume thishas been done, so that we know all the O�ij�;I. One alsoobtains information on the ZO, as first noted in Ref. [22].Away from the chiral limit RO should be automaticallyindependent of y and J up to O�a2�, since there are noadditional operators with coefficients to tune. Note that thecontact term plays no role in this regard. It has the sameoperator �O present in both numerator and denominator,and so is independent of y and J by itself.

Thus, for convenient choices of y and J, evaluating ROaway from the chiral limit allows one to determine onecombination of improvement coefficients for each of thethree independent linear functions of the quark masses,except that one of these is not useful as it is ‘‘contami-nated’’ by the contact term. One complication compared tothe quenched case is that the mass dependence of thecontact term, ~mij, is not simply proportional to mij, buthas an additional part proportional to �rm � 1�trM, as canbe seen from Eq. (49). Because of this, a convenient choiceof basis is ~mij, mi �mj and trM, using which we find

RO �Z�OZAZO

f1� a �b�O � �bO � �bA

� �b�O � bO��2� rm�=6� bA�rm � 1�=3�trM

� a�b�O � bO��mi �mj�=4g

� a�c0O � �b�O � bO � 2bA�

Z�O2ZOZPZm

�~mij

�O�a2�: (75)

The combinations of the improvement coefficients multi-plying trM and mi �mj in this equation can, in principle,be determined. To do so it is sufficient to use a 2� 1 flavortheory: the coefficient of trM can be determined usingmi � mj � mk, and that of mi �mj can then be deter-mined using mi � mj � mk. To simplify the discussionbelow, we note that a quick way of determining the acces-sible combinations of improvement coefficients is to set~mij � 0, so that one can make the substitution mjk �

�trM�rm � 1�=3. The resulting coefficients of mi �mj

and trM are those that can be determined.12

Applying this method to the bilinears in turn, we obtainthe results given in Table II. These allow the determinationof four new quantities: ZA, ZS=ZP, bT and �bA (where forthe latter we have used knowledge of bA and rm from thetwo-point Ward identities). The combinations of S and Pimprovement coefficients that are obtained, however, areall related to those obtained from the two-point Ward

12Note that this is a theoretical device and not a practical tool.Setting ~mij � 0 and considering m1 �m2 � 0 implies that somequark masses are negative. This is undesirable in practice due tothe possible phase structure at m� 0. In practice one wouldlikely need to do a fit using positive quark masses in order toseparate the different mass dependencies.

034504

identities using the relations listed in Table I. Thus weobtain a check of these relations, but no new informationon the constants themselves.

To determine further improvement coefficients we con-sider axial Ward identities involving the transformationof flavor-diagonal bilinears. These have not been consid-ered previously, and, in particular, they are not needed inthe quenched approximation. They involve quark-disconnected contractions in an essential way. The analysisis simplified by the observation that the Ward identitiesconsidered above allow the complete determination of A�ij��

and thus of the improved variation of the action, ��jk�A S�,including its normalization. Thus the only unknown coef-ficients appearing in the three-point axial Ward identitieswe consider below are those in the operator O and its axialvariation �O. We will also make use of the previousdetermination of rm.

We first consider the axial rotation properties of theflavor-singlet operators. The singlet axial current shouldbe invariant under nonsinglet axial transformations, so weenforce

h��12�A S�ctrA��y�J�21��z�i � O�a2�; j � k�: (76)

We choose flavor indices 1 and 2 so that we can use thestandard Gell-Mann basis of SU�3� matrices—permuta-tions of indices are, of course, allowed. Since the right-hand side of Eq. (76) vanishes (at the order we are work-ing), this relation can only determine relative normaliza-tions between independent operators appearing on the left-hand side. Using Eqs. (10) and (25), one finds

ctrA� / trA� � a �cA@� trP� adA tr�MA��

/ trA� � a �cA@� trP� a�dA=6�3 tr��3M�tr��3A��

� tr��08M�tr��08A��: (77)

In the second line we have used

tr �MO� � 13 tr�M�tr�O� � 1

2 tr��3M�tr��3O�

� 16 tr��08M�tr��

08O�; (78)

valid for diagonal mass matrices, to express the result interms of independent operators, and absorbed the contri-bution proportional to trM trA� into the overall constant. Inthe chiral limit only the first two terms in (77) are presentand so �cA can be determined. Away from the chiral limit,we must avoid contact terms, which are proportional to~m12. Following the discussion above, we do so by setting~m12 � 0, leading to tr��08M� � �2rm trM, so that Eq. (77)becomes

ctrA�j ~m12�0 / trA� � a �cA@� trP� a�dA=6�

� 3 tr��3M�tr��3A�� 2rm trM tr��08A��:

(79)

-13

13Using the relations in Table I, �bP � �dS � � �bP � �dP� � � �dP ��bm� � 2� �dm � �bm� � 2dm=3� 2bm=3, and all the coefficientsappearing on the RHS are known.

14Note that all three operators in ctrS, i.e. trS, Tr�F��F��, andtr�MS�, lead to contributions to hctrS�y�i which are separatelydivergent, but each can be combined with its subtraction termand then multiplied by the unknown coefficient to bedetermined.


Thus dA can be determined by tuning the coefficients ofeither tr��3A�� or (assuming rm is known) tr��08A��. Bothrequire nondegenerate quarks [degenerate quarks are notsufficient since the constraint ~m12 � 0 then impliestr��3M� � trM � 0], but 2� 1 flavors suffice [m1 �m2 � m3 to determine the coefficient of tr��08A��, andm1 � m2 � m3 for that of tr��3A��]. The net result is thefirst determination of both �cA and dA.

An almost identical discussion holds for the flavor-singlet vector bilinear, the conclusion from which is thatone can determine �cV (for the first time) and dV (whichchecks the determination from the vector charge).

For the other three bilinears the Ward identities aredifferent, since the singlet bilinears are not invariant:

h��jk�A S�ctrP�y�J�kj��z�i � 2hS�jk��y�J�kj��z�i �O�a2�;

(80)

h��jk�A S�ctrT��y�J�kj��z�i � ��hT�jk�� y�J�kj��z�i

�O�a2�; (81)

h��jk�A S�ctrS�y�J�kj��z�i � 2hP�jk��y�J�kj��z�i �O�a2�:

(82)

The general strategy to enforce these relations is to take theratio of the two sides, and require that the result is unityindependent of y and J. This should also be true indepen-dent of quark masses, as long as one avoids contact termsby keeping ~mjk � 0.

Consider first the transformation of the singlet pseudo-scalar, (80), and set j � 1, k � 2 for convenience. If wework at ~m12 � 0 to avoid contact terms, we find

ctrP � ZPrP1� a� �dP � dP=3�trM�

� ftrP� agP Tr�F�� ~F��

� adPtr��3M�tr��3P�=2� arm trMtr��08P�=3�g

�O�a2�: (83)

This implies that we can determine gP (in the chiral limit)and dP (in two independent ways as for dA above, bothrequiring only 2� 1 flavor simulation). Bringing the over-all factor to the RHS of (80) we obtain (again, with ~m12 �0)

S�12�

ZPrP1� a trM� �dP � dP=3��

�ZSZPrP

1� a trM� �bS � �dP � �rm � 1�bS=3

� dP=3��S�12�: (84)

Thus we can determine ZS=�ZPrP� (which serves as acheck) and the combination multiplying trM (the latteragain requiring only 2� 1 flavors).

034504

The Ward identity just discussed gives the first determi-nation of dP. This then allows the linear combinations ofconstants for the masses and pseudoscalar bilinears deter-mined previously (and given in the second section ofTable II) to be simplified. In particular, we can now sepa-rately determine bP, fP, bm and dm as well as dP. Thisleaves three combinations that cannot yet be disentangled:�bm � �dP, �bP � �dP, and �bm � �dm. Note that, at this stage,the combination multiplying trM in Eq. (84) does notdetermine any further coefficients since �bS � �dP �� bm � �dP�.

The same analysis goes through for the tensor bilinear,leading to the first determination of rT , �cT , dT and �bT ��dT � �rm � 1�bT=3. Since we know bT from previousWard identities, we can extract �bT � �dT .

The analysis for the scalar bilinear is more subtle, due to

the presence of the identity operator in ctrS. The net result,however, is as if this identity component was absent, andwe find that one can determine the same list as for thepseudoscalar after permuting P$ S: gS, dS, ZP=�ZSrS�and (using the previously determined bS) �bP � �dS. Ofthese, only dS is new.13 Using the relation dS �bS � 3 �bS from Table I, and the previously determinedbS � �2bm, we can extract �bS � � �bm. This allows oneto disentangle the remaining linear combinations of M andP improvement coefficients, so that we can determine �bm,�dm, �bP and �dP separately.

We now return to the identity operator contribution onthe LHS of Eq. (82). Formally, this contribution vanishessince it is invariant under axial transformations. However,this fails on the lattice because the variation of the identityoperator does not vanish fast enough to overcome the 1=a3

divergence in its coefficient:

h��jk�A S�J�kj��z�i � O�a2�: (85)

To overcome this one can explicitly subtract the discon-nected contribution

hctrS�y�i � h��jk�A S�J�kj��z�i (86)

from the LHS of Eq. (82). This is equivalent to enforcingthe difference of two continuum Ward identities, withcoefficients chosen so as to completely remove the contri-bution from the eS term, including its mass dependence.The cancellation between the two terms is between con-tributions proportional to 1=a3 leaving a residue that mustbe accurate to O�a2�.14 Thus it will require good statistical

-14


control. On the other hand, the dominant contribution tothe two terms will be correlated (since it involves theidentity operator), which will help the cancellation.15

In addition to allowing the separation of all the coeffi-cients for M and P, this Ward identity gives a new methodfor calculating bg, using the relation bg � 2g2

0gS and thedetermination of gS. Note that this determination can becarried out in the chiral limit, and so one does not need toknow bg a priori. This makes the determination of im-provement coefficients using the Ward identities discussedhere self-contained.

15This situation is similar to the subtraction of power divergentmixing in weak matrix element calculations in the quenchedapproximation, which also benefits from correlations betweenthe two quantities being subtracted, and has been successfullycarried out in practice [23–25].

034504

The final Ward identity giving new information is

h��jk�A S�T�kj�� y�J�z�i � �12��hT

�jj�� y� � T�kk�� y��J�z�i

�O�a2�: (87)

In particular, the RHS contains the diagonal nonsinglet,and thus provides access to fT for the first time. To seewhat we learn from enforcing this identity, we dividethrough by

ZT1� a trM �bT � amjkbT� !~mjk�0

ZT

�1� a trM

��bT �

rm � 1

3bT

��: (88)

Then we know the quantities on the LHS of the Ward identity. The operator on the RHS becomes (again setting j � 1 andk � 2, and still working at ~m12 � 0)

T�11�� T

�22��

ZT1� a trM� �bT �rm�1

3 bT��

�2rT3�N a trM

��trT��I �

�1

3� Pa trM

�tr��08T��

I �Qa tr��3M�tr��3T��; (89)

where

N �2

3

�rT

��dT � �bT �

dT3�rm � 1

3bT

�

� rm

�fT �

bT3

��; (90)

P � 29rm�bT � rTdT�; (91)

Q � 16�bT � 2rTdT�: (92)

Thus in the chiral limit we can determine rT , �cT and cT ,which provide cross-checks, while away from the chirallimit we obtain the three combinations N , P and Q (allobtainable with 2� 1 flavors). These in turn can be com-bined to give bT and dT separately, as well as the quantityrT� �dT � �bT� � rmfT . Given the determination of �dT � �bTfrom the Ward identity (81) above, we can extract fT . Wecannot, however, see any way of disentangling �dT and �bT .

Although we have avoided contact terms by working at~m12 � 0, we note that these terms are more complicatedhere than in the previous Ward identity. The contact termsarise from the fact that, even if one has on shell improvedthe operator P�jk� appearing in ��jk�A S� and the tensor bi-linear T�kj�� , the product of these operators at the sameposition will not be improved. To improve this productone needs to add all operators with the same transformationproperties as the product, and having appropriate dimen-

sion (here dimension 4 because of the overall factor ofmj � mk). In the Ward identity considered above, in whichthe bilinear on the RHS was flavor off diagonal, only asingle operator could appear, namely �O with appropriateflavor indices. By contrast, in the present identity, twooperators are allowed, namely, those appearing on theRHS,

a ~mjk��T�jj�� y� � �T

�kk�� y�� and a ~mjk tr��T��y�:

(93)

The latter arises from Wick contractions in which the quarkand antiquark in P�jk� are both contracted with the corre-sponding antiquark and quark in T�kj�. To understand (93)in terms of operators vanishing by the equations of motionrequires a generalization of the prescription given inRef. [7]. The appearance of a second operator plays animportant role in the discussion of vector Ward identities inAppendix C.

The remaining Ward identities do not provide any newinformation on the improvement coefficients, but do pro-vide several important cross-checks. Consider first

h��jk�A S�S�kj��y�J�z�i � hP�jj��y� � P�kk��y��J�z�i

�O�a2�: (94)

Dividing both sides by

ZS

�1� a trM

��bS �

rm � 1

3bS

��; (95)

the LHS is then a known quantity, while the RHS becomes(picking j � 1, k � 2 and setting ~m12 � 0)

-15


P�11� � P�22�

ZS1� a trM� �bS �rm�1

3 bS��ZPZS

��2rP3�N 0a trM

��trP�I �

�1

3� P 0a trM

�tr��08P�

I �Q0a tr��3M�tr��3P��; (96)

16This corrects the conclusion of Ref. [12] that there were threeundetermined scale-independent quantities.

where

N 0 �2

3

�rP

��dP � �bS �

dP3�rm � 1

3bS

�

� rm

�fP �

bP3

��; (97)

P 0 � 193�

�bP � �bS� � bP � bS � rm�bP � bS � 2rPdP��;

(98)

Q 0 � 16�bP � 2rPdP�: (99)

A similar analysis holds for the Ward identity with S and Pinterchanged, except that one must subtract the discon-nected component from both sides of the equation.Combining these two Ward identities allows the determi-nation of the coefficients listed in Table II.

In fact, the disconnected component on the LHS of theWard identity (94) with S and P interchanged gives accessto the quark condensate:

1

ZPh��jk�A S�P�kj��y�i �

1

ZPhS�jj��y� � S�kk��y�i �O�a2�

~mjk � 0: (100)

Here the idea is to calculate the LHS, and use it as thedefinition of the RHS. We know all the improvement andrenormalization constants appearing on the LHS, and, byworking at ~mjk � 0 we avoid contact terms. In this way wecan calculate the improved condensate, including its de-pendence on ml. This is the O�a� improved version of themethod first suggested in Ref. [22].

The final two Ward identities of this type involve vectorand axial currents, e.g.

h��jk�A S�A�kj�� y�J�z�i � hV�kk�� y� � V

�jj�� y��J�z�i

�O�a2�: (101)

Now only flavor nonsinglet operators appear, so we cannotobtain information about singlet improvement coefficients.In fact, this identity and that with V $ A only differ fromthe previous nonsinglet axial Ward identities, Eq. (69),when mj � mk. In this case there are additional quark-disconnected contractions absent in (69). Since we areassuming that we know the improved off-diagonal axialcurrent, the LHS of this relation is completely known,allowing a complete determination of the constants appear-ing on the RHS. It is a straightforward exercise using theform of the improved diagonal nonsinglet bilinears,Eq. (18), to show that the new information that one obtainshere over that obtained using Eq. (69), is a direct determi-nation of fV and bV . In particular, this Ward identity

034504

provides the only cross-check of the calculation of fV .The identity with V $ A similarly provides an alternativedetermination of fA.

V. SUMMARY AND CONCLUSIONS

Improvement in the presence of nondegenerate dynami-cal quarks is far more complicated than that with degen-erate quarks or that in the quenched approximation.Nevertheless, the considerable number of extra improve-ment coefficients that arise for quark bilinears and quarkmasses can almost all be determined by enforcing thevector and axial transformation properties of these opera-tors. The results from the Ward identities considered aboveare collected in Tables I and II. The only scale-independentquantities which are left undetermined are �dA and�bT � �dT .16 To determine these one must use other meth-ods. The only two that we are aware of are improvednonperturbative renormalization [10,11], and matching toperturbative forms for short distance correlation functionsof bilinears [8]. The scale-dependent quantities ZT , ZSZP,and rA are also undetermined, but this had to be the case asWard identities do not involve a renormalization scale.For these one must use a method like nonperturbativerenormalization.

It is interesting to understand why the two scale-independent quantities cannot be determined using Wardidentities. In the case of �dA, the reason is the lack of anidentity involving trA� that has a nonvanishing variation.Thus the overall normalization factor, which includes �dA,cannot be determined. This is clearly related to the fact thatrA cannot be determined, because it is scale dependent.

The reason is similar for �bT � �dT . Ward identities relatethe flavor-singlet tensor to the flavor nonsinglet tensor.Overall factors thus cancel, and the mass-dependent partof the overall factor is proportional to �bT � �dT . Thus, inessence, this combination cannot be determined becauseZT cannot.

To test these arguments, and for completeness, we haveextended the analysis to two and four nondegenerate fla-vors. These cases are also of phenomenological interest. Asummary of the results is given in Appendixes A and B.With four flavors, one might naively have hoped to deter-mine more coefficients, since the three flavor theory isincluded as a subset. We find, however, that althoughmost Ward identities by themselves allow more combina-tions of coefficients to be determined, so that the analysis iscleaner, the final result is the same as with three flavors. In

-16


particular, �dA and �bT � �dT cannot be determined, for ex-actly the same reasons as for three flavors.

With two flavors, there are, on the one hand, fewercoefficients to determine, but, on the other hand, fewerWard identities. Furthermore, each identity determinesfewer coefficients because there are less masses to varyindependently. The net result is that there are more unde-termined combinations of scale-independent quantitiesthan for three or four flavors (eight in all). Particularlystriking is the result that one cannot determine the im-proved flavor nonsinglet axial current, pseudoscalar den-sity or tensor bilinear away from the chiral limit.

Given the complexity of the calculations we outline, onemight wonder about possible simplifications. Since addingflavors extends (Nf � 2! 3) or simplifies (Nf � 3! 4)the analysis, one might consider using a partially quenchedsimulation with, say 2� 1 flavors of sea quarks, and fouror more flavors of valence quarks. To add additional infor-mation over the unquenched analysis, one must necessarilyconsider theories with differing sea and valence content,which are therefore not unitary. This makes the basis of theSymanzik improvement program less secure. Nevertheless,it is certainly possible, assuming the improvement programremains valid, to use the enlarged graded symmetry groupsof partially quenched theories to constrain the allowedimprovement coefficients, and to generalize the analysispresented here.

Staying within the unquenched three flavor theory, onecan ask whether it is sufficient to use 2� 1 flavor theoriesin which at least two quarks are degenerate. This point hasbeen discussed for each identity considered in the text, andwe find that 2� 1 flavors is sufficient to determine all theallowed coefficients.

Another practical question is whether one needs todetermine all the improvement coefficients for phenom-enologically interesting applications. Particularly interest-ing are (1) matrix elements of the electromagnetic current,(2) matrix elements of flavor off-diagonal vector and axialcurrents (for weak transitions), (3) quark masses and (4)matrix element of the mass term in the action, mjS�jj�. Weconsider these cases in turn. (1) The improvement andrenormalization coefficients for the electromagnetic cur-rent, which is a flavor nonsinglet for Nf � 3, are ZV , bV ,�bV and fV , and these can all be determined by normalizingthe matrix elements of the charge. The determination does,however, require quark-disconnected matrix elements.This also determines the improved flavor off-diagonalvector current. (2) The off-diagonal axial current requiresZA, bA and �bA, which can all be determined from flavor off-diagonal three-point Ward identities, as long as we knowbV . Thus only quark-connected contractions are needed.(3) Improvement of individual quark masses requires Zm,rm, bm, �bm, dm and �dm, i.e. both flavor-singlet and non-singlet coefficients [see Eq. (26)]. To determine theserequires all the types of Ward identity we consider, i.e.

034504

two- and three-point Ward identities involving both flavor-singlet and nonsinglet operators. The same is true for (4),the combination mjS�jj�. Thus we see that for some appli-cations there are considerable simplifications, but forothers there are not. We stress however, that for all of thesequantities one must first determine bg, although, as dis-cussed in the Introduction, a perturbative determination ofthis numerically small coefficient may be sufficient.

Finally, we note that one might consider our work as anadvertisement for other approaches to improvement withWilson-like fermions, namely, ‘‘Wilson averaging’’ [26]and twisted mass QCD [27] at maximal twist [26]. In bothapproaches the O�a� terms are automatically absent in thephysical matrix elements of the operators we consider here,so that no improvement of the operators themselves isnecessary. For this to hold, however, one needs an evennumber of fermions, and so for the physical case one mustsimulate with four flavors. First work in this direction hasbegun [28].

ACKNOWLEDGMENTS

We thank Rainer Sommer for comments on the manu-script. This work was supported in part by the U.S.Department of Energy through Grants No. DE-FG03-96ER40956/A006 and No. KA-04-01010-E161, and bythe BK21 program at Seoul National University, the SNUFoundation & Overhead Research fund and the KoreaResearch Foundation through Grant No. KRF-2002-003-C00033.

APPENDIX A: FOUR NONDEGENERATEFLAVORS

In this section we briefly describe the generalization ofour calculations to Nf � 4. Having one additional flavordoes not change the number of improvement coefficients,but does increase the number of independent masses thatone can vary. This allows the extraction of additionalimprovement coefficients in many of the Ward identities.The final result, however, is the same as for Nf � 3: allscale-independent coefficients can be determined exceptfor �dA and �bT � �dT .

We summarize the results in Tables I and III, and in thefollowing comment on the ways in which the calculationdiffers from that with Nf � 3.

(1) T

-17

he general forms for the improvement of bilinearsand masses, Eqs. (15) and (23)–(25), are unchanged,except that � are now SU�4� generators. This isbecause no special properties of SU�3� have beenused in writing these equations. Thus the numberof improvement coefficients is unchanged fromNf � 3.

(2) T
he use of the vector Ward identity, Eq. (27), is alsounchanged.


(3) T

TABLEcally) f

Ward i

hHjP

~x

@�A�jk��

@��A�jj�

� 2mj

��ij�A V�j

and V��ij�A T�j

��ij�A P�j

and P

��ij�A trA

��ij�A trP

��ij�A trT

��ij�A trS

��ij�A T�j

��ij�A S�j

and S$��ij�A A�j

��ij�A V�j

he enforcement of ‘‘ZSZm � 1,’’ i.e. Eq. (33), issimilar to the three flavor case, although more com-plicated algebraically as it involves determinants of4� 4 matrices. The results are given in Table I;some are identical to those with Nf � 3, whileothers have Nf dependence. There remains a singleconstraint, due to the absence of an ‘‘f term’’ in theimprovement of the quark masses.

(4) T
he two-point axial Ward identities are somewhatmore powerful than for Nf � 3, due to the greaternumber of independent combinations of masses thatcan be constructed, particularly at quadratic order.As can be seen from Table III, one can determine tencombinations of improvement coefficients, all ofwhich are fairly simple, compared to nine whenNf � 3, some of which are complicated (seeTable II). For four flavors the flavor off-diagonalidentity, Eq. (47), becomes redundant, giving noinformation not contained in the flavor-diagonalidentity (54).
(5) T
hree-point Ward identities involving only off-diagonal bilinears, Eq. (69), are also more powerful,because there is one additional combination ofmasses that is independent from that multiplyingthe contact term. This allows the separate determi-nation of the bO from these identities alone, andof �bA, �bV , and �bS � �bP. On the other hand, thenet result, including information from previous
III. Normalization and improvement coefficients determined usinor Nf � 4. Notation as in Table II.

dentity LO

V�jj�4 jHi � QjH ZV , rV

� �mj � mk�P�jk� ZmZP=ZA, rm

[cSW, cA]

� � A�kk�� ZmZP=ZA, rmP�jj� � 2mkP

�kk� rP, gP, [cSW, cA]

k� � A�ik� ZV$ A Z2

A, [cV]k� � T�ik� ZA, [cT]k� � S�ik� ZS=ZP$ S Z2

A

� ��ij�A trV � 0 �cA, �cV� 2S�ij� rPZP=ZS, gP� 2T�ij� rT , �cT� 2P�ij� rSZS=ZP, gSi� � T�ii� � T�jj� rT , �cT , [cT]i� � P�ii� � P�jj� ZP=ZS, rP, gPP rS, gS

i� � V�jj� � V�ii� Not newi� � A�jj� � A�ii� Not new

034504-18

identities, is the same as for Nf � 3: the newlydetermined constants remain ZA, ZS=ZP, �bA andbT .

(6) T
he Ward identities involving axial transformationof flavor singlets, e.g. Eq. (76), give essentially thesame information as for Nf � 3. The extra combi-nation of masses that is available does not multiplynew combinations of coefficients. These identitiescontinue to provide the only determination of dA, �cAand �cV , as well as the first determination of dS. Thelatter, together with the constraint dS � bS � Nf �bS,allows the separate determination of all the massand pseudoscalar improvement coefficients, as forNf � 3.
(7) T
he final type of identity, exemplified by Eq. (87), ismore powerful with Nf � 4, as can be seen from thetables. Nevertheless, the only new information ob-tained is fT .
In summary, with four flavors one has considerably morecross-checks, and the extraction of individual improvementcoefficients is more straightforward, but in the end oneobtains the same set of coefficients as for three flavors.

APPENDIX B: TWO NONDEGENERATE FLAVORS

In this Appendix we describe how our considerationschange when Nf � 2. The first new feature is that there isone less independent improvement coefficient for each

g various Ward identities (which are denoted schemati-

NLO

bV , �bV , fV , dV , �dV

bm, dm, �bm � �dm,bP � bA, �bP � �bA � �dm

bA, fA, bP, fP, dP, bm, dm,�bP � �dP, �bA � �dP � �dm, �bm � �dm

bV , �bVbA, �bA

bT , 4 �bA � bA�rm � 1�

bS, bP, �bP � �bS4 �bA � bA�rm � 1�

dA, dVdP, 4� �bS � �dP� � bS�rm � 1�

dT , 4� �bT � �dT� � bT�rm � 1�

dS, 4� �bP � �dS� � bP�rm � 1�

bT , dT , rT� �dT � �bT� � rmfTbP, bS, dP, dS, �bP � �bS

rP� �dP � �bP� � rmfP, rS� �dS � �bS� � rmfS�bV , bV , fV�bA, bA, fA

TABLE IV. Normalization and improvement coefficients determined using various Ward identities (which are denoted schemati-cally) for Nf � 2. Notation as in Table II.

Ward identity LO NLO

hHjP

~xV�jj�4 jHi � Qj

H ZV , rV �bV , fV , dV , �dV@�A

�12�� m1 � m2�P

�12� rmZmZP=ZA, [cSW, cA] dm, �bA � �bP � �dm@��A

�11�� A�22�

� � rmZmZP=ZA fA, rPdP � rmdm� 2mjP

�11� � 2mkP�22� rm=rP, gP rm�dm � 2 �dm � 2 �bP � 2 �bA�

[cSW, cA] rP�2 �bm � 2 �dP � 2 �bA � dP� � 2rmfP��12�A trA � ��12�

A trV � 0 �cA, �cV dA, dV��12�A trP � 2S�12� rPZPZA=ZS, gP dP��12�A trT � 2T�12� rTZA, �cT , cT dT��12�A trS � 2P�12� rSZSZA=ZP, gS dS��12�A T�21� � T�11� � T�22� ZA=rT , �cT , cT dT��12�A S�21� � P�11� � P�22� rPZP=�ZAZS�, gP dP��12�A P�21� � S�11� � S�22� rSZS=�ZAZP�, gS dS��12�A A�21� � V�22� � V�11� ZV=Z

2A, cV , [cA] fV

��12�A V�21� � A�22� � A�11� ZV , cV , [cA] fA


bilinear and for the quark masses. This follows from the fact that dabc / Tr��af�b; �cg� � 0 in SU�2� (with�a, a � 1; 2; 3the Pauli matrices). Thus one has

f�a;Mg � �a trM� tr��aM�1)�1� tr�f�a;MgO� � trM tr��aO� � tr��aM�trO;�2� tr��aM

2� � tr��aM�trM:(B1)

This means that the bO term in Eq. (15), and the bm term inEq. (24), are not independent and can be absorbed into theother terms by changing their coefficients as follows:

�bO ! b0O ��bO � bO=2;

�fO ! f0O � fO � bO=2;

�bm ! �b0m � �bm � bm:(B2)

In the following, we assume that this has been done, andthat the primes are then dropped.

Although there are thus six less coefficients to deter-mine, it turns out that there are fewer Ward identitiesavailable, and that each is less powerful than for Nf � 3.The results are collected in Tables I and IV, and we discussthe salient features in the following.

(1) T

17Altetable aEq. (B

he use of the vector Ward identity, Eq. (27), isunchanged, aside from the fact that there is one lesscoefficient to determine.

(2) T
he enforcement of ‘‘ZSZm � 1,’’ i.e. Eq. (33), fol-lows the same steps as above, but leads to simplerrelations because of the absence of bm and bS. Theresults in Table I remain valid for Nf � 2 as long asone sets bm � bS � 0.17
rnatively, one can keep the relations as they stand in thend then absorb bm and bS into the other coefficients as in2).

034504

(3) T

-19

he two-point axial Ward identities are less power-ful than for Nf � 3, due to the smaller number ofindependent combinations of masses. Combiningthe flavor-diagonal and off-diagonal identities, oneonly determines rmZmZP=ZA, rP=rm and gP at LO,and fA, dm, dP, and the combinations �dm � �bP � �bAand rP� �bm � �bA � �dP� � rmfP at NLO. Note thatone cannot determine rm and rP separately, norobtain bA, unlike for Nf � 3.

(4) F
or two flavors there are no three-point Ward iden-tities involving only off-diagonal bilinears. Thusone loses what has been one of the central tools inquenched studies. In particular, for Nf � 3 and 4these are the identities that are used to determine cVand cT . Here we need to use other identities for thispurpose, as discussed below and indicated in thetable.
(5) T
he Ward identities involving axial transformationsof the singlet axial and vector currents, e.g. Eq. (76),give the same information as for Nf � 3. However,those involving the tensor, scalar and pseudoscalarbilinears give less information. This is because thereis one less combination of quark masses that isindependent of the contact term. In fact the analysisis more straightforward, because the contact termis proportional to ~m12 / tr�M� �O�a�, and soone does not need to know rm in order to work at~m12 � 0.


At this stage we still have not determined ZA or �bA,so we do not know the normalization of the axialvariation of the action. Note also that the tensorWard identity allows the first determination of cT .

(6) T
he final type of identity is that exemplified byEq. (87). This again is simpler to analyze for Nf �2, since the RHS is a pure flavor singlet (for T, S andP), rather than a mix of singlet and nonsinglet. Inparticular, new information is obtained in the chirallimit, and allows one, combined with previous re-sults, to disentangle ZA and rT .The identities involving the vector bilinears are thefirst to allow a determination of cV . In fact, in thechiral limit, where one works to determine cV , theseidentities involve the same quark contractions as thethree-point Ward identities with only off-diagonalbilinears that are present for Nf � 3. Thus, from acomputational point of view, there is no differencein the method to be used to determine cV for Nf �2. This is not, however, the case for cT .
The following scale-independent constants remain un-determined by the Ward identities: �bA and �dA; �bT , �dT andfT ; ZS=ZP, rP, �bP, �dP and fP; and rS and �dS or equivalentlyrm and �dm, although the following combinations of thesecoefficients are known:

�b A � �bP � �dm; rP� �bA � �bm � �dP� � rmfP;

rPZP=ZS; rPrS:(B3)

Thus in total there are eight undetermined combinations ofscale-independent coefficients. This is six more than forNf � 3, despite the need to determine six fewercoefficients.

What is perhaps most striking about this list is that, evenif one uses nondegenerate quarks when implementingWard identities, one cannot determine all the coefficientsneeded for flavor nonsinglet bilinears composed of degen-erate quarks. In particular, since one cannot separatelydetermine �bA, �bP or �bT , one does not know the overallnormalization of the corresponding bilinears away fromthe chiral limit. To determine this one must use a methodsuch as those proposed in Refs. [8,11].

18The form of the cV and �cV terms for V lat� �x� �=2� is

@�T��x� �� T��x��=2, where the derivative is the sym-metric difference.

APPENDIX C: VECTOR WARD IDENTITIES

In this Appendix we explain the claim made in the textthat, other than the normalization of charges, enforcingvector Ward identities leads to no information on improve-ment coefficients. It is interesting to see how this works indetail, and how we concluded otherwise in Ref. [12]. It alsogives a good example of how the mnemonic introduced inRefs. [7,8] of using equations-of-motion operators to de-termine the form of contact terms needs to be modified.

We begin by recalling some results from the main text.Enforcing the charge of hadrons, Eq. (27), determines theimproved, renormalized diagonal vector currents, V�jj�� ,

034504

aside from the cV and �cV terms. The latter terms do notcontribute to the divergence, so @�V

�jj�� is fully improved.

This implies that the divergences of the off-diagonal cur-rents, @�V

�jk�� , j � k, are also improved, because these

involve the same improvement coefficients as the diagonalcurrents (in particular, ZV , rV , bV and �bV ; the coefficient fVis known but not required). These results hold whatever theprecise form of the underlying lattice current.

Next we recall the form of the exact lattice vector Wardidentities. Making the change of variables �V j � k and�V � k � � � j over a region of the lattice V , one finds

a4XV

�mj �mk�S�jk� � @�Vlat;�jk�� O�y�J�z�

� h�latV O��y�J�z�i: (C1)

Here y is in the region V , while z is not, and �latV O is the

variation of the operator O under the vector transforma-tion. Note that the bare lattice quark masses appear in (C1)irrespective of the presence of the cSW term, and that S�jk� isthe local scalar bilinear. The current V lat;�jk�

� is the usuallattice vector current, also unaffected by the cSW term. It isassociated with a link, and not a site, but its divergence isassociated with a site:

Vlat;�jk��

�x�

�2

��

1

2 � j�x��Ux;� k�x� ��

� � j�x� ��Uyx;� k�x��;

@�Vlat;�jk�� x� �

X�

1

a

�V lat;�jk��

�x�

�2

�

� Vlat;�jk��

�x�

�2

��: (C2)

The results (C1) and (C2) hold both for j � k and j � k.In particular, the diagonal lattice current is conserved, andit generates canonically normalized vector transforma-tions. This implies that the charge constructed from thelattice current is correctly normalized [as can be derivedfrom Eq. (C1)]. In other words, for this current, Zlat

V � 1 �rlatV , and the mass-dependent improvement coefficients all

vanish. As is well known, however, the lattice current is notimproved. For this one must add cV- and �cV-like terms, asin Eq. (6), except that they must be associated with a link,rather than a site. It is possible to do so in such a way thattaking the divergence of the current [in the form thatappears in the Ward identity (C1)] exactly cancels thecV- and �cV-like terms.18 Because of this the cV- and�cV-like terms neither contribute to the vector Ward identi-ties nor affect the values of the other improvement andnormalization constants.

-20


In summary, the divergences of the flavor-diagonal con-served lattice vector currents are automatically improved.As noted above, this also implies that the divergences ofthe flavor off-diagonal lattice vector currents are improved.Thus the @�V

lat;�jk�� term in the vector variation of the

action in the lattice Ward identity (C1) is improved. Thismeans that it can be replaced, up to errors of O�a2�, with@�V

�jk�� , the divergence of the improved local vector cur-

rent. In this replacement we do not have to worry aboutcontact terms as

PV@�V� vanishes except at the surface

of the region V , and by assumption there are no otheroperators there.

The other term in the vector variation of the latticeaction in (C1) involves bare masses and scalar densities.As shown by Eq. (46), this maintains the same form whenwritten in terms of improved masses and scalar densities.

Combining these observations we can rewrite the latticevector Ward identity in terms of the improved bilinears andmasses considered in the main text:

a4XV

�mj � mk�S�jk� � @�V

�jk�� O�y�J�z�

� h�latV O��y�J�z�i �O�a2�: (C3)

This relation leads, among other things, to the normaliza-tion condition for the vector charge, Eq. (27). Note that thisrelation is ‘‘off shell’’ improved, since there are no contactterms of O�a�. These could only enter with the scalardensity term, but the result Eq. (46) is an algebraic identity,so using it does not introduce additional terms. The im-proved bilinear S�jk� is, however, only on shell improved,and this will be crucial in the following.

The result Eq. (C3) shows that, if one uses on-shellimproved local bilinears and masses in the discretizedform of the vector variation of action, then the operatorsin the associated Ward identities automatically transformwith the correct normalization. This is the concrete form ofthe statement made in the text that the vector Ward iden-tities are automatically satisfied. There are some subtleties,however, in the application of this result, and we spend theremainder of this Appendix describing some examples.

1. Two-point vector Ward identities

In quenched studies of improvement, the two-point vec-tor Ward identity

h�mj � mk�S�jk��x� � @�V

�jk�� x��J�kj��0�i � O�a2�;

j � k; x � 0; (C4)

has been used as part of the method employed to determineimprovement coefficients [7,9]. The result Eq. (C3) shows,however, that this identity is automatically satisfied, aslong as one uses the correctly normalized vector current.Thus it serves only to check the normalization of the vectorcurrent, and provides no information on the improvement

034504

coefficients of quark masses and scalar bilinears. Thispoint was not appreciated in Ref. [7].

2. A misleading three-point vector Ward identity

In Ref. [12], we argued that we could determine gS andgP by enforcing the vector transformation properties of theflavor-singlet scalar and pseudoscalar bilinears, respec-tively. This is incorrect, as we now show. It is simplest todo this with the pseudoscalar density as it has no mixingwith the identity operator.

The identity in question is

a4XV

@�V�jk�� ctrP�y�J�kj��z� � O�a2�; (C5)

where we work in the chiral limit so that there are nocontact terms. We recall that, in the chiral limit, the im-proved, normalized pseudoscalar density is

ctrP � ZPrPtrP� agP Tr�F�� ~F��: (C6)

Our previous argument was that, in order for the LHS of(C5) to be ofO�a2�,O�a� contributions from the two termsin ctrPmust cancel against each other, thus determining gP.In fact, it follows from Eq. (C3) that both terms separatelyare automatically of O�a2�, e.g.

a4XV

@�V�jk�� trP�y�J

�kj��z�� O�a2�; (C7)

so that the identity (C5) is satisfied for any value of gP. Inother words, the two terms in ctrP are separately invariantunder vector transformations up to O�a2�.

3. A paradox and its resolution

Another vector Ward identity used in Ref. [12] was


�jk�� x��O

�kj��y�J�z�i

� hO�kk��y� � O�jj��y��J�z�i � contact terms�O�a2�:

(C8)

Here we work at nonzero quark mass, so there are contactterms of O�a� because the scalar density is not off shellimproved. These are proportional to �mj �mk� (withoutany factors involving rm as in the axial case), because mj �

mk / mj �mk �O�a�. If we use the expressions (16) and(18) for the improved bilinears, and divide through by

ZO1� a �bO trM� abOmkj�; (C9)

the Ward identity we are enforcing becomes

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PHYSICAL REVIEW D 73, 034504 (2006)


�jk�� x��O�kj�;I�y�J�z�i

�

�O�kk�;I�y� �O�jj�;I�y��

� abO�mk �mj�

2O�kk��y� �O�jj��y��

� afO�mk �mj�trO�y��J�z�

� contact terms

�O�a2�: (C10)

Previously, we argued that the form of the contact termcould be determined by off-shell improving S�jk� with theaddition of a term proportional to the equations-of-motionoperator (71), as in Refs. [7,8]. This leads to a contact termproportional to

contact term / ha�mk �mj�O�kk��y� �O�jj��y��J�z�i;

(C11)

i.e. of the same form as the bO term. Thus we concludedthat bO could not be determined from this Ward identity,but that fO could, since it multiplies an independentoperator.

To see that this is incorrect, we use the general result(C3), from which it follows that


�jk�� x��O�kj�;I�y�J�z�i

� hO�kk�;I�y� �O�jj�;I�y��J�z�i �O�a2�: (C12)

Note that the improvement terms (those proportional to cO)rotate just as the bare operators, so OI rotates as a whole.Comparing to Eq. (C10) we see that the first term on theRHS is obtained automatically, while the bO and fO termsmust either cancel against contact terms or vanish. Thediscussion above implies that the bO term is canceled, butwe must have fO � 0. This would be a paradoxical con-clusion because we used the vector symmetry in the first

BHATTACHARYA et al.

034504

place to conclude that the fO terms are needed, and yet wehave now used the same symmetry to conclude that theyvanish.

In fact, this result is wrong. The flaw in the argument isthat there is an additional contact term in Eq. (C10) pro-portional to

ha�mk �mj�trO�y�J�z�i; (C13)

and this is of the right form to cancel with the fO term.Thus fO does not need to vanish, and indeed cannot bedetermined from the vector Ward identity, just like bO. Theoperator in the new contact term, trO, is allowed because itappears in the operator product of S�jk� and O�kj�, in addi-tion to the other contact term operator O�kk� �O�jj�. Thenew operator arises from diagrams in which both the quarkand antiquark in the two operators in the product arecontracted together—the closed quark loop then couplingto trO through intermediate gluons. The presence of thisoperator shows that the mnemonic of off-shell improve-ment through the addition of equations-of-motion opera-tors needs to be generalized beyond the considerations ofRefs. [7,8]. A more straightforward approach is simply toenumerate the allowed operators using symmetries.Indeed, one can turn the previous argument around, anduse the fact that symmetry implies the presence of the fOterms to imply the existence of the new contact terms.

Finally, we note that these considerations resolve thepuzzle concerning the counting of improvement coeffi-cients mentioned in the text. The off-diagonal bilinearsrequire one less improvement coefficient than the diagonalbilinears. How can this be consistent with the fact that avector transformation rotates the former into the latter (asin the Ward identities just discussed)? The answer is pro-vided by the presence of the new contact term, whichallows there to be an fO term in the diagonal bilinearsbut not in the off-diagonal ones.

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Improved bilinears in lattice QCD with nondegenerate quarks