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Strange quarks in quenched twisted mass lattice QCD Abdou M. Abdel-Rehim and Randy Lewis Department of Physics, University of Regina, Regina, SK, Canada, S4S 0A2 R. M. Woloshyn and Jackson M. S. Wu TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada, V6T 2A3 (Received 1 February 2006; published 10 July 2006) Two twisted doublets, one containing the up and down quarks and the other containing the strange quark with an SU2-flavor partner, are used for studies in the meson sector. The relevant chiral perturbation theory is presented, and quenched QCD simulations (where the partner of the strange quark is not active) are performed. Pseudoscalar meson masses and decay constants are computed; the vector and scalar mesons are also discussed. A comparison is made to the case of an untwisted strange quark, and some effects due to quenching, discretization, and the definition of maximal twist are explored. DOI: 10.1103/PhysRevD.74.014507 PACS numbers: 12.38.Gc, 12.39.Fe I. INTRODUCTION Twisted mass lattice QCD (tmLQCD) is a variation on the Wilson action —essentially a chiral rotation of quark flavor doublets, acting on quark-mass terms relative to Wilson terms in the action — which produces two desirable features: the removal of unphysical zero modes in quark propagators [1] and the elimination of Oa artifacts (where a denotes lattice spacing) at maximal twist [2]. A number of numerical simulations have been performed for both quenched and dynamical tmLQCD (for a recent re- view, see Ref. [3]). As well, the chiral perturbation theory for tmLQCD (tmPT) has been developed. It differs from continuum PT by discretization effects and is required for the extrapolation of tmLQCD data. The effective theory has also played a vital role in understanding various as- pects of tmLQCD such as Oa improvement, the phase diagram, and the relationships between various definitions of maximal twist [4 11]. With an interest in the phenomenology of hadrons built of u, d and s quarks, our goal in this paper is to explore the usefulness of tmLQCD and tmPT as applied to strange hadrons. There is no unique way to introduce the s quark into the calculation. The method used here, to consider a pair of quark doublets u;d and (‘‘c,’’ s), is similar to the proposal of Pena et al. [12]. For the quenched simulations considered here the partner of the s quark does not play an active role and should not be thought of as the physical charm quark. In this work no mass splitting is introduced within either doublet. The construction of the correspond- ing tmPT formalism is a straightforward generalization of the published one-doublet formalism [7,9]. As noted above, applying a relative chiral twist has some valuable consequences but there are also some less desir- able features that have to be dealt with. The tmLQCD action violates parity conservation so, in general, correla- tion functions contain contributions from states of both parities. Parity mixing can complicate, in particular, the extraction of matrix elements but this can be ameliorated by appropriate tuning of the twist angles. The tmLQCD action also breaks the flavor symmetry. For the version of tmLQCD used in this work the members of the quark doublets are degenerate in mass but are distinguished by having opposite chiral twists. This can lead to mass split- tings within hadron isospin multiplets. It will be seen that charged and neutral kaons can acquire a mass-squared splitting which is roughly proportional to a 2 . To optimize the elimination of Oa lattice discretization errors one has to tune the chiral twist angles [2]. There is not a unique way to achieve maximal twist as has been discussed from the point of view of both effective theory [79,11] and simulation [13 15]. A standard method for defining maximal twist uses a tuning procedure which involves the correlators of the first two isospin components of vector and axial operators with the pseudoscalar density [9,13,16]. Using two variations of this method, we examine the mixing between the third isospin components of scalar and pseudoscalar correlators. Ideally one would like to have a tuning to maximal twist which would banish the physical pseudoscalar meson from appearing in the wrong parity correlator; the scalar meson with its quenched 0 0 contribution would similarly be banished from the other parity correlator. This is seen not to happen in our simula- tions. The mixings observed in actual simulations represent higher order discretization effects which differentiate be- tween vector-axial tuning and scalar-pseudoscalar tuning. In this work, we mainly use maximal twist in the doublet containing the strange quark as well as in the u;d dou- blet. An alternative procedure would be to set the twist angle for the strange quark to zero or equivalently for the quenched theory to consider a twisted u;d doublet and a flavor-singlet Wilson strange quark. The latter approach may be a viable one for doing full dynamical simulations. The twisted and untwisted strange quark actions lead to different patterns of parity mixing and flavor symmetry breaking at nonvanishing lattice spacing. We present some results obtained with an untwisted strange quark action to illustrate some of these differences. PHYSICAL REVIEW D 74, 014507 (2006) 1550-7998= 2006=74(1)=014507(16) 014507-1 © 2006 The American Physical Society

Strange quarks in quenched twisted mass lattice QCD

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Page 1: Strange quarks in quenched twisted mass lattice QCD

Strange quarks in quenched twisted mass lattice QCD

Abdou M. Abdel-Rehim and Randy LewisDepartment of Physics, University of Regina, Regina, SK, Canada, S4S 0A2

R. M. Woloshyn and Jackson M. S. WuTRIUMF, 4004 Wesbrook Mall, Vancouver, BC, Canada, V6T 2A3

(Received 1 February 2006; published 10 July 2006)

Two twisted doublets, one containing the up and down quarks and the other containing the strangequark with an SU�2�-flavor partner, are used for studies in the meson sector. The relevant chiralperturbation theory is presented, and quenched QCD simulations (where the partner of the strange quarkis not active) are performed. Pseudoscalar meson masses and decay constants are computed; the vectorand scalar mesons are also discussed. A comparison is made to the case of an untwisted strange quark, andsome effects due to quenching, discretization, and the definition of maximal twist are explored.

DOI: 10.1103/PhysRevD.74.014507 PACS numbers: 12.38.Gc, 12.39.Fe

I. INTRODUCTION

Twisted mass lattice QCD (tmLQCD) is a variation onthe Wilson action—essentially a chiral rotation of quarkflavor doublets, acting on quark-mass terms relative toWilson terms in the action—which produces two desirablefeatures: the removal of unphysical zero modes in quarkpropagators [1] and the elimination of O�a� artifacts(where a denotes lattice spacing) at maximal twist [2]. Anumber of numerical simulations have been performed forboth quenched and dynamical tmLQCD (for a recent re-view, see Ref. [3]). As well, the chiral perturbation theoryfor tmLQCD (tm�PT) has been developed. It differs fromcontinuum �PT by discretization effects and is required forthe extrapolation of tmLQCD data. The effective theoryhas also played a vital role in understanding various as-pects of tmLQCD such as O�a� improvement, the phasediagram, and the relationships between various definitionsof maximal twist [4–11].

With an interest in the phenomenology of hadrons builtof u, d and s quarks, our goal in this paper is to explore theusefulness of tmLQCD and tm�PT as applied to strangehadrons. There is no unique way to introduce the s quarkinto the calculation. The method used here, to consider apair of quark doublets �u; d� and (‘‘c,’’ s), is similar to theproposal of Pena et al. [12]. For the quenched simulationsconsidered here the partner of the s quark does not play anactive role and should not be thought of as the physicalcharm quark. In this work no mass splitting is introducedwithin either doublet. The construction of the correspond-ing tm�PT formalism is a straightforward generalizationof the published one-doublet formalism [7,9].

As noted above, applying a relative chiral twist has somevaluable consequences but there are also some less desir-able features that have to be dealt with. The tmLQCDaction violates parity conservation so, in general, correla-tion functions contain contributions from states of bothparities. Parity mixing can complicate, in particular, theextraction of matrix elements but this can be ameliorated

by appropriate tuning of the twist angles. The tmLQCDaction also breaks the flavor symmetry. For the version oftmLQCD used in this work the members of the quarkdoublets are degenerate in mass but are distinguished byhaving opposite chiral twists. This can lead to mass split-tings within hadron isospin multiplets. It will be seen thatcharged and neutral kaons can acquire a mass-squaredsplitting which is roughly proportional to a2.

To optimize the elimination ofO�a� lattice discretizationerrors one has to tune the chiral twist angles [2]. There isnot a unique way to achieve maximal twist as has beendiscussed from the point of view of both effective theory[7–9,11] and simulation [13–15]. A standard method fordefining maximal twist uses a tuning procedure whichinvolves the correlators of the first two isospin componentsof vector and axial operators with the pseudoscalar density[9,13,16]. Using two variations of this method, we examinethe mixing between the third isospin components of scalarand pseudoscalar correlators. Ideally one would like tohave a tuning to maximal twist which would banish thephysical pseudoscalar meson from appearing in the wrongparity correlator; the scalar meson with its quenched �0�0

contribution would similarly be banished from the otherparity correlator. This is seen not to happen in our simula-tions. The mixings observed in actual simulations representhigher order discretization effects which differentiate be-tween vector-axial tuning and scalar-pseudoscalar tuning.

In this work, we mainly use maximal twist in the doubletcontaining the strange quark as well as in the �u; d� dou-blet. An alternative procedure would be to set the twistangle for the strange quark to zero or equivalently for thequenched theory to consider a twisted �u; d� doublet and aflavor-singlet Wilson strange quark. The latter approachmay be a viable one for doing full dynamical simulations.The twisted and untwisted strange quark actions lead todifferent patterns of parity mixing and flavor symmetrybreaking at nonvanishing lattice spacing. We present someresults obtained with an untwisted strange quark action toillustrate some of these differences.

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At this point it is worth noting that there exist even otherapproaches for dealing with the strange quark. The pro-posal of Frezzotti and Rossi in Ref. [17] allows for anondegenerate doublet in a way which is suitable fordynamical simulations [18]. In the limit where quarkmasses are degenerate within each doublet, it is equivalentto the scheme used in this paper. However, for nondegen-erate quarks, twist and quark-mass splitting are associatedwith different flavor transformation generators. The fer-mion action contains terms which mix flavors so that flavorsymmetry breaking effects would be more complicated todeal with in simulations and in the effective theory than forthe tmLQCD action considered in this work. A furtherexample is Ref. [19] where options for tmLQCD chosento facilitate the calculation of the so-called kaon bagparameter are discussed.

In addition to meson masses, the pseudoscalar mesondecay constants are also considered. With quark massesfixed by physical meson masses, the decay constants f�and fK become absolute predictions, and are shown tocompare favorably with previous quenched simulationsusing other actions. All results are consistent with tm�PT.

The remainder of the article is organized as follows.Section II defines the effective chiral Lagrangian withtwo twisted flavor doublets, and Sec. III uses thatLagrangian to derive expressions involving the pseudosca-lar masses and decay constants. Section IV presents thetmLQCD action and explains the parameter choices for ournumerical simulations, then Sec. V discusses results ob-tained for the pseudoscalar and vector mesons. Scalar-pseudoscalar mixing is studied in Sec. VI, and a directcomparison to kaons built from untwisted strange quarks isgiven in Sec. VII. Section VIII contains the conclusions ofour work. Details of currents and densities in tm�PT arecollected in the appendix.

II. THE EFFECTIVE CHIRAL LAGRANGIAN

To build four-flavor tm�PT, we begin from tmLQCDwith two quark doublets,

l �ud

� �; h �

cs

� �; (1)

referred to as the light and heavy doublets, respectively.Note that the choice of flavor labels is a convention; inRef. [12] for example, a different choice is made. In thiswork each doublet is taken to be degenerate, so the c quark,which is not active in any of our quenched tmLQCDsimulations, should not be viewed as the physical charmquark. Pena et al. [12] discuss the extension of this case tothe case of a nondegenerate doublet where the quark-masssplitting is aligned with the twist, preserving the favorablefeature of no flavor mixing. The fermion determinant doesnot remain real under this generalization so this would notlead a suitable action for nonquenched simulations.

However, this action may still be useful for valence quarksin a mixed action scenario as discussed, for example, in thecontext of tmLQCD in Ref. [20].

In the so-called ‘‘twisted basis’’ [1,2], the two-doubletlattice action is simply a block-diagonal version of twocopies of the one-doublet theory (the form of which can befound in Refs. [1,2]):

SLF � a4Xx

���x��

1

2

X�

���r?� �r�� �

a2

X�

r��r�

�m0 � i�5�0

���x�; (2)

where r� and r�� are the usual covariant forward andbackward lattice derivatives, respectively, and

� � l h

� �; m0 �

ml;012 00 mh;012

� �;

�0 ��l;0�3 0

0 �h;0�3

� �;

(3)

with 1n the n-by-n identity matrix. The matrix �3 acts in(two-)flavor space and is normalized so that �2

3 � 12. Theparameters mp;0 and �p;0 are the normal bare and twistedmasses, respectively, with p � l, h.

Applying the now familiar two-step procedure ofRef. [21], an effective chiral Lagrangian describing thelow energy physics of tmLQCD with two degenerate quarkdoublets can be built as a straightforward generalization ofthe one-doublet case detailed in Refs. [7,9,22,23]. From asimilar analysis described in Ref. [7], the form of theeffective continuum Lagrangian at the quark level is foundto be identical to that in the one-doublet case:

L eff � Lg � ��� 6D�m� i�5���

� bSWa ��i���F����O�a2�; (4)

where Lg is the continuum gluon Lagrangian, and thephysical normal and twisted mass parameters, m and �,are defined analogously as in the one-doublet case:

m �ml 00 mh

� ��

Zm;l�ml;0 � ~mc;l� 00 Zm;h�mh;0 � ~mc;h�

� �; (5)

� ��l�3 0

0 �h�3

� ��

Z�;l�l;0�3 00 Z�;h�h;0�3

� ��

Z�1P;l�l;0�3 0

0 Z�1P;h�h;0�3

!; (6)

with ZP;l and ZP;h being the matching factors for

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the pseudoscalar density. The quantities ~mc;l and ~mc;h arethe critical masses, aside from an O�a� shift (seeRefs. [7,9,22,23] and discussions below). Lattice symme-tries forbid additive renormalization of�l;0 and�h;0. As anaside, we note that symmetries also cause the ultravioletdivergent parts of ~mc;l and ~mc;h to be identical. One canchoose a definition of maximal twist [it will be called

method (ii) in Sec. IV] for which ~mc;l � ~mc;h, but herewe do not restrict the discussion to that special case.

Working to NLO in the power counting scheme,

ml �mh ��l ��h � p2 � a�2QCD; (7)

the effective chiral Lagrangian found from matching reads

L� �f2

4Tr�D��D��y� �

f2

4Tr��y�� �y�� �

f2

4Tr�Ay���yA� � L1Tr�D��D��y�2 � L2 Tr�D��D��y�

� Tr�D��D��y� � L3 Tr�D��D��y�2 � L4 Tr�D��yD���Tr��y�� �y��

� L5 Tr�D��D��y����y � ��y� � L6Tr��y�� �y��2 � L7Tr��y�� �y��2

� L8 Tr��y�� �y��2 � iL9 Tr�L��D��D��y � R��D��yD��� � L10 Tr�L���R���y�

�W4 Tr�D��yD���Tr�Ay���yA� �W5 Tr�D��D��y��A�y ��Ay�

�W6 Tr��y�� �y��Tr�Ay�� �yA� �W06Tr�Ay���yA�2 �W7 Tr��y�� �y��Tr�Ay���yA�

�W07Tr�Ay���yA�2 �W8 Tr��y���y���Ay�� �yA� �W08 Tr�Ay���yA�2

�W10 Tr�D�AyD���D��yD�A� �H1 Tr�L��L�� � R��R��� �H2 Tr��y�� �H02 Tr�Ay�� �yA�

�H3 Tr�AyA�; (8)

where the � field is now SU�4� matrix-valued, and trans-forms under the chiral group SU�4�L � SU�4�R. Note thatL� has basically the same form as that in the SU�2� theory[7,9,22,23], except that terms linearly dependent underSU�2� are no longer so under SU�4�.

The quantities � and A are spurions for the quark massesand discretization errors, respectively [24]. At the end ofthe analysis they are set to the constant values

� ���! 2B0�m� i��; A ���! 2W0a14; (9)

where B0 and W0 are unspecified constants having dimen-sions [mass] and [mass3], respectively. Notice that A in-volves a single flavor-independent Pauli term for bothdoublets.

The discretization effect due to the Pauli term, i.e. theterm containing bSW in Eq. (4), can be included nonper-turbatively as in Ref. [9] by using the shifted spurion �0 ��� A, which corresponds at the quark level to a redefini-tion of the normal quark mass from m to

m0p � mp � aW0=B0 p � l; h: (10)

This shift in turn corresponds to an O�a� correction to thecritical mass, so that it becomes

mc;p � Zm;p emc;p � aW0=B0 p � l; h: (11)

In terms of �0, the chiral Lagrangian, Eq. (4), can bewritten as

L� �f2

4Tr�D��D��y� �

f2

4Tr��0y�� �y�0� � L1Tr�D��D��y�2 � L2 Tr�D��D��y�Tr�D��D��y�

� L3 Tr�D��D��y�2 � L4 Tr�D��yD���Tr��0y�� �y�0� � L5 Tr�D��D��y���0�y ���0y�

� L6Tr��0y�� �y�0�2 � L7Tr��0y�� �y�0�2 � L8 Tr��0y���y�0�2 � iL9 Tr�L��D��D��y

� R��D��yD��� � eW4 Tr�D��yD���Tr�Ay�� �yA� � eW5 Tr�D��D��y��A�y ��Ay�

� eW6 Tr��0y�� �y�0�Tr�Ay�� �yA� � eW06Tr�Ay�� �yA�2 � eW7 Tr��0y�� �y�0�Tr�Ay�� �yA�

� eW07Tr�Ay�� �yA�2 � eW8 Tr��0y�� �y�0��Ay�� �yA� � eW08 Tr�Ay���yA�2

�W10 Tr�D�AyD���D��yD�A� �H2 Tr��0y�0� � eH02 Tr�Ay�0 � �0yA�; (12)

where terms that lead only to contact terms in correlation functions and are hence not needed below, viz. the L10, H1, andH3 terms, have been dropped. We have also introduced useful combinations

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~W i � Wi � Li; i � 4; 5 ~Wj � Wj � 2Lj;

~W0j � W0j �Wj � 2Lj; j � 6; 7; 8

~H02 � H02 �H2:

(13)

As noted in Ref. [9], theW10 term is redundant. It can betransformed away by the change of variables

� �2W10

f2 ��Ay�� A�: (14)

This transforms the W10 term into the ~W5, ~W8, and ~H02terms with their coefficients shifted to ~W5 �W10, ~W8 �W10=2, and ~H02 �W10, respectively. All physical quantitiesmust depend then only on these combinations and not onW10, ~W5, ~W8, and ~H02 separately. We have kept the W10

term because it provides a useful diagnostic in tm�PTcalculations.

III. CHIRAL PERTURBATION THEORY FORGENERIC SMALL MASSES

In this section, we work out the consequences of theSU�4� effective chiral Lagrangian, Eq. (12), which gener-alizes the results of the SU�2� theory of Ref. [9], and ourfocus will be on the masses and decay constants of kaons:pseudoscalar mesons that involve both flavor doublets. Wework in the ‘‘generic small mass’’ regime defined by

�2� � M0h * M0l * 2W0a; (15)

where �� � 4�f and

M0p � 2B0

��������������������m02p ��

2p

q; p � l; h: (16)

Note that M0p has dimension [mass2]. In the analysis of ournumerical data, quenching effects will also be considered[see Eqs. (43)–(45)].

A. The vacuum

At LO the discretized Lagrangian retains its continuumform, so the LO vacuum expectation value (VEV) of � isthat which cancels out the twists in the shifted mass matrix:

h0j�j0iLO � �0 �exp�i!l;0�3� 0

0 exp�i!h;0�3�

� �; (17)

where !p;0 are defined by

cp;0 � cos�!p;0� � 2B0m0=M0p;

sp;0 � sin�!p;0� � 2B0�p=M0p; p � l; h:

(18)

This provides one definition for the twist angles.At NLO, the VEV of � is realigned by a small amount

from �0. Defining

h0j�j0iNLO � �m �exp�i!l;m�3� 0

0 exp�i!h;m�3�

!;

!p;m � !p;0 � p; p � l; h; (19)

the shifts from LO, l;h, are found from minimizing thepotential to be

p � �16W0asp;0

f2

�2 eW6�M

0l �M

0h�=M

0p � eW8

�4W0aM0p

eW06�cl;0 � ch;0� � eW08cp;0�: (20)

Expanding about the VEV as in Ref. [9], the physicalpion fields are defined by

� � �m�ph�m;

�m �exp�i!l;m�3=2� 0

0 exp�i!h;m�3=2�

!;

�ph � exp�i�=f�;

(21)

where � has the representation

1���2p � �

1���2p

X15

i�1

’i�i �

1��2p �0 � 1��

6p �8 �

1����12p �15 �� D0 K�

�� � 1��2p �0 � 1��

6p �8 �

1����12p �15 D� K0

D0 D� � 3����12p �15 D�s

K� K0 D�s � 2��6p �8 �

1����12p �15

0BBBBB@1CCCCCA:

(22)

Our choices of the 15 generators of SU�4�, �1; . . . ;�15,differ slightly from the conventional ones. Here, all off-diagonal generators and the diagonal �3 are the same asthe conventional ones, but we choose the rest of the diago-nal generators to be such that the (3, 3) and (4, 4) entries ofthe diagonal �8 and �15 are interchanged with respect to

the conventional ones.1 This maintains consistency withthe ordering of quark fields �u; d; c; s�, used throughout thiswork,2 and allows for the standard meson naming conven-tion. In particular, we have for the flavor-diagonal compo-nents of �:

1Note that once the choice is made for �8, �15 is fixed by thenormalization condition Tr��i�j� � 2ij.

2Recall that u and c have a positive twist whereas d and s havea negative twist.

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j�0i �1���2p �ju �ui � jd �di�; (23)

j�8i �1���6p �ju �ui � jd �di � 2js�si�; (24)

j�15i �1������12p �ju �ui � jd �di � js�si � 3jc �ci�: (25)

Note that the c quark is mass-degenerate with the s quark,so pairs ofD’s andK’s related by the interchange of c and squarks are mass-degenerate at LO in the chiral expansion.By inserting the above expansion of � into the chiralLagrangian, Eq. (12), the Feynman rules for the SU�4�theory can be straightforwardly obtained.

B. Defining the twist angle

In the continuum, the twist angle for each degeneratedoublet can be defined unambiguously by

!p � tan�1��p=mp�; p � l; h: (26)

Given the twist angles, the off-diagonal SU�4� componentsof physical currents and densities are related to their coun-terparts in the twisted basis by (with a ‘‘hat’’ denoting thephysical basis)

Va� � cos��a!�Va� � �ab sin��a!�Ab�;

Aa� � cos��a!�Aa� � �ab sin��a!�Vb�;

Sa � cos��a!�Sa � i sin��a!�Pa;

Pa � cos��a!�Pa � i sin��a!�Sa;

a; b 2 KnD; K � f1; . . . ; 15g; D � f3; 8; 15g;

(27)

where

�ab �� 1; for b � a 10; otherwise

(28)

and

�a! � 12�!ia �!ja�; �a! � 1

2�!ia �!ja�;

ia < ja:(29)

The indices ia and ja are the row numbers of the nonzeroentries of the SU�4� generator �a that defines the particularflavor current or density, and

!1 � �!2 � !l; !3 � �!4 � !h: (30)

For a thorough discussion of currents and densities, see theappendix. We note now that Eq. (50) has the form of theinverse transformation of the LO operator, Eq. (A1), exceptthat here the twist angles are !p not !p;m.

On a lattice, discretization errors mean that differentdefinitions for the twist angles will lead to observablesthat differ by O�a�. In this work, we will define !p non-perturbatively as in Refs. [7,13,14,16] by enforcing the

absence of parity breaking in the physical basis. In par-ticular, we will enforce

hVb��x�Pa�y�i � 0; a; b 2 KnD: (31)

From the definitions in Eq. (27), this condition gives

tan!l �hV2

��x�P1�y�i

hA1��x�P

1�y�i; tan!h �

hV14� �x�P

13�y�i

hA13� �x�P

13�y�i:

(32)

The results for !p depend on the distance jx� yj at O�a�.We will enforce the condition in Eq. (31) at long distancewhere the single-meson contribution dominates.

Evaluating Eq. (32) using the results in the appendix, wefind at LO !p � !p;m � !p;0, since only Aa�;LO and Pa �Pa, a 2 f1; 13g, couple to the single-meson state. At NLO,the sole nontrivial contributions surviving in the ratios ofEq. (32) come from theW10 term, just as in the one-doublettheory, and as in Ref. [9] we find

tan!p �sin!p;m

cos!p;m � ; �

8W0a

f2 W10;

p � l; h:

(33)

C. Kaon masses and decay constants

With the Feynman rules in hand, and the twist anglesdefined, we have now all that is needed to calculate pseu-doscalar meson masses and matrix elements. At NLO, wefind that the mass of the neutral kaon is given by

m2K0 � M0 �

M0

6f4Im2

�8 � Im2

�15g

�8

f2 fM02�8L6 � 4L4 � 2L8 � L5�

� M0aW0�cl � ch��8 eW6 � 4 eW4 � 2 eW8 � eW5�

� 2W20a

2�4 eW06 � eW08��cl � ch�2 � eW08�sl � sh�2g;(34)

where

M 0 � �M0l �M0h�=2; (35)

m2�8� �M0l � 2M0h�=3; (36)

m2�15� �M0l � 5M0h�=6; (37)

are, respectively, the LO K, �8, and �15 squared masses,and cp � cos!p, sp � sin!p, which we can use instead ofcos!p;m and sin!p;m respectively at the order we work.Note that the usual continuum one-loop contribution [25]appears in Eq. (34),

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I m2 �m2

32�2f2 lnm2

�2R

; (38)

with �R being the renormalization scale. Note also that, asrequired, the neutral kaon mass depends only on the com-bination 2 ~W8 � ~W5 rather than on ~W8 and eW5 separately,and that the kaon mass is automatically O�a� improved atmaximal twist (cp � 0).

Flavor breaking in the kaon masses at NLO is givensolely by the analytic contribution, just as in the SU�2�theory [9], and it reads

m2K0 �m2

K � �64

f2 W20a

2slsh ~W08 � � ~W0864

f2

W20a

2�l�h

M0lM0h

;

(39)

where the second equality is derived using the fact that wecan replace sp by sp;0 at the order we are working.

Before calculating Eq. (39) explicitly, one might haveanticipated an expression that was quadruply suppressed inour power counting scheme, Eq. (7), due to the requirementof beingO�a2� andO��l�h�. However, Eq. (39) shows thatthe �l�h dependence enters as a ratio with M0lM

0h and the

squared kaon mass difference is therefore nonzero alreadyat NLO.

With the physical axial current defined in the appendix,the K decay constant to NLO is determined to be

fK � f�1�

4

f2 �4L4 � L5�M0 �W0a�cl � ch�

� �4 ~W4 � ~W5 �W10� �3

4Im2

� �3

4Im2

�8

� 2Im2K �

1

2Im2

Ds

�; (40)

where m2� � M0l, m

2K � m2

D � M0, and m2Ds� M0h are the

LO expressions for the pion, kaon and Ds squared masses,respectively. Our tm�PT conventions are such that f� �93 MeV. Note that the one-loop contributions from thepions and the �8 are the same as in the continuum SU�3�theory [25]. Flavor breaking effects enter first at O�a2�,which is NNLO for decay constants. The above resultshows that the decay constant depends only on the combi-nation ~W5 �W10, and is automatically O�a� improved atmaximal twist.

IV. SIMULATION DETAILS

We have performed quenched simulations using theaction of Eq. (2). The ensembles computed in Ref. [14],containing 300 gauge configurations each at � � 5:85 and� � 6:0, have subsequently been extended to include 600configurations each [26]. An additional ensemble at � �6:2 has also been generated, again using a pseudoheatbathalgorithm which acts on all SU�2� subgroups. Quark propa-gators are obtained from a 1-norm quasiminimal residualalgorithm, with periodic boundary conditions in all direc-tions. Simulation parameters are collected in Table I.

To remove O�a� errors, simulations must be done atmaximal twist, and to that end we employ Eq. (32) imple-mented, following Refs. [13,14,16] by

tan!l �

iP~xhV1�i2

4 � ~x; t�P1�i2�0�iP~xhA1�i2

4 � ~x; t�P1�i2�0�i; (41)

tan!h �

iP~xhV13�i14

4 � ~x; t�P13�i14�0�iP~xhA13�i14

4 � ~x; t�P13�i14�0�i: (42)

TABLE I. The parameters used for simulations in this work. Lattice spacings are taken from Ref. [15]. Each �amp;0; a�p;0� pair isthe result of tuning to maximal twist with method (i) as discussed in Sec. IV. The subscript p � l, h is used in the text to distinguish the‘‘light’’ quark doublet from the ‘‘heavy’’ quark doublet, but for purposes of numerical tuning in this table there is no distinction. Thetwist angle was obtained from Eq. (41).

� a [fm] # Sites # Configurations amp;0 a�p;0 Twist angle (degrees)

5.85 0.123 203 � 40 600 �0:8965 0.0376 90:0 0:3�0:9071 0.0188 90:2 0:6�0:9110 0.01252 90:6 0:8�0:9150 0.006 27 90:6 1:6

6.0 0.093 203 � 48 600 �0:8110 0.030 90:4 0:4�0:8170 0.015 91:0 0:7�0:8195 0.010 92:5 1:0�0:8210 0.005 95:5 2:1

6.2 0.068 283 � 56 200 �0:7337 0.021 649 89:1 0:8�0:7367 0.010 825 87:3 1:8�0:7378 0.007 216 86:3 2:8�0:7389 0.003 608 86:4 4:5

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where Oa�ib � �Oa � iOb�=���2p

. This method of tuningseparately at each twisted quark mass is only one possiblechoice. A nice summary of the situation in the genericsmall mass regime [Eq. (15)] that we are using has beengiven by Sharpe in Ref. [11]. In the language of that article,we are using method (i). A variant, called method (ii),extrapolates the results of method (i) to the point of van-ishing twisted mass, and then uses that definition of maxi-mal twist for all values of �p;0 �p � l; h�. Another option,called method (iii), relies on twistings in the scalar-pseudoscalar sector rather than the vector-axial sector.Method (iv) assumes that maximal twist can be sufficientlywell defined by simply holding the hopping parameterfixed at its critical value from the untwisted Wilson theory.

As sketched in Fig. 1 of Ref. [11], methods (i), (ii), and(iii) are all acceptable nonperturbative definitions of maxi-mal twist, and are superior to method (iv). Though most ofour simulations use method (i), we will make frequentcomparisons with results from the �LF Collaboration

[27,28] using method (ii). We will also present our ownresults from method (ii) when discussing aspects of scalarcorrelators in Sec. VI.

Throughout this work, only local operators are used, anderror bars reported in graphs and tables are statistical only.All statistical uncertainties are obtained from the bootstrapmethod with replacement, where the number of bootstrapensembles is 3 times the number of data points in theoriginal ensemble. Masses and decay constants are ob-tained from unconstrained three-state fits to correlators,using all time steps except the source. In the followingsections, each discussion includes references to the rele-vant figures, but we note here that the correspondingnumerical values are collected in Table II.

V. PSEUDOSCALAR AND VECTOR MESONS

Masses of the charged and neutral kaons, i.e. the groundstate pseudoscalar mesons containing one s (anti)quark

TABLE II. Numerical values from our simulations. These are also shown graphically in the figures. The rows with superscripts a andb refer to �a�h;0; a�l;0� � �0:015; 0:005� and (0.010,0.010), respectively.

� a�l;0 � a�h;0 �amPS�2 afPS ZV amV

Charged Neutral Charged Neutral

5.85 0.0752 0.1841(4) 0.2262(12) 0.1127(8) 0.620(4) 0.625(5) 0.622(3)0.0564 0.1388(4) 0.1827(14) 0.1064(8) 0.615(4) 0.591(8) 0.591(5)0.05012 0.1236(4) 0.1683(15) 0.1041(9) 0.614(4) 0.580(10) 0.582(6)0.04387 0.1085(4) 0.1536(19) 0.1019(9) 0.613(4) 0.573(13) 0.575(8)0.0376 0.0937(3) 0.1402(15) 0.0996(9) 0.607(4) 0.553(14) 0.563(7)0.03132 0.0784(10) 0.1259(16) 0.0972(13) 0.602(10) 0.539(18) 0.555(9)0.02507 0.0633(3) 0.1112(19) 0.0947(9) 0.602(5) 0.523(26) 0.548(10)0.02504 0.0633(3) 0.1117(17) 0.0946(9) 0.601(5) 0.522(25) 0.547(10)0.01879 0.0484(2) 0.0969(22) 0.0921(9) 0.601(5) 0.497(37) 0.539(12)0.01254 0.0327(2) 0.0824(33) 0.0892(10) 0.597(7) 0.456(58) 0.527(18)

6.0 0.060 0.1106(4) 0.1260(6) 0.0858(7) 0.661(4) 0.488(4) 0.484(3)0.045 0.0829(4) 0.0986(7) 0.0811(7) 0.656(5) 0.463(6) 0.462(4)0.040 0.0738(4) 0.0898(8) 0.0796(7) 0.654(5) 0.455(7) 0.456(5)0.035 0.0646(5) 0.0801(9) 0.0780(8) 0.654(6) 0.444(10) 0.450(7)0.030 0.0558(4) 0.0724(7) 0.0762(7) 0.649(5) 0.437(8) 0.441(6)0.025 0.0468(4) 0.0641(8) 0.0745(8) 0.647(6) 0.429(10) 0.437(7)0:020a 0.0378(4) 0.0550(10) 0.0726(8) 0.646(7) 0.418(15) 0.432(9)0:020b 0.0378(4) 0.0559(9) 0.0727(8) 0.644(6) 0.421(14) 0.433(9)0.015 0.0290(3) 0.0471(12) 0.0706(8) 0.644(7) 0.412(19) 0.430(11)0.010 0.0198(3) 0.0383(18) 0.0680(9) 0.637(11) 0.407(27) 0.426(15)

6.2 0.043298 0.0585(4) 0.0640(6) 0.0614(7) 0.692(10) 0.362(4) 0.360(3)0.032474 0.0441(4) 0.0497(6) 0.0582(8) 0.689(11) 0.345(6) 0.344(5)0.028865 0.0393(4) 0.0451(7) 0.0571(8) 0.689(11) 0.340(8) 0.340(5)0.025257 0.0346(5) 0.0406(7) 0.0562(8) 0.688(11) 0.335(9) 0.337(6)0.02165 0.0298(4) 0.0358(7) 0.0547(8) 0.686(12) 0.328(9) 0.329(6)0.018041 0.0250(4) 0.0313(7) 0.0536(9) 0.685(13) 0.322(11) 0.325(7)0.014433 0.0203(4) 0.0269(8) 0.0525(10) 0.684(13) 0.317(14) 0.324(8)0.014432 0.0203(4) 0.0268(8) 0.0523(10) 0.684(14) 0.317(14) 0.322(8)0.010824 0.0155(3) 0.0225(9) 0.0510(10) 0.684(16) 0.312(19) 0.321(9)0.007216 0.0107(5) 0.0184(11) 0.0495(14) 0.683(19) 0.305(26) 0.322(11)

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from the heavy doublet and one u or d (anti)quark from thelight doublet, are plotted in Fig. 1. Both twisted masses,a�l;0 and a�h;0, take on all values from Table I and thenormal mass term is tuned accordingly (also shown inTable I) to achieve maximal twist at each particular twistedmass value.

These data are expected to be consistent with a quenchedversion [29–31] of Eq. (34), which at maximal twist reads

m2K0 � M0

�1� quench

�ln

M0l�4�f�2

�M0h

M0h �M0l

lnM0hM0l

���

8M02

f2 �8L6 � 4L4 � 2L8 � L5� �64

f2 a2W2

0eW08(43)

� M0 �64

f2 a2W2

0eW08 �O�M02p � �O�M0quench�;

p � l; h; (44)

m2K � m2

K0 �64

f2 a2W2

0eW08

� M0 �O�M02p � �O�M0quench�; p � l; h; (45)

where quench is the standard coefficient for the quenchedlogarithm. As Fig. 1 indicates, linear fits in M0 to the data ateach � value yield excellent results, so O�M02p � andO�M0quench� effects are largely unnecessary. Notice, inparticular, that our linear fit for the charged kaon did leadto a nonzero (but small) residual kaon mass at M0 � 0, thusreminding us that corrections to the linear form are impor-tant for such details. We have verified, for example, that thefunction AM0 � BM0 lnM0 yields an equally excellent fit toour data, and of course it enforces the absence of anyresidual mass at M0 � 0.

Figure 1 also reveals differences between our resultswith method (i) and results from the �LF Collaboration[27] using equal quark and antiquark masses withmethod (ii). For charged mesons, the method (i) massdifference is smaller than the method (ii) difference, andas expected from tm�PT the chiral limits appear to be verysimilar. For neutral mesons, the chiral limits appear to besomewhat different on the coarser lattices, particularly at� � 6:0, but become consistent at� � 6:2. We recall fromFig. 2 of Ref. [27] that the �LF data at � � 6:0 happen tobe statistically above their fitted a2 extrapolation, so we seeno essential disagreement among any of the data setsdisplayed in our Fig. 1.

The difference between the squared masses of chargedand neutral kaons is plotted directly in Fig. 2, and is foundto be only mildly dependent on (twisted) quark mass overthe range we are studying. This implies that corrections toEq. (39), arising from higher orders in the tm�PT expan-sion, are small but noticeable.

Figure 3 shows the lattice spacing dependence of thesquared mass differences for four mass values that span therange of our available data. At leading-order in tm�PT,Eq. (39) indicates that this quantity should be independent

0 0.02 0.04 0.06 0.08 0.1aµ

l,0+aµ

h,0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

(am

PS)2

charged, method (i)neutral, method (i)charged, method (ii)neutral, method (ii)

(a) β = 5.85

0 0.02 0.04 0.06 0.08aµ

l,0+aµ

h,0

0

0.05

0.1

0.15

0.2

(am

PS)2

charged, method (i)neutral, method (i)charged, method (ii)neutral, method (ii)

(b) β = 6.0

0 0.01 0.02 0.03 0.04 0.05aµ

l,0+aµ

h,0

0

0.02

0.04

0.06

0.08

0.1

(am

PS)2

charged, method (i)neutral, method (i)charged, method (ii)neutral, method (ii)

(c) β = 6.2

FIG. 1 (color online). Pseudoscalar meson mass squared as afunction of the sum of quark and antiquark twisted mass pa-rameters. Subscripts l and h indicate the light and heavy dou-blets. Results labeled by method (i) are from the present work;results labeled by method (ii) are from Ref. [27] and have equalmasses for the quark and antiquark. Straight lines are linear fitsto the data from method (i).

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of mass, linear in a2 and vanishing in the continuum limit.Modulo the unknown higher order effects, Fig. 3 is inreasonable agreement with these expectations. In particu-lar, the approximate mass independence is evident and thedependence on a2 is approximately linear, though a linearfit misses the massless prediction at a � 0 by a few (sta-tistical) standard deviations.

It should be noted from Fig. 3 that even at our smallestlattice spacing, the mass splitting of mK0 �mK �50 MeV is significant relative to the kaon mass itself.However, in terms of the difference of mass squared, ourresults are consistent with the pseudoscalar meson masssplittings in Ref. [27] and compatible with the suggestionof Shindler [3] that flavor breaking effects in tmLQCD areof a magnitude comparable to ‘‘taste’’ symmetry violations

in pseudoscalar meson masses observed with improvedstaggered fermions [32]. The appearance of sizable latticespacing effects like this have led some authors to use apower counting scheme in whichO�a2� effects are taken tobe LO rather than NLO [10,33], but we will follow Eq. (7)throughout the present work.

0 0.05 0.1 0.15 0.2 0.25 0.3(am

PS)2

0.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

afPS

charged, method (i)charged, method (ii)

(a) β = 5.85

0 0.05 0.1 0.15(am

PS)2

0.06

0.065

0.07

0.075

0.08

0.085

0.09

afPS

charged, method (i)charged, method (ii)

(b) β = 6.0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07(am

PS)2

0.045

0.05

0.055

0.06

0.065

afPS

charged, method (i)charged, method (ii)

(c) β = 6.2

FIG. 4. The pseudoscalar meson decay constant as a functionof the squared charged pseudoscalar meson mass. Resultslabeled by method (i) are from the present work; results labeledby method (ii) are from Ref. [28]. Straight lines are linear fits tothe data from method (i).

0 0.005 0.01 0.015 0.02a

2 [fm

2]

0

0.05

0.1

0.15

0.2

∆(m

PS

2 ) [

GeV

2 ]

r0m

PS=1.75

r0m

PS=1.25

r0m

PS=1.00

r0m

PS=0.75

FIG. 3 (color online). The difference between charged andneutral squared pseudoscalar meson masses as a function ofsquared lattice spacing, for selected values of the charged mesonmass.

0 50 100 150µ

l,0+µ

h,0 [MeV]

0

0.05

0.1

0.15∆(

mPS

2 ) [

GeV

2 ]

β=5.85β=6.0β=6.2

FIG. 2 (color online). The difference between charged andneutral squared pseudoscalar meson masses as a function ofthe sum of quark and antiquark twisted masses. Subscripts land h indicate the light and heavy doublets.

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The decay constant of a charged pseudoscalar meson canbe obtained easily from the so-called indirect method[34,35],

fPS ��l;0 ��h;0

m2PS

jh0j �s�5ujK�ij: (46)

where the normalization is such that f� � 130 MeV, i.e.larger than the normalization from our tm�PT conventionsby

���2p

. Unfortunately, the indirect method does not provideeasy access to the neutral pseudoscalar decay constant, dueto mixing with the scalar operator. The neutral decayconstant is not directly accessible in the laboratory due tothe absence of flavor-changing neutral currents in thestandard model, but it is a quantity that appears in theparametrization of some neutral kaon matrix elements(see Ref. [19] for a very recent example in the context oftmLQCD). From the point of view of our work, the com-parison of charged and neutral cases would be able toprovide information about how flavor symmetry breakingin tmLQCD affects the structure of mesons. Note that if wehad chosen a different convention in Eq. (1), i.e. inter-changing the role of c and s quarks as was done in Ref. [12]which focuses on neutral kaons, then the situation would bereversed: the indirect method would have applied to theneutral kaon and not to the charged kaon.

The charged kaon decay constant is plotted in Fig. 4where we continue to define this meson to be the groundstate pseudoscalar meson containing one s (anti)quarkfrom the heavy doublet and one u (anti)quark from thelight doublet. Central values show a hint of curvature, butwithin statistical uncertainties the decay constant is linearin the squared meson mass. This is consistent with thetm�PT expression, i.e. the quenched version of Eq. (40)where the slope has no logarithmic corrections. [We haveverified that a tm�PT calculation of the right-hand side ofEq. (46) also yields Eq. (40).] The computations ofRef. [28], using method (ii), also appear in Fig. 4 and arein agreement with the method (i) results.

As is evident from Fig. 5, there is no visible dependenceof the decay constant on lattice spacing. For the kaon,fitting the data from three � values linearly yields

r0mK � 1:25) fK � 161 5 MeV: (47)

Relying on a linear chiral extrapolation, we similarly ob-tain

f� � 142 4 MeV: (48)

The ratio,

fKf�� 1:136�7�; (49)

agrees nicely with the quenched results from Ref. [31],though the individual decay constants are somewhat larger.

Using the lattice spacings derived from Ref. [36] or [37]would bring us into closer agreement.

There is also a direct method [35] for obtaining thedecay constant, though it requires input of a renormaliza-tion factor for the twisted vector current. Here, we will usethe ratio of results from the direct and indirect methods todetermine this renormalization factor. Figure 6 shows thatthe renormalization factor is essentially mass-independentand that it becomes closer to unity as a! 0. The numericalvalues are comparable to those obtained by the authors ofRef. [28], and those authors also note that ZV is furtherfrom unity in tmLQCD than in both standard and boostedlattice perturbation theory. (See their Table 7.)

Vector meson masses, referred to here as K� massessince the strange (anti)quark from the heavy doublet iscombined with a u or d (anti)quark from the light doublet,are shown in Fig. 7, as computed from local operators ofthe form

P3k�1

� �k . Within the statistical uncertainties,

0 1 2 3 4(r0m

PS)2

0.55

0.6

0.65

0.7

0.75

ZV

β = 5.85β = 6.0β = 6.2

FIG. 6 (color online). The renormalization factor associatedwith the pseudoscalar meson decay constant.

0 0.02 0.04 0.06 0.08 0.1(a/r

0)2

0.3

0.35

0.4

0.45

0.5

r 0f PS

r0m

PS=1.75

r0m

PS=1.25

r0m

PS=1.00

r0m

PS=0.75

FIG. 5 (color online). Scaling of the pseudoscalar decay con-stant for four choices of the quark mass.

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no mass splitting is visible between charged and neutralvector mesons. Comparison with data from the �LFCollaboration reveals that methods (i) and (ii) lead todifferent K� masses on coarser lattices, and that the dis-tinction vanishes as a! 0. The sizable uncertainties make

chiral extrapolations difficult, particularly for charged me-sons. Linear fits to the neutral meson masses at each � aredisplayed in Fig. 7.

It is noteworthy that the neutral K� masses are moreprecise than the charged K�, just as the charged pseudo-scalar masses are more precise than the neutral pseudosca-lar. In both cases, the better precision comes in the channelwhere the interpolating fields are invariant under twisting.Moreover, the absence of large cutoff effects for chargedpseudoscalars has been associated with the existence of anexact lattice axial Ward-Takahashi identity [38].

Scaling of the neutral vector meson mass with a2 isshown in Fig. 8. The nonvanishing dependence on a2 isbarely significant with respect to the uncertainties. A lineara2 fit to the r0mPS � 1:25 data produces

mK� � 970 20 MeV; (50)

and a linear a2 fit to the linear chiral extrapolations fromFig. 7 yields

m � 916 20 MeV: (51)

These quenched values lie above the physical values, butusing the lattice spacings derived from Ref. [36] or [37]would bring us closer to experiment.

VI. SCALAR MESON MASSES AND MIXINGS

Our chosen definition of maximal twist, Eq. (32), tunesthe mixing of vector and axial currents for charged mesons,or more precisely, for mesons built from a quark andantiquark having twist angles of opposite sign. Chargedscalar and pseudoscalar densities do not mix. Conversely,there is mixing of neutral scalar and pseudoscalar densities,while neutral vector and axial currents do not mix. Ourcharged vector-axial tuning to maximal twist can differ

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07(a/r

0)2

1.5

2

2.5

3

r 0m

V

r0m

PS=1.75

r0m

PS=1.25

r0m

PS=1.00

r0m

PS=0.75

FIG. 8 (color online). Scaling of the neutral vector meson massfor four choices of the quark mass. The straight line is a linear fitto the data having r0mPS � 1:25.

0 0.05 0.1 0.15 0.2 0.25(am

PS)2

0.3

0.4

0.5

0.6

0.7

0.8am

V

charged, method (i)neutral, method (i)charged, method (ii)

(a) β = 5.85

0 0.05 0.1 0.15(am

PS)2

0.3

0.35

0.4

0.45

0.5

0.55

0.6

amV

charged, method (i)neutral, method (i)charged, method (ii)

(b) β = 6.0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07(am

PS)2

0.2

0.25

0.3

0.35

0.4

amV

charged, method (i)neutral, method (i)charged, method (ii)

(c) β = 6.2

FIG. 7 (color online). Vector meson mass as a function of thesquared charged pseudoscalar meson mass. Results labeled bymethod (i) are from the present work; results labeled bymethod (ii) are from Ref. [28]. Straight lines are linear fits tothe neutral data from method (i).

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from a neutral scalar-pseudoscalar definition, for example,by differing discretization effects.

Figure 9(a) shows four correlators at � � 6:2 with ourheaviest quark mass: charged and neutral scalar and pseu-doscalar two-point correlators. The charged correlatorscannot mix, and we see a clear ground state exponentialbehavior for the pseudoscalar and scalar mesons, with nocontamination between them. The neutral pseudoscalarcorrelator also provides a clear ground state, where theslope in this log plot is slightly steeper than the chargedcase, as expected since we have already established that theneutral meson is heavier than the charged meson. Theneutral scalar correlator is noticeably different: it maintainssurprisingly small error bars even far from the source, itdisplays a kink (change of slope on the log plot) neartimesteps 12 and 46, and it mirrors the neutral pseudoscalarcurve between these timesteps. Apparently the quickly-decaying scalar signal is being overcome by the pseudo-scalar further from the source, i.e. we are seeing scalar-pseudoscalar mixing.

Figure 9(b) shows the same four correlators but nowwith our lightest quark mass. The effects are now moredramatic, and a new phenomenon is also observed. Thecharged scalar has a brief signal for the scalar meson nearthe source, then the correlator becomes negative. Theneutral scalar similarly has a brief signal, then makes acurious waving shape on the graph. To understand this, seeFig. 10 where the data from Fig. 9(b) are replotted on alinear scale. The negative contribution to the chargedscalar correlator is from the two particle state—quenched�0 and kaon—as discussed in Refs. [39,40]. The neutralscalar has this two particle state as well, but the correlatoris deformed because it apparently also has a mixing withthe neutral pseudoscalar.

One could imagine removing the scalar-pseudoscalarmixing by tuning to maximal twist directly in this sector,called method (iii) in the notation of Ref. [11], but mixingcould then arise between vector and axial currents. Aninteresting observation, sketched in Fig. 1 of Ref. [11], is

0 10 20 30 40 50 60time

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

γ5, charged

γ5, neutral

1, neutral1, charged

FIG. 10 (color online). The data from Fig. 9(b), replotted on alinear scale.

0 10 20 30 40 50 60time

-0.001

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

γ5, charged, method (i)

γ5, charged, method (ii)

γ5, neutral, method (i)

γ5, neutral, method (ii)

1, neutral, method (i)1, neutral, method (ii)1, charged, method (i)1, charged, method (ii)

FIG. 11 (color online). The data from Fig. 10, compared tocomputations using method (ii).

0 10 20 30 40 50 60time

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

γ5, charged

γ5, neutral

1, neutral1, charged

(a) µ = 0.021649

0 10 20 30 40 50 60time

10-6

10-5

10-4

10-3

10-2

10-1

100

γ5, charged

γ5, neutral

1, neutral1, charged

(b) µ = 0.003608

FIG. 9 (color online). Scalar and pseudoscalar correlationfunctions for our (a) heaviest and (b) lightest quarks at � �6:2. The notation �5 and 1 refers to the physical basis, definedusing method (i).

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that method (iii) is distinct from method (i) but identical tomethod (ii) up to O�a2� corrections. Could method (ii) beoptimal for both the vector-axial and the scalar-pseudoscalar sectors?

We have computed 100 quark propagators usingmethod (ii) at � � 6:2 and �l;0 � 0:003 608, allowing adirect comparison to our results from method (i). Themethod (ii) value of normal quark mass, ml;0 ��0:741 546, was obtained from Table 3 in Ref. [27]. Ourfindings are displayed in Fig. 11, and we see that the scalar-pseudoscalar mixing is still apparent, confirming the pres-ence of O�a2� effects.

VII. UNTWISTED STRANGE WITH TWISTED UPAND DOWN QUARKS

If the twist angle, !h, of the heavy doublet is set to zero,then the strange quark becomes a standard Wilson fermionand its partner can be erased from the action. Exceptionalconfigurations are typically not a problem for strangequarks, and O�a� improvement could be accomplishedvia a Sheikholeslami-Wohlert term if desired, though wewill use the unimproved Wilson action here. The �u; d�doublet will be kept at maximal twist.

With this action, the K� and �K0 mesons are exactlydegenerate configuration by configuration since one corre-lator is the Hermitian conjugate of the other. The same istrue for K� and K0. Furthermore, these two pairs arenumerically degenerate in the configuration average. Thiscan easily be seen in the free quark limit, since the twistedmass in one propagator cannot contribute to the correlatorif the other propagator is Wilson, due to the odd number of�5’s. Similarly, our tm�PT expression for the mass differ-ence, Eq. (39), explicitly vanishes when !h � 0.

Numerical results at � � 6:0 for kaon masses obtainedwith a Wilson strange quark are compared to results with amaximally-twisted strange quark in Fig. 12. For this plot,

the twisted strange quark is held fixed at a�h;0 � 0:030and the Wilson strange quark’s hopping parameter, �, istuned such that the pseudoscalar mass (obtained from twoWilson propagators without any twisting) becomes nu-merically equal to the charged twisted pseudoscalarmass, amPS � 0:332�1�. The resulting hopping parameteris � � 0:1545 or equivalently mh;0 � �0:7634. In bothcases, the light quark takes on all four �ml;0; �l;0� valuesfrom Table I.

Figure 12 shows that the Wilson strange quark leads tokaon masses that are numerically between the charged andneutral kaons with a twisted strange quark. All curves arevisibly linear in a�l;0, though the Wilson strange quarktheory has a smaller slope than the twisted strange quarktheory. Recall that method (i), used here, itself has asmaller slope than method (ii).

Untwisting the strange quark has other effects besideseliminating the mass splitting. When an untwisted quarkfield is combined with a maximally-twisted one, parityviolation induces parity mixing in all (charged and neutral)channels [this can be inferred from Eq. (27)]. This is unlikethe situation with complete maximal twisting where insome channels the parity violation in the action inter-changes parity of the correlator but does not mix it. Withan untwisted strange quark the extraction of the decayconstant becomes a much more difficult problem whichis beyond the scope of the present work.

VIII. SUMMARY

Twisted mass lattice QCD is a practical method fornumerical simulations involving light quarks. It has noexceptional configurations, automatic O�a� improvement,and a corresponding version of chiral perturbation theory.However, there are issues of parity and flavor symmetryviolation effects at nonzero lattice spacing that have to beunderstood and dealt with.

Quarks come in pairs in tmLQCD, so the best way toimplement three-flavor simulations requires thought andexploration. In this work, we have considered two doubletsat maximal twist, where in our quenched simulations thefourth quark is benign. This is in line with the two-doublettmLQCD proposed in Ref. [12]. The chiral perturbationtheory is formulated as a natural generalization of theexisting two-flavor formulation, and used to obtain analyticexpressions for masses and decay constants. NumericaltmLQCD results for m�, f�, mK, fK, m , and mK� areobtained from four twisted quark masses at each of threelattice spacings, and are comparable to previous quenchedstudies with other actions.

Though dynamical simulations were not performed inthis study, that is certainly an ultimate goal for QCDphenomenology. Dynamical simulations of the theorywith two twisted doublets would mean the fourth quarkis no longer benign. Identifying it with the physical charmquark requires the introduction of a mass splitting within

0 0.01 0.02 0.03 0.04aµ

l,0

0

0.05

0.1

(am

PS)2

K+,K

- with twisted strange

K0,K

0 with twisted strange

kaons with Wilson strange

pure Wilson mesonmatches this point

FIG. 12 (color online). Squared pseudoscalar meson masseswith one strange quark/antiquark and one light quark/antiquark,plotted as a function of the light quark’s twisted mass.

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the heavy doublet. Progress toward two-doublet dynamicaltwisted mass simulations is reviewed in Ref. [18].Alternatively, one could avoid an active charm quark byusing a mixed action formalism [20], for example, withtwisted �u; d� and untwisted s quarks in the sea (recallSec. VII), and Eq. (2) used for the valence quarks (so thestrange quark’s partner is again benign).

One of the significant twist artifacts found in this work isthe mass difference between charged and neutral kaons,which vanishes in the continuum limit but remains sizableat the lattice spacings studied here, 0:068 fm <a<0:123 fm. This splitting depends upon the particular actionthat has been chosen; it may be different in other variantsof tmLQCD or in nonquenched simulations. For example,if only the up and down quarks, not the strange quark, aretwisted, then this large splitting vanishes however at theprice of a more complicated pattern of parity mixing in thecorrelators. O�a� errors also arise in that scenario, thoughthese could be removed by the addition of a suitable cloveroperator.

Another artifact of twisting is the mixing of scalar andpseudoscalar operators when the standard definitions ofmaximal twist are employed. For sufficiently light quarksin the quenched approximation, this can be studied throughthe appearance of negative correlators that correspond tothe opening of a quenched �0K channel.

Notwithstanding the existence of twisted lattice arti-facts, we see value in the general approach of tmLQCDfor applications involving u, d, and s quarks. There are anumber of options for constructing the action includingstrange quarks, and with systematic studies such as this oneexploring them we can be hopeful that an optimal approachwill be found.

ACKNOWLEDGMENTS

R. L. wishes to thank the TRIUMF Theory Group andthe Jefferson Lab Theory Center for hospitality and supportduring parts of this research. R. L. also thanks NilmaniMathur for a helpful conversation about scalar mesonsand the quenched �0. This work was supported in part bythe Natural Sciences and Engineering Research Council ofCanada, the Canada Foundation for Innovation, the CanadaResearch Chairs Program and the Government ofSaskatchewan.

APPENDIX: CURRENTS AND DENSITIES IN THETWISTED BASIS

With the generators of SU�2� replaced by those ofSU�4�, the currents and densities in the twisted basis ofthe two-doublet theory are defined in the same way as forthe one-doublet theory [7]. At LO, the currents and den-sities have the same form, mutatis mutandis, as in theSU�2� theory. In the physical basis, i.e. in terms of thephysical variable �ph, they take the forms (with a hat

denoting the physical basis quantity),

Va�;LO � cos��a!m�Va�;LO � �ab sin��a!m�A

b�;LO;

V3;8;15�;LO � V3;8;15

�;LO ;

Aa�;LO � cos��a!m�Aa�;LO � �ab sin��a!m�V

b�;LO;

A3;8;15�;LO � A3;8;15

�;LO ;

S0LO �

1

2�cl;m � ch;m�S

0LO � 2isl;mP

3LO

� 2�cl;m � ch;m��

1���3p S8

LO �1���6p S15

LO

� 2ish;m

�1���3p P8

LO �

���2

3

sP15

LO

�;

SaLO � cos��a!m�SaLO � i sin��a!m�P

aLO;

PaLO � cos��a!m�PaLO � i sin��a!m�S

aLO;

a; b 2 KnD; K � f1; . . . ; 15g; D � f3; 8; 15g;

(A1)

where cp;m � cos!p;m, sp;m � sin!p;m, p � l, h, and weuse the notation defined in Eqs. (28)–(30). Note that in theSU�4� theory, P0

LO and SkLO, k 2 K, do not vanish identi-cally in contrast to the SU�2� theory [9].

At NLO, the vector and axial currents are given by

Vk� � Vk�;LO�1� C� � L1;2;3;9 terms

� L51

2Tr��k;��@��y � @���k;���

� ��0�y ���0y� � eW51

2Tr��k;��@��y

� @���k;����A�y ��Ay�;

Ak� � Ak�;LO�1� C� �8aW0

B0f2 W10@�PkLO � L1;2;3;9 terms

� L51

2Tr��k;��@��y � @���k;���

� ��0�y ���0y� � eW51

2Tr��k;��@��y

� @���k;����A�y ��Ay�; (A2)

where k 2 K, and

C �4L4

f2 Tr�0y�� �y�0 �4 eW4

f2 Tr�Ay�� �yA�:

(A3)

We do not give the form of the L1;2;3;9 terms since each hasthe same form as in the continuum SU�2� theory [41].

Dropping terms proportional to the scalar and pseudo-scalar sources, which give rise only to contact terms incorrelation functions, the scalar and pseudoscalar densitiesat NLO are given by

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Sk � SkLO�1�D1� � PkLOD2

� L5B0 Tr@��@��y��k�y ���k�

� L82B0 Tr��k�y � ��k���0�y ���0y�

� eW8B0 Tr��k�y � ��k��A�y � �Ay�;

Pk � PkLO�1�D1� � SkLOD2

� L5B0 Tr@��@��y��k�y ���k�

� L82B0 Tr��k�y � ��k���0�y ���0y�

� eW8B0 Tr��k�y � ��k��A�y � �Ay�

� 4iH2B20 Tr��k��; (A4)

where k 2 K, and

D 1 � �4L4

f2 TrD��D��y �8L6

f2 Tr��0y���y��

�4 eW6

f2 Tr�Ay�� �yA�;

D2 � �8L7

f2 Tr��0y���y�0� �4 eW7

f2 Tr�Ay�� �yA�:

(A5)

To write the NLO currents and densities in the physicalbasis, we need the results

Tr�Dy��D��y� � Tr�Dy��phD��yph�;

Tr��0y� �y�� �

8><>:� 4M0

2B0f2 S0LO �

4�M0

B0f2 �1��3p S8

LO �1��6p S15

LO� ��sign�

4M0

2B0f2 P0LO �

4�M0

B0f2 �1��3p P8

LO �1��6p P15

LO� ��sign��O�M0pp�;

Tr�Ay� �yA� �

8><>:� 8W0a

2B0f2 S0LO ��sign�

8W0a2B0f2 P0

LO ��sign��O�ap�; p � l; h;

(A6)

where

M 0 � �M0l �M0h�=2; �M0 � M0l �M

0h; (A7)

which allow us to express C, D1, and D 2 in terms of the physical fields.

Next we write the L5, L8, eW5, eW8, and H2 terms in the physical basis. For the L5 terms, we need the results

1

2Tr��a;��@��y � @���a;�����0�y � ��0y� �

� cos��a!m�VaL5� �ab sin��a!m�A

bL5�upper sign�

cos��a!m�AaL5� �ab sin��a!m�V

bL5�lower sign�

;

Tr@��@��y��a�y ��a� �

� cos��a!m�SaL5� i sin��a!m�P

aL5��sign�

cos��a!m�PaL5� i sin��a!m�S

aL5��sign�

; (A8)

where a, b 2 KnD, and

VaL5� 1

2h�a��ph@�yph; �0tw�y � �ph�

0ytw� � ��$ �y; �0 $ �0y��i;

�0tw � �ym�0�ym

AaL5� 1

2h�a��yph@�ph; �

0ytw���yph�

0tw� � ��$ �y; �0 $ �0y��i;

SaL5� �h�a�@��ph�yph@��ph � H:c:�i;

PaL5� h�a�@��ph�yph@��ph � H:c:�i;

(A9)

and for the L8 terms we need the results

� Tr��a�y ��a���0�y � ��0y� �

�cos��a!m�S

aL8� i sin��a!m�P

aL8��sign�

cos��a!m�PaL8� i sin��a!m�S

aL8��sign�

;

SaL8� �h�a��ph�

0ytw�ph � H:c:�i; PaL8

� h�a��ph�0ytw�ph � H:c:�i; a 2 KnD:

(A10)

By replacing �0 with A, and L5;8 with eW5;8, the eW5 and eW8 terms can be expressed in the physical basis using the sameresults above for the L5 and L8 terms.

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Lastly, theH2 term contributes only in the flavor-diagonal case, i.e. when the flavor index k � 3, 8, 15. Since we will notbe using the flavor-diagonal currents and densities, we do not give results for the flavor-diagonal cases here.

To conclude this appendix, we provide the explicit expression for the axial current in the physics basis at NLO, using thetwist angles determined in Sec. III B:

Aa� � Aa�;LO �8W0a

f2 W10 cos��a!m� sin��a!m��abVb�;LO�1� C� � L1;2;3;9 terms�

8W0a

B0f2 W10 cos��a!m�

� cos��a!m�@�PaLO � i sin��a!m�@�S

aLO � L5

�AaL5�

8W0a

f2 W10 cos��a!m� sin��a!m��abVbL5

�� L5 $ eW5: (A11)

The term C is the same as in Eq. (A3), but now given in the physical basis.

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