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PHYSICAL REVIEW D 72, 071501(R) (2005)

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Continuum physics with quenched overlap fermions

Stephan Durr1 and Christian Hoelbling2

1Universitat Bern, ITP, Sidlerstr. 5, CH-3012 Bern, Switzerland2Universitat Wuppertal, Gaussstr. 20, D-42119, Wuppertal, Germany

(Received 19 August 2005; published 25 October 2005)

1550-7998=20

We calculate mud � �mu �md�=2, ms, f� and fK in the quenched continuum limit with UV-filteredoverlap fermions. We see rather small scaling violations on lattices as coarse as a�1 ’ 1 GeV andconjecture that similar advantages would be manifest in unquenched studies.

DOI: 10.1103/PhysRevD.72.071501 PACS numbers: 11.15.Ha, 12.38.Gc

TABLE I. Simulation parameters and statistics. The box vol-ume in physical units L3 � T � �1:5 fm�3 � 3:0 fm (based onr0 � 0:5 fm) is kept fixed.

N3L � NT 83 � 16 103 � 20 123 � 24 163 � 32

� 5.66 5.76 5.84 6.0a [fm] 0.188 0.149 0.125 0.093# conf. 30 30 30 30

I. INTRODUCTION

Overlap fermions [1] satisfy the Ginsparg-Wilson rela-tion [2]

�5D�D�5 � 0; �5 � �5

�1�

1

�D�

(1)

which implies exact chiral symmetry at finite lattice spac-ing [3,4]. While theoretically clean, calculations with over-lap fermions are considered a computational challenge. Ina recent investigation [5] it has been conjectured that a UV-filtered (smeared) Wilson kernel operator, as previouslysuggested in [6–9], might substantially reduce the compu-tational cost associated with obtaining continuum physics[10]. In [5] the focus was on technical aspects, but it is clearthat the point relevant in practical applications is whetherUV-filtered (‘‘thick link’’) overlap fermions would extendthe scaling region to significantly coarser lattices, even ifone refrains from (�-specific) tuning and sets � � 1. Thepresent note addresses this question by studying the con-tinuum limit of the pseudoscalar masses and decay con-stants with quenched UV-filtered overlap quarks. Nosystematic comparison to plain (‘‘thin link’’) overlap fer-mions is made, because we could not obtain reasonablesignals for plain overlap fermions with � � 1 on ourcoarser lattices. Given the exploratory nature of the presentinvestigation, we restrict ourselves to pseudoscalar mesoncorrelators in the p-regime of chiral perturbation theory(XPT) [11]. Our main results are the values of thequenched strange quark mass and K decay constant inthe continuum

ms�MS; 2 GeV� � 119�10��7� MeV;

fK � 170�10��2� MeV(2)

as well as the quark mass and decay constant ratios

ms

mud� 23:3�7:1��4:5�; fK=f� � 1:17�4��2� (3)

where mud � �mu �md�=2 and no numerical input fromXPT has been used. Throughout this paper the first error isstatistical and the second systematic, but the latter does notinclude any estimate of the quenching effect.

05=72(7)=071501(5)$23.00 071501

II. TECHNICAL SETUP

We use the Wilson gauge action. The massless overlapoperator is [1]

Dov � ��1� �DW � ��DyW � ��DW � ��1=2� (4)

with DW the massless Wilson operator. The UV-filteredoverlap is constructed by evaluating the Wilson operator onAPE [12] or HYP [13,14] smeared gauge configurations,resulting in an O�a2� redefinition of the fermion action[5,15]. We use 1 or 3 iterations with smearing parameters�APE � 0:5 or �HYP � �0:75; 0:6; 0:3� and the shift pa-rameter � � 1 is kept fixed. Based on (4) the massiveoperator is defined through

Dov;m �

�1�

am2�

�Dov �m: (5)

We set the lattice spacing with the Sommer parameter[16], i.e. we give all intermediate results in appropriatepowers of r0 to facilitate comparison with other quenchedstudies. Only the final result will be converted into physicalunits assuming a standard value for r0. We have data at 4different lattice spacings in matched boxes of physicalvolume L3 � 2Lwith L ’ 3r0. The couplings were chosenwith the interpolation formula given in [17]. The details ofthe simulation are summarized in Table I.

For each coupling and filtering level we use 4 bare quarkmasses. Ideally one would choose them such as to alwaysobtain the same 4 renormalized quark masses in r0 units (orthe same 4 pseudoscalar masses), but for this one wouldhave to know the renormalization factor Zm � Z�1

S before-hand. Our values as summarized in Table II are not bad; ourrenormalized masses are roughly in the region

-1 © 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.071501

TABLE II. The 4 regularly spaced bare quark masses percoupling and filtering.

83 � 16 103 � 20 123 � 24 163 � 32

None 0:16 0:4 0:16 0:4 0:16 0:4 0:08 0:21 APE 0:08 0:2 0:08 0:2 0:04 0:1 0:04 0:13 APE 0:04 0:1 0:04 0:1 0:03 0:075 0:03 0:0751 HYP 0:08 0:2 0:08 0:2 0:04 0:1 0:04 0:13 HYP 0:04 0:1 0:04 0:1 0:03 0:075 0:03 0:075

TABLE III. Renormalization constant ZMSS �2 GeV� for Wilson

glue and � � 1. The main effect of filtering is to bring it muchcloser to its tree-level value 1.

� � 5:66 � � 5:76 � � 5:84 � � 6:0

None ill-def. ill-def. 6.14(15)(38) 2.54(6)(12)1 APE 3.05(7)(34) 1.79(4)(6) 1.60(3)(8) 1.26(3)(8)3 APE 2.05(7)(43) 1.35(4)(8) 1.25(2)(8) 1.04(2)(7)1 HYP 1.71(4)(24) 1.24(3)(6) 1.23(2)(6) 1.03(2)(7)3 HYP 1.57(6)(25) 1.16(3)(5) 1.10(2)(7) 0.98(2)(6)

STEPHAN DURR AND CHRISTIAN HOELBLING PHYSICAL REVIEW D 72, 071501 (2005)


13m

physs . . .mphys

s .In the course of this calculation we will need both ZS �

ZP (to determine the renormalized quark masses) andZV � ZA (for the decay constants), where the allegedidentity is specific for the massless overlap operator.

To compute the scalar and pseudoscalar renormalizationconstant we follow the nonperturbative RI-MOM proce-dure as defined in [18] and first applied to overlap fermionsin [19,20]. We compute both ZS�� and ZP��; in the lattercase a 1=m term is used to extrapolate the result to thechiral limit. It turns out that the values obtained are in goodagreement even on our coarsest lattice. We fit the scalarrenormalization constant to the form

ZRI�MOMS �� U��ZRGI

S � const�a��2; (6)

where U�� is the 4-loop running in the RI-MOM schemeas given in [21] and the second term is introduced to

0 10 20 30

µ2r02

0

0.1

0.2

0.3

0.4

0.5

ZSIR

( µ)/

U(µ

)

0 10 20 30 40 50 60 70 80 90 100

µ2r02

0

0.1

0.2

0.3

0.4

0.5

ZSRI

(µ)/

U(µ

)

FIG. 1 (color online). ZRI�MOMS ��=U�� on our coarsest and

finest (� � 5:66, 6.0) lattice using the 1 HYP overlap operatorwith � � 1. The solid line indicates a linear fit with range 3 �r0 5 and 5 �r0 9, respectively.

071501

account for discretization effects. In other words the scalarrenormalization constant after dividing out the 4-loop per-turbative running should be flat, up to discretization ef-fects, and Fig. 1 shows that the latter are indeed non-negligible. The wiggles in the data signal rotational sym-metry breaking on the lattice. We checked that withinerrors the slope disappears in proportion to �a=r0�

2. Thephenomenological analysis below is based onZRI�MOMS �2 GeV�, where � � 2 GeV is realized through�r0 � 5:067 73. The systematic error is estimated by vary-ing the fit range, by including additional 1=p2 terms intothe fit, and by comparing to the ZRI�MOM

P �2 GeV� data. Asummary of our results, after conversion to �MS; 2 GeV�conventions, is presented in Table III. Choosing a fixed� > 1 would delay the breakdown of the unfiltered ver-sion—see [5] for details. Note that the ZS factors of allUV-filtered operators are much closer to 1, even whencompared to the unfiltered overlap operator with tuned �[19,22,23]. This suggests that one should be able to com-pute renormalization constants perturbatively, as was donein [9] in a slightly different setup.

The second ingredient is the axial-vector renormaliza-tion constant ZA. Here we use the values given in [5](coming from a PCAC renormalization condition), com-plemented by a � � 5:76 column included in [24]. It turnsout that the values in this column are in fair agreement withthe prediction by the Pade curve given in [5], which isanother indication that for the filtered overlap operatorlattice perturbation theory might work rather well.

III. PHYSICAL RESULTS

To extract meson masses and decay constants we com-pute the correlators

C�1;�2�t� �

Xxh � 1�0��1 2�0� � 2�x; t��2 1�x; t�i (7)

where

�1� �5�5

2 (8)

denotes the ‘‘chirally rotated’’ quark field [4]. Specifically,we consider

-2

0 0.2 0.4 0.6 0.8 1(m1+m2) r0

Mπ2 r0

2

1

MK2 r0

22

3

4

5

6

M2 P r

02

0 0.2 0.4 0.6 0.8 1(m1+m2) r0

Mπ2 r0

2

1

MK2 r0

22

3

4

5

6

M2 P r

02

FIG. 2 (color online). MPr0 versus the bare quark mass �m1 �m2�r0 on our coarsest and finest (� � 5:66, 6.0) lattice for the 1HYP operator. The masses come from a fit to the hA0A0icorrelator in the range 5NT=16 t 11NT=16. The solid curverepresents a fit to the functional form (10) and the horizontallines indicate the physical M�r0 and MKr0 values. The latter areused to read off the fitted 2mudr0 and �ms �mud�r0, respectively.

TABLE IV. �ms �mud�r0 in �MS; 2 GeV� conventions. Forthe � � 1 thin link action at � � 5:66, 5.76 no Zm � Z�1

S isavailable (cf. Table III).

� � 5:66 � � 5:76 � � 5:84 � � 6:0


TABLE V. 2mudr0 in �MS; 2 GeV� conventions. For the � � 1thin link action at � � 5:66, 5.76 no Zm � Z�1

S is available (cf.Table III).

� � 5:66 � � 5:76 � � 5:84 � � 6:0


CONTINUUM PHYSICS WITH QUENCHED OVERLAP FERMIONS PHYSICAL REVIEW D 72, 071501 (2005)


hSSi�t� � C1;1�t�; hPPi�t� � C�5;�5�t�;

hA0A0i�t� � C�5�0;�5�0�t�

(9)

and we extract the mass and decay constant of the pseu-doscalar meson from fits to the hA0A0i and hPPi � hSSi[25] channels. We do not consider hPPi, since it is knownto be contaminated by zero mode contributions [20,26].

Generically, we see plateaus in the effective mass fromabout NT=4 on, with the exception of the unfiltered opera-tor, where the plateau sets in later and is less pronounced.Our central values stem from a fit to the hA0A0i correlatorin the range 5NT=16 t 11NT=16. The theoretical erroris dominated by the comparison to hPPi � hSSi. Includingonly variations of the fit range (and below the error on ZS),it would be much smaller.

In Fig. 2 the pseudoscalar meson mass squared is plottedversus the bare quark mass. Using regularly spaced quarkmasses, each m1 �m2 combination is realized in severalways and the pertinent MP are in excellent agreement. Inother words, isospin breaking effects are completely neg-ligible. Therefore, we fit the quark mass dependence of thepseudoscalar meson mass with the resummed quenchedXPT expression [27]

M2P � A�m1 �m2�

1=�1�� B�m1 �m2�2 (10)

071501

which is strictly true only for degenerate masses. Our �fluctuates wildly and is thus trivially consistent with 0.2[27].

The next step requires some experimental input. Sincewe do not see any isospin breaking effects and electromag-netic corrections are of the order of 1%, while our statis-tical errors turn out to be roughly 10%, we decided to usethe charged meson masses as input. Sticking to the identi-fication r0 � 0:5 fm we use MKr0 � 493:7 MeV 0:5 fm � 1:251 and M�r0 � 139:6 MeV 0:5 fm �0:3537 as input. This gives the bare �m1 �m2�r0 valuesindicated by a horizontal error bar in Fig. 2. Multiplyingthe bare masses with the Zm � Z�1

S obtained earlier, weextract the renormalized quark masses. Our results forms �mud and 2mud are reported in Tables IV and V,respectively.

To finally push to the continuum we perform a combinedfit (including all couplings and smearing levels) with acommon continuum value and individual O�a2� terms.We do this for the quantities �ms �mud�r0, 2mudr0 (asshown in Fig. 3) as well as for msr0 and ms=mud. Our finalresult is

ms�MS; 2 GeV�r0 � 0:301�25��7� (11)

in the continuum [�2=d:o:f: � 2:1] which, upon usingr0 � 0:5 fm, leads to the result given in (2). For the ratioms=mud we find the value quoted in (3) [�2=d:o:f: � 0:32].In either case the generic comment on statistical andsystematic errors, as stated above, applies. Evidently, anyresult involving mud benefits from our choice to use (10) atfinite lattice spacing, since it restricts the curve in Fig. 2 togo through zero. The absence of curvature in our data is thereason why this ratio is in rather good agreement with the

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TABLE VI. fr0 on all lattices used in the continuum extrapo-lation and the unfiltered. With the � � 1 thin link action at � �5:66, 5.76 there is no signal.

� � 5:66 � � 5:76 � � 5:84 � � 6:0


Mπ2 r0

2 1 MK2 r0

2 2 3 4 5 6

M2P r0

2

0.3

0.4

0.5

0.6

0.7

f P r0

Mπ2 r0

2 1 MK2 r0

2 2 3 4 5 6

M2P r0

2

0.3

0.4

0.5

0.6

0.7

f P r0

FIG. 4 (color online). Pseudoscalar decay constant fPr0 versus�MPr0�

2 on our coarsest and finest �� 5:66, 6.0) lattice for the1 HYP overlap operator. MP and fP are obtained from a fit to thehA0A0i correlator in the region T � f5NT=16; . . . ; 11NT=16g.The solid curve is a linear fit, the dotted vertical lines indicatethe physical M�r0 and MKr0, respectively.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

a2/r

0

2

0

0.1

0.2

0.3

0.4

0.5

(m1+

m2)

r 0

1 APE3 APE1 HYP3 HYP

FIG. 3 (color online). Renormalized �MS; 2 GeV� masses�ms �mud�r0 and 2mudr0 versus �a=r0�

2. Errors are statistical,the 4 filterings are correlated (same configurations).

STEPHAN DURR AND CHRISTIAN HOELBLING PHYSICAL REVIEW D 72, 071501 (2005)


XPT result ms=mud � 24:4�1:5� [28], in spite of the limi-tations mentioned and in spite of the latter value referringto a different theory (full QCD).

The pseudoscalar decay constant fP is extracted fromthe hA0A0i correlator. Generically, we see plateaus for theeffective decay constant from about NT=4 on, again withthe exception of the unfiltered operator where they are lesspronounced. We use the ZA values discussed earlier. Ourcentral values stem from the interval 5NT=16 t 11NT=16 and the error estimate comes from comparingto the hPPi � hSSi channel, from varying the fit range andfrom the error on ZA.

In Fig. 4 fP is plotted versus the meson mass squared.We see a strictly linear behavior at all couplings, with thelattice spacing having only mild effects on intercept andslope. Since (quenched) XPT predicts a linear dependenceon M2

P at lowest order and (quenched) chiral logs enter at1-loop level only, we decided to stick to the leading orderand do a linear fit. The intercept with �MKr0�

2 � 1:2512

and �M�r0�2 � 0:35372 defines the fKr0 and f�r0 at finite

lattice spacing reported in Tables VI and VII, respectively.To extrapolate to the continuum, the same recipe is used

as before. A combined fit to all filtering levels with acommon continuum limit and individual O�a2� terms isshown in Fig. 5 for fK and f�. We obtain

fKr0 � 0:430�24��5� (12)
TABLE VII. f�r0 on all lattices used in the continuum ex-trapolation and the unfiltered. With the � � 1 thin link action at� � 5:66, 5.76 there is no signal.
� � 5:66 � � 5:76 � � 5:84 � � 6:0


in the continuum [�2=d:o:f: � 0:96] which, via r0 �0:5 fm, yields (2). Applying the same strategy to fK=f�we find the value quoted in (3) [�2=d:o:f: � 0:52]. Giventhe large errors it is not so surprising that both fK andfK=f� are in reasonable agreement with the experimentalfexpK � 159:8�1:5� MeV and fexp

K =fexp� � 1:223�12�, re-

spectively [29]. Especially in the latter case, higher preci-sion would likely reveal a genuine quenching error [30].

071501-4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

a2/r

0

2

0.3

0.35

0.4

0.45

0.5

f P r0

1 APE3 APE1 HYP3 HYP

FIG. 5 (color online). Pseudoscalar decay constants f�r0 andfKr0 versus �a=r0�

2. Errors are statistical, the 4 filterings arecorrelated (same configurations).

CONTINUUM PHYSICS WITH QUENCHED OVERLAP FERMIONS PHYSICAL REVIEW D 72, 071501 (2005)


IV. CONCLUSION

We have presented an exploratory scaling study withUV-filtered overlap quarks, concentrating on the quenchedcontinuum limit of the light quark masses and the � and Kdecay constants. The focus is not so much on the numerical

071501

values obtained; the point is that a nice scaling behaviorhas been observed down to lattices as coarse as a�1 ’1 GeV. As anticipated in [5] the details of the filteringrecipe play a minor role and therefore we conjecture thatwith analytic gauge link smearing techniques [31] theseproperties will carry over to dynamical overlap calcula-tions (cf. [32–34]).

For the time being let us mention that the simulationswere done on a few of today’s standard PCs (3 GHzPentium 4), using an equivalent of �120 days on onesuch PC for the 1 APE operator (including all couplings)and of �60 days for any of the other filtered operators.With a complete scaling study based on overlap valencequarks requiring less than 1 PC year one is tempted to askwhether for some phenomenological questions the overlapmight actually represent the cheapest approach to obtain acontinuum result.

ACKNOWLEDGMENTS

This work is based on an earlier Collaboration with UrsWenger that lead to [5]. S. D. was supported by the SwissNSF.

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