Wilson chiral perturbation theory, Wilson-Dirac operator eigenvalues and clover improvement

0.13 MB·English

by  Poul H. Damgaard

Wilson chiral perturbation theory, Wilson-Dirac
operator eigenvalues and clover improvement∗
3
1
0
2

n
a
J
PoulH.Damgaard
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1 NielsBohrInternationalAcademyandDiscoveryCenter,NielsBohrInstitute
UniversityofCopenhagen

]
t Blegdamsvej17,DK-2100CopenhagenØ,Denmark
a
-l Urs M.Heller†
p
AmericanPhysicalSociety,OneResearchRoad,Ridge,NY11961,USA
e
h E-mail:[email protected]
[

Kim Splittorff
1
v DiscoveryCenter,NielsBohrInstitute,UniversityofCopenhagen
9 Blegdamsvej17,DK-2100CopenhagenØ,Denmark
9
0
3 Chiral perturbation theory for eigenvalue distributions, and equivalently random matrix theory,
.
1 has recently been extended to include lattice effects for Wilson fermions. We test the predic-
0
tions by comparison to eigenvalue distributions of the Hermitian Wilson-Dirac operator from
3
1 pure gauge (quenched)ensembles. We show that the lattice effects are diminished when using
:
v cloverimprovementfortheDiracoperator. We demonstratethattheleadingWilsonlow-energy
i
X constantsassociatedwithWilson(clover)fermionscanbedeterminedusingspectralinformation
r oftherespectiveDiracoperatoratfinitevolume.
a
XthQuarkConfinementandtheHadronSpectrum
October8-12,2012
TUMCampusGarching,Munich,Germany
∗ToappearasPoS(ConfinementX)077
†Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Wilsonchiralperturbationtheory,Wilson-Diracoperatoreigenvalues... UrsM.Heller
1. Introduction
The low-energy behavior of QCD, the spontaneous breaking of chiral symmetry, including
c
the explicit breaking bythe quark masses, isdescribed bychiral perturbation theory ( PT).Inthe
lattice regularization of QCD lattice artifacts can contribute to the breaking of chiral symmetry,
for Wilson fermions, or its partial breaking, in the case of staggered fermions. These effects can
c
beincluded inthe PTapproach, leading tonew,lattice discretization dependent low-energy con-
stants. We consider Wilson fermions in this contribution, for which the effective theory, Wilson
c c
PT(W PT)wasintroducedandworkedoutin[1]. ThenewtermsinthechiralLagrangianaffect
the low-lying spectrum of the (Hermitian) Wilson-Dirac operator [2]. For a recent review with
additional references, see Ref. [3]. Here, we test and verify the predictions for the distribution of
thelow-lyingeigenvalueswithlatticeQCDsimulations[4,5]andshow[6]thattheycanbeusedto
c
obtainthenewlow-energyconstantsintroducedinW PT.Wealsodemonstratetheeffectofclover
improving theWilson-Dirac operator.
c
2. The W PTand WilsonRMTframework
Wewillbeconcernedwiththee -regimeofWc PTwherethezeromomentummodesdominate
– the system size is such that mp L≪1. In addition we adopt the power counting with m∼a2.
Hence,dropping thekineticpartofthechiralLagrangian, weconsider
1 1
L =− mS Tr U+U† − zS Tr U−U† +a2V . (2.1)
2 2
(cid:0) (cid:1) (cid:0) (cid:1)
The second term, representing a yg¯ y term, is introduced for later convenience. V describes the
5
latticeartifacts [1]
V =W Tr U2+U†2 +W Tr U+U† 2+W Tr U−U† 2 . (2.2)
8 6 7
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3) (cid:2) (cid:0) (cid:1)(cid:3)
Atlarge N ,thetwo-tracetermsaresuppressed.
c
Thefinitesizescalingconsidered issuchthat
mˆ =mS V , zˆ=zS V and aˆ2=a2W V for j=6,7,8
j j
areheldfixed. HereS isthecondensate andV thevolume.
c
This leading order in W PT can equivalently be described by a chiral random matrix theory
(RMT).ForWilsonfermions,including theone-tracetermwithlow-energyconstantW ,theDirac
8
operatorisrepresented inWilsonRMT(WRMT)as[2]
a˜A iW
D = , (2.3)
W iW† a˜B
!
with W a random (n+n )×n complex matrix, and A and B random Hermitian matrices of size
(n+n )×(n+n ) andn×n, respectively. Asusual intheRMTcontext, weconsider afixedindex
n . Weuseachiralbasiswithg =diag(1,...,1,−1,...,−1). AandBrepresentthechiralsymmetry
5
breaking termcorresponding totheWilsontermintheWilson-Dirac operator.
2
Wilsonchiralperturbationtheory,Wilson-Diracoperatoreigenvalues... UrsM.Heller
Ens b r /a a[fm] size L[fm] |Q|=0,1,2cfgs
Iw 0
A 2.635 5.37 0.093 164 1.5 1279, 2257,1530
B 2.635 5.37 0.093 204 1.9 401, 682, 644
C 2.79 6.70 0.075 204 1.5 1207, 2130,1448
Table 1: The ensemblesused. The scale is set by r =0.5 fm. The r /a valuescome from interpolation
0 0
formulaein[10]. Qisthetopologicalcharge(seetext).
Thetwo-tracetermscanbeincorporated inWRMTviatwoGaussianintegrations
Zn (mˆ,zˆ;aˆ6,aˆ7,aˆ8)= 16p 1aˆ6aˆ7Z−¥¥ dy6dy7e−1y626aˆ26−1y627aˆ27 Zn (mˆ −y6,zˆ−y7;0,0,aˆ8). (2.4)
Here,
Zn (mˆ,zˆ;0,0,aˆ )= dUdetn Ue−VL(W6=W7=0) (2.5)
8
Z
isthefixed-indexpartition functionwiththeone-trace O(a2)termincluded.
3. Index oftheWilson-Diracoperator
Asindicatedabove,RMTpredictions applytogaugefieldsectorswithafixedindex,or,inthe
continuum, fixedtopological charge. FortheWilson-Dirac operator, theindexcanbedefinedby
n ≡ (cid:229) ′sign(hk|g |ki) (3.1)
5
k
with|kithek’theigenstateoftheWilson-Diracoperator, D . Onlyeigenvectorswithrealeigenval-
W
uescontribute, andthe′indicatesthatonlytherealeigenvaluesinthebranchnearzero,witheigen-
values <r , are kept. Introducing the Hermitian Wilson-Dirac operator D (m )=g (D +m )
cut 5 0 5 W 0
andusing
D (m )|y i=0 ⇒ D |y i=−m |y i (3.2)
5 0 W 0
the index can equivalently be obtained from the zero crossings of the spectral flow of D (m ) up
5 0
to m =−r [7]. It corresponds to the index of an overlap operator [8] with kernel D (m ).
cut cut 5 cut
Because of the dependence on the choice of r , the index of the Wilson-Dirac operator is not
cut
unique.
4. The numerical simulations
Forournumericaltests,inthequenchedcase,wegeneratedthreeensemblesusingtheIwasaki
gauge action [9], which suppresses dislocations and gives a fairly unique index n or topological
charge Q. Theensembles arecharacterized inTable1.
ThetopologicalchargelistedinTable1wasobtainedaftersixstepsofHYPsmearing[11]with
an improved lattice FF˜ operator [12]. On the configurations with |Q|≤1, as well as the |Q|=2
configurations of ensemble A, we also did the much more expensive computation of the index
3
Wilsonchiralperturbationtheory,Wilson-Diracoperatoreigenvalues... UrsM.Heller
50
00..2139045 200 EEvv__rreeaall__22663355__1166__HHYYPP__cQl_1Qan1daQndmQ1m1
40 Ev_real_2635_16_HYP_Q1andQm1
a) Ev_real_279_20_HYP_Q1andQm1
m=0; 30 W) 150
a
Wla , l(areal100
(real20 n=1r
=1
n
r 10 50
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
a l W al W
Figure1: DistributionoftherealeigenvaluesoftheWilson-Diracoperatorforthen =1configurationsof
the two L=1.5 fm ensembles A and C (left) and comparisonof the distribution with and without clover
improvementata=0.093fm(ensembleA,right).
from the spectral flow. We first applied one HYP smearing before constructing the Wilson-Dirac
operator. Thetopological charge andtheindexagreedonmostconfigurations, withtheagreement
improving at smaller lattice spacing and becoming worse for the larger volume ensemble C, for
whichitwasabout97%.
The crossing points in the spectral flow are the real eigenvalues, whose distribution is shown
in Fig. 1 (left). The dashed vertical lines are estimates of (minus) the critical mass. Some real
eigenvalues aresmaller, onso-called “exceptional” configurations.
Forensemble A,wealsocomputed the spectral flowwithclover improving theWilson-Dirac
operator, again after one HYPsmearing. Theclover coefficient wasset to 1, which is expected to
beclosetothenonperturbative valueaftertheHYPsmearing [13]. Theresultingdistributionofthe
real eigenvalues is compared to the unimproved case in Fig. 1 (right). The improvement is quite
dramatic, besides the expected reduced shift away from zero, the distribution is much narrower
and more symmetric. The width is determined by the O(a2) terms in Eq. (2.2), so with clover
improvementthecoefficients aremuchsmaller.
5. Wilsoneigenvaluedistributions andWRMT
We next computed the lowest 20, in magnitude, eigenvalues of the Hermitian Wilson-Dirac
operator D (m )withbaremass am =−0.216forensembles AandBtocompare toeigenvalues
5 0 0
distributions obtained fromWRMT[2].
Without clover improvement, we considered only contributions from the two-trace term in
Eq.(2.2)[4]. Weusedthen =0histogrammedeigenvaluedistributionsofensembleAtodetermine
the WRMT parameters mˆ and aˆ = aˆ and the eigenvalue rescaling factor S V. Using the same
8
parameterswethengetapredictionforthen =1distributionthatcanbecomparedtothenumerical
data(seeFig.2top).
Ensemble B differs from ensemble A only in the volume. Using volume scaling, mˆ =
B
mˆ (V /V ) and aˆ =aˆ V /V , we obtain predictions for the distributions for ensemble B (see
A B A B A B A
Fig.2bottom). Ascanbeseen,theWRMTpredictions workwell.
p
4
Wilsonchiralperturbationtheory,Wilson-Diracoperatoreigenvalues... UrsM.Heller
Ev_2635_16_HYP_Q0_m216 Ev_2635_16_HYP_Q1_Qm1_m216
0.6 m=3.5 a=0.35 n=0 0.6 m=3.5 a=0.35 n=1
FIT PREDICTION
0.5 0.5
a) a)
m; 0.4 m, 0.4
5, 5,
(l (l
n=050.3 n=150.3
r r
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
l 5 l 5
Ev_2635_20_HYP_Q0_m216 Ev_2635_20_HYP_Q1_Qm1_m216
0.6 m=8.5 a=0.55 n=0 0.6 m=8.5 a=0.55 n=1
PREDICTION PREDICTION
0.5 0.5
a) a)
m; 0.4 m, 0.4
5, 5,
(l (l
n=050.3 n=150.3
r r
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
l 5 l 5
Figure 2: Comparisonof the histogrammedeigenvaluedistributionswith WRMT. Then =0 distribution
ofensembleA(topleft)wasusedtoobtaintheparameters. ThepredictionsforensembleB(bottom)used
volumescalingoftheparameters.
ForensembleC,atthesmallerlatticespacing,wecomputedtheeigenvalueswithtwodifferent
bare masses am =−0.178 and −0.184. Weused the histogrammed n =0 distribution with bare
0
mass am = −0.184 to the determine the WRMT parameters, and used “mass scaling”, D mˆ =
0
D m S V for predictions for the distributions with the other bare mass am =−0.178, as shown in
0 0
Fig.3. Again,theWRMTpredictions workwell.
6. Cloverimprovedeigenvalue distributions and WRMT
Wehavealready seen fromthedistribution oftherealeigenvalues inFig.1(right) thatclover
improvement not only, as expected, decreases the additive mass renormalization (the real eigen-
valuepeakismuchclosertozero)butalsothesizeoftheO(a2)low-energyconstantsconsiderably
(the distribution becomes much narrower). Here weconsider the effects on the distribution of the
20 lowest, in magnitude, eigenvalues of the Hermitian Wilson-Dirac operator D (m ) with clover
5 0
improvement for ensemble A using a bare mass am =−0.03. The comparison with WRMT is
0
showninthefirstthreepanelsofFig.4.
For|aˆ |≪1the lattice effects affect, to leading order, only the index peak of the topological
j
modes. These are the lowest eigenvalues with almost chiral eigenvectors which correspond to the
real eigenvalues shifted bythebaremass. AscanbeseeninFig.4(bottom right)thedistributions
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Wilsonchiralperturbationtheory,Wilson-Diracoperatoreigenvalues... UrsM.Heller
Ev_279_20_HYP_Q0_m184 Ev_279_20_HYP_Q1andQm1_m184
0.6 m=3.5 a=0.35 n=0 0.6 m=3.5 a=0.35 n=1
FIT PREDICTION
0.5 0.5
a) a)
m; 0.4 m; 0.4
5, 5,
(l (l
n=050.3 n=150.3
r r
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
l 5 l 5
Ev_279_20_HYP_Q0_m178 Ev_279_20_HYP_Q1andQm1_m178
0.6 m=4.8 a=0.35 n=0 0.6 m=4.8 a=0.35 n=1
PREDICTION PREDICTION
0.5 0.5
a) a)
m; 0.4 m; 0.4
5, 5,
(l (l
n=050.3 n=150.3
r r
0.2 0.2
0.1 0.1
0 0
-12 -8 -4 0 4 8 12 -12 -8 -4 0 4 8 12
l 5 l 5
Figure3:ComparisonofthehistogrammedeigenvaluedistributionswithWRMTforensembleC.Then =0
distributionwithbaremassam =−0.184(topleft)wasusedtoobtaintheparameters. Thepredictionsfor
0
baremassam =−0.178(bottom)used“massscaling”,D mˆ =D m S V.
0 0
matchalmostperfectly. Theeigenvalue density, for|n |>0,ontheopposite sideoftheindexpeak
is almost continuum like and allows determination of mˆ and S V. We use the |n |=1 eigenvalue
distribution forthis. The aˆ are then obtained from theireffect on theindex peak. Weuse the fact
j
that the low-energy constant have fixed signsW <0,W <0 andW >0 [2, 14, 15] and that the
6 7 8
distribution dependsonlyonthecombination |W |+|W |[2]allowingtotakeW =0. Wefindthat
6 7 7
witheither aˆ 6=0oraˆ 6=0wecanreproduce thehistogrammed |n |=0and |n |=1distributions
8 6
inFig.4(top)equallywell. Butonlywithaˆ 6=0canwereproducethe|n |=2distribution, too,as
6
showninFig.4(bottomleft).
We can explain the drastically different effect of W and W on the analytic prediction for
6 8
|n | =2 by noting that the W -term, in WRMT, corresponds to a Gaussian fluctuating mass, see
6
Eq.(2.4). Thed -functionindexpeakofthecontinuumtheoryisthereforesmearedintoaGaussian
peak with an amplitude that increases with |n |. W , therefore, does not introduce a repulsion
6
c
between eigenvalues. On the contrary, the W -term of W PT is included in the representation of
8
theDiracoperator, Eq.(2.3),ofWRMT,andhenceinducesaneigenvaluerepulsion,ascanbeseen
from the red curve in Fig. 4 (bottom left). This repulsion is seen for all sectors with |n |>1 [2].
Itisthususeful toinclude eigenvalues from configurations with|n |>1foradetermination ofthe
low-energy constants fromfitstoeigenvalue distributions.
We finally note that, with clover improvement the nonvanishing |aˆ | is about a factor 3-4
6
6
Wilsonchiralperturbationtheory,Wilson-Diracoperatoreigenvalues... UrsM.Heller
2
Ev_2635_16_HYPclov_Q0_m03 Ev_2635_16_HYPclov_Q1andm1_m03
1 mm== 22..55 aa66==ii00..01 aa88==00..10 n=n=00 1.5 mm== 22..55 aa66==0i0 a.18 =a08=.10 n= n=11
0.8
a) a)
m; m;
5, 0.6 5, 1
(l5 (l5
r 0.4 r
0.5
0.2
0 0
-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10
l 5 l 5
3.5 500
3 Emmv==_2 22.5.65 3a a566_==1ii060._.01H aaY88==P00c.l.10o vn=n=_22Q2andm2_m03 400 rrr1eaanld|Q2||=Q2|= 2 EEvv__2re6a3l5__21663_5H_1Y6P_cHloYvP_Q_c2la_nQd2Qanmd2Q_mm203
2.5
m;a) 2 300
5,
(l51.5 200
r
1
100
0.5
0-10 -8 -6 -4 -2 0 2 4 6 8 10 00 0.05 0.1 0.15 0.2
l 5 alW and al5-m
0
Figure4: Comparisonofthehistogrammedclover-improvedeigenvaluedistributionswithWRMTforen-
sembleA(topandbottomleft).Then =1distribution(topright)wasusedtoobtaintheWRMTparameters.
TheredcurvesaretheWRMTpredictionswithaˆ 6=0,thebluecurvesthosewithaˆ 6=0. Thebottomright
8 6
plotshowsacomparisonofthedistributionoftherealeigenvalueswiththefirsttwopositiveeigenvaluesof
D (m )shiftedbythebaremassforthen =2configurations.
5 0
smaller than the nonvanishing |aˆ | without the improvement, both after one HYP smearing, illus-
8
trating againthequitedramaticeffectofcloverimprovementontheO(a2)low-energyconstants.
7. Conclusion
Wehavepresented numerical simulations, inthequenched case, ofthelow-lying eigenvalues
oftheHermitianWilson-Dirac operator, both withandwithout cloverimprovement tocompareto
predictions from e -regimeWilson c PTor, equivalently, Wilson RMT.Weusedthe Iwasaki gauge
actionwhichsuppresses dislocations andleadstoafairlyuniqueindexortopological charge. This
is helpful, since the analytical predictions are made for sectors of fixed index. We found that our
eigenvaluedistributionsagreewellwiththeanalyticalpredictions,andverifiedscalingwithvolume
and(bare)mass.
We have also looked at the distribution of the real eigenvalues of the Wilson-Dirac operator,
obtainedfromthespectralflow. Wefoundadramaticdecrease ofboththeadditivemassrenormal-
ization (the real eigenvalue peak is closer to zero) and the O(a2) low-energy constants (the width
ofthedistribution becomesnarrowerandmoresymmetric)withthecloverimprovement.
7
Wilsonchiralperturbationtheory,Wilson-Diracoperatoreigenvalues... UrsM.Heller
Fitstothedistributionofthelow-lyingeigenvaluesoftheHermitianWilson-Diracoperatoral-
lowdeterminationofthelow-energyconstantsofQCDincludingthosethatparameterizetheO(a2)
lattice effects. However,distributions onconfigurations with|n |>1areneededtodisentangle the
effectsofW fromthoseofW andW whenall|aˆ |aresmall.
8 6 7 j
Acknowledgments
UMH thanks the organizers for a stimulating conference and the conveners of section A for
theopportunity topresenttheseresults.
References
[1] S.R.SharpeandR.L.Singleton,Phys.Rev.D58(1998)074501[arXiv:heplat/9804028];
G.RupakandN.Shoresh,Phys.Rev.D66(2002)054503[arXiv:hep-lat/0201019];
O.Bär,G.RupalandN.Shoresh,Phys.Rev.D70(2004)034508[arXiv:heplat/0306021];
S.Aoki,Phys.Rev.D68(2003)054508[arXiv:hep-lat/0306027];
S.AokiandO.Bär,Phys.Rev.D70(2004)116011[arXiv:hep-lat/0409006];
A.Shindler,Phys.Lett.B672(2009)82[arXiv:0812.2251];
O.Bär,S.NeccoandS.Schaefer,JHEP0903(2009)006[arXiv:0812.2403].
[2] P.H.Damgaard,K.SplittorffandJ.J.M.Verbaarschot,Phys.Rev.Lett.105(2010)162002
[arXiv:1001.2937];
G.Akemann,P.H.Damgaard,K.Splittorff,andJ.J.M.Verbaarschot,Phys.Rev.D83(2011)085014
[arXiv:1012.0752];
G.AkemannandT.Nagao,JHEP1110(2011)060[arXiv:1108.3035];
M.Kieburg,J.J.M.VerbaarschotandS.Zafeiropoulos,Phys.Rev.Lett.108(2012)022001
[arXiv:1109.0656].
[3] K.Splittorff,PoS(Lattice 2012)018[arxiv:1211.1803].
[4] P.H.Damgaard,U.M.HellerandK.Splittorff,Phys.Rev.D85(2012)014505[arXiv:1110.2851].
[5] A.Deuzeman,U.WengerandJ.Wuilloud,JHEP1212(2012)109[arXiv:1110.4002].
[6] P.H.Damgaard,U.M.HellerandK.Splittorff,Phys.Rev.D86(2012)094502[arXiv:1206.4786].
[7] R.G.Edwards,U.M.HellerandR.NarayananNucl.Phys.B535(1998)403[arXiv:hep-lat/9802016].
[8] R.NarayananandH.Neuberger,Nucl.Phys.B443(1995)305[arXiv:hep-th/9411108];
H.Neuberger,Phys.Lett.B417(1998)141[arXiv:hep-lat/9707022].
[9] Y.Iwasaki,ReportNo.UTHEP-118,1983[arXiv:1111.7054].
[10] S.Takedaetal.(CP-PACSCollaboration),Phys.Rev.D70(2004)074510[arXiv:hep-lat/0408010];
S.Necco,Nucl.Phys.B683(2004)137[arXiv:hep-lat/0309017].
[11] A.HasenfratzandF.Knechtli,Phys.Rev.D64(2001)034504[arXiv:hep-lat/0103029].
[12] T.A.DeGrand,A.HasenfratzandT.G.Kovacs,Nucl.Phys.B505(1997)417[arXiv:heplat/9705009].
[13] Y.Shamir,B.SvetitskyandE.Yurkovsky,Phys.Rev.D83(2011)097502[arXiv:1012.2819].
[14] M.Kieburg,K.SplittorffandJ.J.M.Verbaarschot,Phys.Rev.D85(2012)094011[arXiv:1202.0620].
[15] M.T.HansenandS.Sharpe,Phys.Rev.D85(2012)014503[arXiv:1111.2404];
Phys.Rev.D85(2012)054504[arXiv:1112.3998].
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