Renormalization constants for Wilson fermion lattice QCD with four dynamical flavours

0.41 MB·English

by  P. Dimopoulos

Renormalization constants for Wilson fermion
lattice QCD with four dynamical flavours
ETM Collaboration
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1
Petros Dimopoulosa, Roberto Frezzottib,c, Gregorio Herdoizad, Karl Jansend,
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2 Vittorio Lubicze, David Palao∗c, Giancarlo Rossib,c

n aDipartimentodiFisica,UniversitàdiRoma“LaSapienza”
a PiazzaleA.Moro,I-00185Rome,Italy
J
bDipartimentodiFisica,UniversitàdiRoma“TorVergata”
0
1 ViadellaRicercaScientifica1,I-00133Rome,Italy
cINFNSezionedi“RomaTorVergata”

]
c/oDipartimentodiFisica,UniversitàdiRoma“TorVergata”
t
a
ViadellaRicercaScientifica1,I-00133Rome,Italy
l
- dDESY,Platanenallee6,D-15738Zeuthen,Germany
p
e eDipartimentodiFisica,UniversitàdiRomaTreandINFN
h
ViadellaVascaNavale84,I-00146Rome,Italy
[
E-mail: [email protected]

1
v
7 Wereportonanongoingnon-perturbativecomputationofRI-MOMschemerenormalizationcon-
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stantsforthelatticeactionwithfourdynamicalflavourscurrentlyinusebyETMC.Forthisgoal
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1 dedicatedsimulationswithfourdegenerateseaquarkflavoursareperformedatseveralvaluesof
.
1 thestandardandtwistedquarkmassparameters.WediscussamethodforremovingpossibleO(a)
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artifactsatallmomentaandextrapolatingrenormalizationconstantestimatorstothechirallimit.
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1 Wegivepreliminaryresultsatonelatticespacing.
:
v
i
X
r
a
TheXXVIIIInternationalSymposiumonLatticeFieldTheory,Lattice2010
June14-19,2010
Villasimius,Italy
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
RCs–fourdynamicalflavours DavidPalao
1. Introduction
Simulations including two degenerate light flavours and a non-degenerate doublet of quarks
arecurrentlybeingperformedbytheEuropeanTwistedMass(ETM)Collaboration. Theinclusion
of n =2+1+1 flavours is a necessary step to move towards a realistic situation. Fermions are
f
described by the maximally twisted mass lattice QCD (MtmLQCD) action [1] and gluons by the
Iwasakiaction[2]. Whilethefirstphysicalresultsareveryencouraging[3],dedicatedsimulations
are required to perform the non-perturbative renormalization of operators in a mass-independent
scheme,whererenormalizationconstants(RCs)aredefinedatzeroquarkmass. Inthestudyofn =
f
2+1+1QCDETMCisadoptingtheRI-MOMscheme[4]. TheRCsareevaluatedbyextrapolating
to the chiral limit the RC estimators computed in the theory with n =4 mass degenerate quarks
f
forarangeofmassvalues1. Herewereportontheprogresswemadeinthisproject.
1.1 Actionandquarkmassparameters
Forthepresentstudyweconsiderthelatticeaction
4 (cid:104) (cid:105)
SL=SIYwMa+a4∑∑ χ¯f γ·∇(cid:101)−2a∇∗∇+m0,f +iγ5rfµf χf(x) (1.1)
x f=1
where χ is a one-flavour quark field in the so-called twisted basis and in this work r is set to
f f
either1or−1. Passingfromthetwistedtothephysicalquarkbasis2
4 (cid:104) (cid:105)
SL=SIYwMa+a4∑∑q¯f γ·∇(cid:101)−iγ5rfeiγ5rfθ0,f(−2a∇∗∇+mcr)+M0,f qf(x). (1.3)
x f=1
Thebaremassparameterscanberewrittenas
(cid:113) m −m µ
M = (m −m )2+µ2, sinθ = 0,f cr, cosθ = f . (1.4)
0,f 0,f cr f 0,f M 0,f M
0,f 0,f
(cid:113)
Their renormalized counterparts read M =Z Mˆ = Z2m2 +µ2 and tanθ = ZAmPCAC. The
f P f A PCAC f f µf
parametrizationintermsofM andθ isconvenientbecausetheleadingtermoftheSymanziklocal
effective Lagrangian involves only M, not θ. As we will see later (see the end of section 2), this
remarkisatthebasisofourmethodtoobtainO(a)-improvedRC-estimatorsatallscalesevenout
ofmaximaltwist. Since, forpracticalreasons, weworkinapartiallyquenchedsetupwithallfour
flavours having equal mass parameters, we will have to consider in our analysis four quark mass
parameters: M ,θ ,M ,θ .
sea sea val val
1ForMonteCarlosimulationsweusedahighlyoptimizedimplementationofaHMC-likealgorithm[5].
2Therelationbetweentwisted(χf fields)andphysical(qf fields)quarkbasisis
χf →qf =exp[2i(π2−θ0,f)γ5rf]χf, χ¯f →q¯f =χ¯fexp[2i(π2−θ0f)γ5rf] (1.2)
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RCs–fourdynamicalflavours DavidPalao
1.2 RI’-MOMschemeandoursetup
Thefocusofthepresentstudyisonflavournon-singletquarkbilinearoperators,O =χ¯ Γχ
Γ f f(cid:48)
(or χ¯ Γχ ), with Γ = S,P,V,A,T, which are written in terms of χ and χ¯ quark fields (i.e. in the
f(cid:48) f
standard quark basis for untwisted Wilson fermions). RCs are named after the expression of the
operatorsinthisbasissoastomatchtheusualnotationintheliteratureaboutWilsonfermions.
Asconvenientinlatticestudies,weadopttheRI’-MOMscheme[6,4],whichisdefinedasfollows.
Afirstconditionfixesthequarkfieldrenormalization,namely
−i (cid:20)Tr(γ S (p)−1)(cid:21)
Z−1 ∑(cid:48) ρ f = 1, anyf, (1.5)
q 12N(p) p˜
ρ ρ p˜2=µ2
where p˜2 =∑ p˜2 , p˜ ≡ 1sinap . Thesum∑(cid:48) onlyrunsovertheLorentzindicesforwhich p
µ µ µ a µ ρ ρ
isdifferentfromzeroandN(p)=∑(cid:48) 1. TherenormalizationconditionfortheoperatorsO reads
ρ Γ
(cid:104) (cid:105)
Z−1Z(ff(cid:48))Tr Λ(ff(cid:48))(p˜,p˜)P = 1, f (cid:54)= f(cid:48). (1.6)
q Γ Γ Γ
p˜2=µ2
AboveSf(p) = a4∑xe−ipx(cid:10)χf(x)χ¯f(0)(cid:11)istheχf fieldpropagatorinmomentumspace,while
Λ(ff(cid:48))(p,p) = S−1(p)G(ff(cid:48))(p,p)S−1(p) (1.7)
Γ f Γ f(cid:48)
denotesthequarkbilinearvertexthatisobtainedby“amputating”theGreenfunction
G(ff(cid:48))(p,p) = a8∑e−ip(x−y)(cid:10)χ (x)(χ¯ Γχ )(0)χ¯ (y)(cid:11) Γ = S,P,V,A,T. (1.8)
Γ f f f(cid:48) f(cid:48)
x,y
Barring cutoff effects, RCs are independent of sign(r ). For practical reasons here we limit our-
f
selvestor =−r inevaluatingZ ,seeeq.(1.6).
f(cid:48) f Γ
2. StrategyforRCsintheN =4theory
f
In order to extract useful information from simulations performed with twisted mass Wilson
fermions one must know the twist angle, ω = π −θ, with good precision. The level of precision
2
requestedforω dependsontheobservableofinterest. Inourcase,afteranexploratorystudyona
few 163×32 lattices [7], and some tests near maximal twist on a 243×48 lattice we have chosen
toworkoutofmaximaltwist.
Figure2illustratesthedifficultiesoftuningtomaximaltwist,i.e.settingm tozero,inthe
PCAC
simulation setup for RC computations, at least if the lattice spacing is not very fine. Specifically,
theslopeofm vs1/(2κ)infigure1(a)suggeststhatnearm =0simulationsareinaregion
PCAC PCAC
withasharpechangeoftheslopeform whereitisdifficulttoextractusefulinformation. On
PCAC
theotherhandfigure1(b)givesamorequantitativeviewofthisproblemshowingresultsfromone
simulationclosetothecriticalpoint(thepointclosesttom =0infigure1(a)). Itappearsthat
PCAC
due to the long fluctuations a precise measurement of the PCAC mass will require for this case a
very large number of Monte Carlo trajectories. In fact, we have observed a similar feature for all
theensembleswith|am |(cid:46)0.01atbothβ =1.95andβ =1.90(i.e.fora≥0.08fm).
PCAC
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RCs–fourdynamicalflavours DavidPalao
In summary, working at maximal twist for the chosen range of twisted masses (see table 1)
wouldimplyaconsiderablefinetuningworkowingtothedifficultiesindeterminingam near
PCAC
am =0. To alleviate the problem one would need to increase the value of the twisted mass,
PCAC
µ , andthusM . Instead, workingawayfrommaximaltwist, onecanavoidthemetastableregion
f f
of parameter space and measure the twist angle with good precision. This comes at the price of
a moderate increase of the quark mass M and of a slightly more involved analysis. In our RC-
f
estimators cutoff effects linear in a are expected to be small and can anyway be removed with
controlledprecisionbyaveragingtheresultsobtainedforagivenM atoppositevaluesofθ .
f f
(a) (b)
Figure1: (a)am versus1/2κ onlattices243×48atβ =1.95;(b)MonteCarlohistoryofam for
PCAC PCAC
themostcriticalcase(correspondingtotheredcrosspoint)inpanel(a).
In fact, from the symmetry of the lattice action S under P×(θ → −θ )×D ×(M →
L 0 0 d 0
−M ) [1, 8, 12] it follows that the O(a2k+1) artifacts occurring in the vacuum expectation values
0
of (multi)local operators O that are invariant under P×(θ → −θ ) are quantities that change
0 0
sign upon changing the sign of θ (or θ). Hence O(a2k+1) cutoff effects vanish in θ-averages:
0
(cid:104) (cid:105)
1 (cid:104)O(cid:105)| +(cid:104)O(cid:105)| . ThesameistrueforoperatorformfactorsinvariantunderP×(θ →−θ )
2 Mˆ,θ Mˆ,−θ 0 0
and,inparticular,forourRC-estimatorsatallvaluesofM and p˜2.
f
3. Currentanalysisandpreliminaryresults
Herewedetailtheanalysisprocedurewefollowedinordertoobtainverypreliminaryresults
on the RCs of interest. Indeed, at this stage our main goal was checking the feasibility of the
project. In particular, the analysis procedure is not yet the optimal one, for instance concerning
the order of the various steps, and some refinements, such as the subtraction of the known cutoff
effects at O(a2g2) [9], are still omitted. While these improvements will be included in the final
analysis,thepresentworkshowsthatthestrategyadvocatedinsection2allowstoextracttheRCs
of the quark field and quark bilinear operators with a ∼ 1% level precision by means of stable
simulations at a lattice spacing (a∼0.08 fm) which is among the coarsest ones explored in the
studyofn =2+1+1QCDbyETMC.
f
In practice, for a sequence of M -values, we produced for each M two ensembles with
sea sea
oppositevaluesofθ . WelabelthemasEp/m,whereE=1,2...andp/mreferstosign(θ ). On
sea sea
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RCs–fourdynamicalflavours DavidPalao
ensemble aµ amsea aMsea θsea aµval amval
sea PCAC 0 PCAC
1m 0.0085 -0.04125(13) 0.03288(10) -1.3093(8) [0.0085,...,0.0298] -0.0216(2)
1p 0.0085 +0.04249(13) 0.03380(10) 1.3166(7) [0.0085,...,0.0298] +0.01947(19)
3m 0.0180 -0.0160(2) 0.02182(9) -0.601(6) [0.0060,...,0.0298] -0.0160(2)
3p 0.0180 +0.0163(2) 0.02195(9) 0.610(6) [0.0060,...,0.0298] +0.0162(2)
2m 0.0085 -0.02091(16) 0.01821(11) -1.085(3) [0.0085,...,0.0298] -0.0213(2)
2p 0.0085 +0.0191(2) 0.01696(16) 1.046(6) [0.0085,...,0.0298] +0.01909(18)
4m 0.0085 -0.01459(13) 0.01409(8) -0.923(4) [0.0060,...,0.0298] -0.01459(13)
4p 0.0085 +0.0151(2) 0.01441(14) 0.940(7) [0.0060,...,0.0298] +0.0151(2)
Table1: Massparametersoftheensemblesanalysedforthiscontribution. Fromtheformulaeinsect. 1.1it
followsthatinthevalencesectorwehave 0.013(cid:46)aMval(cid:46)0.033and0.4(cid:46)|θ |(cid:46)1.2 (θ /mval >0).
val val PCAC
(a) (b)
Figure2: Fortheexampleofensemble4m,(ap˜)2=1.5: (a)subtractionofGoldstonepolecontributionand
valencechiralextrapolationinΓ =Tr[Λ P ];(b)overviewofthevalencechiralextrapolationforallRCs.
P P P
eachensembleEp/m,with(MEp/m,θEp/m)wecomputetheRC-estimatorsforseveralvaluesofthe
sea sea
valencemassparameters(M ,θ )and p˜2 (allcorrespondingto“democratic”momenta p,inthe
val val
sensespecifiedin[10]),assummarizedintable1. Thenweproceedinvariousstepsasfollows.
Valencechirallimit. AfitofRC-estimatorslinearin(MPS)2 turnsouttobenumericallyadequate
val
(seefig.2(b)). ForΓ=P(seefig.2(a))or,duetoO(a2)terms,Γ=S,wehavealsokeptinto
accountthecontribution∝(MPS)−2 comingfromtheGoldstonebosonpole.
val
O(a2p˜2)discretizationerrors. We applied two different methods, following [10]. In the first
method(“M1”),afterbringing,viatheknown[11]perturbativeevolutiontheRC-estimators
to a common renormalization scale (p˜2 =1/a2), we remove the remaining O(a2p˜2) dis-
M1
cretization errors by a linear fit in p˜2. The second method (“M2”) consists in simply taking
thevalueoftheRCsestimatorsatahighmomentumpointfixedinphysicalunits. Wechose
p˜2 = 12.2GeV2. ThetwoapproachesyieldRCresultsdifferingonlybycutoffeffects.
M2
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RCs–fourdynamicalflavours DavidPalao
(a) (b)
Figure 3: Residual p˜2-dependence of RC-estimators at scale 1/a and RC values from method M1 for the
casesofensemble1m(panel(a))and2p(panel(b)).
(a) (b)
Figure 4: Msea-dependence before (empty symbols) and after (full symbols) θ-average of RC-estimators
forafewoperatorsaswellastheirchirallimitvalue. Theleft(right)panelcorrespondstoM1(M2)results.
In(a)wealsoshowZ (WI),whichisobtainedbyexploitinganexactlatticeWard-Takahashiidentity(WI).
V
RemovalofO(a)artifacts. Itisachievedbyθ-average(seesection2)oftheRCsestimators,
1(cid:104) (cid:105)
Z (ME ,|θE |) = (cid:104)Z (MˆEp,θEp;θEp )(cid:105)+(cid:104)Z (MˆEm,θEm;θEm )(cid:105) (3.1)
Γ sea sea 2 Γ sea sea val;eff Γ sea sea val;eff
Ep(m)
whereθ parameterizesthedominatingO(a)effectsinRC-estimatorsthat(inthepresent
val;eff
analysis)arisefromemployingMPS inthevalencechiralextrapolation.
val;Ep(m)
Seachirallimit. ThequantitiesZ (ME ,|θE |)areextrapolatedtoM =0byusingthefitAnsatz
Γ sea sea sea
Z (M ,θ ) = Z + AM2 + BM2 cos(2θ ). (3.2)
Γ sea sea Γ sea sea sea
ThisAnsatzcanbejustifiedbyananalysisàlaSymanzikofthelatticeartifactsinZ (M ,θ )
Γ sea sea
uptoO(M2 )andneglectingchiralspontaneoussymmetrybreakingeffects[12].
sea
Thefirst,verypreliminaryresultsofthisanalysisaresummarizedintable2.
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RCs–fourdynamicalflavours DavidPalao
Method Z Z Z (1/a) Z (1/a) Z /Z Z (1/a) Z (1/a)
A V P S P S T q
M1 0.761(08) 0.630(05) 0.438(08) 0.614(09) 0.716(21) 0.753(07) 0.767(06)
M2 0.771(03) 0.674(03) 0.496(04) 0.647(03) 0.767(08) 0.768(03) 0.813(02)
Table2:PreliminaryRCresultsatβ =1.95fromtheanalysisofsection3.WealsogetZ (WI)=0.612(1).
V
4. Conclusionsandoutlook
WehavedescribedourstrategytocomputeO(a)improvedoperatorRCsfortheN =4lattice
f
action currently used by ETMC. We have shown that the method advocated in this work provides
veryencouragingresultsatonelatticespacing(a∼0.08fm)thatisamongthecoarsestsimulatedin
thestudyofQCDwithn =2+1+1dynamicalflavours. Inparticular,theobserveddependences
f
of RCs on valence and sea quark masses are mild and quite in line with our experience [10] in
n =2QCD.Besidesthetechnicalimprovementsmentionedinsection3,weplantopossiblyadd
f
fewmoreensemblesata∼0.08fm(β =1.95)andtoextendourworktootherlatticespacings.
WethankIDRISandINFN/apeNEXTforgivingusCPUtimenecessaryforthisstudy.
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