The one-loop analysis of the beta-function in the Schroedinger Functional for Moebius Domain Wall Fermions

0.24 MB·

by  Yuko Murakami

The one-loop analysis of the beta-function in the
Schroedinger Functional for Moebius Domain Wall
Fermions
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1
0
2 Yuko Murakami∗

n GraduateschoolofScience,HiroshimaUniversity,Higashi-Hiroshima,Hiroshima739-8526,
a
Japan
J
E-mail: [email protected]
5
2
Ken-Ichi Ishikawa

] GraduateschoolofScience,HiroshimaUniversity,Higashi-Hiroshima,Hiroshima739-8526,
t
a Japan,
l
- CoreofResearchforEnergeticUniverse,HiroshimaUniversity,Higashi-Hiroshima,Hiroshima
p
739-8526,Japan
e
h E-mail: [email protected]
[

1 WeproposedaconstructionoftheSchroedingerfunctionalschemefortheMoebiusdomainwall
v
9 fermions(MDWF)byintroducingaproperboundaryoperatortotheoriginalMDWFinthelast
3 conference. Thespectrumoftheeffectivefour-dimensionaloperatorwasinvestigated. Thisyear
1
7 we investigate the fermionic contribution to the beta-function with the Moebius domain wall
0 fermionwiththeSFboundarytermuptotheone-looplevelandfindthatourconstructionproperly
.
1 reproducetheone-loopbeta-function.
0
7
1
:
v
i
X
r
a
The33rdInternationalSymposiumonLatticeFieldTheory
14-18July2015
KobeInternationalConferenceCenter,Kobe,Japan*
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Theone-loopanalysisofthebeta-functionintheSFforMDWF YukoMurakami
1. Introduction
ThechiralsymmetryisimportantnotonlyinQCDbutalsointhestandardmodel. Toextract
hadronic observables based on the chiral symmetry, we need both the lattice field theory and the
chiral symmetry. The lattice chiral symmetry is now realized by the overlap and domain wall
fermions and extensive efforts have been devoted to the accurate values for hadronic observables
sensitive to the chiral symmetry[1, 2]. In this sense the non-preturbative renormalization for the
latticefermionswiththelatticechiralsymmetrybecomesimportant.
TheSchroedingerfunctional(SF)scheme[3]hasbeensuccessfullyappliedtorenormalizethe
lattice QCD with the Wilson fermions non-perturbatively [4]. Although it was not simple for the
latticechiralfermionstoapplytheSFscheme,theSFmethodsforthelatticechiralfermionshave
beendevelopedinRefs.[5,6,7,8].
In the last lattice conference, we proposed a construction of the SF scheme for Moebius do-
main wall fermions (MDWF) [9]. The operator is modified by adding a temporal boundary op-
erator based on Takeda’s implementation for the standard domain wall fermion [8]. The form of
the boundary operator is determined by the symmetry argument of Ref. [5] so that the term holds
thediscretesymmetries(C,P,T,Γ -Hermiticity)andbreaksthedomainwallchiralsymmetry[10]
5
at the temporal boundary. We investigated the properties of the effective four-dimensional opera-
tor induced from the MDWF with the boundary operator and found that it reproduced the proper
continuumtheorywiththeSFboundaryonlyintheinfinitefifthlatticeextentN →∞.
5
In this paper, we extend our previous work to obtain the proper operator for the SF scheme
even at a finite N . To do this we properly renormalize the effective four-dimensional operator
5
sincetheexplicitbreakingofthelatticechiralsymmetryatafiniteN inducestheadditiveresidual
5
mass[11]asseeninWilsontypefermionsevenatthetree-level. Afterrenormalizingtheeffective
four-dimensional operator we investigate the eigenvalues in the SF boundary condition. In order
to check the consistency of our construction, we also investigate the fermionic contribution to the
one-loopbeta-functionintheSFschemeatafiniteN .
5
Thispaperisorganizedasfollows. Inthenextsection,webrieflyexplaintherenormalization
factors for the MDWF at the tree-level without the SF boundary condition. The renormalization
factorsareappliedtothecaseoftheSFscheme. ThenweintroducetheMDWFwiththeSFbound-
arytermgivenbyourpreviouswork[9]. Thelowesteigenvaluesoftheeffectivefour-dimensional
operatorareinvestigated. Insection3,weconfirmthattheproperlyrenormalizedMDWFoperator
withtheSFboundarytermreproducestheproperone-loopbeta-function. Thescalingviolationon
thestepscalingfunctionisalsoexaminedattheone-looplevel. Wesummarizethisworkinthelast
section.
2. ThesetupoftheSFschemefortheMDWF
The Moebius domain wall fermion (MDWF) [12] is one of the lattice chiral fermion and
2
Theone-loopanalysisofthebeta-functionintheSFforMDWF YukoMurakami
definedasthefifthdimensionaloperatorasfollows.
D (n,s;m,t)=D+(n;m)δ +D−(n;m)M (s;t), (2.1)
MDWF s s,t s 5
D+(n;m)=b D (n;m)+1, (2.2)
s s WF
D−(n;m)=c D (n;m)−1, (s=1,···,N ) (2.3)
s s WF 5
M (s;t)=P δ +P δ −m [P δ δ +P δ δ ], (2.4)
5 L s+1,t R s−1,t f L s,t 1,t R s,1 N5,t
where(s,t)isthelatticeindexinthefifthdirection,(n,m)isthefour-dimensionallatticesiteindex,
m is the mass parameter, D is the Wilson fermion operator with a negative mass (m ), P is
f WF 0 R/L
the chiral projection: P =(1±γ )/2, (b ,c ) are the Moebius parameters, and N is the lattice
R/L 5 s s 5
sizeinthefifthdirection. TheMDWFisageneralizationofthedomainwallfermionsandincludes
thestandarddomainwallfermion(SDWF)[10],theBoricidomainwallfermion(BDWF)[13],the
optimalShamirdomainwallfermion[14]andtheoptimalChiudomainwallfermion(CDWF)[15]
by adjusting (b ,c ). The Moebius parameters for the optimal type domain wall fermions can be
s s
derivedfromtheZolotarevsignfunctionapproximation[16].
Theeffectivefour-dimensionaloperatorcalculatedfromtheMDWFoperatoryieldstheover-
lap fermion operator [17, 18] at the infinite extent of the fifth direction [19, 20] (N →∞). At a
5
finiteN theeffectiveoperatordoesnotsatisfytheGinsparg-Wilsonrelation[21]andanO(a)error
5
is expected, so that the relation to the pole mass and the bare mass m can be differ in this case.
f
(N )
Theeffectivefour-dimensionaloperator,D 5 ,atafiniteN withouttheSFboundaryconditionis
eff 5
evaluatedas
aD(N5)≡PTD−1D (am )P
eff PV MDWF f
1+am 1−am
= f − fγ R (H ), (2.5)
2 2 5 N5 W
R (H )= ∏Ns=51(1+ωsHW)−∏Ns=51(1−ωsHW), (2.6)
N5 ω ∏Ns=51(1+ωsHW)+∏Ns=51(1−ωsHW)
aD
H = WF , (2.7)
W
a D +2
5 WF
a =b −c , ω =b +c , (2.8)
5 s s s s s
P(s)=P δ +P δ , (2.9)
L s,1 R s,N5
whereD =D (am =1.0). Thecoefficientsa andω mustbetunedtoproperlyreproduce
PV MDWF f 5 s
the sign function as R (x) → sign(x) in the limit of N → ∞. The ordering of (b ,c ) in the
N5 5 s s
fifth direction is irrelevant for the effective four-dimensional operator without the SF boundary
condition. Inthecontinuumlimittheeffectivefour-dimensionaloperatorbehavesas
aD(N5)≈Z (aD/+am ), (2.10)
eff N5 res
(1−am )R (α)
f N
Z = 5 , (2.11)
N
5 (am )(2−(am )a )
0 0 5
(cid:20) (cid:21)
1+am 1 (am )(2−(am )a )
f 0 0 5
am = −1 , (2.12)
res
1−am R (α) 2
f N
5
(am )
0
α = , (2.13)
(2−(am )a )
0 5
3
Theone-loopanalysisofthebeta-functionintheSFforMDWF YukoMurakami
where D/ is the Dirac operator in the continuum, Z is the normalization factor and m is the
N5 res
residualmassatthetree-level[11]. Evenatm =0theresidualmassdoesnotvanishasR (α)(cid:54)=1
f N
5
atafiniteN <∞. Inordertoinvestigatethepropertyoftheeffectiveoperatoratthevanishingpole
5
mass,werenormalizetheeffectiveoperatorasfollows,
R (α)−1
aD(N5)=Z−1aD(N5)(am ), where am = N5 . (2.14)
R N5 eff cr cr RN (α)+1
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Now we consider the effective four-dimensional operator in the SF scheme. In our previous
work,weproposedthefollowingoperatorfortheMDWFintheSFscheme[9].
DSF (n,s;m,t)=D (n,s;m,t)+c B (n,s;m,t), (2.15)
MDWF MDWF SF SF
B (n,s;m,t)= f(s)δ δ δ γ (δ P +δ P ), (2.16)
SF s,N5−t+1 nnn,mmm n4,m4 5 n4,1 − n4,T−1 +
(cid:40)
−1 (for1≤s≤N /2)
f(s)= 5 , (2.17)
+1 (forN /2+1≤s≤N )
5 5
where P is the projection: P =(1±γ )/2. We introduce the boundary operator B to satisfy
± ± 4 SF
the SF boundary condition, which explicitly breaks the domain wall chiral symmetry [10] at the
temporalboundary,assuggestedinRefs.[5,8]. InordertokeepthediscretesymmetriesC,P,T,and
Γ -Hermiticity we require the parity symmetry in the fifth direction because the ordering of the
5
coefficients (b ,c ) in the fifth direction is relevant in this case. A quasi optimal choice for ω
s s s
with the parity symmetry are determined according to Ref. [9]. We call the domain wall fermion
operators with the quasi optimal choice for ω having the parity symmetry in the fifth direction
s
asthepalindromicShamirdomainwallfermion(PSDWF)andthepalindromicChiudomainwall
fermion (PCDWF) depending on the choice for the kernel operator. We use the same form as
Eq. (2.14) for the normalization factor and the pole mass with the SF boundary condition, though
wecannotobtaintheeffectivefour-dimensionaloperatorinasimpleclosedformlikeEq.(2.5)and
thecorrespondingbehaviorinthecontinuumlimitlikeEq.(2.10).
We investigate the lowest eigenvalues of
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the effective four-dimensional operator derived
)
5
N 20
from Eq. (2.15) at the tree-level. We use (
R
D
L=T and the standard boundary condition for † 15
)
the gauge field [4] yielding a constant back- )5
(N 10
ground chromoelectric field. The generalized DR
(
periodicboundaryconditionwiththephasean- 2 5
L
gle θ = π/5 is used for the spatial boundary
0
0 0.04 0.08 0.12
condition. In order to study the effect of the a/L
SDWF PSDWF
renormalization with a finite fifth dimensional BDWF Continuum
extent at the tree-level, we set m0 = 1.5 and Figure 1: The scaling behavior of the lowest
(b ,c )=(1.0,0.0)fortheSDWFandm =1.5 fiveeigenvaluesoftheHermitiansquaredeffective
s s 0
and(b ,c )=(1.0,1.0)fortheBDWFandm = four-dimensionaloperator.
s s 0
1.0forthePSDWFatN =8andc =1.0.
5 SF
Figure1showsthelowestfiveeigenvaluesoftheHermitiansquaredeffectivefour-dimensional
operator L2(D(N5))†D(N5) for each operator type. The yellow dot points at the continuum limit
R R
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Theone-loopanalysisofthebeta-functionintheSFforMDWF YukoMurakami
are quoted from Ref [22]. We confirm that the operator involving the tree-level renormalization
properlyreproducesthecontinuumtheory.
3. Theone-loopbeta-function
We calculate the fermoinic contribution to the one-loop beta-function using the MDWF with
theSFboundarycondition. TherenormalizedcouplingconstantintheSFscheme[3]isdefinedby
(cid:12)
1 1∂Γ(cid:12)
= (cid:12) , (3.1)
g2 k∂η(cid:12)
SF η=ν=0
where Γ is the effective action with the SF boundary condition, η and ν are parameters for the
SF boundary condition, and k is a normalization factor based on the tree-level analysis. For the
MDWFintheSFscheme,thefermioniccontributiontotheone-loopbeta-functioncanbeobtained
fromthefollowingterm;
(cid:12) (cid:12)
p1,1= 1k∂∂Γη1(cid:12)(cid:12)(cid:12) = 1k∂∂η (cid:2)lndet(DSMFDWF(DSPFV)−1)(cid:3)(cid:12)(cid:12)(cid:12) , (3.2)
η=ν=0 η=ν=0
whereΓ istheone-loopeffectiveactionofthefermionpartandDSF =DSF (am =1.0). The
1 PV MDWF f
cut-offdependenceof p canbewritteninthefollowingformasymptotically;
1,1
p = ∑(a/L)n[A +B ln(L/a)]=A +B ln(L/a)+(a/L)(A +B ln(L/a))+O(a2). (3.3)
1,1 n n 0 0 1 1
n=0
B contains the information of the fermionic contribution to the one-loop beta-function and it
0
should be B = 2b = −1/(12π2) = −0.008443.... We numerically calculate p using the
0 0,1 1,1
MDWF with the SF boundary of Eq. (2.15) and the proper renormalization at the vanishing pole
mass, and fit them with the function Eq. (3.3) up to O(a). Table 1 shows the result of the fitting.
The values for c are determined with the O(a) improvement procedure based on the PCAC re-
SF
lation [23] at the tree-level. We find that B are consistent with the 2b =−0.008443... within
0 0,1
the fitting error for each DWF type and confirm that the MDWF with the SF boundary term we
constructed at the tree-level properly reproduces with the fermionic contribution to the one-loop
beta-function.
Operator m c N Fitrange[L /a,L /a] B (×10−3)
0 SF 5 min max 0
SDWF 1.5 0.520 8 [18:48] −8.43±0.01
16 [26:48] −8.441±0.004
BDWF 1.5 0.312 8 [18:48] −8.37±0.08
16 [34:48] −8.46±0.05
PSDWF 1.0 0.820 8 [10:48] −8.2±0.3
0.630 16 [16:48] −8.44±0.03
Table1: ThefitresultsforB .
0
We investigate the lattice cut-off dependence for the step scaling function (SSF) [24]. The
deviationoftheSSFatafinitecut-offfromthatinthecontinuumlimitisdefinedby
Σ(s,u,a/L)−σ(s,u)
δ = =δ u+δ u2+···, (3.4)
1 2
σ(s,u)
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Theone-loopanalysisofthebeta-functionintheSFforMDWF YukoMurakami
× 10-3
4.0 SDWF
BDWF
3.0 PSDWF
2.0
1
,
1
δ 1.0
0
-1.0
0 0.05 0.1 0.15 0.2 0.25 0.3
a/L
Figure2: ThedeviationoftheSSFbetweenthelatticeandthecontinuumtheoryasafunctionofthelattice
spacing. (N =16,2b =−1/(12π2))
5 0,1
where u is the SF scheme coupling constant u=g2 (L) renormalized at L, Σ(s,u,a/L)=g2 (sL)
SF SF
is the SSF at a finite lattice cut-off, σ(s,u)=g2 (sL) is that in the continuum theory, and s is a
SF
scale factor. δ can be expanded as a polynomial of u perturbatively as shown in the equation. δ
1
mustvanishinthecontinuumlimitattheone-looplevelanalysis. Thefermioniccontributiontoδ
1
isgivenby
δ =δ +δ n , (3.5)
1 1,0 1,1 f
δ = p (a/(2L))−p (a/L)−2b ln2, (3.6)
1,1 1,1 1,1 0,1
wherewesets=2.
The lattice cut-off dependence of Eq. (3.6) with the MDWF atN =16 is shown in Figure2,
5
where 2b =−1/(12π2) is imposed. Although rather complicated cut-off dependence of δ is
0,1 1,1
seen,itgoestovanishinthecontinuumlimit. ThereforeweconcludethattherenormalizedMDWF
withtheSFschemeisconsistentwiththecontinuummasslessDiracoperatorattheone-looplevel.
Thecut-offdependenceincludestheO(a)errorfromA andB termsofEq.(3.3). Theformercan
1 1
beremovedbytuningtheboundaryoperatorofthegaugeaction,whilethelatterrequiresthebulk
O(a)-improvement term such as the clover term for the Wilson fermion because the lattice chiral
symmetryisbrokenexplicitlyatafiniteN .
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4. Summary
We investigated the Schroedinger Functional (SF) scheme with the Moebius domain wall
fermions (MDWF). The MDWF with the SF boundary term was introduced and the spectrum of
themasslessDiracoperatorwiththeSFboundaryconditionisreproducedfromtheMDWFatthe
tree-level after applying the proper renormalization even at a finite fifth dimensional extent. We
alsoconfirmedthattheMDWFoperatorwiththeSFboundarytermreproducedthefermionicpart
oftheuniversalone-loopbeta-functionintheSFscheme. Fromtheseanalysisattheone-looplevel,
we expectthat the SFscheme is applicable to theMDWF at afiniteN , whichcan be regardedas
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akindofbetterWilsonfermions,byaddingtheSFboundaryterm.
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Theone-loopanalysisofthebeta-functionintheSFforMDWF YukoMurakami
Wewillchecktheconsistencyofthestepscalingfunctionofthecouplingcalculatedfromthe
MDWFwiththeSFboundarytermnon-perturbativelyinthefuturework.
Acknowledgment
WethanktoS.Takedaforthehelpfuladvice. Apartofnumericalcomputationsisperformed
ontheINSAM(InstituteforNumericalSimulationsandAppliedMathematics)GPUClusterat
HiroshimaUniversity. ThisworkwassupportedinpartbyaGiant-in-AidforScientificResearch
(C)(No. 24540276)fromtheJapanSocietyforthePromotionofScience(JSPS)andtheMEXT
programforpromotingtheenhancementofresearchuniversities,Japan.
References
[1] J.M.Flynn,T.Izubuchi,T.Kawanai,C.Lehner,A.Soni,R.S.VandeWaterandO.Witzel,Phys.
Rev.D91,no.7,074510(2015)[arXiv:1501.05373[hep-lat]].
[2] S.Aokietal.,Eur.Phys.J.C74,2890(2014)[arXiv:1310.8555[hep-lat]].
[3] M.Lüscher,Nucl.Phys.B254,52(1985)
[4] M.Luscher,R.Sommer,P.WeiszandU.Wolff,Nucl.Phys.B413,481(1994)[hep-lat/9309005].
[5] M.Luscher,JHEP0605,042(2006)[hep-lat/0603029].
[6] Y.Taniguchi,JHEP0512,037(2005)[hep-lat/0412024].
[7] Y.Taniguchi,JHEP0610,027(2006)[hep-lat/0604002].
[8] S.Takeda,Phys.Rev.D87,no.11,114506(2013)[arXiv:1010.3504[hep-lat]].
[9] Y.MurakamiandK.I.Ishikawa,PoSLATTICE2014,331(2014)[arXiv:1410.8335[hep-lat]].
[10] V.FurmanandY.Shamir,Nucl.Phys.B439,54(1995)[hep-lat/9405004].
[11] S.Capitani,PoSLAT2007,066(2007)[arXiv:0708.3281[hep-lat]].
[12] R.C.Brower,H.NeffandK.Orginos,Nucl.Phys.Proc.Suppl.140,686(2005)[hep-lat/0409118].
[13] A.Borici,hep-lat/9912040.
[14] A.Borici,hep-lat/0211001.
[15] T.W.Chiu,Phys.Rev.Lett.90,071601(2003)[hep-lat/0209153].
[16] T.W.Chiu,T.H.Hsieh,C.H.HuangandT.R.Huang,Phys.Rev.D66,114502(2002)
[hep-lat/0206007].
[17] R.NarayananandH.Neuberger,Nucl.Phys.B443,305(1995)[hep-th/9411108].
[18] H.Neuberger,Phys.Lett.B417(1998)141[hep-lat/9707022].
[19] Y.KikukawaandT.Noguchi,doi:10.1016/S0920-5632(00)91758-4hep-lat/9902022.
[20] A.Borici,hep-lat/0402035.
[21] P.H.GinspargandK.G.Wilson,Phys.Rev.D25,2649(1982).
[22] S.SintandR.Sommer,Nucl.Phys.B465,71(1996)[hep-lat/9508012].
[23] M.Luscher,S.Sint,R.SommerandP.Weisz,Nucl.Phys.B478,365(1996)[hep-lat/9605038].
[24] M.Luscher,P.WeiszandU.Wolff,Nucl.Phys.B359,221(1991)
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