Table Of ContentPolyakov loop analysis with Dirac-mode expansion
3
1
0
2
n
a
J
4 TakumiIritani∗,ShinyaGongyo,andHideoSuganuma
1
DepartmentofPhysics,KyotoUniversity,Kitashirakawaoiwake,Sakyo,Kyoto606-8502,Japan
] E-mail:[email protected]
t
a
l
-
p Inorderto investigatethe directrelationbetweenconfinementandchiralsymmetrybreakingin
e QCD, we investigatethe Polyakovloopin termsof the Dirac eigenmodesin both confinedand
h
[ deconfinedphases. UsingtheDirac-modeexpansionmethodin SU(3)lattice QCD,we analyze
the contribution of low-lying and higher Dirac-modes to the Polyakov loop, respectively. In
1
v theconfinedphasebelowT , afterremovinglow-lyingDirac-modes,thechiralcondensatehq¯qi
c
5
is largely reduced, however, the Polyakov loop remains almost zero and Z -center symmetry
5 3
8 is unbroken. These results indicate that the system is still in the confined phase without low-
2
lying Dirac-modes. By higher Dirac-modes cut, the Polyakov loop also remains almost zero
.
1
belowT . WealsoanalyzethePolyakovloopinthedeconfinedphaseaboveT. Wefindthatthe
0 c c
3 PolyakovloopandZ -symmetrybehaviorareinsensitivetolow-lyingandhigherDirac-modesin
3
1
bothconfinedanddeconfinedphases.
:
v
i
X
r
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani
1. Introduction
Quantum chromodynamics (QCD) is the fundamental theory of the strong interaction, how-
ever,itsnon-perturbative properties suchasconfinementandchiralsymmetrybreaking arenotyet
wellunderstood. Inparticular, toclarify thecorrespondence betweenconfinement andchiralsym-
metrybreakingisoneofdifficultandinteresting subjects[1,2,3,4,5,6,7]. Asanevidenceofthe
close relation between them, lattice QCDcalculation shows that simultaneous deconfinement and
chiralphasetransition atfinitetemperature [8].
As shown in the Banks-Casher relation [9], the chiral condensate hq¯qi is proportional to the
Diraczero-mode densityas
hq¯qi=−lim lim p hr (0)i, (1.1)
m→0V→¥
where r (l ) is the Dirac spectral density. Thus, the low-lying Dirac eigenmodes directly relate to
chiralsymmetrybreaking, however,theirrelation toconfinementisstillunclear.
Therefore, itisinteresting toanalyze confinement interms oftherelevant degreesoffreedom
for chiral symmetry breaking, i.e., the low-lying Dirac eigenmodes. For example, based on the
Gattringer’s formula [1], the Polyakov loop can be investigated by the sum of Dirac spectra with
twisted boundary condition on lattice [2, 3, 4]. In our previous studies [5, 6], we formulated the
Dirac-modeexpansionmethodinlatticeQCD,andinvestigated theDirac-modedependence ofthe
Wilson loopandthe interquark potential. Itisalsoreported thathadrons still existasbound states
without“chiralsymmetrybreaking” byremovinglow-lyingDirac-modes[10,11].
Inthispaper,weinvestigatetheDirac-modedependenceofthePolyakovloopinbothconfined
and deconfined phases at finite temperature, using Dirac-mode expansion method in lattice QCD
[5, 6, 7]. In Sec. 2, we review the Dirac-mode expansion method, and formulate the Dirac-mode
projectedPolyakovloop. InSec.3,weperformthelatticeQCDcalculations forthePolyakovloop
withDirac-modeprojection. Section4isdevoted forthesummary.
2. Dirac-mode expansionmethod inlatticeQCD
Here, weintroduce the Dirac-mode expansion technique inlattice QCD[5, 6, 7], and formu-
lationoftheDirac-modeprojected Polyakovloop.
2.1 Dirac-modeexpansioninlatticeQCD
Usingthelink-variableUm ∈SU(Nc),theDiracoperator D/ =gm Dm isgivenby
D/x,y ≡ 21a (cid:229) 4 gm (cid:2)Um (x)d x+mˆ,y−U−m (x)d x−mˆ,y(cid:3), (2.1)
m =1
with a lattice spacing a, andU−m (x)≡Um†(x−mˆ). Here, g -matrix is defined to be hermitian, i.e.,
gm† =gm . Thus, D/ becomes an antihermitian operator, and Dirac eigenvalues are pure imaginary.
Weintroduce thenormalized Diraceigenstate |ni,whichsatisfies
D/|ni=il |ni, (2.2)
n
2
PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani
withl ∈R,andaneigenfunction y (x)isexpressed as
n n
y (x)≡hx|ni, (2.3)
n
whichsatisfiesD/y =il y .
n n n
We introduce the operator formalism in lattice QCD [5, 6, 7], which is constructed from the
link-variable operatorUˆm . Thelink-variable operatorisdefinedbythematrixelement as
hx|Uˆm |yi=Um (x)d x+mˆ,y, (2.4)
usingtheoriginal link-variableUm (x). WedefinetheDirac-modematrixelement hn|Uˆm |mias
hn|Uˆm |mi = (cid:229) hn|xihx|Uˆm |x+mˆihx+mˆ|mi=(cid:229) y n†(x)Um (x)y m(x+mˆ), (2.5)
x x
withtheDiraceigenfunction y (x).
n
Considering thecompleteness relation (cid:229) |nihn|=1,anyoperator Oˆ canbeexpressed as
n
Oˆ =(cid:229) (cid:229) |nihn|Oˆ|mihm|, (2.6)
n m
using the Dirac-mode basis. Note that this procedure is just the insertion of unity, and it is math-
ematically correct. This expansion (2.6) is the mathematical basis of the Dirac-mode expansion
method [5,6,7]. Next, weconsider the Dirac-mode projection by introducing projection operator
as
Pˆ≡ (cid:229) |nihn|, (2.7)
n∈A
forarbitrary set A ofeigenmodes. Forexample,IRandUVmode-cutoperators are givenby
Pˆ ≡ (cid:229) |nihn|, Pˆ ≡ (cid:229) |nihn|, (2.8)
IR UV
|ln|≥L IR |ln|≤L UV
withtheIR/UVcutoffscaleL andL .
IR UV
Usingtheprojection operator Pˆ,theDirac-modeprojected link-variable operatorisgiven by
UˆmP≡PˆUˆm Pˆ= (cid:229) (cid:229) |nihn|Uˆm |mihm|. (2.9)
n∈A m∈A
WecaninvestigatetheDirac-modedependenceofvariouskindsofquantities,e.g.,theWilsonloop
[5,6],usingtheprojected link-variableUˆmP instead oftheoriginallink-variable operatorUˆm .
2.2 PolyakovloopoperatorandDirac-modeprojection
Next, weformulate the Dirac-mode projected Polyakov loop. Here, weconsider the periodic
SU(3) lattice with the space-time volume V =L3×N and the lattice spacing a. In the operator
t
formalism oflatticeQCD,thePolyakovloopoperatorisgivenby
Lˆ ≡ 1 (cid:213)Nt Uˆ = 1 UˆNt (2.10)
P 3V 4 3V 4
i=1
3
PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani
withthetemporallink-variable operatorUˆ . Bythefunctional trace“Tr”,thePolyakov loopoper-
4
atorcoincides withthestandard definitionas
TrLˆ = 1 Tr{(cid:213)Nt Uˆ }= 1 tr(cid:229) h~x,t|(cid:213)Nt Uˆ |~x,ti
P 4 4
3V 3V
i=1 ~x,t i=1
= 1 tr(cid:229) h~x,t|Uˆ |~x,t+aih~x,t+a|Uˆ |~x,t+2ai···h~x,t+(N −1)a|Uˆ |~x,ti
4 4 t 4
3V
~x,t
1 (cid:229)
= tr U (~x,t)U (~x,t+a)···U (~x,t+(N −1)a)=hL i, (2.11)
4 4 4 t P
3V
~x,t
where“tr”denotes thetraceoverSU(3)colorindex.
WedefinetheDirac-modeprojected PolyakovloophLproj.ias
P
LpProj. ≡ 31VTr{(cid:213)Nt Uˆ4P}= 31VTr(cid:8)PˆUˆ4PˆUˆ4Pˆ···PˆUˆ4Pˆ(cid:9)
i=1
= 1 tr (cid:229) hn |Uˆ |n ihn |Uˆ |n i···hn |Uˆ |n i. (2.12)
3V 1 4 2 2 4 3 Nt 4 1
n1,n2,...,nNt∈A
Inparticular, theIRandtheUVDirac-modeprojected Polyakovlooparedenoted as
hL i ≡ 1 tr (cid:229) hn |Uˆ |n i···hn |Uˆ |n i, (2.13)
P IR 3V 1 4 2 Nt 4 1
|lni|≥L IR
hL i ≡ 1 tr (cid:229) hn |Uˆ |n i···hn |Uˆ |n i, (2.14)
P UV 3V 1 4 2 Nt 4 1
|lni|≤L UV
withtheIR/UVeigenvalue cutoffL andL .
IR UV
3. Lattice QCDcalculationforDirac-modeprojected Polyakovloop
Inthissection,wecalculatetheDirac-modeprojectedPolyakovloopusingSU(3)latticeQCD
at the quenched level. We evaluate the full Dirac eigenmodes using LAPACK [12]. For actual
calculation, weusetheeigenmode basisoftheKogut-Susskind (KS)operatorof
DKx,yS≡ 21a (cid:229) 4 h m (x)(cid:2)Um (x)d x+mˆ,y−U−m (x)d x−mˆ,y(cid:3), (3.1)
m =1
withh 1(x)≡1andh m (x)≡(−1)x1+···+xm−1 (m ≥2)inordertoreducethecomputational cost. The
use of the KS-Dirac operator gives the same result as the original Dirac operator in Eq. (2.1) for
thePolyakovloop.
3.1 Theconfinedphase
First, we analyze the Polyakov loop in the confined phase below T . Here, we use 64 lattice
c
withb =5.6, whichcorresponds tothe lattice spacing a≃0.25 fmand T ≡1/(Na)≃0.13 GeV
t
[5,6,7].
4
PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani
1400 64 lattice with β = 5.6 1400 ΛIR = 0.5a-1 1400 ΛUV = 2.0a-1
1200 1200 1200
1000 1000 1000
ρλVa () [] 468000000 ρλVa () []IR 468000000 ρλVa () []UV 468000000
200 200 200
0 0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
λ [a-1] λ [a-1] λ [a-1]
Figure1: Thespectraldensityr (l )oftheDiracoperatoron64 latticewithb =5.6,i.e.,a=0.25fm: (a)
originalspectraldensity,(b)IR-cutr (l )atL =0.5a−1,(c)UV-cutr (l )atL =2.0a−1.
IR IR UV UV
0.3 0.3 0.3
β = 5.6 IR cut Λ = 0.5a-1 UV cut Λ = 2.0a-1
IR UV
0.2 0.2 0.2
0.1 R 0.1 V 0.1
L hiP 0 L iPI 0 L iPU 0
Im -0.1 m h -0.1 m h -0.1
I I
-0.2 -0.2 -0.2
-0.3 -0.3 -0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
Re hL i Re hL i Re hL i
P P IR P UV
Figure 2: The scatter plot of the Polyakov loop in the confined phase on 64 lattice with b =5.6, i.e.,
a=0.25 fm and T ≡1/(Na)≃0.2 GeV. (a) The original (no Dirac-mode cut) Polyakov loop. (b) The
t
low-lyingDirac-modecutatL =0.5a−1. (c)ThehigherDirac-modecutatL =2.0a−1.
IR UV
Figure 1 shows the lattice QCD results for the Dirac spectral density r (l ), and IR/UV-cut
spectraldensityr (l )≡r (l )q (|l |−L ),r (l )≡r (l )q (L −|l |)withL =0.5a−1 and
IR IR UV UV IR
L =2.0a−1. ThetotalnumberoftheKSDirac-modeisL3×N ×3=3888,andbothmodecuts
UV t
correspond toremovingabout400modes.
Figure 2 shows the scatter plot of the original Polyakov loop, low-lying Dirac-modes cut
at L = 0.5a−1, and the higher Dirac-modes cut at L = 2.0a−1, respectively. As shown in
IR UV
Fig.2(a),thePolyakovloopisalmostzero,i.e.,hL i≃0,whichindicates theconfinedphase.
P
Then, weconsider low-lying Dirac-modes projection, which leads tothe effective restoration
ofchiralsymmetrybreaking[6,9,10,11]. Inthepresence of theIRcutL ,thequarkcondensate
IR
isgivenby
1 (cid:229) 2m
hq¯qi =− . (3.2)
IR V l 2+m2
ln≥L IR n
At the IR cut parameter L =0.5a−1 ≃0.4 GeV, only 2% of the quark condensate remains as
IR
hq¯qi /hq¯qi≃0.02 around the physical region m≃5 MeV [6]. However, as shown in Fig. 2(b),
IR
thePolyakovloophL i remainsalmostzeroandZ -centersymmetryisunbroken,andthesefacts
P IR 3
indicatethatthesystemstillremainsintheconfinedphase,evenwithoutchiralsymmetrybreaking.
In addition to the low-lying mode cut, we show the higher Dirac-modes cut in Fig. 2(c). In
thiscase,thechiralcondensationisalmostunchanged, andthePolyakovloopremainsalmostzero.
Therefore, thePolyakovloopisinsensitive tobothlow-lyingandhigherDiraceigenmodes.
5
PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani
0.3 0.3 0.3
β = 6.0 IR cut Λ = 0.5a-1 UV cut Λ = 2.0a-1
IR UV
0.2 0.2 0.2
0.1 R 0.1 V 0.1
L hiP 0 L iPI 0 L iPU 0
Im -0.1 m h -0.1 m h -0.1
I I
-0.2 -0.2 -0.2
-0.3 -0.3 -0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3
Re hL i Re hL i Re hL i
P P IR P UV
Figure 3: The scatter plot of the Polyakov loop in the deconfinedphase on 63×4 lattice with b =6.0,
i.e.,a=0.10fmandT ≡1/(Na)≃0.5GeV.(a)Theoriginal(noDirac-modecut)Polyakovloop. (b)The
t
low-lyingDirac-modecutatL =0.5a−1. (c)ThehigherDirac-modecutatL =2.0a−1.
IR UV
3.2 Thedeconfinedphaseathightemperature
Next,weinvestigatethePolyakovloopinthedeconfinedphaseathightemperature. Here,we
use 63×4 lattice with b =6.0, which corresponds to a=0.10 fm and T ≡1/(Na)≃0.5 GeV.
t
ThetotalnumberoftheKSDirac-modeisL3×N ×3=2592.
t
Weshow theoriginal Polyakov loop, typical low-lying modecutatL =0.5a−1,and higher
IR
mode cut at L =2.0a−1 in Fig. 3. These mode cuts correspond to removing about 200 eigen-
UV
modes. As shown in Fig. 3(a), the Polyakov loop has non-zero expectation value hL i=6 0, and
P
showsthecentergroupZ structureonthecomplexplane. Thesebehaviorsindicatethedeconfined
3
phase.
As shown in Figs. 3(b) and (c), the Polyakov loop shows the characteristic behaviors in the
deconfined phase even after removing low-lying or higher Dirac-modes. To be strict, the UV-cut
Polyakov loop has a smaller absolute value than the IR-cut Polyakov loop, although the number
of UV-cut modes is comparable to that of the IR-cut case. This suggests that contributions of the
higher Dirac-modes are much larger than low-lying modes [2]. However, apart from the normal-
ization, both IR and UV cut Polyakov loops show the characteristic Z -pattern in the deconfined
3
phase, andhencetheseDirac-modesseemtobeinsensitive tothePolyakovloopproperties.
Figure4showstheDiracspectraldensitiesinconfinedanddeconfinedphaseson63×4lattice
withb =5.6and6.0,respectively. Wealsocomparetheirlow-lyingspectraldensities inFig.4(c).
In the deconfinement phase, the low-lying Dirac-modes are suppressed, which leads to the chiral
restoration. Thechiralcondensate isalsoreducedbytheIR cutoftheDirac-modesasinEq.(3.2).
On the other hand, there seems no clear correspondence between the Dirac spectral densities and
thePolyakovloop.
3.3 b -dependenceoftheDirac-modeprojected Polyakovloop
Finally, we investigate b -dependence of the Dirac-mode projected Polyakov loop. Here, we
adopt63×4latticewithb =5.4∼6.0.
Figure 5 is the absolute values of the Polyakov loop with typical Dirac-mode projections,
and the original Polyakov loop data are also added for comparison. In this lattice volume, the
deconfinement phase transition occurs around b =5.6∼5.7. Asshown in Fig. 5, both low-lying
6
PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani
1000 63 × 4 lattice with β = 5.6 1000 63 × 4 lattice with β = 6.0 116800 6633 ×× 44 llaattttiiccee wwiitthh ββ == 65..06
800 800 140
120
Va [] 600 Va [] 600 Va [] 100
λ) λ) λ) 80
ρ ( 400 ρ ( 400 ρ ( 60
200 200 40
20
0 0 0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5
λ [a-1] λ [a-1] λ [a-1]
Figure4: TheDiracspectraldensitiesinconfinedanddeconfinedphases,respectively(a)63×4latticewith
b =5.6intheconfinedphase. (b)63×4latticewithb =6.0inthedeconfinedphase. (c)Thecomparison
betweenconfinedanddeconfinedphasesonlow-lyingspectraldensities.
0.3 0.3
no cut no cut
Λ = 0.5a-1 Λ = 2.0a-1
IR UV
Λ = 1.0a-1 Λ = 1.7a-1
IR UV
0.2 0.2
|R |V
L iPI L iPU
|h 0.1 |h 0.1
0 0
5.4 5.5 5.6 5.7 5.8 5.9 6 5.4 5.5 5.6 5.7 5.8 5.9 6
β β
Figure5: b -dependenceoftheabsolutevalueofthePolyakovloopon63×4lattice.(a)TheIRDirac-mode
cutwithL =0.5a−1and1.0a−1. (b)TheUVDirac-modecutwithL =2.0a−1and1.7a−1.
IR UV
and higher Dirac-mode projected Polyakov loops show the similar b -dependence as the original
data, apart from a normalization factor. This Dirac-mode insensitivity of the Polyakov loop is
consistent withtheresultsintheprevious subsections.
4. Summary
In this paper, we have analyzed the Polyakov loop in terms of the Dirac eigenmodes using
SU(3) lattice QCD. We have carefully removed relevant degrees of freedom for chiral symmetry
breaking fromthePolyakovloop.
IntheconfinedphasebelowT ,thePolyakovloopisalmostzero,i.e.,hL i≃0. Byremoving
c P
low-lying Dirac-modes, the chiral condensate hq¯qi is largely reduced. However, we have found
that the Polyakov loop remains almost zero as hL i ≃ 0 even without low-lying Dirac-modes,
P IR
andthisfactindicatesthatthesystemstillremainsintheconfinedphase. Wehavealsoinvestigated
contributions from higher Dirac-modes to the Polyakov loop, and have found no change of the
PolyakovloopwithouthigherDirac-modes. Therefore,thereseemsnospecificregionoftheDirac
eigenmodes essential forthePolyakovloop.
We have also investigated the Polyakov loop in the deconfined phase at high temperature. In
thedeconfined phase, thePolyakovloophasnon-zero expectation value, i.e.,hL i=6 0,whichdis-
P
tributesaround Z elementsinthecomplexplane. Thesecharacteristic behaviorsalsoremainwith-
3
7
PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani
out low-lying and higher Dirac eigenmodes. Therefore, the Polyakov loop and the Z -symmetry
3
behaviordonotdependonlow-lyingandhigherDiraceigenmodesinbothconfinementanddecon-
finementphases.
Here, we comment on the related studies about the correspondence between the Dirac eigen-
modes and confinement. In the previous studies [5, 6], we investigated Dirac-mode dependence
oftheWilsonloop, andfound thattheconfining potential survives withoutlow-lying Diraceigen-
modes. TheGrazgroupalsoreportedthathadrons stillremainasboundstateswithoutchiralsym-
metrybreakingbyremovinglow-lyingDirac-modes[10,11],whichseemstosuggesttheexistence
oftheconfiningforce.
These lattice QCD studies suggest that there is no direct relation between chiral symmetry
breaking and confinement through the Dirac eigenmodes. For further investigation of correspon-
dence between these phenomena, it is also interesting to analyze chiral symmetry breaking from
therelevanteigenmodes ofconfinement[13].
Acknowledgements
ThelatticeQCDcalculationshavebeendoneonNEC-SX8andNEC-SX9atOsakaUniversity.
This work is in part supported by a Grant-in-Aid for JSPSFellows [No.23-752, 24-1458] and the
GrantforScientificResearch [(C)No.23540306, PriorityAreas“NewHadrons” (E01:21105006)]
fromtheMinistryofEducation, Culture,ScienceandTechnology(MEXT)ofJapan.
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