Table Of ContentLattice simulation of SU(2) gauge theory with
chirally symmetric fermions
4
1
0
2
n
a HideoMatsufuru∗
J
HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba305-0801,Japan
6
E-mail: [email protected]
2
YoshioKikukawa
]
t InstituteofPhysics,UniversityofTokyo,Tokyo153-8092,Japan
a
l E-mail: [email protected]
-
p
e Kei-ichiNagai
h
Kobayashi-MaskawaInstitutefortheOriginofParticlesandtheUniverse(KMI),Nagoya
[
University,Nagoya,464-8602,Japan
1
E-mail: [email protected]
v
5
Norikazu Yamada
5
HighEnergyAcceleratorResearchOrganization(KEK),Tsukuba305-0801,Japan,
6
6 andGraduateUniversityforAdvancedStudies(Sokendai),Tsukuba305-0801,Japan
1. E-mail: [email protected]
0
4
We numerically study the SU(2) gauge theory with two dynamical flavors of the domain-wall
1
: fermionsinfundamentalrepresentation. Themesonspectraandtheresidualmassaremeasured
v
i onthreelatticevolumesandattwovaluesofgaugecouplingsoastoinvestigatethefinitevolume
X
effect. Ongeneratedconfigurations,eigenvaluesoftheoverlapfermionoperatoraredetermined
r
a
andcomparedto the randommatrixtheory. To quantifytheeffectofviolationoftheexactchi-
ral symmetry, we measurethe correlationbetween the eigenvectorsof the domain-walland the
overlapoperators.
31stInternationalSymposiumonLatticeFieldTheoryLATTICE2013
July29-August3,2013
Mainz,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
LatticesimulationofSU(2)gaugetheorywithchirallysymmetricfermions HideoMatsufuru
1. Introduction
The SU(2) gauge theory has for long time been studied as a prototype theory of QCD that
shares important features with QCD such as chiral symmetry breaking and confinement, while
with less numerical cost. Recently it has been drawn much attention in search for a theory be-
yond the standard model as a candidate of technicolor theory [1, 2, 3, 4]. In this work, we also
focus on the chiral dynamics of SU(2) gauge theory. The pattern of spontaneous chiral symme-
try breaking depends on the gauge group and the fermion representation. QCD, and in general
SU(N)withN ≥3gaugetheory withfundamental fermionsareconsidered tochirally breakdown
as SU(N )×SU(N )→SU(N ), where N is the number of flavors. Since SU(2) group is pseu-
f f f f
doreal, ithasapattern ofchiralsymmetrybreaking different fromQCD:SU(2N )→Sp(2N ). The
f f
adjoint representation of fermions results in another pattern SU(2N )→SO(2N ) for any N. One
f f
ofthemotivations ofthisworkistounderstand theeffectofthesymmetrybreaking pattern onthe
chiraldynamics, inparticular itsdependence onthenumberofflavorsandtemperature.
Toexplorethechiralsymmetryonthelattice,itisimportanttoemploythefermionformulation
thatretainsthechiralsymmetryasexactaspossible. Inthissense,theoverlapfermionoperator,
D =M [1+g signH (−M )], (1.1)
ov 0 5 W 0
is the best solution, where H is the Hermitian Wilson-Dirac operator with M the domain-wall
W 0
height. D satisfies the Ginsparg-Wilson relation, which expresses an exact chiral symmetry on
ov
the lattice. In particular, simulations in the so-called e -regime are an attractive device to extract
the chiral condensate precisely. However, its numerical cost is high and to reduce it down to an
acceptable level, elaborated setup, such as reduction of near-zero modes of H by employing the
W
topologyfixingterm,isdemanded. Suchsituationmaycausethesimulationandanalysisinvolved.
An alternative approach is the domain-wall (DW) fermion. The DW fermion operator is de-
finedasamassivefermioninthefive-dimensionalspace,andlightmodesappearontheboundaries
inthe5thdimension. TheDWactioniswrittenas
S = (cid:229) y¯(x,s)D (x,y;−M )y (y,s)
DW W 0
x,s
−1(cid:229) y¯(x,s)[(1−g )y (x,s+1)+(1+g )y (x,s−1)−2y (x,s)]
5 5
2
x,s
+m[y¯(x,1)P y (x,L )+y¯(x,L )P y (x,1)], (1.2)
R s s L
where m is fermion mass, P = (1±g )/2, L the size of 5th dimension, and D the Wilson
R,L 5 s W
operator withnegativemass,
DW(x,y;−M0)=−M0d x,y−21 (cid:229) 4 (cid:8)(1−gm )Um (x)d x+mˆ,y+(1+gm )Um†(x−mˆ)d x−mˆ,y−4d x,y(cid:9).
m =1
(1.3)
Taking the limit of L →¥ with fixed a L , the action (1.2) accompanied by Pauli-Villars ghost
s s s
term arrives at the overlap fermion operator (1.1). How much the chiral symmetry is violated in
DWfermionactionisprobedbyaquantity
(cid:229) hJ (~x,t)P(0)i
R(t)= ~x 5q , (1.4)
(cid:229) hP(~x,t)P(0)i
~x
2
LatticesimulationofSU(2)gaugetheorywithchirallysymmetricfermions HideoMatsufuru
0.35
Iwasaki (quenched) 83 x 16
0.30 Iwasaki (quenched) 123 x 24
Iwasaki + DW (Nf=2, m=0.02) 83 x 16
0.25
m] 0.20
a(r) [f0 0.15
0.10
0.05
0.00
0.7 0.8 0.9 1.0 1.1 1.2
b
Figure1: b -dependenceofthelatticespacingdefinedthroughtheSommer’shadronicradiusforquenched
andN =2simulationswiththeIwasakigaugeandthedomain-wallfermionactions.
f
where
J (x)=−y¯(x,L /2)P y (x,L /2+1)+y¯(x,L /2+1)P y (x,L /2). (1.5)
5q s L s s R s
AnaverageofR(t)overlarget separation isastandard definitionoftheresidual mass.
Our original strategy is to explore the dynamics of the chiral symmetry breaking with the
overlap fermioninthee -regimeinwhichtheComptonwavelength ofthePSmesonislarger than
the system size [5]. The eigenvalue distribution of the overlap fermion can be compared to the
chiral random matrix theory so as to extract the chiral condensate with high precision. However,
due to aforementioned complication, we decided to perform first the dynamical simulations with
DWfermionsasareference andforsystematicsurvey.
In this paper, we focus on the SU(2) gauge theory with two flavors of fundamental domain-
wallfermions, asabasis toinvestigate theN dependence andtheproperties atfinitetemperature.
f
The setup ofnumerical simulations is described in the next section. Sections 3 and 4describe the
results of measurement with valence domain-wall and overlap operators, respectively. How the
eigenmodes of the former approach tothe latter isdiscussed in Sec.5. Thelast section isdevoted
tosummaryandoutlook.
2. Simulationsetup
Thegaugeconfigurations aregenerated withtheIwasakigaugeactionwithN =2dynamical
f
fermions with the standard DW fermions accompanied by the Pauli-Villars ghosts. We employ
the Hybrid Monte Carlo algorithm with the Omelyan integrator and multi-time-step. The code is
implementedbymodifyingtheBridge++codeset[6]soastoenableSU(2)simulations. Thelattice
sizes are83×16,123×24,and163×32. Astheparameters oftheDWfermions, weuseL =16,
s
andM =1.6throughout presentwork.
0
We first observe the b -dependence of the lattice spacing. Although the lattice spacing in
physical units does not make sense for the present theory, it is often convenient to draw a rough
idea of the scale. Figure 1 displays the b -dependence of the lattice spacing defined through the
Sommer’s hadronic radius r =0.05fm that was determined from the static fermion potential in
0
the quenched and N =2 cases. Based on this result, we decided to adopt b =0.85 and 0.90 as
f
3
LatticesimulationofSU(2)gaugetheorywithchirallysymmetricfermions HideoMatsufuru
2.0 2.0
Iwasaki + Domain-wall (Nf=2), volume dependence IIwwaassaakkii ++ DDoommaaiinn--wwaallll ((NNff==22)),, vvoolluummee ddeeppeennddeennccee
b =0.85, M =1.6, Ns=16, m=0.050 b =0.90, M =1.6, Ns=16, m=0.050
0 0
1.5 1.5
d d
uare uare
q q
mass s1.0 mass s1.0
n n
o o
mes 163 x 32 mes 163 x 32
PS 123 x 24 PS 83 x 16
0.5 83 x 16 0.5
0.0 0.0
0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20
valence quark mass (domain-wall) valence quark mass (domain-wall)
Figure2: ThefinitesizeeffectonthemassesofPSandVmesonsatb =0.85(leftpanel)and0.90(right)
andtheseafermionmassm=0.050.
2.5
Iwasaki + domain-wall (Nf=2), 83 x16 (250 configs)
PS
2.0 b =0.85, M0 =1.6, Ns=16, m=0.20 mV =0.050 VS
A
A4
ass1.5
m
e
v
cti
effe1.0
0.5
0.0
0 2 4 6 8
t
Figure 3: Meson masses in PS, V, S and A channels for the valence fermion mass m =0.050 on 250
V
configurationsgeneratedatb =0.85andtheseafermionmassm=0.20onthe83×16lattice.
ourfirsttargetofnumericalsimulations. Themassofthedynamicaldomain-wallfermionisvaried
fromm=0.02to0.20.
3. Mesonspectrum andresidual mass
Wefirstmeasurethemesonspectrumandtheresidualmassundervariationofvalencefermion
mass. Usingalocalmesonoperatoratboththesourceandsink,statisticsof20-40configurationsis
sufficient todetermine thePSmeson masses, whilenotfortheotherchannels. Figure2showsthe
volume dependence of the PS and V meson masses at b =0.85 and 0.90 with m=0.05. The PS
mesonmassisproportionaltothevalencefermionmass,asisfamiliarinQCD.Thefiniteintercept
indicates the explicit breaking of the chiral symmetry in the DW fermion formulation. The offset
values are consistent with the direct calculation of the residual mass with Eq. (1.4). While the
valuesoftheresidual massisnotnegligibly small,itdecreases asb increases asisexpected.
Theleft panel ofFig. 2compares the meson masses in three volumes at b =0.85. While the
values for L= 12 and 16 are consistent, the result for L = 8 is slightly larger. Extrapolating to
m =0, Lm ≃5.4 for L=12 and 3.6 for L=8, and thus the above results are consistent with
V PS
4
LatticesimulationofSU(2)gaugetheorywithchirallysymmetricfermions HideoMatsufuru
0.5 4
Iwasaki + Domain-wall, Nf=2, 83 x16 Iwasaki + Domain-wall, Nf=2, 83 x16
0.4 b =0.85, Ns=16, M0 =1.6, m=0.20 b =0.85, Ns=16, M0 =1.6, m=0.20
3
zero mode distribution00..23 eigenvalue distribution2
1
0.1
0-10 -5 0 5 10 00 0.2 0.4 0.6 0.8
number of zero modes eigenvalue [lattice unit]
Figure4: Thedistributionofzeromodes(leftpanel)andthedensityoflow-lyingeigenvalues(right)ofthe
valenceoverlapfermionoperatoratb =0.85andm=0.20on83×16lattice.
the wisdom in QCD that Lm >4 is required to avoid large finite volume effect. At b = 0.90
PS
displayed in the right panel of Fig. 2, only L = 8 and 16 cases have been obtained which are
significantly deviate. Since Lm ≃5.0 for L=16 and 3.6 for L=8, to settle the finite volume
PS
effectweneedaresultforL=12thatcomputation isnowinprogress.
Once the finite size effect and size of chiral symmetry breaking are quantified, we can sys-
tematically investigate the meson spectrum and other observables. Forthechannels other than PS
and V, however, the correlators suffer from larger statistical fluctuation. In Figure 3 we show an
example of meson masses on 83×16 lattice at b =0.85 with higher statistics. By applying the
same setup as PSand V mesons, O(250) configurations are not enough toextract the masses of S
andAchannels insufficientprecision. Somekindoftechniques toincrease thesignalisnecessary
suchastheall-to-all propagators.
4. Valenceoverlapfermion operator
It is interesting to measure the eigenmodes of the valence overlap fermion operator on the
configurations generatedwithDWfermions. Asnumericalimplementation oftheoverlapoperator
(1.1), weadopt the Zolotarev’s partial fractional approximation to the sign function with the low-
mode projection of H . The low-lying eigenmodes of massless H =g D are determined by
W ov 5 ov
applying the implicitly restarted Lanczos algorithm. Figure 4 shows an example of measurement
on83×16latticeatb =0.85andm=0.20. Theleftpanelshowsthedistribution ofthenumberof
zero modes whose deviation give thetopological susceptibility. Theright panel showsthe density
ofthe low-lying eigenmodes. Clearly theresult shows theexpected behavior forthebroken chiral
symmetrythroughtheBanks-Casher relation. Quantitativeanalysis isnowunderway.
Once the distribution of the eigenvalues are determined, it can be compared to the prediction
of the random matrix theory. In this context the most interesting quantity is the distribution of
low-lying eigenvalues, that isto be compared with the chiral random matrix theory [7]. However,
our present setup and statistics are not suitable to such an analysis. We instead compare the level
statisticsofthebulkregionoftheeigenvaluespectrumtotherandommatrixtheory. Itisnotrelated
tothechiraldynamics butjustreflectsthesymmetryofthefermionoperator.
5
LatticesimulationofSU(2)gaugetheorywithchirallysymmetricfermions HideoMatsufuru
1.0 1
0.8
0.1
0.6
HLPs 0.01
0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0
GOE
0.2 GUE
GSE
0.0
0 1 2 3 4
s
Figure 5: Unfolded level spacing of the overlap fermion operator at b =0.85 and m=0.20 on 83×16
lattice. ThepredictionsoftherandommatrixtheoryforGOE,GUE,andGSEarealsodisplayed.
Figure 5 shows the density of the unfolded level spacing at b = 0.85 and m = 0.201. The
unfolding isapplied tothe eigenmodes in therange [0.2,0.8] whichare measured on500 configu-
rationsirrespectivetothetopologicalcharge. Inthefigure,predictionsoftherandommatrixtheory
with three types of ensembles, orthogonal (GOE), unitary (GUE), and symplectic (GSE) are also
displayed. The numerical result is well consistent with the GOE curve that corresponds to the
symmetryoffundamental representation ofSU(2)group.
5. Correlationbetween DWand overlapeigenmodes
Sincethedynamics ofthechiral symmetry breaking isencoded inthelow-lying eigenmodes,
these modes can be used to quantify the effect of chiral symmetry violation brought by adopting
thedomain-wall fermionaction. Thefour-dimensional domain-wall operator[8,9],
D(4)(m)={P−1[D (1)]−1D (m)P} (5.1)
DW DW 11
where P =d , is the counterpart of the overlap operator that approaches Eq. (1.1) in
ij j,mod(i+1,Ls)
thelimitL →¥ . Thusweinvestigate thecorrelation betweentheeigenvectors ofEq.(1.1)andof
s
Eq.(5.1)determined onthesameconfigurations.
Figure 6 displays the correlation of the 40 lowest eigenmodes of the 4D domain-wall and
(DW)† (ov)
overlap fermion operators, C =v ·v , on single configuration. The left and right panels
i,j i j
showtheresultsatb =0.85(m ≃0.03)andb =0.90(m ≃0.015),respectively. Asexpected,
res res
near-diagonal components of the correlation increase as the residual mass decreases. The same
tendency isgenerally observed onotherconfigurations, whileaquantitative analysis isunderway.
6. Conclusionand outlook
Weareinvestigating theSU(2)gaugetheorywithdynamicalfermionsinthefundamentalrep-
resentation. Numericalsimulationswithtwo-flavorsofdomain-wallfermionswereperformedasa
basisforextensivestudieswithvariationofnumberofflavorsandatfinitetemperature. Theresults
obtained are feasible for further investigation, while techniques to increase the signal is required
1ThisanalysiswasperformedbyS.M.Nishigaki.
6
LatticesimulationofSU(2)gaugetheorywithchirallysymmetricfermions HideoMatsufuru
40 83 x16, Iwasaki + domainwall (Nf=2) 40 83 x16, Iwasaki + domainwall (Nf=2)
35 beta=0.85, Ns=16, M0 =1.6, m=0.20 35 beta=0.90, Ns=16, M0 =1.6, m=0.02
30 30
mode25 mode25
wall eigen20 wall eigen20
main-15 main-15
do do
10 10
5 5
0 0
0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40
overlap eigenmods overlap eigenmods
Figure6: Correlationbetweentheeigenmodesofthedomain-wallandoverlapoperators.Theleftandright
panelsareatb =0.85and0.90,respectively.
for quantitative computation ofvariety ofmeson sectors. Wealsoobserved the eigenmodes ofthe
valence overlap operator which is useful to explore the effect of the chiral symmetry violation of
the domain-wall fermions. We are preparing for systematic studies of subjects described in this
paper, aswellasforapplying thesameprocedure totheadjointrepresentation offermions.
Acknowledgment
WethankS.M.Nishigakiforusefuldiscussion andforanalysistogenerateFig.5. Numerical
simulations were performed on Hitachi SR16000 and IBM Blue Gene/Q at KEK under a support
of its Large-scale Simulation Program (No.12/13-15) and j computer system at KMI, Nagoya
University. We also thank the Japan Lattice Data Grid which is a grid file system constructed on
a virtual private network SINET4 provided by National Institute of Informatics for efficient data
transfer. This work is supported in part by the Grand-in-Aid for Scientific Research of the Japan
(Nos.20105005, 22224003, 22740183, 25400284).
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