Topological observables in many-flavour QCD

0.39 MB·

by  Yasumichi Aoki

LLNL-PROC-679800
Topological observables in many-flavour QCD
Yasumichi Aokia, Tatsumi Aoyamaa, Ed Bennett∗b†, Masafumi Kurachic, Toshihide
Maskawaa, Kohtaroh Miuraad, Kei-ichi Nagaia, Hiroshi Ohkie, Enrico Rinaldif,
6
1 Akihiro Shibatag, Koichi Yamawakia, Takeshi Yamazakih (LatKMI Collaboration)
0 aNagoyaUniversity,Furo,Chikusa,Nagoya,Aichi,464-8602Japan
2
bSwanseaUniversity,SingletonPark,SwanseaSA28PPUK

n
cKEK,Tsukuba,Ibaraki305-0801Japan
a
J d CPT,CampusdeLuminy,Case907,163AvenuedeLuminy,13288Marseillecedex9,France
8 eRIKENBNLResearchCenter,BrookhavenNationalLaboratory,Upton,NY11973-5000USA
1 f LawrenceLivermoreNationalLaboratory,7000EastAve.,Livermore,CA94550-9234USA

gComputingResearchCenter,HighEnergyAcceleratorResearchOrganization(KEK),
]
t Tsukuba,Ibaraki305-0801Japan
a
l hGraduateSchoolofPureandAppliedSciences,UniversityofTsukuba,Tsukuba,Ibaraki
-
p 305-8571,Japan
e
h
[ †E-mail: [email protected]

1
v
SU(3)gaugetheorywitheightmasslessflavoursisbelievedtobewalking,whilethecorrespond-
7
8 ing twelve- and four-flavour appear IR-conformal and confining respectively. Looking at the
6
simulationsperformedbytheLatKMIcollaborationofthesetheories,weusethetopologicalsus-
4
0 ceptibilityasanadditionalprobeoftheIRdynamics. BydrawingacomparisonwithSU(3)pure
.
1 gaugetheory,weseeadynamicalquenchingeffectemergeatlargernumberofflavours,whichis
0
suggestiveofemergingnear-conformalandconformalbehaviour.
6
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/
Topologicalobservablesinmany-flavourQCD EdBennett
1. Introduction
Foranumberofyears,theLatKMICollaborationhasperformedanongoinginvestigation[1,2]
of QCD with 4, 8, 12, and 16 light flavours on the lattice, with the aim to classify them as con-
formal, near-conformal, or confining and chirally broken, and calculate quantities of phenomeno-
logicalrelevance. Inparticular,N =8QCDisacandidateforaWalkingTechnicolortheory,with
f
the potential to provide a mechanism for Electroweak Symmetry Breaking and at the same time
producea(composite)Higgs.
Inlatticegaugetheoriesnearthecontinuumlimit,itispossibleforthetopologicalchargeQof
acomputationtobecomefrozen,causingthesimulationtobecomenon-ergodic. Sinceobservables
ofphysicalinterestarerelatedtotopologicaleffects,andthusdependonthetopologicalcharge,itis
thusnecessarytocomputethetopologicalchargeinordertoverifywhetherornotourcomputations
havesufficientergodicity.
Having calculated these values, we would like to use them for more than just a check of
ergodicity. Inparticular,itwouldbeusefuliftheycouldcontributetoourongoingworktostudythe
infraredregimeofthesetheories,specificallywhethertheyareconformalorconfiningandchirally
broken,andifthelatter,whethertheyareQCD-likeorexhibitwalking-typedynamics. Bennettand
Lucini have previously performed a study of this kind[3] for SU(2) with N =2 adjoint flavours,
f
andweadoptasimilarprocedurehere. Inthenextsectionwewilldiscussthemethodologyweuse
for this; we will then outline the methods we have used to calculate the quantities discussed, and
presentpreliminaryresultsforthetheoriesmentionedabove.
2. (Near-)conformaltopology
AnIR-conformaltheorybecomesconfiningwhendeformedbyafermionmass. Intheprocess,
a scale is added (the fermion mass) which is intrinsically heavy—thus when considering gluonic
IR observables, the theory should be indistinguishable from the equivalent quenched theory. One
suchobservableisthetopologicalsusceptibilityχ ,whichwecalculateas
top
χ = 1 (cid:0)(cid:104)Q2(cid:105)−(cid:104)Q(cid:105)2(cid:1) (2.1)
top
V
whereV isthelatticevolume,and(cid:104)Q(cid:105)isexpectedtobezero. SinceinIR-conformaltheories,the
deforming mass induces non-trivial RG corrections to the lattice spacing, we must consider only
dimensionless ratios, and since we will compare with results from a theory without fermions, we
considerprimarilypuregluonicobservablestoscaletheresults.
Additionalinformationmaybegainedbyprobingthedependenceofχ onthefermionmass
top
directly—specifically,checkingwhethertheχ obeysthepredictionsofeitherchiralperturbation
top
theory(χPT)orconformalhyperscaling. Fortheformercasewefit
χ =Cm + f(a) ⇒χ1/4=(Cm + f(a))1/4; (2.2)
top f top f
whileforthelatterweuse
χ1/4=Am1/(1+γ∗). (2.3)
top f
Here, C, A, and f(a) are constants to be determined, and γ is the anomalous dimension of the
∗
chiralcondensate.
2
Topologicalobservablesinmany-flavourQCD EdBennett
3. Computations
Thetopologicalchargeisdefinedinthecontinuumas
(cid:90) 1
Q= d4xq(x), q(x)= ε F F .
32π2 µνρσ µν ρσ
To find the lattice equivalent, we replace the integral by a sum, and obtain the equivalent of the
fieldstrengthF bytakingthepath-orderedproductoflinkvariablesaroundaclover-shapedpath.
µν
We encounter a problem however when we apply this to gauge configurations as produced by a
typical Monte Carlo process: namely that ultraviolet fluctuations dominate over the topological
contribution to the charge. In principle these cancel out across the lattice volume, but in practise
thetopologicalcontributionisvastlysmallerthantheprecisionerroroftheUVfluctuations,andso
thesignalislost.
We therefore need to suppress these contributions. We use the gradient flow, as suggested by
Lüscher[4],forthispurpose. ThegradientflowdefinesaflowedfieldB (t,x)atflowtimet as
µ
d (cid:12)
dtBµ =DνGνµ Bµ(cid:12)t=0=Aµ ,
(cid:20) (cid:21)
d
G =∂ B −∂ B +[B ,B ], D =∂ + B , ,
µν µ ν ν µ µ ν µ µ µ
dt
where the flow starts at B (0,x)=A (x), the physical gauge field. Integrating numerically from
µ µ
A allows calculation of B (t,x) at arbitrary t. We do this using the Runge–Kutta-like scheme
µ µ
also outlined by Lüscher [4]. The flow has a smearing effect on the gauge configuration, with a
√
characteristic radius dependent on the flow time as r= 8t; it is this effect that allows extracting
thetopologicalcharge.
As mentioned above, in some cases close to the continuum and chiral limits, the topology
may tend to freeze at a single or a very restricted range of topological charge. This will render
equation (2.1) for the topological susceptibility invalid, as both (cid:104)Q(cid:105) and (cid:104)Q2(cid:105) will be distorted
by the insufficient sampling. In these cases we extract further information from the topological
charge density distribution by considering subvolumes of the full lattice rather than the whole
lattice volume [5]. This allows the excluded volume to act as a reservoir of topological charge, to
andfromwhichinstantonsmaymove.
Furtherinformationmaybeextractedbyprobingthedistributionoftopologicalcharge: peaks
andtroughsinthisdistributioncorrespondtoinstantonsandanti-instantonsrespectively. Theheight
ofthepeak(trough)q isdirectlyrelatedtothecharacteristicsizeρ ofthe(anti)instanton,as[3]
peak
6
q = .
peak π2ρ4
Finally,thegradientflow,havingbeenusedtoaccessthetopologicalcharge,mayalsobeused
atnoextracosttoextractascale[4]t ,definedasthevalueoftheflowtimet forwhicht2(cid:104)E(cid:105)=0.3,
0
where E is a lattice discretized version of the continuum relation E = 1G G . G may be
4 µν µν µν
calculatedintwoways: viatheaverageplaquette,orviaafour-plaquettecloveroperator.
3
Topologicalobservablesinmany-flavourQCD EdBennett
15 15
10 10
5 5
Q 0 Q 0
-5 -5
-10 -10
-15 -15
6000 8000 10000 12000 14000 16000 18000 20000 2000 4000 6000 8000 10000 12000 14000 16000
Trajectory # Trajectory #
N =4,β =3.7,m=0.01,L=20 N =8,β =3.8,m=0.04,L=30
f f
5
3
0
Q Q
-5
2
-10
0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 5000 10000 15000 20000 25000
Trajectory # Trajectory #
N =12,β =3.7,m=0.16,L=18 N =12,β =4.0,m=0.04,L=36
f f
Figure1:SamplehistoriesofthetopologicalchargeQfortheparametersetsshown.N =12,β =4.0shows
f
themostsubstantialfreezingobservedoutsideofN =16.
f
8 3
Trajectory 20002 Trajectory 1300
46 TTrraajjeeccttoorryy 2200000160 2 TTrraajjeeccttoorryy 11450000
2 TTrraajjeeccttoorryy 2200001148 1 TTrraajjeeccttoorryy 11670000
Q Q
0 0
-2
-1
-4
-6 -2
0 0.125 0.5 1 2 4 8 12 18 0 0.005 0.02 0.05 0.08 0.125
(8t)1/2 (8t)1/2
N =8,β =3.7,m=0.06,L=24 N =16,β =12,m=0.015,L=48
f f
Figure2: Historyofthetopologicalchargeasafunctionofthegradientflowscaleforselectedparameter
setsforN =8andN =16. N =16israpidlysuppressedtozero,comparedtoN =8whichtakeslonger
f f f f
toequilibrateandgivesavariationintopologicalcharge.
4. Results
We study pre-existing SU(3) gauge configurations generated by the LatKMI collaboration
with one, two, three, and four staggered flavours, equivalent to N =4 (β =3.8), 8 (β =3.8), 12
f
(β =3.7,4.0),and16(β =12),withthetree-levelSymanzikgaugeactionandtheHISQfermionic
action. Inaddition,puregaugeSU(3)configurationsweregeneratedusingboththeWilson(5.7≤
β ≤6.2) and the tree-level Symanzik gauge (4.0≤β ≤5.0) actions, for comparison. The string
tensionsintheformercasewereobtainedfrom[6]
Asazeroth-orderstep,wereviewedthetopologicalchargehistoriesofallsetsof(N,β,m ,L)
f f
studied. We find that N = 4 is highly ergodic, as are most parameters for N = 8, with mild
f f
4
Topologicalobservablesinmany-flavourQCD EdBennett
10-3 1.4
mf
0.009 1.2
0.012
0.015 1
0.02
0.03 0.8
0.04
10-4 00..0056 n 0.6
00..0078 0.4
00..1102 0.2
0.16
0
10-5 -0.2
0 5 10 15 20
ρ / a
4aχ (b)
0.14
10-6 0.12
0.1
10-7 1/4aχ 00..0068 NNNf ff= == 1 248,,, βββ === 333...778
0.04 χPT, mNff� �=�[ 01.20,1 β2 ,= 0 .40.30]
0.02 HHyypperesrcsaclailning,g ,m mf�f����[�0[.00.1058,, 00..13]]
0 Hyperscaling, mf���[0.04, 0.1]
10-8 8 16 32 64 0 0.05 0.1 0.15 0.2
L / a a mf
(a) (c)
Figure3: (a)Finite-volumeofallavailablevaluesofχ forN =8. (b)Theinstantonsizedistributionfor
top f
N =0,β =5.0. (c)χ asafunctionofm,includingconformalandχPTfits.
f top f
loss in ergodicity emerging at the lowest values of am. N = 12 suffers from more ergodicity
f
problems,withverylongregionsoffrozenQ,particularlyatlowvaluesofam. ExampleQhistories
for flavoured theories are shown in Fig. 1. N = 0 (not shown) was ergodic at low beta, while
f
becoming near-frozen at high β. In the cases where the evolution was insufficiently ergodic, the
subvolume method mentioned above was employed to extract χ . Finally, the topology of N =
top f
16 is completely suppressed, with no local minima or maxima in the topological charge density
distribution indicative of the presence of a topological excitation. (An example set of cooling
historiesisshowninFig.2.) Owingtothislackofdata,N =16willnotbediscussedfurther.
f
Wenowmovetolookatthetopologicalsusceptibility. Asacheckthatfinite-volumeartefacts
wereundercontrol,weinvestigatedhow χ variedasafunctionofthespatiallatticeextentLfor
top
each value of m for N =8. The resulting plot is shown in Fig. 3(a); in the majority of cases the
f f
susceptibilitystabilisesasLincreases.
Fig. 3(b) shows another indication of finite volume effects: the instanton size distribution. In
the case of N =0, β =5.0, it was found that the distribution came very close to spilling over the
f
lattice volume—i.e. the instantons would not fit on the lattice. This ensemble was therefore not
considered further; all lower values of β had distributions that tended to zero in advance of the
latticesize.
1/4
InFig.3(c),aχ forN =4,8,and12isplottedasafunctionofthefermionmassam . We
top f f
observethatforN =4,χ tendstoanon-zerovalueinthelimitm →0,contrarytoexpectations.
f top f
5
Topologicalobservablesinmany-flavourQCD EdBennett
0.5 0.18
0.16
0.4 0.14
0.12
1/41/2χ/σ 00..23 1/21/4χt0 0 .00.81 Pure gauge (Wilson)
Pure gauge (Wilson) 0.06 Pure gauge (Symanzik)
0.1 Pure gauNgfe = ( S4y,m β a=nz 3ik.7) 0.04 NNff == 48,, ββ == 33..78
Nf = 8, β = 3.8 0.02 Nf = 12, β = 3.7
0 Nf = 12, β = 4.0 0 Nf = 12, β = 4.0
0 0.05 0.1 0.15 0.2 0 2 4 6 8 10 12
a2σ t0 / a2
(a) (b)
Figure4: (a) χ scaledbythestringtensionσ, asafunctionofa2σ, and(b) χ scaledbythegradient
top top
flowscalet ,asafunctionoft /a2.
0 0
Thisislikelyduetoastaggeredtastesymmetrybreakingeffectatthelowermasspoints;ifso,then
itwoulddisappearclosertothecontinuumlimit. Workisunderwaytotestwhetherthisisthecase;
inthemeantime,theN =4resultsmustberegardedcautiously.
f
Turning our attention to N =8 and 12, we fit the former with both conformal and χSB an-
f
tätze, and the latter with a conformal ansatz only. For N =8, the fits do not distinguish between
f
conformalandconfiningbehaviour,sincebothfitsareequallyconsistentwiththedata;however,if
weassumetheconformalansatz,thenweobtainavalueofγ =1.04(5),consistentwithestimates
∗
fromspectralhyperscaling[2]. ForN =12, theconformalfitsareconsistentwiththedatagiving
f
γ =0.33(6)and0.47(10)respectivelyforβ =3.7and4.0;thesearealsoconsistentwithestimates
∗
fromspectralhyperscaling.
√
1/4
In Fig. 4(a), χ is plotted as a dimensionless ratio to the string tension, χ / σ. The
top top
observedratioforN =0isconsistentwithflat,withthehintedgradientdependingonwhetherthe
f
Wilson or Symanzik-improved gauge action is used. N =4 has a positive gradient, being close
f
to the pure gauge theory at large σ but diverging as σ decreases. This is unlikely to change with
theremovalofthetaste-symmetrybreakingeffects,asthepositivegradientwouldneedtobecome
morepositivetoallowforzerosusceptibilityinthechirallimit. N =12isconsistentwithflatora
f
shallownegativegradient,andagreeswiththepuregaugetheoryinthelimitσ →0. Asdiscussed
above,thisisatentativesignalofconformality.
N =8isconsistentwithN =12athighσ,butturnsovertohaveapositivegradient(likeN =
f f f
4)atsmallσ. Thiscouldbeconsideredtobeindicativeofnear-conformal/walking-typebehaviour,
withquenchedbehaviouratlargedeformingmasses,turningintochirallybrokenbehaviourasthe
chirallimitisapproached.
Wealsoconsiderthescalingofthesusceptibilityrelativetothegradientflowscalet . Thisis
0
plottedinFig.4(b): weseethatinthequenched(t /a2 →0)limit,alltheoriesbecomeconsistent,
0
as we would expect, while the behaviour change ast /a2 increases. N =4 rapidly diverges from
0 f
the flat behaviour of the pure gauge theory, while N =8 and 12 diverge more slowly. N =12
f f
flattens off (emulating the flat gradient of the pure gauge theory) at highert /a2, a behaviour not
0
seenintheothertheories.
6
Topologicalobservablesinmany-flavourQCD EdBennett
5. Conclusions
WehaveperformedastudyofthetopologicalchargeandsusceptibilityofQCDwith4,8,and
12 flavours. We find that χ for both N =8 and 12 may be fitted using a conformal ansatz, and
top f
givesvaluesforthechiralcondensateanomalousdimension(γ =1.04(5)forN =8,and0.33(6)
∗ f
and0.47(10)forN =12atβ =3.7and4.0respectively)consistentwiththoseobtainedfromother
f
methods (provided the ansatz is presumed to be well-motivated for N =8). Studying the scaling
f
of the susceptibility with the string tension σ and the gradient flow scale t , the N =4 theory is
0 f
qualitativelydistinctfromthepuregaugetheory,aswouldbeexpectedofaconfiningtheory,while
N =8andN =12showsignsofbeingsimilarbothtoeachotherandtothepuregaugetheoryas
f f
we would expect for a conformal theory. However, differences between all three theories can be
seen; furtherstudyisongoingtoclarifythemeaningofthesedifferences. AstudyofN >12that
f
was not volume-constrained would provide useful insight into this, as would data for SU(2) with
fundamentalmatter.
6. Acknowledgements
Computations have been carried out on the ϕ computer at KMI, the CX400 machine at the
Information Technology Center in Nagoya University, and a workstation in Swansea University.
This work is supported by the JSPS Grant-in-Aid for Scientific Research (S) No.22224003, (C)
No.23540300 (K.Y.), for Young Scientists (B) No.25800139 (H.O.) and No.25800138 (T.Y.), and
alsobytheMEXTGrants-in-AidforScientificResearchonInnovativeAreasNo.23105708(T.Y.),
No. 25105011 (M.K.). E.R. acknowledges the support of the U.S. Department of Energy under
Contract DE-AC52-07NA27344 (LLNL). The work of H.O. is supported by the RIKEN Special
PostdoctoralResearcherprogram.
References
[1] LatKMICollaboration,Y.Aoki,T.Aoyama,M.Kurachi,T.Maskawa,K.-i.Nagai,H.Ohki,
A.Shibata,K.Yamawaki,andT.Yamazaki,Latticestudyofconformalityintwelve-flavorQCD,Phys.
Rev.D86(2012)054506,[arXiv:1207.3060].
[2] LatKMICollaboration,Y.Aoki,T.Aoyama,M.Kurachi,T.Maskawa,K.-i.Nagai,H.Ohki,
A.Shibata,K.Yamawaki,andT.Yamazaki,WalkingsignalsinN =8QCDonthelattice,Phys.Rev.
f
D87(2013),no.9094511,[arXiv:1302.6859].
[3] E.BennettandB.Lucini,TopologyofMinimalWalkingTechnicolor,Eur.Phys.J.C73(2013)2426,
[arXiv:1209.5579].
[4] M.Lüscher,PropertiesandusesoftheWilsonflowinlatticeQCD,JHEP08(2010)071,
[arXiv:1006.4518].[Erratum: JHEP03,092(2014)].
[5] LSDCollaboration,R.C.Broweretal.,Maximum-LikelihoodApproachtoTopologicalCharge
FluctuationsinLatticeGaugeTheory,Phys.Rev.D90(2014),no.1014503,[arXiv:1403.2761].
[6] B.LuciniandM.Teper,SU(N)gaugetheoriesinfour-dimensions: ExploringtheapproachtoN=
infinity,JHEP06(2001)050,[hep-lat/0103027].
7