Topological susceptibility in the SU(3) gauge theory

0.16 MB·English

by  Luigi Del Debbio

CERN-PH-TH/2004-117, CPT-2004/P.041
Topological susceptibility in the SU(3) gauge theory
Luigi Del Debbioa, Leonardo Giustib, Claudio Picac
a CERN, Department of Physics, TH Division, CH-1211 Geneva 23, Switzerland
b Centre de Physique Th´eorique, Case 907, CNRS Luminy, F-13288 Marseille Cedex 9, France∗
c Dipartimento di Fisica dell’Universit`a di Pisa and INFN, Via Buonarroti 2, I-56127 Pisa, Italy
(Dated: February 1, 2008)
We compute the topological susceptibility for the SU(3) Yang–Mills theory by employing the
expression of the topological charge density operator suggested by Neuberger’s fermions. In the
continuumlimit we find r04χ=0.059(3), which corresponds toχ=(191±5MeV)4 if FK is used to
′
set thescale. Ourresult supportstheWitten–Veneziano explanation for thelarge mass of theη .
PACSnumbers: 11.15.Ha,11.30.Rd,11.10.Gh,12.38.Gc
5
0
0 I. INTRODUCTION whereγ5 =γ5(1−a¯D), D is the masslessDiracoperator
2 and a¯ is proportional to the lattice spacing (see below).
The topological susceptibility in the pure Yang–Mills b
n The corresponding Jacobian is non-trivial [16], and the
a (YM) gaugetheory canbe formallydefined in Euclidean chiral anomaly is recovered a` la Fujikawa [17] with the
J space-time as topological charge density operator defined as2 [18]:

1
2 χ= d4xhq(x)q(0)i, (1) a¯
Z q(x)=−2Trhγ5D(x,x)i, (5)
2
where the topological charge density q(x) is given by
v where the trace runs over spin and color indices. These
2 1 developments triggered a breakthrough in the under-
q(x)=− ǫ Tr F (x)F (x) . (2)
5 32π2 µνρσ h µν ρσ i standingofthetopologicalpropertiesoftheYMvacuum.
0
They made it possible to find anunambiguousdefinition
7 Besidesitsinterestwithinthepuregaugetheory,χplays
of the topological susceptibility with a finite continuum
0 a crucial rˆole in the QCD-based explanation of the large
4 massofthe η′ mesonproposedby WittenandVeneziano limit[4,19,20],whichisindependentofthedetailsofthe
0 (WV) a long time ago [1, 2]. The WV mechanism pre- lattice definition [20]. If the charge density suggestedby
h/ dicts that at the leading order in Nf/Nc, where Nf and GWfermionsQ≡ xq(x)=n+−n−,withn+ (n−)the
number of zero moPdes of D with positive (negative) chi-
-t Nc are the number of flavorsand colors respectively, the rality ina givenbackground,is employed, the suggestive
p contributionduetotheanomalytothemassoftheUA(1)
formula
e particle is given by [1, 2, 3, 4, 5]
h
Xiv: F2π2Nmf2η′ =χ, (3) χ= Vali→→m∞0 hQV2i (6)
whereF isthecorrespondingpiondecayconstant1. No-
r π is recovered,where V is the volume. An immediate con-
a tice that Eq.(3) is expected to be exactly satisfied if the
sequence is an unambiguous derivation of the WV for-
l.h.s. is computed in full QCD and the r.h.s. in the pure
mula[4]which,thankstonewsimulationalgorithms[21],
gauge theory, both in the ’t Hooft large-N limit [6].
c allows for a non-perturbative investigation of the WV
The lattice formulation of gauge theories is at present
mechanism with controlled systematics.
the only approachwhere non-perturbativecomputations
Inthepastthetopologicalpropertiesofthepuregauge
can be performed with controlled systematic errors. Re-
theory were investigated with fermionic [22, 23] and
cent theoretical developments [7, 8, 9, 10] (for a recent
bosonic methods [24, 25, 26, 27, 28, 29, 30, 31, 32].
review see [11]) led to the discovery of a fermion opera-
These results, however,are affected by model-dependent
tor [12, 13, 14] that satisfies the Ginsparg–Wilson(GW)
systematic errors that are not quantifiable, and their in-
relation[15],andthereforepreservesanexactchiralsym-
terpretation rests on a weak theoretical ground. Several
metry at finite lattice spacing [16]
exploratory computations have already studied the sus-
ψ →γ ψ, ψ¯→ψ¯γ , (4) ceptibility employing the GW definition of the topologi-
5 5
cal charge [33, 34, 35, 36, 37, 38, 39, 40].
b
∗UMR 6207 - Unit´e Mixte de Recherche du CNRS et des Uni-
versit´es Aix-Marseille I, Aix-Marseille II et du Sud Toulon-Var - 2 Weusethesamenotationforanalogousquantitiesinthecontin-
Laboratoireaffili´ea`laFRUMAM uum and on the lattice, since they can be clearly distinguished
1 Inourconventions, thephysicalpiondecayconstantis92MeV. fromthecontext.
2
The aim of this work is to achieve a precise and reli- lat β L/a r0/a L[fm] Nconf hQ2i r04χ
able determination ofχ in the continuumlimit. In order A1 6.0 12 5.368 1.12 2452 1.633(48) 0.0654(22)
to reach a robust estimate of the error on the extrapo-
A2 6.1791 16 7.136 1.12 1138 1.589(76) 0.0629(32)
latedvalue,wesupplementthe mostrecentandaccurate
results [39, 40] with additional simulations, and we per- A3 5.8989 10 4.474 1.12 1460 1.737(72) 0.0696(30)
formadetailedanalysisofthe varioussourcesofsystem- A4 6.0938 14 6.263 1.12 1405 1.535(63) 0.0615(27)
atic uncertainties. The result for the adimensional scal- B0 5.8458 12 4.032 1.49 2918 5.61(16) 0.0715(22)
ing quantity computed on the lattice is r4χ = 0.059(3),
0 B1 6.0 16 5.368 1.49 1001 5.58(28) 0.0707(37)
where r is a low-energy reference scale [41]. In physical
units,it0correspondstoχ=(191±5MeV)4 ifF isused B2 6.1366 20 6.693 1.49 963 4.81(24) 0.0604(32)
K
tosetthescale. OurresultsupportstheWVexplanation B3 5.9249 14 4.697 1.49 1284 5.59(24) 0.0708(33)
for the large mass of the η′ meson within QCD. C0 5.8784 16 4.301 1.86 1109 15.02(72) 0.0784(39)
C1 6.0 20 5.368 1.86 931 12.76(95) 0.0662(50)
D 6.0 14 5.368 1.30 1577 3.01(12) 0.0651(27)
II. LATTICE COMPUTATION
E 5.9 12 4.483 1.34 1349 2.79(12) 0.0543(24)
The numerical computation is performed by standard F 5.95 12 4.917 1.22 1291 1.955(79) 0.0551(24)
Monte Carlo techniques. The ensembles of gauge config- G 6.0 12 5.368 1.12 3586 1.489(37) 0.0596(18)
urations are generated with the standard Wilson action
H 6.1 16 6.324 1.26 962 2.45(13) 0.0599(33)
andperiodicboundaryconditions,usingacombinationof
J 6.2 18 7.360 1.22 1721 2.114(76) 0.0591(24)
heat-bath and over-relaxation updates. More details on
the generation of the gauge configurations can be found
TABLE I: Simulation parameters and results. For lattices
in Refs. [39, 40]. Table I shows the list of simulated lat-
tices, where the bare coupling constant β = 6/g2, the A1–D and E–J, s=0.4 and s=0.0 respectively.
0
linearsize L/aineachdirectionandthe number ofinde-
adopted in Ref. [32]. Based on the experience with cool-
pendent configurations are reported for each lattice.
ing,wherelongerMonteCarlohistoriescanbe analyzed,
ThetopologicalchargedensityisdefinedasinEq.(5),
we estimate τ for all our lattices; for each run we sep-
Q
with D being the massless Neuberger–Dirac operator:
arate subsequent measurements by a number of update
cycles1–2ordersofmagnitude largerthanthe estimated
1
D = a¯h1+γ5sign(H)i (7) τQ at the correspondingvalue of β. Statistical errorsare
a thus computedassumingthat the measurementsaresta-
H = γ5(aDw−1−s), a¯= . (8) tistically independent.
1+s
Besides the statistical errors, the systematic uncer-
Here s is an adjustable parameter in the range |s| < 1, tainties stem from finite-volume effects and from the ex-
and D denotes the standard Wilson–Dirac operator trapolation needed to reach the continuum limit.
w
(the notational conventions not explained here are as in The pure gauge theory has a mass gap, and there-
Ref. [21]). For a given gauge configuration, the topolog- forethetopologicalsusceptibilityapproachestheinfinite-
ical charge is computed by counting the number of zero volume limit exponentially fast with L. Since the mass
modes ofD withthe algorithmproposedinRef.[21]. As of the lightest glueball is around 1.5 GeV, finite-volume
s is varied, D defines a one-parameter family of fermion
discretizations,whichcorrespondtothesamecontinuum 0.08
theory but with different discretization errors at finite
latticespacing. Ouranalysisincludesdatasetscomputed
fors=0.4ands=0.0. Mostofthedataweretakenfrom
Refs. [40] and [39] for s = 0.4 and s = 0.0 respectively. 0.07
Thenumberofconfigurationswereincreased,wherenec-
χ
essary,inorderto achievehomogeneousstatisticalerrors 4r0
of the order of 5% for each data point. Some new lat-
tices were added so as to perform careful studies of the 0.06
systematicuncertaintieswhichwedescribebelow,before
presenting the physical results.
Inordertocomputeitsautocorrelationtime,wemoni-
torthetopologicalchargedeterminedwiththeindexofD 0.05
1 1.2 1.4 1.6 1.8 2
for 500 update cycles (1 heat-bath and 6 over-relaxation L [fm]
of all link variables) for the lattice A . The autocorrela-
1
tion time, τQ, estimated as in Ref. [32], turns out to be FIG. 1: The topological susceptibility, in units of r0−4, as a
compatiblewiththe one obtainedforthe samelattice by function of thelinear lattice size, in fm, at β =6.0.
definingthetopologicalchargewiththecoolingtechnique
3
400 0.09
0.08
300
nf
co 0.07
N
χ
200 4
0
r
0.06
100
0.05
0 0.04
−6 −4 −2 0 2 4 6 0 0.02 0.04 0.06 0.08
2
Q (a/r )
0
FIG. 2: Histogram for the distribution of the topological FIG. 3: Continuum extrapolation of theadimensional prod-
charge Q from thelattice D. uct r04χ. The s = 0.0 and s = 0.4 data sets are represented
by black circles and white squares respectively. The dashed
linesrepresenttheresultsofthecombinedfitdescribedinthe
effects are expected to be far below our statistical errors
text. The filled diamond at a = 0 is the extrapolated value
as soon as L ≥ 1 fm. In order to further verify that no
in the continuumlimit.
sizeable finite-volume effects are present in our data, we
simulatedfourlatticesatβ =6.0butwithdifferentlinear
sizes L = 1.12,1.30,1.49,1.86 fm. The results obtained
a2/r2, show sizeable O(a2) effects for both the s = 0.4
for χ are shown in Fig. 1, where no dependence on L 0
and s = 0.0 samples. For β ≤ 6.0, the difference be-
is visible, hence confirming that finite-volume effects are
tween the two discretizations is statistically significant.
below our statistical errors. In the large-volume regime
Within our statisticalerrors,and in the range where our
the probability distribution of the topological charge is
simulations are performed, our results suggest a linear
expected to be a Gaussian of the form [40]
dependence in a2. For the s = 0.4 sample, the value of
PQ = 1 e−2hQQ22i . (9) χst2anpterbedheagvrieoer,owf hfrieleedaolmin,eχar2dofift, colfetahrleyfodrimsfavors a con-
2πhQ2i
p
We havecheckedthatthis formuladescribes allour data
a 2
samplesverywell;forthelatticeD,theresultsareshown r4χ(s)=c +c (s) (10)
in Fig. 2. Much higher statistics are required in order 0 0 1 (cid:16)r0(cid:17)
to highlightthe deviations froma Gaussiandistribution;
higher momenta of the topological charge distribution
yields a value of c = 0.056(3) with χ2 ≈ 0.79. The
measuredonourdataareallcompatiblewithzerowithin 0 dof
quadratic fit in a2/r2 yields an extrapolated value com-
large statistical errors. 0
patible with that of the linear one, but with an error
Aspointedoutintheintroduction,thetopologicalsus-
three times larger, and the coefficient of the quadratic
ceptibility definedfromthe indexofthe Neubergeroper-
term compatible with zero. For the s = 0.0 sample, all
atorisnotplaguedbypowerdivergencesanddoesnotre-
three fits give good values of χ2 , and for the linear one
quiremultiplicativerenormalization. Thisisadistinctive dof
we obtain c =0.064(4) with χ2 ≈0.68, which is com-
feature of this approach, which is at variance with what 0 dof
patible with the outcome of the same fit for s = 0.4.
happensforotherdefinitionsusedinthepasttocompute
The agreementbetweenthe two extrapolationsindicates
χ. Atfinitelatticespacing,χisaffectedbydiscretization
effectsstartingatO(a2),whicharenotuniversal,and,in thatwe reachedthe scalingregime. This is confirmedby
the compatibility of the results in the two data sets for
our case, depend on the value of s chosen to define the
β > 6.0. A robust estimate of χ in the continuum limit
Neuberger operator. In order to compare results at dif-
can thus be obtained by performing a combined linear
ferent lattice spacings, and to extrapolate them to the
fit of the data. This fit gives a very good value of χ2
continuum limit, we adopt r0 as the reference scale; this dof
when all sets are included, and is very stable if some
choice is motivated by its precise determination in the
points at larger values of a2/r2 are removed. In partic-
range of β explored in this work [41]. The values of the 0
adimensional quantity r4χ that we obtain are reported ular a combined fit of all points with a2/r02 < 0.05 gives
0 c =0.059(3)withχ2 ≈0.73,andthe errorisexpected
in Table I. Data, displayed in Fig. 3 as a function of 0 dof
to be Gaussian.
4
III. PHYSICAL RESULTS to physical units in the pure gauge theory is of the same
order as the neglected terms.
From the previous analysis, our best result for the Our result supports the fact that the bulk of the mass
topologicalsusceptibilityistheoneobtainedfromacom- of the pseudoscalar singlet meson is generated by the
bined fit of the two sets of data with a2/r02 <0.05: anomaly through the Witten–Veneziano mechanism.
r4χ=0.059±0.003, (11)
0
Acknowledgments
which is the main result of this work. Since r is not
0
directly accessible to experiments, we express our result
It is a pleasure to thank M. Lu¨scher, G. C. Rossi,
in physical units by using the lattice determination of
R.Sommer,M.Testa,G.VenezianoandE.Vicariforin-
r F = 0.4146(94) in the pure gauge theory with va-
0 K terestingdiscussions. ManythanksalsotoP.Hern´andez,
lence quarks [42] and, taking F = 160(2) MeV as an
K M. Laine, M. Lu¨scher,P. Weisz and H. Wittig for allow-
experimental input, we obtain
ingustousedataonthetopologicalsusceptibilitygener-
χ=(191±5MeV)4, (12) atedinRefs.[40,43]. Thesimulationswereperformedon
PCclustersattheCyprusUniversity,theFermiInstitute
which has to be compared with [2] ofRomeandatthePisaUniversity. Wewishtothankall
theseinstitutionsforsupportingourprojectandthestaff
F2 of their computer centers (particularly M. Daviniand F.
π m2 +m2 −2m2 ≃(180MeV)4 . (13)
6 (cid:16) η η′ K(cid:17)(cid:12)exp Palombi)for their help. L. G. thanks the CERN Theory
(cid:12) Division, where this work was completed, for the warm
(cid:12)
Noticethat,sinceEq.(3)isvalidonlyattheleadingorder hospitality and acknowledges partial support by the EU
in a N /N expansion, the ambiguity in the conversion under contract HPRN-CT-2002-00311(EURIDICE).
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