Table Of ContentOn the spectrum of QCD-like theories and the
conformal window
Albert Deuzeman
2
AlbertEinsteinCenterforFundamentalPhysics-UniversityofBern,Switzerland
1
0 E-mail: [email protected]
2
Maria Paola Lombardo
n
a INFN-LaboratoriNazionalidiFrascati,I-00044,Frascati(RM),Italy
J E-mail: [email protected]
9
Elisabetta Pallante∗
] CentreforTheoreticalPhysics,UniversityofGroningen,9747AG,Netherlands
t
a E-mail: [email protected]
l
-
p
e We report on the spectrum of the SU(3) gauge theory with twelve flavours in the fundamental
h
[ representation of the gauge group. We isolate distinctive features of the hadronic phase - the
1 one proper of QCD at zero temperature - and the so called conformal phase. The latter should
v emerge at sufficiently large N and before the loss of asymptotic freedom. In particular, we
f
3
analyseavailablelatticedataforthespectrumofN =12andincludeacomparisonwithresults
6 f
8 withN =16;thelattertheory,predictedbytheperturbativeβ-functiontodevelopanIRFPand
f
1
therefore be in the conformal phase, can serve as a paradigm for the study of theories in the
.
1
conformalwindow. Ouranalysissuggeststhatthetheorywithtwelveflavoursisintheconformal
0
2 window,possiblyclosetoitslowerboundary.
1
:
v
i
X
r
a
TheXXIXInternationalSymposiumonLatticeFieldTheory-Lattice2011
July10-16,2011
SquawValley,LakeTahoe,California
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
OnthespectrumofQCD-liketheoriesandtheconformalwindow ElisabettaPallante
1. Introduction
Thestudyin[1]forthetheorywithN =12flavourssupportsascenarioasdepictedinFig.1,
f
wheretheend-pointofthechiralphaseboundarysignalstheopeningoftheconformalwindowand
N =12isinsidetheconformalwindow. Whileconformalityforthesametheoryhasbeenreported
f
byothergroups[2,3],contrastingviewswerepresentedaswell[4].
Anymasslesstheorywithintheconformalwindowhasexactchiralsymmetryanddevelopsan
infraredfixedpoint(IRFP)atwhichthetheoryisconformal. Everywhereintheparameterspaceof
the theory, except at the fixed point, observables will show only remnants of conformality. These
remnantsjointwiththerealizationofexactchiralsymmetryleadtofeaturesofthespectrumdistinct
fromQCD.Inparticular,adimensionalmassratiosarerobustindicatorsofpatternsofsymmetries
[1,6].
The aim of our study, of which this proceeding is a preliminary account, is to isolate those
featuresandusetheminthecomparisonofN =12latticeresultswithtypicallatticeQCDresults.
f
The set of data used here consists of the ensembles partly analysed in [1], where more statistics
has been collected for some points at the lattice bare couplings β =6/g2 =3.8,3.9 and 4.0, on
L
the weak-coupling side of the bulk phase transition [1]. Data are at masses am=0.07,0.06 with
volume163x24,am=0.05withvolume244 andam=0.025withvolume324. Theactionusedis
the tree-level Symanzik-improved gauge action with Asqtad staggered fermions. We also include
thelargestvolumedatareportedin[4]atonevalueofthebarelatticecouplingβ =2.2andwitha
L
differentimprovedlatticeaction;tree-levelSymanzikimprovementandtwostepsofstout-smearing
inthestaggeredfermionmatrix. Wheneverinstructive,wecomparewithresultsforthetheorywith
N =16fundamentalflavoursfrom[5]. AstudyofN =16withthesameactionasN =12in[1]
f f f
is in progress [7]. We defer to future work a more refined estimate of finite volume effects and a
finitesizescalinganalysis.
Insection2weanalysethemassratiom /m ,theEdinburghplotandadditionalratiosuseful
π ρ
to discriminate between a QCD phase and a conformal phase. In section 3 we consider another
indicator of chiral symmetry restoration, the splitting between the vector and the axial ground
states. Finally,insection4weanalyseanotherkeyrelationbetweenthewouldbeGoldstoneboson
andthechiralorderparameter. Weconcludeinsection5.
Figure1: ThephasediagramofQCD-liketheoriesintheT,N plane.
f
2
OnthespectrumofQCD-liketheoriesandtheconformalwindow ElisabettaPallante
2
1
Nf=12: this work
Nf=12: Ref. [2]
0.9 1.8 Nf=16: Ref. [3]
1.6
0.8 mρ
m/mπρ m/N1.4
0.7
1.2
0.6
1
0.5 0 0.2 0.4 0.6 0.8 1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 mπ/mρ
am
Figure 3: Edinburgh plot: N = 12 data from [4]
Figure2:Ratiom /m asafunctionofthebarequark f
π ρ
(redsquares), N =12datafromthisworkandβ =
mass for all existing data for N =12, and N =16: f L
f f
3.8,3.9(bluecircles),N =16datafrom[5](magenta
N = 12 data from [4] (red squares), N = 12 data f
f f
diamonds). TheQCDphysicalpoint(blackstar,left-
from this work and β = 3.8,3.9,4.0 (blue circles),
L
most) and the heavy quark limit (free theory) point
N =16datafrom[5](magentadiamonds).
f
(blackstar,rightmost)areshown.
2. TheEdinburghplotandmassratios.
Onefirstsignificantspectrumobservableistheratiom /m ,betweenthemassofthelightest
π ρ
pseudoscalar state (pion) m and the mass of the lightest vector state (rho) m . In QCD at zero
π ρ
temperature,chiralsymmetryisspontaneouslybrokenandthepionisthe(pseudo)Goldstoneboson
√
ofthebrokensymmetry,implyingthatitsmasswillbehaveasm ∼ m. Instead,thevectormass
π
containsaconstanttermandaleadingcorrectionlinearinthequarkmass,thusm ∼m +bm. In
ρ 0
the chirally broken phase, modulo lattice artefacts, one should thus expect their ratio to behave as
√
m /m ∼ am-asafunctionofthebarelatticequarkmassam.
π ρ
Within the conformal window chiral symmetry is restored. The lightest pseudoscalar state is
notanymoreaGoldstoneboson,andthereisnomassgap. AttheIRFPandatinfinitevolume,the
quarkmassdependenceofallhadronmassesinthespectrumisgovernedbyconformalsymmetry:
atleadingorderinthequarkmassexpansionallmassesfollowapower-lawwithcommonexponent
determinedbytheanomalousdimensionofthefermionmassoperatorattheIRFP.Henceweexpect
a constant ratio. Away from the IRFP, for sufficiently light quarks and finite lattice volumes, the
universal power-law dependence receives corrections, due to the fact that the theory is interacting
but no longer conformal. Hence, the pseudoscalar-vector mass ratio is constant at the IRFP at
infinite volume, and approximately constant in its surroundings and at finite volume, as it is the
situationexploredhere.
Inpractice,thetaskremainstheoneofdiscriminatingbetweenalatticemassratiothatgoesto
√
zeroas amandaratiothatremainsconstantandO(1)overasignificantrangeofmasses.
An important caveat concerns the extraction of mass eigenstates from correlators within the
conformalwindow: correlatorsinthevicinityoftheIRFPwillfollowapowerlawdecayatleading
order,correctedbymasscontributions. However,forsufficientlylargequarkmassesandawayfrom
theIRFPoneexpectslatticecorrelatorstodecayexponentiallyasinQCD,possiblywithsubleading
conformal corrections. Given this caveat, we have analysed all correlators in this work assuming
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OnthespectrumofQCD-liketheoriesandtheconformalwindow ElisabettaPallante
7.5
β=4.0
β=3.9
7 β=3.8
6.5
2(am)/(am)π5.56
5
4.5
4
0.5 0.6 0.7 0.8 0.9 1
(mπ/mρ)2
Figure 4: Ratio (am )2/(am) as a function of Figure 5: Ratio (am )2/(am) as a function of
π π
(m /m )2 for QCD data (quenched staggered) re- (m /m )2 fortheN =12dataatβ =3.8(greendi-
π ρ π ρ f L
ported by the MILC collaboration [8]. Notice that amonds),3.9(redsquares)and4.0(blackcircles).
quenchingshouldnotaffecttheleadingbehaviour.
the standard multi-exponential time dependence. The pseudoscalar staggered correlator could be
fittedwithtwocosinehyperbolicfunctions(fundamentalandexcitedstate)withouttheparity-odd
(oscillating) component. All other staggered correlators could be fitted with a cosine hyperbolic
with an oscillating component at intermediate and late times, with the addition of an excited state
atearlytimes. Thelargestuncertaintiesofthepresentanalysisarerelatedtothenucleon,forwhich
morestatisticsandbettersmearingisneeded. Temporalextentslongerthant =24,32wouldalso
facilitatetheanalysisofcorrelatorsatthelightestmasses.
Fig. 2 shows that the mass ratio for all existing N =12 data is approximately constant over
f
a wide range of bare quark masses, as it should be expected for a chirally symmetric theory. Ob-
viously, on the basis of these numerical evidences, we cannot exclude that a change of trend will
occur at even lower masses. Searching for combined evidences seems to be the optimal strategy
forthiscase,followingthelineadoptedin[1].
The Edinburgh plot, widely used in lattice QCD studies, is constructed in terms of adimen-
sional ratios of masses and offers a powerful way to combine results of lattice calculations per-
formed at different lattice spacings. Fig. 3 shows the Edinburgh plot for all existing data with
N =12. For an instructive comparison, we also show lattice results for the N =16 theory from
f f
[5]; the latter is already known to be in the conformal window by perturbative arguments. The
physicalpointofQCD(leftmostsideoffigure)correspondstom /m (cid:39)0.18andm /m (cid:39)1.21.
π ρ N ρ
On the other side of the figure, a useful theoretical limit is the heavy quark mass limit (rightmost
sideoffigure),whereallmassesinthespectrumaregivenbythesumoftheirvalencequarkmasses,
sothatm /m =1andm /m =3/2. Thislimitisalsoequivalenttothefreetheorylimit. AQCD
π ρ N ρ
scenariowilldrawacurveinthisfigurethatextrapolatestothephysicalpointfordecreasingquark
masses. WhatweobserveinFig.3suggestsinsteadabehaviourthatistobeexpectedfortheories
in the conformal window. The two mass ratios are “stuck" at a tiny corner, despite the fact that
quarkmassesin thereporteddatavary onaratherwiderange: N =12baremasses from[1]and
f
[4]varyfromam=0.01toam=0.07atvariouslatticecouplings. N =16baremassesfrom[5]
f
varyfromam=0.025toam=0.15. AlldataintheEdinburghplotarealsoawayfromtheheavy
quarklimitandallhavem /m ∼0.8,showingthatallsimulatedmassesaresufficientlylightand
π ρ
4
OnthespectrumofQCD-liketheoriesandtheconformalwindow ElisabettaPallante
fermions are dynamical. Fig. 3 also importantly suggests that all existing data for the N = 12
f
spectrumcoverthesamedynamicalregion;acomparisonisthereforejustified.
Another interesting insight, useful to discriminate between a QCD-like and a conformal be-
haviour, can be gained through Figs. 4 and 5. At fixed lattice spacing, one can study the ratio
(am )2/(am) as a function of (m /m )2. This ratio behaves as a constant in QCD to a good ap-
π π ρ
proximation, so that parallel horizontal lines are drawn at different lattice couplings. Fig. 4 also
showsthattheratioincreaseswithdecreasingβ ,asignalthattheβ functionforthistheoryisneg-
L
ative. Withintheconformalwindowthebehaviourshouldbequitedifferent,andinfactanalogous
towhatisobservedinFig.5: theratioshouldbehaveas(am )2/(am)∼(am)2δ/(am)∼(am)2δ−1,
π
with0.5<δ (cid:46)1. Separateconstantlinesatdifferentlatticespacingsarethusnolongerobserved
anddataareconcentratedaroundonevalueofm /m . Ideally,takenthesamevalueofm /m at
π ρ π ρ
twodifferentlatticespacings, anorderingoppositetotheQCD-likecasewouldsuggestapositive
β function. The latter is true at the strong coupling side of the IRFP. A few points in Fig. 5 have
a sufficiently close value of m /m and show indeed the inverted ordering proper of a positive β
π ρ
function. Thisisanotherwaytolookattheresultsfortheβ functionreportedin[1].
3. Thevectorandaxial-vectormasssplitting
In[1]itwasshownthatthelightestpseudoscalarandvectormassesatvarioussimulatedcou-
plings were well fitted by a power-law with close exponents in the range 0.6-0.7, thus excluding
theGoldstonenatureofthepionandshowingthatdataareawayfromtheheavyquarkregime. As
a word of caution, we add that the accuracies of the spectrum data and fits are not comparable, as
oftoday,withthoseachievedbythefitstothechiralcondensatein[1]. Thelatterhavebeenshown
tobeatinfinitevolumewithinstatisticaluncertainties. Forthespectrum, finitevolumeeffectsare
expectedtobepresentandoftheorderofabout10%.
Thevectorandaxial-vectormasssplittingisanotherindicatoroftherestorationofchiralsym-
metry. InFig.6weshowourdataforβ =3.9and4.0,anddatafrom[4]forthelightestvectorρ
L
and axial-vector a . The best fits with zero intercept and free exponent are also reported. Best fit
1
valuesoftheexponentsareδ =0.67(4),δ =0.68(3),atβ =3.9,δ =0.68(7),δ =0.67(3),
a1 ρ L a1 ρ
at β =4.0, and δ =0.79(9),δ =0.72(2) for the data from [4]. All exponents lie around 0.7.
L a1 ρ
Figure 6: Mass of the lightest axial-vector m (top) and the lightest vector m (bottom) as a function of
a1 ρ
thebarequarkmass: β =3.9,4.0andam=0.025,0.05,0.06,0.07(rightmost),β =2.2andam=0.01to
L L
0.035from[4](leftmost).Superimposedcurvesarefitstoapower-lawwithzerointerceptandfreeexponent.
5
OnthespectrumofQCD-liketheoriesandtheconformalwindow ElisabettaPallante
Figure7: Thebarepseudoscalarmasssquaredasafunctionofthebarechiralcondensate(orderparameter)
squaredinachirallysymmetric(top)andchirallybroken(bottom)phase[9].
A power-law fit with free intercept favours a slightly negative intercept with non unit exponent,
thus disfavouring a chirally broken scenario. The goodness of power-law fits with zero intercept
for both vector and axial states suggests their degeneracy in the chiral limit, thus a restored chiral
symmetry. Unfortunately, the N =16 results in [5] seem to be still affected by rather large finite
f
volume effects, and for this reason we omit them here. We are currently simulating N =16 with
f
thesamelatticeactionasN =12[1]inordertomeaningfullycomparethetwotheories.
f
4. Goldstonebosonandchiralorderparameter
It was observed in [1] that an additional powerful discriminator between exact and sponta-
neously broken chiral symmetry is provided by the relation between the would-be Goldstone bo-
son mass and the chiral condensate. The observation was based on the theoretical analysis of [9].
Here, we analyse all available N =12 data in light of these theoretical premises. In a chirally
f
symmetric phase the behaviour of m2 as a function of the chiral condensate is as illustrated in
π
Fig. 7 (upper curve): it has positive curvature, it extrapolates to zero, and the curvature is due to
non zero anomalous dimensions. Importantly, the curvature is opposite to that induced by finite
volume effects. Hence, it provides at the same time a powerful indicator of the presence of fi-
nite volume corrections for lattice data. In a chirally broken phase the curvature is opposite and
it extrapolates to a negative value, Fig. 7 (lower curve). In Fig. 8 we collect our data (right) and
data from [4] (left). Both data sets clearly show a positive curvature, and best fits to a power-law
(amπ)2 =A(a3(cid:104)ψ¯ψ(cid:105))2δχ with zero intercept do not suggest qualitative differences between data
sets. Weobtainabestfitexponentδ =0.66(2)forjointdatasetsatβ =3.9and4.0(seealso[1])
χ L
andδ =0.727(5)forthedatafrom[4].
χ
5. Conclusions
We have reported on the spectrum of the SU(3) gauge theory with twelve flavours in the
fundamental representation. In particular, we have isolated a few signatures that are useful to
discriminatebetweenthehadronicphase,i.e. QCDatzerotemperature,andtheconformalphase-
properoftheorieswithintheconformalwindow. Inthisanalysiswehaveassumedthatallourlattice
data,evenifinsidetheconformalwindow,areawayfromtheIRFPandcorrelatorsfollowaleading
6
OnthespectrumofQCD-liketheoriesandtheconformalwindow ElisabettaPallante
Figure8: (am )2asafunctionof(a3(cid:104)ψ¯ψ(cid:105))2forallexistingN =12data: (right)updateddatafrom[1]for
π f
β =3.9(redsquare)andβ =4.0(blackcircle),(left)datafrom[4]forβ =2.2(largestvolumesonly).
L L L
exponential decay law; this seems to be supported by the obtained results. Further investigations
at weaker coupling, lighter masses (and longer temporal extents) would provide a useful piece of
additional information. We have analysed the ratio m /m , the Edinburgh plot and the ρ−a
π ρ 1
mass splitting for all existing data. Points in the Edinburgh plot stick to a region m /m ∼0.8,
π ρ
despitecoveringawiderangeofbarelatticefermionmasses. Fitstotheρ−a splittingfavourthe
1
degeneracyofthetwostatesinthechirallimit. Finally, wehavere-proposedtherelationbetween
theGoldstonebosonmassandthechiralorderparameterasapowerfulindicatorofrestoredchiral
symmetry. All existing data for the N =12 theory seem to consistently favour chiral symmetry
f
restoration and an almost universal power-law behaviour for all massive states, which is to be
expectedinsidetheconformalwindow.
Acknowledgements
ThisworkwasinpartbasedontheMILCcollaboration’spubliclatticegaugetheorycode.
SimulationswereperformedontheIBMBG/PattheUniversityofGroningenandtheIBM
Power6+HuygensatSARA(Amsterdam),undersupportoftheDutchNCF.
References
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