Dynamical quarks effects on the gluon propagation and chiral symmetry restoration

0.78 MB·

by  A. Bashir

Dynamical quarks effects on the gluon propagation
and chiral symmetry restoration
A. Bashir
4 UniversidadMichoacanadeSanNicolásdeHidalgo,EdificioC-3,CiudadUniversitaria,
1
Morelia,Michoacán58040,México.
0
2 E-mail: [email protected]

n
A. Raya
a
J UniversidadMichoacanadeSanNicolásdeHidalgo,EdificioC-3,CiudadUniversitaria,

7 Morelia,Michoacán58040,México.
&
]
h FacultaddeFísica,PontificialUniversidadCatólicadeChile,Casilla306,Santiago22,Chile
p E-mail: [email protected]
-
p
J. Rodríguez-Quintero∗
e
h DepartamentodeFísicaAplicada,FacultaddeCienciasExperimentales,Universidadde
[
Huelva,Huelva21071,Spain.

1 &
v
CAFPE,UniversidaddeGranada,E-18071,Granda,Spain.
7
4 E-mail: [email protected]
4
1
. Weexploittherecentlatticeresultsfortheinfraredgluonpropagatorwithlightdynamicalquarks
1
0 andsolvethegapequationforthequarkpropagator. Chiralsymmetrybreakingandconfinement
4 (intimately tied with the analytic properties of QCD Schwinger functions) order parameters are
1
: thenstudied.
v
i
X
r
a
QCD-TNT-III-Fromquarksandgluonstohadronicmatter: Abridgetoofar?,
2-6September,2013
EuropeanCentreforTheoreticalStudiesinNuclearPhysicsandRelatedAreas(ECT*),Villazzano,Trento
(Italy)
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
1. Introduction
This contribution is mainly devoted to report on the results of Ref. [1], where we dealt with
the dynamical quarks effects on the gluon propagation and applied this knowledge to the analysis
of the chiral symmetry restoration with increasing number of light fermion flavours. Quantum
chromodynamics (QCD) with large number of massless fermion flavors has seen a resurgence
of interest due to its connection with technicolor models, originally proposed by Weinberg and
Susskind [2], which fall into the category of Beyond the Standard Model Theories. They possess
intrinsicallyattractivefeatures. Theydonotresorttofundamentalscalarstoreconcilelocalgauge
symmetry with massive mediators of interaction and have close resemblance with well-studied
fundamental strong interactions, i.e., QCD. However, their simple versions do not live up to the
experimental electroweak precision constraints, in particular the ones related to flavor changing
neutral currents. Walking models containing a conformal window and an infrared fixed point can
possibly cure this defect and become phenomenologically viable [3]. This scenario motivates the
investigation of QCD for similar characteristics. One looks for such behavior of QCD for large
number of light flavors albeit less than the critical value where asymptotic freedom sets in, i.e.,
Nc1 =16.5, aNobelprizewinningresultknownsincetheadventofQCD,[4]. JustasN dictates
f f
the peculiar behavior of QCD in the ultraviolet, we expect it to determine the onslaught of its
emergingphenomenaintheinfrared,i.e.,chiralsymmetrybreakingandconfinement.
Whereastheselfinteractionofgluonsprovidesanti-screening,theproductionofvirtualquark-
antiquark pairs screens and debilitates the strength of this interaction of non abelian origin. For
realQCD,lightflavorsaresmallinnumberandhenceyieldtothegluonicinfluencewhichtriggers
confinementandchiralsymmetrybreaking. Oneneedstoestablishifthereisanothercriticalvalue
Nc2 <Nc1 whichcansufficientlydilutethegluon-gluoninteractionstorestorechiralsymmetryand
f f
deconfine color degrees of freedom. Such a phase transition lies at the non perturbative boundary
oftheinteractionsunderscrutinyandhencewecannotexpecttoextractsufficientlyreliableinfor-
mationfrommultiloopcalculationsoftheQCDβ-function. Purelynonperturbativetechniquesare
required to tackle the problem. Lattice studies in the infrared indicate that just below Nc1, chiral
f
symmetryremainsunbrokenandcolordegreesoffreedomareunconfined[5]. Belowthisconfor-
mal window, for an 8<Nc2 <12, the evolution of the beta function in the infrared is such that
f
QCDentersthephaseofdynamicalmassgenerationaswellasconfinement.
Modern lattice analyses for this matter appear to strongly argue in favour of a restoration
for the chiral symmetric phase taking place somewhere between N ∼8 and N ∼10 [6, 7]. In
f f
particular, the authors of ref. [7], with their study of the meson spectrum in lattice QCD with
eight light flavours using the Highly Improved Staggered Quark action, gathered some striking
evidences that N =8 QCD still lies in the broken chiral symmetry phase but, at the sime time,
f
suffering the effects from a remnant of the infrared conformality (a large anomalous dimension
forthequarkmassrenormalizationconstant)indicatingthattheunbrokenphaseisrecoverednear
above N ∼8. In the present work, we intend to combine the Schwinger-Dyson machinery, well
f
adjusted to account for QCD phenomenology in the pion sector, with the very last lattice data
including twisted-mass dynamical light flavours in order to provide with a model for the chiral
restorationmechanism,inquantitativeagreementwiththeabovementionedlatticestudies.
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Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
2. ChiralphasetransitionpicturefromSchwinger-DysonandLatticegluon
propagators
In continuum, Schwinger-Dyson equations (SDEs) of QCD provide an ideal framework to
studyitsinfraredproperties,[8]. Thesearethefundamentalequationsofanyquantumfieldtheory,
linkingallitsdefiningGreenfunctionstoeachotherthroughintricatelycouplednonlinearintegral
equations. As their formal derivation through variational principle makes no appeal to the weak-
nessoftheinteractionstrength, theynaturallyconnecttheperturbativeultravioletphysicswithits
emerging non perturbative properties in the infrared sector within the same framework. The sim-
plesttwo-pointquarkpropagatorisabasicobjecttoanalyzedynamicalchiralsymmetrybreaking
andconfinement. WithintheformalismoftheSDEs,theinversequarkpropagatorcanbeexpressed
as
S−1(p)=Z (iγ·p+m)+Σ(p), (2.1)
2
whereΣ(p)isthequarkselfenergy:
(cid:90) d4q λa
Σ(p)=Z g2∆ (p−q) γ S(q)Γa(q,p), (2.2)
1 (2π)4 µν 2 µ ν
whereZ =Z (µ2,Λ2)andZ =Z (µ2,Λ2)aretherenormalizationconstantsassociatedrespec-
1 1 2 2
tivelywiththequark-gluonvertexandthequarkpropagator. Λistheultravioletregulatorand µ is
therenormalizationpoint. Thesolutiontothisequationis
iγ·p+M(p2)
S−1(p)= , (2.3)
Z(p2,µ2)
whereZ(p2,µ2)isthequarkwavefunctionrenormalizationandthequarkmassfunctionM(p2)is
renormalization group invariant. This equation involves the quark-gluon vertex Γa(q,p) and the
ν
gluon propagator ∆ (p−q). Very much attention will be paid in the next subsection to the two-
µν
point function as a crucial input to study the quark propagator. Here, in the following, the only
otheringredient,thethree-pointquark-gluonvertexΓa(q,p),willbebrieflydiscussed.
ν
Significant advances have been made in pinning it down through its key attributes in the ul-
traviolet and infrared domains [9]. More recently, the seeds of the most general ansatz for the
fermion-bosonvertexappearedin[10]anditsfullblownextensionwaspresentedin[11]. Further-
more, given the general nature of constraints and the simplicity of the construction, a straightfor-
wardextensionofthisapproachisexpectedtoyieldanansatzadequatetothetaskofrepresenting
thedressed-quark-gluonvertex. Beforethisisachieved,werestrictourselvestoaneffectivethough
efficacious approach. Following the lead of Maris et. al. [12], we employ the following suitable
ansatz which has sufficient integrated strength in the infrared to achieve dynamical mass genera-
tion:
λa
Z g2∆ (p−q)Γ (p,q)→g2 (q2)∆N (p−q,N ) γ , (2.4)
1 µν ν eff µν f 2 ν
where
D(q2,N ) (cid:20) q q (cid:21)
∆N (q) = f δ − µ ν . (2.5)
µν,Nf q2 µν q2
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Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
In the next subsection, the nonperturbative gluon dressing function, D(q2,N ), will appear mod-
f
elledandaccountingthusfortheeffectsfromthedynamicallightflavoursonthegluonpropagation.
Concerningtheeffectivecoupling,g ,ithasbeenchosentobe
eff
D(q2,N )
g2 (q2)D(q2,N ) = f g2 (q2), (2.6)
eff f D(q2,2) MT
where g2 stands for the effective coupling ansatz proposed by Maris et. al. in ref. [12] and
MT
built there to incorporate also the nonperturbative information from the gluon propagator (as it is
assumedtomultiplythetree-levelone). Thus,Eq.(2.6)fortwodegeneratelightfermionflavours
allowstoreproducecorrectlythestaticaswellasdynamicpropertiesofmesonsbelow1GeV(see
for example review [13] and references therein) while, by fixing the right number of flavours in
theperturbativetailofg2 ,italsoreproducestheappropriateultravioletbehaviour,foranyflavour
MT
number.
2.1 Modellingtheflavourbehaviourforthegluonpropagator
The second ingredient needed to solve Eq. (2.1) with (2.2,2.3) is the two-point gluon Green
function, ∆ , that has been the object of a patient effort, spanning several decades, addressed
µν
to unravel its infrared behaviour. Lattice as well as SDE studies have finally converged on its
massiveorsocalleddecouplingsolution(seeforexample[14]). Afterthegluonpropagatorsolution
in the quenched approximation has been chiselled, we now have the first quantitatively reliable
glimpsesofitsquarkflavordependencebyincorporatingN =0,2lightdynamicalquarkflavors1
f
and 2+1+1 (2 light degenerate quarks, with masses ranging from 20 to 50 [MeV], and two non
degenerate flavors for the strange and the charm quarks, with their respective masses set to 95
[MeV] and 1.51 [GeV]) [21]. As we demonstrate shortly, in this last 2+1+1 case, the number
of light quarks corresponds effectively to 3. This is exactly the result derived from the recently
developed "partially unquenched" approach to incorporate flavor effects in the gluon SDE, [22].
Their work is in agreement with one of [21] when the charm flavor is assumed to decouple from
gluons. ItshouldnotbeworthlesstoremarkthatthesameN =2+1+1gaugefieldshavebeenalso
f
used to compute the running strong coupling and estimate accurately the value of Λ , in very
MS
goodagreementwithexperiments[23].
Moreover, our modern understanding of the flavor dependence of the gluon propagator pro-
videsuswiththesolidbasistousethefollowingnonperturbativemodel[24]:
z(µ2)q2(q2+M2)
D(q2) = (2.7)
q4+q2(cid:0)M2−13g2(cid:104)A2(cid:105)/24(cid:1)+M2m2
0
todescribethegluondressingrenormalizedinMOMschemeatq2=µ2. Thismodelisbasedonthe
tree-levelgluonpropagatorobtainedwiththeRenormalizedGribov-Zwanziger(RGZ)action[25]
which have been shown to describe properly the lattice data in the infrared sector (see refs. [26,
24]). Theoverallfactorz(µ2)isintroducedtoguaranteethemultiplicativeMOMrenormalization
prescription, namely, D(µ2)=1, and implies no physical consequence as the effective coupling,
1Thedynamicalflavorshavebeengenerated,withintheframeworkofETMcollaboration[15,16,17,18],withthe
mass-twistedlatticeaction[19],whileN =0datahavebeenborrowedfrom[20].
f
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Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
7
N=0 3
f β=5.80 (N=0) Linear law
6 β=4.05 (Nff=2) m0 2 Exp. law
β=4.90 (N=2)
5 Nf=2 β=1.95 (Nff=2+1+1) 1/ 1
2) / q 4 NNf==34 ββ==12..9100 ((NNff==24+)1+1) 00 1 2 3 4 5 6 7 8 9 10 11 12
2 f
(q 3 8
D 2 Nf=6 2> 6 ELxinpe.a lra wlaw
A
N=8 < 4
1 f
2
g 2
0 0
0.1 1 10 0 1 2 3 4 5 6 7 8 9 10 11 12
N
p (GeV)
f
Figure1: (Left)Latticegluonpropagatordataintermsofmomentafordifferentnumberoffermionflavors
and fits with Eq. (2.7) and the parameters of Eqs. (2.9,2.10). (Right) Parameters g2 <A2 > and 1/m2 in
0
termsofthenumbersofflavorsandthefitswithEqs.(2.9,2.10). Thebluesquaresstandfortheextrapolated
resultsatN =4weusedforFig.2.
f
g ,isfurtheradjustedtoreproduceproperlythemesonphenomenology. Then,weapplyEq.(2.7)
eff
toreproducethegluonpropagatorlatticedataanalyzedinRef.[21]andthusfititsmassparameters.
M2isrelatedtothecondensateofauxiliaryfields,emergingmerelytopreservelocalityfortheRGZ
action. Afreefitofthelatticedatasuggeststhatitdoesnotdependonthenumberoffermionflavors
(wefindM2=4.85[GeV2]). Dimensiontwogluoncondensate(cid:104)A2(cid:105),[27],and
q2
m2 =z(µ2) lim (2.8)
0 q2→0D(q2)
are flavor dependent and we look for their best fits. In order to cover a wide range of possibilities
within reason, we assume their evolution with the flavor number to be driven either by a simple
linearscalinglaw
m−1(N ) = m−1(0)(1−AN )
0 f 0 f
g2(cid:104)A2(cid:105)(N ) = g2(cid:104)A2(cid:105)(0)(1−BN ), (2.9)
f f
asdataappeartosuggest,orbyanexponentiallaw
m−1(N ) = m−1(0)e−ANf
0 f 0
g2(cid:104)A2(cid:105)(N ) = g2(cid:104)A2(cid:105)(0)e−BNf , (2.10)
f
which allows for the possibility that the gluon propagator becomes infinitely massive only when
the number of light quark flavors tends to infinity. The best-fit of the m and g2(cid:104)A2(cid:105) from lattice
0
data will require m (0)=0.333 GeV and g2(cid:104)A2(cid:105)(0)=7.856, in both cases, A=0.083 and B=
0
0.080, for the linear case, and A=0.095 and B=0.091, for the exponential one. Eq. (2.7) now
provides prediction for the gluon propagator for arbitrarily large N , as can be seen in the right
f
plot Fig. 1, while the right one shows the corresponding gluon propagator along with the lattice
data superimposed [21]. We also include some very recent gluon propagator data obtained from
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Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
latticesimulationswithfourdegeneratelighttwisted-massflavors2. Thesenewdataareratherwell
describedbyEq.(2.7)evaluatedforthemassparametersextrapolatedtoN =4withEq.(2.9)(see
f
the zoomedplot in Fig.2). This observation stronglysupports thatN =2+1+1 gluondata indeed
f
correspondtothreelightflavors.
Figure2:ThesameoftheleftplotofFig.1butonlywiththeparametersforthelinearcaseandincorporating
newsmall-volumelatticedatafor4degeneratefermionflavors.
Thus,wecanefficaciouslymodelthedilutionofthegluon-gluoninteractionswithincreasing
flavor number in order to study the chiral restoration mechanism. We can now employ the gap
equation to provide quantitative details of chiral symmetry breaking in terms of the quark mass
functionforanincreasingnumberoflightquarks.
2.2 Results
Inthefollowing,wemostlydiscusstheresultsobtainedbyemployingthelinearlawandstate
the effect of exponential extrapolation afterwards. Note that we have not considered the flavor
dependencewhichwouldarisefromthequark-gluonvertex(noexplicithandleonthisdependence
is available at the moment). Otherwise said, as can be seen from Eq. (2.6), we take the effective
coupling g , in the IR, to depend on N only through the gluon dressing function. The latter can
eff f
befairlyjustifiedbytheresultsof[21](seeEq.(5.2))whichsuggestthataneffectivecouplingcan
beconstructedsuchthatthereisanabsenceofanyflavordependenceintheinfraredregion, more
preciselystartingfromq2(cid:46)1GeV2.
Once the quark mass function is available for varying light quark flavors (see Fig. 3 for the
linear case), one can investigate any of the interrelated order parameters, namely, the Euclidean
2The gluon propagator lattice data for 4 light flavors have been borrowed from ETMC [28]. Simulated at small
volumes,theyareonlyavailableformomentaabove1.25GeVandhardlyallowforafitwithEq.(2.7). Nevertheless,
theycanbeusedtocheckourmodellingoftheflavorevolution.
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Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
N (cid:61)2
0.5 f
(cid:76)
V 0.4
N (cid:61)4
f
e
G
0.3
(cid:72)
(cid:76) N (cid:187)7.07
2 0.2 Nf (cid:61)6 f
p
(cid:72)
M
0.1
N (cid:61)7
f
0.0
10(cid:45)10 10(cid:45)8 10(cid:45)6 10(cid:45)4 0.01 1
2 2
p GeV
Figure3: Thequarkmassfunctiondiminishesheightforincreasinglightquarkflavors(herewithEq.(2.9)
AboveN ≈7.07,onlythechirallysymmetricsolutionexists.
f
(cid:72) (cid:76)
polemassdefinedasm2 +M2(p2=m2 )=0,thequark-antiquarkcondensatewhichisobtained
dyn dyn
from the trace of the quark propagator or the pion leptonic decay constant f defined through the
π
Pagel-Stokarequation[29],orthroughconsideringtheresidueatthepionpoleofthemesonpropa-
gator. Eachofthesequantitiesinvolvesthequarkwave-functionrenormalization,themassfunction
and/oritsderivativesandishencecalculablefromthesolutionforthefullquarkpropagator. More-
over, these order parameters can help to locate the critical number of flavors above which chiral
symmetryisrestored.
We investigate these three order parameters and choose to present here the Euclidean pole
mass of the quark in Fig. 4 for the linear (exponential) case and show that, at a critical value of
aboutNc≈7.1(Nc≈9.4),chiralsymmetryappearsrestored. Thephasetransitionappearssecond
f f
order,describedbythefollowingmeanfieldbehavior(solidlinesinFig.4):
(cid:113)
m ∼ Nc2−N . (2.11)
dyn f f
It has been established that confinement is related to the analytic properties of QCD Schwinger
functions which are the Euclidean space Green functions, namely, propagators and vertices. One
deducesfromthereconstructiontheorem[30]thattheonlySchwingerfunctionswhichcanbeas-
sociatedwithexpectationvaluesintheHilbertspaceofobservables;namely,thesetofmeasurable
expectationvalues,arethosethatsatisfytheaxiomofreflectionpositivity. Whenthathappens,the
real-axismass-polesplits,movingintopairsofcomplexconjugatesingularities. Nomass-shellcan
be associated with a particle whose propagator exhibits such singularity structure. We define the
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Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
1
(cid:76)
V
0.1
e
G
(cid:72)
n
y
d 0.01
m
0.001
2 4 6 8 10
N
f
Figure 4: Quark pole mass in the Euclidean space clearly demonstrates that chiral symmetry is restored
aboveacriticalnumberofquarkflavors. Blue(red)pointscorrespondtolinear(exponential)case. Thesolid
lineisthemean-fieldscaling,Eq.(2.11)
followingSchwingerfunction:
(cid:90) (cid:90) d4p
∆(t) = d3x ei(p4t+p·x)σ (p2), (2.12)
(2π)4 s
to study the analytic properties of the quark propagator; where σ (p2) is the scalar term for the
s
quarkpropagatorinEq.(2.3),thatcanbewrittenintermsofthequarkwavefunctionrenormaliza-
tion and mass function as Z(p2,µ2)M(p2)/(p2+M(p2)). One can show that if there is a stable
asymptoticstateassociatedwiththispropagator,withamassm,then
∆(t)∼e−mt , (2.13)
whereas two complex conjugate mass-like singularities, with complex masses µ =a±ib lead to
anoscillatingbehaviorofthesort
∆(t)∼e−atcos(bt+δ) (2.14)
forlarget,[31]. Fig.5analyzesthisfunctionforvaryingN ,inthelinearextrapolationcase. The
f
existenceofoscillationsclearlydemonstratesthatthequarkscorrespondtoaconfinedexcitationfor
small N . With increasing N , the onslaught of oscillations moves towards higher values oft and
f f
eventuallynevertakesplaceaboveacriticalN whenquarksdeconfineandcorrespondtoastable
f
asymptotic state. As an order parameter of confinement, we therefore employ ν(N )=1/τ (N ),
f 1 f
where τ (N ) is the location of the first singularity, [32]. The first oscillation is pushed to infinity
1 f
whenconfinementislost. Itisnotablethatwhenthedynamicallygeneratedmassapproacheszero,
ν(N )diminishesrapidly(seerightplotofFig.5). Thishighlightstheintimateconnectionbetween
f
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Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
0
0.200
(cid:68)Logt (cid:45)(cid:45)(cid:45)(cid:45)(cid:45)(cid:45)11208642(cid:200)(cid:72)(cid:76)(cid:200)(cid:230) (cid:230) (cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)N(cid:230)(cid:224)(cid:230)(cid:230)f(cid:230)(cid:236)(cid:230)(cid:224)(cid:230)(cid:61)(cid:230)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)4(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:236)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:236)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:242)(cid:230)(cid:224)(cid:230)(cid:230)N(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:236)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:224)(cid:224)(cid:236)f(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:61)(cid:242)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)6(cid:224)(cid:236)(cid:242)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:242)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)N(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:242)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:236)(cid:224)(cid:242)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)f(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:242)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:242)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:61)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:242)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:236)(cid:224)(cid:224)(cid:224)(cid:236)(cid:224)(cid:242)(cid:224)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)7(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)N(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)(cid:236)(cid:236)(cid:242)(cid:236)f(cid:236)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:187)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)7(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242).(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)0(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)7(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) ΝGeV 0000000.......000000100012501250000(cid:72)(cid:76)
(cid:45)14
(cid:224)
1 10 100 1000
2 3 4 5 6 7 8
(cid:45)1
t GeV N
f
Figure5: (Left)Spatiallya(cid:72)veragedE(cid:76)uclideanspace2-pointSchwingerfunction∆(t)developsoscillations
forlargetimeswhichcorrespondstothenon-existenceofasymptoticallystablefreequarkstates. Forsuf-
ficiently large values of N , the first minimum of these oscillations is pushed all the way to infinity, thus
f
ensuringtheexistenceofapoleonthetime-likeaxis,apropertyoffreeparticlepropagators. (Right)Theor-
derparameterforconfinementν(N )=1/τ (N ),whereτ (N )isthelocationofthefirstzeroofEq.(2.12)
f 1 f 1 f
ComparisonwithFig.4suggeststhatquarksgetdeconfinedwhenchiralsymmetryisrestored.
chiralsymmetryrestorationanddeconfinement. Infact,withinournumericalaccuracy,Ncisfound
f
tobethesameforboththetransitions.
The results with the exponential and linear flavor extrapolations are qualitatively the same,
leading to identical conclusions. They only quantitatively differ by the critical flavor numbers,
although both are pretty much in the same ballpark: Nc (cid:39)7.1 and Nc (cid:39)9.4. Note that both the
f f
parameterizations,sofarapartastohaveaninfinitelymassivegluonatN ≈12orN ⇒∞,restore
f f
chiralsymmetryandtriggerdeconfinementatsosimilarvalueoflightquarkflavors.
3. Conclusions
We have benefited from the latest lattice result for the quark flavor dependence of the gluon
propagator in the infrared, as well as from a RGE-grounded nonperturbative model for this IR
gluonpropagator,inordertoperformaPoincare-covariantSDEanalysisforthequarkpropagator.
Thisprovidedwithanefficaciousmodelfordilutionofthegluon-gluoninteractionwithincreasing
numberoflightquarksandweappliedittostudytheevolutionwiththelight-flavorsnumberofthe
quark chiral behaviour. A nonperturbative picture for the chiral symmetry restoration mechanism
emerges thus from this analysis. The quantitative analysis, following this approach, hints towards
chiralsymmetryrestorationinQCDwhenthenumberoflightquarkflavorsexceedsacriticalvalue
ofNc2 ≈8.2±1.2. Thisisinperfectagreementwiththestate-of-the-artforthedirectlatticeinves-
f
tigations on the chiral symmetry restoration in QCD [6, 7] and supports that the model presented
hereforthechiralrestorationmechanismisproperlycapturingmostoftherelevantphysicsforthe
problemwedealwith.
9
Dynamicalquarkseffectsonthegluonpropagation J.Rodríguez-Quintero
Furthermore,wehavealsostudiedanorderparameterfortheQCDconfinement-deconfinement
transition,whichisbasedontheanalyticpropertiesofthequarktwo-pointSchwingerfunction,and
locatednumericallythecriticalpointatthesamenumberofflavorsaschiralsymmetryrestoration
took place. That chiral symmetry appears restored when quarks get deconfined, within the ap-
proachwefollowed,highlightstheintimateconnectionbetweembothfundamentalphenomenaof
QCD.Thisisamainresultofthiswork.
Acknowledgments We acknowledge D. Schaich for a fruitful communication. This work was
supported by the grants: CIC, UMICH, Mexico, 4.10 and 4.22, CONACyT (Mexico) 82230 and
128534,andMINECO(Spain)researchprojectFPA2011-23781.
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