Table Of ContentCERN-PH-TH-2017-012
Thermodynamics of strongly-coupled lattice QCD
in the chiral limit
Philippe de Forcrand
InstitutfürTheoretischePhysik,ETHZürich,CH-8093Zürich,Switzerland
7
CERN,THDivision,CH-1211Geneva23,Switzerland
1
0 E-mail:[email protected]
2
Paul Romatschke
n
a DepartmentofPhysics,390UCB,UniversityofColoradoatBoulder,Boulder,CO,USA
J E-mail:[email protected]
8
2 Wolfgang Unger
FakultätfürPhysik,UniversitätBielefeld,Universitätstasse25,D33619Bielefeld,Germany
]
t E-mail:[email protected]
a
l
- Hélvio Vairinhos∗
p
e InstitutfürTheoretischePhysik,ETHZürich,CH-8093Zürich,Switzerland
h E-mail:[email protected]
[
1
In the strong coupling limit, n-point functions in lattice QCD with staggered fermions can be
v
4 rewrittenexactlyassumsoverconstrainedconfigurationsofmonomers,dimers,andbaryonloops
2
coveringthespacetimelattice.Wormalgorithmsprovideefficientglobalsamplingmethodsover
3
8 such ensembles, and are particularly efficient in the chiral limit. We study the thermodynamics
0
of strongly-coupled U(3) and SU(3) lattice QCD with one massless staggered fermion using
.
1
such methods, and compare the results with the relativistic pion gas down to low temperatures
0
7 O(15MeV).
1
:
v
i
X
r
a
34thannualInternationalSymposiumonLatticeFieldTheory
24-30July2016
UniversityofSouthampton,UK
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Thermodynamicsofstrongly-coupledlatticeQCDinthechirallimit HélvioVairinhos
1. Introduction
ConsiderU(N) or SU(N) lattice QCD with a single staggered fermion flavour, at finite tem-
perature.Atlowtemperatures,thechiralU(1)symmetryofthemasslessstaggeredfermionisspon-
taneouslybroken,towhichasinglemasslessGoldstonebosonisassociated:thepion.
Thepionisinteracting,butatsufficientlylowtemperaturesthestrengthoftheeffectiveinter-
actions vanishes, i.e. T →0, and the pion is effectively free. In such a regime, the physics is that
F
π
ofanidealpiongas,whoseenergydensity,ε,satisfiestheStefan-Boltzmannlaw(SB)forasingle
bosonicdegreeoffreedom:
π2
ε(T)=ε(0)+ T4 (1.1)
30
Here we summarize our numerical study of the thermal properties ofU(3) and SU(3) lattice
QCD with a single staggered fermion, in the chiral limit, where we test the hypothesis of a (near)
idealpiongasbelowthecriticaltemperatureofthechirally-restoringphasetransition.
Wechoosetoperformsimulationsinthestrongcouplinglimit,β =0,fortherewehaveaccess
toMonteCarlo algorithmsofthewormtype, whichareveryefficient,even inthechirallimitand
at low temperatures. This allows us to determine the equation of state of lattice QCD with high
precision,atunprecedentedlylowtemperatures.
2. Thermodynamicsofafreemasslessbosononthelattice
First,itisinstructivetounderstandthebehaviorofanidealgasofmasslessbosonsonalattice.
Lattice corrections to the ideal gas regime of a free massless boson, on a N3×N lattice with
s t
anisotropyξ =a/a ,havebeenstudiedin[1].Theenergydensityε ofsuchagasisgivenby:
t
ξ3 sin2(πj /N) 3
a4ε(T)=− ∑ 0 t , b2= ∑sin2(πj/N ) (2.1a)
Ns3Nt(cid:126)(cid:54)=(cid:126)0b2+ξ2sin2(πj0/Nt) i=1 i s
ξ3 (cid:16) (cid:112) (cid:17)−1
a4ε(0)=− ∑ b2+ξ2+b b2+ξ2 (2.1b)
N3
s(cid:126)(cid:54)=(cid:126)0
where the lattice temperature is given by aT =ξ/N. Similar expressions can be obtained for the
t
pressure p.Inparticular,theyimplythatthetraceanomalyvanishesonanyfinitelattice[1]:
∆ε−3∆p=0 (2.2)
where∆ε(T)=ε(T)−ε(0),and∆p(T)= p(T)−p(0).
In this system, discretization effects are quite significant (see Fig. 1): from [1], we learn that
latticecorrectionsaresmallforN ≥2N andξ ≥2.Actually,theidealgasbehaviorisonlyexact
s t
inthecontinuoustimelimit,ξ →∞.
WekeepthisinmindwhensimulatinglatticeQCDintheregimewheretheU(1)chiralsym-
metryisspontaneouslybroken,andthepionismassless.Eventhoughpionsarenotfree(F (cid:54)=0),
π
theirinteractionsshouldbenegligibleintheregimeT (cid:28)F (orinthelargeNlimit),andthepicture
π
ofanidealpiongasshouldbecomeagoodapproximation.
1
Thermodynamicsofstrongly-coupledlatticeQCDinthechirallimit HélvioVairinhos
0.50 NNNNNNNNNNNNNNNNNNNttttttttttttttttttt =================== 11212121212121212126646464646464646464
111111111100000000001111111111 0.45 Stefan-BoltNNNNNNNNzmtttttttt ========a 33333333n22222222n
0.40
0.35 ξξξξξξξξξξ ========== 1111111111
4444444444( - ) / T( - ) / T( - ) / T( - ) / T( - ) / T( - ) / T( - ) / T( - ) / T( - ) / T( - ) / Tεεεεεεεεεεεεεεεεεεεε0000000000111111111100000000000000000000 0.300.35 0.40 0.45 0.50 0.55 0.60 0.65 ξξξξξξξξξξ ========== 2222222222
ξξξξξξξξξξ ========== 3333333333
0000000000..........5555555555 1111111111..........0000000000 1111111111..........5555555555 2222222222..........0000000000 2222222222..........5555555555
NNNNNNNNNNtttttttttt ////////// NNNNNNNNNNssssssssss
Figure 1: Energy density of a free massless boson on the lattice: the finite-size effects induce very large
deviations from the Stefan-Boltzmann limit (dotted line), but are significantly suppressed (less than 10%)
whenN (cid:38)3N forξ =1,orN (cid:38)2N forξ ≥2.
s t s t
3. DimerrepresentationoflatticeQCDinthestrongcouplinglimit
ThepartitionfunctionofSU(3)latticeQCDwithN =1staggeredfermions,atβ =0,is:
f
(cid:90)
Z= DUDψDψ¯ e2atmq∑xψ¯xψx+∑x,µγδµ0ηxµ(eatµqψ¯xUxµψx+µˆ−e−atµqψ¯x+µˆUx†µψx) (3.1)
whereµ isthequarkchemicalpotential,γ isthebareanisotropy,m isthebarequarkmass,a (a)
q q t
isthetemporal(spatial)latticespacing,andη =±1arethestaggeredphases.
xµ
Analytic integration of the link variables, followed by the integration of the Grassmann vari-
ables,yieldsthepartitionsumofasystemofmonomers,dimers,andbaryonloops[2]:
(cid:32) (cid:33)
(cid:18) (cid:19)
σ((cid:96)) 3! (3−k )!
Z= ∑ ∏ ∏ xµ (2a m )NMγ2NDt+3N(cid:96)te3Ntatµqw(cid:96) (3.2)
t q
3!|(cid:96)| n ! 3!k !
{n,k,(cid:96)} x x x,µ xµ
where n ,k ∈ {0,1,2,3} are occupation numbers of monomers and dimers, (cid:96) ∈ {0,±1} are
x xµ xµ
occupation numbers of oriented baryonic dimers, and N ,N ,N denote, respectively, the total
M Dt (cid:96)t
numberofmonomers,timelikedimers,andtimelikebaryoniclinks:
N =∑ n , N =∑ k , N =∑ |(cid:96) | (3.3)
M x x Dt x x0 (cid:96)t x x0
w countsthenumberoftimesbaryonloopswraparoundthethermaldirection,andσ((cid:96))=±1is
(cid:96)
asignwhichdependsontheshapeofthebaryonloops(andintroducesasignproblem).
DuetotheGrassmannintegration,theconfigurationswhichcontributetothepartitionfunction
areconstrained,oneachsite,tohaveeitherexactly3monomersand/ordimers,orbetraversedby
anon-self-intersectingorientedbaryonloop:
n +∑ k =! 3, ∑ (cid:96) =! 0, ∀x (3.4)
x ±µ xµ ±µ xµ
Such constrained configurations can be efficiently sampled using variants of the worm algo-
rithm: a “mesonic worm”, which updates the monomer-dimer sector [3], and a “baryonic worm”,
whichupdatesthebaryonicloopsandthe3-dimersector[4,5].
The partition function for U(3) QCD is obtained from (3.2) by removing the baryons, i.e.
(cid:96) =0,∀x,µ.Inthiscase,onlythemesonicwormisneededtosimulateit.
xµ
2
Thermodynamicsofstrongly-coupledlatticeQCDinthechirallimit HélvioVairinhos
4. ThermodynamicsoflatticeQCDinthestrongcouplingandchirallimits
The energy density and pressure in SU(3) lattice QCD, in the strong coupling limit (β =0)
and chiral limit (m =0), are related to the density of hadrons hopping in the time direction (i.e.
q
timelikedimersandtimelikebaryonlinks):
a3at∆ε =µBρB−aV3at ∂∂lTog−1Z(cid:12)(cid:12)(cid:12)(cid:12) = ξγ ddξγ (cid:104)2nDt+3n(cid:96)t(cid:105) (4.1a)
V,µB
(cid:12)
a3at∆p=a3atT ∂logZ(cid:12)(cid:12) = ξ dγ (cid:104)2nDt+3n(cid:96)t(cid:105) (4.1b)
∂V (cid:12) 3γ dξ
T,µB
where µ is the baryon chemical potential, ρ =(cid:104)w (cid:105)/N3 is the baryon density, and ξ(γ)= a is
B B (cid:96) s at
therenormalizedanisotropy,whichparameterizesthephysicalanisotropyofthelattice.Inorderto
obtainthecorrespondingexpressionsinU(3)QCD,itsufficestotakeρ =0andn =0.
B (cid:96)t
Eqs.(4.1a)and(4.1b)implythatthetraceanomalyvanishesforanylatticespacing,cf.(2.2).
5. Anisotropycalibration
An accurate determination of the energy density (4.1a), or pressure (4.1b), requires a precise
knowledgeoftherenormalizedanisotropyξ asafunctionofthebareanisotropyγ,andalsoofits
running,dξ/dγ.Forthispurpose,weusethefluctuationsofcertainconservedcharges,labelledby
spacetimedirections,asprobesforthecalibrationofthelatticeanisotropy.
Inthechirallimit,Grassmannconstraints(3.4)implytheexistenceofconservedcurrents[6]:
(cid:18) (cid:19)
3 3
j =π k − |(cid:96) |− ⇒ ∑ j =0, ∀x (5.1)
xµ x xµ xµ xµ
2 8
±µ
whereπx=(−1)∑µxµ =±1istheparity(bipartitecolor)ofthesitex.Wecanalsodefineconserved
chargesbyintegratingthecurrentsalongcodim-1hyperslicesS perpendiculartothedirectionµˆ:
µ
Q = ∑ j (5.2)
µ xµ
x∈S
µ
(cid:10) (cid:11)
Due to parity symmetry, Q =0, ∀µ. We consider lattices with the same size N in all spatial
µ s
directions,andthuscomparevariationsofthetimelikecharge,Q2 =Q2,andoftheaverageofthe
t 0
spacelikecharges,Q2= 1∑3 Q2.
s 3 i=1 i
Ournon-perturbativerenormalizationcriterionrequiresthefluctuationsoftheconservedcharges
tobeisotropicwhenthephysicalvolumeishypercubic(inthethermodynamiclimit):
(cid:10)Q2(cid:11)(γ )=(cid:10)Q2(cid:11)(γ ) ⇒ aNs =ξ(γ )Ns =1 (5.3)
t np s np a N np N
t t t
whereγ isthenonperturbative,finelytunedvalueofthebareanisotropyforwhichfluctuationsof
np
theconservedchargescoincide.Usingscalingarguments,itisalsoeasytorelatetherunningofthe
renormalizedanisotropytoexpectationvaluesassociatedwiththeseconservedcharges:
(cid:10)Q2(cid:11)
ξ dγ = (cid:16) γnp (cid:17)(cid:12) (5.4)
dξ d (cid:10)Q2(cid:11)− d (cid:104)Q2(cid:105) (cid:12)
dγ t dγ s (cid:12)
γnp
3
Thermodynamicsofstrongly-coupledlatticeQCDinthechirallimit HélvioVairinhos
U(3) on a 163 x 32 lattice
142.5
timelike
142.0 spacelike ∂γ〈Qt2〉γnp
141.5
2Q〉µ114401..50 〈Q2〉γnp
〈
140.0
139.5 ∂γ〈Qs2〉γnp
139.0 γnp
138.5
1.548 1.550 1.552 1.554 1.556 1.558 1.560 1.562 1.564 1.566
γ
Figure2:Fluctuationsoftheconservedchargesinthetimelike(purple)andspacelike(green)directions,as
afunctionofthebareanisotropyγ,inU(3)QCD.Theintersectionpointcorrespondstothecriticalvalueof
γ forwhichthephysicalboxishypercubic,whilethelatticehasananisotropyξ(γ )= Nt =2.
np Ns
7.0 1.10
γ2 = 1.231(4) ξ + 0.285(9) log ξ - 0.230(4) Mean field
6.0 γU2( 3=) 1Q.2C7D62(3) ξ 1.05 Non-perturbative
1.00
5.0
0.95
4.0
ξ3.0 2 / ξγ 0.90
0.85
2.0
0.80
1.0
0.75
0.0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0.70
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
γ2 γ2
Figure3:NonperturbativerelationbetweenthebareandrenormalizedanisotropyinU(3)QCD(left),and
itsdeviationfromthemeanfieldprediction(right).
where (cid:10)Q2(cid:11) is the variance at γ =γ , and the denominator depends only on the difference of
γnp np
their slopes at γ =γ . We determine all these quantities by finding the intersection point of the
np
curvesofQ2 andQ2,whichareconstructedusingmulti-histogramreweighting(Fig.2).
t s
We determine γ for several aspect ratios, Nt = ξ ∈ {2,3,4,5,6}, and for different spatial
np Ns
sizes, inU(3) and SU(3) QCD. In the thermodynamic limit, the functional dependence ξ(γ) ap-
pears to be quadratic for large γ (Fig. 3, left). Mean field arguments also predict a quadratic de-
pendence in the large γ limit: ξ(γ) = γ2 [7], but the non-perturbative prefactor differs from the
mean-fieldoneby≈25%(Fig.3,right).
6. Energydensityvs.temperature
Given the relation between the bare and renormalized anisotropies, and the corresponding
running(Fig.3),theremainingingredientforanaccuratedeterminationoftheenergydensity(4.1a)
inU(3)QCDisaprecisemeasurementofthedensityoftimelikedimers,n .
Dt
In order to determine the dependence of the energy density ε on the temperature T, we first
need to accurately subtract from it the T =0 contribution, ε . We compute ε by taking the ther-
0 0
modynamiclimitofthedensityoftimelikedimersevaluatedonahypercubiclattice:
4
Thermodynamicsofstrongly-coupledlatticeQCDinthechirallimit HélvioVairinhos
4.0 101
a4 ε0 = 0.66(2) ξ - 0.48(5) ξ = 2
3.5 UId(e3a) l QmCaDssless lattice boson gas 100 ξξ == 34
ξ = 5
3.0 10-1 ξ = 6
Mean field
4a ε0 22..05 4T) - ) / Tε0c1100--32 Stefan-Boltzmann
1.5 ((ε
10-4
1.0
10-5
0.5
0.0 10-6 10-4 10-3 10-2 10-1 100
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
ξ (T/Tc)4
Figure 4: Scaling of the energy density with the physical anisotropy, at T =0 (left), and the subtracted
energydensity(ε(T)−ε )asafunctionofthetemperature(right),bothforU(3)QCD.Therearesignificant
0
deviationstotheequationofstateofanidealpiongas,forlowtemperaturesandfortemperaturesnearT .
c
NotethatthelowesttemperaturereachedisO(15MeV).
a4ε0(ξ)= lim ξ2 dγ (cid:104)2nDt(cid:105)(cid:12)(cid:12)(cid:12) (6.1)
Ns→∞ γ dξ (cid:12)Nt=ξNs
We observe that ε scales approximately linearly with ξ, for large ξ (Fig. 4, left), similarly to an
0
idealgasofmasslessscalarbosonsonthelattice[1],butwithadifferentnon-universalprefactor.
Atfinitetemperature,wecomputetheenergydensitya4∆ε(N ,N,ξ),withtheε contribution
s t 0
subtracted, on N3×N lattices, for fixed aT = ξ , and for several spatial sizes. We then take the
s t Nt
thermodynamiclimitofa4∆ε,assumingO(N−3)corrections.1 Weexpressbothenergydensityand
s
temperature in units of the critical temperature of the chiral phase transition, which for U(3) is
aT =1.466andforSU(3)isaT =1.089.2
c c
The dependence of the energy density on the temperature, in U(3) QCD, is given in Fig. 4
(right).Thedatapointsseemtofallonanuniversalcurve,whichdeviatesfromSB(dottedline)at
temperaturesnearT ,andalsoatlowtemperatures.Itisqualitativelyconsistentwiththe(analytical)
c
meanfieldpredictioninthelargeN limit(solidline).3
The surprising deviation from SB at low temperatures may be due to finite size effects: the
data points at the lowest temperatures require large N, but are computed for possibly not large
t
enoughvaluesofN .Simulationswithlargerspatialvolumesarerequiredforabettercontrolofthe
s
thermodynamicallimit.Ontheotherhand,thedeviationfromSBathightemperaturesmaybedue
toanUVcutoffeffect,andsimulationswithlargervaluesofξ arerequiredinordertoincreaseN,
t
atfixedtemperature.Atintermediatevaluesofthetemperature,theenergydensityistheclosestto
SB,withasmalldiscrepancywhichmaybeduetoafinite-ξ effect:SBscalingisonlyexpectedto
beexactintheξ →∞(continuoustime)limit.
1Inspiredbythelessonsoftheidealgasofmasslessscalarbosonsonthelattice(Section2),wetakethethermody-
namiclimit,Ns→∞,byonlyusinglatticesforwhichNs≥2Nt,inordertominimizethefinite-sizecorrections.
2Intheliterature,thevaluesofthecriticaltemperature,namelyaTc=1.8843(1)forU(3)[5]andaTc=1.402(2)
forSU(3)[8],aredeterminedassumingthemeanfieldrelationbetweenthebareandrenormalizedanisotropycouplings,
i.e.aTc=γ2atTc.Usingournon-perturbativemethodforsettingtheanisotropyscale,thecorrespondingvaluesforthe
U(3)andSU(3)criticaltemperatures,aTc=ξ(γ)atTc,deviatefromthoseintheliteratureby≈25%.
3 Inthismeanfieldapproach,thecriticaltemperatureisnoteasytofix.ForcomparisonwiththeU(3)andSU(3)
data,wesetthecriticaltemperaturetotheU(3)meanfieldvalue:aTc=5/2[7].
5
Thermodynamicsofstrongly-coupledlatticeQCDinthechirallimit HélvioVairinhos
101
U(3)
SU(3)
100 U(3) mean field
SU(3) mean field
10-1 Stefan-Boltzmann
4
) / T0c10-2
ε
T) - 10-3
ε(
(
10-4
10-5
10-160-5 10-4 10-3 10-2 10-1 100 101
(T/T)4
c
Figure5:ComparisonoftheenergydensitiesinU(3)QCD(blue)andSU(3)QCD(red),asafunctionof
thetemperature,inunitsofthecriticaltemperaturesoftherespectivechiralphasetransitions.Thelargeerror
barsinSU(3)QCD,atlowtemperatures,areduetolargefluctuationsinthebaryonicsign.Themeanfield
curvesarecomputedanalyticallyinthelargeN limit,andwesetaT =5/2andaT =5/3[7].
c,U(3) c,SU(3)
Using the same approach, we have also computed the energy density as a function of the
temperature in SU(3) lattice QCD. A comparison between theU(3) and SU(3) theories is shown
inFig.5.Thedifferencebetweenthetwocasesistheadditionalcontributionofbaryonloopstothe
SU(3)theory,whichalsointroducesasignproblem,thusincreasingthestatisticalerror,especially
forthelargevolumesrequiredatlowtemperatures.
Athightemperatures,uptoT ,theSU(3)energydensityisconsistentwithSB,andconsistently
c
higherthanthatforU(3),whichweunderstandasbeingduetothethermalexcitationoftheextra
baryonicmodes,whichcontributetothetotalpressure(andenergydensity).
Inconclusion,theidealpiongasisareasonablygoodapproximationforthelowT regimeof
U(3) and SU(3) lattice QCD. Deviations from it, associated with pion interactions and with the
thermal excitation of massive hadrons, can be quantified and deserve further study. The study of
the equation of state can also be extended to the cases of non-zero quark mass and of non-zero
chemicalpotential.
References
[1] J.Engels,F.Karsch,H.Satz,Nucl.Phys.B205(1982)239.
[2] P.Rossi,U.Wolff,Nucl.Phys.B248(1984)105;F.Karsch,K.Mutter,Nucl.Phys.B313(1989)541.
[3] D.H.Adams,S.Chandrasekharan,Nucl.Phys.B662(2003)220.[hep-lat/0303003].
[4] P.deForcrand,M.Fromm,Phys.Rev.Lett.104(2010)112005.[arXiv:0907.1915].
[5] W.Unger,P.deForcrand,PoSLATTICE2011(2011)218.[arXiv:1111.1434].
[6] S.Chandrasekharan,F.-J.Jiang,Phys.Rev.D74(2006)014506.
[7] N.Bilic,F.Karsch,K.Redlich,Phys.Rev.D45(1992)3228.
[8] P.deForcrand,J.Langelage,O.Philipsen,W.Unger,Phys.Rev.Lett. 113(2014)15,152002.
[arXiv:1406.4397].
6
