The curvature of the QCD phase transition line

0.12 MB·English

by  G. Endrodi

9 The curvature of the QCD phase transition line
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G. Endro˝di∗1,Z.Fodor1,2, S.D.Katz1,2,K.K.Szabó2

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t 1InstituteforTheoreticalPhysics,EötvösUniversity,H-1117Budapest,Hungary.
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l 2DepartmentofPhysics,UniversityofWuppertal,D-42097Wuppertal,Germany.
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p E-mail: [email protected]
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We determine the curvatureof the phase transition line in the m −T plane through an analysis

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of various observables, including the Polyakov loop, the quark number susceptibilities and the
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8 susceptibilityofthechiralcondensate. Thesecondderivativeofthesequantitieswithrespectto
1 m wascalculated. ThemeasurementswerecarriedoutonN =4,6,8and10latticesgenerated
0 T
3 withaSymanzikimprovedgaugeandstout-linkimproved2+1flavourstaggeredfermionaction
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1 usingphysicalquarkmasses.
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TheXXVIInternationalSymposiumonLatticeFieldTheory
July14-19,2008
Williamsburg,Virginia,USA
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ThecurvatureoftheQCDphasetransitionline G.Endro˝di
1. Introduction
Quantum chromodynamics (QCD) is the theory of strong interactions. While at low temper-
atures the dominant degrees of freedom are hadrons, due to one of the most important properties
of QCD, asymptotic freedom, at high temperatures it describes a different phase of matter called
quark-gluon plasma (QGP). The phase transition between the hadronic phase of matter and QGP
canbeinvestigated bylattice simulations. Thetransition atzerochemicalpotential –whichrepre-
sentsthecaseofequalnumberofquarksandantiquarks–isofhugeimportance,sinceitisrelevant
for the early Universe and also for high energy collisions. Theregion ofthe phase diagram corre-
sponding to small and moderate m is also interesting, since the cooling down of the QGP after a
heavyioncollision occursinthisarea.
Simulations at nonvanishing chemical potential are burdened by a significant problem: the
fermion determinant here becomes complex, and as a result makes importance sampling impossi-
ble. Apossible solution forthisissueistoexpandobservables intoaTaylor-series inm ,wherethe
coefficientscanbecalculatedatvanishingchemicalpotential. Duetothesymmetryofthepartition
functiondescribingthesystem,thefirsttermintheseexpansionsalwaysvanishes. Thesecondterm
ontheotherhandisrelatedtothecurvatureofthetransition line.
The 2+1 flavour QCD transition was found to be an analytic crossover [1] (instead of a first-
order phase transition), which usually results in different transition temperatures for different ob-
servables [4]. Our aim is to determine the curvature k of the phase transition line for different
observables. These include the Polyakov loop, the strange quark number susceptibility, the chiral
condensate andthechiralsusceptibility.
2. Possiblescenarios
Figure 1: Two possible scenarios aboutthe QCD phase diagram aroundzero m . Transition temperatures
determinedbydifferentobservablesmayconvergeto(leftside,comparetofirstlatticeindicationsonN =
T
4[2])ordivergefrom(rightside,firstlatticeindicationsonN =4[3])eachotherasthechemicalpotential
T
isincreased.Thistendencycanalsohaveaneffectontheexistence(andifitexists,theposition)oftheQCD
criticalendpoint.
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ThecurvatureoftheQCDphasetransitionline G.Endro˝di
As an effect of the crossover character of the transition at zero m , the critical temperature
determined from the chiral susceptibility (Tc(c yy¯ )≈151 MeV) and that from the strange quark
numbersusceptibility(T (c )≈175MeV)arequitedifferent[4]. Thissituationcanchangeatfinite
c s
chemical potential according to two possible scenarios. The strengthening of the transition could
cause these two values to approach each other, while if the transition remains a weak crossover,
the two transition lines can diverge (see figure 1). The two cases are separated by the sign of the
difference k (c s)−k (c yy¯ ) of the curvatures for the two observables. A positive sign (left side)
wouldindicatethatthecrossoverregionshrinks,whichmightsuggestthatacriticalendpointexists
on the phase diagram. On the other hand a negative sign would cause the crossover region to
expand, whichmayresultintheabscence ofthecritical point.
Supposing that there exists a critical endpoint around baryonic chemical potential m ≈360
B
MeV, the difference of the curvatures should be around D k ≈0.02 (see definition below). Here
and also in the following under m wemean the m baryonic chemical potential. There are several
B
works in the literature in which the value of the curvature was computed [5, 6, 7, 8, 9]. In these
N =4works thecurvature wasfound tobe inthe interval k =0.003...0.01. Onthe other hand,
T
the N = 6 results of [3] indicate that the curvature might become less pronounced towards the
T
continuum limit.
3. The curvature
Let us parametrize the transition line in the vicinity of the vertical m = 0 axis as T (m ) =
c
T 1−k ·m 2/T2 . Thisimpliesthatthecurvaturecanbewrittenas
c c
(cid:0) (cid:1)
dT (m )
k =−T c (3.1)
c d(m 2) (cid:12)
(cid:12)m =0
(cid:12)
(cid:12)
Weillustrateourfirstproceduretomeasurethederivativeinquestionusingthequantityc /T2
s
(seedefinition insection 6). Since0≤c /T2 ≤1stands independently of m ,wecandefineT (m )
s c
viatherelation c /T2(T ,m )=0.5. Inthefollowingweusetheshortnotation O ≡c /T2.
s c s
The total derivative of the observable O(T,m 2) may be written as dO = (¶ O/¶ T)·dT +
¶ O/¶ (m 2) ·dm 2. So, since along the T (m ) line dO =0 by definition, for the derivative in the
c
d(cid:0)efinition of(cid:1)thecurvature thisreadsas
dT ¶ O ¶ O
c
=− (3.2)
dm 2 (cid:18)¶m 2(cid:19) (cid:18)¶ T (cid:19)
.
This way it is enough to measure these two derivatives at an arbitrary temperature. Formore
complicated quantities this procedure gives only an approximate value for the curvature. In these
cases one should carry out measurements at various temperatures; then T (m ) can be defined as
c
the inflection point (for e.g. c s/T2) or the maximum value (for e.g. c yy¯ ) of the observable. This
second procedure –whichisclearly more CPU-demanding –is anongoing project. Nevertheless,
tocalculate thederivative ¶ O/¶ (m 2)weneedtomeasurenewoperators. Thesearedetailedinthe
nextsection.
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ThecurvatureoftheQCDphasetransitionline G.Endro˝di
4. Technique
Letusconsider thepartition functioninitsusualform
Z = DUe−Sg(U)(detM)Nf/4 (4.1)
Z
anddenotethederivativewithrespecttom by′. ThederivativesofZ areeasilycalculatedtobe
u,d
(logZ)′ =hn i and (logZ)′′ =hc i, where the light quark number density n and the light
u,d u,d u,d
quarknumbersusceptibility c arethefollowingcombinations:
u,d
N
n = fTr M−1M′
u,d
4
(cid:0) (cid:1)
N
c = n2 + fTr M−1M′−M−1M′M−1M′
u,d u,d 4
(cid:0) (cid:1)
Using these definitions the second derivative of any (possibly m -dependent) observable can be
u,d
determined as
¶ 2hOi
=hOc i−hOihc i+h2O′n +O′′i (4.2)
¶m 2 u,d u,d u,d
u,d
For observables that do not depend explicitly on m (like the Polyakov loop L or c ) the third
u,d s
term in (4.2) vanishes. For m -dependent observables the derivatives O′ and O′′ were calculated
u,d
numerically, usingapurelyimaginarychemicalpotential.
5. Simulationsetup
Weused aSymanzik improved gauge andstout-link improved staggered fermionic lattice ac-
tion in order to reduce taste violation. The configurations were generated with an exact RHMC
algorithm. We determined the line of constant physics (LCP) using physical masses for the light
quarksm aswellasforthestrangequarkm . TheLCPwasfixedbysettingtheratiom /f and
u,d s K K
mK/mp totheirphysical values. Weusedfourdifferent lattice spacings NT =4,6,8,10 andaspect
ratiosN /N of4and3. Thescalewasfixedby f anditsunambiguitycheckedbycalculatingm ,
S T K K
fp and r0. For measuring the operators necessary for the above mentioned derivative the random
noise estimator method was used. The number of random vectors was set to 80, in order for the
error coming from therandom estimator method and that from thefiniteness ofthe statistics tobe
ofthesameextent. Thedetailsofthesimulation setupcanbefoundin[4]or[10].
6. Observables
In order to determine the derivative ¶ T /¶ (m 2) expressed in (3.2), one needs to select some
c
observable O. Forthisroleweusedthefollowingquantities:
ThePolyakovloopisdefinedas
1 (cid:229) N(cid:213)T−1
L= Tr U (x,t) (6.1)
N3 4
S x t=0
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ThecurvatureoftheQCDphasetransitionline G.Endro˝di
In order to be able toextract continuum limit results, an appropriate renormalization is necessary.
ForLthismeansL =Lexp(V(r )/2T),whereV(r)istheT =0staticpotential [4].
r 0
Thestrangesusceptibility
T ¶ 2logZ
c = (6.2)
s V ¶m 2
u,d
needsnorenormalization, sinceitisconnected toaconserved current.
Thechiralcondensate canalsobeexpressed asaderivativeofthepartitionfunction:
T ¶ logZ
yy¯ = (6.3)
V ¶ m
It can be renormalized by subtracting the additive divergences, and then multiplying by the quark
mass,sothatthemultiplicativefactorsalsocancel(herethefactorm4 isusedtogetadimensionless
p
combination):
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yy¯ =(yy¯ −yy¯ (T =0))·m· (6.4)
r m4
p
Finally,thedefinitionofthechiralsusceptibility is
T ¶ 2logZ
c yy¯ = V ¶ m2 (6.5)
andtherenormalization isthefollowing:
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c yy¯ r =(c yy¯ −c yy¯ (T =0))·m2·T4 (6.6)
7. Results
Asalreadymentioned, inourapproach weselectanarbitrarytemperaturetocalculatethecur-
vature of the transition line. In order to increase statistics, we can also average measurements at
different temperatures. Thisprocedure isillustrated on figure2forthe case ofthe strange suscep-
tibility andthechiralcondensate.
Having obtained the renormalized quantities for N =4,6,8 and 10, we are in a position to
T
carry out the continuum extrapolation. The results shown on figure 3 are only preliminary; note
thattheextrapolated valueshavequitelargestatistical errors.
8. Summary
While the curvature values for yy¯ are smaller than those for c /T2 at every lattice spacing,
r s
theybecomeconsistent inthea→0limit. Ouraiminthefirstplaceistodetermine thedifference
k (yy¯ )−k (c /T2),soincreasing thestatisctics inordertoclarifythesituation isveryimportant.
r s
Neverthelesswereportthecontinuumresults: k (yy¯ )=0.0034(40)andk (c /T2)=0.0013(51),
r s
andalsoforthePolyakovloopandthechiralsusceptibility: k (Lr)=−0.0095(93) andk (c yy¯ r)=
−0.0018(34).
Asthenextstepwearelookingforwardtoperformtheabovedescribed proceduretomeasure
¶ O/¶ (m 2)forthewholetemperatureintervalaroundT . Thiswayitbecomespossibletodetermine
c
the critical temperature for the given observables (by locating their inflection point or maximum
point) atnonvanishing chemical potential. Usingthistechnique andmuchlarger statistics weplan
togiveafinalanswerforthecurvatureoftheQCDtransition line.
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ThecurvatureoftheQCDphasetransitionline G.Endro˝di
Figure 2: The curvature determined using the normalized strange susceptibility (left side) and the renor-
malizedchiralcondensate(rightside),respectively.Independentmeasurementsatdifferenttemperatures(b
values)canbeaveragedtodecreasestatisticalerrors.ResultsfromN =8latticesareshown.
T
Figure 3: The curvatureof the transition line as a function of the lattice spacing squared, for the strange
susceptibility (leftside) and the chiral condensate(rightside). As characteristic to the appliedaction, the
N =4 resultsare outof the scaling regime, so the linear extrapolationis carried outusing only the finer
T
lattices. Asitcanbeseen,largerstatisticsisneededinordertoextractareliablecontinuumresult.
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